LMFDB - Embedded newform 8008.2.a.o.1.2

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 12x^{7} + 20x^{6} + 47x^{5} - 55x^{4} - 68x^{3} + 37x^{2} + 21x - 9 $ x^9 - 2x^8 - 12x^7 + 20x^6 + 47x^5 - 55x^4 - 68x^3 + 37x^2 + 21x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.33191$ 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)

$f(q)$	$=$	$q-2.33191 q^{3} -2.59697 q^{5} +1.00000 q^{7} +2.43780 q^{9} +O(q^{10})$	$q-2.33191 q^{3} -2.59697 q^{5} +1.00000 q^{7} +2.43780 q^{9} -1.00000 q^{11} +1.00000 q^{13} +6.05589 q^{15} +0.771345 q^{17} -0.470436 q^{19} -2.33191 q^{21} -7.90862 q^{23} +1.74424 q^{25} +1.31100 q^{27} -3.36178 q^{29} +5.85131 q^{31} +2.33191 q^{33} -2.59697 q^{35} +7.25519 q^{37} -2.33191 q^{39} -10.7531 q^{41} +8.08298 q^{43} -6.33088 q^{45} -1.01253 q^{47} +1.00000 q^{49} -1.79871 q^{51} -2.08412 q^{53} +2.59697 q^{55} +1.09701 q^{57} -2.38104 q^{59} +4.66941 q^{61} +2.43780 q^{63} -2.59697 q^{65} +6.10785 q^{67} +18.4422 q^{69} +6.10833 q^{71} +8.85247 q^{73} -4.06740 q^{75} -1.00000 q^{77} -11.2259 q^{79} -10.3705 q^{81} +1.15206 q^{83} -2.00316 q^{85} +7.83937 q^{87} -12.9985 q^{89} +1.00000 q^{91} -13.6447 q^{93} +1.22171 q^{95} -11.7569 q^{97} -2.43780 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9}+O(q^{10}) $ 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9	$ 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9} - 9 q^{11} + 9 q^{13} - 4 q^{15} - 9 q^{17} + 9 q^{19} - 2 q^{21} - 10 q^{23} - q^{25} - 8 q^{27} - 13 q^{29} + 7 q^{31} + 2 q^{33} - 4 q^{35} - 15 q^{37} - 2 q^{39} - 18 q^{41} + 7 q^{43} + 9 q^{45} - 11 q^{47} + 9 q^{49} + q^{51} - 16 q^{53} + 4 q^{55} - 26 q^{57} - 2 q^{59} + 2 q^{61} + q^{63} - 4 q^{65} + 13 q^{67} - 3 q^{69} - q^{71} - 6 q^{73} - 4 q^{75} - 9 q^{77} - 21 q^{79} - 23 q^{81} - 30 q^{83} + q^{85} + q^{87} - 36 q^{89} + 9 q^{91} - 22 q^{93} - 23 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) $ 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9 - 9 * q^11 + 9 * q^13 - 4 * q^15 - 9 * q^17 + 9 * q^19 - 2 * q^21 - 10 * q^23 - q^25 - 8 * q^27 - 13 * q^29 + 7 * q^31 + 2 * q^33 - 4 * q^35 - 15 * q^37 - 2 * q^39 - 18 * q^41 + 7 * q^43 + 9 * q^45 - 11 * q^47 + 9 * q^49 + q^51 - 16 * q^53 + 4 * q^55 - 26 * q^57 - 2 * q^59 + 2 * q^61 + q^63 - 4 * q^65 + 13 * q^67 - 3 * q^69 - q^71 - 6 * q^73 - 4 * q^75 - 9 * q^77 - 21 * q^79 - 23 * q^81 - 30 * q^83 + q^85 + q^87 - 36 * q^89 + 9 * q^91 - 22 * q^93 - 23 * q^95 - 5 * 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
$3$	−2.33191	−1.34633	−0.673164	−	0.739493i	$-0.735065\pi$
$3$	−2.33191	−1.34633	−0.673164	+	0.739493i	$0.735065\pi$
$4$	0	0
$5$	−2.59697	−1.16140	−0.580699	−	0.814118i	$-0.697221\pi$
$5$	−2.59697	−1.16140	−0.580699	+	0.814118i	$0.697221\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	2.43780	0.812599
$10$	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
$15$	6.05589	1.56362
$16$	0	0
$17$	0.771345	0.187079	0.0935393	−	0.995616i	$-0.470182\pi$
$17$	0.771345	0.187079	0.0935393	+	0.995616i	$0.470182\pi$
$18$	0	0
$19$	−0.470436	−0.107925	−0.0539627	−	0.998543i	$-0.517185\pi$
$19$	−0.470436	−0.107925	−0.0539627	+	0.998543i	$0.517185\pi$
$20$	0	0
$21$	−2.33191	−0.508864
$22$	0	0
$23$	−7.90862	−1.64906	−0.824531	−	0.565817i	$-0.808561\pi$
$23$	−7.90862	−1.64906	−0.824531	+	0.565817i	$0.808561\pi$
$24$	0	0
$25$	1.74424	0.348848
$26$	0	0
$27$	1.31100	0.252303
$28$	0	0
$29$	−3.36178	−0.624267	−0.312133	−	0.950038i	$-0.601044\pi$
$29$	−3.36178	−0.624267	−0.312133	+	0.950038i	$0.601044\pi$
$30$	0	0
$31$	5.85131	1.05093	0.525463	−	0.850816i	$-0.323892\pi$
$31$	5.85131	1.05093	0.525463	+	0.850816i	$0.323892\pi$
$32$	0	0
$33$	2.33191	0.405933
$34$	0	0
$35$	−2.59697	−0.438968
$36$	0	0
$37$	7.25519	1.19275	0.596373	−	0.802707i	$-0.296608\pi$
$37$	7.25519	1.19275	0.596373	+	0.802707i	$0.296608\pi$
$38$	0	0
$39$	−2.33191	−0.373404
$40$	0	0
$41$	−10.7531	−1.67934	−0.839672	−	0.543094i	$-0.817253\pi$
$41$	−10.7531	−1.67934	−0.839672	+	0.543094i	$0.817253\pi$
$42$	0	0
$43$	8.08298	1.23264	0.616322	−	0.787495i	$-0.288622\pi$
$43$	8.08298	1.23264	0.616322	+	0.787495i	$0.288622\pi$
$44$	0	0
$45$	−6.33088	−0.943752
$46$	0	0
$47$	−1.01253	−0.147692	−0.0738461	−	0.997270i	$-0.523527\pi$
$47$	−1.01253	−0.147692	−0.0738461	+	0.997270i	$0.523527\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.79871	−0.251869
$52$	0	0
$53$	−2.08412	−0.286275	−0.143138	−	0.989703i	$-0.545719\pi$
$53$	−2.08412	−0.286275	−0.143138	+	0.989703i	$0.545719\pi$
$54$	0	0
$55$	2.59697	0.350175
$56$	0	0
$57$	1.09701	0.145303
$58$	0	0
$59$	−2.38104	−0.309985	−0.154993	−	0.987916i	$-0.549535\pi$
$59$	−2.38104	−0.309985	−0.154993	+	0.987916i	$0.549535\pi$
$60$	0	0
$61$	4.66941	0.597856	0.298928	−	0.954276i	$-0.403371\pi$
$61$	4.66941	0.597856	0.298928	+	0.954276i	$0.403371\pi$
$62$	0	0
$63$	2.43780	0.307134
$64$	0	0
$65$	−2.59697	−0.322114
$66$	0	0
$67$	6.10785	0.746193	0.373096	−	0.927793i	$-0.378296\pi$
$67$	6.10785	0.746193	0.373096	+	0.927793i	$0.378296\pi$
$68$	0	0
$69$	18.4422	2.22018
$70$	0	0
$71$	6.10833	0.724925	0.362463	−	0.931998i	$-0.381936\pi$
$71$	6.10833	0.724925	0.362463	+	0.931998i	$0.381936\pi$
$72$	0	0
$73$	8.85247	1.03610	0.518052	−	0.855349i	$-0.326657\pi$
$73$	8.85247	1.03610	0.518052	+	0.855349i	$0.326657\pi$
$74$	0	0
$75$	−4.06740	−0.469663
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−11.2259	−1.26302	−0.631508	−	0.775369i	$-0.717564\pi$
$79$	−11.2259	−1.26302	−0.631508	+	0.775369i	$0.717564\pi$
$80$	0	0
$81$	−10.3705	−1.15228
$82$	0	0
$83$	1.15206	0.126455	0.0632273	−	0.997999i	$-0.479861\pi$
$83$	1.15206	0.126455	0.0632273	+	0.997999i	$0.479861\pi$
$84$	0	0
$85$	−2.00316	−0.217273
$86$	0	0
$87$	7.83937	0.840468
$88$	0	0
$89$	−12.9985	−1.37784	−0.688920	−	0.724838i	$-0.741915\pi$
$89$	−12.9985	−1.37784	−0.688920	+	0.724838i	$0.741915\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−13.6447	−1.41489
$94$	0	0
$95$	1.22171	0.125344
$96$	0	0
$97$	−11.7569	−1.19373	−0.596864	−	0.802342i	$-0.703587\pi$
$97$	−11.7569	−1.19373	−0.596864	+	0.802342i	$0.703587\pi$
$98$	0	0
$99$	−2.43780	−0.245008
$100$	0	0
$101$	7.99550	0.795582	0.397791	−	0.917476i	$-0.369777\pi$
$101$	7.99550	0.795582	0.397791	+	0.917476i	$0.369777\pi$
$102$	0	0
$103$	0.107657	0.0106078	0.00530390	−	0.999986i	$-0.498312\pi$
$103$	0.107657	0.0106078	0.00530390	+	0.999986i	$0.498312\pi$
$104$	0	0
$105$	6.05589	0.590994
$106$	0	0
$107$	13.1129	1.26767	0.633835	−	0.773469i	$-0.281480\pi$
$107$	13.1129	1.26767	0.633835	+	0.773469i	$0.281480\pi$
$108$	0	0
$109$	14.2368	1.36364	0.681821	−	0.731519i	$-0.261188\pi$
$109$	14.2368	1.36364	0.681821	+	0.731519i	$0.261188\pi$
$110$	0	0
$111$	−16.9184	−1.60583
$112$	0	0
$113$	13.3317	1.25414	0.627072	−	0.778961i	$-0.284253\pi$
$113$	13.3317	1.25414	0.627072	+	0.778961i	$0.284253\pi$
$114$	0	0
$115$	20.5384	1.91522
$116$	0	0
$117$	2.43780	0.225374
$118$	0	0
$119$	0.771345	0.0707091
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	25.0751	2.26095
$124$	0	0
$125$	8.45511	0.756248
$126$	0	0
$127$	9.80877	0.870387	0.435194	−	0.900337i	$-0.356680\pi$
$127$	9.80877	0.870387	0.435194	+	0.900337i	$0.356680\pi$
$128$	0	0
$129$	−18.8488	−1.65954
$130$	0	0
$131$	−7.88280	−0.688724	−0.344362	−	0.938837i	$-0.611905\pi$
$131$	−7.88280	−0.688724	−0.344362	+	0.938837i	$0.611905\pi$
$132$	0	0
$133$	−0.470436	−0.0407920
$134$	0	0
$135$	−3.40464	−0.293024
$136$	0	0
$137$	0.876921	0.0749204	0.0374602	−	0.999298i	$-0.488073\pi$
$137$	0.876921	0.0749204	0.0374602	+	0.999298i	$0.488073\pi$
$138$	0	0
$139$	2.54093	0.215519	0.107760	−	0.994177i	$-0.465632\pi$
$139$	2.54093	0.215519	0.107760	+	0.994177i	$0.465632\pi$
$140$	0	0
$141$	2.36112	0.198842
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	8.73043	0.725023
$146$	0	0
$147$	−2.33191	−0.192333
$148$	0	0
$149$	3.86290	0.316461	0.158231	−	0.987402i	$-0.449421\pi$
$149$	3.86290	0.316461	0.158231	+	0.987402i	$0.449421\pi$
$150$	0	0
$151$	14.9007	1.21260	0.606299	−	0.795237i	$-0.292654\pi$
$151$	14.9007	1.21260	0.606299	+	0.795237i	$0.292654\pi$
$152$	0	0
$153$	1.88038	0.152020
$154$	0	0
$155$	−15.1957	−1.22055
$156$	0	0
$157$	−2.95676	−0.235975	−0.117988	−	0.993015i	$-0.537644\pi$
$157$	−2.95676	−0.235975	−0.117988	+	0.993015i	$0.537644\pi$
$158$	0	0
$159$	4.85997	0.385421
$160$	0	0
$161$	−7.90862	−0.623286
$162$	0	0
$163$	19.5526	1.53148	0.765739	−	0.643151i	$-0.222373\pi$
$163$	19.5526	1.53148	0.765739	+	0.643151i	$0.222373\pi$
$164$	0	0
$165$	−6.05589	−0.471450
$166$	0	0
$167$	−22.2628	−1.72275	−0.861375	−	0.507970i	$-0.830396\pi$
$167$	−22.2628	−1.72275	−0.861375	+	0.507970i	$0.830396\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−1.14683	−0.0877001
$172$	0	0
$173$	4.40280	0.334739	0.167369	−	0.985894i	$-0.446473\pi$
$173$	4.40280	0.334739	0.167369	+	0.985894i	$0.446473\pi$
$174$	0	0
$175$	1.74424	0.131852
$176$	0	0
$177$	5.55237	0.417342
$178$	0	0
$179$	19.7466	1.47593	0.737967	−	0.674837i	$-0.235786\pi$
$179$	19.7466	1.47593	0.737967	+	0.674837i	$0.235786\pi$
$180$	0	0
$181$	−0.258527	−0.0192161	−0.00960807	−	0.999954i	$-0.503058\pi$
$181$	−0.258527	−0.0192161	−0.00960807	+	0.999954i	$0.503058\pi$
$182$	0	0
$183$	−10.8886	−0.804910
$184$	0	0
$185$	−18.8415	−1.38525
$186$	0	0
$187$	−0.771345	−0.0564063
$188$	0	0
$189$	1.31100	0.0953616
$190$	0	0
$191$	−9.83540	−0.711665	−0.355832	−	0.934550i	$-0.615803\pi$
$191$	−9.83540	−0.711665	−0.355832	+	0.934550i	$0.615803\pi$
$192$	0	0
$193$	1.07371	0.0772872	0.0386436	−	0.999253i	$-0.487696\pi$
$193$	1.07371	0.0772872	0.0386436	+	0.999253i	$0.487696\pi$
$194$	0	0
$195$	6.05589	0.433671
$196$	0	0
$197$	−26.2988	−1.87371	−0.936855	−	0.349719i	$-0.886277\pi$
$197$	−26.2988	−1.87371	−0.936855	+	0.349719i	$0.886277\pi$
$198$	0	0
$199$	2.73467	0.193855	0.0969276	−	0.995291i	$-0.469098\pi$
$199$	2.73467	0.193855	0.0969276	+	0.995291i	$0.469098\pi$
$200$	0	0
$201$	−14.2429	−1.00462
$202$	0	0
$203$	−3.36178	−0.235951
$204$	0	0
$205$	27.9253	1.95039
$206$	0	0
$207$	−19.2796	−1.34003
$208$	0	0
$209$	0.470436	0.0325407
$210$	0	0
$211$	20.4252	1.40613	0.703064	−	0.711127i	$-0.251815\pi$
$211$	20.4252	1.40613	0.703064	+	0.711127i	$0.251815\pi$
$212$	0	0
$213$	−14.2441	−0.975987
$214$	0	0
$215$	−20.9912	−1.43159
$216$	0	0
$217$	5.85131	0.397213
$218$	0	0
$219$	−20.6432	−1.39494
$220$	0	0
$221$	0.771345	0.0518863
$222$	0	0
$223$	−8.74407	−0.585546	−0.292773	−	0.956182i	$-0.594578\pi$
$223$	−8.74407	−0.585546	−0.292773	+	0.956182i	$0.594578\pi$
$224$	0	0
$225$	4.25210	0.283473
$226$	0	0
$227$	8.85719	0.587872	0.293936	−	0.955825i	$-0.405035\pi$
$227$	8.85719	0.587872	0.293936	+	0.955825i	$0.405035\pi$
$228$	0	0
$229$	−24.3488	−1.60902	−0.804508	−	0.593941i	$-0.797571\pi$
$229$	−24.3488	−1.60902	−0.804508	+	0.593941i	$0.797571\pi$
$230$	0	0
$231$	2.33191	0.153428
$232$	0	0
$233$	17.5357	1.14880	0.574400	−	0.818575i	$-0.305236\pi$
$233$	17.5357	1.14880	0.574400	+	0.818575i	$0.305236\pi$
$234$	0	0
$235$	2.62950	0.171530
$236$	0	0
$237$	26.1779	1.70044
$238$	0	0
$239$	24.1032	1.55910	0.779552	−	0.626337i	$-0.215447\pi$
$239$	24.1032	1.55910	0.779552	+	0.626337i	$0.215447\pi$
$240$	0	0
$241$	−16.0009	−1.03071	−0.515354	−	0.856977i	$-0.672340\pi$
$241$	−16.0009	−1.03071	−0.515354	+	0.856977i	$0.672340\pi$
$242$	0	0
$243$	20.2501	1.29905
$244$	0	0
$245$	−2.59697	−0.165914
$246$	0	0
$247$	−0.470436	−0.0299331
$248$	0	0
$249$	−2.68649	−0.170249
$250$	0	0
$251$	−23.3548	−1.47414	−0.737070	−	0.675817i	$-0.763791\pi$
$251$	−23.3548	−1.47414	−0.737070	+	0.675817i	$0.763791\pi$
$252$	0	0
$253$	7.90862	0.497211
$254$	0	0
$255$	4.67118	0.292521
$256$	0	0
$257$	−10.1993	−0.636213	−0.318106	−	0.948055i	$-0.603047\pi$
$257$	−10.1993	−0.636213	−0.318106	+	0.948055i	$0.603047\pi$
$258$	0	0
$259$	7.25519	0.450816
$260$	0	0
$261$	−8.19534	−0.507279
$262$	0	0
$263$	−9.79073	−0.603722	−0.301861	−	0.953352i	$-0.597608\pi$
$263$	−9.79073	−0.603722	−0.301861	+	0.953352i	$0.597608\pi$
$264$	0	0
$265$	5.41238	0.332480
$266$	0	0
$267$	30.3113	1.85502
$268$	0	0
$269$	10.2002	0.621916	0.310958	−	0.950424i	$-0.399350\pi$
$269$	10.2002	0.621916	0.310958	+	0.950424i	$0.399350\pi$
$270$	0	0
$271$	−3.79298	−0.230407	−0.115204	−	0.993342i	$-0.536752\pi$
$271$	−3.79298	−0.230407	−0.115204	+	0.993342i	$0.536752\pi$
$272$	0	0
$273$	−2.33191	−0.141134
$274$	0	0
$275$	−1.74424	−0.105181
$276$	0	0
$277$	−28.1094	−1.68893	−0.844465	−	0.535611i	$-0.820081\pi$
$277$	−28.1094	−1.68893	−0.844465	+	0.535611i	$0.820081\pi$
$278$	0	0
$279$	14.2643	0.853982
$280$	0	0
$281$	−5.35784	−0.319622	−0.159811	−	0.987148i	$-0.551088\pi$
$281$	−5.35784	−0.319622	−0.159811	+	0.987148i	$0.551088\pi$
$282$	0	0
$283$	−26.8721	−1.59738	−0.798692	−	0.601741i	$-0.794474\pi$
$283$	−26.8721	−1.59738	−0.798692	+	0.601741i	$0.794474\pi$
$284$	0	0
$285$	−2.84891	−0.168755
$286$	0	0
$287$	−10.7531	−0.634733
$288$	0	0
$289$	−16.4050	−0.965002
$290$	0	0
$291$	27.4159	1.60715
$292$	0	0
$293$	16.4348	0.960130	0.480065	−	0.877233i	$-0.340613\pi$
$293$	16.4348	0.960130	0.480065	+	0.877233i	$0.340613\pi$
$294$	0	0
$295$	6.18349	0.360017
$296$	0	0
$297$	−1.31100	−0.0760722
$298$	0	0
$299$	−7.90862	−0.457367
$300$	0	0
$301$	8.08298	0.465895
$302$	0	0
$303$	−18.6448	−1.07111
$304$	0	0
$305$	−12.1263	−0.694349
$306$	0	0
$307$	−16.5373	−0.943833	−0.471916	−	0.881643i	$-0.656438\pi$
$307$	−16.5373	−0.943833	−0.471916	+	0.881643i	$0.656438\pi$
$308$	0	0
$309$	−0.251047	−0.0142816
$310$	0	0
$311$	32.6368	1.85066	0.925332	−	0.379157i	$-0.123786\pi$
$311$	32.6368	1.85066	0.925332	+	0.379157i	$0.123786\pi$
$312$	0	0
$313$	5.02426	0.283988	0.141994	−	0.989868i	$-0.454649\pi$
$313$	5.02426	0.283988	0.141994	+	0.989868i	$0.454649\pi$
$314$	0	0
$315$	−6.33088	−0.356705
$316$	0	0
$317$	−32.4005	−1.81979	−0.909896	−	0.414837i	$-0.863839\pi$
$317$	−32.4005	−1.81979	−0.909896	+	0.414837i	$0.863839\pi$
$318$	0	0
$319$	3.36178	0.188224
$320$	0	0
$321$	−30.5780	−1.70670
$322$	0	0
$323$	−0.362868	−0.0201905
$324$	0	0
$325$	1.74424	0.0967529
$326$	0	0
$327$	−33.1990	−1.83591
$328$	0	0
$329$	−1.01253	−0.0558224
$330$	0	0
$331$	−15.2527	−0.838367	−0.419183	−	0.907902i	$-0.637684\pi$
$331$	−15.2527	−0.838367	−0.419183	+	0.907902i	$0.637684\pi$
$332$	0	0
$333$	17.6867	0.969224
$334$	0	0
$335$	−15.8619	−0.866627
$336$	0	0
$337$	−32.4211	−1.76609	−0.883046	−	0.469286i	$-0.844511\pi$
$337$	−32.4211	−1.76609	−0.883046	+	0.469286i	$0.844511\pi$
$338$	0	0
$339$	−31.0884	−1.68849
$340$	0	0
$341$	−5.85131	−0.316866
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−47.8937	−2.57851
$346$	0	0
$347$	26.6938	1.43300	0.716499	−	0.697588i	$-0.245743\pi$
$347$	26.6938	1.43300	0.716499	+	0.697588i	$0.245743\pi$
$348$	0	0
$349$	−16.1267	−0.863240	−0.431620	−	0.902055i	$-0.642058\pi$
$349$	−16.1267	−0.863240	−0.431620	+	0.902055i	$0.642058\pi$
$350$	0	0
$351$	1.31100	0.0699763
$352$	0	0
$353$	−35.9435	−1.91308	−0.956541	−	0.291599i	$-0.905813\pi$
$353$	−35.9435	−1.91308	−0.956541	+	0.291599i	$0.905813\pi$
$354$	0	0
$355$	−15.8631	−0.841927
$356$	0	0
$357$	−1.79871	−0.0951976
$358$	0	0
$359$	7.95678	0.419943	0.209971	−	0.977708i	$-0.432663\pi$
$359$	7.95678	0.419943	0.209971	+	0.977708i	$0.432663\pi$
$360$	0	0
$361$	−18.7787	−0.988352
$362$	0	0
$363$	−2.33191	−0.122393
$364$	0	0
$365$	−22.9896	−1.20333
$366$	0	0
$367$	−16.4625	−0.859335	−0.429667	−	0.902987i	$-0.641369\pi$
$367$	−16.4625	−0.859335	−0.429667	+	0.902987i	$0.641369\pi$
$368$	0	0
$369$	−26.2138	−1.36463
$370$	0	0
$371$	−2.08412	−0.108202
$372$	0	0
$373$	−27.3977	−1.41860	−0.709299	−	0.704907i	$-0.750989\pi$
$373$	−27.3977	−1.41860	−0.709299	+	0.704907i	$0.750989\pi$
$374$	0	0
$375$	−19.7165	−1.01816
$376$	0	0
$377$	−3.36178	−0.173141
$378$	0	0
$379$	−2.38785	−0.122655	−0.0613277	−	0.998118i	$-0.519533\pi$
$379$	−2.38785	−0.122655	−0.0613277	+	0.998118i	$0.519533\pi$
$380$	0	0
$381$	−22.8731	−1.17183
$382$	0	0
$383$	3.52949	0.180348	0.0901742	−	0.995926i	$-0.471258\pi$
$383$	3.52949	0.180348	0.0901742	+	0.995926i	$0.471258\pi$
$384$	0	0
$385$	2.59697	0.132354
$386$	0	0
$387$	19.7047	1.00164
$388$	0	0
$389$	28.4831	1.44415	0.722074	−	0.691815i	$-0.243189\pi$
$389$	28.4831	1.44415	0.722074	+	0.691815i	$0.243189\pi$
$390$	0	0
$391$	−6.10028	−0.308504
$392$	0	0
$393$	18.3820	0.927248
$394$	0	0
$395$	29.1534	1.46687
$396$	0	0
$397$	−25.0274	−1.25609	−0.628045	−	0.778177i	$-0.716145\pi$
$397$	−25.0274	−1.25609	−0.628045	+	0.778177i	$0.716145\pi$
$398$	0	0
$399$	1.09701	0.0549194
$400$	0	0
$401$	−12.0617	−0.602332	−0.301166	−	0.953572i	$-0.597376\pi$
$401$	−12.0617	−0.602332	−0.301166	+	0.953572i	$0.597376\pi$
$402$	0	0
$403$	5.85131	0.291475
$404$	0	0
$405$	26.9319	1.33826
$406$	0	0
$407$	−7.25519	−0.359626
$408$	0	0
$409$	4.59718	0.227316	0.113658	−	0.993520i	$-0.463743\pi$
$409$	4.59718	0.227316	0.113658	+	0.993520i	$0.463743\pi$
$410$	0	0
$411$	−2.04490	−0.100867
$412$	0	0
$413$	−2.38104	−0.117163
$414$	0	0
$415$	−2.99185	−0.146864
$416$	0	0
$417$	−5.92523	−0.290160
$418$	0	0
$419$	32.8432	1.60450	0.802248	−	0.596991i	$-0.203637\pi$
$419$	32.8432	1.60450	0.802248	+	0.596991i	$0.203637\pi$
$420$	0	0
$421$	−13.5457	−0.660179	−0.330089	−	0.943950i	$-0.607079\pi$
$421$	−13.5457	−0.660179	−0.330089	+	0.943950i	$0.607079\pi$
$422$	0	0
$423$	−2.46833	−0.120015
$424$	0	0
$425$	1.34541	0.0652619
$426$	0	0
$427$	4.66941	0.225968
$428$	0	0
$429$	2.33191	0.112586
$430$	0	0
$431$	19.6164	0.944891	0.472445	−	0.881360i	$-0.343371\pi$
$431$	19.6164	0.944891	0.472445	+	0.881360i	$0.343371\pi$
$432$	0	0
$433$	10.5091	0.505033	0.252517	−	0.967593i	$-0.418742\pi$
$433$	10.5091	0.505033	0.252517	+	0.967593i	$0.418742\pi$
$434$	0	0
$435$	−20.3586	−0.976119
$436$	0	0
$437$	3.72050	0.177976
$438$	0	0
$439$	−19.6891	−0.939711	−0.469856	−	0.882743i	$-0.655694\pi$
$439$	−19.6891	−0.939711	−0.469856	+	0.882743i	$0.655694\pi$
$440$	0	0
$441$	2.43780	0.116086
$442$	0	0
$443$	−23.3607	−1.10990	−0.554949	−	0.831884i	$-0.687262\pi$
$443$	−23.3607	−1.10990	−0.554949	+	0.831884i	$0.687262\pi$
$444$	0	0
$445$	33.7567	1.60022
$446$	0	0
$447$	−9.00794	−0.426061
$448$	0	0
$449$	15.8780	0.749330	0.374665	−	0.927160i	$-0.377758\pi$
$449$	15.8780	0.749330	0.374665	+	0.927160i	$0.377758\pi$
$450$	0	0
$451$	10.7531	0.506341
$452$	0	0
$453$	−34.7470	−1.63255
$454$	0	0
$455$	−2.59697	−0.121748
$456$	0	0
$457$	−13.9180	−0.651056	−0.325528	−	0.945532i	$-0.605542\pi$
$457$	−13.9180	−0.651056	−0.325528	+	0.945532i	$0.605542\pi$
$458$	0	0
$459$	1.01124	0.0472005
$460$	0	0
$461$	−33.1053	−1.54187	−0.770934	−	0.636915i	$-0.780210\pi$
$461$	−33.1053	−1.54187	−0.770934	+	0.636915i	$0.780210\pi$
$462$	0	0
$463$	7.30873	0.339666	0.169833	−	0.985473i	$-0.445677\pi$
$463$	7.30873	0.339666	0.169833	+	0.985473i	$0.445677\pi$
$464$	0	0
$465$	35.4349	1.64325
$466$	0	0
$467$	28.9683	1.34049	0.670246	−	0.742139i	$-0.266189\pi$
$467$	28.9683	1.34049	0.670246	+	0.742139i	$0.266189\pi$
$468$	0	0
$469$	6.10785	0.282034
$470$	0	0
$471$	6.89489	0.317700
$472$	0	0
$473$	−8.08298	−0.371656
$474$	0	0
$475$	−0.820552	−0.0376495
$476$	0	0
$477$	−5.08065	−0.232627
$478$	0	0
$479$	−34.6282	−1.58220	−0.791102	−	0.611684i	$-0.790492\pi$
$479$	−34.6282	−1.58220	−0.791102	+	0.611684i	$0.790492\pi$
$480$	0	0
$481$	7.25519	0.330808
$482$	0	0
$483$	18.4422	0.839148
$484$	0	0
$485$	30.5322	1.38640
$486$	0	0
$487$	−10.0703	−0.456327	−0.228163	−	0.973623i	$-0.573272\pi$
$487$	−10.0703	−0.456327	−0.228163	+	0.973623i	$0.573272\pi$
$488$	0	0
$489$	−45.5949	−2.06187
$490$	0	0
$491$	−36.5603	−1.64994	−0.824972	−	0.565174i	$-0.808809\pi$
$491$	−36.5603	−1.64994	−0.824972	+	0.565174i	$0.808809\pi$
$492$	0	0
$493$	−2.59309	−0.116787
$494$	0	0
$495$	6.33088	0.284552
$496$	0	0
$497$	6.10833	0.273996
$498$	0	0
$499$	29.4851	1.31993	0.659966	−	0.751295i	$-0.270571\pi$
$499$	29.4851	1.31993	0.659966	+	0.751295i	$0.270571\pi$
$500$	0	0
$501$	51.9149	2.31939
$502$	0	0
$503$	23.4820	1.04701	0.523505	−	0.852023i	$-0.324624\pi$
$503$	23.4820	1.04701	0.523505	+	0.852023i	$0.324624\pi$
$504$	0	0
$505$	−20.7640	−0.923988
$506$	0	0
$507$	−2.33191	−0.103564
$508$	0	0
$509$	6.69346	0.296682	0.148341	−	0.988936i	$-0.452607\pi$
$509$	6.69346	0.296682	0.148341	+	0.988936i	$0.452607\pi$
$510$	0	0
$511$	8.85247	0.391610
$512$	0	0
$513$	−0.616744	−0.0272299
$514$	0	0
$515$	−0.279583	−0.0123199
$516$	0	0
$517$	1.01253	0.0445309
$518$	0	0
$519$	−10.2669	−0.450668
$520$	0	0
$521$	−16.1422	−0.707200	−0.353600	−	0.935397i	$-0.615043\pi$
$521$	−16.1422	−0.707200	−0.353600	+	0.935397i	$0.615043\pi$
$522$	0	0
$523$	−28.9010	−1.26375	−0.631876	−	0.775069i	$-0.717715\pi$
$523$	−28.9010	−1.26375	−0.631876	+	0.775069i	$0.717715\pi$
$524$	0	0
$525$	−4.06740	−0.177516
$526$	0	0
$527$	4.51338	0.196606
$528$	0	0
$529$	39.5463	1.71940
$530$	0	0
$531$	−5.80450	−0.251894
$532$	0	0
$533$	−10.7531	−0.465766
$534$	0	0
$535$	−34.0537	−1.47227
$536$	0	0
$537$	−46.0474	−1.98709
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	10.1749	0.437454	0.218727	−	0.975786i	$-0.429810\pi$
$541$	10.1749	0.437454	0.218727	+	0.975786i	$0.429810\pi$
$542$	0	0
$543$	0.602861	0.0258712
$544$	0	0
$545$	−36.9726	−1.58373
$546$	0	0
$547$	−9.20588	−0.393615	−0.196808	−	0.980442i	$-0.563057\pi$
$547$	−9.20588	−0.393615	−0.196808	+	0.980442i	$0.563057\pi$
$548$	0	0
$549$	11.3831	0.485817
$550$	0	0
$551$	1.58150	0.0673742
$552$	0	0
$553$	−11.2259	−0.477376
$554$	0	0
$555$	43.9366	1.86501
$556$	0	0
$557$	−12.6135	−0.534450	−0.267225	−	0.963634i	$-0.586107\pi$
$557$	−12.6135	−0.534450	−0.267225	+	0.963634i	$0.586107\pi$
$558$	0	0
$559$	8.08298	0.341874
$560$	0	0
$561$	1.79871	0.0759414
$562$	0	0
$563$	0.359299	0.0151427	0.00757133	−	0.999971i	$-0.497590\pi$
$563$	0.359299	0.0151427	0.00757133	+	0.999971i	$0.497590\pi$
$564$	0	0
$565$	−34.6221	−1.45656
$566$	0	0
$567$	−10.3705	−0.435522
$568$	0	0
$569$	15.1688	0.635911	0.317956	−	0.948106i	$-0.397004\pi$
$569$	15.1688	0.635911	0.317956	+	0.948106i	$0.397004\pi$
$570$	0	0
$571$	−24.8399	−1.03952	−0.519758	−	0.854313i	$-0.673978\pi$
$571$	−24.8399	−1.03952	−0.519758	+	0.854313i	$0.673978\pi$
$572$	0	0
$573$	22.9353	0.958134
$574$	0	0
$575$	−13.7945	−0.575271
$576$	0	0
$577$	6.41416	0.267025	0.133512	−	0.991047i	$-0.457374\pi$
$577$	6.41416	0.267025	0.133512	+	0.991047i	$0.457374\pi$
$578$	0	0
$579$	−2.50379	−0.104054
$580$	0	0
$581$	1.15206	0.0477954
$582$	0	0
$583$	2.08412	0.0863153
$584$	0	0
$585$	−6.33088	−0.261750
$586$	0	0
$587$	−37.5842	−1.55126	−0.775632	−	0.631185i	$-0.782569\pi$
$587$	−37.5842	−1.55126	−0.775632	+	0.631185i	$0.782569\pi$
$588$	0	0
$589$	−2.75267	−0.113422
$590$	0	0
$591$	61.3263	2.52263
$592$	0	0
$593$	−9.05991	−0.372046	−0.186023	−	0.982545i	$-0.559560\pi$
$593$	−9.05991	−0.372046	−0.186023	+	0.982545i	$0.559560\pi$
$594$	0	0
$595$	−2.00316	−0.0821215
$596$	0	0
$597$	−6.37699	−0.260993
$598$	0	0
$599$	−16.4725	−0.673048	−0.336524	−	0.941675i	$-0.609251\pi$
$599$	−16.4725	−0.673048	−0.336524	+	0.941675i	$0.609251\pi$
$600$	0	0
$601$	−2.38835	−0.0974230	−0.0487115	−	0.998813i	$-0.515511\pi$
$601$	−2.38835	−0.0974230	−0.0487115	+	0.998813i	$0.515511\pi$
$602$	0	0
$603$	14.8897	0.606356
$604$	0	0
$605$	−2.59697	−0.105582
$606$	0	0
$607$	24.2696	0.985071	0.492536	−	0.870292i	$-0.336070\pi$
$607$	24.2696	0.985071	0.492536	+	0.870292i	$0.336070\pi$
$608$	0	0
$609$	7.83937	0.317667
$610$	0	0
$611$	−1.01253	−0.0409624
$612$	0	0
$613$	45.1059	1.82181	0.910906	−	0.412615i	$-0.135384\pi$
$613$	45.1059	1.82181	0.910906	+	0.412615i	$0.135384\pi$
$614$	0	0
$615$	−65.1193	−2.62586
$616$	0	0
$617$	9.79991	0.394529	0.197265	−	0.980350i	$-0.436794\pi$
$617$	9.79991	0.394529	0.197265	+	0.980350i	$0.436794\pi$
$618$	0	0
$619$	24.1859	0.972115	0.486057	−	0.873927i	$-0.338435\pi$
$619$	24.1859	0.972115	0.486057	+	0.873927i	$0.338435\pi$
$620$	0	0
$621$	−10.3682	−0.416063
$622$	0	0
$623$	−12.9985	−0.520774
$624$	0	0
$625$	−30.6788	−1.22715
$626$	0	0
$627$	−1.09701	−0.0438105
$628$	0	0
$629$	5.59626	0.223137
$630$	0	0
$631$	−5.42623	−0.216015	−0.108007	−	0.994150i	$-0.534447\pi$
$631$	−5.42623	−0.216015	−0.108007	+	0.994150i	$0.534447\pi$
$632$	0	0
$633$	−47.6296	−1.89311
$634$	0	0
$635$	−25.4730	−1.01087
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	14.8909	0.589074
$640$	0	0
$641$	−1.41642	−0.0559451	−0.0279726	−	0.999609i	$-0.508905\pi$
$641$	−1.41642	−0.0559451	−0.0279726	+	0.999609i	$0.508905\pi$
$642$	0	0
$643$	−8.80153	−0.347098	−0.173549	−	0.984825i	$-0.555524\pi$
$643$	−8.80153	−0.347098	−0.173549	+	0.984825i	$0.555524\pi$
$644$	0	0
$645$	48.9496	1.92739
$646$	0	0
$647$	2.07460	0.0815611	0.0407805	−	0.999168i	$-0.487016\pi$
$647$	2.07460	0.0815611	0.0407805	+	0.999168i	$0.487016\pi$
$648$	0	0
$649$	2.38104	0.0934641
$650$	0	0
$651$	−13.6447	−0.534779
$652$	0	0
$653$	−20.7132	−0.810571	−0.405286	−	0.914190i	$-0.632828\pi$
$653$	−20.7132	−0.810571	−0.405286	+	0.914190i	$0.632828\pi$
$654$	0	0
$655$	20.4714	0.799883
$656$	0	0
$657$	21.5805	0.841937
$658$	0	0
$659$	10.1587	0.395725	0.197863	−	0.980230i	$-0.436600\pi$
$659$	10.1587	0.395725	0.197863	+	0.980230i	$0.436600\pi$
$660$	0	0
$661$	10.8754	0.423004	0.211502	−	0.977378i	$-0.432165\pi$
$661$	10.8754	0.423004	0.211502	+	0.977378i	$0.432165\pi$
$662$	0	0
$663$	−1.79871	−0.0698560
$664$	0	0
$665$	1.22171	0.0473757
$666$	0	0
$667$	26.5870	1.02945
$668$	0	0
$669$	20.3904	0.788337
$670$	0	0
$671$	−4.66941	−0.180260
$672$	0	0
$673$	−19.6094	−0.755887	−0.377944	−	0.925829i	$-0.623369\pi$
$673$	−19.6094	−0.755887	−0.377944	+	0.925829i	$0.623369\pi$
$674$	0	0
$675$	2.28670	0.0880153
$676$	0	0
$677$	−42.1710	−1.62076	−0.810382	−	0.585901i	$-0.800741\pi$
$677$	−42.1710	−1.62076	−0.810382	+	0.585901i	$0.800741\pi$
$678$	0	0
$679$	−11.7569	−0.451187
$680$	0	0
$681$	−20.6542	−0.791469
$682$	0	0
$683$	49.8881	1.90891	0.954457	−	0.298348i	$-0.0964356\pi$
$683$	49.8881	1.90891	0.954457	+	0.298348i	$0.0964356\pi$
$684$	0	0
$685$	−2.27734	−0.0870125
$686$	0	0
$687$	56.7793	2.16626
$688$	0	0
$689$	−2.08412	−0.0793985
$690$	0	0
$691$	38.3442	1.45868	0.729340	−	0.684151i	$-0.239827\pi$
$691$	38.3442	1.45868	0.729340	+	0.684151i	$0.239827\pi$
$692$	0	0
$693$	−2.43780	−0.0926043
$694$	0	0
$695$	−6.59872	−0.250304
$696$	0	0
$697$	−8.29431	−0.314170
$698$	0	0
$699$	−40.8916	−1.54666
$700$	0	0
$701$	−16.9385	−0.639759	−0.319879	−	0.947458i	$-0.603642\pi$
$701$	−16.9385	−0.639759	−0.319879	+	0.947458i	$0.603642\pi$
$702$	0	0
$703$	−3.41310	−0.128728
$704$	0	0
$705$	−6.13175	−0.230935
$706$	0	0
$707$	7.99550	0.300702
$708$	0	0
$709$	−4.33950	−0.162973	−0.0814867	−	0.996674i	$-0.525967\pi$
$709$	−4.33950	−0.162973	−0.0814867	+	0.996674i	$0.525967\pi$
$710$	0	0
$711$	−27.3666	−1.02633
$712$	0	0
$713$	−46.2758	−1.73304
$714$	0	0
$715$	2.59697	0.0971211
$716$	0	0
$717$	−56.2064	−2.09907
$718$	0	0
$719$	−28.1051	−1.04815	−0.524073	−	0.851674i	$-0.675588\pi$
$719$	−28.1051	−1.04815	−0.524073	+	0.851674i	$0.675588\pi$
$720$	0	0
$721$	0.107657	0.00400937
$722$	0	0
$723$	37.3126	1.38767
$724$	0	0
$725$	−5.86374	−0.217774
$726$	0	0
$727$	2.96614	0.110008	0.0550040	−	0.998486i	$-0.482483\pi$
$727$	2.96614	0.110008	0.0550040	+	0.998486i	$0.482483\pi$
$728$	0	0
$729$	−16.1098	−0.596661
$730$	0	0
$731$	6.23477	0.230601
$732$	0	0
$733$	−14.8993	−0.550320	−0.275160	−	0.961398i	$-0.588731\pi$
$733$	−14.8993	−0.550320	−0.275160	+	0.961398i	$0.588731\pi$
$734$	0	0
$735$	6.05589	0.223375
$736$	0	0
$737$	−6.10785	−0.224986
$738$	0	0
$739$	19.5394	0.718770	0.359385	−	0.933189i	$-0.382987\pi$
$739$	19.5394	0.718770	0.359385	+	0.933189i	$0.382987\pi$
$740$	0	0
$741$	1.09701	0.0402998
$742$	0	0
$743$	−15.2317	−0.558798	−0.279399	−	0.960175i	$-0.590135\pi$
$743$	−15.2317	−0.558798	−0.279399	+	0.960175i	$0.590135\pi$
$744$	0	0
$745$	−10.0318	−0.367538
$746$	0	0
$747$	2.80848	0.102757
$748$	0	0
$749$	13.1129	0.479134
$750$	0	0
$751$	6.11116	0.223000	0.111500	−	0.993764i	$-0.464435\pi$
$751$	6.11116	0.223000	0.111500	+	0.993764i	$0.464435\pi$
$752$	0	0
$753$	54.4612	1.98468
$754$	0	0
$755$	−38.6965	−1.40831
$756$	0	0
$757$	−27.8734	−1.01307	−0.506537	−	0.862218i	$-0.669075\pi$
$757$	−27.8734	−1.01307	−0.506537	+	0.862218i	$0.669075\pi$
$758$	0	0
$759$	−18.4422	−0.669409
$760$	0	0
$761$	−32.0158	−1.16057	−0.580286	−	0.814413i	$-0.697059\pi$
$761$	−32.0158	−1.16057	−0.580286	+	0.814413i	$0.697059\pi$
$762$	0	0
$763$	14.2368	0.515408
$764$	0	0
$765$	−4.88329	−0.176556
$766$	0	0
$767$	−2.38104	−0.0859745
$768$	0	0
$769$	31.5881	1.13910	0.569548	−	0.821958i	$-0.307118\pi$
$769$	31.5881	1.13910	0.569548	+	0.821958i	$0.307118\pi$
$770$	0	0
$771$	23.7837	0.856551
$772$	0	0
$773$	−13.8206	−0.497092	−0.248546	−	0.968620i	$-0.579953\pi$
$773$	−13.8206	−0.497092	−0.248546	+	0.968620i	$0.579953\pi$
$774$	0	0
$775$	10.2061	0.366613
$776$	0	0
$777$	−16.9184	−0.606946
$778$	0	0
$779$	5.05862	0.181244
$780$	0	0
$781$	−6.10833	−0.218573
$782$	0	0
$783$	−4.40731	−0.157504
$784$	0	0
$785$	7.67861	0.274061
$786$	0	0
$787$	11.5713	0.412474	0.206237	−	0.978502i	$-0.433878\pi$
$787$	11.5713	0.412474	0.206237	+	0.978502i	$0.433878\pi$
$788$	0	0
$789$	22.8311	0.812808
$790$	0	0
$791$	13.3317	0.474022
$792$	0	0
$793$	4.66941	0.165815
$794$	0	0
$795$	−12.6212	−0.447627
$796$	0	0
$797$	−37.9544	−1.34441	−0.672206	−	0.740364i	$-0.734653\pi$
$797$	−37.9544	−1.34441	−0.672206	+	0.740364i	$0.734653\pi$
$798$	0	0
$799$	−0.781007	−0.0276301
$800$	0	0
$801$	−31.6877	−1.11963
$802$	0	0
$803$	−8.85247	−0.312397
$804$	0	0
$805$	20.5384	0.723884
$806$	0	0
$807$	−23.7859	−0.837303
$808$	0	0
$809$	−27.0381	−0.950609	−0.475305	−	0.879821i	$-0.657662\pi$
$809$	−27.0381	−0.950609	−0.475305	+	0.879821i	$0.657662\pi$
$810$	0	0
$811$	−34.8528	−1.22385	−0.611923	−	0.790917i	$-0.709604\pi$
$811$	−34.8528	−1.22385	−0.611923	+	0.790917i	$0.709604\pi$
$812$	0	0
$813$	8.84489	0.310204
$814$	0	0
$815$	−50.7775	−1.77866
$816$	0	0
$817$	−3.80252	−0.133033
$818$	0	0
$819$	2.43780	0.0851835
$820$	0	0
$821$	−11.0197	−0.384589	−0.192294	−	0.981337i	$-0.561593\pi$
$821$	−11.0197	−0.384589	−0.192294	+	0.981337i	$0.561593\pi$
$822$	0	0
$823$	4.47608	0.156026	0.0780131	−	0.996952i	$-0.475142\pi$
$823$	4.47608	0.156026	0.0780131	+	0.996952i	$0.475142\pi$
$824$	0	0
$825$	4.06740	0.141609
$826$	0	0
$827$	39.3765	1.36926	0.684628	−	0.728893i	$-0.259965\pi$
$827$	39.3765	1.36926	0.684628	+	0.728893i	$0.259965\pi$
$828$	0	0
$829$	−21.4533	−0.745103	−0.372552	−	0.928011i	$-0.621517\pi$
$829$	−21.4533	−0.745103	−0.372552	+	0.928011i	$0.621517\pi$
$830$	0	0
$831$	65.5485	2.27385
$832$	0	0
$833$	0.771345	0.0267255
$834$	0	0
$835$	57.8159	2.00080
$836$	0	0
$837$	7.67110	0.265152
$838$	0	0
$839$	27.8438	0.961275	0.480638	−	0.876919i	$-0.340405\pi$
$839$	27.8438	0.961275	0.480638	+	0.876919i	$0.340405\pi$
$840$	0	0
$841$	−17.6984	−0.610291
$842$	0	0
$843$	12.4940	0.430316
$844$	0	0
$845$	−2.59697	−0.0893384
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	62.6634	2.15060
$850$	0	0
$851$	−57.3785	−1.96691
$852$	0	0
$853$	−21.0154	−0.719553	−0.359776	−	0.933039i	$-0.617147\pi$
$853$	−21.0154	−0.719553	−0.359776	+	0.933039i	$0.617147\pi$
$854$	0	0
$855$	2.97827	0.101855
$856$	0	0
$857$	42.9174	1.46603	0.733016	−	0.680212i	$-0.238112\pi$
$857$	42.9174	1.46603	0.733016	+	0.680212i	$0.238112\pi$
$858$	0	0
$859$	−28.2559	−0.964079	−0.482040	−	0.876149i	$-0.660104\pi$
$859$	−28.2559	−0.964079	−0.482040	+	0.876149i	$0.660104\pi$
$860$	0	0
$861$	25.0751	0.854558
$862$	0	0
$863$	21.1754	0.720821	0.360410	−	0.932794i	$-0.382637\pi$
$863$	21.1754	0.720821	0.360410	+	0.932794i	$0.382637\pi$
$864$	0	0
$865$	−11.4339	−0.388765
$866$	0	0
$867$	38.2550	1.29921
$868$	0	0
$869$	11.2259	0.380814
$870$	0	0
$871$	6.10785	0.206957
$872$	0	0
$873$	−28.6609	−0.970023
$874$	0	0
$875$	8.45511	0.285835
$876$	0	0
$877$	14.0417	0.474154	0.237077	−	0.971491i	$-0.423811\pi$
$877$	14.0417	0.474154	0.237077	+	0.971491i	$0.423811\pi$
$878$	0	0
$879$	−38.3244	−1.29265
$880$	0	0
$881$	41.3395	1.39276	0.696382	−	0.717671i	$-0.254792\pi$
$881$	41.3395	1.39276	0.696382	+	0.717671i	$0.254792\pi$
$882$	0	0
$883$	−15.4178	−0.518849	−0.259424	−	0.965763i	$-0.583533\pi$
$883$	−15.4178	−0.518849	−0.259424	+	0.965763i	$0.583533\pi$
$884$	0	0
$885$	−14.4193	−0.484701
$886$	0	0
$887$	−21.3777	−0.717794	−0.358897	−	0.933377i	$-0.616847\pi$
$887$	−21.3777	−0.717794	−0.358897	+	0.933377i	$0.616847\pi$
$888$	0	0
$889$	9.80877	0.328975
$890$	0	0
$891$	10.3705	0.347426
$892$	0	0
$893$	0.476329	0.0159397
$894$	0	0
$895$	−51.2814	−1.71415
$896$	0	0
$897$	18.4422	0.615766
$898$	0	0
$899$	−19.6708	−0.656059
$900$	0	0
$901$	−1.60757	−0.0535560
$902$	0	0
$903$	−18.8488	−0.627248
$904$	0	0
$905$	0.671386	0.0223176
$906$	0	0
$907$	−21.7174	−0.721116	−0.360558	−	0.932737i	$-0.617414\pi$
$907$	−21.7174	−0.721116	−0.360558	+	0.932737i	$0.617414\pi$
$908$	0	0
$909$	19.4914	0.646489
$910$	0	0
$911$	−13.6899	−0.453567	−0.226784	−	0.973945i	$-0.572821\pi$
$911$	−13.6899	−0.453567	−0.226784	+	0.973945i	$0.572821\pi$
$912$	0	0
$913$	−1.15206	−0.0381275
$914$	0	0
$915$	28.2774	0.934822
$916$	0	0
$917$	−7.88280	−0.260313
$918$	0	0
$919$	8.37264	0.276188	0.138094	−	0.990419i	$-0.455902\pi$
$919$	8.37264	0.276188	0.138094	+	0.990419i	$0.455902\pi$
$920$	0	0
$921$	38.5634	1.27071
$922$	0	0
$923$	6.10833	0.201058
$924$	0	0
$925$	12.6548	0.416087
$926$	0	0
$927$	0.262447	0.00861989
$928$	0	0
$929$	49.5404	1.62537	0.812683	−	0.582706i	$-0.198006\pi$
$929$	49.5404	1.62537	0.812683	+	0.582706i	$0.198006\pi$
$930$	0	0
$931$	−0.470436	−0.0154179
$932$	0	0
$933$	−76.1061	−2.49160
$934$	0	0
$935$	2.00316	0.0655103
$936$	0	0
$937$	−48.3966	−1.58105	−0.790524	−	0.612431i	$-0.790192\pi$
$937$	−48.3966	−1.58105	−0.790524	+	0.612431i	$0.790192\pi$
$938$	0	0
$939$	−11.7161	−0.382341
$940$	0	0
$941$	41.8410	1.36398	0.681989	−	0.731363i	$-0.261115\pi$
$941$	41.8410	1.36398	0.681989	+	0.731363i	$0.261115\pi$
$942$	0	0
$943$	85.0418	2.76934
$944$	0	0
$945$	−3.40464	−0.110753
$946$	0	0
$947$	−15.4694	−0.502687	−0.251344	−	0.967898i	$-0.580872\pi$
$947$	−15.4694	−0.502687	−0.251344	+	0.967898i	$0.580872\pi$
$948$	0	0
$949$	8.85247	0.287363
$950$	0	0
$951$	75.5549	2.45004
$952$	0	0
$953$	25.0371	0.811032	0.405516	−	0.914088i	$-0.367092\pi$
$953$	25.0371	0.811032	0.405516	+	0.914088i	$0.367092\pi$
$954$	0	0
$955$	25.5422	0.826527
$956$	0	0
$957$	−7.83937	−0.253411
$958$	0	0
$959$	0.876921	0.0283173
$960$	0	0
$961$	3.23785	0.104447
$962$	0	0
$963$	31.9665	1.03011
$964$	0	0
$965$	−2.78839	−0.0897613
$966$	0	0
$967$	9.64920	0.310297	0.155149	−	0.987891i	$-0.450414\pi$
$967$	9.64920	0.310297	0.155149	+	0.987891i	$0.450414\pi$
$968$	0	0
$969$	0.846176	0.0271831
$970$	0	0
$971$	−49.2337	−1.57998	−0.789992	−	0.613118i	$-0.789915\pi$
$971$	−49.2337	−1.57998	−0.789992	+	0.613118i	$0.789915\pi$
$972$	0	0
$973$	2.54093	0.0814586
$974$	0	0
$975$	−4.06740	−0.130261
$976$	0	0
$977$	−31.6802	−1.01354	−0.506770	−	0.862081i	$-0.669161\pi$
$977$	−31.6802	−1.01354	−0.506770	+	0.862081i	$0.669161\pi$
$978$	0	0
$979$	12.9985	0.415434
$980$	0	0
$981$	34.7065	1.10809
$982$	0	0
$983$	−28.9485	−0.923313	−0.461657	−	0.887059i	$-0.652745\pi$
$983$	−28.9485	−0.923313	−0.461657	+	0.887059i	$0.652745\pi$
$984$	0	0
$985$	68.2970	2.17612
$986$	0	0
$987$	2.36112	0.0751553
$988$	0	0
$989$	−63.9252	−2.03270
$990$	0	0
$991$	−38.0154	−1.20760	−0.603799	−	0.797136i	$-0.706347\pi$
$991$	−38.0154	−1.20760	−0.603799	+	0.797136i	$0.706347\pi$
$992$	0	0
$993$	35.5680	1.12872
$994$	0	0
$995$	−7.10184	−0.225143
$996$	0	0
$997$	28.3765	0.898691	0.449346	−	0.893358i	$-0.351657\pi$
$997$	28.3765	0.898691	0.449346	+	0.893358i	$0.351657\pi$
$998$	0	0
$999$	9.51159	0.300933

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.o.1.2	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.o.1.2	✓	9	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-2.33191 q^{3} -2.59697 q^{5} +1.00000 q^{7} +2.43780 q^{9} +O(q^{10})\)	\(q-2.33191 q^{3} -2.59697 q^{5} +1.00000 q^{7} +2.43780 q^{9} -1.00000 q^{11} +1.00000 q^{13} +6.05589 q^{15} +0.771345 q^{17} -0.470436 q^{19} -2.33191 q^{21} -7.90862 q^{23} +1.74424 q^{25} +1.31100 q^{27} -3.36178 q^{29} +5.85131 q^{31} +2.33191 q^{33} -2.59697 q^{35} +7.25519 q^{37} -2.33191 q^{39} -10.7531 q^{41} +8.08298 q^{43} -6.33088 q^{45} -1.01253 q^{47} +1.00000 q^{49} -1.79871 q^{51} -2.08412 q^{53} +2.59697 q^{55} +1.09701 q^{57} -2.38104 q^{59} +4.66941 q^{61} +2.43780 q^{63} -2.59697 q^{65} +6.10785 q^{67} +18.4422 q^{69} +6.10833 q^{71} +8.85247 q^{73} -4.06740 q^{75} -1.00000 q^{77} -11.2259 q^{79} -10.3705 q^{81} +1.15206 q^{83} -2.00316 q^{85} +7.83937 q^{87} -12.9985 q^{89} +1.00000 q^{91} -13.6447 q^{93} +1.22171 q^{95} -11.7569 q^{97} -2.43780 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9}+O(q^{10}) \) 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9	\( 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9} - 9 q^{11} + 9 q^{13} - 4 q^{15} - 9 q^{17} + 9 q^{19} - 2 q^{21} - 10 q^{23} - q^{25} - 8 q^{27} - 13 q^{29} + 7 q^{31} + 2 q^{33} - 4 q^{35} - 15 q^{37} - 2 q^{39} - 18 q^{41} + 7 q^{43} + 9 q^{45} - 11 q^{47} + 9 q^{49} + q^{51} - 16 q^{53} + 4 q^{55} - 26 q^{57} - 2 q^{59} + 2 q^{61} + q^{63} - 4 q^{65} + 13 q^{67} - 3 q^{69} - q^{71} - 6 q^{73} - 4 q^{75} - 9 q^{77} - 21 q^{79} - 23 q^{81} - 30 q^{83} + q^{85} + q^{87} - 36 q^{89} + 9 q^{91} - 22 q^{93} - 23 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) \) 9 * q - 2 * q^3 - 4 * q^5 + 9 * q^7 + q^9 - 9 * q^11 + 9 * q^13 - 4 * q^15 - 9 * q^17 + 9 * q^19 - 2 * q^21 - 10 * q^23 - q^25 - 8 * q^27 - 13 * q^29 + 7 * q^31 + 2 * q^33 - 4 * q^35 - 15 * q^37 - 2 * q^39 - 18 * q^41 + 7 * q^43 + 9 * q^45 - 11 * q^47 + 9 * q^49 + q^51 - 16 * q^53 + 4 * q^55 - 26 * q^57 - 2 * q^59 + 2 * q^61 + q^63 - 4 * q^65 + 13 * q^67 - 3 * q^69 - q^71 - 6 * q^73 - 4 * q^75 - 9 * q^77 - 21 * q^79 - 23 * q^81 - 30 * q^83 + q^85 + q^87 - 36 * q^89 + 9 * q^91 - 22 * q^93 - 23 * q^95 - 5 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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