LMFDB - Embedded newform 1575.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1575.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.a (trivial)

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:	yes
Analytic conductor:	$12.5764383184$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.174928.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} + 13 $ x^4 - 9*x^2 + 13
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.68190$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.68190 q^{2} +5.19258 q^{4} -1.00000 q^{7} -8.56218 q^{8} +O(q^{10})$	$q-2.68190 q^{2} +5.19258 q^{4} -1.00000 q^{7} -8.56218 q^{8} +3.19839 q^{11} +6.38516 q^{13} +2.68190 q^{14} +12.5777 q^{16} -5.36380 q^{17} -2.38516 q^{19} -8.57775 q^{22} +3.19839 q^{23} -17.1244 q^{26} -5.19258 q^{28} +2.16541 q^{29} +6.00000 q^{31} -16.6079 q^{32} +14.3852 q^{34} +3.00000 q^{37} +6.39677 q^{38} -10.7276 q^{41} -9.38516 q^{43} +16.6079 q^{44} -8.57775 q^{46} +11.7606 q^{47} +1.00000 q^{49} +33.1555 q^{52} +10.7276 q^{53} +8.56218 q^{56} -5.80742 q^{58} +11.7606 q^{59} -16.0914 q^{62} +19.3852 q^{64} -9.38516 q^{67} -27.8520 q^{68} -13.9260 q^{71} +4.38516 q^{73} -8.04570 q^{74} -12.3852 q^{76} -3.19839 q^{77} -5.38516 q^{79} +28.7703 q^{82} +4.33082 q^{83} +25.1701 q^{86} -27.3852 q^{88} -1.03297 q^{89} -6.38516 q^{91} +16.6079 q^{92} -31.5407 q^{94} -10.7703 q^{97} -2.68190 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 10 q^{4} - 4 q^{7}+O(q^{10}) $ 4 * q + 10 * q^4 - 4 * q^7	$ 4 q + 10 q^{4} - 4 q^{7} + 4 q^{13} + 18 q^{16} + 12 q^{19} - 2 q^{22} - 10 q^{28} + 24 q^{31} + 36 q^{34} + 12 q^{37} - 16 q^{43} - 2 q^{46} + 4 q^{49} + 68 q^{52} - 34 q^{58} + 56 q^{64} - 16 q^{67} - 4 q^{73} - 28 q^{76} + 72 q^{82} - 88 q^{88} - 4 q^{91} - 40 q^{94}+O(q^{100}) $ 4 * q + 10 * q^4 - 4 * q^7 + 4 * q^13 + 18 * q^16 + 12 * q^19 - 2 * q^22 - 10 * q^28 + 24 * q^31 + 36 * q^34 + 12 * q^37 - 16 * q^43 - 2 * q^46 + 4 * q^49 + 68 * q^52 - 34 * q^58 + 56 * q^64 - 16 * q^67 - 4 * q^73 - 28 * q^76 + 72 * q^82 - 88 * q^88 - 4 * q^91 - 40 * q^94

Display 100 coefficients 10 coefficients

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 $

$2$	−2.68190	−1.89639	−0.948194	−	0.317690i	$-0.897093\pi$
$2$	−2.68190	−1.89639	−0.948194	+	0.317690i	$0.897093\pi$
$3$	0	0
$3$	0	0
$4$	5.19258	2.59629
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−8.56218	−3.02719
$9$	0	0
$10$	0	0
$11$	3.19839	0.964350	0.482175	−	0.876075i	$-0.339847\pi$
$11$	3.19839	0.964350	0.482175	+	0.876075i	$0.339847\pi$
$12$	0	0
$13$	6.38516	1.77093	0.885463	−	0.464710i	$-0.153841\pi$
$13$	6.38516	1.77093	0.885463	+	0.464710i	$0.153841\pi$
$14$	2.68190	0.716768
$15$	0	0
$16$	12.5777	3.14444
$17$	−5.36380	−1.30091	−0.650456	−	0.759544i	$-0.725422\pi$
$17$	−5.36380	−1.30091	−0.650456	+	0.759544i	$0.725422\pi$
$18$	0	0
$19$	−2.38516	−0.547194	−0.273597	−	0.961844i	$-0.588214\pi$
$19$	−2.38516	−0.547194	−0.273597	+	0.961844i	$0.588214\pi$
$20$	0	0
$21$	0	0
$22$	−8.57775	−1.82878
$23$	3.19839	0.666909	0.333455	−	0.942766i	$-0.391786\pi$
$23$	3.19839	0.666909	0.333455	+	0.942766i	$0.391786\pi$
$24$	0	0
$25$	0	0
$26$	−17.1244	−3.35836
$27$	0	0
$28$	−5.19258	−0.981306
$29$	2.16541	0.402107	0.201054	−	0.979580i	$-0.435563\pi$
$29$	2.16541	0.402107	0.201054	+	0.979580i	$0.435563\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−16.6079	−2.93589
$33$	0	0
$34$	14.3852	2.46704
$35$	0	0
$36$	0	0
$37$	3.00000	0.493197	0.246598	−	0.969118i	$-0.420687\pi$
$37$	3.00000	0.493197	0.246598	+	0.969118i	$0.420687\pi$
$38$	6.39677	1.03769
$39$	0	0
$40$	0	0
$41$	−10.7276	−1.67537	−0.837685	−	0.546154i	$-0.816091\pi$
$41$	−10.7276	−1.67537	−0.837685	+	0.546154i	$0.816091\pi$
$42$	0	0
$43$	−9.38516	−1.43122	−0.715612	−	0.698498i	$-0.753852\pi$
$43$	−9.38516	−1.43122	−0.715612	+	0.698498i	$0.753852\pi$
$44$	16.6079	2.50373
$45$	0	0
$46$	−8.57775	−1.26472
$47$	11.7606	1.71546	0.857728	−	0.514104i	$-0.171876\pi$
$47$	11.7606	1.71546	0.857728	+	0.514104i	$0.171876\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	33.1555	4.59784
$53$	10.7276	1.47355	0.736774	−	0.676139i	$-0.236348\pi$
$53$	10.7276	1.47355	0.736774	+	0.676139i	$0.236348\pi$
$54$	0	0
$55$	0	0
$56$	8.56218	1.14417
$57$	0	0
$58$	−5.80742	−0.762551
$59$	11.7606	1.53110	0.765548	−	0.643379i	$-0.222468\pi$
$59$	11.7606	1.53110	0.765548	+	0.643379i	$0.222468\pi$
$60$	0	0
$61$	0	0		−	1.00000i	$-0.5\pi$
$61$	0	0			1.00000i	$0.5\pi$
$62$	−16.0914	−2.04361
$63$	0	0
$64$	19.3852	2.42315
$65$	0	0
$66$	0	0
$67$	−9.38516	−1.14658	−0.573290	−	0.819352i	$-0.694333\pi$
$67$	−9.38516	−1.14658	−0.573290	+	0.819352i	$0.694333\pi$
$68$	−27.8520	−3.37755
$69$	0	0
$70$	0	0
$71$	−13.9260	−1.65271	−0.826355	−	0.563150i	$-0.809589\pi$
$71$	−13.9260	−1.65271	−0.826355	+	0.563150i	$0.809589\pi$
$72$	0	0
$73$	4.38516	0.513245	0.256622	−	0.966512i	$-0.417390\pi$
$73$	4.38516	0.513245	0.256622	+	0.966512i	$0.417390\pi$
$74$	−8.04570	−0.935293
$75$	0	0
$76$	−12.3852	−1.42068
$77$	−3.19839	−0.364490
$78$	0	0
$79$	−5.38516	−0.605878	−0.302939	−	0.953010i	$-0.597968\pi$
$79$	−5.38516	−0.605878	−0.302939	+	0.953010i	$0.597968\pi$
$80$	0	0
$81$	0	0
$82$	28.7703	3.17715
$83$	4.33082	0.475370	0.237685	−	0.971342i	$-0.423611\pi$
$83$	4.33082	0.475370	0.237685	+	0.971342i	$0.423611\pi$
$84$	0	0
$85$	0	0
$86$	25.1701	2.71416
$87$	0	0
$88$	−27.3852	−2.91927
$89$	−1.03297	−0.109495	−0.0547475	−	0.998500i	$-0.517435\pi$
$89$	−1.03297	−0.109495	−0.0547475	+	0.998500i	$0.517435\pi$
$90$	0	0
$91$	−6.38516	−0.669347
$92$	16.6079	1.73149
$93$	0	0
$94$	−31.5407	−3.25317
$95$	0	0
$96$	0	0
$97$	−10.7703	−1.09356	−0.546781	−	0.837276i	$-0.684147\pi$
$97$	−10.7703	−1.09356	−0.546781	+	0.837276i	$0.684147\pi$
$98$	−2.68190	−0.270913
$99$	0	0
$100$	0	0
$101$	5.36380	0.533718	0.266859	−	0.963736i	$-0.414014\pi$
$101$	5.36380	0.533718	0.266859	+	0.963736i	$0.414014\pi$
$102$	0	0
$103$	5.61484	0.553246	0.276623	−	0.960978i	$-0.410785\pi$
$103$	5.61484	0.553246	0.276623	+	0.960978i	$0.410785\pi$
$104$	−54.6710	−5.36093
$105$	0	0
$106$	−28.7703	−2.79442
$107$	4.33082	0.418677	0.209338	−	0.977843i	$-0.432869\pi$
$107$	4.33082	0.418677	0.209338	+	0.977843i	$0.432869\pi$
$108$	0	0
$109$	7.00000	0.670478	0.335239	−	0.942133i	$-0.391183\pi$
$109$	7.00000	0.670478	0.335239	+	0.942133i	$0.391183\pi$
$110$	0	0
$111$	0	0
$112$	−12.5777	−1.18849
$113$	−2.16541	−0.203705	−0.101852	−	0.994800i	$-0.532477\pi$
$113$	−2.16541	−0.203705	−0.101852	+	0.994800i	$0.532477\pi$
$114$	0	0
$115$	0	0
$116$	11.2441	1.04399
$117$	0	0
$118$	−31.5407	−2.90355
$119$	5.36380	0.491699
$120$	0	0
$121$	−0.770330	−0.0700300
$122$	0	0
$123$	0	0
$124$	31.1555	2.79785
$125$	0	0
$126$	0	0
$127$	−4.61484	−0.409500	−0.204750	−	0.978814i	$-0.565638\pi$
$127$	−4.61484	−0.409500	−0.204750	+	0.978814i	$0.565638\pi$
$128$	−18.7733	−1.65934
$129$	0	0
$130$	0	0
$131$	5.36380	0.468637	0.234319	−	0.972160i	$-0.424714\pi$
$131$	5.36380	0.468637	0.234319	+	0.972160i	$0.424714\pi$
$132$	0	0
$133$	2.38516	0.206820
$134$	25.1701	2.17436
$135$	0	0
$136$	45.9258	3.93811
$137$	10.7276	0.916520	0.458260	−	0.888818i	$-0.348473\pi$
$137$	10.7276	0.916520	0.458260	+	0.888818i	$0.348473\pi$
$138$	0	0
$139$	−4.38516	−0.371945	−0.185972	−	0.982555i	$-0.559544\pi$
$139$	−4.38516	−0.371945	−0.185972	+	0.982555i	$0.559544\pi$
$140$	0	0
$141$	0	0
$142$	37.3481	3.13418
$143$	20.4222	1.70779
$144$	0	0
$145$	0	0
$146$	−11.7606	−0.973312
$147$	0	0
$148$	15.5777	1.28048
$149$	12.8930	1.05624	0.528118	−	0.849171i	$-0.322898\pi$
$149$	12.8930	1.05624	0.528118	+	0.849171i	$0.322898\pi$
$150$	0	0
$151$	11.3852	0.926512	0.463256	−	0.886225i	$-0.346681\pi$
$151$	11.3852	0.926512	0.463256	+	0.886225i	$0.346681\pi$
$152$	20.4222	1.65646
$153$	0	0
$154$	8.57775	0.691215
$155$	0	0
$156$	0	0
$157$	16.7703	1.33842	0.669209	−	0.743074i	$-0.266633\pi$
$157$	16.7703	1.33842	0.669209	+	0.743074i	$0.266633\pi$
$158$	14.4425	1.14898
$159$	0	0
$160$	0	0
$161$	−3.19839	−0.252068
$162$	0	0
$163$	16.7703	1.31355	0.656777	−	0.754085i	$-0.271919\pi$
$163$	16.7703	1.31355	0.656777	+	0.754085i	$0.271919\pi$
$164$	−55.7039	−4.34975
$165$	0	0
$166$	−11.6148	−0.901486
$167$	1.03297	0.0799339	0.0399669	−	0.999201i	$-0.487275\pi$
$167$	1.03297	0.0799339	0.0399669	+	0.999201i	$0.487275\pi$
$168$	0	0
$169$	27.7703	2.13618
$170$	0	0
$171$	0	0
$172$	−48.7332	−3.71587
$173$	22.4882	1.70974	0.854872	−	0.518839i	$-0.173636\pi$
$173$	22.4882	1.70974	0.854872	+	0.518839i	$0.173636\pi$
$174$	0	0
$175$	0	0
$176$	40.2285	3.03234
$177$	0	0
$178$	2.77033	0.207645
$179$	−4.33082	−0.323701	−0.161851	−	0.986815i	$-0.551746\pi$
$179$	−4.33082	−0.323701	−0.161851	+	0.986815i	$0.551746\pi$
$180$	0	0
$181$	20.3852	1.51522	0.757609	−	0.652709i	$-0.226368\pi$
$181$	20.3852	1.51522	0.757609	+	0.652709i	$0.226368\pi$
$182$	17.1244	1.26934
$183$	0	0
$184$	−27.3852	−2.01886
$185$	0	0
$186$	0	0
$187$	−17.1555	−1.25453
$188$	61.0677	4.45382
$189$	0	0
$190$	0	0
$191$	−6.39677	−0.462854	−0.231427	−	0.972852i	$-0.574339\pi$
$191$	−6.39677	−0.462854	−0.231427	+	0.972852i	$0.574339\pi$
$192$	0	0
$193$	1.00000	0.0719816	0.0359908	−	0.999352i	$-0.488541\pi$
$193$	1.00000	0.0719816	0.0359908	+	0.999352i	$0.488541\pi$
$194$	28.8849	2.07382
$195$	0	0
$196$	5.19258	0.370899
$197$	23.6206	1.68290	0.841449	−	0.540336i	$-0.181703\pi$
$197$	23.6206	1.68290	0.841449	+	0.540336i	$0.181703\pi$
$198$	0	0
$199$	0.770330	0.0546072	0.0273036	−	0.999627i	$-0.491308\pi$
$199$	0.770330	0.0546072	0.0273036	+	0.999627i	$0.491308\pi$
$200$	0	0
$201$	0	0
$202$	−14.3852	−1.01214
$203$	−2.16541	−0.151982
$204$	0	0
$205$	0	0
$206$	−15.0584	−1.04917
$207$	0	0
$208$	80.3110	5.56857
$209$	−7.62868	−0.527687
$210$	0	0
$211$	0	0		−	1.00000i	$-0.5\pi$
$211$	0	0			1.00000i	$0.5\pi$
$212$	55.7039	3.82576
$213$	0	0
$214$	−11.6148	−0.793974
$215$	0	0
$216$	0	0
$217$	−6.00000	−0.407307
$218$	−18.7733	−1.27149
$219$	0	0
$220$	0	0
$221$	−34.2487	−2.30382
$222$	0	0
$223$	3.61484	0.242067	0.121034	−	0.992648i	$-0.461379\pi$
$223$	3.61484	0.242067	0.121034	+	0.992648i	$0.461379\pi$
$224$	16.6079	1.10966
$225$	0	0
$226$	5.80742	0.386304
$227$	−11.7606	−0.780576	−0.390288	−	0.920693i	$-0.627625\pi$
$227$	−11.7606	−0.780576	−0.390288	+	0.920693i	$0.627625\pi$
$228$	0	0
$229$	−15.5407	−1.02696	−0.513478	−	0.858103i	$-0.671643\pi$
$229$	−15.5407	−1.02696	−0.513478	+	0.858103i	$0.671643\pi$
$230$	0	0
$231$	0	0
$232$	−18.5407	−1.21725
$233$	−4.23136	−0.277206	−0.138603	−	0.990348i	$-0.544261\pi$
$233$	−4.23136	−0.277206	−0.138603	+	0.990348i	$0.544261\pi$
$234$	0	0
$235$	0	0
$236$	61.0677	3.97517
$237$	0	0
$238$	−14.3852	−0.932452
$239$	−6.39677	−0.413773	−0.206886	−	0.978365i	$-0.566333\pi$
$239$	−6.39677	−0.413773	−0.206886	+	0.978365i	$0.566333\pi$
$240$	0	0
$241$	18.3852	1.18429	0.592146	−	0.805830i	$-0.298281\pi$
$241$	18.3852	1.18429	0.592146	+	0.805830i	$0.298281\pi$
$242$	2.06595	0.132804
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−15.2297	−0.969041
$248$	−51.3731	−3.26220
$249$	0	0
$250$	0	0
$251$	−3.29785	−0.208159	−0.104079	−	0.994569i	$-0.533190\pi$
$251$	−3.29785	−0.208159	−0.104079	+	0.994569i	$0.533190\pi$
$252$	0	0
$253$	10.2297	0.643134
$254$	12.3765	0.776572
$255$	0	0
$256$	11.5777	0.723609
$257$	−17.1244	−1.06819	−0.534094	−	0.845425i	$-0.679347\pi$
$257$	−17.1244	−1.06819	−0.534094	+	0.845425i	$0.679347\pi$
$258$	0	0
$259$	−3.00000	−0.186411
$260$	0	0
$261$	0	0
$262$	−14.3852	−0.888718
$263$	−20.3228	−1.25315	−0.626577	−	0.779359i	$-0.715545\pi$
$263$	−20.3228	−1.25315	−0.626577	+	0.779359i	$0.715545\pi$
$264$	0	0
$265$	0	0
$266$	−6.39677	−0.392211
$267$	0	0
$268$	−48.7332	−2.97686
$269$	4.33082	0.264055	0.132028	−	0.991246i	$-0.457851\pi$
$269$	4.33082	0.264055	0.132028	+	0.991246i	$0.457851\pi$
$270$	0	0
$271$	15.1555	0.920631	0.460315	−	0.887755i	$-0.347736\pi$
$271$	15.1555	0.920631	0.460315	+	0.887755i	$0.347736\pi$
$272$	−67.4645	−4.09064
$273$	0	0
$274$	−28.7703	−1.73808
$275$	0	0
$276$	0	0
$277$	23.5407	1.41442	0.707211	−	0.707003i	$-0.249953\pi$
$277$	23.5407	1.41442	0.707211	+	0.707003i	$0.249953\pi$
$278$	11.7606	0.705352
$279$	0	0
$280$	0	0
$281$	14.9590	0.892376	0.446188	−	0.894939i	$-0.352781\pi$
$281$	14.9590	0.892376	0.446188	+	0.894939i	$0.352781\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	−72.3118	−4.29092
$285$	0	0
$286$	−54.7703	−3.23864
$287$	10.7276	0.633230
$288$	0	0
$289$	11.7703	0.692372
$290$	0	0
$291$	0	0
$292$	22.7703	1.33253
$293$	11.7606	0.687060	0.343530	−	0.939142i	$-0.388377\pi$
$293$	11.7606	0.687060	0.343530	+	0.939142i	$0.388377\pi$
$294$	0	0
$295$	0	0
$296$	−25.6866	−1.49300
$297$	0	0
$298$	−34.5777	−2.00304
$299$	20.4222	1.18105
$300$	0	0
$301$	9.38516	0.540952
$302$	−30.5339	−1.75703
$303$	0	0
$304$	−30.0000	−1.72062
$305$	0	0
$306$	0	0
$307$	13.1555	0.750824	0.375412	−	0.926858i	$-0.377501\pi$
$307$	13.1555	0.750824	0.375412	+	0.926858i	$0.377501\pi$
$308$	−16.6079	−0.946322
$309$	0	0
$310$	0	0
$311$	26.8190	1.52077	0.760383	−	0.649475i	$-0.225011\pi$
$311$	26.8190	1.52077	0.760383	+	0.649475i	$0.225011\pi$
$312$	0	0
$313$	−23.9258	−1.35237	−0.676184	−	0.736733i	$-0.736367\pi$
$313$	−23.9258	−1.35237	−0.676184	+	0.736733i	$0.736367\pi$
$314$	−44.9763	−2.53816
$315$	0	0
$316$	−27.9629	−1.57304
$317$	−8.56218	−0.480900	−0.240450	−	0.970662i	$-0.577295\pi$
$317$	−8.56218	−0.480900	−0.240450	+	0.970662i	$0.577295\pi$
$318$	0	0
$319$	6.92582	0.387772
$320$	0	0
$321$	0	0
$322$	8.57775	0.478019
$323$	12.7935	0.711852
$324$	0	0
$325$	0	0
$326$	−44.9763	−2.49101
$327$	0	0
$328$	91.8516	5.07166
$329$	−11.7606	−0.648381
$330$	0	0
$331$	10.1555	0.558196	0.279098	−	0.960263i	$-0.409964\pi$
$331$	10.1555	0.558196	0.279098	+	0.960263i	$0.409964\pi$
$332$	22.4882	1.23420
$333$	0	0
$334$	−2.77033	−0.151586
$335$	0	0
$336$	0	0
$337$	−10.7703	−0.586697	−0.293349	−	0.956006i	$-0.594770\pi$
$337$	−10.7703	−0.586697	−0.293349	+	0.956006i	$0.594770\pi$
$338$	−74.4772	−4.05103
$339$	0	0
$340$	0	0
$341$	19.1903	1.03921
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	80.3575	4.33259
$345$	0	0
$346$	−60.3110	−3.24234
$347$	−24.6536	−1.32347	−0.661737	−	0.749736i	$-0.730180\pi$
$347$	−24.6536	−1.32347	−0.661737	+	0.749736i	$0.730180\pi$
$348$	0	0
$349$	2.38516	0.127675	0.0638375	−	0.997960i	$-0.479666\pi$
$349$	2.38516	0.127675	0.0638375	+	0.997960i	$0.479666\pi$
$350$	0	0
$351$	0	0
$352$	−53.1184	−2.83122
$353$	17.1244	0.911438	0.455719	−	0.890124i	$-0.349382\pi$
$353$	17.1244	0.911438	0.455719	+	0.890124i	$0.349382\pi$
$354$	0	0
$355$	0	0
$356$	−5.36380	−0.284281
$357$	0	0
$358$	11.6148	0.613863
$359$	18.2568	0.963557	0.481779	−	0.876293i	$-0.339991\pi$
$359$	18.2568	0.963557	0.481779	+	0.876293i	$0.339991\pi$
$360$	0	0
$361$	−13.3110	−0.700578
$362$	−54.6710	−2.87344
$363$	0	0
$364$	−33.1555	−1.73782
$365$	0	0
$366$	0	0
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	40.2285	2.09705
$369$	0	0
$370$	0	0
$371$	−10.7276	−0.556949
$372$	0	0
$373$	−25.0000	−1.29445	−0.647225	−	0.762299i	$-0.724071\pi$
$373$	−25.0000	−1.29445	−0.647225	+	0.762299i	$0.724071\pi$
$374$	46.0093	2.37908
$375$	0	0
$376$	−100.696	−5.19301
$377$	13.8265	0.712102
$378$	0	0
$379$	−11.3852	−0.584817	−0.292408	−	0.956294i	$-0.594457\pi$
$379$	−11.3852	−0.584817	−0.292408	+	0.956294i	$0.594457\pi$
$380$	0	0
$381$	0	0
$382$	17.1555	0.877751
$383$	−9.69462	−0.495372	−0.247686	−	0.968840i	$-0.579670\pi$
$383$	−9.69462	−0.495372	−0.247686	+	0.968840i	$0.579670\pi$
$384$	0	0
$385$	0	0
$386$	−2.68190	−0.136505
$387$	0	0
$388$	−55.9258	−2.83920
$389$	10.6281	0.538868	0.269434	−	0.963019i	$-0.413163\pi$
$389$	10.6281	0.538868	0.269434	+	0.963019i	$0.413163\pi$
$390$	0	0
$391$	−17.1555	−0.867591
$392$	−8.56218	−0.432456
$393$	0	0
$394$	−63.3481	−3.19143
$395$	0	0
$396$	0	0
$397$	−1.22967	−0.0617154	−0.0308577	−	0.999524i	$-0.509824\pi$
$397$	−1.22967	−0.0617154	−0.0308577	+	0.999524i	$0.509824\pi$
$398$	−2.06595	−0.103557
$399$	0	0
$400$	0	0
$401$	12.8930	0.643846	0.321923	−	0.946766i	$-0.395671\pi$
$401$	12.8930	0.643846	0.321923	+	0.946766i	$0.395671\pi$
$402$	0	0
$403$	38.3110	1.90841
$404$	27.8520	1.38569
$405$	0	0
$406$	5.80742	0.288217
$407$	9.59516	0.475614
$408$	0	0
$409$	22.3852	1.10688	0.553438	−	0.832891i	$-0.313316\pi$
$409$	22.3852	1.10688	0.553438	+	0.832891i	$0.313316\pi$
$410$	0	0
$411$	0	0
$412$	29.1555	1.43639
$413$	−11.7606	−0.578700
$414$	0	0
$415$	0	0
$416$	−106.044	−5.19924
$417$	0	0
$418$	20.4593	1.00070
$419$	−17.1244	−0.836580	−0.418290	−	0.908314i	$-0.637370\pi$
$419$	−17.1244	−0.836580	−0.418290	+	0.908314i	$0.637370\pi$
$420$	0	0
$421$	25.3110	1.23358	0.616791	−	0.787127i	$-0.288432\pi$
$421$	25.3110	1.23358	0.616791	+	0.787127i	$0.288432\pi$
$422$	0	0
$423$	0	0
$424$	−91.8516	−4.46071
$425$	0	0
$426$	0	0
$427$	0	0
$428$	22.4882	1.08701
$429$	0	0
$430$	0	0
$431$	−25.7860	−1.24207	−0.621034	−	0.783783i	$-0.713287\pi$
$431$	−25.7860	−1.24207	−0.621034	+	0.783783i	$0.713287\pi$
$432$	0	0
$433$	−9.54066	−0.458495	−0.229247	−	0.973368i	$-0.573626\pi$
$433$	−9.54066	−0.458495	−0.229247	+	0.973368i	$0.573626\pi$
$434$	16.0914	0.772412
$435$	0	0
$436$	36.3481	1.74076
$437$	−7.62868	−0.364929
$438$	0	0
$439$	19.1555	0.914242	0.457121	−	0.889405i	$-0.348881\pi$
$439$	19.1555	0.914242	0.457121	+	0.889405i	$0.348881\pi$
$440$	0	0
$441$	0	0
$442$	91.8516	4.36894
$443$	19.1903	0.911759	0.455880	−	0.890041i	$-0.349325\pi$
$443$	19.1903	0.911759	0.455880	+	0.890041i	$0.349325\pi$
$444$	0	0
$445$	0	0
$446$	−9.69462	−0.459054
$447$	0	0
$448$	−19.3852	−0.915863
$449$	12.8930	0.608459	0.304229	−	0.952599i	$-0.401601\pi$
$449$	12.8930	0.608459	0.304229	+	0.952599i	$0.401601\pi$
$450$	0	0
$451$	−34.3110	−1.61564
$452$	−11.2441	−0.528877
$453$	0	0
$454$	31.5407	1.48028
$455$	0	0
$456$	0	0
$457$	−28.5407	−1.33508	−0.667538	−	0.744576i	$-0.732652\pi$
$457$	−28.5407	−1.33508	−0.667538	+	0.744576i	$0.732652\pi$
$458$	41.6785	1.94751
$459$	0	0
$460$	0	0
$461$	−1.03297	−0.0481104	−0.0240552	−	0.999711i	$-0.507658\pi$
$461$	−1.03297	−0.0481104	−0.0240552	+	0.999711i	$0.507658\pi$
$462$	0	0
$463$	−8.77033	−0.407592	−0.203796	−	0.979013i	$-0.565328\pi$
$463$	−8.77033	−0.407592	−0.203796	+	0.979013i	$0.565328\pi$
$464$	27.2360	1.26440
$465$	0	0
$466$	11.3481	0.525690
$467$	−6.39677	−0.296007	−0.148004	−	0.988987i	$-0.547285\pi$
$467$	−6.39677	−0.296007	−0.148004	+	0.988987i	$0.547285\pi$
$468$	0	0
$469$	9.38516	0.433367
$470$	0	0
$471$	0	0
$472$	−100.696	−4.63492
$473$	−30.0174	−1.38020
$474$	0	0
$475$	0	0
$476$	27.8520	1.27659
$477$	0	0
$478$	17.1555	0.784674
$479$	−33.2158	−1.51767	−0.758833	−	0.651285i	$-0.774230\pi$
$479$	−33.2158	−1.51767	−0.758833	+	0.651285i	$0.774230\pi$
$480$	0	0
$481$	19.1555	0.873415
$482$	−49.3072	−2.24588
$483$	0	0
$484$	−4.00000	−0.181818
$485$	0	0
$486$	0	0
$487$	−13.3852	−0.606540	−0.303270	−	0.952905i	$-0.598078\pi$
$487$	−13.3852	−0.606540	−0.303270	+	0.952905i	$0.598078\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−35.3812	−1.59673	−0.798365	−	0.602174i	$-0.794301\pi$
$491$	−35.3812	−1.59673	−0.798365	+	0.602174i	$0.794301\pi$
$492$	0	0
$493$	−11.6148	−0.523106
$494$	40.8444	1.83768
$495$	0	0
$496$	75.4665	3.38855
$497$	13.9260	0.624666
$498$	0	0
$499$	29.5407	1.32242	0.661211	−	0.750200i	$-0.270043\pi$
$499$	29.5407	1.32242	0.661211	+	0.750200i	$0.270043\pi$
$500$	0	0
$501$	0	0
$502$	8.84451	0.394750
$503$	−40.6455	−1.81229	−0.906147	−	0.422963i	$-0.860990\pi$
$503$	−40.6455	−1.81229	−0.906147	+	0.422963i	$0.860990\pi$
$504$	0	0
$505$	0	0
$506$	−27.4349	−1.21963
$507$	0	0
$508$	−23.9629	−1.06318
$509$	−9.69462	−0.429707	−0.214853	−	0.976646i	$-0.568927\pi$
$509$	−9.69462	−0.429707	−0.214853	+	0.976646i	$0.568927\pi$
$510$	0	0
$511$	−4.38516	−0.193988
$512$	6.49624	0.287096
$513$	0	0
$514$	45.9258	2.02570
$515$	0	0
$516$	0	0
$517$	37.6148	1.65430
$518$	8.04570	0.353508
$519$	0	0
$520$	0	0
$521$	36.5136	1.59969	0.799845	−	0.600206i	$-0.204915\pi$
$521$	36.5136	1.59969	0.799845	+	0.600206i	$0.204915\pi$
$522$	0	0
$523$	19.2297	0.840855	0.420427	−	0.907326i	$-0.361880\pi$
$523$	19.2297	0.840855	0.420427	+	0.907326i	$0.361880\pi$
$524$	27.8520	1.21672
$525$	0	0
$526$	54.5036	2.37647
$527$	−32.1828	−1.40190
$528$	0	0
$529$	−12.7703	−0.555232
$530$	0	0
$531$	0	0
$532$	12.3852	0.536965
$533$	−68.4975	−2.96695
$534$	0	0
$535$	0	0
$536$	80.3575	3.47092
$537$	0	0
$538$	−11.6148	−0.500751
$539$	3.19839	0.137764
$540$	0	0
$541$	−14.5407	−0.625152	−0.312576	−	0.949893i	$-0.601192\pi$
$541$	−14.5407	−0.625152	−0.312576	+	0.949893i	$0.601192\pi$
$542$	−40.6455	−1.74587
$543$	0	0
$544$	89.0813	3.81933
$545$	0	0
$546$	0	0
$547$	−12.6148	−0.539371	−0.269686	−	0.962948i	$-0.586920\pi$
$547$	−12.6148	−0.539371	−0.269686	+	0.962948i	$0.586920\pi$
$548$	55.7039	2.37955
$549$	0	0
$550$	0	0
$551$	−5.16487	−0.220031
$552$	0	0
$553$	5.38516	0.229001
$554$	−63.1337	−2.68229
$555$	0	0
$556$	−22.7703	−0.965677
$557$	8.56218	0.362791	0.181396	−	0.983410i	$-0.441939\pi$
$557$	8.56218	0.362791	0.181396	+	0.983410i	$0.441939\pi$
$558$	0	0
$559$	−59.9258	−2.53459
$560$	0	0
$561$	0	0
$562$	−40.1184	−1.69229
$563$	−4.33082	−0.182523	−0.0912613	−	0.995827i	$-0.529090\pi$
$563$	−4.33082	−0.182523	−0.0912613	+	0.995827i	$0.529090\pi$
$564$	0	0
$565$	0	0
$566$	−37.5466	−1.57820
$567$	0	0
$568$	119.237	5.00307
$569$	−32.0833	−1.34500	−0.672501	−	0.740096i	$-0.734780\pi$
$569$	−32.0833	−1.34500	−0.672501	+	0.740096i	$0.734780\pi$
$570$	0	0
$571$	35.3852	1.48082	0.740412	−	0.672154i	$-0.234631\pi$
$571$	35.3852	1.48082	0.740412	+	0.672154i	$0.234631\pi$
$572$	106.044	4.43392
$573$	0	0
$574$	−28.7703	−1.20085
$575$	0	0
$576$	0	0
$577$	−47.1555	−1.96311	−0.981554	−	0.191183i	$-0.938768\pi$
$577$	−47.1555	−1.96311	−0.981554	+	0.191183i	$0.938768\pi$
$578$	−31.5668	−1.31301
$579$	0	0
$580$	0	0
$581$	−4.33082	−0.179673
$582$	0	0
$583$	34.3110	1.42102
$584$	−37.5466	−1.55369
$585$	0	0
$586$	−31.5407	−1.30293
$587$	14.0254	0.578892	0.289446	−	0.957194i	$-0.406529\pi$
$587$	14.0254	0.578892	0.289446	+	0.957194i	$0.406529\pi$
$588$	0	0
$589$	−14.3110	−0.589674
$590$	0	0
$591$	0	0
$592$	37.7332	1.55083
$593$	−34.2487	−1.40643	−0.703213	−	0.710979i	$-0.748252\pi$
$593$	−34.2487	−1.40643	−0.703213	+	0.710979i	$0.748252\pi$
$594$	0	0
$595$	0	0
$596$	66.9480	2.74230
$597$	0	0
$598$	−54.7703	−2.23973
$599$	−26.7195	−1.09173	−0.545865	−	0.837873i	$-0.683799\pi$
$599$	−26.7195	−1.09173	−0.545865	+	0.837873i	$0.683799\pi$
$600$	0	0
$601$	41.5407	1.69448	0.847239	−	0.531211i	$-0.178263\pi$
$601$	41.5407	1.69448	0.847239	+	0.531211i	$0.178263\pi$
$602$	−25.1701	−1.02586
$603$	0	0
$604$	59.1184	2.40549
$605$	0	0
$606$	0	0
$607$	−23.1555	−0.939853	−0.469926	−	0.882706i	$-0.655720\pi$
$607$	−23.1555	−0.939853	−0.469926	+	0.882706i	$0.655720\pi$
$608$	39.6125	1.60650
$609$	0	0
$610$	0	0
$611$	75.0932	3.03794
$612$	0	0
$613$	−23.0000	−0.928961	−0.464481	−	0.885583i	$-0.653759\pi$
$613$	−23.0000	−0.928961	−0.464481	+	0.885583i	$0.653759\pi$
$614$	−35.2817	−1.42385
$615$	0	0
$616$	27.3852	1.10338
$617$	−10.6281	−0.427872	−0.213936	−	0.976848i	$-0.568628\pi$
$617$	−10.6281	−0.427872	−0.213936	+	0.976848i	$0.568628\pi$
$618$	0	0
$619$	−27.1555	−1.09147	−0.545736	−	0.837957i	$-0.683750\pi$
$619$	−27.1555	−1.09147	−0.545736	+	0.837957i	$0.683750\pi$
$620$	0	0
$621$	0	0
$622$	−71.9258	−2.88396
$623$	1.03297	0.0413852
$624$	0	0
$625$	0	0
$626$	64.1666	2.56461
$627$	0	0
$628$	87.0813	3.47492
$629$	−16.0914	−0.641606
$630$	0	0
$631$	−5.38516	−0.214380	−0.107190	−	0.994239i	$-0.534185\pi$
$631$	−5.38516	−0.214380	−0.107190	+	0.994239i	$0.534185\pi$
$632$	46.1088	1.83411
$633$	0	0
$634$	22.9629	0.911974
$635$	0	0
$636$	0	0
$637$	6.38516	0.252989
$638$	−18.5744	−0.735366
$639$	0	0
$640$	0	0
$641$	21.3557	0.843500	0.421750	−	0.906712i	$-0.361416\pi$
$641$	21.3557	0.843500	0.421750	+	0.906712i	$0.361416\pi$
$642$	0	0
$643$	6.45934	0.254732	0.127366	−	0.991856i	$-0.459348\pi$
$643$	6.45934	0.254732	0.127366	+	0.991856i	$0.459348\pi$
$644$	−16.6079	−0.654442
$645$	0	0
$646$	−34.3110	−1.34995
$647$	−38.5796	−1.51672	−0.758359	−	0.651837i	$-0.773999\pi$
$647$	−38.5796	−1.51672	−0.758359	+	0.651837i	$0.773999\pi$
$648$	0	0
$649$	37.6148	1.47651
$650$	0	0
$651$	0	0
$652$	87.0813	3.41037
$653$	10.7276	0.419803	0.209902	−	0.977723i	$-0.432686\pi$
$653$	10.7276	0.419803	0.209902	+	0.977723i	$0.432686\pi$
$654$	0	0
$655$	0	0
$656$	−134.929	−5.26809
$657$	0	0
$658$	31.5407	1.22958
$659$	−47.2412	−1.84026	−0.920128	−	0.391617i	$-0.871916\pi$
$659$	−47.2412	−1.84026	−0.920128	+	0.391617i	$0.871916\pi$
$660$	0	0
$661$	31.9258	1.24177	0.620885	−	0.783901i	$-0.286773\pi$
$661$	31.9258	1.24177	0.620885	+	0.783901i	$0.286773\pi$
$662$	−27.2360	−1.05856
$663$	0	0
$664$	−37.0813	−1.43903
$665$	0	0
$666$	0	0
$667$	6.92582	0.268169
$668$	5.36380	0.207532
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	3.54066	0.136482	0.0682412	−	0.997669i	$-0.478261\pi$
$673$	3.54066	0.136482	0.0682412	+	0.997669i	$0.478261\pi$
$674$	28.8849	1.11261
$675$	0	0
$676$	144.200	5.54614
$677$	6.39677	0.245848	0.122924	−	0.992416i	$-0.460773\pi$
$677$	6.39677	0.245848	0.122924	+	0.992416i	$0.460773\pi$
$678$	0	0
$679$	10.7703	0.413327
$680$	0	0
$681$	0	0
$682$	−51.4665	−1.97075
$683$	9.59516	0.367148	0.183574	−	0.983006i	$-0.441233\pi$
$683$	9.59516	0.367148	0.183574	+	0.983006i	$0.441233\pi$
$684$	0	0
$685$	0	0
$686$	2.68190	0.102395
$687$	0	0
$688$	−118.044	−4.50039
$689$	68.4975	2.60955
$690$	0	0
$691$	−15.2297	−0.579364	−0.289682	−	0.957123i	$-0.593550\pi$
$691$	−15.2297	−0.579364	−0.289682	+	0.957123i	$0.593550\pi$
$692$	116.772	4.43899
$693$	0	0
$694$	66.1184	2.50982
$695$	0	0
$696$	0	0
$697$	57.5407	2.17951
$698$	−6.39677	−0.242121
$699$	0	0
$700$	0	0
$701$	−44.9763	−1.69873	−0.849366	−	0.527804i	$-0.823016\pi$
$701$	−44.9763	−1.69873	−0.849366	+	0.527804i	$0.823016\pi$
$702$	0	0
$703$	−7.15549	−0.269875
$704$	62.0012	2.33676
$705$	0	0
$706$	−45.9258	−1.72844
$707$	−5.36380	−0.201726
$708$	0	0
$709$	39.5407	1.48498	0.742490	−	0.669857i	$-0.233645\pi$
$709$	39.5407	1.48498	0.742490	+	0.669857i	$0.233645\pi$
$710$	0	0
$711$	0	0
$712$	8.84451	0.331462
$713$	19.1903	0.718683
$714$	0	0
$715$	0	0
$716$	−22.4882	−0.840422
$717$	0	0
$718$	−48.9629	−1.82728
$719$	−19.3892	−0.723097	−0.361548	−	0.932353i	$-0.617752\pi$
$719$	−19.3892	−0.723097	−0.361548	+	0.932353i	$0.617752\pi$
$720$	0	0
$721$	−5.61484	−0.209107
$722$	35.6987	1.32857
$723$	0	0
$724$	105.852	3.93395
$725$	0	0
$726$	0	0
$727$	32.3852	1.20110	0.600550	−	0.799587i	$-0.294948\pi$
$727$	32.3852	1.20110	0.600550	+	0.799587i	$0.294948\pi$
$728$	54.6710	2.02624
$729$	0	0
$730$	0	0
$731$	50.3401	1.86190
$732$	0	0
$733$	−28.0000	−1.03420	−0.517102	−	0.855924i	$-0.672989\pi$
$733$	−28.0000	−1.03420	−0.517102	+	0.855924i	$0.672989\pi$
$734$	10.7276	0.395963
$735$	0	0
$736$	−53.1184	−1.95797
$737$	−30.0174	−1.10570
$738$	0	0
$739$	−43.3852	−1.59595	−0.797975	−	0.602691i	$-0.794095\pi$
$739$	−43.3852	−1.59595	−0.797975	+	0.602691i	$0.794095\pi$
$740$	0	0
$741$	0	0
$742$	28.7703	1.05619
$743$	−15.0584	−0.552440	−0.276220	−	0.961094i	$-0.589082\pi$
$743$	−15.0584	−0.552440	−0.276220	+	0.961094i	$0.589082\pi$
$744$	0	0
$745$	0	0
$746$	67.0475	2.45478
$747$	0	0
$748$	−89.0813	−3.25714
$749$	−4.33082	−0.158245
$750$	0	0
$751$	−28.7703	−1.04984	−0.524922	−	0.851150i	$-0.675906\pi$
$751$	−28.7703	−1.04984	−0.524922	+	0.851150i	$0.675906\pi$
$752$	147.921	5.39414
$753$	0	0
$754$	−37.0813	−1.35042
$755$	0	0
$756$	0	0
$757$	−11.3110	−0.411105	−0.205552	−	0.978646i	$-0.565899\pi$
$757$	−11.3110	−0.411105	−0.205552	+	0.978646i	$0.565899\pi$
$758$	30.5339	1.10904
$759$	0	0
$760$	0	0
$761$	−12.7935	−0.463766	−0.231883	−	0.972744i	$-0.574489\pi$
$761$	−12.7935	−0.463766	−0.231883	+	0.972744i	$0.574489\pi$
$762$	0	0
$763$	−7.00000	−0.253417
$764$	−33.2158	−1.20170
$765$	0	0
$766$	26.0000	0.939418
$767$	75.0932	2.71146
$768$	0	0
$769$	44.7703	1.61446	0.807230	−	0.590237i	$-0.200966\pi$
$769$	44.7703	1.61446	0.807230	+	0.590237i	$0.200966\pi$
$770$	0	0
$771$	0	0
$772$	5.19258	0.186885
$773$	15.0584	0.541614	0.270807	−	0.962634i	$-0.412710\pi$
$773$	15.0584	0.541614	0.270807	+	0.962634i	$0.412710\pi$
$774$	0	0
$775$	0	0
$776$	92.2175	3.31042
$777$	0	0
$778$	−28.5036	−1.02190
$779$	25.5871	0.916752
$780$	0	0
$781$	−44.5407	−1.59379
$782$	46.0093	1.64529
$783$	0	0
$784$	12.5777	0.449205
$785$	0	0
$786$	0	0
$787$	−0.385165	−0.0137296	−0.00686482	−	0.999976i	$-0.502185\pi$
$787$	−0.385165	−0.0137296	−0.00686482	+	0.999976i	$0.502185\pi$
$788$	122.652	4.36929
$789$	0	0
$790$	0	0
$791$	2.16541	0.0769932
$792$	0	0
$793$	0	0
$794$	3.29785	0.117036
$795$	0	0
$796$	4.00000	0.141776
$797$	−30.9509	−1.09634	−0.548168	−	0.836368i	$-0.684675\pi$
$797$	−30.9509	−1.09634	−0.548168	+	0.836368i	$0.684675\pi$
$798$	0	0
$799$	−63.0813	−2.23166
$800$	0	0
$801$	0	0
$802$	−34.5777	−1.22098
$803$	14.0254	0.494947
$804$	0	0
$805$	0	0
$806$	−102.746	−3.61908
$807$	0	0
$808$	−45.9258	−1.61566
$809$	−36.4141	−1.28025	−0.640127	−	0.768269i	$-0.721118\pi$
$809$	−36.4141	−1.28025	−0.640127	+	0.768269i	$0.721118\pi$
$810$	0	0
$811$	30.7703	1.08049	0.540246	−	0.841507i	$-0.318331\pi$
$811$	30.7703	1.08049	0.540246	+	0.841507i	$0.318331\pi$
$812$	−11.2441	−0.394590
$813$	0	0
$814$	−25.7332	−0.901950
$815$	0	0
$816$	0	0
$817$	22.3852	0.783158
$818$	−60.0348	−2.09907
$819$	0	0
$820$	0	0
$821$	47.0423	1.64179	0.820893	−	0.571081i	$-0.193476\pi$
$821$	47.0423	1.64179	0.820893	+	0.571081i	$0.193476\pi$
$822$	0	0
$823$	24.9258	0.868860	0.434430	−	0.900706i	$-0.356950\pi$
$823$	24.9258	0.868860	0.434430	+	0.900706i	$0.356950\pi$
$824$	−48.0753	−1.67478
$825$	0	0
$826$	31.5407	1.09744
$827$	13.9260	0.484254	0.242127	−	0.970245i	$-0.422155\pi$
$827$	13.9260	0.484254	0.242127	+	0.970245i	$0.422155\pi$
$828$	0	0
$829$	−8.38516	−0.291229	−0.145614	−	0.989341i	$-0.546516\pi$
$829$	−8.38516	−0.291229	−0.145614	+	0.989341i	$0.546516\pi$
$830$	0	0
$831$	0	0
$832$	123.777	4.29121
$833$	−5.36380	−0.185845
$834$	0	0
$835$	0	0
$836$	−39.6125	−1.37003
$837$	0	0
$838$	45.9258	1.58648
$839$	14.0254	0.484212	0.242106	−	0.970250i	$-0.422162\pi$
$839$	14.0254	0.484212	0.242106	+	0.970250i	$0.422162\pi$
$840$	0	0
$841$	−24.3110	−0.838310
$842$	−67.8815	−2.33935
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.770330	0.0264688
$848$	134.929	4.63348
$849$	0	0
$850$	0	0
$851$	9.59516	0.328918
$852$	0	0
$853$	30.3110	1.03783	0.518914	−	0.854826i	$-0.326336\pi$
$853$	30.3110	1.03783	0.518914	+	0.854826i	$0.326336\pi$
$854$	0	0
$855$	0	0
$856$	−37.0813	−1.26741
$857$	−36.3147	−1.24049	−0.620243	−	0.784410i	$-0.712966\pi$
$857$	−36.3147	−1.24049	−0.620243	+	0.784410i	$0.712966\pi$
$858$	0	0
$859$	−27.5407	−0.939675	−0.469838	−	0.882753i	$-0.655688\pi$
$859$	−27.5407	−0.939675	−0.469838	+	0.882753i	$0.655688\pi$
$860$	0	0
$861$	0	0
$862$	69.1555	2.35545
$863$	1.13244	0.0385487	0.0192743	−	0.999814i	$-0.493864\pi$
$863$	1.13244	0.0385487	0.0192743	+	0.999814i	$0.493864\pi$
$864$	0	0
$865$	0	0
$866$	25.5871	0.869485
$867$	0	0
$868$	−31.1555	−1.05749
$869$	−17.2238	−0.584279
$870$	0	0
$871$	−59.9258	−2.03051
$872$	−59.9353	−2.02966
$873$	0	0
$874$	20.4593	0.692048
$875$	0	0
$876$	0	0
$877$	19.5407	0.659841	0.329921	−	0.944009i	$-0.392978\pi$
$877$	19.5407	0.659841	0.329921	+	0.944009i	$0.392978\pi$
$878$	−51.3731	−1.73376
$879$	0	0
$880$	0	0
$881$	−40.6455	−1.36938	−0.684691	−	0.728834i	$-0.740063\pi$
$881$	−40.6455	−1.36938	−0.684691	+	0.728834i	$0.740063\pi$
$882$	0	0
$883$	8.61484	0.289912	0.144956	−	0.989438i	$-0.453696\pi$
$883$	8.61484	0.289912	0.144956	+	0.989438i	$0.453696\pi$
$884$	−177.839	−5.98139
$885$	0	0
$886$	−51.4665	−1.72905
$887$	−10.7276	−0.360197	−0.180099	−	0.983649i	$-0.557642\pi$
$887$	−10.7276	−0.360197	−0.180099	+	0.983649i	$0.557642\pi$
$888$	0	0
$889$	4.61484	0.154777
$890$	0	0
$891$	0	0
$892$	18.7703	0.628477
$893$	−28.0509	−0.938687
$894$	0	0
$895$	0	0
$896$	18.7733	0.627172
$897$	0	0
$898$	−34.5777	−1.15387
$899$	12.9925	0.433323
$900$	0	0
$901$	−57.5407	−1.91696
$902$	92.0186	3.06388
$903$	0	0
$904$	18.5407	0.616653
$905$	0	0
$906$	0	0
$907$	−22.3110	−0.740824	−0.370412	−	0.928868i	$-0.620784\pi$
$907$	−22.3110	−0.740824	−0.370412	+	0.928868i	$0.620784\pi$
$908$	−61.0677	−2.02660
$909$	0	0
$910$	0	0
$911$	5.46326	0.181006	0.0905030	−	0.995896i	$-0.471153\pi$
$911$	5.46326	0.181006	0.0905030	+	0.995896i	$0.471153\pi$
$912$	0	0
$913$	13.8516	0.458423
$914$	76.5432	2.53182
$915$	0	0
$916$	−80.6962	−2.66628
$917$	−5.36380	−0.177128
$918$	0	0
$919$	−49.6962	−1.63932	−0.819662	−	0.572847i	$-0.805839\pi$
$919$	−49.6962	−1.63932	−0.819662	+	0.572847i	$0.805839\pi$
$920$	0	0
$921$	0	0
$922$	2.77033	0.0912359
$923$	−88.9197	−2.92683
$924$	0	0
$925$	0	0
$926$	23.5211	0.772953
$927$	0	0
$928$	−35.9629	−1.18054
$929$	−16.0914	−0.527942	−0.263971	−	0.964531i	$-0.585032\pi$
$929$	−16.0914	−0.527942	−0.263971	+	0.964531i	$0.585032\pi$
$930$	0	0
$931$	−2.38516	−0.0781706
$932$	−21.9717	−0.719706
$933$	0	0
$934$	17.1555	0.561345
$935$	0	0
$936$	0	0
$937$	19.2297	0.628206	0.314103	−	0.949389i	$-0.398296\pi$
$937$	19.2297	0.628206	0.314103	+	0.949389i	$0.398296\pi$
$938$	−25.1701	−0.821832
$939$	0	0
$940$	0	0
$941$	49.3072	1.60737	0.803684	−	0.595057i	$-0.202870\pi$
$941$	49.3072	1.60737	0.803684	+	0.595057i	$0.202870\pi$
$942$	0	0
$943$	−34.3110	−1.11732
$944$	147.921	4.81443
$945$	0	0
$946$	80.5036	2.61740
$947$	42.7115	1.38794	0.693968	−	0.720006i	$-0.255861\pi$
$947$	42.7115	1.38794	0.693968	+	0.720006i	$0.255861\pi$
$948$	0	0
$949$	28.0000	0.908918
$950$	0	0
$951$	0	0
$952$	−45.9258	−1.48846
$953$	−0.0994662	−0.00322203	−0.00161101	−	0.999999i	$-0.500513\pi$
$953$	−0.0994662	−0.00322203	−0.00161101	+	0.999999i	$0.500513\pi$
$954$	0	0
$955$	0	0
$956$	−33.2158	−1.07427
$957$	0	0
$958$	89.0813	2.87809
$959$	−10.7276	−0.346412
$960$	0	0
$961$	5.00000	0.161290
$962$	−51.3731	−1.65634
$963$	0	0
$964$	95.4665	3.07477
$965$	0	0
$966$	0	0
$967$	−37.5407	−1.20723	−0.603613	−	0.797277i	$-0.706273\pi$
$967$	−37.5407	−1.20723	−0.603613	+	0.797277i	$0.706273\pi$
$968$	6.59570	0.211994
$969$	0	0
$970$	0	0
$971$	3.09892	0.0994491	0.0497245	−	0.998763i	$-0.484166\pi$
$971$	3.09892	0.0994491	0.0497245	+	0.998763i	$0.484166\pi$
$972$	0	0
$973$	4.38516	0.140582
$974$	35.8977	1.15024
$975$	0	0
$976$	0	0
$977$	34.1493	1.09253	0.546266	−	0.837612i	$-0.316049\pi$
$977$	34.1493	1.09253	0.546266	+	0.837612i	$0.316049\pi$
$978$	0	0
$979$	−3.30385	−0.105591
$980$	0	0
$981$	0	0
$982$	94.8887	3.02802
$983$	36.5136	1.16460	0.582302	−	0.812973i	$-0.302152\pi$
$983$	36.5136	1.16460	0.582302	+	0.812973i	$0.302152\pi$
$984$	0	0
$985$	0	0
$986$	31.1498	0.992012
$987$	0	0
$988$	−79.0813	−2.51591
$989$	−30.0174	−0.954497
$990$	0	0
$991$	39.6962	1.26099	0.630495	−	0.776193i	$-0.282852\pi$
$991$	39.6962	1.26099	0.630495	+	0.776193i	$0.282852\pi$
$992$	−99.6473	−3.16380
$993$	0	0
$994$	−37.3481	−1.18461
$995$	0	0
$996$	0	0
$997$	24.8445	0.786833	0.393417	−	0.919360i	$-0.371293\pi$
$997$	24.8445	0.786833	0.393417	+	0.919360i	$0.371293\pi$
$998$	−79.2251	−2.50783
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.a.y.1.1	✓	4
3.2	odd	2	inner	1575.2.a.y.1.4	yes	4
5.2	odd	4		1575.2.d.l.1324.1		8
5.3	odd	4		1575.2.d.l.1324.8		8
5.4	even	2		1575.2.a.z.1.4	yes	4
15.2	even	4		1575.2.d.l.1324.7		8
15.8	even	4		1575.2.d.l.1324.2		8
15.14	odd	2		1575.2.a.z.1.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1575.2.a.y.1.1	✓	4	1.1	even	1	trivial
1575.2.a.y.1.4	yes	4	3.2	odd	2	inner
1575.2.a.z.1.1	yes	4	15.14	odd	2
1575.2.a.z.1.4	yes	4	5.4	even	2
1575.2.d.l.1324.1		8	5.2	odd	4
1575.2.d.l.1324.2		8	15.8	even	4
1575.2.d.l.1324.7		8	15.2	even	4
1575.2.d.l.1324.8		8	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q-2.68190 q^{2} +5.19258 q^{4} -1.00000 q^{7} -8.56218 q^{8} +O(q^{10})\)	\(q-2.68190 q^{2} +5.19258 q^{4} -1.00000 q^{7} -8.56218 q^{8} +3.19839 q^{11} +6.38516 q^{13} +2.68190 q^{14} +12.5777 q^{16} -5.36380 q^{17} -2.38516 q^{19} -8.57775 q^{22} +3.19839 q^{23} -17.1244 q^{26} -5.19258 q^{28} +2.16541 q^{29} +6.00000 q^{31} -16.6079 q^{32} +14.3852 q^{34} +3.00000 q^{37} +6.39677 q^{38} -10.7276 q^{41} -9.38516 q^{43} +16.6079 q^{44} -8.57775 q^{46} +11.7606 q^{47} +1.00000 q^{49} +33.1555 q^{52} +10.7276 q^{53} +8.56218 q^{56} -5.80742 q^{58} +11.7606 q^{59} -16.0914 q^{62} +19.3852 q^{64} -9.38516 q^{67} -27.8520 q^{68} -13.9260 q^{71} +4.38516 q^{73} -8.04570 q^{74} -12.3852 q^{76} -3.19839 q^{77} -5.38516 q^{79} +28.7703 q^{82} +4.33082 q^{83} +25.1701 q^{86} -27.3852 q^{88} -1.03297 q^{89} -6.38516 q^{91} +16.6079 q^{92} -31.5407 q^{94} -10.7703 q^{97} -2.68190 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 10 q^{4} - 4 q^{7}+O(q^{10}) \) 4 * q + 10 * q^4 - 4 * q^7	\( 4 q + 10 q^{4} - 4 q^{7} + 4 q^{13} + 18 q^{16} + 12 q^{19} - 2 q^{22} - 10 q^{28} + 24 q^{31} + 36 q^{34} + 12 q^{37} - 16 q^{43} - 2 q^{46} + 4 q^{49} + 68 q^{52} - 34 q^{58} + 56 q^{64} - 16 q^{67} - 4 q^{73} - 28 q^{76} + 72 q^{82} - 88 q^{88} - 4 q^{91} - 40 q^{94}+O(q^{100}) \) 4 * q + 10 * q^4 - 4 * q^7 + 4 * q^13 + 18 * q^16 + 12 * q^19 - 2 * q^22 - 10 * q^28 + 24 * q^31 + 36 * q^34 + 12 * q^37 - 16 * q^43 - 2 * q^46 + 4 * q^49 + 68 * q^52 - 34 * q^58 + 56 * q^64 - 16 * q^67 - 4 * q^73 - 28 * q^76 + 72 * q^82 - 88 * q^88 - 4 * q^91 - 40 * q^94

Introduction

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