LMFDB - Embedded newform 3360.2.ba.f.2591.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3360,2,Mod(2591,3360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3360, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1, 0, 0]))

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

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

chi := DirichletCharacter("3360.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.ba (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.5
Character	$\chi$	$=$	3360.2591
Dual form			3360.2.ba.f.2591.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.55507 - 0.762729i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(1.83649 + 2.37220i) q^{9} +O(q^{10})$	\(q+(-1.55507 - 0.762729i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(1.83649 + 2.37220i) q^{9} -2.85267 q^{11} +3.40449 q^{13} +(0.762729 - 1.55507i) q^{15} -7.12622i q^{17} +8.15169i q^{19} +(-0.762729 + 1.55507i) q^{21} -5.24656 q^{23} -1.00000 q^{25} +(-1.04653 - 5.08967i) q^{27} -0.403889i q^{29} +7.42696i q^{31} +(4.43611 + 2.17582i) q^{33} +1.00000 q^{35} +7.87130 q^{37} +(-5.29422 - 2.59670i) q^{39} -6.25776i q^{41} -2.93995i q^{43} +(-2.37220 + 1.83649i) q^{45} -2.33793 q^{47} -1.00000 q^{49} +(-5.43538 + 11.0818i) q^{51} +5.52406i q^{53} -2.85267i q^{55} +(6.21754 - 12.6765i) q^{57} -0.0930502 q^{59} -2.79545 q^{61} +(2.37220 - 1.83649i) q^{63} +3.40449i q^{65} +2.93417i q^{67} +(8.15877 + 4.00171i) q^{69} -3.29594 q^{71} -12.0056 q^{73} +(1.55507 + 0.762729i) q^{75} +2.85267i q^{77} -16.3334i q^{79} +(-2.25462 + 8.71302i) q^{81} +7.88524 q^{83} +7.12622 q^{85} +(-0.308058 + 0.628076i) q^{87} +9.63287i q^{89} -3.40449i q^{91} +(5.66476 - 11.5494i) q^{93} -8.15169 q^{95} -9.92561 q^{97} +(-5.23890 - 6.76710i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 4 q^{9}+O(q^{10}) $ 24 * q + 4 * q^9	$ 24 q + 4 q^{9} + 8 q^{11} + 8 q^{13} + 4 q^{15} - 4 q^{21} - 24 q^{25} + 8 q^{33} + 24 q^{35} + 8 q^{37} + 36 q^{39} - 24 q^{49} - 12 q^{51} + 8 q^{57} + 48 q^{59} + 56 q^{61} - 88 q^{71} - 40 q^{73} + 44 q^{81} + 24 q^{83} + 32 q^{87} - 48 q^{93} - 24 q^{95} - 8 q^{97} - 52 q^{99}+O(q^{100}) $ 24 * q + 4 * q^9 + 8 * q^11 + 8 * q^13 + 4 * q^15 - 4 * q^21 - 24 * q^25 + 8 * q^33 + 24 * q^35 + 8 * q^37 + 36 * q^39 - 24 * q^49 - 12 * q^51 + 8 * q^57 + 48 * q^59 + 56 * q^61 - 88 * q^71 - 40 * q^73 + 44 * q^81 + 24 * q^83 + 32 * q^87 - 48 * q^93 - 24 * q^95 - 8 * q^97 - 52 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times$.

$n$	$421$	$1121$	$1471$	$1921$	$2017$
$\chi(n)$	$1$	$-1$	$-1$	$1$	$1$

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
$3$	−1.55507	−	0.762729i	−0.897820	−	0.440362i
$3$	−1.55507	−	0.762729i	−0.897820	−	0.440362i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	1.83649	+	2.37220i	0.612163	+	0.790732i
$10$		0			0
$11$	−2.85267			−0.860113			−0.430057	−	0.902802i	$-0.641506\pi$
$11$	−2.85267			−0.860113			−0.430057	+	0.902802i	$0.641506\pi$
$12$		0			0
$13$	3.40449			0.944235			0.472118	−	0.881536i	$-0.343490\pi$
$13$	3.40449			0.944235			0.472118	+	0.881536i	$0.343490\pi$
$14$		0			0
$15$	0.762729	−	1.55507i	0.196936	−	0.401517i
$16$		0			0
$17$		−	7.12622i		−	1.72836i	−0.503180	−	0.864181i	$-0.667837\pi$
$17$		−	7.12622i		−	1.72836i	0.503180	−	0.864181i	$-0.332163\pi$
$18$		0			0
$19$			8.15169i			1.87013i	0.354480	+	0.935063i	$0.384658\pi$
$19$			8.15169i			1.87013i	−0.354480	+	0.935063i	$0.615342\pi$
$20$		0			0
$21$	−0.762729	+	1.55507i	−0.166441	+	0.339344i
$22$		0			0
$23$	−5.24656			−1.09398			−0.546992	−	0.837138i	$-0.684227\pi$
$23$	−5.24656			−1.09398			−0.546992	+	0.837138i	$0.684227\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−1.04653	−	5.08967i	−0.201404	−	0.979508i
$28$		0			0
$29$		−	0.403889i		−	0.0750003i	−0.999297	−	0.0375001i	$-0.988061\pi$
$29$		−	0.403889i		−	0.0750003i	0.999297	−	0.0375001i	$-0.0119395\pi$
$30$		0			0
$31$			7.42696i			1.33392i	0.745093	+	0.666960i	$0.232405\pi$
$31$			7.42696i			1.33392i	−0.745093	+	0.666960i	$0.767595\pi$
$32$		0			0
$33$	4.43611	+	2.17582i	0.772227	+	0.378761i
$34$		0			0
$35$	1.00000			0.169031
$36$		0			0
$37$	7.87130			1.29403			0.647017	−	0.762476i	$-0.276016\pi$
$37$	7.87130			1.29403			0.647017	+	0.762476i	$0.276016\pi$
$38$		0			0
$39$	−5.29422	−	2.59670i	−0.847753	−	0.415805i
$40$		0			0
$41$		−	6.25776i		−	0.977297i	−0.872481	−	0.488649i	$-0.837490\pi$
$41$		−	6.25776i		−	0.977297i	0.872481	−	0.488649i	$-0.162510\pi$
$42$		0			0
$43$		−	2.93995i		−	0.448338i	−0.974550	−	0.224169i	$-0.928033\pi$
$43$		−	2.93995i		−	0.448338i	0.974550	−	0.224169i	$-0.0719667\pi$
$44$		0			0
$45$	−2.37220	+	1.83649i	−0.353626	+	0.273767i
$46$		0			0
$47$	−2.33793			−0.341023			−0.170511	−	0.985356i	$-0.554542\pi$
$47$	−2.33793			−0.341023			−0.170511	+	0.985356i	$0.554542\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−5.43538	+	11.0818i	−0.761105	+	1.55176i
$52$		0			0
$53$			5.52406i			0.758788i	0.925235	+	0.379394i	$0.123867\pi$
$53$			5.52406i			0.758788i	−0.925235	+	0.379394i	$0.876133\pi$
$54$		0			0
$55$		−	2.85267i		−	0.384654i
$56$		0			0
$57$	6.21754	−	12.6765i	0.823533	−	1.67904i
$58$		0			0
$59$	−0.0930502			−0.0121141			−0.00605705	−	0.999982i	$-0.501928\pi$
$59$	−0.0930502			−0.0121141			−0.00605705	+	0.999982i	$0.501928\pi$
$60$		0			0
$61$	−2.79545			−0.357921			−0.178960	−	0.983856i	$-0.557273\pi$
$61$	−2.79545			−0.357921			−0.178960	+	0.983856i	$0.557273\pi$
$62$		0			0
$63$	2.37220	−	1.83649i	0.298869	−	0.231376i
$64$		0			0
$65$			3.40449i			0.422275i
$66$		0			0
$67$			2.93417i			0.358465i	0.983807	+	0.179233i	$0.0573615\pi$
$67$			2.93417i			0.358465i	−0.983807	+	0.179233i	$0.942638\pi$
$68$		0			0
$69$	8.15877	+	4.00171i	0.982201	+	0.481749i
$70$		0			0
$71$	−3.29594			−0.391156			−0.195578	−	0.980688i	$-0.562658\pi$
$71$	−3.29594			−0.391156			−0.195578	+	0.980688i	$0.562658\pi$
$72$		0			0
$73$	−12.0056			−1.40515			−0.702573	−	0.711611i	$-0.747966\pi$
$73$	−12.0056			−1.40515			−0.702573	+	0.711611i	$0.747966\pi$
$74$		0			0
$75$	1.55507	+	0.762729i	0.179564	+	0.0880724i
$76$		0			0
$77$			2.85267i			0.325092i
$78$		0			0
$79$		−	16.3334i		−	1.83765i	−0.394663	−	0.918826i	$-0.629138\pi$
$79$		−	16.3334i		−	1.83765i	0.394663	−	0.918826i	$-0.370862\pi$
$80$		0			0
$81$	−2.25462	+	8.71302i	−0.250514	+	0.968113i
$82$		0			0
$83$	7.88524			0.865517			0.432759	−	0.901510i	$-0.357540\pi$
$83$	7.88524			0.865517			0.432759	+	0.901510i	$0.357540\pi$
$84$		0			0
$85$	7.12622			0.772947
$86$		0			0
$87$	−0.308058	+	0.628076i	−0.0330273	+	0.0673368i
$88$		0			0
$89$			9.63287i			1.02108i	0.859853	+	0.510541i	$0.170555\pi$
$89$			9.63287i			1.02108i	−0.859853	+	0.510541i	$0.829445\pi$
$90$		0			0
$91$		−	3.40449i		−	0.356887i
$92$		0			0
$93$	5.66476	−	11.5494i	0.587408	−	1.19762i
$94$		0			0
$95$	−8.15169			−0.836346
$96$		0			0
$97$	−9.92561			−1.00779			−0.503896	−	0.863764i	$-0.668101\pi$
$97$	−9.92561			−1.00779			−0.503896	+	0.863764i	$0.668101\pi$
$98$		0			0
$99$	−5.23890	−	6.76710i	−0.526529	−	0.680119i
$100$		0			0
$101$			6.49414i			0.646191i	0.946366	+	0.323095i	$0.104723\pi$
$101$			6.49414i			0.646191i	−0.946366	+	0.323095i	$0.895277\pi$
$102$		0			0
$103$		−	1.77511i		−	0.174907i	−0.996169	−	0.0874534i	$-0.972127\pi$
$103$		−	1.77511i		−	0.174907i	0.996169	−	0.0874534i	$-0.0278729\pi$
$104$		0			0
$105$	−1.55507	−	0.762729i	−0.151759	−	0.0744348i
$106$		0			0
$107$	6.12854			0.592468			0.296234	−	0.955115i	$-0.404269\pi$
$107$	6.12854			0.592468			0.296234	+	0.955115i	$0.404269\pi$
$108$		0			0
$109$	−7.17312			−0.687061			−0.343530	−	0.939142i	$-0.611623\pi$
$109$	−7.17312			−0.687061			−0.343530	+	0.939142i	$0.611623\pi$
$110$		0			0
$111$	−12.2404	−	6.00367i	−1.16181	−	0.569843i
$112$		0			0
$113$			17.2514i			1.62288i	0.584437	+	0.811439i	$0.301315\pi$
$113$			17.2514i			1.62288i	−0.584437	+	0.811439i	$0.698685\pi$
$114$		0			0
$115$		−	5.24656i		−	0.489244i
$116$		0			0
$117$	6.25230	+	8.07611i	0.578025	+	0.746637i
$118$		0			0
$119$	−7.12622			−0.653260
$120$		0			0
$121$	−2.86226			−0.260205
$122$		0			0
$123$	−4.77297	+	9.73125i	−0.430365	+	0.877438i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			13.9807i			1.24059i	0.784370	+	0.620293i	$0.212986\pi$
$127$			13.9807i			1.24059i	−0.784370	+	0.620293i	$0.787014\pi$
$128$		0			0
$129$	−2.24238	+	4.57182i	−0.197431	+	0.402527i
$130$		0			0
$131$	−11.2433			−0.982332			−0.491166	−	0.871066i	$-0.663429\pi$
$131$	−11.2433			−0.982332			−0.491166	+	0.871066i	$0.663429\pi$
$132$		0			0
$133$	8.15169			0.706842
$134$		0			0
$135$	5.08967	−	1.04653i	0.438049	−	0.0900705i
$136$		0			0
$137$			10.3029i			0.880233i	0.897941	+	0.440117i	$0.145063\pi$
$137$			10.3029i			0.880233i	−0.897941	+	0.440117i	$0.854937\pi$
$138$		0			0
$139$			19.2951i			1.63659i	0.574799	+	0.818295i	$0.305080\pi$
$139$			19.2951i			1.63659i	−0.574799	+	0.818295i	$0.694920\pi$
$140$		0			0
$141$	3.63565	+	1.78321i	0.306177	+	0.150173i
$142$		0			0
$143$	−9.71189			−0.812149
$144$		0			0
$145$	0.403889			0.0335411
$146$		0			0
$147$	1.55507	+	0.762729i	0.128260	+	0.0629089i
$148$		0			0
$149$			7.59206i			0.621966i	0.950415	+	0.310983i	$0.100658\pi$
$149$			7.59206i			0.621966i	−0.950415	+	0.310983i	$0.899342\pi$
$150$		0			0
$151$		−	2.41418i		−	0.196463i	−0.995164	−	0.0982315i	$-0.968681\pi$
$151$		−	2.41418i		−	0.196463i	0.995164	−	0.0982315i	$-0.0313186\pi$
$152$		0			0
$153$	16.9048	−	13.0872i	1.36667	−	1.05804i
$154$		0			0
$155$	−7.42696			−0.596548
$156$		0			0
$157$	−10.3010			−0.822111			−0.411055	−	0.911610i	$-0.634840\pi$
$157$	−10.3010			−0.822111			−0.411055	+	0.911610i	$0.634840\pi$
$158$		0			0
$159$	4.21336	−	8.59030i	0.334141	−	0.681255i
$160$		0			0
$161$			5.24656i			0.413487i
$162$		0			0
$163$			7.42253i			0.581377i	0.956818	+	0.290689i	$0.0938844\pi$
$163$			7.42253i			0.581377i	−0.956818	+	0.290689i	$0.906116\pi$
$164$		0			0
$165$	−2.17582	+	4.43611i	−0.169387	+	0.345350i
$166$		0			0
$167$	−12.4647			−0.964545			−0.482273	−	0.876021i	$-0.660189\pi$
$167$	−12.4647			−0.964545			−0.482273	+	0.876021i	$0.660189\pi$
$168$		0			0
$169$	−1.40946			−0.108420
$170$		0			0
$171$	−19.3374	+	14.9705i	−1.47877	+	1.14482i
$172$		0			0
$173$			13.0426i			0.991609i	0.868434	+	0.495805i	$0.165127\pi$
$173$			13.0426i			0.991609i	−0.868434	+	0.495805i	$0.834873\pi$
$174$		0			0
$175$			1.00000i			0.0755929i
$176$		0			0
$177$	0.144700	+	0.0709721i	0.0108763	+	0.00533459i
$178$		0			0
$179$	1.09066			0.0815199			0.0407599	−	0.999169i	$-0.487022\pi$
$179$	1.09066			0.0815199			0.0407599	+	0.999169i	$0.487022\pi$
$180$		0			0
$181$	10.0136			0.744304			0.372152	−	0.928172i	$-0.378620\pi$
$181$	10.0136			0.744304			0.372152	+	0.928172i	$0.378620\pi$
$182$		0			0
$183$	4.34713	+	2.13217i	0.321349	+	0.157615i
$184$		0			0
$185$			7.87130i			0.578709i
$186$		0			0
$187$			20.3288i			1.48659i
$188$		0			0
$189$	−5.08967	+	1.04653i	−0.370219	+	0.0761235i
$190$		0			0
$191$	26.8284			1.94123			0.970617	−	0.240629i	$-0.0773536\pi$
$191$	26.8284			1.94123			0.970617	+	0.240629i	$0.0773536\pi$
$192$		0			0
$193$	19.2900			1.38852			0.694261	−	0.719723i	$-0.255731\pi$
$193$	19.2900			1.38852			0.694261	+	0.719723i	$0.255731\pi$
$194$		0			0
$195$	2.59670	−	5.29422i	0.185954	−	0.379127i
$196$		0			0
$197$			2.11841i			0.150930i	0.997148	+	0.0754652i	$0.0240442\pi$
$197$			2.11841i			0.150930i	−0.997148	+	0.0754652i	$0.975956\pi$
$198$		0			0
$199$		−	7.30303i		−	0.517698i	−0.965918	−	0.258849i	$-0.916657\pi$
$199$		−	7.30303i		−	0.517698i	0.965918	−	0.258849i	$-0.0833432\pi$
$200$		0			0
$201$	2.23797	−	4.56284i	0.157855	−	0.321838i
$202$		0			0
$203$	−0.403889			−0.0283474
$204$		0			0
$205$	6.25776			0.437061
$206$		0			0
$207$	−9.63525	−	12.4459i	−0.669696	−	0.865048i
$208$		0			0
$209$		−	23.2541i		−	1.60852i
$210$		0			0
$211$			13.2808i			0.914285i	0.889394	+	0.457142i	$0.151127\pi$
$211$			13.2808i			0.914285i	−0.889394	+	0.457142i	$0.848873\pi$
$212$		0			0
$213$	5.12541	+	2.51391i	0.351188	+	0.172250i
$214$		0			0
$215$	2.93995			0.200503
$216$		0			0
$217$	7.42696			0.504175
$218$		0			0
$219$	18.6695	+	9.15701i	1.26157	+	0.618773i
$220$		0			0
$221$		−	24.2611i		−	1.63198i
$222$		0			0
$223$			10.8055i			0.723591i	0.932257	+	0.361796i	$0.117836\pi$
$223$			10.8055i			0.723591i	−0.932257	+	0.361796i	$0.882164\pi$
$224$		0			0
$225$	−1.83649	−	2.37220i	−0.122433	−	0.158146i
$226$		0			0
$227$	−17.9011			−1.18814			−0.594068	−	0.804415i	$-0.702479\pi$
$227$	−17.9011			−1.18814			−0.594068	+	0.804415i	$0.702479\pi$
$228$		0			0
$229$	−24.7865			−1.63794			−0.818969	−	0.573837i	$-0.805454\pi$
$229$	−24.7865			−1.63794			−0.818969	+	0.573837i	$0.805454\pi$
$230$		0			0
$231$	2.17582	−	4.43611i	0.143158	−	0.291874i
$232$		0			0
$233$			12.9724i			0.849849i	0.905229	+	0.424924i	$0.139699\pi$
$233$			12.9724i			0.849849i	−0.905229	+	0.424924i	$0.860301\pi$
$234$		0			0
$235$		−	2.33793i		−	0.152510i
$236$		0			0
$237$	−12.4580	+	25.3996i	−0.809232	+	1.64988i
$238$		0			0
$239$	0.355292			0.0229819			0.0114910	−	0.999934i	$-0.496342\pi$
$239$	0.355292			0.0229819			0.0114910	+	0.999934i	$0.496342\pi$
$240$		0			0
$241$	9.38404			0.604479			0.302239	−	0.953232i	$-0.402266\pi$
$241$	9.38404			0.604479			0.302239	+	0.953232i	$0.402266\pi$
$242$		0			0
$243$	10.1518	−	11.8297i	0.651236	−	0.758875i
$244$		0			0
$245$		−	1.00000i		−	0.0638877i
$246$		0			0
$247$			27.7523i			1.76584i
$248$		0			0
$249$	−12.2621	−	6.01430i	−0.777079	−	0.381141i
$250$		0			0
$251$	2.99566			0.189084			0.0945422	−	0.995521i	$-0.469861\pi$
$251$	2.99566			0.189084			0.0945422	+	0.995521i	$0.469861\pi$
$252$		0			0
$253$	14.9667			0.940950
$254$		0			0
$255$	−11.0818	−	5.43538i	−0.693968	−	0.340377i
$256$		0			0
$257$			7.31505i			0.456300i	0.973626	+	0.228150i	$0.0732677\pi$
$257$			7.31505i			0.456300i	−0.973626	+	0.228150i	$0.926732\pi$
$258$		0			0
$259$		−	7.87130i		−	0.489099i
$260$		0			0
$261$	0.958103	−	0.741737i	0.0593051	−	0.0459124i
$262$		0			0
$263$	20.1069			1.23985			0.619923	−	0.784663i	$-0.287164\pi$
$263$	20.1069			1.23985			0.619923	+	0.784663i	$0.287164\pi$
$264$		0			0
$265$	−5.52406			−0.339340
$266$		0			0
$267$	7.34727	−	14.9798i	0.449646	−	0.916748i
$268$		0			0
$269$			31.9801i			1.94986i	0.222508	+	0.974931i	$0.428576\pi$
$269$			31.9801i			1.94986i	−0.222508	+	0.974931i	$0.571424\pi$
$270$		0			0
$271$		−	12.6192i		−	0.766565i	−0.923631	−	0.383282i	$-0.874794\pi$
$271$		−	12.6192i		−	0.766565i	0.923631	−	0.383282i	$-0.125206\pi$
$272$		0			0
$273$	−2.59670	+	5.29422i	−0.157160	+	0.320421i
$274$		0			0
$275$	2.85267			0.172023
$276$		0			0
$277$	−11.4382			−0.687254			−0.343627	−	0.939106i	$-0.611656\pi$
$277$	−11.4382			−0.687254			−0.343627	+	0.939106i	$0.611656\pi$
$278$		0			0
$279$	−17.6182	+	13.6395i	−1.05477	+	0.816577i
$280$		0			0
$281$			31.3864i			1.87235i	0.351530	+	0.936177i	$0.385661\pi$
$281$			31.3864i			1.87235i	−0.351530	+	0.936177i	$0.614339\pi$
$282$		0			0
$283$			11.5775i			0.688210i	0.938931	+	0.344105i	$0.111818\pi$
$283$			11.5775i			0.688210i	−0.938931	+	0.344105i	$0.888182\pi$
$284$		0			0
$285$	12.6765	+	6.21754i	0.750889	+	0.368295i
$286$		0			0
$287$	−6.25776			−0.369384
$288$		0			0
$289$	−33.7830			−1.98724
$290$		0			0
$291$	15.4350	+	7.57055i	0.904817	+	0.443794i
$292$		0			0
$293$		−	6.11970i		−	0.357517i	−0.983893	−	0.178758i	$-0.942792\pi$
$293$		−	6.11970i		−	0.357517i	0.983893	−	0.178758i	$-0.0572081\pi$
$294$		0			0
$295$		−	0.0930502i		−	0.00541759i
$296$		0			0
$297$	2.98539	+	14.5192i	0.173230	+	0.842488i
$298$		0			0
$299$	−17.8619			−1.03298
$300$		0			0
$301$	−2.93995			−0.169456
$302$		0			0
$303$	4.95327	−	10.0988i	0.284558	−	0.580163i
$304$		0			0
$305$		−	2.79545i		−	0.160067i
$306$		0			0
$307$			19.9930i			1.14106i	0.821277	+	0.570529i	$0.193262\pi$
$307$			19.9930i			1.14106i	−0.821277	+	0.570529i	$0.806738\pi$
$308$		0			0
$309$	−1.35393	+	2.76042i	−0.0770223	+	0.157035i
$310$		0			0
$311$	−22.6409			−1.28385			−0.641923	−	0.766769i	$-0.721863\pi$
$311$	−22.6409			−1.28385			−0.641923	+	0.766769i	$0.721863\pi$
$312$		0			0
$313$	−11.8378			−0.669110			−0.334555	−	0.942376i	$-0.608586\pi$
$313$	−11.8378			−0.669110			−0.334555	+	0.942376i	$0.608586\pi$
$314$		0			0
$315$	1.83649	+	2.37220i	0.103474	+	0.133658i
$316$		0			0
$317$		−	14.0087i		−	0.786805i	−0.919366	−	0.393403i	$-0.871298\pi$
$317$		−	14.0087i		−	0.786805i	0.919366	−	0.393403i	$-0.128702\pi$
$318$		0			0
$319$			1.15216i			0.0645087i
$320$		0			0
$321$	−9.53031	−	4.67442i	−0.531930	−	0.260901i
$322$		0			0
$323$	58.0908			3.23226
$324$		0			0
$325$	−3.40449			−0.188847
$326$		0			0
$327$	11.1547	+	5.47115i	0.616857	+	0.302555i
$328$		0			0
$329$			2.33793i			0.128894i
$330$		0			0
$331$			30.1565i			1.65755i	0.559583	+	0.828775i	$0.310961\pi$
$331$			30.1565i			1.65755i	−0.559583	+	0.828775i	$0.689039\pi$
$332$		0			0
$333$	14.4555	+	18.6723i	0.792159	+	1.02323i
$334$		0			0
$335$	−2.93417			−0.160311
$336$		0			0
$337$	13.8442			0.754140			0.377070	−	0.926185i	$-0.376932\pi$
$337$	13.8442			0.754140			0.377070	+	0.926185i	$0.376932\pi$
$338$		0			0
$339$	13.1582	−	26.8272i	0.714654	−	1.45705i
$340$		0			0
$341$		−	21.1867i		−	1.14732i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	−4.00171	+	8.15877i	−0.215445	+	0.439254i
$346$		0			0
$347$	−16.0219			−0.860102			−0.430051	−	0.902804i	$-0.641505\pi$
$347$	−16.0219			−0.860102			−0.430051	+	0.902804i	$0.641505\pi$
$348$		0			0
$349$	−14.9350			−0.799451			−0.399726	−	0.916635i	$-0.630895\pi$
$349$	−14.9350			−0.799451			−0.399726	+	0.916635i	$0.630895\pi$
$350$		0			0
$351$	−3.56288	−	17.3277i	−0.190173	−	0.924886i
$352$		0			0
$353$		−	21.4264i		−	1.14041i	−0.821501	−	0.570207i	$-0.806863\pi$
$353$		−	21.4264i		−	1.14041i	0.821501	−	0.570207i	$-0.193137\pi$
$354$		0			0
$355$		−	3.29594i		−	0.174930i
$356$		0			0
$357$	11.0818	+	5.43538i	0.586510	+	0.287671i
$358$		0			0
$359$	−19.4852			−1.02839			−0.514194	−	0.857674i	$-0.671909\pi$
$359$	−19.4852			−1.02839			−0.514194	+	0.857674i	$0.671909\pi$
$360$		0			0
$361$	−47.4501			−2.49737
$362$		0			0
$363$	4.45101	+	2.18313i	0.233618	+	0.114584i
$364$		0			0
$365$		−	12.0056i		−	0.628401i
$366$		0			0
$367$		−	35.2769i		−	1.84144i	−0.390222	−	0.920721i	$-0.627602\pi$
$367$		−	35.2769i		−	1.84144i	0.390222	−	0.920721i	$-0.372398\pi$
$368$		0			0
$369$	14.8446	−	11.4923i	0.772780	−	0.598265i
$370$		0			0
$371$	5.52406			0.286795
$372$		0			0
$373$	21.0035			1.08752			0.543759	−	0.839241i	$-0.317001\pi$
$373$	21.0035			1.08752			0.543759	+	0.839241i	$0.317001\pi$
$374$		0			0
$375$	−0.762729	+	1.55507i	−0.0393872	+	0.0803035i
$376$		0			0
$377$		−	1.37503i		−	0.0708179i
$378$		0			0
$379$			9.19116i			0.472118i	0.971739	+	0.236059i	$0.0758559\pi$
$379$			9.19116i			0.472118i	−0.971739	+	0.236059i	$0.924144\pi$
$380$		0			0
$381$	10.6635	−	21.7410i	0.546307	−	1.11382i
$382$		0			0
$383$	−31.4080			−1.60487			−0.802437	−	0.596737i	$-0.796464\pi$
$383$	−31.4080			−1.60487			−0.802437	+	0.596737i	$0.796464\pi$
$384$		0			0
$385$	−2.85267			−0.145386
$386$		0			0
$387$	6.97413	−	5.39917i	0.354515	−	0.274455i
$388$		0			0
$389$		−	5.95342i		−	0.301850i	−0.988545	−	0.150925i	$-0.951775\pi$
$389$		−	5.95342i		−	0.301850i	0.988545	−	0.150925i	$-0.0482252\pi$
$390$		0			0
$391$			37.3882i			1.89080i
$392$		0			0
$393$	17.4841	+	8.57560i	0.881958	+	0.432582i
$394$		0			0
$395$	16.3334			0.821823
$396$		0			0
$397$	−36.1464			−1.81413			−0.907067	−	0.420987i	$-0.861684\pi$
$397$	−36.1464			−1.81413			−0.907067	+	0.420987i	$0.861684\pi$
$398$		0			0
$399$	−12.6765	−	6.21754i	−0.634617	−	0.311266i
$400$		0			0
$401$		−	10.5734i		−	0.528011i	−0.964521	−	0.264005i	$-0.914956\pi$
$401$		−	10.5734i		−	0.528011i	0.964521	−	0.264005i	$-0.0850436\pi$
$402$		0			0
$403$			25.2850i			1.25953i
$404$		0			0
$405$	−8.71302	−	2.25462i	−0.432953	−	0.112033i
$406$		0			0
$407$	−22.4542			−1.11302
$408$		0			0
$409$	3.89950			0.192818			0.0964088	−	0.995342i	$-0.469264\pi$
$409$	3.89950			0.192818			0.0964088	+	0.995342i	$0.469264\pi$
$410$		0			0
$411$	7.85830	−	16.0217i	0.387621	−	0.790291i
$412$		0			0
$413$			0.0930502i			0.00457870i
$414$		0			0
$415$			7.88524i			0.387071i
$416$		0			0
$417$	14.7169	−	30.0053i	0.720692	−	1.46936i
$418$		0			0
$419$	−38.8771			−1.89927			−0.949635	−	0.313358i	$-0.898546\pi$
$419$	−38.8771			−1.89927			−0.949635	+	0.313358i	$0.898546\pi$
$420$		0			0
$421$	−14.8255			−0.722552			−0.361276	−	0.932459i	$-0.617659\pi$
$421$	−14.8255			−0.722552			−0.361276	+	0.932459i	$0.617659\pi$
$422$		0			0
$423$	−4.29359	−	5.54603i	−0.208761	−	0.269657i
$424$		0			0
$425$			7.12622i			0.345673i
$426$		0			0
$427$			2.79545i			0.135281i
$428$		0			0
$429$	15.1027	+	7.40754i	0.729164	+	0.357640i
$430$		0			0
$431$	3.70567			0.178496			0.0892479	−	0.996009i	$-0.471554\pi$
$431$	3.70567			0.178496			0.0892479	+	0.996009i	$0.471554\pi$
$432$		0			0
$433$	36.5682			1.75736			0.878679	−	0.477414i	$-0.158426\pi$
$433$	36.5682			1.75736			0.878679	+	0.477414i	$0.158426\pi$
$434$		0			0
$435$	−0.628076	−	0.308058i	−0.0301139	−	0.0147702i
$436$		0			0
$437$		−	42.7684i		−	2.04589i
$438$		0			0
$439$		−	35.0158i		−	1.67121i	−0.549330	−	0.835606i	$-0.685117\pi$
$439$		−	35.0158i		−	1.67121i	0.549330	−	0.835606i	$-0.314883\pi$
$440$		0			0
$441$	−1.83649	−	2.37220i	−0.0874518	−	0.112962i
$442$		0			0
$443$	24.3525			1.15702			0.578511	−	0.815674i	$-0.303634\pi$
$443$	24.3525			1.15702			0.578511	+	0.815674i	$0.303634\pi$
$444$		0			0
$445$	−9.63287			−0.456642
$446$		0			0
$447$	5.79069	−	11.8062i	0.273890	−	0.558414i
$448$		0			0
$449$			29.1578i			1.37604i	0.725691	+	0.688021i	$0.241520\pi$
$449$			29.1578i			1.37604i	−0.725691	+	0.688021i	$0.758480\pi$
$450$		0			0
$451$			17.8513i			0.840587i
$452$		0			0
$453$	−1.84136	+	3.75422i	−0.0865148	+	0.176388i
$454$		0			0
$455$	3.40449			0.159605
$456$		0			0
$457$	34.6845			1.62247			0.811236	−	0.584720i	$-0.198795\pi$
$457$	34.6845			1.62247			0.811236	+	0.584720i	$0.198795\pi$
$458$		0			0
$459$	−36.2702	+	7.45777i	−1.69295	+	0.348099i
$460$		0			0
$461$		−	31.9336i		−	1.48729i	−0.668572	−	0.743647i	$-0.733094\pi$
$461$		−	31.9336i		−	1.48729i	0.668572	−	0.743647i	$-0.266906\pi$
$462$		0			0
$463$		−	13.0170i		−	0.604952i	−0.953157	−	0.302476i	$-0.902187\pi$
$463$		−	13.0170i		−	0.604952i	0.953157	−	0.302476i	$-0.0978132\pi$
$464$		0			0
$465$	11.5494	+	5.66476i	0.535593	+	0.262697i
$466$		0			0
$467$	20.6382			0.955024			0.477512	−	0.878625i	$-0.341539\pi$
$467$	20.6382			0.955024			0.477512	+	0.878625i	$0.341539\pi$
$468$		0			0
$469$	2.93417			0.135487
$470$		0			0
$471$	16.0188	+	7.85689i	0.738108	+	0.362026i
$472$		0			0
$473$			8.38670i			0.385621i
$474$		0			0
$475$		−	8.15169i		−	0.374025i
$476$		0			0
$477$	−13.1041	+	10.1449i	−0.599998	+	0.464502i
$478$		0			0
$479$	25.5222			1.16614			0.583069	−	0.812423i	$-0.301852\pi$
$479$	25.5222			1.16614			0.583069	+	0.812423i	$0.301852\pi$
$480$		0			0
$481$	26.7977			1.22187
$482$		0			0
$483$	4.00171	−	8.15877i	0.182084	−	0.371237i
$484$		0			0
$485$		−	9.92561i		−	0.450699i
$486$		0			0
$487$		−	22.0867i		−	1.00085i	−0.865781	−	0.500423i	$-0.833178\pi$
$487$		−	22.0867i		−	1.00085i	0.865781	−	0.500423i	$-0.166822\pi$
$488$		0			0
$489$	5.66138	−	11.5426i	0.256016	−	0.521972i
$490$		0			0
$491$	27.9229			1.26014			0.630071	−	0.776538i	$-0.283026\pi$
$491$	27.9229			1.26014			0.630071	+	0.776538i	$0.283026\pi$
$492$		0			0
$493$	−2.87820			−0.129628
$494$		0			0
$495$	6.76710	−	5.23890i	0.304158	−	0.235471i
$496$		0			0
$497$			3.29594i			0.147843i
$498$		0			0
$499$		−	32.0063i		−	1.43280i	−0.697691	−	0.716399i	$-0.745789\pi$
$499$		−	32.0063i		−	1.43280i	0.697691	−	0.716399i	$-0.254211\pi$
$500$		0			0
$501$	19.3834	+	9.50717i	0.865988	+	0.424749i
$502$		0			0
$503$	−12.7870			−0.570146			−0.285073	−	0.958506i	$-0.592018\pi$
$503$	−12.7870			−0.570146			−0.285073	+	0.958506i	$0.592018\pi$
$504$		0			0
$505$	−6.49414			−0.288985
$506$		0			0
$507$	2.19181	+	1.07504i	0.0973418	+	0.0477441i
$508$		0			0
$509$			37.8589i			1.67807i	0.544080	+	0.839033i	$0.316879\pi$
$509$			37.8589i			1.67807i	−0.544080	+	0.839033i	$0.683121\pi$
$510$		0			0
$511$			12.0056i			0.531095i
$512$		0			0
$513$	41.4895	−	8.53095i	1.83180	−	0.376651i
$514$		0			0
$515$	1.77511			0.0782207
$516$		0			0
$517$	6.66936			0.293318
$518$		0			0
$519$	9.94796	−	20.2821i	0.436667	−	0.890287i
$520$		0			0
$521$			31.7706i			1.39190i	0.718092	+	0.695949i	$0.245016\pi$
$521$			31.7706i			1.39190i	−0.718092	+	0.695949i	$0.754984\pi$
$522$		0			0
$523$			12.0019i			0.524809i	0.964958	+	0.262404i	$0.0845154\pi$
$523$			12.0019i			0.524809i	−0.964958	+	0.262404i	$0.915485\pi$
$524$		0			0
$525$	0.762729	−	1.55507i	0.0332882	−	0.0678688i
$526$		0			0
$527$	52.9261			2.30550
$528$		0			0
$529$	4.52642			0.196801
$530$		0			0
$531$	−0.170886	−	0.220733i	−0.00741580	−	0.00957901i
$532$		0			0
$533$		−	21.3045i		−	0.922799i
$534$		0			0
$535$			6.12854i			0.264960i
$536$		0			0
$537$	−1.69606	−	0.831880i	−0.0731902	−	0.0358983i
$538$		0			0
$539$	2.85267			0.122873
$540$		0			0
$541$	−2.57383			−0.110658			−0.0553288	−	0.998468i	$-0.517621\pi$
$541$	−2.57383			−0.110658			−0.0553288	+	0.998468i	$0.517621\pi$
$542$		0			0
$543$	−15.5718	−	7.63765i	−0.668251	−	0.327763i
$544$		0			0
$545$		−	7.17312i		−	0.307263i
$546$		0			0
$547$			17.2871i			0.739142i	0.929203	+	0.369571i	$0.120495\pi$
$547$			17.2871i			0.739142i	−0.929203	+	0.369571i	$0.879505\pi$
$548$		0			0
$549$	−5.13381	−	6.63136i	−0.219106	−	0.283020i
$550$		0			0
$551$	3.29238			0.140260
$552$		0			0
$553$	−16.3334			−0.694567
$554$		0			0
$555$	6.00367	−	12.2404i	0.254842	−	0.519577i
$556$		0			0
$557$		−	12.4267i		−	0.526535i	−0.964723	−	0.263267i	$-0.915200\pi$
$557$		−	12.4267i		−	0.526535i	0.964723	−	0.263267i	$-0.0848002\pi$
$558$		0			0
$559$		−	10.0090i		−	0.423336i
$560$		0			0
$561$	15.5054	−	31.6127i	0.654637	−	1.33469i
$562$		0			0
$563$	32.2563			1.35944			0.679720	−	0.733472i	$-0.262101\pi$
$563$	32.2563			1.35944			0.679720	+	0.733472i	$0.262101\pi$
$564$		0			0
$565$	−17.2514			−0.725773
$566$		0			0
$567$	8.71302	+	2.25462i	0.365912	+	0.0946853i
$568$		0			0
$569$		−	8.00625i		−	0.335639i	−0.985818	−	0.167820i	$-0.946327\pi$
$569$		−	8.00625i		−	0.335639i	0.985818	−	0.167820i	$-0.0536726\pi$
$570$		0			0
$571$			23.4263i			0.980361i	0.871621	+	0.490180i	$0.163069\pi$
$571$			23.4263i			0.980361i	−0.871621	+	0.490180i	$0.836931\pi$
$572$		0			0
$573$	−41.7200	−	20.4628i	−1.74288	−	0.854846i
$574$		0			0
$575$	5.24656			0.218797
$576$		0			0
$577$	−19.5869			−0.815413			−0.407706	−	0.913113i	$-0.633671\pi$
$577$	−19.5869			−0.815413			−0.407706	+	0.913113i	$0.633671\pi$
$578$		0			0
$579$	−29.9973	−	14.7130i	−1.24664	−	0.611453i
$580$		0			0
$581$		−	7.88524i		−	0.327135i
$582$		0			0
$583$		−	15.7583i		−	0.652644i
$584$		0			0
$585$	−8.07611	+	6.25230i	−0.333906	+	0.258501i
$586$		0			0
$587$	−31.8675			−1.31531			−0.657656	−	0.753318i	$-0.728452\pi$
$587$	−31.8675			−1.31531			−0.657656	+	0.753318i	$0.728452\pi$
$588$		0			0
$589$	−60.5423			−2.49460
$590$		0			0
$591$	1.61577	−	3.29428i	0.0664640	−	0.135508i
$592$		0			0
$593$			0.950261i			0.0390226i	0.999810	+	0.0195113i	$0.00621103\pi$
$593$			0.950261i			0.0390226i	−0.999810	+	0.0195113i	$0.993789\pi$
$594$		0			0
$595$		−	7.12622i		−	0.292147i
$596$		0			0
$597$	−5.57023	+	11.3567i	−0.227974	+	0.464800i
$598$		0			0
$599$	−5.73344			−0.234262			−0.117131	−	0.993116i	$-0.537370\pi$
$599$	−5.73344			−0.234262			−0.117131	+	0.993116i	$0.537370\pi$
$600$		0			0
$601$	−31.0778			−1.26769			−0.633845	−	0.773460i	$-0.718524\pi$
$601$	−31.0778			−1.26769			−0.633845	+	0.773460i	$0.718524\pi$
$602$		0			0
$603$	−6.96042	+	5.38856i	−0.283450	+	0.219439i
$604$		0			0
$605$		−	2.86226i		−	0.116367i
$606$		0			0
$607$		−	38.8661i		−	1.57753i	−0.614696	−	0.788764i	$-0.710721\pi$
$607$		−	38.8661i		−	1.57753i	0.614696	−	0.788764i	$-0.289279\pi$
$608$		0			0
$609$	0.628076	+	0.308058i	0.0254509	+	0.0124831i
$610$		0			0
$611$	−7.95946			−0.322005
$612$		0			0
$613$	−8.74878			−0.353360			−0.176680	−	0.984268i	$-0.556536\pi$
$613$	−8.74878			−0.353360			−0.176680	+	0.984268i	$0.556536\pi$
$614$		0			0
$615$	−9.73125	−	4.77297i	−0.392402	−	0.192465i
$616$		0			0
$617$		−	18.4379i		−	0.742282i	−0.928577	−	0.371141i	$-0.878967\pi$
$617$		−	18.4379i		−	0.742282i	0.928577	−	0.371141i	$-0.121033\pi$
$618$		0			0
$619$		−	28.1189i		−	1.13019i	−0.825025	−	0.565097i	$-0.808839\pi$
$619$		−	28.1189i		−	1.13019i	0.825025	−	0.565097i	$-0.191161\pi$
$620$		0			0
$621$	5.49066	+	26.7033i	0.220333	+	1.07157i
$622$		0			0
$623$	9.63287			0.385933
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−17.7366	+	36.1618i	−0.708331	+	1.44416i
$628$		0			0
$629$		−	56.0926i		−	2.23656i
$630$		0			0
$631$		−	28.9942i		−	1.15424i	−0.816659	−	0.577121i	$-0.804176\pi$
$631$		−	28.9942i		−	1.15424i	0.816659	−	0.577121i	$-0.195824\pi$
$632$		0			0
$633$	10.1296	−	20.6525i	0.402616	−	0.820863i
$634$		0			0
$635$	−13.9807			−0.554807
$636$		0			0
$637$	−3.40449			−0.134891
$638$		0			0
$639$	−6.05295	−	7.81861i	−0.239451	−	0.309299i
$640$		0			0
$641$		−	27.5190i		−	1.08693i	−0.839430	−	0.543467i	$-0.817111\pi$
$641$		−	27.5190i		−	1.08693i	0.839430	−	0.543467i	$-0.182889\pi$
$642$		0			0
$643$		−	12.0923i		−	0.476873i	−0.971158	−	0.238436i	$-0.923365\pi$
$643$		−	12.0923i		−	0.476873i	0.971158	−	0.238436i	$-0.0766349\pi$
$644$		0			0
$645$	−4.57182	−	2.24238i	−0.180015	−	0.0882937i
$646$		0			0
$647$	−17.9743			−0.706641			−0.353320	−	0.935502i	$-0.614947\pi$
$647$	−17.9743			−0.706641			−0.353320	+	0.935502i	$0.614947\pi$
$648$		0			0
$649$	0.265442			0.0104195
$650$		0			0
$651$	−11.5494	−	5.66476i	−0.452658	−	0.222019i
$652$		0			0
$653$		−	26.9683i		−	1.05535i	−0.849446	−	0.527675i	$-0.823064\pi$
$653$		−	26.9683i		−	1.05535i	0.849446	−	0.527675i	$-0.176936\pi$
$654$		0			0
$655$		−	11.2433i		−	0.439312i
$656$		0			0
$657$	−22.0481	−	28.4796i	−0.860178	−	1.11109i
$658$		0			0
$659$	15.9598			0.621706			0.310853	−	0.950458i	$-0.399385\pi$
$659$	15.9598			0.621706			0.310853	+	0.950458i	$0.399385\pi$
$660$		0			0
$661$	28.6891			1.11588			0.557938	−	0.829882i	$-0.311593\pi$
$661$	28.6891			1.11588			0.557938	+	0.829882i	$0.311593\pi$
$662$		0			0
$663$	−18.5047	+	37.7278i	−0.718662	+	1.46523i
$664$		0			0
$665$			8.15169i			0.316109i
$666$		0			0
$667$			2.11903i			0.0820491i
$668$		0			0
$669$	8.24169	−	16.8033i	0.318642	−	0.649655i
$670$		0			0
$671$	7.97451			0.307853
$672$		0			0
$673$	10.5770			0.407714			0.203857	−	0.979001i	$-0.434652\pi$
$673$	10.5770			0.407714			0.203857	+	0.979001i	$0.434652\pi$
$674$		0			0
$675$	1.04653	+	5.08967i	0.0402808	+	0.195902i
$676$		0			0
$677$			33.8589i			1.30130i	0.759377	+	0.650651i	$0.225504\pi$
$677$			33.8589i			1.30130i	−0.759377	+	0.650651i	$0.774496\pi$
$678$		0			0
$679$			9.92561i			0.380910i
$680$		0			0
$681$	27.8374	+	13.6537i	1.06673	+	0.523210i
$682$		0			0
$683$	−36.0839			−1.38071			−0.690356	−	0.723469i	$-0.742546\pi$
$683$	−36.0839			−1.38071			−0.690356	+	0.723469i	$0.742546\pi$
$684$		0			0
$685$	−10.3029			−0.393652
$686$		0			0
$687$	38.5448	+	18.9054i	1.47057	+	0.721286i
$688$		0			0
$689$			18.8066i			0.716474i
$690$		0			0
$691$		−	4.64306i		−	0.176630i	−0.996093	−	0.0883151i	$-0.971852\pi$
$691$		−	4.64306i		−	0.176630i	0.996093	−	0.0883151i	$-0.0281482\pi$
$692$		0			0
$693$	−6.76710	+	5.23890i	−0.257061	+	0.199009i
$694$		0			0
$695$	−19.2951			−0.731905
$696$		0			0
$697$	−44.5942			−1.68912
$698$		0			0
$699$	9.89441	−	20.1730i	0.374241	−	0.763011i
$700$		0			0
$701$		−	6.27482i		−	0.236997i	−0.992954	−	0.118498i	$-0.962192\pi$
$701$		−	6.27482i		−	0.236997i	0.992954	−	0.118498i	$-0.0378081\pi$
$702$		0			0
$703$			64.1644i			2.42001i
$704$		0			0
$705$	−1.78321	+	3.63565i	−0.0671596	+	0.136926i
$706$		0			0
$707$	6.49414			0.244237
$708$		0			0
$709$	31.9558			1.20012			0.600062	−	0.799953i	$-0.295143\pi$
$709$	31.9558			1.20012			0.600062	+	0.799953i	$0.295143\pi$
$710$		0			0
$711$	38.7460	−	29.9961i	1.45309	−	1.12494i
$712$		0			0
$713$		−	38.9660i		−	1.45929i
$714$		0			0
$715$		−	9.71189i		−	0.363204i
$716$		0			0
$717$	−0.552505	−	0.270992i	−0.0206337	−	0.0101204i
$718$		0			0
$719$	40.0464			1.49348			0.746739	−	0.665117i	$-0.231619\pi$
$719$	40.0464			1.49348			0.746739	+	0.665117i	$0.231619\pi$
$720$		0			0
$721$	−1.77511			−0.0661085
$722$		0			0
$723$	−14.5928	−	7.15748i	−0.542713	−	0.266189i
$724$		0			0
$725$			0.403889i			0.0150001i
$726$		0			0
$727$			31.9527i			1.18506i	0.805548	+	0.592531i	$0.201871\pi$
$727$			31.9527i			1.18506i	−0.805548	+	0.592531i	$0.798129\pi$
$728$		0			0
$729$	−24.8096	+	10.6529i	−0.918873	+	0.394553i
$730$		0			0
$731$	−20.9507			−0.774890
$732$		0			0
$733$	19.4357			0.717872			0.358936	−	0.933362i	$-0.383140\pi$
$733$	19.4357			0.717872			0.358936	+	0.933362i	$0.383140\pi$
$734$		0			0
$735$	−0.762729	+	1.55507i	−0.0281337	+	0.0573596i
$736$		0			0
$737$		−	8.37022i		−	0.308321i
$738$		0			0
$739$		−	14.0476i		−	0.516748i	−0.966045	−	0.258374i	$-0.916813\pi$
$739$		−	14.0476i		−	0.516748i	0.966045	−	0.258374i	$-0.0831868\pi$
$740$		0			0
$741$	21.1675	−	43.1568i	0.777609	−	1.58541i
$742$		0			0
$743$	−3.01075			−0.110454			−0.0552269	−	0.998474i	$-0.517588\pi$
$743$	−3.01075			−0.110454			−0.0552269	+	0.998474i	$0.517588\pi$
$744$		0			0
$745$	−7.59206			−0.278152
$746$		0			0
$747$	14.4811	+	18.7053i	0.529837	+	0.684392i
$748$		0			0
$749$		−	6.12854i		−	0.223932i
$750$		0			0
$751$			5.37591i			0.196170i	0.995178	+	0.0980849i	$0.0312717\pi$
$751$			5.37591i			0.196170i	−0.995178	+	0.0980849i	$0.968728\pi$
$752$		0			0
$753$	−4.65846	−	2.28488i	−0.169764	−	0.0832656i
$754$		0			0
$755$	2.41418			0.0878609
$756$		0			0
$757$	38.3898			1.39530			0.697651	−	0.716438i	$-0.254229\pi$
$757$	38.3898			1.39530			0.697651	+	0.716438i	$0.254229\pi$
$758$		0			0
$759$	−23.2743	−	11.4156i	−0.844804	−	0.414359i
$760$		0			0
$761$		−	23.7472i		−	0.860834i	−0.902630	−	0.430417i	$-0.858367\pi$
$761$		−	23.7472i		−	0.860834i	0.902630	−	0.430417i	$-0.141633\pi$
$762$		0			0
$763$			7.17312i			0.259685i
$764$		0			0
$765$	13.0872	+	16.9048i	0.473170	+	0.611194i
$766$		0			0
$767$	−0.316788			−0.0114386
$768$		0			0
$769$	−32.3693			−1.16727			−0.583633	−	0.812018i	$-0.698369\pi$
$769$	−32.3693			−1.16727			−0.583633	+	0.812018i	$0.698369\pi$
$770$		0			0
$771$	5.57940	−	11.3754i	0.200937	−	0.409675i
$772$		0			0
$773$		−	26.1245i		−	0.939633i	−0.882764	−	0.469816i	$-0.844320\pi$
$773$		−	26.1245i		−	0.939633i	0.882764	−	0.469816i	$-0.155680\pi$
$774$		0			0
$775$		−	7.42696i		−	0.266784i
$776$		0			0
$777$	−6.00367	+	12.2404i	−0.215380	+	0.439123i
$778$		0			0
$779$	51.0113			1.82767
$780$		0			0
$781$	9.40223			0.336438
$782$		0			0
$783$	−2.05566	+	0.422680i	−0.0734634	+	0.0151053i
$784$		0			0
$785$		−	10.3010i		−	0.367659i
$786$		0			0
$787$		−	27.0408i		−	0.963901i	−0.876198	−	0.481950i	$-0.839929\pi$
$787$		−	27.0408i		−	0.963901i	0.876198	−	0.481950i	$-0.160071\pi$
$788$		0			0
$789$	−31.2677	−	15.3361i	−1.11316	−	0.545981i
$790$		0			0
$791$	17.2514			0.613390
$792$		0			0
$793$	−9.51708			−0.337962
$794$		0			0
$795$	8.59030	+	4.21336i	0.304667	+	0.149433i
$796$		0			0
$797$		−	20.2687i		−	0.717955i	−0.933346	−	0.358977i	$-0.883126\pi$
$797$		−	20.2687i		−	0.717955i	0.933346	−	0.358977i	$-0.116874\pi$
$798$		0			0
$799$			16.6606i			0.589411i
$800$		0			0
$801$	−22.8510	+	17.6906i	−0.807402	+	0.625068i
$802$		0			0
$803$	34.2480			1.20859
$804$		0			0
$805$	−5.24656			−0.184917
$806$		0			0
$807$	24.3922	−	49.7313i	0.858645	−	1.75063i
$808$		0			0
$809$			45.6166i			1.60380i	0.597461	+	0.801898i	$0.296176\pi$
$809$			45.6166i			1.60380i	−0.597461	+	0.801898i	$0.703824\pi$
$810$		0			0
$811$			0.631591i			0.0221782i	0.999939	+	0.0110891i	$0.00352984\pi$
$811$			0.631591i			0.0221782i	−0.999939	+	0.0110891i	$0.996470\pi$
$812$		0			0
$813$	−9.62507	+	19.6238i	−0.337566	+	0.688237i
$814$		0			0
$815$	−7.42253			−0.260000
$816$		0			0
$817$	23.9655			0.838448
$818$		0			0
$819$	8.07611	−	6.25230i	0.282202	−	0.218473i
$820$		0			0
$821$			8.30574i			0.289872i	0.989441	+	0.144936i	$0.0462977\pi$
$821$			8.30574i			0.289872i	−0.989441	+	0.144936i	$0.953702\pi$
$822$		0			0
$823$			22.5936i			0.787563i	0.919204	+	0.393781i	$0.128833\pi$
$823$			22.5936i			0.787563i	−0.919204	+	0.393781i	$0.871167\pi$
$824$		0			0
$825$	−4.43611	−	2.17582i	−0.154445	−	0.0757522i
$826$		0			0
$827$	−48.4994			−1.68649			−0.843244	−	0.537531i	$-0.819357\pi$
$827$	−48.4994			−1.68649			−0.843244	+	0.537531i	$0.819357\pi$
$828$		0			0
$829$	−28.4911			−0.989538			−0.494769	−	0.869025i	$-0.664747\pi$
$829$	−28.4911			−0.989538			−0.494769	+	0.869025i	$0.664747\pi$
$830$		0			0
$831$	17.7872	+	8.72424i	0.617031	+	0.302641i
$832$		0			0
$833$			7.12622i			0.246909i
$834$		0			0
$835$		−	12.4647i		−	0.431358i
$836$		0			0
$837$	37.8008	−	7.77250i	1.30659	−	0.268657i
$838$		0			0
$839$	10.2451			0.353701			0.176850	−	0.984238i	$-0.443409\pi$
$839$	10.2451			0.353701			0.176850	+	0.984238i	$0.443409\pi$
$840$		0			0
$841$	28.8369			0.994375
$842$		0			0
$843$	23.9393	−	48.8080i	0.824513	−	1.68104i
$844$		0			0
$845$		−	1.40946i		−	0.0484870i
$846$		0			0
$847$			2.86226i			0.0983483i
$848$		0			0
$849$	8.83048	−	18.0038i	0.303061	−	0.617889i
$850$		0			0
$851$	−41.2973			−1.41565
$852$		0			0
$853$	−26.4901			−0.907003			−0.453501	−	0.891256i	$-0.649825\pi$
$853$	−26.4901			−0.907003			−0.453501	+	0.891256i	$0.649825\pi$
$854$		0			0
$855$	−14.9705	−	19.3374i	−0.511980	−	0.661326i
$856$		0			0
$857$		−	20.1322i		−	0.687703i	−0.939024	−	0.343851i	$-0.888268\pi$
$857$		−	20.1322i		−	0.687703i	0.939024	−	0.343851i	$-0.111732\pi$
$858$		0			0
$859$			1.98063i			0.0675784i	0.999429	+	0.0337892i	$0.0107575\pi$
$859$			1.98063i			0.0675784i	−0.999429	+	0.0337892i	$0.989243\pi$
$860$		0			0
$861$	9.73125	+	4.77297i	0.331640	+	0.162663i
$862$		0			0
$863$	15.1714			0.516442			0.258221	−	0.966086i	$-0.416864\pi$
$863$	15.1714			0.516442			0.258221	+	0.966086i	$0.416864\pi$
$864$		0			0
$865$	−13.0426			−0.443461
$866$		0			0
$867$	52.5350	+	25.7673i	1.78418	+	0.875104i
$868$		0			0
$869$			46.5939i			1.58059i
$870$		0			0
$871$			9.98933i			0.338476i
$872$		0			0
$873$	−18.2283	−	23.5455i	−0.616933	−	0.796894i
$874$		0			0
$875$	−1.00000			−0.0338062
$876$		0			0
$877$	−43.3654			−1.46434			−0.732172	−	0.681120i	$-0.761493\pi$
$877$	−43.3654			−1.46434			−0.732172	+	0.681120i	$0.761493\pi$
$878$		0			0
$879$	−4.66768	+	9.51657i	−0.157437	+	0.320986i
$880$		0			0
$881$			46.9146i			1.58059i	0.612724	+	0.790297i	$0.290074\pi$
$881$			46.9146i			1.58059i	−0.612724	+	0.790297i	$0.709926\pi$
$882$		0			0
$883$		−	26.1005i		−	0.878351i	−0.898401	−	0.439175i	$-0.855271\pi$
$883$		−	26.1005i		−	0.878351i	0.898401	−	0.439175i	$-0.144729\pi$
$884$		0			0
$885$	−0.0709721	+	0.144700i	−0.00238570	+	0.00486402i
$886$		0			0
$887$	47.3656			1.59038			0.795190	−	0.606360i	$-0.207371\pi$
$887$	47.3656			1.59038			0.795190	+	0.606360i	$0.207371\pi$
$888$		0			0
$889$	13.9807			0.468898
$890$		0			0
$891$	6.43170	−	24.8554i	0.215470	−	0.832687i
$892$		0			0
$893$		−	19.0581i		−	0.637755i
$894$		0			0
$895$			1.09066i			0.0364568i
$896$		0			0
$897$	27.7764	+	13.6238i	0.927429	+	0.454884i
$898$		0			0
$899$	2.99967			0.100044
$900$		0			0
$901$	39.3657			1.31146
$902$		0			0
$903$	4.57182	+	2.24238i	0.152141	+	0.0746218i
$904$		0			0
$905$			10.0136i			0.332863i
$906$		0			0
$907$		−	6.15263i		−	0.204295i	−0.994769	−	0.102147i	$-0.967429\pi$
$907$		−	6.15263i		−	0.204295i	0.994769	−	0.102147i	$-0.0325713\pi$
$908$		0			0
$909$	−15.4054	+	11.9264i	−0.510964	+	0.395574i
$910$		0			0
$911$	30.8142			1.02092			0.510460	−	0.859902i	$-0.329475\pi$
$911$	30.8142			1.02092			0.510460	+	0.859902i	$0.329475\pi$
$912$		0			0
$913$	−22.4940			−0.744443
$914$		0			0
$915$	−2.13217	+	4.34713i	−0.0704875	+	0.143712i
$916$		0			0
$917$			11.2433i			0.371287i
$918$		0			0
$919$		−	39.3519i		−	1.29810i	−0.760746	−	0.649050i	$-0.775167\pi$
$919$		−	39.3519i		−	1.29810i	0.760746	−	0.649050i	$-0.224833\pi$
$920$		0			0
$921$	15.2492	−	31.0905i	0.502479	−	1.02447i
$922$		0			0
$923$	−11.2210			−0.369343
$924$		0			0
$925$	−7.87130			−0.258807
$926$		0			0
$927$	4.21091	−	3.25997i	0.138304	−	0.107071i
$928$		0			0
$929$			3.94914i			0.129567i	0.997899	+	0.0647835i	$0.0206357\pi$
$929$			3.94914i			0.129567i	−0.997899	+	0.0647835i	$0.979364\pi$
$930$		0			0
$931$		−	8.15169i		−	0.267161i
$932$		0			0
$933$	35.2082	+	17.2689i	1.15266	+	0.565357i
$934$		0			0
$935$	−20.3288			−0.664822
$936$		0			0
$937$	26.0900			0.852322			0.426161	−	0.904647i	$-0.359866\pi$
$937$	26.0900			0.852322			0.426161	+	0.904647i	$0.359866\pi$
$938$		0			0
$939$	18.4086	+	9.02901i	0.600741	+	0.294651i
$940$		0			0
$941$		−	6.16446i		−	0.200956i	−0.994939	−	0.100478i	$-0.967963\pi$
$941$		−	6.16446i		−	0.200956i	0.994939	−	0.100478i	$-0.0320372\pi$
$942$		0			0
$943$			32.8317i			1.06915i
$944$		0			0
$945$	−1.04653	−	5.08967i	−0.0340435	−	0.165567i
$946$		0			0
$947$	−3.97780			−0.129261			−0.0646306	−	0.997909i	$-0.520587\pi$
$947$	−3.97780			−0.129261			−0.0646306	+	0.997909i	$0.520587\pi$
$948$		0			0
$949$	−40.8728			−1.32679
$950$		0			0
$951$	−10.6848	+	21.7845i	−0.346479	+	0.706410i
$952$		0			0
$953$		−	39.7672i		−	1.28819i	−0.764947	−	0.644093i	$-0.777235\pi$
$953$		−	39.7672i		−	1.28819i	0.764947	−	0.644093i	$-0.222765\pi$
$954$		0			0
$955$			26.8284i			0.868146i
$956$		0			0
$957$	0.878788	−	1.79169i	0.0284072	−	0.0579173i
$958$		0			0
$959$	10.3029			0.332697
$960$		0			0
$961$	−24.1597			−0.779345
$962$		0			0
$963$	11.2550	+	14.5381i	0.362687	+	0.468484i
$964$		0			0
$965$			19.2900i			0.620966i
$966$		0			0
$967$			36.6680i			1.17916i	0.807709	+	0.589581i	$0.200707\pi$
$967$			36.6680i			1.17916i	−0.807709	+	0.589581i	$0.799293\pi$
$968$		0			0
$969$	−90.3353	−	44.3075i	−2.90199	−	1.42336i
$970$		0			0
$971$	−7.39035			−0.237168			−0.118584	−	0.992944i	$-0.537835\pi$
$971$	−7.39035			−0.237168			−0.118584	+	0.992944i	$0.537835\pi$
$972$		0			0
$973$	19.2951			0.618573
$974$		0			0
$975$	5.29422	+	2.59670i	0.169551	+	0.0831610i
$976$		0			0
$977$		−	8.84809i		−	0.283075i	−0.989933	−	0.141538i	$-0.954795\pi$
$977$		−	8.84809i		−	0.283075i	0.989933	−	0.141538i	$-0.0452046\pi$
$978$		0			0
$979$		−	27.4794i		−	0.878246i
$980$		0			0
$981$	−13.1734	−	17.0161i	−0.420593	−	0.543281i
$982$		0			0
$983$	−1.25166			−0.0399218			−0.0199609	−	0.999801i	$-0.506354\pi$
$983$	−1.25166			−0.0399218			−0.0199609	+	0.999801i	$0.506354\pi$
$984$		0			0
$985$	−2.11841			−0.0674981
$986$		0			0
$987$	1.78321	−	3.63565i	0.0567602	−	0.115724i
$988$		0			0
$989$			15.4246i			0.490474i
$990$		0			0
$991$		−	36.4936i		−	1.15926i	−0.814881	−	0.579629i	$-0.803198\pi$
$991$		−	36.4936i		−	1.15926i	0.814881	−	0.579629i	$-0.196802\pi$
$992$		0			0
$993$	23.0012	−	46.8954i	0.729922	−	1.48818i
$994$		0			0
$995$	7.30303			0.231521
$996$		0			0
$997$	6.33285			0.200563			0.100282	−	0.994959i	$-0.468026\pi$
$997$	6.33285			0.200563			0.100282	+	0.994959i	$0.468026\pi$
$998$		0			0
$999$	−8.23751	−	40.0623i	−0.260623	−	1.26752i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.ba.f.2591.5	yes	24
3.2	odd	2		3360.2.ba.e.2591.19	✓	24
4.3	odd	2		3360.2.ba.e.2591.20	yes	24
12.11	even	2	inner	3360.2.ba.f.2591.6	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3360.2.ba.e.2591.19	✓	24	3.2	odd	2
3360.2.ba.e.2591.20	yes	24	4.3	odd	2
3360.2.ba.f.2591.5	yes	24	1.1	even	1	trivial
3360.2.ba.f.2591.6	yes	24	12.11	even	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.ba (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.55507 - 0.762729i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(1.83649 + 2.37220i) q^{9} +O(q^{10})\)	\(q+(-1.55507 - 0.762729i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(1.83649 + 2.37220i) q^{9} -2.85267 q^{11} +3.40449 q^{13} +(0.762729 - 1.55507i) q^{15} -7.12622i q^{17} +8.15169i q^{19} +(-0.762729 + 1.55507i) q^{21} -5.24656 q^{23} -1.00000 q^{25} +(-1.04653 - 5.08967i) q^{27} -0.403889i q^{29} +7.42696i q^{31} +(4.43611 + 2.17582i) q^{33} +1.00000 q^{35} +7.87130 q^{37} +(-5.29422 - 2.59670i) q^{39} -6.25776i q^{41} -2.93995i q^{43} +(-2.37220 + 1.83649i) q^{45} -2.33793 q^{47} -1.00000 q^{49} +(-5.43538 + 11.0818i) q^{51} +5.52406i q^{53} -2.85267i q^{55} +(6.21754 - 12.6765i) q^{57} -0.0930502 q^{59} -2.79545 q^{61} +(2.37220 - 1.83649i) q^{63} +3.40449i q^{65} +2.93417i q^{67} +(8.15877 + 4.00171i) q^{69} -3.29594 q^{71} -12.0056 q^{73} +(1.55507 + 0.762729i) q^{75} +2.85267i q^{77} -16.3334i q^{79} +(-2.25462 + 8.71302i) q^{81} +7.88524 q^{83} +7.12622 q^{85} +(-0.308058 + 0.628076i) q^{87} +9.63287i q^{89} -3.40449i q^{91} +(5.66476 - 11.5494i) q^{93} -8.15169 q^{95} -9.92561 q^{97} +(-5.23890 - 6.76710i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 4 q^{9}+O(q^{10}) \) 24 * q + 4 * q^9	\( 24 q + 4 q^{9} + 8 q^{11} + 8 q^{13} + 4 q^{15} - 4 q^{21} - 24 q^{25} + 8 q^{33} + 24 q^{35} + 8 q^{37} + 36 q^{39} - 24 q^{49} - 12 q^{51} + 8 q^{57} + 48 q^{59} + 56 q^{61} - 88 q^{71} - 40 q^{73} + 44 q^{81} + 24 q^{83} + 32 q^{87} - 48 q^{93} - 24 q^{95} - 8 q^{97} - 52 q^{99}+O(q^{100}) \) 24 * q + 4 * q^9 + 8 * q^11 + 8 * q^13 + 4 * q^15 - 4 * q^21 - 24 * q^25 + 8 * q^33 + 24 * q^35 + 8 * q^37 + 36 * q^39 - 24 * q^49 - 12 * q^51 + 8 * q^57 + 48 * q^59 + 56 * q^61 - 88 * q^71 - 40 * q^73 + 44 * q^81 + 24 * q^83 + 32 * q^87 - 48 * q^93 - 24 * q^95 - 8 * q^97 - 52 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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