LMFDB - Embedded newform 8016.2.a.v.1.5

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:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 20x^{5} + 2x^{4} + 87x^{3} + 46x^{2} - 48x - 2 $ x^7 - 2x^6 - 20x^5 + 2x^4 + 87x^3 + 46x^2 - 48x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.55552$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +2.55552 q^{5} -3.69934 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.55552 q^{5} -3.69934 q^{7} +1.00000 q^{9} +5.01712 q^{11} +5.57468 q^{13} -2.55552 q^{15} -1.33774 q^{17} +8.44199 q^{19} +3.69934 q^{21} +6.52392 q^{23} +1.53067 q^{25} -1.00000 q^{27} -10.1039 q^{29} +2.82458 q^{31} -5.01712 q^{33} -9.45374 q^{35} +6.51870 q^{37} -5.57468 q^{39} +7.07422 q^{41} -4.12532 q^{43} +2.55552 q^{45} +13.4216 q^{47} +6.68514 q^{49} +1.33774 q^{51} +4.54132 q^{53} +12.8213 q^{55} -8.44199 q^{57} -8.25552 q^{59} -5.80421 q^{61} -3.69934 q^{63} +14.2462 q^{65} +7.81873 q^{67} -6.52392 q^{69} -9.13517 q^{71} -6.04843 q^{73} -1.53067 q^{75} -18.5600 q^{77} -6.04675 q^{79} +1.00000 q^{81} -3.09196 q^{83} -3.41862 q^{85} +10.1039 q^{87} +16.0433 q^{89} -20.6227 q^{91} -2.82458 q^{93} +21.5737 q^{95} -0.668029 q^{97} +5.01712 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{3} + 5 q^{5} - 7 q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q - 7 * q^3 + 5 * q^5 - 7 * q^7 + 7 * q^9	$ 7 q - 7 q^{3} + 5 q^{5} - 7 q^{7} + 7 q^{9} + 6 q^{13} - 5 q^{15} + 6 q^{17} + 2 q^{19} + 7 q^{21} + 12 q^{25} - 7 q^{27} - 4 q^{29} - 7 q^{31} + 13 q^{35} - 3 q^{37} - 6 q^{39} - 12 q^{41} + 2 q^{43} + 5 q^{45} + 11 q^{47} + 10 q^{49} - 6 q^{51} + q^{53} + 2 q^{55} - 2 q^{57} + 19 q^{59} + 12 q^{61} - 7 q^{63} - 10 q^{65} + 17 q^{67} + 20 q^{71} + 10 q^{73} - 12 q^{75} - 24 q^{77} - 2 q^{79} + 7 q^{81} + 7 q^{83} - 18 q^{85} + 4 q^{87} - 3 q^{89} - 4 q^{91} + 7 q^{93} + 24 q^{95} - 3 q^{97}+O(q^{100}) $ 7 * q - 7 * q^3 + 5 * q^5 - 7 * q^7 + 7 * q^9 + 6 * q^13 - 5 * q^15 + 6 * q^17 + 2 * q^19 + 7 * q^21 + 12 * q^25 - 7 * q^27 - 4 * q^29 - 7 * q^31 + 13 * q^35 - 3 * q^37 - 6 * q^39 - 12 * q^41 + 2 * q^43 + 5 * q^45 + 11 * q^47 + 10 * q^49 - 6 * q^51 + q^53 + 2 * q^55 - 2 * q^57 + 19 * q^59 + 12 * q^61 - 7 * q^63 - 10 * q^65 + 17 * q^67 + 20 * q^71 + 10 * q^73 - 12 * q^75 - 24 * q^77 - 2 * q^79 + 7 * q^81 + 7 * q^83 - 18 * q^85 + 4 * q^87 - 3 * q^89 - 4 * q^91 + 7 * q^93 + 24 * q^95 - 3 * q^97

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
$5$	2.55552	1.14286	0.571431	−	0.820650i	$-0.306388\pi$
$5$	2.55552	1.14286	0.571431	+	0.820650i	$0.306388\pi$
$6$	0	0
$7$	−3.69934	−1.39822	−0.699110	−	0.715014i	$-0.746420\pi$
$7$	−3.69934	−1.39822	−0.699110	+	0.715014i	$0.746420\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.01712	1.51272	0.756359	−	0.654157i	$-0.226977\pi$
$11$	5.01712	1.51272	0.756359	+	0.654157i	$0.226977\pi$
$12$	0	0
$13$	5.57468	1.54614	0.773069	−	0.634322i	$-0.218720\pi$
$13$	5.57468	1.54614	0.773069	+	0.634322i	$0.218720\pi$
$14$	0	0
$15$	−2.55552	−0.659832
$16$	0	0
$17$	−1.33774	−0.324449	−0.162225	−	0.986754i	$-0.551867\pi$
$17$	−1.33774	−0.324449	−0.162225	+	0.986754i	$0.551867\pi$
$18$	0	0
$19$	8.44199	1.93672	0.968362	−	0.249548i	$-0.0802822\pi$
$19$	8.44199	1.93672	0.968362	+	0.249548i	$0.0802822\pi$
$20$	0	0
$21$	3.69934	0.807263
$22$	0	0
$23$	6.52392	1.36033	0.680166	−	0.733058i	$-0.261908\pi$
$23$	6.52392	1.36033	0.680166	+	0.733058i	$0.261908\pi$
$24$	0	0
$25$	1.53067	0.306135
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−10.1039	−1.87624	−0.938119	−	0.346313i	$-0.887433\pi$
$29$	−10.1039	−1.87624	−0.938119	+	0.346313i	$0.887433\pi$
$30$	0	0
$31$	2.82458	0.507309	0.253655	−	0.967295i	$-0.418367\pi$
$31$	2.82458	0.507309	0.253655	+	0.967295i	$0.418367\pi$
$32$	0	0
$33$	−5.01712	−0.873368
$34$	0	0
$35$	−9.45374	−1.59797
$36$	0	0
$37$	6.51870	1.07167	0.535834	−	0.844324i	$-0.319997\pi$
$37$	6.51870	1.07167	0.535834	+	0.844324i	$0.319997\pi$
$38$	0	0
$39$	−5.57468	−0.892664
$40$	0	0
$41$	7.07422	1.10481	0.552404	−	0.833577i	$-0.313711\pi$
$41$	7.07422	1.10481	0.552404	+	0.833577i	$0.313711\pi$
$42$	0	0
$43$	−4.12532	−0.629106	−0.314553	−	0.949240i	$-0.601855\pi$
$43$	−4.12532	−0.629106	−0.314553	+	0.949240i	$0.601855\pi$
$44$	0	0
$45$	2.55552	0.380954
$46$	0	0
$47$	13.4216	1.95774	0.978872	−	0.204472i	$-0.0655477\pi$
$47$	13.4216	1.95774	0.978872	+	0.204472i	$0.0655477\pi$
$48$	0	0
$49$	6.68514	0.955021
$50$	0	0
$51$	1.33774	0.187321
$52$	0	0
$53$	4.54132	0.623798	0.311899	−	0.950115i	$-0.399035\pi$
$53$	4.54132	0.623798	0.311899	+	0.950115i	$0.399035\pi$
$54$	0	0
$55$	12.8213	1.72883
$56$	0	0
$57$	−8.44199	−1.11817
$58$	0	0
$59$	−8.25552	−1.07478	−0.537389	−	0.843335i	$-0.680589\pi$
$59$	−8.25552	−1.07478	−0.537389	+	0.843335i	$0.680589\pi$
$60$	0	0
$61$	−5.80421	−0.743154	−0.371577	−	0.928402i	$-0.621183\pi$
$61$	−5.80421	−0.743154	−0.371577	+	0.928402i	$0.621183\pi$
$62$	0	0
$63$	−3.69934	−0.466074
$64$	0	0
$65$	14.2462	1.76702
$66$	0	0
$67$	7.81873	0.955210	0.477605	−	0.878575i	$-0.341505\pi$
$67$	7.81873	0.955210	0.477605	+	0.878575i	$0.341505\pi$
$68$	0	0
$69$	−6.52392	−0.785388
$70$	0	0
$71$	−9.13517	−1.08414	−0.542072	−	0.840332i	$-0.682360\pi$
$71$	−9.13517	−1.08414	−0.542072	+	0.840332i	$0.682360\pi$
$72$	0	0
$73$	−6.04843	−0.707915	−0.353958	−	0.935261i	$-0.615164\pi$
$73$	−6.04843	−0.707915	−0.353958	+	0.935261i	$0.615164\pi$
$74$	0	0
$75$	−1.53067	−0.176747
$76$	0	0
$77$	−18.5600	−2.11511
$78$	0	0
$79$	−6.04675	−0.680312	−0.340156	−	0.940369i	$-0.610480\pi$
$79$	−6.04675	−0.680312	−0.340156	+	0.940369i	$0.610480\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.09196	−0.339387	−0.169693	−	0.985497i	$-0.554278\pi$
$83$	−3.09196	−0.339387	−0.169693	+	0.985497i	$0.554278\pi$
$84$	0	0
$85$	−3.41862	−0.370801
$86$	0	0
$87$	10.1039	1.08325
$88$	0	0
$89$	16.0433	1.70059	0.850294	−	0.526308i	$-0.176424\pi$
$89$	16.0433	1.70059	0.850294	+	0.526308i	$0.176424\pi$
$90$	0	0
$91$	−20.6227	−2.16184
$92$	0	0
$93$	−2.82458	−0.292895
$94$	0	0
$95$	21.5737	2.21341
$96$	0	0
$97$	−0.668029	−0.0678281	−0.0339140	−	0.999425i	$-0.510797\pi$
$97$	−0.668029	−0.0678281	−0.0339140	+	0.999425i	$0.510797\pi$
$98$	0	0
$99$	5.01712	0.504239
$100$	0	0
$101$	3.01681	0.300184	0.150092	−	0.988672i	$-0.452043\pi$
$101$	3.01681	0.300184	0.150092	+	0.988672i	$0.452043\pi$
$102$	0	0
$103$	−11.4959	−1.13273	−0.566363	−	0.824156i	$-0.691650\pi$
$103$	−11.4959	−1.13273	−0.566363	+	0.824156i	$0.691650\pi$
$104$	0	0
$105$	9.45374	0.922591
$106$	0	0
$107$	−7.85487	−0.759359	−0.379680	−	0.925118i	$-0.623966\pi$
$107$	−7.85487	−0.759359	−0.379680	+	0.925118i	$0.623966\pi$
$108$	0	0
$109$	9.89827	0.948083	0.474041	−	0.880503i	$-0.342795\pi$
$109$	9.89827	0.948083	0.474041	+	0.880503i	$0.342795\pi$
$110$	0	0
$111$	−6.51870	−0.618727
$112$	0	0
$113$	−9.73480	−0.915773	−0.457886	−	0.889011i	$-0.651393\pi$
$113$	−9.73480	−0.915773	−0.457886	+	0.889011i	$0.651393\pi$
$114$	0	0
$115$	16.6720	1.55467
$116$	0	0
$117$	5.57468	0.515380
$118$	0	0
$119$	4.94876	0.453652
$120$	0	0
$121$	14.1715	1.28831
$122$	0	0
$123$	−7.07422	−0.637861
$124$	0	0
$125$	−8.86593	−0.792993
$126$	0	0
$127$	−13.0224	−1.15555	−0.577773	−	0.816197i	$-0.696078\pi$
$127$	−13.0224	−1.15555	−0.577773	+	0.816197i	$0.696078\pi$
$128$	0	0
$129$	4.12532	0.363214
$130$	0	0
$131$	3.84831	0.336228	0.168114	−	0.985768i	$-0.446232\pi$
$131$	3.84831	0.336228	0.168114	+	0.985768i	$0.446232\pi$
$132$	0	0
$133$	−31.2298	−2.70797
$134$	0	0
$135$	−2.55552	−0.219944
$136$	0	0
$137$	4.16029	0.355438	0.177719	−	0.984081i	$-0.443128\pi$
$137$	4.16029	0.355438	0.177719	+	0.984081i	$0.443128\pi$
$138$	0	0
$139$	−5.85164	−0.496330	−0.248165	−	0.968718i	$-0.579827\pi$
$139$	−5.85164	−0.496330	−0.248165	+	0.968718i	$0.579827\pi$
$140$	0	0
$141$	−13.4216	−1.13030
$142$	0	0
$143$	27.9688	2.33887
$144$	0	0
$145$	−25.8206	−2.14428
$146$	0	0
$147$	−6.68514	−0.551381
$148$	0	0
$149$	−2.39324	−0.196062	−0.0980309	−	0.995183i	$-0.531254\pi$
$149$	−2.39324	−0.196062	−0.0980309	+	0.995183i	$0.531254\pi$
$150$	0	0
$151$	5.63737	0.458763	0.229382	−	0.973337i	$-0.426330\pi$
$151$	5.63737	0.458763	0.229382	+	0.973337i	$0.426330\pi$
$152$	0	0
$153$	−1.33774	−0.108150
$154$	0	0
$155$	7.21826	0.579785
$156$	0	0
$157$	−15.1191	−1.20663	−0.603317	−	0.797501i	$-0.706155\pi$
$157$	−15.1191	−1.20663	−0.603317	+	0.797501i	$0.706155\pi$
$158$	0	0
$159$	−4.54132	−0.360150
$160$	0	0
$161$	−24.1342	−1.90204
$162$	0	0
$163$	−6.76561	−0.529923	−0.264962	−	0.964259i	$-0.585359\pi$
$163$	−6.76561	−0.529923	−0.264962	+	0.964259i	$0.585359\pi$
$164$	0	0
$165$	−12.8213	−0.998139
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	18.0771	1.39054
$170$	0	0
$171$	8.44199	0.645575
$172$	0	0
$173$	−25.3814	−1.92971	−0.964855	−	0.262784i	$-0.915359\pi$
$173$	−25.3814	−1.92971	−0.964855	+	0.262784i	$0.915359\pi$
$174$	0	0
$175$	−5.66249	−0.428044
$176$	0	0
$177$	8.25552	0.620523
$178$	0	0
$179$	5.57643	0.416802	0.208401	−	0.978044i	$-0.433174\pi$
$179$	5.57643	0.416802	0.208401	+	0.978044i	$0.433174\pi$
$180$	0	0
$181$	8.57002	0.637005	0.318502	−	0.947922i	$-0.396820\pi$
$181$	8.57002	0.637005	0.318502	+	0.947922i	$0.396820\pi$
$182$	0	0
$183$	5.80421	0.429060
$184$	0	0
$185$	16.6587	1.22477
$186$	0	0
$187$	−6.71159	−0.490800
$188$	0	0
$189$	3.69934	0.269088
$190$	0	0
$191$	12.8759	0.931668	0.465834	−	0.884872i	$-0.345754\pi$
$191$	12.8759	0.931668	0.465834	+	0.884872i	$0.345754\pi$
$192$	0	0
$193$	23.0075	1.65611	0.828057	−	0.560644i	$-0.189446\pi$
$193$	23.0075	1.65611	0.828057	+	0.560644i	$0.189446\pi$
$194$	0	0
$195$	−14.2462	−1.02019
$196$	0	0
$197$	12.2832	0.875139	0.437569	−	0.899185i	$-0.355839\pi$
$197$	12.2832	0.875139	0.437569	+	0.899185i	$0.355839\pi$
$198$	0	0
$199$	−6.53733	−0.463419	−0.231709	−	0.972785i	$-0.574432\pi$
$199$	−6.53733	−0.463419	−0.231709	+	0.972785i	$0.574432\pi$
$200$	0	0
$201$	−7.81873	−0.551491
$202$	0	0
$203$	37.3776	2.62339
$204$	0	0
$205$	18.0783	1.26264
$206$	0	0
$207$	6.52392	0.453444
$208$	0	0
$209$	42.3544	2.92972
$210$	0	0
$211$	−7.05340	−0.485576	−0.242788	−	0.970079i	$-0.578062\pi$
$211$	−7.05340	−0.485576	−0.242788	+	0.970079i	$0.578062\pi$
$212$	0	0
$213$	9.13517	0.625931
$214$	0	0
$215$	−10.5423	−0.718981
$216$	0	0
$217$	−10.4491	−0.709330
$218$	0	0
$219$	6.04843	0.408715
$220$	0	0
$221$	−7.45747	−0.501644
$222$	0	0
$223$	2.07342	0.138846	0.0694232	−	0.997587i	$-0.477884\pi$
$223$	2.07342	0.138846	0.0694232	+	0.997587i	$0.477884\pi$
$224$	0	0
$225$	1.53067	0.102045
$226$	0	0
$227$	14.3929	0.955289	0.477644	−	0.878553i	$-0.341491\pi$
$227$	14.3929	0.955289	0.477644	+	0.878553i	$0.341491\pi$
$228$	0	0
$229$	−10.5878	−0.699664	−0.349832	−	0.936812i	$-0.613761\pi$
$229$	−10.5878	−0.699664	−0.349832	+	0.936812i	$0.613761\pi$
$230$	0	0
$231$	18.5600	1.22116
$232$	0	0
$233$	−1.51168	−0.0990331	−0.0495166	−	0.998773i	$-0.515768\pi$
$233$	−1.51168	−0.0990331	−0.0495166	+	0.998773i	$0.515768\pi$
$234$	0	0
$235$	34.2992	2.23743
$236$	0	0
$237$	6.04675	0.392779
$238$	0	0
$239$	6.97556	0.451212	0.225606	−	0.974219i	$-0.427564\pi$
$239$	6.97556	0.451212	0.225606	+	0.974219i	$0.427564\pi$
$240$	0	0
$241$	−9.13186	−0.588235	−0.294118	−	0.955769i	$-0.595026\pi$
$241$	−9.13186	−0.588235	−0.294118	+	0.955769i	$0.595026\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	17.0840	1.09146
$246$	0	0
$247$	47.0614	2.99444
$248$	0	0
$249$	3.09196	0.195945
$250$	0	0
$251$	3.36411	0.212341	0.106170	−	0.994348i	$-0.466141\pi$
$251$	3.36411	0.212341	0.106170	+	0.994348i	$0.466141\pi$
$252$	0	0
$253$	32.7313	2.05780
$254$	0	0
$255$	3.41862	0.214082
$256$	0	0
$257$	−18.3613	−1.14535	−0.572673	−	0.819784i	$-0.694093\pi$
$257$	−18.3613	−1.14535	−0.572673	+	0.819784i	$0.694093\pi$
$258$	0	0
$259$	−24.1149	−1.49843
$260$	0	0
$261$	−10.1039	−0.625413
$262$	0	0
$263$	7.62570	0.470221	0.235111	−	0.971969i	$-0.424455\pi$
$263$	7.62570	0.470221	0.235111	+	0.971969i	$0.424455\pi$
$264$	0	0
$265$	11.6054	0.712915
$266$	0	0
$267$	−16.0433	−0.981835
$268$	0	0
$269$	19.6881	1.20040	0.600202	−	0.799849i	$-0.295087\pi$
$269$	19.6881	1.20040	0.600202	+	0.799849i	$0.295087\pi$
$270$	0	0
$271$	−20.1204	−1.22223	−0.611113	−	0.791543i	$-0.709278\pi$
$271$	−20.1204	−1.22223	−0.611113	+	0.791543i	$0.709278\pi$
$272$	0	0
$273$	20.6227	1.24814
$274$	0	0
$275$	7.67956	0.463095
$276$	0	0
$277$	−24.1781	−1.45272	−0.726362	−	0.687313i	$-0.758790\pi$
$277$	−24.1781	−1.45272	−0.726362	+	0.687313i	$0.758790\pi$
$278$	0	0
$279$	2.82458	0.169103
$280$	0	0
$281$	17.1465	1.02287	0.511436	−	0.859321i	$-0.329114\pi$
$281$	17.1465	1.02287	0.511436	+	0.859321i	$0.329114\pi$
$282$	0	0
$283$	30.9646	1.84065	0.920326	−	0.391153i	$-0.127924\pi$
$283$	30.9646	1.84065	0.920326	+	0.391153i	$0.127924\pi$
$284$	0	0
$285$	−21.5737	−1.27791
$286$	0	0
$287$	−26.1700	−1.54476
$288$	0	0
$289$	−15.2105	−0.894733
$290$	0	0
$291$	0.668029	0.0391605
$292$	0	0
$293$	3.10940	0.181653	0.0908266	−	0.995867i	$-0.471049\pi$
$293$	3.10940	0.181653	0.0908266	+	0.995867i	$0.471049\pi$
$294$	0	0
$295$	−21.0971	−1.22832
$296$	0	0
$297$	−5.01712	−0.291123
$298$	0	0
$299$	36.3688	2.10326
$300$	0	0
$301$	15.2610	0.879629
$302$	0	0
$303$	−3.01681	−0.173311
$304$	0	0
$305$	−14.8328	−0.849322
$306$	0	0
$307$	−8.87724	−0.506651	−0.253325	−	0.967381i	$-0.581524\pi$
$307$	−8.87724	−0.506651	−0.253325	+	0.967381i	$0.581524\pi$
$308$	0	0
$309$	11.4959	0.653980
$310$	0	0
$311$	8.26937	0.468913	0.234456	−	0.972127i	$-0.424669\pi$
$311$	8.26937	0.468913	0.234456	+	0.972127i	$0.424669\pi$
$312$	0	0
$313$	31.8972	1.80294	0.901469	−	0.432843i	$-0.142490\pi$
$313$	31.8972	1.80294	0.901469	+	0.432843i	$0.142490\pi$
$314$	0	0
$315$	−9.45374	−0.532658
$316$	0	0
$317$	−1.41100	−0.0792499	−0.0396250	−	0.999215i	$-0.512616\pi$
$317$	−1.41100	−0.0792499	−0.0396250	+	0.999215i	$0.512616\pi$
$318$	0	0
$319$	−50.6922	−2.83822
$320$	0	0
$321$	7.85487	0.438416
$322$	0	0
$323$	−11.2932	−0.628369
$324$	0	0
$325$	8.53302	0.473327
$326$	0	0
$327$	−9.89827	−0.547376
$328$	0	0
$329$	−49.6512	−2.73736
$330$	0	0
$331$	11.3622	0.624522	0.312261	−	0.949996i	$-0.398914\pi$
$331$	11.3622	0.624522	0.312261	+	0.949996i	$0.398914\pi$
$332$	0	0
$333$	6.51870	0.357222
$334$	0	0
$335$	19.9809	1.09167
$336$	0	0
$337$	5.02627	0.273799	0.136899	−	0.990585i	$-0.456286\pi$
$337$	5.02627	0.273799	0.136899	+	0.990585i	$0.456286\pi$
$338$	0	0
$339$	9.73480	0.528722
$340$	0	0
$341$	14.1712	0.767416
$342$	0	0
$343$	1.16476	0.0628910
$344$	0	0
$345$	−16.6720	−0.897591
$346$	0	0
$347$	−34.4972	−1.85191	−0.925953	−	0.377639i	$-0.876736\pi$
$347$	−34.4972	−1.85191	−0.925953	+	0.377639i	$0.876736\pi$
$348$	0	0
$349$	−10.4754	−0.560734	−0.280367	−	0.959893i	$-0.590456\pi$
$349$	−10.4754	−0.560734	−0.280367	+	0.959893i	$0.590456\pi$
$350$	0	0
$351$	−5.57468	−0.297555
$352$	0	0
$353$	24.8874	1.32462	0.662311	−	0.749229i	$-0.269576\pi$
$353$	24.8874	1.32462	0.662311	+	0.749229i	$0.269576\pi$
$354$	0	0
$355$	−23.3451	−1.23903
$356$	0	0
$357$	−4.94876	−0.261916
$358$	0	0
$359$	21.5518	1.13746	0.568729	−	0.822525i	$-0.307435\pi$
$359$	21.5518	1.13746	0.568729	+	0.822525i	$0.307435\pi$
$360$	0	0
$361$	52.2671	2.75090
$362$	0	0
$363$	−14.1715	−0.743808
$364$	0	0
$365$	−15.4569	−0.809050
$366$	0	0
$367$	2.34002	0.122148	0.0610740	−	0.998133i	$-0.480547\pi$
$367$	2.34002	0.122148	0.0610740	+	0.998133i	$0.480547\pi$
$368$	0	0
$369$	7.07422	0.368269
$370$	0	0
$371$	−16.7999	−0.872207
$372$	0	0
$373$	34.6890	1.79613	0.898065	−	0.439863i	$-0.144973\pi$
$373$	34.6890	1.79613	0.898065	+	0.439863i	$0.144973\pi$
$374$	0	0
$375$	8.86593	0.457835
$376$	0	0
$377$	−56.3258	−2.90092
$378$	0	0
$379$	−7.00411	−0.359777	−0.179888	−	0.983687i	$-0.557574\pi$
$379$	−7.00411	−0.359777	−0.179888	+	0.983687i	$0.557574\pi$
$380$	0	0
$381$	13.0224	0.667155
$382$	0	0
$383$	−30.0932	−1.53769	−0.768845	−	0.639435i	$-0.779168\pi$
$383$	−30.0932	−1.53769	−0.768845	+	0.639435i	$0.779168\pi$
$384$	0	0
$385$	−47.4305	−2.41728
$386$	0	0
$387$	−4.12532	−0.209702
$388$	0	0
$389$	6.46346	0.327710	0.163855	−	0.986484i	$-0.447607\pi$
$389$	6.46346	0.327710	0.163855	+	0.986484i	$0.447607\pi$
$390$	0	0
$391$	−8.72731	−0.441359
$392$	0	0
$393$	−3.84831	−0.194121
$394$	0	0
$395$	−15.4526	−0.777504
$396$	0	0
$397$	−24.6878	−1.23905	−0.619524	−	0.784978i	$-0.712674\pi$
$397$	−24.6878	−1.23905	−0.619524	+	0.784978i	$0.712674\pi$
$398$	0	0
$399$	31.2298	1.56345
$400$	0	0
$401$	−3.34573	−0.167078	−0.0835389	−	0.996505i	$-0.526622\pi$
$401$	−3.34573	−0.167078	−0.0835389	+	0.996505i	$0.526622\pi$
$402$	0	0
$403$	15.7461	0.784371
$404$	0	0
$405$	2.55552	0.126985
$406$	0	0
$407$	32.7051	1.62113
$408$	0	0
$409$	−15.7145	−0.777032	−0.388516	−	0.921442i	$-0.627012\pi$
$409$	−15.7145	−0.777032	−0.388516	+	0.921442i	$0.627012\pi$
$410$	0	0
$411$	−4.16029	−0.205212
$412$	0	0
$413$	30.5400	1.50278
$414$	0	0
$415$	−7.90156	−0.387872
$416$	0	0
$417$	5.85164	0.286556
$418$	0	0
$419$	14.0423	0.686009	0.343004	−	0.939334i	$-0.388555\pi$
$419$	14.0423	0.686009	0.343004	+	0.939334i	$0.388555\pi$
$420$	0	0
$421$	19.5588	0.953235	0.476618	−	0.879111i	$-0.341863\pi$
$421$	19.5588	0.953235	0.476618	+	0.879111i	$0.341863\pi$
$422$	0	0
$423$	13.4216	0.652582
$424$	0	0
$425$	−2.04764	−0.0993252
$426$	0	0
$427$	21.4718	1.03909
$428$	0	0
$429$	−27.9688	−1.35035
$430$	0	0
$431$	6.55567	0.315776	0.157888	−	0.987457i	$-0.449532\pi$
$431$	6.55567	0.315776	0.157888	+	0.987457i	$0.449532\pi$
$432$	0	0
$433$	5.55313	0.266866	0.133433	−	0.991058i	$-0.457400\pi$
$433$	5.55313	0.266866	0.133433	+	0.991058i	$0.457400\pi$
$434$	0	0
$435$	25.8206	1.23800
$436$	0	0
$437$	55.0749	2.63459
$438$	0	0
$439$	−0.225982	−0.0107855	−0.00539276	−	0.999985i	$-0.501717\pi$
$439$	−0.225982	−0.0107855	−0.00539276	+	0.999985i	$0.501717\pi$
$440$	0	0
$441$	6.68514	0.318340
$442$	0	0
$443$	22.7488	1.08083	0.540415	−	0.841399i	$-0.318267\pi$
$443$	22.7488	1.08083	0.540415	+	0.841399i	$0.318267\pi$
$444$	0	0
$445$	40.9990	1.94354
$446$	0	0
$447$	2.39324	0.113196
$448$	0	0
$449$	−22.1743	−1.04647	−0.523234	−	0.852189i	$-0.675275\pi$
$449$	−22.1743	−1.04647	−0.523234	+	0.852189i	$0.675275\pi$
$450$	0	0
$451$	35.4922	1.67126
$452$	0	0
$453$	−5.63737	−0.264867
$454$	0	0
$455$	−52.7016	−2.47069
$456$	0	0
$457$	−28.9081	−1.35226	−0.676132	−	0.736781i	$-0.736345\pi$
$457$	−28.9081	−1.35226	−0.676132	+	0.736781i	$0.736345\pi$
$458$	0	0
$459$	1.33774	0.0624403
$460$	0	0
$461$	−0.181966	−0.00847501	−0.00423751	−	0.999991i	$-0.501349\pi$
$461$	−0.181966	−0.00847501	−0.00423751	+	0.999991i	$0.501349\pi$
$462$	0	0
$463$	−5.19250	−0.241316	−0.120658	−	0.992694i	$-0.538500\pi$
$463$	−5.19250	−0.241316	−0.120658	+	0.992694i	$0.538500\pi$
$464$	0	0
$465$	−7.21826	−0.334739
$466$	0	0
$467$	−38.2967	−1.77216	−0.886080	−	0.463533i	$-0.846582\pi$
$467$	−38.2967	−1.77216	−0.886080	+	0.463533i	$0.846582\pi$
$468$	0	0
$469$	−28.9242	−1.33559
$470$	0	0
$471$	15.1191	0.696651
$472$	0	0
$473$	−20.6972	−0.951659
$474$	0	0
$475$	12.9219	0.592898
$476$	0	0
$477$	4.54132	0.207933
$478$	0	0
$479$	−27.0554	−1.23619	−0.618096	−	0.786103i	$-0.712096\pi$
$479$	−27.0554	−1.23619	−0.618096	+	0.786103i	$0.712096\pi$
$480$	0	0
$481$	36.3397	1.65695
$482$	0	0
$483$	24.1342	1.09815
$484$	0	0
$485$	−1.70716	−0.0775181
$486$	0	0
$487$	1.47810	0.0669793	0.0334897	−	0.999439i	$-0.489338\pi$
$487$	1.47810	0.0669793	0.0334897	+	0.999439i	$0.489338\pi$
$488$	0	0
$489$	6.76561	0.305951
$490$	0	0
$491$	−14.2904	−0.644918	−0.322459	−	0.946583i	$-0.604509\pi$
$491$	−14.2904	−0.644918	−0.322459	+	0.946583i	$0.604509\pi$
$492$	0	0
$493$	13.5163	0.608744
$494$	0	0
$495$	12.8213	0.576276
$496$	0	0
$497$	33.7941	1.51587
$498$	0	0
$499$	28.0729	1.25671	0.628357	−	0.777925i	$-0.283728\pi$
$499$	28.0729	1.25671	0.628357	+	0.777925i	$0.283728\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−17.2055	−0.767154	−0.383577	−	0.923509i	$-0.625308\pi$
$503$	−17.2055	−0.767154	−0.383577	+	0.923509i	$0.625308\pi$
$504$	0	0
$505$	7.70952	0.343069
$506$	0	0
$507$	−18.0771	−0.802831
$508$	0	0
$509$	24.1002	1.06822	0.534110	−	0.845415i	$-0.320647\pi$
$509$	24.1002	1.06822	0.534110	+	0.845415i	$0.320647\pi$
$510$	0	0
$511$	22.3752	0.989822
$512$	0	0
$513$	−8.44199	−0.372723
$514$	0	0
$515$	−29.3780	−1.29455
$516$	0	0
$517$	67.3378	2.96151
$518$	0	0
$519$	25.3814	1.11412
$520$	0	0
$521$	−12.6267	−0.553185	−0.276592	−	0.960987i	$-0.589205\pi$
$521$	−12.6267	−0.553185	−0.276592	+	0.960987i	$0.589205\pi$
$522$	0	0
$523$	−38.9863	−1.70475	−0.852376	−	0.522930i	$-0.824839\pi$
$523$	−38.9863	−1.70475	−0.852376	+	0.522930i	$0.824839\pi$
$524$	0	0
$525$	5.66249	0.247131
$526$	0	0
$527$	−3.77855	−0.164596
$528$	0	0
$529$	19.5616	0.850503
$530$	0	0
$531$	−8.25552	−0.358259
$532$	0	0
$533$	39.4365	1.70818
$534$	0	0
$535$	−20.0733	−0.867843
$536$	0	0
$537$	−5.57643	−0.240641
$538$	0	0
$539$	33.5401	1.44468
$540$	0	0
$541$	−12.7015	−0.546078	−0.273039	−	0.962003i	$-0.588029\pi$
$541$	−12.7015	−0.546078	−0.273039	+	0.962003i	$0.588029\pi$
$542$	0	0
$543$	−8.57002	−0.367775
$544$	0	0
$545$	25.2952	1.08353
$546$	0	0
$547$	30.0026	1.28282	0.641410	−	0.767198i	$-0.278350\pi$
$547$	30.0026	1.28282	0.641410	+	0.767198i	$0.278350\pi$
$548$	0	0
$549$	−5.80421	−0.247718
$550$	0	0
$551$	−85.2966	−3.63376
$552$	0	0
$553$	22.3690	0.951227
$554$	0	0
$555$	−16.6587	−0.707120
$556$	0	0
$557$	−22.2821	−0.944121	−0.472061	−	0.881566i	$-0.656490\pi$
$557$	−22.2821	−0.944121	−0.472061	+	0.881566i	$0.656490\pi$
$558$	0	0
$559$	−22.9974	−0.972685
$560$	0	0
$561$	6.71159	0.283364
$562$	0	0
$563$	−4.14002	−0.174481	−0.0872405	−	0.996187i	$-0.527805\pi$
$563$	−4.14002	−0.174481	−0.0872405	+	0.996187i	$0.527805\pi$
$564$	0	0
$565$	−24.8774	−1.04660
$566$	0	0
$567$	−3.69934	−0.155358
$568$	0	0
$569$	3.32302	0.139308	0.0696541	−	0.997571i	$-0.477810\pi$
$569$	3.32302	0.139308	0.0696541	+	0.997571i	$0.477810\pi$
$570$	0	0
$571$	37.8109	1.58234	0.791168	−	0.611598i	$-0.209473\pi$
$571$	37.8109	1.58234	0.791168	+	0.611598i	$0.209473\pi$
$572$	0	0
$573$	−12.8759	−0.537899
$574$	0	0
$575$	9.98599	0.416445
$576$	0	0
$577$	−4.28107	−0.178223	−0.0891117	−	0.996022i	$-0.528403\pi$
$577$	−4.28107	−0.178223	−0.0891117	+	0.996022i	$0.528403\pi$
$578$	0	0
$579$	−23.0075	−0.956158
$580$	0	0
$581$	11.4382	0.474537
$582$	0	0
$583$	22.7843	0.943630
$584$	0	0
$585$	14.2462	0.589008
$586$	0	0
$587$	32.6002	1.34556	0.672778	−	0.739845i	$-0.265101\pi$
$587$	32.6002	1.34556	0.672778	+	0.739845i	$0.265101\pi$
$588$	0	0
$589$	23.8451	0.982519
$590$	0	0
$591$	−12.2832	−0.505262
$592$	0	0
$593$	−19.8830	−0.816498	−0.408249	−	0.912871i	$-0.633860\pi$
$593$	−19.8830	−0.816498	−0.408249	+	0.912871i	$0.633860\pi$
$594$	0	0
$595$	12.6466	0.518462
$596$	0	0
$597$	6.53733	0.267555
$598$	0	0
$599$	33.0055	1.34857	0.674285	−	0.738471i	$-0.264452\pi$
$599$	33.0055	1.34857	0.674285	+	0.738471i	$0.264452\pi$
$600$	0	0
$601$	14.8490	0.605702	0.302851	−	0.953038i	$-0.402062\pi$
$601$	14.8490	0.605702	0.302851	+	0.953038i	$0.402062\pi$
$602$	0	0
$603$	7.81873	0.318403
$604$	0	0
$605$	36.2154	1.47237
$606$	0	0
$607$	20.7319	0.841481	0.420741	−	0.907181i	$-0.361770\pi$
$607$	20.7319	0.841481	0.420741	+	0.907181i	$0.361770\pi$
$608$	0	0
$609$	−37.3776	−1.51462
$610$	0	0
$611$	74.8213	3.02695
$612$	0	0
$613$	36.4459	1.47204	0.736019	−	0.676961i	$-0.236703\pi$
$613$	36.4459	1.47204	0.736019	+	0.676961i	$0.236703\pi$
$614$	0	0
$615$	−18.0783	−0.728987
$616$	0	0
$617$	−34.1109	−1.37325	−0.686627	−	0.727010i	$-0.740909\pi$
$617$	−34.1109	−1.37325	−0.686627	+	0.727010i	$0.740909\pi$
$618$	0	0
$619$	37.7558	1.51754	0.758768	−	0.651361i	$-0.225802\pi$
$619$	37.7558	1.51754	0.758768	+	0.651361i	$0.225802\pi$
$620$	0	0
$621$	−6.52392	−0.261796
$622$	0	0
$623$	−59.3497	−2.37780
$624$	0	0
$625$	−30.3104	−1.21242
$626$	0	0
$627$	−42.3544	−1.69147
$628$	0	0
$629$	−8.72032	−0.347702
$630$	0	0
$631$	−21.1769	−0.843039	−0.421519	−	0.906819i	$-0.638503\pi$
$631$	−21.1769	−0.843039	−0.421519	+	0.906819i	$0.638503\pi$
$632$	0	0
$633$	7.05340	0.280348
$634$	0	0
$635$	−33.2789	−1.32063
$636$	0	0
$637$	37.2676	1.47659
$638$	0	0
$639$	−9.13517	−0.361382
$640$	0	0
$641$	27.7110	1.09452	0.547259	−	0.836963i	$-0.315671\pi$
$641$	27.7110	1.09452	0.547259	+	0.836963i	$0.315671\pi$
$642$	0	0
$643$	−2.82851	−0.111545	−0.0557727	−	0.998443i	$-0.517762\pi$
$643$	−2.82851	−0.111545	−0.0557727	+	0.998443i	$0.517762\pi$
$644$	0	0
$645$	10.5423	0.415104
$646$	0	0
$647$	−32.9823	−1.29667	−0.648333	−	0.761357i	$-0.724534\pi$
$647$	−32.9823	−1.29667	−0.648333	+	0.761357i	$0.724534\pi$
$648$	0	0
$649$	−41.4189	−1.62583
$650$	0	0
$651$	10.4491	0.409532
$652$	0	0
$653$	0.297132	0.0116277	0.00581383	−	0.999983i	$-0.498149\pi$
$653$	0.297132	0.0116277	0.00581383	+	0.999983i	$0.498149\pi$
$654$	0	0
$655$	9.83442	0.384263
$656$	0	0
$657$	−6.04843	−0.235972
$658$	0	0
$659$	−2.67076	−0.104038	−0.0520190	−	0.998646i	$-0.516566\pi$
$659$	−2.67076	−0.104038	−0.0520190	+	0.998646i	$0.516566\pi$
$660$	0	0
$661$	35.1840	1.36850	0.684249	−	0.729248i	$-0.260130\pi$
$661$	35.1840	1.36850	0.684249	+	0.729248i	$0.260130\pi$
$662$	0	0
$663$	7.45747	0.289624
$664$	0	0
$665$	−79.8084	−3.09484
$666$	0	0
$667$	−65.9167	−2.55231
$668$	0	0
$669$	−2.07342	−0.0801630
$670$	0	0
$671$	−29.1204	−1.12418
$672$	0	0
$673$	17.4055	0.670933	0.335467	−	0.942052i	$-0.391106\pi$
$673$	17.4055	0.670933	0.335467	+	0.942052i	$0.391106\pi$
$674$	0	0
$675$	−1.53067	−0.0589156
$676$	0	0
$677$	−42.5810	−1.63652	−0.818260	−	0.574849i	$-0.805061\pi$
$677$	−42.5810	−1.63652	−0.818260	+	0.574849i	$0.805061\pi$
$678$	0	0
$679$	2.47127	0.0948386
$680$	0	0
$681$	−14.3929	−0.551536
$682$	0	0
$683$	−22.3370	−0.854703	−0.427351	−	0.904086i	$-0.640553\pi$
$683$	−22.3370	−0.854703	−0.427351	+	0.904086i	$0.640553\pi$
$684$	0	0
$685$	10.6317	0.406216
$686$	0	0
$687$	10.5878	0.403951
$688$	0	0
$689$	25.3164	0.964478
$690$	0	0
$691$	15.6271	0.594481	0.297241	−	0.954803i	$-0.403934\pi$
$691$	15.6271	0.594481	0.297241	+	0.954803i	$0.403934\pi$
$692$	0	0
$693$	−18.5600	−0.705037
$694$	0	0
$695$	−14.9540	−0.567237
$696$	0	0
$697$	−9.46346	−0.358454
$698$	0	0
$699$	1.51168	0.0571768
$700$	0	0
$701$	3.28459	0.124057	0.0620286	−	0.998074i	$-0.480243\pi$
$701$	3.28459	0.124057	0.0620286	+	0.998074i	$0.480243\pi$
$702$	0	0
$703$	55.0308	2.07552
$704$	0	0
$705$	−34.2992	−1.29178
$706$	0	0
$707$	−11.1602	−0.419723
$708$	0	0
$709$	2.47723	0.0930343	0.0465171	−	0.998917i	$-0.485188\pi$
$709$	2.47723	0.0930343	0.0465171	+	0.998917i	$0.485188\pi$
$710$	0	0
$711$	−6.04675	−0.226771
$712$	0	0
$713$	18.4273	0.690109
$714$	0	0
$715$	71.4748	2.67301
$716$	0	0
$717$	−6.97556	−0.260507
$718$	0	0
$719$	13.5673	0.505977	0.252988	−	0.967469i	$-0.418587\pi$
$719$	13.5673	0.505977	0.252988	+	0.967469i	$0.418587\pi$
$720$	0	0
$721$	42.5274	1.58380
$722$	0	0
$723$	9.13186	0.339618
$724$	0	0
$725$	−15.4657	−0.574381
$726$	0	0
$727$	34.7979	1.29058	0.645291	−	0.763937i	$-0.276736\pi$
$727$	34.7979	1.29058	0.645291	+	0.763937i	$0.276736\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.51860	0.204113
$732$	0	0
$733$	−39.7662	−1.46880	−0.734399	−	0.678718i	$-0.762536\pi$
$733$	−39.7662	−1.46880	−0.734399	+	0.678718i	$0.762536\pi$
$734$	0	0
$735$	−17.0840	−0.630153
$736$	0	0
$737$	39.2275	1.44496
$738$	0	0
$739$	14.5340	0.534640	0.267320	−	0.963608i	$-0.413862\pi$
$739$	14.5340	0.534640	0.267320	+	0.963608i	$0.413862\pi$
$740$	0	0
$741$	−47.0614	−1.72884
$742$	0	0
$743$	−38.4389	−1.41018	−0.705092	−	0.709115i	$-0.749095\pi$
$743$	−38.4389	−1.41018	−0.705092	+	0.709115i	$0.749095\pi$
$744$	0	0
$745$	−6.11596	−0.224072
$746$	0	0
$747$	−3.09196	−0.113129
$748$	0	0
$749$	29.0579	1.06175
$750$	0	0
$751$	−17.5571	−0.640666	−0.320333	−	0.947305i	$-0.603795\pi$
$751$	−17.5571	−0.640666	−0.320333	+	0.947305i	$0.603795\pi$
$752$	0	0
$753$	−3.36411	−0.122595
$754$	0	0
$755$	14.4064	0.524303
$756$	0	0
$757$	−25.7367	−0.935415	−0.467707	−	0.883883i	$-0.654920\pi$
$757$	−25.7367	−0.935415	−0.467707	+	0.883883i	$0.654920\pi$
$758$	0	0
$759$	−32.7313	−1.18807
$760$	0	0
$761$	34.6561	1.25628	0.628140	−	0.778100i	$-0.283816\pi$
$761$	34.6561	1.25628	0.628140	+	0.778100i	$0.283816\pi$
$762$	0	0
$763$	−36.6171	−1.32563
$764$	0	0
$765$	−3.41862	−0.123600
$766$	0	0
$767$	−46.0219	−1.66176
$768$	0	0
$769$	37.7382	1.36087	0.680437	−	0.732806i	$-0.261790\pi$
$769$	37.7382	1.36087	0.680437	+	0.732806i	$0.261790\pi$
$770$	0	0
$771$	18.3613	0.661265
$772$	0	0
$773$	−45.1172	−1.62275	−0.811377	−	0.584523i	$-0.801282\pi$
$773$	−45.1172	−1.62275	−0.811377	+	0.584523i	$0.801282\pi$
$774$	0	0
$775$	4.32351	0.155305
$776$	0	0
$777$	24.1149	0.865117
$778$	0	0
$779$	59.7204	2.13971
$780$	0	0
$781$	−45.8322	−1.64000
$782$	0	0
$783$	10.1039	0.361082
$784$	0	0
$785$	−38.6371	−1.37902
$786$	0	0
$787$	−36.0782	−1.28605	−0.643024	−	0.765846i	$-0.722320\pi$
$787$	−36.0782	−1.28605	−0.643024	+	0.765846i	$0.722320\pi$
$788$	0	0
$789$	−7.62570	−0.271482
$790$	0	0
$791$	36.0124	1.28045
$792$	0	0
$793$	−32.3567	−1.14902
$794$	0	0
$795$	−11.6054	−0.411602
$796$	0	0
$797$	−0.359885	−0.0127478	−0.00637389	−	0.999980i	$-0.502029\pi$
$797$	−0.359885	−0.0127478	−0.00637389	+	0.999980i	$0.502029\pi$
$798$	0	0
$799$	−17.9546	−0.635189
$800$	0	0
$801$	16.0433	0.566863
$802$	0	0
$803$	−30.3457	−1.07088
$804$	0	0
$805$	−61.6755	−2.17377
$806$	0	0
$807$	−19.6881	−0.693053
$808$	0	0
$809$	28.3224	0.995762	0.497881	−	0.867245i	$-0.334112\pi$
$809$	28.3224	0.995762	0.497881	+	0.867245i	$0.334112\pi$
$810$	0	0
$811$	30.8113	1.08193	0.540965	−	0.841045i	$-0.318059\pi$
$811$	30.8113	1.08193	0.540965	+	0.841045i	$0.318059\pi$
$812$	0	0
$813$	20.1204	0.705653
$814$	0	0
$815$	−17.2896	−0.605629
$816$	0	0
$817$	−34.8259	−1.21840
$818$	0	0
$819$	−20.6227	−0.720614
$820$	0	0
$821$	32.3061	1.12749	0.563745	−	0.825949i	$-0.309360\pi$
$821$	32.3061	1.12749	0.563745	+	0.825949i	$0.309360\pi$
$822$	0	0
$823$	8.45829	0.294837	0.147419	−	0.989074i	$-0.452904\pi$
$823$	8.45829	0.294837	0.147419	+	0.989074i	$0.452904\pi$
$824$	0	0
$825$	−7.67956	−0.267368
$826$	0	0
$827$	−21.7167	−0.755164	−0.377582	−	0.925976i	$-0.623244\pi$
$827$	−21.7167	−0.755164	−0.377582	+	0.925976i	$0.623244\pi$
$828$	0	0
$829$	−1.53322	−0.0532509	−0.0266255	−	0.999645i	$-0.508476\pi$
$829$	−1.53322	−0.0532509	−0.0266255	+	0.999645i	$0.508476\pi$
$830$	0	0
$831$	24.1781	0.838730
$832$	0	0
$833$	−8.94298	−0.309856
$834$	0	0
$835$	−2.55552	−0.0884374
$836$	0	0
$837$	−2.82458	−0.0976317
$838$	0	0
$839$	19.5934	0.676440	0.338220	−	0.941067i	$-0.390175\pi$
$839$	19.5934	0.676440	0.338220	+	0.941067i	$0.390175\pi$
$840$	0	0
$841$	73.0878	2.52027
$842$	0	0
$843$	−17.1465	−0.590556
$844$	0	0
$845$	46.1963	1.58920
$846$	0	0
$847$	−52.4251	−1.80135
$848$	0	0
$849$	−30.9646	−1.06270
$850$	0	0
$851$	42.5275	1.45782
$852$	0	0
$853$	13.5819	0.465037	0.232518	−	0.972592i	$-0.425303\pi$
$853$	13.5819	0.465037	0.232518	+	0.972592i	$0.425303\pi$
$854$	0	0
$855$	21.5737	0.737803
$856$	0	0
$857$	−28.4003	−0.970135	−0.485067	−	0.874477i	$-0.661205\pi$
$857$	−28.4003	−0.970135	−0.485067	+	0.874477i	$0.661205\pi$
$858$	0	0
$859$	13.2135	0.450840	0.225420	−	0.974262i	$-0.427625\pi$
$859$	13.2135	0.450840	0.225420	+	0.974262i	$0.427625\pi$
$860$	0	0
$861$	26.1700	0.891870
$862$	0	0
$863$	33.7150	1.14767	0.573837	−	0.818970i	$-0.305454\pi$
$863$	33.7150	1.14767	0.573837	+	0.818970i	$0.305454\pi$
$864$	0	0
$865$	−64.8625	−2.20539
$866$	0	0
$867$	15.2105	0.516574
$868$	0	0
$869$	−30.3372	−1.02912
$870$	0	0
$871$	43.5869	1.47689
$872$	0	0
$873$	−0.668029	−0.0226094
$874$	0	0
$875$	32.7981	1.10878
$876$	0	0
$877$	−24.1098	−0.814129	−0.407064	−	0.913399i	$-0.633448\pi$
$877$	−24.1098	−0.814129	−0.407064	+	0.913399i	$0.633448\pi$
$878$	0	0
$879$	−3.10940	−0.104878
$880$	0	0
$881$	−30.8462	−1.03923	−0.519617	−	0.854399i	$-0.673925\pi$
$881$	−30.8462	−1.03923	−0.519617	+	0.854399i	$0.673925\pi$
$882$	0	0
$883$	17.2443	0.580315	0.290158	−	0.956979i	$-0.406292\pi$
$883$	17.2443	0.580315	0.290158	+	0.956979i	$0.406292\pi$
$884$	0	0
$885$	21.0971	0.709173
$886$	0	0
$887$	9.76619	0.327917	0.163958	−	0.986467i	$-0.447574\pi$
$887$	9.76619	0.327917	0.163958	+	0.986467i	$0.447574\pi$
$888$	0	0
$889$	48.1742	1.61571
$890$	0	0
$891$	5.01712	0.168080
$892$	0	0
$893$	113.305	3.79161
$894$	0	0
$895$	14.2507	0.476347
$896$	0	0
$897$	−36.3688	−1.21432
$898$	0	0
$899$	−28.5391	−0.951833
$900$	0	0
$901$	−6.07510	−0.202391
$902$	0	0
$903$	−15.2610	−0.507854
$904$	0	0
$905$	21.9008	0.728009
$906$	0	0
$907$	4.71308	0.156495	0.0782476	−	0.996934i	$-0.475068\pi$
$907$	4.71308	0.156495	0.0782476	+	0.996934i	$0.475068\pi$
$908$	0	0
$909$	3.01681	0.100061
$910$	0	0
$911$	20.5504	0.680864	0.340432	−	0.940269i	$-0.389427\pi$
$911$	20.5504	0.680864	0.340432	+	0.940269i	$0.389427\pi$
$912$	0	0
$913$	−15.5127	−0.513396
$914$	0	0
$915$	14.8328	0.490356
$916$	0	0
$917$	−14.2362	−0.470121
$918$	0	0
$919$	57.3714	1.89251	0.946254	−	0.323425i	$-0.104834\pi$
$919$	57.3714	1.89251	0.946254	+	0.323425i	$0.104834\pi$
$920$	0	0
$921$	8.87724	0.292515
$922$	0	0
$923$	−50.9256	−1.67624
$924$	0	0
$925$	9.97800	0.328075
$926$	0	0
$927$	−11.4959	−0.377576
$928$	0	0
$929$	−22.1268	−0.725957	−0.362979	−	0.931797i	$-0.618240\pi$
$929$	−22.1268	−0.725957	−0.362979	+	0.931797i	$0.618240\pi$
$930$	0	0
$931$	56.4359	1.84961
$932$	0	0
$933$	−8.26937	−0.270727
$934$	0	0
$935$	−17.1516	−0.560917
$936$	0	0
$937$	45.5706	1.48873	0.744363	−	0.667775i	$-0.232753\pi$
$937$	45.5706	1.48873	0.744363	+	0.667775i	$0.232753\pi$
$938$	0	0
$939$	−31.8972	−1.04093
$940$	0	0
$941$	−20.7422	−0.676177	−0.338089	−	0.941114i	$-0.609780\pi$
$941$	−20.7422	−0.676177	−0.338089	+	0.941114i	$0.609780\pi$
$942$	0	0
$943$	46.1516	1.50290
$944$	0	0
$945$	9.45374	0.307530
$946$	0	0
$947$	−6.21993	−0.202120	−0.101060	−	0.994880i	$-0.532223\pi$
$947$	−6.21993	−0.202120	−0.101060	+	0.994880i	$0.532223\pi$
$948$	0	0
$949$	−33.7181	−1.09454
$950$	0	0
$951$	1.41100	0.0457550
$952$	0	0
$953$	−33.9653	−1.10024	−0.550121	−	0.835085i	$-0.685419\pi$
$953$	−33.9653	−1.10024	−0.550121	+	0.835085i	$0.685419\pi$
$954$	0	0
$955$	32.9046	1.06477
$956$	0	0
$957$	50.6922	1.63865
$958$	0	0
$959$	−15.3903	−0.496980
$960$	0	0
$961$	−23.0218	−0.742637
$962$	0	0
$963$	−7.85487	−0.253120
$964$	0	0
$965$	58.7960	1.89271
$966$	0	0
$967$	59.6373	1.91781	0.958904	−	0.283732i	$-0.0915724\pi$
$967$	59.6373	1.91781	0.958904	+	0.283732i	$0.0915724\pi$
$968$	0	0
$969$	11.2932	0.362789
$970$	0	0
$971$	26.8559	0.861847	0.430923	−	0.902389i	$-0.358188\pi$
$971$	26.8559	0.861847	0.430923	+	0.902389i	$0.358188\pi$
$972$	0	0
$973$	21.6472	0.693979
$974$	0	0
$975$	−8.53302	−0.273275
$976$	0	0
$977$	−30.0184	−0.960373	−0.480186	−	0.877167i	$-0.659431\pi$
$977$	−30.0184	−0.960373	−0.480186	+	0.877167i	$0.659431\pi$
$978$	0	0
$979$	80.4912	2.57251
$980$	0	0
$981$	9.89827	0.316028
$982$	0	0
$983$	−56.2834	−1.79516	−0.897581	−	0.440849i	$-0.854677\pi$
$983$	−56.2834	−1.79516	−0.897581	+	0.440849i	$0.854677\pi$
$984$	0	0
$985$	31.3898	1.00016
$986$	0	0
$987$	49.6512	1.58041
$988$	0	0
$989$	−26.9133	−0.855793
$990$	0	0
$991$	29.8040	0.946754	0.473377	−	0.880860i	$-0.343035\pi$
$991$	29.8040	0.946754	0.473377	+	0.880860i	$0.343035\pi$
$992$	0	0
$993$	−11.3622	−0.360568
$994$	0	0
$995$	−16.7063	−0.529624
$996$	0	0
$997$	−61.8445	−1.95863	−0.979317	−	0.202332i	$-0.935148\pi$
$997$	−61.8445	−1.95863	−0.979317	+	0.202332i	$0.935148\pi$
$998$	0	0
$999$	−6.51870	−0.206242

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.v.1.5		7
4.3	odd	2		1002.2.a.k.1.5	✓	7
12.11	even	2		3006.2.a.u.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.k.1.5	✓	7	4.3	odd	2
3006.2.a.u.1.3		7	12.11	even	2
8016.2.a.v.1.5		7	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.55552 q^{5} -3.69934 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.55552 q^{5} -3.69934 q^{7} +1.00000 q^{9} +5.01712 q^{11} +5.57468 q^{13} -2.55552 q^{15} -1.33774 q^{17} +8.44199 q^{19} +3.69934 q^{21} +6.52392 q^{23} +1.53067 q^{25} -1.00000 q^{27} -10.1039 q^{29} +2.82458 q^{31} -5.01712 q^{33} -9.45374 q^{35} +6.51870 q^{37} -5.57468 q^{39} +7.07422 q^{41} -4.12532 q^{43} +2.55552 q^{45} +13.4216 q^{47} +6.68514 q^{49} +1.33774 q^{51} +4.54132 q^{53} +12.8213 q^{55} -8.44199 q^{57} -8.25552 q^{59} -5.80421 q^{61} -3.69934 q^{63} +14.2462 q^{65} +7.81873 q^{67} -6.52392 q^{69} -9.13517 q^{71} -6.04843 q^{73} -1.53067 q^{75} -18.5600 q^{77} -6.04675 q^{79} +1.00000 q^{81} -3.09196 q^{83} -3.41862 q^{85} +10.1039 q^{87} +16.0433 q^{89} -20.6227 q^{91} -2.82458 q^{93} +21.5737 q^{95} -0.668029 q^{97} +5.01712 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{3} + 5 q^{5} - 7 q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q - 7 * q^3 + 5 * q^5 - 7 * q^7 + 7 * q^9	\( 7 q - 7 q^{3} + 5 q^{5} - 7 q^{7} + 7 q^{9} + 6 q^{13} - 5 q^{15} + 6 q^{17} + 2 q^{19} + 7 q^{21} + 12 q^{25} - 7 q^{27} - 4 q^{29} - 7 q^{31} + 13 q^{35} - 3 q^{37} - 6 q^{39} - 12 q^{41} + 2 q^{43} + 5 q^{45} + 11 q^{47} + 10 q^{49} - 6 q^{51} + q^{53} + 2 q^{55} - 2 q^{57} + 19 q^{59} + 12 q^{61} - 7 q^{63} - 10 q^{65} + 17 q^{67} + 20 q^{71} + 10 q^{73} - 12 q^{75} - 24 q^{77} - 2 q^{79} + 7 q^{81} + 7 q^{83} - 18 q^{85} + 4 q^{87} - 3 q^{89} - 4 q^{91} + 7 q^{93} + 24 q^{95} - 3 q^{97}+O(q^{100}) \) 7 * q - 7 * q^3 + 5 * q^5 - 7 * q^7 + 7 * q^9 + 6 * q^13 - 5 * q^15 + 6 * q^17 + 2 * q^19 + 7 * q^21 + 12 * q^25 - 7 * q^27 - 4 * q^29 - 7 * q^31 + 13 * q^35 - 3 * q^37 - 6 * q^39 - 12 * q^41 + 2 * q^43 + 5 * q^45 + 11 * q^47 + 10 * q^49 - 6 * q^51 + q^53 + 2 * q^55 - 2 * q^57 + 19 * q^59 + 12 * q^61 - 7 * q^63 - 10 * q^65 + 17 * q^67 + 20 * q^71 + 10 * q^73 - 12 * q^75 - 24 * q^77 - 2 * q^79 + 7 * q^81 + 7 * q^83 - 18 * q^85 + 4 * q^87 - 3 * q^89 - 4 * q^91 + 7 * q^93 + 24 * q^95 - 3 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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