LMFDB - Embedded newform 8008.2.a.x.1.10

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} - 17 x^{10} + 79 x^{9} + 80 x^{8} - 536 x^{7} - 4 x^{6} + 1484 x^{5} - 682 x^{4} + \cdots - 64 $ x^12 - 4x^11 - 17x^10 + 79x^9 + 80x^8 - 536x^7 - 4x^6 + 1484x^5 - 682x^4 - 1431x^3 + 1069x^2 - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.54627$ 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)

$f(q)$	$=$	$q+2.54627 q^{3} +3.33743 q^{5} +1.00000 q^{7} +3.48349 q^{9} +O(q^{10})$	$q+2.54627 q^{3} +3.33743 q^{5} +1.00000 q^{7} +3.48349 q^{9} +1.00000 q^{11} +1.00000 q^{13} +8.49801 q^{15} -0.279668 q^{17} -0.144523 q^{19} +2.54627 q^{21} +1.48027 q^{23} +6.13846 q^{25} +1.23110 q^{27} +2.85870 q^{29} -8.41231 q^{31} +2.54627 q^{33} +3.33743 q^{35} +8.10930 q^{37} +2.54627 q^{39} +8.72647 q^{41} +4.84928 q^{43} +11.6259 q^{45} +1.66358 q^{47} +1.00000 q^{49} -0.712109 q^{51} -11.4563 q^{53} +3.33743 q^{55} -0.367993 q^{57} -9.08770 q^{59} +1.96296 q^{61} +3.48349 q^{63} +3.33743 q^{65} -3.85444 q^{67} +3.76917 q^{69} -2.63131 q^{71} -12.4639 q^{73} +15.6302 q^{75} +1.00000 q^{77} +7.30617 q^{79} -7.31576 q^{81} +6.55599 q^{83} -0.933372 q^{85} +7.27901 q^{87} +8.48321 q^{89} +1.00000 q^{91} -21.4200 q^{93} -0.482334 q^{95} +6.78593 q^{97} +3.48349 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9}+O(q^{10}) $ 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9	$ 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9} + 12 q^{11} + 12 q^{13} - 3 q^{15} + 16 q^{17} - 2 q^{19} + 4 q^{21} + 9 q^{23} + 14 q^{25} + 7 q^{27} + 15 q^{29} + 10 q^{31} + 4 q^{33} + 6 q^{35} + 18 q^{37} + 4 q^{39} + 24 q^{41} + 15 q^{45} + 5 q^{47} + 12 q^{49} + 4 q^{51} + 15 q^{53} + 6 q^{55} - 4 q^{57} + 15 q^{59} + 17 q^{61} + 14 q^{63} + 6 q^{65} - 7 q^{67} + 9 q^{71} + 32 q^{73} - 8 q^{75} + 12 q^{77} + 20 q^{79} - 4 q^{81} - 5 q^{83} + 25 q^{85} + 19 q^{87} + 16 q^{89} + 12 q^{91} + 21 q^{93} + 8 q^{95} + 10 q^{97} + 14 q^{99}+O(q^{100}) $ 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9 + 12 * q^11 + 12 * q^13 - 3 * q^15 + 16 * q^17 - 2 * q^19 + 4 * q^21 + 9 * q^23 + 14 * q^25 + 7 * q^27 + 15 * q^29 + 10 * q^31 + 4 * q^33 + 6 * q^35 + 18 * q^37 + 4 * q^39 + 24 * q^41 + 15 * q^45 + 5 * q^47 + 12 * q^49 + 4 * q^51 + 15 * q^53 + 6 * q^55 - 4 * q^57 + 15 * q^59 + 17 * q^61 + 14 * q^63 + 6 * q^65 - 7 * q^67 + 9 * q^71 + 32 * q^73 - 8 * q^75 + 12 * q^77 + 20 * q^79 - 4 * q^81 - 5 * q^83 + 25 * q^85 + 19 * q^87 + 16 * q^89 + 12 * q^91 + 21 * q^93 + 8 * q^95 + 10 * 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
$3$	2.54627	1.47009	0.735045	−	0.678018i	$-0.237161\pi$
$3$	2.54627	1.47009	0.735045	+	0.678018i	$0.237161\pi$
$4$	0	0
$5$	3.33743	1.49255	0.746273	−	0.665640i	$-0.231842\pi$
$5$	3.33743	1.49255	0.746273	+	0.665640i	$0.231842\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	3.48349	1.16116
$10$	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
$15$	8.49801	2.19418
$16$	0	0
$17$	−0.279668	−0.0678294	−0.0339147	−	0.999425i	$-0.510797\pi$
$17$	−0.279668	−0.0678294	−0.0339147	+	0.999425i	$0.510797\pi$
$18$	0	0
$19$	−0.144523	−0.0331557	−0.0165779	−	0.999863i	$-0.505277\pi$
$19$	−0.144523	−0.0331557	−0.0165779	+	0.999863i	$0.505277\pi$
$20$	0	0
$21$	2.54627	0.555642
$22$	0	0
$23$	1.48027	0.308658	0.154329	−	0.988019i	$-0.450678\pi$
$23$	1.48027	0.308658	0.154329	+	0.988019i	$0.450678\pi$
$24$	0	0
$25$	6.13846	1.22769
$26$	0	0
$27$	1.23110	0.236925
$28$	0	0
$29$	2.85870	0.530846	0.265423	−	0.964132i	$-0.414488\pi$
$29$	2.85870	0.530846	0.265423	+	0.964132i	$0.414488\pi$
$30$	0	0
$31$	−8.41231	−1.51090	−0.755448	−	0.655209i	$-0.772580\pi$
$31$	−8.41231	−1.51090	−0.755448	+	0.655209i	$0.772580\pi$
$32$	0	0
$33$	2.54627	0.443249
$34$	0	0
$35$	3.33743	0.564129
$36$	0	0
$37$	8.10930	1.33316	0.666580	−	0.745433i	$-0.267757\pi$
$37$	8.10930	1.33316	0.666580	+	0.745433i	$0.267757\pi$
$38$	0	0
$39$	2.54627	0.407730
$40$	0	0
$41$	8.72647	1.36285	0.681423	−	0.731890i	$-0.261362\pi$
$41$	8.72647	1.36285	0.681423	+	0.731890i	$0.261362\pi$
$42$	0	0
$43$	4.84928	0.739508	0.369754	−	0.929130i	$-0.379442\pi$
$43$	4.84928	0.739508	0.369754	+	0.929130i	$0.379442\pi$
$44$	0	0
$45$	11.6259	1.73309
$46$	0	0
$47$	1.66358	0.242658	0.121329	−	0.992612i	$-0.461284\pi$
$47$	1.66358	0.242658	0.121329	+	0.992612i	$0.461284\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.712109	−0.0997153
$52$	0	0
$53$	−11.4563	−1.57364	−0.786820	−	0.617183i	$-0.788274\pi$
$53$	−11.4563	−1.57364	−0.786820	+	0.617183i	$0.788274\pi$
$54$	0	0
$55$	3.33743	0.450019
$56$	0	0
$57$	−0.367993	−0.0487419
$58$	0	0
$59$	−9.08770	−1.18312	−0.591559	−	0.806262i	$-0.701487\pi$
$59$	−9.08770	−1.18312	−0.591559	+	0.806262i	$0.701487\pi$
$60$	0	0
$61$	1.96296	0.251331	0.125666	−	0.992073i	$-0.459893\pi$
$61$	1.96296	0.251331	0.125666	+	0.992073i	$0.459893\pi$
$62$	0	0
$63$	3.48349	0.438879
$64$	0	0
$65$	3.33743	0.413958
$66$	0	0
$67$	−3.85444	−0.470894	−0.235447	−	0.971887i	$-0.575655\pi$
$67$	−3.85444	−0.470894	−0.235447	+	0.971887i	$0.575655\pi$
$68$	0	0
$69$	3.76917	0.453755
$70$	0	0
$71$	−2.63131	−0.312279	−0.156140	−	0.987735i	$-0.549905\pi$
$71$	−2.63131	−0.312279	−0.156140	+	0.987735i	$0.549905\pi$
$72$	0	0
$73$	−12.4639	−1.45878	−0.729392	−	0.684096i	$-0.760197\pi$
$73$	−12.4639	−1.45878	−0.729392	+	0.684096i	$0.760197\pi$
$74$	0	0
$75$	15.6302	1.80482
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	7.30617	0.822009	0.411004	−	0.911633i	$-0.365178\pi$
$79$	7.30617	0.822009	0.411004	+	0.911633i	$0.365178\pi$
$80$	0	0
$81$	−7.31576	−0.812863
$82$	0	0
$83$	6.55599	0.719613	0.359807	−	0.933027i	$-0.382843\pi$
$83$	6.55599	0.719613	0.359807	+	0.933027i	$0.382843\pi$
$84$	0	0
$85$	−0.933372	−0.101238
$86$	0	0
$87$	7.27901	0.780392
$88$	0	0
$89$	8.48321	0.899218	0.449609	−	0.893225i	$-0.351563\pi$
$89$	8.48321	0.899218	0.449609	+	0.893225i	$0.351563\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−21.4200	−2.22115
$94$	0	0
$95$	−0.482334	−0.0494865
$96$	0	0
$97$	6.78593	0.689007	0.344504	−	0.938785i	$-0.388047\pi$
$97$	6.78593	0.689007	0.344504	+	0.938785i	$0.388047\pi$
$98$	0	0
$99$	3.48349	0.350104
$100$	0	0
$101$	6.22971	0.619879	0.309940	−	0.950756i	$-0.399691\pi$
$101$	6.22971	0.619879	0.309940	+	0.950756i	$0.399691\pi$
$102$	0	0
$103$	8.85693	0.872699	0.436349	−	0.899777i	$-0.356271\pi$
$103$	8.85693	0.872699	0.436349	+	0.899777i	$0.356271\pi$
$104$	0	0
$105$	8.49801	0.829320
$106$	0	0
$107$	10.5401	1.01895	0.509477	−	0.860484i	$-0.329839\pi$
$107$	10.5401	1.01895	0.509477	+	0.860484i	$0.329839\pi$
$108$	0	0
$109$	7.31340	0.700497	0.350248	−	0.936657i	$-0.386097\pi$
$109$	7.31340	0.700497	0.350248	+	0.936657i	$0.386097\pi$
$110$	0	0
$111$	20.6485	1.95987
$112$	0	0
$113$	−4.82380	−0.453785	−0.226892	−	0.973920i	$-0.572857\pi$
$113$	−4.82380	−0.453785	−0.226892	+	0.973920i	$0.572857\pi$
$114$	0	0
$115$	4.94031	0.460686
$116$	0	0
$117$	3.48349	0.322049
$118$	0	0
$119$	−0.279668	−0.0256371
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	22.2199	2.00350
$124$	0	0
$125$	3.79953	0.339840
$126$	0	0
$127$	−16.8206	−1.49259	−0.746293	−	0.665617i	$-0.768168\pi$
$127$	−16.8206	−1.49259	−0.746293	+	0.665617i	$0.768168\pi$
$128$	0	0
$129$	12.3476	1.08714
$130$	0	0
$131$	−3.29357	−0.287761	−0.143880	−	0.989595i	$-0.545958\pi$
$131$	−3.29357	−0.287761	−0.143880	+	0.989595i	$0.545958\pi$
$132$	0	0
$133$	−0.144523	−0.0125317
$134$	0	0
$135$	4.10871	0.353621
$136$	0	0
$137$	−7.29727	−0.623448	−0.311724	−	0.950173i	$-0.600906\pi$
$137$	−7.29727	−0.623448	−0.311724	+	0.950173i	$0.600906\pi$
$138$	0	0
$139$	−9.93794	−0.842925	−0.421463	−	0.906846i	$-0.638483\pi$
$139$	−9.93794	−0.842925	−0.421463	+	0.906846i	$0.638483\pi$
$140$	0	0
$141$	4.23592	0.356729
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	9.54071	0.792312
$146$	0	0
$147$	2.54627	0.210013
$148$	0	0
$149$	−3.19115	−0.261429	−0.130715	−	0.991420i	$-0.541727\pi$
$149$	−3.19115	−0.261429	−0.130715	+	0.991420i	$0.541727\pi$
$150$	0	0
$151$	−17.8999	−1.45667	−0.728336	−	0.685220i	$-0.759706\pi$
$151$	−17.8999	−1.45667	−0.728336	+	0.685220i	$0.759706\pi$
$152$	0	0
$153$	−0.974220	−0.0787610
$154$	0	0
$155$	−28.0755	−2.25508
$156$	0	0
$157$	−0.263565	−0.0210348	−0.0105174	−	0.999945i	$-0.503348\pi$
$157$	−0.263565	−0.0210348	−0.0105174	+	0.999945i	$0.503348\pi$
$158$	0	0
$159$	−29.1708	−2.31339
$160$	0	0
$161$	1.48027	0.116662
$162$	0	0
$163$	12.6795	0.993132	0.496566	−	0.867999i	$-0.334594\pi$
$163$	12.6795	0.993132	0.496566	+	0.867999i	$0.334594\pi$
$164$	0	0
$165$	8.49801	0.661569
$166$	0	0
$167$	0.622035	0.0481345	0.0240672	−	0.999710i	$-0.492338\pi$
$167$	0.622035	0.0481345	0.0240672	+	0.999710i	$0.492338\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−0.503443	−0.0384992
$172$	0	0
$173$	9.74907	0.741208	0.370604	−	0.928791i	$-0.379151\pi$
$173$	9.74907	0.741208	0.370604	+	0.928791i	$0.379151\pi$
$174$	0	0
$175$	6.13846	0.464024
$176$	0	0
$177$	−23.1397	−1.73929
$178$	0	0
$179$	−14.7967	−1.10596	−0.552980	−	0.833195i	$-0.686509\pi$
$179$	−14.7967	−1.10596	−0.552980	+	0.833195i	$0.686509\pi$
$180$	0	0
$181$	−4.65625	−0.346096	−0.173048	−	0.984913i	$-0.555362\pi$
$181$	−4.65625	−0.346096	−0.173048	+	0.984913i	$0.555362\pi$
$182$	0	0
$183$	4.99823	0.369480
$184$	0	0
$185$	27.0642	1.98980
$186$	0	0
$187$	−0.279668	−0.0204513
$188$	0	0
$189$	1.23110	0.0895493
$190$	0	0
$191$	8.24940	0.596906	0.298453	−	0.954424i	$-0.403529\pi$
$191$	8.24940	0.596906	0.298453	+	0.954424i	$0.403529\pi$
$192$	0	0
$193$	17.9838	1.29451	0.647253	−	0.762275i	$-0.275918\pi$
$193$	17.9838	1.29451	0.647253	+	0.762275i	$0.275918\pi$
$194$	0	0
$195$	8.49801	0.608555
$196$	0	0
$197$	−14.7743	−1.05262	−0.526311	−	0.850292i	$-0.676425\pi$
$197$	−14.7743	−1.05262	−0.526311	+	0.850292i	$0.676425\pi$
$198$	0	0
$199$	−11.2399	−0.796779	−0.398389	−	0.917216i	$-0.630431\pi$
$199$	−11.2399	−0.796779	−0.398389	+	0.917216i	$0.630431\pi$
$200$	0	0
$201$	−9.81444	−0.692257
$202$	0	0
$203$	2.85870	0.200641
$204$	0	0
$205$	29.1240	2.03411
$206$	0	0
$207$	5.15652	0.358403
$208$	0	0
$209$	−0.144523	−0.00999683
$210$	0	0
$211$	−3.62432	−0.249508	−0.124754	−	0.992188i	$-0.539814\pi$
$211$	−3.62432	−0.249508	−0.124754	+	0.992188i	$0.539814\pi$
$212$	0	0
$213$	−6.70003	−0.459078
$214$	0	0
$215$	16.1841	1.10375
$216$	0	0
$217$	−8.41231	−0.571065
$218$	0	0
$219$	−31.7364	−2.14454
$220$	0	0
$221$	−0.279668	−0.0188125
$222$	0	0
$223$	8.99351	0.602250	0.301125	−	0.953585i	$-0.402638\pi$
$223$	8.99351	0.602250	0.301125	+	0.953585i	$0.402638\pi$
$224$	0	0
$225$	21.3833	1.42555
$226$	0	0
$227$	−10.5995	−0.703515	−0.351757	−	0.936091i	$-0.614416\pi$
$227$	−10.5995	−0.703515	−0.351757	+	0.936091i	$0.614416\pi$
$228$	0	0
$229$	−21.6151	−1.42837	−0.714184	−	0.699958i	$-0.753202\pi$
$229$	−21.6151	−1.42837	−0.714184	+	0.699958i	$0.753202\pi$
$230$	0	0
$231$	2.54627	0.167532
$232$	0	0
$233$	−12.1805	−0.797974	−0.398987	−	0.916957i	$-0.630638\pi$
$233$	−12.1805	−0.797974	−0.398987	+	0.916957i	$0.630638\pi$
$234$	0	0
$235$	5.55208	0.362178
$236$	0	0
$237$	18.6035	1.20843
$238$	0	0
$239$	4.47691	0.289587	0.144794	−	0.989462i	$-0.453748\pi$
$239$	4.47691	0.289587	0.144794	+	0.989462i	$0.453748\pi$
$240$	0	0
$241$	26.7399	1.72247	0.861234	−	0.508209i	$-0.169692\pi$
$241$	26.7399	1.72247	0.861234	+	0.508209i	$0.169692\pi$
$242$	0	0
$243$	−22.3212	−1.43191
$244$	0	0
$245$	3.33743	0.213221
$246$	0	0
$247$	−0.144523	−0.00919575
$248$	0	0
$249$	16.6933	1.05790
$250$	0	0
$251$	22.8126	1.43992	0.719960	−	0.694016i	$-0.244160\pi$
$251$	22.8126	1.43992	0.719960	+	0.694016i	$0.244160\pi$
$252$	0	0
$253$	1.48027	0.0930640
$254$	0	0
$255$	−2.37662	−0.148830
$256$	0	0
$257$	6.67914	0.416633	0.208317	−	0.978061i	$-0.433202\pi$
$257$	6.67914	0.416633	0.208317	+	0.978061i	$0.433202\pi$
$258$	0	0
$259$	8.10930	0.503887
$260$	0	0
$261$	9.95824	0.616400
$262$	0	0
$263$	−10.9322	−0.674106	−0.337053	−	0.941486i	$-0.609430\pi$
$263$	−10.9322	−0.674106	−0.337053	+	0.941486i	$0.609430\pi$
$264$	0	0
$265$	−38.2345	−2.34873
$266$	0	0
$267$	21.6005	1.32193
$268$	0	0
$269$	−8.93684	−0.544889	−0.272444	−	0.962172i	$-0.587832\pi$
$269$	−8.93684	−0.544889	−0.272444	+	0.962172i	$0.587832\pi$
$270$	0	0
$271$	13.2488	0.804805	0.402403	−	0.915463i	$-0.368175\pi$
$271$	13.2488	0.804805	0.402403	+	0.915463i	$0.368175\pi$
$272$	0	0
$273$	2.54627	0.154107
$274$	0	0
$275$	6.13846	0.370163
$276$	0	0
$277$	23.1467	1.39075	0.695376	−	0.718646i	$-0.255238\pi$
$277$	23.1467	1.39075	0.695376	+	0.718646i	$0.255238\pi$
$278$	0	0
$279$	−29.3042	−1.75440
$280$	0	0
$281$	−13.8682	−0.827305	−0.413652	−	0.910435i	$-0.635747\pi$
$281$	−13.8682	−0.827305	−0.413652	+	0.910435i	$0.635747\pi$
$282$	0	0
$283$	−24.4932	−1.45597	−0.727986	−	0.685592i	$-0.759543\pi$
$283$	−24.4932	−1.45597	−0.727986	+	0.685592i	$0.759543\pi$
$284$	0	0
$285$	−1.22815	−0.0727495
$286$	0	0
$287$	8.72647	0.515107
$288$	0	0
$289$	−16.9218	−0.995399
$290$	0	0
$291$	17.2788	1.01290
$292$	0	0
$293$	2.46848	0.144210	0.0721051	−	0.997397i	$-0.477028\pi$
$293$	2.46848	0.144210	0.0721051	+	0.997397i	$0.477028\pi$
$294$	0	0
$295$	−30.3296	−1.76586
$296$	0	0
$297$	1.23110	0.0714356
$298$	0	0
$299$	1.48027	0.0856064
$300$	0	0
$301$	4.84928	0.279508
$302$	0	0
$303$	15.8625	0.911278
$304$	0	0
$305$	6.55125	0.375123
$306$	0	0
$307$	19.0119	1.08507	0.542533	−	0.840034i	$-0.317465\pi$
$307$	19.0119	1.08507	0.542533	+	0.840034i	$0.317465\pi$
$308$	0	0
$309$	22.5521	1.28295
$310$	0	0
$311$	5.12546	0.290638	0.145319	−	0.989385i	$-0.453579\pi$
$311$	5.12546	0.290638	0.145319	+	0.989385i	$0.453579\pi$
$312$	0	0
$313$	−10.1956	−0.576287	−0.288143	−	0.957587i	$-0.593038\pi$
$313$	−10.1956	−0.576287	−0.288143	+	0.957587i	$0.593038\pi$
$314$	0	0
$315$	11.6259	0.655046
$316$	0	0
$317$	−5.33478	−0.299631	−0.149815	−	0.988714i	$-0.547868\pi$
$317$	−5.33478	−0.299631	−0.149815	+	0.988714i	$0.547868\pi$
$318$	0	0
$319$	2.85870	0.160056
$320$	0	0
$321$	26.8380	1.49795
$322$	0	0
$323$	0.0404183	0.00224893
$324$	0	0
$325$	6.13846	0.340500
$326$	0	0
$327$	18.6219	1.02979
$328$	0	0
$329$	1.66358	0.0917160
$330$	0	0
$331$	7.51299	0.412951	0.206476	−	0.978452i	$-0.433801\pi$
$331$	7.51299	0.412951	0.206476	+	0.978452i	$0.433801\pi$
$332$	0	0
$333$	28.2487	1.54802
$334$	0	0
$335$	−12.8639	−0.702831
$336$	0	0
$337$	17.3169	0.943309	0.471655	−	0.881783i	$-0.343657\pi$
$337$	17.3169	0.943309	0.471655	+	0.881783i	$0.343657\pi$
$338$	0	0
$339$	−12.2827	−0.667104
$340$	0	0
$341$	−8.41231	−0.455552
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	12.5794	0.677250
$346$	0	0
$347$	−23.7741	−1.27626	−0.638130	−	0.769928i	$-0.720292\pi$
$347$	−23.7741	−1.27626	−0.638130	+	0.769928i	$0.720292\pi$
$348$	0	0
$349$	35.0381	1.87555	0.937773	−	0.347250i	$-0.112884\pi$
$349$	35.0381	1.87555	0.937773	+	0.347250i	$0.112884\pi$
$350$	0	0
$351$	1.23110	0.0657112
$352$	0	0
$353$	−23.8854	−1.27129	−0.635646	−	0.771980i	$-0.719266\pi$
$353$	−23.8854	−1.27129	−0.635646	+	0.771980i	$0.719266\pi$
$354$	0	0
$355$	−8.78182	−0.466091
$356$	0	0
$357$	−0.712109	−0.0376888
$358$	0	0
$359$	15.9570	0.842177	0.421088	−	0.907020i	$-0.361648\pi$
$359$	15.9570	0.842177	0.421088	+	0.907020i	$0.361648\pi$
$360$	0	0
$361$	−18.9791	−0.998901
$362$	0	0
$363$	2.54627	0.133645
$364$	0	0
$365$	−41.5973	−2.17730
$366$	0	0
$367$	−16.9091	−0.882645	−0.441323	−	0.897348i	$-0.645491\pi$
$367$	−16.9091	−0.882645	−0.441323	+	0.897348i	$0.645491\pi$
$368$	0	0
$369$	30.3986	1.58249
$370$	0	0
$371$	−11.4563	−0.594780
$372$	0	0
$373$	−13.0493	−0.675669	−0.337835	−	0.941206i	$-0.609694\pi$
$373$	−13.0493	−0.675669	−0.337835	+	0.941206i	$0.609694\pi$
$374$	0	0
$375$	9.67462	0.499595
$376$	0	0
$377$	2.85870	0.147230
$378$	0	0
$379$	−17.8169	−0.915194	−0.457597	−	0.889160i	$-0.651290\pi$
$379$	−17.8169	−0.915194	−0.457597	+	0.889160i	$0.651290\pi$
$380$	0	0
$381$	−42.8298	−2.19424
$382$	0	0
$383$	−22.7088	−1.16037	−0.580183	−	0.814486i	$-0.697019\pi$
$383$	−22.7088	−1.16037	−0.580183	+	0.814486i	$0.697019\pi$
$384$	0	0
$385$	3.33743	0.170091
$386$	0	0
$387$	16.8924	0.858690
$388$	0	0
$389$	26.1915	1.32796	0.663980	−	0.747750i	$-0.268866\pi$
$389$	26.1915	1.32796	0.663980	+	0.747750i	$0.268866\pi$
$390$	0	0
$391$	−0.413984	−0.0209361
$392$	0	0
$393$	−8.38632	−0.423034
$394$	0	0
$395$	24.3839	1.22689
$396$	0	0
$397$	−34.1216	−1.71251	−0.856257	−	0.516550i	$-0.827216\pi$
$397$	−34.1216	−1.71251	−0.856257	+	0.516550i	$0.827216\pi$
$398$	0	0
$399$	−0.367993	−0.0184227
$400$	0	0
$401$	−34.0661	−1.70118	−0.850590	−	0.525829i	$-0.823755\pi$
$401$	−34.0661	−1.70118	−0.850590	+	0.525829i	$0.823755\pi$
$402$	0	0
$403$	−8.41231	−0.419047
$404$	0	0
$405$	−24.4159	−1.21323
$406$	0	0
$407$	8.10930	0.401963
$408$	0	0
$409$	−10.6341	−0.525823	−0.262911	−	0.964820i	$-0.584683\pi$
$409$	−10.6341	−0.525823	−0.262911	+	0.964820i	$0.584683\pi$
$410$	0	0
$411$	−18.5808	−0.916524
$412$	0	0
$413$	−9.08770	−0.447176
$414$	0	0
$415$	21.8802	1.07406
$416$	0	0
$417$	−25.3047	−1.23918
$418$	0	0
$419$	−19.3512	−0.945368	−0.472684	−	0.881232i	$-0.656715\pi$
$419$	−19.3512	−0.945368	−0.472684	+	0.881232i	$0.656715\pi$
$420$	0	0
$421$	−6.06261	−0.295473	−0.147737	−	0.989027i	$-0.547199\pi$
$421$	−6.06261	−0.295473	−0.147737	+	0.989027i	$0.547199\pi$
$422$	0	0
$423$	5.79506	0.281765
$424$	0	0
$425$	−1.71673	−0.0832736
$426$	0	0
$427$	1.96296	0.0949943
$428$	0	0
$429$	2.54627	0.122935
$430$	0	0
$431$	−29.5257	−1.42220	−0.711102	−	0.703089i	$-0.751803\pi$
$431$	−29.5257	−1.42220	−0.711102	+	0.703089i	$0.751803\pi$
$432$	0	0
$433$	16.5126	0.793545	0.396773	−	0.917917i	$-0.370130\pi$
$433$	16.5126	0.793545	0.396773	+	0.917917i	$0.370130\pi$
$434$	0	0
$435$	24.2932	1.16477
$436$	0	0
$437$	−0.213933	−0.0102338
$438$	0	0
$439$	12.1035	0.577666	0.288833	−	0.957379i	$-0.406733\pi$
$439$	12.1035	0.577666	0.288833	+	0.957379i	$0.406733\pi$
$440$	0	0
$441$	3.48349	0.165881
$442$	0	0
$443$	13.8553	0.658285	0.329142	−	0.944280i	$-0.393240\pi$
$443$	13.8553	0.658285	0.329142	+	0.944280i	$0.393240\pi$
$444$	0	0
$445$	28.3121	1.34212
$446$	0	0
$447$	−8.12553	−0.384324
$448$	0	0
$449$	32.3863	1.52840	0.764202	−	0.644977i	$-0.223133\pi$
$449$	32.3863	1.52840	0.764202	+	0.644977i	$0.223133\pi$
$450$	0	0
$451$	8.72647	0.410913
$452$	0	0
$453$	−45.5779	−2.14144
$454$	0	0
$455$	3.33743	0.156461
$456$	0	0
$457$	12.5203	0.585676	0.292838	−	0.956162i	$-0.405400\pi$
$457$	12.5203	0.585676	0.292838	+	0.956162i	$0.405400\pi$
$458$	0	0
$459$	−0.344299	−0.0160705
$460$	0	0
$461$	21.0849	0.982022	0.491011	−	0.871154i	$-0.336628\pi$
$461$	21.0849	0.982022	0.491011	+	0.871154i	$0.336628\pi$
$462$	0	0
$463$	17.6396	0.819782	0.409891	−	0.912135i	$-0.365567\pi$
$463$	17.6396	0.819782	0.409891	+	0.912135i	$0.365567\pi$
$464$	0	0
$465$	−71.4878	−3.31517
$466$	0	0
$467$	−29.5249	−1.36625	−0.683126	−	0.730301i	$-0.739380\pi$
$467$	−29.5249	−1.36625	−0.683126	+	0.730301i	$0.739380\pi$
$468$	0	0
$469$	−3.85444	−0.177981
$470$	0	0
$471$	−0.671108	−0.0309230
$472$	0	0
$473$	4.84928	0.222970
$474$	0	0
$475$	−0.887146	−0.0407050
$476$	0	0
$477$	−39.9078	−1.82725
$478$	0	0
$479$	−6.17871	−0.282312	−0.141156	−	0.989987i	$-0.545082\pi$
$479$	−6.17871	−0.282312	−0.141156	+	0.989987i	$0.545082\pi$
$480$	0	0
$481$	8.10930	0.369752
$482$	0	0
$483$	3.76917	0.171503
$484$	0	0
$485$	22.6476	1.02837
$486$	0	0
$487$	−5.23936	−0.237418	−0.118709	−	0.992929i	$-0.537876\pi$
$487$	−5.23936	−0.237418	−0.118709	+	0.992929i	$0.537876\pi$
$488$	0	0
$489$	32.2853	1.45999
$490$	0	0
$491$	9.08410	0.409960	0.204980	−	0.978766i	$-0.434287\pi$
$491$	9.08410	0.409960	0.204980	+	0.978766i	$0.434287\pi$
$492$	0	0
$493$	−0.799485	−0.0360070
$494$	0	0
$495$	11.6259	0.522546
$496$	0	0
$497$	−2.63131	−0.118030
$498$	0	0
$499$	−27.8365	−1.24613	−0.623067	−	0.782168i	$-0.714114\pi$
$499$	−27.8365	−1.24613	−0.623067	+	0.782168i	$0.714114\pi$
$500$	0	0
$501$	1.58387	0.0707620
$502$	0	0
$503$	−6.03650	−0.269154	−0.134577	−	0.990903i	$-0.542968\pi$
$503$	−6.03650	−0.269154	−0.134577	+	0.990903i	$0.542968\pi$
$504$	0	0
$505$	20.7912	0.925198
$506$	0	0
$507$	2.54627	0.113084
$508$	0	0
$509$	−24.5373	−1.08760	−0.543798	−	0.839216i	$-0.683014\pi$
$509$	−24.5373	−1.08760	−0.543798	+	0.839216i	$0.683014\pi$
$510$	0	0
$511$	−12.4639	−0.551369
$512$	0	0
$513$	−0.177922	−0.00785543
$514$	0	0
$515$	29.5594	1.30254
$516$	0	0
$517$	1.66358	0.0731641
$518$	0	0
$519$	24.8238	1.08964
$520$	0	0
$521$	11.9941	0.525469	0.262734	−	0.964868i	$-0.415376\pi$
$521$	11.9941	0.525469	0.262734	+	0.964868i	$0.415376\pi$
$522$	0	0
$523$	−13.0213	−0.569382	−0.284691	−	0.958619i	$-0.591891\pi$
$523$	−13.0213	−0.569382	−0.284691	+	0.958619i	$0.591891\pi$
$524$	0	0
$525$	15.6302	0.682157
$526$	0	0
$527$	2.35265	0.102483
$528$	0	0
$529$	−20.8088	−0.904730
$530$	0	0
$531$	−31.6569	−1.37379
$532$	0	0
$533$	8.72647	0.377985
$534$	0	0
$535$	35.1770	1.52083
$536$	0	0
$537$	−37.6765	−1.62586
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−0.783438	−0.0336826	−0.0168413	−	0.999858i	$-0.505361\pi$
$541$	−0.783438	−0.0336826	−0.0168413	+	0.999858i	$0.505361\pi$
$542$	0	0
$543$	−11.8561	−0.508793
$544$	0	0
$545$	24.4080	1.04552
$546$	0	0
$547$	−25.8656	−1.10594	−0.552968	−	0.833203i	$-0.686505\pi$
$547$	−25.8656	−1.10594	−0.552968	+	0.833203i	$0.686505\pi$
$548$	0	0
$549$	6.83796	0.291837
$550$	0	0
$551$	−0.413146	−0.0176006
$552$	0	0
$553$	7.30617	0.310690
$554$	0	0
$555$	68.9129	2.92519
$556$	0	0
$557$	17.7909	0.753826	0.376913	−	0.926249i	$-0.376986\pi$
$557$	17.7909	0.753826	0.376913	+	0.926249i	$0.376986\pi$
$558$	0	0
$559$	4.84928	0.205103
$560$	0	0
$561$	−0.712109	−0.0300653
$562$	0	0
$563$	37.7025	1.58897	0.794485	−	0.607284i	$-0.207741\pi$
$563$	37.7025	1.58897	0.794485	+	0.607284i	$0.207741\pi$
$564$	0	0
$565$	−16.0991	−0.677294
$566$	0	0
$567$	−7.31576	−0.307233
$568$	0	0
$569$	11.7093	0.490878	0.245439	−	0.969412i	$-0.421068\pi$
$569$	11.7093	0.490878	0.245439	+	0.969412i	$0.421068\pi$
$570$	0	0
$571$	24.1377	1.01013	0.505066	−	0.863081i	$-0.331468\pi$
$571$	24.1377	1.01013	0.505066	+	0.863081i	$0.331468\pi$
$572$	0	0
$573$	21.0052	0.877505
$574$	0	0
$575$	9.08659	0.378937
$576$	0	0
$577$	5.74967	0.239362	0.119681	−	0.992812i	$-0.461813\pi$
$577$	5.74967	0.239362	0.119681	+	0.992812i	$0.461813\pi$
$578$	0	0
$579$	45.7917	1.90304
$580$	0	0
$581$	6.55599	0.271988
$582$	0	0
$583$	−11.4563	−0.474470
$584$	0	0
$585$	11.6259	0.480673
$586$	0	0
$587$	−5.32894	−0.219949	−0.109974	−	0.993934i	$-0.535077\pi$
$587$	−5.32894	−0.219949	−0.109974	+	0.993934i	$0.535077\pi$
$588$	0	0
$589$	1.21577	0.0500948
$590$	0	0
$591$	−37.6192	−1.54745
$592$	0	0
$593$	−21.5910	−0.886637	−0.443318	−	0.896364i	$-0.646199\pi$
$593$	−21.5910	−0.886637	−0.443318	+	0.896364i	$0.646199\pi$
$594$	0	0
$595$	−0.933372	−0.0382645
$596$	0	0
$597$	−28.6199	−1.17134
$598$	0	0
$599$	0.885745	0.0361906	0.0180953	−	0.999836i	$-0.494240\pi$
$599$	0.885745	0.0361906	0.0180953	+	0.999836i	$0.494240\pi$
$600$	0	0
$601$	−1.62930	−0.0664604	−0.0332302	−	0.999448i	$-0.510579\pi$
$601$	−1.62930	−0.0664604	−0.0332302	+	0.999448i	$0.510579\pi$
$602$	0	0
$603$	−13.4269	−0.546786
$604$	0	0
$605$	3.33743	0.135686
$606$	0	0
$607$	−16.9814	−0.689254	−0.344627	−	0.938740i	$-0.611995\pi$
$607$	−16.9814	−0.689254	−0.344627	+	0.938740i	$0.611995\pi$
$608$	0	0
$609$	7.27901	0.294960
$610$	0	0
$611$	1.66358	0.0673012
$612$	0	0
$613$	10.8592	0.438600	0.219300	−	0.975657i	$-0.429623\pi$
$613$	10.8592	0.438600	0.219300	+	0.975657i	$0.429623\pi$
$614$	0	0
$615$	74.1576	2.99032
$616$	0	0
$617$	−44.7203	−1.80037	−0.900186	−	0.435506i	$-0.856569\pi$
$617$	−44.7203	−1.80037	−0.900186	+	0.435506i	$0.856569\pi$
$618$	0	0
$619$	19.0476	0.765588	0.382794	−	0.923834i	$-0.374962\pi$
$619$	19.0476	0.765588	0.382794	+	0.923834i	$0.374962\pi$
$620$	0	0
$621$	1.82236	0.0731289
$622$	0	0
$623$	8.48321	0.339872
$624$	0	0
$625$	−18.0116	−0.720465
$626$	0	0
$627$	−0.367993	−0.0146962
$628$	0	0
$629$	−2.26791	−0.0904275
$630$	0	0
$631$	−38.8980	−1.54851	−0.774253	−	0.632876i	$-0.781874\pi$
$631$	−38.8980	−1.54851	−0.774253	+	0.632876i	$0.781874\pi$
$632$	0	0
$633$	−9.22849	−0.366800
$634$	0	0
$635$	−56.1376	−2.22775
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−9.16615	−0.362607
$640$	0	0
$641$	−21.1755	−0.836383	−0.418192	−	0.908359i	$-0.637336\pi$
$641$	−21.1755	−0.836383	−0.418192	+	0.908359i	$0.637336\pi$
$642$	0	0
$643$	−5.07042	−0.199958	−0.0999789	−	0.994990i	$-0.531878\pi$
$643$	−5.07042	−0.199958	−0.0999789	+	0.994990i	$0.531878\pi$
$644$	0	0
$645$	41.2092	1.62261
$646$	0	0
$647$	12.3857	0.486932	0.243466	−	0.969909i	$-0.421716\pi$
$647$	12.3857	0.486932	0.243466	+	0.969909i	$0.421716\pi$
$648$	0	0
$649$	−9.08770	−0.356723
$650$	0	0
$651$	−21.4200	−0.839516
$652$	0	0
$653$	35.4065	1.38556	0.692781	−	0.721148i	$-0.256385\pi$
$653$	35.4065	1.38556	0.692781	+	0.721148i	$0.256385\pi$
$654$	0	0
$655$	−10.9921	−0.429496
$656$	0	0
$657$	−43.4177	−1.69389
$658$	0	0
$659$	8.24400	0.321140	0.160570	−	0.987024i	$-0.448667\pi$
$659$	8.24400	0.321140	0.160570	+	0.987024i	$0.448667\pi$
$660$	0	0
$661$	−16.6416	−0.647283	−0.323641	−	0.946180i	$-0.604907\pi$
$661$	−16.6416	−0.647283	−0.323641	+	0.946180i	$0.604907\pi$
$662$	0	0
$663$	−0.712109	−0.0276560
$664$	0	0
$665$	−0.482334	−0.0187041
$666$	0	0
$667$	4.23165	0.163850
$668$	0	0
$669$	22.8999	0.885361
$670$	0	0
$671$	1.96296	0.0757793
$672$	0	0
$673$	38.0822	1.46796	0.733981	−	0.679170i	$-0.237660\pi$
$673$	38.0822	1.46796	0.733981	+	0.679170i	$0.237660\pi$
$674$	0	0
$675$	7.55705	0.290871
$676$	0	0
$677$	9.41854	0.361984	0.180992	−	0.983485i	$-0.442069\pi$
$677$	9.41854	0.361984	0.180992	+	0.983485i	$0.442069\pi$
$678$	0	0
$679$	6.78593	0.260420
$680$	0	0
$681$	−26.9892	−1.03423
$682$	0	0
$683$	−2.75310	−0.105345	−0.0526723	−	0.998612i	$-0.516774\pi$
$683$	−2.75310	−0.105345	−0.0526723	+	0.998612i	$0.516774\pi$
$684$	0	0
$685$	−24.3541	−0.930524
$686$	0	0
$687$	−55.0379	−2.09983
$688$	0	0
$689$	−11.4563	−0.436449
$690$	0	0
$691$	−2.09128	−0.0795559	−0.0397780	−	0.999209i	$-0.512665\pi$
$691$	−2.09128	−0.0795559	−0.0397780	+	0.999209i	$0.512665\pi$
$692$	0	0
$693$	3.48349	0.132327
$694$	0	0
$695$	−33.1672	−1.25810
$696$	0	0
$697$	−2.44051	−0.0924410
$698$	0	0
$699$	−31.0150	−1.17309
$700$	0	0
$701$	25.7930	0.974189	0.487094	−	0.873349i	$-0.338057\pi$
$701$	25.7930	0.974189	0.487094	+	0.873349i	$0.338057\pi$
$702$	0	0
$703$	−1.17198	−0.0442019
$704$	0	0
$705$	14.1371	0.532434
$706$	0	0
$707$	6.22971	0.234292
$708$	0	0
$709$	14.7375	0.553479	0.276740	−	0.960945i	$-0.410746\pi$
$709$	14.7375	0.553479	0.276740	+	0.960945i	$0.410746\pi$
$710$	0	0
$711$	25.4510	0.954487
$712$	0	0
$713$	−12.4525	−0.466350
$714$	0	0
$715$	3.33743	0.124813
$716$	0	0
$717$	11.3994	0.425719
$718$	0	0
$719$	4.10945	0.153257	0.0766284	−	0.997060i	$-0.475584\pi$
$719$	4.10945	0.153257	0.0766284	+	0.997060i	$0.475584\pi$
$720$	0	0
$721$	8.85693	0.329849
$722$	0	0
$723$	68.0870	2.53218
$724$	0	0
$725$	17.5480	0.651716
$726$	0	0
$727$	−32.5930	−1.20881	−0.604404	−	0.796678i	$-0.706589\pi$
$727$	−32.5930	−1.20881	−0.604404	+	0.796678i	$0.706589\pi$
$728$	0	0
$729$	−34.8885	−1.29217
$730$	0	0
$731$	−1.35619	−0.0501604
$732$	0	0
$733$	−7.50841	−0.277330	−0.138665	−	0.990339i	$-0.544281\pi$
$733$	−7.50841	−0.277330	−0.138665	+	0.990339i	$0.544281\pi$
$734$	0	0
$735$	8.49801	0.313454
$736$	0	0
$737$	−3.85444	−0.141980
$738$	0	0
$739$	−4.87616	−0.179372	−0.0896861	−	0.995970i	$-0.528586\pi$
$739$	−4.87616	−0.179372	−0.0896861	+	0.995970i	$0.528586\pi$
$740$	0	0
$741$	−0.367993	−0.0135186
$742$	0	0
$743$	−37.8782	−1.38962	−0.694809	−	0.719195i	$-0.744511\pi$
$743$	−37.8782	−1.38962	−0.694809	+	0.719195i	$0.744511\pi$
$744$	0	0
$745$	−10.6502	−0.390195
$746$	0	0
$747$	22.8377	0.835589
$748$	0	0
$749$	10.5401	0.385128
$750$	0	0
$751$	16.0975	0.587407	0.293704	−	0.955897i	$-0.405112\pi$
$751$	16.0975	0.587407	0.293704	+	0.955897i	$0.405112\pi$
$752$	0	0
$753$	58.0871	2.11681
$754$	0	0
$755$	−59.7397	−2.17415
$756$	0	0
$757$	13.8005	0.501588	0.250794	−	0.968040i	$-0.419308\pi$
$757$	13.8005	0.501588	0.250794	+	0.968040i	$0.419308\pi$
$758$	0	0
$759$	3.76917	0.136812
$760$	0	0
$761$	−32.5756	−1.18086	−0.590432	−	0.807088i	$-0.701043\pi$
$761$	−32.5756	−1.18086	−0.590432	+	0.807088i	$0.701043\pi$
$762$	0	0
$763$	7.31340	0.264763
$764$	0	0
$765$	−3.25139	−0.117554
$766$	0	0
$767$	−9.08770	−0.328138
$768$	0	0
$769$	−26.0413	−0.939074	−0.469537	−	0.882913i	$-0.655579\pi$
$769$	−26.0413	−0.939074	−0.469537	+	0.882913i	$0.655579\pi$
$770$	0	0
$771$	17.0069	0.612488
$772$	0	0
$773$	12.7897	0.460014	0.230007	−	0.973189i	$-0.426125\pi$
$773$	12.7897	0.460014	0.230007	+	0.973189i	$0.426125\pi$
$774$	0	0
$775$	−51.6386	−1.85491
$776$	0	0
$777$	20.6485	0.740760
$778$	0	0
$779$	−1.26117	−0.0451862
$780$	0	0
$781$	−2.63131	−0.0941557
$782$	0	0
$783$	3.51934	0.125771
$784$	0	0
$785$	−0.879632	−0.0313954
$786$	0	0
$787$	−35.2985	−1.25825	−0.629127	−	0.777302i	$-0.716588\pi$
$787$	−35.2985	−1.25825	−0.629127	+	0.777302i	$0.716588\pi$
$788$	0	0
$789$	−27.8362	−0.990996
$790$	0	0
$791$	−4.82380	−0.171515
$792$	0	0
$793$	1.96296	0.0697068
$794$	0	0
$795$	−97.3554	−3.45284
$796$	0	0
$797$	41.4594	1.46857	0.734284	−	0.678843i	$-0.237518\pi$
$797$	41.4594	1.46857	0.734284	+	0.678843i	$0.237518\pi$
$798$	0	0
$799$	−0.465249	−0.0164593
$800$	0	0
$801$	29.5512	1.04414
$802$	0	0
$803$	−12.4639	−0.439840
$804$	0	0
$805$	4.94031	0.174123
$806$	0	0
$807$	−22.7556	−0.801035
$808$	0	0
$809$	34.1270	1.19984	0.599921	−	0.800060i	$-0.295199\pi$
$809$	34.1270	1.19984	0.599921	+	0.800060i	$0.295199\pi$
$810$	0	0
$811$	5.90054	0.207196	0.103598	−	0.994619i	$-0.466964\pi$
$811$	5.90054	0.207196	0.103598	+	0.994619i	$0.466964\pi$
$812$	0	0
$813$	33.7349	1.18314
$814$	0	0
$815$	42.3169	1.48229
$816$	0	0
$817$	−0.700830	−0.0245189
$818$	0	0
$819$	3.48349	0.121723
$820$	0	0
$821$	36.5741	1.27644	0.638222	−	0.769853i	$-0.279670\pi$
$821$	36.5741	1.27644	0.638222	+	0.769853i	$0.279670\pi$
$822$	0	0
$823$	34.2638	1.19436	0.597180	−	0.802107i	$-0.296288\pi$
$823$	34.2638	1.19436	0.597180	+	0.802107i	$0.296288\pi$
$824$	0	0
$825$	15.6302	0.544173
$826$	0	0
$827$	2.55901	0.0889856	0.0444928	−	0.999010i	$-0.485833\pi$
$827$	2.55901	0.0889856	0.0444928	+	0.999010i	$0.485833\pi$
$828$	0	0
$829$	12.8550	0.446471	0.223236	−	0.974765i	$-0.428338\pi$
$829$	12.8550	0.446471	0.223236	+	0.974765i	$0.428338\pi$
$830$	0	0
$831$	58.9378	2.04453
$832$	0	0
$833$	−0.279668	−0.00968991
$834$	0	0
$835$	2.07600	0.0718429
$836$	0	0
$837$	−10.3564	−0.357969
$838$	0	0
$839$	27.4211	0.946683	0.473341	−	0.880879i	$-0.343048\pi$
$839$	27.4211	0.946683	0.473341	+	0.880879i	$0.343048\pi$
$840$	0	0
$841$	−20.8279	−0.718202
$842$	0	0
$843$	−35.3121	−1.21621
$844$	0	0
$845$	3.33743	0.114811
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−62.3664	−2.14041
$850$	0	0
$851$	12.0040	0.411491
$852$	0	0
$853$	22.6761	0.776416	0.388208	−	0.921572i	$-0.373094\pi$
$853$	22.6761	0.776416	0.388208	+	0.921572i	$0.373094\pi$
$854$	0	0
$855$	−1.68021	−0.0574619
$856$	0	0
$857$	−5.54319	−0.189352	−0.0946759	−	0.995508i	$-0.530181\pi$
$857$	−5.54319	−0.189352	−0.0946759	+	0.995508i	$0.530181\pi$
$858$	0	0
$859$	24.3806	0.831855	0.415927	−	0.909398i	$-0.363457\pi$
$859$	24.3806	0.831855	0.415927	+	0.909398i	$0.363457\pi$
$860$	0	0
$861$	22.2199	0.757254
$862$	0	0
$863$	−44.5568	−1.51673	−0.758365	−	0.651830i	$-0.774002\pi$
$863$	−44.5568	−1.51673	−0.758365	+	0.651830i	$0.774002\pi$
$864$	0	0
$865$	32.5369	1.10629
$866$	0	0
$867$	−43.0874	−1.46333
$868$	0	0
$869$	7.30617	0.247845
$870$	0	0
$871$	−3.85444	−0.130603
$872$	0	0
$873$	23.6387	0.800050
$874$	0	0
$875$	3.79953	0.128447
$876$	0	0
$877$	29.6650	1.00172	0.500858	−	0.865530i	$-0.333018\pi$
$877$	29.6650	1.00172	0.500858	+	0.865530i	$0.333018\pi$
$878$	0	0
$879$	6.28542	0.212002
$880$	0	0
$881$	22.6074	0.761663	0.380832	−	0.924644i	$-0.375638\pi$
$881$	22.6074	0.761663	0.380832	+	0.924644i	$0.375638\pi$
$882$	0	0
$883$	−22.7451	−0.765434	−0.382717	−	0.923866i	$-0.625012\pi$
$883$	−22.7451	−0.765434	−0.382717	+	0.923866i	$0.625012\pi$
$884$	0	0
$885$	−77.2273	−2.59597
$886$	0	0
$887$	0.721297	0.0242188	0.0121094	−	0.999927i	$-0.496145\pi$
$887$	0.721297	0.0242188	0.0121094	+	0.999927i	$0.496145\pi$
$888$	0	0
$889$	−16.8206	−0.564145
$890$	0	0
$891$	−7.31576	−0.245087
$892$	0	0
$893$	−0.240425	−0.00804550
$894$	0	0
$895$	−49.3831	−1.65069
$896$	0	0
$897$	3.76917	0.125849
$898$	0	0
$899$	−24.0482	−0.802053
$900$	0	0
$901$	3.20395	0.106739
$902$	0	0
$903$	12.3476	0.410901
$904$	0	0
$905$	−15.5399	−0.516564
$906$	0	0
$907$	37.5022	1.24524	0.622621	−	0.782524i	$-0.286068\pi$
$907$	37.5022	1.24524	0.622621	+	0.782524i	$0.286068\pi$
$908$	0	0
$909$	21.7011	0.719781
$910$	0	0
$911$	−0.382476	−0.0126720	−0.00633599	−	0.999980i	$-0.502017\pi$
$911$	−0.382476	−0.0126720	−0.00633599	+	0.999980i	$0.502017\pi$
$912$	0	0
$913$	6.55599	0.216972
$914$	0	0
$915$	16.6813	0.551465
$916$	0	0
$917$	−3.29357	−0.108763
$918$	0	0
$919$	30.5526	1.00784	0.503919	−	0.863751i	$-0.331891\pi$
$919$	30.5526	1.00784	0.503919	+	0.863751i	$0.331891\pi$
$920$	0	0
$921$	48.4094	1.59514
$922$	0	0
$923$	−2.63131	−0.0866106
$924$	0	0
$925$	49.7786	1.63671
$926$	0	0
$927$	30.8530	1.01335
$928$	0	0
$929$	20.4049	0.669463	0.334731	−	0.942314i	$-0.391354\pi$
$929$	20.4049	0.669463	0.334731	+	0.942314i	$0.391354\pi$
$930$	0	0
$931$	−0.144523	−0.00473653
$932$	0	0
$933$	13.0508	0.427264
$934$	0	0
$935$	−0.933372	−0.0305245
$936$	0	0
$937$	−53.0813	−1.73409	−0.867045	−	0.498230i	$-0.833984\pi$
$937$	−53.0813	−1.73409	−0.867045	+	0.498230i	$0.833984\pi$
$938$	0	0
$939$	−25.9606	−0.847193
$940$	0	0
$941$	−30.8332	−1.00513	−0.502567	−	0.864538i	$-0.667611\pi$
$941$	−30.8332	−1.00513	−0.502567	+	0.864538i	$0.667611\pi$
$942$	0	0
$943$	12.9176	0.420653
$944$	0	0
$945$	4.10871	0.133656
$946$	0	0
$947$	20.7338	0.673757	0.336879	−	0.941548i	$-0.390629\pi$
$947$	20.7338	0.673757	0.336879	+	0.941548i	$0.390629\pi$
$948$	0	0
$949$	−12.4639	−0.404594
$950$	0	0
$951$	−13.5838	−0.440484
$952$	0	0
$953$	−22.7696	−0.737580	−0.368790	−	0.929513i	$-0.620228\pi$
$953$	−22.7696	−0.737580	−0.368790	+	0.929513i	$0.620228\pi$
$954$	0	0
$955$	27.5318	0.890909
$956$	0	0
$957$	7.27901	0.235297
$958$	0	0
$959$	−7.29727	−0.235641
$960$	0	0
$961$	39.7669	1.28280
$962$	0	0
$963$	36.7165	1.18317
$964$	0	0
$965$	60.0199	1.93211
$966$	0	0
$967$	28.6038	0.919838	0.459919	−	0.887961i	$-0.347878\pi$
$967$	28.6038	0.919838	0.459919	+	0.887961i	$0.347878\pi$
$968$	0	0
$969$	0.102916	0.00330613
$970$	0	0
$971$	−56.1047	−1.80049	−0.900243	−	0.435387i	$-0.856611\pi$
$971$	−56.1047	−1.80049	−0.900243	+	0.435387i	$0.856611\pi$
$972$	0	0
$973$	−9.93794	−0.318596
$974$	0	0
$975$	15.6302	0.500566
$976$	0	0
$977$	24.2054	0.774398	0.387199	−	0.921996i	$-0.373443\pi$
$977$	24.2054	0.774398	0.387199	+	0.921996i	$0.373443\pi$
$978$	0	0
$979$	8.48321	0.271124
$980$	0	0
$981$	25.4762	0.813392
$982$	0	0
$983$	3.92184	0.125087	0.0625437	−	0.998042i	$-0.480079\pi$
$983$	3.92184	0.125087	0.0625437	+	0.998042i	$0.480079\pi$
$984$	0	0
$985$	−49.3081	−1.57109
$986$	0	0
$987$	4.23592	0.134831
$988$	0	0
$989$	7.17825	0.228255
$990$	0	0
$991$	−48.4192	−1.53809	−0.769044	−	0.639196i	$-0.779267\pi$
$991$	−48.4192	−1.53809	−0.769044	+	0.639196i	$0.779267\pi$
$992$	0	0
$993$	19.1301	0.607075
$994$	0	0
$995$	−37.5126	−1.18923
$996$	0	0
$997$	11.6738	0.369714	0.184857	−	0.982765i	$-0.440818\pi$
$997$	11.6738	0.369714	0.184857	+	0.982765i	$0.440818\pi$
$998$	0	0
$999$	9.98335	0.315859

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.x.1.10	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.x.1.10	✓	12	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)

\(f(q)\)	\(=\)	\(q+2.54627 q^{3} +3.33743 q^{5} +1.00000 q^{7} +3.48349 q^{9} +O(q^{10})\)	\(q+2.54627 q^{3} +3.33743 q^{5} +1.00000 q^{7} +3.48349 q^{9} +1.00000 q^{11} +1.00000 q^{13} +8.49801 q^{15} -0.279668 q^{17} -0.144523 q^{19} +2.54627 q^{21} +1.48027 q^{23} +6.13846 q^{25} +1.23110 q^{27} +2.85870 q^{29} -8.41231 q^{31} +2.54627 q^{33} +3.33743 q^{35} +8.10930 q^{37} +2.54627 q^{39} +8.72647 q^{41} +4.84928 q^{43} +11.6259 q^{45} +1.66358 q^{47} +1.00000 q^{49} -0.712109 q^{51} -11.4563 q^{53} +3.33743 q^{55} -0.367993 q^{57} -9.08770 q^{59} +1.96296 q^{61} +3.48349 q^{63} +3.33743 q^{65} -3.85444 q^{67} +3.76917 q^{69} -2.63131 q^{71} -12.4639 q^{73} +15.6302 q^{75} +1.00000 q^{77} +7.30617 q^{79} -7.31576 q^{81} +6.55599 q^{83} -0.933372 q^{85} +7.27901 q^{87} +8.48321 q^{89} +1.00000 q^{91} -21.4200 q^{93} -0.482334 q^{95} +6.78593 q^{97} +3.48349 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9}+O(q^{10}) \) 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9	\( 12 q + 4 q^{3} + 6 q^{5} + 12 q^{7} + 14 q^{9} + 12 q^{11} + 12 q^{13} - 3 q^{15} + 16 q^{17} - 2 q^{19} + 4 q^{21} + 9 q^{23} + 14 q^{25} + 7 q^{27} + 15 q^{29} + 10 q^{31} + 4 q^{33} + 6 q^{35} + 18 q^{37} + 4 q^{39} + 24 q^{41} + 15 q^{45} + 5 q^{47} + 12 q^{49} + 4 q^{51} + 15 q^{53} + 6 q^{55} - 4 q^{57} + 15 q^{59} + 17 q^{61} + 14 q^{63} + 6 q^{65} - 7 q^{67} + 9 q^{71} + 32 q^{73} - 8 q^{75} + 12 q^{77} + 20 q^{79} - 4 q^{81} - 5 q^{83} + 25 q^{85} + 19 q^{87} + 16 q^{89} + 12 q^{91} + 21 q^{93} + 8 q^{95} + 10 q^{97} + 14 q^{99}+O(q^{100}) \) 12 * q + 4 * q^3 + 6 * q^5 + 12 * q^7 + 14 * q^9 + 12 * q^11 + 12 * q^13 - 3 * q^15 + 16 * q^17 - 2 * q^19 + 4 * q^21 + 9 * q^23 + 14 * q^25 + 7 * q^27 + 15 * q^29 + 10 * q^31 + 4 * q^33 + 6 * q^35 + 18 * q^37 + 4 * q^39 + 24 * q^41 + 15 * q^45 + 5 * q^47 + 12 * q^49 + 4 * q^51 + 15 * q^53 + 6 * q^55 - 4 * q^57 + 15 * q^59 + 17 * q^61 + 14 * q^63 + 6 * q^65 - 7 * q^67 + 9 * q^71 + 32 * q^73 - 8 * q^75 + 12 * q^77 + 20 * q^79 - 4 * q^81 - 5 * q^83 + 25 * q^85 + 19 * q^87 + 16 * q^89 + 12 * q^91 + 21 * q^93 + 8 * q^95 + 10 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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