LMFDB - Embedded newform 725.2.b.b.349.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(349,725)]

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([1, 0]))

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

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

chi := DirichletCharacter("725.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$725 = 5^{2} \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	725.b (of order $2$ , degree $1$ , not minimal)

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:	no
Analytic conductor:	$5.78915414654$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 1$ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2$
Twist minimal:	no (minimal twist has level 29)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	725.349
Dual form			725.2.b.b.349.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421i q^{2} +2.41421i q^{3} -3.82843 q^{4} -5.82843 q^{6} +2.82843i q^{7} -4.41421i q^{8} -2.82843 q^{9} -0.414214 q^{11} -9.24264i q^{12} -3.82843i q^{13} -6.82843 q^{14} +3.00000 q^{16} -0.828427i q^{17} -6.82843i q^{18} -6.00000 q^{19} -6.82843 q^{21} -1.00000i q^{22} +3.65685i q^{23} +10.6569 q^{24} +9.24264 q^{26} +0.414214i q^{27} -10.8284i q^{28} -1.00000 q^{29} +10.0711 q^{31} -1.58579i q^{32} -1.00000i q^{33} +2.00000 q^{34} +10.8284 q^{36} +4.00000i q^{37} -14.4853i q^{38} +9.24264 q^{39} -4.48528 q^{41} -16.4853i q^{42} +3.58579i q^{43} +1.58579 q^{44} -8.82843 q^{46} +3.24264i q^{47} +7.24264i q^{48} -1.00000 q^{49} +2.00000 q^{51} +14.6569i q^{52} +9.48528i q^{53} -1.00000 q^{54} +12.4853 q^{56} -14.4853i q^{57} -2.41421i q^{58} +3.65685 q^{59} -4.82843 q^{61} +24.3137i q^{62} -8.00000i q^{63} +9.82843 q^{64} +2.41421 q^{66} -5.65685i q^{67} +3.17157i q^{68} -8.82843 q^{69} -8.82843 q^{71} +12.4853i q^{72} +4.00000i q^{73} -9.65685 q^{74} +22.9706 q^{76} -1.17157i q^{77} +22.3137i q^{78} +2.41421 q^{79} -9.48528 q^{81} -10.8284i q^{82} +7.65685i q^{83} +26.1421 q^{84} -8.65685 q^{86} -2.41421i q^{87} +1.82843i q^{88} +12.4853 q^{89} +10.8284 q^{91} -14.0000i q^{92} +24.3137i q^{93} -7.82843 q^{94} +3.82843 q^{96} -4.48528i q^{97} -2.41421i q^{98} +1.17157 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{4} - 12 q^{6} + 4 q^{11} - 16 q^{14} + 12 q^{16} - 24 q^{19} - 16 q^{21} + 20 q^{24} + 20 q^{26} - 4 q^{29} + 12 q^{31} + 8 q^{34} + 32 q^{36} + 20 q^{39} + 16 q^{41} + 12 q^{44} - 24 q^{46}+ \cdots + 16 q^{99}+O(q^{100})$ 4 * q - 4 * q^4 - 12 * q^6 + 4 * q^11 - 16 * q^14 + 12 * q^16 - 24 * q^19 - 16 * q^21 + 20 * q^24 + 20 * q^26 - 4 * q^29 + 12 * q^31 + 8 * q^34 + 32 * q^36 + 20 * q^39 + 16 * q^41 + 12 * q^44 - 24 * q^46 - 4 * q^49 + 8 * q^51 - 4 * q^54 + 16 * q^56 - 8 * q^59 - 8 * q^61 + 28 * q^64 + 4 * q^66 - 24 * q^69 - 24 * q^71 - 16 * q^74 + 24 * q^76 + 4 * q^79 - 4 * q^81 + 48 * q^84 - 12 * q^86 + 16 * q^89 + 32 * q^91 - 20 * q^94 + 4 * q^96 + 16 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/725\mathbb{Z}\right)^\times$ .

$n$	$176$	$552$
$\chi(n)$	$1$	$-1$

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

)

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

$2$	2.41421i	1.70711i	0.521005	+	0.853553i	$0.325557\pi$
$2$	2.41421i	1.70711i	−0.521005	+	0.853553i	$0.674443\pi$
$3$	2.41421i	1.39385i	0.717146	+	0.696923i	$0.245448\pi$
$3$	2.41421i	1.39385i	−0.717146	+	0.696923i	$0.754552\pi$
$4$	−3.82843	−1.91421
$5$	0	0
$5$	0	0
$6$	−5.82843	−2.37945
$7$	2.82843i	1.06904i	0.845154	+	0.534522i	$0.179509\pi$
$7$	2.82843i	1.06904i	−0.845154	+	0.534522i	$0.820491\pi$
$8$	− 4.41421i	− 1.56066i
$9$	−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	$0.519890\pi$
$12$	− 9.24264i	− 2.66812i
$13$	− 3.82843i	− 1.06181i	−0.847430	−	0.530907i	$-0.821851\pi$
$13$	− 3.82843i	− 1.06181i	0.847430	−	0.530907i	$-0.178149\pi$
$14$	−6.82843	−1.82497
$15$	0	0
$16$	3.00000	0.750000
$17$	− 0.828427i	− 0.200923i	−0.994941	−	0.100462i	$-0.967968\pi$
$17$	− 0.828427i	− 0.200923i	0.994941	−	0.100462i	$-0.0320319\pi$
$18$	− 6.82843i	− 1.60948i
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	−6.82843	−1.49008
$22$	− 1.00000i	− 0.213201i
$23$	3.65685i	0.762507i	0.924471	+	0.381253i	$0.124507\pi$
$23$	3.65685i	0.762507i	−0.924471	+	0.381253i	$0.875493\pi$
$24$	10.6569	2.17532
$25$	0	0
$26$	9.24264	1.81263
$27$	0.414214i	0.0797154i
$28$	− 10.8284i	− 2.04638i
$29$	−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	$0.140313\pi$
$32$	− 1.58579i	− 0.280330i
$33$	− 1.00000i	− 0.174078i
$34$	2.00000	0.342997
$35$	0	0
$36$	10.8284	1.80474
$37$	4.00000i	0.657596i	0.944400	+	0.328798i	$0.106644\pi$
$37$	4.00000i	0.657596i	−0.944400	+	0.328798i	$0.893356\pi$
$38$	− 14.4853i	− 2.34982i
$39$	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	$0.613901\pi$
$42$	− 16.4853i	− 2.54373i
$43$	3.58579i	0.546827i	0.961897	+	0.273414i	$0.0881528\pi$
$43$	3.58579i	0.546827i	−0.961897	+	0.273414i	$0.911847\pi$
$44$	1.58579	0.239066
$45$	0	0
$46$	−8.82843	−1.30168
$47$	3.24264i	0.472988i	0.971633	+	0.236494i	$0.0759983\pi$
$47$	3.24264i	0.472988i	−0.971633	+	0.236494i	$0.924002\pi$
$48$	7.24264i	1.04539i
$49$	−1.00000	−0.142857
$50$	0	0
$51$	2.00000	0.280056
$52$	14.6569i	2.03254i
$53$	9.48528i	1.30290i	0.758690	+	0.651452i	$0.225840\pi$
$53$	9.48528i	1.30290i	−0.758690	+	0.651452i	$0.774160\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	12.4853	1.66842
$57$	− 14.4853i	− 1.91862i
$58$	− 2.41421i	− 0.317002i
$59$	3.65685	0.476082	0.238041	−	0.971255i	$-0.423495\pi$
$59$	3.65685	0.476082	0.238041	+	0.971255i	$0.423495\pi$
$60$	0	0
$61$	−4.82843	−0.618217	−0.309108	−	0.951027i	$-0.600031\pi$
$61$	−4.82843	−0.618217	−0.309108	+	0.951027i	$0.600031\pi$
$62$	24.3137i	3.08784i
$63$	− 8.00000i	− 1.00791i
$64$	9.82843	1.22855
$65$	0	0
$66$	2.41421	0.297169
$67$	− 5.65685i	− 0.691095i	−0.938401	−	0.345547i	$-0.887693\pi$
$67$	− 5.65685i	− 0.691095i	0.938401	−	0.345547i	$-0.112307\pi$
$68$	3.17157i	0.384610i
$69$	−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	$0.675513\pi$
$72$	12.4853i	1.47140i
$73$	4.00000i	0.468165i	0.972217	+	0.234082i	$0.0752085\pi$
$73$	4.00000i	0.468165i	−0.972217	+	0.234082i	$0.924791\pi$
$74$	−9.65685	−1.12259
$75$	0	0
$76$	22.9706	2.63490
$77$	− 1.17157i	− 0.133513i
$78$	22.3137i	2.52653i
$79$	2.41421	0.271620	0.135810	−	0.990735i	$-0.456636\pi$
$79$	2.41421	0.271620	0.135810	+	0.990735i	$0.456636\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	− 10.8284i	− 1.19580i
$83$	7.65685i	0.840449i	0.907420	+	0.420224i	$0.138049\pi$
$83$	7.65685i	0.840449i	−0.907420	+	0.420224i	$0.861951\pi$
$84$	26.1421	2.85234
$85$	0	0
$86$	−8.65685	−0.933493
$87$	− 2.41421i	− 0.258831i
$88$	1.82843i	0.194911i
$89$	12.4853	1.32344	0.661719	−	0.749752i	$-0.269827\pi$
$89$	12.4853	1.32344	0.661719	+	0.749752i	$0.269827\pi$
$90$	0	0
$91$	10.8284	1.13513
$92$	− 14.0000i	− 1.45960i
$93$	24.3137i	2.52121i
$94$	−7.82843	−0.807441
$95$	0	0
$96$	3.82843	0.390737
$97$	− 4.48528i	− 0.455411i	−0.973730	−	0.227706i	$-0.926878\pi$
$97$	− 4.48528i	− 0.455411i	0.973730	−	0.227706i	$-0.0731224\pi$
$98$	− 2.41421i	− 0.243872i
$99$	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	$0.537192\pi$
$102$	4.82843i	0.478086i
$103$	− 4.82843i	− 0.475759i	−0.971295	−	0.237880i	$-0.923548\pi$
$103$	− 4.82843i	− 0.475759i	0.971295	−	0.237880i	$-0.0764523\pi$
$104$	−16.8995	−1.65713
$105$	0	0
$106$	−22.8995	−2.22420
$107$	14.8284i	1.43352i	0.697321	+	0.716759i	$0.254375\pi$
$107$	14.8284i	1.43352i	−0.697321	+	0.716759i	$0.745625\pi$
$108$	− 1.58579i	− 0.152592i
$109$	−12.6569	−1.21231	−0.606153	−	0.795348i	$-0.707288\pi$
$109$	−12.6569	−1.21231	−0.606153	+	0.795348i	$0.707288\pi$
$110$	0	0
$111$	−9.65685	−0.916588
$112$	8.48528i	0.801784i
$113$	− 13.3137i	− 1.25245i	−0.779643	−	0.626224i	$-0.784599\pi$
$113$	− 13.3137i	− 1.25245i	0.779643	−	0.626224i	$-0.215401\pi$
$114$	34.9706	3.27529
$115$	0	0
$116$	3.82843	0.355461
$117$	10.8284i	1.00109i
$118$	8.82843i	0.812723i
$119$	2.34315	0.214796
$120$	0	0
$121$	−10.8284	−0.984402
$122$	− 11.6569i	− 1.05536i
$123$	− 10.8284i	− 0.976366i
$124$	−38.5563	−3.46246
$125$	0	0
$126$	19.3137	1.72060
$127$	4.34315i	0.385392i	0.981259	+	0.192696i	$0.0617231\pi$
$127$	4.34315i	0.385392i	−0.981259	+	0.192696i	$0.938277\pi$
$128$	20.5563i	1.81694i
$129$	−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	$0.118856\pi$
$132$	3.82843i	0.333222i
$133$	− 16.9706i	− 1.47153i
$134$	13.6569	1.17977
$135$	0	0
$136$	−3.65685	−0.313573
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	− 21.3137i	− 1.81434i
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	−7.82843	−0.659272
$142$	− 21.3137i	− 1.78861i
$143$	1.58579i	0.132610i
$144$	−8.48528	−0.707107
$145$	0	0
$146$	−9.65685	−0.799207
$147$	− 2.41421i	− 0.199121i
$148$	− 15.3137i	− 1.25878i
$149$	7.82843	0.641330	0.320665	−	0.947193i	$-0.396094\pi$
$149$	7.82843	0.641330	0.320665	+	0.947193i	$0.396094\pi$
$150$	0	0
$151$	−14.1421	−1.15087	−0.575435	−	0.817847i	$-0.695167\pi$
$151$	−14.1421	−1.15087	−0.575435	+	0.817847i	$0.695167\pi$
$152$	26.4853i	2.14824i
$153$	2.34315i	0.189432i
$154$	2.82843	0.227921
$155$	0	0
$156$	−35.3848	−2.83305
$157$	− 8.48528i	− 0.677199i	−0.940931	−	0.338600i	$-0.890047\pi$
$157$	− 8.48528i	− 0.677199i	0.940931	−	0.338600i	$-0.109953\pi$
$158$	5.82843i	0.463685i
$159$	−22.8995	−1.81605
$160$	0	0
$161$	−10.3431	−0.815154
$162$	− 22.8995i	− 1.79915i
$163$	3.92893i	0.307738i	0.988091	+	0.153869i	$0.0491733\pi$
$163$	3.92893i	0.307738i	−0.988091	+	0.153869i	$0.950827\pi$
$164$	17.1716	1.34087
$165$	0	0
$166$	−18.4853	−1.43474
$167$	3.17157i	0.245424i	0.992442	+	0.122712i	$0.0391591\pi$
$167$	3.17157i	0.245424i	−0.992442	+	0.122712i	$0.960841\pi$
$168$	30.1421i	2.32552i
$169$	−1.65685	−0.127450
$170$	0	0
$171$	16.9706	1.29777
$172$	− 13.7279i	− 1.04674i
$173$	12.3431i	0.938432i	0.883083	+	0.469216i	$0.155463\pi$
$173$	12.3431i	0.938432i	−0.883083	+	0.469216i	$0.844537\pi$
$174$	5.82843	0.441852
$175$	0	0
$176$	−1.24264	−0.0936676
$177$	8.82843i	0.663585i
$178$	30.1421i	2.25925i
$179$	6.48528	0.484733	0.242366	−	0.970185i	$-0.422076\pi$
$179$	6.48528	0.484733	0.242366	+	0.970185i	$0.422076\pi$
$180$	0	0
$181$	8.31371	0.617953	0.308977	−	0.951070i	$-0.400014\pi$
$181$	8.31371	0.617953	0.308977	+	0.951070i	$0.400014\pi$
$182$	26.1421i	1.93778i
$183$	− 11.6569i	− 0.861699i
$184$	16.1421	1.19001
$185$	0	0
$186$	−58.6985	−4.30398
$187$	0.343146i	0.0250933i
$188$	− 12.4142i	− 0.905400i
$189$	−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	$0.131544\pi$
$192$	23.7279i	1.71242i
$193$	− 5.17157i	− 0.372258i	−0.982525	−	0.186129i	$-0.940406\pi$
$193$	− 5.17157i	− 0.372258i	0.982525	−	0.186129i	$-0.0595942\pi$
$194$	10.8284	0.777436
$195$	0	0
$196$	3.82843	0.273459
$197$	− 2.00000i	− 0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	− 2.00000i	− 0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$	2.82843i	0.201008i
$199$	0.485281	0.0344007	0.0172003	−	0.999852i	$-0.494525\pi$
$199$	0.485281	0.0344007	0.0172003	+	0.999852i	$0.494525\pi$
$200$	0	0
$201$	13.6569	0.963280
$202$	− 5.65685i	− 0.398015i
$203$	− 2.82843i	− 0.198517i
$204$	−7.65685	−0.536087
$205$	0	0
$206$	11.6569	0.812172
$207$	− 10.3431i	− 0.718898i
$208$	− 11.4853i	− 0.796361i
$209$	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	$0.732529\pi$
$212$	− 36.3137i	− 2.49404i
$213$	− 21.3137i	− 1.46039i
$214$	−35.7990	−2.44717
$215$	0	0
$216$	1.82843	0.124409
$217$	28.4853i	1.93371i
$218$	− 30.5563i	− 2.06954i
$219$	−9.65685	−0.652550
$220$	0	0
$221$	−3.17157	−0.213343
$222$	− 23.3137i	− 1.56471i
$223$	− 3.17157i	− 0.212384i	−0.994346	−	0.106192i	$-0.966134\pi$
$223$	− 3.17157i	− 0.212384i	0.994346	−	0.106192i	$-0.0338659\pi$
$224$	4.48528	0.299685
$225$	0	0
$226$	32.1421	2.13806
$227$	8.14214i	0.540413i	0.962802	+	0.270206i	$0.0870919\pi$
$227$	8.14214i	0.540413i	−0.962802	+	0.270206i	$0.912908\pi$
$228$	55.4558i	3.67265i
$229$	3.51472	0.232259	0.116130	−	0.993234i	$-0.462951\pi$
$229$	3.51472	0.232259	0.116130	+	0.993234i	$0.462951\pi$
$230$	0	0
$231$	2.82843	0.186097
$232$	4.41421i	0.289807i
$233$	18.3137i	1.19977i	0.800086	+	0.599885i	$0.204787\pi$
$233$	18.3137i	1.19977i	−0.800086	+	0.599885i	$0.795213\pi$
$234$	−26.1421	−1.70896
$235$	0	0
$236$	−14.0000	−0.911322
$237$	5.82843i	0.378597i
$238$	5.65685i	0.366679i
$239$	19.6569	1.27150	0.635748	−	0.771897i	$-0.280692\pi$
$239$	19.6569	1.27150	0.635748	+	0.771897i	$0.280692\pi$
$240$	0	0
$241$	−18.3137	−1.17969	−0.589845	−	0.807517i	$-0.700811\pi$
$241$	−18.3137	−1.17969	−0.589845	+	0.807517i	$0.700811\pi$
$242$	− 26.1421i	− 1.68048i
$243$	− 21.6569i	− 1.38929i
$244$	18.4853	1.18340
$245$	0	0
$246$	26.1421	1.66676
$247$	22.9706i	1.46158i
$248$	− 44.4558i	− 2.82295i
$249$	−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	$0.281643\pi$
$252$	30.6274i	1.92935i
$253$	− 1.51472i	− 0.0952295i
$254$	−10.4853	−0.657905
$255$	0	0
$256$	−29.9706	−1.87316
$257$	18.1716i	1.13351i	0.823886	+	0.566756i	$0.191802\pi$
$257$	18.1716i	1.13351i	−0.823886	+	0.566756i	$0.808198\pi$
$258$	− 20.8995i	− 1.30115i
$259$	−11.3137	−0.703000
$260$	0	0
$261$	2.82843	0.175075
$262$	51.4558i	3.17895i
$263$	2.75736i	0.170026i	0.996380	+	0.0850130i	$0.0270932\pi$
$263$	2.75736i	0.170026i	−0.996380	+	0.0850130i	$0.972907\pi$
$264$	−4.41421	−0.271676
$265$	0	0
$266$	40.9706	2.51207
$267$	30.1421i	1.84467i
$268$	21.6569i	1.32290i
$269$	−31.4558	−1.91790	−0.958948	−	0.283581i	$-0.908478\pi$
$269$	−31.4558	−1.91790	−0.958948	+	0.283581i	$0.908478\pi$
$270$	0	0
$271$	16.5563	1.00573	0.502863	−	0.864366i	$-0.332280\pi$
$271$	16.5563	1.00573	0.502863	+	0.864366i	$0.332280\pi$
$272$	− 2.48528i	− 0.150692i
$273$	26.1421i	1.58219i
$274$	28.9706	1.75018
$275$	0	0
$276$	33.7990	2.03446
$277$	17.3137i	1.04028i	0.854081	+	0.520140i	$0.174120\pi$
$277$	17.3137i	1.04028i	−0.854081	+	0.520140i	$0.825880\pi$
$278$	− 33.7990i	− 2.02713i
$279$	−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	$0.0973439\pi$
$282$	− 18.8995i	− 1.12545i
$283$	11.6569i	0.692928i	0.938063	+	0.346464i	$0.112618\pi$
$283$	11.6569i	0.692928i	−0.938063	+	0.346464i	$0.887382\pi$
$284$	33.7990	2.00560
$285$	0	0
$286$	−3.82843	−0.226380
$287$	− 12.6863i	− 0.748848i
$288$	4.48528i	0.264298i
$289$	16.3137	0.959630
$290$	0	0
$291$	10.8284	0.634774
$292$	− 15.3137i	− 0.896167i
$293$	7.65685i	0.447318i	0.974667	+	0.223659i	$0.0718002\pi$
$293$	7.65685i	0.447318i	−0.974667	+	0.223659i	$0.928200\pi$
$294$	5.82843	0.339921
$295$	0	0
$296$	17.6569	1.02628
$297$	− 0.171573i	− 0.00995567i
$298$	18.8995i	1.09482i
$299$	14.0000	0.809641
$300$	0	0
$301$	−10.1421	−0.584583
$302$	− 34.1421i	− 1.96466i
$303$	− 5.65685i	− 0.324978i
$304$	−18.0000	−1.03237
$305$	0	0
$306$	−5.65685	−0.323381
$307$	− 2.89949i	− 0.165483i	−0.996571	−	0.0827415i	$-0.973632\pi$
$307$	− 2.89949i	− 0.165483i	0.996571	−	0.0827415i	$-0.0263676\pi$
$308$	4.48528i	0.255573i
$309$	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	$0.475733\pi$
$312$	− 40.7990i	− 2.30979i
$313$	9.82843i	0.555536i	0.960648	+	0.277768i	$0.0895946\pi$
$313$	9.82843i	0.555536i	−0.960648	+	0.277768i	$0.910405\pi$
$314$	20.4853	1.15605
$315$	0	0
$316$	−9.24264	−0.519939
$317$	31.4558i	1.76674i	0.468680	+	0.883368i	$0.344730\pi$
$317$	31.4558i	1.76674i	−0.468680	+	0.883368i	$0.655270\pi$
$318$	− 55.2843i	− 3.10019i
$319$	0.414214	0.0231915
$320$	0	0
$321$	−35.7990	−1.99810
$322$	− 24.9706i	− 1.39156i
$323$	4.97056i	0.276570i
$324$	36.3137	2.01743
$325$	0	0
$326$	−9.48528	−0.525341
$327$	− 30.5563i	− 1.68977i
$328$	19.7990i	1.09322i
$329$	−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	$0.521135\pi$
$332$	− 29.3137i	− 1.60880i
$333$	− 11.3137i	− 0.619987i
$334$	−7.65685	−0.418964
$335$	0	0
$336$	−20.4853	−1.11756
$337$	− 21.7990i	− 1.18747i	−0.804662	−	0.593733i	$-0.797653\pi$
$337$	− 21.7990i	− 1.18747i	0.804662	−	0.593733i	$-0.202347\pi$
$338$	− 4.00000i	− 0.217571i
$339$	32.1421	1.74572
$340$	0	0
$341$	−4.17157	−0.225903
$342$	40.9706i	2.21543i
$343$	16.9706i	0.916324i
$344$	15.8284	0.853412
$345$	0	0
$346$	−29.7990	−1.60200
$347$	− 2.48528i	− 0.133417i	−0.997773	−	0.0667084i	$-0.978750\pi$
$347$	− 2.48528i	− 0.133417i	0.997773	−	0.0667084i	$-0.0212497\pi$
$348$	9.24264i	0.495458i
$349$	5.14214	0.275252	0.137626	−	0.990484i	$-0.456053\pi$
$349$	5.14214	0.275252	0.137626	+	0.990484i	$0.456053\pi$
$350$	0	0
$351$	1.58579	0.0846430
$352$	0.656854i	0.0350104i
$353$	26.9706i	1.43550i	0.696302	+	0.717749i	$0.254828\pi$
$353$	26.9706i	1.43550i	−0.696302	+	0.717749i	$0.745172\pi$
$354$	−21.3137	−1.13281
$355$	0	0
$356$	−47.7990	−2.53334
$357$	5.65685i	0.299392i
$358$	15.6569i	0.827490i
$359$	−3.92893	−0.207361	−0.103681	−	0.994611i	$-0.533062\pi$
$359$	−3.92893	−0.207361	−0.103681	+	0.994611i	$0.533062\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	20.0711i	1.05491i
$363$	− 26.1421i	− 1.37211i
$364$	−41.4558	−2.17288
$365$	0	0
$366$	28.1421	1.47101
$367$	− 18.0000i	− 0.939592i	−0.882775	−	0.469796i	$-0.844327\pi$
$367$	− 18.0000i	− 0.939592i	0.882775	−	0.469796i	$-0.155673\pi$
$368$	10.9706i	0.571880i
$369$	12.6863	0.660422
$370$	0	0
$371$	−26.8284	−1.39286
$372$	− 93.0833i	− 4.82614i
$373$	− 26.3137i	− 1.36247i	−0.732064	−	0.681236i	$-0.761443\pi$
$373$	− 26.3137i	− 1.36247i	0.732064	−	0.681236i	$-0.238557\pi$
$374$	−0.828427	−0.0428369
$375$	0	0
$376$	14.3137	0.738173
$377$	3.82843i	0.197174i
$378$	− 2.82843i	− 0.145479i
$379$	6.97056	0.358054	0.179027	−	0.983844i	$-0.442705\pi$
$379$	6.97056	0.358054	0.179027	+	0.983844i	$0.442705\pi$
$380$	0	0
$381$	−10.4853	−0.537177
$382$	61.1127i	3.12680i
$383$	− 3.51472i	− 0.179594i	−0.995960	−	0.0897969i	$-0.971378\pi$
$383$	− 3.51472i	− 0.179594i	0.995960	−	0.0897969i	$-0.0286218\pi$
$384$	−49.6274	−2.53254
$385$	0	0
$386$	12.4853	0.635484
$387$	− 10.1421i	− 0.515554i
$388$	17.1716i	0.871755i
$389$	−3.02944	−0.153599	−0.0767993	−	0.997047i	$-0.524470\pi$
$389$	−3.02944	−0.153599	−0.0767993	+	0.997047i	$0.524470\pi$
$390$	0	0
$391$	3.02944	0.153205
$392$	4.41421i	0.222951i
$393$	51.4558i	2.59560i
$394$	4.82843	0.243253
$395$	0	0
$396$	−4.48528	−0.225394
$397$	− 19.3431i	− 0.970805i	−0.874291	−	0.485402i	$-0.838673\pi$
$397$	− 19.3431i	− 0.970805i	0.874291	−	0.485402i	$-0.161327\pi$
$398$	1.17157i	0.0587256i
$399$	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	$0.654248\pi$
$402$	32.9706i	1.64442i
$403$	− 38.5563i	− 1.92063i
$404$	8.97056	0.446302
$405$	0	0
$406$	6.82843	0.338889
$407$	− 1.65685i	− 0.0821272i
$408$	− 8.82843i	− 0.437072i
$409$	18.9706	0.938034	0.469017	−	0.883189i	$-0.344608\pi$
$409$	18.9706	0.938034	0.469017	+	0.883189i	$0.344608\pi$
$410$	0	0
$411$	28.9706	1.42901
$412$	18.4853i	0.910704i
$413$	10.3431i	0.508953i
$414$	24.9706	1.22724
$415$	0	0
$416$	−6.07107	−0.297659
$417$	− 33.7990i	− 1.65514i
$418$	6.00000i	0.293470i
$419$	9.51472	0.464824	0.232412	−	0.972617i	$-0.425338\pi$
$419$	9.51472	0.464824	0.232412	+	0.972617i	$0.425338\pi$
$420$	0	0
$421$	37.1127	1.80876	0.904381	−	0.426726i	$-0.140333\pi$
$421$	37.1127	1.80876	0.904381	+	0.426726i	$0.140333\pi$
$422$	− 46.7990i	− 2.27814i
$423$	− 9.17157i	− 0.445937i
$424$	41.8701	2.03339
$425$	0	0
$426$	51.4558	2.49304
$427$	− 13.6569i	− 0.660901i
$428$	− 56.7696i	− 2.74406i
$429$	−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	$0.343020\pi$
$432$	1.24264i	0.0597866i
$433$	30.6274i	1.47186i	0.677058	+	0.735930i	$0.263255\pi$
$433$	30.6274i	1.47186i	−0.677058	+	0.735930i	$0.736745\pi$
$434$	−68.7696	−3.30104
$435$	0	0
$436$	48.4558	2.32061
$437$	− 21.9411i	− 1.04959i
$438$	− 23.3137i	− 1.11397i
$439$	0.343146	0.0163775	0.00818873	−	0.999966i	$-0.497393\pi$
$439$	0.343146	0.0163775	0.00818873	+	0.999966i	$0.497393\pi$
$440$	0	0
$441$	2.82843	0.134687
$442$	− 7.65685i	− 0.364199i
$443$	− 24.3431i	− 1.15658i	−0.815832	−	0.578289i	$-0.803721\pi$
$443$	− 24.3431i	− 1.15658i	0.815832	−	0.578289i	$-0.196279\pi$
$444$	36.9706	1.75455
$445$	0	0
$446$	7.65685	0.362563
$447$	18.8995i	0.893915i
$448$	27.7990i	1.31338i
$449$	34.9706	1.65036	0.825181	−	0.564868i	$-0.191073\pi$
$449$	34.9706	1.65036	0.825181	+	0.564868i	$0.191073\pi$
$450$	0	0
$451$	1.85786	0.0874834
$452$	50.9706i	2.39745i
$453$	− 34.1421i	− 1.60414i
$454$	−19.6569	−0.922542
$455$	0	0
$456$	−63.9411	−2.99432
$457$	− 1.02944i	− 0.0481550i	−0.999710	−	0.0240775i	$-0.992335\pi$
$457$	− 1.02944i	− 0.0481550i	0.999710	−	0.0240775i	$-0.00766485\pi$
$458$	8.48528i	0.396491i
$459$	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	$0.394291\pi$
$462$	6.82843i	0.317687i
$463$	− 26.0000i	− 1.20832i	−0.796862	−	0.604161i	$-0.793508\pi$
$463$	− 26.0000i	− 1.20832i	0.796862	−	0.604161i	$-0.206492\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−44.2132	−2.04814
$467$	38.3553i	1.77487i	0.460930	+	0.887437i	$0.347516\pi$
$467$	38.3553i	1.77487i	−0.460930	+	0.887437i	$0.652484\pi$
$468$	− 41.4558i	− 1.91630i
$469$	16.0000	0.738811
$470$	0	0
$471$	20.4853	0.943912
$472$	− 16.1421i	− 0.743002i
$473$	− 1.48528i	− 0.0682933i
$474$	−14.0711	−0.646306
$475$	0	0
$476$	−8.97056	−0.411165
$477$	− 26.8284i	− 1.22839i
$478$	47.4558i	2.17058i
$479$	−6.89949	−0.315246	−0.157623	−	0.987499i	$-0.550383\pi$
$479$	−6.89949	−0.315246	−0.157623	+	0.987499i	$0.550383\pi$
$480$	0	0
$481$	15.3137	0.698245
$482$	− 44.2132i	− 2.01386i
$483$	− 24.9706i	− 1.13620i
$484$	41.4558	1.88436
$485$	0	0
$486$	52.2843	2.37166
$487$	11.5147i	0.521782i	0.965368	+	0.260891i	$0.0840163\pi$
$487$	11.5147i	0.521782i	−0.965368	+	0.260891i	$0.915984\pi$
$488$	21.3137i	0.964826i
$489$	−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	$0.659122\pi$
$492$	41.4558i	1.86897i
$493$	0.828427i	0.0373105i
$494$	−55.4558	−2.49508
$495$	0	0
$496$	30.2132	1.35661
$497$	− 24.9706i	− 1.12008i
$498$	− 44.6274i	− 1.99980i
$499$	−18.9706	−0.849239	−0.424620	−	0.905372i	$-0.639592\pi$
$499$	−18.9706	−0.849239	−0.424620	+	0.905372i	$0.639592\pi$
$500$	0	0
$501$	−7.65685	−0.342083
$502$	48.4558i	2.16269i
$503$	0.272078i	0.0121314i	0.999982	+	0.00606568i	$0.00193078\pi$
$503$	0.272078i	0.0121314i	−0.999982	+	0.00606568i	$0.998069\pi$
$504$	−35.3137	−1.57300
$505$	0	0
$506$	3.65685	0.162567
$507$	− 4.00000i	− 0.177646i
$508$	− 16.6274i	− 0.737722i
$509$	10.5147	0.466057	0.233028	−	0.972470i	$-0.425137\pi$
$509$	10.5147	0.466057	0.233028	+	0.972470i	$0.425137\pi$
$510$	0	0
$511$	−11.3137	−0.500489
$512$	− 31.2426i	− 1.38074i
$513$	− 2.48528i	− 0.109728i
$514$	−43.8701	−1.93503
$515$	0	0
$516$	33.1421	1.45900
$517$	− 1.34315i	− 0.0590715i
$518$	− 27.3137i	− 1.20010i
$519$	−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	$0.720391\pi$
$522$	6.82843i	0.298872i
$523$	4.68629i	0.204917i	0.994737	+	0.102459i	$0.0326709\pi$
$523$	4.68629i	0.204917i	−0.994737	+	0.102459i	$0.967329\pi$
$524$	−81.5980	−3.56462
$525$	0	0
$526$	−6.65685	−0.290253
$527$	− 8.34315i	− 0.363433i
$528$	− 3.00000i	− 0.130558i
$529$	9.62742	0.418583
$530$	0	0
$531$	−10.3431	−0.448854
$532$	64.9706i	2.81683i
$533$	17.1716i	0.743783i
$534$	−72.7696	−3.14905
$535$	0	0
$536$	−24.9706	−1.07856
$537$	15.6569i	0.675643i
$538$	− 75.9411i	− 3.27405i
$539$	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	$0.571371\pi$
$542$	39.9706i	1.71688i
$543$	20.0711i	0.861332i
$544$	−1.31371	−0.0563248
$545$	0	0
$546$	−63.1127	−2.70097
$547$	− 35.7990i	− 1.53065i	−0.643641	−	0.765327i	$-0.722577\pi$
$547$	− 35.7990i	− 1.53065i	0.643641	−	0.765327i	$-0.277423\pi$
$548$	45.9411i	1.96251i
$549$	13.6569	0.582860
$550$	0	0
$551$	6.00000	0.255609
$552$	38.9706i	1.65870i
$553$	6.82843i	0.290374i
$554$	−41.7990	−1.77587
$555$	0	0
$556$	53.5980	2.27306
$557$	17.3137i	0.733605i	0.930299	+	0.366803i	$0.119548\pi$
$557$	17.3137i	0.733605i	−0.930299	+	0.366803i	$0.880452\pi$
$558$	− 68.7696i	− 2.91125i
$559$	13.7279	0.580629
$560$	0	0
$561$	−0.828427	−0.0349762
$562$	77.1838i	3.25580i
$563$	− 0.757359i	− 0.0319189i	−0.999873	−	0.0159594i	$-0.994920\pi$
$563$	− 0.757359i	− 0.0319189i	0.999873	−	0.0159594i	$-0.00508026\pi$
$564$	29.9706	1.26199
$565$	0	0
$566$	−28.1421	−1.18290
$567$	− 26.8284i	− 1.12669i
$568$	38.9706i	1.63517i
$569$	39.6569	1.66250	0.831251	−	0.555897i	$-0.187625\pi$
$569$	39.6569	1.66250	0.831251	+	0.555897i	$0.187625\pi$
$570$	0	0
$571$	14.6274	0.612138	0.306069	−	0.952009i	$-0.400986\pi$
$571$	14.6274	0.612138	0.306069	+	0.952009i	$0.400986\pi$
$572$	− 6.07107i	− 0.253844i
$573$	61.1127i	2.55302i
$574$	30.6274	1.27836
$575$	0	0
$576$	−27.7990	−1.15829
$577$	29.7990i	1.24055i	0.784385	+	0.620274i	$0.212979\pi$
$577$	29.7990i	1.24055i	−0.784385	+	0.620274i	$0.787021\pi$
$578$	39.3848i	1.63819i
$579$	12.4853	0.518871
$580$	0	0
$581$	−21.6569	−0.898478
$582$	26.1421i	1.08363i
$583$	− 3.92893i	− 0.162720i
$584$	17.6569	0.730646
$585$	0	0
$586$	−18.4853	−0.763620
$587$	− 7.65685i	− 0.316032i	−0.987437	−	0.158016i	$-0.949490\pi$
$587$	− 7.65685i	− 0.316032i	0.987437	−	0.158016i	$-0.0505098\pi$
$588$	9.24264i	0.381160i
$589$	−60.4264	−2.48983
$590$	0	0
$591$	4.82843	0.198615
$592$	12.0000i	0.493197i
$593$	− 19.4853i	− 0.800165i	−0.916479	−	0.400082i	$-0.868982\pi$
$593$	− 19.4853i	− 0.800165i	0.916479	−	0.400082i	$-0.131018\pi$
$594$	0.414214	0.0169954
$595$	0	0
$596$	−29.9706	−1.22764
$597$	1.17157i	0.0479493i
$598$	33.7990i	1.38214i
$599$	−9.87006	−0.403280	−0.201640	−	0.979460i	$-0.564627\pi$
$599$	−9.87006	−0.403280	−0.201640	+	0.979460i	$0.564627\pi$
$600$	0	0
$601$	−17.1716	−0.700443	−0.350222	−	0.936667i	$-0.613894\pi$
$601$	−17.1716	−0.700443	−0.350222	+	0.936667i	$0.613894\pi$
$602$	− 24.4853i	− 0.997946i
$603$	16.0000i	0.651570i
$604$	54.1421	2.20301
$605$	0	0
$606$	13.6569	0.554772
$607$	7.72792i	0.313667i	0.987625	+	0.156833i	$0.0501286\pi$
$607$	7.72792i	0.313667i	−0.987625	+	0.156833i	$0.949871\pi$
$608$	9.51472i	0.385873i
$609$	6.82843	0.276702
$610$	0	0
$611$	12.4142	0.502225
$612$	− 8.97056i	− 0.362614i
$613$	− 9.00000i	− 0.363507i	−0.983344	−	0.181753i	$-0.941823\pi$
$613$	− 9.00000i	− 0.363507i	0.983344	−	0.181753i	$-0.0581772\pi$
$614$	7.00000	0.282497
$615$	0	0
$616$	−5.17157	−0.208369
$617$	− 0.686292i	− 0.0276291i	−0.999905	−	0.0138145i	$-0.995603\pi$
$617$	− 0.686292i	− 0.0276291i	0.999905	−	0.0138145i	$-0.00439744\pi$
$618$	28.1421i	1.13204i
$619$	−33.5858	−1.34993	−0.674963	−	0.737851i	$-0.735841\pi$
$619$	−33.5858	−1.34993	−0.674963	+	0.737851i	$0.735841\pi$
$620$	0	0
$621$	−1.51472	−0.0607836
$622$	6.48528i	0.260036i
$623$	35.3137i	1.41481i
$624$	27.7279	1.11001
$625$	0	0
$626$	−23.7279	−0.948358
$627$	6.00000i	0.239617i
$628$	32.4853i	1.29630i
$629$	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	$0.761908\pi$
$632$	− 10.6569i	− 0.423907i
$633$	− 46.7990i	− 1.86009i
$634$	−75.9411	−3.01601
$635$	0	0
$636$	87.6690	3.47630
$637$	3.82843i	0.151688i
$638$	1.00000i	0.0395904i
$639$	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	$0.385669\pi$
$642$	− 86.4264i	− 3.41098i
$643$	32.4853i	1.28109i	0.767919	+	0.640547i	$0.221292\pi$
$643$	32.4853i	1.28109i	−0.767919	+	0.640547i	$0.778708\pi$
$644$	39.5980	1.56038
$645$	0	0
$646$	−12.0000	−0.472134
$647$	− 39.6569i	− 1.55907i	−0.626358	−	0.779536i	$-0.715455\pi$
$647$	− 39.6569i	− 1.55907i	0.626358	−	0.779536i	$-0.284545\pi$
$648$	41.8701i	1.64481i
$649$	−1.51472	−0.0594579
$650$	0	0
$651$	−68.7696	−2.69529
$652$	− 15.0416i	− 0.589076i
$653$	− 30.1421i	− 1.17955i	−0.807567	−	0.589776i	$-0.799216\pi$
$653$	− 30.1421i	− 1.17955i	0.807567	−	0.589776i	$-0.200784\pi$
$654$	73.7696	2.88462
$655$	0	0
$656$	−13.4558	−0.525362
$657$	− 11.3137i	− 0.441390i
$658$	− 22.1421i	− 0.863190i
$659$	−14.4142	−0.561498	−0.280749	−	0.959781i	$-0.590583\pi$
$659$	−14.4142	−0.561498	−0.280749	+	0.959781i	$0.590583\pi$
$660$	0	0
$661$	33.3137	1.29575	0.647877	−	0.761745i	$-0.275657\pi$
$661$	33.3137	1.29575	0.647877	+	0.761745i	$0.275657\pi$
$662$	− 5.82843i	− 0.226528i
$663$	− 7.65685i	− 0.297368i
$664$	33.7990	1.31166
$665$	0	0
$666$	27.3137	1.05838
$667$	− 3.65685i	− 0.141594i
$668$	− 12.1421i	− 0.469793i
$669$	7.65685	0.296031
$670$	0	0
$671$	2.00000	0.0772091
$672$	10.8284i	0.417716i
$673$	− 21.6274i	− 0.833676i	−0.908981	−	0.416838i	$-0.863138\pi$
$673$	− 21.6274i	− 0.833676i	0.908981	−	0.416838i	$-0.136862\pi$
$674$	52.6274	2.02713
$675$	0	0
$676$	6.34315	0.243967
$677$	22.0000i	0.845529i	0.906240	+	0.422764i	$0.138940\pi$
$677$	22.0000i	0.845529i	−0.906240	+	0.422764i	$0.861060\pi$
$678$	77.5980i	2.98013i
$679$	12.6863	0.486855
$680$	0	0
$681$	−19.6569	−0.753252
$682$	− 10.0711i	− 0.385641i
$683$	20.9706i	0.802416i	0.915987	+	0.401208i	$0.131410\pi$
$683$	20.9706i	0.802416i	−0.915987	+	0.401208i	$0.868590\pi$
$684$	−64.9706	−2.48421
$685$	0	0
$686$	−40.9706	−1.56426
$687$	8.48528i	0.323734i
$688$	10.7574i	0.410120i
$689$	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	$0.133757\pi$
$692$	− 47.2548i	− 1.79636i
$693$	3.31371i	0.125877i
$694$	6.00000	0.227757
$695$	0	0
$696$	−10.6569	−0.403947
$697$	3.71573i	0.140743i
$698$	12.4142i	0.469885i
$699$	−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	$0.773588\pi$
$702$	3.82843i	0.144495i
$703$	− 24.0000i	− 0.905177i
$704$	−4.07107	−0.153434
$705$	0	0
$706$	−65.1127	−2.45055
$707$	− 6.62742i	− 0.249250i
$708$	− 33.7990i	− 1.27024i
$709$	−29.1421	−1.09446	−0.547228	−	0.836984i	$-0.684317\pi$
$709$	−29.1421	−1.09446	−0.547228	+	0.836984i	$0.684317\pi$
$710$	0	0
$711$	−6.82843	−0.256086
$712$	− 55.1127i	− 2.06544i
$713$	36.8284i	1.37924i
$714$	−13.6569	−0.511095
$715$	0	0
$716$	−24.8284	−0.927882
$717$	47.4558i	1.77227i
$718$	− 9.48528i	− 0.353988i
$719$	20.1421	0.751175	0.375587	−	0.926787i	$-0.377441\pi$
$719$	20.1421	0.751175	0.375587	+	0.926787i	$0.377441\pi$
$720$	0	0
$721$	13.6569	0.508608
$722$	41.0416i	1.52741i
$723$	− 44.2132i	− 1.64431i
$724$	−31.8284	−1.18289
$725$	0	0
$726$	63.1127	2.34233
$727$	− 1.31371i	− 0.0487228i	−0.999703	−	0.0243614i	$-0.992245\pi$
$727$	− 1.31371i	− 0.0487228i	0.999703	−	0.0243614i	$-0.00775523\pi$
$728$	− 47.7990i	− 1.77155i
$729$	23.8284	0.882534
$730$	0	0
$731$	2.97056	0.109870
$732$	44.6274i	1.64948i
$733$	− 41.2548i	− 1.52378i	−0.647705	−	0.761891i	$-0.724271\pi$
$733$	− 41.2548i	− 1.52378i	0.647705	−	0.761891i	$-0.275729\pi$
$734$	43.4558	1.60398
$735$	0	0
$736$	5.79899	0.213754
$737$	2.34315i	0.0863109i
$738$	30.6274i	1.12741i
$739$	−4.07107	−0.149757	−0.0748783	−	0.997193i	$-0.523857\pi$
$739$	−4.07107	−0.149757	−0.0748783	+	0.997193i	$0.523857\pi$
$740$	0	0
$741$	−55.4558	−2.03722
$742$	− 64.7696i	− 2.37777i
$743$	23.6569i	0.867886i	0.900940	+	0.433943i	$0.142878\pi$
$743$	23.6569i	0.867886i	−0.900940	+	0.433943i	$0.857122\pi$
$744$	107.326	3.93476
$745$	0	0
$746$	63.5269	2.32589
$747$	− 21.6569i	− 0.792383i
$748$	− 1.31371i	− 0.0480339i
$749$	−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	$0.347184\pi$
$752$	9.72792i	0.354741i
$753$	48.4558i	1.76583i
$754$	−9.24264	−0.336597
$755$	0	0
$756$	4.48528	0.163128
$757$	− 25.5147i	− 0.927348i	−0.886006	−	0.463674i	$-0.846531\pi$
$757$	− 25.5147i	− 0.927348i	0.886006	−	0.463674i	$-0.153469\pi$
$758$	16.8284i	0.611236i
$759$	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	$0.190350\pi$
$762$	− 25.3137i	− 0.917019i
$763$	− 35.7990i	− 1.29601i
$764$	−96.9117	−3.50614
$765$	0	0
$766$	8.48528	0.306586
$767$	− 14.0000i	− 0.505511i
$768$	− 72.3553i	− 2.61090i
$769$	49.1127	1.77105	0.885525	−	0.464592i	$-0.153799\pi$
$769$	49.1127	1.77105	0.885525	+	0.464592i	$0.153799\pi$
$770$	0	0
$771$	−43.8701	−1.57994
$772$	19.7990i	0.712581i
$773$	− 19.5147i	− 0.701896i	−0.936395	−	0.350948i	$-0.885859\pi$
$773$	− 19.5147i	− 0.701896i	0.936395	−	0.350948i	$-0.114141\pi$
$774$	24.4853	0.880105
$775$	0	0
$776$	−19.7990	−0.710742
$777$	− 27.3137i	− 0.979874i
$778$	− 7.31371i	− 0.262209i
$779$	26.9117	0.964211
$780$	0	0
$781$	3.65685	0.130853
$782$	7.31371i	0.261538i
$783$	− 0.414214i	− 0.0148028i
$784$	−3.00000	−0.107143
$785$	0	0
$786$	−124.225	−4.43097
$787$	54.0833i	1.92786i	0.266156	+	0.963930i	$0.414246\pi$
$787$	54.0833i	1.92786i	−0.266156	+	0.963930i	$0.585754\pi$
$788$	7.65685i	0.272764i
$789$	−6.65685	−0.236990
$790$	0	0
$791$	37.6569	1.33892
$792$	− 5.17157i	− 0.183764i
$793$	18.4853i	0.656432i
$794$	46.6985	1.65727
$795$	0	0
$796$	−1.85786	−0.0658503
$797$	− 51.7401i	− 1.83273i	−0.400345	−	0.916364i	$-0.631110\pi$
$797$	− 51.7401i	− 1.83273i	0.400345	−	0.916364i	$-0.368890\pi$
$798$	98.9117i	3.50144i
$799$	2.68629	0.0950342
$800$	0	0
$801$	−35.3137	−1.24775
$802$	− 45.0416i	− 1.59048i
$803$	− 1.65685i	− 0.0584691i
$804$	−52.2843	−1.84392
$805$	0	0
$806$	93.0833	3.27872
$807$	− 75.9411i	− 2.67325i
$808$	10.3431i	0.363871i
$809$	−36.2843	−1.27569	−0.637844	−	0.770166i	$-0.720173\pi$
$809$	−36.2843	−1.27569	−0.637844	+	0.770166i	$0.720173\pi$
$810$	0	0
$811$	10.8284	0.380238	0.190119	−	0.981761i	$-0.439113\pi$
$811$	10.8284	0.380238	0.190119	+	0.981761i	$0.439113\pi$
$812$	10.8284i	0.380003i
$813$	39.9706i	1.40183i
$814$	4.00000	0.140200
$815$	0	0
$816$	6.00000	0.210042
$817$	− 21.5147i	− 0.752705i
$818$	45.7990i	1.60132i
$819$	−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	$0.508251\pi$
$822$	69.9411i	2.43948i
$823$	− 54.2843i	− 1.89223i	−0.323830	−	0.946115i	$-0.604971\pi$
$823$	− 54.2843i	− 1.89223i	0.323830	−	0.946115i	$-0.395029\pi$
$824$	−21.3137	−0.742498
$825$	0	0
$826$	−24.9706	−0.868837
$827$	− 32.8995i	− 1.14403i	−0.820244	−	0.572014i	$-0.806162\pi$
$827$	− 32.8995i	− 1.14403i	0.820244	−	0.572014i	$-0.193838\pi$
$828$	39.5980i	1.37612i
$829$	29.7990	1.03496	0.517481	−	0.855695i	$-0.326870\pi$
$829$	29.7990	1.03496	0.517481	+	0.855695i	$0.326870\pi$
$830$	0	0
$831$	−41.7990	−1.44999
$832$	− 37.6274i	− 1.30450i
$833$	0.828427i	0.0287033i
$834$	81.5980	2.82551
$835$	0	0
$836$	−9.51472	−0.329073
$837$	4.17157i	0.144191i
$838$	22.9706i	0.793505i
$839$	7.92893	0.273737	0.136869	−	0.990589i	$-0.456296\pi$
$839$	7.92893	0.273737	0.136869	+	0.990589i	$0.456296\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	89.5980i	3.08775i
$843$	77.1838i	2.65835i
$844$	74.2132	2.55452
$845$	0	0
$846$	22.1421	0.761262
$847$	− 30.6274i	− 1.05237i
$848$	28.4558i	0.977178i
$849$	−28.1421	−0.965836
$850$	0	0
$851$	−14.6274	−0.501421
$852$	81.5980i	2.79550i
$853$	− 22.9706i	− 0.786497i	−0.919432	−	0.393249i	$-0.871351\pi$
$853$	− 22.9706i	− 0.786497i	0.919432	−	0.393249i	$-0.128649\pi$
$854$	32.9706	1.12823
$855$	0	0
$856$	65.4558	2.23723
$857$	6.17157i	0.210817i	0.994429	+	0.105408i	$0.0336150\pi$
$857$	6.17157i	0.210817i	−0.994429	+	0.105408i	$0.966385\pi$
$858$	− 9.24264i	− 0.315539i
$859$	−19.7279	−0.673108	−0.336554	−	0.941664i	$-0.609261\pi$
$859$	−19.7279	−0.673108	−0.336554	+	0.941664i	$0.609261\pi$
$860$	0	0
$861$	30.6274	1.04378
$862$	47.4558i	1.61635i
$863$	17.1127i	0.582523i	0.956643	+	0.291262i	$0.0940750\pi$
$863$	17.1127i	0.582523i	−0.956643	+	0.291262i	$0.905925\pi$
$864$	0.656854	0.0223466
$865$	0	0
$866$	−73.9411	−2.51262
$867$	39.3848i	1.33758i
$868$	− 109.054i	− 3.70153i
$869$	−1.00000	−0.0339227
$870$	0	0
$871$	−21.6569	−0.733815
$872$	55.8701i	1.89200i
$873$	12.6863i	0.429366i
$874$	52.9706	1.79176
$875$	0	0
$876$	36.9706	1.24912
$877$	37.1421i	1.25420i	0.778938	+	0.627100i	$0.215758\pi$
$877$	37.1421i	1.25420i	−0.778938	+	0.627100i	$0.784242\pi$
$878$	0.828427i	0.0279581i
$879$	−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	$0.424217\pi$
$882$	6.82843i	0.229925i
$883$	38.4264i	1.29315i	0.762850	+	0.646576i	$0.223800\pi$
$883$	38.4264i	1.29315i	−0.762850	+	0.646576i	$0.776200\pi$
$884$	12.1421	0.408384
$885$	0	0
$886$	58.7696	1.97440
$887$	− 17.1005i	− 0.574179i	−0.957904	−	0.287089i	$-0.907312\pi$
$887$	− 17.1005i	− 0.574179i	0.957904	−	0.287089i	$-0.0926877\pi$
$888$	42.6274i	1.43048i
$889$	−12.2843	−0.412001
$890$	0	0
$891$	3.92893	0.131624
$892$	12.1421i	0.406549i
$893$	− 19.4558i	− 0.651065i
$894$	−45.6274	−1.52601
$895$	0	0
$896$	−58.1421	−1.94239
$897$	33.7990i	1.12852i
$898$	84.4264i	2.81735i
$899$	−10.0711	−0.335889
$900$	0	0
$901$	7.85786	0.261783
$902$	4.48528i	0.149344i
$903$	− 24.4853i	− 0.814819i
$904$	−58.7696	−1.95465
$905$	0	0
$906$	82.4264	2.73843
$907$	− 22.2843i	− 0.739937i	−0.929044	−	0.369969i	$-0.879368\pi$
$907$	− 22.2843i	− 0.739937i	0.929044	−	0.369969i	$-0.120632\pi$
$908$	− 31.1716i	− 1.03446i
$909$	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	$0.582351\pi$
$912$	− 43.4558i	− 1.43897i
$913$	− 3.17157i	− 0.104964i
$914$	2.48528	0.0822058
$915$	0	0
$916$	−13.4558	−0.444594
$917$	60.2843i	1.99076i
$918$	0.828427i	0.0273422i
$919$	−8.14214	−0.268584	−0.134292	−	0.990942i	$-0.542876\pi$
$919$	−8.14214	−0.268584	−0.134292	+	0.990942i	$0.542876\pi$
$920$	0	0
$921$	7.00000	0.230658
$922$	33.7990i	1.11311i
$923$	33.7990i	1.11251i
$924$	−10.8284	−0.356229
$925$	0	0
$926$	62.7696	2.06274
$927$	13.6569i	0.448550i
$928$	1.58579i	0.0520560i
$929$	−18.6863	−0.613077	−0.306539	−	0.951858i	$-0.599171\pi$
$929$	−18.6863	−0.613077	−0.306539	+	0.951858i	$0.599171\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	− 70.1127i	− 2.29662i
$933$	6.48528i	0.212319i
$934$	−92.5980	−3.02990
$935$	0	0
$936$	47.7990	1.56236
$937$	16.6274i	0.543194i	0.962411	+	0.271597i	$0.0875518\pi$
$937$	16.6274i	0.543194i	−0.962411	+	0.271597i	$0.912448\pi$
$938$	38.6274i	1.26123i
$939$	−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	$0.873875\pi$
$942$	49.4558i	1.61136i
$943$	− 16.4020i	− 0.534123i
$944$	10.9706	0.357061
$945$	0	0
$946$	3.58579	0.116584
$947$	− 2.61522i	− 0.0849834i	−0.999097	−	0.0424917i	$-0.986470\pi$
$947$	− 2.61522i	− 0.0849834i	0.999097	−	0.0424917i	$-0.0135296\pi$
$948$	− 22.3137i	− 0.724716i
$949$	15.3137	0.497104
$950$	0	0
$951$	−75.9411	−2.46256
$952$	− 10.3431i	− 0.335223i
$953$	− 35.6274i	− 1.15409i	−0.816714	−	0.577043i	$-0.804207\pi$
$953$	− 35.6274i	− 1.15409i	0.816714	−	0.577043i	$-0.195793\pi$
$954$	64.7696	2.09699
$955$	0	0
$956$	−75.2548	−2.43392
$957$	1.00000i	0.0323254i
$958$	− 16.6569i	− 0.538159i
$959$	33.9411	1.09602
$960$	0	0
$961$	70.4264	2.27182
$962$	36.9706i	1.19198i
$963$	− 41.9411i	− 1.35153i
$964$	70.1127	2.25818
$965$	0	0
$966$	60.2843	1.93961
$967$	35.2426i	1.13333i	0.823949	+	0.566663i	$0.191766\pi$
$967$	35.2426i	1.13333i	−0.823949	+	0.566663i	$0.808234\pi$
$968$	47.7990i	1.53632i
$969$	−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	$0.580834\pi$
$972$	82.9117i	2.65939i
$973$	− 39.5980i	− 1.26945i
$974$	−27.7990	−0.890737
$975$	0	0
$976$	−14.4853	−0.463663
$977$	− 36.1716i	− 1.15723i	−0.815600	−	0.578616i	$-0.803593\pi$
$977$	− 36.1716i	− 1.15723i	0.815600	−	0.578616i	$-0.196407\pi$
$978$	− 22.8995i	− 0.732245i
$979$	−5.17157	−0.165284
$980$	0	0
$981$	35.7990	1.14297
$982$	− 51.2843i	− 1.63655i
$983$	− 21.8701i	− 0.697547i	−0.937207	−	0.348773i	$-0.886598\pi$
$983$	− 21.8701i	− 0.697547i	0.937207	−	0.348773i	$-0.113402\pi$
$984$	−47.7990	−1.52378
$985$	0	0
$986$	−2.00000	−0.0636930
$987$	− 22.1421i	− 0.704792i
$988$	− 87.9411i	− 2.79778i
$989$	−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	$0.565314\pi$
$992$	− 15.9706i	− 0.507066i
$993$	− 5.82843i	− 0.184960i
$994$	60.2843	1.91210
$995$	0	0
$996$	70.7696	2.24242
$997$	− 28.2843i	− 0.895772i	−0.894091	−	0.447886i	$-0.852177\pi$
$997$	− 28.2843i	− 0.895772i	0.894091	−	0.447886i	$-0.147823\pi$
$998$	− 45.7990i	− 1.44974i
$999$	−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.b.b.349.4		4
5.2	odd	4		29.2.a.a.1.1	✓	2
5.3	odd	4		725.2.a.b.1.2		2
5.4	even	2	inner	725.2.b.b.349.1		4
15.2	even	4		261.2.a.d.1.2		2
15.8	even	4		6525.2.a.o.1.1		2
20.7	even	4		464.2.a.h.1.1		2
35.27	even	4		1421.2.a.j.1.1		2
40.27	even	4		1856.2.a.w.1.2		2
40.37	odd	4		1856.2.a.r.1.1		2
55.32	even	4		3509.2.a.j.1.2		2
60.47	odd	4		4176.2.a.bq.1.2		2
65.12	odd	4		4901.2.a.g.1.2		2
85.67	odd	4		8381.2.a.e.1.1		2
145.2	even	28		841.2.e.k.236.1		24
145.7	odd	28		841.2.d.j.571.2		12
145.12	even	4		841.2.b.a.840.1		4
145.17	even	4		841.2.b.a.840.4		4
145.22	odd	28		841.2.d.f.571.1		12
145.27	even	28		841.2.e.k.236.4		24
145.32	even	28		841.2.e.k.270.1		24
145.37	even	28		841.2.e.k.267.1		24
145.42	odd	28		841.2.d.f.778.2		12
145.47	even	28		841.2.e.k.63.4		24
145.52	odd	28		841.2.d.j.645.2		12
145.57	odd	4		841.2.a.d.1.2		2
145.62	odd	28		841.2.d.f.190.1		12
145.67	odd	28		841.2.d.f.574.2		12
145.72	even	28		841.2.e.k.196.4		24
145.77	even	28		841.2.e.k.651.4		24
145.82	odd	28		841.2.d.j.605.2		12
145.92	odd	28		841.2.d.f.605.1		12
145.97	even	28		841.2.e.k.651.1		24
145.102	even	28		841.2.e.k.196.1		24
145.107	odd	28		841.2.d.j.574.1		12
145.112	odd	28		841.2.d.j.190.2		12
145.122	odd	28		841.2.d.f.645.1		12
145.127	even	28		841.2.e.k.63.1		24
145.132	odd	28		841.2.d.j.778.1		12
145.137	even	28		841.2.e.k.267.4		24
145.142	even	28		841.2.e.k.270.4		24
435.347	even	4		7569.2.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.2.a.a.1.1	✓	2	5.2	odd	4
261.2.a.d.1.2		2	15.2	even	4
464.2.a.h.1.1		2	20.7	even	4
725.2.a.b.1.2		2	5.3	odd	4
725.2.b.b.349.1		4	5.4	even	2	inner
725.2.b.b.349.4		4	1.1	even	1	trivial
841.2.a.d.1.2		2	145.57	odd	4
841.2.b.a.840.1		4	145.12	even	4
841.2.b.a.840.4		4	145.17	even	4
841.2.d.f.190.1		12	145.62	odd	28
841.2.d.f.571.1		12	145.22	odd	28
841.2.d.f.574.2		12	145.67	odd	28
841.2.d.f.605.1		12	145.92	odd	28
841.2.d.f.645.1		12	145.122	odd	28
841.2.d.f.778.2		12	145.42	odd	28
841.2.d.j.190.2		12	145.112	odd	28
841.2.d.j.571.2		12	145.7	odd	28
841.2.d.j.574.1		12	145.107	odd	28
841.2.d.j.605.2		12	145.82	odd	28
841.2.d.j.645.2		12	145.52	odd	28
841.2.d.j.778.1		12	145.132	odd	28
841.2.e.k.63.1		24	145.127	even	28
841.2.e.k.63.4		24	145.47	even	28
841.2.e.k.196.1		24	145.102	even	28
841.2.e.k.196.4		24	145.72	even	28
841.2.e.k.236.1		24	145.2	even	28
841.2.e.k.236.4		24	145.27	even	28
841.2.e.k.267.1		24	145.37	even	28
841.2.e.k.267.4		24	145.137	even	28
841.2.e.k.270.1		24	145.32	even	28
841.2.e.k.270.4		24	145.142	even	28
841.2.e.k.651.1		24	145.97	even	28
841.2.e.k.651.4		24	145.77	even	28
1421.2.a.j.1.1		2	35.27	even	4
1856.2.a.r.1.1		2	40.37	odd	4
1856.2.a.w.1.2		2	40.27	even	4
3509.2.a.j.1.2		2	55.32	even	4
4176.2.a.bq.1.2		2	60.47	odd	4
4901.2.a.g.1.2		2	65.12	odd	4
6525.2.a.o.1.1		2	15.8	even	4
7569.2.a.c.1.1		2	435.347	even	4
8381.2.a.e.1.1		2	85.67	odd	4

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

Character values

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.b.b.349.4

Level	$725$
Weight	$2$
Character	725.349
Analytic conductor	$5.789$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$