LMFDB - Embedded newform 1503.2.a.e.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,2,Mod(1,1503)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1503.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1503.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:	$12.0015154238$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 28x^{5} + 9x^{4} - 64x^{3} + 17x^{2} + 23x + 1 $ x^8 - 3x^7 - 8x^6 + 28x^5 + 9x^4 - 64x^3 + 17x^2 + 23*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.459587$ of defining polynomial
Character	$\chi$	$=$	1503.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.459587 q^{2} -1.78878 q^{4} +2.63865 q^{5} -0.604857 q^{7} -1.74127 q^{8} +O(q^{10})$	\(q+0.459587 q^{2} -1.78878 q^{4} +2.63865 q^{5} -0.604857 q^{7} -1.74127 q^{8} +1.21269 q^{10} -4.18757 q^{11} -0.588099 q^{13} -0.277984 q^{14} +2.77729 q^{16} -3.13576 q^{17} +3.33284 q^{19} -4.71996 q^{20} -1.92456 q^{22} -2.06079 q^{23} +1.96248 q^{25} -0.270283 q^{26} +1.08196 q^{28} -10.1474 q^{29} -1.00945 q^{31} +4.75896 q^{32} -1.44116 q^{34} -1.59601 q^{35} -0.963871 q^{37} +1.53173 q^{38} -4.59462 q^{40} -10.0988 q^{41} -1.46622 q^{43} +7.49065 q^{44} -0.947114 q^{46} +3.81359 q^{47} -6.63415 q^{49} +0.901929 q^{50} +1.05198 q^{52} +2.52878 q^{53} -11.0495 q^{55} +1.05322 q^{56} -4.66361 q^{58} +3.70087 q^{59} +0.962658 q^{61} -0.463931 q^{62} -3.36743 q^{64} -1.55179 q^{65} -6.12872 q^{67} +5.60919 q^{68} -0.733504 q^{70} -4.34451 q^{71} +13.8707 q^{73} -0.442983 q^{74} -5.96172 q^{76} +2.53288 q^{77} +9.16426 q^{79} +7.32830 q^{80} -4.64129 q^{82} -15.6712 q^{83} -8.27418 q^{85} -0.673854 q^{86} +7.29172 q^{88} -5.57911 q^{89} +0.355716 q^{91} +3.68630 q^{92} +1.75268 q^{94} +8.79421 q^{95} -6.60693 q^{97} -3.04897 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 3 q^{2} + 9 q^{4} - 7 q^{5} - 4 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - 3 * q^2 + 9 * q^4 - 7 * q^5 - 4 * q^7 - 3 * q^8	$ 8 q - 3 q^{2} + 9 q^{4} - 7 q^{5} - 4 q^{7} - 3 q^{8} - q^{10} - 13 q^{11} - 5 q^{14} + 7 q^{16} - 11 q^{17} + 12 q^{19} - 9 q^{20} - 17 q^{22} - 7 q^{23} - 5 q^{25} - 3 q^{26} - 27 q^{28} - q^{29} - 2 q^{31} - 4 q^{32} - 14 q^{34} + 4 q^{35} - 9 q^{37} - 22 q^{40} - 4 q^{41} + 2 q^{43} - 3 q^{44} - 5 q^{46} - 17 q^{47} - 2 q^{49} + 4 q^{50} - 36 q^{52} - 9 q^{53} + 7 q^{55} - 9 q^{56} - 29 q^{58} - 29 q^{59} - 12 q^{61} + 34 q^{62} - 5 q^{64} - 8 q^{65} - 26 q^{68} + 5 q^{70} - 13 q^{71} - 20 q^{73} + 17 q^{74} + 30 q^{76} + 22 q^{77} + 8 q^{79} + 34 q^{80} + 15 q^{82} - 33 q^{83} - 31 q^{85} - 11 q^{86} - 44 q^{88} - 4 q^{89} + q^{91} + 33 q^{92} - 2 q^{94} - 3 q^{95} - 31 q^{97} + 57 q^{98}+O(q^{100}) $ 8 * q - 3 * q^2 + 9 * q^4 - 7 * q^5 - 4 * q^7 - 3 * q^8 - q^10 - 13 * q^11 - 5 * q^14 + 7 * q^16 - 11 * q^17 + 12 * q^19 - 9 * q^20 - 17 * q^22 - 7 * q^23 - 5 * q^25 - 3 * q^26 - 27 * q^28 - q^29 - 2 * q^31 - 4 * q^32 - 14 * q^34 + 4 * q^35 - 9 * q^37 - 22 * q^40 - 4 * q^41 + 2 * q^43 - 3 * q^44 - 5 * q^46 - 17 * q^47 - 2 * q^49 + 4 * q^50 - 36 * q^52 - 9 * q^53 + 7 * q^55 - 9 * q^56 - 29 * q^58 - 29 * q^59 - 12 * q^61 + 34 * q^62 - 5 * q^64 - 8 * q^65 - 26 * q^68 + 5 * q^70 - 13 * q^71 - 20 * q^73 + 17 * q^74 + 30 * q^76 + 22 * q^77 + 8 * q^79 + 34 * q^80 + 15 * q^82 - 33 * q^83 - 31 * q^85 - 11 * q^86 - 44 * q^88 - 4 * q^89 + q^91 + 33 * q^92 - 2 * q^94 - 3 * q^95 - 31 * q^97 + 57 * q^98

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 $

$2$	0.459587	0.324977	0.162489	−	0.986710i	$-0.448048\pi$
$2$	0.459587	0.324977	0.162489	+	0.986710i	$0.448048\pi$
$3$	0	0
$3$	0	0
$4$	−1.78878	−0.894390
$5$	2.63865	1.18004	0.590020	−	0.807388i	$-0.299120\pi$
$5$	2.63865	1.18004	0.590020	+	0.807388i	$0.299120\pi$
$6$	0	0
$7$	−0.604857	−0.228614	−0.114307	−	0.993445i	$-0.536465\pi$
$7$	−0.604857	−0.228614	−0.114307	+	0.993445i	$0.536465\pi$
$8$	−1.74127	−0.615634
$9$	0	0
$10$	1.21269	0.383486
$11$	−4.18757	−1.26260	−0.631301	−	0.775538i	$-0.717479\pi$
$11$	−4.18757	−1.26260	−0.631301	+	0.775538i	$0.717479\pi$
$12$	0	0
$13$	−0.588099	−0.163109	−0.0815547	−	0.996669i	$-0.525989\pi$
$13$	−0.588099	−0.163109	−0.0815547	+	0.996669i	$0.525989\pi$
$14$	−0.277984	−0.0742945
$15$	0	0
$16$	2.77729	0.694323
$17$	−3.13576	−0.760534	−0.380267	−	0.924877i	$-0.624168\pi$
$17$	−3.13576	−0.760534	−0.380267	+	0.924877i	$0.624168\pi$
$18$	0	0
$19$	3.33284	0.764607	0.382303	−	0.924037i	$-0.375131\pi$
$19$	3.33284	0.764607	0.382303	+	0.924037i	$0.375131\pi$
$20$	−4.71996	−1.05542
$21$	0	0
$22$	−1.92456	−0.410317
$23$	−2.06079	−0.429705	−0.214852	−	0.976647i	$-0.568927\pi$
$23$	−2.06079	−0.429705	−0.214852	+	0.976647i	$0.568927\pi$
$24$	0	0
$25$	1.96248	0.392495
$26$	−0.270283	−0.0530069
$27$	0	0
$28$	1.08196	0.204470
$29$	−10.1474	−1.88432	−0.942162	−	0.335158i	$-0.891210\pi$
$29$	−10.1474	−1.88432	−0.942162	+	0.335158i	$0.891210\pi$
$30$	0	0
$31$	−1.00945	−0.181303	−0.0906515	−	0.995883i	$-0.528895\pi$
$31$	−1.00945	−0.181303	−0.0906515	+	0.995883i	$0.528895\pi$
$32$	4.75896	0.841273
$33$	0	0
$34$	−1.44116	−0.247156
$35$	−1.59601	−0.269774
$36$	0	0
$37$	−0.963871	−0.158459	−0.0792297	−	0.996856i	$-0.525246\pi$
$37$	−0.963871	−0.158459	−0.0792297	+	0.996856i	$0.525246\pi$
$38$	1.53173	0.248480
$39$	0	0
$40$	−4.59462	−0.726473
$41$	−10.0988	−1.57717	−0.788585	−	0.614926i	$-0.789186\pi$
$41$	−10.0988	−1.57717	−0.788585	+	0.614926i	$0.789186\pi$
$42$	0	0
$43$	−1.46622	−0.223596	−0.111798	−	0.993731i	$-0.535661\pi$
$43$	−1.46622	−0.223596	−0.111798	+	0.993731i	$0.535661\pi$
$44$	7.49065	1.12926
$45$	0	0
$46$	−0.947114	−0.139644
$47$	3.81359	0.556270	0.278135	−	0.960542i	$-0.410284\pi$
$47$	3.81359	0.556270	0.278135	+	0.960542i	$0.410284\pi$
$48$	0	0
$49$	−6.63415	−0.947735
$50$	0.901929	0.127552
$51$	0	0
$52$	1.05198	0.145883
$53$	2.52878	0.347354	0.173677	−	0.984803i	$-0.444435\pi$
$53$	2.52878	0.347354	0.173677	+	0.984803i	$0.444435\pi$
$54$	0	0
$55$	−11.0495	−1.48992
$56$	1.05322	0.140743
$57$	0	0
$58$	−4.66361	−0.612362
$59$	3.70087	0.481812	0.240906	−	0.970548i	$-0.422555\pi$
$59$	3.70087	0.481812	0.240906	+	0.970548i	$0.422555\pi$
$60$	0	0
$61$	0.962658	0.123256	0.0616279	−	0.998099i	$-0.480371\pi$
$61$	0.962658	0.123256	0.0616279	+	0.998099i	$0.480371\pi$
$62$	−0.463931	−0.0589193
$63$	0	0
$64$	−3.36743	−0.420928
$65$	−1.55179	−0.192476
$66$	0	0
$67$	−6.12872	−0.748742	−0.374371	−	0.927279i	$-0.622141\pi$
$67$	−6.12872	−0.748742	−0.374371	+	0.927279i	$0.622141\pi$
$68$	5.60919	0.680214
$69$	0	0
$70$	−0.733504	−0.0876705
$71$	−4.34451	−0.515599	−0.257799	−	0.966198i	$-0.582997\pi$
$71$	−4.34451	−0.515599	−0.257799	+	0.966198i	$0.582997\pi$
$72$	0	0
$73$	13.8707	1.62344	0.811721	−	0.584046i	$-0.198531\pi$
$73$	13.8707	1.62344	0.811721	+	0.584046i	$0.198531\pi$
$74$	−0.442983	−0.0514957
$75$	0	0
$76$	−5.96172	−0.683857
$77$	2.53288	0.288649
$78$	0	0
$79$	9.16426	1.03106	0.515530	−	0.856872i	$-0.327595\pi$
$79$	9.16426	1.03106	0.515530	+	0.856872i	$0.327595\pi$
$80$	7.32830	0.819329
$81$	0	0
$82$	−4.64129	−0.512544
$83$	−15.6712	−1.72013	−0.860067	−	0.510182i	$-0.829578\pi$
$83$	−15.6712	−1.72013	−0.860067	+	0.510182i	$0.829578\pi$
$84$	0	0
$85$	−8.27418	−0.897461
$86$	−0.673854	−0.0726636
$87$	0	0
$88$	7.29172	0.777300
$89$	−5.57911	−0.591384	−0.295692	−	0.955283i	$-0.595550\pi$
$89$	−5.57911	−0.591384	−0.295692	+	0.955283i	$0.595550\pi$
$90$	0	0
$91$	0.355716	0.0372892
$92$	3.68630	0.384324
$93$	0	0
$94$	1.75268	0.180775
$95$	8.79421	0.902267
$96$	0	0
$97$	−6.60693	−0.670832	−0.335416	−	0.942070i	$-0.608877\pi$
$97$	−6.60693	−0.670832	−0.335416	+	0.942070i	$0.608877\pi$
$98$	−3.04897	−0.307992
$99$	0	0
$100$	−3.51044	−0.351044
$101$	−0.212823	−0.0211767	−0.0105883	−	0.999944i	$-0.503370\pi$
$101$	−0.212823	−0.0211767	−0.0105883	+	0.999944i	$0.503370\pi$
$102$	0	0
$103$	−1.50031	−0.147830	−0.0739150	−	0.997265i	$-0.523549\pi$
$103$	−1.50031	−0.147830	−0.0739150	+	0.997265i	$0.523549\pi$
$104$	1.02404	0.100416
$105$	0	0
$106$	1.16219	0.112882
$107$	7.70835	0.745194	0.372597	−	0.927993i	$-0.378467\pi$
$107$	7.70835	0.745194	0.372597	+	0.927993i	$0.378467\pi$
$108$	0	0
$109$	−19.3144	−1.84998	−0.924991	−	0.379988i	$-0.875928\pi$
$109$	−19.3144	−1.84998	−0.924991	+	0.379988i	$0.875928\pi$
$110$	−5.07823	−0.484190
$111$	0	0
$112$	−1.67986	−0.158732
$113$	−2.91046	−0.273793	−0.136896	−	0.990585i	$-0.543713\pi$
$113$	−2.91046	−0.273793	−0.136896	+	0.990585i	$0.543713\pi$
$114$	0	0
$115$	−5.43771	−0.507069
$116$	18.1515	1.68532
$117$	0	0
$118$	1.70087	0.156578
$119$	1.89669	0.173869
$120$	0	0
$121$	6.53578	0.594162
$122$	0.442425	0.0400553
$123$	0	0
$124$	1.80569	0.162156
$125$	−8.01496	−0.716880
$126$	0	0
$127$	−11.5989	−1.02923	−0.514617	−	0.857420i	$-0.672066\pi$
$127$	−11.5989	−1.02923	−0.514617	+	0.857420i	$0.672066\pi$
$128$	−11.0655	−0.978065
$129$	0	0
$130$	−0.713182	−0.0625502
$131$	8.26258	0.721905	0.360952	−	0.932584i	$-0.382452\pi$
$131$	8.26258	0.721905	0.360952	+	0.932584i	$0.382452\pi$
$132$	0	0
$133$	−2.01589	−0.174800
$134$	−2.81668	−0.243324
$135$	0	0
$136$	5.46022	0.468210
$137$	−6.10912	−0.521937	−0.260969	−	0.965347i	$-0.584042\pi$
$137$	−6.10912	−0.521937	−0.260969	+	0.965347i	$0.584042\pi$
$138$	0	0
$139$	14.8050	1.25574	0.627872	−	0.778317i	$-0.283926\pi$
$139$	14.8050	1.25574	0.627872	+	0.778317i	$0.283926\pi$
$140$	2.85490	0.241283
$141$	0	0
$142$	−1.99668	−0.167558
$143$	2.46271	0.205942
$144$	0	0
$145$	−26.7754	−2.22358
$146$	6.37479	0.527582
$147$	0	0
$148$	1.72415	0.141725
$149$	−7.27551	−0.596033	−0.298017	−	0.954561i	$-0.596325\pi$
$149$	−7.27551	−0.596033	−0.298017	+	0.954561i	$0.596325\pi$
$150$	0	0
$151$	−6.77287	−0.551168	−0.275584	−	0.961277i	$-0.588871\pi$
$151$	−6.77287	−0.551168	−0.275584	+	0.961277i	$0.588871\pi$
$152$	−5.80340	−0.470718
$153$	0	0
$154$	1.16408	0.0938043
$155$	−2.66359	−0.213945
$156$	0	0
$157$	−13.7559	−1.09784	−0.548919	−	0.835875i	$-0.684961\pi$
$157$	−13.7559	−1.09784	−0.548919	+	0.835875i	$0.684961\pi$
$158$	4.21178	0.335071
$159$	0	0
$160$	12.5572	0.992736
$161$	1.24648	0.0982367
$162$	0	0
$163$	10.7400	0.841222	0.420611	−	0.907241i	$-0.361816\pi$
$163$	10.7400	0.841222	0.420611	+	0.907241i	$0.361816\pi$
$164$	18.0646	1.41060
$165$	0	0
$166$	−7.20227	−0.559004
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−12.6541	−0.973395
$170$	−3.80271	−0.291654
$171$	0	0
$172$	2.62274	0.199982
$173$	−1.63924	−0.124629	−0.0623145	−	0.998057i	$-0.519848\pi$
$173$	−1.63924	−0.124629	−0.0623145	+	0.998057i	$0.519848\pi$
$174$	0	0
$175$	−1.18702	−0.0897300
$176$	−11.6301	−0.876653
$177$	0	0
$178$	−2.56409	−0.192186
$179$	19.0432	1.42336	0.711678	−	0.702505i	$-0.247935\pi$
$179$	19.0432	1.42336	0.711678	+	0.702505i	$0.247935\pi$
$180$	0	0
$181$	14.7592	1.09704	0.548520	−	0.836138i	$-0.315192\pi$
$181$	14.7592	1.09704	0.548520	+	0.836138i	$0.315192\pi$
$182$	0.163482	0.0121181
$183$	0	0
$184$	3.58841	0.264541
$185$	−2.54332	−0.186989
$186$	0	0
$187$	13.1312	0.960251
$188$	−6.82168	−0.497522
$189$	0	0
$190$	4.04171	0.293216
$191$	8.24196	0.596367	0.298184	−	0.954508i	$-0.403619\pi$
$191$	8.24196	0.596367	0.298184	+	0.954508i	$0.403619\pi$
$192$	0	0
$193$	12.4363	0.895186	0.447593	−	0.894237i	$-0.352281\pi$
$193$	12.4363	0.895186	0.447593	+	0.894237i	$0.352281\pi$
$194$	−3.03646	−0.218005
$195$	0	0
$196$	11.8670	0.847645
$197$	18.4826	1.31683	0.658415	−	0.752655i	$-0.271227\pi$
$197$	18.4826	1.31683	0.658415	+	0.752655i	$0.271227\pi$
$198$	0	0
$199$	4.83561	0.342787	0.171394	−	0.985203i	$-0.445173\pi$
$199$	4.83561	0.342787	0.171394	+	0.985203i	$0.445173\pi$
$200$	−3.41721	−0.241633
$201$	0	0
$202$	−0.0978107	−0.00688193
$203$	6.13772	0.430783
$204$	0	0
$205$	−26.6472	−1.86112
$206$	−0.689523	−0.0480414
$207$	0	0
$208$	−1.63332	−0.113251
$209$	−13.9565	−0.965394
$210$	0	0
$211$	25.4148	1.74963	0.874814	−	0.484459i	$-0.160984\pi$
$211$	25.4148	1.74963	0.874814	+	0.484459i	$0.160984\pi$
$212$	−4.52342	−0.310670
$213$	0	0
$214$	3.54266	0.242171
$215$	−3.86883	−0.263852
$216$	0	0
$217$	0.610574	0.0414485
$218$	−8.87664	−0.601202
$219$	0	0
$220$	19.7652	1.33257
$221$	1.84414	0.124050
$222$	0	0
$223$	13.4950	0.903692	0.451846	−	0.892096i	$-0.350766\pi$
$223$	13.4950	0.903692	0.451846	+	0.892096i	$0.350766\pi$
$224$	−2.87849	−0.192327
$225$	0	0
$226$	−1.33761	−0.0889765
$227$	−22.1793	−1.47209	−0.736047	−	0.676930i	$-0.763310\pi$
$227$	−22.1793	−1.47209	−0.736047	+	0.676930i	$0.763310\pi$
$228$	0	0
$229$	−8.65236	−0.571764	−0.285882	−	0.958265i	$-0.592287\pi$
$229$	−8.65236	−0.571764	−0.285882	+	0.958265i	$0.592287\pi$
$230$	−2.49910	−0.164786
$231$	0	0
$232$	17.6694	1.16005
$233$	15.2852	1.00136	0.500682	−	0.865631i	$-0.333083\pi$
$233$	15.2852	1.00136	0.500682	+	0.865631i	$0.333083\pi$
$234$	0	0
$235$	10.0627	0.656421
$236$	−6.62004	−0.430928
$237$	0	0
$238$	0.871693	0.0565035
$239$	29.8813	1.93286	0.966430	−	0.256930i	$-0.0827111\pi$
$239$	29.8813	1.93286	0.966430	+	0.256930i	$0.0827111\pi$
$240$	0	0
$241$	17.5171	1.12837	0.564187	−	0.825647i	$-0.309190\pi$
$241$	17.5171	1.12837	0.564187	+	0.825647i	$0.309190\pi$
$242$	3.00376	0.193089
$243$	0	0
$244$	−1.72198	−0.110239
$245$	−17.5052	−1.11837
$246$	0	0
$247$	−1.96004	−0.124715
$248$	1.75773	0.111616
$249$	0	0
$250$	−3.68357	−0.232970
$251$	−11.5902	−0.731567	−0.365784	−	0.930700i	$-0.619199\pi$
$251$	−11.5902	−0.731567	−0.365784	+	0.930700i	$0.619199\pi$
$252$	0	0
$253$	8.62972	0.542546
$254$	−5.33069	−0.334477
$255$	0	0
$256$	1.64927	0.103079
$257$	12.3462	0.770137	0.385069	−	0.922888i	$-0.374178\pi$
$257$	12.3462	0.770137	0.385069	+	0.922888i	$0.374178\pi$
$258$	0	0
$259$	0.583004	0.0362261
$260$	2.77581	0.172148
$261$	0	0
$262$	3.79738	0.234603
$263$	10.9134	0.672948	0.336474	−	0.941693i	$-0.390766\pi$
$263$	10.9134	0.672948	0.336474	+	0.941693i	$0.390766\pi$
$264$	0	0
$265$	6.67256	0.409892
$266$	−0.926479	−0.0568061
$267$	0	0
$268$	10.9629	0.669667
$269$	−2.64700	−0.161391	−0.0806953	−	0.996739i	$-0.525714\pi$
$269$	−2.64700	−0.161391	−0.0806953	+	0.996739i	$0.525714\pi$
$270$	0	0
$271$	15.3534	0.932651	0.466326	−	0.884613i	$-0.345578\pi$
$271$	15.3534	0.932651	0.466326	+	0.884613i	$0.345578\pi$
$272$	−8.70892	−0.528056
$273$	0	0
$274$	−2.80767	−0.169618
$275$	−8.21802	−0.495565
$276$	0	0
$277$	18.8696	1.13376	0.566881	−	0.823800i	$-0.308150\pi$
$277$	18.8696	1.13376	0.566881	+	0.823800i	$0.308150\pi$
$278$	6.80419	0.408088
$279$	0	0
$280$	2.77908	0.166082
$281$	2.41769	0.144227	0.0721136	−	0.997396i	$-0.477026\pi$
$281$	2.41769	0.144227	0.0721136	+	0.997396i	$0.477026\pi$
$282$	0	0
$283$	−17.2449	−1.02510	−0.512551	−	0.858657i	$-0.671299\pi$
$283$	−17.2449	−1.02510	−0.512551	+	0.858657i	$0.671299\pi$
$284$	7.77138	0.461146
$285$	0	0
$286$	1.13183	0.0669265
$287$	6.10834	0.360564
$288$	0	0
$289$	−7.16700	−0.421588
$290$	−12.3056	−0.722612
$291$	0	0
$292$	−24.8116	−1.45199
$293$	3.14330	0.183634	0.0918168	−	0.995776i	$-0.470733\pi$
$293$	3.14330	0.183634	0.0918168	+	0.995776i	$0.470733\pi$
$294$	0	0
$295$	9.76531	0.568558
$296$	1.67836	0.0975530
$297$	0	0
$298$	−3.34373	−0.193697
$299$	1.21195	0.0700889
$300$	0	0
$301$	0.886850	0.0511172
$302$	−3.11273	−0.179117
$303$	0	0
$304$	9.25628	0.530884
$305$	2.54012	0.145447
$306$	0	0
$307$	27.5662	1.57329	0.786643	−	0.617409i	$-0.211818\pi$
$307$	27.5662	1.57329	0.786643	+	0.617409i	$0.211818\pi$
$308$	−4.53077	−0.258164
$309$	0	0
$310$	−1.22415	−0.0695272
$311$	−3.97921	−0.225640	−0.112820	−	0.993615i	$-0.535988\pi$
$311$	−3.97921	−0.225640	−0.112820	+	0.993615i	$0.535988\pi$
$312$	0	0
$313$	−15.9773	−0.903090	−0.451545	−	0.892248i	$-0.649127\pi$
$313$	−15.9773	−0.903090	−0.451545	+	0.892248i	$0.649127\pi$
$314$	−6.32203	−0.356773
$315$	0	0
$316$	−16.3928	−0.922169
$317$	25.4618	1.43008	0.715038	−	0.699086i	$-0.246410\pi$
$317$	25.4618	1.43008	0.715038	+	0.699086i	$0.246410\pi$
$318$	0	0
$319$	42.4930	2.37915
$320$	−8.88546	−0.496712
$321$	0	0
$322$	0.572868	0.0319247
$323$	−10.4510	−0.581509
$324$	0	0
$325$	−1.15413	−0.0640197
$326$	4.93597	0.273378
$327$	0	0
$328$	17.5848	0.970959
$329$	−2.30668	−0.127171
$330$	0	0
$331$	−24.0840	−1.32378	−0.661889	−	0.749602i	$-0.730245\pi$
$331$	−24.0840	−1.32378	−0.661889	+	0.749602i	$0.730245\pi$
$332$	28.0323	1.53847
$333$	0	0
$334$	0.459587	0.0251475
$335$	−16.1715	−0.883546
$336$	0	0
$337$	−7.35478	−0.400641	−0.200320	−	0.979730i	$-0.564198\pi$
$337$	−7.35478	−0.400641	−0.200320	+	0.979730i	$0.564198\pi$
$338$	−5.81568	−0.316331
$339$	0	0
$340$	14.8007	0.802680
$341$	4.22716	0.228913
$342$	0	0
$343$	8.24671	0.445280
$344$	2.55308	0.137653
$345$	0	0
$346$	−0.753373	−0.0405016
$347$	−20.8836	−1.12109	−0.560544	−	0.828125i	$-0.689408\pi$
$347$	−20.8836	−1.12109	−0.560544	+	0.828125i	$0.689408\pi$
$348$	0	0
$349$	−26.6111	−1.42446	−0.712229	−	0.701947i	$-0.752314\pi$
$349$	−26.6111	−1.42446	−0.712229	+	0.701947i	$0.752314\pi$
$350$	−0.545538	−0.0291602
$351$	0	0
$352$	−19.9285	−1.06219
$353$	−2.13148	−0.113447	−0.0567237	−	0.998390i	$-0.518065\pi$
$353$	−2.13148	−0.113447	−0.0567237	+	0.998390i	$0.518065\pi$
$354$	0	0
$355$	−11.4637	−0.608428
$356$	9.97979	0.528928
$357$	0	0
$358$	8.75202	0.462559
$359$	−12.5447	−0.662082	−0.331041	−	0.943616i	$-0.607400\pi$
$359$	−12.5447	−0.662082	−0.331041	+	0.943616i	$0.607400\pi$
$360$	0	0
$361$	−7.89215	−0.415376
$362$	6.78312	0.356513
$363$	0	0
$364$	−0.636297	−0.0333510
$365$	36.5999	1.91573
$366$	0	0
$367$	−18.2125	−0.950686	−0.475343	−	0.879800i	$-0.657676\pi$
$367$	−18.2125	−0.950686	−0.475343	+	0.879800i	$0.657676\pi$
$368$	−5.72342	−0.298354
$369$	0	0
$370$	−1.16888	−0.0607670
$371$	−1.52955	−0.0794102
$372$	0	0
$373$	24.0633	1.24595	0.622976	−	0.782241i	$-0.285923\pi$
$373$	24.0633	1.24595	0.622976	+	0.782241i	$0.285923\pi$
$374$	6.03495	0.312060
$375$	0	0
$376$	−6.64052	−0.342458
$377$	5.96768	0.307351
$378$	0	0
$379$	18.1878	0.934243	0.467122	−	0.884193i	$-0.345291\pi$
$379$	18.1878	0.934243	0.467122	+	0.884193i	$0.345291\pi$
$380$	−15.7309	−0.806978
$381$	0	0
$382$	3.78790	0.193806
$383$	21.9159	1.11985	0.559926	−	0.828542i	$-0.310829\pi$
$383$	21.9159	1.11985	0.559926	+	0.828542i	$0.310829\pi$
$384$	0	0
$385$	6.68339	0.340617
$386$	5.71558	0.290915
$387$	0	0
$388$	11.8183	0.599986
$389$	−6.79088	−0.344311	−0.172156	−	0.985070i	$-0.555073\pi$
$389$	−6.79088	−0.344311	−0.172156	+	0.985070i	$0.555073\pi$
$390$	0	0
$391$	6.46215	0.326805
$392$	11.5519	0.583458
$393$	0	0
$394$	8.49436	0.427940
$395$	24.1813	1.21669
$396$	0	0
$397$	−36.3338	−1.82354	−0.911770	−	0.410701i	$-0.865284\pi$
$397$	−36.3338	−1.82354	−0.911770	+	0.410701i	$0.865284\pi$
$398$	2.22238	0.111398
$399$	0	0
$400$	5.45037	0.272518
$401$	26.5238	1.32453	0.662267	−	0.749268i	$-0.269595\pi$
$401$	26.5238	1.32453	0.662267	+	0.749268i	$0.269595\pi$
$402$	0	0
$403$	0.593658	0.0295722
$404$	0.380693	0.0189402
$405$	0	0
$406$	2.82082	0.139995
$407$	4.03628	0.200071
$408$	0	0
$409$	11.5743	0.572313	0.286157	−	0.958183i	$-0.407622\pi$
$409$	11.5743	0.572313	0.286157	+	0.958183i	$0.407622\pi$
$410$	−12.2467	−0.604823
$411$	0	0
$412$	2.68372	0.132218
$413$	−2.23850	−0.110149
$414$	0	0
$415$	−41.3507	−2.02983
$416$	−2.79874	−0.137220
$417$	0	0
$418$	−6.41424	−0.313731
$419$	−28.7087	−1.40251	−0.701256	−	0.712909i	$-0.747377\pi$
$419$	−28.7087	−1.40251	−0.701256	+	0.712909i	$0.747377\pi$
$420$	0	0
$421$	−31.6417	−1.54212	−0.771061	−	0.636762i	$-0.780274\pi$
$421$	−31.6417	−1.54212	−0.771061	+	0.636762i	$0.780274\pi$
$422$	11.6803	0.568589
$423$	0	0
$424$	−4.40330	−0.213843
$425$	−6.15386	−0.298506
$426$	0	0
$427$	−0.582270	−0.0281780
$428$	−13.7885	−0.666494
$429$	0	0
$430$	−1.77807	−0.0857459
$431$	−27.0921	−1.30498	−0.652490	−	0.757798i	$-0.726275\pi$
$431$	−27.0921	−1.30498	−0.652490	+	0.757798i	$0.726275\pi$
$432$	0	0
$433$	−37.9931	−1.82583	−0.912915	−	0.408150i	$-0.866174\pi$
$433$	−37.9931	−1.82583	−0.912915	+	0.408150i	$0.866174\pi$
$434$	0.280612	0.0134698
$435$	0	0
$436$	34.5492	1.65461
$437$	−6.86830	−0.328555
$438$	0	0
$439$	−27.8044	−1.32703	−0.663516	−	0.748162i	$-0.730937\pi$
$439$	−27.8044	−1.32703	−0.663516	+	0.748162i	$0.730937\pi$
$440$	19.2403	0.917245
$441$	0	0
$442$	0.847543	0.0403135
$443$	7.02886	0.333951	0.166976	−	0.985961i	$-0.446600\pi$
$443$	7.02886	0.333951	0.166976	+	0.985961i	$0.446600\pi$
$444$	0	0
$445$	−14.7213	−0.697857
$446$	6.20213	0.293679
$447$	0	0
$448$	2.03681	0.0962303
$449$	−6.19816	−0.292509	−0.146255	−	0.989247i	$-0.546722\pi$
$449$	−6.19816	−0.292509	−0.146255	+	0.989247i	$0.546722\pi$
$450$	0	0
$451$	42.2895	1.99134
$452$	5.20617	0.244878
$453$	0	0
$454$	−10.1933	−0.478397
$455$	0.938610	0.0440027
$456$	0	0
$457$	−24.9646	−1.16779	−0.583897	−	0.811828i	$-0.698473\pi$
$457$	−24.9646	−1.16779	−0.583897	+	0.811828i	$0.698473\pi$
$458$	−3.97651	−0.185810
$459$	0	0
$460$	9.72687	0.453517
$461$	−33.7554	−1.57215	−0.786073	−	0.618134i	$-0.787889\pi$
$461$	−33.7554	−1.57215	−0.786073	+	0.618134i	$0.787889\pi$
$462$	0	0
$463$	17.1602	0.797502	0.398751	−	0.917059i	$-0.369444\pi$
$463$	17.1602	0.797502	0.398751	+	0.917059i	$0.369444\pi$
$464$	−28.1823	−1.30833
$465$	0	0
$466$	7.02486	0.325420
$467$	−23.4137	−1.08346	−0.541728	−	0.840554i	$-0.682230\pi$
$467$	−23.4137	−1.08346	−0.541728	+	0.840554i	$0.682230\pi$
$468$	0	0
$469$	3.70700	0.171173
$470$	4.62471	0.213322
$471$	0	0
$472$	−6.44423	−0.296620
$473$	6.13989	0.282312
$474$	0	0
$475$	6.54063	0.300105
$476$	−3.39275	−0.155507
$477$	0	0
$478$	13.7331	0.628135
$479$	−12.8820	−0.588594	−0.294297	−	0.955714i	$-0.595086\pi$
$479$	−12.8820	−0.588594	−0.294297	+	0.955714i	$0.595086\pi$
$480$	0	0
$481$	0.566852	0.0258462
$482$	8.05062	0.366696
$483$	0	0
$484$	−11.6911	−0.531412
$485$	−17.4334	−0.791609
$486$	0	0
$487$	−37.7019	−1.70844	−0.854218	−	0.519915i	$-0.825963\pi$
$487$	−37.7019	−1.70844	−0.854218	+	0.519915i	$0.825963\pi$
$488$	−1.67625	−0.0758804
$489$	0	0
$490$	−8.04517	−0.363444
$491$	12.3330	0.556581	0.278290	−	0.960497i	$-0.410232\pi$
$491$	12.3330	0.556581	0.278290	+	0.960497i	$0.410232\pi$
$492$	0	0
$493$	31.8198	1.43309
$494$	−0.900811	−0.0405294
$495$	0	0
$496$	−2.80354	−0.125883
$497$	2.62781	0.117873
$498$	0	0
$499$	11.1753	0.500274	0.250137	−	0.968210i	$-0.419524\pi$
$499$	11.1753	0.500274	0.250137	+	0.968210i	$0.419524\pi$
$500$	14.3370	0.641170
$501$	0	0
$502$	−5.32671	−0.237743
$503$	27.2408	1.21461	0.607303	−	0.794470i	$-0.292252\pi$
$503$	27.2408	1.21461	0.607303	+	0.794470i	$0.292252\pi$
$504$	0	0
$505$	−0.561565	−0.0249893
$506$	3.96611	0.176315
$507$	0	0
$508$	20.7478	0.920536
$509$	−8.83882	−0.391774	−0.195887	−	0.980627i	$-0.562759\pi$
$509$	−8.83882	−0.391774	−0.195887	+	0.980627i	$0.562759\pi$
$510$	0	0
$511$	−8.38978	−0.371142
$512$	22.8891	1.01156
$513$	0	0
$514$	5.67417	0.250277
$515$	−3.95879	−0.174445
$516$	0	0
$517$	−15.9697	−0.702347
$518$	0.267941	0.0117727
$519$	0	0
$520$	2.70209	0.118495
$521$	35.2225	1.54312	0.771562	−	0.636154i	$-0.219476\pi$
$521$	35.2225	1.54312	0.771562	+	0.636154i	$0.219476\pi$
$522$	0	0
$523$	0.251562	0.0110000	0.00550001	−	0.999985i	$-0.498249\pi$
$523$	0.251562	0.0110000	0.00550001	+	0.999985i	$0.498249\pi$
$524$	−14.7799	−0.645664
$525$	0	0
$526$	5.01565	0.218693
$527$	3.16540	0.137887
$528$	0	0
$529$	−18.7531	−0.815354
$530$	3.06662	0.133206
$531$	0	0
$532$	3.60599	0.156339
$533$	5.93911	0.257251
$534$	0	0
$535$	20.3396	0.879359
$536$	10.6718	0.460951
$537$	0	0
$538$	−1.21653	−0.0524483
$539$	27.7810	1.19661
$540$	0	0
$541$	30.5799	1.31473	0.657366	−	0.753572i	$-0.271671\pi$
$541$	30.5799	1.31473	0.657366	+	0.753572i	$0.271671\pi$
$542$	7.05622	0.303090
$543$	0	0
$544$	−14.9230	−0.639816
$545$	−50.9639	−2.18305
$546$	0	0
$547$	20.4268	0.873388	0.436694	−	0.899610i	$-0.356149\pi$
$547$	20.4268	0.873388	0.436694	+	0.899610i	$0.356149\pi$
$548$	10.9279	0.466815
$549$	0	0
$550$	−3.77690	−0.161047
$551$	−33.8197	−1.44077
$552$	0	0
$553$	−5.54306	−0.235715
$554$	8.67221	0.368447
$555$	0	0
$556$	−26.4829	−1.12312
$557$	4.28570	0.181591	0.0907956	−	0.995870i	$-0.471059\pi$
$557$	4.28570	0.181591	0.0907956	+	0.995870i	$0.471059\pi$
$558$	0	0
$559$	0.862281	0.0364706
$560$	−4.43257	−0.187310
$561$	0	0
$562$	1.11114	0.0468706
$563$	34.7819	1.46588	0.732942	−	0.680292i	$-0.238147\pi$
$563$	34.7819	1.46588	0.732942	+	0.680292i	$0.238147\pi$
$564$	0	0
$565$	−7.67968	−0.323087
$566$	−7.92552	−0.333135
$567$	0	0
$568$	7.56499	0.317420
$569$	−1.01037	−0.0423571	−0.0211785	−	0.999776i	$-0.506742\pi$
$569$	−1.01037	−0.0423571	−0.0211785	+	0.999776i	$0.506742\pi$
$570$	0	0
$571$	4.17041	0.174526	0.0872632	−	0.996185i	$-0.472188\pi$
$571$	4.17041	0.174526	0.0872632	+	0.996185i	$0.472188\pi$
$572$	−4.40525	−0.184193
$573$	0	0
$574$	2.80731	0.117175
$575$	−4.04426	−0.168657
$576$	0	0
$577$	−22.0426	−0.917647	−0.458824	−	0.888527i	$-0.651729\pi$
$577$	−22.0426	−0.917647	−0.458824	+	0.888527i	$0.651729\pi$
$578$	−3.29386	−0.137007
$579$	0	0
$580$	47.8953	1.98875
$581$	9.47881	0.393247
$582$	0	0
$583$	−10.5894	−0.438570
$584$	−24.1527	−0.999445
$585$	0	0
$586$	1.44462	0.0596767
$587$	−2.82344	−0.116536	−0.0582678	−	0.998301i	$-0.518558\pi$
$587$	−2.82344	−0.116536	−0.0582678	+	0.998301i	$0.518558\pi$
$588$	0	0
$589$	−3.36435	−0.138626
$590$	4.48801	0.184768
$591$	0	0
$592$	−2.67695	−0.110022
$593$	45.6709	1.87548	0.937740	−	0.347339i	$-0.112915\pi$
$593$	45.6709	1.87548	0.937740	+	0.347339i	$0.112915\pi$
$594$	0	0
$595$	5.00469	0.205172
$596$	13.0143	0.533086
$597$	0	0
$598$	0.556997	0.0227773
$599$	−37.8313	−1.54575	−0.772873	−	0.634561i	$-0.781181\pi$
$599$	−37.8313	−1.54575	−0.772873	+	0.634561i	$0.781181\pi$
$600$	0	0
$601$	−6.83325	−0.278734	−0.139367	−	0.990241i	$-0.544507\pi$
$601$	−6.83325	−0.278734	−0.139367	+	0.990241i	$0.544507\pi$
$602$	0.407585	0.0166119
$603$	0	0
$604$	12.1152	0.492959
$605$	17.2456	0.701135
$606$	0	0
$607$	−22.9026	−0.929587	−0.464794	−	0.885419i	$-0.653872\pi$
$607$	−22.9026	−0.929587	−0.464794	+	0.885419i	$0.653872\pi$
$608$	15.8609	0.643243
$609$	0	0
$610$	1.16741	0.0472669
$611$	−2.24277	−0.0907329
$612$	0	0
$613$	−5.01302	−0.202474	−0.101237	−	0.994862i	$-0.532280\pi$
$613$	−5.01302	−0.202474	−0.101237	+	0.994862i	$0.532280\pi$
$614$	12.6691	0.511282
$615$	0	0
$616$	−4.41044	−0.177702
$617$	21.2112	0.853931	0.426966	−	0.904268i	$-0.359583\pi$
$617$	21.2112	0.853931	0.426966	+	0.904268i	$0.359583\pi$
$618$	0	0
$619$	12.4720	0.501290	0.250645	−	0.968079i	$-0.419357\pi$
$619$	12.4720	0.501290	0.250645	+	0.968079i	$0.419357\pi$
$620$	4.76458	0.191350
$621$	0	0
$622$	−1.82880	−0.0733280
$623$	3.37456	0.135199
$624$	0	0
$625$	−30.9611	−1.23844
$626$	−7.34296	−0.293484
$627$	0	0
$628$	24.6062	0.981896
$629$	3.02247	0.120514
$630$	0	0
$631$	43.7516	1.74172	0.870862	−	0.491528i	$-0.163561\pi$
$631$	43.7516	1.74172	0.870862	+	0.491528i	$0.163561\pi$
$632$	−15.9575	−0.634755
$633$	0	0
$634$	11.7019	0.464742
$635$	−30.6054	−1.21454
$636$	0	0
$637$	3.90154	0.154585
$638$	19.5292	0.773170
$639$	0	0
$640$	−29.1981	−1.15416
$641$	−14.2570	−0.563116	−0.281558	−	0.959544i	$-0.590851\pi$
$641$	−14.2570	−0.563116	−0.281558	+	0.959544i	$0.590851\pi$
$642$	0	0
$643$	−32.7973	−1.29340	−0.646699	−	0.762745i	$-0.723851\pi$
$643$	−32.7973	−1.29340	−0.646699	+	0.762745i	$0.723851\pi$
$644$	−2.22968	−0.0878619
$645$	0	0
$646$	−4.80315	−0.188977
$647$	2.71497	0.106736	0.0533682	−	0.998575i	$-0.483004\pi$
$647$	2.71497	0.106736	0.0533682	+	0.998575i	$0.483004\pi$
$648$	0	0
$649$	−15.4977	−0.608337
$650$	−0.530424	−0.0208049
$651$	0	0
$652$	−19.2115	−0.752381
$653$	−0.282536	−0.0110565	−0.00552825	−	0.999985i	$-0.501760\pi$
$653$	−0.282536	−0.0110565	−0.00552825	+	0.999985i	$0.501760\pi$
$654$	0	0
$655$	21.8021	0.851877
$656$	−28.0474	−1.09507
$657$	0	0
$658$	−1.06012	−0.0413278
$659$	−2.66474	−0.103804	−0.0519018	−	0.998652i	$-0.516528\pi$
$659$	−2.66474	−0.103804	−0.0519018	+	0.998652i	$0.516528\pi$
$660$	0	0
$661$	22.1445	0.861321	0.430660	−	0.902514i	$-0.358281\pi$
$661$	22.1445	0.861321	0.430660	+	0.902514i	$0.358281\pi$
$662$	−11.0687	−0.430198
$663$	0	0
$664$	27.2878	1.05897
$665$	−5.31924	−0.206271
$666$	0	0
$667$	20.9117	0.809703
$668$	−1.78878	−0.0692100
$669$	0	0
$670$	−7.43224	−0.287132
$671$	−4.03120	−0.155623
$672$	0	0
$673$	−13.6686	−0.526886	−0.263443	−	0.964675i	$-0.584858\pi$
$673$	−13.6686	−0.526886	−0.263443	+	0.964675i	$0.584858\pi$
$674$	−3.38016	−0.130199
$675$	0	0
$676$	22.6355	0.870595
$677$	5.36889	0.206343	0.103172	−	0.994664i	$-0.467101\pi$
$677$	5.36889	0.206343	0.103172	+	0.994664i	$0.467101\pi$
$678$	0	0
$679$	3.99625	0.153362
$680$	14.4076	0.552507
$681$	0	0
$682$	1.94275	0.0743916
$683$	35.4099	1.35492	0.677460	−	0.735559i	$-0.263081\pi$
$683$	35.4099	1.35492	0.677460	+	0.735559i	$0.263081\pi$
$684$	0	0
$685$	−16.1198	−0.615907
$686$	3.79008	0.144706
$687$	0	0
$688$	−4.07211	−0.155248
$689$	−1.48717	−0.0566567
$690$	0	0
$691$	2.30779	0.0877924	0.0438962	−	0.999036i	$-0.486023\pi$
$691$	2.30779	0.0877924	0.0438962	+	0.999036i	$0.486023\pi$
$692$	2.93223	0.111467
$693$	0	0
$694$	−9.59782	−0.364328
$695$	39.0652	1.48183
$696$	0	0
$697$	31.6675	1.19949
$698$	−12.2301	−0.462916
$699$	0	0
$700$	2.12331	0.0802536
$701$	18.6519	0.704474	0.352237	−	0.935911i	$-0.385421\pi$
$701$	18.6519	0.704474	0.352237	+	0.935911i	$0.385421\pi$
$702$	0	0
$703$	−3.21243	−0.121159
$704$	14.1014	0.531465
$705$	0	0
$706$	−0.979602	−0.0368678
$707$	0.128727	0.00484129
$708$	0	0
$709$	−8.83611	−0.331847	−0.165924	−	0.986139i	$-0.553061\pi$
$709$	−8.83611	−0.331847	−0.165924	+	0.986139i	$0.553061\pi$
$710$	−5.26855	−0.197725
$711$	0	0
$712$	9.71476	0.364076
$713$	2.08027	0.0779068
$714$	0	0
$715$	6.49823	0.243020
$716$	−34.0641	−1.27304
$717$	0	0
$718$	−5.76537	−0.215162
$719$	−17.8566	−0.665940	−0.332970	−	0.942937i	$-0.608051\pi$
$719$	−17.8566	−0.665940	−0.332970	+	0.942937i	$0.608051\pi$
$720$	0	0
$721$	0.907473	0.0337960
$722$	−3.62713	−0.134988
$723$	0	0
$724$	−26.4009	−0.981181
$725$	−19.9140	−0.739588
$726$	0	0
$727$	−20.2602	−0.751409	−0.375704	−	0.926740i	$-0.622599\pi$
$727$	−20.2602	−0.751409	−0.375704	+	0.926740i	$0.622599\pi$
$728$	−0.619399	−0.0229565
$729$	0	0
$730$	16.8208	0.622567
$731$	4.59770	0.170052
$732$	0	0
$733$	−16.9048	−0.624394	−0.312197	−	0.950017i	$-0.601065\pi$
$733$	−16.9048	−0.624394	−0.312197	+	0.950017i	$0.601065\pi$
$734$	−8.37025	−0.308951
$735$	0	0
$736$	−9.80722	−0.361499
$737$	25.6645	0.945363
$738$	0	0
$739$	−35.1818	−1.29418	−0.647092	−	0.762412i	$-0.724015\pi$
$739$	−35.1818	−1.29418	−0.647092	+	0.762412i	$0.724015\pi$
$740$	4.54944	0.167241
$741$	0	0
$742$	−0.702961	−0.0258065
$743$	−29.8016	−1.09331	−0.546657	−	0.837356i	$-0.684100\pi$
$743$	−29.8016	−1.09331	−0.546657	+	0.837356i	$0.684100\pi$
$744$	0	0
$745$	−19.1975	−0.703343
$746$	11.0592	0.404906
$747$	0	0
$748$	−23.4889	−0.858839
$749$	−4.66245	−0.170362
$750$	0	0
$751$	−41.8463	−1.52699	−0.763497	−	0.645812i	$-0.776519\pi$
$751$	−41.8463	−1.52699	−0.763497	+	0.645812i	$0.776519\pi$
$752$	10.5915	0.386231
$753$	0	0
$754$	2.74267	0.0998821
$755$	−17.8712	−0.650401
$756$	0	0
$757$	20.1667	0.732973	0.366486	−	0.930423i	$-0.380561\pi$
$757$	20.1667	0.732973	0.366486	+	0.930423i	$0.380561\pi$
$758$	8.35887	0.303608
$759$	0	0
$760$	−15.3131	−0.555466
$761$	47.8942	1.73616	0.868082	−	0.496421i	$-0.165353\pi$
$761$	47.8942	1.73616	0.868082	+	0.496421i	$0.165353\pi$
$762$	0	0
$763$	11.6824	0.422933
$764$	−14.7431	−0.533385
$765$	0	0
$766$	10.0723	0.363927
$767$	−2.17648	−0.0785881
$768$	0	0
$769$	12.1859	0.439433	0.219717	−	0.975564i	$-0.429487\pi$
$769$	12.1859	0.439433	0.219717	+	0.975564i	$0.429487\pi$
$770$	3.07160	0.110693
$771$	0	0
$772$	−22.2458	−0.800646
$773$	−47.6169	−1.71266	−0.856331	−	0.516427i	$-0.827262\pi$
$773$	−47.6169	−1.71266	−0.856331	+	0.516427i	$0.827262\pi$
$774$	0	0
$775$	−1.98103	−0.0711606
$776$	11.5045	0.412987
$777$	0	0
$778$	−3.12100	−0.111893
$779$	−33.6578	−1.20592
$780$	0	0
$781$	18.1930	0.650996
$782$	2.96992	0.106204
$783$	0	0
$784$	−18.4250	−0.658034
$785$	−36.2970	−1.29549
$786$	0	0
$787$	−25.4329	−0.906586	−0.453293	−	0.891361i	$-0.649751\pi$
$787$	−25.4329	−0.906586	−0.453293	+	0.891361i	$0.649751\pi$
$788$	−33.0613	−1.17776
$789$	0	0
$790$	11.1134	0.395397
$791$	1.76041	0.0625930
$792$	0	0
$793$	−0.566139	−0.0201042
$794$	−16.6985	−0.592609
$795$	0	0
$796$	−8.64984	−0.306585
$797$	−13.4752	−0.477315	−0.238658	−	0.971104i	$-0.576707\pi$
$797$	−13.4752	−0.477315	−0.238658	+	0.971104i	$0.576707\pi$
$798$	0	0
$799$	−11.9585	−0.423062
$800$	9.33934	0.330196
$801$	0	0
$802$	12.1900	0.430443
$803$	−58.0845	−2.04976
$804$	0	0
$805$	3.28904	0.115923
$806$	0.272838	0.00961030
$807$	0	0
$808$	0.370583	0.0130371
$809$	17.0866	0.600732	0.300366	−	0.953824i	$-0.402891\pi$
$809$	17.0866	0.600732	0.300366	+	0.953824i	$0.402891\pi$
$810$	0	0
$811$	18.7124	0.657080	0.328540	−	0.944490i	$-0.393443\pi$
$811$	18.7124	0.657080	0.328540	+	0.944490i	$0.393443\pi$
$812$	−10.9790	−0.385288
$813$	0	0
$814$	1.85502	0.0650186
$815$	28.3391	0.992676
$816$	0	0
$817$	−4.88667	−0.170963
$818$	5.31941	0.185989
$819$	0	0
$820$	47.6660	1.66457
$821$	−43.7257	−1.52604	−0.763019	−	0.646377i	$-0.776284\pi$
$821$	−43.7257	−1.52604	−0.763019	+	0.646377i	$0.776284\pi$
$822$	0	0
$823$	−20.6687	−0.720466	−0.360233	−	0.932862i	$-0.617303\pi$
$823$	−20.6687	−0.720466	−0.360233	+	0.932862i	$0.617303\pi$
$824$	2.61245	0.0910091
$825$	0	0
$826$	−1.02878	−0.0357960
$827$	−53.5678	−1.86274	−0.931368	−	0.364080i	$-0.881383\pi$
$827$	−53.5678	−1.86274	−0.931368	+	0.364080i	$0.881383\pi$
$828$	0	0
$829$	8.09767	0.281244	0.140622	−	0.990063i	$-0.455090\pi$
$829$	8.09767	0.281244	0.140622	+	0.990063i	$0.455090\pi$
$830$	−19.0043	−0.659648
$831$	0	0
$832$	1.98038	0.0686574
$833$	20.8031	0.720785
$834$	0	0
$835$	2.63865	0.0913143
$836$	24.9652	0.863438
$837$	0	0
$838$	−13.1942	−0.455785
$839$	−35.9455	−1.24098	−0.620489	−	0.784215i	$-0.713066\pi$
$839$	−35.9455	−1.24098	−0.620489	+	0.784215i	$0.713066\pi$
$840$	0	0
$841$	73.9696	2.55068
$842$	−14.5421	−0.501154
$843$	0	0
$844$	−45.4615	−1.56485
$845$	−33.3899	−1.14865
$846$	0	0
$847$	−3.95321	−0.135834
$848$	7.02315	0.241176
$849$	0	0
$850$	−2.82823	−0.0970076
$851$	1.98634	0.0680908
$852$	0	0
$853$	−43.1857	−1.47865	−0.739325	−	0.673348i	$-0.764855\pi$
$853$	−43.1857	−1.47865	−0.739325	+	0.673348i	$0.764855\pi$
$854$	−0.267604	−0.00915722
$855$	0	0
$856$	−13.4224	−0.458767
$857$	−26.1566	−0.893491	−0.446746	−	0.894661i	$-0.647417\pi$
$857$	−26.1566	−0.893491	−0.446746	+	0.894661i	$0.647417\pi$
$858$	0	0
$859$	15.1284	0.516174	0.258087	−	0.966122i	$-0.416908\pi$
$859$	15.1284	0.516174	0.258087	+	0.966122i	$0.416908\pi$
$860$	6.92049	0.235987
$861$	0	0
$862$	−12.4512	−0.424089
$863$	−15.0990	−0.513977	−0.256989	−	0.966414i	$-0.582730\pi$
$863$	−15.0990	−0.513977	−0.256989	+	0.966414i	$0.582730\pi$
$864$	0	0
$865$	−4.32538	−0.147067
$866$	−17.4611	−0.593353
$867$	0	0
$868$	−1.09218	−0.0370711
$869$	−38.3760	−1.30182
$870$	0	0
$871$	3.60430	0.122127
$872$	33.6317	1.13891
$873$	0	0
$874$	−3.15658	−0.106773
$875$	4.84790	0.163889
$876$	0	0
$877$	−43.4986	−1.46884	−0.734422	−	0.678693i	$-0.762547\pi$
$877$	−43.4986	−1.46884	−0.734422	+	0.678693i	$0.762547\pi$
$878$	−12.7786	−0.431255
$879$	0	0
$880$	−30.6878	−1.03449
$881$	−13.7969	−0.464828	−0.232414	−	0.972617i	$-0.574662\pi$
$881$	−13.7969	−0.464828	−0.232414	+	0.972617i	$0.574662\pi$
$882$	0	0
$883$	2.77424	0.0933607	0.0466804	−	0.998910i	$-0.485136\pi$
$883$	2.77424	0.0933607	0.0466804	+	0.998910i	$0.485136\pi$
$884$	−3.29876	−0.110949
$885$	0	0
$886$	3.23038	0.108527
$887$	−42.3348	−1.42146	−0.710732	−	0.703462i	$-0.751636\pi$
$887$	−42.3348	−1.42146	−0.710732	+	0.703462i	$0.751636\pi$
$888$	0	0
$889$	7.01565	0.235298
$890$	−6.76573	−0.226788
$891$	0	0
$892$	−24.1396	−0.808253
$893$	12.7101	0.425328
$894$	0	0
$895$	50.2484	1.67962
$896$	6.69307	0.223600
$897$	0	0
$898$	−2.84859	−0.0950588
$899$	10.2433	0.341634
$900$	0	0
$901$	−7.92964	−0.264175
$902$	19.4357	0.647139
$903$	0	0
$904$	5.06791	0.168556
$905$	38.9443	1.29455
$906$	0	0
$907$	−7.94687	−0.263871	−0.131936	−	0.991258i	$-0.542119\pi$
$907$	−7.94687	−0.263871	−0.131936	+	0.991258i	$0.542119\pi$
$908$	39.6739	1.31663
$909$	0	0
$910$	0.431373	0.0142999
$911$	58.6923	1.94456	0.972281	−	0.233814i	$-0.0751208\pi$
$911$	58.6923	1.94456	0.972281	+	0.233814i	$0.0751208\pi$
$912$	0	0
$913$	65.6242	2.17184
$914$	−11.4734	−0.379506
$915$	0	0
$916$	15.4772	0.511380
$917$	−4.99768	−0.165038
$918$	0	0
$919$	38.5200	1.27066	0.635328	−	0.772242i	$-0.280865\pi$
$919$	38.5200	1.27066	0.635328	+	0.772242i	$0.280865\pi$
$920$	9.46855	0.312169
$921$	0	0
$922$	−15.5136	−0.510912
$923$	2.55501	0.0840990
$924$	0	0
$925$	−1.89157	−0.0621946
$926$	7.88661	0.259170
$927$	0	0
$928$	−48.2910	−1.58523
$929$	−38.6862	−1.26925	−0.634627	−	0.772818i	$-0.718846\pi$
$929$	−38.6862	−1.26925	−0.634627	+	0.772818i	$0.718846\pi$
$930$	0	0
$931$	−22.1106	−0.724645
$932$	−27.3418	−0.895609
$933$	0	0
$934$	−10.7606	−0.352098
$935$	34.6487	1.13313
$936$	0	0
$937$	−50.9642	−1.66493	−0.832463	−	0.554080i	$-0.813070\pi$
$937$	−50.9642	−1.66493	−0.832463	+	0.554080i	$0.813070\pi$
$938$	1.70369	0.0556274
$939$	0	0
$940$	−18.0000	−0.587096
$941$	44.6968	1.45708	0.728538	−	0.685006i	$-0.240200\pi$
$941$	44.6968	1.45708	0.728538	+	0.685006i	$0.240200\pi$
$942$	0	0
$943$	20.8116	0.677718
$944$	10.2784	0.334533
$945$	0	0
$946$	2.82181	0.0917451
$947$	9.32767	0.303109	0.151554	−	0.988449i	$-0.451572\pi$
$947$	9.32767	0.303109	0.151554	+	0.988449i	$0.451572\pi$
$948$	0	0
$949$	−8.15734	−0.264799
$950$	3.00599	0.0975272
$951$	0	0
$952$	−3.30265	−0.107040
$953$	4.95569	0.160530	0.0802652	−	0.996774i	$-0.474423\pi$
$953$	4.95569	0.160530	0.0802652	+	0.996774i	$0.474423\pi$
$954$	0	0
$955$	21.7477	0.703738
$956$	−53.4510	−1.72873
$957$	0	0
$958$	−5.92041	−0.191280
$959$	3.69514	0.119322
$960$	0	0
$961$	−29.9810	−0.967129
$962$	0.260518	0.00839944
$963$	0	0
$964$	−31.3342	−1.00921
$965$	32.8151	1.05636
$966$	0	0
$967$	−31.4033	−1.00986	−0.504931	−	0.863159i	$-0.668482\pi$
$967$	−31.4033	−1.00986	−0.504931	+	0.863159i	$0.668482\pi$
$968$	−11.3806	−0.365786
$969$	0	0
$970$	−8.01216	−0.257255
$971$	38.8079	1.24540	0.622702	−	0.782459i	$-0.286035\pi$
$971$	38.8079	1.24540	0.622702	+	0.782459i	$0.286035\pi$
$972$	0	0
$973$	−8.95490	−0.287081
$974$	−17.3273	−0.555203
$975$	0	0
$976$	2.67358	0.0855793
$977$	−36.8665	−1.17946	−0.589732	−	0.807599i	$-0.700767\pi$
$977$	−36.8665	−1.17946	−0.589732	+	0.807599i	$0.700767\pi$
$978$	0	0
$979$	23.3629	0.746682
$980$	31.3129	1.00026
$981$	0	0
$982$	5.66809	0.180876
$983$	14.6521	0.467330	0.233665	−	0.972317i	$-0.424928\pi$
$983$	14.6521	0.467330	0.233665	+	0.972317i	$0.424928\pi$
$984$	0	0
$985$	48.7691	1.55391
$986$	14.6240	0.465722
$987$	0	0
$988$	3.50609	0.111543
$989$	3.02157	0.0960802
$990$	0	0
$991$	−22.2752	−0.707595	−0.353797	−	0.935322i	$-0.615110\pi$
$991$	−22.2752	−0.707595	−0.353797	+	0.935322i	$0.615110\pi$
$992$	−4.80394	−0.152525
$993$	0	0
$994$	1.20771	0.0383061
$995$	12.7595	0.404503
$996$	0	0
$997$	6.53346	0.206917	0.103458	−	0.994634i	$-0.467009\pi$
$997$	6.53346	0.206917	0.103458	+	0.994634i	$0.467009\pi$
$998$	5.13601	0.162578
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.2.a.e.1.6		8
3.2	odd	2		501.2.a.e.1.3	✓	8
12.11	even	2		8016.2.a.x.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.e.1.3	✓	8	3.2	odd	2
1503.2.a.e.1.6		8	1.1	even	1	trivial
8016.2.a.x.1.1		8	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1503.a (trivial)

\(f(q)\)	\(=\)	\(q+0.459587 q^{2} -1.78878 q^{4} +2.63865 q^{5} -0.604857 q^{7} -1.74127 q^{8} +O(q^{10})\)	\(q+0.459587 q^{2} -1.78878 q^{4} +2.63865 q^{5} -0.604857 q^{7} -1.74127 q^{8} +1.21269 q^{10} -4.18757 q^{11} -0.588099 q^{13} -0.277984 q^{14} +2.77729 q^{16} -3.13576 q^{17} +3.33284 q^{19} -4.71996 q^{20} -1.92456 q^{22} -2.06079 q^{23} +1.96248 q^{25} -0.270283 q^{26} +1.08196 q^{28} -10.1474 q^{29} -1.00945 q^{31} +4.75896 q^{32} -1.44116 q^{34} -1.59601 q^{35} -0.963871 q^{37} +1.53173 q^{38} -4.59462 q^{40} -10.0988 q^{41} -1.46622 q^{43} +7.49065 q^{44} -0.947114 q^{46} +3.81359 q^{47} -6.63415 q^{49} +0.901929 q^{50} +1.05198 q^{52} +2.52878 q^{53} -11.0495 q^{55} +1.05322 q^{56} -4.66361 q^{58} +3.70087 q^{59} +0.962658 q^{61} -0.463931 q^{62} -3.36743 q^{64} -1.55179 q^{65} -6.12872 q^{67} +5.60919 q^{68} -0.733504 q^{70} -4.34451 q^{71} +13.8707 q^{73} -0.442983 q^{74} -5.96172 q^{76} +2.53288 q^{77} +9.16426 q^{79} +7.32830 q^{80} -4.64129 q^{82} -15.6712 q^{83} -8.27418 q^{85} -0.673854 q^{86} +7.29172 q^{88} -5.57911 q^{89} +0.355716 q^{91} +3.68630 q^{92} +1.75268 q^{94} +8.79421 q^{95} -6.60693 q^{97} -3.04897 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{2} + 9 q^{4} - 7 q^{5} - 4 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - 3 * q^2 + 9 * q^4 - 7 * q^5 - 4 * q^7 - 3 * q^8	\( 8 q - 3 q^{2} + 9 q^{4} - 7 q^{5} - 4 q^{7} - 3 q^{8} - q^{10} - 13 q^{11} - 5 q^{14} + 7 q^{16} - 11 q^{17} + 12 q^{19} - 9 q^{20} - 17 q^{22} - 7 q^{23} - 5 q^{25} - 3 q^{26} - 27 q^{28} - q^{29} - 2 q^{31} - 4 q^{32} - 14 q^{34} + 4 q^{35} - 9 q^{37} - 22 q^{40} - 4 q^{41} + 2 q^{43} - 3 q^{44} - 5 q^{46} - 17 q^{47} - 2 q^{49} + 4 q^{50} - 36 q^{52} - 9 q^{53} + 7 q^{55} - 9 q^{56} - 29 q^{58} - 29 q^{59} - 12 q^{61} + 34 q^{62} - 5 q^{64} - 8 q^{65} - 26 q^{68} + 5 q^{70} - 13 q^{71} - 20 q^{73} + 17 q^{74} + 30 q^{76} + 22 q^{77} + 8 q^{79} + 34 q^{80} + 15 q^{82} - 33 q^{83} - 31 q^{85} - 11 q^{86} - 44 q^{88} - 4 q^{89} + q^{91} + 33 q^{92} - 2 q^{94} - 3 q^{95} - 31 q^{97} + 57 q^{98}+O(q^{100}) \) 8 * q - 3 * q^2 + 9 * q^4 - 7 * q^5 - 4 * q^7 - 3 * q^8 - q^10 - 13 * q^11 - 5 * q^14 + 7 * q^16 - 11 * q^17 + 12 * q^19 - 9 * q^20 - 17 * q^22 - 7 * q^23 - 5 * q^25 - 3 * q^26 - 27 * q^28 - q^29 - 2 * q^31 - 4 * q^32 - 14 * q^34 + 4 * q^35 - 9 * q^37 - 22 * q^40 - 4 * q^41 + 2 * q^43 - 3 * q^44 - 5 * q^46 - 17 * q^47 - 2 * q^49 + 4 * q^50 - 36 * q^52 - 9 * q^53 + 7 * q^55 - 9 * q^56 - 29 * q^58 - 29 * q^59 - 12 * q^61 + 34 * q^62 - 5 * q^64 - 8 * q^65 - 26 * q^68 + 5 * q^70 - 13 * q^71 - 20 * q^73 + 17 * q^74 + 30 * q^76 + 22 * q^77 + 8 * q^79 + 34 * q^80 + 15 * q^82 - 33 * q^83 - 31 * q^85 - 11 * q^86 - 44 * q^88 - 4 * q^89 + q^91 + 33 * q^92 - 2 * q^94 - 3 * q^95 - 31 * q^97 + 57 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more