LMFDB - Embedded newform 1050.3.h.a.349.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1050,3,Mod(349,1050)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1050, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1050.349");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1050.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$28.6104277578$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.6
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1050.349
Dual form			1050.3.h.a.349.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -1.73205 q^{3} -2.00000 q^{4} -2.44949i q^{6} +(3.16693 - 6.24264i) q^{7} -2.82843i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.41421i q^{2} -1.73205 q^{3} -2.00000 q^{4} -2.44949i q^{6} +(3.16693 - 6.24264i) q^{7} -2.82843i q^{8} +3.00000 q^{9} -1.75736 q^{11} +3.46410 q^{12} +18.7554 q^{13} +(8.82843 + 4.47871i) q^{14} +4.00000 q^{16} -23.4803 q^{17} +4.24264i q^{18} -23.0600i q^{19} +(-5.48528 + 10.8126i) q^{21} -2.48528i q^{22} +18.7279i q^{23} +4.89898i q^{24} +26.5241i q^{26} -5.19615 q^{27} +(-6.33386 + 12.4853i) q^{28} -30.0000 q^{29} -8.60927i q^{31} +5.65685i q^{32} +3.04384 q^{33} -33.2061i q^{34} -6.00000 q^{36} +70.9117i q^{37} +32.6118 q^{38} -32.4853 q^{39} -41.3951i q^{41} +(-15.2913 - 7.75736i) q^{42} +10.4264i q^{43} +3.51472 q^{44} -26.4853 q^{46} -38.6995 q^{47} -6.92820 q^{48} +(-28.9411 - 39.5400i) q^{49} +40.6690 q^{51} -37.5108 q^{52} -37.0294i q^{53} -7.34847i q^{54} +(-17.6569 - 8.95743i) q^{56} +39.9411i q^{57} -42.4264i q^{58} -97.4872i q^{59} -16.7262i q^{61} +12.1753 q^{62} +(9.50079 - 18.7279i) q^{63} -8.00000 q^{64} +4.30463i q^{66} -60.9706i q^{67} +46.9606 q^{68} -32.4377i q^{69} -110.610 q^{71} -8.48528i q^{72} -56.7585 q^{73} -100.284 q^{74} +46.1200i q^{76} +(-5.56543 + 10.9706i) q^{77} -45.9411i q^{78} +69.8234 q^{79} +9.00000 q^{81} +58.5416 q^{82} +6.43583 q^{83} +(10.9706 - 21.6251i) q^{84} -14.7452 q^{86} +51.9615 q^{87} +4.97056i q^{88} -42.0915i q^{89} +(59.3970 - 117.083i) q^{91} -37.4558i q^{92} +14.9117i q^{93} -54.7293i q^{94} -9.79796i q^{96} -51.7153 q^{97} +(55.9180 - 40.9289i) q^{98} -5.27208 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} + 24 q^{9}+O(q^{10}) $ 8 * q - 16 * q^4 + 24 * q^9	$ 8 q - 16 q^{4} + 24 q^{9} - 48 q^{11} + 48 q^{14} + 32 q^{16} + 24 q^{21} - 240 q^{29} - 48 q^{36} - 192 q^{39} + 96 q^{44} - 144 q^{46} + 40 q^{49} - 48 q^{51} - 96 q^{56} - 64 q^{64} - 240 q^{71} - 576 q^{74} - 256 q^{79} + 72 q^{81} - 48 q^{84} - 480 q^{86} - 144 q^{99}+O(q^{100}) $ 8 * q - 16 * q^4 + 24 * q^9 - 48 * q^11 + 48 * q^14 + 32 * q^16 + 24 * q^21 - 240 * q^29 - 48 * q^36 - 192 * q^39 + 96 * q^44 - 144 * q^46 + 40 * q^49 - 48 * q^51 - 96 * q^56 - 64 * q^64 - 240 * q^71 - 576 * q^74 - 256 * q^79 + 72 * q^81 - 48 * q^84 - 480 * q^86 - 144 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1050\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$701$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			1.41421i			0.707107i
$2$			1.41421i			0.707107i
$3$	−1.73205			−0.577350
$3$	−1.73205			−0.577350
$4$	−2.00000			−0.500000
$5$		0			0
$5$		0			0
$6$		−	2.44949i		−	0.408248i
$7$	3.16693	−	6.24264i	0.452418	−	0.891806i
$7$	3.16693	−	6.24264i	0.452418	−	0.891806i
$8$		−	2.82843i		−	0.353553i
$9$	3.00000			0.333333
$10$		0			0
$11$	−1.75736			−0.159760			−0.0798800	−	0.996804i	$-0.525454\pi$
$11$	−1.75736			−0.159760			−0.0798800	+	0.996804i	$0.525454\pi$
$12$	3.46410			0.288675
$13$	18.7554			1.44272			0.721361	−	0.692559i	$-0.243517\pi$
$13$	18.7554			1.44272			0.721361	+	0.692559i	$0.243517\pi$
$14$	8.82843	+	4.47871i	0.630602	+	0.319908i
$15$		0			0
$16$	4.00000			0.250000
$17$	−23.4803			−1.38119			−0.690597	−	0.723240i	$-0.742652\pi$
$17$	−23.4803			−1.38119			−0.690597	+	0.723240i	$0.742652\pi$
$18$			4.24264i			0.235702i
$19$		−	23.0600i		−	1.21369i	−0.794822	−	0.606843i	$-0.792436\pi$
$19$		−	23.0600i		−	1.21369i	0.794822	−	0.606843i	$-0.207564\pi$
$20$		0			0
$21$	−5.48528	+	10.8126i	−0.261204	+	0.514884i
$22$		−	2.48528i		−	0.112967i
$23$			18.7279i			0.814257i	0.913371	+	0.407129i	$0.133470\pi$
$23$			18.7279i			0.814257i	−0.913371	+	0.407129i	$0.866530\pi$
$24$			4.89898i			0.204124i
$25$		0			0
$26$			26.5241i			1.02016i
$27$	−5.19615			−0.192450
$28$	−6.33386	+	12.4853i	−0.226209	+	0.445903i
$29$	−30.0000			−1.03448			−0.517241	−	0.855840i	$-0.673041\pi$
$29$	−30.0000			−1.03448			−0.517241	+	0.855840i	$0.673041\pi$
$30$		0			0
$31$		−	8.60927i		−	0.277718i	−0.990312	−	0.138859i	$-0.955656\pi$
$31$		−	8.60927i		−	0.277718i	0.990312	−	0.138859i	$-0.0443435\pi$
$32$			5.65685i			0.176777i
$33$	3.04384			0.0922374
$34$		−	33.2061i		−	0.976651i
$35$		0			0
$36$	−6.00000			−0.166667
$37$			70.9117i			1.91653i	0.285878	+	0.958266i	$0.407715\pi$
$37$			70.9117i			1.91653i	−0.285878	+	0.958266i	$0.592285\pi$
$38$	32.6118			0.858205
$39$	−32.4853			−0.832956
$40$		0			0
$41$		−	41.3951i		−	1.00964i	−0.863225	−	0.504819i	$-0.831559\pi$
$41$		−	41.3951i		−	1.00964i	0.863225	−	0.504819i	$-0.168441\pi$
$42$	−15.2913	−	7.75736i	−0.364078	−	0.184699i
$43$			10.4264i			0.242475i	0.992624	+	0.121237i	$0.0386862\pi$
$43$			10.4264i			0.242475i	−0.992624	+	0.121237i	$0.961314\pi$
$44$	3.51472			0.0798800
$45$		0			0
$46$	−26.4853			−0.575767
$47$	−38.6995			−0.823393			−0.411696	−	0.911321i	$-0.635064\pi$
$47$	−38.6995			−0.823393			−0.411696	+	0.911321i	$0.635064\pi$
$48$	−6.92820			−0.144338
$49$	−28.9411	−	39.5400i	−0.590635	−	0.806939i
$50$		0			0
$51$	40.6690			0.797432
$52$	−37.5108			−0.721361
$53$		−	37.0294i		−	0.698669i	−0.936998	−	0.349334i	$-0.886408\pi$
$53$		−	37.0294i		−	0.698669i	0.936998	−	0.349334i	$-0.113592\pi$
$54$		−	7.34847i		−	0.136083i
$55$		0			0
$56$	−17.6569	−	8.95743i	−0.315301	−	0.159954i
$57$			39.9411i			0.700721i
$58$		−	42.4264i		−	0.731490i
$59$		−	97.4872i		−	1.65233i	−0.563431	−	0.826163i	$-0.690519\pi$
$59$		−	97.4872i		−	1.65233i	0.563431	−	0.826163i	$-0.309481\pi$
$60$		0			0
$61$		−	16.7262i		−	0.274199i	−0.990557	−	0.137100i	$-0.956222\pi$
$61$		−	16.7262i		−	0.274199i	0.990557	−	0.137100i	$-0.0437781\pi$
$62$	12.1753			0.196376
$63$	9.50079	−	18.7279i	0.150806	−	0.297269i
$64$	−8.00000			−0.125000
$65$		0			0
$66$			4.30463i			0.0652217i
$67$		−	60.9706i		−	0.910008i	−0.890489	−	0.455004i	$-0.849638\pi$
$67$		−	60.9706i		−	0.910008i	0.890489	−	0.455004i	$-0.150362\pi$
$68$	46.9606			0.690597
$69$		−	32.4377i		−	0.470112i
$70$		0			0
$71$	−110.610			−1.55789			−0.778945	−	0.627092i	$-0.784245\pi$
$71$	−110.610			−1.55789			−0.778945	+	0.627092i	$0.784245\pi$
$72$		−	8.48528i		−	0.117851i
$73$	−56.7585			−0.777514			−0.388757	−	0.921340i	$-0.627095\pi$
$73$	−56.7585			−0.777514			−0.388757	+	0.921340i	$0.627095\pi$
$74$	−100.284			−1.35519
$75$		0			0
$76$			46.1200i			0.606843i
$77$	−5.56543	+	10.9706i	−0.0722783	+	0.142475i
$78$		−	45.9411i		−	0.588989i
$79$	69.8234			0.883840			0.441920	−	0.897054i	$-0.354297\pi$
$79$	69.8234			0.883840			0.441920	+	0.897054i	$0.354297\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$	58.5416			0.713922
$83$	6.43583			0.0775401			0.0387701	−	0.999248i	$-0.487656\pi$
$83$	6.43583			0.0775401			0.0387701	+	0.999248i	$0.487656\pi$
$84$	10.9706	−	21.6251i	0.130602	−	0.257442i
$85$		0			0
$86$	−14.7452			−0.171455
$87$	51.9615			0.597259
$88$			4.97056i			0.0564837i
$89$		−	42.0915i		−	0.472938i	−0.971639	−	0.236469i	$-0.924010\pi$
$89$		−	42.0915i		−	0.472938i	0.971639	−	0.236469i	$-0.0759901\pi$
$90$		0			0
$91$	59.3970	−	117.083i	0.652714	−	1.28663i
$92$		−	37.4558i		−	0.407129i
$93$			14.9117i			0.160341i
$94$		−	54.7293i		−	0.582227i
$95$		0			0
$96$		−	9.79796i		−	0.102062i
$97$	−51.7153			−0.533148			−0.266574	−	0.963814i	$-0.585892\pi$
$97$	−51.7153			−0.533148			−0.266574	+	0.963814i	$0.585892\pi$
$98$	55.9180	−	40.9289i	0.570592	−	0.417642i
$99$	−5.27208			−0.0532533
$100$		0			0
$101$		−	6.60991i		−	0.0654447i	−0.999464	−	0.0327223i	$-0.989582\pi$
$101$		−	6.60991i		−	0.0654447i	0.999464	−	0.0327223i	$-0.0104177\pi$
$102$			57.5147i			0.563870i
$103$	175.871			1.70748			0.853742	−	0.520696i	$-0.174327\pi$
$103$	175.871			1.70748			0.853742	+	0.520696i	$0.174327\pi$
$104$		−	53.0482i		−	0.510079i
$105$		0			0
$106$	52.3675			0.494033
$107$			46.2426i			0.432174i	0.976374	+	0.216087i	$0.0693295\pi$
$107$			46.2426i			0.432174i	−0.976374	+	0.216087i	$0.930670\pi$
$108$	10.3923			0.0962250
$109$	35.9411			0.329735			0.164868	−	0.986316i	$-0.447280\pi$
$109$	35.9411			0.329735			0.164868	+	0.986316i	$0.447280\pi$
$110$		0			0
$111$		−	122.823i		−	1.10651i
$112$	12.6677	−	24.9706i	0.113105	−	0.222951i
$113$		−	73.0294i		−	0.646278i	−0.946351	−	0.323139i	$-0.895262\pi$
$113$		−	73.0294i		−	0.646278i	0.946351	−	0.323139i	$-0.104738\pi$
$114$	−56.4853			−0.495485
$115$		0			0
$116$	60.0000			0.517241
$117$	56.2662			0.480907
$118$	137.868			1.16837
$119$	−74.3604	+	146.579i	−0.624877	+	1.23176i
$120$		0			0
$121$	−117.912			−0.974477
$122$	23.6544			0.193888
$123$			71.6985i			0.582915i
$124$			17.2185i			0.138859i
$125$		0			0
$126$	26.4853	+	13.4361i	0.210201	+	0.106636i
$127$		−	89.9411i		−	0.708198i	−0.935208	−	0.354099i	$-0.884788\pi$
$127$		−	89.9411i		−	0.708198i	0.935208	−	0.354099i	$-0.115212\pi$
$128$		−	11.3137i		−	0.0883883i
$129$		−	18.0591i		−	0.139993i
$130$		0			0
$131$		−	12.1753i		−	0.0929415i	−0.998920	−	0.0464708i	$-0.985203\pi$
$131$		−	12.1753i		−	0.0929415i	0.998920	−	0.0464708i	$-0.0147974\pi$
$132$	−6.08767			−0.0461187
$133$	−143.955	−	73.0294i	−1.08237	−	0.549094i
$134$	86.2254			0.643473
$135$		0			0
$136$			66.4123i			0.488326i
$137$		−	165.765i		−	1.20996i	−0.796241	−	0.604980i	$-0.793181\pi$
$137$		−	165.765i		−	1.20996i	0.796241	−	0.604980i	$-0.206819\pi$
$138$	45.8739			0.332419
$139$		−	220.514i		−	1.58643i	−0.608941	−	0.793215i	$-0.708406\pi$
$139$		−	220.514i		−	1.58643i	0.608941	−	0.793215i	$-0.291594\pi$
$140$		0			0
$141$	67.0294			0.475386
$142$		−	156.426i		−	1.10159i
$143$	−32.9600			−0.230489
$144$	12.0000			0.0833333
$145$		0			0
$146$		−	80.2687i		−	0.549786i
$147$	50.1275	+	68.4853i	0.341003	+	0.465886i
$148$		−	141.823i		−	0.958266i
$149$	−210.853			−1.41512			−0.707560	−	0.706653i	$-0.750204\pi$
$149$	−210.853			−1.41512			−0.707560	+	0.706653i	$0.750204\pi$
$150$		0			0
$151$	−72.3675			−0.479255			−0.239628	−	0.970865i	$-0.577025\pi$
$151$	−72.3675			−0.479255			−0.239628	+	0.970865i	$0.577025\pi$
$152$	−65.2236			−0.429103
$153$	−70.4409			−0.460398
$154$	−15.5147	−	7.87071i	−0.100745	−	0.0511085i
$155$		0			0
$156$	64.9706			0.416478
$157$	−233.674			−1.48837			−0.744184	−	0.667974i	$-0.767162\pi$
$157$	−233.674			−1.48837			−0.744184	+	0.667974i	$0.767162\pi$
$158$			98.7452i			0.624969i
$159$			64.1369i			0.403377i
$160$		0			0
$161$	116.912	+	59.3100i	0.726160	+	0.368385i
$162$			12.7279i			0.0785674i
$163$		−	73.0883i		−	0.448395i	−0.974544	−	0.224197i	$-0.928024\pi$
$163$		−	73.0883i		−	0.448395i	0.974544	−	0.224197i	$-0.0719760\pi$
$164$			82.7903i			0.504819i
$165$		0			0
$166$			9.10164i			0.0548292i
$167$	39.3958			0.235903			0.117951	−	0.993019i	$-0.462367\pi$
$167$	39.3958			0.235903			0.117951	+	0.993019i	$0.462367\pi$
$168$	30.5826	+	15.5147i	0.182039	+	0.0923495i
$169$	182.765			1.08145
$170$		0			0
$171$		−	69.1801i		−	0.404562i
$172$		−	20.8528i		−	0.121237i
$173$	23.8284			0.137737			0.0688683	−	0.997626i	$-0.478061\pi$
$173$	23.8284			0.137737			0.0688683	+	0.997626i	$0.478061\pi$
$174$			73.4847i			0.422326i
$175$		0			0
$176$	−7.02944			−0.0399400
$177$			168.853i			0.953971i
$178$	59.5263			0.334417
$179$	12.9045			0.0720924			0.0360462	−	0.999350i	$-0.488524\pi$
$179$	12.9045			0.0720924			0.0360462	+	0.999350i	$0.488524\pi$
$180$		0			0
$181$		−	65.3678i		−	0.361148i	−0.983561	−	0.180574i	$-0.942204\pi$
$181$		−	65.3678i		−	0.361148i	0.983561	−	0.180574i	$-0.0577955\pi$
$182$	165.581	+	84.0000i	0.909783	+	0.461538i
$183$			28.9706i			0.158309i
$184$	52.9706			0.287883
$185$		0			0
$186$	−21.0883			−0.113378
$187$	41.2633			0.220659
$188$	77.3989			0.411696
$189$	−16.4558	+	32.4377i	−0.0870680	+	0.171628i
$190$		0			0
$191$	−100.066			−0.523906			−0.261953	−	0.965081i	$-0.584367\pi$
$191$	−100.066			−0.523906			−0.261953	+	0.965081i	$0.584367\pi$
$192$	13.8564			0.0721688
$193$			78.9117i			0.408869i	0.978880	+	0.204434i	$0.0655355\pi$
$193$			78.9117i			0.408869i	−0.978880	+	0.204434i	$0.934464\pi$
$194$		−	73.1365i		−	0.376992i
$195$		0			0
$196$	57.8823	+	79.0800i	0.295318	+	0.403469i
$197$			183.941i			0.933711i	0.884334	+	0.466856i	$0.154613\pi$
$197$			183.941i			0.933711i	−0.884334	+	0.466856i	$0.845387\pi$
$198$		−	7.45584i		−	0.0376558i
$199$			170.029i			0.854419i	0.904153	+	0.427210i	$0.140503\pi$
$199$			170.029i			0.854419i	−0.904153	+	0.427210i	$0.859497\pi$
$200$		0			0
$201$			105.604i			0.525394i
$202$	9.34783			0.0462764
$203$	−95.0079	+	187.279i	−0.468019	+	0.922558i
$204$	−81.3381			−0.398716
$205$		0			0
$206$			248.719i			1.20737i
$207$			56.1838i			0.271419i
$208$	75.0215			0.360680
$209$			40.5247i			0.193898i
$210$		0			0
$211$	21.5736			0.102245			0.0511223	−	0.998692i	$-0.483720\pi$
$211$	21.5736			0.102245			0.0511223	+	0.998692i	$0.483720\pi$
$212$			74.0589i			0.349334i
$213$	191.582			0.899448
$214$	−65.3970			−0.305593
$215$		0			0
$216$			14.6969i			0.0680414i
$217$	−53.7446	−	27.2649i	−0.247671	−	0.125645i
$218$			50.8284i			0.233158i
$219$	98.3087			0.448898
$220$		0			0
$221$	−440.382			−1.99268
$222$	173.697			0.782421
$223$	−119.359			−0.535240			−0.267620	−	0.963525i	$-0.586237\pi$
$223$	−119.359			−0.535240			−0.267620	+	0.963525i	$0.586237\pi$
$224$	35.3137	+	17.9149i	0.157650	+	0.0799770i
$225$		0			0
$226$	103.279			0.456988
$227$	169.843			0.748207			0.374103	−	0.927387i	$-0.377951\pi$
$227$	169.843			0.748207			0.374103	+	0.927387i	$0.377951\pi$
$228$		−	79.8823i		−	0.350361i
$229$		−	110.011i		−	0.480396i	−0.970724	−	0.240198i	$-0.922788\pi$
$229$		−	110.011i		−	0.480396i	0.970724	−	0.240198i	$-0.0772124\pi$
$230$		0			0
$231$	9.63961	−	19.0016i	0.0417299	−	0.0822579i
$232$			84.8528i			0.365745i
$233$			57.2649i			0.245772i	0.992421	+	0.122886i	$0.0392150\pi$
$233$			57.2649i			0.245772i	−0.992421	+	0.122886i	$0.960785\pi$
$234$			79.5724i			0.340053i
$235$		0			0
$236$			194.974i			0.826163i
$237$	−120.938			−0.510285
$238$	−207.294	−	105.161i	−0.870983	−	0.441855i
$239$	−281.522			−1.17792			−0.588958	−	0.808164i	$-0.700462\pi$
$239$	−281.522			−1.17792			−0.588958	+	0.808164i	$0.700462\pi$
$240$		0			0
$241$		−	168.306i		−	0.698366i	−0.937055	−	0.349183i	$-0.886459\pi$
$241$		−	168.306i		−	0.698366i	0.937055	−	0.349183i	$-0.113541\pi$
$242$		−	166.752i		−	0.689059i
$243$	−15.5885			−0.0641500
$244$			33.4523i			0.137100i
$245$		0			0
$246$	−101.397			−0.412183
$247$		−	432.500i		−	1.75101i
$248$	−24.3507			−0.0981882
$249$	−11.1472			−0.0447678
$250$		0			0
$251$		−	106.096i		−	0.422695i	−0.977411	−	0.211348i	$-0.932215\pi$
$251$		−	106.096i		−	0.422695i	0.977411	−	0.211348i	$-0.0677852\pi$
$252$	−19.0016	+	37.4558i	−0.0754031	+	0.148634i
$253$		−	32.9117i		−	0.130086i
$254$	127.196			0.500771
$255$		0			0
$256$	16.0000			0.0625000
$257$	290.462			1.13020			0.565102	−	0.825021i	$-0.308837\pi$
$257$	290.462			1.13020			0.565102	+	0.825021i	$0.308837\pi$
$258$	25.5394			0.0989898
$259$	442.676	+	224.572i	1.70917	+	0.867074i
$260$		0			0
$261$	−90.0000			−0.344828
$262$	17.2185			0.0657196
$263$		−	89.0223i		−	0.338488i	−0.985574	−	0.169244i	$-0.945867\pi$
$263$		−	89.0223i		−	0.338488i	0.985574	−	0.169244i	$-0.0541326\pi$
$264$		−	8.60927i		−	0.0326109i
$265$		0			0
$266$	103.279	−	203.584i	0.388268	−	0.765352i
$267$			72.9045i			0.273051i
$268$			121.941i			0.455004i
$269$			191.498i			0.711888i	0.934507	+	0.355944i	$0.115841\pi$
$269$			191.498i			0.711888i	−0.934507	+	0.355944i	$0.884159\pi$
$270$		0			0
$271$			217.440i			0.802362i	0.915999	+	0.401181i	$0.131400\pi$
$271$			217.440i			0.802362i	−0.915999	+	0.401181i	$0.868600\pi$
$272$	−93.9211			−0.345298
$273$	−102.879	+	202.794i	−0.376845	+	0.742835i
$274$	234.426			0.855571
$275$		0			0
$276$			64.8754i			0.235056i
$277$		−	290.676i		−	1.04937i	−0.851296	−	0.524686i	$-0.824183\pi$
$277$		−	290.676i		−	1.04937i	0.851296	−	0.524686i	$-0.175817\pi$
$278$	311.854			1.12178
$279$		−	25.8278i		−	0.0925728i
$280$		0			0
$281$	18.8528			0.0670919			0.0335459	−	0.999437i	$-0.489320\pi$
$281$	18.8528			0.0670919			0.0335459	+	0.999437i	$0.489320\pi$
$282$			94.7939i			0.336149i
$283$	−401.734			−1.41955			−0.709777	−	0.704426i	$-0.751204\pi$
$283$	−401.734			−1.41955			−0.709777	+	0.704426i	$0.751204\pi$
$284$	221.220			0.778945
$285$		0			0
$286$		−	46.6124i		−	0.162980i
$287$	−258.415	−	131.095i	−0.900401	−	0.456779i
$288$			16.9706i			0.0589256i
$289$	262.324			0.907695
$290$		0			0
$291$	89.5736			0.307813
$292$	113.517			0.388757
$293$	−280.893			−0.958679			−0.479340	−	0.877629i	$-0.659124\pi$
$293$	−280.893			−0.958679			−0.479340	+	0.877629i	$0.659124\pi$
$294$	−96.8528	+	70.8910i	−0.329431	+	0.241126i
$295$		0			0
$296$	200.569			0.677596
$297$	9.13151			0.0307458
$298$		−	298.191i		−	1.00064i
$299$			351.249i			1.17475i
$300$		0			0
$301$	65.0883	+	33.0197i	0.216240	+	0.109700i
$302$		−	102.343i		−	0.338885i
$303$			11.4487i			0.0377845i
$304$		−	92.2401i		−	0.303421i
$305$		0			0
$306$		−	99.6184i		−	0.325550i
$307$	−152.318			−0.496151			−0.248076	−	0.968741i	$-0.579798\pi$
$307$	−152.318			−0.496151			−0.248076	+	0.968741i	$0.579798\pi$
$308$	11.1309	−	21.9411i	0.0361392	−	0.0712374i
$309$	−304.617			−0.985817
$310$		0			0
$311$			283.156i			0.910470i	0.890371	+	0.455235i	$0.150445\pi$
$311$			283.156i			0.910470i	−0.890371	+	0.455235i	$0.849555\pi$
$312$			91.8823i			0.294494i
$313$	48.5819			0.155214			0.0776069	−	0.996984i	$-0.475272\pi$
$313$	48.5819			0.155214			0.0776069	+	0.996984i	$0.475272\pi$
$314$		−	330.465i		−	1.05244i
$315$		0			0
$316$	−139.647			−0.441920
$317$			578.029i			1.82343i	0.410819	+	0.911717i	$0.365243\pi$
$317$			578.029i			1.82343i	−0.410819	+	0.911717i	$0.634757\pi$
$318$	−90.7032			−0.285230
$319$	52.7208			0.165269
$320$		0			0
$321$		−	80.0946i		−	0.249516i
$322$	−83.8770	+	165.338i	−0.260488	+	0.513472i
$323$			541.456i			1.67633i
$324$	−18.0000			−0.0555556
$325$		0			0
$326$	103.362			0.317063
$327$	−62.2519			−0.190373
$328$	−117.083			−0.356961
$329$	−122.558	+	241.587i	−0.372518	+	0.734307i
$330$		0			0
$331$	332.368			1.00413			0.502066	−	0.864829i	$-0.332574\pi$
$331$	332.368			1.00413			0.502066	+	0.864829i	$0.332574\pi$
$332$	−12.8717			−0.0387701
$333$			212.735i			0.638844i
$334$			55.7141i			0.166809i
$335$		0			0
$336$	−21.9411	+	43.2503i	−0.0653010	+	0.128721i
$337$		−	88.1766i		−	0.261652i	−0.991405	−	0.130826i	$-0.958237\pi$
$337$		−	88.1766i		−	0.261652i	0.991405	−	0.130826i	$-0.0417629\pi$
$338$			258.468i			0.764698i
$339$			126.491i			0.373129i
$340$		0			0
$341$			15.1296i			0.0443683i
$342$	97.8354			0.286068
$343$	−338.488	+	55.4487i	−0.986847	+	0.161658i
$344$	29.4903			0.0857277
$345$		0			0
$346$			33.6985i			0.0973945i
$347$			320.080i			0.922422i	0.887291	+	0.461211i	$0.152585\pi$
$347$			320.080i			0.922422i	−0.887291	+	0.461211i	$0.847415\pi$
$348$	−103.923			−0.298629
$349$		−	333.046i		−	0.954287i	−0.878825	−	0.477143i	$-0.841672\pi$
$349$		−	333.046i		−	0.954287i	0.878825	−	0.477143i	$-0.158328\pi$
$350$		0			0
$351$	−97.4558			−0.277652
$352$		−	9.94113i		−	0.0282418i
$353$	−655.712			−1.85754			−0.928771	−	0.370654i	$-0.879134\pi$
$353$	−655.712			−1.85754			−0.928771	+	0.370654i	$0.879134\pi$
$354$	−238.794			−0.674559
$355$		0			0
$356$			84.1829i			0.236469i
$357$	128.796	−	253.882i	0.360773	−	0.711155i
$358$			18.2498i			0.0509770i
$359$	97.7574			0.272305			0.136152	−	0.990688i	$-0.456526\pi$
$359$	97.7574			0.272305			0.136152	+	0.990688i	$0.456526\pi$
$360$		0			0
$361$	−170.765			−0.473032
$362$	92.4440			0.255370
$363$	204.229			0.562614
$364$	−118.794	+	234.166i	−0.326357	+	0.643314i
$365$		0			0
$366$	−40.9706			−0.111941
$367$	321.057			0.874815			0.437408	−	0.899263i	$-0.355897\pi$
$367$	321.057			0.874815			0.437408	+	0.899263i	$0.355897\pi$
$368$			74.9117i			0.203564i
$369$		−	124.185i		−	0.336546i
$370$		0			0
$371$	−231.161	−	117.270i	−0.623077	−	0.316091i
$372$		−	29.8234i		−	0.0801704i
$373$			187.470i			0.502601i	0.967909	+	0.251300i	$0.0808582\pi$
$373$			187.470i			0.502601i	−0.967909	+	0.251300i	$0.919142\pi$
$374$			58.3551i			0.156030i
$375$		0			0
$376$			109.459i			0.291113i
$377$	−562.662			−1.49247
$378$	−45.8739	−	23.2721i	−0.121359	−	0.0615663i
$379$	357.103			0.942223			0.471112	−	0.882074i	$-0.343853\pi$
$379$	357.103			0.942223			0.471112	+	0.882074i	$0.343853\pi$
$380$		0			0
$381$			155.783i			0.408878i
$382$		−	141.515i		−	0.370457i
$383$	622.230			1.62462			0.812311	−	0.583225i	$-0.198209\pi$
$383$	622.230			1.62462			0.812311	+	0.583225i	$0.198209\pi$
$384$			19.5959i			0.0510310i
$385$		0			0
$386$	−111.598			−0.289114
$387$			31.2792i			0.0808249i
$388$	103.431			0.266574
$389$	−227.470			−0.584756			−0.292378	−	0.956303i	$-0.594447\pi$
$389$	−227.470			−0.584756			−0.292378	+	0.956303i	$0.594447\pi$
$390$		0			0
$391$		−	439.737i		−	1.12465i
$392$	−111.836	+	81.8579i	−0.285296	+	0.208821i
$393$			21.0883i			0.0536598i
$394$	−260.132			−0.660234
$395$		0			0
$396$	10.5442			0.0266267
$397$	−720.329			−1.81443			−0.907216	−	0.420666i	$-0.861796\pi$
$397$	−720.329			−1.81443			−0.907216	+	0.420666i	$0.861796\pi$
$398$	−240.458			−0.604166
$399$	249.338	+	126.491i	0.624908	+	0.317019i
$400$		0			0
$401$	697.176			1.73859			0.869296	−	0.494291i	$-0.164572\pi$
$401$	697.176			1.73859			0.869296	+	0.494291i	$0.164572\pi$
$402$	−149.347			−0.371509
$403$		−	161.470i		−	0.400670i
$404$			13.2198i			0.0327223i
$405$		0			0
$406$	−264.853	−	134.361i	−0.652347	−	0.330939i
$407$		−	124.617i		−	0.306185i
$408$		−	115.029i		−	0.281935i
$409$			102.386i			0.250333i	0.992136	+	0.125166i	$0.0399465\pi$
$409$			102.386i			0.250333i	−0.992136	+	0.125166i	$0.960053\pi$
$410$		0			0
$411$			287.113i			0.698571i
$412$	−351.742			−0.853742
$413$	−608.578	−	308.735i	−1.47355	−	0.747543i
$414$	−79.4558			−0.191922
$415$		0			0
$416$			106.096i			0.255040i
$417$			381.941i			0.915926i
$418$	−57.3106			−0.137107
$419$			391.426i			0.934191i	0.884207	+	0.467095i	$0.154700\pi$
$419$			391.426i			0.934191i	−0.884207	+	0.467095i	$0.845300\pi$
$420$		0			0
$421$	354.441			0.841902			0.420951	−	0.907083i	$-0.361696\pi$
$421$	354.441			0.841902			0.420951	+	0.907083i	$0.361696\pi$
$422$			30.5097i			0.0722978i
$423$	−116.098			−0.274464
$424$	−104.735			−0.247017
$425$		0			0
$426$			270.938i			0.636006i
$427$	−104.415	−	52.9706i	−0.244533	−	0.124053i
$428$		−	92.4853i		−	0.216087i
$429$	57.0883			0.133073
$430$		0			0
$431$	585.286			1.35797			0.678987	−	0.734151i	$-0.262419\pi$
$431$	585.286			1.35797			0.678987	+	0.734151i	$0.262419\pi$
$432$	−20.7846			−0.0481125
$433$	−392.207			−0.905789			−0.452895	−	0.891564i	$-0.649609\pi$
$433$	−392.207			−0.905789			−0.452895	+	0.891564i	$0.649609\pi$
$434$	38.5584	−	76.0063i	0.0888443	−	0.175130i
$435$		0			0
$436$	−71.8823			−0.164868
$437$	431.866			0.988252
$438$			139.029i			0.317419i
$439$		−	392.513i		−	0.894106i	−0.894507	−	0.447053i	$-0.852473\pi$
$439$		−	392.513i		−	0.894106i	0.894507	−	0.447053i	$-0.147527\pi$
$440$		0			0
$441$	−86.8234	−	118.620i	−0.196878	−	0.268980i
$442$		−	622.794i		−	1.40904i
$443$		−	814.742i		−	1.83915i	−0.392918	−	0.919574i	$-0.628534\pi$
$443$		−	814.742i		−	1.83915i	0.392918	−	0.919574i	$-0.371466\pi$
$444$			245.645i			0.553255i
$445$		0			0
$446$		−	168.798i		−	0.378472i
$447$	365.208			0.817020
$448$	−25.3354	+	49.9411i	−0.0565523	+	0.111476i
$449$	−180.323			−0.401610			−0.200805	−	0.979631i	$-0.564356\pi$
$449$	−180.323			−0.401610			−0.200805	+	0.979631i	$0.564356\pi$
$450$		0			0
$451$			72.7461i			0.161300i
$452$			146.059i			0.323139i
$453$	125.344			0.276698
$454$			240.194i			0.529062i
$455$		0			0
$456$	112.971			0.247742
$457$		−	280.177i		−	0.613078i	−0.951858	−	0.306539i	$-0.900829\pi$
$457$		−	280.177i		−	0.613078i	0.951858	−	0.306539i	$-0.0991710\pi$
$458$	155.579			0.339691
$459$	122.007			0.265811
$460$		0			0
$461$		−	406.297i		−	0.881338i	−0.897670	−	0.440669i	$-0.854741\pi$
$461$		−	406.297i		−	0.881338i	0.897670	−	0.440669i	$-0.145259\pi$
$462$	26.8723	+	13.6325i	0.0581651	+	0.0295075i
$463$			457.470i			0.988056i	0.869446	+	0.494028i	$0.164476\pi$
$463$			457.470i			0.988056i	−0.869446	+	0.494028i	$0.835524\pi$
$464$	−120.000			−0.258621
$465$		0			0
$466$	−80.9848			−0.173787
$467$	643.711			1.37840			0.689198	−	0.724573i	$-0.257963\pi$
$467$	643.711			1.37840			0.689198	+	0.724573i	$0.257963\pi$
$468$	−112.532			−0.240454
$469$	−380.617	−	193.089i	−0.811551	−	0.411705i
$470$		0			0
$471$	404.735			0.859310
$472$	−275.735			−0.584185
$473$		−	18.3229i		−	0.0387377i
$474$		−	171.032i		−	0.360826i
$475$		0			0
$476$	148.721	−	293.158i	0.312439	−	0.615878i
$477$		−	111.088i		−	0.232890i
$478$		−	398.132i		−	0.832912i
$479$			168.535i			0.351847i	0.984404	+	0.175924i	$0.0562912\pi$
$479$			168.535i			0.351847i	−0.984404	+	0.175924i	$0.943709\pi$
$480$		0			0
$481$			1329.98i			2.76502i
$482$	238.021			0.493819
$483$	−202.497	−	102.728i	−0.419248	−	0.212687i
$484$	235.823			0.487238
$485$		0			0
$486$		−	22.0454i		−	0.0453609i
$487$			2.77965i			0.00570771i	0.999996	+	0.00285385i	$0.000908411\pi$
$487$			2.77965i			0.00570771i	−0.999996	+	0.00285385i	$0.999092\pi$
$488$	−47.3087			−0.0969441
$489$			126.593i			0.258881i
$490$		0			0
$491$	−247.477			−0.504027			−0.252014	−	0.967724i	$-0.581093\pi$
$491$	−247.477			−0.504027			−0.252014	+	0.967724i	$0.581093\pi$
$492$		−	143.397i		−	0.291457i
$493$	704.409			1.42882
$494$	611.647			1.23815
$495$		0			0
$496$		−	34.4371i		−	0.0694296i
$497$	−350.295	+	690.500i	−0.704818	+	1.38934i
$498$		−	15.7645i		−	0.0316556i
$499$	483.426			0.968789			0.484394	−	0.874850i	$-0.339040\pi$
$499$	483.426			0.968789			0.484394	+	0.874850i	$0.339040\pi$
$500$		0			0
$501$	−68.2355			−0.136199
$502$	150.043			0.298891
$503$	58.7033			0.116706			0.0583532	−	0.998296i	$-0.481415\pi$
$503$	58.7033			0.116706			0.0583532	+	0.998296i	$0.481415\pi$
$504$	−52.9706	−	26.8723i	−0.105100	−	0.0533180i
$505$		0			0
$506$	46.5442			0.0919845
$507$	−316.557			−0.624374
$508$			179.882i			0.354099i
$509$			68.8793i			0.135323i	0.997708	+	0.0676614i	$0.0215537\pi$
$509$			68.8793i			0.135323i	−0.997708	+	0.0676614i	$0.978446\pi$
$510$		0			0
$511$	−179.750	+	354.323i	−0.351762	+	0.693392i
$512$			22.6274i			0.0441942i
$513$			119.823i			0.233574i
$514$			410.776i			0.799175i
$515$		0			0
$516$			36.1181i			0.0699964i
$517$	68.0089			0.131545
$518$	−317.593	+	626.039i	−0.613114	+	1.20857i
$519$	−41.2721			−0.0795223
$520$		0			0
$521$		−	292.720i		−	0.561843i	−0.959731	−	0.280922i	$-0.909360\pi$
$521$		−	292.720i		−	0.561843i	0.959731	−	0.280922i	$-0.0906401\pi$
$522$		−	127.279i		−	0.243830i
$523$	493.056			0.942746			0.471373	−	0.881934i	$-0.343759\pi$
$523$	493.056			0.942746			0.471373	+	0.881934i	$0.343759\pi$
$524$			24.3507i			0.0464708i
$525$		0			0
$526$	125.897			0.239347
$527$			202.148i			0.383583i
$528$	12.1753			0.0230594
$529$	178.265			0.336985
$530$		0			0
$531$		−	292.462i		−	0.550775i
$532$	287.911	+	146.059i	0.541186	+	0.274547i
$533$		−	776.382i		−	1.45663i
$534$	−103.103			−0.193076
$535$		0			0
$536$	−172.451			−0.321737
$537$	−22.3513			−0.0416226
$538$	−270.819			−0.503381
$539$	50.8600	+	69.4860i	0.0943598	+	0.128916i
$540$		0			0
$541$	−1037.85			−1.91840			−0.959198	−	0.282736i	$-0.908758\pi$
$541$	−1037.85			−1.91840			−0.959198	+	0.282736i	$0.908758\pi$
$542$	−307.507			−0.567356
$543$			113.220i			0.208509i
$544$		−	132.825i		−	0.244163i
$545$		0			0
$546$	−286.794	−	145.492i	−0.525264	−	0.266469i
$547$			130.530i			0.238629i	0.992857	+	0.119314i	$0.0380696\pi$
$547$			130.530i			0.238629i	−0.992857	+	0.119314i	$0.961930\pi$
$548$			331.529i			0.604980i
$549$		−	50.1785i		−	0.0913998i
$550$		0			0
$551$			691.801i			1.25554i
$552$	−91.7477			−0.166210
$553$	221.126	−	435.882i	0.399866	−	0.788214i
$554$	411.078			0.742018
$555$		0			0
$556$			441.028i			0.793215i
$557$		−	665.147i		−	1.19416i	−0.802182	−	0.597080i	$-0.796327\pi$
$557$		−	665.147i		−	1.19416i	0.802182	−	0.597080i	$-0.203673\pi$
$558$	36.5260			0.0654588
$559$			195.551i			0.349823i
$560$		0			0
$561$	−71.4701			−0.127398
$562$			26.6619i			0.0474411i
$563$	829.295			1.47299			0.736497	−	0.676441i	$-0.236479\pi$
$563$	829.295			1.47299			0.736497	+	0.676441i	$0.236479\pi$
$564$	−134.059			−0.237693
$565$		0			0
$566$		−	568.137i		−	1.00378i
$567$	28.5024	−	56.1838i	0.0502687	−	0.0990895i
$568$			312.853i			0.550797i
$569$	−706.971			−1.24248			−0.621240	−	0.783621i	$-0.713371\pi$
$569$	−706.971			−1.24248			−0.621240	+	0.783621i	$0.713371\pi$
$570$		0			0
$571$	−366.912			−0.642577			−0.321289	−	0.946981i	$-0.604116\pi$
$571$	−366.912			−0.642577			−0.321289	+	0.946981i	$0.604116\pi$
$572$	65.9199			0.115245
$573$	173.319			0.302477
$574$	185.397	−	365.454i	0.322991	−	0.636679i
$575$		0			0
$576$	−24.0000			−0.0416667
$577$	390.357			0.676528			0.338264	−	0.941051i	$-0.390160\pi$
$577$	390.357			0.676528			0.338264	+	0.941051i	$0.390160\pi$
$578$			370.982i			0.641837i
$579$		−	136.679i		−	0.236061i
$580$		0			0
$581$	20.3818	−	40.1766i	0.0350806	−	0.0691507i
$582$			126.676i			0.217657i
$583$			65.0740i			0.111619i
$584$			160.537i			0.274893i
$585$		0			0
$586$		−	397.243i		−	0.677889i
$587$	702.499			1.19676			0.598381	−	0.801212i	$-0.295811\pi$
$587$	702.499			1.19676			0.598381	+	0.801212i	$0.295811\pi$
$588$	−100.255	−	136.971i	−0.170502	−	0.232943i
$589$	−198.530			−0.337063
$590$		0			0
$591$		−	318.595i		−	0.539078i
$592$			283.647i			0.479133i
$593$	942.519			1.58941			0.794704	−	0.606997i	$-0.207626\pi$
$593$	942.519			1.58941			0.794704	+	0.606997i	$0.207626\pi$
$594$			12.9139i			0.0217406i
$595$		0			0
$596$	421.706			0.707560
$597$		−	294.500i		−	0.493299i
$598$	−496.742			−0.830672
$599$	952.109			1.58950			0.794749	−	0.606939i	$-0.207603\pi$
$599$	952.109			1.58950			0.794749	+	0.606939i	$0.207603\pi$
$600$		0			0
$601$		−	729.804i		−	1.21432i	−0.794581	−	0.607158i	$-0.792310\pi$
$601$		−	729.804i		−	1.21432i	0.794581	−	0.607158i	$-0.207690\pi$
$602$	−46.6969	+	92.0488i	−0.0775696	+	0.152905i
$603$		−	182.912i		−	0.303336i
$604$	144.735			0.239628
$605$		0			0
$606$	−16.1909			−0.0267177
$607$	1006.34			1.65789			0.828944	−	0.559332i	$-0.188942\pi$
$607$	1006.34			1.65789			0.828944	+	0.559332i	$0.188942\pi$
$608$	130.447			0.214551
$609$	164.558	−	324.377i	0.270211	−	0.532639i
$610$		0			0
$611$	−725.823			−1.18793
$612$	140.882			0.230199
$613$		−	199.588i		−	0.325592i	−0.986660	−	0.162796i	$-0.947949\pi$
$613$		−	199.588i		−	0.325592i	0.986660	−	0.162796i	$-0.0520512\pi$
$614$		−	215.411i		−	0.350832i
$615$		0			0
$616$	31.0294	+	15.7414i	0.0503725	+	0.0255542i
$617$			353.294i			0.572599i	0.958140	+	0.286299i	$0.0924252\pi$
$617$			353.294i			0.572599i	−0.958140	+	0.286299i	$0.907575\pi$
$618$		−	430.794i		−	0.697078i
$619$		−	56.2064i		−	0.0908020i	−0.998969	−	0.0454010i	$-0.985543\pi$
$619$		−	56.2064i		−	0.0908020i	0.998969	−	0.0454010i	$-0.0144566\pi$
$620$		0			0
$621$		−	97.3131i		−	0.156704i
$622$	−400.443			−0.643799
$623$	−262.762	−	133.301i	−0.421769	−	0.213966i
$624$	−129.941			−0.208239
$625$		0			0
$626$			68.7052i			0.109753i
$627$		−	70.1909i		−	0.111947i
$628$	467.348			0.744184
$629$		−	1665.03i		−	2.64710i
$630$		0			0
$631$	807.322			1.27943			0.639716	−	0.768611i	$-0.279052\pi$
$631$	807.322			1.27943			0.639716	+	0.768611i	$0.279052\pi$
$632$		−	197.490i		−	0.312485i
$633$	−37.3666			−0.0590309
$634$	−817.456			−1.28936
$635$		0			0
$636$		−	128.274i		−	0.201688i
$637$	−542.802	−	741.588i	−0.852122	−	1.16419i
$638$			74.5584i			0.116863i
$639$	−331.831			−0.519297
$640$		0			0
$641$	1016.35			1.58557			0.792786	−	0.609500i	$-0.208630\pi$
$641$	1016.35			1.58557			0.792786	+	0.609500i	$0.208630\pi$
$642$	113.271			0.176434
$643$	−404.688			−0.629375			−0.314687	−	0.949195i	$-0.601900\pi$
$643$	−404.688			−0.629375			−0.314687	+	0.949195i	$0.601900\pi$
$644$	−233.823	−	118.620i	−0.363080	−	0.184193i
$645$		0			0
$646$	−765.734			−1.18535
$647$	940.604			1.45379			0.726896	−	0.686747i	$-0.240962\pi$
$647$	940.604			1.45379			0.726896	+	0.686747i	$0.240962\pi$
$648$		−	25.4558i		−	0.0392837i
$649$			171.320i			0.263975i
$650$		0			0
$651$	93.0883	+	47.2243i	0.142993	+	0.0725411i
$652$			146.177i			0.224197i
$653$			731.970i			1.12093i	0.828177	+	0.560467i	$0.189378\pi$
$653$			731.970i			1.12093i	−0.828177	+	0.560467i	$0.810622\pi$
$654$		−	88.0374i		−	0.134614i
$655$		0			0
$656$		−	165.581i		−	0.252409i
$657$	−170.276			−0.259171
$658$	−341.655	−	173.324i	−0.519233	−	0.263410i
$659$	−904.316			−1.37225			−0.686127	−	0.727481i	$-0.740691\pi$
$659$	−904.316			−1.37225			−0.686127	+	0.727481i	$0.740691\pi$
$660$		0			0
$661$			335.881i			0.508141i	0.967186	+	0.254070i	$0.0817695\pi$
$661$			335.881i			0.508141i	−0.967186	+	0.254070i	$0.918231\pi$
$662$			470.039i			0.710028i
$663$	762.764			1.15047
$664$		−	18.2033i		−	0.0274146i
$665$		0			0
$666$	−300.853			−0.451731
$667$		−	561.838i		−	0.842335i
$668$	−78.7916			−0.117951
$669$	206.735			0.309021
$670$		0			0
$671$			29.3939i			0.0438061i
$672$	−61.1651	−	31.0294i	−0.0910195	−	0.0461748i
$673$			1191.44i			1.77034i	0.465266	+	0.885171i	$0.345959\pi$
$673$			1191.44i			1.77034i	−0.465266	+	0.885171i	$0.654041\pi$
$674$	124.701			0.185016
$675$		0			0
$676$	−365.529			−0.540723
$677$	−1211.07			−1.78888			−0.894441	−	0.447186i	$-0.852426\pi$
$677$	−1211.07			−1.78888			−0.894441	+	0.447186i	$0.852426\pi$
$678$	−178.885			−0.263842
$679$	−163.779	+	322.840i	−0.241206	+	0.475464i
$680$		0			0
$681$	−294.177			−0.431977
$682$	−21.3965			−0.0313731
$683$		−	1233.89i		−	1.80657i	−0.429038	−	0.903287i	$-0.641147\pi$
$683$		−	1233.89i		−	1.80657i	0.429038	−	0.903287i	$-0.358853\pi$
$684$			138.360i			0.202281i
$685$		0			0
$686$	−78.4163	−	478.695i	−0.114309	−	0.697806i
$687$			190.544i			0.277357i
$688$			41.7056i			0.0606186i
$689$		−	694.501i		−	1.00798i
$690$		0			0
$691$			86.7045i			0.125477i	0.998030	+	0.0627384i	$0.0199834\pi$
$691$			86.7045i			0.125477i	−0.998030	+	0.0627384i	$0.980017\pi$
$692$	−47.6569			−0.0688683
$693$	−16.6963	+	32.9117i	−0.0240928	+	0.0474916i
$694$	−452.662			−0.652251
$695$		0			0
$696$		−	146.969i		−	0.211163i
$697$			971.970i			1.39450i
$698$	470.998			0.674783
$699$		−	99.1858i		−	0.141897i
$700$		0			0
$701$	−149.147			−0.212763			−0.106382	−	0.994325i	$-0.533927\pi$
$701$	−149.147			−0.212763			−0.106382	+	0.994325i	$0.533927\pi$
$702$		−	137.823i		−	0.196330i
$703$	1635.22			2.32607
$704$	14.0589			0.0199700
$705$		0			0
$706$		−	927.317i		−	1.31348i
$707$	−41.2633	−	20.9331i	−0.0583639	−	0.0296084i
$708$		−	337.706i		−	0.476985i
$709$	−189.647			−0.267485			−0.133742	−	0.991016i	$-0.542699\pi$
$709$	−189.647			−0.267485			−0.133742	+	0.991016i	$0.542699\pi$
$710$		0			0
$711$	209.470			0.294613
$712$	−119.053			−0.167209
$713$	161.234			0.226134
$714$	359.044	+	182.145i	0.502862	+	0.255105i
$715$		0			0
$716$	−25.8091			−0.0360462
$717$	487.610			0.680070
$718$			138.250i			0.192548i
$719$		−	12.0064i		−	0.0166987i	−0.999965	−	0.00834937i	$-0.997342\pi$
$719$		−	12.0064i		−	0.0166987i	0.999965	−	0.00834937i	$-0.00265772\pi$
$720$		0			0
$721$	556.971	−	1097.90i	0.772497	−	1.52274i
$722$		−	241.497i		−	0.334484i
$723$			291.515i			0.403202i
$724$			130.736i			0.180574i
$725$		0			0
$726$			288.823i			0.397828i
$727$	417.169			0.573823			0.286911	−	0.957957i	$-0.407371\pi$
$727$	417.169			0.573823			0.286911	+	0.957957i	$0.407371\pi$
$728$	−331.161	−	168.000i	−0.454892	−	0.230769i
$729$	27.0000			0.0370370
$730$		0			0
$731$		−	244.815i		−	0.334904i
$732$		−	57.9411i		−	0.0791545i
$733$	1286.99			1.75578			0.877892	−	0.478859i	$-0.158949\pi$
$733$	1286.99			1.75578			0.877892	+	0.478859i	$0.158949\pi$
$734$			454.043i			0.618588i
$735$		0			0
$736$	−105.941			−0.143942
$737$			107.147i			0.145383i
$738$	175.625			0.237974
$739$	−1333.47			−1.80443			−0.902213	−	0.431292i	$-0.858058\pi$
$739$	−1333.47			−1.80443			−0.902213	+	0.431292i	$0.858058\pi$
$740$		0			0
$741$			749.111i			1.01095i
$742$	165.844	−	326.912i	0.223510	−	0.440582i
$743$		−	776.476i		−	1.04506i	−0.852622	−	0.522528i	$-0.824989\pi$
$743$		−	776.476i		−	1.04506i	0.852622	−	0.522528i	$-0.175011\pi$
$744$	42.1766			0.0566890
$745$		0			0
$746$	−265.123			−0.355392
$747$	19.3075			0.0258467
$748$	−82.5266			−0.110330
$749$	288.676	+	146.447i	0.385415	+	0.195524i
$750$		0			0
$751$	48.8385			0.0650313			0.0325157	−	0.999471i	$-0.489648\pi$
$751$	48.8385			0.0650313			0.0325157	+	0.999471i	$0.489648\pi$
$752$	−154.798			−0.205848
$753$			183.765i			0.244043i
$754$		−	795.724i		−	1.05534i
$755$		0			0
$756$	32.9117	−	64.8754i	0.0435340	−	0.0858141i
$757$			1279.47i			1.69019i	0.534620	+	0.845093i	$0.320455\pi$
$757$			1279.47i			1.69019i	−0.534620	+	0.845093i	$0.679545\pi$
$758$			505.019i			0.666252i
$759$			57.0047i			0.0751050i
$760$		0			0
$761$			1316.12i			1.72947i	0.502231	+	0.864734i	$0.332513\pi$
$761$			1316.12i			1.72947i	−0.502231	+	0.864734i	$0.667487\pi$
$762$	−220.310			−0.289121
$763$	113.823	−	224.368i	0.149178	−	0.294060i
$764$	200.132			0.261953
$765$		0			0
$766$			879.966i			1.14878i
$767$		−	1828.41i		−	2.38385i
$768$	−27.7128			−0.0360844
$769$		−	110.324i		−	0.143464i	−0.997424	−	0.0717320i	$-0.977147\pi$
$769$		−	110.324i		−	0.143464i	0.997424	−	0.0717320i	$-0.0228526\pi$
$770$		0			0
$771$	−503.095			−0.652523
$772$		−	157.823i		−	0.204434i
$773$	−717.634			−0.928375			−0.464187	−	0.885737i	$-0.653654\pi$
$773$	−717.634			−0.928375			−0.464187	+	0.885737i	$0.653654\pi$
$774$	−44.2355			−0.0571518
$775$		0			0
$776$			146.273i			0.188496i
$777$	−766.738	−	388.971i	−0.986792	−	0.500606i
$778$		−	321.691i		−	0.413485i
$779$	−954.573			−1.22538
$780$		0			0
$781$	194.382			0.248888
$782$	621.882			0.795245
$783$	155.885			0.199086
$784$	−115.765	−	158.160i	−0.147659	−	0.201735i
$785$		0			0
$786$	−29.8234			−0.0379432
$787$	347.191			0.441158			0.220579	−	0.975369i	$-0.429205\pi$
$787$	347.191			0.441158			0.220579	+	0.975369i	$0.429205\pi$
$788$		−	367.882i		−	0.466856i
$789$			154.191i			0.195426i
$790$		0			0
$791$	−455.897	−	231.279i	−0.576355	−	0.292388i
$792$			14.9117i			0.0188279i
$793$		−	313.706i		−	0.395593i
$794$		−	1018.70i		−	1.28300i
$795$		0			0
$796$		−	340.059i		−	0.427210i
$797$	−361.246			−0.453257			−0.226629	−	0.973981i	$-0.572770\pi$
$797$	−361.246			−0.453257			−0.226629	+	0.973981i	$0.572770\pi$
$798$	−178.885	+	352.617i	−0.224166	+	0.441876i
$799$	908.674			1.13726
$800$		0			0
$801$		−	126.274i		−	0.157646i
$802$			985.955i			1.22937i
$803$	99.7451			0.124216
$804$		−	211.208i		−	0.262697i
$805$		0			0
$806$	228.353			0.283317
$807$		−	331.684i		−	0.411009i
$808$	−18.6957			−0.0231382
$809$	113.147			0.139861			0.0699303	−	0.997552i	$-0.477722\pi$
$809$	113.147			0.139861			0.0699303	+	0.997552i	$0.477722\pi$
$810$		0			0
$811$		−	134.182i		−	0.165453i	−0.996572	−	0.0827264i	$-0.973637\pi$
$811$		−	134.182i		−	0.165453i	0.996572	−	0.0827264i	$-0.0263628\pi$
$812$	190.016	−	374.558i	0.234010	−	0.461279i
$813$		−	376.617i		−	0.463244i
$814$	176.235			0.216506
$815$		0			0
$816$	162.676			0.199358
$817$	240.433			0.294288
$818$	−144.796			−0.177012
$819$	178.191	−	351.249i	0.217571	−	0.428876i
$820$		0			0
$821$	491.677			0.598876			0.299438	−	0.954116i	$-0.403201\pi$
$821$	491.677			0.598876			0.299438	+	0.954116i	$0.403201\pi$
$822$	−406.038			−0.493964
$823$		−	941.852i		−	1.14441i	−0.820110	−	0.572207i	$-0.806088\pi$
$823$		−	941.852i		−	1.14441i	0.820110	−	0.572207i	$-0.193912\pi$
$824$		−	497.438i		−	0.603687i
$825$		0			0
$826$	436.617	−	860.659i	0.528592	−	1.04196i
$827$		−	966.978i		−	1.16926i	−0.811300	−	0.584630i	$-0.801240\pi$
$827$		−	966.978i		−	1.16926i	0.811300	−	0.584630i	$-0.198760\pi$
$828$		−	112.368i		−	0.135710i
$829$		−	1185.53i		−	1.43007i	−0.699088	−	0.715035i	$-0.746411\pi$
$829$		−	1185.53i		−	1.43007i	0.699088	−	0.715035i	$-0.253589\pi$
$830$		0			0
$831$			503.466i			0.605856i
$832$	−150.043			−0.180340
$833$	679.546	+	928.410i	0.815781	+	1.11454i
$834$	−540.146			−0.647657
$835$		0			0
$836$		−	81.0495i		−	0.0969491i
$837$			44.7351i			0.0534469i
$838$	−553.560			−0.660573
$839$			1376.91i			1.64113i	0.571550	+	0.820567i	$0.306342\pi$
$839$			1376.91i			1.64113i	−0.571550	+	0.820567i	$0.693658\pi$
$840$		0			0
$841$	59.0000			0.0701546
$842$			501.255i			0.595315i
$843$	−32.6540			−0.0387355
$844$	−43.1472			−0.0511223
$845$		0			0
$846$		−	164.188i		−	0.194076i
$847$	−373.418	+	736.080i	−0.440871	+	0.869044i
$848$		−	148.118i		−	0.174667i
$849$	695.823			0.819580
$850$		0			0
$851$	−1328.03			−1.56055
$852$	−383.165			−0.449724
$853$	−175.006			−0.205165			−0.102582	−	0.994725i	$-0.532711\pi$
$853$	−175.006			−0.205165			−0.102582	+	0.994725i	$0.532711\pi$
$854$	74.9117	−	147.666i	0.0877186	−	0.172911i
$855$		0			0
$856$	130.794			0.152797
$857$	61.7471			0.0720503			0.0360252	−	0.999351i	$-0.488530\pi$
$857$	61.7471			0.0720503			0.0360252	+	0.999351i	$0.488530\pi$
$858$			80.7351i			0.0940968i
$859$		−	191.306i		−	0.222708i	−0.993781	−	0.111354i	$-0.964481\pi$
$859$		−	191.306i		−	0.222708i	0.993781	−	0.111354i	$-0.0355188\pi$
$860$		0			0
$861$	447.588	+	227.064i	0.519847	+	0.263721i
$862$			827.720i			0.960232i
$863$		−	494.507i		−	0.573009i	−0.958079	−	0.286504i	$-0.907507\pi$
$863$		−	494.507i		−	0.573009i	0.958079	−	0.286504i	$-0.0924933\pi$
$864$		−	29.3939i		−	0.0340207i
$865$		0			0
$866$		−	554.664i		−	0.640490i
$867$	−454.358			−0.524058
$868$	107.489	+	54.5299i	0.123835	+	0.0628224i
$869$	−122.705			−0.141202
$870$		0			0
$871$		−	1143.53i		−	1.31289i
$872$		−	101.657i		−	0.116579i
$873$	−155.146			−0.177716
$874$			610.751i			0.698800i
$875$		0			0
$876$	−196.617			−0.224449
$877$			454.353i			0.518077i	0.965867	+	0.259038i	$0.0834056\pi$
$877$			454.353i			0.518077i	−0.965867	+	0.259038i	$0.916594\pi$
$878$	555.097			0.632229
$879$	486.521			0.553494
$880$		0			0
$881$		−	143.493i		−	0.162875i	−0.996678	−	0.0814375i	$-0.974049\pi$
$881$		−	143.493i		−	0.162875i	0.996678	−	0.0814375i	$-0.0259511\pi$
$882$	167.754	−	122.787i	0.190197	−	0.139214i
$883$		−	927.986i		−	1.05095i	−0.850810	−	0.525473i	$-0.823888\pi$
$883$		−	927.986i		−	1.05095i	0.850810	−	0.525473i	$-0.176112\pi$
$884$	880.764			0.996339
$885$		0			0
$886$	1152.22			1.30047
$887$	−958.265			−1.08034			−0.540172	−	0.841555i	$-0.681641\pi$
$887$	−958.265			−1.08034			−0.540172	+	0.841555i	$0.681641\pi$
$888$	−347.395			−0.391210
$889$	−561.470	−	284.837i	−0.631575	−	0.320402i
$890$		0			0
$891$	−15.8162			−0.0177511
$892$	238.717			0.267620
$893$			892.410i			0.999340i
$894$			516.482i			0.577720i
$895$		0			0
$896$	−70.6274	−	35.8297i	−0.0788252	−	0.0399885i
$897$		−	608.382i		−	0.678241i
$898$		−	255.015i		−	0.283981i
$899$			258.278i			0.287295i
$900$		0			0
$901$			869.462i			0.964996i
$902$	−102.879			−0.114056
$903$	−112.736	−	57.1918i	−0.124846	−	0.0633353i
$904$	−206.558			−0.228494
$905$		0			0
$906$			177.264i			0.195655i
$907$			552.721i			0.609394i	0.952449	+	0.304697i	$0.0985553\pi$
$907$			552.721i			0.609394i	−0.952449	+	0.304697i	$0.901445\pi$
$908$	−339.686			−0.374103
$909$		−	19.8297i		−	0.0218149i
$910$		0			0
$911$	142.742			0.156687			0.0783437	−	0.996926i	$-0.475037\pi$
$911$	142.742			0.156687			0.0783437	+	0.996926i	$0.475037\pi$
$912$			159.765i			0.175180i
$913$	−11.3101			−0.0123878
$914$	396.230			0.433512
$915$		0			0
$916$			220.021i			0.240198i
$917$	−76.0063	−	38.5584i	−0.0828858	−	0.0420485i
$918$			172.544i			0.187957i
$919$	−1086.45			−1.18221			−0.591107	−	0.806593i	$-0.701309\pi$
$919$	−1086.45			−1.18221			−0.591107	+	0.806593i	$0.701309\pi$
$920$		0			0
$921$	263.823			0.286453
$922$	574.591			0.623200
$923$	−2074.54			−2.24760
$924$	−19.2792	+	38.0031i	−0.0208650	+	0.0411289i
$925$		0			0
$926$	−646.960			−0.698661
$927$	527.613			0.569161
$928$		−	169.706i		−	0.182872i
$929$		−	1550.75i		−	1.66927i	−0.550806	−	0.834633i	$-0.685680\pi$
$929$		−	1550.75i		−	1.66927i	0.550806	−	0.834633i	$-0.314320\pi$
$930$		0			0
$931$	−911.793	+	667.383i	−0.979370	+	0.716845i
$932$		−	114.530i		−	0.122886i
$933$		−	490.441i		−	0.525660i
$934$			910.345i			0.974673i
$935$		0			0
$936$		−	159.145i		−	0.170026i
$937$	−759.317			−0.810370			−0.405185	−	0.914235i	$-0.632793\pi$
$937$	−759.317			−0.810370			−0.405185	+	0.914235i	$0.632793\pi$
$938$	273.070	−	538.274i	0.291119	−	0.573853i
$939$	−84.1463			−0.0896127
$940$		0			0
$941$		−	36.3519i		−	0.0386312i	−0.999813	−	0.0193156i	$-0.993851\pi$
$941$		−	36.3519i		−	0.0386312i	0.999813	−	0.0193156i	$-0.00614873\pi$
$942$			572.382i			0.607624i
$943$	775.245			0.822105
$944$		−	389.949i		−	0.413081i
$945$		0			0
$946$	25.9126			0.0273917
$947$		−	92.5370i		−	0.0977160i	−0.998806	−	0.0488580i	$-0.984442\pi$
$947$		−	92.5370i		−	0.0977160i	0.998806	−	0.0488580i	$-0.0155582\pi$
$948$	241.875			0.255143
$949$	−1064.53			−1.12174
$950$		0			0
$951$		−	1001.17i		−	1.05276i
$952$	414.588	+	210.323i	0.435492	+	0.220927i
$953$		−	1361.29i		−	1.42843i	−0.699927	−	0.714215i	$-0.746784\pi$
$953$		−	1361.29i		−	1.42843i	0.699927	−	0.714215i	$-0.253216\pi$
$954$	157.103			0.164678
$955$		0			0
$956$	563.044			0.588958
$957$	−91.3151			−0.0954180
$958$	−238.344			−0.248794
$959$	−1034.81	−	524.964i	−1.07905	−	0.547408i
$960$		0			0
$961$	886.881			0.922873
$962$	−1880.87			−1.95517
$963$			138.728i			0.144058i
$964$			336.612i			0.349183i
$965$		0			0
$966$	145.279	−	286.374i	0.150393	−	0.296453i
$967$		−	481.677i		−	0.498115i	−0.968489	−	0.249057i	$-0.919879\pi$
$967$		−	481.677i		−	0.498115i	0.968489	−	0.249057i	$-0.0801208\pi$
$968$			333.505i			0.344530i
$969$		−	937.829i		−	0.967832i
$970$		0			0
$971$			1427.34i			1.46997i	0.678082	+	0.734987i	$0.262812\pi$
$971$			1427.34i			1.46997i	−0.678082	+	0.734987i	$0.737188\pi$
$972$	31.1769			0.0320750
$973$	−1376.59	−	698.352i	−1.41479	−	0.717730i
$974$	−3.93102			−0.00403596
$975$		0			0
$976$		−	66.9046i		−	0.0685498i
$977$			1356.26i			1.38819i	0.719883	+	0.694095i	$0.244195\pi$
$977$			1356.26i			1.38819i	−0.719883	+	0.694095i	$0.755805\pi$
$978$	−179.029			−0.183056
$979$			73.9698i			0.0755565i
$980$		0			0
$981$	107.823			0.109912
$982$		−	349.986i		−	0.356401i
$983$	1315.51			1.33826			0.669129	−	0.743146i	$-0.266667\pi$
$983$	1315.51			1.33826			0.669129	+	0.743146i	$0.266667\pi$
$984$	202.794			0.206091
$985$		0			0
$986$			996.184i			1.01033i
$987$	212.277	−	418.441i	0.215073	−	0.423952i
$988$			864.999i			0.875505i
$989$	−195.265			−0.197437
$990$		0			0
$991$	1043.28			1.05275			0.526377	−	0.850251i	$-0.323550\pi$
$991$	1043.28			1.05275			0.526377	+	0.850251i	$0.323550\pi$
$992$	48.7014			0.0490941
$993$	−575.677			−0.579736
$994$	−976.514	−	495.391i	−0.982408	−	0.498382i
$995$		0			0
$996$	22.2944			0.0223839
$997$	−820.328			−0.822796			−0.411398	−	0.911456i	$-0.634959\pi$
$997$	−820.328			−0.822796			−0.411398	+	0.911456i	$0.634959\pi$
$998$			683.667i			0.685037i
$999$		−	368.468i		−	0.368837i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.3.h.a.349.6		8
5.2	odd	4		42.3.c.a.13.2	yes	4
5.3	odd	4		1050.3.f.a.601.3		4
5.4	even	2	inner	1050.3.h.a.349.3		8
7.6	odd	2	inner	1050.3.h.a.349.7		8
15.2	even	4		126.3.c.b.55.3		4
20.7	even	4		336.3.f.c.97.2		4
35.2	odd	12		294.3.g.b.31.2		4
35.12	even	12		294.3.g.c.31.2		4
35.13	even	4		1050.3.f.a.601.4		4
35.17	even	12		294.3.g.b.19.2		4
35.27	even	4		42.3.c.a.13.1	✓	4
35.32	odd	12		294.3.g.c.19.2		4
35.34	odd	2	inner	1050.3.h.a.349.2		8
40.27	even	4		1344.3.f.e.769.3		4
40.37	odd	4		1344.3.f.f.769.1		4
60.47	odd	4		1008.3.f.g.433.1		4
105.2	even	12		882.3.n.a.325.1		4
105.17	odd	12		882.3.n.a.19.1		4
105.32	even	12		882.3.n.d.19.1		4
105.47	odd	12		882.3.n.d.325.1		4
105.62	odd	4		126.3.c.b.55.4		4
140.27	odd	4		336.3.f.c.97.3		4
280.27	odd	4		1344.3.f.e.769.2		4
280.237	even	4		1344.3.f.f.769.4		4
420.167	even	4		1008.3.f.g.433.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.3.c.a.13.1	✓	4	35.27	even	4
42.3.c.a.13.2	yes	4	5.2	odd	4
126.3.c.b.55.3		4	15.2	even	4
126.3.c.b.55.4		4	105.62	odd	4
294.3.g.b.19.2		4	35.17	even	12
294.3.g.b.31.2		4	35.2	odd	12
294.3.g.c.19.2		4	35.32	odd	12
294.3.g.c.31.2		4	35.12	even	12
336.3.f.c.97.2		4	20.7	even	4
336.3.f.c.97.3		4	140.27	odd	4
882.3.n.a.19.1		4	105.17	odd	12
882.3.n.a.325.1		4	105.2	even	12
882.3.n.d.19.1		4	105.32	even	12
882.3.n.d.325.1		4	105.47	odd	12
1008.3.f.g.433.1		4	60.47	odd	4
1008.3.f.g.433.4		4	420.167	even	4
1050.3.f.a.601.3		4	5.3	odd	4
1050.3.f.a.601.4		4	35.13	even	4
1050.3.h.a.349.2		8	35.34	odd	2	inner
1050.3.h.a.349.3		8	5.4	even	2	inner
1050.3.h.a.349.6		8	1.1	even	1	trivial
1050.3.h.a.349.7		8	7.6	odd	2	inner
1344.3.f.e.769.2		4	280.27	odd	4
1344.3.f.e.769.3		4	40.27	even	4
1344.3.f.f.769.1		4	40.37	odd	4
1344.3.f.f.769.4		4	280.237	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1050.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -1.73205 q^{3} -2.00000 q^{4} -2.44949i q^{6} +(3.16693 - 6.24264i) q^{7} -2.82843i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} -1.73205 q^{3} -2.00000 q^{4} -2.44949i q^{6} +(3.16693 - 6.24264i) q^{7} -2.82843i q^{8} +3.00000 q^{9} -1.75736 q^{11} +3.46410 q^{12} +18.7554 q^{13} +(8.82843 + 4.47871i) q^{14} +4.00000 q^{16} -23.4803 q^{17} +4.24264i q^{18} -23.0600i q^{19} +(-5.48528 + 10.8126i) q^{21} -2.48528i q^{22} +18.7279i q^{23} +4.89898i q^{24} +26.5241i q^{26} -5.19615 q^{27} +(-6.33386 + 12.4853i) q^{28} -30.0000 q^{29} -8.60927i q^{31} +5.65685i q^{32} +3.04384 q^{33} -33.2061i q^{34} -6.00000 q^{36} +70.9117i q^{37} +32.6118 q^{38} -32.4853 q^{39} -41.3951i q^{41} +(-15.2913 - 7.75736i) q^{42} +10.4264i q^{43} +3.51472 q^{44} -26.4853 q^{46} -38.6995 q^{47} -6.92820 q^{48} +(-28.9411 - 39.5400i) q^{49} +40.6690 q^{51} -37.5108 q^{52} -37.0294i q^{53} -7.34847i q^{54} +(-17.6569 - 8.95743i) q^{56} +39.9411i q^{57} -42.4264i q^{58} -97.4872i q^{59} -16.7262i q^{61} +12.1753 q^{62} +(9.50079 - 18.7279i) q^{63} -8.00000 q^{64} +4.30463i q^{66} -60.9706i q^{67} +46.9606 q^{68} -32.4377i q^{69} -110.610 q^{71} -8.48528i q^{72} -56.7585 q^{73} -100.284 q^{74} +46.1200i q^{76} +(-5.56543 + 10.9706i) q^{77} -45.9411i q^{78} +69.8234 q^{79} +9.00000 q^{81} +58.5416 q^{82} +6.43583 q^{83} +(10.9706 - 21.6251i) q^{84} -14.7452 q^{86} +51.9615 q^{87} +4.97056i q^{88} -42.0915i q^{89} +(59.3970 - 117.083i) q^{91} -37.4558i q^{92} +14.9117i q^{93} -54.7293i q^{94} -9.79796i q^{96} -51.7153 q^{97} +(55.9180 - 40.9289i) q^{98} -5.27208 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 24 q^{9}+O(q^{10}) \) 8 * q - 16 * q^4 + 24 * q^9	\( 8 q - 16 q^{4} + 24 q^{9} - 48 q^{11} + 48 q^{14} + 32 q^{16} + 24 q^{21} - 240 q^{29} - 48 q^{36} - 192 q^{39} + 96 q^{44} - 144 q^{46} + 40 q^{49} - 48 q^{51} - 96 q^{56} - 64 q^{64} - 240 q^{71} - 576 q^{74} - 256 q^{79} + 72 q^{81} - 48 q^{84} - 480 q^{86} - 144 q^{99}+O(q^{100}) \) 8 * q - 16 * q^4 + 24 * q^9 - 48 * q^11 + 48 * q^14 + 32 * q^16 + 24 * q^21 - 240 * q^29 - 48 * q^36 - 192 * q^39 + 96 * q^44 - 144 * q^46 + 40 * q^49 - 48 * q^51 - 96 * q^56 - 64 * q^64 - 240 * q^71 - 576 * q^74 - 256 * q^79 + 72 * q^81 - 48 * q^84 - 480 * q^86 - 144 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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