LMFDB - Embedded newform 4600.2.a.bk.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(1,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.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:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 13x^{6} + 38x^{5} + 41x^{4} - 123x^{3} + 15x^{2} + 32x - 8 $ x^8 - 3x^7 - 13x^6 + 38x^5 + 41x^4 - 123x^3 + 15x^2 + 32*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.540724$ of defining polynomial
Character	$\chi$	$=$	4600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.540724 q^{3} +1.15693 q^{7} -2.70762 q^{9} +O(q^{10})$	$q-0.540724 q^{3} +1.15693 q^{7} -2.70762 q^{9} +2.10688 q^{11} +5.59296 q^{13} +0.244199 q^{17} +1.45043 q^{19} -0.625580 q^{21} -1.00000 q^{23} +3.08625 q^{27} +4.29653 q^{29} +3.19717 q^{31} -1.13924 q^{33} -0.807940 q^{37} -3.02425 q^{39} -1.59546 q^{41} -5.98219 q^{43} -0.624940 q^{47} -5.66151 q^{49} -0.132044 q^{51} -0.536087 q^{53} -0.784283 q^{57} -1.03806 q^{59} +3.77846 q^{61} -3.13253 q^{63} -4.61571 q^{67} +0.540724 q^{69} +6.77489 q^{71} -8.87290 q^{73} +2.43751 q^{77} +14.8521 q^{79} +6.45405 q^{81} +6.11896 q^{83} -2.32323 q^{87} +5.79942 q^{89} +6.47066 q^{91} -1.72879 q^{93} -2.38582 q^{97} -5.70462 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{3} + 7 q^{7} + 11 q^{9}+O(q^{10}) $ 8 * q + 3 * q^3 + 7 * q^7 + 11 * q^9	$ 8 q + 3 q^{3} + 7 q^{7} + 11 q^{9} + 7 q^{11} - 11 q^{13} + 7 q^{17} + 11 q^{19} - 8 q^{23} + 12 q^{27} + 22 q^{29} + 9 q^{31} + 9 q^{33} - 4 q^{37} + 7 q^{41} + 22 q^{43} + 4 q^{47} + 39 q^{49} - 19 q^{51} - 4 q^{53} + 32 q^{59} + 17 q^{61} + 44 q^{63} - 4 q^{67} - 3 q^{69} + 15 q^{71} - 6 q^{73} - 18 q^{77} - 2 q^{79} + 24 q^{81} + 36 q^{83} - 4 q^{87} + 46 q^{89} - 35 q^{91} - 20 q^{93} - 3 q^{97} + 61 q^{99}+O(q^{100}) $ 8 * q + 3 * q^3 + 7 * q^7 + 11 * q^9 + 7 * q^11 - 11 * q^13 + 7 * q^17 + 11 * q^19 - 8 * q^23 + 12 * q^27 + 22 * q^29 + 9 * q^31 + 9 * q^33 - 4 * q^37 + 7 * q^41 + 22 * q^43 + 4 * q^47 + 39 * q^49 - 19 * q^51 - 4 * q^53 + 32 * q^59 + 17 * q^61 + 44 * q^63 - 4 * q^67 - 3 * q^69 + 15 * q^71 - 6 * q^73 - 18 * q^77 - 2 * q^79 + 24 * q^81 + 36 * q^83 - 4 * q^87 + 46 * q^89 - 35 * q^91 - 20 * q^93 - 3 * q^97 + 61 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−0.540724	−0.312187	−0.156094	−	0.987742i	$-0.549890\pi$
$3$	−0.540724	−0.312187	−0.156094	+	0.987742i	$0.549890\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.15693	0.437279	0.218639	−	0.975806i	$-0.429838\pi$
$7$	1.15693	0.437279	0.218639	+	0.975806i	$0.429838\pi$
$8$	0	0
$9$	−2.70762	−0.902539
$10$	0	0
$11$	2.10688	0.635247	0.317624	−	0.948217i	$-0.397115\pi$
$11$	2.10688	0.635247	0.317624	+	0.948217i	$0.397115\pi$
$12$	0	0
$13$	5.59296	1.55121	0.775604	−	0.631220i	$-0.217446\pi$
$13$	5.59296	1.55121	0.775604	+	0.631220i	$0.217446\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.244199	0.0592268	0.0296134	−	0.999561i	$-0.490572\pi$
$17$	0.244199	0.0592268	0.0296134	+	0.999561i	$0.490572\pi$
$18$	0	0
$19$	1.45043	0.332752	0.166376	−	0.986062i	$-0.446794\pi$
$19$	1.45043	0.332752	0.166376	+	0.986062i	$0.446794\pi$
$20$	0	0
$21$	−0.625580	−0.136513
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3.08625	0.593948
$28$	0	0
$29$	4.29653	0.797845	0.398922	−	0.916985i	$-0.369384\pi$
$29$	4.29653	0.797845	0.398922	+	0.916985i	$0.369384\pi$
$30$	0	0
$31$	3.19717	0.574229	0.287114	−	0.957896i	$-0.407304\pi$
$31$	3.19717	0.574229	0.287114	+	0.957896i	$0.407304\pi$
$32$	0	0
$33$	−1.13924	−0.198316
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.807940	−0.132824	−0.0664122	−	0.997792i	$-0.521155\pi$
$37$	−0.807940	−0.132824	−0.0664122	+	0.997792i	$0.521155\pi$
$38$	0	0
$39$	−3.02425	−0.484267
$40$	0	0
$41$	−1.59546	−0.249169	−0.124585	−	0.992209i	$-0.539760\pi$
$41$	−1.59546	−0.249169	−0.124585	+	0.992209i	$0.539760\pi$
$42$	0	0
$43$	−5.98219	−0.912275	−0.456138	−	0.889909i	$-0.650768\pi$
$43$	−5.98219	−0.912275	−0.456138	+	0.889909i	$0.650768\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.624940	−0.0911569	−0.0455784	−	0.998961i	$-0.514513\pi$
$47$	−0.624940	−0.0911569	−0.0455784	+	0.998961i	$0.514513\pi$
$48$	0	0
$49$	−5.66151	−0.808787
$50$	0	0
$51$	−0.132044	−0.0184899
$52$	0	0
$53$	−0.536087	−0.0736372	−0.0368186	−	0.999322i	$-0.511722\pi$
$53$	−0.536087	−0.0736372	−0.0368186	+	0.999322i	$0.511722\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.784283	−0.103881
$58$	0	0
$59$	−1.03806	−0.135144	−0.0675721	−	0.997714i	$-0.521525\pi$
$59$	−1.03806	−0.135144	−0.0675721	+	0.997714i	$0.521525\pi$
$60$	0	0
$61$	3.77846	0.483783	0.241891	−	0.970303i	$-0.422232\pi$
$61$	3.77846	0.483783	0.241891	+	0.970303i	$0.422232\pi$
$62$	0	0
$63$	−3.13253	−0.394661
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.61571	−0.563899	−0.281949	−	0.959429i	$-0.590981\pi$
$67$	−4.61571	−0.563899	−0.281949	+	0.959429i	$0.590981\pi$
$68$	0	0
$69$	0.540724	0.0650955
$70$	0	0
$71$	6.77489	0.804032	0.402016	−	0.915633i	$-0.368310\pi$
$71$	6.77489	0.804032	0.402016	+	0.915633i	$0.368310\pi$
$72$	0	0
$73$	−8.87290	−1.03849	−0.519247	−	0.854624i	$-0.673788\pi$
$73$	−8.87290	−1.03849	−0.519247	+	0.854624i	$0.673788\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.43751	0.277780
$78$	0	0
$79$	14.8521	1.67099	0.835494	−	0.549500i	$-0.185182\pi$
$79$	14.8521	1.67099	0.835494	+	0.549500i	$0.185182\pi$
$80$	0	0
$81$	6.45405	0.717116
$82$	0	0
$83$	6.11896	0.671644	0.335822	−	0.941926i	$-0.390986\pi$
$83$	6.11896	0.671644	0.335822	+	0.941926i	$0.390986\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.32323	−0.249077
$88$	0	0
$89$	5.79942	0.614737	0.307369	−	0.951591i	$-0.400552\pi$
$89$	5.79942	0.614737	0.307369	+	0.951591i	$0.400552\pi$
$90$	0	0
$91$	6.47066	0.678310
$92$	0	0
$93$	−1.72879	−0.179267
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.38582	−0.242244	−0.121122	−	0.992638i	$-0.538649\pi$
$97$	−2.38582	−0.242244	−0.121122	+	0.992638i	$0.538649\pi$
$98$	0	0
$99$	−5.70462	−0.573336
$100$	0	0
$101$	8.33379	0.829243	0.414621	−	0.909994i	$-0.363914\pi$
$101$	8.33379	0.829243	0.414621	+	0.909994i	$0.363914\pi$
$102$	0	0
$103$	−12.4250	−1.22427	−0.612135	−	0.790754i	$-0.709689\pi$
$103$	−12.4250	−1.22427	−0.612135	+	0.790754i	$0.709689\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.5804	1.50621	0.753106	−	0.657899i	$-0.228555\pi$
$107$	15.5804	1.50621	0.753106	+	0.657899i	$0.228555\pi$
$108$	0	0
$109$	9.71396	0.930428	0.465214	−	0.885198i	$-0.345977\pi$
$109$	9.71396	0.930428	0.465214	+	0.885198i	$0.345977\pi$
$110$	0	0
$111$	0.436872	0.0414661
$112$	0	0
$113$	−2.69724	−0.253735	−0.126867	−	0.991920i	$-0.540492\pi$
$113$	−2.69724	−0.253735	−0.126867	+	0.991920i	$0.540492\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−15.1436	−1.40003
$118$	0	0
$119$	0.282521	0.0258986
$120$	0	0
$121$	−6.56107	−0.596461
$122$	0	0
$123$	0.862705	0.0777875
$124$	0	0
$125$	0	0
$126$	0	0
$127$	4.40424	0.390813	0.195406	−	0.980722i	$-0.437397\pi$
$127$	4.40424	0.390813	0.195406	+	0.980722i	$0.437397\pi$
$128$	0	0
$129$	3.23471	0.284801
$130$	0	0
$131$	1.65815	0.144873	0.0724367	−	0.997373i	$-0.476922\pi$
$131$	1.65815	0.144873	0.0724367	+	0.997373i	$0.476922\pi$
$132$	0	0
$133$	1.67805	0.145505
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.70894	−0.231440	−0.115720	−	0.993282i	$-0.536918\pi$
$137$	−2.70894	−0.231440	−0.115720	+	0.993282i	$0.536918\pi$
$138$	0	0
$139$	8.99503	0.762949	0.381474	−	0.924379i	$-0.375416\pi$
$139$	8.99503	0.762949	0.381474	+	0.924379i	$0.375416\pi$
$140$	0	0
$141$	0.337920	0.0284580
$142$	0	0
$143$	11.7837	0.985400
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.06132	0.252493
$148$	0	0
$149$	9.54172	0.781688	0.390844	−	0.920457i	$-0.372183\pi$
$149$	9.54172	0.781688	0.390844	+	0.920457i	$0.372183\pi$
$150$	0	0
$151$	−12.1471	−0.988514	−0.494257	−	0.869316i	$-0.664560\pi$
$151$	−12.1471	−0.988514	−0.494257	+	0.869316i	$0.664560\pi$
$152$	0	0
$153$	−0.661196	−0.0534545
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−19.1756	−1.53038	−0.765189	−	0.643806i	$-0.777354\pi$
$157$	−19.1756	−1.53038	−0.765189	+	0.643806i	$0.777354\pi$
$158$	0	0
$159$	0.289875	0.0229886
$160$	0	0
$161$	−1.15693	−0.0911789
$162$	0	0
$163$	17.3397	1.35815	0.679073	−	0.734071i	$-0.262382\pi$
$163$	17.3397	1.35815	0.679073	+	0.734071i	$0.262382\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.582642	−0.0450862	−0.0225431	−	0.999746i	$-0.507176\pi$
$167$	−0.582642	−0.0450862	−0.0225431	+	0.999746i	$0.507176\pi$
$168$	0	0
$169$	18.2812	1.40624
$170$	0	0
$171$	−3.92721	−0.300321
$172$	0	0
$173$	12.0591	0.916840	0.458420	−	0.888736i	$-0.348416\pi$
$173$	12.0591	0.916840	0.458420	+	0.888736i	$0.348416\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0.561305	0.0421903
$178$	0	0
$179$	18.0142	1.34645	0.673223	−	0.739439i	$-0.264909\pi$
$179$	18.0142	1.34645	0.673223	+	0.739439i	$0.264909\pi$
$180$	0	0
$181$	3.33577	0.247946	0.123973	−	0.992286i	$-0.460436\pi$
$181$	3.33577	0.247946	0.123973	+	0.992286i	$0.460436\pi$
$182$	0	0
$183$	−2.04311	−0.151031
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.514496	0.0376237
$188$	0	0
$189$	3.57057	0.259721
$190$	0	0
$191$	−2.90013	−0.209846	−0.104923	−	0.994480i	$-0.533460\pi$
$191$	−2.90013	−0.209846	−0.104923	+	0.994480i	$0.533460\pi$
$192$	0	0
$193$	−11.6127	−0.835898	−0.417949	−	0.908470i	$-0.637251\pi$
$193$	−11.6127	−0.835898	−0.417949	+	0.908470i	$0.637251\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.156603	0.0111575	0.00557876	−	0.999984i	$-0.498224\pi$
$197$	0.156603	0.0111575	0.00557876	+	0.999984i	$0.498224\pi$
$198$	0	0
$199$	23.3474	1.65506	0.827528	−	0.561425i	$-0.189747\pi$
$199$	23.3474	1.65506	0.827528	+	0.561425i	$0.189747\pi$
$200$	0	0
$201$	2.49583	0.176042
$202$	0	0
$203$	4.97078	0.348880
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.70762	0.188192
$208$	0	0
$209$	3.05588	0.211380
$210$	0	0
$211$	−21.2620	−1.46374	−0.731870	−	0.681445i	$-0.761352\pi$
$211$	−21.2620	−1.46374	−0.731870	+	0.681445i	$0.761352\pi$
$212$	0	0
$213$	−3.66335	−0.251008
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.69890	0.251098
$218$	0	0
$219$	4.79779	0.324205
$220$	0	0
$221$	1.36579	0.0918731
$222$	0	0
$223$	28.7628	1.92610	0.963049	−	0.269328i	$-0.0868014\pi$
$223$	28.7628	1.92610	0.963049	+	0.269328i	$0.0868014\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6.08990	0.404201	0.202100	−	0.979365i	$-0.435223\pi$
$227$	6.08990	0.404201	0.202100	+	0.979365i	$0.435223\pi$
$228$	0	0
$229$	19.9294	1.31697	0.658486	−	0.752593i	$-0.271197\pi$
$229$	19.9294	1.31697	0.658486	+	0.752593i	$0.271197\pi$
$230$	0	0
$231$	−1.31802	−0.0867194
$232$	0	0
$233$	−9.63207	−0.631018	−0.315509	−	0.948923i	$-0.602175\pi$
$233$	−9.63207	−0.631018	−0.315509	+	0.948923i	$0.602175\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.03087	−0.521661
$238$	0	0
$239$	13.2583	0.857607	0.428803	−	0.903398i	$-0.358935\pi$
$239$	13.2583	0.857607	0.428803	+	0.903398i	$0.358935\pi$
$240$	0	0
$241$	−14.4250	−0.929195	−0.464597	−	0.885522i	$-0.653801\pi$
$241$	−14.4250	−0.929195	−0.464597	+	0.885522i	$0.653801\pi$
$242$	0	0
$243$	−12.7486	−0.817823
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.11220	0.516167
$248$	0	0
$249$	−3.30867	−0.209678
$250$	0	0
$251$	18.6789	1.17900	0.589500	−	0.807768i	$-0.299325\pi$
$251$	18.6789	1.17900	0.589500	+	0.807768i	$0.299325\pi$
$252$	0	0
$253$	−2.10688	−0.132458
$254$	0	0
$255$	0	0
$256$	0	0
$257$	25.5035	1.59087	0.795433	−	0.606042i	$-0.207244\pi$
$257$	25.5035	1.59087	0.795433	+	0.606042i	$0.207244\pi$
$258$	0	0
$259$	−0.934730	−0.0580813
$260$	0	0
$261$	−11.6333	−0.720086
$262$	0	0
$263$	18.4521	1.13780	0.568902	−	0.822405i	$-0.307368\pi$
$263$	18.4521	1.13780	0.568902	+	0.822405i	$0.307368\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3.13589	−0.191913
$268$	0	0
$269$	2.02723	0.123602	0.0618012	−	0.998088i	$-0.480316\pi$
$269$	2.02723	0.123602	0.0618012	+	0.998088i	$0.480316\pi$
$270$	0	0
$271$	26.7402	1.62435	0.812175	−	0.583414i	$-0.198284\pi$
$271$	26.7402	1.62435	0.812175	+	0.583414i	$0.198284\pi$
$272$	0	0
$273$	−3.49884	−0.211760
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−18.8601	−1.13320	−0.566598	−	0.823995i	$-0.691741\pi$
$277$	−18.8601	−1.13320	−0.566598	+	0.823995i	$0.691741\pi$
$278$	0	0
$279$	−8.65672	−0.518264
$280$	0	0
$281$	6.09240	0.363442	0.181721	−	0.983350i	$-0.441833\pi$
$281$	6.09240	0.363442	0.181721	+	0.983350i	$0.441833\pi$
$282$	0	0
$283$	15.6971	0.933095	0.466548	−	0.884496i	$-0.345498\pi$
$283$	15.6971	0.933095	0.466548	+	0.884496i	$0.345498\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.84584	−0.108956
$288$	0	0
$289$	−16.9404	−0.996492
$290$	0	0
$291$	1.29007	0.0756254
$292$	0	0
$293$	−20.3629	−1.18962	−0.594808	−	0.803868i	$-0.702772\pi$
$293$	−20.3629	−1.18962	−0.594808	+	0.803868i	$0.702772\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	6.50234	0.377304
$298$	0	0
$299$	−5.59296	−0.323449
$300$	0	0
$301$	−6.92098	−0.398918
$302$	0	0
$303$	−4.50628	−0.258879
$304$	0	0
$305$	0	0
$306$	0	0
$307$	6.31386	0.360351	0.180176	−	0.983634i	$-0.442333\pi$
$307$	6.31386	0.360351	0.180176	+	0.983634i	$0.442333\pi$
$308$	0	0
$309$	6.71848	0.382201
$310$	0	0
$311$	−2.26901	−0.128664	−0.0643318	−	0.997929i	$-0.520492\pi$
$311$	−2.26901	−0.128664	−0.0643318	+	0.997929i	$0.520492\pi$
$312$	0	0
$313$	−25.8886	−1.46331	−0.731654	−	0.681676i	$-0.761251\pi$
$313$	−25.8886	−1.46331	−0.731654	+	0.681676i	$0.761251\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	0.952783	0.0535136	0.0267568	−	0.999642i	$-0.491482\pi$
$317$	0.952783	0.0535136	0.0267568	+	0.999642i	$0.491482\pi$
$318$	0	0
$319$	9.05225	0.506829
$320$	0	0
$321$	−8.42468	−0.470220
$322$	0	0
$323$	0.354193	0.0197078
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.25257	−0.290468
$328$	0	0
$329$	−0.723012	−0.0398609
$330$	0	0
$331$	−14.7188	−0.809017	−0.404509	−	0.914534i	$-0.632557\pi$
$331$	−14.7188	−0.809017	−0.404509	+	0.914534i	$0.632557\pi$
$332$	0	0
$333$	2.18759	0.119879
$334$	0	0
$335$	0	0
$336$	0	0
$337$	23.7550	1.29402	0.647010	−	0.762482i	$-0.276019\pi$
$337$	23.7550	1.29402	0.647010	+	0.762482i	$0.276019\pi$
$338$	0	0
$339$	1.45846	0.0792127
$340$	0	0
$341$	6.73605	0.364777
$342$	0	0
$343$	−14.6485	−0.790944
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.8557	−1.11959	−0.559797	−	0.828630i	$-0.689121\pi$
$347$	−20.8557	−1.11959	−0.559797	+	0.828630i	$0.689121\pi$
$348$	0	0
$349$	−12.8328	−0.686922	−0.343461	−	0.939167i	$-0.611599\pi$
$349$	−12.8328	−0.686922	−0.343461	+	0.939167i	$0.611599\pi$
$350$	0	0
$351$	17.2612	0.921337
$352$	0	0
$353$	−30.4888	−1.62275	−0.811377	−	0.584523i	$-0.801282\pi$
$353$	−30.4888	−1.62275	−0.811377	+	0.584523i	$0.801282\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.152766	−0.00808522
$358$	0	0
$359$	11.0247	0.581864	0.290932	−	0.956744i	$-0.406035\pi$
$359$	11.0247	0.581864	0.290932	+	0.956744i	$0.406035\pi$
$360$	0	0
$361$	−16.8963	−0.889276
$362$	0	0
$363$	3.54773	0.186207
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−18.5124	−0.966340	−0.483170	−	0.875527i	$-0.660515\pi$
$367$	−18.5124	−0.966340	−0.483170	+	0.875527i	$0.660515\pi$
$368$	0	0
$369$	4.31990	0.224885
$370$	0	0
$371$	−0.620215	−0.0322000
$372$	0	0
$373$	26.4707	1.37060	0.685301	−	0.728260i	$-0.259670\pi$
$373$	26.4707	1.37060	0.685301	+	0.728260i	$0.259670\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.0303	1.23762
$378$	0	0
$379$	10.8785	0.558791	0.279395	−	0.960176i	$-0.409866\pi$
$379$	10.8785	0.558791	0.279395	+	0.960176i	$0.409866\pi$
$380$	0	0
$381$	−2.38148	−0.122007
$382$	0	0
$383$	21.6026	1.10384	0.551920	−	0.833897i	$-0.313895\pi$
$383$	21.6026	1.10384	0.551920	+	0.833897i	$0.313895\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.1975	0.823364
$388$	0	0
$389$	−29.4117	−1.49123	−0.745617	−	0.666374i	$-0.767845\pi$
$389$	−29.4117	−1.49123	−0.745617	+	0.666374i	$0.767845\pi$
$390$	0	0
$391$	−0.244199	−0.0123497
$392$	0	0
$393$	−0.896603	−0.0452276
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−33.2753	−1.67004	−0.835021	−	0.550219i	$-0.814544\pi$
$397$	−33.2753	−1.67004	−0.835021	+	0.550219i	$0.814544\pi$
$398$	0	0
$399$	−0.907360	−0.0454248
$400$	0	0
$401$	33.7280	1.68429	0.842147	−	0.539248i	$-0.181292\pi$
$401$	33.7280	1.68429	0.842147	+	0.539248i	$0.181292\pi$
$402$	0	0
$403$	17.8816	0.890748
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.70223	−0.0843764
$408$	0	0
$409$	22.6282	1.11889	0.559445	−	0.828867i	$-0.311014\pi$
$409$	22.6282	1.11889	0.559445	+	0.828867i	$0.311014\pi$
$410$	0	0
$411$	1.46479	0.0722527
$412$	0	0
$413$	−1.20097	−0.0590956
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−4.86383	−0.238183
$418$	0	0
$419$	−34.6846	−1.69446	−0.847228	−	0.531230i	$-0.821730\pi$
$419$	−34.6846	−1.69446	−0.847228	+	0.531230i	$0.821730\pi$
$420$	0	0
$421$	−0.348044	−0.0169626	−0.00848131	−	0.999964i	$-0.502700\pi$
$421$	−0.348044	−0.0169626	−0.00848131	+	0.999964i	$0.502700\pi$
$422$	0	0
$423$	1.69210	0.0822726
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.37142	0.211548
$428$	0	0
$429$	−6.37172	−0.307629
$430$	0	0
$431$	21.2559	1.02386	0.511930	−	0.859027i	$-0.328931\pi$
$431$	21.2559	1.02386	0.511930	+	0.859027i	$0.328931\pi$
$432$	0	0
$433$	−29.3363	−1.40981	−0.704906	−	0.709301i	$-0.749011\pi$
$433$	−29.3363	−1.40981	−0.704906	+	0.709301i	$0.749011\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.45043	−0.0693835
$438$	0	0
$439$	20.2210	0.965096	0.482548	−	0.875870i	$-0.339711\pi$
$439$	20.2210	0.965096	0.482548	+	0.875870i	$0.339711\pi$
$440$	0	0
$441$	15.3292	0.729962
$442$	0	0
$443$	33.8124	1.60648	0.803239	−	0.595657i	$-0.203108\pi$
$443$	33.8124	1.60648	0.803239	+	0.595657i	$0.203108\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−5.15944	−0.244033
$448$	0	0
$449$	17.7383	0.837120	0.418560	−	0.908189i	$-0.362535\pi$
$449$	17.7383	0.837120	0.418560	+	0.908189i	$0.362535\pi$
$450$	0	0
$451$	−3.36144	−0.158284
$452$	0	0
$453$	6.56821	0.308601
$454$	0	0
$455$	0	0
$456$	0	0
$457$	31.7861	1.48689	0.743445	−	0.668797i	$-0.233190\pi$
$457$	31.7861	1.48689	0.743445	+	0.668797i	$0.233190\pi$
$458$	0	0
$459$	0.753657	0.0351777
$460$	0	0
$461$	17.2206	0.802044	0.401022	−	0.916068i	$-0.368655\pi$
$461$	17.2206	0.802044	0.401022	+	0.916068i	$0.368655\pi$
$462$	0	0
$463$	40.0085	1.85935	0.929677	−	0.368376i	$-0.120086\pi$
$463$	40.0085	1.85935	0.929677	+	0.368376i	$0.120086\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11.9947	−0.555049	−0.277524	−	0.960719i	$-0.589514\pi$
$467$	−11.9947	−0.555049	−0.277524	+	0.960719i	$0.589514\pi$
$468$	0	0
$469$	−5.34006	−0.246581
$470$	0	0
$471$	10.3687	0.477764
$472$	0	0
$473$	−12.6037	−0.579520
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.45152	0.0664604
$478$	0	0
$479$	−20.8249	−0.951514	−0.475757	−	0.879577i	$-0.657826\pi$
$479$	−20.8249	−0.951514	−0.475757	+	0.879577i	$0.657826\pi$
$480$	0	0
$481$	−4.51877	−0.206038
$482$	0	0
$483$	0.625580	0.0284649
$484$	0	0
$485$	0	0
$486$	0	0
$487$	16.5640	0.750584	0.375292	−	0.926907i	$-0.377542\pi$
$487$	16.5640	0.750584	0.375292	+	0.926907i	$0.377542\pi$
$488$	0	0
$489$	−9.37597	−0.423996
$490$	0	0
$491$	28.8494	1.30196	0.650978	−	0.759097i	$-0.274359\pi$
$491$	28.8494	1.30196	0.650978	+	0.759097i	$0.274359\pi$
$492$	0	0
$493$	1.04921	0.0472538
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.83808	0.351586
$498$	0	0
$499$	−14.0289	−0.628021	−0.314011	−	0.949419i	$-0.601673\pi$
$499$	−14.0289	−0.628021	−0.314011	+	0.949419i	$0.601673\pi$
$500$	0	0
$501$	0.315048	0.0140753
$502$	0	0
$503$	−23.2157	−1.03514	−0.517568	−	0.855642i	$-0.673162\pi$
$503$	−23.2157	−1.03514	−0.517568	+	0.855642i	$0.673162\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−9.88507	−0.439011
$508$	0	0
$509$	−17.6290	−0.781394	−0.390697	−	0.920519i	$-0.627766\pi$
$509$	−17.6290	−0.781394	−0.390697	+	0.920519i	$0.627766\pi$
$510$	0	0
$511$	−10.2653	−0.454111
$512$	0	0
$513$	4.47638	0.197637
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.31667	−0.0579072
$518$	0	0
$519$	−6.52067	−0.286226
$520$	0	0
$521$	−1.64734	−0.0721711	−0.0360855	−	0.999349i	$-0.511489\pi$
$521$	−1.64734	−0.0721711	−0.0360855	+	0.999349i	$0.511489\pi$
$522$	0	0
$523$	−40.1998	−1.75781	−0.878907	−	0.476994i	$-0.841726\pi$
$523$	−40.1998	−1.75781	−0.878907	+	0.476994i	$0.841726\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.780744	0.0340098
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	2.81067	0.121973
$532$	0	0
$533$	−8.92336	−0.386513
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−9.74072	−0.420343
$538$	0	0
$539$	−11.9281	−0.513780
$540$	0	0
$541$	26.8419	1.15403	0.577013	−	0.816735i	$-0.304218\pi$
$541$	26.8419	1.15403	0.577013	+	0.816735i	$0.304218\pi$
$542$	0	0
$543$	−1.80373	−0.0774055
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20.7472	0.887087	0.443544	−	0.896253i	$-0.353721\pi$
$547$	20.7472	0.887087	0.443544	+	0.896253i	$0.353721\pi$
$548$	0	0
$549$	−10.2306	−0.436633
$550$	0	0
$551$	6.23181	0.265484
$552$	0	0
$553$	17.1828	0.730687
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18.1711	0.769933	0.384966	−	0.922931i	$-0.374213\pi$
$557$	18.1711	0.769933	0.384966	+	0.922931i	$0.374213\pi$
$558$	0	0
$559$	−33.4581	−1.41513
$560$	0	0
$561$	−0.278201	−0.0117456
$562$	0	0
$563$	22.9947	0.969110	0.484555	−	0.874761i	$-0.338982\pi$
$563$	22.9947	0.969110	0.484555	+	0.874761i	$0.338982\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	7.46688	0.313580
$568$	0	0
$569$	27.0829	1.13537	0.567687	−	0.823245i	$-0.307838\pi$
$569$	27.0829	1.13537	0.567687	+	0.823245i	$0.307838\pi$
$570$	0	0
$571$	−10.5631	−0.442053	−0.221027	−	0.975268i	$-0.570941\pi$
$571$	−10.5631	−0.442053	−0.221027	+	0.975268i	$0.570941\pi$
$572$	0	0
$573$	1.56817	0.0655112
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−2.51016	−0.104499	−0.0522496	−	0.998634i	$-0.516639\pi$
$577$	−2.51016	−0.104499	−0.0522496	+	0.998634i	$0.516639\pi$
$578$	0	0
$579$	6.27925	0.260957
$580$	0	0
$581$	7.07922	0.293695
$582$	0	0
$583$	−1.12947	−0.0467778
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−43.0923	−1.77861	−0.889305	−	0.457315i	$-0.848811\pi$
$587$	−43.0923	−1.77861	−0.889305	+	0.457315i	$0.848811\pi$
$588$	0	0
$589$	4.63727	0.191076
$590$	0	0
$591$	−0.0846792	−0.00348324
$592$	0	0
$593$	29.7306	1.22089	0.610444	−	0.792059i	$-0.290991\pi$
$593$	29.7306	1.22089	0.610444	+	0.792059i	$0.290991\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−12.6245	−0.516687
$598$	0	0
$599$	−22.9491	−0.937674	−0.468837	−	0.883285i	$-0.655327\pi$
$599$	−22.9491	−0.937674	−0.468837	+	0.883285i	$0.655327\pi$
$600$	0	0
$601$	3.49920	0.142735	0.0713677	−	0.997450i	$-0.477264\pi$
$601$	3.49920	0.142735	0.0713677	+	0.997450i	$0.477264\pi$
$602$	0	0
$603$	12.4976	0.508941
$604$	0	0
$605$	0	0
$606$	0	0
$607$	42.5670	1.72774	0.863871	−	0.503714i	$-0.168033\pi$
$607$	42.5670	1.72774	0.863871	+	0.503714i	$0.168033\pi$
$608$	0	0
$609$	−2.68782	−0.108916
$610$	0	0
$611$	−3.49526	−0.141403
$612$	0	0
$613$	−28.0072	−1.13120	−0.565600	−	0.824679i	$-0.691356\pi$
$613$	−28.0072	−1.13120	−0.565600	+	0.824679i	$0.691356\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.14856	0.126756	0.0633781	−	0.997990i	$-0.479813\pi$
$617$	3.14856	0.126756	0.0633781	+	0.997990i	$0.479813\pi$
$618$	0	0
$619$	13.1433	0.528275	0.264137	−	0.964485i	$-0.414913\pi$
$619$	13.1433	0.528275	0.264137	+	0.964485i	$0.414913\pi$
$620$	0	0
$621$	−3.08625	−0.123847
$622$	0	0
$623$	6.70953	0.268812
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1.65239	−0.0659900
$628$	0	0
$629$	−0.197298	−0.00786677
$630$	0	0
$631$	−35.4545	−1.41142	−0.705711	−	0.708500i	$-0.749372\pi$
$631$	−35.4545	−1.41142	−0.705711	+	0.708500i	$0.749372\pi$
$632$	0	0
$633$	11.4969	0.456961
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−31.6646	−1.25460
$638$	0	0
$639$	−18.3438	−0.725670
$640$	0	0
$641$	9.79611	0.386923	0.193462	−	0.981108i	$-0.438029\pi$
$641$	9.79611	0.386923	0.193462	+	0.981108i	$0.438029\pi$
$642$	0	0
$643$	−15.6613	−0.617621	−0.308810	−	0.951124i	$-0.599931\pi$
$643$	−15.6613	−0.617621	−0.308810	+	0.951124i	$0.599931\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−38.6953	−1.52127	−0.760634	−	0.649180i	$-0.775112\pi$
$647$	−38.6953	−1.52127	−0.760634	+	0.649180i	$0.775112\pi$
$648$	0	0
$649$	−2.18707	−0.0858500
$650$	0	0
$651$	−2.00009	−0.0783896
$652$	0	0
$653$	−37.8261	−1.48025	−0.740125	−	0.672469i	$-0.765234\pi$
$653$	−37.8261	−1.48025	−0.740125	+	0.672469i	$0.765234\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	24.0244	0.937282
$658$	0	0
$659$	9.00852	0.350922	0.175461	−	0.984486i	$-0.443858\pi$
$659$	9.00852	0.350922	0.175461	+	0.984486i	$0.443858\pi$
$660$	0	0
$661$	−26.0490	−1.01319	−0.506594	−	0.862185i	$-0.669096\pi$
$661$	−26.0490	−1.01319	−0.506594	+	0.862185i	$0.669096\pi$
$662$	0	0
$663$	−0.738517	−0.0286816
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.29653	−0.166362
$668$	0	0
$669$	−15.5527	−0.601303
$670$	0	0
$671$	7.96076	0.307322
$672$	0	0
$673$	4.68835	0.180722	0.0903612	−	0.995909i	$-0.471198\pi$
$673$	4.68835	0.180722	0.0903612	+	0.995909i	$0.471198\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−31.1370	−1.19669	−0.598346	−	0.801238i	$-0.704175\pi$
$677$	−31.1370	−1.19669	−0.598346	+	0.801238i	$0.704175\pi$
$678$	0	0
$679$	−2.76023	−0.105928
$680$	0	0
$681$	−3.29296	−0.126186
$682$	0	0
$683$	−15.2671	−0.584180	−0.292090	−	0.956391i	$-0.594351\pi$
$683$	−15.2671	−0.584180	−0.292090	+	0.956391i	$0.594351\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−10.7763	−0.411142
$688$	0	0
$689$	−2.99831	−0.114227
$690$	0	0
$691$	−30.3229	−1.15354	−0.576768	−	0.816908i	$-0.695686\pi$
$691$	−30.3229	−1.15354	−0.576768	+	0.816908i	$0.695686\pi$
$692$	0	0
$693$	−6.59985	−0.250707
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−0.389610	−0.0147575
$698$	0	0
$699$	5.20829	0.196996
$700$	0	0
$701$	−19.7545	−0.746118	−0.373059	−	0.927808i	$-0.621691\pi$
$701$	−19.7545	−0.746118	−0.373059	+	0.927808i	$0.621691\pi$
$702$	0	0
$703$	−1.17186	−0.0441975
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.64161	0.362610
$708$	0	0
$709$	7.45728	0.280064	0.140032	−	0.990147i	$-0.455279\pi$
$709$	7.45728	0.280064	0.140032	+	0.990147i	$0.455279\pi$
$710$	0	0
$711$	−40.2137	−1.50813
$712$	0	0
$713$	−3.19717	−0.119735
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−7.16907	−0.267734
$718$	0	0
$719$	48.4683	1.80756	0.903781	−	0.427995i	$-0.140780\pi$
$719$	48.4683	1.80756	0.903781	+	0.427995i	$0.140780\pi$
$720$	0	0
$721$	−14.3748	−0.535347
$722$	0	0
$723$	7.79993	0.290083
$724$	0	0
$725$	0	0
$726$	0	0
$727$	49.7498	1.84512	0.922559	−	0.385856i	$-0.126094\pi$
$727$	49.7498	1.84512	0.922559	+	0.385856i	$0.126094\pi$
$728$	0	0
$729$	−12.4687	−0.461802
$730$	0	0
$731$	−1.46084	−0.0540312
$732$	0	0
$733$	−12.1579	−0.449063	−0.224531	−	0.974467i	$-0.572085\pi$
$733$	−12.1579	−0.449063	−0.224531	+	0.974467i	$0.572085\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.72474	−0.358215
$738$	0	0
$739$	−20.5807	−0.757075	−0.378537	−	0.925586i	$-0.623573\pi$
$739$	−20.5807	−0.757075	−0.378537	+	0.925586i	$0.623573\pi$
$740$	0	0
$741$	−4.38646	−0.161141
$742$	0	0
$743$	−22.7862	−0.835944	−0.417972	−	0.908460i	$-0.637259\pi$
$743$	−22.7862	−0.835944	−0.417972	+	0.908460i	$0.637259\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−16.5678	−0.606185
$748$	0	0
$749$	18.0254	0.658634
$750$	0	0
$751$	1.69147	0.0617226	0.0308613	−	0.999524i	$-0.490175\pi$
$751$	1.69147	0.0617226	0.0308613	+	0.999524i	$0.490175\pi$
$752$	0	0
$753$	−10.1001	−0.368069
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−25.7514	−0.935952	−0.467976	−	0.883741i	$-0.655017\pi$
$757$	−25.7514	−0.935952	−0.467976	+	0.883741i	$0.655017\pi$
$758$	0	0
$759$	1.13924	0.0413518
$760$	0	0
$761$	41.5286	1.50541	0.752706	−	0.658357i	$-0.228748\pi$
$761$	41.5286	1.50541	0.752706	+	0.658357i	$0.228748\pi$
$762$	0	0
$763$	11.2384	0.406856
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.80584	−0.209637
$768$	0	0
$769$	0.962676	0.0347150	0.0173575	−	0.999849i	$-0.494475\pi$
$769$	0.962676	0.0347150	0.0173575	+	0.999849i	$0.494475\pi$
$770$	0	0
$771$	−13.7904	−0.496648
$772$	0	0
$773$	−42.5570	−1.53067	−0.765334	−	0.643633i	$-0.777426\pi$
$773$	−42.5570	−1.53067	−0.765334	+	0.643633i	$0.777426\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0.505431	0.0181322
$778$	0	0
$779$	−2.31411	−0.0829115
$780$	0	0
$781$	14.2739	0.510759
$782$	0	0
$783$	13.2601	0.473879
$784$	0	0
$785$	0	0
$786$	0	0
$787$	32.0542	1.14261	0.571304	−	0.820738i	$-0.306438\pi$
$787$	32.0542	1.14261	0.571304	+	0.820738i	$0.306438\pi$
$788$	0	0
$789$	−9.97749	−0.355208
$790$	0	0
$791$	−3.12051	−0.110953
$792$	0	0
$793$	21.1328	0.750447
$794$	0	0
$795$	0	0
$796$	0	0
$797$	42.6339	1.51017	0.755084	−	0.655628i	$-0.227596\pi$
$797$	42.6339	1.51017	0.755084	+	0.655628i	$0.227596\pi$
$798$	0	0
$799$	−0.152609	−0.00539893
$800$	0	0
$801$	−15.7026	−0.554825
$802$	0	0
$803$	−18.6941	−0.659701
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.09617	−0.0385871
$808$	0	0
$809$	−5.90921	−0.207757	−0.103878	−	0.994590i	$-0.533125\pi$
$809$	−5.90921	−0.207757	−0.103878	+	0.994590i	$0.533125\pi$
$810$	0	0
$811$	−38.1150	−1.33840	−0.669199	−	0.743084i	$-0.733362\pi$
$811$	−38.1150	−1.33840	−0.669199	+	0.743084i	$0.733362\pi$
$812$	0	0
$813$	−14.4591	−0.507101
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−8.67675	−0.303561
$818$	0	0
$819$	−17.5201	−0.612201
$820$	0	0
$821$	−31.4861	−1.09887	−0.549437	−	0.835535i	$-0.685158\pi$
$821$	−31.4861	−1.09887	−0.549437	+	0.835535i	$0.685158\pi$
$822$	0	0
$823$	2.04415	0.0712546	0.0356273	−	0.999365i	$-0.488657\pi$
$823$	2.04415	0.0712546	0.0356273	+	0.999365i	$0.488657\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−34.9513	−1.21538	−0.607689	−	0.794175i	$-0.707903\pi$
$827$	−34.9513	−1.21538	−0.607689	+	0.794175i	$0.707903\pi$
$828$	0	0
$829$	−2.48455	−0.0862919	−0.0431460	−	0.999069i	$-0.513738\pi$
$829$	−2.48455	−0.0862919	−0.0431460	+	0.999069i	$0.513738\pi$
$830$	0	0
$831$	10.1981	0.353769
$832$	0	0
$833$	−1.38253	−0.0479019
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.86726	0.341062
$838$	0	0
$839$	−46.5747	−1.60794	−0.803969	−	0.594672i	$-0.797282\pi$
$839$	−46.5747	−1.60794	−0.803969	+	0.594672i	$0.797282\pi$
$840$	0	0
$841$	−10.5399	−0.363444
$842$	0	0
$843$	−3.29431	−0.113462
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−7.59070	−0.260820
$848$	0	0
$849$	−8.48780	−0.291300
$850$	0	0
$851$	0.807940	0.0276958
$852$	0	0
$853$	−1.74956	−0.0599036	−0.0299518	−	0.999551i	$-0.509535\pi$
$853$	−1.74956	−0.0599036	−0.0299518	+	0.999551i	$0.509535\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−5.88603	−0.201063	−0.100532	−	0.994934i	$-0.532054\pi$
$857$	−5.88603	−0.201063	−0.100532	+	0.994934i	$0.532054\pi$
$858$	0	0
$859$	−38.4929	−1.31336	−0.656680	−	0.754169i	$-0.728040\pi$
$859$	−38.4929	−1.31336	−0.656680	+	0.754169i	$0.728040\pi$
$860$	0	0
$861$	0.998090	0.0340148
$862$	0	0
$863$	36.8687	1.25503	0.627513	−	0.778606i	$-0.284073\pi$
$863$	36.8687	1.25503	0.627513	+	0.778606i	$0.284073\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	9.16006	0.311092
$868$	0	0
$869$	31.2915	1.06149
$870$	0	0
$871$	−25.8155	−0.874724
$872$	0	0
$873$	6.45990	0.218634
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−46.7182	−1.57756	−0.788780	−	0.614676i	$-0.789287\pi$
$877$	−46.7182	−1.57756	−0.788780	+	0.614676i	$0.789287\pi$
$878$	0	0
$879$	11.0107	0.371383
$880$	0	0
$881$	−13.8791	−0.467600	−0.233800	−	0.972285i	$-0.575116\pi$
$881$	−13.8791	−0.467600	−0.233800	+	0.972285i	$0.575116\pi$
$882$	0	0
$883$	−23.8941	−0.804100	−0.402050	−	0.915618i	$-0.631702\pi$
$883$	−23.8941	−0.804100	−0.402050	+	0.915618i	$0.631702\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.36292	0.0793391	0.0396695	−	0.999213i	$-0.487369\pi$
$887$	2.36292	0.0793391	0.0396695	+	0.999213i	$0.487369\pi$
$888$	0	0
$889$	5.09540	0.170894
$890$	0	0
$891$	13.5979	0.455546
$892$	0	0
$893$	−0.906432	−0.0303326
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3.02425	0.100977
$898$	0	0
$899$	13.7367	0.458145
$900$	0	0
$901$	−0.130912	−0.00436130
$902$	0	0
$903$	3.74234	0.124537
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−23.1649	−0.769179	−0.384590	−	0.923088i	$-0.625657\pi$
$907$	−23.1649	−0.769179	−0.384590	+	0.923088i	$0.625657\pi$
$908$	0	0
$909$	−22.5647	−0.748424
$910$	0	0
$911$	−43.6423	−1.44593	−0.722967	−	0.690882i	$-0.757222\pi$
$911$	−43.6423	−1.44593	−0.722967	+	0.690882i	$0.757222\pi$
$912$	0	0
$913$	12.8919	0.426660
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.91837	0.0633500
$918$	0	0
$919$	−10.3644	−0.341890	−0.170945	−	0.985281i	$-0.554682\pi$
$919$	−10.3644	−0.341890	−0.170945	+	0.985281i	$0.554682\pi$
$920$	0	0
$921$	−3.41406	−0.112497
$922$	0	0
$923$	37.8917	1.24722
$924$	0	0
$925$	0	0
$926$	0	0
$927$	33.6421	1.10495
$928$	0	0
$929$	−38.1805	−1.25266	−0.626331	−	0.779557i	$-0.715444\pi$
$929$	−38.1805	−1.25266	−0.626331	+	0.779557i	$0.715444\pi$
$930$	0	0
$931$	−8.21163	−0.269125
$932$	0	0
$933$	1.22691	0.0401671
$934$	0	0
$935$	0	0
$936$	0	0
$937$	12.1405	0.396613	0.198306	−	0.980140i	$-0.436456\pi$
$937$	12.1405	0.396613	0.198306	+	0.980140i	$0.436456\pi$
$938$	0	0
$939$	13.9986	0.456826
$940$	0	0
$941$	−18.2871	−0.596141	−0.298071	−	0.954544i	$-0.596343\pi$
$941$	−18.2871	−0.596141	−0.298071	+	0.954544i	$0.596343\pi$
$942$	0	0
$943$	1.59546	0.0519554
$944$	0	0
$945$	0	0
$946$	0	0
$947$	51.7376	1.68125	0.840623	−	0.541621i	$-0.182189\pi$
$947$	51.7376	1.68125	0.840623	+	0.541621i	$0.182189\pi$
$948$	0	0
$949$	−49.6257	−1.61092
$950$	0	0
$951$	−0.515193	−0.0167063
$952$	0	0
$953$	58.1675	1.88423	0.942114	−	0.335292i	$-0.108835\pi$
$953$	58.1675	1.88423	0.942114	+	0.335292i	$0.108835\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−4.89477	−0.158225
$958$	0	0
$959$	−3.13406	−0.101204
$960$	0	0
$961$	−20.7781	−0.670261
$962$	0	0
$963$	−42.1857	−1.35942
$964$	0	0
$965$	0	0
$966$	0	0
$967$	12.1138	0.389555	0.194777	−	0.980847i	$-0.437602\pi$
$967$	12.1138	0.389555	0.194777	+	0.980847i	$0.437602\pi$
$968$	0	0
$969$	−0.191521	−0.00615253
$970$	0	0
$971$	36.5922	1.17430	0.587149	−	0.809479i	$-0.300250\pi$
$971$	36.5922	1.17430	0.587149	+	0.809479i	$0.300250\pi$
$972$	0	0
$973$	10.4066	0.333621
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−22.3327	−0.714487	−0.357243	−	0.934011i	$-0.616283\pi$
$977$	−22.3327	−0.714487	−0.357243	+	0.934011i	$0.616283\pi$
$978$	0	0
$979$	12.2187	0.390510
$980$	0	0
$981$	−26.3017	−0.839748
$982$	0	0
$983$	−49.8139	−1.58882	−0.794408	−	0.607384i	$-0.792219\pi$
$983$	−49.8139	−1.58882	−0.794408	+	0.607384i	$0.792219\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.390950	0.0124441
$988$	0	0
$989$	5.98219	0.190223
$990$	0	0
$991$	−14.3340	−0.455334	−0.227667	−	0.973739i	$-0.573110\pi$
$991$	−14.3340	−0.455334	−0.227667	+	0.973739i	$0.573110\pi$
$992$	0	0
$993$	7.95880	0.252565
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−51.7021	−1.63742	−0.818711	−	0.574205i	$-0.805311\pi$
$997$	−51.7021	−1.63742	−0.818711	+	0.574205i	$0.805311\pi$
$998$	0	0
$999$	−2.49350	−0.0788909

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.a.bk.1.3		8
4.3	odd	2		9200.2.a.dd.1.6		8
5.2	odd	4		920.2.e.c.369.11	yes	16
5.3	odd	4		920.2.e.c.369.6	✓	16
5.4	even	2		4600.2.a.bj.1.6		8
20.3	even	4		1840.2.e.h.369.11		16
20.7	even	4		1840.2.e.h.369.6		16
20.19	odd	2		9200.2.a.de.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.c.369.6	✓	16	5.3	odd	4
920.2.e.c.369.11	yes	16	5.2	odd	4
1840.2.e.h.369.6		16	20.7	even	4
1840.2.e.h.369.11		16	20.3	even	4
4600.2.a.bj.1.6		8	5.4	even	2
4600.2.a.bk.1.3		8	1.1	even	1	trivial
9200.2.a.dd.1.6		8	4.3	odd	2
9200.2.a.de.1.3		8	20.19	odd	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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.a (trivial)

\(f(q)\)	\(=\)	\(q-0.540724 q^{3} +1.15693 q^{7} -2.70762 q^{9} +O(q^{10})\)	\(q-0.540724 q^{3} +1.15693 q^{7} -2.70762 q^{9} +2.10688 q^{11} +5.59296 q^{13} +0.244199 q^{17} +1.45043 q^{19} -0.625580 q^{21} -1.00000 q^{23} +3.08625 q^{27} +4.29653 q^{29} +3.19717 q^{31} -1.13924 q^{33} -0.807940 q^{37} -3.02425 q^{39} -1.59546 q^{41} -5.98219 q^{43} -0.624940 q^{47} -5.66151 q^{49} -0.132044 q^{51} -0.536087 q^{53} -0.784283 q^{57} -1.03806 q^{59} +3.77846 q^{61} -3.13253 q^{63} -4.61571 q^{67} +0.540724 q^{69} +6.77489 q^{71} -8.87290 q^{73} +2.43751 q^{77} +14.8521 q^{79} +6.45405 q^{81} +6.11896 q^{83} -2.32323 q^{87} +5.79942 q^{89} +6.47066 q^{91} -1.72879 q^{93} -2.38582 q^{97} -5.70462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{3} + 7 q^{7} + 11 q^{9}+O(q^{10}) \) 8 * q + 3 * q^3 + 7 * q^7 + 11 * q^9	\( 8 q + 3 q^{3} + 7 q^{7} + 11 q^{9} + 7 q^{11} - 11 q^{13} + 7 q^{17} + 11 q^{19} - 8 q^{23} + 12 q^{27} + 22 q^{29} + 9 q^{31} + 9 q^{33} - 4 q^{37} + 7 q^{41} + 22 q^{43} + 4 q^{47} + 39 q^{49} - 19 q^{51} - 4 q^{53} + 32 q^{59} + 17 q^{61} + 44 q^{63} - 4 q^{67} - 3 q^{69} + 15 q^{71} - 6 q^{73} - 18 q^{77} - 2 q^{79} + 24 q^{81} + 36 q^{83} - 4 q^{87} + 46 q^{89} - 35 q^{91} - 20 q^{93} - 3 q^{97} + 61 q^{99}+O(q^{100}) \) 8 * q + 3 * q^3 + 7 * q^7 + 11 * q^9 + 7 * q^11 - 11 * q^13 + 7 * q^17 + 11 * q^19 - 8 * q^23 + 12 * q^27 + 22 * q^29 + 9 * q^31 + 9 * q^33 - 4 * q^37 + 7 * q^41 + 22 * q^43 + 4 * q^47 + 39 * q^49 - 19 * q^51 - 4 * q^53 + 32 * q^59 + 17 * q^61 + 44 * q^63 - 4 * q^67 - 3 * q^69 + 15 * q^71 - 6 * q^73 - 18 * q^77 - 2 * q^79 + 24 * q^81 + 36 * q^83 - 4 * q^87 + 46 * q^89 - 35 * q^91 - 20 * q^93 - 3 * q^97 + 61 * q^99

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