LMFDB - Embedded newform 4016.2.a.j.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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 27 x^{12} + 79 x^{11} + 274 x^{10} - 747 x^{9} - 1422 x^{8} + 3287 x^{7} + \cdots - 196 $ x^14 - 3x^13 - 27x^12 + 79x^11 + 274x^10 - 747x^9 - 1422x^8 + 3287x^7 + 4161x^6 - 6861x^5 - 6676x^4 + 5599x^3 + 4627x^2 - 359*x - 196
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1004)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.788858$ 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)

$f(q)$	$=$	$q+0.788858 q^{3} +0.735420 q^{5} -1.11520 q^{7} -2.37770 q^{9} +O(q^{10})$	$q+0.788858 q^{3} +0.735420 q^{5} -1.11520 q^{7} -2.37770 q^{9} +1.83964 q^{11} +0.684684 q^{13} +0.580141 q^{15} +6.42611 q^{17} -3.93182 q^{19} -0.879731 q^{21} -5.20848 q^{23} -4.45916 q^{25} -4.24224 q^{27} -6.50863 q^{29} -7.80694 q^{31} +1.45122 q^{33} -0.820138 q^{35} +4.51085 q^{37} +0.540118 q^{39} +0.951324 q^{41} -1.44119 q^{43} -1.74861 q^{45} -4.97913 q^{47} -5.75634 q^{49} +5.06928 q^{51} -1.03644 q^{53} +1.35291 q^{55} -3.10165 q^{57} +2.68231 q^{59} +12.0662 q^{61} +2.65161 q^{63} +0.503530 q^{65} -1.00402 q^{67} -4.10875 q^{69} -9.70346 q^{71} +15.1651 q^{73} -3.51764 q^{75} -2.05156 q^{77} +7.46004 q^{79} +3.78659 q^{81} -14.5638 q^{83} +4.72589 q^{85} -5.13438 q^{87} +7.56919 q^{89} -0.763557 q^{91} -6.15856 q^{93} -2.89154 q^{95} -3.72438 q^{97} -4.37413 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9}+O(q^{10}) $ 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9	$ 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9} - 9 q^{11} - q^{13} - 14 q^{15} - 27 q^{19} - 3 q^{21} - 13 q^{23} + 26 q^{25} - 15 q^{27} - 25 q^{31} + 16 q^{33} - 21 q^{35} - q^{37} - 33 q^{39} + 10 q^{41} - 35 q^{43} - 4 q^{45} - 6 q^{47} + 36 q^{49} - 48 q^{51} - q^{53} - 41 q^{55} + 14 q^{57} - 30 q^{59} + 3 q^{61} - 31 q^{63} + 7 q^{65} - 22 q^{67} - 17 q^{69} - 6 q^{71} + 5 q^{73} - 4 q^{75} - 14 q^{77} - 56 q^{79} + 26 q^{81} + 28 q^{83} - 23 q^{85} - 11 q^{87} - 24 q^{89} - 38 q^{91} - 55 q^{93} + 4 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100}) $ 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9 - 9 * q^11 - q^13 - 14 * q^15 - 27 * q^19 - 3 * q^21 - 13 * q^23 + 26 * q^25 - 15 * q^27 - 25 * q^31 + 16 * q^33 - 21 * q^35 - q^37 - 33 * q^39 + 10 * q^41 - 35 * q^43 - 4 * q^45 - 6 * q^47 + 36 * q^49 - 48 * q^51 - q^53 - 41 * q^55 + 14 * q^57 - 30 * q^59 + 3 * q^61 - 31 * q^63 + 7 * q^65 - 22 * q^67 - 17 * q^69 - 6 * q^71 + 5 * q^73 - 4 * q^75 - 14 * q^77 - 56 * q^79 + 26 * q^81 + 28 * q^83 - 23 * q^85 - 11 * q^87 - 24 * q^89 - 38 * q^91 - 55 * q^93 + 4 * q^95 + 6 * q^97 - 12 * 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
$3$	0.788858	0.455447	0.227724	−	0.973726i	$-0.426872\pi$
$3$	0.788858	0.455447	0.227724	+	0.973726i	$0.426872\pi$
$4$	0	0
$5$	0.735420	0.328890	0.164445	−	0.986386i	$-0.447417\pi$
$5$	0.735420	0.328890	0.164445	+	0.986386i	$0.447417\pi$
$6$	0	0
$7$	−1.11520	−0.421505	−0.210752	−	0.977539i	$-0.567591\pi$
$7$	−1.11520	−0.421505	−0.210752	+	0.977539i	$0.567591\pi$
$8$	0	0
$9$	−2.37770	−0.792568
$10$	0	0
$11$	1.83964	0.554673	0.277337	−	0.960773i	$-0.410548\pi$
$11$	1.83964	0.554673	0.277337	+	0.960773i	$0.410548\pi$
$12$	0	0
$13$	0.684684	0.189897	0.0949485	−	0.995482i	$-0.469731\pi$
$13$	0.684684	0.189897	0.0949485	+	0.995482i	$0.469731\pi$
$14$	0	0
$15$	0.580141	0.149792
$16$	0	0
$17$	6.42611	1.55856	0.779280	−	0.626676i	$-0.215585\pi$
$17$	6.42611	1.55856	0.779280	+	0.626676i	$0.215585\pi$
$18$	0	0
$19$	−3.93182	−0.902022	−0.451011	−	0.892518i	$-0.648936\pi$
$19$	−3.93182	−0.902022	−0.451011	+	0.892518i	$0.648936\pi$
$20$	0	0
$21$	−0.879731	−0.191973
$22$	0	0
$23$	−5.20848	−1.08604	−0.543021	−	0.839719i	$-0.682720\pi$
$23$	−5.20848	−1.08604	−0.543021	+	0.839719i	$0.682720\pi$
$24$	0	0
$25$	−4.45916	−0.891832
$26$	0	0
$27$	−4.24224	−0.816420
$28$	0	0
$29$	−6.50863	−1.20862	−0.604311	−	0.796748i	$-0.706552\pi$
$29$	−6.50863	−1.20862	−0.604311	+	0.796748i	$0.706552\pi$
$30$	0	0
$31$	−7.80694	−1.40217	−0.701084	−	0.713079i	$-0.747300\pi$
$31$	−7.80694	−1.40217	−0.701084	+	0.713079i	$0.747300\pi$
$32$	0	0
$33$	1.45122	0.252624
$34$	0	0
$35$	−0.820138	−0.138629
$36$	0	0
$37$	4.51085	0.741579	0.370789	−	0.928717i	$-0.379087\pi$
$37$	4.51085	0.741579	0.370789	+	0.928717i	$0.379087\pi$
$38$	0	0
$39$	0.540118	0.0864881
$40$	0	0
$41$	0.951324	0.148572	0.0742859	−	0.997237i	$-0.476332\pi$
$41$	0.951324	0.148572	0.0742859	+	0.997237i	$0.476332\pi$
$42$	0	0
$43$	−1.44119	−0.219779	−0.109890	−	0.993944i	$-0.535050\pi$
$43$	−1.44119	−0.219779	−0.109890	+	0.993944i	$0.535050\pi$
$44$	0	0
$45$	−1.74861	−0.260667
$46$	0	0
$47$	−4.97913	−0.726281	−0.363141	−	0.931734i	$-0.618296\pi$
$47$	−4.97913	−0.726281	−0.363141	+	0.931734i	$0.618296\pi$
$48$	0	0
$49$	−5.75634	−0.822334
$50$	0	0
$51$	5.06928	0.709842
$52$	0	0
$53$	−1.03644	−0.142365	−0.0711827	−	0.997463i	$-0.522677\pi$
$53$	−1.03644	−0.142365	−0.0711827	+	0.997463i	$0.522677\pi$
$54$	0	0
$55$	1.35291	0.182426
$56$	0	0
$57$	−3.10165	−0.410823
$58$	0	0
$59$	2.68231	0.349207	0.174603	−	0.984639i	$-0.444136\pi$
$59$	2.68231	0.349207	0.174603	+	0.984639i	$0.444136\pi$
$60$	0	0
$61$	12.0662	1.54492	0.772462	−	0.635061i	$-0.219025\pi$
$61$	12.0662	1.54492	0.772462	+	0.635061i	$0.219025\pi$
$62$	0	0
$63$	2.65161	0.334071
$64$	0	0
$65$	0.503530	0.0624552
$66$	0	0
$67$	−1.00402	−0.122661	−0.0613304	−	0.998118i	$-0.519534\pi$
$67$	−1.00402	−0.122661	−0.0613304	+	0.998118i	$0.519534\pi$
$68$	0	0
$69$	−4.10875	−0.494635
$70$	0	0
$71$	−9.70346	−1.15159	−0.575795	−	0.817594i	$-0.695307\pi$
$71$	−9.70346	−1.15159	−0.575795	+	0.817594i	$0.695307\pi$
$72$	0	0
$73$	15.1651	1.77494	0.887471	−	0.460863i	$-0.152460\pi$
$73$	15.1651	1.77494	0.887471	+	0.460863i	$0.152460\pi$
$74$	0	0
$75$	−3.51764	−0.406182
$76$	0	0
$77$	−2.05156	−0.233797
$78$	0	0
$79$	7.46004	0.839320	0.419660	−	0.907681i	$-0.362149\pi$
$79$	7.46004	0.839320	0.419660	+	0.907681i	$0.362149\pi$
$80$	0	0
$81$	3.78659	0.420732
$82$	0	0
$83$	−14.5638	−1.59858	−0.799291	−	0.600945i	$-0.794791\pi$
$83$	−14.5638	−1.59858	−0.799291	+	0.600945i	$0.794791\pi$
$84$	0	0
$85$	4.72589	0.512594
$86$	0	0
$87$	−5.13438	−0.550464
$88$	0	0
$89$	7.56919	0.802332	0.401166	−	0.916005i	$-0.368605\pi$
$89$	7.56919	0.802332	0.401166	+	0.916005i	$0.368605\pi$
$90$	0	0
$91$	−0.763557	−0.0800425
$92$	0	0
$93$	−6.15856	−0.638613
$94$	0	0
$95$	−2.89154	−0.296666
$96$	0	0
$97$	−3.72438	−0.378153	−0.189077	−	0.981962i	$-0.560549\pi$
$97$	−3.72438	−0.378153	−0.189077	+	0.981962i	$0.560549\pi$
$98$	0	0
$99$	−4.37413	−0.439616
$100$	0	0
$101$	−16.4769	−1.63951	−0.819754	−	0.572716i	$-0.805890\pi$
$101$	−16.4769	−1.63951	−0.819754	+	0.572716i	$0.805890\pi$
$102$	0	0
$103$	−12.5251	−1.23413	−0.617065	−	0.786912i	$-0.711679\pi$
$103$	−12.5251	−1.23413	−0.617065	+	0.786912i	$0.711679\pi$
$104$	0	0
$105$	−0.646972	−0.0631380
$106$	0	0
$107$	7.86767	0.760596	0.380298	−	0.924864i	$-0.375821\pi$
$107$	7.86767	0.760596	0.380298	+	0.924864i	$0.375821\pi$
$108$	0	0
$109$	13.4636	1.28957	0.644787	−	0.764362i	$-0.276946\pi$
$109$	13.4636	1.28957	0.644787	+	0.764362i	$0.276946\pi$
$110$	0	0
$111$	3.55842	0.337750
$112$	0	0
$113$	−18.5724	−1.74714	−0.873571	−	0.486697i	$-0.838201\pi$
$113$	−18.5724	−1.74714	−0.873571	+	0.486697i	$0.838201\pi$
$114$	0	0
$115$	−3.83042	−0.357188
$116$	0	0
$117$	−1.62797	−0.150506
$118$	0	0
$119$	−7.16638	−0.656941
$120$	0	0
$121$	−7.61571	−0.692338
$122$	0	0
$123$	0.750459	0.0676666
$124$	0	0
$125$	−6.95645	−0.622204
$126$	0	0
$127$	8.84233	0.784630	0.392315	−	0.919831i	$-0.371674\pi$
$127$	8.84233	0.784630	0.392315	+	0.919831i	$0.371674\pi$
$128$	0	0
$129$	−1.13689	−0.100098
$130$	0	0
$131$	3.68966	0.322367	0.161183	−	0.986924i	$-0.448469\pi$
$131$	3.68966	0.322367	0.161183	+	0.986924i	$0.448469\pi$
$132$	0	0
$133$	4.38475	0.380206
$134$	0	0
$135$	−3.11983	−0.268512
$136$	0	0
$137$	−8.77433	−0.749641	−0.374821	−	0.927097i	$-0.622296\pi$
$137$	−8.77433	−0.749641	−0.374821	+	0.927097i	$0.622296\pi$
$138$	0	0
$139$	6.39080	0.542060	0.271030	−	0.962571i	$-0.412636\pi$
$139$	6.39080	0.542060	0.271030	+	0.962571i	$0.412636\pi$
$140$	0	0
$141$	−3.92783	−0.330783
$142$	0	0
$143$	1.25957	0.105331
$144$	0	0
$145$	−4.78657	−0.397503
$146$	0	0
$147$	−4.54093	−0.374530
$148$	0	0
$149$	−15.2022	−1.24541	−0.622705	−	0.782457i	$-0.713966\pi$
$149$	−15.2022	−1.24541	−0.622705	+	0.782457i	$0.713966\pi$
$150$	0	0
$151$	−11.6853	−0.950935	−0.475468	−	0.879733i	$-0.657721\pi$
$151$	−11.6853	−0.950935	−0.475468	+	0.879733i	$0.657721\pi$
$152$	0	0
$153$	−15.2794	−1.23526
$154$	0	0
$155$	−5.74138	−0.461158
$156$	0	0
$157$	−17.7339	−1.41532	−0.707660	−	0.706553i	$-0.750249\pi$
$157$	−17.7339	−1.41532	−0.707660	+	0.706553i	$0.750249\pi$
$158$	0	0
$159$	−0.817600	−0.0648399
$160$	0	0
$161$	5.80848	0.457772
$162$	0	0
$163$	−11.9706	−0.937607	−0.468804	−	0.883302i	$-0.655315\pi$
$163$	−11.9706	−0.937607	−0.468804	+	0.883302i	$0.655315\pi$
$164$	0	0
$165$	1.06725	0.0830855
$166$	0	0
$167$	−14.0367	−1.08619	−0.543094	−	0.839672i	$-0.682747\pi$
$167$	−14.0367	−1.08619	−0.543094	+	0.839672i	$0.682747\pi$
$168$	0	0
$169$	−12.5312	−0.963939
$170$	0	0
$171$	9.34871	0.714913
$172$	0	0
$173$	−9.28586	−0.705991	−0.352995	−	0.935625i	$-0.614837\pi$
$173$	−9.28586	−0.705991	−0.352995	+	0.935625i	$0.614837\pi$
$174$	0	0
$175$	4.97284	0.375911
$176$	0	0
$177$	2.11596	0.159045
$178$	0	0
$179$	14.6990	1.09865	0.549326	−	0.835608i	$-0.314884\pi$
$179$	14.6990	1.09865	0.549326	+	0.835608i	$0.314884\pi$
$180$	0	0
$181$	10.2473	0.761674	0.380837	−	0.924642i	$-0.375636\pi$
$181$	10.2473	0.761674	0.380837	+	0.924642i	$0.375636\pi$
$182$	0	0
$183$	9.51855	0.703631
$184$	0	0
$185$	3.31736	0.243898
$186$	0	0
$187$	11.8217	0.864492
$188$	0	0
$189$	4.73093	0.344125
$190$	0	0
$191$	15.7046	1.13635	0.568173	−	0.822909i	$-0.307650\pi$
$191$	15.7046	1.13635	0.568173	+	0.822909i	$0.307650\pi$
$192$	0	0
$193$	−23.6708	−1.70386	−0.851931	−	0.523655i	$-0.824568\pi$
$193$	−23.6708	−1.70386	−0.851931	+	0.523655i	$0.824568\pi$
$194$	0	0
$195$	0.397213	0.0284450
$196$	0	0
$197$	17.7262	1.26294	0.631468	−	0.775402i	$-0.282453\pi$
$197$	17.7262	1.26294	0.631468	+	0.775402i	$0.282453\pi$
$198$	0	0
$199$	−22.0052	−1.55991	−0.779953	−	0.625838i	$-0.784757\pi$
$199$	−22.0052	−1.55991	−0.779953	+	0.625838i	$0.784757\pi$
$200$	0	0
$201$	−0.792031	−0.0558655
$202$	0	0
$203$	7.25840	0.509440
$204$	0	0
$205$	0.699622	0.0488638
$206$	0	0
$207$	12.3842	0.860762
$208$	0	0
$209$	−7.23315	−0.500327
$210$	0	0
$211$	0.368554	0.0253723	0.0126862	−	0.999920i	$-0.495962\pi$
$211$	0.368554	0.0253723	0.0126862	+	0.999920i	$0.495962\pi$
$212$	0	0
$213$	−7.65465	−0.524488
$214$	0	0
$215$	−1.05988	−0.0722830
$216$	0	0
$217$	8.70627	0.591020
$218$	0	0
$219$	11.9631	0.808393
$220$	0	0
$221$	4.39985	0.295966
$222$	0	0
$223$	2.61090	0.174839	0.0874193	−	0.996172i	$-0.472138\pi$
$223$	2.61090	0.174839	0.0874193	+	0.996172i	$0.472138\pi$
$224$	0	0
$225$	10.6026	0.706837
$226$	0	0
$227$	10.0697	0.668350	0.334175	−	0.942511i	$-0.391542\pi$
$227$	10.0697	0.668350	0.334175	+	0.942511i	$0.391542\pi$
$228$	0	0
$229$	−10.1566	−0.671164	−0.335582	−	0.942011i	$-0.608933\pi$
$229$	−10.1566	−0.671164	−0.335582	+	0.942011i	$0.608933\pi$
$230$	0	0
$231$	−1.61839	−0.106482
$232$	0	0
$233$	−0.730367	−0.0478480	−0.0239240	−	0.999714i	$-0.507616\pi$
$233$	−0.730367	−0.0478480	−0.0239240	+	0.999714i	$0.507616\pi$
$234$	0	0
$235$	−3.66175	−0.238866
$236$	0	0
$237$	5.88491	0.382266
$238$	0	0
$239$	0.0433292	0.00280273	0.00140137	−	0.999999i	$-0.499554\pi$
$239$	0.0433292	0.00280273	0.00140137	+	0.999999i	$0.499554\pi$
$240$	0	0
$241$	23.8137	1.53397	0.766987	−	0.641663i	$-0.221755\pi$
$241$	23.8137	1.53397	0.766987	+	0.641663i	$0.221755\pi$
$242$	0	0
$243$	15.7138	1.00804
$244$	0	0
$245$	−4.23332	−0.270457
$246$	0	0
$247$	−2.69205	−0.171291
$248$	0	0
$249$	−11.4887	−0.728069
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−9.58174	−0.602399
$254$	0	0
$255$	3.72805	0.233460
$256$	0	0
$257$	14.8070	0.923633	0.461817	−	0.886975i	$-0.347198\pi$
$257$	14.8070	0.923633	0.461817	+	0.886975i	$0.347198\pi$
$258$	0	0
$259$	−5.03048	−0.312579
$260$	0	0
$261$	15.4756	0.957915
$262$	0	0
$263$	−0.946488	−0.0583630	−0.0291815	−	0.999574i	$-0.509290\pi$
$263$	−0.946488	−0.0583630	−0.0291815	+	0.999574i	$0.509290\pi$
$264$	0	0
$265$	−0.762215	−0.0468225
$266$	0	0
$267$	5.97101	0.365420
$268$	0	0
$269$	−5.00518	−0.305171	−0.152586	−	0.988290i	$-0.548760\pi$
$269$	−5.00518	−0.305171	−0.152586	+	0.988290i	$0.548760\pi$
$270$	0	0
$271$	23.5141	1.42838	0.714190	−	0.699952i	$-0.246795\pi$
$271$	23.5141	1.42838	0.714190	+	0.699952i	$0.246795\pi$
$272$	0	0
$273$	−0.602338	−0.0364551
$274$	0	0
$275$	−8.20326	−0.494675
$276$	0	0
$277$	−9.78785	−0.588095	−0.294047	−	0.955791i	$-0.595002\pi$
$277$	−9.78785	−0.588095	−0.294047	+	0.955791i	$0.595002\pi$
$278$	0	0
$279$	18.5626	1.11131
$280$	0	0
$281$	20.8837	1.24582	0.622908	−	0.782295i	$-0.285951\pi$
$281$	20.8837	1.24582	0.622908	+	0.782295i	$0.285951\pi$
$282$	0	0
$283$	3.55183	0.211134	0.105567	−	0.994412i	$-0.466334\pi$
$283$	3.55183	0.211134	0.105567	+	0.994412i	$0.466334\pi$
$284$	0	0
$285$	−2.28101	−0.135115
$286$	0	0
$287$	−1.06091	−0.0626237
$288$	0	0
$289$	24.2949	1.42911
$290$	0	0
$291$	−2.93800	−0.172229
$292$	0	0
$293$	−28.1298	−1.64336	−0.821682	−	0.569947i	$-0.806964\pi$
$293$	−28.1298	−1.64336	−0.821682	+	0.569947i	$0.806964\pi$
$294$	0	0
$295$	1.97262	0.114850
$296$	0	0
$297$	−7.80421	−0.452846
$298$	0	0
$299$	−3.56616	−0.206236
$300$	0	0
$301$	1.60721	0.0926379
$302$	0	0
$303$	−12.9979	−0.746709
$304$	0	0
$305$	8.87376	0.508110
$306$	0	0
$307$	−21.9399	−1.25218	−0.626088	−	0.779753i	$-0.715345\pi$
$307$	−21.9399	−1.25218	−0.626088	+	0.779753i	$0.715345\pi$
$308$	0	0
$309$	−9.88049	−0.562081
$310$	0	0
$311$	−25.0002	−1.41763	−0.708817	−	0.705392i	$-0.750771\pi$
$311$	−25.0002	−1.41763	−0.708817	+	0.705392i	$0.750771\pi$
$312$	0	0
$313$	−14.8605	−0.839965	−0.419983	−	0.907532i	$-0.637964\pi$
$313$	−14.8605	−0.839965	−0.419983	+	0.907532i	$0.637964\pi$
$314$	0	0
$315$	1.95004	0.109873
$316$	0	0
$317$	−25.5892	−1.43723	−0.718617	−	0.695406i	$-0.755224\pi$
$317$	−25.5892	−1.43723	−0.718617	+	0.695406i	$0.755224\pi$
$318$	0	0
$319$	−11.9736	−0.670391
$320$	0	0
$321$	6.20647	0.346411
$322$	0	0
$323$	−25.2663	−1.40586
$324$	0	0
$325$	−3.05311	−0.169356
$326$	0	0
$327$	10.6208	0.587333
$328$	0	0
$329$	5.55271	0.306131
$330$	0	0
$331$	11.1314	0.611835	0.305918	−	0.952058i	$-0.401037\pi$
$331$	11.1314	0.611835	0.305918	+	0.952058i	$0.401037\pi$
$332$	0	0
$333$	−10.7255	−0.587751
$334$	0	0
$335$	−0.738378	−0.0403419
$336$	0	0
$337$	2.62997	0.143264	0.0716319	−	0.997431i	$-0.477179\pi$
$337$	2.62997	0.143264	0.0716319	+	0.997431i	$0.477179\pi$
$338$	0	0
$339$	−14.6510	−0.795731
$340$	0	0
$341$	−14.3620	−0.777745
$342$	0	0
$343$	14.2258	0.768122
$344$	0	0
$345$	−3.02165	−0.162680
$346$	0	0
$347$	34.4780	1.85087	0.925437	−	0.378901i	$-0.123698\pi$
$347$	34.4780	1.85087	0.925437	+	0.378901i	$0.123698\pi$
$348$	0	0
$349$	−9.38252	−0.502235	−0.251117	−	0.967957i	$-0.580798\pi$
$349$	−9.38252	−0.502235	−0.251117	+	0.967957i	$0.580798\pi$
$350$	0	0
$351$	−2.90459	−0.155036
$352$	0	0
$353$	24.1142	1.28347	0.641735	−	0.766926i	$-0.278215\pi$
$353$	24.1142	1.28347	0.641735	+	0.766926i	$0.278215\pi$
$354$	0	0
$355$	−7.13612	−0.378746
$356$	0	0
$357$	−5.65325	−0.299202
$358$	0	0
$359$	26.4981	1.39852	0.699258	−	0.714869i	$-0.253514\pi$
$359$	26.4981	1.39852	0.699258	+	0.714869i	$0.253514\pi$
$360$	0	0
$361$	−3.54078	−0.186357
$362$	0	0
$363$	−6.00771	−0.315323
$364$	0	0
$365$	11.1527	0.583760
$366$	0	0
$367$	−37.0039	−1.93159	−0.965793	−	0.259314i	$-0.916504\pi$
$367$	−37.0039	−1.93159	−0.965793	+	0.259314i	$0.916504\pi$
$368$	0	0
$369$	−2.26197	−0.117753
$370$	0	0
$371$	1.15583	0.0600076
$372$	0	0
$373$	−0.0619923	−0.00320984	−0.00160492	−	0.999999i	$-0.500511\pi$
$373$	−0.0619923	−0.00320984	−0.00160492	+	0.999999i	$0.500511\pi$
$374$	0	0
$375$	−5.48765	−0.283381
$376$	0	0
$377$	−4.45635	−0.229514
$378$	0	0
$379$	−34.5019	−1.77224	−0.886122	−	0.463453i	$-0.846610\pi$
$379$	−34.5019	−1.77224	−0.886122	+	0.463453i	$0.846610\pi$
$380$	0	0
$381$	6.97534	0.357358
$382$	0	0
$383$	18.0188	0.920718	0.460359	−	0.887733i	$-0.347721\pi$
$383$	18.0188	0.920718	0.460359	+	0.887733i	$0.347721\pi$
$384$	0	0
$385$	−1.50876	−0.0768936
$386$	0	0
$387$	3.42672	0.174190
$388$	0	0
$389$	8.82215	0.447301	0.223650	−	0.974669i	$-0.428203\pi$
$389$	8.82215	0.447301	0.223650	+	0.974669i	$0.428203\pi$
$390$	0	0
$391$	−33.4702	−1.69266
$392$	0	0
$393$	2.91061	0.146821
$394$	0	0
$395$	5.48626	0.276044
$396$	0	0
$397$	−9.89089	−0.496409	−0.248205	−	0.968708i	$-0.579841\pi$
$397$	−9.89089	−0.496409	−0.248205	+	0.968708i	$0.579841\pi$
$398$	0	0
$399$	3.45895	0.173164
$400$	0	0
$401$	−18.1431	−0.906024	−0.453012	−	0.891504i	$-0.649651\pi$
$401$	−18.1431	−0.906024	−0.453012	+	0.891504i	$0.649651\pi$
$402$	0	0
$403$	−5.34528	−0.266267
$404$	0	0
$405$	2.78473	0.138374
$406$	0	0
$407$	8.29835	0.411334
$408$	0	0
$409$	24.1746	1.19535	0.597677	−	0.801737i	$-0.296090\pi$
$409$	24.1746	1.19535	0.597677	+	0.801737i	$0.296090\pi$
$410$	0	0
$411$	−6.92169	−0.341422
$412$	0	0
$413$	−2.99130	−0.147192
$414$	0	0
$415$	−10.7105	−0.525757
$416$	0	0
$417$	5.04143	0.246880
$418$	0	0
$419$	32.0839	1.56740	0.783699	−	0.621140i	$-0.213330\pi$
$419$	32.0839	1.56740	0.783699	+	0.621140i	$0.213330\pi$
$420$	0	0
$421$	38.4909	1.87593	0.937965	−	0.346729i	$-0.112708\pi$
$421$	38.4909	1.87593	0.937965	+	0.346729i	$0.112708\pi$
$422$	0	0
$423$	11.8389	0.575627
$424$	0	0
$425$	−28.6550	−1.38997
$426$	0	0
$427$	−13.4562	−0.651193
$428$	0	0
$429$	0.993624	0.0479726
$430$	0	0
$431$	−3.24015	−0.156072	−0.0780362	−	0.996951i	$-0.524865\pi$
$431$	−3.24015	−0.156072	−0.0780362	+	0.996951i	$0.524865\pi$
$432$	0	0
$433$	22.8106	1.09621	0.548104	−	0.836410i	$-0.315350\pi$
$433$	22.8106	1.09621	0.548104	+	0.836410i	$0.315350\pi$
$434$	0	0
$435$	−3.77593	−0.181042
$436$	0	0
$437$	20.4788	0.979634
$438$	0	0
$439$	−3.27318	−0.156220	−0.0781101	−	0.996945i	$-0.524889\pi$
$439$	−3.27318	−0.156220	−0.0781101	+	0.996945i	$0.524889\pi$
$440$	0	0
$441$	13.6869	0.651755
$442$	0	0
$443$	7.10731	0.337678	0.168839	−	0.985644i	$-0.445998\pi$
$443$	7.10731	0.337678	0.168839	+	0.985644i	$0.445998\pi$
$444$	0	0
$445$	5.56653	0.263879
$446$	0	0
$447$	−11.9923	−0.567218
$448$	0	0
$449$	5.77466	0.272523	0.136262	−	0.990673i	$-0.456491\pi$
$449$	5.77466	0.272523	0.136262	+	0.990673i	$0.456491\pi$
$450$	0	0
$451$	1.75010	0.0824089
$452$	0	0
$453$	−9.21803	−0.433101
$454$	0	0
$455$	−0.561535	−0.0263252
$456$	0	0
$457$	35.2501	1.64893	0.824465	−	0.565912i	$-0.191476\pi$
$457$	35.2501	1.64893	0.824465	+	0.565912i	$0.191476\pi$
$458$	0	0
$459$	−27.2611	−1.27244
$460$	0	0
$461$	−20.1828	−0.940006	−0.470003	−	0.882665i	$-0.655747\pi$
$461$	−20.1828	−0.940006	−0.470003	+	0.882665i	$0.655747\pi$
$462$	0	0
$463$	26.8195	1.24641	0.623204	−	0.782059i	$-0.285831\pi$
$463$	26.8195	1.24641	0.623204	+	0.782059i	$0.285831\pi$
$464$	0	0
$465$	−4.52913	−0.210033
$466$	0	0
$467$	−12.8592	−0.595054	−0.297527	−	0.954713i	$-0.596162\pi$
$467$	−12.8592	−0.595054	−0.297527	+	0.954713i	$0.596162\pi$
$468$	0	0
$469$	1.11968	0.0517021
$470$	0	0
$471$	−13.9895	−0.644603
$472$	0	0
$473$	−2.65127	−0.121906
$474$	0	0
$475$	17.5326	0.804451
$476$	0	0
$477$	2.46434	0.112834
$478$	0	0
$479$	−0.354905	−0.0162160	−0.00810802	−	0.999967i	$-0.502581\pi$
$479$	−0.354905	−0.0162160	−0.00810802	+	0.999967i	$0.502581\pi$
$480$	0	0
$481$	3.08850	0.140824
$482$	0	0
$483$	4.58206	0.208491
$484$	0	0
$485$	−2.73898	−0.124371
$486$	0	0
$487$	−10.6542	−0.482789	−0.241395	−	0.970427i	$-0.577605\pi$
$487$	−10.6542	−0.482789	−0.241395	+	0.970427i	$0.577605\pi$
$488$	0	0
$489$	−9.44307	−0.427030
$490$	0	0
$491$	1.35677	0.0612303	0.0306151	−	0.999531i	$-0.490253\pi$
$491$	1.35677	0.0612303	0.0306151	+	0.999531i	$0.490253\pi$
$492$	0	0
$493$	−41.8252	−1.88371
$494$	0	0
$495$	−3.21682	−0.144585
$496$	0	0
$497$	10.8213	0.485400
$498$	0	0
$499$	−7.48305	−0.334987	−0.167494	−	0.985873i	$-0.553567\pi$
$499$	−7.48305	−0.334987	−0.167494	+	0.985873i	$0.553567\pi$
$500$	0	0
$501$	−11.0729	−0.494702
$502$	0	0
$503$	44.0455	1.96389	0.981947	−	0.189159i	$-0.0605761\pi$
$503$	44.0455	1.96389	0.981947	+	0.189159i	$0.0605761\pi$
$504$	0	0
$505$	−12.1174	−0.539217
$506$	0	0
$507$	−9.88534	−0.439023
$508$	0	0
$509$	−36.7840	−1.63042	−0.815212	−	0.579163i	$-0.803379\pi$
$509$	−36.7840	−1.63042	−0.815212	+	0.579163i	$0.803379\pi$
$510$	0	0
$511$	−16.9121	−0.748147
$512$	0	0
$513$	16.6797	0.736428
$514$	0	0
$515$	−9.21117	−0.405893
$516$	0	0
$517$	−9.15983	−0.402849
$518$	0	0
$519$	−7.32522	−0.321541
$520$	0	0
$521$	20.6246	0.903580	0.451790	−	0.892124i	$-0.350786\pi$
$521$	20.6246	0.903580	0.451790	+	0.892124i	$0.350786\pi$
$522$	0	0
$523$	7.36895	0.322222	0.161111	−	0.986936i	$-0.448492\pi$
$523$	7.36895	0.322222	0.161111	+	0.986936i	$0.448492\pi$
$524$	0	0
$525$	3.92286	0.171208
$526$	0	0
$527$	−50.1682	−2.18536
$528$	0	0
$529$	4.12822	0.179488
$530$	0	0
$531$	−6.37773	−0.276770
$532$	0	0
$533$	0.651356	0.0282134
$534$	0	0
$535$	5.78604	0.250152
$536$	0	0
$537$	11.5954	0.500378
$538$	0	0
$539$	−10.5896	−0.456127
$540$	0	0
$541$	3.64585	0.156747	0.0783736	−	0.996924i	$-0.475027\pi$
$541$	3.64585	0.156747	0.0783736	+	0.996924i	$0.475027\pi$
$542$	0	0
$543$	8.08364	0.346902
$544$	0	0
$545$	9.90136	0.424128
$546$	0	0
$547$	33.1164	1.41595	0.707977	−	0.706235i	$-0.249608\pi$
$547$	33.1164	1.41595	0.707977	+	0.706235i	$0.249608\pi$
$548$	0	0
$549$	−28.6900	−1.22446
$550$	0	0
$551$	25.5908	1.09020
$552$	0	0
$553$	−8.31941	−0.353777
$554$	0	0
$555$	2.61693	0.111082
$556$	0	0
$557$	10.5771	0.448168	0.224084	−	0.974570i	$-0.428061\pi$
$557$	10.5771	0.448168	0.224084	+	0.974570i	$0.428061\pi$
$558$	0	0
$559$	−0.986757	−0.0417354
$560$	0	0
$561$	9.32568	0.393730
$562$	0	0
$563$	−23.0595	−0.971842	−0.485921	−	0.874003i	$-0.661516\pi$
$563$	−23.0595	−0.971842	−0.485921	+	0.874003i	$0.661516\pi$
$564$	0	0
$565$	−13.6585	−0.574617
$566$	0	0
$567$	−4.22279	−0.177340
$568$	0	0
$569$	39.0065	1.63524	0.817619	−	0.575760i	$-0.195294\pi$
$569$	39.0065	1.63524	0.817619	+	0.575760i	$0.195294\pi$
$570$	0	0
$571$	−30.8138	−1.28952	−0.644759	−	0.764386i	$-0.723042\pi$
$571$	−30.8138	−1.28952	−0.644759	+	0.764386i	$0.723042\pi$
$572$	0	0
$573$	12.3887	0.517546
$574$	0	0
$575$	23.2254	0.968567
$576$	0	0
$577$	9.35620	0.389504	0.194752	−	0.980853i	$-0.437610\pi$
$577$	9.35620	0.389504	0.194752	+	0.980853i	$0.437610\pi$
$578$	0	0
$579$	−18.6729	−0.776019
$580$	0	0
$581$	16.2415	0.673810
$582$	0	0
$583$	−1.90667	−0.0789662
$584$	0	0
$585$	−1.19724	−0.0495000
$586$	0	0
$587$	30.4175	1.25546	0.627732	−	0.778430i	$-0.283983\pi$
$587$	30.4175	1.25546	0.627732	+	0.778430i	$0.283983\pi$
$588$	0	0
$589$	30.6955	1.26479
$590$	0	0
$591$	13.9834	0.575201
$592$	0	0
$593$	−6.01366	−0.246951	−0.123476	−	0.992348i	$-0.539404\pi$
$593$	−6.01366	−0.246951	−0.123476	+	0.992348i	$0.539404\pi$
$594$	0	0
$595$	−5.27029	−0.216061
$596$	0	0
$597$	−17.3590	−0.710455
$598$	0	0
$599$	−15.6252	−0.638429	−0.319214	−	0.947683i	$-0.603419\pi$
$599$	−15.6252	−0.638429	−0.319214	+	0.947683i	$0.603419\pi$
$600$	0	0
$601$	13.6151	0.555373	0.277686	−	0.960672i	$-0.410432\pi$
$601$	13.6151	0.555373	0.277686	+	0.960672i	$0.410432\pi$
$602$	0	0
$603$	2.38727	0.0972171
$604$	0	0
$605$	−5.60074	−0.227703
$606$	0	0
$607$	46.6601	1.89387	0.946937	−	0.321420i	$-0.104160\pi$
$607$	46.6601	1.89387	0.946937	+	0.321420i	$0.104160\pi$
$608$	0	0
$609$	5.72585	0.232023
$610$	0	0
$611$	−3.40913	−0.137919
$612$	0	0
$613$	−27.1692	−1.09735	−0.548677	−	0.836035i	$-0.684868\pi$
$613$	−27.1692	−1.09735	−0.548677	+	0.836035i	$0.684868\pi$
$614$	0	0
$615$	0.551902	0.0222549
$616$	0	0
$617$	46.1646	1.85852	0.929259	−	0.369429i	$-0.120447\pi$
$617$	46.1646	1.85852	0.929259	+	0.369429i	$0.120447\pi$
$618$	0	0
$619$	−29.3967	−1.18155	−0.590776	−	0.806835i	$-0.701179\pi$
$619$	−29.3967	−1.18155	−0.590776	+	0.806835i	$0.701179\pi$
$620$	0	0
$621$	22.0956	0.886667
$622$	0	0
$623$	−8.44113	−0.338187
$624$	0	0
$625$	17.1799	0.687195
$626$	0	0
$627$	−5.70592	−0.227873
$628$	0	0
$629$	28.9872	1.15579
$630$	0	0
$631$	−43.3425	−1.72544	−0.862719	−	0.505683i	$-0.831240\pi$
$631$	−43.3425	−1.72544	−0.862719	+	0.505683i	$0.831240\pi$
$632$	0	0
$633$	0.290737	0.0115558
$634$	0	0
$635$	6.50283	0.258057
$636$	0	0
$637$	−3.94127	−0.156159
$638$	0	0
$639$	23.0720	0.912713
$640$	0	0
$641$	−14.7038	−0.580764	−0.290382	−	0.956911i	$-0.593782\pi$
$641$	−14.7038	−0.580764	−0.290382	+	0.956911i	$0.593782\pi$
$642$	0	0
$643$	−35.7547	−1.41003	−0.705013	−	0.709194i	$-0.749059\pi$
$643$	−35.7547	−1.41003	−0.705013	+	0.709194i	$0.749059\pi$
$644$	0	0
$645$	−0.836092	−0.0329211
$646$	0	0
$647$	0.245478	0.00965075	0.00482537	−	0.999988i	$-0.498464\pi$
$647$	0.245478	0.00965075	0.00482537	+	0.999988i	$0.498464\pi$
$648$	0	0
$649$	4.93449	0.193696
$650$	0	0
$651$	6.86801	0.269178
$652$	0	0
$653$	5.56580	0.217807	0.108903	−	0.994052i	$-0.465266\pi$
$653$	5.56580	0.217807	0.108903	+	0.994052i	$0.465266\pi$
$654$	0	0
$655$	2.71345	0.106023
$656$	0	0
$657$	−36.0582	−1.40676
$658$	0	0
$659$	39.3637	1.53339	0.766697	−	0.642010i	$-0.221899\pi$
$659$	39.3637	1.53339	0.766697	+	0.642010i	$0.221899\pi$
$660$	0	0
$661$	13.2111	0.513851	0.256925	−	0.966431i	$-0.417291\pi$
$661$	13.2111	0.513851	0.256925	+	0.966431i	$0.417291\pi$
$662$	0	0
$663$	3.47086	0.134797
$664$	0	0
$665$	3.22463	0.125046
$666$	0	0
$667$	33.9000	1.31262
$668$	0	0
$669$	2.05963	0.0796297
$670$	0	0
$671$	22.1976	0.856928
$672$	0	0
$673$	−0.950627	−0.0366440	−0.0183220	−	0.999832i	$-0.505832\pi$
$673$	−0.950627	−0.0366440	−0.0183220	+	0.999832i	$0.505832\pi$
$674$	0	0
$675$	18.9168	0.728109
$676$	0	0
$677$	−17.8717	−0.686867	−0.343433	−	0.939177i	$-0.611590\pi$
$677$	−17.8717	−0.686867	−0.343433	+	0.939177i	$0.611590\pi$
$678$	0	0
$679$	4.15341	0.159393
$680$	0	0
$681$	7.94356	0.304398
$682$	0	0
$683$	−11.5503	−0.441959	−0.220979	−	0.975279i	$-0.570925\pi$
$683$	−11.5503	−0.441959	−0.220979	+	0.975279i	$0.570925\pi$
$684$	0	0
$685$	−6.45281	−0.246549
$686$	0	0
$687$	−8.01207	−0.305680
$688$	0	0
$689$	−0.709630	−0.0270348
$690$	0	0
$691$	−32.5179	−1.23704	−0.618520	−	0.785769i	$-0.712268\pi$
$691$	−32.5179	−1.23704	−0.618520	+	0.785769i	$0.712268\pi$
$692$	0	0
$693$	4.87801	0.185300
$694$	0	0
$695$	4.69992	0.178278
$696$	0	0
$697$	6.11331	0.231558
$698$	0	0
$699$	−0.576156	−0.0217922
$700$	0	0
$701$	−32.3461	−1.22169	−0.610847	−	0.791748i	$-0.709171\pi$
$701$	−32.3461	−1.22169	−0.610847	+	0.791748i	$0.709171\pi$
$702$	0	0
$703$	−17.7358	−0.668920
$704$	0	0
$705$	−2.88860	−0.108791
$706$	0	0
$707$	18.3749	0.691060
$708$	0	0
$709$	22.1004	0.829997	0.414999	−	0.909822i	$-0.363782\pi$
$709$	22.1004	0.829997	0.414999	+	0.909822i	$0.363782\pi$
$710$	0	0
$711$	−17.7378	−0.665218
$712$	0	0
$713$	40.6623	1.52281
$714$	0	0
$715$	0.926315	0.0346422
$716$	0	0
$717$	0.0341805	0.00127650
$718$	0	0
$719$	−20.1162	−0.750207	−0.375104	−	0.926983i	$-0.622393\pi$
$719$	−20.1162	−0.750207	−0.375104	+	0.926983i	$0.622393\pi$
$720$	0	0
$721$	13.9679	0.520192
$722$	0	0
$723$	18.7856	0.698644
$724$	0	0
$725$	29.0230	1.07789
$726$	0	0
$727$	4.76915	0.176878	0.0884390	−	0.996082i	$-0.471812\pi$
$727$	4.76915	0.176878	0.0884390	+	0.996082i	$0.471812\pi$
$728$	0	0
$729$	1.03619	0.0383775
$730$	0	0
$731$	−9.26123	−0.342539
$732$	0	0
$733$	−18.4243	−0.680516	−0.340258	−	0.940332i	$-0.610514\pi$
$733$	−18.4243	−0.680516	−0.340258	+	0.940332i	$0.610514\pi$
$734$	0	0
$735$	−3.33949	−0.123179
$736$	0	0
$737$	−1.84704	−0.0680367
$738$	0	0
$739$	33.3551	1.22699	0.613494	−	0.789699i	$-0.289763\pi$
$739$	33.3551	1.22699	0.613494	+	0.789699i	$0.289763\pi$
$740$	0	0
$741$	−2.12365	−0.0780141
$742$	0	0
$743$	−9.17424	−0.336570	−0.168285	−	0.985738i	$-0.553823\pi$
$743$	−9.17424	−0.336570	−0.168285	+	0.985738i	$0.553823\pi$
$744$	0	0
$745$	−11.1800	−0.409602
$746$	0	0
$747$	34.6283	1.26698
$748$	0	0
$749$	−8.77400	−0.320595
$750$	0	0
$751$	13.8848	0.506665	0.253333	−	0.967379i	$-0.418473\pi$
$751$	13.8848	0.506665	0.253333	+	0.967379i	$0.418473\pi$
$752$	0	0
$753$	0.788858	0.0287476
$754$	0	0
$755$	−8.59359	−0.312753
$756$	0	0
$757$	−9.28513	−0.337474	−0.168737	−	0.985661i	$-0.553969\pi$
$757$	−9.28513	−0.337474	−0.168737	+	0.985661i	$0.553969\pi$
$758$	0	0
$759$	−7.55863	−0.274361
$760$	0	0
$761$	27.1126	0.982832	0.491416	−	0.870925i	$-0.336479\pi$
$761$	27.1126	0.982832	0.491416	+	0.870925i	$0.336479\pi$
$762$	0	0
$763$	−15.0145	−0.543562
$764$	0	0
$765$	−11.2368	−0.406266
$766$	0	0
$767$	1.83653	0.0663133
$768$	0	0
$769$	−3.07152	−0.110762	−0.0553809	−	0.998465i	$-0.517637\pi$
$769$	−3.07152	−0.110762	−0.0553809	+	0.998465i	$0.517637\pi$
$770$	0	0
$771$	11.6806	0.420666
$772$	0	0
$773$	8.76972	0.315425	0.157712	−	0.987485i	$-0.449588\pi$
$773$	8.76972	0.315425	0.157712	+	0.987485i	$0.449588\pi$
$774$	0	0
$775$	34.8124	1.25050
$776$	0	0
$777$	−3.96833	−0.142363
$778$	0	0
$779$	−3.74044	−0.134015
$780$	0	0
$781$	−17.8509	−0.638756
$782$	0	0
$783$	27.6112	0.986743
$784$	0	0
$785$	−13.0419	−0.465484
$786$	0	0
$787$	−14.6980	−0.523928	−0.261964	−	0.965078i	$-0.584370\pi$
$787$	−14.6980	−0.523928	−0.261964	+	0.965078i	$0.584370\pi$
$788$	0	0
$789$	−0.746644	−0.0265812
$790$	0	0
$791$	20.7118	0.736428
$792$	0	0
$793$	8.26156	0.293377
$794$	0	0
$795$	−0.601279	−0.0213252
$796$	0	0
$797$	37.2251	1.31858	0.659291	−	0.751888i	$-0.270857\pi$
$797$	37.2251	1.31858	0.659291	+	0.751888i	$0.270857\pi$
$798$	0	0
$799$	−31.9965	−1.13195
$800$	0	0
$801$	−17.9973	−0.635903
$802$	0	0
$803$	27.8984	0.984513
$804$	0	0
$805$	4.27167	0.150556
$806$	0	0
$807$	−3.94837	−0.138989
$808$	0	0
$809$	−5.23774	−0.184149	−0.0920746	−	0.995752i	$-0.529350\pi$
$809$	−5.23774	−0.184149	−0.0920746	+	0.995752i	$0.529350\pi$
$810$	0	0
$811$	1.89704	0.0666143	0.0333071	−	0.999445i	$-0.489396\pi$
$811$	1.89704	0.0666143	0.0333071	+	0.999445i	$0.489396\pi$
$812$	0	0
$813$	18.5493	0.650552
$814$	0	0
$815$	−8.80339	−0.308369
$816$	0	0
$817$	5.66649	0.198245
$818$	0	0
$819$	1.81551	0.0634391
$820$	0	0
$821$	−4.83902	−0.168883	−0.0844414	−	0.996428i	$-0.526911\pi$
$821$	−4.83902	−0.168883	−0.0844414	+	0.996428i	$0.526911\pi$
$822$	0	0
$823$	−51.4425	−1.79317	−0.896586	−	0.442869i	$-0.853961\pi$
$823$	−51.4425	−1.79317	−0.896586	+	0.442869i	$0.853961\pi$
$824$	0	0
$825$	−6.47120	−0.225298
$826$	0	0
$827$	24.4310	0.849548	0.424774	−	0.905299i	$-0.360354\pi$
$827$	24.4310	0.849548	0.424774	+	0.905299i	$0.360354\pi$
$828$	0	0
$829$	−27.1864	−0.944222	−0.472111	−	0.881539i	$-0.656508\pi$
$829$	−27.1864	−0.944222	−0.472111	+	0.881539i	$0.656508\pi$
$830$	0	0
$831$	−7.72122	−0.267846
$832$	0	0
$833$	−36.9908	−1.28166
$834$	0	0
$835$	−10.3228	−0.357236
$836$	0	0
$837$	33.1189	1.14476
$838$	0	0
$839$	30.1665	1.04146	0.520732	−	0.853720i	$-0.325659\pi$
$839$	30.1665	1.04146	0.520732	+	0.853720i	$0.325659\pi$
$840$	0	0
$841$	13.3623	0.460768
$842$	0	0
$843$	16.4742	0.567403
$844$	0	0
$845$	−9.21570	−0.317030
$846$	0	0
$847$	8.49302	0.291824
$848$	0	0
$849$	2.80189	0.0961606
$850$	0	0
$851$	−23.4946	−0.805386
$852$	0	0
$853$	−34.3039	−1.17454	−0.587271	−	0.809390i	$-0.699798\pi$
$853$	−34.3039	−1.17454	−0.587271	+	0.809390i	$0.699798\pi$
$854$	0	0
$855$	6.87522	0.235128
$856$	0	0
$857$	21.5127	0.734859	0.367430	−	0.930051i	$-0.380238\pi$
$857$	21.5127	0.734859	0.367430	+	0.930051i	$0.380238\pi$
$858$	0	0
$859$	10.5914	0.361373	0.180686	−	0.983541i	$-0.442168\pi$
$859$	10.5914	0.361373	0.180686	+	0.983541i	$0.442168\pi$
$860$	0	0
$861$	−0.836910	−0.0285218
$862$	0	0
$863$	39.2724	1.33685	0.668424	−	0.743780i	$-0.266969\pi$
$863$	39.2724	1.33685	0.668424	+	0.743780i	$0.266969\pi$
$864$	0	0
$865$	−6.82900	−0.232193
$866$	0	0
$867$	19.1652	0.650884
$868$	0	0
$869$	13.7238	0.465548
$870$	0	0
$871$	−0.687438	−0.0232929
$872$	0	0
$873$	8.85547	0.299712
$874$	0	0
$875$	7.75781	0.262262
$876$	0	0
$877$	−2.11786	−0.0715151	−0.0357576	−	0.999360i	$-0.511384\pi$
$877$	−2.11786	−0.0715151	−0.0357576	+	0.999360i	$0.511384\pi$
$878$	0	0
$879$	−22.1904	−0.748465
$880$	0	0
$881$	−41.4875	−1.39775	−0.698874	−	0.715244i	$-0.746315\pi$
$881$	−41.4875	−1.39775	−0.698874	+	0.715244i	$0.746315\pi$
$882$	0	0
$883$	18.1798	0.611800	0.305900	−	0.952064i	$-0.401043\pi$
$883$	18.1798	0.611800	0.305900	+	0.952064i	$0.401043\pi$
$884$	0	0
$885$	1.55612	0.0523083
$886$	0	0
$887$	34.7083	1.16539	0.582695	−	0.812691i	$-0.301998\pi$
$887$	34.7083	1.16539	0.582695	+	0.812691i	$0.301998\pi$
$888$	0	0
$889$	−9.86094	−0.330725
$890$	0	0
$891$	6.96597	0.233369
$892$	0	0
$893$	19.5771	0.655122
$894$	0	0
$895$	10.8099	0.361335
$896$	0	0
$897$	−2.81319	−0.0939297
$898$	0	0
$899$	50.8125	1.69469
$900$	0	0
$901$	−6.66024	−0.221885
$902$	0	0
$903$	1.26786	0.0421917
$904$	0	0
$905$	7.53604	0.250507
$906$	0	0
$907$	6.63577	0.220337	0.110169	−	0.993913i	$-0.464861\pi$
$907$	6.63577	0.220337	0.110169	+	0.993913i	$0.464861\pi$
$908$	0	0
$909$	39.1771	1.29942
$910$	0	0
$911$	−1.88535	−0.0624643	−0.0312321	−	0.999512i	$-0.509943\pi$
$911$	−1.88535	−0.0624643	−0.0312321	+	0.999512i	$0.509943\pi$
$912$	0	0
$913$	−26.7921	−0.886690
$914$	0	0
$915$	7.00013	0.231417
$916$	0	0
$917$	−4.11469	−0.135879
$918$	0	0
$919$	42.2779	1.39462	0.697310	−	0.716769i	$-0.254380\pi$
$919$	42.2779	1.39462	0.697310	+	0.716769i	$0.254380\pi$
$920$	0	0
$921$	−17.3074	−0.570300
$922$	0	0
$923$	−6.64380	−0.218683
$924$	0	0
$925$	−20.1146	−0.661363
$926$	0	0
$927$	29.7809	0.978132
$928$	0	0
$929$	−6.00665	−0.197072	−0.0985359	−	0.995133i	$-0.531416\pi$
$929$	−6.00665	−0.197072	−0.0985359	+	0.995133i	$0.531416\pi$
$930$	0	0
$931$	22.6329	0.741763
$932$	0	0
$933$	−19.7216	−0.645657
$934$	0	0
$935$	8.69395	0.284322
$936$	0	0
$937$	35.4638	1.15855	0.579275	−	0.815132i	$-0.303336\pi$
$937$	35.4638	1.15855	0.579275	+	0.815132i	$0.303336\pi$
$938$	0	0
$939$	−11.7228	−0.382560
$940$	0	0
$941$	24.7662	0.807355	0.403677	−	0.914901i	$-0.367732\pi$
$941$	24.7662	0.807355	0.403677	+	0.914901i	$0.367732\pi$
$942$	0	0
$943$	−4.95495	−0.161355
$944$	0	0
$945$	3.47922	0.113179
$946$	0	0
$947$	31.3260	1.01796	0.508978	−	0.860779i	$-0.330023\pi$
$947$	31.3260	1.01796	0.508978	+	0.860779i	$0.330023\pi$
$948$	0	0
$949$	10.3833	0.337056
$950$	0	0
$951$	−20.1863	−0.654584
$952$	0	0
$953$	−30.0853	−0.974557	−0.487279	−	0.873247i	$-0.662010\pi$
$953$	−30.0853	−0.974557	−0.487279	+	0.873247i	$0.662010\pi$
$954$	0	0
$955$	11.5495	0.373733
$956$	0	0
$957$	−9.44543	−0.305327
$958$	0	0
$959$	9.78510	0.315977
$960$	0	0
$961$	29.9483	0.966074
$962$	0	0
$963$	−18.7070	−0.602824
$964$	0	0
$965$	−17.4080	−0.560382
$966$	0	0
$967$	8.09637	0.260362	0.130181	−	0.991490i	$-0.458444\pi$
$967$	8.09637	0.260362	0.130181	+	0.991490i	$0.458444\pi$
$968$	0	0
$969$	−19.9315	−0.640293
$970$	0	0
$971$	−44.2126	−1.41885	−0.709424	−	0.704782i	$-0.751045\pi$
$971$	−44.2126	−1.41885	−0.709424	+	0.704782i	$0.751045\pi$
$972$	0	0
$973$	−7.12700	−0.228481
$974$	0	0
$975$	−2.40847	−0.0771328
$976$	0	0
$977$	5.78747	0.185158	0.0925788	−	0.995705i	$-0.470489\pi$
$977$	5.78747	0.185158	0.0925788	+	0.995705i	$0.470489\pi$
$978$	0	0
$979$	13.9246	0.445032
$980$	0	0
$981$	−32.0123	−1.02208
$982$	0	0
$983$	−40.5198	−1.29238	−0.646191	−	0.763176i	$-0.723639\pi$
$983$	−40.5198	−1.29238	−0.646191	+	0.763176i	$0.723639\pi$
$984$	0	0
$985$	13.0362	0.415367
$986$	0	0
$987$	4.38030	0.139426
$988$	0	0
$989$	7.50639	0.238689
$990$	0	0
$991$	50.6653	1.60944	0.804718	−	0.593657i	$-0.202317\pi$
$991$	50.6653	1.60944	0.804718	+	0.593657i	$0.202317\pi$
$992$	0	0
$993$	8.78107	0.278659
$994$	0	0
$995$	−16.1830	−0.513037
$996$	0	0
$997$	11.0831	0.351004	0.175502	−	0.984479i	$-0.443845\pi$
$997$	11.0831	0.351004	0.175502	+	0.984479i	$0.443845\pi$
$998$	0	0
$999$	−19.1361	−0.605439

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.j.1.9		14
4.3	odd	2		1004.2.a.b.1.6	✓	14
12.11	even	2		9036.2.a.m.1.6		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1004.2.a.b.1.6	✓	14	4.3	odd	2
4016.2.a.j.1.9		14	1.1	even	1	trivial
9036.2.a.m.1.6		14	12.11	even	2

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)

\(f(q)\)	\(=\)	\(q+0.788858 q^{3} +0.735420 q^{5} -1.11520 q^{7} -2.37770 q^{9} +O(q^{10})\)	\(q+0.788858 q^{3} +0.735420 q^{5} -1.11520 q^{7} -2.37770 q^{9} +1.83964 q^{11} +0.684684 q^{13} +0.580141 q^{15} +6.42611 q^{17} -3.93182 q^{19} -0.879731 q^{21} -5.20848 q^{23} -4.45916 q^{25} -4.24224 q^{27} -6.50863 q^{29} -7.80694 q^{31} +1.45122 q^{33} -0.820138 q^{35} +4.51085 q^{37} +0.540118 q^{39} +0.951324 q^{41} -1.44119 q^{43} -1.74861 q^{45} -4.97913 q^{47} -5.75634 q^{49} +5.06928 q^{51} -1.03644 q^{53} +1.35291 q^{55} -3.10165 q^{57} +2.68231 q^{59} +12.0662 q^{61} +2.65161 q^{63} +0.503530 q^{65} -1.00402 q^{67} -4.10875 q^{69} -9.70346 q^{71} +15.1651 q^{73} -3.51764 q^{75} -2.05156 q^{77} +7.46004 q^{79} +3.78659 q^{81} -14.5638 q^{83} +4.72589 q^{85} -5.13438 q^{87} +7.56919 q^{89} -0.763557 q^{91} -6.15856 q^{93} -2.89154 q^{95} -3.72438 q^{97} -4.37413 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9}+O(q^{10}) \) 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9	\( 14 q - 3 q^{3} - 2 q^{5} - 8 q^{7} + 21 q^{9} - 9 q^{11} - q^{13} - 14 q^{15} - 27 q^{19} - 3 q^{21} - 13 q^{23} + 26 q^{25} - 15 q^{27} - 25 q^{31} + 16 q^{33} - 21 q^{35} - q^{37} - 33 q^{39} + 10 q^{41} - 35 q^{43} - 4 q^{45} - 6 q^{47} + 36 q^{49} - 48 q^{51} - q^{53} - 41 q^{55} + 14 q^{57} - 30 q^{59} + 3 q^{61} - 31 q^{63} + 7 q^{65} - 22 q^{67} - 17 q^{69} - 6 q^{71} + 5 q^{73} - 4 q^{75} - 14 q^{77} - 56 q^{79} + 26 q^{81} + 28 q^{83} - 23 q^{85} - 11 q^{87} - 24 q^{89} - 38 q^{91} - 55 q^{93} + 4 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100}) \) 14 * q - 3 * q^3 - 2 * q^5 - 8 * q^7 + 21 * q^9 - 9 * q^11 - q^13 - 14 * q^15 - 27 * q^19 - 3 * q^21 - 13 * q^23 + 26 * q^25 - 15 * q^27 - 25 * q^31 + 16 * q^33 - 21 * q^35 - q^37 - 33 * q^39 + 10 * q^41 - 35 * q^43 - 4 * q^45 - 6 * q^47 + 36 * q^49 - 48 * q^51 - q^53 - 41 * q^55 + 14 * q^57 - 30 * q^59 + 3 * q^61 - 31 * q^63 + 7 * q^65 - 22 * q^67 - 17 * q^69 - 6 * q^71 + 5 * q^73 - 4 * q^75 - 14 * q^77 - 56 * q^79 + 26 * q^81 + 28 * q^83 - 23 * q^85 - 11 * q^87 - 24 * q^89 - 38 * q^91 - 55 * q^93 + 4 * q^95 + 6 * q^97 - 12 * q^99

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