LMFDB - Embedded newform 1216.3.e.n.1025.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1216,3,Mod(1025,1216)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1216, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1216.1025");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1216 = 2^{6} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1216.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$33.1336001462$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 34x^{6} + 345x^{4} + 1064x^{2} + 256 $ x^8 + 34x^6 + 345x^4 + 1064*x^2 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1025.2
Root			$-3.20945i$ of defining polynomial
Character	$\chi$	$=$	1216.1025
Dual form			1216.3.e.n.1025.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.20945i q^{3} +3.06310 q^{5} +12.3151 q^{7} -1.30054 q^{9} +O(q^{10})$	$q-3.20945i q^{3} +3.06310 q^{5} +12.3151 q^{7} -1.30054 q^{9} -5.53798 q^{11} -6.70944i q^{13} -9.83087i q^{15} -5.36365 q^{17} +(1.17433 - 18.9637i) q^{19} -39.5247i q^{21} +9.72688 q^{23} -15.6174 q^{25} -24.7110i q^{27} +14.4351i q^{29} +36.0875i q^{31} +17.7738i q^{33} +37.7225 q^{35} -32.4102i q^{37} -21.5336 q^{39} -60.2390i q^{41} +60.5760 q^{43} -3.98369 q^{45} +58.8019 q^{47} +102.662 q^{49} +17.2143i q^{51} -24.4833i q^{53} -16.9634 q^{55} +(-60.8629 - 3.76895i) q^{57} +107.312i q^{59} -45.9998 q^{61} -16.0163 q^{63} -20.5517i q^{65} -42.6265i q^{67} -31.2179i q^{69} +70.3493i q^{71} +52.3089 q^{73} +50.1232i q^{75} -68.2008 q^{77} -2.68432i q^{79} -91.0135 q^{81} -138.135 q^{83} -16.4294 q^{85} +46.3288 q^{87} -106.061i q^{89} -82.6276i q^{91} +115.821 q^{93} +(3.59710 - 58.0877i) q^{95} -79.3904i q^{97} +7.20236 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 14 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) $ 8 * q + 14 * q^5 + 6 * q^7 + 4 * q^9	$ 8 q + 14 q^{5} + 6 q^{7} + 4 q^{9} - 26 q^{11} - 18 q^{17} + 16 q^{19} - 12 q^{23} + 34 q^{25} - 50 q^{35} + 108 q^{39} + 62 q^{43} - 162 q^{45} - 22 q^{47} + 22 q^{49} - 174 q^{55} + 4 q^{57} + 158 q^{61} + 2 q^{63} - 170 q^{73} - 82 q^{77} - 256 q^{81} - 64 q^{83} - 410 q^{85} + 332 q^{87} + 344 q^{93} - 222 q^{95} + 526 q^{99}+O(q^{100}) $ 8 * q + 14 * q^5 + 6 * q^7 + 4 * q^9 - 26 * q^11 - 18 * q^17 + 16 * q^19 - 12 * q^23 + 34 * q^25 - 50 * q^35 + 108 * q^39 + 62 * q^43 - 162 * q^45 - 22 * q^47 + 22 * q^49 - 174 * q^55 + 4 * q^57 + 158 * q^61 + 2 * q^63 - 170 * q^73 - 82 * q^77 - 256 * q^81 - 64 * q^83 - 410 * q^85 + 332 * q^87 + 344 * q^93 - 222 * q^95 + 526 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1216\mathbb{Z}\right)^\times$.

$n$	$191$	$705$	$837$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$		0			0
$2$		0			0
$3$		−	3.20945i		−	1.06982i	−0.844911	−	0.534908i	$-0.820346\pi$
$3$		−	3.20945i		−	1.06982i	0.844911	−	0.534908i	$-0.179654\pi$
$4$		0			0
$5$	3.06310			0.612621			0.306310	−	0.951932i	$-0.400905\pi$
$5$	3.06310			0.612621			0.306310	+	0.951932i	$0.400905\pi$
$6$		0			0
$7$	12.3151			1.75930			0.879651	−	0.475619i	$-0.157776\pi$
$7$	12.3151			1.75930			0.879651	+	0.475619i	$0.157776\pi$
$8$		0			0
$9$	−1.30054			−0.144504
$10$		0			0
$11$	−5.53798			−0.503452			−0.251726	−	0.967798i	$-0.580998\pi$
$11$	−5.53798			−0.503452			−0.251726	+	0.967798i	$0.580998\pi$
$12$		0			0
$13$		−	6.70944i		−	0.516111i	−0.966130	−	0.258055i	$-0.916918\pi$
$13$		−	6.70944i		−	0.516111i	0.966130	−	0.258055i	$-0.0830817\pi$
$14$		0			0
$15$		−	9.83087i		−	0.655391i
$16$		0			0
$17$	−5.36365			−0.315509			−0.157754	−	0.987478i	$-0.550425\pi$
$17$	−5.36365			−0.315509			−0.157754	+	0.987478i	$0.550425\pi$
$18$		0			0
$19$	1.17433	−	18.9637i	0.0618069	−	0.998088i
$19$	1.17433	−	18.9637i	0.0618069	−	0.998088i
$20$		0			0
$21$		−	39.5247i		−	1.88213i
$22$		0			0
$23$	9.72688			0.422908			0.211454	−	0.977388i	$-0.432180\pi$
$23$	9.72688			0.422908			0.211454	+	0.977388i	$0.432180\pi$
$24$		0			0
$25$	−15.6174			−0.624696
$26$		0			0
$27$		−	24.7110i		−	0.915222i
$28$		0			0
$29$			14.4351i			0.497763i	0.968534	+	0.248882i	$0.0800630\pi$
$29$			14.4351i			0.497763i	−0.968534	+	0.248882i	$0.919937\pi$
$30$		0			0
$31$			36.0875i			1.16411i	0.813148	+	0.582057i	$0.197752\pi$
$31$			36.0875i			1.16411i	−0.813148	+	0.582057i	$0.802248\pi$
$32$		0			0
$33$			17.7738i			0.538601i
$34$		0			0
$35$	37.7225			1.07779
$36$		0			0
$37$		−	32.4102i		−	0.875951i	−0.898987	−	0.437976i	$-0.855696\pi$
$37$		−	32.4102i		−	0.875951i	0.898987	−	0.437976i	$-0.144304\pi$
$38$		0			0
$39$	−21.5336			−0.552143
$40$		0			0
$41$		−	60.2390i		−	1.46924i	−0.678476	−	0.734622i	$-0.737359\pi$
$41$		−	60.2390i		−	1.46924i	0.678476	−	0.734622i	$-0.262641\pi$
$42$		0			0
$43$	60.5760			1.40874			0.704372	−	0.709831i	$-0.251229\pi$
$43$	60.5760			1.40874			0.704372	+	0.709831i	$0.251229\pi$
$44$		0			0
$45$	−3.98369			−0.0885265
$46$		0			0
$47$	58.8019			1.25110			0.625552	−	0.780182i	$-0.284874\pi$
$47$	58.8019			1.25110			0.625552	+	0.780182i	$0.284874\pi$
$48$		0			0
$49$	102.662			2.09515
$50$		0			0
$51$			17.2143i			0.337536i
$52$		0			0
$53$		−	24.4833i		−	0.461949i	−0.972960	−	0.230974i	$-0.925809\pi$
$53$		−	24.4833i		−	0.461949i	0.972960	−	0.230974i	$-0.0741913\pi$
$54$		0			0
$55$	−16.9634			−0.308425
$56$		0			0
$57$	−60.8629	−	3.76895i	−1.06777	−	0.0661219i
$58$		0			0
$59$			107.312i			1.81885i	0.415869	+	0.909424i	$0.363477\pi$
$59$			107.312i			1.81885i	−0.415869	+	0.909424i	$0.636523\pi$
$60$		0			0
$61$	−45.9998			−0.754095			−0.377048	−	0.926194i	$-0.623061\pi$
$61$	−45.9998			−0.754095			−0.377048	+	0.926194i	$0.623061\pi$
$62$		0			0
$63$	−16.0163			−0.254227
$64$		0			0
$65$		−	20.5517i		−	0.316180i
$66$		0			0
$67$		−	42.6265i		−	0.636216i	−0.948054	−	0.318108i	$-0.896952\pi$
$67$		−	42.6265i		−	0.636216i	0.948054	−	0.318108i	$-0.103048\pi$
$68$		0			0
$69$		−	31.2179i		−	0.452433i
$70$		0			0
$71$			70.3493i			0.990835i	0.868655	+	0.495418i	$0.164985\pi$
$71$			70.3493i			0.990835i	−0.868655	+	0.495418i	$0.835015\pi$
$72$		0			0
$73$	52.3089			0.716560			0.358280	−	0.933614i	$-0.383363\pi$
$73$	52.3089			0.716560			0.358280	+	0.933614i	$0.383363\pi$
$74$		0			0
$75$			50.1232i			0.668309i
$76$		0			0
$77$	−68.2008			−0.885725
$78$		0			0
$79$		−	2.68432i		−	0.0339787i	−0.999856	−	0.0169893i	$-0.994592\pi$
$79$		−	2.68432i		−	0.0339787i	0.999856	−	0.0169893i	$-0.00540814\pi$
$80$		0			0
$81$	−91.0135			−1.12362
$82$		0			0
$83$	−138.135			−1.66428			−0.832140	−	0.554566i	$-0.812884\pi$
$83$	−138.135			−1.66428			−0.832140	+	0.554566i	$0.812884\pi$
$84$		0			0
$85$	−16.4294			−0.193287
$86$		0			0
$87$	46.3288			0.532515
$88$		0			0
$89$		−	106.061i		−	1.19170i	−0.803096	−	0.595850i	$-0.796815\pi$
$89$		−	106.061i		−	1.19170i	0.803096	−	0.595850i	$-0.203185\pi$
$90$		0			0
$91$		−	82.6276i		−	0.907995i
$92$		0			0
$93$	115.821			1.24539
$94$		0			0
$95$	3.59710	−	58.0877i	0.0378642	−	0.611450i
$96$		0			0
$97$		−	79.3904i		−	0.818457i	−0.912432	−	0.409229i	$-0.865798\pi$
$97$		−	79.3904i		−	0.818457i	0.912432	−	0.409229i	$-0.134202\pi$
$98$		0			0
$99$	7.20236			0.0727511
$100$		0			0
$101$	170.345			1.68658			0.843290	−	0.537459i	$-0.180616\pi$
$101$	170.345			1.68658			0.843290	+	0.537459i	$0.180616\pi$
$102$		0			0
$103$		−	185.850i		−	1.80437i	−0.431350	−	0.902185i	$-0.641963\pi$
$103$		−	185.850i		−	1.80437i	0.431350	−	0.902185i	$-0.358037\pi$
$104$		0			0
$105$		−	121.068i		−	1.15303i
$106$		0			0
$107$			84.0434i			0.785452i	0.919655	+	0.392726i	$0.128468\pi$
$107$			84.0434i			0.785452i	−0.919655	+	0.392726i	$0.871532\pi$
$108$		0			0
$109$			63.4658i			0.582255i	0.956684	+	0.291127i	$0.0940304\pi$
$109$			63.4658i			0.582255i	−0.956684	+	0.291127i	$0.905970\pi$
$110$		0			0
$111$	−104.019			−0.937106
$112$		0			0
$113$			81.8004i			0.723897i	0.932198	+	0.361949i	$0.117888\pi$
$113$			81.8004i			0.723897i	−0.932198	+	0.361949i	$0.882112\pi$
$114$		0			0
$115$	29.7945			0.259082
$116$		0			0
$117$			8.72590i			0.0745803i
$118$		0			0
$119$	−66.0539			−0.555075
$120$		0			0
$121$	−90.3308			−0.746536
$122$		0			0
$123$	−193.334			−1.57182
$124$		0			0
$125$	−124.415			−0.995323
$126$		0			0
$127$		−	148.454i		−	1.16893i	−0.811420	−	0.584464i	$-0.801305\pi$
$127$		−	148.454i		−	1.16893i	0.811420	−	0.584464i	$-0.198695\pi$
$128$		0			0
$129$		−	194.415i		−	1.50710i
$130$		0			0
$131$	53.1636			0.405829			0.202915	−	0.979196i	$-0.434959\pi$
$131$	53.1636			0.405829			0.202915	+	0.979196i	$0.434959\pi$
$132$		0			0
$133$	14.4620	−	233.540i	0.108737	−	1.75594i
$134$		0			0
$135$		−	75.6924i		−	0.560684i
$136$		0			0
$137$	−152.424			−1.11258			−0.556292	−	0.830987i	$-0.687776\pi$
$137$	−152.424			−1.11258			−0.556292	+	0.830987i	$0.687776\pi$
$138$		0			0
$139$	−17.9384			−0.129053			−0.0645267	−	0.997916i	$-0.520554\pi$
$139$	−17.9384			−0.129053			−0.0645267	+	0.997916i	$0.520554\pi$
$140$		0			0
$141$		−	188.722i		−	1.33845i
$142$		0			0
$143$			37.1567i			0.259837i
$144$		0			0
$145$			44.2163i			0.304940i
$146$		0			0
$147$		−	329.489i		−	2.24142i
$148$		0			0
$149$	55.8636			0.374923			0.187462	−	0.982272i	$-0.439974\pi$
$149$	55.8636			0.374923			0.187462	+	0.982272i	$0.439974\pi$
$150$		0			0
$151$			190.710i			1.26298i	0.775383	+	0.631491i	$0.217557\pi$
$151$			190.710i			1.26298i	−0.775383	+	0.631491i	$0.782443\pi$
$152$		0			0
$153$	6.97564			0.0455924
$154$		0			0
$155$			110.540i			0.713161i
$156$		0			0
$157$	−218.585			−1.39226			−0.696131	−	0.717915i	$-0.745097\pi$
$157$	−218.585			−1.39226			−0.696131	+	0.717915i	$0.745097\pi$
$158$		0			0
$159$	−78.5777			−0.494200
$160$		0			0
$161$	119.788			0.744023
$162$		0			0
$163$	43.3239			0.265791			0.132895	−	0.991130i	$-0.457573\pi$
$163$	43.3239			0.265791			0.132895	+	0.991130i	$0.457573\pi$
$164$		0			0
$165$			54.4431i			0.329958i
$166$		0			0
$167$			162.852i			0.975159i	0.873078	+	0.487580i	$0.162120\pi$
$167$			162.852i			0.975159i	−0.873078	+	0.487580i	$0.837880\pi$
$168$		0			0
$169$	123.983			0.733630
$170$		0			0
$171$	−1.52726	+	24.6630i	−0.00893137	+	0.144228i
$172$		0			0
$173$		−	64.2618i		−	0.371456i	−0.982601	−	0.185728i	$-0.940536\pi$
$173$		−	64.2618i		−	0.371456i	0.982601	−	0.185728i	$-0.0594643\pi$
$174$		0			0
$175$	−192.330			−1.09903
$176$		0			0
$177$	344.412			1.94583
$178$		0			0
$179$			103.185i			0.576455i	0.957562	+	0.288228i	$0.0930659\pi$
$179$			103.185i			0.576455i	−0.957562	+	0.288228i	$0.906934\pi$
$180$		0			0
$181$			199.029i			1.09961i	0.835294	+	0.549804i	$0.185298\pi$
$181$			199.029i			1.09961i	−0.835294	+	0.549804i	$0.814702\pi$
$182$		0			0
$183$			147.634i			0.806742i
$184$		0			0
$185$		−	99.2758i		−	0.536626i
$186$		0			0
$187$	29.7037			0.158844
$188$		0			0
$189$		−	304.319i		−	1.61015i
$190$		0			0
$191$	180.081			0.942834			0.471417	−	0.881911i	$-0.343743\pi$
$191$	180.081			0.942834			0.471417	+	0.881911i	$0.343743\pi$
$192$		0			0
$193$		−	320.709i		−	1.66170i	−0.556493	−	0.830852i	$-0.687854\pi$
$193$		−	320.709i		−	1.66170i	0.556493	−	0.830852i	$-0.312146\pi$
$194$		0			0
$195$	−65.9596			−0.338255
$196$		0			0
$197$	−126.793			−0.643619			−0.321809	−	0.946804i	$-0.604291\pi$
$197$	−126.793			−0.643619			−0.321809	+	0.946804i	$0.604291\pi$
$198$		0			0
$199$	−94.6913			−0.475836			−0.237918	−	0.971285i	$-0.576465\pi$
$199$	−94.6913			−0.475836			−0.237918	+	0.971285i	$0.576465\pi$
$200$		0			0
$201$	−136.807			−0.680634
$202$		0			0
$203$			177.770i			0.875716i
$204$		0			0
$205$		−	184.518i		−	0.900090i
$206$		0			0
$207$	−12.6502			−0.0611121
$208$		0			0
$209$	−6.50342	+	105.020i	−0.0311168	+	0.502490i
$210$		0			0
$211$			69.7789i			0.330706i	0.986234	+	0.165353i	$0.0528763\pi$
$211$			69.7789i			0.330706i	−0.986234	+	0.165353i	$0.947124\pi$
$212$		0			0
$213$	225.782			1.06001
$214$		0			0
$215$	185.551			0.863026
$216$		0			0
$217$			444.422i			2.04803i
$218$		0			0
$219$		−	167.883i		−	0.766587i
$220$		0			0
$221$			35.9871i			0.162837i
$222$		0			0
$223$			152.781i			0.685117i	0.939496	+	0.342559i	$0.111294\pi$
$223$			152.781i			0.685117i	−0.939496	+	0.342559i	$0.888706\pi$
$224$		0			0
$225$	20.3110			0.0902713
$226$		0			0
$227$			28.2692i			0.124534i	0.998060	+	0.0622670i	$0.0198330\pi$
$227$			28.2692i			0.124534i	−0.998060	+	0.0622670i	$0.980167\pi$
$228$		0			0
$229$	−308.715			−1.34810			−0.674050	−	0.738686i	$-0.735447\pi$
$229$	−308.715			−1.34810			−0.674050	+	0.738686i	$0.735447\pi$
$230$		0			0
$231$			218.887i			0.947562i
$232$		0			0
$233$	−289.375			−1.24195			−0.620977	−	0.783828i	$-0.713264\pi$
$233$	−289.375			−1.24195			−0.620977	+	0.783828i	$0.713264\pi$
$234$		0			0
$235$	180.116			0.766453
$236$		0			0
$237$	−8.61517			−0.0363509
$238$		0			0
$239$	62.7214			0.262433			0.131216	−	0.991354i	$-0.458112\pi$
$239$	62.7214			0.262433			0.131216	+	0.991354i	$0.458112\pi$
$240$		0			0
$241$			285.131i			1.18311i	0.806263	+	0.591557i	$0.201487\pi$
$241$			285.131i			1.18311i	−0.806263	+	0.591557i	$0.798513\pi$
$242$		0			0
$243$			69.7038i			0.286847i
$244$		0			0
$245$	314.465			1.28353
$246$		0			0
$247$	−127.236	−	7.87911i	−0.515124	−	0.0318992i
$248$		0			0
$249$			443.337i			1.78047i
$250$		0			0
$251$	444.455			1.77074			0.885370	−	0.464888i	$-0.153905\pi$
$251$	444.455			1.77074			0.885370	+	0.464888i	$0.153905\pi$
$252$		0			0
$253$	−53.8673			−0.212914
$254$		0			0
$255$			52.7293i			0.206782i
$256$		0			0
$257$		−	361.131i		−	1.40518i	−0.711595	−	0.702590i	$-0.752027\pi$
$257$		−	361.131i		−	1.40518i	0.711595	−	0.702590i	$-0.247973\pi$
$258$		0			0
$259$		−	399.135i		−	1.54106i
$260$		0			0
$261$		−	18.7735i		−	0.0719290i
$262$		0			0
$263$	283.913			1.07952			0.539759	−	0.841820i	$-0.318515\pi$
$263$	283.913			1.07952			0.539759	+	0.841820i	$0.318515\pi$
$264$		0			0
$265$		−	74.9948i		−	0.282999i
$266$		0			0
$267$	−340.398			−1.27490
$268$		0			0
$269$			405.876i			1.50883i	0.656396	+	0.754416i	$0.272080\pi$
$269$			405.876i			1.50883i	−0.656396	+	0.754416i	$0.727920\pi$
$270$		0			0
$271$	528.781			1.95122			0.975610	−	0.219511i	$-0.0704463\pi$
$271$	528.781			1.95122			0.975610	+	0.219511i	$0.0704463\pi$
$272$		0			0
$273$	−265.189			−0.971387
$274$		0			0
$275$	86.4887			0.314504
$276$		0			0
$277$	406.620			1.46794			0.733971	−	0.679181i	$-0.237665\pi$
$277$	406.620			1.46794			0.733971	+	0.679181i	$0.237665\pi$
$278$		0			0
$279$		−	46.9333i		−	0.168220i
$280$		0			0
$281$			18.0317i			0.0641697i	0.999485	+	0.0320849i	$0.0102147\pi$
$281$			18.0317i			0.0641697i	−0.999485	+	0.0320849i	$0.989785\pi$
$282$		0			0
$283$	106.244			0.375422			0.187711	−	0.982224i	$-0.439893\pi$
$283$	106.244			0.375422			0.187711	+	0.982224i	$0.439893\pi$
$284$		0			0
$285$	−186.429	−	11.5447i	−0.654138	−	0.0405077i
$286$		0			0
$287$		−	741.851i		−	2.58485i
$288$		0			0
$289$	−260.231			−0.900454
$290$		0			0
$291$	−254.799			−0.875598
$292$		0			0
$293$			473.502i			1.61605i	0.589149	+	0.808024i	$0.299463\pi$
$293$			473.502i			1.61605i	−0.589149	+	0.808024i	$0.700537\pi$
$294$		0			0
$295$			328.708i			1.11426i
$296$		0			0
$297$			136.849i			0.460771i
$298$		0			0
$299$		−	65.2620i		−	0.218267i
$300$		0			0
$301$	746.001			2.47841
$302$		0			0
$303$		−	546.712i		−	1.80433i
$304$		0			0
$305$	−140.902			−0.461974
$306$		0			0
$307$		−	481.865i		−	1.56959i	−0.619754	−	0.784796i	$-0.712768\pi$
$307$		−	481.865i		−	1.56959i	0.619754	−	0.784796i	$-0.287232\pi$
$308$		0			0
$309$	−596.476			−1.93034
$310$		0			0
$311$	−529.837			−1.70366			−0.851828	−	0.523822i	$-0.824506\pi$
$311$	−529.837			−1.70366			−0.851828	+	0.523822i	$0.824506\pi$
$312$		0			0
$313$	261.542			0.835598			0.417799	−	0.908539i	$-0.362802\pi$
$313$	261.542			0.835598			0.417799	+	0.908539i	$0.362802\pi$
$314$		0			0
$315$	−49.0596			−0.155745
$316$		0			0
$317$			265.379i			0.837158i	0.908180	+	0.418579i	$0.137472\pi$
$317$			265.379i			0.837158i	−0.908180	+	0.418579i	$0.862528\pi$
$318$		0			0
$319$		−	79.9414i		−	0.250600i
$320$		0			0
$321$	269.733			0.840289
$322$		0			0
$323$	−6.29869	+	101.714i	−0.0195006	+	0.314905i
$324$		0			0
$325$			104.784i			0.322412i
$326$		0			0
$327$	203.690			0.622905
$328$		0			0
$329$	724.152			2.20107
$330$		0			0
$331$			605.094i			1.82808i	0.405626	+	0.914039i	$0.367053\pi$
$331$			605.094i			1.82808i	−0.405626	+	0.914039i	$0.632947\pi$
$332$		0			0
$333$			42.1508i			0.126579i
$334$		0			0
$335$		−	130.569i		−	0.389759i
$336$		0			0
$337$			349.555i			1.03726i	0.855000	+	0.518628i	$0.173557\pi$
$337$			349.555i			1.03726i	−0.855000	+	0.518628i	$0.826443\pi$
$338$		0			0
$339$	262.534			0.774436
$340$		0			0
$341$		−	199.852i		−	0.586076i
$342$		0			0
$343$	660.856			1.92669
$344$		0			0
$345$		−	95.6237i		−	0.277170i
$346$		0			0
$347$	162.204			0.467447			0.233724	−	0.972303i	$-0.424909\pi$
$347$	162.204			0.467447			0.233724	+	0.972303i	$0.424909\pi$
$348$		0			0
$349$	−273.918			−0.784865			−0.392433	−	0.919781i	$-0.628366\pi$
$349$	−273.918			−0.784865			−0.392433	+	0.919781i	$0.628366\pi$
$350$		0			0
$351$	−165.797			−0.472356
$352$		0			0
$353$	90.1452			0.255369			0.127684	−	0.991815i	$-0.459246\pi$
$353$	90.1452			0.255369			0.127684	+	0.991815i	$0.459246\pi$
$354$		0			0
$355$			215.487i			0.607006i
$356$		0			0
$357$			211.996i			0.593828i
$358$		0			0
$359$	−342.459			−0.953925			−0.476962	−	0.878924i	$-0.658262\pi$
$359$	−342.459			−0.953925			−0.476962	+	0.878924i	$0.658262\pi$
$360$		0			0
$361$	−358.242	−	44.5393i	−0.992360	−	0.123377i
$362$		0			0
$363$			289.912i			0.798655i
$364$		0			0
$365$	160.228			0.438980
$366$		0			0
$367$	203.266			0.553858			0.276929	−	0.960890i	$-0.410683\pi$
$367$	203.266			0.553858			0.276929	+	0.960890i	$0.410683\pi$
$368$		0			0
$369$			78.3433i			0.212312i
$370$		0			0
$371$		−	301.514i		−	0.812707i
$372$		0			0
$373$			458.615i			1.22953i	0.788710	+	0.614765i	$0.210749\pi$
$373$			458.615i			1.22953i	−0.788710	+	0.614765i	$0.789251\pi$
$374$		0			0
$375$			399.304i			1.06481i
$376$		0			0
$377$	96.8517			0.256901
$378$		0			0
$379$		−	114.433i		−	0.301934i	−0.988539	−	0.150967i	$-0.951761\pi$
$379$		−	114.433i		−	0.301934i	0.988539	−	0.150967i	$-0.0482388\pi$
$380$		0			0
$381$	−476.454			−1.25054
$382$		0			0
$383$		−	357.223i		−	0.932696i	−0.884601	−	0.466348i	$-0.845569\pi$
$383$		−	357.223i		−	0.932696i	0.884601	−	0.466348i	$-0.154431\pi$
$384$		0			0
$385$	−208.906			−0.542614
$386$		0			0
$387$	−78.7815			−0.203570
$388$		0			0
$389$	−336.952			−0.866200			−0.433100	−	0.901346i	$-0.642580\pi$
$389$	−336.952			−0.866200			−0.433100	+	0.901346i	$0.642580\pi$
$390$		0			0
$391$	−52.1716			−0.133431
$392$		0			0
$393$		−	170.626i		−	0.434162i
$394$		0			0
$395$		−	8.22234i		−	0.0208161i
$396$		0			0
$397$	225.208			0.567274			0.283637	−	0.958932i	$-0.408459\pi$
$397$	225.208			0.567274			0.283637	+	0.958932i	$0.408459\pi$
$398$		0			0
$399$	−749.534	−	46.4151i	−1.87853	−	0.116329i
$400$		0			0
$401$			488.538i			1.21830i	0.793055	+	0.609149i	$0.208489\pi$
$401$			488.538i			1.21830i	−0.793055	+	0.609149i	$0.791511\pi$
$402$		0			0
$403$	242.127			0.600812
$404$		0			0
$405$	−278.784			−0.688355
$406$		0			0
$407$			179.487i			0.441000i
$408$		0			0
$409$		−	125.524i		−	0.306905i	−0.988156	−	0.153452i	$-0.950961\pi$
$409$		−	125.524i		−	0.306905i	0.988156	−	0.153452i	$-0.0490391\pi$
$410$		0			0
$411$			489.196i			1.19026i
$412$		0			0
$413$			1321.56i			3.19991i
$414$		0			0
$415$	−423.122			−1.01957
$416$		0			0
$417$			57.5724i			0.138063i
$418$		0			0
$419$	187.698			0.447967			0.223984	−	0.974593i	$-0.428094\pi$
$419$	187.698			0.447967			0.223984	+	0.974593i	$0.428094\pi$
$420$		0			0
$421$		−	146.182i		−	0.347227i	−0.984814	−	0.173613i	$-0.944456\pi$
$421$		−	146.182i		−	0.347227i	0.984814	−	0.173613i	$-0.0555443\pi$
$422$		0			0
$423$	−76.4743			−0.180790
$424$		0			0
$425$	83.7661			0.197097
$426$		0			0
$427$	−566.493			−1.32668
$428$		0			0
$429$	119.252			0.277978
$430$		0			0
$431$			470.285i			1.09115i	0.838063	+	0.545574i	$0.183688\pi$
$431$			470.285i			1.09115i	−0.838063	+	0.545574i	$0.816312\pi$
$432$		0			0
$433$		−	436.928i		−	1.00907i	−0.863391	−	0.504536i	$-0.831664\pi$
$433$		−	436.928i		−	1.00907i	0.863391	−	0.504536i	$-0.168336\pi$
$434$		0			0
$435$	141.910			0.326230
$436$		0			0
$437$	11.4226	−	184.457i	0.0261386	−	0.422099i
$438$		0			0
$439$			518.542i			1.18119i	0.806968	+	0.590595i	$0.201107\pi$
$439$			518.542i			1.18119i	−0.806968	+	0.590595i	$0.798893\pi$
$440$		0			0
$441$	−133.516			−0.302758
$442$		0			0
$443$	−107.645			−0.242991			−0.121495	−	0.992592i	$-0.538769\pi$
$443$	−107.645			−0.242991			−0.121495	+	0.992592i	$0.538769\pi$
$444$		0			0
$445$		−	324.877i		−	0.730060i
$446$		0			0
$447$		−	179.291i		−	0.401099i
$448$		0			0
$449$			643.806i			1.43387i	0.697142	+	0.716933i	$0.254455\pi$
$449$			643.806i			1.43387i	−0.697142	+	0.716933i	$0.745545\pi$
$450$		0			0
$451$			333.602i			0.739695i
$452$		0			0
$453$	612.074			1.35116
$454$		0			0
$455$		−	253.097i		−	0.556257i
$456$		0			0
$457$	5.99715			0.0131229			0.00656144	−	0.999978i	$-0.497911\pi$
$457$	5.99715			0.0131229			0.00656144	+	0.999978i	$0.497911\pi$
$458$		0			0
$459$			132.541i			0.288760i
$460$		0			0
$461$	400.082			0.867858			0.433929	−	0.900947i	$-0.357127\pi$
$461$	400.082			0.867858			0.433929	+	0.900947i	$0.357127\pi$
$462$		0			0
$463$	−83.5840			−0.180527			−0.0902635	−	0.995918i	$-0.528771\pi$
$463$	−83.5840			−0.180527			−0.0902635	+	0.995918i	$0.528771\pi$
$464$		0			0
$465$	354.772			0.762950
$466$		0			0
$467$	−601.132			−1.28722			−0.643610	−	0.765354i	$-0.722564\pi$
$467$	−601.132			−1.28722			−0.643610	+	0.765354i	$0.722564\pi$
$468$		0			0
$469$		−	524.950i		−	1.11930i
$470$		0			0
$471$			701.537i			1.48946i
$472$		0			0
$473$	−335.468			−0.709236
$474$		0			0
$475$	−18.3400	+	296.163i	−0.0386105	+	0.623501i
$476$		0			0
$477$			31.8415i			0.0667536i
$478$		0			0
$479$	−394.710			−0.824030			−0.412015	−	0.911177i	$-0.635175\pi$
$479$	−394.710			−0.824030			−0.412015	+	0.911177i	$0.635175\pi$
$480$		0			0
$481$	−217.454			−0.452088
$482$		0			0
$483$		−	384.452i		−	0.795967i
$484$		0			0
$485$		−	243.181i		−	0.501404i
$486$		0			0
$487$		−	454.916i		−	0.934119i	−0.884226	−	0.467060i	$-0.845313\pi$
$487$		−	454.916i		−	0.934119i	0.884226	−	0.467060i	$-0.154687\pi$
$488$		0			0
$489$		−	139.046i		−	0.284347i
$490$		0			0
$491$	115.066			0.234351			0.117175	−	0.993111i	$-0.462616\pi$
$491$	115.066			0.234351			0.117175	+	0.993111i	$0.462616\pi$
$492$		0			0
$493$		−	77.4249i		−	0.157049i
$494$		0			0
$495$	22.0616			0.0445689
$496$		0			0
$497$			866.360i			1.74318i
$498$		0			0
$499$	−196.082			−0.392950			−0.196475	−	0.980509i	$-0.562950\pi$
$499$	−196.082			−0.392950			−0.196475	+	0.980509i	$0.562950\pi$
$500$		0			0
$501$	522.663			1.04324
$502$		0			0
$503$	504.642			1.00327			0.501633	−	0.865081i	$-0.332733\pi$
$503$	504.642			1.00327			0.501633	+	0.865081i	$0.332733\pi$
$504$		0			0
$505$	521.783			1.03323
$506$		0			0
$507$		−	397.918i		−	0.784848i
$508$		0			0
$509$		−	48.7384i		−	0.0957533i	−0.998853	−	0.0478766i	$-0.984755\pi$
$509$		−	48.7384i		−	0.0957533i	0.998853	−	0.0478766i	$-0.0152454\pi$
$510$		0			0
$511$	644.190			1.26065
$512$		0			0
$513$	−468.611	−	29.0189i	−0.913472	−	0.0565670i
$514$		0			0
$515$		−	569.278i		−	1.10539i
$516$		0			0
$517$	−325.644			−0.629872
$518$		0			0
$519$	−206.245			−0.397389
$520$		0			0
$521$			646.673i			1.24121i	0.784122	+	0.620607i	$0.213114\pi$
$521$			646.673i			1.24121i	−0.784122	+	0.620607i	$0.786886\pi$
$522$		0			0
$523$			257.895i			0.493108i	0.969129	+	0.246554i	$0.0792982\pi$
$523$			257.895i			0.493108i	−0.969129	+	0.246554i	$0.920702\pi$
$524$		0			0
$525$			617.273i			1.17576i
$526$		0			0
$527$		−	193.561i		−	0.367288i
$528$		0			0
$529$	−434.388			−0.821149
$530$		0			0
$531$		−	139.564i		−	0.262832i
$532$		0			0
$533$	−404.170			−0.758293
$534$		0			0
$535$			257.434i			0.481184i
$536$		0			0
$537$	331.168			0.616700
$538$		0			0
$539$	−568.540			−1.05481
$540$		0			0
$541$	−158.185			−0.292393			−0.146197	−	0.989256i	$-0.546703\pi$
$541$	−158.185			−0.292393			−0.146197	+	0.989256i	$0.546703\pi$
$542$		0			0
$543$	638.773			1.17638
$544$		0			0
$545$			194.402i			0.356701i
$546$		0			0
$547$		−	483.829i		−	0.884513i	−0.896889	−	0.442257i	$-0.854178\pi$
$547$		−	483.829i		−	0.884513i	0.896889	−	0.442257i	$-0.145822\pi$
$548$		0			0
$549$	59.8246			0.108970
$550$		0			0
$551$	273.743	+	16.9516i	0.496812	+	0.0307652i
$552$		0			0
$553$		−	33.0577i		−	0.0597788i
$554$		0			0
$555$	−318.620			−0.574091
$556$		0			0
$557$	763.214			1.37022			0.685111	−	0.728438i	$-0.259754\pi$
$557$	763.214			1.37022			0.685111	+	0.728438i	$0.259754\pi$
$558$		0			0
$559$		−	406.431i		−	0.727068i
$560$		0			0
$561$		−	95.3325i		−	0.169933i
$562$		0			0
$563$		−	414.128i		−	0.735573i	−0.929910	−	0.367787i	$-0.880116\pi$
$563$		−	414.128i		−	0.735573i	0.929910	−	0.367787i	$-0.119884\pi$
$564$		0			0
$565$			250.563i			0.443475i
$566$		0			0
$567$	−1120.84			−1.97679
$568$		0			0
$569$		−	728.193i		−	1.27978i	−0.768468	−	0.639888i	$-0.778981\pi$
$569$		−	728.193i		−	1.27978i	0.768468	−	0.639888i	$-0.221019\pi$
$570$		0			0
$571$	−532.481			−0.932541			−0.466271	−	0.884642i	$-0.654403\pi$
$571$	−532.481			−0.932541			−0.466271	+	0.884642i	$0.654403\pi$
$572$		0			0
$573$		−	577.961i		−	1.00866i
$574$		0			0
$575$	−151.909			−0.264189
$576$		0			0
$577$	−164.020			−0.284264			−0.142132	−	0.989848i	$-0.545396\pi$
$577$	−164.020			−0.284264			−0.142132	+	0.989848i	$0.545396\pi$
$578$		0			0
$579$	−1029.30			−1.77772
$580$		0			0
$581$	−1701.15			−2.92797
$582$		0			0
$583$			135.588i			0.232569i
$584$		0			0
$585$			26.7283i			0.0456895i
$586$		0			0
$587$	934.356			1.59175			0.795874	−	0.605463i	$-0.207012\pi$
$587$	934.356			1.59175			0.795874	+	0.605463i	$0.207012\pi$
$588$		0			0
$589$	684.352	+	42.3787i	1.16189	+	0.0719503i
$590$		0			0
$591$			406.935i			0.688553i
$592$		0			0
$593$	−387.798			−0.653960			−0.326980	−	0.945031i	$-0.606031\pi$
$593$	−387.798			−0.653960			−0.326980	+	0.945031i	$0.606031\pi$
$594$		0			0
$595$	−202.330			−0.340051
$596$		0			0
$597$			303.906i			0.509056i
$598$		0			0
$599$			255.369i			0.426325i	0.977017	+	0.213162i	$0.0683763\pi$
$599$			255.369i			0.426325i	−0.977017	+	0.213162i	$0.931624\pi$
$600$		0			0
$601$			585.508i			0.974223i	0.873340	+	0.487112i	$0.161950\pi$
$601$			585.508i			0.974223i	−0.873340	+	0.487112i	$0.838050\pi$
$602$		0			0
$603$			55.4375i			0.0919361i
$604$		0			0
$605$	−276.693			−0.457343
$606$		0			0
$607$			438.164i			0.721852i	0.932594	+	0.360926i	$0.117539\pi$
$607$			438.164i			0.721852i	−0.932594	+	0.360926i	$0.882461\pi$
$608$		0			0
$609$	570.544			0.936854
$610$		0			0
$611$		−	394.528i		−	0.645709i
$612$		0			0
$613$	174.723			0.285030			0.142515	−	0.989793i	$-0.454481\pi$
$613$	174.723			0.285030			0.142515	+	0.989793i	$0.454481\pi$
$614$		0			0
$615$	−592.202			−0.962930
$616$		0			0
$617$	494.815			0.801969			0.400984	−	0.916085i	$-0.368668\pi$
$617$	494.815			0.801969			0.400984	+	0.916085i	$0.368668\pi$
$618$		0			0
$619$	788.206			1.27335			0.636677	−	0.771131i	$-0.280308\pi$
$619$	788.206			1.27335			0.636677	+	0.771131i	$0.280308\pi$
$620$		0			0
$621$		−	240.361i		−	0.387055i
$622$		0			0
$623$		−	1306.16i		−	2.09656i
$624$		0			0
$625$	9.33758			0.0149401
$626$		0			0
$627$	337.057	+	20.8724i	0.537571	+	0.0332893i
$628$		0			0
$629$			173.837i			0.276370i
$630$		0			0
$631$	371.925			0.589421			0.294710	−	0.955587i	$-0.404777\pi$
$631$	371.925			0.589421			0.294710	+	0.955587i	$0.404777\pi$
$632$		0			0
$633$	223.952			0.353794
$634$		0			0
$635$		−	454.729i		−	0.716109i
$636$		0			0
$637$		−	688.806i		−	1.08133i
$638$		0			0
$639$		−	91.4921i		−	0.143180i
$640$		0			0
$641$			1239.81i			1.93419i	0.254423	+	0.967093i	$0.418114\pi$
$641$			1239.81i			1.93419i	−0.254423	+	0.967093i	$0.581886\pi$
$642$		0			0
$643$	545.751			0.848757			0.424378	−	0.905485i	$-0.360493\pi$
$643$	545.751			0.848757			0.424378	+	0.905485i	$0.360493\pi$
$644$		0			0
$645$		−	595.515i		−	0.923278i
$646$		0			0
$647$	402.170			0.621593			0.310796	−	0.950477i	$-0.399404\pi$
$647$	402.170			0.621593			0.310796	+	0.950477i	$0.399404\pi$
$648$		0			0
$649$		−	594.292i		−	0.915704i
$650$		0			0
$651$	1426.35			2.19101
$652$		0			0
$653$	−59.6101			−0.0912866			−0.0456433	−	0.998958i	$-0.514534\pi$
$653$	−59.6101			−0.0912866			−0.0456433	+	0.998958i	$0.514534\pi$
$654$		0			0
$655$	162.846			0.248619
$656$		0			0
$657$	−68.0298			−0.103546
$658$		0			0
$659$		−	556.858i		−	0.845004i	−0.906362	−	0.422502i	$-0.861152\pi$
$659$		−	556.858i		−	0.845004i	0.906362	−	0.422502i	$-0.138848\pi$
$660$		0			0
$661$		−	282.675i		−	0.427647i	−0.976872	−	0.213824i	$-0.931408\pi$
$661$		−	282.675i		−	0.427647i	0.976872	−	0.213824i	$-0.0685918\pi$
$662$		0			0
$663$	115.499			0.174206
$664$		0			0
$665$	44.2987	−	715.357i	0.0666146	−	1.07573i
$666$		0			0
$667$			140.409i			0.210508i
$668$		0			0
$669$	490.343			0.732949
$670$		0			0
$671$	254.746			0.379651
$672$		0			0
$673$		−	580.225i		−	0.862147i	−0.902317	−	0.431073i	$-0.858135\pi$
$673$		−	580.225i		−	0.862147i	0.902317	−	0.431073i	$-0.141865\pi$
$674$		0			0
$675$			385.921i			0.571735i
$676$		0			0
$677$			669.862i			0.989457i	0.869048	+	0.494728i	$0.164732\pi$
$677$			669.862i			0.989457i	−0.869048	+	0.494728i	$0.835268\pi$
$678$		0			0
$679$		−	977.702i		−	1.43991i
$680$		0			0
$681$	90.7284			0.133228
$682$		0			0
$683$			938.331i			1.37384i	0.726734	+	0.686919i	$0.241037\pi$
$683$			938.331i			1.37384i	−0.726734	+	0.686919i	$0.758963\pi$
$684$		0			0
$685$	−466.890			−0.681592
$686$		0			0
$687$			990.803i			1.44222i
$688$		0			0
$689$	−164.269			−0.238417
$690$		0			0
$691$	−952.665			−1.37868			−0.689338	−	0.724440i	$-0.742098\pi$
$691$	−952.665			−1.37868			−0.689338	+	0.724440i	$0.742098\pi$
$692$		0			0
$693$	88.6979			0.127991
$694$		0			0
$695$	−54.9472			−0.0790608
$696$		0			0
$697$			323.101i			0.463559i
$698$		0			0
$699$			928.735i			1.32866i
$700$		0			0
$701$	99.3497			0.141726			0.0708628	−	0.997486i	$-0.477425\pi$
$701$	99.3497			0.141726			0.0708628	+	0.997486i	$0.477425\pi$
$702$		0			0
$703$	−614.616	−	38.0603i	−0.874277	−	0.0541398i
$704$		0			0
$705$		−	578.074i		−	0.819963i
$706$		0			0
$707$	2097.81			2.96720
$708$		0			0
$709$	14.3478			0.0202366			0.0101183	−	0.999949i	$-0.496779\pi$
$709$	14.3478			0.0202366			0.0101183	+	0.999949i	$0.496779\pi$
$710$		0			0
$711$			3.49106i			0.00491007i
$712$		0			0
$713$			351.019i			0.492313i
$714$		0			0
$715$			113.815i			0.159182i
$716$		0			0
$717$		−	201.301i		−	0.280755i
$718$		0			0
$719$	−68.9065			−0.0958365			−0.0479183	−	0.998851i	$-0.515259\pi$
$719$	−68.9065			−0.0958365			−0.0479183	+	0.998851i	$0.515259\pi$
$720$		0			0
$721$		−	2288.77i		−	3.17443i
$722$		0			0
$723$	915.111			1.26571
$724$		0			0
$725$		−	225.439i		−	0.310950i
$726$		0			0
$727$	250.027			0.343916			0.171958	−	0.985104i	$-0.444991\pi$
$727$	250.027			0.343916			0.171958	+	0.985104i	$0.444991\pi$
$728$		0			0
$729$	−595.411			−0.816750
$730$		0			0
$731$	−324.908			−0.444471
$732$		0			0
$733$	−65.2029			−0.0889535			−0.0444768	−	0.999010i	$-0.514162\pi$
$733$	−65.2029			−0.0889535			−0.0444768	+	0.999010i	$0.514162\pi$
$734$		0			0
$735$		−	1009.26i		−	1.37314i
$736$		0			0
$737$			236.064i			0.320305i
$738$		0			0
$739$	108.182			0.146389			0.0731945	−	0.997318i	$-0.476681\pi$
$739$	108.182			0.146389			0.0731945	+	0.997318i	$0.476681\pi$
$740$		0			0
$741$	−25.2876	+	408.356i	−0.0341263	+	0.551088i
$742$		0			0
$743$			824.780i			1.11007i	0.831828	+	0.555034i	$0.187295\pi$
$743$			824.780i			1.11007i	−0.831828	+	0.555034i	$0.812705\pi$
$744$		0			0
$745$	171.116			0.229686
$746$		0			0
$747$	179.650			0.240496
$748$		0			0
$749$			1035.00i			1.38185i
$750$		0			0
$751$			529.661i			0.705274i	0.935760	+	0.352637i	$0.114715\pi$
$751$			529.661i			0.705274i	−0.935760	+	0.352637i	$0.885285\pi$
$752$		0			0
$753$		−	1426.46i		−	1.89436i
$754$		0			0
$755$			584.166i			0.773729i
$756$		0			0
$757$	−808.792			−1.06842			−0.534209	−	0.845352i	$-0.679390\pi$
$757$	−808.792			−1.06842			−0.534209	+	0.845352i	$0.679390\pi$
$758$		0			0
$759$			172.884i			0.227779i
$760$		0			0
$761$	749.782			0.985259			0.492629	−	0.870239i	$-0.336036\pi$
$761$	749.782			0.985259			0.492629	+	0.870239i	$0.336036\pi$
$762$		0			0
$763$			781.589i			1.02436i
$764$		0			0
$765$	21.3671			0.0279309
$766$		0			0
$767$	720.004			0.938728
$768$		0			0
$769$	500.547			0.650906			0.325453	−	0.945558i	$-0.394483\pi$
$769$	500.547			0.650906			0.325453	+	0.945558i	$0.394483\pi$
$770$		0			0
$771$	−1159.03			−1.50328
$772$		0			0
$773$		−	208.556i		−	0.269801i	−0.990859	−	0.134900i	$-0.956929\pi$
$773$		−	208.556i		−	0.269801i	0.990859	−	0.134900i	$-0.0430715\pi$
$774$		0			0
$775$		−	563.593i		−	0.727217i
$776$		0			0
$777$	−1281.00			−1.64865
$778$		0			0
$779$	−1142.35	−	70.7405i	−1.46644	−	0.0908094i
$780$		0			0
$781$		−	389.593i		−	0.498838i
$782$		0			0
$783$	356.707			0.455564
$784$		0			0
$785$	−669.549			−0.852928
$786$		0			0
$787$		−	400.393i		−	0.508758i	−0.967105	−	0.254379i	$-0.918129\pi$
$787$		−	400.393i		−	0.508758i	0.967105	−	0.254379i	$-0.0818711\pi$
$788$		0			0
$789$		−	911.204i		−	1.15488i
$790$		0			0
$791$			1007.38i			1.27355i
$792$		0			0
$793$			308.633i			0.389197i
$794$		0			0
$795$	−240.692			−0.302757
$796$		0			0
$797$			211.793i			0.265738i	0.991134	+	0.132869i	$0.0424190\pi$
$797$			211.793i			0.265738i	−0.991134	+	0.132869i	$0.957581\pi$
$798$		0			0
$799$	−315.393			−0.394734
$800$		0			0
$801$			137.937i			0.172206i
$802$		0			0
$803$	−289.685			−0.360754
$804$		0			0
$805$	366.922			0.455804
$806$		0			0
$807$	1302.64			1.61417
$808$		0			0
$809$	111.836			0.138240			0.0691199	−	0.997608i	$-0.477981\pi$
$809$	111.836			0.138240			0.0691199	+	0.997608i	$0.477981\pi$
$810$		0			0
$811$		−	643.996i		−	0.794077i	−0.917802	−	0.397038i	$-0.870038\pi$
$811$		−	643.996i		−	0.794077i	0.917802	−	0.397038i	$-0.129962\pi$
$812$		0			0
$813$		−	1697.09i		−	2.08744i
$814$		0			0
$815$	132.706			0.162829
$816$		0			0
$817$	71.1363	−	1148.74i	0.0870701	−	1.40605i
$818$		0			0
$819$			107.460i			0.131209i
$820$		0			0
$821$	1540.49			1.87636			0.938182	−	0.346144i	$-0.112509\pi$
$821$	1540.49			1.87636			0.938182	+	0.346144i	$0.112509\pi$
$822$		0			0
$823$	789.808			0.959670			0.479835	−	0.877359i	$-0.340697\pi$
$823$	789.808			0.959670			0.479835	+	0.877359i	$0.340697\pi$
$824$		0			0
$825$		−	277.581i		−	0.336462i
$826$		0			0
$827$			329.463i			0.398384i	0.979961	+	0.199192i	$0.0638317\pi$
$827$			329.463i			0.398384i	−0.979961	+	0.199192i	$0.936168\pi$
$828$		0			0
$829$		−	727.661i		−	0.877757i	−0.898546	−	0.438879i	$-0.855376\pi$
$829$		−	727.661i		−	0.877757i	0.898546	−	0.438879i	$-0.144624\pi$
$830$		0			0
$831$		−	1305.02i		−	1.57043i
$832$		0			0
$833$	−550.643			−0.661036
$834$		0			0
$835$			498.832i			0.597403i
$836$		0			0
$837$	891.759			1.06542
$838$		0			0
$839$		−	593.731i		−	0.707665i	−0.935309	−	0.353833i	$-0.884878\pi$
$839$		−	593.731i		−	0.707665i	0.935309	−	0.353833i	$-0.115122\pi$
$840$		0			0
$841$	632.627			0.752232
$842$		0			0
$843$	57.8717			0.0686498
$844$		0			0
$845$	379.774			0.449437
$846$		0			0
$847$	−1112.43			−1.31338
$848$		0			0
$849$		−	340.985i		−	0.401632i
$850$		0			0
$851$		−	315.250i		−	0.370447i
$852$		0			0
$853$	−675.336			−0.791719			−0.395860	−	0.918311i	$-0.629553\pi$
$853$	−675.336			−0.791719			−0.395860	+	0.918311i	$0.629553\pi$
$854$		0			0
$855$	−4.67817	+	75.5454i	−0.00547155	+	0.0883572i
$856$		0			0
$857$			69.5952i			0.0812079i	0.999175	+	0.0406040i	$0.0129282\pi$
$857$			69.5952i			0.0812079i	−0.999175	+	0.0406040i	$0.987072\pi$
$858$		0			0
$859$	388.824			0.452647			0.226324	−	0.974052i	$-0.427329\pi$
$859$	388.824			0.452647			0.226324	+	0.974052i	$0.427329\pi$
$860$		0			0
$861$	−2380.93			−2.76531
$862$		0			0
$863$			671.642i			0.778264i	0.921182	+	0.389132i	$0.127225\pi$
$863$			671.642i			0.778264i	−0.921182	+	0.389132i	$0.872775\pi$
$864$		0			0
$865$		−	196.841i		−	0.227561i
$866$		0			0
$867$			835.198i			0.963320i
$868$		0			0
$869$			14.8657i			0.0171067i
$870$		0			0
$871$	−286.000			−0.328358
$872$		0			0
$873$			103.250i			0.118271i
$874$		0			0
$875$	−1532.19			−1.75107
$876$		0			0
$877$		−	291.049i		−	0.331869i	−0.986137	−	0.165934i	$-0.946936\pi$
$877$		−	291.049i		−	0.331869i	0.986137	−	0.165934i	$-0.0530640\pi$
$878$		0			0
$879$	1519.68			1.72887
$880$		0			0
$881$	1021.90			1.15993			0.579965	−	0.814641i	$-0.303066\pi$
$881$	1021.90			1.15993			0.579965	+	0.814641i	$0.303066\pi$
$882$		0			0
$883$	1029.33			1.16572			0.582859	−	0.812574i	$-0.301934\pi$
$883$	1029.33			1.16572			0.582859	+	0.812574i	$0.301934\pi$
$884$		0			0
$885$	1054.97			1.19206
$886$		0			0
$887$			316.096i			0.356366i	0.983997	+	0.178183i	$0.0570218\pi$
$887$			316.096i			0.356366i	−0.983997	+	0.178183i	$0.942978\pi$
$888$		0			0
$889$		−	1828.23i		−	2.05650i
$890$		0			0
$891$	504.030			0.565691
$892$		0			0
$893$	69.0529	−	1115.10i	0.0773269	−	1.24871i
$894$		0			0
$895$			316.068i			0.353148i
$896$		0			0
$897$	−209.455			−0.233506
$898$		0			0
$899$	−520.928			−0.579453
$900$		0			0
$901$			131.320i			0.145749i
$902$		0			0
$903$		−	2394.25i		−	2.65144i
$904$		0			0
$905$			609.647i			0.673643i
$906$		0			0
$907$			805.455i			0.888043i	0.896016	+	0.444021i	$0.146449\pi$
$907$			805.455i			0.888043i	−0.896016	+	0.444021i	$0.853551\pi$
$908$		0			0
$909$	−221.540			−0.243718
$910$		0			0
$911$			1340.38i			1.47132i	0.677348	+	0.735662i	$0.263129\pi$
$911$			1340.38i			1.47132i	−0.677348	+	0.735662i	$0.736871\pi$
$912$		0			0
$913$	764.989			0.837885
$914$		0			0
$915$			452.218i			0.494227i
$916$		0			0
$917$	654.716			0.713976
$918$		0			0
$919$	−1708.81			−1.85942			−0.929710	−	0.368291i	$-0.879943\pi$
$919$	−1708.81			−1.85942			−0.929710	+	0.368291i	$0.879943\pi$
$920$		0			0
$921$	−1546.52			−1.67917
$922$		0			0
$923$	472.005			0.511381
$924$		0			0
$925$			506.163i			0.547203i
$926$		0			0
$927$			241.706i			0.260739i
$928$		0			0
$929$	−976.889			−1.05155			−0.525774	−	0.850624i	$-0.676224\pi$
$929$	−976.889			−1.05155			−0.525774	+	0.850624i	$0.676224\pi$
$930$		0			0
$931$	120.559	−	1946.85i	0.129494	−	2.09114i
$932$		0			0
$933$			1700.48i			1.82260i
$934$		0			0
$935$	90.9857			0.0973109
$936$		0			0
$937$	−765.004			−0.816439			−0.408220	−	0.912884i	$-0.633850\pi$
$937$	−765.004			−0.816439			−0.408220	+	0.912884i	$0.633850\pi$
$938$		0			0
$939$		−	839.406i		−	0.893936i
$940$		0			0
$941$		−	720.166i		−	0.765320i	−0.923889	−	0.382660i	$-0.875008\pi$
$941$		−	720.166i		−	0.765320i	0.923889	−	0.382660i	$-0.124992\pi$
$942$		0			0
$943$		−	585.938i		−	0.621355i
$944$		0			0
$945$		−	932.160i		−	0.986413i
$946$		0			0
$947$	−1659.17			−1.75203			−0.876015	−	0.482284i	$-0.839807\pi$
$947$	−1659.17			−1.75203			−0.876015	+	0.482284i	$0.839807\pi$
$948$		0			0
$949$		−	350.963i		−	0.369824i
$950$		0			0
$951$	851.720			0.895605
$952$		0			0
$953$			422.810i			0.443662i	0.975085	+	0.221831i	$0.0712033\pi$
$953$			422.810i			0.443662i	−0.975085	+	0.221831i	$0.928797\pi$
$954$		0			0
$955$	551.608			0.577600
$956$		0			0
$957$	−256.568			−0.268096
$958$		0			0
$959$	−1877.12			−1.95737
$960$		0			0
$961$	−341.310			−0.355161
$962$		0			0
$963$		−	109.302i		−	0.113501i
$964$		0			0
$965$		−	982.365i		−	1.01799i
$966$		0			0
$967$	−786.215			−0.813046			−0.406523	−	0.913641i	$-0.633259\pi$
$967$	−786.215			−0.813046			−0.406523	+	0.913641i	$0.633259\pi$
$968$		0			0
$969$	326.447	+	20.2153i	0.336890	+	0.0208620i
$970$		0			0
$971$		−	1397.18i		−	1.43891i	−0.694538	−	0.719456i	$-0.744391\pi$
$971$		−	1397.18i		−	1.43891i	0.694538	−	0.719456i	$-0.255609\pi$
$972$		0			0
$973$	−220.914			−0.227044
$974$		0			0
$975$	336.298			0.344921
$976$		0			0
$977$			396.287i			0.405616i	0.979218	+	0.202808i	$0.0650068\pi$
$977$			396.287i			0.405616i	−0.979218	+	0.202808i	$0.934993\pi$
$978$		0			0
$979$			587.365i			0.599964i
$980$		0			0
$981$		−	82.5398i		−	0.0841384i
$982$		0			0
$983$			1271.91i			1.29391i	0.762530	+	0.646953i	$0.223957\pi$
$983$			1271.91i			1.29391i	−0.762530	+	0.646953i	$0.776043\pi$
$984$		0			0
$985$	−388.380			−0.394294
$986$		0			0
$987$		−	2324.13i		−	2.35474i
$988$		0			0
$989$	589.216			0.595769
$990$		0			0
$991$		−	1721.90i		−	1.73753i	−0.495220	−	0.868767i	$-0.664913\pi$
$991$		−	1721.90i		−	1.73753i	0.495220	−	0.868767i	$-0.335087\pi$
$992$		0			0
$993$	1942.02			1.95571
$994$		0			0
$995$	−290.049			−0.291507
$996$		0			0
$997$	−1153.53			−1.15700			−0.578500	−	0.815683i	$-0.696362\pi$
$997$	−1153.53			−1.15700			−0.578500	+	0.815683i	$0.696362\pi$
$998$		0			0
$999$	−800.888			−0.801690

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.e.n.1025.2		8
4.3	odd	2		1216.3.e.m.1025.7		8
8.3	odd	2		152.3.e.b.113.2	✓	8
8.5	even	2		304.3.e.g.113.7		8
19.18	odd	2	inner	1216.3.e.n.1025.7		8
24.5	odd	2		2736.3.o.p.721.6		8
24.11	even	2		1368.3.o.b.721.6		8
76.75	even	2		1216.3.e.m.1025.2		8
152.37	odd	2		304.3.e.g.113.2		8
152.75	even	2		152.3.e.b.113.7	yes	8
456.227	odd	2		1368.3.o.b.721.5		8
456.341	even	2		2736.3.o.p.721.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.3.e.b.113.2	✓	8	8.3	odd	2
152.3.e.b.113.7	yes	8	152.75	even	2
304.3.e.g.113.2		8	152.37	odd	2
304.3.e.g.113.7		8	8.5	even	2
1216.3.e.m.1025.2		8	76.75	even	2
1216.3.e.m.1025.7		8	4.3	odd	2
1216.3.e.n.1025.2		8	1.1	even	1	trivial
1216.3.e.n.1025.7		8	19.18	odd	2	inner
1368.3.o.b.721.5		8	456.227	odd	2
1368.3.o.b.721.6		8	24.11	even	2
2736.3.o.p.721.5		8	456.341	even	2
2736.3.o.p.721.6		8	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1216 = 2^{6} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1216.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.20945i q^{3} +3.06310 q^{5} +12.3151 q^{7} -1.30054 q^{9} +O(q^{10})\)	\(q-3.20945i q^{3} +3.06310 q^{5} +12.3151 q^{7} -1.30054 q^{9} -5.53798 q^{11} -6.70944i q^{13} -9.83087i q^{15} -5.36365 q^{17} +(1.17433 - 18.9637i) q^{19} -39.5247i q^{21} +9.72688 q^{23} -15.6174 q^{25} -24.7110i q^{27} +14.4351i q^{29} +36.0875i q^{31} +17.7738i q^{33} +37.7225 q^{35} -32.4102i q^{37} -21.5336 q^{39} -60.2390i q^{41} +60.5760 q^{43} -3.98369 q^{45} +58.8019 q^{47} +102.662 q^{49} +17.2143i q^{51} -24.4833i q^{53} -16.9634 q^{55} +(-60.8629 - 3.76895i) q^{57} +107.312i q^{59} -45.9998 q^{61} -16.0163 q^{63} -20.5517i q^{65} -42.6265i q^{67} -31.2179i q^{69} +70.3493i q^{71} +52.3089 q^{73} +50.1232i q^{75} -68.2008 q^{77} -2.68432i q^{79} -91.0135 q^{81} -138.135 q^{83} -16.4294 q^{85} +46.3288 q^{87} -106.061i q^{89} -82.6276i q^{91} +115.821 q^{93} +(3.59710 - 58.0877i) q^{95} -79.3904i q^{97} +7.20236 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 14 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) 8 * q + 14 * q^5 + 6 * q^7 + 4 * q^9	\( 8 q + 14 q^{5} + 6 q^{7} + 4 q^{9} - 26 q^{11} - 18 q^{17} + 16 q^{19} - 12 q^{23} + 34 q^{25} - 50 q^{35} + 108 q^{39} + 62 q^{43} - 162 q^{45} - 22 q^{47} + 22 q^{49} - 174 q^{55} + 4 q^{57} + 158 q^{61} + 2 q^{63} - 170 q^{73} - 82 q^{77} - 256 q^{81} - 64 q^{83} - 410 q^{85} + 332 q^{87} + 344 q^{93} - 222 q^{95} + 526 q^{99}+O(q^{100}) \) 8 * q + 14 * q^5 + 6 * q^7 + 4 * q^9 - 26 * q^11 - 18 * q^17 + 16 * q^19 - 12 * q^23 + 34 * q^25 - 50 * q^35 + 108 * q^39 + 62 * q^43 - 162 * q^45 - 22 * q^47 + 22 * q^49 - 174 * q^55 + 4 * q^57 + 158 * q^61 + 2 * q^63 - 170 * q^73 - 82 * q^77 - 256 * q^81 - 64 * q^83 - 410 * q^85 + 332 * q^87 + 344 * q^93 - 222 * q^95 + 526 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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