LMFDB - Embedded newform 8008.2.a.u.1.1

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 16x^{8} + 26x^{7} + 93x^{6} - 113x^{5} - 230x^{4} + 197x^{3} + 201x^{2} - 115x + 8 $ x^10 - 2x^9 - 16x^8 + 26x^7 + 93x^6 - 113x^5 - 230x^4 + 197x^3 + 201x^2 - 115*x + 8
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			1.1
Root			$-2.44592$ 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)

$f(q)$	$=$	$q-2.44592 q^{3} +0.430286 q^{5} -1.00000 q^{7} +2.98251 q^{9} +O(q^{10})$	$q-2.44592 q^{3} +0.430286 q^{5} -1.00000 q^{7} +2.98251 q^{9} -1.00000 q^{11} -1.00000 q^{13} -1.05244 q^{15} -2.46017 q^{17} +2.25598 q^{19} +2.44592 q^{21} +0.471995 q^{23} -4.81485 q^{25} +0.0427829 q^{27} +3.50972 q^{29} +2.03486 q^{31} +2.44592 q^{33} -0.430286 q^{35} -3.29838 q^{37} +2.44592 q^{39} +4.77742 q^{41} +6.41707 q^{43} +1.28333 q^{45} +4.46079 q^{47} +1.00000 q^{49} +6.01736 q^{51} -14.0776 q^{53} -0.430286 q^{55} -5.51793 q^{57} +2.10449 q^{59} -12.5768 q^{61} -2.98251 q^{63} -0.430286 q^{65} +9.60207 q^{67} -1.15446 q^{69} +3.80453 q^{71} -8.00847 q^{73} +11.7767 q^{75} +1.00000 q^{77} +7.77702 q^{79} -9.05217 q^{81} +16.6675 q^{83} -1.05858 q^{85} -8.58448 q^{87} +4.07691 q^{89} +1.00000 q^{91} -4.97709 q^{93} +0.970714 q^{95} +9.75068 q^{97} -2.98251 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9	$ 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} - 3 q^{17} + 13 q^{19} - 2 q^{21} + 6 q^{25} + 14 q^{27} - 3 q^{29} - q^{31} - 2 q^{33} + 4 q^{35} - 5 q^{37} - 2 q^{39} + 4 q^{41} + 19 q^{43} - 11 q^{45} + q^{47} + 10 q^{49} + 15 q^{51} - 6 q^{53} + 4 q^{55} - 6 q^{57} + 4 q^{59} - 18 q^{61} - 6 q^{63} + 4 q^{65} + q^{67} - 11 q^{69} - 21 q^{71} - 10 q^{73} + 10 q^{77} + 3 q^{79} - 30 q^{81} + 6 q^{83} - 33 q^{85} - 9 q^{87} - 22 q^{89} + 10 q^{91} - 34 q^{93} - 3 q^{95} - 9 q^{97} - 6 q^{99}+O(q^{100}) $ 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 - 3 * q^17 + 13 * q^19 - 2 * q^21 + 6 * q^25 + 14 * q^27 - 3 * q^29 - q^31 - 2 * q^33 + 4 * q^35 - 5 * q^37 - 2 * q^39 + 4 * q^41 + 19 * q^43 - 11 * q^45 + q^47 + 10 * q^49 + 15 * q^51 - 6 * q^53 + 4 * q^55 - 6 * q^57 + 4 * q^59 - 18 * q^61 - 6 * q^63 + 4 * q^65 + q^67 - 11 * q^69 - 21 * q^71 - 10 * q^73 + 10 * q^77 + 3 * q^79 - 30 * q^81 + 6 * q^83 - 33 * q^85 - 9 * q^87 - 22 * q^89 + 10 * q^91 - 34 * q^93 - 3 * q^95 - 9 * q^97 - 6 * 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
$3$	−2.44592	−1.41215	−0.706075	−	0.708137i	$-0.749536\pi$
$3$	−2.44592	−1.41215	−0.706075	+	0.708137i	$0.749536\pi$
$4$	0	0
$5$	0.430286	0.192430	0.0962148	−	0.995361i	$-0.469326\pi$
$5$	0.430286	0.192430	0.0962148	+	0.995361i	$0.469326\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	2.98251	0.994169
$10$	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
$15$	−1.05244	−0.271740
$16$	0	0
$17$	−2.46017	−0.596678	−0.298339	−	0.954460i	$-0.596433\pi$
$17$	−2.46017	−0.596678	−0.298339	+	0.954460i	$0.596433\pi$
$18$	0	0
$19$	2.25598	0.517556	0.258778	−	0.965937i	$-0.416680\pi$
$19$	2.25598	0.517556	0.258778	+	0.965937i	$0.416680\pi$
$20$	0	0
$21$	2.44592	0.533743
$22$	0	0
$23$	0.471995	0.0984177	0.0492088	−	0.998789i	$-0.484330\pi$
$23$	0.471995	0.0984177	0.0492088	+	0.998789i	$0.484330\pi$
$24$	0	0
$25$	−4.81485	−0.962971
$26$	0	0
$27$	0.0427829	0.00823358
$28$	0	0
$29$	3.50972	0.651739	0.325869	−	0.945415i	$-0.394343\pi$
$29$	3.50972	0.651739	0.325869	+	0.945415i	$0.394343\pi$
$30$	0	0
$31$	2.03486	0.365471	0.182735	−	0.983162i	$-0.441505\pi$
$31$	2.03486	0.365471	0.182735	+	0.983162i	$0.441505\pi$
$32$	0	0
$33$	2.44592	0.425779
$34$	0	0
$35$	−0.430286	−0.0727316
$36$	0	0
$37$	−3.29838	−0.542251	−0.271126	−	0.962544i	$-0.587396\pi$
$37$	−3.29838	−0.542251	−0.271126	+	0.962544i	$0.587396\pi$
$38$	0	0
$39$	2.44592	0.391660
$40$	0	0
$41$	4.77742	0.746108	0.373054	−	0.927810i	$-0.378311\pi$
$41$	4.77742	0.746108	0.373054	+	0.927810i	$0.378311\pi$
$42$	0	0
$43$	6.41707	0.978593	0.489297	−	0.872117i	$-0.337254\pi$
$43$	6.41707	0.978593	0.489297	+	0.872117i	$0.337254\pi$
$44$	0	0
$45$	1.28333	0.191308
$46$	0	0
$47$	4.46079	0.650673	0.325337	−	0.945598i	$-0.394522\pi$
$47$	4.46079	0.650673	0.325337	+	0.945598i	$0.394522\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.01736	0.842600
$52$	0	0
$53$	−14.0776	−1.93371	−0.966854	−	0.255330i	$-0.917816\pi$
$53$	−14.0776	−1.93371	−0.966854	+	0.255330i	$0.917816\pi$
$54$	0	0
$55$	−0.430286	−0.0580197
$56$	0	0
$57$	−5.51793	−0.730867
$58$	0	0
$59$	2.10449	0.273981	0.136991	−	0.990572i	$-0.456257\pi$
$59$	2.10449	0.273981	0.136991	+	0.990572i	$0.456257\pi$
$60$	0	0
$61$	−12.5768	−1.61029	−0.805144	−	0.593079i	$-0.797912\pi$
$61$	−12.5768	−1.61029	−0.805144	+	0.593079i	$0.797912\pi$
$62$	0	0
$63$	−2.98251	−0.375761
$64$	0	0
$65$	−0.430286	−0.0533704
$66$	0	0
$67$	9.60207	1.17308	0.586540	−	0.809920i	$-0.300490\pi$
$67$	9.60207	1.17308	0.586540	+	0.809920i	$0.300490\pi$
$68$	0	0
$69$	−1.15446	−0.138981
$70$	0	0
$71$	3.80453	0.451515	0.225757	−	0.974184i	$-0.427514\pi$
$71$	3.80453	0.451515	0.225757	+	0.974184i	$0.427514\pi$
$72$	0	0
$73$	−8.00847	−0.937320	−0.468660	−	0.883379i	$-0.655263\pi$
$73$	−8.00847	−0.937320	−0.468660	+	0.883379i	$0.655263\pi$
$74$	0	0
$75$	11.7767	1.35986
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	7.77702	0.874983	0.437491	−	0.899223i	$-0.355867\pi$
$79$	7.77702	0.874983	0.437491	+	0.899223i	$0.355867\pi$
$80$	0	0
$81$	−9.05217	−1.00580
$82$	0	0
$83$	16.6675	1.82950	0.914749	−	0.404022i	$-0.132388\pi$
$83$	16.6675	1.82950	0.914749	+	0.404022i	$0.132388\pi$
$84$	0	0
$85$	−1.05858	−0.114819
$86$	0	0
$87$	−8.58448	−0.920353
$88$	0	0
$89$	4.07691	0.432152	0.216076	−	0.976377i	$-0.430674\pi$
$89$	4.07691	0.432152	0.216076	+	0.976377i	$0.430674\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	−4.97709	−0.516100
$94$	0	0
$95$	0.970714	0.0995932
$96$	0	0
$97$	9.75068	0.990032	0.495016	−	0.868884i	$-0.335162\pi$
$97$	9.75068	0.990032	0.495016	+	0.868884i	$0.335162\pi$
$98$	0	0
$99$	−2.98251	−0.299753
$100$	0	0
$101$	−4.32610	−0.430463	−0.215232	−	0.976563i	$-0.569051\pi$
$101$	−4.32610	−0.430463	−0.215232	+	0.976563i	$0.569051\pi$
$102$	0	0
$103$	−13.3418	−1.31460	−0.657302	−	0.753627i	$-0.728303\pi$
$103$	−13.3418	−1.31460	−0.657302	+	0.753627i	$0.728303\pi$
$104$	0	0
$105$	1.05244	0.102708
$106$	0	0
$107$	−0.401094	−0.0387752	−0.0193876	−	0.999812i	$-0.506172\pi$
$107$	−0.401094	−0.0387752	−0.0193876	+	0.999812i	$0.506172\pi$
$108$	0	0
$109$	10.2616	0.982881	0.491441	−	0.870911i	$-0.336471\pi$
$109$	10.2616	0.982881	0.491441	+	0.870911i	$0.336471\pi$
$110$	0	0
$111$	8.06757	0.765740
$112$	0	0
$113$	14.6946	1.38236	0.691178	−	0.722685i	$-0.257092\pi$
$113$	14.6946	1.38236	0.691178	+	0.722685i	$0.257092\pi$
$114$	0	0
$115$	0.203093	0.0189385
$116$	0	0
$117$	−2.98251	−0.275733
$118$	0	0
$119$	2.46017	0.225523
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−11.6852	−1.05362
$124$	0	0
$125$	−4.22319	−0.377734
$126$	0	0
$127$	−9.61745	−0.853410	−0.426705	−	0.904391i	$-0.640326\pi$
$127$	−9.61745	−0.853410	−0.426705	+	0.904391i	$0.640326\pi$
$128$	0	0
$129$	−15.6956	−1.38192
$130$	0	0
$131$	−20.5214	−1.79296	−0.896482	−	0.443080i	$-0.853886\pi$
$131$	−20.5214	−1.79296	−0.896482	+	0.443080i	$0.853886\pi$
$132$	0	0
$133$	−2.25598	−0.195618
$134$	0	0
$135$	0.0184089	0.00158439
$136$	0	0
$137$	−16.6648	−1.42377	−0.711886	−	0.702295i	$-0.752159\pi$
$137$	−16.6648	−1.42377	−0.711886	+	0.702295i	$0.752159\pi$
$138$	0	0
$139$	−12.5834	−1.06731	−0.533654	−	0.845703i	$-0.679182\pi$
$139$	−12.5834	−1.06731	−0.533654	+	0.845703i	$0.679182\pi$
$140$	0	0
$141$	−10.9107	−0.918848
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	1.51018	0.125414
$146$	0	0
$147$	−2.44592	−0.201736
$148$	0	0
$149$	−1.08465	−0.0888582	−0.0444291	−	0.999013i	$-0.514147\pi$
$149$	−1.08465	−0.0888582	−0.0444291	+	0.999013i	$0.514147\pi$
$150$	0	0
$151$	11.4279	0.929987	0.464994	−	0.885314i	$-0.346057\pi$
$151$	11.4279	0.929987	0.464994	+	0.885314i	$0.346057\pi$
$152$	0	0
$153$	−7.33747	−0.593199
$154$	0	0
$155$	0.875570	0.0703275
$156$	0	0
$157$	15.8207	1.26263	0.631316	−	0.775526i	$-0.282515\pi$
$157$	15.8207	1.26263	0.631316	+	0.775526i	$0.282515\pi$
$158$	0	0
$159$	34.4327	2.73069
$160$	0	0
$161$	−0.471995	−0.0371984
$162$	0	0
$163$	−18.4982	−1.44889	−0.724445	−	0.689333i	$-0.757904\pi$
$163$	−18.4982	−1.44889	−0.724445	+	0.689333i	$0.757904\pi$
$164$	0	0
$165$	1.05244	0.0819326
$166$	0	0
$167$	11.5024	0.890082	0.445041	−	0.895510i	$-0.353189\pi$
$167$	11.5024	0.890082	0.445041	+	0.895510i	$0.353189\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.72847	0.514539
$172$	0	0
$173$	2.45503	0.186652	0.0933261	−	0.995636i	$-0.470250\pi$
$173$	2.45503	0.186652	0.0933261	+	0.995636i	$0.470250\pi$
$174$	0	0
$175$	4.81485	0.363969
$176$	0	0
$177$	−5.14740	−0.386903
$178$	0	0
$179$	24.9468	1.86461	0.932306	−	0.361671i	$-0.117794\pi$
$179$	24.9468	1.86461	0.932306	+	0.361671i	$0.117794\pi$
$180$	0	0
$181$	20.4478	1.51987	0.759937	−	0.649996i	$-0.225230\pi$
$181$	20.4478	1.51987	0.759937	+	0.649996i	$0.225230\pi$
$182$	0	0
$183$	30.7617	2.27397
$184$	0	0
$185$	−1.41925	−0.104345
$186$	0	0
$187$	2.46017	0.179905
$188$	0	0
$189$	−0.0427829	−0.00311200
$190$	0	0
$191$	−16.3252	−1.18125	−0.590624	−	0.806947i	$-0.701118\pi$
$191$	−16.3252	−1.18125	−0.590624	+	0.806947i	$0.701118\pi$
$192$	0	0
$193$	5.58042	0.401688	0.200844	−	0.979623i	$-0.435632\pi$
$193$	5.58042	0.401688	0.200844	+	0.979623i	$0.435632\pi$
$194$	0	0
$195$	1.05244	0.0753670
$196$	0	0
$197$	−12.8425	−0.914991	−0.457495	−	0.889212i	$-0.651253\pi$
$197$	−12.8425	−0.914991	−0.457495	+	0.889212i	$0.651253\pi$
$198$	0	0
$199$	−10.9559	−0.776646	−0.388323	−	0.921523i	$-0.626946\pi$
$199$	−10.9559	−0.776646	−0.388323	+	0.921523i	$0.626946\pi$
$200$	0	0
$201$	−23.4859	−1.65657
$202$	0	0
$203$	−3.50972	−0.246334
$204$	0	0
$205$	2.05566	0.143573
$206$	0	0
$207$	1.40773	0.0978439
$208$	0	0
$209$	−2.25598	−0.156049
$210$	0	0
$211$	1.62870	0.112124	0.0560622	−	0.998427i	$-0.482146\pi$
$211$	1.62870	0.112124	0.0560622	+	0.998427i	$0.482146\pi$
$212$	0	0
$213$	−9.30557	−0.637607
$214$	0	0
$215$	2.76117	0.188310
$216$	0	0
$217$	−2.03486	−0.138135
$218$	0	0
$219$	19.5880	1.32364
$220$	0	0
$221$	2.46017	0.165489
$222$	0	0
$223$	1.61551	0.108182	0.0540911	−	0.998536i	$-0.482774\pi$
$223$	1.61551	0.108182	0.0540911	+	0.998536i	$0.482774\pi$
$224$	0	0
$225$	−14.3603	−0.957356
$226$	0	0
$227$	20.1498	1.33739	0.668693	−	0.743539i	$-0.266854\pi$
$227$	20.1498	1.33739	0.668693	+	0.743539i	$0.266854\pi$
$228$	0	0
$229$	22.4708	1.48492	0.742458	−	0.669893i	$-0.233660\pi$
$229$	22.4708	1.48492	0.742458	+	0.669893i	$0.233660\pi$
$230$	0	0
$231$	−2.44592	−0.160930
$232$	0	0
$233$	−23.3179	−1.52761	−0.763803	−	0.645450i	$-0.776670\pi$
$233$	−23.3179	−1.52761	−0.763803	+	0.645450i	$0.776670\pi$
$234$	0	0
$235$	1.91941	0.125209
$236$	0	0
$237$	−19.0219	−1.23561
$238$	0	0
$239$	−12.2926	−0.795143	−0.397572	−	0.917571i	$-0.630147\pi$
$239$	−12.2926	−0.795143	−0.397572	+	0.917571i	$0.630147\pi$
$240$	0	0
$241$	−1.36081	−0.0876576	−0.0438288	−	0.999039i	$-0.513956\pi$
$241$	−1.36081	−0.0876576	−0.0438288	+	0.999039i	$0.513956\pi$
$242$	0	0
$243$	22.0125	1.41210
$244$	0	0
$245$	0.430286	0.0274900
$246$	0	0
$247$	−2.25598	−0.143544
$248$	0	0
$249$	−40.7674	−2.58353
$250$	0	0
$251$	18.4003	1.16142	0.580710	−	0.814111i	$-0.302775\pi$
$251$	18.4003	1.16142	0.580710	+	0.814111i	$0.302775\pi$
$252$	0	0
$253$	−0.471995	−0.0296741
$254$	0	0
$255$	2.58919	0.162141
$256$	0	0
$257$	−9.86941	−0.615637	−0.307819	−	0.951445i	$-0.599599\pi$
$257$	−9.86941	−0.615637	−0.307819	+	0.951445i	$0.599599\pi$
$258$	0	0
$259$	3.29838	0.204952
$260$	0	0
$261$	10.4678	0.647939
$262$	0	0
$263$	−30.2415	−1.86477	−0.932386	−	0.361463i	$-0.882277\pi$
$263$	−30.2415	−1.86477	−0.932386	+	0.361463i	$0.882277\pi$
$264$	0	0
$265$	−6.05739	−0.372103
$266$	0	0
$267$	−9.97179	−0.610264
$268$	0	0
$269$	−8.75389	−0.533734	−0.266867	−	0.963733i	$-0.585988\pi$
$269$	−8.75389	−0.533734	−0.266867	+	0.963733i	$0.585988\pi$
$270$	0	0
$271$	11.6433	0.707278	0.353639	−	0.935382i	$-0.384944\pi$
$271$	11.6433	0.707278	0.353639	+	0.935382i	$0.384944\pi$
$272$	0	0
$273$	−2.44592	−0.148034
$274$	0	0
$275$	4.81485	0.290347
$276$	0	0
$277$	−13.7762	−0.827729	−0.413864	−	0.910339i	$-0.635821\pi$
$277$	−13.7762	−0.827729	−0.413864	+	0.910339i	$0.635821\pi$
$278$	0	0
$279$	6.06898	0.363340
$280$	0	0
$281$	20.0948	1.19875	0.599377	−	0.800467i	$-0.295415\pi$
$281$	20.0948	1.19875	0.599377	+	0.800467i	$0.295415\pi$
$282$	0	0
$283$	−19.3879	−1.15249	−0.576245	−	0.817277i	$-0.695483\pi$
$283$	−19.3879	−1.15249	−0.576245	+	0.817277i	$0.695483\pi$
$284$	0	0
$285$	−2.37429	−0.140641
$286$	0	0
$287$	−4.77742	−0.282002
$288$	0	0
$289$	−10.9476	−0.643975
$290$	0	0
$291$	−23.8494	−1.39807
$292$	0	0
$293$	22.0144	1.28609	0.643047	−	0.765827i	$-0.277670\pi$
$293$	22.0144	1.28609	0.643047	+	0.765827i	$0.277670\pi$
$294$	0	0
$295$	0.905532	0.0527221
$296$	0	0
$297$	−0.0427829	−0.00248252
$298$	0	0
$299$	−0.471995	−0.0272962
$300$	0	0
$301$	−6.41707	−0.369874
$302$	0	0
$303$	10.5813	0.607879
$304$	0	0
$305$	−5.41160	−0.309867
$306$	0	0
$307$	2.00157	0.114236	0.0571178	−	0.998367i	$-0.481809\pi$
$307$	2.00157	0.114236	0.0571178	+	0.998367i	$0.481809\pi$
$308$	0	0
$309$	32.6329	1.85642
$310$	0	0
$311$	29.1021	1.65023	0.825115	−	0.564964i	$-0.191110\pi$
$311$	29.1021	1.65023	0.825115	+	0.564964i	$0.191110\pi$
$312$	0	0
$313$	17.4016	0.983597	0.491798	−	0.870709i	$-0.336340\pi$
$313$	17.4016	0.983597	0.491798	+	0.870709i	$0.336340\pi$
$314$	0	0
$315$	−1.28333	−0.0723075
$316$	0	0
$317$	−5.63617	−0.316559	−0.158279	−	0.987394i	$-0.550595\pi$
$317$	−5.63617	−0.316559	−0.158279	+	0.987394i	$0.550595\pi$
$318$	0	0
$319$	−3.50972	−0.196507
$320$	0	0
$321$	0.981042	0.0547564
$322$	0	0
$323$	−5.55008	−0.308815
$324$	0	0
$325$	4.81485	0.267080
$326$	0	0
$327$	−25.0990	−1.38798
$328$	0	0
$329$	−4.46079	−0.245931
$330$	0	0
$331$	−4.40982	−0.242385	−0.121193	−	0.992629i	$-0.538672\pi$
$331$	−4.40982	−0.242385	−0.121193	+	0.992629i	$0.538672\pi$
$332$	0	0
$333$	−9.83746	−0.539089
$334$	0	0
$335$	4.13164	0.225735
$336$	0	0
$337$	0.228993	0.0124740	0.00623701	−	0.999981i	$-0.498015\pi$
$337$	0.228993	0.0124740	0.00623701	+	0.999981i	$0.498015\pi$
$338$	0	0
$339$	−35.9419	−1.95210
$340$	0	0
$341$	−2.03486	−0.110194
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−0.496748	−0.0267440
$346$	0	0
$347$	−22.0734	−1.18496	−0.592482	−	0.805584i	$-0.701852\pi$
$347$	−22.0734	−1.18496	−0.592482	+	0.805584i	$0.701852\pi$
$348$	0	0
$349$	−37.0083	−1.98101	−0.990505	−	0.137473i	$-0.956102\pi$
$349$	−37.0083	−1.98101	−0.990505	+	0.137473i	$0.956102\pi$
$350$	0	0
$351$	−0.0427829	−0.00228358
$352$	0	0
$353$	−33.8429	−1.80128	−0.900638	−	0.434570i	$-0.856900\pi$
$353$	−33.8429	−1.80128	−0.900638	+	0.434570i	$0.856900\pi$
$354$	0	0
$355$	1.63704	0.0868849
$356$	0	0
$357$	−6.01736	−0.318473
$358$	0	0
$359$	−35.9056	−1.89503	−0.947513	−	0.319718i	$-0.896412\pi$
$359$	−35.9056	−1.89503	−0.947513	+	0.319718i	$0.896412\pi$
$360$	0	0
$361$	−13.9106	−0.732136
$362$	0	0
$363$	−2.44592	−0.128377
$364$	0	0
$365$	−3.44593	−0.180368
$366$	0	0
$367$	−22.3371	−1.16599	−0.582995	−	0.812476i	$-0.698119\pi$
$367$	−22.3371	−1.16599	−0.582995	+	0.812476i	$0.698119\pi$
$368$	0	0
$369$	14.2487	0.741757
$370$	0	0
$371$	14.0776	0.730873
$372$	0	0
$373$	−30.0439	−1.55561	−0.777806	−	0.628504i	$-0.783667\pi$
$373$	−30.0439	−1.55561	−0.777806	+	0.628504i	$0.783667\pi$
$374$	0	0
$375$	10.3296	0.533417
$376$	0	0
$377$	−3.50972	−0.180760
$378$	0	0
$379$	14.4689	0.743219	0.371609	−	0.928389i	$-0.378806\pi$
$379$	14.4689	0.743219	0.371609	+	0.928389i	$0.378806\pi$
$380$	0	0
$381$	23.5235	1.20514
$382$	0	0
$383$	−25.4535	−1.30061	−0.650306	−	0.759672i	$-0.725360\pi$
$383$	−25.4535	−1.30061	−0.650306	+	0.759672i	$0.725360\pi$
$384$	0	0
$385$	0.430286	0.0219294
$386$	0	0
$387$	19.1390	0.972888
$388$	0	0
$389$	6.42095	0.325555	0.162777	−	0.986663i	$-0.447955\pi$
$389$	6.42095	0.325555	0.162777	+	0.986663i	$0.447955\pi$
$390$	0	0
$391$	−1.16119	−0.0587237
$392$	0	0
$393$	50.1937	2.53194
$394$	0	0
$395$	3.34634	0.168373
$396$	0	0
$397$	−19.1594	−0.961580	−0.480790	−	0.876836i	$-0.659650\pi$
$397$	−19.1594	−0.961580	−0.480790	+	0.876836i	$0.659650\pi$
$398$	0	0
$399$	5.51793	0.276242
$400$	0	0
$401$	−6.39880	−0.319541	−0.159770	−	0.987154i	$-0.551075\pi$
$401$	−6.39880	−0.319541	−0.159770	+	0.987154i	$0.551075\pi$
$402$	0	0
$403$	−2.03486	−0.101363
$404$	0	0
$405$	−3.89502	−0.193545
$406$	0	0
$407$	3.29838	0.163495
$408$	0	0
$409$	33.3123	1.64719	0.823594	−	0.567180i	$-0.191966\pi$
$409$	33.3123	1.64719	0.823594	+	0.567180i	$0.191966\pi$
$410$	0	0
$411$	40.7608	2.01058
$412$	0	0
$413$	−2.10449	−0.103555
$414$	0	0
$415$	7.17180	0.352050
$416$	0	0
$417$	30.7779	1.50720
$418$	0	0
$419$	15.0157	0.733566	0.366783	−	0.930306i	$-0.380459\pi$
$419$	15.0157	0.733566	0.366783	+	0.930306i	$0.380459\pi$
$420$	0	0
$421$	−39.5704	−1.92854	−0.964271	−	0.264917i	$-0.914655\pi$
$421$	−39.5704	−1.92854	−0.964271	+	0.264917i	$0.914655\pi$
$422$	0	0
$423$	13.3043	0.646879
$424$	0	0
$425$	11.8453	0.574584
$426$	0	0
$427$	12.5768	0.608632
$428$	0	0
$429$	−2.44592	−0.118090
$430$	0	0
$431$	−10.0512	−0.484150	−0.242075	−	0.970258i	$-0.577828\pi$
$431$	−10.0512	−0.484150	−0.242075	+	0.970258i	$0.577828\pi$
$432$	0	0
$433$	−23.2951	−1.11949	−0.559745	−	0.828665i	$-0.689101\pi$
$433$	−23.2951	−1.11949	−0.559745	+	0.828665i	$0.689101\pi$
$434$	0	0
$435$	−3.69378	−0.177103
$436$	0	0
$437$	1.06481	0.0509367
$438$	0	0
$439$	31.8872	1.52189	0.760947	−	0.648814i	$-0.224735\pi$
$439$	31.8872	1.52189	0.760947	+	0.648814i	$0.224735\pi$
$440$	0	0
$441$	2.98251	0.142024
$442$	0	0
$443$	−29.9351	−1.42226	−0.711130	−	0.703061i	$-0.751816\pi$
$443$	−29.9351	−1.42226	−0.711130	+	0.703061i	$0.751816\pi$
$444$	0	0
$445$	1.75424	0.0831588
$446$	0	0
$447$	2.65297	0.125481
$448$	0	0
$449$	−37.0024	−1.74625	−0.873125	−	0.487497i	$-0.837910\pi$
$449$	−37.0024	−1.74625	−0.873125	+	0.487497i	$0.837910\pi$
$450$	0	0
$451$	−4.77742	−0.224960
$452$	0	0
$453$	−27.9516	−1.31328
$454$	0	0
$455$	0.430286	0.0201721
$456$	0	0
$457$	13.3568	0.624803	0.312401	−	0.949950i	$-0.398867\pi$
$457$	13.3568	0.624803	0.312401	+	0.949950i	$0.398867\pi$
$458$	0	0
$459$	−0.105253	−0.00491280
$460$	0	0
$461$	11.7343	0.546521	0.273261	−	0.961940i	$-0.411898\pi$
$461$	11.7343	0.546521	0.273261	+	0.961940i	$0.411898\pi$
$462$	0	0
$463$	−15.9157	−0.739663	−0.369832	−	0.929099i	$-0.620585\pi$
$463$	−15.9157	−0.739663	−0.369832	+	0.929099i	$0.620585\pi$
$464$	0	0
$465$	−2.14157	−0.0993130
$466$	0	0
$467$	28.7261	1.32929	0.664644	−	0.747160i	$-0.268583\pi$
$467$	28.7261	1.32929	0.664644	+	0.747160i	$0.268583\pi$
$468$	0	0
$469$	−9.60207	−0.443383
$470$	0	0
$471$	−38.6962	−1.78303
$472$	0	0
$473$	−6.41707	−0.295057
$474$	0	0
$475$	−10.8622	−0.498392
$476$	0	0
$477$	−41.9866	−1.92243
$478$	0	0
$479$	−11.8871	−0.543137	−0.271568	−	0.962419i	$-0.587542\pi$
$479$	−11.8871	−0.543137	−0.271568	+	0.962419i	$0.587542\pi$
$480$	0	0
$481$	3.29838	0.150393
$482$	0	0
$483$	1.15446	0.0525297
$484$	0	0
$485$	4.19558	0.190511
$486$	0	0
$487$	−30.3669	−1.37605	−0.688027	−	0.725685i	$-0.741523\pi$
$487$	−30.3669	−1.37605	−0.688027	+	0.725685i	$0.741523\pi$
$488$	0	0
$489$	45.2450	2.04605
$490$	0	0
$491$	−6.05583	−0.273296	−0.136648	−	0.990620i	$-0.543633\pi$
$491$	−6.05583	−0.273296	−0.136648	+	0.990620i	$0.543633\pi$
$492$	0	0
$493$	−8.63450	−0.388878
$494$	0	0
$495$	−1.28333	−0.0576814
$496$	0	0
$497$	−3.80453	−0.170657
$498$	0	0
$499$	−37.7665	−1.69066	−0.845330	−	0.534245i	$-0.820596\pi$
$499$	−37.7665	−1.69066	−0.845330	+	0.534245i	$0.820596\pi$
$500$	0	0
$501$	−28.1339	−1.25693
$502$	0	0
$503$	−40.0601	−1.78619	−0.893096	−	0.449867i	$-0.851472\pi$
$503$	−40.0601	−1.78619	−0.893096	+	0.449867i	$0.851472\pi$
$504$	0	0
$505$	−1.86146	−0.0828339
$506$	0	0
$507$	−2.44592	−0.108627
$508$	0	0
$509$	16.6237	0.736831	0.368415	−	0.929661i	$-0.379900\pi$
$509$	16.6237	0.736831	0.368415	+	0.929661i	$0.379900\pi$
$510$	0	0
$511$	8.00847	0.354274
$512$	0	0
$513$	0.0965173	0.00426134
$514$	0	0
$515$	−5.74078	−0.252969
$516$	0	0
$517$	−4.46079	−0.196185
$518$	0	0
$519$	−6.00479	−0.263581
$520$	0	0
$521$	−28.8994	−1.26611	−0.633053	−	0.774108i	$-0.718198\pi$
$521$	−28.8994	−1.26611	−0.633053	+	0.774108i	$0.718198\pi$
$522$	0	0
$523$	12.0050	0.524940	0.262470	−	0.964940i	$-0.415463\pi$
$523$	12.0050	0.524940	0.262470	+	0.964940i	$0.415463\pi$
$524$	0	0
$525$	−11.7767	−0.513979
$526$	0	0
$527$	−5.00609	−0.218069
$528$	0	0
$529$	−22.7772	−0.990314
$530$	0	0
$531$	6.27666	0.272384
$532$	0	0
$533$	−4.77742	−0.206933
$534$	0	0
$535$	−0.172585	−0.00746150
$536$	0	0
$537$	−61.0178	−2.63311
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	37.0483	1.59283	0.796415	−	0.604750i	$-0.206727\pi$
$541$	37.0483	1.59283	0.796415	+	0.604750i	$0.206727\pi$
$542$	0	0
$543$	−50.0137	−2.14629
$544$	0	0
$545$	4.41541	0.189135
$546$	0	0
$547$	10.9024	0.466154	0.233077	−	0.972458i	$-0.425121\pi$
$547$	10.9024	0.466154	0.233077	+	0.972458i	$0.425121\pi$
$548$	0	0
$549$	−37.5103	−1.60090
$550$	0	0
$551$	7.91784	0.337311
$552$	0	0
$553$	−7.77702	−0.330712
$554$	0	0
$555$	3.47136	0.147351
$556$	0	0
$557$	33.1124	1.40302	0.701508	−	0.712661i	$-0.252510\pi$
$557$	33.1124	1.40302	0.701508	+	0.712661i	$0.252510\pi$
$558$	0	0
$559$	−6.41707	−0.271413
$560$	0	0
$561$	−6.01736	−0.254053
$562$	0	0
$563$	−34.4266	−1.45091	−0.725455	−	0.688270i	$-0.758370\pi$
$563$	−34.4266	−1.45091	−0.725455	+	0.688270i	$0.758370\pi$
$564$	0	0
$565$	6.32290	0.266006
$566$	0	0
$567$	9.05217	0.380155
$568$	0	0
$569$	−44.4569	−1.86373	−0.931865	−	0.362805i	$-0.881819\pi$
$569$	−44.4569	−1.86373	−0.931865	+	0.362805i	$0.881819\pi$
$570$	0	0
$571$	−2.06974	−0.0866158	−0.0433079	−	0.999062i	$-0.513790\pi$
$571$	−2.06974	−0.0866158	−0.0433079	+	0.999062i	$0.513790\pi$
$572$	0	0
$573$	39.9300	1.66810
$574$	0	0
$575$	−2.27259	−0.0947734
$576$	0	0
$577$	−32.1771	−1.33955	−0.669775	−	0.742564i	$-0.733610\pi$
$577$	−32.1771	−1.33955	−0.669775	+	0.742564i	$0.733610\pi$
$578$	0	0
$579$	−13.6492	−0.567243
$580$	0	0
$581$	−16.6675	−0.691486
$582$	0	0
$583$	14.0776	0.583035
$584$	0	0
$585$	−1.28333	−0.0530592
$586$	0	0
$587$	−21.4640	−0.885914	−0.442957	−	0.896543i	$-0.646070\pi$
$587$	−21.4640	−0.885914	−0.442957	+	0.896543i	$0.646070\pi$
$588$	0	0
$589$	4.59059	0.189152
$590$	0	0
$591$	31.4117	1.29211
$592$	0	0
$593$	−19.9275	−0.818325	−0.409162	−	0.912462i	$-0.634179\pi$
$593$	−19.9275	−0.818325	−0.409162	+	0.912462i	$0.634179\pi$
$594$	0	0
$595$	1.05858	0.0433973
$596$	0	0
$597$	26.7973	1.09674
$598$	0	0
$599$	32.8162	1.34083	0.670417	−	0.741985i	$-0.266115\pi$
$599$	32.8162	1.34083	0.670417	+	0.741985i	$0.266115\pi$
$600$	0	0
$601$	−31.7770	−1.29621	−0.648105	−	0.761551i	$-0.724438\pi$
$601$	−31.7770	−1.29621	−0.648105	+	0.761551i	$0.724438\pi$
$602$	0	0
$603$	28.6383	1.16624
$604$	0	0
$605$	0.430286	0.0174936
$606$	0	0
$607$	−34.2966	−1.39206	−0.696028	−	0.718014i	$-0.745051\pi$
$607$	−34.2966	−1.39206	−0.696028	+	0.718014i	$0.745051\pi$
$608$	0	0
$609$	8.58448	0.347861
$610$	0	0
$611$	−4.46079	−0.180464
$612$	0	0
$613$	15.9777	0.645333	0.322666	−	0.946513i	$-0.395421\pi$
$613$	15.9777	0.645333	0.322666	+	0.946513i	$0.395421\pi$
$614$	0	0
$615$	−5.02796	−0.202747
$616$	0	0
$617$	2.93761	0.118264	0.0591318	−	0.998250i	$-0.481167\pi$
$617$	2.93761	0.118264	0.0591318	+	0.998250i	$0.481167\pi$
$618$	0	0
$619$	29.3199	1.17846	0.589232	−	0.807964i	$-0.299430\pi$
$619$	29.3199	1.17846	0.589232	+	0.807964i	$0.299430\pi$
$620$	0	0
$621$	0.0201933	0.000810330 0
$622$	0	0
$623$	−4.07691	−0.163338
$624$	0	0
$625$	22.2571	0.890284
$626$	0	0
$627$	5.51793	0.220365
$628$	0	0
$629$	8.11458	0.323549
$630$	0	0
$631$	−26.5661	−1.05758	−0.528791	−	0.848752i	$-0.677354\pi$
$631$	−26.5661	−1.05758	−0.528791	+	0.848752i	$0.677354\pi$
$632$	0	0
$633$	−3.98366	−0.158336
$634$	0	0
$635$	−4.13825	−0.164221
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	11.3470	0.448882
$640$	0	0
$641$	−6.59794	−0.260603	−0.130301	−	0.991474i	$-0.541595\pi$
$641$	−6.59794	−0.260603	−0.130301	+	0.991474i	$0.541595\pi$
$642$	0	0
$643$	42.4949	1.67584	0.837918	−	0.545795i	$-0.183772\pi$
$643$	42.4949	1.67584	0.837918	+	0.545795i	$0.183772\pi$
$644$	0	0
$645$	−6.75360	−0.265923
$646$	0	0
$647$	−13.1085	−0.515347	−0.257673	−	0.966232i	$-0.582956\pi$
$647$	−13.1085	−0.515347	−0.257673	+	0.966232i	$0.582956\pi$
$648$	0	0
$649$	−2.10449	−0.0826084
$650$	0	0
$651$	4.97709	0.195067
$652$	0	0
$653$	29.2231	1.14359	0.571794	−	0.820397i	$-0.306248\pi$
$653$	29.2231	1.14359	0.571794	+	0.820397i	$0.306248\pi$
$654$	0	0
$655$	−8.83007	−0.345019
$656$	0	0
$657$	−23.8853	−0.931855
$658$	0	0
$659$	12.7328	0.495998	0.247999	−	0.968760i	$-0.420227\pi$
$659$	12.7328	0.495998	0.247999	+	0.968760i	$0.420227\pi$
$660$	0	0
$661$	19.2055	0.747007	0.373504	−	0.927629i	$-0.378156\pi$
$661$	19.2055	0.747007	0.373504	+	0.927629i	$0.378156\pi$
$662$	0	0
$663$	−6.01736	−0.233695
$664$	0	0
$665$	−0.970714	−0.0376427
$666$	0	0
$667$	1.65657	0.0641426
$668$	0	0
$669$	−3.95139	−0.152770
$670$	0	0
$671$	12.5768	0.485520
$672$	0	0
$673$	11.5766	0.446244	0.223122	−	0.974791i	$-0.428375\pi$
$673$	11.5766	0.446244	0.223122	+	0.974791i	$0.428375\pi$
$674$	0	0
$675$	−0.205994	−0.00792870
$676$	0	0
$677$	29.9180	1.14984	0.574920	−	0.818209i	$-0.305033\pi$
$677$	29.9180	1.14984	0.574920	+	0.818209i	$0.305033\pi$
$678$	0	0
$679$	−9.75068	−0.374197
$680$	0	0
$681$	−49.2846	−1.88859
$682$	0	0
$683$	−46.7734	−1.78974	−0.894868	−	0.446331i	$-0.852730\pi$
$683$	−46.7734	−1.78974	−0.894868	+	0.446331i	$0.852730\pi$
$684$	0	0
$685$	−7.17064	−0.273976
$686$	0	0
$687$	−54.9618	−2.09692
$688$	0	0
$689$	14.0776	0.536314
$690$	0	0
$691$	16.2567	0.618435	0.309218	−	0.950991i	$-0.399933\pi$
$691$	16.2567	0.618435	0.309218	+	0.950991i	$0.399933\pi$
$692$	0	0
$693$	2.98251	0.113296
$694$	0	0
$695$	−5.41445	−0.205382
$696$	0	0
$697$	−11.7533	−0.445186
$698$	0	0
$699$	57.0336	2.15721
$700$	0	0
$701$	−20.7743	−0.784633	−0.392317	−	0.919830i	$-0.628326\pi$
$701$	−20.7743	−0.784633	−0.392317	+	0.919830i	$0.628326\pi$
$702$	0	0
$703$	−7.44107	−0.280645
$704$	0	0
$705$	−4.69473	−0.176814
$706$	0	0
$707$	4.32610	0.162700
$708$	0	0
$709$	16.3500	0.614038	0.307019	−	0.951703i	$-0.400668\pi$
$709$	16.3500	0.614038	0.307019	+	0.951703i	$0.400668\pi$
$710$	0	0
$711$	23.1950	0.869881
$712$	0	0
$713$	0.960441	0.0359688
$714$	0	0
$715$	0.430286	0.0160918
$716$	0	0
$717$	30.0667	1.12286
$718$	0	0
$719$	31.1224	1.16067	0.580334	−	0.814379i	$-0.302922\pi$
$719$	31.1224	1.16067	0.580334	+	0.814379i	$0.302922\pi$
$720$	0	0
$721$	13.3418	0.496874
$722$	0	0
$723$	3.32843	0.123786
$724$	0	0
$725$	−16.8988	−0.627605
$726$	0	0
$727$	16.0190	0.594114	0.297057	−	0.954860i	$-0.403995\pi$
$727$	16.0190	0.594114	0.297057	+	0.954860i	$0.403995\pi$
$728$	0	0
$729$	−26.6842	−0.988305
$730$	0	0
$731$	−15.7871	−0.583905
$732$	0	0
$733$	−29.7086	−1.09731	−0.548657	−	0.836048i	$-0.684861\pi$
$733$	−29.7086	−1.09731	−0.548657	+	0.836048i	$0.684861\pi$
$734$	0	0
$735$	−1.05244	−0.0388200
$736$	0	0
$737$	−9.60207	−0.353697
$738$	0	0
$739$	7.31827	0.269207	0.134603	−	0.990900i	$-0.457024\pi$
$739$	7.31827	0.269207	0.134603	+	0.990900i	$0.457024\pi$
$740$	0	0
$741$	5.51793	0.202706
$742$	0	0
$743$	5.18777	0.190321	0.0951603	−	0.995462i	$-0.469664\pi$
$743$	5.18777	0.190321	0.0951603	+	0.995462i	$0.469664\pi$
$744$	0	0
$745$	−0.466711	−0.0170990
$746$	0	0
$747$	49.7110	1.81883
$748$	0	0
$749$	0.401094	0.0146557
$750$	0	0
$751$	−53.2637	−1.94362	−0.971811	−	0.235763i	$-0.924241\pi$
$751$	−53.2637	−1.94362	−0.971811	+	0.235763i	$0.924241\pi$
$752$	0	0
$753$	−45.0057	−1.64010
$754$	0	0
$755$	4.91725	0.178957
$756$	0	0
$757$	−12.9677	−0.471317	−0.235659	−	0.971836i	$-0.575725\pi$
$757$	−12.9677	−0.471317	−0.235659	+	0.971836i	$0.575725\pi$
$758$	0	0
$759$	1.15446	0.0419042
$760$	0	0
$761$	20.0232	0.725841	0.362921	−	0.931820i	$-0.381780\pi$
$761$	20.0232	0.725841	0.362921	+	0.931820i	$0.381780\pi$
$762$	0	0
$763$	−10.2616	−0.371494
$764$	0	0
$765$	−3.15721	−0.114149
$766$	0	0
$767$	−2.10449	−0.0759887
$768$	0	0
$769$	−26.5877	−0.958777	−0.479389	−	0.877603i	$-0.659142\pi$
$769$	−26.5877	−0.958777	−0.479389	+	0.877603i	$0.659142\pi$
$770$	0	0
$771$	24.1398	0.869372
$772$	0	0
$773$	−20.3699	−0.732654	−0.366327	−	0.930486i	$-0.619385\pi$
$773$	−20.3699	−0.732654	−0.366327	+	0.930486i	$0.619385\pi$
$774$	0	0
$775$	−9.79754	−0.351938
$776$	0	0
$777$	−8.06757	−0.289423
$778$	0	0
$779$	10.7777	0.386153
$780$	0	0
$781$	−3.80453	−0.136137
$782$	0	0
$783$	0.150156	0.00536614
$784$	0	0
$785$	6.80743	0.242968
$786$	0	0
$787$	−31.0433	−1.10657	−0.553287	−	0.832991i	$-0.686627\pi$
$787$	−31.0433	−1.10657	−0.553287	+	0.832991i	$0.686627\pi$
$788$	0	0
$789$	73.9683	2.63334
$790$	0	0
$791$	−14.6946	−0.522481
$792$	0	0
$793$	12.5768	0.446614
$794$	0	0
$795$	14.8159	0.525465
$796$	0	0
$797$	−40.8431	−1.44674	−0.723369	−	0.690461i	$-0.757408\pi$
$797$	−40.8431	−1.44674	−0.723369	+	0.690461i	$0.757408\pi$
$798$	0	0
$799$	−10.9743	−0.388242
$800$	0	0
$801$	12.1594	0.429632
$802$	0	0
$803$	8.00847	0.282613
$804$	0	0
$805$	−0.203093	−0.00715807
$806$	0	0
$807$	21.4113	0.753713
$808$	0	0
$809$	−15.3240	−0.538763	−0.269382	−	0.963034i	$-0.586819\pi$
$809$	−15.3240	−0.538763	−0.269382	+	0.963034i	$0.586819\pi$
$810$	0	0
$811$	6.59135	0.231453	0.115727	−	0.993281i	$-0.463080\pi$
$811$	6.59135	0.231453	0.115727	+	0.993281i	$0.463080\pi$
$812$	0	0
$813$	−28.4784	−0.998782
$814$	0	0
$815$	−7.95951	−0.278809
$816$	0	0
$817$	14.4767	0.506477
$818$	0	0
$819$	2.98251	0.104217
$820$	0	0
$821$	−5.27759	−0.184189	−0.0920946	−	0.995750i	$-0.529356\pi$
$821$	−5.27759	−0.184189	−0.0920946	+	0.995750i	$0.529356\pi$
$822$	0	0
$823$	55.7225	1.94236	0.971182	−	0.238341i	$-0.0766034\pi$
$823$	55.7225	1.94236	0.971182	+	0.238341i	$0.0766034\pi$
$824$	0	0
$825$	−11.7767	−0.410013
$826$	0	0
$827$	−21.4517	−0.745948	−0.372974	−	0.927842i	$-0.621662\pi$
$827$	−21.4517	−0.745948	−0.372974	+	0.927842i	$0.621662\pi$
$828$	0	0
$829$	−30.1857	−1.04839	−0.524197	−	0.851597i	$-0.675634\pi$
$829$	−30.1857	−1.04839	−0.524197	+	0.851597i	$0.675634\pi$
$830$	0	0
$831$	33.6953	1.16888
$832$	0	0
$833$	−2.46017	−0.0852398
$834$	0	0
$835$	4.94932	0.171278
$836$	0	0
$837$	0.0870571	0.00300913
$838$	0	0
$839$	−43.9728	−1.51811	−0.759055	−	0.651026i	$-0.774339\pi$
$839$	−43.9728	−1.51811	−0.759055	+	0.651026i	$0.774339\pi$
$840$	0	0
$841$	−16.6819	−0.575237
$842$	0	0
$843$	−49.1502	−1.69282
$844$	0	0
$845$	0.430286	0.0148023
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	47.4211	1.62749
$850$	0	0
$851$	−1.55682	−0.0533671
$852$	0	0
$853$	1.52390	0.0521774	0.0260887	−	0.999660i	$-0.491695\pi$
$853$	1.52390	0.0521774	0.0260887	+	0.999660i	$0.491695\pi$
$854$	0	0
$855$	2.89516	0.0990125
$856$	0	0
$857$	42.2986	1.44489	0.722447	−	0.691426i	$-0.243017\pi$
$857$	42.2986	1.44489	0.722447	+	0.691426i	$0.243017\pi$
$858$	0	0
$859$	45.4285	1.55000	0.775000	−	0.631961i	$-0.217750\pi$
$859$	45.4285	1.55000	0.775000	+	0.631961i	$0.217750\pi$
$860$	0	0
$861$	11.6852	0.398230
$862$	0	0
$863$	16.2079	0.551723	0.275862	−	0.961197i	$-0.411037\pi$
$863$	16.2079	0.551723	0.275862	+	0.961197i	$0.411037\pi$
$864$	0	0
$865$	1.05636	0.0359174
$866$	0	0
$867$	26.7769	0.909390
$868$	0	0
$869$	−7.77702	−0.263817
$870$	0	0
$871$	−9.60207	−0.325354
$872$	0	0
$873$	29.0815	0.984259
$874$	0	0
$875$	4.22319	0.142770
$876$	0	0
$877$	13.2048	0.445894	0.222947	−	0.974831i	$-0.428432\pi$
$877$	13.2048	0.445894	0.222947	+	0.974831i	$0.428432\pi$
$878$	0	0
$879$	−53.8454	−1.81616
$880$	0	0
$881$	−36.0770	−1.21546	−0.607732	−	0.794142i	$-0.707921\pi$
$881$	−36.0770	−1.21546	−0.607732	+	0.794142i	$0.707921\pi$
$882$	0	0
$883$	40.5857	1.36582	0.682909	−	0.730503i	$-0.260714\pi$
$883$	40.5857	1.36582	0.682909	+	0.730503i	$0.260714\pi$
$884$	0	0
$885$	−2.21485	−0.0744515
$886$	0	0
$887$	−17.6532	−0.592738	−0.296369	−	0.955074i	$-0.595776\pi$
$887$	−17.6532	−0.592738	−0.296369	+	0.955074i	$0.595776\pi$
$888$	0	0
$889$	9.61745	0.322559
$890$	0	0
$891$	9.05217	0.303259
$892$	0	0
$893$	10.0634	0.336760
$894$	0	0
$895$	10.7343	0.358807
$896$	0	0
$897$	1.15446	0.0385463
$898$	0	0
$899$	7.14177	0.238192
$900$	0	0
$901$	34.6333	1.15380
$902$	0	0
$903$	15.6956	0.522317
$904$	0	0
$905$	8.79841	0.292469
$906$	0	0
$907$	19.5922	0.650547	0.325274	−	0.945620i	$-0.394544\pi$
$907$	19.5922	0.650547	0.325274	+	0.945620i	$0.394544\pi$
$908$	0	0
$909$	−12.9026	−0.427953
$910$	0	0
$911$	44.6191	1.47830	0.739148	−	0.673543i	$-0.235228\pi$
$911$	44.6191	1.47830	0.739148	+	0.673543i	$0.235228\pi$
$912$	0	0
$913$	−16.6675	−0.551615
$914$	0	0
$915$	13.2363	0.437579
$916$	0	0
$917$	20.5214	0.677677
$918$	0	0
$919$	29.5773	0.975666	0.487833	−	0.872937i	$-0.337787\pi$
$919$	29.5773	0.975666	0.487833	+	0.872937i	$0.337787\pi$
$920$	0	0
$921$	−4.89567	−0.161318
$922$	0	0
$923$	−3.80453	−0.125228
$924$	0	0
$925$	15.8812	0.522172
$926$	0	0
$927$	−39.7920	−1.30694
$928$	0	0
$929$	−9.63852	−0.316230	−0.158115	−	0.987421i	$-0.550542\pi$
$929$	−9.63852	−0.316230	−0.158115	+	0.987421i	$0.550542\pi$
$930$	0	0
$931$	2.25598	0.0739366
$932$	0	0
$933$	−71.1814	−2.33037
$934$	0	0
$935$	1.05858	0.0346191
$936$	0	0
$937$	17.4870	0.571275	0.285637	−	0.958338i	$-0.407795\pi$
$937$	17.4870	0.571275	0.285637	+	0.958338i	$0.407795\pi$
$938$	0	0
$939$	−42.5629	−1.38899
$940$	0	0
$941$	27.9745	0.911941	0.455971	−	0.889995i	$-0.349292\pi$
$941$	27.9745	0.911941	0.455971	+	0.889995i	$0.349292\pi$
$942$	0	0
$943$	2.25492	0.0734302
$944$	0	0
$945$	−0.0184089	−0.000598841 0
$946$	0	0
$947$	−55.6986	−1.80996	−0.904981	−	0.425452i	$-0.860115\pi$
$947$	−55.6986	−1.80996	−0.904981	+	0.425452i	$0.860115\pi$
$948$	0	0
$949$	8.00847	0.259966
$950$	0	0
$951$	13.7856	0.447029
$952$	0	0
$953$	−14.9657	−0.484786	−0.242393	−	0.970178i	$-0.577932\pi$
$953$	−14.9657	−0.484786	−0.242393	+	0.970178i	$0.577932\pi$
$954$	0	0
$955$	−7.02449	−0.227307
$956$	0	0
$957$	8.58448	0.277497
$958$	0	0
$959$	16.6648	0.538135
$960$	0	0
$961$	−26.8594	−0.866431
$962$	0	0
$963$	−1.19627	−0.0385491
$964$	0	0
$965$	2.40118	0.0772966
$966$	0	0
$967$	−49.3773	−1.58787	−0.793933	−	0.608005i	$-0.791970\pi$
$967$	−49.3773	−1.58787	−0.793933	+	0.608005i	$0.791970\pi$
$968$	0	0
$969$	13.5750	0.436093
$970$	0	0
$971$	9.20796	0.295498	0.147749	−	0.989025i	$-0.452797\pi$
$971$	9.20796	0.295498	0.147749	+	0.989025i	$0.452797\pi$
$972$	0	0
$973$	12.5834	0.403405
$974$	0	0
$975$	−11.7767	−0.377157
$976$	0	0
$977$	−3.55557	−0.113753	−0.0568764	−	0.998381i	$-0.518114\pi$
$977$	−3.55557	−0.113753	−0.0568764	+	0.998381i	$0.518114\pi$
$978$	0	0
$979$	−4.07691	−0.130299
$980$	0	0
$981$	30.6052	0.977150
$982$	0	0
$983$	−40.8259	−1.30214	−0.651072	−	0.759016i	$-0.725680\pi$
$983$	−40.8259	−1.30214	−0.651072	+	0.759016i	$0.725680\pi$
$984$	0	0
$985$	−5.52595	−0.176071
$986$	0	0
$987$	10.9107	0.347292
$988$	0	0
$989$	3.02882	0.0963109
$990$	0	0
$991$	21.5379	0.684174	0.342087	−	0.939668i	$-0.388866\pi$
$991$	21.5379	0.684174	0.342087	+	0.939668i	$0.388866\pi$
$992$	0	0
$993$	10.7860	0.342285
$994$	0	0
$995$	−4.71419	−0.149450
$996$	0	0
$997$	−17.3745	−0.550256	−0.275128	−	0.961408i	$-0.588720\pi$
$997$	−17.3745	−0.550256	−0.275128	+	0.961408i	$0.588720\pi$
$998$	0	0
$999$	−0.141115	−0.00446467

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.u.1.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.u.1.1	✓	10	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)

\(f(q)\)	\(=\)	\(q-2.44592 q^{3} +0.430286 q^{5} -1.00000 q^{7} +2.98251 q^{9} +O(q^{10})\)	\(q-2.44592 q^{3} +0.430286 q^{5} -1.00000 q^{7} +2.98251 q^{9} -1.00000 q^{11} -1.00000 q^{13} -1.05244 q^{15} -2.46017 q^{17} +2.25598 q^{19} +2.44592 q^{21} +0.471995 q^{23} -4.81485 q^{25} +0.0427829 q^{27} +3.50972 q^{29} +2.03486 q^{31} +2.44592 q^{33} -0.430286 q^{35} -3.29838 q^{37} +2.44592 q^{39} +4.77742 q^{41} +6.41707 q^{43} +1.28333 q^{45} +4.46079 q^{47} +1.00000 q^{49} +6.01736 q^{51} -14.0776 q^{53} -0.430286 q^{55} -5.51793 q^{57} +2.10449 q^{59} -12.5768 q^{61} -2.98251 q^{63} -0.430286 q^{65} +9.60207 q^{67} -1.15446 q^{69} +3.80453 q^{71} -8.00847 q^{73} +11.7767 q^{75} +1.00000 q^{77} +7.77702 q^{79} -9.05217 q^{81} +16.6675 q^{83} -1.05858 q^{85} -8.58448 q^{87} +4.07691 q^{89} +1.00000 q^{91} -4.97709 q^{93} +0.970714 q^{95} +9.75068 q^{97} -2.98251 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9	\( 10 q + 2 q^{3} - 4 q^{5} - 10 q^{7} + 6 q^{9} - 10 q^{11} - 10 q^{13} - 4 q^{15} - 3 q^{17} + 13 q^{19} - 2 q^{21} + 6 q^{25} + 14 q^{27} - 3 q^{29} - q^{31} - 2 q^{33} + 4 q^{35} - 5 q^{37} - 2 q^{39} + 4 q^{41} + 19 q^{43} - 11 q^{45} + q^{47} + 10 q^{49} + 15 q^{51} - 6 q^{53} + 4 q^{55} - 6 q^{57} + 4 q^{59} - 18 q^{61} - 6 q^{63} + 4 q^{65} + q^{67} - 11 q^{69} - 21 q^{71} - 10 q^{73} + 10 q^{77} + 3 q^{79} - 30 q^{81} + 6 q^{83} - 33 q^{85} - 9 q^{87} - 22 q^{89} + 10 q^{91} - 34 q^{93} - 3 q^{95} - 9 q^{97} - 6 q^{99}+O(q^{100}) \) 10 * q + 2 * q^3 - 4 * q^5 - 10 * q^7 + 6 * q^9 - 10 * q^11 - 10 * q^13 - 4 * q^15 - 3 * q^17 + 13 * q^19 - 2 * q^21 + 6 * q^25 + 14 * q^27 - 3 * q^29 - q^31 - 2 * q^33 + 4 * q^35 - 5 * q^37 - 2 * q^39 + 4 * q^41 + 19 * q^43 - 11 * q^45 + q^47 + 10 * q^49 + 15 * q^51 - 6 * q^53 + 4 * q^55 - 6 * q^57 + 4 * q^59 - 18 * q^61 - 6 * q^63 + 4 * q^65 + q^67 - 11 * q^69 - 21 * q^71 - 10 * q^73 + 10 * q^77 + 3 * q^79 - 30 * q^81 + 6 * q^83 - 33 * q^85 - 9 * q^87 - 22 * q^89 + 10 * q^91 - 34 * q^93 - 3 * q^95 - 9 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more