LMFDB - Embedded newform 8016.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.21432 q^{5} -0.525428 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.21432 q^{5} -0.525428 q^{7} +1.00000 q^{9} -2.28100 q^{11} -0.474572 q^{13} -3.21432 q^{15} -3.09679 q^{17} +7.33185 q^{19} -0.525428 q^{21} +3.52543 q^{23} +5.33185 q^{25} +1.00000 q^{27} -2.28100 q^{29} +7.56199 q^{31} -2.28100 q^{33} +1.68889 q^{35} +5.02074 q^{37} -0.474572 q^{39} -9.05086 q^{41} -5.95407 q^{43} -3.21432 q^{45} +7.57628 q^{47} -6.72393 q^{49} -3.09679 q^{51} -13.9240 q^{53} +7.33185 q^{55} +7.33185 q^{57} -0.458751 q^{59} +7.61285 q^{61} -0.525428 q^{63} +1.52543 q^{65} -7.77631 q^{67} +3.52543 q^{69} +7.95407 q^{71} +13.0049 q^{73} +5.33185 q^{75} +1.19850 q^{77} -6.19358 q^{79} +1.00000 q^{81} -1.06668 q^{83} +9.95407 q^{85} -2.28100 q^{87} +2.13828 q^{89} +0.249353 q^{91} +7.56199 q^{93} -23.5669 q^{95} -4.95407 q^{97} -2.28100 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} + 5 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^5 + 5 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{5} + 5 q^{7} + 3 q^{9} - 8 q^{13} - 3 q^{15} - 16 q^{17} + 2 q^{19} + 5 q^{21} + 4 q^{23} - 4 q^{25} + 3 q^{27} + 9 q^{31} + 5 q^{35} - 5 q^{37} - 8 q^{39} - 14 q^{41} + 2 q^{43} - 3 q^{45} + 3 q^{47} + 6 q^{49} - 16 q^{51} - 15 q^{53} + 2 q^{55} + 2 q^{57} + 5 q^{59} - 4 q^{61} + 5 q^{63} - 2 q^{65} - 3 q^{67} + 4 q^{69} + 4 q^{71} + 6 q^{73} - 4 q^{75} - 16 q^{77} - 32 q^{79} + 3 q^{81} - 3 q^{83} + 10 q^{85} - 27 q^{89} - 32 q^{91} + 9 q^{93} - 24 q^{95} + 5 q^{97}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 + 5 * q^7 + 3 * q^9 - 8 * q^13 - 3 * q^15 - 16 * q^17 + 2 * q^19 + 5 * q^21 + 4 * q^23 - 4 * q^25 + 3 * q^27 + 9 * q^31 + 5 * q^35 - 5 * q^37 - 8 * q^39 - 14 * q^41 + 2 * q^43 - 3 * q^45 + 3 * q^47 + 6 * q^49 - 16 * q^51 - 15 * q^53 + 2 * q^55 + 2 * q^57 + 5 * q^59 - 4 * q^61 + 5 * q^63 - 2 * q^65 - 3 * q^67 + 4 * q^69 + 4 * q^71 + 6 * q^73 - 4 * q^75 - 16 * q^77 - 32 * q^79 + 3 * q^81 - 3 * q^83 + 10 * q^85 - 27 * q^89 - 32 * q^91 + 9 * q^93 - 24 * q^95 + 5 * q^97

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
$5$	−3.21432	−1.43749	−0.718744	−	0.695275i	$-0.755283\pi$
$5$	−3.21432	−1.43749	−0.718744	+	0.695275i	$0.755283\pi$
$6$	0	0
$7$	−0.525428	−0.198593	−0.0992965	−	0.995058i	$-0.531659\pi$
$7$	−0.525428	−0.198593	−0.0992965	+	0.995058i	$0.531659\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.28100	−0.687746	−0.343873	−	0.939016i	$-0.611739\pi$
$11$	−2.28100	−0.687746	−0.343873	+	0.939016i	$0.611739\pi$
$12$	0	0
$13$	−0.474572	−0.131623	−0.0658114	−	0.997832i	$-0.520964\pi$
$13$	−0.474572	−0.131623	−0.0658114	+	0.997832i	$0.520964\pi$
$14$	0	0
$15$	−3.21432	−0.829934
$16$	0	0
$17$	−3.09679	−0.751081	−0.375541	−	0.926806i	$-0.622543\pi$
$17$	−3.09679	−0.751081	−0.375541	+	0.926806i	$0.622543\pi$
$18$	0	0
$19$	7.33185	1.68204	0.841021	−	0.541002i	$-0.181955\pi$
$19$	7.33185	1.68204	0.841021	+	0.541002i	$0.181955\pi$
$20$	0	0
$21$	−0.525428	−0.114658
$22$	0	0
$23$	3.52543	0.735102	0.367551	−	0.930003i	$-0.380196\pi$
$23$	3.52543	0.735102	0.367551	+	0.930003i	$0.380196\pi$
$24$	0	0
$25$	5.33185	1.06637
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.28100	−0.423570	−0.211785	−	0.977316i	$-0.567928\pi$
$29$	−2.28100	−0.423570	−0.211785	+	0.977316i	$0.567928\pi$
$30$	0	0
$31$	7.56199	1.35817	0.679087	−	0.734058i	$-0.262376\pi$
$31$	7.56199	1.35817	0.679087	+	0.734058i	$0.262376\pi$
$32$	0	0
$33$	−2.28100	−0.397070
$34$	0	0
$35$	1.68889	0.285475
$36$	0	0
$37$	5.02074	0.825405	0.412703	−	0.910866i	$-0.364585\pi$
$37$	5.02074	0.825405	0.412703	+	0.910866i	$0.364585\pi$
$38$	0	0
$39$	−0.474572	−0.0759924
$40$	0	0
$41$	−9.05086	−1.41351	−0.706753	−	0.707460i	$-0.749841\pi$
$41$	−9.05086	−1.41351	−0.706753	+	0.707460i	$0.749841\pi$
$42$	0	0
$43$	−5.95407	−0.907987	−0.453993	−	0.891005i	$-0.650001\pi$
$43$	−5.95407	−0.907987	−0.453993	+	0.891005i	$0.650001\pi$
$44$	0	0
$45$	−3.21432	−0.479162
$46$	0	0
$47$	7.57628	1.10511	0.552557	−	0.833475i	$-0.313652\pi$
$47$	7.57628	1.10511	0.552557	+	0.833475i	$0.313652\pi$
$48$	0	0
$49$	−6.72393	−0.960561
$50$	0	0
$51$	−3.09679	−0.433637
$52$	0	0
$53$	−13.9240	−1.91260	−0.956301	−	0.292383i	$-0.905552\pi$
$53$	−13.9240	−1.91260	−0.956301	+	0.292383i	$0.905552\pi$
$54$	0	0
$55$	7.33185	0.988627
$56$	0	0
$57$	7.33185	0.971127
$58$	0	0
$59$	−0.458751	−0.0597243	−0.0298621	−	0.999554i	$-0.509507\pi$
$59$	−0.458751	−0.0597243	−0.0298621	+	0.999554i	$0.509507\pi$
$60$	0	0
$61$	7.61285	0.974725	0.487363	−	0.873200i	$-0.337959\pi$
$61$	7.61285	0.974725	0.487363	+	0.873200i	$0.337959\pi$
$62$	0	0
$63$	−0.525428	−0.0661977
$64$	0	0
$65$	1.52543	0.189206
$66$	0	0
$67$	−7.77631	−0.950028	−0.475014	−	0.879978i	$-0.657557\pi$
$67$	−7.77631	−0.950028	−0.475014	+	0.879978i	$0.657557\pi$
$68$	0	0
$69$	3.52543	0.424412
$70$	0	0
$71$	7.95407	0.943974	0.471987	−	0.881605i	$-0.343537\pi$
$71$	7.95407	0.943974	0.471987	+	0.881605i	$0.343537\pi$
$72$	0	0
$73$	13.0049	1.52211	0.761056	−	0.648687i	$-0.224681\pi$
$73$	13.0049	1.52211	0.761056	+	0.648687i	$0.224681\pi$
$74$	0	0
$75$	5.33185	0.615669
$76$	0	0
$77$	1.19850	0.136582
$78$	0	0
$79$	−6.19358	−0.696832	−0.348416	−	0.937340i	$-0.613280\pi$
$79$	−6.19358	−0.696832	−0.348416	+	0.937340i	$0.613280\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−1.06668	−0.117083	−0.0585415	−	0.998285i	$-0.518645\pi$
$83$	−1.06668	−0.117083	−0.0585415	+	0.998285i	$0.518645\pi$
$84$	0	0
$85$	9.95407	1.07967
$86$	0	0
$87$	−2.28100	−0.244548
$88$	0	0
$89$	2.13828	0.226657	0.113328	−	0.993558i	$-0.463849\pi$
$89$	2.13828	0.226657	0.113328	+	0.993558i	$0.463849\pi$
$90$	0	0
$91$	0.249353	0.0261393
$92$	0	0
$93$	7.56199	0.784142
$94$	0	0
$95$	−23.5669	−2.41791
$96$	0	0
$97$	−4.95407	−0.503009	−0.251505	−	0.967856i	$-0.580925\pi$
$97$	−4.95407	−0.503009	−0.251505	+	0.967856i	$0.580925\pi$
$98$	0	0
$99$	−2.28100	−0.229249
$100$	0	0
$101$	12.2192	1.21586	0.607930	−	0.793991i	$-0.292000\pi$
$101$	12.2192	1.21586	0.607930	+	0.793991i	$0.292000\pi$
$102$	0	0
$103$	10.0415	0.989417	0.494709	−	0.869059i	$-0.335275\pi$
$103$	10.0415	0.989417	0.494709	+	0.869059i	$0.335275\pi$
$104$	0	0
$105$	1.68889	0.164819
$106$	0	0
$107$	−0.0602231	−0.00582198	−0.00291099	−	0.999996i	$-0.500927\pi$
$107$	−0.0602231	−0.00582198	−0.00291099	+	0.999996i	$0.500927\pi$
$108$	0	0
$109$	12.7699	1.22313	0.611565	−	0.791194i	$-0.290540\pi$
$109$	12.7699	1.22313	0.611565	+	0.791194i	$0.290540\pi$
$110$	0	0
$111$	5.02074	0.476548
$112$	0	0
$113$	−17.1526	−1.61358	−0.806789	−	0.590840i	$-0.798796\pi$
$113$	−17.1526	−1.61358	−0.806789	+	0.590840i	$0.798796\pi$
$114$	0	0
$115$	−11.3319	−1.05670
$116$	0	0
$117$	−0.474572	−0.0438742
$118$	0	0
$119$	1.62714	0.149159
$120$	0	0
$121$	−5.79706	−0.527005
$122$	0	0
$123$	−9.05086	−0.816088
$124$	0	0
$125$	−1.06668	−0.0954065
$126$	0	0
$127$	−0.184208	−0.0163458	−0.00817292	−	0.999967i	$-0.502602\pi$
$127$	−0.184208	−0.0163458	−0.00817292	+	0.999967i	$0.502602\pi$
$128$	0	0
$129$	−5.95407	−0.524226
$130$	0	0
$131$	−17.9541	−1.56865	−0.784327	−	0.620348i	$-0.786992\pi$
$131$	−17.9541	−1.56865	−0.784327	+	0.620348i	$0.786992\pi$
$132$	0	0
$133$	−3.85236	−0.334042
$134$	0	0
$135$	−3.21432	−0.276645
$136$	0	0
$137$	6.95407	0.594126	0.297063	−	0.954858i	$-0.403993\pi$
$137$	6.95407	0.594126	0.297063	+	0.954858i	$0.403993\pi$
$138$	0	0
$139$	−7.34767	−0.623221	−0.311611	−	0.950210i	$-0.600868\pi$
$139$	−7.34767	−0.623221	−0.311611	+	0.950210i	$0.600868\pi$
$140$	0	0
$141$	7.57628	0.638038
$142$	0	0
$143$	1.08250	0.0905230
$144$	0	0
$145$	7.33185	0.608877
$146$	0	0
$147$	−6.72393	−0.554580
$148$	0	0
$149$	−13.4035	−1.09805	−0.549027	−	0.835805i	$-0.685001\pi$
$149$	−13.4035	−1.09805	−0.549027	+	0.835805i	$0.685001\pi$
$150$	0	0
$151$	−16.3368	−1.32947	−0.664734	−	0.747080i	$-0.731455\pi$
$151$	−16.3368	−1.32947	−0.664734	+	0.747080i	$0.731455\pi$
$152$	0	0
$153$	−3.09679	−0.250360
$154$	0	0
$155$	−24.3067	−1.95236
$156$	0	0
$157$	−19.2859	−1.53918	−0.769592	−	0.638536i	$-0.779540\pi$
$157$	−19.2859	−1.53918	−0.769592	+	0.638536i	$0.779540\pi$
$158$	0	0
$159$	−13.9240	−1.10424
$160$	0	0
$161$	−1.85236	−0.145986
$162$	0	0
$163$	8.97481	0.702961	0.351481	−	0.936195i	$-0.385678\pi$
$163$	8.97481	0.702961	0.351481	+	0.936195i	$0.385678\pi$
$164$	0	0
$165$	7.33185	0.570784
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−12.7748	−0.982675
$170$	0	0
$171$	7.33185	0.560681
$172$	0	0
$173$	9.84791	0.748723	0.374361	−	0.927283i	$-0.377862\pi$
$173$	9.84791	0.748723	0.374361	+	0.927283i	$0.377862\pi$
$174$	0	0
$175$	−2.80150	−0.211774
$176$	0	0
$177$	−0.458751	−0.0344818
$178$	0	0
$179$	−6.84299	−0.511469	−0.255735	−	0.966747i	$-0.582317\pi$
$179$	−6.84299	−0.511469	−0.255735	+	0.966747i	$0.582317\pi$
$180$	0	0
$181$	−14.5161	−1.07897	−0.539485	−	0.841995i	$-0.681381\pi$
$181$	−14.5161	−1.07897	−0.539485	+	0.841995i	$0.681381\pi$
$182$	0	0
$183$	7.61285	0.562758
$184$	0	0
$185$	−16.1383	−1.18651
$186$	0	0
$187$	7.06376	0.516553
$188$	0	0
$189$	−0.525428	−0.0382192
$190$	0	0
$191$	7.66370	0.554526	0.277263	−	0.960794i	$-0.410573\pi$
$191$	7.66370	0.554526	0.277263	+	0.960794i	$0.410573\pi$
$192$	0	0
$193$	−17.3733	−1.25056	−0.625280	−	0.780400i	$-0.715015\pi$
$193$	−17.3733	−1.25056	−0.625280	+	0.780400i	$0.715015\pi$
$194$	0	0
$195$	1.52543	0.109238
$196$	0	0
$197$	−22.4558	−1.59991	−0.799956	−	0.600059i	$-0.795144\pi$
$197$	−22.4558	−1.59991	−0.799956	+	0.600059i	$0.795144\pi$
$198$	0	0
$199$	−12.9906	−0.920881	−0.460441	−	0.887690i	$-0.652309\pi$
$199$	−12.9906	−0.920881	−0.460441	+	0.887690i	$0.652309\pi$
$200$	0	0
$201$	−7.77631	−0.548499
$202$	0	0
$203$	1.19850	0.0841181
$204$	0	0
$205$	29.0923	2.03190
$206$	0	0
$207$	3.52543	0.245034
$208$	0	0
$209$	−16.7239	−1.15682
$210$	0	0
$211$	−26.4701	−1.82228	−0.911139	−	0.412098i	$-0.864796\pi$
$211$	−26.4701	−1.82228	−0.911139	+	0.412098i	$0.864796\pi$
$212$	0	0
$213$	7.95407	0.545004
$214$	0	0
$215$	19.1383	1.30522
$216$	0	0
$217$	−3.97328	−0.269724
$218$	0	0
$219$	13.0049	0.878791
$220$	0	0
$221$	1.46965	0.0988594
$222$	0	0
$223$	3.23951	0.216934	0.108467	−	0.994100i	$-0.465406\pi$
$223$	3.23951	0.216934	0.108467	+	0.994100i	$0.465406\pi$
$224$	0	0
$225$	5.33185	0.355457
$226$	0	0
$227$	−18.6336	−1.23675	−0.618377	−	0.785881i	$-0.712210\pi$
$227$	−18.6336	−1.23675	−0.618377	+	0.785881i	$0.712210\pi$
$228$	0	0
$229$	3.73329	0.246703	0.123351	−	0.992363i	$-0.460636\pi$
$229$	3.73329	0.246703	0.123351	+	0.992363i	$0.460636\pi$
$230$	0	0
$231$	1.19850	0.0788554
$232$	0	0
$233$	−15.1334	−0.991419	−0.495709	−	0.868488i	$-0.665092\pi$
$233$	−15.1334	−0.991419	−0.495709	+	0.868488i	$0.665092\pi$
$234$	0	0
$235$	−24.3526	−1.58859
$236$	0	0
$237$	−6.19358	−0.402316
$238$	0	0
$239$	−11.0923	−0.717504	−0.358752	−	0.933433i	$-0.616798\pi$
$239$	−11.0923	−0.717504	−0.358752	+	0.933433i	$0.616798\pi$
$240$	0	0
$241$	14.8430	0.956121	0.478060	−	0.878327i	$-0.341340\pi$
$241$	14.8430	0.956121	0.478060	+	0.878327i	$0.341340\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	21.6128	1.38079
$246$	0	0
$247$	−3.47949	−0.221395
$248$	0	0
$249$	−1.06668	−0.0675979
$250$	0	0
$251$	−7.16992	−0.452561	−0.226281	−	0.974062i	$-0.572657\pi$
$251$	−7.16992	−0.452561	−0.226281	+	0.974062i	$0.572657\pi$
$252$	0	0
$253$	−8.04149	−0.505564
$254$	0	0
$255$	9.95407	0.623348
$256$	0	0
$257$	22.9447	1.43125	0.715626	−	0.698484i	$-0.246142\pi$
$257$	22.9447	1.43125	0.715626	+	0.698484i	$0.246142\pi$
$258$	0	0
$259$	−2.63804	−0.163920
$260$	0	0
$261$	−2.28100	−0.141190
$262$	0	0
$263$	25.6494	1.58161	0.790805	−	0.612068i	$-0.209662\pi$
$263$	25.6494	1.58161	0.790805	+	0.612068i	$0.209662\pi$
$264$	0	0
$265$	44.7560	2.74934
$266$	0	0
$267$	2.13828	0.130860
$268$	0	0
$269$	−4.03011	−0.245720	−0.122860	−	0.992424i	$-0.539207\pi$
$269$	−4.03011	−0.245720	−0.122860	+	0.992424i	$0.539207\pi$
$270$	0	0
$271$	−17.2716	−1.04918	−0.524588	−	0.851356i	$-0.675781\pi$
$271$	−17.2716	−1.04918	−0.524588	+	0.851356i	$0.675781\pi$
$272$	0	0
$273$	0.249353	0.0150916
$274$	0	0
$275$	−12.1619	−0.733392
$276$	0	0
$277$	−0.0760445	−0.00456907	−0.00228453	−	0.999997i	$-0.500727\pi$
$277$	−0.0760445	−0.00456907	−0.00228453	+	0.999997i	$0.500727\pi$
$278$	0	0
$279$	7.56199	0.452725
$280$	0	0
$281$	2.93978	0.175372	0.0876862	−	0.996148i	$-0.472053\pi$
$281$	2.93978	0.175372	0.0876862	+	0.996148i	$0.472053\pi$
$282$	0	0
$283$	20.3082	1.20720	0.603598	−	0.797289i	$-0.293733\pi$
$283$	20.3082	1.20720	0.603598	+	0.797289i	$0.293733\pi$
$284$	0	0
$285$	−23.5669	−1.39598
$286$	0	0
$287$	4.75557	0.280712
$288$	0	0
$289$	−7.40990	−0.435877
$290$	0	0
$291$	−4.95407	−0.290413
$292$	0	0
$293$	−1.53972	−0.0899513	−0.0449756	−	0.998988i	$-0.514321\pi$
$293$	−1.53972	−0.0899513	−0.0449756	+	0.998988i	$0.514321\pi$
$294$	0	0
$295$	1.47457	0.0858529
$296$	0	0
$297$	−2.28100	−0.132357
$298$	0	0
$299$	−1.67307	−0.0967562
$300$	0	0
$301$	3.12843	0.180320
$302$	0	0
$303$	12.2192	0.701977
$304$	0	0
$305$	−24.4701	−1.40116
$306$	0	0
$307$	−1.68889	−0.0963902	−0.0481951	−	0.998838i	$-0.515347\pi$
$307$	−1.68889	−0.0963902	−0.0481951	+	0.998838i	$0.515347\pi$
$308$	0	0
$309$	10.0415	0.571240
$310$	0	0
$311$	−2.67307	−0.151576	−0.0757880	−	0.997124i	$-0.524147\pi$
$311$	−2.67307	−0.151576	−0.0757880	+	0.997124i	$0.524147\pi$
$312$	0	0
$313$	−21.9541	−1.24092	−0.620459	−	0.784239i	$-0.713054\pi$
$313$	−21.9541	−1.24092	−0.620459	+	0.784239i	$0.713054\pi$
$314$	0	0
$315$	1.68889	0.0951583
$316$	0	0
$317$	−13.2400	−0.743632	−0.371816	−	0.928307i	$-0.621265\pi$
$317$	−13.2400	−0.743632	−0.371816	+	0.928307i	$0.621265\pi$
$318$	0	0
$319$	5.20294	0.291309
$320$	0	0
$321$	−0.0602231	−0.00336132
$322$	0	0
$323$	−22.7052	−1.26335
$324$	0	0
$325$	−2.53035	−0.140359
$326$	0	0
$327$	12.7699	0.706175
$328$	0	0
$329$	−3.98079	−0.219468
$330$	0	0
$331$	1.92549	0.105834	0.0529172	−	0.998599i	$-0.483148\pi$
$331$	1.92549	0.105834	0.0529172	+	0.998599i	$0.483148\pi$
$332$	0	0
$333$	5.02074	0.275135
$334$	0	0
$335$	24.9956	1.36565
$336$	0	0
$337$	13.5620	0.738769	0.369384	−	0.929277i	$-0.379569\pi$
$337$	13.5620	0.738769	0.369384	+	0.929277i	$0.379569\pi$
$338$	0	0
$339$	−17.1526	−0.931599
$340$	0	0
$341$	−17.2489	−0.934079
$342$	0	0
$343$	7.21093	0.389354
$344$	0	0
$345$	−11.3319	−0.610086
$346$	0	0
$347$	−15.6015	−0.837531	−0.418765	−	0.908094i	$-0.637537\pi$
$347$	−15.6015	−0.837531	−0.418765	+	0.908094i	$0.637537\pi$
$348$	0	0
$349$	−22.2652	−1.19183	−0.595914	−	0.803048i	$-0.703210\pi$
$349$	−22.2652	−1.19183	−0.595914	+	0.803048i	$0.703210\pi$
$350$	0	0
$351$	−0.474572	−0.0253308
$352$	0	0
$353$	−6.14272	−0.326944	−0.163472	−	0.986548i	$-0.552269\pi$
$353$	−6.14272	−0.326944	−0.163472	+	0.986548i	$0.552269\pi$
$354$	0	0
$355$	−25.5669	−1.35695
$356$	0	0
$357$	1.62714	0.0861173
$358$	0	0
$359$	21.9684	1.15945	0.579723	−	0.814814i	$-0.303161\pi$
$359$	21.9684	1.15945	0.579723	+	0.814814i	$0.303161\pi$
$360$	0	0
$361$	34.7560	1.82927
$362$	0	0
$363$	−5.79706	−0.304267
$364$	0	0
$365$	−41.8020	−2.18802
$366$	0	0
$367$	−11.0923	−0.579016	−0.289508	−	0.957176i	$-0.593492\pi$
$367$	−11.0923	−0.579016	−0.289508	+	0.957176i	$0.593492\pi$
$368$	0	0
$369$	−9.05086	−0.471169
$370$	0	0
$371$	7.31603	0.379829
$372$	0	0
$373$	18.0988	0.937120	0.468560	−	0.883432i	$-0.344773\pi$
$373$	18.0988	0.937120	0.468560	+	0.883432i	$0.344773\pi$
$374$	0	0
$375$	−1.06668	−0.0550829
$376$	0	0
$377$	1.08250	0.0557515
$378$	0	0
$379$	12.4543	0.639735	0.319867	−	0.947462i	$-0.396362\pi$
$379$	12.4543	0.639735	0.319867	+	0.947462i	$0.396362\pi$
$380$	0	0
$381$	−0.184208	−0.00943727
$382$	0	0
$383$	−8.42864	−0.430683	−0.215342	−	0.976539i	$-0.569087\pi$
$383$	−8.42864	−0.430683	−0.215342	+	0.976539i	$0.569087\pi$
$384$	0	0
$385$	−3.85236	−0.196334
$386$	0	0
$387$	−5.95407	−0.302662
$388$	0	0
$389$	−30.5892	−1.55093	−0.775467	−	0.631388i	$-0.782485\pi$
$389$	−30.5892	−1.55093	−0.775467	+	0.631388i	$0.782485\pi$
$390$	0	0
$391$	−10.9175	−0.552122
$392$	0	0
$393$	−17.9541	−0.905663
$394$	0	0
$395$	19.9081	1.00169
$396$	0	0
$397$	−17.3733	−0.871943	−0.435971	−	0.899961i	$-0.643595\pi$
$397$	−17.3733	−0.871943	−0.435971	+	0.899961i	$0.643595\pi$
$398$	0	0
$399$	−3.85236	−0.192859
$400$	0	0
$401$	23.2672	1.16191	0.580954	−	0.813937i	$-0.302680\pi$
$401$	23.2672	1.16191	0.580954	+	0.813937i	$0.302680\pi$
$402$	0	0
$403$	−3.58871	−0.178767
$404$	0	0
$405$	−3.21432	−0.159721
$406$	0	0
$407$	−11.4523	−0.567669
$408$	0	0
$409$	28.5718	1.41279	0.706393	−	0.707820i	$-0.250321\pi$
$409$	28.5718	1.41279	0.706393	+	0.707820i	$0.250321\pi$
$410$	0	0
$411$	6.95407	0.343019
$412$	0	0
$413$	0.241040	0.0118608
$414$	0	0
$415$	3.42864	0.168305
$416$	0	0
$417$	−7.34767	−0.359817
$418$	0	0
$419$	−8.47013	−0.413793	−0.206896	−	0.978363i	$-0.566336\pi$
$419$	−8.47013	−0.413793	−0.206896	+	0.978363i	$0.566336\pi$
$420$	0	0
$421$	−24.9032	−1.21371	−0.606854	−	0.794813i	$-0.707569\pi$
$421$	−24.9032	−1.21371	−0.606854	+	0.794813i	$0.707569\pi$
$422$	0	0
$423$	7.57628	0.368371
$424$	0	0
$425$	−16.5116	−0.800931
$426$	0	0
$427$	−4.00000	−0.193574
$428$	0	0
$429$	1.08250	0.0522635
$430$	0	0
$431$	−24.8163	−1.19536	−0.597679	−	0.801736i	$-0.703910\pi$
$431$	−24.8163	−1.19536	−0.597679	+	0.801736i	$0.703910\pi$
$432$	0	0
$433$	−10.2761	−0.493837	−0.246918	−	0.969036i	$-0.579418\pi$
$433$	−10.2761	−0.493837	−0.246918	+	0.969036i	$0.579418\pi$
$434$	0	0
$435$	7.33185	0.351535
$436$	0	0
$437$	25.8479	1.23647
$438$	0	0
$439$	12.9906	0.620009	0.310005	−	0.950735i	$-0.399669\pi$
$439$	12.9906	0.620009	0.310005	+	0.950735i	$0.399669\pi$
$440$	0	0
$441$	−6.72393	−0.320187
$442$	0	0
$443$	−12.1318	−0.576400	−0.288200	−	0.957570i	$-0.593057\pi$
$443$	−12.1318	−0.576400	−0.288200	+	0.957570i	$0.593057\pi$
$444$	0	0
$445$	−6.87310	−0.325816
$446$	0	0
$447$	−13.4035	−0.633961
$448$	0	0
$449$	2.43356	0.114847	0.0574234	−	0.998350i	$-0.481711\pi$
$449$	2.43356	0.114847	0.0574234	+	0.998350i	$0.481711\pi$
$450$	0	0
$451$	20.6450	0.972134
$452$	0	0
$453$	−16.3368	−0.767569
$454$	0	0
$455$	−0.801502	−0.0375750
$456$	0	0
$457$	1.43801	0.0672671	0.0336336	−	0.999434i	$-0.489292\pi$
$457$	1.43801	0.0672671	0.0336336	+	0.999434i	$0.489292\pi$
$458$	0	0
$459$	−3.09679	−0.144546
$460$	0	0
$461$	−10.8015	−0.503076	−0.251538	−	0.967847i	$-0.580936\pi$
$461$	−10.8015	−0.503076	−0.251538	+	0.967847i	$0.580936\pi$
$462$	0	0
$463$	19.9813	0.928608	0.464304	−	0.885676i	$-0.346304\pi$
$463$	19.9813	0.928608	0.464304	+	0.885676i	$0.346304\pi$
$464$	0	0
$465$	−24.3067	−1.12719
$466$	0	0
$467$	6.00000	0.277647	0.138823	−	0.990317i	$-0.455668\pi$
$467$	6.00000	0.277647	0.138823	+	0.990317i	$0.455668\pi$
$468$	0	0
$469$	4.08589	0.188669
$470$	0	0
$471$	−19.2859	−0.888648
$472$	0	0
$473$	13.5812	0.624464
$474$	0	0
$475$	39.0923	1.79368
$476$	0	0
$477$	−13.9240	−0.637534
$478$	0	0
$479$	−21.3145	−0.973884	−0.486942	−	0.873434i	$-0.661888\pi$
$479$	−21.3145	−0.973884	−0.486942	+	0.873434i	$0.661888\pi$
$480$	0	0
$481$	−2.38271	−0.108642
$482$	0	0
$483$	−1.85236	−0.0842852
$484$	0	0
$485$	15.9240	0.723070
$486$	0	0
$487$	−12.5575	−0.569037	−0.284518	−	0.958671i	$-0.591834\pi$
$487$	−12.5575	−0.569037	−0.284518	+	0.958671i	$0.591834\pi$
$488$	0	0
$489$	8.97481	0.405855
$490$	0	0
$491$	23.7877	1.07352	0.536762	−	0.843734i	$-0.319647\pi$
$491$	23.7877	1.07352	0.536762	+	0.843734i	$0.319647\pi$
$492$	0	0
$493$	7.06376	0.318136
$494$	0	0
$495$	7.33185	0.329542
$496$	0	0
$497$	−4.17929	−0.187467
$498$	0	0
$499$	14.4242	0.645716	0.322858	−	0.946447i	$-0.395356\pi$
$499$	14.4242	0.645716	0.322858	+	0.946447i	$0.395356\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	22.9175	1.02184	0.510920	−	0.859628i	$-0.329305\pi$
$503$	22.9175	1.02184	0.510920	+	0.859628i	$0.329305\pi$
$504$	0	0
$505$	−39.2766	−1.74778
$506$	0	0
$507$	−12.7748	−0.567348
$508$	0	0
$509$	28.0415	1.24292	0.621459	−	0.783447i	$-0.286540\pi$
$509$	28.0415	1.24292	0.621459	+	0.783447i	$0.286540\pi$
$510$	0	0
$511$	−6.83314	−0.302281
$512$	0	0
$513$	7.33185	0.323709
$514$	0	0
$515$	−32.2766	−1.42227
$516$	0	0
$517$	−17.2815	−0.760038
$518$	0	0
$519$	9.84791	0.432275
$520$	0	0
$521$	−1.25872	−0.0551456	−0.0275728	−	0.999620i	$-0.508778\pi$
$521$	−1.25872	−0.0551456	−0.0275728	+	0.999620i	$0.508778\pi$
$522$	0	0
$523$	6.23951	0.272835	0.136417	−	0.990651i	$-0.456441\pi$
$523$	6.23951	0.272835	0.136417	+	0.990651i	$0.456441\pi$
$524$	0	0
$525$	−2.80150	−0.122268
$526$	0	0
$527$	−23.4179	−1.02010
$528$	0	0
$529$	−10.5714	−0.459624
$530$	0	0
$531$	−0.458751	−0.0199081
$532$	0	0
$533$	4.29529	0.186050
$534$	0	0
$535$	0.193576	0.00836903
$536$	0	0
$537$	−6.84299	−0.295297
$538$	0	0
$539$	15.3372	0.660622
$540$	0	0
$541$	−2.67952	−0.115202	−0.0576009	−	0.998340i	$-0.518345\pi$
$541$	−2.67952	−0.115202	−0.0576009	+	0.998340i	$0.518345\pi$
$542$	0	0
$543$	−14.5161	−0.622944
$544$	0	0
$545$	−41.0464	−1.75823
$546$	0	0
$547$	4.84590	0.207196	0.103598	−	0.994619i	$-0.466964\pi$
$547$	4.84590	0.207196	0.103598	+	0.994619i	$0.466964\pi$
$548$	0	0
$549$	7.61285	0.324908
$550$	0	0
$551$	−16.7239	−0.712463
$552$	0	0
$553$	3.25428	0.138386
$554$	0	0
$555$	−16.1383	−0.685032
$556$	0	0
$557$	−6.73822	−0.285507	−0.142754	−	0.989758i	$-0.545596\pi$
$557$	−6.73822	−0.285507	−0.142754	+	0.989758i	$0.545596\pi$
$558$	0	0
$559$	2.82564	0.119512
$560$	0	0
$561$	7.06376	0.298232
$562$	0	0
$563$	−9.99555	−0.421262	−0.210631	−	0.977566i	$-0.567552\pi$
$563$	−9.99555	−0.421262	−0.210631	+	0.977566i	$0.567552\pi$
$564$	0	0
$565$	55.1338	2.31950
$566$	0	0
$567$	−0.525428	−0.0220659
$568$	0	0
$569$	−11.0968	−0.465202	−0.232601	−	0.972572i	$-0.574724\pi$
$569$	−11.0968	−0.465202	−0.232601	+	0.972572i	$0.574724\pi$
$570$	0	0
$571$	−23.4449	−0.981140	−0.490570	−	0.871402i	$-0.663211\pi$
$571$	−23.4449	−0.981140	−0.490570	+	0.871402i	$0.663211\pi$
$572$	0	0
$573$	7.66370	0.320156
$574$	0	0
$575$	18.7971	0.783891
$576$	0	0
$577$	9.16055	0.381359	0.190679	−	0.981652i	$-0.438931\pi$
$577$	9.16055	0.381359	0.190679	+	0.981652i	$0.438931\pi$
$578$	0	0
$579$	−17.3733	−0.722011
$580$	0	0
$581$	0.560461	0.0232519
$582$	0	0
$583$	31.7605	1.31539
$584$	0	0
$585$	1.52543	0.0630687
$586$	0	0
$587$	−41.1684	−1.69920	−0.849601	−	0.527427i	$-0.823157\pi$
$587$	−41.1684	−1.69920	−0.849601	+	0.527427i	$0.823157\pi$
$588$	0	0
$589$	55.4434	2.28451
$590$	0	0
$591$	−22.4558	−0.923710
$592$	0	0
$593$	−22.6780	−0.931274	−0.465637	−	0.884976i	$-0.654175\pi$
$593$	−22.6780	−0.931274	−0.465637	+	0.884976i	$0.654175\pi$
$594$	0	0
$595$	−5.23014	−0.214415
$596$	0	0
$597$	−12.9906	−0.531671
$598$	0	0
$599$	44.2627	1.80853	0.904263	−	0.426975i	$-0.140421\pi$
$599$	44.2627	1.80853	0.904263	+	0.426975i	$0.140421\pi$
$600$	0	0
$601$	−34.3230	−1.40006	−0.700031	−	0.714112i	$-0.746831\pi$
$601$	−34.3230	−1.40006	−0.700031	+	0.714112i	$0.746831\pi$
$602$	0	0
$603$	−7.77631	−0.316676
$604$	0	0
$605$	18.6336	0.757563
$606$	0	0
$607$	45.9782	1.86620	0.933099	−	0.359620i	$-0.117094\pi$
$607$	45.9782	1.86620	0.933099	+	0.359620i	$0.117094\pi$
$608$	0	0
$609$	1.19850	0.0485656
$610$	0	0
$611$	−3.59549	−0.145458
$612$	0	0
$613$	−30.3412	−1.22547	−0.612735	−	0.790288i	$-0.709931\pi$
$613$	−30.3412	−1.22547	−0.612735	+	0.790288i	$0.709931\pi$
$614$	0	0
$615$	29.0923	1.17312
$616$	0	0
$617$	23.0192	0.926719	0.463359	−	0.886171i	$-0.346644\pi$
$617$	23.0192	0.926719	0.463359	+	0.886171i	$0.346644\pi$
$618$	0	0
$619$	−25.2212	−1.01373	−0.506864	−	0.862026i	$-0.669195\pi$
$619$	−25.2212	−1.01373	−0.506864	+	0.862026i	$0.669195\pi$
$620$	0	0
$621$	3.52543	0.141471
$622$	0	0
$623$	−1.12351	−0.0450124
$624$	0	0
$625$	−23.2306	−0.929225
$626$	0	0
$627$	−16.7239	−0.667889
$628$	0	0
$629$	−15.5482	−0.619946
$630$	0	0
$631$	−12.4608	−0.496055	−0.248027	−	0.968753i	$-0.579782\pi$
$631$	−12.4608	−0.496055	−0.248027	+	0.968753i	$0.579782\pi$
$632$	0	0
$633$	−26.4701	−1.05209
$634$	0	0
$635$	0.592104	0.0234969
$636$	0	0
$637$	3.19099	0.126432
$638$	0	0
$639$	7.95407	0.314658
$640$	0	0
$641$	−2.08297	−0.0822725	−0.0411363	−	0.999154i	$-0.513098\pi$
$641$	−2.08297	−0.0822725	−0.0411363	+	0.999154i	$0.513098\pi$
$642$	0	0
$643$	13.9541	0.550295	0.275147	−	0.961402i	$-0.411273\pi$
$643$	13.9541	0.550295	0.275147	+	0.961402i	$0.411273\pi$
$644$	0	0
$645$	19.1383	0.753569
$646$	0	0
$647$	17.2859	0.679580	0.339790	−	0.940501i	$-0.389644\pi$
$647$	17.2859	0.679580	0.339790	+	0.940501i	$0.389644\pi$
$648$	0	0
$649$	1.04641	0.0410752
$650$	0	0
$651$	−3.97328	−0.155725
$652$	0	0
$653$	−37.5022	−1.46758	−0.733788	−	0.679378i	$-0.762250\pi$
$653$	−37.5022	−1.46758	−0.733788	+	0.679378i	$0.762250\pi$
$654$	0	0
$655$	57.7101	2.25492
$656$	0	0
$657$	13.0049	0.507370
$658$	0	0
$659$	2.84146	0.110687	0.0553437	−	0.998467i	$-0.482375\pi$
$659$	2.84146	0.110687	0.0553437	+	0.998467i	$0.482375\pi$
$660$	0	0
$661$	−38.9545	−1.51516	−0.757578	−	0.652745i	$-0.773617\pi$
$661$	−38.9545	−1.51516	−0.757578	+	0.652745i	$0.773617\pi$
$662$	0	0
$663$	1.46965	0.0570765
$664$	0	0
$665$	12.3827	0.480181
$666$	0	0
$667$	−8.04149	−0.311368
$668$	0	0
$669$	3.23951	0.125247
$670$	0	0
$671$	−17.3649	−0.670364
$672$	0	0
$673$	8.74128	0.336952	0.168476	−	0.985706i	$-0.446116\pi$
$673$	8.74128	0.336952	0.168476	+	0.985706i	$0.446116\pi$
$674$	0	0
$675$	5.33185	0.205223
$676$	0	0
$677$	20.8113	0.799845	0.399923	−	0.916549i	$-0.369037\pi$
$677$	20.8113	0.799845	0.399923	+	0.916549i	$0.369037\pi$
$678$	0	0
$679$	2.60300	0.0998941
$680$	0	0
$681$	−18.6336	−0.714041
$682$	0	0
$683$	0.493308	0.0188759	0.00943796	−	0.999955i	$-0.496996\pi$
$683$	0.493308	0.0188759	0.00943796	+	0.999955i	$0.496996\pi$
$684$	0	0
$685$	−22.3526	−0.854049
$686$	0	0
$687$	3.73329	0.142434
$688$	0	0
$689$	6.60793	0.251742
$690$	0	0
$691$	11.4652	0.436157	0.218078	−	0.975931i	$-0.430021\pi$
$691$	11.4652	0.436157	0.218078	+	0.975931i	$0.430021\pi$
$692$	0	0
$693$	1.19850	0.0455272
$694$	0	0
$695$	23.6178	0.895873
$696$	0	0
$697$	28.0286	1.06166
$698$	0	0
$699$	−15.1334	−0.572396
$700$	0	0
$701$	16.7368	0.632141	0.316071	−	0.948736i	$-0.397636\pi$
$701$	16.7368	0.632141	0.316071	+	0.948736i	$0.397636\pi$
$702$	0	0
$703$	36.8113	1.38837
$704$	0	0
$705$	−24.3526	−0.917172
$706$	0	0
$707$	−6.42033	−0.241461
$708$	0	0
$709$	16.8227	0.631791	0.315895	−	0.948794i	$-0.397695\pi$
$709$	16.8227	0.631791	0.315895	+	0.948794i	$0.397695\pi$
$710$	0	0
$711$	−6.19358	−0.232277
$712$	0	0
$713$	26.6593	0.998397
$714$	0	0
$715$	−3.47949	−0.130126
$716$	0	0
$717$	−11.0923	−0.414251
$718$	0	0
$719$	−30.3684	−1.13255	−0.566275	−	0.824216i	$-0.691616\pi$
$719$	−30.3684	−1.13255	−0.566275	+	0.824216i	$0.691616\pi$
$720$	0	0
$721$	−5.27607	−0.196491
$722$	0	0
$723$	14.8430	0.552017
$724$	0	0
$725$	−12.1619	−0.451683
$726$	0	0
$727$	43.5669	1.61581	0.807904	−	0.589315i	$-0.200602\pi$
$727$	43.5669	1.61581	0.807904	+	0.589315i	$0.200602\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	18.4385	0.681972
$732$	0	0
$733$	−1.85236	−0.0684184	−0.0342092	−	0.999415i	$-0.510891\pi$
$733$	−1.85236	−0.0684184	−0.0342092	+	0.999415i	$0.510891\pi$
$734$	0	0
$735$	21.6128	0.797202
$736$	0	0
$737$	17.7377	0.653378
$738$	0	0
$739$	−3.89231	−0.143181	−0.0715905	−	0.997434i	$-0.522807\pi$
$739$	−3.89231	−0.143181	−0.0715905	+	0.997434i	$0.522807\pi$
$740$	0	0
$741$	−3.47949	−0.127822
$742$	0	0
$743$	−0.496847	−0.0182276	−0.00911378	−	0.999958i	$-0.502901\pi$
$743$	−0.496847	−0.0182276	−0.00911378	+	0.999958i	$0.502901\pi$
$744$	0	0
$745$	43.0830	1.57844
$746$	0	0
$747$	−1.06668	−0.0390277
$748$	0	0
$749$	0.0316429	0.00115620
$750$	0	0
$751$	−42.6593	−1.55666	−0.778329	−	0.627856i	$-0.783933\pi$
$751$	−42.6593	−1.55666	−0.778329	+	0.627856i	$0.783933\pi$
$752$	0	0
$753$	−7.16992	−0.261286
$754$	0	0
$755$	52.5116	1.91109
$756$	0	0
$757$	26.4143	0.960046	0.480023	−	0.877256i	$-0.340628\pi$
$757$	26.4143	0.960046	0.480023	+	0.877256i	$0.340628\pi$
$758$	0	0
$759$	−8.04149	−0.291887
$760$	0	0
$761$	−15.9491	−0.578156	−0.289078	−	0.957306i	$-0.593349\pi$
$761$	−15.9491	−0.578156	−0.289078	+	0.957306i	$0.593349\pi$
$762$	0	0
$763$	−6.70964	−0.242905
$764$	0	0
$765$	9.95407	0.359890
$766$	0	0
$767$	0.217711	0.00786107
$768$	0	0
$769$	−10.3541	−0.373379	−0.186690	−	0.982419i	$-0.559776\pi$
$769$	−10.3541	−0.373379	−0.186690	+	0.982419i	$0.559776\pi$
$770$	0	0
$771$	22.9447	0.826333
$772$	0	0
$773$	−31.0163	−1.11558	−0.557789	−	0.829983i	$-0.688350\pi$
$773$	−31.0163	−1.11558	−0.557789	+	0.829983i	$0.688350\pi$
$774$	0	0
$775$	40.3194	1.44832
$776$	0	0
$777$	−2.63804	−0.0946391
$778$	0	0
$779$	−66.3595	−2.37758
$780$	0	0
$781$	−18.1432	−0.649215
$782$	0	0
$783$	−2.28100	−0.0815162
$784$	0	0
$785$	61.9911	2.21256
$786$	0	0
$787$	18.0370	0.642951	0.321476	−	0.946918i	$-0.395821\pi$
$787$	18.0370	0.642951	0.321476	+	0.946918i	$0.395821\pi$
$788$	0	0
$789$	25.6494	0.913143
$790$	0	0
$791$	9.01243	0.320445
$792$	0	0
$793$	−3.61285	−0.128296
$794$	0	0
$795$	44.7560	1.58733
$796$	0	0
$797$	−48.4943	−1.71775	−0.858877	−	0.512181i	$-0.828838\pi$
$797$	−48.4943	−1.71775	−0.858877	+	0.512181i	$0.828838\pi$
$798$	0	0
$799$	−23.4621	−0.830031
$800$	0	0
$801$	2.13828	0.0755522
$802$	0	0
$803$	−29.6642	−1.04683
$804$	0	0
$805$	5.95407	0.209853
$806$	0	0
$807$	−4.03011	−0.141867
$808$	0	0
$809$	44.0781	1.54970	0.774851	−	0.632144i	$-0.217825\pi$
$809$	44.0781	1.54970	0.774851	+	0.632144i	$0.217825\pi$
$810$	0	0
$811$	18.6523	0.654972	0.327486	−	0.944856i	$-0.393799\pi$
$811$	18.6523	0.654972	0.327486	+	0.944856i	$0.393799\pi$
$812$	0	0
$813$	−17.2716	−0.605742
$814$	0	0
$815$	−28.8479	−1.01050
$816$	0	0
$817$	−43.6543	−1.52727
$818$	0	0
$819$	0.249353	0.00871311
$820$	0	0
$821$	23.1669	0.808529	0.404264	−	0.914642i	$-0.367528\pi$
$821$	23.1669	0.808529	0.404264	+	0.914642i	$0.367528\pi$
$822$	0	0
$823$	−50.2623	−1.75203	−0.876016	−	0.482282i	$-0.839808\pi$
$823$	−50.2623	−1.75203	−0.876016	+	0.482282i	$0.839808\pi$
$824$	0	0
$825$	−12.1619	−0.423424
$826$	0	0
$827$	−49.5036	−1.72141	−0.860705	−	0.509104i	$-0.829977\pi$
$827$	−49.5036	−1.72141	−0.860705	+	0.509104i	$0.829977\pi$
$828$	0	0
$829$	35.0765	1.21826	0.609129	−	0.793071i	$-0.291519\pi$
$829$	35.0765	1.21826	0.609129	+	0.793071i	$0.291519\pi$
$830$	0	0
$831$	−0.0760445	−0.00263795
$832$	0	0
$833$	20.8226	0.721459
$834$	0	0
$835$	3.21432	0.111236
$836$	0	0
$837$	7.56199	0.261381
$838$	0	0
$839$	16.1971	0.559187	0.279593	−	0.960119i	$-0.409800\pi$
$839$	16.1971	0.559187	0.279593	+	0.960119i	$0.409800\pi$
$840$	0	0
$841$	−23.7971	−0.820588
$842$	0	0
$843$	2.93978	0.101251
$844$	0	0
$845$	41.0622	1.41258
$846$	0	0
$847$	3.04593	0.104659
$848$	0	0
$849$	20.3082	0.696975
$850$	0	0
$851$	17.7003	0.606757
$852$	0	0
$853$	40.3497	1.38155	0.690773	−	0.723071i	$-0.257270\pi$
$853$	40.3497	1.38155	0.690773	+	0.723071i	$0.257270\pi$
$854$	0	0
$855$	−23.5669	−0.805971
$856$	0	0
$857$	−13.9590	−0.476830	−0.238415	−	0.971163i	$-0.576628\pi$
$857$	−13.9590	−0.476830	−0.238415	+	0.971163i	$0.576628\pi$
$858$	0	0
$859$	−11.3220	−0.386302	−0.193151	−	0.981169i	$-0.561871\pi$
$859$	−11.3220	−0.386302	−0.193151	+	0.981169i	$0.561871\pi$
$860$	0	0
$861$	4.75557	0.162069
$862$	0	0
$863$	30.7556	1.04693	0.523466	−	0.852047i	$-0.324639\pi$
$863$	30.7556	1.04693	0.523466	+	0.852047i	$0.324639\pi$
$864$	0	0
$865$	−31.6543	−1.07628
$866$	0	0
$867$	−7.40990	−0.251654
$868$	0	0
$869$	14.1275	0.479243
$870$	0	0
$871$	3.69042	0.125045
$872$	0	0
$873$	−4.95407	−0.167670
$874$	0	0
$875$	0.560461	0.0189470
$876$	0	0
$877$	0.566438	0.0191273	0.00956363	−	0.999954i	$-0.496956\pi$
$877$	0.566438	0.0191273	0.00956363	+	0.999954i	$0.496956\pi$
$878$	0	0
$879$	−1.53972	−0.0519334
$880$	0	0
$881$	−32.9906	−1.11148	−0.555741	−	0.831355i	$-0.687566\pi$
$881$	−32.9906	−1.11148	−0.555741	+	0.831355i	$0.687566\pi$
$882$	0	0
$883$	55.1798	1.85695	0.928473	−	0.371399i	$-0.121122\pi$
$883$	55.1798	1.85695	0.928473	+	0.371399i	$0.121122\pi$
$884$	0	0
$885$	1.47457	0.0495672
$886$	0	0
$887$	30.8988	1.03748	0.518740	−	0.854932i	$-0.326401\pi$
$887$	30.8988	1.03748	0.518740	+	0.854932i	$0.326401\pi$
$888$	0	0
$889$	0.0967881	0.00324617
$890$	0	0
$891$	−2.28100	−0.0764163
$892$	0	0
$893$	55.5482	1.85885
$894$	0	0
$895$	21.9956	0.735230
$896$	0	0
$897$	−1.67307	−0.0558622
$898$	0	0
$899$	−17.2489	−0.575282
$900$	0	0
$901$	43.1195	1.43652
$902$	0	0
$903$	3.12843	0.104108
$904$	0	0
$905$	46.6593	1.55101
$906$	0	0
$907$	43.3131	1.43819	0.719094	−	0.694913i	$-0.244557\pi$
$907$	43.3131	1.43819	0.719094	+	0.694913i	$0.244557\pi$
$908$	0	0
$909$	12.2192	0.405287
$910$	0	0
$911$	57.2578	1.89704	0.948518	−	0.316723i	$-0.102583\pi$
$911$	57.2578	1.89704	0.948518	+	0.316723i	$0.102583\pi$
$912$	0	0
$913$	2.43309	0.0805234
$914$	0	0
$915$	−24.4701	−0.808957
$916$	0	0
$917$	9.43356	0.311524
$918$	0	0
$919$	52.0879	1.71822	0.859111	−	0.511790i	$-0.171017\pi$
$919$	52.0879	1.71822	0.859111	+	0.511790i	$0.171017\pi$
$920$	0	0
$921$	−1.68889	−0.0556509
$922$	0	0
$923$	−3.77478	−0.124248
$924$	0	0
$925$	26.7699	0.880188
$926$	0	0
$927$	10.0415	0.329806
$928$	0	0
$929$	55.7338	1.82857	0.914283	−	0.405076i	$-0.132755\pi$
$929$	55.7338	1.82857	0.914283	+	0.405076i	$0.132755\pi$
$930$	0	0
$931$	−49.2988	−1.61570
$932$	0	0
$933$	−2.67307	−0.0875124
$934$	0	0
$935$	−22.7052	−0.742539
$936$	0	0
$937$	58.2405	1.90263	0.951316	−	0.308216i	$-0.0997319\pi$
$937$	58.2405	1.90263	0.951316	+	0.308216i	$0.0997319\pi$
$938$	0	0
$939$	−21.9541	−0.716444
$940$	0	0
$941$	−29.9224	−0.975443	−0.487722	−	0.872999i	$-0.662172\pi$
$941$	−29.9224	−0.975443	−0.487722	+	0.872999i	$0.662172\pi$
$942$	0	0
$943$	−31.9081	−1.03907
$944$	0	0
$945$	1.68889	0.0549397
$946$	0	0
$947$	−9.58427	−0.311447	−0.155723	−	0.987801i	$-0.549771\pi$
$947$	−9.58427	−0.311447	−0.155723	+	0.987801i	$0.549771\pi$
$948$	0	0
$949$	−6.17178	−0.200344
$950$	0	0
$951$	−13.2400	−0.429336
$952$	0	0
$953$	9.20834	0.298287	0.149144	−	0.988816i	$-0.452348\pi$
$953$	9.20834	0.298287	0.149144	+	0.988816i	$0.452348\pi$
$954$	0	0
$955$	−24.6336	−0.797124
$956$	0	0
$957$	5.20294	0.168187
$958$	0	0
$959$	−3.65386	−0.117989
$960$	0	0
$961$	26.1837	0.844637
$962$	0	0
$963$	−0.0602231	−0.00194066
$964$	0	0
$965$	55.8435	1.79766
$966$	0	0
$967$	−15.2257	−0.489625	−0.244813	−	0.969570i	$-0.578726\pi$
$967$	−15.2257	−0.489625	−0.244813	+	0.969570i	$0.578726\pi$
$968$	0	0
$969$	−22.7052	−0.729396
$970$	0	0
$971$	−23.6889	−0.760213	−0.380106	−	0.924943i	$-0.624113\pi$
$971$	−23.6889	−0.760213	−0.380106	+	0.924943i	$0.624113\pi$
$972$	0	0
$973$	3.86067	0.123767
$974$	0	0
$975$	−2.53035	−0.0810360
$976$	0	0
$977$	36.3827	1.16399	0.581993	−	0.813194i	$-0.302273\pi$
$977$	36.3827	1.16399	0.581993	+	0.813194i	$0.302273\pi$
$978$	0	0
$979$	−4.87740	−0.155882
$980$	0	0
$981$	12.7699	0.407710
$982$	0	0
$983$	−30.6222	−0.976697	−0.488348	−	0.872649i	$-0.662401\pi$
$983$	−30.6222	−0.976697	−0.488348	+	0.872649i	$0.662401\pi$
$984$	0	0
$985$	72.1802	2.29985
$986$	0	0
$987$	−3.98079	−0.126710
$988$	0	0
$989$	−20.9906	−0.667463
$990$	0	0
$991$	−41.7605	−1.32657	−0.663283	−	0.748369i	$-0.730837\pi$
$991$	−41.7605	−1.32657	−0.663283	+	0.748369i	$0.730837\pi$
$992$	0	0
$993$	1.92549	0.0611035
$994$	0	0
$995$	41.7560	1.32376
$996$	0	0
$997$	7.33185	0.232202	0.116101	−	0.993237i	$-0.462960\pi$
$997$	7.33185	0.232202	0.116101	+	0.993237i	$0.462960\pi$
$998$	0	0
$999$	5.02074	0.158849

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.l.1.1		3
4.3	odd	2		1002.2.a.h.1.1	✓	3
12.11	even	2		3006.2.a.o.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.h.1.1	✓	3	4.3	odd	2
3006.2.a.o.1.3		3	12.11	even	2
8016.2.a.l.1.1		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.21432 q^{5} -0.525428 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.21432 q^{5} -0.525428 q^{7} +1.00000 q^{9} -2.28100 q^{11} -0.474572 q^{13} -3.21432 q^{15} -3.09679 q^{17} +7.33185 q^{19} -0.525428 q^{21} +3.52543 q^{23} +5.33185 q^{25} +1.00000 q^{27} -2.28100 q^{29} +7.56199 q^{31} -2.28100 q^{33} +1.68889 q^{35} +5.02074 q^{37} -0.474572 q^{39} -9.05086 q^{41} -5.95407 q^{43} -3.21432 q^{45} +7.57628 q^{47} -6.72393 q^{49} -3.09679 q^{51} -13.9240 q^{53} +7.33185 q^{55} +7.33185 q^{57} -0.458751 q^{59} +7.61285 q^{61} -0.525428 q^{63} +1.52543 q^{65} -7.77631 q^{67} +3.52543 q^{69} +7.95407 q^{71} +13.0049 q^{73} +5.33185 q^{75} +1.19850 q^{77} -6.19358 q^{79} +1.00000 q^{81} -1.06668 q^{83} +9.95407 q^{85} -2.28100 q^{87} +2.13828 q^{89} +0.249353 q^{91} +7.56199 q^{93} -23.5669 q^{95} -4.95407 q^{97} -2.28100 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} + 5 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^5 + 5 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{5} + 5 q^{7} + 3 q^{9} - 8 q^{13} - 3 q^{15} - 16 q^{17} + 2 q^{19} + 5 q^{21} + 4 q^{23} - 4 q^{25} + 3 q^{27} + 9 q^{31} + 5 q^{35} - 5 q^{37} - 8 q^{39} - 14 q^{41} + 2 q^{43} - 3 q^{45} + 3 q^{47} + 6 q^{49} - 16 q^{51} - 15 q^{53} + 2 q^{55} + 2 q^{57} + 5 q^{59} - 4 q^{61} + 5 q^{63} - 2 q^{65} - 3 q^{67} + 4 q^{69} + 4 q^{71} + 6 q^{73} - 4 q^{75} - 16 q^{77} - 32 q^{79} + 3 q^{81} - 3 q^{83} + 10 q^{85} - 27 q^{89} - 32 q^{91} + 9 q^{93} - 24 q^{95} + 5 q^{97}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 + 5 * q^7 + 3 * q^9 - 8 * q^13 - 3 * q^15 - 16 * q^17 + 2 * q^19 + 5 * q^21 + 4 * q^23 - 4 * q^25 + 3 * q^27 + 9 * q^31 + 5 * q^35 - 5 * q^37 - 8 * q^39 - 14 * q^41 + 2 * q^43 - 3 * q^45 + 3 * q^47 + 6 * q^49 - 16 * q^51 - 15 * q^53 + 2 * q^55 + 2 * q^57 + 5 * q^59 - 4 * q^61 + 5 * q^63 - 2 * q^65 - 3 * q^67 + 4 * q^69 + 4 * q^71 + 6 * q^73 - 4 * q^75 - 16 * q^77 - 32 * q^79 + 3 * q^81 - 3 * q^83 + 10 * q^85 - 27 * q^89 - 32 * q^91 + 9 * q^93 - 24 * q^95 + 5 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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