LMFDB - Embedded newform 2523.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2523,2,Mod(1,2523)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2523, 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("2523.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2523 = 3 \cdot 29^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2523.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:	$20.1462564300$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{9} - 4x^{8} - 6x^{7} + 33x^{6} + 6x^{5} - 90x^{4} + 21x^{3} + 84x^{2} - 36x - 8$ x^9 - 4x^8 - 6x^7 + 33x^6 + 6x^5 - 90x^4 + 21x^3 + 84x^2 - 36x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 87)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.21072$ of defining polynomial
Character	$\chi$	$=$	2523.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.21072 q^{2} +1.00000 q^{3} -0.534161 q^{4} +1.79844 q^{5} -1.21072 q^{6} +4.27693 q^{7} +3.06816 q^{8} +1.00000 q^{9} -2.17740 q^{10} +1.05626 q^{11} -0.534161 q^{12} -5.28193 q^{13} -5.17816 q^{14} +1.79844 q^{15} -2.64635 q^{16} +5.61769 q^{17} -1.21072 q^{18} +6.41175 q^{19} -0.960655 q^{20} +4.27693 q^{21} -1.27883 q^{22} +2.04633 q^{23} +3.06816 q^{24} -1.76563 q^{25} +6.39492 q^{26} +1.00000 q^{27} -2.28457 q^{28} -2.17740 q^{30} +3.82471 q^{31} -2.93233 q^{32} +1.05626 q^{33} -6.80144 q^{34} +7.69178 q^{35} -0.534161 q^{36} -0.350297 q^{37} -7.76282 q^{38} -5.28193 q^{39} +5.51788 q^{40} -3.62240 q^{41} -5.17816 q^{42} +1.74900 q^{43} -0.564212 q^{44} +1.79844 q^{45} -2.47753 q^{46} +6.98005 q^{47} -2.64635 q^{48} +11.2921 q^{49} +2.13768 q^{50} +5.61769 q^{51} +2.82140 q^{52} -7.81869 q^{53} -1.21072 q^{54} +1.89961 q^{55} +13.1223 q^{56} +6.41175 q^{57} +0.382668 q^{59} -0.960655 q^{60} -5.46644 q^{61} -4.63064 q^{62} +4.27693 q^{63} +8.84292 q^{64} -9.49920 q^{65} -1.27883 q^{66} +7.81166 q^{67} -3.00075 q^{68} +2.04633 q^{69} -9.31258 q^{70} -15.6115 q^{71} +3.06816 q^{72} -2.54289 q^{73} +0.424110 q^{74} -1.76563 q^{75} -3.42491 q^{76} +4.51754 q^{77} +6.39492 q^{78} +10.2680 q^{79} -4.75929 q^{80} +1.00000 q^{81} +4.38571 q^{82} +1.52139 q^{83} -2.28457 q^{84} +10.1030 q^{85} -2.11755 q^{86} +3.24076 q^{88} -17.9444 q^{89} -2.17740 q^{90} -22.5904 q^{91} -1.09307 q^{92} +3.82471 q^{93} -8.45087 q^{94} +11.5311 q^{95} -2.93233 q^{96} -6.80968 q^{97} -13.6716 q^{98} +1.05626 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$9 q + 5 q^{2} + 9 q^{3} + 11 q^{4} - 4 q^{5} + 5 q^{6} + 5 q^{7} + 24 q^{8} + 9 q^{9} - q^{11} + 11 q^{12} + q^{13} + 9 q^{14} - 4 q^{15} + 35 q^{16} + 2 q^{17} + 5 q^{18} + 9 q^{19} - 18 q^{20} + 5 q^{21}+ \cdots - q^{99}+O(q^{100})$ 9 * q + 5 * q^2 + 9 * q^3 + 11 * q^4 - 4 * q^5 + 5 * q^6 + 5 * q^7 + 24 * q^8 + 9 * q^9 - q^11 + 11 * q^12 + q^13 + 9 * q^14 - 4 * q^15 + 35 * q^16 + 2 * q^17 + 5 * q^18 + 9 * q^19 - 18 * q^20 + 5 * q^21 - 4 * q^22 - 4 * q^23 + 24 * q^24 + q^25 - 8 * q^26 + 9 * q^27 + 40 * q^28 + 8 * q^31 + 43 * q^32 - q^33 - 4 * q^34 + 22 * q^35 + 11 * q^36 + 27 * q^37 - 30 * q^38 + q^39 - 29 * q^40 + 12 * q^41 + 9 * q^42 + 16 * q^43 + 37 * q^44 - 4 * q^45 - 22 * q^46 - 8 * q^47 + 35 * q^48 - 6 * q^49 - 7 * q^50 + 2 * q^51 + 33 * q^52 - 8 * q^53 + 5 * q^54 + 9 * q^55 + 40 * q^56 + 9 * q^57 - 16 * q^59 - 18 * q^60 + 21 * q^61 - 32 * q^62 + 5 * q^63 + 36 * q^64 - 31 * q^65 - 4 * q^66 + 3 * q^67 + 33 * q^68 - 4 * q^69 - 6 * q^70 - 33 * q^71 + 24 * q^72 + 3 * q^73 + 28 * q^74 + q^75 - 26 * q^76 + 24 * q^77 - 8 * q^78 + 3 * q^79 - 64 * q^80 + 9 * q^81 + 13 * q^82 + 13 * q^83 + 40 * q^84 + 6 * q^85 + 58 * q^86 + 27 * q^88 + 6 * q^89 + q^91 - 29 * q^92 + 8 * q^93 - 18 * q^94 + 48 * q^95 + 43 * q^96 + 4 * q^97 - 30 * q^98 - q^99

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

)

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

$2$	−1.21072	−0.856107	−0.428054	−	0.903753i	$-0.640801\pi$
$2$	−1.21072	−0.856107	−0.428054	+	0.903753i	$0.640801\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−0.534161	−0.267081
$5$	1.79844	0.804285	0.402142	−	0.915577i	$-0.368266\pi$
$5$	1.79844	0.804285	0.402142	+	0.915577i	$0.368266\pi$
$6$	−1.21072	−0.494274
$7$	4.27693	1.61653	0.808264	−	0.588821i	$-0.200408\pi$
$7$	4.27693	1.61653	0.808264	+	0.588821i	$0.200408\pi$
$8$	3.06816	1.08476
$9$	1.00000	0.333333
$10$	−2.17740	−0.688554
$11$	1.05626	0.318474	0.159237	−	0.987240i	$-0.449097\pi$
$11$	1.05626	0.318474	0.159237	+	0.987240i	$0.449097\pi$
$12$	−0.534161	−0.154199
$13$	−5.28193	−1.46494	−0.732471	−	0.680798i	$-0.761633\pi$
$13$	−5.28193	−1.46494	−0.732471	+	0.680798i	$0.761633\pi$
$14$	−5.17816	−1.38392
$15$	1.79844	0.464354
$16$	−2.64635	−0.661587
$17$	5.61769	1.36249	0.681245	−	0.732056i	$-0.261439\pi$
$17$	5.61769	1.36249	0.681245	+	0.732056i	$0.261439\pi$
$18$	−1.21072	−0.285369
$19$	6.41175	1.47096	0.735478	−	0.677548i	$-0.236958\pi$
$19$	6.41175	1.47096	0.735478	+	0.677548i	$0.236958\pi$
$20$	−0.960655	−0.214809
$21$	4.27693	0.933303
$22$	−1.27883	−0.272648
$23$	2.04633	0.426689	0.213345	−	0.976977i	$-0.431564\pi$
$23$	2.04633	0.426689	0.213345	+	0.976977i	$0.431564\pi$
$24$	3.06816	0.626285
$25$	−1.76563	−0.353126
$26$	6.39492	1.25415
$27$	1.00000	0.192450
$28$	−2.28457	−0.431743
$29$	0	0
$29$	0	0
$30$	−2.17740	−0.397537
$31$	3.82471	0.686937	0.343469	−	0.939164i	$-0.388398\pi$
$31$	3.82471	0.686937	0.343469	+	0.939164i	$0.388398\pi$
$32$	−2.93233	−0.518367
$33$	1.05626	0.183871
$34$	−6.80144	−1.16644
$35$	7.69178	1.30015
$36$	−0.534161	−0.0890269
$37$	−0.350297	−0.0575884	−0.0287942	−	0.999585i	$-0.509167\pi$
$37$	−0.350297	−0.0575884	−0.0287942	+	0.999585i	$0.509167\pi$
$38$	−7.76282	−1.25930
$39$	−5.28193	−0.845785
$40$	5.51788	0.872453
$41$	−3.62240	−0.565725	−0.282862	−	0.959161i	$-0.591284\pi$
$41$	−3.62240	−0.565725	−0.282862	+	0.959161i	$0.591284\pi$
$42$	−5.17816	−0.799007
$43$	1.74900	0.266721	0.133360	−	0.991068i	$-0.457423\pi$
$43$	1.74900	0.266721	0.133360	+	0.991068i	$0.457423\pi$
$44$	−0.564212	−0.0850582
$45$	1.79844	0.268095
$46$	−2.47753	−0.365292
$47$	6.98005	1.01814	0.509072	−	0.860724i	$-0.329989\pi$
$47$	6.98005	1.01814	0.509072	+	0.860724i	$0.329989\pi$
$48$	−2.64635	−0.381968
$49$	11.2921	1.61316
$50$	2.13768	0.302314
$51$	5.61769	0.786634
$52$	2.82140	0.391258
$53$	−7.81869	−1.07398	−0.536990	−	0.843589i	$-0.680439\pi$
$53$	−7.81869	−1.07398	−0.536990	+	0.843589i	$0.680439\pi$
$54$	−1.21072	−0.164758
$55$	1.89961	0.256144
$56$	13.1223	1.75354
$57$	6.41175	0.849257
$58$	0	0
$59$	0.382668	0.0498191	0.0249096	−	0.999690i	$-0.492070\pi$
$59$	0.382668	0.0498191	0.0249096	+	0.999690i	$0.492070\pi$
$60$	−0.960655	−0.124020
$61$	−5.46644	−0.699906	−0.349953	−	0.936767i	$-0.613803\pi$
$61$	−5.46644	−0.699906	−0.349953	+	0.936767i	$0.613803\pi$
$62$	−4.63064	−0.588092
$63$	4.27693	0.538843
$64$	8.84292	1.10537
$65$	−9.49920	−1.17823
$66$	−1.27883	−0.157413
$67$	7.81166	0.954346	0.477173	−	0.878809i	$-0.341662\pi$
$67$	7.81166	0.954346	0.477173	+	0.878809i	$0.341662\pi$
$68$	−3.00075	−0.363895
$69$	2.04633	0.246349
$70$	−9.31258	−1.11307
$71$	−15.6115	−1.85274	−0.926370	−	0.376615i	$-0.877088\pi$
$71$	−15.6115	−1.85274	−0.926370	+	0.376615i	$0.877088\pi$
$72$	3.06816	0.361586
$73$	−2.54289	−0.297623	−0.148811	−	0.988866i	$-0.547545\pi$
$73$	−2.54289	−0.297623	−0.148811	+	0.988866i	$0.547545\pi$
$74$	0.424110	0.0493018
$75$	−1.76563	−0.203877
$76$	−3.42491	−0.392864
$77$	4.51754	0.514822
$78$	6.39492	0.724083
$79$	10.2680	1.15524	0.577619	−	0.816306i	$-0.303982\pi$
$79$	10.2680	1.15524	0.577619	+	0.816306i	$0.303982\pi$
$80$	−4.75929	−0.532104
$81$	1.00000	0.111111
$82$	4.38571	0.484321
$83$	1.52139	0.166994	0.0834971	−	0.996508i	$-0.473391\pi$
$83$	1.52139	0.166994	0.0834971	+	0.996508i	$0.473391\pi$
$84$	−2.28457	−0.249267
$85$	10.1030	1.09583
$86$	−2.11755	−0.228341
$87$	0	0
$88$	3.24076	0.345467
$89$	−17.9444	−1.90210	−0.951049	−	0.309041i	$-0.899992\pi$
$89$	−17.9444	−1.90210	−0.951049	+	0.309041i	$0.899992\pi$
$90$	−2.17740	−0.229518
$91$	−22.5904	−2.36812
$92$	−1.09307	−0.113960
$93$	3.82471	0.396604
$94$	−8.45087	−0.871641
$95$	11.5311	1.18307
$96$	−2.93233	−0.299279
$97$	−6.80968	−0.691418	−0.345709	−	0.938342i	$-0.612362\pi$
$97$	−6.80968	−0.691418	−0.345709	+	0.938342i	$0.612362\pi$
$98$	−13.6716	−1.38104
$99$	1.05626	0.106158
$100$	0.943132	0.0943132
$101$	4.72573	0.470228	0.235114	−	0.971968i	$-0.424454\pi$
$101$	4.72573	0.470228	0.235114	+	0.971968i	$0.424454\pi$
$102$	−6.80144	−0.673443
$103$	3.49491	0.344364	0.172182	−	0.985065i	$-0.444918\pi$
$103$	3.49491	0.344364	0.172182	+	0.985065i	$0.444918\pi$
$104$	−16.2058	−1.58911
$105$	7.69178	0.750641
$106$	9.46624	0.919442
$107$	−4.92192	−0.475820	−0.237910	−	0.971287i	$-0.576462\pi$
$107$	−4.92192	−0.475820	−0.237910	+	0.971287i	$0.576462\pi$
$108$	−0.534161	−0.0513997
$109$	−14.8847	−1.42569	−0.712846	−	0.701321i	$-0.752594\pi$
$109$	−14.8847	−1.42569	−0.712846	+	0.701321i	$0.752594\pi$
$110$	−2.29989	−0.219286
$111$	−0.350297	−0.0332487
$112$	−11.3182	−1.06947
$113$	−6.34278	−0.596678	−0.298339	−	0.954460i	$-0.596433\pi$
$113$	−6.34278	−0.596678	−0.298339	+	0.954460i	$0.596433\pi$
$114$	−7.76282	−0.727055
$115$	3.68019	0.343180
$116$	0	0
$117$	−5.28193	−0.488314
$118$	−0.463303	−0.0426505
$119$	24.0265	2.20250
$120$	5.51788	0.503711
$121$	−9.88432	−0.898574
$122$	6.61832	0.599195
$123$	−3.62240	−0.326621
$124$	−2.04301	−0.183468
$125$	−12.1675	−1.08830
$126$	−5.17816	−0.461307
$127$	−4.46569	−0.396266	−0.198133	−	0.980175i	$-0.563488\pi$
$127$	−4.46569	−0.396266	−0.198133	+	0.980175i	$0.563488\pi$
$128$	−4.84163	−0.427944
$129$	1.74900	0.153991
$130$	11.5009	1.00869
$131$	0.00995943	0.000870160 0	0.000435080	−	1.00000i	$-0.499862\pi$
$131$	0.00995943	0.000870160 0	0.000435080		1.00000i	$0.499862\pi$
$132$	−0.564212	−0.0491084
$133$	27.4226	2.37784
$134$	−9.45772	−0.817022
$135$	1.79844	0.154785
$136$	17.2359	1.47797
$137$	12.2134	1.04346	0.521732	−	0.853109i	$-0.325286\pi$
$137$	12.2134	1.04346	0.521732	+	0.853109i	$0.325286\pi$
$138$	−2.47753	−0.210901
$139$	4.94561	0.419481	0.209741	−	0.977757i	$-0.432738\pi$
$139$	4.94561	0.419481	0.209741	+	0.977757i	$0.432738\pi$
$140$	−4.10865	−0.347245
$141$	6.98005	0.587826
$142$	18.9011	1.58614
$143$	−5.57908	−0.466546
$144$	−2.64635	−0.220529
$145$	0	0
$146$	3.07873	0.254797
$147$	11.2921	0.931359
$148$	0.187115	0.0153808
$149$	14.0140	1.14807	0.574035	−	0.818831i	$-0.305377\pi$
$149$	14.0140	1.14807	0.574035	+	0.818831i	$0.305377\pi$
$150$	2.13768	0.174541
$151$	2.93392	0.238759	0.119379	−	0.992849i	$-0.461910\pi$
$151$	2.93392	0.238759	0.119379	+	0.992849i	$0.461910\pi$
$152$	19.6722	1.59563
$153$	5.61769	0.454163
$154$	−5.46947	−0.440742
$155$	6.87849	0.552493
$156$	2.82140	0.225893
$157$	16.1919	1.29226	0.646128	−	0.763229i	$-0.276387\pi$
$157$	16.1919	1.29226	0.646128	+	0.763229i	$0.276387\pi$
$158$	−12.4316	−0.989008
$159$	−7.81869	−0.620063
$160$	−5.27360	−0.416915
$161$	8.75201	0.689755
$162$	−1.21072	−0.0951230
$163$	13.5734	1.06315	0.531575	−	0.847011i	$-0.321600\pi$
$163$	13.5734	1.06315	0.531575	+	0.847011i	$0.321600\pi$
$164$	1.93495	0.151094
$165$	1.89961	0.147885
$166$	−1.84197	−0.142965
$167$	−16.3616	−1.26610	−0.633050	−	0.774111i	$-0.718197\pi$
$167$	−16.3616	−1.26610	−0.633050	+	0.774111i	$0.718197\pi$
$168$	13.1223	1.01241
$169$	14.8987	1.14606
$170$	−12.2319	−0.938147
$171$	6.41175	0.490319
$172$	−0.934250	−0.0712359
$173$	−4.46374	−0.339372	−0.169686	−	0.985498i	$-0.554275\pi$
$173$	−4.46374	−0.339372	−0.169686	+	0.985498i	$0.554275\pi$
$174$	0	0
$175$	−7.55148	−0.570838
$176$	−2.79523	−0.210698
$177$	0.382668	0.0287631
$178$	21.7256	1.62840
$179$	3.40063	0.254175	0.127088	−	0.991891i	$-0.459437\pi$
$179$	3.40063	0.254175	0.127088	+	0.991891i	$0.459437\pi$
$180$	−0.960655	−0.0716030
$181$	6.62134	0.492160	0.246080	−	0.969250i	$-0.420857\pi$
$181$	6.62134	0.492160	0.246080	+	0.969250i	$0.420857\pi$
$182$	27.3506	2.02736
$183$	−5.46644	−0.404091
$184$	6.27846	0.462854
$185$	−0.629986	−0.0463175
$186$	−4.63064	−0.339535
$187$	5.93373	0.433917
$188$	−3.72847	−0.271927
$189$	4.27693	0.311101
$190$	−13.9609	−1.01283
$191$	17.6196	1.27491	0.637454	−	0.770489i	$-0.279988\pi$
$191$	17.6196	1.27491	0.637454	+	0.770489i	$0.279988\pi$
$192$	8.84292	0.638183
$193$	−13.3986	−0.964454	−0.482227	−	0.876046i	$-0.660172\pi$
$193$	−13.3986	−0.964454	−0.482227	+	0.876046i	$0.660172\pi$
$194$	8.24460	0.591928
$195$	−9.49920	−0.680252
$196$	−6.03182	−0.430844
$197$	−8.20476	−0.584565	−0.292282	−	0.956332i	$-0.594415\pi$
$197$	−8.20476	−0.584565	−0.292282	+	0.956332i	$0.594415\pi$
$198$	−1.27883	−0.0908825
$199$	24.4954	1.73643	0.868216	−	0.496187i	$-0.165267\pi$
$199$	24.4954	1.73643	0.868216	+	0.496187i	$0.165267\pi$
$200$	−5.41723	−0.383056
$201$	7.81166	0.550992
$202$	−5.72153	−0.402565
$203$	0	0
$204$	−3.00075	−0.210095
$205$	−6.51466	−0.455004
$206$	−4.23135	−0.294812
$207$	2.04633	0.142230
$208$	13.9778	0.969187
$209$	6.77246	0.468461
$210$	−9.31258	−0.642629
$211$	−2.21143	−0.152241	−0.0761205	−	0.997099i	$-0.524253\pi$
$211$	−2.21143	−0.152241	−0.0761205	+	0.997099i	$0.524253\pi$
$212$	4.17644	0.286839
$213$	−15.6115	−1.06968
$214$	5.95905	0.407353
$215$	3.14547	0.214519
$216$	3.06816	0.208762
$217$	16.3580	1.11045
$218$	18.0211	1.22054
$219$	−2.54289	−0.171833
$220$	−1.01470	−0.0684110
$221$	−29.6722	−1.99597
$222$	0.424110	0.0284644
$223$	2.70437	0.181098	0.0905491	−	0.995892i	$-0.471138\pi$
$223$	2.70437	0.181098	0.0905491	+	0.995892i	$0.471138\pi$
$224$	−12.5414	−0.837955
$225$	−1.76563	−0.117709
$226$	7.67932	0.510820
$227$	14.4225	0.957258	0.478629	−	0.878017i	$-0.341134\pi$
$227$	14.4225	0.957258	0.478629	+	0.878017i	$0.341134\pi$
$228$	−3.42491	−0.226820
$229$	25.7467	1.70139	0.850694	−	0.525662i	$-0.176182\pi$
$229$	25.7467	1.70139	0.850694	+	0.525662i	$0.176182\pi$
$230$	−4.45567	−0.293798
$231$	4.51754	0.297232
$232$	0	0
$233$	−22.6039	−1.48083	−0.740417	−	0.672148i	$-0.765372\pi$
$233$	−22.6039	−1.48083	−0.740417	+	0.672148i	$0.765372\pi$
$234$	6.39492	0.418049
$235$	12.5532	0.818878
$236$	−0.204406	−0.0133057
$237$	10.2680	0.666977
$238$	−29.0893	−1.88558
$239$	23.4233	1.51513	0.757565	−	0.652760i	$-0.226389\pi$
$239$	23.4233	1.51513	0.757565	+	0.652760i	$0.226389\pi$
$240$	−4.75929	−0.307211
$241$	22.3826	1.44179	0.720896	−	0.693043i	$-0.243730\pi$
$241$	22.3826	1.44179	0.720896	+	0.693043i	$0.243730\pi$
$242$	11.9671	0.769276
$243$	1.00000	0.0641500
$244$	2.91996	0.186932
$245$	20.3082	1.29744
$246$	4.38571	0.279623
$247$	−33.8664	−2.15487
$248$	11.7348	0.745160
$249$	1.52139	0.0964141
$250$	14.7315	0.931700
$251$	−5.95981	−0.376180	−0.188090	−	0.982152i	$-0.560230\pi$
$251$	−5.95981	−0.376180	−0.188090	+	0.982152i	$0.560230\pi$
$252$	−2.28457	−0.143914
$253$	2.16145	0.135889
$254$	5.40669	0.339246
$255$	10.1030	0.632677
$256$	−11.8240	−0.739000
$257$	−15.0723	−0.940185	−0.470092	−	0.882617i	$-0.655779\pi$
$257$	−15.0723	−0.940185	−0.470092	+	0.882617i	$0.655779\pi$
$258$	−2.11755	−0.131833
$259$	−1.49819	−0.0930932
$260$	5.07411	0.314683
$261$	0	0
$262$	−0.0120581	−0.000744950 0
$263$	−18.7252	−1.15465	−0.577324	−	0.816515i	$-0.695903\pi$
$263$	−18.7252	−1.15465	−0.577324	+	0.816515i	$0.695903\pi$
$264$	3.24076	0.199455
$265$	−14.0614	−0.863786
$266$	−33.2010	−2.03569
$267$	−17.9444	−1.09818
$268$	−4.17269	−0.254887
$269$	−23.1259	−1.41001	−0.705006	−	0.709202i	$-0.749056\pi$
$269$	−23.1259	−1.41001	−0.705006	+	0.709202i	$0.749056\pi$
$270$	−2.17740	−0.132512
$271$	21.3469	1.29673	0.648365	−	0.761330i	$-0.275453\pi$
$271$	21.3469	1.29673	0.648365	+	0.761330i	$0.275453\pi$
$272$	−14.8664	−0.901406
$273$	−22.5904	−1.36723
$274$	−14.7870	−0.893317
$275$	−1.86496	−0.112461
$276$	−1.09307	−0.0657951
$277$	−8.32181	−0.500009	−0.250004	−	0.968245i	$-0.580432\pi$
$277$	−8.32181	−0.500009	−0.250004	+	0.968245i	$0.580432\pi$
$278$	−5.98774	−0.359121
$279$	3.82471	0.228979
$280$	23.5996	1.41034
$281$	−28.9202	−1.72524	−0.862618	−	0.505856i	$-0.831177\pi$
$281$	−28.9202	−1.72524	−0.862618	+	0.505856i	$0.831177\pi$
$282$	−8.45087	−0.503242
$283$	29.9799	1.78212	0.891060	−	0.453885i	$-0.149962\pi$
$283$	29.9799	1.78212	0.891060	+	0.453885i	$0.149962\pi$
$284$	8.33904	0.494831
$285$	11.5311	0.683044
$286$	6.75469	0.399413
$287$	−15.4928	−0.914509
$288$	−2.93233	−0.172789
$289$	14.5584	0.856378
$290$	0	0
$291$	−6.80968	−0.399191
$292$	1.35831	0.0794893
$293$	8.75406	0.511418	0.255709	−	0.966754i	$-0.417691\pi$
$293$	8.75406	0.511418	0.255709	+	0.966754i	$0.417691\pi$
$294$	−13.6716	−0.797343
$295$	0.688204	0.0400688
$296$	−1.07476	−0.0624694
$297$	1.05626	0.0612903
$298$	−16.9670	−0.982871
$299$	−10.8086	−0.625075
$300$	0.943132	0.0544517
$301$	7.48036	0.431161
$302$	−3.55215	−0.204403
$303$	4.72573	0.271486
$304$	−16.9677	−0.973166
$305$	−9.83105	−0.562924
$306$	−6.80144	−0.388812
$307$	−33.9161	−1.93570	−0.967848	−	0.251536i	$-0.919064\pi$
$307$	−33.9161	−1.93570	−0.967848	+	0.251536i	$0.919064\pi$
$308$	−2.41310	−0.137499
$309$	3.49491	0.198819
$310$	−8.32791	−0.472993
$311$	8.12574	0.460769	0.230384	−	0.973100i	$-0.426002\pi$
$311$	8.12574	0.460769	0.230384	+	0.973100i	$0.426002\pi$
$312$	−16.2058	−0.917471
$313$	2.52738	0.142856	0.0714280	−	0.997446i	$-0.477244\pi$
$313$	2.52738	0.142856	0.0714280	+	0.997446i	$0.477244\pi$
$314$	−19.6039	−1.10631
$315$	7.69178	0.433383
$316$	−5.48476	−0.308542
$317$	16.6383	0.934499	0.467250	−	0.884125i	$-0.345245\pi$
$317$	16.6383	0.934499	0.467250	+	0.884125i	$0.345245\pi$
$318$	9.46624	0.530840
$319$	0	0
$320$	15.9034	0.889028
$321$	−4.92192	−0.274715
$322$	−10.5962	−0.590504
$323$	36.0192	2.00416
$324$	−0.534161	−0.0296756
$325$	9.32593	0.517309
$326$	−16.4336	−0.910170
$327$	−14.8847	−0.823123
$328$	−11.1141	−0.613673
$329$	29.8532	1.64586
$330$	−2.29989	−0.126605
$331$	12.9286	0.710619	0.355309	−	0.934749i	$-0.384375\pi$
$331$	12.9286	0.710619	0.355309	+	0.934749i	$0.384375\pi$
$332$	−0.812667	−0.0446009
$333$	−0.350297	−0.0191961
$334$	19.8093	1.08392
$335$	14.0488	0.767566
$336$	−11.3182	−0.617461
$337$	−5.56273	−0.303021	−0.151511	−	0.988456i	$-0.548414\pi$
$337$	−5.56273	−0.303021	−0.151511	+	0.988456i	$0.548414\pi$
$338$	−18.0382	−0.981148
$339$	−6.34278	−0.344492
$340$	−5.39666	−0.292675
$341$	4.03988	0.218772
$342$	−7.76282	−0.419765
$343$	18.3571	0.991192
$344$	5.36621	0.289327
$345$	3.68019	0.198135
$346$	5.40433	0.290539
$347$	−8.78686	−0.471703	−0.235852	−	0.971789i	$-0.575788\pi$
$347$	−8.78686	−0.471703	−0.235852	+	0.971789i	$0.575788\pi$
$348$	0	0
$349$	15.1943	0.813332	0.406666	−	0.913577i	$-0.366691\pi$
$349$	15.1943	0.813332	0.406666	+	0.913577i	$0.366691\pi$
$350$	9.14271	0.488698
$351$	−5.28193	−0.281928
$352$	−3.09729	−0.165086
$353$	−12.3667	−0.658214	−0.329107	−	0.944293i	$-0.606748\pi$
$353$	−12.3667	−0.658214	−0.329107	+	0.944293i	$0.606748\pi$
$354$	−0.463303	−0.0246243
$355$	−28.0762	−1.49013
$356$	9.58518	0.508014
$357$	24.0265	1.27161
$358$	−4.11721	−0.217601
$359$	−16.0164	−0.845312	−0.422656	−	0.906290i	$-0.638902\pi$
$359$	−16.0164	−0.845312	−0.422656	+	0.906290i	$0.638902\pi$
$360$	5.51788	0.290818
$361$	22.1105	1.16371
$362$	−8.01657	−0.421342
$363$	−9.88432	−0.518792
$364$	12.0669	0.632479
$365$	−4.57323	−0.239374
$366$	6.61832	0.345945
$367$	−22.2447	−1.16116	−0.580581	−	0.814203i	$-0.697174\pi$
$367$	−22.2447	−1.16116	−0.580581	+	0.814203i	$0.697174\pi$
$368$	−5.41530	−0.282292
$369$	−3.62240	−0.188575
$370$	0.762735	0.0396527
$371$	−33.4400	−1.73612
$372$	−2.04301	−0.105925
$373$	6.93029	0.358837	0.179418	−	0.983773i	$-0.442578\pi$
$373$	6.93029	0.358837	0.179418	+	0.983773i	$0.442578\pi$
$374$	−7.18407	−0.371480
$375$	−12.1675	−0.628330
$376$	21.4159	1.10444
$377$	0	0
$378$	−5.17816	−0.266336
$379$	−7.78848	−0.400067	−0.200034	−	0.979789i	$-0.564105\pi$
$379$	−7.78848	−0.400067	−0.200034	+	0.979789i	$0.564105\pi$
$380$	−6.15948	−0.315975
$381$	−4.46569	−0.228784
$382$	−21.3323	−1.09146
$383$	25.4625	1.30107	0.650537	−	0.759475i	$-0.274544\pi$
$383$	25.4625	1.30107	0.650537	+	0.759475i	$0.274544\pi$
$384$	−4.84163	−0.247073
$385$	8.12451	0.414063
$386$	16.2220	0.825676
$387$	1.74900	0.0889068
$388$	3.63747	0.184664
$389$	10.9838	0.556902	0.278451	−	0.960450i	$-0.410179\pi$
$389$	10.9838	0.556902	0.278451	+	0.960450i	$0.410179\pi$
$390$	11.5009	0.582369
$391$	11.4956	0.581359
$392$	34.6460	1.74989
$393$	0.00995943	0.000502387 0
$394$	9.93365	0.500450
$395$	18.4663	0.929140
$396$	−0.564212	−0.0283527
$397$	5.10047	0.255985	0.127993	−	0.991775i	$-0.459147\pi$
$397$	5.10047	0.255985	0.127993	+	0.991775i	$0.459147\pi$
$398$	−29.6570	−1.48657
$399$	27.4226	1.37285
$400$	4.67247	0.233624
$401$	−19.5421	−0.975886	−0.487943	−	0.872875i	$-0.662253\pi$
$401$	−19.5421	−0.975886	−0.487943	+	0.872875i	$0.662253\pi$
$402$	−9.45772	−0.471708
$403$	−20.2018	−1.00632
$404$	−2.52430	−0.125589
$405$	1.79844	0.0893650
$406$	0	0
$407$	−0.370004	−0.0183404
$408$	17.2359	0.853306
$409$	−17.9244	−0.886305	−0.443152	−	0.896446i	$-0.646140\pi$
$409$	−17.9244	−0.886305	−0.443152	+	0.896446i	$0.646140\pi$
$410$	7.88742	0.389532
$411$	12.2134	0.602444
$412$	−1.86685	−0.0919730
$413$	1.63664	0.0805340
$414$	−2.47753	−0.121764
$415$	2.73612	0.134311
$416$	15.4883	0.759378
$417$	4.94561	0.242188
$418$	−8.19954	−0.401053
$419$	3.73821	0.182623	0.0913116	−	0.995822i	$-0.470894\pi$
$419$	3.73821	0.182623	0.0913116	+	0.995822i	$0.470894\pi$
$420$	−4.10865	−0.200482
$421$	8.28779	0.403922	0.201961	−	0.979394i	$-0.435269\pi$
$421$	8.28779	0.403922	0.201961	+	0.979394i	$0.435269\pi$
$422$	2.67742	0.130335
$423$	6.98005	0.339382
$424$	−23.9890	−1.16501
$425$	−9.91876	−0.481131
$426$	18.9011	0.915761
$427$	−23.3796	−1.13142
$428$	2.62910	0.127082
$429$	−5.57908	−0.269360
$430$	−3.80828	−0.183651
$431$	19.1294	0.921432	0.460716	−	0.887548i	$-0.347593\pi$
$431$	19.1294	0.921432	0.460716	+	0.887548i	$0.347593\pi$
$432$	−2.64635	−0.127323
$433$	−16.1509	−0.776163	−0.388081	−	0.921625i	$-0.626862\pi$
$433$	−16.1509	−0.776163	−0.388081	+	0.921625i	$0.626862\pi$
$434$	−19.8049	−0.950667
$435$	0	0
$436$	7.95081	0.380775
$437$	13.1206	0.627641
$438$	3.07873	0.147107
$439$	2.96836	0.141672	0.0708360	−	0.997488i	$-0.477433\pi$
$439$	2.96836	0.141672	0.0708360	+	0.997488i	$0.477433\pi$
$440$	5.82830	0.277853
$441$	11.2921	0.537720
$442$	35.9247	1.70876
$443$	10.4916	0.498471	0.249235	−	0.968443i	$-0.419821\pi$
$443$	10.4916	0.498471	0.249235	+	0.968443i	$0.419821\pi$
$444$	0.187115	0.00888008
$445$	−32.2718	−1.52983
$446$	−3.27423	−0.155039
$447$	14.0140	0.662839
$448$	37.8206	1.78685
$449$	−4.40325	−0.207802	−0.103901	−	0.994588i	$-0.533133\pi$
$449$	−4.40325	−0.207802	−0.103901	+	0.994588i	$0.533133\pi$
$450$	2.13768	0.100771
$451$	−3.82619	−0.180168
$452$	3.38807	0.159361
$453$	2.93392	0.137847
$454$	−17.4616	−0.819515
$455$	−40.6274	−1.90464
$456$	19.6722	0.921237
$457$	11.3485	0.530860	0.265430	−	0.964130i	$-0.414486\pi$
$457$	11.3485	0.530860	0.265430	+	0.964130i	$0.414486\pi$
$458$	−31.1720	−1.45657
$459$	5.61769	0.262211
$460$	−1.96582	−0.0916566
$461$	19.7092	0.917947	0.458973	−	0.888450i	$-0.348217\pi$
$461$	19.7092	0.917947	0.458973	+	0.888450i	$0.348217\pi$
$462$	−5.46947	−0.254463
$463$	0.753315	0.0350095	0.0175048	−	0.999847i	$-0.494428\pi$
$463$	0.753315	0.0350095	0.0175048	+	0.999847i	$0.494428\pi$
$464$	0	0
$465$	6.87849	0.318982
$466$	27.3670	1.26775
$467$	10.4450	0.483339	0.241670	−	0.970359i	$-0.422305\pi$
$467$	10.4450	0.483339	0.241670	+	0.970359i	$0.422305\pi$
$468$	2.82140	0.130419
$469$	33.4099	1.54273
$470$	−15.1983	−0.701047
$471$	16.1919	0.746085
$472$	1.17408	0.0540416
$473$	1.84740	0.0849435
$474$	−12.4316	−0.571004
$475$	−11.3208	−0.519433
$476$	−12.8340	−0.588246
$477$	−7.81869	−0.357993
$478$	−28.3591	−1.29711
$479$	−13.9781	−0.638673	−0.319337	−	0.947641i	$-0.603460\pi$
$479$	−13.9781	−0.638673	−0.319337	+	0.947641i	$0.603460\pi$
$480$	−5.27360	−0.240706
$481$	1.85024	0.0843637
$482$	−27.0991	−1.23433
$483$	8.75201	0.398230
$484$	5.27982	0.239992
$485$	−12.2468	−0.556097
$486$	−1.21072	−0.0549193
$487$	−1.46028	−0.0661718	−0.0330859	−	0.999453i	$-0.510533\pi$
$487$	−1.46028	−0.0661718	−0.0330859	+	0.999453i	$0.510533\pi$
$488$	−16.7719	−0.759228
$489$	13.5734	0.613810
$490$	−24.5875	−1.11075
$491$	−12.1522	−0.548421	−0.274210	−	0.961670i	$-0.588416\pi$
$491$	−12.1522	−0.548421	−0.274210	+	0.961670i	$0.588416\pi$
$492$	1.93495	0.0872342
$493$	0	0
$494$	41.0027	1.84480
$495$	1.89961	0.0853812
$496$	−10.1215	−0.454469
$497$	−66.7691	−2.99501
$498$	−1.84197	−0.0825408
$499$	−35.9679	−1.61014	−0.805071	−	0.593178i	$-0.797873\pi$
$499$	−35.9679	−1.61014	−0.805071	+	0.593178i	$0.797873\pi$
$500$	6.49943	0.290664
$501$	−16.3616	−0.730983
$502$	7.21565	0.322050
$503$	−14.3546	−0.640041	−0.320020	−	0.947411i	$-0.603690\pi$
$503$	−14.3546	−0.640041	−0.320020	+	0.947411i	$0.603690\pi$
$504$	13.1223	0.584513
$505$	8.49892	0.378197
$506$	−2.61691	−0.116336
$507$	14.8987	0.661676
$508$	2.38540	0.105835
$509$	−31.0049	−1.37427	−0.687135	−	0.726530i	$-0.741132\pi$
$509$	−31.0049	−1.37427	−0.687135	+	0.726530i	$0.741132\pi$
$510$	−12.2319	−0.541640
$511$	−10.8758	−0.481116
$512$	23.9988	1.06061
$513$	6.41175	0.283086
$514$	18.2483	0.804899
$515$	6.28537	0.276967
$516$	−0.934250	−0.0411281
$517$	7.37273	0.324252
$518$	1.81389	0.0796978
$519$	−4.46374	−0.195936
$520$	−29.1450	−1.27809
$521$	−14.9084	−0.653150	−0.326575	−	0.945171i	$-0.605895\pi$
$521$	−14.9084	−0.653150	−0.326575	+	0.945171i	$0.605895\pi$
$522$	0	0
$523$	−12.9505	−0.566284	−0.283142	−	0.959078i	$-0.591377\pi$
$523$	−12.9505	−0.566284	−0.283142	+	0.959078i	$0.591377\pi$
$524$	−0.00531995	−0.000232403 0
$525$	−7.55148	−0.329573
$526$	22.6710	0.988502
$527$	21.4860	0.935945
$528$	−2.79523	−0.121647
$529$	−18.8125	−0.817936
$530$	17.0244	0.739493
$531$	0.382668	0.0166064
$532$	−14.6481	−0.635076
$533$	19.1333	0.828754
$534$	21.7256	0.940157
$535$	−8.85175	−0.382694
$536$	23.9674	1.03523
$537$	3.40063	0.146748
$538$	27.9990	1.20712
$539$	11.9274	0.513750
$540$	−0.960655	−0.0413400
$541$	10.8732	0.467475	0.233737	−	0.972300i	$-0.424904\pi$
$541$	10.8732	0.467475	0.233737	+	0.972300i	$0.424904\pi$
$542$	−25.8450	−1.11014
$543$	6.62134	0.284149
$544$	−16.4729	−0.706270
$545$	−26.7691	−1.14666
$546$	27.3506	1.17050
$547$	−23.1301	−0.988970	−0.494485	−	0.869186i	$-0.664643\pi$
$547$	−23.1301	−0.988970	−0.494485	+	0.869186i	$0.664643\pi$
$548$	−6.52394	−0.278689
$549$	−5.46644	−0.233302
$550$	2.25794	0.0962790
$551$	0	0
$552$	6.27846	0.267229
$553$	43.9154	1.86747
$554$	10.0754	0.428061
$555$	−0.629986	−0.0267414
$556$	−2.64175	−0.112035
$557$	−43.1297	−1.82746	−0.913731	−	0.406319i	$-0.866812\pi$
$557$	−43.1297	−1.82746	−0.913731	+	0.406319i	$0.866812\pi$
$558$	−4.63064	−0.196031
$559$	−9.23811	−0.390730
$560$	−20.3551	−0.860162
$561$	5.93373	0.250522
$562$	35.0142	1.47699
$563$	34.1036	1.43729	0.718647	−	0.695375i	$-0.244762\pi$
$563$	34.1036	1.43729	0.718647	+	0.695375i	$0.244762\pi$
$564$	−3.72847	−0.156997
$565$	−11.4071	−0.479899
$566$	−36.2972	−1.52569
$567$	4.27693	0.179614
$568$	−47.8984	−2.00977
$569$	−19.0810	−0.799917	−0.399958	−	0.916533i	$-0.630975\pi$
$569$	−19.0810	−0.799917	−0.399958	+	0.916533i	$0.630975\pi$
$570$	−13.9609	−0.584759
$571$	−6.56641	−0.274796	−0.137398	−	0.990516i	$-0.543874\pi$
$571$	−6.56641	−0.274796	−0.137398	+	0.990516i	$0.543874\pi$
$572$	2.98013	0.124605
$573$	17.6196	0.736068
$574$	18.7574	0.782918
$575$	−3.61306	−0.150675
$576$	8.84292	0.368455
$577$	−20.5602	−0.855934	−0.427967	−	0.903794i	$-0.640770\pi$
$577$	−20.5602	−0.855934	−0.427967	+	0.903794i	$0.640770\pi$
$578$	−17.6261	−0.733151
$579$	−13.3986	−0.556828
$580$	0	0
$581$	6.50687	0.269951
$582$	8.24460	0.341750
$583$	−8.25856	−0.342035
$584$	−7.80199	−0.322849
$585$	−9.49920	−0.392744
$586$	−10.5987	−0.437828
$587$	−9.69142	−0.400008	−0.200004	−	0.979795i	$-0.564095\pi$
$587$	−9.69142	−0.400008	−0.200004	+	0.979795i	$0.564095\pi$
$588$	−6.03182	−0.248748
$589$	24.5231	1.01046
$590$	−0.833221	−0.0343032
$591$	−8.20476	−0.337499
$592$	0.927007	0.0380997
$593$	−13.1810	−0.541279	−0.270639	−	0.962681i	$-0.587235\pi$
$593$	−13.1810	−0.541279	−0.270639	+	0.962681i	$0.587235\pi$
$594$	−1.27883	−0.0524711
$595$	43.2100	1.77144
$596$	−7.48573	−0.306627
$597$	24.4954	1.00253
$598$	13.0861	0.535131
$599$	−14.4397	−0.589991	−0.294995	−	0.955499i	$-0.595318\pi$
$599$	−14.4397	−0.589991	−0.294995	+	0.955499i	$0.595318\pi$
$600$	−5.41723	−0.221157
$601$	4.88823	0.199395	0.0996976	−	0.995018i	$-0.468212\pi$
$601$	4.88823	0.199395	0.0996976	+	0.995018i	$0.468212\pi$
$602$	−9.05661	−0.369120
$603$	7.81166	0.318115
$604$	−1.56718	−0.0637678
$605$	−17.7763	−0.722710
$606$	−5.72153	−0.232421
$607$	−12.5548	−0.509583	−0.254792	−	0.966996i	$-0.582007\pi$
$607$	−12.5548	−0.509583	−0.254792	+	0.966996i	$0.582007\pi$
$608$	−18.8014	−0.762496
$609$	0	0
$610$	11.9026	0.481923
$611$	−36.8681	−1.49152
$612$	−3.00075	−0.121298
$613$	3.49761	0.141267	0.0706336	−	0.997502i	$-0.477498\pi$
$613$	3.49761	0.141267	0.0706336	+	0.997502i	$0.477498\pi$
$614$	41.0629	1.65716
$615$	−6.51466	−0.262696
$616$	13.8605	0.558456
$617$	−27.2896	−1.09864	−0.549318	−	0.835613i	$-0.685113\pi$
$617$	−27.2896	−1.09864	−0.549318	+	0.835613i	$0.685113\pi$
$618$	−4.23135	−0.170210
$619$	−25.5410	−1.02658	−0.513289	−	0.858216i	$-0.671573\pi$
$619$	−25.5410	−1.02658	−0.513289	+	0.858216i	$0.671573\pi$
$620$	−3.67422	−0.147560
$621$	2.04633	0.0821164
$622$	−9.83798	−0.394467
$623$	−76.7467	−3.07479
$624$	13.9778	0.559561
$625$	−13.0544	−0.522176
$626$	−3.05995	−0.122300
$627$	6.77246	0.270466
$628$	−8.64911	−0.345137
$629$	−1.96786	−0.0784636
$630$	−9.31258	−0.371022
$631$	20.7956	0.827858	0.413929	−	0.910309i	$-0.364156\pi$
$631$	20.7956	0.827858	0.413929	+	0.910309i	$0.364156\pi$
$632$	31.5038	1.25315
$633$	−2.21143	−0.0878963
$634$	−20.1443	−0.800031
$635$	−8.03126	−0.318711
$636$	4.17644	0.165607
$637$	−59.6442	−2.36319
$638$	0	0
$639$	−15.6115	−0.617580
$640$	−8.70736	−0.344188
$641$	−14.1393	−0.558468	−0.279234	−	0.960223i	$-0.590080\pi$
$641$	−14.1393	−0.558468	−0.279234	+	0.960223i	$0.590080\pi$
$642$	5.95905	0.235185
$643$	35.4943	1.39976	0.699878	−	0.714262i	$-0.253238\pi$
$643$	35.4943	1.39976	0.699878	+	0.714262i	$0.253238\pi$
$644$	−4.67498	−0.184220
$645$	3.14547	0.123853
$646$	−43.6091	−1.71578
$647$	−20.0626	−0.788743	−0.394371	−	0.918951i	$-0.629038\pi$
$647$	−20.0626	−0.788743	−0.394371	+	0.918951i	$0.629038\pi$
$648$	3.06816	0.120529
$649$	0.404196	0.0158661
$650$	−11.2911	−0.442872
$651$	16.3580	0.641121
$652$	−7.25038	−0.283947
$653$	13.3537	0.522569	0.261285	−	0.965262i	$-0.415854\pi$
$653$	13.3537	0.522569	0.261285	+	0.965262i	$0.415854\pi$
$654$	18.0211	0.704682
$655$	0.0179114	0.000699856 0
$656$	9.58615	0.374276
$657$	−2.54289	−0.0992077
$658$	−36.1438	−1.40903
$659$	11.3791	0.443267	0.221633	−	0.975130i	$-0.428861\pi$
$659$	11.3791	0.443267	0.221633	+	0.975130i	$0.428861\pi$
$660$	−1.01470	−0.0394971
$661$	−37.0298	−1.44029	−0.720145	−	0.693823i	$-0.755925\pi$
$661$	−37.0298	−1.44029	−0.720145	+	0.693823i	$0.755925\pi$
$662$	−15.6529	−0.608366
$663$	−29.6722	−1.15237
$664$	4.66786	0.181148
$665$	49.3178	1.91246
$666$	0.424110	0.0164339
$667$	0	0
$668$	8.73974	0.338151
$669$	2.70437	0.104557
$670$	−17.0091	−0.657119
$671$	−5.77398	−0.222902
$672$	−12.5414	−0.483794
$673$	37.2436	1.43563	0.717817	−	0.696232i	$-0.245141\pi$
$673$	37.2436	1.43563	0.717817	+	0.696232i	$0.245141\pi$
$674$	6.73490	0.259419
$675$	−1.76563	−0.0679591
$676$	−7.95833	−0.306090
$677$	44.9350	1.72699	0.863496	−	0.504356i	$-0.168270\pi$
$677$	44.9350	1.72699	0.863496	+	0.504356i	$0.168270\pi$
$678$	7.67932	0.294922
$679$	−29.1245	−1.11770
$680$	30.9977	1.18871
$681$	14.4225	0.552673
$682$	−4.89115	−0.187292
$683$	22.6022	0.864848	0.432424	−	0.901670i	$-0.357658\pi$
$683$	22.6022	0.864848	0.432424	+	0.901670i	$0.357658\pi$
$684$	−3.42491	−0.130955
$685$	21.9651	0.839242
$686$	−22.2253	−0.848567
$687$	25.7467	0.982296
$688$	−4.62847	−0.176459
$689$	41.2978	1.57332
$690$	−4.45567	−0.169625
$691$	−38.8727	−1.47879	−0.739394	−	0.673273i	$-0.764888\pi$
$691$	−38.8727	−1.47879	−0.739394	+	0.673273i	$0.764888\pi$
$692$	2.38436	0.0906397
$693$	4.51754	0.171607
$694$	10.6384	0.403828
$695$	8.89436	0.337382
$696$	0	0
$697$	−20.3495	−0.770794
$698$	−18.3960	−0.696299
$699$	−22.6039	−0.854960
$700$	4.03371	0.152460
$701$	−39.4427	−1.48973	−0.744865	−	0.667215i	$-0.767486\pi$
$701$	−39.4427	−1.48973	−0.744865	+	0.667215i	$0.767486\pi$
$702$	6.39492	0.241361
$703$	−2.24601	−0.0847100
$704$	9.34041	0.352030
$705$	12.5532	0.472780
$706$	14.9726	0.563501
$707$	20.2116	0.760136
$708$	−0.204406	−0.00768207
$709$	−4.07382	−0.152996	−0.0764978	−	0.997070i	$-0.524374\pi$
$709$	−4.07382	−0.152996	−0.0764978	+	0.997070i	$0.524374\pi$
$710$	33.9924	1.27571
$711$	10.2680	0.385079
$712$	−55.0561	−2.06331
$713$	7.82661	0.293109
$714$	−29.0893	−1.08864
$715$	−10.0336	−0.375236
$716$	−1.81649	−0.0678853
$717$	23.4233	0.874760
$718$	19.3913	0.723677
$719$	−18.4303	−0.687335	−0.343667	−	0.939091i	$-0.611669\pi$
$719$	−18.4303	−0.687335	−0.343667	+	0.939091i	$0.611669\pi$
$720$	−4.75929	−0.177368
$721$	14.9475	0.556674
$722$	−26.7696	−0.996263
$723$	22.3826	0.832419
$724$	−3.53686	−0.131446
$725$	0	0
$726$	11.9671	0.444142
$727$	17.6150	0.653306	0.326653	−	0.945144i	$-0.394079\pi$
$727$	17.6150	0.653306	0.326653	+	0.945144i	$0.394079\pi$
$728$	−69.3109	−2.56883
$729$	1.00000	0.0370370
$730$	5.53689	0.204929
$731$	9.82535	0.363404
$732$	2.91996	0.107925
$733$	−8.22724	−0.303880	−0.151940	−	0.988390i	$-0.548552\pi$
$733$	−8.22724	−0.303880	−0.151940	+	0.988390i	$0.548552\pi$
$734$	26.9320	0.994079
$735$	20.3082	0.749078
$736$	−6.00051	−0.221182
$737$	8.25113	0.303934
$738$	4.38571	0.161440
$739$	35.8226	1.31776	0.658878	−	0.752250i	$-0.271031\pi$
$739$	35.8226	1.31776	0.658878	+	0.752250i	$0.271031\pi$
$740$	0.336514	0.0123705
$741$	−33.8664	−1.24411
$742$	40.4864	1.48630
$743$	9.29461	0.340986	0.170493	−	0.985359i	$-0.445464\pi$
$743$	9.29461	0.340986	0.170493	+	0.985359i	$0.445464\pi$
$744$	11.7348	0.430218
$745$	25.2032	0.923375
$746$	−8.39063	−0.307203
$747$	1.52139	0.0556647
$748$	−3.16957	−0.115891
$749$	−21.0507	−0.769175
$750$	14.7315	0.537917
$751$	−44.8712	−1.63737	−0.818687	−	0.574240i	$-0.805298\pi$
$751$	−44.8712	−1.63737	−0.818687	+	0.574240i	$0.805298\pi$
$752$	−18.4716	−0.673592
$753$	−5.95981	−0.217188
$754$	0	0
$755$	5.27646	0.192030
$756$	−2.28457	−0.0830890
$757$	21.6178	0.785712	0.392856	−	0.919600i	$-0.371487\pi$
$757$	21.6178	0.785712	0.392856	+	0.919600i	$0.371487\pi$
$758$	9.42966	0.342501
$759$	2.16145	0.0784557
$760$	35.3793	1.28334
$761$	−39.3758	−1.42737	−0.713685	−	0.700466i	$-0.752975\pi$
$761$	−39.3758	−1.42737	−0.713685	+	0.700466i	$0.752975\pi$
$762$	5.40669	0.195864
$763$	−63.6606	−2.30467
$764$	−9.41169	−0.340503
$765$	10.1030	0.365276
$766$	−30.8279	−1.11386
$767$	−2.02122	−0.0729822
$768$	−11.8240	−0.426662
$769$	4.29550	0.154900	0.0774499	−	0.996996i	$-0.475322\pi$
$769$	4.29550	0.154900	0.0774499	+	0.996996i	$0.475322\pi$
$770$	−9.83649	−0.354482
$771$	−15.0723	−0.542816
$772$	7.15703	0.257587
$773$	27.9002	1.00350	0.501751	−	0.865012i	$-0.332690\pi$
$773$	27.9002	1.00350	0.501751	+	0.865012i	$0.332690\pi$
$774$	−2.11755	−0.0761138
$775$	−6.75302	−0.242576
$776$	−20.8932	−0.750021
$777$	−1.49819	−0.0537474
$778$	−13.2983	−0.476768
$779$	−23.2260	−0.832156
$780$	5.07411	0.181682
$781$	−16.4897	−0.590049
$782$	−13.9180	−0.497706
$783$	0	0
$784$	−29.8829	−1.06725
$785$	29.1201	1.03934
$786$	−0.0120581	−0.000430097 0
$787$	−5.90689	−0.210558	−0.105279	−	0.994443i	$-0.533574\pi$
$787$	−5.90689	−0.210558	−0.105279	+	0.994443i	$0.533574\pi$
$788$	4.38266	0.156126
$789$	−18.7252	−0.666636
$790$	−22.3575	−0.795444
$791$	−27.1276	−0.964547
$792$	3.24076	0.115156
$793$	28.8734	1.02532
$794$	−6.17523	−0.219151
$795$	−14.0614	−0.498707
$796$	−13.0845	−0.463767
$797$	−7.59899	−0.269170	−0.134585	−	0.990902i	$-0.542970\pi$
$797$	−7.59899	−0.269170	−0.134585	+	0.990902i	$0.542970\pi$
$798$	−33.2010	−1.17530
$799$	39.2117	1.38721
$800$	5.17741	0.183049
$801$	−17.9444	−0.634032
$802$	23.6600	0.835463
$803$	−2.68595	−0.0947851
$804$	−4.17269	−0.147159
$805$	15.7399	0.554759
$806$	24.4587	0.861521
$807$	−23.1259	−0.814071
$808$	14.4993	0.510083
$809$	−9.85027	−0.346317	−0.173159	−	0.984894i	$-0.555397\pi$
$809$	−9.85027	−0.346317	−0.173159	+	0.984894i	$0.555397\pi$
$810$	−2.17740	−0.0765060
$811$	46.3250	1.62669	0.813346	−	0.581780i	$-0.197644\pi$
$811$	46.3250	1.62669	0.813346	+	0.581780i	$0.197644\pi$
$812$	0	0
$813$	21.3469	0.748667
$814$	0.447970	0.0157013
$815$	24.4109	0.855075
$816$	−14.8664	−0.520427
$817$	11.2142	0.392334
$818$	21.7014	0.758772
$819$	−22.5904	−0.789373
$820$	3.47988	0.121523
$821$	18.0554	0.630136	0.315068	−	0.949069i	$-0.397973\pi$
$821$	18.0554	0.630136	0.315068	+	0.949069i	$0.397973\pi$
$822$	−14.7870	−0.515757
$823$	−33.6347	−1.17243	−0.586216	−	0.810155i	$-0.699383\pi$
$823$	−33.6347	−1.17243	−0.586216	+	0.810155i	$0.699383\pi$
$824$	10.7229	0.373551
$825$	−1.86496	−0.0649296
$826$	−1.98151	−0.0689457
$827$	41.1338	1.43036	0.715181	−	0.698939i	$-0.246344\pi$
$827$	41.1338	1.43036	0.715181	+	0.698939i	$0.246344\pi$
$828$	−1.09307	−0.0379868
$829$	7.95542	0.276303	0.138152	−	0.990411i	$-0.455884\pi$
$829$	7.95542	0.276303	0.138152	+	0.990411i	$0.455884\pi$
$830$	−3.31267	−0.114984
$831$	−8.32181	−0.288680
$832$	−46.7077	−1.61930
$833$	63.4357	2.19792
$834$	−5.98774	−0.207339
$835$	−29.4253	−1.01830
$836$	−3.61759	−0.125117
$837$	3.82471	0.132201
$838$	−4.52591	−0.156345
$839$	−11.3228	−0.390908	−0.195454	−	0.980713i	$-0.562618\pi$
$839$	−11.3228	−0.390908	−0.195454	+	0.980713i	$0.562618\pi$
$840$	23.5996	0.814263
$841$	0	0
$842$	−10.0342	−0.345801
$843$	−28.9202	−0.996065
$844$	1.18126	0.0406606
$845$	26.7944	0.921756
$846$	−8.45087	−0.290547
$847$	−42.2745	−1.45257
$848$	20.6910	0.710532
$849$	29.9799	1.02891
$850$	12.0088	0.411899
$851$	−0.716822	−0.0245723
$852$	8.33904	0.285691
$853$	14.5146	0.496971	0.248486	−	0.968636i	$-0.420067\pi$
$853$	14.5146	0.496971	0.248486	+	0.968636i	$0.420067\pi$
$854$	28.3061	0.968615
$855$	11.5311	0.394356
$856$	−15.1012	−0.516148
$857$	−10.7169	−0.366083	−0.183042	−	0.983105i	$-0.558594\pi$
$857$	−10.7169	−0.366083	−0.183042	+	0.983105i	$0.558594\pi$
$858$	6.75469	0.230601
$859$	15.7139	0.536150	0.268075	−	0.963398i	$-0.413612\pi$
$859$	15.7139	0.536150	0.268075	+	0.963398i	$0.413612\pi$
$860$	−1.68019	−0.0572939
$861$	−15.4928	−0.527992
$862$	−23.1603	−0.788844
$863$	−14.4146	−0.490679	−0.245340	−	0.969437i	$-0.578899\pi$
$863$	−14.4146	−0.490679	−0.245340	+	0.969437i	$0.578899\pi$
$864$	−2.93233	−0.0997598
$865$	−8.02775	−0.272952
$866$	19.5542	0.664478
$867$	14.5584	0.494430
$868$	−8.73781	−0.296581
$869$	10.8456	0.367913
$870$	0	0
$871$	−41.2606	−1.39806
$872$	−45.6684	−1.54653
$873$	−6.80968	−0.230473
$874$	−15.8853	−0.537328
$875$	−52.0398	−1.75926
$876$	1.35831	0.0458932
$877$	13.5829	0.458662	0.229331	−	0.973348i	$-0.426346\pi$
$877$	13.5829	0.458662	0.229331	+	0.973348i	$0.426346\pi$
$878$	−3.59384	−0.121286
$879$	8.75406	0.295267
$880$	−5.02703	−0.169461
$881$	21.3996	0.720972	0.360486	−	0.932765i	$-0.382611\pi$
$881$	21.3996	0.720972	0.360486	+	0.932765i	$0.382611\pi$
$882$	−13.6716	−0.460346
$883$	−11.1221	−0.374288	−0.187144	−	0.982332i	$-0.559923\pi$
$883$	−11.1221	−0.374288	−0.187144	+	0.982332i	$0.559923\pi$
$884$	15.8498	0.533085
$885$	0.688204	0.0231337
$886$	−12.7024	−0.426744
$887$	−1.53319	−0.0514795	−0.0257397	−	0.999669i	$-0.508194\pi$
$887$	−1.53319	−0.0514795	−0.0257397	+	0.999669i	$0.508194\pi$
$888$	−1.07476	−0.0360667
$889$	−19.0995	−0.640575
$890$	39.0720	1.30970
$891$	1.05626	0.0353860
$892$	−1.44457	−0.0483678
$893$	44.7543	1.49765
$894$	−16.9670	−0.567461
$895$	6.11582	0.204429
$896$	−20.7073	−0.691783
$897$	−10.8086	−0.360887
$898$	5.33110	0.177901
$899$	0	0
$900$	0.943132	0.0314377
$901$	−43.9230	−1.46329
$902$	4.63244	0.154243
$903$	7.48036	0.248931
$904$	−19.4606	−0.647251
$905$	11.9080	0.395837
$906$	−3.55215	−0.118012
$907$	7.68948	0.255325	0.127663	−	0.991818i	$-0.459253\pi$
$907$	7.68948	0.255325	0.127663	+	0.991818i	$0.459253\pi$
$908$	−7.70396	−0.255665
$909$	4.72573	0.156743
$910$	49.1884	1.63058
$911$	−29.5727	−0.979788	−0.489894	−	0.871782i	$-0.662964\pi$
$911$	−29.5727	−0.979788	−0.489894	+	0.871782i	$0.662964\pi$
$912$	−16.9677	−0.561858
$913$	1.60698	0.0531832
$914$	−13.7398	−0.454473
$915$	−9.83105	−0.325004
$916$	−13.7529	−0.454408
$917$	0.0425958	0.00140664
$918$	−6.80144	−0.224481
$919$	−34.2295	−1.12913	−0.564564	−	0.825389i	$-0.690956\pi$
$919$	−34.2295	−1.12913	−0.564564	+	0.825389i	$0.690956\pi$
$920$	11.2914	0.372266
$921$	−33.9161	−1.11757
$922$	−23.8622	−0.785861
$923$	82.4586	2.71416
$924$	−2.41310	−0.0793850
$925$	0.618494	0.0203360
$926$	−0.912053	−0.0299719
$927$	3.49491	0.114788
$928$	0	0
$929$	48.1477	1.57967	0.789837	−	0.613317i	$-0.210165\pi$
$929$	48.1477	1.57967	0.789837	+	0.613317i	$0.210165\pi$
$930$	−8.32791	−0.273083
$931$	72.4023	2.37289
$932$	12.0742	0.395502
$933$	8.12574	0.266025
$934$	−12.6460	−0.413790
$935$	10.6714	0.348993
$936$	−16.2058	−0.529702
$937$	−13.7375	−0.448784	−0.224392	−	0.974499i	$-0.572040\pi$
$937$	−13.7375	−0.448784	−0.224392	+	0.974499i	$0.572040\pi$
$938$	−40.4500	−1.32074
$939$	2.52738	0.0824780
$940$	−6.70542	−0.218707
$941$	−37.2514	−1.21436	−0.607181	−	0.794563i	$-0.707700\pi$
$941$	−37.2514	−1.21436	−0.607181	+	0.794563i	$0.707700\pi$
$942$	−19.6039	−0.638729
$943$	−7.41263	−0.241389
$944$	−1.01267	−0.0329597
$945$	7.69178	0.250214
$946$	−2.23668	−0.0727207
$947$	16.4787	0.535486	0.267743	−	0.963490i	$-0.413722\pi$
$947$	16.4787	0.535486	0.267743	+	0.963490i	$0.413722\pi$
$948$	−5.48476	−0.178137
$949$	13.4314	0.436001
$950$	13.7063	0.444690
$951$	16.6383	0.539533
$952$	73.7169	2.38918
$953$	−27.7913	−0.900247	−0.450124	−	0.892966i	$-0.648620\pi$
$953$	−27.7913	−0.900247	−0.450124	+	0.892966i	$0.648620\pi$
$954$	9.46624	0.306481
$955$	31.6877	1.02539
$956$	−12.5118	−0.404662
$957$	0	0
$958$	16.9235	0.546773
$959$	52.2360	1.68679
$960$	15.9034	0.513281
$961$	−16.3716	−0.528117
$962$	−2.24012	−0.0722244
$963$	−4.92192	−0.158607
$964$	−11.9559	−0.385075
$965$	−24.0966	−0.775696
$966$	−10.5962	−0.340928
$967$	−51.9876	−1.67181	−0.835904	−	0.548876i	$-0.815056\pi$
$967$	−51.9876	−1.67181	−0.835904	+	0.548876i	$0.815056\pi$
$968$	−30.3266	−0.974735
$969$	36.0192	1.15710
$970$	14.8274	0.476079
$971$	26.6145	0.854099	0.427049	−	0.904228i	$-0.359553\pi$
$971$	26.6145	0.854099	0.427049	+	0.904228i	$0.359553\pi$
$972$	−0.534161	−0.0171332
$973$	21.1520	0.678103
$974$	1.76799	0.0566502
$975$	9.32593	0.298669
$976$	14.4661	0.463049
$977$	60.8959	1.94823	0.974116	−	0.226050i	$-0.0725813\pi$
$977$	60.8959	1.94823	0.974116	+	0.226050i	$0.0725813\pi$
$978$	−16.4336	−0.525487
$979$	−18.9539	−0.605768
$980$	−10.8478	−0.346521
$981$	−14.8847	−0.475231
$982$	14.7129	0.469507
$983$	46.9308	1.49686	0.748430	−	0.663214i	$-0.230808\pi$
$983$	46.9308	1.49686	0.748430	+	0.663214i	$0.230808\pi$
$984$	−11.1141	−0.354305
$985$	−14.7557	−0.470156
$986$	0	0
$987$	29.8532	0.950237
$988$	18.0901	0.575523
$989$	3.57904	0.113807
$990$	−2.29989	−0.0730954
$991$	50.7905	1.61341	0.806707	−	0.590952i	$-0.201248\pi$
$991$	50.7905	1.61341	0.806707	+	0.590952i	$0.201248\pi$
$992$	−11.2153	−0.356086
$993$	12.9286	0.410276
$994$	80.8386	2.56405
$995$	44.0534	1.39659
$996$	−0.812667	−0.0257503
$997$	−45.8087	−1.45078	−0.725389	−	0.688339i	$-0.758340\pi$
$997$	−45.8087	−1.45078	−0.725389	+	0.688339i	$0.758340\pi$
$998$	43.5469	1.37845
$999$	−0.350297	−0.0110829

Display

a_p

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p

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a_n

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a_n

instead) (See

a_n

instead) Display

a_n

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n

up to: 50 250 1000 (See only

a_p

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a_p

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a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2523.2.a.r.1.2		9
3.2	odd	2		7569.2.a.bj.1.8		9
29.16	even	7		87.2.g.a.82.1	yes	18
29.20	even	7		87.2.g.a.52.1	✓	18
29.28	even	2		2523.2.a.o.1.8		9
87.20	odd	14		261.2.k.c.226.3		18
87.74	odd	14		261.2.k.c.82.3		18
87.86	odd	2		7569.2.a.bm.1.2		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.a.52.1	✓	18	29.20	even	7
87.2.g.a.82.1	yes	18	29.16	even	7
261.2.k.c.82.3		18	87.74	odd	14
261.2.k.c.226.3		18	87.20	odd	14
2523.2.a.o.1.8		9	29.28	even	2
2523.2.a.r.1.2		9	1.1	even	1	trivial
7569.2.a.bj.1.8		9	3.2	odd	2
7569.2.a.bm.1.2		9	87.86	odd	2

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2523.2.a.r.1.2

Level	$2523$
Weight	$2$
Character	2523.1
Self dual	yes
Analytic conductor	$20.146$
Analytic rank	$0$
Dimension	$9$
CM	no
Inner twists	$1$