LMFDB - Embedded newform 1216.3.e.n.1025.5

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			1025.5
Root			$0.512197i$ of defining polynomial
Character	$\chi$	$=$	1216.1025
Dual form			1216.3.e.n.1025.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.512197i q^{3} +2.30665 q^{5} -4.59559 q^{7} +8.73765 q^{9} +O(q^{10})$	$q+0.512197i q^{3} +2.30665 q^{5} -4.59559 q^{7} +8.73765 q^{9} +13.7820 q^{11} -22.2405i q^{13} +1.18146i q^{15} +5.43101 q^{17} +(-7.35095 - 17.5204i) q^{19} -2.35385i q^{21} -24.9908 q^{23} -19.6794 q^{25} +9.08517i q^{27} -34.7357i q^{29} +41.2508i q^{31} +7.05907i q^{33} -10.6004 q^{35} +20.5969i q^{37} +11.3915 q^{39} -27.8883i q^{41} -10.8491 q^{43} +20.1547 q^{45} +33.8609 q^{47} -27.8805 q^{49} +2.78174i q^{51} -29.2996i q^{53} +31.7901 q^{55} +(8.97388 - 3.76513i) q^{57} -65.8291i q^{59} +107.373 q^{61} -40.1547 q^{63} -51.3010i q^{65} -1.78720i q^{67} -12.8002i q^{69} -95.8822i q^{71} +65.4556 q^{73} -10.0797i q^{75} -63.3362 q^{77} -37.2168i q^{79} +73.9855 q^{81} -32.6916 q^{83} +12.5274 q^{85} +17.7915 q^{87} -74.3815i q^{89} +102.208i q^{91} -21.1285 q^{93} +(-16.9560 - 40.4133i) q^{95} -128.116i q^{97} +120.422 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$			0.512197i			0.170732i	0.996350	+	0.0853661i	$0.0272060\pi$
$3$			0.512197i			0.170732i	−0.996350	+	0.0853661i	$0.972794\pi$
$4$		0			0
$5$	2.30665			0.461329			0.230665	−	0.973033i	$-0.425910\pi$
$5$	2.30665			0.461329			0.230665	+	0.973033i	$0.425910\pi$
$6$		0			0
$7$	−4.59559			−0.656513			−0.328257	−	0.944589i	$-0.606461\pi$
$7$	−4.59559			−0.656513			−0.328257	+	0.944589i	$0.606461\pi$
$8$		0			0
$9$	8.73765			0.970851
$10$		0			0
$11$	13.7820			1.25291			0.626453	−	0.779460i	$-0.284506\pi$
$11$	13.7820			1.25291			0.626453	+	0.779460i	$0.284506\pi$
$12$		0			0
$13$		−	22.2405i		−	1.71081i	−0.517960	−	0.855405i	$-0.673308\pi$
$13$		−	22.2405i		−	1.71081i	0.517960	−	0.855405i	$-0.326692\pi$
$14$		0			0
$15$			1.18146i			0.0787638i
$16$		0			0
$17$	5.43101			0.319471			0.159736	−	0.987160i	$-0.448936\pi$
$17$	5.43101			0.319471			0.159736	+	0.987160i	$0.448936\pi$
$18$		0			0
$19$	−7.35095	−	17.5204i	−0.386892	−	0.922125i
$19$	−7.35095	−	17.5204i	−0.386892	−	0.922125i
$20$		0			0
$21$		−	2.35385i		−	0.112088i
$22$		0			0
$23$	−24.9908			−1.08656			−0.543279	−	0.839552i	$-0.682817\pi$
$23$	−24.9908			−1.08656			−0.543279	+	0.839552i	$0.682817\pi$
$24$		0			0
$25$	−19.6794			−0.787175
$26$		0			0
$27$			9.08517i			0.336488i
$28$		0			0
$29$		−	34.7357i		−	1.19778i	−0.800830	−	0.598891i	$-0.795608\pi$
$29$		−	34.7357i		−	1.19778i	0.800830	−	0.598891i	$-0.204392\pi$
$30$		0			0
$31$			41.2508i			1.33067i	0.746544	+	0.665336i	$0.231712\pi$
$31$			41.2508i			1.33067i	−0.746544	+	0.665336i	$0.768288\pi$
$32$		0			0
$33$			7.05907i			0.213911i
$34$		0			0
$35$	−10.6004			−0.302869
$36$		0			0
$37$			20.5969i			0.556674i	0.960483	+	0.278337i	$0.0897832\pi$
$37$			20.5969i			0.556674i	−0.960483	+	0.278337i	$0.910217\pi$
$38$		0			0
$39$	11.3915			0.292090
$40$		0			0
$41$		−	27.8883i		−	0.680203i	−0.940389	−	0.340101i	$-0.889539\pi$
$41$		−	27.8883i		−	0.680203i	0.940389	−	0.340101i	$-0.110461\pi$
$42$		0			0
$43$	−10.8491			−0.252305			−0.126153	−	0.992011i	$-0.540263\pi$
$43$	−10.8491			−0.252305			−0.126153	+	0.992011i	$0.540263\pi$
$44$		0			0
$45$	20.1547			0.447882
$46$		0			0
$47$	33.8609			0.720445			0.360223	−	0.932866i	$-0.382701\pi$
$47$	33.8609			0.720445			0.360223	+	0.932866i	$0.382701\pi$
$48$		0			0
$49$	−27.8805			−0.568991
$50$		0			0
$51$			2.78174i			0.0545440i
$52$		0			0
$53$		−	29.2996i		−	0.552823i	−0.961039	−	0.276411i	$-0.910855\pi$
$53$		−	29.2996i		−	0.552823i	0.961039	−	0.276411i	$-0.0891452\pi$
$54$		0			0
$55$	31.7901			0.578002
$56$		0			0
$57$	8.97388	−	3.76513i	0.157436	−	0.0660549i
$58$		0			0
$59$		−	65.8291i		−	1.11575i	−0.829926	−	0.557874i	$-0.811617\pi$
$59$		−	65.8291i		−	1.11575i	0.829926	−	0.557874i	$-0.188383\pi$
$60$		0			0
$61$	107.373			1.76021			0.880103	−	0.474782i	$-0.157473\pi$
$61$	107.373			1.76021			0.880103	+	0.474782i	$0.157473\pi$
$62$		0			0
$63$	−40.1547			−0.637376
$64$		0			0
$65$		−	51.3010i		−	0.789247i
$66$		0			0
$67$		−	1.78720i		−	0.0266746i	−0.999911	−	0.0133373i	$-0.995754\pi$
$67$		−	1.78720i		−	0.0266746i	0.999911	−	0.0133373i	$-0.00424552\pi$
$68$		0			0
$69$		−	12.8002i		−	0.185511i
$70$		0			0
$71$		−	95.8822i		−	1.35045i	−0.737610	−	0.675227i	$-0.764046\pi$
$71$		−	95.8822i		−	1.35045i	0.737610	−	0.675227i	$-0.235954\pi$
$72$		0			0
$73$	65.4556			0.896651			0.448326	−	0.893870i	$-0.352020\pi$
$73$	65.4556			0.896651			0.448326	+	0.893870i	$0.352020\pi$
$74$		0			0
$75$		−	10.0797i		−	0.134396i
$76$		0			0
$77$	−63.3362			−0.822549
$78$		0			0
$79$		−	37.2168i		−	0.471098i	−0.971862	−	0.235549i	$-0.924311\pi$
$79$		−	37.2168i		−	0.471098i	0.971862	−	0.235549i	$-0.0756888\pi$
$80$		0			0
$81$	73.9855			0.913401
$82$		0			0
$83$	−32.6916			−0.393874			−0.196937	−	0.980416i	$-0.563099\pi$
$83$	−32.6916			−0.393874			−0.196937	+	0.980416i	$0.563099\pi$
$84$		0			0
$85$	12.5274			0.147381
$86$		0			0
$87$	17.7915			0.204500
$88$		0			0
$89$		−	74.3815i		−	0.835747i	−0.908505	−	0.417874i	$-0.862775\pi$
$89$		−	74.3815i		−	0.835747i	0.908505	−	0.417874i	$-0.137225\pi$
$90$		0			0
$91$			102.208i			1.12317i
$92$		0			0
$93$	−21.1285			−0.227188
$94$		0			0
$95$	−16.9560	−	40.4133i	−0.178485	−	0.425403i
$96$		0			0
$97$		−	128.116i		−	1.32078i	−0.750921	−	0.660392i	$-0.770390\pi$
$97$		−	128.116i		−	1.32078i	0.750921	−	0.660392i	$-0.229610\pi$
$98$		0			0
$99$	120.422			1.21638
$100$		0			0
$101$	−66.1257			−0.654710			−0.327355	−	0.944902i	$-0.606157\pi$
$101$	−66.1257			−0.654710			−0.327355	+	0.944902i	$0.606157\pi$
$102$		0			0
$103$			68.6107i			0.666123i	0.942905	+	0.333061i	$0.108082\pi$
$103$			68.6107i			0.666123i	−0.942905	+	0.333061i	$0.891918\pi$
$104$		0			0
$105$		−	5.42949i		−	0.0517095i
$106$		0			0
$107$			163.182i			1.52507i	0.646949	+	0.762534i	$0.276045\pi$
$107$			163.182i			1.52507i	−0.646949	+	0.762534i	$0.723955\pi$
$108$		0			0
$109$		−	98.8345i		−	0.906738i	−0.891323	−	0.453369i	$-0.850222\pi$
$109$		−	98.8345i		−	0.906738i	0.891323	−	0.453369i	$-0.149778\pi$
$110$		0			0
$111$	−10.5497			−0.0950422
$112$		0			0
$113$		−	94.2226i		−	0.833829i	−0.908946	−	0.416914i	$-0.863112\pi$
$113$		−	94.2226i		−	0.833829i	0.908946	−	0.416914i	$-0.136888\pi$
$114$		0			0
$115$	−57.6451			−0.501261
$116$		0			0
$117$		−	194.330i		−	1.66094i
$118$		0			0
$119$	−24.9587			−0.209737
$120$		0			0
$121$	68.9423			0.569771
$122$		0			0
$123$	14.2843			0.116133
$124$		0			0
$125$	−103.060			−0.824477
$126$		0			0
$127$			119.017i			0.937146i	0.883425	+	0.468573i	$0.155232\pi$
$127$			119.017i			0.937146i	−0.883425	+	0.468573i	$0.844768\pi$
$128$		0			0
$129$		−	5.55689i		−	0.0430766i
$130$		0			0
$131$	126.661			0.966878			0.483439	−	0.875378i	$-0.339387\pi$
$131$	126.661			0.966878			0.483439	+	0.875378i	$0.339387\pi$
$132$		0			0
$133$	33.7820	+	80.5165i	0.254000	+	0.605387i
$134$		0			0
$135$			20.9563i			0.155232i
$136$		0			0
$137$	48.2645			0.352296			0.176148	−	0.984364i	$-0.443636\pi$
$137$	48.2645			0.352296			0.176148	+	0.984364i	$0.443636\pi$
$138$		0			0
$139$	−243.120			−1.74907			−0.874534	−	0.484965i	$-0.838832\pi$
$139$	−243.120			−1.74907			−0.874534	+	0.484965i	$0.838832\pi$
$140$		0			0
$141$			17.3435i			0.123003i
$142$		0			0
$143$		−	306.518i		−	2.14348i
$144$		0			0
$145$		−	80.1230i		−	0.552572i
$146$		0			0
$147$		−	14.2803i		−	0.0971450i
$148$		0			0
$149$	131.025			0.879361			0.439681	−	0.898154i	$-0.355092\pi$
$149$	131.025			0.879361			0.439681	+	0.898154i	$0.355092\pi$
$150$		0			0
$151$			173.521i			1.14915i	0.818453	+	0.574574i	$0.194832\pi$
$151$			173.521i			1.14915i	−0.818453	+	0.574574i	$0.805168\pi$
$152$		0			0
$153$	47.4543			0.310159
$154$		0			0
$155$			95.1511i			0.613878i
$156$		0			0
$157$	259.490			1.65280			0.826402	−	0.563081i	$-0.190384\pi$
$157$	259.490			1.65280			0.826402	+	0.563081i	$0.190384\pi$
$158$		0			0
$159$	15.0072			0.0943846
$160$		0			0
$161$	114.848			0.713340
$162$		0			0
$163$	62.7264			0.384825			0.192412	−	0.981314i	$-0.438369\pi$
$163$	62.7264			0.384825			0.192412	+	0.981314i	$0.438369\pi$
$164$		0			0
$165$			16.2828i			0.0986836i
$166$		0			0
$167$			134.688i			0.806518i	0.915086	+	0.403259i	$0.132123\pi$
$167$			134.688i			0.806518i	−0.915086	+	0.403259i	$0.867877\pi$
$168$		0			0
$169$	−325.641			−1.92687
$170$		0			0
$171$	−64.2301	−	153.087i	−0.375614	−	0.895246i
$172$		0			0
$173$			90.4432i			0.522793i	0.965232	+	0.261397i	$0.0841830\pi$
$173$			90.4432i			0.522793i	−0.965232	+	0.261397i	$0.915817\pi$
$174$		0			0
$175$	90.4384			0.516791
$176$		0			0
$177$	33.7174			0.190494
$178$		0			0
$179$		−	322.972i		−	1.80431i	−0.431409	−	0.902156i	$-0.641984\pi$
$179$		−	322.972i		−	1.80431i	0.431409	−	0.902156i	$-0.358016\pi$
$180$		0			0
$181$		−	241.679i		−	1.33524i	−0.744502	−	0.667621i	$-0.767313\pi$
$181$		−	241.679i		−	1.33524i	0.744502	−	0.667621i	$-0.232687\pi$
$182$		0			0
$183$			54.9959i			0.300524i
$184$		0			0
$185$			47.5099i			0.256810i
$186$		0			0
$187$	74.8499			0.400267
$188$		0			0
$189$		−	41.7517i		−	0.220909i
$190$		0			0
$191$	115.733			0.605931			0.302966	−	0.953001i	$-0.402023\pi$
$191$	115.733			0.605931			0.302966	+	0.953001i	$0.402023\pi$
$192$		0			0
$193$			277.929i			1.44005i	0.693951	+	0.720023i	$0.255869\pi$
$193$			277.929i			1.44005i	−0.693951	+	0.720023i	$0.744131\pi$
$194$		0			0
$195$	26.2762			0.134750
$196$		0			0
$197$	78.7308			0.399649			0.199824	−	0.979832i	$-0.435963\pi$
$197$	78.7308			0.399649			0.199824	+	0.979832i	$0.435963\pi$
$198$		0			0
$199$	93.6642			0.470674			0.235337	−	0.971914i	$-0.424381\pi$
$199$	93.6642			0.470674			0.235337	+	0.971914i	$0.424381\pi$
$200$		0			0
$201$	0.915397			0.00455422
$202$		0			0
$203$			159.631i			0.786360i
$204$		0			0
$205$		−	64.3285i		−	0.313798i
$206$		0			0
$207$	−218.361			−1.05489
$208$		0			0
$209$	−101.310	−	241.465i	−0.484739	−	1.15534i
$210$		0			0
$211$		−	362.818i		−	1.71952i	−0.510701	−	0.859758i	$-0.670614\pi$
$211$		−	362.818i		−	1.71952i	0.510701	−	0.859758i	$-0.329386\pi$
$212$		0			0
$213$	49.1106			0.230566
$214$		0			0
$215$	−25.0251			−0.116396
$216$		0			0
$217$		−	189.572i		−	0.873603i
$218$		0			0
$219$			33.5261i			0.153087i
$220$		0			0
$221$		−	120.788i		−	0.546554i
$222$		0			0
$223$		−	221.055i		−	0.991277i	−0.868529	−	0.495639i	$-0.834934\pi$
$223$		−	221.055i		−	0.991277i	0.868529	−	0.495639i	$-0.165066\pi$
$224$		0			0
$225$	−171.952			−0.764229
$226$		0			0
$227$			46.2298i			0.203656i	0.994802	+	0.101828i	$0.0324691\pi$
$227$			46.2298i			0.203656i	−0.994802	+	0.101828i	$0.967531\pi$
$228$		0			0
$229$	−44.5210			−0.194415			−0.0972075	−	0.995264i	$-0.530991\pi$
$229$	−44.5210			−0.194415			−0.0972075	+	0.995264i	$0.530991\pi$
$230$		0			0
$231$		−	32.4406i		−	0.140436i
$232$		0			0
$233$	276.616			1.18719			0.593596	−	0.804763i	$-0.297708\pi$
$233$	276.616			1.18719			0.593596	+	0.804763i	$0.297708\pi$
$234$		0			0
$235$	78.1052			0.332363
$236$		0			0
$237$	19.0623			0.0804316
$238$		0			0
$239$	431.105			1.80379			0.901893	−	0.431959i	$-0.142177\pi$
$239$	431.105			1.80379			0.901893	+	0.431959i	$0.142177\pi$
$240$		0			0
$241$			90.5900i			0.375892i	0.982179	+	0.187946i	$0.0601830\pi$
$241$			90.5900i			0.375892i	−0.982179	+	0.187946i	$0.939817\pi$
$242$		0			0
$243$			119.662i			0.492435i
$244$		0			0
$245$	−64.3106			−0.262492
$246$		0			0
$247$	−389.662	+	163.489i	−1.57758	+	0.661899i
$248$		0			0
$249$		−	16.7445i		−	0.0672470i
$250$		0			0
$251$	165.585			0.659702			0.329851	−	0.944033i	$-0.393001\pi$
$251$	165.585			0.659702			0.329851	+	0.944033i	$0.393001\pi$
$252$		0			0
$253$	−344.423			−1.36135
$254$		0			0
$255$			6.41650i			0.0251628i
$256$		0			0
$257$			500.700i			1.94825i	0.226011	+	0.974125i	$0.427431\pi$
$257$			500.700i			1.94825i	−0.226011	+	0.974125i	$0.572569\pi$
$258$		0			0
$259$		−	94.6551i		−	0.365464i
$260$		0			0
$261$		−	303.508i		−	1.16287i
$262$		0			0
$263$	−370.994			−1.41062			−0.705312	−	0.708897i	$-0.749193\pi$
$263$	−370.994			−1.41062			−0.705312	+	0.708897i	$0.749193\pi$
$264$		0			0
$265$		−	67.5838i		−	0.255033i
$266$		0			0
$267$	38.0980			0.142689
$268$		0			0
$269$			273.479i			1.01665i	0.861165	+	0.508326i	$0.169735\pi$
$269$			273.479i			1.01665i	−0.861165	+	0.508326i	$0.830265\pi$
$270$		0			0
$271$	−82.1566			−0.303161			−0.151580	−	0.988445i	$-0.548436\pi$
$271$	−82.1566			−0.303161			−0.151580	+	0.988445i	$0.548436\pi$
$272$		0			0
$273$	−52.3508			−0.191761
$274$		0			0
$275$	−271.220			−0.986256
$276$		0			0
$277$	−228.921			−0.826430			−0.413215	−	0.910633i	$-0.635594\pi$
$277$	−228.921			−0.826430			−0.413215	+	0.910633i	$0.635594\pi$
$278$		0			0
$279$			360.435i			1.29188i
$280$		0			0
$281$			170.230i			0.605800i	0.953022	+	0.302900i	$0.0979549\pi$
$281$			170.230i			0.605800i	−0.953022	+	0.302900i	$0.902045\pi$
$282$		0			0
$283$	37.0280			0.130841			0.0654205	−	0.997858i	$-0.479161\pi$
$283$	37.0280			0.130841			0.0654205	+	0.997858i	$0.479161\pi$
$284$		0			0
$285$	20.6996	−	8.68483i	0.0726301	−	0.0304731i
$286$		0			0
$287$			128.163i			0.446562i
$288$		0			0
$289$	−259.504			−0.897938
$290$		0			0
$291$	65.6206			0.225500
$292$		0			0
$293$			379.390i			1.29485i	0.762130	+	0.647424i	$0.224153\pi$
$293$			379.390i			1.29485i	−0.762130	+	0.647424i	$0.775847\pi$
$294$		0			0
$295$		−	151.844i		−	0.514727i
$296$		0			0
$297$			125.211i			0.421587i
$298$		0			0
$299$			555.809i			1.85889i
$300$		0			0
$301$	49.8582			0.165642
$302$		0			0
$303$		−	33.8693i		−	0.111780i
$304$		0			0
$305$	247.671			0.812035
$306$		0			0
$307$			207.485i			0.675846i	0.941174	+	0.337923i	$0.109724\pi$
$307$			207.485i			0.675846i	−0.941174	+	0.337923i	$0.890276\pi$
$308$		0			0
$309$	−35.1421			−0.113729
$310$		0			0
$311$	−250.191			−0.804474			−0.402237	−	0.915536i	$-0.631767\pi$
$311$	−250.191			−0.804474			−0.402237	+	0.915536i	$0.631767\pi$
$312$		0			0
$313$	490.013			1.56554			0.782768	−	0.622313i	$-0.213807\pi$
$313$	490.013			1.56554			0.782768	+	0.622313i	$0.213807\pi$
$314$		0			0
$315$	−92.6227			−0.294040
$316$		0			0
$317$			193.572i			0.610637i	0.952250	+	0.305319i	$0.0987630\pi$
$317$			193.572i			0.610637i	−0.952250	+	0.305319i	$0.901237\pi$
$318$		0			0
$319$		−	478.726i		−	1.50071i
$320$		0			0
$321$	−83.5814			−0.260378
$322$		0			0
$323$	−39.9231	−	95.1533i	−0.123601	−	0.294592i
$324$		0			0
$325$			437.680i			1.34671i
$326$		0			0
$327$	50.6227			0.154809
$328$		0			0
$329$	−155.611			−0.472982
$330$		0			0
$331$			77.3740i			0.233758i	0.993146	+	0.116879i	$0.0372890\pi$
$331$			77.3740i			0.233758i	−0.993146	+	0.116879i	$0.962711\pi$
$332$		0			0
$333$			179.969i			0.540447i
$334$		0			0
$335$		−	4.12244i		−	0.0123058i
$336$		0			0
$337$			307.055i			0.911143i	0.890199	+	0.455571i	$0.150565\pi$
$337$			307.055i			0.911143i	−0.890199	+	0.455571i	$0.849435\pi$
$338$		0			0
$339$	48.2605			0.142361
$340$		0			0
$341$			568.517i			1.66720i
$342$		0			0
$343$	353.312			1.03006
$344$		0			0
$345$		−	29.5256i		−	0.0855815i
$346$		0			0
$347$	−343.072			−0.988679			−0.494340	−	0.869269i	$-0.664590\pi$
$347$	−343.072			−0.988679			−0.494340	+	0.869269i	$0.664590\pi$
$348$		0			0
$349$	135.313			0.387716			0.193858	−	0.981030i	$-0.437900\pi$
$349$	135.313			0.387716			0.193858	+	0.981030i	$0.437900\pi$
$350$		0			0
$351$	202.059			0.575666
$352$		0			0
$353$	95.3649			0.270156			0.135078	−	0.990835i	$-0.456872\pi$
$353$	95.3649			0.270156			0.135078	+	0.990835i	$0.456872\pi$
$354$		0			0
$355$		−	221.167i		−	0.623004i
$356$		0			0
$357$		−	12.7838i		−	0.0358088i
$358$		0			0
$359$	339.918			0.946846			0.473423	−	0.880835i	$-0.343018\pi$
$359$	339.918			0.946846			0.473423	+	0.880835i	$0.343018\pi$
$360$		0			0
$361$	−252.927	+	257.583i	−0.700629	+	0.713526i
$362$		0			0
$363$			35.3120i			0.0972783i
$364$		0			0
$365$	150.983			0.413652
$366$		0			0
$367$	207.153			0.564449			0.282225	−	0.959348i	$-0.408928\pi$
$367$	207.153			0.564449			0.282225	+	0.959348i	$0.408928\pi$
$368$		0			0
$369$		−	243.678i		−	0.660375i
$370$		0			0
$371$			134.649i			0.362935i
$372$		0			0
$373$			221.266i			0.593207i	0.955001	+	0.296604i	$0.0958540\pi$
$373$			221.266i			0.593207i	−0.955001	+	0.296604i	$0.904146\pi$
$374$		0			0
$375$		−	52.7868i		−	0.140765i
$376$		0			0
$377$	−772.540			−2.04918
$378$		0			0
$379$		−	251.511i		−	0.663619i	−0.943346	−	0.331809i	$-0.892341\pi$
$379$		−	251.511i		−	0.663619i	0.943346	−	0.331809i	$-0.107659\pi$
$380$		0			0
$381$	−60.9604			−0.160001
$382$		0			0
$383$		−	367.906i		−	0.960590i	−0.877107	−	0.480295i	$-0.840530\pi$
$383$		−	367.906i		−	0.960590i	0.877107	−	0.480295i	$-0.159470\pi$
$384$		0			0
$385$	−146.094			−0.379466
$386$		0			0
$387$	−94.7959			−0.244951
$388$		0			0
$389$	−480.911			−1.23628			−0.618138	−	0.786070i	$-0.712113\pi$
$389$	−480.911			−1.23628			−0.618138	+	0.786070i	$0.712113\pi$
$390$		0			0
$391$	−135.725			−0.347124
$392$		0			0
$393$			64.8754i			0.165077i
$394$		0			0
$395$		−	85.8459i		−	0.217331i
$396$		0			0
$397$	−432.964			−1.09059			−0.545294	−	0.838245i	$-0.683582\pi$
$397$	−432.964			−1.09059			−0.545294	+	0.838245i	$0.683582\pi$
$398$		0			0
$399$	−41.2403	+	17.3030i	−0.103359	+	0.0433659i
$400$		0			0
$401$		−	314.930i		−	0.785361i	−0.919675	−	0.392680i	$-0.871548\pi$
$401$		−	314.930i		−	0.785361i	0.919675	−	0.392680i	$-0.128452\pi$
$402$		0			0
$403$	917.440			2.27653
$404$		0			0
$405$	170.658			0.421379
$406$		0			0
$407$			283.866i			0.697460i
$408$		0			0
$409$		−	13.2424i		−	0.0323775i	−0.999869	−	0.0161888i	$-0.994847\pi$
$409$		−	13.2424i		−	0.0323775i	0.999869	−	0.0161888i	$-0.00515327\pi$
$410$		0			0
$411$			24.7209i			0.0601483i
$412$		0			0
$413$			302.524i			0.732503i
$414$		0			0
$415$	−75.4079			−0.181706
$416$		0			0
$417$		−	124.525i		−	0.298622i
$418$		0			0
$419$	−667.426			−1.59290			−0.796451	−	0.604703i	$-0.793292\pi$
$419$	−667.426			−1.59290			−0.796451	+	0.604703i	$0.793292\pi$
$420$		0			0
$421$		−	437.582i		−	1.03939i	−0.854352	−	0.519694i	$-0.826046\pi$
$421$		−	437.582i		−	1.03939i	0.854352	−	0.519694i	$-0.173954\pi$
$422$		0			0
$423$	295.865			0.699445
$424$		0			0
$425$	−106.879			−0.251480
$426$		0			0
$427$	−493.441			−1.15560
$428$		0			0
$429$	156.997			0.365961
$430$		0			0
$431$		−	680.592i		−	1.57910i	−0.613686	−	0.789550i	$-0.710314\pi$
$431$		−	680.592i		−	1.57910i	0.613686	−	0.789550i	$-0.289686\pi$
$432$		0			0
$433$			391.348i			0.903807i	0.892067	+	0.451904i	$0.149255\pi$
$433$			391.348i			0.903807i	−0.892067	+	0.451904i	$0.850745\pi$
$434$		0			0
$435$	41.0387			0.0943419
$436$		0			0
$437$	183.706	+	437.849i	0.420381	+	1.00194i
$438$		0			0
$439$			433.093i			0.986544i	0.869875	+	0.493272i	$0.164199\pi$
$439$			433.093i			0.986544i	−0.869875	+	0.493272i	$0.835801\pi$
$440$		0			0
$441$	−243.611			−0.552405
$442$		0			0
$443$	−372.634			−0.841160			−0.420580	−	0.907256i	$-0.638173\pi$
$443$	−372.634			−0.841160			−0.420580	+	0.907256i	$0.638173\pi$
$444$		0			0
$445$		−	171.572i		−	0.385555i
$446$		0			0
$447$			67.1105i			0.150135i
$448$		0			0
$449$			96.4188i			0.214741i	0.994219	+	0.107371i	$0.0342431\pi$
$449$			96.4188i			0.214741i	−0.994219	+	0.107371i	$0.965757\pi$
$450$		0			0
$451$		−	384.355i		−	0.852229i
$452$		0			0
$453$	−88.8770			−0.196196
$454$		0			0
$455$			235.759i			0.518151i
$456$		0			0
$457$	741.777			1.62314			0.811572	−	0.584253i	$-0.198612\pi$
$457$	741.777			1.62314			0.811572	+	0.584253i	$0.198612\pi$
$458$		0			0
$459$			49.3416i			0.107498i
$460$		0			0
$461$	−40.2016			−0.0872053			−0.0436026	−	0.999049i	$-0.513884\pi$
$461$	−40.2016			−0.0872053			−0.0436026	+	0.999049i	$0.513884\pi$
$462$		0			0
$463$	−534.775			−1.15502			−0.577510	−	0.816383i	$-0.695976\pi$
$463$	−534.775			−1.15502			−0.577510	+	0.816383i	$0.695976\pi$
$464$		0			0
$465$	−48.7361			−0.104809
$466$		0			0
$467$	390.715			0.836649			0.418325	−	0.908298i	$-0.362617\pi$
$467$	390.715			0.836649			0.418325	+	0.908298i	$0.362617\pi$
$468$		0			0
$469$			8.21324i			0.0175122i
$470$		0			0
$471$			132.910i			0.282187i
$472$		0			0
$473$	−149.522			−0.316115
$474$		0			0
$475$	144.662	+	344.790i	0.304552	+	0.725874i
$476$		0			0
$477$		−	256.010i		−	0.536708i
$478$		0			0
$479$	4.79318			0.0100066			0.00500332	−	0.999987i	$-0.498407\pi$
$479$	4.79318			0.0100066			0.00500332	+	0.999987i	$0.498407\pi$
$480$		0			0
$481$	458.087			0.952364
$482$		0			0
$483$			58.8246i			0.121790i
$484$		0			0
$485$		−	295.518i		−	0.609316i
$486$		0			0
$487$		−	173.322i		−	0.355898i	−0.984040	−	0.177949i	$-0.943054\pi$
$487$		−	173.322i		−	0.355898i	0.984040	−	0.177949i	$-0.0569462\pi$
$488$		0			0
$489$			32.1283i			0.0657020i
$490$		0			0
$491$	522.057			1.06325			0.531626	−	0.846979i	$-0.321581\pi$
$491$	522.057			1.06325			0.531626	+	0.846979i	$0.321581\pi$
$492$		0			0
$493$		−	188.650i		−	0.382657i
$494$		0			0
$495$	277.771			0.561154
$496$		0			0
$497$			440.636i			0.886591i
$498$		0			0
$499$	806.635			1.61650			0.808251	−	0.588838i	$-0.200414\pi$
$499$	806.635			1.61650			0.808251	+	0.588838i	$0.200414\pi$
$500$		0			0
$501$	−68.9870			−0.137699
$502$		0			0
$503$	−148.884			−0.295992			−0.147996	−	0.988988i	$-0.547282\pi$
$503$	−148.884			−0.295992			−0.147996	+	0.988988i	$0.547282\pi$
$504$		0			0
$505$	−152.529			−0.302037
$506$		0			0
$507$		−	166.792i		−	0.328979i
$508$		0			0
$509$			469.470i			0.922339i	0.887312	+	0.461169i	$0.152570\pi$
$509$			469.470i			0.922339i	−0.887312	+	0.461169i	$0.847430\pi$
$510$		0			0
$511$	−300.807			−0.588663
$512$		0			0
$513$	159.176	−	66.7846i	0.310284	−	0.130184i
$514$		0			0
$515$			158.261i			0.307302i
$516$		0			0
$517$	466.670			0.902650
$518$		0			0
$519$	−46.3247			−0.0892576
$520$		0			0
$521$			952.247i			1.82773i	0.406020	+	0.913864i	$0.366916\pi$
$521$			952.247i			1.82773i	−0.406020	+	0.913864i	$0.633084\pi$
$522$		0			0
$523$			883.789i			1.68985i	0.534889	+	0.844923i	$0.320354\pi$
$523$			883.789i			1.68985i	−0.534889	+	0.844923i	$0.679646\pi$
$524$		0			0
$525$			46.3222i			0.0882328i
$526$		0			0
$527$			224.033i			0.425111i
$528$		0			0
$529$	95.5422			0.180609
$530$		0			0
$531$		−	575.192i		−	1.08322i
$532$		0			0
$533$	−620.251			−1.16370
$534$		0			0
$535$			376.404i			0.703558i
$536$		0			0
$537$	165.425			0.308054
$538$		0			0
$539$	−384.248			−0.712891
$540$		0			0
$541$	−370.690			−0.685193			−0.342597	−	0.939483i	$-0.611306\pi$
$541$	−370.690			−0.685193			−0.342597	+	0.939483i	$0.611306\pi$
$542$		0			0
$543$	123.787			0.227969
$544$		0			0
$545$		−	227.976i		−	0.418305i
$546$		0			0
$547$		−	361.146i		−	0.660230i	−0.943941	−	0.330115i	$-0.892912\pi$
$547$		−	361.146i		−	0.660230i	0.943941	−	0.330115i	$-0.107088\pi$
$548$		0			0
$549$	938.185			1.70890
$550$		0			0
$551$	−608.582	+	255.340i	−1.10451	+	0.463413i
$552$		0			0
$553$			171.033i			0.309282i
$554$		0			0
$555$	−24.3344			−0.0438458
$556$		0			0
$557$	−935.225			−1.67904			−0.839519	−	0.543330i	$-0.817163\pi$
$557$	−935.225			−1.67904			−0.839519	+	0.543330i	$0.817163\pi$
$558$		0			0
$559$			241.290i			0.431646i
$560$		0			0
$561$			38.3379i			0.0683384i
$562$		0			0
$563$		−	288.854i		−	0.513062i	−0.966536	−	0.256531i	$-0.917420\pi$
$563$		−	288.854i		−	0.513062i	0.966536	−	0.256531i	$-0.0825796\pi$
$564$		0			0
$565$		−	217.338i		−	0.384670i
$566$		0			0
$567$	−340.007			−0.599660
$568$		0			0
$569$			1109.10i			1.94921i	0.223936	+	0.974604i	$0.428109\pi$
$569$			1109.10i			1.94921i	−0.223936	+	0.974604i	$0.571891\pi$
$570$		0			0
$571$	81.8698			0.143380			0.0716898	−	0.997427i	$-0.477161\pi$
$571$	81.8698			0.143380			0.0716898	+	0.997427i	$0.477161\pi$
$572$		0			0
$573$			59.2780i			0.103452i
$574$		0			0
$575$	491.804			0.855312
$576$		0			0
$577$	−282.577			−0.489735			−0.244867	−	0.969557i	$-0.578744\pi$
$577$	−282.577			−0.489735			−0.244867	+	0.969557i	$0.578744\pi$
$578$		0			0
$579$	−142.354			−0.245862
$580$		0			0
$581$	150.237			0.258584
$582$		0			0
$583$		−	403.806i		−	0.692634i
$584$		0			0
$585$		−	448.251i		−	0.766241i
$586$		0			0
$587$	732.036			1.24708			0.623540	−	0.781792i	$-0.285694\pi$
$587$	732.036			1.24708			0.623540	+	0.781792i	$0.285694\pi$
$588$		0			0
$589$	722.730	−	303.233i	1.22705	−	0.514826i
$590$		0			0
$591$			40.3256i			0.0682329i
$592$		0			0
$593$	−669.125			−1.12837			−0.564187	−	0.825647i	$-0.690810\pi$
$593$	−669.125			−1.12837			−0.564187	+	0.825647i	$0.690810\pi$
$594$		0			0
$595$	−57.5709			−0.0967578
$596$		0			0
$597$			47.9745i			0.0803592i
$598$		0			0
$599$			852.366i			1.42298i	0.702695	+	0.711491i	$0.251980\pi$
$599$			852.366i			1.42298i	−0.702695	+	0.711491i	$0.748020\pi$
$600$		0			0
$601$		−	265.429i		−	0.441646i	−0.975314	−	0.220823i	$-0.929126\pi$
$601$		−	265.429i		−	0.441646i	0.975314	−	0.220823i	$-0.0708743\pi$
$602$		0			0
$603$		−	15.6159i		−	0.0258971i
$604$		0			0
$605$	159.026			0.262852
$606$		0			0
$607$		−	751.473i		−	1.23801i	−0.785387	−	0.619005i	$-0.787536\pi$
$607$		−	751.473i		−	1.23801i	0.785387	−	0.619005i	$-0.212464\pi$
$608$		0			0
$609$	−81.7625			−0.134257
$610$		0			0
$611$		−	753.085i		−	1.23254i
$612$		0			0
$613$	−606.487			−0.989376			−0.494688	−	0.869071i	$-0.664718\pi$
$613$	−606.487			−0.989376			−0.494688	+	0.869071i	$0.664718\pi$
$614$		0			0
$615$	32.9488			0.0535754
$616$		0			0
$617$	−1020.39			−1.65380			−0.826900	−	0.562349i	$-0.809898\pi$
$617$	−1020.39			−1.65380			−0.826900	+	0.562349i	$0.809898\pi$
$618$		0			0
$619$	−886.720			−1.43250			−0.716252	−	0.697842i	$-0.754144\pi$
$619$	−886.720			−1.43250			−0.716252	+	0.697842i	$0.754144\pi$
$620$		0			0
$621$		−	227.046i		−	0.365613i
$622$		0			0
$623$			341.827i			0.548679i
$624$		0			0
$625$	254.262			0.406820
$626$		0			0
$627$	123.678	−	51.8909i	0.197253	−	0.0827606i
$628$		0			0
$629$			111.862i			0.177841i
$630$		0			0
$631$	−737.311			−1.16848			−0.584240	−	0.811581i	$-0.698607\pi$
$631$	−737.311			−1.16848			−0.584240	+	0.811581i	$0.698607\pi$
$632$		0			0
$633$	185.834			0.293577
$634$		0			0
$635$			274.531i			0.432333i
$636$		0			0
$637$			620.078i			0.973435i
$638$		0			0
$639$		−	837.786i		−	1.31109i
$640$		0			0
$641$			267.570i			0.417426i	0.977977	+	0.208713i	$0.0669274\pi$
$641$			267.570i			0.417426i	−0.977977	+	0.208713i	$0.933073\pi$
$642$		0			0
$643$	−567.693			−0.882881			−0.441441	−	0.897290i	$-0.645532\pi$
$643$	−567.693			−0.882881			−0.441441	+	0.897290i	$0.645532\pi$
$644$		0			0
$645$		−	12.8178i		−	0.0198725i
$646$		0			0
$647$	543.577			0.840150			0.420075	−	0.907489i	$-0.362004\pi$
$647$	543.577			0.840150			0.420075	+	0.907489i	$0.362004\pi$
$648$		0			0
$649$		−	907.254i		−	1.39793i
$650$		0			0
$651$	97.0981			0.149152
$652$		0			0
$653$	39.9308			0.0611498			0.0305749	−	0.999532i	$-0.490266\pi$
$653$	39.9308			0.0611498			0.0305749	+	0.999532i	$0.490266\pi$
$654$		0			0
$655$	292.162			0.446049
$656$		0			0
$657$	571.928			0.870515
$658$		0			0
$659$		−	779.029i		−	1.18214i	−0.806621	−	0.591069i	$-0.798706\pi$
$659$		−	779.029i		−	1.18214i	0.806621	−	0.591069i	$-0.201294\pi$
$660$		0			0
$661$		−	419.496i		−	0.634638i	−0.948319	−	0.317319i	$-0.897217\pi$
$661$		−	419.496i		−	0.634638i	0.948319	−	0.317319i	$-0.102783\pi$
$662$		0			0
$663$	61.8674			0.0933144
$664$		0			0
$665$	77.9231	+	185.723i	0.117178	+	0.279283i
$666$		0			0
$667$			868.074i			1.30146i
$668$		0			0
$669$	113.224			0.169243
$670$		0			0
$671$	1479.80			2.20537
$672$		0			0
$673$		−	997.533i		−	1.48222i	−0.671385	−	0.741109i	$-0.734300\pi$
$673$		−	997.533i		−	1.48222i	0.671385	−	0.741109i	$-0.265700\pi$
$674$		0			0
$675$		−	178.790i		−	0.264875i
$676$		0			0
$677$		−	441.407i		−	0.652005i	−0.945369	−	0.326003i	$-0.894298\pi$
$677$		−	441.407i		−	0.652005i	0.945369	−	0.326003i	$-0.105702\pi$
$678$		0			0
$679$			588.769i			0.867111i
$680$		0			0
$681$	−23.6788			−0.0347706
$682$		0			0
$683$			907.324i			1.32844i	0.747537	+	0.664220i	$0.231236\pi$
$683$			907.324i			1.32844i	−0.747537	+	0.664220i	$0.768764\pi$
$684$		0			0
$685$	111.329			0.162525
$686$		0			0
$687$		−	22.8035i		−	0.0331929i
$688$		0			0
$689$	−651.638			−0.945774
$690$		0			0
$691$	−928.930			−1.34433			−0.672164	−	0.740403i	$-0.734635\pi$
$691$	−928.930			−1.34433			−0.672164	+	0.740403i	$0.734635\pi$
$692$		0			0
$693$	−553.410			−0.798572
$694$		0			0
$695$	−560.793			−0.806896
$696$		0			0
$697$		−	151.462i		−	0.217305i
$698$		0			0
$699$			141.682i			0.202692i
$700$		0			0
$701$	1104.52			1.57563			0.787814	−	0.615913i	$-0.211213\pi$
$701$	1104.52			1.57563			0.787814	+	0.615913i	$0.211213\pi$
$702$		0			0
$703$	360.866	−	151.407i	0.513323	−	0.215373i
$704$		0			0
$705$			40.0052i			0.0567450i
$706$		0			0
$707$	303.886			0.429825
$708$		0			0
$709$	382.002			0.538790			0.269395	−	0.963030i	$-0.413176\pi$
$709$	382.002			0.538790			0.269395	+	0.963030i	$0.413176\pi$
$710$		0			0
$711$		−	325.187i		−	0.457366i
$712$		0			0
$713$		−	1030.89i		−	1.44585i
$714$		0			0
$715$		−	707.029i		−	0.988851i
$716$		0			0
$717$			220.811i			0.307964i
$718$		0			0
$719$	−784.961			−1.09174			−0.545870	−	0.837870i	$-0.683801\pi$
$719$	−784.961			−1.09174			−0.545870	+	0.837870i	$0.683801\pi$
$720$		0			0
$721$		−	315.307i		−	0.437318i
$722$		0			0
$723$	−46.3999			−0.0641769
$724$		0			0
$725$			683.577i			0.942865i
$726$		0			0
$727$	776.412			1.06797			0.533983	−	0.845495i	$-0.320694\pi$
$727$	776.412			1.06797			0.533983	+	0.845495i	$0.320694\pi$
$728$		0			0
$729$	604.579			0.829327
$730$		0			0
$731$	−58.9217			−0.0806042
$732$		0			0
$733$	−263.990			−0.360150			−0.180075	−	0.983653i	$-0.557634\pi$
$733$	−263.990			−0.360150			−0.180075	+	0.983653i	$0.557634\pi$
$734$		0			0
$735$		−	32.9397i		−	0.0448159i
$736$		0			0
$737$		−	24.6311i		−	0.0334208i
$738$		0			0
$739$	696.550			0.942558			0.471279	−	0.881984i	$-0.343792\pi$
$739$	696.550			0.942558			0.471279	+	0.881984i	$0.343792\pi$
$740$		0			0
$741$	−83.7385	−	199.584i	−0.113007	−	0.269344i
$742$		0			0
$743$		−	544.317i		−	0.732593i	−0.930498	−	0.366297i	$-0.880626\pi$
$743$		−	544.317i		−	0.732593i	0.930498	−	0.366297i	$-0.119374\pi$
$744$		0			0
$745$	302.228			0.405675
$746$		0			0
$747$	−285.648			−0.382393
$748$		0			0
$749$		−	749.919i		−	1.00123i
$750$		0			0
$751$			648.270i			0.863209i	0.902063	+	0.431605i	$0.142052\pi$
$751$			648.270i			0.863209i	−0.902063	+	0.431605i	$0.857948\pi$
$752$		0			0
$753$			84.8122i			0.112632i
$754$		0			0
$755$			400.252i			0.530136i
$756$		0			0
$757$	726.694			0.959965			0.479983	−	0.877278i	$-0.340643\pi$
$757$	726.694			0.959965			0.479983	+	0.877278i	$0.340643\pi$
$758$		0			0
$759$		−	176.412i		−	0.232427i
$760$		0			0
$761$	425.877			0.559629			0.279814	−	0.960054i	$-0.409727\pi$
$761$	425.877			0.559629			0.279814	+	0.960054i	$0.409727\pi$
$762$		0			0
$763$			454.203i			0.595286i
$764$		0			0
$765$	109.460			0.143085
$766$		0			0
$767$	−1464.07			−1.90883
$768$		0			0
$769$	1060.37			1.37890			0.689448	−	0.724335i	$-0.257853\pi$
$769$	1060.37			1.37890			0.689448	+	0.724335i	$0.257853\pi$
$770$		0			0
$771$	−256.457			−0.332629
$772$		0			0
$773$			5.03627i			0.00651523i	0.999995	+	0.00325761i	$0.00103693\pi$
$773$			5.03627i			0.00651523i	−0.999995	+	0.00325761i	$0.998963\pi$
$774$		0			0
$775$		−	811.790i		−	1.04747i
$776$		0			0
$777$	48.4820			0.0623964
$778$		0			0
$779$	−488.614	+	205.006i	−0.627232	+	0.263165i
$780$		0			0
$781$		−	1321.44i		−	1.69199i
$782$		0			0
$783$	315.580			0.403039
$784$		0			0
$785$	598.552			0.762487
$786$		0			0
$787$			284.035i			0.360909i	0.983583	+	0.180454i	$0.0577568\pi$
$787$			284.035i			0.360909i	−0.983583	+	0.180454i	$0.942243\pi$
$788$		0			0
$789$		−	190.022i		−	0.240839i
$790$		0			0
$791$			433.009i			0.547419i
$792$		0			0
$793$		−	2388.02i		−	3.01138i
$794$		0			0
$795$	34.6162			0.0435424
$796$		0			0
$797$		−	262.594i		−	0.329479i	−0.986337	−	0.164739i	$-0.947322\pi$
$797$		−	262.594i		−	0.329479i	0.986337	−	0.164739i	$-0.0526783\pi$
$798$		0			0
$799$	183.899			0.230161
$800$		0			0
$801$		−	649.920i		−	0.811386i
$802$		0			0
$803$	902.106			1.12342
$804$		0			0
$805$	264.913			0.329085
$806$		0			0
$807$	−140.075			−0.173575
$808$		0			0
$809$	1302.72			1.61028			0.805142	−	0.593082i	$-0.202089\pi$
$809$	1302.72			1.61028			0.805142	+	0.593082i	$0.202089\pi$
$810$		0			0
$811$			603.977i			0.744731i	0.928086	+	0.372365i	$0.121453\pi$
$811$			603.977i			0.744731i	−0.928086	+	0.372365i	$0.878547\pi$
$812$		0			0
$813$		−	42.0803i		−	0.0517593i
$814$		0			0
$815$	144.688			0.177531
$816$		0			0
$817$	79.7514	+	190.081i	0.0976149	+	0.232657i
$818$		0			0
$819$			893.061i			1.09043i
$820$		0			0
$821$	−137.526			−0.167510			−0.0837550	−	0.996486i	$-0.526691\pi$
$821$	−137.526			−0.167510			−0.0837550	+	0.996486i	$0.526691\pi$
$822$		0			0
$823$	−18.1953			−0.0221085			−0.0110543	−	0.999939i	$-0.503519\pi$
$823$	−18.1953			−0.0221085			−0.0110543	+	0.999939i	$0.503519\pi$
$824$		0			0
$825$		−	138.918i		−	0.168386i
$826$		0			0
$827$		−	1257.34i		−	1.52036i	−0.649710	−	0.760182i	$-0.725110\pi$
$827$		−	1257.34i		−	1.52036i	0.649710	−	0.760182i	$-0.274890\pi$
$828$		0			0
$829$		−	103.574i		−	0.124939i	−0.998047	−	0.0624695i	$-0.980102\pi$
$829$		−	103.574i		−	0.124939i	0.998047	−	0.0624695i	$-0.0198976\pi$
$830$		0			0
$831$		−	117.253i		−	0.141098i
$832$		0			0
$833$	−151.419			−0.181776
$834$		0			0
$835$			310.679i			0.372070i
$836$		0			0
$837$	−374.770			−0.447754
$838$		0			0
$839$		−	343.417i		−	0.409317i	−0.978833	−	0.204659i	$-0.934392\pi$
$839$		−	343.417i		−	0.409317i	0.978833	−	0.204659i	$-0.0656084\pi$
$840$		0			0
$841$	−365.568			−0.434683
$842$		0			0
$843$	−87.1912			−0.103430
$844$		0			0
$845$	−751.139			−0.888922
$846$		0			0
$847$	−316.831			−0.374062
$848$		0			0
$849$			18.9656i			0.0223388i
$850$		0			0
$851$		−	514.735i		−	0.604859i
$852$		0			0
$853$	260.020			0.304830			0.152415	−	0.988317i	$-0.451295\pi$
$853$	260.020			0.304830			0.152415	+	0.988317i	$0.451295\pi$
$854$		0			0
$855$	−148.156	−	353.118i	−0.173282	−	0.413003i
$856$		0			0
$857$			1420.47i			1.65749i	0.559623	+	0.828747i	$0.310946\pi$
$857$			1420.47i			1.65749i	−0.559623	+	0.828747i	$0.689054\pi$
$858$		0			0
$859$	856.333			0.996895			0.498447	−	0.866920i	$-0.333904\pi$
$859$	856.333			0.996895			0.498447	+	0.866920i	$0.333904\pi$
$860$		0			0
$861$	−65.6448			−0.0762425
$862$		0			0
$863$		−	1477.83i		−	1.71243i	−0.516616	−	0.856217i	$-0.672808\pi$
$863$		−	1477.83i		−	1.71243i	0.516616	−	0.856217i	$-0.327192\pi$
$864$		0			0
$865$			208.621i			0.241180i
$866$		0			0
$867$		−	132.917i		−	0.153307i
$868$		0			0
$869$		−	512.920i		−	0.590241i
$870$		0			0
$871$	−39.7483			−0.0456352
$872$		0			0
$873$		−	1119.43i		−	1.28228i
$874$		0			0
$875$	473.620			0.541280
$876$		0			0
$877$			160.901i			0.183467i	0.995784	+	0.0917337i	$0.0292408\pi$
$877$			160.901i			0.183467i	−0.995784	+	0.0917337i	$0.970759\pi$
$878$		0			0
$879$	−194.322			−0.221072
$880$		0			0
$881$	−711.737			−0.807874			−0.403937	−	0.914787i	$-0.632359\pi$
$881$	−711.737			−0.807874			−0.403937	+	0.914787i	$0.632359\pi$
$882$		0			0
$883$	−1690.91			−1.91496			−0.957481	−	0.288496i	$-0.906845\pi$
$883$	−1690.91			−1.91496			−0.957481	+	0.288496i	$0.906845\pi$
$884$		0			0
$885$	77.7742			0.0878805
$886$		0			0
$887$			52.1528i			0.0587968i	0.999568	+	0.0293984i	$0.00935916\pi$
$887$			52.1528i			0.0587968i	−0.999568	+	0.0293984i	$0.990641\pi$
$888$		0			0
$889$		−	546.956i		−	0.615248i
$890$		0			0
$891$	1019.66			1.14441
$892$		0			0
$893$	−248.910	−	593.256i	−0.278735	−	0.664341i
$894$		0			0
$895$		−	744.982i		−	0.832383i
$896$		0			0
$897$	−284.684			−0.317373
$898$		0			0
$899$	1432.88			1.59385
$900$		0			0
$901$		−	159.126i		−	0.176611i
$902$		0			0
$903$			25.5372i			0.0282804i
$904$		0			0
$905$		−	557.467i		−	0.615986i
$906$		0			0
$907$		−	1447.89i		−	1.59635i	−0.602425	−	0.798176i	$-0.705799\pi$
$907$		−	1447.89i		−	1.59635i	0.602425	−	0.798176i	$-0.294201\pi$
$908$		0			0
$909$	−577.783			−0.635625
$910$		0			0
$911$			946.717i			1.03921i	0.854408	+	0.519603i	$0.173920\pi$
$911$			946.717i			1.03921i	−0.854408	+	0.519603i	$0.826080\pi$
$912$		0			0
$913$	−450.554			−0.493487
$914$		0			0
$915$			126.856i			0.138641i
$916$		0			0
$917$	−582.082			−0.634768
$918$		0			0
$919$	236.916			0.257797			0.128899	−	0.991658i	$-0.458856\pi$
$919$	236.916			0.257797			0.128899	+	0.991658i	$0.458856\pi$
$920$		0			0
$921$	−106.273			−0.115389
$922$		0			0
$923$	−2132.47			−2.31037
$924$		0			0
$925$		−	405.335i		−	0.438200i
$926$		0			0
$927$			599.496i			0.646706i
$928$		0			0
$929$	1000.34			1.07680			0.538398	−	0.842691i	$-0.319030\pi$
$929$	1000.34			1.07680			0.538398	+	0.842691i	$0.319030\pi$
$930$		0			0
$931$	204.948	+	488.478i	0.220138	+	0.524681i
$932$		0			0
$933$		−	128.147i		−	0.137350i
$934$		0			0
$935$	172.652			0.184655
$936$		0			0
$937$	−440.075			−0.469664			−0.234832	−	0.972036i	$-0.575454\pi$
$937$	−440.075			−0.469664			−0.234832	+	0.972036i	$0.575454\pi$
$938$		0			0
$939$			250.983i			0.267287i
$940$		0			0
$941$			1176.64i			1.25042i	0.780457	+	0.625209i	$0.214986\pi$
$941$			1176.64i			1.25042i	−0.780457	+	0.625209i	$0.785014\pi$
$942$		0			0
$943$			696.952i			0.739080i
$944$		0			0
$945$		−	96.3065i		−	0.101912i
$946$		0			0
$947$	225.120			0.237719			0.118860	−	0.992911i	$-0.462076\pi$
$947$	225.120			0.237719			0.118860	+	0.992911i	$0.462076\pi$
$948$		0			0
$949$		−	1455.77i		−	1.53400i
$950$		0			0
$951$	−99.1469			−0.104255
$952$		0			0
$953$			589.549i			0.618624i	0.950961	+	0.309312i	$0.100099\pi$
$953$			589.549i			0.618624i	−0.950961	+	0.309312i	$0.899901\pi$
$954$		0			0
$955$	266.955			0.279534
$956$		0			0
$957$	245.202			0.256219
$958$		0			0
$959$	−221.804			−0.231287
$960$		0			0
$961$	−740.629			−0.770686
$962$		0			0
$963$			1425.83i			1.48061i
$964$		0			0
$965$			641.084i			0.664335i
$966$		0			0
$967$	1641.09			1.69709			0.848545	−	0.529124i	$-0.177479\pi$
$967$	1641.09			1.69709			0.848545	+	0.529124i	$0.177479\pi$
$968$		0			0
$969$	48.7372	−	20.4485i	0.0502964	−	0.0211026i
$970$		0			0
$971$		−	721.099i		−	0.742636i	−0.928506	−	0.371318i	$-0.878906\pi$
$971$		−	721.099i		−	0.742636i	0.928506	−	0.371318i	$-0.121094\pi$
$972$		0			0
$973$	1117.28			1.14829
$974$		0			0
$975$	−224.178			−0.229926
$976$		0			0
$977$			1222.67i			1.25145i	0.780043	+	0.625725i	$0.215197\pi$
$977$			1222.67i			1.25145i	−0.780043	+	0.625725i	$0.784803\pi$
$978$		0			0
$979$		−	1025.12i		−	1.04711i
$980$		0			0
$981$		−	863.582i		−	0.880308i
$982$		0			0
$983$			1493.88i			1.51971i	0.650090	+	0.759857i	$0.274731\pi$
$983$			1493.88i			1.51971i	−0.650090	+	0.759857i	$0.725269\pi$
$984$		0			0
$985$	181.604			0.184370
$986$		0			0
$987$		−	79.7034i		−	0.0807532i
$988$		0			0
$989$	271.129			0.274144
$990$		0			0
$991$			1461.21i			1.47448i	0.675631	+	0.737240i	$0.263871\pi$
$991$			1461.21i			1.47448i	−0.675631	+	0.737240i	$0.736129\pi$
$992$		0			0
$993$	−39.6307			−0.0399101
$994$		0			0
$995$	216.050			0.217136
$996$		0			0
$997$	963.569			0.966469			0.483234	−	0.875491i	$-0.339462\pi$
$997$	963.569			0.966469			0.483234	+	0.875491i	$0.339462\pi$
$998$		0			0
$999$	−187.127			−0.187314

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1216.3.e.n.1025.5		8
4.3	odd	2		1216.3.e.m.1025.4		8
8.3	odd	2		152.3.e.b.113.5	yes	8
8.5	even	2		304.3.e.g.113.4		8
19.18	odd	2	inner	1216.3.e.n.1025.4		8
24.5	odd	2		2736.3.o.p.721.4		8
24.11	even	2		1368.3.o.b.721.4		8
76.75	even	2		1216.3.e.m.1025.5		8
152.37	odd	2		304.3.e.g.113.5		8
152.75	even	2		152.3.e.b.113.4	✓	8
456.227	odd	2		1368.3.o.b.721.3		8
456.341	even	2		2736.3.o.p.721.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.3.e.b.113.4	✓	8	152.75	even	2
152.3.e.b.113.5	yes	8	8.3	odd	2
304.3.e.g.113.4		8	8.5	even	2
304.3.e.g.113.5		8	152.37	odd	2
1216.3.e.m.1025.4		8	4.3	odd	2
1216.3.e.m.1025.5		8	76.75	even	2
1216.3.e.n.1025.4		8	19.18	odd	2	inner
1216.3.e.n.1025.5		8	1.1	even	1	trivial
1368.3.o.b.721.3		8	456.227	odd	2
1368.3.o.b.721.4		8	24.11	even	2
2736.3.o.p.721.3		8	456.341	even	2
2736.3.o.p.721.4		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)

\(f(q)\)	\(=\)	\(q+0.512197i q^{3} +2.30665 q^{5} -4.59559 q^{7} +8.73765 q^{9} +O(q^{10})\)	\(q+0.512197i q^{3} +2.30665 q^{5} -4.59559 q^{7} +8.73765 q^{9} +13.7820 q^{11} -22.2405i q^{13} +1.18146i q^{15} +5.43101 q^{17} +(-7.35095 - 17.5204i) q^{19} -2.35385i q^{21} -24.9908 q^{23} -19.6794 q^{25} +9.08517i q^{27} -34.7357i q^{29} +41.2508i q^{31} +7.05907i q^{33} -10.6004 q^{35} +20.5969i q^{37} +11.3915 q^{39} -27.8883i q^{41} -10.8491 q^{43} +20.1547 q^{45} +33.8609 q^{47} -27.8805 q^{49} +2.78174i q^{51} -29.2996i q^{53} +31.7901 q^{55} +(8.97388 - 3.76513i) q^{57} -65.8291i q^{59} +107.373 q^{61} -40.1547 q^{63} -51.3010i q^{65} -1.78720i q^{67} -12.8002i q^{69} -95.8822i q^{71} +65.4556 q^{73} -10.0797i q^{75} -63.3362 q^{77} -37.2168i q^{79} +73.9855 q^{81} -32.6916 q^{83} +12.5274 q^{85} +17.7915 q^{87} -74.3815i q^{89} +102.208i q^{91} -21.1285 q^{93} +(-16.9560 - 40.4133i) q^{95} -128.116i q^{97} +120.422 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 14 q^{5} + 6 q^{7} + 4 q^{9}+O(q^{10}) \) 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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