LMFDB - Embedded newform 8008.2.a.y.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 27 x^{12} + 78 x^{11} + 273 x^{10} - 750 x^{9} - 1306 x^{8} + 3378 x^{7} + \cdots - 128 $ x^14 - 3x^13 - 27x^12 + 78x^11 + 273x^10 - 750x^9 - 1306x^8 + 3378x^7 + 2996x^6 - 7275x^5 - 2804x^4 + 6417x^3 + 538x^2 - 1032*x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.53687$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.53687 q^{3} -0.216819 q^{5} +1.00000 q^{7} -0.638026 q^{9} +O(q^{10})$	$q-1.53687 q^{3} -0.216819 q^{5} +1.00000 q^{7} -0.638026 q^{9} -1.00000 q^{11} -1.00000 q^{13} +0.333223 q^{15} +1.18307 q^{17} +7.88361 q^{19} -1.53687 q^{21} -1.22149 q^{23} -4.95299 q^{25} +5.59118 q^{27} -5.20441 q^{29} -4.80232 q^{31} +1.53687 q^{33} -0.216819 q^{35} +8.61831 q^{37} +1.53687 q^{39} -7.95896 q^{41} -3.91246 q^{43} +0.138336 q^{45} +3.51062 q^{47} +1.00000 q^{49} -1.81822 q^{51} +6.35917 q^{53} +0.216819 q^{55} -12.1161 q^{57} -4.23350 q^{59} -0.624073 q^{61} -0.638026 q^{63} +0.216819 q^{65} +4.74600 q^{67} +1.87727 q^{69} -13.7254 q^{71} +16.1174 q^{73} +7.61211 q^{75} -1.00000 q^{77} +13.0838 q^{79} -6.67884 q^{81} -3.48532 q^{83} -0.256512 q^{85} +7.99850 q^{87} +3.11011 q^{89} -1.00000 q^{91} +7.38054 q^{93} -1.70932 q^{95} -3.26128 q^{97} +0.638026 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) $ 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9	$ 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) $ 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9 - 14 * q^11 - 14 * q^13 - 6 * q^15 - 6 * q^17 - 13 * q^19 - 3 * q^21 - 9 * q^23 + 22 * q^25 - 18 * q^27 + 2 * q^29 - 2 * q^31 + 3 * q^33 - 6 * q^35 - q^37 + 3 * q^39 - 16 * q^41 - 15 * q^43 - 44 * q^45 - 8 * q^47 + 14 * q^49 - 14 * q^51 - 6 * q^53 + 6 * q^55 - 10 * q^57 - 36 * q^59 - 19 * q^61 + 21 * q^63 + 6 * q^65 - 34 * q^67 - q^69 - 10 * q^71 + 9 * q^73 - 44 * q^75 - 14 * q^77 - q^79 + 42 * q^81 - 56 * q^83 + 21 * q^85 - 5 * q^87 - 14 * q^89 - 14 * q^91 - 20 * q^93 + q^95 - 14 * q^97 - 21 * 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
$3$	−1.53687	−0.887313	−0.443657	−	0.896197i	$-0.646319\pi$
$3$	−1.53687	−0.887313	−0.443657	+	0.896197i	$0.646319\pi$
$4$	0	0
$5$	−0.216819	−0.0969644	−0.0484822	−	0.998824i	$-0.515438\pi$
$5$	−0.216819	−0.0969644	−0.0484822	+	0.998824i	$0.515438\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−0.638026	−0.212675
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0.333223	0.0860378
$16$	0	0
$17$	1.18307	0.286936	0.143468	−	0.989655i	$-0.454175\pi$
$17$	1.18307	0.286936	0.143468	+	0.989655i	$0.454175\pi$
$18$	0	0
$19$	7.88361	1.80862	0.904312	−	0.426872i	$-0.140385\pi$
$19$	7.88361	1.80862	0.904312	+	0.426872i	$0.140385\pi$
$20$	0	0
$21$	−1.53687	−0.335373
$22$	0	0
$23$	−1.22149	−0.254698	−0.127349	−	0.991858i	$-0.540647\pi$
$23$	−1.22149	−0.254698	−0.127349	+	0.991858i	$0.540647\pi$
$24$	0	0
$25$	−4.95299	−0.990598
$26$	0	0
$27$	5.59118	1.07602
$28$	0	0
$29$	−5.20441	−0.966434	−0.483217	−	0.875501i	$-0.660532\pi$
$29$	−5.20441	−0.966434	−0.483217	+	0.875501i	$0.660532\pi$
$30$	0	0
$31$	−4.80232	−0.862522	−0.431261	−	0.902227i	$-0.641931\pi$
$31$	−4.80232	−0.862522	−0.431261	+	0.902227i	$0.641931\pi$
$32$	0	0
$33$	1.53687	0.267535
$34$	0	0
$35$	−0.216819	−0.0366491
$36$	0	0
$37$	8.61831	1.41684	0.708420	−	0.705791i	$-0.249408\pi$
$37$	8.61831	1.41684	0.708420	+	0.705791i	$0.249408\pi$
$38$	0	0
$39$	1.53687	0.246096
$40$	0	0
$41$	−7.95896	−1.24298	−0.621490	−	0.783422i	$-0.713472\pi$
$41$	−7.95896	−1.24298	−0.621490	+	0.783422i	$0.713472\pi$
$42$	0	0
$43$	−3.91246	−0.596644	−0.298322	−	0.954465i	$-0.596427\pi$
$43$	−3.91246	−0.596644	−0.298322	+	0.954465i	$0.596427\pi$
$44$	0	0
$45$	0.138336	0.0206219
$46$	0	0
$47$	3.51062	0.512077	0.256038	−	0.966667i	$-0.417583\pi$
$47$	3.51062	0.512077	0.256038	+	0.966667i	$0.417583\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.81822	−0.254602
$52$	0	0
$53$	6.35917	0.873499	0.436750	−	0.899583i	$-0.356130\pi$
$53$	6.35917	0.873499	0.436750	+	0.899583i	$0.356130\pi$
$54$	0	0
$55$	0.216819	0.0292359
$56$	0	0
$57$	−12.1161	−1.60482
$58$	0	0
$59$	−4.23350	−0.551155	−0.275577	−	0.961279i	$-0.588869\pi$
$59$	−4.23350	−0.551155	−0.275577	+	0.961279i	$0.588869\pi$
$60$	0	0
$61$	−0.624073	−0.0799043	−0.0399522	−	0.999202i	$-0.512721\pi$
$61$	−0.624073	−0.0799043	−0.0399522	+	0.999202i	$0.512721\pi$
$62$	0	0
$63$	−0.638026	−0.0803837
$64$	0	0
$65$	0.216819	0.0268931
$66$	0	0
$67$	4.74600	0.579817	0.289908	−	0.957054i	$-0.406375\pi$
$67$	4.74600	0.579817	0.289908	+	0.957054i	$0.406375\pi$
$68$	0	0
$69$	1.87727	0.225997
$70$	0	0
$71$	−13.7254	−1.62890	−0.814452	−	0.580231i	$-0.802962\pi$
$71$	−13.7254	−1.62890	−0.814452	+	0.580231i	$0.802962\pi$
$72$	0	0
$73$	16.1174	1.88640	0.943202	−	0.332220i	$-0.107798\pi$
$73$	16.1174	1.88640	0.943202	+	0.332220i	$0.107798\pi$
$74$	0	0
$75$	7.61211	0.878971
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	13.0838	1.47204	0.736019	−	0.676960i	$-0.236703\pi$
$79$	13.0838	1.47204	0.736019	+	0.676960i	$0.236703\pi$
$80$	0	0
$81$	−6.67884	−0.742094
$82$	0	0
$83$	−3.48532	−0.382564	−0.191282	−	0.981535i	$-0.561265\pi$
$83$	−3.48532	−0.382564	−0.191282	+	0.981535i	$0.561265\pi$
$84$	0	0
$85$	−0.256512	−0.0278226
$86$	0	0
$87$	7.99850	0.857530
$88$	0	0
$89$	3.11011	0.329671	0.164836	−	0.986321i	$-0.447291\pi$
$89$	3.11011	0.329671	0.164836	+	0.986321i	$0.447291\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	7.38054	0.765327
$94$	0	0
$95$	−1.70932	−0.175372
$96$	0	0
$97$	−3.26128	−0.331133	−0.165567	−	0.986199i	$-0.552945\pi$
$97$	−3.26128	−0.331133	−0.165567	+	0.986199i	$0.552945\pi$
$98$	0	0
$99$	0.638026	0.0641240
$100$	0	0
$101$	11.7866	1.17281	0.586406	−	0.810018i	$-0.300542\pi$
$101$	11.7866	1.17281	0.586406	+	0.810018i	$0.300542\pi$
$102$	0	0
$103$	−15.9517	−1.57177	−0.785884	−	0.618373i	$-0.787792\pi$
$103$	−15.9517	−1.57177	−0.785884	+	0.618373i	$0.787792\pi$
$104$	0	0
$105$	0.333223	0.0325192
$106$	0	0
$107$	−4.08272	−0.394691	−0.197346	−	0.980334i	$-0.563232\pi$
$107$	−4.08272	−0.394691	−0.197346	+	0.980334i	$0.563232\pi$
$108$	0	0
$109$	−8.93859	−0.856162	−0.428081	−	0.903740i	$-0.640810\pi$
$109$	−8.93859	−0.856162	−0.428081	+	0.903740i	$0.640810\pi$
$110$	0	0
$111$	−13.2452	−1.25718
$112$	0	0
$113$	7.43235	0.699177	0.349588	−	0.936903i	$-0.386321\pi$
$113$	7.43235	0.699177	0.349588	+	0.936903i	$0.386321\pi$
$114$	0	0
$115$	0.264842	0.0246967
$116$	0	0
$117$	0.638026	0.0589855
$118$	0	0
$119$	1.18307	0.108452
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	12.2319	1.10291
$124$	0	0
$125$	2.15800	0.193017
$126$	0	0
$127$	1.10500	0.0980525	0.0490263	−	0.998797i	$-0.484388\pi$
$127$	1.10500	0.0980525	0.0490263	+	0.998797i	$0.484388\pi$
$128$	0	0
$129$	6.01294	0.529410
$130$	0	0
$131$	−13.4517	−1.17528	−0.587642	−	0.809121i	$-0.699943\pi$
$131$	−13.4517	−1.17528	−0.587642	+	0.809121i	$0.699943\pi$
$132$	0	0
$133$	7.88361	0.683596
$134$	0	0
$135$	−1.21227	−0.104336
$136$	0	0
$137$	5.13711	0.438893	0.219447	−	0.975625i	$-0.429575\pi$
$137$	5.13711	0.438893	0.219447	+	0.975625i	$0.429575\pi$
$138$	0	0
$139$	3.18510	0.270156	0.135078	−	0.990835i	$-0.456871\pi$
$139$	3.18510	0.270156	0.135078	+	0.990835i	$0.456871\pi$
$140$	0	0
$141$	−5.39537	−0.454372
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	1.12841	0.0937097
$146$	0	0
$147$	−1.53687	−0.126759
$148$	0	0
$149$	−9.25902	−0.758529	−0.379264	−	0.925288i	$-0.623823\pi$
$149$	−9.25902	−0.758529	−0.379264	+	0.925288i	$0.623823\pi$
$150$	0	0
$151$	−6.48990	−0.528141	−0.264070	−	0.964503i	$-0.585065\pi$
$151$	−6.48990	−0.528141	−0.264070	+	0.964503i	$0.585065\pi$
$152$	0	0
$153$	−0.754828	−0.0610242
$154$	0	0
$155$	1.04123	0.0836339
$156$	0	0
$157$	10.3997	0.829989	0.414995	−	0.909824i	$-0.363783\pi$
$157$	10.3997	0.829989	0.414995	+	0.909824i	$0.363783\pi$
$158$	0	0
$159$	−9.77323	−0.775068
$160$	0	0
$161$	−1.22149	−0.0962670
$162$	0	0
$163$	−15.6197	−1.22343	−0.611716	−	0.791077i	$-0.709521\pi$
$163$	−15.6197	−1.22343	−0.611716	+	0.791077i	$0.709521\pi$
$164$	0	0
$165$	−0.333223	−0.0259414
$166$	0	0
$167$	17.3584	1.34323	0.671617	−	0.740898i	$-0.265600\pi$
$167$	17.3584	1.34323	0.671617	+	0.740898i	$0.265600\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−5.02995	−0.384650
$172$	0	0
$173$	−23.0133	−1.74967	−0.874834	−	0.484422i	$-0.839030\pi$
$173$	−23.0133	−1.74967	−0.874834	+	0.484422i	$0.839030\pi$
$174$	0	0
$175$	−4.95299	−0.374411
$176$	0	0
$177$	6.50635	0.489047
$178$	0	0
$179$	−15.2298	−1.13833	−0.569163	−	0.822225i	$-0.692733\pi$
$179$	−15.2298	−1.13833	−0.569163	+	0.822225i	$0.692733\pi$
$180$	0	0
$181$	−7.68362	−0.571119	−0.285559	−	0.958361i	$-0.592179\pi$
$181$	−7.68362	−0.571119	−0.285559	+	0.958361i	$0.592179\pi$
$182$	0	0
$183$	0.959120	0.0709002
$184$	0	0
$185$	−1.86861	−0.137383
$186$	0	0
$187$	−1.18307	−0.0865145
$188$	0	0
$189$	5.59118	0.406698
$190$	0	0
$191$	−22.5121	−1.62892	−0.814459	−	0.580221i	$-0.802966\pi$
$191$	−22.5121	−1.62892	−0.814459	+	0.580221i	$0.802966\pi$
$192$	0	0
$193$	19.9282	1.43446	0.717230	−	0.696837i	$-0.245410\pi$
$193$	19.9282	1.43446	0.717230	+	0.696837i	$0.245410\pi$
$194$	0	0
$195$	−0.333223	−0.0238626
$196$	0	0
$197$	20.3502	1.44989	0.724947	−	0.688804i	$-0.241864\pi$
$197$	20.3502	1.44989	0.724947	+	0.688804i	$0.241864\pi$
$198$	0	0
$199$	9.68637	0.686648	0.343324	−	0.939217i	$-0.388447\pi$
$199$	9.68637	0.686648	0.343324	+	0.939217i	$0.388447\pi$
$200$	0	0
$201$	−7.29400	−0.514479
$202$	0	0
$203$	−5.20441	−0.365278
$204$	0	0
$205$	1.72565	0.120525
$206$	0	0
$207$	0.779343	0.0541681
$208$	0	0
$209$	−7.88361	−0.545321
$210$	0	0
$211$	−8.97542	−0.617893	−0.308947	−	0.951079i	$-0.599976\pi$
$211$	−8.97542	−0.617893	−0.308947	+	0.951079i	$0.599976\pi$
$212$	0	0
$213$	21.0942	1.44535
$214$	0	0
$215$	0.848295	0.0578532
$216$	0	0
$217$	−4.80232	−0.326003
$218$	0	0
$219$	−24.7704	−1.67383
$220$	0	0
$221$	−1.18307	−0.0795817
$222$	0	0
$223$	14.5413	0.973760	0.486880	−	0.873469i	$-0.338135\pi$
$223$	14.5413	0.973760	0.486880	+	0.873469i	$0.338135\pi$
$224$	0	0
$225$	3.16014	0.210676
$226$	0	0
$227$	−7.69756	−0.510905	−0.255453	−	0.966822i	$-0.582224\pi$
$227$	−7.69756	−0.510905	−0.255453	+	0.966822i	$0.582224\pi$
$228$	0	0
$229$	21.9995	1.45377	0.726883	−	0.686761i	$-0.240968\pi$
$229$	21.9995	1.45377	0.726883	+	0.686761i	$0.240968\pi$
$230$	0	0
$231$	1.53687	0.101119
$232$	0	0
$233$	19.1946	1.25748	0.628739	−	0.777617i	$-0.283571\pi$
$233$	19.1946	1.25748	0.628739	+	0.777617i	$0.283571\pi$
$234$	0	0
$235$	−0.761169	−0.0496532
$236$	0	0
$237$	−20.1081	−1.30616
$238$	0	0
$239$	6.74464	0.436274	0.218137	−	0.975918i	$-0.430002\pi$
$239$	6.74464	0.436274	0.218137	+	0.975918i	$0.430002\pi$
$240$	0	0
$241$	−16.1910	−1.04296	−0.521478	−	0.853264i	$-0.674619\pi$
$241$	−16.1910	−1.04296	−0.521478	+	0.853264i	$0.674619\pi$
$242$	0	0
$243$	−6.50901	−0.417553
$244$	0	0
$245$	−0.216819	−0.0138521
$246$	0	0
$247$	−7.88361	−0.501622
$248$	0	0
$249$	5.35650	0.339454
$250$	0	0
$251$	−8.71405	−0.550026	−0.275013	−	0.961440i	$-0.588682\pi$
$251$	−8.71405	−0.550026	−0.275013	+	0.961440i	$0.588682\pi$
$252$	0	0
$253$	1.22149	0.0767945
$254$	0	0
$255$	0.394225	0.0246873
$256$	0	0
$257$	−21.6186	−1.34853	−0.674265	−	0.738489i	$-0.735540\pi$
$257$	−21.6186	−1.34853	−0.674265	+	0.738489i	$0.735540\pi$
$258$	0	0
$259$	8.61831	0.535515
$260$	0	0
$261$	3.32055	0.205537
$262$	0	0
$263$	−16.8633	−1.03984	−0.519918	−	0.854216i	$-0.674038\pi$
$263$	−16.8633	−1.03984	−0.519918	+	0.854216i	$0.674038\pi$
$264$	0	0
$265$	−1.37879	−0.0846984
$266$	0	0
$267$	−4.77984	−0.292522
$268$	0	0
$269$	7.95831	0.485227	0.242613	−	0.970123i	$-0.421995\pi$
$269$	7.95831	0.485227	0.242613	+	0.970123i	$0.421995\pi$
$270$	0	0
$271$	2.64947	0.160944	0.0804719	−	0.996757i	$-0.474357\pi$
$271$	2.64947	0.160944	0.0804719	+	0.996757i	$0.474357\pi$
$272$	0	0
$273$	1.53687	0.0930157
$274$	0	0
$275$	4.95299	0.298677
$276$	0	0
$277$	−15.8295	−0.951103	−0.475551	−	0.879688i	$-0.657751\pi$
$277$	−15.8295	−0.951103	−0.475551	+	0.879688i	$0.657751\pi$
$278$	0	0
$279$	3.06400	0.183437
$280$	0	0
$281$	13.7268	0.818871	0.409436	−	0.912339i	$-0.365726\pi$
$281$	13.7268	0.818871	0.409436	+	0.912339i	$0.365726\pi$
$282$	0	0
$283$	−33.1130	−1.96836	−0.984181	−	0.177169i	$-0.943306\pi$
$283$	−33.1130	−1.96836	−0.984181	+	0.177169i	$0.943306\pi$
$284$	0	0
$285$	2.62700	0.155610
$286$	0	0
$287$	−7.95896	−0.469802
$288$	0	0
$289$	−15.6004	−0.917668
$290$	0	0
$291$	5.01217	0.293819
$292$	0	0
$293$	−30.8767	−1.80384	−0.901919	−	0.431905i	$-0.857842\pi$
$293$	−30.8767	−1.80384	−0.901919	+	0.431905i	$0.857842\pi$
$294$	0	0
$295$	0.917903	0.0534424
$296$	0	0
$297$	−5.59118	−0.324433
$298$	0	0
$299$	1.22149	0.0706406
$300$	0	0
$301$	−3.91246	−0.225510
$302$	0	0
$303$	−18.1145	−1.04065
$304$	0	0
$305$	0.135311	0.00774788
$306$	0	0
$307$	−30.9074	−1.76398	−0.881990	−	0.471268i	$-0.843796\pi$
$307$	−30.9074	−1.76398	−0.881990	+	0.471268i	$0.843796\pi$
$308$	0	0
$309$	24.5157	1.39465
$310$	0	0
$311$	21.5426	1.22157	0.610785	−	0.791796i	$-0.290854\pi$
$311$	21.5426	1.22157	0.610785	+	0.791796i	$0.290854\pi$
$312$	0	0
$313$	−26.1297	−1.47694	−0.738470	−	0.674286i	$-0.764451\pi$
$313$	−26.1297	−1.47694	−0.738470	+	0.674286i	$0.764451\pi$
$314$	0	0
$315$	0.138336	0.00779436
$316$	0	0
$317$	−17.7543	−0.997179	−0.498590	−	0.866838i	$-0.666148\pi$
$317$	−17.7543	−0.997179	−0.498590	+	0.866838i	$0.666148\pi$
$318$	0	0
$319$	5.20441	0.291391
$320$	0	0
$321$	6.27461	0.350215
$322$	0	0
$323$	9.32684	0.518959
$324$	0	0
$325$	4.95299	0.274742
$326$	0	0
$327$	13.7375	0.759684
$328$	0	0
$329$	3.51062	0.193547
$330$	0	0
$331$	6.34621	0.348819	0.174410	−	0.984673i	$-0.444198\pi$
$331$	6.34621	0.348819	0.174410	+	0.984673i	$0.444198\pi$
$332$	0	0
$333$	−5.49870	−0.301327
$334$	0	0
$335$	−1.02902	−0.0562216
$336$	0	0
$337$	−24.4535	−1.33206	−0.666032	−	0.745923i	$-0.732009\pi$
$337$	−24.4535	−1.33206	−0.666032	+	0.745923i	$0.732009\pi$
$338$	0	0
$339$	−11.4226	−0.620389
$340$	0	0
$341$	4.80232	0.260060
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−0.407029	−0.0219137
$346$	0	0
$347$	−15.7975	−0.848057	−0.424028	−	0.905649i	$-0.639384\pi$
$347$	−15.7975	−0.848057	−0.424028	+	0.905649i	$0.639384\pi$
$348$	0	0
$349$	−11.3768	−0.608988	−0.304494	−	0.952514i	$-0.598487\pi$
$349$	−11.3768	−0.608988	−0.304494	+	0.952514i	$0.598487\pi$
$350$	0	0
$351$	−5.59118	−0.298435
$352$	0	0
$353$	25.1377	1.33795	0.668973	−	0.743287i	$-0.266734\pi$
$353$	25.1377	1.33795	0.668973	+	0.743287i	$0.266734\pi$
$354$	0	0
$355$	2.97593	0.157946
$356$	0	0
$357$	−1.81822	−0.0962306
$358$	0	0
$359$	19.7034	1.03991	0.519954	−	0.854195i	$-0.325949\pi$
$359$	19.7034	1.03991	0.519954	+	0.854195i	$0.325949\pi$
$360$	0	0
$361$	43.1513	2.27112
$362$	0	0
$363$	−1.53687	−0.0806648
$364$	0	0
$365$	−3.49457	−0.182914
$366$	0	0
$367$	3.32182	0.173398	0.0866988	−	0.996235i	$-0.472368\pi$
$367$	3.32182	0.173398	0.0866988	+	0.996235i	$0.472368\pi$
$368$	0	0
$369$	5.07802	0.264351
$370$	0	0
$371$	6.35917	0.330152
$372$	0	0
$373$	3.60634	0.186729	0.0933645	−	0.995632i	$-0.470238\pi$
$373$	3.60634	0.186729	0.0933645	+	0.995632i	$0.470238\pi$
$374$	0	0
$375$	−3.31656	−0.171267
$376$	0	0
$377$	5.20441	0.268041
$378$	0	0
$379$	6.21587	0.319288	0.159644	−	0.987175i	$-0.448965\pi$
$379$	6.21587	0.319288	0.159644	+	0.987175i	$0.448965\pi$
$380$	0	0
$381$	−1.69824	−0.0870033
$382$	0	0
$383$	−11.4018	−0.582607	−0.291303	−	0.956631i	$-0.594089\pi$
$383$	−11.4018	−0.582607	−0.291303	+	0.956631i	$0.594089\pi$
$384$	0	0
$385$	0.216819	0.0110501
$386$	0	0
$387$	2.49625	0.126891
$388$	0	0
$389$	−13.4648	−0.682692	−0.341346	−	0.939938i	$-0.610883\pi$
$389$	−13.4648	−0.682692	−0.341346	+	0.939938i	$0.610883\pi$
$390$	0	0
$391$	−1.44511	−0.0730822
$392$	0	0
$393$	20.6736	1.04284
$394$	0	0
$395$	−2.83681	−0.142735
$396$	0	0
$397$	15.9250	0.799254	0.399627	−	0.916678i	$-0.369140\pi$
$397$	15.9250	0.799254	0.399627	+	0.916678i	$0.369140\pi$
$398$	0	0
$399$	−12.1161	−0.606563
$400$	0	0
$401$	7.83195	0.391109	0.195555	−	0.980693i	$-0.437349\pi$
$401$	7.83195	0.391109	0.195555	+	0.980693i	$0.437349\pi$
$402$	0	0
$403$	4.80232	0.239220
$404$	0	0
$405$	1.44810	0.0719567
$406$	0	0
$407$	−8.61831	−0.427194
$408$	0	0
$409$	−31.3616	−1.55073	−0.775367	−	0.631511i	$-0.782435\pi$
$409$	−31.3616	−1.55073	−0.775367	+	0.631511i	$0.782435\pi$
$410$	0	0
$411$	−7.89508	−0.389436
$412$	0	0
$413$	−4.23350	−0.208317
$414$	0	0
$415$	0.755685	0.0370951
$416$	0	0
$417$	−4.89508	−0.239713
$418$	0	0
$419$	32.0041	1.56350	0.781751	−	0.623591i	$-0.214327\pi$
$419$	32.0041	1.56350	0.781751	+	0.623591i	$0.214327\pi$
$420$	0	0
$421$	−11.0636	−0.539208	−0.269604	−	0.962971i	$-0.586893\pi$
$421$	−11.0636	−0.539208	−0.269604	+	0.962971i	$0.586893\pi$
$422$	0	0
$423$	−2.23987	−0.108906
$424$	0	0
$425$	−5.85972	−0.284238
$426$	0	0
$427$	−0.624073	−0.0302010
$428$	0	0
$429$	−1.53687	−0.0742009
$430$	0	0
$431$	−17.5456	−0.845142	−0.422571	−	0.906330i	$-0.638872\pi$
$431$	−17.5456	−0.845142	−0.422571	+	0.906330i	$0.638872\pi$
$432$	0	0
$433$	36.7819	1.76763	0.883814	−	0.467839i	$-0.154967\pi$
$433$	36.7819	1.76763	0.883814	+	0.467839i	$0.154967\pi$
$434$	0	0
$435$	−1.73423	−0.0831499
$436$	0	0
$437$	−9.62975	−0.460654
$438$	0	0
$439$	−4.35920	−0.208053	−0.104027	−	0.994575i	$-0.533173\pi$
$439$	−4.35920	−0.208053	−0.104027	+	0.994575i	$0.533173\pi$
$440$	0	0
$441$	−0.638026	−0.0303822
$442$	0	0
$443$	−9.75163	−0.463314	−0.231657	−	0.972798i	$-0.574415\pi$
$443$	−9.75163	−0.463314	−0.231657	+	0.972798i	$0.574415\pi$
$444$	0	0
$445$	−0.674331	−0.0319664
$446$	0	0
$447$	14.2299	0.673052
$448$	0	0
$449$	−7.07433	−0.333858	−0.166929	−	0.985969i	$-0.553385\pi$
$449$	−7.07433	−0.333858	−0.166929	+	0.985969i	$0.553385\pi$
$450$	0	0
$451$	7.95896	0.374773
$452$	0	0
$453$	9.97415	0.468626
$454$	0	0
$455$	0.216819	0.0101646
$456$	0	0
$457$	18.9154	0.884823	0.442412	−	0.896812i	$-0.354123\pi$
$457$	18.9154	0.884823	0.442412	+	0.896812i	$0.354123\pi$
$458$	0	0
$459$	6.61474	0.308750
$460$	0	0
$461$	−8.96871	−0.417714	−0.208857	−	0.977946i	$-0.566974\pi$
$461$	−8.96871	−0.417714	−0.208857	+	0.977946i	$0.566974\pi$
$462$	0	0
$463$	−10.7098	−0.497728	−0.248864	−	0.968538i	$-0.580057\pi$
$463$	−10.7098	−0.497728	−0.248864	+	0.968538i	$0.580057\pi$
$464$	0	0
$465$	−1.60024	−0.0742094
$466$	0	0
$467$	−28.1611	−1.30314	−0.651571	−	0.758588i	$-0.725890\pi$
$467$	−28.1611	−1.30314	−0.651571	+	0.758588i	$0.725890\pi$
$468$	0	0
$469$	4.74600	0.219150
$470$	0	0
$471$	−15.9831	−0.736460
$472$	0	0
$473$	3.91246	0.179895
$474$	0	0
$475$	−39.0474	−1.79162
$476$	0	0
$477$	−4.05732	−0.185772
$478$	0	0
$479$	2.12403	0.0970494	0.0485247	−	0.998822i	$-0.484548\pi$
$479$	2.12403	0.0970494	0.0485247	+	0.998822i	$0.484548\pi$
$480$	0	0
$481$	−8.61831	−0.392961
$482$	0	0
$483$	1.87727	0.0854189
$484$	0	0
$485$	0.707108	0.0321081
$486$	0	0
$487$	20.5415	0.930822	0.465411	−	0.885095i	$-0.345907\pi$
$487$	20.5415	0.930822	0.465411	+	0.885095i	$0.345907\pi$
$488$	0	0
$489$	24.0055	1.08557
$490$	0	0
$491$	−6.18566	−0.279155	−0.139577	−	0.990211i	$-0.544574\pi$
$491$	−6.18566	−0.279155	−0.139577	+	0.990211i	$0.544574\pi$
$492$	0	0
$493$	−6.15717	−0.277305
$494$	0	0
$495$	−0.138336	−0.00621775
$496$	0	0
$497$	−13.7254	−0.615668
$498$	0	0
$499$	−5.87923	−0.263190	−0.131595	−	0.991304i	$-0.542010\pi$
$499$	−5.87923	−0.263190	−0.131595	+	0.991304i	$0.542010\pi$
$500$	0	0
$501$	−26.6777	−1.19187
$502$	0	0
$503$	−40.7941	−1.81892	−0.909460	−	0.415791i	$-0.863505\pi$
$503$	−40.7941	−1.81892	−0.909460	+	0.415791i	$0.863505\pi$
$504$	0	0
$505$	−2.55556	−0.113721
$506$	0	0
$507$	−1.53687	−0.0682549
$508$	0	0
$509$	−25.7132	−1.13972	−0.569858	−	0.821743i	$-0.693002\pi$
$509$	−25.7132	−1.13972	−0.569858	+	0.821743i	$0.693002\pi$
$510$	0	0
$511$	16.1174	0.712994
$512$	0	0
$513$	44.0787	1.94612
$514$	0	0
$515$	3.45863	0.152406
$516$	0	0
$517$	−3.51062	−0.154397
$518$	0	0
$519$	35.3685	1.55250
$520$	0	0
$521$	−35.0574	−1.53589	−0.767947	−	0.640513i	$-0.778722\pi$
$521$	−35.0574	−1.53589	−0.767947	+	0.640513i	$0.778722\pi$
$522$	0	0
$523$	−27.2540	−1.19174	−0.595868	−	0.803082i	$-0.703192\pi$
$523$	−27.2540	−1.19174	−0.595868	+	0.803082i	$0.703192\pi$
$524$	0	0
$525$	7.61211	0.332220
$526$	0	0
$527$	−5.68147	−0.247489
$528$	0	0
$529$	−21.5080	−0.935129
$530$	0	0
$531$	2.70108	0.117217
$532$	0	0
$533$	7.95896	0.344741
$534$	0	0
$535$	0.885210	0.0382710
$536$	0	0
$537$	23.4062	1.01005
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−40.6761	−1.74880	−0.874401	−	0.485204i	$-0.838745\pi$
$541$	−40.6761	−1.74880	−0.874401	+	0.485204i	$0.838745\pi$
$542$	0	0
$543$	11.8087	0.506761
$544$	0	0
$545$	1.93806	0.0830172
$546$	0	0
$547$	35.1108	1.50123	0.750615	−	0.660740i	$-0.229758\pi$
$547$	35.1108	1.50123	0.750615	+	0.660740i	$0.229758\pi$
$548$	0	0
$549$	0.398175	0.0169937
$550$	0	0
$551$	−41.0295	−1.74792
$552$	0	0
$553$	13.0838	0.556378
$554$	0	0
$555$	2.87182	0.121902
$556$	0	0
$557$	−38.7242	−1.64080	−0.820399	−	0.571791i	$-0.806249\pi$
$557$	−38.7242	−1.64080	−0.820399	+	0.571791i	$0.806249\pi$
$558$	0	0
$559$	3.91246	0.165479
$560$	0	0
$561$	1.81822	0.0767654
$562$	0	0
$563$	−27.6682	−1.16608	−0.583038	−	0.812445i	$-0.698136\pi$
$563$	−27.6682	−1.16608	−0.583038	+	0.812445i	$0.698136\pi$
$564$	0	0
$565$	−1.61147	−0.0677952
$566$	0	0
$567$	−6.67884	−0.280485
$568$	0	0
$569$	24.3371	1.02026	0.510132	−	0.860096i	$-0.329597\pi$
$569$	24.3371	1.02026	0.510132	+	0.860096i	$0.329597\pi$
$570$	0	0
$571$	−4.19074	−0.175377	−0.0876886	−	0.996148i	$-0.527948\pi$
$571$	−4.19074	−0.175377	−0.0876886	+	0.996148i	$0.527948\pi$
$572$	0	0
$573$	34.5982	1.44536
$574$	0	0
$575$	6.05003	0.252304
$576$	0	0
$577$	39.6102	1.64900	0.824498	−	0.565866i	$-0.191458\pi$
$577$	39.6102	1.64900	0.824498	+	0.565866i	$0.191458\pi$
$578$	0	0
$579$	−30.6270	−1.27282
$580$	0	0
$581$	−3.48532	−0.144596
$582$	0	0
$583$	−6.35917	−0.263370
$584$	0	0
$585$	−0.138336	−0.00571950
$586$	0	0
$587$	−45.5309	−1.87926	−0.939631	−	0.342188i	$-0.888832\pi$
$587$	−45.5309	−1.87926	−0.939631	+	0.342188i	$0.888832\pi$
$588$	0	0
$589$	−37.8596	−1.55998
$590$	0	0
$591$	−31.2757	−1.28651
$592$	0	0
$593$	−14.1339	−0.580408	−0.290204	−	0.956965i	$-0.593723\pi$
$593$	−14.1339	−0.580408	−0.290204	+	0.956965i	$0.593723\pi$
$594$	0	0
$595$	−0.256512	−0.0105159
$596$	0	0
$597$	−14.8867	−0.609272
$598$	0	0
$599$	18.9941	0.776077	0.388039	−	0.921643i	$-0.373153\pi$
$599$	18.9941	0.776077	0.388039	+	0.921643i	$0.373153\pi$
$600$	0	0
$601$	−24.8536	−1.01380	−0.506899	−	0.862005i	$-0.669208\pi$
$601$	−24.8536	−1.01380	−0.506899	+	0.862005i	$0.669208\pi$
$602$	0	0
$603$	−3.02807	−0.123313
$604$	0	0
$605$	−0.216819	−0.00881495
$606$	0	0
$607$	−32.1691	−1.30570	−0.652852	−	0.757486i	$-0.726427\pi$
$607$	−32.1691	−1.30570	−0.652852	+	0.757486i	$0.726427\pi$
$608$	0	0
$609$	7.99850	0.324116
$610$	0	0
$611$	−3.51062	−0.142025
$612$	0	0
$613$	38.8049	1.56732	0.783658	−	0.621192i	$-0.213351\pi$
$613$	38.8049	1.56732	0.783658	+	0.621192i	$0.213351\pi$
$614$	0	0
$615$	−2.65211	−0.106943
$616$	0	0
$617$	10.1541	0.408788	0.204394	−	0.978889i	$-0.434478\pi$
$617$	10.1541	0.408788	0.204394	+	0.978889i	$0.434478\pi$
$618$	0	0
$619$	1.67915	0.0674908	0.0337454	−	0.999430i	$-0.489256\pi$
$619$	1.67915	0.0674908	0.0337454	+	0.999430i	$0.489256\pi$
$620$	0	0
$621$	−6.82957	−0.274061
$622$	0	0
$623$	3.11011	0.124604
$624$	0	0
$625$	24.2971	0.971882
$626$	0	0
$627$	12.1161	0.483870
$628$	0	0
$629$	10.1960	0.406543
$630$	0	0
$631$	−21.3599	−0.850323	−0.425161	−	0.905118i	$-0.639783\pi$
$631$	−21.3599	−0.850323	−0.425161	+	0.905118i	$0.639783\pi$
$632$	0	0
$633$	13.7941	0.548265
$634$	0	0
$635$	−0.239584	−0.00950761
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	8.75716	0.346428
$640$	0	0
$641$	15.7163	0.620755	0.310378	−	0.950613i	$-0.399544\pi$
$641$	15.7163	0.620755	0.310378	+	0.950613i	$0.399544\pi$
$642$	0	0
$643$	10.1370	0.399766	0.199883	−	0.979820i	$-0.435944\pi$
$643$	10.1370	0.399766	0.199883	+	0.979820i	$0.435944\pi$
$644$	0	0
$645$	−1.30372	−0.0513339
$646$	0	0
$647$	−36.0162	−1.41594	−0.707972	−	0.706241i	$-0.750390\pi$
$647$	−36.0162	−1.41594	−0.707972	+	0.706241i	$0.750390\pi$
$648$	0	0
$649$	4.23350	0.166179
$650$	0	0
$651$	7.38054	0.289266
$652$	0	0
$653$	−23.4509	−0.917705	−0.458853	−	0.888512i	$-0.651739\pi$
$653$	−23.4509	−0.917705	−0.458853	+	0.888512i	$0.651739\pi$
$654$	0	0
$655$	2.91659	0.113961
$656$	0	0
$657$	−10.2833	−0.401192
$658$	0	0
$659$	−43.1008	−1.67897	−0.839484	−	0.543384i	$-0.817143\pi$
$659$	−43.1008	−1.67897	−0.839484	+	0.543384i	$0.817143\pi$
$660$	0	0
$661$	−40.7391	−1.58457	−0.792285	−	0.610152i	$-0.791108\pi$
$661$	−40.7391	−1.58457	−0.792285	+	0.610152i	$0.791108\pi$
$662$	0	0
$663$	1.81822	0.0706139
$664$	0	0
$665$	−1.70932	−0.0662844
$666$	0	0
$667$	6.35713	0.246149
$668$	0	0
$669$	−22.3482	−0.864030
$670$	0	0
$671$	0.624073	0.0240921
$672$	0	0
$673$	−10.4837	−0.404118	−0.202059	−	0.979373i	$-0.564763\pi$
$673$	−10.4837	−0.404118	−0.202059	+	0.979373i	$0.564763\pi$
$674$	0	0
$675$	−27.6930	−1.06591
$676$	0	0
$677$	−50.8887	−1.95581	−0.977905	−	0.209049i	$-0.932963\pi$
$677$	−50.8887	−1.95581	−0.977905	+	0.209049i	$0.932963\pi$
$678$	0	0
$679$	−3.26128	−0.125157
$680$	0	0
$681$	11.8302	0.453333
$682$	0	0
$683$	31.7447	1.21468	0.607338	−	0.794443i	$-0.292237\pi$
$683$	31.7447	1.21468	0.607338	+	0.794443i	$0.292237\pi$
$684$	0	0
$685$	−1.11382	−0.0425570
$686$	0	0
$687$	−33.8104	−1.28995
$688$	0	0
$689$	−6.35917	−0.242265
$690$	0	0
$691$	−23.9350	−0.910532	−0.455266	−	0.890356i	$-0.650456\pi$
$691$	−23.9350	−0.910532	−0.455266	+	0.890356i	$0.650456\pi$
$692$	0	0
$693$	0.638026	0.0242366
$694$	0	0
$695$	−0.690589	−0.0261956
$696$	0	0
$697$	−9.41598	−0.356656
$698$	0	0
$699$	−29.4996	−1.11578
$700$	0	0
$701$	40.4956	1.52950	0.764748	−	0.644329i	$-0.222863\pi$
$701$	40.4956	1.52950	0.764748	+	0.644329i	$0.222863\pi$
$702$	0	0
$703$	67.9433	2.56253
$704$	0	0
$705$	1.16982	0.0440579
$706$	0	0
$707$	11.7866	0.443281
$708$	0	0
$709$	41.3601	1.55331	0.776655	−	0.629926i	$-0.216915\pi$
$709$	41.3601	1.55331	0.776655	+	0.629926i	$0.216915\pi$
$710$	0	0
$711$	−8.34778	−0.313066
$712$	0	0
$713$	5.86599	0.219683
$714$	0	0
$715$	−0.216819	−0.00810857
$716$	0	0
$717$	−10.3656	−0.387112
$718$	0	0
$719$	−35.2601	−1.31498	−0.657490	−	0.753464i	$-0.728382\pi$
$719$	−35.2601	−1.31498	−0.657490	+	0.753464i	$0.728382\pi$
$720$	0	0
$721$	−15.9517	−0.594073
$722$	0	0
$723$	24.8836	0.925429
$724$	0	0
$725$	25.7774	0.957348
$726$	0	0
$727$	−52.5398	−1.94859	−0.974297	−	0.225268i	$-0.927674\pi$
$727$	−52.5398	−1.94859	−0.974297	+	0.225268i	$0.927674\pi$
$728$	0	0
$729$	30.0400	1.11259
$730$	0	0
$731$	−4.62870	−0.171199
$732$	0	0
$733$	−51.6413	−1.90742	−0.953708	−	0.300734i	$-0.902768\pi$
$733$	−51.6413	−1.90742	−0.953708	+	0.300734i	$0.902768\pi$
$734$	0	0
$735$	0.333223	0.0122911
$736$	0	0
$737$	−4.74600	−0.174821
$738$	0	0
$739$	−31.1117	−1.14446	−0.572231	−	0.820093i	$-0.693922\pi$
$739$	−31.1117	−1.14446	−0.572231	+	0.820093i	$0.693922\pi$
$740$	0	0
$741$	12.1161	0.445096
$742$	0	0
$743$	10.4366	0.382883	0.191442	−	0.981504i	$-0.438684\pi$
$743$	10.4366	0.382883	0.191442	+	0.981504i	$0.438684\pi$
$744$	0	0
$745$	2.00753	0.0735503
$746$	0	0
$747$	2.22373	0.0813620
$748$	0	0
$749$	−4.08272	−0.149179
$750$	0	0
$751$	−33.0188	−1.20487	−0.602437	−	0.798166i	$-0.705804\pi$
$751$	−33.0188	−1.20487	−0.602437	+	0.798166i	$0.705804\pi$
$752$	0	0
$753$	13.3924	0.488045
$754$	0	0
$755$	1.40713	0.0512109
$756$	0	0
$757$	33.0313	1.20054	0.600272	−	0.799796i	$-0.295059\pi$
$757$	33.0313	1.20054	0.600272	+	0.799796i	$0.295059\pi$
$758$	0	0
$759$	−1.87727	−0.0681407
$760$	0	0
$761$	47.9565	1.73842	0.869211	−	0.494441i	$-0.164627\pi$
$761$	47.9565	1.73842	0.869211	+	0.494441i	$0.164627\pi$
$762$	0	0
$763$	−8.93859	−0.323599
$764$	0	0
$765$	0.163661	0.00591718
$766$	0	0
$767$	4.23350	0.152863
$768$	0	0
$769$	−31.6963	−1.14300	−0.571498	−	0.820603i	$-0.693638\pi$
$769$	−31.6963	−1.14300	−0.571498	+	0.820603i	$0.693638\pi$
$770$	0	0
$771$	33.2250	1.19657
$772$	0	0
$773$	−4.39679	−0.158141	−0.0790707	−	0.996869i	$-0.525195\pi$
$773$	−4.39679	−0.158141	−0.0790707	+	0.996869i	$0.525195\pi$
$774$	0	0
$775$	23.7858	0.854412
$776$	0	0
$777$	−13.2452	−0.475170
$778$	0	0
$779$	−62.7453	−2.24808
$780$	0	0
$781$	13.7254	0.491133
$782$	0	0
$783$	−29.0988	−1.03991
$784$	0	0
$785$	−2.25486	−0.0804794
$786$	0	0
$787$	32.2163	1.14839	0.574193	−	0.818720i	$-0.305316\pi$
$787$	32.2163	1.14839	0.574193	+	0.818720i	$0.305316\pi$
$788$	0	0
$789$	25.9168	0.922661
$790$	0	0
$791$	7.43235	0.264264
$792$	0	0
$793$	0.624073	0.0221615
$794$	0	0
$795$	2.11902	0.0751540
$796$	0	0
$797$	9.19579	0.325731	0.162866	−	0.986648i	$-0.447926\pi$
$797$	9.19579	0.325731	0.162866	+	0.986648i	$0.447926\pi$
$798$	0	0
$799$	4.15330	0.146933
$800$	0	0
$801$	−1.98433	−0.0701129
$802$	0	0
$803$	−16.1174	−0.568772
$804$	0	0
$805$	0.264842	0.00933447
$806$	0	0
$807$	−12.2309	−0.430548
$808$	0	0
$809$	31.6701	1.11346	0.556731	−	0.830693i	$-0.312055\pi$
$809$	31.6701	1.11346	0.556731	+	0.830693i	$0.312055\pi$
$810$	0	0
$811$	22.2343	0.780752	0.390376	−	0.920655i	$-0.372345\pi$
$811$	22.2343	0.780752	0.390376	+	0.920655i	$0.372345\pi$
$812$	0	0
$813$	−4.07189	−0.142808
$814$	0	0
$815$	3.38666	0.118629
$816$	0	0
$817$	−30.8443	−1.07910
$818$	0	0
$819$	0.638026	0.0222944
$820$	0	0
$821$	22.0362	0.769068	0.384534	−	0.923111i	$-0.374362\pi$
$821$	22.0362	0.769068	0.384534	+	0.923111i	$0.374362\pi$
$822$	0	0
$823$	7.29612	0.254327	0.127163	−	0.991882i	$-0.459413\pi$
$823$	7.29612	0.254327	0.127163	+	0.991882i	$0.459413\pi$
$824$	0	0
$825$	−7.61211	−0.265020
$826$	0	0
$827$	13.7818	0.479242	0.239621	−	0.970867i	$-0.422977\pi$
$827$	13.7818	0.479242	0.239621	+	0.970867i	$0.422977\pi$
$828$	0	0
$829$	−10.1029	−0.350888	−0.175444	−	0.984489i	$-0.556136\pi$
$829$	−10.1029	−0.350888	−0.175444	+	0.984489i	$0.556136\pi$
$830$	0	0
$831$	24.3279	0.843926
$832$	0	0
$833$	1.18307	0.0409909
$834$	0	0
$835$	−3.76363	−0.130246
$836$	0	0
$837$	−26.8506	−0.928093
$838$	0	0
$839$	−9.01500	−0.311232	−0.155616	−	0.987818i	$-0.549736\pi$
$839$	−9.01500	−0.311232	−0.155616	+	0.987818i	$0.549736\pi$
$840$	0	0
$841$	−1.91415	−0.0660051
$842$	0	0
$843$	−21.0963	−0.726595
$844$	0	0
$845$	−0.216819	−0.00745880
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	50.8904	1.74655
$850$	0	0
$851$	−10.5272	−0.360867
$852$	0	0
$853$	−49.8042	−1.70526	−0.852631	−	0.522514i	$-0.824994\pi$
$853$	−49.8042	−1.70526	−0.852631	+	0.522514i	$0.824994\pi$
$854$	0	0
$855$	1.09059	0.0372973
$856$	0	0
$857$	−10.8500	−0.370630	−0.185315	−	0.982679i	$-0.559331\pi$
$857$	−10.8500	−0.370630	−0.185315	+	0.982679i	$0.559331\pi$
$858$	0	0
$859$	12.2324	0.417364	0.208682	−	0.977984i	$-0.433083\pi$
$859$	12.2324	0.417364	0.208682	+	0.977984i	$0.433083\pi$
$860$	0	0
$861$	12.2319	0.416862
$862$	0	0
$863$	−8.41335	−0.286394	−0.143197	−	0.989694i	$-0.545738\pi$
$863$	−8.41335	−0.286394	−0.143197	+	0.989694i	$0.545738\pi$
$864$	0	0
$865$	4.98972	0.169656
$866$	0	0
$867$	23.9757	0.814259
$868$	0	0
$869$	−13.0838	−0.443836
$870$	0	0
$871$	−4.74600	−0.160812
$872$	0	0
$873$	2.08078	0.0704239
$874$	0	0
$875$	2.15800	0.0729536
$876$	0	0
$877$	0.242393	0.00818503	0.00409252	−	0.999992i	$-0.498697\pi$
$877$	0.242393	0.00818503	0.00409252	+	0.999992i	$0.498697\pi$
$878$	0	0
$879$	47.4536	1.60057
$880$	0	0
$881$	−17.3393	−0.584176	−0.292088	−	0.956392i	$-0.594350\pi$
$881$	−17.3393	−0.584176	−0.292088	+	0.956392i	$0.594350\pi$
$882$	0	0
$883$	45.8842	1.54413	0.772063	−	0.635547i	$-0.219225\pi$
$883$	45.8842	1.54413	0.772063	+	0.635547i	$0.219225\pi$
$884$	0	0
$885$	−1.41070	−0.0474201
$886$	0	0
$887$	−53.7338	−1.80420	−0.902102	−	0.431523i	$-0.857976\pi$
$887$	−53.7338	−1.80420	−0.902102	+	0.431523i	$0.857976\pi$
$888$	0	0
$889$	1.10500	0.0370604
$890$	0	0
$891$	6.67884	0.223750
$892$	0	0
$893$	27.6764	0.926154
$894$	0	0
$895$	3.30210	0.110377
$896$	0	0
$897$	−1.87727	−0.0626804
$898$	0	0
$899$	24.9932	0.833570
$900$	0	0
$901$	7.52333	0.250638
$902$	0	0
$903$	6.01294	0.200098
$904$	0	0
$905$	1.66595	0.0553782
$906$	0	0
$907$	−3.15037	−0.104606	−0.0523032	−	0.998631i	$-0.516656\pi$
$907$	−3.15037	−0.104606	−0.0523032	+	0.998631i	$0.516656\pi$
$908$	0	0
$909$	−7.52016	−0.249428
$910$	0	0
$911$	40.0568	1.32714	0.663571	−	0.748114i	$-0.269040\pi$
$911$	40.0568	1.32714	0.663571	+	0.748114i	$0.269040\pi$
$912$	0	0
$913$	3.48532	0.115347
$914$	0	0
$915$	−0.207955	−0.00687479
$916$	0	0
$917$	−13.4517	−0.444215
$918$	0	0
$919$	10.5917	0.349387	0.174694	−	0.984623i	$-0.444107\pi$
$919$	10.5917	0.349387	0.174694	+	0.984623i	$0.444107\pi$
$920$	0	0
$921$	47.5007	1.56520
$922$	0	0
$923$	13.7254	0.451777
$924$	0	0
$925$	−42.6864	−1.40352
$926$	0	0
$927$	10.1776	0.334277
$928$	0	0
$929$	3.65036	0.119764	0.0598822	−	0.998205i	$-0.480927\pi$
$929$	3.65036	0.119764	0.0598822	+	0.998205i	$0.480927\pi$
$930$	0	0
$931$	7.88361	0.258375
$932$	0	0
$933$	−33.1082	−1.08392
$934$	0	0
$935$	0.256512	0.00838882
$936$	0	0
$937$	34.2893	1.12018	0.560091	−	0.828431i	$-0.310766\pi$
$937$	34.2893	1.12018	0.560091	+	0.828431i	$0.310766\pi$
$938$	0	0
$939$	40.1581	1.31051
$940$	0	0
$941$	32.1415	1.04778	0.523891	−	0.851785i	$-0.324480\pi$
$941$	32.1415	1.04778	0.523891	+	0.851785i	$0.324480\pi$
$942$	0	0
$943$	9.72179	0.316585
$944$	0	0
$945$	−1.21227	−0.0394353
$946$	0	0
$947$	−23.8830	−0.776093	−0.388047	−	0.921640i	$-0.626850\pi$
$947$	−23.8830	−0.776093	−0.388047	+	0.921640i	$0.626850\pi$
$948$	0	0
$949$	−16.1174	−0.523194
$950$	0	0
$951$	27.2860	0.884810
$952$	0	0
$953$	−2.13519	−0.0691655	−0.0345828	−	0.999402i	$-0.511010\pi$
$953$	−2.13519	−0.0691655	−0.0345828	+	0.999402i	$0.511010\pi$
$954$	0	0
$955$	4.88105	0.157947
$956$	0	0
$957$	−7.99850	−0.258555
$958$	0	0
$959$	5.13711	0.165886
$960$	0	0
$961$	−7.93775	−0.256057
$962$	0	0
$963$	2.60488	0.0839411
$964$	0	0
$965$	−4.32080	−0.139092
$966$	0	0
$967$	−47.5940	−1.53052	−0.765261	−	0.643720i	$-0.777390\pi$
$967$	−47.5940	−1.53052	−0.765261	+	0.643720i	$0.777390\pi$
$968$	0	0
$969$	−14.3342	−0.460479
$970$	0	0
$971$	−6.66802	−0.213987	−0.106993	−	0.994260i	$-0.534122\pi$
$971$	−6.66802	−0.213987	−0.106993	+	0.994260i	$0.534122\pi$
$972$	0	0
$973$	3.18510	0.102110
$974$	0	0
$975$	−7.61211	−0.243783
$976$	0	0
$977$	−8.35855	−0.267414	−0.133707	−	0.991021i	$-0.542688\pi$
$977$	−8.35855	−0.267414	−0.133707	+	0.991021i	$0.542688\pi$
$978$	0	0
$979$	−3.11011	−0.0993996
$980$	0	0
$981$	5.70306	0.182085
$982$	0	0
$983$	15.9127	0.507536	0.253768	−	0.967265i	$-0.418330\pi$
$983$	15.9127	0.507536	0.253768	+	0.967265i	$0.418330\pi$
$984$	0	0
$985$	−4.41232	−0.140588
$986$	0	0
$987$	−5.39537	−0.171737
$988$	0	0
$989$	4.77903	0.151964
$990$	0	0
$991$	38.5861	1.22573	0.612863	−	0.790189i	$-0.290018\pi$
$991$	38.5861	1.22573	0.612863	+	0.790189i	$0.290018\pi$
$992$	0	0
$993$	−9.75331	−0.309512
$994$	0	0
$995$	−2.10019	−0.0665805
$996$	0	0
$997$	52.4700	1.66174	0.830871	−	0.556465i	$-0.187843\pi$
$997$	52.4700	1.66174	0.830871	+	0.556465i	$0.187843\pi$
$998$	0	0
$999$	48.1865	1.52455

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.y.1.5	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.y.1.5	✓	14	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.53687 q^{3} -0.216819 q^{5} +1.00000 q^{7} -0.638026 q^{9} +O(q^{10})\)	\(q-1.53687 q^{3} -0.216819 q^{5} +1.00000 q^{7} -0.638026 q^{9} -1.00000 q^{11} -1.00000 q^{13} +0.333223 q^{15} +1.18307 q^{17} +7.88361 q^{19} -1.53687 q^{21} -1.22149 q^{23} -4.95299 q^{25} +5.59118 q^{27} -5.20441 q^{29} -4.80232 q^{31} +1.53687 q^{33} -0.216819 q^{35} +8.61831 q^{37} +1.53687 q^{39} -7.95896 q^{41} -3.91246 q^{43} +0.138336 q^{45} +3.51062 q^{47} +1.00000 q^{49} -1.81822 q^{51} +6.35917 q^{53} +0.216819 q^{55} -12.1161 q^{57} -4.23350 q^{59} -0.624073 q^{61} -0.638026 q^{63} +0.216819 q^{65} +4.74600 q^{67} +1.87727 q^{69} -13.7254 q^{71} +16.1174 q^{73} +7.61211 q^{75} -1.00000 q^{77} +13.0838 q^{79} -6.67884 q^{81} -3.48532 q^{83} -0.256512 q^{85} +7.99850 q^{87} +3.11011 q^{89} -1.00000 q^{91} +7.38054 q^{93} -1.70932 q^{95} -3.26128 q^{97} +0.638026 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9}+O(q^{10}) \) 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9	\( 14 q - 3 q^{3} - 6 q^{5} + 14 q^{7} + 21 q^{9} - 14 q^{11} - 14 q^{13} - 6 q^{15} - 6 q^{17} - 13 q^{19} - 3 q^{21} - 9 q^{23} + 22 q^{25} - 18 q^{27} + 2 q^{29} - 2 q^{31} + 3 q^{33} - 6 q^{35} - q^{37} + 3 q^{39} - 16 q^{41} - 15 q^{43} - 44 q^{45} - 8 q^{47} + 14 q^{49} - 14 q^{51} - 6 q^{53} + 6 q^{55} - 10 q^{57} - 36 q^{59} - 19 q^{61} + 21 q^{63} + 6 q^{65} - 34 q^{67} - q^{69} - 10 q^{71} + 9 q^{73} - 44 q^{75} - 14 q^{77} - q^{79} + 42 q^{81} - 56 q^{83} + 21 q^{85} - 5 q^{87} - 14 q^{89} - 14 q^{91} - 20 q^{93} + q^{95} - 14 q^{97} - 21 q^{99}+O(q^{100}) \) 14 * q - 3 * q^3 - 6 * q^5 + 14 * q^7 + 21 * q^9 - 14 * q^11 - 14 * q^13 - 6 * q^15 - 6 * q^17 - 13 * q^19 - 3 * q^21 - 9 * q^23 + 22 * q^25 - 18 * q^27 + 2 * q^29 - 2 * q^31 + 3 * q^33 - 6 * q^35 - q^37 + 3 * q^39 - 16 * q^41 - 15 * q^43 - 44 * q^45 - 8 * q^47 + 14 * q^49 - 14 * q^51 - 6 * q^53 + 6 * q^55 - 10 * q^57 - 36 * q^59 - 19 * q^61 + 21 * q^63 + 6 * q^65 - 34 * q^67 - q^69 - 10 * q^71 + 9 * q^73 - 44 * q^75 - 14 * q^77 - q^79 + 42 * q^81 - 56 * q^83 + 21 * q^85 - 5 * q^87 - 14 * q^89 - 14 * q^91 - 20 * q^93 + q^95 - 14 * q^97 - 21 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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