LMFDB - Embedded newform 8004.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8004,2,Mod(1,8004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8004, 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("8004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8004.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:	$63.9122617778$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 5 x^{12} - 27 x^{11} + 158 x^{10} + 180 x^{9} - 1652 x^{8} + 65 x^{7} + 7388 x^{6} + \cdots - 5832 $ x^13 - 5x^12 - 27x^11 + 158x^10 + 180x^9 - 1652x^8 + 65x^7 + 7388x^6 - 3259x^5 - 15445x^4 + 7832x^3 + 15102x^2 - 5184x - 5832
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.03097$ of defining polynomial
Character	$\chi$	$=$	8004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -4.03097 q^{5} -3.90486 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -4.03097 q^{5} -3.90486 q^{7} +1.00000 q^{9} +4.96079 q^{11} +0.829895 q^{13} +4.03097 q^{15} +0.147001 q^{17} -2.16848 q^{19} +3.90486 q^{21} -1.00000 q^{23} +11.2487 q^{25} -1.00000 q^{27} -1.00000 q^{29} -9.11610 q^{31} -4.96079 q^{33} +15.7404 q^{35} +6.18358 q^{37} -0.829895 q^{39} -4.10485 q^{41} +1.53731 q^{43} -4.03097 q^{45} -6.98483 q^{47} +8.24795 q^{49} -0.147001 q^{51} -9.48267 q^{53} -19.9968 q^{55} +2.16848 q^{57} -4.01956 q^{59} -13.6833 q^{61} -3.90486 q^{63} -3.34528 q^{65} -6.37725 q^{67} +1.00000 q^{69} -7.54807 q^{71} -1.99373 q^{73} -11.2487 q^{75} -19.3712 q^{77} +13.3147 q^{79} +1.00000 q^{81} -15.7448 q^{83} -0.592557 q^{85} +1.00000 q^{87} +13.4691 q^{89} -3.24063 q^{91} +9.11610 q^{93} +8.74106 q^{95} -4.85389 q^{97} +4.96079 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9}+O(q^{10}) $ 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9	$ 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9} + q^{11} + q^{13} - 5 q^{15} - 2 q^{17} - 10 q^{19} + 8 q^{21} - 13 q^{23} + 14 q^{25} - 13 q^{27} - 13 q^{29} - 26 q^{31} - q^{33} + 19 q^{35} + 15 q^{37} - q^{39} + 21 q^{41} - 6 q^{43} + 5 q^{45} + 16 q^{47} + 19 q^{49} + 2 q^{51} + 7 q^{53} + 15 q^{55} + 10 q^{57} - 11 q^{59} + 19 q^{61} - 8 q^{63} + 6 q^{65} - 13 q^{67} + 13 q^{69} + 9 q^{73} - 14 q^{75} + 10 q^{77} - 25 q^{79} + 13 q^{81} + 3 q^{83} + 14 q^{85} + 13 q^{87} + 23 q^{89} + 19 q^{91} + 26 q^{93} + 7 q^{95} + 25 q^{97} + q^{99}+O(q^{100}) $ 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9 + q^11 + q^13 - 5 * q^15 - 2 * q^17 - 10 * q^19 + 8 * q^21 - 13 * q^23 + 14 * q^25 - 13 * q^27 - 13 * q^29 - 26 * q^31 - q^33 + 19 * q^35 + 15 * q^37 - q^39 + 21 * q^41 - 6 * q^43 + 5 * q^45 + 16 * q^47 + 19 * q^49 + 2 * q^51 + 7 * q^53 + 15 * q^55 + 10 * q^57 - 11 * q^59 + 19 * q^61 - 8 * q^63 + 6 * q^65 - 13 * q^67 + 13 * q^69 + 9 * q^73 - 14 * q^75 + 10 * q^77 - 25 * q^79 + 13 * q^81 + 3 * q^83 + 14 * q^85 + 13 * q^87 + 23 * q^89 + 19 * q^91 + 26 * q^93 + 7 * q^95 + 25 * q^97 + q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−4.03097	−1.80270	−0.901352	−	0.433087i	$-0.857424\pi$
$5$	−4.03097	−1.80270	−0.901352	+	0.433087i	$0.857424\pi$
$6$	0	0
$7$	−3.90486	−1.47590	−0.737950	−	0.674856i	$-0.764206\pi$
$7$	−3.90486	−1.47590	−0.737950	+	0.674856i	$0.764206\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.96079	1.49573	0.747867	−	0.663849i	$-0.231078\pi$
$11$	4.96079	1.49573	0.747867	+	0.663849i	$0.231078\pi$
$12$	0	0
$13$	0.829895	0.230172	0.115086	−	0.993356i	$-0.463286\pi$
$13$	0.829895	0.230172	0.115086	+	0.993356i	$0.463286\pi$
$14$	0	0
$15$	4.03097	1.04079
$16$	0	0
$17$	0.147001	0.0356530	0.0178265	−	0.999841i	$-0.494325\pi$
$17$	0.147001	0.0356530	0.0178265	+	0.999841i	$0.494325\pi$
$18$	0	0
$19$	−2.16848	−0.497482	−0.248741	−	0.968570i	$-0.580017\pi$
$19$	−2.16848	−0.497482	−0.248741	+	0.968570i	$0.580017\pi$
$20$	0	0
$21$	3.90486	0.852111
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	11.2487	2.24974
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−9.11610	−1.63730	−0.818650	−	0.574293i	$-0.805277\pi$
$31$	−9.11610	−1.63730	−0.818650	+	0.574293i	$0.805277\pi$
$32$	0	0
$33$	−4.96079	−0.863562
$34$	0	0
$35$	15.7404	2.66061
$36$	0	0
$37$	6.18358	1.01657	0.508287	−	0.861188i	$-0.330279\pi$
$37$	6.18358	1.01657	0.508287	+	0.861188i	$0.330279\pi$
$38$	0	0
$39$	−0.829895	−0.132890
$40$	0	0
$41$	−4.10485	−0.641071	−0.320535	−	0.947237i	$-0.603863\pi$
$41$	−4.10485	−0.641071	−0.320535	+	0.947237i	$0.603863\pi$
$42$	0	0
$43$	1.53731	0.234438	0.117219	−	0.993106i	$-0.462602\pi$
$43$	1.53731	0.234438	0.117219	+	0.993106i	$0.462602\pi$
$44$	0	0
$45$	−4.03097	−0.600901
$46$	0	0
$47$	−6.98483	−1.01884	−0.509421	−	0.860517i	$-0.670140\pi$
$47$	−6.98483	−1.01884	−0.509421	+	0.860517i	$0.670140\pi$
$48$	0	0
$49$	8.24795	1.17828
$50$	0	0
$51$	−0.147001	−0.0205843
$52$	0	0
$53$	−9.48267	−1.30254	−0.651272	−	0.758844i	$-0.725764\pi$
$53$	−9.48267	−1.30254	−0.651272	+	0.758844i	$0.725764\pi$
$54$	0	0
$55$	−19.9968	−2.69636
$56$	0	0
$57$	2.16848	0.287222
$58$	0	0
$59$	−4.01956	−0.523302	−0.261651	−	0.965163i	$-0.584267\pi$
$59$	−4.01956	−0.523302	−0.261651	+	0.965163i	$0.584267\pi$
$60$	0	0
$61$	−13.6833	−1.75197	−0.875985	−	0.482338i	$-0.839788\pi$
$61$	−13.6833	−1.75197	−0.875985	+	0.482338i	$0.839788\pi$
$62$	0	0
$63$	−3.90486	−0.491966
$64$	0	0
$65$	−3.34528	−0.414931
$66$	0	0
$67$	−6.37725	−0.779105	−0.389553	−	0.921004i	$-0.627370\pi$
$67$	−6.37725	−0.779105	−0.389553	+	0.921004i	$0.627370\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−7.54807	−0.895792	−0.447896	−	0.894086i	$-0.647826\pi$
$71$	−7.54807	−0.895792	−0.447896	+	0.894086i	$0.647826\pi$
$72$	0	0
$73$	−1.99373	−0.233348	−0.116674	−	0.993170i	$-0.537223\pi$
$73$	−1.99373	−0.233348	−0.116674	+	0.993170i	$0.537223\pi$
$74$	0	0
$75$	−11.2487	−1.29889
$76$	0	0
$77$	−19.3712	−2.20755
$78$	0	0
$79$	13.3147	1.49802	0.749008	−	0.662561i	$-0.230531\pi$
$79$	13.3147	1.49802	0.749008	+	0.662561i	$0.230531\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.7448	−1.72822	−0.864110	−	0.503303i	$-0.832118\pi$
$83$	−15.7448	−1.72822	−0.864110	+	0.503303i	$0.832118\pi$
$84$	0	0
$85$	−0.592557	−0.0642719
$86$	0	0
$87$	1.00000	0.107211
$88$	0	0
$89$	13.4691	1.42772	0.713861	−	0.700287i	$-0.246945\pi$
$89$	13.4691	1.42772	0.713861	+	0.700287i	$0.246945\pi$
$90$	0	0
$91$	−3.24063	−0.339710
$92$	0	0
$93$	9.11610	0.945296
$94$	0	0
$95$	8.74106	0.896813
$96$	0	0
$97$	−4.85389	−0.492838	−0.246419	−	0.969163i	$-0.579254\pi$
$97$	−4.85389	−0.492838	−0.246419	+	0.969163i	$0.579254\pi$
$98$	0	0
$99$	4.96079	0.498578
$100$	0	0
$101$	−1.52395	−0.151638	−0.0758192	−	0.997122i	$-0.524157\pi$
$101$	−1.52395	−0.151638	−0.0758192	+	0.997122i	$0.524157\pi$
$102$	0	0
$103$	6.81435	0.671438	0.335719	−	0.941962i	$-0.391021\pi$
$103$	6.81435	0.671438	0.335719	+	0.941962i	$0.391021\pi$
$104$	0	0
$105$	−15.7404	−1.53610
$106$	0	0
$107$	3.19721	0.309086	0.154543	−	0.987986i	$-0.450609\pi$
$107$	3.19721	0.309086	0.154543	+	0.987986i	$0.450609\pi$
$108$	0	0
$109$	−6.08806	−0.583130	−0.291565	−	0.956551i	$-0.594176\pi$
$109$	−6.08806	−0.583130	−0.291565	+	0.956551i	$0.594176\pi$
$110$	0	0
$111$	−6.18358	−0.586919
$112$	0	0
$113$	17.0154	1.60068	0.800339	−	0.599547i	$-0.204653\pi$
$113$	17.0154	1.60068	0.800339	+	0.599547i	$0.204653\pi$
$114$	0	0
$115$	4.03097	0.375890
$116$	0	0
$117$	0.829895	0.0767239
$118$	0	0
$119$	−0.574020	−0.0526203
$120$	0	0
$121$	13.6094	1.23722
$122$	0	0
$123$	4.10485	0.370122
$124$	0	0
$125$	−25.1883	−2.25291
$126$	0	0
$127$	−5.07441	−0.450281	−0.225141	−	0.974326i	$-0.572284\pi$
$127$	−5.07441	−0.450281	−0.225141	+	0.974326i	$0.572284\pi$
$128$	0	0
$129$	−1.53731	−0.135353
$130$	0	0
$131$	15.2269	1.33038	0.665192	−	0.746672i	$-0.268350\pi$
$131$	15.2269	1.33038	0.665192	+	0.746672i	$0.268350\pi$
$132$	0	0
$133$	8.46760	0.734234
$134$	0	0
$135$	4.03097	0.346931
$136$	0	0
$137$	−18.7628	−1.60302	−0.801509	−	0.597983i	$-0.795969\pi$
$137$	−18.7628	−1.60302	−0.801509	+	0.597983i	$0.795969\pi$
$138$	0	0
$139$	5.21984	0.442741	0.221370	−	0.975190i	$-0.428947\pi$
$139$	5.21984	0.442741	0.221370	+	0.975190i	$0.428947\pi$
$140$	0	0
$141$	6.98483	0.588229
$142$	0	0
$143$	4.11693	0.344275
$144$	0	0
$145$	4.03097	0.334754
$146$	0	0
$147$	−8.24795	−0.680279
$148$	0	0
$149$	6.87639	0.563336	0.281668	−	0.959512i	$-0.409112\pi$
$149$	6.87639	0.563336	0.281668	+	0.959512i	$0.409112\pi$
$150$	0	0
$151$	−18.3390	−1.49241	−0.746203	−	0.665718i	$-0.768125\pi$
$151$	−18.3390	−1.49241	−0.746203	+	0.665718i	$0.768125\pi$
$152$	0	0
$153$	0.147001	0.0118843
$154$	0	0
$155$	36.7467	2.95157
$156$	0	0
$157$	−10.9650	−0.875105	−0.437553	−	0.899193i	$-0.644155\pi$
$157$	−10.9650	−0.875105	−0.437553	+	0.899193i	$0.644155\pi$
$158$	0	0
$159$	9.48267	0.752025
$160$	0	0
$161$	3.90486	0.307746
$162$	0	0
$163$	−13.9769	−1.09476	−0.547379	−	0.836885i	$-0.684374\pi$
$163$	−13.9769	−1.09476	−0.547379	+	0.836885i	$0.684374\pi$
$164$	0	0
$165$	19.9968	1.55675
$166$	0	0
$167$	4.64616	0.359531	0.179765	−	0.983710i	$-0.442466\pi$
$167$	4.64616	0.359531	0.179765	+	0.983710i	$0.442466\pi$
$168$	0	0
$169$	−12.3113	−0.947021
$170$	0	0
$171$	−2.16848	−0.165827
$172$	0	0
$173$	−9.16388	−0.696717	−0.348358	−	0.937361i	$-0.613261\pi$
$173$	−9.16388	−0.696717	−0.348358	+	0.937361i	$0.613261\pi$
$174$	0	0
$175$	−43.9246	−3.32039
$176$	0	0
$177$	4.01956	0.302129
$178$	0	0
$179$	11.7124	0.875426	0.437713	−	0.899115i	$-0.355789\pi$
$179$	11.7124	0.875426	0.437713	+	0.899115i	$0.355789\pi$
$180$	0	0
$181$	−6.52725	−0.485167	−0.242583	−	0.970131i	$-0.577995\pi$
$181$	−6.52725	−0.485167	−0.242583	+	0.970131i	$0.577995\pi$
$182$	0	0
$183$	13.6833	1.01150
$184$	0	0
$185$	−24.9258	−1.83258
$186$	0	0
$187$	0.729242	0.0533275
$188$	0	0
$189$	3.90486	0.284037
$190$	0	0
$191$	−4.40460	−0.318706	−0.159353	−	0.987222i	$-0.550941\pi$
$191$	−4.40460	−0.318706	−0.159353	+	0.987222i	$0.550941\pi$
$192$	0	0
$193$	9.96239	0.717108	0.358554	−	0.933509i	$-0.383270\pi$
$193$	9.96239	0.717108	0.358554	+	0.933509i	$0.383270\pi$
$194$	0	0
$195$	3.34528	0.239561
$196$	0	0
$197$	7.93648	0.565451	0.282725	−	0.959201i	$-0.408762\pi$
$197$	7.93648	0.565451	0.282725	+	0.959201i	$0.408762\pi$
$198$	0	0
$199$	−3.76032	−0.266562	−0.133281	−	0.991078i	$-0.542551\pi$
$199$	−3.76032	−0.266562	−0.133281	+	0.991078i	$0.542551\pi$
$200$	0	0
$201$	6.37725	0.449817
$202$	0	0
$203$	3.90486	0.274068
$204$	0	0
$205$	16.5465	1.15566
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−10.7573	−0.744101
$210$	0	0
$211$	10.2194	0.703533	0.351766	−	0.936088i	$-0.385581\pi$
$211$	10.2194	0.703533	0.351766	+	0.936088i	$0.385581\pi$
$212$	0	0
$213$	7.54807	0.517186
$214$	0	0
$215$	−6.19685	−0.422622
$216$	0	0
$217$	35.5971	2.41649
$218$	0	0
$219$	1.99373	0.134724
$220$	0	0
$221$	0.121996	0.00820632
$222$	0	0
$223$	−12.1127	−0.811126	−0.405563	−	0.914067i	$-0.632925\pi$
$223$	−12.1127	−0.811126	−0.405563	+	0.914067i	$0.632925\pi$
$224$	0	0
$225$	11.2487	0.749914
$226$	0	0
$227$	−14.9527	−0.992447	−0.496224	−	0.868195i	$-0.665280\pi$
$227$	−14.9527	−0.992447	−0.496224	+	0.868195i	$0.665280\pi$
$228$	0	0
$229$	−8.53624	−0.564090	−0.282045	−	0.959401i	$-0.591013\pi$
$229$	−8.53624	−0.564090	−0.282045	+	0.959401i	$0.591013\pi$
$230$	0	0
$231$	19.3712	1.27453
$232$	0	0
$233$	−25.5531	−1.67404	−0.837020	−	0.547172i	$-0.815704\pi$
$233$	−25.5531	−1.67404	−0.837020	+	0.547172i	$0.815704\pi$
$234$	0	0
$235$	28.1556	1.83667
$236$	0	0
$237$	−13.3147	−0.864880
$238$	0	0
$239$	20.3948	1.31923	0.659614	−	0.751604i	$-0.270720\pi$
$239$	20.3948	1.31923	0.659614	+	0.751604i	$0.270720\pi$
$240$	0	0
$241$	26.1445	1.68412	0.842058	−	0.539388i	$-0.181344\pi$
$241$	26.1445	1.68412	0.842058	+	0.539388i	$0.181344\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−33.2472	−2.12409
$246$	0	0
$247$	−1.79961	−0.114506
$248$	0	0
$249$	15.7448	0.997788
$250$	0	0
$251$	17.1143	1.08024	0.540122	−	0.841587i	$-0.318378\pi$
$251$	17.1143	1.08024	0.540122	+	0.841587i	$0.318378\pi$
$252$	0	0
$253$	−4.96079	−0.311882
$254$	0	0
$255$	0.592557	0.0371074
$256$	0	0
$257$	25.1527	1.56898	0.784490	−	0.620141i	$-0.212925\pi$
$257$	25.1527	1.56898	0.784490	+	0.620141i	$0.212925\pi$
$258$	0	0
$259$	−24.1460	−1.50036
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	13.4949	0.832130	0.416065	−	0.909335i	$-0.363409\pi$
$263$	13.4949	0.832130	0.416065	+	0.909335i	$0.363409\pi$
$264$	0	0
$265$	38.2243	2.34810
$266$	0	0
$267$	−13.4691	−0.824296
$268$	0	0
$269$	29.9542	1.82634	0.913169	−	0.407581i	$-0.133628\pi$
$269$	29.9542	1.82634	0.913169	+	0.407581i	$0.133628\pi$
$270$	0	0
$271$	−1.86261	−0.113146	−0.0565729	−	0.998398i	$-0.518017\pi$
$271$	−1.86261	−0.113146	−0.0565729	+	0.998398i	$0.518017\pi$
$272$	0	0
$273$	3.24063	0.196132
$274$	0	0
$275$	55.8024	3.36501
$276$	0	0
$277$	29.8432	1.79310	0.896551	−	0.442941i	$-0.146065\pi$
$277$	29.8432	1.79310	0.896551	+	0.442941i	$0.146065\pi$
$278$	0	0
$279$	−9.11610	−0.545767
$280$	0	0
$281$	8.73980	0.521373	0.260686	−	0.965424i	$-0.416051\pi$
$281$	8.73980	0.521373	0.260686	+	0.965424i	$0.416051\pi$
$282$	0	0
$283$	17.0153	1.01146	0.505728	−	0.862693i	$-0.331224\pi$
$283$	17.0153	1.01146	0.505728	+	0.862693i	$0.331224\pi$
$284$	0	0
$285$	−8.74106	−0.517775
$286$	0	0
$287$	16.0289	0.946155
$288$	0	0
$289$	−16.9784	−0.998729
$290$	0	0
$291$	4.85389	0.284540
$292$	0	0
$293$	10.4295	0.609300	0.304650	−	0.952464i	$-0.401461\pi$
$293$	10.4295	0.609300	0.304650	+	0.952464i	$0.401461\pi$
$294$	0	0
$295$	16.2027	0.943359
$296$	0	0
$297$	−4.96079	−0.287854
$298$	0	0
$299$	−0.829895	−0.0479941
$300$	0	0
$301$	−6.00299	−0.346006
$302$	0	0
$303$	1.52395	0.0875484
$304$	0	0
$305$	55.1571	3.15828
$306$	0	0
$307$	−32.9207	−1.87889	−0.939443	−	0.342706i	$-0.888657\pi$
$307$	−32.9207	−1.87889	−0.939443	+	0.342706i	$0.888657\pi$
$308$	0	0
$309$	−6.81435	−0.387655
$310$	0	0
$311$	10.5860	0.600279	0.300139	−	0.953895i	$-0.402967\pi$
$311$	10.5860	0.600279	0.300139	+	0.953895i	$0.402967\pi$
$312$	0	0
$313$	−6.05323	−0.342149	−0.171075	−	0.985258i	$-0.554724\pi$
$313$	−6.05323	−0.342149	−0.171075	+	0.985258i	$0.554724\pi$
$314$	0	0
$315$	15.7404	0.886870
$316$	0	0
$317$	−7.74177	−0.434821	−0.217410	−	0.976080i	$-0.569761\pi$
$317$	−7.74177	−0.434821	−0.217410	+	0.976080i	$0.569761\pi$
$318$	0	0
$319$	−4.96079	−0.277751
$320$	0	0
$321$	−3.19721	−0.178451
$322$	0	0
$323$	−0.318769	−0.0177368
$324$	0	0
$325$	9.33525	0.517826
$326$	0	0
$327$	6.08806	0.336670
$328$	0	0
$329$	27.2748	1.50371
$330$	0	0
$331$	3.41328	0.187611	0.0938053	−	0.995591i	$-0.470097\pi$
$331$	3.41328	0.187611	0.0938053	+	0.995591i	$0.470097\pi$
$332$	0	0
$333$	6.18358	0.338858
$334$	0	0
$335$	25.7065	1.40450
$336$	0	0
$337$	1.64344	0.0895237	0.0447618	−	0.998998i	$-0.485747\pi$
$337$	1.64344	0.0895237	0.0447618	+	0.998998i	$0.485747\pi$
$338$	0	0
$339$	−17.0154	−0.924152
$340$	0	0
$341$	−45.2230	−2.44897
$342$	0	0
$343$	−4.87307	−0.263121
$344$	0	0
$345$	−4.03097	−0.217020
$346$	0	0
$347$	11.7854	0.632674	0.316337	−	0.948647i	$-0.397547\pi$
$347$	11.7854	0.632674	0.316337	+	0.948647i	$0.397547\pi$
$348$	0	0
$349$	−5.26298	−0.281721	−0.140860	−	0.990029i	$-0.544987\pi$
$349$	−5.26298	−0.281721	−0.140860	+	0.990029i	$0.544987\pi$
$350$	0	0
$351$	−0.829895	−0.0442965
$352$	0	0
$353$	11.2153	0.596932	0.298466	−	0.954420i	$-0.403525\pi$
$353$	11.2153	0.596932	0.298466	+	0.954420i	$0.403525\pi$
$354$	0	0
$355$	30.4260	1.61485
$356$	0	0
$357$	0.574020	0.0303803
$358$	0	0
$359$	20.7645	1.09591	0.547953	−	0.836509i	$-0.315407\pi$
$359$	20.7645	1.09591	0.547953	+	0.836509i	$0.315407\pi$
$360$	0	0
$361$	−14.2977	−0.752511
$362$	0	0
$363$	−13.6094	−0.714309
$364$	0	0
$365$	8.03665	0.420657
$366$	0	0
$367$	−22.6854	−1.18417	−0.592085	−	0.805875i	$-0.701695\pi$
$367$	−22.6854	−1.18417	−0.592085	+	0.805875i	$0.701695\pi$
$368$	0	0
$369$	−4.10485	−0.213690
$370$	0	0
$371$	37.0285	1.92243
$372$	0	0
$373$	7.31491	0.378751	0.189376	−	0.981905i	$-0.439354\pi$
$373$	7.31491	0.378751	0.189376	+	0.981905i	$0.439354\pi$
$374$	0	0
$375$	25.1883	1.30072
$376$	0	0
$377$	−0.829895	−0.0427418
$378$	0	0
$379$	−24.9633	−1.28228	−0.641140	−	0.767424i	$-0.721538\pi$
$379$	−24.9633	−1.28228	−0.641140	+	0.767424i	$0.721538\pi$
$380$	0	0
$381$	5.07441	0.259970
$382$	0	0
$383$	−5.87282	−0.300087	−0.150043	−	0.988679i	$-0.547941\pi$
$383$	−5.87282	−0.300087	−0.150043	+	0.988679i	$0.547941\pi$
$384$	0	0
$385$	78.0847	3.97956
$386$	0	0
$387$	1.53731	0.0781459
$388$	0	0
$389$	−1.89909	−0.0962878	−0.0481439	−	0.998840i	$-0.515331\pi$
$389$	−1.89909	−0.0962878	−0.0481439	+	0.998840i	$0.515331\pi$
$390$	0	0
$391$	−0.147001	−0.00743417
$392$	0	0
$393$	−15.2269	−0.768098
$394$	0	0
$395$	−53.6709	−2.70048
$396$	0	0
$397$	0.774834	0.0388878	0.0194439	−	0.999811i	$-0.493810\pi$
$397$	0.774834	0.0388878	0.0194439	+	0.999811i	$0.493810\pi$
$398$	0	0
$399$	−8.46760	−0.423910
$400$	0	0
$401$	2.41544	0.120621	0.0603106	−	0.998180i	$-0.480791\pi$
$401$	2.41544	0.120621	0.0603106	+	0.998180i	$0.480791\pi$
$402$	0	0
$403$	−7.56541	−0.376860
$404$	0	0
$405$	−4.03097	−0.200300
$406$	0	0
$407$	30.6754	1.52052
$408$	0	0
$409$	11.8714	0.587004	0.293502	−	0.955959i	$-0.405179\pi$
$409$	11.8714	0.587004	0.293502	+	0.955959i	$0.405179\pi$
$410$	0	0
$411$	18.7628	0.925502
$412$	0	0
$413$	15.6958	0.772341
$414$	0	0
$415$	63.4669	3.11547
$416$	0	0
$417$	−5.21984	−0.255617
$418$	0	0
$419$	18.3072	0.894364	0.447182	−	0.894443i	$-0.352428\pi$
$419$	18.3072	0.894364	0.447182	+	0.894443i	$0.352428\pi$
$420$	0	0
$421$	12.5315	0.610749	0.305375	−	0.952232i	$-0.401218\pi$
$421$	12.5315	0.610749	0.305375	+	0.952232i	$0.401218\pi$
$422$	0	0
$423$	−6.98483	−0.339614
$424$	0	0
$425$	1.65357	0.0802101
$426$	0	0
$427$	53.4315	2.58573
$428$	0	0
$429$	−4.11693	−0.198767
$430$	0	0
$431$	10.6824	0.514555	0.257277	−	0.966338i	$-0.417175\pi$
$431$	10.6824	0.514555	0.257277	+	0.966338i	$0.417175\pi$
$432$	0	0
$433$	0.573366	0.0275542	0.0137771	−	0.999905i	$-0.495614\pi$
$433$	0.573366	0.0275542	0.0137771	+	0.999905i	$0.495614\pi$
$434$	0	0
$435$	−4.03097	−0.193270
$436$	0	0
$437$	2.16848	0.103732
$438$	0	0
$439$	−38.2927	−1.82761	−0.913805	−	0.406153i	$-0.866870\pi$
$439$	−38.2927	−1.82761	−0.913805	+	0.406153i	$0.866870\pi$
$440$	0	0
$441$	8.24795	0.392759
$442$	0	0
$443$	−21.2516	−1.00969	−0.504847	−	0.863209i	$-0.668451\pi$
$443$	−21.2516	−1.00969	−0.504847	+	0.863209i	$0.668451\pi$
$444$	0	0
$445$	−54.2935	−2.57376
$446$	0	0
$447$	−6.87639	−0.325242
$448$	0	0
$449$	−15.1730	−0.716060	−0.358030	−	0.933710i	$-0.616552\pi$
$449$	−15.1730	−0.716060	−0.358030	+	0.933710i	$0.616552\pi$
$450$	0	0
$451$	−20.3633	−0.958871
$452$	0	0
$453$	18.3390	0.861641
$454$	0	0
$455$	13.0629	0.612397
$456$	0	0
$457$	−30.8406	−1.44266	−0.721331	−	0.692591i	$-0.756469\pi$
$457$	−30.8406	−1.44266	−0.721331	+	0.692591i	$0.756469\pi$
$458$	0	0
$459$	−0.147001	−0.00686143
$460$	0	0
$461$	21.2697	0.990630	0.495315	−	0.868714i	$-0.335053\pi$
$461$	21.2697	0.990630	0.495315	+	0.868714i	$0.335053\pi$
$462$	0	0
$463$	1.24681	0.0579441	0.0289721	−	0.999580i	$-0.490777\pi$
$463$	1.24681	0.0579441	0.0289721	+	0.999580i	$0.490777\pi$
$464$	0	0
$465$	−36.7467	−1.70409
$466$	0	0
$467$	30.2601	1.40027	0.700134	−	0.714011i	$-0.253123\pi$
$467$	30.2601	1.40027	0.700134	+	0.714011i	$0.253123\pi$
$468$	0	0
$469$	24.9023	1.14988
$470$	0	0
$471$	10.9650	0.505242
$472$	0	0
$473$	7.62627	0.350656
$474$	0	0
$475$	−24.3925	−1.11921
$476$	0	0
$477$	−9.48267	−0.434182
$478$	0	0
$479$	−36.2396	−1.65583	−0.827915	−	0.560854i	$-0.810473\pi$
$479$	−36.2396	−1.65583	−0.827915	+	0.560854i	$0.810473\pi$
$480$	0	0
$481$	5.13172	0.233986
$482$	0	0
$483$	−3.90486	−0.177677
$484$	0	0
$485$	19.5659	0.888441
$486$	0	0
$487$	−3.03406	−0.137487	−0.0687433	−	0.997634i	$-0.521899\pi$
$487$	−3.03406	−0.137487	−0.0687433	+	0.997634i	$0.521899\pi$
$488$	0	0
$489$	13.9769	0.632059
$490$	0	0
$491$	−11.9779	−0.540554	−0.270277	−	0.962783i	$-0.587115\pi$
$491$	−11.9779	−0.540554	−0.270277	+	0.962783i	$0.587115\pi$
$492$	0	0
$493$	−0.147001	−0.00662060
$494$	0	0
$495$	−19.9968	−0.898788
$496$	0	0
$497$	29.4742	1.32210
$498$	0	0
$499$	−22.4638	−1.00562	−0.502809	−	0.864398i	$-0.667700\pi$
$499$	−22.4638	−1.00562	−0.502809	+	0.864398i	$0.667700\pi$
$500$	0	0
$501$	−4.64616	−0.207575
$502$	0	0
$503$	10.4505	0.465967	0.232983	−	0.972481i	$-0.425151\pi$
$503$	10.4505	0.465967	0.232983	+	0.972481i	$0.425151\pi$
$504$	0	0
$505$	6.14298	0.273359
$506$	0	0
$507$	12.3113	0.546763
$508$	0	0
$509$	28.5002	1.26325	0.631624	−	0.775275i	$-0.282389\pi$
$509$	28.5002	1.26325	0.631624	+	0.775275i	$0.282389\pi$
$510$	0	0
$511$	7.78523	0.344398
$512$	0	0
$513$	2.16848	0.0957405
$514$	0	0
$515$	−27.4684	−1.21040
$516$	0	0
$517$	−34.6503	−1.52392
$518$	0	0
$519$	9.16388	0.402250
$520$	0	0
$521$	38.9395	1.70597	0.852985	−	0.521935i	$-0.174790\pi$
$521$	38.9395	1.70597	0.852985	+	0.521935i	$0.174790\pi$
$522$	0	0
$523$	−10.9394	−0.478348	−0.239174	−	0.970977i	$-0.576877\pi$
$523$	−10.9394	−0.478348	−0.239174	+	0.970977i	$0.576877\pi$
$524$	0	0
$525$	43.9246	1.91703
$526$	0	0
$527$	−1.34008	−0.0583747
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−4.01956	−0.174434
$532$	0	0
$533$	−3.40660	−0.147556
$534$	0	0
$535$	−12.8879	−0.557191
$536$	0	0
$537$	−11.7124	−0.505427
$538$	0	0
$539$	40.9163	1.76239
$540$	0	0
$541$	22.2251	0.955532	0.477766	−	0.878487i	$-0.341447\pi$
$541$	22.2251	0.955532	0.477766	+	0.878487i	$0.341447\pi$
$542$	0	0
$543$	6.52725	0.280111
$544$	0	0
$545$	24.5408	1.05121
$546$	0	0
$547$	−0.560610	−0.0239700	−0.0119850	−	0.999928i	$-0.503815\pi$
$547$	−0.560610	−0.0239700	−0.0119850	+	0.999928i	$0.503815\pi$
$548$	0	0
$549$	−13.6833	−0.583990
$550$	0	0
$551$	2.16848	0.0923802
$552$	0	0
$553$	−51.9919	−2.21092
$554$	0	0
$555$	24.9258	1.05804
$556$	0	0
$557$	37.7776	1.60069	0.800344	−	0.599542i	$-0.204650\pi$
$557$	37.7776	1.60069	0.800344	+	0.599542i	$0.204650\pi$
$558$	0	0
$559$	1.27581	0.0539609
$560$	0	0
$561$	−0.729242	−0.0307886
$562$	0	0
$563$	−32.6315	−1.37525	−0.687627	−	0.726064i	$-0.741348\pi$
$563$	−32.6315	−1.37525	−0.687627	+	0.726064i	$0.741348\pi$
$564$	0	0
$565$	−68.5887	−2.88555
$566$	0	0
$567$	−3.90486	−0.163989
$568$	0	0
$569$	−14.7589	−0.618724	−0.309362	−	0.950944i	$-0.600115\pi$
$569$	−14.7589	−0.618724	−0.309362	+	0.950944i	$0.600115\pi$
$570$	0	0
$571$	42.5411	1.78029	0.890146	−	0.455676i	$-0.150602\pi$
$571$	42.5411	1.78029	0.890146	+	0.455676i	$0.150602\pi$
$572$	0	0
$573$	4.40460	0.184005
$574$	0	0
$575$	−11.2487	−0.469103
$576$	0	0
$577$	−15.6892	−0.653148	−0.326574	−	0.945172i	$-0.605894\pi$
$577$	−15.6892	−0.653148	−0.326574	+	0.945172i	$0.605894\pi$
$578$	0	0
$579$	−9.96239	−0.414023
$580$	0	0
$581$	61.4814	2.55068
$582$	0	0
$583$	−47.0415	−1.94826
$584$	0	0
$585$	−3.34528	−0.138310
$586$	0	0
$587$	23.8218	0.983229	0.491615	−	0.870813i	$-0.336407\pi$
$587$	23.8218	0.983229	0.491615	+	0.870813i	$0.336407\pi$
$588$	0	0
$589$	19.7680	0.814528
$590$	0	0
$591$	−7.93648	−0.326463
$592$	0	0
$593$	−34.8866	−1.43262	−0.716310	−	0.697782i	$-0.754171\pi$
$593$	−34.8866	−1.43262	−0.716310	+	0.697782i	$0.754171\pi$
$594$	0	0
$595$	2.31385	0.0948588
$596$	0	0
$597$	3.76032	0.153900
$598$	0	0
$599$	15.3555	0.627409	0.313705	−	0.949521i	$-0.398430\pi$
$599$	15.3555	0.627409	0.313705	+	0.949521i	$0.398430\pi$
$600$	0	0
$601$	5.80170	0.236656	0.118328	−	0.992975i	$-0.462247\pi$
$601$	5.80170	0.236656	0.118328	+	0.992975i	$0.462247\pi$
$602$	0	0
$603$	−6.37725	−0.259702
$604$	0	0
$605$	−54.8591	−2.23034
$606$	0	0
$607$	−28.0828	−1.13985	−0.569923	−	0.821698i	$-0.693027\pi$
$607$	−28.0828	−1.13985	−0.569923	+	0.821698i	$0.693027\pi$
$608$	0	0
$609$	−3.90486	−0.158233
$610$	0	0
$611$	−5.79668	−0.234509
$612$	0	0
$613$	9.45164	0.381748	0.190874	−	0.981615i	$-0.438868\pi$
$613$	9.45164	0.381748	0.190874	+	0.981615i	$0.438868\pi$
$614$	0	0
$615$	−16.5465	−0.667221
$616$	0	0
$617$	34.7059	1.39721	0.698603	−	0.715510i	$-0.253806\pi$
$617$	34.7059	1.39721	0.698603	+	0.715510i	$0.253806\pi$
$618$	0	0
$619$	−40.7134	−1.63641	−0.818206	−	0.574926i	$-0.805031\pi$
$619$	−40.7134	−1.63641	−0.818206	+	0.574926i	$0.805031\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−52.5950	−2.10717
$624$	0	0
$625$	45.2898	1.81159
$626$	0	0
$627$	10.7573	0.429607
$628$	0	0
$629$	0.908993	0.0362439
$630$	0	0
$631$	33.5231	1.33453	0.667267	−	0.744818i	$-0.267464\pi$
$631$	33.5231	1.33453	0.667267	+	0.744818i	$0.267464\pi$
$632$	0	0
$633$	−10.2194	−0.406185
$634$	0	0
$635$	20.4548	0.811724
$636$	0	0
$637$	6.84494	0.271206
$638$	0	0
$639$	−7.54807	−0.298597
$640$	0	0
$641$	6.84476	0.270352	0.135176	−	0.990822i	$-0.456840\pi$
$641$	6.84476	0.270352	0.135176	+	0.990822i	$0.456840\pi$
$642$	0	0
$643$	32.8310	1.29473	0.647363	−	0.762181i	$-0.275872\pi$
$643$	32.8310	1.29473	0.647363	+	0.762181i	$0.275872\pi$
$644$	0	0
$645$	6.19685	0.244001
$646$	0	0
$647$	15.0945	0.593427	0.296714	−	0.954966i	$-0.404109\pi$
$647$	15.0945	0.593427	0.296714	+	0.954966i	$0.404109\pi$
$648$	0	0
$649$	−19.9402	−0.782721
$650$	0	0
$651$	−35.5971	−1.39516
$652$	0	0
$653$	40.5264	1.58592	0.792961	−	0.609272i	$-0.208538\pi$
$653$	40.5264	1.58592	0.792961	+	0.609272i	$0.208538\pi$
$654$	0	0
$655$	−61.3793	−2.39829
$656$	0	0
$657$	−1.99373	−0.0777827
$658$	0	0
$659$	22.2279	0.865875	0.432938	−	0.901424i	$-0.357477\pi$
$659$	22.2279	0.865875	0.432938	+	0.901424i	$0.357477\pi$
$660$	0	0
$661$	−16.2576	−0.632348	−0.316174	−	0.948701i	$-0.602398\pi$
$661$	−16.2576	−0.632348	−0.316174	+	0.948701i	$0.602398\pi$
$662$	0	0
$663$	−0.121996	−0.00473792
$664$	0	0
$665$	−34.1326	−1.32361
$666$	0	0
$667$	1.00000	0.0387202
$668$	0	0
$669$	12.1127	0.468304
$670$	0	0
$671$	−67.8801	−2.62048
$672$	0	0
$673$	−18.4106	−0.709675	−0.354838	−	0.934928i	$-0.615464\pi$
$673$	−18.4106	−0.709675	−0.354838	+	0.934928i	$0.615464\pi$
$674$	0	0
$675$	−11.2487	−0.432963
$676$	0	0
$677$	34.3896	1.32170	0.660850	−	0.750518i	$-0.270196\pi$
$677$	34.3896	1.32170	0.660850	+	0.750518i	$0.270196\pi$
$678$	0	0
$679$	18.9538	0.727379
$680$	0	0
$681$	14.9527	0.572990
$682$	0	0
$683$	−34.5638	−1.32255	−0.661274	−	0.750144i	$-0.729984\pi$
$683$	−34.5638	−1.32255	−0.661274	+	0.750144i	$0.729984\pi$
$684$	0	0
$685$	75.6324	2.88976
$686$	0	0
$687$	8.53624	0.325678
$688$	0	0
$689$	−7.86962	−0.299809
$690$	0	0
$691$	1.08087	0.0411181	0.0205591	−	0.999789i	$-0.493455\pi$
$691$	1.08087	0.0411181	0.0205591	+	0.999789i	$0.493455\pi$
$692$	0	0
$693$	−19.3712	−0.735851
$694$	0	0
$695$	−21.0410	−0.798131
$696$	0	0
$697$	−0.603419	−0.0228561
$698$	0	0
$699$	25.5531	0.966508
$700$	0	0
$701$	−41.2475	−1.55790	−0.778948	−	0.627088i	$-0.784247\pi$
$701$	−41.2475	−1.55790	−0.778948	+	0.627088i	$0.784247\pi$
$702$	0	0
$703$	−13.4089	−0.505727
$704$	0	0
$705$	−28.1556	−1.06040
$706$	0	0
$707$	5.95080	0.223803
$708$	0	0
$709$	41.4970	1.55845	0.779227	−	0.626742i	$-0.215612\pi$
$709$	41.4970	1.55845	0.779227	+	0.626742i	$0.215612\pi$
$710$	0	0
$711$	13.3147	0.499339
$712$	0	0
$713$	9.11610	0.341401
$714$	0	0
$715$	−16.5952	−0.620627
$716$	0	0
$717$	−20.3948	−0.761657
$718$	0	0
$719$	−26.6451	−0.993696	−0.496848	−	0.867838i	$-0.665509\pi$
$719$	−26.6451	−0.993696	−0.496848	+	0.867838i	$0.665509\pi$
$720$	0	0
$721$	−26.6091	−0.990975
$722$	0	0
$723$	−26.1445	−0.972324
$724$	0	0
$725$	−11.2487	−0.417766
$726$	0	0
$727$	−1.12096	−0.0415742	−0.0207871	−	0.999784i	$-0.506617\pi$
$727$	−1.12096	−0.0415742	−0.0207871	+	0.999784i	$0.506617\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.225987	0.00835842
$732$	0	0
$733$	21.9912	0.812265	0.406132	−	0.913814i	$-0.366877\pi$
$733$	21.9912	0.812265	0.406132	+	0.913814i	$0.366877\pi$
$734$	0	0
$735$	33.2472	1.22634
$736$	0	0
$737$	−31.6362	−1.16533
$738$	0	0
$739$	−16.0801	−0.591515	−0.295757	−	0.955263i	$-0.595572\pi$
$739$	−16.0801	−0.591515	−0.295757	+	0.955263i	$0.595572\pi$
$740$	0	0
$741$	1.79961	0.0661103
$742$	0	0
$743$	−16.2773	−0.597156	−0.298578	−	0.954385i	$-0.596512\pi$
$743$	−16.2773	−0.597156	−0.298578	+	0.954385i	$0.596512\pi$
$744$	0	0
$745$	−27.7185	−1.01553
$746$	0	0
$747$	−15.7448	−0.576073
$748$	0	0
$749$	−12.4847	−0.456180
$750$	0	0
$751$	−35.5283	−1.29645	−0.648223	−	0.761450i	$-0.724488\pi$
$751$	−35.5283	−1.29645	−0.648223	+	0.761450i	$0.724488\pi$
$752$	0	0
$753$	−17.1143	−0.623679
$754$	0	0
$755$	73.9239	2.69037
$756$	0	0
$757$	35.3643	1.28534	0.642669	−	0.766144i	$-0.277827\pi$
$757$	35.3643	1.28534	0.642669	+	0.766144i	$0.277827\pi$
$758$	0	0
$759$	4.96079	0.180065
$760$	0	0
$761$	−6.67008	−0.241790	−0.120895	−	0.992665i	$-0.538576\pi$
$761$	−6.67008	−0.241790	−0.120895	+	0.992665i	$0.538576\pi$
$762$	0	0
$763$	23.7730	0.860641
$764$	0	0
$765$	−0.592557	−0.0214240
$766$	0	0
$767$	−3.33582	−0.120449
$768$	0	0
$769$	−5.37674	−0.193890	−0.0969451	−	0.995290i	$-0.530907\pi$
$769$	−5.37674	−0.193890	−0.0969451	+	0.995290i	$0.530907\pi$
$770$	0	0
$771$	−25.1527	−0.905852
$772$	0	0
$773$	−53.1000	−1.90988	−0.954938	−	0.296805i	$-0.904079\pi$
$773$	−53.1000	−1.90988	−0.954938	+	0.296805i	$0.904079\pi$
$774$	0	0
$775$	−102.544	−3.68350
$776$	0	0
$777$	24.1460	0.866233
$778$	0	0
$779$	8.90128	0.318921
$780$	0	0
$781$	−37.4444	−1.33987
$782$	0	0
$783$	1.00000	0.0357371
$784$	0	0
$785$	44.1997	1.57756
$786$	0	0
$787$	9.33014	0.332584	0.166292	−	0.986077i	$-0.446821\pi$
$787$	9.33014	0.332584	0.166292	+	0.986077i	$0.446821\pi$
$788$	0	0
$789$	−13.4949	−0.480431
$790$	0	0
$791$	−66.4430	−2.36244
$792$	0	0
$793$	−11.3557	−0.403254
$794$	0	0
$795$	−38.2243	−1.35568
$796$	0	0
$797$	34.2215	1.21219	0.606094	−	0.795393i	$-0.292735\pi$
$797$	34.2215	1.21219	0.606094	+	0.795393i	$0.292735\pi$
$798$	0	0
$799$	−1.02678	−0.0363248
$800$	0	0
$801$	13.4691	0.475908
$802$	0	0
$803$	−9.89045	−0.349027
$804$	0	0
$805$	−15.7404	−0.554775
$806$	0	0
$807$	−29.9542	−1.05444
$808$	0	0
$809$	−28.8744	−1.01517	−0.507585	−	0.861601i	$-0.669462\pi$
$809$	−28.8744	−1.01517	−0.507585	+	0.861601i	$0.669462\pi$
$810$	0	0
$811$	24.1457	0.847871	0.423935	−	0.905692i	$-0.360648\pi$
$811$	24.1457	0.847871	0.423935	+	0.905692i	$0.360648\pi$
$812$	0	0
$813$	1.86261	0.0653247
$814$	0	0
$815$	56.3406	1.97352
$816$	0	0
$817$	−3.33362	−0.116629
$818$	0	0
$819$	−3.24063	−0.113237
$820$	0	0
$821$	37.2427	1.29978	0.649889	−	0.760029i	$-0.274815\pi$
$821$	37.2427	1.29978	0.649889	+	0.760029i	$0.274815\pi$
$822$	0	0
$823$	10.8654	0.378745	0.189372	−	0.981905i	$-0.439355\pi$
$823$	10.8654	0.378745	0.189372	+	0.981905i	$0.439355\pi$
$824$	0	0
$825$	−55.8024	−1.94279
$826$	0	0
$827$	−38.2007	−1.32837	−0.664183	−	0.747570i	$-0.731221\pi$
$827$	−38.2007	−1.32837	−0.664183	+	0.747570i	$0.731221\pi$
$828$	0	0
$829$	12.2516	0.425516	0.212758	−	0.977105i	$-0.431755\pi$
$829$	12.2516	0.425516	0.212758	+	0.977105i	$0.431755\pi$
$830$	0	0
$831$	−29.8432	−1.03525
$832$	0	0
$833$	1.21246	0.0420092
$834$	0	0
$835$	−18.7285	−0.648127
$836$	0	0
$837$	9.11610	0.315099
$838$	0	0
$839$	35.6628	1.23121	0.615607	−	0.788053i	$-0.288911\pi$
$839$	35.6628	1.23121	0.615607	+	0.788053i	$0.288911\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−8.73980	−0.301015
$844$	0	0
$845$	49.6264	1.70720
$846$	0	0
$847$	−53.1429	−1.82601
$848$	0	0
$849$	−17.0153	−0.583965
$850$	0	0
$851$	−6.18358	−0.211970
$852$	0	0
$853$	13.9763	0.478540	0.239270	−	0.970953i	$-0.423092\pi$
$853$	13.9763	0.478540	0.239270	+	0.970953i	$0.423092\pi$
$854$	0	0
$855$	8.74106	0.298938
$856$	0	0
$857$	−13.7131	−0.468432	−0.234216	−	0.972185i	$-0.575252\pi$
$857$	−13.7131	−0.468432	−0.234216	+	0.972185i	$0.575252\pi$
$858$	0	0
$859$	−38.0918	−1.29968	−0.649838	−	0.760073i	$-0.725163\pi$
$859$	−38.0918	−1.29968	−0.649838	+	0.760073i	$0.725163\pi$
$860$	0	0
$861$	−16.0289	−0.546263
$862$	0	0
$863$	−39.0177	−1.32818	−0.664089	−	0.747654i	$-0.731180\pi$
$863$	−39.0177	−1.32818	−0.664089	+	0.747654i	$0.731180\pi$
$864$	0	0
$865$	36.9393	1.25597
$866$	0	0
$867$	16.9784	0.576616
$868$	0	0
$869$	66.0512	2.24063
$870$	0	0
$871$	−5.29245	−0.179328
$872$	0	0
$873$	−4.85389	−0.164279
$874$	0	0
$875$	98.3570	3.32507
$876$	0	0
$877$	45.9512	1.55166	0.775831	−	0.630941i	$-0.217331\pi$
$877$	45.9512	1.55166	0.775831	+	0.630941i	$0.217331\pi$
$878$	0	0
$879$	−10.4295	−0.351779
$880$	0	0
$881$	30.2706	1.01984	0.509920	−	0.860222i	$-0.329675\pi$
$881$	30.2706	1.01984	0.509920	+	0.860222i	$0.329675\pi$
$882$	0	0
$883$	−1.95233	−0.0657010	−0.0328505	−	0.999460i	$-0.510459\pi$
$883$	−1.95233	−0.0657010	−0.0328505	+	0.999460i	$0.510459\pi$
$884$	0	0
$885$	−16.2027	−0.544649
$886$	0	0
$887$	11.5661	0.388351	0.194176	−	0.980967i	$-0.437797\pi$
$887$	11.5661	0.388351	0.194176	+	0.980967i	$0.437797\pi$
$888$	0	0
$889$	19.8149	0.664570
$890$	0	0
$891$	4.96079	0.166193
$892$	0	0
$893$	15.1464	0.506856
$894$	0	0
$895$	−47.2123	−1.57813
$896$	0	0
$897$	0.829895	0.0277094
$898$	0	0
$899$	9.11610	0.304039
$900$	0	0
$901$	−1.39396	−0.0464397
$902$	0	0
$903$	6.00299	0.199767
$904$	0	0
$905$	26.3111	0.874612
$906$	0	0
$907$	34.6836	1.15165	0.575825	−	0.817573i	$-0.304681\pi$
$907$	34.6836	1.15165	0.575825	+	0.817573i	$0.304681\pi$
$908$	0	0
$909$	−1.52395	−0.0505461
$910$	0	0
$911$	−25.5610	−0.846873	−0.423436	−	0.905926i	$-0.639176\pi$
$911$	−25.5610	−0.846873	−0.423436	+	0.905926i	$0.639176\pi$
$912$	0	0
$913$	−78.1068	−2.58496
$914$	0	0
$915$	−55.1571	−1.82344
$916$	0	0
$917$	−59.4591	−1.96351
$918$	0	0
$919$	3.95660	0.130516	0.0652581	−	0.997868i	$-0.479213\pi$
$919$	3.95660	0.130516	0.0652581	+	0.997868i	$0.479213\pi$
$920$	0	0
$921$	32.9207	1.08478
$922$	0	0
$923$	−6.26411	−0.206186
$924$	0	0
$925$	69.5572	2.28703
$926$	0	0
$927$	6.81435	0.223813
$928$	0	0
$929$	−1.48452	−0.0487055	−0.0243527	−	0.999703i	$-0.507752\pi$
$929$	−1.48452	−0.0487055	−0.0243527	+	0.999703i	$0.507752\pi$
$930$	0	0
$931$	−17.8855	−0.586173
$932$	0	0
$933$	−10.5860	−0.346571
$934$	0	0
$935$	−2.93955	−0.0961336
$936$	0	0
$937$	54.2332	1.77172	0.885860	−	0.463952i	$-0.153569\pi$
$937$	54.2332	1.77172	0.885860	+	0.463952i	$0.153569\pi$
$938$	0	0
$939$	6.05323	0.197540
$940$	0	0
$941$	39.8925	1.30046	0.650230	−	0.759738i	$-0.274673\pi$
$941$	39.8925	1.30046	0.650230	+	0.759738i	$0.274673\pi$
$942$	0	0
$943$	4.10485	0.133672
$944$	0	0
$945$	−15.7404	−0.512034
$946$	0	0
$947$	−5.94190	−0.193086	−0.0965429	−	0.995329i	$-0.530778\pi$
$947$	−5.94190	−0.193086	−0.0965429	+	0.995329i	$0.530778\pi$
$948$	0	0
$949$	−1.65458	−0.0537101
$950$	0	0
$951$	7.74177	0.251044
$952$	0	0
$953$	12.7427	0.412775	0.206388	−	0.978470i	$-0.433829\pi$
$953$	12.7427	0.412775	0.206388	+	0.978470i	$0.433829\pi$
$954$	0	0
$955$	17.7548	0.574532
$956$	0	0
$957$	4.96079	0.160359
$958$	0	0
$959$	73.2663	2.36589
$960$	0	0
$961$	52.1033	1.68075
$962$	0	0
$963$	3.19721	0.103029
$964$	0	0
$965$	−40.1581	−1.29273
$966$	0	0
$967$	3.67016	0.118025	0.0590123	−	0.998257i	$-0.481205\pi$
$967$	3.67016	0.118025	0.0590123	+	0.998257i	$0.481205\pi$
$968$	0	0
$969$	0.318769	0.0102403
$970$	0	0
$971$	5.44430	0.174716	0.0873580	−	0.996177i	$-0.472158\pi$
$971$	5.44430	0.174716	0.0873580	+	0.996177i	$0.472158\pi$
$972$	0	0
$973$	−20.3828	−0.653441
$974$	0	0
$975$	−9.33525	−0.298967
$976$	0	0
$977$	−23.3595	−0.747336	−0.373668	−	0.927563i	$-0.621900\pi$
$977$	−23.3595	−0.747336	−0.373668	+	0.927563i	$0.621900\pi$
$978$	0	0
$979$	66.8174	2.13549
$980$	0	0
$981$	−6.08806	−0.194377
$982$	0	0
$983$	22.9356	0.731534	0.365767	−	0.930707i	$-0.380807\pi$
$983$	22.9356	0.731534	0.365767	+	0.930707i	$0.380807\pi$
$984$	0	0
$985$	−31.9917	−1.01934
$986$	0	0
$987$	−27.2748	−0.868166
$988$	0	0
$989$	−1.53731	−0.0488836
$990$	0	0
$991$	36.2226	1.15065	0.575325	−	0.817925i	$-0.304876\pi$
$991$	36.2226	1.15065	0.575325	+	0.817925i	$0.304876\pi$
$992$	0	0
$993$	−3.41328	−0.108317
$994$	0	0
$995$	15.1577	0.480532
$996$	0	0
$997$	1.15952	0.0367224	0.0183612	−	0.999831i	$-0.494155\pi$
$997$	1.15952	0.0367224	0.0183612	+	0.999831i	$0.494155\pi$
$998$	0	0
$999$	−6.18358	−0.195640

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8004.2.a.h.1.1	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8004.2.a.h.1.1	✓	13	1.1	even	1	trivial

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8004.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.03097 q^{5} -3.90486 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -4.03097 q^{5} -3.90486 q^{7} +1.00000 q^{9} +4.96079 q^{11} +0.829895 q^{13} +4.03097 q^{15} +0.147001 q^{17} -2.16848 q^{19} +3.90486 q^{21} -1.00000 q^{23} +11.2487 q^{25} -1.00000 q^{27} -1.00000 q^{29} -9.11610 q^{31} -4.96079 q^{33} +15.7404 q^{35} +6.18358 q^{37} -0.829895 q^{39} -4.10485 q^{41} +1.53731 q^{43} -4.03097 q^{45} -6.98483 q^{47} +8.24795 q^{49} -0.147001 q^{51} -9.48267 q^{53} -19.9968 q^{55} +2.16848 q^{57} -4.01956 q^{59} -13.6833 q^{61} -3.90486 q^{63} -3.34528 q^{65} -6.37725 q^{67} +1.00000 q^{69} -7.54807 q^{71} -1.99373 q^{73} -11.2487 q^{75} -19.3712 q^{77} +13.3147 q^{79} +1.00000 q^{81} -15.7448 q^{83} -0.592557 q^{85} +1.00000 q^{87} +13.4691 q^{89} -3.24063 q^{91} +9.11610 q^{93} +8.74106 q^{95} -4.85389 q^{97} +4.96079 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9}+O(q^{10}) \) 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9	\( 13 q - 13 q^{3} + 5 q^{5} - 8 q^{7} + 13 q^{9} + q^{11} + q^{13} - 5 q^{15} - 2 q^{17} - 10 q^{19} + 8 q^{21} - 13 q^{23} + 14 q^{25} - 13 q^{27} - 13 q^{29} - 26 q^{31} - q^{33} + 19 q^{35} + 15 q^{37} - q^{39} + 21 q^{41} - 6 q^{43} + 5 q^{45} + 16 q^{47} + 19 q^{49} + 2 q^{51} + 7 q^{53} + 15 q^{55} + 10 q^{57} - 11 q^{59} + 19 q^{61} - 8 q^{63} + 6 q^{65} - 13 q^{67} + 13 q^{69} + 9 q^{73} - 14 q^{75} + 10 q^{77} - 25 q^{79} + 13 q^{81} + 3 q^{83} + 14 q^{85} + 13 q^{87} + 23 q^{89} + 19 q^{91} + 26 q^{93} + 7 q^{95} + 25 q^{97} + q^{99}+O(q^{100}) \) 13 * q - 13 * q^3 + 5 * q^5 - 8 * q^7 + 13 * q^9 + q^11 + q^13 - 5 * q^15 - 2 * q^17 - 10 * q^19 + 8 * q^21 - 13 * q^23 + 14 * q^25 - 13 * q^27 - 13 * q^29 - 26 * q^31 - q^33 + 19 * q^35 + 15 * q^37 - q^39 + 21 * q^41 - 6 * q^43 + 5 * q^45 + 16 * q^47 + 19 * q^49 + 2 * q^51 + 7 * q^53 + 15 * q^55 + 10 * q^57 - 11 * q^59 + 19 * q^61 - 8 * q^63 + 6 * q^65 - 13 * q^67 + 13 * q^69 + 9 * q^73 - 14 * q^75 + 10 * q^77 - 25 * q^79 + 13 * q^81 + 3 * q^83 + 14 * q^85 + 13 * q^87 + 23 * q^89 + 19 * q^91 + 26 * q^93 + 7 * q^95 + 25 * q^97 + q^99

Introduction

L-functions

Modular forms

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Database

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