LMFDB - Embedded newform 8016.2.a.bg.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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 2 x^{12} - 49 x^{11} + 99 x^{10} + 901 x^{9} - 1879 x^{8} - 7582 x^{7} + 16968 x^{6} + \cdots + 12144 $ x^13 - 2x^12 - 49x^11 + 99x^10 + 901x^9 - 1879x^8 - 7582x^7 + 16968x^6 + 26911x^5 - 72240x^4 - 14532x^3 + 112850x^2 - 72184x + 12144
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.04519$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -2.04519 q^{5} -1.81275 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.04519 q^{5} -1.81275 q^{7} +1.00000 q^{9} +1.62116 q^{11} -3.43477 q^{13} +2.04519 q^{15} -4.77393 q^{17} -5.04637 q^{19} +1.81275 q^{21} -3.71784 q^{23} -0.817207 q^{25} -1.00000 q^{27} +5.69513 q^{29} -4.23616 q^{31} -1.62116 q^{33} +3.70741 q^{35} -2.11506 q^{37} +3.43477 q^{39} +2.68365 q^{41} -12.6956 q^{43} -2.04519 q^{45} +8.44574 q^{47} -3.71395 q^{49} +4.77393 q^{51} -2.62431 q^{53} -3.31557 q^{55} +5.04637 q^{57} -0.400667 q^{59} +0.131727 q^{61} -1.81275 q^{63} +7.02474 q^{65} -6.63043 q^{67} +3.71784 q^{69} -12.2723 q^{71} +13.2063 q^{73} +0.817207 q^{75} -2.93875 q^{77} -2.93266 q^{79} +1.00000 q^{81} -15.4039 q^{83} +9.76358 q^{85} -5.69513 q^{87} +5.43002 q^{89} +6.22636 q^{91} +4.23616 q^{93} +10.3208 q^{95} +9.62800 q^{97} +1.62116 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9}+O(q^{10}) $ 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9	$ 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9} - 11 q^{11} + 12 q^{13} - 2 q^{15} + 15 q^{17} - 14 q^{19} + q^{21} - 9 q^{23} + 37 q^{25} - 13 q^{27} - 3 q^{29} + 17 q^{31} + 11 q^{33} - 15 q^{35} + 16 q^{37} - 12 q^{39} + 12 q^{41} - 20 q^{43} + 2 q^{45} + 6 q^{47} + 26 q^{49} - 15 q^{51} - 12 q^{53} - 7 q^{55} + 14 q^{57} - 14 q^{59} + 24 q^{61} - q^{63} + 8 q^{65} - 3 q^{67} + 9 q^{69} - 17 q^{71} + 34 q^{73} - 37 q^{75} + 30 q^{77} - 10 q^{79} + 13 q^{81} - 44 q^{83} + 25 q^{85} + 3 q^{87} + 25 q^{89} - 29 q^{91} - 17 q^{93} + 15 q^{95} + 38 q^{97} - 11 q^{99}+O(q^{100}) $ 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9 - 11 * q^11 + 12 * q^13 - 2 * q^15 + 15 * q^17 - 14 * q^19 + q^21 - 9 * q^23 + 37 * q^25 - 13 * q^27 - 3 * q^29 + 17 * q^31 + 11 * q^33 - 15 * q^35 + 16 * q^37 - 12 * q^39 + 12 * q^41 - 20 * q^43 + 2 * q^45 + 6 * q^47 + 26 * q^49 - 15 * q^51 - 12 * q^53 - 7 * q^55 + 14 * q^57 - 14 * q^59 + 24 * q^61 - q^63 + 8 * q^65 - 3 * q^67 + 9 * q^69 - 17 * q^71 + 34 * q^73 - 37 * q^75 + 30 * q^77 - 10 * q^79 + 13 * q^81 - 44 * q^83 + 25 * q^85 + 3 * q^87 + 25 * q^89 - 29 * q^91 - 17 * q^93 + 15 * q^95 + 38 * q^97 - 11 * 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
$5$	−2.04519	−0.914636	−0.457318	−	0.889303i	$-0.651190\pi$
$5$	−2.04519	−0.914636	−0.457318	+	0.889303i	$0.651190\pi$
$6$	0	0
$7$	−1.81275	−0.685154	−0.342577	−	0.939490i	$-0.611300\pi$
$7$	−1.81275	−0.685154	−0.342577	+	0.939490i	$0.611300\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.62116	0.488797	0.244398	−	0.969675i	$-0.421410\pi$
$11$	1.62116	0.488797	0.244398	+	0.969675i	$0.421410\pi$
$12$	0	0
$13$	−3.43477	−0.952633	−0.476316	−	0.879274i	$-0.658028\pi$
$13$	−3.43477	−0.952633	−0.476316	+	0.879274i	$0.658028\pi$
$14$	0	0
$15$	2.04519	0.528065
$16$	0	0
$17$	−4.77393	−1.15785	−0.578924	−	0.815381i	$-0.696527\pi$
$17$	−4.77393	−1.15785	−0.578924	+	0.815381i	$0.696527\pi$
$18$	0	0
$19$	−5.04637	−1.15772	−0.578859	−	0.815428i	$-0.696502\pi$
$19$	−5.04637	−1.15772	−0.578859	+	0.815428i	$0.696502\pi$
$20$	0	0
$21$	1.81275	0.395574
$22$	0	0
$23$	−3.71784	−0.775222	−0.387611	−	0.921823i	$-0.626700\pi$
$23$	−3.71784	−0.775222	−0.387611	+	0.921823i	$0.626700\pi$
$24$	0	0
$25$	−0.817207	−0.163441
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.69513	1.05756	0.528780	−	0.848759i	$-0.322650\pi$
$29$	5.69513	1.05756	0.528780	+	0.848759i	$0.322650\pi$
$30$	0	0
$31$	−4.23616	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$31$	−4.23616	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$32$	0	0
$33$	−1.62116	−0.282207
$34$	0	0
$35$	3.70741	0.626666
$36$	0	0
$37$	−2.11506	−0.347714	−0.173857	−	0.984771i	$-0.555623\pi$
$37$	−2.11506	−0.347714	−0.173857	+	0.984771i	$0.555623\pi$
$38$	0	0
$39$	3.43477	0.550003
$40$	0	0
$41$	2.68365	0.419116	0.209558	−	0.977796i	$-0.432797\pi$
$41$	2.68365	0.419116	0.209558	+	0.977796i	$0.432797\pi$
$42$	0	0
$43$	−12.6956	−1.93606	−0.968030	−	0.250834i	$-0.919295\pi$
$43$	−12.6956	−1.93606	−0.968030	+	0.250834i	$0.919295\pi$
$44$	0	0
$45$	−2.04519	−0.304879
$46$	0	0
$47$	8.44574	1.23194	0.615969	−	0.787771i	$-0.288765\pi$
$47$	8.44574	1.23194	0.615969	+	0.787771i	$0.288765\pi$
$48$	0	0
$49$	−3.71395	−0.530564
$50$	0	0
$51$	4.77393	0.668484
$52$	0	0
$53$	−2.62431	−0.360476	−0.180238	−	0.983623i	$-0.557687\pi$
$53$	−2.62431	−0.360476	−0.180238	+	0.983623i	$0.557687\pi$
$54$	0	0
$55$	−3.31557	−0.447071
$56$	0	0
$57$	5.04637	0.668409
$58$	0	0
$59$	−0.400667	−0.0521624	−0.0260812	−	0.999660i	$-0.508303\pi$
$59$	−0.400667	−0.0521624	−0.0260812	+	0.999660i	$0.508303\pi$
$60$	0	0
$61$	0.131727	0.0168660	0.00843298	−	0.999964i	$-0.497316\pi$
$61$	0.131727	0.0168660	0.00843298	+	0.999964i	$0.497316\pi$
$62$	0	0
$63$	−1.81275	−0.228385
$64$	0	0
$65$	7.02474	0.871312
$66$	0	0
$67$	−6.63043	−0.810036	−0.405018	−	0.914309i	$-0.632735\pi$
$67$	−6.63043	−0.810036	−0.405018	+	0.914309i	$0.632735\pi$
$68$	0	0
$69$	3.71784	0.447575
$70$	0	0
$71$	−12.2723	−1.45646	−0.728228	−	0.685335i	$-0.759656\pi$
$71$	−12.2723	−1.45646	−0.728228	+	0.685335i	$0.759656\pi$
$72$	0	0
$73$	13.2063	1.54568	0.772840	−	0.634601i	$-0.218836\pi$
$73$	13.2063	1.54568	0.772840	+	0.634601i	$0.218836\pi$
$74$	0	0
$75$	0.817207	0.0943629
$76$	0	0
$77$	−2.93875	−0.334901
$78$	0	0
$79$	−2.93266	−0.329950	−0.164975	−	0.986298i	$-0.552754\pi$
$79$	−2.93266	−0.329950	−0.164975	+	0.986298i	$0.552754\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.4039	−1.69080	−0.845399	−	0.534135i	$-0.820637\pi$
$83$	−15.4039	−1.69080	−0.845399	+	0.534135i	$0.820637\pi$
$84$	0	0
$85$	9.76358	1.05901
$86$	0	0
$87$	−5.69513	−0.610582
$88$	0	0
$89$	5.43002	0.575581	0.287791	−	0.957693i	$-0.407079\pi$
$89$	5.43002	0.575581	0.287791	+	0.957693i	$0.407079\pi$
$90$	0	0
$91$	6.22636	0.652700
$92$	0	0
$93$	4.23616	0.439269
$94$	0	0
$95$	10.3208	1.05889
$96$	0	0
$97$	9.62800	0.977575	0.488788	−	0.872403i	$-0.337439\pi$
$97$	9.62800	0.977575	0.488788	+	0.872403i	$0.337439\pi$
$98$	0	0
$99$	1.62116	0.162932
$100$	0	0
$101$	−11.8915	−1.18325	−0.591625	−	0.806213i	$-0.701513\pi$
$101$	−11.8915	−1.18325	−0.591625	+	0.806213i	$0.701513\pi$
$102$	0	0
$103$	−6.78324	−0.668372	−0.334186	−	0.942507i	$-0.608461\pi$
$103$	−6.78324	−0.668372	−0.334186	+	0.942507i	$0.608461\pi$
$104$	0	0
$105$	−3.70741	−0.361806
$106$	0	0
$107$	3.79200	0.366586	0.183293	−	0.983058i	$-0.441324\pi$
$107$	3.79200	0.366586	0.183293	+	0.983058i	$0.441324\pi$
$108$	0	0
$109$	13.8797	1.32944	0.664719	−	0.747093i	$-0.268551\pi$
$109$	13.8797	1.32944	0.664719	+	0.747093i	$0.268551\pi$
$110$	0	0
$111$	2.11506	0.200753
$112$	0	0
$113$	−19.5403	−1.83820	−0.919099	−	0.394027i	$-0.871082\pi$
$113$	−19.5403	−1.83820	−0.919099	+	0.394027i	$0.871082\pi$
$114$	0	0
$115$	7.60367	0.709046
$116$	0	0
$117$	−3.43477	−0.317544
$118$	0	0
$119$	8.65393	0.793304
$120$	0	0
$121$	−8.37185	−0.761077
$122$	0	0
$123$	−2.68365	−0.241977
$124$	0	0
$125$	11.8973	1.06413
$126$	0	0
$127$	4.29936	0.381507	0.190753	−	0.981638i	$-0.438907\pi$
$127$	4.29936	0.381507	0.190753	+	0.981638i	$0.438907\pi$
$128$	0	0
$129$	12.6956	1.11778
$130$	0	0
$131$	−15.0965	−1.31898	−0.659492	−	0.751712i	$-0.729229\pi$
$131$	−15.0965	−1.31898	−0.659492	+	0.751712i	$0.729229\pi$
$132$	0	0
$133$	9.14780	0.793215
$134$	0	0
$135$	2.04519	0.176022
$136$	0	0
$137$	−1.35974	−0.116170	−0.0580852	−	0.998312i	$-0.518499\pi$
$137$	−1.35974	−0.116170	−0.0580852	+	0.998312i	$0.518499\pi$
$138$	0	0
$139$	19.9242	1.68995	0.844973	−	0.534808i	$-0.179616\pi$
$139$	19.9242	1.68995	0.844973	+	0.534808i	$0.179616\pi$
$140$	0	0
$141$	−8.44574	−0.711259
$142$	0	0
$143$	−5.56829	−0.465644
$144$	0	0
$145$	−11.6476	−0.967282
$146$	0	0
$147$	3.71395	0.306321
$148$	0	0
$149$	−7.38010	−0.604601	−0.302301	−	0.953213i	$-0.597755\pi$
$149$	−7.38010	−0.604601	−0.302301	+	0.953213i	$0.597755\pi$
$150$	0	0
$151$	−15.5284	−1.26368	−0.631841	−	0.775098i	$-0.717700\pi$
$151$	−15.5284	−1.26368	−0.631841	+	0.775098i	$0.717700\pi$
$152$	0	0
$153$	−4.77393	−0.385949
$154$	0	0
$155$	8.66373	0.695888
$156$	0	0
$157$	19.4332	1.55094	0.775470	−	0.631385i	$-0.217513\pi$
$157$	19.4332	1.55094	0.775470	+	0.631385i	$0.217513\pi$
$158$	0	0
$159$	2.62431	0.208121
$160$	0	0
$161$	6.73949	0.531147
$162$	0	0
$163$	−9.15822	−0.717327	−0.358664	−	0.933467i	$-0.616767\pi$
$163$	−9.15822	−0.717327	−0.358664	+	0.933467i	$0.616767\pi$
$164$	0	0
$165$	3.31557	0.258117
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−1.20239	−0.0924912
$170$	0	0
$171$	−5.04637	−0.385906
$172$	0	0
$173$	−18.5002	−1.40655	−0.703274	−	0.710919i	$-0.748279\pi$
$173$	−18.5002	−1.40655	−0.703274	+	0.710919i	$0.748279\pi$
$174$	0	0
$175$	1.48139	0.111982
$176$	0	0
$177$	0.400667	0.0301160
$178$	0	0
$179$	−19.2383	−1.43794	−0.718969	−	0.695042i	$-0.755386\pi$
$179$	−19.2383	−1.43794	−0.718969	+	0.695042i	$0.755386\pi$
$180$	0	0
$181$	26.1135	1.94100	0.970501	−	0.241098i	$-0.0775076\pi$
$181$	26.1135	1.94100	0.970501	+	0.241098i	$0.0775076\pi$
$182$	0	0
$183$	−0.131727	−0.00973757
$184$	0	0
$185$	4.32569	0.318031
$186$	0	0
$187$	−7.73929	−0.565953
$188$	0	0
$189$	1.81275	0.131858
$190$	0	0
$191$	26.0096	1.88199	0.940993	−	0.338425i	$-0.109894\pi$
$191$	26.0096	1.88199	0.940993	+	0.338425i	$0.109894\pi$
$192$	0	0
$193$	6.70727	0.482800	0.241400	−	0.970426i	$-0.422393\pi$
$193$	6.70727	0.482800	0.241400	+	0.970426i	$0.422393\pi$
$194$	0	0
$195$	−7.02474	−0.503052
$196$	0	0
$197$	−20.2795	−1.44486	−0.722429	−	0.691445i	$-0.756974\pi$
$197$	−20.2795	−1.44486	−0.722429	+	0.691445i	$0.756974\pi$
$198$	0	0
$199$	20.4915	1.45260	0.726301	−	0.687377i	$-0.241238\pi$
$199$	20.4915	1.45260	0.726301	+	0.687377i	$0.241238\pi$
$200$	0	0
$201$	6.63043	0.467675
$202$	0	0
$203$	−10.3238	−0.724591
$204$	0	0
$205$	−5.48857	−0.383339
$206$	0	0
$207$	−3.71784	−0.258407
$208$	0	0
$209$	−8.18096	−0.565889
$210$	0	0
$211$	9.74296	0.670733	0.335366	−	0.942088i	$-0.391140\pi$
$211$	9.74296	0.670733	0.335366	+	0.942088i	$0.391140\pi$
$212$	0	0
$213$	12.2723	0.840885
$214$	0	0
$215$	25.9649	1.77079
$216$	0	0
$217$	7.67908	0.521290
$218$	0	0
$219$	−13.2063	−0.892398
$220$	0	0
$221$	16.3973	1.10300
$222$	0	0
$223$	−19.0423	−1.27516	−0.637582	−	0.770382i	$-0.720065\pi$
$223$	−19.0423	−1.27516	−0.637582	+	0.770382i	$0.720065\pi$
$224$	0	0
$225$	−0.817207	−0.0544804
$226$	0	0
$227$	−24.0274	−1.59476	−0.797378	−	0.603480i	$-0.793780\pi$
$227$	−24.0274	−1.59476	−0.797378	+	0.603480i	$0.793780\pi$
$228$	0	0
$229$	−12.6375	−0.835109	−0.417555	−	0.908652i	$-0.637113\pi$
$229$	−12.6375	−0.835109	−0.417555	+	0.908652i	$0.637113\pi$
$230$	0	0
$231$	2.93875	0.193355
$232$	0	0
$233$	25.0038	1.63805	0.819027	−	0.573754i	$-0.194514\pi$
$233$	25.0038	1.63805	0.819027	+	0.573754i	$0.194514\pi$
$234$	0	0
$235$	−17.2731	−1.12677
$236$	0	0
$237$	2.93266	0.190497
$238$	0	0
$239$	−12.9032	−0.834638	−0.417319	−	0.908760i	$-0.637030\pi$
$239$	−12.9032	−0.834638	−0.417319	+	0.908760i	$0.637030\pi$
$240$	0	0
$241$	−10.1834	−0.655972	−0.327986	−	0.944683i	$-0.606370\pi$
$241$	−10.1834	−0.655972	−0.327986	+	0.944683i	$0.606370\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	7.59572	0.485273
$246$	0	0
$247$	17.3331	1.10288
$248$	0	0
$249$	15.4039	0.976183
$250$	0	0
$251$	−0.514670	−0.0324857	−0.0162428	−	0.999868i	$-0.505170\pi$
$251$	−0.514670	−0.0324857	−0.0162428	+	0.999868i	$0.505170\pi$
$252$	0	0
$253$	−6.02719	−0.378926
$254$	0	0
$255$	−9.76358	−0.611419
$256$	0	0
$257$	22.3692	1.39535	0.697677	−	0.716412i	$-0.254217\pi$
$257$	22.3692	1.39535	0.697677	+	0.716412i	$0.254217\pi$
$258$	0	0
$259$	3.83407	0.238237
$260$	0	0
$261$	5.69513	0.352520
$262$	0	0
$263$	15.7692	0.972368	0.486184	−	0.873857i	$-0.338389\pi$
$263$	15.7692	0.972368	0.486184	+	0.873857i	$0.338389\pi$
$264$	0	0
$265$	5.36720	0.329705
$266$	0	0
$267$	−5.43002	−0.332312
$268$	0	0
$269$	14.4917	0.883575	0.441788	−	0.897120i	$-0.354345\pi$
$269$	14.4917	0.883575	0.441788	+	0.897120i	$0.354345\pi$
$270$	0	0
$271$	28.2465	1.71585	0.857927	−	0.513772i	$-0.171752\pi$
$271$	28.2465	1.71585	0.857927	+	0.513772i	$0.171752\pi$
$272$	0	0
$273$	−6.22636	−0.376836
$274$	0	0
$275$	−1.32482	−0.0798896
$276$	0	0
$277$	−5.04261	−0.302981	−0.151490	−	0.988459i	$-0.548407\pi$
$277$	−5.04261	−0.302981	−0.151490	+	0.988459i	$0.548407\pi$
$278$	0	0
$279$	−4.23616	−0.253612
$280$	0	0
$281$	−16.8548	−1.00547	−0.502736	−	0.864440i	$-0.667673\pi$
$281$	−16.8548	−1.00547	−0.502736	+	0.864440i	$0.667673\pi$
$282$	0	0
$283$	−23.2569	−1.38248	−0.691238	−	0.722627i	$-0.742934\pi$
$283$	−23.2569	−1.38248	−0.691238	+	0.722627i	$0.742934\pi$
$284$	0	0
$285$	−10.3208	−0.611350
$286$	0	0
$287$	−4.86478	−0.287159
$288$	0	0
$289$	5.79041	0.340612
$290$	0	0
$291$	−9.62800	−0.564403
$292$	0	0
$293$	9.38017	0.547995	0.273998	−	0.961730i	$-0.411654\pi$
$293$	9.38017	0.547995	0.273998	+	0.961730i	$0.411654\pi$
$294$	0	0
$295$	0.819440	0.0477096
$296$	0	0
$297$	−1.62116	−0.0940690
$298$	0	0
$299$	12.7699	0.738502
$300$	0	0
$301$	23.0139	1.32650
$302$	0	0
$303$	11.8915	0.683150
$304$	0	0
$305$	−0.269407	−0.0154262
$306$	0	0
$307$	−23.3987	−1.33543	−0.667716	−	0.744416i	$-0.732728\pi$
$307$	−23.3987	−1.33543	−0.667716	+	0.744416i	$0.732728\pi$
$308$	0	0
$309$	6.78324	0.385885
$310$	0	0
$311$	4.38962	0.248912	0.124456	−	0.992225i	$-0.460281\pi$
$311$	4.38962	0.248912	0.124456	+	0.992225i	$0.460281\pi$
$312$	0	0
$313$	30.5307	1.72570	0.862848	−	0.505464i	$-0.168679\pi$
$313$	30.5307	1.72570	0.862848	+	0.505464i	$0.168679\pi$
$314$	0	0
$315$	3.70741	0.208889
$316$	0	0
$317$	23.0063	1.29216	0.646082	−	0.763268i	$-0.276406\pi$
$317$	23.0063	1.29216	0.646082	+	0.763268i	$0.276406\pi$
$318$	0	0
$319$	9.23270	0.516932
$320$	0	0
$321$	−3.79200	−0.211649
$322$	0	0
$323$	24.0910	1.34046
$324$	0	0
$325$	2.80691	0.155700
$326$	0	0
$327$	−13.8797	−0.767552
$328$	0	0
$329$	−15.3100	−0.844067
$330$	0	0
$331$	−21.5613	−1.18512	−0.592559	−	0.805527i	$-0.701882\pi$
$331$	−21.5613	−1.18512	−0.592559	+	0.805527i	$0.701882\pi$
$332$	0	0
$333$	−2.11506	−0.115905
$334$	0	0
$335$	13.5605	0.740888
$336$	0	0
$337$	−3.17835	−0.173136	−0.0865679	−	0.996246i	$-0.527590\pi$
$337$	−3.17835	−0.173136	−0.0865679	+	0.996246i	$0.527590\pi$
$338$	0	0
$339$	19.5403	1.06128
$340$	0	0
$341$	−6.86747	−0.371894
$342$	0	0
$343$	19.4217	1.04867
$344$	0	0
$345$	−7.60367	−0.409368
$346$	0	0
$347$	2.98591	0.160292	0.0801460	−	0.996783i	$-0.474461\pi$
$347$	2.98591	0.160292	0.0801460	+	0.996783i	$0.474461\pi$
$348$	0	0
$349$	23.1887	1.24126	0.620631	−	0.784102i	$-0.286876\pi$
$349$	23.1887	1.24126	0.620631	+	0.784102i	$0.286876\pi$
$350$	0	0
$351$	3.43477	0.183334
$352$	0	0
$353$	20.7847	1.10626	0.553128	−	0.833096i	$-0.313434\pi$
$353$	20.7847	1.10626	0.553128	+	0.833096i	$0.313434\pi$
$354$	0	0
$355$	25.0992	1.33213
$356$	0	0
$357$	−8.65393	−0.458014
$358$	0	0
$359$	8.44829	0.445884	0.222942	−	0.974832i	$-0.428434\pi$
$359$	8.44829	0.445884	0.222942	+	0.974832i	$0.428434\pi$
$360$	0	0
$361$	6.46589	0.340310
$362$	0	0
$363$	8.37185	0.439408
$364$	0	0
$365$	−27.0093	−1.41373
$366$	0	0
$367$	−2.08409	−0.108789	−0.0543944	−	0.998520i	$-0.517323\pi$
$367$	−2.08409	−0.108789	−0.0543944	+	0.998520i	$0.517323\pi$
$368$	0	0
$369$	2.68365	0.139705
$370$	0	0
$371$	4.75721	0.246982
$372$	0	0
$373$	−17.7296	−0.918005	−0.459002	−	0.888435i	$-0.651793\pi$
$373$	−17.7296	−0.918005	−0.459002	+	0.888435i	$0.651793\pi$
$374$	0	0
$375$	−11.8973	−0.614373
$376$	0	0
$377$	−19.5614	−1.00747
$378$	0	0
$379$	20.5594	1.05607	0.528034	−	0.849223i	$-0.322929\pi$
$379$	20.5594	1.05607	0.528034	+	0.849223i	$0.322929\pi$
$380$	0	0
$381$	−4.29936	−0.220263
$382$	0	0
$383$	20.1341	1.02880	0.514401	−	0.857550i	$-0.328014\pi$
$383$	20.1341	1.02880	0.514401	+	0.857550i	$0.328014\pi$
$384$	0	0
$385$	6.01029	0.306313
$386$	0	0
$387$	−12.6956	−0.645353
$388$	0	0
$389$	−30.1426	−1.52829	−0.764146	−	0.645043i	$-0.776839\pi$
$389$	−30.1426	−1.52829	−0.764146	+	0.645043i	$0.776839\pi$
$390$	0	0
$391$	17.7487	0.897590
$392$	0	0
$393$	15.0965	0.761516
$394$	0	0
$395$	5.99784	0.301784
$396$	0	0
$397$	15.2249	0.764116	0.382058	−	0.924138i	$-0.375215\pi$
$397$	15.2249	0.764116	0.382058	+	0.924138i	$0.375215\pi$
$398$	0	0
$399$	−9.14780	−0.457963
$400$	0	0
$401$	11.0171	0.550166	0.275083	−	0.961421i	$-0.411295\pi$
$401$	11.0171	0.550166	0.275083	+	0.961421i	$0.411295\pi$
$402$	0	0
$403$	14.5502	0.724797
$404$	0	0
$405$	−2.04519	−0.101626
$406$	0	0
$407$	−3.42884	−0.169961
$408$	0	0
$409$	18.1892	0.899400	0.449700	−	0.893180i	$-0.351531\pi$
$409$	18.1892	0.899400	0.449700	+	0.893180i	$0.351531\pi$
$410$	0	0
$411$	1.35974	0.0670710
$412$	0	0
$413$	0.726308	0.0357393
$414$	0	0
$415$	31.5039	1.54646
$416$	0	0
$417$	−19.9242	−0.975691
$418$	0	0
$419$	19.1465	0.935370	0.467685	−	0.883895i	$-0.345088\pi$
$419$	19.1465	0.935370	0.467685	+	0.883895i	$0.345088\pi$
$420$	0	0
$421$	34.1391	1.66384	0.831918	−	0.554899i	$-0.187243\pi$
$421$	34.1391	1.66384	0.831918	+	0.554899i	$0.187243\pi$
$422$	0	0
$423$	8.44574	0.410646
$424$	0	0
$425$	3.90129	0.189240
$426$	0	0
$427$	−0.238788	−0.0115558
$428$	0	0
$429$	5.56829	0.268840
$430$	0	0
$431$	37.9439	1.82769	0.913846	−	0.406061i	$-0.133098\pi$
$431$	37.9439	1.82769	0.913846	+	0.406061i	$0.133098\pi$
$432$	0	0
$433$	31.0410	1.49174	0.745869	−	0.666093i	$-0.232035\pi$
$433$	31.0410	1.49174	0.745869	+	0.666093i	$0.232035\pi$
$434$	0	0
$435$	11.6476	0.558460
$436$	0	0
$437$	18.7616	0.897488
$438$	0	0
$439$	13.8058	0.658915	0.329458	−	0.944170i	$-0.393134\pi$
$439$	13.8058	0.658915	0.329458	+	0.944170i	$0.393134\pi$
$440$	0	0
$441$	−3.71395	−0.176855
$442$	0	0
$443$	−27.9941	−1.33004	−0.665021	−	0.746825i	$-0.731577\pi$
$443$	−27.9941	−1.33004	−0.665021	+	0.746825i	$0.731577\pi$
$444$	0	0
$445$	−11.1054	−0.526447
$446$	0	0
$447$	7.38010	0.349067
$448$	0	0
$449$	17.3679	0.819644	0.409822	−	0.912166i	$-0.365591\pi$
$449$	17.3679	0.819644	0.409822	+	0.912166i	$0.365591\pi$
$450$	0	0
$451$	4.35062	0.204863
$452$	0	0
$453$	15.5284	0.729587
$454$	0	0
$455$	−12.7341	−0.596983
$456$	0	0
$457$	−41.6155	−1.94669	−0.973345	−	0.229347i	$-0.926341\pi$
$457$	−41.6155	−1.94669	−0.973345	+	0.229347i	$0.926341\pi$
$458$	0	0
$459$	4.77393	0.222828
$460$	0	0
$461$	−4.29941	−0.200243	−0.100122	−	0.994975i	$-0.531923\pi$
$461$	−4.29941	−0.200243	−0.100122	+	0.994975i	$0.531923\pi$
$462$	0	0
$463$	37.3526	1.73592	0.867960	−	0.496634i	$-0.165431\pi$
$463$	37.3526	1.73592	0.867960	+	0.496634i	$0.165431\pi$
$464$	0	0
$465$	−8.66373	−0.401771
$466$	0	0
$467$	−28.1745	−1.30376	−0.651881	−	0.758321i	$-0.726020\pi$
$467$	−28.1745	−1.30376	−0.651881	+	0.758321i	$0.726020\pi$
$468$	0	0
$469$	12.0193	0.554999
$470$	0	0
$471$	−19.4332	−0.895435
$472$	0	0
$473$	−20.5815	−0.946340
$474$	0	0
$475$	4.12393	0.189219
$476$	0	0
$477$	−2.62431	−0.120159
$478$	0	0
$479$	9.39801	0.429406	0.214703	−	0.976679i	$-0.431122\pi$
$479$	9.39801	0.429406	0.214703	+	0.976679i	$0.431122\pi$
$480$	0	0
$481$	7.26473	0.331243
$482$	0	0
$483$	−6.73949	−0.306658
$484$	0	0
$485$	−19.6911	−0.894125
$486$	0	0
$487$	−10.5408	−0.477650	−0.238825	−	0.971063i	$-0.576762\pi$
$487$	−10.5408	−0.477650	−0.238825	+	0.971063i	$0.576762\pi$
$488$	0	0
$489$	9.15822	0.414149
$490$	0	0
$491$	14.3694	0.648482	0.324241	−	0.945975i	$-0.394891\pi$
$491$	14.3694	0.648482	0.324241	+	0.945975i	$0.394891\pi$
$492$	0	0
$493$	−27.1882	−1.22449
$494$	0	0
$495$	−3.31557	−0.149024
$496$	0	0
$497$	22.2466	0.997896
$498$	0	0
$499$	29.1966	1.30702	0.653511	−	0.756917i	$-0.273296\pi$
$499$	29.1966	1.30702	0.653511	+	0.756917i	$0.273296\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	−24.1817	−1.07821	−0.539104	−	0.842239i	$-0.681237\pi$
$503$	−24.1817	−1.07821	−0.539104	+	0.842239i	$0.681237\pi$
$504$	0	0
$505$	24.3204	1.08224
$506$	0	0
$507$	1.20239	0.0533998
$508$	0	0
$509$	−33.3635	−1.47881	−0.739405	−	0.673260i	$-0.764893\pi$
$509$	−33.3635	−1.47881	−0.739405	+	0.673260i	$0.764893\pi$
$510$	0	0
$511$	−23.9397	−1.05903
$512$	0	0
$513$	5.04637	0.222803
$514$	0	0
$515$	13.8730	0.611317
$516$	0	0
$517$	13.6919	0.602167
$518$	0	0
$519$	18.5002	0.812071
$520$	0	0
$521$	−0.496462	−0.0217504	−0.0108752	−	0.999941i	$-0.503462\pi$
$521$	−0.496462	−0.0217504	−0.0108752	+	0.999941i	$0.503462\pi$
$522$	0	0
$523$	23.9410	1.04687	0.523434	−	0.852066i	$-0.324651\pi$
$523$	23.9410	1.04687	0.523434	+	0.852066i	$0.324651\pi$
$524$	0	0
$525$	−1.48139	−0.0646531
$526$	0	0
$527$	20.2231	0.880932
$528$	0	0
$529$	−9.17770	−0.399030
$530$	0	0
$531$	−0.400667	−0.0173875
$532$	0	0
$533$	−9.21772	−0.399264
$534$	0	0
$535$	−7.75535	−0.335293
$536$	0	0
$537$	19.2383	0.830194
$538$	0	0
$539$	−6.02089	−0.259338
$540$	0	0
$541$	−19.0931	−0.820876	−0.410438	−	0.911889i	$-0.634624\pi$
$541$	−19.0931	−0.820876	−0.410438	+	0.911889i	$0.634624\pi$
$542$	0	0
$543$	−26.1135	−1.12064
$544$	0	0
$545$	−28.3867	−1.21595
$546$	0	0
$547$	−4.49603	−0.192236	−0.0961181	−	0.995370i	$-0.530643\pi$
$547$	−4.49603	−0.192236	−0.0961181	+	0.995370i	$0.530643\pi$
$548$	0	0
$549$	0.131727	0.00562199
$550$	0	0
$551$	−28.7398	−1.22436
$552$	0	0
$553$	5.31617	0.226066
$554$	0	0
$555$	−4.32569	−0.183615
$556$	0	0
$557$	−30.0454	−1.27306	−0.636532	−	0.771250i	$-0.719632\pi$
$557$	−30.0454	−1.27306	−0.636532	+	0.771250i	$0.719632\pi$
$558$	0	0
$559$	43.6064	1.84435
$560$	0	0
$561$	7.73929	0.326753
$562$	0	0
$563$	32.3325	1.36265	0.681326	−	0.731980i	$-0.261403\pi$
$563$	32.3325	1.36265	0.681326	+	0.731980i	$0.261403\pi$
$564$	0	0
$565$	39.9636	1.68128
$566$	0	0
$567$	−1.81275	−0.0761282
$568$	0	0
$569$	20.1979	0.846738	0.423369	−	0.905957i	$-0.360847\pi$
$569$	20.1979	0.846738	0.423369	+	0.905957i	$0.360847\pi$
$570$	0	0
$571$	−33.2137	−1.38995	−0.694976	−	0.719033i	$-0.744585\pi$
$571$	−33.2137	−1.38995	−0.694976	+	0.719033i	$0.744585\pi$
$572$	0	0
$573$	−26.0096	−1.08657
$574$	0	0
$575$	3.03824	0.126703
$576$	0	0
$577$	−34.6552	−1.44271	−0.721356	−	0.692564i	$-0.756481\pi$
$577$	−34.6552	−1.44271	−0.721356	+	0.692564i	$0.756481\pi$
$578$	0	0
$579$	−6.70727	−0.278745
$580$	0	0
$581$	27.9234	1.15846
$582$	0	0
$583$	−4.25441	−0.176200
$584$	0	0
$585$	7.02474	0.290437
$586$	0	0
$587$	−3.82995	−0.158079	−0.0790394	−	0.996871i	$-0.525185\pi$
$587$	−3.82995	−0.158079	−0.0790394	+	0.996871i	$0.525185\pi$
$588$	0	0
$589$	21.3772	0.880833
$590$	0	0
$591$	20.2795	0.834189
$592$	0	0
$593$	−21.3949	−0.878581	−0.439291	−	0.898345i	$-0.644770\pi$
$593$	−21.3949	−0.878581	−0.439291	+	0.898345i	$0.644770\pi$
$594$	0	0
$595$	−17.6989	−0.725584
$596$	0	0
$597$	−20.4915	−0.838660
$598$	0	0
$599$	−8.79373	−0.359302	−0.179651	−	0.983730i	$-0.557497\pi$
$599$	−8.79373	−0.359302	−0.179651	+	0.983730i	$0.557497\pi$
$600$	0	0
$601$	−0.891749	−0.0363752	−0.0181876	−	0.999835i	$-0.505790\pi$
$601$	−0.891749	−0.0363752	−0.0181876	+	0.999835i	$0.505790\pi$
$602$	0	0
$603$	−6.63043	−0.270012
$604$	0	0
$605$	17.1220	0.696109
$606$	0	0
$607$	20.2852	0.823353	0.411676	−	0.911330i	$-0.364943\pi$
$607$	20.2852	0.823353	0.411676	+	0.911330i	$0.364943\pi$
$608$	0	0
$609$	10.3238	0.418343
$610$	0	0
$611$	−29.0091	−1.17358
$612$	0	0
$613$	11.1524	0.450441	0.225221	−	0.974308i	$-0.427690\pi$
$613$	11.1524	0.450441	0.225221	+	0.974308i	$0.427690\pi$
$614$	0	0
$615$	5.48857	0.221321
$616$	0	0
$617$	−33.1142	−1.33313	−0.666564	−	0.745447i	$-0.732236\pi$
$617$	−33.1142	−1.33313	−0.666564	+	0.745447i	$0.732236\pi$
$618$	0	0
$619$	24.3281	0.977828	0.488914	−	0.872332i	$-0.337393\pi$
$619$	24.3281	0.977828	0.488914	+	0.872332i	$0.337393\pi$
$620$	0	0
$621$	3.71784	0.149192
$622$	0	0
$623$	−9.84326	−0.394362
$624$	0	0
$625$	−20.2461	−0.809846
$626$	0	0
$627$	8.18096	0.326716
$628$	0	0
$629$	10.0971	0.402599
$630$	0	0
$631$	20.3393	0.809693	0.404846	−	0.914385i	$-0.367325\pi$
$631$	20.3393	0.809693	0.404846	+	0.914385i	$0.367325\pi$
$632$	0	0
$633$	−9.74296	−0.387248
$634$	0	0
$635$	−8.79300	−0.348940
$636$	0	0
$637$	12.7565	0.505433
$638$	0	0
$639$	−12.2723	−0.485485
$640$	0	0
$641$	−12.8194	−0.506338	−0.253169	−	0.967422i	$-0.581473\pi$
$641$	−12.8194	−0.506338	−0.253169	+	0.967422i	$0.581473\pi$
$642$	0	0
$643$	11.5937	0.457211	0.228605	−	0.973519i	$-0.426583\pi$
$643$	11.5937	0.457211	0.228605	+	0.973519i	$0.426583\pi$
$644$	0	0
$645$	−25.9649	−1.02237
$646$	0	0
$647$	−3.44127	−0.135290	−0.0676452	−	0.997709i	$-0.521549\pi$
$647$	−3.44127	−0.135290	−0.0676452	+	0.997709i	$0.521549\pi$
$648$	0	0
$649$	−0.649544	−0.0254968
$650$	0	0
$651$	−7.67908	−0.300967
$652$	0	0
$653$	23.9286	0.936400	0.468200	−	0.883622i	$-0.344903\pi$
$653$	23.9286	0.936400	0.468200	+	0.883622i	$0.344903\pi$
$654$	0	0
$655$	30.8751	1.20639
$656$	0	0
$657$	13.2063	0.515226
$658$	0	0
$659$	−20.8812	−0.813417	−0.406708	−	0.913558i	$-0.633323\pi$
$659$	−20.8812	−0.813417	−0.406708	+	0.913558i	$0.633323\pi$
$660$	0	0
$661$	29.7368	1.15663	0.578314	−	0.815814i	$-0.303711\pi$
$661$	29.7368	1.15663	0.578314	+	0.815814i	$0.303711\pi$
$662$	0	0
$663$	−16.3973	−0.636820
$664$	0	0
$665$	−18.7090	−0.725503
$666$	0	0
$667$	−21.1736	−0.819844
$668$	0	0
$669$	19.0423	0.736217
$670$	0	0
$671$	0.213551	0.00824403
$672$	0	0
$673$	4.94319	0.190546	0.0952730	−	0.995451i	$-0.469628\pi$
$673$	4.94319	0.190546	0.0952730	+	0.995451i	$0.469628\pi$
$674$	0	0
$675$	0.817207	0.0314543
$676$	0	0
$677$	14.2784	0.548764	0.274382	−	0.961621i	$-0.411527\pi$
$677$	14.2784	0.548764	0.274382	+	0.961621i	$0.411527\pi$
$678$	0	0
$679$	−17.4531	−0.669789
$680$	0	0
$681$	24.0274	0.920733
$682$	0	0
$683$	−23.5913	−0.902694	−0.451347	−	0.892348i	$-0.649056\pi$
$683$	−23.5913	−0.902694	−0.451347	+	0.892348i	$0.649056\pi$
$684$	0	0
$685$	2.78092	0.106254
$686$	0	0
$687$	12.6375	0.482151
$688$	0	0
$689$	9.01388	0.343402
$690$	0	0
$691$	−23.3287	−0.887467	−0.443733	−	0.896159i	$-0.646346\pi$
$691$	−23.3287	−0.887467	−0.443733	+	0.896159i	$0.646346\pi$
$692$	0	0
$693$	−2.93875	−0.111634
$694$	0	0
$695$	−40.7487	−1.54569
$696$	0	0
$697$	−12.8116	−0.485273
$698$	0	0
$699$	−25.0038	−0.945731
$700$	0	0
$701$	30.0234	1.13397	0.566984	−	0.823729i	$-0.308110\pi$
$701$	30.0234	1.13397	0.566984	+	0.823729i	$0.308110\pi$
$702$	0	0
$703$	10.6734	0.402554
$704$	0	0
$705$	17.2731	0.650543
$706$	0	0
$707$	21.5563	0.810709
$708$	0	0
$709$	−15.6410	−0.587410	−0.293705	−	0.955896i	$-0.594888\pi$
$709$	−15.6410	−0.587410	−0.293705	+	0.955896i	$0.594888\pi$
$710$	0	0
$711$	−2.93266	−0.109983
$712$	0	0
$713$	15.7493	0.589817
$714$	0	0
$715$	11.3882	0.425895
$716$	0	0
$717$	12.9032	0.481878
$718$	0	0
$719$	−8.66277	−0.323067	−0.161533	−	0.986867i	$-0.551644\pi$
$719$	−8.66277	−0.323067	−0.161533	+	0.986867i	$0.551644\pi$
$720$	0	0
$721$	12.2963	0.457938
$722$	0	0
$723$	10.1834	0.378726
$724$	0	0
$725$	−4.65410	−0.172849
$726$	0	0
$727$	−24.1368	−0.895183	−0.447592	−	0.894238i	$-0.647718\pi$
$727$	−24.1368	−0.895183	−0.447592	+	0.894238i	$0.647718\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	60.6079	2.24166
$732$	0	0
$733$	−38.9993	−1.44047	−0.720235	−	0.693730i	$-0.755966\pi$
$733$	−38.9993	−1.44047	−0.720235	+	0.693730i	$0.755966\pi$
$734$	0	0
$735$	−7.59572	−0.280172
$736$	0	0
$737$	−10.7490	−0.395943
$738$	0	0
$739$	30.1884	1.11050	0.555250	−	0.831684i	$-0.312623\pi$
$739$	30.1884	1.11050	0.555250	+	0.831684i	$0.312623\pi$
$740$	0	0
$741$	−17.3331	−0.636748
$742$	0	0
$743$	−10.1790	−0.373432	−0.186716	−	0.982414i	$-0.559784\pi$
$743$	−10.1790	−0.373432	−0.186716	+	0.982414i	$0.559784\pi$
$744$	0	0
$745$	15.0937	0.552990
$746$	0	0
$747$	−15.4039	−0.563599
$748$	0	0
$749$	−6.87394	−0.251168
$750$	0	0
$751$	−14.6388	−0.534179	−0.267089	−	0.963672i	$-0.586062\pi$
$751$	−14.6388	−0.534179	−0.267089	+	0.963672i	$0.586062\pi$
$752$	0	0
$753$	0.514670	0.0187556
$754$	0	0
$755$	31.7584	1.15581
$756$	0	0
$757$	−40.6501	−1.47745	−0.738727	−	0.674004i	$-0.764573\pi$
$757$	−40.6501	−1.47745	−0.738727	+	0.674004i	$0.764573\pi$
$758$	0	0
$759$	6.02719	0.218773
$760$	0	0
$761$	−33.7343	−1.22287	−0.611434	−	0.791295i	$-0.709407\pi$
$761$	−33.7343	−1.22287	−0.611434	+	0.791295i	$0.709407\pi$
$762$	0	0
$763$	−25.1605	−0.910870
$764$	0	0
$765$	9.76358	0.353003
$766$	0	0
$767$	1.37620	0.0496916
$768$	0	0
$769$	−9.28736	−0.334911	−0.167455	−	0.985880i	$-0.553555\pi$
$769$	−9.28736	−0.334911	−0.167455	+	0.985880i	$0.553555\pi$
$770$	0	0
$771$	−22.3692	−0.805608
$772$	0	0
$773$	−23.0520	−0.829122	−0.414561	−	0.910021i	$-0.636065\pi$
$773$	−23.0520	−0.829122	−0.414561	+	0.910021i	$0.636065\pi$
$774$	0	0
$775$	3.46181	0.124352
$776$	0	0
$777$	−3.83407	−0.137546
$778$	0	0
$779$	−13.5427	−0.485218
$780$	0	0
$781$	−19.8953	−0.711911
$782$	0	0
$783$	−5.69513	−0.203527
$784$	0	0
$785$	−39.7446	−1.41854
$786$	0	0
$787$	−16.4746	−0.587255	−0.293627	−	0.955920i	$-0.594863\pi$
$787$	−16.4746	−0.587255	−0.293627	+	0.955920i	$0.594863\pi$
$788$	0	0
$789$	−15.7692	−0.561397
$790$	0	0
$791$	35.4216	1.25945
$792$	0	0
$793$	−0.452453	−0.0160671
$794$	0	0
$795$	−5.36720	−0.190355
$796$	0	0
$797$	−44.0201	−1.55927	−0.779636	−	0.626233i	$-0.784596\pi$
$797$	−44.0201	−1.55927	−0.779636	+	0.626233i	$0.784596\pi$
$798$	0	0
$799$	−40.3194	−1.42640
$800$	0	0
$801$	5.43002	0.191860
$802$	0	0
$803$	21.4095	0.755523
$804$	0	0
$805$	−13.7835	−0.485806
$806$	0	0
$807$	−14.4917	−0.510132
$808$	0	0
$809$	2.92181	0.102726	0.0513628	−	0.998680i	$-0.483644\pi$
$809$	2.92181	0.102726	0.0513628	+	0.998680i	$0.483644\pi$
$810$	0	0
$811$	22.6747	0.796217	0.398109	−	0.917338i	$-0.369667\pi$
$811$	22.6747	0.796217	0.398109	+	0.917338i	$0.369667\pi$
$812$	0	0
$813$	−28.2465	−0.990649
$814$	0	0
$815$	18.7303	0.656093
$816$	0	0
$817$	64.0667	2.24141
$818$	0	0
$819$	6.22636	0.217567
$820$	0	0
$821$	11.1758	0.390038	0.195019	−	0.980799i	$-0.437523\pi$
$821$	11.1758	0.390038	0.195019	+	0.980799i	$0.437523\pi$
$822$	0	0
$823$	−14.9066	−0.519613	−0.259806	−	0.965661i	$-0.583659\pi$
$823$	−14.9066	−0.519613	−0.259806	+	0.965661i	$0.583659\pi$
$824$	0	0
$825$	1.32482	0.0461243
$826$	0	0
$827$	10.7483	0.373754	0.186877	−	0.982383i	$-0.440163\pi$
$827$	10.7483	0.373754	0.186877	+	0.982383i	$0.440163\pi$
$828$	0	0
$829$	42.3443	1.47068	0.735339	−	0.677700i	$-0.237023\pi$
$829$	42.3443	1.47068	0.735339	+	0.677700i	$0.237023\pi$
$830$	0	0
$831$	5.04261	0.174926
$832$	0	0
$833$	17.7301	0.614313
$834$	0	0
$835$	−2.04519	−0.0707766
$836$	0	0
$837$	4.23616	0.146423
$838$	0	0
$839$	9.10935	0.314490	0.157245	−	0.987560i	$-0.449739\pi$
$839$	9.10935	0.314490	0.157245	+	0.987560i	$0.449739\pi$
$840$	0	0
$841$	3.43453	0.118432
$842$	0	0
$843$	16.8548	0.580510
$844$	0	0
$845$	2.45911	0.0845958
$846$	0	0
$847$	15.1760	0.521455
$848$	0	0
$849$	23.2569	0.798173
$850$	0	0
$851$	7.86344	0.269555
$852$	0	0
$853$	−20.1422	−0.689656	−0.344828	−	0.938666i	$-0.612063\pi$
$853$	−20.1422	−0.689656	−0.344828	+	0.938666i	$0.612063\pi$
$854$	0	0
$855$	10.3208	0.352963
$856$	0	0
$857$	12.3338	0.421316	0.210658	−	0.977560i	$-0.432439\pi$
$857$	12.3338	0.421316	0.210658	+	0.977560i	$0.432439\pi$
$858$	0	0
$859$	−37.6878	−1.28589	−0.642945	−	0.765912i	$-0.722288\pi$
$859$	−37.6878	−1.28589	−0.642945	+	0.765912i	$0.722288\pi$
$860$	0	0
$861$	4.86478	0.165791
$862$	0	0
$863$	18.5076	0.630005	0.315003	−	0.949091i	$-0.397995\pi$
$863$	18.5076	0.630005	0.315003	+	0.949091i	$0.397995\pi$
$864$	0	0
$865$	37.8365	1.28648
$866$	0	0
$867$	−5.79041	−0.196653
$868$	0	0
$869$	−4.75430	−0.161278
$870$	0	0
$871$	22.7740	0.771667
$872$	0	0
$873$	9.62800	0.325858
$874$	0	0
$875$	−21.5668	−0.729090
$876$	0	0
$877$	−2.78287	−0.0939710	−0.0469855	−	0.998896i	$-0.514961\pi$
$877$	−2.78287	−0.0939710	−0.0469855	+	0.998896i	$0.514961\pi$
$878$	0	0
$879$	−9.38017	−0.316385
$880$	0	0
$881$	−49.2793	−1.66026	−0.830131	−	0.557568i	$-0.811735\pi$
$881$	−49.2793	−1.66026	−0.830131	+	0.557568i	$0.811735\pi$
$882$	0	0
$883$	22.1741	0.746219	0.373110	−	0.927787i	$-0.378292\pi$
$883$	22.1741	0.746219	0.373110	+	0.927787i	$0.378292\pi$
$884$	0	0
$885$	−0.819440	−0.0275452
$886$	0	0
$887$	11.7418	0.394250	0.197125	−	0.980378i	$-0.436840\pi$
$887$	11.7418	0.394250	0.197125	+	0.980378i	$0.436840\pi$
$888$	0	0
$889$	−7.79365	−0.261391
$890$	0	0
$891$	1.62116	0.0543108
$892$	0	0
$893$	−42.6204	−1.42624
$894$	0	0
$895$	39.3459	1.31519
$896$	0	0
$897$	−12.7699	−0.426374
$898$	0	0
$899$	−24.1255	−0.804629
$900$	0	0
$901$	12.5283	0.417377
$902$	0	0
$903$	−23.0139	−0.765855
$904$	0	0
$905$	−53.4070	−1.77531
$906$	0	0
$907$	27.7655	0.921938	0.460969	−	0.887416i	$-0.347502\pi$
$907$	27.7655	0.921938	0.460969	+	0.887416i	$0.347502\pi$
$908$	0	0
$909$	−11.8915	−0.394417
$910$	0	0
$911$	39.1106	1.29579	0.647896	−	0.761729i	$-0.275649\pi$
$911$	39.1106	1.29579	0.647896	+	0.761729i	$0.275649\pi$
$912$	0	0
$913$	−24.9721	−0.826457
$914$	0	0
$915$	0.269407	0.00890633
$916$	0	0
$917$	27.3661	0.903707
$918$	0	0
$919$	−4.66627	−0.153926	−0.0769630	−	0.997034i	$-0.524522\pi$
$919$	−4.66627	−0.153926	−0.0769630	+	0.997034i	$0.524522\pi$
$920$	0	0
$921$	23.3987	0.771012
$922$	0	0
$923$	42.1525	1.38747
$924$	0	0
$925$	1.72844	0.0568308
$926$	0	0
$927$	−6.78324	−0.222791
$928$	0	0
$929$	2.00389	0.0657454	0.0328727	−	0.999460i	$-0.489534\pi$
$929$	2.00389	0.0657454	0.0328727	+	0.999460i	$0.489534\pi$
$930$	0	0
$931$	18.7420	0.614243
$932$	0	0
$933$	−4.38962	−0.143710
$934$	0	0
$935$	15.8283	0.517641
$936$	0	0
$937$	41.0177	1.33999	0.669994	−	0.742366i	$-0.266297\pi$
$937$	41.0177	1.33999	0.669994	+	0.742366i	$0.266297\pi$
$938$	0	0
$939$	−30.5307	−0.996331
$940$	0	0
$941$	−28.9672	−0.944302	−0.472151	−	0.881518i	$-0.656522\pi$
$941$	−28.9672	−0.944302	−0.472151	+	0.881518i	$0.656522\pi$
$942$	0	0
$943$	−9.97738	−0.324908
$944$	0	0
$945$	−3.70741	−0.120602
$946$	0	0
$947$	−56.9248	−1.84981	−0.924905	−	0.380199i	$-0.875855\pi$
$947$	−56.9248	−1.84981	−0.924905	+	0.380199i	$0.875855\pi$
$948$	0	0
$949$	−45.3605	−1.47246
$950$	0	0
$951$	−23.0063	−0.746031
$952$	0	0
$953$	6.31738	0.204640	0.102320	−	0.994752i	$-0.467373\pi$
$953$	6.31738	0.204640	0.102320	+	0.994752i	$0.467373\pi$
$954$	0	0
$955$	−53.1945	−1.72133
$956$	0	0
$957$	−9.23270	−0.298451
$958$	0	0
$959$	2.46486	0.0795946
$960$	0	0
$961$	−13.0550	−0.421129
$962$	0	0
$963$	3.79200	0.122195
$964$	0	0
$965$	−13.7176	−0.441586
$966$	0	0
$967$	−3.63830	−0.117000	−0.0585000	−	0.998287i	$-0.518632\pi$
$967$	−3.63830	−0.117000	−0.0585000	+	0.998287i	$0.518632\pi$
$968$	0	0
$969$	−24.0910	−0.773916
$970$	0	0
$971$	56.4885	1.81280	0.906402	−	0.422417i	$-0.138818\pi$
$971$	56.4885	1.81280	0.906402	+	0.422417i	$0.138818\pi$
$972$	0	0
$973$	−36.1175	−1.15787
$974$	0	0
$975$	−2.80691	−0.0898932
$976$	0	0
$977$	29.4542	0.942322	0.471161	−	0.882047i	$-0.343835\pi$
$977$	29.4542	0.942322	0.471161	+	0.882047i	$0.343835\pi$
$978$	0	0
$979$	8.80292	0.281342
$980$	0	0
$981$	13.8797	0.443146
$982$	0	0
$983$	−44.6845	−1.42521	−0.712607	−	0.701564i	$-0.752486\pi$
$983$	−44.6845	−1.42521	−0.712607	+	0.701564i	$0.752486\pi$
$984$	0	0
$985$	41.4755	1.32152
$986$	0	0
$987$	15.3100	0.487322
$988$	0	0
$989$	47.2001	1.50088
$990$	0	0
$991$	−53.8233	−1.70975	−0.854877	−	0.518831i	$-0.826367\pi$
$991$	−53.8233	−1.70975	−0.854877	+	0.518831i	$0.826367\pi$
$992$	0	0
$993$	21.5613	0.684229
$994$	0	0
$995$	−41.9089	−1.32860
$996$	0	0
$997$	32.4015	1.02617	0.513083	−	0.858339i	$-0.328503\pi$
$997$	32.4015	1.02617	0.513083	+	0.858339i	$0.328503\pi$
$998$	0	0
$999$	2.11506	0.0669175

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.bg.1.5		13
4.3	odd	2		4008.2.a.m.1.5	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.m.1.5	✓	13	4.3	odd	2
8016.2.a.bg.1.5		13	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.04519 q^{5} -1.81275 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.04519 q^{5} -1.81275 q^{7} +1.00000 q^{9} +1.62116 q^{11} -3.43477 q^{13} +2.04519 q^{15} -4.77393 q^{17} -5.04637 q^{19} +1.81275 q^{21} -3.71784 q^{23} -0.817207 q^{25} -1.00000 q^{27} +5.69513 q^{29} -4.23616 q^{31} -1.62116 q^{33} +3.70741 q^{35} -2.11506 q^{37} +3.43477 q^{39} +2.68365 q^{41} -12.6956 q^{43} -2.04519 q^{45} +8.44574 q^{47} -3.71395 q^{49} +4.77393 q^{51} -2.62431 q^{53} -3.31557 q^{55} +5.04637 q^{57} -0.400667 q^{59} +0.131727 q^{61} -1.81275 q^{63} +7.02474 q^{65} -6.63043 q^{67} +3.71784 q^{69} -12.2723 q^{71} +13.2063 q^{73} +0.817207 q^{75} -2.93875 q^{77} -2.93266 q^{79} +1.00000 q^{81} -15.4039 q^{83} +9.76358 q^{85} -5.69513 q^{87} +5.43002 q^{89} +6.22636 q^{91} +4.23616 q^{93} +10.3208 q^{95} +9.62800 q^{97} +1.62116 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9}+O(q^{10}) \) 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9	\( 13 q - 13 q^{3} + 2 q^{5} - q^{7} + 13 q^{9} - 11 q^{11} + 12 q^{13} - 2 q^{15} + 15 q^{17} - 14 q^{19} + q^{21} - 9 q^{23} + 37 q^{25} - 13 q^{27} - 3 q^{29} + 17 q^{31} + 11 q^{33} - 15 q^{35} + 16 q^{37} - 12 q^{39} + 12 q^{41} - 20 q^{43} + 2 q^{45} + 6 q^{47} + 26 q^{49} - 15 q^{51} - 12 q^{53} - 7 q^{55} + 14 q^{57} - 14 q^{59} + 24 q^{61} - q^{63} + 8 q^{65} - 3 q^{67} + 9 q^{69} - 17 q^{71} + 34 q^{73} - 37 q^{75} + 30 q^{77} - 10 q^{79} + 13 q^{81} - 44 q^{83} + 25 q^{85} + 3 q^{87} + 25 q^{89} - 29 q^{91} - 17 q^{93} + 15 q^{95} + 38 q^{97} - 11 q^{99}+O(q^{100}) \) 13 * q - 13 * q^3 + 2 * q^5 - q^7 + 13 * q^9 - 11 * q^11 + 12 * q^13 - 2 * q^15 + 15 * q^17 - 14 * q^19 + q^21 - 9 * q^23 + 37 * q^25 - 13 * q^27 - 3 * q^29 + 17 * q^31 + 11 * q^33 - 15 * q^35 + 16 * q^37 - 12 * q^39 + 12 * q^41 - 20 * q^43 + 2 * q^45 + 6 * q^47 + 26 * q^49 - 15 * q^51 - 12 * q^53 - 7 * q^55 + 14 * q^57 - 14 * q^59 + 24 * q^61 - q^63 + 8 * q^65 - 3 * q^67 + 9 * q^69 - 17 * q^71 + 34 * q^73 - 37 * q^75 + 30 * q^77 - 10 * q^79 + 13 * q^81 - 44 * q^83 + 25 * q^85 + 3 * q^87 + 25 * q^89 - 29 * q^91 - 17 * q^93 + 15 * q^95 + 38 * q^97 - 11 * q^99

Introduction

L-functions

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Varieties

Fields

Representations

Groups

Database

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