LMFDB - Embedded newform 3808.2.a.g.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:	6.6.147697840.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 10x^{4} + 10x^{3} + 18x^{2} - 16x + 3 $ x^6 - 2x^5 - 10x^4 + 10x^3 + 18x^2 - 16*x + 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			$1.46371$ 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)

$f(q)$	$=$	$q+0.463711 q^{3} -0.183002 q^{5} +1.00000 q^{7} -2.78497 q^{9} +4.93124 q^{11} -2.19585 q^{13} -0.0848598 q^{15} +1.00000 q^{17} -7.89068 q^{19} +0.463711 q^{21} +3.58009 q^{23} -4.96651 q^{25} -2.68255 q^{27} -2.53954 q^{29} -1.62496 q^{31} +2.28667 q^{33} -0.183002 q^{35} -9.29680 q^{37} -1.01824 q^{39} -0.851594 q^{41} +0.271676 q^{43} +0.509655 q^{45} -10.3374 q^{47} +1.00000 q^{49} +0.463711 q^{51} +13.0049 q^{53} -0.902425 q^{55} -3.65899 q^{57} +2.95312 q^{59} +5.62282 q^{61} -2.78497 q^{63} +0.401844 q^{65} -4.20488 q^{67} +1.66013 q^{69} +6.04234 q^{71} -13.9383 q^{73} -2.30302 q^{75} +4.93124 q^{77} -1.79400 q^{79} +7.11099 q^{81} -10.8384 q^{83} -0.183002 q^{85} -1.17761 q^{87} -1.13342 q^{89} -2.19585 q^{91} -0.753513 q^{93} +1.44401 q^{95} +6.87312 q^{97} -13.7334 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} - 4 q^{5} + 6 q^{7} + 8 q^{9} - 8 q^{11} - 4 q^{13} + 6 q^{17} - 18 q^{19} - 4 q^{21} - 6 q^{23} + 12 q^{25} - 10 q^{27} + 4 q^{29} + 8 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} + 18 q^{39}+ \cdots - 32 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 - 4 * q^5 + 6 * q^7 + 8 * q^9 - 8 * q^11 - 4 * q^13 + 6 * q^17 - 18 * q^19 - 4 * q^21 - 6 * q^23 + 12 * q^25 - 10 * q^27 + 4 * q^29 + 8 * q^31 - 6 * q^33 - 4 * q^35 - 8 * q^37 + 18 * q^39 - 4 * q^41 - 20 * q^43 - 24 * q^45 - 6 * q^47 + 6 * q^49 - 4 * q^51 + 14 * q^53 - 4 * q^55 - 4 * q^57 - 20 * q^59 + 2 * q^61 + 8 * q^63 + 6 * q^65 - 28 * q^67 - 4 * q^69 - 18 * q^71 + 4 * q^73 - 20 * q^75 - 8 * q^77 + 2 * q^79 + 34 * q^81 - 20 * q^83 - 4 * q^85 - 22 * q^87 - 6 * q^89 - 4 * q^91 + 40 * q^93 + 6 * q^95 + 30 * q^97 - 32 * 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
$3$	0.463711	0.267723	0.133862	−	0.991000i	$-0.457262\pi$
$3$	0.463711	0.267723	0.133862	+	0.991000i	$0.457262\pi$
$4$	0	0
$5$	−0.183002	−0.0818408	−0.0409204	−	0.999162i	$-0.513029\pi$
$5$	−0.183002	−0.0818408	−0.0409204	+	0.999162i	$0.513029\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.78497	−0.928324
$10$	0	0
$11$	4.93124	1.48682	0.743412	−	0.668834i	$-0.233206\pi$
$11$	4.93124	1.48682	0.743412	+	0.668834i	$0.233206\pi$
$12$	0	0
$13$	−2.19585	−0.609019	−0.304509	−	0.952509i	$-0.598493\pi$
$13$	−2.19585	−0.609019	−0.304509	+	0.952509i	$0.598493\pi$
$14$	0	0
$15$	−0.0848598	−0.0219107
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−7.89068	−1.81025	−0.905123	−	0.425149i	$-0.860222\pi$
$19$	−7.89068	−1.81025	−0.905123	+	0.425149i	$0.860222\pi$
$20$	0	0
$21$	0.463711	0.101190
$22$	0	0
$23$	3.58009	0.746501	0.373250	−	0.927731i	$-0.378243\pi$
$23$	3.58009	0.746501	0.373250	+	0.927731i	$0.378243\pi$
$24$	0	0
$25$	−4.96651	−0.993302
$26$	0	0
$27$	−2.68255	−0.516258
$28$	0	0
$29$	−2.53954	−0.471580	−0.235790	−	0.971804i	$-0.575768\pi$
$29$	−2.53954	−0.471580	−0.235790	+	0.971804i	$0.575768\pi$
$30$	0	0
$31$	−1.62496	−0.291852	−0.145926	−	0.989295i	$-0.546616\pi$
$31$	−1.62496	−0.291852	−0.145926	+	0.989295i	$0.546616\pi$
$32$	0	0
$33$	2.28667	0.398058
$34$	0	0
$35$	−0.183002	−0.0309329
$36$	0	0
$37$	−9.29680	−1.52839	−0.764193	−	0.644988i	$-0.776862\pi$
$37$	−9.29680	−1.52839	−0.764193	+	0.644988i	$0.776862\pi$
$38$	0	0
$39$	−1.01824	−0.163049
$40$	0	0
$41$	−0.851594	−0.132997	−0.0664983	−	0.997787i	$-0.521183\pi$
$41$	−0.851594	−0.132997	−0.0664983	+	0.997787i	$0.521183\pi$
$42$	0	0
$43$	0.271676	0.0414303	0.0207151	−	0.999785i	$-0.493406\pi$
$43$	0.271676	0.0414303	0.0207151	+	0.999785i	$0.493406\pi$
$44$	0	0
$45$	0.509655	0.0759748
$46$	0	0
$47$	−10.3374	−1.50786	−0.753929	−	0.656956i	$-0.771844\pi$
$47$	−10.3374	−1.50786	−0.753929	+	0.656956i	$0.771844\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.463711	0.0649325
$52$	0	0
$53$	13.0049	1.78636	0.893182	−	0.449696i	$-0.148468\pi$
$53$	13.0049	1.78636	0.893182	+	0.449696i	$0.148468\pi$
$54$	0	0
$55$	−0.902425	−0.121683
$56$	0	0
$57$	−3.65899	−0.484645
$58$	0	0
$59$	2.95312	0.384463	0.192231	−	0.981350i	$-0.438428\pi$
$59$	2.95312	0.384463	0.192231	+	0.981350i	$0.438428\pi$
$60$	0	0
$61$	5.62282	0.719929	0.359964	−	0.932966i	$-0.382789\pi$
$61$	5.62282	0.719929	0.359964	+	0.932966i	$0.382789\pi$
$62$	0	0
$63$	−2.78497	−0.350874
$64$	0	0
$65$	0.401844	0.0498426
$66$	0	0
$67$	−4.20488	−0.513708	−0.256854	−	0.966450i	$-0.582686\pi$
$67$	−4.20488	−0.513708	−0.256854	+	0.966450i	$0.582686\pi$
$68$	0	0
$69$	1.66013	0.199856
$70$	0	0
$71$	6.04234	0.717094	0.358547	−	0.933512i	$-0.383272\pi$
$71$	6.04234	0.717094	0.358547	+	0.933512i	$0.383272\pi$
$72$	0	0
$73$	−13.9383	−1.63135	−0.815677	−	0.578507i	$-0.803635\pi$
$73$	−13.9383	−1.63135	−0.815677	+	0.578507i	$0.803635\pi$
$74$	0	0
$75$	−2.30302	−0.265930
$76$	0	0
$77$	4.93124	0.561967
$78$	0	0
$79$	−1.79400	−0.201841	−0.100921	−	0.994894i	$-0.532179\pi$
$79$	−1.79400	−0.201841	−0.100921	+	0.994894i	$0.532179\pi$
$80$	0	0
$81$	7.11099	0.790110
$82$	0	0
$83$	−10.8384	−1.18967	−0.594833	−	0.803849i	$-0.702782\pi$
$83$	−10.8384	−1.18967	−0.594833	+	0.803849i	$0.702782\pi$
$84$	0	0
$85$	−0.183002	−0.0198493
$86$	0	0
$87$	−1.17761	−0.126253
$88$	0	0
$89$	−1.13342	−0.120142	−0.0600709	−	0.998194i	$-0.519133\pi$
$89$	−1.13342	−0.120142	−0.0600709	+	0.998194i	$0.519133\pi$
$90$	0	0
$91$	−2.19585	−0.230188
$92$	0	0
$93$	−0.753513	−0.0781357
$94$	0	0
$95$	1.44401	0.148152
$96$	0	0
$97$	6.87312	0.697860	0.348930	−	0.937149i	$-0.386545\pi$
$97$	6.87312	0.697860	0.348930	+	0.937149i	$0.386545\pi$
$98$	0	0
$99$	−13.7334	−1.38025
$100$	0	0
$101$	16.1192	1.60392	0.801958	−	0.597380i	$-0.203792\pi$
$101$	16.1192	1.60392	0.801958	+	0.597380i	$0.203792\pi$
$102$	0	0
$103$	−14.6646	−1.44495	−0.722473	−	0.691400i	$-0.756994\pi$
$103$	−14.6646	−1.44495	−0.722473	+	0.691400i	$0.756994\pi$
$104$	0	0
$105$	−0.0848598	−0.00828147
$106$	0	0
$107$	−16.4813	−1.59331	−0.796656	−	0.604433i	$-0.793400\pi$
$107$	−16.4813	−1.59331	−0.796656	+	0.604433i	$0.793400\pi$
$108$	0	0
$109$	−6.62607	−0.634662	−0.317331	−	0.948315i	$-0.602787\pi$
$109$	−6.62607	−0.634662	−0.317331	+	0.948315i	$0.602787\pi$
$110$	0	0
$111$	−4.31103	−0.409185
$112$	0	0
$113$	−6.45254	−0.607003	−0.303502	−	0.952831i	$-0.598156\pi$
$113$	−6.45254	−0.607003	−0.303502	+	0.952831i	$0.598156\pi$
$114$	0	0
$115$	−0.655163	−0.0610942
$116$	0	0
$117$	6.11538	0.565367
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	13.3171	1.21064
$122$	0	0
$123$	−0.394893	−0.0356063
$124$	0	0
$125$	1.82389	0.163134
$126$	0	0
$127$	18.7804	1.66649	0.833247	−	0.552901i	$-0.186479\pi$
$127$	18.7804	1.66649	0.833247	+	0.552901i	$0.186479\pi$
$128$	0	0
$129$	0.125979	0.0110919
$130$	0	0
$131$	−8.85120	−0.773333	−0.386667	−	0.922220i	$-0.626374\pi$
$131$	−8.85120	−0.773333	−0.386667	+	0.922220i	$0.626374\pi$
$132$	0	0
$133$	−7.89068	−0.684209
$134$	0	0
$135$	0.490912	0.0422510
$136$	0	0
$137$	11.5919	0.990360	0.495180	−	0.868790i	$-0.335102\pi$
$137$	11.5919	0.990360	0.495180	+	0.868790i	$0.335102\pi$
$138$	0	0
$139$	−12.4056	−1.05223	−0.526115	−	0.850413i	$-0.676352\pi$
$139$	−12.4056	−1.05223	−0.526115	+	0.850413i	$0.676352\pi$
$140$	0	0
$141$	−4.79354	−0.403689
$142$	0	0
$143$	−10.8283	−0.905504
$144$	0	0
$145$	0.464740	0.0385945
$146$	0	0
$147$	0.463711	0.0382462
$148$	0	0
$149$	−9.99221	−0.818593	−0.409297	−	0.912401i	$-0.634226\pi$
$149$	−9.99221	−0.818593	−0.409297	+	0.912401i	$0.634226\pi$
$150$	0	0
$151$	3.27257	0.266318	0.133159	−	0.991095i	$-0.457488\pi$
$151$	3.27257	0.266318	0.133159	+	0.991095i	$0.457488\pi$
$152$	0	0
$153$	−2.78497	−0.225152
$154$	0	0
$155$	0.297371	0.0238854
$156$	0	0
$157$	−22.5564	−1.80020	−0.900099	−	0.435686i	$-0.856506\pi$
$157$	−22.5564	−1.80020	−0.900099	+	0.435686i	$0.856506\pi$
$158$	0	0
$159$	6.03052	0.478251
$160$	0	0
$161$	3.58009	0.282151
$162$	0	0
$163$	−7.06625	−0.553471	−0.276736	−	0.960946i	$-0.589253\pi$
$163$	−7.06625	−0.553471	−0.276736	+	0.960946i	$0.589253\pi$
$164$	0	0
$165$	−0.418464	−0.0325774
$166$	0	0
$167$	11.4098	0.882917	0.441458	−	0.897282i	$-0.354461\pi$
$167$	11.4098	0.882917	0.441458	+	0.897282i	$0.354461\pi$
$168$	0	0
$169$	−8.17825	−0.629096
$170$	0	0
$171$	21.9753	1.68050
$172$	0	0
$173$	−6.55453	−0.498332	−0.249166	−	0.968461i	$-0.580156\pi$
$173$	−6.55453	−0.498332	−0.249166	+	0.968461i	$0.580156\pi$
$174$	0	0
$175$	−4.96651	−0.375433
$176$	0	0
$177$	1.36939	0.102930
$178$	0	0
$179$	−9.80943	−0.733191	−0.366596	−	0.930380i	$-0.619477\pi$
$179$	−9.80943	−0.733191	−0.366596	+	0.930380i	$0.619477\pi$
$180$	0	0
$181$	−12.2344	−0.909374	−0.454687	−	0.890651i	$-0.650249\pi$
$181$	−12.2344	−0.909374	−0.454687	+	0.890651i	$0.650249\pi$
$182$	0	0
$183$	2.60736	0.192742
$184$	0	0
$185$	1.70133	0.125084
$186$	0	0
$187$	4.93124	0.360608
$188$	0	0
$189$	−2.68255	−0.195127
$190$	0	0
$191$	−16.1122	−1.16584	−0.582919	−	0.812530i	$-0.698090\pi$
$191$	−16.1122	−1.16584	−0.582919	+	0.812530i	$0.698090\pi$
$192$	0	0
$193$	20.8268	1.49915	0.749573	−	0.661922i	$-0.230259\pi$
$193$	20.8268	1.49915	0.749573	+	0.661922i	$0.230259\pi$
$194$	0	0
$195$	0.186339	0.0133440
$196$	0	0
$197$	12.6076	0.898252	0.449126	−	0.893468i	$-0.351735\pi$
$197$	12.6076	0.898252	0.449126	+	0.893468i	$0.351735\pi$
$198$	0	0
$199$	−4.26153	−0.302092	−0.151046	−	0.988527i	$-0.548264\pi$
$199$	−4.26153	−0.302092	−0.151046	+	0.988527i	$0.548264\pi$
$200$	0	0
$201$	−1.94985	−0.137532
$202$	0	0
$203$	−2.53954	−0.178241
$204$	0	0
$205$	0.155843	0.0108846
$206$	0	0
$207$	−9.97045	−0.692995
$208$	0	0
$209$	−38.9108	−2.69152
$210$	0	0
$211$	−19.6784	−1.35471	−0.677357	−	0.735654i	$-0.736875\pi$
$211$	−19.6784	−1.35471	−0.677357	+	0.735654i	$0.736875\pi$
$212$	0	0
$213$	2.80190	0.191983
$214$	0	0
$215$	−0.0497172	−0.00339069
$216$	0	0
$217$	−1.62496	−0.110310
$218$	0	0
$219$	−6.46334	−0.436752
$220$	0	0
$221$	−2.19585	−0.147709
$222$	0	0
$223$	15.6272	1.04647	0.523237	−	0.852187i	$-0.324724\pi$
$223$	15.6272	1.04647	0.523237	+	0.852187i	$0.324724\pi$
$224$	0	0
$225$	13.8316	0.922106
$226$	0	0
$227$	−6.40663	−0.425223	−0.212611	−	0.977137i	$-0.568197\pi$
$227$	−6.40663	−0.425223	−0.212611	+	0.977137i	$0.568197\pi$
$228$	0	0
$229$	−15.3865	−1.01677	−0.508383	−	0.861131i	$-0.669756\pi$
$229$	−15.3865	−1.01677	−0.508383	+	0.861131i	$0.669756\pi$
$230$	0	0
$231$	2.28667	0.150452
$232$	0	0
$233$	−10.1129	−0.662517	−0.331259	−	0.943540i	$-0.607473\pi$
$233$	−10.1129	−0.662517	−0.331259	+	0.943540i	$0.607473\pi$
$234$	0	0
$235$	1.89175	0.123404
$236$	0	0
$237$	−0.831899	−0.0540377
$238$	0	0
$239$	−3.25451	−0.210517	−0.105258	−	0.994445i	$-0.533567\pi$
$239$	−3.25451	−0.210517	−0.105258	+	0.994445i	$0.533567\pi$
$240$	0	0
$241$	−1.31260	−0.0845522	−0.0422761	−	0.999106i	$-0.513461\pi$
$241$	−1.31260	−0.0845522	−0.0422761	+	0.999106i	$0.513461\pi$
$242$	0	0
$243$	11.3451	0.727789
$244$	0	0
$245$	−0.183002	−0.0116915
$246$	0	0
$247$	17.3267	1.10247
$248$	0	0
$249$	−5.02587	−0.318501
$250$	0	0
$251$	−16.2386	−1.02497	−0.512486	−	0.858695i	$-0.671275\pi$
$251$	−16.2386	−1.02497	−0.512486	+	0.858695i	$0.671275\pi$
$252$	0	0
$253$	17.6543	1.10991
$254$	0	0
$255$	−0.0848598	−0.00531413
$256$	0	0
$257$	21.4009	1.33495	0.667477	−	0.744631i	$-0.267374\pi$
$257$	21.4009	1.33495	0.667477	+	0.744631i	$0.267374\pi$
$258$	0	0
$259$	−9.29680	−0.577675
$260$	0	0
$261$	7.07254	0.437779
$262$	0	0
$263$	13.8465	0.853810	0.426905	−	0.904297i	$-0.359604\pi$
$263$	13.8465	0.853810	0.426905	+	0.904297i	$0.359604\pi$
$264$	0	0
$265$	−2.37992	−0.146197
$266$	0	0
$267$	−0.525577	−0.0321648
$268$	0	0
$269$	21.6111	1.31765	0.658826	−	0.752295i	$-0.271053\pi$
$269$	21.6111	1.31765	0.658826	+	0.752295i	$0.271053\pi$
$270$	0	0
$271$	12.9784	0.788382	0.394191	−	0.919029i	$-0.371025\pi$
$271$	12.9784	0.788382	0.394191	+	0.919029i	$0.371025\pi$
$272$	0	0
$273$	−1.01824	−0.0616266
$274$	0	0
$275$	−24.4910	−1.47686
$276$	0	0
$277$	9.78877	0.588150	0.294075	−	0.955782i	$-0.404988\pi$
$277$	9.78877	0.588150	0.294075	+	0.955782i	$0.404988\pi$
$278$	0	0
$279$	4.52548	0.270933
$280$	0	0
$281$	5.70608	0.340396	0.170198	−	0.985410i	$-0.445559\pi$
$281$	5.70608	0.340396	0.170198	+	0.985410i	$0.445559\pi$
$282$	0	0
$283$	−12.4924	−0.742598	−0.371299	−	0.928513i	$-0.621088\pi$
$283$	−12.4924	−0.742598	−0.371299	+	0.928513i	$0.621088\pi$
$284$	0	0
$285$	0.669602	0.0396638
$286$	0	0
$287$	−0.851594	−0.0502680
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	3.18714	0.186833
$292$	0	0
$293$	−1.15192	−0.0672956	−0.0336478	−	0.999434i	$-0.510712\pi$
$293$	−1.15192	−0.0672956	−0.0336478	+	0.999434i	$0.510712\pi$
$294$	0	0
$295$	−0.540425	−0.0314648
$296$	0	0
$297$	−13.2283	−0.767584
$298$	0	0
$299$	−7.86134	−0.454633
$300$	0	0
$301$	0.271676	0.0156592
$302$	0	0
$303$	7.47462	0.429406
$304$	0	0
$305$	−1.02899	−0.0589196
$306$	0	0
$307$	−11.0721	−0.631921	−0.315960	−	0.948772i	$-0.602327\pi$
$307$	−11.0721	−0.631921	−0.315960	+	0.948772i	$0.602327\pi$
$308$	0	0
$309$	−6.80013	−0.386846
$310$	0	0
$311$	4.77752	0.270908	0.135454	−	0.990784i	$-0.456751\pi$
$311$	4.77752	0.270908	0.135454	+	0.990784i	$0.456751\pi$
$312$	0	0
$313$	27.1742	1.53598	0.767988	−	0.640464i	$-0.221258\pi$
$313$	27.1742	1.53598	0.767988	+	0.640464i	$0.221258\pi$
$314$	0	0
$315$	0.509655	0.0287158
$316$	0	0
$317$	−0.373669	−0.0209873	−0.0104937	−	0.999945i	$-0.503340\pi$
$317$	−0.373669	−0.0209873	−0.0104937	+	0.999945i	$0.503340\pi$
$318$	0	0
$319$	−12.5231	−0.701157
$320$	0	0
$321$	−7.64258	−0.426567
$322$	0	0
$323$	−7.89068	−0.439049
$324$	0	0
$325$	10.9057	0.604940
$326$	0	0
$327$	−3.07258	−0.169914
$328$	0	0
$329$	−10.3374	−0.569917
$330$	0	0
$331$	−8.15819	−0.448415	−0.224207	−	0.974541i	$-0.571979\pi$
$331$	−8.15819	−0.448415	−0.224207	+	0.974541i	$0.571979\pi$
$332$	0	0
$333$	25.8913	1.41884
$334$	0	0
$335$	0.769500	0.0420423
$336$	0	0
$337$	−36.3869	−1.98212	−0.991060	−	0.133416i	$-0.957405\pi$
$337$	−36.3869	−1.98212	−0.991060	+	0.133416i	$0.957405\pi$
$338$	0	0
$339$	−2.99211	−0.162509
$340$	0	0
$341$	−8.01308	−0.433933
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−0.303806	−0.0163564
$346$	0	0
$347$	−28.6177	−1.53628	−0.768139	−	0.640283i	$-0.778817\pi$
$347$	−28.6177	−1.53628	−0.768139	+	0.640283i	$0.778817\pi$
$348$	0	0
$349$	29.0704	1.55610	0.778052	−	0.628200i	$-0.216208\pi$
$349$	29.0704	1.55610	0.778052	+	0.628200i	$0.216208\pi$
$350$	0	0
$351$	5.89048	0.314411
$352$	0	0
$353$	−0.921995	−0.0490728	−0.0245364	−	0.999699i	$-0.507811\pi$
$353$	−0.921995	−0.0490728	−0.0245364	+	0.999699i	$0.507811\pi$
$354$	0	0
$355$	−1.10576	−0.0586875
$356$	0	0
$357$	0.463711	0.0245422
$358$	0	0
$359$	−2.33598	−0.123288	−0.0616442	−	0.998098i	$-0.519634\pi$
$359$	−2.33598	−0.123288	−0.0616442	+	0.998098i	$0.519634\pi$
$360$	0	0
$361$	43.2629	2.27699
$362$	0	0
$363$	6.17528	0.324118
$364$	0	0
$365$	2.55073	0.133511
$366$	0	0
$367$	−13.3212	−0.695360	−0.347680	−	0.937613i	$-0.613030\pi$
$367$	−13.3212	−0.695360	−0.347680	+	0.937613i	$0.613030\pi$
$368$	0	0
$369$	2.37167	0.123464
$370$	0	0
$371$	13.0049	0.675182
$372$	0	0
$373$	17.1148	0.886170	0.443085	−	0.896480i	$-0.353884\pi$
$373$	17.1148	0.886170	0.443085	+	0.896480i	$0.353884\pi$
$374$	0	0
$375$	0.845756	0.0436747
$376$	0	0
$377$	5.57644	0.287201
$378$	0	0
$379$	−2.94066	−0.151052	−0.0755258	−	0.997144i	$-0.524064\pi$
$379$	−2.94066	−0.151052	−0.0755258	+	0.997144i	$0.524064\pi$
$380$	0	0
$381$	8.70868	0.446159
$382$	0	0
$383$	4.75655	0.243048	0.121524	−	0.992588i	$-0.461222\pi$
$383$	4.75655	0.243048	0.121524	+	0.992588i	$0.461222\pi$
$384$	0	0
$385$	−0.902425	−0.0459918
$386$	0	0
$387$	−0.756611	−0.0384607
$388$	0	0
$389$	−19.5985	−0.993685	−0.496842	−	0.867841i	$-0.665507\pi$
$389$	−19.5985	−0.993685	−0.496842	+	0.867841i	$0.665507\pi$
$390$	0	0
$391$	3.58009	0.181053
$392$	0	0
$393$	−4.10440	−0.207039
$394$	0	0
$395$	0.328306	0.0165189
$396$	0	0
$397$	6.03784	0.303031	0.151515	−	0.988455i	$-0.451585\pi$
$397$	6.03784	0.303031	0.151515	+	0.988455i	$0.451585\pi$
$398$	0	0
$399$	−3.65899	−0.183179
$400$	0	0
$401$	−20.1813	−1.00781	−0.503904	−	0.863760i	$-0.668103\pi$
$401$	−20.1813	−1.00781	−0.503904	+	0.863760i	$0.668103\pi$
$402$	0	0
$403$	3.56818	0.177744
$404$	0	0
$405$	−1.30132	−0.0646633
$406$	0	0
$407$	−45.8447	−2.27244
$408$	0	0
$409$	−5.39688	−0.266859	−0.133429	−	0.991058i	$-0.542599\pi$
$409$	−5.39688	−0.266859	−0.133429	+	0.991058i	$0.542599\pi$
$410$	0	0
$411$	5.37527	0.265143
$412$	0	0
$413$	2.95312	0.145313
$414$	0	0
$415$	1.98344	0.0973633
$416$	0	0
$417$	−5.75262	−0.281707
$418$	0	0
$419$	21.5680	1.05367	0.526833	−	0.849969i	$-0.323379\pi$
$419$	21.5680	1.05367	0.526833	+	0.849969i	$0.323379\pi$
$420$	0	0
$421$	6.23653	0.303950	0.151975	−	0.988384i	$-0.451437\pi$
$421$	6.23653	0.303950	0.151975	+	0.988384i	$0.451437\pi$
$422$	0	0
$423$	28.7893	1.39978
$424$	0	0
$425$	−4.96651	−0.240911
$426$	0	0
$427$	5.62282	0.272107
$428$	0	0
$429$	−5.02117	−0.242425
$430$	0	0
$431$	1.11677	0.0537931	0.0268965	−	0.999638i	$-0.491438\pi$
$431$	1.11677	0.0537931	0.0268965	+	0.999638i	$0.491438\pi$
$432$	0	0
$433$	−15.9211	−0.765118	−0.382559	−	0.923931i	$-0.624957\pi$
$433$	−15.9211	−0.765118	−0.382559	+	0.923931i	$0.624957\pi$
$434$	0	0
$435$	0.215505	0.0103327
$436$	0	0
$437$	−28.2494	−1.35135
$438$	0	0
$439$	33.5445	1.60099	0.800496	−	0.599338i	$-0.204569\pi$
$439$	33.5445	1.60099	0.800496	+	0.599338i	$0.204569\pi$
$440$	0	0
$441$	−2.78497	−0.132618
$442$	0	0
$443$	−7.65293	−0.363602	−0.181801	−	0.983335i	$-0.558193\pi$
$443$	−7.65293	−0.363602	−0.181801	+	0.983335i	$0.558193\pi$
$444$	0	0
$445$	0.207417	0.00983251
$446$	0	0
$447$	−4.63349	−0.219157
$448$	0	0
$449$	−34.5304	−1.62959	−0.814795	−	0.579749i	$-0.803151\pi$
$449$	−34.5304	−1.62959	−0.814795	+	0.579749i	$0.803151\pi$
$450$	0	0
$451$	−4.19941	−0.197743
$452$	0	0
$453$	1.51753	0.0712996
$454$	0	0
$455$	0.401844	0.0188387
$456$	0	0
$457$	30.1151	1.40872	0.704362	−	0.709841i	$-0.251233\pi$
$457$	30.1151	1.40872	0.704362	+	0.709841i	$0.251233\pi$
$458$	0	0
$459$	−2.68255	−0.125211
$460$	0	0
$461$	−11.0758	−0.515851	−0.257926	−	0.966165i	$-0.583039\pi$
$461$	−11.0758	−0.515851	−0.257926	+	0.966165i	$0.583039\pi$
$462$	0	0
$463$	30.0852	1.39818	0.699090	−	0.715034i	$-0.253589\pi$
$463$	30.0852	1.39818	0.699090	+	0.715034i	$0.253589\pi$
$464$	0	0
$465$	0.137894	0.00639469
$466$	0	0
$467$	−3.37320	−0.156093	−0.0780465	−	0.996950i	$-0.524868\pi$
$467$	−3.37320	−0.156093	−0.0780465	+	0.996950i	$0.524868\pi$
$468$	0	0
$469$	−4.20488	−0.194163
$470$	0	0
$471$	−10.4596	−0.481955
$472$	0	0
$473$	1.33970	0.0615995
$474$	0	0
$475$	39.1892	1.79812
$476$	0	0
$477$	−36.2183	−1.65832
$478$	0	0
$479$	−35.1850	−1.60764	−0.803822	−	0.594870i	$-0.797204\pi$
$479$	−35.1850	−1.60764	−0.803822	+	0.594870i	$0.797204\pi$
$480$	0	0
$481$	20.4144	0.930816
$482$	0	0
$483$	1.66013	0.0755384
$484$	0	0
$485$	−1.25779	−0.0571134
$486$	0	0
$487$	−2.59431	−0.117559	−0.0587797	−	0.998271i	$-0.518721\pi$
$487$	−2.59431	−0.117559	−0.0587797	+	0.998271i	$0.518721\pi$
$488$	0	0
$489$	−3.27669	−0.148177
$490$	0	0
$491$	28.4561	1.28420	0.642102	−	0.766619i	$-0.278063\pi$
$491$	28.4561	1.28420	0.642102	+	0.766619i	$0.278063\pi$
$492$	0	0
$493$	−2.53954	−0.114375
$494$	0	0
$495$	2.51323	0.112961
$496$	0	0
$497$	6.04234	0.271036
$498$	0	0
$499$	31.1134	1.39283	0.696414	−	0.717640i	$-0.254778\pi$
$499$	31.1134	1.39283	0.696414	+	0.717640i	$0.254778\pi$
$500$	0	0
$501$	5.29084	0.236377
$502$	0	0
$503$	19.0754	0.850532	0.425266	−	0.905068i	$-0.360181\pi$
$503$	19.0754	0.850532	0.425266	+	0.905068i	$0.360181\pi$
$504$	0	0
$505$	−2.94983	−0.131266
$506$	0	0
$507$	−3.79234	−0.168424
$508$	0	0
$509$	18.5666	0.822948	0.411474	−	0.911421i	$-0.365014\pi$
$509$	18.5666	0.822948	0.411474	+	0.911421i	$0.365014\pi$
$510$	0	0
$511$	−13.9383	−0.616594
$512$	0	0
$513$	21.1672	0.934554
$514$	0	0
$515$	2.68365	0.118256
$516$	0	0
$517$	−50.9760	−2.24192
$518$	0	0
$519$	−3.03940	−0.133415
$520$	0	0
$521$	−26.3149	−1.15287	−0.576437	−	0.817141i	$-0.695558\pi$
$521$	−26.3149	−1.15287	−0.576437	+	0.817141i	$0.695558\pi$
$522$	0	0
$523$	−1.74845	−0.0764545	−0.0382273	−	0.999269i	$-0.512171\pi$
$523$	−1.74845	−0.0764545	−0.0382273	+	0.999269i	$0.512171\pi$
$524$	0	0
$525$	−2.30302	−0.100512
$526$	0	0
$527$	−1.62496	−0.0707845
$528$	0	0
$529$	−10.1829	−0.442737
$530$	0	0
$531$	−8.22435	−0.356906
$532$	0	0
$533$	1.86997	0.0809975
$534$	0	0
$535$	3.01611	0.130398
$536$	0	0
$537$	−4.54874	−0.196292
$538$	0	0
$539$	4.93124	0.212403
$540$	0	0
$541$	2.54424	0.109386	0.0546928	−	0.998503i	$-0.482582\pi$
$541$	2.54424	0.109386	0.0546928	+	0.998503i	$0.482582\pi$
$542$	0	0
$543$	−5.67321	−0.243461
$544$	0	0
$545$	1.21258	0.0519413
$546$	0	0
$547$	−31.3222	−1.33924	−0.669621	−	0.742703i	$-0.733543\pi$
$547$	−31.3222	−1.33924	−0.669621	+	0.742703i	$0.733543\pi$
$548$	0	0
$549$	−15.6594	−0.668327
$550$	0	0
$551$	20.0387	0.853677
$552$	0	0
$553$	−1.79400	−0.0762889
$554$	0	0
$555$	0.788925	0.0334880
$556$	0	0
$557$	−16.6278	−0.704543	−0.352271	−	0.935898i	$-0.614591\pi$
$557$	−16.6278	−0.704543	−0.352271	+	0.935898i	$0.614591\pi$
$558$	0	0
$559$	−0.596560	−0.0252318
$560$	0	0
$561$	2.28667	0.0965431
$562$	0	0
$563$	−37.8498	−1.59518	−0.797590	−	0.603200i	$-0.793892\pi$
$563$	−37.8498	−1.59518	−0.797590	+	0.603200i	$0.793892\pi$
$564$	0	0
$565$	1.18082	0.0496777
$566$	0	0
$567$	7.11099	0.298633
$568$	0	0
$569$	41.8257	1.75343	0.876713	−	0.481013i	$-0.159731\pi$
$569$	41.8257	1.75343	0.876713	+	0.481013i	$0.159731\pi$
$570$	0	0
$571$	27.2584	1.14073	0.570365	−	0.821391i	$-0.306802\pi$
$571$	27.2584	1.14073	0.570365	+	0.821391i	$0.306802\pi$
$572$	0	0
$573$	−7.47140	−0.312122
$574$	0	0
$575$	−17.7806	−0.741501
$576$	0	0
$577$	9.98450	0.415660	0.207830	−	0.978165i	$-0.433360\pi$
$577$	9.98450	0.415660	0.207830	+	0.978165i	$0.433360\pi$
$578$	0	0
$579$	9.65761	0.401357
$580$	0	0
$581$	−10.8384	−0.449651
$582$	0	0
$583$	64.1303	2.65601
$584$	0	0
$585$	−1.11912	−0.0462701
$586$	0	0
$587$	−15.1058	−0.623485	−0.311742	−	0.950167i	$-0.600913\pi$
$587$	−15.1058	−0.623485	−0.311742	+	0.950167i	$0.600913\pi$
$588$	0	0
$589$	12.8221	0.528324
$590$	0	0
$591$	5.84626	0.240483
$592$	0	0
$593$	0.0549998	0.00225857	0.00112929	−	0.999999i	$-0.499641\pi$
$593$	0.0549998	0.00225857	0.00112929	+	0.999999i	$0.499641\pi$
$594$	0	0
$595$	−0.183002	−0.00750234
$596$	0	0
$597$	−1.97612	−0.0808771
$598$	0	0
$599$	25.7172	1.05078	0.525388	−	0.850863i	$-0.323920\pi$
$599$	25.7172	1.05078	0.525388	+	0.850863i	$0.323920\pi$
$600$	0	0
$601$	27.4462	1.11955	0.559777	−	0.828643i	$-0.310887\pi$
$601$	27.4462	1.11955	0.559777	+	0.828643i	$0.310887\pi$
$602$	0	0
$603$	11.7105	0.476888
$604$	0	0
$605$	−2.43705	−0.0990802
$606$	0	0
$607$	−5.87264	−0.238363	−0.119181	−	0.992872i	$-0.538027\pi$
$607$	−5.87264	−0.238363	−0.119181	+	0.992872i	$0.538027\pi$
$608$	0	0
$609$	−1.17761	−0.0477192
$610$	0	0
$611$	22.6993	0.918314
$612$	0	0
$613$	24.7178	0.998343	0.499172	−	0.866503i	$-0.333638\pi$
$613$	24.7178	0.998343	0.499172	+	0.866503i	$0.333638\pi$
$614$	0	0
$615$	0.0722661	0.00291405
$616$	0	0
$617$	35.3322	1.42242	0.711211	−	0.702978i	$-0.248147\pi$
$617$	35.3322	1.42242	0.711211	+	0.702978i	$0.248147\pi$
$618$	0	0
$619$	−34.5313	−1.38793	−0.693965	−	0.720009i	$-0.744138\pi$
$619$	−34.5313	−1.38793	−0.693965	+	0.720009i	$0.744138\pi$
$620$	0	0
$621$	−9.60378	−0.385387
$622$	0	0
$623$	−1.13342	−0.0454094
$624$	0	0
$625$	24.4988	0.979951
$626$	0	0
$627$	−18.0434	−0.720582
$628$	0	0
$629$	−9.29680	−0.370688
$630$	0	0
$631$	−12.7939	−0.509319	−0.254659	−	0.967031i	$-0.581963\pi$
$631$	−12.7939	−0.509319	−0.254659	+	0.967031i	$0.581963\pi$
$632$	0	0
$633$	−9.12506	−0.362689
$634$	0	0
$635$	−3.43685	−0.136387
$636$	0	0
$637$	−2.19585	−0.0870027
$638$	0	0
$639$	−16.8277	−0.665695
$640$	0	0
$641$	25.0501	0.989421	0.494711	−	0.869058i	$-0.335274\pi$
$641$	25.0501	0.989421	0.494711	+	0.869058i	$0.335274\pi$
$642$	0	0
$643$	28.6582	1.13017	0.565084	−	0.825033i	$-0.308844\pi$
$643$	28.6582	1.13017	0.565084	+	0.825033i	$0.308844\pi$
$644$	0	0
$645$	−0.0230544	−0.000907767 0
$646$	0	0
$647$	27.5782	1.08421	0.542105	−	0.840311i	$-0.317627\pi$
$647$	27.5782	1.08421	0.542105	+	0.840311i	$0.317627\pi$
$648$	0	0
$649$	14.5625	0.571629
$650$	0	0
$651$	−0.753513	−0.0295325
$652$	0	0
$653$	23.4365	0.917142	0.458571	−	0.888658i	$-0.348361\pi$
$653$	23.4365	0.917142	0.458571	+	0.888658i	$0.348361\pi$
$654$	0	0
$655$	1.61978	0.0632902
$656$	0	0
$657$	38.8178	1.51443
$658$	0	0
$659$	−14.1301	−0.550432	−0.275216	−	0.961382i	$-0.588749\pi$
$659$	−14.1301	−0.550432	−0.275216	+	0.961382i	$0.588749\pi$
$660$	0	0
$661$	−43.7488	−1.70163	−0.850815	−	0.525465i	$-0.823891\pi$
$661$	−43.7488	−1.70163	−0.850815	+	0.525465i	$0.823891\pi$
$662$	0	0
$663$	−1.01824	−0.0395451
$664$	0	0
$665$	1.44401	0.0559962
$666$	0	0
$667$	−9.09178	−0.352035
$668$	0	0
$669$	7.24650	0.280166
$670$	0	0
$671$	27.7275	1.07041
$672$	0	0
$673$	−38.4550	−1.48233	−0.741166	−	0.671322i	$-0.765727\pi$
$673$	−38.4550	−1.48233	−0.741166	+	0.671322i	$0.765727\pi$
$674$	0	0
$675$	13.3229	0.512800
$676$	0	0
$677$	9.19792	0.353505	0.176752	−	0.984255i	$-0.443441\pi$
$677$	9.19792	0.353505	0.176752	+	0.984255i	$0.443441\pi$
$678$	0	0
$679$	6.87312	0.263766
$680$	0	0
$681$	−2.97082	−0.113842
$682$	0	0
$683$	29.8086	1.14059	0.570297	−	0.821439i	$-0.306828\pi$
$683$	29.8086	1.14059	0.570297	+	0.821439i	$0.306828\pi$
$684$	0	0
$685$	−2.12133	−0.0810519
$686$	0	0
$687$	−7.13486	−0.272212
$688$	0	0
$689$	−28.5568	−1.08793
$690$	0	0
$691$	−38.5017	−1.46467	−0.732337	−	0.680943i	$-0.761570\pi$
$691$	−38.5017	−1.46467	−0.732337	+	0.680943i	$0.761570\pi$
$692$	0	0
$693$	−13.7334	−0.521687
$694$	0	0
$695$	2.27025	0.0861154
$696$	0	0
$697$	−0.851594	−0.0322564
$698$	0	0
$699$	−4.68945	−0.177371
$700$	0	0
$701$	27.2127	1.02781	0.513904	−	0.857848i	$-0.328199\pi$
$701$	27.2127	1.02781	0.513904	+	0.857848i	$0.328199\pi$
$702$	0	0
$703$	73.3581	2.76675
$704$	0	0
$705$	0.877226	0.0330383
$706$	0	0
$707$	16.1192	0.606223
$708$	0	0
$709$	19.7388	0.741304	0.370652	−	0.928772i	$-0.379134\pi$
$709$	19.7388	0.741304	0.370652	+	0.928772i	$0.379134\pi$
$710$	0	0
$711$	4.99625	0.187374
$712$	0	0
$713$	−5.81752	−0.217868
$714$	0	0
$715$	1.98159	0.0741072
$716$	0	0
$717$	−1.50915	−0.0563602
$718$	0	0
$719$	1.02686	0.0382955	0.0191477	−	0.999817i	$-0.493905\pi$
$719$	1.02686	0.0382955	0.0191477	+	0.999817i	$0.493905\pi$
$720$	0	0
$721$	−14.6646	−0.546138
$722$	0	0
$723$	−0.608668	−0.0226366
$724$	0	0
$725$	12.6126	0.468422
$726$	0	0
$727$	−1.92475	−0.0713849	−0.0356924	−	0.999363i	$-0.511364\pi$
$727$	−1.92475	−0.0713849	−0.0356924	+	0.999363i	$0.511364\pi$
$728$	0	0
$729$	−16.0721	−0.595264
$730$	0	0
$731$	0.271676	0.0100483
$732$	0	0
$733$	−45.0103	−1.66249	−0.831246	−	0.555904i	$-0.812372\pi$
$733$	−45.0103	−1.66249	−0.831246	+	0.555904i	$0.812372\pi$
$734$	0	0
$735$	−0.0848598	−0.00313010
$736$	0	0
$737$	−20.7353	−0.763793
$738$	0	0
$739$	39.2233	1.44285	0.721427	−	0.692491i	$-0.243487\pi$
$739$	39.2233	1.44285	0.721427	+	0.692491i	$0.243487\pi$
$740$	0	0
$741$	8.03460	0.295158
$742$	0	0
$743$	−22.9672	−0.842586	−0.421293	−	0.906924i	$-0.638424\pi$
$743$	−22.9672	−0.842586	−0.421293	+	0.906924i	$0.638424\pi$
$744$	0	0
$745$	1.82859	0.0669944
$746$	0	0
$747$	30.1846	1.10440
$748$	0	0
$749$	−16.4813	−0.602215
$750$	0	0
$751$	45.8102	1.67164	0.835818	−	0.549006i	$-0.184994\pi$
$751$	45.8102	1.67164	0.835818	+	0.549006i	$0.184994\pi$
$752$	0	0
$753$	−7.53002	−0.274409
$754$	0	0
$755$	−0.598887	−0.0217957
$756$	0	0
$757$	−26.7597	−0.972597	−0.486299	−	0.873793i	$-0.661653\pi$
$757$	−26.7597	−0.972597	−0.486299	+	0.873793i	$0.661653\pi$
$758$	0	0
$759$	8.18647	0.297150
$760$	0	0
$761$	−6.10079	−0.221154	−0.110577	−	0.993868i	$-0.535270\pi$
$761$	−6.10079	−0.221154	−0.110577	+	0.993868i	$0.535270\pi$
$762$	0	0
$763$	−6.62607	−0.239880
$764$	0	0
$765$	0.509655	0.0184266
$766$	0	0
$767$	−6.48460	−0.234145
$768$	0	0
$769$	32.4699	1.17090	0.585448	−	0.810710i	$-0.300919\pi$
$769$	32.4699	1.17090	0.585448	+	0.810710i	$0.300919\pi$
$770$	0	0
$771$	9.92384	0.357398
$772$	0	0
$773$	9.02047	0.324444	0.162222	−	0.986754i	$-0.448134\pi$
$773$	9.02047	0.324444	0.162222	+	0.986754i	$0.448134\pi$
$774$	0	0
$775$	8.07040	0.289897
$776$	0	0
$777$	−4.31103	−0.154657
$778$	0	0
$779$	6.71966	0.240757
$780$	0	0
$781$	29.7962	1.06619
$782$	0	0
$783$	6.81245	0.243457
$784$	0	0
$785$	4.12786	0.147330
$786$	0	0
$787$	28.4077	1.01263	0.506313	−	0.862350i	$-0.331008\pi$
$787$	28.4077	1.01263	0.506313	+	0.862350i	$0.331008\pi$
$788$	0	0
$789$	6.42075	0.228585
$790$	0	0
$791$	−6.45254	−0.229426
$792$	0	0
$793$	−12.3469	−0.438450
$794$	0	0
$795$	−1.10360	−0.0391405
$796$	0	0
$797$	−41.4002	−1.46647	−0.733235	−	0.679975i	$-0.761991\pi$
$797$	−41.4002	−1.46647	−0.733235	+	0.679975i	$0.761991\pi$
$798$	0	0
$799$	−10.3374	−0.365709
$800$	0	0
$801$	3.15653	0.111531
$802$	0	0
$803$	−68.7330	−2.42554
$804$	0	0
$805$	−0.655163	−0.0230915
$806$	0	0
$807$	10.0213	0.352767
$808$	0	0
$809$	18.3561	0.645368	0.322684	−	0.946507i	$-0.395415\pi$
$809$	18.3561	0.645368	0.322684	+	0.946507i	$0.395415\pi$
$810$	0	0
$811$	29.5985	1.03934	0.519671	−	0.854366i	$-0.326054\pi$
$811$	29.5985	1.03934	0.519671	+	0.854366i	$0.326054\pi$
$812$	0	0
$813$	6.01822	0.211068
$814$	0	0
$815$	1.29314	0.0452966
$816$	0	0
$817$	−2.14371	−0.0749990
$818$	0	0
$819$	6.11538	0.213689
$820$	0	0
$821$	34.8892	1.21764	0.608821	−	0.793308i	$-0.291643\pi$
$821$	34.8892	1.21764	0.608821	+	0.793308i	$0.291643\pi$
$822$	0	0
$823$	16.0577	0.559735	0.279867	−	0.960039i	$-0.409709\pi$
$823$	16.0577	0.559735	0.279867	+	0.960039i	$0.409709\pi$
$824$	0	0
$825$	−11.3568	−0.395391
$826$	0	0
$827$	52.5866	1.82861	0.914307	−	0.405022i	$-0.132736\pi$
$827$	52.5866	1.82861	0.914307	+	0.405022i	$0.132736\pi$
$828$	0	0
$829$	15.8591	0.550810	0.275405	−	0.961328i	$-0.411188\pi$
$829$	15.8591	0.550810	0.275405	+	0.961328i	$0.411188\pi$
$830$	0	0
$831$	4.53916	0.157462
$832$	0	0
$833$	1.00000	0.0346479
$834$	0	0
$835$	−2.08801	−0.0722586
$836$	0	0
$837$	4.35905	0.150671
$838$	0	0
$839$	−9.73445	−0.336071	−0.168035	−	0.985781i	$-0.553742\pi$
$839$	−9.73445	−0.336071	−0.168035	+	0.985781i	$0.553742\pi$
$840$	0	0
$841$	−22.5507	−0.777612
$842$	0	0
$843$	2.64597	0.0911320
$844$	0	0
$845$	1.49663	0.0514857
$846$	0	0
$847$	13.3171	0.457581
$848$	0	0
$849$	−5.79287	−0.198811
$850$	0	0
$851$	−33.2834	−1.14094
$852$	0	0
$853$	0.384669	0.0131708	0.00658541	−	0.999978i	$-0.497904\pi$
$853$	0.384669	0.0131708	0.00658541	+	0.999978i	$0.497904\pi$
$854$	0	0
$855$	−4.02152	−0.137533
$856$	0	0
$857$	−9.71692	−0.331924	−0.165962	−	0.986132i	$-0.553073\pi$
$857$	−9.71692	−0.331924	−0.165962	+	0.986132i	$0.553073\pi$
$858$	0	0
$859$	−24.0428	−0.820328	−0.410164	−	0.912012i	$-0.634529\pi$
$859$	−24.0428	−0.820328	−0.410164	+	0.912012i	$0.634529\pi$
$860$	0	0
$861$	−0.394893	−0.0134579
$862$	0	0
$863$	−58.2764	−1.98375	−0.991875	−	0.127213i	$-0.959397\pi$
$863$	−58.2764	−1.98375	−0.991875	+	0.127213i	$0.959397\pi$
$864$	0	0
$865$	1.19949	0.0407839
$866$	0	0
$867$	0.463711	0.0157484
$868$	0	0
$869$	−8.84666	−0.300102
$870$	0	0
$871$	9.23329	0.312858
$872$	0	0
$873$	−19.1415	−0.647840
$874$	0	0
$875$	1.82389	0.0616587
$876$	0	0
$877$	−40.6895	−1.37399	−0.686994	−	0.726663i	$-0.741070\pi$
$877$	−40.6895	−1.37399	−0.686994	+	0.726663i	$0.741070\pi$
$878$	0	0
$879$	−0.534156	−0.0180166
$880$	0	0
$881$	−13.8474	−0.466530	−0.233265	−	0.972413i	$-0.574941\pi$
$881$	−13.8474	−0.466530	−0.233265	+	0.972413i	$0.574941\pi$
$882$	0	0
$883$	−56.1576	−1.88985	−0.944927	−	0.327282i	$-0.893867\pi$
$883$	−56.1576	−1.88985	−0.944927	+	0.327282i	$0.893867\pi$
$884$	0	0
$885$	−0.250601	−0.00842386
$886$	0	0
$887$	−27.9890	−0.939777	−0.469889	−	0.882726i	$-0.655706\pi$
$887$	−27.9890	−0.939777	−0.469889	+	0.882726i	$0.655706\pi$
$888$	0	0
$889$	18.7804	0.629875
$890$	0	0
$891$	35.0660	1.17475
$892$	0	0
$893$	81.5688	2.72960
$894$	0	0
$895$	1.79514	0.0600050
$896$	0	0
$897$	−3.64539	−0.121716
$898$	0	0
$899$	4.12666	0.137632
$900$	0	0
$901$	13.0049	0.433257
$902$	0	0
$903$	0.125979	0.00419233
$904$	0	0
$905$	2.23891	0.0744239
$906$	0	0
$907$	−6.31795	−0.209784	−0.104892	−	0.994484i	$-0.533450\pi$
$907$	−6.31795	−0.209784	−0.104892	+	0.994484i	$0.533450\pi$
$908$	0	0
$909$	−44.8914	−1.48895
$910$	0	0
$911$	57.1761	1.89433	0.947165	−	0.320747i	$-0.103934\pi$
$911$	57.1761	1.89433	0.947165	+	0.320747i	$0.103934\pi$
$912$	0	0
$913$	−53.4466	−1.76882
$914$	0	0
$915$	−0.477152	−0.0157741
$916$	0	0
$917$	−8.85120	−0.292292
$918$	0	0
$919$	12.9388	0.426813	0.213407	−	0.976963i	$-0.431544\pi$
$919$	12.9388	0.426813	0.213407	+	0.976963i	$0.431544\pi$
$920$	0	0
$921$	−5.13427	−0.169180
$922$	0	0
$923$	−13.2681	−0.436724
$924$	0	0
$925$	46.1727	1.51815
$926$	0	0
$927$	40.8405	1.34138
$928$	0	0
$929$	46.8304	1.53646	0.768228	−	0.640177i	$-0.221139\pi$
$929$	46.8304	1.53646	0.768228	+	0.640177i	$0.221139\pi$
$930$	0	0
$931$	−7.89068	−0.258607
$932$	0	0
$933$	2.21538	0.0725284
$934$	0	0
$935$	−0.902425	−0.0295124
$936$	0	0
$937$	−37.1054	−1.21218	−0.606090	−	0.795396i	$-0.707263\pi$
$937$	−37.1054	−1.21218	−0.606090	+	0.795396i	$0.707263\pi$
$938$	0	0
$939$	12.6010	0.411217
$940$	0	0
$941$	−29.8654	−0.973585	−0.486792	−	0.873518i	$-0.661833\pi$
$941$	−29.8654	−0.973585	−0.486792	+	0.873518i	$0.661833\pi$
$942$	0	0
$943$	−3.04878	−0.0992820
$944$	0	0
$945$	0.490912	0.0159694
$946$	0	0
$947$	34.5751	1.12354	0.561769	−	0.827294i	$-0.310121\pi$
$947$	34.5751	1.12354	0.561769	+	0.827294i	$0.310121\pi$
$948$	0	0
$949$	30.6064	0.993526
$950$	0	0
$951$	−0.173274	−0.00561880
$952$	0	0
$953$	−21.7992	−0.706147	−0.353073	−	0.935596i	$-0.614863\pi$
$953$	−21.7992	−0.706147	−0.353073	+	0.935596i	$0.614863\pi$
$954$	0	0
$955$	2.94856	0.0954132
$956$	0	0
$957$	−5.80708	−0.187716
$958$	0	0
$959$	11.5919	0.374321
$960$	0	0
$961$	−28.3595	−0.914822
$962$	0	0
$963$	45.9001	1.47911
$964$	0	0
$965$	−3.81134	−0.122691
$966$	0	0
$967$	36.9088	1.18691	0.593453	−	0.804868i	$-0.297764\pi$
$967$	36.9088	1.18691	0.593453	+	0.804868i	$0.297764\pi$
$968$	0	0
$969$	−3.65899	−0.117544
$970$	0	0
$971$	32.7964	1.05249	0.526243	−	0.850334i	$-0.323600\pi$
$971$	32.7964	1.05249	0.526243	+	0.850334i	$0.323600\pi$
$972$	0	0
$973$	−12.4056	−0.397706
$974$	0	0
$975$	5.05709	0.161957
$976$	0	0
$977$	46.5869	1.49045	0.745223	−	0.666816i	$-0.232343\pi$
$977$	46.5869	1.49045	0.745223	+	0.666816i	$0.232343\pi$
$978$	0	0
$979$	−5.58914	−0.178630
$980$	0	0
$981$	18.4534	0.589172
$982$	0	0
$983$	−44.6023	−1.42259	−0.711296	−	0.702892i	$-0.751892\pi$
$983$	−44.6023	−1.42259	−0.711296	+	0.702892i	$0.751892\pi$
$984$	0	0
$985$	−2.30721	−0.0735137
$986$	0	0
$987$	−4.79354	−0.152580
$988$	0	0
$989$	0.972626	0.0309277
$990$	0	0
$991$	−4.43880	−0.141003	−0.0705016	−	0.997512i	$-0.522460\pi$
$991$	−4.43880	−0.141003	−0.0705016	+	0.997512i	$0.522460\pi$
$992$	0	0
$993$	−3.78304	−0.120051
$994$	0	0
$995$	0.779867	0.0247235
$996$	0	0
$997$	42.7786	1.35481	0.677406	−	0.735609i	$-0.263104\pi$
$997$	42.7786	1.35481	0.677406	+	0.735609i	$0.263104\pi$
$998$	0	0
$999$	24.9392	0.789040

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3808.2.a.g.1.5	✓	6
4.3	odd	2		3808.2.a.o.1.2	yes	6
8.3	odd	2		7616.2.a.bv.1.5		6
8.5	even	2		7616.2.a.cd.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.g.1.5	✓	6	1.1	even	1	trivial
3808.2.a.o.1.2	yes	6	4.3	odd	2
7616.2.a.bv.1.5		6	8.3	odd	2
7616.2.a.cd.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)

\(f(q)\)	\(=\)	\(q+0.463711 q^{3} -0.183002 q^{5} +1.00000 q^{7} -2.78497 q^{9} +4.93124 q^{11} -2.19585 q^{13} -0.0848598 q^{15} +1.00000 q^{17} -7.89068 q^{19} +0.463711 q^{21} +3.58009 q^{23} -4.96651 q^{25} -2.68255 q^{27} -2.53954 q^{29} -1.62496 q^{31} +2.28667 q^{33} -0.183002 q^{35} -9.29680 q^{37} -1.01824 q^{39} -0.851594 q^{41} +0.271676 q^{43} +0.509655 q^{45} -10.3374 q^{47} +1.00000 q^{49} +0.463711 q^{51} +13.0049 q^{53} -0.902425 q^{55} -3.65899 q^{57} +2.95312 q^{59} +5.62282 q^{61} -2.78497 q^{63} +0.401844 q^{65} -4.20488 q^{67} +1.66013 q^{69} +6.04234 q^{71} -13.9383 q^{73} -2.30302 q^{75} +4.93124 q^{77} -1.79400 q^{79} +7.11099 q^{81} -10.8384 q^{83} -0.183002 q^{85} -1.17761 q^{87} -1.13342 q^{89} -2.19585 q^{91} -0.753513 q^{93} +1.44401 q^{95} +6.87312 q^{97} -13.7334 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} - 4 q^{5} + 6 q^{7} + 8 q^{9} - 8 q^{11} - 4 q^{13} + 6 q^{17} - 18 q^{19} - 4 q^{21} - 6 q^{23} + 12 q^{25} - 10 q^{27} + 4 q^{29} + 8 q^{31} - 6 q^{33} - 4 q^{35} - 8 q^{37} + 18 q^{39}+ \cdots - 32 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 - 4 * q^5 + 6 * q^7 + 8 * q^9 - 8 * q^11 - 4 * q^13 + 6 * q^17 - 18 * q^19 - 4 * q^21 - 6 * q^23 + 12 * q^25 - 10 * q^27 + 4 * q^29 + 8 * q^31 - 6 * q^33 - 4 * q^35 - 8 * q^37 + 18 * q^39 - 4 * q^41 - 20 * q^43 - 24 * q^45 - 6 * q^47 + 6 * q^49 - 4 * q^51 + 14 * q^53 - 4 * q^55 - 4 * q^57 - 20 * q^59 + 2 * q^61 + 8 * q^63 + 6 * q^65 - 28 * q^67 - 4 * q^69 - 18 * q^71 + 4 * q^73 - 20 * q^75 - 8 * q^77 + 2 * q^79 + 34 * q^81 - 20 * q^83 - 4 * q^85 - 22 * q^87 - 6 * q^89 - 4 * q^91 + 40 * q^93 + 6 * q^95 + 30 * q^97 - 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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