LMFDB - Embedded newform 1452.2.b.e.725.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1452,2,Mod(725,1452)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1452, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1452.725"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1452 = 2^{2} \cdot 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1452.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,6,0,0,0,0,0,-10,0,0,0,0,0,-16,0,0,0,0,0,0,0,0,0,-32,0,6, 0,0,0,40,0,0,0,0,0,36,0,0,0,0,0,0,0,2,0,0,0,-28,0,0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,44,0,16,0,0,0,0,0,-84,0,0,0,0,0,-58,0,0,0,0,0,0,0,0,0,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

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

Self dual:	no
Analytic conductor:	$11.5942783735$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} + 8 x^{14} - 12 x^{13} + 23 x^{12} - 72 x^{11} + 146 x^{10} - 176 x^{9} + 223 x^{8} + \cdots + 6561 $ x^16 - 4x^15 + 8x^14 - 12x^13 + 23x^12 - 72x^11 + 146x^10 - 176x^9 + 223x^8 - 528x^7 + 1314x^6 - 1944x^5 + 1863x^4 - 2916x^3 + 5832x^2 - 8748*x + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 132)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			725.10
Root			$-1.53089 + 0.810175i$ of defining polynomial
Character	$\chi$	$=$	1452.725
Dual form			1452.2.b.e.725.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.24359 - 1.20560i) q^{3} -3.05881i q^{5} +3.65040i q^{7} +(0.0930440 - 2.99856i) q^{9} -1.08050i q^{13} +(-3.68771 - 3.80392i) q^{15} -1.23182 q^{17} -5.17993i q^{19} +(4.40094 + 4.53961i) q^{21} -6.43947i q^{23} -4.35634 q^{25} +(-3.49936 - 3.84116i) q^{27} -7.42093 q^{29} +3.91913 q^{31} +11.1659 q^{35} +0.211719 q^{37} +(-1.30266 - 1.34371i) q^{39} +6.43039 q^{41} -11.6128i q^{43} +(-9.17202 - 0.284604i) q^{45} +3.18566i q^{47} -6.32545 q^{49} +(-1.53188 + 1.48509i) q^{51} +8.20307i q^{53} +(-6.24494 - 6.44173i) q^{57} -6.23422i q^{59} -6.10661i q^{61} +(10.9459 + 0.339648i) q^{63} -3.30506 q^{65} -3.24458 q^{67} +(-7.76344 - 8.00808i) q^{69} +0.332103i q^{71} +13.6200i q^{73} +(-5.41751 + 5.25201i) q^{75} -1.32789i q^{79} +(-8.98269 - 0.557995i) q^{81} +7.81134 q^{83} +3.76791i q^{85} +(-9.22861 + 8.94669i) q^{87} +10.3772i q^{89} +3.94427 q^{91} +(4.87380 - 4.72491i) q^{93} -15.8444 q^{95} +6.35634 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 6 q^{3} - 10 q^{9} - 16 q^{15} - 32 q^{25} + 6 q^{27} + 40 q^{31} + 36 q^{37} + 2 q^{45} - 28 q^{49} + 44 q^{67} + 16 q^{69} - 84 q^{75} - 58 q^{81} - 80 q^{91} + 12 q^{93} + 64 q^{97}+O(q^{100}) $ 16 * q + 6 * q^3 - 10 * q^9 - 16 * q^15 - 32 * q^25 + 6 * q^27 + 40 * q^31 + 36 * q^37 + 2 * q^45 - 28 * q^49 + 44 * q^67 + 16 * q^69 - 84 * q^75 - 58 * q^81 - 80 * q^91 + 12 * q^93 + 64 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1452\mathbb{Z}\right)^\times$.

$n$	$485$	$727$	$1333$
$\chi(n)$	$-1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	1.24359	−	1.20560i	0.717988	−	0.696055i
$3$	1.24359	−	1.20560i	0.717988	−	0.696055i
$4$		0			0
$5$		−	3.05881i		−	1.36794i	−0.729509	−	0.683971i	$-0.760251\pi$
$5$		−	3.05881i		−	1.36794i	0.729509	−	0.683971i	$-0.239749\pi$
$6$		0			0
$7$			3.65040i			1.37972i	0.723941	+	0.689861i	$0.242329\pi$
$7$			3.65040i			1.37972i	−0.723941	+	0.689861i	$0.757671\pi$
$8$		0			0
$9$	0.0930440	−	2.99856i	0.0310147	−	0.999519i
$10$		0			0
$11$		0			0
$11$		0			0
$12$		0			0
$13$		−	1.08050i		−	0.299678i	−0.988710	−	0.149839i	$-0.952125\pi$
$13$		−	1.08050i		−	0.299678i	0.988710	−	0.149839i	$-0.0478755\pi$
$14$		0			0
$15$	−3.68771	−	3.80392i	−0.952163	−	0.982167i
$16$		0			0
$17$	−1.23182			−0.298761			−0.149380	−	0.988780i	$-0.547728\pi$
$17$	−1.23182			−0.298761			−0.149380	+	0.988780i	$0.547728\pi$
$18$		0			0
$19$		−	5.17993i		−	1.18836i	−0.804333	−	0.594179i	$-0.797477\pi$
$19$		−	5.17993i		−	1.18836i	0.804333	−	0.594179i	$-0.202523\pi$
$20$		0			0
$21$	4.40094	+	4.53961i	0.960363	+	0.990625i
$22$		0			0
$23$		−	6.43947i		−	1.34272i	−0.741130	−	0.671361i	$-0.765710\pi$
$23$		−	6.43947i		−	1.34272i	0.741130	−	0.671361i	$-0.234290\pi$
$24$		0			0
$25$	−4.35634			−0.871267
$26$		0			0
$27$	−3.49936	−	3.84116i	−0.673452	−	0.739231i
$28$		0			0
$29$	−7.42093			−1.37803			−0.689016	−	0.724746i	$-0.741957\pi$
$29$	−7.42093			−1.37803			−0.689016	+	0.724746i	$0.741957\pi$
$30$		0			0
$31$	3.91913			0.703897			0.351948	−	0.936019i	$-0.385519\pi$
$31$	3.91913			0.703897			0.351948	+	0.936019i	$0.385519\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	11.1659			1.88738
$36$		0			0
$37$	0.211719			0.0348063			0.0174032	−	0.999849i	$-0.494460\pi$
$37$	0.211719			0.0348063			0.0174032	+	0.999849i	$0.494460\pi$
$38$		0			0
$39$	−1.30266	−	1.34371i	−0.208592	−	0.215165i
$40$		0			0
$41$	6.43039			1.00426			0.502129	−	0.864793i	$-0.332550\pi$
$41$	6.43039			1.00426			0.502129	+	0.864793i	$0.332550\pi$
$42$		0			0
$43$		−	11.6128i		−	1.77094i	−0.464699	−	0.885469i	$-0.653837\pi$
$43$		−	11.6128i		−	1.77094i	0.464699	−	0.885469i	$-0.346163\pi$
$44$		0			0
$45$	−9.17202	−	0.284604i	−1.36728	−	0.0424263i
$46$		0			0
$47$			3.18566i			0.464677i	0.972635	+	0.232338i	$0.0746377\pi$
$47$			3.18566i			0.464677i	−0.972635	+	0.232338i	$0.925362\pi$
$48$		0			0
$49$	−6.32545			−0.903635
$50$		0			0
$51$	−1.53188	+	1.48509i	−0.214507	+	0.207954i
$52$		0			0
$53$			8.20307i			1.12678i	0.826192	+	0.563389i	$0.190503\pi$
$53$			8.20307i			1.12678i	−0.826192	+	0.563389i	$0.809497\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−6.24494	−	6.44173i	−0.827163	−	0.853228i
$58$		0			0
$59$		−	6.23422i		−	0.811626i	−0.913956	−	0.405813i	$-0.866988\pi$
$59$		−	6.23422i		−	0.811626i	0.913956	−	0.405813i	$-0.133012\pi$
$60$		0			0
$61$		−	6.10661i		−	0.781872i	−0.920418	−	0.390936i	$-0.872151\pi$
$61$		−	6.10661i		−	0.781872i	0.920418	−	0.390936i	$-0.127849\pi$
$62$		0			0
$63$	10.9459	+	0.339648i	1.37906	+	0.0427916i
$64$		0			0
$65$	−3.30506			−0.409942
$66$		0			0
$67$	−3.24458			−0.396388			−0.198194	−	0.980163i	$-0.563508\pi$
$67$	−3.24458			−0.396388			−0.198194	+	0.980163i	$0.563508\pi$
$68$		0			0
$69$	−7.76344	−	8.00808i	−0.934609	−	0.964059i
$70$		0			0
$71$			0.332103i			0.0394134i	0.999806	+	0.0197067i	$0.00627324\pi$
$71$			0.332103i			0.0394134i	−0.999806	+	0.0197067i	$0.993727\pi$
$72$		0			0
$73$			13.6200i			1.59410i	0.603913	+	0.797050i	$0.293607\pi$
$73$			13.6200i			1.59410i	−0.603913	+	0.797050i	$0.706393\pi$
$74$		0			0
$75$	−5.41751	+	5.25201i	−0.625560	+	0.606450i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	1.32789i		−	0.149399i	−0.997206	−	0.0746995i	$-0.976200\pi$
$79$		−	1.32789i		−	0.149399i	0.997206	−	0.0746995i	$-0.0237998\pi$
$80$		0			0
$81$	−8.98269	−	0.557995i	−0.998076	−	0.0619995i
$82$		0			0
$83$	7.81134			0.857406			0.428703	−	0.903446i	$-0.358971\pi$
$83$	7.81134			0.857406			0.428703	+	0.903446i	$0.358971\pi$
$84$		0			0
$85$			3.76791i			0.408688i
$86$		0			0
$87$	−9.22861	+	8.94669i	−0.989411	+	0.959186i
$88$		0			0
$89$			10.3772i			1.09998i	0.835172	+	0.549989i	$0.185368\pi$
$89$			10.3772i			1.09998i	−0.835172	+	0.549989i	$0.814632\pi$
$90$		0			0
$91$	3.94427			0.413472
$92$		0			0
$93$	4.87380	−	4.72491i	0.505390	−	0.489951i
$94$		0			0
$95$	−15.8444			−1.62561
$96$		0			0
$97$	6.35634			0.645388			0.322694	−	0.946503i	$-0.395412\pi$
$97$	6.35634			0.645388			0.322694	+	0.946503i	$0.395412\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.81134			−0.777257			−0.388629	−	0.921395i	$-0.627051\pi$
$101$	−7.81134			−0.777257			−0.388629	+	0.921395i	$0.627051\pi$
$102$		0			0
$103$	0.438486			0.0432053			0.0216026	−	0.999767i	$-0.493123\pi$
$103$	0.438486			0.0432053			0.0216026	+	0.999767i	$0.493123\pi$
$104$		0			0
$105$	13.8858	−	13.4616i	1.35512	−	1.31372i
$106$		0			0
$107$	−10.4167			−1.00702			−0.503508	−	0.863991i	$-0.667958\pi$
$107$	−10.4167			−1.00702			−0.503508	+	0.863991i	$0.667958\pi$
$108$		0			0
$109$			2.29088i			0.219427i	0.993963	+	0.109713i	$0.0349933\pi$
$109$			2.29088i			0.219427i	−0.993963	+	0.109713i	$0.965007\pi$
$110$		0			0
$111$	0.263292	−	0.255248i	0.0249905	−	0.0242271i
$112$		0			0
$113$		−	11.9879i		−	1.12773i	−0.825868	−	0.563863i	$-0.809315\pi$
$113$		−	11.9879i		−	1.12773i	0.825868	−	0.563863i	$-0.190685\pi$
$114$		0			0
$115$	−19.6971			−1.83677
$116$		0			0
$117$	−3.23995	−	0.100534i	−0.299533	−	0.00929440i
$118$		0			0
$119$		−	4.49665i		−	0.412207i
$120$		0			0
$121$		0			0
$122$		0			0
$123$	7.99679	−	7.75250i	0.721046	−	0.699019i
$124$		0			0
$125$		−	1.96885i		−	0.176099i
$126$		0			0
$127$			19.9941i			1.77419i	0.461585	+	0.887096i	$0.347281\pi$
$127$			19.9941i			1.77419i	−0.461585	+	0.887096i	$0.652719\pi$
$128$		0			0
$129$	−14.0004	−	14.4416i	−1.23267	−	1.27151i
$130$		0			0
$131$	6.04743			0.528367			0.264183	−	0.964472i	$-0.414898\pi$
$131$	6.04743			0.528367			0.264183	+	0.964472i	$0.414898\pi$
$132$		0			0
$133$	18.9089			1.63961
$134$		0			0
$135$	−11.7494	+	10.7039i	−1.01123	+	0.921244i
$136$		0			0
$137$			3.57957i			0.305823i	0.988240	+	0.152912i	$0.0488650\pi$
$137$			3.57957i			0.305823i	−0.988240	+	0.152912i	$0.951135\pi$
$138$		0			0
$139$			6.62302i			0.561757i	0.959743	+	0.280878i	$0.0906258\pi$
$139$			6.62302i			0.561757i	−0.959743	+	0.280878i	$0.909374\pi$
$140$		0			0
$141$	3.84065	+	3.96167i	0.323441	+	0.333633i
$142$		0			0
$143$		0			0
$144$		0			0
$145$			22.6992i			1.88507i
$146$		0			0
$147$	−7.86628	+	7.62598i	−0.648800	+	0.628980i
$148$		0			0
$149$	15.4615			1.26665			0.633327	−	0.773884i	$-0.281689\pi$
$149$	15.4615			1.26665			0.633327	+	0.773884i	$0.281689\pi$
$150$		0			0
$151$		−	6.50933i		−	0.529722i	−0.964287	−	0.264861i	$-0.914674\pi$
$151$		−	6.50933i		−	0.529722i	0.964287	−	0.264861i	$-0.0853260\pi$
$152$		0			0
$153$	−0.114614	+	3.69369i	−0.00926597	+	0.298617i
$154$		0			0
$155$		−	11.9879i		−	0.962890i
$156$		0			0
$157$	20.8780			1.66624			0.833121	−	0.553090i	$-0.186552\pi$
$157$	20.8780			1.66624			0.833121	+	0.553090i	$0.186552\pi$
$158$		0			0
$159$	9.88964	+	10.2013i	0.784300	+	0.809014i
$160$		0			0
$161$	23.5067			1.85258
$162$		0			0
$163$	1.64041			0.128487			0.0642436	−	0.997934i	$-0.479537\pi$
$163$	1.64041			0.128487			0.0642436	+	0.997934i	$0.479537\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.96934			−0.384539			−0.192270	−	0.981342i	$-0.561585\pi$
$167$	−4.96934			−0.384539			−0.192270	+	0.981342i	$0.561585\pi$
$168$		0			0
$169$	11.8325			0.910193
$170$		0			0
$171$	−15.5323	−	0.481962i	−1.18779	−	0.0368565i
$172$		0			0
$173$	7.29132			0.554349			0.277174	−	0.960820i	$-0.410602\pi$
$173$	7.29132			0.554349			0.277174	+	0.960820i	$0.410602\pi$
$174$		0			0
$175$		−	15.9024i		−	1.20211i
$176$		0			0
$177$	−7.51599	−	7.75283i	−0.564937	−	0.582738i
$178$		0			0
$179$		−	9.10438i		−	0.680493i	−0.940336	−	0.340247i	$-0.889489\pi$
$179$		−	9.10438i		−	0.680493i	0.940336	−	0.340247i	$-0.110511\pi$
$180$		0			0
$181$	22.0567			1.63946			0.819731	−	0.572748i	$-0.194123\pi$
$181$	22.0567			1.63946			0.819731	+	0.572748i	$0.194123\pi$
$182$		0			0
$183$	−7.36215	−	7.59414i	−0.544226	−	0.561375i
$184$		0			0
$185$		−	0.647607i		−	0.0476130i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	14.0218	−	12.7741i	1.01993	−	0.929177i
$190$		0			0
$191$			9.65116i			0.698334i	0.937061	+	0.349167i	$0.113535\pi$
$191$			9.65116i			0.698334i	−0.937061	+	0.349167i	$0.886465\pi$
$192$		0			0
$193$			2.25700i			0.162463i	0.996695	+	0.0812314i	$0.0258853\pi$
$193$			2.25700i			0.162463i	−0.996695	+	0.0812314i	$0.974115\pi$
$194$		0			0
$195$	−4.11014	+	3.98458i	−0.294333	+	0.285342i
$196$		0			0
$197$	−5.10641			−0.363816			−0.181908	−	0.983316i	$-0.558227\pi$
$197$	−5.10641			−0.363816			−0.181908	+	0.983316i	$0.558227\pi$
$198$		0			0
$199$	−9.78375			−0.693552			−0.346776	−	0.937948i	$-0.612724\pi$
$199$	−9.78375			−0.693552			−0.346776	+	0.937948i	$0.612724\pi$
$200$		0			0
$201$	−4.03493	+	3.91167i	−0.284602	+	0.275908i
$202$		0			0
$203$		−	27.0894i		−	1.90130i
$204$		0			0
$205$		−	19.6694i		−	1.37377i
$206$		0			0
$207$	−19.3091	−	0.599154i	−1.34208	−	0.0416441i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	7.35956i		−	0.506653i	−0.967381	−	0.253327i	$-0.918475\pi$
$211$		−	7.35956i		−	0.506653i	0.967381	−	0.253327i	$-0.0815247\pi$
$212$		0			0
$213$	0.400384	+	0.413001i	0.0274339	+	0.0282983i
$214$		0			0
$215$	−35.5214			−2.42254
$216$		0			0
$217$			14.3064i			0.971182i
$218$		0			0
$219$	16.4203	+	16.9377i	1.10958	+	1.14455i
$220$		0			0
$221$			1.33099i			0.0895319i
$222$		0			0
$223$	3.93694			0.263637			0.131819	−	0.991274i	$-0.457918\pi$
$223$	3.93694			0.263637			0.131819	+	0.991274i	$0.457918\pi$
$224$		0			0
$225$	−0.405331	+	13.0627i	−0.0270221	+	0.870848i
$226$		0			0
$227$	28.7622			1.90901			0.954507	−	0.298189i	$-0.0963825\pi$
$227$	28.7622			1.90901			0.954507	+	0.298189i	$0.0963825\pi$
$228$		0			0
$229$	12.7265			0.840991			0.420495	−	0.907295i	$-0.361856\pi$
$229$	12.7265			0.840991			0.420495	+	0.907295i	$0.361856\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−19.0453			−1.24770			−0.623849	−	0.781545i	$-0.714432\pi$
$233$	−19.0453			−1.24770			−0.623849	+	0.781545i	$0.714432\pi$
$234$		0			0
$235$	9.74435			0.635651
$236$		0			0
$237$	−1.60090	−	1.65135i	−0.103990	−	0.107267i
$238$		0			0
$239$	20.4999			1.32603			0.663013	−	0.748608i	$-0.269278\pi$
$239$	20.4999			1.32603			0.663013	+	0.748608i	$0.269278\pi$
$240$		0			0
$241$		−	18.9036i		−	1.21769i	−0.793290	−	0.608845i	$-0.791633\pi$
$241$		−	18.9036i		−	1.21769i	0.793290	−	0.608845i	$-0.208367\pi$
$242$		0			0
$243$	−11.8435	+	10.1356i	−0.759762	+	0.650201i
$244$		0			0
$245$			19.3484i			1.23612i
$246$		0			0
$247$	−5.59693			−0.356124
$248$		0			0
$249$	9.71412	−	9.41737i	0.615607	−	0.596802i
$250$		0			0
$251$		−	25.5144i		−	1.61046i	−0.592965	−	0.805228i	$-0.702043\pi$
$251$		−	25.5144i		−	1.61046i	0.592965	−	0.805228i	$-0.297957\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	4.54261	+	4.68575i	0.284469	+	0.293433i
$256$		0			0
$257$		−	6.80152i		−	0.424267i	−0.977241	−	0.212134i	$-0.931959\pi$
$257$		−	6.80152i		−	0.424267i	0.977241	−	0.212134i	$-0.0680412\pi$
$258$		0			0
$259$			0.772858i			0.0480231i
$260$		0			0
$261$	−0.690473	+	22.2521i	−0.0427392	+	1.37737i
$262$		0			0
$263$	17.9177			1.10485			0.552426	−	0.833562i	$-0.313702\pi$
$263$	17.9177			1.10485			0.552426	+	0.833562i	$0.313702\pi$
$264$		0			0
$265$	25.0916			1.54137
$266$		0			0
$267$	12.5107	+	12.9050i	0.765645	+	0.789771i
$268$		0			0
$269$			28.6051i			1.74408i	0.489431	+	0.872042i	$0.337204\pi$
$269$			28.6051i			1.74408i	−0.489431	+	0.872042i	$0.662796\pi$
$270$		0			0
$271$			3.19670i			0.194186i	0.995275	+	0.0970928i	$0.0309544\pi$
$271$			3.19670i			0.194186i	−0.995275	+	0.0970928i	$0.969046\pi$
$272$		0			0
$273$	4.90507	−	4.75523i	0.296868	−	0.287799i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			21.4310i			1.28767i	0.765166	+	0.643833i	$0.222657\pi$
$277$			21.4310i			1.28767i	−0.765166	+	0.643833i	$0.777343\pi$
$278$		0			0
$279$	0.364652	−	11.7517i	0.0218311	−	0.703558i
$280$		0			0
$281$	−10.3245			−0.615907			−0.307954	−	0.951401i	$-0.599644\pi$
$281$	−10.3245			−0.615907			−0.307954	+	0.951401i	$0.599644\pi$
$282$		0			0
$283$			6.04879i			0.359563i	0.983707	+	0.179782i	$0.0575391\pi$
$283$			6.04879i			0.359563i	−0.983707	+	0.179782i	$0.942461\pi$
$284$		0			0
$285$	−19.7040	+	19.1021i	−1.16717	+	1.13151i
$286$		0			0
$287$			23.4735i			1.38560i
$288$		0			0
$289$	−15.4826			−0.910742
$290$		0			0
$291$	7.90469	−	7.66322i	0.463381	−	0.449226i
$292$		0			0
$293$	−28.0204			−1.63697			−0.818485	−	0.574528i	$-0.805185\pi$
$293$	−28.0204			−1.63697			−0.818485	+	0.574528i	$0.805185\pi$
$294$		0			0
$295$	−19.0693			−1.11026
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.95787			−0.402384
$300$		0			0
$301$	42.3915			2.44340
$302$		0			0
$303$	−9.71412	+	9.41737i	−0.558062	+	0.541014i
$304$		0			0
$305$	−18.6790			−1.06956
$306$		0			0
$307$		−	7.37725i		−	0.421042i	−0.977589	−	0.210521i	$-0.932484\pi$
$307$		−	7.37725i		−	0.421042i	0.977589	−	0.210521i	$-0.0675160\pi$
$308$		0			0
$309$	0.545298	−	0.528640i	0.0310209	−	0.0300733i
$310$		0			0
$311$			11.3364i			0.642826i	0.946939	+	0.321413i	$0.104158\pi$
$311$			11.3364i			0.642826i	−0.946939	+	0.321413i	$0.895842\pi$
$312$		0			0
$313$	1.67902			0.0949039			0.0474519	−	0.998874i	$-0.484890\pi$
$313$	1.67902			0.0949039			0.0474519	+	0.998874i	$0.484890\pi$
$314$		0			0
$315$	1.03892	−	33.4816i	0.0585365	−	1.88647i
$316$		0			0
$317$		−	0.852858i		−	0.0479013i	−0.999713	−	0.0239506i	$-0.992376\pi$
$317$		−	0.852858i		−	0.0479013i	0.999713	−	0.0239506i	$-0.00762446\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−12.9541	+	12.5583i	−0.723026	+	0.700938i
$322$		0			0
$323$			6.38076i			0.355035i
$324$		0			0
$325$			4.70703i			0.261099i
$326$		0			0
$327$	2.76189	+	2.84892i	0.152733	+	0.157546i
$328$		0			0
$329$	−11.6290			−0.641125
$330$		0			0
$331$	21.1263			1.16121			0.580603	−	0.814187i	$-0.302817\pi$
$331$	21.1263			1.16121			0.580603	+	0.814187i	$0.302817\pi$
$332$		0			0
$333$	0.0196991	−	0.634850i	0.00107951	−	0.0347896i
$334$		0			0
$335$			9.92455i			0.542236i
$336$		0			0
$337$		−	12.0494i		−	0.656373i	−0.944613	−	0.328186i	$-0.893563\pi$
$337$		−	12.0494i		−	0.656373i	0.944613	−	0.328186i	$-0.106437\pi$
$338$		0			0
$339$	−14.4526	−	14.9080i	−0.784959	−	0.809694i
$340$		0			0
$341$		0			0
$342$		0			0
$343$			2.46239i			0.132957i
$344$		0			0
$345$	−24.4952	+	23.7469i	−1.31878	+	1.27849i
$346$		0			0
$347$	2.09275			0.112345			0.0561723	−	0.998421i	$-0.482110\pi$
$347$	2.09275			0.112345			0.0561723	+	0.998421i	$0.482110\pi$
$348$		0			0
$349$			12.5572i			0.672171i	0.941832	+	0.336085i	$0.109103\pi$
$349$			12.5572i			0.672171i	−0.941832	+	0.336085i	$0.890897\pi$
$350$		0			0
$351$	−4.15038	+	3.78107i	−0.221531	+	0.201818i
$352$		0			0
$353$			11.9737i			0.637296i	0.947873	+	0.318648i	$0.103229\pi$
$353$			11.9737i			0.637296i	−0.947873	+	0.318648i	$0.896771\pi$
$354$		0			0
$355$	1.01584			0.0539152
$356$		0			0
$357$	−5.42117	−	5.59200i	−0.286919	−	0.295960i
$358$		0			0
$359$	28.2301			1.48993			0.744964	−	0.667104i	$-0.232467\pi$
$359$	28.2301			1.48993			0.744964	+	0.667104i	$0.232467\pi$
$360$		0			0
$361$	−7.83172			−0.412196
$362$		0			0
$363$		0			0
$364$		0			0
$365$	41.6610			2.18064
$366$		0			0
$367$	−13.7951			−0.720100			−0.360050	−	0.932933i	$-0.617240\pi$
$367$	−13.7951			−0.720100			−0.360050	+	0.932933i	$0.617240\pi$
$368$		0			0
$369$	0.598309	−	19.2819i	0.0311467	−	1.00378i
$370$		0			0
$371$	−29.9445			−1.55464
$372$		0			0
$373$		−	4.34739i		−	0.225099i	−0.993646	−	0.112550i	$-0.964098\pi$
$373$		−	4.34739i		−	0.225099i	0.993646	−	0.112550i	$-0.0359017\pi$
$374$		0			0
$375$	−2.37365	−	2.44845i	−0.122575	−	0.126437i
$376$		0			0
$377$			8.01833i			0.412965i
$378$		0			0
$379$	22.4289			1.15209			0.576047	−	0.817416i	$-0.304594\pi$
$379$	22.4289			1.15209			0.576047	+	0.817416i	$0.304594\pi$
$380$		0			0
$381$	24.1050	+	24.8645i	1.23494	+	1.27385i
$382$		0			0
$383$			18.1642i			0.928148i	0.885796	+	0.464074i	$0.153613\pi$
$383$			18.1642i			0.928148i	−0.885796	+	0.464074i	$0.846387\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−34.8217	−	1.08050i	−1.77009	−	0.0549251i
$388$		0			0
$389$			25.3194i			1.28375i	0.766811	+	0.641873i	$0.221843\pi$
$389$			25.3194i			1.28375i	−0.766811	+	0.641873i	$0.778157\pi$
$390$		0			0
$391$			7.93228i			0.401153i
$392$		0			0
$393$	7.52054	−	7.29080i	0.379361	−	0.367772i
$394$		0			0
$395$	−4.06176			−0.204369
$396$		0			0
$397$	−1.96507			−0.0986240			−0.0493120	−	0.998783i	$-0.515703\pi$
$397$	−1.96507			−0.0986240			−0.0493120	+	0.998783i	$0.515703\pi$
$398$		0			0
$399$	23.5149	−	22.7966i	1.17722	−	1.14126i
$400$		0			0
$401$			18.5121i			0.924450i	0.886763	+	0.462225i	$0.152949\pi$
$401$			18.5121i			0.924450i	−0.886763	+	0.462225i	$0.847051\pi$
$402$		0			0
$403$		−	4.23463i		−	0.210942i
$404$		0			0
$405$	−1.70680	+	27.4764i	−0.0848118	+	1.36531i
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	11.6205i		−	0.574597i	−0.957841	−	0.287298i	$-0.907243\pi$
$409$		−	11.6205i		−	0.574597i	0.957841	−	0.287298i	$-0.0927572\pi$
$410$		0			0
$411$	4.31554	+	4.45152i	0.212870	+	0.219577i
$412$		0			0
$413$	22.7574			1.11982
$414$		0			0
$415$		−	23.8934i		−	1.17288i
$416$		0			0
$417$	7.98473	+	8.23633i	0.391014	+	0.403335i
$418$		0			0
$419$		−	29.2430i		−	1.42861i	−0.699833	−	0.714306i	$-0.746742\pi$
$419$		−	29.2430i		−	1.42861i	0.699833	−	0.714306i	$-0.253258\pi$
$420$		0			0
$421$	18.1210			0.883161			0.441581	−	0.897222i	$-0.354418\pi$
$421$	18.1210			0.883161			0.441581	+	0.897222i	$0.354418\pi$
$422$		0			0
$423$	9.55240	+	0.296407i	0.464453	+	0.0144118i
$424$		0			0
$425$	5.36623			0.260300
$426$		0			0
$427$	22.2916			1.07877
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−14.3909			−0.693183			−0.346592	−	0.938016i	$-0.612661\pi$
$431$	−14.3909			−0.693183			−0.346592	+	0.938016i	$0.612661\pi$
$432$		0			0
$433$	20.3440			0.977673			0.488836	−	0.872376i	$-0.337422\pi$
$433$	20.3440			0.977673			0.488836	+	0.872376i	$0.337422\pi$
$434$		0			0
$435$	27.3663	+	28.2286i	1.31211	+	1.35346i
$436$		0			0
$437$	−33.3560			−1.59564
$438$		0			0
$439$		−	4.46676i		−	0.213187i	−0.994303	−	0.106593i	$-0.966006\pi$
$439$		−	4.46676i		−	0.213187i	0.994303	−	0.106593i	$-0.0339943\pi$
$440$		0			0
$441$	−0.588545	+	18.9672i	−0.0280259	+	0.903200i
$442$		0			0
$443$		−	12.4792i		−	0.592905i	−0.955048	−	0.296453i	$-0.904196\pi$
$443$		−	12.4792i		−	0.592905i	0.955048	−	0.296453i	$-0.0958037\pi$
$444$		0			0
$445$	31.7418			1.50471
$446$		0			0
$447$	19.2278	−	18.6404i	0.909443	−	0.881662i
$448$		0			0
$449$			13.6637i			0.644828i	0.946599	+	0.322414i	$0.104494\pi$
$449$			13.6637i			0.644828i	−0.946599	+	0.322414i	$0.895506\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−7.84766	−	8.09495i	−0.368715	−	0.380334i
$454$		0			0
$455$		−	12.0648i		−	0.565606i
$456$		0			0
$457$			8.90829i			0.416712i	0.978053	+	0.208356i	$0.0668113\pi$
$457$			8.90829i			0.416712i	−0.978053	+	0.208356i	$0.933189\pi$
$458$		0			0
$459$	4.31059	+	4.73162i	0.201201	+	0.220853i
$460$		0			0
$461$	−12.5468			−0.584365			−0.292182	−	0.956363i	$-0.594381\pi$
$461$	−12.5468			−0.584365			−0.292182	+	0.956363i	$0.594381\pi$
$462$		0			0
$463$	−5.12033			−0.237962			−0.118981	−	0.992897i	$-0.537963\pi$
$463$	−5.12033			−0.237962			−0.118981	+	0.992897i	$0.537963\pi$
$464$		0			0
$465$	−14.4526	−	14.9080i	−0.670225	−	0.691344i
$466$		0			0
$467$			21.4504i			0.992606i	0.868149	+	0.496303i	$0.165309\pi$
$467$			21.4504i			0.992606i	−0.868149	+	0.496303i	$0.834691\pi$
$468$		0			0
$469$		−	11.8440i		−	0.546906i
$470$		0			0
$471$	25.9637	−	25.1705i	1.19634	−	1.15980i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			22.5655i			1.03538i
$476$		0			0
$477$	24.5974	+	0.763246i	1.12624	+	0.0349466i
$478$		0			0
$479$	7.48249			0.341884			0.170942	−	0.985281i	$-0.445319\pi$
$479$	7.48249			0.341884			0.170942	+	0.985281i	$0.445319\pi$
$480$		0			0
$481$		−	0.228762i		−	0.0104307i
$482$		0			0
$483$	29.2327	−	28.3397i	1.33013	−	1.28950i
$484$		0			0
$485$		−	19.4428i		−	0.882854i
$486$		0			0
$487$	11.2665			0.510532			0.255266	−	0.966871i	$-0.417837\pi$
$487$	11.2665			0.510532			0.255266	+	0.966871i	$0.417837\pi$
$488$		0			0
$489$	2.04001	−	1.97769i	0.0922523	−	0.0894341i
$490$		0			0
$491$	−13.7517			−0.620606			−0.310303	−	0.950638i	$-0.600430\pi$
$491$	−13.7517			−0.620606			−0.310303	+	0.950638i	$0.600430\pi$
$492$		0			0
$493$	9.14126			0.411702
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.21231			−0.0543795
$498$		0			0
$499$	−26.2330			−1.17435			−0.587176	−	0.809460i	$-0.699760\pi$
$499$	−26.2330			−1.17435			−0.587176	+	0.809460i	$0.699760\pi$
$500$		0			0
$501$	−6.17984	+	5.99105i	−0.276095	+	0.267661i
$502$		0			0
$503$	19.5399			0.871242			0.435621	−	0.900130i	$-0.356529\pi$
$503$	19.5399			0.871242			0.435621	+	0.900130i	$0.356529\pi$
$504$		0			0
$505$			23.8934i			1.06324i
$506$		0			0
$507$	14.7148	−	14.2653i	0.653508	−	0.633545i
$508$		0			0
$509$		−	0.289477i		−	0.0128308i	−0.999979	−	0.00641542i	$-0.997958\pi$
$509$		−	0.289477i		−	0.0128308i	0.999979	−	0.00641542i	$-0.00204211\pi$
$510$		0			0
$511$	−49.7185			−2.19942
$512$		0			0
$513$	−19.8969	+	18.1265i	−0.878471	+	0.800302i
$514$		0			0
$515$		−	1.34125i		−	0.0591024i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	9.06743	−	8.79043i	0.398016	−	0.385857i
$520$		0			0
$521$			36.7929i			1.61193i	0.591965	+	0.805963i	$0.298352\pi$
$521$			36.7929i			1.61193i	−0.591965	+	0.805963i	$0.701648\pi$
$522$		0			0
$523$		−	5.06868i		−	0.221638i	−0.993841	−	0.110819i	$-0.964653\pi$
$523$		−	5.06868i		−	0.221638i	0.993841	−	0.110819i	$-0.0353474\pi$
$524$		0			0
$525$	−19.1720	−	19.7761i	−0.836733	−	0.863099i
$526$		0			0
$527$	−4.82767			−0.210297
$528$		0			0
$529$	−18.4668			−0.802903
$530$		0			0
$531$	−18.6937	−	0.580057i	−0.811236	−	0.0251723i
$532$		0			0
$533$		−	6.94806i		−	0.300954i
$534$		0			0
$535$			31.8626i			1.37754i
$536$		0			0
$537$	−10.9763	−	11.3221i	−0.473661	−	0.488586i
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	21.2060i		−	0.911718i	−0.890052	−	0.455859i	$-0.849332\pi$
$541$		−	21.2060i		−	0.911718i	0.890052	−	0.455859i	$-0.150668\pi$
$542$		0			0
$543$	27.4296	−	26.5916i	1.17712	−	1.14116i
$544$		0			0
$545$	7.00737			0.300163
$546$		0			0
$547$		−	0.965430i		−	0.0412788i	−0.999787	−	0.0206394i	$-0.993430\pi$
$547$		−	0.965430i		−	0.0412788i	0.999787	−	0.0206394i	$-0.00657019\pi$
$548$		0			0
$549$	−18.3110	−	0.568184i	−0.781496	−	0.0242495i
$550$		0			0
$551$			38.4399i			1.63760i
$552$		0			0
$553$	4.84732			0.206129
$554$		0			0
$555$	−0.780757	−	0.805359i	−0.0331413	−	0.0341856i
$556$		0			0
$557$	−14.1607			−0.600006			−0.300003	−	0.953938i	$-0.596988\pi$
$557$	−14.1607			−0.600006			−0.300003	+	0.953938i	$0.596988\pi$
$558$		0			0
$559$	−12.5477			−0.530710
$560$		0			0
$561$		0			0
$562$		0			0
$563$	10.3740			0.437212			0.218606	−	0.975813i	$-0.429849\pi$
$563$	10.3740			0.437212			0.218606	+	0.975813i	$0.429849\pi$
$564$		0			0
$565$	−36.6687			−1.54266
$566$		0			0
$567$	2.03691	−	32.7904i	0.0855421	−	1.37707i
$568$		0			0
$569$	12.9057			0.541034			0.270517	−	0.962715i	$-0.412805\pi$
$569$	12.9057			0.541034			0.270517	+	0.962715i	$0.412805\pi$
$570$		0			0
$571$			29.4474i			1.23234i	0.787614	+	0.616169i	$0.211316\pi$
$571$			29.4474i			1.23234i	−0.787614	+	0.616169i	$0.788684\pi$
$572$		0			0
$573$	11.6355	+	12.0021i	0.486079	+	0.501395i
$574$		0			0
$575$			28.0525i			1.16987i
$576$		0			0
$577$	−4.04137			−0.168244			−0.0841222	−	0.996455i	$-0.526809\pi$
$577$	−4.04137			−0.168244			−0.0841222	+	0.996455i	$0.526809\pi$
$578$		0			0
$579$	2.72105	+	2.80679i	0.113083	+	0.116646i
$580$		0			0
$581$			28.5145i			1.18298i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−0.307516	+	9.91040i	−0.0127142	+	0.409745i
$586$		0			0
$587$		−	20.8322i		−	0.859838i	−0.902867	−	0.429919i	$-0.858542\pi$
$587$		−	20.8322i		−	0.859838i	0.902867	−	0.429919i	$-0.141458\pi$
$588$		0			0
$589$		−	20.3008i		−	0.836482i
$590$		0			0
$591$	−6.35029	+	6.15630i	−0.261216	+	0.253236i
$592$		0			0
$593$	12.8061			0.525882			0.262941	−	0.964812i	$-0.415308\pi$
$593$	12.8061			0.525882			0.262941	+	0.964812i	$0.415308\pi$
$594$		0			0
$595$	−13.7544			−0.563876
$596$		0			0
$597$	−12.1670	+	11.7953i	−0.497962	+	0.482750i
$598$		0			0
$599$		−	33.7538i		−	1.37914i	−0.724218	−	0.689571i	$-0.757799\pi$
$599$		−	33.7538i		−	1.37914i	0.724218	−	0.689571i	$-0.242201\pi$
$600$		0			0
$601$		−	40.4651i		−	1.65061i	−0.564690	−	0.825303i	$-0.691004\pi$
$601$		−	40.4651i		−	1.65061i	0.564690	−	0.825303i	$-0.308996\pi$
$602$		0			0
$603$	−0.301888	+	9.72905i	−0.0122938	+	0.396197i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		−	22.5565i		−	0.915538i	−0.889071	−	0.457769i	$-0.848649\pi$
$607$		−	22.5565i		−	0.915538i	0.889071	−	0.457769i	$-0.151351\pi$
$608$		0			0
$609$	−32.6590	−	33.6881i	−1.32341	−	1.36511i
$610$		0			0
$611$	3.44212			0.139253
$612$		0			0
$613$			27.2539i			1.10078i	0.834909	+	0.550388i	$0.185520\pi$
$613$			27.2539i			1.10078i	−0.834909	+	0.550388i	$0.814480\pi$
$614$		0			0
$615$	−23.7134	−	24.4607i	−0.956218	−	0.986350i
$616$		0			0
$617$			6.44731i			0.259559i	0.991543	+	0.129779i	$0.0414269\pi$
$617$			6.44731i			0.259559i	−0.991543	+	0.129779i	$0.958573\pi$
$618$		0			0
$619$	−46.7893			−1.88062			−0.940311	−	0.340316i	$-0.889466\pi$
$619$	−46.7893			−1.88062			−0.940311	+	0.340316i	$0.889466\pi$
$620$		0			0
$621$	−24.7350	+	22.5340i	−0.992582	+	0.904259i
$622$		0			0
$623$	−37.8809			−1.51766
$624$		0			0
$625$	−27.8040			−1.11216
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−0.260800			−0.0103988
$630$		0			0
$631$	−12.2283			−0.486801			−0.243401	−	0.969926i	$-0.578263\pi$
$631$	−12.2283			−0.486801			−0.243401	+	0.969926i	$0.578263\pi$
$632$		0			0
$633$	−8.87271	−	9.15230i	−0.352658	−	0.363771i
$634$		0			0
$635$	61.1583			2.42699
$636$		0			0
$637$			6.83466i			0.270799i
$638$		0			0
$639$	0.995829	+	0.0309002i	0.0393944	+	0.00122239i
$640$		0			0
$641$		−	20.3115i		−	0.802255i	−0.916022	−	0.401127i	$-0.868618\pi$
$641$		−	20.3115i		−	0.802255i	0.916022	−	0.401127i	$-0.131382\pi$
$642$		0			0
$643$	−23.7532			−0.936737			−0.468368	−	0.883533i	$-0.655158\pi$
$643$	−23.7532			−0.936737			−0.468368	+	0.883533i	$0.655158\pi$
$644$		0			0
$645$	−44.1742	+	42.8247i	−1.73936	+	1.68622i
$646$		0			0
$647$		−	1.94916i		−	0.0766295i	−0.999266	−	0.0383147i	$-0.987801\pi$
$647$		−	1.94916i		−	0.0766295i	0.999266	−	0.0383147i	$-0.0121989\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	17.2478	+	17.7913i	0.675996	+	0.697298i
$652$		0			0
$653$		−	23.0875i		−	0.903482i	−0.892149	−	0.451741i	$-0.850803\pi$
$653$		−	23.0875i		−	0.903482i	0.892149	−	0.451741i	$-0.149197\pi$
$654$		0			0
$655$		−	18.4980i		−	0.722776i
$656$		0			0
$657$	40.8403	+	1.26726i	1.59333	+	0.0494405i
$658$		0			0
$659$	−44.0671			−1.71661			−0.858305	−	0.513139i	$-0.828482\pi$
$659$	−44.0671			−1.71661			−0.858305	+	0.513139i	$0.828482\pi$
$660$		0			0
$661$	−31.5126			−1.22570			−0.612848	−	0.790201i	$-0.709976\pi$
$661$	−31.5126			−1.22570			−0.612848	+	0.790201i	$0.709976\pi$
$662$		0			0
$663$	1.60464	+	1.65521i	0.0623191	+	0.0642829i
$664$		0			0
$665$		−	57.8386i		−	2.24289i
$666$		0			0
$667$			47.7868i			1.85031i
$668$		0			0
$669$	4.89595	−	4.74639i	0.189288	−	0.183506i
$670$		0			0
$671$		0			0
$672$		0			0
$673$			39.7858i			1.53363i	0.641868	+	0.766815i	$0.278160\pi$
$673$			39.7858i			1.53363i	−0.641868	+	0.766815i	$0.721840\pi$
$674$		0			0
$675$	15.2444	+	16.7334i	0.586757	+	0.644068i
$676$		0			0
$677$	−39.7490			−1.52768			−0.763839	−	0.645407i	$-0.776688\pi$
$677$	−39.7490			−1.52768			−0.763839	+	0.645407i	$0.776688\pi$
$678$		0			0
$679$			23.2032i			0.890457i
$680$		0			0
$681$	35.7684	−	34.6758i	1.37065	−	1.32878i
$682$		0			0
$683$		−	24.4150i		−	0.934216i	−0.884200	−	0.467108i	$-0.845296\pi$
$683$		−	24.4150i		−	0.934216i	0.884200	−	0.467108i	$-0.154704\pi$
$684$		0			0
$685$	10.9492			0.418349
$686$		0			0
$687$	15.8266	−	15.3431i	0.603822	−	0.585376i
$688$		0			0
$689$	8.86344			0.337670
$690$		0			0
$691$	−30.1233			−1.14595			−0.572973	−	0.819574i	$-0.694210\pi$
$691$	−30.1233			−1.14595			−0.572973	+	0.819574i	$0.694210\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	20.2586			0.768451
$696$		0			0
$697$	−7.92110			−0.300033
$698$		0			0
$699$	−23.6846	+	22.9610i	−0.895833	+	0.868467i
$700$		0			0
$701$	8.51569			0.321633			0.160817	−	0.986984i	$-0.448587\pi$
$701$	8.51569			0.321633			0.160817	+	0.986984i	$0.448587\pi$
$702$		0			0
$703$		−	1.09669i		−	0.0413624i
$704$		0			0
$705$	12.1180	−	11.7478i	0.456390	−	0.442448i
$706$		0			0
$707$		−	28.5145i		−	1.07240i
$708$		0			0
$709$	−1.23558			−0.0464031			−0.0232016	−	0.999731i	$-0.507386\pi$
$709$	−1.23558			−0.0464031			−0.0232016	+	0.999731i	$0.507386\pi$
$710$		0			0
$711$	−3.98174	−	0.123552i	−0.149327	−	0.00463356i
$712$		0			0
$713$		−	25.2371i		−	0.945138i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	25.4935	−	24.7147i	0.952071	−	0.922987i
$718$		0			0
$719$		−	20.9164i		−	0.780052i	−0.920804	−	0.390026i	$-0.872466\pi$
$719$		−	20.9164i		−	0.780052i	0.920804	−	0.390026i	$-0.127534\pi$
$720$		0			0
$721$			1.60065i			0.0596113i
$722$		0			0
$723$	−22.7903	−	23.5084i	−0.847579	−	0.874287i
$724$		0			0
$725$	32.3281			1.20063
$726$		0			0
$727$	−5.62159			−0.208493			−0.104247	−	0.994551i	$-0.533243\pi$
$727$	−5.62159			−0.208493			−0.104247	+	0.994551i	$0.533243\pi$
$728$		0			0
$729$	−2.50897	+	26.8832i	−0.0929247	+	0.995673i
$730$		0			0
$731$			14.3049i			0.529087i
$732$		0			0
$733$			18.1073i			0.668807i	0.942430	+	0.334404i	$0.108535\pi$
$733$			18.1073i			0.668807i	−0.942430	+	0.334404i	$0.891465\pi$
$734$		0			0
$735$	23.3264	+	24.0615i	0.860408	+	0.887521i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	49.7282i		−	1.82928i	−0.404270	−	0.914640i	$-0.632474\pi$
$739$		−	49.7282i		−	1.82928i	0.404270	−	0.914640i	$-0.367526\pi$
$740$		0			0
$741$	−6.96031	+	6.74768i	−0.255693	+	0.247882i
$742$		0			0
$743$	−15.3013			−0.561350			−0.280675	−	0.959803i	$-0.590558\pi$
$743$	−15.3013			−0.561350			−0.280675	+	0.959803i	$0.590558\pi$
$744$		0			0
$745$		−	47.2938i		−	1.73271i
$746$		0			0
$747$	0.726798	−	23.4227i	0.0265922	−	0.856993i
$748$		0			0
$749$		−	38.0250i		−	1.38940i
$750$		0			0
$751$	1.18083			0.0430891			0.0215445	−	0.999768i	$-0.493142\pi$
$751$	1.18083			0.0430891			0.0215445	+	0.999768i	$0.493142\pi$
$752$		0			0
$753$	−30.7603	−	31.7296i	−1.12097	−	1.15629i
$754$		0			0
$755$	−19.9108			−0.724629
$756$		0			0
$757$	54.9853			1.99848			0.999238	−	0.0390247i	$-0.0124251\pi$
$757$	54.9853			1.99848			0.999238	+	0.0390247i	$0.0124251\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−36.7383			−1.33176			−0.665882	−	0.746057i	$-0.731945\pi$
$761$	−36.7383			−1.33176			−0.665882	+	0.746057i	$0.731945\pi$
$762$		0			0
$763$	−8.36264			−0.302748
$764$		0			0
$765$	11.2983	+	0.350582i	0.408491	+	0.0126753i
$766$		0			0
$767$	−6.73609			−0.243226
$768$		0			0
$769$			33.5717i			1.21063i	0.795988	+	0.605313i	$0.206952\pi$
$769$			33.5717i			1.21063i	−0.795988	+	0.605313i	$0.793048\pi$
$770$		0			0
$771$	−8.19993	−	8.45832i	−0.295313	−	0.304619i
$772$		0			0
$773$			7.97762i			0.286935i	0.989655	+	0.143467i	$0.0458252\pi$
$773$			7.97762i			0.286935i	−0.989655	+	0.143467i	$0.954175\pi$
$774$		0			0
$775$	−17.0730			−0.613282
$776$		0			0
$777$	0.931760	+	0.961120i	0.0334267	+	0.0344800i
$778$		0			0
$779$		−	33.3090i		−	1.19342i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	25.9685	+	28.5049i	0.928038	+	1.01868i
$784$		0			0
$785$		−	63.8618i		−	2.27932i
$786$		0			0
$787$		−	23.4104i		−	0.834492i	−0.908794	−	0.417246i	$-0.862995\pi$
$787$		−	23.4104i		−	0.834492i	0.908794	−	0.417246i	$-0.137005\pi$
$788$		0			0
$789$	22.2823	−	21.6016i	0.793271	−	0.769038i
$790$		0			0
$791$	43.7606			1.55595
$792$		0			0
$793$	−6.59822			−0.234310
$794$		0			0
$795$	31.2038	−	30.2506i	1.10668	−	1.07288i
$796$		0			0
$797$		−	10.6500i		−	0.377244i	−0.982050	−	0.188622i	$-0.939598\pi$
$797$		−	10.6500i		−	0.377244i	0.982050	−	0.188622i	$-0.0604021\pi$
$798$		0			0
$799$		−	3.92417i		−	0.138827i
$800$		0			0
$801$	31.1165	+	0.965533i	1.09945	+	0.0341154i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		−	71.9025i		−	2.53423i
$806$		0			0
$807$	34.4864	+	35.5731i	1.21398	+	1.25223i
$808$		0			0
$809$	39.6943			1.39558			0.697788	−	0.716304i	$-0.254168\pi$
$809$	39.6943			1.39558			0.697788	+	0.716304i	$0.254168\pi$
$810$		0			0
$811$		−	39.7461i		−	1.39568i	−0.716256	−	0.697838i	$-0.754146\pi$
$811$		−	39.7461i		−	1.39568i	0.716256	−	0.697838i	$-0.245854\pi$
$812$		0			0
$813$	3.85395	+	3.97539i	0.135164	+	0.139423i
$814$		0			0
$815$		−	5.01772i		−	0.175763i
$816$		0			0
$817$	−60.1536			−2.10451
$818$		0			0
$819$	0.366991	−	11.8271i	0.0128237	−	0.413273i
$820$		0			0
$821$	36.4514			1.27216			0.636081	−	0.771622i	$-0.280554\pi$
$821$	36.4514			1.27216			0.636081	+	0.771622i	$0.280554\pi$
$822$		0			0
$823$	39.8859			1.39034			0.695168	−	0.718847i	$-0.255330\pi$
$823$	39.8859			1.39034			0.695168	+	0.718847i	$0.255330\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	14.1727			0.492833			0.246417	−	0.969164i	$-0.420747\pi$
$827$	14.1727			0.492833			0.246417	+	0.969164i	$0.420747\pi$
$828$		0			0
$829$	−39.5015			−1.37194			−0.685971	−	0.727629i	$-0.740622\pi$
$829$	−39.5015			−1.37194			−0.685971	+	0.727629i	$0.740622\pi$
$830$		0			0
$831$	25.8373	+	26.6515i	0.896286	+	0.924529i
$832$		0			0
$833$	7.79183			0.269971
$834$		0			0
$835$			15.2003i			0.526028i
$836$		0			0
$837$	−13.7144	−	15.0540i	−0.474041	−	0.520342i
$838$		0			0
$839$		−	17.3687i		−	0.599635i	−0.953997	−	0.299818i	$-0.903074\pi$
$839$		−	17.3687i		−	0.599635i	0.953997	−	0.299818i	$-0.0969258\pi$
$840$		0			0
$841$	26.0702			0.898971
$842$		0			0
$843$	−12.8395	+	12.4472i	−0.442214	+	0.428705i
$844$		0			0
$845$		−	36.1934i		−	1.24509i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	7.29244	+	7.52223i	0.250276	+	0.258162i
$850$		0			0
$851$		−	1.36335i		−	0.0467352i
$852$		0			0
$853$			9.24212i			0.316444i	0.987404	+	0.158222i	$0.0505762\pi$
$853$			9.24212i			0.316444i	−0.987404	+	0.158222i	$0.949424\pi$
$854$		0			0
$855$	−1.47423	+	47.5105i	−0.0504176	+	1.62482i
$856$		0			0
$857$	56.2505			1.92148			0.960740	−	0.277449i	$-0.0894889\pi$
$857$	56.2505			1.92148			0.960740	+	0.277449i	$0.0894889\pi$
$858$		0			0
$859$	−38.7547			−1.32229			−0.661147	−	0.750257i	$-0.729930\pi$
$859$	−38.7547			−1.32229			−0.661147	+	0.750257i	$0.729930\pi$
$860$		0			0
$861$	28.2997	+	29.1915i	0.964453	+	0.994844i
$862$		0			0
$863$		−	5.59929i		−	0.190602i	−0.995448	−	0.0953011i	$-0.969619\pi$
$863$		−	5.59929i		−	0.190602i	0.995448	−	0.0953011i	$-0.0303814\pi$
$864$		0			0
$865$		−	22.3028i		−	0.758317i
$866$		0			0
$867$	−19.2541	+	18.6659i	−0.653902	+	0.633927i
$868$		0			0
$869$		0			0
$870$		0			0
$871$			3.50578i			0.118789i
$872$		0			0
$873$	0.591419	−	19.0598i	0.0200165	−	0.645078i
$874$		0			0
$875$	7.18709			0.242968
$876$		0			0
$877$		−	34.4472i		−	1.16320i	−0.813475	−	0.581600i	$-0.802427\pi$
$877$		−	34.4472i		−	1.16320i	0.813475	−	0.581600i	$-0.197573\pi$
$878$		0			0
$879$	−34.8460	+	33.7815i	−1.17532	+	1.13942i
$880$		0			0
$881$		−	8.23935i		−	0.277591i	−0.990321	−	0.138795i	$-0.955677\pi$
$881$		−	8.23935i		−	0.277591i	0.990321	−	0.138795i	$-0.0443231\pi$
$882$		0			0
$883$	7.69807			0.259061			0.129530	−	0.991575i	$-0.458653\pi$
$883$	7.69807			0.259061			0.129530	+	0.991575i	$0.458653\pi$
$884$		0			0
$885$	−23.7144	+	22.9900i	−0.797152	+	0.772801i
$886$		0			0
$887$	−12.6960			−0.426289			−0.213144	−	0.977021i	$-0.568371\pi$
$887$	−12.6960			−0.426289			−0.213144	+	0.977021i	$0.568371\pi$
$888$		0			0
$889$	−72.9866			−2.44789
$890$		0			0
$891$		0			0
$892$		0			0
$893$	16.5015			0.552203
$894$		0			0
$895$	−27.8486			−0.930876
$896$		0			0
$897$	−8.65275	+	8.38842i	−0.288907	+	0.280081i
$898$		0			0
$899$	−29.0836			−0.969992
$900$		0			0
$901$		−	10.1047i		−	0.336637i
$902$		0			0
$903$	52.7177	−	51.1073i	1.75434	−	1.70074i
$904$		0			0
$905$		−	67.4674i		−	2.24269i
$906$		0			0
$907$	44.5008			1.47763			0.738813	−	0.673910i	$-0.235387\pi$
$907$	44.5008			1.47763			0.738813	+	0.673910i	$0.235387\pi$
$908$		0			0
$909$	−0.726798	+	23.4227i	−0.0241064	+	0.776883i
$910$		0			0
$911$			52.1022i			1.72622i	0.505013	+	0.863112i	$0.331488\pi$
$911$			52.1022i			1.72622i	−0.505013	+	0.863112i	$0.668512\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	−23.2291	+	22.5194i	−0.767929	+	0.744470i
$916$		0			0
$917$			22.0756i			0.729000i
$918$		0			0
$919$		−	45.7680i		−	1.50975i	−0.655870	−	0.754874i	$-0.727698\pi$
$919$		−	45.7680i		−	1.50975i	0.655870	−	0.754874i	$-0.272302\pi$
$920$		0			0
$921$	−8.89404	−	9.17430i	−0.293068	−	0.302303i
$922$		0			0
$923$	0.358838			0.0118113
$924$		0			0
$925$	−0.922317			−0.0303256
$926$		0			0
$927$	0.0407985	−	1.31482i	0.00134000	−	0.0431845i
$928$		0			0
$929$		−	24.9892i		−	0.819871i	−0.912115	−	0.409935i	$-0.865551\pi$
$929$		−	24.9892i		−	0.819871i	0.912115	−	0.409935i	$-0.134449\pi$
$930$		0			0
$931$			32.7654i			1.07384i
$932$		0			0
$933$	13.6671	+	14.0978i	0.447442	+	0.461542i
$934$		0			0
$935$		0			0
$936$		0			0
$937$			6.80770i			0.222398i	0.993798	+	0.111199i	$0.0354691\pi$
$937$			6.80770i			0.222398i	−0.993798	+	0.111199i	$0.964531\pi$
$938$		0			0
$939$	2.08802	−	2.02423i	0.0681399	−	0.0660583i
$940$		0			0
$941$	−0.266675			−0.00869335			−0.00434667	−	0.999991i	$-0.501384\pi$
$941$	−0.266675			−0.00869335			−0.00434667	+	0.999991i	$0.501384\pi$
$942$		0			0
$943$		−	41.4083i		−	1.34844i
$944$		0			0
$945$	−39.0735	−	42.8900i	−1.27106	−	1.39521i
$946$		0			0
$947$			33.2727i			1.08122i	0.841274	+	0.540609i	$0.181806\pi$
$947$			33.2727i			1.08122i	−0.841274	+	0.540609i	$0.818194\pi$
$948$		0			0
$949$	14.7165			0.477716
$950$		0			0
$951$	−1.02821	−	1.06061i	−0.0333419	−	0.0343926i
$952$		0			0
$953$	−34.9634			−1.13257			−0.566287	−	0.824208i	$-0.691621\pi$
$953$	−34.9634			−1.13257			−0.566287	+	0.824208i	$0.691621\pi$
$954$		0			0
$955$	29.5211			0.955280
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−13.0669			−0.421951
$960$		0			0
$961$	−15.6404			−0.504529
$962$		0			0
$963$	−0.969207	+	31.2349i	−0.0312323	+	1.00653i
$964$		0			0
$965$	6.90375			0.222240
$966$		0			0
$967$			28.9143i			0.929821i	0.885358	+	0.464910i	$0.153913\pi$
$967$			28.9143i			0.929821i	−0.885358	+	0.464910i	$0.846087\pi$
$968$		0			0
$969$	7.69266	+	7.93506i	0.247124	+	0.254911i
$970$		0			0
$971$			30.1285i			0.966870i	0.875380	+	0.483435i	$0.160611\pi$
$971$			30.1285i			0.966870i	−0.875380	+	0.483435i	$0.839389\pi$
$972$		0			0
$973$	−24.1767			−0.775069
$974$		0			0
$975$	5.67481	+	5.85363i	0.181739	+	0.187466i
$976$		0			0
$977$		−	21.3778i		−	0.683937i	−0.939711	−	0.341969i	$-0.888906\pi$
$977$		−	21.3778i		−	0.683937i	0.939711	−	0.341969i	$-0.111094\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	6.86933	+	0.213153i	0.219321	+	0.00680544i
$982$		0			0
$983$			50.2823i			1.60376i	0.597487	+	0.801879i	$0.296166\pi$
$983$			50.2823i			1.60376i	−0.597487	+	0.801879i	$0.703834\pi$
$984$		0			0
$985$			15.6195i			0.497680i
$986$		0			0
$987$	−14.4617	+	14.0199i	−0.460321	+	0.446259i
$988$		0			0
$989$	−74.7804			−2.37788
$990$		0			0
$991$	26.0847			0.828609			0.414305	−	0.910138i	$-0.364025\pi$
$991$	26.0847			0.828609			0.414305	+	0.910138i	$0.364025\pi$
$992$		0			0
$993$	26.2725	−	25.4699i	0.833732	−	0.808263i
$994$		0			0
$995$			29.9267i			0.948739i
$996$		0			0
$997$			22.5902i			0.715439i	0.933829	+	0.357720i	$0.116446\pi$
$997$			22.5902i			0.715439i	−0.933829	+	0.357720i	$0.883554\pi$
$998$		0			0
$999$	−0.740879	−	0.813244i	−0.0234404	−	0.0257299i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1452.2.b.e.725.10		16
3.2	odd	2	inner	1452.2.b.e.725.12		16
11.4	even	5		132.2.p.a.17.1	✓	16
11.8	odd	10		132.2.p.a.101.2	yes	16
11.10	odd	2	inner	1452.2.b.e.725.9		16
33.8	even	10		132.2.p.a.101.1	yes	16
33.26	odd	10		132.2.p.a.17.2	yes	16
33.32	even	2	inner	1452.2.b.e.725.11		16
44.15	odd	10		528.2.bn.d.17.4		16
44.19	even	10		528.2.bn.d.497.3		16
132.59	even	10		528.2.bn.d.17.3		16
132.107	odd	10		528.2.bn.d.497.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
132.2.p.a.17.1	✓	16	11.4	even	5
132.2.p.a.17.2	yes	16	33.26	odd	10
132.2.p.a.101.1	yes	16	33.8	even	10
132.2.p.a.101.2	yes	16	11.8	odd	10
528.2.bn.d.17.3		16	132.59	even	10
528.2.bn.d.17.4		16	44.15	odd	10
528.2.bn.d.497.3		16	44.19	even	10
528.2.bn.d.497.4		16	132.107	odd	10
1452.2.b.e.725.9		16	11.10	odd	2	inner
1452.2.b.e.725.10		16	1.1	even	1	trivial
1452.2.b.e.725.11		16	33.32	even	2	inner
1452.2.b.e.725.12		16	3.2	odd	2	inner

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1452 = 2^{2} \cdot 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1452.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.24359 - 1.20560i) q^{3} -3.05881i q^{5} +3.65040i q^{7} +(0.0930440 - 2.99856i) q^{9} -1.08050i q^{13} +(-3.68771 - 3.80392i) q^{15} -1.23182 q^{17} -5.17993i q^{19} +(4.40094 + 4.53961i) q^{21} -6.43947i q^{23} -4.35634 q^{25} +(-3.49936 - 3.84116i) q^{27} -7.42093 q^{29} +3.91913 q^{31} +11.1659 q^{35} +0.211719 q^{37} +(-1.30266 - 1.34371i) q^{39} +6.43039 q^{41} -11.6128i q^{43} +(-9.17202 - 0.284604i) q^{45} +3.18566i q^{47} -6.32545 q^{49} +(-1.53188 + 1.48509i) q^{51} +8.20307i q^{53} +(-6.24494 - 6.44173i) q^{57} -6.23422i q^{59} -6.10661i q^{61} +(10.9459 + 0.339648i) q^{63} -3.30506 q^{65} -3.24458 q^{67} +(-7.76344 - 8.00808i) q^{69} +0.332103i q^{71} +13.6200i q^{73} +(-5.41751 + 5.25201i) q^{75} -1.32789i q^{79} +(-8.98269 - 0.557995i) q^{81} +7.81134 q^{83} +3.76791i q^{85} +(-9.22861 + 8.94669i) q^{87} +10.3772i q^{89} +3.94427 q^{91} +(4.87380 - 4.72491i) q^{93} -15.8444 q^{95} +6.35634 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 6 q^{3} - 10 q^{9} - 16 q^{15} - 32 q^{25} + 6 q^{27} + 40 q^{31} + 36 q^{37} + 2 q^{45} - 28 q^{49} + 44 q^{67} + 16 q^{69} - 84 q^{75} - 58 q^{81} - 80 q^{91} + 12 q^{93} + 64 q^{97}+O(q^{100}) \) 16 * q + 6 * q^3 - 10 * q^9 - 16 * q^15 - 32 * q^25 + 6 * q^27 + 40 * q^31 + 36 * q^37 + 2 * q^45 - 28 * q^49 + 44 * q^67 + 16 * q^69 - 84 * q^75 - 58 * q^81 - 80 * q^91 + 12 * q^93 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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