LMFDB - Embedded newform 2368.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,2,Mod(1,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, 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("2368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2368.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:	$18.9085751986$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 296)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	2368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.47283 q^{3} -1.47283 q^{5} +2.64207 q^{7} +3.11491 q^{9} +O(q^{10})$	$q-2.47283 q^{3} -1.47283 q^{5} +2.64207 q^{7} +3.11491 q^{9} +6.34472 q^{11} +5.34472 q^{13} +3.64207 q^{15} -0.715853 q^{17} -3.28415 q^{19} -6.53341 q^{21} +0.885092 q^{23} -2.83076 q^{25} -0.284147 q^{27} +0.885092 q^{29} +7.47283 q^{31} -15.6894 q^{33} -3.89134 q^{35} +1.00000 q^{37} -13.2166 q^{39} -7.75698 q^{41} -10.1212 q^{43} -4.58774 q^{45} +11.8176 q^{47} -0.0194469 q^{49} +1.77018 q^{51} -4.15604 q^{53} -9.34472 q^{55} +8.12115 q^{57} +12.1212 q^{59} -3.81131 q^{61} +8.22982 q^{63} -7.87189 q^{65} +5.32528 q^{67} -2.18869 q^{69} +2.87189 q^{71} -8.34472 q^{73} +7.00000 q^{75} +16.7632 q^{77} -1.83076 q^{79} -8.64207 q^{81} -2.15604 q^{83} +1.05433 q^{85} -2.18869 q^{87} -1.89134 q^{89} +14.1212 q^{91} -18.4791 q^{93} +4.83700 q^{95} +15.0668 q^{97} +19.7632 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + q^{5} + 7 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 + q^5 + 7 * q^7 + 3 * q^9	$ 3 q - 2 q^{3} + q^{5} + 7 q^{7} + 3 q^{9} - 3 q^{13} + 10 q^{15} - 4 q^{17} - 8 q^{19} + 3 q^{21} + 9 q^{23} - 4 q^{25} + q^{27} + 9 q^{29} + 17 q^{31} - 9 q^{33} + 10 q^{35} + 3 q^{37} - 7 q^{39} - 16 q^{41} + 4 q^{43} - 2 q^{45} + 11 q^{47} + 8 q^{49} + 18 q^{51} + 3 q^{53} - 9 q^{55} - 10 q^{57} + 2 q^{59} - 15 q^{61} + 12 q^{63} - 10 q^{65} + 5 q^{67} - 3 q^{69} - 5 q^{71} - 6 q^{73} + 21 q^{75} + 15 q^{77} - q^{79} - 25 q^{81} + 9 q^{83} + 14 q^{85} - 3 q^{87} + 16 q^{89} + 8 q^{91} - 22 q^{93} - 18 q^{95} + 24 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + q^5 + 7 * q^7 + 3 * q^9 - 3 * q^13 + 10 * q^15 - 4 * q^17 - 8 * q^19 + 3 * q^21 + 9 * q^23 - 4 * q^25 + q^27 + 9 * q^29 + 17 * q^31 - 9 * q^33 + 10 * q^35 + 3 * q^37 - 7 * q^39 - 16 * q^41 + 4 * q^43 - 2 * q^45 + 11 * q^47 + 8 * q^49 + 18 * q^51 + 3 * q^53 - 9 * q^55 - 10 * q^57 + 2 * q^59 - 15 * q^61 + 12 * q^63 - 10 * q^65 + 5 * q^67 - 3 * q^69 - 5 * q^71 - 6 * q^73 + 21 * q^75 + 15 * q^77 - q^79 - 25 * q^81 + 9 * q^83 + 14 * q^85 - 3 * q^87 + 16 * q^89 + 8 * q^91 - 22 * q^93 - 18 * q^95 + 24 * 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
$3$	−2.47283	−1.42769	−0.713846	−	0.700303i	$-0.753048\pi$
$3$	−2.47283	−1.42769	−0.713846	+	0.700303i	$0.753048\pi$
$4$	0	0
$5$	−1.47283	−0.658671	−0.329336	−	0.944213i	$-0.606825\pi$
$5$	−1.47283	−0.658671	−0.329336	+	0.944213i	$0.606825\pi$
$6$	0	0
$7$	2.64207	0.998610	0.499305	−	0.866426i	$-0.333589\pi$
$7$	2.64207	0.998610	0.499305	+	0.866426i	$0.333589\pi$
$8$	0	0
$9$	3.11491	1.03830
$10$	0	0
$11$	6.34472	1.91301	0.956503	−	0.291723i	$-0.0942285\pi$
$11$	6.34472	1.91301	0.956503	+	0.291723i	$0.0942285\pi$
$12$	0	0
$13$	5.34472	1.48236	0.741180	−	0.671307i	$-0.234267\pi$
$13$	5.34472	1.48236	0.741180	+	0.671307i	$0.234267\pi$
$14$	0	0
$15$	3.64207	0.940379
$16$	0	0
$17$	−0.715853	−0.173620	−0.0868099	−	0.996225i	$-0.527667\pi$
$17$	−0.715853	−0.173620	−0.0868099	+	0.996225i	$0.527667\pi$
$18$	0	0
$19$	−3.28415	−0.753435	−0.376718	−	0.926328i	$-0.622947\pi$
$19$	−3.28415	−0.753435	−0.376718	+	0.926328i	$0.622947\pi$
$20$	0	0
$21$	−6.53341	−1.42571
$22$	0	0
$23$	0.885092	0.184555	0.0922773	−	0.995733i	$-0.470585\pi$
$23$	0.885092	0.184555	0.0922773	+	0.995733i	$0.470585\pi$
$24$	0	0
$25$	−2.83076	−0.566152
$26$	0	0
$27$	−0.284147	−0.0546842
$28$	0	0
$29$	0.885092	0.164358	0.0821788	−	0.996618i	$-0.473812\pi$
$29$	0.885092	0.164358	0.0821788	+	0.996618i	$0.473812\pi$
$30$	0	0
$31$	7.47283	1.34216	0.671080	−	0.741385i	$-0.265831\pi$
$31$	7.47283	1.34216	0.671080	+	0.741385i	$0.265831\pi$
$32$	0	0
$33$	−15.6894	−2.73118
$34$	0	0
$35$	−3.89134	−0.657756
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	−13.2166	−2.11635
$40$	0	0
$41$	−7.75698	−1.21144	−0.605718	−	0.795679i	$-0.707114\pi$
$41$	−7.75698	−1.21144	−0.605718	+	0.795679i	$0.707114\pi$
$42$	0	0
$43$	−10.1212	−1.54346	−0.771731	−	0.635950i	$-0.780609\pi$
$43$	−10.1212	−1.54346	−0.771731	+	0.635950i	$0.780609\pi$
$44$	0	0
$45$	−4.58774	−0.683900
$46$	0	0
$47$	11.8176	1.72377	0.861884	−	0.507106i	$-0.169285\pi$
$47$	11.8176	1.72377	0.861884	+	0.507106i	$0.169285\pi$
$48$	0	0
$49$	−0.0194469	−0.00277813
$50$	0	0
$51$	1.77018	0.247875
$52$	0	0
$53$	−4.15604	−0.570875	−0.285438	−	0.958397i	$-0.592139\pi$
$53$	−4.15604	−0.570875	−0.285438	+	0.958397i	$0.592139\pi$
$54$	0	0
$55$	−9.34472	−1.26004
$56$	0	0
$57$	8.12115	1.07567
$58$	0	0
$59$	12.1212	1.57804	0.789020	−	0.614368i	$-0.210589\pi$
$59$	12.1212	1.57804	0.789020	+	0.614368i	$0.210589\pi$
$60$	0	0
$61$	−3.81131	−0.487989	−0.243994	−	0.969777i	$-0.578458\pi$
$61$	−3.81131	−0.487989	−0.243994	+	0.969777i	$0.578458\pi$
$62$	0	0
$63$	8.22982	1.03686
$64$	0	0
$65$	−7.87189	−0.976388
$66$	0	0
$67$	5.32528	0.650586	0.325293	−	0.945613i	$-0.394537\pi$
$67$	5.32528	0.650586	0.325293	+	0.945613i	$0.394537\pi$
$68$	0	0
$69$	−2.18869	−0.263487
$70$	0	0
$71$	2.87189	0.340830	0.170415	−	0.985372i	$-0.445489\pi$
$71$	2.87189	0.340830	0.170415	+	0.985372i	$0.445489\pi$
$72$	0	0
$73$	−8.34472	−0.976676	−0.488338	−	0.872655i	$-0.662397\pi$
$73$	−8.34472	−0.976676	−0.488338	+	0.872655i	$0.662397\pi$
$74$	0	0
$75$	7.00000	0.808290
$76$	0	0
$77$	16.7632	1.91035
$78$	0	0
$79$	−1.83076	−0.205977	−0.102988	−	0.994683i	$-0.532840\pi$
$79$	−1.83076	−0.205977	−0.102988	+	0.994683i	$0.532840\pi$
$80$	0	0
$81$	−8.64207	−0.960230
$82$	0	0
$83$	−2.15604	−0.236656	−0.118328	−	0.992975i	$-0.537753\pi$
$83$	−2.15604	−0.236656	−0.118328	+	0.992975i	$0.537753\pi$
$84$	0	0
$85$	1.05433	0.114358
$86$	0	0
$87$	−2.18869	−0.234652
$88$	0	0
$89$	−1.89134	−0.200481	−0.100241	−	0.994963i	$-0.531961\pi$
$89$	−1.89134	−0.200481	−0.100241	+	0.994963i	$0.531961\pi$
$90$	0	0
$91$	14.1212	1.48030
$92$	0	0
$93$	−18.4791	−1.91619
$94$	0	0
$95$	4.83700	0.496266
$96$	0	0
$97$	15.0668	1.52980	0.764902	−	0.644147i	$-0.222787\pi$
$97$	15.0668	1.52980	0.764902	+	0.644147i	$0.222787\pi$
$98$	0	0
$99$	19.7632	1.98628
$100$	0	0
$101$	8.15604	0.811556	0.405778	−	0.913972i	$-0.367001\pi$
$101$	8.15604	0.811556	0.405778	+	0.913972i	$0.367001\pi$
$102$	0	0
$103$	14.6483	1.44334	0.721671	−	0.692236i	$-0.243374\pi$
$103$	14.6483	1.44334	0.721671	+	0.692236i	$0.243374\pi$
$104$	0	0
$105$	9.62263	0.939072
$106$	0	0
$107$	6.20813	0.600163	0.300081	−	0.953914i	$-0.402986\pi$
$107$	6.20813	0.600163	0.300081	+	0.953914i	$0.402986\pi$
$108$	0	0
$109$	−6.82452	−0.653670	−0.326835	−	0.945081i	$-0.605982\pi$
$109$	−6.82452	−0.653670	−0.326835	+	0.945081i	$0.605982\pi$
$110$	0	0
$111$	−2.47283	−0.234711
$112$	0	0
$113$	12.3510	1.16188	0.580941	−	0.813946i	$-0.302685\pi$
$113$	12.3510	1.16188	0.580941	+	0.813946i	$0.302685\pi$
$114$	0	0
$115$	−1.30359	−0.121561
$116$	0	0
$117$	16.6483	1.53914
$118$	0	0
$119$	−1.89134	−0.173378
$120$	0	0
$121$	29.2555	2.65959
$122$	0	0
$123$	19.1817	1.72956
$124$	0	0
$125$	11.5334	1.03158
$126$	0	0
$127$	−18.5070	−1.64223	−0.821115	−	0.570762i	$-0.806648\pi$
$127$	−18.5070	−1.64223	−0.821115	+	0.570762i	$0.806648\pi$
$128$	0	0
$129$	25.0279	2.20359
$130$	0	0
$131$	14.6894	1.28342	0.641711	−	0.766946i	$-0.278225\pi$
$131$	14.6894	1.28342	0.641711	+	0.766946i	$0.278225\pi$
$132$	0	0
$133$	−8.67696	−0.752388
$134$	0	0
$135$	0.418502	0.0360189
$136$	0	0
$137$	−3.32528	−0.284097	−0.142049	−	0.989860i	$-0.545369\pi$
$137$	−3.32528	−0.284097	−0.142049	+	0.989860i	$0.545369\pi$
$138$	0	0
$139$	−9.23606	−0.783392	−0.391696	−	0.920095i	$-0.628112\pi$
$139$	−9.23606	−0.783392	−0.391696	+	0.920095i	$0.628112\pi$
$140$	0	0
$141$	−29.2229	−2.46101
$142$	0	0
$143$	33.9108	2.83576
$144$	0	0
$145$	−1.30359	−0.108258
$146$	0	0
$147$	0.0480890	0.00396631
$148$	0	0
$149$	2.64207	0.216447	0.108224	−	0.994127i	$-0.465484\pi$
$149$	2.64207	0.216447	0.108224	+	0.994127i	$0.465484\pi$
$150$	0	0
$151$	17.9736	1.46267	0.731335	−	0.682018i	$-0.238898\pi$
$151$	17.9736	1.46267	0.731335	+	0.682018i	$0.238898\pi$
$152$	0	0
$153$	−2.22982	−0.180270
$154$	0	0
$155$	−11.0062	−0.884043
$156$	0	0
$157$	4.15604	0.331688	0.165844	−	0.986152i	$-0.446965\pi$
$157$	4.15604	0.331688	0.165844	+	0.986152i	$0.446965\pi$
$158$	0	0
$159$	10.2772	0.815034
$160$	0	0
$161$	2.33848	0.184298
$162$	0	0
$163$	−15.8913	−1.24471	−0.622353	−	0.782737i	$-0.713823\pi$
$163$	−15.8913	−1.24471	−0.622353	+	0.782737i	$0.713823\pi$
$164$	0	0
$165$	23.1079	1.79895
$166$	0	0
$167$	−16.7695	−1.29766	−0.648830	−	0.760933i	$-0.724741\pi$
$167$	−16.7695	−1.29766	−0.648830	+	0.760933i	$0.724741\pi$
$168$	0	0
$169$	15.5661	1.19739
$170$	0	0
$171$	−10.2298	−0.782294
$172$	0	0
$173$	2.41226	0.183401	0.0917003	−	0.995787i	$-0.470770\pi$
$173$	2.41226	0.183401	0.0917003	+	0.995787i	$0.470770\pi$
$174$	0	0
$175$	−7.47908	−0.565365
$176$	0	0
$177$	−29.9736	−2.25295
$178$	0	0
$179$	−24.0125	−1.79478	−0.897389	−	0.441241i	$-0.854538\pi$
$179$	−24.0125	−1.79478	−0.897389	+	0.441241i	$0.854538\pi$
$180$	0	0
$181$	−17.3315	−1.28824	−0.644121	−	0.764924i	$-0.722777\pi$
$181$	−17.3315	−1.28824	−0.644121	+	0.764924i	$0.722777\pi$
$182$	0	0
$183$	9.42474	0.696697
$184$	0	0
$185$	−1.47283	−0.108285
$186$	0	0
$187$	−4.54189	−0.332136
$188$	0	0
$189$	−0.750738	−0.0546082
$190$	0	0
$191$	−7.55509	−0.546667	−0.273334	−	0.961919i	$-0.588126\pi$
$191$	−7.55509	−0.546667	−0.273334	+	0.961919i	$0.588126\pi$
$192$	0	0
$193$	5.74378	0.413446	0.206723	−	0.978399i	$-0.433720\pi$
$193$	5.74378	0.413446	0.206723	+	0.978399i	$0.433720\pi$
$194$	0	0
$195$	19.4659	1.39398
$196$	0	0
$197$	16.1560	1.15107	0.575535	−	0.817777i	$-0.304794\pi$
$197$	16.1560	1.15107	0.575535	+	0.817777i	$0.304794\pi$
$198$	0	0
$199$	−7.17548	−0.508656	−0.254328	−	0.967118i	$-0.581854\pi$
$199$	−7.17548	−0.508656	−0.254328	+	0.967118i	$0.581854\pi$
$200$	0	0
$201$	−13.1685	−0.928836
$202$	0	0
$203$	2.33848	0.164129
$204$	0	0
$205$	11.4247	0.797939
$206$	0	0
$207$	2.75698	0.191623
$208$	0	0
$209$	−20.8370	−1.44133
$210$	0	0
$211$	−15.1692	−1.04429	−0.522147	−	0.852856i	$-0.674869\pi$
$211$	−15.1692	−1.04429	−0.522147	+	0.852856i	$0.674869\pi$
$212$	0	0
$213$	−7.10170	−0.486601
$214$	0	0
$215$	14.9068	1.01663
$216$	0	0
$217$	19.7438	1.34029
$218$	0	0
$219$	20.6351	1.39439
$220$	0	0
$221$	−3.82603	−0.257367
$222$	0	0
$223$	0.789632	0.0528777	0.0264388	−	0.999650i	$-0.491583\pi$
$223$	0.789632	0.0528777	0.0264388	+	0.999650i	$0.491583\pi$
$224$	0	0
$225$	−8.81756	−0.587837
$226$	0	0
$227$	1.05433	0.0699785	0.0349892	−	0.999388i	$-0.488860\pi$
$227$	1.05433	0.0699785	0.0349892	+	0.999388i	$0.488860\pi$
$228$	0	0
$229$	29.7089	1.96322	0.981609	−	0.190900i	$-0.0611407\pi$
$229$	29.7089	1.96322	0.981609	+	0.190900i	$0.0611407\pi$
$230$	0	0
$231$	−41.4527	−2.72739
$232$	0	0
$233$	18.9798	1.24341	0.621705	−	0.783251i	$-0.286440\pi$
$233$	18.9798	1.24341	0.621705	+	0.783251i	$0.286440\pi$
$234$	0	0
$235$	−17.4053	−1.13540
$236$	0	0
$237$	4.52717	0.294071
$238$	0	0
$239$	0.756981	0.0489650	0.0244825	−	0.999700i	$-0.492206\pi$
$239$	0.756981	0.0489650	0.0244825	+	0.999700i	$0.492206\pi$
$240$	0	0
$241$	0.147558	0.00950506	0.00475253	−	0.999989i	$-0.498487\pi$
$241$	0.147558	0.00950506	0.00475253	+	0.999989i	$0.498487\pi$
$242$	0	0
$243$	22.2229	1.42560
$244$	0	0
$245$	0.0286421	0.00182988
$246$	0	0
$247$	−17.5529	−1.11686
$248$	0	0
$249$	5.33152	0.337871
$250$	0	0
$251$	15.1755	0.957868	0.478934	−	0.877851i	$-0.341023\pi$
$251$	15.1755	0.957868	0.478934	+	0.877851i	$0.341023\pi$
$252$	0	0
$253$	5.61567	0.353054
$254$	0	0
$255$	−2.60719	−0.163268
$256$	0	0
$257$	23.5264	1.46754	0.733770	−	0.679398i	$-0.237759\pi$
$257$	23.5264	1.46754	0.733770	+	0.679398i	$0.237759\pi$
$258$	0	0
$259$	2.64207	0.164170
$260$	0	0
$261$	2.75698	0.170653
$262$	0	0
$263$	24.7368	1.52534	0.762669	−	0.646789i	$-0.223889\pi$
$263$	24.7368	1.52534	0.762669	+	0.646789i	$0.223889\pi$
$264$	0	0
$265$	6.12115	0.376019
$266$	0	0
$267$	4.67696	0.286225
$268$	0	0
$269$	−14.0389	−0.855966	−0.427983	−	0.903787i	$-0.640776\pi$
$269$	−14.0389	−0.855966	−0.427983	+	0.903787i	$0.640776\pi$
$270$	0	0
$271$	4.04737	0.245860	0.122930	−	0.992415i	$-0.460771\pi$
$271$	4.04737	0.245860	0.122930	+	0.992415i	$0.460771\pi$
$272$	0	0
$273$	−34.9193	−2.11341
$274$	0	0
$275$	−17.9604	−1.08305
$276$	0	0
$277$	16.2709	0.977626	0.488813	−	0.872389i	$-0.337430\pi$
$277$	16.2709	0.977626	0.488813	+	0.872389i	$0.337430\pi$
$278$	0	0
$279$	23.2772	1.39357
$280$	0	0
$281$	−12.9193	−0.770698	−0.385349	−	0.922771i	$-0.625919\pi$
$281$	−12.9193	−0.770698	−0.385349	+	0.922771i	$0.625919\pi$
$282$	0	0
$283$	1.97359	0.117318	0.0586589	−	0.998278i	$-0.481318\pi$
$283$	1.97359	0.117318	0.0586589	+	0.998278i	$0.481318\pi$
$284$	0	0
$285$	−11.9611	−0.708515
$286$	0	0
$287$	−20.4945	−1.20975
$288$	0	0
$289$	−16.4876	−0.969856
$290$	0	0
$291$	−37.2577	−2.18409
$292$	0	0
$293$	34.1212	1.99338	0.996689	−	0.0813027i	$-0.0259081\pi$
$293$	34.1212	1.99338	0.996689	+	0.0813027i	$0.0259081\pi$
$294$	0	0
$295$	−17.8524	−1.03941
$296$	0	0
$297$	−1.80284	−0.104611
$298$	0	0
$299$	4.73057	0.273576
$300$	0	0
$301$	−26.7408	−1.54132
$302$	0	0
$303$	−20.1685	−1.15865
$304$	0	0
$305$	5.61343	0.321424
$306$	0	0
$307$	20.3642	1.16224	0.581122	−	0.813816i	$-0.302614\pi$
$307$	20.3642	1.16224	0.581122	+	0.813816i	$0.302614\pi$
$308$	0	0
$309$	−36.2229	−2.06065
$310$	0	0
$311$	28.0606	1.59117	0.795585	−	0.605842i	$-0.207164\pi$
$311$	28.0606	1.59117	0.795585	+	0.605842i	$0.207164\pi$
$312$	0	0
$313$	−14.3774	−0.812657	−0.406329	−	0.913727i	$-0.633191\pi$
$313$	−14.3774	−0.812657	−0.406329	+	0.913727i	$0.633191\pi$
$314$	0	0
$315$	−12.1212	−0.682949
$316$	0	0
$317$	−20.3121	−1.14084	−0.570420	−	0.821353i	$-0.693219\pi$
$317$	−20.3121	−1.14084	−0.570420	+	0.821353i	$0.693219\pi$
$318$	0	0
$319$	5.61567	0.314417
$320$	0	0
$321$	−15.3517	−0.856847
$322$	0	0
$323$	2.35097	0.130811
$324$	0	0
$325$	−15.1296	−0.839241
$326$	0	0
$327$	16.8759	0.933239
$328$	0	0
$329$	31.2229	1.72137
$330$	0	0
$331$	13.7438	0.755426	0.377713	−	0.925923i	$-0.376711\pi$
$331$	13.7438	0.755426	0.377713	+	0.925923i	$0.376711\pi$
$332$	0	0
$333$	3.11491	0.170696
$334$	0	0
$335$	−7.84325	−0.428522
$336$	0	0
$337$	−24.3447	−1.32614	−0.663071	−	0.748557i	$-0.730747\pi$
$337$	−24.3447	−1.32614	−0.663071	+	0.748557i	$0.730747\pi$
$338$	0	0
$339$	−30.5419	−1.65881
$340$	0	0
$341$	47.4131	2.56756
$342$	0	0
$343$	−18.5459	−1.00138
$344$	0	0
$345$	3.22357	0.173551
$346$	0	0
$347$	23.7702	1.27605	0.638025	−	0.770015i	$-0.279752\pi$
$347$	23.7702	1.27605	0.638025	+	0.770015i	$0.279752\pi$
$348$	0	0
$349$	−11.7702	−0.630044	−0.315022	−	0.949084i	$-0.602012\pi$
$349$	−11.7702	−0.630044	−0.315022	+	0.949084i	$0.602012\pi$
$350$	0	0
$351$	−1.51869	−0.0810616
$352$	0	0
$353$	5.05433	0.269015	0.134507	−	0.990913i	$-0.457055\pi$
$353$	5.05433	0.269015	0.134507	+	0.990913i	$0.457055\pi$
$354$	0	0
$355$	−4.22982	−0.224495
$356$	0	0
$357$	4.67696	0.247531
$358$	0	0
$359$	−3.93871	−0.207877	−0.103939	−	0.994584i	$-0.533145\pi$
$359$	−3.93871	−0.207877	−0.103939	+	0.994584i	$0.533145\pi$
$360$	0	0
$361$	−8.21438	−0.432336
$362$	0	0
$363$	−72.3440	−3.79708
$364$	0	0
$365$	12.2904	0.643308
$366$	0	0
$367$	−2.90677	−0.151732	−0.0758662	−	0.997118i	$-0.524172\pi$
$367$	−2.90677	−0.151732	−0.0758662	+	0.997118i	$0.524172\pi$
$368$	0	0
$369$	−24.1623	−1.25784
$370$	0	0
$371$	−10.9806	−0.570082
$372$	0	0
$373$	−8.95415	−0.463628	−0.231814	−	0.972760i	$-0.574466\pi$
$373$	−8.95415	−0.463628	−0.231814	+	0.972760i	$0.574466\pi$
$374$	0	0
$375$	−28.5202	−1.47278
$376$	0	0
$377$	4.73057	0.243637
$378$	0	0
$379$	34.9325	1.79436	0.897180	−	0.441665i	$-0.145612\pi$
$379$	34.9325	1.79436	0.897180	+	0.441665i	$0.145612\pi$
$380$	0	0
$381$	45.7647	2.34460
$382$	0	0
$383$	−19.4876	−0.995768	−0.497884	−	0.867244i	$-0.665889\pi$
$383$	−19.4876	−0.995768	−0.497884	+	0.867244i	$0.665889\pi$
$384$	0	0
$385$	−24.6894	−1.25829
$386$	0	0
$387$	−31.5264	−1.60258
$388$	0	0
$389$	−21.8308	−1.10686	−0.553432	−	0.832895i	$-0.686682\pi$
$389$	−21.8308	−1.10686	−0.553432	+	0.832895i	$0.686682\pi$
$390$	0	0
$391$	−0.633596	−0.0320423
$392$	0	0
$393$	−36.3246	−1.83233
$394$	0	0
$395$	2.69641	0.135671
$396$	0	0
$397$	−17.5613	−0.881378	−0.440689	−	0.897660i	$-0.645266\pi$
$397$	−17.5613	−0.881378	−0.440689	+	0.897660i	$0.645266\pi$
$398$	0	0
$399$	21.4567	1.07418
$400$	0	0
$401$	5.62263	0.280781	0.140390	−	0.990096i	$-0.455164\pi$
$401$	5.62263	0.280781	0.140390	+	0.990096i	$0.455164\pi$
$402$	0	0
$403$	39.9402	1.98956
$404$	0	0
$405$	12.7283	0.632476
$406$	0	0
$407$	6.34472	0.314496
$408$	0	0
$409$	−12.3246	−0.609410	−0.304705	−	0.952447i	$-0.598558\pi$
$409$	−12.3246	−0.609410	−0.304705	+	0.952447i	$0.598558\pi$
$410$	0	0
$411$	8.22285	0.405604
$412$	0	0
$413$	32.0250	1.57585
$414$	0	0
$415$	3.17548	0.155878
$416$	0	0
$417$	22.8392	1.11844
$418$	0	0
$419$	−16.2166	−0.792233	−0.396117	−	0.918200i	$-0.629642\pi$
$419$	−16.2166	−0.792233	−0.396117	+	0.918200i	$0.629642\pi$
$420$	0	0
$421$	−32.0078	−1.55996	−0.779981	−	0.625803i	$-0.784772\pi$
$421$	−32.0078	−1.55996	−0.779981	+	0.625803i	$0.784772\pi$
$422$	0	0
$423$	36.8106	1.78979
$424$	0	0
$425$	2.02641	0.0982952
$426$	0	0
$427$	−10.0698	−0.487310
$428$	0	0
$429$	−83.8557	−4.04859
$430$	0	0
$431$	27.4178	1.32067	0.660334	−	0.750972i	$-0.270415\pi$
$431$	27.4178	1.32067	0.660334	+	0.750972i	$0.270415\pi$
$432$	0	0
$433$	28.4589	1.36765	0.683824	−	0.729647i	$-0.260316\pi$
$433$	28.4589	1.36765	0.683824	+	0.729647i	$0.260316\pi$
$434$	0	0
$435$	3.22357	0.154558
$436$	0	0
$437$	−2.90677	−0.139050
$438$	0	0
$439$	13.4923	0.643951	0.321976	−	0.946748i	$-0.395653\pi$
$439$	13.4923	0.643951	0.321976	+	0.946748i	$0.395653\pi$
$440$	0	0
$441$	−0.0605754	−0.00288454
$442$	0	0
$443$	1.43795	0.0683190	0.0341595	−	0.999416i	$-0.489125\pi$
$443$	1.43795	0.0683190	0.0341595	+	0.999416i	$0.489125\pi$
$444$	0	0
$445$	2.78562	0.132051
$446$	0	0
$447$	−6.53341	−0.309020
$448$	0	0
$449$	−12.4985	−0.589842	−0.294921	−	0.955522i	$-0.595293\pi$
$449$	−12.4985	−0.589842	−0.294921	+	0.955522i	$0.595293\pi$
$450$	0	0
$451$	−49.2159	−2.31749
$452$	0	0
$453$	−44.4457	−2.08824
$454$	0	0
$455$	−20.7981	−0.975030
$456$	0	0
$457$	12.4860	0.584072	0.292036	−	0.956407i	$-0.405667\pi$
$457$	12.4860	0.584072	0.292036	+	0.956407i	$0.405667\pi$
$458$	0	0
$459$	0.203408	0.00949425
$460$	0	0
$461$	−24.1336	−1.12402	−0.562008	−	0.827132i	$-0.689971\pi$
$461$	−24.1336	−1.12402	−0.562008	+	0.827132i	$0.689971\pi$
$462$	0	0
$463$	10.2640	0.477008	0.238504	−	0.971142i	$-0.423343\pi$
$463$	10.2640	0.477008	0.238504	+	0.971142i	$0.423343\pi$
$464$	0	0
$465$	27.2166	1.26214
$466$	0	0
$467$	28.5544	1.32134	0.660669	−	0.750677i	$-0.270273\pi$
$467$	28.5544	1.32134	0.660669	+	0.750677i	$0.270273\pi$
$468$	0	0
$469$	14.0698	0.649682
$470$	0	0
$471$	−10.2772	−0.473548
$472$	0	0
$473$	−64.2159	−2.95265
$474$	0	0
$475$	9.29663	0.426559
$476$	0	0
$477$	−12.9457	−0.592741
$478$	0	0
$479$	3.79187	0.173255	0.0866274	−	0.996241i	$-0.472391\pi$
$479$	3.79187	0.173255	0.0866274	+	0.996241i	$0.472391\pi$
$480$	0	0
$481$	5.34472	0.243698
$482$	0	0
$483$	−5.78267	−0.263121
$484$	0	0
$485$	−22.1909	−1.00764
$486$	0	0
$487$	−6.10866	−0.276810	−0.138405	−	0.990376i	$-0.544198\pi$
$487$	−6.10866	−0.276810	−0.138405	+	0.990376i	$0.544198\pi$
$488$	0	0
$489$	39.2966	1.77706
$490$	0	0
$491$	−15.1274	−0.682690	−0.341345	−	0.939938i	$-0.610882\pi$
$491$	−15.1274	−0.682690	−0.341345	+	0.939938i	$0.610882\pi$
$492$	0	0
$493$	−0.633596	−0.0285357
$494$	0	0
$495$	−29.1079	−1.30830
$496$	0	0
$497$	7.58774	0.340357
$498$	0	0
$499$	10.6894	0.478525	0.239263	−	0.970955i	$-0.423094\pi$
$499$	10.6894	0.478525	0.239263	+	0.970955i	$0.423094\pi$
$500$	0	0
$501$	41.4681	1.85266
$502$	0	0
$503$	0.0675359	0.00301128	0.00150564	−	0.999999i	$-0.499521\pi$
$503$	0.0675359	0.00301128	0.00150564	+	0.999999i	$0.499521\pi$
$504$	0	0
$505$	−12.0125	−0.534549
$506$	0	0
$507$	−38.4923	−1.70950
$508$	0	0
$509$	−7.39682	−0.327858	−0.163929	−	0.986472i	$-0.552417\pi$
$509$	−7.39682	−0.327858	−0.163929	+	0.986472i	$0.552417\pi$
$510$	0	0
$511$	−22.0474	−0.975318
$512$	0	0
$513$	0.933181	0.0412010
$514$	0	0
$515$	−21.5745	−0.950688
$516$	0	0
$517$	74.9791	3.29758
$518$	0	0
$519$	−5.96511	−0.261839
$520$	0	0
$521$	20.9806	0.919175	0.459587	−	0.888133i	$-0.347997\pi$
$521$	20.9806	0.919175	0.459587	+	0.888133i	$0.347997\pi$
$522$	0	0
$523$	−4.43322	−0.193851	−0.0969256	−	0.995292i	$-0.530901\pi$
$523$	−4.43322	−0.193851	−0.0969256	+	0.995292i	$0.530901\pi$
$524$	0	0
$525$	18.4945	0.807167
$526$	0	0
$527$	−5.34945	−0.233026
$528$	0	0
$529$	−22.2166	−0.965940
$530$	0	0
$531$	37.7563	1.63848
$532$	0	0
$533$	−41.4589	−1.79578
$534$	0	0
$535$	−9.14355	−0.395310
$536$	0	0
$537$	59.3789	2.56239
$538$	0	0
$539$	−0.123385	−0.00531458
$540$	0	0
$541$	−3.81131	−0.163861	−0.0819306	−	0.996638i	$-0.526109\pi$
$541$	−3.81131	−0.163861	−0.0819306	+	0.996638i	$0.526109\pi$
$542$	0	0
$543$	42.8580	1.83921
$544$	0	0
$545$	10.0514	0.430554
$546$	0	0
$547$	28.8370	1.23298	0.616491	−	0.787362i	$-0.288554\pi$
$547$	28.8370	1.23298	0.616491	+	0.787362i	$0.288554\pi$
$548$	0	0
$549$	−11.8719	−0.506680
$550$	0	0
$551$	−2.90677	−0.123833
$552$	0	0
$553$	−4.83700	−0.205690
$554$	0	0
$555$	3.64207	0.154597
$556$	0	0
$557$	29.7151	1.25907	0.629535	−	0.776972i	$-0.283245\pi$
$557$	29.7151	1.25907	0.629535	+	0.776972i	$0.283245\pi$
$558$	0	0
$559$	−54.0947	−2.28796
$560$	0	0
$561$	11.2313	0.474187
$562$	0	0
$563$	−22.6241	−0.953494	−0.476747	−	0.879041i	$-0.658184\pi$
$563$	−22.6241	−0.953494	−0.476747	+	0.879041i	$0.658184\pi$
$564$	0	0
$565$	−18.1909	−0.765298
$566$	0	0
$567$	−22.8330	−0.958896
$568$	0	0
$569$	31.7827	1.33240	0.666199	−	0.745774i	$-0.267920\pi$
$569$	31.7827	1.33240	0.666199	+	0.745774i	$0.267920\pi$
$570$	0	0
$571$	−23.2974	−0.974964	−0.487482	−	0.873133i	$-0.662084\pi$
$571$	−23.2974	−0.974964	−0.487482	+	0.873133i	$0.662084\pi$
$572$	0	0
$573$	18.6825	0.780472
$574$	0	0
$575$	−2.50548	−0.104486
$576$	0	0
$577$	4.33848	0.180613	0.0903066	−	0.995914i	$-0.471215\pi$
$577$	4.33848	0.180613	0.0903066	+	0.995914i	$0.471215\pi$
$578$	0	0
$579$	−14.2034	−0.590273
$580$	0	0
$581$	−5.69641	−0.236327
$582$	0	0
$583$	−26.3689	−1.09209
$584$	0	0
$585$	−24.5202	−1.01379
$586$	0	0
$587$	−1.68793	−0.0696682	−0.0348341	−	0.999393i	$-0.511090\pi$
$587$	−1.68793	−0.0696682	−0.0348341	+	0.999393i	$0.511090\pi$
$588$	0	0
$589$	−24.5419	−1.01123
$590$	0	0
$591$	−39.9512	−1.64337
$592$	0	0
$593$	−21.0217	−0.863257	−0.431628	−	0.902051i	$-0.642061\pi$
$593$	−21.0217	−0.863257	−0.431628	+	0.902051i	$0.642061\pi$
$594$	0	0
$595$	2.78562	0.114199
$596$	0	0
$597$	17.7438	0.726204
$598$	0	0
$599$	−13.2926	−0.543122	−0.271561	−	0.962421i	$-0.587540\pi$
$599$	−13.2926	−0.543122	−0.271561	+	0.962421i	$0.587540\pi$
$600$	0	0
$601$	−16.2974	−0.664783	−0.332391	−	0.943142i	$-0.607855\pi$
$601$	−16.2974	−0.664783	−0.332391	+	0.943142i	$0.607855\pi$
$602$	0	0
$603$	16.5877	0.675505
$604$	0	0
$605$	−43.0885	−1.75180
$606$	0	0
$607$	11.9394	0.484606	0.242303	−	0.970201i	$-0.422097\pi$
$607$	11.9394	0.484606	0.242303	+	0.970201i	$0.422097\pi$
$608$	0	0
$609$	−5.78267	−0.234326
$610$	0	0
$611$	63.1616	2.55524
$612$	0	0
$613$	44.5195	1.79813	0.899063	−	0.437820i	$-0.144249\pi$
$613$	44.5195	1.79813	0.899063	+	0.437820i	$0.144249\pi$
$614$	0	0
$615$	−28.2515	−1.13921
$616$	0	0
$617$	−43.3921	−1.74690	−0.873450	−	0.486914i	$-0.838123\pi$
$617$	−43.3921	−1.74690	−0.873450	+	0.486914i	$0.838123\pi$
$618$	0	0
$619$	−34.3572	−1.38093	−0.690466	−	0.723364i	$-0.742595\pi$
$619$	−34.3572	−1.38093	−0.690466	+	0.723364i	$0.742595\pi$
$620$	0	0
$621$	−0.251497	−0.0100922
$622$	0	0
$623$	−4.99705	−0.200203
$624$	0	0
$625$	−2.83299	−0.113320
$626$	0	0
$627$	51.5264	2.05777
$628$	0	0
$629$	−0.715853	−0.0285429
$630$	0	0
$631$	−28.3991	−1.13055	−0.565274	−	0.824903i	$-0.691230\pi$
$631$	−28.3991	−1.13055	−0.565274	+	0.824903i	$0.691230\pi$
$632$	0	0
$633$	37.5110	1.49093
$634$	0	0
$635$	27.2577	1.08169
$636$	0	0
$637$	−0.103938	−0.00411819
$638$	0	0
$639$	8.94567	0.353885
$640$	0	0
$641$	18.5940	0.734418	0.367209	−	0.930138i	$-0.380313\pi$
$641$	18.5940	0.734418	0.367209	+	0.930138i	$0.380313\pi$
$642$	0	0
$643$	7.51396	0.296322	0.148161	−	0.988963i	$-0.452665\pi$
$643$	7.51396	0.296322	0.148161	+	0.988963i	$0.452665\pi$
$644$	0	0
$645$	−36.8620	−1.45144
$646$	0	0
$647$	18.2904	0.719069	0.359535	−	0.933132i	$-0.382935\pi$
$647$	18.2904	0.719069	0.359535	+	0.933132i	$0.382935\pi$
$648$	0	0
$649$	76.9053	3.01880
$650$	0	0
$651$	−48.8231	−1.91353
$652$	0	0
$653$	1.70961	0.0669022	0.0334511	−	0.999440i	$-0.489350\pi$
$653$	1.70961	0.0669022	0.0334511	+	0.999440i	$0.489350\pi$
$654$	0	0
$655$	−21.6351	−0.845354
$656$	0	0
$657$	−25.9930	−1.01409
$658$	0	0
$659$	21.4310	0.834833	0.417416	−	0.908715i	$-0.362936\pi$
$659$	21.4310	0.834833	0.417416	+	0.908715i	$0.362936\pi$
$660$	0	0
$661$	45.5606	1.77210	0.886051	−	0.463587i	$-0.153438\pi$
$661$	45.5606	1.77210	0.886051	+	0.463587i	$0.153438\pi$
$662$	0	0
$663$	9.46115	0.367441
$664$	0	0
$665$	12.7797	0.495576
$666$	0	0
$667$	0.783389	0.0303329
$668$	0	0
$669$	−1.95263	−0.0754930
$670$	0	0
$671$	−24.1817	−0.933525
$672$	0	0
$673$	−45.4310	−1.75124	−0.875618	−	0.483004i	$-0.839546\pi$
$673$	−45.4310	−1.75124	−0.875618	+	0.483004i	$0.839546\pi$
$674$	0	0
$675$	0.804353	0.0309596
$676$	0	0
$677$	13.4357	0.516376	0.258188	−	0.966095i	$-0.416875\pi$
$677$	13.4357	0.516376	0.258188	+	0.966095i	$0.416875\pi$
$678$	0	0
$679$	39.8076	1.52768
$680$	0	0
$681$	−2.60719	−0.0999077
$682$	0	0
$683$	−18.4985	−0.707826	−0.353913	−	0.935278i	$-0.615149\pi$
$683$	−18.4985	−0.707826	−0.353913	+	0.935278i	$0.615149\pi$
$684$	0	0
$685$	4.89758	0.187127
$686$	0	0
$687$	−73.4652	−2.80287
$688$	0	0
$689$	−22.2129	−0.846243
$690$	0	0
$691$	11.4876	0.437007	0.218504	−	0.975836i	$-0.429882\pi$
$691$	11.4876	0.437007	0.218504	+	0.975836i	$0.429882\pi$
$692$	0	0
$693$	52.2159	1.98352
$694$	0	0
$695$	13.6032	0.515998
$696$	0	0
$697$	5.55286	0.210329
$698$	0	0
$699$	−46.9340	−1.77521
$700$	0	0
$701$	−22.0800	−0.833951	−0.416975	−	0.908918i	$-0.636910\pi$
$701$	−22.0800	−0.833951	−0.416975	+	0.908918i	$0.636910\pi$
$702$	0	0
$703$	−3.28415	−0.123864
$704$	0	0
$705$	43.0404	1.62100
$706$	0	0
$707$	21.5488	0.810428
$708$	0	0
$709$	18.7570	0.704433	0.352217	−	0.935919i	$-0.385428\pi$
$709$	18.7570	0.704433	0.352217	+	0.935919i	$0.385428\pi$
$710$	0	0
$711$	−5.70265	−0.213866
$712$	0	0
$713$	6.61415	0.247702
$714$	0	0
$715$	−49.9450	−1.86784
$716$	0	0
$717$	−1.87189	−0.0699070
$718$	0	0
$719$	31.9776	1.19256	0.596282	−	0.802775i	$-0.296644\pi$
$719$	31.9776	1.19256	0.596282	+	0.802775i	$0.296644\pi$
$720$	0	0
$721$	38.7019	1.44134
$722$	0	0
$723$	−0.364887	−0.0135703
$724$	0	0
$725$	−2.50548	−0.0930514
$726$	0	0
$727$	37.9061	1.40586	0.702929	−	0.711260i	$-0.251875\pi$
$727$	37.9061	1.40586	0.702929	+	0.711260i	$0.251875\pi$
$728$	0	0
$729$	−29.0272	−1.07508
$730$	0	0
$731$	7.24525	0.267975
$732$	0	0
$733$	17.8051	0.657645	0.328823	−	0.944392i	$-0.393348\pi$
$733$	17.8051	0.657645	0.328823	+	0.944392i	$0.393348\pi$
$734$	0	0
$735$	−0.0708271	−0.00261250
$736$	0	0
$737$	33.7874	1.24457
$738$	0	0
$739$	−29.3851	−1.08095	−0.540475	−	0.841360i	$-0.681755\pi$
$739$	−29.3851	−1.08095	−0.540475	+	0.841360i	$0.681755\pi$
$740$	0	0
$741$	43.4053	1.59453
$742$	0	0
$743$	3.72281	0.136577	0.0682884	−	0.997666i	$-0.478246\pi$
$743$	3.72281	0.136577	0.0682884	+	0.997666i	$0.478246\pi$
$744$	0	0
$745$	−3.89134	−0.142568
$746$	0	0
$747$	−6.71585	−0.245720
$748$	0	0
$749$	16.4023	0.599329
$750$	0	0
$751$	−7.47908	−0.272915	−0.136458	−	0.990646i	$-0.543572\pi$
$751$	−7.47908	−0.272915	−0.136458	+	0.990646i	$0.543572\pi$
$752$	0	0
$753$	−37.5264	−1.36754
$754$	0	0
$755$	−26.4721	−0.963419
$756$	0	0
$757$	15.1513	0.550684	0.275342	−	0.961346i	$-0.411209\pi$
$757$	15.1513	0.550684	0.275342	+	0.961346i	$0.411209\pi$
$758$	0	0
$759$	−13.8866	−0.504052
$760$	0	0
$761$	−32.1902	−1.16689	−0.583447	−	0.812151i	$-0.698296\pi$
$761$	−32.1902	−1.16689	−0.583447	+	0.812151i	$0.698296\pi$
$762$	0	0
$763$	−18.0309	−0.652762
$764$	0	0
$765$	3.28415	0.118739
$766$	0	0
$767$	64.7842	2.33922
$768$	0	0
$769$	6.91926	0.249515	0.124757	−	0.992187i	$-0.460185\pi$
$769$	6.91926	0.249515	0.124757	+	0.992187i	$0.460185\pi$
$770$	0	0
$771$	−58.1770	−2.09519
$772$	0	0
$773$	−45.2617	−1.62795	−0.813976	−	0.580899i	$-0.802701\pi$
$773$	−45.2617	−1.62795	−0.813976	+	0.580899i	$0.802701\pi$
$774$	0	0
$775$	−21.1538	−0.759867
$776$	0	0
$777$	−6.53341	−0.234385
$778$	0	0
$779$	25.4751	0.912739
$780$	0	0
$781$	18.2213	0.652011
$782$	0	0
$783$	−0.251497	−0.00898776
$784$	0	0
$785$	−6.12115	−0.218473
$786$	0	0
$787$	14.2338	0.507381	0.253691	−	0.967285i	$-0.418356\pi$
$787$	14.2338	0.507381	0.253691	+	0.967285i	$0.418356\pi$
$788$	0	0
$789$	−61.1700	−2.17771
$790$	0	0
$791$	32.6322	1.16027
$792$	0	0
$793$	−20.3704	−0.723375
$794$	0	0
$795$	−15.1366	−0.536839
$796$	0	0
$797$	34.5077	1.22233	0.611163	−	0.791505i	$-0.290702\pi$
$797$	34.5077	1.22233	0.611163	+	0.791505i	$0.290702\pi$
$798$	0	0
$799$	−8.45963	−0.299280
$800$	0	0
$801$	−5.89134	−0.208160
$802$	0	0
$803$	−52.9450	−1.86839
$804$	0	0
$805$	−3.44419	−0.121392
$806$	0	0
$807$	34.7159	1.22206
$808$	0	0
$809$	4.65055	0.163505	0.0817523	−	0.996653i	$-0.473948\pi$
$809$	4.65055	0.163505	0.0817523	+	0.996653i	$0.473948\pi$
$810$	0	0
$811$	−26.8697	−0.943521	−0.471761	−	0.881727i	$-0.656381\pi$
$811$	−26.8697	−0.943521	−0.471761	+	0.881727i	$0.656381\pi$
$812$	0	0
$813$	−10.0085	−0.351013
$814$	0	0
$815$	23.4053	0.819852
$816$	0	0
$817$	33.2393	1.16290
$818$	0	0
$819$	43.9861	1.53700
$820$	0	0
$821$	−7.86092	−0.274348	−0.137174	−	0.990547i	$-0.543802\pi$
$821$	−7.86092	−0.274348	−0.137174	+	0.990547i	$0.543802\pi$
$822$	0	0
$823$	−32.9193	−1.14749	−0.573747	−	0.819033i	$-0.694511\pi$
$823$	−32.9193	−1.14749	−0.573747	+	0.819033i	$0.694511\pi$
$824$	0	0
$825$	44.4131	1.54626
$826$	0	0
$827$	20.2687	0.704812	0.352406	−	0.935847i	$-0.385364\pi$
$827$	20.2687	0.704812	0.352406	+	0.935847i	$0.385364\pi$
$828$	0	0
$829$	1.83076	0.0635849	0.0317925	−	0.999494i	$-0.489878\pi$
$829$	1.83076	0.0635849	0.0317925	+	0.999494i	$0.489878\pi$
$830$	0	0
$831$	−40.2353	−1.39575
$832$	0	0
$833$	0.0139211	0.000482339 0
$834$	0	0
$835$	24.6986	0.854732
$836$	0	0
$837$	−2.12339	−0.0733949
$838$	0	0
$839$	−7.64760	−0.264024	−0.132012	−	0.991248i	$-0.542144\pi$
$839$	−7.64760	−0.264024	−0.132012	+	0.991248i	$0.542144\pi$
$840$	0	0
$841$	−28.2166	−0.972987
$842$	0	0
$843$	31.9472	1.10032
$844$	0	0
$845$	−22.9262	−0.788686
$846$	0	0
$847$	77.2952	2.65589
$848$	0	0
$849$	−4.88037	−0.167494
$850$	0	0
$851$	0.885092	0.0303406
$852$	0	0
$853$	−16.7431	−0.573271	−0.286636	−	0.958040i	$-0.592537\pi$
$853$	−16.7431	−0.573271	−0.286636	+	0.958040i	$0.592537\pi$
$854$	0	0
$855$	15.0668	0.515274
$856$	0	0
$857$	−15.4487	−0.527716	−0.263858	−	0.964562i	$-0.584995\pi$
$857$	−15.4487	−0.527716	−0.263858	+	0.964562i	$0.584995\pi$
$858$	0	0
$859$	38.4766	1.31280	0.656402	−	0.754411i	$-0.272078\pi$
$859$	38.4766	1.31280	0.656402	+	0.754411i	$0.272078\pi$
$860$	0	0
$861$	50.6795	1.72715
$862$	0	0
$863$	−17.5962	−0.598982	−0.299491	−	0.954099i	$-0.596817\pi$
$863$	−17.5962	−0.598982	−0.299491	+	0.954099i	$0.596817\pi$
$864$	0	0
$865$	−3.55286	−0.120801
$866$	0	0
$867$	40.7710	1.38466
$868$	0	0
$869$	−11.6157	−0.394034
$870$	0	0
$871$	28.4621	0.964402
$872$	0	0
$873$	46.9317	1.58840
$874$	0	0
$875$	30.4721	1.03015
$876$	0	0
$877$	20.1336	0.679865	0.339932	−	0.940450i	$-0.389596\pi$
$877$	20.1336	0.679865	0.339932	+	0.940450i	$0.389596\pi$
$878$	0	0
$879$	−84.3759	−2.84593
$880$	0	0
$881$	−8.91703	−0.300422	−0.150211	−	0.988654i	$-0.547995\pi$
$881$	−8.91703	−0.300422	−0.150211	+	0.988654i	$0.547995\pi$
$882$	0	0
$883$	5.59622	0.188328	0.0941639	−	0.995557i	$-0.469982\pi$
$883$	5.59622	0.188328	0.0941639	+	0.995557i	$0.469982\pi$
$884$	0	0
$885$	44.1461	1.48396
$886$	0	0
$887$	42.3594	1.42229	0.711145	−	0.703045i	$-0.248177\pi$
$887$	42.3594	1.42229	0.711145	+	0.703045i	$0.248177\pi$
$888$	0	0
$889$	−48.8969	−1.63995
$890$	0	0
$891$	−54.8316	−1.83693
$892$	0	0
$893$	−38.8106	−1.29875
$894$	0	0
$895$	35.3664	1.18217
$896$	0	0
$897$	−11.6979	−0.390582
$898$	0	0
$899$	6.61415	0.220594
$900$	0	0
$901$	2.97511	0.0991153
$902$	0	0
$903$	66.1256	2.20052
$904$	0	0
$905$	25.5264	0.848528
$906$	0	0
$907$	−22.3121	−0.740860	−0.370430	−	0.928860i	$-0.620790\pi$
$907$	−22.3121	−0.740860	−0.370430	+	0.928860i	$0.620790\pi$
$908$	0	0
$909$	25.4053	0.842641
$910$	0	0
$911$	20.0947	0.665769	0.332884	−	0.942968i	$-0.391978\pi$
$911$	20.0947	0.665769	0.332884	+	0.942968i	$0.391978\pi$
$912$	0	0
$913$	−13.6795	−0.452724
$914$	0	0
$915$	−13.8811	−0.458894
$916$	0	0
$917$	38.8106	1.28164
$918$	0	0
$919$	−3.47060	−0.114485	−0.0572423	−	0.998360i	$-0.518231\pi$
$919$	−3.47060	−0.114485	−0.0572423	+	0.998360i	$0.518231\pi$
$920$	0	0
$921$	−50.3572	−1.65933
$922$	0	0
$923$	15.3494	0.505233
$924$	0	0
$925$	−2.83076	−0.0930748
$926$	0	0
$927$	45.6282	1.49863
$928$	0	0
$929$	−1.59398	−0.0522969	−0.0261485	−	0.999658i	$-0.508324\pi$
$929$	−1.59398	−0.0522969	−0.0261485	+	0.999658i	$0.508324\pi$
$930$	0	0
$931$	0.0638666	0.00209314
$932$	0	0
$933$	−69.3891	−2.27170
$934$	0	0
$935$	6.68945	0.218768
$936$	0	0
$937$	13.0675	0.426898	0.213449	−	0.976954i	$-0.431530\pi$
$937$	13.0675	0.426898	0.213449	+	0.976954i	$0.431530\pi$
$938$	0	0
$939$	35.5529	1.16022
$940$	0	0
$941$	18.2857	0.596096	0.298048	−	0.954551i	$-0.403665\pi$
$941$	18.2857	0.596096	0.298048	+	0.954551i	$0.403665\pi$
$942$	0	0
$943$	−6.86565	−0.223576
$944$	0	0
$945$	1.10571	0.0359688
$946$	0	0
$947$	−6.86341	−0.223031	−0.111515	−	0.993763i	$-0.535570\pi$
$947$	−6.86341	−0.223031	−0.111515	+	0.993763i	$0.535570\pi$
$948$	0	0
$949$	−44.6002	−1.44778
$950$	0	0
$951$	50.2284	1.62877
$952$	0	0
$953$	20.1319	0.652135	0.326068	−	0.945346i	$-0.394276\pi$
$953$	20.1319	0.652135	0.326068	+	0.945346i	$0.394276\pi$
$954$	0	0
$955$	11.1274	0.360074
$956$	0	0
$957$	−13.8866	−0.448890
$958$	0	0
$959$	−8.78562	−0.283703
$960$	0	0
$961$	24.8432	0.801395
$962$	0	0
$963$	19.3378	0.623151
$964$	0	0
$965$	−8.45963	−0.272325
$966$	0	0
$967$	−36.6219	−1.17768	−0.588841	−	0.808249i	$-0.700415\pi$
$967$	−36.6219	−1.17768	−0.588841	+	0.808249i	$0.700415\pi$
$968$	0	0
$969$	−5.81355	−0.186758
$970$	0	0
$971$	−49.3502	−1.58372	−0.791862	−	0.610700i	$-0.790888\pi$
$971$	−49.3502	−1.58372	−0.791862	+	0.610700i	$0.790888\pi$
$972$	0	0
$973$	−24.4023	−0.782303
$974$	0	0
$975$	37.4131	1.19818
$976$	0	0
$977$	−33.6351	−1.07608	−0.538041	−	0.842918i	$-0.680836\pi$
$977$	−33.6351	−1.07608	−0.538041	+	0.842918i	$0.680836\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	0	0
$981$	−21.2577	−0.678707
$982$	0	0
$983$	−4.35944	−0.139045	−0.0695223	−	0.997580i	$-0.522148\pi$
$983$	−4.35944	−0.139045	−0.0695223	+	0.997580i	$0.522148\pi$
$984$	0	0
$985$	−23.7952	−0.758177
$986$	0	0
$987$	−77.2089	−2.45759
$988$	0	0
$989$	−8.95815	−0.284853
$990$	0	0
$991$	−53.1957	−1.68982	−0.844909	−	0.534910i	$-0.820345\pi$
$991$	−53.1957	−1.68982	−0.844909	+	0.534910i	$0.820345\pi$
$992$	0	0
$993$	−33.9861	−1.07852
$994$	0	0
$995$	10.5683	0.335037
$996$	0	0
$997$	−2.00000	−0.0633406	−0.0316703	−	0.999498i	$-0.510083\pi$
$997$	−2.00000	−0.0633406	−0.0316703	+	0.999498i	$0.510083\pi$
$998$	0	0
$999$	−0.284147	−0.00899002

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.bb.1.1		3
4.3	odd	2		2368.2.a.be.1.3		3
8.3	odd	2		592.2.a.i.1.1		3
8.5	even	2		296.2.a.c.1.3	✓	3
24.5	odd	2		2664.2.a.p.1.1		3
24.11	even	2		5328.2.a.bn.1.1		3
40.29	even	2		7400.2.a.k.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
296.2.a.c.1.3	✓	3	8.5	even	2
592.2.a.i.1.1		3	8.3	odd	2
2368.2.a.bb.1.1		3	1.1	even	1	trivial
2368.2.a.be.1.3		3	4.3	odd	2
2664.2.a.p.1.1		3	24.5	odd	2
5328.2.a.bn.1.1		3	24.11	even	2
7400.2.a.k.1.1		3	40.29	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:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.a (trivial)

\(f(q)\)	\(=\)	\(q-2.47283 q^{3} -1.47283 q^{5} +2.64207 q^{7} +3.11491 q^{9} +O(q^{10})\)	\(q-2.47283 q^{3} -1.47283 q^{5} +2.64207 q^{7} +3.11491 q^{9} +6.34472 q^{11} +5.34472 q^{13} +3.64207 q^{15} -0.715853 q^{17} -3.28415 q^{19} -6.53341 q^{21} +0.885092 q^{23} -2.83076 q^{25} -0.284147 q^{27} +0.885092 q^{29} +7.47283 q^{31} -15.6894 q^{33} -3.89134 q^{35} +1.00000 q^{37} -13.2166 q^{39} -7.75698 q^{41} -10.1212 q^{43} -4.58774 q^{45} +11.8176 q^{47} -0.0194469 q^{49} +1.77018 q^{51} -4.15604 q^{53} -9.34472 q^{55} +8.12115 q^{57} +12.1212 q^{59} -3.81131 q^{61} +8.22982 q^{63} -7.87189 q^{65} +5.32528 q^{67} -2.18869 q^{69} +2.87189 q^{71} -8.34472 q^{73} +7.00000 q^{75} +16.7632 q^{77} -1.83076 q^{79} -8.64207 q^{81} -2.15604 q^{83} +1.05433 q^{85} -2.18869 q^{87} -1.89134 q^{89} +14.1212 q^{91} -18.4791 q^{93} +4.83700 q^{95} +15.0668 q^{97} +19.7632 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + q^{5} + 7 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 + q^5 + 7 * q^7 + 3 * q^9	\( 3 q - 2 q^{3} + q^{5} + 7 q^{7} + 3 q^{9} - 3 q^{13} + 10 q^{15} - 4 q^{17} - 8 q^{19} + 3 q^{21} + 9 q^{23} - 4 q^{25} + q^{27} + 9 q^{29} + 17 q^{31} - 9 q^{33} + 10 q^{35} + 3 q^{37} - 7 q^{39} - 16 q^{41} + 4 q^{43} - 2 q^{45} + 11 q^{47} + 8 q^{49} + 18 q^{51} + 3 q^{53} - 9 q^{55} - 10 q^{57} + 2 q^{59} - 15 q^{61} + 12 q^{63} - 10 q^{65} + 5 q^{67} - 3 q^{69} - 5 q^{71} - 6 q^{73} + 21 q^{75} + 15 q^{77} - q^{79} - 25 q^{81} + 9 q^{83} + 14 q^{85} - 3 q^{87} + 16 q^{89} + 8 q^{91} - 22 q^{93} - 18 q^{95} + 24 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + q^5 + 7 * q^7 + 3 * q^9 - 3 * q^13 + 10 * q^15 - 4 * q^17 - 8 * q^19 + 3 * q^21 + 9 * q^23 - 4 * q^25 + q^27 + 9 * q^29 + 17 * q^31 - 9 * q^33 + 10 * q^35 + 3 * q^37 - 7 * q^39 - 16 * q^41 + 4 * q^43 - 2 * q^45 + 11 * q^47 + 8 * q^49 + 18 * q^51 + 3 * q^53 - 9 * q^55 - 10 * q^57 + 2 * q^59 - 15 * q^61 + 12 * q^63 - 10 * q^65 + 5 * q^67 - 3 * q^69 - 5 * q^71 - 6 * q^73 + 21 * q^75 + 15 * q^77 - q^79 - 25 * q^81 + 9 * q^83 + 14 * q^85 - 3 * q^87 + 16 * q^89 + 8 * q^91 - 22 * q^93 - 18 * q^95 + 24 * q^99

Introduction

L-functions

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