LMFDB - Embedded newform 3600.2.o.c.3599.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3600,2,Mod(3599,3600)]

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([1, 0, 1, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3600.3599");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3600.o (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:	$28.7461447277$
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:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 720)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3599.2
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	3600.3599
Dual form			3600.2.o.c.3599.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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
$7$	−2.44949	−0.925820	−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.01461	−0.305917	−0.152958	−	0.988233i	$-0.548880\pi$
$11$	−1.01461	−0.305917	−0.152958	+	0.988233i	$0.548880\pi$
$12$	0	0
$13$	2.24264i	0.621997i	0.950410	+	0.310998i	$0.100663\pi$
$13$	2.24264i	0.621997i	−0.950410	+	0.310998i	$0.899337\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	4.89898i	1.12390i	0.827170	+	0.561951i	$0.189949\pi$
$19$	4.89898i	1.12390i	−0.827170	+	0.561951i	$0.810051\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 8.36308i	− 1.74382i	−0.489664	−	0.871911i	$-0.662880\pi$
$23$	− 8.36308i	− 1.74382i	0.489664	−	0.871911i	$-0.337120\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.00000i	1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$	6.00000i	1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	0	0
$31$	− 8.36308i	− 1.50205i	−0.660272	−	0.751027i	$-0.729559\pi$
$31$	− 8.36308i	− 1.50205i	0.660272	−	0.751027i	$-0.270441\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 6.24264i	− 1.02628i	−0.858304	−	0.513142i	$-0.828481\pi$
$37$	− 6.24264i	− 1.02628i	0.858304	−	0.513142i	$-0.171519\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	4.24264i	0.662589i	0.943527	+	0.331295i	$0.107485\pi$
$41$	4.24264i	0.662589i	−0.943527	+	0.331295i	$0.892515\pi$
$42$	0	0
$43$	−2.02922	−0.309454	−0.154727	−	0.987957i	$-0.549450\pi$
$43$	−2.02922	−0.309454	−0.154727	+	0.987957i	$0.549450\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.02922i	0.295993i	0.988988	+	0.147996i	$0.0472824\pi$
$47$	2.02922i	0.295993i	−0.988988	+	0.147996i	$0.952718\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.48528	1.16554	0.582772	−	0.812636i	$-0.301968\pi$
$53$	8.48528	1.16554	0.582772	+	0.812636i	$0.301968\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.01461	0.132091	0.0660456	−	0.997817i	$-0.478962\pi$
$59$	1.01461	0.132091	0.0660456	+	0.997817i	$0.478962\pi$
$60$	0	0
$61$	10.4853	1.34250	0.671251	−	0.741230i	$-0.265757\pi$
$61$	10.4853	1.34250	0.671251	+	0.741230i	$0.265757\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−6.92820	−0.846415	−0.423207	−	0.906033i	$-0.639096\pi$
$67$	−6.92820	−0.846415	−0.423207	+	0.906033i	$0.639096\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	16.7262	1.98503	0.992515	−	0.122122i	$-0.0389698\pi$
$71$	16.7262	1.98503	0.992515	+	0.122122i	$0.0389698\pi$
$72$	0	0
$73$	− 10.4853i	− 1.22721i	−0.789613	−	0.613605i	$-0.789719\pi$
$73$	− 10.4853i	− 1.22721i	0.789613	−	0.613605i	$-0.210281\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.48528	0.283224
$78$	0	0
$79$	− 1.43488i	− 0.161436i	−0.996737	−	0.0807182i	$-0.974279\pi$
$79$	− 1.43488i	− 0.161436i	0.996737	−	0.0807182i	$-0.0257214\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 14.6969i	− 1.61320i	−0.591099	−	0.806599i	$-0.701306\pi$
$83$	− 14.6969i	− 1.61320i	0.591099	−	0.806599i	$-0.298694\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 16.2426i	− 1.72172i	−0.508845	−	0.860858i	$-0.669927\pi$
$89$	− 16.2426i	− 1.72172i	0.508845	−	0.860858i	$-0.330073\pi$
$90$	0	0
$91$	− 5.49333i	− 0.575857i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	10.4853i	1.06462i	0.846550	+	0.532310i	$0.178676\pi$
$97$	10.4853i	1.06462i	−0.846550	+	0.532310i	$0.821324\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.48528i	0.247295i	0.992326	+	0.123647i	$0.0394592\pi$
$101$	2.48528i	0.247295i	−0.992326	+	0.123647i	$0.960541\pi$
$102$	0	0
$103$	−16.3059	−1.60667	−0.803334	−	0.595529i	$-0.796943\pi$
$103$	−16.3059	−1.60667	−0.803334	+	0.595529i	$0.796943\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 18.7554i	− 1.81315i	−0.422043	−	0.906576i	$-0.638687\pi$
$107$	− 18.7554i	− 1.81315i	0.422043	−	0.906576i	$-0.361313\pi$
$108$	0	0
$109$	−6.48528	−0.621177	−0.310589	−	0.950544i	$-0.600526\pi$
$109$	−6.48528	−0.621177	−0.310589	+	0.950544i	$0.600526\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.48528	0.798228	0.399114	−	0.916901i	$-0.369318\pi$
$113$	8.48528	0.798228	0.399114	+	0.916901i	$0.369318\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−9.97056	−0.906415
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	7.34847	0.652071	0.326036	−	0.945357i	$-0.394287\pi$
$127$	7.34847	0.652071	0.326036	+	0.945357i	$0.394287\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13.6823	1.19543	0.597715	−	0.801709i	$-0.296075\pi$
$131$	13.6823	1.19543	0.597715	+	0.801709i	$0.296075\pi$
$132$	0	0
$133$	− 12.0000i	− 1.04053i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.51472	0.300283	0.150141	−	0.988665i	$-0.452027\pi$
$137$	3.51472	0.300283	0.150141	+	0.988665i	$0.452027\pi$
$138$	0	0
$139$	− 13.2621i	− 1.12487i	−0.826840	−	0.562437i	$-0.809864\pi$
$139$	− 13.2621i	− 1.12487i	0.826840	−	0.562437i	$-0.190136\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 2.27541i	− 0.190279i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.48528i	0.203602i	0.994805	+	0.101801i	$0.0324605\pi$
$149$	2.48528i	0.203602i	−0.994805	+	0.101801i	$0.967539\pi$
$150$	0	0
$151$	− 0.594346i	− 0.0483672i	−0.999708	−	0.0241836i	$-0.992301\pi$
$151$	− 0.594346i	− 0.0483672i	0.999708	−	0.0241836i	$-0.00769863\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 18.2426i	− 1.45592i	−0.685619	−	0.727961i	$-0.740468\pi$
$157$	− 18.2426i	− 1.45592i	0.685619	−	0.727961i	$-0.259532\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	20.4853i	1.61447i
$162$	0	0
$163$	−21.6251	−1.69381	−0.846906	−	0.531743i	$-0.821537\pi$
$163$	−21.6251	−1.69381	−0.846906	+	0.531743i	$0.821537\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 8.36308i	− 0.647155i	−0.946202	−	0.323577i	$-0.895114\pi$
$167$	− 8.36308i	− 0.647155i	0.946202	−	0.323577i	$-0.104886\pi$
$168$	0	0
$169$	7.97056	0.613120
$170$	0	0
$171$	0	0
$172$	0	0
$173$	20.4853	1.55747	0.778734	−	0.627355i	$-0.215862\pi$
$173$	20.4853	1.55747	0.778734	+	0.627355i	$0.215862\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−19.7700	−1.47768	−0.738840	−	0.673881i	$-0.764626\pi$
$179$	−19.7700	−1.47768	−0.738840	+	0.673881i	$0.764626\pi$
$180$	0	0
$181$	1.51472	0.112588	0.0562941	−	0.998414i	$-0.482072\pi$
$181$	1.51472	0.112588	0.0562941	+	0.998414i	$0.482072\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.6969	1.06343	0.531717	−	0.846922i	$-0.321547\pi$
$191$	14.6969	1.06343	0.531717	+	0.846922i	$0.321547\pi$
$192$	0	0
$193$	− 10.0000i	− 0.719816i	−0.932988	−	0.359908i	$-0.882808\pi$
$193$	− 10.0000i	− 0.719816i	0.932988	−	0.359908i	$-0.117192\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	22.9706	1.63658	0.818292	−	0.574802i	$-0.194921\pi$
$197$	22.9706	1.63658	0.818292	+	0.574802i	$0.194921\pi$
$198$	0	0
$199$	3.46410i	0.245564i	0.992434	+	0.122782i	$0.0391815\pi$
$199$	3.46410i	0.245564i	−0.992434	+	0.122782i	$0.960818\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 14.6969i	− 1.03152i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 4.97056i	− 0.343821i
$210$	0	0
$211$	− 3.46410i	− 0.238479i	−0.992866	−	0.119239i	$-0.961954\pi$
$211$	− 3.46410i	− 0.238479i	0.992866	−	0.119239i	$-0.0380456\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	20.4853i	1.39063i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−9.37769	−0.627977	−0.313988	−	0.949427i	$-0.601665\pi$
$223$	−9.37769	−0.627977	−0.313988	+	0.949427i	$0.601665\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.05845i	0.269369i	0.990889	+	0.134684i	$0.0430020\pi$
$227$	4.05845i	0.269369i	−0.990889	+	0.134684i	$0.956998\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−24.0000	−1.57229	−0.786146	−	0.618041i	$-0.787927\pi$
$233$	−24.0000	−1.57229	−0.786146	+	0.618041i	$0.787927\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.02922	0.131260	0.0656298	−	0.997844i	$-0.479094\pi$
$239$	2.02922	0.131260	0.0656298	+	0.997844i	$0.479094\pi$
$240$	0	0
$241$	20.9706	1.35083	0.675416	−	0.737437i	$-0.263964\pi$
$241$	20.9706	1.35083	0.675416	+	0.737437i	$0.263964\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−10.9867	−0.699064
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.7700	1.24787	0.623936	−	0.781476i	$-0.285533\pi$
$251$	19.7700	1.24787	0.623936	+	0.781476i	$0.285533\pi$
$252$	0	0
$253$	8.48528i	0.533465i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−20.4853	−1.27784	−0.638918	−	0.769275i	$-0.720618\pi$
$257$	−20.4853	−1.27784	−0.638918	+	0.769275i	$0.720618\pi$
$258$	0	0
$259$	15.2913i	0.950154i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 29.1477i	− 1.79732i	−0.438642	−	0.898662i	$-0.644540\pi$
$263$	− 29.1477i	− 1.79732i	0.438642	−	0.898662i	$-0.355460\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.9706i	1.40054i	0.713878	+	0.700270i	$0.246937\pi$
$269$	22.9706i	1.40054i	−0.713878	+	0.700270i	$0.753063\pi$
$270$	0	0
$271$	− 13.2621i	− 0.805613i	−0.915285	−	0.402806i	$-0.868035\pi$
$271$	− 13.2621i	− 0.805613i	0.915285	−	0.402806i	$-0.131965\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 5.75736i	− 0.345926i	−0.984928	−	0.172963i	$-0.944666\pi$
$277$	− 5.75736i	− 0.345926i	0.984928	−	0.172963i	$-0.0553341\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.24264i	0.253095i	0.991961	+	0.126547i	$0.0403896\pi$
$281$	4.24264i	0.253095i	−0.991961	+	0.126547i	$0.959610\pi$
$282$	0	0
$283$	−4.05845	−0.241250	−0.120625	−	0.992698i	$-0.538490\pi$
$283$	−4.05845	−0.241250	−0.120625	+	0.992698i	$0.538490\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 10.3923i	− 0.613438i
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	10.9706	0.640907	0.320454	−	0.947264i	$-0.396165\pi$
$293$	10.9706	0.640907	0.320454	+	0.947264i	$0.396165\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	18.7554	1.08465
$300$	0	0
$301$	4.97056	0.286498
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	4.05845	0.231628	0.115814	−	0.993271i	$-0.463052\pi$
$307$	4.05845	0.231628	0.115814	+	0.993271i	$0.463052\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−18.7554	−1.06352	−0.531760	−	0.846895i	$-0.678469\pi$
$311$	−18.7554	−1.06352	−0.531760	+	0.846895i	$0.678469\pi$
$312$	0	0
$313$	14.0000i	0.791327i	0.918396	+	0.395663i	$0.129485\pi$
$313$	14.0000i	0.791327i	−0.918396	+	0.395663i	$0.870515\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.02944	−0.0578190	−0.0289095	−	0.999582i	$-0.509203\pi$
$317$	−1.02944	−0.0578190	−0.0289095	+	0.999582i	$0.509203\pi$
$318$	0	0
$319$	− 6.08767i	− 0.340844i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 4.97056i	− 0.274036i
$330$	0	0
$331$	− 11.8272i	− 0.650081i	−0.945700	−	0.325040i	$-0.894622\pi$
$331$	− 11.8272i	− 0.650081i	0.945700	−	0.325040i	$-0.105378\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 23.4558i	− 1.27772i	−0.769322	−	0.638861i	$-0.779406\pi$
$337$	− 23.4558i	− 1.27772i	0.769322	−	0.638861i	$-0.220594\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.48528i	0.459504i
$342$	0	0
$343$	19.5959	1.05808
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.7846i	1.11578i	0.829916	+	0.557888i	$0.188388\pi$
$347$	20.7846i	1.11578i	−0.829916	+	0.557888i	$0.811612\pi$
$348$	0	0
$349$	1.51472	0.0810810	0.0405405	−	0.999178i	$-0.487092\pi$
$349$	1.51472	0.0810810	0.0405405	+	0.999178i	$0.487092\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.0000	0.638696	0.319348	−	0.947638i	$-0.396536\pi$
$353$	12.0000	0.638696	0.319348	+	0.947638i	$0.396536\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.7554	−0.989871	−0.494936	−	0.868930i	$-0.664808\pi$
$359$	−18.7554	−0.989871	−0.494936	+	0.868930i	$0.664808\pi$
$360$	0	0
$361$	−5.00000	−0.263158
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−15.1172	−0.789112	−0.394556	−	0.918872i	$-0.629102\pi$
$367$	−15.1172	−0.789112	−0.394556	+	0.918872i	$0.629102\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−20.7846	−1.07908
$372$	0	0
$373$	− 22.7279i	− 1.17681i	−0.808567	−	0.588404i	$-0.799757\pi$
$373$	− 22.7279i	− 1.17681i	0.808567	−	0.588404i	$-0.200243\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13.4558	−0.693011
$378$	0	0
$379$	17.3205i	0.889695i	0.895606	+	0.444847i	$0.146742\pi$
$379$	17.3205i	0.889695i	−0.895606	+	0.444847i	$0.853258\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 2.02922i	− 0.103688i	−0.998655	−	0.0518442i	$-0.983490\pi$
$383$	− 2.02922i	− 0.103688i	0.998655	−	0.0518442i	$-0.0165099\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	10.9706i	0.556230i	0.960548	+	0.278115i	$0.0897096\pi$
$389$	10.9706i	0.556230i	−0.960548	+	0.278115i	$0.910290\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 14.7279i	− 0.739173i	−0.929196	−	0.369587i	$-0.879499\pi$
$397$	− 14.7279i	− 0.739173i	0.929196	−	0.369587i	$-0.120501\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 19.7574i	− 0.986635i	−0.869849	−	0.493318i	$-0.835784\pi$
$401$	− 19.7574i	− 0.986635i	0.869849	−	0.493318i	$-0.164216\pi$
$402$	0	0
$403$	18.7554	0.934272
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.33386i	0.313958i
$408$	0	0
$409$	−14.9706	−0.740247	−0.370123	−	0.928983i	$-0.620685\pi$
$409$	−14.9706	−0.740247	−0.370123	+	0.928983i	$0.620685\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.48528	−0.122293
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	17.7408	0.866694	0.433347	−	0.901227i	$-0.357333\pi$
$419$	17.7408	0.866694	0.433347	+	0.901227i	$0.357333\pi$
$420$	0	0
$421$	−6.97056	−0.339724	−0.169862	−	0.985468i	$-0.554332\pi$
$421$	−6.97056	−0.339724	−0.169862	+	0.985468i	$0.554332\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−25.6836	−1.24292
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.02922	−0.0977443	−0.0488721	−	0.998805i	$-0.515563\pi$
$431$	−2.02922	−0.0977443	−0.0488721	+	0.998805i	$0.515563\pi$
$432$	0	0
$433$	27.4558i	1.31944i	0.751510	+	0.659722i	$0.229326\pi$
$433$	27.4558i	1.31944i	−0.751510	+	0.659722i	$0.770674\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	40.9706	1.95989
$438$	0	0
$439$	23.0600i	1.10059i	0.834969	+	0.550297i	$0.185486\pi$
$439$	23.0600i	1.10059i	−0.834969	+	0.550297i	$0.814514\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.7262i	0.794684i	0.917671	+	0.397342i	$0.130067\pi$
$443$	16.7262i	0.794684i	−0.917671	+	0.397342i	$0.869933\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 24.7279i	− 1.16698i	−0.812119	−	0.583491i	$-0.801686\pi$
$449$	− 24.7279i	− 1.16698i	0.812119	−	0.583491i	$-0.198314\pi$
$450$	0	0
$451$	− 4.30463i	− 0.202697i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	34.4853i	1.61315i	0.591129	+	0.806577i	$0.298682\pi$
$457$	34.4853i	1.61315i	−0.591129	+	0.806577i	$0.701318\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 31.4558i	− 1.46504i	−0.680743	−	0.732522i	$-0.738343\pi$
$461$	− 31.4558i	− 1.46504i	0.680743	−	0.732522i	$-0.261657\pi$
$462$	0	0
$463$	4.47871	0.208143	0.104072	−	0.994570i	$-0.466813\pi$
$463$	4.47871	0.208143	0.104072	+	0.994570i	$0.466813\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 2.02922i	− 0.0939013i	−0.998897	−	0.0469506i	$-0.985050\pi$
$467$	− 2.02922i	− 0.0939013i	0.998897	−	0.0469506i	$-0.0149503\pi$
$468$	0	0
$469$	16.9706	0.783628
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.05887	0.0946672
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	20.7846	0.949673	0.474837	−	0.880074i	$-0.342507\pi$
$479$	20.7846	0.949673	0.474837	+	0.880074i	$0.342507\pi$
$480$	0	0
$481$	14.0000	0.638345
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	20.3643	0.922796	0.461398	−	0.887193i	$-0.347348\pi$
$487$	20.3643	0.922796	0.461398	+	0.887193i	$0.347348\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.04384	0.137366	0.0686832	−	0.997639i	$-0.478120\pi$
$491$	3.04384	0.137366	0.0686832	+	0.997639i	$0.478120\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−40.9706	−1.83778
$498$	0	0
$499$	− 39.5400i	− 1.77005i	−0.465540	−	0.885027i	$-0.654140\pi$
$499$	− 39.5400i	− 1.77005i	0.465540	−	0.885027i	$-0.345860\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	35.4815i	1.58204i	0.611788	+	0.791022i	$0.290451\pi$
$503$	35.4815i	1.58204i	−0.611788	+	0.791022i	$0.709549\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	2.48528i	0.110158i	0.998482	+	0.0550791i	$0.0175411\pi$
$509$	2.48528i	0.110158i	−0.998482	+	0.0550791i	$0.982459\pi$
$510$	0	0
$511$	25.6836i	1.13618i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 2.05887i	− 0.0905492i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 9.21320i	− 0.403638i	−0.979423	−	0.201819i	$-0.935315\pi$
$521$	− 9.21320i	− 0.403638i	0.979423	−	0.201819i	$-0.0646852\pi$
$522$	0	0
$523$	25.6836	1.12306	0.561532	−	0.827455i	$-0.310212\pi$
$523$	25.6836	1.12306	0.561532	+	0.827455i	$0.310212\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−46.9411	−2.04092
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−9.51472	−0.412128
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.01461	0.0437024
$540$	0	0
$541$	−22.4853	−0.966718	−0.483359	−	0.875422i	$-0.660583\pi$
$541$	−22.4853	−0.966718	−0.483359	+	0.875422i	$0.660583\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	38.3513	1.63978	0.819892	−	0.572519i	$-0.194034\pi$
$547$	38.3513	1.63978	0.819892	+	0.572519i	$0.194034\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−29.3939	−1.25222
$552$	0	0
$553$	3.51472i	0.149461i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.48528	0.359533	0.179766	−	0.983709i	$-0.442466\pi$
$557$	8.48528	0.359533	0.179766	+	0.983709i	$0.442466\pi$
$558$	0	0
$559$	− 4.55082i	− 0.192479i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	12.6677i	0.533881i	0.963713	+	0.266940i	$0.0860126\pi$
$563$	12.6677i	0.533881i	−0.963713	+	0.266940i	$0.913987\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 19.7574i	− 0.828272i	−0.910215	−	0.414136i	$-0.864084\pi$
$569$	− 19.7574i	− 0.828272i	0.910215	−	0.414136i	$-0.135916\pi$
$570$	0	0
$571$	15.8856i	0.664793i	0.943140	+	0.332396i	$0.107857\pi$
$571$	15.8856i	0.664793i	−0.943140	+	0.332396i	$0.892143\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13.5147i	0.562625i	0.959616	+	0.281313i	$0.0907698\pi$
$577$	13.5147i	0.562625i	−0.959616	+	0.281313i	$0.909230\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	36.0000i	1.49353i
$582$	0	0
$583$	−8.60927	−0.356559
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.6969i	0.606608i	0.952894	+	0.303304i	$0.0980897\pi$
$587$	14.6969i	0.606608i	−0.952894	+	0.303304i	$0.901910\pi$
$588$	0	0
$589$	40.9706	1.68816
$590$	0	0
$591$	0	0
$592$	0	0
$593$	3.51472	0.144332	0.0721661	−	0.997393i	$-0.477009\pi$
$593$	3.51472	0.144332	0.0721661	+	0.997393i	$0.477009\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.08767	−0.248736	−0.124368	−	0.992236i	$-0.539690\pi$
$599$	−6.08767	−0.248736	−0.124368	+	0.992236i	$0.539690\pi$
$600$	0	0
$601$	−28.0000	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$601$	−28.0000	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−25.2633	−1.02541	−0.512703	−	0.858566i	$-0.671356\pi$
$607$	−25.2633	−1.02541	−0.512703	+	0.858566i	$0.671356\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.55082	−0.184106
$612$	0	0
$613$	− 19.2132i	− 0.776014i	−0.921656	−	0.388007i	$-0.873164\pi$
$613$	− 19.2132i	− 0.776014i	0.921656	−	0.388007i	$-0.126836\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−45.9411	−1.84952	−0.924760	−	0.380551i	$-0.875734\pi$
$617$	−45.9411	−1.84952	−0.924760	+	0.380551i	$0.875734\pi$
$618$	0	0
$619$	− 20.1903i	− 0.811515i	−0.913981	−	0.405758i	$-0.867008\pi$
$619$	− 20.1903i	− 0.811515i	0.913981	−	0.405758i	$-0.132992\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	39.7862i	1.59400i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	18.1610i	0.722980i	0.932376	+	0.361490i	$0.117732\pi$
$631$	18.1610i	0.722980i	−0.932376	+	0.361490i	$0.882268\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 2.24264i	− 0.0888567i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0.727922i	0.0287512i	0.999897	+	0.0143756i	$0.00457605\pi$
$641$	0.727922i	0.0287512i	−0.999897	+	0.0143756i	$0.995424\pi$
$642$	0	0
$643$	−14.6969	−0.579591	−0.289795	−	0.957089i	$-0.593587\pi$
$643$	−14.6969	−0.579591	−0.289795	+	0.957089i	$0.593587\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.02922i	0.0797770i	0.999204	+	0.0398885i	$0.0127003\pi$
$647$	2.02922i	0.0797770i	−0.999204	+	0.0398885i	$0.987300\pi$
$648$	0	0
$649$	−1.02944	−0.0404089
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−46.9706	−1.83810	−0.919050	−	0.394141i	$-0.871042\pi$
$653$	−46.9706	−1.83810	−0.919050	+	0.394141i	$0.871042\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−32.4377	−1.26359	−0.631797	−	0.775134i	$-0.717682\pi$
$659$	−32.4377	−1.26359	−0.631797	+	0.775134i	$0.717682\pi$
$660$	0	0
$661$	−30.4853	−1.18574	−0.592870	−	0.805298i	$-0.702005\pi$
$661$	−30.4853	−1.18574	−0.592870	+	0.805298i	$0.702005\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	50.1785	1.94292
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.6385	−0.410694
$672$	0	0
$673$	17.5147i	0.675143i	0.941300	+	0.337571i	$0.109605\pi$
$673$	17.5147i	0.675143i	−0.941300	+	0.337571i	$0.890395\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	39.9411	1.53506	0.767531	−	0.641012i	$-0.221485\pi$
$677$	39.9411	1.53506	0.767531	+	0.641012i	$0.221485\pi$
$678$	0	0
$679$	− 25.6836i	− 0.985646i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	27.3647i	1.04708i	0.852001	+	0.523540i	$0.175389\pi$
$683$	27.3647i	1.04708i	−0.852001	+	0.523540i	$0.824611\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	19.0294i	0.724964i
$690$	0	0
$691$	14.6969i	0.559098i	0.960131	+	0.279549i	$0.0901849\pi$
$691$	14.6969i	0.559098i	−0.960131	+	0.279549i	$0.909815\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 31.4558i	− 1.18807i	−0.804439	−	0.594035i	$-0.797534\pi$
$701$	− 31.4558i	− 1.18807i	0.804439	−	0.594035i	$-0.202466\pi$
$702$	0	0
$703$	30.5826	1.15344
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 6.08767i	− 0.228950i
$708$	0	0
$709$	35.9411	1.34980	0.674899	−	0.737910i	$-0.264187\pi$
$709$	35.9411	1.34980	0.674899	+	0.737910i	$0.264187\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−69.9411	−2.61932
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10.6385	0.396749	0.198374	−	0.980126i	$-0.436434\pi$
$719$	10.6385	0.396749	0.198374	+	0.980126i	$0.436434\pi$
$720$	0	0
$721$	39.9411	1.48749
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	44.8592	1.66374	0.831869	−	0.554973i	$-0.187271\pi$
$727$	44.8592	1.66374	0.831869	+	0.554973i	$0.187271\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	26.2426i	0.969294i	0.874710	+	0.484647i	$0.161052\pi$
$733$	26.2426i	0.969294i	−0.874710	+	0.484647i	$0.838948\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.02944	0.258933
$738$	0	0
$739$	35.4815i	1.30521i	0.757698	+	0.652605i	$0.226324\pi$
$739$	35.4815i	1.30521i	−0.757698	+	0.652605i	$0.773676\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31.4231i	1.15280i	0.817167	+	0.576401i	$0.195543\pi$
$743$	31.4231i	1.15280i	−0.817167	+	0.576401i	$0.804457\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	45.9411i	1.67865i
$750$	0	0
$751$	51.6134i	1.88340i	0.336456	+	0.941699i	$0.390772\pi$
$751$	51.6134i	1.88340i	−0.336456	+	0.941699i	$0.609228\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	35.2132i	1.27985i	0.768439	+	0.639923i	$0.221034\pi$
$757$	35.2132i	1.27985i	−0.768439	+	0.639923i	$0.778966\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 41.6985i	− 1.51157i	−0.654820	−	0.755784i	$-0.727256\pi$
$761$	− 41.6985i	− 1.51157i	0.654820	−	0.755784i	$-0.272744\pi$
$762$	0	0
$763$	15.8856	0.575098
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.27541i	0.0821603i
$768$	0	0
$769$	16.0000	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	10.9706	0.394584	0.197292	−	0.980345i	$-0.436785\pi$
$773$	10.9706	0.394584	0.197292	+	0.980345i	$0.436785\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−20.7846	−0.744686
$780$	0	0
$781$	−16.9706	−0.607254
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−46.4682	−1.65641	−0.828206	−	0.560423i	$-0.810638\pi$
$787$	−46.4682	−1.65641	−0.828206	+	0.560423i	$0.810638\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−20.7846	−0.739016
$792$	0	0
$793$	23.5147i	0.835032i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.5147	0.974621	0.487311	−	0.873229i	$-0.337978\pi$
$797$	27.5147	0.974621	0.487311	+	0.873229i	$0.337978\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	10.6385i	0.375424i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 17.6985i	− 0.622246i	−0.950370	−	0.311123i	$-0.899295\pi$
$809$	− 17.6985i	− 0.622246i	0.950370	−	0.311123i	$-0.100705\pi$
$810$	0	0
$811$	43.8446i	1.53959i	0.638289	+	0.769797i	$0.279642\pi$
$811$	43.8446i	1.53959i	−0.638289	+	0.769797i	$0.720358\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 9.94113i	− 0.347796i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	18.0000i	0.628204i	0.949389	+	0.314102i	$0.101703\pi$
$821$	18.0000i	0.628204i	−0.949389	+	0.314102i	$0.898297\pi$
$822$	0	0
$823$	−0.420266	−0.0146496	−0.00732478	−	0.999973i	$-0.502332\pi$
$823$	−0.420266	−0.0146496	−0.00732478	+	0.999973i	$0.502332\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.6677i	0.440500i	0.975444	+	0.220250i	$0.0706873\pi$
$827$	12.6677i	0.440500i	−0.975444	+	0.220250i	$0.929313\pi$
$828$	0	0
$829$	−6.48528	−0.225243	−0.112622	−	0.993638i	$-0.535925\pi$
$829$	−6.48528	−0.225243	−0.112622	+	0.993638i	$0.535925\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−18.7554	−0.647508	−0.323754	−	0.946141i	$-0.604945\pi$
$839$	−18.7554	−0.647508	−0.323754	+	0.946141i	$0.604945\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	24.4228	0.839177
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−52.2077	−1.78966
$852$	0	0
$853$	− 26.2426i	− 0.898531i	−0.893398	−	0.449265i	$-0.851686\pi$
$853$	− 26.2426i	− 0.898531i	0.893398	−	0.449265i	$-0.148314\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−28.9706	−0.989616	−0.494808	−	0.869002i	$-0.664762\pi$
$857$	−28.9706	−0.989616	−0.494808	+	0.869002i	$0.664762\pi$
$858$	0	0
$859$	− 4.65279i	− 0.158751i	−0.996845	−	0.0793756i	$-0.974707\pi$
$859$	− 4.65279i	− 0.158751i	0.996845	−	0.0793756i	$-0.0252927\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.30463i	0.146531i	0.997312	+	0.0732657i	$0.0233421\pi$
$863$	4.30463i	0.146531i	−0.997312	+	0.0732657i	$0.976658\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.45584i	0.0493861i
$870$	0	0
$871$	− 15.5375i	− 0.526467i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 21.7574i	− 0.734694i	−0.930084	−	0.367347i	$-0.880266\pi$
$877$	− 21.7574i	− 0.734694i	0.930084	−	0.367347i	$-0.119734\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	9.21320i	0.310401i	0.987883	+	0.155200i	$0.0496023\pi$
$881$	9.21320i	0.310401i	−0.987883	+	0.155200i	$0.950398\pi$
$882$	0	0
$883$	15.8856	0.534594	0.267297	−	0.963614i	$-0.413870\pi$
$883$	15.8856	0.534594	0.267297	+	0.963614i	$0.413870\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	29.1477i	0.978684i	0.872092	+	0.489342i	$0.162763\pi$
$887$	29.1477i	0.978684i	−0.872092	+	0.489342i	$0.837237\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9.94113	−0.332667
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	50.1785	1.67355
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	3.21792	0.106849	0.0534246	−	0.998572i	$-0.482986\pi$
$907$	3.21792	0.106849	0.0534246	+	0.998572i	$0.482986\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−25.3354	−0.839400	−0.419700	−	0.907663i	$-0.637865\pi$
$911$	−25.3354	−0.839400	−0.419700	+	0.907663i	$0.637865\pi$
$912$	0	0
$913$	14.9117i	0.493505i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−33.5147	−1.10675
$918$	0	0
$919$	− 8.36308i	− 0.275873i	−0.990441	−	0.137936i	$-0.955953\pi$
$919$	− 8.36308i	− 0.275873i	0.990441	−	0.137936i	$-0.0440469\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	37.5108i	1.23468i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 24.7279i	− 0.811297i	−0.914029	−	0.405648i	$-0.867046\pi$
$929$	− 24.7279i	− 0.811297i	0.914029	−	0.405648i	$-0.132954\pi$
$930$	0	0
$931$	− 4.89898i	− 0.160558i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38.9706i	1.27311i	0.771230	+	0.636556i	$0.219642\pi$
$937$	38.9706i	1.27311i	−0.771230	+	0.636556i	$0.780358\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 22.9706i	− 0.748819i	−0.927263	−	0.374409i	$-0.877845\pi$
$941$	− 22.9706i	− 0.748819i	0.927263	−	0.374409i	$-0.122155\pi$
$942$	0	0
$943$	35.4815	1.15544
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 20.7846i	− 0.675409i	−0.941252	−	0.337705i	$-0.890350\pi$
$947$	− 20.7846i	− 0.675409i	0.941252	−	0.337705i	$-0.109650\pi$
$948$	0	0
$949$	23.5147	0.763320
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−30.4264	−0.985608	−0.492804	−	0.870140i	$-0.664028\pi$
$953$	−30.4264	−0.985608	−0.492804	+	0.870140i	$0.664028\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.60927	−0.278008
$960$	0	0
$961$	−38.9411	−1.25617
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−51.7874	−1.66537	−0.832686	−	0.553745i	$-0.813198\pi$
$967$	−51.7874	−1.66537	−0.832686	+	0.553745i	$0.813198\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−57.2808	−1.83823	−0.919114	−	0.393992i	$-0.871094\pi$
$971$	−57.2808	−1.83823	−0.919114	+	0.393992i	$0.871094\pi$
$972$	0	0
$973$	32.4853i	1.04143i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−28.9706	−0.926850	−0.463425	−	0.886136i	$-0.653380\pi$
$977$	−28.9706	−0.926850	−0.463425	+	0.886136i	$0.653380\pi$
$978$	0	0
$979$	16.4800i	0.526702i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 2.02922i	− 0.0647222i	−0.999476	−	0.0323611i	$-0.989697\pi$
$983$	− 2.02922i	− 0.0647222i	0.999476	−	0.0323611i	$-0.0103027\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.9706i	0.539633i
$990$	0	0
$991$	− 49.5841i	− 1.57509i	−0.616256	−	0.787546i	$-0.711351\pi$
$991$	− 49.5841i	− 1.57509i	0.616256	−	0.787546i	$-0.288649\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 47.2132i	− 1.49526i	−0.664117	−	0.747629i	$-0.731192\pi$
$997$	− 47.2132i	− 1.49526i	0.664117	−	0.747629i	$-0.268808\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3600.2.o.c.3599.2		8
3.2	odd	2		3600.2.o.d.3599.4		8
4.3	odd	2	inner	3600.2.o.c.3599.8		8
5.2	odd	4		720.2.h.a.431.1	✓	8
5.3	odd	4		3600.2.h.j.1151.6		8
5.4	even	2		3600.2.o.d.3599.5		8
12.11	even	2		3600.2.o.d.3599.6		8
15.2	even	4		720.2.h.a.431.6	yes	8
15.8	even	4		3600.2.h.j.1151.7		8
15.14	odd	2	inner	3600.2.o.c.3599.7		8
20.3	even	4		3600.2.h.j.1151.3		8
20.7	even	4		720.2.h.a.431.4	yes	8
20.19	odd	2		3600.2.o.d.3599.3		8
40.27	even	4		2880.2.h.f.1151.7		8
40.37	odd	4		2880.2.h.f.1151.6		8
60.23	odd	4		3600.2.h.j.1151.2		8
60.47	odd	4		720.2.h.a.431.7	yes	8
60.59	even	2	inner	3600.2.o.c.3599.1		8
120.77	even	4		2880.2.h.f.1151.1		8
120.107	odd	4		2880.2.h.f.1151.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
720.2.h.a.431.1	✓	8	5.2	odd	4
720.2.h.a.431.4	yes	8	20.7	even	4
720.2.h.a.431.6	yes	8	15.2	even	4
720.2.h.a.431.7	yes	8	60.47	odd	4
2880.2.h.f.1151.1		8	120.77	even	4
2880.2.h.f.1151.4		8	120.107	odd	4
2880.2.h.f.1151.6		8	40.37	odd	4
2880.2.h.f.1151.7		8	40.27	even	4
3600.2.h.j.1151.2		8	60.23	odd	4
3600.2.h.j.1151.3		8	20.3	even	4
3600.2.h.j.1151.6		8	5.3	odd	4
3600.2.h.j.1151.7		8	15.8	even	4
3600.2.o.c.3599.1		8	60.59	even	2	inner
3600.2.o.c.3599.2		8	1.1	even	1	trivial
3600.2.o.c.3599.7		8	15.14	odd	2	inner
3600.2.o.c.3599.8		8	4.3	odd	2	inner
3600.2.o.d.3599.3		8	20.19	odd	2
3600.2.o.d.3599.4		8	3.2	odd	2
3600.2.o.d.3599.5		8	5.4	even	2
3600.2.o.d.3599.6		8	12.11	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.44949 q^{7} -1.01461 q^{11} +2.24264i q^{13} +4.89898i q^{19} -8.36308i q^{23} +6.00000i q^{29} -8.36308i q^{31} -6.24264i q^{37} +4.24264i q^{41} -2.02922 q^{43} +2.02922i q^{47} -1.00000 q^{49} +8.48528 q^{53} +1.01461 q^{59} +10.4853 q^{61} -6.92820 q^{67} +16.7262 q^{71} -10.4853i q^{73} +2.48528 q^{77} -1.43488i q^{79} -14.6969i q^{83} -16.2426i q^{89} -5.49333i q^{91} +10.4853i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{49} + 16 q^{61} - 48 q^{77}+O(q^{100}) \) 8 * q - 8 * q^49 + 16 * q^61 - 48 * q^77

Introduction

L-functions

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