LMFDB - Embedded newform 1050.3.f.a.601.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1050,3,Mod(601,1050)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1050.601"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1050.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,8,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$28.6104277578$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2x^{2} + 4 $ x^4 + 2*x^2 + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			601.2
Root			$-0.707107 - 1.22474i$ of defining polynomial
Character	$\chi$	$=$	1050.601
Dual form			1050.3.f.a.601.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} -2.44949i q^{6} +(2.24264 - 6.63103i) q^{7} -2.82843 q^{8} -3.00000 q^{9} -10.2426 q^{11} +3.46410i q^{12} -8.95743i q^{13} +(-3.17157 + 9.37769i) q^{14} +4.00000 q^{16} +30.4085i q^{17} +4.24264 q^{18} +16.1318i q^{19} +(11.4853 + 3.88437i) q^{21} +14.4853 q^{22} +6.72792 q^{23} -4.89898i q^{24} +12.6677i q^{26} -5.19615i q^{27} +(4.48528 - 13.2621i) q^{28} +30.0000 q^{29} -50.1785i q^{31} -5.65685 q^{32} -17.7408i q^{33} -43.0041i q^{34} -6.00000 q^{36} -30.9117 q^{37} -22.8138i q^{38} +15.5147 q^{39} +7.10228i q^{41} +(-16.2426 - 5.49333i) q^{42} +74.4264 q^{43} -20.4853 q^{44} -9.51472 q^{46} -58.2954i q^{47} +6.92820i q^{48} +(-38.9411 - 29.7420i) q^{49} -52.6690 q^{51} -17.9149i q^{52} +70.9706 q^{53} +7.34847i q^{54} +(-6.34315 + 18.7554i) q^{56} -27.9411 q^{57} -42.4264 q^{58} +0.492372i q^{59} -2.86976i q^{61} +70.9631i q^{62} +(-6.72792 + 19.8931i) q^{63} +8.00000 q^{64} +25.0892i q^{66} -27.0294 q^{67} +60.8170i q^{68} +11.6531i q^{69} +50.6102 q^{71} +8.48528 q^{72} -70.6149i q^{73} +43.7157 q^{74} +32.2636i q^{76} +(-22.9706 + 67.9193i) q^{77} -21.9411 q^{78} +133.823 q^{79} +9.00000 q^{81} -10.0441i q^{82} -104.415i q^{83} +(22.9706 + 7.76874i) q^{84} -105.255 q^{86} +51.9615i q^{87} +28.9706 q^{88} -144.970i q^{89} +(-59.3970 - 20.0883i) q^{91} +13.4558 q^{92} +86.9117 q^{93} +82.4421i q^{94} -9.79796i q^{96} -100.705i q^{97} +(55.0711 + 42.0616i) q^{98} +30.7279 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} - 8 q^{7} - 12 q^{9} - 24 q^{11} - 24 q^{14} + 16 q^{16} + 12 q^{21} + 24 q^{22} - 24 q^{23} - 16 q^{28} + 120 q^{29} - 24 q^{36} + 80 q^{37} + 96 q^{39} - 48 q^{42} + 128 q^{43} - 48 q^{44}+ \cdots + 72 q^{99}+O(q^{100}) $ 4 * q + 8 * q^4 - 8 * q^7 - 12 * q^9 - 24 * q^11 - 24 * q^14 + 16 * q^16 + 12 * q^21 + 24 * q^22 - 24 * q^23 - 16 * q^28 + 120 * q^29 - 24 * q^36 + 80 * q^37 + 96 * q^39 - 48 * q^42 + 128 * q^43 - 48 * q^44 - 72 * q^46 - 20 * q^49 - 24 * q^51 + 216 * q^53 - 48 * q^56 + 24 * q^57 + 24 * q^63 + 32 * q^64 - 176 * q^67 - 120 * q^71 + 288 * q^74 - 24 * q^77 + 48 * q^78 + 128 * q^79 + 36 * q^81 + 24 * q^84 - 240 * q^86 + 48 * q^88 - 48 * q^92 + 144 * q^93 + 192 * q^98 + 72 * q^99

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 $

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$			1.73205i			0.577350i
$3$			1.73205i			0.577350i
$4$	2.00000			0.500000
$5$		0			0
$5$		0			0
$6$		−	2.44949i		−	0.408248i
$7$	2.24264	−	6.63103i	0.320377	−	0.947290i
$7$	2.24264	−	6.63103i	0.320377	−	0.947290i
$8$	−2.82843			−0.353553
$9$	−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	$0.654152\pi$
$12$			3.46410i			0.288675i
$13$		−	8.95743i		−	0.689033i	−0.938780	−	0.344516i	$-0.888043\pi$
$13$		−	8.95743i		−	0.689033i	0.938780	−	0.344516i	$-0.111957\pi$
$14$	−3.17157	+	9.37769i	−0.226541	+	0.669835i
$15$		0			0
$16$	4.00000			0.250000
$17$			30.4085i			1.78873i	0.447333	+	0.894367i	$0.352374\pi$
$17$			30.4085i			1.78873i	−0.447333	+	0.894367i	$0.647626\pi$
$18$	4.24264			0.235702
$19$			16.1318i			0.849043i	0.905418	+	0.424521i	$0.139558\pi$
$19$			16.1318i			0.849043i	−0.905418	+	0.424521i	$0.860442\pi$
$20$		0			0
$21$	11.4853	+	3.88437i	0.546918	+	0.184970i
$22$	14.4853			0.658422
$23$	6.72792			0.292518			0.146259	−	0.989246i	$-0.453277\pi$
$23$	6.72792			0.292518			0.146259	+	0.989246i	$0.453277\pi$
$24$		−	4.89898i		−	0.204124i
$25$		0			0
$26$			12.6677i			0.487220i
$27$		−	5.19615i		−	0.192450i
$28$	4.48528	−	13.2621i	0.160189	−	0.473645i
$29$	30.0000			1.03448			0.517241	−	0.855840i	$-0.326959\pi$
$29$	30.0000			1.03448			0.517241	+	0.855840i	$0.326959\pi$
$30$		0			0
$31$		−	50.1785i		−	1.61866i	−0.587354	−	0.809330i	$-0.699830\pi$
$31$		−	50.1785i		−	1.61866i	0.587354	−	0.809330i	$-0.300170\pi$
$32$	−5.65685			−0.176777
$33$		−	17.7408i		−	0.537599i
$34$		−	43.0041i		−	1.26483i
$35$		0			0
$36$	−6.00000			−0.166667
$37$	−30.9117			−0.835451			−0.417726	−	0.908573i	$-0.637173\pi$
$37$	−30.9117			−0.835451			−0.417726	+	0.908573i	$0.637173\pi$
$38$		−	22.8138i		−	0.600364i
$39$	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	$0.972396\pi$
$42$	−16.2426	−	5.49333i	−0.386730	−	0.130793i
$43$	74.4264			1.73085			0.865423	−	0.501041i	$-0.167050\pi$
$43$	74.4264			1.73085			0.865423	+	0.501041i	$0.167050\pi$
$44$	−20.4853			−0.465575
$45$		0			0
$46$	−9.51472			−0.206842
$47$		−	58.2954i		−	1.24033i	−0.784472	−	0.620164i	$-0.787066\pi$
$47$		−	58.2954i		−	1.24033i	0.784472	−	0.620164i	$-0.212934\pi$
$48$			6.92820i			0.144338i
$49$	−38.9411	−	29.7420i	−0.794717	−	0.606980i
$50$		0			0
$51$	−52.6690			−1.03273
$52$		−	17.9149i		−	0.344516i
$53$	70.9706			1.33907			0.669534	−	0.742782i	$-0.266494\pi$
$53$	70.9706			1.33907			0.669534	+	0.742782i	$0.266494\pi$
$54$			7.34847i			0.136083i
$55$		0			0
$56$	−6.34315	+	18.7554i	−0.113270	+	0.334918i
$57$	−27.9411			−0.490195
$58$	−42.4264			−0.731490
$59$			0.492372i			0.00834529i	0.999991	+	0.00417265i	$0.00132820\pi$
$59$			0.492372i			0.00834529i	−0.999991	+	0.00417265i	$0.998672\pi$
$60$		0			0
$61$		−	2.86976i		−	0.0470452i	−0.999723	−	0.0235226i	$-0.992512\pi$
$61$		−	2.86976i		−	0.0470452i	0.999723	−	0.0235226i	$-0.00748816\pi$
$62$			70.9631i			1.14457i
$63$	−6.72792	+	19.8931i	−0.106792	+	0.315763i
$64$	8.00000			0.125000
$65$		0			0
$66$			25.0892i			0.380140i
$67$	−27.0294			−0.403424			−0.201712	−	0.979445i	$-0.564651\pi$
$67$	−27.0294			−0.403424			−0.201712	+	0.979445i	$0.564651\pi$
$68$			60.8170i			0.894367i
$69$			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	$0.384001\pi$
$72$	8.48528			0.117851
$73$		−	70.6149i		−	0.967328i	−0.875254	−	0.483664i	$-0.839306\pi$
$73$		−	70.6149i		−	0.967328i	0.875254	−	0.483664i	$-0.160694\pi$
$74$	43.7157			0.590753
$75$		0			0
$76$			32.2636i			0.424521i
$77$	−22.9706	+	67.9193i	−0.298319	+	0.882068i
$78$	−21.9411			−0.281296
$79$	133.823			1.69397			0.846983	−	0.531619i	$-0.178416\pi$
$79$	133.823			1.69397			0.846983	+	0.531619i	$0.178416\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		−	10.0441i		−	0.122490i
$83$		−	104.415i		−	1.25802i	−0.777398	−	0.629009i	$-0.783461\pi$
$83$		−	104.415i		−	1.25802i	0.777398	−	0.629009i	$-0.216539\pi$
$84$	22.9706	+	7.76874i	0.273459	+	0.0924849i
$85$		0			0
$86$	−105.255			−1.22389
$87$			51.9615i			0.597259i
$88$	28.9706			0.329211
$89$		−	144.970i		−	1.62888i	−0.580250	−	0.814438i	$-0.697045\pi$
$89$		−	144.970i		−	1.62888i	0.580250	−	0.814438i	$-0.302955\pi$
$90$		0			0
$91$	−59.3970	−	20.0883i	−0.652714	−	0.220750i
$92$	13.4558			0.146259
$93$	86.9117			0.934534
$94$			82.4421i			0.877044i
$95$		0			0
$96$		−	9.79796i		−	0.102062i
$97$		−	100.705i		−	1.03820i	−0.854714	−	0.519099i	$-0.826268\pi$
$97$		−	100.705i		−	1.03820i	0.854714	−	0.519099i	$-0.173732\pi$
$98$	55.0711	+	42.0616i	0.561950	+	0.429200i
$99$	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	$0.758692\pi$
$102$	74.4853			0.730248
$103$			78.8760i			0.765787i	0.923793	+	0.382893i	$0.125072\pi$
$103$			78.8760i			0.765787i	−0.923793	+	0.382893i	$0.874928\pi$
$104$			25.3354i			0.243610i
$105$		0			0
$106$	−100.368			−0.946864
$107$	37.7574			0.352873			0.176436	−	0.984312i	$-0.443543\pi$
$107$	37.7574			0.352873			0.176436	+	0.984312i	$0.443543\pi$
$108$		−	10.3923i		−	0.0962250i
$109$	31.9411			0.293038			0.146519	−	0.989208i	$-0.453193\pi$
$109$	31.9411			0.293038			0.146519	+	0.989208i	$0.453193\pi$
$110$		0			0
$111$		−	53.5406i		−	0.482348i
$112$	8.97056	−	26.5241i	0.0800943	−	0.236823i
$113$	106.971			0.946642			0.473321	−	0.880890i	$-0.343055\pi$
$113$	106.971			0.946642			0.473321	+	0.880890i	$0.343055\pi$
$114$	39.5147			0.346620
$115$		0			0
$116$	60.0000			0.517241
$117$			26.8723i			0.229678i
$118$		−	0.696320i		−	0.00590101i
$119$	201.640	+	68.1953i	1.69445	+	0.573070i
$120$		0			0
$121$	−16.0883			−0.132961
$122$			4.05845i			0.0332660i
$123$	−12.3015			−0.100012
$124$		−	100.357i		−	0.809330i
$125$		0			0
$126$	9.51472	−	28.1331i	0.0755136	−	0.223278i
$127$	−22.0589			−0.173692			−0.0868460	−	0.996222i	$-0.527679\pi$
$127$	−22.0589			−0.173692			−0.0868460	+	0.996222i	$0.527679\pi$
$128$	−11.3137			−0.0883883
$129$			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	$0.912695\pi$
$132$		−	35.4815i		−	0.268800i
$133$	106.971	+	36.1779i	0.804290	+	0.272014i
$134$	38.2254			0.285264
$135$		0			0
$136$		−	86.0082i		−	0.632413i
$137$	105.765			0.772004			0.386002	−	0.922498i	$-0.373856\pi$
$137$	105.765			0.772004			0.386002	+	0.922498i	$0.373856\pi$
$138$		−	16.4800i		−	0.119420i
$139$		−	181.322i		−	1.30447i	−0.758015	−	0.652237i	$-0.773831\pi$
$139$		−	181.322i		−	1.30447i	0.758015	−	0.652237i	$-0.226169\pi$
$140$		0			0
$141$	100.971			0.716103
$142$	−71.5736			−0.504039
$143$			91.7477i			0.641592i
$144$	−12.0000			−0.0833333
$145$		0			0
$146$			99.8646i			0.684004i
$147$	51.5147	−	67.4480i	0.350440	−	0.458830i
$148$	−61.8234			−0.417726
$149$	41.1472			0.276156			0.138078	−	0.990421i	$-0.455908\pi$
$149$	41.1472			0.276156			0.138078	+	0.990421i	$0.455908\pi$
$150$		0			0
$151$	80.3675			0.532235			0.266118	−	0.963941i	$-0.414259\pi$
$151$	80.3675			0.532235			0.266118	+	0.963941i	$0.414259\pi$
$152$		−	45.6277i		−	0.300182i
$153$		−	91.2255i		−	0.596245i
$154$	32.4853	−	96.0523i	0.210943	−	0.623716i
$155$		0			0
$156$	31.0294			0.198907
$157$		−	57.3106i		−	0.365036i	−0.983203	−	0.182518i	$-0.941575\pi$
$157$		−	57.3106i		−	0.365036i	0.983203	−	0.182518i	$-0.0584248\pi$
$158$	−189.255			−1.19782
$159$			122.925i			0.773111i
$160$		0			0
$161$	15.0883	−	44.6131i	0.0937162	−	0.277100i
$162$	−12.7279			−0.0785674
$163$	174.912			1.07308			0.536539	−	0.843876i	$-0.319731\pi$
$163$	174.912			1.07308			0.536539	+	0.843876i	$0.319731\pi$
$164$			14.2046i			0.0866132i
$165$		0			0
$166$			147.666i			0.889552i
$167$			196.163i			1.17463i	0.809359	+	0.587315i	$0.199815\pi$
$167$			196.163i			1.17463i	−0.809359	+	0.587315i	$0.800185\pi$
$168$	−32.4853	−	10.9867i	−0.193365	−	0.0653967i
$169$	88.7645			0.525234
$170$		0			0
$171$		−	48.3954i		−	0.283014i
$172$	148.853			0.865423
$173$		−	38.5254i		−	0.222690i	−0.993782	−	0.111345i	$-0.964484\pi$
$173$		−	38.5254i		−	0.222690i	0.993782	−	0.111345i	$-0.0355159\pi$
$174$		−	73.4847i		−	0.422326i
$175$		0			0
$176$	−40.9706			−0.232787
$177$	−0.852814			−0.00481816
$178$			205.019i			1.15179i
$179$	−191.095			−1.06757			−0.533786	−	0.845619i	$-0.679231\pi$
$179$	−191.095			−1.06757			−0.533786	+	0.845619i	$0.679231\pi$
$180$		0			0
$181$		−	120.793i		−	0.667367i	−0.942685	−	0.333683i	$-0.891708\pi$
$181$		−	120.793i		−	0.667367i	0.942685	−	0.333683i	$-0.108292\pi$
$182$	84.0000	+	28.4091i	0.461538	+	0.156094i
$183$	4.97056			0.0271615
$184$	−19.0294			−0.103421
$185$		0			0
$186$	−122.912			−0.660816
$187$		−	311.463i		−	1.66558i
$188$		−	116.591i		−	0.620164i
$189$	−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	$0.405224\pi$
$192$			13.8564i			0.0721688i
$193$	22.9117			0.118713			0.0593567	−	0.998237i	$-0.481095\pi$
$193$	22.9117			0.118713			0.0593567	+	0.998237i	$0.481095\pi$
$194$			142.419i			0.734116i
$195$		0			0
$196$	−77.8823	−	59.4841i	−0.397358	−	0.303490i
$197$	116.059			0.589131			0.294566	−	0.955631i	$-0.404825\pi$
$197$	116.059			0.589131			0.294566	+	0.955631i	$0.404825\pi$
$198$	−43.4558			−0.219474
$199$		−	163.101i		−	0.819604i	−0.912174	−	0.409802i	$-0.865598\pi$
$199$		−	163.101i		−	0.819604i	0.912174	−	0.409802i	$-0.134402\pi$
$200$		0			0
$201$		−	46.8164i		−	0.232917i
$202$		−	196.409i		−	0.972323i
$203$	67.2792	−	198.931i	0.331425	−	0.979955i
$204$	−105.338			−0.516363
$205$		0			0
$206$		−	111.548i		−	0.541493i
$207$	−20.1838			−0.0975061
$208$		−	35.8297i		−	0.172258i
$209$		−	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	$0.418847\pi$
$212$	141.941			0.669534
$213$			87.6594i			0.411546i
$214$	−53.3970			−0.249519
$215$		0			0
$216$			14.6969i			0.0680414i
$217$	−332.735	−	112.532i	−1.53334	−	0.518582i
$218$	−45.1716			−0.207209
$219$	122.309			0.558487
$220$		0			0
$221$	272.382			1.23250
$222$			75.7179i			0.341071i
$223$		−	57.0047i		−	0.255627i	−0.991798	−	0.127813i	$-0.959204\pi$
$223$		−	57.0047i		−	0.255627i	0.991798	−	0.127813i	$-0.0407958\pi$
$224$	−12.6863	+	37.5108i	−0.0566352	+	0.167459i
$225$		0			0
$226$	−151.279			−0.669377
$227$			287.418i			1.26616i	0.774086	+	0.633080i	$0.218210\pi$
$227$			287.418i			1.26616i	−0.774086	+	0.633080i	$0.781790\pi$
$228$	−55.8823			−0.245098
$229$		−	139.405i		−	0.608754i	−0.952552	−	0.304377i	$-0.901552\pi$
$229$		−	139.405i		−	0.608754i	0.952552	−	0.304377i	$-0.0984482\pi$
$230$		0			0
$231$	−117.640	−	39.7862i	−0.509262	−	0.172235i
$232$	−84.8528			−0.365745
$233$	−362.735			−1.55680			−0.778401	−	0.627767i	$-0.783969\pi$
$233$	−362.735			−1.55680			−0.778401	+	0.627767i	$0.783969\pi$
$234$		−	38.0031i		−	0.162407i
$235$		0			0
$236$			0.984744i			0.00417265i
$237$			231.789i			0.978012i
$238$	−285.161	−	96.4427i	−1.19816	−	0.405222i
$239$	18.4781			0.0773144			0.0386572	−	0.999253i	$-0.487692\pi$
$239$	18.4781			0.0773144			0.0386572	+	0.999253i	$0.487692\pi$
$240$		0			0
$241$			178.104i			0.739021i	0.929227	+	0.369510i	$0.120475\pi$
$241$			178.104i			0.739021i	−0.929227	+	0.369510i	$0.879525\pi$
$242$	22.7523			0.0940178
$243$			15.5885i			0.0641500i
$244$		−	5.73951i		−	0.0235226i
$245$		0			0
$246$	17.3970			0.0707194
$247$	144.500			0.585018
$248$			141.926i			0.572283i
$249$	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	$-0.0321844\pi$
$252$	−13.4558	+	39.7862i	−0.0533962	+	0.157882i
$253$	−68.9117			−0.272378
$254$	31.1960			0.122819
$255$		0			0
$256$	16.0000			0.0625000
$257$			187.584i			0.729898i	0.931028	+	0.364949i	$0.118914\pi$
$257$			187.584i			0.729898i	−0.931028	+	0.364949i	$0.881086\pi$
$258$		−	182.307i		−	0.706615i
$259$	−69.3238	+	204.976i	−0.267659	+	0.791414i
$260$		0			0
$261$	−90.0000			−0.344828
$262$		−	100.357i		−	0.383042i
$263$	402.978			1.53223			0.766117	−	0.642701i	$-0.222186\pi$
$263$	402.978			1.53223			0.766117	+	0.642701i	$0.222186\pi$
$264$			50.1785i			0.190070i
$265$		0			0
$266$	−151.279	−	51.1632i	−0.568719	−	0.192343i
$267$	251.095			0.940432
$268$	−54.0589			−0.201712
$269$			480.538i			1.78639i	0.449674	+	0.893193i	$0.351540\pi$
$269$			480.538i			1.78639i	−0.449674	+	0.893193i	$0.648460\pi$
$270$		0			0
$271$			37.3068i			0.137664i	0.997628	+	0.0688318i	$0.0219272\pi$
$271$			37.3068i			0.137664i	−0.997628	+	0.0688318i	$0.978073\pi$
$272$			121.634i			0.447184i
$273$	34.7939	−	102.879i	0.127450	−	0.376845i
$274$	−149.574			−0.545889
$275$		0			0
$276$			23.3062i			0.0844428i
$277$	82.6762			0.298470			0.149235	−	0.988802i	$-0.452319\pi$
$277$	82.6762			0.298470			0.149235	+	0.988802i	$0.452319\pi$
$278$			256.428i			0.922403i
$279$			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	$0.586502\pi$
$282$	−142.794			−0.506361
$283$			284.158i			1.00409i	0.864841	+	0.502046i	$0.167419\pi$
$283$			284.158i			1.00409i	−0.864841	+	0.502046i	$0.832581\pi$
$284$	101.220			0.356410
$285$		0			0
$286$		−	129.751i		−	0.453674i
$287$	47.0955	+	15.9279i	0.164096	+	0.0554978i
$288$	16.9706			0.0589256
$289$	−635.676			−2.19957
$290$		0			0
$291$	174.426			0.599403
$292$		−	141.230i		−	0.483664i
$293$		−	537.237i		−	1.83357i	−0.399379	−	0.916786i	$-0.630774\pi$
$293$		−	537.237i		−	1.83357i	0.399379	−	0.916786i	$-0.369226\pi$
$294$	−72.8528	+	95.3859i	−0.247799	+	0.324442i
$295$		0			0
$296$	87.4315			0.295377
$297$			53.2223i			0.179200i
$298$	−58.1909			−0.195272
$299$		−	60.2649i		−	0.201555i
$300$		0			0
$301$	166.912	−	493.524i	0.554524	−	1.63961i
$302$	−113.657			−0.376347
$303$	−240.551			−0.793899
$304$			64.5273i			0.212261i
$305$		0			0
$306$			129.012i			0.421609i
$307$		−	34.7430i		−	0.113169i	−0.998398	−	0.0565847i	$-0.981979\pi$
$307$		−	34.7430i		−	0.113169i	0.998398	−	0.0565847i	$-0.0180211\pi$
$308$	−45.9411	+	135.839i	−0.149159	+	0.441034i
$309$	−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	$0.954211\pi$
$312$	−43.8823			−0.140648
$313$			519.700i			1.66038i	0.557479	+	0.830191i	$0.311769\pi$
$313$			519.700i			1.66038i	−0.557479	+	0.830191i	$0.688231\pi$
$314$			81.0495i			0.258119i
$315$		0			0
$316$	267.647			0.846983
$317$	−542.029			−1.70987			−0.854935	−	0.518736i	$-0.826403\pi$
$317$	−542.029			−1.70987			−0.854935	+	0.518736i	$0.826403\pi$
$318$		−	173.842i		−	0.546672i
$319$	−307.279			−0.963258
$320$		0			0
$321$			65.3977i			0.203731i
$322$	−21.3381	+	63.0924i	−0.0662674	+	0.195939i
$323$	−490.544			−1.51871
$324$	18.0000			0.0555556
$325$		0			0
$326$	−247.362			−0.758781
$327$			55.3237i			0.169185i
$328$		−	20.0883i		−	0.0612448i
$329$	−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	$0.412531\pi$
$332$		−	208.831i		−	0.629009i
$333$	92.7351			0.278484
$334$		−	277.417i		−	0.830588i
$335$		0			0
$336$	45.9411	+	15.5375i	0.136730	+	0.0462425i
$337$	−291.823			−0.865945			−0.432972	−	0.901407i	$-0.642535\pi$
$337$	−291.823			−0.865945			−0.432972	+	0.901407i	$0.642535\pi$
$338$	−125.532			−0.371396
$339$			185.278i			0.546544i
$340$		0			0
$341$			513.960i			1.50721i
$342$			68.4415i			0.200121i
$343$	−284.551	+	191.519i	−0.829596	+	0.558365i
$344$	−210.510			−0.611947
$345$		0			0
$346$			54.4831i			0.157466i
$347$	−452.080			−1.30283			−0.651413	−	0.758724i	$-0.725823\pi$
$347$	−452.080			−1.30283			−0.651413	+	0.758724i	$0.725823\pi$
$348$			103.923i			0.298629i
$349$		−	235.067i		−	0.673543i	−0.941586	−	0.336772i	$-0.890665\pi$
$349$		−	235.067i		−	0.673543i	0.941586	−	0.336772i	$-0.109335\pi$
$350$		0			0
$351$	−46.5442			−0.132604
$352$	57.9411			0.164605
$353$		−	25.2458i		−	0.0715179i	−0.999360	−	0.0357590i	$-0.988615\pi$
$353$		−	25.2458i		−	0.0715179i	0.999360	−	0.0357590i	$-0.0113849\pi$
$354$	1.20606			0.00340695
$355$		0			0
$356$		−	289.940i		−	0.814438i
$357$	−118.118	+	349.250i	−0.330862	+	0.978291i
$358$	270.250			0.754888
$359$	−106.243			−0.295941			−0.147970	−	0.988992i	$-0.547274\pi$
$359$	−106.243			−0.295941			−0.147970	+	0.988992i	$0.547274\pi$
$360$		0			0
$361$	100.765			0.279126
$362$			170.828i			0.471900i
$363$		−	27.8658i		−	0.0767652i
$364$	−118.794	−	40.1766i	−0.326357	−	0.110375i
$365$		0			0
$366$	−7.02944			−0.0192061
$367$		−	286.416i		−	0.780426i	−0.920725	−	0.390213i	$-0.872401\pi$
$367$		−	286.416i		−	0.780426i	0.920725	−	0.390213i	$-0.127599\pi$
$368$	26.9117			0.0731296
$369$		−	21.3068i		−	0.0577421i
$370$		0			0
$371$	159.161	−	470.608i	0.429007	−	1.26849i
$372$	173.823			0.467267
$373$	423.470			1.13531			0.567654	−	0.823267i	$-0.307851\pi$
$373$	423.470			1.13531			0.567654	+	0.823267i	$0.307851\pi$
$374$			440.476i			1.17774i
$375$		0			0
$376$			164.884i			0.438522i
$377$		−	268.723i		−	0.712793i
$378$	48.7279	+	16.4800i	0.128910	+	0.0435978i
$379$	101.103			0.266761			0.133381	−	0.991065i	$-0.457417\pi$
$379$	101.103			0.266761			0.133381	+	0.991065i	$0.457417\pi$
$380$		0			0
$381$		−	38.2071i		−	0.100281i
$382$	−158.485			−0.414883
$383$			220.394i			0.575442i	0.957714	+	0.287721i	$0.0928976\pi$
$383$			220.394i			0.575442i	−0.957714	+	0.287721i	$0.907102\pi$
$384$		−	19.5959i		−	0.0510310i
$385$		0			0
$386$	−32.4020			−0.0839431
$387$	−223.279			−0.576949
$388$		−	201.410i		−	0.519099i
$389$	−383.470			−0.985784			−0.492892	−	0.870090i	$-0.664060\pi$
$389$	−383.470			−0.985784			−0.492892	+	0.870090i	$0.664060\pi$
$390$		0			0
$391$			204.586i			0.523238i
$392$	110.142	+	84.1232i	0.280975	+	0.214600i
$393$	−122.912			−0.312752
$394$	−164.132			−0.416579
$395$		0			0
$396$	61.4558			0.155192
$397$		−	485.178i		−	1.22211i	−0.791588	−	0.611056i	$-0.790745\pi$
$397$		−	485.178i		−	1.22211i	0.791588	−	0.611056i	$-0.209255\pi$
$398$			230.660i			0.579548i
$399$	−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	$0.602233\pi$
$402$			66.2083i			0.164697i
$403$	−449.470			−1.11531
$404$			277.765i			0.687536i
$405$		0			0
$406$	−95.1472	+	281.331i	−0.234353	+	0.692933i
$407$	316.617			0.777930
$408$	148.971			0.365124
$409$		−	5.39135i		−	0.0131818i	−0.999978	−	0.00659089i	$-0.997902\pi$
$409$		−	5.39135i		−	0.0131818i	0.999978	−	0.00659089i	$-0.00209796\pi$
$410$		0			0
$411$			183.189i			0.445717i
$412$			157.752i			0.382893i
$413$	3.26494	+	1.10421i	0.00790541	+	0.00267364i
$414$	28.5442			0.0689472
$415$		0			0
$416$			50.6709i			0.121805i
$417$	314.059			0.753139
$418$			233.674i			0.559028i
$419$		−	294.431i		−	0.702700i	−0.936244	−	0.351350i	$-0.885723\pi$
$419$		−	294.431i		−	0.702700i	0.936244	−	0.351350i	$-0.114277\pi$
$420$		0			0
$421$	−290.441			−0.689883			−0.344941	−	0.938624i	$-0.612101\pi$
$421$	−290.441			−0.689883			−0.344941	+	0.938624i	$0.612101\pi$
$422$	−150.510			−0.356658
$423$			174.886i			0.413442i
$424$	−200.735			−0.473432
$425$		0			0
$426$		−	123.969i		−	0.291007i
$427$	−19.0294	−	6.43583i	−0.0445654	−	0.0150722i
$428$	75.5147			0.176436
$429$	−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	$0.481262\pi$
$432$		−	20.7846i		−	0.0481125i
$433$		−	724.761i		−	1.67381i	−0.547347	−	0.836906i	$-0.684362\pi$
$433$		−	724.761i		−	1.67381i	0.547347	−	0.836906i	$-0.315638\pi$
$434$	470.558	+	159.145i	1.08424	+	0.366693i
$435$		0			0
$436$	63.8823			0.146519
$437$			108.534i			0.248361i
$438$	−172.971			−0.394910
$439$			371.728i			0.846761i	0.905952	+	0.423381i	$0.139157\pi$
$439$			371.728i			0.846761i	−0.905952	+	0.423381i	$0.860843\pi$
$440$		0			0
$441$	116.823	+	89.2261i	0.264906	+	0.202327i
$442$	−385.206			−0.871507
$443$	229.258			0.517512			0.258756	−	0.965943i	$-0.416687\pi$
$443$	229.258			0.517512			0.258756	+	0.965943i	$0.416687\pi$
$444$		−	107.081i		−	0.241174i
$445$		0			0
$446$			80.6168i			0.180755i
$447$			71.2690i			0.159439i
$448$	17.9411	−	53.0482i	0.0400472	−	0.118411i
$449$	−600.323			−1.33702			−0.668511	−	0.743702i	$-0.733068\pi$
$449$	−600.323			−1.33702			−0.668511	+	0.743702i	$0.733068\pi$
$450$		0			0
$451$		−	72.7461i		−	0.161300i
$452$	213.941			0.473321
$453$			139.201i			0.307286i
$454$		−	406.471i		−	0.895311i
$455$		0			0
$456$	79.0294			0.173310
$457$	−483.823			−1.05869			−0.529347	−	0.848405i	$-0.677563\pi$
$457$	−483.823			−1.05869			−0.529347	+	0.848405i	$0.677563\pi$
$458$			197.148i			0.430454i
$459$	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	$-0.0962852\pi$
$462$	166.368	+	56.2662i	0.360103	+	0.121788i
$463$	153.470			0.331469			0.165734	−	0.986170i	$-0.447001\pi$
$463$	153.470			0.331469			0.165734	+	0.986170i	$0.447001\pi$
$464$	120.000			0.258621
$465$		0			0
$466$	512.985			1.10083
$467$		−	61.7420i		−	0.132210i	−0.997813	−	0.0661049i	$-0.978943\pi$
$467$		−	61.7420i		−	0.132210i	0.997813	−	0.0661049i	$-0.0210572\pi$
$468$			53.7446i			0.114839i
$469$	−60.6173	+	179.233i	−0.129248	+	0.382160i
$470$		0			0
$471$	99.2649			0.210754
$472$		−	1.39264i		−	0.00295051i
$473$	−762.323			−1.61168
$474$		−	327.799i		−	0.691559i
$475$		0			0
$476$	403.279	+	136.391i	0.847225	+	0.286535i
$477$	−212.912			−0.446356
$478$	−26.1320			−0.0546695
$479$		−	556.514i		−	1.16182i	−0.813966	−	0.580912i	$-0.802696\pi$
$479$		−	556.514i		−	1.16182i	0.813966	−	0.580912i	$-0.197304\pi$
$480$		0			0
$481$			276.889i			0.575653i
$482$		−	251.877i		−	0.522567i
$483$	77.2721	+	26.1337i	0.159984	+	0.0541071i
$484$	−32.1766			−0.0664806
$485$		0			0
$486$		−	22.0454i		−	0.0453609i
$487$	325.220			0.667804			0.333902	−	0.942608i	$-0.391635\pi$
$487$	325.220			0.667804			0.333902	+	0.942608i	$0.391635\pi$
$488$			8.11689i			0.0166330i
$489$			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	$0.272554\pi$
$492$	−24.6030			−0.0500062
$493$			912.255i			1.85042i
$494$	−204.353			−0.413671
$495$		0			0
$496$		−	200.714i		−	0.404665i
$497$	113.500	−	335.598i	0.228371	−	0.675247i
$498$	−255.765			−0.513583
$499$	755.426			1.51388			0.756939	−	0.653485i	$-0.226694\pi$
$499$	755.426			1.51388			0.756939	+	0.653485i	$0.226694\pi$
$500$		0			0
$501$	−339.765			−0.678173
$502$			71.6594i			0.142748i
$503$		−	509.409i		−	1.01274i	−0.862316	−	0.506371i	$-0.830987\pi$
$503$		−	509.409i		−	1.01274i	0.862316	−	0.506371i	$-0.169013\pi$
$504$	19.0294	−	56.2662i	0.0377568	−	0.111639i
$505$		0			0
$506$	97.4558			0.192600
$507$			153.745i			0.303244i
$508$	−44.1177			−0.0868460
$509$			769.433i			1.51166i	0.654770	+	0.755828i	$0.272766\pi$
$509$			769.433i			1.51166i	−0.654770	+	0.755828i	$0.727234\pi$
$510$		0			0
$511$	−468.250	−	158.364i	−0.916340	−	0.309910i
$512$	−22.6274			−0.0441942
$513$	83.8234			0.163398
$514$		−	265.283i		−	0.516116i
$515$		0			0
$516$			257.821i			0.499652i
$517$			597.099i			1.15493i
$518$	98.0387	−	289.880i	0.189264	−	0.559615i
$519$	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	$-0.171701\pi$
$522$	127.279			0.243830
$523$			839.466i			1.60510i	0.596586	+	0.802549i	$0.296523\pi$
$523$			839.466i			1.60510i	−0.596586	+	0.802549i	$0.703477\pi$
$524$			141.926i			0.270852i
$525$		0			0
$526$	−569.897			−1.08345
$527$	1525.85			2.89535
$528$		−	70.9631i		−	0.134400i
$529$	−483.735			−0.914433
$530$		0			0
$531$		−	1.47712i		−	0.00278176i
$532$	213.941	+	72.3557i	0.402145	+	0.136007i
$533$	63.6182			0.119359
$534$	−355.103			−0.664986
$535$		0			0
$536$	76.4508			0.142632
$537$		−	330.987i		−	0.616363i
$538$		−	679.583i		−	1.26317i
$539$	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	$0.414896\pi$
$542$		−	52.7598i		−	0.0973428i
$543$	209.220			0.385305
$544$		−	172.016i		−	0.316207i
$545$		0			0
$546$	−49.2061	+	145.492i	−0.0901210	+	0.266469i
$547$	741.470			1.35552			0.677761	−	0.735283i	$-0.262951\pi$
$547$	741.470			1.35552			0.677761	+	0.735283i	$0.262951\pi$
$548$	211.529			0.386002
$549$			8.60927i			0.0156817i
$550$		0			0
$551$			483.954i			0.878320i
$552$		−	32.9600i		−	0.0597101i
$553$	300.118	−	887.387i	0.542708	−	1.60468i
$554$	−116.922			−0.211050
$555$		0			0
$556$		−	362.644i		−	0.652237i
$557$	−834.853			−1.49884			−0.749419	−	0.662096i	$-0.769667\pi$
$557$	−834.853			−1.49884			−0.749419	+	0.662096i	$0.769667\pi$
$558$		−	212.889i		−	0.381522i
$559$		−	666.669i		−	1.19261i
$560$		0			0
$561$	539.470			0.961622
$562$	213.338			0.379605
$563$		−	417.781i		−	0.742062i	−0.928620	−	0.371031i	$-0.879004\pi$
$563$		−	417.781i		−	0.742062i	0.928620	−	0.371031i	$-0.120996\pi$
$564$	201.941			0.358052
$565$		0			0
$566$		−	401.861i		−	0.710001i
$567$	20.1838	−	59.6793i	0.0355975	−	0.105254i
$568$	−143.147			−0.252020
$569$	673.029			1.18283			0.591414	−	0.806368i	$-0.298570\pi$
$569$	673.029			1.18283			0.591414	+	0.806368i	$0.298570\pi$
$570$		0			0
$571$	−265.088			−0.464253			−0.232126	−	0.972686i	$-0.574568\pi$
$571$	−265.088			−0.464253			−0.232126	+	0.972686i	$0.574568\pi$
$572$			183.495i			0.320796i
$573$			194.104i			0.338750i
$574$	−66.6030	−	22.5254i	−0.116033	−	0.0392429i
$575$		0			0
$576$	−24.0000			−0.0416667
$577$			468.740i			0.812375i	0.913790	+	0.406188i	$0.133142\pi$
$577$			468.740i			0.812375i	−0.913790	+	0.406188i	$0.866858\pi$
$578$	898.982			1.55533
$579$			39.6842i			0.0685392i
$580$		0			0
$581$	−692.382	−	234.166i	−1.19171	−	0.403040i
$582$	−246.676			−0.423842
$583$	−726.926			−1.24687
$584$			199.729i			0.342002i
$585$		0			0
$586$			759.767i			1.29653i
$587$		−	120.530i		−	0.205332i	−0.994716	−	0.102666i	$-0.967263\pi$
$587$		−	120.530i		−	0.205332i	0.994716	−	0.102666i	$-0.0327372\pi$
$588$	103.029	−	134.896i	0.175220	−	0.229415i
$589$	809.470			1.37431
$590$		0			0
$591$			201.020i			0.340135i
$592$	−123.647			−0.208863
$593$			561.468i			0.946826i	0.880841	+	0.473413i	$0.156978\pi$
$593$			561.468i			0.946826i	−0.880841	+	0.473413i	$0.843022\pi$
$594$		−	75.2677i		−	0.126713i
$595$		0			0
$596$	82.2944			0.138078
$597$	282.500			0.473199
$598$			85.2274i			0.142521i
$599$	940.109			1.56946			0.784732	−	0.619835i	$-0.212801\pi$
$599$	940.109			1.56946			0.784732	+	0.619835i	$0.212801\pi$
$600$		0			0
$601$		−	563.527i		−	0.937649i	−0.883291	−	0.468824i	$-0.844678\pi$
$601$		−	563.527i		−	0.937649i	0.883291	−	0.468824i	$-0.155322\pi$
$602$	−236.049	+	697.948i	−0.392108	+	1.15938i
$603$	81.0883			0.134475
$604$	160.735			0.266118
$605$		0			0
$606$	340.191			0.561371
$607$		−	306.589i		−	0.505089i	−0.967585	−	0.252544i	$-0.918733\pi$
$607$		−	306.589i		−	0.505089i	0.967585	−	0.252544i	$-0.0812674\pi$
$608$		−	91.2553i		−	0.150091i
$609$	344.558	+	116.531i	0.565777	+	0.191348i
$610$		0			0
$611$	−522.177			−0.854626
$612$		−	182.451i		−	0.298122i
$613$	−275.588			−0.449572			−0.224786	−	0.974408i	$-0.572168\pi$
$613$	−275.588			−0.449572			−0.224786	+	0.974408i	$0.572168\pi$
$614$			49.1340i			0.0800228i
$615$		0			0
$616$	64.9706	−	192.105i	0.105472	−	0.311858i
$617$	−461.294			−0.747639			−0.373820	−	0.927501i	$-0.621952\pi$
$617$	−461.294			−0.747639			−0.373820	+	0.927501i	$0.621952\pi$
$618$	193.206			0.312631
$619$			374.904i			0.605660i	0.953045	+	0.302830i	$0.0979315\pi$
$619$			374.904i			0.605660i	−0.953045	+	0.302830i	$0.902068\pi$
$620$		0			0
$621$		−	34.9593i		−	0.0562952i
$622$		−	126.100i		−	0.202734i
$623$	−961.301	−	325.116i	−1.54302	−	0.521855i
$624$	62.0589			0.0994533
$625$		0			0
$626$		−	734.966i		−	1.17407i
$627$	286.191			0.456445
$628$		−	114.621i		−	0.182518i
$629$		−	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	$0.851603\pi$
$632$	−378.510			−0.598908
$633$			184.336i			0.291210i
$634$	766.544			1.20906
$635$		0			0
$636$			245.849i			0.386555i
$637$	−266.412	+	348.812i	−0.418229	+	0.547586i
$638$	434.558			0.681126
$639$	−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	$0.742311\pi$
$642$		−	92.4863i		−	0.144060i
$643$		−	300.765i		−	0.467753i	−0.972266	−	0.233876i	$-0.924859\pi$
$643$		−	300.765i		−	0.467753i	0.972266	−	0.233876i	$-0.0751411\pi$
$644$	30.1766	−	89.2261i	0.0468581	−	0.138550i
$645$		0			0
$646$	693.734			1.07389
$647$		−	940.604i		−	1.45379i	−0.686747	−	0.726896i	$-0.740962\pi$
$647$		−	940.604i		−	1.45379i	0.686747	−	0.726896i	$-0.259038\pi$
$648$	−25.4558			−0.0392837
$649$		−	5.04319i		−	0.00777071i
$650$		0			0
$651$	194.912	−	576.314i	0.299404	−	0.885275i
$652$	349.823			0.536539
$653$	455.970			0.698269			0.349135	−	0.937073i	$-0.386476\pi$
$653$	455.970			0.698269			0.349135	+	0.937073i	$0.386476\pi$
$654$		−	78.2395i		−	0.119632i
$655$		0			0
$656$			28.4091i			0.0433066i
$657$			211.845i			0.322443i
$658$	546.676	+	184.888i	0.830815	+	0.280985i
$659$	403.684			0.612571			0.306285	−	0.951940i	$-0.400914\pi$
$659$	403.684			0.612571			0.306285	+	0.951940i	$0.400914\pi$
$660$		0			0
$661$			1153.41i			1.74495i	0.488663	+	0.872473i	$0.337485\pi$
$661$			1153.41i			1.74495i	−0.488663	+	0.872473i	$0.662515\pi$
$662$	−254.039			−0.383744
$663$			471.779i			0.711582i
$664$			295.331i			0.444776i
$665$		0			0
$666$	−131.147			−0.196918
$667$	201.838			0.302605
$668$			392.326i			0.587315i
$669$	98.7351			0.147586
$670$		0			0
$671$			29.3939i			0.0438061i
$672$	−64.9706	−	21.9733i	−0.0966824	−	0.0326984i
$673$	607.440			0.902585			0.451293	−	0.892376i	$-0.350963\pi$
$673$	607.440			0.902585			0.451293	+	0.892376i	$0.350963\pi$
$674$	412.701			0.612315
$675$		0			0
$676$	177.529			0.262617
$677$		−	1137.59i		−	1.68034i	−0.542326	−	0.840168i	$-0.682456\pi$
$677$		−	1137.59i		−	1.68034i	0.542326	−	0.840168i	$-0.317544\pi$
$678$		−	262.023i		−	0.386465i
$679$	−667.779	−	225.845i	−0.983474	−	0.332615i
$680$		0			0
$681$	−497.823			−0.731018
$682$		−	726.849i		−	1.06576i
$683$	818.111			1.19782			0.598910	−	0.800817i	$-0.295601\pi$
$683$	818.111			1.19782			0.598910	+	0.800817i	$0.295601\pi$
$684$		−	96.7909i		−	0.141507i
$685$		0			0
$686$	402.416	−	270.849i	0.586613	−	0.394823i
$687$	241.456			0.351464
$688$	297.706			0.432712
$689$		−	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	$-0.0472239\pi$
$692$		−	77.0508i		−	0.111345i
$693$	68.9117	−	203.758i	0.0994397	−	0.294023i
$694$	639.338			0.921236
$695$		0			0
$696$		−	146.969i		−	0.211163i
$697$	−215.970			−0.309856
$698$			332.434i			0.476267i
$699$		−	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	$0.573031\pi$
$702$	65.8234			0.0937655
$703$		−	498.662i		−	0.709334i
$704$	−81.9411			−0.116394
$705$		0			0
$706$			35.7030i			0.0505708i
$707$	920.933	+	311.463i	1.30259	+	0.440542i
$708$	−1.70563			−0.00240908
$709$	−217.647			−0.306977			−0.153489	−	0.988150i	$-0.549051\pi$
$709$	−217.647			−0.306977			−0.153489	+	0.988150i	$0.549051\pi$
$710$		0			0
$711$	−401.470			−0.564656
$712$			410.037i			0.575895i
$713$		−	337.597i		−	0.473488i
$714$	167.044	−	493.914i	0.233955	−	0.691757i
$715$		0			0
$716$	−382.191			−0.533786
$717$			32.0051i			0.0446375i
$718$	150.250			0.209262
$719$		−	1207.36i		−	1.67922i	−0.543192	−	0.839609i	$-0.682784\pi$
$719$		−	1207.36i		−	1.67922i	0.543192	−	0.839609i	$-0.317216\pi$
$720$		0			0
$721$	523.029	+	176.891i	0.725422	+	0.245341i
$722$	−142.503			−0.197372
$723$	−308.485			−0.426674
$724$		−	241.587i		−	0.333683i
$725$		0			0
$726$			39.4082i			0.0542812i
$727$			123.231i			0.169506i	0.996402	+	0.0847528i	$0.0270100\pi$
$727$			123.231i			0.169506i	−0.996402	+	0.0847528i	$0.972990\pi$
$728$	168.000	+	56.8183i	0.230769	+	0.0780471i
$729$	−27.0000			−0.0370370
$730$		0			0
$731$			2263.19i			3.09603i
$732$	9.94113			0.0135808
$733$		−	375.779i		−	0.512659i	−0.966589	−	0.256330i	$-0.917487\pi$
$733$		−	375.779i		−	0.512659i	0.966589	−	0.256330i	$-0.0825133\pi$
$734$			405.054i			0.551844i
$735$		0			0
$736$	−38.0589			−0.0517104
$737$	276.853			0.375648
$738$			30.1324i			0.0408299i
$739$	722.530			0.977713			0.488856	−	0.872364i	$-0.337414\pi$
$739$	722.530			0.977713			0.488856	+	0.872364i	$0.337414\pi$
$740$		0			0
$741$			250.281i			0.337761i
$742$	−225.088	+	665.540i	−0.303354	+	0.896954i
$743$	−1268.48			−1.70724			−0.853618	−	0.520899i	$-0.825597\pi$
$743$	−1268.48			−1.70724			−0.853618	+	0.520899i	$0.825597\pi$
$744$	−245.823			−0.330408
$745$		0			0
$746$	−598.877			−0.802784
$747$			313.246i			0.419339i
$748$		−	622.926i		−	0.832789i
$749$	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	$0.405551\pi$
$752$		−	233.182i		−	0.310082i
$753$	87.7645			0.116553
$754$			380.031i			0.504020i
$755$		0			0
$756$	−68.9117	−	23.3062i	−0.0911530	−	0.0308283i
$757$	668.530			0.883131			0.441565	−	0.897229i	$-0.354423\pi$
$757$	668.530			0.883131			0.441565	+	0.897229i	$0.354423\pi$
$758$	−142.981			−0.188629
$759$		−	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	$-0.196269\pi$
$762$			54.0330i			0.0709094i
$763$	71.6325	−	211.803i	0.0938827	−	0.277592i
$764$	224.132			0.293367
$765$		0			0
$766$		−	311.685i		−	0.406899i
$767$	4.41039			0.00575018
$768$			27.7128i			0.0360844i
$769$			1163.41i			1.51289i	0.654059	+	0.756444i	$0.273065\pi$
$769$			1163.41i			1.51289i	−0.654059	+	0.756444i	$0.726935\pi$
$770$		0			0
$771$	−324.905			−0.421407
$772$	45.8234			0.0593567
$773$			536.371i			0.693883i	0.937887	+	0.346941i	$0.112780\pi$
$773$			536.371i			0.693883i	−0.937887	+	0.346941i	$0.887220\pi$
$774$	315.765			0.407964
$775$		0			0
$776$			284.837i			0.367058i
$777$	−355.029	−	120.072i	−0.456923	−	0.154533i
$778$	542.309			0.697055
$779$	−114.573			−0.147077
$780$		0			0
$781$	−518.382			−0.663741
$782$		−	289.328i		−	0.369985i
$783$		−	155.885i		−	0.199086i
$784$	−155.765	−	118.968i	−0.198679	−	0.151745i
$785$		0			0
$786$	173.823			0.221149
$787$		−	83.9192i		−	0.106632i	−0.998578	−	0.0533159i	$-0.983021\pi$
$787$		−	83.9192i		−	0.106632i	0.998578	−	0.0533159i	$-0.0169790\pi$
$788$	232.118			0.294566
$789$			697.978i			0.884636i
$790$		0			0
$791$	239.897	−	709.325i	0.303283	−	0.896745i
$792$	−86.9117			−0.109737
$793$	−25.7056			−0.0324157
$794$			686.146i			0.864163i
$795$		0			0
$796$		−	326.202i		−	0.409802i
$797$			1005.57i			1.26169i	0.775908	+	0.630846i	$0.217292\pi$
$797$			1005.57i			1.26169i	−0.775908	+	0.630846i	$0.782708\pi$
$798$	88.6173	−	262.023i	0.111049	−	0.328350i
$799$	1772.67			2.21862
$800$		0			0
$801$			434.910i			0.542959i
$802$	358.045			0.446440
$803$			723.283i			0.900727i
$804$		−	93.6327i		−	0.116459i
$805$		0			0
$806$	635.647			0.788644
$807$	−832.316			−1.03137
$808$		−	392.819i		−	0.486162i
$809$	−282.853			−0.349633			−0.174816	−	0.984601i	$-0.555933\pi$
$809$	−282.853			−0.349633			−0.174816	+	0.984601i	$0.555933\pi$
$810$		0			0
$811$		−	923.997i		−	1.13933i	−0.821877	−	0.569665i	$-0.807073\pi$
$811$		−	923.997i		−	1.13933i	0.821877	−	0.569665i	$-0.192927\pi$
$812$	134.558	−	397.862i	0.165712	−	0.489978i
$813$	−64.6173			−0.0794801
$814$	−447.765			−0.550079
$815$		0			0
$816$	−210.676			−0.258182
$817$			1200.63i			1.46956i
$818$			7.62452i			0.00932093i
$819$	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	$0.217819\pi$
$822$		−	259.069i		−	0.315169i
$823$	−381.852			−0.463976			−0.231988	−	0.972719i	$-0.574523\pi$
$823$	−381.852			−0.463976			−0.231988	+	0.972719i	$0.574523\pi$
$824$		−	223.095i		−	0.270747i
$825$		0			0
$826$	−4.61732	−	1.56159i	−0.00558997	−	0.00189055i
$827$	−653.022			−0.789628			−0.394814	−	0.918761i	$-0.629191\pi$
$827$	−653.022			−0.789628			−0.394814	+	0.918761i	$0.629191\pi$
$828$	−40.3675			−0.0487531
$829$			215.580i			0.260048i	0.991511	+	0.130024i	$0.0415054\pi$
$829$			215.580i			0.260048i	−0.991511	+	0.130024i	$0.958495\pi$
$830$		0			0
$831$			143.199i			0.172322i
$832$		−	71.6594i		−	0.0861291i
$833$	904.410	−	1184.14i	1.08573	−	1.42154i
$834$	−444.146			−0.532550
$835$		0			0
$836$		−	330.465i		−	0.395293i
$837$	−260.735			−0.311511
$838$			416.389i			0.496884i
$839$			632.267i			0.753596i	0.926295	+	0.376798i	$0.122975\pi$
$839$			632.267i			0.753596i	−0.926295	+	0.376798i	$0.877025\pi$
$840$		0			0
$841$	59.0000			0.0701546
$842$	410.745			0.487821
$843$		−	261.285i		−	0.309946i
$844$	212.853			0.252195
$845$		0			0
$846$		−	247.326i		−	0.292348i
$847$	−36.0803	+	106.682i	−0.0425978	+	0.125953i
$848$	283.882			0.334767
$849$	−492.177			−0.579713
$850$		0			0
$851$	−207.971			−0.244385
$852$			175.319i			0.205773i
$853$			919.650i			1.07814i	0.842262	+	0.539068i	$0.181224\pi$
$853$			919.650i			1.07814i	−0.842262	+	0.539068i	$0.818776\pi$
$854$	26.9117	+	9.10164i	0.0315125	+	0.0106577i
$855$		0			0
$856$	−106.794			−0.124759
$857$			527.150i			0.615111i	0.951530	+	0.307556i	$0.0995110\pi$
$857$			527.150i			0.615111i	−0.951530	+	0.307556i	$0.900489\pi$
$858$	224.735			0.261929
$859$			239.804i			0.279166i	0.990210	+	0.139583i	$0.0445762\pi$
$859$			239.804i			0.279166i	−0.990210	+	0.139583i	$0.955424\pi$
$860$		0			0
$861$	−27.5879	+	81.5717i	−0.0320417	+	0.0947407i
$862$	−71.7199			−0.0832018
$863$	−362.507			−0.420054			−0.210027	−	0.977696i	$-0.567355\pi$
$863$	−362.507			−0.420054			−0.210027	+	0.977696i	$0.567355\pi$
$864$			29.3939i			0.0340207i
$865$		0			0
$866$			1024.97i			1.18356i
$867$		−	1101.02i		−	1.26992i
$868$	−665.470	−	225.065i	−0.766671	−	0.259291i
$869$	−1370.70			−1.57734
$870$		0			0
$871$			242.114i			0.277973i
$872$	−90.3431			−0.103605
$873$			302.115i			0.346066i
$874$		−	153.490i		−	0.175617i
$875$		0			0
$876$	244.617			0.279244
$877$	861.647			0.982493			0.491247	−	0.871020i	$-0.336541\pi$
$877$	861.647			0.982493			0.491247	+	0.871020i	$0.336541\pi$
$878$		−	525.703i		−	0.598750i
$879$	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	$0.939193\pi$
$882$	−165.213	−	126.185i	−0.187317	−	0.143067i
$883$	1488.01			1.68518			0.842590	−	0.538555i	$-0.181030\pi$
$883$	1488.01			1.68518			0.842590	+	0.538555i	$0.181030\pi$
$884$	544.764			0.616248
$885$		0			0
$886$	−324.219			−0.365936
$887$		−	801.498i		−	0.903606i	−0.892118	−	0.451803i	$-0.850781\pi$
$887$		−	801.498i		−	0.903606i	0.892118	−	0.451803i	$-0.149219\pi$
$888$			151.436i			0.170536i
$889$	−49.4701	+	146.273i	−0.0556469	+	0.164537i
$890$		0			0
$891$	−92.1838			−0.103461
$892$		−	114.009i		−	0.127813i
$893$	940.410			1.05309
$894$		−	100.790i		−	0.112740i
$895$		0			0
$896$	−25.3726	+	75.0215i	−0.0283176	+	0.0837294i
$897$	104.382			0.116368
$898$	848.985			0.945417
$899$		−	1505.35i		−	1.67448i
$900$		0			0
$901$			2158.11i			2.39524i
$902$			102.879i			0.114056i
$903$	854.808	+	289.100i	0.946631	+	0.320155i
$904$	−302.558			−0.334689
$905$		0			0
$906$		−	196.859i		−	0.217284i
$907$	807.279			0.890054			0.445027	−	0.895517i	$-0.353194\pi$
$907$	807.279			0.890054			0.445027	+	0.895517i	$0.353194\pi$
$908$			574.837i			0.633080i
$909$		−	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	$0.578131\pi$
$912$	−111.765			−0.122549
$913$			1069.49i			1.17140i
$914$	684.230			0.748610
$915$		0			0
$916$		−	278.809i		−	0.304377i
$917$	470.558	+	159.145i	0.513150	+	0.173549i
$918$	−223.456			−0.243416
$919$	−118.455			−0.128896			−0.0644478	−	0.997921i	$-0.520529\pi$
$919$	−118.455			−0.128896			−0.0644478	+	0.997921i	$0.520529\pi$
$920$		0			0
$921$	60.1766			0.0653384
$922$			388.430i			0.421290i
$923$		−	453.337i		−	0.491156i
$924$	−235.279	−	79.5724i	−0.254631	−	0.0861173i
$925$		0			0
$926$	−217.040			−0.234384
$927$		−	236.628i		−	0.255262i
$928$	−169.706			−0.182872
$929$			629.297i			0.677392i	0.940896	+	0.338696i	$0.109986\pi$
$929$			629.297i			0.677392i	−0.940896	+	0.338696i	$0.890014\pi$
$930$		0			0
$931$	479.793	−	628.191i	0.515352	−	0.674749i
$932$	−725.470			−0.778401
$933$	−154.441			−0.165531
$934$			87.3164i			0.0934865i
$935$		0			0
$936$		−	76.0063i		−	0.0812033i
$937$		−	210.631i		−	0.224793i	−0.993663	−	0.112397i	$-0.964147\pi$
$937$		−	210.631i		−	0.224793i	0.993663	−	0.112397i	$-0.0358527\pi$
$938$	85.7258	−	253.474i	0.0913921	−	0.270228i
$939$	−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	$0.969777\pi$
$942$	−140.382			−0.149025
$943$			47.7836i			0.0506719i
$944$			1.96949i			0.00208632i
$945$		0			0
$946$	1078.09			1.13963
$947$	−423.463			−0.447163			−0.223581	−	0.974685i	$-0.571775\pi$
$947$	−423.463			−0.447163			−0.223581	+	0.974685i	$0.571775\pi$
$948$			463.578i			0.489006i
$949$	−632.528			−0.666521
$950$		0			0
$951$		−	938.821i		−	0.987194i
$952$	−570.323	−	192.885i	−0.599079	−	0.202611i
$953$	546.706			0.573669			0.286834	−	0.957980i	$-0.407397\pi$
$953$	546.706			0.573669			0.286834	+	0.957980i	$0.407397\pi$
$954$	301.103			0.315621
$955$		0			0
$956$	36.9563			0.0386572
$957$		−	532.223i		−	0.556137i
$958$			787.030i			0.821534i
$959$	237.192	−	701.328i	0.247332	−	0.731311i
$960$		0			0
$961$	−1556.88			−1.62006
$962$		−	391.580i		−	0.407048i
$963$	−113.272			−0.117624
$964$			356.208i			0.369510i
$965$		0			0
$966$	−109.279	−	36.9587i	−0.113125	−	0.0382595i
$967$	−1262.32			−1.30540			−0.652701	−	0.757616i	$-0.726364\pi$
$967$	−1262.32			−1.30540			−0.652701	+	0.757616i	$0.726364\pi$
$968$	45.5046			0.0470089
$969$		−	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	$0.812102\pi$
$972$			31.1769i			0.0320750i
$973$	−1202.35	−	406.640i	−1.23572	−	0.417924i
$974$	−459.931			−0.472208
$975$		0			0
$976$		−	11.4790i		−	0.0117613i
$977$	−1800.26			−1.84264			−0.921322	−	0.388801i	$-0.872889\pi$
$977$	−1800.26			−1.84264			−0.921322	+	0.388801i	$0.872889\pi$
$978$		−	428.444i		−	0.438082i
$979$			1484.88i			1.51673i
$980$		0			0
$981$	−95.8234			−0.0976793
$982$	−910.014			−0.926695
$983$			996.810i			1.01405i	0.861932	+	0.507025i	$0.169255\pi$
$983$			996.810i			1.01405i	−0.861932	+	0.507025i	$0.830745\pi$
$984$	34.7939			0.0353597
$985$		0			0
$986$		−	1290.12i		−	1.30844i
$987$	226.441	−	669.539i	0.229423	−	0.678358i
$988$	288.999			0.292509
$989$	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	$0.369725\pi$
$992$			283.852i			0.286142i
$993$			311.133i			0.313326i
$994$	−160.514	+	474.607i	−0.161483	+	0.477471i
$995$		0			0
$996$	361.706			0.363158
$997$			1942.70i			1.94854i	0.225379	+	0.974271i	$0.427638\pi$
$997$			1942.70i			1.94854i	−0.225379	+	0.974271i	$0.572362\pi$
$998$	−1068.33			−1.07047
$999$			160.622i			0.160783i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1050.3.f.a.601.2		4
5.2	odd	4		1050.3.h.a.349.4		8
5.3	odd	4		1050.3.h.a.349.5		8
5.4	even	2		42.3.c.a.13.3	✓	4
7.6	odd	2	inner	1050.3.f.a.601.1		4
15.14	odd	2		126.3.c.b.55.2		4
20.19	odd	2		336.3.f.c.97.4		4
35.4	even	6		294.3.g.b.19.1		4
35.9	even	6		294.3.g.c.31.1		4
35.13	even	4		1050.3.h.a.349.8		8
35.19	odd	6		294.3.g.b.31.1		4
35.24	odd	6		294.3.g.c.19.1		4
35.27	even	4		1050.3.h.a.349.1		8
35.34	odd	2		42.3.c.a.13.4	yes	4
40.19	odd	2		1344.3.f.e.769.1		4
40.29	even	2		1344.3.f.f.769.3		4
60.59	even	2		1008.3.f.g.433.3		4
105.44	odd	6		882.3.n.d.325.2		4
105.59	even	6		882.3.n.d.19.2		4
105.74	odd	6		882.3.n.a.19.2		4
105.89	even	6		882.3.n.a.325.2		4
105.104	even	2		126.3.c.b.55.1		4
140.139	even	2		336.3.f.c.97.1		4
280.69	odd	2		1344.3.f.f.769.2		4
280.139	even	2		1344.3.f.e.769.4		4
420.419	odd	2		1008.3.f.g.433.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.3.c.a.13.3	✓	4	5.4	even	2
42.3.c.a.13.4	yes	4	35.34	odd	2
126.3.c.b.55.1		4	105.104	even	2
126.3.c.b.55.2		4	15.14	odd	2
294.3.g.b.19.1		4	35.4	even	6
294.3.g.b.31.1		4	35.19	odd	6
294.3.g.c.19.1		4	35.24	odd	6
294.3.g.c.31.1		4	35.9	even	6
336.3.f.c.97.1		4	140.139	even	2
336.3.f.c.97.4		4	20.19	odd	2
882.3.n.a.19.2		4	105.74	odd	6
882.3.n.a.325.2		4	105.89	even	6
882.3.n.d.19.2		4	105.59	even	6
882.3.n.d.325.2		4	105.44	odd	6
1008.3.f.g.433.2		4	420.419	odd	2
1008.3.f.g.433.3		4	60.59	even	2
1050.3.f.a.601.1		4	7.6	odd	2	inner
1050.3.f.a.601.2		4	1.1	even	1	trivial
1050.3.h.a.349.1		8	35.27	even	4
1050.3.h.a.349.4		8	5.2	odd	4
1050.3.h.a.349.5		8	5.3	odd	4
1050.3.h.a.349.8		8	35.13	even	4
1344.3.f.e.769.1		4	40.19	odd	2
1344.3.f.e.769.4		4	280.139	even	2
1344.3.f.f.769.2		4	280.69	odd	2
1344.3.f.f.769.3		4	40.29	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1050.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +1.73205i q^{3} +2.00000 q^{4} -2.44949i q^{6} +(2.24264 - 6.63103i) q^{7} -2.82843 q^{8} -3.00000 q^{9} -10.2426 q^{11} +3.46410i q^{12} -8.95743i q^{13} +(-3.17157 + 9.37769i) q^{14} +4.00000 q^{16} +30.4085i q^{17} +4.24264 q^{18} +16.1318i q^{19} +(11.4853 + 3.88437i) q^{21} +14.4853 q^{22} +6.72792 q^{23} -4.89898i q^{24} +12.6677i q^{26} -5.19615i q^{27} +(4.48528 - 13.2621i) q^{28} +30.0000 q^{29} -50.1785i q^{31} -5.65685 q^{32} -17.7408i q^{33} -43.0041i q^{34} -6.00000 q^{36} -30.9117 q^{37} -22.8138i q^{38} +15.5147 q^{39} +7.10228i q^{41} +(-16.2426 - 5.49333i) q^{42} +74.4264 q^{43} -20.4853 q^{44} -9.51472 q^{46} -58.2954i q^{47} +6.92820i q^{48} +(-38.9411 - 29.7420i) q^{49} -52.6690 q^{51} -17.9149i q^{52} +70.9706 q^{53} +7.34847i q^{54} +(-6.34315 + 18.7554i) q^{56} -27.9411 q^{57} -42.4264 q^{58} +0.492372i q^{59} -2.86976i q^{61} +70.9631i q^{62} +(-6.72792 + 19.8931i) q^{63} +8.00000 q^{64} +25.0892i q^{66} -27.0294 q^{67} +60.8170i q^{68} +11.6531i q^{69} +50.6102 q^{71} +8.48528 q^{72} -70.6149i q^{73} +43.7157 q^{74} +32.2636i q^{76} +(-22.9706 + 67.9193i) q^{77} -21.9411 q^{78} +133.823 q^{79} +9.00000 q^{81} -10.0441i q^{82} -104.415i q^{83} +(22.9706 + 7.76874i) q^{84} -105.255 q^{86} +51.9615i q^{87} +28.9706 q^{88} -144.970i q^{89} +(-59.3970 - 20.0883i) q^{91} +13.4558 q^{92} +86.9117 q^{93} +82.4421i q^{94} -9.79796i q^{96} -100.705i q^{97} +(55.0711 + 42.0616i) q^{98} +30.7279 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 8 q^{7} - 12 q^{9} - 24 q^{11} - 24 q^{14} + 16 q^{16} + 12 q^{21} + 24 q^{22} - 24 q^{23} - 16 q^{28} + 120 q^{29} - 24 q^{36} + 80 q^{37} + 96 q^{39} - 48 q^{42} + 128 q^{43} - 48 q^{44}+ \cdots + 72 q^{99}+O(q^{100}) \) 4 * q + 8 * q^4 - 8 * q^7 - 12 * q^9 - 24 * q^11 - 24 * q^14 + 16 * q^16 + 12 * q^21 + 24 * q^22 - 24 * q^23 - 16 * q^28 + 120 * q^29 - 24 * q^36 + 80 * q^37 + 96 * q^39 - 48 * q^42 + 128 * q^43 - 48 * q^44 - 72 * q^46 - 20 * q^49 - 24 * q^51 + 216 * q^53 - 48 * q^56 + 24 * q^57 + 24 * q^63 + 32 * q^64 - 176 * q^67 - 120 * q^71 + 288 * q^74 - 24 * q^77 + 48 * q^78 + 128 * q^79 + 36 * q^81 + 24 * q^84 - 240 * q^86 + 48 * q^88 - 48 * q^92 + 144 * q^93 + 192 * q^98 + 72 * q^99

Introduction

L-functions

Modular forms

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