LMFDB - Embedded newform 8008.2.a.w.1.7

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:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 3 x^{10} - 19 x^{9} + 55 x^{8} + 128 x^{7} - 361 x^{6} - 343 x^{5} + 1012 x^{4} + 215 x^{3} + \cdots + 160 $ x^11 - 3x^10 - 19x^9 + 55x^8 + 128x^7 - 361x^6 - 343x^5 + 1012x^4 + 215x^3 - 1090x^2 + 240x + 160
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.12948$ 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)

$f(q)$	$=$	$q+1.12948 q^{3} +2.32569 q^{5} -1.00000 q^{7} -1.72428 q^{9} +O(q^{10})$	$q+1.12948 q^{3} +2.32569 q^{5} -1.00000 q^{7} -1.72428 q^{9} +1.00000 q^{11} +1.00000 q^{13} +2.62682 q^{15} -0.181028 q^{17} -3.85091 q^{19} -1.12948 q^{21} +5.88552 q^{23} +0.408833 q^{25} -5.33597 q^{27} +5.30964 q^{29} -0.646249 q^{31} +1.12948 q^{33} -2.32569 q^{35} -4.51899 q^{37} +1.12948 q^{39} +0.0197976 q^{41} +9.58543 q^{43} -4.01013 q^{45} +0.114598 q^{47} +1.00000 q^{49} -0.204467 q^{51} +5.72704 q^{53} +2.32569 q^{55} -4.34952 q^{57} +8.82125 q^{59} +7.79478 q^{61} +1.72428 q^{63} +2.32569 q^{65} +1.53967 q^{67} +6.64758 q^{69} +6.91200 q^{71} -1.11217 q^{73} +0.461769 q^{75} -1.00000 q^{77} +12.5563 q^{79} -0.854050 q^{81} -1.28048 q^{83} -0.421015 q^{85} +5.99713 q^{87} -14.3129 q^{89} -1.00000 q^{91} -0.729925 q^{93} -8.95602 q^{95} +10.4936 q^{97} -1.72428 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9}+O(q^{10}) $ 11 * q + 3 * q^3 - 2 * q^5 - 11 * q^7 + 14 * q^9	$ 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9} + 11 q^{11} + 11 q^{13} + 7 q^{15} + 9 q^{17} + 20 q^{19} - 3 q^{21} + 12 q^{23} + 13 q^{25} + 15 q^{27} + 8 q^{29} + 7 q^{31} + 3 q^{33} + 2 q^{35} - 10 q^{37} + 3 q^{39} - 2 q^{41} + 24 q^{43} - 6 q^{45} + 2 q^{47} + 11 q^{49} + 17 q^{51} + 3 q^{53} - 2 q^{55} - 16 q^{57} + q^{59} - 22 q^{61} - 14 q^{63} - 2 q^{65} + 14 q^{67} - 22 q^{69} + 6 q^{71} + 3 q^{73} - 11 q^{77} + 8 q^{79} - 9 q^{81} + 29 q^{83} - 9 q^{85} + 19 q^{87} + 20 q^{89} - 11 q^{91} - q^{93} + 18 q^{95} - 25 q^{97} + 14 q^{99}+O(q^{100}) $ 11 * q + 3 * q^3 - 2 * q^5 - 11 * q^7 + 14 * q^9 + 11 * q^11 + 11 * q^13 + 7 * q^15 + 9 * q^17 + 20 * q^19 - 3 * q^21 + 12 * q^23 + 13 * q^25 + 15 * q^27 + 8 * q^29 + 7 * q^31 + 3 * q^33 + 2 * q^35 - 10 * q^37 + 3 * q^39 - 2 * q^41 + 24 * q^43 - 6 * q^45 + 2 * q^47 + 11 * q^49 + 17 * q^51 + 3 * q^53 - 2 * q^55 - 16 * q^57 + q^59 - 22 * q^61 - 14 * q^63 - 2 * q^65 + 14 * q^67 - 22 * q^69 + 6 * q^71 + 3 * q^73 - 11 * q^77 + 8 * q^79 - 9 * q^81 + 29 * q^83 - 9 * q^85 + 19 * q^87 + 20 * q^89 - 11 * q^91 - q^93 + 18 * q^95 - 25 * q^97 + 14 * 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
$3$	1.12948	0.652106	0.326053	−	0.945352i	$-0.394281\pi$
$3$	1.12948	0.652106	0.326053	+	0.945352i	$0.394281\pi$
$4$	0	0
$5$	2.32569	1.04008	0.520040	−	0.854142i	$-0.325917\pi$
$5$	2.32569	1.04008	0.520040	+	0.854142i	$0.325917\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.72428	−0.574758
$10$	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
$15$	2.62682	0.678242
$16$	0	0
$17$	−0.181028	−0.0439057	−0.0219529	−	0.999759i	$-0.506988\pi$
$17$	−0.181028	−0.0439057	−0.0219529	+	0.999759i	$0.506988\pi$
$18$	0	0
$19$	−3.85091	−0.883459	−0.441730	−	0.897148i	$-0.645635\pi$
$19$	−3.85091	−0.883459	−0.441730	+	0.897148i	$0.645635\pi$
$20$	0	0
$21$	−1.12948	−0.246473
$22$	0	0
$23$	5.88552	1.22722	0.613608	−	0.789611i	$-0.289717\pi$
$23$	5.88552	1.22722	0.613608	+	0.789611i	$0.289717\pi$
$24$	0	0
$25$	0.408833	0.0817666
$26$	0	0
$27$	−5.33597	−1.02691
$28$	0	0
$29$	5.30964	0.985976	0.492988	−	0.870036i	$-0.335905\pi$
$29$	5.30964	0.985976	0.492988	+	0.870036i	$0.335905\pi$
$30$	0	0
$31$	−0.646249	−0.116070	−0.0580349	−	0.998315i	$-0.518483\pi$
$31$	−0.646249	−0.116070	−0.0580349	+	0.998315i	$0.518483\pi$
$32$	0	0
$33$	1.12948	0.196617
$34$	0	0
$35$	−2.32569	−0.393113
$36$	0	0
$37$	−4.51899	−0.742918	−0.371459	−	0.928449i	$-0.621142\pi$
$37$	−4.51899	−0.742918	−0.371459	+	0.928449i	$0.621142\pi$
$38$	0	0
$39$	1.12948	0.180862
$40$	0	0
$41$	0.0197976	0.00309186	0.00154593	−	0.999999i	$-0.499508\pi$
$41$	0.0197976	0.00309186	0.00154593	+	0.999999i	$0.499508\pi$
$42$	0	0
$43$	9.58543	1.46176	0.730882	−	0.682504i	$-0.239109\pi$
$43$	9.58543	1.46176	0.730882	+	0.682504i	$0.239109\pi$
$44$	0	0
$45$	−4.01013	−0.597795
$46$	0	0
$47$	0.114598	0.0167158	0.00835791	−	0.999965i	$-0.497340\pi$
$47$	0.114598	0.0167158	0.00835791	+	0.999965i	$0.497340\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.204467	−0.0286312
$52$	0	0
$53$	5.72704	0.786669	0.393334	−	0.919395i	$-0.371321\pi$
$53$	5.72704	0.786669	0.393334	+	0.919395i	$0.371321\pi$
$54$	0	0
$55$	2.32569	0.313596
$56$	0	0
$57$	−4.34952	−0.576109
$58$	0	0
$59$	8.82125	1.14843	0.574214	−	0.818705i	$-0.305308\pi$
$59$	8.82125	1.14843	0.574214	+	0.818705i	$0.305308\pi$
$60$	0	0
$61$	7.79478	0.998019	0.499010	−	0.866596i	$-0.333697\pi$
$61$	7.79478	0.998019	0.499010	+	0.866596i	$0.333697\pi$
$62$	0	0
$63$	1.72428	0.217238
$64$	0	0
$65$	2.32569	0.288466
$66$	0	0
$67$	1.53967	0.188101	0.0940504	−	0.995567i	$-0.470019\pi$
$67$	1.53967	0.188101	0.0940504	+	0.995567i	$0.470019\pi$
$68$	0	0
$69$	6.64758	0.800275
$70$	0	0
$71$	6.91200	0.820304	0.410152	−	0.912017i	$-0.365476\pi$
$71$	6.91200	0.820304	0.410152	+	0.912017i	$0.365476\pi$
$72$	0	0
$73$	−1.11217	−0.130169	−0.0650845	−	0.997880i	$-0.520732\pi$
$73$	−1.11217	−0.130169	−0.0650845	+	0.997880i	$0.520732\pi$
$74$	0	0
$75$	0.461769	0.0533205
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	12.5563	1.41270	0.706350	−	0.707863i	$-0.250341\pi$
$79$	12.5563	1.41270	0.706350	+	0.707863i	$0.250341\pi$
$80$	0	0
$81$	−0.854050	−0.0948944
$82$	0	0
$83$	−1.28048	−0.140551	−0.0702755	−	0.997528i	$-0.522388\pi$
$83$	−1.28048	−0.140551	−0.0702755	+	0.997528i	$0.522388\pi$
$84$	0	0
$85$	−0.421015	−0.0456655
$86$	0	0
$87$	5.99713	0.642960
$88$	0	0
$89$	−14.3129	−1.51717	−0.758583	−	0.651577i	$-0.774108\pi$
$89$	−14.3129	−1.51717	−0.758583	+	0.651577i	$0.774108\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−0.729925	−0.0756897
$94$	0	0
$95$	−8.95602	−0.918868
$96$	0	0
$97$	10.4936	1.06547	0.532734	−	0.846283i	$-0.321165\pi$
$97$	10.4936	1.06547	0.532734	+	0.846283i	$0.321165\pi$
$98$	0	0
$99$	−1.72428	−0.173296
$100$	0	0
$101$	−0.889266	−0.0884852	−0.0442426	−	0.999021i	$-0.514087\pi$
$101$	−0.889266	−0.0884852	−0.0442426	+	0.999021i	$0.514087\pi$
$102$	0	0
$103$	−2.20543	−0.217307	−0.108653	−	0.994080i	$-0.534654\pi$
$103$	−2.20543	−0.217307	−0.108653	+	0.994080i	$0.534654\pi$
$104$	0	0
$105$	−2.62682	−0.256351
$106$	0	0
$107$	1.52520	0.147447	0.0737235	−	0.997279i	$-0.476512\pi$
$107$	1.52520	0.147447	0.0737235	+	0.997279i	$0.476512\pi$
$108$	0	0
$109$	16.7075	1.60029	0.800146	−	0.599805i	$-0.204755\pi$
$109$	16.7075	1.60029	0.800146	+	0.599805i	$0.204755\pi$
$110$	0	0
$111$	−5.10411	−0.484461
$112$	0	0
$113$	8.39079	0.789339	0.394670	−	0.918823i	$-0.370859\pi$
$113$	8.39079	0.789339	0.394670	+	0.918823i	$0.370859\pi$
$114$	0	0
$115$	13.6879	1.27640
$116$	0	0
$117$	−1.72428	−0.159409
$118$	0	0
$119$	0.181028	0.0165948
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0.0223610	0.00201622
$124$	0	0
$125$	−10.6776	−0.955036
$126$	0	0
$127$	9.24327	0.820207	0.410104	−	0.912039i	$-0.365493\pi$
$127$	9.24327	0.820207	0.410104	+	0.912039i	$0.365493\pi$
$128$	0	0
$129$	10.8266	0.953225
$130$	0	0
$131$	−15.5287	−1.35675	−0.678374	−	0.734716i	$-0.737315\pi$
$131$	−15.5287	−1.35675	−0.678374	+	0.734716i	$0.737315\pi$
$132$	0	0
$133$	3.85091	0.333916
$134$	0	0
$135$	−12.4098	−1.06807
$136$	0	0
$137$	−8.66947	−0.740683	−0.370342	−	0.928896i	$-0.620759\pi$
$137$	−8.66947	−0.740683	−0.370342	+	0.928896i	$0.620759\pi$
$138$	0	0
$139$	−9.28409	−0.787466	−0.393733	−	0.919225i	$-0.628817\pi$
$139$	−9.28409	−0.787466	−0.393733	+	0.919225i	$0.628817\pi$
$140$	0	0
$141$	0.129436	0.0109005
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	12.3486	1.02549
$146$	0	0
$147$	1.12948	0.0931579
$148$	0	0
$149$	−7.56309	−0.619593	−0.309796	−	0.950803i	$-0.600261\pi$
$149$	−7.56309	−0.619593	−0.309796	+	0.950803i	$0.600261\pi$
$150$	0	0
$151$	18.5620	1.51055	0.755276	−	0.655407i	$-0.227503\pi$
$151$	18.5620	1.51055	0.755276	+	0.655407i	$0.227503\pi$
$152$	0	0
$153$	0.312142	0.0252352
$154$	0	0
$155$	−1.50297	−0.120722
$156$	0	0
$157$	2.12979	0.169976	0.0849880	−	0.996382i	$-0.472915\pi$
$157$	2.12979	0.169976	0.0849880	+	0.996382i	$0.472915\pi$
$158$	0	0
$159$	6.46857	0.512991
$160$	0	0
$161$	−5.88552	−0.463844
$162$	0	0
$163$	21.7858	1.70640	0.853199	−	0.521585i	$-0.174659\pi$
$163$	21.7858	1.70640	0.853199	+	0.521585i	$0.174659\pi$
$164$	0	0
$165$	2.62682	0.204498
$166$	0	0
$167$	−1.92422	−0.148900	−0.0744501	−	0.997225i	$-0.523720\pi$
$167$	−1.92422	−0.148900	−0.0744501	+	0.997225i	$0.523720\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.64003	0.507776
$172$	0	0
$173$	24.2897	1.84671	0.923356	−	0.383946i	$-0.125435\pi$
$173$	24.2897	1.84671	0.923356	+	0.383946i	$0.125435\pi$
$174$	0	0
$175$	−0.408833	−0.0309049
$176$	0	0
$177$	9.96342	0.748897
$178$	0	0
$179$	0.169868	0.0126965	0.00634826	−	0.999980i	$-0.497979\pi$
$179$	0.169868	0.0126965	0.00634826	+	0.999980i	$0.497979\pi$
$180$	0	0
$181$	−1.51035	−0.112263	−0.0561316	−	0.998423i	$-0.517877\pi$
$181$	−1.51035	−0.112263	−0.0561316	+	0.998423i	$0.517877\pi$
$182$	0	0
$183$	8.80405	0.650814
$184$	0	0
$185$	−10.5098	−0.772694
$186$	0	0
$187$	−0.181028	−0.0132381
$188$	0	0
$189$	5.33597	0.388135
$190$	0	0
$191$	4.18807	0.303038	0.151519	−	0.988454i	$-0.451584\pi$
$191$	4.18807	0.303038	0.151519	+	0.988454i	$0.451584\pi$
$192$	0	0
$193$	−24.0576	−1.73170	−0.865851	−	0.500302i	$-0.833222\pi$
$193$	−24.0576	−1.73170	−0.865851	+	0.500302i	$0.833222\pi$
$194$	0	0
$195$	2.62682	0.188110
$196$	0	0
$197$	−5.94877	−0.423832	−0.211916	−	0.977288i	$-0.567970\pi$
$197$	−5.94877	−0.423832	−0.211916	+	0.977288i	$0.567970\pi$
$198$	0	0
$199$	9.67753	0.686022	0.343011	−	0.939331i	$-0.388553\pi$
$199$	9.67753	0.686022	0.343011	+	0.939331i	$0.388553\pi$
$200$	0	0
$201$	1.73903	0.122662
$202$	0	0
$203$	−5.30964	−0.372664
$204$	0	0
$205$	0.0460430	0.00321578
$206$	0	0
$207$	−10.1483	−0.705353
$208$	0	0
$209$	−3.85091	−0.266373
$210$	0	0
$211$	22.9291	1.57850	0.789250	−	0.614071i	$-0.210469\pi$
$211$	22.9291	1.57850	0.789250	+	0.614071i	$0.210469\pi$
$212$	0	0
$213$	7.80697	0.534925
$214$	0	0
$215$	22.2927	1.52035
$216$	0	0
$217$	0.646249	0.0438702
$218$	0	0
$219$	−1.25617	−0.0848840
$220$	0	0
$221$	−0.181028	−0.0121773
$222$	0	0
$223$	−2.03513	−0.136282	−0.0681412	−	0.997676i	$-0.521707\pi$
$223$	−2.03513	−0.136282	−0.0681412	+	0.997676i	$0.521707\pi$
$224$	0	0
$225$	−0.704941	−0.0469960
$226$	0	0
$227$	−7.06436	−0.468878	−0.234439	−	0.972131i	$-0.575325\pi$
$227$	−7.06436	−0.468878	−0.234439	+	0.972131i	$0.575325\pi$
$228$	0	0
$229$	−10.9648	−0.724572	−0.362286	−	0.932067i	$-0.618004\pi$
$229$	−10.9648	−0.724572	−0.362286	+	0.932067i	$0.618004\pi$
$230$	0	0
$231$	−1.12948	−0.0743143
$232$	0	0
$233$	12.5230	0.820411	0.410205	−	0.911993i	$-0.365457\pi$
$233$	12.5230	0.820411	0.410205	+	0.911993i	$0.365457\pi$
$234$	0	0
$235$	0.266519	0.0173858
$236$	0	0
$237$	14.1821	0.921229
$238$	0	0
$239$	18.2027	1.17744	0.588719	−	0.808338i	$-0.299632\pi$
$239$	18.2027	1.17744	0.588719	+	0.808338i	$0.299632\pi$
$240$	0	0
$241$	8.50213	0.547670	0.273835	−	0.961777i	$-0.411708\pi$
$241$	8.50213	0.547670	0.273835	+	0.961777i	$0.411708\pi$
$242$	0	0
$243$	15.0433	0.965027
$244$	0	0
$245$	2.32569	0.148583
$246$	0	0
$247$	−3.85091	−0.245027
$248$	0	0
$249$	−1.44628	−0.0916541
$250$	0	0
$251$	6.77285	0.427498	0.213749	−	0.976889i	$-0.431432\pi$
$251$	6.77285	0.427498	0.213749	+	0.976889i	$0.431432\pi$
$252$	0	0
$253$	5.88552	0.370020
$254$	0	0
$255$	−0.475528	−0.0297787
$256$	0	0
$257$	−4.54257	−0.283357	−0.141679	−	0.989913i	$-0.545250\pi$
$257$	−4.54257	−0.283357	−0.141679	+	0.989913i	$0.545250\pi$
$258$	0	0
$259$	4.51899	0.280797
$260$	0	0
$261$	−9.15528	−0.566698
$262$	0	0
$263$	4.55549	0.280904	0.140452	−	0.990088i	$-0.455145\pi$
$263$	4.55549	0.280904	0.140452	+	0.990088i	$0.455145\pi$
$264$	0	0
$265$	13.3193	0.818199
$266$	0	0
$267$	−16.1661	−0.989352
$268$	0	0
$269$	−8.83020	−0.538387	−0.269193	−	0.963086i	$-0.586757\pi$
$269$	−8.83020	−0.538387	−0.269193	+	0.963086i	$0.586757\pi$
$270$	0	0
$271$	−30.7156	−1.86584	−0.932920	−	0.360084i	$-0.882748\pi$
$271$	−30.7156	−1.86584	−0.932920	+	0.360084i	$0.882748\pi$
$272$	0	0
$273$	−1.12948	−0.0683592
$274$	0	0
$275$	0.408833	0.0246536
$276$	0	0
$277$	−18.0823	−1.08646	−0.543229	−	0.839584i	$-0.682799\pi$
$277$	−18.0823	−1.08646	−0.543229	+	0.839584i	$0.682799\pi$
$278$	0	0
$279$	1.11431	0.0667121
$280$	0	0
$281$	−8.92636	−0.532502	−0.266251	−	0.963904i	$-0.585785\pi$
$281$	−8.92636	−0.532502	−0.266251	+	0.963904i	$0.585785\pi$
$282$	0	0
$283$	−12.9055	−0.767155	−0.383578	−	0.923509i	$-0.625308\pi$
$283$	−12.9055	−0.767155	−0.383578	+	0.923509i	$0.625308\pi$
$284$	0	0
$285$	−10.1156	−0.599199
$286$	0	0
$287$	−0.0197976	−0.00116861
$288$	0	0
$289$	−16.9672	−0.998072
$290$	0	0
$291$	11.8524	0.694797
$292$	0	0
$293$	23.6380	1.38094	0.690472	−	0.723359i	$-0.257403\pi$
$293$	23.6380	1.38094	0.690472	+	0.723359i	$0.257403\pi$
$294$	0	0
$295$	20.5155	1.19446
$296$	0	0
$297$	−5.33597	−0.309625
$298$	0	0
$299$	5.88552	0.340369
$300$	0	0
$301$	−9.58543	−0.552495
$302$	0	0
$303$	−1.00441	−0.0577017
$304$	0	0
$305$	18.1282	1.03802
$306$	0	0
$307$	−8.61108	−0.491460	−0.245730	−	0.969338i	$-0.579028\pi$
$307$	−8.61108	−0.491460	−0.245730	+	0.969338i	$0.579028\pi$
$308$	0	0
$309$	−2.49098	−0.141707
$310$	0	0
$311$	−15.0722	−0.854666	−0.427333	−	0.904094i	$-0.640547\pi$
$311$	−15.0722	−0.854666	−0.427333	+	0.904094i	$0.640547\pi$
$312$	0	0
$313$	−6.99314	−0.395276	−0.197638	−	0.980275i	$-0.563327\pi$
$313$	−6.99314	−0.395276	−0.197638	+	0.980275i	$0.563327\pi$
$314$	0	0
$315$	4.01013	0.225945
$316$	0	0
$317$	−30.2317	−1.69798	−0.848992	−	0.528406i	$-0.822790\pi$
$317$	−30.2317	−1.69798	−0.848992	+	0.528406i	$0.822790\pi$
$318$	0	0
$319$	5.30964	0.297283
$320$	0	0
$321$	1.72269	0.0961510
$322$	0	0
$323$	0.697122	0.0387889
$324$	0	0
$325$	0.408833	0.0226780
$326$	0	0
$327$	18.8708	1.04356
$328$	0	0
$329$	−0.114598	−0.00631799
$330$	0	0
$331$	8.59758	0.472566	0.236283	−	0.971684i	$-0.424071\pi$
$331$	8.59758	0.472566	0.236283	+	0.971684i	$0.424071\pi$
$332$	0	0
$333$	7.79199	0.426998
$334$	0	0
$335$	3.58080	0.195640
$336$	0	0
$337$	8.11013	0.441787	0.220893	−	0.975298i	$-0.429103\pi$
$337$	8.11013	0.441787	0.220893	+	0.975298i	$0.429103\pi$
$338$	0	0
$339$	9.47723	0.514732
$340$	0	0
$341$	−0.646249	−0.0349963
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	15.4602	0.832350
$346$	0	0
$347$	14.1979	0.762183	0.381091	−	0.924537i	$-0.375548\pi$
$347$	14.1979	0.762183	0.381091	+	0.924537i	$0.375548\pi$
$348$	0	0
$349$	12.8431	0.687475	0.343738	−	0.939066i	$-0.388307\pi$
$349$	12.8431	0.687475	0.343738	+	0.939066i	$0.388307\pi$
$350$	0	0
$351$	−5.33597	−0.284813
$352$	0	0
$353$	10.3307	0.549847	0.274923	−	0.961466i	$-0.411348\pi$
$353$	10.3307	0.549847	0.274923	+	0.961466i	$0.411348\pi$
$354$	0	0
$355$	16.0752	0.853182
$356$	0	0
$357$	0.204467	0.0108216
$358$	0	0
$359$	17.4351	0.920190	0.460095	−	0.887870i	$-0.347815\pi$
$359$	17.4351	0.920190	0.460095	+	0.887870i	$0.347815\pi$
$360$	0	0
$361$	−4.17050	−0.219500
$362$	0	0
$363$	1.12948	0.0592823
$364$	0	0
$365$	−2.58655	−0.135386
$366$	0	0
$367$	15.7713	0.823257	0.411629	−	0.911352i	$-0.364960\pi$
$367$	15.7713	0.823257	0.411629	+	0.911352i	$0.364960\pi$
$368$	0	0
$369$	−0.0341365	−0.00177707
$370$	0	0
$371$	−5.72704	−0.297333
$372$	0	0
$373$	−5.45895	−0.282654	−0.141327	−	0.989963i	$-0.545137\pi$
$373$	−5.45895	−0.282654	−0.141327	+	0.989963i	$0.545137\pi$
$374$	0	0
$375$	−12.0602	−0.622784
$376$	0	0
$377$	5.30964	0.273460
$378$	0	0
$379$	−20.7251	−1.06458	−0.532288	−	0.846563i	$-0.678668\pi$
$379$	−20.7251	−1.06458	−0.532288	+	0.846563i	$0.678668\pi$
$380$	0	0
$381$	10.4401	0.534862
$382$	0	0
$383$	18.4802	0.944295	0.472148	−	0.881520i	$-0.343479\pi$
$383$	18.4802	0.944295	0.472148	+	0.881520i	$0.343479\pi$
$384$	0	0
$385$	−2.32569	−0.118528
$386$	0	0
$387$	−16.5279	−0.840161
$388$	0	0
$389$	1.74996	0.0887264	0.0443632	−	0.999015i	$-0.485874\pi$
$389$	1.74996	0.0887264	0.0443632	+	0.999015i	$0.485874\pi$
$390$	0	0
$391$	−1.06544	−0.0538818
$392$	0	0
$393$	−17.5394	−0.884743
$394$	0	0
$395$	29.2022	1.46932
$396$	0	0
$397$	2.56525	0.128746	0.0643732	−	0.997926i	$-0.479495\pi$
$397$	2.56525	0.128746	0.0643732	+	0.997926i	$0.479495\pi$
$398$	0	0
$399$	4.34952	0.217749
$400$	0	0
$401$	−24.5137	−1.22416	−0.612078	−	0.790797i	$-0.709666\pi$
$401$	−24.5137	−1.22416	−0.612078	+	0.790797i	$0.709666\pi$
$402$	0	0
$403$	−0.646249	−0.0321920
$404$	0	0
$405$	−1.98625	−0.0986978
$406$	0	0
$407$	−4.51899	−0.223998
$408$	0	0
$409$	29.3062	1.44910	0.724549	−	0.689223i	$-0.242048\pi$
$409$	29.3062	1.44910	0.724549	+	0.689223i	$0.242048\pi$
$410$	0	0
$411$	−9.79200	−0.483004
$412$	0	0
$413$	−8.82125	−0.434065
$414$	0	0
$415$	−2.97800	−0.146184
$416$	0	0
$417$	−10.4862	−0.513511
$418$	0	0
$419$	−23.8693	−1.16609	−0.583046	−	0.812439i	$-0.698139\pi$
$419$	−23.8693	−1.16609	−0.583046	+	0.812439i	$0.698139\pi$
$420$	0	0
$421$	−23.6161	−1.15098	−0.575490	−	0.817809i	$-0.695189\pi$
$421$	−23.6161	−1.15098	−0.575490	+	0.817809i	$0.695189\pi$
$422$	0	0
$423$	−0.197598	−0.00960756
$424$	0	0
$425$	−0.0740102	−0.00359002
$426$	0	0
$427$	−7.79478	−0.377216
$428$	0	0
$429$	1.12948	0.0545318
$430$	0	0
$431$	−17.2702	−0.831877	−0.415938	−	0.909393i	$-0.636547\pi$
$431$	−17.2702	−0.831877	−0.415938	+	0.909393i	$0.636547\pi$
$432$	0	0
$433$	−13.6495	−0.655955	−0.327978	−	0.944686i	$-0.606367\pi$
$433$	−13.6495	−0.655955	−0.327978	+	0.944686i	$0.606367\pi$
$434$	0	0
$435$	13.9475	0.668730
$436$	0	0
$437$	−22.6646	−1.08420
$438$	0	0
$439$	31.1295	1.48573	0.742865	−	0.669441i	$-0.233466\pi$
$439$	31.1295	1.48573	0.742865	+	0.669441i	$0.233466\pi$
$440$	0	0
$441$	−1.72428	−0.0821083
$442$	0	0
$443$	−23.1587	−1.10030	−0.550151	−	0.835065i	$-0.685430\pi$
$443$	−23.1587	−1.10030	−0.550151	+	0.835065i	$0.685430\pi$
$444$	0	0
$445$	−33.2874	−1.57797
$446$	0	0
$447$	−8.54236	−0.404040
$448$	0	0
$449$	1.39757	0.0659552	0.0329776	−	0.999456i	$-0.489501\pi$
$449$	1.39757	0.0659552	0.0329776	+	0.999456i	$0.489501\pi$
$450$	0	0
$451$	0.0197976	0.000932231 0
$452$	0	0
$453$	20.9654	0.985040
$454$	0	0
$455$	−2.32569	−0.109030
$456$	0	0
$457$	19.5052	0.912415	0.456207	−	0.889874i	$-0.349208\pi$
$457$	19.5052	0.912415	0.456207	+	0.889874i	$0.349208\pi$
$458$	0	0
$459$	0.965960	0.0450871
$460$	0	0
$461$	0.713378	0.0332253	0.0166127	−	0.999862i	$-0.494712\pi$
$461$	0.713378	0.0332253	0.0166127	+	0.999862i	$0.494712\pi$
$462$	0	0
$463$	3.77374	0.175380	0.0876902	−	0.996148i	$-0.472051\pi$
$463$	3.77374	0.175380	0.0876902	+	0.996148i	$0.472051\pi$
$464$	0	0
$465$	−1.69758	−0.0787234
$466$	0	0
$467$	−2.04854	−0.0947953	−0.0473977	−	0.998876i	$-0.515093\pi$
$467$	−2.04854	−0.0947953	−0.0473977	+	0.998876i	$0.515093\pi$
$468$	0	0
$469$	−1.53967	−0.0710954
$470$	0	0
$471$	2.40556	0.110842
$472$	0	0
$473$	9.58543	0.440739
$474$	0	0
$475$	−1.57438	−0.0722375
$476$	0	0
$477$	−9.87499	−0.452145
$478$	0	0
$479$	41.9508	1.91678	0.958391	−	0.285458i	$-0.0921458\pi$
$479$	41.9508	1.91678	0.958391	+	0.285458i	$0.0921458\pi$
$480$	0	0
$481$	−4.51899	−0.206048
$482$	0	0
$483$	−6.64758	−0.302475
$484$	0	0
$485$	24.4049	1.10817
$486$	0	0
$487$	−15.5442	−0.704377	−0.352188	−	0.935929i	$-0.614562\pi$
$487$	−15.5442	−0.704377	−0.352188	+	0.935929i	$0.614562\pi$
$488$	0	0
$489$	24.6067	1.11275
$490$	0	0
$491$	−23.0932	−1.04218	−0.521091	−	0.853501i	$-0.674475\pi$
$491$	−23.0932	−1.04218	−0.521091	+	0.853501i	$0.674475\pi$
$492$	0	0
$493$	−0.961193	−0.0432900
$494$	0	0
$495$	−4.01013	−0.180242
$496$	0	0
$497$	−6.91200	−0.310046
$498$	0	0
$499$	6.61723	0.296228	0.148114	−	0.988970i	$-0.452680\pi$
$499$	6.61723	0.296228	0.148114	+	0.988970i	$0.452680\pi$
$500$	0	0
$501$	−2.17336	−0.0970987
$502$	0	0
$503$	1.25028	0.0557472	0.0278736	−	0.999611i	$-0.491126\pi$
$503$	1.25028	0.0557472	0.0278736	+	0.999611i	$0.491126\pi$
$504$	0	0
$505$	−2.06816	−0.0920317
$506$	0	0
$507$	1.12948	0.0501620
$508$	0	0
$509$	−22.3819	−0.992059	−0.496030	−	0.868306i	$-0.665209\pi$
$509$	−22.3819	−0.992059	−0.496030	+	0.868306i	$0.665209\pi$
$510$	0	0
$511$	1.11217	0.0491993
$512$	0	0
$513$	20.5483	0.907232
$514$	0	0
$515$	−5.12913	−0.226017
$516$	0	0
$517$	0.114598	0.00504001
$518$	0	0
$519$	27.4347	1.20425
$520$	0	0
$521$	6.37854	0.279449	0.139725	−	0.990190i	$-0.455378\pi$
$521$	6.37854	0.279449	0.139725	+	0.990190i	$0.455378\pi$
$522$	0	0
$523$	−0.982462	−0.0429601	−0.0214800	−	0.999769i	$-0.506838\pi$
$523$	−0.982462	−0.0429601	−0.0214800	+	0.999769i	$0.506838\pi$
$524$	0	0
$525$	−0.461769	−0.0201532
$526$	0	0
$527$	0.116989	0.00509612
$528$	0	0
$529$	11.6394	0.506060
$530$	0	0
$531$	−15.2103	−0.660069
$532$	0	0
$533$	0.0197976	0.000857528 0
$534$	0	0
$535$	3.54715	0.153357
$536$	0	0
$537$	0.191862	0.00827947
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	17.8219	0.766225	0.383113	−	0.923702i	$-0.374852\pi$
$541$	17.8219	0.766225	0.383113	+	0.923702i	$0.374852\pi$
$542$	0	0
$543$	−1.70591	−0.0732074
$544$	0	0
$545$	38.8566	1.66443
$546$	0	0
$547$	−28.6943	−1.22688	−0.613440	−	0.789741i	$-0.710215\pi$
$547$	−28.6943	−1.22688	−0.613440	+	0.789741i	$0.710215\pi$
$548$	0	0
$549$	−13.4403	−0.573620
$550$	0	0
$551$	−20.4469	−0.871069
$552$	0	0
$553$	−12.5563	−0.533950
$554$	0	0
$555$	−11.8706	−0.503878
$556$	0	0
$557$	−31.7780	−1.34648	−0.673239	−	0.739425i	$-0.735097\pi$
$557$	−31.7780	−1.34648	−0.673239	+	0.739425i	$0.735097\pi$
$558$	0	0
$559$	9.58543	0.405421
$560$	0	0
$561$	−0.204467	−0.00863262
$562$	0	0
$563$	−0.120614	−0.00508327	−0.00254164	−	0.999997i	$-0.500809\pi$
$563$	−0.120614	−0.00508327	−0.00254164	+	0.999997i	$0.500809\pi$
$564$	0	0
$565$	19.5144	0.820976
$566$	0	0
$567$	0.854050	0.0358667
$568$	0	0
$569$	4.55279	0.190863	0.0954314	−	0.995436i	$-0.469577\pi$
$569$	4.55279	0.190863	0.0954314	+	0.995436i	$0.469577\pi$
$570$	0	0
$571$	15.8869	0.664846	0.332423	−	0.943130i	$-0.392134\pi$
$571$	15.8869	0.664846	0.332423	+	0.943130i	$0.392134\pi$
$572$	0	0
$573$	4.73034	0.197613
$574$	0	0
$575$	2.40620	0.100345
$576$	0	0
$577$	27.9386	1.16310	0.581550	−	0.813510i	$-0.302446\pi$
$577$	27.9386	1.16310	0.581550	+	0.813510i	$0.302446\pi$
$578$	0	0
$579$	−27.1725	−1.12925
$580$	0	0
$581$	1.28048	0.0531233
$582$	0	0
$583$	5.72704	0.237190
$584$	0	0
$585$	−4.01013	−0.165798
$586$	0	0
$587$	−0.713410	−0.0294456	−0.0147228	−	0.999892i	$-0.504687\pi$
$587$	−0.713410	−0.0294456	−0.0147228	+	0.999892i	$0.504687\pi$
$588$	0	0
$589$	2.48865	0.102543
$590$	0	0
$591$	−6.71901	−0.276383
$592$	0	0
$593$	16.7860	0.689316	0.344658	−	0.938728i	$-0.387995\pi$
$593$	16.7860	0.689316	0.344658	+	0.938728i	$0.387995\pi$
$594$	0	0
$595$	0.421015	0.0172599
$596$	0	0
$597$	10.9306	0.447359
$598$	0	0
$599$	−36.5454	−1.49321	−0.746603	−	0.665270i	$-0.768316\pi$
$599$	−36.5454	−1.49321	−0.746603	+	0.665270i	$0.768316\pi$
$600$	0	0
$601$	−35.1046	−1.43195	−0.715973	−	0.698128i	$-0.754017\pi$
$601$	−35.1046	−1.43195	−0.715973	+	0.698128i	$0.754017\pi$
$602$	0	0
$603$	−2.65482	−0.108112
$604$	0	0
$605$	2.32569	0.0945527
$606$	0	0
$607$	−43.0665	−1.74801	−0.874007	−	0.485913i	$-0.838487\pi$
$607$	−43.0665	−1.74801	−0.874007	+	0.485913i	$0.838487\pi$
$608$	0	0
$609$	−5.99713	−0.243016
$610$	0	0
$611$	0.114598	0.00463614
$612$	0	0
$613$	−30.7926	−1.24370	−0.621850	−	0.783136i	$-0.713619\pi$
$613$	−30.7926	−1.24370	−0.621850	+	0.783136i	$0.713619\pi$
$614$	0	0
$615$	0.0520046	0.00209703
$616$	0	0
$617$	7.33418	0.295263	0.147632	−	0.989042i	$-0.452835\pi$
$617$	7.33418	0.295263	0.147632	+	0.989042i	$0.452835\pi$
$618$	0	0
$619$	30.6060	1.23016	0.615080	−	0.788465i	$-0.289124\pi$
$619$	30.6060	1.23016	0.615080	+	0.788465i	$0.289124\pi$
$620$	0	0
$621$	−31.4050	−1.26024
$622$	0	0
$623$	14.3129	0.573435
$624$	0	0
$625$	−26.8770	−1.07508
$626$	0	0
$627$	−4.34952	−0.173703
$628$	0	0
$629$	0.818064	0.0326183
$630$	0	0
$631$	36.9149	1.46956	0.734779	−	0.678306i	$-0.237286\pi$
$631$	36.9149	1.46956	0.734779	+	0.678306i	$0.237286\pi$
$632$	0	0
$633$	25.8979	1.02935
$634$	0	0
$635$	21.4970	0.853081
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−11.9182	−0.471476
$640$	0	0
$641$	−7.10967	−0.280815	−0.140407	−	0.990094i	$-0.544841\pi$
$641$	−7.10967	−0.280815	−0.140407	+	0.990094i	$0.544841\pi$
$642$	0	0
$643$	−22.2334	−0.876799	−0.438399	−	0.898780i	$-0.644454\pi$
$643$	−22.2334	−0.876799	−0.438399	+	0.898780i	$0.644454\pi$
$644$	0	0
$645$	25.1792	0.991430
$646$	0	0
$647$	−2.13592	−0.0839716	−0.0419858	−	0.999118i	$-0.513368\pi$
$647$	−2.13592	−0.0839716	−0.0419858	+	0.999118i	$0.513368\pi$
$648$	0	0
$649$	8.82125	0.346264
$650$	0	0
$651$	0.729925	0.0286080
$652$	0	0
$653$	−16.8729	−0.660286	−0.330143	−	0.943931i	$-0.607097\pi$
$653$	−16.8729	−0.660286	−0.330143	+	0.943931i	$0.607097\pi$
$654$	0	0
$655$	−36.1149	−1.41113
$656$	0	0
$657$	1.91768	0.0748158
$658$	0	0
$659$	6.73540	0.262374	0.131187	−	0.991358i	$-0.458121\pi$
$659$	6.73540	0.262374	0.131187	+	0.991358i	$0.458121\pi$
$660$	0	0
$661$	−25.8584	−1.00578	−0.502888	−	0.864352i	$-0.667729\pi$
$661$	−25.8584	−1.00578	−0.502888	+	0.864352i	$0.667729\pi$
$662$	0	0
$663$	−0.204467	−0.00794085
$664$	0	0
$665$	8.95602	0.347300
$666$	0	0
$667$	31.2500	1.21001
$668$	0	0
$669$	−2.29864	−0.0888705
$670$	0	0
$671$	7.79478	0.300914
$672$	0	0
$673$	26.9396	1.03845	0.519223	−	0.854639i	$-0.326221\pi$
$673$	26.9396	1.03845	0.519223	+	0.854639i	$0.326221\pi$
$674$	0	0
$675$	−2.18152	−0.0839668
$676$	0	0
$677$	−18.7267	−0.719724	−0.359862	−	0.933006i	$-0.617176\pi$
$677$	−18.7267	−0.719724	−0.359862	+	0.933006i	$0.617176\pi$
$678$	0	0
$679$	−10.4936	−0.402709
$680$	0	0
$681$	−7.97905	−0.305758
$682$	0	0
$683$	−25.2787	−0.967263	−0.483631	−	0.875272i	$-0.660682\pi$
$683$	−25.2787	−0.967263	−0.483631	+	0.875272i	$0.660682\pi$
$684$	0	0
$685$	−20.1625	−0.770370
$686$	0	0
$687$	−12.3845	−0.472498
$688$	0	0
$689$	5.72704	0.218183
$690$	0	0
$691$	30.9255	1.17646	0.588230	−	0.808693i	$-0.299825\pi$
$691$	30.9255	1.17646	0.588230	+	0.808693i	$0.299825\pi$
$692$	0	0
$693$	1.72428	0.0654998
$694$	0	0
$695$	−21.5919	−0.819028
$696$	0	0
$697$	−0.00358391	−0.000135750 0
$698$	0	0
$699$	14.1445	0.534994
$700$	0	0
$701$	28.3634	1.07127	0.535636	−	0.844449i	$-0.320072\pi$
$701$	28.3634	1.07127	0.535636	+	0.844449i	$0.320072\pi$
$702$	0	0
$703$	17.4022	0.656337
$704$	0	0
$705$	0.301028	0.0113374
$706$	0	0
$707$	0.889266	0.0334443
$708$	0	0
$709$	−44.2817	−1.66303	−0.831517	−	0.555499i	$-0.812527\pi$
$709$	−44.2817	−1.66303	−0.831517	+	0.555499i	$0.812527\pi$
$710$	0	0
$711$	−21.6506	−0.811961
$712$	0	0
$713$	−3.80351	−0.142443
$714$	0	0
$715$	2.32569	0.0869759
$716$	0	0
$717$	20.5596	0.767814
$718$	0	0
$719$	31.9159	1.19026	0.595131	−	0.803629i	$-0.297100\pi$
$719$	31.9159	1.19026	0.595131	+	0.803629i	$0.297100\pi$
$720$	0	0
$721$	2.20543	0.0821343
$722$	0	0
$723$	9.60298	0.357139
$724$	0	0
$725$	2.17076	0.0806199
$726$	0	0
$727$	17.3382	0.643037	0.321518	−	0.946903i	$-0.395807\pi$
$727$	17.3382	0.643037	0.321518	+	0.946903i	$0.395807\pi$
$728$	0	0
$729$	19.5532	0.724194
$730$	0	0
$731$	−1.73523	−0.0641798
$732$	0	0
$733$	−21.7959	−0.805051	−0.402526	−	0.915409i	$-0.631868\pi$
$733$	−21.7959	−0.805051	−0.402526	+	0.915409i	$0.631868\pi$
$734$	0	0
$735$	2.62682	0.0968917
$736$	0	0
$737$	1.53967	0.0567145
$738$	0	0
$739$	33.2083	1.22159	0.610793	−	0.791790i	$-0.290851\pi$
$739$	33.2083	1.22159	0.610793	+	0.791790i	$0.290851\pi$
$740$	0	0
$741$	−4.34952	−0.159784
$742$	0	0
$743$	−2.50762	−0.0919955	−0.0459977	−	0.998942i	$-0.514647\pi$
$743$	−2.50762	−0.0919955	−0.0459977	+	0.998942i	$0.514647\pi$
$744$	0	0
$745$	−17.5894	−0.644426
$746$	0	0
$747$	2.20790	0.0807829
$748$	0	0
$749$	−1.52520	−0.0557297
$750$	0	0
$751$	14.3197	0.522533	0.261266	−	0.965267i	$-0.415860\pi$
$751$	14.3197	0.522533	0.261266	+	0.965267i	$0.415860\pi$
$752$	0	0
$753$	7.64980	0.278774
$754$	0	0
$755$	43.1694	1.57110
$756$	0	0
$757$	10.5357	0.382927	0.191464	−	0.981500i	$-0.438677\pi$
$757$	10.5357	0.382927	0.191464	+	0.981500i	$0.438677\pi$
$758$	0	0
$759$	6.64758	0.241292
$760$	0	0
$761$	40.9248	1.48352	0.741761	−	0.670664i	$-0.233991\pi$
$761$	40.9248	1.48352	0.741761	+	0.670664i	$0.233991\pi$
$762$	0	0
$763$	−16.7075	−0.604854
$764$	0	0
$765$	0.725945	0.0262466
$766$	0	0
$767$	8.82125	0.318517
$768$	0	0
$769$	−8.46379	−0.305212	−0.152606	−	0.988287i	$-0.548767\pi$
$769$	−8.46379	−0.305212	−0.152606	+	0.988287i	$0.548767\pi$
$770$	0	0
$771$	−5.13074	−0.184779
$772$	0	0
$773$	−54.9250	−1.97551	−0.987757	−	0.155999i	$-0.950140\pi$
$773$	−54.9250	−1.97551	−0.987757	+	0.155999i	$0.950140\pi$
$774$	0	0
$775$	−0.264208	−0.00949063
$776$	0	0
$777$	5.10411	0.183109
$778$	0	0
$779$	−0.0762386	−0.00273153
$780$	0	0
$781$	6.91200	0.247331
$782$	0	0
$783$	−28.3321	−1.01251
$784$	0	0
$785$	4.95324	0.176789
$786$	0	0
$787$	30.3957	1.08349	0.541746	−	0.840543i	$-0.317764\pi$
$787$	30.3957	1.08349	0.541746	+	0.840543i	$0.317764\pi$
$788$	0	0
$789$	5.14533	0.183179
$790$	0	0
$791$	−8.39079	−0.298342
$792$	0	0
$793$	7.79478	0.276801
$794$	0	0
$795$	15.0439	0.533552
$796$	0	0
$797$	−42.2973	−1.49825	−0.749123	−	0.662431i	$-0.769525\pi$
$797$	−42.2973	−1.49825	−0.749123	+	0.662431i	$0.769525\pi$
$798$	0	0
$799$	−0.0207454	−0.000733920 0
$800$	0	0
$801$	24.6794	0.872003
$802$	0	0
$803$	−1.11217	−0.0392475
$804$	0	0
$805$	−13.6879	−0.482435
$806$	0	0
$807$	−9.97353	−0.351085
$808$	0	0
$809$	−7.53375	−0.264873	−0.132436	−	0.991192i	$-0.542280\pi$
$809$	−7.53375	−0.264873	−0.132436	+	0.991192i	$0.542280\pi$
$810$	0	0
$811$	−2.05203	−0.0720565	−0.0360282	−	0.999351i	$-0.511471\pi$
$811$	−2.05203	−0.0720565	−0.0360282	+	0.999351i	$0.511471\pi$
$812$	0	0
$813$	−34.6927	−1.21672
$814$	0	0
$815$	50.6671	1.77479
$816$	0	0
$817$	−36.9126	−1.29141
$818$	0	0
$819$	1.72428	0.0602510
$820$	0	0
$821$	51.0274	1.78087	0.890434	−	0.455112i	$-0.150401\pi$
$821$	51.0274	1.78087	0.890434	+	0.455112i	$0.150401\pi$
$822$	0	0
$823$	−3.12344	−0.108876	−0.0544381	−	0.998517i	$-0.517337\pi$
$823$	−3.12344	−0.108876	−0.0544381	+	0.998517i	$0.517337\pi$
$824$	0	0
$825$	0.461769	0.0160767
$826$	0	0
$827$	43.8924	1.52629	0.763144	−	0.646229i	$-0.223655\pi$
$827$	43.8924	1.52629	0.763144	+	0.646229i	$0.223655\pi$
$828$	0	0
$829$	−2.39254	−0.0830963	−0.0415481	−	0.999137i	$-0.513229\pi$
$829$	−2.39254	−0.0830963	−0.0415481	+	0.999137i	$0.513229\pi$
$830$	0	0
$831$	−20.4236	−0.708486
$832$	0	0
$833$	−0.181028	−0.00627224
$834$	0	0
$835$	−4.47513	−0.154868
$836$	0	0
$837$	3.44837	0.119193
$838$	0	0
$839$	−18.6951	−0.645427	−0.322714	−	0.946497i	$-0.604595\pi$
$839$	−18.6951	−0.645427	−0.322714	+	0.946497i	$0.604595\pi$
$840$	0	0
$841$	−0.807713	−0.0278522
$842$	0	0
$843$	−10.0821	−0.347247
$844$	0	0
$845$	2.32569	0.0800062
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−14.5766	−0.500266
$850$	0	0
$851$	−26.5966	−0.911721
$852$	0	0
$853$	−0.817001	−0.0279736	−0.0139868	−	0.999902i	$-0.504452\pi$
$853$	−0.817001	−0.0279736	−0.0139868	+	0.999902i	$0.504452\pi$
$854$	0	0
$855$	15.4426	0.528127
$856$	0	0
$857$	13.1986	0.450856	0.225428	−	0.974260i	$-0.427622\pi$
$857$	13.1986	0.450856	0.225428	+	0.974260i	$0.427622\pi$
$858$	0	0
$859$	33.5795	1.14572	0.572858	−	0.819654i	$-0.305835\pi$
$859$	33.5795	1.14572	0.572858	+	0.819654i	$0.305835\pi$
$860$	0	0
$861$	−0.0223610	−0.000762059 0
$862$	0	0
$863$	16.9653	0.577506	0.288753	−	0.957404i	$-0.406759\pi$
$863$	16.9653	0.577506	0.288753	+	0.957404i	$0.406759\pi$
$864$	0	0
$865$	56.4903	1.92073
$866$	0	0
$867$	−19.1641	−0.650848
$868$	0	0
$869$	12.5563	0.425945
$870$	0	0
$871$	1.53967	0.0521698
$872$	0	0
$873$	−18.0939	−0.612386
$874$	0	0
$875$	10.6776	0.360970
$876$	0	0
$877$	−40.2154	−1.35798	−0.678988	−	0.734149i	$-0.737581\pi$
$877$	−40.2154	−1.35798	−0.678988	+	0.734149i	$0.737581\pi$
$878$	0	0
$879$	26.6986	0.900521
$880$	0	0
$881$	10.6695	0.359466	0.179733	−	0.983715i	$-0.442477\pi$
$881$	10.6695	0.359466	0.179733	+	0.983715i	$0.442477\pi$
$882$	0	0
$883$	31.3450	1.05484	0.527421	−	0.849604i	$-0.323159\pi$
$883$	31.3450	1.05484	0.527421	+	0.849604i	$0.323159\pi$
$884$	0	0
$885$	23.1718	0.778913
$886$	0	0
$887$	14.3035	0.480264	0.240132	−	0.970740i	$-0.422809\pi$
$887$	14.3035	0.480264	0.240132	+	0.970740i	$0.422809\pi$
$888$	0	0
$889$	−9.24327	−0.310009
$890$	0	0
$891$	−0.854050	−0.0286117
$892$	0	0
$893$	−0.441306	−0.0147678
$894$	0	0
$895$	0.395060	0.0132054
$896$	0	0
$897$	6.64758	0.221956
$898$	0	0
$899$	−3.43135	−0.114442
$900$	0	0
$901$	−1.03675	−0.0345393
$902$	0	0
$903$	−10.8266	−0.360285
$904$	0	0
$905$	−3.51260	−0.116763
$906$	0	0
$907$	39.6570	1.31679	0.658395	−	0.752673i	$-0.271236\pi$
$907$	39.6570	1.31679	0.658395	+	0.752673i	$0.271236\pi$
$908$	0	0
$909$	1.53334	0.0508576
$910$	0	0
$911$	14.8931	0.493430	0.246715	−	0.969088i	$-0.420649\pi$
$911$	14.8931	0.493430	0.246715	+	0.969088i	$0.420649\pi$
$912$	0	0
$913$	−1.28048	−0.0423777
$914$	0	0
$915$	20.4755	0.676899
$916$	0	0
$917$	15.5287	0.512803
$918$	0	0
$919$	−1.17538	−0.0387721	−0.0193861	−	0.999812i	$-0.506171\pi$
$919$	−1.17538	−0.0387721	−0.0193861	+	0.999812i	$0.506171\pi$
$920$	0	0
$921$	−9.72604	−0.320484
$922$	0	0
$923$	6.91200	0.227511
$924$	0	0
$925$	−1.84751	−0.0607459
$926$	0	0
$927$	3.80276	0.124899
$928$	0	0
$929$	1.01581	0.0333275	0.0166638	−	0.999861i	$-0.494696\pi$
$929$	1.01581	0.0333275	0.0166638	+	0.999861i	$0.494696\pi$
$930$	0	0
$931$	−3.85091	−0.126208
$932$	0	0
$933$	−17.0237	−0.557332
$934$	0	0
$935$	−0.421015	−0.0137687
$936$	0	0
$937$	17.1280	0.559547	0.279773	−	0.960066i	$-0.409741\pi$
$937$	17.1280	0.559547	0.279773	+	0.960066i	$0.409741\pi$
$938$	0	0
$939$	−7.89861	−0.257762
$940$	0	0
$941$	27.7968	0.906149	0.453074	−	0.891473i	$-0.350327\pi$
$941$	27.7968	0.906149	0.453074	+	0.891473i	$0.350327\pi$
$942$	0	0
$943$	0.116519	0.00379438
$944$	0	0
$945$	12.4098	0.403691
$946$	0	0
$947$	30.1775	0.980638	0.490319	−	0.871543i	$-0.336880\pi$
$947$	30.1775	0.980638	0.490319	+	0.871543i	$0.336880\pi$
$948$	0	0
$949$	−1.11217	−0.0361024
$950$	0	0
$951$	−34.1461	−1.10726
$952$	0	0
$953$	11.6094	0.376065	0.188033	−	0.982163i	$-0.439789\pi$
$953$	11.6094	0.376065	0.188033	+	0.982163i	$0.439789\pi$
$954$	0	0
$955$	9.74015	0.315184
$956$	0	0
$957$	5.99713	0.193860
$958$	0	0
$959$	8.66947	0.279952
$960$	0	0
$961$	−30.5824	−0.986528
$962$	0	0
$963$	−2.62987	−0.0847464
$964$	0	0
$965$	−55.9505	−1.80111
$966$	0	0
$967$	−32.8688	−1.05699	−0.528495	−	0.848936i	$-0.677243\pi$
$967$	−32.8688	−1.05699	−0.528495	+	0.848936i	$0.677243\pi$
$968$	0	0
$969$	0.787385	0.0252945
$970$	0	0
$971$	−15.7976	−0.506968	−0.253484	−	0.967340i	$-0.581577\pi$
$971$	−15.7976	−0.506968	−0.253484	+	0.967340i	$0.581577\pi$
$972$	0	0
$973$	9.28409	0.297634
$974$	0	0
$975$	0.461769	0.0147884
$976$	0	0
$977$	20.0921	0.642802	0.321401	−	0.946943i	$-0.395846\pi$
$977$	20.0921	0.642802	0.321401	+	0.946943i	$0.395846\pi$
$978$	0	0
$979$	−14.3129	−0.457442
$980$	0	0
$981$	−28.8084	−0.919782
$982$	0	0
$983$	−41.3990	−1.32042	−0.660212	−	0.751079i	$-0.729534\pi$
$983$	−41.3990	−1.32042	−0.660212	+	0.751079i	$0.729534\pi$
$984$	0	0
$985$	−13.8350	−0.440819
$986$	0	0
$987$	−0.129436	−0.00412000
$988$	0	0
$989$	56.4153	1.79390
$990$	0	0
$991$	41.6094	1.32177	0.660883	−	0.750489i	$-0.270182\pi$
$991$	41.6094	1.32177	0.660883	+	0.750489i	$0.270182\pi$
$992$	0	0
$993$	9.71079	0.308163
$994$	0	0
$995$	22.5069	0.713518
$996$	0	0
$997$	−18.3593	−0.581446	−0.290723	−	0.956807i	$-0.593896\pi$
$997$	−18.3593	−0.581446	−0.290723	+	0.956807i	$0.593896\pi$
$998$	0	0
$999$	24.1132	0.762909

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.w.1.7	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.w.1.7	✓	11	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)

\(f(q)\)	\(=\)	\(q+1.12948 q^{3} +2.32569 q^{5} -1.00000 q^{7} -1.72428 q^{9} +O(q^{10})\)	\(q+1.12948 q^{3} +2.32569 q^{5} -1.00000 q^{7} -1.72428 q^{9} +1.00000 q^{11} +1.00000 q^{13} +2.62682 q^{15} -0.181028 q^{17} -3.85091 q^{19} -1.12948 q^{21} +5.88552 q^{23} +0.408833 q^{25} -5.33597 q^{27} +5.30964 q^{29} -0.646249 q^{31} +1.12948 q^{33} -2.32569 q^{35} -4.51899 q^{37} +1.12948 q^{39} +0.0197976 q^{41} +9.58543 q^{43} -4.01013 q^{45} +0.114598 q^{47} +1.00000 q^{49} -0.204467 q^{51} +5.72704 q^{53} +2.32569 q^{55} -4.34952 q^{57} +8.82125 q^{59} +7.79478 q^{61} +1.72428 q^{63} +2.32569 q^{65} +1.53967 q^{67} +6.64758 q^{69} +6.91200 q^{71} -1.11217 q^{73} +0.461769 q^{75} -1.00000 q^{77} +12.5563 q^{79} -0.854050 q^{81} -1.28048 q^{83} -0.421015 q^{85} +5.99713 q^{87} -14.3129 q^{89} -1.00000 q^{91} -0.729925 q^{93} -8.95602 q^{95} +10.4936 q^{97} -1.72428 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9}+O(q^{10}) \) 11 * q + 3 * q^3 - 2 * q^5 - 11 * q^7 + 14 * q^9	\( 11 q + 3 q^{3} - 2 q^{5} - 11 q^{7} + 14 q^{9} + 11 q^{11} + 11 q^{13} + 7 q^{15} + 9 q^{17} + 20 q^{19} - 3 q^{21} + 12 q^{23} + 13 q^{25} + 15 q^{27} + 8 q^{29} + 7 q^{31} + 3 q^{33} + 2 q^{35} - 10 q^{37} + 3 q^{39} - 2 q^{41} + 24 q^{43} - 6 q^{45} + 2 q^{47} + 11 q^{49} + 17 q^{51} + 3 q^{53} - 2 q^{55} - 16 q^{57} + q^{59} - 22 q^{61} - 14 q^{63} - 2 q^{65} + 14 q^{67} - 22 q^{69} + 6 q^{71} + 3 q^{73} - 11 q^{77} + 8 q^{79} - 9 q^{81} + 29 q^{83} - 9 q^{85} + 19 q^{87} + 20 q^{89} - 11 q^{91} - q^{93} + 18 q^{95} - 25 q^{97} + 14 q^{99}+O(q^{100}) \) 11 * q + 3 * q^3 - 2 * q^5 - 11 * q^7 + 14 * q^9 + 11 * q^11 + 11 * q^13 + 7 * q^15 + 9 * q^17 + 20 * q^19 - 3 * q^21 + 12 * q^23 + 13 * q^25 + 15 * q^27 + 8 * q^29 + 7 * q^31 + 3 * q^33 + 2 * q^35 - 10 * q^37 + 3 * q^39 - 2 * q^41 + 24 * q^43 - 6 * q^45 + 2 * q^47 + 11 * q^49 + 17 * q^51 + 3 * q^53 - 2 * q^55 - 16 * q^57 + q^59 - 22 * q^61 - 14 * q^63 - 2 * q^65 + 14 * q^67 - 22 * q^69 + 6 * q^71 + 3 * q^73 - 11 * q^77 + 8 * q^79 - 9 * q^81 + 29 * q^83 - 9 * q^85 + 19 * q^87 + 20 * q^89 - 11 * q^91 - q^93 + 18 * q^95 - 25 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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