LMFDB - Embedded newform 3808.2.a.i.1.1

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			1.1
Root			$1.76333$ 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)

$f(q)$	$=$	$q-2.64896 q^{3} -3.47027 q^{5} +1.00000 q^{7} +4.01699 q^{9} -3.20072 q^{11} -1.54672 q^{13} +9.19261 q^{15} -1.00000 q^{17} +0.0113771 q^{19} -2.64896 q^{21} +6.16132 q^{23} +7.04278 q^{25} -2.69397 q^{27} -5.20072 q^{29} -2.66511 q^{31} +8.47858 q^{33} -3.47027 q^{35} +3.06788 q^{37} +4.09720 q^{39} -2.97490 q^{41} +7.60088 q^{43} -13.9401 q^{45} +4.94604 q^{47} +1.00000 q^{49} +2.64896 q^{51} -0.535608 q^{53} +11.1074 q^{55} -0.0301375 q^{57} +8.21252 q^{59} +14.9370 q^{61} +4.01699 q^{63} +5.36754 q^{65} +9.29109 q^{67} -16.3211 q^{69} -1.53804 q^{71} -9.37634 q^{73} -18.6561 q^{75} -3.20072 q^{77} +8.74908 q^{79} -4.91475 q^{81} -1.44402 q^{83} +3.47027 q^{85} +13.7765 q^{87} -6.23128 q^{89} -1.54672 q^{91} +7.05977 q^{93} -0.0394816 q^{95} -3.03705 q^{97} -12.8573 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.64896	−1.52938	−0.764689	−	0.644399i	$-0.777107\pi$
$3$	−2.64896	−1.52938	−0.764689	+	0.644399i	$0.777107\pi$
$4$	0	0
$5$	−3.47027	−1.55195	−0.775976	−	0.630762i	$-0.782742\pi$
$5$	−3.47027	−1.55195	−0.775976	+	0.630762i	$0.782742\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	4.01699	1.33900
$10$	0	0
$11$	−3.20072	−0.965053	−0.482527	−	0.875881i	$-0.660281\pi$
$11$	−3.20072	−0.965053	−0.482527	+	0.875881i	$0.660281\pi$
$12$	0	0
$13$	−1.54672	−0.428983	−0.214492	−	0.976726i	$-0.568809\pi$
$13$	−1.54672	−0.428983	−0.214492	+	0.976726i	$0.568809\pi$
$14$	0	0
$15$	9.19261	2.37352
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	0.0113771	0.00261008	0.00130504	−	0.999999i	$-0.499585\pi$
$19$	0.0113771	0.00261008	0.00130504	+	0.999999i	$0.499585\pi$
$20$	0	0
$21$	−2.64896	−0.578051
$22$	0	0
$23$	6.16132	1.28472	0.642362	−	0.766401i	$-0.277954\pi$
$23$	6.16132	1.28472	0.642362	+	0.766401i	$0.277954\pi$
$24$	0	0
$25$	7.04278	1.40856
$26$	0	0
$27$	−2.69397	−0.518455
$28$	0	0
$29$	−5.20072	−0.965749	−0.482875	−	0.875689i	$-0.660407\pi$
$29$	−5.20072	−0.965749	−0.482875	+	0.875689i	$0.660407\pi$
$30$	0	0
$31$	−2.66511	−0.478668	−0.239334	−	0.970937i	$-0.576929\pi$
$31$	−2.66511	−0.478668	−0.239334	+	0.970937i	$0.576929\pi$
$32$	0	0
$33$	8.47858	1.47593
$34$	0	0
$35$	−3.47027	−0.586583
$36$	0	0
$37$	3.06788	0.504357	0.252178	−	0.967681i	$-0.418853\pi$
$37$	3.06788	0.504357	0.252178	+	0.967681i	$0.418853\pi$
$38$	0	0
$39$	4.09720	0.656077
$40$	0	0
$41$	−2.97490	−0.464601	−0.232301	−	0.972644i	$-0.574625\pi$
$41$	−2.97490	−0.464601	−0.232301	+	0.972644i	$0.574625\pi$
$42$	0	0
$43$	7.60088	1.15912	0.579562	−	0.814928i	$-0.303224\pi$
$43$	7.60088	1.15912	0.579562	+	0.814928i	$0.303224\pi$
$44$	0	0
$45$	−13.9401	−2.07806
$46$	0	0
$47$	4.94604	0.721454	0.360727	−	0.932671i	$-0.382529\pi$
$47$	4.94604	0.721454	0.360727	+	0.932671i	$0.382529\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.64896	0.370929
$52$	0	0
$53$	−0.535608	−0.0735714	−0.0367857	−	0.999323i	$-0.511712\pi$
$53$	−0.535608	−0.0735714	−0.0367857	+	0.999323i	$0.511712\pi$
$54$	0	0
$55$	11.1074	1.49772
$56$	0	0
$57$	−0.0301375	−0.00399181
$58$	0	0
$59$	8.21252	1.06918	0.534590	−	0.845112i	$-0.320466\pi$
$59$	8.21252	1.06918	0.534590	+	0.845112i	$0.320466\pi$
$60$	0	0
$61$	14.9370	1.91248	0.956242	−	0.292576i	$-0.0945125\pi$
$61$	14.9370	1.91248	0.956242	+	0.292576i	$0.0945125\pi$
$62$	0	0
$63$	4.01699	0.506093
$64$	0	0
$65$	5.36754	0.665761
$66$	0	0
$67$	9.29109	1.13509	0.567544	−	0.823343i	$-0.307894\pi$
$67$	9.29109	1.13509	0.567544	+	0.823343i	$0.307894\pi$
$68$	0	0
$69$	−16.3211	−1.96483
$70$	0	0
$71$	−1.53804	−0.182531	−0.0912657	−	0.995827i	$-0.529091\pi$
$71$	−1.53804	−0.182531	−0.0912657	+	0.995827i	$0.529091\pi$
$72$	0	0
$73$	−9.37634	−1.09742	−0.548709	−	0.836014i	$-0.684880\pi$
$73$	−9.37634	−1.09742	−0.548709	+	0.836014i	$0.684880\pi$
$74$	0	0
$75$	−18.6561	−2.15422
$76$	0	0
$77$	−3.20072	−0.364756
$78$	0	0
$79$	8.74908	0.984349	0.492174	−	0.870497i	$-0.336202\pi$
$79$	8.74908	0.984349	0.492174	+	0.870497i	$0.336202\pi$
$80$	0	0
$81$	−4.91475	−0.546084
$82$	0	0
$83$	−1.44402	−0.158502	−0.0792509	−	0.996855i	$-0.525253\pi$
$83$	−1.44402	−0.158502	−0.0792509	+	0.996855i	$0.525253\pi$
$84$	0	0
$85$	3.47027	0.376404
$86$	0	0
$87$	13.7765	1.47700
$88$	0	0
$89$	−6.23128	−0.660515	−0.330257	−	0.943891i	$-0.607136\pi$
$89$	−6.23128	−0.660515	−0.330257	+	0.943891i	$0.607136\pi$
$90$	0	0
$91$	−1.54672	−0.162140
$92$	0	0
$93$	7.05977	0.732064
$94$	0	0
$95$	−0.0394816	−0.00405073
$96$	0	0
$97$	−3.03705	−0.308366	−0.154183	−	0.988042i	$-0.549275\pi$
$97$	−3.03705	−0.308366	−0.154183	+	0.988042i	$0.549275\pi$
$98$	0	0
$99$	−12.8573	−1.29220
$100$	0	0
$101$	−5.64392	−0.561591	−0.280796	−	0.959768i	$-0.590598\pi$
$101$	−5.64392	−0.561591	−0.280796	+	0.959768i	$0.590598\pi$
$102$	0	0
$103$	−9.19907	−0.906411	−0.453206	−	0.891406i	$-0.649720\pi$
$103$	−9.19907	−0.906411	−0.453206	+	0.891406i	$0.649720\pi$
$104$	0	0
$105$	9.19261	0.897107
$106$	0	0
$107$	10.6372	1.02834	0.514169	−	0.857689i	$-0.328100\pi$
$107$	10.6372	1.02834	0.514169	+	0.857689i	$0.328100\pi$
$108$	0	0
$109$	3.82950	0.366800	0.183400	−	0.983038i	$-0.441290\pi$
$109$	3.82950	0.366800	0.183400	+	0.983038i	$0.441290\pi$
$110$	0	0
$111$	−8.12670	−0.771352
$112$	0	0
$113$	−6.55468	−0.616612	−0.308306	−	0.951287i	$-0.599762\pi$
$113$	−6.55468	−0.616612	−0.308306	+	0.951287i	$0.599762\pi$
$114$	0	0
$115$	−21.3815	−1.99383
$116$	0	0
$117$	−6.21316	−0.574407
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−0.755397	−0.0686725
$122$	0	0
$123$	7.88039	0.710551
$124$	0	0
$125$	−7.08901	−0.634060
$126$	0	0
$127$	−7.45151	−0.661215	−0.330607	−	0.943768i	$-0.607254\pi$
$127$	−7.45151	−0.661215	−0.330607	+	0.943768i	$0.607254\pi$
$128$	0	0
$129$	−20.1344	−1.77274
$130$	0	0
$131$	−22.8184	−1.99366	−0.996828	−	0.0795834i	$-0.974641\pi$
$131$	−22.8184	−1.99366	−0.996828	+	0.0795834i	$0.974641\pi$
$132$	0	0
$133$	0.0113771	0.000986519 0
$134$	0	0
$135$	9.34881	0.804618
$136$	0	0
$137$	6.63694	0.567032	0.283516	−	0.958967i	$-0.408499\pi$
$137$	6.63694	0.567032	0.283516	+	0.958967i	$0.408499\pi$
$138$	0	0
$139$	1.53989	0.130612	0.0653059	−	0.997865i	$-0.479198\pi$
$139$	1.53989	0.130612	0.0653059	+	0.997865i	$0.479198\pi$
$140$	0	0
$141$	−13.1019	−1.10338
$142$	0	0
$143$	4.95062	0.413992
$144$	0	0
$145$	18.0479	1.49880
$146$	0	0
$147$	−2.64896	−0.218483
$148$	0	0
$149$	8.99432	0.736843	0.368422	−	0.929659i	$-0.379898\pi$
$149$	8.99432	0.736843	0.368422	+	0.929659i	$0.379898\pi$
$150$	0	0
$151$	20.4497	1.66417	0.832087	−	0.554645i	$-0.187146\pi$
$151$	20.4497	1.66417	0.832087	+	0.554645i	$0.187146\pi$
$152$	0	0
$153$	−4.01699	−0.324755
$154$	0	0
$155$	9.24866	0.742870
$156$	0	0
$157$	9.20876	0.734939	0.367470	−	0.930036i	$-0.380224\pi$
$157$	9.20876	0.734939	0.367470	+	0.930036i	$0.380224\pi$
$158$	0	0
$159$	1.41880	0.112519
$160$	0	0
$161$	6.16132	0.485580
$162$	0	0
$163$	1.97242	0.154492	0.0772458	−	0.997012i	$-0.475387\pi$
$163$	1.97242	0.154492	0.0772458	+	0.997012i	$0.475387\pi$
$164$	0	0
$165$	−29.4230	−2.29057
$166$	0	0
$167$	2.91168	0.225313	0.112656	−	0.993634i	$-0.464064\pi$
$167$	2.91168	0.225313	0.112656	+	0.993634i	$0.464064\pi$
$168$	0	0
$169$	−10.6077	−0.815973
$170$	0	0
$171$	0.0457017	0.00349490
$172$	0	0
$173$	−24.5219	−1.86436	−0.932182	−	0.361990i	$-0.882098\pi$
$173$	−24.5219	−1.86436	−0.932182	+	0.361990i	$0.882098\pi$
$174$	0	0
$175$	7.04278	0.532384
$176$	0	0
$177$	−21.7546	−1.63518
$178$	0	0
$179$	14.9089	1.11434	0.557171	−	0.830398i	$-0.311887\pi$
$179$	14.9089	1.11434	0.557171	+	0.830398i	$0.311887\pi$
$180$	0	0
$181$	−25.2389	−1.87599	−0.937997	−	0.346642i	$-0.887322\pi$
$181$	−25.2389	−1.87599	−0.937997	+	0.346642i	$0.887322\pi$
$182$	0	0
$183$	−39.5675	−2.92491
$184$	0	0
$185$	−10.6464	−0.782738
$186$	0	0
$187$	3.20072	0.234060
$188$	0	0
$189$	−2.69397	−0.195958
$190$	0	0
$191$	16.6866	1.20740	0.603698	−	0.797213i	$-0.293693\pi$
$191$	16.6866	1.20740	0.603698	+	0.797213i	$0.293693\pi$
$192$	0	0
$193$	−24.5256	−1.76539	−0.882695	−	0.469946i	$-0.844273\pi$
$193$	−24.5256	−1.76539	−0.882695	+	0.469946i	$0.844273\pi$
$194$	0	0
$195$	−14.2184	−1.01820
$196$	0	0
$197$	−20.2406	−1.44208	−0.721042	−	0.692891i	$-0.756337\pi$
$197$	−20.2406	−1.44208	−0.721042	+	0.692891i	$0.756337\pi$
$198$	0	0
$199$	−13.6553	−0.967996	−0.483998	−	0.875069i	$-0.660816\pi$
$199$	−13.6553	−0.967996	−0.483998	+	0.875069i	$0.660816\pi$
$200$	0	0
$201$	−24.6117	−1.73598
$202$	0	0
$203$	−5.20072	−0.365019
$204$	0	0
$205$	10.3237	0.721039
$206$	0	0
$207$	24.7500	1.72024
$208$	0	0
$209$	−0.0364149	−0.00251887
$210$	0	0
$211$	2.38354	0.164090	0.0820448	−	0.996629i	$-0.473855\pi$
$211$	2.38354	0.164090	0.0820448	+	0.996629i	$0.473855\pi$
$212$	0	0
$213$	4.07420	0.279159
$214$	0	0
$215$	−26.3771	−1.79890
$216$	0	0
$217$	−2.66511	−0.180920
$218$	0	0
$219$	24.8376	1.67837
$220$	0	0
$221$	1.54672	0.104044
$222$	0	0
$223$	12.4786	0.835627	0.417814	−	0.908533i	$-0.362796\pi$
$223$	12.4786	0.835627	0.417814	+	0.908533i	$0.362796\pi$
$224$	0	0
$225$	28.2908	1.88605
$226$	0	0
$227$	−28.2900	−1.87767	−0.938837	−	0.344362i	$-0.888095\pi$
$227$	−28.2900	−1.87767	−0.938837	+	0.344362i	$0.888095\pi$
$228$	0	0
$229$	19.6231	1.29673	0.648366	−	0.761329i	$-0.275453\pi$
$229$	19.6231	1.29673	0.648366	+	0.761329i	$0.275453\pi$
$230$	0	0
$231$	8.47858	0.557850
$232$	0	0
$233$	−11.8839	−0.778540	−0.389270	−	0.921124i	$-0.627273\pi$
$233$	−11.8839	−0.778540	−0.389270	+	0.921124i	$0.627273\pi$
$234$	0	0
$235$	−17.1641	−1.11966
$236$	0	0
$237$	−23.1760	−1.50544
$238$	0	0
$239$	−22.4124	−1.44974	−0.724870	−	0.688886i	$-0.758100\pi$
$239$	−22.4124	−1.44974	−0.724870	+	0.688886i	$0.758100\pi$
$240$	0	0
$241$	9.23371	0.594796	0.297398	−	0.954754i	$-0.403881\pi$
$241$	9.23371	0.594796	0.297398	+	0.954754i	$0.403881\pi$
$242$	0	0
$243$	21.1009	1.35362
$244$	0	0
$245$	−3.47027	−0.221707
$246$	0	0
$247$	−0.0175972	−0.00111968
$248$	0	0
$249$	3.82515	0.242409
$250$	0	0
$251$	−7.60242	−0.479860	−0.239930	−	0.970790i	$-0.577125\pi$
$251$	−7.60242	−0.479860	−0.239930	+	0.970790i	$0.577125\pi$
$252$	0	0
$253$	−19.7207	−1.23983
$254$	0	0
$255$	−9.19261	−0.575664
$256$	0	0
$257$	16.3597	1.02049	0.510244	−	0.860030i	$-0.329555\pi$
$257$	16.3597	1.02049	0.510244	+	0.860030i	$0.329555\pi$
$258$	0	0
$259$	3.06788	0.190629
$260$	0	0
$261$	−20.8912	−1.29314
$262$	0	0
$263$	−17.0815	−1.05329	−0.526644	−	0.850086i	$-0.676550\pi$
$263$	−17.0815	−1.05329	−0.526644	+	0.850086i	$0.676550\pi$
$264$	0	0
$265$	1.85871	0.114179
$266$	0	0
$267$	16.5064	1.01018
$268$	0	0
$269$	16.1464	0.984464	0.492232	−	0.870464i	$-0.336181\pi$
$269$	16.1464	0.984464	0.492232	+	0.870464i	$0.336181\pi$
$270$	0	0
$271$	−0.0670645	−0.00407388	−0.00203694	−	0.999998i	$-0.500648\pi$
$271$	−0.0670645	−0.00407388	−0.00203694	+	0.999998i	$0.500648\pi$
$272$	0	0
$273$	4.09720	0.247974
$274$	0	0
$275$	−22.5420	−1.35933
$276$	0	0
$277$	0.875160	0.0525833	0.0262916	−	0.999654i	$-0.491630\pi$
$277$	0.875160	0.0525833	0.0262916	+	0.999654i	$0.491630\pi$
$278$	0	0
$279$	−10.7057	−0.640935
$280$	0	0
$281$	16.7138	0.997062	0.498531	−	0.866872i	$-0.333873\pi$
$281$	16.7138	0.997062	0.498531	+	0.866872i	$0.333873\pi$
$282$	0	0
$283$	3.60884	0.214523	0.107262	−	0.994231i	$-0.465792\pi$
$283$	3.60884	0.214523	0.107262	+	0.994231i	$0.465792\pi$
$284$	0	0
$285$	0.104585	0.00619509
$286$	0	0
$287$	−2.97490	−0.175603
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	8.04503	0.471608
$292$	0	0
$293$	−21.8592	−1.27703	−0.638513	−	0.769611i	$-0.720450\pi$
$293$	−21.8592	−1.27703	−0.638513	+	0.769611i	$0.720450\pi$
$294$	0	0
$295$	−28.4997	−1.65932
$296$	0	0
$297$	8.62265	0.500337
$298$	0	0
$299$	−9.52985	−0.551125
$300$	0	0
$301$	7.60088	0.438107
$302$	0	0
$303$	14.9505	0.858885
$304$	0	0
$305$	−51.8354	−2.96809
$306$	0	0
$307$	−30.1814	−1.72254	−0.861272	−	0.508145i	$-0.830332\pi$
$307$	−30.1814	−1.72254	−0.861272	+	0.508145i	$0.830332\pi$
$308$	0	0
$309$	24.3680	1.38625
$310$	0	0
$311$	−3.27625	−0.185779	−0.0928895	−	0.995676i	$-0.529610\pi$
$311$	−3.27625	−0.185779	−0.0928895	+	0.995676i	$0.529610\pi$
$312$	0	0
$313$	17.6038	0.995024	0.497512	−	0.867457i	$-0.334247\pi$
$313$	17.6038	0.995024	0.497512	+	0.867457i	$0.334247\pi$
$314$	0	0
$315$	−13.9401	−0.785433
$316$	0	0
$317$	1.37930	0.0774691	0.0387346	−	0.999250i	$-0.487667\pi$
$317$	1.37930	0.0774691	0.0387346	+	0.999250i	$0.487667\pi$
$318$	0	0
$319$	16.6460	0.931999
$320$	0	0
$321$	−28.1775	−1.57272
$322$	0	0
$323$	−0.0113771	−0.000633038 0
$324$	0	0
$325$	−10.8932	−0.604247
$326$	0	0
$327$	−10.1442	−0.560976
$328$	0	0
$329$	4.94604	0.272684
$330$	0	0
$331$	−33.9723	−1.86728	−0.933642	−	0.358207i	$-0.883388\pi$
$331$	−33.9723	−1.86728	−0.933642	+	0.358207i	$0.883388\pi$
$332$	0	0
$333$	12.3237	0.675332
$334$	0	0
$335$	−32.2426	−1.76160
$336$	0	0
$337$	−7.43345	−0.404926	−0.202463	−	0.979290i	$-0.564895\pi$
$337$	−7.43345	−0.404926	−0.202463	+	0.979290i	$0.564895\pi$
$338$	0	0
$339$	17.3631	0.943033
$340$	0	0
$341$	8.53027	0.461940
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	56.6386	3.04932
$346$	0	0
$347$	−25.7875	−1.38435	−0.692173	−	0.721731i	$-0.743347\pi$
$347$	−25.7875	−1.38435	−0.692173	+	0.721731i	$0.743347\pi$
$348$	0	0
$349$	−1.82999	−0.0979572	−0.0489786	−	0.998800i	$-0.515597\pi$
$349$	−1.82999	−0.0979572	−0.0489786	+	0.998800i	$0.515597\pi$
$350$	0	0
$351$	4.16682	0.222408
$352$	0	0
$353$	−24.8569	−1.32300	−0.661499	−	0.749946i	$-0.730079\pi$
$353$	−24.8569	−1.32300	−0.661499	+	0.749946i	$0.730079\pi$
$354$	0	0
$355$	5.33740	0.283280
$356$	0	0
$357$	2.64896	0.140198
$358$	0	0
$359$	20.5506	1.08462	0.542309	−	0.840179i	$-0.317550\pi$
$359$	20.5506	1.08462	0.542309	+	0.840179i	$0.317550\pi$
$360$	0	0
$361$	−18.9999	−0.999993
$362$	0	0
$363$	2.00102	0.105026
$364$	0	0
$365$	32.5384	1.70314
$366$	0	0
$367$	6.59274	0.344138	0.172069	−	0.985085i	$-0.444955\pi$
$367$	6.59274	0.344138	0.172069	+	0.985085i	$0.444955\pi$
$368$	0	0
$369$	−11.9502	−0.622100
$370$	0	0
$371$	−0.535608	−0.0278074
$372$	0	0
$373$	15.1702	0.785484	0.392742	−	0.919649i	$-0.371527\pi$
$373$	15.1702	0.785484	0.392742	+	0.919649i	$0.371527\pi$
$374$	0	0
$375$	18.7785	0.969718
$376$	0	0
$377$	8.04406	0.414290
$378$	0	0
$379$	−2.30255	−0.118274	−0.0591371	−	0.998250i	$-0.518835\pi$
$379$	−2.30255	−0.118274	−0.0591371	+	0.998250i	$0.518835\pi$
$380$	0	0
$381$	19.7388	1.01125
$382$	0	0
$383$	−10.2770	−0.525131	−0.262566	−	0.964914i	$-0.584569\pi$
$383$	−10.2770	−0.525131	−0.262566	+	0.964914i	$0.584569\pi$
$384$	0	0
$385$	11.1074	0.566084
$386$	0	0
$387$	30.5327	1.55206
$388$	0	0
$389$	−11.5364	−0.584920	−0.292460	−	0.956278i	$-0.594474\pi$
$389$	−11.5364	−0.584920	−0.292460	+	0.956278i	$0.594474\pi$
$390$	0	0
$391$	−6.16132	−0.311592
$392$	0	0
$393$	60.4451	3.04905
$394$	0	0
$395$	−30.3617	−1.52766
$396$	0	0
$397$	33.2102	1.66677	0.833385	−	0.552693i	$-0.186400\pi$
$397$	33.2102	1.66677	0.833385	+	0.552693i	$0.186400\pi$
$398$	0	0
$399$	−0.0301375	−0.00150876
$400$	0	0
$401$	−12.4388	−0.621162	−0.310581	−	0.950547i	$-0.600524\pi$
$401$	−12.4388	−0.621162	−0.310581	+	0.950547i	$0.600524\pi$
$402$	0	0
$403$	4.12218	0.205341
$404$	0	0
$405$	17.0555	0.847496
$406$	0	0
$407$	−9.81943	−0.486731
$408$	0	0
$409$	36.3620	1.79799	0.898993	−	0.437963i	$-0.144300\pi$
$409$	36.3620	1.79799	0.898993	+	0.437963i	$0.144300\pi$
$410$	0	0
$411$	−17.5810	−0.867207
$412$	0	0
$413$	8.21252	0.404112
$414$	0	0
$415$	5.01114	0.245987
$416$	0	0
$417$	−4.07911	−0.199755
$418$	0	0
$419$	2.66361	0.130126	0.0650629	−	0.997881i	$-0.479275\pi$
$419$	2.66361	0.130126	0.0650629	+	0.997881i	$0.479275\pi$
$420$	0	0
$421$	37.2328	1.81461	0.907307	−	0.420468i	$-0.138134\pi$
$421$	37.2328	1.81461	0.907307	+	0.420468i	$0.138134\pi$
$422$	0	0
$423$	19.8682	0.966025
$424$	0	0
$425$	−7.04278	−0.341625
$426$	0	0
$427$	14.9370	0.722851
$428$	0	0
$429$	−13.1140	−0.633150
$430$	0	0
$431$	37.0726	1.78573	0.892863	−	0.450329i	$-0.148693\pi$
$431$	37.0726	1.78573	0.892863	+	0.450329i	$0.148693\pi$
$432$	0	0
$433$	6.12435	0.294317	0.147159	−	0.989113i	$-0.452987\pi$
$433$	6.12435	0.294317	0.147159	+	0.989113i	$0.452987\pi$
$434$	0	0
$435$	−47.8082	−2.29223
$436$	0	0
$437$	0.0700980	0.00335324
$438$	0	0
$439$	−18.2035	−0.868804	−0.434402	−	0.900719i	$-0.643040\pi$
$439$	−18.2035	−0.868804	−0.434402	+	0.900719i	$0.643040\pi$
$440$	0	0
$441$	4.01699	0.191285
$442$	0	0
$443$	−12.9973	−0.617519	−0.308760	−	0.951140i	$-0.599914\pi$
$443$	−12.9973	−0.617519	−0.308760	+	0.951140i	$0.599914\pi$
$444$	0	0
$445$	21.6242	1.02509
$446$	0	0
$447$	−23.8256	−1.12691
$448$	0	0
$449$	−4.06157	−0.191677	−0.0958386	−	0.995397i	$-0.530553\pi$
$449$	−4.06157	−0.191677	−0.0958386	+	0.995397i	$0.530553\pi$
$450$	0	0
$451$	9.52182	0.448365
$452$	0	0
$453$	−54.1705	−2.54515
$454$	0	0
$455$	5.36754	0.251634
$456$	0	0
$457$	2.30315	0.107737	0.0538685	−	0.998548i	$-0.482845\pi$
$457$	2.30315	0.107737	0.0538685	+	0.998548i	$0.482845\pi$
$458$	0	0
$459$	2.69397	0.125744
$460$	0	0
$461$	14.9436	0.695994	0.347997	−	0.937496i	$-0.386862\pi$
$461$	14.9436	0.695994	0.347997	+	0.937496i	$0.386862\pi$
$462$	0	0
$463$	7.78313	0.361713	0.180856	−	0.983510i	$-0.442113\pi$
$463$	7.78313	0.361713	0.180856	+	0.983510i	$0.442113\pi$
$464$	0	0
$465$	−24.4993	−1.13613
$466$	0	0
$467$	13.9498	0.645520	0.322760	−	0.946481i	$-0.395389\pi$
$467$	13.9498	0.645520	0.322760	+	0.946481i	$0.395389\pi$
$468$	0	0
$469$	9.29109	0.429023
$470$	0	0
$471$	−24.3936	−1.12400
$472$	0	0
$473$	−24.3283	−1.11862
$474$	0	0
$475$	0.0801264	0.00367645
$476$	0	0
$477$	−2.15153	−0.0985119
$478$	0	0
$479$	25.1103	1.14732	0.573660	−	0.819094i	$-0.305523\pi$
$479$	25.1103	1.14732	0.573660	+	0.819094i	$0.305523\pi$
$480$	0	0
$481$	−4.74516	−0.216360
$482$	0	0
$483$	−16.3211	−0.742636
$484$	0	0
$485$	10.5394	0.478569
$486$	0	0
$487$	−0.495306	−0.0224445	−0.0112222	−	0.999937i	$-0.503572\pi$
$487$	−0.495306	−0.0224445	−0.0112222	+	0.999937i	$0.503572\pi$
$488$	0	0
$489$	−5.22486	−0.236276
$490$	0	0
$491$	−27.8965	−1.25895	−0.629475	−	0.777021i	$-0.716730\pi$
$491$	−27.8965	−1.25895	−0.629475	+	0.777021i	$0.716730\pi$
$492$	0	0
$493$	5.20072	0.234229
$494$	0	0
$495$	44.6182	2.00544
$496$	0	0
$497$	−1.53804	−0.0689904
$498$	0	0
$499$	−34.8309	−1.55924	−0.779622	−	0.626251i	$-0.784589\pi$
$499$	−34.8309	−1.55924	−0.779622	+	0.626251i	$0.784589\pi$
$500$	0	0
$501$	−7.71293	−0.344588
$502$	0	0
$503$	40.7250	1.81584	0.907920	−	0.419144i	$-0.137670\pi$
$503$	40.7250	1.81584	0.907920	+	0.419144i	$0.137670\pi$
$504$	0	0
$505$	19.5859	0.871563
$506$	0	0
$507$	28.0993	1.24793
$508$	0	0
$509$	36.2516	1.60682	0.803412	−	0.595424i	$-0.203016\pi$
$509$	36.2516	1.60682	0.803412	+	0.595424i	$0.203016\pi$
$510$	0	0
$511$	−9.37634	−0.414785
$512$	0	0
$513$	−0.0306496	−0.00135321
$514$	0	0
$515$	31.9233	1.40671
$516$	0	0
$517$	−15.8309	−0.696242
$518$	0	0
$519$	64.9575	2.85132
$520$	0	0
$521$	−21.3665	−0.936083	−0.468042	−	0.883706i	$-0.655040\pi$
$521$	−21.3665	−0.936083	−0.468042	+	0.883706i	$0.655040\pi$
$522$	0	0
$523$	21.3287	0.932641	0.466321	−	0.884616i	$-0.345579\pi$
$523$	21.3287	0.932641	0.466321	+	0.884616i	$0.345579\pi$
$524$	0	0
$525$	−18.6561	−0.814217
$526$	0	0
$527$	2.66511	0.116094
$528$	0	0
$529$	14.9619	0.650518
$530$	0	0
$531$	32.9896	1.43163
$532$	0	0
$533$	4.60134	0.199306
$534$	0	0
$535$	−36.9140	−1.59593
$536$	0	0
$537$	−39.4930	−1.70425
$538$	0	0
$539$	−3.20072	−0.137865
$540$	0	0
$541$	37.0080	1.59110	0.795549	−	0.605889i	$-0.207182\pi$
$541$	37.0080	1.59110	0.795549	+	0.605889i	$0.207182\pi$
$542$	0	0
$543$	66.8569	2.86911
$544$	0	0
$545$	−13.2894	−0.569256
$546$	0	0
$547$	5.62675	0.240582	0.120291	−	0.992739i	$-0.461617\pi$
$547$	5.62675	0.240582	0.120291	+	0.992739i	$0.461617\pi$
$548$	0	0
$549$	60.0017	2.56081
$550$	0	0
$551$	−0.0591691	−0.00252069
$552$	0	0
$553$	8.74908	0.372049
$554$	0	0
$555$	28.2018	1.19710
$556$	0	0
$557$	−18.7841	−0.795908	−0.397954	−	0.917405i	$-0.630280\pi$
$557$	−18.7841	−0.795908	−0.397954	+	0.917405i	$0.630280\pi$
$558$	0	0
$559$	−11.7564	−0.497244
$560$	0	0
$561$	−8.47858	−0.357966
$562$	0	0
$563$	−0.743293	−0.0313261	−0.0156630	−	0.999877i	$-0.504986\pi$
$563$	−0.743293	−0.0313261	−0.0156630	+	0.999877i	$0.504986\pi$
$564$	0	0
$565$	22.7465	0.956953
$566$	0	0
$567$	−4.91475	−0.206400
$568$	0	0
$569$	−10.0137	−0.419794	−0.209897	−	0.977723i	$-0.567313\pi$
$569$	−10.0137	−0.419794	−0.209897	+	0.977723i	$0.567313\pi$
$570$	0	0
$571$	−27.2153	−1.13893	−0.569463	−	0.822017i	$-0.692849\pi$
$571$	−27.2153	−1.13893	−0.569463	+	0.822017i	$0.692849\pi$
$572$	0	0
$573$	−44.2020	−1.84657
$574$	0	0
$575$	43.3929	1.80961
$576$	0	0
$577$	−30.9186	−1.28716	−0.643580	−	0.765379i	$-0.722551\pi$
$577$	−30.9186	−1.28716	−0.643580	+	0.765379i	$0.722551\pi$
$578$	0	0
$579$	64.9673	2.69995
$580$	0	0
$581$	−1.44402	−0.0599081
$582$	0	0
$583$	1.71433	0.0710003
$584$	0	0
$585$	21.5614	0.891453
$586$	0	0
$587$	−36.5625	−1.50910	−0.754548	−	0.656245i	$-0.772144\pi$
$587$	−36.5625	−1.50910	−0.754548	+	0.656245i	$0.772144\pi$
$588$	0	0
$589$	−0.0303212	−0.00124936
$590$	0	0
$591$	53.6166	2.20549
$592$	0	0
$593$	−28.3634	−1.16474	−0.582372	−	0.812922i	$-0.697875\pi$
$593$	−28.3634	−1.16474	−0.582372	+	0.812922i	$0.697875\pi$
$594$	0	0
$595$	3.47027	0.142267
$596$	0	0
$597$	36.1723	1.48043
$598$	0	0
$599$	−10.9449	−0.447196	−0.223598	−	0.974681i	$-0.571780\pi$
$599$	−10.9449	−0.447196	−0.223598	+	0.974681i	$0.571780\pi$
$600$	0	0
$601$	39.5183	1.61198	0.805992	−	0.591927i	$-0.201632\pi$
$601$	39.5183	1.61198	0.805992	+	0.591927i	$0.201632\pi$
$602$	0	0
$603$	37.3222	1.51988
$604$	0	0
$605$	2.62143	0.106576
$606$	0	0
$607$	−27.1078	−1.10027	−0.550137	−	0.835075i	$-0.685424\pi$
$607$	−27.1078	−1.10027	−0.550137	+	0.835075i	$0.685424\pi$
$608$	0	0
$609$	13.7765	0.558252
$610$	0	0
$611$	−7.65014	−0.309492
$612$	0	0
$613$	9.09919	0.367513	0.183756	−	0.982972i	$-0.441174\pi$
$613$	9.09919	0.367513	0.183756	+	0.982972i	$0.441174\pi$
$614$	0	0
$615$	−27.3471	−1.10274
$616$	0	0
$617$	−24.1649	−0.972844	−0.486422	−	0.873724i	$-0.661698\pi$
$617$	−24.1649	−0.972844	−0.486422	+	0.873724i	$0.661698\pi$
$618$	0	0
$619$	−12.9334	−0.519838	−0.259919	−	0.965630i	$-0.583696\pi$
$619$	−12.9334	−0.519838	−0.259919	+	0.965630i	$0.583696\pi$
$620$	0	0
$621$	−16.5984	−0.666072
$622$	0	0
$623$	−6.23128	−0.249651
$624$	0	0
$625$	−10.6131	−0.424525
$626$	0	0
$627$	0.0964616	0.00385230
$628$	0	0
$629$	−3.06788	−0.122324
$630$	0	0
$631$	−23.0233	−0.916542	−0.458271	−	0.888812i	$-0.651531\pi$
$631$	−23.0233	−0.916542	−0.458271	+	0.888812i	$0.651531\pi$
$632$	0	0
$633$	−6.31390	−0.250955
$634$	0	0
$635$	25.8588	1.02617
$636$	0	0
$637$	−1.54672	−0.0612833
$638$	0	0
$639$	−6.17828	−0.244409
$640$	0	0
$641$	−3.82902	−0.151237	−0.0756187	−	0.997137i	$-0.524093\pi$
$641$	−3.82902	−0.151237	−0.0756187	+	0.997137i	$0.524093\pi$
$642$	0	0
$643$	46.9509	1.85156	0.925782	−	0.378058i	$-0.123408\pi$
$643$	46.9509	1.85156	0.925782	+	0.378058i	$0.123408\pi$
$644$	0	0
$645$	69.8719	2.75120
$646$	0	0
$647$	0.107512	0.00422672	0.00211336	−	0.999998i	$-0.499327\pi$
$647$	0.107512	0.00422672	0.00211336	+	0.999998i	$0.499327\pi$
$648$	0	0
$649$	−26.2860	−1.03181
$650$	0	0
$651$	7.05977	0.276694
$652$	0	0
$653$	−4.79227	−0.187536	−0.0937680	−	0.995594i	$-0.529891\pi$
$653$	−4.79227	−0.187536	−0.0937680	+	0.995594i	$0.529891\pi$
$654$	0	0
$655$	79.1862	3.09406
$656$	0	0
$657$	−37.6647	−1.46944
$658$	0	0
$659$	16.6176	0.647330	0.323665	−	0.946172i	$-0.395085\pi$
$659$	16.6176	0.647330	0.323665	+	0.946172i	$0.395085\pi$
$660$	0	0
$661$	−29.8500	−1.16103	−0.580514	−	0.814250i	$-0.697148\pi$
$661$	−29.8500	−1.16103	−0.580514	+	0.814250i	$0.697148\pi$
$662$	0	0
$663$	−4.09720	−0.159122
$664$	0	0
$665$	−0.0394816	−0.00153103
$666$	0	0
$667$	−32.0433	−1.24072
$668$	0	0
$669$	−33.0553	−1.27799
$670$	0	0
$671$	−47.8091	−1.84565
$672$	0	0
$673$	13.9438	0.537496	0.268748	−	0.963211i	$-0.413390\pi$
$673$	13.9438	0.537496	0.268748	+	0.963211i	$0.413390\pi$
$674$	0	0
$675$	−18.9731	−0.730273
$676$	0	0
$677$	−34.2451	−1.31614	−0.658072	−	0.752955i	$-0.728628\pi$
$677$	−34.2451	−1.31614	−0.658072	+	0.752955i	$0.728628\pi$
$678$	0	0
$679$	−3.03705	−0.116551
$680$	0	0
$681$	74.9391	2.87167
$682$	0	0
$683$	−1.75015	−0.0669676	−0.0334838	−	0.999439i	$-0.510660\pi$
$683$	−1.75015	−0.0669676	−0.0334838	+	0.999439i	$0.510660\pi$
$684$	0	0
$685$	−23.0320	−0.880007
$686$	0	0
$687$	−51.9808	−1.98319
$688$	0	0
$689$	0.828436	0.0315609
$690$	0	0
$691$	22.4798	0.855174	0.427587	−	0.903974i	$-0.359364\pi$
$691$	22.4798	0.855174	0.427587	+	0.903974i	$0.359364\pi$
$692$	0	0
$693$	−12.8573	−0.488407
$694$	0	0
$695$	−5.34384	−0.202703
$696$	0	0
$697$	2.97490	0.112682
$698$	0	0
$699$	31.4800	1.19068
$700$	0	0
$701$	24.7339	0.934188	0.467094	−	0.884208i	$-0.345301\pi$
$701$	24.7339	0.934188	0.467094	+	0.884208i	$0.345301\pi$
$702$	0	0
$703$	0.0349036	0.00131641
$704$	0	0
$705$	45.4670	1.71239
$706$	0	0
$707$	−5.64392	−0.212262
$708$	0	0
$709$	−26.3461	−0.989449	−0.494724	−	0.869050i	$-0.664731\pi$
$709$	−26.3461	−0.989449	−0.494724	+	0.869050i	$0.664731\pi$
$710$	0	0
$711$	35.1450	1.31804
$712$	0	0
$713$	−16.4206	−0.614957
$714$	0	0
$715$	−17.1800	−0.642495
$716$	0	0
$717$	59.3696	2.21720
$718$	0	0
$719$	29.7454	1.10932	0.554658	−	0.832079i	$-0.312849\pi$
$719$	29.7454	1.10932	0.554658	+	0.832079i	$0.312849\pi$
$720$	0	0
$721$	−9.19907	−0.342591
$722$	0	0
$723$	−24.4597	−0.909667
$724$	0	0
$725$	−36.6275	−1.36031
$726$	0	0
$727$	−42.6322	−1.58114	−0.790571	−	0.612371i	$-0.790216\pi$
$727$	−42.6322	−1.58114	−0.790571	+	0.612371i	$0.790216\pi$
$728$	0	0
$729$	−41.1512	−1.52412
$730$	0	0
$731$	−7.60088	−0.281129
$732$	0	0
$733$	50.0667	1.84926	0.924629	−	0.380870i	$-0.124375\pi$
$733$	50.0667	1.84926	0.924629	+	0.380870i	$0.124375\pi$
$734$	0	0
$735$	9.19261	0.339075
$736$	0	0
$737$	−29.7382	−1.09542
$738$	0	0
$739$	−7.71366	−0.283751	−0.141876	−	0.989884i	$-0.545313\pi$
$739$	−7.71366	−0.283751	−0.141876	+	0.989884i	$0.545313\pi$
$740$	0	0
$741$	0.0466143	0.00171242
$742$	0	0
$743$	3.79522	0.139233	0.0696166	−	0.997574i	$-0.477822\pi$
$743$	3.79522	0.139233	0.0696166	+	0.997574i	$0.477822\pi$
$744$	0	0
$745$	−31.2127	−1.14355
$746$	0	0
$747$	−5.80062	−0.212234
$748$	0	0
$749$	10.6372	0.388675
$750$	0	0
$751$	38.0086	1.38695	0.693477	−	0.720479i	$-0.256078\pi$
$751$	38.0086	1.38695	0.693477	+	0.720479i	$0.256078\pi$
$752$	0	0
$753$	20.1385	0.733888
$754$	0	0
$755$	−70.9661	−2.58272
$756$	0	0
$757$	−30.9647	−1.12543	−0.562715	−	0.826651i	$-0.690243\pi$
$757$	−30.9647	−1.12543	−0.562715	+	0.826651i	$0.690243\pi$
$758$	0	0
$759$	52.2393	1.89617
$760$	0	0
$761$	39.0647	1.41609	0.708047	−	0.706165i	$-0.249576\pi$
$761$	39.0647	1.41609	0.708047	+	0.706165i	$0.249576\pi$
$762$	0	0
$763$	3.82950	0.138637
$764$	0	0
$765$	13.9401	0.504004
$766$	0	0
$767$	−12.7025	−0.458660
$768$	0	0
$769$	40.9950	1.47832	0.739158	−	0.673532i	$-0.235224\pi$
$769$	40.9950	1.47832	0.739158	+	0.673532i	$0.235224\pi$
$770$	0	0
$771$	−43.3361	−1.56071
$772$	0	0
$773$	−43.5267	−1.56555	−0.782774	−	0.622307i	$-0.786196\pi$
$773$	−43.5267	−1.56555	−0.782774	+	0.622307i	$0.786196\pi$
$774$	0	0
$775$	−18.7698	−0.674231
$776$	0	0
$777$	−8.12670	−0.291544
$778$	0	0
$779$	−0.0338457	−0.00121265
$780$	0	0
$781$	4.92282	0.176152
$782$	0	0
$783$	14.0106	0.500698
$784$	0	0
$785$	−31.9569	−1.14059
$786$	0	0
$787$	−42.5862	−1.51803	−0.759017	−	0.651070i	$-0.774320\pi$
$787$	−42.5862	−1.51803	−0.759017	+	0.651070i	$0.774320\pi$
$788$	0	0
$789$	45.2481	1.61088
$790$	0	0
$791$	−6.55468	−0.233058
$792$	0	0
$793$	−23.1033	−0.820424
$794$	0	0
$795$	−4.92364	−0.174623
$796$	0	0
$797$	−34.9978	−1.23968	−0.619842	−	0.784726i	$-0.712803\pi$
$797$	−34.9978	−1.23968	−0.619842	+	0.784726i	$0.712803\pi$
$798$	0	0
$799$	−4.94604	−0.174978
$800$	0	0
$801$	−25.0310	−0.884427
$802$	0	0
$803$	30.0110	1.05907
$804$	0	0
$805$	−21.3815	−0.753598
$806$	0	0
$807$	−42.7712	−1.50562
$808$	0	0
$809$	2.14216	0.0753145	0.0376572	−	0.999291i	$-0.488010\pi$
$809$	2.14216	0.0753145	0.0376572	+	0.999291i	$0.488010\pi$
$810$	0	0
$811$	−18.4844	−0.649074	−0.324537	−	0.945873i	$-0.605208\pi$
$811$	−18.4844	−0.649074	−0.324537	+	0.945873i	$0.605208\pi$
$812$	0	0
$813$	0.177651	0.00623050
$814$	0	0
$815$	−6.84482	−0.239764
$816$	0	0
$817$	0.0864759	0.00302541
$818$	0	0
$819$	−6.21316	−0.217106
$820$	0	0
$821$	−27.9732	−0.976270	−0.488135	−	0.872768i	$-0.662323\pi$
$821$	−27.9732	−0.976270	−0.488135	+	0.872768i	$0.662323\pi$
$822$	0	0
$823$	−15.8037	−0.550881	−0.275441	−	0.961318i	$-0.588824\pi$
$823$	−15.8037	−0.550881	−0.275441	+	0.961318i	$0.588824\pi$
$824$	0	0
$825$	59.7128	2.07893
$826$	0	0
$827$	21.9192	0.762206	0.381103	−	0.924533i	$-0.375544\pi$
$827$	21.9192	0.762206	0.381103	+	0.924533i	$0.375544\pi$
$828$	0	0
$829$	12.7382	0.442417	0.221209	−	0.975227i	$-0.429000\pi$
$829$	12.7382	0.442417	0.221209	+	0.975227i	$0.429000\pi$
$830$	0	0
$831$	−2.31827	−0.0804197
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	−10.1043	−0.349675
$836$	0	0
$837$	7.17973	0.248168
$838$	0	0
$839$	19.5781	0.675912	0.337956	−	0.941162i	$-0.390265\pi$
$839$	19.5781	0.675912	0.337956	+	0.941162i	$0.390265\pi$
$840$	0	0
$841$	−1.95252	−0.0673283
$842$	0	0
$843$	−44.2742	−1.52488
$844$	0	0
$845$	36.8114	1.26635
$846$	0	0
$847$	−0.755397	−0.0259558
$848$	0	0
$849$	−9.55968	−0.328087
$850$	0	0
$851$	18.9022	0.647959
$852$	0	0
$853$	3.13675	0.107400	0.0537002	−	0.998557i	$-0.482898\pi$
$853$	3.13675	0.107400	0.0537002	+	0.998557i	$0.482898\pi$
$854$	0	0
$855$	−0.158597	−0.00542391
$856$	0	0
$857$	−32.1680	−1.09884	−0.549418	−	0.835547i	$-0.685151\pi$
$857$	−32.1680	−1.09884	−0.549418	+	0.835547i	$0.685151\pi$
$858$	0	0
$859$	10.7404	0.366456	0.183228	−	0.983070i	$-0.441345\pi$
$859$	10.7404	0.366456	0.183228	+	0.983070i	$0.441345\pi$
$860$	0	0
$861$	7.88039	0.268563
$862$	0	0
$863$	34.7776	1.18384	0.591921	−	0.805996i	$-0.298370\pi$
$863$	34.7776	1.18384	0.591921	+	0.805996i	$0.298370\pi$
$864$	0	0
$865$	85.0976	2.89340
$866$	0	0
$867$	−2.64896	−0.0899634
$868$	0	0
$869$	−28.0034	−0.949949
$870$	0	0
$871$	−14.3707	−0.486933
$872$	0	0
$873$	−12.1998	−0.412901
$874$	0	0
$875$	−7.08901	−0.239652
$876$	0	0
$877$	39.4027	1.33054	0.665268	−	0.746605i	$-0.268317\pi$
$877$	39.4027	1.33054	0.665268	+	0.746605i	$0.268317\pi$
$878$	0	0
$879$	57.9041	1.95306
$880$	0	0
$881$	−42.6074	−1.43548	−0.717739	−	0.696312i	$-0.754823\pi$
$881$	−42.6074	−1.43548	−0.717739	+	0.696312i	$0.754823\pi$
$882$	0	0
$883$	−39.2162	−1.31973	−0.659866	−	0.751384i	$-0.729387\pi$
$883$	−39.2162	−1.31973	−0.659866	+	0.751384i	$0.729387\pi$
$884$	0	0
$885$	75.4945	2.53772
$886$	0	0
$887$	16.7917	0.563809	0.281905	−	0.959442i	$-0.409034\pi$
$887$	16.7917	0.563809	0.281905	+	0.959442i	$0.409034\pi$
$888$	0	0
$889$	−7.45151	−0.249916
$890$	0	0
$891$	15.7307	0.527000
$892$	0	0
$893$	0.0562716	0.00188306
$894$	0	0
$895$	−51.7378	−1.72941
$896$	0	0
$897$	25.2442	0.842879
$898$	0	0
$899$	13.8605	0.462273
$900$	0	0
$901$	0.535608	0.0178437
$902$	0	0
$903$	−20.1344	−0.670032
$904$	0	0
$905$	87.5859	2.91145
$906$	0	0
$907$	20.3496	0.675696	0.337848	−	0.941201i	$-0.390301\pi$
$907$	20.3496	0.675696	0.337848	+	0.941201i	$0.390301\pi$
$908$	0	0
$909$	−22.6716	−0.751969
$910$	0	0
$911$	7.65810	0.253724	0.126862	−	0.991920i	$-0.459509\pi$
$911$	7.65810	0.253724	0.126862	+	0.991920i	$0.459509\pi$
$912$	0	0
$913$	4.62191	0.152963
$914$	0	0
$915$	137.310	4.53932
$916$	0	0
$917$	−22.8184	−0.753531
$918$	0	0
$919$	23.6786	0.781086	0.390543	−	0.920585i	$-0.372287\pi$
$919$	23.6786	0.781086	0.390543	+	0.920585i	$0.372287\pi$
$920$	0	0
$921$	79.9493	2.63442
$922$	0	0
$923$	2.37891	0.0783029
$924$	0	0
$925$	21.6064	0.710415
$926$	0	0
$927$	−36.9526	−1.21368
$928$	0	0
$929$	51.3901	1.68605	0.843027	−	0.537871i	$-0.180771\pi$
$929$	51.3901	1.68605	0.843027	+	0.537871i	$0.180771\pi$
$930$	0	0
$931$	0.0113771	0.000372869 0
$932$	0	0
$933$	8.67865	0.284126
$934$	0	0
$935$	−11.1074	−0.363250
$936$	0	0
$937$	−45.0616	−1.47210	−0.736050	−	0.676927i	$-0.763311\pi$
$937$	−45.0616	−1.47210	−0.736050	+	0.676927i	$0.763311\pi$
$938$	0	0
$939$	−46.6317	−1.52177
$940$	0	0
$941$	31.8818	1.03932	0.519658	−	0.854374i	$-0.326059\pi$
$941$	31.8818	1.03932	0.519658	+	0.854374i	$0.326059\pi$
$942$	0	0
$943$	−18.3293	−0.596885
$944$	0	0
$945$	9.34881	0.304117
$946$	0	0
$947$	0.924158	0.0300311	0.0150156	−	0.999887i	$-0.495220\pi$
$947$	0.924158	0.0300311	0.0150156	+	0.999887i	$0.495220\pi$
$948$	0	0
$949$	14.5026	0.470774
$950$	0	0
$951$	−3.65371	−0.118480
$952$	0	0
$953$	−4.18494	−0.135564	−0.0677818	−	0.997700i	$-0.521592\pi$
$953$	−4.18494	−0.135564	−0.0677818	+	0.997700i	$0.521592\pi$
$954$	0	0
$955$	−57.9069	−1.87382
$956$	0	0
$957$	−44.0947	−1.42538
$958$	0	0
$959$	6.63694	0.214318
$960$	0	0
$961$	−23.8972	−0.770877
$962$	0	0
$963$	42.7296	1.37694
$964$	0	0
$965$	85.1104	2.73980
$966$	0	0
$967$	28.7146	0.923399	0.461699	−	0.887037i	$-0.347240\pi$
$967$	28.7146	0.923399	0.461699	+	0.887037i	$0.347240\pi$
$968$	0	0
$969$	0.0301375	0.000968155 0
$970$	0	0
$971$	27.4138	0.879750	0.439875	−	0.898059i	$-0.355023\pi$
$971$	27.4138	0.879750	0.439875	+	0.898059i	$0.355023\pi$
$972$	0	0
$973$	1.53989	0.0493666
$974$	0	0
$975$	28.8557	0.924122
$976$	0	0
$977$	42.2346	1.35121	0.675603	−	0.737266i	$-0.263884\pi$
$977$	42.2346	1.35121	0.675603	+	0.737266i	$0.263884\pi$
$978$	0	0
$979$	19.9446	0.637432
$980$	0	0
$981$	15.3831	0.491144
$982$	0	0
$983$	−11.0260	−0.351676	−0.175838	−	0.984419i	$-0.556263\pi$
$983$	−11.0260	−0.351676	−0.175838	+	0.984419i	$0.556263\pi$
$984$	0	0
$985$	70.2405	2.23805
$986$	0	0
$987$	−13.1019	−0.417037
$988$	0	0
$989$	46.8315	1.48915
$990$	0	0
$991$	−30.5359	−0.970006	−0.485003	−	0.874513i	$-0.661181\pi$
$991$	−30.5359	−0.970006	−0.485003	+	0.874513i	$0.661181\pi$
$992$	0	0
$993$	89.9912	2.85578
$994$	0	0
$995$	47.3875	1.50228
$996$	0	0
$997$	−32.1217	−1.01731	−0.508653	−	0.860972i	$-0.669856\pi$
$997$	−32.1217	−1.01731	−0.508653	+	0.860972i	$0.669856\pi$
$998$	0	0
$999$	−8.26479	−0.261486

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.i.1.1	✓	6
4.3	odd	2		3808.2.a.m.1.6	yes	6
8.3	odd	2		7616.2.a.bx.1.1		6
8.5	even	2		7616.2.a.cb.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.i.1.1	✓	6	1.1	even	1	trivial
3808.2.a.m.1.6	yes	6	4.3	odd	2
7616.2.a.bx.1.1		6	8.3	odd	2
7616.2.a.cb.1.6		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)

\(f(q)\)	\(=\)	\(q-2.64896 q^{3} -3.47027 q^{5} +1.00000 q^{7} +4.01699 q^{9} -3.20072 q^{11} -1.54672 q^{13} +9.19261 q^{15} -1.00000 q^{17} +0.0113771 q^{19} -2.64896 q^{21} +6.16132 q^{23} +7.04278 q^{25} -2.69397 q^{27} -5.20072 q^{29} -2.66511 q^{31} +8.47858 q^{33} -3.47027 q^{35} +3.06788 q^{37} +4.09720 q^{39} -2.97490 q^{41} +7.60088 q^{43} -13.9401 q^{45} +4.94604 q^{47} +1.00000 q^{49} +2.64896 q^{51} -0.535608 q^{53} +11.1074 q^{55} -0.0301375 q^{57} +8.21252 q^{59} +14.9370 q^{61} +4.01699 q^{63} +5.36754 q^{65} +9.29109 q^{67} -16.3211 q^{69} -1.53804 q^{71} -9.37634 q^{73} -18.6561 q^{75} -3.20072 q^{77} +8.74908 q^{79} -4.91475 q^{81} -1.44402 q^{83} +3.47027 q^{85} +13.7765 q^{87} -6.23128 q^{89} -1.54672 q^{91} +7.05977 q^{93} -0.0394816 q^{95} -3.03705 q^{97} -12.8573 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 6 q^{5} + 6 q^{7} + 4 q^{9} - 2 q^{11} - 4 q^{13} + 2 q^{15} - 6 q^{17} - 10 q^{19} - 2 q^{21} + 4 q^{23} + 4 q^{25} - 8 q^{27} - 14 q^{29} + 8 q^{31} - 8 q^{33} - 6 q^{35} - 4 q^{37} + 14 q^{39}+ \cdots - 4 q^{99}+O(q^{100}) \) 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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