LMFDB - Embedded newform 1305.2.d.f.811.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1305,2,Mod(811,1305)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1305.811"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1305, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1305.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,-6,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$10.4204774638$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 13x^{8} + 56x^{6} + 89x^{4} + 41x^{2} + 4 $ x^10 + 13x^8 + 56x^6 + 89x^4 + 41x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			811.7
Root			$0.732188i$ of defining polynomial
Character	$\chi$	$=$	1305.811
Dual form			1305.2.d.f.811.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.732188i q^{2} +1.46390 q^{4} +1.00000 q^{5} -4.08783 q^{7} +2.53623i q^{8} +0.732188i q^{10} -2.45619i q^{11} -0.393095 q^{13} -2.99306i q^{14} +1.07081 q^{16} +2.58997i q^{17} +7.38707i q^{19} +1.46390 q^{20} +1.79839 q^{22} +8.05499 q^{23} +1.00000 q^{25} -0.287819i q^{26} -5.98418 q^{28} +(0.825542 + 5.32151i) q^{29} +6.86403i q^{31} +5.85649i q^{32} -1.89635 q^{34} -4.08783 q^{35} +8.71766i q^{37} -5.40873 q^{38} +2.53623i q^{40} -11.3595i q^{41} +7.75140i q^{43} -3.59561i q^{44} +5.89776i q^{46} +2.06693i q^{47} +9.71037 q^{49} +0.732188i q^{50} -0.575452 q^{52} -3.02923 q^{53} -2.45619i q^{55} -10.3677i q^{56} +(-3.89635 + 0.604452i) q^{58} +8.03405 q^{59} -11.5529i q^{61} -5.02576 q^{62} -2.14644 q^{64} -0.393095 q^{65} -12.8083 q^{67} +3.79146i q^{68} -2.99306i q^{70} +16.4929 q^{71} +3.80149i q^{73} -6.38297 q^{74} +10.8139i q^{76} +10.0405i q^{77} -7.38707i q^{79} +1.07081 q^{80} +8.31728 q^{82} -8.43795 q^{83} +2.58997i q^{85} -5.67548 q^{86} +6.22944 q^{88} -6.04177i q^{89} +1.60691 q^{91} +11.7917 q^{92} -1.51338 q^{94} +7.38707i q^{95} -10.4496i q^{97} +7.10982i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 6 q^{4} + 10 q^{5} - 4 q^{7} + 4 q^{13} - 2 q^{16} - 6 q^{20} + 10 q^{22} + 10 q^{23} + 10 q^{25} - 2 q^{28} + 2 q^{34} - 4 q^{35} + 32 q^{38} - 10 q^{49} - 34 q^{52} - 14 q^{53} - 18 q^{58} + 32 q^{59}+ \cdots - 26 q^{94}+O(q^{100}) $ 10 * q - 6 * q^4 + 10 * q^5 - 4 * q^7 + 4 * q^13 - 2 * q^16 - 6 * q^20 + 10 * q^22 + 10 * q^23 + 10 * q^25 - 2 * q^28 + 2 * q^34 - 4 * q^35 + 32 * q^38 - 10 * q^49 - 34 * q^52 - 14 * q^53 - 18 * q^58 + 32 * q^59 + 4 * q^62 + 36 * q^64 + 4 * q^65 - 8 * q^67 - 8 * q^71 - 32 * q^74 - 2 * q^80 - 16 * q^82 + 18 * q^83 - 56 * q^86 + 24 * q^91 - 26 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times$.

$n$	$146$	$262$	$901$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$			0.732188i			0.517735i	0.965913	+	0.258868i	$0.0833493\pi$
$2$			0.732188i			0.517735i	−0.965913	+	0.258868i	$0.916651\pi$
$3$		0			0
$3$		0			0
$4$	1.46390			0.731950
$5$	1.00000			0.447214
$5$	1.00000			0.447214
$6$		0			0
$7$	−4.08783			−1.54506			−0.772528	−	0.634981i	$-0.781008\pi$
$7$	−4.08783			−1.54506			−0.772528	+	0.634981i	$0.781008\pi$
$8$			2.53623i			0.896692i
$9$		0			0
$10$			0.732188i			0.231538i
$11$		−	2.45619i		−	0.740568i	−0.928919	−	0.370284i	$-0.879260\pi$
$11$		−	2.45619i		−	0.740568i	0.928919	−	0.370284i	$-0.120740\pi$
$12$		0			0
$13$	−0.393095			−0.109025			−0.0545124	−	0.998513i	$-0.517360\pi$
$13$	−0.393095			−0.109025			−0.0545124	+	0.998513i	$0.517360\pi$
$14$		−	2.99306i		−	0.799929i
$15$		0			0
$16$	1.07081			0.267701
$17$			2.58997i			0.628161i	0.949396	+	0.314080i	$0.101696\pi$
$17$			2.58997i			0.628161i	−0.949396	+	0.314080i	$0.898304\pi$
$18$		0			0
$19$			7.38707i			1.69471i	0.531026	+	0.847355i	$0.321807\pi$
$19$			7.38707i			1.69471i	−0.531026	+	0.847355i	$0.678193\pi$
$20$	1.46390			0.327338
$21$		0			0
$22$	1.79839			0.383418
$23$	8.05499			1.67958			0.839790	−	0.542911i	$-0.182678\pi$
$23$	8.05499			1.67958			0.839790	+	0.542911i	$0.182678\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		−	0.287819i		−	0.0564460i
$27$		0			0
$28$	−5.98418			−1.13090
$29$	0.825542	+	5.32151i	0.153299	+	0.988180i
$29$	0.825542	+	5.32151i	0.153299	+	0.988180i
$30$		0			0
$31$			6.86403i			1.23282i	0.787427	+	0.616408i	$0.211413\pi$
$31$			6.86403i			1.23282i	−0.787427	+	0.616408i	$0.788587\pi$
$32$			5.85649i			1.03529i
$33$		0			0
$34$	−1.89635			−0.325221
$35$	−4.08783			−0.690970
$36$		0			0
$37$			8.71766i			1.43317i	0.697497	+	0.716587i	$0.254297\pi$
$37$			8.71766i			1.43317i	−0.697497	+	0.716587i	$0.745703\pi$
$38$	−5.40873			−0.877412
$39$		0			0
$40$			2.53623i			0.401013i
$41$		−	11.3595i		−	1.77405i	−0.461719	−	0.887026i	$-0.652767\pi$
$41$		−	11.3595i		−	1.77405i	0.461719	−	0.887026i	$-0.347233\pi$
$42$		0			0
$43$			7.75140i			1.18208i	0.806643	+	0.591039i	$0.201282\pi$
$43$			7.75140i			1.18208i	−0.806643	+	0.591039i	$0.798718\pi$
$44$		−	3.59561i		−	0.542059i
$45$		0			0
$46$			5.89776i			0.869578i
$47$			2.06693i			0.301492i	0.988573	+	0.150746i	$0.0481676\pi$
$47$			2.06693i			0.301492i	−0.988573	+	0.150746i	$0.951832\pi$
$48$		0			0
$49$	9.71037			1.38720
$50$			0.732188i			0.103547i
$51$		0			0
$52$	−0.575452			−0.0798008
$53$	−3.02923			−0.416096			−0.208048	−	0.978119i	$-0.566711\pi$
$53$	−3.02923			−0.416096			−0.208048	+	0.978119i	$0.566711\pi$
$54$		0			0
$55$		−	2.45619i		−	0.331192i
$56$		−	10.3677i		−	1.38544i
$57$		0			0
$58$	−3.89635	+	0.604452i	−0.511615	+	0.0793684i
$59$	8.03405			1.04594			0.522972	−	0.852350i	$-0.324823\pi$
$59$	8.03405			1.04594			0.522972	+	0.852350i	$0.324823\pi$
$60$		0			0
$61$		−	11.5529i		−	1.47920i	−0.673049	−	0.739598i	$-0.735016\pi$
$61$		−	11.5529i		−	1.47920i	0.673049	−	0.739598i	$-0.264984\pi$
$62$	−5.02576			−0.638272
$63$		0			0
$64$	−2.14644			−0.268305
$65$	−0.393095			−0.0487574
$66$		0			0
$67$	−12.8083			−1.56479			−0.782393	−	0.622785i	$-0.786001\pi$
$67$	−12.8083			−1.56479			−0.782393	+	0.622785i	$0.786001\pi$
$68$			3.79146i			0.459782i
$69$		0			0
$70$		−	2.99306i		−	0.357739i
$71$	16.4929			1.95735			0.978676	−	0.205411i	$-0.0658530\pi$
$71$	16.4929			1.95735			0.978676	+	0.205411i	$0.0658530\pi$
$72$		0			0
$73$			3.80149i			0.444930i	0.974941	+	0.222465i	$0.0714104\pi$
$73$			3.80149i			0.444930i	−0.974941	+	0.222465i	$0.928590\pi$
$74$	−6.38297			−0.742005
$75$		0			0
$76$			10.8139i			1.24044i
$77$			10.0405i			1.14422i
$78$		0			0
$79$		−	7.38707i		−	0.831111i	−0.909568	−	0.415555i	$-0.863587\pi$
$79$		−	7.38707i		−	0.831111i	0.909568	−	0.415555i	$-0.136413\pi$
$80$	1.07081			0.119720
$81$		0			0
$82$	8.31728			0.918489
$83$	−8.43795			−0.926186			−0.463093	−	0.886310i	$-0.653260\pi$
$83$	−8.43795			−0.926186			−0.463093	+	0.886310i	$0.653260\pi$
$84$		0			0
$85$			2.58997i			0.280922i
$86$	−5.67548			−0.612003
$87$		0			0
$88$	6.22944			0.664061
$89$		−	6.04177i		−	0.640427i	−0.947345	−	0.320213i	$-0.896245\pi$
$89$		−	6.04177i		−	0.640427i	0.947345	−	0.320213i	$-0.103755\pi$
$90$		0			0
$91$	1.60691			0.168449
$92$	11.7917			1.22937
$93$		0			0
$94$	−1.51338			−0.156093
$95$			7.38707i			0.757898i
$96$		0			0
$97$		−	10.4496i		−	1.06100i	−0.847686	−	0.530499i	$-0.822005\pi$
$97$		−	10.4496i		−	1.06100i	0.847686	−	0.530499i	$-0.177995\pi$
$98$			7.10982i			0.718200i
$99$		0			0
$100$	1.46390			0.146390
$101$			0.249057i			0.0247821i	0.999923	+	0.0123911i	$0.00394430\pi$
$101$			0.249057i			0.0247821i	−0.999923	+	0.0123911i	$0.996056\pi$
$102$		0			0
$103$	7.40873			0.730004			0.365002	−	0.931007i	$-0.381068\pi$
$103$	7.40873			0.730004			0.365002	+	0.931007i	$0.381068\pi$
$104$		−	0.996977i		−	0.0977617i
$105$		0			0
$106$		−	2.21796i		−	0.215428i
$107$	−7.05981			−0.682498			−0.341249	−	0.939973i	$-0.610850\pi$
$107$	−7.05981			−0.682498			−0.341249	+	0.939973i	$0.610850\pi$
$108$		0			0
$109$	−10.8118			−1.03558			−0.517791	−	0.855507i	$-0.673246\pi$
$109$	−10.8118			−1.03558			−0.517791	+	0.855507i	$0.673246\pi$
$110$	1.79839			0.171470
$111$		0			0
$112$	−4.37727			−0.413614
$113$		−	20.7872i		−	1.95549i	−0.209788	−	0.977747i	$-0.567278\pi$
$113$		−	20.7872i		−	1.95549i	0.209788	−	0.977747i	$-0.432722\pi$
$114$		0			0
$115$	8.05499			0.751131
$116$	1.20851	+	7.79016i	0.112207	+	0.723298i
$117$		0			0
$118$			5.88244i			0.541522i
$119$		−	10.5874i		−	0.970543i
$120$		0			0
$121$	4.96715			0.451559
$122$	8.45889			0.765832
$123$		0			0
$124$			10.0483i			0.902360i
$125$	1.00000			0.0894427
$126$		0			0
$127$			8.27445i			0.734239i	0.930174	+	0.367119i	$0.119656\pi$
$127$			8.27445i			0.734239i	−0.930174	+	0.367119i	$0.880344\pi$
$128$			10.1414i			0.896379i
$129$		0			0
$130$		−	0.287819i		−	0.0252434i
$131$			6.75579i			0.590256i	0.955458	+	0.295128i	$0.0953623\pi$
$131$			6.75579i			0.590256i	−0.955458	+	0.295128i	$0.904638\pi$
$132$		0			0
$133$		−	30.1971i		−	2.61842i
$134$		−	9.37811i		−	0.810145i
$135$		0			0
$136$	−6.56876			−0.563266
$137$			10.6430i			0.909295i	0.890672	+	0.454647i	$0.150235\pi$
$137$			10.6430i			0.909295i	−0.890672	+	0.454647i	$0.849765\pi$
$138$		0			0
$139$	−13.8568			−1.17532			−0.587660	−	0.809108i	$-0.699951\pi$
$139$	−13.8568			−1.17532			−0.587660	+	0.809108i	$0.699951\pi$
$140$	−5.98418			−0.505755
$141$		0			0
$142$			12.0759i			1.01339i
$143$			0.965514i			0.0807403i
$144$		0			0
$145$	0.825542	+	5.32151i	0.0685575	+	0.441927i
$146$	−2.78340			−0.230356
$147$		0			0
$148$			12.7618i			1.04901i
$149$	−14.7834			−1.21110			−0.605552	−	0.795806i	$-0.707048\pi$
$149$	−14.7834			−1.21110			−0.605552	+	0.795806i	$0.707048\pi$
$150$		0			0
$151$	7.00483			0.570045			0.285022	−	0.958521i	$-0.407999\pi$
$151$	7.00483			0.570045			0.285022	+	0.958521i	$0.407999\pi$
$152$	−18.7353			−1.51963
$153$		0			0
$154$	−7.35152			−0.592402
$155$			6.86403i			0.551332i
$156$		0			0
$157$		−	14.2511i		−	1.13736i	−0.822558	−	0.568681i	$-0.807454\pi$
$157$		−	14.2511i		−	1.13736i	0.822558	−	0.568681i	$-0.192546\pi$
$158$	5.40873			0.430295
$159$		0			0
$160$			5.85649i			0.462996i
$161$	−32.9274			−2.59504
$162$		0			0
$163$			9.68392i			0.758503i	0.925294	+	0.379252i	$0.123819\pi$
$163$			9.68392i			0.758503i	−0.925294	+	0.379252i	$0.876181\pi$
$164$		−	16.6291i		−	1.29852i
$165$		0			0
$166$		−	6.17817i		−	0.479519i
$167$	−20.7693			−1.60718			−0.803590	−	0.595183i	$-0.797080\pi$
$167$	−20.7693			−1.60718			−0.803590	+	0.595183i	$0.797080\pi$
$168$		0			0
$169$	−12.8455			−0.988114
$170$	−1.89635			−0.145443
$171$		0			0
$172$			11.3473i			0.865222i
$173$	13.4637			1.02363			0.511814	−	0.859097i	$-0.328974\pi$
$173$	13.4637			1.02363			0.511814	+	0.859097i	$0.328974\pi$
$174$		0			0
$175$	−4.08783			−0.309011
$176$		−	2.63010i		−	0.198251i
$177$		0			0
$178$	4.42371			0.331571
$179$	10.4345			0.779910			0.389955	−	0.920834i	$-0.372491\pi$
$179$	10.4345			0.779910			0.389955	+	0.920834i	$0.372491\pi$
$180$		0			0
$181$	5.61824			0.417600			0.208800	−	0.977958i	$-0.433044\pi$
$181$	5.61824			0.417600			0.208800	+	0.977958i	$0.433044\pi$
$182$			1.17656i			0.0872122i
$183$		0			0
$184$			20.4293i			1.50607i
$185$			8.71766i			0.640935i
$186$		0			0
$187$	6.36145			0.465196
$188$			3.02577i			0.220677i
$189$		0			0
$190$	−5.40873			−0.392390
$191$			10.2787i			0.743740i	0.928285	+	0.371870i	$0.121283\pi$
$191$			10.2787i			0.743740i	−0.928285	+	0.371870i	$0.878717\pi$
$192$		0			0
$193$		−	3.87008i		−	0.278575i	−0.990252	−	0.139287i	$-0.955519\pi$
$193$		−	3.87008i		−	0.278575i	0.990252	−	0.139287i	$-0.0444811\pi$
$194$	7.65108			0.549316
$195$		0			0
$196$	14.2150			1.01536
$197$	17.8126			1.26910			0.634549	−	0.772883i	$-0.281186\pi$
$197$	17.8126			1.26910			0.634549	+	0.772883i	$0.281186\pi$
$198$		0			0
$199$	19.3907			1.37457			0.687284	−	0.726388i	$-0.258803\pi$
$199$	19.3907			1.37457			0.687284	+	0.726388i	$0.258803\pi$
$200$			2.53623i			0.179338i
$201$		0			0
$202$	−0.182357			−0.0128306
$203$	−3.37468	−	21.7534i	−0.236856	−	1.52679i
$204$		0			0
$205$		−	11.3595i		−	0.793380i
$206$			5.42458i			0.377949i
$207$		0			0
$208$	−0.420928			−0.0291861
$209$	18.1440			1.25505
$210$		0			0
$211$			12.2234i			0.841495i	0.907178	+	0.420747i	$0.138232\pi$
$211$			12.2234i			0.841495i	−0.907178	+	0.420747i	$0.861768\pi$
$212$	−4.43449			−0.304562
$213$		0			0
$214$		−	5.16911i		−	0.353353i
$215$			7.75140i			0.528641i
$216$		0			0
$217$		−	28.0590i		−	1.90477i
$218$		−	7.91627i		−	0.536157i
$219$		0			0
$220$		−	3.59561i		−	0.242416i
$221$		−	1.01810i		−	0.0684851i
$222$		0			0
$223$	−15.8262			−1.05980			−0.529901	−	0.848059i	$-0.677771\pi$
$223$	−15.8262			−1.05980			−0.529901	+	0.848059i	$0.677771\pi$
$224$		−	23.9403i		−	1.59958i
$225$		0			0
$226$	15.2201			1.01243
$227$	15.1873			1.00802			0.504008	−	0.863699i	$-0.331858\pi$
$227$	15.1873			1.00802			0.504008	+	0.863699i	$0.331858\pi$
$228$		0			0
$229$		−	20.1335i		−	1.33046i	−0.746638	−	0.665231i	$-0.768333\pi$
$229$		−	20.1335i		−	1.33046i	0.746638	−	0.665231i	$-0.231667\pi$
$230$			5.89776i			0.388887i
$231$		0			0
$232$	−13.4966	+	2.09376i	−0.886093	+	0.137462i
$233$	18.7144			1.22602			0.613009	−	0.790076i	$-0.289959\pi$
$233$	18.7144			1.22602			0.613009	+	0.790076i	$0.289959\pi$
$234$		0			0
$235$			2.06693i			0.134831i
$236$	11.7611			0.765579
$237$		0			0
$238$	7.75195			0.502484
$239$	−20.4589			−1.32338			−0.661688	−	0.749780i	$-0.730160\pi$
$239$	−20.4589			−1.32338			−0.661688	+	0.749780i	$0.730160\pi$
$240$		0			0
$241$	16.7558			1.07933			0.539666	−	0.841879i	$-0.318550\pi$
$241$	16.7558			1.07933			0.539666	+	0.841879i	$0.318550\pi$
$242$			3.63689i			0.233788i
$243$		0			0
$244$		−	16.9123i		−	1.08270i
$245$	9.71037			0.620373
$246$		0			0
$247$		−	2.90382i		−	0.184766i
$248$	−17.4087			−1.10546
$249$		0			0
$250$			0.732188i			0.0463076i
$251$			16.7970i			1.06022i	0.847930	+	0.530109i	$0.177849\pi$
$251$			16.7970i			1.06022i	−0.847930	+	0.530109i	$0.822151\pi$
$252$		0			0
$253$		−	19.7845i		−	1.24384i
$254$	−6.05845			−0.380141
$255$		0			0
$256$	−11.7183			−0.732392
$257$	−9.49777			−0.592454			−0.296227	−	0.955118i	$-0.595729\pi$
$257$	−9.49777			−0.592454			−0.296227	+	0.955118i	$0.595729\pi$
$258$		0			0
$259$		−	35.6363i		−	2.21433i
$260$	−0.575452			−0.0356880
$261$		0			0
$262$	−4.94651			−0.305596
$263$		−	17.8221i		−	1.09896i	−0.835507	−	0.549480i	$-0.814826\pi$
$263$		−	17.8221i		−	1.09896i	0.835507	−	0.549480i	$-0.185174\pi$
$264$		0			0
$265$	−3.02923			−0.186084
$266$	22.1100			1.35565
$267$		0			0
$268$	−18.7501			−1.14535
$269$		−	12.3387i		−	0.752303i	−0.926558	−	0.376151i	$-0.877247\pi$
$269$		−	12.3387i		−	0.752303i	0.926558	−	0.376151i	$-0.122753\pi$
$270$		0			0
$271$		−	12.2234i		−	0.742520i	−0.928529	−	0.371260i	$-0.878926\pi$
$271$		−	12.2234i		−	0.742520i	0.928529	−	0.371260i	$-0.121074\pi$
$272$			2.77336i			0.168160i
$273$		0			0
$274$	−7.79269			−0.470774
$275$		−	2.45619i		−	0.148114i
$276$		0			0
$277$	−5.81142			−0.349175			−0.174587	−	0.984642i	$-0.555859\pi$
$277$	−5.81142			−0.349175			−0.174587	+	0.984642i	$0.555859\pi$
$278$		−	10.1458i		−	0.608504i
$279$		0			0
$280$		−	10.3677i		−	0.619587i
$281$	16.4345			0.980399			0.490200	−	0.871610i	$-0.336924\pi$
$281$	16.4345			0.980399			0.490200	+	0.871610i	$0.336924\pi$
$282$		0			0
$283$	13.4672			0.800541			0.400270	−	0.916397i	$-0.368916\pi$
$283$	13.4672			0.800541			0.400270	+	0.916397i	$0.368916\pi$
$284$	24.1440			1.43268
$285$		0			0
$286$	−0.706938			−0.0418021
$287$			46.4356i			2.74101i
$288$		0			0
$289$	10.2920			0.605414
$290$	−3.89635	+	0.604452i	−0.228801	+	0.0354946i
$291$		0			0
$292$			5.56500i			0.325667i
$293$		−	19.7986i		−	1.15664i	−0.815808	−	0.578322i	$-0.803708\pi$
$293$		−	19.7986i		−	1.15664i	0.815808	−	0.578322i	$-0.196292\pi$
$294$		0			0
$295$	8.03405			0.467761
$296$	−22.1100			−1.28512
$297$		0			0
$298$		−	10.8242i		−	0.627031i
$299$	−3.16637			−0.183116
$300$		0			0
$301$		−	31.6864i		−	1.82638i
$302$			5.12885i			0.295132i
$303$		0			0
$304$			7.91012i			0.453677i
$305$		−	11.5529i		−	0.661516i
$306$		0			0
$307$		−	13.6338i		−	0.778124i	−0.921212	−	0.389062i	$-0.872799\pi$
$307$		−	13.6338i		−	0.778124i	0.921212	−	0.389062i	$-0.127201\pi$
$308$			14.6983i			0.837511i
$309$		0			0
$310$	−5.02576			−0.285444
$311$			12.3180i			0.698488i	0.937032	+	0.349244i	$0.113561\pi$
$311$			12.3180i			0.698488i	−0.937032	+	0.349244i	$0.886439\pi$
$312$		0			0
$313$	3.93662			0.222511			0.111255	−	0.993792i	$-0.464513\pi$
$313$	3.93662			0.222511			0.111255	+	0.993792i	$0.464513\pi$
$314$	10.4345			0.588852
$315$		0			0
$316$		−	10.8139i		−	0.608332i
$317$			10.9075i			0.612628i	0.951930	+	0.306314i	$0.0990958\pi$
$317$			10.9075i			0.612628i	−0.951930	+	0.306314i	$0.900904\pi$
$318$		0			0
$319$	13.0706	−	2.02768i	0.731814	−	0.113528i
$320$	−2.14644			−0.119989
$321$		0			0
$322$		−	24.1091i		−	1.34355i
$323$	−19.1323			−1.06455
$324$		0			0
$325$	−0.393095			−0.0218050
$326$	−7.09045			−0.392704
$327$		0			0
$328$	28.8102			1.59078
$329$		−	8.44924i		−	0.465822i
$330$		0			0
$331$		−	6.86403i		−	0.377281i	−0.982046	−	0.188640i	$-0.939592\pi$
$331$		−	6.86403i		−	0.377281i	0.982046	−	0.188640i	$-0.0604081\pi$
$332$	−12.3523			−0.677922
$333$		0			0
$334$		−	15.2071i		−	0.832094i
$335$	−12.8083			−0.699794
$336$		0			0
$337$			5.35939i			0.291945i	0.989289	+	0.145972i	$0.0466310\pi$
$337$			5.35939i			0.291945i	−0.989289	+	0.145972i	$0.953369\pi$
$338$		−	9.40531i		−	0.511581i
$339$		0			0
$340$			3.79146i			0.205621i
$341$	16.8593			0.912983
$342$		0			0
$343$	−11.0795			−0.598239
$344$	−19.6593			−1.05996
$345$		0			0
$346$			9.85797i			0.529968i
$347$	20.8628			1.11997			0.559987	−	0.828502i	$-0.310806\pi$
$347$	20.8628			1.11997			0.559987	+	0.828502i	$0.310806\pi$
$348$		0			0
$349$	18.7659			1.00451			0.502257	−	0.864718i	$-0.332503\pi$
$349$	18.7659			1.00451			0.502257	+	0.864718i	$0.332503\pi$
$350$		−	2.99306i		−	0.159986i
$351$		0			0
$352$	14.3846			0.766702
$353$	16.8175			0.895103			0.447551	−	0.894258i	$-0.352296\pi$
$353$	16.8175			0.895103			0.447551	+	0.894258i	$0.352296\pi$
$354$		0			0
$355$	16.4929			0.875354
$356$		−	8.84455i		−	0.468760i
$357$		0			0
$358$			7.64001i			0.403787i
$359$		−	34.0437i		−	1.79676i	−0.439219	−	0.898380i	$-0.644745\pi$
$359$		−	34.0437i		−	1.79676i	0.439219	−	0.898380i	$-0.355255\pi$
$360$		0			0
$361$	−35.5689			−1.87205
$362$			4.11361i			0.216206i
$363$		0			0
$364$	2.35235			0.123297
$365$			3.80149i			0.198979i
$366$		0			0
$367$		−	16.3321i		−	0.852526i	−0.904599	−	0.426263i	$-0.859830\pi$
$367$		−	16.3321i		−	0.852526i	0.904599	−	0.426263i	$-0.140170\pi$
$368$	8.62532			0.449626
$369$		0			0
$370$	−6.38297			−0.331835
$371$	12.3830			0.642892
$372$		0			0
$373$	−14.8759			−0.770245			−0.385123	−	0.922865i	$-0.625841\pi$
$373$	−14.8759			−0.770245			−0.385123	+	0.922865i	$0.625841\pi$
$374$			4.65778i			0.240848i
$375$		0			0
$376$	−5.24219			−0.270345
$377$	−0.324516	−	2.09186i	−0.0167134	−	0.107736i
$378$		0			0
$379$			27.7633i			1.42610i	0.701112	+	0.713051i	$0.252687\pi$
$379$			27.7633i			1.42610i	−0.701112	+	0.713051i	$0.747313\pi$
$380$			10.8139i			0.554743i
$381$		0			0
$382$	−7.52594			−0.385061
$383$	4.08904			0.208940			0.104470	−	0.994528i	$-0.466685\pi$
$383$	4.08904			0.208940			0.104470	+	0.994528i	$0.466685\pi$
$384$		0			0
$385$			10.0405i			0.511710i
$386$	2.83363			0.144228
$387$		0			0
$388$		−	15.2972i		−	0.776597i
$389$			9.61272i			0.487384i	0.969853	+	0.243692i	$0.0783586\pi$
$389$			9.61272i			0.487384i	−0.969853	+	0.243692i	$0.921641\pi$
$390$		0			0
$391$			20.8622i			1.05505i
$392$			24.6277i			1.24389i
$393$		0			0
$394$			13.0422i			0.657056i
$395$		−	7.38707i		−	0.371684i
$396$		0			0
$397$	−4.74935			−0.238363			−0.119182	−	0.992872i	$-0.538027\pi$
$397$	−4.74935			−0.238363			−0.119182	+	0.992872i	$0.538027\pi$
$398$			14.1976i			0.711663i
$399$		0			0
$400$	1.07081			0.0535403
$401$	−25.8536			−1.29107			−0.645533	−	0.763733i	$-0.723365\pi$
$401$	−25.8536			−1.29107			−0.645533	+	0.763733i	$0.723365\pi$
$402$		0			0
$403$		−	2.69821i		−	0.134408i
$404$			0.364595i			0.0181393i
$405$		0			0
$406$	15.9276	−	2.47090i	0.790474	−	0.122629i
$407$	21.4122			1.06136
$408$		0			0
$409$			24.0835i			1.19085i	0.803411	+	0.595425i	$0.203016\pi$
$409$			24.0835i			1.19085i	−0.803411	+	0.595425i	$0.796984\pi$
$410$	8.31728			0.410761
$411$		0			0
$412$	10.8456			0.534326
$413$	−32.8419			−1.61604
$414$		0			0
$415$	−8.43795			−0.414203
$416$		−	2.30215i		−	0.112872i
$417$		0			0
$418$			13.2848i			0.649783i
$419$	6.46854			0.316009			0.158004	−	0.987438i	$-0.449494\pi$
$419$	6.46854			0.316009			0.158004	+	0.987438i	$0.449494\pi$
$420$		0			0
$421$		−	8.26321i		−	0.402724i	−0.979517	−	0.201362i	$-0.935463\pi$
$421$		−	8.26321i		−	0.402724i	0.979517	−	0.201362i	$-0.0645368\pi$
$422$	−8.94984			−0.435671
$423$		0			0
$424$		−	7.68281i		−	0.373110i
$425$			2.58997i			0.125632i
$426$		0			0
$427$			47.2263i			2.28544i
$428$	−10.3349			−0.499554
$429$		0			0
$430$	−5.67548			−0.273696
$431$	−4.10997			−0.197970			−0.0989852	−	0.995089i	$-0.531560\pi$
$431$	−4.10997			−0.197970			−0.0989852	+	0.995089i	$0.531560\pi$
$432$		0			0
$433$		−	14.3197i		−	0.688161i	−0.938940	−	0.344080i	$-0.888191\pi$
$433$		−	14.3197i		−	0.688161i	0.938940	−	0.344080i	$-0.111809\pi$
$434$	20.5445			0.986165
$435$		0			0
$436$	−15.8274			−0.757995
$437$			59.5028i			2.84640i
$438$		0			0
$439$	−33.0589			−1.57782			−0.788908	−	0.614511i	$-0.789353\pi$
$439$	−33.0589			−1.57782			−0.788908	+	0.614511i	$0.789353\pi$
$440$	6.22944			0.296977
$441$		0			0
$442$	0.745444			0.0354572
$443$		−	21.8045i		−	1.03596i	−0.855391	−	0.517982i	$-0.826683\pi$
$443$		−	21.8045i		−	1.03596i	0.855391	−	0.517982i	$-0.173317\pi$
$444$		0			0
$445$		−	6.04177i		−	0.286408i
$446$		−	11.5878i		−	0.548697i
$447$		0			0
$448$	8.77428			0.414546
$449$			16.0693i			0.758357i	0.925324	+	0.379179i	$0.123793\pi$
$449$			16.0693i			0.758357i	−0.925324	+	0.379179i	$0.876207\pi$
$450$		0			0
$451$	−27.9010			−1.31381
$452$		−	30.4304i		−	1.43132i
$453$		0			0
$454$			11.1200i			0.521886i
$455$	1.60691			0.0753329
$456$		0			0
$457$	−1.20090			−0.0561757			−0.0280878	−	0.999605i	$-0.508942\pi$
$457$	−1.20090			−0.0561757			−0.0280878	+	0.999605i	$0.508942\pi$
$458$	14.7415			0.688827
$459$		0			0
$460$	11.7917			0.549791
$461$		−	27.0201i		−	1.25845i	−0.777223	−	0.629225i	$-0.783373\pi$
$461$		−	27.0201i		−	1.25845i	0.777223	−	0.629225i	$-0.216627\pi$
$462$		0			0
$463$	−0.766462			−0.0356205			−0.0178103	−	0.999841i	$-0.505669\pi$
$463$	−0.766462			−0.0356205			−0.0178103	+	0.999841i	$0.505669\pi$
$464$	0.883995	+	5.69830i	0.0410384	+	0.264537i
$465$		0			0
$466$			13.7024i			0.634753i
$467$			17.0485i			0.788911i	0.918915	+	0.394455i	$0.129067\pi$
$467$			17.0485i			0.788911i	−0.918915	+	0.394455i	$0.870933\pi$
$468$		0			0
$469$	52.3583			2.41768
$470$	−1.51338			−0.0698069
$471$		0			0
$472$			20.3762i			0.937890i
$473$	19.0389			0.875409
$474$		0			0
$475$			7.38707i			0.338942i
$476$		−	15.4989i		−	0.710389i
$477$		0			0
$478$		−	14.9798i		−	0.685158i
$479$			6.44017i			0.294259i	0.989117	+	0.147130i	$0.0470034\pi$
$479$			6.44017i			0.294259i	−0.989117	+	0.147130i	$0.952997\pi$
$480$		0			0
$481$		−	3.42687i		−	0.156252i
$482$			12.2684i			0.558809i
$483$		0			0
$484$	7.27142			0.330519
$485$		−	10.4496i		−	0.474493i
$486$		0			0
$487$	14.1581			0.641564			0.320782	−	0.947153i	$-0.396054\pi$
$487$	14.1581			0.641564			0.320782	+	0.947153i	$0.396054\pi$
$488$	29.3007			1.32638
$489$		0			0
$490$			7.10982i			0.321189i
$491$			7.48627i			0.337851i	0.985629	+	0.168925i	$0.0540297\pi$
$491$			7.48627i			0.337851i	−0.985629	+	0.168925i	$0.945970\pi$
$492$		0			0
$493$	−13.7826	+	2.13813i	−0.620736	+	0.0962965i
$494$	2.12614			0.0956597
$495$		0			0
$496$			7.35004i			0.330026i
$497$	−67.4204			−3.02422
$498$		0			0
$499$	−12.4742			−0.558422			−0.279211	−	0.960230i	$-0.590073\pi$
$499$	−12.4742			−0.558422			−0.279211	+	0.960230i	$0.590073\pi$
$500$	1.46390			0.0654676
$501$		0			0
$502$	−12.2986			−0.548912
$503$		−	40.6320i		−	1.81169i	−0.423607	−	0.905846i	$-0.639236\pi$
$503$		−	40.6320i		−	1.81169i	0.423607	−	0.905846i	$-0.360764\pi$
$504$		0			0
$505$			0.249057i			0.0110829i
$506$	14.4860			0.643981
$507$		0			0
$508$			12.1130i			0.537426i
$509$	−12.3664			−0.548130			−0.274065	−	0.961711i	$-0.588368\pi$
$509$	−12.3664			−0.548130			−0.274065	+	0.961711i	$0.588368\pi$
$510$		0			0
$511$		−	15.5398i		−	0.687442i
$512$			11.7028i			0.517194i
$513$		0			0
$514$		−	6.95415i		−	0.306734i
$515$	7.40873			0.326468
$516$		0			0
$517$	5.07675			0.223275
$518$	26.0925			1.14644
$519$		0			0
$520$		−	0.996977i		−	0.0437204i
$521$	7.19077			0.315033			0.157517	−	0.987516i	$-0.449651\pi$
$521$	7.19077			0.315033			0.157517	+	0.987516i	$0.449651\pi$
$522$		0			0
$523$	−30.2075			−1.32088			−0.660440	−	0.750879i	$-0.729630\pi$
$523$	−30.2075			−1.32088			−0.660440	+	0.750879i	$0.729630\pi$
$524$			9.88980i			0.432038i
$525$		0			0
$526$	13.0492			0.568970
$527$	−17.7776			−0.774406
$528$		0			0
$529$	41.8828			1.82099
$530$		−	2.21796i		−	0.0963422i
$531$		0			0
$532$		−	44.2056i		−	1.91655i
$533$			4.46535i			0.193416i
$534$		0			0
$535$	−7.05981			−0.305222
$536$		−	32.4848i		−	1.40313i
$537$		0			0
$538$	9.03424			0.389494
$539$		−	23.8505i		−	1.02731i
$540$		0			0
$541$			5.95085i			0.255847i	0.991784	+	0.127924i	$0.0408312\pi$
$541$			5.95085i			0.255847i	−0.991784	+	0.127924i	$0.959169\pi$
$542$	8.94984			0.384429
$543$		0			0
$544$	−15.1681			−0.650329
$545$	−10.8118			−0.463126
$546$		0			0
$547$	28.2579			1.20822			0.604111	−	0.796900i	$-0.293528\pi$
$547$	28.2579			1.20822			0.604111	+	0.796900i	$0.293528\pi$
$548$			15.5803i			0.665559i
$549$		0			0
$550$	1.79839			0.0766836
$551$	−39.3104	+	6.09834i	−1.67468	+	0.259798i
$552$		0			0
$553$			30.1971i			1.28411i
$554$		−	4.25506i		−	0.180780i
$555$		0			0
$556$	−20.2850			−0.860275
$557$	5.38780			0.228288			0.114144	−	0.993464i	$-0.463587\pi$
$557$	5.38780			0.228288			0.114144	+	0.993464i	$0.463587\pi$
$558$		0			0
$559$		−	3.04703i		−	0.128876i
$560$	−4.37727			−0.184974
$561$		0			0
$562$			12.0331i			0.507587i
$563$		−	13.1414i		−	0.553844i	−0.960892	−	0.276922i	$-0.910686\pi$
$563$		−	13.1414i		−	0.553844i	0.960892	−	0.276922i	$-0.0893144\pi$
$564$		0			0
$565$		−	20.7872i		−	0.874523i
$566$			9.86051i			0.414468i
$567$		0			0
$568$			41.8298i			1.75514i
$569$			16.7529i			0.702320i	0.936315	+	0.351160i	$0.114213\pi$
$569$			16.7529i			0.702320i	−0.936315	+	0.351160i	$0.885787\pi$
$570$		0			0
$571$	3.74452			0.156703			0.0783517	−	0.996926i	$-0.475034\pi$
$571$	3.74452			0.156703			0.0783517	+	0.996926i	$0.475034\pi$
$572$			1.41342i			0.0590979i
$573$		0			0
$574$	−33.9996			−1.41912
$575$	8.05499			0.335916
$576$		0			0
$577$			1.52634i			0.0635425i	0.999495	+	0.0317713i	$0.0101148\pi$
$577$			1.52634i			0.0635425i	−0.999495	+	0.0317713i	$0.989885\pi$
$578$			7.53571i			0.313444i
$579$		0			0
$580$	1.20851	+	7.79016i	0.0501807	+	0.323469i
$581$	34.4929			1.43101
$582$		0			0
$583$			7.44034i			0.308147i
$584$	−9.64143			−0.398965
$585$		0			0
$586$	14.4963			0.598836
$587$	−25.4262			−1.04945			−0.524726	−	0.851271i	$-0.675832\pi$
$587$	−25.4262			−1.04945			−0.524726	+	0.851271i	$0.675832\pi$
$588$		0			0
$589$	−50.7051			−2.08927
$590$			5.88244i			0.242176i
$591$		0			0
$592$			9.33492i			0.383663i
$593$	−4.67343			−0.191915			−0.0959574	−	0.995385i	$-0.530591\pi$
$593$	−4.67343			−0.191915			−0.0959574	+	0.995385i	$0.530591\pi$
$594$		0			0
$595$		−	10.5874i		−	0.434040i
$596$	−21.6414			−0.886468
$597$		0			0
$598$		−	2.31838i		−	0.0948056i
$599$		−	27.8345i		−	1.13729i	−0.822585	−	0.568643i	$-0.807469\pi$
$599$		−	27.8345i		−	1.13729i	0.822585	−	0.568643i	$-0.192531\pi$
$600$		0			0
$601$			46.9466i			1.91499i	0.288448	+	0.957496i	$0.406861\pi$
$601$			46.9466i			1.91499i	−0.288448	+	0.957496i	$0.593139\pi$
$602$	23.2004			0.945579
$603$		0			0
$604$	10.2544			0.417244
$605$	4.96715			0.201943
$606$		0			0
$607$			9.65740i			0.391982i	0.980606	+	0.195991i	$0.0627923\pi$
$607$			9.65740i			0.391982i	−0.980606	+	0.195991i	$0.937208\pi$
$608$	−43.2623			−1.75452
$609$		0			0
$610$	8.45889			0.342490
$611$		−	0.812497i		−	0.0328701i
$612$		0			0
$613$	25.1614			1.01626			0.508131	−	0.861280i	$-0.330337\pi$
$613$	25.1614			1.01626			0.508131	+	0.861280i	$0.330337\pi$
$614$	9.98254			0.402862
$615$		0			0
$616$	−25.4649			−1.02601
$617$			10.1628i			0.409138i	0.978852	+	0.204569i	$0.0655793\pi$
$617$			10.1628i			0.409138i	−0.978852	+	0.204569i	$0.934421\pi$
$618$		0			0
$619$		−	12.1864i		−	0.489812i	−0.969547	−	0.244906i	$-0.921243\pi$
$619$		−	12.1864i		−	0.489812i	0.969547	−	0.244906i	$-0.0787571\pi$
$620$			10.0483i			0.403547i
$621$		0			0
$622$	−9.01907			−0.361632
$623$			24.6978i			0.989495i
$624$		0			0
$625$	1.00000			0.0400000
$626$			2.88235i			0.115202i
$627$		0			0
$628$		−	20.8622i		−	0.832492i
$629$	−22.5785			−0.900264
$630$		0			0
$631$	26.8317			1.06815			0.534077	−	0.845436i	$-0.320659\pi$
$631$	26.8317			1.06815			0.534077	+	0.845436i	$0.320659\pi$
$632$	18.7353			0.745250
$633$		0			0
$634$	−7.98637			−0.317179
$635$			8.27445i			0.328361i
$636$		0			0
$637$	−3.81710			−0.151239
$638$	1.48465	+	9.57015i	0.0587777	+	0.378886i
$639$		0			0
$640$			10.1414i			0.400873i
$641$		−	14.3162i		−	0.565457i	−0.959200	−	0.282728i	$-0.908760\pi$
$641$		−	14.3162i		−	0.565457i	0.959200	−	0.282728i	$-0.0912395\pi$
$642$		0			0
$643$	−17.5584			−0.692437			−0.346218	−	0.938154i	$-0.612534\pi$
$643$	−17.5584			−0.692437			−0.346218	+	0.938154i	$0.612534\pi$
$644$	−48.2025			−1.89944
$645$		0			0
$646$		−	14.0085i		−	0.551155i
$647$	18.7284			0.736290			0.368145	−	0.929768i	$-0.379993\pi$
$647$	18.7284			0.736290			0.368145	+	0.929768i	$0.379993\pi$
$648$		0			0
$649$		−	19.7331i		−	0.774593i
$650$		−	0.287819i		−	0.0112892i
$651$		0			0
$652$			14.1763i			0.555187i
$653$		−	20.4186i		−	0.799043i	−0.916724	−	0.399522i	$-0.869176\pi$
$653$		−	20.4186i		−	0.799043i	0.916724	−	0.399522i	$-0.130824\pi$
$654$		0			0
$655$			6.75579i			0.263970i
$656$		−	12.1638i		−	0.474916i
$657$		0			0
$658$	6.18644			0.241172
$659$		−	19.3511i		−	0.753813i	−0.926251	−	0.376907i	$-0.876988\pi$
$659$		−	19.3511i		−	0.753813i	0.926251	−	0.376907i	$-0.123012\pi$
$660$		0			0
$661$	8.37452			0.325731			0.162866	−	0.986648i	$-0.447926\pi$
$661$	8.37452			0.325731			0.162866	+	0.986648i	$0.447926\pi$
$662$	5.02576			0.195332
$663$		0			0
$664$		−	21.4006i		−	0.830503i
$665$		−	30.1971i		−	1.17099i
$666$		0			0
$667$	6.64973	+	42.8647i	0.257478	+	1.65973i
$668$	−30.4043			−1.17638
$669$		0			0
$670$		−	9.37811i		−	0.362308i
$671$	−28.3760			−1.09544
$672$		0			0
$673$	18.4958			0.712962			0.356481	−	0.934303i	$-0.383977\pi$
$673$	18.4958			0.712962			0.356481	+	0.934303i	$0.383977\pi$
$674$	−3.92408			−0.151150
$675$		0			0
$676$	−18.8045			−0.723250
$677$		−	31.0373i		−	1.19286i	−0.802665	−	0.596430i	$-0.796585\pi$
$677$		−	31.0373i		−	1.19286i	0.802665	−	0.596430i	$-0.203415\pi$
$678$		0			0
$679$			42.7163i			1.63930i
$680$	−6.56876			−0.251900
$681$		0			0
$682$			12.3442i			0.472684i
$683$	−19.9192			−0.762187			−0.381093	−	0.924537i	$-0.624452\pi$
$683$	−19.9192			−0.762187			−0.381093	+	0.924537i	$0.624452\pi$
$684$		0			0
$685$			10.6430i			0.406649i
$686$		−	8.11231i		−	0.309729i
$687$		0			0
$688$			8.30025i			0.316444i
$689$	1.19077			0.0453648
$690$		0			0
$691$	6.37284			0.242434			0.121217	−	0.992626i	$-0.461320\pi$
$691$	6.37284			0.242434			0.121217	+	0.992626i	$0.461320\pi$
$692$	19.7095			0.749244
$693$		0			0
$694$			15.2755i			0.579850i
$695$	−13.8568			−0.525619
$696$		0			0
$697$	29.4207			1.11439
$698$			13.7402i			0.520073i
$699$		0			0
$700$	−5.98418			−0.226181
$701$	7.13232			0.269384			0.134692	−	0.990888i	$-0.456995\pi$
$701$	7.13232			0.269384			0.134692	+	0.990888i	$0.456995\pi$
$702$		0			0
$703$	−64.3980			−2.42882
$704$			5.27205i			0.198698i
$705$		0			0
$706$			12.3135i			0.463426i
$707$		−	1.01810i		−	0.0382898i
$708$		0			0
$709$	8.75623			0.328847			0.164424	−	0.986390i	$-0.447424\pi$
$709$	8.75623			0.328847			0.164424	+	0.986390i	$0.447424\pi$
$710$			12.0759i			0.453202i
$711$		0			0
$712$	15.3233			0.574265
$713$			55.2896i			2.07061i
$714$		0			0
$715$			0.965514i			0.0361082i
$716$	15.2750			0.570855
$717$		0			0
$718$	24.9264			0.930246
$719$	4.61498			0.172110			0.0860549	−	0.996290i	$-0.472574\pi$
$719$	4.61498			0.172110			0.0860549	+	0.996290i	$0.472574\pi$
$720$		0			0
$721$	−30.2856			−1.12790
$722$		−	26.0431i		−	0.969224i
$723$		0			0
$724$	8.22454			0.305663
$725$	0.825542	+	5.32151i	0.0306598	+	0.197636i
$726$		0			0
$727$		−	49.9559i		−	1.85276i	−0.376590	−	0.926380i	$-0.622903\pi$
$727$		−	49.9559i		−	1.85276i	0.376590	−	0.926380i	$-0.377097\pi$
$728$			4.07548i			0.151047i
$729$		0			0
$730$	−2.78340			−0.103018
$731$	−20.0759			−0.742535
$732$		0			0
$733$			12.4507i			0.459878i	0.973205	+	0.229939i	$0.0738527\pi$
$733$			12.4507i			0.459878i	−0.973205	+	0.229939i	$0.926147\pi$
$734$	11.9581			0.441383
$735$		0			0
$736$			47.1739i			1.73885i
$737$			31.4596i			1.15883i
$738$		0			0
$739$			40.9683i			1.50704i	0.657424	+	0.753521i	$0.271646\pi$
$739$			40.9683i			1.50704i	−0.657424	+	0.753521i	$0.728354\pi$
$740$			12.7618i			0.469133i
$741$		0			0
$742$			9.06666i			0.332848i
$743$		−	1.25137i		−	0.0459083i	−0.999737	−	0.0229542i	$-0.992693\pi$
$743$		−	1.25137i		−	0.0459083i	0.999737	−	0.0229542i	$-0.00730718\pi$
$744$		0			0
$745$	−14.7834			−0.541622
$746$		−	10.8920i		−	0.398783i
$747$		0			0
$748$	9.31254			0.340500
$749$	28.8593			1.05450
$750$		0			0
$751$			2.55073i			0.0930775i	0.998916	+	0.0465387i	$0.0148191\pi$
$751$			2.55073i			0.0930775i	−0.998916	+	0.0465387i	$0.985181\pi$
$752$			2.21328i			0.0807099i
$753$		0			0
$754$	1.53163	−	0.237607i	0.0557788	−	0.00865313i
$755$	7.00483			0.254932
$756$		0			0
$757$			43.4707i			1.57997i	0.613126	+	0.789985i	$0.289912\pi$
$757$			43.4707i			1.57997i	−0.613126	+	0.789985i	$0.710088\pi$
$758$	−20.3279			−0.738343
$759$		0			0
$760$	−18.7353			−0.679601
$761$	6.90996			0.250486			0.125243	−	0.992126i	$-0.460029\pi$
$761$	6.90996			0.250486			0.125243	+	0.992126i	$0.460029\pi$
$762$		0			0
$763$	44.1968			1.60003
$764$			15.0470i			0.544381i
$765$		0			0
$766$			2.99394i			0.108176i
$767$	−3.15814			−0.114034
$768$		0			0
$769$			6.95919i			0.250955i	0.992096	+	0.125477i	$0.0400462\pi$
$769$			6.95919i			0.250955i	−0.992096	+	0.125477i	$0.959954\pi$
$770$	−7.35152			−0.264930
$771$		0			0
$772$		−	5.66541i		−	0.203903i
$773$		−	12.4628i		−	0.448254i	−0.974560	−	0.224127i	$-0.928047\pi$
$773$		−	12.4628i		−	0.448254i	0.974560	−	0.224127i	$-0.0719531\pi$
$774$		0			0
$775$			6.86403i			0.246563i
$776$	26.5026			0.951388
$777$		0			0
$778$	−7.03832			−0.252336
$779$	83.9133			3.00651
$780$		0			0
$781$		−	40.5097i		−	1.44955i
$782$	−15.2750			−0.546235
$783$		0			0
$784$	10.3979			0.371354
$785$		−	14.2511i		−	0.508644i
$786$		0			0
$787$	43.8418			1.56279			0.781396	−	0.624036i	$-0.214508\pi$
$787$	43.8418			1.56279			0.781396	+	0.624036i	$0.214508\pi$
$788$	26.0759			0.928916
$789$		0			0
$790$	5.40873			0.192434
$791$			84.9745i			3.02135i
$792$		0			0
$793$			4.54138i			0.161269i
$794$		−	3.47742i		−	0.123409i
$795$		0			0
$796$	28.3860			1.00612
$797$		−	36.3456i		−	1.28743i	−0.765266	−	0.643714i	$-0.777393\pi$
$797$		−	36.3456i		−	1.28743i	0.765266	−	0.643714i	$-0.222607\pi$
$798$		0			0
$799$	−5.35328			−0.189385
$800$			5.85649i			0.207058i
$801$		0			0
$802$		−	18.9297i		−	0.668430i
$803$	9.33715			0.329501
$804$		0			0
$805$	−32.9274			−1.16054
$806$	1.97560			0.0695875
$807$		0			0
$808$	−0.631666			−0.0222219
$809$		−	18.2835i		−	0.642812i	−0.946941	−	0.321406i	$-0.895845\pi$
$809$		−	18.2835i		−	0.642812i	0.946941	−	0.321406i	$-0.104155\pi$
$810$		0			0
$811$	−36.5484			−1.28339			−0.641695	−	0.766960i	$-0.721768\pi$
$811$	−36.5484			−1.28339			−0.641695	+	0.766960i	$0.721768\pi$
$812$	−4.94019	−	31.8449i	−0.173367	−	1.11754i
$813$		0			0
$814$			15.6778i			0.549505i
$815$			9.68392i			0.339213i
$816$		0			0
$817$	−57.2602			−2.00328
$818$	−17.6336			−0.616545
$819$		0			0
$820$		−	16.6291i		−	0.580715i
$821$	24.7659			0.864337			0.432169	−	0.901793i	$-0.357749\pi$
$821$	24.7659			0.864337			0.432169	+	0.901793i	$0.357749\pi$
$822$		0			0
$823$			26.3641i			0.918994i	0.888179	+	0.459497i	$0.151970\pi$
$823$			26.3641i			0.918994i	−0.888179	+	0.459497i	$0.848030\pi$
$824$			18.7902i			0.654588i
$825$		0			0
$826$		−	24.0464i		−	0.836682i
$827$		−	41.5422i		−	1.44456i	−0.691598	−	0.722282i	$-0.743093\pi$
$827$		−	41.5422i		−	1.44456i	0.691598	−	0.722282i	$-0.256907\pi$
$828$		0			0
$829$		−	20.0965i		−	0.697980i	−0.937126	−	0.348990i	$-0.886525\pi$
$829$		−	20.0965i		−	0.697980i	0.937126	−	0.348990i	$-0.113475\pi$
$830$		−	6.17817i		−	0.214447i
$831$		0			0
$832$	0.843753			0.0292519
$833$			25.1496i			0.871382i
$834$		0			0
$835$	−20.7693			−0.718753
$836$	26.5610			0.918633
$837$		0			0
$838$			4.73619i			0.163609i
$839$			26.9404i			0.930087i	0.885288	+	0.465043i	$0.153961\pi$
$839$			26.9404i			0.930087i	−0.885288	+	0.465043i	$0.846039\pi$
$840$		0			0
$841$	−27.6370	+	8.78626i	−0.952999	+	0.302974i
$842$	6.05022			0.208505
$843$		0			0
$844$			17.8939i			0.615932i
$845$	−12.8455			−0.441898
$846$		0			0
$847$	−20.3049			−0.697684
$848$	−3.24371			−0.111390
$849$		0			0
$850$	−1.89635			−0.0650442
$851$			70.2206i			2.40713i
$852$		0			0
$853$			8.47502i			0.290179i	0.989419	+	0.145090i	$0.0463470\pi$
$853$			8.47502i			0.290179i	−0.989419	+	0.145090i	$0.953653\pi$
$854$	−34.5785			−1.18325
$855$		0			0
$856$		−	17.9053i		−	0.611990i
$857$	−13.7027			−0.468074			−0.234037	−	0.972228i	$-0.575194\pi$
$857$	−13.7027			−0.468074			−0.234037	+	0.972228i	$0.575194\pi$
$858$		0			0
$859$			28.9779i			0.988714i	0.869259	+	0.494357i	$0.164596\pi$
$859$			28.9779i			0.988714i	−0.869259	+	0.494357i	$0.835404\pi$
$860$			11.3473i			0.386939i
$861$		0			0
$862$		−	3.00927i		−	0.102496i
$863$	31.2038			1.06219			0.531095	−	0.847312i	$-0.321781\pi$
$863$	31.2038			1.06219			0.531095	+	0.847312i	$0.321781\pi$
$864$		0			0
$865$	13.4637			0.457780
$866$	10.4847			0.356285
$867$		0			0
$868$		−	41.0756i		−	1.39420i
$869$	−18.1440			−0.615494
$870$		0			0
$871$	5.03489			0.170601
$872$		−	27.4212i		−	0.928598i
$873$		0			0
$874$	−43.5672			−1.47368
$875$	−4.08783			−0.138194
$876$		0			0
$877$	2.69090			0.0908652			0.0454326	−	0.998967i	$-0.485533\pi$
$877$	2.69090			0.0908652			0.0454326	+	0.998967i	$0.485533\pi$
$878$		−	24.2053i		−	0.816891i
$879$		0			0
$880$		−	2.63010i		−	0.0886606i
$881$			32.4956i			1.09480i	0.836870	+	0.547402i	$0.184383\pi$
$881$			32.4956i			1.09480i	−0.836870	+	0.547402i	$0.815617\pi$
$882$		0			0
$883$	43.0340			1.44821			0.724104	−	0.689690i	$-0.242253\pi$
$883$	43.0340			1.44821			0.724104	+	0.689690i	$0.242253\pi$
$884$		−	1.49040i		−	0.0501277i
$885$		0			0
$886$	15.9650			0.536355
$887$		−	32.0176i		−	1.07505i	−0.843249	−	0.537523i	$-0.819360\pi$
$887$		−	32.0176i		−	1.07505i	0.843249	−	0.537523i	$-0.180640\pi$
$888$		0			0
$889$		−	33.8246i		−	1.13444i
$890$	4.42371			0.148283
$891$		0			0
$892$	−23.1680			−0.775723
$893$	−15.2685			−0.510942
$894$		0			0
$895$	10.4345			0.348786
$896$		−	41.4562i		−	1.38496i
$897$		0			0
$898$	−11.7658			−0.392628
$899$	−36.5270	+	5.66654i	−1.21824	+	0.188990i
$900$		0			0
$901$		−	7.84561i		−	0.261375i
$902$		−	20.4288i		−	0.680204i
$903$		0			0
$904$	52.7210			1.75347
$905$	5.61824			0.186756
$906$		0			0
$907$		−	7.61421i		−	0.252826i	−0.991978	−	0.126413i	$-0.959654\pi$
$907$		−	7.61421i		−	0.252826i	0.991978	−	0.126413i	$-0.0403464\pi$
$908$	22.2327			0.737818
$909$		0			0
$910$			1.17656i			0.0390025i
$911$		−	44.9668i		−	1.48982i	−0.667166	−	0.744909i	$-0.732493\pi$
$911$		−	44.9668i		−	1.48982i	0.667166	−	0.744909i	$-0.267507\pi$
$912$		0			0
$913$			20.7252i			0.685903i
$914$		−	0.879284i		−	0.0290841i
$915$		0			0
$916$		−	29.4735i		−	0.973832i
$917$		−	27.6165i		−	0.911978i
$918$		0			0
$919$	45.3421			1.49570			0.747850	−	0.663868i	$-0.231086\pi$
$919$	45.3421			1.49570			0.747850	+	0.663868i	$0.231086\pi$
$920$			20.4293i			0.673533i
$921$		0			0
$922$	19.7838			0.651544
$923$	−6.48329			−0.213400
$924$		0			0
$925$			8.71766i			0.286635i
$926$		−	0.561195i		−	0.0184420i
$927$		0			0
$928$	−31.1654	+	4.83477i	−1.02305	+	0.158709i
$929$	9.34687			0.306661			0.153330	−	0.988175i	$-0.451000\pi$
$929$	9.34687			0.306661			0.153330	+	0.988175i	$0.451000\pi$
$930$		0			0
$931$			71.7312i			2.35090i
$932$	27.3960			0.897385
$933$		0			0
$934$	−12.4827			−0.408447
$935$	6.36145			0.208042
$936$		0			0
$937$	24.7740			0.809332			0.404666	−	0.914465i	$-0.367388\pi$
$937$	24.7740			0.809332			0.404666	+	0.914465i	$0.367388\pi$
$938$			38.3361i			1.25172i
$939$		0			0
$940$			3.02577i			0.0986898i
$941$	−57.0130			−1.85857			−0.929285	−	0.369363i	$-0.879576\pi$
$941$	−57.0130			−1.85857			−0.929285	+	0.369363i	$0.879576\pi$
$942$		0			0
$943$		−	91.5004i		−	2.97966i
$944$	8.60291			0.280001
$945$		0			0
$946$			13.9400i			0.453230i
$947$		−	5.48495i		−	0.178237i	−0.996021	−	0.0891185i	$-0.971595\pi$
$947$		−	5.48495i		−	0.178237i	0.996021	−	0.0891185i	$-0.0284050\pi$
$948$		0			0
$949$		−	1.49434i		−	0.0485085i
$950$	−5.40873			−0.175482
$951$		0			0
$952$	26.8520			0.870278
$953$	17.6170			0.570672			0.285336	−	0.958428i	$-0.407895\pi$
$953$	17.6170			0.570672			0.285336	+	0.958428i	$0.407895\pi$
$954$		0			0
$955$			10.2787i			0.332611i
$956$	−29.9498			−0.968645
$957$		0			0
$958$	−4.71542			−0.152348
$959$		−	43.5069i		−	1.40491i
$960$		0			0
$961$	−16.1149			−0.519834
$962$	2.50911			0.0808970
$963$		0			0
$964$	24.5288			0.790018
$965$		−	3.87008i		−	0.124582i
$966$		0			0
$967$			25.6556i			0.825027i	0.910951	+	0.412514i	$0.135349\pi$
$967$			25.6556i			0.825027i	−0.910951	+	0.412514i	$0.864651\pi$
$968$			12.5978i			0.404910i
$969$		0			0
$970$	7.65108			0.245661
$971$			7.75471i			0.248860i	0.992228	+	0.124430i	$0.0397103\pi$
$971$			7.75471i			0.248860i	−0.992228	+	0.124430i	$0.960290\pi$
$972$		0			0
$973$	56.6443			1.81593
$974$			10.3664i			0.332160i
$975$		0			0
$976$		−	12.3709i		−	0.395983i
$977$	43.9323			1.40552			0.702759	−	0.711428i	$-0.251951\pi$
$977$	43.9323			1.40552			0.702759	+	0.711428i	$0.251951\pi$
$978$		0			0
$979$	−14.8397			−0.474279
$980$	14.2150			0.454082
$981$		0			0
$982$	−5.48136			−0.174917
$983$			44.7786i			1.42822i	0.700035	+	0.714108i	$0.253168\pi$
$983$			44.7786i			1.42822i	−0.700035	+	0.714108i	$0.746832\pi$
$984$		0			0
$985$	17.8126			0.567558
$986$	−1.56551	−	10.0914i	−0.0498561	−	0.321377i
$987$		0			0
$988$		−	4.25090i		−	0.135239i
$989$			62.4374i			1.98539i
$990$		0			0
$991$	−27.4494			−0.871960			−0.435980	−	0.899956i	$-0.643598\pi$
$991$	−27.4494			−0.871960			−0.435980	+	0.899956i	$0.643598\pi$
$992$	−40.1991			−1.27632
$993$		0			0
$994$		−	49.3644i		−	1.56574i
$995$	19.3907			0.614726
$996$		0			0
$997$		−	5.29583i		−	0.167721i	−0.996478	−	0.0838603i	$-0.973275\pi$
$997$		−	5.29583i		−	0.167721i	0.996478	−	0.0838603i	$-0.0267250\pi$
$998$		−	9.13347i		−	0.289115i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.d.f.811.7	yes	10
3.2	odd	2		1305.2.d.e.811.4	✓	10
29.28	even	2	inner	1305.2.d.f.811.4	yes	10
87.86	odd	2		1305.2.d.e.811.7	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1305.2.d.e.811.4	✓	10	3.2	odd	2
1305.2.d.e.811.7	yes	10	87.86	odd	2
1305.2.d.f.811.4	yes	10	29.28	even	2	inner
1305.2.d.f.811.7	yes	10	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1305.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.732188i q^{2} +1.46390 q^{4} +1.00000 q^{5} -4.08783 q^{7} +2.53623i q^{8} +0.732188i q^{10} -2.45619i q^{11} -0.393095 q^{13} -2.99306i q^{14} +1.07081 q^{16} +2.58997i q^{17} +7.38707i q^{19} +1.46390 q^{20} +1.79839 q^{22} +8.05499 q^{23} +1.00000 q^{25} -0.287819i q^{26} -5.98418 q^{28} +(0.825542 + 5.32151i) q^{29} +6.86403i q^{31} +5.85649i q^{32} -1.89635 q^{34} -4.08783 q^{35} +8.71766i q^{37} -5.40873 q^{38} +2.53623i q^{40} -11.3595i q^{41} +7.75140i q^{43} -3.59561i q^{44} +5.89776i q^{46} +2.06693i q^{47} +9.71037 q^{49} +0.732188i q^{50} -0.575452 q^{52} -3.02923 q^{53} -2.45619i q^{55} -10.3677i q^{56} +(-3.89635 + 0.604452i) q^{58} +8.03405 q^{59} -11.5529i q^{61} -5.02576 q^{62} -2.14644 q^{64} -0.393095 q^{65} -12.8083 q^{67} +3.79146i q^{68} -2.99306i q^{70} +16.4929 q^{71} +3.80149i q^{73} -6.38297 q^{74} +10.8139i q^{76} +10.0405i q^{77} -7.38707i q^{79} +1.07081 q^{80} +8.31728 q^{82} -8.43795 q^{83} +2.58997i q^{85} -5.67548 q^{86} +6.22944 q^{88} -6.04177i q^{89} +1.60691 q^{91} +11.7917 q^{92} -1.51338 q^{94} +7.38707i q^{95} -10.4496i q^{97} +7.10982i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 6 q^{4} + 10 q^{5} - 4 q^{7} + 4 q^{13} - 2 q^{16} - 6 q^{20} + 10 q^{22} + 10 q^{23} + 10 q^{25} - 2 q^{28} + 2 q^{34} - 4 q^{35} + 32 q^{38} - 10 q^{49} - 34 q^{52} - 14 q^{53} - 18 q^{58} + 32 q^{59}+ \cdots - 26 q^{94}+O(q^{100}) \) 10 * q - 6 * q^4 + 10 * q^5 - 4 * q^7 + 4 * q^13 - 2 * q^16 - 6 * q^20 + 10 * q^22 + 10 * q^23 + 10 * q^25 - 2 * q^28 + 2 * q^34 - 4 * q^35 + 32 * q^38 - 10 * q^49 - 34 * q^52 - 14 * q^53 - 18 * q^58 + 32 * q^59 + 4 * q^62 + 36 * q^64 + 4 * q^65 - 8 * q^67 - 8 * q^71 - 32 * q^74 - 2 * q^80 - 16 * q^82 + 18 * q^83 - 56 * q^86 + 24 * q^91 - 26 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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