LMFDB - Embedded newform 8008.2.a.f.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q-0.618034 q^{3} +1.61803 q^{5} -1.00000 q^{7} -2.61803 q^{9} +O(q^{10})$	$q-0.618034 q^{3} +1.61803 q^{5} -1.00000 q^{7} -2.61803 q^{9} +1.00000 q^{11} -1.00000 q^{13} -1.00000 q^{15} -5.85410 q^{17} -4.09017 q^{19} +0.618034 q^{21} -6.47214 q^{23} -2.38197 q^{25} +3.47214 q^{27} -0.618034 q^{33} -1.61803 q^{35} +7.23607 q^{37} +0.618034 q^{39} +1.70820 q^{41} -7.09017 q^{43} -4.23607 q^{45} -4.00000 q^{47} +1.00000 q^{49} +3.61803 q^{51} +5.38197 q^{53} +1.61803 q^{55} +2.52786 q^{57} +6.00000 q^{59} +4.14590 q^{61} +2.61803 q^{63} -1.61803 q^{65} +9.38197 q^{67} +4.00000 q^{69} +1.90983 q^{71} +7.70820 q^{73} +1.47214 q^{75} -1.00000 q^{77} +2.38197 q^{79} +5.70820 q^{81} +12.8541 q^{83} -9.47214 q^{85} -7.85410 q^{89} +1.00000 q^{91} -6.61803 q^{95} +2.94427 q^{97} -2.61803 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) $ 2 * q + q^3 + q^5 - 2 * q^7 - 3 * q^9	$ 2 q + q^{3} + q^{5} - 2 q^{7} - 3 q^{9} + 2 q^{11} - 2 q^{13} - 2 q^{15} - 5 q^{17} + 3 q^{19} - q^{21} - 4 q^{23} - 7 q^{25} - 2 q^{27} + q^{33} - q^{35} + 10 q^{37} - q^{39} - 10 q^{41} - 3 q^{43} - 4 q^{45} - 8 q^{47} + 2 q^{49} + 5 q^{51} + 13 q^{53} + q^{55} + 14 q^{57} + 12 q^{59} + 15 q^{61} + 3 q^{63} - q^{65} + 21 q^{67} + 8 q^{69} + 15 q^{71} + 2 q^{73} - 6 q^{75} - 2 q^{77} + 7 q^{79} - 2 q^{81} + 19 q^{83} - 10 q^{85} - 9 q^{89} + 2 q^{91} - 11 q^{95} - 12 q^{97} - 3 q^{99}+O(q^{100}) $ 2 * q + q^3 + q^5 - 2 * q^7 - 3 * q^9 + 2 * q^11 - 2 * q^13 - 2 * q^15 - 5 * q^17 + 3 * q^19 - q^21 - 4 * q^23 - 7 * q^25 - 2 * q^27 + q^33 - q^35 + 10 * q^37 - q^39 - 10 * q^41 - 3 * q^43 - 4 * q^45 - 8 * q^47 + 2 * q^49 + 5 * q^51 + 13 * q^53 + q^55 + 14 * q^57 + 12 * q^59 + 15 * q^61 + 3 * q^63 - q^65 + 21 * q^67 + 8 * q^69 + 15 * q^71 + 2 * q^73 - 6 * q^75 - 2 * q^77 + 7 * q^79 - 2 * q^81 + 19 * q^83 - 10 * q^85 - 9 * q^89 + 2 * q^91 - 11 * q^95 - 12 * q^97 - 3 * 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
$3$	−0.618034	−0.356822	−0.178411	−	0.983956i	$-0.557096\pi$
$3$	−0.618034	−0.356822	−0.178411	+	0.983956i	$0.557096\pi$
$4$	0	0
$5$	1.61803	0.723607	0.361803	−	0.932254i	$-0.382161\pi$
$5$	1.61803	0.723607	0.361803	+	0.932254i	$0.382161\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.61803	−0.872678
$10$	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
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−5.85410	−1.41983	−0.709914	−	0.704288i	$-0.751266\pi$
$17$	−5.85410	−1.41983	−0.709914	+	0.704288i	$0.751266\pi$
$18$	0	0
$19$	−4.09017	−0.938349	−0.469175	−	0.883105i	$-0.655449\pi$
$19$	−4.09017	−0.938349	−0.469175	+	0.883105i	$0.655449\pi$
$20$	0	0
$21$	0.618034	0.134866
$22$	0	0
$23$	−6.47214	−1.34953	−0.674767	−	0.738031i	$-0.735756\pi$
$23$	−6.47214	−1.34953	−0.674767	+	0.738031i	$0.735756\pi$
$24$	0	0
$25$	−2.38197	−0.476393
$26$	0	0
$27$	3.47214	0.668213
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−0.618034	−0.107586
$34$	0	0
$35$	−1.61803	−0.273498
$36$	0	0
$37$	7.23607	1.18960	0.594801	−	0.803873i	$-0.297231\pi$
$37$	7.23607	1.18960	0.594801	+	0.803873i	$0.297231\pi$
$38$	0	0
$39$	0.618034	0.0989646
$40$	0	0
$41$	1.70820	0.266777	0.133388	−	0.991064i	$-0.457414\pi$
$41$	1.70820	0.266777	0.133388	+	0.991064i	$0.457414\pi$
$42$	0	0
$43$	−7.09017	−1.08124	−0.540620	−	0.841267i	$-0.681810\pi$
$43$	−7.09017	−1.08124	−0.540620	+	0.841267i	$0.681810\pi$
$44$	0	0
$45$	−4.23607	−0.631476
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.61803	0.506626
$52$	0	0
$53$	5.38197	0.739270	0.369635	−	0.929177i	$-0.379483\pi$
$53$	5.38197	0.739270	0.369635	+	0.929177i	$0.379483\pi$
$54$	0	0
$55$	1.61803	0.218176
$56$	0	0
$57$	2.52786	0.334824
$58$	0	0
$59$	6.00000	0.781133	0.390567	−	0.920575i	$-0.372279\pi$
$59$	6.00000	0.781133	0.390567	+	0.920575i	$0.372279\pi$
$60$	0	0
$61$	4.14590	0.530828	0.265414	−	0.964135i	$-0.414491\pi$
$61$	4.14590	0.530828	0.265414	+	0.964135i	$0.414491\pi$
$62$	0	0
$63$	2.61803	0.329841
$64$	0	0
$65$	−1.61803	−0.200692
$66$	0	0
$67$	9.38197	1.14619	0.573095	−	0.819489i	$-0.305743\pi$
$67$	9.38197	1.14619	0.573095	+	0.819489i	$0.305743\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	1.90983	0.226655	0.113328	−	0.993558i	$-0.463849\pi$
$71$	1.90983	0.226655	0.113328	+	0.993558i	$0.463849\pi$
$72$	0	0
$73$	7.70820	0.902177	0.451089	−	0.892479i	$-0.351036\pi$
$73$	7.70820	0.902177	0.451089	+	0.892479i	$0.351036\pi$
$74$	0	0
$75$	1.47214	0.169988
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	2.38197	0.267992	0.133996	−	0.990982i	$-0.457219\pi$
$79$	2.38197	0.267992	0.133996	+	0.990982i	$0.457219\pi$
$80$	0	0
$81$	5.70820	0.634245
$82$	0	0
$83$	12.8541	1.41092	0.705460	−	0.708749i	$-0.250740\pi$
$83$	12.8541	1.41092	0.705460	+	0.708749i	$0.250740\pi$
$84$	0	0
$85$	−9.47214	−1.02740
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.85410	−0.832533	−0.416267	−	0.909243i	$-0.636662\pi$
$89$	−7.85410	−0.832533	−0.416267	+	0.909243i	$0.636662\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.61803	−0.678996
$96$	0	0
$97$	2.94427	0.298946	0.149473	−	0.988766i	$-0.452242\pi$
$97$	2.94427	0.298946	0.149473	+	0.988766i	$0.452242\pi$
$98$	0	0
$99$	−2.61803	−0.263122
$100$	0	0
$101$	18.0902	1.80004	0.900020	−	0.435849i	$-0.143552\pi$
$101$	18.0902	1.80004	0.900020	+	0.435849i	$0.143552\pi$
$102$	0	0
$103$	−10.4721	−1.03185	−0.515925	−	0.856634i	$-0.672552\pi$
$103$	−10.4721	−1.03185	−0.515925	+	0.856634i	$0.672552\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	20.3262	1.96501	0.982506	−	0.186232i	$-0.0596276\pi$
$107$	20.3262	1.96501	0.982506	+	0.186232i	$0.0596276\pi$
$108$	0	0
$109$	−6.38197	−0.611281	−0.305641	−	0.952147i	$-0.598871\pi$
$109$	−6.38197	−0.611281	−0.305641	+	0.952147i	$0.598871\pi$
$110$	0	0
$111$	−4.47214	−0.424476
$112$	0	0
$113$	8.85410	0.832924	0.416462	−	0.909153i	$-0.363270\pi$
$113$	8.85410	0.832924	0.416462	+	0.909153i	$0.363270\pi$
$114$	0	0
$115$	−10.4721	−0.976532
$116$	0	0
$117$	2.61803	0.242037
$118$	0	0
$119$	5.85410	0.536645
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.05573	−0.0951918
$124$	0	0
$125$	−11.9443	−1.06833
$126$	0	0
$127$	19.5066	1.73093	0.865464	−	0.500970i	$-0.167023\pi$
$127$	19.5066	1.73093	0.865464	+	0.500970i	$0.167023\pi$
$128$	0	0
$129$	4.38197	0.385811
$130$	0	0
$131$	−13.4164	−1.17220	−0.586098	−	0.810240i	$-0.699337\pi$
$131$	−13.4164	−1.17220	−0.586098	+	0.810240i	$0.699337\pi$
$132$	0	0
$133$	4.09017	0.354663
$134$	0	0
$135$	5.61803	0.483523
$136$	0	0
$137$	−15.8885	−1.35745	−0.678725	−	0.734393i	$-0.737467\pi$
$137$	−15.8885	−1.35745	−0.678725	+	0.734393i	$0.737467\pi$
$138$	0	0
$139$	15.4164	1.30760	0.653801	−	0.756666i	$-0.273173\pi$
$139$	15.4164	1.30760	0.653801	+	0.756666i	$0.273173\pi$
$140$	0	0
$141$	2.47214	0.208191
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.618034	−0.0509746
$148$	0	0
$149$	21.7984	1.78579	0.892896	−	0.450263i	$-0.148670\pi$
$149$	21.7984	1.78579	0.892896	+	0.450263i	$0.148670\pi$
$150$	0	0
$151$	−23.4164	−1.90560	−0.952800	−	0.303598i	$-0.901812\pi$
$151$	−23.4164	−1.90560	−0.952800	+	0.303598i	$0.901812\pi$
$152$	0	0
$153$	15.3262	1.23905
$154$	0	0
$155$	0	0
$156$	0	0
$157$	7.52786	0.600789	0.300394	−	0.953815i	$-0.402882\pi$
$157$	7.52786	0.600789	0.300394	+	0.953815i	$0.402882\pi$
$158$	0	0
$159$	−3.32624	−0.263788
$160$	0	0
$161$	6.47214	0.510076
$162$	0	0
$163$	−13.7984	−1.08077	−0.540386	−	0.841417i	$-0.681722\pi$
$163$	−13.7984	−1.08077	−0.540386	+	0.841417i	$0.681722\pi$
$164$	0	0
$165$	−1.00000	−0.0778499
$166$	0	0
$167$	−7.09017	−0.548654	−0.274327	−	0.961636i	$-0.588455\pi$
$167$	−7.09017	−0.548654	−0.274327	+	0.961636i	$0.588455\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	10.7082	0.818877
$172$	0	0
$173$	17.1459	1.30358	0.651789	−	0.758400i	$-0.274019\pi$
$173$	17.1459	1.30358	0.651789	+	0.758400i	$0.274019\pi$
$174$	0	0
$175$	2.38197	0.180060
$176$	0	0
$177$	−3.70820	−0.278726
$178$	0	0
$179$	19.4164	1.45125	0.725625	−	0.688090i	$-0.241551\pi$
$179$	19.4164	1.45125	0.725625	+	0.688090i	$0.241551\pi$
$180$	0	0
$181$	−15.4164	−1.14589	−0.572946	−	0.819593i	$-0.694200\pi$
$181$	−15.4164	−1.14589	−0.572946	+	0.819593i	$0.694200\pi$
$182$	0	0
$183$	−2.56231	−0.189411
$184$	0	0
$185$	11.7082	0.860804
$186$	0	0
$187$	−5.85410	−0.428094
$188$	0	0
$189$	−3.47214	−0.252561
$190$	0	0
$191$	−26.9443	−1.94962	−0.974810	−	0.223039i	$-0.928402\pi$
$191$	−26.9443	−1.94962	−0.974810	+	0.223039i	$0.928402\pi$
$192$	0	0
$193$	20.7984	1.49710	0.748550	−	0.663079i	$-0.230750\pi$
$193$	20.7984	1.49710	0.748550	+	0.663079i	$0.230750\pi$
$194$	0	0
$195$	1.00000	0.0716115
$196$	0	0
$197$	3.09017	0.220165	0.110083	−	0.993922i	$-0.464888\pi$
$197$	3.09017	0.220165	0.110083	+	0.993922i	$0.464888\pi$
$198$	0	0
$199$	12.8541	0.911203	0.455602	−	0.890184i	$-0.349424\pi$
$199$	12.8541	0.911203	0.455602	+	0.890184i	$0.349424\pi$
$200$	0	0
$201$	−5.79837	−0.408986
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.76393	0.193041
$206$	0	0
$207$	16.9443	1.17771
$208$	0	0
$209$	−4.09017	−0.282923
$210$	0	0
$211$	4.61803	0.317919	0.158959	−	0.987285i	$-0.449186\pi$
$211$	4.61803	0.317919	0.158959	+	0.987285i	$0.449186\pi$
$212$	0	0
$213$	−1.18034	−0.0808756
$214$	0	0
$215$	−11.4721	−0.782393
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.76393	−0.321917
$220$	0	0
$221$	5.85410	0.393790
$222$	0	0
$223$	−29.1246	−1.95033	−0.975164	−	0.221483i	$-0.928910\pi$
$223$	−29.1246	−1.95033	−0.975164	+	0.221483i	$0.928910\pi$
$224$	0	0
$225$	6.23607	0.415738
$226$	0	0
$227$	−5.09017	−0.337846	−0.168923	−	0.985629i	$-0.554029\pi$
$227$	−5.09017	−0.337846	−0.168923	+	0.985629i	$0.554029\pi$
$228$	0	0
$229$	10.3820	0.686060	0.343030	−	0.939325i	$-0.388547\pi$
$229$	10.3820	0.686060	0.343030	+	0.939325i	$0.388547\pi$
$230$	0	0
$231$	0.618034	0.0406637
$232$	0	0
$233$	−14.9443	−0.979032	−0.489516	−	0.871994i	$-0.662826\pi$
$233$	−14.9443	−0.979032	−0.489516	+	0.871994i	$0.662826\pi$
$234$	0	0
$235$	−6.47214	−0.422196
$236$	0	0
$237$	−1.47214	−0.0956255
$238$	0	0
$239$	−2.94427	−0.190449	−0.0952246	−	0.995456i	$-0.530357\pi$
$239$	−2.94427	−0.190449	−0.0952246	+	0.995456i	$0.530357\pi$
$240$	0	0
$241$	23.4164	1.50838	0.754192	−	0.656654i	$-0.228029\pi$
$241$	23.4164	1.50838	0.754192	+	0.656654i	$0.228029\pi$
$242$	0	0
$243$	−13.9443	−0.894525
$244$	0	0
$245$	1.61803	0.103372
$246$	0	0
$247$	4.09017	0.260251
$248$	0	0
$249$	−7.94427	−0.503448
$250$	0	0
$251$	2.85410	0.180149	0.0900747	−	0.995935i	$-0.471289\pi$
$251$	2.85410	0.180149	0.0900747	+	0.995935i	$0.471289\pi$
$252$	0	0
$253$	−6.47214	−0.406900
$254$	0	0
$255$	5.85410	0.366598
$256$	0	0
$257$	1.70820	0.106555	0.0532774	−	0.998580i	$-0.483033\pi$
$257$	1.70820	0.106555	0.0532774	+	0.998580i	$0.483033\pi$
$258$	0	0
$259$	−7.23607	−0.449627
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7.50658	−0.462875	−0.231438	−	0.972850i	$-0.574343\pi$
$263$	−7.50658	−0.462875	−0.231438	+	0.972850i	$0.574343\pi$
$264$	0	0
$265$	8.70820	0.534941
$266$	0	0
$267$	4.85410	0.297066
$268$	0	0
$269$	3.41641	0.208302	0.104151	−	0.994561i	$-0.466787\pi$
$269$	3.41641	0.208302	0.104151	+	0.994561i	$0.466787\pi$
$270$	0	0
$271$	−16.9787	−1.03138	−0.515692	−	0.856774i	$-0.672465\pi$
$271$	−16.9787	−1.03138	−0.515692	+	0.856774i	$0.672465\pi$
$272$	0	0
$273$	−0.618034	−0.0374051
$274$	0	0
$275$	−2.38197	−0.143638
$276$	0	0
$277$	10.7639	0.646742	0.323371	−	0.946272i	$-0.395184\pi$
$277$	10.7639	0.646742	0.323371	+	0.946272i	$0.395184\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−24.1459	−1.44042	−0.720212	−	0.693754i	$-0.755955\pi$
$281$	−24.1459	−1.44042	−0.720212	+	0.693754i	$0.755955\pi$
$282$	0	0
$283$	−16.7639	−0.996512	−0.498256	−	0.867030i	$-0.666026\pi$
$283$	−16.7639	−0.996512	−0.498256	+	0.867030i	$0.666026\pi$
$284$	0	0
$285$	4.09017	0.242281
$286$	0	0
$287$	−1.70820	−0.100832
$288$	0	0
$289$	17.2705	1.01591
$290$	0	0
$291$	−1.81966	−0.106670
$292$	0	0
$293$	−13.8885	−0.811377	−0.405689	−	0.914011i	$-0.632968\pi$
$293$	−13.8885	−0.811377	−0.405689	+	0.914011i	$0.632968\pi$
$294$	0	0
$295$	9.70820	0.565233
$296$	0	0
$297$	3.47214	0.201474
$298$	0	0
$299$	6.47214	0.374293
$300$	0	0
$301$	7.09017	0.408671
$302$	0	0
$303$	−11.1803	−0.642294
$304$	0	0
$305$	6.70820	0.384111
$306$	0	0
$307$	13.3820	0.763749	0.381875	−	0.924214i	$-0.375279\pi$
$307$	13.3820	0.763749	0.381875	+	0.924214i	$0.375279\pi$
$308$	0	0
$309$	6.47214	0.368187
$310$	0	0
$311$	−9.03444	−0.512296	−0.256148	−	0.966638i	$-0.582453\pi$
$311$	−9.03444	−0.512296	−0.256148	+	0.966638i	$0.582453\pi$
$312$	0	0
$313$	5.41641	0.306153	0.153077	−	0.988214i	$-0.451082\pi$
$313$	5.41641	0.306153	0.153077	+	0.988214i	$0.451082\pi$
$314$	0	0
$315$	4.23607	0.238675
$316$	0	0
$317$	16.6525	0.935296	0.467648	−	0.883915i	$-0.345101\pi$
$317$	16.6525	0.935296	0.467648	+	0.883915i	$0.345101\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−12.5623	−0.701160
$322$	0	0
$323$	23.9443	1.33229
$324$	0	0
$325$	2.38197	0.132128
$326$	0	0
$327$	3.94427	0.218119
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	−1.03444	−0.0568581	−0.0284290	−	0.999596i	$-0.509050\pi$
$331$	−1.03444	−0.0568581	−0.0284290	+	0.999596i	$0.509050\pi$
$332$	0	0
$333$	−18.9443	−1.03814
$334$	0	0
$335$	15.1803	0.829391
$336$	0	0
$337$	20.2918	1.10536	0.552682	−	0.833392i	$-0.313604\pi$
$337$	20.2918	1.10536	0.552682	+	0.833392i	$0.313604\pi$
$338$	0	0
$339$	−5.47214	−0.297206
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	6.47214	0.348448
$346$	0	0
$347$	−7.43769	−0.399276	−0.199638	−	0.979870i	$-0.563977\pi$
$347$	−7.43769	−0.399276	−0.199638	+	0.979870i	$0.563977\pi$
$348$	0	0
$349$	16.7639	0.897353	0.448676	−	0.893694i	$-0.351896\pi$
$349$	16.7639	0.897353	0.448676	+	0.893694i	$0.351896\pi$
$350$	0	0
$351$	−3.47214	−0.185329
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	3.09017	0.164009
$356$	0	0
$357$	−3.61803	−0.191487
$358$	0	0
$359$	0.180340	0.00951798	0.00475899	−	0.999989i	$-0.498485\pi$
$359$	0.180340	0.00951798	0.00475899	+	0.999989i	$0.498485\pi$
$360$	0	0
$361$	−2.27051	−0.119501
$362$	0	0
$363$	−0.618034	−0.0324384
$364$	0	0
$365$	12.4721	0.652821
$366$	0	0
$367$	23.2705	1.21471	0.607355	−	0.794430i	$-0.292230\pi$
$367$	23.2705	1.21471	0.607355	+	0.794430i	$0.292230\pi$
$368$	0	0
$369$	−4.47214	−0.232810
$370$	0	0
$371$	−5.38197	−0.279418
$372$	0	0
$373$	19.2361	0.996006	0.498003	−	0.867175i	$-0.334067\pi$
$373$	19.2361	0.996006	0.498003	+	0.867175i	$0.334067\pi$
$374$	0	0
$375$	7.38197	0.381203
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−20.2148	−1.03836	−0.519182	−	0.854664i	$-0.673763\pi$
$379$	−20.2148	−1.03836	−0.519182	+	0.854664i	$0.673763\pi$
$380$	0	0
$381$	−12.0557	−0.617634
$382$	0	0
$383$	−3.52786	−0.180265	−0.0901327	−	0.995930i	$-0.528729\pi$
$383$	−3.52786	−0.180265	−0.0901327	+	0.995930i	$0.528729\pi$
$384$	0	0
$385$	−1.61803	−0.0824626
$386$	0	0
$387$	18.5623	0.943575
$388$	0	0
$389$	−8.56231	−0.434126	−0.217063	−	0.976158i	$-0.569648\pi$
$389$	−8.56231	−0.434126	−0.217063	+	0.976158i	$0.569648\pi$
$390$	0	0
$391$	37.8885	1.91611
$392$	0	0
$393$	8.29180	0.418266
$394$	0	0
$395$	3.85410	0.193921
$396$	0	0
$397$	20.9787	1.05289	0.526446	−	0.850209i	$-0.323524\pi$
$397$	20.9787	1.05289	0.526446	+	0.850209i	$0.323524\pi$
$398$	0	0
$399$	−2.52786	−0.126551
$400$	0	0
$401$	−9.81966	−0.490370	−0.245185	−	0.969476i	$-0.578849\pi$
$401$	−9.81966	−0.490370	−0.245185	+	0.969476i	$0.578849\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	9.23607	0.458944
$406$	0	0
$407$	7.23607	0.358679
$408$	0	0
$409$	−27.5967	−1.36457	−0.682286	−	0.731086i	$-0.739014\pi$
$409$	−27.5967	−1.36457	−0.682286	+	0.731086i	$0.739014\pi$
$410$	0	0
$411$	9.81966	0.484368
$412$	0	0
$413$	−6.00000	−0.295241
$414$	0	0
$415$	20.7984	1.02095
$416$	0	0
$417$	−9.52786	−0.466582
$418$	0	0
$419$	−4.32624	−0.211351	−0.105675	−	0.994401i	$-0.533700\pi$
$419$	−4.32624	−0.211351	−0.105675	+	0.994401i	$0.533700\pi$
$420$	0	0
$421$	10.2918	0.501591	0.250796	−	0.968040i	$-0.419308\pi$
$421$	10.2918	0.501591	0.250796	+	0.968040i	$0.419308\pi$
$422$	0	0
$423$	10.4721	0.509173
$424$	0	0
$425$	13.9443	0.676397
$426$	0	0
$427$	−4.14590	−0.200634
$428$	0	0
$429$	0.618034	0.0298390
$430$	0	0
$431$	−18.1803	−0.875716	−0.437858	−	0.899044i	$-0.644263\pi$
$431$	−18.1803	−0.875716	−0.437858	+	0.899044i	$0.644263\pi$
$432$	0	0
$433$	−33.2361	−1.59722	−0.798612	−	0.601847i	$-0.794432\pi$
$433$	−33.2361	−1.59722	−0.798612	+	0.601847i	$0.794432\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	26.4721	1.26633
$438$	0	0
$439$	−35.8885	−1.71287	−0.856433	−	0.516258i	$-0.827325\pi$
$439$	−35.8885	−1.71287	−0.856433	+	0.516258i	$0.827325\pi$
$440$	0	0
$441$	−2.61803	−0.124668
$442$	0	0
$443$	14.1803	0.673728	0.336864	−	0.941553i	$-0.390634\pi$
$443$	14.1803	0.673728	0.336864	+	0.941553i	$0.390634\pi$
$444$	0	0
$445$	−12.7082	−0.602427
$446$	0	0
$447$	−13.4721	−0.637210
$448$	0	0
$449$	23.4164	1.10509	0.552544	−	0.833484i	$-0.313657\pi$
$449$	23.4164	1.10509	0.552544	+	0.833484i	$0.313657\pi$
$450$	0	0
$451$	1.70820	0.0804362
$452$	0	0
$453$	14.4721	0.679960
$454$	0	0
$455$	1.61803	0.0758546
$456$	0	0
$457$	40.9230	1.91430	0.957148	−	0.289598i	$-0.0935217\pi$
$457$	40.9230	1.91430	0.957148	+	0.289598i	$0.0935217\pi$
$458$	0	0
$459$	−20.3262	−0.948748
$460$	0	0
$461$	33.5279	1.56155	0.780774	−	0.624813i	$-0.214825\pi$
$461$	33.5279	1.56155	0.780774	+	0.624813i	$0.214825\pi$
$462$	0	0
$463$	10.4721	0.486681	0.243341	−	0.969941i	$-0.421757\pi$
$463$	10.4721	0.486681	0.243341	+	0.969941i	$0.421757\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−22.4377	−1.03829	−0.519146	−	0.854685i	$-0.673750\pi$
$467$	−22.4377	−1.03829	−0.519146	+	0.854685i	$0.673750\pi$
$468$	0	0
$469$	−9.38197	−0.433219
$470$	0	0
$471$	−4.65248	−0.214375
$472$	0	0
$473$	−7.09017	−0.326006
$474$	0	0
$475$	9.74265	0.447023
$476$	0	0
$477$	−14.0902	−0.645145
$478$	0	0
$479$	−2.49342	−0.113927	−0.0569637	−	0.998376i	$-0.518142\pi$
$479$	−2.49342	−0.113927	−0.0569637	+	0.998376i	$0.518142\pi$
$480$	0	0
$481$	−7.23607	−0.329936
$482$	0	0
$483$	−4.00000	−0.182006
$484$	0	0
$485$	4.76393	0.216319
$486$	0	0
$487$	−30.4721	−1.38082	−0.690412	−	0.723416i	$-0.742571\pi$
$487$	−30.4721	−1.38082	−0.690412	+	0.723416i	$0.742571\pi$
$488$	0	0
$489$	8.52786	0.385643
$490$	0	0
$491$	−30.8328	−1.39147	−0.695733	−	0.718301i	$-0.744920\pi$
$491$	−30.8328	−1.39147	−0.695733	+	0.718301i	$0.744920\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−4.23607	−0.190397
$496$	0	0
$497$	−1.90983	−0.0856676
$498$	0	0
$499$	11.9098	0.533157	0.266579	−	0.963813i	$-0.414107\pi$
$499$	11.9098	0.533157	0.266579	+	0.963813i	$0.414107\pi$
$500$	0	0
$501$	4.38197	0.195772
$502$	0	0
$503$	−8.47214	−0.377754	−0.188877	−	0.982001i	$-0.560485\pi$
$503$	−8.47214	−0.377754	−0.188877	+	0.982001i	$0.560485\pi$
$504$	0	0
$505$	29.2705	1.30252
$506$	0	0
$507$	−0.618034	−0.0274479
$508$	0	0
$509$	−11.5279	−0.510964	−0.255482	−	0.966814i	$-0.582234\pi$
$509$	−11.5279	−0.510964	−0.255482	+	0.966814i	$0.582234\pi$
$510$	0	0
$511$	−7.70820	−0.340991
$512$	0	0
$513$	−14.2016	−0.627017
$514$	0	0
$515$	−16.9443	−0.746654
$516$	0	0
$517$	−4.00000	−0.175920
$518$	0	0
$519$	−10.5967	−0.465146
$520$	0	0
$521$	8.47214	0.371171	0.185586	−	0.982628i	$-0.440582\pi$
$521$	8.47214	0.371171	0.185586	+	0.982628i	$0.440582\pi$
$522$	0	0
$523$	37.1246	1.62335	0.811673	−	0.584112i	$-0.198557\pi$
$523$	37.1246	1.62335	0.811673	+	0.584112i	$0.198557\pi$
$524$	0	0
$525$	−1.47214	−0.0642493
$526$	0	0
$527$	0	0
$528$	0	0
$529$	18.8885	0.821241
$530$	0	0
$531$	−15.7082	−0.681678
$532$	0	0
$533$	−1.70820	−0.0739905
$534$	0	0
$535$	32.8885	1.42190
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−30.9443	−1.33040	−0.665199	−	0.746666i	$-0.731653\pi$
$541$	−30.9443	−1.33040	−0.665199	+	0.746666i	$0.731653\pi$
$542$	0	0
$543$	9.52786	0.408880
$544$	0	0
$545$	−10.3262	−0.442327
$546$	0	0
$547$	−24.9443	−1.06654	−0.533270	−	0.845945i	$-0.679037\pi$
$547$	−24.9443	−1.06654	−0.533270	+	0.845945i	$0.679037\pi$
$548$	0	0
$549$	−10.8541	−0.463242
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−2.38197	−0.101291
$554$	0	0
$555$	−7.23607	−0.307154
$556$	0	0
$557$	24.8541	1.05310	0.526551	−	0.850144i	$-0.323485\pi$
$557$	24.8541	1.05310	0.526551	+	0.850144i	$0.323485\pi$
$558$	0	0
$559$	7.09017	0.299882
$560$	0	0
$561$	3.61803	0.152754
$562$	0	0
$563$	27.4164	1.15546	0.577732	−	0.816227i	$-0.303938\pi$
$563$	27.4164	1.15546	0.577732	+	0.816227i	$0.303938\pi$
$564$	0	0
$565$	14.3262	0.602709
$566$	0	0
$567$	−5.70820	−0.239722
$568$	0	0
$569$	−4.00000	−0.167689	−0.0838444	−	0.996479i	$-0.526720\pi$
$569$	−4.00000	−0.167689	−0.0838444	+	0.996479i	$0.526720\pi$
$570$	0	0
$571$	24.1591	1.01102	0.505512	−	0.862819i	$-0.331303\pi$
$571$	24.1591	1.01102	0.505512	+	0.862819i	$0.331303\pi$
$572$	0	0
$573$	16.6525	0.695667
$574$	0	0
$575$	15.4164	0.642909
$576$	0	0
$577$	10.3262	0.429887	0.214943	−	0.976626i	$-0.431043\pi$
$577$	10.3262	0.429887	0.214943	+	0.976626i	$0.431043\pi$
$578$	0	0
$579$	−12.8541	−0.534198
$580$	0	0
$581$	−12.8541	−0.533278
$582$	0	0
$583$	5.38197	0.222898
$584$	0	0
$585$	4.23607	0.175140
$586$	0	0
$587$	−47.0132	−1.94044	−0.970220	−	0.242224i	$-0.922123\pi$
$587$	−47.0132	−1.94044	−0.970220	+	0.242224i	$0.922123\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−1.90983	−0.0785599
$592$	0	0
$593$	−27.7771	−1.14067	−0.570334	−	0.821413i	$-0.693186\pi$
$593$	−27.7771	−1.14067	−0.570334	+	0.821413i	$0.693186\pi$
$594$	0	0
$595$	9.47214	0.388320
$596$	0	0
$597$	−7.94427	−0.325137
$598$	0	0
$599$	40.6525	1.66102	0.830508	−	0.557007i	$-0.188050\pi$
$599$	40.6525	1.66102	0.830508	+	0.557007i	$0.188050\pi$
$600$	0	0
$601$	−25.5623	−1.04271	−0.521354	−	0.853340i	$-0.674573\pi$
$601$	−25.5623	−1.04271	−0.521354	+	0.853340i	$0.674573\pi$
$602$	0	0
$603$	−24.5623	−1.00025
$604$	0	0
$605$	1.61803	0.0657824
$606$	0	0
$607$	44.6525	1.81239	0.906194	−	0.422862i	$-0.138975\pi$
$607$	44.6525	1.81239	0.906194	+	0.422862i	$0.138975\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.00000	0.161823
$612$	0	0
$613$	22.0000	0.888572	0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000	0.888572	0.444286	+	0.895885i	$0.353457\pi$
$614$	0	0
$615$	−1.70820	−0.0688814
$616$	0	0
$617$	15.3475	0.617868	0.308934	−	0.951083i	$-0.400028\pi$
$617$	15.3475	0.617868	0.308934	+	0.951083i	$0.400028\pi$
$618$	0	0
$619$	−13.8197	−0.555459	−0.277730	−	0.960659i	$-0.589582\pi$
$619$	−13.8197	−0.555459	−0.277730	+	0.960659i	$0.589582\pi$
$620$	0	0
$621$	−22.4721	−0.901776
$622$	0	0
$623$	7.85410	0.314668
$624$	0	0
$625$	−7.41641	−0.296656
$626$	0	0
$627$	2.52786	0.100953
$628$	0	0
$629$	−42.3607	−1.68903
$630$	0	0
$631$	−49.2705	−1.96143	−0.980714	−	0.195448i	$-0.937384\pi$
$631$	−49.2705	−1.96143	−0.980714	+	0.195448i	$0.937384\pi$
$632$	0	0
$633$	−2.85410	−0.113440
$634$	0	0
$635$	31.5623	1.25251
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−5.00000	−0.197797
$640$	0	0
$641$	14.2148	0.561450	0.280725	−	0.959788i	$-0.409425\pi$
$641$	14.2148	0.561450	0.280725	+	0.959788i	$0.409425\pi$
$642$	0	0
$643$	−4.29180	−0.169252	−0.0846260	−	0.996413i	$-0.526970\pi$
$643$	−4.29180	−0.169252	−0.0846260	+	0.996413i	$0.526970\pi$
$644$	0	0
$645$	7.09017	0.279175
$646$	0	0
$647$	14.8328	0.583138	0.291569	−	0.956550i	$-0.405823\pi$
$647$	14.8328	0.583138	0.291569	+	0.956550i	$0.405823\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−9.45085	−0.369840	−0.184920	−	0.982754i	$-0.559203\pi$
$653$	−9.45085	−0.369840	−0.184920	+	0.982754i	$0.559203\pi$
$654$	0	0
$655$	−21.7082	−0.848210
$656$	0	0
$657$	−20.1803	−0.787310
$658$	0	0
$659$	23.6738	0.922199	0.461099	−	0.887348i	$-0.347455\pi$
$659$	23.6738	0.922199	0.461099	+	0.887348i	$0.347455\pi$
$660$	0	0
$661$	3.90983	0.152075	0.0760374	−	0.997105i	$-0.475773\pi$
$661$	3.90983	0.152075	0.0760374	+	0.997105i	$0.475773\pi$
$662$	0	0
$663$	−3.61803	−0.140513
$664$	0	0
$665$	6.61803	0.256636
$666$	0	0
$667$	0	0
$668$	0	0
$669$	18.0000	0.695920
$670$	0	0
$671$	4.14590	0.160051
$672$	0	0
$673$	31.8885	1.22921	0.614607	−	0.788834i	$-0.289315\pi$
$673$	31.8885	1.22921	0.614607	+	0.788834i	$0.289315\pi$
$674$	0	0
$675$	−8.27051	−0.318332
$676$	0	0
$677$	−18.0344	−0.693120	−0.346560	−	0.938028i	$-0.612650\pi$
$677$	−18.0344	−0.693120	−0.346560	+	0.938028i	$0.612650\pi$
$678$	0	0
$679$	−2.94427	−0.112991
$680$	0	0
$681$	3.14590	0.120551
$682$	0	0
$683$	42.9787	1.64453	0.822267	−	0.569101i	$-0.192709\pi$
$683$	42.9787	1.64453	0.822267	+	0.569101i	$0.192709\pi$
$684$	0	0
$685$	−25.7082	−0.982260
$686$	0	0
$687$	−6.41641	−0.244801
$688$	0	0
$689$	−5.38197	−0.205037
$690$	0	0
$691$	15.4164	0.586468	0.293234	−	0.956041i	$-0.405269\pi$
$691$	15.4164	0.586468	0.293234	+	0.956041i	$0.405269\pi$
$692$	0	0
$693$	2.61803	0.0994509
$694$	0	0
$695$	24.9443	0.946190
$696$	0	0
$697$	−10.0000	−0.378777
$698$	0	0
$699$	9.23607	0.349340
$700$	0	0
$701$	32.5410	1.22906	0.614529	−	0.788894i	$-0.289346\pi$
$701$	32.5410	1.22906	0.614529	+	0.788894i	$0.289346\pi$
$702$	0	0
$703$	−29.5967	−1.11626
$704$	0	0
$705$	4.00000	0.150649
$706$	0	0
$707$	−18.0902	−0.680351
$708$	0	0
$709$	7.41641	0.278529	0.139265	−	0.990255i	$-0.455526\pi$
$709$	7.41641	0.278529	0.139265	+	0.990255i	$0.455526\pi$
$710$	0	0
$711$	−6.23607	−0.233871
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−1.61803	−0.0605110
$716$	0	0
$717$	1.81966	0.0679565
$718$	0	0
$719$	31.6312	1.17964	0.589822	−	0.807533i	$-0.299198\pi$
$719$	31.6312	1.17964	0.589822	+	0.807533i	$0.299198\pi$
$720$	0	0
$721$	10.4721	0.390003
$722$	0	0
$723$	−14.4721	−0.538225
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23.5066	0.871811	0.435905	−	0.899993i	$-0.356428\pi$
$727$	23.5066	0.871811	0.435905	+	0.899993i	$0.356428\pi$
$728$	0	0
$729$	−8.50658	−0.315058
$730$	0	0
$731$	41.5066	1.53518
$732$	0	0
$733$	−17.1246	−0.632512	−0.316256	−	0.948674i	$-0.602426\pi$
$733$	−17.1246	−0.632512	−0.316256	+	0.948674i	$0.602426\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	9.38197	0.345589
$738$	0	0
$739$	17.8885	0.658041	0.329020	−	0.944323i	$-0.393282\pi$
$739$	17.8885	0.658041	0.329020	+	0.944323i	$0.393282\pi$
$740$	0	0
$741$	−2.52786	−0.0928634
$742$	0	0
$743$	−22.5410	−0.826950	−0.413475	−	0.910516i	$-0.635685\pi$
$743$	−22.5410	−0.826950	−0.413475	+	0.910516i	$0.635685\pi$
$744$	0	0
$745$	35.2705	1.29221
$746$	0	0
$747$	−33.6525	−1.23128
$748$	0	0
$749$	−20.3262	−0.742705
$750$	0	0
$751$	6.83282	0.249333	0.124666	−	0.992199i	$-0.460214\pi$
$751$	6.83282	0.249333	0.124666	+	0.992199i	$0.460214\pi$
$752$	0	0
$753$	−1.76393	−0.0642813
$754$	0	0
$755$	−37.8885	−1.37891
$756$	0	0
$757$	16.9656	0.616624	0.308312	−	0.951285i	$-0.400236\pi$
$757$	16.9656	0.616624	0.308312	+	0.951285i	$0.400236\pi$
$758$	0	0
$759$	4.00000	0.145191
$760$	0	0
$761$	18.0000	0.652499	0.326250	−	0.945284i	$-0.394215\pi$
$761$	18.0000	0.652499	0.326250	+	0.945284i	$0.394215\pi$
$762$	0	0
$763$	6.38197	0.231043
$764$	0	0
$765$	24.7984	0.896587
$766$	0	0
$767$	−6.00000	−0.216647
$768$	0	0
$769$	32.0689	1.15643	0.578217	−	0.815883i	$-0.303749\pi$
$769$	32.0689	1.15643	0.578217	+	0.815883i	$0.303749\pi$
$770$	0	0
$771$	−1.05573	−0.0380211
$772$	0	0
$773$	3.45085	0.124118	0.0620592	−	0.998072i	$-0.480233\pi$
$773$	3.45085	0.124118	0.0620592	+	0.998072i	$0.480233\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.47214	0.160437
$778$	0	0
$779$	−6.98684	−0.250330
$780$	0	0
$781$	1.90983	0.0683391
$782$	0	0
$783$	0	0
$784$	0	0
$785$	12.1803	0.434735
$786$	0	0
$787$	−32.7984	−1.16914	−0.584568	−	0.811345i	$-0.698736\pi$
$787$	−32.7984	−1.16914	−0.584568	+	0.811345i	$0.698736\pi$
$788$	0	0
$789$	4.63932	0.165164
$790$	0	0
$791$	−8.85410	−0.314816
$792$	0	0
$793$	−4.14590	−0.147225
$794$	0	0
$795$	−5.38197	−0.190879
$796$	0	0
$797$	−31.1246	−1.10249	−0.551245	−	0.834343i	$-0.685847\pi$
$797$	−31.1246	−1.10249	−0.551245	+	0.834343i	$0.685847\pi$
$798$	0	0
$799$	23.4164	0.828413
$800$	0	0
$801$	20.5623	0.726533
$802$	0	0
$803$	7.70820	0.272017
$804$	0	0
$805$	10.4721	0.369094
$806$	0	0
$807$	−2.11146	−0.0743268
$808$	0	0
$809$	24.2918	0.854054	0.427027	−	0.904239i	$-0.359561\pi$
$809$	24.2918	0.854054	0.427027	+	0.904239i	$0.359561\pi$
$810$	0	0
$811$	40.4508	1.42042	0.710211	−	0.703989i	$-0.248600\pi$
$811$	40.4508	1.42042	0.710211	+	0.703989i	$0.248600\pi$
$812$	0	0
$813$	10.4934	0.368020
$814$	0	0
$815$	−22.3262	−0.782054
$816$	0	0
$817$	29.0000	1.01458
$818$	0	0
$819$	−2.61803	−0.0914815
$820$	0	0
$821$	−52.0344	−1.81601	−0.908007	−	0.418954i	$-0.862397\pi$
$821$	−52.0344	−1.81601	−0.908007	+	0.418954i	$0.862397\pi$
$822$	0	0
$823$	−4.94427	−0.172346	−0.0861732	−	0.996280i	$-0.527464\pi$
$823$	−4.94427	−0.172346	−0.0861732	+	0.996280i	$0.527464\pi$
$824$	0	0
$825$	1.47214	0.0512532
$826$	0	0
$827$	52.3607	1.82076	0.910380	−	0.413774i	$-0.135790\pi$
$827$	52.3607	1.82076	0.910380	+	0.413774i	$0.135790\pi$
$828$	0	0
$829$	−32.8328	−1.14033	−0.570165	−	0.821530i	$-0.693121\pi$
$829$	−32.8328	−1.14033	−0.570165	+	0.821530i	$0.693121\pi$
$830$	0	0
$831$	−6.65248	−0.230772
$832$	0	0
$833$	−5.85410	−0.202833
$834$	0	0
$835$	−11.4721	−0.397010
$836$	0	0
$837$	0	0
$838$	0	0
$839$	20.4721	0.706777	0.353388	−	0.935477i	$-0.385029\pi$
$839$	20.4721	0.706777	0.353388	+	0.935477i	$0.385029\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	14.9230	0.513975
$844$	0	0
$845$	1.61803	0.0556621
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	10.3607	0.355578
$850$	0	0
$851$	−46.8328	−1.60541
$852$	0	0
$853$	−24.7639	−0.847901	−0.423950	−	0.905685i	$-0.639357\pi$
$853$	−24.7639	−0.847901	−0.423950	+	0.905685i	$0.639357\pi$
$854$	0	0
$855$	17.3262	0.592545
$856$	0	0
$857$	8.85410	0.302450	0.151225	−	0.988499i	$-0.451678\pi$
$857$	8.85410	0.302450	0.151225	+	0.988499i	$0.451678\pi$
$858$	0	0
$859$	14.8328	0.506089	0.253045	−	0.967455i	$-0.418568\pi$
$859$	14.8328	0.506089	0.253045	+	0.967455i	$0.418568\pi$
$860$	0	0
$861$	1.05573	0.0359791
$862$	0	0
$863$	50.2492	1.71050	0.855252	−	0.518213i	$-0.173402\pi$
$863$	50.2492	1.71050	0.855252	+	0.518213i	$0.173402\pi$
$864$	0	0
$865$	27.7426	0.943278
$866$	0	0
$867$	−10.6738	−0.362500
$868$	0	0
$869$	2.38197	0.0808027
$870$	0	0
$871$	−9.38197	−0.317896
$872$	0	0
$873$	−7.70820	−0.260883
$874$	0	0
$875$	11.9443	0.403790
$876$	0	0
$877$	−14.5066	−0.489852	−0.244926	−	0.969542i	$-0.578764\pi$
$877$	−14.5066	−0.489852	−0.244926	+	0.969542i	$0.578764\pi$
$878$	0	0
$879$	8.58359	0.289517
$880$	0	0
$881$	−24.0000	−0.808581	−0.404290	−	0.914631i	$-0.632481\pi$
$881$	−24.0000	−0.808581	−0.404290	+	0.914631i	$0.632481\pi$
$882$	0	0
$883$	−26.0000	−0.874970	−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000	−0.874970	−0.437485	+	0.899226i	$0.644131\pi$
$884$	0	0
$885$	−6.00000	−0.201688
$886$	0	0
$887$	1.70820	0.0573559	0.0286779	−	0.999589i	$-0.490870\pi$
$887$	1.70820	0.0573559	0.0286779	+	0.999589i	$0.490870\pi$
$888$	0	0
$889$	−19.5066	−0.654230
$890$	0	0
$891$	5.70820	0.191232
$892$	0	0
$893$	16.3607	0.547489
$894$	0	0
$895$	31.4164	1.05013
$896$	0	0
$897$	−4.00000	−0.133556
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−31.5066	−1.04964
$902$	0	0
$903$	−4.38197	−0.145823
$904$	0	0
$905$	−24.9443	−0.829176
$906$	0	0
$907$	39.7082	1.31849	0.659245	−	0.751929i	$-0.270876\pi$
$907$	39.7082	1.31849	0.659245	+	0.751929i	$0.270876\pi$
$908$	0	0
$909$	−47.3607	−1.57085
$910$	0	0
$911$	12.9443	0.428863	0.214431	−	0.976739i	$-0.431210\pi$
$911$	12.9443	0.428863	0.214431	+	0.976739i	$0.431210\pi$
$912$	0	0
$913$	12.8541	0.425409
$914$	0	0
$915$	−4.14590	−0.137059
$916$	0	0
$917$	13.4164	0.443049
$918$	0	0
$919$	12.2148	0.402928	0.201464	−	0.979496i	$-0.435430\pi$
$919$	12.2148	0.402928	0.201464	+	0.979496i	$0.435430\pi$
$920$	0	0
$921$	−8.27051	−0.272523
$922$	0	0
$923$	−1.90983	−0.0628628
$924$	0	0
$925$	−17.2361	−0.566718
$926$	0	0
$927$	27.4164	0.900473
$928$	0	0
$929$	−4.21478	−0.138283	−0.0691413	−	0.997607i	$-0.522026\pi$
$929$	−4.21478	−0.138283	−0.0691413	+	0.997607i	$0.522026\pi$
$930$	0	0
$931$	−4.09017	−0.134050
$932$	0	0
$933$	5.58359	0.182799
$934$	0	0
$935$	−9.47214	−0.309772
$936$	0	0
$937$	34.3394	1.12182	0.560910	−	0.827877i	$-0.310452\pi$
$937$	34.3394	1.12182	0.560910	+	0.827877i	$0.310452\pi$
$938$	0	0
$939$	−3.34752	−0.109242
$940$	0	0
$941$	39.7082	1.29445	0.647225	−	0.762299i	$-0.275929\pi$
$941$	39.7082	1.29445	0.647225	+	0.762299i	$0.275929\pi$
$942$	0	0
$943$	−11.0557	−0.360024
$944$	0	0
$945$	−5.61803	−0.182755
$946$	0	0
$947$	21.7426	0.706541	0.353271	−	0.935521i	$-0.385070\pi$
$947$	21.7426	0.706541	0.353271	+	0.935521i	$0.385070\pi$
$948$	0	0
$949$	−7.70820	−0.250219
$950$	0	0
$951$	−10.2918	−0.333734
$952$	0	0
$953$	−8.76393	−0.283892	−0.141946	−	0.989874i	$-0.545336\pi$
$953$	−8.76393	−0.283892	−0.141946	+	0.989874i	$0.545336\pi$
$954$	0	0
$955$	−43.5967	−1.41076
$956$	0	0
$957$	0	0
$958$	0	0
$959$	15.8885	0.513068
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−53.2148	−1.71482
$964$	0	0
$965$	33.6525	1.08331
$966$	0	0
$967$	−31.5967	−1.01608	−0.508041	−	0.861333i	$-0.669630\pi$
$967$	−31.5967	−1.01608	−0.508041	+	0.861333i	$0.669630\pi$
$968$	0	0
$969$	−14.7984	−0.475392
$970$	0	0
$971$	−39.2148	−1.25846	−0.629231	−	0.777218i	$-0.716630\pi$
$971$	−39.2148	−1.25846	−0.629231	+	0.777218i	$0.716630\pi$
$972$	0	0
$973$	−15.4164	−0.494227
$974$	0	0
$975$	−1.47214	−0.0471461
$976$	0	0
$977$	−49.8885	−1.59608	−0.798038	−	0.602607i	$-0.794129\pi$
$977$	−49.8885	−1.59608	−0.798038	+	0.602607i	$0.794129\pi$
$978$	0	0
$979$	−7.85410	−0.251018
$980$	0	0
$981$	16.7082	0.533452
$982$	0	0
$983$	38.0689	1.21421	0.607104	−	0.794622i	$-0.292331\pi$
$983$	38.0689	1.21421	0.607104	+	0.794622i	$0.292331\pi$
$984$	0	0
$985$	5.00000	0.159313
$986$	0	0
$987$	−2.47214	−0.0786890
$988$	0	0
$989$	45.8885	1.45917
$990$	0	0
$991$	46.7214	1.48415	0.742076	−	0.670315i	$-0.233841\pi$
$991$	46.7214	1.48415	0.742076	+	0.670315i	$0.233841\pi$
$992$	0	0
$993$	0.639320	0.0202882
$994$	0	0
$995$	20.7984	0.659353
$996$	0	0
$997$	46.2705	1.46540	0.732701	−	0.680551i	$-0.238259\pi$
$997$	46.2705	1.46540	0.732701	+	0.680551i	$0.238259\pi$
$998$	0	0
$999$	25.1246	0.794908

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.f.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.f.1.1	✓	2	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)

\(f(q)\)	\(=\)	\(q-0.618034 q^{3} +1.61803 q^{5} -1.00000 q^{7} -2.61803 q^{9} +O(q^{10})\)	\(q-0.618034 q^{3} +1.61803 q^{5} -1.00000 q^{7} -2.61803 q^{9} +1.00000 q^{11} -1.00000 q^{13} -1.00000 q^{15} -5.85410 q^{17} -4.09017 q^{19} +0.618034 q^{21} -6.47214 q^{23} -2.38197 q^{25} +3.47214 q^{27} -0.618034 q^{33} -1.61803 q^{35} +7.23607 q^{37} +0.618034 q^{39} +1.70820 q^{41} -7.09017 q^{43} -4.23607 q^{45} -4.00000 q^{47} +1.00000 q^{49} +3.61803 q^{51} +5.38197 q^{53} +1.61803 q^{55} +2.52786 q^{57} +6.00000 q^{59} +4.14590 q^{61} +2.61803 q^{63} -1.61803 q^{65} +9.38197 q^{67} +4.00000 q^{69} +1.90983 q^{71} +7.70820 q^{73} +1.47214 q^{75} -1.00000 q^{77} +2.38197 q^{79} +5.70820 q^{81} +12.8541 q^{83} -9.47214 q^{85} -7.85410 q^{89} +1.00000 q^{91} -6.61803 q^{95} +2.94427 q^{97} -2.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + q^{5} - 2 q^{7} - 3 q^{9}+O(q^{10}) \) 2 * q + q^3 + q^5 - 2 * q^7 - 3 * q^9	\( 2 q + q^{3} + q^{5} - 2 q^{7} - 3 q^{9} + 2 q^{11} - 2 q^{13} - 2 q^{15} - 5 q^{17} + 3 q^{19} - q^{21} - 4 q^{23} - 7 q^{25} - 2 q^{27} + q^{33} - q^{35} + 10 q^{37} - q^{39} - 10 q^{41} - 3 q^{43} - 4 q^{45} - 8 q^{47} + 2 q^{49} + 5 q^{51} + 13 q^{53} + q^{55} + 14 q^{57} + 12 q^{59} + 15 q^{61} + 3 q^{63} - q^{65} + 21 q^{67} + 8 q^{69} + 15 q^{71} + 2 q^{73} - 6 q^{75} - 2 q^{77} + 7 q^{79} - 2 q^{81} + 19 q^{83} - 10 q^{85} - 9 q^{89} + 2 q^{91} - 11 q^{95} - 12 q^{97} - 3 q^{99}+O(q^{100}) \) 2 * q + q^3 + q^5 - 2 * q^7 - 3 * q^9 + 2 * q^11 - 2 * q^13 - 2 * q^15 - 5 * q^17 + 3 * q^19 - q^21 - 4 * q^23 - 7 * q^25 - 2 * q^27 + q^33 - q^35 + 10 * q^37 - q^39 - 10 * q^41 - 3 * q^43 - 4 * q^45 - 8 * q^47 + 2 * q^49 + 5 * q^51 + 13 * q^53 + q^55 + 14 * q^57 + 12 * q^59 + 15 * q^61 + 3 * q^63 - q^65 + 21 * q^67 + 8 * q^69 + 15 * q^71 + 2 * q^73 - 6 * q^75 - 2 * q^77 + 7 * q^79 - 2 * q^81 + 19 * q^83 - 10 * q^85 - 9 * q^89 + 2 * q^91 - 11 * q^95 - 12 * q^97 - 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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