LMFDB - Embedded newform 3808.2.a.i.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3808,2,Mod(1,3808)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3808, 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("3808.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3808 = 2^{5} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3808.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:	$30.4070330897$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.80686992.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 15x^{3} + 8x^{2} - 15x + 3 $ x^6 - x^5 - 10x^4 + 15x^3 + 8x^2 - 15x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.39657$ of defining polynomial
Character	$\chi$	$=$	3808.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.20648 q^{3} +0.863883 q^{5} +1.00000 q^{7} -1.54442 q^{9} +0.0275234 q^{11} -0.319466 q^{13} +1.04225 q^{15} -1.00000 q^{17} -1.03217 q^{19} +1.20648 q^{21} -8.22897 q^{23} -4.25371 q^{25} -5.48273 q^{27} -1.97248 q^{29} -8.12128 q^{31} +0.0332063 q^{33} +0.863883 q^{35} -8.86790 q^{37} -0.385427 q^{39} -3.61419 q^{41} -5.96641 q^{43} -1.33420 q^{45} +8.25275 q^{47} +1.00000 q^{49} -1.20648 q^{51} +8.14880 q^{53} +0.0237770 q^{55} -1.24529 q^{57} +7.34565 q^{59} -0.736541 q^{61} -1.54442 q^{63} -0.275981 q^{65} +0.540671 q^{67} -9.92805 q^{69} -1.76097 q^{71} -3.55915 q^{73} -5.13199 q^{75} +0.0275234 q^{77} +13.2559 q^{79} -1.98152 q^{81} +7.27562 q^{83} -0.863883 q^{85} -2.37974 q^{87} -7.62312 q^{89} -0.319466 q^{91} -9.79812 q^{93} -0.891677 q^{95} +5.01810 q^{97} -0.0425076 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 4 q^{9} - 2 q^{11} - 4 q^{13} + 2 q^{15} - 6 q^{17} - 10 q^{19} - 2 q^{21} + 4 q^{23} + 4 q^{25} - 8 q^{27} - 14 q^{29} + 8 q^{31} - 8 q^{33} - 6 q^{35} - 4 q^{37} + 14 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 6 * q^5 + 6 * q^7 + 4 * q^9 - 2 * q^11 - 4 * q^13 + 2 * q^15 - 6 * q^17 - 10 * q^19 - 2 * q^21 + 4 * q^23 + 4 * q^25 - 8 * q^27 - 14 * q^29 + 8 * q^31 - 8 * q^33 - 6 * q^35 - 4 * q^37 + 14 * q^39 - 2 * q^41 - 8 * q^43 - 8 * q^45 - 2 * q^47 + 6 * q^49 + 2 * q^51 - 10 * q^53 + 2 * q^55 - 18 * q^57 - 14 * q^59 - 20 * q^61 + 4 * q^63 - 20 * q^65 - 6 * q^67 - 14 * q^69 + 20 * q^71 - 6 * q^73 - 10 * q^75 - 2 * q^77 + 38 * q^79 - 18 * q^81 - 16 * q^83 + 6 * q^85 - 4 * q^87 - 2 * q^89 - 4 * q^91 - 16 * q^93 + 42 * q^95 - 12 * q^97 - 4 * 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
$3$	1.20648	0.696559	0.348279	−	0.937391i	$-0.386766\pi$
$3$	1.20648	0.696559	0.348279	+	0.937391i	$0.386766\pi$
$4$	0	0
$5$	0.863883	0.386340	0.193170	−	0.981165i	$-0.438123\pi$
$5$	0.863883	0.386340	0.193170	+	0.981165i	$0.438123\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.54442	−0.514806
$10$	0	0
$11$	0.0275234	0.00829861	0.00414930	−	0.999991i	$-0.498679\pi$
$11$	0.0275234	0.00829861	0.00414930	+	0.999991i	$0.498679\pi$
$12$	0	0
$13$	−0.319466	−0.0886038	−0.0443019	−	0.999018i	$-0.514106\pi$
$13$	−0.319466	−0.0886038	−0.0443019	+	0.999018i	$0.514106\pi$
$14$	0	0
$15$	1.04225	0.269109
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−1.03217	−0.236797	−0.118398	−	0.992966i	$-0.537776\pi$
$19$	−1.03217	−0.236797	−0.118398	+	0.992966i	$0.537776\pi$
$20$	0	0
$21$	1.20648	0.263275
$22$	0	0
$23$	−8.22897	−1.71586	−0.857929	−	0.513768i	$-0.828249\pi$
$23$	−8.22897	−1.71586	−0.857929	+	0.513768i	$0.828249\pi$
$24$	0	0
$25$	−4.25371	−0.850741
$26$	0	0
$27$	−5.48273	−1.05515
$28$	0	0
$29$	−1.97248	−0.366280	−0.183140	−	0.983087i	$-0.558626\pi$
$29$	−1.97248	−0.366280	−0.183140	+	0.983087i	$0.558626\pi$
$30$	0	0
$31$	−8.12128	−1.45862	−0.729312	−	0.684181i	$-0.760160\pi$
$31$	−8.12128	−1.45862	−0.729312	+	0.684181i	$0.760160\pi$
$32$	0	0
$33$	0.0332063	0.00578047
$34$	0	0
$35$	0.863883	0.146023
$36$	0	0
$37$	−8.86790	−1.45787	−0.728937	−	0.684581i	$-0.759985\pi$
$37$	−8.86790	−1.45787	−0.728937	+	0.684581i	$0.759985\pi$
$38$	0	0
$39$	−0.385427	−0.0617178
$40$	0	0
$41$	−3.61419	−0.564442	−0.282221	−	0.959349i	$-0.591071\pi$
$41$	−3.61419	−0.564442	−0.282221	+	0.959349i	$0.591071\pi$
$42$	0	0
$43$	−5.96641	−0.909870	−0.454935	−	0.890525i	$-0.650337\pi$
$43$	−5.96641	−0.909870	−0.454935	+	0.890525i	$0.650337\pi$
$44$	0	0
$45$	−1.33420	−0.198890
$46$	0	0
$47$	8.25275	1.20379	0.601893	−	0.798576i	$-0.294413\pi$
$47$	8.25275	1.20379	0.601893	+	0.798576i	$0.294413\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.20648	−0.168940
$52$	0	0
$53$	8.14880	1.11932	0.559662	−	0.828721i	$-0.310931\pi$
$53$	8.14880	1.11932	0.559662	+	0.828721i	$0.310931\pi$
$54$	0	0
$55$	0.0237770	0.00320609
$56$	0	0
$57$	−1.24529	−0.164943
$58$	0	0
$59$	7.34565	0.956322	0.478161	−	0.878272i	$-0.341304\pi$
$59$	7.34565	0.956322	0.478161	+	0.878272i	$0.341304\pi$
$60$	0	0
$61$	−0.736541	−0.0943045	−0.0471522	−	0.998888i	$-0.515015\pi$
$61$	−0.736541	−0.0943045	−0.0471522	+	0.998888i	$0.515015\pi$
$62$	0	0
$63$	−1.54442	−0.194578
$64$	0	0
$65$	−0.275981	−0.0342312
$66$	0	0
$67$	0.540671	0.0660535	0.0330267	−	0.999454i	$-0.489485\pi$
$67$	0.540671	0.0660535	0.0330267	+	0.999454i	$0.489485\pi$
$68$	0	0
$69$	−9.92805	−1.19520
$70$	0	0
$71$	−1.76097	−0.208989	−0.104494	−	0.994525i	$-0.533322\pi$
$71$	−1.76097	−0.208989	−0.104494	+	0.994525i	$0.533322\pi$
$72$	0	0
$73$	−3.55915	−0.416567	−0.208283	−	0.978069i	$-0.566788\pi$
$73$	−3.55915	−0.416567	−0.208283	+	0.978069i	$0.566788\pi$
$74$	0	0
$75$	−5.13199	−0.592591
$76$	0	0
$77$	0.0275234	0.00313658
$78$	0	0
$79$	13.2559	1.49141	0.745703	−	0.666278i	$-0.232114\pi$
$79$	13.2559	1.49141	0.745703	+	0.666278i	$0.232114\pi$
$80$	0	0
$81$	−1.98152	−0.220169
$82$	0	0
$83$	7.27562	0.798603	0.399301	−	0.916820i	$-0.369253\pi$
$83$	7.27562	0.798603	0.399301	+	0.916820i	$0.369253\pi$
$84$	0	0
$85$	−0.863883	−0.0937012
$86$	0	0
$87$	−2.37974	−0.255135
$88$	0	0
$89$	−7.62312	−0.808049	−0.404024	−	0.914748i	$-0.632389\pi$
$89$	−7.62312	−0.808049	−0.404024	+	0.914748i	$0.632389\pi$
$90$	0	0
$91$	−0.319466	−0.0334891
$92$	0	0
$93$	−9.79812	−1.01602
$94$	0	0
$95$	−0.891677	−0.0914842
$96$	0	0
$97$	5.01810	0.509511	0.254755	−	0.967006i	$-0.418005\pi$
$97$	5.01810	0.509511	0.254755	+	0.967006i	$0.418005\pi$
$98$	0	0
$99$	−0.0425076	−0.00427217
$100$	0	0
$101$	0.0659618	0.00656344	0.00328172	−	0.999995i	$-0.498955\pi$
$101$	0.0659618	0.00656344	0.00328172	+	0.999995i	$0.498955\pi$
$102$	0	0
$103$	18.3422	1.80731	0.903653	−	0.428264i	$-0.140875\pi$
$103$	18.3422	1.80731	0.903653	+	0.428264i	$0.140875\pi$
$104$	0	0
$105$	1.04225	0.101714
$106$	0	0
$107$	−12.6051	−1.21858	−0.609290	−	0.792947i	$-0.708546\pi$
$107$	−12.6051	−1.21858	−0.609290	+	0.792947i	$0.708546\pi$
$108$	0	0
$109$	−2.03695	−0.195105	−0.0975523	−	0.995230i	$-0.531101\pi$
$109$	−2.03695	−0.195105	−0.0975523	+	0.995230i	$0.531101\pi$
$110$	0	0
$111$	−10.6989	−1.01549
$112$	0	0
$113$	−20.0268	−1.88396	−0.941980	−	0.335669i	$-0.891037\pi$
$113$	−20.0268	−1.88396	−0.941980	+	0.335669i	$0.891037\pi$
$114$	0	0
$115$	−7.10887	−0.662905
$116$	0	0
$117$	0.493388	0.0456137
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−10.9992	−0.999931
$122$	0	0
$123$	−4.36044	−0.393167
$124$	0	0
$125$	−7.99412	−0.715016
$126$	0	0
$127$	−0.858651	−0.0761929	−0.0380965	−	0.999274i	$-0.512129\pi$
$127$	−0.858651	−0.0761929	−0.0380965	+	0.999274i	$0.512129\pi$
$128$	0	0
$129$	−7.19833	−0.633778
$130$	0	0
$131$	−10.2387	−0.894561	−0.447281	−	0.894394i	$-0.647607\pi$
$131$	−10.2387	−0.894561	−0.447281	+	0.894394i	$0.647607\pi$
$132$	0	0
$133$	−1.03217	−0.0895008
$134$	0	0
$135$	−4.73643	−0.407647
$136$	0	0
$137$	9.66474	0.825714	0.412857	−	0.910796i	$-0.364531\pi$
$137$	9.66474	0.825714	0.412857	+	0.910796i	$0.364531\pi$
$138$	0	0
$139$	−0.726913	−0.0616559	−0.0308280	−	0.999525i	$-0.509814\pi$
$139$	−0.726913	−0.0616559	−0.0308280	+	0.999525i	$0.509814\pi$
$140$	0	0
$141$	9.95673	0.838508
$142$	0	0
$143$	−0.00879277	−0.000735288 0
$144$	0	0
$145$	−1.70399	−0.141509
$146$	0	0
$147$	1.20648	0.0995084
$148$	0	0
$149$	12.9796	1.06333	0.531663	−	0.846956i	$-0.321567\pi$
$149$	12.9796	1.06333	0.531663	+	0.846956i	$0.321567\pi$
$150$	0	0
$151$	14.6718	1.19397	0.596985	−	0.802252i	$-0.296365\pi$
$151$	14.6718	1.19397	0.596985	+	0.802252i	$0.296365\pi$
$152$	0	0
$153$	1.54442	0.124859
$154$	0	0
$155$	−7.01583	−0.563525
$156$	0	0
$157$	10.3700	0.827617	0.413808	−	0.910364i	$-0.364198\pi$
$157$	10.3700	0.827617	0.413808	+	0.910364i	$0.364198\pi$
$158$	0	0
$159$	9.83133	0.779675
$160$	0	0
$161$	−8.22897	−0.648534
$162$	0	0
$163$	8.52240	0.667526	0.333763	−	0.942657i	$-0.391682\pi$
$163$	8.52240	0.667526	0.333763	+	0.942657i	$0.391682\pi$
$164$	0	0
$165$	0.0286863	0.00223323
$166$	0	0
$167$	−3.08921	−0.239050	−0.119525	−	0.992831i	$-0.538137\pi$
$167$	−3.08921	−0.239050	−0.119525	+	0.992831i	$0.538137\pi$
$168$	0	0
$169$	−12.8979	−0.992149
$170$	0	0
$171$	1.59411	0.121904
$172$	0	0
$173$	−13.2257	−1.00554	−0.502768	−	0.864422i	$-0.667685\pi$
$173$	−13.2257	−1.00554	−0.502768	+	0.864422i	$0.667685\pi$
$174$	0	0
$175$	−4.25371	−0.321550
$176$	0	0
$177$	8.86235	0.666135
$178$	0	0
$179$	−2.66039	−0.198847	−0.0994235	−	0.995045i	$-0.531700\pi$
$179$	−2.66039	−0.198847	−0.0994235	+	0.995045i	$0.531700\pi$
$180$	0	0
$181$	26.4437	1.96554	0.982772	−	0.184824i	$-0.0591714\pi$
$181$	26.4437	1.96554	0.982772	+	0.184824i	$0.0591714\pi$
$182$	0	0
$183$	−0.888619	−0.0656886
$184$	0	0
$185$	−7.66083	−0.563235
$186$	0	0
$187$	−0.0275234	−0.00201271
$188$	0	0
$189$	−5.48273	−0.398810
$190$	0	0
$191$	9.45741	0.684314	0.342157	−	0.939643i	$-0.388843\pi$
$191$	9.45741	0.684314	0.342157	+	0.939643i	$0.388843\pi$
$192$	0	0
$193$	−12.8762	−0.926847	−0.463423	−	0.886137i	$-0.653379\pi$
$193$	−12.8762	−0.926847	−0.463423	+	0.886137i	$0.653379\pi$
$194$	0	0
$195$	−0.332964	−0.0238440
$196$	0	0
$197$	−21.2695	−1.51539	−0.757695	−	0.652609i	$-0.773674\pi$
$197$	−21.2695	−1.51539	−0.757695	+	0.652609i	$0.773674\pi$
$198$	0	0
$199$	−0.186186	−0.0131984	−0.00659919	−	0.999978i	$-0.502101\pi$
$199$	−0.186186	−0.0131984	−0.00659919	+	0.999978i	$0.502101\pi$
$200$	0	0
$201$	0.652306	0.0460101
$202$	0	0
$203$	−1.97248	−0.138441
$204$	0	0
$205$	−3.12224	−0.218067
$206$	0	0
$207$	12.7090	0.883334
$208$	0	0
$209$	−0.0284089	−0.00196509
$210$	0	0
$211$	15.8289	1.08970	0.544852	−	0.838533i	$-0.316586\pi$
$211$	15.8289	1.08970	0.544852	+	0.838533i	$0.316586\pi$
$212$	0	0
$213$	−2.12457	−0.145573
$214$	0	0
$215$	−5.15428	−0.351519
$216$	0	0
$217$	−8.12128	−0.551308
$218$	0	0
$219$	−4.29402	−0.290163
$220$	0	0
$221$	0.319466	0.0214896
$222$	0	0
$223$	4.03321	0.270083	0.135042	−	0.990840i	$-0.456883\pi$
$223$	4.03321	0.270083	0.135042	+	0.990840i	$0.456883\pi$
$224$	0	0
$225$	6.56950	0.437967
$226$	0	0
$227$	−6.62369	−0.439630	−0.219815	−	0.975542i	$-0.570545\pi$
$227$	−6.62369	−0.439630	−0.219815	+	0.975542i	$0.570545\pi$
$228$	0	0
$229$	−24.4539	−1.61596	−0.807979	−	0.589212i	$-0.799438\pi$
$229$	−24.4539	−1.61596	−0.807979	+	0.589212i	$0.799438\pi$
$230$	0	0
$231$	0.0332063	0.00218481
$232$	0	0
$233$	−6.11565	−0.400649	−0.200325	−	0.979730i	$-0.564200\pi$
$233$	−6.11565	−0.400649	−0.200325	+	0.979730i	$0.564200\pi$
$234$	0	0
$235$	7.12941	0.465071
$236$	0	0
$237$	15.9929	1.03885
$238$	0	0
$239$	20.8687	1.34988	0.674941	−	0.737871i	$-0.264169\pi$
$239$	20.8687	1.34988	0.674941	+	0.737871i	$0.264169\pi$
$240$	0	0
$241$	19.5329	1.25822	0.629112	−	0.777315i	$-0.283419\pi$
$241$	19.5329	1.25822	0.629112	+	0.777315i	$0.283419\pi$
$242$	0	0
$243$	14.0575	0.901790
$244$	0	0
$245$	0.863883	0.0551914
$246$	0	0
$247$	0.329744	0.0209811
$248$	0	0
$249$	8.77785	0.556274
$250$	0	0
$251$	−17.0716	−1.07755	−0.538775	−	0.842450i	$-0.681113\pi$
$251$	−17.0716	−1.07755	−0.538775	+	0.842450i	$0.681113\pi$
$252$	0	0
$253$	−0.226489	−0.0142392
$254$	0	0
$255$	−1.04225	−0.0652684
$256$	0	0
$257$	−9.42232	−0.587748	−0.293874	−	0.955844i	$-0.594945\pi$
$257$	−9.42232	−0.587748	−0.293874	+	0.955844i	$0.594945\pi$
$258$	0	0
$259$	−8.86790	−0.551025
$260$	0	0
$261$	3.04633	0.188563
$262$	0	0
$263$	17.1982	1.06048	0.530242	−	0.847846i	$-0.322101\pi$
$263$	17.1982	1.06048	0.530242	+	0.847846i	$0.322101\pi$
$264$	0	0
$265$	7.03961	0.432440
$266$	0	0
$267$	−9.19710	−0.562854
$268$	0	0
$269$	−12.8406	−0.782903	−0.391451	−	0.920199i	$-0.628027\pi$
$269$	−12.8406	−0.782903	−0.391451	+	0.920199i	$0.628027\pi$
$270$	0	0
$271$	5.63072	0.342042	0.171021	−	0.985267i	$-0.445293\pi$
$271$	5.63072	0.342042	0.171021	+	0.985267i	$0.445293\pi$
$272$	0	0
$273$	−0.385427	−0.0233271
$274$	0	0
$275$	−0.117076	−0.00705997
$276$	0	0
$277$	−31.8414	−1.91316	−0.956582	−	0.291464i	$-0.905858\pi$
$277$	−31.8414	−1.91316	−0.956582	+	0.291464i	$0.905858\pi$
$278$	0	0
$279$	12.5426	0.750908
$280$	0	0
$281$	−10.7801	−0.643087	−0.321544	−	0.946895i	$-0.604202\pi$
$281$	−10.7801	−0.643087	−0.321544	+	0.946895i	$0.604202\pi$
$282$	0	0
$283$	−13.0285	−0.774464	−0.387232	−	0.921982i	$-0.626569\pi$
$283$	−13.0285	−0.774464	−0.387232	+	0.921982i	$0.626569\pi$
$284$	0	0
$285$	−1.07579	−0.0637241
$286$	0	0
$287$	−3.61419	−0.213339
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	6.05421	0.354904
$292$	0	0
$293$	23.0193	1.34480	0.672401	−	0.740187i	$-0.265263\pi$
$293$	23.0193	1.34480	0.672401	+	0.740187i	$0.265263\pi$
$294$	0	0
$295$	6.34578	0.369466
$296$	0	0
$297$	−0.150903	−0.00875629
$298$	0	0
$299$	2.62887	0.152032
$300$	0	0
$301$	−5.96641	−0.343898
$302$	0	0
$303$	0.0795812	0.00457182
$304$	0	0
$305$	−0.636285	−0.0364336
$306$	0	0
$307$	−20.2243	−1.15426	−0.577130	−	0.816652i	$-0.695828\pi$
$307$	−20.2243	−1.15426	−0.577130	+	0.816652i	$0.695828\pi$
$308$	0	0
$309$	22.1294	1.25890
$310$	0	0
$311$	17.0862	0.968871	0.484435	−	0.874827i	$-0.339025\pi$
$311$	17.0862	0.968871	0.484435	+	0.874827i	$0.339025\pi$
$312$	0	0
$313$	23.0644	1.30368	0.651838	−	0.758358i	$-0.273998\pi$
$313$	23.0644	1.30368	0.651838	+	0.758358i	$0.273998\pi$
$314$	0	0
$315$	−1.33420	−0.0751734
$316$	0	0
$317$	−30.3014	−1.70189	−0.850947	−	0.525252i	$-0.823971\pi$
$317$	−30.3014	−1.70189	−0.850947	+	0.525252i	$0.823971\pi$
$318$	0	0
$319$	−0.0542892	−0.00303961
$320$	0	0
$321$	−15.2077	−0.848813
$322$	0	0
$323$	1.03217	0.0574317
$324$	0	0
$325$	1.35891	0.0753789
$326$	0	0
$327$	−2.45753	−0.135902
$328$	0	0
$329$	8.25275	0.454989
$330$	0	0
$331$	−20.1929	−1.10990	−0.554950	−	0.831884i	$-0.687263\pi$
$331$	−20.1929	−1.10990	−0.554950	+	0.831884i	$0.687263\pi$
$332$	0	0
$333$	13.6957	0.750522
$334$	0	0
$335$	0.467076	0.0255191
$336$	0	0
$337$	−8.83676	−0.481369	−0.240684	−	0.970603i	$-0.577372\pi$
$337$	−8.83676	−0.481369	−0.240684	+	0.970603i	$0.577372\pi$
$338$	0	0
$339$	−24.1618	−1.31229
$340$	0	0
$341$	−0.223525	−0.0121046
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−8.57667	−0.461752
$346$	0	0
$347$	20.4777	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$347$	20.4777	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$348$	0	0
$349$	−17.6705	−0.945879	−0.472939	−	0.881095i	$-0.656807\pi$
$349$	−17.6705	−0.945879	−0.472939	+	0.881095i	$0.656807\pi$
$350$	0	0
$351$	1.75154	0.0934904
$352$	0	0
$353$	−21.6850	−1.15418	−0.577089	−	0.816681i	$-0.695811\pi$
$353$	−21.6850	−1.15418	−0.577089	+	0.816681i	$0.695811\pi$
$354$	0	0
$355$	−1.52127	−0.0807408
$356$	0	0
$357$	−1.20648	−0.0638534
$358$	0	0
$359$	−20.6693	−1.09088	−0.545441	−	0.838149i	$-0.683638\pi$
$359$	−20.6693	−1.09088	−0.545441	+	0.838149i	$0.683638\pi$
$360$	0	0
$361$	−17.9346	−0.943927
$362$	0	0
$363$	−13.2703	−0.696511
$364$	0	0
$365$	−3.07469	−0.160936
$366$	0	0
$367$	24.3930	1.27330	0.636651	−	0.771152i	$-0.280319\pi$
$367$	24.3930	1.27330	0.636651	+	0.771152i	$0.280319\pi$
$368$	0	0
$369$	5.58182	0.290578
$370$	0	0
$371$	8.14880	0.423065
$372$	0	0
$373$	−9.70361	−0.502434	−0.251217	−	0.967931i	$-0.580831\pi$
$373$	−9.70361	−0.502434	−0.251217	+	0.967931i	$0.580831\pi$
$374$	0	0
$375$	−9.64471	−0.498050
$376$	0	0
$377$	0.630138	0.0324538
$378$	0	0
$379$	−30.0747	−1.54483	−0.772417	−	0.635115i	$-0.780953\pi$
$379$	−30.0747	−1.54483	−0.772417	+	0.635115i	$0.780953\pi$
$380$	0	0
$381$	−1.03594	−0.0530729
$382$	0	0
$383$	25.4911	1.30253	0.651267	−	0.758849i	$-0.274238\pi$
$383$	25.4911	1.30253	0.651267	+	0.758849i	$0.274238\pi$
$384$	0	0
$385$	0.0237770	0.00121179
$386$	0	0
$387$	9.21463	0.468406
$388$	0	0
$389$	3.38598	0.171676	0.0858381	−	0.996309i	$-0.472643\pi$
$389$	3.38598	0.171676	0.0858381	+	0.996309i	$0.472643\pi$
$390$	0	0
$391$	8.22897	0.416157
$392$	0	0
$393$	−12.3528	−0.623115
$394$	0	0
$395$	11.4516	0.576190
$396$	0	0
$397$	−8.34551	−0.418849	−0.209425	−	0.977825i	$-0.567159\pi$
$397$	−8.34551	−0.418849	−0.209425	+	0.977825i	$0.567159\pi$
$398$	0	0
$399$	−1.24529	−0.0623426
$400$	0	0
$401$	10.5736	0.528020	0.264010	−	0.964520i	$-0.414955\pi$
$401$	10.5736	0.528020	0.264010	+	0.964520i	$0.414955\pi$
$402$	0	0
$403$	2.59447	0.129240
$404$	0	0
$405$	−1.71180	−0.0850602
$406$	0	0
$407$	−0.244074	−0.0120983
$408$	0	0
$409$	−2.72811	−0.134896	−0.0674481	−	0.997723i	$-0.521486\pi$
$409$	−2.72811	−0.134896	−0.0674481	+	0.997723i	$0.521486\pi$
$410$	0	0
$411$	11.6603	0.575159
$412$	0	0
$413$	7.34565	0.361456
$414$	0	0
$415$	6.28528	0.308532
$416$	0	0
$417$	−0.877003	−0.0429470
$418$	0	0
$419$	−8.20105	−0.400647	−0.200324	−	0.979730i	$-0.564199\pi$
$419$	−8.20105	−0.400647	−0.200324	+	0.979730i	$0.564199\pi$
$420$	0	0
$421$	24.8388	1.21057	0.605285	−	0.796009i	$-0.293059\pi$
$421$	24.8388	1.21057	0.605285	+	0.796009i	$0.293059\pi$
$422$	0	0
$423$	−12.7457	−0.619716
$424$	0	0
$425$	4.25371	0.206335
$426$	0	0
$427$	−0.736541	−0.0356437
$428$	0	0
$429$	−0.0106083	−0.000512172 0
$430$	0	0
$431$	−3.74898	−0.180582	−0.0902909	−	0.995915i	$-0.528780\pi$
$431$	−3.74898	−0.180582	−0.0902909	+	0.995915i	$0.528780\pi$
$432$	0	0
$433$	22.0047	1.05748	0.528739	−	0.848785i	$-0.322665\pi$
$433$	22.0047	1.05748	0.528739	+	0.848785i	$0.322665\pi$
$434$	0	0
$435$	−2.05582	−0.0985690
$436$	0	0
$437$	8.49373	0.406310
$438$	0	0
$439$	1.64364	0.0784466	0.0392233	−	0.999230i	$-0.487512\pi$
$439$	1.64364	0.0784466	0.0392233	+	0.999230i	$0.487512\pi$
$440$	0	0
$441$	−1.54442	−0.0735437
$442$	0	0
$443$	8.88095	0.421947	0.210973	−	0.977492i	$-0.432337\pi$
$443$	8.88095	0.421947	0.210973	+	0.977492i	$0.432337\pi$
$444$	0	0
$445$	−6.58548	−0.312182
$446$	0	0
$447$	15.6595	0.740669
$448$	0	0
$449$	13.6112	0.642354	0.321177	−	0.947019i	$-0.395922\pi$
$449$	13.6112	0.642354	0.321177	+	0.947019i	$0.395922\pi$
$450$	0	0
$451$	−0.0994748	−0.00468409
$452$	0	0
$453$	17.7011	0.831671
$454$	0	0
$455$	−0.275981	−0.0129382
$456$	0	0
$457$	15.0680	0.704851	0.352426	−	0.935840i	$-0.385357\pi$
$457$	15.0680	0.704851	0.352426	+	0.935840i	$0.385357\pi$
$458$	0	0
$459$	5.48273	0.255912
$460$	0	0
$461$	−18.3116	−0.852858	−0.426429	−	0.904521i	$-0.640229\pi$
$461$	−18.3116	−0.852858	−0.426429	+	0.904521i	$0.640229\pi$
$462$	0	0
$463$	−25.9548	−1.20622	−0.603112	−	0.797657i	$-0.706073\pi$
$463$	−25.9548	−1.20622	−0.603112	+	0.797657i	$0.706073\pi$
$464$	0	0
$465$	−8.46443	−0.392529
$466$	0	0
$467$	30.5791	1.41503	0.707515	−	0.706698i	$-0.249816\pi$
$467$	30.5791	1.41503	0.707515	+	0.706698i	$0.249816\pi$
$468$	0	0
$469$	0.540671	0.0249659
$470$	0	0
$471$	12.5112	0.576484
$472$	0	0
$473$	−0.164216	−0.00755065
$474$	0	0
$475$	4.39057	0.201453
$476$	0	0
$477$	−12.5852	−0.576235
$478$	0	0
$479$	−11.8367	−0.540834	−0.270417	−	0.962743i	$-0.587162\pi$
$479$	−11.8367	−0.540834	−0.270417	+	0.962743i	$0.587162\pi$
$480$	0	0
$481$	2.83299	0.129173
$482$	0	0
$483$	−9.92805	−0.451742
$484$	0	0
$485$	4.33505	0.196844
$486$	0	0
$487$	5.01077	0.227060	0.113530	−	0.993535i	$-0.463784\pi$
$487$	5.01077	0.227060	0.113530	+	0.993535i	$0.463784\pi$
$488$	0	0
$489$	10.2821	0.464971
$490$	0	0
$491$	−24.4346	−1.10272	−0.551360	−	0.834268i	$-0.685891\pi$
$491$	−24.4346	−1.10272	−0.551360	+	0.834268i	$0.685891\pi$
$492$	0	0
$493$	1.97248	0.0888359
$494$	0	0
$495$	−0.0367216	−0.00165051
$496$	0	0
$497$	−1.76097	−0.0789904
$498$	0	0
$499$	38.2375	1.71175	0.855874	−	0.517185i	$-0.173020\pi$
$499$	38.2375	1.71175	0.855874	+	0.517185i	$0.173020\pi$
$500$	0	0
$501$	−3.72706	−0.166513
$502$	0	0
$503$	−10.3160	−0.459967	−0.229984	−	0.973194i	$-0.573867\pi$
$503$	−10.3160	−0.459967	−0.229984	+	0.973194i	$0.573867\pi$
$504$	0	0
$505$	0.0569832	0.00253572
$506$	0	0
$507$	−15.5610	−0.691090
$508$	0	0
$509$	−42.6986	−1.89258	−0.946292	−	0.323315i	$-0.895203\pi$
$509$	−42.6986	−1.89258	−0.946292	+	0.323315i	$0.895203\pi$
$510$	0	0
$511$	−3.55915	−0.157447
$512$	0	0
$513$	5.65913	0.249857
$514$	0	0
$515$	15.8455	0.698235
$516$	0	0
$517$	0.227143	0.00998975
$518$	0	0
$519$	−15.9565	−0.700414
$520$	0	0
$521$	21.1453	0.926394	0.463197	−	0.886255i	$-0.346702\pi$
$521$	21.1453	0.926394	0.463197	+	0.886255i	$0.346702\pi$
$522$	0	0
$523$	−38.1388	−1.66769	−0.833846	−	0.551998i	$-0.813866\pi$
$523$	−38.1388	−1.66769	−0.833846	+	0.551998i	$0.813866\pi$
$524$	0	0
$525$	−5.13199	−0.223978
$526$	0	0
$527$	8.12128	0.353769
$528$	0	0
$529$	44.7159	1.94417
$530$	0	0
$531$	−11.3447	−0.492320
$532$	0	0
$533$	1.15461	0.0500117
$534$	0	0
$535$	−10.8893	−0.470787
$536$	0	0
$537$	−3.20970	−0.138509
$538$	0	0
$539$	0.0275234	0.00118552
$540$	0	0
$541$	4.57525	0.196705	0.0983526	−	0.995152i	$-0.468643\pi$
$541$	4.57525	0.196705	0.0983526	+	0.995152i	$0.468643\pi$
$542$	0	0
$543$	31.9037	1.36912
$544$	0	0
$545$	−1.75969	−0.0753768
$546$	0	0
$547$	8.53985	0.365138	0.182569	−	0.983193i	$-0.441559\pi$
$547$	8.53985	0.365138	0.182569	+	0.983193i	$0.441559\pi$
$548$	0	0
$549$	1.13753	0.0485485
$550$	0	0
$551$	2.03594	0.0867339
$552$	0	0
$553$	13.2559	0.563699
$554$	0	0
$555$	−9.24260	−0.392326
$556$	0	0
$557$	11.7382	0.497365	0.248682	−	0.968585i	$-0.420002\pi$
$557$	11.7382	0.497365	0.248682	+	0.968585i	$0.420002\pi$
$558$	0	0
$559$	1.90606	0.0806179
$560$	0	0
$561$	−0.0332063	−0.00140197
$562$	0	0
$563$	−28.1802	−1.18765	−0.593827	−	0.804592i	$-0.702384\pi$
$563$	−28.1802	−1.18765	−0.593827	+	0.804592i	$0.702384\pi$
$564$	0	0
$565$	−17.3008	−0.727849
$566$	0	0
$567$	−1.98152	−0.0832162
$568$	0	0
$569$	−27.6547	−1.15934	−0.579672	−	0.814850i	$-0.696819\pi$
$569$	−27.6547	−1.15934	−0.579672	+	0.814850i	$0.696819\pi$
$570$	0	0
$571$	22.2631	0.931682	0.465841	−	0.884869i	$-0.345752\pi$
$571$	22.2631	0.931682	0.465841	+	0.884869i	$0.345752\pi$
$572$	0	0
$573$	11.4101	0.476665
$574$	0	0
$575$	35.0036	1.45975
$576$	0	0
$577$	20.6422	0.859346	0.429673	−	0.902985i	$-0.358629\pi$
$577$	20.6422	0.859346	0.429673	+	0.902985i	$0.358629\pi$
$578$	0	0
$579$	−15.5348	−0.645603
$580$	0	0
$581$	7.27562	0.301844
$582$	0	0
$583$	0.224283	0.00928883
$584$	0	0
$585$	0.426230	0.0176224
$586$	0	0
$587$	3.43818	0.141909	0.0709544	−	0.997480i	$-0.477396\pi$
$587$	3.43818	0.141909	0.0709544	+	0.997480i	$0.477396\pi$
$588$	0	0
$589$	8.38257	0.345398
$590$	0	0
$591$	−25.6611	−1.05556
$592$	0	0
$593$	2.39809	0.0984779	0.0492390	−	0.998787i	$-0.484320\pi$
$593$	2.39809	0.0984779	0.0492390	+	0.998787i	$0.484320\pi$
$594$	0	0
$595$	−0.863883	−0.0354157
$596$	0	0
$597$	−0.224629	−0.00919345
$598$	0	0
$599$	−7.01935	−0.286803	−0.143401	−	0.989665i	$-0.545804\pi$
$599$	−7.01935	−0.286803	−0.143401	+	0.989665i	$0.545804\pi$
$600$	0	0
$601$	−22.5322	−0.919109	−0.459555	−	0.888149i	$-0.651991\pi$
$601$	−22.5322	−0.919109	−0.459555	+	0.888149i	$0.651991\pi$
$602$	0	0
$603$	−0.835021	−0.0340047
$604$	0	0
$605$	−9.50206	−0.386314
$606$	0	0
$607$	46.9338	1.90498	0.952492	−	0.304564i	$-0.0985107\pi$
$607$	46.9338	1.90498	0.952492	+	0.304564i	$0.0985107\pi$
$608$	0	0
$609$	−2.37974	−0.0964321
$610$	0	0
$611$	−2.63647	−0.106660
$612$	0	0
$613$	40.4369	1.63323	0.816615	−	0.577183i	$-0.195848\pi$
$613$	40.4369	1.63323	0.816615	+	0.577183i	$0.195848\pi$
$614$	0	0
$615$	−3.76691	−0.151896
$616$	0	0
$617$	−30.4224	−1.22476	−0.612380	−	0.790563i	$-0.709788\pi$
$617$	−30.4224	−1.22476	−0.612380	+	0.790563i	$0.709788\pi$
$618$	0	0
$619$	−20.1168	−0.808561	−0.404280	−	0.914635i	$-0.632478\pi$
$619$	−20.1168	−0.808561	−0.404280	+	0.914635i	$0.632478\pi$
$620$	0	0
$621$	45.1172	1.81049
$622$	0	0
$623$	−7.62312	−0.305414
$624$	0	0
$625$	14.3626	0.574502
$626$	0	0
$627$	−0.0342746	−0.00136880
$628$	0	0
$629$	8.86790	0.353586
$630$	0	0
$631$	−1.72730	−0.0687627	−0.0343813	−	0.999409i	$-0.510946\pi$
$631$	−1.72730	−0.0687627	−0.0343813	+	0.999409i	$0.510946\pi$
$632$	0	0
$633$	19.0971	0.759042
$634$	0	0
$635$	−0.741774	−0.0294364
$636$	0	0
$637$	−0.319466	−0.0126577
$638$	0	0
$639$	2.71968	0.107589
$640$	0	0
$641$	49.1915	1.94295	0.971474	−	0.237146i	$-0.0762120\pi$
$641$	49.1915	1.94295	0.971474	+	0.237146i	$0.0762120\pi$
$642$	0	0
$643$	−31.4107	−1.23872	−0.619358	−	0.785109i	$-0.712607\pi$
$643$	−31.4107	−1.23872	−0.619358	+	0.785109i	$0.712607\pi$
$644$	0	0
$645$	−6.21852	−0.244854
$646$	0	0
$647$	−33.6520	−1.32300	−0.661498	−	0.749947i	$-0.730079\pi$
$647$	−33.6520	−1.32300	−0.661498	+	0.749947i	$0.730079\pi$
$648$	0	0
$649$	0.202177	0.00793614
$650$	0	0
$651$	−9.79812	−0.384019
$652$	0	0
$653$	−5.17182	−0.202389	−0.101194	−	0.994867i	$-0.532266\pi$
$653$	−5.17182	−0.202389	−0.101194	+	0.994867i	$0.532266\pi$
$654$	0	0
$655$	−8.84506	−0.345605
$656$	0	0
$657$	5.49681	0.214451
$658$	0	0
$659$	−11.7798	−0.458875	−0.229438	−	0.973323i	$-0.573689\pi$
$659$	−11.7798	−0.458875	−0.229438	+	0.973323i	$0.573689\pi$
$660$	0	0
$661$	11.9276	0.463929	0.231964	−	0.972724i	$-0.425485\pi$
$661$	11.9276	0.463929	0.231964	+	0.972724i	$0.425485\pi$
$662$	0	0
$663$	0.385427	0.0149688
$664$	0	0
$665$	−0.891677	−0.0345778
$666$	0	0
$667$	16.2314	0.628484
$668$	0	0
$669$	4.86596	0.188129
$670$	0	0
$671$	−0.0202721	−0.000782596 0
$672$	0	0
$673$	−21.6291	−0.833740	−0.416870	−	0.908966i	$-0.636873\pi$
$673$	−21.6291	−0.833740	−0.416870	+	0.908966i	$0.636873\pi$
$674$	0	0
$675$	23.3219	0.897661
$676$	0	0
$677$	−19.0156	−0.730831	−0.365415	−	0.930845i	$-0.619073\pi$
$677$	−19.0156	−0.730831	−0.365415	+	0.930845i	$0.619073\pi$
$678$	0	0
$679$	5.01810	0.192577
$680$	0	0
$681$	−7.99132	−0.306228
$682$	0	0
$683$	−3.82222	−0.146253	−0.0731266	−	0.997323i	$-0.523298\pi$
$683$	−3.82222	−0.146253	−0.0731266	+	0.997323i	$0.523298\pi$
$684$	0	0
$685$	8.34920	0.319007
$686$	0	0
$687$	−29.5030	−1.12561
$688$	0	0
$689$	−2.60326	−0.0991764
$690$	0	0
$691$	2.56638	0.0976297	0.0488149	−	0.998808i	$-0.484456\pi$
$691$	2.56638	0.0976297	0.0488149	+	0.998808i	$0.484456\pi$
$692$	0	0
$693$	−0.0425076	−0.00161473
$694$	0	0
$695$	−0.627968	−0.0238202
$696$	0	0
$697$	3.61419	0.136897
$698$	0	0
$699$	−7.37838	−0.279076
$700$	0	0
$701$	26.1915	0.989239	0.494620	−	0.869110i	$-0.335307\pi$
$701$	26.1915	0.989239	0.494620	+	0.869110i	$0.335307\pi$
$702$	0	0
$703$	9.15322	0.345220
$704$	0	0
$705$	8.60145	0.323949
$706$	0	0
$707$	0.0659618	0.00248075
$708$	0	0
$709$	−45.1855	−1.69698	−0.848489	−	0.529212i	$-0.822487\pi$
$709$	−45.1855	−1.69698	−0.848489	+	0.529212i	$0.822487\pi$
$710$	0	0
$711$	−20.4727	−0.767785
$712$	0	0
$713$	66.8298	2.50279
$714$	0	0
$715$	−0.00759592	−0.000284071 0
$716$	0	0
$717$	25.1776	0.940273
$718$	0	0
$719$	−47.8520	−1.78458	−0.892290	−	0.451463i	$-0.850902\pi$
$719$	−47.8520	−1.78458	−0.892290	+	0.451463i	$0.850902\pi$
$720$	0	0
$721$	18.3422	0.683098
$722$	0	0
$723$	23.5660	0.876427
$724$	0	0
$725$	8.39034	0.311609
$726$	0	0
$727$	−0.509972	−0.0189138	−0.00945690	−	0.999955i	$-0.503010\pi$
$727$	−0.509972	−0.0189138	−0.00945690	+	0.999955i	$0.503010\pi$
$728$	0	0
$729$	22.9046	0.848319
$730$	0	0
$731$	5.96641	0.220676
$732$	0	0
$733$	37.5811	1.38809	0.694045	−	0.719932i	$-0.255827\pi$
$733$	37.5811	1.38809	0.694045	+	0.719932i	$0.255827\pi$
$734$	0	0
$735$	1.04225	0.0384441
$736$	0	0
$737$	0.0148811	0.000548152 0
$738$	0	0
$739$	−5.34764	−0.196716	−0.0983581	−	0.995151i	$-0.531359\pi$
$739$	−5.34764	−0.196716	−0.0983581	+	0.995151i	$0.531359\pi$
$740$	0	0
$741$	0.397828	0.0146146
$742$	0	0
$743$	28.6060	1.04945	0.524727	−	0.851271i	$-0.324167\pi$
$743$	28.6060	1.04945	0.524727	+	0.851271i	$0.324167\pi$
$744$	0	0
$745$	11.2128	0.410806
$746$	0	0
$747$	−11.2366	−0.411125
$748$	0	0
$749$	−12.6051	−0.460580
$750$	0	0
$751$	32.3236	1.17950	0.589752	−	0.807584i	$-0.299225\pi$
$751$	32.3236	1.17950	0.589752	+	0.807584i	$0.299225\pi$
$752$	0	0
$753$	−20.5965	−0.750576
$754$	0	0
$755$	12.6747	0.461279
$756$	0	0
$757$	9.95916	0.361972	0.180986	−	0.983486i	$-0.442071\pi$
$757$	9.95916	0.361972	0.180986	+	0.983486i	$0.442071\pi$
$758$	0	0
$759$	−0.273253	−0.00991847
$760$	0	0
$761$	47.4810	1.72119	0.860593	−	0.509293i	$-0.170093\pi$
$761$	47.4810	1.72119	0.860593	+	0.509293i	$0.170093\pi$
$762$	0	0
$763$	−2.03695	−0.0737426
$764$	0	0
$765$	1.33420	0.0482379
$766$	0	0
$767$	−2.34668	−0.0847338
$768$	0	0
$769$	54.7024	1.97262	0.986310	−	0.164901i	$-0.0527305\pi$
$769$	54.7024	1.97262	0.986310	+	0.164901i	$0.0527305\pi$
$770$	0	0
$771$	−11.3678	−0.409401
$772$	0	0
$773$	−13.0118	−0.468001	−0.234000	−	0.972237i	$-0.575182\pi$
$773$	−13.0118	−0.468001	−0.234000	+	0.972237i	$0.575182\pi$
$774$	0	0
$775$	34.5455	1.24091
$776$	0	0
$777$	−10.6989	−0.383821
$778$	0	0
$779$	3.73048	0.133658
$780$	0	0
$781$	−0.0484679	−0.00173432
$782$	0	0
$783$	10.8146	0.386481
$784$	0	0
$785$	8.95847	0.319742
$786$	0	0
$787$	−18.4700	−0.658385	−0.329193	−	0.944263i	$-0.606777\pi$
$787$	−18.4700	−0.658385	−0.329193	+	0.944263i	$0.606777\pi$
$788$	0	0
$789$	20.7492	0.738690
$790$	0	0
$791$	−20.0268	−0.712070
$792$	0	0
$793$	0.235300	0.00835573
$794$	0	0
$795$	8.49312	0.301220
$796$	0	0
$797$	−34.8265	−1.23362	−0.616809	−	0.787113i	$-0.711575\pi$
$797$	−34.8265	−1.23362	−0.616809	+	0.787113i	$0.711575\pi$
$798$	0	0
$799$	−8.25275	−0.291961
$800$	0	0
$801$	11.7733	0.415988
$802$	0	0
$803$	−0.0979597	−0.00345692
$804$	0	0
$805$	−7.10887	−0.250555
$806$	0	0
$807$	−15.4918	−0.545338
$808$	0	0
$809$	−26.3022	−0.924737	−0.462369	−	0.886688i	$-0.653000\pi$
$809$	−26.3022	−0.924737	−0.462369	+	0.886688i	$0.653000\pi$
$810$	0	0
$811$	−47.8836	−1.68142	−0.840710	−	0.541486i	$-0.817862\pi$
$811$	−47.8836	−1.68142	−0.840710	+	0.541486i	$0.817862\pi$
$812$	0	0
$813$	6.79332	0.238252
$814$	0	0
$815$	7.36235	0.257892
$816$	0	0
$817$	6.15838	0.215454
$818$	0	0
$819$	0.493388	0.0172404
$820$	0	0
$821$	−30.1712	−1.05298	−0.526492	−	0.850180i	$-0.676493\pi$
$821$	−30.1712	−1.05298	−0.526492	+	0.850180i	$0.676493\pi$
$822$	0	0
$823$	34.0486	1.18686	0.593430	−	0.804886i	$-0.297773\pi$
$823$	34.0486	1.18686	0.593430	+	0.804886i	$0.297773\pi$
$824$	0	0
$825$	−0.141250	−0.00491768
$826$	0	0
$827$	−38.4326	−1.33643	−0.668217	−	0.743966i	$-0.732942\pi$
$827$	−38.4326	−1.33643	−0.668217	+	0.743966i	$0.732942\pi$
$828$	0	0
$829$	1.28720	0.0447063	0.0223531	−	0.999750i	$-0.492884\pi$
$829$	1.28720	0.0447063	0.0223531	+	0.999750i	$0.492884\pi$
$830$	0	0
$831$	−38.4159	−1.33263
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−2.66872	−0.0923548
$836$	0	0
$837$	44.5268	1.53907
$838$	0	0
$839$	−52.6767	−1.81860	−0.909301	−	0.416138i	$-0.863383\pi$
$839$	−52.6767	−1.81860	−0.909301	+	0.416138i	$0.863383\pi$
$840$	0	0
$841$	−25.1093	−0.865839
$842$	0	0
$843$	−13.0059	−0.447948
$844$	0	0
$845$	−11.1423	−0.383307
$846$	0	0
$847$	−10.9992	−0.377938
$848$	0	0
$849$	−15.7186	−0.539460
$850$	0	0
$851$	72.9737	2.50151
$852$	0	0
$853$	−17.7304	−0.607079	−0.303539	−	0.952819i	$-0.598168\pi$
$853$	−17.7304	−0.607079	−0.303539	+	0.952819i	$0.598168\pi$
$854$	0	0
$855$	1.37712	0.0470966
$856$	0	0
$857$	16.7675	0.572766	0.286383	−	0.958115i	$-0.407547\pi$
$857$	16.7675	0.572766	0.286383	+	0.958115i	$0.407547\pi$
$858$	0	0
$859$	−16.5719	−0.565424	−0.282712	−	0.959205i	$-0.591234\pi$
$859$	−16.5719	−0.565424	−0.282712	+	0.959205i	$0.591234\pi$
$860$	0	0
$861$	−4.36044	−0.148603
$862$	0	0
$863$	−16.3388	−0.556178	−0.278089	−	0.960555i	$-0.589701\pi$
$863$	−16.3388	−0.556178	−0.278089	+	0.960555i	$0.589701\pi$
$864$	0	0
$865$	−11.4255	−0.388479
$866$	0	0
$867$	1.20648	0.0409741
$868$	0	0
$869$	0.364847	0.0123766
$870$	0	0
$871$	−0.172726	−0.00585259
$872$	0	0
$873$	−7.75004	−0.262299
$874$	0	0
$875$	−7.99412	−0.270251
$876$	0	0
$877$	43.1610	1.45744	0.728722	−	0.684810i	$-0.240115\pi$
$877$	43.1610	1.45744	0.728722	+	0.684810i	$0.240115\pi$
$878$	0	0
$879$	27.7722	0.936733
$880$	0	0
$881$	14.0920	0.474770	0.237385	−	0.971416i	$-0.423710\pi$
$881$	14.0920	0.474770	0.237385	+	0.971416i	$0.423710\pi$
$882$	0	0
$883$	−5.71528	−0.192334	−0.0961672	−	0.995365i	$-0.530658\pi$
$883$	−5.71528	−0.192334	−0.0961672	+	0.995365i	$0.530658\pi$
$884$	0	0
$885$	7.65603	0.257355
$886$	0	0
$887$	32.1785	1.08045	0.540225	−	0.841521i	$-0.318339\pi$
$887$	32.1785	1.08045	0.540225	+	0.841521i	$0.318339\pi$
$888$	0	0
$889$	−0.858651	−0.0287982
$890$	0	0
$891$	−0.0545382	−0.00182710
$892$	0	0
$893$	−8.51827	−0.285053
$894$	0	0
$895$	−2.29827	−0.0768226
$896$	0	0
$897$	3.17167	0.105899
$898$	0	0
$899$	16.0190	0.534265
$900$	0	0
$901$	−8.14880	−0.271476
$902$	0	0
$903$	−7.19833	−0.239546
$904$	0	0
$905$	22.8442	0.759368
$906$	0	0
$907$	44.9177	1.49147	0.745734	−	0.666244i	$-0.232099\pi$
$907$	44.9177	1.49147	0.745734	+	0.666244i	$0.232099\pi$
$908$	0	0
$909$	−0.101872	−0.00337890
$910$	0	0
$911$	12.4730	0.413250	0.206625	−	0.978420i	$-0.433752\pi$
$911$	12.4730	0.413250	0.206625	+	0.978420i	$0.433752\pi$
$912$	0	0
$913$	0.200250	0.00662729
$914$	0	0
$915$	−0.767663	−0.0253781
$916$	0	0
$917$	−10.2387	−0.338112
$918$	0	0
$919$	11.1290	0.367113	0.183557	−	0.983009i	$-0.441239\pi$
$919$	11.1290	0.367113	0.183557	+	0.983009i	$0.441239\pi$
$920$	0	0
$921$	−24.4001	−0.804010
$922$	0	0
$923$	0.562570	0.0185172
$924$	0	0
$925$	37.7214	1.24027
$926$	0	0
$927$	−28.3280	−0.930412
$928$	0	0
$929$	−25.6400	−0.841221	−0.420610	−	0.907241i	$-0.638184\pi$
$929$	−25.6400	−0.841221	−0.420610	+	0.907241i	$0.638184\pi$
$930$	0	0
$931$	−1.03217	−0.0338281
$932$	0	0
$933$	20.6141	0.674876
$934$	0	0
$935$	−0.0237770	−0.000777590 0
$936$	0	0
$937$	28.1337	0.919089	0.459545	−	0.888155i	$-0.348013\pi$
$937$	28.1337	0.919089	0.459545	+	0.888155i	$0.348013\pi$
$938$	0	0
$939$	27.8266	0.908087
$940$	0	0
$941$	−14.4410	−0.470762	−0.235381	−	0.971903i	$-0.575634\pi$
$941$	−14.4410	−0.470762	−0.235381	+	0.971903i	$0.575634\pi$
$942$	0	0
$943$	29.7411	0.968503
$944$	0	0
$945$	−4.73643	−0.154076
$946$	0	0
$947$	43.8291	1.42426	0.712128	−	0.702050i	$-0.247732\pi$
$947$	43.8291	1.42426	0.712128	+	0.702050i	$0.247732\pi$
$948$	0	0
$949$	1.13702	0.0369094
$950$	0	0
$951$	−36.5578	−1.18547
$952$	0	0
$953$	−57.7440	−1.87051	−0.935257	−	0.353971i	$-0.884831\pi$
$953$	−57.7440	−1.87051	−0.935257	+	0.353971i	$0.884831\pi$
$954$	0	0
$955$	8.17009	0.264378
$956$	0	0
$957$	−0.0654986	−0.00211727
$958$	0	0
$959$	9.66474	0.312091
$960$	0	0
$961$	34.9552	1.12759
$962$	0	0
$963$	19.4675	0.627332
$964$	0	0
$965$	−11.1235	−0.358078
$966$	0	0
$967$	−13.9404	−0.448294	−0.224147	−	0.974555i	$-0.571960\pi$
$967$	−13.9404	−0.448294	−0.224147	+	0.974555i	$0.571960\pi$
$968$	0	0
$969$	1.24529	0.0400046
$970$	0	0
$971$	6.31724	0.202730	0.101365	−	0.994849i	$-0.467679\pi$
$971$	6.31724	0.202730	0.101365	+	0.994849i	$0.467679\pi$
$972$	0	0
$973$	−0.726913	−0.0233038
$974$	0	0
$975$	1.63949	0.0525059
$976$	0	0
$977$	−39.5447	−1.26515	−0.632573	−	0.774501i	$-0.718001\pi$
$977$	−39.5447	−1.26515	−0.632573	+	0.774501i	$0.718001\pi$
$978$	0	0
$979$	−0.209814	−0.00670568
$980$	0	0
$981$	3.14590	0.100441
$982$	0	0
$983$	−33.0799	−1.05508	−0.527542	−	0.849529i	$-0.676886\pi$
$983$	−33.0799	−1.05508	−0.527542	+	0.849529i	$0.676886\pi$
$984$	0	0
$985$	−18.3744	−0.585456
$986$	0	0
$987$	9.95673	0.316926
$988$	0	0
$989$	49.0974	1.56121
$990$	0	0
$991$	23.6516	0.751316	0.375658	−	0.926758i	$-0.377417\pi$
$991$	23.6516	0.751316	0.375658	+	0.926758i	$0.377417\pi$
$992$	0	0
$993$	−24.3622	−0.773111
$994$	0	0
$995$	−0.160843	−0.00509906
$996$	0	0
$997$	39.7065	1.25752	0.628759	−	0.777601i	$-0.283563\pi$
$997$	39.7065	1.25752	0.628759	+	0.777601i	$0.283563\pi$
$998$	0	0
$999$	48.6203	1.53828

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.i.1.5	✓	6
4.3	odd	2		3808.2.a.m.1.2	yes	6
8.3	odd	2		7616.2.a.bx.1.5		6
8.5	even	2		7616.2.a.cb.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.i.1.5	✓	6	1.1	even	1	trivial
3808.2.a.m.1.2	yes	6	4.3	odd	2
7616.2.a.bx.1.5		6	8.3	odd	2
7616.2.a.cb.1.2		6	8.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3808 = 2^{5} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3808.a (trivial)

\(f(q)\)	\(=\)	\(q+1.20648 q^{3} +0.863883 q^{5} +1.00000 q^{7} -1.54442 q^{9} +0.0275234 q^{11} -0.319466 q^{13} +1.04225 q^{15} -1.00000 q^{17} -1.03217 q^{19} +1.20648 q^{21} -8.22897 q^{23} -4.25371 q^{25} -5.48273 q^{27} -1.97248 q^{29} -8.12128 q^{31} +0.0332063 q^{33} +0.863883 q^{35} -8.86790 q^{37} -0.385427 q^{39} -3.61419 q^{41} -5.96641 q^{43} -1.33420 q^{45} +8.25275 q^{47} +1.00000 q^{49} -1.20648 q^{51} +8.14880 q^{53} +0.0237770 q^{55} -1.24529 q^{57} +7.34565 q^{59} -0.736541 q^{61} -1.54442 q^{63} -0.275981 q^{65} +0.540671 q^{67} -9.92805 q^{69} -1.76097 q^{71} -3.55915 q^{73} -5.13199 q^{75} +0.0275234 q^{77} +13.2559 q^{79} -1.98152 q^{81} +7.27562 q^{83} -0.863883 q^{85} -2.37974 q^{87} -7.62312 q^{89} -0.319466 q^{91} -9.79812 q^{93} -0.891677 q^{95} +5.01810 q^{97} -0.0425076 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 4 q^{9} - 2 q^{11} - 4 q^{13} + 2 q^{15} - 6 q^{17} - 10 q^{19} - 2 q^{21} + 4 q^{23} + 4 q^{25} - 8 q^{27} - 14 q^{29} + 8 q^{31} - 8 q^{33} - 6 q^{35} - 4 q^{37} + 14 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 6 * q^5 + 6 * q^7 + 4 * q^9 - 2 * q^11 - 4 * q^13 + 2 * q^15 - 6 * q^17 - 10 * q^19 - 2 * q^21 + 4 * q^23 + 4 * q^25 - 8 * q^27 - 14 * q^29 + 8 * q^31 - 8 * q^33 - 6 * q^35 - 4 * q^37 + 14 * q^39 - 2 * q^41 - 8 * q^43 - 8 * q^45 - 2 * q^47 + 6 * q^49 + 2 * q^51 - 10 * q^53 + 2 * q^55 - 18 * q^57 - 14 * q^59 - 20 * q^61 + 4 * q^63 - 20 * q^65 - 6 * q^67 - 14 * q^69 + 20 * q^71 - 6 * q^73 - 10 * q^75 - 2 * q^77 + 38 * q^79 - 18 * q^81 - 16 * q^83 + 6 * q^85 - 4 * q^87 - 2 * q^89 - 4 * q^91 - 16 * q^93 + 42 * q^95 - 12 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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