LMFDB - Embedded newform 8016.2.a.l.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:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.48119$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.675131 q^{5} +5.15633 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.675131 q^{5} +5.15633 q^{7} +1.00000 q^{9} -3.76845 q^{11} -6.15633 q^{13} +0.675131 q^{15} -5.19394 q^{17} -2.54420 q^{19} +5.15633 q^{21} -2.15633 q^{23} -4.54420 q^{25} +1.00000 q^{27} -3.76845 q^{29} +10.5369 q^{31} -3.76845 q^{33} +3.48119 q^{35} -3.06300 q^{37} -6.15633 q^{39} +2.31265 q^{41} +7.50659 q^{43} +0.675131 q^{45} -9.46898 q^{47} +19.5877 q^{49} -5.19394 q^{51} -3.74306 q^{53} -2.54420 q^{55} -2.54420 q^{57} +10.5999 q^{59} -0.775746 q^{61} +5.15633 q^{63} -4.15633 q^{65} -6.86177 q^{67} -2.15633 q^{69} -5.50659 q^{71} -11.8192 q^{73} -4.54420 q^{75} -19.4314 q^{77} -10.3879 q^{79} +1.00000 q^{81} -6.44358 q^{83} -3.50659 q^{85} -3.76845 q^{87} -11.9321 q^{89} -31.7440 q^{91} +10.5369 q^{93} -1.71767 q^{95} +8.50659 q^{97} -3.76845 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} + 5 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^5 + 5 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{5} + 5 q^{7} + 3 q^{9} - 8 q^{13} - 3 q^{15} - 16 q^{17} + 2 q^{19} + 5 q^{21} + 4 q^{23} - 4 q^{25} + 3 q^{27} + 9 q^{31} + 5 q^{35} - 5 q^{37} - 8 q^{39} - 14 q^{41} + 2 q^{43} - 3 q^{45} + 3 q^{47} + 6 q^{49} - 16 q^{51} - 15 q^{53} + 2 q^{55} + 2 q^{57} + 5 q^{59} - 4 q^{61} + 5 q^{63} - 2 q^{65} - 3 q^{67} + 4 q^{69} + 4 q^{71} + 6 q^{73} - 4 q^{75} - 16 q^{77} - 32 q^{79} + 3 q^{81} - 3 q^{83} + 10 q^{85} - 27 q^{89} - 32 q^{91} + 9 q^{93} - 24 q^{95} + 5 q^{97}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 + 5 * q^7 + 3 * q^9 - 8 * q^13 - 3 * q^15 - 16 * q^17 + 2 * q^19 + 5 * q^21 + 4 * q^23 - 4 * q^25 + 3 * q^27 + 9 * q^31 + 5 * q^35 - 5 * q^37 - 8 * q^39 - 14 * q^41 + 2 * q^43 - 3 * q^45 + 3 * q^47 + 6 * q^49 - 16 * q^51 - 15 * q^53 + 2 * q^55 + 2 * q^57 + 5 * q^59 - 4 * q^61 + 5 * q^63 - 2 * q^65 - 3 * q^67 + 4 * q^69 + 4 * q^71 + 6 * q^73 - 4 * q^75 - 16 * q^77 - 32 * q^79 + 3 * q^81 - 3 * q^83 + 10 * q^85 - 27 * q^89 - 32 * q^91 + 9 * q^93 - 24 * q^95 + 5 * 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
$5$	0.675131	0.301928	0.150964	−	0.988539i	$-0.451762\pi$
$5$	0.675131	0.301928	0.150964	+	0.988539i	$0.451762\pi$
$6$	0	0
$7$	5.15633	1.94891	0.974454	−	0.224588i	$-0.0721035\pi$
$7$	5.15633	1.94891	0.974454	+	0.224588i	$0.0721035\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.76845	−1.13623	−0.568116	−	0.822949i	$-0.692327\pi$
$11$	−3.76845	−1.13623	−0.568116	+	0.822949i	$0.692327\pi$
$12$	0	0
$13$	−6.15633	−1.70746	−0.853729	−	0.520718i	$-0.825664\pi$
$13$	−6.15633	−1.70746	−0.853729	+	0.520718i	$0.825664\pi$
$14$	0	0
$15$	0.675131	0.174318
$16$	0	0
$17$	−5.19394	−1.25971	−0.629857	−	0.776711i	$-0.716887\pi$
$17$	−5.19394	−1.25971	−0.629857	+	0.776711i	$0.716887\pi$
$18$	0	0
$19$	−2.54420	−0.583679	−0.291840	−	0.956467i	$-0.594267\pi$
$19$	−2.54420	−0.583679	−0.291840	+	0.956467i	$0.594267\pi$
$20$	0	0
$21$	5.15633	1.12520
$22$	0	0
$23$	−2.15633	−0.449625	−0.224812	−	0.974402i	$-0.572177\pi$
$23$	−2.15633	−0.449625	−0.224812	+	0.974402i	$0.572177\pi$
$24$	0	0
$25$	−4.54420	−0.908840
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.76845	−0.699784	−0.349892	−	0.936790i	$-0.613782\pi$
$29$	−3.76845	−0.699784	−0.349892	+	0.936790i	$0.613782\pi$
$30$	0	0
$31$	10.5369	1.89248	0.946242	−	0.323460i	$-0.104846\pi$
$31$	10.5369	1.89248	0.946242	+	0.323460i	$0.104846\pi$
$32$	0	0
$33$	−3.76845	−0.656003
$34$	0	0
$35$	3.48119	0.588429
$36$	0	0
$37$	−3.06300	−0.503555	−0.251777	−	0.967785i	$-0.581015\pi$
$37$	−3.06300	−0.503555	−0.251777	+	0.967785i	$0.581015\pi$
$38$	0	0
$39$	−6.15633	−0.985801
$40$	0	0
$41$	2.31265	0.361175	0.180588	−	0.983559i	$-0.442200\pi$
$41$	2.31265	0.361175	0.180588	+	0.983559i	$0.442200\pi$
$42$	0	0
$43$	7.50659	1.14474	0.572372	−	0.819994i	$-0.306023\pi$
$43$	7.50659	1.14474	0.572372	+	0.819994i	$0.306023\pi$
$44$	0	0
$45$	0.675131	0.100643
$46$	0	0
$47$	−9.46898	−1.38119	−0.690596	−	0.723241i	$-0.742652\pi$
$47$	−9.46898	−1.38119	−0.690596	+	0.723241i	$0.742652\pi$
$48$	0	0
$49$	19.5877	2.79824
$50$	0	0
$51$	−5.19394	−0.727297
$52$	0	0
$53$	−3.74306	−0.514149	−0.257074	−	0.966392i	$-0.582759\pi$
$53$	−3.74306	−0.514149	−0.257074	+	0.966392i	$0.582759\pi$
$54$	0	0
$55$	−2.54420	−0.343060
$56$	0	0
$57$	−2.54420	−0.336987
$58$	0	0
$59$	10.5999	1.37999	0.689995	−	0.723814i	$-0.257613\pi$
$59$	10.5999	1.37999	0.689995	+	0.723814i	$0.257613\pi$
$60$	0	0
$61$	−0.775746	−0.0993241	−0.0496621	−	0.998766i	$-0.515814\pi$
$61$	−0.775746	−0.0993241	−0.0496621	+	0.998766i	$0.515814\pi$
$62$	0	0
$63$	5.15633	0.649636
$64$	0	0
$65$	−4.15633	−0.515529
$66$	0	0
$67$	−6.86177	−0.838299	−0.419150	−	0.907917i	$-0.637672\pi$
$67$	−6.86177	−0.838299	−0.419150	+	0.907917i	$0.637672\pi$
$68$	0	0
$69$	−2.15633	−0.259591
$70$	0	0
$71$	−5.50659	−0.653512	−0.326756	−	0.945109i	$-0.605955\pi$
$71$	−5.50659	−0.653512	−0.326756	+	0.945109i	$0.605955\pi$
$72$	0	0
$73$	−11.8192	−1.38334	−0.691669	−	0.722215i	$-0.743124\pi$
$73$	−11.8192	−1.38334	−0.691669	+	0.722215i	$0.743124\pi$
$74$	0	0
$75$	−4.54420	−0.524719
$76$	0	0
$77$	−19.4314	−2.21441
$78$	0	0
$79$	−10.3879	−1.16873	−0.584364	−	0.811492i	$-0.698656\pi$
$79$	−10.3879	−1.16873	−0.584364	+	0.811492i	$0.698656\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.44358	−0.707275	−0.353638	−	0.935383i	$-0.615055\pi$
$83$	−6.44358	−0.707275	−0.353638	+	0.935383i	$0.615055\pi$
$84$	0	0
$85$	−3.50659	−0.380343
$86$	0	0
$87$	−3.76845	−0.404020
$88$	0	0
$89$	−11.9321	−1.26480	−0.632399	−	0.774643i	$-0.717929\pi$
$89$	−11.9321	−1.26480	−0.632399	+	0.774643i	$0.717929\pi$
$90$	0	0
$91$	−31.7440	−3.32768
$92$	0	0
$93$	10.5369	1.09263
$94$	0	0
$95$	−1.71767	−0.176229
$96$	0	0
$97$	8.50659	0.863713	0.431857	−	0.901942i	$-0.357859\pi$
$97$	8.50659	0.863713	0.431857	+	0.901942i	$0.357859\pi$
$98$	0	0
$99$	−3.76845	−0.378744
$100$	0	0
$101$	−16.4944	−1.64125	−0.820625	−	0.571466i	$-0.806375\pi$
$101$	−16.4944	−1.64125	−0.820625	+	0.571466i	$0.806375\pi$
$102$	0	0
$103$	−6.12601	−0.603614	−0.301807	−	0.953369i	$-0.597590\pi$
$103$	−6.12601	−0.603614	−0.301807	+	0.953369i	$0.597590\pi$
$104$	0	0
$105$	3.48119	0.339730
$106$	0	0
$107$	6.49929	0.628310	0.314155	−	0.949372i	$-0.398279\pi$
$107$	6.49929	0.628310	0.314155	+	0.949372i	$0.398279\pi$
$108$	0	0
$109$	−0.0811024	−0.00776820	−0.00388410	−	0.999992i	$-0.501236\pi$
$109$	−0.0811024	−0.00776820	−0.00388410	+	0.999992i	$0.501236\pi$
$110$	0	0
$111$	−3.06300	−0.290727
$112$	0	0
$113$	16.9380	1.59339	0.796694	−	0.604383i	$-0.206580\pi$
$113$	16.9380	1.59339	0.796694	+	0.604383i	$0.206580\pi$
$114$	0	0
$115$	−1.45580	−0.135754
$116$	0	0
$117$	−6.15633	−0.569152
$118$	0	0
$119$	−26.7816	−2.45507
$120$	0	0
$121$	3.20123	0.291021
$122$	0	0
$123$	2.31265	0.208525
$124$	0	0
$125$	−6.44358	−0.576332
$126$	0	0
$127$	0.425485	0.0377556	0.0188778	−	0.999822i	$-0.493991\pi$
$127$	0.425485	0.0377556	0.0188778	+	0.999822i	$0.493991\pi$
$128$	0	0
$129$	7.50659	0.660918
$130$	0	0
$131$	−4.49341	−0.392591	−0.196296	−	0.980545i	$-0.562891\pi$
$131$	−4.49341	−0.392591	−0.196296	+	0.980545i	$0.562891\pi$
$132$	0	0
$133$	−13.1187	−1.13754
$134$	0	0
$135$	0.675131	0.0581060
$136$	0	0
$137$	−6.50659	−0.555895	−0.277948	−	0.960596i	$-0.589654\pi$
$137$	−6.50659	−0.555895	−0.277948	+	0.960596i	$0.589654\pi$
$138$	0	0
$139$	−14.2120	−1.20545	−0.602725	−	0.797949i	$-0.705918\pi$
$139$	−14.2120	−1.20545	−0.602725	+	0.797949i	$0.705918\pi$
$140$	0	0
$141$	−9.46898	−0.797432
$142$	0	0
$143$	23.1998	1.94007
$144$	0	0
$145$	−2.54420	−0.211284
$146$	0	0
$147$	19.5877	1.61557
$148$	0	0
$149$	15.9199	1.30421	0.652103	−	0.758131i	$-0.273887\pi$
$149$	15.9199	1.30421	0.652103	+	0.758131i	$0.273887\pi$
$150$	0	0
$151$	18.3634	1.49440	0.747198	−	0.664602i	$-0.231399\pi$
$151$	18.3634	1.49440	0.747198	+	0.664602i	$0.231399\pi$
$152$	0	0
$153$	−5.19394	−0.419905
$154$	0	0
$155$	7.11379	0.571393
$156$	0	0
$157$	4.05079	0.323288	0.161644	−	0.986849i	$-0.448320\pi$
$157$	4.05079	0.323288	0.161644	+	0.986849i	$0.448320\pi$
$158$	0	0
$159$	−3.74306	−0.296844
$160$	0	0
$161$	−11.1187	−0.876277
$162$	0	0
$163$	−12.5696	−0.984526	−0.492263	−	0.870446i	$-0.663830\pi$
$163$	−12.5696	−0.984526	−0.492263	+	0.870446i	$0.663830\pi$
$164$	0	0
$165$	−2.54420	−0.198066
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	24.9003	1.91541
$170$	0	0
$171$	−2.54420	−0.194560
$172$	0	0
$173$	−10.5139	−0.799356	−0.399678	−	0.916656i	$-0.630878\pi$
$173$	−10.5139	−0.799356	−0.399678	+	0.916656i	$0.630878\pi$
$174$	0	0
$175$	−23.4314	−1.77124
$176$	0	0
$177$	10.5999	0.796738
$178$	0	0
$179$	−11.3054	−0.845002	−0.422501	−	0.906362i	$-0.638848\pi$
$179$	−11.3054	−0.845002	−0.422501	+	0.906362i	$0.638848\pi$
$180$	0	0
$181$	−4.03032	−0.299571	−0.149786	−	0.988719i	$-0.547858\pi$
$181$	−4.03032	−0.299571	−0.149786	+	0.988719i	$0.547858\pi$
$182$	0	0
$183$	−0.775746	−0.0573448
$184$	0	0
$185$	−2.06793	−0.152037
$186$	0	0
$187$	19.5731	1.43133
$188$	0	0
$189$	5.15633	0.375067
$190$	0	0
$191$	−12.0884	−0.874686	−0.437343	−	0.899295i	$-0.644080\pi$
$191$	−12.0884	−0.874686	−0.437343	+	0.899295i	$0.644080\pi$
$192$	0	0
$193$	8.67021	0.624095	0.312048	−	0.950066i	$-0.398985\pi$
$193$	8.67021	0.624095	0.312048	+	0.950066i	$0.398985\pi$
$194$	0	0
$195$	−4.15633	−0.297641
$196$	0	0
$197$	−18.5296	−1.32018	−0.660090	−	0.751187i	$-0.729482\pi$
$197$	−18.5296	−1.32018	−0.660090	+	0.751187i	$0.729482\pi$
$198$	0	0
$199$	−8.18664	−0.580336	−0.290168	−	0.956976i	$-0.593711\pi$
$199$	−8.18664	−0.580336	−0.290168	+	0.956976i	$0.593711\pi$
$200$	0	0
$201$	−6.86177	−0.483992
$202$	0	0
$203$	−19.4314	−1.36381
$204$	0	0
$205$	1.56134	0.109049
$206$	0	0
$207$	−2.15633	−0.149875
$208$	0	0
$209$	9.58769	0.663194
$210$	0	0
$211$	−2.52373	−0.173741	−0.0868704	−	0.996220i	$-0.527687\pi$
$211$	−2.52373	−0.173741	−0.0868704	+	0.996220i	$0.527687\pi$
$212$	0	0
$213$	−5.50659	−0.377305
$214$	0	0
$215$	5.06793	0.345630
$216$	0	0
$217$	54.3317	3.68828
$218$	0	0
$219$	−11.8192	−0.798670
$220$	0	0
$221$	31.9756	2.15091
$222$	0	0
$223$	20.8945	1.39920	0.699598	−	0.714536i	$-0.253362\pi$
$223$	20.8945	1.39920	0.699598	+	0.714536i	$0.253362\pi$
$224$	0	0
$225$	−4.54420	−0.302947
$226$	0	0
$227$	−2.16125	−0.143447	−0.0717236	−	0.997425i	$-0.522850\pi$
$227$	−2.16125	−0.143447	−0.0717236	+	0.997425i	$0.522850\pi$
$228$	0	0
$229$	−17.7743	−1.17456	−0.587280	−	0.809384i	$-0.699801\pi$
$229$	−17.7743	−1.17456	−0.587280	+	0.809384i	$0.699801\pi$
$230$	0	0
$231$	−19.4314	−1.27849
$232$	0	0
$233$	−25.8872	−1.69592	−0.847962	−	0.530057i	$-0.822171\pi$
$233$	−25.8872	−1.69592	−0.847962	+	0.530057i	$0.822171\pi$
$234$	0	0
$235$	−6.39280	−0.417020
$236$	0	0
$237$	−10.3879	−0.674765
$238$	0	0
$239$	16.4387	1.06333	0.531664	−	0.846955i	$-0.321567\pi$
$239$	16.4387	1.06333	0.531664	+	0.846955i	$0.321567\pi$
$240$	0	0
$241$	19.3054	1.24357	0.621784	−	0.783189i	$-0.286408\pi$
$241$	19.3054	1.24357	0.621784	+	0.783189i	$0.286408\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	13.2243	0.844867
$246$	0	0
$247$	15.6629	0.996607
$248$	0	0
$249$	−6.44358	−0.408345
$250$	0	0
$251$	−26.5804	−1.67774	−0.838870	−	0.544332i	$-0.816783\pi$
$251$	−26.5804	−1.67774	−0.838870	+	0.544332i	$0.816783\pi$
$252$	0	0
$253$	8.12601	0.510878
$254$	0	0
$255$	−3.50659	−0.219591
$256$	0	0
$257$	4.68006	0.291934	0.145967	−	0.989289i	$-0.453371\pi$
$257$	4.68006	0.291934	0.145967	+	0.989289i	$0.453371\pi$
$258$	0	0
$259$	−15.7938	−0.981382
$260$	0	0
$261$	−3.76845	−0.233261
$262$	0	0
$263$	25.9175	1.59814	0.799070	−	0.601238i	$-0.205326\pi$
$263$	25.9175	1.59814	0.799070	+	0.601238i	$0.205326\pi$
$264$	0	0
$265$	−2.52705	−0.155236
$266$	0	0
$267$	−11.9321	−0.730231
$268$	0	0
$269$	−0.750354	−0.0457499	−0.0228749	−	0.999738i	$-0.507282\pi$
$269$	−0.750354	−0.0457499	−0.0228749	+	0.999738i	$0.507282\pi$
$270$	0	0
$271$	−13.9551	−0.847712	−0.423856	−	0.905730i	$-0.639324\pi$
$271$	−13.9551	−0.847712	−0.423856	+	0.905730i	$0.639324\pi$
$272$	0	0
$273$	−31.7440	−1.92124
$274$	0	0
$275$	17.1246	1.03265
$276$	0	0
$277$	−10.2569	−0.616280	−0.308140	−	0.951341i	$-0.599706\pi$
$277$	−10.2569	−0.616280	−0.308140	+	0.951341i	$0.599706\pi$
$278$	0	0
$279$	10.5369	0.630828
$280$	0	0
$281$	9.49929	0.566680	0.283340	−	0.959019i	$-0.408557\pi$
$281$	9.49929	0.566680	0.283340	+	0.959019i	$0.408557\pi$
$282$	0	0
$283$	25.6483	1.52463	0.762317	−	0.647203i	$-0.224062\pi$
$283$	25.6483	1.52463	0.762317	+	0.647203i	$0.224062\pi$
$284$	0	0
$285$	−1.71767	−0.101746
$286$	0	0
$287$	11.9248	0.703897
$288$	0	0
$289$	9.97698	0.586881
$290$	0	0
$291$	8.50659	0.498665
$292$	0	0
$293$	24.1622	1.41157	0.705786	−	0.708426i	$-0.250594\pi$
$293$	24.1622	1.41157	0.705786	+	0.708426i	$0.250594\pi$
$294$	0	0
$295$	7.15633	0.416657
$296$	0	0
$297$	−3.76845	−0.218668
$298$	0	0
$299$	13.2750	0.767715
$300$	0	0
$301$	38.7064	2.23100
$302$	0	0
$303$	−16.4944	−0.947577
$304$	0	0
$305$	−0.523730	−0.0299887
$306$	0	0
$307$	−3.48119	−0.198682	−0.0993411	−	0.995053i	$-0.531674\pi$
$307$	−3.48119	−0.198682	−0.0993411	+	0.995053i	$0.531674\pi$
$308$	0	0
$309$	−6.12601	−0.348496
$310$	0	0
$311$	12.2750	0.696054	0.348027	−	0.937485i	$-0.386852\pi$
$311$	12.2750	0.696054	0.348027	+	0.937485i	$0.386852\pi$
$312$	0	0
$313$	−8.49341	−0.480076	−0.240038	−	0.970763i	$-0.577160\pi$
$313$	−8.49341	−0.480076	−0.240038	+	0.970763i	$0.577160\pi$
$314$	0	0
$315$	3.48119	0.196143
$316$	0	0
$317$	23.5574	1.32311	0.661557	−	0.749895i	$-0.269896\pi$
$317$	23.5574	1.32311	0.661557	+	0.749895i	$0.269896\pi$
$318$	0	0
$319$	14.2012	0.795116
$320$	0	0
$321$	6.49929	0.362755
$322$	0	0
$323$	13.2144	0.735269
$324$	0	0
$325$	27.9756	1.55180
$326$	0	0
$327$	−0.0811024	−0.00448497
$328$	0	0
$329$	−48.8251	−2.69182
$330$	0	0
$331$	28.5052	1.56679	0.783393	−	0.621527i	$-0.213487\pi$
$331$	28.5052	1.56679	0.783393	+	0.621527i	$0.213487\pi$
$332$	0	0
$333$	−3.06300	−0.167852
$334$	0	0
$335$	−4.63259	−0.253106
$336$	0	0
$337$	16.5369	0.900823	0.450411	−	0.892821i	$-0.351277\pi$
$337$	16.5369	0.900823	0.450411	+	0.892821i	$0.351277\pi$
$338$	0	0
$339$	16.9380	0.919943
$340$	0	0
$341$	−39.7078	−2.15030
$342$	0	0
$343$	64.9062	3.50461
$344$	0	0
$345$	−1.45580	−0.0783777
$346$	0	0
$347$	−20.1006	−1.07906	−0.539529	−	0.841967i	$-0.681398\pi$
$347$	−20.1006	−1.07906	−0.539529	+	0.841967i	$0.681398\pi$
$348$	0	0
$349$	−7.01222	−0.375355	−0.187678	−	0.982231i	$-0.560096\pi$
$349$	−7.01222	−0.375355	−0.187678	+	0.982231i	$0.560096\pi$
$350$	0	0
$351$	−6.15633	−0.328600
$352$	0	0
$353$	−21.7005	−1.15500	−0.577501	−	0.816390i	$-0.695972\pi$
$353$	−21.7005	−1.15500	−0.577501	+	0.816390i	$0.695972\pi$
$354$	0	0
$355$	−3.71767	−0.197313
$356$	0	0
$357$	−26.7816	−1.41743
$358$	0	0
$359$	−11.5125	−0.607605	−0.303802	−	0.952735i	$-0.598256\pi$
$359$	−11.5125	−0.607605	−0.303802	+	0.952735i	$0.598256\pi$
$360$	0	0
$361$	−12.5271	−0.659319
$362$	0	0
$363$	3.20123	0.168021
$364$	0	0
$365$	−7.97953	−0.417668
$366$	0	0
$367$	16.4387	0.858091	0.429045	−	0.903283i	$-0.358850\pi$
$367$	16.4387	0.858091	0.429045	+	0.903283i	$0.358850\pi$
$368$	0	0
$369$	2.31265	0.120392
$370$	0	0
$371$	−19.3004	−1.00203
$372$	0	0
$373$	2.50422	0.129663	0.0648317	−	0.997896i	$-0.479349\pi$
$373$	2.50422	0.129663	0.0648317	+	0.997896i	$0.479349\pi$
$374$	0	0
$375$	−6.44358	−0.332745
$376$	0	0
$377$	23.1998	1.19485
$378$	0	0
$379$	−28.2325	−1.45021	−0.725103	−	0.688640i	$-0.758208\pi$
$379$	−28.2325	−1.45021	−0.725103	+	0.688640i	$0.758208\pi$
$380$	0	0
$381$	0.425485	0.0217982
$382$	0	0
$383$	−0.649738	−0.0332001	−0.0166000	−	0.999862i	$-0.505284\pi$
$383$	−0.649738	−0.0332001	−0.0166000	+	0.999862i	$0.505284\pi$
$384$	0	0
$385$	−13.1187	−0.668592
$386$	0	0
$387$	7.50659	0.381581
$388$	0	0
$389$	−37.4168	−1.89711	−0.948553	−	0.316619i	$-0.897452\pi$
$389$	−37.4168	−1.89711	−0.948553	+	0.316619i	$0.897452\pi$
$390$	0	0
$391$	11.1998	0.566399
$392$	0	0
$393$	−4.49341	−0.226663
$394$	0	0
$395$	−7.01317	−0.352871
$396$	0	0
$397$	8.67021	0.435145	0.217573	−	0.976044i	$-0.430186\pi$
$397$	8.67021	0.435145	0.217573	+	0.976044i	$0.430186\pi$
$398$	0	0
$399$	−13.1187	−0.656757
$400$	0	0
$401$	−9.67750	−0.483271	−0.241636	−	0.970367i	$-0.577684\pi$
$401$	−9.67750	−0.483271	−0.241636	+	0.970367i	$0.577684\pi$
$402$	0	0
$403$	−64.8686	−3.23134
$404$	0	0
$405$	0.675131	0.0335475
$406$	0	0
$407$	11.5428	0.572155
$408$	0	0
$409$	−18.1016	−0.895065	−0.447533	−	0.894268i	$-0.647697\pi$
$409$	−18.1016	−0.895065	−0.447533	+	0.894268i	$0.647697\pi$
$410$	0	0
$411$	−6.50659	−0.320946
$412$	0	0
$413$	54.6566	2.68947
$414$	0	0
$415$	−4.35026	−0.213546
$416$	0	0
$417$	−14.2120	−0.695966
$418$	0	0
$419$	15.4763	0.756065	0.378033	−	0.925792i	$-0.376601\pi$
$419$	15.4763	0.756065	0.378033	+	0.925792i	$0.376601\pi$
$420$	0	0
$421$	−22.8061	−1.11150	−0.555750	−	0.831350i	$-0.687569\pi$
$421$	−22.8061	−1.11150	−0.555750	+	0.831350i	$0.687569\pi$
$422$	0	0
$423$	−9.46898	−0.460397
$424$	0	0
$425$	23.6023	1.14488
$426$	0	0
$427$	−4.00000	−0.193574
$428$	0	0
$429$	23.1998	1.12010
$430$	0	0
$431$	29.0263	1.39815	0.699075	−	0.715048i	$-0.253595\pi$
$431$	29.0263	1.39815	0.699075	+	0.715048i	$0.253595\pi$
$432$	0	0
$433$	−36.5877	−1.75829	−0.879146	−	0.476552i	$-0.841886\pi$
$433$	−36.5877	−1.75829	−0.879146	+	0.476552i	$0.841886\pi$
$434$	0	0
$435$	−2.54420	−0.121985
$436$	0	0
$437$	5.48612	0.262437
$438$	0	0
$439$	8.18664	0.390727	0.195364	−	0.980731i	$-0.437411\pi$
$439$	8.18664	0.390727	0.195364	+	0.980731i	$0.437411\pi$
$440$	0	0
$441$	19.5877	0.932747
$442$	0	0
$443$	13.8749	0.659219	0.329609	−	0.944117i	$-0.393083\pi$
$443$	13.8749	0.659219	0.329609	+	0.944117i	$0.393083\pi$
$444$	0	0
$445$	−8.05571	−0.381877
$446$	0	0
$447$	15.9199	0.752983
$448$	0	0
$449$	−30.1695	−1.42379	−0.711893	−	0.702288i	$-0.752162\pi$
$449$	−30.1695	−1.42379	−0.711893	+	0.702288i	$0.752162\pi$
$450$	0	0
$451$	−8.71511	−0.410379
$452$	0	0
$453$	18.3634	0.862789
$454$	0	0
$455$	−21.4314	−1.00472
$456$	0	0
$457$	−1.53690	−0.0718933	−0.0359467	−	0.999354i	$-0.511445\pi$
$457$	−1.53690	−0.0718933	−0.0359467	+	0.999354i	$0.511445\pi$
$458$	0	0
$459$	−5.19394	−0.242432
$460$	0	0
$461$	−31.4314	−1.46390	−0.731952	−	0.681356i	$-0.761391\pi$
$461$	−31.4314	−1.46390	−0.731952	+	0.681356i	$0.761391\pi$
$462$	0	0
$463$	10.3733	0.482087	0.241044	−	0.970514i	$-0.422510\pi$
$463$	10.3733	0.482087	0.241044	+	0.970514i	$0.422510\pi$
$464$	0	0
$465$	7.11379	0.329894
$466$	0	0
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	−35.3815	−1.63377
$470$	0	0
$471$	4.05079	0.186650
$472$	0	0
$473$	−28.2882	−1.30069
$474$	0	0
$475$	11.5613	0.530471
$476$	0	0
$477$	−3.74306	−0.171383
$478$	0	0
$479$	42.0625	1.92189	0.960943	−	0.276745i	$-0.0892558\pi$
$479$	42.0625	1.92189	0.960943	+	0.276745i	$0.0892558\pi$
$480$	0	0
$481$	18.8568	0.859798
$482$	0	0
$483$	−11.1187	−0.505919
$484$	0	0
$485$	5.74306	0.260779
$486$	0	0
$487$	14.0957	0.638737	0.319368	−	0.947631i	$-0.396529\pi$
$487$	14.0957	0.638737	0.319368	+	0.947631i	$0.396529\pi$
$488$	0	0
$489$	−12.5696	−0.568417
$490$	0	0
$491$	9.98541	0.450635	0.225318	−	0.974285i	$-0.427658\pi$
$491$	9.98541	0.450635	0.225318	+	0.974285i	$0.427658\pi$
$492$	0	0
$493$	19.5731	0.881528
$494$	0	0
$495$	−2.54420	−0.114353
$496$	0	0
$497$	−28.3938	−1.27363
$498$	0	0
$499$	−22.9829	−1.02885	−0.514427	−	0.857534i	$-0.671995\pi$
$499$	−22.9829	−1.02885	−0.514427	+	0.857534i	$0.671995\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	0.800184	0.0356784	0.0178392	−	0.999841i	$-0.494321\pi$
$503$	0.800184	0.0356784	0.0178392	+	0.999841i	$0.494321\pi$
$504$	0	0
$505$	−11.1359	−0.495539
$506$	0	0
$507$	24.9003	1.10586
$508$	0	0
$509$	11.8740	0.526305	0.263153	−	0.964754i	$-0.415238\pi$
$509$	11.8740	0.526305	0.263153	+	0.964754i	$0.415238\pi$
$510$	0	0
$511$	−60.9438	−2.69600
$512$	0	0
$513$	−2.54420	−0.112329
$514$	0	0
$515$	−4.13586	−0.182248
$516$	0	0
$517$	35.6834	1.56935
$518$	0	0
$519$	−10.5139	−0.461508
$520$	0	0
$521$	25.9307	1.13604	0.568021	−	0.823014i	$-0.307709\pi$
$521$	25.9307	1.13604	0.568021	+	0.823014i	$0.307709\pi$
$522$	0	0
$523$	23.8945	1.04483	0.522416	−	0.852691i	$-0.325031\pi$
$523$	23.8945	1.04483	0.522416	+	0.852691i	$0.325031\pi$
$524$	0	0
$525$	−23.4314	−1.02263
$526$	0	0
$527$	−54.7280	−2.38399
$528$	0	0
$529$	−18.3503	−0.797837
$530$	0	0
$531$	10.5999	0.459997
$532$	0	0
$533$	−14.2374	−0.616691
$534$	0	0
$535$	4.38787	0.189704
$536$	0	0
$537$	−11.3054	−0.487862
$538$	0	0
$539$	−73.8153	−3.17945
$540$	0	0
$541$	0.332163	0.0142808	0.00714041	−	0.999975i	$-0.497727\pi$
$541$	0.332163	0.0142808	0.00714041	+	0.999975i	$0.497727\pi$
$542$	0	0
$543$	−4.03032	−0.172957
$544$	0	0
$545$	−0.0547547	−0.00234543
$546$	0	0
$547$	2.17584	0.0930321	0.0465161	−	0.998918i	$-0.485188\pi$
$547$	2.17584	0.0930321	0.0465161	+	0.998918i	$0.485188\pi$
$548$	0	0
$549$	−0.775746	−0.0331080
$550$	0	0
$551$	9.58769	0.408449
$552$	0	0
$553$	−53.5633	−2.27774
$554$	0	0
$555$	−2.06793	−0.0877787
$556$	0	0
$557$	39.5936	1.67763	0.838817	−	0.544414i	$-0.183248\pi$
$557$	39.5936	1.67763	0.838817	+	0.544414i	$0.183248\pi$
$558$	0	0
$559$	−46.2130	−1.95460
$560$	0	0
$561$	19.5731	0.826377
$562$	0	0
$563$	19.6326	0.827415	0.413708	−	0.910410i	$-0.364234\pi$
$563$	19.6326	0.827415	0.413708	+	0.910410i	$0.364234\pi$
$564$	0	0
$565$	11.4353	0.481088
$566$	0	0
$567$	5.15633	0.216545
$568$	0	0
$569$	−13.1939	−0.553119	−0.276559	−	0.960997i	$-0.589194\pi$
$569$	−13.1939	−0.553119	−0.276559	+	0.960997i	$0.589194\pi$
$570$	0	0
$571$	22.0459	0.922591	0.461295	−	0.887247i	$-0.347385\pi$
$571$	22.0459	0.922591	0.461295	+	0.887247i	$0.347385\pi$
$572$	0	0
$573$	−12.0884	−0.505000
$574$	0	0
$575$	9.79877	0.408637
$576$	0	0
$577$	23.7670	0.989435	0.494717	−	0.869054i	$-0.335272\pi$
$577$	23.7670	0.989435	0.494717	+	0.869054i	$0.335272\pi$
$578$	0	0
$579$	8.67021	0.360321
$580$	0	0
$581$	−33.2252	−1.37841
$582$	0	0
$583$	14.1055	0.584192
$584$	0	0
$585$	−4.15633	−0.171843
$586$	0	0
$587$	−23.8183	−0.983086	−0.491543	−	0.870853i	$-0.663567\pi$
$587$	−23.8183	−0.983086	−0.491543	+	0.870853i	$0.663567\pi$
$588$	0	0
$589$	−26.8080	−1.10460
$590$	0	0
$591$	−18.5296	−0.762206
$592$	0	0
$593$	17.0943	0.701978	0.350989	−	0.936380i	$-0.385845\pi$
$593$	17.0943	0.701978	0.350989	+	0.936380i	$0.385845\pi$
$594$	0	0
$595$	−18.0811	−0.741253
$596$	0	0
$597$	−8.18664	−0.335057
$598$	0	0
$599$	−18.3101	−0.748130	−0.374065	−	0.927402i	$-0.622036\pi$
$599$	−18.3101	−0.748130	−0.374065	+	0.927402i	$0.622036\pi$
$600$	0	0
$601$	34.8094	1.41990	0.709952	−	0.704250i	$-0.248716\pi$
$601$	34.8094	1.41990	0.709952	+	0.704250i	$0.248716\pi$
$602$	0	0
$603$	−6.86177	−0.279433
$604$	0	0
$605$	2.16125	0.0878673
$606$	0	0
$607$	−37.1509	−1.50791	−0.753955	−	0.656926i	$-0.771856\pi$
$607$	−37.1509	−1.50791	−0.753955	+	0.656926i	$0.771856\pi$
$608$	0	0
$609$	−19.4314	−0.787399
$610$	0	0
$611$	58.2941	2.35833
$612$	0	0
$613$	−25.2692	−1.02061	−0.510306	−	0.859993i	$-0.670468\pi$
$613$	−25.2692	−1.02061	−0.510306	+	0.859993i	$0.670468\pi$
$614$	0	0
$615$	1.56134	0.0629594
$616$	0	0
$617$	−21.8251	−0.878646	−0.439323	−	0.898329i	$-0.644782\pi$
$617$	−21.8251	−0.878646	−0.439323	+	0.898329i	$0.644782\pi$
$618$	0	0
$619$	21.1841	0.851460	0.425730	−	0.904850i	$-0.360017\pi$
$619$	21.1841	0.851460	0.425730	+	0.904850i	$0.360017\pi$
$620$	0	0
$621$	−2.15633	−0.0865303
$622$	0	0
$623$	−61.5256	−2.46497
$624$	0	0
$625$	18.3707	0.734829
$626$	0	0
$627$	9.58769	0.382895
$628$	0	0
$629$	15.9090	0.634335
$630$	0	0
$631$	16.2896	0.648480	0.324240	−	0.945975i	$-0.394891\pi$
$631$	16.2896	0.648480	0.324240	+	0.945975i	$0.394891\pi$
$632$	0	0
$633$	−2.52373	−0.100309
$634$	0	0
$635$	0.287258	0.0113995
$636$	0	0
$637$	−120.588	−4.77788
$638$	0	0
$639$	−5.50659	−0.217837
$640$	0	0
$641$	30.2520	1.19488	0.597441	−	0.801913i	$-0.296184\pi$
$641$	30.2520	1.19488	0.597441	+	0.801913i	$0.296184\pi$
$642$	0	0
$643$	0.493413	0.0194583	0.00972916	−	0.999953i	$-0.496903\pi$
$643$	0.493413	0.0194583	0.00972916	+	0.999953i	$0.496903\pi$
$644$	0	0
$645$	5.06793	0.199549
$646$	0	0
$647$	−6.05079	−0.237881	−0.118940	−	0.992901i	$-0.537950\pi$
$647$	−6.05079	−0.237881	−0.118940	+	0.992901i	$0.537950\pi$
$648$	0	0
$649$	−39.9452	−1.56799
$650$	0	0
$651$	54.3317	2.12943
$652$	0	0
$653$	7.41564	0.290196	0.145098	−	0.989417i	$-0.453650\pi$
$653$	7.41564	0.290196	0.145098	+	0.989417i	$0.453650\pi$
$654$	0	0
$655$	−3.03364	−0.118534
$656$	0	0
$657$	−11.8192	−0.461112
$658$	0	0
$659$	−29.4568	−1.14747	−0.573736	−	0.819040i	$-0.694507\pi$
$659$	−29.4568	−1.14747	−0.573736	+	0.819040i	$0.694507\pi$
$660$	0	0
$661$	28.9584	1.12635	0.563176	−	0.826337i	$-0.309579\pi$
$661$	28.9584	1.12635	0.563176	+	0.826337i	$0.309579\pi$
$662$	0	0
$663$	31.9756	1.24183
$664$	0	0
$665$	−8.85685	−0.343454
$666$	0	0
$667$	8.12601	0.314640
$668$	0	0
$669$	20.8945	0.807826
$670$	0	0
$671$	2.92336	0.112855
$672$	0	0
$673$	35.9307	1.38503	0.692513	−	0.721406i	$-0.256504\pi$
$673$	35.9307	1.38503	0.692513	+	0.721406i	$0.256504\pi$
$674$	0	0
$675$	−4.54420	−0.174906
$676$	0	0
$677$	−8.20711	−0.315425	−0.157712	−	0.987485i	$-0.550412\pi$
$677$	−8.20711	−0.315425	−0.157712	+	0.987485i	$0.550412\pi$
$678$	0	0
$679$	43.8627	1.68330
$680$	0	0
$681$	−2.16125	−0.0828193
$682$	0	0
$683$	15.7830	0.603921	0.301961	−	0.953320i	$-0.402359\pi$
$683$	15.7830	0.603921	0.301961	+	0.953320i	$0.402359\pi$
$684$	0	0
$685$	−4.39280	−0.167840
$686$	0	0
$687$	−17.7743	−0.678133
$688$	0	0
$689$	23.0435	0.877887
$690$	0	0
$691$	12.3430	0.469549	0.234774	−	0.972050i	$-0.424565\pi$
$691$	12.3430	0.469549	0.234774	+	0.972050i	$0.424565\pi$
$692$	0	0
$693$	−19.4314	−0.738136
$694$	0	0
$695$	−9.59498	−0.363958
$696$	0	0
$697$	−12.0118	−0.454978
$698$	0	0
$699$	−25.8872	−0.979143
$700$	0	0
$701$	14.2981	0.540030	0.270015	−	0.962856i	$-0.412971\pi$
$701$	14.2981	0.540030	0.270015	+	0.962856i	$0.412971\pi$
$702$	0	0
$703$	7.79289	0.293914
$704$	0	0
$705$	−6.39280	−0.240767
$706$	0	0
$707$	−85.0503	−3.19865
$708$	0	0
$709$	−25.0835	−0.942030	−0.471015	−	0.882125i	$-0.656112\pi$
$709$	−25.0835	−0.942030	−0.471015	+	0.882125i	$0.656112\pi$
$710$	0	0
$711$	−10.3879	−0.389576
$712$	0	0
$713$	−22.7210	−0.850908
$714$	0	0
$715$	15.6629	0.585760
$716$	0	0
$717$	16.4387	0.613913
$718$	0	0
$719$	−29.1490	−1.08708	−0.543538	−	0.839385i	$-0.682916\pi$
$719$	−29.1490	−1.08708	−0.543538	+	0.839385i	$0.682916\pi$
$720$	0	0
$721$	−31.5877	−1.17639
$722$	0	0
$723$	19.3054	0.717974
$724$	0	0
$725$	17.1246	0.635991
$726$	0	0
$727$	21.7177	0.805464	0.402732	−	0.915318i	$-0.368061\pi$
$727$	21.7177	0.805464	0.402732	+	0.915318i	$0.368061\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−38.9887	−1.44205
$732$	0	0
$733$	−11.1187	−0.410679	−0.205340	−	0.978691i	$-0.565830\pi$
$733$	−11.1187	−0.410679	−0.205340	+	0.978691i	$0.565830\pi$
$734$	0	0
$735$	13.2243	0.487784
$736$	0	0
$737$	25.8583	0.952501
$738$	0	0
$739$	39.7694	1.46294	0.731471	−	0.681873i	$-0.238834\pi$
$739$	39.7694	1.46294	0.731471	+	0.681873i	$0.238834\pi$
$740$	0	0
$741$	15.6629	0.575391
$742$	0	0
$743$	−34.8554	−1.27872	−0.639361	−	0.768907i	$-0.720801\pi$
$743$	−34.8554	−1.27872	−0.639361	+	0.768907i	$0.720801\pi$
$744$	0	0
$745$	10.7480	0.393776
$746$	0	0
$747$	−6.44358	−0.235758
$748$	0	0
$749$	33.5125	1.22452
$750$	0	0
$751$	6.72099	0.245252	0.122626	−	0.992453i	$-0.460868\pi$
$751$	6.72099	0.245252	0.122626	+	0.992453i	$0.460868\pi$
$752$	0	0
$753$	−26.5804	−0.968643
$754$	0	0
$755$	12.3977	0.451199
$756$	0	0
$757$	38.6556	1.40496	0.702481	−	0.711702i	$-0.252076\pi$
$757$	38.6556	1.40496	0.702481	+	0.711702i	$0.252076\pi$
$758$	0	0
$759$	8.12601	0.294955
$760$	0	0
$761$	−27.3127	−0.990083	−0.495041	−	0.868869i	$-0.664847\pi$
$761$	−27.3127	−0.990083	−0.495041	+	0.868869i	$0.664847\pi$
$762$	0	0
$763$	−0.418190	−0.0151395
$764$	0	0
$765$	−3.50659	−0.126781
$766$	0	0
$767$	−65.2565	−2.35627
$768$	0	0
$769$	−29.1549	−1.05135	−0.525676	−	0.850685i	$-0.676188\pi$
$769$	−29.1549	−1.05135	−0.525676	+	0.850685i	$0.676188\pi$
$770$	0	0
$771$	4.68006	0.168548
$772$	0	0
$773$	6.69560	0.240824	0.120412	−	0.992724i	$-0.461578\pi$
$773$	6.69560	0.240824	0.120412	+	0.992724i	$0.461578\pi$
$774$	0	0
$775$	−47.8818	−1.71996
$776$	0	0
$777$	−15.7938	−0.566601
$778$	0	0
$779$	−5.88384	−0.210810
$780$	0	0
$781$	20.7513	0.742540
$782$	0	0
$783$	−3.76845	−0.134673
$784$	0	0
$785$	2.73481	0.0976096
$786$	0	0
$787$	−27.7586	−0.989487	−0.494744	−	0.869039i	$-0.664738\pi$
$787$	−27.7586	−0.989487	−0.494744	+	0.869039i	$0.664738\pi$
$788$	0	0
$789$	25.9175	0.922687
$790$	0	0
$791$	87.3376	3.10537
$792$	0	0
$793$	4.77575	0.169592
$794$	0	0
$795$	−2.52705	−0.0896254
$796$	0	0
$797$	45.1206	1.59825	0.799127	−	0.601162i	$-0.205295\pi$
$797$	45.1206	1.59825	0.799127	+	0.601162i	$0.205295\pi$
$798$	0	0
$799$	49.1813	1.73991
$800$	0	0
$801$	−11.9321	−0.421599
$802$	0	0
$803$	44.5402	1.57179
$804$	0	0
$805$	−7.50659	−0.264572
$806$	0	0
$807$	−0.750354	−0.0264137
$808$	0	0
$809$	36.5672	1.28564	0.642818	−	0.766019i	$-0.277765\pi$
$809$	36.5672	1.28564	0.642818	+	0.766019i	$0.277765\pi$
$810$	0	0
$811$	11.7880	0.413931	0.206966	−	0.978348i	$-0.433641\pi$
$811$	11.7880	0.413931	0.206966	+	0.978348i	$0.433641\pi$
$812$	0	0
$813$	−13.9551	−0.489427
$814$	0	0
$815$	−8.48612	−0.297256
$816$	0	0
$817$	−19.0982	−0.668163
$818$	0	0
$819$	−31.7440	−1.10923
$820$	0	0
$821$	−30.9438	−1.07995	−0.539974	−	0.841682i	$-0.681566\pi$
$821$	−30.9438	−1.07995	−0.539974	+	0.841682i	$0.681566\pi$
$822$	0	0
$823$	−42.1417	−1.46897	−0.734484	−	0.678626i	$-0.762576\pi$
$823$	−42.1417	−1.46897	−0.734484	+	0.678626i	$0.762576\pi$
$824$	0	0
$825$	17.1246	0.596202
$826$	0	0
$827$	39.3073	1.36685	0.683424	−	0.730022i	$-0.260490\pi$
$827$	39.3073	1.36685	0.683424	+	0.730022i	$0.260490\pi$
$828$	0	0
$829$	−9.19489	−0.319352	−0.159676	−	0.987169i	$-0.551045\pi$
$829$	−9.19489	−0.319352	−0.159676	+	0.987169i	$0.551045\pi$
$830$	0	0
$831$	−10.2569	−0.355809
$832$	0	0
$833$	−101.737	−3.52499
$834$	0	0
$835$	−0.675131	−0.0233639
$836$	0	0
$837$	10.5369	0.364209
$838$	0	0
$839$	39.4603	1.36232	0.681160	−	0.732135i	$-0.261476\pi$
$839$	39.4603	1.36232	0.681160	+	0.732135i	$0.261476\pi$
$840$	0	0
$841$	−14.7988	−0.510302
$842$	0	0
$843$	9.49929	0.327173
$844$	0	0
$845$	16.8110	0.578316
$846$	0	0
$847$	16.5066	0.567173
$848$	0	0
$849$	25.6483	0.880248
$850$	0	0
$851$	6.60483	0.226411
$852$	0	0
$853$	29.5223	1.01082	0.505412	−	0.862878i	$-0.331340\pi$
$853$	29.5223	1.01082	0.505412	+	0.862878i	$0.331340\pi$
$854$	0	0
$855$	−1.71767	−0.0587430
$856$	0	0
$857$	24.3258	0.830954	0.415477	−	0.909604i	$-0.363615\pi$
$857$	24.3258	0.830954	0.415477	+	0.909604i	$0.363615\pi$
$858$	0	0
$859$	−51.0943	−1.74331	−0.871657	−	0.490116i	$-0.836954\pi$
$859$	−51.0943	−1.74331	−0.871657	+	0.490116i	$0.836954\pi$
$860$	0	0
$861$	11.9248	0.406395
$862$	0	0
$863$	37.9248	1.29097	0.645487	−	0.763771i	$-0.276654\pi$
$863$	37.9248	1.29097	0.645487	+	0.763771i	$0.276654\pi$
$864$	0	0
$865$	−7.09825	−0.241348
$866$	0	0
$867$	9.97698	0.338836
$868$	0	0
$869$	39.1462	1.32794
$870$	0	0
$871$	42.2433	1.43136
$872$	0	0
$873$	8.50659	0.287904
$874$	0	0
$875$	−33.2252	−1.12322
$876$	0	0
$877$	33.1695	1.12005	0.560027	−	0.828474i	$-0.310791\pi$
$877$	33.1695	1.12005	0.560027	+	0.828474i	$0.310791\pi$
$878$	0	0
$879$	24.1622	0.814971
$880$	0	0
$881$	−28.1866	−0.949632	−0.474816	−	0.880085i	$-0.657485\pi$
$881$	−28.1866	−0.949632	−0.474816	+	0.880085i	$0.657485\pi$
$882$	0	0
$883$	24.9419	0.839362	0.419681	−	0.907672i	$-0.362142\pi$
$883$	24.9419	0.839362	0.419681	+	0.907672i	$0.362142\pi$
$884$	0	0
$885$	7.15633	0.240557
$886$	0	0
$887$	−0.826531	−0.0277522	−0.0138761	−	0.999904i	$-0.504417\pi$
$887$	−0.826531	−0.0277522	−0.0138761	+	0.999904i	$0.504417\pi$
$888$	0	0
$889$	2.19394	0.0735823
$890$	0	0
$891$	−3.76845	−0.126248
$892$	0	0
$893$	24.0910	0.806173
$894$	0	0
$895$	−7.63259	−0.255130
$896$	0	0
$897$	13.2750	0.443241
$898$	0	0
$899$	−39.7078	−1.32433
$900$	0	0
$901$	19.4412	0.647681
$902$	0	0
$903$	38.7064	1.28807
$904$	0	0
$905$	−2.72099	−0.0904488
$906$	0	0
$907$	23.8291	0.791232	0.395616	−	0.918416i	$-0.370531\pi$
$907$	23.8291	0.791232	0.395616	+	0.918416i	$0.370531\pi$
$908$	0	0
$909$	−16.4944	−0.547084
$910$	0	0
$911$	19.5091	0.646367	0.323183	−	0.946336i	$-0.395247\pi$
$911$	19.5091	0.646367	0.323183	+	0.946336i	$0.395247\pi$
$912$	0	0
$913$	24.2823	0.803628
$914$	0	0
$915$	−0.523730	−0.0173140
$916$	0	0
$917$	−23.1695	−0.765124
$918$	0	0
$919$	−5.07125	−0.167285	−0.0836426	−	0.996496i	$-0.526655\pi$
$919$	−5.07125	−0.167285	−0.0836426	+	0.996496i	$0.526655\pi$
$920$	0	0
$921$	−3.48119	−0.114709
$922$	0	0
$923$	33.9003	1.11584
$924$	0	0
$925$	13.9189	0.457651
$926$	0	0
$927$	−6.12601	−0.201205
$928$	0	0
$929$	−20.2262	−0.663599	−0.331799	−	0.943350i	$-0.607656\pi$
$929$	−20.2262	−0.663599	−0.331799	+	0.943350i	$0.607656\pi$
$930$	0	0
$931$	−49.8350	−1.63328
$932$	0	0
$933$	12.2750	0.401867
$934$	0	0
$935$	13.2144	0.432157
$936$	0	0
$937$	−33.0092	−1.07836	−0.539182	−	0.842189i	$-0.681266\pi$
$937$	−33.0092	−1.07836	−0.539182	+	0.842189i	$0.681266\pi$
$938$	0	0
$939$	−8.49341	−0.277172
$940$	0	0
$941$	17.0191	0.554805	0.277403	−	0.960754i	$-0.410526\pi$
$941$	17.0191	0.554805	0.277403	+	0.960754i	$0.410526\pi$
$942$	0	0
$943$	−4.98683	−0.162393
$944$	0	0
$945$	3.48119	0.113243
$946$	0	0
$947$	−41.2360	−1.33999	−0.669995	−	0.742365i	$-0.733704\pi$
$947$	−41.2360	−1.33999	−0.669995	+	0.742365i	$0.733704\pi$
$948$	0	0
$949$	72.7631	2.36199
$950$	0	0
$951$	23.5574	0.763900
$952$	0	0
$953$	−61.0698	−1.97825	−0.989123	−	0.147090i	$-0.953009\pi$
$953$	−61.0698	−1.97825	−0.989123	+	0.147090i	$0.953009\pi$
$954$	0	0
$955$	−8.16125	−0.264092
$956$	0	0
$957$	14.2012	0.459061
$958$	0	0
$959$	−33.5501	−1.08339
$960$	0	0
$961$	80.0263	2.58150
$962$	0	0
$963$	6.49929	0.209437
$964$	0	0
$965$	5.85352	0.188432
$966$	0	0
$967$	1.55149	0.0498926	0.0249463	−	0.999689i	$-0.492059\pi$
$967$	1.55149	0.0498926	0.0249463	+	0.999689i	$0.492059\pi$
$968$	0	0
$969$	13.2144	0.424508
$970$	0	0
$971$	−25.4812	−0.817730	−0.408865	−	0.912595i	$-0.634075\pi$
$971$	−25.4812	−0.817730	−0.408865	+	0.912595i	$0.634075\pi$
$972$	0	0
$973$	−73.2819	−2.34931
$974$	0	0
$975$	27.9756	0.895935
$976$	0	0
$977$	15.1432	0.484472	0.242236	−	0.970217i	$-0.422119\pi$
$977$	15.1432	0.484472	0.242236	+	0.970217i	$0.422119\pi$
$978$	0	0
$979$	44.9654	1.43710
$980$	0	0
$981$	−0.0811024	−0.00258940
$982$	0	0
$983$	−27.0376	−0.862366	−0.431183	−	0.902265i	$-0.641904\pi$
$983$	−27.0376	−0.862366	−0.431183	+	0.902265i	$0.641904\pi$
$984$	0	0
$985$	−12.5099	−0.398599
$986$	0	0
$987$	−48.8251	−1.55412
$988$	0	0
$989$	−16.1866	−0.514705
$990$	0	0
$991$	−24.1055	−0.765738	−0.382869	−	0.923803i	$-0.625064\pi$
$991$	−24.1055	−0.765738	−0.382869	+	0.923803i	$0.625064\pi$
$992$	0	0
$993$	28.5052	0.904584
$994$	0	0
$995$	−5.52705	−0.175219
$996$	0	0
$997$	−2.54420	−0.0805756	−0.0402878	−	0.999188i	$-0.512827\pi$
$997$	−2.54420	−0.0805756	−0.0402878	+	0.999188i	$0.512827\pi$
$998$	0	0
$999$	−3.06300	−0.0969092

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.l.1.3		3
4.3	odd	2		1002.2.a.h.1.3	✓	3
12.11	even	2		3006.2.a.o.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.h.1.3	✓	3	4.3	odd	2
3006.2.a.o.1.1		3	12.11	even	2
8016.2.a.l.1.3		3	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.675131 q^{5} +5.15633 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.675131 q^{5} +5.15633 q^{7} +1.00000 q^{9} -3.76845 q^{11} -6.15633 q^{13} +0.675131 q^{15} -5.19394 q^{17} -2.54420 q^{19} +5.15633 q^{21} -2.15633 q^{23} -4.54420 q^{25} +1.00000 q^{27} -3.76845 q^{29} +10.5369 q^{31} -3.76845 q^{33} +3.48119 q^{35} -3.06300 q^{37} -6.15633 q^{39} +2.31265 q^{41} +7.50659 q^{43} +0.675131 q^{45} -9.46898 q^{47} +19.5877 q^{49} -5.19394 q^{51} -3.74306 q^{53} -2.54420 q^{55} -2.54420 q^{57} +10.5999 q^{59} -0.775746 q^{61} +5.15633 q^{63} -4.15633 q^{65} -6.86177 q^{67} -2.15633 q^{69} -5.50659 q^{71} -11.8192 q^{73} -4.54420 q^{75} -19.4314 q^{77} -10.3879 q^{79} +1.00000 q^{81} -6.44358 q^{83} -3.50659 q^{85} -3.76845 q^{87} -11.9321 q^{89} -31.7440 q^{91} +10.5369 q^{93} -1.71767 q^{95} +8.50659 q^{97} -3.76845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} + 5 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^5 + 5 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{5} + 5 q^{7} + 3 q^{9} - 8 q^{13} - 3 q^{15} - 16 q^{17} + 2 q^{19} + 5 q^{21} + 4 q^{23} - 4 q^{25} + 3 q^{27} + 9 q^{31} + 5 q^{35} - 5 q^{37} - 8 q^{39} - 14 q^{41} + 2 q^{43} - 3 q^{45} + 3 q^{47} + 6 q^{49} - 16 q^{51} - 15 q^{53} + 2 q^{55} + 2 q^{57} + 5 q^{59} - 4 q^{61} + 5 q^{63} - 2 q^{65} - 3 q^{67} + 4 q^{69} + 4 q^{71} + 6 q^{73} - 4 q^{75} - 16 q^{77} - 32 q^{79} + 3 q^{81} - 3 q^{83} + 10 q^{85} - 27 q^{89} - 32 q^{91} + 9 q^{93} - 24 q^{95} + 5 q^{97}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 + 5 * q^7 + 3 * q^9 - 8 * q^13 - 3 * q^15 - 16 * q^17 + 2 * q^19 + 5 * q^21 + 4 * q^23 - 4 * q^25 + 3 * q^27 + 9 * q^31 + 5 * q^35 - 5 * q^37 - 8 * q^39 - 14 * q^41 + 2 * q^43 - 3 * q^45 + 3 * q^47 + 6 * q^49 - 16 * q^51 - 15 * q^53 + 2 * q^55 + 2 * q^57 + 5 * q^59 - 4 * q^61 + 5 * q^63 - 2 * q^65 - 3 * q^67 + 4 * q^69 + 4 * q^71 + 6 * q^73 - 4 * q^75 - 16 * q^77 - 32 * q^79 + 3 * q^81 - 3 * q^83 + 10 * q^85 - 27 * q^89 - 32 * q^91 + 9 * q^93 - 24 * q^95 + 5 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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