LMFDB - Embedded newform 1254.2.a.r.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1254,2,Mod(1,1254)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1254, 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("1254.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1254 = 2 \cdot 3 \cdot 11 \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1254.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:	$10.0132404135$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.23377.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} - 7x^{2} + 6x + 7$ x^4 - x^3 - 7x^2 + 6x + 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$2^{2}$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.35438$ of defining polynomial
Character	$\chi$	$=$	1254.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +2.27870 q^{5} -1.00000 q^{6} +4.70876 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.27870 q^{10} +1.00000 q^{11} +1.00000 q^{12} +5.08623 q^{13} -4.70876 q^{14} +2.27870 q^{15} +1.00000 q^{16} -5.36493 q^{17} -1.00000 q^{18} +1.00000 q^{19} +2.27870 q^{20} +4.70876 q^{21} -1.00000 q^{22} +7.64363 q^{23} -1.00000 q^{24} +0.192470 q^{25} -5.08623 q^{26} +1.00000 q^{27} +4.70876 q^{28} -5.36493 q^{29} -2.27870 q^{30} -8.35239 q^{31} -1.00000 q^{32} +1.00000 q^{33} +5.36493 q^{34} +10.7299 q^{35} +1.00000 q^{36} +1.56994 q^{37} -1.00000 q^{38} +5.08623 q^{39} -2.27870 q^{40} -11.4175 q^{41} -4.70876 q^{42} -3.36493 q^{43} +1.00000 q^{44} +2.27870 q^{45} -7.64363 q^{46} -7.64363 q^{47} +1.00000 q^{48} +15.1725 q^{49} -0.192470 q^{50} -5.36493 q^{51} +5.08623 q^{52} -7.26616 q^{53} -1.00000 q^{54} +2.27870 q^{55} -4.70876 q^{56} +1.00000 q^{57} +5.36493 q^{58} +2.27870 q^{60} -10.6311 q^{61} +8.35239 q^{62} +4.70876 q^{63} +1.00000 q^{64} +11.5900 q^{65} -1.00000 q^{66} -7.92233 q^{67} -5.36493 q^{68} +7.64363 q^{69} -10.7299 q^{70} +4.15137 q^{71} -1.00000 q^{72} +6.55740 q^{73} -1.56994 q^{74} +0.192470 q^{75} +1.00000 q^{76} +4.70876 q^{77} -5.08623 q^{78} -17.0612 q^{79} +2.27870 q^{80} +1.00000 q^{81} +11.4175 q^{82} +0.860130 q^{83} +4.70876 q^{84} -12.2251 q^{85} +3.36493 q^{86} -5.36493 q^{87} -1.00000 q^{88} +5.36493 q^{89} -2.27870 q^{90} +23.9499 q^{91} +7.64363 q^{92} -8.35239 q^{93} +7.64363 q^{94} +2.27870 q^{95} -1.00000 q^{96} +18.9023 q^{97} -15.1725 q^{98} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{2} + 4 q^{3} + 4 q^{4} + 3 q^{5} - 4 q^{6} + 2 q^{7} - 4 q^{8} + 4 q^{9} - 3 q^{10} + 4 q^{11} + 4 q^{12} + 6 q^{13} - 2 q^{14} + 3 q^{15} + 4 q^{16} - q^{17} - 4 q^{18} + 4 q^{19} + 3 q^{20}+ \cdots + 4 q^{99}+O(q^{100})$ 4 * q - 4 * q^2 + 4 * q^3 + 4 * q^4 + 3 * q^5 - 4 * q^6 + 2 * q^7 - 4 * q^8 + 4 * q^9 - 3 * q^10 + 4 * q^11 + 4 * q^12 + 6 * q^13 - 2 * q^14 + 3 * q^15 + 4 * q^16 - q^17 - 4 * q^18 + 4 * q^19 + 3 * q^20 + 2 * q^21 - 4 * q^22 + 4 * q^23 - 4 * q^24 + 9 * q^25 - 6 * q^26 + 4 * q^27 + 2 * q^28 - q^29 - 3 * q^30 + 10 * q^31 - 4 * q^32 + 4 * q^33 + q^34 + 2 * q^35 + 4 * q^36 + 17 * q^37 - 4 * q^38 + 6 * q^39 - 3 * q^40 - 12 * q^41 - 2 * q^42 + 7 * q^43 + 4 * q^44 + 3 * q^45 - 4 * q^46 - 4 * q^47 + 4 * q^48 + 32 * q^49 - 9 * q^50 - q^51 + 6 * q^52 - 4 * q^54 + 3 * q^55 - 2 * q^56 + 4 * q^57 + q^58 + 3 * q^60 + 7 * q^61 - 10 * q^62 + 2 * q^63 + 4 * q^64 - 16 * q^65 - 4 * q^66 + q^67 - q^68 + 4 * q^69 - 2 * q^70 + 12 * q^71 - 4 * q^72 + 14 * q^73 - 17 * q^74 + 9 * q^75 + 4 * q^76 + 2 * q^77 - 6 * q^78 - 8 * q^79 + 3 * q^80 + 4 * q^81 + 12 * q^82 - 18 * q^83 + 2 * q^84 - 7 * q^85 - 7 * q^86 - q^87 - 4 * q^88 + q^89 - 3 * q^90 + 4 * q^91 + 4 * q^92 + 10 * q^93 + 4 * q^94 + 3 * q^95 - 4 * q^96 + 6 * q^97 - 32 * q^98 + 4 * 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

)

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	2.27870	1.01907	0.509533	−	0.860451i	$-0.329818\pi$
$5$	2.27870	1.01907	0.509533	+	0.860451i	$0.329818\pi$
$6$	−1.00000	−0.408248
$7$	4.70876	1.77975	0.889873	−	0.456209i	$-0.150793\pi$
$7$	4.70876	1.77975	0.889873	+	0.456209i	$0.150793\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−2.27870	−0.720588
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	1.00000	0.288675
$13$	5.08623	1.41067	0.705333	−	0.708876i	$-0.250797\pi$
$13$	5.08623	1.41067	0.705333	+	0.708876i	$0.250797\pi$
$14$	−4.70876	−1.25847
$15$	2.27870	0.588358
$16$	1.00000	0.250000
$17$	−5.36493	−1.30119	−0.650593	−	0.759426i	$-0.725480\pi$
$17$	−5.36493	−1.30119	−0.650593	+	0.759426i	$0.725480\pi$
$18$	−1.00000	−0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.27870	0.509533
$21$	4.70876	1.02754
$22$	−1.00000	−0.213201
$23$	7.64363	1.59381	0.796903	−	0.604107i	$-0.206470\pi$
$23$	7.64363	1.59381	0.796903	+	0.604107i	$0.206470\pi$
$24$	−1.00000	−0.204124
$25$	0.192470	0.0384940
$26$	−5.08623	−0.997492
$27$	1.00000	0.192450
$28$	4.70876	0.889873
$29$	−5.36493	−0.996242	−0.498121	−	0.867107i	$-0.665977\pi$
$29$	−5.36493	−0.996242	−0.498121	+	0.867107i	$0.665977\pi$
$30$	−2.27870	−0.416032
$31$	−8.35239	−1.50013	−0.750067	−	0.661362i	$-0.769979\pi$
$31$	−8.35239	−1.50013	−0.750067	+	0.661362i	$0.769979\pi$
$32$	−1.00000	−0.176777
$33$	1.00000	0.174078
$34$	5.36493	0.920078
$35$	10.7299	1.81368
$36$	1.00000	0.166667
$37$	1.56994	0.258096	0.129048	−	0.991638i	$-0.458808\pi$
$37$	1.56994	0.258096	0.129048	+	0.991638i	$0.458808\pi$
$38$	−1.00000	−0.162221
$39$	5.08623	0.814448
$40$	−2.27870	−0.360294
$41$	−11.4175	−1.78312	−0.891559	−	0.452904i	$-0.850388\pi$
$41$	−11.4175	−1.78312	−0.891559	+	0.452904i	$0.850388\pi$
$42$	−4.70876	−0.726578
$43$	−3.36493	−0.513147	−0.256573	−	0.966525i	$-0.582594\pi$
$43$	−3.36493	−0.513147	−0.256573	+	0.966525i	$0.582594\pi$
$44$	1.00000	0.150756
$45$	2.27870	0.339688
$46$	−7.64363	−1.12699
$47$	−7.64363	−1.11494	−0.557469	−	0.830198i	$-0.688227\pi$
$47$	−7.64363	−1.11494	−0.557469	+	0.830198i	$0.688227\pi$
$48$	1.00000	0.144338
$49$	15.1725	2.16749
$50$	−0.192470	−0.0272194
$51$	−5.36493	−0.751240
$52$	5.08623	0.705333
$53$	−7.26616	−0.998084	−0.499042	−	0.866578i	$-0.666315\pi$
$53$	−7.26616	−0.998084	−0.499042	+	0.866578i	$0.666315\pi$
$54$	−1.00000	−0.136083
$55$	2.27870	0.307260
$56$	−4.70876	−0.629235
$57$	1.00000	0.132453
$58$	5.36493	0.704450
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	2.27870	0.294179
$61$	−10.6311	−1.36117	−0.680586	−	0.732668i	$-0.738275\pi$
$61$	−10.6311	−1.36117	−0.680586	+	0.732668i	$0.738275\pi$
$62$	8.35239	1.06075
$63$	4.70876	0.593249
$64$	1.00000	0.125000
$65$	11.5900	1.43756
$66$	−1.00000	−0.123091
$67$	−7.92233	−0.967866	−0.483933	−	0.875105i	$-0.660792\pi$
$67$	−7.92233	−0.967866	−0.483933	+	0.875105i	$0.660792\pi$
$68$	−5.36493	−0.650593
$69$	7.64363	0.920185
$70$	−10.7299	−1.28246
$71$	4.15137	0.492676	0.246338	−	0.969184i	$-0.420773\pi$
$71$	4.15137	0.492676	0.246338	+	0.969184i	$0.420773\pi$
$72$	−1.00000	−0.117851
$73$	6.55740	0.767485	0.383743	−	0.923440i	$-0.374635\pi$
$73$	6.55740	0.767485	0.383743	+	0.923440i	$0.374635\pi$
$74$	−1.56994	−0.182501
$75$	0.192470	0.0222245
$76$	1.00000	0.114708
$77$	4.70876	0.536613
$78$	−5.08623	−0.575902
$79$	−17.0612	−1.91953	−0.959765	−	0.280805i	$-0.909399\pi$
$79$	−17.0612	−1.91953	−0.959765	+	0.280805i	$0.909399\pi$
$80$	2.27870	0.254766
$81$	1.00000	0.111111
$82$	11.4175	1.26086
$83$	0.860130	0.0944115	0.0472057	−	0.998885i	$-0.484968\pi$
$83$	0.860130	0.0944115	0.0472057	+	0.998885i	$0.484968\pi$
$84$	4.70876	0.513768
$85$	−12.2251	−1.32599
$86$	3.36493	0.362850
$87$	−5.36493	−0.575181
$88$	−1.00000	−0.106600
$89$	5.36493	0.568681	0.284341	−	0.958723i	$-0.408225\pi$
$89$	5.36493	0.568681	0.284341	+	0.958723i	$0.408225\pi$
$90$	−2.27870	−0.240196
$91$	23.9499	2.51063
$92$	7.64363	0.796903
$93$	−8.35239	−0.866103
$94$	7.64363	0.788380
$95$	2.27870	0.233790
$96$	−1.00000	−0.102062
$97$	18.9023	1.91924	0.959620	−	0.281301i	$-0.0907659\pi$
$97$	18.9023	1.91924	0.959620	+	0.281301i	$0.0907659\pi$
$98$	−15.1725	−1.53265
$99$	1.00000	0.100504
$100$	0.192470	0.0192470
$101$	−6.54992	−0.651742	−0.325871	−	0.945414i	$-0.605657\pi$
$101$	−6.54992	−0.651742	−0.325871	+	0.945414i	$0.605657\pi$
$102$	5.36493	0.531207
$103$	−2.17993	−0.214795	−0.107398	−	0.994216i	$-0.534252\pi$
$103$	−2.17993	−0.214795	−0.107398	+	0.994216i	$0.534252\pi$
$104$	−5.08623	−0.498746
$105$	10.7299	1.04713
$106$	7.26616	0.705752
$107$	−5.19247	−0.501975	−0.250988	−	0.967990i	$-0.580755\pi$
$107$	−5.19247	−0.501975	−0.250988	+	0.967990i	$0.580755\pi$
$108$	1.00000	0.0962250
$109$	4.22610	0.404787	0.202393	−	0.979304i	$-0.435128\pi$
$109$	4.22610	0.404787	0.202393	+	0.979304i	$0.435128\pi$
$110$	−2.27870	−0.217265
$111$	1.56994	0.149012
$112$	4.70876	0.444936
$113$	9.92233	0.933414	0.466707	−	0.884412i	$-0.345440\pi$
$113$	9.92233	0.933414	0.466707	+	0.884412i	$0.345440\pi$
$114$	−1.00000	−0.0936586
$115$	17.4175	1.62419
$116$	−5.36493	−0.498121
$117$	5.08623	0.470222
$118$	0	0
$119$	−25.2622	−2.31578
$120$	−2.27870	−0.208016
$121$	1.00000	0.0909091
$122$	10.6311	0.962494
$123$	−11.4175	−1.02948
$124$	−8.35239	−0.750067
$125$	−10.9549	−0.979837
$126$	−4.70876	−0.419490
$127$	9.69623	0.860401	0.430201	−	0.902733i	$-0.358443\pi$
$127$	9.69623	0.860401	0.430201	+	0.902733i	$0.358443\pi$
$128$	−1.00000	−0.0883883
$129$	−3.36493	−0.296265
$130$	−11.5900	−1.01651
$131$	0.860130	0.0751499	0.0375749	−	0.999294i	$-0.488037\pi$
$131$	0.860130	0.0751499	0.0375749	+	0.999294i	$0.488037\pi$
$132$	1.00000	0.0870388
$133$	4.70876	0.408302
$134$	7.92233	0.684385
$135$	2.27870	0.196119
$136$	5.36493	0.460039
$137$	0.687671	0.0587517	0.0293759	−	0.999568i	$-0.490648\pi$
$137$	0.687671	0.0587517	0.0293759	+	0.999568i	$0.490648\pi$
$138$	−7.64363	−0.650669
$139$	16.4797	1.39779	0.698896	−	0.715223i	$-0.253675\pi$
$139$	16.4797	1.39779	0.698896	+	0.715223i	$0.253675\pi$
$140$	10.7299	0.906838
$141$	−7.64363	−0.643710
$142$	−4.15137	−0.348375
$143$	5.08623	0.425332
$144$	1.00000	0.0833333
$145$	−12.2251	−1.01524
$146$	−6.55740	−0.542694
$147$	15.1725	1.25140
$148$	1.56994	0.129048
$149$	−11.2125	−0.918566	−0.459283	−	0.888290i	$-0.651894\pi$
$149$	−11.2125	−0.918566	−0.459283	+	0.888290i	$0.651894\pi$
$150$	−0.192470	−0.0157151
$151$	−0.278699	−0.0226802	−0.0113401	−	0.999936i	$-0.503610\pi$
$151$	−0.278699	−0.0226802	−0.0113401	+	0.999936i	$0.503610\pi$
$152$	−1.00000	−0.0811107
$153$	−5.36493	−0.433729
$154$	−4.70876	−0.379443
$155$	−19.0326	−1.52873
$156$	5.08623	0.407224
$157$	6.55740	0.523337	0.261669	−	0.965158i	$-0.415727\pi$
$157$	6.55740	0.523337	0.261669	+	0.965158i	$0.415727\pi$
$158$	17.0612	1.35731
$159$	−7.26616	−0.576244
$160$	−2.27870	−0.180147
$161$	35.9920	2.83657
$162$	−1.00000	−0.0785674
$163$	19.5900	1.53441	0.767203	−	0.641404i	$-0.221648\pi$
$163$	19.5900	1.53441	0.767203	+	0.641404i	$0.221648\pi$
$164$	−11.4175	−0.891559
$165$	2.27870	0.177396
$166$	−0.860130	−0.0667590
$167$	15.2873	1.18296	0.591482	−	0.806318i	$-0.298543\pi$
$167$	15.2873	1.18296	0.591482	+	0.806318i	$0.298543\pi$
$168$	−4.70876	−0.363289
$169$	12.8697	0.989979
$170$	12.2251	0.937619
$171$	1.00000	0.0764719
$172$	−3.36493	−0.256573
$173$	3.33985	0.253924	0.126962	−	0.991908i	$-0.459477\pi$
$173$	3.33985	0.253924	0.126962	+	0.991908i	$0.459477\pi$
$174$	5.36493	0.406714
$175$	0.906296	0.0685095
$176$	1.00000	0.0753778
$177$	0	0
$178$	−5.36493	−0.402118
$179$	−23.6471	−1.76747	−0.883734	−	0.467989i	$-0.844979\pi$
$179$	−23.6471	−1.76747	−0.883734	+	0.467989i	$0.844979\pi$
$180$	2.27870	0.169844
$181$	19.8476	1.47526	0.737630	−	0.675205i	$-0.235945\pi$
$181$	19.8476	1.47526	0.737630	+	0.675205i	$0.235945\pi$
$182$	−23.9499	−1.77528
$183$	−10.6311	−0.785873
$184$	−7.64363	−0.563496
$185$	3.57741	0.263016
$186$	8.35239	0.612427
$187$	−5.36493	−0.392322
$188$	−7.64363	−0.557469
$189$	4.70876	0.342512
$190$	−2.27870	−0.165314
$191$	15.9464	1.15384	0.576919	−	0.816801i	$-0.304255\pi$
$191$	15.9464	1.15384	0.576919	+	0.816801i	$0.304255\pi$
$192$	1.00000	0.0721688
$193$	−13.9223	−1.00215	−0.501076	−	0.865404i	$-0.667062\pi$
$193$	−13.9223	−1.00215	−0.501076	+	0.865404i	$0.667062\pi$
$194$	−18.9023	−1.35711
$195$	11.5900	0.829976
$196$	15.1725	1.08375
$197$	6.65512	0.474158	0.237079	−	0.971490i	$-0.423810\pi$
$197$	6.65512	0.474158	0.237079	+	0.971490i	$0.423810\pi$
$198$	−1.00000	−0.0710669
$199$	16.0697	1.13915	0.569576	−	0.821939i	$-0.307107\pi$
$199$	16.0697	1.13915	0.569576	+	0.821939i	$0.307107\pi$
$200$	−0.192470	−0.0136097
$201$	−7.92233	−0.558798
$202$	6.54992	0.460851
$203$	−25.2622	−1.77306
$204$	−5.36493	−0.375620
$205$	−26.0171	−1.81711
$206$	2.17993	0.151883
$207$	7.64363	0.531269
$208$	5.08623	0.352667
$209$	1.00000	0.0691714
$210$	−10.7299	−0.740430
$211$	−13.0577	−0.898926	−0.449463	−	0.893299i	$-0.648385\pi$
$211$	−13.0577	−0.898926	−0.449463	+	0.893299i	$0.648385\pi$
$212$	−7.26616	−0.499042
$213$	4.15137	0.284447
$214$	5.19247	0.354950
$215$	−7.66766	−0.522930
$216$	−1.00000	−0.0680414
$217$	−39.3294	−2.66986
$218$	−4.22610	−0.286228
$219$	6.55740	0.443108
$220$	2.27870	0.153630
$221$	−27.2873	−1.83554
$222$	−1.56994	−0.105367
$223$	−21.9675	−1.47105	−0.735525	−	0.677498i	$-0.763064\pi$
$223$	−21.9675	−1.47105	−0.735525	+	0.677498i	$0.763064\pi$
$224$	−4.70876	−0.314618
$225$	0.192470	0.0128313
$226$	−9.92233	−0.660023
$227$	−22.2000	−1.47346	−0.736732	−	0.676184i	$-0.763632\pi$
$227$	−22.2000	−1.47346	−0.736732	+	0.676184i	$0.763632\pi$
$228$	1.00000	0.0662266
$229$	8.68767	0.574097	0.287049	−	0.957916i	$-0.407326\pi$
$229$	8.68767	0.574097	0.287049	+	0.957916i	$0.407326\pi$
$230$	−17.4175	−1.14848
$231$	4.70876	0.309814
$232$	5.36493	0.352225
$233$	−5.24507	−0.343616	−0.171808	−	0.985130i	$-0.554961\pi$
$233$	−5.24507	−0.343616	−0.171808	+	0.985130i	$0.554961\pi$
$234$	−5.08623	−0.332497
$235$	−17.4175	−1.13619
$236$	0	0
$237$	−17.0612	−1.10824
$238$	25.2622	1.63750
$239$	4.88226	0.315807	0.157904	−	0.987455i	$-0.449526\pi$
$239$	4.88226	0.315807	0.157904	+	0.987455i	$0.449526\pi$
$240$	2.27870	0.147089
$241$	−4.05260	−0.261051	−0.130525	−	0.991445i	$-0.541666\pi$
$241$	−4.05260	−0.261051	−0.130525	+	0.991445i	$0.541666\pi$
$242$	−1.00000	−0.0642824
$243$	1.00000	0.0641500
$244$	−10.6311	−0.680586
$245$	34.5735	2.20882
$246$	11.4175	0.727955
$247$	5.08623	0.323629
$248$	8.35239	0.530377
$249$	0.860130	0.0545085
$250$	10.9549	0.692850
$251$	20.0697	1.26679	0.633394	−	0.773829i	$-0.281661\pi$
$251$	20.0697	1.26679	0.633394	+	0.773829i	$0.281661\pi$
$252$	4.70876	0.296624
$253$	7.64363	0.480551
$254$	−9.69623	−0.608395
$255$	−12.2251	−0.765563
$256$	1.00000	0.0625000
$257$	26.5849	1.65832	0.829161	−	0.559010i	$-0.188819\pi$
$257$	26.5849	1.65832	0.829161	+	0.559010i	$0.188819\pi$
$258$	3.36493	0.209491
$259$	7.39245	0.459345
$260$	11.5900	0.718780
$261$	−5.36493	−0.332081
$262$	−0.860130	−0.0531390
$263$	−15.4923	−0.955294	−0.477647	−	0.878552i	$-0.658510\pi$
$263$	−15.4923	−0.955294	−0.477647	+	0.878552i	$0.658510\pi$
$264$	−1.00000	−0.0615457
$265$	−16.5574	−1.01711
$266$	−4.70876	−0.288713
$267$	5.36493	0.328328
$268$	−7.92233	−0.483933
$269$	13.3334	0.812953	0.406477	−	0.913661i	$-0.366757\pi$
$269$	13.3334	0.812953	0.406477	+	0.913661i	$0.366757\pi$
$270$	−2.27870	−0.138677
$271$	−16.5112	−1.00299	−0.501493	−	0.865162i	$-0.667216\pi$
$271$	−16.5112	−1.00299	−0.501493	+	0.865162i	$0.667216\pi$
$272$	−5.36493	−0.325297
$273$	23.9499	1.44951
$274$	−0.687671	−0.0415437
$275$	0.192470	0.0116064
$276$	7.64363	0.460092
$277$	13.8190	0.830305	0.415152	−	0.909752i	$-0.363728\pi$
$277$	13.8190	0.830305	0.415152	+	0.909752i	$0.363728\pi$
$278$	−16.4797	−0.988388
$279$	−8.35239	−0.500045
$280$	−10.7299	−0.641232
$281$	16.2146	0.967285	0.483642	−	0.875266i	$-0.339314\pi$
$281$	16.2146	0.967285	0.483642	+	0.875266i	$0.339314\pi$
$282$	7.64363	0.455171
$283$	−6.88520	−0.409283	−0.204641	−	0.978837i	$-0.565603\pi$
$283$	−6.88520	−0.409283	−0.204641	+	0.978837i	$0.565603\pi$
$284$	4.15137	0.246338
$285$	2.27870	0.134978
$286$	−5.08623	−0.300755
$287$	−53.7624	−3.17350
$288$	−1.00000	−0.0589256
$289$	11.7825	0.693086
$290$	12.2251	0.717880
$291$	18.9023	1.10807
$292$	6.55740	0.383743
$293$	9.92233	0.579669	0.289834	−	0.957077i	$-0.406400\pi$
$293$	9.92233	0.579669	0.289834	+	0.957077i	$0.406400\pi$
$294$	−15.1725	−0.884876
$295$	0	0
$296$	−1.56994	−0.0912506
$297$	1.00000	0.0580259
$298$	11.2125	0.649524
$299$	38.8772	2.24833
$300$	0.192470	0.0111123
$301$	−15.8447	−0.913271
$302$	0.278699	0.0160373
$303$	−6.54992	−0.376283
$304$	1.00000	0.0573539
$305$	−24.2251	−1.38712
$306$	5.36493	0.306693
$307$	−28.3449	−1.61773	−0.808865	−	0.587995i	$-0.799918\pi$
$307$	−28.3449	−1.61773	−0.808865	+	0.587995i	$0.799918\pi$
$308$	4.70876	0.268307
$309$	−2.17993	−0.124012
$310$	19.0326	1.08098
$311$	−14.6762	−0.832212	−0.416106	−	0.909316i	$-0.636605\pi$
$311$	−14.6762	−0.832212	−0.416106	+	0.909316i	$0.636605\pi$
$312$	−5.08623	−0.287951
$313$	−23.0852	−1.30485	−0.652426	−	0.757852i	$-0.726249\pi$
$313$	−23.0852	−1.30485	−0.652426	+	0.757852i	$0.726249\pi$
$314$	−6.55740	−0.370055
$315$	10.7299	0.604559
$316$	−17.0612	−0.959765
$317$	−1.09370	−0.0614285	−0.0307143	−	0.999528i	$-0.509778\pi$
$317$	−1.09370	−0.0614285	−0.0307143	+	0.999528i	$0.509778\pi$
$318$	7.26616	0.407466
$319$	−5.36493	−0.300378
$320$	2.27870	0.127383
$321$	−5.19247	−0.289815
$322$	−35.9920	−2.00576
$323$	−5.36493	−0.298513
$324$	1.00000	0.0555556
$325$	0.978947	0.0543022
$326$	−19.5900	−1.08499
$327$	4.22610	0.233704
$328$	11.4175	0.630428
$329$	−35.9920	−1.98431
$330$	−2.27870	−0.125438
$331$	−21.1925	−1.16484	−0.582422	−	0.812887i	$-0.697895\pi$
$331$	−21.1925	−1.16484	−0.582422	+	0.812887i	$0.697895\pi$
$332$	0.860130	0.0472057
$333$	1.56994	0.0860319
$334$	−15.2873	−0.836481
$335$	−18.0526	−0.986319
$336$	4.70876	0.256884
$337$	11.2347	0.611991	0.305995	−	0.952033i	$-0.401011\pi$
$337$	11.2347	0.611991	0.305995	+	0.952033i	$0.401011\pi$
$338$	−12.8697	−0.700021
$339$	9.92233	0.538907
$340$	−12.2251	−0.662997
$341$	−8.35239	−0.452307
$342$	−1.00000	−0.0540738
$343$	38.4822	2.07784
$344$	3.36493	0.181425
$345$	17.4175	0.937728
$346$	−3.33985	−0.179552
$347$	19.9499	1.07096	0.535482	−	0.844547i	$-0.320130\pi$
$347$	19.9499	1.07096	0.535482	+	0.844547i	$0.320130\pi$
$348$	−5.36493	−0.287590
$349$	−4.45863	−0.238665	−0.119333	−	0.992854i	$-0.538075\pi$
$349$	−4.45863	−0.238665	−0.119333	+	0.992854i	$0.538075\pi$
$350$	−0.906296	−0.0484436
$351$	5.08623	0.271483
$352$	−1.00000	−0.0533002
$353$	6.86013	0.365128	0.182564	−	0.983194i	$-0.441560\pi$
$353$	6.86013	0.365128	0.182564	+	0.983194i	$0.441560\pi$
$354$	0	0
$355$	9.45971	0.502069
$356$	5.36493	0.284341
$357$	−25.2622	−1.33702
$358$	23.6471	1.24979
$359$	−32.4998	−1.71527	−0.857636	−	0.514257i	$-0.828068\pi$
$359$	−32.4998	−1.71527	−0.857636	+	0.514257i	$0.828068\pi$
$360$	−2.27870	−0.120098
$361$	1.00000	0.0526316
$362$	−19.8476	−1.04317
$363$	1.00000	0.0524864
$364$	23.9499	1.25531
$365$	14.9423	0.782118
$366$	10.6311	0.555696
$367$	−10.9127	−0.569640	−0.284820	−	0.958581i	$-0.591934\pi$
$367$	−10.9127	−0.569640	−0.284820	+	0.958581i	$0.591934\pi$
$368$	7.64363	0.398452
$369$	−11.4175	−0.594373
$370$	−3.57741	−0.185981
$371$	−34.2146	−1.77634
$372$	−8.35239	−0.433051
$373$	14.2010	0.735301	0.367651	−	0.929964i	$-0.380162\pi$
$373$	14.2010	0.735301	0.367651	+	0.929964i	$0.380162\pi$
$374$	5.36493	0.277414
$375$	−10.9549	−0.565709
$376$	7.64363	0.394190
$377$	−27.2873	−1.40537
$378$	−4.70876	−0.242193
$379$	33.1423	1.70241	0.851203	−	0.524836i	$-0.175873\pi$
$379$	33.1423	1.70241	0.851203	+	0.524836i	$0.175873\pi$
$380$	2.27870	0.116895
$381$	9.69623	0.496753
$382$	−15.9464	−0.815887
$383$	30.2883	1.54766	0.773831	−	0.633392i	$-0.218338\pi$
$383$	30.2883	1.54766	0.773831	+	0.633392i	$0.218338\pi$
$384$	−1.00000	−0.0510310
$385$	10.7299	0.546844
$386$	13.9223	0.708628
$387$	−3.36493	−0.171049
$388$	18.9023	0.959620
$389$	17.3409	0.879218	0.439609	−	0.898189i	$-0.355117\pi$
$389$	17.3409	0.879218	0.439609	+	0.898189i	$0.355117\pi$
$390$	−11.5900	−0.586882
$391$	−41.0075	−2.07384
$392$	−15.1725	−0.766325
$393$	0.860130	0.0433878
$394$	−6.65512	−0.335280
$395$	−38.8772	−1.95613
$396$	1.00000	0.0502519
$397$	−25.2873	−1.26913	−0.634565	−	0.772869i	$-0.718821\pi$
$397$	−25.2873	−1.26913	−0.634565	+	0.772869i	$0.718821\pi$
$398$	−16.0697	−0.805502
$399$	4.70876	0.235733
$400$	0.192470	0.00962350
$401$	−21.1103	−1.05420	−0.527098	−	0.849804i	$-0.676720\pi$
$401$	−21.1103	−1.05420	−0.527098	+	0.849804i	$0.676720\pi$
$402$	7.92233	0.395130
$403$	−42.4822	−2.11619
$404$	−6.54992	−0.325871
$405$	2.27870	0.113229
$406$	25.2622	1.25374
$407$	1.56994	0.0778188
$408$	5.36493	0.265604
$409$	−27.4096	−1.35532	−0.677658	−	0.735377i	$-0.737005\pi$
$409$	−27.4096	−1.35532	−0.677658	+	0.735377i	$0.737005\pi$
$410$	26.0171	1.28489
$411$	0.687671	0.0339203
$412$	−2.17993	−0.107398
$413$	0	0
$414$	−7.64363	−0.375664
$415$	1.95998	0.0962115
$416$	−5.08623	−0.249373
$417$	16.4797	0.807016
$418$	−1.00000	−0.0489116
$419$	−8.88974	−0.434292	−0.217146	−	0.976139i	$-0.569675\pi$
$419$	−8.88974	−0.434292	−0.217146	+	0.976139i	$0.569675\pi$
$420$	10.7299	0.523563
$421$	7.96745	0.388310	0.194155	−	0.980971i	$-0.437804\pi$
$421$	7.96745	0.388310	0.194155	+	0.980971i	$0.437804\pi$
$422$	13.0577	0.635637
$423$	−7.64363	−0.371646
$424$	7.26616	0.352876
$425$	−1.03259	−0.0500879
$426$	−4.15137	−0.201134
$427$	−50.0593	−2.42254
$428$	−5.19247	−0.250988
$429$	5.08623	0.245565
$430$	7.66766	0.369767
$431$	−22.3198	−1.07511	−0.537555	−	0.843229i	$-0.680652\pi$
$431$	−22.3198	−1.07511	−0.537555	+	0.843229i	$0.680652\pi$
$432$	1.00000	0.0481125
$433$	1.58999	0.0764099	0.0382049	−	0.999270i	$-0.487836\pi$
$433$	1.58999	0.0764099	0.0382049	+	0.999270i	$0.487836\pi$
$434$	39.3294	1.88787
$435$	−12.2251	−0.586147
$436$	4.22610	0.202393
$437$	7.64363	0.365644
$438$	−6.55740	−0.313325
$439$	1.85157	0.0883708	0.0441854	−	0.999023i	$-0.485931\pi$
$439$	1.85157	0.0883708	0.0441854	+	0.999023i	$0.485931\pi$
$440$	−2.27870	−0.108633
$441$	15.1725	0.722498
$442$	27.2873	1.29792
$443$	1.73492	0.0824285	0.0412142	−	0.999150i	$-0.486877\pi$
$443$	1.73492	0.0824285	0.0412142	+	0.999150i	$0.486877\pi$
$444$	1.56994	0.0745058
$445$	12.2251	0.579523
$446$	21.9675	1.04019
$447$	−11.2125	−0.530334
$448$	4.70876	0.222468
$449$	4.49014	0.211903	0.105951	−	0.994371i	$-0.466211\pi$
$449$	4.49014	0.211903	0.105951	+	0.994371i	$0.466211\pi$
$450$	−0.192470	−0.00907312
$451$	−11.4175	−0.537630
$452$	9.92233	0.466707
$453$	−0.278699	−0.0130944
$454$	22.2000	1.04190
$455$	54.5745	2.55849
$456$	−1.00000	−0.0468293
$457$	2.81207	0.131543	0.0657714	−	0.997835i	$-0.479049\pi$
$457$	2.81207	0.131543	0.0657714	+	0.997835i	$0.479049\pi$
$458$	−8.68767	−0.405948
$459$	−5.36493	−0.250413
$460$	17.4175	0.812096
$461$	25.5825	1.19150	0.595748	−	0.803171i	$-0.296856\pi$
$461$	25.5825	1.19150	0.595748	+	0.803171i	$0.296856\pi$
$462$	−4.70876	−0.219072
$463$	−25.3399	−1.17764	−0.588821	−	0.808263i	$-0.700408\pi$
$463$	−25.3399	−1.17764	−0.588821	+	0.808263i	$0.700408\pi$
$464$	−5.36493	−0.249061
$465$	−19.0326	−0.882615
$466$	5.24507	0.242973
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	5.08623	0.235111
$469$	−37.3044	−1.72256
$470$	17.4175	0.803411
$471$	6.55740	0.302149
$472$	0	0
$473$	−3.36493	−0.154720
$474$	17.0612	0.783645
$475$	0.192470	0.00883113
$476$	−25.2622	−1.15789
$477$	−7.26616	−0.332695
$478$	−4.88226	−0.223310
$479$	30.7649	1.40568	0.702841	−	0.711347i	$-0.251914\pi$
$479$	30.7649	1.40568	0.702841	+	0.711347i	$0.251914\pi$
$480$	−2.27870	−0.104008
$481$	7.98505	0.364087
$482$	4.05260	0.184591
$483$	35.9920	1.63769
$484$	1.00000	0.0454545
$485$	43.0727	1.95583
$486$	−1.00000	−0.0453609
$487$	−7.95034	−0.360264	−0.180132	−	0.983642i	$-0.557653\pi$
$487$	−7.95034	−0.360264	−0.180132	+	0.983642i	$0.557653\pi$
$488$	10.6311	0.481247
$489$	19.5900	0.885890
$490$	−34.5735	−1.56187
$491$	32.8100	1.48069	0.740347	−	0.672225i	$-0.234661\pi$
$491$	32.8100	1.48069	0.740347	+	0.672225i	$0.234661\pi$
$492$	−11.4175	−0.514742
$493$	28.7825	1.29630
$494$	−5.08623	−0.228840
$495$	2.27870	0.102420
$496$	−8.35239	−0.375033
$497$	19.5478	0.876839
$498$	−0.860130	−0.0385433
$499$	−8.04219	−0.360018	−0.180009	−	0.983665i	$-0.557613\pi$
$499$	−8.04219	−0.360018	−0.180009	+	0.983665i	$0.557613\pi$
$500$	−10.9549	−0.489919
$501$	15.2873	0.682984
$502$	−20.0697	−0.895755
$503$	−5.56540	−0.248149	−0.124074	−	0.992273i	$-0.539596\pi$
$503$	−5.56540	−0.248149	−0.124074	+	0.992273i	$0.539596\pi$
$504$	−4.70876	−0.209745
$505$	−14.9253	−0.664167
$506$	−7.64363	−0.339801
$507$	12.8697	0.571565
$508$	9.69623	0.430201
$509$	5.24905	0.232660	0.116330	−	0.993211i	$-0.462887\pi$
$509$	5.24905	0.232660	0.116330	+	0.993211i	$0.462887\pi$
$510$	12.2251	0.541335
$511$	30.8772	1.36593
$512$	−1.00000	−0.0441942
$513$	1.00000	0.0441511
$514$	−26.5849	−1.17261
$515$	−4.96741	−0.218890
$516$	−3.36493	−0.148133
$517$	−7.64363	−0.336166
$518$	−7.39245	−0.324806
$519$	3.33985	0.146603
$520$	−11.5900	−0.508255
$521$	10.4375	0.457277	0.228638	−	0.973511i	$-0.426573\pi$
$521$	10.4375	0.457277	0.228638	+	0.973511i	$0.426573\pi$
$522$	5.36493	0.234817
$523$	−19.0475	−0.832891	−0.416445	−	0.909161i	$-0.636724\pi$
$523$	−19.0475	−0.832891	−0.416445	+	0.909161i	$0.636724\pi$
$524$	0.860130	0.0375749
$525$	0.906296	0.0395540
$526$	15.4923	0.675495
$527$	44.8100	1.95195
$528$	1.00000	0.0435194
$529$	35.4250	1.54022
$530$	16.5574	0.719207
$531$	0	0
$532$	4.70876	0.204151
$533$	−58.0722	−2.51538
$534$	−5.36493	−0.232163
$535$	−11.8321	−0.511545
$536$	7.92233	0.342192
$537$	−23.6471	−1.02045
$538$	−13.3334	−0.574845
$539$	15.1725	0.653524
$540$	2.27870	0.0980596
$541$	33.7835	1.45247	0.726234	−	0.687448i	$-0.241269\pi$
$541$	33.7835	1.45247	0.726234	+	0.687448i	$0.241269\pi$
$542$	16.5112	0.709218
$543$	19.8476	0.851742
$544$	5.36493	0.230019
$545$	9.63001	0.412504
$546$	−23.9499	−1.02496
$547$	15.8447	0.677468	0.338734	−	0.940882i	$-0.390001\pi$
$547$	15.8447	0.677468	0.338734	+	0.940882i	$0.390001\pi$
$548$	0.687671	0.0293759
$549$	−10.6311	−0.453724
$550$	−0.192470	−0.00820695
$551$	−5.36493	−0.228554
$552$	−7.64363	−0.325334
$553$	−80.3370	−3.41627
$554$	−13.8190	−0.587114
$555$	3.57741	0.151853
$556$	16.4797	0.698896
$557$	25.5825	1.08397	0.541983	−	0.840390i	$-0.317674\pi$
$557$	25.5825	1.08397	0.541983	+	0.840390i	$0.317674\pi$
$558$	8.35239	0.353585
$559$	−17.1148	−0.723879
$560$	10.7299	0.453419
$561$	−5.36493	−0.226507
$562$	−16.2146	−0.683973
$563$	−11.3169	−0.476949	−0.238474	−	0.971149i	$-0.576647\pi$
$563$	−11.3169	−0.476949	−0.238474	+	0.971149i	$0.576647\pi$
$564$	−7.64363	−0.321855
$565$	22.6100	0.951210
$566$	6.88520	0.289407
$567$	4.70876	0.197750
$568$	−4.15137	−0.174187
$569$	24.5174	1.02782	0.513911	−	0.857844i	$-0.328196\pi$
$569$	24.5174	1.02782	0.513911	+	0.857844i	$0.328196\pi$
$570$	−2.27870	−0.0954442
$571$	12.3323	0.516092	0.258046	−	0.966133i	$-0.416921\pi$
$571$	12.3323	0.516092	0.258046	+	0.966133i	$0.416921\pi$
$572$	5.08623	0.212666
$573$	15.9464	0.666169
$574$	53.7624	2.24400
$575$	1.47117	0.0613520
$576$	1.00000	0.0416667
$577$	3.04509	0.126769	0.0633843	−	0.997989i	$-0.479811\pi$
$577$	3.04509	0.126769	0.0633843	+	0.997989i	$0.479811\pi$
$578$	−11.7825	−0.490086
$579$	−13.9223	−0.578592
$580$	−12.2251	−0.507618
$581$	4.05015	0.168028
$582$	−18.9023	−0.783526
$583$	−7.26616	−0.300934
$584$	−6.55740	−0.271347
$585$	11.5900	0.479187
$586$	−9.92233	−0.409888
$587$	13.6004	0.561349	0.280674	−	0.959803i	$-0.409442\pi$
$587$	13.6004	0.561349	0.280674	+	0.959803i	$0.409442\pi$
$588$	15.1725	0.625702
$589$	−8.35239	−0.344154
$590$	0	0
$591$	6.65512	0.273755
$592$	1.56994	0.0645239
$593$	35.0075	1.43759	0.718793	−	0.695224i	$-0.244695\pi$
$593$	35.0075	1.43759	0.718793	+	0.695224i	$0.244695\pi$
$594$	−1.00000	−0.0410305
$595$	−57.5649	−2.35993
$596$	−11.2125	−0.459283
$597$	16.0697	0.657690
$598$	−38.8772	−1.58981
$599$	38.3033	1.56503	0.782515	−	0.622632i	$-0.213937\pi$
$599$	38.3033	1.56503	0.782515	+	0.622632i	$0.213937\pi$
$600$	−0.192470	−0.00785756
$601$	34.3519	1.40124	0.700622	−	0.713533i	$-0.252906\pi$
$601$	34.3519	1.40124	0.700622	+	0.713533i	$0.252906\pi$
$602$	15.8447	0.645780
$603$	−7.92233	−0.322622
$604$	−0.278699	−0.0113401
$605$	2.27870	0.0926423
$606$	6.54992	0.266072
$607$	19.2838	0.782704	0.391352	−	0.920241i	$-0.372008\pi$
$607$	19.2838	0.782704	0.391352	+	0.920241i	$0.372008\pi$
$608$	−1.00000	−0.0405554
$609$	−25.2622	−1.02368
$610$	24.2251	0.980844
$611$	−38.8772	−1.57280
$612$	−5.36493	−0.216864
$613$	0.790973	0.0319471	0.0159735	−	0.999872i	$-0.494915\pi$
$613$	0.790973	0.0319471	0.0159735	+	0.999872i	$0.494915\pi$
$614$	28.3449	1.14391
$615$	−26.0171	−1.04911
$616$	−4.70876	−0.189722
$617$	40.4250	1.62745	0.813725	−	0.581249i	$-0.197436\pi$
$617$	40.4250	1.62745	0.813725	+	0.581249i	$0.197436\pi$
$618$	2.17993	0.0876898
$619$	37.0075	1.48746	0.743729	−	0.668481i	$-0.233055\pi$
$619$	37.0075	1.48746	0.743729	+	0.668481i	$0.233055\pi$
$620$	−19.0326	−0.764367
$621$	7.64363	0.306728
$622$	14.6762	0.588463
$623$	25.2622	1.01211
$624$	5.08623	0.203612
$625$	−25.9253	−1.03701
$626$	23.0852	0.922670
$627$	1.00000	0.0399362
$628$	6.55740	0.261669
$629$	−8.42259	−0.335831
$630$	−10.7299	−0.427488
$631$	−40.6271	−1.61734	−0.808670	−	0.588263i	$-0.799812\pi$
$631$	−40.6271	−1.61734	−0.808670	+	0.588263i	$0.799812\pi$
$632$	17.0612	0.678656
$633$	−13.0577	−0.518995
$634$	1.09370	0.0434365
$635$	22.0948	0.876805
$636$	−7.26616	−0.288122
$637$	77.1706	3.05761
$638$	5.36493	0.212400
$639$	4.15137	0.164225
$640$	−2.27870	−0.0900735
$641$	−22.8543	−0.902689	−0.451344	−	0.892350i	$-0.649055\pi$
$641$	−22.8543	−0.902689	−0.451344	+	0.892350i	$0.649055\pi$
$642$	5.19247	0.204930
$643$	−37.5649	−1.48142	−0.740708	−	0.671827i	$-0.765510\pi$
$643$	−37.5649	−1.48142	−0.740708	+	0.671827i	$0.765510\pi$
$644$	35.9920	1.41828
$645$	−7.66766	−0.301914
$646$	5.36493	0.211080
$647$	−36.3062	−1.42734	−0.713672	−	0.700480i	$-0.752970\pi$
$647$	−36.3062	−1.42734	−0.713672	+	0.700480i	$0.752970\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	−0.978947	−0.0383974
$651$	−39.3294	−1.54144
$652$	19.5900	0.767203
$653$	21.7931	0.852830	0.426415	−	0.904528i	$-0.359776\pi$
$653$	21.7931	0.852830	0.426415	+	0.904528i	$0.359776\pi$
$654$	−4.22610	−0.165254
$655$	1.95998	0.0765826
$656$	−11.4175	−0.445780
$657$	6.55740	0.255828
$658$	35.9920	1.40312
$659$	−7.13779	−0.278049	−0.139024	−	0.990289i	$-0.544397\pi$
$659$	−7.13779	−0.278049	−0.139024	+	0.990289i	$0.544397\pi$
$660$	2.27870	0.0886982
$661$	16.2702	0.632837	0.316418	−	0.948620i	$-0.397520\pi$
$661$	16.2702	0.632837	0.316418	+	0.948620i	$0.397520\pi$
$662$	21.1925	0.823669
$663$	−27.2873	−1.05975
$664$	−0.860130	−0.0333795
$665$	10.7299	0.416086
$666$	−1.56994	−0.0608338
$667$	−41.0075	−1.58782
$668$	15.2873	0.591482
$669$	−21.9675	−0.849311
$670$	18.0526	0.697433
$671$	−10.6311	−0.410409
$672$	−4.70876	−0.181645
$673$	8.80753	0.339505	0.169753	−	0.985487i	$-0.445703\pi$
$673$	8.80753	0.339505	0.169753	+	0.985487i	$0.445703\pi$
$674$	−11.2347	−0.432743
$675$	0.192470	0.00740817
$676$	12.8697	0.494990
$677$	3.81259	0.146530	0.0732649	−	0.997313i	$-0.476658\pi$
$677$	3.81259	0.146530	0.0732649	+	0.997313i	$0.476658\pi$
$678$	−9.92233	−0.381065
$679$	89.0065	3.41576
$680$	12.2251	0.468810
$681$	−22.2000	−0.850705
$682$	8.35239	0.319830
$683$	−40.2526	−1.54022	−0.770111	−	0.637910i	$-0.779799\pi$
$683$	−40.2526	−1.54022	−0.770111	+	0.637910i	$0.779799\pi$
$684$	1.00000	0.0382360
$685$	1.56700	0.0598718
$686$	−38.4822	−1.46926
$687$	8.68767	0.331455
$688$	−3.36493	−0.128287
$689$	−36.9574	−1.40796
$690$	−17.4175	−0.663074
$691$	39.7875	1.51359	0.756794	−	0.653653i	$-0.226764\pi$
$691$	39.7875	1.51359	0.756794	+	0.653653i	$0.226764\pi$
$692$	3.33985	0.126962
$693$	4.70876	0.178871
$694$	−19.9499	−0.757286
$695$	37.5523	1.42444
$696$	5.36493	0.203357
$697$	61.2542	2.32017
$698$	4.45863	0.168762
$699$	−5.24507	−0.198387
$700$	0.906296	0.0342548
$701$	−33.0822	−1.24950	−0.624750	−	0.780825i	$-0.714799\pi$
$701$	−33.0822	−1.24950	−0.624750	+	0.780825i	$0.714799\pi$
$702$	−5.08623	−0.191967
$703$	1.56994	0.0592112
$704$	1.00000	0.0376889
$705$	−17.4175	−0.655982
$706$	−6.86013	−0.258184
$707$	−30.8420	−1.15993
$708$	0	0
$709$	−37.2222	−1.39791	−0.698954	−	0.715167i	$-0.746351\pi$
$709$	−37.2222	−1.39791	−0.698954	+	0.715167i	$0.746351\pi$
$710$	−9.45971	−0.355017
$711$	−17.0612	−0.639843
$712$	−5.36493	−0.201059
$713$	−63.8426	−2.39092
$714$	25.2622	0.945413
$715$	11.5900	0.433441
$716$	−23.6471	−0.883734
$717$	4.88226	0.182331
$718$	32.4998	1.21288
$719$	26.6191	0.992724	0.496362	−	0.868116i	$-0.334669\pi$
$719$	26.6191	0.992724	0.496362	+	0.868116i	$0.334669\pi$
$720$	2.27870	0.0849221
$721$	−10.2648	−0.382281
$722$	−1.00000	−0.0372161
$723$	−4.05260	−0.150718
$724$	19.8476	0.737630
$725$	−1.03259	−0.0383494
$726$	−1.00000	−0.0371135
$727$	−30.7002	−1.13861	−0.569305	−	0.822127i	$-0.692787\pi$
$727$	−30.7002	−1.13861	−0.569305	+	0.822127i	$0.692787\pi$
$728$	−23.9499	−0.887641
$729$	1.00000	0.0370370
$730$	−14.9423	−0.553041
$731$	18.0526	0.667700
$732$	−10.6311	−0.392936
$733$	15.2515	0.563327	0.281664	−	0.959513i	$-0.409114\pi$
$733$	15.2515	0.563327	0.281664	+	0.959513i	$0.409114\pi$
$734$	10.9127	0.402796
$735$	34.5735	1.27526
$736$	−7.64363	−0.281748
$737$	−7.92233	−0.291823
$738$	11.4175	0.420285
$739$	18.6247	0.685119	0.342560	−	0.939496i	$-0.388706\pi$
$739$	18.6247	0.685119	0.342560	+	0.939496i	$0.388706\pi$
$740$	3.57741	0.131508
$741$	5.08623	0.186847
$742$	34.2146	1.25606
$743$	−5.72234	−0.209932	−0.104966	−	0.994476i	$-0.533473\pi$
$743$	−5.72234	−0.209932	−0.104966	+	0.994476i	$0.533473\pi$
$744$	8.35239	0.306214
$745$	−25.5500	−0.936078
$746$	−14.2010	−0.519937
$747$	0.860130	0.0314705
$748$	−5.36493	−0.196161
$749$	−24.4501	−0.893388
$750$	10.9549	0.400017
$751$	9.01707	0.329038	0.164519	−	0.986374i	$-0.447393\pi$
$751$	9.01707	0.329038	0.164519	+	0.986374i	$0.447393\pi$
$752$	−7.64363	−0.278734
$753$	20.0697	0.731381
$754$	27.2873	0.993743
$755$	−0.635072	−0.0231126
$756$	4.70876	0.171256
$757$	19.4747	0.707819	0.353909	−	0.935280i	$-0.384852\pi$
$757$	19.4747	0.707819	0.353909	+	0.935280i	$0.384852\pi$
$758$	−33.1423	−1.20378
$759$	7.64363	0.277446
$760$	−2.27870	−0.0826571
$761$	26.7194	0.968579	0.484290	−	0.874908i	$-0.339078\pi$
$761$	26.7194	0.968579	0.484290	+	0.874908i	$0.339078\pi$
$762$	−9.69623	−0.351257
$763$	19.8997	0.720418
$764$	15.9464	0.576919
$765$	−12.2251	−0.441998
$766$	−30.2883	−1.09436
$767$	0	0
$768$	1.00000	0.0360844
$769$	−17.4346	−0.628709	−0.314355	−	0.949306i	$-0.601788\pi$
$769$	−17.4346	−0.628709	−0.314355	+	0.949306i	$0.601788\pi$
$770$	−10.7299	−0.386677
$771$	26.5849	0.957433
$772$	−13.9223	−0.501076
$773$	12.0633	0.433886	0.216943	−	0.976184i	$-0.430391\pi$
$773$	12.0633	0.433886	0.216943	+	0.976184i	$0.430391\pi$
$774$	3.36493	0.120950
$775$	−1.60759	−0.0577462
$776$	−18.9023	−0.678554
$777$	7.39245	0.265203
$778$	−17.3409	−0.621701
$779$	−11.4175	−0.409075
$780$	11.5900	0.414988
$781$	4.15137	0.148548
$782$	41.0075	1.46643
$783$	−5.36493	−0.191727
$784$	15.1725	0.541874
$785$	14.9423	0.533315
$786$	−0.860130	−0.0306798
$787$	−34.2777	−1.22187	−0.610933	−	0.791682i	$-0.709206\pi$
$787$	−34.2777	−1.22187	−0.610933	+	0.791682i	$0.709206\pi$
$788$	6.65512	0.237079
$789$	−15.4923	−0.551539
$790$	38.8772	1.38319
$791$	46.7219	1.66124
$792$	−1.00000	−0.0355335
$793$	−54.0722	−1.92016
$794$	25.2873	0.897411
$795$	−16.5574	−0.587230
$796$	16.0697	0.569576
$797$	−17.4808	−0.619202	−0.309601	−	0.950867i	$-0.600195\pi$
$797$	−17.4808	−0.619202	−0.309601	+	0.950867i	$0.600195\pi$
$798$	−4.70876	−0.166688
$799$	41.0075	1.45074
$800$	−0.192470	−0.00680484
$801$	5.36493	0.189560
$802$	21.1103	0.745429
$803$	6.55740	0.231406
$804$	−7.92233	−0.279399
$805$	82.0150	2.89065
$806$	42.4822	1.49637
$807$	13.3334	0.469359
$808$	6.54992	0.230426
$809$	1.73945	0.0611560	0.0305780	−	0.999532i	$-0.490265\pi$
$809$	1.73945	0.0611560	0.0305780	+	0.999532i	$0.490265\pi$
$810$	−2.27870	−0.0800653
$811$	−25.0577	−0.879894	−0.439947	−	0.898024i	$-0.645003\pi$
$811$	−25.0577	−0.879894	−0.439947	+	0.898024i	$0.645003\pi$
$812$	−25.2622	−0.886529
$813$	−16.5112	−0.579074
$814$	−1.56994	−0.0550262
$815$	44.6397	1.56366
$816$	−5.36493	−0.187810
$817$	−3.36493	−0.117724
$818$	27.4096	0.958353
$819$	23.9499	0.836876
$820$	−26.0171	−0.908557
$821$	42.3503	1.47804	0.739018	−	0.673686i	$-0.235290\pi$
$821$	42.3503	1.47804	0.739018	+	0.673686i	$0.235290\pi$
$822$	−0.687671	−0.0239853
$823$	−43.4071	−1.51308	−0.756538	−	0.653949i	$-0.773111\pi$
$823$	−43.4071	−1.51308	−0.756538	+	0.653949i	$0.773111\pi$
$824$	2.17993	0.0759416
$825$	0.192470	0.00670095
$826$	0	0
$827$	−7.54780	−0.262463	−0.131231	−	0.991352i	$-0.541893\pi$
$827$	−7.54780	−0.262463	−0.131231	+	0.991352i	$0.541893\pi$
$828$	7.64363	0.265634
$829$	−34.3650	−1.19354	−0.596772	−	0.802411i	$-0.703550\pi$
$829$	−34.3650	−1.19354	−0.596772	+	0.802411i	$0.703550\pi$
$830$	−1.95998	−0.0680318
$831$	13.8190	0.479377
$832$	5.08623	0.176333
$833$	−81.3992	−2.82031
$834$	−16.4797	−0.570646
$835$	34.8351	1.20552
$836$	1.00000	0.0345857
$837$	−8.35239	−0.288701
$838$	8.88974	0.307091
$839$	−28.1581	−0.972124	−0.486062	−	0.873924i	$-0.661567\pi$
$839$	−28.1581	−0.972124	−0.486062	+	0.873924i	$0.661567\pi$
$840$	−10.7299	−0.370215
$841$	−0.217544	−0.00750151
$842$	−7.96745	−0.274577
$843$	16.2146	0.558462
$844$	−13.0577	−0.449463
$845$	29.3262	1.00885
$846$	7.64363	0.262793
$847$	4.70876	0.161795
$848$	−7.26616	−0.249521
$849$	−6.88520	−0.236300
$850$	1.03259	0.0354175
$851$	12.0000	0.411355
$852$	4.15137	0.142223
$853$	−56.3012	−1.92772	−0.963858	−	0.266416i	$-0.914161\pi$
$853$	−56.3012	−1.92772	−0.963858	+	0.266416i	$0.914161\pi$
$854$	50.0593	1.71299
$855$	2.27870	0.0779299
$856$	5.19247	0.177475
$857$	21.6893	0.740893	0.370446	−	0.928854i	$-0.379205\pi$
$857$	21.6893	0.740893	0.370446	+	0.928854i	$0.379205\pi$
$858$	−5.08623	−0.173641
$859$	53.1068	1.81198	0.905991	−	0.423297i	$-0.139127\pi$
$859$	53.1068	1.81198	0.905991	+	0.423297i	$0.139127\pi$
$860$	−7.66766	−0.261465
$861$	−53.7624	−1.83222
$862$	22.3198	0.760217
$863$	2.83316	0.0964418	0.0482209	−	0.998837i	$-0.484645\pi$
$863$	2.83316	0.0964418	0.0482209	+	0.998837i	$0.484645\pi$
$864$	−1.00000	−0.0340207
$865$	7.61052	0.258766
$866$	−1.58999	−0.0540299
$867$	11.7825	0.400153
$868$	−39.3294	−1.33493
$869$	−17.0612	−0.578760
$870$	12.2251	0.414468
$871$	−40.2948	−1.36534
$872$	−4.22610	−0.143114
$873$	18.9023	0.639746
$874$	−7.64363	−0.258550
$875$	−51.5841	−1.74386
$876$	6.55740	0.221554
$877$	3.81609	0.128860	0.0644300	−	0.997922i	$-0.479477\pi$
$877$	3.81609	0.128860	0.0644300	+	0.997922i	$0.479477\pi$
$878$	−1.85157	−0.0624876
$879$	9.92233	0.334672
$880$	2.27870	0.0768149
$881$	−52.7790	−1.77817	−0.889085	−	0.457741i	$-0.848659\pi$
$881$	−52.7790	−1.77817	−0.889085	+	0.457741i	$0.848659\pi$
$882$	−15.1725	−0.510883
$883$	−49.6220	−1.66991	−0.834957	−	0.550315i	$-0.814508\pi$
$883$	−49.6220	−1.66991	−0.834957	+	0.550315i	$0.814508\pi$
$884$	−27.2873	−0.917770
$885$	0	0
$886$	−1.73492	−0.0582857
$887$	37.7774	1.26844	0.634220	−	0.773152i	$-0.281321\pi$
$887$	37.7774	1.26844	0.634220	+	0.773152i	$0.281321\pi$
$888$	−1.56994	−0.0526836
$889$	45.6572	1.53129
$890$	−12.2251	−0.409785
$891$	1.00000	0.0335013
$892$	−21.9675	−0.735525
$893$	−7.64363	−0.255784
$894$	11.2125	0.375003
$895$	−53.8847	−1.80117
$896$	−4.70876	−0.157309
$897$	38.8772	1.29807
$898$	−4.49014	−0.149838
$899$	44.8100	1.49450
$900$	0.192470	0.00641567
$901$	38.9824	1.29869
$902$	11.4175	0.380162
$903$	−15.8447	−0.527277
$904$	−9.92233	−0.330012
$905$	45.2267	1.50339
$906$	0.278699	0.00925916
$907$	24.1327	0.801314	0.400657	−	0.916228i	$-0.368782\pi$
$907$	24.1327	0.801314	0.400657	+	0.916228i	$0.368782\pi$
$908$	−22.2000	−0.736732
$909$	−6.54992	−0.217247
$910$	−54.5745	−1.80913
$911$	−26.8666	−0.890129	−0.445064	−	0.895499i	$-0.646819\pi$
$911$	−26.8666	−0.890129	−0.445064	+	0.895499i	$0.646819\pi$
$912$	1.00000	0.0331133
$913$	0.860130	0.0284661
$914$	−2.81207	−0.0930149
$915$	−24.2251	−0.800856
$916$	8.68767	0.287049
$917$	4.05015	0.133748
$918$	5.36493	0.177069
$919$	7.28915	0.240447	0.120223	−	0.992747i	$-0.461639\pi$
$919$	7.28915	0.240447	0.120223	+	0.992747i	$0.461639\pi$
$920$	−17.4175	−0.574239
$921$	−28.3449	−0.933997
$922$	−25.5825	−0.842515
$923$	21.1148	0.695002
$924$	4.70876	0.154907
$925$	0.302165	0.00993514
$926$	25.3399	0.832719
$927$	−2.17993	−0.0715984
$928$	5.36493	0.176112
$929$	−58.9515	−1.93414	−0.967068	−	0.254519i	$-0.918083\pi$
$929$	−58.9515	−1.93414	−0.967068	+	0.254519i	$0.918083\pi$
$930$	19.0326	0.624103
$931$	15.1725	0.497257
$932$	−5.24507	−0.171808
$933$	−14.6762	−0.480478
$934$	−24.0000	−0.785304
$935$	−12.2251	−0.399802
$936$	−5.08623	−0.166249
$937$	−12.9423	−0.422808	−0.211404	−	0.977399i	$-0.567804\pi$
$937$	−12.9423	−0.422808	−0.211404	+	0.977399i	$0.567804\pi$
$938$	37.3044	1.21803
$939$	−23.0852	−0.753357
$940$	−17.4175	−0.568097
$941$	16.5048	0.538041	0.269021	−	0.963134i	$-0.413300\pi$
$941$	16.5048	0.538041	0.269021	+	0.963134i	$0.413300\pi$
$942$	−6.55740	−0.213652
$943$	−87.2713	−2.84195
$944$	0	0
$945$	10.7299	0.349042
$946$	3.36493	0.109403
$947$	−39.5123	−1.28398	−0.641989	−	0.766714i	$-0.721890\pi$
$947$	−39.5123	−1.28398	−0.641989	+	0.766714i	$0.721890\pi$
$948$	−17.0612	−0.554121
$949$	33.3524	1.08267
$950$	−0.192470	−0.00624455
$951$	−1.09370	−0.0354658
$952$	25.2622	0.818752
$953$	−32.4271	−1.05042	−0.525209	−	0.850973i	$-0.676013\pi$
$953$	−32.4271	−1.05042	−0.525209	+	0.850973i	$0.676013\pi$
$954$	7.26616	0.235251
$955$	36.3370	1.17584
$956$	4.88226	0.157904
$957$	−5.36493	−0.173424
$958$	−30.7649	−0.993967
$959$	3.23808	0.104563
$960$	2.27870	0.0735447
$961$	38.7624	1.25040
$962$	−7.98505	−0.257448
$963$	−5.19247	−0.167325
$964$	−4.05260	−0.130525
$965$	−31.7248	−1.02126
$966$	−35.9920	−1.15802
$967$	34.4712	1.10852	0.554260	−	0.832344i	$-0.313001\pi$
$967$	34.4712	1.10852	0.554260	+	0.832344i	$0.313001\pi$
$968$	−1.00000	−0.0321412
$969$	−5.36493	−0.172346
$970$	−43.0727	−1.38298
$971$	−9.67807	−0.310584	−0.155292	−	0.987869i	$-0.549632\pi$
$971$	−9.67807	−0.310584	−0.155292	+	0.987869i	$0.549632\pi$
$972$	1.00000	0.0320750
$973$	77.5991	2.48771
$974$	7.95034	0.254745
$975$	0.978947	0.0313514
$976$	−10.6311	−0.340293
$977$	20.1223	0.643770	0.321885	−	0.946779i	$-0.395684\pi$
$977$	20.1223	0.643770	0.321885	+	0.946779i	$0.395684\pi$
$978$	−19.5900	−0.626419
$979$	5.36493	0.171464
$980$	34.5735	1.10441
$981$	4.22610	0.134929
$982$	−32.8100	−1.04701
$983$	−34.0281	−1.08533	−0.542664	−	0.839950i	$-0.682584\pi$
$983$	−34.0281	−1.08533	−0.542664	+	0.839950i	$0.682584\pi$
$984$	11.4175	0.363977
$985$	15.1650	0.483198
$986$	−28.7825	−0.916620
$987$	−35.9920	−1.14564
$988$	5.08623	0.161815
$989$	−25.7203	−0.817857
$990$	−2.27870	−0.0724218
$991$	−24.9579	−0.792812	−0.396406	−	0.918075i	$-0.629743\pi$
$991$	−24.9579	−0.792812	−0.396406	+	0.918075i	$0.629743\pi$
$992$	8.35239	0.265189
$993$	−21.1925	−0.672523
$994$	−19.5478	−0.620019
$995$	36.6180	1.16087
$996$	0.860130	0.0272542
$997$	3.41943	0.108294	0.0541471	−	0.998533i	$-0.482756\pi$
$997$	3.41943	0.108294	0.0541471	+	0.998533i	$0.482756\pi$
$998$	8.04219	0.254571
$999$	1.56994	0.0496706

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instead) (See

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a_n

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1254.2.a.r.1.3	✓	4
3.2	odd	2		3762.2.a.bh.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1254.2.a.r.1.3	✓	4	1.1	even	1	trivial
3762.2.a.bh.1.2		4	3.2	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

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Fields

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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	1254.2.a.r.1.3

Level	$1254$
Weight	$2$
Character	1254.1
Self dual	yes
Analytic conductor	$10.013$
Analytic rank	$0$
Dimension	$4$
CM	no
Inner twists	$1$