LMFDB - Embedded newform 374.2.a.c.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [374,2,Mod(1,374)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(374, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("374.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$374 = 2 \cdot 11 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	374.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$2.98640503560$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{3} - x^{2} - 4x + 3$ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.19869$ of defining polynomial
Character	$\chi$	$=$	374.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +2.83424 q^{3} +1.00000 q^{4} -4.03293 q^{5} +2.83424 q^{6} +4.19869 q^{7} +1.00000 q^{8} +5.03293 q^{9} -4.03293 q^{10} -1.00000 q^{11} +2.83424 q^{12} -2.83424 q^{13} +4.19869 q^{14} -11.4303 q^{15} +1.00000 q^{16} -1.00000 q^{17} +5.03293 q^{18} -3.86718 q^{19} -4.03293 q^{20} +11.9001 q^{21} -1.00000 q^{22} +5.12628 q^{23} +2.83424 q^{24} +11.2646 q^{25} -2.83424 q^{26} +5.76183 q^{27} +4.19869 q^{28} -5.12628 q^{29} -11.4303 q^{30} -10.4633 q^{31} +1.00000 q^{32} -2.83424 q^{33} -1.00000 q^{34} -16.9330 q^{35} +5.03293 q^{36} +2.59607 q^{37} -3.86718 q^{38} -8.03293 q^{39} -4.03293 q^{40} +3.30404 q^{41} +11.9001 q^{42} +5.30404 q^{43} -1.00000 q^{44} -20.2975 q^{45} +5.12628 q^{46} -10.7618 q^{47} +2.83424 q^{48} +10.6290 q^{49} +11.2646 q^{50} -2.83424 q^{51} -2.83424 q^{52} -7.33697 q^{53} +5.76183 q^{54} +4.03293 q^{55} +4.19869 q^{56} -10.9605 q^{57} -5.12628 q^{58} -6.72890 q^{59} -11.4303 q^{60} +5.66849 q^{61} -10.4633 q^{62} +21.1317 q^{63} +1.00000 q^{64} +11.4303 q^{65} -2.83424 q^{66} +4.21069 q^{67} -1.00000 q^{68} +14.5291 q^{69} -16.9330 q^{70} -2.93959 q^{71} +5.03293 q^{72} +5.23163 q^{73} +2.59607 q^{74} +31.9265 q^{75} -3.86718 q^{76} -4.19869 q^{77} -8.03293 q^{78} +8.29204 q^{79} -4.03293 q^{80} +1.23163 q^{81} +3.30404 q^{82} -0.132822 q^{83} +11.9001 q^{84} +4.03293 q^{85} +5.30404 q^{86} -14.5291 q^{87} -1.00000 q^{88} -2.92759 q^{89} -20.2975 q^{90} -11.9001 q^{91} +5.12628 q^{92} -29.6554 q^{93} -10.7618 q^{94} +15.5961 q^{95} +2.83424 q^{96} +15.3370 q^{97} +10.6290 q^{98} -5.03293 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - q^{5} + 3 q^{6} + 7 q^{7} + 3 q^{8} + 4 q^{9} - q^{10} - 3 q^{11} + 3 q^{12} - 3 q^{13} + 7 q^{14} - 12 q^{15} + 3 q^{16} - 3 q^{17} + 4 q^{18} + 5 q^{19} - q^{20}+ \cdots - 4 q^{99}+O(q^{100})$ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 - q^5 + 3 * q^6 + 7 * q^7 + 3 * q^8 + 4 * q^9 - q^10 - 3 * q^11 + 3 * q^12 - 3 * q^13 + 7 * q^14 - 12 * q^15 + 3 * q^16 - 3 * q^17 + 4 * q^18 + 5 * q^19 - q^20 + 8 * q^21 - 3 * q^22 + 4 * q^23 + 3 * q^24 + 6 * q^25 - 3 * q^26 + 6 * q^27 + 7 * q^28 - 4 * q^29 - 12 * q^30 + 2 * q^31 + 3 * q^32 - 3 * q^33 - 3 * q^34 - 12 * q^35 + 4 * q^36 - 9 * q^37 + 5 * q^38 - 13 * q^39 - q^40 - q^41 + 8 * q^42 + 5 * q^43 - 3 * q^44 - 22 * q^45 + 4 * q^46 - 21 * q^47 + 3 * q^48 + 4 * q^49 + 6 * q^50 - 3 * q^51 - 3 * q^52 + 6 * q^54 + q^55 + 7 * q^56 - 16 * q^57 - 4 * q^58 - 20 * q^59 - 12 * q^60 + 6 * q^61 + 2 * q^62 + 19 * q^63 + 3 * q^64 + 12 * q^65 - 3 * q^66 + 2 * q^67 - 3 * q^68 - 12 * q^69 - 12 * q^70 + 2 * q^71 + 4 * q^72 - q^73 - 9 * q^74 + 29 * q^75 + 5 * q^76 - 7 * q^77 - 13 * q^78 + 19 * q^79 - q^80 - 13 * q^81 - q^82 - 17 * q^83 + 8 * q^84 + q^85 + 5 * q^86 + 12 * q^87 - 3 * q^88 - 3 * q^89 - 22 * q^90 - 8 * q^91 + 4 * q^92 - 22 * q^93 - 21 * q^94 + 30 * q^95 + 3 * q^96 + 24 * q^97 + 4 * 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
$3$	2.83424	1.63635	0.818176	−	0.574969i	$-0.194986\pi$
$3$	2.83424	1.63635	0.818176	+	0.574969i	$0.194986\pi$
$4$	1.00000	0.500000
$5$	−4.03293	−1.80358	−0.901792	−	0.432171i	$-0.857748\pi$
$5$	−4.03293	−1.80358	−0.901792	+	0.432171i	$0.857748\pi$
$6$	2.83424	1.15707
$7$	4.19869	1.58696	0.793478	−	0.608599i	$-0.208268\pi$
$7$	4.19869	1.58696	0.793478	+	0.608599i	$0.208268\pi$
$8$	1.00000	0.353553
$9$	5.03293	1.67764
$10$	−4.03293	−1.27533
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	2.83424	0.818176
$13$	−2.83424	−0.786078	−0.393039	−	0.919522i	$-0.628576\pi$
$13$	−2.83424	−0.786078	−0.393039	+	0.919522i	$0.628576\pi$
$14$	4.19869	1.12215
$15$	−11.4303	−2.95130
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	5.03293	1.18627
$19$	−3.86718	−0.887191	−0.443596	−	0.896227i	$-0.646297\pi$
$19$	−3.86718	−0.887191	−0.443596	+	0.896227i	$0.646297\pi$
$20$	−4.03293	−0.901792
$21$	11.9001	2.59682
$22$	−1.00000	−0.213201
$23$	5.12628	1.06890	0.534451	−	0.845199i	$-0.320518\pi$
$23$	5.12628	1.06890	0.534451	+	0.845199i	$0.320518\pi$
$24$	2.83424	0.578537
$25$	11.2646	2.25291
$26$	−2.83424	−0.555841
$27$	5.76183	1.10886
$28$	4.19869	0.793478
$29$	−5.12628	−0.951926	−0.475963	−	0.879465i	$-0.657900\pi$
$29$	−5.12628	−0.951926	−0.475963	+	0.879465i	$0.657900\pi$
$30$	−11.4303	−2.08688
$31$	−10.4633	−1.87926	−0.939628	−	0.342198i	$-0.888828\pi$
$31$	−10.4633	−1.87926	−0.939628	+	0.342198i	$0.888828\pi$
$32$	1.00000	0.176777
$33$	−2.83424	−0.493378
$34$	−1.00000	−0.171499
$35$	−16.9330	−2.86221
$36$	5.03293	0.838822
$37$	2.59607	0.426792	0.213396	−	0.976966i	$-0.431548\pi$
$37$	2.59607	0.426792	0.213396	+	0.976966i	$0.431548\pi$
$38$	−3.86718	−0.627339
$39$	−8.03293	−1.28630
$40$	−4.03293	−0.637663
$41$	3.30404	0.516004	0.258002	−	0.966144i	$-0.416936\pi$
$41$	3.30404	0.516004	0.258002	+	0.966144i	$0.416936\pi$
$42$	11.9001	1.83623
$43$	5.30404	0.808858	0.404429	−	0.914569i	$-0.367470\pi$
$43$	5.30404	0.808858	0.404429	+	0.914569i	$0.367470\pi$
$44$	−1.00000	−0.150756
$45$	−20.2975	−3.02577
$46$	5.12628	0.755829
$47$	−10.7618	−1.56977	−0.784887	−	0.619639i	$-0.787279\pi$
$47$	−10.7618	−1.56977	−0.784887	+	0.619639i	$0.787279\pi$
$48$	2.83424	0.409088
$49$	10.6290	1.51843
$50$	11.2646	1.59305
$51$	−2.83424	−0.396873
$52$	−2.83424	−0.393039
$53$	−7.33697	−1.00781	−0.503905	−	0.863759i	$-0.668104\pi$
$53$	−7.33697	−1.00781	−0.503905	+	0.863759i	$0.668104\pi$
$54$	5.76183	0.784086
$55$	4.03293	0.543801
$56$	4.19869	0.561074
$57$	−10.9605	−1.45176
$58$	−5.12628	−0.673113
$59$	−6.72890	−0.876028	−0.438014	−	0.898968i	$-0.644318\pi$
$59$	−6.72890	−0.876028	−0.438014	+	0.898968i	$0.644318\pi$
$60$	−11.4303	−1.47565
$61$	5.66849	0.725775	0.362888	−	0.931833i	$-0.381791\pi$
$61$	5.66849	0.725775	0.362888	+	0.931833i	$0.381791\pi$
$62$	−10.4633	−1.32883
$63$	21.1317	2.66235
$64$	1.00000	0.125000
$65$	11.4303	1.41776
$66$	−2.83424	−0.348871
$67$	4.21069	0.514418	0.257209	−	0.966356i	$-0.417197\pi$
$67$	4.21069	0.514418	0.257209	+	0.966356i	$0.417197\pi$
$68$	−1.00000	−0.121268
$69$	14.5291	1.74910
$70$	−16.9330	−2.02389
$71$	−2.93959	−0.348865	−0.174433	−	0.984669i	$-0.555809\pi$
$71$	−2.93959	−0.348865	−0.174433	+	0.984669i	$0.555809\pi$
$72$	5.03293	0.593137
$73$	5.23163	0.612315	0.306158	−	0.951981i	$-0.400957\pi$
$73$	5.23163	0.612315	0.306158	+	0.951981i	$0.400957\pi$
$74$	2.59607	0.301787
$75$	31.9265	3.68655
$76$	−3.86718	−0.443596
$77$	−4.19869	−0.478485
$78$	−8.03293	−0.909551
$79$	8.29204	0.932927	0.466463	−	0.884540i	$-0.345528\pi$
$79$	8.29204	0.932927	0.466463	+	0.884540i	$0.345528\pi$
$80$	−4.03293	−0.450896
$81$	1.23163	0.136847
$82$	3.30404	0.364870
$83$	−0.132822	−0.0145792	−0.00728958	−	0.999973i	$-0.502320\pi$
$83$	−0.132822	−0.0145792	−0.00728958	+	0.999973i	$0.502320\pi$
$84$	11.9001	1.29841
$85$	4.03293	0.437433
$86$	5.30404	0.571949
$87$	−14.5291	−1.55769
$88$	−1.00000	−0.106600
$89$	−2.92759	−0.310324	−0.155162	−	0.987889i	$-0.549590\pi$
$89$	−2.92759	−0.310324	−0.155162	+	0.987889i	$0.549590\pi$
$90$	−20.2975	−2.13954
$91$	−11.9001	−1.24747
$92$	5.12628	0.534451
$93$	−29.6554	−3.07512
$94$	−10.7618	−1.11000
$95$	15.5961	1.60012
$96$	2.83424	0.289269
$97$	15.3370	1.55723	0.778617	−	0.627500i	$-0.215922\pi$
$97$	15.3370	1.55723	0.778617	+	0.627500i	$0.215922\pi$
$98$	10.6290	1.07369
$99$	−5.03293	−0.505829
$100$	11.2646	1.12646
$101$	−0.509273	−0.0506746	−0.0253373	−	0.999679i	$-0.508066\pi$
$101$	−0.509273	−0.0506746	−0.0253373	+	0.999679i	$0.508066\pi$
$102$	−2.83424	−0.280632
$103$	5.23163	0.515487	0.257744	−	0.966213i	$-0.417021\pi$
$103$	5.23163	0.515487	0.257744	+	0.966213i	$0.417021\pi$
$104$	−2.83424	−0.277920
$105$	−47.9924	−4.68358
$106$	−7.33697	−0.712630
$107$	13.3370	1.28933	0.644667	−	0.764464i	$-0.276996\pi$
$107$	13.3370	1.28933	0.644667	+	0.764464i	$0.276996\pi$
$108$	5.76183	0.554432
$109$	−14.1317	−1.35357	−0.676787	−	0.736178i	$-0.736628\pi$
$109$	−14.1317	−1.35357	−0.676787	+	0.736178i	$0.736628\pi$
$110$	4.03293	0.384525
$111$	7.35790	0.698381
$112$	4.19869	0.396739
$113$	0.728896	0.0685688	0.0342844	−	0.999412i	$-0.489085\pi$
$113$	0.728896	0.0685688	0.0342844	+	0.999412i	$0.489085\pi$
$114$	−10.9605	−1.02655
$115$	−20.6739	−1.92786
$116$	−5.12628	−0.475963
$117$	−14.2646	−1.31876
$118$	−6.72890	−0.619445
$119$	−4.19869	−0.384893
$120$	−11.4303	−1.04344
$121$	1.00000	0.0909091
$122$	5.66849	0.513201
$123$	9.36445	0.844364
$124$	−10.4633	−0.939628
$125$	−25.2646	−2.25973
$126$	21.1317	1.88256
$127$	7.12628	0.632355	0.316177	−	0.948700i	$-0.397600\pi$
$127$	7.12628	0.632355	0.316177	+	0.948700i	$0.397600\pi$
$128$	1.00000	0.0883883
$129$	15.0329	1.32358
$130$	11.4303	1.00251
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−2.83424	−0.246689
$133$	−16.2371	−1.40793
$134$	4.21069	0.363749
$135$	−23.2371	−1.99993
$136$	−1.00000	−0.0857493
$137$	17.2975	1.47782	0.738912	−	0.673802i	$-0.235340\pi$
$137$	17.2975	1.47782	0.738912	+	0.673802i	$0.235340\pi$
$138$	14.5291	1.23680
$139$	−3.12628	−0.265167	−0.132584	−	0.991172i	$-0.542327\pi$
$139$	−3.12628	−0.265167	−0.132584	+	0.991172i	$0.542327\pi$
$140$	−16.9330	−1.43110
$141$	−30.5016	−2.56870
$142$	−2.93959	−0.246685
$143$	2.83424	0.237011
$144$	5.03293	0.419411
$145$	20.6739	1.71688
$146$	5.23163	0.432972
$147$	30.1252	2.48468
$148$	2.59607	0.213396
$149$	−2.03948	−0.167081	−0.0835403	−	0.996504i	$-0.526623\pi$
$149$	−2.03948	−0.167081	−0.0835403	+	0.996504i	$0.526623\pi$
$150$	31.9265	2.60679
$151$	5.20524	0.423596	0.211798	−	0.977313i	$-0.432068\pi$
$151$	5.20524	0.423596	0.211798	+	0.977313i	$0.432068\pi$
$152$	−3.86718	−0.313670
$153$	−5.03293	−0.406889
$154$	−4.19869	−0.338340
$155$	42.1976	3.38939
$156$	−8.03293	−0.643149
$157$	5.66849	0.452394	0.226197	−	0.974082i	$-0.427371\pi$
$157$	5.66849	0.452394	0.226197	+	0.974082i	$0.427371\pi$
$158$	8.29204	0.659679
$159$	−20.7948	−1.64913
$160$	−4.03293	−0.318831
$161$	21.5237	1.69630
$162$	1.23163	0.0967657
$163$	−13.6225	−1.06699	−0.533497	−	0.845802i	$-0.679122\pi$
$163$	−13.6225	−1.06699	−0.533497	+	0.845802i	$0.679122\pi$
$164$	3.30404	0.258002
$165$	11.4303	0.889849
$166$	−0.132822	−0.0103090
$167$	21.9121	1.69561	0.847805	−	0.530308i	$-0.177924\pi$
$167$	21.9121	1.69561	0.847805	+	0.530308i	$0.177924\pi$
$168$	11.9001	0.918114
$169$	−4.96707	−0.382082
$170$	4.03293	0.309312
$171$	−19.4633	−1.48839
$172$	5.30404	0.404429
$173$	4.13174	0.314130	0.157065	−	0.987588i	$-0.449797\pi$
$173$	4.13174	0.314130	0.157065	+	0.987588i	$0.449797\pi$
$174$	−14.5291	−1.10145
$175$	47.2964	3.57527
$176$	−1.00000	−0.0753778
$177$	−19.0713	−1.43349
$178$	−2.92759	−0.219432
$179$	−17.8552	−1.33456	−0.667279	−	0.744808i	$-0.732541\pi$
$179$	−17.8552	−1.33456	−0.667279	+	0.744808i	$0.732541\pi$
$180$	−20.2975	−1.51289
$181$	−2.68942	−0.199903	−0.0999514	−	0.994992i	$-0.531869\pi$
$181$	−2.68942	−0.199903	−0.0999514	+	0.994992i	$0.531869\pi$
$182$	−11.9001	−0.882095
$183$	16.0659	1.18762
$184$	5.12628	0.377914
$185$	−10.4698	−0.769755
$186$	−29.6554	−2.17444
$187$	1.00000	0.0731272
$188$	−10.7618	−0.784887
$189$	24.1921	1.75972
$190$	15.5961	1.13146
$191$	−25.3634	−1.83523	−0.917614	−	0.397472i	$-0.869888\pi$
$191$	−25.3634	−1.83523	−0.917614	+	0.397472i	$0.869888\pi$
$192$	2.83424	0.204544
$193$	6.68942	0.481515	0.240757	−	0.970585i	$-0.422604\pi$
$193$	6.68942	0.481515	0.240757	+	0.970585i	$0.422604\pi$
$194$	15.3370	1.10113
$195$	32.3963	2.31995
$196$	10.6290	0.759215
$197$	23.7344	1.69100	0.845501	−	0.533973i	$-0.179302\pi$
$197$	23.7344	1.69100	0.845501	+	0.533973i	$0.179302\pi$
$198$	−5.03293	−0.357675
$199$	3.60262	0.255383	0.127691	−	0.991814i	$-0.459243\pi$
$199$	3.60262	0.255383	0.127691	+	0.991814i	$0.459243\pi$
$200$	11.2646	0.796525
$201$	11.9341	0.841769
$202$	−0.509273	−0.0358323
$203$	−21.5237	−1.51066
$204$	−2.83424	−0.198437
$205$	−13.3250	−0.930656
$206$	5.23163	0.364505
$207$	25.8002	1.79324
$208$	−2.83424	−0.196519
$209$	3.86718	0.267498
$210$	−47.9924	−3.31179
$211$	11.2052	0.771400	0.385700	−	0.922624i	$-0.373960\pi$
$211$	11.2052	0.771400	0.385700	+	0.922624i	$0.373960\pi$
$212$	−7.33697	−0.503905
$213$	−8.33151	−0.570866
$214$	13.3370	0.911697
$215$	−21.3908	−1.45884
$216$	5.76183	0.392043
$217$	−43.9320	−2.98230
$218$	−14.1317	−0.957122
$219$	14.8277	1.00196
$220$	4.03293	0.271900
$221$	2.83424	0.190652
$222$	7.35790	0.493830
$223$	−11.2042	−0.750285	−0.375143	−	0.926967i	$-0.622406\pi$
$223$	−11.2042	−0.750285	−0.375143	+	0.926967i	$0.622406\pi$
$224$	4.19869	0.280537
$225$	56.6938	3.77959
$226$	0.728896	0.0484854
$227$	26.5291	1.76080	0.880400	−	0.474232i	$-0.157274\pi$
$227$	26.5291	1.76080	0.880400	+	0.474232i	$0.157274\pi$
$228$	−10.9605	−0.725878
$229$	11.0604	0.730892	0.365446	−	0.930832i	$-0.380916\pi$
$229$	11.0604	0.730892	0.365446	+	0.930832i	$0.380916\pi$
$230$	−20.6739	−1.36320
$231$	−11.9001	−0.782970
$232$	−5.12628	−0.336557
$233$	−14.2371	−0.932702	−0.466351	−	0.884600i	$-0.654432\pi$
$233$	−14.2371	−0.932702	−0.466351	+	0.884600i	$0.654432\pi$
$234$	−14.2646	−0.932503
$235$	43.4018	2.83122
$236$	−6.72890	−0.438014
$237$	23.5016	1.52660
$238$	−4.19869	−0.272161
$239$	1.45779	0.0942967	0.0471484	−	0.998888i	$-0.484987\pi$
$239$	1.45779	0.0942967	0.0471484	+	0.998888i	$0.484987\pi$
$240$	−11.4303	−0.737824
$241$	−22.9265	−1.47683	−0.738413	−	0.674349i	$-0.764424\pi$
$241$	−22.9265	−1.47683	−0.738413	+	0.674349i	$0.764424\pi$
$242$	1.00000	0.0642824
$243$	−13.7948	−0.884935
$244$	5.66849	0.362888
$245$	−42.8661	−2.73861
$246$	9.36445	0.597055
$247$	10.9605	0.697401
$248$	−10.4633	−0.664417
$249$	−0.376451	−0.0238566
$250$	−25.2646	−1.59787
$251$	3.66849	0.231553	0.115776	−	0.993275i	$-0.463064\pi$
$251$	3.66849	0.231553	0.115776	+	0.993275i	$0.463064\pi$
$252$	21.1317	1.33117
$253$	−5.12628	−0.322286
$254$	7.12628	0.447142
$255$	11.4303	0.715794
$256$	1.00000	0.0625000
$257$	−7.32497	−0.456919	−0.228460	−	0.973553i	$-0.573369\pi$
$257$	−7.32497	−0.456919	−0.228460	+	0.973553i	$0.573369\pi$
$258$	15.0329	0.935910
$259$	10.9001	0.677300
$260$	11.4303	0.708878
$261$	−25.8002	−1.59699
$262$	12.0000	0.741362
$263$	−11.7344	−0.723571	−0.361786	−	0.932261i	$-0.617833\pi$
$263$	−11.7344	−0.723571	−0.361786	+	0.932261i	$0.617833\pi$
$264$	−2.83424	−0.174436
$265$	29.5895	1.81767
$266$	−16.2371	−0.995560
$267$	−8.29749	−0.507798
$268$	4.21069	0.257209
$269$	−3.68049	−0.224403	−0.112202	−	0.993685i	$-0.535790\pi$
$269$	−3.68049	−0.224403	−0.112202	+	0.993685i	$0.535790\pi$
$270$	−23.2371	−1.41416
$271$	−32.0109	−1.94452	−0.972262	−	0.233893i	$-0.924854\pi$
$271$	−32.0109	−1.94452	−0.972262	+	0.233893i	$0.924854\pi$
$272$	−1.00000	−0.0606339
$273$	−33.7278	−2.04130
$274$	17.2975	1.04498
$275$	−11.2646	−0.679279
$276$	14.5291	0.874550
$277$	17.4028	1.04564	0.522818	−	0.852445i	$-0.324881\pi$
$277$	17.4028	1.04564	0.522818	+	0.852445i	$0.324881\pi$
$278$	−3.12628	−0.187502
$279$	−52.6609	−3.15272
$280$	−16.9330	−1.01194
$281$	17.5895	1.04930	0.524652	−	0.851317i	$-0.324196\pi$
$281$	17.5895	1.04930	0.524652	+	0.851317i	$0.324196\pi$
$282$	−30.5016	−1.81635
$283$	−6.67395	−0.396725	−0.198362	−	0.980129i	$-0.563562\pi$
$283$	−6.67395	−0.396725	−0.198362	+	0.980129i	$0.563562\pi$
$284$	−2.93959	−0.174433
$285$	44.2031	2.61836
$286$	2.83424	0.167592
$287$	13.8726	0.818876
$288$	5.03293	0.296569
$289$	1.00000	0.0588235
$290$	20.6739	1.21402
$291$	43.4687	2.54818
$292$	5.23163	0.306158
$293$	12.7069	0.742344	0.371172	−	0.928564i	$-0.378956\pi$
$293$	12.7069	0.742344	0.371172	+	0.928564i	$0.378956\pi$
$294$	30.1252	1.75694
$295$	27.1372	1.57999
$296$	2.59607	0.150894
$297$	−5.76183	−0.334335
$298$	−2.03948	−0.118144
$299$	−14.5291	−0.840241
$300$	31.9265	1.84328
$301$	22.2700	1.28362
$302$	5.20524	0.299528
$303$	−1.44340	−0.0829214
$304$	−3.86718	−0.221798
$305$	−22.8606	−1.30900
$306$	−5.03293	−0.287714
$307$	−14.1053	−0.805035	−0.402517	−	0.915412i	$-0.631865\pi$
$307$	−14.1053	−0.805035	−0.402517	+	0.915412i	$0.631865\pi$
$308$	−4.19869	−0.239243
$309$	14.8277	0.843518
$310$	42.1976	2.39666
$311$	20.3843	1.15589	0.577944	−	0.816077i	$-0.303855\pi$
$311$	20.3843	1.15589	0.577944	+	0.816077i	$0.303855\pi$
$312$	−8.03293	−0.454775
$313$	5.52366	0.312216	0.156108	−	0.987740i	$-0.450105\pi$
$313$	5.52366	0.312216	0.156108	+	0.987740i	$0.450105\pi$
$314$	5.66849	0.319891
$315$	−85.2229	−4.80177
$316$	8.29204	0.466463
$317$	5.46871	0.307153	0.153577	−	0.988137i	$-0.450921\pi$
$317$	5.46871	0.307153	0.153577	+	0.988137i	$0.450921\pi$
$318$	−20.7948	−1.16611
$319$	5.12628	0.287017
$320$	−4.03293	−0.225448
$321$	37.8002	2.10980
$322$	21.5237	1.19947
$323$	3.86718	0.215176
$324$	1.23163	0.0684236
$325$	−31.9265	−1.77096
$326$	−13.6225	−0.754479
$327$	−40.0528	−2.21492
$328$	3.30404	0.182435
$329$	−45.1856	−2.49116
$330$	11.4303	0.629218
$331$	−9.58953	−0.527088	−0.263544	−	0.964647i	$-0.584891\pi$
$331$	−9.58953	−0.527088	−0.263544	+	0.964647i	$0.584891\pi$
$332$	−0.132822	−0.00728958
$333$	13.0659	0.716005
$334$	21.9121	1.19898
$335$	−16.9815	−0.927796
$336$	11.9001	0.649204
$337$	9.11735	0.496654	0.248327	−	0.968676i	$-0.420119\pi$
$337$	9.11735	0.496654	0.248327	+	0.968676i	$0.420119\pi$
$338$	−4.96707	−0.270173
$339$	2.06587	0.112203
$340$	4.03293	0.218717
$341$	10.4633	0.566617
$342$	−19.4633	−1.05245
$343$	15.2371	0.822725
$344$	5.30404	0.285975
$345$	−58.5950	−3.15465
$346$	4.13174	0.222124
$347$	−20.6739	−1.10984	−0.554918	−	0.831905i	$-0.687250\pi$
$347$	−20.6739	−1.10984	−0.554918	+	0.831905i	$0.687250\pi$
$348$	−14.5291	−0.778843
$349$	3.45779	0.185091	0.0925457	−	0.995708i	$-0.470500\pi$
$349$	3.45779	0.185091	0.0925457	+	0.995708i	$0.470500\pi$
$350$	47.2964	2.52810
$351$	−16.3304	−0.871654
$352$	−1.00000	−0.0533002
$353$	−9.30404	−0.495204	−0.247602	−	0.968862i	$-0.579643\pi$
$353$	−9.30404	−0.495204	−0.247602	+	0.968862i	$0.579643\pi$
$354$	−19.0713	−1.01363
$355$	11.8552	0.629207
$356$	−2.92759	−0.155162
$357$	−11.9001	−0.629821
$358$	−17.8552	−0.943675
$359$	11.1263	0.587223	0.293611	−	0.955925i	$-0.405143\pi$
$359$	11.1263	0.587223	0.293611	+	0.955925i	$0.405143\pi$
$360$	−20.2975	−1.06977
$361$	−4.04494	−0.212891
$362$	−2.68942	−0.141353
$363$	2.83424	0.148759
$364$	−11.9001	−0.623735
$365$	−21.0988	−1.10436
$366$	16.0659	0.839776
$367$	−23.8002	−1.24236	−0.621181	−	0.783667i	$-0.713347\pi$
$367$	−23.8002	−1.24236	−0.621181	+	0.783667i	$0.713347\pi$
$368$	5.12628	0.267226
$369$	16.6290	0.865672
$370$	−10.4698	−0.544299
$371$	−30.8057	−1.59935
$372$	−29.6554	−1.53756
$373$	−11.6146	−0.601382	−0.300691	−	0.953722i	$-0.597217\pi$
$373$	−11.6146	−0.601382	−0.300691	+	0.953722i	$0.597217\pi$
$374$	1.00000	0.0517088
$375$	−71.6059	−3.69771
$376$	−10.7618	−0.554999
$377$	14.5291	0.748288
$378$	24.1921	1.24431
$379$	2.70688	0.139043	0.0695215	−	0.997580i	$-0.477853\pi$
$379$	2.70688	0.139043	0.0695215	+	0.997580i	$0.477853\pi$
$380$	15.5961	0.800062
$381$	20.1976	1.03475
$382$	−25.3634	−1.29770
$383$	27.6004	1.41032	0.705159	−	0.709050i	$-0.250876\pi$
$383$	27.6004	1.41032	0.705159	+	0.709050i	$0.250876\pi$
$384$	2.83424	0.144634
$385$	16.9330	0.862988
$386$	6.68942	0.340482
$387$	26.6949	1.35698
$388$	15.3370	0.778617
$389$	−39.2820	−1.99168	−0.995839	−	0.0911323i	$-0.970951\pi$
$389$	−39.2820	−1.99168	−0.995839	+	0.0911323i	$0.970951\pi$
$390$	32.3963	1.64045
$391$	−5.12628	−0.259247
$392$	10.6290	0.536846
$393$	34.0109	1.71562
$394$	23.7344	1.19572
$395$	−33.4412	−1.68261
$396$	−5.03293	−0.252914
$397$	23.1252	1.16062	0.580310	−	0.814396i	$-0.302931\pi$
$397$	23.1252	1.16062	0.580310	+	0.814396i	$0.302931\pi$
$398$	3.60262	0.180583
$399$	−46.0198	−2.30387
$400$	11.2646	0.563228
$401$	4.68703	0.234059	0.117030	−	0.993128i	$-0.462663\pi$
$401$	4.68703	0.234059	0.117030	+	0.993128i	$0.462663\pi$
$402$	11.9341	0.595220
$403$	29.6554	1.47724
$404$	−0.509273	−0.0253373
$405$	−4.96707	−0.246815
$406$	−21.5237	−1.06820
$407$	−2.59607	−0.128683
$408$	−2.83424	−0.140316
$409$	−24.2635	−1.19975	−0.599876	−	0.800093i	$-0.704783\pi$
$409$	−24.2635	−1.19975	−0.599876	+	0.800093i	$0.704783\pi$
$410$	−13.3250	−0.658073
$411$	49.0253	2.41824
$412$	5.23163	0.257744
$413$	−28.2526	−1.39022
$414$	25.8002	1.26801
$415$	0.535664	0.0262947
$416$	−2.83424	−0.138960
$417$	−8.86063	−0.433907
$418$	3.86718	0.189150
$419$	25.2569	1.23388	0.616941	−	0.787009i	$-0.288372\pi$
$419$	25.2569	1.23388	0.616941	+	0.787009i	$0.288372\pi$
$420$	−47.9924	−2.34179
$421$	−28.9265	−1.40979	−0.704896	−	0.709311i	$-0.749006\pi$
$421$	−28.9265	−1.40979	−0.704896	+	0.709311i	$0.749006\pi$
$422$	11.2052	0.545462
$423$	−54.1636	−2.63352
$424$	−7.33697	−0.356315
$425$	−11.2646	−0.546411
$426$	−8.33151	−0.403663
$427$	23.8002	1.15177
$428$	13.3370	0.644667
$429$	8.03293	0.387834
$430$	−21.3908	−1.03156
$431$	15.5357	0.748327	0.374163	−	0.927363i	$-0.377930\pi$
$431$	15.5357	0.748327	0.374163	+	0.927363i	$0.377930\pi$
$432$	5.76183	0.277216
$433$	2.82770	0.135891	0.0679453	−	0.997689i	$-0.478356\pi$
$433$	2.82770	0.135891	0.0679453	+	0.997689i	$0.478356\pi$
$434$	−43.9320	−2.10880
$435$	58.5950	2.80941
$436$	−14.1317	−0.676787
$437$	−19.8242	−0.948322
$438$	14.8277	0.708495
$439$	−20.7618	−0.990908	−0.495454	−	0.868634i	$-0.664998\pi$
$439$	−20.7618	−0.990908	−0.495454	+	0.868634i	$0.664998\pi$
$440$	4.03293	0.192263
$441$	53.4951	2.54739
$442$	2.83424	0.134811
$443$	−3.32605	−0.158026	−0.0790128	−	0.996874i	$-0.525177\pi$
$443$	−3.32605	−0.158026	−0.0790128	+	0.996874i	$0.525177\pi$
$444$	7.35790	0.349191
$445$	11.8068	0.559694
$446$	−11.2042	−0.530532
$447$	−5.78038	−0.273402
$448$	4.19869	0.198370
$449$	−37.3898	−1.76453	−0.882266	−	0.470752i	$-0.843983\pi$
$449$	−37.3898	−1.76453	−0.882266	+	0.470752i	$0.843983\pi$
$450$	56.6938	2.67257
$451$	−3.30404	−0.155581
$452$	0.728896	0.0342844
$453$	14.7529	0.693152
$454$	26.5291	1.24507
$455$	47.9924	2.24992
$456$	−10.9605	−0.513273
$457$	−27.2031	−1.27251	−0.636253	−	0.771481i	$-0.719516\pi$
$457$	−27.2031	−1.27251	−0.636253	+	0.771481i	$0.719516\pi$
$458$	11.0604	0.516819
$459$	−5.76183	−0.268939
$460$	−20.6739	−0.963928
$461$	9.08680	0.423214	0.211607	−	0.977355i	$-0.432130\pi$
$461$	9.08680	0.423214	0.211607	+	0.977355i	$0.432130\pi$
$462$	−11.9001	−0.553643
$463$	−17.8947	−0.831635	−0.415818	−	0.909448i	$-0.636505\pi$
$463$	−17.8947	−0.831635	−0.415818	+	0.909448i	$0.636505\pi$
$464$	−5.12628	−0.237982
$465$	119.598	5.54624
$466$	−14.2371	−0.659520
$467$	24.1736	1.11862	0.559310	−	0.828958i	$-0.311066\pi$
$467$	24.1736	1.11862	0.559310	+	0.828958i	$0.311066\pi$
$468$	−14.2646	−0.659380
$469$	17.6794	0.816359
$470$	43.4018	2.00197
$471$	16.0659	0.740276
$472$	−6.72890	−0.309723
$473$	−5.30404	−0.243880
$474$	23.5016	1.07947
$475$	−43.5621	−1.99876
$476$	−4.19869	−0.192447
$477$	−36.9265	−1.69075
$478$	1.45779	0.0666779
$479$	−2.03948	−0.0931861	−0.0465931	−	0.998914i	$-0.514836\pi$
$479$	−2.03948	−0.0931861	−0.0465931	+	0.998914i	$0.514836\pi$
$480$	−11.4303	−0.521720
$481$	−7.35790	−0.335492
$482$	−22.9265	−1.04427
$483$	61.0033	2.77575
$484$	1.00000	0.0454545
$485$	−61.8530	−2.80860
$486$	−13.7948	−0.625743
$487$	−27.7871	−1.25916	−0.629578	−	0.776938i	$-0.716772\pi$
$487$	−27.7871	−1.25916	−0.629578	+	0.776938i	$0.716772\pi$
$488$	5.66849	0.256600
$489$	−38.6094	−1.74598
$490$	−42.8661	−1.93649
$491$	−3.91211	−0.176551	−0.0882756	−	0.996096i	$-0.528136\pi$
$491$	−3.91211	−0.176551	−0.0882756	+	0.996096i	$0.528136\pi$
$492$	9.36445	0.422182
$493$	5.12628	0.230876
$494$	10.9605	0.493137
$495$	20.2975	0.912305
$496$	−10.4633	−0.469814
$497$	−12.3424	−0.553634
$498$	−0.376451	−0.0168692
$499$	−36.5555	−1.63645	−0.818225	−	0.574899i	$-0.805042\pi$
$499$	−36.5555	−1.63645	−0.818225	+	0.574899i	$0.805042\pi$
$500$	−25.2646	−1.12987
$501$	62.1043	2.77461
$502$	3.66849	0.163733
$503$	1.12736	0.0502666	0.0251333	−	0.999684i	$-0.491999\pi$
$503$	1.12736	0.0502666	0.0251333	+	0.999684i	$0.491999\pi$
$504$	21.1317	0.941282
$505$	2.05387	0.0913958
$506$	−5.12628	−0.227891
$507$	−14.0779	−0.625220
$508$	7.12628	0.316177
$509$	9.86609	0.437307	0.218653	−	0.975803i	$-0.429834\pi$
$509$	9.86609	0.437307	0.218653	+	0.975803i	$0.429834\pi$
$510$	11.4303	0.506143
$511$	21.9660	0.971718
$512$	1.00000	0.0441942
$513$	−22.2820	−0.983775
$514$	−7.32497	−0.323091
$515$	−21.0988	−0.929724
$516$	15.0329	0.661788
$517$	10.7618	0.473305
$518$	10.9001	0.478923
$519$	11.7103	0.514027
$520$	11.4303	0.501253
$521$	1.01855	0.0446233	0.0223117	−	0.999751i	$-0.492897\pi$
$521$	1.01855	0.0446233	0.0223117	+	0.999751i	$0.492897\pi$
$522$	−25.8002	−1.12925
$523$	−31.6225	−1.38275	−0.691377	−	0.722494i	$-0.742996\pi$
$523$	−31.6225	−1.38275	−0.691377	+	0.722494i	$0.742996\pi$
$524$	12.0000	0.524222
$525$	134.050	5.85040
$526$	−11.7344	−0.511642
$527$	10.4633	0.455786
$528$	−2.83424	−0.123345
$529$	3.27873	0.142554
$530$	29.5895	1.28529
$531$	−33.8661	−1.46966
$532$	−16.2371	−0.703967
$533$	−9.36445	−0.405619
$534$	−8.29749	−0.359068
$535$	−53.7871	−2.32542
$536$	4.21069	0.181874
$537$	−50.6059	−2.18381
$538$	−3.68049	−0.158677
$539$	−10.6290	−0.457824
$540$	−23.2371	−0.999965
$541$	−11.3610	−0.488447	−0.244223	−	0.969719i	$-0.578533\pi$
$541$	−11.3610	−0.488447	−0.244223	+	0.969719i	$0.578533\pi$
$542$	−32.0109	−1.37499
$543$	−7.62246	−0.327111
$544$	−1.00000	−0.0428746
$545$	56.9924	2.44128
$546$	−33.7278	−1.44342
$547$	−14.1317	−0.604229	−0.302115	−	0.953272i	$-0.597693\pi$
$547$	−14.1317	−0.604229	−0.302115	+	0.953272i	$0.597693\pi$
$548$	17.2975	0.738912
$549$	28.5291	1.21759
$550$	−11.2646	−0.480322
$551$	19.8242	0.844541
$552$	14.5291	0.618400
$553$	34.8157	1.48051
$554$	17.4028	0.739376
$555$	−29.6739	−1.25959
$556$	−3.12628	−0.132584
$557$	−10.4722	−0.443721	−0.221860	−	0.975078i	$-0.571213\pi$
$557$	−10.4722	−0.443721	−0.221860	+	0.975078i	$0.571213\pi$
$558$	−52.6609	−2.22931
$559$	−15.0329	−0.635825
$560$	−16.9330	−0.715552
$561$	2.83424	0.119662
$562$	17.5895	0.741970
$563$	−0.948519	−0.0399753	−0.0199877	−	0.999800i	$-0.506363\pi$
$563$	−0.948519	−0.0399753	−0.0199877	+	0.999800i	$0.506363\pi$
$564$	−30.5016	−1.28435
$565$	−2.93959	−0.123669
$566$	−6.67395	−0.280527
$567$	5.17122	0.217171
$568$	−2.93959	−0.123342
$569$	20.2107	0.847276	0.423638	−	0.905831i	$-0.360753\pi$
$569$	20.2107	0.847276	0.423638	+	0.905831i	$0.360753\pi$
$570$	44.2031	1.85146
$571$	40.5291	1.69609	0.848045	−	0.529924i	$-0.177779\pi$
$571$	40.5291	1.69609	0.848045	+	0.529924i	$0.177779\pi$
$572$	2.83424	0.118506
$573$	−71.8859	−3.00308
$574$	13.8726	0.579033
$575$	57.7453	2.40814
$576$	5.03293	0.209706
$577$	−2.53020	−0.105334	−0.0526669	−	0.998612i	$-0.516772\pi$
$577$	−2.53020	−0.105334	−0.0526669	+	0.998612i	$0.516772\pi$
$578$	1.00000	0.0415945
$579$	18.9594	0.787927
$580$	20.6739	0.858439
$581$	−0.557680	−0.0231365
$582$	43.4687	1.80184
$583$	7.33697	0.303866
$584$	5.23163	0.216486
$585$	57.5280	2.37849
$586$	12.7069	0.524916
$587$	−2.21069	−0.0912451	−0.0456226	−	0.998959i	$-0.514527\pi$
$587$	−2.21069	−0.0912451	−0.0456226	+	0.998959i	$0.514527\pi$
$588$	30.1252	1.24234
$589$	40.4633	1.66726
$590$	27.1372	1.11722
$591$	67.2689	2.76707
$592$	2.59607	0.106698
$593$	−48.0528	−1.97329	−0.986646	−	0.162882i	$-0.947921\pi$
$593$	−48.0528	−1.97329	−0.986646	+	0.162882i	$0.947921\pi$
$594$	−5.76183	−0.236411
$595$	16.9330	0.694187
$596$	−2.03948	−0.0835403
$597$	10.2107	0.417896
$598$	−14.5291	−0.594140
$599$	32.5051	1.32812	0.664061	−	0.747678i	$-0.268831\pi$
$599$	32.5051	1.32812	0.664061	+	0.747678i	$0.268831\pi$
$600$	31.9265	1.30339
$601$	36.6070	1.49323	0.746615	−	0.665256i	$-0.231678\pi$
$601$	36.6070	1.49323	0.746615	+	0.665256i	$0.231678\pi$
$602$	22.2700	0.907658
$603$	21.1921	0.863011
$604$	5.20524	0.211798
$605$	−4.03293	−0.163962
$606$	−1.44340	−0.0586343
$607$	−5.55660	−0.225535	−0.112768	−	0.993621i	$-0.535972\pi$
$607$	−5.55660	−0.225535	−0.112768	+	0.993621i	$0.535972\pi$
$608$	−3.86718	−0.156835
$609$	−61.0033	−2.47198
$610$	−22.8606	−0.925600
$611$	30.5016	1.23396
$612$	−5.03293	−0.203444
$613$	−37.9989	−1.53476	−0.767381	−	0.641192i	$-0.778440\pi$
$613$	−37.9989	−1.53476	−0.767381	+	0.641192i	$0.778440\pi$
$614$	−14.1053	−0.569245
$615$	−37.7662	−1.52288
$616$	−4.19869	−0.169170
$617$	−8.52912	−0.343369	−0.171685	−	0.985152i	$-0.554921\pi$
$617$	−8.52912	−0.343369	−0.171685	+	0.985152i	$0.554921\pi$
$618$	14.8277	0.596458
$619$	1.88157	0.0756265	0.0378132	−	0.999285i	$-0.487961\pi$
$619$	1.88157	0.0756265	0.0378132	+	0.999285i	$0.487961\pi$
$620$	42.1976	1.69470
$621$	29.5367	1.18527
$622$	20.3843	0.817336
$623$	−12.2920	−0.492470
$624$	−8.03293	−0.321575
$625$	45.5675	1.82270
$626$	5.52366	0.220770
$627$	10.9605	0.437721
$628$	5.66849	0.226197
$629$	−2.59607	−0.103512
$630$	−85.2229	−3.39536
$631$	29.8157	1.18694	0.593472	−	0.804855i	$-0.297757\pi$
$631$	29.8157	1.18694	0.593472	+	0.804855i	$0.297757\pi$
$632$	8.29204	0.329839
$633$	31.7584	1.26228
$634$	5.46871	0.217190
$635$	−28.7398	−1.14050
$636$	−20.7948	−0.824566
$637$	−30.1252	−1.19360
$638$	5.12628	0.202951
$639$	−14.7948	−0.585272
$640$	−4.03293	−0.159416
$641$	−21.2580	−0.839641	−0.419821	−	0.907607i	$-0.637907\pi$
$641$	−21.2580	−0.839641	−0.419821	+	0.907607i	$0.637907\pi$
$642$	37.8002	1.49186
$643$	−4.53129	−0.178697	−0.0893483	−	0.996000i	$-0.528478\pi$
$643$	−4.53129	−0.178697	−0.0893483	+	0.996000i	$0.528478\pi$
$644$	21.5237	0.848151
$645$	−60.6268	−2.38718
$646$	3.86718	0.152152
$647$	42.0264	1.65223	0.826114	−	0.563503i	$-0.190547\pi$
$647$	42.0264	1.65223	0.826114	+	0.563503i	$0.190547\pi$
$648$	1.23163	0.0483828
$649$	6.72890	0.264132
$650$	−31.9265	−1.25226
$651$	−124.514	−4.88008
$652$	−13.6225	−0.533497
$653$	15.3668	0.601351	0.300675	−	0.953727i	$-0.402788\pi$
$653$	15.3668	0.601351	0.300675	+	0.953727i	$0.402788\pi$
$654$	−40.0528	−1.56619
$655$	−48.3952	−1.89096
$656$	3.30404	0.129001
$657$	26.3304	1.02725
$658$	−45.1856	−1.76152
$659$	−27.7333	−1.08033	−0.540167	−	0.841558i	$-0.681639\pi$
$659$	−27.7333	−1.08033	−0.540167	+	0.841558i	$0.681639\pi$
$660$	11.4303	0.444924
$661$	24.8846	0.967900	0.483950	−	0.875096i	$-0.339202\pi$
$661$	24.8846	0.967900	0.483950	+	0.875096i	$0.339202\pi$
$662$	−9.58953	−0.372708
$663$	8.03293	0.311973
$664$	−0.132822	−0.00515451
$665$	65.4831	2.53933
$666$	13.0659	0.506292
$667$	−26.2787	−1.01752
$668$	21.9121	0.847805
$669$	−31.7553	−1.22773
$670$	−16.9815	−0.656051
$671$	−5.66849	−0.218829
$672$	11.9001	0.459057
$673$	17.0912	0.658816	0.329408	−	0.944188i	$-0.393151\pi$
$673$	17.0912	0.658816	0.329408	+	0.944188i	$0.393151\pi$
$674$	9.11735	0.351187
$675$	64.9045	2.49817
$676$	−4.96707	−0.191041
$677$	28.0109	1.07655	0.538274	−	0.842770i	$-0.319077\pi$
$677$	28.0109	1.07655	0.538274	+	0.842770i	$0.319077\pi$
$678$	2.06587	0.0793392
$679$	64.3952	2.47126
$680$	4.03293	0.154656
$681$	75.1900	2.88129
$682$	10.4633	0.400659
$683$	11.1723	0.427496	0.213748	−	0.976889i	$-0.431433\pi$
$683$	11.1723	0.427496	0.213748	+	0.976889i	$0.431433\pi$
$684$	−19.4633	−0.744196
$685$	−69.7597	−2.66538
$686$	15.2371	0.581755
$687$	31.3479	1.19600
$688$	5.30404	0.202215
$689$	20.7948	0.792217
$690$	−58.5950	−2.23067
$691$	−38.6070	−1.46868	−0.734340	−	0.678782i	$-0.762508\pi$
$691$	−38.6070	−1.46868	−0.734340	+	0.678782i	$0.762508\pi$
$692$	4.13174	0.157065
$693$	−21.1317	−0.802728
$694$	−20.6739	−0.784772
$695$	12.6081	0.478252
$696$	−14.5291	−0.550725
$697$	−3.30404	−0.125149
$698$	3.45779	0.130879
$699$	−40.3514	−1.52623
$700$	47.2964	1.78764
$701$	43.0833	1.62723	0.813617	−	0.581401i	$-0.197495\pi$
$701$	43.0833	1.62723	0.813617	+	0.581401i	$0.197495\pi$
$702$	−16.3304	−0.616352
$703$	−10.0395	−0.378646
$704$	−1.00000	−0.0376889
$705$	123.011	4.63287
$706$	−9.30404	−0.350162
$707$	−2.13828	−0.0804184
$708$	−19.0713	−0.716744
$709$	17.2460	0.647688	0.323844	−	0.946111i	$-0.395025\pi$
$709$	17.2460	0.647688	0.323844	+	0.946111i	$0.395025\pi$
$710$	11.8552	0.444917
$711$	41.7333	1.56512
$712$	−2.92759	−0.109716
$713$	−53.6375	−2.00874
$714$	−11.9001	−0.445351
$715$	−11.4303	−0.427470
$716$	−17.8552	−0.667279
$717$	4.13174	0.154303
$718$	11.1263	0.415229
$719$	4.51820	0.168501	0.0842503	−	0.996445i	$-0.473150\pi$
$719$	4.51820	0.168501	0.0842503	+	0.996445i	$0.473150\pi$
$720$	−20.2975	−0.756443
$721$	21.9660	0.818056
$722$	−4.04494	−0.150537
$723$	−64.9793	−2.41661
$724$	−2.68942	−0.0999514
$725$	−57.7453	−2.14461
$726$	2.83424	0.105189
$727$	38.9529	1.44468	0.722341	−	0.691537i	$-0.243066\pi$
$727$	38.9529	1.44468	0.722341	+	0.691537i	$0.243066\pi$
$728$	−11.9001	−0.441047
$729$	−42.7926	−1.58491
$730$	−21.0988	−0.780902
$731$	−5.30404	−0.196177
$732$	16.0659	0.593812
$733$	0.882650	0.0326014	0.0163007	−	0.999867i	$-0.494811\pi$
$733$	0.882650	0.0326014	0.0163007	+	0.999867i	$0.494811\pi$
$734$	−23.8002	−0.878482
$735$	−121.493	−4.48133
$736$	5.12628	0.188957
$737$	−4.21069	−0.155103
$738$	16.6290	0.612122
$739$	40.7531	1.49913	0.749564	−	0.661932i	$-0.230263\pi$
$739$	40.7531	1.49913	0.749564	+	0.661932i	$0.230263\pi$
$740$	−10.4698	−0.384877
$741$	31.0648	1.14119
$742$	−30.8057	−1.13091
$743$	−12.0988	−0.443862	−0.221931	−	0.975062i	$-0.571236\pi$
$743$	−12.0988	−0.443862	−0.221931	+	0.975062i	$0.571236\pi$
$744$	−29.6554	−1.08722
$745$	8.22508	0.301344
$746$	−11.6146	−0.425241
$747$	−0.668486	−0.0244586
$748$	1.00000	0.0365636
$749$	55.9978	2.04612
$750$	−71.6059	−2.61468
$751$	−12.4523	−0.454392	−0.227196	−	0.973849i	$-0.572956\pi$
$751$	−12.4523	−0.454392	−0.227196	+	0.973849i	$0.572956\pi$
$752$	−10.7618	−0.392444
$753$	10.3974	0.378902
$754$	14.5291	0.529119
$755$	−20.9924	−0.763991
$756$	24.1921	0.879860
$757$	39.9450	1.45183	0.725914	−	0.687786i	$-0.241417\pi$
$757$	39.9450	1.45183	0.725914	+	0.687786i	$0.241417\pi$
$758$	2.70688	0.0983183
$759$	−14.5291	−0.527374
$760$	15.5961	0.565729
$761$	−34.7638	−1.26019	−0.630094	−	0.776519i	$-0.716984\pi$
$761$	−34.7638	−1.26019	−0.630094	+	0.776519i	$0.716984\pi$
$762$	20.1976	0.731682
$763$	−59.3348	−2.14806
$764$	−25.3634	−0.917614
$765$	20.2975	0.733857
$766$	27.6004	0.997245
$767$	19.0713	0.688626
$768$	2.83424	0.102272
$769$	−29.3130	−1.05705	−0.528526	−	0.848917i	$-0.677255\pi$
$769$	−29.3130	−1.05705	−0.528526	+	0.848917i	$0.677255\pi$
$770$	16.9330	0.610225
$771$	−20.7607	−0.747680
$772$	6.68942	0.240757
$773$	−32.5531	−1.17085	−0.585427	−	0.810725i	$-0.699073\pi$
$773$	−32.5531	−1.17085	−0.585427	+	0.810725i	$0.699073\pi$
$774$	26.6949	0.959528
$775$	−117.864	−4.23380
$776$	15.3370	0.550565
$777$	30.8936	1.10830
$778$	−39.2820	−1.40833
$779$	−12.7773	−0.457794
$780$	32.3963	1.15997
$781$	2.93959	0.105187
$782$	−5.12628	−0.183315
$783$	−29.5367	−1.05556
$784$	10.6290	0.379607
$785$	−22.8606	−0.815931
$786$	34.0109	1.21313
$787$	−42.4323	−1.51255	−0.756274	−	0.654255i	$-0.772982\pi$
$787$	−42.4323	−1.51255	−0.756274	+	0.654255i	$0.772982\pi$
$788$	23.7344	0.845501
$789$	−33.2580	−1.18402
$790$	−33.4412	−1.18979
$791$	3.06041	0.108816
$792$	−5.03293	−0.178838
$793$	−16.0659	−0.570516
$794$	23.1252	0.820682
$795$	83.8639	2.97435
$796$	3.60262	0.127691
$797$	−33.8421	−1.19875	−0.599374	−	0.800469i	$-0.704584\pi$
$797$	−33.8421	−1.19875	−0.599374	+	0.800469i	$0.704584\pi$
$798$	−46.0198	−1.62908
$799$	10.7618	0.380726
$800$	11.2646	0.398262
$801$	−14.7344	−0.520613
$802$	4.68703	0.165505
$803$	−5.23163	−0.184620
$804$	11.9341	0.420884
$805$	−86.8035	−3.05942
$806$	29.6554	1.04457
$807$	−10.4314	−0.367203
$808$	−0.509273	−0.0179162
$809$	13.8683	0.487582	0.243791	−	0.969828i	$-0.421609\pi$
$809$	13.8683	0.487582	0.243791	+	0.969828i	$0.421609\pi$
$810$	−4.96707	−0.174525
$811$	−1.20524	−0.0423215	−0.0211608	−	0.999776i	$-0.506736\pi$
$811$	−1.20524	−0.0423215	−0.0211608	+	0.999776i	$0.506736\pi$
$812$	−21.5237	−0.755332
$813$	−90.7267	−3.18193
$814$	−2.59607	−0.0909923
$815$	54.9385	1.92441
$816$	−2.83424	−0.0992184
$817$	−20.5117	−0.717612
$818$	−24.2635	−0.848353
$819$	−59.8925	−2.09281
$820$	−13.3250	−0.465328
$821$	33.6004	1.17266	0.586332	−	0.810071i	$-0.300571\pi$
$821$	33.6004	1.17266	0.586332	+	0.810071i	$0.300571\pi$
$822$	49.0253	1.70995
$823$	−54.7267	−1.90765	−0.953827	−	0.300357i	$-0.902894\pi$
$823$	−54.7267	−1.90765	−0.953827	+	0.300357i	$0.902894\pi$
$824$	5.23163	0.182252
$825$	−31.9265	−1.11154
$826$	−28.2526	−0.983032
$827$	6.26564	0.217878	0.108939	−	0.994048i	$-0.465255\pi$
$827$	6.26564	0.217878	0.108939	+	0.994048i	$0.465255\pi$
$828$	25.8002	0.896620
$829$	−3.27110	−0.113610	−0.0568051	−	0.998385i	$-0.518091\pi$
$829$	−3.27110	−0.113610	−0.0568051	+	0.998385i	$0.518091\pi$
$830$	0.535664	0.0185932
$831$	49.3239	1.71103
$832$	−2.83424	−0.0982597
$833$	−10.6290	−0.368273
$834$	−8.86063	−0.306819
$835$	−88.3701	−3.05817
$836$	3.86718	0.133749
$837$	−60.2875	−2.08384
$838$	25.2569	0.872486
$839$	−5.56075	−0.191979	−0.0959893	−	0.995382i	$-0.530601\pi$
$839$	−5.56075	−0.191979	−0.0959893	+	0.995382i	$0.530601\pi$
$840$	−47.9924	−1.65589
$841$	−2.72127	−0.0938368
$842$	−28.9265	−0.996873
$843$	49.8530	1.71703
$844$	11.2052	0.385700
$845$	20.0318	0.689117
$846$	−54.1636	−1.86218
$847$	4.19869	0.144269
$848$	−7.33697	−0.251953
$849$	−18.9156	−0.649181
$850$	−11.2646	−0.386371
$851$	13.3082	0.456199
$852$	−8.33151	−0.285433
$853$	37.1791	1.27299	0.636493	−	0.771282i	$-0.280384\pi$
$853$	37.1791	1.27299	0.636493	+	0.771282i	$0.280384\pi$
$854$	23.8002	0.814427
$855$	78.4940	2.68444
$856$	13.3370	0.455848
$857$	6.06478	0.207169	0.103585	−	0.994621i	$-0.466969\pi$
$857$	6.06478	0.207169	0.103585	+	0.994621i	$0.466969\pi$
$858$	8.03293	0.274240
$859$	40.9924	1.39864	0.699321	−	0.714808i	$-0.253486\pi$
$859$	40.9924	1.39864	0.699321	+	0.714808i	$0.253486\pi$
$860$	−21.3908	−0.729422
$861$	39.3184	1.33997
$862$	15.5357	0.529147
$863$	35.5775	1.21107	0.605537	−	0.795817i	$-0.292959\pi$
$863$	35.5775	1.21107	0.605537	+	0.795817i	$0.292959\pi$
$864$	5.76183	0.196021
$865$	−16.6630	−0.566560
$866$	2.82770	0.0960891
$867$	2.83424	0.0962559
$868$	−43.9320	−1.49115
$869$	−8.29204	−0.281288
$870$	58.5950	1.98656
$871$	−11.9341	−0.404373
$872$	−14.1317	−0.478561
$873$	77.1900	2.61248
$874$	−19.8242	−0.670565
$875$	−106.078	−3.58609
$876$	14.8277	0.500982
$877$	−9.65757	−0.326113	−0.163056	−	0.986617i	$-0.552135\pi$
$877$	−9.65757	−0.326113	−0.163056	+	0.986617i	$0.552135\pi$
$878$	−20.7618	−0.700678
$879$	36.0144	1.21474
$880$	4.03293	0.135950
$881$	−9.13720	−0.307840	−0.153920	−	0.988083i	$-0.549190\pi$
$881$	−9.13720	−0.307840	−0.153920	+	0.988083i	$0.549190\pi$
$882$	53.4951	1.80127
$883$	23.6925	0.797316	0.398658	−	0.917100i	$-0.369476\pi$
$883$	23.6925	0.797316	0.398658	+	0.917100i	$0.369476\pi$
$884$	2.83424	0.0953259
$885$	76.9134	2.58542
$886$	−3.32605	−0.111741
$887$	−7.64340	−0.256640	−0.128320	−	0.991733i	$-0.540958\pi$
$887$	−7.64340	−0.256640	−0.128320	+	0.991733i	$0.540958\pi$
$888$	7.35790	0.246915
$889$	29.9210	1.00352
$890$	11.8068	0.395764
$891$	−1.23163	−0.0412610
$892$	−11.2042	−0.375143
$893$	41.6179	1.39269
$894$	−5.78038	−0.193325
$895$	72.0087	2.40699
$896$	4.19869	0.140268
$897$	−41.1791	−1.37493
$898$	−37.3898	−1.24771
$899$	53.6375	1.78891
$900$	56.6938	1.88979
$901$	7.33697	0.244430
$902$	−3.30404	−0.110012
$903$	63.1187	2.10046
$904$	0.728896	0.0242427
$905$	10.8462	0.360541
$906$	14.7529	0.490132
$907$	−17.1658	−0.569980	−0.284990	−	0.958531i	$-0.591990\pi$
$907$	−17.1658	−0.569980	−0.284990	+	0.958531i	$0.591990\pi$
$908$	26.5291	0.880400
$909$	−2.56314	−0.0850140
$910$	47.9924	1.59093
$911$	4.25256	0.140894	0.0704468	−	0.997516i	$-0.477558\pi$
$911$	4.25256	0.140894	0.0704468	+	0.997516i	$0.477558\pi$
$912$	−10.9605	−0.362939
$913$	0.132822	0.00439578
$914$	−27.2031	−0.899797
$915$	−64.7926	−2.14198
$916$	11.0604	0.365446
$917$	50.3843	1.66384
$918$	−5.76183	−0.190169
$919$	45.2162	1.49154	0.745772	−	0.666202i	$-0.232081\pi$
$919$	45.2162	1.49154	0.745772	+	0.666202i	$0.232081\pi$
$920$	−20.6739	−0.681600
$921$	−39.9780	−1.31732
$922$	9.08680	0.299258
$923$	8.33151	0.274235
$924$	−11.9001	−0.391485
$925$	29.2436	0.961525
$926$	−17.8947	−0.588055
$927$	26.3304	0.864805
$928$	−5.12628	−0.168278
$929$	23.6135	0.774735	0.387367	−	0.921925i	$-0.373384\pi$
$929$	23.6135	0.774735	0.387367	+	0.921925i	$0.373384\pi$
$930$	119.598	3.92178
$931$	−41.1043	−1.34714
$932$	−14.2371	−0.466351
$933$	57.7740	1.89144
$934$	24.1736	0.790984
$935$	−4.03293	−0.131891
$936$	−14.2646	−0.466252
$937$	5.72127	0.186906	0.0934528	−	0.995624i	$-0.470210\pi$
$937$	5.72127	0.186906	0.0934528	+	0.995624i	$0.470210\pi$
$938$	17.6794	0.577253
$939$	15.6554	0.510895
$940$	43.4018	1.41561
$941$	−35.2558	−1.14931	−0.574654	−	0.818397i	$-0.694863\pi$
$941$	−35.2558	−1.14931	−0.574654	+	0.818397i	$0.694863\pi$
$942$	16.0659	0.523454
$943$	16.9374	0.551558
$944$	−6.72890	−0.219007
$945$	−97.5653	−3.17380
$946$	−5.30404	−0.172449
$947$	−15.7913	−0.513148	−0.256574	−	0.966525i	$-0.582594\pi$
$947$	−15.7913	−0.513148	−0.256574	+	0.966525i	$0.582594\pi$
$948$	23.5016	0.763298
$949$	−14.8277	−0.481327
$950$	−43.5621	−1.41334
$951$	15.4997	0.502611
$952$	−4.19869	−0.136080
$953$	9.47088	0.306792	0.153396	−	0.988165i	$-0.450979\pi$
$953$	9.47088	0.306792	0.153396	+	0.988165i	$0.450979\pi$
$954$	−36.9265	−1.19554
$955$	102.289	3.30999
$956$	1.45779	0.0471484
$957$	14.5291	0.469660
$958$	−2.03948	−0.0658926
$959$	72.6268	2.34524
$960$	−11.4303	−0.368912
$961$	78.4796	2.53160
$962$	−7.35790	−0.237228
$963$	67.1241	2.16304
$964$	−22.9265	−0.738413
$965$	−26.9780	−0.868452
$966$	61.0033	1.96275
$967$	52.9924	1.70412	0.852060	−	0.523444i	$-0.175353\pi$
$967$	52.9924	1.70412	0.852060	+	0.523444i	$0.175353\pi$
$968$	1.00000	0.0321412
$969$	10.9605	0.352103
$970$	−61.8530	−1.98598
$971$	−51.8901	−1.66523	−0.832616	−	0.553850i	$-0.813158\pi$
$971$	−51.8901	−1.66523	−0.832616	+	0.553850i	$0.813158\pi$
$972$	−13.7948	−0.442467
$973$	−13.1263	−0.420809
$974$	−27.7871	−0.890357
$975$	−90.4875	−2.89792
$976$	5.66849	0.181444
$977$	−21.7433	−0.695629	−0.347815	−	0.937563i	$-0.613076\pi$
$977$	−21.7433	−0.695629	−0.347815	+	0.937563i	$0.613076\pi$
$978$	−38.6094	−1.23459
$979$	2.92759	0.0935661
$980$	−42.8661	−1.36931
$981$	−71.1241	−2.27082
$982$	−3.91211	−0.124841
$983$	41.0295	1.30864	0.654318	−	0.756219i	$-0.272956\pi$
$983$	41.0295	1.30864	0.654318	+	0.756219i	$0.272956\pi$
$984$	9.36445	0.298528
$985$	−95.7191	−3.04986
$986$	5.12628	0.163254
$987$	−128.067	−4.07642
$988$	10.9605	0.348701
$989$	27.1900	0.864591
$990$	20.2975	0.645097
$991$	44.9505	1.42790	0.713950	−	0.700197i	$-0.246904\pi$
$991$	44.9505	1.42790	0.713950	+	0.700197i	$0.246904\pi$
$992$	−10.4633	−0.332209
$993$	−27.1791	−0.862501
$994$	−12.3424	−0.391478
$995$	−14.5291	−0.460604
$996$	−0.376451	−0.0119283
$997$	−27.4159	−0.868271	−0.434136	−	0.900848i	$-0.642946\pi$
$997$	−27.4159	−0.868271	−0.434136	+	0.900848i	$0.642946\pi$
$998$	−36.5555	−1.15714
$999$	14.9581	0.473255

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	374.2.a.c.1.3	✓	3
3.2	odd	2		3366.2.a.ba.1.3		3
4.3	odd	2		2992.2.a.o.1.1		3
5.4	even	2		9350.2.a.bw.1.1		3
11.10	odd	2		4114.2.a.p.1.3		3
17.16	even	2		6358.2.a.r.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
374.2.a.c.1.3	✓	3	1.1	even	1	trivial
2992.2.a.o.1.1		3	4.3	odd	2
3366.2.a.ba.1.3		3	3.2	odd	2
4114.2.a.p.1.3		3	11.10	odd	2
6358.2.a.r.1.1		3	17.16	even	2
9350.2.a.bw.1.1		3	5.4	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 $\F_{q}$
Belyi maps

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	374.2.a.c.1.3

Level	$374$
Weight	$2$
Character	374.1
Self dual	yes
Analytic conductor	$2.986$
Analytic rank	$0$
Dimension	$3$
CM	no
Inner twists	$1$