LMFDB - Embedded newform 4008.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4008,2,Mod(1,4008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4008 = 2^{3} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4008.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:	$32.0040411301$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 26x^{8} - 3x^{7} + 220x^{6} + 42x^{5} - 675x^{4} - 67x^{3} + 628x^{2} - 48x - 2 $ x^10 - 26x^8 - 3x^7 + 220x^6 + 42x^5 - 675x^4 - 67x^3 + 628x^2 - 48x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.20371$ of defining polynomial
Character	$\chi$	$=$	4008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -4.20371 q^{5} -3.93441 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -4.20371 q^{5} -3.93441 q^{7} +1.00000 q^{9} +2.04067 q^{11} +2.48726 q^{13} +4.20371 q^{15} -7.89415 q^{17} +4.59249 q^{19} +3.93441 q^{21} -3.05415 q^{23} +12.6712 q^{25} -1.00000 q^{27} +10.2765 q^{29} +4.68949 q^{31} -2.04067 q^{33} +16.5391 q^{35} +9.76889 q^{37} -2.48726 q^{39} -7.20906 q^{41} -4.73158 q^{43} -4.20371 q^{45} +8.18355 q^{47} +8.47956 q^{49} +7.89415 q^{51} -6.41734 q^{53} -8.57837 q^{55} -4.59249 q^{57} -9.60011 q^{59} -8.53771 q^{61} -3.93441 q^{63} -10.4557 q^{65} -0.838729 q^{67} +3.05415 q^{69} +11.4409 q^{71} -3.72057 q^{73} -12.6712 q^{75} -8.02881 q^{77} +13.8405 q^{79} +1.00000 q^{81} -8.17588 q^{83} +33.1847 q^{85} -10.2765 q^{87} +9.03864 q^{89} -9.78589 q^{91} -4.68949 q^{93} -19.3055 q^{95} -18.4044 q^{97} +2.04067 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) $ 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9	$ 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9} - q^{11} - 6 q^{13} + 10 q^{15} - 9 q^{17} + 2 q^{19} - q^{21} + 7 q^{23} + 12 q^{25} - 10 q^{27} - 13 q^{29} + 23 q^{31} + q^{33} + q^{35} - 6 q^{37} + 6 q^{39} - 12 q^{41} - 10 q^{45} + 10 q^{47} + 7 q^{49} + 9 q^{51} - 26 q^{53} + 11 q^{55} - 2 q^{57} - 10 q^{59} - 10 q^{61} + q^{63} - 22 q^{65} - 5 q^{67} - 7 q^{69} + 25 q^{71} - 8 q^{73} - 12 q^{75} - 46 q^{77} + 26 q^{79} + 10 q^{81} - 14 q^{83} + 9 q^{85} + 13 q^{87} - 31 q^{89} - 3 q^{91} - 23 q^{93} - 5 q^{95} - 32 q^{97} - q^{99}+O(q^{100}) $ 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9 - q^11 - 6 * q^13 + 10 * q^15 - 9 * q^17 + 2 * q^19 - q^21 + 7 * q^23 + 12 * q^25 - 10 * q^27 - 13 * q^29 + 23 * q^31 + q^33 + q^35 - 6 * q^37 + 6 * q^39 - 12 * q^41 - 10 * q^45 + 10 * q^47 + 7 * q^49 + 9 * q^51 - 26 * q^53 + 11 * q^55 - 2 * q^57 - 10 * q^59 - 10 * q^61 + q^63 - 22 * q^65 - 5 * q^67 - 7 * q^69 + 25 * q^71 - 8 * q^73 - 12 * q^75 - 46 * q^77 + 26 * q^79 + 10 * q^81 - 14 * q^83 + 9 * q^85 + 13 * q^87 - 31 * q^89 - 3 * q^91 - 23 * q^93 - 5 * q^95 - 32 * q^97 - q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−4.20371	−1.87996	−0.939979	−	0.341232i	$-0.889156\pi$
$5$	−4.20371	−1.87996	−0.939979	+	0.341232i	$0.889156\pi$
$6$	0	0
$7$	−3.93441	−1.48707	−0.743533	−	0.668699i	$-0.766851\pi$
$7$	−3.93441	−1.48707	−0.743533	+	0.668699i	$0.766851\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.04067	0.615284	0.307642	−	0.951502i	$-0.400460\pi$
$11$	2.04067	0.615284	0.307642	+	0.951502i	$0.400460\pi$
$12$	0	0
$13$	2.48726	0.689842	0.344921	−	0.938632i	$-0.387906\pi$
$13$	2.48726	0.689842	0.344921	+	0.938632i	$0.387906\pi$
$14$	0	0
$15$	4.20371	1.08539
$16$	0	0
$17$	−7.89415	−1.91461	−0.957306	−	0.289077i	$-0.906652\pi$
$17$	−7.89415	−1.91461	−0.957306	+	0.289077i	$0.906652\pi$
$18$	0	0
$19$	4.59249	1.05359	0.526795	−	0.849992i	$-0.323393\pi$
$19$	4.59249	1.05359	0.526795	+	0.849992i	$0.323393\pi$
$20$	0	0
$21$	3.93441	0.858558
$22$	0	0
$23$	−3.05415	−0.636835	−0.318417	−	0.947951i	$-0.603151\pi$
$23$	−3.05415	−0.636835	−0.318417	+	0.947951i	$0.603151\pi$
$24$	0	0
$25$	12.6712	2.53424
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	10.2765	1.90830	0.954148	−	0.299336i	$-0.0967650\pi$
$29$	10.2765	1.90830	0.954148	+	0.299336i	$0.0967650\pi$
$30$	0	0
$31$	4.68949	0.842257	0.421129	−	0.907001i	$-0.361634\pi$
$31$	4.68949	0.842257	0.421129	+	0.907001i	$0.361634\pi$
$32$	0	0
$33$	−2.04067	−0.355234
$34$	0	0
$35$	16.5391	2.79562
$36$	0	0
$37$	9.76889	1.60600	0.802998	−	0.595982i	$-0.203237\pi$
$37$	9.76889	1.60600	0.802998	+	0.595982i	$0.203237\pi$
$38$	0	0
$39$	−2.48726	−0.398280
$40$	0	0
$41$	−7.20906	−1.12587	−0.562933	−	0.826502i	$-0.690327\pi$
$41$	−7.20906	−1.12587	−0.562933	+	0.826502i	$0.690327\pi$
$42$	0	0
$43$	−4.73158	−0.721559	−0.360780	−	0.932651i	$-0.617489\pi$
$43$	−4.73158	−0.721559	−0.360780	+	0.932651i	$0.617489\pi$
$44$	0	0
$45$	−4.20371	−0.626653
$46$	0	0
$47$	8.18355	1.19369	0.596847	−	0.802355i	$-0.296420\pi$
$47$	8.18355	1.19369	0.596847	+	0.802355i	$0.296420\pi$
$48$	0	0
$49$	8.47956	1.21137
$50$	0	0
$51$	7.89415	1.10540
$52$	0	0
$53$	−6.41734	−0.881490	−0.440745	−	0.897632i	$-0.645286\pi$
$53$	−6.41734	−0.881490	−0.440745	+	0.897632i	$0.645286\pi$
$54$	0	0
$55$	−8.57837	−1.15671
$56$	0	0
$57$	−4.59249	−0.608291
$58$	0	0
$59$	−9.60011	−1.24983	−0.624914	−	0.780693i	$-0.714866\pi$
$59$	−9.60011	−1.24983	−0.624914	+	0.780693i	$0.714866\pi$
$60$	0	0
$61$	−8.53771	−1.09314	−0.546571	−	0.837413i	$-0.684067\pi$
$61$	−8.53771	−1.09314	−0.546571	+	0.837413i	$0.684067\pi$
$62$	0	0
$63$	−3.93441	−0.495689
$64$	0	0
$65$	−10.4557	−1.29687
$66$	0	0
$67$	−0.838729	−0.102467	−0.0512335	−	0.998687i	$-0.516315\pi$
$67$	−0.838729	−0.102467	−0.0512335	+	0.998687i	$0.516315\pi$
$68$	0	0
$69$	3.05415	0.367677
$70$	0	0
$71$	11.4409	1.35778	0.678892	−	0.734238i	$-0.262461\pi$
$71$	11.4409	1.35778	0.678892	+	0.734238i	$0.262461\pi$
$72$	0	0
$73$	−3.72057	−0.435460	−0.217730	−	0.976009i	$-0.569865\pi$
$73$	−3.72057	−0.435460	−0.217730	+	0.976009i	$0.569865\pi$
$74$	0	0
$75$	−12.6712	−1.46315
$76$	0	0
$77$	−8.02881	−0.914968
$78$	0	0
$79$	13.8405	1.55718	0.778591	−	0.627532i	$-0.215935\pi$
$79$	13.8405	1.55718	0.778591	+	0.627532i	$0.215935\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.17588	−0.897419	−0.448710	−	0.893678i	$-0.648116\pi$
$83$	−8.17588	−0.897419	−0.448710	+	0.893678i	$0.648116\pi$
$84$	0	0
$85$	33.1847	3.59939
$86$	0	0
$87$	−10.2765	−1.10176
$88$	0	0
$89$	9.03864	0.958093	0.479047	−	0.877789i	$-0.340982\pi$
$89$	9.03864	0.958093	0.479047	+	0.877789i	$0.340982\pi$
$90$	0	0
$91$	−9.78589	−1.02584
$92$	0	0
$93$	−4.68949	−0.486277
$94$	0	0
$95$	−19.3055	−1.98071
$96$	0	0
$97$	−18.4044	−1.86868	−0.934340	−	0.356382i	$-0.884010\pi$
$97$	−18.4044	−1.86868	−0.934340	+	0.356382i	$0.884010\pi$
$98$	0	0
$99$	2.04067	0.205095
$100$	0	0
$101$	4.62417	0.460123	0.230061	−	0.973176i	$-0.426107\pi$
$101$	4.62417	0.460123	0.230061	+	0.973176i	$0.426107\pi$
$102$	0	0
$103$	−6.98676	−0.688426	−0.344213	−	0.938892i	$-0.611854\pi$
$103$	−6.98676	−0.688426	−0.344213	+	0.938892i	$0.611854\pi$
$104$	0	0
$105$	−16.5391	−1.61405
$106$	0	0
$107$	5.18557	0.501308	0.250654	−	0.968077i	$-0.419354\pi$
$107$	5.18557	0.501308	0.250654	+	0.968077i	$0.419354\pi$
$108$	0	0
$109$	1.56406	0.149810	0.0749048	−	0.997191i	$-0.476135\pi$
$109$	1.56406	0.149810	0.0749048	+	0.997191i	$0.476135\pi$
$110$	0	0
$111$	−9.76889	−0.927222
$112$	0	0
$113$	14.0539	1.32208	0.661039	−	0.750352i	$-0.270116\pi$
$113$	14.0539	1.32208	0.661039	+	0.750352i	$0.270116\pi$
$114$	0	0
$115$	12.8388	1.19722
$116$	0	0
$117$	2.48726	0.229947
$118$	0	0
$119$	31.0588	2.84715
$120$	0	0
$121$	−6.83569	−0.621426
$122$	0	0
$123$	7.20906	0.650019
$124$	0	0
$125$	−32.2476	−2.88431
$126$	0	0
$127$	12.6561	1.12305	0.561525	−	0.827460i	$-0.310215\pi$
$127$	12.6561	1.12305	0.561525	+	0.827460i	$0.310215\pi$
$128$	0	0
$129$	4.73158	0.416592
$130$	0	0
$131$	14.6772	1.28235	0.641175	−	0.767394i	$-0.278447\pi$
$131$	14.6772	1.28235	0.641175	+	0.767394i	$0.278447\pi$
$132$	0	0
$133$	−18.0687	−1.56676
$134$	0	0
$135$	4.20371	0.361798
$136$	0	0
$137$	−9.48155	−0.810063	−0.405032	−	0.914303i	$-0.632740\pi$
$137$	−9.48155	−0.810063	−0.405032	+	0.914303i	$0.632740\pi$
$138$	0	0
$139$	−17.3602	−1.47247	−0.736237	−	0.676723i	$-0.763399\pi$
$139$	−17.3602	−1.47247	−0.736237	+	0.676723i	$0.763399\pi$
$140$	0	0
$141$	−8.18355	−0.689179
$142$	0	0
$143$	5.07566	0.424448
$144$	0	0
$145$	−43.1994	−3.58752
$146$	0	0
$147$	−8.47956	−0.699382
$148$	0	0
$149$	−19.7233	−1.61579	−0.807897	−	0.589324i	$-0.799394\pi$
$149$	−19.7233	−1.61579	−0.807897	+	0.589324i	$0.799394\pi$
$150$	0	0
$151$	−1.58963	−0.129362	−0.0646811	−	0.997906i	$-0.520603\pi$
$151$	−1.58963	−0.129362	−0.0646811	+	0.997906i	$0.520603\pi$
$152$	0	0
$153$	−7.89415	−0.638204
$154$	0	0
$155$	−19.7133	−1.58341
$156$	0	0
$157$	−0.551860	−0.0440432	−0.0220216	−	0.999757i	$-0.507010\pi$
$157$	−0.551860	−0.0440432	−0.0220216	+	0.999757i	$0.507010\pi$
$158$	0	0
$159$	6.41734	0.508928
$160$	0	0
$161$	12.0163	0.947015
$162$	0	0
$163$	−0.941980	−0.0737815	−0.0368908	−	0.999319i	$-0.511745\pi$
$163$	−0.941980	−0.0737815	−0.0368908	+	0.999319i	$0.511745\pi$
$164$	0	0
$165$	8.57837	0.667825
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−6.81354	−0.524118
$170$	0	0
$171$	4.59249	0.351197
$172$	0	0
$173$	−10.2114	−0.776357	−0.388179	−	0.921584i	$-0.626896\pi$
$173$	−10.2114	−0.776357	−0.388179	+	0.921584i	$0.626896\pi$
$174$	0	0
$175$	−49.8537	−3.76859
$176$	0	0
$177$	9.60011	0.721589
$178$	0	0
$179$	−18.3373	−1.37059	−0.685295	−	0.728265i	$-0.740327\pi$
$179$	−18.3373	−1.37059	−0.685295	+	0.728265i	$0.740327\pi$
$180$	0	0
$181$	12.8187	0.952809	0.476405	−	0.879226i	$-0.341940\pi$
$181$	12.8187	0.952809	0.476405	+	0.879226i	$0.341940\pi$
$182$	0	0
$183$	8.53771	0.631126
$184$	0	0
$185$	−41.0656	−3.01920
$186$	0	0
$187$	−16.1093	−1.17803
$188$	0	0
$189$	3.93441	0.286186
$190$	0	0
$191$	0.710128	0.0513831	0.0256915	−	0.999670i	$-0.491821\pi$
$191$	0.710128	0.0513831	0.0256915	+	0.999670i	$0.491821\pi$
$192$	0	0
$193$	23.0087	1.65620	0.828100	−	0.560580i	$-0.189422\pi$
$193$	23.0087	1.65620	0.828100	+	0.560580i	$0.189422\pi$
$194$	0	0
$195$	10.4557	0.748750
$196$	0	0
$197$	−13.9561	−0.994331	−0.497165	−	0.867656i	$-0.665626\pi$
$197$	−13.9561	−0.994331	−0.497165	+	0.867656i	$0.665626\pi$
$198$	0	0
$199$	0.807605	0.0572496	0.0286248	−	0.999590i	$-0.490887\pi$
$199$	0.807605	0.0572496	0.0286248	+	0.999590i	$0.490887\pi$
$200$	0	0
$201$	0.838729	0.0591594
$202$	0	0
$203$	−40.4319	−2.83776
$204$	0	0
$205$	30.3048	2.11658
$206$	0	0
$207$	−3.05415	−0.212278
$208$	0	0
$209$	9.37174	0.648257
$210$	0	0
$211$	17.1904	1.18343	0.591717	−	0.806146i	$-0.298450\pi$
$211$	17.1904	1.18343	0.591717	+	0.806146i	$0.298450\pi$
$212$	0	0
$213$	−11.4409	−0.783917
$214$	0	0
$215$	19.8902	1.35650
$216$	0	0
$217$	−18.4504	−1.25249
$218$	0	0
$219$	3.72057	0.251413
$220$	0	0
$221$	−19.6348	−1.32078
$222$	0	0
$223$	21.8784	1.46509	0.732544	−	0.680720i	$-0.238333\pi$
$223$	21.8784	1.46509	0.732544	+	0.680720i	$0.238333\pi$
$224$	0	0
$225$	12.6712	0.844748
$226$	0	0
$227$	13.0916	0.868921	0.434461	−	0.900691i	$-0.356939\pi$
$227$	13.0916	0.868921	0.434461	+	0.900691i	$0.356939\pi$
$228$	0	0
$229$	−24.4167	−1.61350	−0.806751	−	0.590891i	$-0.798776\pi$
$229$	−24.4167	−1.61350	−0.806751	+	0.590891i	$0.798776\pi$
$230$	0	0
$231$	8.02881	0.528257
$232$	0	0
$233$	−15.1811	−0.994546	−0.497273	−	0.867594i	$-0.665665\pi$
$233$	−15.1811	−0.994546	−0.497273	+	0.867594i	$0.665665\pi$
$234$	0	0
$235$	−34.4013	−2.24409
$236$	0	0
$237$	−13.8405	−0.899039
$238$	0	0
$239$	−5.18054	−0.335101	−0.167551	−	0.985863i	$-0.553586\pi$
$239$	−5.18054	−0.335101	−0.167551	+	0.985863i	$0.553586\pi$
$240$	0	0
$241$	−3.97604	−0.256119	−0.128060	−	0.991766i	$-0.540875\pi$
$241$	−3.97604	−0.256119	−0.128060	+	0.991766i	$0.540875\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−35.6457	−2.27732
$246$	0	0
$247$	11.4227	0.726811
$248$	0	0
$249$	8.17588	0.518125
$250$	0	0
$251$	−5.44503	−0.343687	−0.171844	−	0.985124i	$-0.554972\pi$
$251$	−5.44503	−0.343687	−0.171844	+	0.985124i	$0.554972\pi$
$252$	0	0
$253$	−6.23250	−0.391834
$254$	0	0
$255$	−33.1847	−2.07811
$256$	0	0
$257$	−10.3024	−0.642646	−0.321323	−	0.946970i	$-0.604128\pi$
$257$	−10.3024	−0.642646	−0.321323	+	0.946970i	$0.604128\pi$
$258$	0	0
$259$	−38.4348	−2.38822
$260$	0	0
$261$	10.2765	0.636099
$262$	0	0
$263$	−21.2288	−1.30903	−0.654513	−	0.756050i	$-0.727126\pi$
$263$	−21.2288	−1.30903	−0.654513	+	0.756050i	$0.727126\pi$
$264$	0	0
$265$	26.9767	1.65716
$266$	0	0
$267$	−9.03864	−0.553156
$268$	0	0
$269$	6.65889	0.405999	0.203000	−	0.979179i	$-0.434931\pi$
$269$	6.65889	0.405999	0.203000	+	0.979179i	$0.434931\pi$
$270$	0	0
$271$	19.2960	1.17215	0.586075	−	0.810257i	$-0.300673\pi$
$271$	19.2960	1.17215	0.586075	+	0.810257i	$0.300673\pi$
$272$	0	0
$273$	9.78589	0.592269
$274$	0	0
$275$	25.8577	1.55928
$276$	0	0
$277$	18.9532	1.13879	0.569393	−	0.822066i	$-0.307178\pi$
$277$	18.9532	1.13879	0.569393	+	0.822066i	$0.307178\pi$
$278$	0	0
$279$	4.68949	0.280752
$280$	0	0
$281$	−15.2503	−0.909757	−0.454879	−	0.890553i	$-0.650317\pi$
$281$	−15.2503	−0.909757	−0.454879	+	0.890553i	$0.650317\pi$
$282$	0	0
$283$	2.16727	0.128831	0.0644153	−	0.997923i	$-0.479482\pi$
$283$	2.16727	0.128831	0.0644153	+	0.997923i	$0.479482\pi$
$284$	0	0
$285$	19.3055	1.14356
$286$	0	0
$287$	28.3634	1.67424
$288$	0	0
$289$	45.3175	2.66574
$290$	0	0
$291$	18.4044	1.07888
$292$	0	0
$293$	−23.5794	−1.37752	−0.688762	−	0.724987i	$-0.741846\pi$
$293$	−23.5794	−1.37752	−0.688762	+	0.724987i	$0.741846\pi$
$294$	0	0
$295$	40.3561	2.34963
$296$	0	0
$297$	−2.04067	−0.118411
$298$	0	0
$299$	−7.59647	−0.439315
$300$	0	0
$301$	18.6160	1.07301
$302$	0	0
$303$	−4.62417	−0.265652
$304$	0	0
$305$	35.8901	2.05506
$306$	0	0
$307$	−8.57792	−0.489568	−0.244784	−	0.969578i	$-0.578717\pi$
$307$	−8.57792	−0.489568	−0.244784	+	0.969578i	$0.578717\pi$
$308$	0	0
$309$	6.98676	0.397463
$310$	0	0
$311$	−0.215903	−0.0122427	−0.00612136	−	0.999981i	$-0.501949\pi$
$311$	−0.215903	−0.0122427	−0.00612136	+	0.999981i	$0.501949\pi$
$312$	0	0
$313$	3.76380	0.212743	0.106371	−	0.994326i	$-0.466077\pi$
$313$	3.76380	0.212743	0.106371	+	0.994326i	$0.466077\pi$
$314$	0	0
$315$	16.5391	0.931874
$316$	0	0
$317$	14.5894	0.819421	0.409710	−	0.912216i	$-0.365630\pi$
$317$	14.5894	0.819421	0.409710	+	0.912216i	$0.365630\pi$
$318$	0	0
$319$	20.9709	1.17414
$320$	0	0
$321$	−5.18557	−0.289430
$322$	0	0
$323$	−36.2538	−2.01722
$324$	0	0
$325$	31.5166	1.74823
$326$	0	0
$327$	−1.56406	−0.0864926
$328$	0	0
$329$	−32.1974	−1.77510
$330$	0	0
$331$	21.1090	1.16025	0.580127	−	0.814526i	$-0.303003\pi$
$331$	21.1090	1.16025	0.580127	+	0.814526i	$0.303003\pi$
$332$	0	0
$333$	9.76889	0.535332
$334$	0	0
$335$	3.52578	0.192634
$336$	0	0
$337$	−18.5365	−1.00975	−0.504875	−	0.863192i	$-0.668461\pi$
$337$	−18.5365	−1.00975	−0.504875	+	0.863192i	$0.668461\pi$
$338$	0	0
$339$	−14.0539	−0.763302
$340$	0	0
$341$	9.56968	0.518227
$342$	0	0
$343$	−5.82119	−0.314315
$344$	0	0
$345$	−12.8388	−0.691217
$346$	0	0
$347$	−2.02686	−0.108807	−0.0544037	−	0.998519i	$-0.517326\pi$
$347$	−2.02686	−0.108807	−0.0544037	+	0.998519i	$0.517326\pi$
$348$	0	0
$349$	0.926361	0.0495870	0.0247935	−	0.999693i	$-0.492107\pi$
$349$	0.926361	0.0495870	0.0247935	+	0.999693i	$0.492107\pi$
$350$	0	0
$351$	−2.48726	−0.132760
$352$	0	0
$353$	0.255400	0.0135936	0.00679679	−	0.999977i	$-0.497836\pi$
$353$	0.255400	0.0135936	0.00679679	+	0.999977i	$0.497836\pi$
$354$	0	0
$355$	−48.0942	−2.55258
$356$	0	0
$357$	−31.0588	−1.64381
$358$	0	0
$359$	6.37470	0.336444	0.168222	−	0.985749i	$-0.446197\pi$
$359$	6.37470	0.336444	0.168222	+	0.985749i	$0.446197\pi$
$360$	0	0
$361$	2.09100	0.110053
$362$	0	0
$363$	6.83569	0.358780
$364$	0	0
$365$	15.6402	0.818646
$366$	0	0
$367$	34.3428	1.79268	0.896340	−	0.443368i	$-0.146217\pi$
$367$	34.3428	1.79268	0.896340	+	0.443368i	$0.146217\pi$
$368$	0	0
$369$	−7.20906	−0.375289
$370$	0	0
$371$	25.2484	1.31083
$372$	0	0
$373$	−17.8131	−0.922327	−0.461164	−	0.887315i	$-0.652568\pi$
$373$	−17.8131	−0.922327	−0.461164	+	0.887315i	$0.652568\pi$
$374$	0	0
$375$	32.2476	1.66526
$376$	0	0
$377$	25.5603	1.31642
$378$	0	0
$379$	−29.4274	−1.51158	−0.755792	−	0.654812i	$-0.772748\pi$
$379$	−29.4274	−1.51158	−0.755792	+	0.654812i	$0.772748\pi$
$380$	0	0
$381$	−12.6561	−0.648394
$382$	0	0
$383$	8.91557	0.455564	0.227782	−	0.973712i	$-0.426853\pi$
$383$	8.91557	0.455564	0.227782	+	0.973712i	$0.426853\pi$
$384$	0	0
$385$	33.7508	1.72010
$386$	0	0
$387$	−4.73158	−0.240520
$388$	0	0
$389$	−35.3858	−1.79413	−0.897066	−	0.441898i	$-0.854305\pi$
$389$	−35.3858	−1.79413	−0.897066	+	0.441898i	$0.854305\pi$
$390$	0	0
$391$	24.1099	1.21929
$392$	0	0
$393$	−14.6772	−0.740365
$394$	0	0
$395$	−58.1817	−2.92744
$396$	0	0
$397$	−4.91158	−0.246505	−0.123253	−	0.992375i	$-0.539333\pi$
$397$	−4.91158	−0.246505	−0.123253	+	0.992375i	$0.539333\pi$
$398$	0	0
$399$	18.0687	0.904569
$400$	0	0
$401$	−17.3914	−0.868486	−0.434243	−	0.900796i	$-0.642984\pi$
$401$	−17.3914	−0.868486	−0.434243	+	0.900796i	$0.642984\pi$
$402$	0	0
$403$	11.6640	0.581024
$404$	0	0
$405$	−4.20371	−0.208884
$406$	0	0
$407$	19.9350	0.988143
$408$	0	0
$409$	11.9333	0.590063	0.295032	−	0.955487i	$-0.404670\pi$
$409$	11.9333	0.590063	0.295032	+	0.955487i	$0.404670\pi$
$410$	0	0
$411$	9.48155	0.467690
$412$	0	0
$413$	37.7708	1.85858
$414$	0	0
$415$	34.3691	1.68711
$416$	0	0
$417$	17.3602	0.850134
$418$	0	0
$419$	12.7381	0.622295	0.311147	−	0.950362i	$-0.399287\pi$
$419$	12.7381	0.622295	0.311147	+	0.950362i	$0.399287\pi$
$420$	0	0
$421$	−0.792643	−0.0386311	−0.0193155	−	0.999813i	$-0.506149\pi$
$421$	−0.792643	−0.0386311	−0.0193155	+	0.999813i	$0.506149\pi$
$422$	0	0
$423$	8.18355	0.397898
$424$	0	0
$425$	−100.028	−4.85209
$426$	0	0
$427$	33.5908	1.62557
$428$	0	0
$429$	−5.07566	−0.245055
$430$	0	0
$431$	−2.88489	−0.138960	−0.0694802	−	0.997583i	$-0.522134\pi$
$431$	−2.88489	−0.138960	−0.0694802	+	0.997583i	$0.522134\pi$
$432$	0	0
$433$	−30.5682	−1.46902	−0.734508	−	0.678600i	$-0.762587\pi$
$433$	−30.5682	−1.46902	−0.734508	+	0.678600i	$0.762587\pi$
$434$	0	0
$435$	43.1994	2.07125
$436$	0	0
$437$	−14.0262	−0.670963
$438$	0	0
$439$	5.61783	0.268125	0.134062	−	0.990973i	$-0.457198\pi$
$439$	5.61783	0.268125	0.134062	+	0.990973i	$0.457198\pi$
$440$	0	0
$441$	8.47956	0.403789
$442$	0	0
$443$	−6.05621	−0.287739	−0.143870	−	0.989597i	$-0.545955\pi$
$443$	−6.05621	−0.287739	−0.143870	+	0.989597i	$0.545955\pi$
$444$	0	0
$445$	−37.9958	−1.80118
$446$	0	0
$447$	19.7233	0.932879
$448$	0	0
$449$	−21.6596	−1.02218	−0.511090	−	0.859527i	$-0.670758\pi$
$449$	−21.6596	−1.02218	−0.511090	+	0.859527i	$0.670758\pi$
$450$	0	0
$451$	−14.7113	−0.692727
$452$	0	0
$453$	1.58963	0.0746873
$454$	0	0
$455$	41.1371	1.92854
$456$	0	0
$457$	−19.3739	−0.906275	−0.453138	−	0.891441i	$-0.649695\pi$
$457$	−19.3739	−0.906275	−0.453138	+	0.891441i	$0.649695\pi$
$458$	0	0
$459$	7.89415	0.368467
$460$	0	0
$461$	−1.88576	−0.0878288	−0.0439144	−	0.999035i	$-0.513983\pi$
$461$	−1.88576	−0.0878288	−0.0439144	+	0.999035i	$0.513983\pi$
$462$	0	0
$463$	−35.2730	−1.63928	−0.819638	−	0.572882i	$-0.805825\pi$
$463$	−35.2730	−1.63928	−0.819638	+	0.572882i	$0.805825\pi$
$464$	0	0
$465$	19.7133	0.914181
$466$	0	0
$467$	11.4516	0.529915	0.264957	−	0.964260i	$-0.414642\pi$
$467$	11.4516	0.529915	0.264957	+	0.964260i	$0.414642\pi$
$468$	0	0
$469$	3.29990	0.152375
$470$	0	0
$471$	0.551860	0.0254284
$472$	0	0
$473$	−9.65557	−0.443964
$474$	0	0
$475$	58.1925	2.67005
$476$	0	0
$477$	−6.41734	−0.293830
$478$	0	0
$479$	−2.05633	−0.0939562	−0.0469781	−	0.998896i	$-0.514959\pi$
$479$	−2.05633	−0.0939562	−0.0469781	+	0.998896i	$0.514959\pi$
$480$	0	0
$481$	24.2978	1.10788
$482$	0	0
$483$	−12.0163	−0.546760
$484$	0	0
$485$	77.3667	3.51304
$486$	0	0
$487$	−3.72895	−0.168975	−0.0844873	−	0.996425i	$-0.526925\pi$
$487$	−3.72895	−0.168975	−0.0844873	+	0.996425i	$0.526925\pi$
$488$	0	0
$489$	0.941980	0.0425978
$490$	0	0
$491$	−23.5064	−1.06083	−0.530415	−	0.847738i	$-0.677964\pi$
$491$	−23.5064	−1.06083	−0.530415	+	0.847738i	$0.677964\pi$
$492$	0	0
$493$	−81.1241	−3.65365
$494$	0	0
$495$	−8.57837	−0.385569
$496$	0	0
$497$	−45.0131	−2.01911
$498$	0	0
$499$	18.0250	0.806909	0.403454	−	0.915000i	$-0.367809\pi$
$499$	18.0250	0.806909	0.403454	+	0.915000i	$0.367809\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−3.21136	−0.143187	−0.0715936	−	0.997434i	$-0.522808\pi$
$503$	−3.21136	−0.143187	−0.0715936	+	0.997434i	$0.522808\pi$
$504$	0	0
$505$	−19.4387	−0.865011
$506$	0	0
$507$	6.81354	0.302600
$508$	0	0
$509$	30.7643	1.36360	0.681802	−	0.731537i	$-0.261197\pi$
$509$	30.7643	1.36360	0.681802	+	0.731537i	$0.261197\pi$
$510$	0	0
$511$	14.6382	0.647557
$512$	0	0
$513$	−4.59249	−0.202764
$514$	0	0
$515$	29.3703	1.29421
$516$	0	0
$517$	16.6999	0.734460
$518$	0	0
$519$	10.2114	0.448230
$520$	0	0
$521$	17.3478	0.760019	0.380010	−	0.924983i	$-0.375921\pi$
$521$	17.3478	0.760019	0.380010	+	0.924983i	$0.375921\pi$
$522$	0	0
$523$	19.5209	0.853591	0.426795	−	0.904348i	$-0.359642\pi$
$523$	19.5209	0.853591	0.426795	+	0.904348i	$0.359642\pi$
$524$	0	0
$525$	49.8537	2.17579
$526$	0	0
$527$	−37.0195	−1.61260
$528$	0	0
$529$	−13.6722	−0.594441
$530$	0	0
$531$	−9.60011	−0.416609
$532$	0	0
$533$	−17.9308	−0.776669
$534$	0	0
$535$	−21.7987	−0.942439
$536$	0	0
$537$	18.3373	0.791311
$538$	0	0
$539$	17.3039	0.745334
$540$	0	0
$541$	−36.2542	−1.55869	−0.779344	−	0.626596i	$-0.784448\pi$
$541$	−36.2542	−1.55869	−0.779344	+	0.626596i	$0.784448\pi$
$542$	0	0
$543$	−12.8187	−0.550105
$544$	0	0
$545$	−6.57485	−0.281636
$546$	0	0
$547$	−7.06291	−0.301988	−0.150994	−	0.988535i	$-0.548247\pi$
$547$	−7.06291	−0.301988	−0.150994	+	0.988535i	$0.548247\pi$
$548$	0	0
$549$	−8.53771	−0.364381
$550$	0	0
$551$	47.1947	2.01056
$552$	0	0
$553$	−54.4543	−2.31563
$554$	0	0
$555$	41.0656	1.74314
$556$	0	0
$557$	−30.7205	−1.30167	−0.650836	−	0.759219i	$-0.725581\pi$
$557$	−30.7205	−1.30167	−0.650836	+	0.759219i	$0.725581\pi$
$558$	0	0
$559$	−11.7687	−0.497762
$560$	0	0
$561$	16.1093	0.680135
$562$	0	0
$563$	−2.12348	−0.0894940	−0.0447470	−	0.998998i	$-0.514248\pi$
$563$	−2.12348	−0.0894940	−0.0447470	+	0.998998i	$0.514248\pi$
$564$	0	0
$565$	−59.0785	−2.48545
$566$	0	0
$567$	−3.93441	−0.165230
$568$	0	0
$569$	21.7660	0.912479	0.456239	−	0.889857i	$-0.349196\pi$
$569$	21.7660	0.912479	0.456239	+	0.889857i	$0.349196\pi$
$570$	0	0
$571$	−18.5047	−0.774399	−0.387199	−	0.921996i	$-0.626558\pi$
$571$	−18.5047	−0.774399	−0.387199	+	0.921996i	$0.626558\pi$
$572$	0	0
$573$	−0.710128	−0.0296660
$574$	0	0
$575$	−38.6998	−1.61389
$576$	0	0
$577$	7.52657	0.313335	0.156668	−	0.987651i	$-0.449925\pi$
$577$	7.52657	0.313335	0.156668	+	0.987651i	$0.449925\pi$
$578$	0	0
$579$	−23.0087	−0.956208
$580$	0	0
$581$	32.1672	1.33452
$582$	0	0
$583$	−13.0957	−0.542366
$584$	0	0
$585$	−10.4557	−0.432291
$586$	0	0
$587$	15.1211	0.624116	0.312058	−	0.950063i	$-0.398982\pi$
$587$	15.1211	0.624116	0.312058	+	0.950063i	$0.398982\pi$
$588$	0	0
$589$	21.5365	0.887394
$590$	0	0
$591$	13.9561	0.574077
$592$	0	0
$593$	19.1396	0.785970	0.392985	−	0.919545i	$-0.371442\pi$
$593$	19.1396	0.785970	0.392985	+	0.919545i	$0.371442\pi$
$594$	0	0
$595$	−130.562	−5.35253
$596$	0	0
$597$	−0.807605	−0.0330531
$598$	0	0
$599$	39.9237	1.63124	0.815620	−	0.578588i	$-0.196396\pi$
$599$	39.9237	1.63124	0.815620	+	0.578588i	$0.196396\pi$
$600$	0	0
$601$	1.45936	0.0595286	0.0297643	−	0.999557i	$-0.490524\pi$
$601$	1.45936	0.0595286	0.0297643	+	0.999557i	$0.490524\pi$
$602$	0	0
$603$	−0.838729	−0.0341557
$604$	0	0
$605$	28.7353	1.16825
$606$	0	0
$607$	30.3050	1.23004	0.615020	−	0.788511i	$-0.289148\pi$
$607$	30.3050	1.23004	0.615020	+	0.788511i	$0.289148\pi$
$608$	0	0
$609$	40.4319	1.63838
$610$	0	0
$611$	20.3546	0.823459
$612$	0	0
$613$	1.99804	0.0806999	0.0403499	−	0.999186i	$-0.487153\pi$
$613$	1.99804	0.0806999	0.0403499	+	0.999186i	$0.487153\pi$
$614$	0	0
$615$	−30.3048	−1.22201
$616$	0	0
$617$	5.21803	0.210070	0.105035	−	0.994469i	$-0.466505\pi$
$617$	5.21803	0.210070	0.105035	+	0.994469i	$0.466505\pi$
$618$	0	0
$619$	21.2927	0.855825	0.427912	−	0.903820i	$-0.359249\pi$
$619$	21.2927	0.855825	0.427912	+	0.903820i	$0.359249\pi$
$620$	0	0
$621$	3.05415	0.122559
$622$	0	0
$623$	−35.5617	−1.42475
$624$	0	0
$625$	72.2036	2.88815
$626$	0	0
$627$	−9.37174	−0.374271
$628$	0	0
$629$	−77.1170	−3.07486
$630$	0	0
$631$	13.7205	0.546205	0.273103	−	0.961985i	$-0.411950\pi$
$631$	13.7205	0.546205	0.273103	+	0.961985i	$0.411950\pi$
$632$	0	0
$633$	−17.1904	−0.683256
$634$	0	0
$635$	−53.2028	−2.11129
$636$	0	0
$637$	21.0909	0.835651
$638$	0	0
$639$	11.4409	0.452595
$640$	0	0
$641$	21.4675	0.847916	0.423958	−	0.905682i	$-0.360640\pi$
$641$	21.4675	0.847916	0.423958	+	0.905682i	$0.360640\pi$
$642$	0	0
$643$	27.3260	1.07763	0.538816	−	0.842423i	$-0.318872\pi$
$643$	27.3260	1.07763	0.538816	+	0.842423i	$0.318872\pi$
$644$	0	0
$645$	−19.8902	−0.783176
$646$	0	0
$647$	−30.6029	−1.20312	−0.601561	−	0.798827i	$-0.705454\pi$
$647$	−30.6029	−1.20312	−0.601561	+	0.798827i	$0.705454\pi$
$648$	0	0
$649$	−19.5906	−0.768999
$650$	0	0
$651$	18.4504	0.723127
$652$	0	0
$653$	−4.08287	−0.159775	−0.0798875	−	0.996804i	$-0.525456\pi$
$653$	−4.08287	−0.159775	−0.0798875	+	0.996804i	$0.525456\pi$
$654$	0	0
$655$	−61.6986	−2.41077
$656$	0	0
$657$	−3.72057	−0.145153
$658$	0	0
$659$	−9.96598	−0.388220	−0.194110	−	0.980980i	$-0.562182\pi$
$659$	−9.96598	−0.388220	−0.194110	+	0.980980i	$0.562182\pi$
$660$	0	0
$661$	−18.7932	−0.730971	−0.365486	−	0.930817i	$-0.619097\pi$
$661$	−18.7932	−0.730971	−0.365486	+	0.930817i	$0.619097\pi$
$662$	0	0
$663$	19.6348	0.762552
$664$	0	0
$665$	75.9558	2.94544
$666$	0	0
$667$	−31.3860	−1.21527
$668$	0	0
$669$	−21.8784	−0.845869
$670$	0	0
$671$	−17.4226	−0.672592
$672$	0	0
$673$	9.14536	0.352528	0.176264	−	0.984343i	$-0.443599\pi$
$673$	9.14536	0.352528	0.176264	+	0.984343i	$0.443599\pi$
$674$	0	0
$675$	−12.6712	−0.487715
$676$	0	0
$677$	23.5702	0.905877	0.452939	−	0.891542i	$-0.350376\pi$
$677$	23.5702	0.905877	0.452939	+	0.891542i	$0.350376\pi$
$678$	0	0
$679$	72.4103	2.77885
$680$	0	0
$681$	−13.0916	−0.501672
$682$	0	0
$683$	34.8488	1.33345	0.666727	−	0.745302i	$-0.267695\pi$
$683$	34.8488	1.33345	0.666727	+	0.745302i	$0.267695\pi$
$684$	0	0
$685$	39.8577	1.52289
$686$	0	0
$687$	24.4167	0.931556
$688$	0	0
$689$	−15.9616	−0.608089
$690$	0	0
$691$	−12.9252	−0.491697	−0.245848	−	0.969308i	$-0.579067\pi$
$691$	−12.9252	−0.491697	−0.245848	+	0.969308i	$0.579067\pi$
$692$	0	0
$693$	−8.02881	−0.304989
$694$	0	0
$695$	72.9774	2.76819
$696$	0	0
$697$	56.9094	2.15560
$698$	0	0
$699$	15.1811	0.574201
$700$	0	0
$701$	−33.8900	−1.28001	−0.640003	−	0.768372i	$-0.721067\pi$
$701$	−33.8900	−1.28001	−0.640003	+	0.768372i	$0.721067\pi$
$702$	0	0
$703$	44.8636	1.69206
$704$	0	0
$705$	34.4013	1.29563
$706$	0	0
$707$	−18.1934	−0.684233
$708$	0	0
$709$	−5.56580	−0.209028	−0.104514	−	0.994523i	$-0.533329\pi$
$709$	−5.56580	−0.209028	−0.104514	+	0.994523i	$0.533329\pi$
$710$	0	0
$711$	13.8405	0.519061
$712$	0	0
$713$	−14.3224	−0.536379
$714$	0	0
$715$	−21.3366	−0.797945
$716$	0	0
$717$	5.18054	0.193471
$718$	0	0
$719$	−40.0628	−1.49409	−0.747045	−	0.664773i	$-0.768528\pi$
$719$	−40.0628	−1.49409	−0.747045	+	0.664773i	$0.768528\pi$
$720$	0	0
$721$	27.4888	1.02374
$722$	0	0
$723$	3.97604	0.147871
$724$	0	0
$725$	130.216	4.83609
$726$	0	0
$727$	−14.6765	−0.544320	−0.272160	−	0.962252i	$-0.587738\pi$
$727$	−14.6765	−0.544320	−0.272160	+	0.962252i	$0.587738\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	37.3518	1.38151
$732$	0	0
$733$	−27.9341	−1.03177	−0.515885	−	0.856658i	$-0.672537\pi$
$733$	−27.9341	−1.03177	−0.515885	+	0.856658i	$0.672537\pi$
$734$	0	0
$735$	35.6457	1.31481
$736$	0	0
$737$	−1.71156	−0.0630463
$738$	0	0
$739$	−30.6304	−1.12676	−0.563379	−	0.826199i	$-0.690499\pi$
$739$	−30.6304	−1.12676	−0.563379	+	0.826199i	$0.690499\pi$
$740$	0	0
$741$	−11.4227	−0.419624
$742$	0	0
$743$	−36.0049	−1.32089	−0.660446	−	0.750874i	$-0.729633\pi$
$743$	−36.0049	−1.32089	−0.660446	+	0.750874i	$0.729633\pi$
$744$	0	0
$745$	82.9110	3.03762
$746$	0	0
$747$	−8.17588	−0.299140
$748$	0	0
$749$	−20.4022	−0.745478
$750$	0	0
$751$	−5.21628	−0.190345	−0.0951724	−	0.995461i	$-0.530340\pi$
$751$	−5.21628	−0.190345	−0.0951724	+	0.995461i	$0.530340\pi$
$752$	0	0
$753$	5.44503	0.198428
$754$	0	0
$755$	6.68235	0.243196
$756$	0	0
$757$	6.94496	0.252419	0.126210	−	0.992004i	$-0.459719\pi$
$757$	6.94496	0.252419	0.126210	+	0.992004i	$0.459719\pi$
$758$	0	0
$759$	6.23250	0.226225
$760$	0	0
$761$	35.0181	1.26941	0.634703	−	0.772756i	$-0.281123\pi$
$761$	35.0181	1.26941	0.634703	+	0.772756i	$0.281123\pi$
$762$	0	0
$763$	−6.15364	−0.222777
$764$	0	0
$765$	33.1847	1.19980
$766$	0	0
$767$	−23.8780	−0.862184
$768$	0	0
$769$	−7.65227	−0.275948	−0.137974	−	0.990436i	$-0.544059\pi$
$769$	−7.65227	−0.275948	−0.137974	+	0.990436i	$0.544059\pi$
$770$	0	0
$771$	10.3024	0.371032
$772$	0	0
$773$	19.2150	0.691115	0.345557	−	0.938398i	$-0.387690\pi$
$773$	19.2150	0.691115	0.345557	+	0.938398i	$0.387690\pi$
$774$	0	0
$775$	59.4215	2.13448
$776$	0	0
$777$	38.4348	1.37884
$778$	0	0
$779$	−33.1076	−1.18620
$780$	0	0
$781$	23.3470	0.835422
$782$	0	0
$783$	−10.2765	−0.367252
$784$	0	0
$785$	2.31986	0.0827994
$786$	0	0
$787$	16.2180	0.578109	0.289054	−	0.957313i	$-0.406659\pi$
$787$	16.2180	0.578109	0.289054	+	0.957313i	$0.406659\pi$
$788$	0	0
$789$	21.2288	0.755767
$790$	0	0
$791$	−55.2937	−1.96602
$792$	0	0
$793$	−21.2355	−0.754095
$794$	0	0
$795$	−26.9767	−0.956764
$796$	0	0
$797$	−42.2046	−1.49496	−0.747482	−	0.664282i	$-0.768737\pi$
$797$	−42.2046	−1.49496	−0.747482	+	0.664282i	$0.768737\pi$
$798$	0	0
$799$	−64.6021	−2.28546
$800$	0	0
$801$	9.03864	0.319364
$802$	0	0
$803$	−7.59243	−0.267931
$804$	0	0
$805$	−50.5130	−1.78035
$806$	0	0
$807$	−6.65889	−0.234404
$808$	0	0
$809$	−17.5769	−0.617969	−0.308985	−	0.951067i	$-0.599989\pi$
$809$	−17.5769	−0.617969	−0.308985	+	0.951067i	$0.599989\pi$
$810$	0	0
$811$	−45.2311	−1.58828	−0.794139	−	0.607736i	$-0.792078\pi$
$811$	−45.2311	−1.58828	−0.794139	+	0.607736i	$0.792078\pi$
$812$	0	0
$813$	−19.2960	−0.676741
$814$	0	0
$815$	3.95981	0.138706
$816$	0	0
$817$	−21.7298	−0.760228
$818$	0	0
$819$	−9.78589	−0.341947
$820$	0	0
$821$	29.9555	1.04545	0.522727	−	0.852500i	$-0.324915\pi$
$821$	29.9555	1.04545	0.522727	+	0.852500i	$0.324915\pi$
$822$	0	0
$823$	32.6625	1.13854	0.569272	−	0.822149i	$-0.307225\pi$
$823$	32.6625	1.13854	0.569272	+	0.822149i	$0.307225\pi$
$824$	0	0
$825$	−25.8577	−0.900250
$826$	0	0
$827$	−55.6345	−1.93460	−0.967300	−	0.253635i	$-0.918374\pi$
$827$	−55.6345	−1.93460	−0.967300	+	0.253635i	$0.918374\pi$
$828$	0	0
$829$	−30.9256	−1.07409	−0.537045	−	0.843554i	$-0.680459\pi$
$829$	−30.9256	−1.07409	−0.537045	+	0.843554i	$0.680459\pi$
$830$	0	0
$831$	−18.9532	−0.657478
$832$	0	0
$833$	−66.9389	−2.31929
$834$	0	0
$835$	4.20371	0.145476
$836$	0	0
$837$	−4.68949	−0.162092
$838$	0	0
$839$	−30.4980	−1.05291	−0.526453	−	0.850204i	$-0.676479\pi$
$839$	−30.4980	−1.05291	−0.526453	+	0.850204i	$0.676479\pi$
$840$	0	0
$841$	76.6062	2.64159
$842$	0	0
$843$	15.2503	0.525249
$844$	0	0
$845$	28.6422	0.985321
$846$	0	0
$847$	26.8944	0.924102
$848$	0	0
$849$	−2.16727	−0.0743804
$850$	0	0
$851$	−29.8357	−1.02275
$852$	0	0
$853$	2.81114	0.0962516	0.0481258	−	0.998841i	$-0.484675\pi$
$853$	2.81114	0.0962516	0.0481258	+	0.998841i	$0.484675\pi$
$854$	0	0
$855$	−19.3055	−0.660235
$856$	0	0
$857$	0.590454	0.0201695	0.0100848	−	0.999949i	$-0.496790\pi$
$857$	0.590454	0.0201695	0.0100848	+	0.999949i	$0.496790\pi$
$858$	0	0
$859$	−40.1740	−1.37072	−0.685360	−	0.728204i	$-0.740355\pi$
$859$	−40.1740	−1.37072	−0.685360	+	0.728204i	$0.740355\pi$
$860$	0	0
$861$	−28.3634	−0.966621
$862$	0	0
$863$	50.1452	1.70696	0.853481	−	0.521125i	$-0.174487\pi$
$863$	50.1452	1.70696	0.853481	+	0.521125i	$0.174487\pi$
$864$	0	0
$865$	42.9258	1.45952
$866$	0	0
$867$	−45.3175	−1.53906
$868$	0	0
$869$	28.2439	0.958108
$870$	0	0
$871$	−2.08614	−0.0706860
$872$	0	0
$873$	−18.4044	−0.622893
$874$	0	0
$875$	126.875	4.28916
$876$	0	0
$877$	18.8495	0.636501	0.318251	−	0.948007i	$-0.396905\pi$
$877$	18.8495	0.636501	0.318251	+	0.948007i	$0.396905\pi$
$878$	0	0
$879$	23.5794	0.795314
$880$	0	0
$881$	3.30498	0.111348	0.0556739	−	0.998449i	$-0.482269\pi$
$881$	3.30498	0.111348	0.0556739	+	0.998449i	$0.482269\pi$
$882$	0	0
$883$	−20.1301	−0.677430	−0.338715	−	0.940889i	$-0.609992\pi$
$883$	−20.1301	−0.677430	−0.338715	+	0.940889i	$0.609992\pi$
$884$	0	0
$885$	−40.3561	−1.35656
$886$	0	0
$887$	−27.5842	−0.926186	−0.463093	−	0.886310i	$-0.653260\pi$
$887$	−27.5842	−0.926186	−0.463093	+	0.886310i	$0.653260\pi$
$888$	0	0
$889$	−49.7944	−1.67005
$890$	0	0
$891$	2.04067	0.0683649
$892$	0	0
$893$	37.5829	1.25766
$894$	0	0
$895$	77.0846	2.57665
$896$	0	0
$897$	7.59647	0.253639
$898$	0	0
$899$	48.1915	1.60728
$900$	0	0
$901$	50.6594	1.68771
$902$	0	0
$903$	−18.6160	−0.619500
$904$	0	0
$905$	−53.8863	−1.79124
$906$	0	0
$907$	−21.1400	−0.701941	−0.350970	−	0.936387i	$-0.614148\pi$
$907$	−21.1400	−0.701941	−0.350970	+	0.936387i	$0.614148\pi$
$908$	0	0
$909$	4.62417	0.153374
$910$	0	0
$911$	32.7997	1.08670	0.543351	−	0.839506i	$-0.317155\pi$
$911$	32.7997	1.08670	0.543351	+	0.839506i	$0.317155\pi$
$912$	0	0
$913$	−16.6842	−0.552167
$914$	0	0
$915$	−35.8901	−1.18649
$916$	0	0
$917$	−57.7460	−1.90694
$918$	0	0
$919$	−52.6430	−1.73653	−0.868266	−	0.496100i	$-0.834765\pi$
$919$	−52.6430	−1.73653	−0.868266	+	0.496100i	$0.834765\pi$
$920$	0	0
$921$	8.57792	0.282652
$922$	0	0
$923$	28.4565	0.936656
$924$	0	0
$925$	123.784	4.06998
$926$	0	0
$927$	−6.98676	−0.229475
$928$	0	0
$929$	20.6578	0.677761	0.338880	−	0.940829i	$-0.389952\pi$
$929$	20.6578	0.677761	0.338880	+	0.940829i	$0.389952\pi$
$930$	0	0
$931$	38.9423	1.27628
$932$	0	0
$933$	0.215903	0.00706834
$934$	0	0
$935$	67.7189	2.21465
$936$	0	0
$937$	−47.5611	−1.55375	−0.776877	−	0.629653i	$-0.783197\pi$
$937$	−47.5611	−1.55375	−0.776877	+	0.629653i	$0.783197\pi$
$938$	0	0
$939$	−3.76380	−0.122827
$940$	0	0
$941$	23.1626	0.755078	0.377539	−	0.925994i	$-0.376770\pi$
$941$	23.1626	0.755078	0.377539	+	0.925994i	$0.376770\pi$
$942$	0	0
$943$	22.0176	0.716991
$944$	0	0
$945$	−16.5391	−0.538018
$946$	0	0
$947$	48.7861	1.58533	0.792667	−	0.609654i	$-0.208692\pi$
$947$	48.7861	1.58533	0.792667	+	0.609654i	$0.208692\pi$
$948$	0	0
$949$	−9.25402	−0.300398
$950$	0	0
$951$	−14.5894	−0.473093
$952$	0	0
$953$	58.9833	1.91066	0.955329	−	0.295546i	$-0.0955015\pi$
$953$	58.9833	1.91066	0.955329	+	0.295546i	$0.0955015\pi$
$954$	0	0
$955$	−2.98518	−0.0965980
$956$	0	0
$957$	−20.9709	−0.677892
$958$	0	0
$959$	37.3043	1.20462
$960$	0	0
$961$	−9.00869	−0.290603
$962$	0	0
$963$	5.18557	0.167103
$964$	0	0
$965$	−96.7219	−3.11359
$966$	0	0
$967$	−43.6000	−1.40208	−0.701040	−	0.713122i	$-0.747281\pi$
$967$	−43.6000	−1.40208	−0.701040	+	0.713122i	$0.747281\pi$
$968$	0	0
$969$	36.2538	1.16464
$970$	0	0
$971$	−5.18657	−0.166445	−0.0832224	−	0.996531i	$-0.526521\pi$
$971$	−5.18657	−0.166445	−0.0832224	+	0.996531i	$0.526521\pi$
$972$	0	0
$973$	68.3022	2.18967
$974$	0	0
$975$	−31.5166	−1.00934
$976$	0	0
$977$	2.45910	0.0786737	0.0393368	−	0.999226i	$-0.487475\pi$
$977$	2.45910	0.0786737	0.0393368	+	0.999226i	$0.487475\pi$
$978$	0	0
$979$	18.4448	0.589499
$980$	0	0
$981$	1.56406	0.0499365
$982$	0	0
$983$	−3.69215	−0.117761	−0.0588807	−	0.998265i	$-0.518753\pi$
$983$	−3.69215	−0.117761	−0.0588807	+	0.998265i	$0.518753\pi$
$984$	0	0
$985$	58.6675	1.86930
$986$	0	0
$987$	32.1974	1.02486
$988$	0	0
$989$	14.4510	0.459514
$990$	0	0
$991$	−47.3494	−1.50410	−0.752052	−	0.659104i	$-0.770936\pi$
$991$	−47.3494	−1.50410	−0.752052	+	0.659104i	$0.770936\pi$
$992$	0	0
$993$	−21.1090	−0.669874
$994$	0	0
$995$	−3.39494	−0.107627
$996$	0	0
$997$	−15.2972	−0.484467	−0.242233	−	0.970218i	$-0.577880\pi$
$997$	−15.2972	−0.484467	−0.242233	+	0.970218i	$0.577880\pi$
$998$	0	0
$999$	−9.76889	−0.309074

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4008.2.a.j.1.2	✓	10
4.3	odd	2		8016.2.a.bd.1.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.j.1.2	✓	10	1.1	even	1	trivial
8016.2.a.bd.1.2		10	4.3	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:	\( N \)	\(=\)	\( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4008.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -4.20371 q^{5} -3.93441 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -4.20371 q^{5} -3.93441 q^{7} +1.00000 q^{9} +2.04067 q^{11} +2.48726 q^{13} +4.20371 q^{15} -7.89415 q^{17} +4.59249 q^{19} +3.93441 q^{21} -3.05415 q^{23} +12.6712 q^{25} -1.00000 q^{27} +10.2765 q^{29} +4.68949 q^{31} -2.04067 q^{33} +16.5391 q^{35} +9.76889 q^{37} -2.48726 q^{39} -7.20906 q^{41} -4.73158 q^{43} -4.20371 q^{45} +8.18355 q^{47} +8.47956 q^{49} +7.89415 q^{51} -6.41734 q^{53} -8.57837 q^{55} -4.59249 q^{57} -9.60011 q^{59} -8.53771 q^{61} -3.93441 q^{63} -10.4557 q^{65} -0.838729 q^{67} +3.05415 q^{69} +11.4409 q^{71} -3.72057 q^{73} -12.6712 q^{75} -8.02881 q^{77} +13.8405 q^{79} +1.00000 q^{81} -8.17588 q^{83} +33.1847 q^{85} -10.2765 q^{87} +9.03864 q^{89} -9.78589 q^{91} -4.68949 q^{93} -19.3055 q^{95} -18.4044 q^{97} +2.04067 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9}+O(q^{10}) \) 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9	\( 10 q - 10 q^{3} - 10 q^{5} + q^{7} + 10 q^{9} - q^{11} - 6 q^{13} + 10 q^{15} - 9 q^{17} + 2 q^{19} - q^{21} + 7 q^{23} + 12 q^{25} - 10 q^{27} - 13 q^{29} + 23 q^{31} + q^{33} + q^{35} - 6 q^{37} + 6 q^{39} - 12 q^{41} - 10 q^{45} + 10 q^{47} + 7 q^{49} + 9 q^{51} - 26 q^{53} + 11 q^{55} - 2 q^{57} - 10 q^{59} - 10 q^{61} + q^{63} - 22 q^{65} - 5 q^{67} - 7 q^{69} + 25 q^{71} - 8 q^{73} - 12 q^{75} - 46 q^{77} + 26 q^{79} + 10 q^{81} - 14 q^{83} + 9 q^{85} + 13 q^{87} - 31 q^{89} - 3 q^{91} - 23 q^{93} - 5 q^{95} - 32 q^{97} - q^{99}+O(q^{100}) \) 10 * q - 10 * q^3 - 10 * q^5 + q^7 + 10 * q^9 - q^11 - 6 * q^13 + 10 * q^15 - 9 * q^17 + 2 * q^19 - q^21 + 7 * q^23 + 12 * q^25 - 10 * q^27 - 13 * q^29 + 23 * q^31 + q^33 + q^35 - 6 * q^37 + 6 * q^39 - 12 * q^41 - 10 * q^45 + 10 * q^47 + 7 * q^49 + 9 * q^51 - 26 * q^53 + 11 * q^55 - 2 * q^57 - 10 * q^59 - 10 * q^61 + q^63 - 22 * q^65 - 5 * q^67 - 7 * q^69 + 25 * q^71 - 8 * q^73 - 12 * q^75 - 46 * q^77 + 26 * q^79 + 10 * q^81 - 14 * q^83 + 9 * q^85 + 13 * q^87 - 31 * q^89 - 3 * q^91 - 23 * q^93 - 5 * q^95 - 32 * q^97 - q^99

Introduction

L-functions

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