LMFDB - Embedded newform 8016.2.a.t.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.149169.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} - 3x^{2} + 4x + 1 $ x^5 - 6x^3 - 3x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2004)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.25464$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.171230 q^{5} -2.92708 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.171230 q^{5} -2.92708 q^{7} +1.00000 q^{9} -0.171230 q^{11} -3.29824 q^{13} -0.171230 q^{15} -3.91520 q^{17} +5.81055 q^{19} -2.92708 q^{21} +2.35674 q^{23} -4.97068 q^{25} +1.00000 q^{27} +8.91568 q^{29} +1.15936 q^{31} -0.171230 q^{33} +0.501205 q^{35} +6.88598 q^{37} -3.29824 q^{39} -6.93593 q^{41} +11.1872 q^{43} -0.171230 q^{45} -2.22165 q^{47} +1.56778 q^{49} -3.91520 q^{51} +10.5783 q^{53} +0.0293199 q^{55} +5.81055 q^{57} -1.88853 q^{59} -9.31996 q^{61} -2.92708 q^{63} +0.564760 q^{65} -7.04904 q^{67} +2.35674 q^{69} -3.10327 q^{71} -11.0274 q^{73} -4.97068 q^{75} +0.501205 q^{77} +5.12636 q^{79} +1.00000 q^{81} +5.60127 q^{83} +0.670402 q^{85} +8.91568 q^{87} -13.0682 q^{89} +9.65422 q^{91} +1.15936 q^{93} -0.994944 q^{95} -13.2166 q^{97} -0.171230 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9	$ 5 q + 5 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9} - 3 q^{11} - 4 q^{13} - 3 q^{15} - 7 q^{17} + 2 q^{19} + 2 q^{21} - 13 q^{23} - 2 q^{25} + 5 q^{27} - 3 q^{29} + 12 q^{31} - 3 q^{33} - 10 q^{35} - 7 q^{37} - 4 q^{39} - 16 q^{41} - 3 q^{45} - q^{47} - 17 q^{49} - 7 q^{51} + 3 q^{53} + 23 q^{55} + 2 q^{57} - q^{59} - 22 q^{61} + 2 q^{63} - 20 q^{65} - 2 q^{67} - 13 q^{69} - 9 q^{71} - 28 q^{73} - 2 q^{75} - 10 q^{77} + 28 q^{79} + 5 q^{81} - 7 q^{83} - 11 q^{85} - 3 q^{87} - 30 q^{89} + 13 q^{91} + 12 q^{93} - 3 q^{95} - 33 q^{97} - 3 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9 - 3 * q^11 - 4 * q^13 - 3 * q^15 - 7 * q^17 + 2 * q^19 + 2 * q^21 - 13 * q^23 - 2 * q^25 + 5 * q^27 - 3 * q^29 + 12 * q^31 - 3 * q^33 - 10 * q^35 - 7 * q^37 - 4 * q^39 - 16 * q^41 - 3 * q^45 - q^47 - 17 * q^49 - 7 * q^51 + 3 * q^53 + 23 * q^55 + 2 * q^57 - q^59 - 22 * q^61 + 2 * q^63 - 20 * q^65 - 2 * q^67 - 13 * q^69 - 9 * q^71 - 28 * q^73 - 2 * q^75 - 10 * q^77 + 28 * q^79 + 5 * q^81 - 7 * q^83 - 11 * q^85 - 3 * q^87 - 30 * q^89 + 13 * q^91 + 12 * q^93 - 3 * q^95 - 33 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−0.171230	−0.0765766	−0.0382883	−	0.999267i	$-0.512191\pi$
$5$	−0.171230	−0.0765766	−0.0382883	+	0.999267i	$0.512191\pi$
$6$	0	0
$7$	−2.92708	−1.10633	−0.553166	−	0.833071i	$-0.686580\pi$
$7$	−2.92708	−1.10633	−0.553166	+	0.833071i	$0.686580\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.171230	−0.0516279	−0.0258140	−	0.999667i	$-0.508218\pi$
$11$	−0.171230	−0.0516279	−0.0258140	+	0.999667i	$0.508218\pi$
$12$	0	0
$13$	−3.29824	−0.914769	−0.457384	−	0.889269i	$-0.651214\pi$
$13$	−3.29824	−0.914769	−0.457384	+	0.889269i	$0.651214\pi$
$14$	0	0
$15$	−0.171230	−0.0442115
$16$	0	0
$17$	−3.91520	−0.949577	−0.474788	−	0.880100i	$-0.657475\pi$
$17$	−3.91520	−0.949577	−0.474788	+	0.880100i	$0.657475\pi$
$18$	0	0
$19$	5.81055	1.33303	0.666516	−	0.745491i	$-0.267785\pi$
$19$	5.81055	1.33303	0.666516	+	0.745491i	$0.267785\pi$
$20$	0	0
$21$	−2.92708	−0.638741
$22$	0	0
$23$	2.35674	0.491415	0.245708	−	0.969344i	$-0.420980\pi$
$23$	2.35674	0.491415	0.245708	+	0.969344i	$0.420980\pi$
$24$	0	0
$25$	−4.97068	−0.994136
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.91568	1.65560	0.827800	−	0.561024i	$-0.189592\pi$
$29$	8.91568	1.65560	0.827800	+	0.561024i	$0.189592\pi$
$30$	0	0
$31$	1.15936	0.208227	0.104113	−	0.994565i	$-0.466800\pi$
$31$	1.15936	0.208227	0.104113	+	0.994565i	$0.466800\pi$
$32$	0	0
$33$	−0.171230	−0.0298074
$34$	0	0
$35$	0.501205	0.0847191
$36$	0	0
$37$	6.88598	1.13205	0.566024	−	0.824389i	$-0.308481\pi$
$37$	6.88598	1.13205	0.566024	+	0.824389i	$0.308481\pi$
$38$	0	0
$39$	−3.29824	−0.528142
$40$	0	0
$41$	−6.93593	−1.08321	−0.541605	−	0.840633i	$-0.682183\pi$
$41$	−6.93593	−1.08321	−0.541605	+	0.840633i	$0.682183\pi$
$42$	0	0
$43$	11.1872	1.70604	0.853020	−	0.521879i	$-0.174769\pi$
$43$	11.1872	1.70604	0.853020	+	0.521879i	$0.174769\pi$
$44$	0	0
$45$	−0.171230	−0.0255255
$46$	0	0
$47$	−2.22165	−0.324061	−0.162030	−	0.986786i	$-0.551804\pi$
$47$	−2.22165	−0.324061	−0.162030	+	0.986786i	$0.551804\pi$
$48$	0	0
$49$	1.56778	0.223969
$50$	0	0
$51$	−3.91520	−0.548238
$52$	0	0
$53$	10.5783	1.45304	0.726519	−	0.687147i	$-0.241137\pi$
$53$	10.5783	1.45304	0.726519	+	0.687147i	$0.241137\pi$
$54$	0	0
$55$	0.0293199	0.00395349
$56$	0	0
$57$	5.81055	0.769627
$58$	0	0
$59$	−1.88853	−0.245866	−0.122933	−	0.992415i	$-0.539230\pi$
$59$	−1.88853	−0.245866	−0.122933	+	0.992415i	$0.539230\pi$
$60$	0	0
$61$	−9.31996	−1.19330	−0.596649	−	0.802502i	$-0.703502\pi$
$61$	−9.31996	−1.19330	−0.596649	+	0.802502i	$0.703502\pi$
$62$	0	0
$63$	−2.92708	−0.368777
$64$	0	0
$65$	0.564760	0.0700499
$66$	0	0
$67$	−7.04904	−0.861177	−0.430588	−	0.902548i	$-0.641694\pi$
$67$	−7.04904	−0.861177	−0.430588	+	0.902548i	$0.641694\pi$
$68$	0	0
$69$	2.35674	0.283719
$70$	0	0
$71$	−3.10327	−0.368290	−0.184145	−	0.982899i	$-0.558952\pi$
$71$	−3.10327	−0.368290	−0.184145	+	0.982899i	$0.558952\pi$
$72$	0	0
$73$	−11.0274	−1.29066	−0.645331	−	0.763903i	$-0.723280\pi$
$73$	−11.0274	−1.29066	−0.645331	+	0.763903i	$0.723280\pi$
$74$	0	0
$75$	−4.97068	−0.573965
$76$	0	0
$77$	0.501205	0.0571176
$78$	0	0
$79$	5.12636	0.576761	0.288381	−	0.957516i	$-0.406883\pi$
$79$	5.12636	0.576761	0.288381	+	0.957516i	$0.406883\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.60127	0.614820	0.307410	−	0.951577i	$-0.400538\pi$
$83$	5.60127	0.614820	0.307410	+	0.951577i	$0.400538\pi$
$84$	0	0
$85$	0.670402	0.0727153
$86$	0	0
$87$	8.91568	0.955861
$88$	0	0
$89$	−13.0682	−1.38523	−0.692614	−	0.721308i	$-0.743541\pi$
$89$	−13.0682	−1.38523	−0.692614	+	0.721308i	$0.743541\pi$
$90$	0	0
$91$	9.65422	1.01204
$92$	0	0
$93$	1.15936	0.120220
$94$	0	0
$95$	−0.994944	−0.102079
$96$	0	0
$97$	−13.2166	−1.34194	−0.670970	−	0.741485i	$-0.734122\pi$
$97$	−13.2166	−1.34194	−0.670970	+	0.741485i	$0.734122\pi$
$98$	0	0
$99$	−0.171230	−0.0172093
$100$	0	0
$101$	−9.31616	−0.926993	−0.463497	−	0.886099i	$-0.653405\pi$
$101$	−9.31616	−0.926993	−0.463497	+	0.886099i	$0.653405\pi$
$102$	0	0
$103$	3.73046	0.367573	0.183787	−	0.982966i	$-0.441164\pi$
$103$	3.73046	0.367573	0.183787	+	0.982966i	$0.441164\pi$
$104$	0	0
$105$	0.501205	0.0489126
$106$	0	0
$107$	−8.04546	−0.777784	−0.388892	−	0.921283i	$-0.627142\pi$
$107$	−8.04546	−0.777784	−0.388892	+	0.921283i	$0.627142\pi$
$108$	0	0
$109$	−12.2314	−1.17155	−0.585776	−	0.810473i	$-0.699210\pi$
$109$	−12.2314	−1.17155	−0.585776	+	0.810473i	$0.699210\pi$
$110$	0	0
$111$	6.88598	0.653588
$112$	0	0
$113$	−7.18970	−0.676350	−0.338175	−	0.941083i	$-0.609810\pi$
$113$	−7.18970	−0.676350	−0.338175	+	0.941083i	$0.609810\pi$
$114$	0	0
$115$	−0.403546	−0.0376309
$116$	0	0
$117$	−3.29824	−0.304923
$118$	0	0
$119$	11.4601	1.05055
$120$	0	0
$121$	−10.9707	−0.997335
$122$	0	0
$123$	−6.93593	−0.625391
$124$	0	0
$125$	1.70728	0.152704
$126$	0	0
$127$	−10.3571	−0.919044	−0.459522	−	0.888166i	$-0.651979\pi$
$127$	−10.3571	−0.919044	−0.459522	+	0.888166i	$0.651979\pi$
$128$	0	0
$129$	11.1872	0.984982
$130$	0	0
$131$	−8.29366	−0.724621	−0.362310	−	0.932058i	$-0.618012\pi$
$131$	−8.29366	−0.724621	−0.362310	+	0.932058i	$0.618012\pi$
$132$	0	0
$133$	−17.0079	−1.47478
$134$	0	0
$135$	−0.171230	−0.0147372
$136$	0	0
$137$	−15.6198	−1.33449	−0.667247	−	0.744837i	$-0.732527\pi$
$137$	−15.6198	−1.33449	−0.667247	+	0.744837i	$0.732527\pi$
$138$	0	0
$139$	−12.0894	−1.02541	−0.512705	−	0.858565i	$-0.671357\pi$
$139$	−12.0894	−1.02541	−0.512705	+	0.858565i	$0.671357\pi$
$140$	0	0
$141$	−2.22165	−0.187097
$142$	0	0
$143$	0.564760	0.0472276
$144$	0	0
$145$	−1.52664	−0.126780
$146$	0	0
$147$	1.56778	0.129309
$148$	0	0
$149$	2.07824	0.170256	0.0851278	−	0.996370i	$-0.472870\pi$
$149$	2.07824	0.170256	0.0851278	+	0.996370i	$0.472870\pi$
$150$	0	0
$151$	−13.7897	−1.12219	−0.561094	−	0.827752i	$-0.689620\pi$
$151$	−13.7897	−1.12219	−0.561094	+	0.827752i	$0.689620\pi$
$152$	0	0
$153$	−3.91520	−0.316526
$154$	0	0
$155$	−0.198517	−0.0159453
$156$	0	0
$157$	6.05129	0.482945	0.241473	−	0.970408i	$-0.422370\pi$
$157$	6.05129	0.482945	0.241473	+	0.970408i	$0.422370\pi$
$158$	0	0
$159$	10.5783	0.838912
$160$	0	0
$161$	−6.89837	−0.543668
$162$	0	0
$163$	2.43528	0.190746	0.0953728	−	0.995442i	$-0.469596\pi$
$163$	2.43528	0.190746	0.0953728	+	0.995442i	$0.469596\pi$
$164$	0	0
$165$	0.0293199	0.00228255
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−2.12158	−0.163199
$170$	0	0
$171$	5.81055	0.444344
$172$	0	0
$173$	4.68421	0.356134	0.178067	−	0.984018i	$-0.443016\pi$
$173$	4.68421	0.356134	0.178067	+	0.984018i	$0.443016\pi$
$174$	0	0
$175$	14.5496	1.09984
$176$	0	0
$177$	−1.88853	−0.141951
$178$	0	0
$179$	−6.53179	−0.488209	−0.244104	−	0.969749i	$-0.578494\pi$
$179$	−6.53179	−0.488209	−0.244104	+	0.969749i	$0.578494\pi$
$180$	0	0
$181$	−9.03552	−0.671605	−0.335803	−	0.941932i	$-0.609007\pi$
$181$	−9.03552	−0.671605	−0.335803	+	0.941932i	$0.609007\pi$
$182$	0	0
$183$	−9.31996	−0.688951
$184$	0	0
$185$	−1.17909	−0.0866884
$186$	0	0
$187$	0.670402	0.0490247
$188$	0	0
$189$	−2.92708	−0.212914
$190$	0	0
$191$	19.4848	1.40987	0.704935	−	0.709272i	$-0.250976\pi$
$191$	19.4848	1.40987	0.704935	+	0.709272i	$0.250976\pi$
$192$	0	0
$193$	11.5049	0.828139	0.414070	−	0.910245i	$-0.364107\pi$
$193$	11.5049	0.828139	0.414070	+	0.910245i	$0.364107\pi$
$194$	0	0
$195$	0.564760	0.0404433
$196$	0	0
$197$	−13.0763	−0.931644	−0.465822	−	0.884878i	$-0.654241\pi$
$197$	−13.0763	−0.931644	−0.465822	+	0.884878i	$0.654241\pi$
$198$	0	0
$199$	18.9013	1.33987	0.669937	−	0.742418i	$-0.266321\pi$
$199$	18.9013	1.33987	0.669937	+	0.742418i	$0.266321\pi$
$200$	0	0
$201$	−7.04904	−0.497201
$202$	0	0
$203$	−26.0969	−1.83164
$204$	0	0
$205$	1.18764	0.0829485
$206$	0	0
$207$	2.35674	0.163805
$208$	0	0
$209$	−0.994944	−0.0688217
$210$	0	0
$211$	−0.276376	−0.0190265	−0.00951324	−	0.999955i	$-0.503028\pi$
$211$	−0.276376	−0.0190265	−0.00951324	+	0.999955i	$0.503028\pi$
$212$	0	0
$213$	−3.10327	−0.212632
$214$	0	0
$215$	−1.91560	−0.130643
$216$	0	0
$217$	−3.39353	−0.230368
$218$	0	0
$219$	−11.0274	−0.745164
$220$	0	0
$221$	12.9133	0.868643
$222$	0	0
$223$	−11.1248	−0.744974	−0.372487	−	0.928037i	$-0.621495\pi$
$223$	−11.1248	−0.744974	−0.372487	+	0.928037i	$0.621495\pi$
$224$	0	0
$225$	−4.97068	−0.331379
$226$	0	0
$227$	−13.5523	−0.899496	−0.449748	−	0.893155i	$-0.648486\pi$
$227$	−13.5523	−0.899496	−0.449748	+	0.893155i	$0.648486\pi$
$228$	0	0
$229$	−5.31695	−0.351354	−0.175677	−	0.984448i	$-0.556211\pi$
$229$	−5.31695	−0.351354	−0.175677	+	0.984448i	$0.556211\pi$
$230$	0	0
$231$	0.501205	0.0329769
$232$	0	0
$233$	23.9680	1.57020	0.785098	−	0.619371i	$-0.212612\pi$
$233$	23.9680	1.57020	0.785098	+	0.619371i	$0.212612\pi$
$234$	0	0
$235$	0.380414	0.0248155
$236$	0	0
$237$	5.12636	0.332993
$238$	0	0
$239$	−25.2667	−1.63437	−0.817184	−	0.576376i	$-0.804466\pi$
$239$	−25.2667	−1.63437	−0.817184	+	0.576376i	$0.804466\pi$
$240$	0	0
$241$	−12.3538	−0.795780	−0.397890	−	0.917433i	$-0.630257\pi$
$241$	−12.3538	−0.795780	−0.397890	+	0.917433i	$0.630257\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−0.268452	−0.0171508
$246$	0	0
$247$	−19.1646	−1.21942
$248$	0	0
$249$	5.60127	0.354966
$250$	0	0
$251$	−11.7102	−0.739141	−0.369570	−	0.929203i	$-0.620495\pi$
$251$	−11.7102	−0.739141	−0.369570	+	0.929203i	$0.620495\pi$
$252$	0	0
$253$	−0.403546	−0.0253707
$254$	0	0
$255$	0.670402	0.0419822
$256$	0	0
$257$	−23.3845	−1.45868	−0.729342	−	0.684149i	$-0.760174\pi$
$257$	−23.3845	−1.45868	−0.729342	+	0.684149i	$0.760174\pi$
$258$	0	0
$259$	−20.1558	−1.25242
$260$	0	0
$261$	8.91568	0.551867
$262$	0	0
$263$	23.2296	1.43240	0.716201	−	0.697894i	$-0.245880\pi$
$263$	23.2296	1.43240	0.716201	+	0.697894i	$0.245880\pi$
$264$	0	0
$265$	−1.81132	−0.111269
$266$	0	0
$267$	−13.0682	−0.799762
$268$	0	0
$269$	18.4648	1.12582	0.562910	−	0.826518i	$-0.309682\pi$
$269$	18.4648	1.12582	0.562910	+	0.826518i	$0.309682\pi$
$270$	0	0
$271$	4.12219	0.250406	0.125203	−	0.992131i	$-0.460042\pi$
$271$	4.12219	0.250406	0.125203	+	0.992131i	$0.460042\pi$
$272$	0	0
$273$	9.65422	0.584300
$274$	0	0
$275$	0.851132	0.0513252
$276$	0	0
$277$	13.6333	0.819143	0.409571	−	0.912278i	$-0.365678\pi$
$277$	13.6333	0.819143	0.409571	+	0.912278i	$0.365678\pi$
$278$	0	0
$279$	1.15936	0.0694089
$280$	0	0
$281$	−7.52562	−0.448941	−0.224471	−	0.974481i	$-0.572065\pi$
$281$	−7.52562	−0.448941	−0.224471	+	0.974481i	$0.572065\pi$
$282$	0	0
$283$	25.7323	1.52963	0.764813	−	0.644252i	$-0.222831\pi$
$283$	25.7323	1.52963	0.764813	+	0.644252i	$0.222831\pi$
$284$	0	0
$285$	−0.994944	−0.0589354
$286$	0	0
$287$	20.3020	1.19839
$288$	0	0
$289$	−1.67117	−0.0983042
$290$	0	0
$291$	−13.2166	−0.774769
$292$	0	0
$293$	7.13069	0.416579	0.208290	−	0.978067i	$-0.433210\pi$
$293$	7.13069	0.416579	0.208290	+	0.978067i	$0.433210\pi$
$294$	0	0
$295$	0.323374	0.0188276
$296$	0	0
$297$	−0.171230	−0.00993580
$298$	0	0
$299$	−7.77312	−0.449531
$300$	0	0
$301$	−32.7459	−1.88744
$302$	0	0
$303$	−9.31616	−0.535200
$304$	0	0
$305$	1.59586	0.0913787
$306$	0	0
$307$	−27.5691	−1.57345	−0.786725	−	0.617303i	$-0.788225\pi$
$307$	−27.5691	−1.57345	−0.786725	+	0.617303i	$0.788225\pi$
$308$	0	0
$309$	3.73046	0.212219
$310$	0	0
$311$	−30.4457	−1.72642	−0.863208	−	0.504848i	$-0.831548\pi$
$311$	−30.4457	−1.72642	−0.863208	+	0.504848i	$0.831548\pi$
$312$	0	0
$313$	15.8279	0.894643	0.447322	−	0.894373i	$-0.352378\pi$
$313$	15.8279	0.894643	0.447322	+	0.894373i	$0.352378\pi$
$314$	0	0
$315$	0.501205	0.0282397
$316$	0	0
$317$	−17.2233	−0.967359	−0.483680	−	0.875245i	$-0.660700\pi$
$317$	−17.2233	−0.967359	−0.483680	+	0.875245i	$0.660700\pi$
$318$	0	0
$319$	−1.52664	−0.0854752
$320$	0	0
$321$	−8.04546	−0.449054
$322$	0	0
$323$	−22.7495	−1.26582
$324$	0	0
$325$	16.3945	0.909404
$326$	0	0
$327$	−12.2314	−0.676396
$328$	0	0
$329$	6.50294	0.358519
$330$	0	0
$331$	22.3052	1.22600	0.613002	−	0.790081i	$-0.289962\pi$
$331$	22.3052	1.22600	0.613002	+	0.790081i	$0.289962\pi$
$332$	0	0
$333$	6.88598	0.377349
$334$	0	0
$335$	1.20701	0.0659460
$336$	0	0
$337$	3.90740	0.212850	0.106425	−	0.994321i	$-0.466060\pi$
$337$	3.90740	0.212850	0.106425	+	0.994321i	$0.466060\pi$
$338$	0	0
$339$	−7.18970	−0.390491
$340$	0	0
$341$	−0.198517	−0.0107503
$342$	0	0
$343$	15.9005	0.858547
$344$	0	0
$345$	−0.403546	−0.0217262
$346$	0	0
$347$	−0.363481	−0.0195127	−0.00975633	−	0.999952i	$-0.503106\pi$
$347$	−0.363481	−0.0195127	−0.00975633	+	0.999952i	$0.503106\pi$
$348$	0	0
$349$	29.9181	1.60148	0.800740	−	0.599012i	$-0.204440\pi$
$349$	29.9181	1.60148	0.800740	+	0.599012i	$0.204440\pi$
$350$	0	0
$351$	−3.29824	−0.176047
$352$	0	0
$353$	−6.51556	−0.346788	−0.173394	−	0.984853i	$-0.555473\pi$
$353$	−6.51556	−0.346788	−0.173394	+	0.984853i	$0.555473\pi$
$354$	0	0
$355$	0.531374	0.0282024
$356$	0	0
$357$	11.4601	0.606533
$358$	0	0
$359$	33.4888	1.76747	0.883736	−	0.467985i	$-0.155020\pi$
$359$	33.4888	1.76747	0.883736	+	0.467985i	$0.155020\pi$
$360$	0	0
$361$	14.7625	0.776975
$362$	0	0
$363$	−10.9707	−0.575811
$364$	0	0
$365$	1.88823	0.0988345
$366$	0	0
$367$	16.4075	0.856465	0.428233	−	0.903669i	$-0.359136\pi$
$367$	16.4075	0.856465	0.428233	+	0.903669i	$0.359136\pi$
$368$	0	0
$369$	−6.93593	−0.361070
$370$	0	0
$371$	−30.9634	−1.60754
$372$	0	0
$373$	11.7315	0.607434	0.303717	−	0.952762i	$-0.401772\pi$
$373$	11.7315	0.607434	0.303717	+	0.952762i	$0.401772\pi$
$374$	0	0
$375$	1.70728	0.0881638
$376$	0	0
$377$	−29.4061	−1.51449
$378$	0	0
$379$	26.0199	1.33655	0.668275	−	0.743914i	$-0.267033\pi$
$379$	26.0199	1.33655	0.668275	+	0.743914i	$0.267033\pi$
$380$	0	0
$381$	−10.3571	−0.530610
$382$	0	0
$383$	−37.3246	−1.90720	−0.953599	−	0.301081i	$-0.902653\pi$
$383$	−37.3246	−1.90720	−0.953599	+	0.301081i	$0.902653\pi$
$384$	0	0
$385$	−0.0858215	−0.00437387
$386$	0	0
$387$	11.1872	0.568680
$388$	0	0
$389$	16.5360	0.838410	0.419205	−	0.907892i	$-0.362309\pi$
$389$	16.5360	0.838410	0.419205	+	0.907892i	$0.362309\pi$
$390$	0	0
$391$	−9.22713	−0.466636
$392$	0	0
$393$	−8.29366	−0.418360
$394$	0	0
$395$	−0.877790	−0.0441664
$396$	0	0
$397$	−14.0055	−0.702916	−0.351458	−	0.936204i	$-0.614314\pi$
$397$	−14.0055	−0.702916	−0.351458	+	0.936204i	$0.614314\pi$
$398$	0	0
$399$	−17.0079	−0.851462
$400$	0	0
$401$	−6.97675	−0.348402	−0.174201	−	0.984710i	$-0.555734\pi$
$401$	−6.97675	−0.348402	−0.174201	+	0.984710i	$0.555734\pi$
$402$	0	0
$403$	−3.82385	−0.190479
$404$	0	0
$405$	−0.171230	−0.00850851
$406$	0	0
$407$	−1.17909	−0.0584453
$408$	0	0
$409$	−5.27762	−0.260961	−0.130481	−	0.991451i	$-0.541652\pi$
$409$	−5.27762	−0.260961	−0.130481	+	0.991451i	$0.541652\pi$
$410$	0	0
$411$	−15.6198	−0.770470
$412$	0	0
$413$	5.52788	0.272009
$414$	0	0
$415$	−0.959109	−0.0470808
$416$	0	0
$417$	−12.0894	−0.592021
$418$	0	0
$419$	−1.33784	−0.0653579	−0.0326789	−	0.999466i	$-0.510404\pi$
$419$	−1.33784	−0.0653579	−0.0326789	+	0.999466i	$0.510404\pi$
$420$	0	0
$421$	−27.6500	−1.34758	−0.673790	−	0.738923i	$-0.735335\pi$
$421$	−27.6500	−1.34758	−0.673790	+	0.738923i	$0.735335\pi$
$422$	0	0
$423$	−2.22165	−0.108020
$424$	0	0
$425$	19.4612	0.944008
$426$	0	0
$427$	27.2802	1.32018
$428$	0	0
$429$	0.564760	0.0272669
$430$	0	0
$431$	−4.87617	−0.234877	−0.117438	−	0.993080i	$-0.537468\pi$
$431$	−4.87617	−0.234877	−0.117438	+	0.993080i	$0.537468\pi$
$432$	0	0
$433$	−28.0176	−1.34644	−0.673221	−	0.739441i	$-0.735090\pi$
$433$	−28.0176	−1.34644	−0.673221	+	0.739441i	$0.735090\pi$
$434$	0	0
$435$	−1.52664	−0.0731966
$436$	0	0
$437$	13.6940	0.655072
$438$	0	0
$439$	−14.2469	−0.679967	−0.339984	−	0.940431i	$-0.610422\pi$
$439$	−14.2469	−0.679967	−0.339984	+	0.940431i	$0.610422\pi$
$440$	0	0
$441$	1.56778	0.0746563
$442$	0	0
$443$	22.1345	1.05164	0.525821	−	0.850595i	$-0.323758\pi$
$443$	22.1345	1.05164	0.525821	+	0.850595i	$0.323758\pi$
$444$	0	0
$445$	2.23768	0.106076
$446$	0	0
$447$	2.07824	0.0982971
$448$	0	0
$449$	38.7231	1.82746	0.913728	−	0.406325i	$-0.133190\pi$
$449$	38.7231	1.82746	0.913728	+	0.406325i	$0.133190\pi$
$450$	0	0
$451$	1.18764	0.0559239
$452$	0	0
$453$	−13.7897	−0.647896
$454$	0	0
$455$	−1.65310	−0.0774984
$456$	0	0
$457$	34.0042	1.59065	0.795326	−	0.606183i	$-0.207300\pi$
$457$	34.0042	1.59065	0.795326	+	0.606183i	$0.207300\pi$
$458$	0	0
$459$	−3.91520	−0.182746
$460$	0	0
$461$	6.72427	0.313181	0.156590	−	0.987664i	$-0.449950\pi$
$461$	6.72427	0.313181	0.156590	+	0.987664i	$0.449950\pi$
$462$	0	0
$463$	−9.18126	−0.426689	−0.213345	−	0.976977i	$-0.568436\pi$
$463$	−9.18126	−0.426689	−0.213345	+	0.976977i	$0.568436\pi$
$464$	0	0
$465$	−0.198517	−0.00920602
$466$	0	0
$467$	−35.4583	−1.64081	−0.820406	−	0.571781i	$-0.806253\pi$
$467$	−35.4583	−1.64081	−0.820406	+	0.571781i	$0.806253\pi$
$468$	0	0
$469$	20.6331	0.952747
$470$	0	0
$471$	6.05129	0.278829
$472$	0	0
$473$	−1.91560	−0.0880793
$474$	0	0
$475$	−28.8824	−1.32522
$476$	0	0
$477$	10.5783	0.484346
$478$	0	0
$479$	−14.5746	−0.665930	−0.332965	−	0.942939i	$-0.608049\pi$
$479$	−14.5746	−0.665930	−0.332965	+	0.942939i	$0.608049\pi$
$480$	0	0
$481$	−22.7117	−1.03556
$482$	0	0
$483$	−6.89837	−0.313887
$484$	0	0
$485$	2.26308	0.102761
$486$	0	0
$487$	−1.54755	−0.0701262	−0.0350631	−	0.999385i	$-0.511163\pi$
$487$	−1.54755	−0.0701262	−0.0350631	+	0.999385i	$0.511163\pi$
$488$	0	0
$489$	2.43528	0.110127
$490$	0	0
$491$	−13.2473	−0.597843	−0.298921	−	0.954278i	$-0.596627\pi$
$491$	−13.2473	−0.597843	−0.298921	+	0.954278i	$0.596627\pi$
$492$	0	0
$493$	−34.9067	−1.57212
$494$	0	0
$495$	0.0293199	0.00131783
$496$	0	0
$497$	9.08351	0.407451
$498$	0	0
$499$	1.13932	0.0510030	0.0255015	−	0.999675i	$-0.491882\pi$
$499$	1.13932	0.0510030	0.0255015	+	0.999675i	$0.491882\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−34.3692	−1.53245	−0.766224	−	0.642574i	$-0.777867\pi$
$503$	−34.3692	−1.53245	−0.766224	+	0.642574i	$0.777867\pi$
$504$	0	0
$505$	1.59521	0.0709860
$506$	0	0
$507$	−2.12158	−0.0942227
$508$	0	0
$509$	−40.7043	−1.80419	−0.902093	−	0.431541i	$-0.857970\pi$
$509$	−40.7043	−1.80419	−0.902093	+	0.431541i	$0.857970\pi$
$510$	0	0
$511$	32.2781	1.42790
$512$	0	0
$513$	5.81055	0.256542
$514$	0	0
$515$	−0.638769	−0.0281475
$516$	0	0
$517$	0.380414	0.0167306
$518$	0	0
$519$	4.68421	0.205614
$520$	0	0
$521$	−9.15675	−0.401164	−0.200582	−	0.979677i	$-0.564283\pi$
$521$	−9.15675	−0.401164	−0.200582	+	0.979677i	$0.564283\pi$
$522$	0	0
$523$	13.9466	0.609844	0.304922	−	0.952377i	$-0.401370\pi$
$523$	13.9466	0.609844	0.304922	+	0.952377i	$0.401370\pi$
$524$	0	0
$525$	14.5496	0.634995
$526$	0	0
$527$	−4.53912	−0.197727
$528$	0	0
$529$	−17.4458	−0.758511
$530$	0	0
$531$	−1.88853	−0.0819553
$532$	0	0
$533$	22.8764	0.990886
$534$	0	0
$535$	1.37763	0.0595600
$536$	0	0
$537$	−6.53179	−0.281867
$538$	0	0
$539$	−0.268452	−0.0115631
$540$	0	0
$541$	−1.88638	−0.0811017	−0.0405509	−	0.999177i	$-0.512911\pi$
$541$	−1.88638	−0.0811017	−0.0405509	+	0.999177i	$0.512911\pi$
$542$	0	0
$543$	−9.03552	−0.387751
$544$	0	0
$545$	2.09438	0.0897135
$546$	0	0
$547$	−2.23617	−0.0956119	−0.0478059	−	0.998857i	$-0.515223\pi$
$547$	−2.23617	−0.0956119	−0.0478059	+	0.998857i	$0.515223\pi$
$548$	0	0
$549$	−9.31996	−0.397766
$550$	0	0
$551$	51.8050	2.20697
$552$	0	0
$553$	−15.0053	−0.638089
$554$	0	0
$555$	−1.17909	−0.0500496
$556$	0	0
$557$	44.5503	1.88766	0.943828	−	0.330437i	$-0.107196\pi$
$557$	44.5503	1.88766	0.943828	+	0.330437i	$0.107196\pi$
$558$	0	0
$559$	−36.8983	−1.56063
$560$	0	0
$561$	0.670402	0.0283044
$562$	0	0
$563$	44.6653	1.88242	0.941210	−	0.337823i	$-0.109691\pi$
$563$	44.6653	1.88242	0.941210	+	0.337823i	$0.109691\pi$
$564$	0	0
$565$	1.23110	0.0517926
$566$	0	0
$567$	−2.92708	−0.122926
$568$	0	0
$569$	25.7100	1.07782	0.538910	−	0.842364i	$-0.318836\pi$
$569$	25.7100	1.07782	0.538910	+	0.842364i	$0.318836\pi$
$570$	0	0
$571$	33.9580	1.42110	0.710550	−	0.703647i	$-0.248446\pi$
$571$	33.9580	1.42110	0.710550	+	0.703647i	$0.248446\pi$
$572$	0	0
$573$	19.4848	0.813989
$574$	0	0
$575$	−11.7146	−0.488533
$576$	0	0
$577$	46.3514	1.92963	0.964817	−	0.262922i	$-0.0846860\pi$
$577$	46.3514	1.92963	0.964817	+	0.262922i	$0.0846860\pi$
$578$	0	0
$579$	11.5049	0.478126
$580$	0	0
$581$	−16.3954	−0.680194
$582$	0	0
$583$	−1.81132	−0.0750173
$584$	0	0
$585$	0.564760	0.0233500
$586$	0	0
$587$	30.0610	1.24075	0.620376	−	0.784304i	$-0.286980\pi$
$587$	30.0610	1.24075	0.620376	+	0.784304i	$0.286980\pi$
$588$	0	0
$589$	6.73651	0.277573
$590$	0	0
$591$	−13.0763	−0.537885
$592$	0	0
$593$	1.11204	0.0456661	0.0228330	−	0.999739i	$-0.492731\pi$
$593$	1.11204	0.0456661	0.0228330	+	0.999739i	$0.492731\pi$
$594$	0	0
$595$	−1.96232	−0.0804473
$596$	0	0
$597$	18.9013	0.773577
$598$	0	0
$599$	−20.6799	−0.844959	−0.422480	−	0.906372i	$-0.638840\pi$
$599$	−20.6799	−0.844959	−0.422480	+	0.906372i	$0.638840\pi$
$600$	0	0
$601$	−10.3801	−0.423415	−0.211707	−	0.977333i	$-0.567902\pi$
$601$	−10.3801	−0.423415	−0.211707	+	0.977333i	$0.567902\pi$
$602$	0	0
$603$	−7.04904	−0.287059
$604$	0	0
$605$	1.87851	0.0763725
$606$	0	0
$607$	1.51460	0.0614758	0.0307379	−	0.999527i	$-0.490214\pi$
$607$	1.51460	0.0614758	0.0307379	+	0.999527i	$0.490214\pi$
$608$	0	0
$609$	−26.0969	−1.05750
$610$	0	0
$611$	7.32754	0.296441
$612$	0	0
$613$	47.5024	1.91860	0.959301	−	0.282385i	$-0.0911256\pi$
$613$	47.5024	1.91860	0.959301	+	0.282385i	$0.0911256\pi$
$614$	0	0
$615$	1.18764	0.0478903
$616$	0	0
$617$	−15.5169	−0.624686	−0.312343	−	0.949969i	$-0.601114\pi$
$617$	−15.5169	−0.624686	−0.312343	+	0.949969i	$0.601114\pi$
$618$	0	0
$619$	−19.3528	−0.777857	−0.388928	−	0.921268i	$-0.627155\pi$
$619$	−19.3528	−0.777857	−0.388928	+	0.921268i	$0.627155\pi$
$620$	0	0
$621$	2.35674	0.0945729
$622$	0	0
$623$	38.2517	1.53252
$624$	0	0
$625$	24.5611	0.982442
$626$	0	0
$627$	−0.994944	−0.0397342
$628$	0	0
$629$	−26.9600	−1.07497
$630$	0	0
$631$	−24.0573	−0.957705	−0.478852	−	0.877895i	$-0.658947\pi$
$631$	−24.0573	−0.957705	−0.478852	+	0.877895i	$0.658947\pi$
$632$	0	0
$633$	−0.276376	−0.0109849
$634$	0	0
$635$	1.77345	0.0703772
$636$	0	0
$637$	−5.17093	−0.204880
$638$	0	0
$639$	−3.10327	−0.122763
$640$	0	0
$641$	7.13349	0.281756	0.140878	−	0.990027i	$-0.455007\pi$
$641$	7.13349	0.281756	0.140878	+	0.990027i	$0.455007\pi$
$642$	0	0
$643$	13.5006	0.532411	0.266205	−	0.963916i	$-0.414230\pi$
$643$	13.5006	0.532411	0.266205	+	0.963916i	$0.414230\pi$
$644$	0	0
$645$	−1.91560	−0.0754266
$646$	0	0
$647$	−46.7829	−1.83923	−0.919613	−	0.392825i	$-0.871498\pi$
$647$	−46.7829	−1.83923	−0.919613	+	0.392825i	$0.871498\pi$
$648$	0	0
$649$	0.323374	0.0126935
$650$	0	0
$651$	−3.39353	−0.133003
$652$	0	0
$653$	−10.3435	−0.404772	−0.202386	−	0.979306i	$-0.564870\pi$
$653$	−10.3435	−0.404772	−0.202386	+	0.979306i	$0.564870\pi$
$654$	0	0
$655$	1.42013	0.0554890
$656$	0	0
$657$	−11.0274	−0.430221
$658$	0	0
$659$	−29.6392	−1.15458	−0.577289	−	0.816540i	$-0.695889\pi$
$659$	−29.6392	−1.15458	−0.577289	+	0.816540i	$0.695889\pi$
$660$	0	0
$661$	4.65561	0.181082	0.0905412	−	0.995893i	$-0.471140\pi$
$661$	4.65561	0.181082	0.0905412	+	0.995893i	$0.471140\pi$
$662$	0	0
$663$	12.9133	0.501511
$664$	0	0
$665$	2.91228	0.112933
$666$	0	0
$667$	21.0120	0.813587
$668$	0	0
$669$	−11.1248	−0.430111
$670$	0	0
$671$	1.59586	0.0616075
$672$	0	0
$673$	33.2261	1.28077	0.640387	−	0.768053i	$-0.278774\pi$
$673$	33.2261	1.28077	0.640387	+	0.768053i	$0.278774\pi$
$674$	0	0
$675$	−4.97068	−0.191322
$676$	0	0
$677$	−35.0466	−1.34695	−0.673475	−	0.739210i	$-0.735199\pi$
$677$	−35.0466	−1.34695	−0.673475	+	0.739210i	$0.735199\pi$
$678$	0	0
$679$	38.6859	1.48463
$680$	0	0
$681$	−13.5523	−0.519324
$682$	0	0
$683$	−3.83934	−0.146908	−0.0734542	−	0.997299i	$-0.523402\pi$
$683$	−3.83934	−0.146908	−0.0734542	+	0.997299i	$0.523402\pi$
$684$	0	0
$685$	2.67459	0.102191
$686$	0	0
$687$	−5.31695	−0.202854
$688$	0	0
$689$	−34.8897	−1.32919
$690$	0	0
$691$	−35.1473	−1.33707	−0.668534	−	0.743682i	$-0.733078\pi$
$691$	−35.1473	−1.33707	−0.668534	+	0.743682i	$0.733078\pi$
$692$	0	0
$693$	0.501205	0.0190392
$694$	0	0
$695$	2.07008	0.0785225
$696$	0	0
$697$	27.1556	1.02859
$698$	0	0
$699$	23.9680	0.906553
$700$	0	0
$701$	−12.0199	−0.453986	−0.226993	−	0.973896i	$-0.572889\pi$
$701$	−12.0199	−0.453986	−0.226993	+	0.973896i	$0.572889\pi$
$702$	0	0
$703$	40.0114	1.50906
$704$	0	0
$705$	0.380414	0.0143272
$706$	0	0
$707$	27.2691	1.02556
$708$	0	0
$709$	−30.8183	−1.15741	−0.578703	−	0.815538i	$-0.696441\pi$
$709$	−30.8183	−1.15741	−0.578703	+	0.815538i	$0.696441\pi$
$710$	0	0
$711$	5.12636	0.192254
$712$	0	0
$713$	2.73231	0.102326
$714$	0	0
$715$	−0.0967041	−0.00361653
$716$	0	0
$717$	−25.2667	−0.943603
$718$	0	0
$719$	−35.2346	−1.31403	−0.657015	−	0.753878i	$-0.728181\pi$
$719$	−35.2346	−1.31403	−0.657015	+	0.753878i	$0.728181\pi$
$720$	0	0
$721$	−10.9194	−0.406658
$722$	0	0
$723$	−12.3538	−0.459444
$724$	0	0
$725$	−44.3170	−1.64589
$726$	0	0
$727$	43.9229	1.62901	0.814505	−	0.580156i	$-0.197009\pi$
$727$	43.9229	1.62901	0.814505	+	0.580156i	$0.197009\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−43.8004	−1.62002
$732$	0	0
$733$	43.1615	1.59421	0.797103	−	0.603844i	$-0.206365\pi$
$733$	43.1615	1.59421	0.797103	+	0.603844i	$0.206365\pi$
$734$	0	0
$735$	−0.268452	−0.00990201
$736$	0	0
$737$	1.20701	0.0444608
$738$	0	0
$739$	−24.8598	−0.914481	−0.457241	−	0.889343i	$-0.651162\pi$
$739$	−24.8598	−0.914481	−0.457241	+	0.889343i	$0.651162\pi$
$740$	0	0
$741$	−19.1646	−0.704030
$742$	0	0
$743$	33.3742	1.22438	0.612191	−	0.790710i	$-0.290288\pi$
$743$	33.3742	1.22438	0.612191	+	0.790710i	$0.290288\pi$
$744$	0	0
$745$	−0.355857	−0.0130376
$746$	0	0
$747$	5.60127	0.204940
$748$	0	0
$749$	23.5497	0.860487
$750$	0	0
$751$	−6.60294	−0.240945	−0.120472	−	0.992717i	$-0.538441\pi$
$751$	−6.60294	−0.240945	−0.120472	+	0.992717i	$0.538441\pi$
$752$	0	0
$753$	−11.7102	−0.426743
$754$	0	0
$755$	2.36121	0.0859334
$756$	0	0
$757$	−2.07219	−0.0753149	−0.0376575	−	0.999291i	$-0.511990\pi$
$757$	−2.07219	−0.0753149	−0.0376575	+	0.999291i	$0.511990\pi$
$758$	0	0
$759$	−0.403546	−0.0146478
$760$	0	0
$761$	−7.37348	−0.267288	−0.133644	−	0.991029i	$-0.542668\pi$
$761$	−7.37348	−0.267288	−0.133644	+	0.991029i	$0.542668\pi$
$762$	0	0
$763$	35.8022	1.29613
$764$	0	0
$765$	0.670402	0.0242384
$766$	0	0
$767$	6.22884	0.224910
$768$	0	0
$769$	0.805300	0.0290399	0.0145199	−	0.999895i	$-0.495378\pi$
$769$	0.805300	0.0290399	0.0145199	+	0.999895i	$0.495378\pi$
$770$	0	0
$771$	−23.3845	−0.842171
$772$	0	0
$773$	8.43414	0.303355	0.151678	−	0.988430i	$-0.451532\pi$
$773$	8.43414	0.303355	0.151678	+	0.988430i	$0.451532\pi$
$774$	0	0
$775$	−5.76280	−0.207006
$776$	0	0
$777$	−20.1558	−0.723085
$778$	0	0
$779$	−40.3016	−1.44395
$780$	0	0
$781$	0.531374	0.0190141
$782$	0	0
$783$	8.91568	0.318620
$784$	0	0
$785$	−1.03617	−0.0369823
$786$	0	0
$787$	−19.7383	−0.703594	−0.351797	−	0.936076i	$-0.614429\pi$
$787$	−19.7383	−0.703594	−0.351797	+	0.936076i	$0.614429\pi$
$788$	0	0
$789$	23.2296	0.826997
$790$	0	0
$791$	21.0448	0.748268
$792$	0	0
$793$	30.7395	1.09159
$794$	0	0
$795$	−1.81132	−0.0642410
$796$	0	0
$797$	−25.9278	−0.918408	−0.459204	−	0.888331i	$-0.651865\pi$
$797$	−25.9278	−0.918408	−0.459204	+	0.888331i	$0.651865\pi$
$798$	0	0
$799$	8.69821	0.307721
$800$	0	0
$801$	−13.0682	−0.461743
$802$	0	0
$803$	1.88823	0.0666342
$804$	0	0
$805$	1.18121	0.0416322
$806$	0	0
$807$	18.4648	0.649993
$808$	0	0
$809$	−29.5937	−1.04046	−0.520229	−	0.854027i	$-0.674153\pi$
$809$	−29.5937	−1.04046	−0.520229	+	0.854027i	$0.674153\pi$
$810$	0	0
$811$	−6.59256	−0.231496	−0.115748	−	0.993279i	$-0.536927\pi$
$811$	−6.59256	−0.231496	−0.115748	+	0.993279i	$0.536927\pi$
$812$	0	0
$813$	4.12219	0.144572
$814$	0	0
$815$	−0.416993	−0.0146066
$816$	0	0
$817$	65.0041	2.27421
$818$	0	0
$819$	9.65422	0.337346
$820$	0	0
$821$	−38.5957	−1.34700	−0.673500	−	0.739188i	$-0.735210\pi$
$821$	−38.5957	−1.34700	−0.673500	+	0.739188i	$0.735210\pi$
$822$	0	0
$823$	29.2512	1.01963	0.509817	−	0.860283i	$-0.329713\pi$
$823$	29.2512	1.01963	0.509817	+	0.860283i	$0.329713\pi$
$824$	0	0
$825$	0.851132	0.0296326
$826$	0	0
$827$	−36.9174	−1.28374	−0.641872	−	0.766811i	$-0.721842\pi$
$827$	−36.9174	−1.28374	−0.641872	+	0.766811i	$0.721842\pi$
$828$	0	0
$829$	−4.73793	−0.164555	−0.0822775	−	0.996609i	$-0.526219\pi$
$829$	−4.73793	−0.164555	−0.0822775	+	0.996609i	$0.526219\pi$
$830$	0	0
$831$	13.6333	0.472932
$832$	0	0
$833$	−6.13819	−0.212676
$834$	0	0
$835$	0.171230	0.00592567
$836$	0	0
$837$	1.15936	0.0400733
$838$	0	0
$839$	33.1457	1.14432	0.572158	−	0.820143i	$-0.306106\pi$
$839$	33.1457	1.14432	0.572158	+	0.820143i	$0.306106\pi$
$840$	0	0
$841$	50.4893	1.74101
$842$	0	0
$843$	−7.52562	−0.259196
$844$	0	0
$845$	0.363279	0.0124972
$846$	0	0
$847$	32.1120	1.10338
$848$	0	0
$849$	25.7323	0.883130
$850$	0	0
$851$	16.2285	0.556306
$852$	0	0
$853$	19.2882	0.660416	0.330208	−	0.943908i	$-0.392881\pi$
$853$	19.2882	0.660416	0.330208	+	0.943908i	$0.392881\pi$
$854$	0	0
$855$	−0.994944	−0.0340264
$856$	0	0
$857$	16.3356	0.558012	0.279006	−	0.960289i	$-0.409995\pi$
$857$	16.3356	0.558012	0.279006	+	0.960289i	$0.409995\pi$
$858$	0	0
$859$	37.9235	1.29393	0.646967	−	0.762518i	$-0.276037\pi$
$859$	37.9235	1.29393	0.646967	+	0.762518i	$0.276037\pi$
$860$	0	0
$861$	20.3020	0.691890
$862$	0	0
$863$	−23.6597	−0.805385	−0.402692	−	0.915335i	$-0.631926\pi$
$863$	−23.6597	−0.805385	−0.402692	+	0.915335i	$0.631926\pi$
$864$	0	0
$865$	−0.802080	−0.0272715
$866$	0	0
$867$	−1.67117	−0.0567560
$868$	0	0
$869$	−0.877790	−0.0297770
$870$	0	0
$871$	23.2494	0.787777
$872$	0	0
$873$	−13.2166	−0.447313
$874$	0	0
$875$	−4.99735	−0.168941
$876$	0	0
$877$	17.8832	0.603871	0.301936	−	0.953328i	$-0.402367\pi$
$877$	17.8832	0.603871	0.301936	+	0.953328i	$0.402367\pi$
$878$	0	0
$879$	7.13069	0.240512
$880$	0	0
$881$	4.78045	0.161058	0.0805288	−	0.996752i	$-0.474339\pi$
$881$	4.78045	0.161058	0.0805288	+	0.996752i	$0.474339\pi$
$882$	0	0
$883$	−38.3981	−1.29220	−0.646100	−	0.763253i	$-0.723601\pi$
$883$	−38.3981	−1.29220	−0.646100	+	0.763253i	$0.723601\pi$
$884$	0	0
$885$	0.323374	0.0108701
$886$	0	0
$887$	−34.1197	−1.14563	−0.572814	−	0.819686i	$-0.694148\pi$
$887$	−34.1197	−1.14563	−0.572814	+	0.819686i	$0.694148\pi$
$888$	0	0
$889$	30.3160	1.01677
$890$	0	0
$891$	−0.171230	−0.00573644
$892$	0	0
$893$	−12.9090	−0.431984
$894$	0	0
$895$	1.11844	0.0373854
$896$	0	0
$897$	−7.77312	−0.259537
$898$	0	0
$899$	10.3365	0.344740
$900$	0	0
$901$	−41.4161	−1.37977
$902$	0	0
$903$	−32.7459	−1.08972
$904$	0	0
$905$	1.54716	0.0514292
$906$	0	0
$907$	25.9706	0.862340	0.431170	−	0.902271i	$-0.358101\pi$
$907$	25.9706	0.862340	0.431170	+	0.902271i	$0.358101\pi$
$908$	0	0
$909$	−9.31616	−0.308998
$910$	0	0
$911$	23.0338	0.763145	0.381572	−	0.924339i	$-0.375383\pi$
$911$	23.0338	0.763145	0.381572	+	0.924339i	$0.375383\pi$
$912$	0	0
$913$	−0.959109	−0.0317419
$914$	0	0
$915$	1.59586	0.0527575
$916$	0	0
$917$	24.2762	0.801670
$918$	0	0
$919$	47.0032	1.55049	0.775246	−	0.631659i	$-0.217626\pi$
$919$	47.0032	1.55049	0.775246	+	0.631659i	$0.217626\pi$
$920$	0	0
$921$	−27.5691	−0.908432
$922$	0	0
$923$	10.2353	0.336900
$924$	0	0
$925$	−34.2280	−1.12541
$926$	0	0
$927$	3.73046	0.122524
$928$	0	0
$929$	−44.1553	−1.44869	−0.724344	−	0.689439i	$-0.757857\pi$
$929$	−44.1553	−1.44869	−0.724344	+	0.689439i	$0.757857\pi$
$930$	0	0
$931$	9.10969	0.298558
$932$	0	0
$933$	−30.4457	−0.996747
$934$	0	0
$935$	−0.114793	−0.00375414
$936$	0	0
$937$	12.9269	0.422305	0.211152	−	0.977453i	$-0.432278\pi$
$937$	12.9269	0.422305	0.211152	+	0.977453i	$0.432278\pi$
$938$	0	0
$939$	15.8279	0.516523
$940$	0	0
$941$	−49.8258	−1.62427	−0.812137	−	0.583467i	$-0.801696\pi$
$941$	−49.8258	−1.62427	−0.812137	+	0.583467i	$0.801696\pi$
$942$	0	0
$943$	−16.3462	−0.532306
$944$	0	0
$945$	0.501205	0.0163042
$946$	0	0
$947$	25.5458	0.830128	0.415064	−	0.909792i	$-0.363759\pi$
$947$	25.5458	0.830128	0.415064	+	0.909792i	$0.363759\pi$
$948$	0	0
$949$	36.3711	1.18066
$950$	0	0
$951$	−17.2233	−0.558505
$952$	0	0
$953$	−48.6379	−1.57553	−0.787767	−	0.615973i	$-0.788763\pi$
$953$	−48.6379	−1.57553	−0.787767	+	0.615973i	$0.788763\pi$
$954$	0	0
$955$	−3.33639	−0.107963
$956$	0	0
$957$	−1.52664	−0.0493491
$958$	0	0
$959$	45.7205	1.47639
$960$	0	0
$961$	−29.6559	−0.956642
$962$	0	0
$963$	−8.04546	−0.259261
$964$	0	0
$965$	−1.96999	−0.0634161
$966$	0	0
$967$	−0.339589	−0.0109204	−0.00546022	−	0.999985i	$-0.501738\pi$
$967$	−0.339589	−0.0109204	−0.00546022	+	0.999985i	$0.501738\pi$
$968$	0	0
$969$	−22.7495	−0.730819
$970$	0	0
$971$	15.0126	0.481777	0.240888	−	0.970553i	$-0.422561\pi$
$971$	15.0126	0.481777	0.240888	+	0.970553i	$0.422561\pi$
$972$	0	0
$973$	35.3867	1.13444
$974$	0	0
$975$	16.3945	0.525045
$976$	0	0
$977$	−26.2407	−0.839515	−0.419758	−	0.907636i	$-0.637885\pi$
$977$	−26.2407	−0.839515	−0.419758	+	0.907636i	$0.637885\pi$
$978$	0	0
$979$	2.23768	0.0715165
$980$	0	0
$981$	−12.2314	−0.390518
$982$	0	0
$983$	−20.5220	−0.654551	−0.327275	−	0.944929i	$-0.606130\pi$
$983$	−20.5220	−0.654551	−0.327275	+	0.944929i	$0.606130\pi$
$984$	0	0
$985$	2.23905	0.0713422
$986$	0	0
$987$	6.50294	0.206991
$988$	0	0
$989$	26.3655	0.838373
$990$	0	0
$991$	11.1930	0.355559	0.177779	−	0.984070i	$-0.443109\pi$
$991$	11.1930	0.355559	0.177779	+	0.984070i	$0.443109\pi$
$992$	0	0
$993$	22.3052	0.707834
$994$	0	0
$995$	−3.23647	−0.102603
$996$	0	0
$997$	−18.6207	−0.589724	−0.294862	−	0.955540i	$-0.595274\pi$
$997$	−18.6207	−0.589724	−0.294862	+	0.955540i	$0.595274\pi$
$998$	0	0
$999$	6.88598	0.217863

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.t.1.3		5
4.3	odd	2		2004.2.a.a.1.3	✓	5
12.11	even	2		6012.2.a.e.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.a.1.3	✓	5	4.3	odd	2
6012.2.a.e.1.3		5	12.11	even	2
8016.2.a.t.1.3		5	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.171230 q^{5} -2.92708 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.171230 q^{5} -2.92708 q^{7} +1.00000 q^{9} -0.171230 q^{11} -3.29824 q^{13} -0.171230 q^{15} -3.91520 q^{17} +5.81055 q^{19} -2.92708 q^{21} +2.35674 q^{23} -4.97068 q^{25} +1.00000 q^{27} +8.91568 q^{29} +1.15936 q^{31} -0.171230 q^{33} +0.501205 q^{35} +6.88598 q^{37} -3.29824 q^{39} -6.93593 q^{41} +11.1872 q^{43} -0.171230 q^{45} -2.22165 q^{47} +1.56778 q^{49} -3.91520 q^{51} +10.5783 q^{53} +0.0293199 q^{55} +5.81055 q^{57} -1.88853 q^{59} -9.31996 q^{61} -2.92708 q^{63} +0.564760 q^{65} -7.04904 q^{67} +2.35674 q^{69} -3.10327 q^{71} -11.0274 q^{73} -4.97068 q^{75} +0.501205 q^{77} +5.12636 q^{79} +1.00000 q^{81} +5.60127 q^{83} +0.670402 q^{85} +8.91568 q^{87} -13.0682 q^{89} +9.65422 q^{91} +1.15936 q^{93} -0.994944 q^{95} -13.2166 q^{97} -0.171230 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9	\( 5 q + 5 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9} - 3 q^{11} - 4 q^{13} - 3 q^{15} - 7 q^{17} + 2 q^{19} + 2 q^{21} - 13 q^{23} - 2 q^{25} + 5 q^{27} - 3 q^{29} + 12 q^{31} - 3 q^{33} - 10 q^{35} - 7 q^{37} - 4 q^{39} - 16 q^{41} - 3 q^{45} - q^{47} - 17 q^{49} - 7 q^{51} + 3 q^{53} + 23 q^{55} + 2 q^{57} - q^{59} - 22 q^{61} + 2 q^{63} - 20 q^{65} - 2 q^{67} - 13 q^{69} - 9 q^{71} - 28 q^{73} - 2 q^{75} - 10 q^{77} + 28 q^{79} + 5 q^{81} - 7 q^{83} - 11 q^{85} - 3 q^{87} - 30 q^{89} + 13 q^{91} + 12 q^{93} - 3 q^{95} - 33 q^{97} - 3 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9 - 3 * q^11 - 4 * q^13 - 3 * q^15 - 7 * q^17 + 2 * q^19 + 2 * q^21 - 13 * q^23 - 2 * q^25 + 5 * q^27 - 3 * q^29 + 12 * q^31 - 3 * q^33 - 10 * q^35 - 7 * q^37 - 4 * q^39 - 16 * q^41 - 3 * q^45 - q^47 - 17 * q^49 - 7 * q^51 + 3 * q^53 + 23 * q^55 + 2 * q^57 - q^59 - 22 * q^61 + 2 * q^63 - 20 * q^65 - 2 * q^67 - 13 * q^69 - 9 * q^71 - 28 * q^73 - 2 * q^75 - 10 * q^77 + 28 * q^79 + 5 * q^81 - 7 * q^83 - 11 * q^85 - 3 * q^87 - 30 * q^89 + 13 * q^91 + 12 * q^93 - 3 * q^95 - 33 * q^97 - 3 * q^99

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