LMFDB - Embedded newform 4016.2.a.k.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:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 2 x^{16} - 28 x^{15} + 54 x^{14} + 317 x^{13} - 582 x^{12} - 1867 x^{11} + 3178 x^{10} + \cdots + 256 $ x^17 - 2x^16 - 28x^15 + 54x^14 + 317x^13 - 582x^12 - 1867x^11 + 3178x^10 + 6186x^9 - 9216x^8 - 11921x^7 + 13680x^6 + 13752x^5 - 9400x^4 - 8800x^3 + 1920x^2 + 2240x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 251)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-2.51582$ 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)

$f(q)$	$=$	$q+0.505139 q^{3} -3.97764 q^{5} -1.36760 q^{7} -2.74483 q^{9} +O(q^{10})$	$q+0.505139 q^{3} -3.97764 q^{5} -1.36760 q^{7} -2.74483 q^{9} -4.72173 q^{11} -3.07300 q^{13} -2.00926 q^{15} -2.19516 q^{17} -5.12915 q^{19} -0.690830 q^{21} -5.63883 q^{23} +10.8217 q^{25} -2.90194 q^{27} +8.38748 q^{29} -5.97694 q^{31} -2.38513 q^{33} +5.43984 q^{35} -0.0197683 q^{37} -1.55229 q^{39} -8.14255 q^{41} -1.90290 q^{43} +10.9180 q^{45} +1.09502 q^{47} -5.12966 q^{49} -1.10886 q^{51} +7.79503 q^{53} +18.7813 q^{55} -2.59094 q^{57} -1.16638 q^{59} +7.58084 q^{61} +3.75385 q^{63} +12.2233 q^{65} +7.25841 q^{67} -2.84839 q^{69} -13.8131 q^{71} -12.6464 q^{73} +5.46644 q^{75} +6.45745 q^{77} +4.79645 q^{79} +6.76862 q^{81} +3.89784 q^{83} +8.73156 q^{85} +4.23684 q^{87} -13.4186 q^{89} +4.20265 q^{91} -3.01919 q^{93} +20.4019 q^{95} +0.906726 q^{97} +12.9604 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * 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
$3$	0.505139	0.291642	0.145821	−	0.989311i	$-0.453418\pi$
$3$	0.505139	0.291642	0.145821	+	0.989311i	$0.453418\pi$
$4$	0	0
$5$	−3.97764	−1.77886	−0.889428	−	0.457075i	$-0.848897\pi$
$5$	−3.97764	−1.77886	−0.889428	+	0.457075i	$0.848897\pi$
$6$	0	0
$7$	−1.36760	−0.516906	−0.258453	−	0.966024i	$-0.583213\pi$
$7$	−1.36760	−0.516906	−0.258453	+	0.966024i	$0.583213\pi$
$8$	0	0
$9$	−2.74483	−0.914945
$10$	0	0
$11$	−4.72173	−1.42365	−0.711827	−	0.702355i	$-0.752132\pi$
$11$	−4.72173	−1.42365	−0.711827	+	0.702355i	$0.752132\pi$
$12$	0	0
$13$	−3.07300	−0.852297	−0.426149	−	0.904653i	$-0.640130\pi$
$13$	−3.07300	−0.852297	−0.426149	+	0.904653i	$0.640130\pi$
$14$	0	0
$15$	−2.00926	−0.518790
$16$	0	0
$17$	−2.19516	−0.532404	−0.266202	−	0.963917i	$-0.585769\pi$
$17$	−2.19516	−0.532404	−0.266202	+	0.963917i	$0.585769\pi$
$18$	0	0
$19$	−5.12915	−1.17671	−0.588354	−	0.808603i	$-0.700224\pi$
$19$	−5.12915	−1.17671	−0.588354	+	0.808603i	$0.700224\pi$
$20$	0	0
$21$	−0.690830	−0.150752
$22$	0	0
$23$	−5.63883	−1.17578	−0.587888	−	0.808942i	$-0.700041\pi$
$23$	−5.63883	−1.17578	−0.587888	+	0.808942i	$0.700041\pi$
$24$	0	0
$25$	10.8217	2.16433
$26$	0	0
$27$	−2.90194	−0.558479
$28$	0	0
$29$	8.38748	1.55752	0.778758	−	0.627325i	$-0.215850\pi$
$29$	8.38748	1.55752	0.778758	+	0.627325i	$0.215850\pi$
$30$	0	0
$31$	−5.97694	−1.07349	−0.536745	−	0.843744i	$-0.680346\pi$
$31$	−5.97694	−1.07349	−0.536745	+	0.843744i	$0.680346\pi$
$32$	0	0
$33$	−2.38513	−0.415198
$34$	0	0
$35$	5.43984	0.919501
$36$	0	0
$37$	−0.0197683	−0.00324988	−0.00162494	−	0.999999i	$-0.500517\pi$
$37$	−0.0197683	−0.00324988	−0.00162494	+	0.999999i	$0.500517\pi$
$38$	0	0
$39$	−1.55229	−0.248566
$40$	0	0
$41$	−8.14255	−1.27165	−0.635826	−	0.771832i	$-0.719340\pi$
$41$	−8.14255	−1.27165	−0.635826	+	0.771832i	$0.719340\pi$
$42$	0	0
$43$	−1.90290	−0.290189	−0.145095	−	0.989418i	$-0.546349\pi$
$43$	−1.90290	−0.290189	−0.145095	+	0.989418i	$0.546349\pi$
$44$	0	0
$45$	10.9180	1.62756
$46$	0	0
$47$	1.09502	0.159725	0.0798624	−	0.996806i	$-0.474552\pi$
$47$	1.09502	0.159725	0.0798624	+	0.996806i	$0.474552\pi$
$48$	0	0
$49$	−5.12966	−0.732808
$50$	0	0
$51$	−1.10886	−0.155271
$52$	0	0
$53$	7.79503	1.07073	0.535365	−	0.844621i	$-0.320174\pi$
$53$	7.79503	1.07073	0.535365	+	0.844621i	$0.320174\pi$
$54$	0	0
$55$	18.7813	2.53248
$56$	0	0
$57$	−2.59094	−0.343178
$58$	0	0
$59$	−1.16638	−0.151850	−0.0759248	−	0.997114i	$-0.524191\pi$
$59$	−1.16638	−0.151850	−0.0759248	+	0.997114i	$0.524191\pi$
$60$	0	0
$61$	7.58084	0.970627	0.485313	−	0.874340i	$-0.338705\pi$
$61$	7.58084	0.970627	0.485313	+	0.874340i	$0.338705\pi$
$62$	0	0
$63$	3.75385	0.472940
$64$	0	0
$65$	12.2233	1.51611
$66$	0	0
$67$	7.25841	0.886755	0.443378	−	0.896335i	$-0.353780\pi$
$67$	7.25841	0.886755	0.443378	+	0.896335i	$0.353780\pi$
$68$	0	0
$69$	−2.84839	−0.342906
$70$	0	0
$71$	−13.8131	−1.63931	−0.819655	−	0.572858i	$-0.805835\pi$
$71$	−13.8131	−1.63931	−0.819655	+	0.572858i	$0.805835\pi$
$72$	0	0
$73$	−12.6464	−1.48015	−0.740074	−	0.672526i	$-0.765209\pi$
$73$	−12.6464	−1.48015	−0.740074	+	0.672526i	$0.765209\pi$
$74$	0	0
$75$	5.46644	0.631210
$76$	0	0
$77$	6.45745	0.735895
$78$	0	0
$79$	4.79645	0.539643	0.269821	−	0.962910i	$-0.413035\pi$
$79$	4.79645	0.539643	0.269821	+	0.962910i	$0.413035\pi$
$80$	0	0
$81$	6.76862	0.752069
$82$	0	0
$83$	3.89784	0.427843	0.213922	−	0.976851i	$-0.431376\pi$
$83$	3.89784	0.427843	0.213922	+	0.976851i	$0.431376\pi$
$84$	0	0
$85$	8.73156	0.947070
$86$	0	0
$87$	4.23684	0.454237
$88$	0	0
$89$	−13.4186	−1.42237	−0.711183	−	0.703007i	$-0.751840\pi$
$89$	−13.4186	−1.42237	−0.711183	+	0.703007i	$0.751840\pi$
$90$	0	0
$91$	4.20265	0.440557
$92$	0	0
$93$	−3.01919	−0.313075
$94$	0	0
$95$	20.4019	2.09320
$96$	0	0
$97$	0.906726	0.0920641	0.0460320	−	0.998940i	$-0.485342\pi$
$97$	0.906726	0.0920641	0.0460320	+	0.998940i	$0.485342\pi$
$98$	0	0
$99$	12.9604	1.30256
$100$	0	0
$101$	10.6420	1.05892	0.529458	−	0.848336i	$-0.322395\pi$
$101$	10.6420	1.05892	0.529458	+	0.848336i	$0.322395\pi$
$102$	0	0
$103$	−17.0735	−1.68231	−0.841153	−	0.540797i	$-0.818123\pi$
$103$	−17.0735	−1.68231	−0.841153	+	0.540797i	$0.818123\pi$
$104$	0	0
$105$	2.74788	0.268165
$106$	0	0
$107$	−4.72904	−0.457174	−0.228587	−	0.973524i	$-0.573410\pi$
$107$	−4.72904	−0.457174	−0.228587	+	0.973524i	$0.573410\pi$
$108$	0	0
$109$	12.7089	1.21729	0.608644	−	0.793444i	$-0.291714\pi$
$109$	12.7089	1.21729	0.608644	+	0.793444i	$0.291714\pi$
$110$	0	0
$111$	−0.00998573	−0.000947803 0
$112$	0	0
$113$	4.13398	0.388893	0.194446	−	0.980913i	$-0.437709\pi$
$113$	4.13398	0.388893	0.194446	+	0.980913i	$0.437709\pi$
$114$	0	0
$115$	22.4292	2.09154
$116$	0	0
$117$	8.43488	0.779805
$118$	0	0
$119$	3.00211	0.275203
$120$	0	0
$121$	11.2947	1.02679
$122$	0	0
$123$	−4.11312	−0.370868
$124$	0	0
$125$	−23.1565	−2.07118
$126$	0	0
$127$	11.4022	1.01178	0.505891	−	0.862598i	$-0.331164\pi$
$127$	11.4022	1.01178	0.505891	+	0.862598i	$0.331164\pi$
$128$	0	0
$129$	−0.961229	−0.0846315
$130$	0	0
$131$	−20.7099	−1.80943	−0.904714	−	0.426019i	$-0.859916\pi$
$131$	−20.7099	−1.80943	−0.904714	+	0.426019i	$0.859916\pi$
$132$	0	0
$133$	7.01465	0.608247
$134$	0	0
$135$	11.5429	0.993454
$136$	0	0
$137$	−7.34033	−0.627126	−0.313563	−	0.949567i	$-0.601523\pi$
$137$	−7.34033	−0.627126	−0.313563	+	0.949567i	$0.601523\pi$
$138$	0	0
$139$	−8.30015	−0.704010	−0.352005	−	0.935998i	$-0.614500\pi$
$139$	−8.30015	−0.704010	−0.352005	+	0.935998i	$0.614500\pi$
$140$	0	0
$141$	0.553136	0.0465825
$142$	0	0
$143$	14.5099	1.21338
$144$	0	0
$145$	−33.3624	−2.77060
$146$	0	0
$147$	−2.59119	−0.213718
$148$	0	0
$149$	−19.5843	−1.60441	−0.802206	−	0.597047i	$-0.796341\pi$
$149$	−19.5843	−1.60441	−0.802206	+	0.597047i	$0.796341\pi$
$150$	0	0
$151$	−5.78022	−0.470388	−0.235194	−	0.971948i	$-0.575573\pi$
$151$	−5.78022	−0.470388	−0.235194	+	0.971948i	$0.575573\pi$
$152$	0	0
$153$	6.02534	0.487120
$154$	0	0
$155$	23.7741	1.90959
$156$	0	0
$157$	8.87250	0.708102	0.354051	−	0.935226i	$-0.384804\pi$
$157$	8.87250	0.708102	0.354051	+	0.935226i	$0.384804\pi$
$158$	0	0
$159$	3.93757	0.312270
$160$	0	0
$161$	7.71168	0.607766
$162$	0	0
$163$	5.65852	0.443209	0.221605	−	0.975137i	$-0.428871\pi$
$163$	5.65852	0.443209	0.221605	+	0.975137i	$0.428871\pi$
$164$	0	0
$165$	9.48719	0.738577
$166$	0	0
$167$	−17.7207	−1.37127	−0.685634	−	0.727947i	$-0.740475\pi$
$167$	−17.7207	−1.37127	−0.685634	+	0.727947i	$0.740475\pi$
$168$	0	0
$169$	−3.55666	−0.273590
$170$	0	0
$171$	14.0787	1.07662
$172$	0	0
$173$	−13.2092	−1.00428	−0.502138	−	0.864788i	$-0.667453\pi$
$173$	−13.2092	−1.00428	−0.502138	+	0.864788i	$0.667453\pi$
$174$	0	0
$175$	−14.7997	−1.11875
$176$	0	0
$177$	−0.589184	−0.0442858
$178$	0	0
$179$	5.58502	0.417444	0.208722	−	0.977975i	$-0.433070\pi$
$179$	5.58502	0.417444	0.208722	+	0.977975i	$0.433070\pi$
$180$	0	0
$181$	9.56280	0.710797	0.355399	−	0.934715i	$-0.384345\pi$
$181$	9.56280	0.710797	0.355399	+	0.934715i	$0.384345\pi$
$182$	0	0
$183$	3.82938	0.283076
$184$	0	0
$185$	0.0786311	0.00578107
$186$	0	0
$187$	10.3649	0.757959
$188$	0	0
$189$	3.96871	0.288681
$190$	0	0
$191$	−4.01683	−0.290647	−0.145324	−	0.989384i	$-0.546422\pi$
$191$	−4.01683	−0.290647	−0.145324	+	0.989384i	$0.546422\pi$
$192$	0	0
$193$	17.8205	1.28275	0.641374	−	0.767229i	$-0.278365\pi$
$193$	17.8205	1.28275	0.641374	+	0.767229i	$0.278365\pi$
$194$	0	0
$195$	6.17447	0.442163
$196$	0	0
$197$	−2.20498	−0.157098	−0.0785492	−	0.996910i	$-0.525029\pi$
$197$	−2.20498	−0.157098	−0.0785492	+	0.996910i	$0.525029\pi$
$198$	0	0
$199$	4.66253	0.330518	0.165259	−	0.986250i	$-0.447154\pi$
$199$	4.66253	0.330518	0.165259	+	0.986250i	$0.447154\pi$
$200$	0	0
$201$	3.66651	0.258615
$202$	0	0
$203$	−11.4707	−0.805088
$204$	0	0
$205$	32.3882	2.26209
$206$	0	0
$207$	15.4776	1.07577
$208$	0	0
$209$	24.2185	1.67523
$210$	0	0
$211$	−8.28137	−0.570113	−0.285057	−	0.958511i	$-0.592012\pi$
$211$	−8.28137	−0.570113	−0.285057	+	0.958511i	$0.592012\pi$
$212$	0	0
$213$	−6.97752	−0.478092
$214$	0	0
$215$	7.56906	0.516205
$216$	0	0
$217$	8.17409	0.554893
$218$	0	0
$219$	−6.38818	−0.431673
$220$	0	0
$221$	6.74572	0.453766
$222$	0	0
$223$	1.39692	0.0935447	0.0467724	−	0.998906i	$-0.485106\pi$
$223$	1.39692	0.0935447	0.0467724	+	0.998906i	$0.485106\pi$
$224$	0	0
$225$	−29.7036	−1.98024
$226$	0	0
$227$	−19.1150	−1.26871	−0.634353	−	0.773043i	$-0.718733\pi$
$227$	−19.1150	−1.26871	−0.634353	+	0.773043i	$0.718733\pi$
$228$	0	0
$229$	−0.507073	−0.0335083	−0.0167542	−	0.999860i	$-0.505333\pi$
$229$	−0.507073	−0.0335083	−0.0167542	+	0.999860i	$0.505333\pi$
$230$	0	0
$231$	3.26191	0.214618
$232$	0	0
$233$	3.93106	0.257532	0.128766	−	0.991675i	$-0.458898\pi$
$233$	3.93106	0.257532	0.128766	+	0.991675i	$0.458898\pi$
$234$	0	0
$235$	−4.35559	−0.284127
$236$	0	0
$237$	2.42288	0.157383
$238$	0	0
$239$	13.2390	0.856359	0.428179	−	0.903694i	$-0.359155\pi$
$239$	13.2390	0.856359	0.428179	+	0.903694i	$0.359155\pi$
$240$	0	0
$241$	4.33722	0.279385	0.139693	−	0.990195i	$-0.455389\pi$
$241$	4.33722	0.279385	0.139693	+	0.990195i	$0.455389\pi$
$242$	0	0
$243$	12.1249	0.777814
$244$	0	0
$245$	20.4040	1.30356
$246$	0	0
$247$	15.7619	1.00291
$248$	0	0
$249$	1.96895	0.124777
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	26.6250	1.67390
$254$	0	0
$255$	4.41065	0.276206
$256$	0	0
$257$	6.08097	0.379320	0.189660	−	0.981850i	$-0.439261\pi$
$257$	6.08097	0.379320	0.189660	+	0.981850i	$0.439261\pi$
$258$	0	0
$259$	0.0270352	0.00167988
$260$	0	0
$261$	−23.0222	−1.42504
$262$	0	0
$263$	−23.4826	−1.44800	−0.724000	−	0.689800i	$-0.757698\pi$
$263$	−23.4826	−1.44800	−0.724000	+	0.689800i	$0.757698\pi$
$264$	0	0
$265$	−31.0058	−1.90467
$266$	0	0
$267$	−6.77825	−0.414822
$268$	0	0
$269$	−28.1750	−1.71786	−0.858929	−	0.512094i	$-0.828870\pi$
$269$	−28.1750	−1.71786	−0.858929	+	0.512094i	$0.828870\pi$
$270$	0	0
$271$	24.9255	1.51411	0.757057	−	0.653348i	$-0.226636\pi$
$271$	24.9255	1.51411	0.757057	+	0.653348i	$0.226636\pi$
$272$	0	0
$273$	2.12292	0.128485
$274$	0	0
$275$	−51.0969	−3.08126
$276$	0	0
$277$	26.9040	1.61651	0.808253	−	0.588835i	$-0.200413\pi$
$277$	26.9040	1.61651	0.808253	+	0.588835i	$0.200413\pi$
$278$	0	0
$279$	16.4057	0.982184
$280$	0	0
$281$	−8.10390	−0.483438	−0.241719	−	0.970346i	$-0.577711\pi$
$281$	−8.10390	−0.483438	−0.241719	+	0.970346i	$0.577711\pi$
$282$	0	0
$283$	−13.9448	−0.828933	−0.414467	−	0.910064i	$-0.636032\pi$
$283$	−13.9448	−0.828933	−0.414467	+	0.910064i	$0.636032\pi$
$284$	0	0
$285$	10.3058	0.610464
$286$	0	0
$287$	11.1358	0.657324
$288$	0	0
$289$	−12.1813	−0.716546
$290$	0	0
$291$	0.458023	0.0268498
$292$	0	0
$293$	−10.5482	−0.616231	−0.308115	−	0.951349i	$-0.599698\pi$
$293$	−10.5482	−0.616231	−0.308115	+	0.951349i	$0.599698\pi$
$294$	0	0
$295$	4.63944	0.270119
$296$	0	0
$297$	13.7022	0.795081
$298$	0	0
$299$	17.3281	1.00211
$300$	0	0
$301$	2.60241	0.150001
$302$	0	0
$303$	5.37568	0.308825
$304$	0	0
$305$	−30.1539	−1.72661
$306$	0	0
$307$	−14.6005	−0.833293	−0.416646	−	0.909069i	$-0.636795\pi$
$307$	−14.6005	−0.833293	−0.416646	+	0.909069i	$0.636795\pi$
$308$	0	0
$309$	−8.62451	−0.490631
$310$	0	0
$311$	22.3335	1.26642	0.633209	−	0.773981i	$-0.281737\pi$
$311$	22.3335	1.26642	0.633209	+	0.773981i	$0.281737\pi$
$312$	0	0
$313$	−9.61104	−0.543248	−0.271624	−	0.962403i	$-0.587561\pi$
$313$	−9.61104	−0.543248	−0.271624	+	0.962403i	$0.587561\pi$
$314$	0	0
$315$	−14.9315	−0.841293
$316$	0	0
$317$	17.3538	0.974686	0.487343	−	0.873211i	$-0.337966\pi$
$317$	17.3538	0.974686	0.487343	+	0.873211i	$0.337966\pi$
$318$	0	0
$319$	−39.6034	−2.21736
$320$	0	0
$321$	−2.38882	−0.133331
$322$	0	0
$323$	11.2593	0.626484
$324$	0	0
$325$	−33.2550	−1.84465
$326$	0	0
$327$	6.41974	0.355012
$328$	0	0
$329$	−1.49755	−0.0825626
$330$	0	0
$331$	7.38143	0.405720	0.202860	−	0.979208i	$-0.434976\pi$
$331$	7.38143	0.405720	0.202860	+	0.979208i	$0.434976\pi$
$332$	0	0
$333$	0.0542606	0.00297346
$334$	0	0
$335$	−28.8714	−1.57741
$336$	0	0
$337$	−32.3740	−1.76352	−0.881761	−	0.471696i	$-0.843642\pi$
$337$	−32.3740	−1.76352	−0.881761	+	0.471696i	$0.843642\pi$
$338$	0	0
$339$	2.08824	0.113418
$340$	0	0
$341$	28.2215	1.52828
$342$	0	0
$343$	16.5886	0.895699
$344$	0	0
$345$	11.3299	0.609981
$346$	0	0
$347$	−20.0439	−1.07601	−0.538007	−	0.842940i	$-0.680823\pi$
$347$	−20.0439	−1.07601	−0.538007	+	0.842940i	$0.680823\pi$
$348$	0	0
$349$	20.3545	1.08955	0.544777	−	0.838581i	$-0.316614\pi$
$349$	20.3545	1.08955	0.544777	+	0.838581i	$0.316614\pi$
$350$	0	0
$351$	8.91767	0.475990
$352$	0	0
$353$	2.04037	0.108598	0.0542991	−	0.998525i	$-0.482708\pi$
$353$	2.04037	0.108598	0.0542991	+	0.998525i	$0.482708\pi$
$354$	0	0
$355$	54.9435	2.91610
$356$	0	0
$357$	1.51648	0.0802607
$358$	0	0
$359$	−3.72830	−0.196772	−0.0983861	−	0.995148i	$-0.531368\pi$
$359$	−3.72830	−0.196772	−0.0983861	+	0.995148i	$0.531368\pi$
$360$	0	0
$361$	7.30822	0.384643
$362$	0	0
$363$	5.70539	0.299456
$364$	0	0
$365$	50.3028	2.63297
$366$	0	0
$367$	−19.5910	−1.02264	−0.511322	−	0.859389i	$-0.670844\pi$
$367$	−19.5910	−1.02264	−0.511322	+	0.859389i	$0.670844\pi$
$368$	0	0
$369$	22.3499	1.16349
$370$	0	0
$371$	−10.6605	−0.553466
$372$	0	0
$373$	19.6526	1.01757	0.508786	−	0.860893i	$-0.330094\pi$
$373$	19.6526	1.01757	0.508786	+	0.860893i	$0.330094\pi$
$374$	0	0
$375$	−11.6972	−0.604043
$376$	0	0
$377$	−25.7747	−1.32747
$378$	0	0
$379$	−5.77854	−0.296824	−0.148412	−	0.988926i	$-0.547416\pi$
$379$	−5.77854	−0.296824	−0.148412	+	0.988926i	$0.547416\pi$
$380$	0	0
$381$	5.75970	0.295078
$382$	0	0
$383$	−12.3411	−0.630602	−0.315301	−	0.948992i	$-0.602106\pi$
$383$	−12.3411	−0.630602	−0.315301	+	0.948992i	$0.602106\pi$
$384$	0	0
$385$	−25.6854	−1.30905
$386$	0	0
$387$	5.22314	0.265507
$388$	0	0
$389$	−20.9174	−1.06056	−0.530278	−	0.847824i	$-0.677912\pi$
$389$	−20.9174	−1.06056	−0.530278	+	0.847824i	$0.677912\pi$
$390$	0	0
$391$	12.3781	0.625988
$392$	0	0
$393$	−10.4614	−0.527706
$394$	0	0
$395$	−19.0786	−0.959947
$396$	0	0
$397$	−23.6278	−1.18584	−0.592922	−	0.805260i	$-0.702026\pi$
$397$	−23.6278	−1.18584	−0.592922	+	0.805260i	$0.702026\pi$
$398$	0	0
$399$	3.54338	0.177391
$400$	0	0
$401$	−37.0125	−1.84832	−0.924158	−	0.382010i	$-0.875232\pi$
$401$	−37.0125	−1.84832	−0.924158	+	0.382010i	$0.875232\pi$
$402$	0	0
$403$	18.3671	0.914933
$404$	0	0
$405$	−26.9232	−1.33782
$406$	0	0
$407$	0.0933403	0.00462671
$408$	0	0
$409$	−7.04773	−0.348488	−0.174244	−	0.984703i	$-0.555748\pi$
$409$	−7.04773	−0.348488	−0.174244	+	0.984703i	$0.555748\pi$
$410$	0	0
$411$	−3.70789	−0.182897
$412$	0	0
$413$	1.59515	0.0784920
$414$	0	0
$415$	−15.5042	−0.761071
$416$	0	0
$417$	−4.19273	−0.205319
$418$	0	0
$419$	23.8457	1.16494	0.582470	−	0.812852i	$-0.302086\pi$
$419$	23.8457	1.16494	0.582470	+	0.812852i	$0.302086\pi$
$420$	0	0
$421$	4.77392	0.232667	0.116333	−	0.993210i	$-0.462886\pi$
$421$	4.77392	0.232667	0.116333	+	0.993210i	$0.462886\pi$
$422$	0	0
$423$	−3.00564	−0.146139
$424$	0	0
$425$	−23.7552	−1.15230
$426$	0	0
$427$	−10.3676	−0.501723
$428$	0	0
$429$	7.32950	0.353872
$430$	0	0
$431$	−32.8399	−1.58184	−0.790921	−	0.611918i	$-0.790398\pi$
$431$	−32.8399	−1.58184	−0.790921	+	0.611918i	$0.790398\pi$
$432$	0	0
$433$	28.2791	1.35901	0.679503	−	0.733673i	$-0.262195\pi$
$433$	28.2791	1.35901	0.679503	+	0.733673i	$0.262195\pi$
$434$	0	0
$435$	−16.8527	−0.808023
$436$	0	0
$437$	28.9224	1.38355
$438$	0	0
$439$	20.9875	1.00168	0.500839	−	0.865540i	$-0.333025\pi$
$439$	20.9875	1.00168	0.500839	+	0.865540i	$0.333025\pi$
$440$	0	0
$441$	14.0801	0.670479
$442$	0	0
$443$	−20.8346	−0.989883	−0.494942	−	0.868926i	$-0.664811\pi$
$443$	−20.8346	−0.989883	−0.494942	+	0.868926i	$0.664811\pi$
$444$	0	0
$445$	53.3743	2.53018
$446$	0	0
$447$	−9.89282	−0.467914
$448$	0	0
$449$	10.5961	0.500059	0.250029	−	0.968238i	$-0.419560\pi$
$449$	10.5961	0.500059	0.250029	+	0.968238i	$0.419560\pi$
$450$	0	0
$451$	38.4469	1.81039
$452$	0	0
$453$	−2.91982	−0.137185
$454$	0	0
$455$	−16.7166	−0.783688
$456$	0	0
$457$	10.1463	0.474624	0.237312	−	0.971433i	$-0.423734\pi$
$457$	10.1463	0.474624	0.237312	+	0.971433i	$0.423734\pi$
$458$	0	0
$459$	6.37022	0.297336
$460$	0	0
$461$	−12.9020	−0.600908	−0.300454	−	0.953796i	$-0.597138\pi$
$461$	−12.9020	−0.600908	−0.300454	+	0.953796i	$0.597138\pi$
$462$	0	0
$463$	13.3413	0.620023	0.310011	−	0.950733i	$-0.399667\pi$
$463$	13.3413	0.620023	0.310011	+	0.950733i	$0.399667\pi$
$464$	0	0
$465$	12.0093	0.556916
$466$	0	0
$467$	2.89976	0.134185	0.0670924	−	0.997747i	$-0.478628\pi$
$467$	2.89976	0.134185	0.0670924	+	0.997747i	$0.478628\pi$
$468$	0	0
$469$	−9.92663	−0.458369
$470$	0	0
$471$	4.48185	0.206513
$472$	0	0
$473$	8.98497	0.413129
$474$	0	0
$475$	−55.5059	−2.54679
$476$	0	0
$477$	−21.3961	−0.979658
$478$	0	0
$479$	10.8229	0.494513	0.247256	−	0.968950i	$-0.420471\pi$
$479$	10.8229	0.494513	0.247256	+	0.968950i	$0.420471\pi$
$480$	0	0
$481$	0.0607479	0.00276987
$482$	0	0
$483$	3.89547	0.177250
$484$	0	0
$485$	−3.60663	−0.163769
$486$	0	0
$487$	16.1712	0.732787	0.366394	−	0.930460i	$-0.380592\pi$
$487$	16.1712	0.732787	0.366394	+	0.930460i	$0.380592\pi$
$488$	0	0
$489$	2.85834	0.129259
$490$	0	0
$491$	−22.6126	−1.02049	−0.510247	−	0.860028i	$-0.670446\pi$
$491$	−22.6126	−1.02049	−0.510247	+	0.860028i	$0.670446\pi$
$492$	0	0
$493$	−18.4118	−0.829227
$494$	0	0
$495$	−51.5517	−2.31708
$496$	0	0
$497$	18.8908	0.847368
$498$	0	0
$499$	−0.500607	−0.0224103	−0.0112051	−	0.999937i	$-0.503567\pi$
$499$	−0.500607	−0.0224103	−0.0112051	+	0.999937i	$0.503567\pi$
$500$	0	0
$501$	−8.95141	−0.399920
$502$	0	0
$503$	4.14861	0.184977	0.0924887	−	0.995714i	$-0.470518\pi$
$503$	4.14861	0.184977	0.0924887	+	0.995714i	$0.470518\pi$
$504$	0	0
$505$	−42.3300	−1.88366
$506$	0	0
$507$	−1.79661	−0.0797903
$508$	0	0
$509$	−10.0094	−0.443659	−0.221829	−	0.975085i	$-0.571203\pi$
$509$	−10.0094	−0.443659	−0.221829	+	0.975085i	$0.571203\pi$
$510$	0	0
$511$	17.2952	0.765097
$512$	0	0
$513$	14.8845	0.657167
$514$	0	0
$515$	67.9125	2.99258
$516$	0	0
$517$	−5.17037	−0.227393
$518$	0	0
$519$	−6.67247	−0.292889
$520$	0	0
$521$	19.2906	0.845137	0.422569	−	0.906331i	$-0.361129\pi$
$521$	19.2906	0.845137	0.422569	+	0.906331i	$0.361129\pi$
$522$	0	0
$523$	−20.5030	−0.896534	−0.448267	−	0.893900i	$-0.647959\pi$
$523$	−20.5030	−0.896534	−0.448267	+	0.893900i	$0.647959\pi$
$524$	0	0
$525$	−7.47593	−0.326276
$526$	0	0
$527$	13.1203	0.571530
$528$	0	0
$529$	8.79638	0.382451
$530$	0	0
$531$	3.20152	0.138934
$532$	0	0
$533$	25.0221	1.08383
$534$	0	0
$535$	18.8104	0.813246
$536$	0	0
$537$	2.82121	0.121744
$538$	0	0
$539$	24.2208	1.04327
$540$	0	0
$541$	−39.2010	−1.68538	−0.842691	−	0.538397i	$-0.819030\pi$
$541$	−39.2010	−1.68538	−0.842691	+	0.538397i	$0.819030\pi$
$542$	0	0
$543$	4.83055	0.207299
$544$	0	0
$545$	−50.5513	−2.16538
$546$	0	0
$547$	21.4696	0.917972	0.458986	−	0.888443i	$-0.348213\pi$
$547$	21.4696	0.917972	0.458986	+	0.888443i	$0.348213\pi$
$548$	0	0
$549$	−20.8081	−0.888070
$550$	0	0
$551$	−43.0207	−1.83274
$552$	0	0
$553$	−6.55965	−0.278944
$554$	0	0
$555$	0.0397197	0.00168601
$556$	0	0
$557$	−38.1174	−1.61509	−0.807543	−	0.589808i	$-0.799203\pi$
$557$	−38.1174	−1.61509	−0.807543	+	0.589808i	$0.799203\pi$
$558$	0	0
$559$	5.84761	0.247328
$560$	0	0
$561$	5.23573	0.221053
$562$	0	0
$563$	18.0624	0.761239	0.380620	−	0.924732i	$-0.375711\pi$
$563$	18.0624	0.761239	0.380620	+	0.924732i	$0.375711\pi$
$564$	0	0
$565$	−16.4435	−0.691784
$566$	0	0
$567$	−9.25679	−0.388749
$568$	0	0
$569$	32.9861	1.38285	0.691425	−	0.722448i	$-0.256983\pi$
$569$	32.9861	1.38285	0.691425	+	0.722448i	$0.256983\pi$
$570$	0	0
$571$	16.3992	0.686284	0.343142	−	0.939283i	$-0.388509\pi$
$571$	16.3992	0.686284	0.343142	+	0.939283i	$0.388509\pi$
$572$	0	0
$573$	−2.02906	−0.0847651
$574$	0	0
$575$	−61.0214	−2.54477
$576$	0	0
$577$	−19.2734	−0.802364	−0.401182	−	0.915998i	$-0.631400\pi$
$577$	−19.2734	−0.802364	−0.401182	+	0.915998i	$0.631400\pi$
$578$	0	0
$579$	9.00183	0.374103
$580$	0	0
$581$	−5.33070	−0.221155
$582$	0	0
$583$	−36.8060	−1.52435
$584$	0	0
$585$	−33.5509	−1.38716
$586$	0	0
$587$	36.1900	1.49372	0.746861	−	0.664980i	$-0.231560\pi$
$587$	36.1900	1.49372	0.746861	+	0.664980i	$0.231560\pi$
$588$	0	0
$589$	30.6566	1.26319
$590$	0	0
$591$	−1.11382	−0.0458165
$592$	0	0
$593$	16.0288	0.658225	0.329112	−	0.944291i	$-0.393250\pi$
$593$	16.0288	0.658225	0.329112	+	0.944291i	$0.393250\pi$
$594$	0	0
$595$	−11.9413	−0.489546
$596$	0	0
$597$	2.35523	0.0963930
$598$	0	0
$599$	12.4369	0.508156	0.254078	−	0.967184i	$-0.418228\pi$
$599$	12.4369	0.508156	0.254078	+	0.967184i	$0.418228\pi$
$600$	0	0
$601$	4.74225	0.193440	0.0967202	−	0.995312i	$-0.469165\pi$
$601$	4.74225	0.193440	0.0967202	+	0.995312i	$0.469165\pi$
$602$	0	0
$603$	−19.9231	−0.811332
$604$	0	0
$605$	−44.9263	−1.82651
$606$	0	0
$607$	−44.0250	−1.78692	−0.893459	−	0.449144i	$-0.851729\pi$
$607$	−44.0250	−1.78692	−0.893459	+	0.449144i	$0.851729\pi$
$608$	0	0
$609$	−5.79432	−0.234798
$610$	0	0
$611$	−3.36499	−0.136133
$612$	0	0
$613$	34.5917	1.39715	0.698574	−	0.715538i	$-0.253818\pi$
$613$	34.5917	1.39715	0.698574	+	0.715538i	$0.253818\pi$
$614$	0	0
$615$	16.3605	0.659720
$616$	0	0
$617$	−14.3336	−0.577048	−0.288524	−	0.957473i	$-0.593164\pi$
$617$	−14.3336	−0.577048	−0.288524	+	0.957473i	$0.593164\pi$
$618$	0	0
$619$	−29.8003	−1.19777	−0.598887	−	0.800833i	$-0.704390\pi$
$619$	−29.8003	−1.19777	−0.598887	+	0.800833i	$0.704390\pi$
$620$	0	0
$621$	16.3635	0.656646
$622$	0	0
$623$	18.3513	0.735229
$624$	0	0
$625$	37.9999	1.52000
$626$	0	0
$627$	12.2337	0.488567
$628$	0	0
$629$	0.0433945	0.00173025
$630$	0	0
$631$	−37.9163	−1.50943	−0.754713	−	0.656056i	$-0.772224\pi$
$631$	−37.9163	−1.50943	−0.754713	+	0.656056i	$0.772224\pi$
$632$	0	0
$633$	−4.18325	−0.166269
$634$	0	0
$635$	−45.3539	−1.79981
$636$	0	0
$637$	15.7634	0.624571
$638$	0	0
$639$	37.9146	1.49988
$640$	0	0
$641$	7.60808	0.300501	0.150251	−	0.988648i	$-0.451992\pi$
$641$	7.60808	0.300501	0.150251	+	0.988648i	$0.451992\pi$
$642$	0	0
$643$	−37.8524	−1.49275	−0.746377	−	0.665523i	$-0.768208\pi$
$643$	−37.8524	−1.49275	−0.746377	+	0.665523i	$0.768208\pi$
$644$	0	0
$645$	3.82343	0.150547
$646$	0	0
$647$	16.9159	0.665032	0.332516	−	0.943098i	$-0.392102\pi$
$647$	16.9159	0.665032	0.332516	+	0.943098i	$0.392102\pi$
$648$	0	0
$649$	5.50732	0.216181
$650$	0	0
$651$	4.12905	0.161830
$652$	0	0
$653$	−11.9551	−0.467840	−0.233920	−	0.972256i	$-0.575155\pi$
$653$	−11.9551	−0.467840	−0.233920	+	0.972256i	$0.575155\pi$
$654$	0	0
$655$	82.3764	3.21871
$656$	0	0
$657$	34.7122	1.35425
$658$	0	0
$659$	0.662703	0.0258152	0.0129076	−	0.999917i	$-0.495891\pi$
$659$	0.662703	0.0258152	0.0129076	+	0.999917i	$0.495891\pi$
$660$	0	0
$661$	8.18593	0.318396	0.159198	−	0.987247i	$-0.449109\pi$
$661$	8.18593	0.318396	0.159198	+	0.987247i	$0.449109\pi$
$662$	0	0
$663$	3.40753	0.132337
$664$	0	0
$665$	−27.9018	−1.08199
$666$	0	0
$667$	−47.2955	−1.83129
$668$	0	0
$669$	0.705639	0.0272816
$670$	0	0
$671$	−35.7946	−1.38184
$672$	0	0
$673$	0.789156	0.0304197	0.0152099	−	0.999884i	$-0.495158\pi$
$673$	0.789156	0.0304197	0.0152099	+	0.999884i	$0.495158\pi$
$674$	0	0
$675$	−31.4038	−1.20873
$676$	0	0
$677$	21.7846	0.837251	0.418626	−	0.908159i	$-0.362512\pi$
$677$	21.7846	0.837251	0.418626	+	0.908159i	$0.362512\pi$
$678$	0	0
$679$	−1.24004	−0.0475885
$680$	0	0
$681$	−9.65573	−0.370008
$682$	0	0
$683$	45.5637	1.74345	0.871723	−	0.489999i	$-0.163003\pi$
$683$	45.5637	1.74345	0.871723	+	0.489999i	$0.163003\pi$
$684$	0	0
$685$	29.1972	1.11557
$686$	0	0
$687$	−0.256142	−0.00977244
$688$	0	0
$689$	−23.9541	−0.912579
$690$	0	0
$691$	12.2846	0.467329	0.233664	−	0.972317i	$-0.424928\pi$
$691$	12.2846	0.467329	0.233664	+	0.972317i	$0.424928\pi$
$692$	0	0
$693$	−17.7246	−0.673303
$694$	0	0
$695$	33.0151	1.25233
$696$	0	0
$697$	17.8742	0.677033
$698$	0	0
$699$	1.98573	0.0751072
$700$	0	0
$701$	−15.1051	−0.570512	−0.285256	−	0.958451i	$-0.592079\pi$
$701$	−15.1051	−0.570512	−0.285256	+	0.958451i	$0.592079\pi$
$702$	0	0
$703$	0.101394	0.00382417
$704$	0	0
$705$	−2.20018	−0.0828636
$706$	0	0
$707$	−14.5540	−0.547360
$708$	0	0
$709$	−10.0038	−0.375702	−0.187851	−	0.982198i	$-0.560152\pi$
$709$	−10.0038	−0.375702	−0.187851	+	0.982198i	$0.560152\pi$
$710$	0	0
$711$	−13.1655	−0.493743
$712$	0	0
$713$	33.7029	1.26218
$714$	0	0
$715$	−57.7151	−2.15842
$716$	0	0
$717$	6.68753	0.249750
$718$	0	0
$719$	−24.5893	−0.917026	−0.458513	−	0.888688i	$-0.651618\pi$
$719$	−24.5893	−0.917026	−0.458513	+	0.888688i	$0.651618\pi$
$720$	0	0
$721$	23.3498	0.869594
$722$	0	0
$723$	2.19090	0.0814805
$724$	0	0
$725$	90.7663	3.37098
$726$	0	0
$727$	27.3274	1.01352	0.506759	−	0.862088i	$-0.330844\pi$
$727$	27.3274	1.01352	0.506759	+	0.862088i	$0.330844\pi$
$728$	0	0
$729$	−14.1811	−0.525225
$730$	0	0
$731$	4.17717	0.154498
$732$	0	0
$733$	5.18800	0.191623	0.0958117	−	0.995399i	$-0.469455\pi$
$733$	5.18800	0.191623	0.0958117	+	0.995399i	$0.469455\pi$
$734$	0	0
$735$	10.3068	0.380173
$736$	0	0
$737$	−34.2722	−1.26243
$738$	0	0
$739$	−21.8947	−0.805411	−0.402706	−	0.915330i	$-0.631930\pi$
$739$	−21.8947	−0.805411	−0.402706	+	0.915330i	$0.631930\pi$
$740$	0	0
$741$	7.96195	0.292490
$742$	0	0
$743$	26.9515	0.988754	0.494377	−	0.869248i	$-0.335396\pi$
$743$	26.9515	0.988754	0.494377	+	0.869248i	$0.335396\pi$
$744$	0	0
$745$	77.8996	2.85402
$746$	0	0
$747$	−10.6989	−0.391453
$748$	0	0
$749$	6.46746	0.236316
$750$	0	0
$751$	31.2592	1.14066	0.570332	−	0.821414i	$-0.306814\pi$
$751$	31.2592	1.14066	0.570332	+	0.821414i	$0.306814\pi$
$752$	0	0
$753$	−0.505139	−0.0184083
$754$	0	0
$755$	22.9917	0.836752
$756$	0	0
$757$	−43.8693	−1.59446	−0.797229	−	0.603677i	$-0.793702\pi$
$757$	−43.8693	−1.59446	−0.797229	+	0.603677i	$0.793702\pi$
$758$	0	0
$759$	13.4493	0.488180
$760$	0	0
$761$	−36.9515	−1.33949	−0.669746	−	0.742590i	$-0.733597\pi$
$761$	−36.9515	−1.33949	−0.669746	+	0.742590i	$0.733597\pi$
$762$	0	0
$763$	−17.3807	−0.629223
$764$	0	0
$765$	−23.9667	−0.866517
$766$	0	0
$767$	3.58429	0.129421
$768$	0	0
$769$	15.5825	0.561921	0.280960	−	0.959719i	$-0.409347\pi$
$769$	15.5825	0.561921	0.280960	+	0.959719i	$0.409347\pi$
$770$	0	0
$771$	3.07174	0.110626
$772$	0	0
$773$	−27.0918	−0.974424	−0.487212	−	0.873284i	$-0.661986\pi$
$773$	−27.0918	−0.974424	−0.487212	+	0.873284i	$0.661986\pi$
$774$	0	0
$775$	−64.6804	−2.32339
$776$	0	0
$777$	0.0136565	0.000489925 0
$778$	0	0
$779$	41.7644	1.49636
$780$	0	0
$781$	65.2215	2.33381
$782$	0	0
$783$	−24.3400	−0.869839
$784$	0	0
$785$	−35.2916	−1.25961
$786$	0	0
$787$	10.3014	0.367207	0.183603	−	0.983000i	$-0.441224\pi$
$787$	10.3014	0.367207	0.183603	+	0.983000i	$0.441224\pi$
$788$	0	0
$789$	−11.8620	−0.422298
$790$	0	0
$791$	−5.65365	−0.201021
$792$	0	0
$793$	−23.2959	−0.827263
$794$	0	0
$795$	−15.6623	−0.555483
$796$	0	0
$797$	−7.31908	−0.259255	−0.129628	−	0.991563i	$-0.541378\pi$
$797$	−7.31908	−0.259255	−0.129628	+	0.991563i	$0.541378\pi$
$798$	0	0
$799$	−2.40374	−0.0850381
$800$	0	0
$801$	36.8318	1.30139
$802$	0	0
$803$	59.7128	2.10722
$804$	0	0
$805$	−30.6743	−1.08113
$806$	0	0
$807$	−14.2323	−0.501000
$808$	0	0
$809$	0.619286	0.0217729	0.0108865	−	0.999941i	$-0.496535\pi$
$809$	0.619286	0.0217729	0.0108865	+	0.999941i	$0.496535\pi$
$810$	0	0
$811$	19.2262	0.675123	0.337561	−	0.941304i	$-0.390398\pi$
$811$	19.2262	0.675123	0.337561	+	0.941304i	$0.390398\pi$
$812$	0	0
$813$	12.5908	0.441580
$814$	0	0
$815$	−22.5076	−0.788406
$816$	0	0
$817$	9.76027	0.341468
$818$	0	0
$819$	−11.5356	−0.403086
$820$	0	0
$821$	−25.2605	−0.881598	−0.440799	−	0.897606i	$-0.645305\pi$
$821$	−25.2605	−0.881598	−0.440799	+	0.897606i	$0.645305\pi$
$822$	0	0
$823$	36.4975	1.27222	0.636111	−	0.771598i	$-0.280542\pi$
$823$	36.4975	1.27222	0.636111	+	0.771598i	$0.280542\pi$
$824$	0	0
$825$	−25.8110	−0.898625
$826$	0	0
$827$	−5.39180	−0.187491	−0.0937457	−	0.995596i	$-0.529884\pi$
$827$	−5.39180	−0.187491	−0.0937457	+	0.995596i	$0.529884\pi$
$828$	0	0
$829$	−13.7810	−0.478633	−0.239317	−	0.970942i	$-0.576923\pi$
$829$	−13.7810	−0.478633	−0.239317	+	0.970942i	$0.576923\pi$
$830$	0	0
$831$	13.5903	0.471442
$832$	0	0
$833$	11.2604	0.390150
$834$	0	0
$835$	70.4866	2.43929
$836$	0	0
$837$	17.3447	0.599522
$838$	0	0
$839$	−23.9006	−0.825139	−0.412569	−	0.910926i	$-0.635369\pi$
$839$	−23.9006	−0.825139	−0.412569	+	0.910926i	$0.635369\pi$
$840$	0	0
$841$	41.3497	1.42585
$842$	0	0
$843$	−4.09360	−0.140991
$844$	0	0
$845$	14.1471	0.486676
$846$	0	0
$847$	−15.4467	−0.530754
$848$	0	0
$849$	−7.04407	−0.241752
$850$	0	0
$851$	0.111470	0.00382114
$852$	0	0
$853$	6.04070	0.206829	0.103415	−	0.994638i	$-0.467023\pi$
$853$	6.04070	0.206829	0.103415	+	0.994638i	$0.467023\pi$
$854$	0	0
$855$	−56.0000	−1.91516
$856$	0	0
$857$	−49.4330	−1.68860	−0.844300	−	0.535871i	$-0.819983\pi$
$857$	−49.4330	−1.68860	−0.844300	+	0.535871i	$0.819983\pi$
$858$	0	0
$859$	42.1477	1.43806	0.719031	−	0.694978i	$-0.244586\pi$
$859$	42.1477	1.43806	0.719031	+	0.694978i	$0.244586\pi$
$860$	0	0
$861$	5.62512	0.191704
$862$	0	0
$863$	47.6313	1.62139	0.810694	−	0.585470i	$-0.199090\pi$
$863$	47.6313	1.62139	0.810694	+	0.585470i	$0.199090\pi$
$864$	0	0
$865$	52.5414	1.78646
$866$	0	0
$867$	−6.15324	−0.208975
$868$	0	0
$869$	−22.6475	−0.768265
$870$	0	0
$871$	−22.3051	−0.755779
$872$	0	0
$873$	−2.48881	−0.0842336
$874$	0	0
$875$	31.6689	1.07060
$876$	0	0
$877$	29.7998	1.00627	0.503135	−	0.864208i	$-0.332180\pi$
$877$	29.7998	1.00627	0.503135	+	0.864208i	$0.332180\pi$
$878$	0	0
$879$	−5.32829	−0.179719
$880$	0	0
$881$	15.0651	0.507555	0.253778	−	0.967263i	$-0.418327\pi$
$881$	15.0651	0.507555	0.253778	+	0.967263i	$0.418327\pi$
$882$	0	0
$883$	29.6528	0.997895	0.498948	−	0.866632i	$-0.333720\pi$
$883$	29.6528	0.997895	0.498948	+	0.866632i	$0.333720\pi$
$884$	0	0
$885$	2.34356	0.0787781
$886$	0	0
$887$	−26.5999	−0.893137	−0.446569	−	0.894749i	$-0.647354\pi$
$887$	−26.5999	−0.893137	−0.446569	+	0.894749i	$0.647354\pi$
$888$	0	0
$889$	−15.5937	−0.522995
$890$	0	0
$891$	−31.9596	−1.07069
$892$	0	0
$893$	−5.61651	−0.187949
$894$	0	0
$895$	−22.2152	−0.742572
$896$	0	0
$897$	8.75311	0.292258
$898$	0	0
$899$	−50.1314	−1.67198
$900$	0	0
$901$	−17.1113	−0.570060
$902$	0	0
$903$	1.31458	0.0437465
$904$	0	0
$905$	−38.0374	−1.26441
$906$	0	0
$907$	−34.0602	−1.13095	−0.565475	−	0.824766i	$-0.691307\pi$
$907$	−34.0602	−1.13095	−0.565475	+	0.824766i	$0.691307\pi$
$908$	0	0
$909$	−29.2104	−0.968849
$910$	0	0
$911$	−45.8154	−1.51793	−0.758967	−	0.651130i	$-0.774295\pi$
$911$	−45.8154	−1.51793	−0.758967	+	0.651130i	$0.774295\pi$
$912$	0	0
$913$	−18.4045	−0.609100
$914$	0	0
$915$	−15.2319	−0.503551
$916$	0	0
$917$	28.3229	0.935304
$918$	0	0
$919$	−16.0622	−0.529843	−0.264921	−	0.964270i	$-0.585346\pi$
$919$	−16.0622	−0.529843	−0.264921	+	0.964270i	$0.585346\pi$
$920$	0	0
$921$	−7.37527	−0.243023
$922$	0	0
$923$	42.4476	1.39718
$924$	0	0
$925$	−0.213925	−0.00703382
$926$	0	0
$927$	46.8640	1.53922
$928$	0	0
$929$	12.7150	0.417166	0.208583	−	0.978005i	$-0.433115\pi$
$929$	12.7150	0.417166	0.208583	+	0.978005i	$0.433115\pi$
$930$	0	0
$931$	26.3108	0.862302
$932$	0	0
$933$	11.2815	0.369341
$934$	0	0
$935$	−41.2280	−1.34830
$936$	0	0
$937$	−27.1316	−0.886352	−0.443176	−	0.896435i	$-0.646148\pi$
$937$	−27.1316	−0.886352	−0.443176	+	0.896435i	$0.646148\pi$
$938$	0	0
$939$	−4.85491	−0.158434
$940$	0	0
$941$	−13.7253	−0.447432	−0.223716	−	0.974654i	$-0.571819\pi$
$941$	−13.7253	−0.447432	−0.223716	+	0.974654i	$0.571819\pi$
$942$	0	0
$943$	45.9144	1.49518
$944$	0	0
$945$	−15.7861	−0.513522
$946$	0	0
$947$	−2.81417	−0.0914481	−0.0457240	−	0.998954i	$-0.514559\pi$
$947$	−2.81417	−0.0914481	−0.0457240	+	0.998954i	$0.514559\pi$
$948$	0	0
$949$	38.8623	1.26153
$950$	0	0
$951$	8.76608	0.284260
$952$	0	0
$953$	4.21604	0.136571	0.0682855	−	0.997666i	$-0.478247\pi$
$953$	4.21604	0.136571	0.0682855	+	0.997666i	$0.478247\pi$
$954$	0	0
$955$	15.9775	0.517020
$956$	0	0
$957$	−20.0052	−0.646677
$958$	0	0
$959$	10.0387	0.324165
$960$	0	0
$961$	4.72382	0.152381
$962$	0	0
$963$	12.9804	0.418289
$964$	0	0
$965$	−70.8836	−2.28182
$966$	0	0
$967$	−5.40164	−0.173705	−0.0868526	−	0.996221i	$-0.527681\pi$
$967$	−5.40164	−0.173705	−0.0868526	+	0.996221i	$0.527681\pi$
$968$	0	0
$969$	5.68752	0.182709
$970$	0	0
$971$	−5.51980	−0.177139	−0.0885694	−	0.996070i	$-0.528230\pi$
$971$	−5.51980	−0.177139	−0.0885694	+	0.996070i	$0.528230\pi$
$972$	0	0
$973$	11.3513	0.363907
$974$	0	0
$975$	−16.7984	−0.537979
$976$	0	0
$977$	−11.6053	−0.371286	−0.185643	−	0.982617i	$-0.559437\pi$
$977$	−11.6053	−0.371286	−0.185643	+	0.982617i	$0.559437\pi$
$978$	0	0
$979$	63.3588	2.02496
$980$	0	0
$981$	−34.8837	−1.11375
$982$	0	0
$983$	2.29946	0.0733412	0.0366706	−	0.999327i	$-0.488325\pi$
$983$	2.29946	0.0733412	0.0366706	+	0.999327i	$0.488325\pi$
$984$	0	0
$985$	8.77063	0.279455
$986$	0	0
$987$	−0.756471	−0.0240788
$988$	0	0
$989$	10.7301	0.341198
$990$	0	0
$991$	−38.0074	−1.20734	−0.603672	−	0.797233i	$-0.706296\pi$
$991$	−38.0074	−1.20734	−0.603672	+	0.797233i	$0.706296\pi$
$992$	0	0
$993$	3.72865	0.118325
$994$	0	0
$995$	−18.5459	−0.587944
$996$	0	0
$997$	−14.3031	−0.452985	−0.226492	−	0.974013i	$-0.572726\pi$
$997$	−14.3031	−0.452985	−0.226492	+	0.974013i	$0.572726\pi$
$998$	0	0
$999$	0.0573663	0.00181499

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.k.1.9		17
4.3	odd	2		251.2.a.b.1.2	✓	17
12.11	even	2		2259.2.a.k.1.16		17
20.19	odd	2		6275.2.a.e.1.16		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
251.2.a.b.1.2	✓	17	4.3	odd	2
2259.2.a.k.1.16		17	12.11	even	2
4016.2.a.k.1.9		17	1.1	even	1	trivial
6275.2.a.e.1.16		17	20.19	odd	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)

\(f(q)\)	\(=\)	\(q+0.505139 q^{3} -3.97764 q^{5} -1.36760 q^{7} -2.74483 q^{9} +O(q^{10})\)	\(q+0.505139 q^{3} -3.97764 q^{5} -1.36760 q^{7} -2.74483 q^{9} -4.72173 q^{11} -3.07300 q^{13} -2.00926 q^{15} -2.19516 q^{17} -5.12915 q^{19} -0.690830 q^{21} -5.63883 q^{23} +10.8217 q^{25} -2.90194 q^{27} +8.38748 q^{29} -5.97694 q^{31} -2.38513 q^{33} +5.43984 q^{35} -0.0197683 q^{37} -1.55229 q^{39} -8.14255 q^{41} -1.90290 q^{43} +10.9180 q^{45} +1.09502 q^{47} -5.12966 q^{49} -1.10886 q^{51} +7.79503 q^{53} +18.7813 q^{55} -2.59094 q^{57} -1.16638 q^{59} +7.58084 q^{61} +3.75385 q^{63} +12.2233 q^{65} +7.25841 q^{67} -2.84839 q^{69} -13.8131 q^{71} -12.6464 q^{73} +5.46644 q^{75} +6.45745 q^{77} +4.79645 q^{79} +6.76862 q^{81} +3.89784 q^{83} +8.73156 q^{85} +4.23684 q^{87} -13.4186 q^{89} +4.20265 q^{91} -3.01919 q^{93} +20.4019 q^{95} +0.906726 q^{97} +12.9604 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * q^99

Introduction

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