LMFDB - Embedded newform 6069.2.a.bd.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6069,2,Mod(1,6069)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6069, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6069.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$6069 = 3 \cdot 7 \cdot 17^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	6069.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:	$48.4612089867$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{10} - 4x^{9} - 8x^{8} + 44x^{7} + 5x^{6} - 144x^{5} + 48x^{4} + 160x^{3} - 44x^{2} - 64x - 2$ x^10 - 4x^9 - 8x^8 + 44x^7 + 5x^6 - 144x^5 + 48x^4 + 160x^3 - 44x^2 - 64*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 357)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.661473$ of defining polynomial
Character	$\chi$	$=$	6069.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.661473 q^{2} -1.00000 q^{3} -1.56245 q^{4} +2.35286 q^{5} +0.661473 q^{6} +1.00000 q^{7} +2.35647 q^{8} +1.00000 q^{9} -1.55635 q^{10} +5.25085 q^{11} +1.56245 q^{12} -4.49854 q^{13} -0.661473 q^{14} -2.35286 q^{15} +1.56617 q^{16} -0.661473 q^{18} +3.58834 q^{19} -3.67623 q^{20} -1.00000 q^{21} -3.47329 q^{22} +0.603575 q^{23} -2.35647 q^{24} +0.535927 q^{25} +2.97566 q^{26} -1.00000 q^{27} -1.56245 q^{28} +5.75604 q^{29} +1.55635 q^{30} -0.199739 q^{31} -5.74891 q^{32} -5.25085 q^{33} +2.35286 q^{35} -1.56245 q^{36} +7.65696 q^{37} -2.37359 q^{38} +4.49854 q^{39} +5.54442 q^{40} +6.09517 q^{41} +0.661473 q^{42} +9.93667 q^{43} -8.20421 q^{44} +2.35286 q^{45} -0.399248 q^{46} -6.16149 q^{47} -1.56617 q^{48} +1.00000 q^{49} -0.354501 q^{50} +7.02876 q^{52} -9.47974 q^{53} +0.661473 q^{54} +12.3545 q^{55} +2.35647 q^{56} -3.58834 q^{57} -3.80746 q^{58} -1.92313 q^{59} +3.67623 q^{60} +10.5442 q^{61} +0.132122 q^{62} +1.00000 q^{63} +0.670404 q^{64} -10.5844 q^{65} +3.47329 q^{66} -7.47259 q^{67} -0.603575 q^{69} -1.55635 q^{70} -1.54424 q^{71} +2.35647 q^{72} +9.89000 q^{73} -5.06487 q^{74} -0.535927 q^{75} -5.60662 q^{76} +5.25085 q^{77} -2.97566 q^{78} +10.4711 q^{79} +3.68497 q^{80} +1.00000 q^{81} -4.03178 q^{82} -8.09294 q^{83} +1.56245 q^{84} -6.57283 q^{86} -5.75604 q^{87} +12.3734 q^{88} +12.3039 q^{89} -1.55635 q^{90} -4.49854 q^{91} -0.943058 q^{92} +0.199739 q^{93} +4.07566 q^{94} +8.44285 q^{95} +5.74891 q^{96} -8.25850 q^{97} -0.661473 q^{98} +5.25085 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$10 q + 4 q^{2} - 10 q^{3} + 12 q^{4} + 6 q^{5} - 4 q^{6} + 10 q^{7} + 12 q^{8} + 10 q^{9} + 2 q^{11} - 12 q^{12} + 6 q^{13} + 4 q^{14} - 6 q^{15} + 20 q^{16} + 4 q^{18} + 6 q^{19} + 16 q^{20} - 10 q^{21}+ \cdots + 2 q^{99}+O(q^{100})$ 10 * q + 4 * q^2 - 10 * q^3 + 12 * q^4 + 6 * q^5 - 4 * q^6 + 10 * q^7 + 12 * q^8 + 10 * q^9 + 2 * q^11 - 12 * q^12 + 6 * q^13 + 4 * q^14 - 6 * q^15 + 20 * q^16 + 4 * q^18 + 6 * q^19 + 16 * q^20 - 10 * q^21 - 24 * q^22 - 18 * q^23 - 12 * q^24 + 4 * q^25 + 12 * q^26 - 10 * q^27 + 12 * q^28 + 12 * q^29 + 8 * q^31 + 28 * q^32 - 2 * q^33 + 6 * q^35 + 12 * q^36 - 24 * q^37 + 32 * q^38 - 6 * q^39 + 36 * q^40 + 2 * q^41 - 4 * q^42 + 6 * q^43 - 28 * q^44 + 6 * q^45 + 28 * q^46 + 16 * q^47 - 20 * q^48 + 10 * q^49 + 52 * q^50 + 24 * q^52 - 12 * q^53 - 4 * q^54 + 18 * q^55 + 12 * q^56 - 6 * q^57 - 20 * q^58 + 20 * q^59 - 16 * q^60 + 4 * q^61 + 24 * q^62 + 10 * q^63 + 56 * q^64 + 38 * q^65 + 24 * q^66 + 20 * q^67 + 18 * q^69 + 20 * q^71 + 12 * q^72 - 20 * q^73 - 36 * q^74 - 4 * q^75 - 8 * q^76 + 2 * q^77 - 12 * q^78 - 20 * q^79 + 92 * q^80 + 10 * q^81 - 8 * q^82 + 36 * q^83 - 12 * q^84 - 12 * q^87 - 92 * q^88 + 32 * q^89 + 6 * q^91 - 40 * q^92 - 8 * q^93 + 60 * q^94 + 14 * q^95 - 28 * q^96 - 36 * q^97 + 4 * q^98 + 2 * 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

)

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

$2$	−0.661473	−0.467732	−0.233866	−	0.972269i	$-0.575138\pi$
$2$	−0.661473	−0.467732	−0.233866	+	0.972269i	$0.575138\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.56245	−0.781227
$5$	2.35286	1.05223	0.526114	−	0.850414i	$-0.323648\pi$
$5$	2.35286	1.05223	0.526114	+	0.850414i	$0.323648\pi$
$6$	0.661473	0.270045
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.35647	0.833136
$9$	1.00000	0.333333
$10$	−1.55635	−0.492161
$11$	5.25085	1.58319	0.791595	−	0.611046i	$-0.209251\pi$
$11$	5.25085	1.58319	0.791595	+	0.611046i	$0.209251\pi$
$12$	1.56245	0.451042
$13$	−4.49854	−1.24767	−0.623835	−	0.781556i	$-0.714426\pi$
$13$	−4.49854	−1.24767	−0.623835	+	0.781556i	$0.714426\pi$
$14$	−0.661473	−0.176786
$15$	−2.35286	−0.607505
$16$	1.56617	0.391543
$17$	0	0
$17$	0	0
$18$	−0.661473	−0.155911
$19$	3.58834	0.823222	0.411611	−	0.911360i	$-0.364966\pi$
$19$	3.58834	0.823222	0.411611	+	0.911360i	$0.364966\pi$
$20$	−3.67623	−0.822030
$21$	−1.00000	−0.218218
$22$	−3.47329	−0.740508
$23$	0.603575	0.125854	0.0629271	−	0.998018i	$-0.479956\pi$
$23$	0.603575	0.125854	0.0629271	+	0.998018i	$0.479956\pi$
$24$	−2.35647	−0.481012
$25$	0.535927	0.107185
$26$	2.97566	0.583575
$27$	−1.00000	−0.192450
$28$	−1.56245	−0.295276
$29$	5.75604	1.06887	0.534435	−	0.845210i	$-0.320524\pi$
$29$	5.75604	1.06887	0.534435	+	0.845210i	$0.320524\pi$
$30$	1.55635	0.284149
$31$	−0.199739	−0.0358741	−0.0179371	−	0.999839i	$-0.505710\pi$
$31$	−0.199739	−0.0358741	−0.0179371	+	0.999839i	$0.505710\pi$
$32$	−5.74891	−1.01627
$33$	−5.25085	−0.914055
$34$	0	0
$35$	2.35286	0.397705
$36$	−1.56245	−0.260409
$37$	7.65696	1.25880	0.629398	−	0.777083i	$-0.283301\pi$
$37$	7.65696	1.25880	0.629398	+	0.777083i	$0.283301\pi$
$38$	−2.37359	−0.385047
$39$	4.49854	0.720343
$40$	5.54442	0.876650
$41$	6.09517	0.951905	0.475952	−	0.879471i	$-0.342103\pi$
$41$	6.09517	0.951905	0.475952	+	0.879471i	$0.342103\pi$
$42$	0.661473	0.102067
$43$	9.93667	1.51533	0.757664	−	0.652645i	$-0.226341\pi$
$43$	9.93667	1.51533	0.757664	+	0.652645i	$0.226341\pi$
$44$	−8.20421	−1.23683
$45$	2.35286	0.350743
$46$	−0.399248	−0.0588660
$47$	−6.16149	−0.898745	−0.449373	−	0.893344i	$-0.648352\pi$
$47$	−6.16149	−0.898745	−0.449373	+	0.893344i	$0.648352\pi$
$48$	−1.56617	−0.226057
$49$	1.00000	0.142857
$50$	−0.354501	−0.0501340
$51$	0	0
$52$	7.02876	0.974714
$53$	−9.47974	−1.30214	−0.651071	−	0.759017i	$-0.725680\pi$
$53$	−9.47974	−1.30214	−0.651071	+	0.759017i	$0.725680\pi$
$54$	0.661473	0.0900150
$55$	12.3545	1.66588
$56$	2.35647	0.314896
$57$	−3.58834	−0.475288
$58$	−3.80746	−0.499945
$59$	−1.92313	−0.250370	−0.125185	−	0.992133i	$-0.539952\pi$
$59$	−1.92313	−0.250370	−0.125185	+	0.992133i	$0.539952\pi$
$60$	3.67623	0.474599
$61$	10.5442	1.35005	0.675023	−	0.737797i	$-0.264134\pi$
$61$	10.5442	1.35005	0.675023	+	0.737797i	$0.264134\pi$
$62$	0.132122	0.0167795
$63$	1.00000	0.125988
$64$	0.670404	0.0838006
$65$	−10.5844	−1.31283
$66$	3.47329	0.427533
$67$	−7.47259	−0.912922	−0.456461	−	0.889743i	$-0.650883\pi$
$67$	−7.47259	−0.912922	−0.456461	+	0.889743i	$0.650883\pi$
$68$	0	0
$69$	−0.603575	−0.0726619
$70$	−1.55635	−0.186019
$71$	−1.54424	−0.183267	−0.0916337	−	0.995793i	$-0.529209\pi$
$71$	−1.54424	−0.183267	−0.0916337	+	0.995793i	$0.529209\pi$
$72$	2.35647	0.277712
$73$	9.89000	1.15754	0.578769	−	0.815492i	$-0.303533\pi$
$73$	9.89000	1.15754	0.578769	+	0.815492i	$0.303533\pi$
$74$	−5.06487	−0.588779
$75$	−0.535927	−0.0618835
$76$	−5.60662	−0.643123
$77$	5.25085	0.598390
$78$	−2.97566	−0.336927
$79$	10.4711	1.17809	0.589043	−	0.808102i	$-0.299505\pi$
$79$	10.4711	1.17809	0.589043	+	0.808102i	$0.299505\pi$
$80$	3.68497	0.411993
$81$	1.00000	0.111111
$82$	−4.03178	−0.445236
$83$	−8.09294	−0.888316	−0.444158	−	0.895949i	$-0.646497\pi$
$83$	−8.09294	−0.888316	−0.444158	+	0.895949i	$0.646497\pi$
$84$	1.56245	0.170478
$85$	0	0
$86$	−6.57283	−0.708767
$87$	−5.75604	−0.617113
$88$	12.3734	1.31901
$89$	12.3039	1.30421	0.652105	−	0.758129i	$-0.273886\pi$
$89$	12.3039	1.30421	0.652105	+	0.758129i	$0.273886\pi$
$90$	−1.55635	−0.164054
$91$	−4.49854	−0.471575
$92$	−0.943058	−0.0983206
$93$	0.199739	0.0207119
$94$	4.07566	0.420372
$95$	8.44285	0.866218
$96$	5.74891	0.586746
$97$	−8.25850	−0.838523	−0.419262	−	0.907865i	$-0.637711\pi$
$97$	−8.25850	−0.838523	−0.419262	+	0.907865i	$0.637711\pi$
$98$	−0.661473	−0.0668188
$99$	5.25085	0.527730
$100$	−0.837361	−0.0837361
$101$	12.3613	1.22999	0.614995	−	0.788531i	$-0.289158\pi$
$101$	12.3613	1.22999	0.614995	+	0.788531i	$0.289158\pi$
$102$	0	0
$103$	3.48146	0.343039	0.171519	−	0.985181i	$-0.445132\pi$
$103$	3.48146	0.343039	0.171519	+	0.985181i	$0.445132\pi$
$104$	−10.6007	−1.03948
$105$	−2.35286	−0.229615
$106$	6.27059	0.609053
$107$	4.64675	0.449218	0.224609	−	0.974449i	$-0.427889\pi$
$107$	4.64675	0.449218	0.224609	+	0.974449i	$0.427889\pi$
$108$	1.56245	0.150347
$109$	−13.9330	−1.33453	−0.667267	−	0.744818i	$-0.732536\pi$
$109$	−13.9330	−1.33453	−0.667267	+	0.744818i	$0.732536\pi$
$110$	−8.17215	−0.779184
$111$	−7.65696	−0.726767
$112$	1.56617	0.147989
$113$	−9.88305	−0.929719	−0.464860	−	0.885384i	$-0.653895\pi$
$113$	−9.88305	−0.929719	−0.464860	+	0.885384i	$0.653895\pi$
$114$	2.37359	0.222307
$115$	1.42012	0.132427
$116$	−8.99355	−0.835030
$117$	−4.49854	−0.415890
$118$	1.27210	0.117106
$119$	0	0
$120$	−5.54442	−0.506134
$121$	16.5714	1.50649
$122$	−6.97469	−0.631459
$123$	−6.09517	−0.549583
$124$	0.312083	0.0280258
$125$	−10.5033	−0.939445
$126$	−0.661473	−0.0589287
$127$	−14.8193	−1.31500	−0.657499	−	0.753456i	$-0.728386\pi$
$127$	−14.8193	−1.31500	−0.657499	+	0.753456i	$0.728386\pi$
$128$	11.0544	0.977077
$129$	−9.93667	−0.874875
$130$	7.00130	0.614054
$131$	−9.13853	−0.798437	−0.399219	−	0.916856i	$-0.630719\pi$
$131$	−9.13853	−0.798437	−0.399219	+	0.916856i	$0.630719\pi$
$132$	8.20421	0.714085
$133$	3.58834	0.311149
$134$	4.94291	0.427003
$135$	−2.35286	−0.202502
$136$	0	0
$137$	−7.53516	−0.643772	−0.321886	−	0.946778i	$-0.604317\pi$
$137$	−7.53516	−0.643772	−0.321886	+	0.946778i	$0.604317\pi$
$138$	0.399248	0.0339863
$139$	−0.515904	−0.0437584	−0.0218792	−	0.999761i	$-0.506965\pi$
$139$	−0.515904	−0.0437584	−0.0218792	+	0.999761i	$0.506965\pi$
$140$	−3.67623	−0.310698
$141$	6.16149	0.518891
$142$	1.02147	0.0857200
$143$	−23.6211	−1.97530
$144$	1.56617	0.130514
$145$	13.5431	1.12470
$146$	−6.54197	−0.541417
$147$	−1.00000	−0.0824786
$148$	−11.9637	−0.983406
$149$	−2.24948	−0.184284	−0.0921422	−	0.995746i	$-0.529371\pi$
$149$	−2.24948	−0.184284	−0.0921422	+	0.995746i	$0.529371\pi$
$150$	0.354501	0.0289449
$151$	−10.1044	−0.822286	−0.411143	−	0.911571i	$-0.634870\pi$
$151$	−10.1044	−0.822286	−0.411143	+	0.911571i	$0.634870\pi$
$152$	8.45580	0.685856
$153$	0	0
$154$	−3.47329	−0.279886
$155$	−0.469956	−0.0377478
$156$	−7.02876	−0.562751
$157$	23.6643	1.88862	0.944310	−	0.329058i	$-0.106731\pi$
$157$	23.6643	1.88862	0.944310	+	0.329058i	$0.106731\pi$
$158$	−6.92631	−0.551028
$159$	9.47974	0.751792
$160$	−13.5264	−1.06935
$161$	0.603575	0.0475684
$162$	−0.661473	−0.0519702
$163$	−23.9501	−1.87592	−0.937960	−	0.346743i	$-0.887288\pi$
$163$	−23.9501	−1.87592	−0.937960	+	0.346743i	$0.887288\pi$
$164$	−9.52342	−0.743654
$165$	−12.3545	−0.961795
$166$	5.35326	0.415493
$167$	25.1742	1.94803	0.974017	−	0.226474i	$-0.0727197\pi$
$167$	25.1742	1.94803	0.974017	+	0.226474i	$0.0727197\pi$
$168$	−2.35647	−0.181805
$169$	7.23686	0.556681
$170$	0	0
$171$	3.58834	0.274407
$172$	−15.5256	−1.18381
$173$	11.1650	0.848857	0.424428	−	0.905462i	$-0.360475\pi$
$173$	11.1650	0.848857	0.424428	+	0.905462i	$0.360475\pi$
$174$	3.80746	0.288643
$175$	0.535927	0.0405123
$176$	8.22373	0.619887
$177$	1.92313	0.144551
$178$	−8.13869	−0.610020
$179$	−2.39571	−0.179064	−0.0895320	−	0.995984i	$-0.528537\pi$
$179$	−2.39571	−0.179064	−0.0895320	+	0.995984i	$0.528537\pi$
$180$	−3.67623	−0.274010
$181$	−19.6381	−1.45969	−0.729845	−	0.683613i	$-0.760408\pi$
$181$	−19.6381	−1.45969	−0.729845	+	0.683613i	$0.760408\pi$
$182$	2.97566	0.220571
$183$	−10.5442	−0.779449
$184$	1.42230	0.104854
$185$	18.0157	1.32454
$186$	−0.132122	−0.00968763
$187$	0	0
$188$	9.62704	0.702124
$189$	−1.00000	−0.0727393
$190$	−5.58471	−0.405158
$191$	−5.22705	−0.378216	−0.189108	−	0.981956i	$-0.560560\pi$
$191$	−5.22705	−0.378216	−0.189108	+	0.981956i	$0.560560\pi$
$192$	−0.670404	−0.0483823
$193$	−21.6970	−1.56178	−0.780891	−	0.624667i	$-0.785235\pi$
$193$	−21.6970	−1.56178	−0.780891	+	0.624667i	$0.785235\pi$
$194$	5.46277	0.392204
$195$	10.5844	0.757965
$196$	−1.56245	−0.111604
$197$	−6.69620	−0.477085	−0.238542	−	0.971132i	$-0.576670\pi$
$197$	−6.69620	−0.477085	−0.238542	+	0.971132i	$0.576670\pi$
$198$	−3.47329	−0.246836
$199$	6.71661	0.476128	0.238064	−	0.971250i	$-0.423487\pi$
$199$	6.71661	0.476128	0.238064	+	0.971250i	$0.423487\pi$
$200$	1.26289	0.0893000
$201$	7.47259	0.527076
$202$	−8.17663	−0.575306
$203$	5.75604	0.403995
$204$	0	0
$205$	14.3410	1.00162
$206$	−2.30289	−0.160450
$207$	0.603575	0.0419514
$208$	−7.04548	−0.488516
$209$	18.8418	1.30332
$210$	1.55635	0.107398
$211$	13.1733	0.906889	0.453444	−	0.891285i	$-0.350195\pi$
$211$	13.1733	0.906889	0.453444	+	0.891285i	$0.350195\pi$
$212$	14.8117	1.01727
$213$	1.54424	0.105809
$214$	−3.07370	−0.210114
$215$	23.3795	1.59447
$216$	−2.35647	−0.160337
$217$	−0.199739	−0.0135591
$218$	9.21627	0.624204
$219$	−9.89000	−0.668305
$220$	−19.3033	−1.30143
$221$	0	0
$222$	5.06487	0.339932
$223$	12.0749	0.808596	0.404298	−	0.914627i	$-0.367516\pi$
$223$	12.0749	0.808596	0.404298	+	0.914627i	$0.367516\pi$
$224$	−5.74891	−0.384115
$225$	0.535927	0.0357285
$226$	6.53737	0.434859
$227$	9.96214	0.661211	0.330605	−	0.943769i	$-0.392747\pi$
$227$	9.96214	0.661211	0.330605	+	0.943769i	$0.392747\pi$
$228$	5.60662	0.371307
$229$	0.341524	0.0225685	0.0112843	−	0.999936i	$-0.496408\pi$
$229$	0.341524	0.0225685	0.0112843	+	0.999936i	$0.496408\pi$
$230$	−0.939374	−0.0619405
$231$	−5.25085	−0.345480
$232$	13.5639	0.890515
$233$	−15.8961	−1.04139	−0.520693	−	0.853744i	$-0.674326\pi$
$233$	−15.8961	−1.04139	−0.520693	+	0.853744i	$0.674326\pi$
$234$	2.97566	0.194525
$235$	−14.4971	−0.945686
$236$	3.00480	0.195596
$237$	−10.4711	−0.680168
$238$	0	0
$239$	−1.16131	−0.0751187	−0.0375594	−	0.999294i	$-0.511958\pi$
$239$	−1.16131	−0.0751187	−0.0375594	+	0.999294i	$0.511958\pi$
$240$	−3.68497	−0.237864
$241$	9.12543	0.587821	0.293910	−	0.955833i	$-0.405043\pi$
$241$	9.12543	0.587821	0.293910	+	0.955833i	$0.405043\pi$
$242$	−10.9615	−0.704634
$243$	−1.00000	−0.0641500
$244$	−16.4748	−1.05469
$245$	2.35286	0.150318
$246$	4.03178	0.257057
$247$	−16.1423	−1.02711
$248$	−0.470677	−0.0298880
$249$	8.09294	0.512869
$250$	6.94766	0.439408
$251$	2.02464	0.127794	0.0638970	−	0.997956i	$-0.479647\pi$
$251$	2.02464	0.127794	0.0638970	+	0.997956i	$0.479647\pi$
$252$	−1.56245	−0.0984254
$253$	3.16928	0.199251
$254$	9.80254	0.615066
$255$	0	0
$256$	−8.65297	−0.540811
$257$	−19.5756	−1.22109	−0.610545	−	0.791982i	$-0.709049\pi$
$257$	−19.5756	−1.22109	−0.610545	+	0.791982i	$0.709049\pi$
$258$	6.57283	0.409207
$259$	7.65696	0.475780
$260$	16.5377	1.02562
$261$	5.75604	0.356290
$262$	6.04489	0.373454
$263$	−23.4356	−1.44510	−0.722550	−	0.691319i	$-0.757030\pi$
$263$	−23.4356	−1.44510	−0.722550	+	0.691319i	$0.757030\pi$
$264$	−12.3734	−0.761533
$265$	−22.3044	−1.37015
$266$	−2.37359	−0.145534
$267$	−12.3039	−0.752986
$268$	11.6756	0.713200
$269$	8.00374	0.487997	0.243998	−	0.969776i	$-0.421541\pi$
$269$	8.00374	0.487997	0.243998	+	0.969776i	$0.421541\pi$
$270$	1.55635	0.0947164
$271$	24.5987	1.49427	0.747133	−	0.664675i	$-0.231430\pi$
$271$	24.5987	1.49427	0.747133	+	0.664675i	$0.231430\pi$
$272$	0	0
$273$	4.49854	0.272264
$274$	4.98430	0.301112
$275$	2.81407	0.169695
$276$	0.943058	0.0567654
$277$	26.7647	1.60814	0.804069	−	0.594536i	$-0.202664\pi$
$277$	26.7647	1.60814	0.804069	+	0.594536i	$0.202664\pi$
$278$	0.341257	0.0204672
$279$	−0.199739	−0.0119580
$280$	5.54442	0.331343
$281$	24.2978	1.44948	0.724742	−	0.689020i	$-0.241959\pi$
$281$	24.2978	1.44948	0.724742	+	0.689020i	$0.241959\pi$
$282$	−4.07566	−0.242702
$283$	−9.16677	−0.544908	−0.272454	−	0.962169i	$-0.587835\pi$
$283$	−9.16677	−0.544908	−0.272454	+	0.962169i	$0.587835\pi$
$284$	2.41280	0.143173
$285$	−8.44285	−0.500111
$286$	15.6247	0.923910
$287$	6.09517	0.359786
$288$	−5.74891	−0.338758
$289$	0	0
$290$	−8.95841	−0.526056
$291$	8.25850	0.484122
$292$	−15.4527	−0.904300
$293$	−2.19698	−0.128349	−0.0641746	−	0.997939i	$-0.520441\pi$
$293$	−2.19698	−0.128349	−0.0641746	+	0.997939i	$0.520441\pi$
$294$	0.661473	0.0385779
$295$	−4.52485	−0.263447
$296$	18.0434	1.04875
$297$	−5.25085	−0.304685
$298$	1.48797	0.0861956
$299$	−2.71521	−0.157024
$300$	0.837361	0.0483451
$301$	9.93667	0.572740
$302$	6.68379	0.384609
$303$	−12.3613	−0.710136
$304$	5.61996	0.322327
$305$	24.8090	1.42056
$306$	0	0
$307$	21.0887	1.20360	0.601798	−	0.798648i	$-0.294451\pi$
$307$	21.0887	1.20360	0.601798	+	0.798648i	$0.294451\pi$
$308$	−8.20421	−0.467478
$309$	−3.48146	−0.198053
$310$	0.310863	0.0176558
$311$	11.2787	0.639556	0.319778	−	0.947492i	$-0.396392\pi$
$311$	11.2787	0.639556	0.319778	+	0.947492i	$0.396392\pi$
$312$	10.6007	0.600144
$313$	19.9287	1.12644	0.563218	−	0.826308i	$-0.309563\pi$
$313$	19.9287	1.12644	0.563218	+	0.826308i	$0.309563\pi$
$314$	−15.6533	−0.883367
$315$	2.35286	0.132568
$316$	−16.3605	−0.920352
$317$	−30.5111	−1.71367	−0.856837	−	0.515588i	$-0.827574\pi$
$317$	−30.5111	−1.71367	−0.856837	+	0.515588i	$0.827574\pi$
$318$	−6.27059	−0.351637
$319$	30.2241	1.69222
$320$	1.57736	0.0881774
$321$	−4.64675	−0.259356
$322$	−0.399248	−0.0222492
$323$	0	0
$324$	−1.56245	−0.0868030
$325$	−2.41089	−0.133732
$326$	15.8424	0.877427
$327$	13.9330	0.770494
$328$	14.3630	0.793067
$329$	−6.16149	−0.339694
$330$	8.17215	0.449862
$331$	24.7998	1.36312	0.681560	−	0.731763i	$-0.261302\pi$
$331$	24.7998	1.36312	0.681560	+	0.731763i	$0.261302\pi$
$332$	12.6448	0.693976
$333$	7.65696	0.419599
$334$	−16.6520	−0.911158
$335$	−17.5819	−0.960603
$336$	−1.56617	−0.0854416
$337$	8.73557	0.475857	0.237928	−	0.971283i	$-0.423532\pi$
$337$	8.73557	0.475857	0.237928	+	0.971283i	$0.423532\pi$
$338$	−4.78698	−0.260378
$339$	9.88305	0.536774
$340$	0	0
$341$	−1.04880	−0.0567956
$342$	−2.37359	−0.128349
$343$	1.00000	0.0539949
$344$	23.4154	1.26247
$345$	−1.42012	−0.0764570
$346$	−7.38532	−0.397037
$347$	7.99169	0.429016	0.214508	−	0.976722i	$-0.431185\pi$
$347$	7.99169	0.429016	0.214508	+	0.976722i	$0.431185\pi$
$348$	8.99355	0.482105
$349$	−20.2193	−1.08232	−0.541158	−	0.840921i	$-0.682014\pi$
$349$	−20.2193	−1.08232	−0.541158	+	0.840921i	$0.682014\pi$
$350$	−0.354501	−0.0189489
$351$	4.49854	0.240114
$352$	−30.1867	−1.60895
$353$	−0.363744	−0.0193601	−0.00968007	−	0.999953i	$-0.503081\pi$
$353$	−0.363744	−0.0193601	−0.00968007	+	0.999953i	$0.503081\pi$
$354$	−1.27210	−0.0676113
$355$	−3.63337	−0.192839
$356$	−19.2243	−1.01888
$357$	0	0
$358$	1.58470	0.0837539
$359$	26.3254	1.38940	0.694701	−	0.719299i	$-0.255537\pi$
$359$	26.3254	1.38940	0.694701	+	0.719299i	$0.255537\pi$
$360$	5.54442	0.292217
$361$	−6.12380	−0.322305
$362$	12.9901	0.682743
$363$	−16.5714	−0.869773
$364$	7.02876	0.368407
$365$	23.2697	1.21799
$366$	6.97469	0.364573
$367$	9.89556	0.516544	0.258272	−	0.966072i	$-0.416847\pi$
$367$	9.89556	0.516544	0.258272	+	0.966072i	$0.416847\pi$
$368$	0.945302	0.0492773
$369$	6.09517	0.317302
$370$	−11.9169	−0.619530
$371$	−9.47974	−0.492163
$372$	−0.312083	−0.0161807
$373$	12.9157	0.668751	0.334376	−	0.942440i	$-0.391475\pi$
$373$	12.9157	0.668751	0.334376	+	0.942440i	$0.391475\pi$
$374$	0	0
$375$	10.5033	0.542389
$376$	−14.5193	−0.748778
$377$	−25.8938	−1.33360
$378$	0.661473	0.0340225
$379$	−15.9141	−0.817452	−0.408726	−	0.912657i	$-0.634027\pi$
$379$	−15.9141	−0.817452	−0.408726	+	0.912657i	$0.634027\pi$
$380$	−13.1916	−0.676713
$381$	14.8193	0.759214
$382$	3.45755	0.176904
$383$	19.1824	0.980176	0.490088	−	0.871673i	$-0.336965\pi$
$383$	19.1824	0.980176	0.490088	+	0.871673i	$0.336965\pi$
$384$	−11.0544	−0.564116
$385$	12.3545	0.629643
$386$	14.3520	0.730495
$387$	9.93667	0.505109
$388$	12.9035	0.655077
$389$	24.5986	1.24720	0.623598	−	0.781745i	$-0.285670\pi$
$389$	24.5986	1.24720	0.623598	+	0.781745i	$0.285670\pi$
$390$	−7.00130	−0.354524
$391$	0	0
$392$	2.35647	0.119019
$393$	9.13853	0.460978
$394$	4.42936	0.223148
$395$	24.6369	1.23962
$396$	−8.20421	−0.412277
$397$	−22.3328	−1.12085	−0.560425	−	0.828205i	$-0.689362\pi$
$397$	−22.3328	−1.12085	−0.560425	+	0.828205i	$0.689362\pi$
$398$	−4.44285	−0.222700
$399$	−3.58834	−0.179642
$400$	0.839353	0.0419676
$401$	25.8421	1.29049	0.645247	−	0.763974i	$-0.276754\pi$
$401$	25.8421	1.29049	0.645247	+	0.763974i	$0.276754\pi$
$402$	−4.94291	−0.246530
$403$	0.898533	0.0447591
$404$	−19.3139	−0.960902
$405$	2.35286	0.116914
$406$	−3.80746	−0.188961
$407$	40.2055	1.99291
$408$	0	0
$409$	15.7611	0.779339	0.389669	−	0.920955i	$-0.372589\pi$
$409$	15.7611	0.779339	0.389669	+	0.920955i	$0.372589\pi$
$410$	−9.48620	−0.468490
$411$	7.53516	0.371682
$412$	−5.43962	−0.267991
$413$	−1.92313	−0.0946311
$414$	−0.399248	−0.0196220
$415$	−19.0415	−0.934711
$416$	25.8617	1.26797
$417$	0.515904	0.0252640
$418$	−12.4634	−0.609603
$419$	13.4203	0.655624	0.327812	−	0.944743i	$-0.393689\pi$
$419$	13.4203	0.655624	0.327812	+	0.944743i	$0.393689\pi$
$420$	3.67623	0.179382
$421$	10.2759	0.500818	0.250409	−	0.968140i	$-0.419435\pi$
$421$	10.2759	0.500818	0.250409	+	0.968140i	$0.419435\pi$
$422$	−8.71379	−0.424181
$423$	−6.16149	−0.299582
$424$	−22.3387	−1.08486
$425$	0	0
$426$	−1.02147	−0.0494904
$427$	10.5442	0.510269
$428$	−7.26034	−0.350942
$429$	23.6211	1.14044
$430$	−15.4649	−0.745785
$431$	28.8097	1.38771	0.693856	−	0.720114i	$-0.255910\pi$
$431$	28.8097	1.38771	0.693856	+	0.720114i	$0.255910\pi$
$432$	−1.56617	−0.0753524
$433$	−25.4783	−1.22441	−0.612205	−	0.790699i	$-0.709717\pi$
$433$	−25.4783	−1.22441	−0.612205	+	0.790699i	$0.709717\pi$
$434$	0.132122	0.00634204
$435$	−13.5431	−0.649344
$436$	21.7696	1.04257
$437$	2.16583	0.103606
$438$	6.54197	0.312587
$439$	−33.8768	−1.61685	−0.808425	−	0.588599i	$-0.799680\pi$
$439$	−33.8768	−1.61685	−0.808425	+	0.588599i	$0.799680\pi$
$440$	29.1129	1.38790
$441$	1.00000	0.0476190
$442$	0	0
$443$	−10.6561	−0.506289	−0.253144	−	0.967429i	$-0.581465\pi$
$443$	−10.6561	−0.506289	−0.253144	+	0.967429i	$0.581465\pi$
$444$	11.9637	0.567770
$445$	28.9493	1.37233
$446$	−7.98722	−0.378206
$447$	2.24948	0.106397
$448$	0.670404	0.0316736
$449$	−41.1981	−1.94426	−0.972130	−	0.234444i	$-0.924673\pi$
$449$	−41.1981	−1.94426	−0.972130	+	0.234444i	$0.924673\pi$
$450$	−0.354501	−0.0167113
$451$	32.0048	1.50705
$452$	15.4418	0.726322
$453$	10.1044	0.474747
$454$	−6.58968	−0.309269
$455$	−10.5844	−0.496205
$456$	−8.45580	−0.395979
$457$	42.1677	1.97252	0.986260	−	0.165202i	$-0.0528275\pi$
$457$	42.1677	1.97252	0.986260	+	0.165202i	$0.0528275\pi$
$458$	−0.225909	−0.0105560
$459$	0	0
$460$	−2.21888	−0.103456
$461$	8.08845	0.376717	0.188358	−	0.982100i	$-0.439683\pi$
$461$	8.08845	0.376717	0.188358	+	0.982100i	$0.439683\pi$
$462$	3.47329	0.161592
$463$	−31.1333	−1.44688	−0.723442	−	0.690385i	$-0.757441\pi$
$463$	−31.1333	−1.44688	−0.723442	+	0.690385i	$0.757441\pi$
$464$	9.01495	0.418508
$465$	0.469956	0.0217937
$466$	10.5148	0.487089
$467$	7.39951	0.342408	0.171204	−	0.985236i	$-0.445234\pi$
$467$	7.39951	0.342408	0.171204	+	0.985236i	$0.445234\pi$
$468$	7.02876	0.324905
$469$	−7.47259	−0.345052
$470$	9.58943	0.442327
$471$	−23.6643	−1.09040
$472$	−4.53179	−0.208593
$473$	52.1759	2.39905
$474$	6.92631	0.318136
$475$	1.92309	0.0882374
$476$	0	0
$477$	−9.47974	−0.434047
$478$	0.768173	0.0351354
$479$	27.1142	1.23888	0.619439	−	0.785045i	$-0.287360\pi$
$479$	27.1142	1.23888	0.619439	+	0.785045i	$0.287360\pi$
$480$	13.5264	0.617391
$481$	−34.4451	−1.57056
$482$	−6.03622	−0.274942
$483$	−0.603575	−0.0274636
$484$	−25.8921	−1.17691
$485$	−19.4310	−0.882318
$486$	0.661473	0.0300050
$487$	31.4999	1.42740	0.713699	−	0.700452i	$-0.247018\pi$
$487$	31.4999	1.42740	0.713699	+	0.700452i	$0.247018\pi$
$488$	24.8470	1.12477
$489$	23.9501	1.08306
$490$	−1.55635	−0.0703087
$491$	−3.17369	−0.143227	−0.0716133	−	0.997432i	$-0.522815\pi$
$491$	−3.17369	−0.143227	−0.0716133	+	0.997432i	$0.522815\pi$
$492$	9.52342	0.429349
$493$	0	0
$494$	10.6777	0.480412
$495$	12.3545	0.555293
$496$	−0.312825	−0.0140463
$497$	−1.54424	−0.0692686
$498$	−5.35326	−0.239885
$499$	−10.0220	−0.448648	−0.224324	−	0.974515i	$-0.572017\pi$
$499$	−10.0220	−0.448648	−0.224324	+	0.974515i	$0.572017\pi$
$500$	16.4110	0.733920
$501$	−25.1742	−1.12470
$502$	−1.33924	−0.0597734
$503$	32.6493	1.45576	0.727881	−	0.685703i	$-0.240505\pi$
$503$	32.6493	1.45576	0.727881	+	0.685703i	$0.240505\pi$
$504$	2.35647	0.104965
$505$	29.0842	1.29423
$506$	−2.09639	−0.0931960
$507$	−7.23686	−0.321400
$508$	23.1544	1.02731
$509$	−6.78115	−0.300569	−0.150285	−	0.988643i	$-0.548019\pi$
$509$	−6.78115	−0.300569	−0.150285	+	0.988643i	$0.548019\pi$
$510$	0	0
$511$	9.89000	0.437508
$512$	−16.3850	−0.724123
$513$	−3.58834	−0.158429
$514$	12.9487	0.571142
$515$	8.19137	0.360955
$516$	15.5256	0.683476
$517$	−32.3530	−1.42289
$518$	−5.06487	−0.222538
$519$	−11.1650	−0.490088
$520$	−24.9418	−1.09377
$521$	13.5823	0.595052	0.297526	−	0.954714i	$-0.403839\pi$
$521$	13.5823	0.595052	0.297526	+	0.954714i	$0.403839\pi$
$522$	−3.80746	−0.166648
$523$	43.7951	1.91502	0.957512	−	0.288392i	$-0.0931207\pi$
$523$	43.7951	1.91502	0.957512	+	0.288392i	$0.0931207\pi$
$524$	14.2785	0.623761
$525$	−0.535927	−0.0233898
$526$	15.5020	0.675919
$527$	0	0
$528$	−8.22373	−0.357892
$529$	−22.6357	−0.984161
$530$	14.7538	0.640863
$531$	−1.92313	−0.0834568
$532$	−5.60662	−0.243078
$533$	−27.4193	−1.18766
$534$	8.13869	0.352195
$535$	10.9331	0.472681
$536$	−17.6089	−0.760589
$537$	2.39571	0.103383
$538$	−5.29425	−0.228251
$539$	5.25085	0.226170
$540$	3.67623	0.158200
$541$	−1.88211	−0.0809180	−0.0404590	−	0.999181i	$-0.512882\pi$
$541$	−1.88211	−0.0809180	−0.0404590	+	0.999181i	$0.512882\pi$
$542$	−16.2714	−0.698915
$543$	19.6381	0.842752
$544$	0	0
$545$	−32.7822	−1.40424
$546$	−2.97566	−0.127347
$547$	−18.0180	−0.770394	−0.385197	−	0.922834i	$-0.625866\pi$
$547$	−18.0180	−0.770394	−0.385197	+	0.922834i	$0.625866\pi$
$548$	11.7733	0.502932
$549$	10.5442	0.450015
$550$	−1.86143	−0.0793716
$551$	20.6546	0.879918
$552$	−1.42230	−0.0605373
$553$	10.4711	0.445275
$554$	−17.7041	−0.752177
$555$	−18.0157	−0.764725
$556$	0.806077	0.0341853
$557$	19.5144	0.826851	0.413426	−	0.910538i	$-0.364332\pi$
$557$	19.5144	0.826851	0.413426	+	0.910538i	$0.364332\pi$
$558$	0.132122	0.00559316
$559$	−44.7005	−1.89063
$560$	3.68497	0.155719
$561$	0	0
$562$	−16.0723	−0.677970
$563$	−32.4814	−1.36893	−0.684465	−	0.729046i	$-0.739964\pi$
$563$	−32.4814	−1.36893	−0.684465	+	0.729046i	$0.739964\pi$
$564$	−9.62704	−0.405372
$565$	−23.2534	−0.978278
$566$	6.06357	0.254871
$567$	1.00000	0.0419961
$568$	−3.63894	−0.152687
$569$	−32.6889	−1.37039	−0.685194	−	0.728361i	$-0.740283\pi$
$569$	−32.6889	−1.37039	−0.685194	+	0.728361i	$0.740283\pi$
$570$	5.58471	0.233918
$571$	−19.6459	−0.822157	−0.411079	−	0.911600i	$-0.634848\pi$
$571$	−19.6459	−0.822157	−0.411079	+	0.911600i	$0.634848\pi$
$572$	36.9070	1.54316
$573$	5.22705	0.218363
$574$	−4.03178	−0.168283
$575$	0.323472	0.0134897
$576$	0.670404	0.0279335
$577$	−22.6080	−0.941182	−0.470591	−	0.882351i	$-0.655959\pi$
$577$	−22.6080	−0.941182	−0.470591	+	0.882351i	$0.655959\pi$
$578$	0	0
$579$	21.6970	0.901696
$580$	−21.1605	−0.878643
$581$	−8.09294	−0.335752
$582$	−5.46277	−0.226439
$583$	−49.7767	−2.06154
$584$	23.3055	0.964387
$585$	−10.5844	−0.437612
$586$	1.45324	0.0600330
$587$	1.89079	0.0780414	0.0390207	−	0.999238i	$-0.487576\pi$
$587$	1.89079	0.0780414	0.0390207	+	0.999238i	$0.487576\pi$
$588$	1.56245	0.0644345
$589$	−0.716731	−0.0295324
$590$	2.99306	0.123222
$591$	6.69620	0.275445
$592$	11.9921	0.492873
$593$	33.0067	1.35542	0.677711	−	0.735329i	$-0.262972\pi$
$593$	33.0067	1.35542	0.677711	+	0.735329i	$0.262972\pi$
$594$	3.47329	0.142511
$595$	0	0
$596$	3.51470	0.143968
$597$	−6.71661	−0.274892
$598$	1.79603	0.0734453
$599$	12.9658	0.529767	0.264884	−	0.964280i	$-0.414666\pi$
$599$	12.9658	0.529767	0.264884	+	0.964280i	$0.414666\pi$
$600$	−1.26289	−0.0515574
$601$	34.7211	1.41630	0.708152	−	0.706060i	$-0.249529\pi$
$601$	34.7211	1.41630	0.708152	+	0.706060i	$0.249529\pi$
$602$	−6.57283	−0.267889
$603$	−7.47259	−0.304307
$604$	15.7877	0.642392
$605$	38.9901	1.58517
$606$	8.17663	0.332153
$607$	−20.3530	−0.826104	−0.413052	−	0.910708i	$-0.635537\pi$
$607$	−20.3530	−0.826104	−0.413052	+	0.910708i	$0.635537\pi$
$608$	−20.6291	−0.836619
$609$	−5.75604	−0.233247
$610$	−16.4104	−0.664439
$611$	27.7177	1.12134
$612$	0	0
$613$	34.0233	1.37419	0.687095	−	0.726568i	$-0.258886\pi$
$613$	34.0233	1.37419	0.687095	+	0.726568i	$0.258886\pi$
$614$	−13.9496	−0.562960
$615$	−14.3410	−0.578287
$616$	12.3734	0.498540
$617$	29.5230	1.18855	0.594276	−	0.804261i	$-0.297439\pi$
$617$	29.5230	1.18855	0.594276	+	0.804261i	$0.297439\pi$
$618$	2.30289	0.0926358
$619$	−6.20342	−0.249337	−0.124668	−	0.992198i	$-0.539787\pi$
$619$	−6.20342	−0.249337	−0.124668	+	0.992198i	$0.539787\pi$
$620$	0.734285	0.0294896
$621$	−0.603575	−0.0242206
$622$	−7.46055	−0.299141
$623$	12.3039	0.492945
$624$	7.04548	0.282045
$625$	−27.3924	−1.09570
$626$	−13.1823	−0.526870
$627$	−18.8418	−0.752471
$628$	−36.9744	−1.47544
$629$	0	0
$630$	−1.55635	−0.0620064
$631$	5.22637	0.208059	0.104029	−	0.994574i	$-0.466826\pi$
$631$	5.22637	0.208059	0.104029	+	0.994574i	$0.466826\pi$
$632$	24.6747	0.981506
$633$	−13.1733	−0.523592
$634$	20.1822	0.801539
$635$	−34.8676	−1.38368
$636$	−14.8117	−0.587320
$637$	−4.49854	−0.178239
$638$	−19.9924	−0.791507
$639$	−1.54424	−0.0610891
$640$	26.0093	1.02811
$641$	22.9866	0.907917	0.453958	−	0.891023i	$-0.350011\pi$
$641$	22.9866	0.907917	0.453958	+	0.891023i	$0.350011\pi$
$642$	3.07370	0.121309
$643$	−3.61099	−0.142404	−0.0712018	−	0.997462i	$-0.522683\pi$
$643$	−3.61099	−0.142404	−0.0712018	+	0.997462i	$0.522683\pi$
$644$	−0.943058	−0.0371617
$645$	−23.3795	−0.920568
$646$	0	0
$647$	7.56712	0.297494	0.148747	−	0.988875i	$-0.452476\pi$
$647$	7.56712	0.297494	0.148747	+	0.988875i	$0.452476\pi$
$648$	2.35647	0.0925707
$649$	−10.0981	−0.396384
$650$	1.59474	0.0625507
$651$	0.199739	0.00782838
$652$	37.4210	1.46552
$653$	16.6225	0.650487	0.325243	−	0.945630i	$-0.394554\pi$
$653$	16.6225	0.650487	0.325243	+	0.945630i	$0.394554\pi$
$654$	−9.21627	−0.360384
$655$	−21.5016	−0.840139
$656$	9.54607	0.372711
$657$	9.89000	0.385846
$658$	4.07566	0.158886
$659$	20.8412	0.811859	0.405929	−	0.913904i	$-0.366948\pi$
$659$	20.8412	0.811859	0.405929	+	0.913904i	$0.366948\pi$
$660$	19.3033	0.751380
$661$	−8.01914	−0.311908	−0.155954	−	0.987764i	$-0.549845\pi$
$661$	−8.01914	−0.311908	−0.155954	+	0.987764i	$0.549845\pi$
$662$	−16.4044	−0.637574
$663$	0	0
$664$	−19.0707	−0.740088
$665$	8.44285	0.327400
$666$	−5.06487	−0.196260
$667$	3.47420	0.134522
$668$	−39.3335	−1.52186
$669$	−12.0749	−0.466843
$670$	11.6300	0.449305
$671$	55.3659	2.13738
$672$	5.74891	0.221769
$673$	2.90453	0.111962	0.0559808	−	0.998432i	$-0.482171\pi$
$673$	2.90453	0.111962	0.0559808	+	0.998432i	$0.482171\pi$
$674$	−5.77834	−0.222573
$675$	−0.535927	−0.0206278
$676$	−11.3073	−0.434894
$677$	−15.1204	−0.581124	−0.290562	−	0.956856i	$-0.593842\pi$
$677$	−15.1204	−0.581124	−0.290562	+	0.956856i	$0.593842\pi$
$678$	−6.53737	−0.251066
$679$	−8.25850	−0.316932
$680$	0	0
$681$	−9.96214	−0.381750
$682$	0.693751	0.0265651
$683$	−20.5579	−0.786625	−0.393313	−	0.919405i	$-0.628671\pi$
$683$	−20.5579	−0.786625	−0.393313	+	0.919405i	$0.628671\pi$
$684$	−5.60662	−0.214374
$685$	−17.7291	−0.677395
$686$	−0.661473	−0.0252551
$687$	−0.341524	−0.0130300
$688$	15.5625	0.593315
$689$	42.6450	1.62464
$690$	0.939374	0.0357613
$691$	−14.9243	−0.567748	−0.283874	−	0.958862i	$-0.591620\pi$
$691$	−14.9243	−0.567748	−0.283874	+	0.958862i	$0.591620\pi$
$692$	−17.4447	−0.663150
$693$	5.25085	0.199463
$694$	−5.28628	−0.200665
$695$	−1.21385	−0.0460439
$696$	−13.5639	−0.514139
$697$	0	0
$698$	13.3745	0.506234
$699$	15.8961	0.601245
$700$	−0.837361	−0.0316493
$701$	−42.2529	−1.59587	−0.797936	−	0.602743i	$-0.794075\pi$
$701$	−42.2529	−1.59587	−0.797936	+	0.602743i	$0.794075\pi$
$702$	−2.97566	−0.112309
$703$	27.4758	1.03627
$704$	3.52019	0.132672
$705$	14.4971	0.545992
$706$	0.240607	0.00905535
$707$	12.3613	0.464893
$708$	−3.00480	−0.112927
$709$	−41.7973	−1.56973	−0.784865	−	0.619667i	$-0.787268\pi$
$709$	−41.7973	−1.56973	−0.784865	+	0.619667i	$0.787268\pi$
$710$	2.40337	0.0901970
$711$	10.4711	0.392695
$712$	28.9937	1.08658
$713$	−0.120557	−0.00451491
$714$	0	0
$715$	−55.5771	−2.07847
$716$	3.74319	0.139890
$717$	1.16131	0.0433698
$718$	−17.4135	−0.649867
$719$	26.1251	0.974303	0.487152	−	0.873317i	$-0.338036\pi$
$719$	26.1251	0.974303	0.487152	+	0.873317i	$0.338036\pi$
$720$	3.68497	0.137331
$721$	3.48146	0.129656
$722$	4.05073	0.150752
$723$	−9.12543	−0.339379
$724$	30.6836	1.14035
$725$	3.08482	0.114567
$726$	10.9615	0.406820
$727$	18.5323	0.687325	0.343663	−	0.939093i	$-0.388332\pi$
$727$	18.5323	0.687325	0.343663	+	0.939093i	$0.388332\pi$
$728$	−10.6007	−0.392886
$729$	1.00000	0.0370370
$730$	−15.3923	−0.569695
$731$	0	0
$732$	16.4748	0.608927
$733$	−42.9060	−1.58477	−0.792385	−	0.610021i	$-0.791161\pi$
$733$	−42.9060	−1.58477	−0.792385	+	0.610021i	$0.791161\pi$
$734$	−6.54564	−0.241604
$735$	−2.35286	−0.0867864
$736$	−3.46990	−0.127902
$737$	−39.2374	−1.44533
$738$	−4.03178	−0.148412
$739$	19.1622	0.704893	0.352446	−	0.935832i	$-0.385350\pi$
$739$	19.1622	0.704893	0.352446	+	0.935832i	$0.385350\pi$
$740$	−28.1487	−1.03477
$741$	16.1423	0.593002
$742$	6.27059	0.230200
$743$	28.6338	1.05047	0.525236	−	0.850956i	$-0.323977\pi$
$743$	28.6338	1.05047	0.525236	+	0.850956i	$0.323977\pi$
$744$	0.470677	0.0172559
$745$	−5.29269	−0.193909
$746$	−8.54340	−0.312796
$747$	−8.09294	−0.296105
$748$	0	0
$749$	4.64675	0.169789
$750$	−6.94766	−0.253693
$751$	−9.25310	−0.337650	−0.168825	−	0.985646i	$-0.553997\pi$
$751$	−9.25310	−0.337650	−0.168825	+	0.985646i	$0.553997\pi$
$752$	−9.64994	−0.351897
$753$	−2.02464	−0.0737820
$754$	17.1280	0.623766
$755$	−23.7742	−0.865233
$756$	1.56245	0.0568259
$757$	30.7343	1.11706	0.558529	−	0.829485i	$-0.311366\pi$
$757$	30.7343	1.11706	0.558529	+	0.829485i	$0.311366\pi$
$758$	10.5267	0.382348
$759$	−3.16928	−0.115038
$760$	19.8953	0.721678
$761$	−2.54666	−0.0923164	−0.0461582	−	0.998934i	$-0.514698\pi$
$761$	−2.54666	−0.0923164	−0.0461582	+	0.998934i	$0.514698\pi$
$762$	−9.80254	−0.355109
$763$	−13.9330	−0.504407
$764$	8.16703	0.295473
$765$	0	0
$766$	−12.6886	−0.458459
$767$	8.65128	0.312380
$768$	8.65297	0.312237
$769$	−8.31448	−0.299828	−0.149914	−	0.988699i	$-0.547900\pi$
$769$	−8.31448	−0.299828	−0.149914	+	0.988699i	$0.547900\pi$
$770$	−8.17215	−0.294504
$771$	19.5756	0.704996
$772$	33.9005	1.22011
$773$	−37.8279	−1.36058	−0.680288	−	0.732945i	$-0.738145\pi$
$773$	−37.8279	−1.36058	−0.680288	+	0.732945i	$0.738145\pi$
$774$	−6.57283	−0.236256
$775$	−0.107045	−0.00384518
$776$	−19.4609	−0.698604
$777$	−7.65696	−0.274692
$778$	−16.2713	−0.583353
$779$	21.8715	0.783629
$780$	−16.5377	−0.592143
$781$	−8.10856	−0.290147
$782$	0	0
$783$	−5.75604	−0.205704
$784$	1.56617	0.0559347
$785$	55.6788	1.98726
$786$	−6.04489	−0.215614
$787$	45.1711	1.61018	0.805088	−	0.593155i	$-0.202118\pi$
$787$	45.1711	1.61018	0.805088	+	0.593155i	$0.202118\pi$
$788$	10.4625	0.372712
$789$	23.4356	0.834329
$790$	−16.2966	−0.579808
$791$	−9.88305	−0.351401
$792$	12.3734	0.439671
$793$	−47.4335	−1.68441
$794$	14.7725	0.524257
$795$	22.3044	0.791057
$796$	−10.4944	−0.371964
$797$	−6.59399	−0.233571	−0.116786	−	0.993157i	$-0.537259\pi$
$797$	−6.59399	−0.233571	−0.116786	+	0.993157i	$0.537259\pi$
$798$	2.37359	0.0840242
$799$	0	0
$800$	−3.08099	−0.108930
$801$	12.3039	0.434737
$802$	−17.0939	−0.603605
$803$	51.9309	1.83260
$804$	−11.6756	−0.411766
$805$	1.42012	0.0500528
$806$	−0.594355	−0.0209352
$807$	−8.00374	−0.281745
$808$	29.1289	1.02475
$809$	40.7345	1.43215	0.716075	−	0.698024i	$-0.245937\pi$
$809$	40.7345	1.43215	0.716075	+	0.698024i	$0.245937\pi$
$810$	−1.55635	−0.0546845
$811$	−17.5707	−0.616991	−0.308496	−	0.951226i	$-0.599825\pi$
$811$	−17.5707	−0.616991	−0.308496	+	0.951226i	$0.599825\pi$
$812$	−8.99355	−0.315612
$813$	−24.5987	−0.862714
$814$	−26.5949	−0.932149
$815$	−56.3512	−1.97390
$816$	0	0
$817$	35.6562	1.24745
$818$	−10.4256	−0.364521
$819$	−4.49854	−0.157192
$820$	−22.4072	−0.782494
$821$	3.39184	0.118376	0.0591880	−	0.998247i	$-0.481149\pi$
$821$	3.39184	0.118376	0.0591880	+	0.998247i	$0.481149\pi$
$822$	−4.98430	−0.173847
$823$	25.3479	0.883574	0.441787	−	0.897120i	$-0.354345\pi$
$823$	25.3479	0.883574	0.441787	+	0.897120i	$0.354345\pi$
$824$	8.20394	0.285798
$825$	−2.81407	−0.0979733
$826$	1.27210	0.0442620
$827$	8.39886	0.292057	0.146029	−	0.989280i	$-0.453351\pi$
$827$	8.39886	0.292057	0.146029	+	0.989280i	$0.453351\pi$
$828$	−0.943058	−0.0327735
$829$	−0.720245	−0.0250151	−0.0125076	−	0.999922i	$-0.503981\pi$
$829$	−0.720245	−0.0250151	−0.0125076	+	0.999922i	$0.503981\pi$
$830$	12.5954	0.437194
$831$	−26.7647	−0.928458
$832$	−3.01584	−0.104555
$833$	0	0
$834$	−0.341257	−0.0118168
$835$	59.2311	2.04978
$836$	−29.4395	−1.01819
$837$	0.199739	0.00690398
$838$	−8.87716	−0.306656
$839$	−44.7395	−1.54458	−0.772289	−	0.635271i	$-0.780888\pi$
$839$	−44.7395	−1.54458	−0.772289	+	0.635271i	$0.780888\pi$
$840$	−5.54442	−0.191301
$841$	4.13202	0.142484
$842$	−6.79724	−0.234248
$843$	−24.2978	−0.836860
$844$	−20.5827	−0.708486
$845$	17.0273	0.585756
$846$	4.07566	0.140124
$847$	16.5714	0.569400
$848$	−14.8469	−0.509844
$849$	9.16677	0.314603
$850$	0	0
$851$	4.62155	0.158425
$852$	−2.41280	−0.0826612
$853$	6.86208	0.234953	0.117477	−	0.993076i	$-0.462519\pi$
$853$	6.86208	0.234953	0.117477	+	0.993076i	$0.462519\pi$
$854$	−6.97469	−0.238669
$855$	8.44285	0.288739
$856$	10.9499	0.374260
$857$	−20.6553	−0.705573	−0.352787	−	0.935704i	$-0.614766\pi$
$857$	−20.6553	−0.705573	−0.352787	+	0.935704i	$0.614766\pi$
$858$	−15.6247	−0.533420
$859$	5.76996	0.196868	0.0984342	−	0.995144i	$-0.468617\pi$
$859$	5.76996	0.196868	0.0984342	+	0.995144i	$0.468617\pi$
$860$	−36.5294	−1.24564
$861$	−6.09517	−0.207723
$862$	−19.0568	−0.649077
$863$	−45.6981	−1.55558	−0.777790	−	0.628524i	$-0.783659\pi$
$863$	−45.6981	−1.55558	−0.777790	+	0.628524i	$0.783659\pi$
$864$	5.74891	0.195582
$865$	26.2696	0.893191
$866$	16.8532	0.572696
$867$	0	0
$868$	0.312083	0.0105928
$869$	54.9819	1.86513
$870$	8.95841	0.303719
$871$	33.6158	1.13903
$872$	−32.8325	−1.11185
$873$	−8.25850	−0.279508
$874$	−1.43264	−0.0484598
$875$	−10.5033	−0.355077
$876$	15.4527	0.522098
$877$	−5.89643	−0.199108	−0.0995541	−	0.995032i	$-0.531742\pi$
$877$	−5.89643	−0.199108	−0.0995541	+	0.995032i	$0.531742\pi$
$878$	22.4086	0.756252
$879$	2.19698	0.0741024
$880$	19.3492	0.652263
$881$	−28.3117	−0.953846	−0.476923	−	0.878945i	$-0.658248\pi$
$881$	−28.3117	−0.953846	−0.476923	+	0.878945i	$0.658248\pi$
$882$	−0.661473	−0.0222729
$883$	−39.7453	−1.33754	−0.668768	−	0.743471i	$-0.733178\pi$
$883$	−39.7453	−1.33754	−0.668768	+	0.743471i	$0.733178\pi$
$884$	0	0
$885$	4.52485	0.152101
$886$	7.04875	0.236807
$887$	12.7691	0.428746	0.214373	−	0.976752i	$-0.431229\pi$
$887$	12.7691	0.428746	0.214373	+	0.976752i	$0.431229\pi$
$888$	−18.0434	−0.605496
$889$	−14.8193	−0.497022
$890$	−19.1491	−0.641881
$891$	5.25085	0.175910
$892$	−18.8665	−0.631697
$893$	−22.1095	−0.739867
$894$	−1.48797	−0.0497651
$895$	−5.63677	−0.188416
$896$	11.0544	0.369300
$897$	2.71521	0.0906581
$898$	27.2514	0.909392
$899$	−1.14970	−0.0383448
$900$	−0.837361	−0.0279120
$901$	0	0
$902$	−21.1703	−0.704893
$903$	−9.93667	−0.330672
$904$	−23.2891	−0.774583
$905$	−46.2056	−1.53593
$906$	−6.68379	−0.222054
$907$	12.4016	0.411787	0.205894	−	0.978574i	$-0.433990\pi$
$907$	12.4016	0.411787	0.205894	+	0.978574i	$0.433990\pi$
$908$	−15.5654	−0.516556
$909$	12.3613	0.409997
$910$	7.00130	0.232091
$911$	5.75035	0.190517	0.0952587	−	0.995453i	$-0.469632\pi$
$911$	5.75035	0.190517	0.0952587	+	0.995453i	$0.469632\pi$
$912$	−5.61996	−0.186095
$913$	−42.4948	−1.40637
$914$	−27.8927	−0.922610
$915$	−24.8090	−0.820159
$916$	−0.533616	−0.0176312
$917$	−9.13853	−0.301781
$918$	0	0
$919$	−42.2895	−1.39500	−0.697500	−	0.716585i	$-0.745704\pi$
$919$	−42.2895	−1.39500	−0.697500	+	0.716585i	$0.745704\pi$
$920$	3.34648	0.110330
$921$	−21.0887	−0.694897
$922$	−5.35029	−0.176202
$923$	6.94682	0.228657
$924$	8.20421	0.269899
$925$	4.10357	0.134925
$926$	20.5938	0.676754
$927$	3.48146	0.114346
$928$	−33.0910	−1.08626
$929$	11.7820	0.386556	0.193278	−	0.981144i	$-0.438088\pi$
$929$	11.7820	0.386556	0.193278	+	0.981144i	$0.438088\pi$
$930$	−0.310863	−0.0101936
$931$	3.58834	0.117603
$932$	24.8369	0.813559
$933$	−11.2787	−0.369248
$934$	−4.89457	−0.160155
$935$	0	0
$936$	−10.6007	−0.346493
$937$	39.3993	1.28712	0.643560	−	0.765396i	$-0.277457\pi$
$937$	39.3993	1.28712	0.643560	+	0.765396i	$0.277457\pi$
$938$	4.94291	0.161392
$939$	−19.9287	−0.650348
$940$	22.6510	0.738795
$941$	−1.70550	−0.0555977	−0.0277989	−	0.999614i	$-0.508850\pi$
$941$	−1.70550	−0.0555977	−0.0277989	+	0.999614i	$0.508850\pi$
$942$	15.6533	0.510012
$943$	3.67889	0.119801
$944$	−3.01195	−0.0980307
$945$	−2.35286	−0.0765384
$946$	−34.5129	−1.12211
$947$	38.3272	1.24547	0.622733	−	0.782434i	$-0.286022\pi$
$947$	38.3272	1.24547	0.622733	+	0.782434i	$0.286022\pi$
$948$	16.3605	0.531366
$949$	−44.4906	−1.44423
$950$	−1.27207	−0.0412714
$951$	30.5111	0.989390
$952$	0	0
$953$	−43.5262	−1.40995	−0.704975	−	0.709232i	$-0.749042\pi$
$953$	−43.5262	−1.40995	−0.704975	+	0.709232i	$0.749042\pi$
$954$	6.27059	0.203018
$955$	−12.2985	−0.397970
$956$	1.81449	0.0586848
$957$	−30.2241	−0.977006
$958$	−17.9353	−0.579463
$959$	−7.53516	−0.243323
$960$	−1.57736	−0.0509092
$961$	−30.9601	−0.998713
$962$	22.7845	0.734602
$963$	4.64675	0.149739
$964$	−14.2581	−0.459222
$965$	−51.0498	−1.64335
$966$	0.399248	0.0128456
$967$	9.46568	0.304396	0.152198	−	0.988350i	$-0.451365\pi$
$967$	9.46568	0.304396	0.152198	+	0.988350i	$0.451365\pi$
$968$	39.0499	1.25511
$969$	0	0
$970$	12.8531	0.412688
$971$	−1.89708	−0.0608801	−0.0304400	−	0.999537i	$-0.509691\pi$
$971$	−1.89708	−0.0608801	−0.0304400	+	0.999537i	$0.509691\pi$
$972$	1.56245	0.0501157
$973$	−0.515904	−0.0165391
$974$	−20.8363	−0.667639
$975$	2.41089	0.0772102
$976$	16.5140	0.528600
$977$	13.9325	0.445739	0.222869	−	0.974848i	$-0.428458\pi$
$977$	13.9325	0.445739	0.222869	+	0.974848i	$0.428458\pi$
$978$	−15.8424	−0.506583
$979$	64.6059	2.06481
$980$	−3.67623	−0.117433
$981$	−13.9330	−0.444845
$982$	2.09931	0.0669917
$983$	−25.4582	−0.811991	−0.405995	−	0.913875i	$-0.633075\pi$
$983$	−25.4582	−0.811991	−0.405995	+	0.913875i	$0.633075\pi$
$984$	−14.3630	−0.457877
$985$	−15.7552	−0.502002
$986$	0	0
$987$	6.16149	0.196122
$988$	25.2216	0.802406
$989$	5.99752	0.190710
$990$	−8.17215	−0.259728
$991$	−30.2935	−0.962305	−0.481152	−	0.876637i	$-0.659782\pi$
$991$	−30.2935	−0.962305	−0.481152	+	0.876637i	$0.659782\pi$
$992$	1.14828	0.0364579
$993$	−24.7998	−0.786997
$994$	1.02147	0.0323991
$995$	15.8032	0.500995
$996$	−12.6448	−0.400667
$997$	29.7213	0.941282	0.470641	−	0.882325i	$-0.344023\pi$
$997$	29.7213	0.941282	0.470641	+	0.882325i	$0.344023\pi$
$998$	6.62931	0.209847
$999$	−7.65696	−0.242256

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6069.2.a.bd.1.4		10
17.2	even	8		357.2.k.b.106.4	yes	20
17.9	even	8		357.2.k.b.64.7	✓	20
17.16	even	2		6069.2.a.be.1.4		10
51.2	odd	8		1071.2.n.b.820.7		20
51.26	odd	8		1071.2.n.b.64.4		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
357.2.k.b.64.7	✓	20	17.9	even	8
357.2.k.b.106.4	yes	20	17.2	even	8
1071.2.n.b.64.4		20	51.26	odd	8
1071.2.n.b.820.7		20	51.2	odd	8
6069.2.a.bd.1.4		10	1.1	even	1	trivial
6069.2.a.be.1.4		10	17.16	even	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	6069.2.a.bd.1.4

Level	$6069$
Weight	$2$
Character	6069.1
Self dual	yes
Analytic conductor	$48.461$
Analytic rank	$0$
Dimension	$10$
CM	no
Inner twists	$1$