LMFDB - Embedded newform 3808.2.a.h.1.6

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:	6.6.109859312.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 7x^{4} + 15x^{3} + 13x^{2} - 9x - 2 $ x^6 - 3x^5 - 7x^4 + 15x^3 + 13x^2 - 9*x - 2
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			1.6
Root			$0.635879$ 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)

$f(q)$	$=$	$q+1.84197 q^{3} +3.17127 q^{5} -1.00000 q^{7} +0.392864 q^{9} -4.90224 q^{11} -5.51984 q^{13} +5.84140 q^{15} -1.00000 q^{17} -2.99670 q^{19} -1.84197 q^{21} -4.99670 q^{23} +5.05696 q^{25} -4.80227 q^{27} -8.89564 q^{29} +3.56686 q^{31} -9.02979 q^{33} -3.17127 q^{35} +5.95240 q^{37} -10.1674 q^{39} +9.28773 q^{41} +1.21777 q^{43} +1.24588 q^{45} -11.8579 q^{47} +1.00000 q^{49} -1.84197 q^{51} -5.42595 q^{53} -15.5463 q^{55} -5.51984 q^{57} +0.469674 q^{59} +10.4856 q^{61} -0.392864 q^{63} -17.5049 q^{65} -14.6501 q^{67} -9.20379 q^{69} +11.0352 q^{71} -2.08016 q^{73} +9.31479 q^{75} +4.90224 q^{77} +12.3222 q^{79} -10.0243 q^{81} +2.17844 q^{83} -3.17127 q^{85} -16.3855 q^{87} -6.04687 q^{89} +5.51984 q^{91} +6.57006 q^{93} -9.50335 q^{95} -7.90496 q^{97} -1.92591 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} + 4 q^{5} - 6 q^{7} + 8 q^{9} - 8 q^{11} + 8 q^{13} - 6 q^{17} - 6 q^{19} + 4 q^{21} - 18 q^{23} + 12 q^{25} - 22 q^{27} - 8 q^{29} + 4 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} - 14 q^{39}+ \cdots - 28 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 + 4 * q^5 - 6 * q^7 + 8 * q^9 - 8 * q^11 + 8 * q^13 - 6 * q^17 - 6 * q^19 + 4 * q^21 - 18 * q^23 + 12 * q^25 - 22 * q^27 - 8 * q^29 + 4 * q^31 - 6 * q^33 - 4 * q^35 - 8 * q^37 - 14 * q^39 + 16 * q^41 - 12 * q^43 + 4 * q^45 - 18 * q^47 + 6 * q^49 + 4 * q^51 - 2 * q^53 + 8 * q^55 + 8 * q^57 - 16 * q^59 + 6 * q^61 - 8 * q^63 - 22 * q^65 - 12 * q^67 + 16 * q^69 + 2 * q^71 + 8 * q^73 - 24 * q^75 + 8 * q^77 + 18 * q^79 + 18 * q^81 - 28 * q^83 - 4 * q^85 + 2 * q^87 - 2 * q^89 - 8 * q^91 - 32 * q^93 - 26 * q^95 - 18 * q^97 - 28 * 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
$3$	1.84197	1.06346	0.531732	−	0.846913i	$-0.321541\pi$
$3$	1.84197	1.06346	0.531732	+	0.846913i	$0.321541\pi$
$4$	0	0
$5$	3.17127	1.41824	0.709118	−	0.705090i	$-0.249093\pi$
$5$	3.17127	1.41824	0.709118	+	0.705090i	$0.249093\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0.392864	0.130955
$10$	0	0
$11$	−4.90224	−1.47808	−0.739040	−	0.673661i	$-0.764721\pi$
$11$	−4.90224	−1.47808	−0.739040	+	0.673661i	$0.764721\pi$
$12$	0	0
$13$	−5.51984	−1.53093	−0.765464	−	0.643479i	$-0.777491\pi$
$13$	−5.51984	−1.53093	−0.765464	+	0.643479i	$0.777491\pi$
$14$	0	0
$15$	5.84140	1.50824
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−2.99670	−0.687490	−0.343745	−	0.939063i	$-0.611696\pi$
$19$	−2.99670	−0.687490	−0.343745	+	0.939063i	$0.611696\pi$
$20$	0	0
$21$	−1.84197	−0.401951
$22$	0	0
$23$	−4.99670	−1.04188	−0.520942	−	0.853592i	$-0.674419\pi$
$23$	−4.99670	−1.04188	−0.520942	+	0.853592i	$0.674419\pi$
$24$	0	0
$25$	5.05696	1.01139
$26$	0	0
$27$	−4.80227	−0.924198
$28$	0	0
$29$	−8.89564	−1.65188	−0.825939	−	0.563759i	$-0.809355\pi$
$29$	−8.89564	−1.65188	−0.825939	+	0.563759i	$0.809355\pi$
$30$	0	0
$31$	3.56686	0.640627	0.320313	−	0.947312i	$-0.396212\pi$
$31$	3.56686	0.640627	0.320313	+	0.947312i	$0.396212\pi$
$32$	0	0
$33$	−9.02979	−1.57188
$34$	0	0
$35$	−3.17127	−0.536043
$36$	0	0
$37$	5.95240	0.978569	0.489285	−	0.872124i	$-0.337258\pi$
$37$	5.95240	0.978569	0.489285	+	0.872124i	$0.337258\pi$
$38$	0	0
$39$	−10.1674	−1.62809
$40$	0	0
$41$	9.28773	1.45050	0.725250	−	0.688486i	$-0.241724\pi$
$41$	9.28773	1.45050	0.725250	+	0.688486i	$0.241724\pi$
$42$	0	0
$43$	1.21777	0.185708	0.0928541	−	0.995680i	$-0.470401\pi$
$43$	1.21777	0.185708	0.0928541	+	0.995680i	$0.470401\pi$
$44$	0	0
$45$	1.24588	0.185725
$46$	0	0
$47$	−11.8579	−1.72966	−0.864829	−	0.502066i	$-0.832573\pi$
$47$	−11.8579	−1.72966	−0.864829	+	0.502066i	$0.832573\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.84197	−0.257928
$52$	0	0
$53$	−5.42595	−0.745312	−0.372656	−	0.927970i	$-0.621553\pi$
$53$	−5.42595	−0.745312	−0.372656	+	0.927970i	$0.621553\pi$
$54$	0	0
$55$	−15.5463	−2.09627
$56$	0	0
$57$	−5.51984	−0.731121
$58$	0	0
$59$	0.469674	0.0611464	0.0305732	−	0.999533i	$-0.490267\pi$
$59$	0.469674	0.0611464	0.0305732	+	0.999533i	$0.490267\pi$
$60$	0	0
$61$	10.4856	1.34255	0.671275	−	0.741209i	$-0.265747\pi$
$61$	10.4856	1.34255	0.671275	+	0.741209i	$0.265747\pi$
$62$	0	0
$63$	−0.392864	−0.0494962
$64$	0	0
$65$	−17.5049	−2.17122
$66$	0	0
$67$	−14.6501	−1.78980	−0.894898	−	0.446270i	$-0.852752\pi$
$67$	−14.6501	−1.78980	−0.894898	+	0.446270i	$0.852752\pi$
$68$	0	0
$69$	−9.20379	−1.10801
$70$	0	0
$71$	11.0352	1.30964	0.654821	−	0.755784i	$-0.272744\pi$
$71$	11.0352	1.30964	0.654821	+	0.755784i	$0.272744\pi$
$72$	0	0
$73$	−2.08016	−0.243465	−0.121732	−	0.992563i	$-0.538845\pi$
$73$	−2.08016	−0.243465	−0.121732	+	0.992563i	$0.538845\pi$
$74$	0	0
$75$	9.31479	1.07558
$76$	0	0
$77$	4.90224	0.558662
$78$	0	0
$79$	12.3222	1.38635	0.693176	−	0.720768i	$-0.256211\pi$
$79$	12.3222	1.38635	0.693176	+	0.720768i	$0.256211\pi$
$80$	0	0
$81$	−10.0243	−1.11381
$82$	0	0
$83$	2.17844	0.239115	0.119557	−	0.992827i	$-0.461852\pi$
$83$	2.17844	0.239115	0.119557	+	0.992827i	$0.461852\pi$
$84$	0	0
$85$	−3.17127	−0.343973
$86$	0	0
$87$	−16.3855	−1.75671
$88$	0	0
$89$	−6.04687	−0.640966	−0.320483	−	0.947254i	$-0.603845\pi$
$89$	−6.04687	−0.640966	−0.320483	+	0.947254i	$0.603845\pi$
$90$	0	0
$91$	5.51984	0.578636
$92$	0	0
$93$	6.57006	0.681283
$94$	0	0
$95$	−9.50335	−0.975023
$96$	0	0
$97$	−7.90496	−0.802627	−0.401314	−	0.915941i	$-0.631446\pi$
$97$	−7.90496	−0.802627	−0.401314	+	0.915941i	$0.631446\pi$
$98$	0	0
$99$	−1.92591	−0.193562
$100$	0	0
$101$	19.6872	1.95895	0.979477	−	0.201558i	$-0.0646005\pi$
$101$	19.6872	1.95895	0.979477	+	0.201558i	$0.0646005\pi$
$102$	0	0
$103$	−6.97632	−0.687398	−0.343699	−	0.939080i	$-0.611680\pi$
$103$	−6.97632	−0.687398	−0.343699	+	0.939080i	$0.611680\pi$
$104$	0	0
$105$	−5.84140	−0.570062
$106$	0	0
$107$	−2.54320	−0.245861	−0.122930	−	0.992415i	$-0.539229\pi$
$107$	−2.54320	−0.245861	−0.122930	+	0.992415i	$0.539229\pi$
$108$	0	0
$109$	5.74699	0.550462	0.275231	−	0.961378i	$-0.411246\pi$
$109$	5.74699	0.550462	0.275231	+	0.961378i	$0.411246\pi$
$110$	0	0
$111$	10.9642	1.04067
$112$	0	0
$113$	−12.6768	−1.19253	−0.596265	−	0.802788i	$-0.703349\pi$
$113$	−12.6768	−1.19253	−0.596265	+	0.802788i	$0.703349\pi$
$114$	0	0
$115$	−15.8459	−1.47764
$116$	0	0
$117$	−2.16855	−0.200482
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	13.0319	1.18472
$122$	0	0
$123$	17.1077	1.54255
$124$	0	0
$125$	0.180651	0.0161579
$126$	0	0
$127$	−4.84564	−0.429981	−0.214990	−	0.976616i	$-0.568972\pi$
$127$	−4.84564	−0.429981	−0.214990	+	0.976616i	$0.568972\pi$
$128$	0	0
$129$	2.24310	0.197494
$130$	0	0
$131$	−8.29155	−0.724436	−0.362218	−	0.932093i	$-0.617980\pi$
$131$	−8.29155	−0.724436	−0.362218	+	0.932093i	$0.617980\pi$
$132$	0	0
$133$	2.99670	0.259847
$134$	0	0
$135$	−15.2293	−1.31073
$136$	0	0
$137$	−1.47221	−0.125779	−0.0628896	−	0.998020i	$-0.520032\pi$
$137$	−1.47221	−0.125779	−0.0628896	+	0.998020i	$0.520032\pi$
$138$	0	0
$139$	−9.31500	−0.790088	−0.395044	−	0.918662i	$-0.629271\pi$
$139$	−9.31500	−0.790088	−0.395044	+	0.918662i	$0.629271\pi$
$140$	0	0
$141$	−21.8420	−1.83943
$142$	0	0
$143$	27.0596	2.26283
$144$	0	0
$145$	−28.2105	−2.34275
$146$	0	0
$147$	1.84197	0.151923
$148$	0	0
$149$	8.51304	0.697415	0.348708	−	0.937232i	$-0.386621\pi$
$149$	8.51304	0.697415	0.348708	+	0.937232i	$0.386621\pi$
$150$	0	0
$151$	8.22551	0.669383	0.334691	−	0.942328i	$-0.391368\pi$
$151$	8.22551	0.669383	0.334691	+	0.942328i	$0.391368\pi$
$152$	0	0
$153$	−0.392864	−0.0317612
$154$	0	0
$155$	11.3115	0.908560
$156$	0	0
$157$	21.4614	1.71281	0.856405	−	0.516305i	$-0.172693\pi$
$157$	21.4614	1.71281	0.856405	+	0.516305i	$0.172693\pi$
$158$	0	0
$159$	−9.99446	−0.792612
$160$	0	0
$161$	4.99670	0.393795
$162$	0	0
$163$	−6.88019	−0.538898	−0.269449	−	0.963015i	$-0.586842\pi$
$163$	−6.88019	−0.538898	−0.269449	+	0.963015i	$0.586842\pi$
$164$	0	0
$165$	−28.6359	−2.22930
$166$	0	0
$167$	10.5845	0.819057	0.409528	−	0.912297i	$-0.365693\pi$
$167$	10.5845	0.819057	0.409528	+	0.912297i	$0.365693\pi$
$168$	0	0
$169$	17.4686	1.34374
$170$	0	0
$171$	−1.17730	−0.0900301
$172$	0	0
$173$	−9.03031	−0.686562	−0.343281	−	0.939233i	$-0.611538\pi$
$173$	−9.03031	−0.686562	−0.343281	+	0.939233i	$0.611538\pi$
$174$	0	0
$175$	−5.05696	−0.382271
$176$	0	0
$177$	0.865127	0.0650270
$178$	0	0
$179$	12.3724	0.924758	0.462379	−	0.886682i	$-0.346996\pi$
$179$	12.3724	0.924758	0.462379	+	0.886682i	$0.346996\pi$
$180$	0	0
$181$	1.52057	0.113023	0.0565116	−	0.998402i	$-0.482002\pi$
$181$	1.52057	0.113023	0.0565116	+	0.998402i	$0.482002\pi$
$182$	0	0
$183$	19.3143	1.42775
$184$	0	0
$185$	18.8767	1.38784
$186$	0	0
$187$	4.90224	0.358487
$188$	0	0
$189$	4.80227	0.349314
$190$	0	0
$191$	5.28778	0.382610	0.191305	−	0.981531i	$-0.438728\pi$
$191$	5.28778	0.382610	0.191305	+	0.981531i	$0.438728\pi$
$192$	0	0
$193$	20.0160	1.44078	0.720392	−	0.693567i	$-0.243962\pi$
$193$	20.0160	1.44078	0.720392	+	0.693567i	$0.243962\pi$
$194$	0	0
$195$	−32.2436	−2.30901
$196$	0	0
$197$	14.8559	1.05844	0.529219	−	0.848485i	$-0.322485\pi$
$197$	14.8559	1.05844	0.529219	+	0.848485i	$0.322485\pi$
$198$	0	0
$199$	20.5820	1.45902	0.729508	−	0.683972i	$-0.239749\pi$
$199$	20.5820	1.45902	0.729508	+	0.683972i	$0.239749\pi$
$200$	0	0
$201$	−26.9851	−1.90338
$202$	0	0
$203$	8.89564	0.624351
$204$	0	0
$205$	29.4539	2.05715
$206$	0	0
$207$	−1.96302	−0.136440
$208$	0	0
$209$	14.6905	1.01617
$210$	0	0
$211$	−6.68855	−0.460459	−0.230229	−	0.973136i	$-0.573948\pi$
$211$	−6.68855	−0.460459	−0.230229	+	0.973136i	$0.573948\pi$
$212$	0	0
$213$	20.3266	1.39276
$214$	0	0
$215$	3.86188	0.263378
$216$	0	0
$217$	−3.56686	−0.242134
$218$	0	0
$219$	−3.83161	−0.258916
$220$	0	0
$221$	5.51984	0.371305
$222$	0	0
$223$	−20.8277	−1.39472	−0.697362	−	0.716719i	$-0.745643\pi$
$223$	−20.8277	−1.39472	−0.697362	+	0.716719i	$0.745643\pi$
$224$	0	0
$225$	1.98670	0.132447
$226$	0	0
$227$	−3.52257	−0.233802	−0.116901	−	0.993144i	$-0.537296\pi$
$227$	−3.52257	−0.233802	−0.116901	+	0.993144i	$0.537296\pi$
$228$	0	0
$229$	−25.4372	−1.68094	−0.840470	−	0.541858i	$-0.817721\pi$
$229$	−25.4372	−1.68094	−0.840470	+	0.541858i	$0.817721\pi$
$230$	0	0
$231$	9.02979	0.594117
$232$	0	0
$233$	12.7192	0.833261	0.416631	−	0.909076i	$-0.363211\pi$
$233$	12.7192	0.833261	0.416631	+	0.909076i	$0.363211\pi$
$234$	0	0
$235$	−37.6048	−2.45306
$236$	0	0
$237$	22.6971	1.47434
$238$	0	0
$239$	−11.1182	−0.719175	−0.359587	−	0.933111i	$-0.617083\pi$
$239$	−11.1182	−0.719175	−0.359587	+	0.933111i	$0.617083\pi$
$240$	0	0
$241$	−2.47056	−0.159143	−0.0795715	−	0.996829i	$-0.525355\pi$
$241$	−2.47056	−0.159143	−0.0795715	+	0.996829i	$0.525355\pi$
$242$	0	0
$243$	−4.05758	−0.260294
$244$	0	0
$245$	3.17127	0.202605
$246$	0	0
$247$	16.5413	1.05250
$248$	0	0
$249$	4.01262	0.254290
$250$	0	0
$251$	−8.48616	−0.535642	−0.267821	−	0.963469i	$-0.586304\pi$
$251$	−8.48616	−0.535642	−0.267821	+	0.963469i	$0.586304\pi$
$252$	0	0
$253$	24.4950	1.53999
$254$	0	0
$255$	−5.84140	−0.365802
$256$	0	0
$257$	0.592363	0.0369506	0.0184753	−	0.999829i	$-0.494119\pi$
$257$	0.592363	0.0369506	0.0184753	+	0.999829i	$0.494119\pi$
$258$	0	0
$259$	−5.95240	−0.369864
$260$	0	0
$261$	−3.49478	−0.216321
$262$	0	0
$263$	−14.0265	−0.864910	−0.432455	−	0.901655i	$-0.642353\pi$
$263$	−14.0265	−0.864910	−0.432455	+	0.901655i	$0.642353\pi$
$264$	0	0
$265$	−17.2072	−1.05703
$266$	0	0
$267$	−11.1382	−0.681644
$268$	0	0
$269$	−5.43754	−0.331532	−0.165766	−	0.986165i	$-0.553010\pi$
$269$	−5.43754	−0.331532	−0.165766	+	0.986165i	$0.553010\pi$
$270$	0	0
$271$	−7.58030	−0.460471	−0.230235	−	0.973135i	$-0.573950\pi$
$271$	−7.58030	−0.460471	−0.230235	+	0.973135i	$0.573950\pi$
$272$	0	0
$273$	10.1674	0.615359
$274$	0	0
$275$	−24.7904	−1.49492
$276$	0	0
$277$	−5.03325	−0.302419	−0.151209	−	0.988502i	$-0.548317\pi$
$277$	−5.03325	−0.302419	−0.151209	+	0.988502i	$0.548317\pi$
$278$	0	0
$279$	1.40129	0.0838931
$280$	0	0
$281$	14.5131	0.865779	0.432889	−	0.901447i	$-0.357494\pi$
$281$	14.5131	0.865779	0.432889	+	0.901447i	$0.357494\pi$
$282$	0	0
$283$	−27.9114	−1.65916	−0.829580	−	0.558388i	$-0.811420\pi$
$283$	−27.9114	−1.65916	−0.829580	+	0.558388i	$0.811420\pi$
$284$	0	0
$285$	−17.5049	−1.03690
$286$	0	0
$287$	−9.28773	−0.548237
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−14.5607	−0.853565
$292$	0	0
$293$	17.4516	1.01953	0.509767	−	0.860312i	$-0.329731\pi$
$293$	17.4516	1.01953	0.509767	+	0.860312i	$0.329731\pi$
$294$	0	0
$295$	1.48946	0.0867200
$296$	0	0
$297$	23.5419	1.36604
$298$	0	0
$299$	27.5810	1.59505
$300$	0	0
$301$	−1.21777	−0.0701911
$302$	0	0
$303$	36.2634	2.08328
$304$	0	0
$305$	33.2528	1.90405
$306$	0	0
$307$	−31.5184	−1.79885	−0.899424	−	0.437076i	$-0.856014\pi$
$307$	−31.5184	−1.79885	−0.899424	+	0.437076i	$0.856014\pi$
$308$	0	0
$309$	−12.8502	−0.731022
$310$	0	0
$311$	−28.1292	−1.59506	−0.797530	−	0.603279i	$-0.793861\pi$
$311$	−28.1292	−1.59506	−0.797530	+	0.603279i	$0.793861\pi$
$312$	0	0
$313$	23.3258	1.31845	0.659227	−	0.751944i	$-0.270884\pi$
$313$	23.3258	1.31845	0.659227	+	0.751944i	$0.270884\pi$
$314$	0	0
$315$	−1.24588	−0.0701973
$316$	0	0
$317$	−20.8972	−1.17370	−0.586852	−	0.809694i	$-0.699633\pi$
$317$	−20.8972	−1.17370	−0.586852	+	0.809694i	$0.699633\pi$
$318$	0	0
$319$	43.6085	2.44161
$320$	0	0
$321$	−4.68451	−0.261464
$322$	0	0
$323$	2.99670	0.166741
$324$	0	0
$325$	−27.9136	−1.54837
$326$	0	0
$327$	10.5858	0.585396
$328$	0	0
$329$	11.8579	0.653749
$330$	0	0
$331$	−25.2015	−1.38520	−0.692599	−	0.721323i	$-0.743534\pi$
$331$	−25.2015	−1.38520	−0.692599	+	0.721323i	$0.743534\pi$
$332$	0	0
$333$	2.33849	0.128148
$334$	0	0
$335$	−46.4595	−2.53835
$336$	0	0
$337$	9.27489	0.505236	0.252618	−	0.967566i	$-0.418708\pi$
$337$	9.27489	0.505236	0.252618	+	0.967566i	$0.418708\pi$
$338$	0	0
$339$	−23.3502	−1.26821
$340$	0	0
$341$	−17.4856	−0.946898
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−29.1877	−1.57141
$346$	0	0
$347$	0.00975457	0.000523653 0	0.000261826	−	1.00000i	$-0.499917\pi$
$347$	0.00975457	0.000523653 0	0.000261826		1.00000i	$0.499917\pi$
$348$	0	0
$349$	−29.3717	−1.57223	−0.786117	−	0.618078i	$-0.787911\pi$
$349$	−29.3717	−1.57223	−0.786117	+	0.618078i	$0.787911\pi$
$350$	0	0
$351$	26.5078	1.41488
$352$	0	0
$353$	−30.8899	−1.64410	−0.822051	−	0.569414i	$-0.807170\pi$
$353$	−30.8899	−1.64410	−0.822051	+	0.569414i	$0.807170\pi$
$354$	0	0
$355$	34.9957	1.85738
$356$	0	0
$357$	1.84197	0.0974875
$358$	0	0
$359$	36.4947	1.92612	0.963058	−	0.269295i	$-0.0867906\pi$
$359$	36.4947	1.92612	0.963058	+	0.269295i	$0.0867906\pi$
$360$	0	0
$361$	−10.0198	−0.527357
$362$	0	0
$363$	24.0045	1.25991
$364$	0	0
$365$	−6.59677	−0.345291
$366$	0	0
$367$	2.45668	0.128238	0.0641189	−	0.997942i	$-0.479576\pi$
$367$	2.45668	0.128238	0.0641189	+	0.997942i	$0.479576\pi$
$368$	0	0
$369$	3.64882	0.189950
$370$	0	0
$371$	5.42595	0.281702
$372$	0	0
$373$	−29.9063	−1.54849	−0.774244	−	0.632887i	$-0.781870\pi$
$373$	−29.9063	−1.54849	−0.774244	+	0.632887i	$0.781870\pi$
$374$	0	0
$375$	0.332754	0.0171834
$376$	0	0
$377$	49.1025	2.52891
$378$	0	0
$379$	−13.9784	−0.718021	−0.359010	−	0.933334i	$-0.616886\pi$
$379$	−13.9784	−0.718021	−0.359010	+	0.933334i	$0.616886\pi$
$380$	0	0
$381$	−8.92554	−0.457269
$382$	0	0
$383$	−21.8393	−1.11594	−0.557968	−	0.829863i	$-0.688419\pi$
$383$	−21.8393	−1.11594	−0.557968	+	0.829863i	$0.688419\pi$
$384$	0	0
$385$	15.5463	0.792314
$386$	0	0
$387$	0.478419	0.0243194
$388$	0	0
$389$	−9.40502	−0.476854	−0.238427	−	0.971160i	$-0.576632\pi$
$389$	−9.40502	−0.476854	−0.238427	+	0.971160i	$0.576632\pi$
$390$	0	0
$391$	4.99670	0.252694
$392$	0	0
$393$	−15.2728	−0.770411
$394$	0	0
$395$	39.0769	1.96617
$396$	0	0
$397$	34.2892	1.72092	0.860462	−	0.509514i	$-0.170175\pi$
$397$	34.2892	1.72092	0.860462	+	0.509514i	$0.170175\pi$
$398$	0	0
$399$	5.51984	0.276338
$400$	0	0
$401$	−24.4847	−1.22271	−0.611354	−	0.791357i	$-0.709375\pi$
$401$	−24.4847	−1.22271	−0.611354	+	0.791357i	$0.709375\pi$
$402$	0	0
$403$	−19.6885	−0.980754
$404$	0	0
$405$	−31.7896	−1.57964
$406$	0	0
$407$	−29.1801	−1.44640
$408$	0	0
$409$	−18.4377	−0.911686	−0.455843	−	0.890060i	$-0.650662\pi$
$409$	−18.4377	−0.911686	−0.455843	+	0.890060i	$0.650662\pi$
$410$	0	0
$411$	−2.71177	−0.133762
$412$	0	0
$413$	−0.469674	−0.0231112
$414$	0	0
$415$	6.90842	0.339121
$416$	0	0
$417$	−17.1580	−0.840230
$418$	0	0
$419$	−9.05186	−0.442212	−0.221106	−	0.975250i	$-0.570967\pi$
$419$	−9.05186	−0.442212	−0.221106	+	0.975250i	$0.570967\pi$
$420$	0	0
$421$	19.2513	0.938252	0.469126	−	0.883131i	$-0.344569\pi$
$421$	19.2513	0.938252	0.469126	+	0.883131i	$0.344569\pi$
$422$	0	0
$423$	−4.65856	−0.226507
$424$	0	0
$425$	−5.05696	−0.245299
$426$	0	0
$427$	−10.4856	−0.507436
$428$	0	0
$429$	49.8430	2.40644
$430$	0	0
$431$	38.5919	1.85890	0.929452	−	0.368943i	$-0.120280\pi$
$431$	38.5919	1.85890	0.929452	+	0.368943i	$0.120280\pi$
$432$	0	0
$433$	13.7242	0.659543	0.329771	−	0.944061i	$-0.393028\pi$
$433$	13.7242	0.659543	0.329771	+	0.944061i	$0.393028\pi$
$434$	0	0
$435$	−51.9629	−2.49143
$436$	0	0
$437$	14.9736	0.716285
$438$	0	0
$439$	13.4302	0.640988	0.320494	−	0.947251i	$-0.396151\pi$
$439$	13.4302	0.640988	0.320494	+	0.947251i	$0.396151\pi$
$440$	0	0
$441$	0.392864	0.0187078
$442$	0	0
$443$	−37.2253	−1.76863	−0.884315	−	0.466891i	$-0.845374\pi$
$443$	−37.2253	−1.76863	−0.884315	+	0.466891i	$0.845374\pi$
$444$	0	0
$445$	−19.1763	−0.909042
$446$	0	0
$447$	15.6808	0.741676
$448$	0	0
$449$	16.5573	0.781389	0.390695	−	0.920520i	$-0.372235\pi$
$449$	16.5573	0.781389	0.390695	+	0.920520i	$0.372235\pi$
$450$	0	0
$451$	−45.5306	−2.14395
$452$	0	0
$453$	15.1512	0.711864
$454$	0	0
$455$	17.5049	0.820643
$456$	0	0
$457$	−15.3451	−0.717815	−0.358908	−	0.933373i	$-0.616851\pi$
$457$	−15.3451	−0.717815	−0.358908	+	0.933373i	$0.616851\pi$
$458$	0	0
$459$	4.80227	0.224151
$460$	0	0
$461$	−1.73553	−0.0808315	−0.0404158	−	0.999183i	$-0.512868\pi$
$461$	−1.73553	−0.0808315	−0.0404158	+	0.999183i	$0.512868\pi$
$462$	0	0
$463$	41.5997	1.93330	0.966651	−	0.256098i	$-0.0824369\pi$
$463$	41.5997	1.93330	0.966651	+	0.256098i	$0.0824369\pi$
$464$	0	0
$465$	20.8354	0.966221
$466$	0	0
$467$	−10.0876	−0.466799	−0.233400	−	0.972381i	$-0.574985\pi$
$467$	−10.0876	−0.466799	−0.233400	+	0.972381i	$0.574985\pi$
$468$	0	0
$469$	14.6501	0.676479
$470$	0	0
$471$	39.5314	1.82151
$472$	0	0
$473$	−5.96980	−0.274492
$474$	0	0
$475$	−15.1542	−0.695323
$476$	0	0
$477$	−2.13166	−0.0976022
$478$	0	0
$479$	−37.4618	−1.71167	−0.855837	−	0.517246i	$-0.826957\pi$
$479$	−37.4618	−1.71167	−0.855837	+	0.517246i	$0.826957\pi$
$480$	0	0
$481$	−32.8563	−1.49812
$482$	0	0
$483$	9.20379	0.418787
$484$	0	0
$485$	−25.0688	−1.13831
$486$	0	0
$487$	−5.47825	−0.248243	−0.124122	−	0.992267i	$-0.539611\pi$
$487$	−5.47825	−0.248243	−0.124122	+	0.992267i	$0.539611\pi$
$488$	0	0
$489$	−12.6731	−0.573098
$490$	0	0
$491$	17.5417	0.791647	0.395824	−	0.918327i	$-0.370459\pi$
$491$	17.5417	0.791647	0.395824	+	0.918327i	$0.370459\pi$
$492$	0	0
$493$	8.89564	0.400639
$494$	0	0
$495$	−6.10760	−0.274516
$496$	0	0
$497$	−11.0352	−0.494998
$498$	0	0
$499$	8.99853	0.402829	0.201415	−	0.979506i	$-0.435446\pi$
$499$	8.99853	0.402829	0.201415	+	0.979506i	$0.435446\pi$
$500$	0	0
$501$	19.4964	0.871037
$502$	0	0
$503$	−38.0025	−1.69445	−0.847223	−	0.531237i	$-0.821727\pi$
$503$	−38.0025	−1.69445	−0.847223	+	0.531237i	$0.821727\pi$
$504$	0	0
$505$	62.4336	2.77826
$506$	0	0
$507$	32.1767	1.42902
$508$	0	0
$509$	1.46181	0.0647936	0.0323968	−	0.999475i	$-0.489686\pi$
$509$	1.46181	0.0647936	0.0323968	+	0.999475i	$0.489686\pi$
$510$	0	0
$511$	2.08016	0.0920211
$512$	0	0
$513$	14.3910	0.635377
$514$	0	0
$515$	−22.1238	−0.974892
$516$	0	0
$517$	58.1305	2.55657
$518$	0	0
$519$	−16.6336	−0.730134
$520$	0	0
$521$	−18.0860	−0.792363	−0.396181	−	0.918172i	$-0.629665\pi$
$521$	−18.0860	−0.792363	−0.396181	+	0.918172i	$0.629665\pi$
$522$	0	0
$523$	−16.4637	−0.719906	−0.359953	−	0.932971i	$-0.617207\pi$
$523$	−16.4637	−0.719906	−0.359953	+	0.932971i	$0.617207\pi$
$524$	0	0
$525$	−9.31479	−0.406531
$526$	0	0
$527$	−3.56686	−0.155375
$528$	0	0
$529$	1.96701	0.0855221
$530$	0	0
$531$	0.184518	0.00800741
$532$	0	0
$533$	−51.2668	−2.22061
$534$	0	0
$535$	−8.06519	−0.348689
$536$	0	0
$537$	22.7897	0.983447
$538$	0	0
$539$	−4.90224	−0.211154
$540$	0	0
$541$	6.00015	0.257967	0.128983	−	0.991647i	$-0.458829\pi$
$541$	6.00015	0.257967	0.128983	+	0.991647i	$0.458829\pi$
$542$	0	0
$543$	2.80085	0.120196
$544$	0	0
$545$	18.2253	0.780685
$546$	0	0
$547$	39.0995	1.67178	0.835888	−	0.548901i	$-0.184953\pi$
$547$	39.0995	1.67178	0.835888	+	0.548901i	$0.184953\pi$
$548$	0	0
$549$	4.11943	0.175813
$550$	0	0
$551$	26.6576	1.13565
$552$	0	0
$553$	−12.3222	−0.523992
$554$	0	0
$555$	34.7704	1.47592
$556$	0	0
$557$	−23.1412	−0.980523	−0.490261	−	0.871576i	$-0.663099\pi$
$557$	−23.1412	−0.980523	−0.490261	+	0.871576i	$0.663099\pi$
$558$	0	0
$559$	−6.72190	−0.284306
$560$	0	0
$561$	9.02979	0.381238
$562$	0	0
$563$	19.2670	0.812007	0.406003	−	0.913872i	$-0.366922\pi$
$563$	19.2670	0.812007	0.406003	+	0.913872i	$0.366922\pi$
$564$	0	0
$565$	−40.2015	−1.69129
$566$	0	0
$567$	10.0243	0.420979
$568$	0	0
$569$	11.3499	0.475811	0.237906	−	0.971288i	$-0.423539\pi$
$569$	11.3499	0.475811	0.237906	+	0.971288i	$0.423539\pi$
$570$	0	0
$571$	7.69413	0.321990	0.160995	−	0.986955i	$-0.448530\pi$
$571$	7.69413	0.321990	0.160995	+	0.986955i	$0.448530\pi$
$572$	0	0
$573$	9.73995	0.406892
$574$	0	0
$575$	−25.2681	−1.05375
$576$	0	0
$577$	−38.2373	−1.59184	−0.795919	−	0.605403i	$-0.793012\pi$
$577$	−38.2373	−1.59184	−0.795919	+	0.605403i	$0.793012\pi$
$578$	0	0
$579$	36.8689	1.53222
$580$	0	0
$581$	−2.17844	−0.0903768
$582$	0	0
$583$	26.5993	1.10163
$584$	0	0
$585$	−6.87705	−0.284331
$586$	0	0
$587$	12.3596	0.510135	0.255068	−	0.966923i	$-0.417902\pi$
$587$	12.3596	0.510135	0.255068	+	0.966923i	$0.417902\pi$
$588$	0	0
$589$	−10.6888	−0.440425
$590$	0	0
$591$	27.3642	1.12561
$592$	0	0
$593$	−17.2903	−0.710029	−0.355015	−	0.934861i	$-0.615524\pi$
$593$	−17.2903	−0.710029	−0.355015	+	0.934861i	$0.615524\pi$
$594$	0	0
$595$	3.17127	0.130009
$596$	0	0
$597$	37.9114	1.55161
$598$	0	0
$599$	−28.4052	−1.16061	−0.580303	−	0.814401i	$-0.697066\pi$
$599$	−28.4052	−1.16061	−0.580303	+	0.814401i	$0.697066\pi$
$600$	0	0
$601$	8.11364	0.330962	0.165481	−	0.986213i	$-0.447082\pi$
$601$	8.11364	0.330962	0.165481	+	0.986213i	$0.447082\pi$
$602$	0	0
$603$	−5.75551	−0.234382
$604$	0	0
$605$	41.3278	1.68021
$606$	0	0
$607$	18.6683	0.757724	0.378862	−	0.925453i	$-0.376315\pi$
$607$	18.6683	0.757724	0.378862	+	0.925453i	$0.376315\pi$
$608$	0	0
$609$	16.3855	0.663975
$610$	0	0
$611$	65.4539	2.64798
$612$	0	0
$613$	−8.90771	−0.359779	−0.179889	−	0.983687i	$-0.557574\pi$
$613$	−8.90771	−0.359779	−0.179889	+	0.983687i	$0.557574\pi$
$614$	0	0
$615$	54.2533	2.18770
$616$	0	0
$617$	−46.7962	−1.88394	−0.941972	−	0.335691i	$-0.891030\pi$
$617$	−46.7962	−1.88394	−0.941972	+	0.335691i	$0.891030\pi$
$618$	0	0
$619$	19.8021	0.795913	0.397956	−	0.917404i	$-0.369720\pi$
$619$	19.8021	0.795913	0.397956	+	0.917404i	$0.369720\pi$
$620$	0	0
$621$	23.9955	0.962907
$622$	0	0
$623$	6.04687	0.242263
$624$	0	0
$625$	−24.7119	−0.988477
$626$	0	0
$627$	27.0596	1.08066
$628$	0	0
$629$	−5.95240	−0.237338
$630$	0	0
$631$	−29.5498	−1.17636	−0.588180	−	0.808730i	$-0.700155\pi$
$631$	−29.5498	−1.17636	−0.588180	+	0.808730i	$0.700155\pi$
$632$	0	0
$633$	−12.3201	−0.489681
$634$	0	0
$635$	−15.3668	−0.609814
$636$	0	0
$637$	−5.51984	−0.218704
$638$	0	0
$639$	4.33535	0.171504
$640$	0	0
$641$	0.304018	0.0120080	0.00600400	−	0.999982i	$-0.498089\pi$
$641$	0.304018	0.0120080	0.00600400	+	0.999982i	$0.498089\pi$
$642$	0	0
$643$	10.3117	0.406652	0.203326	−	0.979111i	$-0.434825\pi$
$643$	10.3117	0.406652	0.203326	+	0.979111i	$0.434825\pi$
$644$	0	0
$645$	7.11348	0.280093
$646$	0	0
$647$	−8.43641	−0.331669	−0.165835	−	0.986154i	$-0.553032\pi$
$647$	−8.43641	−0.331669	−0.165835	+	0.986154i	$0.553032\pi$
$648$	0	0
$649$	−2.30246	−0.0903793
$650$	0	0
$651$	−6.57006	−0.257501
$652$	0	0
$653$	−6.46767	−0.253099	−0.126550	−	0.991960i	$-0.540390\pi$
$653$	−6.46767	−0.253099	−0.126550	+	0.991960i	$0.540390\pi$
$654$	0	0
$655$	−26.2948	−1.02742
$656$	0	0
$657$	−0.817222	−0.0318829
$658$	0	0
$659$	−17.7519	−0.691516	−0.345758	−	0.938324i	$-0.612378\pi$
$659$	−17.7519	−0.691516	−0.345758	+	0.938324i	$0.612378\pi$
$660$	0	0
$661$	−36.8875	−1.43476	−0.717379	−	0.696683i	$-0.754659\pi$
$661$	−36.8875	−1.43476	−0.717379	+	0.696683i	$0.754659\pi$
$662$	0	0
$663$	10.1674	0.394869
$664$	0	0
$665$	9.50335	0.368524
$666$	0	0
$667$	44.4488	1.72107
$668$	0	0
$669$	−38.3640	−1.48324
$670$	0	0
$671$	−51.4031	−1.98440
$672$	0	0
$673$	−15.7077	−0.605486	−0.302743	−	0.953072i	$-0.597902\pi$
$673$	−15.7077	−0.605486	−0.302743	+	0.953072i	$0.597902\pi$
$674$	0	0
$675$	−24.2849	−0.934727
$676$	0	0
$677$	−16.1705	−0.621481	−0.310741	−	0.950495i	$-0.600577\pi$
$677$	−16.1705	−0.621481	−0.310741	+	0.950495i	$0.600577\pi$
$678$	0	0
$679$	7.90496	0.303365
$680$	0	0
$681$	−6.48849	−0.248639
$682$	0	0
$683$	−40.4512	−1.54782	−0.773912	−	0.633294i	$-0.781703\pi$
$683$	−40.4512	−1.54782	−0.773912	+	0.633294i	$0.781703\pi$
$684$	0	0
$685$	−4.66877	−0.178385
$686$	0	0
$687$	−46.8547	−1.78762
$688$	0	0
$689$	29.9504	1.14102
$690$	0	0
$691$	26.2429	0.998329	0.499164	−	0.866507i	$-0.333640\pi$
$691$	26.2429	0.998329	0.499164	+	0.866507i	$0.333640\pi$
$692$	0	0
$693$	1.92591	0.0731594
$694$	0	0
$695$	−29.5404	−1.12053
$696$	0	0
$697$	−9.28773	−0.351798
$698$	0	0
$699$	23.4284	0.886143
$700$	0	0
$701$	−52.1592	−1.97002	−0.985012	−	0.172485i	$-0.944820\pi$
$701$	−52.1592	−1.97002	−0.985012	+	0.172485i	$0.944820\pi$
$702$	0	0
$703$	−17.8376	−0.672757
$704$	0	0
$705$	−69.2669	−2.60874
$706$	0	0
$707$	−19.6872	−0.740415
$708$	0	0
$709$	−2.56196	−0.0962165	−0.0481083	−	0.998842i	$-0.515319\pi$
$709$	−2.56196	−0.0962165	−0.0481083	+	0.998842i	$0.515319\pi$
$710$	0	0
$711$	4.84094	0.181549
$712$	0	0
$713$	−17.8225	−0.667459
$714$	0	0
$715$	85.8132	3.20923
$716$	0	0
$717$	−20.4794	−0.764816
$718$	0	0
$719$	28.3880	1.05869	0.529346	−	0.848406i	$-0.322437\pi$
$719$	28.3880	1.05869	0.529346	+	0.848406i	$0.322437\pi$
$720$	0	0
$721$	6.97632	0.259812
$722$	0	0
$723$	−4.55071	−0.169243
$724$	0	0
$725$	−44.9849	−1.67070
$726$	0	0
$727$	38.5198	1.42862	0.714311	−	0.699829i	$-0.246740\pi$
$727$	38.5198	1.42862	0.714311	+	0.699829i	$0.246740\pi$
$728$	0	0
$729$	22.5988	0.836993
$730$	0	0
$731$	−1.21777	−0.0450409
$732$	0	0
$733$	−45.8656	−1.69408	−0.847042	−	0.531526i	$-0.821619\pi$
$733$	−45.8656	−1.69408	−0.847042	+	0.531526i	$0.821619\pi$
$734$	0	0
$735$	5.84140	0.215463
$736$	0	0
$737$	71.8184	2.64546
$738$	0	0
$739$	4.45947	0.164044	0.0820220	−	0.996631i	$-0.473862\pi$
$739$	4.45947	0.164044	0.0820220	+	0.996631i	$0.473862\pi$
$740$	0	0
$741$	30.4686	1.11929
$742$	0	0
$743$	28.0409	1.02872	0.514361	−	0.857574i	$-0.328029\pi$
$743$	28.0409	1.02872	0.514361	+	0.857574i	$0.328029\pi$
$744$	0	0
$745$	26.9972	0.989100
$746$	0	0
$747$	0.855830	0.0313132
$748$	0	0
$749$	2.54320	0.0929267
$750$	0	0
$751$	−12.4653	−0.454866	−0.227433	−	0.973794i	$-0.573033\pi$
$751$	−12.4653	−0.454866	−0.227433	+	0.973794i	$0.573033\pi$
$752$	0	0
$753$	−15.6313	−0.569636
$754$	0	0
$755$	26.0853	0.949343
$756$	0	0
$757$	26.6426	0.968340	0.484170	−	0.874974i	$-0.339122\pi$
$757$	26.6426	0.968340	0.484170	+	0.874974i	$0.339122\pi$
$758$	0	0
$759$	45.1191	1.63772
$760$	0	0
$761$	23.9301	0.867467	0.433734	−	0.901041i	$-0.357196\pi$
$761$	23.9301	0.867467	0.433734	+	0.901041i	$0.357196\pi$
$762$	0	0
$763$	−5.74699	−0.208055
$764$	0	0
$765$	−1.24588	−0.0450449
$766$	0	0
$767$	−2.59253	−0.0936107
$768$	0	0
$769$	13.6047	0.490599	0.245299	−	0.969447i	$-0.421114\pi$
$769$	13.6047	0.490599	0.245299	+	0.969447i	$0.421114\pi$
$770$	0	0
$771$	1.09112	0.0392956
$772$	0	0
$773$	49.2804	1.77249	0.886246	−	0.463215i	$-0.153304\pi$
$773$	49.2804	1.77249	0.886246	+	0.463215i	$0.153304\pi$
$774$	0	0
$775$	18.0375	0.647926
$776$	0	0
$777$	−10.9642	−0.393337
$778$	0	0
$779$	−27.8325	−0.997204
$780$	0	0
$781$	−54.0974	−1.93576
$782$	0	0
$783$	42.7193	1.52666
$784$	0	0
$785$	68.0601	2.42917
$786$	0	0
$787$	−24.5769	−0.876073	−0.438036	−	0.898957i	$-0.644326\pi$
$787$	−24.5769	−0.876073	−0.438036	+	0.898957i	$0.644326\pi$
$788$	0	0
$789$	−25.8364	−0.919801
$790$	0	0
$791$	12.6768	0.450734
$792$	0	0
$793$	−57.8791	−2.05535
$794$	0	0
$795$	−31.6952	−1.12411
$796$	0	0
$797$	24.2244	0.858073	0.429036	−	0.903287i	$-0.358853\pi$
$797$	24.2244	0.858073	0.429036	+	0.903287i	$0.358853\pi$
$798$	0	0
$799$	11.8579	0.419504
$800$	0	0
$801$	−2.37560	−0.0839376
$802$	0	0
$803$	10.1975	0.359861
$804$	0	0
$805$	15.8459	0.558494
$806$	0	0
$807$	−10.0158	−0.352573
$808$	0	0
$809$	−24.5413	−0.862826	−0.431413	−	0.902155i	$-0.641985\pi$
$809$	−24.5413	−0.862826	−0.431413	+	0.902155i	$0.641985\pi$
$810$	0	0
$811$	34.6372	1.21628	0.608139	−	0.793831i	$-0.291916\pi$
$811$	34.6372	1.21628	0.608139	+	0.793831i	$0.291916\pi$
$812$	0	0
$813$	−13.9627	−0.489694
$814$	0	0
$815$	−21.8190	−0.764285
$816$	0	0
$817$	−3.64929	−0.127673
$818$	0	0
$819$	2.16855	0.0757752
$820$	0	0
$821$	−24.5819	−0.857913	−0.428957	−	0.903325i	$-0.641119\pi$
$821$	−24.5819	−0.857913	−0.428957	+	0.903325i	$0.641119\pi$
$822$	0	0
$823$	56.4683	1.96836	0.984181	−	0.177168i	$-0.0566935\pi$
$823$	56.4683	1.96836	0.984181	+	0.177168i	$0.0566935\pi$
$824$	0	0
$825$	−45.6633	−1.58979
$826$	0	0
$827$	28.7035	0.998119	0.499059	−	0.866568i	$-0.333679\pi$
$827$	28.7035	0.998119	0.499059	+	0.866568i	$0.333679\pi$
$828$	0	0
$829$	8.72844	0.303151	0.151576	−	0.988446i	$-0.451565\pi$
$829$	8.72844	0.303151	0.151576	+	0.988446i	$0.451565\pi$
$830$	0	0
$831$	−9.27112	−0.321612
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	33.5665	1.16162
$836$	0	0
$837$	−17.1290	−0.592066
$838$	0	0
$839$	−1.48258	−0.0511842	−0.0255921	−	0.999672i	$-0.508147\pi$
$839$	−1.48258	−0.0511842	−0.0255921	+	0.999672i	$0.508147\pi$
$840$	0	0
$841$	50.1324	1.72870
$842$	0	0
$843$	26.7327	0.920724
$844$	0	0
$845$	55.3978	1.90574
$846$	0	0
$847$	−13.0319	−0.447783
$848$	0	0
$849$	−51.4120	−1.76446
$850$	0	0
$851$	−29.7424	−1.01956
$852$	0	0
$853$	−14.5460	−0.498045	−0.249023	−	0.968498i	$-0.580109\pi$
$853$	−14.5460	−0.498045	−0.249023	+	0.968498i	$0.580109\pi$
$854$	0	0
$855$	−3.73353	−0.127684
$856$	0	0
$857$	27.9212	0.953770	0.476885	−	0.878966i	$-0.341766\pi$
$857$	27.9212	0.953770	0.476885	+	0.878966i	$0.341766\pi$
$858$	0	0
$859$	−1.71562	−0.0585361	−0.0292680	−	0.999572i	$-0.509318\pi$
$859$	−1.71562	−0.0585361	−0.0292680	+	0.999572i	$0.509318\pi$
$860$	0	0
$861$	−17.1077	−0.583030
$862$	0	0
$863$	−40.3399	−1.37319	−0.686593	−	0.727042i	$-0.740894\pi$
$863$	−40.3399	−1.37319	−0.686593	+	0.727042i	$0.740894\pi$
$864$	0	0
$865$	−28.6376	−0.973707
$866$	0	0
$867$	1.84197	0.0625567
$868$	0	0
$869$	−60.4062	−2.04914
$870$	0	0
$871$	80.8663	2.74005
$872$	0	0
$873$	−3.10558	−0.105108
$874$	0	0
$875$	−0.180651	−0.00610712
$876$	0	0
$877$	−15.2424	−0.514699	−0.257350	−	0.966318i	$-0.582849\pi$
$877$	−15.2424	−0.514699	−0.257350	+	0.966318i	$0.582849\pi$
$878$	0	0
$879$	32.1454	1.08424
$880$	0	0
$881$	0.652152	0.0219716	0.0109858	−	0.999940i	$-0.496503\pi$
$881$	0.652152	0.0219716	0.0109858	+	0.999940i	$0.496503\pi$
$882$	0	0
$883$	31.8504	1.07185	0.535926	−	0.844265i	$-0.319963\pi$
$883$	31.8504	1.07185	0.535926	+	0.844265i	$0.319963\pi$
$884$	0	0
$885$	2.74355	0.0922236
$886$	0	0
$887$	27.7046	0.930229	0.465114	−	0.885251i	$-0.346013\pi$
$887$	27.7046	0.930229	0.465114	+	0.885251i	$0.346013\pi$
$888$	0	0
$889$	4.84564	0.162518
$890$	0	0
$891$	49.1413	1.64629
$892$	0	0
$893$	35.5347	1.18912
$894$	0	0
$895$	39.2363	1.31153
$896$	0	0
$897$	50.8034	1.69628
$898$	0	0
$899$	−31.7295	−1.05824
$900$	0	0
$901$	5.42595	0.180765
$902$	0	0
$903$	−2.24310	−0.0746457
$904$	0	0
$905$	4.82214	0.160293
$906$	0	0
$907$	−22.2364	−0.738349	−0.369174	−	0.929360i	$-0.620360\pi$
$907$	−22.2364	−0.738349	−0.369174	+	0.929360i	$0.620360\pi$
$908$	0	0
$909$	7.73441	0.256534
$910$	0	0
$911$	−5.55623	−0.184086	−0.0920430	−	0.995755i	$-0.529340\pi$
$911$	−5.55623	−0.184086	−0.0920430	+	0.995755i	$0.529340\pi$
$912$	0	0
$913$	−10.6792	−0.353431
$914$	0	0
$915$	61.2508	2.02489
$916$	0	0
$917$	8.29155	0.273811
$918$	0	0
$919$	15.9238	0.525278	0.262639	−	0.964894i	$-0.415407\pi$
$919$	15.9238	0.525278	0.262639	+	0.964894i	$0.415407\pi$
$920$	0	0
$921$	−58.0560	−1.91301
$922$	0	0
$923$	−60.9127	−2.00497
$924$	0	0
$925$	30.1011	0.989718
$926$	0	0
$927$	−2.74075	−0.0900180
$928$	0	0
$929$	−25.3790	−0.832659	−0.416329	−	0.909214i	$-0.636684\pi$
$929$	−25.3790	−0.832659	−0.416329	+	0.909214i	$0.636684\pi$
$930$	0	0
$931$	−2.99670	−0.0982129
$932$	0	0
$933$	−51.8132	−1.69629
$934$	0	0
$935$	15.5463	0.508419
$936$	0	0
$937$	48.0849	1.57087	0.785433	−	0.618946i	$-0.212440\pi$
$937$	48.0849	1.57087	0.785433	+	0.618946i	$0.212440\pi$
$938$	0	0
$939$	42.9656	1.40213
$940$	0	0
$941$	6.41915	0.209258	0.104629	−	0.994511i	$-0.466634\pi$
$941$	6.41915	0.209258	0.104629	+	0.994511i	$0.466634\pi$
$942$	0	0
$943$	−46.4080	−1.51125
$944$	0	0
$945$	15.2293	0.495410
$946$	0	0
$947$	−45.6234	−1.48256	−0.741280	−	0.671196i	$-0.765781\pi$
$947$	−45.6234	−1.48256	−0.741280	+	0.671196i	$0.765781\pi$
$948$	0	0
$949$	11.4822	0.372727
$950$	0	0
$951$	−38.4921	−1.24819
$952$	0	0
$953$	42.1500	1.36537	0.682686	−	0.730712i	$-0.260812\pi$
$953$	42.1500	1.36537	0.682686	+	0.730712i	$0.260812\pi$
$954$	0	0
$955$	16.7690	0.542632
$956$	0	0
$957$	80.3257	2.59656
$958$	0	0
$959$	1.47221	0.0475401
$960$	0	0
$961$	−18.2775	−0.589597
$962$	0	0
$963$	−0.999134	−0.0321966
$964$	0	0
$965$	63.4762	2.04337
$966$	0	0
$967$	27.3993	0.881103	0.440551	−	0.897727i	$-0.354783\pi$
$967$	27.3993	0.881103	0.440551	+	0.897727i	$0.354783\pi$
$968$	0	0
$969$	5.51984	0.177323
$970$	0	0
$971$	11.8114	0.379045	0.189522	−	0.981876i	$-0.439306\pi$
$971$	11.8114	0.379045	0.189522	+	0.981876i	$0.439306\pi$
$972$	0	0
$973$	9.31500	0.298625
$974$	0	0
$975$	−51.4162	−1.64663
$976$	0	0
$977$	17.6992	0.566247	0.283123	−	0.959084i	$-0.408629\pi$
$977$	17.6992	0.566247	0.283123	+	0.959084i	$0.408629\pi$
$978$	0	0
$979$	29.6432	0.947400
$980$	0	0
$981$	2.25779	0.0720856
$982$	0	0
$983$	29.9484	0.955207	0.477604	−	0.878575i	$-0.341506\pi$
$983$	29.9484	0.955207	0.477604	+	0.878575i	$0.341506\pi$
$984$	0	0
$985$	47.1121	1.50112
$986$	0	0
$987$	21.8420	0.695239
$988$	0	0
$989$	−6.08483	−0.193486
$990$	0	0
$991$	16.4355	0.522092	0.261046	−	0.965326i	$-0.415933\pi$
$991$	16.4355	0.522092	0.261046	+	0.965326i	$0.415933\pi$
$992$	0	0
$993$	−46.4204	−1.47311
$994$	0	0
$995$	65.2710	2.06923
$996$	0	0
$997$	26.6437	0.843813	0.421907	−	0.906639i	$-0.361361\pi$
$997$	26.6437	0.843813	0.421907	+	0.906639i	$0.361361\pi$
$998$	0	0
$999$	−28.5851	−0.904392

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.h.1.6	✓	6
4.3	odd	2		3808.2.a.p.1.1	yes	6
8.3	odd	2		7616.2.a.bu.1.6		6
8.5	even	2		7616.2.a.cc.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.h.1.6	✓	6	1.1	even	1	trivial
3808.2.a.p.1.1	yes	6	4.3	odd	2
7616.2.a.bu.1.6		6	8.3	odd	2
7616.2.a.cc.1.1		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)

\(f(q)\)	\(=\)	\(q+1.84197 q^{3} +3.17127 q^{5} -1.00000 q^{7} +0.392864 q^{9} -4.90224 q^{11} -5.51984 q^{13} +5.84140 q^{15} -1.00000 q^{17} -2.99670 q^{19} -1.84197 q^{21} -4.99670 q^{23} +5.05696 q^{25} -4.80227 q^{27} -8.89564 q^{29} +3.56686 q^{31} -9.02979 q^{33} -3.17127 q^{35} +5.95240 q^{37} -10.1674 q^{39} +9.28773 q^{41} +1.21777 q^{43} +1.24588 q^{45} -11.8579 q^{47} +1.00000 q^{49} -1.84197 q^{51} -5.42595 q^{53} -15.5463 q^{55} -5.51984 q^{57} +0.469674 q^{59} +10.4856 q^{61} -0.392864 q^{63} -17.5049 q^{65} -14.6501 q^{67} -9.20379 q^{69} +11.0352 q^{71} -2.08016 q^{73} +9.31479 q^{75} +4.90224 q^{77} +12.3222 q^{79} -10.0243 q^{81} +2.17844 q^{83} -3.17127 q^{85} -16.3855 q^{87} -6.04687 q^{89} +5.51984 q^{91} +6.57006 q^{93} -9.50335 q^{95} -7.90496 q^{97} -1.92591 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + 4 q^{5} - 6 q^{7} + 8 q^{9} - 8 q^{11} + 8 q^{13} - 6 q^{17} - 6 q^{19} + 4 q^{21} - 18 q^{23} + 12 q^{25} - 22 q^{27} - 8 q^{29} + 4 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} - 14 q^{39}+ \cdots - 28 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 + 4 * q^5 - 6 * q^7 + 8 * q^9 - 8 * q^11 + 8 * q^13 - 6 * q^17 - 6 * q^19 + 4 * q^21 - 18 * q^23 + 12 * q^25 - 22 * q^27 - 8 * q^29 + 4 * q^31 - 6 * q^33 - 4 * q^35 - 8 * q^37 - 14 * q^39 + 16 * q^41 - 12 * q^43 + 4 * q^45 - 18 * q^47 + 6 * q^49 + 4 * q^51 - 2 * q^53 + 8 * q^55 + 8 * q^57 - 16 * q^59 + 6 * q^61 - 8 * q^63 - 22 * q^65 - 12 * q^67 + 16 * q^69 + 2 * q^71 + 8 * q^73 - 24 * q^75 + 8 * q^77 + 18 * q^79 + 18 * q^81 - 28 * q^83 - 4 * q^85 + 2 * q^87 - 2 * q^89 - 8 * q^91 - 32 * q^93 - 26 * q^95 - 18 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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