LMFDB - Embedded newform 8664.2.a.br.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8664,2,Mod(1,8664)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8664, 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("8664.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8664 = 2^{3} \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8664.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:	$69.1823883112$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} - 37 x^{10} + 52 x^{9} + 526 x^{8} - 414 x^{7} - 3501 x^{6} + 832 x^{5} + 10258 x^{4} + \cdots + 361 $ x^12 - 2x^11 - 37x^10 + 52x^9 + 526x^8 - 414x^7 - 3501x^6 + 832x^5 + 10258x^4 + 1506x^3 - 8617x^2 - 684*x + 361
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.78590$ of defining polynomial
Character	$\chi$	$=$	8664.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -2.78590 q^{5} -4.08002 q^{7} +1.00000 q^{9} +5.94993 q^{11} -6.50317 q^{13} +2.78590 q^{15} -7.02253 q^{17} +4.08002 q^{21} +3.16242 q^{23} +2.76126 q^{25} -1.00000 q^{27} -2.36833 q^{29} -1.04900 q^{31} -5.94993 q^{33} +11.3666 q^{35} -4.23995 q^{37} +6.50317 q^{39} +0.889974 q^{41} -9.87935 q^{43} -2.78590 q^{45} -6.29928 q^{47} +9.64658 q^{49} +7.02253 q^{51} -7.08537 q^{53} -16.5759 q^{55} +6.50012 q^{59} -12.7346 q^{61} -4.08002 q^{63} +18.1172 q^{65} -1.33860 q^{67} -3.16242 q^{69} -9.70111 q^{71} -13.8287 q^{73} -2.76126 q^{75} -24.2758 q^{77} -11.8181 q^{79} +1.00000 q^{81} -1.31483 q^{83} +19.5641 q^{85} +2.36833 q^{87} -15.5577 q^{89} +26.5331 q^{91} +1.04900 q^{93} -2.18852 q^{97} +5.94993 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{3} + 2 q^{5} + 4 q^{7} + 12 q^{9} + 14 q^{11} + 4 q^{13} - 2 q^{15} - 4 q^{21} + 26 q^{23} + 18 q^{25} - 12 q^{27} - 10 q^{29} + 8 q^{31} - 14 q^{33} + 30 q^{35} - 8 q^{37} - 4 q^{39} + 18 q^{41}+ \cdots + 14 q^{99}+O(q^{100}) $ 12 * q - 12 * q^3 + 2 * q^5 + 4 * q^7 + 12 * q^9 + 14 * q^11 + 4 * q^13 - 2 * q^15 - 4 * q^21 + 26 * q^23 + 18 * q^25 - 12 * q^27 - 10 * q^29 + 8 * q^31 - 14 * q^33 + 30 * q^35 - 8 * q^37 - 4 * q^39 + 18 * q^41 - 8 * q^43 + 2 * q^45 + 48 * q^49 - 28 * q^53 + 2 * q^55 + 6 * q^59 + 38 * q^61 + 4 * q^63 + 24 * q^65 - 16 * q^67 - 26 * q^69 - 38 * q^71 + 28 * q^73 - 18 * q^75 + 4 * q^77 - 40 * q^79 + 12 * q^81 + 18 * q^83 + 54 * q^85 + 10 * q^87 + 24 * q^89 + 20 * q^91 - 8 * q^93 + 24 * q^97 + 14 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−2.78590	−1.24589	−0.622947	−	0.782264i	$-0.714065\pi$
$5$	−2.78590	−1.24589	−0.622947	+	0.782264i	$0.714065\pi$
$6$	0	0
$7$	−4.08002	−1.54210	−0.771052	−	0.636772i	$-0.780269\pi$
$7$	−4.08002	−1.54210	−0.771052	+	0.636772i	$0.780269\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.94993	1.79397	0.896986	−	0.442060i	$-0.145752\pi$
$11$	5.94993	1.79397	0.896986	+	0.442060i	$0.145752\pi$
$12$	0	0
$13$	−6.50317	−1.80366	−0.901828	−	0.432096i	$-0.857774\pi$
$13$	−6.50317	−1.80366	−0.901828	+	0.432096i	$0.857774\pi$
$14$	0	0
$15$	2.78590	0.719317
$16$	0	0
$17$	−7.02253	−1.70321	−0.851607	−	0.524182i	$-0.824371\pi$
$17$	−7.02253	−1.70321	−0.851607	+	0.524182i	$0.824371\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	4.08002	0.890334
$22$	0	0
$23$	3.16242	0.659410	0.329705	−	0.944084i	$-0.393051\pi$
$23$	3.16242	0.659410	0.329705	+	0.944084i	$0.393051\pi$
$24$	0	0
$25$	2.76126	0.552252
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.36833	−0.439787	−0.219894	−	0.975524i	$-0.570571\pi$
$29$	−2.36833	−0.439787	−0.219894	+	0.975524i	$0.570571\pi$
$30$	0	0
$31$	−1.04900	−0.188406	−0.0942028	−	0.995553i	$-0.530030\pi$
$31$	−1.04900	−0.188406	−0.0942028	+	0.995553i	$0.530030\pi$
$32$	0	0
$33$	−5.94993	−1.03575
$34$	0	0
$35$	11.3666	1.92130
$36$	0	0
$37$	−4.23995	−0.697043	−0.348522	−	0.937301i	$-0.613316\pi$
$37$	−4.23995	−0.697043	−0.348522	+	0.937301i	$0.613316\pi$
$38$	0	0
$39$	6.50317	1.04134
$40$	0	0
$41$	0.889974	0.138991	0.0694953	−	0.997582i	$-0.477861\pi$
$41$	0.889974	0.138991	0.0694953	+	0.997582i	$0.477861\pi$
$42$	0	0
$43$	−9.87935	−1.50659	−0.753294	−	0.657684i	$-0.771536\pi$
$43$	−9.87935	−1.50659	−0.753294	+	0.657684i	$0.771536\pi$
$44$	0	0
$45$	−2.78590	−0.415298
$46$	0	0
$47$	−6.29928	−0.918844	−0.459422	−	0.888218i	$-0.651943\pi$
$47$	−6.29928	−0.918844	−0.459422	+	0.888218i	$0.651943\pi$
$48$	0	0
$49$	9.64658	1.37808
$50$	0	0
$51$	7.02253	0.983351
$52$	0	0
$53$	−7.08537	−0.973251	−0.486625	−	0.873611i	$-0.661772\pi$
$53$	−7.08537	−0.973251	−0.486625	+	0.873611i	$0.661772\pi$
$54$	0	0
$55$	−16.5759	−2.23510
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.50012	0.846244	0.423122	−	0.906073i	$-0.360934\pi$
$59$	6.50012	0.846244	0.423122	+	0.906073i	$0.360934\pi$
$60$	0	0
$61$	−12.7346	−1.63049	−0.815246	−	0.579114i	$-0.803399\pi$
$61$	−12.7346	−1.63049	−0.815246	+	0.579114i	$0.803399\pi$
$62$	0	0
$63$	−4.08002	−0.514035
$64$	0	0
$65$	18.1172	2.24716
$66$	0	0
$67$	−1.33860	−0.163537	−0.0817683	−	0.996651i	$-0.526057\pi$
$67$	−1.33860	−0.163537	−0.0817683	+	0.996651i	$0.526057\pi$
$68$	0	0
$69$	−3.16242	−0.380710
$70$	0	0
$71$	−9.70111	−1.15131	−0.575655	−	0.817693i	$-0.695253\pi$
$71$	−9.70111	−1.15131	−0.575655	+	0.817693i	$0.695253\pi$
$72$	0	0
$73$	−13.8287	−1.61853	−0.809264	−	0.587445i	$-0.800134\pi$
$73$	−13.8287	−1.61853	−0.809264	+	0.587445i	$0.800134\pi$
$74$	0	0
$75$	−2.76126	−0.318843
$76$	0	0
$77$	−24.2758	−2.76649
$78$	0	0
$79$	−11.8181	−1.32964	−0.664820	−	0.747003i	$-0.731492\pi$
$79$	−11.8181	−1.32964	−0.664820	+	0.747003i	$0.731492\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.31483	−0.144321	−0.0721605	−	0.997393i	$-0.522989\pi$
$83$	−1.31483	−0.144321	−0.0721605	+	0.997393i	$0.522989\pi$
$84$	0	0
$85$	19.5641	2.12202
$86$	0	0
$87$	2.36833	0.253911
$88$	0	0
$89$	−15.5577	−1.64911	−0.824555	−	0.565782i	$-0.808574\pi$
$89$	−15.5577	−1.64911	−0.824555	+	0.565782i	$0.808574\pi$
$90$	0	0
$91$	26.5331	2.78142
$92$	0	0
$93$	1.04900	0.108776
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.18852	−0.222211	−0.111105	−	0.993809i	$-0.535439\pi$
$97$	−2.18852	−0.222211	−0.111105	+	0.993809i	$0.535439\pi$
$98$	0	0
$99$	5.94993	0.597990
$100$	0	0
$101$	0.288061	0.0286631	0.0143316	−	0.999897i	$-0.495438\pi$
$101$	0.288061	0.0286631	0.0143316	+	0.999897i	$0.495438\pi$
$102$	0	0
$103$	−7.11901	−0.701457	−0.350728	−	0.936477i	$-0.614066\pi$
$103$	−7.11901	−0.701457	−0.350728	+	0.936477i	$0.614066\pi$
$104$	0	0
$105$	−11.3666	−1.10926
$106$	0	0
$107$	8.13769	0.786701	0.393350	−	0.919389i	$-0.371316\pi$
$107$	8.13769	0.786701	0.393350	+	0.919389i	$0.371316\pi$
$108$	0	0
$109$	−1.45359	−0.139229	−0.0696144	−	0.997574i	$-0.522177\pi$
$109$	−1.45359	−0.139229	−0.0696144	+	0.997574i	$0.522177\pi$
$110$	0	0
$111$	4.23995	0.402438
$112$	0	0
$113$	7.67440	0.721947	0.360973	−	0.932576i	$-0.382444\pi$
$113$	7.67440	0.721947	0.360973	+	0.932576i	$0.382444\pi$
$114$	0	0
$115$	−8.81019	−0.821555
$116$	0	0
$117$	−6.50317	−0.601219
$118$	0	0
$119$	28.6521	2.62653
$120$	0	0
$121$	24.4017	2.21833
$122$	0	0
$123$	−0.889974	−0.0802463
$124$	0	0
$125$	6.23691	0.557846
$126$	0	0
$127$	−1.19191	−0.105765	−0.0528826	−	0.998601i	$-0.516841\pi$
$127$	−1.19191	−0.105765	−0.0528826	+	0.998601i	$0.516841\pi$
$128$	0	0
$129$	9.87935	0.869828
$130$	0	0
$131$	−2.87735	−0.251395	−0.125697	−	0.992069i	$-0.540117\pi$
$131$	−2.87735	−0.251395	−0.125697	+	0.992069i	$0.540117\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.78590	0.239772
$136$	0	0
$137$	5.85903	0.500570	0.250285	−	0.968172i	$-0.419476\pi$
$137$	5.85903	0.500570	0.250285	+	0.968172i	$0.419476\pi$
$138$	0	0
$139$	−9.54067	−0.809229	−0.404615	−	0.914487i	$-0.632594\pi$
$139$	−9.54067	−0.809229	−0.404615	+	0.914487i	$0.632594\pi$
$140$	0	0
$141$	6.29928	0.530495
$142$	0	0
$143$	−38.6934	−3.23571
$144$	0	0
$145$	6.59793	0.547928
$146$	0	0
$147$	−9.64658	−0.795637
$148$	0	0
$149$	10.4298	0.854443	0.427222	−	0.904147i	$-0.359492\pi$
$149$	10.4298	0.854443	0.427222	+	0.904147i	$0.359492\pi$
$150$	0	0
$151$	−8.51391	−0.692852	−0.346426	−	0.938077i	$-0.612605\pi$
$151$	−8.51391	−0.692852	−0.346426	+	0.938077i	$0.612605\pi$
$152$	0	0
$153$	−7.02253	−0.567738
$154$	0	0
$155$	2.92241	0.234733
$156$	0	0
$157$	5.11635	0.408329	0.204165	−	0.978937i	$-0.434552\pi$
$157$	5.11635	0.408329	0.204165	+	0.978937i	$0.434552\pi$
$158$	0	0
$159$	7.08537	0.561907
$160$	0	0
$161$	−12.9027	−1.01688
$162$	0	0
$163$	1.52781	0.119667	0.0598337	−	0.998208i	$-0.480943\pi$
$163$	1.52781	0.119667	0.0598337	+	0.998208i	$0.480943\pi$
$164$	0	0
$165$	16.5759	1.29043
$166$	0	0
$167$	0.0473249	0.00366211	0.00183106	−	0.999998i	$-0.499417\pi$
$167$	0.0473249	0.00366211	0.00183106	+	0.999998i	$0.499417\pi$
$168$	0	0
$169$	29.2913	2.25317
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−16.6260	−1.26405	−0.632025	−	0.774948i	$-0.717776\pi$
$173$	−16.6260	−1.26405	−0.632025	+	0.774948i	$0.717776\pi$
$174$	0	0
$175$	−11.2660	−0.851630
$176$	0	0
$177$	−6.50012	−0.488579
$178$	0	0
$179$	−23.1274	−1.72862	−0.864310	−	0.502960i	$-0.832244\pi$
$179$	−23.1274	−1.72862	−0.864310	+	0.502960i	$0.832244\pi$
$180$	0	0
$181$	6.56843	0.488227	0.244114	−	0.969747i	$-0.421503\pi$
$181$	6.56843	0.488227	0.244114	+	0.969747i	$0.421503\pi$
$182$	0	0
$183$	12.7346	0.941365
$184$	0	0
$185$	11.8121	0.868442
$186$	0	0
$187$	−41.7835	−3.05552
$188$	0	0
$189$	4.08002	0.296778
$190$	0	0
$191$	6.87840	0.497704	0.248852	−	0.968542i	$-0.419947\pi$
$191$	6.87840	0.497704	0.248852	+	0.968542i	$0.419947\pi$
$192$	0	0
$193$	13.7087	0.986774	0.493387	−	0.869810i	$-0.335759\pi$
$193$	13.7087	0.986774	0.493387	+	0.869810i	$0.335759\pi$
$194$	0	0
$195$	−18.1172	−1.29740
$196$	0	0
$197$	3.18942	0.227236	0.113618	−	0.993524i	$-0.463756\pi$
$197$	3.18942	0.227236	0.113618	+	0.993524i	$0.463756\pi$
$198$	0	0
$199$	25.4804	1.80625	0.903127	−	0.429373i	$-0.141266\pi$
$199$	25.4804	1.80625	0.903127	+	0.429373i	$0.141266\pi$
$200$	0	0
$201$	1.33860	0.0944179
$202$	0	0
$203$	9.66282	0.678197
$204$	0	0
$205$	−2.47938	−0.173168
$206$	0	0
$207$	3.16242	0.219803
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−26.0450	−1.79301	−0.896505	−	0.443033i	$-0.853903\pi$
$211$	−26.0450	−1.79301	−0.896505	+	0.443033i	$0.853903\pi$
$212$	0	0
$213$	9.70111	0.664709
$214$	0	0
$215$	27.5229	1.87705
$216$	0	0
$217$	4.27994	0.290541
$218$	0	0
$219$	13.8287	0.934458
$220$	0	0
$221$	45.6687	3.07201
$222$	0	0
$223$	0.708891	0.0474709	0.0237354	−	0.999718i	$-0.492444\pi$
$223$	0.708891	0.0474709	0.0237354	+	0.999718i	$0.492444\pi$
$224$	0	0
$225$	2.76126	0.184084
$226$	0	0
$227$	−4.07740	−0.270627	−0.135313	−	0.990803i	$-0.543204\pi$
$227$	−4.07740	−0.270627	−0.135313	+	0.990803i	$0.543204\pi$
$228$	0	0
$229$	−24.7578	−1.63604	−0.818021	−	0.575188i	$-0.804929\pi$
$229$	−24.7578	−1.63604	−0.818021	+	0.575188i	$0.804929\pi$
$230$	0	0
$231$	24.2758	1.59723
$232$	0	0
$233$	−6.71057	−0.439624	−0.219812	−	0.975542i	$-0.570544\pi$
$233$	−6.71057	−0.439624	−0.219812	+	0.975542i	$0.570544\pi$
$234$	0	0
$235$	17.5492	1.14478
$236$	0	0
$237$	11.8181	0.767668
$238$	0	0
$239$	−17.8550	−1.15494	−0.577472	−	0.816411i	$-0.695961\pi$
$239$	−17.8550	−1.15494	−0.577472	+	0.816411i	$0.695961\pi$
$240$	0	0
$241$	−19.2356	−1.23907	−0.619537	−	0.784967i	$-0.712680\pi$
$241$	−19.2356	−1.23907	−0.619537	+	0.784967i	$0.712680\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−26.8745	−1.71695
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1.31483	0.0833237
$250$	0	0
$251$	16.7225	1.05551	0.527757	−	0.849396i	$-0.323033\pi$
$251$	16.7225	1.05551	0.527757	+	0.849396i	$0.323033\pi$
$252$	0	0
$253$	18.8162	1.18296
$254$	0	0
$255$	−19.5641	−1.22515
$256$	0	0
$257$	−2.13312	−0.133060	−0.0665302	−	0.997784i	$-0.521193\pi$
$257$	−2.13312	−0.133060	−0.0665302	+	0.997784i	$0.521193\pi$
$258$	0	0
$259$	17.2991	1.07491
$260$	0	0
$261$	−2.36833	−0.146596
$262$	0	0
$263$	−2.68398	−0.165501	−0.0827506	−	0.996570i	$-0.526370\pi$
$263$	−2.68398	−0.165501	−0.0827506	+	0.996570i	$0.526370\pi$
$264$	0	0
$265$	19.7392	1.21257
$266$	0	0
$267$	15.5577	0.952114
$268$	0	0
$269$	−1.41658	−0.0863704	−0.0431852	−	0.999067i	$-0.513751\pi$
$269$	−1.41658	−0.0863704	−0.0431852	+	0.999067i	$0.513751\pi$
$270$	0	0
$271$	−3.58596	−0.217832	−0.108916	−	0.994051i	$-0.534738\pi$
$271$	−3.58596	−0.217832	−0.108916	+	0.994051i	$0.534738\pi$
$272$	0	0
$273$	−26.5331	−1.60586
$274$	0	0
$275$	16.4293	0.990725
$276$	0	0
$277$	13.6785	0.821860	0.410930	−	0.911667i	$-0.365204\pi$
$277$	13.6785	0.821860	0.410930	+	0.911667i	$0.365204\pi$
$278$	0	0
$279$	−1.04900	−0.0628019
$280$	0	0
$281$	8.58284	0.512009	0.256005	−	0.966676i	$-0.417594\pi$
$281$	8.58284	0.512009	0.256005	+	0.966676i	$0.417594\pi$
$282$	0	0
$283$	−11.7942	−0.701091	−0.350545	−	0.936546i	$-0.614004\pi$
$283$	−11.7942	−0.701091	−0.350545	+	0.936546i	$0.614004\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.63112	−0.214338
$288$	0	0
$289$	32.3159	1.90094
$290$	0	0
$291$	2.18852	0.128294
$292$	0	0
$293$	−9.96188	−0.581979	−0.290990	−	0.956726i	$-0.593985\pi$
$293$	−9.96188	−0.581979	−0.290990	+	0.956726i	$0.593985\pi$
$294$	0	0
$295$	−18.1087	−1.05433
$296$	0	0
$297$	−5.94993	−0.345250
$298$	0	0
$299$	−20.5658	−1.18935
$300$	0	0
$301$	40.3080	2.32331
$302$	0	0
$303$	−0.288061	−0.0165487
$304$	0	0
$305$	35.4773	2.03142
$306$	0	0
$307$	7.02029	0.400669	0.200335	−	0.979728i	$-0.435797\pi$
$307$	7.02029	0.400669	0.200335	+	0.979728i	$0.435797\pi$
$308$	0	0
$309$	7.11901	0.404986
$310$	0	0
$311$	2.49946	0.141731	0.0708657	−	0.997486i	$-0.477424\pi$
$311$	2.49946	0.141731	0.0708657	+	0.997486i	$0.477424\pi$
$312$	0	0
$313$	12.7406	0.720142	0.360071	−	0.932925i	$-0.382752\pi$
$313$	12.7406	0.720142	0.360071	+	0.932925i	$0.382752\pi$
$314$	0	0
$315$	11.3666	0.640433
$316$	0	0
$317$	2.22620	0.125036	0.0625179	−	0.998044i	$-0.480087\pi$
$317$	2.22620	0.125036	0.0625179	+	0.998044i	$0.480087\pi$
$318$	0	0
$319$	−14.0914	−0.788965
$320$	0	0
$321$	−8.13769	−0.454202
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−17.9570	−0.996073
$326$	0	0
$327$	1.45359	0.0803838
$328$	0	0
$329$	25.7012	1.41695
$330$	0	0
$331$	15.4633	0.849942	0.424971	−	0.905207i	$-0.360284\pi$
$331$	15.4633	0.849942	0.424971	+	0.905207i	$0.360284\pi$
$332$	0	0
$333$	−4.23995	−0.232348
$334$	0	0
$335$	3.72922	0.203749
$336$	0	0
$337$	−26.7323	−1.45620	−0.728100	−	0.685471i	$-0.759596\pi$
$337$	−26.7323	−1.45620	−0.728100	+	0.685471i	$0.759596\pi$
$338$	0	0
$339$	−7.67440	−0.416816
$340$	0	0
$341$	−6.24146	−0.337994
$342$	0	0
$343$	−10.7981	−0.583044
$344$	0	0
$345$	8.81019	0.474325
$346$	0	0
$347$	−3.27507	−0.175815	−0.0879075	−	0.996129i	$-0.528018\pi$
$347$	−3.27507	−0.175815	−0.0879075	+	0.996129i	$0.528018\pi$
$348$	0	0
$349$	−13.7099	−0.733876	−0.366938	−	0.930245i	$-0.619594\pi$
$349$	−13.7099	−0.733876	−0.366938	+	0.930245i	$0.619594\pi$
$350$	0	0
$351$	6.50317	0.347114
$352$	0	0
$353$	3.16539	0.168477	0.0842385	−	0.996446i	$-0.473154\pi$
$353$	3.16539	0.168477	0.0842385	+	0.996446i	$0.473154\pi$
$354$	0	0
$355$	27.0264	1.43441
$356$	0	0
$357$	−28.6521	−1.51643
$358$	0	0
$359$	7.48651	0.395123	0.197561	−	0.980291i	$-0.436698\pi$
$359$	7.48651	0.395123	0.197561	+	0.980291i	$0.436698\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−24.4017	−1.28075
$364$	0	0
$365$	38.5255	2.01651
$366$	0	0
$367$	19.3834	1.01181	0.505903	−	0.862591i	$-0.331159\pi$
$367$	19.3834	1.01181	0.505903	+	0.862591i	$0.331159\pi$
$368$	0	0
$369$	0.889974	0.0463302
$370$	0	0
$371$	28.9085	1.50085
$372$	0	0
$373$	−1.19704	−0.0619802	−0.0309901	−	0.999520i	$-0.509866\pi$
$373$	−1.19704	−0.0619802	−0.0309901	+	0.999520i	$0.509866\pi$
$374$	0	0
$375$	−6.23691	−0.322073
$376$	0	0
$377$	15.4016	0.793224
$378$	0	0
$379$	−29.0179	−1.49055	−0.745275	−	0.666757i	$-0.767682\pi$
$379$	−29.0179	−1.49055	−0.745275	+	0.666757i	$0.767682\pi$
$380$	0	0
$381$	1.19191	0.0610636
$382$	0	0
$383$	−14.3727	−0.734410	−0.367205	−	0.930140i	$-0.619685\pi$
$383$	−14.3727	−0.734410	−0.367205	+	0.930140i	$0.619685\pi$
$384$	0	0
$385$	67.6302	3.44675
$386$	0	0
$387$	−9.87935	−0.502196
$388$	0	0
$389$	18.9939	0.963028	0.481514	−	0.876438i	$-0.340087\pi$
$389$	18.9939	0.963028	0.481514	+	0.876438i	$0.340087\pi$
$390$	0	0
$391$	−22.2082	−1.12312
$392$	0	0
$393$	2.87735	0.145143
$394$	0	0
$395$	32.9241	1.65659
$396$	0	0
$397$	30.3149	1.52146	0.760731	−	0.649067i	$-0.224841\pi$
$397$	30.3149	1.52146	0.760731	+	0.649067i	$0.224841\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.7329	−1.23510	−0.617550	−	0.786532i	$-0.711875\pi$
$401$	−24.7329	−1.23510	−0.617550	+	0.786532i	$0.711875\pi$
$402$	0	0
$403$	6.82182	0.339819
$404$	0	0
$405$	−2.78590	−0.138433
$406$	0	0
$407$	−25.2274	−1.25048
$408$	0	0
$409$	15.7141	0.777011	0.388505	−	0.921446i	$-0.372991\pi$
$409$	15.7141	0.777011	0.388505	+	0.921446i	$0.372991\pi$
$410$	0	0
$411$	−5.85903	−0.289004
$412$	0	0
$413$	−26.5206	−1.30500
$414$	0	0
$415$	3.66298	0.179809
$416$	0	0
$417$	9.54067	0.467209
$418$	0	0
$419$	13.5283	0.660899	0.330450	−	0.943824i	$-0.392800\pi$
$419$	13.5283	0.660899	0.330450	+	0.943824i	$0.392800\pi$
$420$	0	0
$421$	16.9424	0.825720	0.412860	−	0.910795i	$-0.364530\pi$
$421$	16.9424	0.825720	0.412860	+	0.910795i	$0.364530\pi$
$422$	0	0
$423$	−6.29928	−0.306281
$424$	0	0
$425$	−19.3910	−0.940603
$426$	0	0
$427$	51.9573	2.51439
$428$	0	0
$429$	38.6934	1.86814
$430$	0	0
$431$	−29.6535	−1.42836	−0.714178	−	0.699964i	$-0.753199\pi$
$431$	−29.6535	−1.42836	−0.714178	+	0.699964i	$0.753199\pi$
$432$	0	0
$433$	27.8381	1.33781	0.668907	−	0.743346i	$-0.266762\pi$
$433$	27.8381	1.33781	0.668907	+	0.743346i	$0.266762\pi$
$434$	0	0
$435$	−6.59793	−0.316346
$436$	0	0
$437$	0	0
$438$	0	0
$439$	36.8561	1.75905	0.879523	−	0.475857i	$-0.157862\pi$
$439$	36.8561	1.75905	0.879523	+	0.475857i	$0.157862\pi$
$440$	0	0
$441$	9.64658	0.459361
$442$	0	0
$443$	−2.14225	−0.101781	−0.0508906	−	0.998704i	$-0.516206\pi$
$443$	−2.14225	−0.101781	−0.0508906	+	0.998704i	$0.516206\pi$
$444$	0	0
$445$	43.3422	2.05462
$446$	0	0
$447$	−10.4298	−0.493313
$448$	0	0
$449$	7.35267	0.346994	0.173497	−	0.984834i	$-0.444493\pi$
$449$	7.35267	0.346994	0.173497	+	0.984834i	$0.444493\pi$
$450$	0	0
$451$	5.29528	0.249345
$452$	0	0
$453$	8.51391	0.400018
$454$	0	0
$455$	−73.9186	−3.46536
$456$	0	0
$457$	21.3157	0.997107	0.498553	−	0.866859i	$-0.333865\pi$
$457$	21.3157	0.997107	0.498553	+	0.866859i	$0.333865\pi$
$458$	0	0
$459$	7.02253	0.327784
$460$	0	0
$461$	−17.2827	−0.804938	−0.402469	−	0.915434i	$-0.631848\pi$
$461$	−17.2827	−0.804938	−0.402469	+	0.915434i	$0.631848\pi$
$462$	0	0
$463$	25.5416	1.18702	0.593509	−	0.804828i	$-0.297742\pi$
$463$	25.5416	1.18702	0.593509	+	0.804828i	$0.297742\pi$
$464$	0	0
$465$	−2.92241	−0.135523
$466$	0	0
$467$	−19.6071	−0.907308	−0.453654	−	0.891178i	$-0.649880\pi$
$467$	−19.6071	−0.907308	−0.453654	+	0.891178i	$0.649880\pi$
$468$	0	0
$469$	5.46154	0.252190
$470$	0	0
$471$	−5.11635	−0.235749
$472$	0	0
$473$	−58.7815	−2.70277
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−7.08537	−0.324417
$478$	0	0
$479$	17.6604	0.806926	0.403463	−	0.914996i	$-0.367806\pi$
$479$	17.6604	0.806926	0.403463	+	0.914996i	$0.367806\pi$
$480$	0	0
$481$	27.5731	1.25723
$482$	0	0
$483$	12.9027	0.587095
$484$	0	0
$485$	6.09702	0.276851
$486$	0	0
$487$	10.4736	0.474606	0.237303	−	0.971436i	$-0.423737\pi$
$487$	10.4736	0.474606	0.237303	+	0.971436i	$0.423737\pi$
$488$	0	0
$489$	−1.52781	−0.0690900
$490$	0	0
$491$	29.7515	1.34267	0.671333	−	0.741156i	$-0.265722\pi$
$491$	29.7515	1.34267	0.671333	+	0.741156i	$0.265722\pi$
$492$	0	0
$493$	16.6316	0.749051
$494$	0	0
$495$	−16.5759	−0.745033
$496$	0	0
$497$	39.5808	1.77544
$498$	0	0
$499$	12.7752	0.571896	0.285948	−	0.958245i	$-0.407692\pi$
$499$	12.7752	0.571896	0.285948	+	0.958245i	$0.407692\pi$
$500$	0	0
$501$	−0.0473249	−0.00211432
$502$	0	0
$503$	7.61172	0.339390	0.169695	−	0.985497i	$-0.445722\pi$
$503$	7.61172	0.339390	0.169695	+	0.985497i	$0.445722\pi$
$504$	0	0
$505$	−0.802510	−0.0357112
$506$	0	0
$507$	−29.2913	−1.30087
$508$	0	0
$509$	−40.9674	−1.81585	−0.907924	−	0.419134i	$-0.862334\pi$
$509$	−40.9674	−1.81585	−0.907924	+	0.419134i	$0.862334\pi$
$510$	0	0
$511$	56.4214	2.49594
$512$	0	0
$513$	0	0
$514$	0	0
$515$	19.8329	0.873941
$516$	0	0
$517$	−37.4803	−1.64838
$518$	0	0
$519$	16.6260	0.729799
$520$	0	0
$521$	1.63438	0.0716033	0.0358016	−	0.999359i	$-0.488602\pi$
$521$	1.63438	0.0716033	0.0358016	+	0.999359i	$0.488602\pi$
$522$	0	0
$523$	7.75496	0.339101	0.169550	−	0.985522i	$-0.445768\pi$
$523$	7.75496	0.339101	0.169550	+	0.985522i	$0.445768\pi$
$524$	0	0
$525$	11.2660	0.491689
$526$	0	0
$527$	7.36662	0.320895
$528$	0	0
$529$	−12.9991	−0.565179
$530$	0	0
$531$	6.50012	0.282081
$532$	0	0
$533$	−5.78766	−0.250691
$534$	0	0
$535$	−22.6708	−0.980146
$536$	0	0
$537$	23.1274	0.998019
$538$	0	0
$539$	57.3965	2.47224
$540$	0	0
$541$	−28.5190	−1.22613	−0.613063	−	0.790034i	$-0.710063\pi$
$541$	−28.5190	−1.22613	−0.613063	+	0.790034i	$0.710063\pi$
$542$	0	0
$543$	−6.56843	−0.281878
$544$	0	0
$545$	4.04957	0.173464
$546$	0	0
$547$	28.0509	1.19937	0.599685	−	0.800236i	$-0.295292\pi$
$547$	28.0509	1.19937	0.599685	+	0.800236i	$0.295292\pi$
$548$	0	0
$549$	−12.7346	−0.543498
$550$	0	0
$551$	0	0
$552$	0	0
$553$	48.2181	2.05044
$554$	0	0
$555$	−11.8121	−0.501395
$556$	0	0
$557$	20.6767	0.876099	0.438049	−	0.898951i	$-0.355670\pi$
$557$	20.6767	0.876099	0.438049	+	0.898951i	$0.355670\pi$
$558$	0	0
$559$	64.2471	2.71736
$560$	0	0
$561$	41.7835	1.76410
$562$	0	0
$563$	−43.6876	−1.84121	−0.920606	−	0.390493i	$-0.872304\pi$
$563$	−43.6876	−1.84121	−0.920606	+	0.390493i	$0.872304\pi$
$564$	0	0
$565$	−21.3801	−0.899469
$566$	0	0
$567$	−4.08002	−0.171345
$568$	0	0
$569$	−31.3932	−1.31607	−0.658035	−	0.752987i	$-0.728612\pi$
$569$	−31.3932	−1.31607	−0.658035	+	0.752987i	$0.728612\pi$
$570$	0	0
$571$	−31.5929	−1.32212	−0.661062	−	0.750332i	$-0.729894\pi$
$571$	−31.5929	−1.32212	−0.661062	+	0.750332i	$0.729894\pi$
$572$	0	0
$573$	−6.87840	−0.287349
$574$	0	0
$575$	8.73226	0.364161
$576$	0	0
$577$	17.3989	0.724325	0.362163	−	0.932115i	$-0.382038\pi$
$577$	17.3989	0.724325	0.362163	+	0.932115i	$0.382038\pi$
$578$	0	0
$579$	−13.7087	−0.569714
$580$	0	0
$581$	5.36452	0.222558
$582$	0	0
$583$	−42.1575	−1.74598
$584$	0	0
$585$	18.1172	0.749055
$586$	0	0
$587$	26.9657	1.11299	0.556497	−	0.830850i	$-0.312145\pi$
$587$	26.9657	1.11299	0.556497	+	0.830850i	$0.312145\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−3.18942	−0.131195
$592$	0	0
$593$	−32.8354	−1.34839	−0.674195	−	0.738554i	$-0.735509\pi$
$593$	−32.8354	−1.34839	−0.674195	+	0.738554i	$0.735509\pi$
$594$	0	0
$595$	−79.8219	−3.27238
$596$	0	0
$597$	−25.4804	−1.04284
$598$	0	0
$599$	−41.6188	−1.70050	−0.850249	−	0.526380i	$-0.823549\pi$
$599$	−41.6188	−1.70050	−0.850249	+	0.526380i	$0.823549\pi$
$600$	0	0
$601$	39.3982	1.60708	0.803542	−	0.595248i	$-0.202946\pi$
$601$	39.3982	1.60708	0.803542	+	0.595248i	$0.202946\pi$
$602$	0	0
$603$	−1.33860	−0.0545122
$604$	0	0
$605$	−67.9807	−2.76381
$606$	0	0
$607$	3.10672	0.126098	0.0630490	−	0.998010i	$-0.479918\pi$
$607$	3.10672	0.126098	0.0630490	+	0.998010i	$0.479918\pi$
$608$	0	0
$609$	−9.66282	−0.391557
$610$	0	0
$611$	40.9653	1.65728
$612$	0	0
$613$	−5.75470	−0.232430	−0.116215	−	0.993224i	$-0.537076\pi$
$613$	−5.75470	−0.232430	−0.116215	+	0.993224i	$0.537076\pi$
$614$	0	0
$615$	2.47938	0.0999784
$616$	0	0
$617$	30.8052	1.24017	0.620086	−	0.784534i	$-0.287098\pi$
$617$	30.8052	1.24017	0.620086	+	0.784534i	$0.287098\pi$
$618$	0	0
$619$	12.3636	0.496937	0.248468	−	0.968640i	$-0.420073\pi$
$619$	12.3636	0.496937	0.248468	+	0.968640i	$0.420073\pi$
$620$	0	0
$621$	−3.16242	−0.126903
$622$	0	0
$623$	63.4756	2.54310
$624$	0	0
$625$	−31.1817	−1.24727
$626$	0	0
$627$	0	0
$628$	0	0
$629$	29.7752	1.18721
$630$	0	0
$631$	23.6601	0.941892	0.470946	−	0.882162i	$-0.343913\pi$
$631$	23.6601	0.941892	0.470946	+	0.882162i	$0.343913\pi$
$632$	0	0
$633$	26.0450	1.03520
$634$	0	0
$635$	3.32056	0.131772
$636$	0	0
$637$	−62.7334	−2.48559
$638$	0	0
$639$	−9.70111	−0.383770
$640$	0	0
$641$	3.52353	0.139171	0.0695855	−	0.997576i	$-0.477832\pi$
$641$	3.52353	0.139171	0.0695855	+	0.997576i	$0.477832\pi$
$642$	0	0
$643$	−6.53942	−0.257889	−0.128945	−	0.991652i	$-0.541159\pi$
$643$	−6.53942	−0.257889	−0.128945	+	0.991652i	$0.541159\pi$
$644$	0	0
$645$	−27.5229	−1.08371
$646$	0	0
$647$	−12.8400	−0.504792	−0.252396	−	0.967624i	$-0.581218\pi$
$647$	−12.8400	−0.504792	−0.252396	+	0.967624i	$0.581218\pi$
$648$	0	0
$649$	38.6753	1.51814
$650$	0	0
$651$	−4.27994	−0.167744
$652$	0	0
$653$	−42.3347	−1.65668	−0.828342	−	0.560223i	$-0.810716\pi$
$653$	−42.3347	−1.65668	−0.828342	+	0.560223i	$0.810716\pi$
$654$	0	0
$655$	8.01601	0.313212
$656$	0	0
$657$	−13.8287	−0.539509
$658$	0	0
$659$	6.51944	0.253961	0.126981	−	0.991905i	$-0.459471\pi$
$659$	6.51944	0.253961	0.126981	+	0.991905i	$0.459471\pi$
$660$	0	0
$661$	48.4750	1.88546	0.942729	−	0.333558i	$-0.108249\pi$
$661$	48.4750	1.88546	0.942729	+	0.333558i	$0.108249\pi$
$662$	0	0
$663$	−45.6687	−1.77363
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−7.48964	−0.290000
$668$	0	0
$669$	−0.708891	−0.0274073
$670$	0	0
$671$	−75.7697	−2.92506
$672$	0	0
$673$	25.2740	0.974241	0.487120	−	0.873335i	$-0.338047\pi$
$673$	25.2740	0.974241	0.487120	+	0.873335i	$0.338047\pi$
$674$	0	0
$675$	−2.76126	−0.106281
$676$	0	0
$677$	−3.47921	−0.133717	−0.0668585	−	0.997762i	$-0.521298\pi$
$677$	−3.47921	−0.133717	−0.0668585	+	0.997762i	$0.521298\pi$
$678$	0	0
$679$	8.92923	0.342672
$680$	0	0
$681$	4.07740	0.156246
$682$	0	0
$683$	29.5183	1.12949	0.564744	−	0.825266i	$-0.308975\pi$
$683$	29.5183	1.12949	0.564744	+	0.825266i	$0.308975\pi$
$684$	0	0
$685$	−16.3227	−0.623658
$686$	0	0
$687$	24.7578	0.944569
$688$	0	0
$689$	46.0774	1.75541
$690$	0	0
$691$	−38.2711	−1.45590	−0.727951	−	0.685629i	$-0.759527\pi$
$691$	−38.2711	−1.45590	−0.727951	+	0.685629i	$0.759527\pi$
$692$	0	0
$693$	−24.2758	−0.922163
$694$	0	0
$695$	26.5794	1.00821
$696$	0	0
$697$	−6.24987	−0.236731
$698$	0	0
$699$	6.71057	0.253817
$700$	0	0
$701$	−6.33265	−0.239181	−0.119590	−	0.992823i	$-0.538158\pi$
$701$	−6.33265	−0.239181	−0.119590	+	0.992823i	$0.538158\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−17.5492	−0.660940
$706$	0	0
$707$	−1.17529	−0.0442015
$708$	0	0
$709$	−19.7945	−0.743398	−0.371699	−	0.928353i	$-0.621225\pi$
$709$	−19.7945	−0.743398	−0.371699	+	0.928353i	$0.621225\pi$
$710$	0	0
$711$	−11.8181	−0.443214
$712$	0	0
$713$	−3.31737	−0.124237
$714$	0	0
$715$	107.796	4.03135
$716$	0	0
$717$	17.8550	0.666807
$718$	0	0
$719$	25.4290	0.948341	0.474171	−	0.880433i	$-0.342748\pi$
$719$	25.4290	0.948341	0.474171	+	0.880433i	$0.342748\pi$
$720$	0	0
$721$	29.0457	1.08172
$722$	0	0
$723$	19.2356	0.715380
$724$	0	0
$725$	−6.53957	−0.242873
$726$	0	0
$727$	11.3695	0.421670	0.210835	−	0.977522i	$-0.432382\pi$
$727$	11.3695	0.421670	0.210835	+	0.977522i	$0.432382\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	69.3780	2.56604
$732$	0	0
$733$	11.3712	0.420006	0.210003	−	0.977701i	$-0.432653\pi$
$733$	11.3712	0.420006	0.210003	+	0.977701i	$0.432653\pi$
$734$	0	0
$735$	26.8745	0.991279
$736$	0	0
$737$	−7.96460	−0.293380
$738$	0	0
$739$	15.3000	0.562820	0.281410	−	0.959588i	$-0.409198\pi$
$739$	15.3000	0.562820	0.281410	+	0.959588i	$0.409198\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−35.0696	−1.28658	−0.643289	−	0.765623i	$-0.722431\pi$
$743$	−35.0696	−1.28658	−0.643289	+	0.765623i	$0.722431\pi$
$744$	0	0
$745$	−29.0564	−1.06455
$746$	0	0
$747$	−1.31483	−0.0481070
$748$	0	0
$749$	−33.2020	−1.21317
$750$	0	0
$751$	6.29180	0.229591	0.114796	−	0.993389i	$-0.463379\pi$
$751$	6.29180	0.229591	0.114796	+	0.993389i	$0.463379\pi$
$752$	0	0
$753$	−16.7225	−0.609401
$754$	0	0
$755$	23.7189	0.863221
$756$	0	0
$757$	15.8615	0.576494	0.288247	−	0.957556i	$-0.406928\pi$
$757$	15.8615	0.576494	0.288247	+	0.957556i	$0.406928\pi$
$758$	0	0
$759$	−18.8162	−0.682984
$760$	0	0
$761$	−23.4325	−0.849427	−0.424714	−	0.905328i	$-0.639625\pi$
$761$	−23.4325	−0.849427	−0.424714	+	0.905328i	$0.639625\pi$
$762$	0	0
$763$	5.93069	0.214705
$764$	0	0
$765$	19.5641	0.707341
$766$	0	0
$767$	−42.2714	−1.52633
$768$	0	0
$769$	−36.0085	−1.29850	−0.649250	−	0.760575i	$-0.724917\pi$
$769$	−36.0085	−1.29850	−0.649250	+	0.760575i	$0.724917\pi$
$770$	0	0
$771$	2.13312	0.0768224
$772$	0	0
$773$	−49.9914	−1.79806	−0.899032	−	0.437882i	$-0.855729\pi$
$773$	−49.9914	−1.79806	−0.899032	+	0.437882i	$0.855729\pi$
$774$	0	0
$775$	−2.89656	−0.104047
$776$	0	0
$777$	−17.2991	−0.620601
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−57.7209	−2.06542
$782$	0	0
$783$	2.36833	0.0846371
$784$	0	0
$785$	−14.2537	−0.508735
$786$	0	0
$787$	−19.2999	−0.687966	−0.343983	−	0.938976i	$-0.611776\pi$
$787$	−19.2999	−0.687966	−0.343983	+	0.938976i	$0.611776\pi$
$788$	0	0
$789$	2.68398	0.0955522
$790$	0	0
$791$	−31.3117	−1.11332
$792$	0	0
$793$	82.8150	2.94085
$794$	0	0
$795$	−19.7392	−0.700076
$796$	0	0
$797$	−1.93075	−0.0683906	−0.0341953	−	0.999415i	$-0.510887\pi$
$797$	−1.93075	−0.0683906	−0.0341953	+	0.999415i	$0.510887\pi$
$798$	0	0
$799$	44.2368	1.56499
$800$	0	0
$801$	−15.5577	−0.549703
$802$	0	0
$803$	−82.2798	−2.90359
$804$	0	0
$805$	35.9458	1.26692
$806$	0	0
$807$	1.41658	0.0498660
$808$	0	0
$809$	4.69817	0.165179	0.0825895	−	0.996584i	$-0.473681\pi$
$809$	4.69817	0.165179	0.0825895	+	0.996584i	$0.473681\pi$
$810$	0	0
$811$	−39.6396	−1.39194	−0.695968	−	0.718073i	$-0.745024\pi$
$811$	−39.6396	−1.39194	−0.695968	+	0.718073i	$0.745024\pi$
$812$	0	0
$813$	3.58596	0.125765
$814$	0	0
$815$	−4.25634	−0.149093
$816$	0	0
$817$	0	0
$818$	0	0
$819$	26.5331	0.927141
$820$	0	0
$821$	−29.7394	−1.03791	−0.518957	−	0.854800i	$-0.673679\pi$
$821$	−29.7394	−1.03791	−0.518957	+	0.854800i	$0.673679\pi$
$822$	0	0
$823$	−32.6170	−1.13696	−0.568479	−	0.822698i	$-0.692468\pi$
$823$	−32.6170	−1.13696	−0.568479	+	0.822698i	$0.692468\pi$
$824$	0	0
$825$	−16.4293	−0.571995
$826$	0	0
$827$	−33.0538	−1.14939	−0.574697	−	0.818366i	$-0.694880\pi$
$827$	−33.0538	−1.14939	−0.574697	+	0.818366i	$0.694880\pi$
$828$	0	0
$829$	−3.66392	−0.127253	−0.0636266	−	0.997974i	$-0.520267\pi$
$829$	−3.66392	−0.127253	−0.0636266	+	0.997974i	$0.520267\pi$
$830$	0	0
$831$	−13.6785	−0.474501
$832$	0	0
$833$	−67.7434	−2.34717
$834$	0	0
$835$	−0.131843	−0.00456260
$836$	0	0
$837$	1.04900	0.0362587
$838$	0	0
$839$	−36.5289	−1.26112	−0.630558	−	0.776142i	$-0.717174\pi$
$839$	−36.5289	−1.26112	−0.630558	+	0.776142i	$0.717174\pi$
$840$	0	0
$841$	−23.3910	−0.806587
$842$	0	0
$843$	−8.58284	−0.295609
$844$	0	0
$845$	−81.6026	−2.80722
$846$	0	0
$847$	−99.5593	−3.42090
$848$	0	0
$849$	11.7942	0.404775
$850$	0	0
$851$	−13.4085	−0.459637
$852$	0	0
$853$	−57.2479	−1.96013	−0.980066	−	0.198673i	$-0.936337\pi$
$853$	−57.2479	−1.96013	−0.980066	+	0.198673i	$0.936337\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.67922	0.330636	0.165318	−	0.986240i	$-0.447135\pi$
$857$	9.67922	0.330636	0.165318	+	0.986240i	$0.447135\pi$
$858$	0	0
$859$	−11.5237	−0.393182	−0.196591	−	0.980486i	$-0.562987\pi$
$859$	−11.5237	−0.393182	−0.196591	+	0.980486i	$0.562987\pi$
$860$	0	0
$861$	3.63112	0.123748
$862$	0	0
$863$	−11.5861	−0.394397	−0.197198	−	0.980364i	$-0.563184\pi$
$863$	−11.5861	−0.394397	−0.197198	+	0.980364i	$0.563184\pi$
$864$	0	0
$865$	46.3184	1.57487
$866$	0	0
$867$	−32.3159	−1.09751
$868$	0	0
$869$	−70.3169	−2.38534
$870$	0	0
$871$	8.70517	0.294964
$872$	0	0
$873$	−2.18852	−0.0740703
$874$	0	0
$875$	−25.4467	−0.860257
$876$	0	0
$877$	−12.8240	−0.433034	−0.216517	−	0.976279i	$-0.569470\pi$
$877$	−12.8240	−0.433034	−0.216517	+	0.976279i	$0.569470\pi$
$878$	0	0
$879$	9.96188	0.336006
$880$	0	0
$881$	20.6998	0.697394	0.348697	−	0.937236i	$-0.386624\pi$
$881$	20.6998	0.697394	0.348697	+	0.937236i	$0.386624\pi$
$882$	0	0
$883$	3.73167	0.125581	0.0627904	−	0.998027i	$-0.480000\pi$
$883$	3.73167	0.125581	0.0627904	+	0.998027i	$0.480000\pi$
$884$	0	0
$885$	18.1087	0.608718
$886$	0	0
$887$	8.92721	0.299746	0.149873	−	0.988705i	$-0.452113\pi$
$887$	8.92721	0.299746	0.149873	+	0.988705i	$0.452113\pi$
$888$	0	0
$889$	4.86304	0.163101
$890$	0	0
$891$	5.94993	0.199330
$892$	0	0
$893$	0	0
$894$	0	0
$895$	64.4306	2.15368
$896$	0	0
$897$	20.5658	0.686670
$898$	0	0
$899$	2.48437	0.0828584
$900$	0	0
$901$	49.7572	1.65765
$902$	0	0
$903$	−40.3080	−1.34137
$904$	0	0
$905$	−18.2990	−0.608280
$906$	0	0
$907$	−40.3490	−1.33977	−0.669884	−	0.742466i	$-0.733656\pi$
$907$	−40.3490	−1.33977	−0.669884	+	0.742466i	$0.733656\pi$
$908$	0	0
$909$	0.288061	0.00955437
$910$	0	0
$911$	20.6148	0.682998	0.341499	−	0.939882i	$-0.389065\pi$
$911$	20.6148	0.682998	0.341499	+	0.939882i	$0.389065\pi$
$912$	0	0
$913$	−7.82312	−0.258908
$914$	0	0
$915$	−35.4773	−1.17284
$916$	0	0
$917$	11.7396	0.387677
$918$	0	0
$919$	22.3348	0.736757	0.368379	−	0.929676i	$-0.379913\pi$
$919$	22.3348	0.736757	0.368379	+	0.929676i	$0.379913\pi$
$920$	0	0
$921$	−7.02029	−0.231326
$922$	0	0
$923$	63.0880	2.07657
$924$	0	0
$925$	−11.7076	−0.384944
$926$	0	0
$927$	−7.11901	−0.233819
$928$	0	0
$929$	−20.4139	−0.669759	−0.334880	−	0.942261i	$-0.608696\pi$
$929$	−20.4139	−0.669759	−0.334880	+	0.942261i	$0.608696\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−2.49946	−0.0818287
$934$	0	0
$935$	116.405	3.80685
$936$	0	0
$937$	−28.5544	−0.932833	−0.466417	−	0.884565i	$-0.654455\pi$
$937$	−28.5544	−0.932833	−0.466417	+	0.884565i	$0.654455\pi$
$938$	0	0
$939$	−12.7406	−0.415774
$940$	0	0
$941$	40.8481	1.33161	0.665804	−	0.746127i	$-0.268089\pi$
$941$	40.8481	1.33161	0.665804	+	0.746127i	$0.268089\pi$
$942$	0	0
$943$	2.81447	0.0916518
$944$	0	0
$945$	−11.3666	−0.369754
$946$	0	0
$947$	−0.182758	−0.00593883	−0.00296941	−	0.999996i	$-0.500945\pi$
$947$	−0.182758	−0.00593883	−0.00296941	+	0.999996i	$0.500945\pi$
$948$	0	0
$949$	89.9305	2.91927
$950$	0	0
$951$	−2.22620	−0.0721895
$952$	0	0
$953$	54.0511	1.75089	0.875444	−	0.483320i	$-0.160569\pi$
$953$	54.0511	1.75089	0.875444	+	0.483320i	$0.160569\pi$
$954$	0	0
$955$	−19.1626	−0.620086
$956$	0	0
$957$	14.0914	0.455509
$958$	0	0
$959$	−23.9050	−0.771931
$960$	0	0
$961$	−29.8996	−0.964503
$962$	0	0
$963$	8.13769	0.262234
$964$	0	0
$965$	−38.1911	−1.22942
$966$	0	0
$967$	−1.12580	−0.0362033	−0.0181016	−	0.999836i	$-0.505762\pi$
$967$	−1.12580	−0.0362033	−0.0181016	+	0.999836i	$0.505762\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−21.0600	−0.675848	−0.337924	−	0.941173i	$-0.609725\pi$
$971$	−21.0600	−0.675848	−0.337924	+	0.941173i	$0.609725\pi$
$972$	0	0
$973$	38.9262	1.24792
$974$	0	0
$975$	17.9570	0.575083
$976$	0	0
$977$	25.8203	0.826064	0.413032	−	0.910717i	$-0.364470\pi$
$977$	25.8203	0.826064	0.413032	+	0.910717i	$0.364470\pi$
$978$	0	0
$979$	−92.5670	−2.95845
$980$	0	0
$981$	−1.45359	−0.0464096
$982$	0	0
$983$	−44.9028	−1.43218	−0.716089	−	0.698009i	$-0.754069\pi$
$983$	−44.9028	−1.43218	−0.716089	+	0.698009i	$0.754069\pi$
$984$	0	0
$985$	−8.88541	−0.283113
$986$	0	0
$987$	−25.7012	−0.818078
$988$	0	0
$989$	−31.2426	−0.993458
$990$	0	0
$991$	−5.86794	−0.186401	−0.0932006	−	0.995647i	$-0.529710\pi$
$991$	−5.86794	−0.186401	−0.0932006	+	0.995647i	$0.529710\pi$
$992$	0	0
$993$	−15.4633	−0.490714
$994$	0	0
$995$	−70.9858	−2.25040
$996$	0	0
$997$	37.9629	1.20230	0.601149	−	0.799137i	$-0.294710\pi$
$997$	37.9629	1.20230	0.601149	+	0.799137i	$0.294710\pi$
$998$	0	0
$999$	4.23995	0.134146

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8664.2.a.br.1.2	✓	12
19.18	odd	2		8664.2.a.bs.1.2	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8664.2.a.br.1.2	✓	12	1.1	even	1	trivial
8664.2.a.bs.1.2	yes	12	19.18	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:	\( N \)	\(=\)	\( 8664 = 2^{3} \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8664.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.78590 q^{5} -4.08002 q^{7} +1.00000 q^{9} +5.94993 q^{11} -6.50317 q^{13} +2.78590 q^{15} -7.02253 q^{17} +4.08002 q^{21} +3.16242 q^{23} +2.76126 q^{25} -1.00000 q^{27} -2.36833 q^{29} -1.04900 q^{31} -5.94993 q^{33} +11.3666 q^{35} -4.23995 q^{37} +6.50317 q^{39} +0.889974 q^{41} -9.87935 q^{43} -2.78590 q^{45} -6.29928 q^{47} +9.64658 q^{49} +7.02253 q^{51} -7.08537 q^{53} -16.5759 q^{55} +6.50012 q^{59} -12.7346 q^{61} -4.08002 q^{63} +18.1172 q^{65} -1.33860 q^{67} -3.16242 q^{69} -9.70111 q^{71} -13.8287 q^{73} -2.76126 q^{75} -24.2758 q^{77} -11.8181 q^{79} +1.00000 q^{81} -1.31483 q^{83} +19.5641 q^{85} +2.36833 q^{87} -15.5577 q^{89} +26.5331 q^{91} +1.04900 q^{93} -2.18852 q^{97} +5.94993 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{3} + 2 q^{5} + 4 q^{7} + 12 q^{9} + 14 q^{11} + 4 q^{13} - 2 q^{15} - 4 q^{21} + 26 q^{23} + 18 q^{25} - 12 q^{27} - 10 q^{29} + 8 q^{31} - 14 q^{33} + 30 q^{35} - 8 q^{37} - 4 q^{39} + 18 q^{41}+ \cdots + 14 q^{99}+O(q^{100}) \) 12 * q - 12 * q^3 + 2 * q^5 + 4 * q^7 + 12 * q^9 + 14 * q^11 + 4 * q^13 - 2 * q^15 - 4 * q^21 + 26 * q^23 + 18 * q^25 - 12 * q^27 - 10 * q^29 + 8 * q^31 - 14 * q^33 + 30 * q^35 - 8 * q^37 - 4 * q^39 + 18 * q^41 - 8 * q^43 + 2 * q^45 + 48 * q^49 - 28 * q^53 + 2 * q^55 + 6 * q^59 + 38 * q^61 + 4 * q^63 + 24 * q^65 - 16 * q^67 - 26 * q^69 - 38 * q^71 + 28 * q^73 - 18 * q^75 + 4 * q^77 - 40 * q^79 + 12 * q^81 + 18 * q^83 + 54 * q^85 + 10 * q^87 + 24 * q^89 + 20 * q^91 - 8 * q^93 + 24 * q^97 + 14 * q^99

Introduction

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