LMFDB - Embedded newform 7605.2.a.ct.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7605,2,Mod(1,7605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7605, 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("7605.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7605 = 3^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7605.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:	$60.7262307372$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 19x^{8} + 102x^{6} - 202x^{4} + 133x^{2} - 27 $ x^10 - 19x^8 + 102x^6 - 202x^4 + 133x^2 - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 585)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.635993$ of defining polynomial
Character	$\chi$	$=$	7605.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.794363 q^{2} -1.36899 q^{4} +1.00000 q^{5} -3.59800 q^{7} +2.67620 q^{8} +O(q^{10})$	$q-0.794363 q^{2} -1.36899 q^{4} +1.00000 q^{5} -3.59800 q^{7} +2.67620 q^{8} -0.794363 q^{10} -3.69958 q^{11} +2.85812 q^{14} +0.612105 q^{16} +7.56145 q^{17} -6.09683 q^{19} -1.36899 q^{20} +2.93881 q^{22} +4.84309 q^{23} +1.00000 q^{25} +4.92562 q^{28} -3.06590 q^{29} -5.25369 q^{31} -5.83863 q^{32} -6.00654 q^{34} -3.59800 q^{35} +4.14524 q^{37} +4.84309 q^{38} +2.67620 q^{40} +3.38988 q^{41} -2.62852 q^{43} +5.06468 q^{44} -3.84717 q^{46} +8.99327 q^{47} +5.94559 q^{49} -0.794363 q^{50} +5.92401 q^{53} -3.69958 q^{55} -9.62896 q^{56} +2.43543 q^{58} -7.80115 q^{59} -13.4449 q^{61} +4.17334 q^{62} +3.41378 q^{64} +4.88824 q^{67} -10.3515 q^{68} +2.85812 q^{70} +15.1420 q^{71} +12.6568 q^{73} -3.29282 q^{74} +8.34648 q^{76} +13.3111 q^{77} +0.207617 q^{79} +0.612105 q^{80} -2.69279 q^{82} -8.67620 q^{83} +7.56145 q^{85} +2.08800 q^{86} -9.90080 q^{88} -13.2508 q^{89} -6.63013 q^{92} -7.14392 q^{94} -6.09683 q^{95} -11.3969 q^{97} -4.72296 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 4 q^{2} + 12 q^{4} + 10 q^{5} - 12 q^{8}+O(q^{10}) $ 10 * q - 4 * q^2 + 12 * q^4 + 10 * q^5 - 12 * q^8	$ 10 q - 4 q^{2} + 12 q^{4} + 10 q^{5} - 12 q^{8} - 4 q^{10} - 24 q^{11} + 24 q^{16} + 12 q^{20} - 12 q^{22} + 10 q^{25} - 68 q^{32} - 12 q^{40} - 4 q^{41} - 14 q^{43} - 60 q^{44} - 4 q^{47} - 8 q^{49} - 4 q^{50} - 24 q^{55} - 48 q^{59} + 10 q^{61} + 88 q^{64} - 44 q^{71} - 14 q^{79} + 24 q^{80} - 40 q^{82} - 48 q^{83} + 56 q^{86} + 48 q^{88} - 72 q^{89} + 32 q^{94} - 60 q^{98}+O(q^{100}) $ 10 * q - 4 * q^2 + 12 * q^4 + 10 * q^5 - 12 * q^8 - 4 * q^10 - 24 * q^11 + 24 * q^16 + 12 * q^20 - 12 * q^22 + 10 * q^25 - 68 * q^32 - 12 * q^40 - 4 * q^41 - 14 * q^43 - 60 * q^44 - 4 * q^47 - 8 * q^49 - 4 * q^50 - 24 * q^55 - 48 * q^59 + 10 * q^61 + 88 * q^64 - 44 * q^71 - 14 * q^79 + 24 * q^80 - 40 * q^82 - 48 * q^83 + 56 * q^86 + 48 * q^88 - 72 * q^89 + 32 * q^94 - 60 * q^98

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 $

$2$	−0.794363	−0.561699	−0.280850	−	0.959752i	$-0.590616\pi$
$2$	−0.794363	−0.561699	−0.280850	+	0.959752i	$0.590616\pi$
$3$	0	0
$3$	0	0
$4$	−1.36899	−0.684494
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.59800	−1.35992	−0.679958	−	0.733251i	$-0.738002\pi$
$7$	−3.59800	−1.35992	−0.679958	+	0.733251i	$0.738002\pi$
$8$	2.67620	0.946179
$9$	0	0
$10$	−0.794363	−0.251200
$11$	−3.69958	−1.11546	−0.557732	−	0.830021i	$-0.688328\pi$
$11$	−3.69958	−1.11546	−0.557732	+	0.830021i	$0.688328\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	2.85812	0.763863
$15$	0	0
$16$	0.612105	0.153026
$17$	7.56145	1.83392	0.916961	−	0.398977i	$-0.130635\pi$
$17$	7.56145	1.83392	0.916961	+	0.398977i	$0.130635\pi$
$18$	0	0
$19$	−6.09683	−1.39871	−0.699354	−	0.714776i	$-0.746529\pi$
$19$	−6.09683	−1.39871	−0.699354	+	0.714776i	$0.746529\pi$
$20$	−1.36899	−0.306115
$21$	0	0
$22$	2.93881	0.626556
$23$	4.84309	1.00985	0.504927	−	0.863162i	$-0.331519\pi$
$23$	4.84309	1.00985	0.504927	+	0.863162i	$0.331519\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	4.92562	0.930854
$29$	−3.06590	−0.569322	−0.284661	−	0.958628i	$-0.591881\pi$
$29$	−3.06590	−0.569322	−0.284661	+	0.958628i	$0.591881\pi$
$30$	0	0
$31$	−5.25369	−0.943591	−0.471796	−	0.881708i	$-0.656394\pi$
$31$	−5.25369	−0.943591	−0.471796	+	0.881708i	$0.656394\pi$
$32$	−5.83863	−1.03213
$33$	0	0
$34$	−6.00654	−1.03011
$35$	−3.59800	−0.608173
$36$	0	0
$37$	4.14524	0.681473	0.340737	−	0.940159i	$-0.389324\pi$
$37$	4.14524	0.681473	0.340737	+	0.940159i	$0.389324\pi$
$38$	4.84309	0.785653
$39$	0	0
$40$	2.67620	0.423144
$41$	3.38988	0.529410	0.264705	−	0.964329i	$-0.414725\pi$
$41$	3.38988	0.529410	0.264705	+	0.964329i	$0.414725\pi$
$42$	0	0
$43$	−2.62852	−0.400845	−0.200423	−	0.979710i	$-0.564232\pi$
$43$	−2.62852	−0.400845	−0.200423	+	0.979710i	$0.564232\pi$
$44$	5.06468	0.763529
$45$	0	0
$46$	−3.84717	−0.567234
$47$	8.99327	1.31180	0.655902	−	0.754846i	$-0.272289\pi$
$47$	8.99327	1.31180	0.655902	+	0.754846i	$0.272289\pi$
$48$	0	0
$49$	5.94559	0.849370
$50$	−0.794363	−0.112340
$51$	0	0
$52$	0	0
$53$	5.92401	0.813725	0.406863	−	0.913489i	$-0.366623\pi$
$53$	5.92401	0.813725	0.406863	+	0.913489i	$0.366623\pi$
$54$	0	0
$55$	−3.69958	−0.498851
$56$	−9.62896	−1.28672
$57$	0	0
$58$	2.43543	0.319788
$59$	−7.80115	−1.01562	−0.507812	−	0.861468i	$-0.669545\pi$
$59$	−7.80115	−1.01562	−0.507812	+	0.861468i	$0.669545\pi$
$60$	0	0
$61$	−13.4449	−1.72145	−0.860723	−	0.509073i	$-0.829988\pi$
$61$	−13.4449	−1.72145	−0.860723	+	0.509073i	$0.829988\pi$
$62$	4.17334	0.530014
$63$	0	0
$64$	3.41378	0.426722
$65$	0	0
$66$	0	0
$67$	4.88824	0.597193	0.298597	−	0.954379i	$-0.403482\pi$
$67$	4.88824	0.597193	0.298597	+	0.954379i	$0.403482\pi$
$68$	−10.3515	−1.25531
$69$	0	0
$70$	2.85812	0.341610
$71$	15.1420	1.79703	0.898513	−	0.438947i	$-0.144648\pi$
$71$	15.1420	1.79703	0.898513	+	0.438947i	$0.144648\pi$
$72$	0	0
$73$	12.6568	1.48137	0.740685	−	0.671852i	$-0.234501\pi$
$73$	12.6568	1.48137	0.740685	+	0.671852i	$0.234501\pi$
$74$	−3.29282	−0.382783
$75$	0	0
$76$	8.34648	0.957407
$77$	13.3111	1.51694
$78$	0	0
$79$	0.207617	0.0233588	0.0116794	−	0.999932i	$-0.496282\pi$
$79$	0.207617	0.0233588	0.0116794	+	0.999932i	$0.496282\pi$
$80$	0.612105	0.0684354
$81$	0	0
$82$	−2.69279	−0.297369
$83$	−8.67620	−0.952336	−0.476168	−	0.879354i	$-0.657975\pi$
$83$	−8.67620	−0.952336	−0.476168	+	0.879354i	$0.657975\pi$
$84$	0	0
$85$	7.56145	0.820155
$86$	2.08800	0.225155
$87$	0	0
$88$	−9.90080	−1.05543
$89$	−13.2508	−1.40458	−0.702292	−	0.711889i	$-0.747840\pi$
$89$	−13.2508	−1.40458	−0.702292	+	0.711889i	$0.747840\pi$
$90$	0	0
$91$	0	0
$92$	−6.63013	−0.691239
$93$	0	0
$94$	−7.14392	−0.736839
$95$	−6.09683	−0.625521
$96$	0	0
$97$	−11.3969	−1.15718	−0.578588	−	0.815620i	$-0.696396\pi$
$97$	−11.3969	−1.15718	−0.578588	+	0.815620i	$0.696396\pi$
$98$	−4.72296	−0.477091
$99$	0	0
$100$	−1.36899	−0.136899
$101$	18.6380	1.85455	0.927276	−	0.374379i	$-0.122144\pi$
$101$	18.6380	1.85455	0.927276	+	0.374379i	$0.122144\pi$
$102$	0	0
$103$	12.6811	1.24950	0.624752	−	0.780823i	$-0.285200\pi$
$103$	12.6811	1.24950	0.624752	+	0.780823i	$0.285200\pi$
$104$	0	0
$105$	0	0
$106$	−4.70581	−0.457069
$107$	5.89062	0.569468	0.284734	−	0.958607i	$-0.408095\pi$
$107$	5.89062	0.569468	0.284734	+	0.958607i	$0.408095\pi$
$108$	0	0
$109$	3.00497	0.287824	0.143912	−	0.989590i	$-0.454032\pi$
$109$	3.00497	0.287824	0.143912	+	0.989590i	$0.454032\pi$
$110$	2.93881	0.280204
$111$	0	0
$112$	−2.20235	−0.208103
$113$	0.261568	0.0246062	0.0123031	−	0.999924i	$-0.496084\pi$
$113$	0.261568	0.0246062	0.0123031	+	0.999924i	$0.496084\pi$
$114$	0	0
$115$	4.84309	0.451621
$116$	4.19717	0.389698
$117$	0	0
$118$	6.19694	0.570475
$119$	−27.2061	−2.49398
$120$	0	0
$121$	2.68687	0.244261
$122$	10.6801	0.966935
$123$	0	0
$124$	7.19224	0.645883
$125$	1.00000	0.0894427
$126$	0	0
$127$	−6.51732	−0.578319	−0.289159	−	0.957281i	$-0.593376\pi$
$127$	−6.51732	−0.578319	−0.289159	+	0.957281i	$0.593376\pi$
$128$	8.96548	0.792444
$129$	0	0
$130$	0	0
$131$	−9.04369	−0.790151	−0.395076	−	0.918649i	$-0.629282\pi$
$131$	−9.04369	−0.790151	−0.395076	+	0.918649i	$0.629282\pi$
$132$	0	0
$133$	21.9364	1.90212
$134$	−3.88303	−0.335443
$135$	0	0
$136$	20.2360	1.73522
$137$	6.45844	0.551782	0.275891	−	0.961189i	$-0.411027\pi$
$137$	6.45844	0.551782	0.275891	+	0.961189i	$0.411027\pi$
$138$	0	0
$139$	18.8139	1.59578	0.797888	−	0.602806i	$-0.205951\pi$
$139$	18.8139	1.59578	0.797888	+	0.602806i	$0.205951\pi$
$140$	4.92562	0.416291
$141$	0	0
$142$	−12.0282	−1.00939
$143$	0	0
$144$	0	0
$145$	−3.06590	−0.254609
$146$	−10.0541	−0.832084
$147$	0	0
$148$	−5.67478	−0.466464
$149$	−9.08017	−0.743877	−0.371938	−	0.928257i	$-0.621307\pi$
$149$	−9.08017	−0.743877	−0.371938	+	0.928257i	$0.621307\pi$
$150$	0	0
$151$	7.40002	0.602205	0.301103	−	0.953592i	$-0.402645\pi$
$151$	7.40002	0.602205	0.301103	+	0.953592i	$0.402645\pi$
$152$	−16.3163	−1.32343
$153$	0	0
$154$	−10.5738	−0.852063
$155$	−5.25369	−0.421987
$156$	0	0
$157$	−11.3117	−0.902775	−0.451388	−	0.892328i	$-0.649071\pi$
$157$	−11.3117	−0.902775	−0.451388	+	0.892328i	$0.649071\pi$
$158$	−0.164923	−0.0131206
$159$	0	0
$160$	−5.83863	−0.461584
$161$	−17.4254	−1.37332
$162$	0	0
$163$	2.44658	0.191631	0.0958154	−	0.995399i	$-0.469454\pi$
$163$	2.44658	0.191631	0.0958154	+	0.995399i	$0.469454\pi$
$164$	−4.64070	−0.362378
$165$	0	0
$166$	6.89205	0.534927
$167$	−6.18957	−0.478963	−0.239482	−	0.970901i	$-0.576977\pi$
$167$	−6.18957	−0.478963	−0.239482	+	0.970901i	$0.576977\pi$
$168$	0	0
$169$	0	0
$170$	−6.00654	−0.460680
$171$	0	0
$172$	3.59841	0.274376
$173$	−5.46204	−0.415271	−0.207635	−	0.978206i	$-0.566577\pi$
$173$	−5.46204	−0.415271	−0.207635	+	0.978206i	$0.566577\pi$
$174$	0	0
$175$	−3.59800	−0.271983
$176$	−2.26453	−0.170695
$177$	0	0
$178$	10.5260	0.788954
$179$	−1.88699	−0.141040	−0.0705200	−	0.997510i	$-0.522466\pi$
$179$	−1.88699	−0.141040	−0.0705200	+	0.997510i	$0.522466\pi$
$180$	0	0
$181$	14.5109	1.07859	0.539295	−	0.842117i	$-0.318691\pi$
$181$	14.5109	1.07859	0.539295	+	0.842117i	$0.318691\pi$
$182$	0	0
$183$	0	0
$184$	12.9611	0.955503
$185$	4.14524	0.304764
$186$	0	0
$187$	−27.9742	−2.04568
$188$	−12.3117	−0.897922
$189$	0	0
$190$	4.84309	0.351355
$191$	−13.2359	−0.957718	−0.478859	−	0.877892i	$-0.658949\pi$
$191$	−13.2359	−0.957718	−0.478859	+	0.877892i	$0.658949\pi$
$192$	0	0
$193$	−8.48680	−0.610893	−0.305447	−	0.952209i	$-0.598806\pi$
$193$	−8.48680	−0.610893	−0.305447	+	0.952209i	$0.598806\pi$
$194$	9.05325	0.649985
$195$	0	0
$196$	−8.13945	−0.581389
$197$	−11.5659	−0.824038	−0.412019	−	0.911175i	$-0.635176\pi$
$197$	−11.5659	−0.824038	−0.412019	+	0.911175i	$0.635176\pi$
$198$	0	0
$199$	−5.57661	−0.395315	−0.197658	−	0.980271i	$-0.563333\pi$
$199$	−5.57661	−0.395315	−0.197658	+	0.980271i	$0.563333\pi$
$200$	2.67620	0.189236
$201$	0	0
$202$	−14.8053	−1.04170
$203$	11.0311	0.774230
$204$	0	0
$205$	3.38988	0.236759
$206$	−10.0734	−0.701845
$207$	0	0
$208$	0	0
$209$	22.5557	1.56021
$210$	0	0
$211$	−10.0454	−0.691555	−0.345778	−	0.938316i	$-0.612385\pi$
$211$	−10.0454	−0.691555	−0.345778	+	0.938316i	$0.612385\pi$
$212$	−8.10990	−0.556990
$213$	0	0
$214$	−4.67929	−0.319870
$215$	−2.62852	−0.179264
$216$	0	0
$217$	18.9028	1.28320
$218$	−2.38704	−0.161671
$219$	0	0
$220$	5.06468	0.341460
$221$	0	0
$222$	0	0
$223$	−7.01943	−0.470056	−0.235028	−	0.971989i	$-0.575518\pi$
$223$	−7.01943	−0.470056	−0.235028	+	0.971989i	$0.575518\pi$
$224$	21.0074	1.40361
$225$	0	0
$226$	−0.207780	−0.0138213
$227$	−21.3200	−1.41506	−0.707528	−	0.706685i	$-0.750190\pi$
$227$	−21.3200	−1.41506	−0.707528	+	0.706685i	$0.750190\pi$
$228$	0	0
$229$	−19.8980	−1.31490	−0.657448	−	0.753500i	$-0.728364\pi$
$229$	−19.8980	−1.31490	−0.657448	+	0.753500i	$0.728364\pi$
$230$	−3.84717	−0.253675
$231$	0	0
$232$	−8.20494	−0.538681
$233$	2.37696	0.155720	0.0778598	−	0.996964i	$-0.475191\pi$
$233$	2.37696	0.155720	0.0778598	+	0.996964i	$0.475191\pi$
$234$	0	0
$235$	8.99327	0.586656
$236$	10.6797	0.695188
$237$	0	0
$238$	21.6115	1.40087
$239$	−24.1120	−1.55968	−0.779838	−	0.625981i	$-0.784699\pi$
$239$	−24.1120	−1.55968	−0.779838	+	0.625981i	$0.784699\pi$
$240$	0	0
$241$	18.3520	1.18215	0.591077	−	0.806615i	$-0.298703\pi$
$241$	18.3520	1.18215	0.591077	+	0.806615i	$0.298703\pi$
$242$	−2.13435	−0.137201
$243$	0	0
$244$	18.4059	1.17832
$245$	5.94559	0.379850
$246$	0	0
$247$	0	0
$248$	−14.0599	−0.892806
$249$	0	0
$250$	−0.794363	−0.0502399
$251$	0.624848	0.0394401	0.0197200	−	0.999806i	$-0.493723\pi$
$251$	0.624848	0.0394401	0.0197200	+	0.999806i	$0.493723\pi$
$252$	0	0
$253$	−17.9174	−1.12646
$254$	5.17711	0.324841
$255$	0	0
$256$	−13.9494	−0.871838
$257$	22.7480	1.41898	0.709491	−	0.704715i	$-0.248925\pi$
$257$	22.7480	1.41898	0.709491	+	0.704715i	$0.248925\pi$
$258$	0	0
$259$	−14.9146	−0.926746
$260$	0	0
$261$	0	0
$262$	7.18397	0.443827
$263$	−21.0388	−1.29731	−0.648654	−	0.761083i	$-0.724668\pi$
$263$	−21.0388	−1.29731	−0.648654	+	0.761083i	$0.724668\pi$
$264$	0	0
$265$	5.92401	0.363909
$266$	−17.4254	−1.06842
$267$	0	0
$268$	−6.69194	−0.408775
$269$	−19.0059	−1.15881	−0.579406	−	0.815039i	$-0.696716\pi$
$269$	−19.0059	−1.15881	−0.579406	+	0.815039i	$0.696716\pi$
$270$	0	0
$271$	−0.520078	−0.0315925	−0.0157963	−	0.999875i	$-0.505028\pi$
$271$	−0.520078	−0.0315925	−0.0157963	+	0.999875i	$0.505028\pi$
$272$	4.62840	0.280638
$273$	0	0
$274$	−5.13034	−0.309935
$275$	−3.69958	−0.223093
$276$	0	0
$277$	−24.5333	−1.47406	−0.737032	−	0.675857i	$-0.763774\pi$
$277$	−24.5333	−1.47406	−0.737032	+	0.675857i	$0.763774\pi$
$278$	−14.9451	−0.896346
$279$	0	0
$280$	−9.62896	−0.575440
$281$	9.57446	0.571164	0.285582	−	0.958354i	$-0.407813\pi$
$281$	9.57446	0.571164	0.285582	+	0.958354i	$0.407813\pi$
$282$	0	0
$283$	−3.40705	−0.202528	−0.101264	−	0.994860i	$-0.532289\pi$
$283$	−3.40705	−0.202528	−0.101264	+	0.994860i	$0.532289\pi$
$284$	−20.7292	−1.23005
$285$	0	0
$286$	0	0
$287$	−12.1968	−0.719952
$288$	0	0
$289$	40.1756	2.36327
$290$	2.43543	0.143014
$291$	0	0
$292$	−17.3271	−1.01399
$293$	17.1722	1.00321	0.501605	−	0.865097i	$-0.332743\pi$
$293$	17.1722	1.00321	0.501605	+	0.865097i	$0.332743\pi$
$294$	0	0
$295$	−7.80115	−0.454201
$296$	11.0935	0.644796
$297$	0	0
$298$	7.21295	0.417835
$299$	0	0
$300$	0	0
$301$	9.45741	0.545116
$302$	−5.87830	−0.338258
$303$	0	0
$304$	−3.73190	−0.214039
$305$	−13.4449	−0.769854
$306$	0	0
$307$	−1.06057	−0.0605298	−0.0302649	−	0.999542i	$-0.509635\pi$
$307$	−1.06057	−0.0605298	−0.0302649	+	0.999542i	$0.509635\pi$
$308$	−18.2227	−1.03833
$309$	0	0
$310$	4.17334	0.237030
$311$	−23.8274	−1.35113	−0.675563	−	0.737302i	$-0.736099\pi$
$311$	−23.8274	−1.35113	−0.675563	+	0.737302i	$0.736099\pi$
$312$	0	0
$313$	−6.82935	−0.386018	−0.193009	−	0.981197i	$-0.561825\pi$
$313$	−6.82935	−0.386018	−0.193009	+	0.981197i	$0.561825\pi$
$314$	8.98562	0.507088
$315$	0	0
$316$	−0.284225	−0.0159889
$317$	−20.6430	−1.15943	−0.579714	−	0.814820i	$-0.696836\pi$
$317$	−20.6430	−1.15943	−0.579714	+	0.814820i	$0.696836\pi$
$318$	0	0
$319$	11.3425	0.635059
$320$	3.41378	0.190836
$321$	0	0
$322$	13.8421	0.771391
$323$	−46.1009	−2.56512
$324$	0	0
$325$	0	0
$326$	−1.94347	−0.107639
$327$	0	0
$328$	9.07198	0.500916
$329$	−32.3578	−1.78394
$330$	0	0
$331$	29.4206	1.61710	0.808552	−	0.588425i	$-0.200252\pi$
$331$	29.4206	1.61710	0.808552	+	0.588425i	$0.200252\pi$
$332$	11.8776	0.651869
$333$	0	0
$334$	4.91676	0.269033
$335$	4.88824	0.267073
$336$	0	0
$337$	−0.899872	−0.0490191	−0.0245096	−	0.999700i	$-0.507802\pi$
$337$	−0.899872	−0.0490191	−0.0245096	+	0.999700i	$0.507802\pi$
$338$	0	0
$339$	0	0
$340$	−10.3515	−0.561391
$341$	19.4364	1.05254
$342$	0	0
$343$	3.79375	0.204843
$344$	−7.03444	−0.379272
$345$	0	0
$346$	4.33884	0.233257
$347$	2.46613	0.132389	0.0661945	−	0.997807i	$-0.478914\pi$
$347$	2.46613	0.132389	0.0661945	+	0.997807i	$0.478914\pi$
$348$	0	0
$349$	2.19745	0.117627	0.0588135	−	0.998269i	$-0.481268\pi$
$349$	2.19745	0.117627	0.0588135	+	0.998269i	$0.481268\pi$
$350$	2.85812	0.152773
$351$	0	0
$352$	21.6005	1.15131
$353$	1.67765	0.0892925	0.0446462	−	0.999003i	$-0.485784\pi$
$353$	1.67765	0.0892925	0.0446462	+	0.999003i	$0.485784\pi$
$354$	0	0
$355$	15.1420	0.803654
$356$	18.1402	0.961430
$357$	0	0
$358$	1.49895	0.0792221
$359$	−4.19387	−0.221344	−0.110672	−	0.993857i	$-0.535300\pi$
$359$	−4.19387	−0.221344	−0.110672	+	0.993857i	$0.535300\pi$
$360$	0	0
$361$	18.1713	0.956384
$362$	−11.5269	−0.605843
$363$	0	0
$364$	0	0
$365$	12.6568	0.662489
$366$	0	0
$367$	−23.5414	−1.22885	−0.614427	−	0.788974i	$-0.710613\pi$
$367$	−23.5414	−1.22885	−0.614427	+	0.788974i	$0.710613\pi$
$368$	2.96448	0.154534
$369$	0	0
$370$	−3.29282	−0.171186
$371$	−21.3146	−1.10660
$372$	0	0
$373$	−7.07537	−0.366349	−0.183174	−	0.983080i	$-0.558637\pi$
$373$	−7.07537	−0.366349	−0.183174	+	0.983080i	$0.558637\pi$
$374$	22.2216	1.14905
$375$	0	0
$376$	24.0678	1.24120
$377$	0	0
$378$	0	0
$379$	−1.92966	−0.0991199	−0.0495600	−	0.998771i	$-0.515782\pi$
$379$	−1.92966	−0.0991199	−0.0495600	+	0.998771i	$0.515782\pi$
$380$	8.34648	0.428166
$381$	0	0
$382$	10.5141	0.537949
$383$	19.6962	1.00643	0.503215	−	0.864161i	$-0.332150\pi$
$383$	19.6962	1.00643	0.503215	+	0.864161i	$0.332150\pi$
$384$	0	0
$385$	13.3111	0.678395
$386$	6.74159	0.343138
$387$	0	0
$388$	15.6022	0.792081
$389$	1.29947	0.0658859	0.0329430	−	0.999457i	$-0.489512\pi$
$389$	1.29947	0.0658859	0.0329430	+	0.999457i	$0.489512\pi$
$390$	0	0
$391$	36.6208	1.85199
$392$	15.9116	0.803656
$393$	0	0
$394$	9.18754	0.462862
$395$	0.207617	0.0104464
$396$	0	0
$397$	−1.72956	−0.0868040	−0.0434020	−	0.999058i	$-0.513820\pi$
$397$	−1.72956	−0.0868040	−0.0434020	+	0.999058i	$0.513820\pi$
$398$	4.42985	0.222048
$399$	0	0
$400$	0.612105	0.0306052
$401$	1.10645	0.0552536	0.0276268	−	0.999618i	$-0.491205\pi$
$401$	1.10645	0.0552536	0.0276268	+	0.999618i	$0.491205\pi$
$402$	0	0
$403$	0	0
$404$	−25.5152	−1.26943
$405$	0	0
$406$	−8.76268	−0.434885
$407$	−15.3356	−0.760159
$408$	0	0
$409$	33.3878	1.65092	0.825459	−	0.564462i	$-0.190916\pi$
$409$	33.3878	1.65092	0.825459	+	0.564462i	$0.190916\pi$
$410$	−2.69279	−0.132987
$411$	0	0
$412$	−17.3602	−0.855278
$413$	28.0685	1.38116
$414$	0	0
$415$	−8.67620	−0.425898
$416$	0	0
$417$	0	0
$418$	−17.9174	−0.876368
$419$	−18.2033	−0.889290	−0.444645	−	0.895707i	$-0.646670\pi$
$419$	−18.2033	−0.889290	−0.444645	+	0.895707i	$0.646670\pi$
$420$	0	0
$421$	−1.47202	−0.0717419	−0.0358709	−	0.999356i	$-0.511421\pi$
$421$	−1.47202	−0.0717419	−0.0358709	+	0.999356i	$0.511421\pi$
$422$	7.97971	0.388446
$423$	0	0
$424$	15.8538	0.769930
$425$	7.56145	0.366784
$426$	0	0
$427$	48.3748	2.34102
$428$	−8.06419	−0.389797
$429$	0	0
$430$	2.08800	0.100692
$431$	−25.9078	−1.24794	−0.623968	−	0.781450i	$-0.714480\pi$
$431$	−25.9078	−1.24794	−0.623968	+	0.781450i	$0.714480\pi$
$432$	0	0
$433$	23.8732	1.14727	0.573636	−	0.819110i	$-0.305532\pi$
$433$	23.8732	1.14727	0.573636	+	0.819110i	$0.305532\pi$
$434$	−15.0157	−0.720775
$435$	0	0
$436$	−4.11377	−0.197014
$437$	−29.5275	−1.41249
$438$	0	0
$439$	−28.3754	−1.35428	−0.677142	−	0.735853i	$-0.736782\pi$
$439$	−28.3754	−1.35428	−0.677142	+	0.735853i	$0.736782\pi$
$440$	−9.90080	−0.472002
$441$	0	0
$442$	0	0
$443$	−12.5716	−0.597293	−0.298646	−	0.954364i	$-0.596535\pi$
$443$	−12.5716	−0.597293	−0.298646	+	0.954364i	$0.596535\pi$
$444$	0	0
$445$	−13.2508	−0.628149
$446$	5.57597	0.264030
$447$	0	0
$448$	−12.2828	−0.580306
$449$	−36.8053	−1.73695	−0.868474	−	0.495734i	$-0.834899\pi$
$449$	−36.8053	−1.73695	−0.868474	+	0.495734i	$0.834899\pi$
$450$	0	0
$451$	−12.5411	−0.590538
$452$	−0.358083	−0.0168428
$453$	0	0
$454$	16.9358	0.794836
$455$	0	0
$456$	0	0
$457$	−6.73250	−0.314933	−0.157467	−	0.987524i	$-0.550333\pi$
$457$	−6.73250	−0.314933	−0.157467	+	0.987524i	$0.550333\pi$
$458$	15.8062	0.738575
$459$	0	0
$460$	−6.63013	−0.309132
$461$	−30.0493	−1.39953	−0.699767	−	0.714372i	$-0.746713\pi$
$461$	−30.0493	−1.39953	−0.699767	+	0.714372i	$0.746713\pi$
$462$	0	0
$463$	−18.0496	−0.838835	−0.419418	−	0.907793i	$-0.637766\pi$
$463$	−18.0496	−0.838835	−0.419418	+	0.907793i	$0.637766\pi$
$464$	−1.87665	−0.0871212
$465$	0	0
$466$	−1.88816	−0.0874676
$467$	26.2888	1.21650	0.608251	−	0.793745i	$-0.291872\pi$
$467$	26.2888	1.21650	0.608251	+	0.793745i	$0.291872\pi$
$468$	0	0
$469$	−17.5879	−0.812132
$470$	−7.14392	−0.329524
$471$	0	0
$472$	−20.8774	−0.960961
$473$	9.72441	0.447129
$474$	0	0
$475$	−6.09683	−0.279742
$476$	37.2448	1.70711
$477$	0	0
$478$	19.1537	0.876069
$479$	3.11197	0.142190	0.0710948	−	0.997470i	$-0.477351\pi$
$479$	3.11197	0.142190	0.0710948	+	0.997470i	$0.477351\pi$
$480$	0	0
$481$	0	0
$482$	−14.5781	−0.664015
$483$	0	0
$484$	−3.67830	−0.167195
$485$	−11.3969	−0.517505
$486$	0	0
$487$	−5.23540	−0.237238	−0.118619	−	0.992940i	$-0.537847\pi$
$487$	−5.23540	−0.237238	−0.118619	+	0.992940i	$0.537847\pi$
$488$	−35.9813	−1.62880
$489$	0	0
$490$	−4.72296	−0.213361
$491$	6.17499	0.278673	0.139337	−	0.990245i	$-0.455503\pi$
$491$	6.17499	0.278673	0.139337	+	0.990245i	$0.455503\pi$
$492$	0	0
$493$	−23.1826	−1.04409
$494$	0	0
$495$	0	0
$496$	−3.21581	−0.144394
$497$	−54.4809	−2.44380
$498$	0	0
$499$	−10.5068	−0.470348	−0.235174	−	0.971953i	$-0.575566\pi$
$499$	−10.5068	−0.470348	−0.235174	+	0.971953i	$0.575566\pi$
$500$	−1.36899	−0.0612230
$501$	0	0
$502$	−0.496356	−0.0221535
$503$	22.4351	1.00033	0.500166	−	0.865929i	$-0.333272\pi$
$503$	22.4351	1.00033	0.500166	+	0.865929i	$0.333272\pi$
$504$	0	0
$505$	18.6380	0.829381
$506$	14.2329	0.632730
$507$	0	0
$508$	8.92213	0.395856
$509$	7.07738	0.313699	0.156850	−	0.987622i	$-0.449866\pi$
$509$	7.07738	0.313699	0.156850	+	0.987622i	$0.449866\pi$
$510$	0	0
$511$	−45.5393	−2.01454
$512$	−6.85008	−0.302734
$513$	0	0
$514$	−18.0702	−0.797041
$515$	12.6811	0.558795
$516$	0	0
$517$	−33.2713	−1.46327
$518$	11.8476	0.520552
$519$	0	0
$520$	0	0
$521$	37.7073	1.65199	0.825993	−	0.563680i	$-0.190615\pi$
$521$	37.7073	1.65199	0.825993	+	0.563680i	$0.190615\pi$
$522$	0	0
$523$	−27.1143	−1.18563	−0.592814	−	0.805340i	$-0.701983\pi$
$523$	−27.1143	−1.18563	−0.592814	+	0.805340i	$0.701983\pi$
$524$	12.3807	0.540854
$525$	0	0
$526$	16.7124	0.728697
$527$	−39.7256	−1.73047
$528$	0	0
$529$	0.455529	0.0198056
$530$	−4.70581	−0.204407
$531$	0	0
$532$	−30.0306	−1.30199
$533$	0	0
$534$	0	0
$535$	5.89062	0.254674
$536$	13.0819	0.565051
$537$	0	0
$538$	15.0976	0.650904
$539$	−21.9962	−0.947443
$540$	0	0
$541$	−9.60518	−0.412959	−0.206479	−	0.978451i	$-0.566201\pi$
$541$	−9.60518	−0.412959	−0.206479	+	0.978451i	$0.566201\pi$
$542$	0.413131	0.0177455
$543$	0	0
$544$	−44.1485	−1.89285
$545$	3.00497	0.128719
$546$	0	0
$547$	−8.74709	−0.373999	−0.186999	−	0.982360i	$-0.559876\pi$
$547$	−8.74709	−0.373999	−0.186999	+	0.982360i	$0.559876\pi$
$548$	−8.84153	−0.377691
$549$	0	0
$550$	2.93881	0.125311
$551$	18.6922	0.796316
$552$	0	0
$553$	−0.747006	−0.0317659
$554$	19.4884	0.827981
$555$	0	0
$556$	−25.7560	−1.09230
$557$	−34.4692	−1.46051	−0.730254	−	0.683176i	$-0.760598\pi$
$557$	−34.4692	−1.46051	−0.730254	+	0.683176i	$0.760598\pi$
$558$	0	0
$559$	0	0
$560$	−2.20235	−0.0930663
$561$	0	0
$562$	−7.60559	−0.320823
$563$	1.03521	0.0436289	0.0218145	−	0.999762i	$-0.493056\pi$
$563$	1.03521	0.0436289	0.0218145	+	0.999762i	$0.493056\pi$
$564$	0	0
$565$	0.261568	0.0110042
$566$	2.70643	0.113760
$567$	0	0
$568$	40.5230	1.70031
$569$	8.23150	0.345083	0.172541	−	0.985002i	$-0.444802\pi$
$569$	8.23150	0.345083	0.172541	+	0.985002i	$0.444802\pi$
$570$	0	0
$571$	−41.6948	−1.74487	−0.872436	−	0.488728i	$-0.837461\pi$
$571$	−41.6948	−1.74487	−0.872436	+	0.488728i	$0.837461\pi$
$572$	0	0
$573$	0	0
$574$	9.68865	0.404397
$575$	4.84309	0.201971
$576$	0	0
$577$	−4.80474	−0.200024	−0.100012	−	0.994986i	$-0.531888\pi$
$577$	−4.80474	−0.200024	−0.100012	+	0.994986i	$0.531888\pi$
$578$	−31.9140	−1.32745
$579$	0	0
$580$	4.19717	0.174278
$581$	31.2169	1.29510
$582$	0	0
$583$	−21.9163	−0.907682
$584$	33.8722	1.40164
$585$	0	0
$586$	−13.6409	−0.563502
$587$	−29.6737	−1.22476	−0.612382	−	0.790562i	$-0.709789\pi$
$587$	−29.6737	−1.22476	−0.612382	+	0.790562i	$0.709789\pi$
$588$	0	0
$589$	32.0309	1.31981
$590$	6.19694	0.255124
$591$	0	0
$592$	2.53732	0.104283
$593$	−24.7381	−1.01587	−0.507936	−	0.861395i	$-0.669591\pi$
$593$	−24.7381	−1.01587	−0.507936	+	0.861395i	$0.669591\pi$
$594$	0	0
$595$	−27.2061	−1.11534
$596$	12.4306	0.509179
$597$	0	0
$598$	0	0
$599$	−29.6358	−1.21088	−0.605442	−	0.795889i	$-0.707004\pi$
$599$	−29.6358	−1.21088	−0.605442	+	0.795889i	$0.707004\pi$
$600$	0	0
$601$	−7.69070	−0.313710	−0.156855	−	0.987622i	$-0.550136\pi$
$601$	−7.69070	−0.313710	−0.156855	+	0.987622i	$0.550136\pi$
$602$	−7.51261	−0.306191
$603$	0	0
$604$	−10.1305	−0.412206
$605$	2.68687	0.109237
$606$	0	0
$607$	−23.0449	−0.935364	−0.467682	−	0.883897i	$-0.654911\pi$
$607$	−23.0449	−0.935364	−0.467682	+	0.883897i	$0.654911\pi$
$608$	35.5971	1.44365
$609$	0	0
$610$	10.6801	0.432426
$611$	0	0
$612$	0	0
$613$	−3.61531	−0.146021	−0.0730106	−	0.997331i	$-0.523261\pi$
$613$	−3.61531	−0.146021	−0.0730106	+	0.997331i	$0.523261\pi$
$614$	0.842475	0.0339995
$615$	0	0
$616$	35.6231	1.43529
$617$	25.2725	1.01743	0.508717	−	0.860934i	$-0.330120\pi$
$617$	25.2725	1.01743	0.508717	+	0.860934i	$0.330120\pi$
$618$	0	0
$619$	−5.67627	−0.228149	−0.114074	−	0.993472i	$-0.536390\pi$
$619$	−5.67627	−0.228149	−0.114074	+	0.993472i	$0.536390\pi$
$620$	7.19224	0.288847
$621$	0	0
$622$	18.9276	0.758927
$623$	47.6764	1.91012
$624$	0	0
$625$	1.00000	0.0400000
$626$	5.42498	0.216826
$627$	0	0
$628$	15.4856	0.617944
$629$	31.3440	1.24977
$630$	0	0
$631$	7.05103	0.280697	0.140349	−	0.990102i	$-0.455178\pi$
$631$	7.05103	0.280697	0.140349	+	0.990102i	$0.455178\pi$
$632$	0.555625	0.0221016
$633$	0	0
$634$	16.3980	0.651249
$635$	−6.51732	−0.258632
$636$	0	0
$637$	0	0
$638$	−9.01007	−0.356712
$639$	0	0
$640$	8.96548	0.354392
$641$	−21.9797	−0.868145	−0.434073	−	0.900878i	$-0.642924\pi$
$641$	−21.9797	−0.868145	−0.434073	+	0.900878i	$0.642924\pi$
$642$	0	0
$643$	−20.6748	−0.815334	−0.407667	−	0.913131i	$-0.633657\pi$
$643$	−20.6748	−0.815334	−0.407667	+	0.913131i	$0.633657\pi$
$644$	23.8552	0.940027
$645$	0	0
$646$	36.6208	1.44083
$647$	−13.2573	−0.521198	−0.260599	−	0.965447i	$-0.583920\pi$
$647$	−13.2573	−0.521198	−0.260599	+	0.965447i	$0.583920\pi$
$648$	0	0
$649$	28.8610	1.13289
$650$	0	0
$651$	0	0
$652$	−3.34934	−0.131170
$653$	20.9720	0.820699	0.410349	−	0.911928i	$-0.365407\pi$
$653$	20.9720	0.820699	0.410349	+	0.911928i	$0.365407\pi$
$654$	0	0
$655$	−9.04369	−0.353366
$656$	2.07496	0.0810135
$657$	0	0
$658$	25.7038	1.00204
$659$	18.4646	0.719278	0.359639	−	0.933092i	$-0.382900\pi$
$659$	18.4646	0.719278	0.359639	+	0.933092i	$0.382900\pi$
$660$	0	0
$661$	−38.0221	−1.47889	−0.739444	−	0.673218i	$-0.764912\pi$
$661$	−38.0221	−1.47889	−0.739444	+	0.673218i	$0.764912\pi$
$662$	−23.3706	−0.908326
$663$	0	0
$664$	−23.2192	−0.901081
$665$	21.9364	0.850656
$666$	0	0
$667$	−14.8484	−0.574933
$668$	8.47345	0.327848
$669$	0	0
$670$	−3.88303	−0.150015
$671$	49.7405	1.92021
$672$	0	0
$673$	27.0828	1.04396	0.521982	−	0.852956i	$-0.325193\pi$
$673$	27.0828	1.04396	0.521982	+	0.852956i	$0.325193\pi$
$674$	0.714824	0.0275340
$675$	0	0
$676$	0	0
$677$	35.5804	1.36747	0.683733	−	0.729732i	$-0.260355\pi$
$677$	35.5804	1.36747	0.683733	+	0.729732i	$0.260355\pi$
$678$	0	0
$679$	41.0059	1.57366
$680$	20.2360	0.776013
$681$	0	0
$682$	−15.4396	−0.591212
$683$	16.2500	0.621790	0.310895	−	0.950444i	$-0.399371\pi$
$683$	16.2500	0.621790	0.310895	+	0.950444i	$0.399371\pi$
$684$	0	0
$685$	6.45844	0.246764
$686$	−3.01362	−0.115060
$687$	0	0
$688$	−1.60893	−0.0613398
$689$	0	0
$690$	0	0
$691$	−3.65243	−0.138945	−0.0694724	−	0.997584i	$-0.522132\pi$
$691$	−3.65243	−0.138945	−0.0694724	+	0.997584i	$0.522132\pi$
$692$	7.47746	0.284251
$693$	0	0
$694$	−1.95901	−0.0743628
$695$	18.8139	0.713652
$696$	0	0
$697$	25.6324	0.970896
$698$	−1.74557	−0.0660710
$699$	0	0
$700$	4.92562	0.186171
$701$	−7.49092	−0.282928	−0.141464	−	0.989943i	$-0.545181\pi$
$701$	−7.49092	−0.282928	−0.141464	+	0.989943i	$0.545181\pi$
$702$	0	0
$703$	−25.2728	−0.953182
$704$	−12.6295	−0.475994
$705$	0	0
$706$	−1.33266	−0.0501555
$707$	−67.0595	−2.52203
$708$	0	0
$709$	15.7402	0.591137	0.295568	−	0.955322i	$-0.404491\pi$
$709$	15.7402	0.591137	0.295568	+	0.955322i	$0.404491\pi$
$710$	−12.0282	−0.451412
$711$	0	0
$712$	−35.4618	−1.32899
$713$	−25.4441	−0.952890
$714$	0	0
$715$	0	0
$716$	2.58326	0.0965411
$717$	0	0
$718$	3.33145	0.124329
$719$	22.5451	0.840790	0.420395	−	0.907341i	$-0.361892\pi$
$719$	22.5451	0.840790	0.420395	+	0.907341i	$0.361892\pi$
$720$	0	0
$721$	−45.6265	−1.69922
$722$	−14.4346	−0.537200
$723$	0	0
$724$	−19.8653	−0.738288
$725$	−3.06590	−0.113864
$726$	0	0
$727$	−27.6723	−1.02631	−0.513155	−	0.858296i	$-0.671523\pi$
$727$	−27.6723	−1.02631	−0.513155	+	0.858296i	$0.671523\pi$
$728$	0	0
$729$	0	0
$730$	−10.0541	−0.372119
$731$	−19.8754	−0.735119
$732$	0	0
$733$	33.5438	1.23897	0.619483	−	0.785010i	$-0.287342\pi$
$733$	33.5438	1.23897	0.619483	+	0.785010i	$0.287342\pi$
$734$	18.7004	0.690246
$735$	0	0
$736$	−28.2770	−1.04230
$737$	−18.0844	−0.666148
$738$	0	0
$739$	−50.5719	−1.86032	−0.930158	−	0.367160i	$-0.880330\pi$
$739$	−50.5719	−1.86032	−0.930158	+	0.367160i	$0.880330\pi$
$740$	−5.67478	−0.208609
$741$	0	0
$742$	16.9315	0.621575
$743$	−1.69152	−0.0620557	−0.0310279	−	0.999519i	$-0.509878\pi$
$743$	−1.69152	−0.0620557	−0.0310279	+	0.999519i	$0.509878\pi$
$744$	0	0
$745$	−9.08017	−0.332672
$746$	5.62041	0.205778
$747$	0	0
$748$	38.2963	1.40025
$749$	−21.1944	−0.774428
$750$	0	0
$751$	33.6681	1.22857	0.614283	−	0.789086i	$-0.289445\pi$
$751$	33.6681	1.22857	0.614283	+	0.789086i	$0.289445\pi$
$752$	5.50482	0.200740
$753$	0	0
$754$	0	0
$755$	7.40002	0.269314
$756$	0	0
$757$	−49.8344	−1.81126	−0.905631	−	0.424067i	$-0.860602\pi$
$757$	−49.8344	−1.81126	−0.905631	+	0.424067i	$0.860602\pi$
$758$	1.53285	0.0556756
$759$	0	0
$760$	−16.3163	−0.591855
$761$	−21.3347	−0.773383	−0.386691	−	0.922209i	$-0.626382\pi$
$761$	−21.3347	−0.773383	−0.386691	+	0.922209i	$0.626382\pi$
$762$	0	0
$763$	−10.8119	−0.391417
$764$	18.1198	0.655552
$765$	0	0
$766$	−15.6459	−0.565311
$767$	0	0
$768$	0	0
$769$	−10.8212	−0.390223	−0.195111	−	0.980781i	$-0.562507\pi$
$769$	−10.8212	−0.390223	−0.195111	+	0.980781i	$0.562507\pi$
$770$	−10.5738	−0.381054
$771$	0	0
$772$	11.6183	0.418153
$773$	9.05930	0.325840	0.162920	−	0.986639i	$-0.447909\pi$
$773$	9.05930	0.325840	0.162920	+	0.986639i	$0.447909\pi$
$774$	0	0
$775$	−5.25369	−0.188718
$776$	−30.5003	−1.09490
$777$	0	0
$778$	−1.03225	−0.0370081
$779$	−20.6675	−0.740489
$780$	0	0
$781$	−56.0190	−2.00452
$782$	−29.0902	−1.04026
$783$	0	0
$784$	3.63932	0.129976
$785$	−11.3117	−0.403733
$786$	0	0
$787$	−52.4800	−1.87071	−0.935355	−	0.353712i	$-0.884919\pi$
$787$	−52.4800	−1.87071	−0.935355	+	0.353712i	$0.884919\pi$
$788$	15.8336	0.564049
$789$	0	0
$790$	−0.164923	−0.00586771
$791$	−0.941121	−0.0334624
$792$	0	0
$793$	0	0
$794$	1.37390	0.0487577
$795$	0	0
$796$	7.63431	0.270591
$797$	18.6687	0.661279	0.330640	−	0.943757i	$-0.392736\pi$
$797$	18.6687	0.661279	0.330640	+	0.943757i	$0.392736\pi$
$798$	0	0
$799$	68.0022	2.40575
$800$	−5.83863	−0.206427
$801$	0	0
$802$	−0.878924	−0.0310359
$803$	−46.8249	−1.65242
$804$	0	0
$805$	−17.4254	−0.614166
$806$	0	0
$807$	0	0
$808$	49.8790	1.75474
$809$	−40.1781	−1.41259	−0.706293	−	0.707920i	$-0.749634\pi$
$809$	−40.1781	−1.41259	−0.706293	+	0.707920i	$0.749634\pi$
$810$	0	0
$811$	−20.8571	−0.732390	−0.366195	−	0.930538i	$-0.619340\pi$
$811$	−20.8571	−0.732390	−0.366195	+	0.930538i	$0.619340\pi$
$812$	−15.1014	−0.529956
$813$	0	0
$814$	12.1821	0.426981
$815$	2.44658	0.0856999
$816$	0	0
$817$	16.0256	0.560666
$818$	−26.5220	−0.927319
$819$	0	0
$820$	−4.64070	−0.162060
$821$	38.5261	1.34457	0.672286	−	0.740292i	$-0.265313\pi$
$821$	38.5261	1.34457	0.672286	+	0.740292i	$0.265313\pi$
$822$	0	0
$823$	−28.5502	−0.995196	−0.497598	−	0.867408i	$-0.665785\pi$
$823$	−28.5502	−0.995196	−0.497598	+	0.867408i	$0.665785\pi$
$824$	33.9371	1.18225
$825$	0	0
$826$	−22.2966	−0.775797
$827$	−21.5230	−0.748428	−0.374214	−	0.927342i	$-0.622087\pi$
$827$	−21.5230	−0.748428	−0.374214	+	0.927342i	$0.622087\pi$
$828$	0	0
$829$	−24.0960	−0.836889	−0.418445	−	0.908242i	$-0.637425\pi$
$829$	−24.0960	−0.836889	−0.418445	+	0.908242i	$0.637425\pi$
$830$	6.89205	0.239226
$831$	0	0
$832$	0	0
$833$	44.9573	1.55768
$834$	0	0
$835$	−6.18957	−0.214199
$836$	−30.8785	−1.06795
$837$	0	0
$838$	14.4600	0.499513
$839$	42.0604	1.45209	0.726043	−	0.687649i	$-0.241357\pi$
$839$	42.0604	1.45209	0.726043	+	0.687649i	$0.241357\pi$
$840$	0	0
$841$	−19.6003	−0.675872
$842$	1.16932	0.0402974
$843$	0	0
$844$	13.7521	0.473366
$845$	0	0
$846$	0	0
$847$	−9.66737	−0.332175
$848$	3.62611	0.124521
$849$	0	0
$850$	−6.00654	−0.206023
$851$	20.0758	0.688189
$852$	0	0
$853$	−18.5814	−0.636214	−0.318107	−	0.948055i	$-0.603047\pi$
$853$	−18.5814	−0.636214	−0.318107	+	0.948055i	$0.603047\pi$
$854$	−38.4271	−1.31495
$855$	0	0
$856$	15.7645	0.538819
$857$	20.5905	0.703359	0.351679	−	0.936120i	$-0.385611\pi$
$857$	20.5905	0.703359	0.351679	+	0.936120i	$0.385611\pi$
$858$	0	0
$859$	−33.4044	−1.13974	−0.569872	−	0.821733i	$-0.693007\pi$
$859$	−33.4044	−1.13974	−0.569872	+	0.821733i	$0.693007\pi$
$860$	3.59841	0.122705
$861$	0	0
$862$	20.5802	0.700965
$863$	7.73166	0.263189	0.131594	−	0.991304i	$-0.457990\pi$
$863$	7.73166	0.263189	0.131594	+	0.991304i	$0.457990\pi$
$864$	0	0
$865$	−5.46204	−0.185715
$866$	−18.9640	−0.644422
$867$	0	0
$868$	−25.8777	−0.878346
$869$	−0.768096	−0.0260559
$870$	0	0
$871$	0	0
$872$	8.04190	0.272333
$873$	0	0
$874$	23.4555	0.793395
$875$	−3.59800	−0.121635
$876$	0	0
$877$	18.5450	0.626221	0.313111	−	0.949717i	$-0.398629\pi$
$877$	18.5450	0.626221	0.313111	+	0.949717i	$0.398629\pi$
$878$	22.5403	0.760700
$879$	0	0
$880$	−2.26453	−0.0763372
$881$	14.7031	0.495362	0.247681	−	0.968842i	$-0.420332\pi$
$881$	14.7031	0.495362	0.247681	+	0.968842i	$0.420332\pi$
$882$	0	0
$883$	32.7770	1.10303	0.551517	−	0.834163i	$-0.314049\pi$
$883$	32.7770	1.10303	0.551517	+	0.834163i	$0.314049\pi$
$884$	0	0
$885$	0	0
$886$	9.98638	0.335499
$887$	11.6346	0.390653	0.195326	−	0.980738i	$-0.437423\pi$
$887$	11.6346	0.390653	0.195326	+	0.980738i	$0.437423\pi$
$888$	0	0
$889$	23.4493	0.786464
$890$	10.5260	0.352831
$891$	0	0
$892$	9.60951	0.321750
$893$	−54.8304	−1.83483
$894$	0	0
$895$	−1.88699	−0.0630750
$896$	−32.2578	−1.07766
$897$	0	0
$898$	29.2367	0.975642
$899$	16.1073	0.537208
$900$	0	0
$901$	44.7941	1.49231
$902$	9.96219	0.331704
$903$	0	0
$904$	0.700008	0.0232819
$905$	14.5109	0.482360
$906$	0	0
$907$	51.2083	1.70034	0.850171	−	0.526506i	$-0.176498\pi$
$907$	51.2083	1.70034	0.850171	+	0.526506i	$0.176498\pi$
$908$	29.1868	0.968598
$909$	0	0
$910$	0	0
$911$	−34.9806	−1.15896	−0.579480	−	0.814986i	$-0.696744\pi$
$911$	−34.9806	−1.15896	−0.579480	+	0.814986i	$0.696744\pi$
$912$	0	0
$913$	32.0983	1.06230
$914$	5.34805	0.176898
$915$	0	0
$916$	27.2401	0.900038
$917$	32.5392	1.07454
$918$	0	0
$919$	−26.5092	−0.874458	−0.437229	−	0.899350i	$-0.644040\pi$
$919$	−26.5092	−0.874458	−0.437229	+	0.899350i	$0.644040\pi$
$920$	12.9611	0.427314
$921$	0	0
$922$	23.8700	0.786117
$923$	0	0
$924$	0	0
$925$	4.14524	0.136295
$926$	14.3379	0.471173
$927$	0	0
$928$	17.9006	0.587617
$929$	11.6119	0.380975	0.190487	−	0.981690i	$-0.438993\pi$
$929$	11.6119	0.380975	0.190487	+	0.981690i	$0.438993\pi$
$930$	0	0
$931$	−36.2493	−1.18802
$932$	−3.25402	−0.106589
$933$	0	0
$934$	−20.8829	−0.683308
$935$	−27.9742	−0.914854
$936$	0	0
$937$	53.0400	1.73274	0.866370	−	0.499403i	$-0.166447\pi$
$937$	53.0400	1.73274	0.866370	+	0.499403i	$0.166447\pi$
$938$	13.9711	0.456174
$939$	0	0
$940$	−12.3117	−0.401563
$941$	−28.2633	−0.921356	−0.460678	−	0.887567i	$-0.652394\pi$
$941$	−28.2633	−0.921356	−0.460678	+	0.887567i	$0.652394\pi$
$942$	0	0
$943$	16.4175	0.534627
$944$	−4.77512	−0.155417
$945$	0	0
$946$	−7.72471	−0.251152
$947$	−14.7748	−0.480115	−0.240057	−	0.970759i	$-0.577166\pi$
$947$	−14.7748	−0.480115	−0.240057	+	0.970759i	$0.577166\pi$
$948$	0	0
$949$	0	0
$950$	4.84309	0.157131
$951$	0	0
$952$	−72.8089	−2.35975
$953$	−30.5951	−0.991072	−0.495536	−	0.868587i	$-0.665028\pi$
$953$	−30.5951	−0.991072	−0.495536	+	0.868587i	$0.665028\pi$
$954$	0	0
$955$	−13.2359	−0.428304
$956$	33.0091	1.06759
$957$	0	0
$958$	−2.47203	−0.0798678
$959$	−23.2375	−0.750377
$960$	0	0
$961$	−3.39870	−0.109636
$962$	0	0
$963$	0	0
$964$	−25.1236	−0.809177
$965$	−8.48680	−0.273200
$966$	0	0
$967$	29.3381	0.943451	0.471726	−	0.881745i	$-0.343631\pi$
$967$	29.3381	0.943451	0.471726	+	0.881745i	$0.343631\pi$
$968$	7.19061	0.231115
$969$	0	0
$970$	9.05325	0.290682
$971$	−35.0961	−1.12629	−0.563143	−	0.826359i	$-0.690408\pi$
$971$	−35.0961	−1.12629	−0.563143	+	0.826359i	$0.690408\pi$
$972$	0	0
$973$	−67.6924	−2.17012
$974$	4.15880	0.133257
$975$	0	0
$976$	−8.22970	−0.263426
$977$	23.4663	0.750754	0.375377	−	0.926872i	$-0.377513\pi$
$977$	23.4663	0.750754	0.375377	+	0.926872i	$0.377513\pi$
$978$	0	0
$979$	49.0224	1.56676
$980$	−8.13945	−0.260005
$981$	0	0
$982$	−4.90518	−0.156531
$983$	−19.0785	−0.608511	−0.304255	−	0.952591i	$-0.598408\pi$
$983$	−19.0785	−0.608511	−0.304255	+	0.952591i	$0.598408\pi$
$984$	0	0
$985$	−11.5659	−0.368521
$986$	18.4154	0.586466
$987$	0	0
$988$	0	0
$989$	−12.7302	−0.404796
$990$	0	0
$991$	23.7934	0.755822	0.377911	−	0.925842i	$-0.376643\pi$
$991$	23.7934	0.755822	0.377911	+	0.925842i	$0.376643\pi$
$992$	30.6744	0.973912
$993$	0	0
$994$	43.2776	1.37268
$995$	−5.57661	−0.176790
$996$	0	0
$997$	−54.0849	−1.71289	−0.856444	−	0.516241i	$-0.827331\pi$
$997$	−54.0849	−1.71289	−0.856444	+	0.516241i	$0.827331\pi$
$998$	8.34620	0.264194
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7605.2.a.ct.1.5		10
3.2	odd	2		7605.2.a.cu.1.5		10
13.2	odd	12		585.2.bu.e.316.5	✓	20
13.7	odd	12		585.2.bu.e.361.5	yes	20
13.12	even	2		7605.2.a.cu.1.6		10
39.2	even	12		585.2.bu.e.316.6	yes	20
39.20	even	12		585.2.bu.e.361.6	yes	20
39.38	odd	2	inner	7605.2.a.ct.1.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
585.2.bu.e.316.5	✓	20	13.2	odd	12
585.2.bu.e.316.6	yes	20	39.2	even	12
585.2.bu.e.361.5	yes	20	13.7	odd	12
585.2.bu.e.361.6	yes	20	39.20	even	12
7605.2.a.ct.1.5		10	1.1	even	1	trivial
7605.2.a.ct.1.6		10	39.38	odd	2	inner
7605.2.a.cu.1.5		10	3.2	odd	2
7605.2.a.cu.1.6		10	13.12	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:	\( N \)	\(=\)	\( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7605.a (trivial)

\(f(q)\)	\(=\)	\(q-0.794363 q^{2} -1.36899 q^{4} +1.00000 q^{5} -3.59800 q^{7} +2.67620 q^{8} +O(q^{10})\)	\(q-0.794363 q^{2} -1.36899 q^{4} +1.00000 q^{5} -3.59800 q^{7} +2.67620 q^{8} -0.794363 q^{10} -3.69958 q^{11} +2.85812 q^{14} +0.612105 q^{16} +7.56145 q^{17} -6.09683 q^{19} -1.36899 q^{20} +2.93881 q^{22} +4.84309 q^{23} +1.00000 q^{25} +4.92562 q^{28} -3.06590 q^{29} -5.25369 q^{31} -5.83863 q^{32} -6.00654 q^{34} -3.59800 q^{35} +4.14524 q^{37} +4.84309 q^{38} +2.67620 q^{40} +3.38988 q^{41} -2.62852 q^{43} +5.06468 q^{44} -3.84717 q^{46} +8.99327 q^{47} +5.94559 q^{49} -0.794363 q^{50} +5.92401 q^{53} -3.69958 q^{55} -9.62896 q^{56} +2.43543 q^{58} -7.80115 q^{59} -13.4449 q^{61} +4.17334 q^{62} +3.41378 q^{64} +4.88824 q^{67} -10.3515 q^{68} +2.85812 q^{70} +15.1420 q^{71} +12.6568 q^{73} -3.29282 q^{74} +8.34648 q^{76} +13.3111 q^{77} +0.207617 q^{79} +0.612105 q^{80} -2.69279 q^{82} -8.67620 q^{83} +7.56145 q^{85} +2.08800 q^{86} -9.90080 q^{88} -13.2508 q^{89} -6.63013 q^{92} -7.14392 q^{94} -6.09683 q^{95} -11.3969 q^{97} -4.72296 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 4 q^{2} + 12 q^{4} + 10 q^{5} - 12 q^{8}+O(q^{10}) \) 10 * q - 4 * q^2 + 12 * q^4 + 10 * q^5 - 12 * q^8	\( 10 q - 4 q^{2} + 12 q^{4} + 10 q^{5} - 12 q^{8} - 4 q^{10} - 24 q^{11} + 24 q^{16} + 12 q^{20} - 12 q^{22} + 10 q^{25} - 68 q^{32} - 12 q^{40} - 4 q^{41} - 14 q^{43} - 60 q^{44} - 4 q^{47} - 8 q^{49} - 4 q^{50} - 24 q^{55} - 48 q^{59} + 10 q^{61} + 88 q^{64} - 44 q^{71} - 14 q^{79} + 24 q^{80} - 40 q^{82} - 48 q^{83} + 56 q^{86} + 48 q^{88} - 72 q^{89} + 32 q^{94} - 60 q^{98}+O(q^{100}) \) 10 * q - 4 * q^2 + 12 * q^4 + 10 * q^5 - 12 * q^8 - 4 * q^10 - 24 * q^11 + 24 * q^16 + 12 * q^20 - 12 * q^22 + 10 * q^25 - 68 * q^32 - 12 * q^40 - 4 * q^41 - 14 * q^43 - 60 * q^44 - 4 * q^47 - 8 * q^49 - 4 * q^50 - 24 * q^55 - 48 * q^59 + 10 * q^61 + 88 * q^64 - 44 * q^71 - 14 * q^79 + 24 * q^80 - 40 * q^82 - 48 * q^83 + 56 * q^86 + 48 * q^88 - 72 * q^89 + 32 * q^94 - 60 * q^98

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