LMFDB - Embedded newform 8008.2.a.q.1.4

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:	$9$
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} - 3x^{8} - 15x^{7} + 45x^{6} + 64x^{5} - 201x^{4} - 53x^{3} + 252x^{2} - 69x - 26 $ x^9 - 3x^8 - 15x^7 + 45x^6 + 64x^5 - 201x^4 - 53x^3 + 252x^2 - 69x - 26
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.211829$ 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)

$f(q)$	$=$	$q-0.211829 q^{3} +1.56322 q^{5} +1.00000 q^{7} -2.95513 q^{9} +O(q^{10})$	$q-0.211829 q^{3} +1.56322 q^{5} +1.00000 q^{7} -2.95513 q^{9} +1.00000 q^{11} -1.00000 q^{13} -0.331136 q^{15} -0.0280022 q^{17} +2.10303 q^{19} -0.211829 q^{21} -2.23481 q^{23} -2.55635 q^{25} +1.26147 q^{27} +7.08521 q^{29} +3.30079 q^{31} -0.211829 q^{33} +1.56322 q^{35} -0.372253 q^{37} +0.211829 q^{39} +1.81617 q^{41} -10.5854 q^{43} -4.61951 q^{45} +2.23481 q^{47} +1.00000 q^{49} +0.00593169 q^{51} +9.11321 q^{53} +1.56322 q^{55} -0.445484 q^{57} -9.02094 q^{59} +4.49296 q^{61} -2.95513 q^{63} -1.56322 q^{65} +8.78254 q^{67} +0.473397 q^{69} +10.5481 q^{71} +5.30539 q^{73} +0.541510 q^{75} +1.00000 q^{77} -1.81890 q^{79} +8.59817 q^{81} -11.3736 q^{83} -0.0437735 q^{85} -1.50085 q^{87} -0.279991 q^{89} -1.00000 q^{91} -0.699204 q^{93} +3.28750 q^{95} +7.49583 q^{97} -2.95513 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9	$ 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + q^{17} + 16 q^{19} + 3 q^{21} + 6 q^{23} + 11 q^{25} + 9 q^{27} - 2 q^{29} + 3 q^{31} + 3 q^{33} + 8 q^{35} - 4 q^{37} - 3 q^{39} + 20 q^{41} + 20 q^{43} + 8 q^{45} - 6 q^{47} + 9 q^{49} + 15 q^{51} + 15 q^{53} + 8 q^{55} + 16 q^{57} + 21 q^{59} + 12 q^{63} - 8 q^{65} + 18 q^{67} + 10 q^{69} + 12 q^{71} + 29 q^{73} + 12 q^{75} + 9 q^{77} - 6 q^{79} + 5 q^{81} + 25 q^{83} + 13 q^{85} + 17 q^{87} + 32 q^{89} - 9 q^{91} - 13 q^{93} + 38 q^{95} + 17 q^{97} + 12 q^{99}+O(q^{100}) $ 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + q^17 + 16 * q^19 + 3 * q^21 + 6 * q^23 + 11 * q^25 + 9 * q^27 - 2 * q^29 + 3 * q^31 + 3 * q^33 + 8 * q^35 - 4 * q^37 - 3 * q^39 + 20 * q^41 + 20 * q^43 + 8 * q^45 - 6 * q^47 + 9 * q^49 + 15 * q^51 + 15 * q^53 + 8 * q^55 + 16 * q^57 + 21 * q^59 + 12 * q^63 - 8 * q^65 + 18 * q^67 + 10 * q^69 + 12 * q^71 + 29 * q^73 + 12 * q^75 + 9 * q^77 - 6 * q^79 + 5 * q^81 + 25 * q^83 + 13 * q^85 + 17 * q^87 + 32 * q^89 - 9 * q^91 - 13 * q^93 + 38 * q^95 + 17 * q^97 + 12 * 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
$3$	−0.211829	−0.122300	−0.0611499	−	0.998129i	$-0.519477\pi$
$3$	−0.211829	−0.122300	−0.0611499	+	0.998129i	$0.519477\pi$
$4$	0	0
$5$	1.56322	0.699092	0.349546	−	0.936919i	$-0.386336\pi$
$5$	1.56322	0.699092	0.349546	+	0.936919i	$0.386336\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.95513	−0.985043
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−0.331136	−0.0854988
$16$	0	0
$17$	−0.0280022	−0.00679153	−0.00339576	−	0.999994i	$-0.501081\pi$
$17$	−0.0280022	−0.00679153	−0.00339576	+	0.999994i	$0.501081\pi$
$18$	0	0
$19$	2.10303	0.482469	0.241234	−	0.970467i	$-0.422448\pi$
$19$	2.10303	0.482469	0.241234	+	0.970467i	$0.422448\pi$
$20$	0	0
$21$	−0.211829	−0.0462250
$22$	0	0
$23$	−2.23481	−0.465989	−0.232995	−	0.972478i	$-0.574852\pi$
$23$	−2.23481	−0.465989	−0.232995	+	0.972478i	$0.574852\pi$
$24$	0	0
$25$	−2.55635	−0.511270
$26$	0	0
$27$	1.26147	0.242770
$28$	0	0
$29$	7.08521	1.31569	0.657845	−	0.753154i	$-0.271468\pi$
$29$	7.08521	1.31569	0.657845	+	0.753154i	$0.271468\pi$
$30$	0	0
$31$	3.30079	0.592839	0.296420	−	0.955058i	$-0.404207\pi$
$31$	3.30079	0.592839	0.296420	+	0.955058i	$0.404207\pi$
$32$	0	0
$33$	−0.211829	−0.0368748
$34$	0	0
$35$	1.56322	0.264232
$36$	0	0
$37$	−0.372253	−0.0611981	−0.0305990	−	0.999532i	$-0.509741\pi$
$37$	−0.372253	−0.0611981	−0.0305990	+	0.999532i	$0.509741\pi$
$38$	0	0
$39$	0.211829	0.0339198
$40$	0	0
$41$	1.81617	0.283639	0.141819	−	0.989893i	$-0.454705\pi$
$41$	1.81617	0.283639	0.141819	+	0.989893i	$0.454705\pi$
$42$	0	0
$43$	−10.5854	−1.61425	−0.807127	−	0.590378i	$-0.798979\pi$
$43$	−10.5854	−1.61425	−0.807127	+	0.590378i	$0.798979\pi$
$44$	0	0
$45$	−4.61951	−0.688636
$46$	0	0
$47$	2.23481	0.325980	0.162990	−	0.986628i	$-0.447886\pi$
$47$	2.23481	0.325980	0.162990	+	0.986628i	$0.447886\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.00593169	0.000830602 0
$52$	0	0
$53$	9.11321	1.25180	0.625898	−	0.779905i	$-0.284733\pi$
$53$	9.11321	1.25180	0.625898	+	0.779905i	$0.284733\pi$
$54$	0	0
$55$	1.56322	0.210784
$56$	0	0
$57$	−0.445484	−0.0590058
$58$	0	0
$59$	−9.02094	−1.17443	−0.587213	−	0.809432i	$-0.699775\pi$
$59$	−9.02094	−1.17443	−0.587213	+	0.809432i	$0.699775\pi$
$60$	0	0
$61$	4.49296	0.575264	0.287632	−	0.957741i	$-0.407132\pi$
$61$	4.49296	0.575264	0.287632	+	0.957741i	$0.407132\pi$
$62$	0	0
$63$	−2.95513	−0.372311
$64$	0	0
$65$	−1.56322	−0.193893
$66$	0	0
$67$	8.78254	1.07296	0.536479	−	0.843914i	$-0.319754\pi$
$67$	8.78254	1.07296	0.536479	+	0.843914i	$0.319754\pi$
$68$	0	0
$69$	0.473397	0.0569903
$70$	0	0
$71$	10.5481	1.25183	0.625915	−	0.779891i	$-0.284726\pi$
$71$	10.5481	1.25183	0.625915	+	0.779891i	$0.284726\pi$
$72$	0	0
$73$	5.30539	0.620949	0.310474	−	0.950582i	$-0.399512\pi$
$73$	5.30539	0.620949	0.310474	+	0.950582i	$0.399512\pi$
$74$	0	0
$75$	0.541510	0.0625282
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−1.81890	−0.204642	−0.102321	−	0.994751i	$-0.532627\pi$
$79$	−1.81890	−0.204642	−0.102321	+	0.994751i	$0.532627\pi$
$80$	0	0
$81$	8.59817	0.955352
$82$	0	0
$83$	−11.3736	−1.24842	−0.624210	−	0.781257i	$-0.714579\pi$
$83$	−11.3736	−1.24842	−0.624210	+	0.781257i	$0.714579\pi$
$84$	0	0
$85$	−0.0437735	−0.00474791
$86$	0	0
$87$	−1.50085	−0.160909
$88$	0	0
$89$	−0.279991	−0.0296790	−0.0148395	−	0.999890i	$-0.504724\pi$
$89$	−0.279991	−0.0296790	−0.0148395	+	0.999890i	$0.504724\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−0.699204	−0.0725041
$94$	0	0
$95$	3.28750	0.337290
$96$	0	0
$97$	7.49583	0.761086	0.380543	−	0.924763i	$-0.375737\pi$
$97$	7.49583	0.761086	0.380543	+	0.924763i	$0.375737\pi$
$98$	0	0
$99$	−2.95513	−0.297002
$100$	0	0
$101$	−8.56451	−0.852200	−0.426100	−	0.904676i	$-0.640113\pi$
$101$	−8.56451	−0.852200	−0.426100	+	0.904676i	$0.640113\pi$
$102$	0	0
$103$	7.44823	0.733896	0.366948	−	0.930242i	$-0.380403\pi$
$103$	7.44823	0.733896	0.366948	+	0.930242i	$0.380403\pi$
$104$	0	0
$105$	−0.331136	−0.0323155
$106$	0	0
$107$	13.9564	1.34922	0.674609	−	0.738175i	$-0.264312\pi$
$107$	13.9564	1.34922	0.674609	+	0.738175i	$0.264312\pi$
$108$	0	0
$109$	4.16339	0.398781	0.199390	−	0.979920i	$-0.436104\pi$
$109$	4.16339	0.398781	0.199390	+	0.979920i	$0.436104\pi$
$110$	0	0
$111$	0.0788542	0.00748451
$112$	0	0
$113$	−19.7708	−1.85988	−0.929939	−	0.367714i	$-0.880141\pi$
$113$	−19.7708	−1.85988	−0.929939	+	0.367714i	$0.880141\pi$
$114$	0	0
$115$	−3.49349	−0.325769
$116$	0	0
$117$	2.95513	0.273202
$118$	0	0
$119$	−0.0280022	−0.00256696
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.384719	−0.0346889
$124$	0	0
$125$	−11.8122	−1.05652
$126$	0	0
$127$	15.0551	1.33592	0.667962	−	0.744196i	$-0.267167\pi$
$127$	15.0551	1.33592	0.667962	+	0.744196i	$0.267167\pi$
$128$	0	0
$129$	2.24229	0.197423
$130$	0	0
$131$	2.36366	0.206514	0.103257	−	0.994655i	$-0.467074\pi$
$131$	2.36366	0.206514	0.103257	+	0.994655i	$0.467074\pi$
$132$	0	0
$133$	2.10303	0.182356
$134$	0	0
$135$	1.97195	0.169719
$136$	0	0
$137$	−0.756348	−0.0646192	−0.0323096	−	0.999478i	$-0.510286\pi$
$137$	−0.756348	−0.0646192	−0.0323096	+	0.999478i	$0.510286\pi$
$138$	0	0
$139$	16.5552	1.40419	0.702097	−	0.712081i	$-0.252247\pi$
$139$	16.5552	1.40419	0.702097	+	0.712081i	$0.252247\pi$
$140$	0	0
$141$	−0.473397	−0.0398672
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	11.0757	0.919789
$146$	0	0
$147$	−0.211829	−0.0174714
$148$	0	0
$149$	8.16575	0.668964	0.334482	−	0.942402i	$-0.391439\pi$
$149$	8.16575	0.668964	0.334482	+	0.942402i	$0.391439\pi$
$150$	0	0
$151$	−6.57109	−0.534748	−0.267374	−	0.963593i	$-0.586156\pi$
$151$	−6.57109	−0.534748	−0.267374	+	0.963593i	$0.586156\pi$
$152$	0	0
$153$	0.0827501	0.00668995
$154$	0	0
$155$	5.15985	0.414449
$156$	0	0
$157$	18.3061	1.46098	0.730492	−	0.682921i	$-0.239291\pi$
$157$	18.3061	1.46098	0.730492	+	0.682921i	$0.239291\pi$
$158$	0	0
$159$	−1.93045	−0.153094
$160$	0	0
$161$	−2.23481	−0.176127
$162$	0	0
$163$	18.1427	1.42105	0.710523	−	0.703674i	$-0.248459\pi$
$163$	18.1427	1.42105	0.710523	+	0.703674i	$0.248459\pi$
$164$	0	0
$165$	−0.331136	−0.0257789
$166$	0	0
$167$	25.0943	1.94185	0.970927	−	0.239377i	$-0.0769433\pi$
$167$	25.0943	1.94185	0.970927	+	0.239377i	$0.0769433\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.21473	−0.475253
$172$	0	0
$173$	−6.71625	−0.510627	−0.255314	−	0.966858i	$-0.582179\pi$
$173$	−6.71625	−0.510627	−0.255314	+	0.966858i	$0.582179\pi$
$174$	0	0
$175$	−2.55635	−0.193242
$176$	0	0
$177$	1.91090	0.143632
$178$	0	0
$179$	−6.71797	−0.502125	−0.251062	−	0.967971i	$-0.580780\pi$
$179$	−6.71797	−0.502125	−0.251062	+	0.967971i	$0.580780\pi$
$180$	0	0
$181$	−17.8711	−1.32835	−0.664174	−	0.747578i	$-0.731217\pi$
$181$	−17.8711	−1.32835	−0.664174	+	0.747578i	$0.731217\pi$
$182$	0	0
$183$	−0.951740	−0.0703547
$184$	0	0
$185$	−0.581913	−0.0427831
$186$	0	0
$187$	−0.0280022	−0.00204772
$188$	0	0
$189$	1.26147	0.0917585
$190$	0	0
$191$	20.5351	1.48587	0.742933	−	0.669366i	$-0.233434\pi$
$191$	20.5351	1.48587	0.742933	+	0.669366i	$0.233434\pi$
$192$	0	0
$193$	12.5484	0.903256	0.451628	−	0.892206i	$-0.350844\pi$
$193$	12.5484	0.903256	0.451628	+	0.892206i	$0.350844\pi$
$194$	0	0
$195$	0.331136	0.0237131
$196$	0	0
$197$	−3.06486	−0.218362	−0.109181	−	0.994022i	$-0.534823\pi$
$197$	−3.06486	−0.218362	−0.109181	+	0.994022i	$0.534823\pi$
$198$	0	0
$199$	−3.54347	−0.251190	−0.125595	−	0.992082i	$-0.540084\pi$
$199$	−3.54347	−0.251190	−0.125595	+	0.992082i	$0.540084\pi$
$200$	0	0
$201$	−1.86040	−0.131222
$202$	0	0
$203$	7.08521	0.497284
$204$	0	0
$205$	2.83907	0.198290
$206$	0	0
$207$	6.60414	0.459019
$208$	0	0
$209$	2.10303	0.145470
$210$	0	0
$211$	−7.46034	−0.513591	−0.256795	−	0.966466i	$-0.582667\pi$
$211$	−7.46034	−0.513591	−0.256795	+	0.966466i	$0.582667\pi$
$212$	0	0
$213$	−2.23440	−0.153099
$214$	0	0
$215$	−16.5472	−1.12851
$216$	0	0
$217$	3.30079	0.224072
$218$	0	0
$219$	−1.12384	−0.0759419
$220$	0	0
$221$	0.0280022	0.00188363
$222$	0	0
$223$	8.17036	0.547128	0.273564	−	0.961854i	$-0.411797\pi$
$223$	8.17036	0.547128	0.273564	+	0.961854i	$0.411797\pi$
$224$	0	0
$225$	7.55434	0.503623
$226$	0	0
$227$	−20.5698	−1.36527	−0.682633	−	0.730761i	$-0.739165\pi$
$227$	−20.5698	−1.36527	−0.682633	+	0.730761i	$0.739165\pi$
$228$	0	0
$229$	−11.8266	−0.781526	−0.390763	−	0.920491i	$-0.627789\pi$
$229$	−11.8266	−0.781526	−0.390763	+	0.920491i	$0.627789\pi$
$230$	0	0
$231$	−0.211829	−0.0139373
$232$	0	0
$233$	−11.9593	−0.783478	−0.391739	−	0.920076i	$-0.628126\pi$
$233$	−11.9593	−0.783478	−0.391739	+	0.920076i	$0.628126\pi$
$234$	0	0
$235$	3.49349	0.227890
$236$	0	0
$237$	0.385295	0.0250276
$238$	0	0
$239$	17.5868	1.13760	0.568799	−	0.822476i	$-0.307408\pi$
$239$	17.5868	1.13760	0.568799	+	0.822476i	$0.307408\pi$
$240$	0	0
$241$	−21.9097	−1.41133	−0.705664	−	0.708547i	$-0.749351\pi$
$241$	−21.9097	−1.41133	−0.705664	+	0.708547i	$0.749351\pi$
$242$	0	0
$243$	−5.60576	−0.359610
$244$	0	0
$245$	1.56322	0.0998704
$246$	0	0
$247$	−2.10303	−0.133813
$248$	0	0
$249$	2.40927	0.152681
$250$	0	0
$251$	24.6169	1.55380	0.776901	−	0.629623i	$-0.216791\pi$
$251$	24.6169	1.55380	0.776901	+	0.629623i	$0.216791\pi$
$252$	0	0
$253$	−2.23481	−0.140501
$254$	0	0
$255$	0.00927252	0.000580668 0
$256$	0	0
$257$	19.4461	1.21302	0.606508	−	0.795077i	$-0.292570\pi$
$257$	19.4461	1.21302	0.606508	+	0.795077i	$0.292570\pi$
$258$	0	0
$259$	−0.372253	−0.0231307
$260$	0	0
$261$	−20.9377	−1.29601
$262$	0	0
$263$	24.0598	1.48359	0.741797	−	0.670625i	$-0.233974\pi$
$263$	24.0598	1.48359	0.741797	+	0.670625i	$0.233974\pi$
$264$	0	0
$265$	14.2459	0.875121
$266$	0	0
$267$	0.0593103	0.00362973
$268$	0	0
$269$	−4.95361	−0.302027	−0.151014	−	0.988532i	$-0.548254\pi$
$269$	−4.95361	−0.302027	−0.151014	+	0.988532i	$0.548254\pi$
$270$	0	0
$271$	−1.89192	−0.114926	−0.0574629	−	0.998348i	$-0.518301\pi$
$271$	−1.89192	−0.114926	−0.0574629	+	0.998348i	$0.518301\pi$
$272$	0	0
$273$	0.211829	0.0128205
$274$	0	0
$275$	−2.55635	−0.154154
$276$	0	0
$277$	−6.69219	−0.402095	−0.201047	−	0.979582i	$-0.564435\pi$
$277$	−6.69219	−0.402095	−0.201047	+	0.979582i	$0.564435\pi$
$278$	0	0
$279$	−9.75425	−0.583972
$280$	0	0
$281$	−12.6041	−0.751896	−0.375948	−	0.926641i	$-0.622683\pi$
$281$	−12.6041	−0.751896	−0.375948	+	0.926641i	$0.622683\pi$
$282$	0	0
$283$	12.4856	0.742191	0.371096	−	0.928595i	$-0.378982\pi$
$283$	12.4856	0.742191	0.371096	+	0.928595i	$0.378982\pi$
$284$	0	0
$285$	−0.696389	−0.0412505
$286$	0	0
$287$	1.81617	0.107205
$288$	0	0
$289$	−16.9992	−0.999954
$290$	0	0
$291$	−1.58784	−0.0930806
$292$	0	0
$293$	−15.1764	−0.886615	−0.443308	−	0.896370i	$-0.646195\pi$
$293$	−15.1764	−0.886615	−0.443308	+	0.896370i	$0.646195\pi$
$294$	0	0
$295$	−14.1017	−0.821033
$296$	0	0
$297$	1.26147	0.0731980
$298$	0	0
$299$	2.23481	0.129242
$300$	0	0
$301$	−10.5854	−0.610130
$302$	0	0
$303$	1.81421	0.104224
$304$	0	0
$305$	7.02347	0.402163
$306$	0	0
$307$	−15.1648	−0.865504	−0.432752	−	0.901513i	$-0.642457\pi$
$307$	−15.1648	−0.865504	−0.432752	+	0.901513i	$0.642457\pi$
$308$	0	0
$309$	−1.57775	−0.0897553
$310$	0	0
$311$	22.5969	1.28135	0.640677	−	0.767811i	$-0.278654\pi$
$311$	22.5969	1.28135	0.640677	+	0.767811i	$0.278654\pi$
$312$	0	0
$313$	24.4754	1.38343	0.691717	−	0.722169i	$-0.256855\pi$
$313$	24.4754	1.38343	0.691717	+	0.722169i	$0.256855\pi$
$314$	0	0
$315$	−4.61951	−0.260280
$316$	0	0
$317$	5.06397	0.284421	0.142210	−	0.989836i	$-0.454579\pi$
$317$	5.06397	0.284421	0.142210	+	0.989836i	$0.454579\pi$
$318$	0	0
$319$	7.08521	0.396695
$320$	0	0
$321$	−2.95638	−0.165009
$322$	0	0
$323$	−0.0588895	−0.00327670
$324$	0	0
$325$	2.55635	0.141801
$326$	0	0
$327$	−0.881929	−0.0487708
$328$	0	0
$329$	2.23481	0.123209
$330$	0	0
$331$	17.6872	0.972175	0.486087	−	0.873910i	$-0.338424\pi$
$331$	17.6872	0.972175	0.486087	+	0.873910i	$0.338424\pi$
$332$	0	0
$333$	1.10006	0.0602827
$334$	0	0
$335$	13.7290	0.750097
$336$	0	0
$337$	−3.86778	−0.210691	−0.105346	−	0.994436i	$-0.533595\pi$
$337$	−3.86778	−0.210691	−0.105346	+	0.994436i	$0.533595\pi$
$338$	0	0
$339$	4.18803	0.227463
$340$	0	0
$341$	3.30079	0.178748
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.740023	0.0398415
$346$	0	0
$347$	26.6176	1.42891	0.714453	−	0.699683i	$-0.246676\pi$
$347$	26.6176	1.42891	0.714453	+	0.699683i	$0.246676\pi$
$348$	0	0
$349$	0.615686	0.0329569	0.0164785	−	0.999864i	$-0.494755\pi$
$349$	0.615686	0.0329569	0.0164785	+	0.999864i	$0.494755\pi$
$350$	0	0
$351$	−1.26147	−0.0673323
$352$	0	0
$353$	−20.6380	−1.09845	−0.549224	−	0.835675i	$-0.685076\pi$
$353$	−20.6380	−1.09845	−0.549224	+	0.835675i	$0.685076\pi$
$354$	0	0
$355$	16.4890	0.875145
$356$	0	0
$357$	0.00593169	0.000313938 0
$358$	0	0
$359$	−21.9518	−1.15857	−0.579287	−	0.815124i	$-0.696669\pi$
$359$	−21.9518	−1.15857	−0.579287	+	0.815124i	$0.696669\pi$
$360$	0	0
$361$	−14.5773	−0.767224
$362$	0	0
$363$	−0.211829	−0.0111182
$364$	0	0
$365$	8.29348	0.434101
$366$	0	0
$367$	−16.6478	−0.869009	−0.434505	−	0.900670i	$-0.643077\pi$
$367$	−16.6478	−0.869009	−0.434505	+	0.900670i	$0.643077\pi$
$368$	0	0
$369$	−5.36702	−0.279396
$370$	0	0
$371$	9.11321	0.473134
$372$	0	0
$373$	−9.15576	−0.474067	−0.237034	−	0.971501i	$-0.576175\pi$
$373$	−9.15576	−0.474067	−0.237034	+	0.971501i	$0.576175\pi$
$374$	0	0
$375$	2.50218	0.129212
$376$	0	0
$377$	−7.08521	−0.364907
$378$	0	0
$379$	−23.1478	−1.18902	−0.594511	−	0.804087i	$-0.702654\pi$
$379$	−23.1478	−1.18902	−0.594511	+	0.804087i	$0.702654\pi$
$380$	0	0
$381$	−3.18911	−0.163383
$382$	0	0
$383$	31.7879	1.62428	0.812142	−	0.583460i	$-0.198301\pi$
$383$	31.7879	1.62428	0.812142	+	0.583460i	$0.198301\pi$
$384$	0	0
$385$	1.56322	0.0796690
$386$	0	0
$387$	31.2811	1.59011
$388$	0	0
$389$	22.5864	1.14518	0.572588	−	0.819843i	$-0.305939\pi$
$389$	22.5864	1.14518	0.572588	+	0.819843i	$0.305939\pi$
$390$	0	0
$391$	0.0625794	0.00316478
$392$	0	0
$393$	−0.500693	−0.0252566
$394$	0	0
$395$	−2.84333	−0.143063
$396$	0	0
$397$	30.0518	1.50826	0.754128	−	0.656728i	$-0.228060\pi$
$397$	30.0518	1.50826	0.754128	+	0.656728i	$0.228060\pi$
$398$	0	0
$399$	−0.445484	−0.0223021
$400$	0	0
$401$	−15.5180	−0.774934	−0.387467	−	0.921884i	$-0.626650\pi$
$401$	−15.5180	−0.774934	−0.387467	+	0.921884i	$0.626650\pi$
$402$	0	0
$403$	−3.30079	−0.164424
$404$	0	0
$405$	13.4408	0.667879
$406$	0	0
$407$	−0.372253	−0.0184519
$408$	0	0
$409$	21.9209	1.08392	0.541960	−	0.840404i	$-0.317682\pi$
$409$	21.9209	1.08392	0.541960	+	0.840404i	$0.317682\pi$
$410$	0	0
$411$	0.160217	0.00790291
$412$	0	0
$413$	−9.02094	−0.443892
$414$	0	0
$415$	−17.7795	−0.872761
$416$	0	0
$417$	−3.50688	−0.171733
$418$	0	0
$419$	29.1908	1.42606	0.713032	−	0.701131i	$-0.247321\pi$
$419$	29.1908	1.42606	0.713032	+	0.701131i	$0.247321\pi$
$420$	0	0
$421$	2.46835	0.120300	0.0601500	−	0.998189i	$-0.480842\pi$
$421$	2.46835	0.120300	0.0601500	+	0.998189i	$0.480842\pi$
$422$	0	0
$423$	−6.60414	−0.321104
$424$	0	0
$425$	0.0715833	0.00347230
$426$	0	0
$427$	4.49296	0.217429
$428$	0	0
$429$	0.211829	0.0102272
$430$	0	0
$431$	−15.5201	−0.747577	−0.373788	−	0.927514i	$-0.621941\pi$
$431$	−15.5201	−0.747577	−0.373788	+	0.927514i	$0.621941\pi$
$432$	0	0
$433$	−4.65738	−0.223820	−0.111910	−	0.993718i	$-0.535697\pi$
$433$	−4.65738	−0.223820	−0.111910	+	0.993718i	$0.535697\pi$
$434$	0	0
$435$	−2.34616	−0.112490
$436$	0	0
$437$	−4.69987	−0.224825
$438$	0	0
$439$	20.0861	0.958655	0.479328	−	0.877636i	$-0.340881\pi$
$439$	20.0861	0.958655	0.479328	+	0.877636i	$0.340881\pi$
$440$	0	0
$441$	−2.95513	−0.140720
$442$	0	0
$443$	14.8473	0.705419	0.352709	−	0.935733i	$-0.385260\pi$
$443$	14.8473	0.705419	0.352709	+	0.935733i	$0.385260\pi$
$444$	0	0
$445$	−0.437687	−0.0207483
$446$	0	0
$447$	−1.72975	−0.0818142
$448$	0	0
$449$	40.7034	1.92091	0.960457	−	0.278428i	$-0.0898135\pi$
$449$	40.7034	1.92091	0.960457	+	0.278428i	$0.0898135\pi$
$450$	0	0
$451$	1.81617	0.0855202
$452$	0	0
$453$	1.39195	0.0653995
$454$	0	0
$455$	−1.56322	−0.0732848
$456$	0	0
$457$	−41.2946	−1.93168	−0.965839	−	0.259143i	$-0.916560\pi$
$457$	−41.2946	−1.93168	−0.965839	+	0.259143i	$0.916560\pi$
$458$	0	0
$459$	−0.0353239	−0.00164878
$460$	0	0
$461$	−3.48171	−0.162159	−0.0810797	−	0.996708i	$-0.525837\pi$
$461$	−3.48171	−0.162159	−0.0810797	+	0.996708i	$0.525837\pi$
$462$	0	0
$463$	10.5575	0.490649	0.245325	−	0.969441i	$-0.421105\pi$
$463$	10.5575	0.490649	0.245325	+	0.969441i	$0.421105\pi$
$464$	0	0
$465$	−1.09301	−0.0506871
$466$	0	0
$467$	−27.1181	−1.25488	−0.627438	−	0.778667i	$-0.715896\pi$
$467$	−27.1181	−1.25488	−0.627438	+	0.778667i	$0.715896\pi$
$468$	0	0
$469$	8.78254	0.405540
$470$	0	0
$471$	−3.87776	−0.178678
$472$	0	0
$473$	−10.5854	−0.486716
$474$	0	0
$475$	−5.37609	−0.246672
$476$	0	0
$477$	−26.9307	−1.23307
$478$	0	0
$479$	35.3410	1.61477	0.807387	−	0.590023i	$-0.200881\pi$
$479$	35.3410	1.61477	0.807387	+	0.590023i	$0.200881\pi$
$480$	0	0
$481$	0.372253	0.0169733
$482$	0	0
$483$	0.473397	0.0215403
$484$	0	0
$485$	11.7176	0.532070
$486$	0	0
$487$	−0.481930	−0.0218384	−0.0109192	−	0.999940i	$-0.503476\pi$
$487$	−0.481930	−0.0218384	−0.0109192	+	0.999940i	$0.503476\pi$
$488$	0	0
$489$	−3.84316	−0.173794
$490$	0	0
$491$	−23.1840	−1.04628	−0.523139	−	0.852248i	$-0.675239\pi$
$491$	−23.1840	−1.04628	−0.523139	+	0.852248i	$0.675239\pi$
$492$	0	0
$493$	−0.198401	−0.00893554
$494$	0	0
$495$	−4.61951	−0.207632
$496$	0	0
$497$	10.5481	0.473148
$498$	0	0
$499$	−37.0510	−1.65863	−0.829316	−	0.558780i	$-0.811270\pi$
$499$	−37.0510	−1.65863	−0.829316	+	0.558780i	$0.811270\pi$
$500$	0	0
$501$	−5.31570	−0.237488
$502$	0	0
$503$	−16.6152	−0.740833	−0.370417	−	0.928866i	$-0.620785\pi$
$503$	−16.6152	−0.740833	−0.370417	+	0.928866i	$0.620785\pi$
$504$	0	0
$505$	−13.3882	−0.595767
$506$	0	0
$507$	−0.211829	−0.00940767
$508$	0	0
$509$	−31.0589	−1.37666	−0.688330	−	0.725398i	$-0.741656\pi$
$509$	−31.0589	−1.37666	−0.688330	+	0.725398i	$0.741656\pi$
$510$	0	0
$511$	5.30539	0.234697
$512$	0	0
$513$	2.65292	0.117129
$514$	0	0
$515$	11.6432	0.513061
$516$	0	0
$517$	2.23481	0.0982866
$518$	0	0
$519$	1.42270	0.0624496
$520$	0	0
$521$	−26.8787	−1.17758	−0.588788	−	0.808288i	$-0.700395\pi$
$521$	−26.8787	−1.17758	−0.588788	+	0.808288i	$0.700395\pi$
$522$	0	0
$523$	21.3161	0.932090	0.466045	−	0.884761i	$-0.345678\pi$
$523$	21.3161	0.932090	0.466045	+	0.884761i	$0.345678\pi$
$524$	0	0
$525$	0.541510	0.0236334
$526$	0	0
$527$	−0.0924293	−0.00402628
$528$	0	0
$529$	−18.0056	−0.782854
$530$	0	0
$531$	26.6580	1.15686
$532$	0	0
$533$	−1.81617	−0.0786672
$534$	0	0
$535$	21.8169	0.943228
$536$	0	0
$537$	1.42306	0.0614097
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	41.6119	1.78903	0.894517	−	0.447034i	$-0.147520\pi$
$541$	41.6119	1.78903	0.894517	+	0.447034i	$0.147520\pi$
$542$	0	0
$543$	3.78562	0.162457
$544$	0	0
$545$	6.50829	0.278785
$546$	0	0
$547$	41.5484	1.77648	0.888240	−	0.459379i	$-0.151928\pi$
$547$	41.5484	1.77648	0.888240	+	0.459379i	$0.151928\pi$
$548$	0	0
$549$	−13.2773	−0.566660
$550$	0	0
$551$	14.9004	0.634779
$552$	0	0
$553$	−1.81890	−0.0773473
$554$	0	0
$555$	0.123266	0.00523236
$556$	0	0
$557$	16.1398	0.683867	0.341934	−	0.939724i	$-0.388918\pi$
$557$	16.1398	0.683867	0.341934	+	0.939724i	$0.388918\pi$
$558$	0	0
$559$	10.5854	0.447713
$560$	0	0
$561$	0.00593169	0.000250436 0
$562$	0	0
$563$	−2.39951	−0.101127	−0.0505636	−	0.998721i	$-0.516102\pi$
$563$	−2.39951	−0.101127	−0.0505636	+	0.998721i	$0.516102\pi$
$564$	0	0
$565$	−30.9060	−1.30023
$566$	0	0
$567$	8.59817	0.361089
$568$	0	0
$569$	20.3940	0.854961	0.427480	−	0.904025i	$-0.359401\pi$
$569$	20.3940	0.854961	0.427480	+	0.904025i	$0.359401\pi$
$570$	0	0
$571$	32.4950	1.35987	0.679937	−	0.733270i	$-0.262007\pi$
$571$	32.4950	1.35987	0.679937	+	0.733270i	$0.262007\pi$
$572$	0	0
$573$	−4.34993	−0.181721
$574$	0	0
$575$	5.71294	0.238246
$576$	0	0
$577$	−16.0199	−0.666916	−0.333458	−	0.942765i	$-0.608215\pi$
$577$	−16.0199	−0.666916	−0.333458	+	0.942765i	$0.608215\pi$
$578$	0	0
$579$	−2.65813	−0.110468
$580$	0	0
$581$	−11.3736	−0.471858
$582$	0	0
$583$	9.11321	0.377430
$584$	0	0
$585$	4.61951	0.190993
$586$	0	0
$587$	−2.73869	−0.113038	−0.0565189	−	0.998402i	$-0.518000\pi$
$587$	−2.73869	−0.113038	−0.0565189	+	0.998402i	$0.518000\pi$
$588$	0	0
$589$	6.94167	0.286026
$590$	0	0
$591$	0.649227	0.0267056
$592$	0	0
$593$	8.82453	0.362380	0.181190	−	0.983448i	$-0.442005\pi$
$593$	8.82453	0.362380	0.181190	+	0.983448i	$0.442005\pi$
$594$	0	0
$595$	−0.0437735	−0.00179454
$596$	0	0
$597$	0.750611	0.0307205
$598$	0	0
$599$	−34.6618	−1.41624	−0.708120	−	0.706092i	$-0.750457\pi$
$599$	−34.6618	−1.41624	−0.708120	+	0.706092i	$0.750457\pi$
$600$	0	0
$601$	−22.4532	−0.915885	−0.457943	−	0.888982i	$-0.651413\pi$
$601$	−22.4532	−0.915885	−0.457943	+	0.888982i	$0.651413\pi$
$602$	0	0
$603$	−25.9535	−1.05691
$604$	0	0
$605$	1.56322	0.0635539
$606$	0	0
$607$	−3.29218	−0.133626	−0.0668128	−	0.997766i	$-0.521283\pi$
$607$	−3.29218	−0.133626	−0.0668128	+	0.997766i	$0.521283\pi$
$608$	0	0
$609$	−1.50085	−0.0608177
$610$	0	0
$611$	−2.23481	−0.0904105
$612$	0	0
$613$	4.11907	0.166368	0.0831839	−	0.996534i	$-0.473491\pi$
$613$	4.11907	0.166368	0.0831839	+	0.996534i	$0.473491\pi$
$614$	0	0
$615$	−0.601399	−0.0242508
$616$	0	0
$617$	−0.547283	−0.0220328	−0.0110164	−	0.999939i	$-0.503507\pi$
$617$	−0.547283	−0.0220328	−0.0110164	+	0.999939i	$0.503507\pi$
$618$	0	0
$619$	−37.4219	−1.50411	−0.752057	−	0.659098i	$-0.770938\pi$
$619$	−37.4219	−1.50411	−0.752057	+	0.659098i	$0.770938\pi$
$620$	0	0
$621$	−2.81914	−0.113128
$622$	0	0
$623$	−0.279991	−0.0112176
$624$	0	0
$625$	−5.68334	−0.227334
$626$	0	0
$627$	−0.445484	−0.0177909
$628$	0	0
$629$	0.0104239	0.000415628 0
$630$	0	0
$631$	−4.56330	−0.181662	−0.0908310	−	0.995866i	$-0.528952\pi$
$631$	−4.56330	−0.181662	−0.0908310	+	0.995866i	$0.528952\pi$
$632$	0	0
$633$	1.58032	0.0628120
$634$	0	0
$635$	23.5344	0.933934
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−31.1710	−1.23311
$640$	0	0
$641$	31.5008	1.24421	0.622104	−	0.782934i	$-0.286278\pi$
$641$	31.5008	1.24421	0.622104	+	0.782934i	$0.286278\pi$
$642$	0	0
$643$	9.69301	0.382255	0.191128	−	0.981565i	$-0.438786\pi$
$643$	9.69301	0.382255	0.191128	+	0.981565i	$0.438786\pi$
$644$	0	0
$645$	3.50519	0.138017
$646$	0	0
$647$	−30.6882	−1.20648	−0.603239	−	0.797560i	$-0.706124\pi$
$647$	−30.6882	−1.20648	−0.603239	+	0.797560i	$0.706124\pi$
$648$	0	0
$649$	−9.02094	−0.354103
$650$	0	0
$651$	−0.699204	−0.0274040
$652$	0	0
$653$	38.9877	1.52571	0.762853	−	0.646571i	$-0.223798\pi$
$653$	38.9877	1.52571	0.762853	+	0.646571i	$0.223798\pi$
$654$	0	0
$655$	3.69492	0.144373
$656$	0	0
$657$	−15.6781	−0.611661
$658$	0	0
$659$	23.0165	0.896597	0.448298	−	0.893884i	$-0.352030\pi$
$659$	23.0165	0.896597	0.448298	+	0.893884i	$0.352030\pi$
$660$	0	0
$661$	45.3123	1.76244	0.881222	−	0.472702i	$-0.156721\pi$
$661$	45.3123	1.76244	0.881222	+	0.472702i	$0.156721\pi$
$662$	0	0
$663$	−0.00593169	−0.000230368 0
$664$	0	0
$665$	3.28750	0.127484
$666$	0	0
$667$	−15.8341	−0.613097
$668$	0	0
$669$	−1.73072	−0.0669136
$670$	0	0
$671$	4.49296	0.173449
$672$	0	0
$673$	28.0457	1.08108	0.540541	−	0.841318i	$-0.318220\pi$
$673$	28.0457	1.08108	0.540541	+	0.841318i	$0.318220\pi$
$674$	0	0
$675$	−3.22476	−0.124121
$676$	0	0
$677$	29.9240	1.15007	0.575037	−	0.818127i	$-0.304988\pi$
$677$	29.9240	1.15007	0.575037	+	0.818127i	$0.304988\pi$
$678$	0	0
$679$	7.49583	0.287663
$680$	0	0
$681$	4.35729	0.166972
$682$	0	0
$683$	−21.1496	−0.809266	−0.404633	−	0.914479i	$-0.632601\pi$
$683$	−21.1496	−0.809266	−0.404633	+	0.914479i	$0.632601\pi$
$684$	0	0
$685$	−1.18234	−0.0451748
$686$	0	0
$687$	2.50523	0.0955804
$688$	0	0
$689$	−9.11321	−0.347186
$690$	0	0
$691$	22.5147	0.856500	0.428250	−	0.903660i	$-0.359130\pi$
$691$	22.5147	0.856500	0.428250	+	0.903660i	$0.359130\pi$
$692$	0	0
$693$	−2.95513	−0.112256
$694$	0	0
$695$	25.8794	0.981662
$696$	0	0
$697$	−0.0508568	−0.00192634
$698$	0	0
$699$	2.53333	0.0958192
$700$	0	0
$701$	16.4465	0.621178	0.310589	−	0.950544i	$-0.399474\pi$
$701$	16.4465	0.621178	0.310589	+	0.950544i	$0.399474\pi$
$702$	0	0
$703$	−0.782861	−0.0295262
$704$	0	0
$705$	−0.740023	−0.0278709
$706$	0	0
$707$	−8.56451	−0.322101
$708$	0	0
$709$	−4.56153	−0.171312	−0.0856559	−	0.996325i	$-0.527299\pi$
$709$	−4.56153	−0.171312	−0.0856559	+	0.996325i	$0.527299\pi$
$710$	0	0
$711$	5.37507	0.201581
$712$	0	0
$713$	−7.37662	−0.276257
$714$	0	0
$715$	−1.56322	−0.0584611
$716$	0	0
$717$	−3.72541	−0.139128
$718$	0	0
$719$	21.8867	0.816237	0.408119	−	0.912929i	$-0.366185\pi$
$719$	21.8867	0.816237	0.408119	+	0.912929i	$0.366185\pi$
$720$	0	0
$721$	7.44823	0.277387
$722$	0	0
$723$	4.64112	0.172605
$724$	0	0
$725$	−18.1123	−0.672672
$726$	0	0
$727$	23.8866	0.885905	0.442953	−	0.896545i	$-0.353931\pi$
$727$	23.8866	0.885905	0.442953	+	0.896545i	$0.353931\pi$
$728$	0	0
$729$	−24.6070	−0.911372
$730$	0	0
$731$	0.296413	0.0109632
$732$	0	0
$733$	8.21340	0.303369	0.151684	−	0.988429i	$-0.451530\pi$
$733$	8.21340	0.303369	0.151684	+	0.988429i	$0.451530\pi$
$734$	0	0
$735$	−0.331136	−0.0122141
$736$	0	0
$737$	8.78254	0.323509
$738$	0	0
$739$	−36.3387	−1.33674	−0.668370	−	0.743829i	$-0.733008\pi$
$739$	−36.3387	−1.33674	−0.668370	+	0.743829i	$0.733008\pi$
$740$	0	0
$741$	0.445484	0.0163653
$742$	0	0
$743$	12.2631	0.449890	0.224945	−	0.974371i	$-0.427780\pi$
$743$	12.2631	0.449890	0.224945	+	0.974371i	$0.427780\pi$
$744$	0	0
$745$	12.7648	0.467668
$746$	0	0
$747$	33.6106	1.22975
$748$	0	0
$749$	13.9564	0.509956
$750$	0	0
$751$	−23.6437	−0.862772	−0.431386	−	0.902168i	$-0.641975\pi$
$751$	−23.6437	−0.862772	−0.431386	+	0.902168i	$0.641975\pi$
$752$	0	0
$753$	−5.21457	−0.190030
$754$	0	0
$755$	−10.2721	−0.373838
$756$	0	0
$757$	36.9543	1.34313	0.671563	−	0.740947i	$-0.265623\pi$
$757$	36.9543	1.34313	0.671563	+	0.740947i	$0.265623\pi$
$758$	0	0
$759$	0.473397	0.0171832
$760$	0	0
$761$	21.3080	0.772413	0.386206	−	0.922412i	$-0.373785\pi$
$761$	21.3080	0.772413	0.386206	+	0.922412i	$0.373785\pi$
$762$	0	0
$763$	4.16339	0.150725
$764$	0	0
$765$	0.129356	0.00467689
$766$	0	0
$767$	9.02094	0.325727
$768$	0	0
$769$	26.8029	0.966536	0.483268	−	0.875472i	$-0.339450\pi$
$769$	26.8029	0.966536	0.483268	+	0.875472i	$0.339450\pi$
$770$	0	0
$771$	−4.11926	−0.148352
$772$	0	0
$773$	−53.1399	−1.91131	−0.955655	−	0.294487i	$-0.904851\pi$
$773$	−53.1399	−1.91131	−0.955655	+	0.294487i	$0.904851\pi$
$774$	0	0
$775$	−8.43797	−0.303101
$776$	0	0
$777$	0.0788542	0.00282888
$778$	0	0
$779$	3.81947	0.136847
$780$	0	0
$781$	10.5481	0.377441
$782$	0	0
$783$	8.93778	0.319410
$784$	0	0
$785$	28.6164	1.02136
$786$	0	0
$787$	49.7843	1.77462	0.887310	−	0.461173i	$-0.152571\pi$
$787$	49.7843	1.77462	0.887310	+	0.461173i	$0.152571\pi$
$788$	0	0
$789$	−5.09658	−0.181443
$790$	0	0
$791$	−19.7708	−0.702968
$792$	0	0
$793$	−4.49296	−0.159550
$794$	0	0
$795$	−3.01771	−0.107027
$796$	0	0
$797$	25.2703	0.895120	0.447560	−	0.894254i	$-0.352293\pi$
$797$	25.2703	0.895120	0.447560	+	0.894254i	$0.352293\pi$
$798$	0	0
$799$	−0.0625794	−0.00221390
$800$	0	0
$801$	0.827409	0.0292350
$802$	0	0
$803$	5.30539	0.187223
$804$	0	0
$805$	−3.49349	−0.123129
$806$	0	0
$807$	1.04932	0.0369378
$808$	0	0
$809$	−16.3553	−0.575023	−0.287511	−	0.957777i	$-0.592828\pi$
$809$	−16.3553	−0.575023	−0.287511	+	0.957777i	$0.592828\pi$
$810$	0	0
$811$	22.1195	0.776720	0.388360	−	0.921508i	$-0.373042\pi$
$811$	22.1195	0.776720	0.388360	+	0.921508i	$0.373042\pi$
$812$	0	0
$813$	0.400764	0.0140554
$814$	0	0
$815$	28.3610	0.993442
$816$	0	0
$817$	−22.2614	−0.778827
$818$	0	0
$819$	2.95513	0.103261
$820$	0	0
$821$	−4.29972	−0.150061	−0.0750306	−	0.997181i	$-0.523905\pi$
$821$	−4.29972	−0.150061	−0.0750306	+	0.997181i	$0.523905\pi$
$822$	0	0
$823$	6.94434	0.242064	0.121032	−	0.992649i	$-0.461380\pi$
$823$	6.94434	0.242064	0.121032	+	0.992649i	$0.461380\pi$
$824$	0	0
$825$	0.541510	0.0188529
$826$	0	0
$827$	−11.4771	−0.399097	−0.199548	−	0.979888i	$-0.563948\pi$
$827$	−11.4771	−0.399097	−0.199548	+	0.979888i	$0.563948\pi$
$828$	0	0
$829$	44.8391	1.55733	0.778664	−	0.627441i	$-0.215898\pi$
$829$	44.8391	1.55733	0.778664	+	0.627441i	$0.215898\pi$
$830$	0	0
$831$	1.41760	0.0491761
$832$	0	0
$833$	−0.0280022	−0.000970218 0
$834$	0	0
$835$	39.2278	1.35753
$836$	0	0
$837$	4.16385	0.143924
$838$	0	0
$839$	21.0161	0.725556	0.362778	−	0.931875i	$-0.381828\pi$
$839$	21.0161	0.725556	0.362778	+	0.931875i	$0.381828\pi$
$840$	0	0
$841$	21.2001	0.731039
$842$	0	0
$843$	2.66991	0.0919567
$844$	0	0
$845$	1.56322	0.0537763
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−2.64482	−0.0907698
$850$	0	0
$851$	0.831914	0.0285176
$852$	0	0
$853$	−29.5256	−1.01094	−0.505469	−	0.862845i	$-0.668681\pi$
$853$	−29.5256	−1.01094	−0.505469	+	0.862845i	$0.668681\pi$
$854$	0	0
$855$	−9.71499	−0.332245
$856$	0	0
$857$	−5.28389	−0.180494	−0.0902472	−	0.995919i	$-0.528766\pi$
$857$	−5.28389	−0.180494	−0.0902472	+	0.995919i	$0.528766\pi$
$858$	0	0
$859$	−31.8446	−1.08652	−0.543261	−	0.839564i	$-0.682811\pi$
$859$	−31.8446	−1.08652	−0.543261	+	0.839564i	$0.682811\pi$
$860$	0	0
$861$	−0.384719	−0.0131112
$862$	0	0
$863$	45.2595	1.54065	0.770325	−	0.637651i	$-0.220094\pi$
$863$	45.2595	1.54065	0.770325	+	0.637651i	$0.220094\pi$
$864$	0	0
$865$	−10.4990	−0.356976
$866$	0	0
$867$	3.60093	0.122294
$868$	0	0
$869$	−1.81890	−0.0617018
$870$	0	0
$871$	−8.78254	−0.297585
$872$	0	0
$873$	−22.1511	−0.749702
$874$	0	0
$875$	−11.8122	−0.399326
$876$	0	0
$877$	−25.4599	−0.859721	−0.429860	−	0.902895i	$-0.641437\pi$
$877$	−25.4599	−0.859721	−0.429860	+	0.902895i	$0.641437\pi$
$878$	0	0
$879$	3.21481	0.108433
$880$	0	0
$881$	47.5973	1.60359	0.801797	−	0.597597i	$-0.203878\pi$
$881$	47.5973	1.60359	0.801797	+	0.597597i	$0.203878\pi$
$882$	0	0
$883$	−34.7562	−1.16964	−0.584820	−	0.811163i	$-0.698835\pi$
$883$	−34.7562	−1.16964	−0.584820	+	0.811163i	$0.698835\pi$
$884$	0	0
$885$	2.98716	0.100412
$886$	0	0
$887$	−2.88305	−0.0968034	−0.0484017	−	0.998828i	$-0.515413\pi$
$887$	−2.88305	−0.0968034	−0.0484017	+	0.998828i	$0.515413\pi$
$888$	0	0
$889$	15.0551	0.504932
$890$	0	0
$891$	8.59817	0.288049
$892$	0	0
$893$	4.69987	0.157275
$894$	0	0
$895$	−10.5017	−0.351031
$896$	0	0
$897$	−0.473397	−0.0158063
$898$	0	0
$899$	23.3868	0.779992
$900$	0	0
$901$	−0.255190	−0.00850160
$902$	0	0
$903$	2.24229	0.0746188
$904$	0	0
$905$	−27.9364	−0.928639
$906$	0	0
$907$	23.2041	0.770478	0.385239	−	0.922817i	$-0.374119\pi$
$907$	23.2041	0.770478	0.385239	+	0.922817i	$0.374119\pi$
$908$	0	0
$909$	25.3092	0.839454
$910$	0	0
$911$	−53.2240	−1.76339	−0.881695	−	0.471819i	$-0.843598\pi$
$911$	−53.2240	−1.76339	−0.881695	+	0.471819i	$0.843598\pi$
$912$	0	0
$913$	−11.3736	−0.376413
$914$	0	0
$915$	−1.48778	−0.0491844
$916$	0	0
$917$	2.36366	0.0780550
$918$	0	0
$919$	39.6471	1.30784	0.653918	−	0.756565i	$-0.273124\pi$
$919$	39.6471	1.30784	0.653918	+	0.756565i	$0.273124\pi$
$920$	0	0
$921$	3.21236	0.105851
$922$	0	0
$923$	−10.5481	−0.347195
$924$	0	0
$925$	0.951609	0.0312887
$926$	0	0
$927$	−22.0105	−0.722919
$928$	0	0
$929$	−37.8523	−1.24189	−0.620947	−	0.783852i	$-0.713252\pi$
$929$	−37.8523	−1.24189	−0.620947	+	0.783852i	$0.713252\pi$
$930$	0	0
$931$	2.10303	0.0689241
$932$	0	0
$933$	−4.78669	−0.156709
$934$	0	0
$935$	−0.0437735	−0.00143155
$936$	0	0
$937$	−40.2652	−1.31541	−0.657704	−	0.753277i	$-0.728472\pi$
$937$	−40.2652	−1.31541	−0.657704	+	0.753277i	$0.728472\pi$
$938$	0	0
$939$	−5.18462	−0.169194
$940$	0	0
$941$	−18.1034	−0.590155	−0.295078	−	0.955473i	$-0.595345\pi$
$941$	−18.1034	−0.590155	−0.295078	+	0.955473i	$0.595345\pi$
$942$	0	0
$943$	−4.05879	−0.132172
$944$	0	0
$945$	1.97195	0.0641477
$946$	0	0
$947$	−5.19830	−0.168922	−0.0844611	−	0.996427i	$-0.526917\pi$
$947$	−5.19830	−0.168922	−0.0844611	+	0.996427i	$0.526917\pi$
$948$	0	0
$949$	−5.30539	−0.172220
$950$	0	0
$951$	−1.07270	−0.0347846
$952$	0	0
$953$	−56.6086	−1.83373	−0.916867	−	0.399193i	$-0.869290\pi$
$953$	−56.6086	−1.83373	−0.916867	+	0.399193i	$0.869290\pi$
$954$	0	0
$955$	32.1008	1.03876
$956$	0	0
$957$	−1.50085	−0.0485157
$958$	0	0
$959$	−0.756348	−0.0244238
$960$	0	0
$961$	−20.1048	−0.648542
$962$	0	0
$963$	−41.2430	−1.32904
$964$	0	0
$965$	19.6159	0.631459
$966$	0	0
$967$	−20.9111	−0.672456	−0.336228	−	0.941781i	$-0.609151\pi$
$967$	−20.9111	−0.672456	−0.336228	+	0.941781i	$0.609151\pi$
$968$	0	0
$969$	0.0124745	0.000400740 0
$970$	0	0
$971$	−34.0822	−1.09375	−0.546875	−	0.837215i	$-0.684183\pi$
$971$	−34.0822	−1.09375	−0.546875	+	0.837215i	$0.684183\pi$
$972$	0	0
$973$	16.5552	0.530736
$974$	0	0
$975$	−0.541510	−0.0173422
$976$	0	0
$977$	−13.1437	−0.420503	−0.210252	−	0.977647i	$-0.567428\pi$
$977$	−13.1437	−0.420503	−0.210252	+	0.977647i	$0.567428\pi$
$978$	0	0
$979$	−0.279991	−0.00894854
$980$	0	0
$981$	−12.3034	−0.392816
$982$	0	0
$983$	−59.0140	−1.88225	−0.941127	−	0.338052i	$-0.890232\pi$
$983$	−59.0140	−1.88225	−0.941127	+	0.338052i	$0.890232\pi$
$984$	0	0
$985$	−4.79104	−0.152655
$986$	0	0
$987$	−0.473397	−0.0150684
$988$	0	0
$989$	23.6562	0.752224
$990$	0	0
$991$	−50.5637	−1.60621	−0.803104	−	0.595838i	$-0.796820\pi$
$991$	−50.5637	−1.60621	−0.803104	+	0.595838i	$0.796820\pi$
$992$	0	0
$993$	−3.74666	−0.118897
$994$	0	0
$995$	−5.53922	−0.175605
$996$	0	0
$997$	−29.7036	−0.940722	−0.470361	−	0.882474i	$-0.655876\pi$
$997$	−29.7036	−0.940722	−0.470361	+	0.882474i	$0.655876\pi$
$998$	0	0
$999$	−0.469587	−0.0148571

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.q.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.q.1.4	✓	9	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)

\(f(q)\)	\(=\)	\(q-0.211829 q^{3} +1.56322 q^{5} +1.00000 q^{7} -2.95513 q^{9} +O(q^{10})\)	\(q-0.211829 q^{3} +1.56322 q^{5} +1.00000 q^{7} -2.95513 q^{9} +1.00000 q^{11} -1.00000 q^{13} -0.331136 q^{15} -0.0280022 q^{17} +2.10303 q^{19} -0.211829 q^{21} -2.23481 q^{23} -2.55635 q^{25} +1.26147 q^{27} +7.08521 q^{29} +3.30079 q^{31} -0.211829 q^{33} +1.56322 q^{35} -0.372253 q^{37} +0.211829 q^{39} +1.81617 q^{41} -10.5854 q^{43} -4.61951 q^{45} +2.23481 q^{47} +1.00000 q^{49} +0.00593169 q^{51} +9.11321 q^{53} +1.56322 q^{55} -0.445484 q^{57} -9.02094 q^{59} +4.49296 q^{61} -2.95513 q^{63} -1.56322 q^{65} +8.78254 q^{67} +0.473397 q^{69} +10.5481 q^{71} +5.30539 q^{73} +0.541510 q^{75} +1.00000 q^{77} -1.81890 q^{79} +8.59817 q^{81} -11.3736 q^{83} -0.0437735 q^{85} -1.50085 q^{87} -0.279991 q^{89} -1.00000 q^{91} -0.699204 q^{93} +3.28750 q^{95} +7.49583 q^{97} -2.95513 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9	\( 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + q^{17} + 16 q^{19} + 3 q^{21} + 6 q^{23} + 11 q^{25} + 9 q^{27} - 2 q^{29} + 3 q^{31} + 3 q^{33} + 8 q^{35} - 4 q^{37} - 3 q^{39} + 20 q^{41} + 20 q^{43} + 8 q^{45} - 6 q^{47} + 9 q^{49} + 15 q^{51} + 15 q^{53} + 8 q^{55} + 16 q^{57} + 21 q^{59} + 12 q^{63} - 8 q^{65} + 18 q^{67} + 10 q^{69} + 12 q^{71} + 29 q^{73} + 12 q^{75} + 9 q^{77} - 6 q^{79} + 5 q^{81} + 25 q^{83} + 13 q^{85} + 17 q^{87} + 32 q^{89} - 9 q^{91} - 13 q^{93} + 38 q^{95} + 17 q^{97} + 12 q^{99}+O(q^{100}) \) 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + q^17 + 16 * q^19 + 3 * q^21 + 6 * q^23 + 11 * q^25 + 9 * q^27 - 2 * q^29 + 3 * q^31 + 3 * q^33 + 8 * q^35 - 4 * q^37 - 3 * q^39 + 20 * q^41 + 20 * q^43 + 8 * q^45 - 6 * q^47 + 9 * q^49 + 15 * q^51 + 15 * q^53 + 8 * q^55 + 16 * q^57 + 21 * q^59 + 12 * q^63 - 8 * q^65 + 18 * q^67 + 10 * q^69 + 12 * q^71 + 29 * q^73 + 12 * q^75 + 9 * q^77 - 6 * q^79 + 5 * q^81 + 25 * q^83 + 13 * q^85 + 17 * q^87 + 32 * q^89 - 9 * q^91 - 13 * q^93 + 38 * q^95 + 17 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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