LMFDB - Embedded newform 1050.3.h.a.349.4

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			349.4
Root			$-0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1050.349
Dual form			1050.3.h.a.349.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421i q^{2} +1.73205 q^{3} -2.00000 q^{4} -2.44949i q^{6} +(6.63103 + 2.24264i) q^{7} +2.82843i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q-1.41421i q^{2} +1.73205 q^{3} -2.00000 q^{4} -2.44949i q^{6} +(6.63103 + 2.24264i) q^{7} +2.82843i q^{8} +3.00000 q^{9} -10.2426 q^{11} -3.46410 q^{12} -8.95743 q^{13} +(3.17157 - 9.37769i) q^{14} +4.00000 q^{16} -30.4085 q^{17} -4.24264i q^{18} -16.1318i q^{19} +(11.4853 + 3.88437i) q^{21} +14.4853i q^{22} -6.72792i q^{23} +4.89898i q^{24} +12.6677i q^{26} +5.19615 q^{27} +(-13.2621 - 4.48528i) q^{28} -30.0000 q^{29} -50.1785i q^{31} -5.65685i q^{32} -17.7408 q^{33} +43.0041i q^{34} -6.00000 q^{36} -30.9117i q^{37} -22.8138 q^{38} -15.5147 q^{39} +7.10228i q^{41} +(5.49333 - 16.2426i) q^{42} -74.4264i q^{43} +20.4853 q^{44} -9.51472 q^{46} +58.2954 q^{47} +6.92820 q^{48} +(38.9411 + 29.7420i) q^{49} -52.6690 q^{51} +17.9149 q^{52} -70.9706i q^{53} -7.34847i q^{54} +(-6.34315 + 18.7554i) q^{56} -27.9411i q^{57} +42.4264i q^{58} -0.492372i q^{59} -2.86976i q^{61} -70.9631 q^{62} +(19.8931 + 6.72792i) q^{63} -8.00000 q^{64} +25.0892i q^{66} -27.0294i q^{67} +60.8170 q^{68} -11.6531i q^{69} +50.6102 q^{71} +8.48528i q^{72} -70.6149 q^{73} -43.7157 q^{74} +32.2636i q^{76} +(-67.9193 - 22.9706i) q^{77} +21.9411i q^{78} -133.823 q^{79} +9.00000 q^{81} +10.0441 q^{82} -104.415 q^{83} +(-22.9706 - 7.76874i) q^{84} -105.255 q^{86} -51.9615 q^{87} -28.9706i q^{88} +144.970i q^{89} +(-59.3970 - 20.0883i) q^{91} +13.4558i q^{92} -86.9117i q^{93} -82.4421i q^{94} -9.79796i q^{96} +100.705 q^{97} +(42.0616 - 55.0711i) q^{98} -30.7279 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	6.63103	+	2.24264i	0.947290	+	0.320377i
$7$	6.63103	+	2.24264i	0.947290	+	0.320377i
$8$			2.82843i			0.353553i
$9$	3.00000			0.333333
$10$		0			0
$11$	−10.2426			−0.931149			−0.465575	−	0.885009i	$-0.654152\pi$
$11$	−10.2426			−0.931149			−0.465575	+	0.885009i	$0.654152\pi$
$12$	−3.46410			−0.288675
$13$	−8.95743			−0.689033			−0.344516	−	0.938780i	$-0.611957\pi$
$13$	−8.95743			−0.689033			−0.344516	+	0.938780i	$0.611957\pi$
$14$	3.17157	−	9.37769i	0.226541	−	0.669835i
$15$		0			0
$16$	4.00000			0.250000
$17$	−30.4085			−1.78873			−0.894367	−	0.447333i	$-0.852374\pi$
$17$	−30.4085			−1.78873			−0.894367	+	0.447333i	$0.852374\pi$
$18$		−	4.24264i		−	0.235702i
$19$		−	16.1318i		−	0.849043i	−0.905418	−	0.424521i	$-0.860442\pi$
$19$		−	16.1318i		−	0.849043i	0.905418	−	0.424521i	$-0.139558\pi$
$20$		0			0
$21$	11.4853	+	3.88437i	0.546918	+	0.184970i
$22$			14.4853i			0.658422i
$23$		−	6.72792i		−	0.292518i	−0.989246	−	0.146259i	$-0.953277\pi$
$23$		−	6.72792i		−	0.292518i	0.989246	−	0.146259i	$-0.0467233\pi$
$24$			4.89898i			0.204124i
$25$		0			0
$26$			12.6677i			0.487220i
$27$	5.19615			0.192450
$28$	−13.2621	−	4.48528i	−0.473645	−	0.160189i
$29$	−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
$31$		−	50.1785i		−	1.61866i	−0.587354	−	0.809330i	$-0.699830\pi$
$31$		−	50.1785i		−	1.61866i	0.587354	−	0.809330i	$-0.300170\pi$
$32$		−	5.65685i		−	0.176777i
$33$	−17.7408			−0.537599
$34$			43.0041i			1.26483i
$35$		0			0
$36$	−6.00000			−0.166667
$37$		−	30.9117i		−	0.835451i	−0.908573	−	0.417726i	$-0.862827\pi$
$37$		−	30.9117i		−	0.835451i	0.908573	−	0.417726i	$-0.137173\pi$
$38$	−22.8138			−0.600364
$39$	−15.5147			−0.397813
$40$		0			0
$41$			7.10228i			0.173226i	0.996242	+	0.0866132i	$0.0276044\pi$
$41$			7.10228i			0.173226i	−0.996242	+	0.0866132i	$0.972396\pi$
$42$	5.49333	−	16.2426i	0.130793	−	0.386730i
$43$		−	74.4264i		−	1.73085i	−0.501041	−	0.865423i	$-0.667050\pi$
$43$		−	74.4264i		−	1.73085i	0.501041	−	0.865423i	$-0.332950\pi$
$44$	20.4853			0.465575
$45$		0			0
$46$	−9.51472			−0.206842
$47$	58.2954			1.24033			0.620164	−	0.784472i	$-0.287066\pi$
$47$	58.2954			1.24033			0.620164	+	0.784472i	$0.287066\pi$
$48$	6.92820			0.144338
$49$	38.9411	+	29.7420i	0.794717	+	0.606980i
$50$		0			0
$51$	−52.6690			−1.03273
$52$	17.9149			0.344516
$53$		−	70.9706i		−	1.33907i	−0.742782	−	0.669534i	$-0.766494\pi$
$53$		−	70.9706i		−	1.33907i	0.742782	−	0.669534i	$-0.233506\pi$
$54$		−	7.34847i		−	0.136083i
$55$		0			0
$56$	−6.34315	+	18.7554i	−0.113270	+	0.334918i
$57$		−	27.9411i		−	0.490195i
$58$			42.4264i			0.731490i
$59$		−	0.492372i		−	0.00834529i	−0.999991	−	0.00417265i	$-0.998672\pi$
$59$		−	0.492372i		−	0.00834529i	0.999991	−	0.00417265i	$-0.00132820\pi$
$60$		0			0
$61$		−	2.86976i		−	0.0470452i	−0.999723	−	0.0235226i	$-0.992512\pi$
$61$		−	2.86976i		−	0.0470452i	0.999723	−	0.0235226i	$-0.00748816\pi$
$62$	−70.9631			−1.14457
$63$	19.8931	+	6.72792i	0.315763	+	0.106792i
$64$	−8.00000			−0.125000
$65$		0			0
$66$			25.0892i			0.380140i
$67$		−	27.0294i		−	0.403424i	−0.979445	−	0.201712i	$-0.935349\pi$
$67$		−	27.0294i		−	0.403424i	0.979445	−	0.201712i	$-0.0646506\pi$
$68$	60.8170			0.894367
$69$		−	11.6531i		−	0.168886i
$70$		0			0
$71$	50.6102			0.712819			0.356410	−	0.934330i	$-0.384001\pi$
$71$	50.6102			0.712819			0.356410	+	0.934330i	$0.384001\pi$
$72$			8.48528i			0.117851i
$73$	−70.6149			−0.967328			−0.483664	−	0.875254i	$-0.660694\pi$
$73$	−70.6149			−0.967328			−0.483664	+	0.875254i	$0.660694\pi$
$74$	−43.7157			−0.590753
$75$		0			0
$76$			32.2636i			0.424521i
$77$	−67.9193	−	22.9706i	−0.882068	−	0.298319i
$78$			21.9411i			0.281296i
$79$	−133.823			−1.69397			−0.846983	−	0.531619i	$-0.821584\pi$
$79$	−133.823			−1.69397			−0.846983	+	0.531619i	$0.821584\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$	10.0441			0.122490
$83$	−104.415			−1.25802			−0.629009	−	0.777398i	$-0.716539\pi$
$83$	−104.415			−1.25802			−0.629009	+	0.777398i	$0.716539\pi$
$84$	−22.9706	−	7.76874i	−0.273459	−	0.0924849i
$85$		0			0
$86$	−105.255			−1.22389
$87$	−51.9615			−0.597259
$88$		−	28.9706i		−	0.329211i
$89$			144.970i			1.62888i	0.580250	+	0.814438i	$0.302955\pi$
$89$			144.970i			1.62888i	−0.580250	+	0.814438i	$0.697045\pi$
$90$		0			0
$91$	−59.3970	−	20.0883i	−0.652714	−	0.220750i
$92$			13.4558i			0.146259i
$93$		−	86.9117i		−	0.934534i
$94$		−	82.4421i		−	0.877044i
$95$		0			0
$96$		−	9.79796i		−	0.102062i
$97$	100.705			1.03820			0.519099	−	0.854714i	$-0.326268\pi$
$97$	100.705			1.03820			0.519099	+	0.854714i	$0.326268\pi$
$98$	42.0616	−	55.0711i	0.429200	−	0.561950i
$99$	−30.7279			−0.310383
$100$		0			0
$101$			138.882i			1.37507i	0.726150	+	0.687536i	$0.241308\pi$
$101$			138.882i			1.37507i	−0.726150	+	0.687536i	$0.758692\pi$
$102$			74.4853i			0.730248i
$103$	78.8760			0.765787			0.382893	−	0.923793i	$-0.374928\pi$
$103$	78.8760			0.765787			0.382893	+	0.923793i	$0.374928\pi$
$104$		−	25.3354i		−	0.243610i
$105$		0			0
$106$	−100.368			−0.946864
$107$			37.7574i			0.352873i	0.984312	+	0.176436i	$0.0564569\pi$
$107$			37.7574i			0.352873i	−0.984312	+	0.176436i	$0.943543\pi$
$108$	−10.3923			−0.0962250
$109$	−31.9411			−0.293038			−0.146519	−	0.989208i	$-0.546807\pi$
$109$	−31.9411			−0.293038			−0.146519	+	0.989208i	$0.546807\pi$
$110$		0			0
$111$		−	53.5406i		−	0.482348i
$112$	26.5241	+	8.97056i	0.236823	+	0.0800943i
$113$		−	106.971i		−	0.946642i	−0.880890	−	0.473321i	$-0.843055\pi$
$113$		−	106.971i		−	0.946642i	0.880890	−	0.473321i	$-0.156945\pi$
$114$	−39.5147			−0.346620
$115$		0			0
$116$	60.0000			0.517241
$117$	−26.8723			−0.229678
$118$	−0.696320			−0.00590101
$119$	−201.640	−	68.1953i	−1.69445	−	0.573070i
$120$		0			0
$121$	−16.0883			−0.132961
$122$	−4.05845			−0.0332660
$123$			12.3015i			0.100012i
$124$			100.357i			0.809330i
$125$		0			0
$126$	9.51472	−	28.1331i	0.0755136	−	0.223278i
$127$		−	22.0589i		−	0.173692i	−0.996222	−	0.0868460i	$-0.972321\pi$
$127$		−	22.0589i		−	0.173692i	0.996222	−	0.0868460i	$-0.0276788\pi$
$128$			11.3137i			0.0883883i
$129$		−	128.910i		−	0.999305i
$130$		0			0
$131$			70.9631i			0.541703i	0.962621	+	0.270852i	$0.0873052\pi$
$131$			70.9631i			0.541703i	−0.962621	+	0.270852i	$0.912695\pi$
$132$	35.4815			0.268800
$133$	36.1779	−	106.971i	0.272014	−	0.804290i
$134$	−38.2254			−0.285264
$135$		0			0
$136$		−	86.0082i		−	0.632413i
$137$			105.765i			0.772004i	0.922498	+	0.386002i	$0.126144\pi$
$137$			105.765i			0.772004i	−0.922498	+	0.386002i	$0.873856\pi$
$138$	−16.4800			−0.119420
$139$			181.322i			1.30447i	0.758015	+	0.652237i	$0.226169\pi$
$139$			181.322i			1.30447i	−0.758015	+	0.652237i	$0.773831\pi$
$140$		0			0
$141$	100.971			0.716103
$142$		−	71.5736i		−	0.504039i
$143$	91.7477			0.641592
$144$	12.0000			0.0833333
$145$		0			0
$146$			99.8646i			0.684004i
$147$	67.4480	+	51.5147i	0.458830	+	0.350440i
$148$			61.8234i			0.417726i
$149$	−41.1472			−0.276156			−0.138078	−	0.990421i	$-0.544092\pi$
$149$	−41.1472			−0.276156			−0.138078	+	0.990421i	$0.544092\pi$
$150$		0			0
$151$	80.3675			0.532235			0.266118	−	0.963941i	$-0.414259\pi$
$151$	80.3675			0.532235			0.266118	+	0.963941i	$0.414259\pi$
$152$	45.6277			0.300182
$153$	−91.2255			−0.596245
$154$	−32.4853	+	96.0523i	−0.210943	+	0.623716i
$155$		0			0
$156$	31.0294			0.198907
$157$	57.3106			0.365036			0.182518	−	0.983203i	$-0.441575\pi$
$157$	57.3106			0.365036			0.182518	+	0.983203i	$0.441575\pi$
$158$			189.255i			1.19782i
$159$		−	122.925i		−	0.773111i
$160$		0			0
$161$	15.0883	−	44.6131i	0.0937162	−	0.277100i
$162$		−	12.7279i		−	0.0785674i
$163$		−	174.912i		−	1.07308i	−0.843876	−	0.536539i	$-0.819731\pi$
$163$		−	174.912i		−	1.07308i	0.843876	−	0.536539i	$-0.180269\pi$
$164$		−	14.2046i		−	0.0866132i
$165$		0			0
$166$			147.666i			0.889552i
$167$	−196.163			−1.17463			−0.587315	−	0.809359i	$-0.699815\pi$
$167$	−196.163			−1.17463			−0.587315	+	0.809359i	$0.699815\pi$
$168$	−10.9867	+	32.4853i	−0.0653967	+	0.193365i
$169$	−88.7645			−0.525234
$170$		0			0
$171$		−	48.3954i		−	0.283014i
$172$			148.853i			0.865423i
$173$	−38.5254			−0.222690			−0.111345	−	0.993782i	$-0.535516\pi$
$173$	−38.5254			−0.222690			−0.111345	+	0.993782i	$0.535516\pi$
$174$			73.4847i			0.422326i
$175$		0			0
$176$	−40.9706			−0.232787
$177$		−	0.852814i		−	0.00481816i
$178$	205.019			1.15179
$179$	191.095			1.06757			0.533786	−	0.845619i	$-0.320769\pi$
$179$	191.095			1.06757			0.533786	+	0.845619i	$0.320769\pi$
$180$		0			0
$181$		−	120.793i		−	0.667367i	−0.942685	−	0.333683i	$-0.891708\pi$
$181$		−	120.793i		−	0.667367i	0.942685	−	0.333683i	$-0.108292\pi$
$182$	−28.4091	+	84.0000i	−0.156094	+	0.461538i
$183$		−	4.97056i		−	0.0271615i
$184$	19.0294			0.103421
$185$		0			0
$186$	−122.912			−0.660816
$187$	311.463			1.66558
$188$	−116.591			−0.620164
$189$	34.4558	+	11.6531i	0.182306	+	0.0616566i
$190$		0			0
$191$	112.066			0.586733			0.293367	−	0.956000i	$-0.405224\pi$
$191$	112.066			0.586733			0.293367	+	0.956000i	$0.405224\pi$
$192$	−13.8564			−0.0721688
$193$		−	22.9117i		−	0.118713i	−0.998237	−	0.0593567i	$-0.981095\pi$
$193$		−	22.9117i		−	0.118713i	0.998237	−	0.0593567i	$-0.0189049\pi$
$194$		−	142.419i		−	0.734116i
$195$		0			0
$196$	−77.8823	−	59.4841i	−0.397358	−	0.303490i
$197$			116.059i			0.589131i	0.955631	+	0.294566i	$0.0951750\pi$
$197$			116.059i			0.589131i	−0.955631	+	0.294566i	$0.904825\pi$
$198$			43.4558i			0.219474i
$199$			163.101i			0.819604i	0.912174	+	0.409802i	$0.134402\pi$
$199$			163.101i			0.819604i	−0.912174	+	0.409802i	$0.865598\pi$
$200$		0			0
$201$		−	46.8164i		−	0.232917i
$202$	196.409			0.972323
$203$	−198.931	−	67.2792i	−0.979955	−	0.331425i
$204$	105.338			0.516363
$205$		0			0
$206$		−	111.548i		−	0.541493i
$207$		−	20.1838i		−	0.0975061i
$208$	−35.8297			−0.172258
$209$			165.232i			0.790586i
$210$		0			0
$211$	106.426			0.504391			0.252195	−	0.967676i	$-0.418847\pi$
$211$	106.426			0.504391			0.252195	+	0.967676i	$0.418847\pi$
$212$			141.941i			0.669534i
$213$	87.6594			0.411546
$214$	53.3970			0.249519
$215$		0			0
$216$			14.6969i			0.0680414i
$217$	112.532	−	332.735i	0.518582	−	1.53334i
$218$			45.1716i			0.207209i
$219$	−122.309			−0.558487
$220$		0			0
$221$	272.382			1.23250
$222$	−75.7179			−0.341071
$223$	−57.0047			−0.255627			−0.127813	−	0.991798i	$-0.540796\pi$
$223$	−57.0047			−0.255627			−0.127813	+	0.991798i	$0.540796\pi$
$224$	12.6863	−	37.5108i	0.0566352	−	0.167459i
$225$		0			0
$226$	−151.279			−0.669377
$227$	−287.418			−1.26616			−0.633080	−	0.774086i	$-0.718210\pi$
$227$	−287.418			−1.26616			−0.633080	+	0.774086i	$0.718210\pi$
$228$			55.8823i			0.245098i
$229$			139.405i			0.608754i	0.952552	+	0.304377i	$0.0984482\pi$
$229$			139.405i			0.608754i	−0.952552	+	0.304377i	$0.901552\pi$
$230$		0			0
$231$	−117.640	−	39.7862i	−0.509262	−	0.172235i
$232$		−	84.8528i		−	0.365745i
$233$			362.735i			1.55680i	0.627767	+	0.778401i	$0.283969\pi$
$233$			362.735i			1.55680i	−0.627767	+	0.778401i	$0.716031\pi$
$234$			38.0031i			0.162407i
$235$		0			0
$236$			0.984744i			0.00417265i
$237$	−231.789			−0.978012
$238$	−96.4427	+	285.161i	−0.405222	+	1.19816i
$239$	−18.4781			−0.0773144			−0.0386572	−	0.999253i	$-0.512308\pi$
$239$	−18.4781			−0.0773144			−0.0386572	+	0.999253i	$0.512308\pi$
$240$		0			0
$241$			178.104i			0.739021i	0.929227	+	0.369510i	$0.120475\pi$
$241$			178.104i			0.739021i	−0.929227	+	0.369510i	$0.879525\pi$
$242$			22.7523i			0.0940178i
$243$	15.5885			0.0641500
$244$			5.73951i			0.0235226i
$245$		0			0
$246$	17.3970			0.0707194
$247$			144.500i			0.585018i
$248$	141.926			0.572283
$249$	−180.853			−0.726317
$250$		0			0
$251$		−	50.6709i		−	0.201876i	−0.994893	−	0.100938i	$-0.967816\pi$
$251$		−	50.6709i		−	0.201876i	0.994893	−	0.100938i	$-0.0321844\pi$
$252$	−39.7862	−	13.4558i	−0.157882	−	0.0533962i
$253$			68.9117i			0.272378i
$254$	−31.1960			−0.122819
$255$		0			0
$256$	16.0000			0.0625000
$257$	−187.584			−0.729898			−0.364949	−	0.931028i	$-0.618914\pi$
$257$	−187.584			−0.729898			−0.364949	+	0.931028i	$0.618914\pi$
$258$	−182.307			−0.706615
$259$	69.3238	−	204.976i	0.267659	−	0.791414i
$260$		0			0
$261$	−90.0000			−0.344828
$262$	100.357			0.383042
$263$		−	402.978i		−	1.53223i	−0.642701	−	0.766117i	$-0.722186\pi$
$263$		−	402.978i		−	1.53223i	0.642701	−	0.766117i	$-0.277814\pi$
$264$		−	50.1785i		−	0.190070i
$265$		0			0
$266$	−151.279	−	51.1632i	−0.568719	−	0.192343i
$267$			251.095i			0.940432i
$268$			54.0589i			0.201712i
$269$		−	480.538i		−	1.78639i	−0.449674	−	0.893193i	$-0.648460\pi$
$269$		−	480.538i		−	1.78639i	0.449674	−	0.893193i	$-0.351540\pi$
$270$		0			0
$271$			37.3068i			0.137664i	0.997628	+	0.0688318i	$0.0219272\pi$
$271$			37.3068i			0.137664i	−0.997628	+	0.0688318i	$0.978073\pi$
$272$	−121.634			−0.447184
$273$	−102.879	−	34.7939i	−0.376845	−	0.127450i
$274$	149.574			0.545889
$275$		0			0
$276$			23.3062i			0.0844428i
$277$			82.6762i			0.298470i	0.988802	+	0.149235i	$0.0476811\pi$
$277$			82.6762i			0.298470i	−0.988802	+	0.149235i	$0.952319\pi$
$278$	256.428			0.922403
$279$		−	150.535i		−	0.539554i
$280$		0			0
$281$	−150.853			−0.536843			−0.268421	−	0.963302i	$-0.586502\pi$
$281$	−150.853			−0.536843			−0.268421	+	0.963302i	$0.586502\pi$
$282$		−	142.794i		−	0.506361i
$283$	284.158			1.00409			0.502046	−	0.864841i	$-0.332581\pi$
$283$	284.158			1.00409			0.502046	+	0.864841i	$0.332581\pi$
$284$	−101.220			−0.356410
$285$		0			0
$286$		−	129.751i		−	0.453674i
$287$	−15.9279	+	47.0955i	−0.0554978	+	0.164096i
$288$		−	16.9706i		−	0.0589256i
$289$	635.676			2.19957
$290$		0			0
$291$	174.426			0.599403
$292$	141.230			0.483664
$293$	−537.237			−1.83357			−0.916786	−	0.399379i	$-0.869226\pi$
$293$	−537.237			−1.83357			−0.916786	+	0.399379i	$0.869226\pi$
$294$	72.8528	−	95.3859i	0.247799	−	0.324442i
$295$		0			0
$296$	87.4315			0.295377
$297$	−53.2223			−0.179200
$298$			58.1909i			0.195272i
$299$			60.2649i			0.201555i
$300$		0			0
$301$	166.912	−	493.524i	0.554524	−	1.63961i
$302$		−	113.657i		−	0.376347i
$303$			240.551i			0.793899i
$304$		−	64.5273i		−	0.212261i
$305$		0			0
$306$			129.012i			0.421609i
$307$	34.7430			0.113169			0.0565847	−	0.998398i	$-0.481979\pi$
$307$	34.7430			0.113169			0.0565847	+	0.998398i	$0.481979\pi$
$308$	135.839	+	45.9411i	0.441034	+	0.149159i
$309$	136.617			0.442127
$310$		0			0
$311$			89.1664i			0.286709i	0.989671	+	0.143354i	$0.0457888\pi$
$311$			89.1664i			0.286709i	−0.989671	+	0.143354i	$0.954211\pi$
$312$		−	43.8823i		−	0.140648i
$313$	519.700			1.66038			0.830191	−	0.557479i	$-0.188231\pi$
$313$	519.700			1.66038			0.830191	+	0.557479i	$0.188231\pi$
$314$		−	81.0495i		−	0.258119i
$315$		0			0
$316$	267.647			0.846983
$317$		−	542.029i		−	1.70987i	−0.518736	−	0.854935i	$-0.673597\pi$
$317$		−	542.029i		−	1.70987i	0.518736	−	0.854935i	$-0.326403\pi$
$318$	−173.842			−0.546672
$319$	307.279			0.963258
$320$		0			0
$321$			65.3977i			0.203731i
$322$	−63.0924	−	21.3381i	−0.195939	−	0.0662674i
$323$			490.544i			1.51871i
$324$	−18.0000			−0.0555556
$325$		0			0
$326$	−247.362			−0.758781
$327$	−55.3237			−0.169185
$328$	−20.0883			−0.0612448
$329$	386.558	+	130.736i	1.17495	+	0.397373i
$330$		0			0
$331$	179.632			0.542696			0.271348	−	0.962481i	$-0.412531\pi$
$331$	179.632			0.542696			0.271348	+	0.962481i	$0.412531\pi$
$332$	208.831			0.629009
$333$		−	92.7351i		−	0.278484i
$334$			277.417i			0.830588i
$335$		0			0
$336$	45.9411	+	15.5375i	0.136730	+	0.0462425i
$337$		−	291.823i		−	0.865945i	−0.901407	−	0.432972i	$-0.857465\pi$
$337$		−	291.823i		−	0.865945i	0.901407	−	0.432972i	$-0.142535\pi$
$338$			125.532i			0.371396i
$339$		−	185.278i		−	0.546544i
$340$		0			0
$341$			513.960i			1.50721i
$342$	−68.4415			−0.200121
$343$	191.519	+	284.551i	0.558365	+	0.829596i
$344$	210.510			0.611947
$345$		0			0
$346$			54.4831i			0.157466i
$347$		−	452.080i		−	1.30283i	−0.758724	−	0.651413i	$-0.774177\pi$
$347$		−	452.080i		−	1.30283i	0.758724	−	0.651413i	$-0.225823\pi$
$348$	103.923			0.298629
$349$			235.067i			0.673543i	0.941586	+	0.336772i	$0.109335\pi$
$349$			235.067i			0.673543i	−0.941586	+	0.336772i	$0.890665\pi$
$350$		0			0
$351$	−46.5442			−0.132604
$352$			57.9411i			0.164605i
$353$	−25.2458			−0.0715179			−0.0357590	−	0.999360i	$-0.511385\pi$
$353$	−25.2458			−0.0715179			−0.0357590	+	0.999360i	$0.511385\pi$
$354$	−1.20606			−0.00340695
$355$		0			0
$356$		−	289.940i		−	0.814438i
$357$	−349.250	−	118.118i	−0.978291	−	0.330862i
$358$		−	270.250i		−	0.754888i
$359$	106.243			0.295941			0.147970	−	0.988992i	$-0.452726\pi$
$359$	106.243			0.295941			0.147970	+	0.988992i	$0.452726\pi$
$360$		0			0
$361$	100.765			0.279126
$362$	−170.828			−0.471900
$363$	−27.8658			−0.0767652
$364$	118.794	+	40.1766i	0.326357	+	0.110375i
$365$		0			0
$366$	−7.02944			−0.0192061
$367$	286.416			0.780426			0.390213	−	0.920725i	$-0.372401\pi$
$367$	286.416			0.780426			0.390213	+	0.920725i	$0.372401\pi$
$368$		−	26.9117i		−	0.0731296i
$369$			21.3068i			0.0577421i
$370$		0			0
$371$	159.161	−	470.608i	0.429007	−	1.26849i
$372$			173.823i			0.467267i
$373$		−	423.470i		−	1.13531i	−0.823267	−	0.567654i	$-0.807851\pi$
$373$		−	423.470i		−	1.13531i	0.823267	−	0.567654i	$-0.192149\pi$
$374$		−	440.476i		−	1.17774i
$375$		0			0
$376$			164.884i			0.438522i
$377$	268.723			0.712793
$378$	16.4800	−	48.7279i	0.0435978	−	0.128910i
$379$	−101.103			−0.266761			−0.133381	−	0.991065i	$-0.542583\pi$
$379$	−101.103			−0.266761			−0.133381	+	0.991065i	$0.542583\pi$
$380$		0			0
$381$		−	38.2071i		−	0.100281i
$382$		−	158.485i		−	0.414883i
$383$	220.394			0.575442			0.287721	−	0.957714i	$-0.407102\pi$
$383$	220.394			0.575442			0.287721	+	0.957714i	$0.407102\pi$
$384$			19.5959i			0.0510310i
$385$		0			0
$386$	−32.4020			−0.0839431
$387$		−	223.279i		−	0.576949i
$388$	−201.410			−0.519099
$389$	383.470			0.985784			0.492892	−	0.870090i	$-0.335940\pi$
$389$	383.470			0.985784			0.492892	+	0.870090i	$0.335940\pi$
$390$		0			0
$391$			204.586i			0.523238i
$392$	−84.1232	+	110.142i	−0.214600	+	0.280975i
$393$			122.912i			0.312752i
$394$	164.132			0.416579
$395$		0			0
$396$	61.4558			0.155192
$397$	485.178			1.22211			0.611056	−	0.791588i	$-0.290745\pi$
$397$	485.178			1.22211			0.611056	+	0.791588i	$0.290745\pi$
$398$	230.660			0.579548
$399$	62.6619	−	185.278i	0.157047	−	0.464357i
$400$		0			0
$401$	−253.176			−0.631361			−0.315680	−	0.948866i	$-0.602233\pi$
$401$	−253.176			−0.631361			−0.315680	+	0.948866i	$0.602233\pi$
$402$	−66.2083			−0.164697
$403$			449.470i			1.11531i
$404$		−	277.765i		−	0.687536i
$405$		0			0
$406$	−95.1472	+	281.331i	−0.234353	+	0.692933i
$407$			316.617i			0.777930i
$408$		−	148.971i		−	0.365124i
$409$			5.39135i			0.0131818i	0.999978	+	0.00659089i	$0.00209796\pi$
$409$			5.39135i			0.0131818i	−0.999978	+	0.00659089i	$0.997902\pi$
$410$		0			0
$411$			183.189i			0.445717i
$412$	−157.752			−0.382893
$413$	1.10421	−	3.26494i	0.00267364	−	0.00790541i
$414$	−28.5442			−0.0689472
$415$		0			0
$416$			50.6709i			0.121805i
$417$			314.059i			0.753139i
$418$	233.674			0.559028
$419$			294.431i			0.702700i	0.936244	+	0.351350i	$0.114277\pi$
$419$			294.431i			0.702700i	−0.936244	+	0.351350i	$0.885723\pi$
$420$		0			0
$421$	−290.441			−0.689883			−0.344941	−	0.938624i	$-0.612101\pi$
$421$	−290.441			−0.689883			−0.344941	+	0.938624i	$0.612101\pi$
$422$		−	150.510i		−	0.356658i
$423$	174.886			0.413442
$424$	200.735			0.473432
$425$		0			0
$426$		−	123.969i		−	0.291007i
$427$	6.43583	−	19.0294i	0.0150722	−	0.0445654i
$428$		−	75.5147i		−	0.176436i
$429$	158.912			0.370424
$430$		0			0
$431$	50.7136			0.117665			0.0588325	−	0.998268i	$-0.481262\pi$
$431$	50.7136			0.117665			0.0588325	+	0.998268i	$0.481262\pi$
$432$	20.7846			0.0481125
$433$	−724.761			−1.67381			−0.836906	−	0.547347i	$-0.815638\pi$
$433$	−724.761			−1.67381			−0.836906	+	0.547347i	$0.815638\pi$
$434$	−470.558	−	159.145i	−1.08424	−	0.366693i
$435$		0			0
$436$	63.8823			0.146519
$437$	−108.534			−0.248361
$438$			172.971i			0.394910i
$439$		−	371.728i		−	0.846761i	−0.905952	−	0.423381i	$-0.860843\pi$
$439$		−	371.728i		−	0.846761i	0.905952	−	0.423381i	$-0.139157\pi$
$440$		0			0
$441$	116.823	+	89.2261i	0.264906	+	0.202327i
$442$		−	385.206i		−	0.871507i
$443$		−	229.258i		−	0.517512i	−0.965943	−	0.258756i	$-0.916687\pi$
$443$		−	229.258i		−	0.517512i	0.965943	−	0.258756i	$-0.0833125\pi$
$444$			107.081i			0.241174i
$445$		0			0
$446$			80.6168i			0.180755i
$447$	−71.2690			−0.159439
$448$	−53.0482	−	17.9411i	−0.118411	−	0.0400472i
$449$	600.323			1.33702			0.668511	−	0.743702i	$-0.266932\pi$
$449$	600.323			1.33702			0.668511	+	0.743702i	$0.266932\pi$
$450$		0			0
$451$		−	72.7461i		−	0.161300i
$452$			213.941i			0.473321i
$453$	139.201			0.307286
$454$			406.471i			0.895311i
$455$		0			0
$456$	79.0294			0.173310
$457$		−	483.823i		−	1.05869i	−0.848405	−	0.529347i	$-0.822437\pi$
$457$		−	483.823i		−	1.05869i	0.848405	−	0.529347i	$-0.177563\pi$
$458$	197.148			0.430454
$459$	−158.007			−0.344242
$460$		0			0
$461$		−	274.661i		−	0.595794i	−0.954598	−	0.297897i	$-0.903715\pi$
$461$		−	274.661i		−	0.595794i	0.954598	−	0.297897i	$-0.0962852\pi$
$462$	−56.2662	+	166.368i	−0.121788	+	0.360103i
$463$		−	153.470i		−	0.331469i	−0.986170	−	0.165734i	$-0.947001\pi$
$463$		−	153.470i		−	0.331469i	0.986170	−	0.165734i	$-0.0529995\pi$
$464$	−120.000			−0.258621
$465$		0			0
$466$	512.985			1.10083
$467$	61.7420			0.132210			0.0661049	−	0.997813i	$-0.478943\pi$
$467$	61.7420			0.132210			0.0661049	+	0.997813i	$0.478943\pi$
$468$	53.7446			0.114839
$469$	60.6173	−	179.233i	0.129248	−	0.382160i
$470$		0			0
$471$	99.2649			0.210754
$472$	1.39264			0.00295051
$473$			762.323i			1.61168i
$474$			327.799i			0.691559i
$475$		0			0
$476$	403.279	+	136.391i	0.847225	+	0.286535i
$477$		−	212.912i		−	0.446356i
$478$			26.1320i			0.0546695i
$479$			556.514i			1.16182i	0.813966	+	0.580912i	$0.197304\pi$
$479$			556.514i			1.16182i	−0.813966	+	0.580912i	$0.802696\pi$
$480$		0			0
$481$			276.889i			0.575653i
$482$	251.877			0.522567
$483$	26.1337	−	77.2721i	0.0541071	−	0.159984i
$484$	32.1766			0.0664806
$485$		0			0
$486$		−	22.0454i		−	0.0453609i
$487$			325.220i			0.667804i	0.942608	+	0.333902i	$0.108365\pi$
$487$			325.220i			0.667804i	−0.942608	+	0.333902i	$0.891635\pi$
$488$	8.11689			0.0166330
$489$		−	302.956i		−	0.619542i
$490$		0			0
$491$	643.477			1.31054			0.655272	−	0.755393i	$-0.272554\pi$
$491$	643.477			1.31054			0.655272	+	0.755393i	$0.272554\pi$
$492$		−	24.6030i		−	0.0500062i
$493$	912.255			1.85042
$494$	204.353			0.413671
$495$		0			0
$496$		−	200.714i		−	0.404665i
$497$	335.598	+	113.500i	0.675247	+	0.228371i
$498$			255.765i			0.513583i
$499$	−755.426			−1.51388			−0.756939	−	0.653485i	$-0.773306\pi$
$499$	−755.426			−1.51388			−0.756939	+	0.653485i	$0.773306\pi$
$500$		0			0
$501$	−339.765			−0.678173
$502$	−71.6594			−0.142748
$503$	−509.409			−1.01274			−0.506371	−	0.862316i	$-0.669013\pi$
$503$	−509.409			−1.01274			−0.506371	+	0.862316i	$0.669013\pi$
$504$	−19.0294	+	56.2662i	−0.0377568	+	0.111639i
$505$		0			0
$506$	97.4558			0.192600
$507$	−153.745			−0.303244
$508$			44.1177i			0.0868460i
$509$		−	769.433i		−	1.51166i	−0.654770	−	0.755828i	$-0.727234\pi$
$509$		−	769.433i		−	1.51166i	0.654770	−	0.755828i	$-0.272766\pi$
$510$		0			0
$511$	−468.250	−	158.364i	−0.916340	−	0.309910i
$512$		−	22.6274i		−	0.0441942i
$513$		−	83.8234i		−	0.163398i
$514$			265.283i			0.516116i
$515$		0			0
$516$			257.821i			0.499652i
$517$	−597.099			−1.15493
$518$	−289.880	−	98.0387i	−0.559615	−	0.189264i
$519$	−66.7279			−0.128570
$520$		0			0
$521$		−	535.207i		−	1.02727i	−0.858009	−	0.513635i	$-0.828299\pi$
$521$		−	535.207i		−	1.02727i	0.858009	−	0.513635i	$-0.171701\pi$
$522$			127.279i			0.243830i
$523$	839.466			1.60510			0.802549	−	0.596586i	$-0.203477\pi$
$523$	839.466			1.60510			0.802549	+	0.596586i	$0.203477\pi$
$524$		−	141.926i		−	0.270852i
$525$		0			0
$526$	−569.897			−1.08345
$527$			1525.85i			2.89535i
$528$	−70.9631			−0.134400
$529$	483.735			0.914433
$530$		0			0
$531$		−	1.47712i		−	0.00278176i
$532$	−72.3557	+	213.941i	−0.136007	+	0.402145i
$533$		−	63.6182i		−	0.119359i
$534$	355.103			0.664986
$535$		0			0
$536$	76.4508			0.142632
$537$	330.987			0.616363
$538$	−679.583			−1.26317
$539$	−398.860	−	304.637i	−0.740000	−	0.565189i
$540$		0			0
$541$	285.852			0.528377			0.264188	−	0.964471i	$-0.414896\pi$
$541$	285.852			0.528377			0.264188	+	0.964471i	$0.414896\pi$
$542$	52.7598			0.0973428
$543$		−	209.220i		−	0.385305i
$544$			172.016i			0.316207i
$545$		0			0
$546$	−49.2061	+	145.492i	−0.0901210	+	0.266469i
$547$			741.470i			1.35552i	0.735283	+	0.677761i	$0.237049\pi$
$547$			741.470i			1.35552i	−0.735283	+	0.677761i	$0.762951\pi$
$548$		−	211.529i		−	0.386002i
$549$		−	8.60927i		−	0.0156817i
$550$		0			0
$551$			483.954i			0.878320i
$552$	32.9600			0.0597101
$553$	−887.387	−	300.118i	−1.60468	−	0.542708i
$554$	116.922			0.211050
$555$		0			0
$556$		−	362.644i		−	0.652237i
$557$		−	834.853i		−	1.49884i	−0.662096	−	0.749419i	$-0.730333\pi$
$557$		−	834.853i		−	1.49884i	0.662096	−	0.749419i	$-0.269667\pi$
$558$	−212.889			−0.381522
$559$			666.669i			1.19261i
$560$		0			0
$561$	539.470			0.961622
$562$			213.338i			0.379605i
$563$	−417.781			−0.742062			−0.371031	−	0.928620i	$-0.620996\pi$
$563$	−417.781			−0.742062			−0.371031	+	0.928620i	$0.620996\pi$
$564$	−201.941			−0.358052
$565$		0			0
$566$		−	401.861i		−	0.710001i
$567$	59.6793	+	20.1838i	0.105254	+	0.0355975i
$568$			143.147i			0.252020i
$569$	−673.029			−1.18283			−0.591414	−	0.806368i	$-0.701430\pi$
$569$	−673.029			−1.18283			−0.591414	+	0.806368i	$0.701430\pi$
$570$		0			0
$571$	−265.088			−0.464253			−0.232126	−	0.972686i	$-0.574568\pi$
$571$	−265.088			−0.464253			−0.232126	+	0.972686i	$0.574568\pi$
$572$	−183.495			−0.320796
$573$	194.104			0.338750
$574$	66.6030	+	22.5254i	0.116033	+	0.0392429i
$575$		0			0
$576$	−24.0000			−0.0416667
$577$	−468.740			−0.812375			−0.406188	−	0.913790i	$-0.633142\pi$
$577$	−468.740			−0.812375			−0.406188	+	0.913790i	$0.633142\pi$
$578$		−	898.982i		−	1.55533i
$579$		−	39.6842i		−	0.0685392i
$580$		0			0
$581$	−692.382	−	234.166i	−1.19171	−	0.403040i
$582$		−	246.676i		−	0.423842i
$583$			726.926i			1.24687i
$584$		−	199.729i		−	0.342002i
$585$		0			0
$586$			759.767i			1.29653i
$587$	120.530			0.205332			0.102666	−	0.994716i	$-0.467263\pi$
$587$	120.530			0.205332			0.102666	+	0.994716i	$0.467263\pi$
$588$	−134.896	−	103.029i	−0.229415	−	0.175220i
$589$	−809.470			−1.37431
$590$		0			0
$591$			201.020i			0.340135i
$592$		−	123.647i		−	0.208863i
$593$	561.468			0.946826			0.473413	−	0.880841i	$-0.343022\pi$
$593$	561.468			0.946826			0.473413	+	0.880841i	$0.343022\pi$
$594$			75.2677i			0.126713i
$595$		0			0
$596$	82.2944			0.138078
$597$			282.500i			0.473199i
$598$	85.2274			0.142521
$599$	−940.109			−1.56946			−0.784732	−	0.619835i	$-0.787199\pi$
$599$	−940.109			−1.56946			−0.784732	+	0.619835i	$0.787199\pi$
$600$		0			0
$601$		−	563.527i		−	0.937649i	−0.883291	−	0.468824i	$-0.844678\pi$
$601$		−	563.527i		−	0.937649i	0.883291	−	0.468824i	$-0.155322\pi$
$602$	−697.948	−	236.049i	−1.15938	−	0.392108i
$603$		−	81.0883i		−	0.134475i
$604$	−160.735			−0.266118
$605$		0			0
$606$	340.191			0.561371
$607$	306.589			0.505089			0.252544	−	0.967585i	$-0.418733\pi$
$607$	306.589			0.505089			0.252544	+	0.967585i	$0.418733\pi$
$608$	−91.2553			−0.150091
$609$	−344.558	−	116.531i	−0.565777	−	0.191348i
$610$		0			0
$611$	−522.177			−0.854626
$612$	182.451			0.298122
$613$			275.588i			0.449572i	0.974408	+	0.224786i	$0.0721684\pi$
$613$			275.588i			0.449572i	−0.974408	+	0.224786i	$0.927832\pi$
$614$		−	49.1340i		−	0.0800228i
$615$		0			0
$616$	64.9706	−	192.105i	0.105472	−	0.311858i
$617$		−	461.294i		−	0.747639i	−0.927501	−	0.373820i	$-0.878048\pi$
$617$		−	461.294i		−	0.747639i	0.927501	−	0.373820i	$-0.121952\pi$
$618$		−	193.206i		−	0.312631i
$619$		−	374.904i		−	0.605660i	−0.953045	−	0.302830i	$-0.902068\pi$
$619$		−	374.904i		−	0.605660i	0.953045	−	0.302830i	$-0.0979315\pi$
$620$		0			0
$621$		−	34.9593i		−	0.0562952i
$622$	126.100			0.202734
$623$	−325.116	+	961.301i	−0.521855	+	1.54302i
$624$	−62.0589			−0.0994533
$625$		0			0
$626$		−	734.966i		−	1.17407i
$627$			286.191i			0.456445i
$628$	−114.621			−0.182518
$629$			939.978i			1.49440i
$630$		0			0
$631$	−1127.32			−1.78656			−0.893282	−	0.449496i	$-0.851603\pi$
$631$	−1127.32			−1.78656			−0.893282	+	0.449496i	$0.851603\pi$
$632$		−	378.510i		−	0.598908i
$633$	184.336			0.291210
$634$	−766.544			−1.20906
$635$		0			0
$636$			245.849i			0.386555i
$637$	−348.812	−	266.412i	−0.547586	−	0.418229i
$638$		−	434.558i		−	0.681126i
$639$	151.831			0.237606
$640$		0			0
$641$	−884.352			−1.37964			−0.689822	−	0.723979i	$-0.742311\pi$
$641$	−884.352			−1.37964			−0.689822	+	0.723979i	$0.742311\pi$
$642$	92.4863			0.144060
$643$	−300.765			−0.467753			−0.233876	−	0.972266i	$-0.575141\pi$
$643$	−300.765			−0.467753			−0.233876	+	0.972266i	$0.575141\pi$
$644$	−30.1766	+	89.2261i	−0.0468581	+	0.138550i
$645$		0			0
$646$	693.734			1.07389
$647$	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
$649$			5.04319i			0.00777071i
$650$		0			0
$651$	194.912	−	576.314i	0.299404	−	0.885275i
$652$			349.823i			0.536539i
$653$		−	455.970i		−	0.698269i	−0.937073	−	0.349135i	$-0.886476\pi$
$653$		−	455.970i		−	0.698269i	0.937073	−	0.349135i	$-0.113524\pi$
$654$			78.2395i			0.119632i
$655$		0			0
$656$			28.4091i			0.0433066i
$657$	−211.845			−0.322443
$658$	184.888	−	546.676i	0.280985	−	0.830815i
$659$	−403.684			−0.612571			−0.306285	−	0.951940i	$-0.599086\pi$
$659$	−403.684			−0.612571			−0.306285	+	0.951940i	$0.599086\pi$
$660$		0			0
$661$			1153.41i			1.74495i	0.488663	+	0.872473i	$0.337485\pi$
$661$			1153.41i			1.74495i	−0.488663	+	0.872473i	$0.662515\pi$
$662$		−	254.039i		−	0.383744i
$663$	471.779			0.711582
$664$		−	295.331i		−	0.444776i
$665$		0			0
$666$	−131.147			−0.196918
$667$			201.838i			0.302605i
$668$	392.326			0.587315
$669$	−98.7351			−0.147586
$670$		0			0
$671$			29.3939i			0.0438061i
$672$	21.9733	−	64.9706i	0.0326984	−	0.0966824i
$673$		−	607.440i		−	0.902585i	−0.892376	−	0.451293i	$-0.850963\pi$
$673$		−	607.440i		−	0.902585i	0.892376	−	0.451293i	$-0.149037\pi$
$674$	−412.701			−0.612315
$675$		0			0
$676$	177.529			0.262617
$677$	1137.59			1.68034			0.840168	−	0.542326i	$-0.182456\pi$
$677$	1137.59			1.68034			0.840168	+	0.542326i	$0.182456\pi$
$678$	−262.023			−0.386465
$679$	667.779	+	225.845i	0.983474	+	0.332615i
$680$		0			0
$681$	−497.823			−0.731018
$682$	726.849			1.06576
$683$		−	818.111i		−	1.19782i	−0.800817	−	0.598910i	$-0.795601\pi$
$683$		−	818.111i		−	1.19782i	0.800817	−	0.598910i	$-0.204399\pi$
$684$			96.7909i			0.141507i
$685$		0			0
$686$	402.416	−	270.849i	0.586613	−	0.394823i
$687$			241.456i			0.351464i
$688$		−	297.706i		−	0.432712i
$689$			635.714i			0.922661i
$690$		0			0
$691$		−	204.280i		−	0.295630i	−0.989015	−	0.147815i	$-0.952776\pi$
$691$		−	204.280i		−	0.295630i	0.989015	−	0.147815i	$-0.0472239\pi$
$692$	77.0508			0.111345
$693$	−203.758	−	68.9117i	−0.294023	−	0.0994397i
$694$	−639.338			−0.921236
$695$		0			0
$696$		−	146.969i		−	0.211163i
$697$		−	215.970i		−	0.309856i
$698$	332.434			0.476267
$699$			628.276i			0.898821i
$700$		0			0
$701$	−318.853			−0.454854			−0.227427	−	0.973795i	$-0.573031\pi$
$701$	−318.853			−0.454854			−0.227427	+	0.973795i	$0.573031\pi$
$702$			65.8234i			0.0937655i
$703$	−498.662			−0.709334
$704$	81.9411			0.116394
$705$		0			0
$706$			35.7030i			0.0505708i
$707$	−311.463	+	920.933i	−0.440542	+	1.30259i
$708$			1.70563i			0.00240908i
$709$	217.647			0.306977			0.153489	−	0.988150i	$-0.450949\pi$
$709$	217.647			0.306977			0.153489	+	0.988150i	$0.450949\pi$
$710$		0			0
$711$	−401.470			−0.564656
$712$	−410.037			−0.575895
$713$	−337.597			−0.473488
$714$	−167.044	+	493.914i	−0.233955	+	0.691757i
$715$		0			0
$716$	−382.191			−0.533786
$717$	−32.0051			−0.0446375
$718$		−	150.250i		−	0.209262i
$719$			1207.36i			1.67922i	0.543192	+	0.839609i	$0.317216\pi$
$719$			1207.36i			1.67922i	−0.543192	+	0.839609i	$0.682784\pi$
$720$		0			0
$721$	523.029	+	176.891i	0.725422	+	0.245341i
$722$		−	142.503i		−	0.197372i
$723$			308.485i			0.426674i
$724$			241.587i			0.333683i
$725$		0			0
$726$			39.4082i			0.0542812i
$727$	−123.231			−0.169506			−0.0847528	−	0.996402i	$-0.527010\pi$
$727$	−123.231			−0.169506			−0.0847528	+	0.996402i	$0.527010\pi$
$728$	56.8183	−	168.000i	0.0780471	−	0.230769i
$729$	27.0000			0.0370370
$730$		0			0
$731$			2263.19i			3.09603i
$732$			9.94113i			0.0135808i
$733$	−375.779			−0.512659			−0.256330	−	0.966589i	$-0.582513\pi$
$733$	−375.779			−0.512659			−0.256330	+	0.966589i	$0.582513\pi$
$734$		−	405.054i		−	0.551844i
$735$		0			0
$736$	−38.0589			−0.0517104
$737$			276.853i			0.375648i
$738$	30.1324			0.0408299
$739$	−722.530			−0.977713			−0.488856	−	0.872364i	$-0.662586\pi$
$739$	−722.530			−0.977713			−0.488856	+	0.872364i	$0.662586\pi$
$740$		0			0
$741$			250.281i			0.337761i
$742$	−665.540	−	225.088i	−0.896954	−	0.303354i
$743$			1268.48i			1.70724i	0.520899	+	0.853618i	$0.325597\pi$
$743$			1268.48i			1.70724i	−0.520899	+	0.853618i	$0.674403\pi$
$744$	245.823			0.330408
$745$		0			0
$746$	−598.877			−0.802784
$747$	−313.246			−0.419339
$748$	−622.926			−0.832789
$749$	−84.6762	+	250.370i	−0.113052	+	0.334273i
$750$		0			0
$751$	439.161			0.584769			0.292384	−	0.956301i	$-0.405551\pi$
$751$	439.161			0.584769			0.292384	+	0.956301i	$0.405551\pi$
$752$	233.182			0.310082
$753$		−	87.7645i		−	0.116553i
$754$		−	380.031i		−	0.504020i
$755$		0			0
$756$	−68.9117	−	23.3062i	−0.0911530	−	0.0308283i
$757$			668.530i			0.883131i	0.897229	+	0.441565i	$0.145577\pi$
$757$			668.530i			0.883131i	−0.897229	+	0.441565i	$0.854423\pi$
$758$			142.981i			0.188629i
$759$			119.359i			0.157258i
$760$		0			0
$761$		−	880.116i		−	1.15653i	−0.815851	−	0.578263i	$-0.803731\pi$
$761$		−	880.116i		−	1.15653i	0.815851	−	0.578263i	$-0.196269\pi$
$762$	−54.0330			−0.0709094
$763$	−211.803	−	71.6325i	−0.277592	−	0.0938827i
$764$	−224.132			−0.293367
$765$		0			0
$766$		−	311.685i		−	0.406899i
$767$			4.41039i			0.00575018i
$768$	27.7128			0.0360844
$769$		−	1163.41i		−	1.51289i	−0.654059	−	0.756444i	$-0.726935\pi$
$769$		−	1163.41i		−	1.51289i	0.654059	−	0.756444i	$-0.273065\pi$
$770$		0			0
$771$	−324.905			−0.421407
$772$			45.8234i			0.0593567i
$773$	536.371			0.693883			0.346941	−	0.937887i	$-0.387220\pi$
$773$	536.371			0.693883			0.346941	+	0.937887i	$0.387220\pi$
$774$	−315.765			−0.407964
$775$		0			0
$776$			284.837i			0.367058i
$777$	120.072	−	355.029i	0.154533	−	0.456923i
$778$		−	542.309i		−	0.697055i
$779$	114.573			0.147077
$780$		0			0
$781$	−518.382			−0.663741
$782$	289.328			0.369985
$783$	−155.885			−0.199086
$784$	155.765	+	118.968i	0.198679	+	0.151745i
$785$		0			0
$786$	173.823			0.221149
$787$	83.9192			0.106632			0.0533159	−	0.998578i	$-0.483021\pi$
$787$	83.9192			0.106632			0.0533159	+	0.998578i	$0.483021\pi$
$788$		−	232.118i		−	0.294566i
$789$		−	697.978i		−	0.884636i
$790$		0			0
$791$	239.897	−	709.325i	0.303283	−	0.896745i
$792$		−	86.9117i		−	0.109737i
$793$			25.7056i			0.0324157i
$794$		−	686.146i		−	0.864163i
$795$		0			0
$796$		−	326.202i		−	0.409802i
$797$	−1005.57			−1.26169			−0.630846	−	0.775908i	$-0.717292\pi$
$797$	−1005.57			−1.26169			−0.630846	+	0.775908i	$0.717292\pi$
$798$	−262.023	−	88.6173i	−0.328350	−	0.111049i
$799$	−1772.67			−2.21862
$800$		0			0
$801$			434.910i			0.542959i
$802$			358.045i			0.446440i
$803$	723.283			0.900727
$804$			93.6327i			0.116459i
$805$		0			0
$806$	635.647			0.788644
$807$		−	832.316i		−	1.03137i
$808$	−392.819			−0.486162
$809$	282.853			0.349633			0.174816	−	0.984601i	$-0.444067\pi$
$809$	282.853			0.349633			0.174816	+	0.984601i	$0.444067\pi$
$810$		0			0
$811$		−	923.997i		−	1.13933i	−0.821877	−	0.569665i	$-0.807073\pi$
$811$		−	923.997i		−	1.13933i	0.821877	−	0.569665i	$-0.192927\pi$
$812$	397.862	+	134.558i	0.489978	+	0.165712i
$813$			64.6173i			0.0794801i
$814$	447.765			0.550079
$815$		0			0
$816$	−210.676			−0.258182
$817$	−1200.63			−1.46956
$818$	7.62452			0.00932093
$819$	−178.191	−	60.2649i	−0.217571	−	0.0735835i
$820$		0			0
$821$	1272.32			1.54972			0.774862	−	0.632131i	$-0.217819\pi$
$821$	1272.32			1.54972			0.774862	+	0.632131i	$0.217819\pi$
$822$	259.069			0.315169
$823$			381.852i			0.463976i	0.972719	+	0.231988i	$0.0745230\pi$
$823$			381.852i			0.463976i	−0.972719	+	0.231988i	$0.925477\pi$
$824$			223.095i			0.270747i
$825$		0			0
$826$	−4.61732	−	1.56159i	−0.00558997	−	0.00189055i
$827$		−	653.022i		−	0.789628i	−0.918761	−	0.394814i	$-0.870809\pi$
$827$		−	653.022i		−	0.789628i	0.918761	−	0.394814i	$-0.129191\pi$
$828$			40.3675i			0.0487531i
$829$		−	215.580i		−	0.260048i	−0.991511	−	0.130024i	$-0.958495\pi$
$829$		−	215.580i		−	0.260048i	0.991511	−	0.130024i	$-0.0415054\pi$
$830$		0			0
$831$			143.199i			0.172322i
$832$	71.6594			0.0861291
$833$	−1184.14	−	904.410i	−1.42154	−	1.08573i
$834$	444.146			0.532550
$835$		0			0
$836$		−	330.465i		−	0.395293i
$837$		−	260.735i		−	0.311511i
$838$	416.389			0.496884
$839$		−	632.267i		−	0.753596i	−0.926295	−	0.376798i	$-0.877025\pi$
$839$		−	632.267i		−	0.753596i	0.926295	−	0.376798i	$-0.122975\pi$
$840$		0			0
$841$	59.0000			0.0701546
$842$			410.745i			0.487821i
$843$	−261.285			−0.309946
$844$	−212.853			−0.252195
$845$		0			0
$846$		−	247.326i		−	0.292348i
$847$	−106.682	−	36.0803i	−0.125953	−	0.0425978i
$848$		−	283.882i		−	0.334767i
$849$	492.177			0.579713
$850$		0			0
$851$	−207.971			−0.244385
$852$	−175.319			−0.205773
$853$	919.650			1.07814			0.539068	−	0.842262i	$-0.318776\pi$
$853$	919.650			1.07814			0.539068	+	0.842262i	$0.318776\pi$
$854$	−26.9117	−	9.10164i	−0.0315125	−	0.0106577i
$855$		0			0
$856$	−106.794			−0.124759
$857$	−527.150			−0.615111			−0.307556	−	0.951530i	$-0.599511\pi$
$857$	−527.150			−0.615111			−0.307556	+	0.951530i	$0.599511\pi$
$858$		−	224.735i		−	0.261929i
$859$		−	239.804i		−	0.279166i	−0.990210	−	0.139583i	$-0.955424\pi$
$859$		−	239.804i		−	0.279166i	0.990210	−	0.139583i	$-0.0445762\pi$
$860$		0			0
$861$	−27.5879	+	81.5717i	−0.0320417	+	0.0947407i
$862$		−	71.7199i		−	0.0832018i
$863$			362.507i			0.420054i	0.977696	+	0.210027i	$0.0673552\pi$
$863$			362.507i			0.420054i	−0.977696	+	0.210027i	$0.932645\pi$
$864$		−	29.3939i		−	0.0340207i
$865$		0			0
$866$			1024.97i			1.18356i
$867$	1101.02			1.26992
$868$	−225.065	+	665.470i	−0.259291	+	0.766671i
$869$	1370.70			1.57734
$870$		0			0
$871$			242.114i			0.277973i
$872$		−	90.3431i		−	0.103605i
$873$	302.115			0.346066
$874$			153.490i			0.175617i
$875$		0			0
$876$	244.617			0.279244
$877$			861.647i			0.982493i	0.871020	+	0.491247i	$0.163459\pi$
$877$			861.647i			0.982493i	−0.871020	+	0.491247i	$0.836541\pi$
$878$	−525.703			−0.598750
$879$	−930.521			−1.05861
$880$		0			0
$881$			334.553i			0.379742i	0.981809	+	0.189871i	$0.0608071\pi$
$881$			334.553i			0.379742i	−0.981809	+	0.189871i	$0.939193\pi$
$882$	126.185	−	165.213i	0.143067	−	0.187317i
$883$		−	1488.01i		−	1.68518i	−0.538555	−	0.842590i	$-0.681030\pi$
$883$		−	1488.01i		−	1.68518i	0.538555	−	0.842590i	$-0.318970\pi$
$884$	−544.764			−0.616248
$885$		0			0
$886$	−324.219			−0.365936
$887$	801.498			0.903606			0.451803	−	0.892118i	$-0.350781\pi$
$887$	801.498			0.903606			0.451803	+	0.892118i	$0.350781\pi$
$888$	151.436			0.170536
$889$	49.4701	−	146.273i	0.0556469	−	0.164537i
$890$		0			0
$891$	−92.1838			−0.103461
$892$	114.009			0.127813
$893$		−	940.410i		−	1.05309i
$894$			100.790i			0.112740i
$895$		0			0
$896$	−25.3726	+	75.0215i	−0.0283176	+	0.0837294i
$897$			104.382i			0.116368i
$898$		−	848.985i		−	0.945417i
$899$			1505.35i			1.67448i
$900$		0			0
$901$			2158.11i			2.39524i
$902$	−102.879			−0.114056
$903$	289.100	−	854.808i	0.320155	−	0.946631i
$904$	302.558			0.334689
$905$		0			0
$906$		−	196.859i		−	0.217284i
$907$			807.279i			0.890054i	0.895517	+	0.445027i	$0.146806\pi$
$907$			807.279i			0.890054i	−0.895517	+	0.445027i	$0.853194\pi$
$908$	574.837			0.633080
$909$			416.647i			0.458358i
$910$		0			0
$911$	−442.742			−0.485996			−0.242998	−	0.970027i	$-0.578131\pi$
$911$	−442.742			−0.485996			−0.242998	+	0.970027i	$0.578131\pi$
$912$		−	111.765i		−	0.122549i
$913$	1069.49			1.17140
$914$	−684.230			−0.748610
$915$		0			0
$916$		−	278.809i		−	0.304377i
$917$	−159.145	+	470.558i	−0.173549	+	0.513150i
$918$			223.456i			0.243416i
$919$	118.455			0.128896			0.0644478	−	0.997921i	$-0.479471\pi$
$919$	118.455			0.128896			0.0644478	+	0.997921i	$0.479471\pi$
$920$		0			0
$921$	60.1766			0.0653384
$922$	−388.430			−0.421290
$923$	−453.337			−0.491156
$924$	235.279	+	79.5724i	0.254631	+	0.0861173i
$925$		0			0
$926$	−217.040			−0.234384
$927$	236.628			0.255262
$928$			169.706i			0.182872i
$929$		−	629.297i		−	0.677392i	−0.940896	−	0.338696i	$-0.890014\pi$
$929$		−	629.297i		−	0.677392i	0.940896	−	0.338696i	$-0.109986\pi$
$930$		0			0
$931$	479.793	−	628.191i	0.515352	−	0.674749i
$932$		−	725.470i		−	0.778401i
$933$			154.441i			0.165531i
$934$		−	87.3164i		−	0.0934865i
$935$		0			0
$936$		−	76.0063i		−	0.0812033i
$937$	210.631			0.224793			0.112397	−	0.993663i	$-0.464147\pi$
$937$	210.631			0.224793			0.112397	+	0.993663i	$0.464147\pi$
$938$	−253.474	−	85.7258i	−0.270228	−	0.0913921i
$939$	900.146			0.958622
$940$		0			0
$941$			178.422i			0.189609i	0.995496	+	0.0948047i	$0.0302226\pi$
$941$			178.422i			0.189609i	−0.995496	+	0.0948047i	$0.969777\pi$
$942$		−	140.382i		−	0.149025i
$943$	47.7836			0.0506719
$944$		−	1.96949i		−	0.00208632i
$945$		0			0
$946$	1078.09			1.13963
$947$		−	423.463i		−	0.447163i	−0.974685	−	0.223581i	$-0.928225\pi$
$947$		−	423.463i		−	0.447163i	0.974685	−	0.223581i	$-0.0717748\pi$
$948$	463.578			0.489006
$949$	632.528			0.666521
$950$		0			0
$951$		−	938.821i		−	0.987194i
$952$	192.885	−	570.323i	0.202611	−	0.599079i
$953$		−	546.706i		−	0.573669i	−0.957980	−	0.286834i	$-0.907397\pi$
$953$		−	546.706i		−	0.573669i	0.957980	−	0.286834i	$-0.0926030\pi$
$954$	−301.103			−0.315621
$955$		0			0
$956$	36.9563			0.0386572
$957$	532.223			0.556137
$958$	787.030			0.821534
$959$	−237.192	+	701.328i	−0.247332	+	0.731311i
$960$		0			0
$961$	−1556.88			−1.62006
$962$	391.580			0.407048
$963$			113.272i			0.117624i
$964$		−	356.208i		−	0.369510i
$965$		0			0
$966$	−109.279	−	36.9587i	−0.113125	−	0.0382595i
$967$		−	1262.32i		−	1.30540i	−0.757616	−	0.652701i	$-0.773636\pi$
$967$		−	1262.32i		−	1.30540i	0.757616	−	0.652701i	$-0.226364\pi$
$968$		−	45.5046i		−	0.0470089i
$969$			849.647i			0.876829i
$970$		0			0
$971$			1080.93i			1.11322i	0.830775	+	0.556608i	$0.187898\pi$
$971$			1080.93i			1.11322i	−0.830775	+	0.556608i	$0.812102\pi$
$972$	−31.1769			−0.0320750
$973$	−406.640	+	1202.35i	−0.417924	+	1.23572i
$974$	459.931			0.472208
$975$		0			0
$976$		−	11.4790i		−	0.0117613i
$977$		−	1800.26i		−	1.84264i	−0.388801	−	0.921322i	$-0.627111\pi$
$977$		−	1800.26i		−	1.84264i	0.388801	−	0.921322i	$-0.372889\pi$
$978$	−428.444			−0.438082
$979$		−	1484.88i		−	1.51673i
$980$		0			0
$981$	−95.8234			−0.0976793
$982$		−	910.014i		−	0.926695i
$983$	996.810			1.01405			0.507025	−	0.861932i	$-0.330745\pi$
$983$	996.810			1.01405			0.507025	+	0.861932i	$0.330745\pi$
$984$	−34.7939			−0.0353597
$985$		0			0
$986$		−	1290.12i		−	1.30844i
$987$	669.539	+	226.441i	0.678358	+	0.229423i
$988$		−	288.999i		−	0.292509i
$989$	−500.735			−0.506304
$990$		0			0
$991$	788.721			0.795884			0.397942	−	0.917411i	$-0.369725\pi$
$991$	788.721			0.795884			0.397942	+	0.917411i	$0.369725\pi$
$992$	−283.852			−0.286142
$993$	311.133			0.313326
$994$	160.514	−	474.607i	0.161483	−	0.477471i
$995$		0			0
$996$	361.706			0.363158
$997$	−1942.70			−1.94854			−0.974271	−	0.225379i	$-0.927638\pi$
$997$	−1942.70			−1.94854			−0.974271	+	0.225379i	$0.927638\pi$
$998$			1068.33i			1.07047i
$999$		−	160.622i		−	0.160783i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.3.h.a.349.4		8
5.2	odd	4		42.3.c.a.13.3	✓	4
5.3	odd	4		1050.3.f.a.601.2		4
5.4	even	2	inner	1050.3.h.a.349.5		8
7.6	odd	2	inner	1050.3.h.a.349.1		8
15.2	even	4		126.3.c.b.55.2		4
20.7	even	4		336.3.f.c.97.4		4
35.2	odd	12		294.3.g.c.31.1		4
35.12	even	12		294.3.g.b.31.1		4
35.13	even	4		1050.3.f.a.601.1		4
35.17	even	12		294.3.g.c.19.1		4
35.27	even	4		42.3.c.a.13.4	yes	4
35.32	odd	12		294.3.g.b.19.1		4
35.34	odd	2	inner	1050.3.h.a.349.8		8
40.27	even	4		1344.3.f.e.769.1		4
40.37	odd	4		1344.3.f.f.769.3		4
60.47	odd	4		1008.3.f.g.433.3		4
105.2	even	12		882.3.n.d.325.2		4
105.17	odd	12		882.3.n.d.19.2		4
105.32	even	12		882.3.n.a.19.2		4
105.47	odd	12		882.3.n.a.325.2		4
105.62	odd	4		126.3.c.b.55.1		4
140.27	odd	4		336.3.f.c.97.1		4
280.27	odd	4		1344.3.f.e.769.4		4
280.237	even	4		1344.3.f.f.769.2		4
420.167	even	4		1008.3.f.g.433.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.3.c.a.13.3	✓	4	5.2	odd	4
42.3.c.a.13.4	yes	4	35.27	even	4
126.3.c.b.55.1		4	105.62	odd	4
126.3.c.b.55.2		4	15.2	even	4
294.3.g.b.19.1		4	35.32	odd	12
294.3.g.b.31.1		4	35.12	even	12
294.3.g.c.19.1		4	35.17	even	12
294.3.g.c.31.1		4	35.2	odd	12
336.3.f.c.97.1		4	140.27	odd	4
336.3.f.c.97.4		4	20.7	even	4
882.3.n.a.19.2		4	105.32	even	12
882.3.n.a.325.2		4	105.47	odd	12
882.3.n.d.19.2		4	105.17	odd	12
882.3.n.d.325.2		4	105.2	even	12
1008.3.f.g.433.2		4	420.167	even	4
1008.3.f.g.433.3		4	60.47	odd	4
1050.3.f.a.601.1		4	35.13	even	4
1050.3.f.a.601.2		4	5.3	odd	4
1050.3.h.a.349.1		8	7.6	odd	2	inner
1050.3.h.a.349.4		8	1.1	even	1	trivial
1050.3.h.a.349.5		8	5.4	even	2	inner
1050.3.h.a.349.8		8	35.34	odd	2	inner
1344.3.f.e.769.1		4	40.27	even	4
1344.3.f.e.769.4		4	280.27	odd	4
1344.3.f.f.769.2		4	280.237	even	4
1344.3.f.f.769.3		4	40.37	odd	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)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} +1.73205 q^{3} -2.00000 q^{4} -2.44949i q^{6} +(6.63103 + 2.24264i) q^{7} +2.82843i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q-1.41421i q^{2} +1.73205 q^{3} -2.00000 q^{4} -2.44949i q^{6} +(6.63103 + 2.24264i) q^{7} +2.82843i q^{8} +3.00000 q^{9} -10.2426 q^{11} -3.46410 q^{12} -8.95743 q^{13} +(3.17157 - 9.37769i) q^{14} +4.00000 q^{16} -30.4085 q^{17} -4.24264i q^{18} -16.1318i q^{19} +(11.4853 + 3.88437i) q^{21} +14.4853i q^{22} -6.72792i q^{23} +4.89898i q^{24} +12.6677i q^{26} +5.19615 q^{27} +(-13.2621 - 4.48528i) q^{28} -30.0000 q^{29} -50.1785i q^{31} -5.65685i q^{32} -17.7408 q^{33} +43.0041i q^{34} -6.00000 q^{36} -30.9117i q^{37} -22.8138 q^{38} -15.5147 q^{39} +7.10228i q^{41} +(5.49333 - 16.2426i) q^{42} -74.4264i q^{43} +20.4853 q^{44} -9.51472 q^{46} +58.2954 q^{47} +6.92820 q^{48} +(38.9411 + 29.7420i) q^{49} -52.6690 q^{51} +17.9149 q^{52} -70.9706i q^{53} -7.34847i q^{54} +(-6.34315 + 18.7554i) q^{56} -27.9411i q^{57} +42.4264i q^{58} -0.492372i q^{59} -2.86976i q^{61} -70.9631 q^{62} +(19.8931 + 6.72792i) q^{63} -8.00000 q^{64} +25.0892i q^{66} -27.0294i q^{67} +60.8170 q^{68} -11.6531i q^{69} +50.6102 q^{71} +8.48528i q^{72} -70.6149 q^{73} -43.7157 q^{74} +32.2636i q^{76} +(-67.9193 - 22.9706i) q^{77} +21.9411i q^{78} -133.823 q^{79} +9.00000 q^{81} +10.0441 q^{82} -104.415 q^{83} +(-22.9706 - 7.76874i) q^{84} -105.255 q^{86} -51.9615 q^{87} -28.9706i q^{88} +144.970i q^{89} +(-59.3970 - 20.0883i) q^{91} +13.4558i q^{92} -86.9117i q^{93} -82.4421i q^{94} -9.79796i q^{96} +100.705 q^{97} +(42.0616 - 55.0711i) q^{98} -30.7279 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 24 q^{9}+O(q^{10}) \) 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

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Database

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