LMFDB - Embedded newform 3600.3.l.q.1601.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,3,Mod(1601,3600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3600.1601");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	3600.l (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:	$98.0928951697$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1601.4
Root			$1.58114 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	3600.1601
Dual form			3600.3.l.q.1601.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.16228 q^{7} +O(q^{10})$	$q+3.16228 q^{7} +3.05792i q^{11} -5.16228 q^{13} +17.8885i q^{17} -10.9737 q^{19} -10.5880i q^{23} -42.6187i q^{29} -33.6228 q^{31} +3.48683 q^{37} -34.3629i q^{41} +80.2719 q^{43} -0.458991i q^{47} -39.0000 q^{49} -12.3062i q^{53} +12.5357i q^{59} -6.27189 q^{61} +71.8947 q^{67} -77.7445i q^{71} +31.5701 q^{73} +9.67000i q^{77} +125.623 q^{79} +74.3826i q^{83} -68.8375i q^{89} -16.3246 q^{91} +39.5701 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 8 q^{13} + 32 q^{19} - 8 q^{31} - 24 q^{37} + 144 q^{43} - 156 q^{49} + 152 q^{61} - 16 q^{67} - 152 q^{73} + 376 q^{79} - 40 q^{91} - 120 q^{97}+O(q^{100}) $ 4 * q - 8 * q^13 + 32 * q^19 - 8 * q^31 - 24 * q^37 + 144 * q^43 - 156 * q^49 + 152 * q^61 - 16 * q^67 - 152 * q^73 + 376 * q^79 - 40 * q^91 - 120 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$2801$	$3151$
$\chi(n)$	$1$	$1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.16228	0.451754	0.225877	−	0.974156i	$-0.427475\pi$
$7$	3.16228	0.451754	0.225877	+	0.974156i	$0.427475\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.05792i	0.277993i	0.990293	+	0.138996i	$0.0443877\pi$
$11$	3.05792i	0.277993i	−0.990293	+	0.138996i	$0.955612\pi$
$12$	0	0
$13$	−5.16228	−0.397098	−0.198549	−	0.980091i	$-0.563623\pi$
$13$	−5.16228	−0.397098	−0.198549	+	0.980091i	$0.563623\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	17.8885i	1.05227i	0.850402	+	0.526134i	$0.176359\pi$
$17$	17.8885i	1.05227i	−0.850402	+	0.526134i	$0.823641\pi$
$18$	0	0
$19$	−10.9737	−0.577561	−0.288781	−	0.957395i	$-0.593250\pi$
$19$	−10.9737	−0.577561	−0.288781	+	0.957395i	$0.593250\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 10.5880i	− 0.460347i	−0.973150	−	0.230173i	$-0.926071\pi$
$23$	− 10.5880i	− 0.460347i	0.973150	−	0.230173i	$-0.0739294\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 42.6187i	− 1.46961i	−0.678279	−	0.734804i	$-0.737274\pi$
$29$	− 42.6187i	− 1.46961i	0.678279	−	0.734804i	$-0.262726\pi$
$30$	0	0
$31$	−33.6228	−1.08461	−0.542303	−	0.840183i	$-0.682447\pi$
$31$	−33.6228	−1.08461	−0.542303	+	0.840183i	$0.682447\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.48683	0.0942387	0.0471194	−	0.998889i	$-0.484996\pi$
$37$	3.48683	0.0942387	0.0471194	+	0.998889i	$0.484996\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 34.3629i	− 0.838119i	−0.907959	−	0.419059i	$-0.862360\pi$
$41$	− 34.3629i	− 0.838119i	0.907959	−	0.419059i	$-0.137640\pi$
$42$	0	0
$43$	80.2719	1.86679	0.933394	−	0.358853i	$-0.116832\pi$
$43$	80.2719	1.86679	0.933394	+	0.358853i	$0.116832\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 0.458991i	− 0.00976576i	−0.999988	−	0.00488288i	$-0.998446\pi$
$47$	− 0.458991i	− 0.00976576i	0.999988	−	0.00488288i	$-0.00155427\pi$
$48$	0	0
$49$	−39.0000	−0.795918
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 12.3062i	− 0.232192i	−0.993238	−	0.116096i	$-0.962962\pi$
$53$	− 12.3062i	− 0.232192i	0.993238	−	0.116096i	$-0.0370380\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	12.5357i	0.212469i	0.994341	+	0.106234i	$0.0338794\pi$
$59$	12.5357i	0.212469i	−0.994341	+	0.106234i	$0.966121\pi$
$60$	0	0
$61$	−6.27189	−0.102818	−0.0514089	−	0.998678i	$-0.516371\pi$
$61$	−6.27189	−0.102818	−0.0514089	+	0.998678i	$0.516371\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	71.8947	1.07305	0.536527	−	0.843883i	$-0.319736\pi$
$67$	71.8947	1.07305	0.536527	+	0.843883i	$0.319736\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 77.7445i	− 1.09499i	−0.836808	−	0.547497i	$-0.815581\pi$
$71$	− 77.7445i	− 1.09499i	0.836808	−	0.547497i	$-0.184419\pi$
$72$	0	0
$73$	31.5701	0.432467	0.216234	−	0.976342i	$-0.430623\pi$
$73$	31.5701	0.432467	0.216234	+	0.976342i	$0.430623\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	9.67000i	0.125584i
$78$	0	0
$79$	125.623	1.59016	0.795081	−	0.606503i	$-0.207428\pi$
$79$	125.623	1.59016	0.795081	+	0.606503i	$0.207428\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	74.3826i	0.896176i	0.893989	+	0.448088i	$0.147895\pi$
$83$	74.3826i	0.896176i	−0.893989	+	0.448088i	$0.852105\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 68.8375i	− 0.773455i	−0.922194	−	0.386727i	$-0.873605\pi$
$89$	− 68.8375i	− 0.773455i	0.922194	−	0.386727i	$-0.126395\pi$
$90$	0	0
$91$	−16.3246	−0.179391
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	39.5701	0.407939	0.203970	−	0.978977i	$-0.434616\pi$
$97$	39.5701	0.407939	0.203970	+	0.978977i	$0.434616\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	21.3682i	0.211566i	0.994389	+	0.105783i	$0.0337350\pi$
$101$	21.3682i	0.211566i	−0.994389	+	0.105783i	$0.966265\pi$
$102$	0	0
$103$	107.540	1.04407	0.522036	−	0.852923i	$-0.325173\pi$
$103$	107.540	1.04407	0.522036	+	0.852923i	$0.325173\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	173.303i	1.61965i	0.586668	+	0.809827i	$0.300439\pi$
$107$	173.303i	1.61965i	−0.586668	+	0.809827i	$0.699561\pi$
$108$	0	0
$109$	−212.868	−1.95292	−0.976460	−	0.215698i	$-0.930797\pi$
$109$	−212.868	−1.95292	−0.976460	+	0.215698i	$0.930797\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 197.915i	− 1.75146i	−0.482798	−	0.875732i	$-0.660379\pi$
$113$	− 197.915i	− 1.75146i	0.482798	−	0.875732i	$-0.339621\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	56.5685i	0.475366i
$120$	0	0
$121$	111.649	0.922720
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−132.627	−1.04431	−0.522154	−	0.852851i	$-0.674871\pi$
$127$	−132.627	−1.04431	−0.522154	+	0.852851i	$0.674871\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 213.664i	− 1.63102i	−0.578741	−	0.815512i	$-0.696456\pi$
$131$	− 213.664i	− 1.63102i	0.578741	−	0.815512i	$-0.303544\pi$
$132$	0	0
$133$	−34.7018	−0.260916
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 59.7815i	− 0.436361i	−0.975908	−	0.218181i	$-0.929988\pi$
$137$	− 59.7815i	− 0.436361i	0.975908	−	0.218181i	$-0.0700122\pi$
$138$	0	0
$139$	36.1580	0.260130	0.130065	−	0.991506i	$-0.458481\pi$
$139$	36.1580	0.260130	0.130065	+	0.991506i	$0.458481\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 15.7858i	− 0.110391i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 47.9655i	− 0.321916i	−0.986961	−	0.160958i	$-0.948542\pi$
$149$	− 47.9655i	− 0.321916i	0.986961	−	0.160958i	$-0.0514584\pi$
$150$	0	0
$151$	166.438	1.10224	0.551121	−	0.834426i	$-0.314200\pi$
$151$	166.438	1.10224	0.551121	+	0.834426i	$0.314200\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−138.355	−0.881243	−0.440622	−	0.897693i	$-0.645242\pi$
$157$	−138.355	−0.881243	−0.440622	+	0.897693i	$0.645242\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 33.4821i	− 0.207964i
$162$	0	0
$163$	196.377	1.20477	0.602384	−	0.798206i	$-0.294218\pi$
$163$	196.377	1.20477	0.602384	+	0.798206i	$0.294218\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 132.520i	− 0.793535i	−0.917919	−	0.396768i	$-0.870132\pi$
$167$	− 132.520i	− 0.793535i	0.917919	−	0.396768i	$-0.129868\pi$
$168$	0	0
$169$	−142.351	−0.842313
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 285.138i	− 1.64819i	−0.566448	−	0.824097i	$-0.691683\pi$
$173$	− 285.138i	− 1.64819i	0.566448	−	0.824097i	$-0.308317\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	111.072i	0.620512i	0.950653	+	0.310256i	$0.100415\pi$
$179$	111.072i	0.620512i	−0.950653	+	0.310256i	$0.899585\pi$
$180$	0	0
$181$	152.974	0.845158	0.422579	−	0.906326i	$-0.361125\pi$
$181$	152.974	0.845158	0.422579	+	0.906326i	$0.361125\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−54.7018	−0.292523
$188$	0	0
$189$	0	0
$190$	0	0
$191$	98.6970i	0.516738i	0.966046	+	0.258369i	$0.0831851\pi$
$191$	98.6970i	0.516738i	−0.966046	+	0.258369i	$0.916815\pi$
$192$	0	0
$193$	−268.280	−1.39005	−0.695027	−	0.718984i	$-0.744608\pi$
$193$	−268.280	−1.39005	−0.695027	+	0.718984i	$0.744608\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	167.454i	0.850020i	0.905189	+	0.425010i	$0.139729\pi$
$197$	167.454i	0.850020i	−0.905189	+	0.425010i	$0.860271\pi$
$198$	0	0
$199$	384.386	1.93159	0.965793	−	0.259313i	$-0.0834961\pi$
$199$	384.386	1.93159	0.965793	+	0.259313i	$0.0834961\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 134.772i	− 0.663902i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 33.5566i	− 0.160558i
$210$	0	0
$211$	−202.982	−0.962001	−0.481001	−	0.876720i	$-0.659726\pi$
$211$	−202.982	−0.962001	−0.481001	+	0.876720i	$0.659726\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−106.325	−0.489975
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 92.3456i	− 0.417854i
$222$	0	0
$223$	27.1011	0.121529	0.0607647	−	0.998152i	$-0.480646\pi$
$223$	27.1011	0.121529	0.0607647	+	0.998152i	$0.480646\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	93.5615i	0.412165i	0.978535	+	0.206083i	$0.0660716\pi$
$227$	93.5615i	0.412165i	−0.978535	+	0.206083i	$0.933928\pi$
$228$	0	0
$229$	272.175	1.18854	0.594269	−	0.804267i	$-0.297442\pi$
$229$	272.175	1.18854	0.594269	+	0.804267i	$0.297442\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 70.0161i	− 0.300498i	−0.988648	−	0.150249i	$-0.951992\pi$
$233$	− 70.0161i	− 0.300498i	0.988648	−	0.150249i	$-0.0480076\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	340.341i	1.42402i	0.702168	+	0.712011i	$0.252215\pi$
$239$	340.341i	1.42402i	−0.702168	+	0.712011i	$0.747785\pi$
$240$	0	0
$241$	353.737	1.46779	0.733893	−	0.679265i	$-0.237701\pi$
$241$	353.737	1.46779	0.733893	+	0.679265i	$0.237701\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	56.6491	0.229349
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 263.199i	− 1.04860i	−0.851533	−	0.524300i	$-0.824327\pi$
$251$	− 263.199i	− 1.04860i	0.851533	−	0.524300i	$-0.175673\pi$
$252$	0	0
$253$	32.3772	0.127973
$254$	0	0
$255$	0	0
$256$	0	0
$257$	46.7808i	0.182026i	0.995850	+	0.0910132i	$0.0290105\pi$
$257$	46.7808i	0.182026i	−0.995850	+	0.0910132i	$0.970989\pi$
$258$	0	0
$259$	11.0263	0.0425727
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 418.886i	− 1.59272i	−0.604821	−	0.796361i	$-0.706755\pi$
$263$	− 418.886i	− 1.59272i	0.604821	−	0.796361i	$-0.293245\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 283.729i	− 1.05476i	−0.849631	−	0.527378i	$-0.823175\pi$
$269$	− 283.729i	− 1.05476i	0.849631	−	0.527378i	$-0.176825\pi$
$270$	0	0
$271$	400.930	1.47944	0.739722	−	0.672912i	$-0.234957\pi$
$271$	400.930	1.47944	0.739722	+	0.672912i	$0.234957\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−213.978	−0.772484	−0.386242	−	0.922398i	$-0.626227\pi$
$277$	−213.978	−0.772484	−0.386242	+	0.922398i	$0.626227\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 328.519i	− 1.16911i	−0.811355	−	0.584554i	$-0.801270\pi$
$281$	− 328.519i	− 1.16911i	0.811355	−	0.584554i	$-0.198730\pi$
$282$	0	0
$283$	532.228	1.88066	0.940332	−	0.340259i	$-0.110515\pi$
$283$	532.228	1.88066	0.940332	+	0.340259i	$0.110515\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 108.665i	− 0.378624i
$288$	0	0
$289$	−31.0000	−0.107266
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 284.796i	− 0.972002i	−0.873958	−	0.486001i	$-0.838455\pi$
$293$	− 284.796i	− 0.972002i	0.873958	−	0.486001i	$-0.161545\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	54.6581i	0.182803i
$300$	0	0
$301$	253.842	0.843329
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−162.053	−0.527859	−0.263929	−	0.964542i	$-0.585019\pi$
$307$	−162.053	−0.527859	−0.263929	+	0.964542i	$0.585019\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 64.4338i	− 0.207183i	−0.994620	−	0.103591i	$-0.966967\pi$
$311$	− 64.4338i	− 0.207183i	0.994620	−	0.103591i	$-0.0330334\pi$
$312$	0	0
$313$	363.088	1.16002	0.580012	−	0.814608i	$-0.303048\pi$
$313$	363.088	1.16002	0.580012	+	0.814608i	$0.303048\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 111.419i	− 0.351479i	−0.984437	−	0.175740i	$-0.943768\pi$
$317$	− 111.419i	− 0.351479i	0.984437	−	0.175740i	$-0.0562317\pi$
$318$	0	0
$319$	130.325	0.408541
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 196.303i	− 0.607749i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 1.45146i	− 0.00441172i
$330$	0	0
$331$	79.7281	0.240870	0.120435	−	0.992721i	$-0.461571\pi$
$331$	79.7281	0.240870	0.120435	+	0.992721i	$0.461571\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−234.605	−0.696157	−0.348079	−	0.937465i	$-0.613166\pi$
$337$	−234.605	−0.696157	−0.348079	+	0.937465i	$0.613166\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 102.816i	− 0.301513i
$342$	0	0
$343$	−278.280	−0.811313
$344$	0	0
$345$	0	0
$346$	0	0
$347$	265.190i	0.764235i	0.924114	+	0.382118i	$0.124805\pi$
$347$	265.190i	0.764235i	−0.924114	+	0.382118i	$0.875195\pi$
$348$	0	0
$349$	−199.254	−0.570929	−0.285464	−	0.958389i	$-0.592148\pi$
$349$	−199.254	−0.570929	−0.285464	+	0.958389i	$0.592148\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 508.285i	− 1.43990i	−0.694025	−	0.719951i	$-0.744164\pi$
$353$	− 508.285i	− 1.43990i	0.694025	−	0.719951i	$-0.255836\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 561.492i	− 1.56404i	−0.623250	−	0.782022i	$-0.714188\pi$
$359$	− 561.492i	− 1.56404i	0.623250	−	0.782022i	$-0.285812\pi$
$360$	0	0
$361$	−240.579	−0.666423
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−78.5744	−0.214099	−0.107050	−	0.994254i	$-0.534140\pi$
$367$	−78.5744	−0.214099	−0.107050	+	0.994254i	$0.534140\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 38.9155i	− 0.104894i
$372$	0	0
$373$	168.241	0.451049	0.225525	−	0.974237i	$-0.427590\pi$
$373$	168.241	0.451049	0.225525	+	0.974237i	$0.427590\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	220.009i	0.583579i
$378$	0	0
$379$	−334.000	−0.881266	−0.440633	−	0.897687i	$-0.645246\pi$
$379$	−334.000	−0.881266	−0.440633	+	0.897687i	$0.645246\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 155.179i	− 0.405167i	−0.979265	−	0.202584i	$-0.935066\pi$
$383$	− 155.179i	− 0.405167i	0.979265	−	0.202584i	$-0.0649338\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	414.755i	1.06621i	0.846050	+	0.533104i	$0.178975\pi$
$389$	414.755i	1.06621i	−0.846050	+	0.533104i	$0.821025\pi$
$390$	0	0
$391$	189.404	0.484408
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	418.478	1.05410	0.527050	−	0.849834i	$-0.323298\pi$
$397$	418.478	1.05410	0.527050	+	0.849834i	$0.323298\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	196.203i	0.489285i	0.969613	+	0.244642i	$0.0786706\pi$
$401$	196.203i	0.489285i	−0.969613	+	0.244642i	$0.921329\pi$
$402$	0	0
$403$	173.570	0.430695
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.6625i	0.0261977i
$408$	0	0
$409$	244.158	0.596963	0.298482	−	0.954415i	$-0.403520\pi$
$409$	244.158	0.596963	0.298482	+	0.954415i	$0.403520\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	39.6413i	0.0959837i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 286.297i	− 0.683287i	−0.939830	−	0.341643i	$-0.889016\pi$
$419$	− 286.297i	− 0.683287i	0.939830	−	0.341643i	$-0.110984\pi$
$420$	0	0
$421$	−155.842	−0.370171	−0.185086	−	0.982722i	$-0.559256\pi$
$421$	−155.842	−0.370171	−0.185086	+	0.982722i	$0.559256\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−19.8334	−0.0464484
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 684.342i	− 1.58780i	−0.608047	−	0.793901i	$-0.708047\pi$
$431$	− 684.342i	− 1.58780i	0.608047	−	0.793901i	$-0.291953\pi$
$432$	0	0
$433$	637.956	1.47334	0.736670	−	0.676253i	$-0.236397\pi$
$433$	637.956	1.47334	0.736670	+	0.676253i	$0.236397\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	116.189i	0.265879i
$438$	0	0
$439$	−78.1053	−0.177916	−0.0889582	−	0.996035i	$-0.528354\pi$
$439$	−78.1053	−0.177916	−0.0889582	+	0.996035i	$0.528354\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 304.317i	− 0.686945i	−0.939163	−	0.343472i	$-0.888397\pi$
$443$	− 304.317i	− 0.686945i	0.939163	−	0.343472i	$-0.111603\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 37.4268i	− 0.0833560i	−0.999131	−	0.0416780i	$-0.986730\pi$
$449$	− 37.4268i	− 0.0833560i	0.999131	−	0.0416780i	$-0.0132704\pi$
$450$	0	0
$451$	105.079	0.232991
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	504.236	1.10336	0.551681	−	0.834055i	$-0.313987\pi$
$457$	504.236	1.10336	0.551681	+	0.834055i	$0.313987\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 515.115i	− 1.11739i	−0.829375	−	0.558693i	$-0.811303\pi$
$461$	− 515.115i	− 1.11739i	0.829375	−	0.558693i	$-0.188697\pi$
$462$	0	0
$463$	−101.531	−0.219289	−0.109645	−	0.993971i	$-0.534971\pi$
$463$	−101.531	−0.219289	−0.109645	+	0.993971i	$0.534971\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	786.166i	1.68344i	0.539915	+	0.841719i	$0.318456\pi$
$467$	786.166i	1.68344i	−0.539915	+	0.841719i	$0.681544\pi$
$468$	0	0
$469$	227.351	0.484757
$470$	0	0
$471$	0	0
$472$	0	0
$473$	245.465i	0.518954i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 622.787i	− 1.30018i	−0.759856	−	0.650091i	$-0.774731\pi$
$479$	− 622.787i	− 1.30018i	0.759856	−	0.650091i	$-0.225269\pi$
$480$	0	0
$481$	−18.0000	−0.0374220
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−79.8114	−0.163884	−0.0819419	−	0.996637i	$-0.526112\pi$
$487$	−79.8114	−0.163884	−0.0819419	+	0.996637i	$0.526112\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 152.580i	− 0.310754i	−0.987855	−	0.155377i	$-0.950341\pi$
$491$	− 152.580i	− 0.310754i	0.987855	−	0.155377i	$-0.0496592\pi$
$492$	0	0
$493$	762.386	1.54642
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 245.850i	− 0.494667i
$498$	0	0
$499$	−712.605	−1.42807	−0.714033	−	0.700112i	$-0.753133\pi$
$499$	−712.605	−1.42807	−0.714033	+	0.700112i	$0.753133\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	311.908i	0.620096i	0.950721	+	0.310048i	$0.100345\pi$
$503$	311.908i	0.620096i	−0.950721	+	0.310048i	$0.899655\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	111.493i	0.219044i	0.993984	+	0.109522i	$0.0349320\pi$
$509$	111.493i	0.219044i	−0.993984	+	0.109522i	$0.965068\pi$
$510$	0	0
$511$	99.8334	0.195369
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1.40356	0.00271481
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 621.323i	− 1.19256i	−0.802777	−	0.596279i	$-0.796645\pi$
$521$	− 621.323i	− 1.19256i	0.802777	−	0.596279i	$-0.203355\pi$
$522$	0	0
$523$	−6.42989	−0.0122942	−0.00614712	−	0.999981i	$-0.501957\pi$
$523$	−6.42989	−0.0122942	−0.00614712	+	0.999981i	$0.501957\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 601.463i	− 1.14130i
$528$	0	0
$529$	416.895	0.788081
$530$	0	0
$531$	0	0
$532$	0	0
$533$	177.391i	0.332816i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 119.259i	− 0.221260i
$540$	0	0
$541$	−405.096	−0.748791	−0.374396	−	0.927269i	$-0.622150\pi$
$541$	−405.096	−0.748791	−0.374396	+	0.927269i	$0.622150\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−92.9039	−0.169843	−0.0849213	−	0.996388i	$-0.527064\pi$
$547$	−92.9039	−0.169843	−0.0849213	+	0.996388i	$0.527064\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	467.683i	0.848789i
$552$	0	0
$553$	397.254	0.718362
$554$	0	0
$555$	0	0
$556$	0	0
$557$	803.372i	1.44232i	0.692769	+	0.721160i	$0.256391\pi$
$557$	803.372i	1.44232i	−0.692769	+	0.721160i	$0.743609\pi$
$558$	0	0
$559$	−414.386	−0.741298
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 638.542i	− 1.13418i	−0.823657	−	0.567089i	$-0.808070\pi$
$563$	− 638.542i	− 1.13418i	0.823657	−	0.567089i	$-0.191930\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	405.110i	0.711969i	0.934492	+	0.355984i	$0.115854\pi$
$569$	405.110i	0.711969i	−0.934492	+	0.355984i	$0.884146\pi$
$570$	0	0
$571$	45.1317	0.0790397	0.0395199	−	0.999219i	$-0.487417\pi$
$571$	45.1317	0.0790397	0.0395199	+	0.999219i	$0.487417\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−662.938	−1.14894	−0.574470	−	0.818526i	$-0.694792\pi$
$577$	−662.938	−1.14894	−0.574470	+	0.818526i	$0.694792\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	235.218i	0.404851i
$582$	0	0
$583$	37.6313	0.0645477
$584$	0	0
$585$	0	0
$586$	0	0
$587$	416.226i	0.709073i	0.935042	+	0.354536i	$0.115361\pi$
$587$	416.226i	0.709073i	−0.935042	+	0.354536i	$0.884639\pi$
$588$	0	0
$589$	368.965	0.626426
$590$	0	0
$591$	0	0
$592$	0	0
$593$	534.659i	0.901618i	0.892621	+	0.450809i	$0.148864\pi$
$593$	534.659i	0.901618i	−0.892621	+	0.450809i	$0.851136\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 74.5940i	− 0.124531i	−0.998060	−	0.0622654i	$-0.980167\pi$
$599$	− 74.5940i	− 0.124531i	0.998060	−	0.0622654i	$-0.0198325\pi$
$600$	0	0
$601$	−302.596	−0.503488	−0.251744	−	0.967794i	$-0.581004\pi$
$601$	−302.596	−0.503488	−0.251744	+	0.967794i	$0.581004\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	785.276	1.29370	0.646850	−	0.762617i	$-0.276086\pi$
$607$	785.276	1.29370	0.646850	+	0.762617i	$0.276086\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.36944i	0.00387796i
$612$	0	0
$613$	−67.5039	−0.110121	−0.0550603	−	0.998483i	$-0.517535\pi$
$613$	−67.5039	−0.110121	−0.0550603	+	0.998483i	$0.517535\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	678.462i	1.09961i	0.835292	+	0.549807i	$0.185299\pi$
$617$	678.462i	1.09961i	−0.835292	+	0.549807i	$0.814701\pi$
$618$	0	0
$619$	−847.842	−1.36970	−0.684848	−	0.728686i	$-0.740131\pi$
$619$	−847.842	−1.36970	−0.684848	+	0.728686i	$0.740131\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 217.683i	− 0.349411i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	62.3744i	0.0991643i
$630$	0	0
$631$	−521.973	−0.827215	−0.413608	−	0.910455i	$-0.635732\pi$
$631$	−521.973	−0.827215	−0.413608	+	0.910455i	$0.635732\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	201.329	0.316058
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 450.055i	− 0.702114i	−0.936354	−	0.351057i	$-0.885822\pi$
$641$	− 450.055i	− 0.702114i	0.936354	−	0.351057i	$-0.114178\pi$
$642$	0	0
$643$	−868.394	−1.35054	−0.675268	−	0.737573i	$-0.735972\pi$
$643$	−868.394	−1.35054	−0.675268	+	0.737573i	$0.735972\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 773.761i	− 1.19592i	−0.801525	−	0.597961i	$-0.795978\pi$
$647$	− 773.761i	− 1.19592i	0.801525	−	0.597961i	$-0.204022\pi$
$648$	0	0
$649$	−38.3331	−0.0590649
$650$	0	0
$651$	0	0
$652$	0	0
$653$	417.894i	0.639960i	0.947424	+	0.319980i	$0.103676\pi$
$653$	417.894i	0.639960i	−0.947424	+	0.319980i	$0.896324\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1002.83i	1.52175i	0.648900	+	0.760874i	$0.275229\pi$
$659$	1002.83i	1.52175i	−0.648900	+	0.760874i	$0.724771\pi$
$660$	0	0
$661$	325.535	0.492488	0.246244	−	0.969208i	$-0.420804\pi$
$661$	325.535	0.492488	0.246244	+	0.969208i	$0.420804\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−451.246	−0.676530
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 19.1789i	− 0.0285826i
$672$	0	0
$673$	−540.464	−0.803067	−0.401533	−	0.915844i	$-0.631523\pi$
$673$	−540.464	−0.803067	−0.401533	+	0.915844i	$0.631523\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1253.36i	− 1.85134i	−0.378329	−	0.925671i	$-0.623501\pi$
$677$	− 1253.36i	− 1.85134i	0.378329	−	0.925671i	$-0.376499\pi$
$678$	0	0
$679$	125.132	0.184288
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 456.381i	− 0.668201i	−0.942537	−	0.334101i	$-0.891567\pi$
$683$	− 456.381i	− 0.668201i	0.942537	−	0.334101i	$-0.108433\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	63.5279i	0.0922030i
$690$	0	0
$691$	685.482	0.992014	0.496007	−	0.868318i	$-0.334799\pi$
$691$	685.482	0.992014	0.496007	+	0.868318i	$0.334799\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	614.702	0.881925
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 853.037i	− 1.21689i	−0.793598	−	0.608443i	$-0.791795\pi$
$701$	− 853.037i	− 1.21689i	0.793598	−	0.608443i	$-0.208205\pi$
$702$	0	0
$703$	−38.2633	−0.0544286
$704$	0	0
$705$	0	0
$706$	0	0
$707$	67.5722i	0.0955760i
$708$	0	0
$709$	440.280	0.620988	0.310494	−	0.950575i	$-0.399506\pi$
$709$	440.280	0.620988	0.310494	+	0.950575i	$0.399506\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	355.997i	0.499295i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 832.872i	− 1.15838i	−0.815194	−	0.579188i	$-0.803370\pi$
$719$	− 832.872i	− 1.15838i	0.815194	−	0.579188i	$-0.196630\pi$
$720$	0	0
$721$	340.070	0.471664
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	596.250	0.820151	0.410076	−	0.912052i	$-0.365502\pi$
$727$	596.250	0.820151	0.410076	+	0.912052i	$0.365502\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1435.95i	1.96436i
$732$	0	0
$733$	250.206	0.341345	0.170672	−	0.985328i	$-0.445406\pi$
$733$	250.206	0.341345	0.170672	+	0.985328i	$0.445406\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	219.848i	0.298302i
$738$	0	0
$739$	−522.535	−0.707084	−0.353542	−	0.935419i	$-0.615023\pi$
$739$	−522.535	−0.707084	−0.353542	+	0.935419i	$0.615023\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 311.908i	− 0.419796i	−0.977723	−	0.209898i	$-0.932687\pi$
$743$	− 311.908i	− 0.419796i	0.977723	−	0.209898i	$-0.0673131\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	548.032i	0.731685i
$750$	0	0
$751$	172.552	0.229763	0.114882	−	0.993379i	$-0.463351\pi$
$751$	172.552	0.229763	0.114882	+	0.993379i	$0.463351\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−845.890	−1.11742	−0.558712	−	0.829362i	$-0.688704\pi$
$757$	−845.890	−1.11742	−0.558712	+	0.829362i	$0.688704\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1336.07i	1.75568i	0.478955	+	0.877839i	$0.341016\pi$
$761$	1336.07i	1.75568i	−0.478955	+	0.877839i	$0.658984\pi$
$762$	0	0
$763$	−673.149	−0.882240
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 64.7126i	− 0.0843711i
$768$	0	0
$769$	674.175	0.876691	0.438345	−	0.898807i	$-0.355565\pi$
$769$	674.175	0.876691	0.438345	+	0.898807i	$0.355565\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	130.437i	0.168741i	0.996434	+	0.0843705i	$0.0268879\pi$
$773$	130.437i	0.168741i	−0.996434	+	0.0843705i	$0.973112\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	377.087i	0.484065i
$780$	0	0
$781$	237.737	0.304400
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−1266.25	−1.60896	−0.804481	−	0.593978i	$-0.797557\pi$
$787$	−1266.25	−1.60896	−0.804481	+	0.593978i	$0.797557\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 625.863i	− 0.791231i
$792$	0	0
$793$	32.3772	0.0408288
$794$	0	0
$795$	0	0
$796$	0	0
$797$	122.752i	0.154017i	0.997030	+	0.0770086i	$0.0245369\pi$
$797$	122.752i	0.154017i	−0.997030	+	0.0770086i	$0.975463\pi$
$798$	0	0
$799$	8.21067	0.0102762
$800$	0	0
$801$	0	0
$802$	0	0
$803$	96.5389i	0.120223i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 409.712i	− 0.506443i	−0.967408	−	0.253221i	$-0.918510\pi$
$809$	− 409.712i	− 0.506443i	0.967408	−	0.253221i	$-0.0814901\pi$
$810$	0	0
$811$	271.859	0.335215	0.167607	−	0.985854i	$-0.446396\pi$
$811$	271.859	0.335215	0.167607	+	0.985854i	$0.446396\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−880.877	−1.07818
$818$	0	0
$819$	0	0
$820$	0	0
$821$	518.203i	0.631185i	0.948895	+	0.315593i	$0.102203\pi$
$821$	518.203i	0.631185i	−0.948895	+	0.315593i	$0.897797\pi$
$822$	0	0
$823$	−723.978	−0.879682	−0.439841	−	0.898076i	$-0.644965\pi$
$823$	−723.978	−0.879682	−0.439841	+	0.898076i	$0.644965\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 174.853i	− 0.211431i	−0.994396	−	0.105715i	$-0.966287\pi$
$827$	− 174.853i	− 0.211431i	0.994396	−	0.105715i	$-0.0337132\pi$
$828$	0	0
$829$	63.9032	0.0770847	0.0385423	−	0.999257i	$-0.487729\pi$
$829$	63.9032	0.0770847	0.0385423	+	0.999257i	$0.487729\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 697.653i	− 0.837519i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 1650.68i	− 1.96744i	−0.179701	−	0.983721i	$-0.557513\pi$
$839$	− 1650.68i	− 1.96744i	0.179701	−	0.983721i	$-0.442487\pi$
$840$	0	0
$841$	−975.350	−1.15975
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	353.065	0.416842
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 36.9185i	− 0.0433825i
$852$	0	0
$853$	−906.083	−1.06223	−0.531116	−	0.847299i	$-0.678227\pi$
$853$	−906.083	−1.06223	−0.531116	+	0.847299i	$0.678227\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 953.639i	− 1.11276i	−0.830927	−	0.556382i	$-0.812189\pi$
$857$	− 953.639i	− 1.11276i	0.830927	−	0.556382i	$-0.187811\pi$
$858$	0	0
$859$	1670.38	1.94457	0.972284	−	0.233802i	$-0.0751167\pi$
$859$	1670.38	1.94457	0.972284	+	0.233802i	$0.0751167\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1720.99i	1.99420i	0.0761280	+	0.997098i	$0.475744\pi$
$863$	1720.99i	1.99420i	−0.0761280	+	0.997098i	$0.524256\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	384.145i	0.442054i
$870$	0	0
$871$	−371.140	−0.426108
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1349.20	1.53842	0.769211	−	0.638995i	$-0.220649\pi$
$877$	1349.20	1.53842	0.769211	+	0.638995i	$0.220649\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1109.50i	1.25936i	0.776853	+	0.629682i	$0.216815\pi$
$881$	1109.50i	1.25936i	−0.776853	+	0.629682i	$0.783185\pi$
$882$	0	0
$883$	271.939	0.307971	0.153986	−	0.988073i	$-0.450789\pi$
$883$	271.939	0.307971	0.153986	+	0.988073i	$0.450789\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1220.92i	− 1.37646i	−0.725493	−	0.688230i	$-0.758388\pi$
$887$	− 1220.92i	− 1.37646i	0.725493	−	0.688230i	$-0.241612\pi$
$888$	0	0
$889$	−419.404	−0.471770
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.03681i	0.00564032i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1432.96i	1.59395i
$900$	0	0
$901$	220.140	0.244328
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−1486.36	−1.63876	−0.819382	−	0.573248i	$-0.805683\pi$
$907$	−1486.36	−1.63876	−0.819382	+	0.573248i	$0.805683\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1083.51i	1.18936i	0.803961	+	0.594682i	$0.202722\pi$
$911$	1083.51i	1.18936i	−0.803961	+	0.594682i	$0.797278\pi$
$912$	0	0
$913$	−227.456	−0.249131
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 675.665i	− 0.736821i
$918$	0	0
$919$	328.535	0.357492	0.178746	−	0.983895i	$-0.442796\pi$
$919$	328.535	0.357492	0.178746	+	0.983895i	$0.442796\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	401.339i	0.434820i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1306.82i	1.40669i	0.710847	+	0.703347i	$0.248312\pi$
$929$	1306.82i	1.40669i	−0.710847	+	0.703347i	$0.751688\pi$
$930$	0	0
$931$	427.973	0.459692
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	725.701	0.774494	0.387247	−	0.921976i	$-0.373426\pi$
$937$	725.701	0.774494	0.387247	+	0.921976i	$0.373426\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	883.121i	0.938492i	0.883068	+	0.469246i	$0.155474\pi$
$941$	883.121i	0.938492i	−0.883068	+	0.469246i	$0.844526\pi$
$942$	0	0
$943$	−363.833	−0.385826
$944$	0	0
$945$	0	0
$946$	0	0
$947$	954.557i	1.00798i	0.863710	+	0.503990i	$0.168135\pi$
$947$	954.557i	1.00798i	−0.863710	+	0.503990i	$0.831865\pi$
$948$	0	0
$949$	−162.974	−0.171732
$950$	0	0
$951$	0	0
$952$	0	0
$953$	560.959i	0.588624i	0.955709	+	0.294312i	$0.0950905\pi$
$953$	560.959i	0.588624i	−0.955709	+	0.294312i	$0.904910\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 189.046i	− 0.197128i
$960$	0	0
$961$	169.491	0.176370
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1602.35	1.65703	0.828514	−	0.559968i	$-0.189187\pi$
$967$	1602.35	1.65703	0.828514	+	0.559968i	$0.189187\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	325.424i	0.335143i	0.985860	+	0.167572i	$0.0535926\pi$
$971$	325.424i	0.335143i	−0.985860	+	0.167572i	$0.946407\pi$
$972$	0	0
$973$	114.342	0.117515
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 223.160i	− 0.228413i	−0.993457	−	0.114207i	$-0.963567\pi$
$977$	− 223.160i	− 0.228413i	0.993457	−	0.114207i	$-0.0364326\pi$
$978$	0	0
$979$	210.500	0.215015
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1065.46i	1.08389i	0.840415	+	0.541944i	$0.182311\pi$
$983$	1065.46i	1.08389i	−0.840415	+	0.541944i	$0.817689\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 849.917i	− 0.859370i
$990$	0	0
$991$	−821.579	−0.829040	−0.414520	−	0.910040i	$-0.636051\pi$
$991$	−821.579	−0.829040	−0.414520	+	0.910040i	$0.636051\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−544.986	−0.546626	−0.273313	−	0.961925i	$-0.588119\pi$
$997$	−544.986	−0.546626	−0.273313	+	0.961925i	$0.588119\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.3.l.q.1601.4		4
3.2	odd	2	inner	3600.3.l.q.1601.3		4
4.3	odd	2		1800.3.l.d.1601.1		4
5.2	odd	4		3600.3.c.h.449.7		8
5.3	odd	4		3600.3.c.h.449.3		8
5.4	even	2		720.3.l.b.161.1		4
12.11	even	2		1800.3.l.d.1601.2		4
15.2	even	4		3600.3.c.h.449.6		8
15.8	even	4		3600.3.c.h.449.2		8
15.14	odd	2		720.3.l.b.161.3		4
20.3	even	4		1800.3.c.d.449.6		8
20.7	even	4		1800.3.c.d.449.2		8
20.19	odd	2		360.3.l.b.161.2	✓	4
40.19	odd	2		2880.3.l.e.1601.4		4
40.29	even	2		2880.3.l.d.1601.3		4
60.23	odd	4		1800.3.c.d.449.7		8
60.47	odd	4		1800.3.c.d.449.3		8
60.59	even	2		360.3.l.b.161.4	yes	4
120.29	odd	2		2880.3.l.d.1601.1		4
120.59	even	2		2880.3.l.e.1601.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.3.l.b.161.2	✓	4	20.19	odd	2
360.3.l.b.161.4	yes	4	60.59	even	2
720.3.l.b.161.1		4	5.4	even	2
720.3.l.b.161.3		4	15.14	odd	2
1800.3.c.d.449.2		8	20.7	even	4
1800.3.c.d.449.3		8	60.47	odd	4
1800.3.c.d.449.6		8	20.3	even	4
1800.3.c.d.449.7		8	60.23	odd	4
1800.3.l.d.1601.1		4	4.3	odd	2
1800.3.l.d.1601.2		4	12.11	even	2
2880.3.l.d.1601.1		4	120.29	odd	2
2880.3.l.d.1601.3		4	40.29	even	2
2880.3.l.e.1601.2		4	120.59	even	2
2880.3.l.e.1601.4		4	40.19	odd	2
3600.3.c.h.449.2		8	15.8	even	4
3600.3.c.h.449.3		8	5.3	odd	4
3600.3.c.h.449.6		8	15.2	even	4
3600.3.c.h.449.7		8	5.2	odd	4
3600.3.l.q.1601.3		4	3.2	odd	2	inner
3600.3.l.q.1601.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	3600.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+3.16228 q^{7} +O(q^{10})\)	\(q+3.16228 q^{7} +3.05792i q^{11} -5.16228 q^{13} +17.8885i q^{17} -10.9737 q^{19} -10.5880i q^{23} -42.6187i q^{29} -33.6228 q^{31} +3.48683 q^{37} -34.3629i q^{41} +80.2719 q^{43} -0.458991i q^{47} -39.0000 q^{49} -12.3062i q^{53} +12.5357i q^{59} -6.27189 q^{61} +71.8947 q^{67} -77.7445i q^{71} +31.5701 q^{73} +9.67000i q^{77} +125.623 q^{79} +74.3826i q^{83} -68.8375i q^{89} -16.3246 q^{91} +39.5701 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 8 q^{13} + 32 q^{19} - 8 q^{31} - 24 q^{37} + 144 q^{43} - 156 q^{49} + 152 q^{61} - 16 q^{67} - 152 q^{73} + 376 q^{79} - 40 q^{91} - 120 q^{97}+O(q^{100}) \) 4 * q - 8 * q^13 + 32 * q^19 - 8 * q^31 - 24 * q^37 + 144 * q^43 - 156 * q^49 + 152 * q^61 - 16 * q^67 - 152 * q^73 + 376 * q^79 - 40 * q^91 - 120 * q^97

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