LMFDB - Embedded newform 725.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(1,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("725.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$725 = 5^{2} \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	725.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:	$5.78915414654$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 2$ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 29)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	725.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{2} -2.41421 q^{3} +3.82843 q^{4} -5.82843 q^{6} +2.82843 q^{7} +4.41421 q^{8} +2.82843 q^{9} -0.414214 q^{11} -9.24264 q^{12} +3.82843 q^{13} +6.82843 q^{14} +3.00000 q^{16} -0.828427 q^{17} +6.82843 q^{18} +6.00000 q^{19} -6.82843 q^{21} -1.00000 q^{22} -3.65685 q^{23} -10.6569 q^{24} +9.24264 q^{26} +0.414214 q^{27} +10.8284 q^{28} +1.00000 q^{29} +10.0711 q^{31} -1.58579 q^{32} +1.00000 q^{33} -2.00000 q^{34} +10.8284 q^{36} +4.00000 q^{37} +14.4853 q^{38} -9.24264 q^{39} -4.48528 q^{41} -16.4853 q^{42} -3.58579 q^{43} -1.58579 q^{44} -8.82843 q^{46} +3.24264 q^{47} -7.24264 q^{48} +1.00000 q^{49} +2.00000 q^{51} +14.6569 q^{52} -9.48528 q^{53} +1.00000 q^{54} +12.4853 q^{56} -14.4853 q^{57} +2.41421 q^{58} -3.65685 q^{59} -4.82843 q^{61} +24.3137 q^{62} +8.00000 q^{63} -9.82843 q^{64} +2.41421 q^{66} -5.65685 q^{67} -3.17157 q^{68} +8.82843 q^{69} -8.82843 q^{71} +12.4853 q^{72} -4.00000 q^{73} +9.65685 q^{74} +22.9706 q^{76} -1.17157 q^{77} -22.3137 q^{78} -2.41421 q^{79} -9.48528 q^{81} -10.8284 q^{82} -7.65685 q^{83} -26.1421 q^{84} -8.65685 q^{86} -2.41421 q^{87} -1.82843 q^{88} -12.4853 q^{89} +10.8284 q^{91} -14.0000 q^{92} -24.3137 q^{93} +7.82843 q^{94} +3.82843 q^{96} -4.48528 q^{97} +2.41421 q^{98} -1.17157 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 6 q^{6} + 6 q^{8} + 2 q^{11} - 10 q^{12} + 2 q^{13} + 8 q^{14} + 6 q^{16} + 4 q^{17} + 8 q^{18} + 12 q^{19} - 8 q^{21} - 2 q^{22} + 4 q^{23} - 10 q^{24} + 10 q^{26} - 2 q^{27}+ \cdots - 8 q^{99}+O(q^{100})$ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 6 * q^6 + 6 * q^8 + 2 * q^11 - 10 * q^12 + 2 * q^13 + 8 * q^14 + 6 * q^16 + 4 * q^17 + 8 * q^18 + 12 * q^19 - 8 * q^21 - 2 * q^22 + 4 * q^23 - 10 * q^24 + 10 * q^26 - 2 * q^27 + 16 * q^28 + 2 * q^29 + 6 * q^31 - 6 * q^32 + 2 * q^33 - 4 * q^34 + 16 * q^36 + 8 * q^37 + 12 * q^38 - 10 * q^39 + 8 * q^41 - 16 * q^42 - 10 * q^43 - 6 * q^44 - 12 * q^46 - 2 * q^47 - 6 * q^48 + 2 * q^49 + 4 * q^51 + 18 * q^52 - 2 * q^53 + 2 * q^54 + 8 * q^56 - 12 * q^57 + 2 * q^58 + 4 * q^59 - 4 * q^61 + 26 * q^62 + 16 * q^63 - 14 * q^64 + 2 * q^66 - 12 * q^68 + 12 * q^69 - 12 * q^71 + 8 * q^72 - 8 * q^73 + 8 * q^74 + 12 * q^76 - 8 * q^77 - 22 * q^78 - 2 * q^79 - 2 * q^81 - 16 * q^82 - 4 * q^83 - 24 * q^84 - 6 * q^86 - 2 * q^87 + 2 * q^88 - 8 * q^89 + 16 * q^91 - 28 * q^92 - 26 * q^93 + 10 * q^94 + 2 * q^96 + 8 * q^97 + 2 * q^98 - 8 * 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

)

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

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	−2.41421	−1.39385	−0.696923	−	0.717146i	$-0.745448\pi$
$3$	−2.41421	−1.39385	−0.696923	+	0.717146i	$0.745448\pi$
$4$	3.82843	1.91421
$5$	0	0
$5$	0	0
$6$	−5.82843	−2.37945
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	4.41421	1.56066
$9$	2.82843	0.942809
$10$	0	0
$11$	−0.414214	−0.124890	−0.0624450	−	0.998048i	$-0.519890\pi$
$11$	−0.414214	−0.124890	−0.0624450	+	0.998048i	$0.519890\pi$
$12$	−9.24264	−2.66812
$13$	3.82843	1.06181	0.530907	−	0.847430i	$-0.321851\pi$
$13$	3.82843	1.06181	0.530907	+	0.847430i	$0.321851\pi$
$14$	6.82843	1.82497
$15$	0	0
$16$	3.00000	0.750000
$17$	−0.828427	−0.200923	−0.100462	−	0.994941i	$-0.532032\pi$
$17$	−0.828427	−0.200923	−0.100462	+	0.994941i	$0.532032\pi$
$18$	6.82843	1.60948
$19$	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
$21$	−6.82843	−1.49008
$22$	−1.00000	−0.213201
$23$	−3.65685	−0.762507	−0.381253	−	0.924471i	$-0.624507\pi$
$23$	−3.65685	−0.762507	−0.381253	+	0.924471i	$0.624507\pi$
$24$	−10.6569	−2.17532
$25$	0	0
$26$	9.24264	1.81263
$27$	0.414214	0.0797154
$28$	10.8284	2.04638
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	10.0711	1.80882	0.904409	−	0.426667i	$-0.140313\pi$
$31$	10.0711	1.80882	0.904409	+	0.426667i	$0.140313\pi$
$32$	−1.58579	−0.280330
$33$	1.00000	0.174078
$34$	−2.00000	−0.342997
$35$	0	0
$36$	10.8284	1.80474
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	14.4853	2.34982
$39$	−9.24264	−1.48001
$40$	0	0
$41$	−4.48528	−0.700483	−0.350242	−	0.936659i	$-0.613901\pi$
$41$	−4.48528	−0.700483	−0.350242	+	0.936659i	$0.613901\pi$
$42$	−16.4853	−2.54373
$43$	−3.58579	−0.546827	−0.273414	−	0.961897i	$-0.588153\pi$
$43$	−3.58579	−0.546827	−0.273414	+	0.961897i	$0.588153\pi$
$44$	−1.58579	−0.239066
$45$	0	0
$46$	−8.82843	−1.30168
$47$	3.24264	0.472988	0.236494	−	0.971633i	$-0.424002\pi$
$47$	3.24264	0.472988	0.236494	+	0.971633i	$0.424002\pi$
$48$	−7.24264	−1.04539
$49$	1.00000	0.142857
$50$	0	0
$51$	2.00000	0.280056
$52$	14.6569	2.03254
$53$	−9.48528	−1.30290	−0.651452	−	0.758690i	$-0.725840\pi$
$53$	−9.48528	−1.30290	−0.651452	+	0.758690i	$0.725840\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	12.4853	1.66842
$57$	−14.4853	−1.91862
$58$	2.41421	0.317002
$59$	−3.65685	−0.476082	−0.238041	−	0.971255i	$-0.576505\pi$
$59$	−3.65685	−0.476082	−0.238041	+	0.971255i	$0.576505\pi$
$60$	0	0
$61$	−4.82843	−0.618217	−0.309108	−	0.951027i	$-0.600031\pi$
$61$	−4.82843	−0.618217	−0.309108	+	0.951027i	$0.600031\pi$
$62$	24.3137	3.08784
$63$	8.00000	1.00791
$64$	−9.82843	−1.22855
$65$	0	0
$66$	2.41421	0.297169
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	−3.17157	−0.384610
$69$	8.82843	1.06282
$70$	0	0
$71$	−8.82843	−1.04774	−0.523871	−	0.851798i	$-0.675513\pi$
$71$	−8.82843	−1.04774	−0.523871	+	0.851798i	$0.675513\pi$
$72$	12.4853	1.47140
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	9.65685	1.12259
$75$	0	0
$76$	22.9706	2.63490
$77$	−1.17157	−0.133513
$78$	−22.3137	−2.52653
$79$	−2.41421	−0.271620	−0.135810	−	0.990735i	$-0.543364\pi$
$79$	−2.41421	−0.271620	−0.135810	+	0.990735i	$0.543364\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	−10.8284	−1.19580
$83$	−7.65685	−0.840449	−0.420224	−	0.907420i	$-0.638049\pi$
$83$	−7.65685	−0.840449	−0.420224	+	0.907420i	$0.638049\pi$
$84$	−26.1421	−2.85234
$85$	0	0
$86$	−8.65685	−0.933493
$87$	−2.41421	−0.258831
$88$	−1.82843	−0.194911
$89$	−12.4853	−1.32344	−0.661719	−	0.749752i	$-0.730173\pi$
$89$	−12.4853	−1.32344	−0.661719	+	0.749752i	$0.730173\pi$
$90$	0	0
$91$	10.8284	1.13513
$92$	−14.0000	−1.45960
$93$	−24.3137	−2.52121
$94$	7.82843	0.807441
$95$	0	0
$96$	3.82843	0.390737
$97$	−4.48528	−0.455411	−0.227706	−	0.973730i	$-0.573122\pi$
$97$	−4.48528	−0.455411	−0.227706	+	0.973730i	$0.573122\pi$
$98$	2.41421	0.243872
$99$	−1.17157	−0.117748
$100$	0	0
$101$	−2.34315	−0.233152	−0.116576	−	0.993182i	$-0.537192\pi$
$101$	−2.34315	−0.233152	−0.116576	+	0.993182i	$0.537192\pi$
$102$	4.82843	0.478086
$103$	4.82843	0.475759	0.237880	−	0.971295i	$-0.423548\pi$
$103$	4.82843	0.475759	0.237880	+	0.971295i	$0.423548\pi$
$104$	16.8995	1.65713
$105$	0	0
$106$	−22.8995	−2.22420
$107$	14.8284	1.43352	0.716759	−	0.697321i	$-0.245625\pi$
$107$	14.8284	1.43352	0.716759	+	0.697321i	$0.245625\pi$
$108$	1.58579	0.152592
$109$	12.6569	1.21231	0.606153	−	0.795348i	$-0.292712\pi$
$109$	12.6569	1.21231	0.606153	+	0.795348i	$0.292712\pi$
$110$	0	0
$111$	−9.65685	−0.916588
$112$	8.48528	0.801784
$113$	13.3137	1.25245	0.626224	−	0.779643i	$-0.284599\pi$
$113$	13.3137	1.25245	0.626224	+	0.779643i	$0.284599\pi$
$114$	−34.9706	−3.27529
$115$	0	0
$116$	3.82843	0.355461
$117$	10.8284	1.00109
$118$	−8.82843	−0.812723
$119$	−2.34315	−0.214796
$120$	0	0
$121$	−10.8284	−0.984402
$122$	−11.6569	−1.05536
$123$	10.8284	0.976366
$124$	38.5563	3.46246
$125$	0	0
$126$	19.3137	1.72060
$127$	4.34315	0.385392	0.192696	−	0.981259i	$-0.438277\pi$
$127$	4.34315	0.385392	0.192696	+	0.981259i	$0.438277\pi$
$128$	−20.5563	−1.81694
$129$	8.65685	0.762194
$130$	0	0
$131$	21.3137	1.86219	0.931094	−	0.364780i	$-0.118856\pi$
$131$	21.3137	1.86219	0.931094	+	0.364780i	$0.118856\pi$
$132$	3.82843	0.333222
$133$	16.9706	1.47153
$134$	−13.6569	−1.17977
$135$	0	0
$136$	−3.65685	−0.313573
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	21.3137	1.81434
$139$	14.0000	1.18746	0.593732	−	0.804663i	$-0.297654\pi$
$139$	14.0000	1.18746	0.593732	+	0.804663i	$0.297654\pi$
$140$	0	0
$141$	−7.82843	−0.659272
$142$	−21.3137	−1.78861
$143$	−1.58579	−0.132610
$144$	8.48528	0.707107
$145$	0	0
$146$	−9.65685	−0.799207
$147$	−2.41421	−0.199121
$148$	15.3137	1.25878
$149$	−7.82843	−0.641330	−0.320665	−	0.947193i	$-0.603906\pi$
$149$	−7.82843	−0.641330	−0.320665	+	0.947193i	$0.603906\pi$
$150$	0	0
$151$	−14.1421	−1.15087	−0.575435	−	0.817847i	$-0.695167\pi$
$151$	−14.1421	−1.15087	−0.575435	+	0.817847i	$0.695167\pi$
$152$	26.4853	2.14824
$153$	−2.34315	−0.189432
$154$	−2.82843	−0.227921
$155$	0	0
$156$	−35.3848	−2.83305
$157$	−8.48528	−0.677199	−0.338600	−	0.940931i	$-0.609953\pi$
$157$	−8.48528	−0.677199	−0.338600	+	0.940931i	$0.609953\pi$
$158$	−5.82843	−0.463685
$159$	22.8995	1.81605
$160$	0	0
$161$	−10.3431	−0.815154
$162$	−22.8995	−1.79915
$163$	−3.92893	−0.307738	−0.153869	−	0.988091i	$-0.549173\pi$
$163$	−3.92893	−0.307738	−0.153869	+	0.988091i	$0.549173\pi$
$164$	−17.1716	−1.34087
$165$	0	0
$166$	−18.4853	−1.43474
$167$	3.17157	0.245424	0.122712	−	0.992442i	$-0.460841\pi$
$167$	3.17157	0.245424	0.122712	+	0.992442i	$0.460841\pi$
$168$	−30.1421	−2.32552
$169$	1.65685	0.127450
$170$	0	0
$171$	16.9706	1.29777
$172$	−13.7279	−1.04674
$173$	−12.3431	−0.938432	−0.469216	−	0.883083i	$-0.655463\pi$
$173$	−12.3431	−0.938432	−0.469216	+	0.883083i	$0.655463\pi$
$174$	−5.82843	−0.441852
$175$	0	0
$176$	−1.24264	−0.0936676
$177$	8.82843	0.663585
$178$	−30.1421	−2.25925
$179$	−6.48528	−0.484733	−0.242366	−	0.970185i	$-0.577924\pi$
$179$	−6.48528	−0.484733	−0.242366	+	0.970185i	$0.577924\pi$
$180$	0	0
$181$	8.31371	0.617953	0.308977	−	0.951070i	$-0.400014\pi$
$181$	8.31371	0.617953	0.308977	+	0.951070i	$0.400014\pi$
$182$	26.1421	1.93778
$183$	11.6569	0.861699
$184$	−16.1421	−1.19001
$185$	0	0
$186$	−58.6985	−4.30398
$187$	0.343146	0.0250933
$188$	12.4142	0.905400
$189$	1.17157	0.0852194
$190$	0	0
$191$	25.3137	1.83164	0.915818	−	0.401594i	$-0.131544\pi$
$191$	25.3137	1.83164	0.915818	+	0.401594i	$0.131544\pi$
$192$	23.7279	1.71242
$193$	5.17157	0.372258	0.186129	−	0.982525i	$-0.440406\pi$
$193$	5.17157	0.372258	0.186129	+	0.982525i	$0.440406\pi$
$194$	−10.8284	−0.777436
$195$	0	0
$196$	3.82843	0.273459
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	−2.82843	−0.201008
$199$	−0.485281	−0.0344007	−0.0172003	−	0.999852i	$-0.505475\pi$
$199$	−0.485281	−0.0344007	−0.0172003	+	0.999852i	$0.505475\pi$
$200$	0	0
$201$	13.6569	0.963280
$202$	−5.65685	−0.398015
$203$	2.82843	0.198517
$204$	7.65685	0.536087
$205$	0	0
$206$	11.6569	0.812172
$207$	−10.3431	−0.718898
$208$	11.4853	0.796361
$209$	−2.48528	−0.171911
$210$	0	0
$211$	−19.3848	−1.33450	−0.667252	−	0.744832i	$-0.732529\pi$
$211$	−19.3848	−1.33450	−0.667252	+	0.744832i	$0.732529\pi$
$212$	−36.3137	−2.49404
$213$	21.3137	1.46039
$214$	35.7990	2.44717
$215$	0	0
$216$	1.82843	0.124409
$217$	28.4853	1.93371
$218$	30.5563	2.06954
$219$	9.65685	0.652550
$220$	0	0
$221$	−3.17157	−0.213343
$222$	−23.3137	−1.56471
$223$	3.17157	0.212384	0.106192	−	0.994346i	$-0.466134\pi$
$223$	3.17157	0.212384	0.106192	+	0.994346i	$0.466134\pi$
$224$	−4.48528	−0.299685
$225$	0	0
$226$	32.1421	2.13806
$227$	8.14214	0.540413	0.270206	−	0.962802i	$-0.412908\pi$
$227$	8.14214	0.540413	0.270206	+	0.962802i	$0.412908\pi$
$228$	−55.4558	−3.67265
$229$	−3.51472	−0.232259	−0.116130	−	0.993234i	$-0.537049\pi$
$229$	−3.51472	−0.232259	−0.116130	+	0.993234i	$0.537049\pi$
$230$	0	0
$231$	2.82843	0.186097
$232$	4.41421	0.289807
$233$	−18.3137	−1.19977	−0.599885	−	0.800086i	$-0.704787\pi$
$233$	−18.3137	−1.19977	−0.599885	+	0.800086i	$0.704787\pi$
$234$	26.1421	1.70896
$235$	0	0
$236$	−14.0000	−0.911322
$237$	5.82843	0.378597
$238$	−5.65685	−0.366679
$239$	−19.6569	−1.27150	−0.635748	−	0.771897i	$-0.719308\pi$
$239$	−19.6569	−1.27150	−0.635748	+	0.771897i	$0.719308\pi$
$240$	0	0
$241$	−18.3137	−1.17969	−0.589845	−	0.807517i	$-0.700811\pi$
$241$	−18.3137	−1.17969	−0.589845	+	0.807517i	$0.700811\pi$
$242$	−26.1421	−1.68048
$243$	21.6569	1.38929
$244$	−18.4853	−1.18340
$245$	0	0
$246$	26.1421	1.66676
$247$	22.9706	1.46158
$248$	44.4558	2.82295
$249$	18.4853	1.17146
$250$	0	0
$251$	20.0711	1.26687	0.633437	−	0.773794i	$-0.281643\pi$
$251$	20.0711	1.26687	0.633437	+	0.773794i	$0.281643\pi$
$252$	30.6274	1.92935
$253$	1.51472	0.0952295
$254$	10.4853	0.657905
$255$	0	0
$256$	−29.9706	−1.87316
$257$	18.1716	1.13351	0.566756	−	0.823886i	$-0.308198\pi$
$257$	18.1716	1.13351	0.566756	+	0.823886i	$0.308198\pi$
$258$	20.8995	1.30115
$259$	11.3137	0.703000
$260$	0	0
$261$	2.82843	0.175075
$262$	51.4558	3.17895
$263$	−2.75736	−0.170026	−0.0850130	−	0.996380i	$-0.527093\pi$
$263$	−2.75736	−0.170026	−0.0850130	+	0.996380i	$0.527093\pi$
$264$	4.41421	0.271676
$265$	0	0
$266$	40.9706	2.51207
$267$	30.1421	1.84467
$268$	−21.6569	−1.32290
$269$	31.4558	1.91790	0.958948	−	0.283581i	$-0.0915224\pi$
$269$	31.4558	1.91790	0.958948	+	0.283581i	$0.0915224\pi$
$270$	0	0
$271$	16.5563	1.00573	0.502863	−	0.864366i	$-0.332280\pi$
$271$	16.5563	1.00573	0.502863	+	0.864366i	$0.332280\pi$
$272$	−2.48528	−0.150692
$273$	−26.1421	−1.58219
$274$	−28.9706	−1.75018
$275$	0	0
$276$	33.7990	2.03446
$277$	17.3137	1.04028	0.520140	−	0.854081i	$-0.325880\pi$
$277$	17.3137	1.04028	0.520140	+	0.854081i	$0.325880\pi$
$278$	33.7990	2.02713
$279$	28.4853	1.70537
$280$	0	0
$281$	31.9706	1.90720	0.953602	−	0.301070i	$-0.0973439\pi$
$281$	31.9706	1.90720	0.953602	+	0.301070i	$0.0973439\pi$
$282$	−18.8995	−1.12545
$283$	−11.6569	−0.692928	−0.346464	−	0.938063i	$-0.612618\pi$
$283$	−11.6569	−0.692928	−0.346464	+	0.938063i	$0.612618\pi$
$284$	−33.7990	−2.00560
$285$	0	0
$286$	−3.82843	−0.226380
$287$	−12.6863	−0.748848
$288$	−4.48528	−0.264298
$289$	−16.3137	−0.959630
$290$	0	0
$291$	10.8284	0.634774
$292$	−15.3137	−0.896167
$293$	−7.65685	−0.447318	−0.223659	−	0.974667i	$-0.571800\pi$
$293$	−7.65685	−0.447318	−0.223659	+	0.974667i	$0.571800\pi$
$294$	−5.82843	−0.339921
$295$	0	0
$296$	17.6569	1.02628
$297$	−0.171573	−0.00995567
$298$	−18.8995	−1.09482
$299$	−14.0000	−0.809641
$300$	0	0
$301$	−10.1421	−0.584583
$302$	−34.1421	−1.96466
$303$	5.65685	0.324978
$304$	18.0000	1.03237
$305$	0	0
$306$	−5.65685	−0.323381
$307$	−2.89949	−0.165483	−0.0827415	−	0.996571i	$-0.526368\pi$
$307$	−2.89949	−0.165483	−0.0827415	+	0.996571i	$0.526368\pi$
$308$	−4.48528	−0.255573
$309$	−11.6569	−0.663135
$310$	0	0
$311$	2.68629	0.152326	0.0761628	−	0.997095i	$-0.475733\pi$
$311$	2.68629	0.152326	0.0761628	+	0.997095i	$0.475733\pi$
$312$	−40.7990	−2.30979
$313$	−9.82843	−0.555536	−0.277768	−	0.960648i	$-0.589595\pi$
$313$	−9.82843	−0.555536	−0.277768	+	0.960648i	$0.589595\pi$
$314$	−20.4853	−1.15605
$315$	0	0
$316$	−9.24264	−0.519939
$317$	31.4558	1.76674	0.883368	−	0.468680i	$-0.155270\pi$
$317$	31.4558	1.76674	0.883368	+	0.468680i	$0.155270\pi$
$318$	55.2843	3.10019
$319$	−0.414214	−0.0231915
$320$	0	0
$321$	−35.7990	−1.99810
$322$	−24.9706	−1.39156
$323$	−4.97056	−0.276570
$324$	−36.3137	−2.01743
$325$	0	0
$326$	−9.48528	−0.525341
$327$	−30.5563	−1.68977
$328$	−19.7990	−1.09322
$329$	9.17157	0.505645
$330$	0	0
$331$	−2.41421	−0.132697	−0.0663486	−	0.997797i	$-0.521135\pi$
$331$	−2.41421	−0.132697	−0.0663486	+	0.997797i	$0.521135\pi$
$332$	−29.3137	−1.60880
$333$	11.3137	0.619987
$334$	7.65685	0.418964
$335$	0	0
$336$	−20.4853	−1.11756
$337$	−21.7990	−1.18747	−0.593733	−	0.804662i	$-0.702347\pi$
$337$	−21.7990	−1.18747	−0.593733	+	0.804662i	$0.702347\pi$
$338$	4.00000	0.217571
$339$	−32.1421	−1.74572
$340$	0	0
$341$	−4.17157	−0.225903
$342$	40.9706	2.21543
$343$	−16.9706	−0.916324
$344$	−15.8284	−0.853412
$345$	0	0
$346$	−29.7990	−1.60200
$347$	−2.48528	−0.133417	−0.0667084	−	0.997773i	$-0.521250\pi$
$347$	−2.48528	−0.133417	−0.0667084	+	0.997773i	$0.521250\pi$
$348$	−9.24264	−0.495458
$349$	−5.14214	−0.275252	−0.137626	−	0.990484i	$-0.543947\pi$
$349$	−5.14214	−0.275252	−0.137626	+	0.990484i	$0.543947\pi$
$350$	0	0
$351$	1.58579	0.0846430
$352$	0.656854	0.0350104
$353$	−26.9706	−1.43550	−0.717749	−	0.696302i	$-0.754828\pi$
$353$	−26.9706	−1.43550	−0.717749	+	0.696302i	$0.754828\pi$
$354$	21.3137	1.13281
$355$	0	0
$356$	−47.7990	−2.53334
$357$	5.65685	0.299392
$358$	−15.6569	−0.827490
$359$	3.92893	0.207361	0.103681	−	0.994611i	$-0.466938\pi$
$359$	3.92893	0.207361	0.103681	+	0.994611i	$0.466938\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	20.0711	1.05491
$363$	26.1421	1.37211
$364$	41.4558	2.17288
$365$	0	0
$366$	28.1421	1.47101
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	−10.9706	−0.571880
$369$	−12.6863	−0.660422
$370$	0	0
$371$	−26.8284	−1.39286
$372$	−93.0833	−4.82614
$373$	26.3137	1.36247	0.681236	−	0.732064i	$-0.261443\pi$
$373$	26.3137	1.36247	0.681236	+	0.732064i	$0.261443\pi$
$374$	0.828427	0.0428369
$375$	0	0
$376$	14.3137	0.738173
$377$	3.82843	0.197174
$378$	2.82843	0.145479
$379$	−6.97056	−0.358054	−0.179027	−	0.983844i	$-0.557295\pi$
$379$	−6.97056	−0.358054	−0.179027	+	0.983844i	$0.557295\pi$
$380$	0	0
$381$	−10.4853	−0.537177
$382$	61.1127	3.12680
$383$	3.51472	0.179594	0.0897969	−	0.995960i	$-0.471378\pi$
$383$	3.51472	0.179594	0.0897969	+	0.995960i	$0.471378\pi$
$384$	49.6274	2.53254
$385$	0	0
$386$	12.4853	0.635484
$387$	−10.1421	−0.515554
$388$	−17.1716	−0.871755
$389$	3.02944	0.153599	0.0767993	−	0.997047i	$-0.475530\pi$
$389$	3.02944	0.153599	0.0767993	+	0.997047i	$0.475530\pi$
$390$	0	0
$391$	3.02944	0.153205
$392$	4.41421	0.222951
$393$	−51.4558	−2.59560
$394$	−4.82843	−0.243253
$395$	0	0
$396$	−4.48528	−0.225394
$397$	−19.3431	−0.970805	−0.485402	−	0.874291i	$-0.661327\pi$
$397$	−19.3431	−0.970805	−0.485402	+	0.874291i	$0.661327\pi$
$398$	−1.17157	−0.0587256
$399$	−40.9706	−2.05109
$400$	0	0
$401$	−18.6569	−0.931679	−0.465839	−	0.884869i	$-0.654248\pi$
$401$	−18.6569	−0.931679	−0.465839	+	0.884869i	$0.654248\pi$
$402$	32.9706	1.64442
$403$	38.5563	1.92063
$404$	−8.97056	−0.446302
$405$	0	0
$406$	6.82843	0.338889
$407$	−1.65685	−0.0821272
$408$	8.82843	0.437072
$409$	−18.9706	−0.938034	−0.469017	−	0.883189i	$-0.655392\pi$
$409$	−18.9706	−0.938034	−0.469017	+	0.883189i	$0.655392\pi$
$410$	0	0
$411$	28.9706	1.42901
$412$	18.4853	0.910704
$413$	−10.3431	−0.508953
$414$	−24.9706	−1.22724
$415$	0	0
$416$	−6.07107	−0.297659
$417$	−33.7990	−1.65514
$418$	−6.00000	−0.293470
$419$	−9.51472	−0.464824	−0.232412	−	0.972617i	$-0.574662\pi$
$419$	−9.51472	−0.464824	−0.232412	+	0.972617i	$0.574662\pi$
$420$	0	0
$421$	37.1127	1.80876	0.904381	−	0.426726i	$-0.140333\pi$
$421$	37.1127	1.80876	0.904381	+	0.426726i	$0.140333\pi$
$422$	−46.7990	−2.27814
$423$	9.17157	0.445937
$424$	−41.8701	−2.03339
$425$	0	0
$426$	51.4558	2.49304
$427$	−13.6569	−0.660901
$428$	56.7696	2.74406
$429$	3.82843	0.184838
$430$	0	0
$431$	19.6569	0.946837	0.473419	−	0.880838i	$-0.343020\pi$
$431$	19.6569	0.946837	0.473419	+	0.880838i	$0.343020\pi$
$432$	1.24264	0.0597866
$433$	−30.6274	−1.47186	−0.735930	−	0.677058i	$-0.763255\pi$
$433$	−30.6274	−1.47186	−0.735930	+	0.677058i	$0.763255\pi$
$434$	68.7696	3.30104
$435$	0	0
$436$	48.4558	2.32061
$437$	−21.9411	−1.04959
$438$	23.3137	1.11397
$439$	−0.343146	−0.0163775	−0.00818873	−	0.999966i	$-0.502607\pi$
$439$	−0.343146	−0.0163775	−0.00818873	+	0.999966i	$0.502607\pi$
$440$	0	0
$441$	2.82843	0.134687
$442$	−7.65685	−0.364199
$443$	24.3431	1.15658	0.578289	−	0.815832i	$-0.303721\pi$
$443$	24.3431	1.15658	0.578289	+	0.815832i	$0.303721\pi$
$444$	−36.9706	−1.75455
$445$	0	0
$446$	7.65685	0.362563
$447$	18.8995	0.893915
$448$	−27.7990	−1.31338
$449$	−34.9706	−1.65036	−0.825181	−	0.564868i	$-0.808927\pi$
$449$	−34.9706	−1.65036	−0.825181	+	0.564868i	$0.808927\pi$
$450$	0	0
$451$	1.85786	0.0874834
$452$	50.9706	2.39745
$453$	34.1421	1.60414
$454$	19.6569	0.922542
$455$	0	0
$456$	−63.9411	−2.99432
$457$	−1.02944	−0.0481550	−0.0240775	−	0.999710i	$-0.507665\pi$
$457$	−1.02944	−0.0481550	−0.0240775	+	0.999710i	$0.507665\pi$
$458$	−8.48528	−0.396491
$459$	−0.343146	−0.0160167
$460$	0	0
$461$	14.0000	0.652045	0.326023	−	0.945362i	$-0.394291\pi$
$461$	14.0000	0.652045	0.326023	+	0.945362i	$0.394291\pi$
$462$	6.82843	0.317687
$463$	26.0000	1.20832	0.604161	−	0.796862i	$-0.293508\pi$
$463$	26.0000	1.20832	0.604161	+	0.796862i	$0.293508\pi$
$464$	3.00000	0.139272
$465$	0	0
$466$	−44.2132	−2.04814
$467$	38.3553	1.77487	0.887437	−	0.460930i	$-0.152484\pi$
$467$	38.3553	1.77487	0.887437	+	0.460930i	$0.152484\pi$
$468$	41.4558	1.91630
$469$	−16.0000	−0.738811
$470$	0	0
$471$	20.4853	0.943912
$472$	−16.1421	−0.743002
$473$	1.48528	0.0682933
$474$	14.0711	0.646306
$475$	0	0
$476$	−8.97056	−0.411165
$477$	−26.8284	−1.22839
$478$	−47.4558	−2.17058
$479$	6.89949	0.315246	0.157623	−	0.987499i	$-0.449617\pi$
$479$	6.89949	0.315246	0.157623	+	0.987499i	$0.449617\pi$
$480$	0	0
$481$	15.3137	0.698245
$482$	−44.2132	−2.01386
$483$	24.9706	1.13620
$484$	−41.4558	−1.88436
$485$	0	0
$486$	52.2843	2.37166
$487$	11.5147	0.521782	0.260891	−	0.965368i	$-0.415984\pi$
$487$	11.5147	0.521782	0.260891	+	0.965368i	$0.415984\pi$
$488$	−21.3137	−0.964826
$489$	9.48528	0.428939
$490$	0	0
$491$	−21.2426	−0.958667	−0.479333	−	0.877633i	$-0.659122\pi$
$491$	−21.2426	−0.958667	−0.479333	+	0.877633i	$0.659122\pi$
$492$	41.4558	1.86897
$493$	−0.828427	−0.0373105
$494$	55.4558	2.49508
$495$	0	0
$496$	30.2132	1.35661
$497$	−24.9706	−1.12008
$498$	44.6274	1.99980
$499$	18.9706	0.849239	0.424620	−	0.905372i	$-0.360408\pi$
$499$	18.9706	0.849239	0.424620	+	0.905372i	$0.360408\pi$
$500$	0	0
$501$	−7.65685	−0.342083
$502$	48.4558	2.16269
$503$	−0.272078	−0.0121314	−0.00606568	−	0.999982i	$-0.501931\pi$
$503$	−0.272078	−0.0121314	−0.00606568	+	0.999982i	$0.501931\pi$
$504$	35.3137	1.57300
$505$	0	0
$506$	3.65685	0.162567
$507$	−4.00000	−0.177646
$508$	16.6274	0.737722
$509$	−10.5147	−0.466057	−0.233028	−	0.972470i	$-0.574863\pi$
$509$	−10.5147	−0.466057	−0.233028	+	0.972470i	$0.574863\pi$
$510$	0	0
$511$	−11.3137	−0.500489
$512$	−31.2426	−1.38074
$513$	2.48528	0.109728
$514$	43.8701	1.93503
$515$	0	0
$516$	33.1421	1.45900
$517$	−1.34315	−0.0590715
$518$	27.3137	1.20010
$519$	29.7990	1.30803
$520$	0	0
$521$	−29.1421	−1.27674	−0.638370	−	0.769730i	$-0.720391\pi$
$521$	−29.1421	−1.27674	−0.638370	+	0.769730i	$0.720391\pi$
$522$	6.82843	0.298872
$523$	−4.68629	−0.204917	−0.102459	−	0.994737i	$-0.532671\pi$
$523$	−4.68629	−0.204917	−0.102459	+	0.994737i	$0.532671\pi$
$524$	81.5980	3.56462
$525$	0	0
$526$	−6.65685	−0.290253
$527$	−8.34315	−0.363433
$528$	3.00000	0.130558
$529$	−9.62742	−0.418583
$530$	0	0
$531$	−10.3431	−0.448854
$532$	64.9706	2.81683
$533$	−17.1716	−0.743783
$534$	72.7696	3.14905
$535$	0	0
$536$	−24.9706	−1.07856
$537$	15.6569	0.675643
$538$	75.9411	3.27405
$539$	−0.414214	−0.0178414
$540$	0	0
$541$	−10.3431	−0.444687	−0.222343	−	0.974968i	$-0.571371\pi$
$541$	−10.3431	−0.444687	−0.222343	+	0.974968i	$0.571371\pi$
$542$	39.9706	1.71688
$543$	−20.0711	−0.861332
$544$	1.31371	0.0563248
$545$	0	0
$546$	−63.1127	−2.70097
$547$	−35.7990	−1.53065	−0.765327	−	0.643641i	$-0.777423\pi$
$547$	−35.7990	−1.53065	−0.765327	+	0.643641i	$0.777423\pi$
$548$	−45.9411	−1.96251
$549$	−13.6569	−0.582860
$550$	0	0
$551$	6.00000	0.255609
$552$	38.9706	1.65870
$553$	−6.82843	−0.290374
$554$	41.7990	1.77587
$555$	0	0
$556$	53.5980	2.27306
$557$	17.3137	0.733605	0.366803	−	0.930299i	$-0.380452\pi$
$557$	17.3137	0.733605	0.366803	+	0.930299i	$0.380452\pi$
$558$	68.7696	2.91125
$559$	−13.7279	−0.580629
$560$	0	0
$561$	−0.828427	−0.0349762
$562$	77.1838	3.25580
$563$	0.757359	0.0319189	0.0159594	−	0.999873i	$-0.494920\pi$
$563$	0.757359	0.0319189	0.0159594	+	0.999873i	$0.494920\pi$
$564$	−29.9706	−1.26199
$565$	0	0
$566$	−28.1421	−1.18290
$567$	−26.8284	−1.12669
$568$	−38.9706	−1.63517
$569$	−39.6569	−1.66250	−0.831251	−	0.555897i	$-0.812375\pi$
$569$	−39.6569	−1.66250	−0.831251	+	0.555897i	$0.812375\pi$
$570$	0	0
$571$	14.6274	0.612138	0.306069	−	0.952009i	$-0.400986\pi$
$571$	14.6274	0.612138	0.306069	+	0.952009i	$0.400986\pi$
$572$	−6.07107	−0.253844
$573$	−61.1127	−2.55302
$574$	−30.6274	−1.27836
$575$	0	0
$576$	−27.7990	−1.15829
$577$	29.7990	1.24055	0.620274	−	0.784385i	$-0.287021\pi$
$577$	29.7990	1.24055	0.620274	+	0.784385i	$0.287021\pi$
$578$	−39.3848	−1.63819
$579$	−12.4853	−0.518871
$580$	0	0
$581$	−21.6569	−0.898478
$582$	26.1421	1.08363
$583$	3.92893	0.162720
$584$	−17.6569	−0.730646
$585$	0	0
$586$	−18.4853	−0.763620
$587$	−7.65685	−0.316032	−0.158016	−	0.987437i	$-0.550510\pi$
$587$	−7.65685	−0.316032	−0.158016	+	0.987437i	$0.550510\pi$
$588$	−9.24264	−0.381160
$589$	60.4264	2.48983
$590$	0	0
$591$	4.82843	0.198615
$592$	12.0000	0.493197
$593$	19.4853	0.800165	0.400082	−	0.916479i	$-0.368982\pi$
$593$	19.4853	0.800165	0.400082	+	0.916479i	$0.368982\pi$
$594$	−0.414214	−0.0169954
$595$	0	0
$596$	−29.9706	−1.22764
$597$	1.17157	0.0479493
$598$	−33.7990	−1.38214
$599$	9.87006	0.403280	0.201640	−	0.979460i	$-0.435373\pi$
$599$	9.87006	0.403280	0.201640	+	0.979460i	$0.435373\pi$
$600$	0	0
$601$	−17.1716	−0.700443	−0.350222	−	0.936667i	$-0.613894\pi$
$601$	−17.1716	−0.700443	−0.350222	+	0.936667i	$0.613894\pi$
$602$	−24.4853	−0.997946
$603$	−16.0000	−0.651570
$604$	−54.1421	−2.20301
$605$	0	0
$606$	13.6569	0.554772
$607$	7.72792	0.313667	0.156833	−	0.987625i	$-0.449871\pi$
$607$	7.72792	0.313667	0.156833	+	0.987625i	$0.449871\pi$
$608$	−9.51472	−0.385873
$609$	−6.82843	−0.276702
$610$	0	0
$611$	12.4142	0.502225
$612$	−8.97056	−0.362614
$613$	9.00000	0.363507	0.181753	−	0.983344i	$-0.441823\pi$
$613$	9.00000	0.363507	0.181753	+	0.983344i	$0.441823\pi$
$614$	−7.00000	−0.282497
$615$	0	0
$616$	−5.17157	−0.208369
$617$	−0.686292	−0.0276291	−0.0138145	−	0.999905i	$-0.504397\pi$
$617$	−0.686292	−0.0276291	−0.0138145	+	0.999905i	$0.504397\pi$
$618$	−28.1421	−1.13204
$619$	33.5858	1.34993	0.674963	−	0.737851i	$-0.264159\pi$
$619$	33.5858	1.34993	0.674963	+	0.737851i	$0.264159\pi$
$620$	0	0
$621$	−1.51472	−0.0607836
$622$	6.48528	0.260036
$623$	−35.3137	−1.41481
$624$	−27.7279	−1.11001
$625$	0	0
$626$	−23.7279	−0.948358
$627$	6.00000	0.239617
$628$	−32.4853	−1.29630
$629$	−3.31371	−0.132126
$630$	0	0
$631$	−36.8284	−1.46612	−0.733058	−	0.680166i	$-0.761908\pi$
$631$	−36.8284	−1.46612	−0.733058	+	0.680166i	$0.761908\pi$
$632$	−10.6569	−0.423907
$633$	46.7990	1.86009
$634$	75.9411	3.01601
$635$	0	0
$636$	87.6690	3.47630
$637$	3.82843	0.151688
$638$	−1.00000	−0.0395904
$639$	−24.9706	−0.987820
$640$	0	0
$641$	17.7990	0.703018	0.351509	−	0.936185i	$-0.385669\pi$
$641$	17.7990	0.703018	0.351509	+	0.936185i	$0.385669\pi$
$642$	−86.4264	−3.41098
$643$	−32.4853	−1.28109	−0.640547	−	0.767919i	$-0.721292\pi$
$643$	−32.4853	−1.28109	−0.640547	+	0.767919i	$0.721292\pi$
$644$	−39.5980	−1.56038
$645$	0	0
$646$	−12.0000	−0.472134
$647$	−39.6569	−1.55907	−0.779536	−	0.626358i	$-0.784545\pi$
$647$	−39.6569	−1.55907	−0.779536	+	0.626358i	$0.784545\pi$
$648$	−41.8701	−1.64481
$649$	1.51472	0.0594579
$650$	0	0
$651$	−68.7696	−2.69529
$652$	−15.0416	−0.589076
$653$	30.1421	1.17955	0.589776	−	0.807567i	$-0.299216\pi$
$653$	30.1421	1.17955	0.589776	+	0.807567i	$0.299216\pi$
$654$	−73.7696	−2.88462
$655$	0	0
$656$	−13.4558	−0.525362
$657$	−11.3137	−0.441390
$658$	22.1421	0.863190
$659$	14.4142	0.561498	0.280749	−	0.959781i	$-0.409417\pi$
$659$	14.4142	0.561498	0.280749	+	0.959781i	$0.409417\pi$
$660$	0	0
$661$	33.3137	1.29575	0.647877	−	0.761745i	$-0.275657\pi$
$661$	33.3137	1.29575	0.647877	+	0.761745i	$0.275657\pi$
$662$	−5.82843	−0.226528
$663$	7.65685	0.297368
$664$	−33.7990	−1.31166
$665$	0	0
$666$	27.3137	1.05838
$667$	−3.65685	−0.141594
$668$	12.1421	0.469793
$669$	−7.65685	−0.296031
$670$	0	0
$671$	2.00000	0.0772091
$672$	10.8284	0.417716
$673$	21.6274	0.833676	0.416838	−	0.908981i	$-0.363138\pi$
$673$	21.6274	0.833676	0.416838	+	0.908981i	$0.363138\pi$
$674$	−52.6274	−2.02713
$675$	0	0
$676$	6.34315	0.243967
$677$	22.0000	0.845529	0.422764	−	0.906240i	$-0.361060\pi$
$677$	22.0000	0.845529	0.422764	+	0.906240i	$0.361060\pi$
$678$	−77.5980	−2.98013
$679$	−12.6863	−0.486855
$680$	0	0
$681$	−19.6569	−0.753252
$682$	−10.0711	−0.385641
$683$	−20.9706	−0.802416	−0.401208	−	0.915987i	$-0.631410\pi$
$683$	−20.9706	−0.802416	−0.401208	+	0.915987i	$0.631410\pi$
$684$	64.9706	2.48421
$685$	0	0
$686$	−40.9706	−1.56426
$687$	8.48528	0.323734
$688$	−10.7574	−0.410120
$689$	−36.3137	−1.38344
$690$	0	0
$691$	48.0000	1.82601	0.913003	−	0.407953i	$-0.133757\pi$
$691$	48.0000	1.82601	0.913003	+	0.407953i	$0.133757\pi$
$692$	−47.2548	−1.79636
$693$	−3.31371	−0.125877
$694$	−6.00000	−0.227757
$695$	0	0
$696$	−10.6569	−0.403947
$697$	3.71573	0.140743
$698$	−12.4142	−0.469885
$699$	44.2132	1.67230
$700$	0	0
$701$	−40.1127	−1.51504	−0.757518	−	0.652814i	$-0.773588\pi$
$701$	−40.1127	−1.51504	−0.757518	+	0.652814i	$0.773588\pi$
$702$	3.82843	0.144495
$703$	24.0000	0.905177
$704$	4.07107	0.153434
$705$	0	0
$706$	−65.1127	−2.45055
$707$	−6.62742	−0.249250
$708$	33.7990	1.27024
$709$	29.1421	1.09446	0.547228	−	0.836984i	$-0.315683\pi$
$709$	29.1421	1.09446	0.547228	+	0.836984i	$0.315683\pi$
$710$	0	0
$711$	−6.82843	−0.256086
$712$	−55.1127	−2.06544
$713$	−36.8284	−1.37924
$714$	13.6569	0.511095
$715$	0	0
$716$	−24.8284	−0.927882
$717$	47.4558	1.77227
$718$	9.48528	0.353988
$719$	−20.1421	−0.751175	−0.375587	−	0.926787i	$-0.622559\pi$
$719$	−20.1421	−0.751175	−0.375587	+	0.926787i	$0.622559\pi$
$720$	0	0
$721$	13.6569	0.508608
$722$	41.0416	1.52741
$723$	44.2132	1.64431
$724$	31.8284	1.18289
$725$	0	0
$726$	63.1127	2.34233
$727$	−1.31371	−0.0487228	−0.0243614	−	0.999703i	$-0.507755\pi$
$727$	−1.31371	−0.0487228	−0.0243614	+	0.999703i	$0.507755\pi$
$728$	47.7990	1.77155
$729$	−23.8284	−0.882534
$730$	0	0
$731$	2.97056	0.109870
$732$	44.6274	1.64948
$733$	41.2548	1.52378	0.761891	−	0.647705i	$-0.224271\pi$
$733$	41.2548	1.52378	0.761891	+	0.647705i	$0.224271\pi$
$734$	−43.4558	−1.60398
$735$	0	0
$736$	5.79899	0.213754
$737$	2.34315	0.0863109
$738$	−30.6274	−1.12741
$739$	4.07107	0.149757	0.0748783	−	0.997193i	$-0.476143\pi$
$739$	4.07107	0.149757	0.0748783	+	0.997193i	$0.476143\pi$
$740$	0	0
$741$	−55.4558	−2.03722
$742$	−64.7696	−2.37777
$743$	−23.6569	−0.867886	−0.433943	−	0.900940i	$-0.642878\pi$
$743$	−23.6569	−0.867886	−0.433943	+	0.900940i	$0.642878\pi$
$744$	−107.326	−3.93476
$745$	0	0
$746$	63.5269	2.32589
$747$	−21.6569	−0.792383
$748$	1.31371	0.0480339
$749$	41.9411	1.53250
$750$	0	0
$751$	25.3137	0.923710	0.461855	−	0.886955i	$-0.347184\pi$
$751$	25.3137	0.923710	0.461855	+	0.886955i	$0.347184\pi$
$752$	9.72792	0.354741
$753$	−48.4558	−1.76583
$754$	9.24264	0.336597
$755$	0	0
$756$	4.48528	0.163128
$757$	−25.5147	−0.927348	−0.463674	−	0.886006i	$-0.653469\pi$
$757$	−25.5147	−0.927348	−0.463674	+	0.886006i	$0.653469\pi$
$758$	−16.8284	−0.611236
$759$	−3.65685	−0.132735
$760$	0	0
$761$	45.5980	1.65293	0.826463	−	0.562991i	$-0.190350\pi$
$761$	45.5980	1.65293	0.826463	+	0.562991i	$0.190350\pi$
$762$	−25.3137	−0.917019
$763$	35.7990	1.29601
$764$	96.9117	3.50614
$765$	0	0
$766$	8.48528	0.306586
$767$	−14.0000	−0.505511
$768$	72.3553	2.61090
$769$	−49.1127	−1.77105	−0.885525	−	0.464592i	$-0.846201\pi$
$769$	−49.1127	−1.77105	−0.885525	+	0.464592i	$0.846201\pi$
$770$	0	0
$771$	−43.8701	−1.57994
$772$	19.7990	0.712581
$773$	19.5147	0.701896	0.350948	−	0.936395i	$-0.385859\pi$
$773$	19.5147	0.701896	0.350948	+	0.936395i	$0.385859\pi$
$774$	−24.4853	−0.880105
$775$	0	0
$776$	−19.7990	−0.710742
$777$	−27.3137	−0.979874
$778$	7.31371	0.262209
$779$	−26.9117	−0.964211
$780$	0	0
$781$	3.65685	0.130853
$782$	7.31371	0.261538
$783$	0.414214	0.0148028
$784$	3.00000	0.107143
$785$	0	0
$786$	−124.225	−4.43097
$787$	54.0833	1.92786	0.963930	−	0.266156i	$-0.0857536\pi$
$787$	54.0833	1.92786	0.963930	+	0.266156i	$0.0857536\pi$
$788$	−7.65685	−0.272764
$789$	6.65685	0.236990
$790$	0	0
$791$	37.6569	1.33892
$792$	−5.17157	−0.183764
$793$	−18.4853	−0.656432
$794$	−46.6985	−1.65727
$795$	0	0
$796$	−1.85786	−0.0658503
$797$	−51.7401	−1.83273	−0.916364	−	0.400345i	$-0.868890\pi$
$797$	−51.7401	−1.83273	−0.916364	+	0.400345i	$0.868890\pi$
$798$	−98.9117	−3.50144
$799$	−2.68629	−0.0950342
$800$	0	0
$801$	−35.3137	−1.24775
$802$	−45.0416	−1.59048
$803$	1.65685	0.0584691
$804$	52.2843	1.84392
$805$	0	0
$806$	93.0833	3.27872
$807$	−75.9411	−2.67325
$808$	−10.3431	−0.363871
$809$	36.2843	1.27569	0.637844	−	0.770166i	$-0.279827\pi$
$809$	36.2843	1.27569	0.637844	+	0.770166i	$0.279827\pi$
$810$	0	0
$811$	10.8284	0.380238	0.190119	−	0.981761i	$-0.439113\pi$
$811$	10.8284	0.380238	0.190119	+	0.981761i	$0.439113\pi$
$812$	10.8284	0.380003
$813$	−39.9706	−1.40183
$814$	−4.00000	−0.140200
$815$	0	0
$816$	6.00000	0.210042
$817$	−21.5147	−0.752705
$818$	−45.7990	−1.60132
$819$	30.6274	1.07021
$820$	0	0
$821$	−1.48528	−0.0518367	−0.0259183	−	0.999664i	$-0.508251\pi$
$821$	−1.48528	−0.0518367	−0.0259183	+	0.999664i	$0.508251\pi$
$822$	69.9411	2.43948
$823$	54.2843	1.89223	0.946115	−	0.323830i	$-0.104971\pi$
$823$	54.2843	1.89223	0.946115	+	0.323830i	$0.104971\pi$
$824$	21.3137	0.742498
$825$	0	0
$826$	−24.9706	−0.868837
$827$	−32.8995	−1.14403	−0.572014	−	0.820244i	$-0.693838\pi$
$827$	−32.8995	−1.14403	−0.572014	+	0.820244i	$0.693838\pi$
$828$	−39.5980	−1.37612
$829$	−29.7990	−1.03496	−0.517481	−	0.855695i	$-0.673130\pi$
$829$	−29.7990	−1.03496	−0.517481	+	0.855695i	$0.673130\pi$
$830$	0	0
$831$	−41.7990	−1.44999
$832$	−37.6274	−1.30450
$833$	−0.828427	−0.0287033
$834$	−81.5980	−2.82551
$835$	0	0
$836$	−9.51472	−0.329073
$837$	4.17157	0.144191
$838$	−22.9706	−0.793505
$839$	−7.92893	−0.273737	−0.136869	−	0.990589i	$-0.543704\pi$
$839$	−7.92893	−0.273737	−0.136869	+	0.990589i	$0.543704\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	89.5980	3.08775
$843$	−77.1838	−2.65835
$844$	−74.2132	−2.55452
$845$	0	0
$846$	22.1421	0.761262
$847$	−30.6274	−1.05237
$848$	−28.4558	−0.977178
$849$	28.1421	0.965836
$850$	0	0
$851$	−14.6274	−0.501421
$852$	81.5980	2.79550
$853$	22.9706	0.786497	0.393249	−	0.919432i	$-0.371351\pi$
$853$	22.9706	0.786497	0.393249	+	0.919432i	$0.371351\pi$
$854$	−32.9706	−1.12823
$855$	0	0
$856$	65.4558	2.23723
$857$	6.17157	0.210817	0.105408	−	0.994429i	$-0.466385\pi$
$857$	6.17157	0.210817	0.105408	+	0.994429i	$0.466385\pi$
$858$	9.24264	0.315539
$859$	19.7279	0.673108	0.336554	−	0.941664i	$-0.390739\pi$
$859$	19.7279	0.673108	0.336554	+	0.941664i	$0.390739\pi$
$860$	0	0
$861$	30.6274	1.04378
$862$	47.4558	1.61635
$863$	−17.1127	−0.582523	−0.291262	−	0.956643i	$-0.594075\pi$
$863$	−17.1127	−0.582523	−0.291262	+	0.956643i	$0.594075\pi$
$864$	−0.656854	−0.0223466
$865$	0	0
$866$	−73.9411	−2.51262
$867$	39.3848	1.33758
$868$	109.054	3.70153
$869$	1.00000	0.0339227
$870$	0	0
$871$	−21.6569	−0.733815
$872$	55.8701	1.89200
$873$	−12.6863	−0.429366
$874$	−52.9706	−1.79176
$875$	0	0
$876$	36.9706	1.24912
$877$	37.1421	1.25420	0.627100	−	0.778938i	$-0.284242\pi$
$877$	37.1421	1.25420	0.627100	+	0.778938i	$0.284242\pi$
$878$	−0.828427	−0.0279581
$879$	18.4853	0.623493
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	6.82843	0.229925
$883$	−38.4264	−1.29315	−0.646576	−	0.762850i	$-0.723800\pi$
$883$	−38.4264	−1.29315	−0.646576	+	0.762850i	$0.723800\pi$
$884$	−12.1421	−0.408384
$885$	0	0
$886$	58.7696	1.97440
$887$	−17.1005	−0.574179	−0.287089	−	0.957904i	$-0.592688\pi$
$887$	−17.1005	−0.574179	−0.287089	+	0.957904i	$0.592688\pi$
$888$	−42.6274	−1.43048
$889$	12.2843	0.412001
$890$	0	0
$891$	3.92893	0.131624
$892$	12.1421	0.406549
$893$	19.4558	0.651065
$894$	45.6274	1.52601
$895$	0	0
$896$	−58.1421	−1.94239
$897$	33.7990	1.12852
$898$	−84.4264	−2.81735
$899$	10.0711	0.335889
$900$	0	0
$901$	7.85786	0.261783
$902$	4.48528	0.149344
$903$	24.4853	0.814819
$904$	58.7696	1.95465
$905$	0	0
$906$	82.4264	2.73843
$907$	−22.2843	−0.739937	−0.369969	−	0.929044i	$-0.620632\pi$
$907$	−22.2843	−0.739937	−0.369969	+	0.929044i	$0.620632\pi$
$908$	31.1716	1.03446
$909$	−6.62742	−0.219818
$910$	0	0
$911$	−15.4437	−0.511671	−0.255835	−	0.966720i	$-0.582351\pi$
$911$	−15.4437	−0.511671	−0.255835	+	0.966720i	$0.582351\pi$
$912$	−43.4558	−1.43897
$913$	3.17157	0.104964
$914$	−2.48528	−0.0822058
$915$	0	0
$916$	−13.4558	−0.444594
$917$	60.2843	1.99076
$918$	−0.828427	−0.0273422
$919$	8.14214	0.268584	0.134292	−	0.990942i	$-0.457124\pi$
$919$	8.14214	0.268584	0.134292	+	0.990942i	$0.457124\pi$
$920$	0	0
$921$	7.00000	0.230658
$922$	33.7990	1.11311
$923$	−33.7990	−1.11251
$924$	10.8284	0.356229
$925$	0	0
$926$	62.7696	2.06274
$927$	13.6569	0.448550
$928$	−1.58579	−0.0520560
$929$	18.6863	0.613077	0.306539	−	0.951858i	$-0.400829\pi$
$929$	18.6863	0.613077	0.306539	+	0.951858i	$0.400829\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	−70.1127	−2.29662
$933$	−6.48528	−0.212319
$934$	92.5980	3.02990
$935$	0	0
$936$	47.7990	1.56236
$937$	16.6274	0.543194	0.271597	−	0.962411i	$-0.412448\pi$
$937$	16.6274	0.543194	0.271597	+	0.962411i	$0.412448\pi$
$938$	−38.6274	−1.26123
$939$	23.7279	0.774331
$940$	0	0
$941$	−56.5980	−1.84504	−0.922521	−	0.385948i	$-0.873875\pi$
$941$	−56.5980	−1.84504	−0.922521	+	0.385948i	$0.873875\pi$
$942$	49.4558	1.61136
$943$	16.4020	0.534123
$944$	−10.9706	−0.357061
$945$	0	0
$946$	3.58579	0.116584
$947$	−2.61522	−0.0849834	−0.0424917	−	0.999097i	$-0.513530\pi$
$947$	−2.61522	−0.0849834	−0.0424917	+	0.999097i	$0.513530\pi$
$948$	22.3137	0.724716
$949$	−15.3137	−0.497104
$950$	0	0
$951$	−75.9411	−2.46256
$952$	−10.3431	−0.335223
$953$	35.6274	1.15409	0.577043	−	0.816714i	$-0.304207\pi$
$953$	35.6274	1.15409	0.577043	+	0.816714i	$0.304207\pi$
$954$	−64.7696	−2.09699
$955$	0	0
$956$	−75.2548	−2.43392
$957$	1.00000	0.0323254
$958$	16.6569	0.538159
$959$	−33.9411	−1.09602
$960$	0	0
$961$	70.4264	2.27182
$962$	36.9706	1.19198
$963$	41.9411	1.35153
$964$	−70.1127	−2.25818
$965$	0	0
$966$	60.2843	1.93961
$967$	35.2426	1.13333	0.566663	−	0.823949i	$-0.308234\pi$
$967$	35.2426	1.13333	0.566663	+	0.823949i	$0.308234\pi$
$968$	−47.7990	−1.53632
$969$	12.0000	0.385496
$970$	0	0
$971$	−15.6569	−0.502452	−0.251226	−	0.967928i	$-0.580834\pi$
$971$	−15.6569	−0.502452	−0.251226	+	0.967928i	$0.580834\pi$
$972$	82.9117	2.65939
$973$	39.5980	1.26945
$974$	27.7990	0.890737
$975$	0	0
$976$	−14.4853	−0.463663
$977$	−36.1716	−1.15723	−0.578616	−	0.815600i	$-0.696407\pi$
$977$	−36.1716	−1.15723	−0.578616	+	0.815600i	$0.696407\pi$
$978$	22.8995	0.732245
$979$	5.17157	0.165284
$980$	0	0
$981$	35.7990	1.14297
$982$	−51.2843	−1.63655
$983$	21.8701	0.697547	0.348773	−	0.937207i	$-0.386598\pi$
$983$	21.8701	0.697547	0.348773	+	0.937207i	$0.386598\pi$
$984$	47.7990	1.52378
$985$	0	0
$986$	−2.00000	−0.0636930
$987$	−22.1421	−0.704792
$988$	87.9411	2.79778
$989$	13.1127	0.416960
$990$	0	0
$991$	−12.8284	−0.407508	−0.203754	−	0.979022i	$-0.565314\pi$
$991$	−12.8284	−0.407508	−0.203754	+	0.979022i	$0.565314\pi$
$992$	−15.9706	−0.507066
$993$	5.82843	0.184960
$994$	−60.2843	−1.91210
$995$	0	0
$996$	70.7696	2.24242
$997$	−28.2843	−0.895772	−0.447886	−	0.894091i	$-0.647823\pi$
$997$	−28.2843	−0.895772	−0.447886	+	0.894091i	$0.647823\pi$
$998$	45.7990	1.44974
$999$	1.65685	0.0524205

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	725.2.a.b.1.2		2
3.2	odd	2		6525.2.a.o.1.1		2
5.2	odd	4		725.2.b.b.349.4		4
5.3	odd	4		725.2.b.b.349.1		4
5.4	even	2		29.2.a.a.1.1	✓	2
15.14	odd	2		261.2.a.d.1.2		2
20.19	odd	2		464.2.a.h.1.1		2
35.34	odd	2		1421.2.a.j.1.1		2
40.19	odd	2		1856.2.a.w.1.2		2
40.29	even	2		1856.2.a.r.1.1		2
55.54	odd	2		3509.2.a.j.1.2		2
60.59	even	2		4176.2.a.bq.1.2		2
65.64	even	2		4901.2.a.g.1.2		2
85.84	even	2		8381.2.a.e.1.1		2
145.4	even	14		841.2.d.f.190.1		12
145.9	even	14		841.2.d.f.574.2		12
145.14	odd	28		841.2.e.k.196.4		24
145.19	odd	28		841.2.e.k.651.4		24
145.24	even	14		841.2.d.j.605.2		12
145.34	even	14		841.2.d.f.605.1		12
145.39	odd	28		841.2.e.k.651.1		24
145.44	odd	28		841.2.e.k.196.1		24
145.49	even	14		841.2.d.j.574.1		12
145.54	even	14		841.2.d.j.190.2		12
145.64	even	14		841.2.d.f.645.1		12
145.69	odd	28		841.2.e.k.63.1		24
145.74	even	14		841.2.d.j.778.1		12
145.79	odd	28		841.2.e.k.267.4		24
145.84	odd	28		841.2.e.k.270.4		24
145.89	odd	28		841.2.e.k.236.1		24
145.94	even	14		841.2.d.j.571.2		12
145.99	odd	4		841.2.b.a.840.1		4
145.104	odd	4		841.2.b.a.840.4		4
145.109	even	14		841.2.d.f.571.1		12
145.114	odd	28		841.2.e.k.236.4		24
145.119	odd	28		841.2.e.k.270.1		24
145.124	odd	28		841.2.e.k.267.1		24
145.129	even	14		841.2.d.f.778.2		12
145.134	odd	28		841.2.e.k.63.4		24
145.139	even	14		841.2.d.j.645.2		12
145.144	even	2		841.2.a.d.1.2		2
435.434	odd	2		7569.2.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.2.a.a.1.1	✓	2	5.4	even	2
261.2.a.d.1.2		2	15.14	odd	2
464.2.a.h.1.1		2	20.19	odd	2
725.2.a.b.1.2		2	1.1	even	1	trivial
725.2.b.b.349.1		4	5.3	odd	4
725.2.b.b.349.4		4	5.2	odd	4
841.2.a.d.1.2		2	145.144	even	2
841.2.b.a.840.1		4	145.99	odd	4
841.2.b.a.840.4		4	145.104	odd	4
841.2.d.f.190.1		12	145.4	even	14
841.2.d.f.571.1		12	145.109	even	14
841.2.d.f.574.2		12	145.9	even	14
841.2.d.f.605.1		12	145.34	even	14
841.2.d.f.645.1		12	145.64	even	14
841.2.d.f.778.2		12	145.129	even	14
841.2.d.j.190.2		12	145.54	even	14
841.2.d.j.571.2		12	145.94	even	14
841.2.d.j.574.1		12	145.49	even	14
841.2.d.j.605.2		12	145.24	even	14
841.2.d.j.645.2		12	145.139	even	14
841.2.d.j.778.1		12	145.74	even	14
841.2.e.k.63.1		24	145.69	odd	28
841.2.e.k.63.4		24	145.134	odd	28
841.2.e.k.196.1		24	145.44	odd	28
841.2.e.k.196.4		24	145.14	odd	28
841.2.e.k.236.1		24	145.89	odd	28
841.2.e.k.236.4		24	145.114	odd	28
841.2.e.k.267.1		24	145.124	odd	28
841.2.e.k.267.4		24	145.79	odd	28
841.2.e.k.270.1		24	145.119	odd	28
841.2.e.k.270.4		24	145.84	odd	28
841.2.e.k.651.1		24	145.39	odd	28
841.2.e.k.651.4		24	145.19	odd	28
1421.2.a.j.1.1		2	35.34	odd	2
1856.2.a.r.1.1		2	40.29	even	2
1856.2.a.w.1.2		2	40.19	odd	2
3509.2.a.j.1.2		2	55.54	odd	2
4176.2.a.bq.1.2		2	60.59	even	2
4901.2.a.g.1.2		2	65.64	even	2
6525.2.a.o.1.1		2	3.2	odd	2
7569.2.a.c.1.1		2	435.434	odd	2
8381.2.a.e.1.1		2	85.84	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	725.2.a.b.1.2

Level	$725$
Weight	$2$
Character	725.1
Self dual	yes
Analytic conductor	$5.789$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$