LMFDB - Embedded newform 750.2.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [750,2,Mod(1,750)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(750, 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("750.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$750 = 2 \cdot 3 \cdot 5^{3}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	750.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-2,2,2,0,-2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.98878015160$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 1$ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	750.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.23607 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.38197 q^{11} +1.00000 q^{12} +2.85410 q^{13} +1.23607 q^{14} +1.00000 q^{16} -4.85410 q^{17} -1.00000 q^{18} +6.00000 q^{19} -1.23607 q^{21} -1.38197 q^{22} +3.09017 q^{23} -1.00000 q^{24} -2.85410 q^{26} +1.00000 q^{27} -1.23607 q^{28} +9.32624 q^{29} +2.14590 q^{31} -1.00000 q^{32} +1.38197 q^{33} +4.85410 q^{34} +1.00000 q^{36} +7.85410 q^{37} -6.00000 q^{38} +2.85410 q^{39} -3.23607 q^{41} +1.23607 q^{42} +0.145898 q^{43} +1.38197 q^{44} -3.09017 q^{46} -10.8541 q^{47} +1.00000 q^{48} -5.47214 q^{49} -4.85410 q^{51} +2.85410 q^{52} -3.23607 q^{53} -1.00000 q^{54} +1.23607 q^{56} +6.00000 q^{57} -9.32624 q^{58} +6.38197 q^{59} +13.4164 q^{61} -2.14590 q^{62} -1.23607 q^{63} +1.00000 q^{64} -1.38197 q^{66} +2.38197 q^{67} -4.85410 q^{68} +3.09017 q^{69} -8.94427 q^{71} -1.00000 q^{72} +12.0000 q^{73} -7.85410 q^{74} +6.00000 q^{76} -1.70820 q^{77} -2.85410 q^{78} -1.14590 q^{79} +1.00000 q^{81} +3.23607 q^{82} +8.18034 q^{83} -1.23607 q^{84} -0.145898 q^{86} +9.32624 q^{87} -1.38197 q^{88} -9.23607 q^{89} -3.52786 q^{91} +3.09017 q^{92} +2.14590 q^{93} +10.8541 q^{94} -1.00000 q^{96} +18.1803 q^{97} +5.47214 q^{98} +1.38197 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{7} - 2 q^{8} + 2 q^{9} + 5 q^{11} + 2 q^{12} - q^{13} - 2 q^{14} + 2 q^{16} - 3 q^{17} - 2 q^{18} + 12 q^{19} + 2 q^{21} - 5 q^{22} - 5 q^{23} - 2 q^{24}+ \cdots + 5 q^{99}+O(q^{100})$ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^7 - 2 * q^8 + 2 * q^9 + 5 * q^11 + 2 * q^12 - q^13 - 2 * q^14 + 2 * q^16 - 3 * q^17 - 2 * q^18 + 12 * q^19 + 2 * q^21 - 5 * q^22 - 5 * q^23 - 2 * q^24 + q^26 + 2 * q^27 + 2 * q^28 + 3 * q^29 + 11 * q^31 - 2 * q^32 + 5 * q^33 + 3 * q^34 + 2 * q^36 + 9 * q^37 - 12 * q^38 - q^39 - 2 * q^41 - 2 * q^42 + 7 * q^43 + 5 * q^44 + 5 * q^46 - 15 * q^47 + 2 * q^48 - 2 * q^49 - 3 * q^51 - q^52 - 2 * q^53 - 2 * q^54 - 2 * q^56 + 12 * q^57 - 3 * q^58 + 15 * q^59 - 11 * q^62 + 2 * q^63 + 2 * q^64 - 5 * q^66 + 7 * q^67 - 3 * q^68 - 5 * q^69 - 2 * q^72 + 24 * q^73 - 9 * q^74 + 12 * q^76 + 10 * q^77 + q^78 - 9 * q^79 + 2 * q^81 + 2 * q^82 - 6 * q^83 + 2 * q^84 - 7 * q^86 + 3 * q^87 - 5 * q^88 - 14 * q^89 - 16 * q^91 - 5 * q^92 + 11 * q^93 + 15 * q^94 - 2 * q^96 + 14 * q^97 + 2 * q^98 + 5 * 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
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	1.38197	0.416678	0.208339	−	0.978057i	$-0.433194\pi$
$11$	1.38197	0.416678	0.208339	+	0.978057i	$0.433194\pi$
$12$	1.00000	0.288675
$13$	2.85410	0.791585	0.395793	−	0.918340i	$-0.370470\pi$
$13$	2.85410	0.791585	0.395793	+	0.918340i	$0.370470\pi$
$14$	1.23607	0.330353
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.85410	−1.17729	−0.588646	−	0.808391i	$-0.700339\pi$
$17$	−4.85410	−1.17729	−0.588646	+	0.808391i	$0.700339\pi$
$18$	−1.00000	−0.235702
$19$	6.00000	1.37649	0.688247	−	0.725476i	$-0.258380\pi$
$19$	6.00000	1.37649	0.688247	+	0.725476i	$0.258380\pi$
$20$	0	0
$21$	−1.23607	−0.269732
$22$	−1.38197	−0.294636
$23$	3.09017	0.644345	0.322172	−	0.946681i	$-0.395587\pi$
$23$	3.09017	0.644345	0.322172	+	0.946681i	$0.395587\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−2.85410	−0.559735
$27$	1.00000	0.192450
$28$	−1.23607	−0.233595
$29$	9.32624	1.73184	0.865919	−	0.500183i	$-0.166734\pi$
$29$	9.32624	1.73184	0.865919	+	0.500183i	$0.166734\pi$
$30$	0	0
$31$	2.14590	0.385415	0.192707	−	0.981256i	$-0.438273\pi$
$31$	2.14590	0.385415	0.192707	+	0.981256i	$0.438273\pi$
$32$	−1.00000	−0.176777
$33$	1.38197	0.240569
$34$	4.85410	0.832472
$35$	0	0
$36$	1.00000	0.166667
$37$	7.85410	1.29121	0.645603	−	0.763673i	$-0.276606\pi$
$37$	7.85410	1.29121	0.645603	+	0.763673i	$0.276606\pi$
$38$	−6.00000	−0.973329
$39$	2.85410	0.457022
$40$	0	0
$41$	−3.23607	−0.505389	−0.252694	−	0.967546i	$-0.581317\pi$
$41$	−3.23607	−0.505389	−0.252694	+	0.967546i	$0.581317\pi$
$42$	1.23607	0.190729
$43$	0.145898	0.0222492	0.0111246	−	0.999938i	$-0.496459\pi$
$43$	0.145898	0.0222492	0.0111246	+	0.999938i	$0.496459\pi$
$44$	1.38197	0.208339
$45$	0	0
$46$	−3.09017	−0.455621
$47$	−10.8541	−1.58323	−0.791617	−	0.611018i	$-0.790760\pi$
$47$	−10.8541	−1.58323	−0.791617	+	0.611018i	$0.790760\pi$
$48$	1.00000	0.144338
$49$	−5.47214	−0.781734
$50$	0	0
$51$	−4.85410	−0.679710
$52$	2.85410	0.395793
$53$	−3.23607	−0.444508	−0.222254	−	0.974989i	$-0.571341\pi$
$53$	−3.23607	−0.444508	−0.222254	+	0.974989i	$0.571341\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	1.23607	0.165177
$57$	6.00000	0.794719
$58$	−9.32624	−1.22460
$59$	6.38197	0.830861	0.415431	−	0.909625i	$-0.363631\pi$
$59$	6.38197	0.830861	0.415431	+	0.909625i	$0.363631\pi$
$60$	0	0
$61$	13.4164	1.71780	0.858898	−	0.512148i	$-0.171150\pi$
$61$	13.4164	1.71780	0.858898	+	0.512148i	$0.171150\pi$
$62$	−2.14590	−0.272529
$63$	−1.23607	−0.155730
$64$	1.00000	0.125000
$65$	0	0
$66$	−1.38197	−0.170108
$67$	2.38197	0.291003	0.145502	−	0.989358i	$-0.453520\pi$
$67$	2.38197	0.291003	0.145502	+	0.989358i	$0.453520\pi$
$68$	−4.85410	−0.588646
$69$	3.09017	0.372013
$70$	0	0
$71$	−8.94427	−1.06149	−0.530745	−	0.847532i	$-0.678088\pi$
$71$	−8.94427	−1.06149	−0.530745	+	0.847532i	$0.678088\pi$
$72$	−1.00000	−0.117851
$73$	12.0000	1.40449	0.702247	−	0.711934i	$-0.252180\pi$
$73$	12.0000	1.40449	0.702247	+	0.711934i	$0.252180\pi$
$74$	−7.85410	−0.913021
$75$	0	0
$76$	6.00000	0.688247
$77$	−1.70820	−0.194668
$78$	−2.85410	−0.323163
$79$	−1.14590	−0.128924	−0.0644618	−	0.997920i	$-0.520533\pi$
$79$	−1.14590	−0.128924	−0.0644618	+	0.997920i	$0.520533\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	3.23607	0.357364
$83$	8.18034	0.897909	0.448954	−	0.893555i	$-0.351797\pi$
$83$	8.18034	0.897909	0.448954	+	0.893555i	$0.351797\pi$
$84$	−1.23607	−0.134866
$85$	0	0
$86$	−0.145898	−0.0157326
$87$	9.32624	0.999878
$88$	−1.38197	−0.147318
$89$	−9.23607	−0.979021	−0.489511	−	0.871997i	$-0.662825\pi$
$89$	−9.23607	−0.979021	−0.489511	+	0.871997i	$0.662825\pi$
$90$	0	0
$91$	−3.52786	−0.369821
$92$	3.09017	0.322172
$93$	2.14590	0.222519
$94$	10.8541	1.11952
$95$	0	0
$96$	−1.00000	−0.102062
$97$	18.1803	1.84593	0.922967	−	0.384879i	$-0.125757\pi$
$97$	18.1803	1.84593	0.922967	+	0.384879i	$0.125757\pi$
$98$	5.47214	0.552769
$99$	1.38197	0.138893
$100$	0	0
$101$	−2.32624	−0.231469	−0.115735	−	0.993280i	$-0.536922\pi$
$101$	−2.32624	−0.231469	−0.115735	+	0.993280i	$0.536922\pi$
$102$	4.85410	0.480628
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−2.85410	−0.279868
$105$	0	0
$106$	3.23607	0.314315
$107$	−8.18034	−0.790823	−0.395412	−	0.918504i	$-0.629398\pi$
$107$	−8.18034	−0.790823	−0.395412	+	0.918504i	$0.629398\pi$
$108$	1.00000	0.0962250
$109$	−13.4164	−1.28506	−0.642529	−	0.766261i	$-0.722115\pi$
$109$	−13.4164	−1.28506	−0.642529	+	0.766261i	$0.722115\pi$
$110$	0	0
$111$	7.85410	0.745478
$112$	−1.23607	−0.116797
$113$	12.3262	1.15955	0.579777	−	0.814775i	$-0.303139\pi$
$113$	12.3262	1.15955	0.579777	+	0.814775i	$0.303139\pi$
$114$	−6.00000	−0.561951
$115$	0	0
$116$	9.32624	0.865919
$117$	2.85410	0.263862
$118$	−6.38197	−0.587508
$119$	6.00000	0.550019
$120$	0	0
$121$	−9.09017	−0.826379
$122$	−13.4164	−1.21466
$123$	−3.23607	−0.291786
$124$	2.14590	0.192707
$125$	0	0
$126$	1.23607	0.110118
$127$	−13.4164	−1.19051	−0.595257	−	0.803535i	$-0.702950\pi$
$127$	−13.4164	−1.19051	−0.595257	+	0.803535i	$0.702950\pi$
$128$	−1.00000	−0.0883883
$129$	0.145898	0.0128456
$130$	0	0
$131$	17.8885	1.56293	0.781465	−	0.623949i	$-0.214473\pi$
$131$	17.8885	1.56293	0.781465	+	0.623949i	$0.214473\pi$
$132$	1.38197	0.120285
$133$	−7.41641	−0.643084
$134$	−2.38197	−0.205771
$135$	0	0
$136$	4.85410	0.416236
$137$	4.61803	0.394545	0.197273	−	0.980349i	$-0.436792\pi$
$137$	4.61803	0.394545	0.197273	+	0.980349i	$0.436792\pi$
$138$	−3.09017	−0.263053
$139$	−21.2361	−1.80122	−0.900610	−	0.434628i	$-0.856880\pi$
$139$	−21.2361	−1.80122	−0.900610	+	0.434628i	$0.856880\pi$
$140$	0	0
$141$	−10.8541	−0.914080
$142$	8.94427	0.750587
$143$	3.94427	0.329837
$144$	1.00000	0.0833333
$145$	0	0
$146$	−12.0000	−0.993127
$147$	−5.47214	−0.451334
$148$	7.85410	0.645603
$149$	−20.4721	−1.67714	−0.838571	−	0.544792i	$-0.816609\pi$
$149$	−20.4721	−1.67714	−0.838571	+	0.544792i	$0.816609\pi$
$150$	0	0
$151$	−17.8541	−1.45295	−0.726473	−	0.687195i	$-0.758842\pi$
$151$	−17.8541	−1.45295	−0.726473	+	0.687195i	$0.758842\pi$
$152$	−6.00000	−0.486664
$153$	−4.85410	−0.392431
$154$	1.70820	0.137651
$155$	0	0
$156$	2.85410	0.228511
$157$	−0.437694	−0.0349318	−0.0174659	−	0.999847i	$-0.505560\pi$
$157$	−0.437694	−0.0349318	−0.0174659	+	0.999847i	$0.505560\pi$
$158$	1.14590	0.0911628
$159$	−3.23607	−0.256637
$160$	0	0
$161$	−3.81966	−0.301031
$162$	−1.00000	−0.0785674
$163$	20.2705	1.58771	0.793854	−	0.608108i	$-0.208071\pi$
$163$	20.2705	1.58771	0.793854	+	0.608108i	$0.208071\pi$
$164$	−3.23607	−0.252694
$165$	0	0
$166$	−8.18034	−0.634918
$167$	−23.0902	−1.78677	−0.893385	−	0.449291i	$-0.851677\pi$
$167$	−23.0902	−1.78677	−0.893385	+	0.449291i	$0.851677\pi$
$168$	1.23607	0.0953647
$169$	−4.85410	−0.373392
$170$	0	0
$171$	6.00000	0.458831
$172$	0.145898	0.0111246
$173$	−9.23607	−0.702205	−0.351103	−	0.936337i	$-0.614193\pi$
$173$	−9.23607	−0.702205	−0.351103	+	0.936337i	$0.614193\pi$
$174$	−9.32624	−0.707020
$175$	0	0
$176$	1.38197	0.104170
$177$	6.38197	0.479698
$178$	9.23607	0.692273
$179$	7.41641	0.554328	0.277164	−	0.960823i	$-0.410605\pi$
$179$	7.41641	0.554328	0.277164	+	0.960823i	$0.410605\pi$
$180$	0	0
$181$	7.41641	0.551257	0.275629	−	0.961264i	$-0.411114\pi$
$181$	7.41641	0.551257	0.275629	+	0.961264i	$0.411114\pi$
$182$	3.52786	0.261503
$183$	13.4164	0.991769
$184$	−3.09017	−0.227810
$185$	0	0
$186$	−2.14590	−0.157345
$187$	−6.70820	−0.490552
$188$	−10.8541	−0.791617
$189$	−1.23607	−0.0899107
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	1.00000	0.0721688
$193$	−17.1246	−1.23266	−0.616328	−	0.787489i	$-0.711381\pi$
$193$	−17.1246	−1.23266	−0.616328	+	0.787489i	$0.711381\pi$
$194$	−18.1803	−1.30527
$195$	0	0
$196$	−5.47214	−0.390867
$197$	−1.52786	−0.108856	−0.0544279	−	0.998518i	$-0.517334\pi$
$197$	−1.52786	−0.108856	−0.0544279	+	0.998518i	$0.517334\pi$
$198$	−1.38197	−0.0982120
$199$	−15.6180	−1.10713	−0.553567	−	0.832805i	$-0.686734\pi$
$199$	−15.6180	−1.10713	−0.553567	+	0.832805i	$0.686734\pi$
$200$	0	0
$201$	2.38197	0.168011
$202$	2.32624	0.163674
$203$	−11.5279	−0.809097
$204$	−4.85410	−0.339855
$205$	0	0
$206$	4.00000	0.278693
$207$	3.09017	0.214782
$208$	2.85410	0.197896
$209$	8.29180	0.573556
$210$	0	0
$211$	3.41641	0.235195	0.117598	−	0.993061i	$-0.462481\pi$
$211$	3.41641	0.235195	0.117598	+	0.993061i	$0.462481\pi$
$212$	−3.23607	−0.222254
$213$	−8.94427	−0.612851
$214$	8.18034	0.559197
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−2.65248	−0.180062
$218$	13.4164	0.908674
$219$	12.0000	0.810885
$220$	0	0
$221$	−13.8541	−0.931928
$222$	−7.85410	−0.527133
$223$	−21.7082	−1.45369	−0.726844	−	0.686802i	$-0.759014\pi$
$223$	−21.7082	−1.45369	−0.726844	+	0.686802i	$0.759014\pi$
$224$	1.23607	0.0825883
$225$	0	0
$226$	−12.3262	−0.819929
$227$	−16.4721	−1.09329	−0.546647	−	0.837363i	$-0.684096\pi$
$227$	−16.4721	−1.09329	−0.546647	+	0.837363i	$0.684096\pi$
$228$	6.00000	0.397360
$229$	−1.70820	−0.112881	−0.0564406	−	0.998406i	$-0.517975\pi$
$229$	−1.70820	−0.112881	−0.0564406	+	0.998406i	$0.517975\pi$
$230$	0	0
$231$	−1.70820	−0.112392
$232$	−9.32624	−0.612298
$233$	7.32624	0.479958	0.239979	−	0.970778i	$-0.422859\pi$
$233$	7.32624	0.479958	0.239979	+	0.970778i	$0.422859\pi$
$234$	−2.85410	−0.186578
$235$	0	0
$236$	6.38197	0.415431
$237$	−1.14590	−0.0744341
$238$	−6.00000	−0.388922
$239$	−26.6525	−1.72401	−0.862003	−	0.506904i	$-0.830790\pi$
$239$	−26.6525	−1.72401	−0.862003	+	0.506904i	$0.830790\pi$
$240$	0	0
$241$	0.0901699	0.00580836	0.00290418	−	0.999996i	$-0.499076\pi$
$241$	0.0901699	0.00580836	0.00290418	+	0.999996i	$0.499076\pi$
$242$	9.09017	0.584338
$243$	1.00000	0.0641500
$244$	13.4164	0.858898
$245$	0	0
$246$	3.23607	0.206324
$247$	17.1246	1.08961
$248$	−2.14590	−0.136265
$249$	8.18034	0.518408
$250$	0	0
$251$	7.90983	0.499264	0.249632	−	0.968341i	$-0.419690\pi$
$251$	7.90983	0.499264	0.249632	+	0.968341i	$0.419690\pi$
$252$	−1.23607	−0.0778650
$253$	4.27051	0.268485
$254$	13.4164	0.841820
$255$	0	0
$256$	1.00000	0.0625000
$257$	23.5066	1.46630	0.733150	−	0.680067i	$-0.238049\pi$
$257$	23.5066	1.46630	0.733150	+	0.680067i	$0.238049\pi$
$258$	−0.145898	−0.00908321
$259$	−9.70820	−0.603238
$260$	0	0
$261$	9.32624	0.577280
$262$	−17.8885	−1.10516
$263$	−15.9787	−0.985290	−0.492645	−	0.870230i	$-0.663970\pi$
$263$	−15.9787	−0.985290	−0.492645	+	0.870230i	$0.663970\pi$
$264$	−1.38197	−0.0850541
$265$	0	0
$266$	7.41641	0.454729
$267$	−9.23607	−0.565238
$268$	2.38197	0.145502
$269$	22.6180	1.37905	0.689523	−	0.724264i	$-0.257820\pi$
$269$	22.6180	1.37905	0.689523	+	0.724264i	$0.257820\pi$
$270$	0	0
$271$	−8.56231	−0.520123	−0.260062	−	0.965592i	$-0.583743\pi$
$271$	−8.56231	−0.520123	−0.260062	+	0.965592i	$0.583743\pi$
$272$	−4.85410	−0.294323
$273$	−3.52786	−0.213516
$274$	−4.61803	−0.278986
$275$	0	0
$276$	3.09017	0.186006
$277$	14.3607	0.862850	0.431425	−	0.902149i	$-0.358011\pi$
$277$	14.3607	0.862850	0.431425	+	0.902149i	$0.358011\pi$
$278$	21.2361	1.27365
$279$	2.14590	0.128472
$280$	0	0
$281$	22.4721	1.34058	0.670288	−	0.742101i	$-0.266171\pi$
$281$	22.4721	1.34058	0.670288	+	0.742101i	$0.266171\pi$
$282$	10.8541	0.646352
$283$	−21.0902	−1.25368	−0.626840	−	0.779148i	$-0.715652\pi$
$283$	−21.0902	−1.25368	−0.626840	+	0.779148i	$0.715652\pi$
$284$	−8.94427	−0.530745
$285$	0	0
$286$	−3.94427	−0.233230
$287$	4.00000	0.236113
$288$	−1.00000	−0.0589256
$289$	6.56231	0.386018
$290$	0	0
$291$	18.1803	1.06575
$292$	12.0000	0.702247
$293$	−15.5967	−0.911172	−0.455586	−	0.890192i	$-0.650570\pi$
$293$	−15.5967	−0.911172	−0.455586	+	0.890192i	$0.650570\pi$
$294$	5.47214	0.319141
$295$	0	0
$296$	−7.85410	−0.456510
$297$	1.38197	0.0801898
$298$	20.4721	1.18592
$299$	8.81966	0.510054
$300$	0	0
$301$	−0.180340	−0.0103946
$302$	17.8541	1.02739
$303$	−2.32624	−0.133639
$304$	6.00000	0.344124
$305$	0	0
$306$	4.85410	0.277491
$307$	−33.0902	−1.88856	−0.944278	−	0.329149i	$-0.893238\pi$
$307$	−33.0902	−1.88856	−0.944278	+	0.329149i	$0.893238\pi$
$308$	−1.70820	−0.0973340
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−4.47214	−0.253592	−0.126796	−	0.991929i	$-0.540469\pi$
$311$	−4.47214	−0.253592	−0.126796	+	0.991929i	$0.540469\pi$
$312$	−2.85410	−0.161582
$313$	30.9443	1.74907	0.874537	−	0.484959i	$-0.161166\pi$
$313$	30.9443	1.74907	0.874537	+	0.484959i	$0.161166\pi$
$314$	0.437694	0.0247005
$315$	0	0
$316$	−1.14590	−0.0644618
$317$	9.70820	0.545267	0.272634	−	0.962118i	$-0.412105\pi$
$317$	9.70820	0.545267	0.272634	+	0.962118i	$0.412105\pi$
$318$	3.23607	0.181470
$319$	12.8885	0.721620
$320$	0	0
$321$	−8.18034	−0.456582
$322$	3.81966	0.212861
$323$	−29.1246	−1.62054
$324$	1.00000	0.0555556
$325$	0	0
$326$	−20.2705	−1.12268
$327$	−13.4164	−0.741929
$328$	3.23607	0.178682
$329$	13.4164	0.739671
$330$	0	0
$331$	−21.4164	−1.17715	−0.588576	−	0.808442i	$-0.700311\pi$
$331$	−21.4164	−1.17715	−0.588576	+	0.808442i	$0.700311\pi$
$332$	8.18034	0.448954
$333$	7.85410	0.430402
$334$	23.0902	1.26344
$335$	0	0
$336$	−1.23607	−0.0674330
$337$	−12.9443	−0.705119	−0.352560	−	0.935789i	$-0.614689\pi$
$337$	−12.9443	−0.705119	−0.352560	+	0.935789i	$0.614689\pi$
$338$	4.85410	0.264028
$339$	12.3262	0.669469
$340$	0	0
$341$	2.96556	0.160594
$342$	−6.00000	−0.324443
$343$	15.4164	0.832408
$344$	−0.145898	−0.00786629
$345$	0	0
$346$	9.23607	0.496534
$347$	24.3607	1.30775	0.653875	−	0.756603i	$-0.273142\pi$
$347$	24.3607	1.30775	0.653875	+	0.756603i	$0.273142\pi$
$348$	9.32624	0.499939
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	2.85410	0.152341
$352$	−1.38197	−0.0736590
$353$	1.20163	0.0639561	0.0319781	−	0.999489i	$-0.489819\pi$
$353$	1.20163	0.0639561	0.0319781	+	0.999489i	$0.489819\pi$
$354$	−6.38197	−0.339198
$355$	0	0
$356$	−9.23607	−0.489511
$357$	6.00000	0.317554
$358$	−7.41641	−0.391969
$359$	23.8885	1.26079	0.630395	−	0.776275i	$-0.282893\pi$
$359$	23.8885	1.26079	0.630395	+	0.776275i	$0.282893\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	−7.41641	−0.389798
$363$	−9.09017	−0.477110
$364$	−3.52786	−0.184910
$365$	0	0
$366$	−13.4164	−0.701287
$367$	10.2918	0.537227	0.268614	−	0.963248i	$-0.413434\pi$
$367$	10.2918	0.537227	0.268614	+	0.963248i	$0.413434\pi$
$368$	3.09017	0.161086
$369$	−3.23607	−0.168463
$370$	0	0
$371$	4.00000	0.207670
$372$	2.14590	0.111260
$373$	−32.7426	−1.69535	−0.847675	−	0.530516i	$-0.821998\pi$
$373$	−32.7426	−1.69535	−0.847675	+	0.530516i	$0.821998\pi$
$374$	6.70820	0.346873
$375$	0	0
$376$	10.8541	0.559758
$377$	26.6180	1.37090
$378$	1.23607	0.0635765
$379$	−1.23607	−0.0634925	−0.0317463	−	0.999496i	$-0.510107\pi$
$379$	−1.23607	−0.0634925	−0.0317463	+	0.999496i	$0.510107\pi$
$380$	0	0
$381$	−13.4164	−0.687343
$382$	−6.00000	−0.306987
$383$	17.8885	0.914062	0.457031	−	0.889451i	$-0.348913\pi$
$383$	17.8885	0.914062	0.457031	+	0.889451i	$0.348913\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	17.1246	0.871620
$387$	0.145898	0.00741641
$388$	18.1803	0.922967
$389$	18.2705	0.926352	0.463176	−	0.886266i	$-0.346710\pi$
$389$	18.2705	0.926352	0.463176	+	0.886266i	$0.346710\pi$
$390$	0	0
$391$	−15.0000	−0.758583
$392$	5.47214	0.276385
$393$	17.8885	0.902358
$394$	1.52786	0.0769727
$395$	0	0
$396$	1.38197	0.0694464
$397$	32.4721	1.62973	0.814865	−	0.579651i	$-0.196811\pi$
$397$	32.4721	1.62973	0.814865	+	0.579651i	$0.196811\pi$
$398$	15.6180	0.782861
$399$	−7.41641	−0.371285
$400$	0	0
$401$	−10.4721	−0.522954	−0.261477	−	0.965210i	$-0.584209\pi$
$401$	−10.4721	−0.522954	−0.261477	+	0.965210i	$0.584209\pi$
$402$	−2.38197	−0.118802
$403$	6.12461	0.305089
$404$	−2.32624	−0.115735
$405$	0	0
$406$	11.5279	0.572118
$407$	10.8541	0.538018
$408$	4.85410	0.240314
$409$	13.5623	0.670613	0.335306	−	0.942109i	$-0.391160\pi$
$409$	13.5623	0.670613	0.335306	+	0.942109i	$0.391160\pi$
$410$	0	0
$411$	4.61803	0.227791
$412$	−4.00000	−0.197066
$413$	−7.88854	−0.388170
$414$	−3.09017	−0.151874
$415$	0	0
$416$	−2.85410	−0.139934
$417$	−21.2361	−1.03993
$418$	−8.29180	−0.405565
$419$	−8.94427	−0.436956	−0.218478	−	0.975842i	$-0.570109\pi$
$419$	−8.94427	−0.436956	−0.218478	+	0.975842i	$0.570109\pi$
$420$	0	0
$421$	14.4721	0.705329	0.352664	−	0.935750i	$-0.385276\pi$
$421$	14.4721	0.705329	0.352664	+	0.935750i	$0.385276\pi$
$422$	−3.41641	−0.166308
$423$	−10.8541	−0.527744
$424$	3.23607	0.157157
$425$	0	0
$426$	8.94427	0.433351
$427$	−16.5836	−0.802536
$428$	−8.18034	−0.395412
$429$	3.94427	0.190431
$430$	0	0
$431$	−21.7082	−1.04565	−0.522824	−	0.852441i	$-0.675121\pi$
$431$	−21.7082	−1.04565	−0.522824	+	0.852441i	$0.675121\pi$
$432$	1.00000	0.0481125
$433$	−0.180340	−0.00866658	−0.00433329	−	0.999991i	$-0.501379\pi$
$433$	−0.180340	−0.00866658	−0.00433329	+	0.999991i	$0.501379\pi$
$434$	2.65248	0.127323
$435$	0	0
$436$	−13.4164	−0.642529
$437$	18.5410	0.886937
$438$	−12.0000	−0.573382
$439$	28.3262	1.35194	0.675969	−	0.736930i	$-0.263725\pi$
$439$	28.3262	1.35194	0.675969	+	0.736930i	$0.263725\pi$
$440$	0	0
$441$	−5.47214	−0.260578
$442$	13.8541	0.658972
$443$	32.6525	1.55137	0.775683	−	0.631123i	$-0.217406\pi$
$443$	32.6525	1.55137	0.775683	+	0.631123i	$0.217406\pi$
$444$	7.85410	0.372739
$445$	0	0
$446$	21.7082	1.02791
$447$	−20.4721	−0.968299
$448$	−1.23607	−0.0583987
$449$	2.94427	0.138949	0.0694744	−	0.997584i	$-0.477868\pi$
$449$	2.94427	0.138949	0.0694744	+	0.997584i	$0.477868\pi$
$450$	0	0
$451$	−4.47214	−0.210585
$452$	12.3262	0.579777
$453$	−17.8541	−0.838859
$454$	16.4721	0.773076
$455$	0	0
$456$	−6.00000	−0.280976
$457$	−22.9443	−1.07329	−0.536644	−	0.843809i	$-0.680308\pi$
$457$	−22.9443	−1.07329	−0.536644	+	0.843809i	$0.680308\pi$
$458$	1.70820	0.0798191
$459$	−4.85410	−0.226570
$460$	0	0
$461$	−15.9787	−0.744203	−0.372101	−	0.928192i	$-0.621363\pi$
$461$	−15.9787	−0.744203	−0.372101	+	0.928192i	$0.621363\pi$
$462$	1.70820	0.0794728
$463$	−2.29180	−0.106509	−0.0532544	−	0.998581i	$-0.516959\pi$
$463$	−2.29180	−0.106509	−0.0532544	+	0.998581i	$0.516959\pi$
$464$	9.32624	0.432960
$465$	0	0
$466$	−7.32624	−0.339381
$467$	2.29180	0.106052	0.0530258	−	0.998593i	$-0.483113\pi$
$467$	2.29180	0.106052	0.0530258	+	0.998593i	$0.483113\pi$
$468$	2.85410	0.131931
$469$	−2.94427	−0.135954
$470$	0	0
$471$	−0.437694	−0.0201679
$472$	−6.38197	−0.293754
$473$	0.201626	0.00927078
$474$	1.14590	0.0526328
$475$	0	0
$476$	6.00000	0.275010
$477$	−3.23607	−0.148169
$478$	26.6525	1.21906
$479$	6.65248	0.303959	0.151980	−	0.988384i	$-0.451435\pi$
$479$	6.65248	0.303959	0.151980	+	0.988384i	$0.451435\pi$
$480$	0	0
$481$	22.4164	1.02210
$482$	−0.0901699	−0.00410713
$483$	−3.81966	−0.173801
$484$	−9.09017	−0.413190
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−17.5967	−0.797385	−0.398692	−	0.917085i	$-0.630536\pi$
$487$	−17.5967	−0.797385	−0.398692	+	0.917085i	$0.630536\pi$
$488$	−13.4164	−0.607332
$489$	20.2705	0.916664
$490$	0	0
$491$	2.61803	0.118150	0.0590751	−	0.998254i	$-0.481185\pi$
$491$	2.61803	0.118150	0.0590751	+	0.998254i	$0.481185\pi$
$492$	−3.23607	−0.145893
$493$	−45.2705	−2.03888
$494$	−17.1246	−0.770473
$495$	0	0
$496$	2.14590	0.0963537
$497$	11.0557	0.495917
$498$	−8.18034	−0.366570
$499$	13.2361	0.592528	0.296264	−	0.955106i	$-0.404259\pi$
$499$	13.2361	0.592528	0.296264	+	0.955106i	$0.404259\pi$
$500$	0	0
$501$	−23.0902	−1.03159
$502$	−7.90983	−0.353033
$503$	−10.4721	−0.466929	−0.233465	−	0.972365i	$-0.575006\pi$
$503$	−10.4721	−0.466929	−0.233465	+	0.972365i	$0.575006\pi$
$504$	1.23607	0.0550588
$505$	0	0
$506$	−4.27051	−0.189847
$507$	−4.85410	−0.215578
$508$	−13.4164	−0.595257
$509$	2.94427	0.130503	0.0652513	−	0.997869i	$-0.479215\pi$
$509$	2.94427	0.130503	0.0652513	+	0.997869i	$0.479215\pi$
$510$	0	0
$511$	−14.8328	−0.656165
$512$	−1.00000	−0.0441942
$513$	6.00000	0.264906
$514$	−23.5066	−1.03683
$515$	0	0
$516$	0.145898	0.00642280
$517$	−15.0000	−0.659699
$518$	9.70820	0.426554
$519$	−9.23607	−0.405418
$520$	0	0
$521$	−8.18034	−0.358387	−0.179194	−	0.983814i	$-0.557349\pi$
$521$	−8.18034	−0.358387	−0.179194	+	0.983814i	$0.557349\pi$
$522$	−9.32624	−0.408198
$523$	11.9098	0.520781	0.260390	−	0.965503i	$-0.416149\pi$
$523$	11.9098	0.520781	0.260390	+	0.965503i	$0.416149\pi$
$524$	17.8885	0.781465
$525$	0	0
$526$	15.9787	0.696705
$527$	−10.4164	−0.453746
$528$	1.38197	0.0601424
$529$	−13.4508	−0.584820
$530$	0	0
$531$	6.38197	0.276954
$532$	−7.41641	−0.321542
$533$	−9.23607	−0.400059
$534$	9.23607	0.399684
$535$	0	0
$536$	−2.38197	−0.102885
$537$	7.41641	0.320042
$538$	−22.6180	−0.975133
$539$	−7.56231	−0.325732
$540$	0	0
$541$	11.7082	0.503375	0.251688	−	0.967809i	$-0.419014\pi$
$541$	11.7082	0.503375	0.251688	+	0.967809i	$0.419014\pi$
$542$	8.56231	0.367783
$543$	7.41641	0.318269
$544$	4.85410	0.208118
$545$	0	0
$546$	3.52786	0.150979
$547$	−5.43769	−0.232499	−0.116250	−	0.993220i	$-0.537087\pi$
$547$	−5.43769	−0.232499	−0.116250	+	0.993220i	$0.537087\pi$
$548$	4.61803	0.197273
$549$	13.4164	0.572598
$550$	0	0
$551$	55.9574	2.38387
$552$	−3.09017	−0.131526
$553$	1.41641	0.0602318
$554$	−14.3607	−0.610127
$555$	0	0
$556$	−21.2361	−0.900610
$557$	10.3607	0.438996	0.219498	−	0.975613i	$-0.429558\pi$
$557$	10.3607	0.438996	0.219498	+	0.975613i	$0.429558\pi$
$558$	−2.14590	−0.0908431
$559$	0.416408	0.0176122
$560$	0	0
$561$	−6.70820	−0.283221
$562$	−22.4721	−0.947930
$563$	−12.6525	−0.533238	−0.266619	−	0.963802i	$-0.585907\pi$
$563$	−12.6525	−0.533238	−0.266619	+	0.963802i	$0.585907\pi$
$564$	−10.8541	−0.457040
$565$	0	0
$566$	21.0902	0.886486
$567$	−1.23607	−0.0519100
$568$	8.94427	0.375293
$569$	−13.0557	−0.547325	−0.273662	−	0.961826i	$-0.588235\pi$
$569$	−13.0557	−0.547325	−0.273662	+	0.961826i	$0.588235\pi$
$570$	0	0
$571$	12.3607	0.517278	0.258639	−	0.965974i	$-0.416726\pi$
$571$	12.3607	0.517278	0.258639	+	0.965974i	$0.416726\pi$
$572$	3.94427	0.164918
$573$	6.00000	0.250654
$574$	−4.00000	−0.166957
$575$	0	0
$576$	1.00000	0.0416667
$577$	4.11146	0.171162	0.0855811	−	0.996331i	$-0.472725\pi$
$577$	4.11146	0.171162	0.0855811	+	0.996331i	$0.472725\pi$
$578$	−6.56231	−0.272956
$579$	−17.1246	−0.711675
$580$	0	0
$581$	−10.1115	−0.419494
$582$	−18.1803	−0.753599
$583$	−4.47214	−0.185217
$584$	−12.0000	−0.496564
$585$	0	0
$586$	15.5967	0.644296
$587$	18.6525	0.769870	0.384935	−	0.922944i	$-0.374224\pi$
$587$	18.6525	0.769870	0.384935	+	0.922944i	$0.374224\pi$
$588$	−5.47214	−0.225667
$589$	12.8754	0.530521
$590$	0	0
$591$	−1.52786	−0.0628479
$592$	7.85410	0.322802
$593$	34.7984	1.42900	0.714499	−	0.699636i	$-0.246655\pi$
$593$	34.7984	1.42900	0.714499	+	0.699636i	$0.246655\pi$
$594$	−1.38197	−0.0567028
$595$	0	0
$596$	−20.4721	−0.838571
$597$	−15.6180	−0.639204
$598$	−8.81966	−0.360663
$599$	−31.8885	−1.30293	−0.651465	−	0.758678i	$-0.725845\pi$
$599$	−31.8885	−1.30293	−0.651465	+	0.758678i	$0.725845\pi$
$600$	0	0
$601$	40.6869	1.65965	0.829827	−	0.558021i	$-0.188439\pi$
$601$	40.6869	1.65965	0.829827	+	0.558021i	$0.188439\pi$
$602$	0.180340	0.00735011
$603$	2.38197	0.0970012
$604$	−17.8541	−0.726473
$605$	0	0
$606$	2.32624	0.0944970
$607$	−47.4164	−1.92457	−0.962286	−	0.272039i	$-0.912302\pi$
$607$	−47.4164	−1.92457	−0.962286	+	0.272039i	$0.912302\pi$
$608$	−6.00000	−0.243332
$609$	−11.5279	−0.467133
$610$	0	0
$611$	−30.9787	−1.25326
$612$	−4.85410	−0.196215
$613$	37.7771	1.52580	0.762901	−	0.646515i	$-0.223774\pi$
$613$	37.7771	1.52580	0.762901	+	0.646515i	$0.223774\pi$
$614$	33.0902	1.33541
$615$	0	0
$616$	1.70820	0.0688255
$617$	−31.3050	−1.26029	−0.630145	−	0.776478i	$-0.717005\pi$
$617$	−31.3050	−1.26029	−0.630145	+	0.776478i	$0.717005\pi$
$618$	4.00000	0.160904
$619$	−47.7082	−1.91755	−0.958777	−	0.284159i	$-0.908286\pi$
$619$	−47.7082	−1.91755	−0.958777	+	0.284159i	$0.908286\pi$
$620$	0	0
$621$	3.09017	0.124004
$622$	4.47214	0.179316
$623$	11.4164	0.457389
$624$	2.85410	0.114256
$625$	0	0
$626$	−30.9443	−1.23678
$627$	8.29180	0.331142
$628$	−0.437694	−0.0174659
$629$	−38.1246	−1.52013
$630$	0	0
$631$	−15.0344	−0.598512	−0.299256	−	0.954173i	$-0.596738\pi$
$631$	−15.0344	−0.598512	−0.299256	+	0.954173i	$0.596738\pi$
$632$	1.14590	0.0455814
$633$	3.41641	0.135790
$634$	−9.70820	−0.385562
$635$	0	0
$636$	−3.23607	−0.128318
$637$	−15.6180	−0.618809
$638$	−12.8885	−0.510262
$639$	−8.94427	−0.353830
$640$	0	0
$641$	−30.6525	−1.21070	−0.605350	−	0.795959i	$-0.706967\pi$
$641$	−30.6525	−1.21070	−0.605350	+	0.795959i	$0.706967\pi$
$642$	8.18034	0.322852
$643$	−31.5623	−1.24470	−0.622348	−	0.782741i	$-0.713821\pi$
$643$	−31.5623	−1.24470	−0.622348	+	0.782741i	$0.713821\pi$
$644$	−3.81966	−0.150516
$645$	0	0
$646$	29.1246	1.14589
$647$	12.6180	0.496066	0.248033	−	0.968752i	$-0.420216\pi$
$647$	12.6180	0.496066	0.248033	+	0.968752i	$0.420216\pi$
$648$	−1.00000	−0.0392837
$649$	8.81966	0.346202
$650$	0	0
$651$	−2.65248	−0.103959
$652$	20.2705	0.793854
$653$	−9.88854	−0.386969	−0.193484	−	0.981103i	$-0.561979\pi$
$653$	−9.88854	−0.386969	−0.193484	+	0.981103i	$0.561979\pi$
$654$	13.4164	0.524623
$655$	0	0
$656$	−3.23607	−0.126347
$657$	12.0000	0.468165
$658$	−13.4164	−0.523026
$659$	2.02129	0.0787381	0.0393691	−	0.999225i	$-0.487465\pi$
$659$	2.02129	0.0787381	0.0393691	+	0.999225i	$0.487465\pi$
$660$	0	0
$661$	5.70820	0.222023	0.111012	−	0.993819i	$-0.464591\pi$
$661$	5.70820	0.222023	0.111012	+	0.993819i	$0.464591\pi$
$662$	21.4164	0.832372
$663$	−13.8541	−0.538049
$664$	−8.18034	−0.317459
$665$	0	0
$666$	−7.85410	−0.304340
$667$	28.8197	1.11590
$668$	−23.0902	−0.893385
$669$	−21.7082	−0.839288
$670$	0	0
$671$	18.5410	0.715768
$672$	1.23607	0.0476824
$673$	47.3050	1.82347	0.911736	−	0.410777i	$-0.134742\pi$
$673$	47.3050	1.82347	0.911736	+	0.410777i	$0.134742\pi$
$674$	12.9443	0.498595
$675$	0	0
$676$	−4.85410	−0.186696
$677$	31.2361	1.20050	0.600250	−	0.799813i	$-0.295068\pi$
$677$	31.2361	1.20050	0.600250	+	0.799813i	$0.295068\pi$
$678$	−12.3262	−0.473386
$679$	−22.4721	−0.862401
$680$	0	0
$681$	−16.4721	−0.631214
$682$	−2.96556	−0.113557
$683$	10.7639	0.411870	0.205935	−	0.978566i	$-0.433976\pi$
$683$	10.7639	0.411870	0.205935	+	0.978566i	$0.433976\pi$
$684$	6.00000	0.229416
$685$	0	0
$686$	−15.4164	−0.588601
$687$	−1.70820	−0.0651720
$688$	0.145898	0.00556231
$689$	−9.23607	−0.351866
$690$	0	0
$691$	26.7639	1.01815	0.509074	−	0.860723i	$-0.329988\pi$
$691$	26.7639	1.01815	0.509074	+	0.860723i	$0.329988\pi$
$692$	−9.23607	−0.351103
$693$	−1.70820	−0.0648893
$694$	−24.3607	−0.924719
$695$	0	0
$696$	−9.32624	−0.353510
$697$	15.7082	0.594991
$698$	0	0
$699$	7.32624	0.277104
$700$	0	0
$701$	12.2148	0.461346	0.230673	−	0.973031i	$-0.425907\pi$
$701$	12.2148	0.461346	0.230673	+	0.973031i	$0.425907\pi$
$702$	−2.85410	−0.107721
$703$	47.1246	1.77734
$704$	1.38197	0.0520848
$705$	0	0
$706$	−1.20163	−0.0452238
$707$	2.87539	0.108140
$708$	6.38197	0.239849
$709$	−6.87539	−0.258211	−0.129105	−	0.991631i	$-0.541211\pi$
$709$	−6.87539	−0.258211	−0.129105	+	0.991631i	$0.541211\pi$
$710$	0	0
$711$	−1.14590	−0.0429745
$712$	9.23607	0.346136
$713$	6.63119	0.248340
$714$	−6.00000	−0.224544
$715$	0	0
$716$	7.41641	0.277164
$717$	−26.6525	−0.995355
$718$	−23.8885	−0.891513
$719$	−7.23607	−0.269860	−0.134930	−	0.990855i	$-0.543081\pi$
$719$	−7.23607	−0.269860	−0.134930	+	0.990855i	$0.543081\pi$
$720$	0	0
$721$	4.94427	0.184134
$722$	−17.0000	−0.632674
$723$	0.0901699	0.00335346
$724$	7.41641	0.275629
$725$	0	0
$726$	9.09017	0.337368
$727$	−7.34752	−0.272505	−0.136252	−	0.990674i	$-0.543506\pi$
$727$	−7.34752	−0.272505	−0.136252	+	0.990674i	$0.543506\pi$
$728$	3.52786	0.130751
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.708204	−0.0261939
$732$	13.4164	0.495885
$733$	−34.6869	−1.28119	−0.640595	−	0.767879i	$-0.721312\pi$
$733$	−34.6869	−1.28119	−0.640595	+	0.767879i	$0.721312\pi$
$734$	−10.2918	−0.379877
$735$	0	0
$736$	−3.09017	−0.113905
$737$	3.29180	0.121255
$738$	3.23607	0.119121
$739$	8.65248	0.318286	0.159143	−	0.987256i	$-0.449127\pi$
$739$	8.65248	0.318286	0.159143	+	0.987256i	$0.449127\pi$
$740$	0	0
$741$	17.1246	0.629088
$742$	−4.00000	−0.146845
$743$	19.6180	0.719716	0.359858	−	0.933007i	$-0.382825\pi$
$743$	19.6180	0.719716	0.359858	+	0.933007i	$0.382825\pi$
$744$	−2.14590	−0.0786724
$745$	0	0
$746$	32.7426	1.19879
$747$	8.18034	0.299303
$748$	−6.70820	−0.245276
$749$	10.1115	0.369465
$750$	0	0
$751$	−43.4164	−1.58429	−0.792144	−	0.610335i	$-0.791035\pi$
$751$	−43.4164	−1.58429	−0.792144	+	0.610335i	$0.791035\pi$
$752$	−10.8541	−0.395808
$753$	7.90983	0.288250
$754$	−26.6180	−0.969372
$755$	0	0
$756$	−1.23607	−0.0449554
$757$	−4.83282	−0.175652	−0.0878258	−	0.996136i	$-0.527992\pi$
$757$	−4.83282	−0.175652	−0.0878258	+	0.996136i	$0.527992\pi$
$758$	1.23607	0.0448960
$759$	4.27051	0.155010
$760$	0	0
$761$	−32.6525	−1.18365	−0.591826	−	0.806066i	$-0.701593\pi$
$761$	−32.6525	−1.18365	−0.591826	+	0.806066i	$0.701593\pi$
$762$	13.4164	0.486025
$763$	16.5836	0.600366
$764$	6.00000	0.217072
$765$	0	0
$766$	−17.8885	−0.646339
$767$	18.2148	0.657698
$768$	1.00000	0.0360844
$769$	−22.9230	−0.826624	−0.413312	−	0.910589i	$-0.635628\pi$
$769$	−22.9230	−0.826624	−0.413312	+	0.910589i	$0.635628\pi$
$770$	0	0
$771$	23.5066	0.846569
$772$	−17.1246	−0.616328
$773$	6.47214	0.232787	0.116393	−	0.993203i	$-0.462867\pi$
$773$	6.47214	0.232787	0.116393	+	0.993203i	$0.462867\pi$
$774$	−0.145898	−0.00524420
$775$	0	0
$776$	−18.1803	−0.652636
$777$	−9.70820	−0.348280
$778$	−18.2705	−0.655030
$779$	−19.4164	−0.695665
$780$	0	0
$781$	−12.3607	−0.442300
$782$	15.0000	0.536399
$783$	9.32624	0.333293
$784$	−5.47214	−0.195433
$785$	0	0
$786$	−17.8885	−0.638063
$787$	38.5623	1.37460	0.687299	−	0.726375i	$-0.258796\pi$
$787$	38.5623	1.37460	0.687299	+	0.726375i	$0.258796\pi$
$788$	−1.52786	−0.0544279
$789$	−15.9787	−0.568857
$790$	0	0
$791$	−15.2361	−0.541732
$792$	−1.38197	−0.0491060
$793$	38.2918	1.35978
$794$	−32.4721	−1.15239
$795$	0	0
$796$	−15.6180	−0.553567
$797$	−34.4721	−1.22107	−0.610533	−	0.791991i	$-0.709045\pi$
$797$	−34.4721	−1.22107	−0.610533	+	0.791991i	$0.709045\pi$
$798$	7.41641	0.262538
$799$	52.6869	1.86393
$800$	0	0
$801$	−9.23607	−0.326340
$802$	10.4721	0.369784
$803$	16.5836	0.585222
$804$	2.38197	0.0840055
$805$	0	0
$806$	−6.12461	−0.215730
$807$	22.6180	0.796193
$808$	2.32624	0.0818368
$809$	−29.5279	−1.03814	−0.519072	−	0.854730i	$-0.673722\pi$
$809$	−29.5279	−1.03814	−0.519072	+	0.854730i	$0.673722\pi$
$810$	0	0
$811$	−34.0000	−1.19390	−0.596951	−	0.802278i	$-0.703621\pi$
$811$	−34.0000	−1.19390	−0.596951	+	0.802278i	$0.703621\pi$
$812$	−11.5279	−0.404549
$813$	−8.56231	−0.300293
$814$	−10.8541	−0.380436
$815$	0	0
$816$	−4.85410	−0.169928
$817$	0.875388	0.0306260
$818$	−13.5623	−0.474195
$819$	−3.52786	−0.123274
$820$	0	0
$821$	36.2148	1.26390	0.631952	−	0.775007i	$-0.282254\pi$
$821$	36.2148	1.26390	0.631952	+	0.775007i	$0.282254\pi$
$822$	−4.61803	−0.161072
$823$	−15.1246	−0.527211	−0.263605	−	0.964631i	$-0.584912\pi$
$823$	−15.1246	−0.527211	−0.263605	+	0.964631i	$0.584912\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	7.88854	0.274478
$827$	−18.0000	−0.625921	−0.312961	−	0.949766i	$-0.601321\pi$
$827$	−18.0000	−0.625921	−0.312961	+	0.949766i	$0.601321\pi$
$828$	3.09017	0.107391
$829$	34.1803	1.18713	0.593566	−	0.804785i	$-0.297720\pi$
$829$	34.1803	1.18713	0.593566	+	0.804785i	$0.297720\pi$
$830$	0	0
$831$	14.3607	0.498166
$832$	2.85410	0.0989482
$833$	26.5623	0.920329
$834$	21.2361	0.735345
$835$	0	0
$836$	8.29180	0.286778
$837$	2.14590	0.0741731
$838$	8.94427	0.308975
$839$	−11.1246	−0.384064	−0.192032	−	0.981389i	$-0.561508\pi$
$839$	−11.1246	−0.384064	−0.192032	+	0.981389i	$0.561508\pi$
$840$	0	0
$841$	57.9787	1.99927
$842$	−14.4721	−0.498743
$843$	22.4721	0.773981
$844$	3.41641	0.117598
$845$	0	0
$846$	10.8541	0.373172
$847$	11.2361	0.386076
$848$	−3.23607	−0.111127
$849$	−21.0902	−0.723813
$850$	0	0
$851$	24.2705	0.831982
$852$	−8.94427	−0.306426
$853$	−29.0902	−0.996028	−0.498014	−	0.867169i	$-0.665937\pi$
$853$	−29.0902	−0.996028	−0.498014	+	0.867169i	$0.665937\pi$
$854$	16.5836	0.567479
$855$	0	0
$856$	8.18034	0.279598
$857$	12.3262	0.421056	0.210528	−	0.977588i	$-0.432482\pi$
$857$	12.3262	0.421056	0.210528	+	0.977588i	$0.432482\pi$
$858$	−3.94427	−0.134655
$859$	−15.7082	−0.535957	−0.267979	−	0.963425i	$-0.586356\pi$
$859$	−15.7082	−0.535957	−0.267979	+	0.963425i	$0.586356\pi$
$860$	0	0
$861$	4.00000	0.136320
$862$	21.7082	0.739384
$863$	−8.56231	−0.291464	−0.145732	−	0.989324i	$-0.546554\pi$
$863$	−8.56231	−0.291464	−0.145732	+	0.989324i	$0.546554\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	0.180340	0.00612820
$867$	6.56231	0.222868
$868$	−2.65248	−0.0900309
$869$	−1.58359	−0.0537197
$870$	0	0
$871$	6.79837	0.230354
$872$	13.4164	0.454337
$873$	18.1803	0.615311
$874$	−18.5410	−0.627159
$875$	0	0
$876$	12.0000	0.405442
$877$	26.3951	0.891300	0.445650	−	0.895207i	$-0.352973\pi$
$877$	26.3951	0.891300	0.445650	+	0.895207i	$0.352973\pi$
$878$	−28.3262	−0.955964
$879$	−15.5967	−0.526065
$880$	0	0
$881$	−5.05573	−0.170332	−0.0851659	−	0.996367i	$-0.527142\pi$
$881$	−5.05573	−0.170332	−0.0851659	+	0.996367i	$0.527142\pi$
$882$	5.47214	0.184256
$883$	46.9787	1.58096	0.790480	−	0.612488i	$-0.209831\pi$
$883$	46.9787	1.58096	0.790480	+	0.612488i	$0.209831\pi$
$884$	−13.8541	−0.465964
$885$	0	0
$886$	−32.6525	−1.09698
$887$	54.4721	1.82900	0.914498	−	0.404591i	$-0.132586\pi$
$887$	54.4721	1.82900	0.914498	+	0.404591i	$0.132586\pi$
$888$	−7.85410	−0.263566
$889$	16.5836	0.556196
$890$	0	0
$891$	1.38197	0.0462976
$892$	−21.7082	−0.726844
$893$	−65.1246	−2.17931
$894$	20.4721	0.684691
$895$	0	0
$896$	1.23607	0.0412941
$897$	8.81966	0.294480
$898$	−2.94427	−0.0982516
$899$	20.0132	0.667476
$900$	0	0
$901$	15.7082	0.523316
$902$	4.47214	0.148906
$903$	−0.180340	−0.00600134
$904$	−12.3262	−0.409965
$905$	0	0
$906$	17.8541	0.593163
$907$	23.1459	0.768547	0.384273	−	0.923219i	$-0.374452\pi$
$907$	23.1459	0.768547	0.384273	+	0.923219i	$0.374452\pi$
$908$	−16.4721	−0.546647
$909$	−2.32624	−0.0771564
$910$	0	0
$911$	37.5279	1.24335	0.621677	−	0.783274i	$-0.286452\pi$
$911$	37.5279	1.24335	0.621677	+	0.783274i	$0.286452\pi$
$912$	6.00000	0.198680
$913$	11.3050	0.374139
$914$	22.9443	0.758929
$915$	0	0
$916$	−1.70820	−0.0564406
$917$	−22.1115	−0.730185
$918$	4.85410	0.160209
$919$	21.8885	0.722036	0.361018	−	0.932559i	$-0.382429\pi$
$919$	21.8885	0.722036	0.361018	+	0.932559i	$0.382429\pi$
$920$	0	0
$921$	−33.0902	−1.09036
$922$	15.9787	0.526231
$923$	−25.5279	−0.840260
$924$	−1.70820	−0.0561958
$925$	0	0
$926$	2.29180	0.0753131
$927$	−4.00000	−0.131377
$928$	−9.32624	−0.306149
$929$	−24.5410	−0.805165	−0.402582	−	0.915384i	$-0.631887\pi$
$929$	−24.5410	−0.805165	−0.402582	+	0.915384i	$0.631887\pi$
$930$	0	0
$931$	−32.8328	−1.07605
$932$	7.32624	0.239979
$933$	−4.47214	−0.146411
$934$	−2.29180	−0.0749899
$935$	0	0
$936$	−2.85410	−0.0932892
$937$	−27.4164	−0.895655	−0.447828	−	0.894120i	$-0.647802\pi$
$937$	−27.4164	−0.895655	−0.447828	+	0.894120i	$0.647802\pi$
$938$	2.94427	0.0961339
$939$	30.9443	1.00983
$940$	0	0
$941$	−22.4508	−0.731877	−0.365938	−	0.930639i	$-0.619252\pi$
$941$	−22.4508	−0.731877	−0.365938	+	0.930639i	$0.619252\pi$
$942$	0.437694	0.0142608
$943$	−10.0000	−0.325645
$944$	6.38197	0.207715
$945$	0	0
$946$	−0.201626	−0.00655543
$947$	−55.3050	−1.79717	−0.898585	−	0.438800i	$-0.855404\pi$
$947$	−55.3050	−1.79717	−0.898585	+	0.438800i	$0.855404\pi$
$948$	−1.14590	−0.0372170
$949$	34.2492	1.11178
$950$	0	0
$951$	9.70820	0.314810
$952$	−6.00000	−0.194461
$953$	−1.63932	−0.0531028	−0.0265514	−	0.999647i	$-0.508453\pi$
$953$	−1.63932	−0.0531028	−0.0265514	+	0.999647i	$0.508453\pi$
$954$	3.23607	0.104772
$955$	0	0
$956$	−26.6525	−0.862003
$957$	12.8885	0.416627
$958$	−6.65248	−0.214932
$959$	−5.70820	−0.184328
$960$	0	0
$961$	−26.3951	−0.851456
$962$	−22.4164	−0.722734
$963$	−8.18034	−0.263608
$964$	0.0901699	0.00290418
$965$	0	0
$966$	3.81966	0.122896
$967$	39.0132	1.25458	0.627289	−	0.778786i	$-0.284164\pi$
$967$	39.0132	1.25458	0.627289	+	0.778786i	$0.284164\pi$
$968$	9.09017	0.292169
$969$	−29.1246	−0.935617
$970$	0	0
$971$	52.0902	1.67165	0.835827	−	0.548994i	$-0.184989\pi$
$971$	52.0902	1.67165	0.835827	+	0.548994i	$0.184989\pi$
$972$	1.00000	0.0320750
$973$	26.2492	0.841511
$974$	17.5967	0.563836
$975$	0	0
$976$	13.4164	0.429449
$977$	−54.2837	−1.73669	−0.868344	−	0.495962i	$-0.834815\pi$
$977$	−54.2837	−1.73669	−0.868344	+	0.495962i	$0.834815\pi$
$978$	−20.2705	−0.648179
$979$	−12.7639	−0.407937
$980$	0	0
$981$	−13.4164	−0.428353
$982$	−2.61803	−0.0835448
$983$	−24.9787	−0.796697	−0.398349	−	0.917234i	$-0.630417\pi$
$983$	−24.9787	−0.796697	−0.398349	+	0.917234i	$0.630417\pi$
$984$	3.23607	0.103162
$985$	0	0
$986$	45.2705	1.44171
$987$	13.4164	0.427049
$988$	17.1246	0.544806
$989$	0.450850	0.0143362
$990$	0	0
$991$	−8.68692	−0.275949	−0.137975	−	0.990436i	$-0.544059\pi$
$991$	−8.68692	−0.275949	−0.137975	+	0.990436i	$0.544059\pi$
$992$	−2.14590	−0.0681323
$993$	−21.4164	−0.679629
$994$	−11.0557	−0.350666
$995$	0	0
$996$	8.18034	0.259204
$997$	−38.2705	−1.21204	−0.606020	−	0.795450i	$-0.707235\pi$
$997$	−38.2705	−1.21204	−0.606020	+	0.795450i	$0.707235\pi$
$998$	−13.2361	−0.418980
$999$	7.85410	0.248493

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	750.2.a.c.1.1	✓	2
3.2	odd	2		2250.2.a.n.1.1		2
4.3	odd	2		6000.2.a.d.1.2		2
5.2	odd	4		750.2.c.b.499.1		4
5.3	odd	4		750.2.c.b.499.4		4
5.4	even	2		750.2.a.f.1.2	yes	2
15.2	even	4		2250.2.c.b.1999.3		4
15.8	even	4		2250.2.c.b.1999.2		4
15.14	odd	2		2250.2.a.c.1.2		2
20.3	even	4		6000.2.f.a.1249.1		4
20.7	even	4		6000.2.f.a.1249.4		4
20.19	odd	2		6000.2.a.y.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
750.2.a.c.1.1	✓	2	1.1	even	1	trivial
750.2.a.f.1.2	yes	2	5.4	even	2
750.2.c.b.499.1		4	5.2	odd	4
750.2.c.b.499.4		4	5.3	odd	4
2250.2.a.c.1.2		2	15.14	odd	2
2250.2.a.n.1.1		2	3.2	odd	2
2250.2.c.b.1999.2		4	15.8	even	4
2250.2.c.b.1999.3		4	15.2	even	4
6000.2.a.d.1.2		2	4.3	odd	2
6000.2.a.y.1.1		2	20.19	odd	2
6000.2.f.a.1249.1		4	20.3	even	4
6000.2.f.a.1249.4		4	20.7	even	4

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

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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	750.2.a.c.1.1

Level	$750$
Weight	$2$
Character	750.1
Self dual	yes
Analytic conductor	$5.989$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$