LMFDB - Embedded newform 4016.2.a.i.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 17 x^{10} + 49 x^{9} + 106 x^{8} - 277 x^{7} - 317 x^{6} + 644 x^{5} + 537 x^{4} + \cdots + 104 $ x^12 - 3x^11 - 17x^10 + 49x^9 + 106x^8 - 277x^7 - 317x^6 + 644x^5 + 537x^4 - 601x^3 - 462x^2 + 136*x + 104
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-1.18772$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.18772 q^{3} +0.261605 q^{5} -3.40145 q^{7} -1.58932 q^{9} +O(q^{10})$	$q+1.18772 q^{3} +0.261605 q^{5} -3.40145 q^{7} -1.58932 q^{9} -2.11467 q^{11} +4.74687 q^{13} +0.310714 q^{15} +3.91468 q^{17} +1.18917 q^{19} -4.03997 q^{21} +2.60787 q^{23} -4.93156 q^{25} -5.45083 q^{27} +0.684263 q^{29} -2.40908 q^{31} -2.51164 q^{33} -0.889836 q^{35} -3.28777 q^{37} +5.63796 q^{39} +2.97822 q^{41} +2.26666 q^{43} -0.415773 q^{45} -5.45453 q^{47} +4.56986 q^{49} +4.64955 q^{51} -9.66594 q^{53} -0.553207 q^{55} +1.41240 q^{57} -10.2950 q^{59} +2.71511 q^{61} +5.40599 q^{63} +1.24181 q^{65} -3.58877 q^{67} +3.09742 q^{69} -9.71644 q^{71} -7.72016 q^{73} -5.85732 q^{75} +7.19294 q^{77} +1.26726 q^{79} -1.70611 q^{81} +0.519602 q^{83} +1.02410 q^{85} +0.812713 q^{87} +5.74537 q^{89} -16.1463 q^{91} -2.86131 q^{93} +0.311093 q^{95} +10.7536 q^{97} +3.36088 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) $ 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9	$ 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9} - 10 q^{11} + 3 q^{13} - 11 q^{15} + 2 q^{17} - 15 q^{19} + 3 q^{21} - 20 q^{23} - 3 q^{25} - 15 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{33} - 16 q^{35} + 5 q^{37} - 21 q^{39} - 21 q^{43} + 10 q^{45} - 27 q^{47} - 13 q^{49} - 19 q^{51} + 22 q^{53} - 24 q^{55} + q^{57} - 23 q^{59} + 4 q^{61} - 21 q^{63} - q^{65} - 26 q^{67} + 10 q^{69} - 23 q^{71} - 8 q^{73} - 16 q^{75} + 22 q^{77} - 37 q^{79} - 20 q^{81} - 30 q^{83} + 2 q^{85} - 16 q^{87} + 3 q^{89} - 8 q^{91} + 20 q^{93} - 33 q^{95} - 4 q^{97} - 15 q^{99}+O(q^{100}) $ 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9 - 10 * q^11 + 3 * q^13 - 11 * q^15 + 2 * q^17 - 15 * q^19 + 3 * q^21 - 20 * q^23 - 3 * q^25 - 15 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^33 - 16 * q^35 + 5 * q^37 - 21 * q^39 - 21 * q^43 + 10 * q^45 - 27 * q^47 - 13 * q^49 - 19 * q^51 + 22 * q^53 - 24 * q^55 + q^57 - 23 * q^59 + 4 * q^61 - 21 * q^63 - q^65 - 26 * q^67 + 10 * q^69 - 23 * q^71 - 8 * q^73 - 16 * q^75 + 22 * q^77 - 37 * q^79 - 20 * q^81 - 30 * q^83 + 2 * q^85 - 16 * q^87 + 3 * q^89 - 8 * q^91 + 20 * q^93 - 33 * q^95 - 4 * q^97 - 15 * 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
$3$	1.18772	0.685731	0.342866	−	0.939384i	$-0.388602\pi$
$3$	1.18772	0.685731	0.342866	+	0.939384i	$0.388602\pi$
$4$	0	0
$5$	0.261605	0.116993	0.0584966	−	0.998288i	$-0.481369\pi$
$5$	0.261605	0.116993	0.0584966	+	0.998288i	$0.481369\pi$
$6$	0	0
$7$	−3.40145	−1.28563	−0.642814	−	0.766023i	$-0.722233\pi$
$7$	−3.40145	−1.28563	−0.642814	+	0.766023i	$0.722233\pi$
$8$	0	0
$9$	−1.58932	−0.529773
$10$	0	0
$11$	−2.11467	−0.637596	−0.318798	−	0.947823i	$-0.603279\pi$
$11$	−2.11467	−0.637596	−0.318798	+	0.947823i	$0.603279\pi$
$12$	0	0
$13$	4.74687	1.31655	0.658273	−	0.752779i	$-0.271287\pi$
$13$	4.74687	1.31655	0.658273	+	0.752779i	$0.271287\pi$
$14$	0	0
$15$	0.310714	0.0802259
$16$	0	0
$17$	3.91468	0.949450	0.474725	−	0.880134i	$-0.342548\pi$
$17$	3.91468	0.949450	0.474725	+	0.880134i	$0.342548\pi$
$18$	0	0
$19$	1.18917	0.272815	0.136407	−	0.990653i	$-0.456444\pi$
$19$	1.18917	0.272815	0.136407	+	0.990653i	$0.456444\pi$
$20$	0	0
$21$	−4.03997	−0.881595
$22$	0	0
$23$	2.60787	0.543778	0.271889	−	0.962329i	$-0.412352\pi$
$23$	2.60787	0.543778	0.271889	+	0.962329i	$0.412352\pi$
$24$	0	0
$25$	−4.93156	−0.986313
$26$	0	0
$27$	−5.45083	−1.04901
$28$	0	0
$29$	0.684263	0.127064	0.0635322	−	0.997980i	$-0.479763\pi$
$29$	0.684263	0.127064	0.0635322	+	0.997980i	$0.479763\pi$
$30$	0	0
$31$	−2.40908	−0.432683	−0.216341	−	0.976318i	$-0.569412\pi$
$31$	−2.40908	−0.432683	−0.216341	+	0.976318i	$0.569412\pi$
$32$	0	0
$33$	−2.51164	−0.437220
$34$	0	0
$35$	−0.889836	−0.150410
$36$	0	0
$37$	−3.28777	−0.540507	−0.270253	−	0.962789i	$-0.587107\pi$
$37$	−3.28777	−0.540507	−0.270253	+	0.962789i	$0.587107\pi$
$38$	0	0
$39$	5.63796	0.902797
$40$	0	0
$41$	2.97822	0.465120	0.232560	−	0.972582i	$-0.425290\pi$
$41$	2.97822	0.465120	0.232560	+	0.972582i	$0.425290\pi$
$42$	0	0
$43$	2.26666	0.345663	0.172831	−	0.984951i	$-0.444708\pi$
$43$	2.26666	0.345663	0.172831	+	0.984951i	$0.444708\pi$
$44$	0	0
$45$	−0.415773	−0.0619798
$46$	0	0
$47$	−5.45453	−0.795625	−0.397813	−	0.917467i	$-0.630231\pi$
$47$	−5.45453	−0.795625	−0.397813	+	0.917467i	$0.630231\pi$
$48$	0	0
$49$	4.56986	0.652838
$50$	0	0
$51$	4.64955	0.651067
$52$	0	0
$53$	−9.66594	−1.32772	−0.663860	−	0.747857i	$-0.731083\pi$
$53$	−9.66594	−1.32772	−0.663860	+	0.747857i	$0.731083\pi$
$54$	0	0
$55$	−0.553207	−0.0745945
$56$	0	0
$57$	1.41240	0.187077
$58$	0	0
$59$	−10.2950	−1.34029	−0.670145	−	0.742230i	$-0.733768\pi$
$59$	−10.2950	−1.34029	−0.670145	+	0.742230i	$0.733768\pi$
$60$	0	0
$61$	2.71511	0.347635	0.173817	−	0.984778i	$-0.444390\pi$
$61$	2.71511	0.347635	0.173817	+	0.984778i	$0.444390\pi$
$62$	0	0
$63$	5.40599	0.681090
$64$	0	0
$65$	1.24181	0.154027
$66$	0	0
$67$	−3.58877	−0.438439	−0.219219	−	0.975676i	$-0.570351\pi$
$67$	−3.58877	−0.438439	−0.219219	+	0.975676i	$0.570351\pi$
$68$	0	0
$69$	3.09742	0.372885
$70$	0	0
$71$	−9.71644	−1.15313	−0.576565	−	0.817052i	$-0.695607\pi$
$71$	−9.71644	−1.15313	−0.576565	+	0.817052i	$0.695607\pi$
$72$	0	0
$73$	−7.72016	−0.903576	−0.451788	−	0.892125i	$-0.649214\pi$
$73$	−7.72016	−0.903576	−0.451788	+	0.892125i	$0.649214\pi$
$74$	0	0
$75$	−5.85732	−0.676345
$76$	0	0
$77$	7.19294	0.819711
$78$	0	0
$79$	1.26726	0.142578	0.0712890	−	0.997456i	$-0.477289\pi$
$79$	1.26726	0.142578	0.0712890	+	0.997456i	$0.477289\pi$
$80$	0	0
$81$	−1.70611	−0.189568
$82$	0	0
$83$	0.519602	0.0570337	0.0285169	−	0.999593i	$-0.490922\pi$
$83$	0.519602	0.0570337	0.0285169	+	0.999593i	$0.490922\pi$
$84$	0	0
$85$	1.02410	0.111079
$86$	0	0
$87$	0.812713	0.0871320
$88$	0	0
$89$	5.74537	0.609008	0.304504	−	0.952511i	$-0.401509\pi$
$89$	5.74537	0.609008	0.304504	+	0.952511i	$0.401509\pi$
$90$	0	0
$91$	−16.1463	−1.69259
$92$	0	0
$93$	−2.86131	−0.296704
$94$	0	0
$95$	0.311093	0.0319175
$96$	0	0
$97$	10.7536	1.09186	0.545931	−	0.837830i	$-0.316176\pi$
$97$	10.7536	1.09186	0.545931	+	0.837830i	$0.316176\pi$
$98$	0	0
$99$	3.36088	0.337781
$100$	0	0
$101$	4.91129	0.488691	0.244346	−	0.969688i	$-0.421427\pi$
$101$	4.91129	0.488691	0.244346	+	0.969688i	$0.421427\pi$
$102$	0	0
$103$	−9.62385	−0.948266	−0.474133	−	0.880453i	$-0.657238\pi$
$103$	−9.62385	−0.948266	−0.474133	+	0.880453i	$0.657238\pi$
$104$	0	0
$105$	−1.05688	−0.103141
$106$	0	0
$107$	−20.2733	−1.95989	−0.979945	−	0.199270i	$-0.936143\pi$
$107$	−20.2733	−1.95989	−0.979945	+	0.199270i	$0.936143\pi$
$108$	0	0
$109$	−12.9442	−1.23983	−0.619916	−	0.784668i	$-0.712833\pi$
$109$	−12.9442	−1.23983	−0.619916	+	0.784668i	$0.712833\pi$
$110$	0	0
$111$	−3.90496	−0.370642
$112$	0	0
$113$	−6.26795	−0.589640	−0.294820	−	0.955553i	$-0.595260\pi$
$113$	−6.26795	−0.589640	−0.294820	+	0.955553i	$0.595260\pi$
$114$	0	0
$115$	0.682230	0.0636183
$116$	0	0
$117$	−7.54429	−0.697470
$118$	0	0
$119$	−13.3156	−1.22064
$120$	0	0
$121$	−6.52818	−0.593471
$122$	0	0
$123$	3.53730	0.318947
$124$	0	0
$125$	−2.59815	−0.232385
$126$	0	0
$127$	14.7581	1.30957	0.654784	−	0.755816i	$-0.272760\pi$
$127$	14.7581	1.30957	0.654784	+	0.755816i	$0.272760\pi$
$128$	0	0
$129$	2.69216	0.237032
$130$	0	0
$131$	−17.5272	−1.53136	−0.765681	−	0.643221i	$-0.777598\pi$
$131$	−17.5272	−1.53136	−0.765681	+	0.643221i	$0.777598\pi$
$132$	0	0
$133$	−4.04491	−0.350738
$134$	0	0
$135$	−1.42596	−0.122727
$136$	0	0
$137$	−14.1998	−1.21317	−0.606583	−	0.795020i	$-0.707460\pi$
$137$	−14.1998	−1.21317	−0.606583	+	0.795020i	$0.707460\pi$
$138$	0	0
$139$	−21.6360	−1.83514	−0.917572	−	0.397570i	$-0.869854\pi$
$139$	−21.6360	−1.83514	−0.917572	+	0.397570i	$0.869854\pi$
$140$	0	0
$141$	−6.47846	−0.545585
$142$	0	0
$143$	−10.0381	−0.839425
$144$	0	0
$145$	0.179006	0.0148657
$146$	0	0
$147$	5.42772	0.447671
$148$	0	0
$149$	13.0851	1.07197	0.535987	−	0.844226i	$-0.319940\pi$
$149$	13.0851	1.07197	0.535987	+	0.844226i	$0.319940\pi$
$150$	0	0
$151$	3.36525	0.273860	0.136930	−	0.990581i	$-0.456276\pi$
$151$	3.36525	0.273860	0.136930	+	0.990581i	$0.456276\pi$
$152$	0	0
$153$	−6.22168	−0.502993
$154$	0	0
$155$	−0.630226	−0.0506210
$156$	0	0
$157$	−9.12478	−0.728236	−0.364118	−	0.931353i	$-0.618630\pi$
$157$	−9.12478	−0.728236	−0.364118	+	0.931353i	$0.618630\pi$
$158$	0	0
$159$	−11.4804	−0.910458
$160$	0	0
$161$	−8.87053	−0.699095
$162$	0	0
$163$	−18.4665	−1.44641	−0.723204	−	0.690634i	$-0.757331\pi$
$163$	−18.4665	−1.44641	−0.723204	+	0.690634i	$0.757331\pi$
$164$	0	0
$165$	−0.657056	−0.0511518
$166$	0	0
$167$	5.16684	0.399822	0.199911	−	0.979814i	$-0.435935\pi$
$167$	5.16684	0.399822	0.199911	+	0.979814i	$0.435935\pi$
$168$	0	0
$169$	9.53281	0.733293
$170$	0	0
$171$	−1.88997	−0.144530
$172$	0	0
$173$	19.0550	1.44872	0.724361	−	0.689421i	$-0.242135\pi$
$173$	19.0550	1.44872	0.724361	+	0.689421i	$0.242135\pi$
$174$	0	0
$175$	16.7745	1.26803
$176$	0	0
$177$	−12.2275	−0.919079
$178$	0	0
$179$	7.43813	0.555952	0.277976	−	0.960588i	$-0.410336\pi$
$179$	7.43813	0.555952	0.277976	+	0.960588i	$0.410336\pi$
$180$	0	0
$181$	−7.14098	−0.530785	−0.265393	−	0.964140i	$-0.585502\pi$
$181$	−7.14098	−0.530785	−0.265393	+	0.964140i	$0.585502\pi$
$182$	0	0
$183$	3.22480	0.238384
$184$	0	0
$185$	−0.860098	−0.0632356
$186$	0	0
$187$	−8.27825	−0.605366
$188$	0	0
$189$	18.5407	1.34864
$190$	0	0
$191$	−19.9630	−1.44447	−0.722235	−	0.691648i	$-0.756885\pi$
$191$	−19.9630	−1.44447	−0.722235	+	0.691648i	$0.756885\pi$
$192$	0	0
$193$	17.2082	1.23867	0.619335	−	0.785127i	$-0.287402\pi$
$193$	17.2082	1.23867	0.619335	+	0.785127i	$0.287402\pi$
$194$	0	0
$195$	1.47492	0.105621
$196$	0	0
$197$	23.1209	1.64730	0.823649	−	0.567100i	$-0.191935\pi$
$197$	23.1209	1.64730	0.823649	+	0.567100i	$0.191935\pi$
$198$	0	0
$199$	15.8582	1.12416	0.562080	−	0.827083i	$-0.310001\pi$
$199$	15.8582	1.12416	0.562080	+	0.827083i	$0.310001\pi$
$200$	0	0
$201$	−4.26246	−0.300651
$202$	0	0
$203$	−2.32749	−0.163357
$204$	0	0
$205$	0.779117	0.0544159
$206$	0	0
$207$	−4.14473	−0.288079
$208$	0	0
$209$	−2.51470	−0.173946
$210$	0	0
$211$	−3.89795	−0.268346	−0.134173	−	0.990958i	$-0.542838\pi$
$211$	−3.89795	−0.268346	−0.134173	+	0.990958i	$0.542838\pi$
$212$	0	0
$213$	−11.5404	−0.790737
$214$	0	0
$215$	0.592970	0.0404402
$216$	0	0
$217$	8.19435	0.556269
$218$	0	0
$219$	−9.16940	−0.619610
$220$	0	0
$221$	18.5825	1.24999
$222$	0	0
$223$	0.680442	0.0455657	0.0227829	−	0.999740i	$-0.492747\pi$
$223$	0.680442	0.0455657	0.0227829	+	0.999740i	$0.492747\pi$
$224$	0	0
$225$	7.83782	0.522522
$226$	0	0
$227$	6.45762	0.428607	0.214304	−	0.976767i	$-0.431252\pi$
$227$	6.45762	0.428607	0.214304	+	0.976767i	$0.431252\pi$
$228$	0	0
$229$	12.1167	0.800691	0.400346	−	0.916364i	$-0.368890\pi$
$229$	12.1167	0.800691	0.400346	+	0.916364i	$0.368890\pi$
$230$	0	0
$231$	8.54320	0.562102
$232$	0	0
$233$	−12.3118	−0.806575	−0.403288	−	0.915073i	$-0.632133\pi$
$233$	−12.3118	−0.806575	−0.403288	+	0.915073i	$0.632133\pi$
$234$	0	0
$235$	−1.42693	−0.0930828
$236$	0	0
$237$	1.50515	0.0977702
$238$	0	0
$239$	12.8265	0.829676	0.414838	−	0.909895i	$-0.363838\pi$
$239$	12.8265	0.829676	0.414838	+	0.909895i	$0.363838\pi$
$240$	0	0
$241$	−16.4983	−1.06275	−0.531376	−	0.847136i	$-0.678325\pi$
$241$	−16.4983	−1.06275	−0.531376	+	0.847136i	$0.678325\pi$
$242$	0	0
$243$	14.3261	0.919020
$244$	0	0
$245$	1.19550	0.0763776
$246$	0	0
$247$	5.64485	0.359173
$248$	0	0
$249$	0.617142	0.0391098
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−5.51477	−0.346711
$254$	0	0
$255$	1.21635	0.0761705
$256$	0	0
$257$	−1.54952	−0.0966565	−0.0483283	−	0.998832i	$-0.515389\pi$
$257$	−1.54952	−0.0966565	−0.0483283	+	0.998832i	$0.515389\pi$
$258$	0	0
$259$	11.1832	0.694890
$260$	0	0
$261$	−1.08751	−0.0673153
$262$	0	0
$263$	17.8180	1.09870	0.549351	−	0.835592i	$-0.314875\pi$
$263$	17.8180	1.09870	0.549351	+	0.835592i	$0.314875\pi$
$264$	0	0
$265$	−2.52866	−0.155334
$266$	0	0
$267$	6.82389	0.417616
$268$	0	0
$269$	8.06782	0.491904	0.245952	−	0.969282i	$-0.420899\pi$
$269$	8.06782	0.491904	0.245952	+	0.969282i	$0.420899\pi$
$270$	0	0
$271$	15.4917	0.941053	0.470526	−	0.882386i	$-0.344064\pi$
$271$	15.4917	0.941053	0.470526	+	0.882386i	$0.344064\pi$
$272$	0	0
$273$	−19.1772	−1.16066
$274$	0	0
$275$	10.4286	0.628869
$276$	0	0
$277$	−3.18037	−0.191090	−0.0955449	−	0.995425i	$-0.530459\pi$
$277$	−3.18037	−0.191090	−0.0955449	+	0.995425i	$0.530459\pi$
$278$	0	0
$279$	3.82879	0.229224
$280$	0	0
$281$	4.06888	0.242729	0.121365	−	0.992608i	$-0.461273\pi$
$281$	4.06888	0.242729	0.121365	+	0.992608i	$0.461273\pi$
$282$	0	0
$283$	9.93369	0.590496	0.295248	−	0.955421i	$-0.404598\pi$
$283$	9.93369	0.590496	0.295248	+	0.955421i	$0.404598\pi$
$284$	0	0
$285$	0.369492	0.0218868
$286$	0	0
$287$	−10.1303	−0.597971
$288$	0	0
$289$	−1.67526	−0.0985446
$290$	0	0
$291$	12.7723	0.748723
$292$	0	0
$293$	−28.2049	−1.64775	−0.823875	−	0.566772i	$-0.808192\pi$
$293$	−28.2049	−1.64775	−0.823875	+	0.566772i	$0.808192\pi$
$294$	0	0
$295$	−2.69321	−0.156805
$296$	0	0
$297$	11.5267	0.668847
$298$	0	0
$299$	12.3792	0.715908
$300$	0	0
$301$	−7.70994	−0.444393
$302$	0	0
$303$	5.83324	0.335111
$304$	0	0
$305$	0.710287	0.0406709
$306$	0	0
$307$	−14.6791	−0.837780	−0.418890	−	0.908037i	$-0.637581\pi$
$307$	−14.6791	−0.837780	−0.418890	+	0.908037i	$0.637581\pi$
$308$	0	0
$309$	−11.4304	−0.650255
$310$	0	0
$311$	2.13066	0.120819	0.0604093	−	0.998174i	$-0.480759\pi$
$311$	2.13066	0.120819	0.0604093	+	0.998174i	$0.480759\pi$
$312$	0	0
$313$	−14.5124	−0.820287	−0.410143	−	0.912021i	$-0.634521\pi$
$313$	−14.5124	−0.820287	−0.410143	+	0.912021i	$0.634521\pi$
$314$	0	0
$315$	1.41423	0.0796830
$316$	0	0
$317$	12.1594	0.682939	0.341469	−	0.939893i	$-0.389075\pi$
$317$	12.1594	0.682939	0.341469	+	0.939893i	$0.389075\pi$
$318$	0	0
$319$	−1.44699	−0.0810158
$320$	0	0
$321$	−24.0790	−1.34396
$322$	0	0
$323$	4.65523	0.259024
$324$	0	0
$325$	−23.4095	−1.29853
$326$	0	0
$327$	−15.3741	−0.850192
$328$	0	0
$329$	18.5533	1.02288
$330$	0	0
$331$	7.31699	0.402178	0.201089	−	0.979573i	$-0.435552\pi$
$331$	7.31699	0.402178	0.201089	+	0.979573i	$0.435552\pi$
$332$	0	0
$333$	5.22532	0.286346
$334$	0	0
$335$	−0.938841	−0.0512944
$336$	0	0
$337$	−2.68405	−0.146210	−0.0731048	−	0.997324i	$-0.523291\pi$
$337$	−2.68405	−0.146210	−0.0731048	+	0.997324i	$0.523291\pi$
$338$	0	0
$339$	−7.44458	−0.404334
$340$	0	0
$341$	5.09440	0.275877
$342$	0	0
$343$	8.26599	0.446322
$344$	0	0
$345$	0.810300	0.0436251
$346$	0	0
$347$	2.45798	0.131951	0.0659757	−	0.997821i	$-0.478984\pi$
$347$	2.45798	0.131951	0.0659757	+	0.997821i	$0.478984\pi$
$348$	0	0
$349$	27.1822	1.45503	0.727515	−	0.686092i	$-0.240675\pi$
$349$	27.1822	1.45503	0.727515	+	0.686092i	$0.240675\pi$
$350$	0	0
$351$	−25.8744	−1.38107
$352$	0	0
$353$	−6.15795	−0.327754	−0.163877	−	0.986481i	$-0.552400\pi$
$353$	−6.15795	−0.327754	−0.163877	+	0.986481i	$0.552400\pi$
$354$	0	0
$355$	−2.54187	−0.134908
$356$	0	0
$357$	−15.8152	−0.837030
$358$	0	0
$359$	25.8781	1.36579	0.682897	−	0.730515i	$-0.260720\pi$
$359$	25.8781	1.36579	0.682897	+	0.730515i	$0.260720\pi$
$360$	0	0
$361$	−17.5859	−0.925572
$362$	0	0
$363$	−7.75366	−0.406961
$364$	0	0
$365$	−2.01963	−0.105712
$366$	0	0
$367$	5.98447	0.312387	0.156193	−	0.987727i	$-0.450078\pi$
$367$	5.98447	0.312387	0.156193	+	0.987727i	$0.450078\pi$
$368$	0	0
$369$	−4.73334	−0.246408
$370$	0	0
$371$	32.8782	1.70695
$372$	0	0
$373$	−15.5404	−0.804649	−0.402325	−	0.915497i	$-0.631798\pi$
$373$	−15.5404	−0.804649	−0.402325	+	0.915497i	$0.631798\pi$
$374$	0	0
$375$	−3.08587	−0.159354
$376$	0	0
$377$	3.24811	0.167286
$378$	0	0
$379$	22.2085	1.14077	0.570387	−	0.821376i	$-0.306793\pi$
$379$	22.2085	1.14077	0.570387	+	0.821376i	$0.306793\pi$
$380$	0	0
$381$	17.5285	0.898011
$382$	0	0
$383$	−12.5337	−0.640440	−0.320220	−	0.947343i	$-0.603757\pi$
$383$	−12.5337	−0.640440	−0.320220	+	0.947343i	$0.603757\pi$
$384$	0	0
$385$	1.88171	0.0959007
$386$	0	0
$387$	−3.60245	−0.183123
$388$	0	0
$389$	−6.95863	−0.352817	−0.176408	−	0.984317i	$-0.556448\pi$
$389$	−6.95863	−0.352817	−0.176408	+	0.984317i	$0.556448\pi$
$390$	0	0
$391$	10.2090	0.516290
$392$	0	0
$393$	−20.8175	−1.05010
$394$	0	0
$395$	0.331522	0.0166807
$396$	0	0
$397$	11.7325	0.588839	0.294419	−	0.955676i	$-0.404874\pi$
$397$	11.7325	0.588839	0.294419	+	0.955676i	$0.404874\pi$
$398$	0	0
$399$	−4.80422	−0.240512
$400$	0	0
$401$	−13.0523	−0.651803	−0.325902	−	0.945404i	$-0.605668\pi$
$401$	−13.0523	−0.651803	−0.325902	+	0.945404i	$0.605668\pi$
$402$	0	0
$403$	−11.4356	−0.569647
$404$	0	0
$405$	−0.446327	−0.0221782
$406$	0	0
$407$	6.95255	0.344625
$408$	0	0
$409$	1.57296	0.0777779	0.0388890	−	0.999244i	$-0.487618\pi$
$409$	1.57296	0.0777779	0.0388890	+	0.999244i	$0.487618\pi$
$410$	0	0
$411$	−16.8653	−0.831906
$412$	0	0
$413$	35.0178	1.72311
$414$	0	0
$415$	0.135930	0.00667256
$416$	0	0
$417$	−25.6976	−1.25842
$418$	0	0
$419$	20.8586	1.01901	0.509504	−	0.860468i	$-0.329829\pi$
$419$	20.8586	1.01901	0.509504	+	0.860468i	$0.329829\pi$
$420$	0	0
$421$	−4.03040	−0.196430	−0.0982149	−	0.995165i	$-0.531313\pi$
$421$	−4.03040	−0.196430	−0.0982149	+	0.995165i	$0.531313\pi$
$422$	0	0
$423$	8.66899	0.421500
$424$	0	0
$425$	−19.3055	−0.936455
$426$	0	0
$427$	−9.23532	−0.446928
$428$	0	0
$429$	−11.9224	−0.575620
$430$	0	0
$431$	8.83441	0.425538	0.212769	−	0.977102i	$-0.431752\pi$
$431$	8.83441	0.425538	0.212769	+	0.977102i	$0.431752\pi$
$432$	0	0
$433$	−11.9470	−0.574134	−0.287067	−	0.957911i	$-0.592680\pi$
$433$	−11.9470	−0.574134	−0.287067	+	0.957911i	$0.592680\pi$
$434$	0	0
$435$	0.212610	0.0101939
$436$	0	0
$437$	3.10120	0.148350
$438$	0	0
$439$	−8.07722	−0.385505	−0.192752	−	0.981247i	$-0.561741\pi$
$439$	−8.07722	−0.385505	−0.192752	+	0.981247i	$0.561741\pi$
$440$	0	0
$441$	−7.26297	−0.345856
$442$	0	0
$443$	11.3946	0.541374	0.270687	−	0.962667i	$-0.412749\pi$
$443$	11.3946	0.541374	0.270687	+	0.962667i	$0.412749\pi$
$444$	0	0
$445$	1.50302	0.0712498
$446$	0	0
$447$	15.5415	0.735086
$448$	0	0
$449$	−8.23352	−0.388564	−0.194282	−	0.980946i	$-0.562238\pi$
$449$	−8.23352	−0.388564	−0.194282	+	0.980946i	$0.562238\pi$
$450$	0	0
$451$	−6.29795	−0.296559
$452$	0	0
$453$	3.99698	0.187794
$454$	0	0
$455$	−4.22394	−0.198021
$456$	0	0
$457$	−9.97676	−0.466693	−0.233347	−	0.972394i	$-0.574968\pi$
$457$	−9.97676	−0.466693	−0.233347	+	0.972394i	$0.574968\pi$
$458$	0	0
$459$	−21.3383	−0.995985
$460$	0	0
$461$	15.1621	0.706169	0.353084	−	0.935592i	$-0.385133\pi$
$461$	15.1621	0.706169	0.353084	+	0.935592i	$0.385133\pi$
$462$	0	0
$463$	−34.7706	−1.61592	−0.807962	−	0.589234i	$-0.799430\pi$
$463$	−34.7706	−1.61592	−0.807962	+	0.589234i	$0.799430\pi$
$464$	0	0
$465$	−0.748533	−0.0347124
$466$	0	0
$467$	−13.6390	−0.631138	−0.315569	−	0.948903i	$-0.602195\pi$
$467$	−13.6390	−0.631138	−0.315569	+	0.948903i	$0.602195\pi$
$468$	0	0
$469$	12.2070	0.563669
$470$	0	0
$471$	−10.8377	−0.499374
$472$	0	0
$473$	−4.79324	−0.220393
$474$	0	0
$475$	−5.86447	−0.269080
$476$	0	0
$477$	15.3623	0.703389
$478$	0	0
$479$	11.4011	0.520932	0.260466	−	0.965483i	$-0.416124\pi$
$479$	11.4011	0.520932	0.260466	+	0.965483i	$0.416124\pi$
$480$	0	0
$481$	−15.6066	−0.711602
$482$	0	0
$483$	−10.5357	−0.479392
$484$	0	0
$485$	2.81319	0.127740
$486$	0	0
$487$	−8.48404	−0.384449	−0.192224	−	0.981351i	$-0.561570\pi$
$487$	−8.48404	−0.384449	−0.192224	+	0.981351i	$0.561570\pi$
$488$	0	0
$489$	−21.9331	−0.991847
$490$	0	0
$491$	−6.72620	−0.303549	−0.151775	−	0.988415i	$-0.548499\pi$
$491$	−6.72620	−0.303549	−0.151775	+	0.988415i	$0.548499\pi$
$492$	0	0
$493$	2.67867	0.120641
$494$	0	0
$495$	0.879223	0.0395181
$496$	0	0
$497$	33.0500	1.48249
$498$	0	0
$499$	35.8196	1.60350	0.801752	−	0.597657i	$-0.203902\pi$
$499$	35.8196	1.60350	0.801752	+	0.597657i	$0.203902\pi$
$500$	0	0
$501$	6.13676	0.274170
$502$	0	0
$503$	−25.4839	−1.13627	−0.568135	−	0.822936i	$-0.692335\pi$
$503$	−25.4839	−1.13627	−0.568135	+	0.822936i	$0.692335\pi$
$504$	0	0
$505$	1.28482	0.0571736
$506$	0	0
$507$	11.3223	0.502842
$508$	0	0
$509$	−23.6782	−1.04952	−0.524760	−	0.851250i	$-0.675845\pi$
$509$	−23.6782	−1.04952	−0.524760	+	0.851250i	$0.675845\pi$
$510$	0	0
$511$	26.2597	1.16166
$512$	0	0
$513$	−6.48197	−0.286186
$514$	0	0
$515$	−2.51765	−0.110941
$516$	0	0
$517$	11.5345	0.507288
$518$	0	0
$519$	22.6320	0.993434
$520$	0	0
$521$	−20.1916	−0.884610	−0.442305	−	0.896865i	$-0.645839\pi$
$521$	−20.1916	−0.884610	−0.442305	+	0.896865i	$0.645839\pi$
$522$	0	0
$523$	−0.207699	−0.00908203	−0.00454101	−	0.999990i	$-0.501445\pi$
$523$	−0.207699	−0.00908203	−0.00454101	+	0.999990i	$0.501445\pi$
$524$	0	0
$525$	19.9234	0.869528
$526$	0	0
$527$	−9.43077	−0.410811
$528$	0	0
$529$	−16.1990	−0.704306
$530$	0	0
$531$	16.3620	0.710049
$532$	0	0
$533$	14.1372	0.612352
$534$	0	0
$535$	−5.30358	−0.229294
$536$	0	0
$537$	8.83443	0.381234
$538$	0	0
$539$	−9.66374	−0.416247
$540$	0	0
$541$	36.7952	1.58195	0.790976	−	0.611848i	$-0.209573\pi$
$541$	36.7952	1.58195	0.790976	+	0.611848i	$0.209573\pi$
$542$	0	0
$543$	−8.48150	−0.363976
$544$	0	0
$545$	−3.38627	−0.145052
$546$	0	0
$547$	−12.4507	−0.532353	−0.266176	−	0.963924i	$-0.585760\pi$
$547$	−12.4507	−0.532353	−0.266176	+	0.963924i	$0.585760\pi$
$548$	0	0
$549$	−4.31518	−0.184167
$550$	0	0
$551$	0.813706	0.0346650
$552$	0	0
$553$	−4.31053	−0.183302
$554$	0	0
$555$	−1.02156	−0.0433626
$556$	0	0
$557$	−12.3204	−0.522032	−0.261016	−	0.965335i	$-0.584058\pi$
$557$	−12.3204	−0.522032	−0.261016	+	0.965335i	$0.584058\pi$
$558$	0	0
$559$	10.7596	0.455081
$560$	0	0
$561$	−9.83226	−0.415118
$562$	0	0
$563$	−3.92821	−0.165554	−0.0827771	−	0.996568i	$-0.526379\pi$
$563$	−3.92821	−0.165554	−0.0827771	+	0.996568i	$0.526379\pi$
$564$	0	0
$565$	−1.63973	−0.0689839
$566$	0	0
$567$	5.80326	0.243714
$568$	0	0
$569$	32.0858	1.34511	0.672554	−	0.740048i	$-0.265197\pi$
$569$	32.0858	1.34511	0.672554	+	0.740048i	$0.265197\pi$
$570$	0	0
$571$	13.8302	0.578775	0.289387	−	0.957212i	$-0.406548\pi$
$571$	13.8302	0.578775	0.289387	+	0.957212i	$0.406548\pi$
$572$	0	0
$573$	−23.7104	−0.990518
$574$	0	0
$575$	−12.8609	−0.536335
$576$	0	0
$577$	−34.9599	−1.45540	−0.727700	−	0.685895i	$-0.759411\pi$
$577$	−34.9599	−1.45540	−0.727700	+	0.685895i	$0.759411\pi$
$578$	0	0
$579$	20.4385	0.849395
$580$	0	0
$581$	−1.76740	−0.0733241
$582$	0	0
$583$	20.4403	0.846549
$584$	0	0
$585$	−1.97362	−0.0815993
$586$	0	0
$587$	−1.31023	−0.0540788	−0.0270394	−	0.999634i	$-0.508608\pi$
$587$	−1.31023	−0.0540788	−0.0270394	+	0.999634i	$0.508608\pi$
$588$	0	0
$589$	−2.86480	−0.118042
$590$	0	0
$591$	27.4612	1.12960
$592$	0	0
$593$	9.94255	0.408292	0.204146	−	0.978941i	$-0.434558\pi$
$593$	9.94255	0.408292	0.204146	+	0.978941i	$0.434558\pi$
$594$	0	0
$595$	−3.48343	−0.142807
$596$	0	0
$597$	18.8352	0.770871
$598$	0	0
$599$	−46.3649	−1.89442	−0.947209	−	0.320617i	$-0.896110\pi$
$599$	−46.3649	−1.89442	−0.947209	+	0.320617i	$0.896110\pi$
$600$	0	0
$601$	−4.25518	−0.173572	−0.0867862	−	0.996227i	$-0.527660\pi$
$601$	−4.25518	−0.173572	−0.0867862	+	0.996227i	$0.527660\pi$
$602$	0	0
$603$	5.70371	0.232273
$604$	0	0
$605$	−1.70780	−0.0694321
$606$	0	0
$607$	−7.99001	−0.324305	−0.162152	−	0.986766i	$-0.551844\pi$
$607$	−7.99001	−0.324305	−0.162152	+	0.986766i	$0.551844\pi$
$608$	0	0
$609$	−2.76440	−0.112019
$610$	0	0
$611$	−25.8920	−1.04748
$612$	0	0
$613$	28.7857	1.16264	0.581322	−	0.813674i	$-0.302536\pi$
$613$	28.7857	1.16264	0.581322	+	0.813674i	$0.302536\pi$
$614$	0	0
$615$	0.925374	0.0373147
$616$	0	0
$617$	14.9558	0.602099	0.301050	−	0.953608i	$-0.402663\pi$
$617$	14.9558	0.602099	0.301050	+	0.953608i	$0.402663\pi$
$618$	0	0
$619$	28.8152	1.15818	0.579090	−	0.815264i	$-0.303408\pi$
$619$	28.8152	1.15818	0.579090	+	0.815264i	$0.303408\pi$
$620$	0	0
$621$	−14.2150	−0.570430
$622$	0	0
$623$	−19.5426	−0.782957
$624$	0	0
$625$	23.9781	0.959125
$626$	0	0
$627$	−2.98676	−0.119280
$628$	0	0
$629$	−12.8706	−0.513184
$630$	0	0
$631$	48.9852	1.95007	0.975034	−	0.222055i	$-0.0712766\pi$
$631$	48.9852	1.95007	0.975034	+	0.222055i	$0.0712766\pi$
$632$	0	0
$633$	−4.62968	−0.184013
$634$	0	0
$635$	3.86079	0.153211
$636$	0	0
$637$	21.6926	0.859491
$638$	0	0
$639$	15.4425	0.610896
$640$	0	0
$641$	−12.3995	−0.489750	−0.244875	−	0.969555i	$-0.578747\pi$
$641$	−12.3995	−0.489750	−0.244875	+	0.969555i	$0.578747\pi$
$642$	0	0
$643$	−7.46910	−0.294553	−0.147276	−	0.989095i	$-0.547051\pi$
$643$	−7.46910	−0.294553	−0.147276	+	0.989095i	$0.547051\pi$
$644$	0	0
$645$	0.704283	0.0277311
$646$	0	0
$647$	−3.95227	−0.155380	−0.0776899	−	0.996978i	$-0.524754\pi$
$647$	−3.95227	−0.155380	−0.0776899	+	0.996978i	$0.524754\pi$
$648$	0	0
$649$	21.7704	0.854564
$650$	0	0
$651$	9.73261	0.381451
$652$	0	0
$653$	−40.8957	−1.60037	−0.800187	−	0.599751i	$-0.795266\pi$
$653$	−40.8957	−1.60037	−0.800187	+	0.599751i	$0.795266\pi$
$654$	0	0
$655$	−4.58521	−0.179159
$656$	0	0
$657$	12.2698	0.478690
$658$	0	0
$659$	45.9216	1.78885	0.894426	−	0.447217i	$-0.147585\pi$
$659$	45.9216	1.78885	0.894426	+	0.447217i	$0.147585\pi$
$660$	0	0
$661$	−43.0264	−1.67353	−0.836767	−	0.547559i	$-0.815557\pi$
$661$	−43.0264	−1.67353	−0.836767	+	0.547559i	$0.815557\pi$
$662$	0	0
$663$	22.0708	0.857160
$664$	0	0
$665$	−1.05817	−0.0410340
$666$	0	0
$667$	1.78447	0.0690948
$668$	0	0
$669$	0.808175	0.0312459
$670$	0	0
$671$	−5.74156	−0.221651
$672$	0	0
$673$	10.1862	0.392650	0.196325	−	0.980539i	$-0.437099\pi$
$673$	10.1862	0.392650	0.196325	+	0.980539i	$0.437099\pi$
$674$	0	0
$675$	26.8811	1.03465
$676$	0	0
$677$	1.35261	0.0519852	0.0259926	−	0.999662i	$-0.491725\pi$
$677$	1.35261	0.0519852	0.0259926	+	0.999662i	$0.491725\pi$
$678$	0	0
$679$	−36.5778	−1.40373
$680$	0	0
$681$	7.66985	0.293909
$682$	0	0
$683$	16.4917	0.631037	0.315519	−	0.948919i	$-0.397822\pi$
$683$	16.4917	0.631037	0.315519	+	0.948919i	$0.397822\pi$
$684$	0	0
$685$	−3.71472	−0.141932
$686$	0	0
$687$	14.3912	0.549059
$688$	0	0
$689$	−45.8830	−1.74800
$690$	0	0
$691$	16.5806	0.630756	0.315378	−	0.948966i	$-0.397869\pi$
$691$	16.5806	0.630756	0.315378	+	0.948966i	$0.397869\pi$
$692$	0	0
$693$	−11.4319	−0.434261
$694$	0	0
$695$	−5.66009	−0.214699
$696$	0	0
$697$	11.6588	0.441608
$698$	0	0
$699$	−14.6230	−0.553094
$700$	0	0
$701$	17.5181	0.661651	0.330826	−	0.943692i	$-0.392673\pi$
$701$	17.5181	0.661651	0.330826	+	0.943692i	$0.392673\pi$
$702$	0	0
$703$	−3.90973	−0.147458
$704$	0	0
$705$	−1.69480	−0.0638298
$706$	0	0
$707$	−16.7055	−0.628275
$708$	0	0
$709$	−15.8878	−0.596681	−0.298340	−	0.954460i	$-0.596433\pi$
$709$	−15.8878	−0.596681	−0.298340	+	0.954460i	$0.596433\pi$
$710$	0	0
$711$	−2.01408	−0.0755340
$712$	0	0
$713$	−6.28255	−0.235283
$714$	0	0
$715$	−2.62601	−0.0982071
$716$	0	0
$717$	15.2343	0.568935
$718$	0	0
$719$	−9.51038	−0.354677	−0.177339	−	0.984150i	$-0.556749\pi$
$719$	−9.51038	−0.354677	−0.177339	+	0.984150i	$0.556749\pi$
$720$	0	0
$721$	32.7350	1.21912
$722$	0	0
$723$	−19.5954	−0.728762
$724$	0	0
$725$	−3.37449	−0.125325
$726$	0	0
$727$	−3.76509	−0.139639	−0.0698197	−	0.997560i	$-0.522242\pi$
$727$	−3.76509	−0.139639	−0.0698197	+	0.997560i	$0.522242\pi$
$728$	0	0
$729$	22.1338	0.819769
$730$	0	0
$731$	8.87326	0.328190
$732$	0	0
$733$	−35.3718	−1.30649	−0.653243	−	0.757148i	$-0.726592\pi$
$733$	−35.3718	−1.30649	−0.653243	+	0.757148i	$0.726592\pi$
$734$	0	0
$735$	1.41992	0.0523745
$736$	0	0
$737$	7.58907	0.279547
$738$	0	0
$739$	−32.0898	−1.18044	−0.590222	−	0.807241i	$-0.700960\pi$
$739$	−32.0898	−1.18044	−0.590222	+	0.807241i	$0.700960\pi$
$740$	0	0
$741$	6.70450	0.246296
$742$	0	0
$743$	17.0919	0.627039	0.313520	−	0.949582i	$-0.398492\pi$
$743$	17.0919	0.627039	0.313520	+	0.949582i	$0.398492\pi$
$744$	0	0
$745$	3.42313	0.125414
$746$	0	0
$747$	−0.825813	−0.0302149
$748$	0	0
$749$	68.9585	2.51969
$750$	0	0
$751$	−20.2534	−0.739056	−0.369528	−	0.929220i	$-0.620481\pi$
$751$	−20.2534	−0.739056	−0.369528	+	0.929220i	$0.620481\pi$
$752$	0	0
$753$	−1.18772	−0.0432830
$754$	0	0
$755$	0.880365	0.0320398
$756$	0	0
$757$	18.2267	0.662461	0.331231	−	0.943550i	$-0.392536\pi$
$757$	18.2267	0.662461	0.331231	+	0.943550i	$0.392536\pi$
$758$	0	0
$759$	−6.55001	−0.237750
$760$	0	0
$761$	−41.8080	−1.51554	−0.757769	−	0.652523i	$-0.773711\pi$
$761$	−41.8080	−1.51554	−0.757769	+	0.652523i	$0.773711\pi$
$762$	0	0
$763$	44.0292	1.59396
$764$	0	0
$765$	−1.62762	−0.0588468
$766$	0	0
$767$	−48.8689	−1.76455
$768$	0	0
$769$	44.3163	1.59809	0.799043	−	0.601274i	$-0.205340\pi$
$769$	44.3163	1.59809	0.799043	+	0.601274i	$0.205340\pi$
$770$	0	0
$771$	−1.84040	−0.0662804
$772$	0	0
$773$	5.07207	0.182430	0.0912148	−	0.995831i	$-0.470925\pi$
$773$	5.07207	0.182430	0.0912148	+	0.995831i	$0.470925\pi$
$774$	0	0
$775$	11.8805	0.426761
$776$	0	0
$777$	13.2825	0.476508
$778$	0	0
$779$	3.54161	0.126891
$780$	0	0
$781$	20.5470	0.735231
$782$	0	0
$783$	−3.72980	−0.133292
$784$	0	0
$785$	−2.38709	−0.0851987
$786$	0	0
$787$	−11.3657	−0.405144	−0.202572	−	0.979267i	$-0.564930\pi$
$787$	−11.3657	−0.405144	−0.202572	+	0.979267i	$0.564930\pi$
$788$	0	0
$789$	21.1628	0.753414
$790$	0	0
$791$	21.3201	0.758057
$792$	0	0
$793$	12.8883	0.457677
$794$	0	0
$795$	−3.00334	−0.106517
$796$	0	0
$797$	5.75334	0.203794	0.101897	−	0.994795i	$-0.467509\pi$
$797$	5.75334	0.203794	0.101897	+	0.994795i	$0.467509\pi$
$798$	0	0
$799$	−21.3528	−0.755406
$800$	0	0
$801$	−9.13122	−0.322636
$802$	0	0
$803$	16.3256	0.576117
$804$	0	0
$805$	−2.32057	−0.0817894
$806$	0	0
$807$	9.58232	0.337314
$808$	0	0
$809$	39.1298	1.37573	0.687866	−	0.725838i	$-0.258548\pi$
$809$	39.1298	1.37573	0.687866	+	0.725838i	$0.258548\pi$
$810$	0	0
$811$	−38.5227	−1.35271	−0.676357	−	0.736574i	$-0.736442\pi$
$811$	−38.5227	−1.35271	−0.676357	+	0.736574i	$0.736442\pi$
$812$	0	0
$813$	18.3998	0.645309
$814$	0	0
$815$	−4.83093	−0.169220
$816$	0	0
$817$	2.69545	0.0943018
$818$	0	0
$819$	25.6615	0.896687
$820$	0	0
$821$	44.4061	1.54979	0.774893	−	0.632093i	$-0.217804\pi$
$821$	44.4061	1.54979	0.774893	+	0.632093i	$0.217804\pi$
$822$	0	0
$823$	9.78509	0.341087	0.170543	−	0.985350i	$-0.445448\pi$
$823$	9.78509	0.341087	0.170543	+	0.985350i	$0.445448\pi$
$824$	0	0
$825$	12.3863	0.431235
$826$	0	0
$827$	31.5065	1.09559	0.547794	−	0.836613i	$-0.315468\pi$
$827$	31.5065	1.09559	0.547794	+	0.836613i	$0.315468\pi$
$828$	0	0
$829$	−46.9063	−1.62912	−0.814562	−	0.580076i	$-0.803023\pi$
$829$	−46.9063	−1.62912	−0.814562	+	0.580076i	$0.803023\pi$
$830$	0	0
$831$	−3.77739	−0.131036
$832$	0	0
$833$	17.8896	0.619837
$834$	0	0
$835$	1.35167	0.0467765
$836$	0	0
$837$	13.1315	0.453890
$838$	0	0
$839$	−13.0317	−0.449905	−0.224952	−	0.974370i	$-0.572223\pi$
$839$	−13.0317	−0.449905	−0.224952	+	0.974370i	$0.572223\pi$
$840$	0	0
$841$	−28.5318	−0.983855
$842$	0	0
$843$	4.83270	0.166447
$844$	0	0
$845$	2.49383	0.0857904
$846$	0	0
$847$	22.2053	0.762982
$848$	0	0
$849$	11.7985	0.404922
$850$	0	0
$851$	−8.57407	−0.293915
$852$	0	0
$853$	7.72095	0.264360	0.132180	−	0.991226i	$-0.457802\pi$
$853$	7.72095	0.264360	0.132180	+	0.991226i	$0.457802\pi$
$854$	0	0
$855$	−0.494426	−0.0169090
$856$	0	0
$857$	26.8291	0.916464	0.458232	−	0.888833i	$-0.348483\pi$
$857$	26.8291	0.916464	0.458232	+	0.888833i	$0.348483\pi$
$858$	0	0
$859$	30.1017	1.02706	0.513529	−	0.858072i	$-0.328338\pi$
$859$	30.1017	1.02706	0.513529	+	0.858072i	$0.328338\pi$
$860$	0	0
$861$	−12.0319	−0.410047
$862$	0	0
$863$	−15.0060	−0.510810	−0.255405	−	0.966834i	$-0.582209\pi$
$863$	−15.0060	−0.510810	−0.255405	+	0.966834i	$0.582209\pi$
$864$	0	0
$865$	4.98487	0.169491
$866$	0	0
$867$	−1.98974	−0.0675751
$868$	0	0
$869$	−2.67984	−0.0909072
$870$	0	0
$871$	−17.0355	−0.577225
$872$	0	0
$873$	−17.0909	−0.578438
$874$	0	0
$875$	8.83746	0.298761
$876$	0	0
$877$	23.7071	0.800531	0.400265	−	0.916399i	$-0.368918\pi$
$877$	23.7071	0.800531	0.400265	+	0.916399i	$0.368918\pi$
$878$	0	0
$879$	−33.4996	−1.12991
$880$	0	0
$881$	12.1393	0.408984	0.204492	−	0.978868i	$-0.434446\pi$
$881$	12.1393	0.408984	0.204492	+	0.978868i	$0.434446\pi$
$882$	0	0
$883$	−37.1871	−1.25145	−0.625723	−	0.780045i	$-0.715196\pi$
$883$	−37.1871	−1.25145	−0.625723	+	0.780045i	$0.715196\pi$
$884$	0	0
$885$	−3.19879	−0.107526
$886$	0	0
$887$	−44.4474	−1.49240	−0.746199	−	0.665723i	$-0.768123\pi$
$887$	−44.4474	−1.49240	−0.746199	+	0.665723i	$0.768123\pi$
$888$	0	0
$889$	−50.1989	−1.68362
$890$	0	0
$891$	3.60786	0.120868
$892$	0	0
$893$	−6.48637	−0.217058
$894$	0	0
$895$	1.94585	0.0650426
$896$	0	0
$897$	14.7031	0.490921
$898$	0	0
$899$	−1.64844	−0.0549786
$900$	0	0
$901$	−37.8391	−1.26060
$902$	0	0
$903$	−9.15726	−0.304734
$904$	0	0
$905$	−1.86812	−0.0620983
$906$	0	0
$907$	−6.40303	−0.212609	−0.106305	−	0.994334i	$-0.533902\pi$
$907$	−6.40303	−0.212609	−0.106305	+	0.994334i	$0.533902\pi$
$908$	0	0
$909$	−7.80560	−0.258895
$910$	0	0
$911$	−5.63600	−0.186729	−0.0933646	−	0.995632i	$-0.529762\pi$
$911$	−5.63600	−0.186729	−0.0933646	+	0.995632i	$0.529762\pi$
$912$	0	0
$913$	−1.09879	−0.0363645
$914$	0	0
$915$	0.843623	0.0278893
$916$	0	0
$917$	59.6180	1.96876
$918$	0	0
$919$	6.73747	0.222249	0.111124	−	0.993807i	$-0.464555\pi$
$919$	6.73747	0.222249	0.111124	+	0.993807i	$0.464555\pi$
$920$	0	0
$921$	−17.4347	−0.574492
$922$	0	0
$923$	−46.1227	−1.51815
$924$	0	0
$925$	16.2139	0.533108
$926$	0	0
$927$	15.2954	0.502365
$928$	0	0
$929$	46.4077	1.52259	0.761294	−	0.648406i	$-0.224564\pi$
$929$	46.4077	1.52259	0.761294	+	0.648406i	$0.224564\pi$
$930$	0	0
$931$	5.43435	0.178104
$932$	0	0
$933$	2.53063	0.0828491
$934$	0	0
$935$	−2.16563	−0.0708237
$936$	0	0
$937$	−42.9005	−1.40150	−0.700750	−	0.713407i	$-0.747151\pi$
$937$	−42.9005	−1.40150	−0.700750	+	0.713407i	$0.747151\pi$
$938$	0	0
$939$	−17.2366	−0.562496
$940$	0	0
$941$	16.8623	0.549695	0.274848	−	0.961488i	$-0.411373\pi$
$941$	16.8623	0.549695	0.274848	+	0.961488i	$0.411373\pi$
$942$	0	0
$943$	7.76680	0.252922
$944$	0	0
$945$	4.85035	0.157782
$946$	0	0
$947$	37.1003	1.20560	0.602799	−	0.797893i	$-0.294052\pi$
$947$	37.1003	1.20560	0.602799	+	0.797893i	$0.294052\pi$
$948$	0	0
$949$	−36.6466	−1.18960
$950$	0	0
$951$	14.4420	0.468312
$952$	0	0
$953$	51.2148	1.65901	0.829505	−	0.558499i	$-0.188623\pi$
$953$	51.2148	1.65901	0.829505	+	0.558499i	$0.188623\pi$
$954$	0	0
$955$	−5.22241	−0.168993
$956$	0	0
$957$	−1.71862	−0.0555551
$958$	0	0
$959$	48.2997	1.55968
$960$	0	0
$961$	−25.1964	−0.812786
$962$	0	0
$963$	32.2207	1.03830
$964$	0	0
$965$	4.50174	0.144916
$966$	0	0
$967$	27.0288	0.869188	0.434594	−	0.900626i	$-0.356892\pi$
$967$	27.0288	0.869188	0.434594	+	0.900626i	$0.356892\pi$
$968$	0	0
$969$	5.52911	0.177621
$970$	0	0
$971$	32.3342	1.03765	0.518827	−	0.854879i	$-0.326369\pi$
$971$	32.3342	1.03765	0.518827	+	0.854879i	$0.326369\pi$
$972$	0	0
$973$	73.5939	2.35931
$974$	0	0
$975$	−27.8040	−0.890440
$976$	0	0
$977$	26.9387	0.861845	0.430923	−	0.902389i	$-0.358188\pi$
$977$	26.9387	0.861845	0.430923	+	0.902389i	$0.358188\pi$
$978$	0	0
$979$	−12.1495	−0.388301
$980$	0	0
$981$	20.5725	0.656830
$982$	0	0
$983$	32.9702	1.05159	0.525793	−	0.850612i	$-0.323768\pi$
$983$	32.9702	1.05159	0.525793	+	0.850612i	$0.323768\pi$
$984$	0	0
$985$	6.04855	0.192723
$986$	0	0
$987$	22.0362	0.701419
$988$	0	0
$989$	5.91115	0.187964
$990$	0	0
$991$	−8.35248	−0.265325	−0.132663	−	0.991161i	$-0.542353\pi$
$991$	−8.35248	−0.265325	−0.132663	+	0.991161i	$0.542353\pi$
$992$	0	0
$993$	8.69054	0.275786
$994$	0	0
$995$	4.14859	0.131519
$996$	0	0
$997$	49.4250	1.56531	0.782653	−	0.622458i	$-0.213866\pi$
$997$	49.4250	1.56531	0.782653	+	0.622458i	$0.213866\pi$
$998$	0	0
$999$	17.9211	0.566998

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.i.1.9		12
4.3	odd	2		2008.2.a.b.1.4	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.b.1.4	✓	12	4.3	odd	2
4016.2.a.i.1.9		12	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.18772 q^{3} +0.261605 q^{5} -3.40145 q^{7} -1.58932 q^{9} +O(q^{10})\)	\(q+1.18772 q^{3} +0.261605 q^{5} -3.40145 q^{7} -1.58932 q^{9} -2.11467 q^{11} +4.74687 q^{13} +0.310714 q^{15} +3.91468 q^{17} +1.18917 q^{19} -4.03997 q^{21} +2.60787 q^{23} -4.93156 q^{25} -5.45083 q^{27} +0.684263 q^{29} -2.40908 q^{31} -2.51164 q^{33} -0.889836 q^{35} -3.28777 q^{37} +5.63796 q^{39} +2.97822 q^{41} +2.26666 q^{43} -0.415773 q^{45} -5.45453 q^{47} +4.56986 q^{49} +4.64955 q^{51} -9.66594 q^{53} -0.553207 q^{55} +1.41240 q^{57} -10.2950 q^{59} +2.71511 q^{61} +5.40599 q^{63} +1.24181 q^{65} -3.58877 q^{67} +3.09742 q^{69} -9.71644 q^{71} -7.72016 q^{73} -5.85732 q^{75} +7.19294 q^{77} +1.26726 q^{79} -1.70611 q^{81} +0.519602 q^{83} +1.02410 q^{85} +0.812713 q^{87} +5.74537 q^{89} -16.1463 q^{91} -2.86131 q^{93} +0.311093 q^{95} +10.7536 q^{97} +3.36088 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) \) 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9	\( 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9} - 10 q^{11} + 3 q^{13} - 11 q^{15} + 2 q^{17} - 15 q^{19} + 3 q^{21} - 20 q^{23} - 3 q^{25} - 15 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{33} - 16 q^{35} + 5 q^{37} - 21 q^{39} - 21 q^{43} + 10 q^{45} - 27 q^{47} - 13 q^{49} - 19 q^{51} + 22 q^{53} - 24 q^{55} + q^{57} - 23 q^{59} + 4 q^{61} - 21 q^{63} - q^{65} - 26 q^{67} + 10 q^{69} - 23 q^{71} - 8 q^{73} - 16 q^{75} + 22 q^{77} - 37 q^{79} - 20 q^{81} - 30 q^{83} + 2 q^{85} - 16 q^{87} + 3 q^{89} - 8 q^{91} + 20 q^{93} - 33 q^{95} - 4 q^{97} - 15 q^{99}+O(q^{100}) \) 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9 - 10 * q^11 + 3 * q^13 - 11 * q^15 + 2 * q^17 - 15 * q^19 + 3 * q^21 - 20 * q^23 - 3 * q^25 - 15 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^33 - 16 * q^35 + 5 * q^37 - 21 * q^39 - 21 * q^43 + 10 * q^45 - 27 * q^47 - 13 * q^49 - 19 * q^51 + 22 * q^53 - 24 * q^55 + q^57 - 23 * q^59 + 4 * q^61 - 21 * q^63 - q^65 - 26 * q^67 + 10 * q^69 - 23 * q^71 - 8 * q^73 - 16 * q^75 + 22 * q^77 - 37 * q^79 - 20 * q^81 - 30 * q^83 + 2 * q^85 - 16 * q^87 + 3 * q^89 - 8 * q^91 + 20 * q^93 - 33 * q^95 - 4 * q^97 - 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more