LMFDB - Embedded newform 760.2.d.e.609.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [760,2,Mod(609,760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(760, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("760.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 760 = 2^{3} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	760.d (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:	$6.06863055362$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 4 x^{11} + 9 x^{10} - 8 x^{9} - 11 x^{8} + 60 x^{7} - 126 x^{6} + 180 x^{5} - 99 x^{4} + \cdots + 729 $ x^12 - 4x^11 + 9x^10 - 8x^9 - 11x^8 + 60x^7 - 126x^6 + 180x^5 - 99x^4 - 216x^3 + 729x^2 - 972*x + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.11
Root			$1.35119 - 1.08364i$ of defining polynomial
Character	$\chi$	$=$	760.609
Dual form			760.2.d.e.609.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.84742i q^{3} +(0.691595 + 2.12643i) q^{5} -0.145034i q^{7} -5.10782 q^{9} -5.71774 q^{11} -5.24345i q^{13} +(-6.05484 + 1.96926i) q^{15} +7.15378i q^{17} -1.00000 q^{19} +0.412974 q^{21} +0.622762i q^{23} +(-4.04339 + 2.94125i) q^{25} -6.00186i q^{27} +5.46811 q^{29} -3.77513 q^{31} -16.2808i q^{33} +(0.308405 - 0.100305i) q^{35} -5.03553i q^{37} +14.9303 q^{39} +5.77513 q^{41} +3.32308i q^{43} +(-3.53254 - 10.8614i) q^{45} +5.85683i q^{47} +6.97897 q^{49} -20.3699 q^{51} +6.97219i q^{53} +(-3.95436 - 12.1584i) q^{55} -2.84742i q^{57} -9.09089 q^{59} -8.16707 q^{61} +0.740810i q^{63} +(11.1498 - 3.62634i) q^{65} +13.6583i q^{67} -1.77327 q^{69} +2.41484 q^{71} -7.44385i q^{73} +(-8.37499 - 11.5133i) q^{75} +0.829270i q^{77} +9.69485 q^{79} +1.76638 q^{81} +2.17116i q^{83} +(-15.2120 + 4.94752i) q^{85} +15.5700i q^{87} -3.90782 q^{89} -0.760481 q^{91} -10.7494i q^{93} +(-0.691595 - 2.12643i) q^{95} +4.98328i q^{97} +29.2052 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{5} - 16 q^{9} - 4 q^{11} - 12 q^{15} - 12 q^{19} + 36 q^{21} + 18 q^{25} + 4 q^{29} + 16 q^{31} + 6 q^{35} + 36 q^{39} + 8 q^{41} - 2 q^{45} - 4 q^{49} - 68 q^{51} - 18 q^{55} + 4 q^{59} + 20 q^{61}+ \cdots + 44 q^{99}+O(q^{100}) $ 12 * q + 6 * q^5 - 16 * q^9 - 4 * q^11 - 12 * q^15 - 12 * q^19 + 36 * q^21 + 18 * q^25 + 4 * q^29 + 16 * q^31 + 6 * q^35 + 36 * q^39 + 8 * q^41 - 2 * q^45 - 4 * q^49 - 68 * q^51 - 18 * q^55 + 4 * q^59 + 20 * q^61 + 20 * q^65 - 36 * q^69 - 16 * q^71 - 16 * q^75 + 40 * q^79 + 12 * q^81 + 6 * q^85 - 64 * q^89 + 20 * q^91 - 6 * q^95 + 44 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/760\mathbb{Z}\right)^\times$.

$n$	$191$	$381$	$401$	$457$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$			2.84742i			1.64396i	0.569516	+	0.821980i	$0.307131\pi$
$3$			2.84742i			1.64396i	−0.569516	+	0.821980i	$0.692869\pi$
$4$		0			0
$5$	0.691595	+	2.12643i	0.309291	+	0.950968i
$5$	0.691595	+	2.12643i	0.309291	+	0.950968i
$6$		0			0
$7$		−	0.145034i		−	0.0548179i	−0.999624	−	0.0274089i	$-0.991274\pi$
$7$		−	0.145034i		−	0.0548179i	0.999624	−	0.0274089i	$-0.00872563\pi$
$8$		0			0
$9$	−5.10782			−1.70261
$10$		0			0
$11$	−5.71774			−1.72396			−0.861982	−	0.506938i	$-0.830777\pi$
$11$	−5.71774			−1.72396			−0.861982	+	0.506938i	$0.830777\pi$
$12$		0			0
$13$		−	5.24345i		−	1.45427i	−0.686493	−	0.727136i	$-0.740851\pi$
$13$		−	5.24345i		−	1.45427i	0.686493	−	0.727136i	$-0.259149\pi$
$14$		0			0
$15$	−6.05484	+	1.96926i	−1.56335	+	0.508462i
$16$		0			0
$17$			7.15378i			1.73505i	0.497396	+	0.867524i	$0.334290\pi$
$17$			7.15378i			1.73505i	−0.497396	+	0.867524i	$0.665710\pi$
$18$		0			0
$19$	−1.00000			−0.229416
$19$	−1.00000			−0.229416
$20$		0			0
$21$	0.412974			0.0901184
$22$		0			0
$23$			0.622762i			0.129855i	0.997890	+	0.0649274i	$0.0206816\pi$
$23$			0.622762i			0.129855i	−0.997890	+	0.0649274i	$0.979318\pi$
$24$		0			0
$25$	−4.04339	+	2.94125i	−0.808679	+	0.588251i
$26$		0			0
$27$		−	6.00186i		−	1.15506i
$28$		0			0
$29$	5.46811			1.01540			0.507702	−	0.861533i	$-0.330495\pi$
$29$	5.46811			1.01540			0.507702	+	0.861533i	$0.330495\pi$
$30$		0			0
$31$	−3.77513			−0.678033			−0.339017	−	0.940780i	$-0.610094\pi$
$31$	−3.77513			−0.678033			−0.339017	+	0.940780i	$0.610094\pi$
$32$		0			0
$33$		−	16.2808i		−	2.83413i
$34$		0			0
$35$	0.308405	−	0.100305i	0.0521300	−	0.0169546i
$36$		0			0
$37$		−	5.03553i		−	0.827836i	−0.910314	−	0.413918i	$-0.864160\pi$
$37$		−	5.03553i		−	0.827836i	0.910314	−	0.413918i	$-0.135840\pi$
$38$		0			0
$39$	14.9303			2.39077
$40$		0			0
$41$	5.77513			0.901924			0.450962	−	0.892543i	$-0.351081\pi$
$41$	5.77513			0.901924			0.450962	+	0.892543i	$0.351081\pi$
$42$		0			0
$43$			3.32308i			0.506765i	0.967366	+	0.253382i	$0.0815431\pi$
$43$			3.32308i			0.506765i	−0.967366	+	0.253382i	$0.918457\pi$
$44$		0			0
$45$	−3.53254	−	10.8614i	−0.526600	−	1.61912i
$46$		0			0
$47$			5.85683i			0.854306i	0.904179	+	0.427153i	$0.140483\pi$
$47$			5.85683i			0.854306i	−0.904179	+	0.427153i	$0.859517\pi$
$48$		0			0
$49$	6.97897			0.996995
$50$		0			0
$51$	−20.3699			−2.85235
$52$		0			0
$53$			6.97219i			0.957704i	0.877896	+	0.478852i	$0.158947\pi$
$53$			6.97219i			0.957704i	−0.877896	+	0.478852i	$0.841053\pi$
$54$		0			0
$55$	−3.95436	−	12.1584i	−0.533206	−	1.63943i
$56$		0			0
$57$		−	2.84742i		−	0.377150i
$58$		0			0
$59$	−9.09089			−1.18353			−0.591767	−	0.806109i	$-0.701569\pi$
$59$	−9.09089			−1.18353			−0.591767	+	0.806109i	$0.701569\pi$
$60$		0			0
$61$	−8.16707			−1.04569			−0.522843	−	0.852429i	$-0.675129\pi$
$61$	−8.16707			−1.04569			−0.522843	+	0.852429i	$0.675129\pi$
$62$		0			0
$63$			0.740810i			0.0933333i
$64$		0			0
$65$	11.1498	−	3.62634i	1.38297	−	0.449793i
$66$		0			0
$67$			13.6583i			1.66863i	0.551291	+	0.834313i	$0.314135\pi$
$67$			13.6583i			1.66863i	−0.551291	+	0.834313i	$0.685865\pi$
$68$		0			0
$69$	−1.77327			−0.213476
$70$		0			0
$71$	2.41484			0.286588			0.143294	−	0.989680i	$-0.454230\pi$
$71$	2.41484			0.286588			0.143294	+	0.989680i	$0.454230\pi$
$72$		0			0
$73$		−	7.44385i		−	0.871237i	−0.900131	−	0.435619i	$-0.856530\pi$
$73$		−	7.44385i		−	0.871237i	0.900131	−	0.435619i	$-0.143470\pi$
$74$		0			0
$75$	−8.37499	−	11.5133i	−0.967061	−	1.32944i
$76$		0			0
$77$			0.829270i			0.0945041i
$78$		0			0
$79$	9.69485			1.09076			0.545378	−	0.838190i	$-0.316386\pi$
$79$	9.69485			1.09076			0.545378	+	0.838190i	$0.316386\pi$
$80$		0			0
$81$	1.76638			0.196264
$82$		0			0
$83$			2.17116i			0.238316i	0.992875	+	0.119158i	$0.0380194\pi$
$83$			2.17116i			0.238316i	−0.992875	+	0.119158i	$0.961981\pi$
$84$		0			0
$85$	−15.2120	+	4.94752i	−1.64997	+	0.536634i
$86$		0			0
$87$			15.5700i			1.66928i
$88$		0			0
$89$	−3.90782			−0.414228			−0.207114	−	0.978317i	$-0.566407\pi$
$89$	−3.90782			−0.414228			−0.207114	+	0.978317i	$0.566407\pi$
$90$		0			0
$91$	−0.760481			−0.0797201
$92$		0			0
$93$		−	10.7494i		−	1.11466i
$94$		0			0
$95$	−0.691595	−	2.12643i	−0.0709561	−	0.218167i
$96$		0			0
$97$			4.98328i			0.505976i	0.967469	+	0.252988i	$0.0814132\pi$
$97$			4.98328i			0.505976i	−0.967469	+	0.252988i	$0.918587\pi$
$98$		0			0
$99$	29.2052			2.93524
$100$		0			0
$101$	−5.35148			−0.532493			−0.266246	−	0.963905i	$-0.585783\pi$
$101$	−5.35148			−0.532493			−0.266246	+	0.963905i	$0.585783\pi$
$102$		0			0
$103$			3.59495i			0.354221i	0.984191	+	0.177111i	$0.0566750\pi$
$103$			3.59495i			0.354221i	−0.984191	+	0.177111i	$0.943325\pi$
$104$		0			0
$105$	0.285611	+	0.878160i	0.0278728	+	0.0856997i
$106$		0			0
$107$		−	15.3518i		−	1.48412i	−0.670335	−	0.742059i	$-0.733850\pi$
$107$		−	15.3518i		−	1.48412i	0.670335	−	0.742059i	$-0.266150\pi$
$108$		0			0
$109$	−2.20105			−0.210823			−0.105411	−	0.994429i	$-0.533616\pi$
$109$	−2.20105			−0.210823			−0.105411	+	0.994429i	$0.533616\pi$
$110$		0			0
$111$	14.3383			1.36093
$112$		0			0
$113$			12.3613i			1.16286i	0.813598	+	0.581428i	$0.197506\pi$
$113$			12.3613i			1.16286i	−0.813598	+	0.581428i	$0.802494\pi$
$114$		0			0
$115$	−1.32426	+	0.430699i	−0.123488	+	0.0401629i
$116$		0			0
$117$			26.7826i			2.47605i
$118$		0			0
$119$	1.03754			0.0951116
$120$		0			0
$121$	21.6926			1.97205
$122$		0			0
$123$			16.4442i			1.48273i
$124$		0			0
$125$	−9.05075	−	6.56383i	−0.809524	−	0.587087i
$126$		0			0
$127$			18.0907i			1.60529i	0.596459	+	0.802644i	$0.296574\pi$
$127$			18.0907i			1.60529i	−0.596459	+	0.802644i	$0.703426\pi$
$128$		0			0
$129$	−9.46222			−0.833102
$130$		0			0
$131$	12.1244			1.05931			0.529655	−	0.848213i	$-0.322321\pi$
$131$	12.1244			1.05931			0.529655	+	0.848213i	$0.322321\pi$
$132$		0			0
$133$			0.145034i			0.0125761i
$134$		0			0
$135$	12.7625	−	4.15086i	1.09842	−	0.357249i
$136$		0			0
$137$		−	20.0385i		−	1.71201i	−0.516969	−	0.856004i	$-0.672940\pi$
$137$		−	20.0385i		−	1.71201i	0.516969	−	0.856004i	$-0.327060\pi$
$138$		0			0
$139$	−11.3996			−0.966905			−0.483452	−	0.875371i	$-0.660617\pi$
$139$	−11.3996			−0.966905			−0.483452	+	0.875371i	$0.660617\pi$
$140$		0			0
$141$	−16.6769			−1.40445
$142$		0			0
$143$			29.9807i			2.50711i
$144$		0			0
$145$	3.78172	+	11.6276i	0.314055	+	0.965616i
$146$		0			0
$147$			19.8721i			1.63902i
$148$		0			0
$149$	15.5818			1.27651			0.638257	−	0.769823i	$-0.279656\pi$
$149$	15.5818			1.27651			0.638257	+	0.769823i	$0.279656\pi$
$150$		0			0
$151$	22.3717			1.82058			0.910292	−	0.413966i	$-0.135857\pi$
$151$	22.3717			1.82058			0.910292	+	0.413966i	$0.135857\pi$
$152$		0			0
$153$		−	36.5403i		−	2.95410i
$154$		0			0
$155$	−2.61086	−	8.02754i	−0.209709	−	0.644788i
$156$		0			0
$157$			23.1047i			1.84395i	0.387246	+	0.921976i	$0.373426\pi$
$157$			23.1047i			1.84395i	−0.387246	+	0.921976i	$0.626574\pi$
$158$		0			0
$159$	−19.8528			−1.57443
$160$		0			0
$161$	0.0903219			0.00711836
$162$		0			0
$163$		−	11.3524i		−	0.889186i	−0.895733	−	0.444593i	$-0.853348\pi$
$163$		−	11.3524i		−	0.889186i	0.895733	−	0.444593i	$-0.146652\pi$
$164$		0			0
$165$	34.6200	−	11.2597i	2.69517	−	0.876570i
$166$		0			0
$167$			6.09989i			0.472024i	0.971750	+	0.236012i	$0.0758405\pi$
$167$			6.09989i			0.472024i	−0.971750	+	0.236012i	$0.924160\pi$
$168$		0			0
$169$	−14.4938			−1.11491
$170$		0			0
$171$	5.10782			0.390605
$172$		0			0
$173$			10.8951i			0.828338i	0.910200	+	0.414169i	$0.135928\pi$
$173$			10.8951i			0.828338i	−0.910200	+	0.414169i	$0.864072\pi$
$174$		0			0
$175$	0.426583	+	0.586431i	0.0322466	+	0.0443300i
$176$		0			0
$177$		−	25.8856i		−	1.94568i
$178$		0			0
$179$	−4.46123			−0.333448			−0.166724	−	0.986004i	$-0.553319\pi$
$179$	−4.46123			−0.333448			−0.166724	+	0.986004i	$0.553319\pi$
$180$		0			0
$181$	3.58784			0.266682			0.133341	−	0.991070i	$-0.457429\pi$
$181$	3.58784			0.266682			0.133341	+	0.991070i	$0.457429\pi$
$182$		0			0
$183$		−	23.2551i		−	1.71907i
$184$		0			0
$185$	10.7077	−	3.48254i	0.787245	−	0.256042i
$186$		0			0
$187$		−	40.9035i		−	2.99116i
$188$		0			0
$189$	−0.870477			−0.0633179
$190$		0			0
$191$	8.00777			0.579422			0.289711	−	0.957114i	$-0.406441\pi$
$191$	8.00777			0.579422			0.289711	+	0.957114i	$0.406441\pi$
$192$		0			0
$193$			1.45641i			0.104835i	0.998625	+	0.0524173i	$0.0166926\pi$
$193$			1.45641i			0.104835i	−0.998625	+	0.0524173i	$0.983307\pi$
$194$		0			0
$195$	10.3257	+	31.7483i	0.739441	+	2.27354i
$196$		0			0
$197$			26.3510i			1.87743i	0.344692	+	0.938716i	$0.387983\pi$
$197$			26.3510i			1.87743i	−0.344692	+	0.938716i	$0.612017\pi$
$198$		0			0
$199$	−12.1739			−0.862985			−0.431492	−	0.902117i	$-0.642013\pi$
$199$	−12.1739			−0.862985			−0.431492	+	0.902117i	$0.642013\pi$
$200$		0			0
$201$	−38.8909			−2.74316
$202$		0			0
$203$		−	0.793065i		−	0.0556622i
$204$		0			0
$205$	3.99405	+	12.2804i	0.278956	+	0.857700i
$206$		0			0
$207$		−	3.18096i		−	0.221092i
$208$		0			0
$209$	5.71774			0.395505
$210$		0			0
$211$	−19.5088			−1.34304			−0.671521	−	0.740985i	$-0.734359\pi$
$211$	−19.5088			−1.34304			−0.671521	+	0.740985i	$0.734359\pi$
$212$		0			0
$213$			6.87606i			0.471140i
$214$		0			0
$215$	−7.06629	+	2.29822i	−0.481917	+	0.156738i
$216$		0			0
$217$			0.547524i			0.0371683i
$218$		0			0
$219$	21.1958			1.43228
$220$		0			0
$221$	37.5105			2.52323
$222$		0			0
$223$		−	14.9208i		−	0.999168i	−0.866265	−	0.499584i	$-0.833486\pi$
$223$		−	14.9208i		−	0.999168i	0.866265	−	0.499584i	$-0.166514\pi$
$224$		0			0
$225$	20.6529	−	15.0234i	1.37686	−	1.00156i
$226$		0			0
$227$			13.4422i			0.892192i	0.894985	+	0.446096i	$0.147186\pi$
$227$			13.4422i			0.892192i	−0.894985	+	0.446096i	$0.852814\pi$
$228$		0			0
$229$	7.36620			0.486772			0.243386	−	0.969929i	$-0.421742\pi$
$229$	7.36620			0.486772			0.243386	+	0.969929i	$0.421742\pi$
$230$		0			0
$231$	−2.36128			−0.155361
$232$		0			0
$233$		−	4.24405i		−	0.278037i	−0.990290	−	0.139018i	$-0.955605\pi$
$233$		−	4.24405i		−	0.278037i	0.990290	−	0.139018i	$-0.0443947\pi$
$234$		0			0
$235$	−12.4541	+	4.05055i	−0.812417	+	0.264229i
$236$		0			0
$237$			27.6053i			1.79316i
$238$		0			0
$239$	15.3387			0.992176			0.496088	−	0.868272i	$-0.334769\pi$
$239$	15.3387			0.992176			0.496088	+	0.868272i	$0.334769\pi$
$240$		0			0
$241$	26.7026			1.72006			0.860032	−	0.510240i	$-0.170443\pi$
$241$	26.7026			1.72006			0.860032	+	0.510240i	$0.170443\pi$
$242$		0			0
$243$		−	12.9760i		−	0.832408i
$244$		0			0
$245$	4.82662	+	14.8403i	0.308361	+	0.948110i
$246$		0			0
$247$			5.24345i			0.333633i
$248$		0			0
$249$	−6.18221			−0.391781
$250$		0			0
$251$	3.64463			0.230047			0.115023	−	0.993363i	$-0.463306\pi$
$251$	3.64463			0.230047			0.115023	+	0.993363i	$0.463306\pi$
$252$		0			0
$253$		−	3.56079i		−	0.223865i
$254$		0			0
$255$	−14.0877	−	43.3150i	−0.882205	−	2.71249i
$256$		0			0
$257$			9.71103i			0.605757i	0.953029	+	0.302879i	$0.0979477\pi$
$257$			9.71103i			0.605757i	−0.953029	+	0.302879i	$0.902052\pi$
$258$		0			0
$259$	−0.730325			−0.0453802
$260$		0			0
$261$	−27.9302			−1.72883
$262$		0			0
$263$			1.82052i			0.112258i	0.998424	+	0.0561290i	$0.0178758\pi$
$263$			1.82052i			0.112258i	−0.998424	+	0.0561290i	$0.982124\pi$
$264$		0			0
$265$	−14.8259	+	4.82193i	−0.910746	+	0.296209i
$266$		0			0
$267$		−	11.1272i		−	0.680974i
$268$		0			0
$269$	−21.6353			−1.31913			−0.659563	−	0.751649i	$-0.729259\pi$
$269$	−21.6353			−1.31913			−0.659563	+	0.751649i	$0.729259\pi$
$270$		0			0
$271$	−7.54436			−0.458287			−0.229144	−	0.973393i	$-0.573593\pi$
$271$	−7.54436			−0.458287			−0.229144	+	0.973393i	$0.573593\pi$
$272$		0			0
$273$		−	2.16541i		−	0.131057i
$274$		0			0
$275$	23.1191	−	16.8173i	1.39413	−	1.01412i
$276$		0			0
$277$		−	12.5559i		−	0.754410i	−0.926130	−	0.377205i	$-0.876885\pi$
$277$		−	12.5559i		−	0.754410i	0.926130	−	0.377205i	$-0.123115\pi$
$278$		0			0
$279$	19.2827			1.15442
$280$		0			0
$281$	22.1947			1.32403			0.662013	−	0.749493i	$-0.269703\pi$
$281$	22.1947			1.32403			0.662013	+	0.749493i	$0.269703\pi$
$282$		0			0
$283$		−	20.6171i		−	1.22556i	−0.790254	−	0.612779i	$-0.790052\pi$
$283$		−	20.6171i		−	1.22556i	0.790254	−	0.612779i	$-0.209948\pi$
$284$		0			0
$285$	6.05484	−	1.96926i	0.358658	−	0.116649i
$286$		0			0
$287$		−	0.837592i		−	0.0494415i
$288$		0			0
$289$	−34.1766			−2.01039
$290$		0			0
$291$	−14.1895			−0.831804
$292$		0			0
$293$		−	22.1421i		−	1.29356i	−0.762679	−	0.646778i	$-0.776116\pi$
$293$		−	22.1421i		−	1.29356i	0.762679	−	0.646778i	$-0.223884\pi$
$294$		0			0
$295$	−6.28721	−	19.3311i	−0.366056	−	1.12550i
$296$		0			0
$297$			34.3171i			1.99128i
$298$		0			0
$299$	3.26542			0.188844
$300$		0			0
$301$	0.481961			0.0277798
$302$		0			0
$303$		−	15.2379i		−	0.875397i
$304$		0			0
$305$	−5.64830	−	17.3667i	−0.323421	−	0.994413i
$306$		0			0
$307$		−	1.80060i		−	0.102766i	−0.998679	−	0.0513828i	$-0.983637\pi$
$307$		−	1.80060i		−	0.102766i	0.998679	−	0.0513828i	$-0.0163629\pi$
$308$		0			0
$309$	−10.2364			−0.582326
$310$		0			0
$311$	−9.26554			−0.525401			−0.262700	−	0.964877i	$-0.584613\pi$
$311$	−9.26554			−0.525401			−0.262700	+	0.964877i	$0.584613\pi$
$312$		0			0
$313$			4.29779i			0.242925i	0.992596	+	0.121463i	$0.0387585\pi$
$313$			4.29779i			0.242925i	−0.992596	+	0.121463i	$0.961242\pi$
$314$		0			0
$315$	−1.57528	+	0.512340i	−0.0887569	+	0.0288671i
$316$		0			0
$317$		−	4.61065i		−	0.258960i	−0.991582	−	0.129480i	$-0.958669\pi$
$317$		−	4.61065i		−	0.258960i	0.991582	−	0.129480i	$-0.0413308\pi$
$318$		0			0
$319$	−31.2653			−1.75052
$320$		0			0
$321$	43.7132			2.43983
$322$		0			0
$323$		−	7.15378i		−	0.398047i
$324$		0			0
$325$	15.4223	+	21.2013i	0.855476	+	1.17604i
$326$		0			0
$327$		−	6.26734i		−	0.346584i
$328$		0			0
$329$	0.849442			0.0468312
$330$		0			0
$331$	−0.934366			−0.0513574			−0.0256787	−	0.999670i	$-0.508175\pi$
$331$	−0.934366			−0.0513574			−0.0256787	+	0.999670i	$0.508175\pi$
$332$		0			0
$333$			25.7206i			1.40948i
$334$		0			0
$335$	−29.0434	+	9.44600i	−1.58681	+	0.516090i
$336$		0			0
$337$			29.8588i			1.62651i	0.581905	+	0.813256i	$0.302307\pi$
$337$			29.8588i			1.62651i	−0.581905	+	0.813256i	$0.697693\pi$
$338$		0			0
$339$	−35.1980			−1.91169
$340$		0			0
$341$	21.5852			1.16891
$342$		0			0
$343$		−	2.02743i		−	0.109471i
$344$		0			0
$345$	−1.22638	−	3.77072i	−0.0660262	−	0.203009i
$346$		0			0
$347$			10.6502i			0.571732i	0.958270	+	0.285866i	$0.0922813\pi$
$347$			10.6502i			0.571732i	−0.958270	+	0.285866i	$0.907719\pi$
$348$		0			0
$349$	−29.3479			−1.57096			−0.785480	−	0.618888i	$-0.787584\pi$
$349$	−29.3479			−1.57096			−0.785480	+	0.618888i	$0.787584\pi$
$350$		0			0
$351$	−31.4705			−1.67977
$352$		0			0
$353$			14.3847i			0.765619i	0.923827	+	0.382810i	$0.125043\pi$
$353$			14.3847i			0.765619i	−0.923827	+	0.382810i	$0.874957\pi$
$354$		0			0
$355$	1.67009	+	5.13498i	0.0886391	+	0.272536i
$356$		0			0
$357$			2.95433i			0.156360i
$358$		0			0
$359$	2.23773			0.118103			0.0590514	−	0.998255i	$-0.481192\pi$
$359$	2.23773			0.118103			0.0590514	+	0.998255i	$0.481192\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$			61.7680i			3.24198i
$364$		0			0
$365$	15.8288	−	5.14813i	0.828518	−	0.269465i
$366$		0			0
$367$		−	0.636873i		−	0.0332445i	−0.999862	−	0.0166222i	$-0.994709\pi$
$367$		−	0.636873i		−	0.0332445i	0.999862	−	0.0166222i	$-0.00529127\pi$
$368$		0			0
$369$	−29.4983			−1.53562
$370$		0			0
$371$	1.01121			0.0524993
$372$		0			0
$373$		−	16.2358i		−	0.840655i	−0.907372	−	0.420328i	$-0.861915\pi$
$373$		−	16.2358i		−	0.840655i	0.907372	−	0.420328i	$-0.138085\pi$
$374$		0			0
$375$	18.6900	−	25.7713i	0.965148	−	1.33083i
$376$		0			0
$377$		−	28.6718i		−	1.47667i
$378$		0			0
$379$	17.9624			0.922667			0.461333	−	0.887227i	$-0.347371\pi$
$379$	17.9624			0.922667			0.461333	+	0.887227i	$0.347371\pi$
$380$		0			0
$381$	−51.5118			−2.63903
$382$		0			0
$383$		−	4.06305i		−	0.207612i	−0.994598	−	0.103806i	$-0.966898\pi$
$383$		−	4.06305i		−	0.207612i	0.994598	−	0.103806i	$-0.0331021\pi$
$384$		0			0
$385$	−1.76338	+	0.573519i	−0.0898703	+	0.0292292i
$386$		0			0
$387$		−	16.9737i		−	0.862822i
$388$		0			0
$389$	28.7507			1.45772			0.728859	−	0.684663i	$-0.240051\pi$
$389$	28.7507			1.45772			0.728859	+	0.684663i	$0.240051\pi$
$390$		0			0
$391$	−4.45510			−0.225304
$392$		0			0
$393$			34.5232i			1.74147i
$394$		0			0
$395$	6.70491	+	20.6154i	0.337360	+	1.03727i
$396$		0			0
$397$		−	11.6518i		−	0.584786i	−0.956298	−	0.292393i	$-0.905548\pi$
$397$		−	11.6518i		−	0.584786i	0.956298	−	0.292393i	$-0.0944515\pi$
$398$		0			0
$399$	−0.412974			−0.0206746
$400$		0			0
$401$	1.98190			0.0989716			0.0494858	−	0.998775i	$-0.484242\pi$
$401$	1.98190			0.0989716			0.0494858	+	0.998775i	$0.484242\pi$
$402$		0			0
$403$			19.7947i			0.986045i
$404$		0			0
$405$	1.22162	+	3.75608i	0.0607027	+	0.186641i
$406$		0			0
$407$			28.7919i			1.42716i
$408$		0			0
$409$	−26.3984			−1.30532			−0.652660	−	0.757651i	$-0.726347\pi$
$409$	−26.3984			−1.30532			−0.652660	+	0.757651i	$0.726347\pi$
$410$		0			0
$411$	57.0582			2.81447
$412$		0			0
$413$			1.31849i			0.0648788i
$414$		0			0
$415$	−4.61681	+	1.50156i	−0.226630	+	0.0737087i
$416$		0			0
$417$		−	32.4596i		−	1.58955i
$418$		0			0
$419$	16.3134			0.796963			0.398481	−	0.917176i	$-0.369537\pi$
$419$	16.3134			0.796963			0.398481	+	0.917176i	$0.369537\pi$
$420$		0			0
$421$	−1.90815			−0.0929976			−0.0464988	−	0.998918i	$-0.514806\pi$
$421$	−1.90815			−0.0929976			−0.0464988	+	0.998918i	$0.514806\pi$
$422$		0			0
$423$		−	29.9156i		−	1.45455i
$424$		0			0
$425$	−21.0411	−	28.9256i	−1.02064	−	1.40310i
$426$		0			0
$427$			1.18451i			0.0573223i
$428$		0			0
$429$	−85.3678			−4.12160
$430$		0			0
$431$	13.9013			0.669600			0.334800	−	0.942289i	$-0.391331\pi$
$431$	13.9013			0.669600			0.334800	+	0.942289i	$0.391331\pi$
$432$		0			0
$433$			4.93885i			0.237346i	0.992933	+	0.118673i	$0.0378640\pi$
$433$			4.93885i			0.237346i	−0.992933	+	0.118673i	$0.962136\pi$
$434$		0			0
$435$	−33.1086	+	10.7682i	−1.58743	+	0.516294i
$436$		0			0
$437$		−	0.622762i		−	0.0297907i
$438$		0			0
$439$	18.0918			0.863473			0.431736	−	0.902000i	$-0.357901\pi$
$439$	18.0918			0.863473			0.431736	+	0.902000i	$0.357901\pi$
$440$		0			0
$441$	−35.6473			−1.69749
$442$		0			0
$443$		−	8.13690i		−	0.386596i	−0.981140	−	0.193298i	$-0.938082\pi$
$443$		−	8.13690i		−	0.386596i	0.981140	−	0.193298i	$-0.0619184\pi$
$444$		0			0
$445$	−2.70262	−	8.30969i	−0.128117	−	0.393917i
$446$		0			0
$447$			44.3681i			2.09854i
$448$		0			0
$449$	−19.9187			−0.940020			−0.470010	−	0.882661i	$-0.655750\pi$
$449$	−19.9187			−0.940020			−0.470010	+	0.882661i	$0.655750\pi$
$450$		0			0
$451$	−33.0207			−1.55488
$452$		0			0
$453$			63.7018i			2.99297i
$454$		0			0
$455$	−0.525945	−	1.61711i	−0.0246567	−	0.0758112i
$456$		0			0
$457$			21.4434i			1.00308i	0.865134	+	0.501540i	$0.167233\pi$
$457$			21.4434i			1.00308i	−0.865134	+	0.501540i	$0.832767\pi$
$458$		0			0
$459$	42.9360			2.00408
$460$		0			0
$461$	11.2325			0.523148			0.261574	−	0.965183i	$-0.415758\pi$
$461$	11.2325			0.523148			0.261574	+	0.965183i	$0.415758\pi$
$462$		0			0
$463$		−	37.8262i		−	1.75793i	−0.476882	−	0.878967i	$-0.658233\pi$
$463$		−	37.8262i		−	1.75793i	0.476882	−	0.878967i	$-0.341767\pi$
$464$		0			0
$465$	22.8578	−	7.43422i	1.06001	−	0.344754i
$466$		0			0
$467$		−	30.3388i		−	1.40391i	−0.712221	−	0.701955i	$-0.752311\pi$
$467$		−	30.3388i		−	1.40391i	0.712221	−	0.701955i	$-0.247689\pi$
$468$		0			0
$469$	1.98092			0.0914705
$470$		0			0
$471$	−65.7888			−3.03139
$472$		0			0
$473$		−	19.0005i		−	0.873645i
$474$		0			0
$475$	4.04339	−	2.94125i	0.185524	−	0.134954i
$476$		0			0
$477$		−	35.6127i		−	1.63059i
$478$		0			0
$479$	35.3121			1.61345			0.806725	−	0.590927i	$-0.201238\pi$
$479$	35.3121			1.61345			0.806725	+	0.590927i	$0.201238\pi$
$480$		0			0
$481$	−26.4035			−1.20390
$482$		0			0
$483$			0.257185i			0.0117023i
$484$		0			0
$485$	−10.5966	+	3.44641i	−0.481166	+	0.156493i
$486$		0			0
$487$			6.88482i			0.311981i	0.987759	+	0.155991i	$0.0498569\pi$
$487$			6.88482i			0.311981i	−0.987759	+	0.155991i	$0.950143\pi$
$488$		0			0
$489$	32.3250			1.46179
$490$		0			0
$491$	−10.1116			−0.456328			−0.228164	−	0.973623i	$-0.573272\pi$
$491$	−10.1116			−0.456328			−0.228164	+	0.973623i	$0.573272\pi$
$492$		0			0
$493$			39.1177i			1.76177i
$494$		0			0
$495$	20.1982	+	62.1028i	0.907841	+	2.79131i
$496$		0			0
$497$		−	0.350234i		−	0.0157102i
$498$		0			0
$499$	36.4594			1.63215			0.816074	−	0.577947i	$-0.196146\pi$
$499$	36.4594			1.63215			0.816074	+	0.577947i	$0.196146\pi$
$500$		0			0
$501$	−17.3690			−0.775989
$502$		0			0
$503$		−	9.00269i		−	0.401410i	−0.979652	−	0.200705i	$-0.935677\pi$
$503$		−	9.00269i		−	0.401410i	0.979652	−	0.200705i	$-0.0643233\pi$
$504$		0			0
$505$	−3.70106	−	11.3795i	−0.164695	−	0.506383i
$506$		0			0
$507$		−	41.2700i		−	1.83286i
$508$		0			0
$509$	−10.3542			−0.458943			−0.229471	−	0.973315i	$-0.573700\pi$
$509$	−10.3542			−0.458943			−0.229471	+	0.973315i	$0.573700\pi$
$510$		0			0
$511$	−1.07961			−0.0477594
$512$		0			0
$513$			6.00186i			0.264989i
$514$		0			0
$515$	−7.64441	+	2.48625i	−0.336853	+	0.109557i
$516$		0			0
$517$		−	33.4878i		−	1.47279i
$518$		0			0
$519$	−31.0229			−1.36175
$520$		0			0
$521$	11.5209			0.504738			0.252369	−	0.967631i	$-0.418790\pi$
$521$	11.5209			0.504738			0.252369	+	0.967631i	$0.418790\pi$
$522$		0			0
$523$		−	24.0131i		−	1.05002i	−0.851097	−	0.525009i	$-0.824062\pi$
$523$		−	24.0131i		−	1.05002i	0.851097	−	0.525009i	$-0.175938\pi$
$524$		0			0
$525$	−1.66982	+	1.21466i	−0.0728768	+	0.0530122i
$526$		0			0
$527$		−	27.0065i		−	1.17642i
$528$		0			0
$529$	22.6122			0.983138
$530$		0			0
$531$	46.4347			2.01509
$532$		0			0
$533$		−	30.2816i		−	1.31164i
$534$		0			0
$535$	32.6446	−	10.6172i	1.41135	−	0.459023i
$536$		0			0
$537$		−	12.7030i		−	0.548175i
$538$		0			0
$539$	−39.9039			−1.71878
$540$		0			0
$541$	−34.5131			−1.48383			−0.741916	−	0.670492i	$-0.766083\pi$
$541$	−34.5131			−1.48383			−0.741916	+	0.670492i	$0.766083\pi$
$542$		0			0
$543$			10.2161i			0.438415i
$544$		0			0
$545$	−1.52224	−	4.68039i	−0.0652055	−	0.200486i
$546$		0			0
$547$		−	3.89383i		−	0.166488i	−0.996529	−	0.0832441i	$-0.973472\pi$
$547$		−	3.89383i		−	0.166488i	0.996529	−	0.0832441i	$-0.0265281\pi$
$548$		0			0
$549$	41.7159			1.78039
$550$		0			0
$551$	−5.46811			−0.232949
$552$		0			0
$553$		−	1.40609i		−	0.0597929i
$554$		0			0
$555$	9.91628	+	30.4893i	0.420923	+	1.29420i
$556$		0			0
$557$		−	11.1816i		−	0.473779i	−0.971537	−	0.236889i	$-0.923872\pi$
$557$		−	11.1816i		−	0.473779i	0.971537	−	0.236889i	$-0.0761279\pi$
$558$		0			0
$559$	17.4244			0.736974
$560$		0			0
$561$	116.470			4.91735
$562$		0			0
$563$		−	2.54455i		−	0.107240i	−0.998561	−	0.0536200i	$-0.982924\pi$
$563$		−	2.54455i		−	0.107240i	0.998561	−	0.0536200i	$-0.0170760\pi$
$564$		0			0
$565$	−26.2855	+	8.54903i	−1.10584	+	0.359660i
$566$		0			0
$567$		−	0.256186i		−	0.0107588i
$568$		0			0
$569$	−2.09511			−0.0878314			−0.0439157	−	0.999035i	$-0.513983\pi$
$569$	−2.09511			−0.0878314			−0.0439157	+	0.999035i	$0.513983\pi$
$570$		0			0
$571$	36.2358			1.51642			0.758211	−	0.652009i	$-0.226074\pi$
$571$	36.2358			1.51642			0.758211	+	0.652009i	$0.226074\pi$
$572$		0			0
$573$			22.8015i			0.952547i
$574$		0			0
$575$	−1.83170	−	2.51807i	−0.0763872	−	0.105011i
$576$		0			0
$577$		−	15.5756i		−	0.648419i	−0.945985	−	0.324209i	$-0.894902\pi$
$577$		−	15.5756i		−	0.648419i	0.945985	−	0.324209i	$-0.105098\pi$
$578$		0			0
$579$	−4.14702			−0.172344
$580$		0			0
$581$	0.314893			0.0130639
$582$		0			0
$583$		−	39.8652i		−	1.65105i
$584$		0			0
$585$	−56.9513	+	18.5227i	−2.35465	+	0.765820i
$586$		0			0
$587$		−	8.54894i		−	0.352853i	−0.984314	−	0.176426i	$-0.943546\pi$
$587$		−	8.54894i		−	0.352853i	0.984314	−	0.176426i	$-0.0564537\pi$
$588$		0			0
$589$	3.77513			0.155551
$590$		0			0
$591$	−75.0325			−3.08642
$592$		0			0
$593$			25.1005i			1.03075i	0.856964	+	0.515377i	$0.172348\pi$
$593$			25.1005i			1.03075i	−0.856964	+	0.515377i	$0.827652\pi$
$594$		0			0
$595$	0.717561	+	2.20626i	0.0294171	+	0.0904480i
$596$		0			0
$597$		−	34.6642i		−	1.41871i
$598$		0			0
$599$	24.9528			1.01954			0.509772	−	0.860310i	$-0.329730\pi$
$599$	24.9528			1.01954			0.509772	+	0.860310i	$0.329730\pi$
$600$		0			0
$601$	11.9744			0.488445			0.244223	−	0.969719i	$-0.421467\pi$
$601$	11.9744			0.488445			0.244223	+	0.969719i	$0.421467\pi$
$602$		0			0
$603$		−	69.7641i		−	2.84101i
$604$		0			0
$605$	15.0025	+	46.1278i	0.609938	+	1.87536i
$606$		0			0
$607$		−	6.89412i		−	0.279824i	−0.990164	−	0.139912i	$-0.955318\pi$
$607$		−	6.89412i		−	0.279824i	0.990164	−	0.139912i	$-0.0446819\pi$
$608$		0			0
$609$	2.25819			0.0915065
$610$		0			0
$611$	30.7100			1.24239
$612$		0			0
$613$		−	25.8193i		−	1.04283i	−0.853303	−	0.521416i	$-0.825404\pi$
$613$		−	25.8193i		−	1.04283i	0.853303	−	0.521416i	$-0.174596\pi$
$614$		0			0
$615$	−34.9675	+	11.3727i	−1.41003	+	0.458594i
$616$		0			0
$617$		−	15.7584i		−	0.634408i	−0.948357	−	0.317204i	$-0.897256\pi$
$617$		−	15.7584i		−	0.634408i	0.948357	−	0.317204i	$-0.102744\pi$
$618$		0			0
$619$	25.9111			1.04145			0.520727	−	0.853723i	$-0.325661\pi$
$619$	25.9111			1.04145			0.520727	+	0.853723i	$0.325661\pi$
$620$		0			0
$621$	3.73773			0.149990
$622$		0			0
$623$			0.566768i			0.0227071i
$624$		0			0
$625$	7.69806	−	23.7853i	0.307922	−	0.951411i
$626$		0			0
$627$			16.2808i			0.650194i
$628$		0			0
$629$	36.0231			1.43633
$630$		0			0
$631$	−6.86287			−0.273207			−0.136603	−	0.990626i	$-0.543619\pi$
$631$	−6.86287			−0.273207			−0.136603	+	0.990626i	$0.543619\pi$
$632$		0			0
$633$		−	55.5499i		−	2.20791i
$634$		0			0
$635$	−38.4685	+	12.5114i	−1.52658	+	0.496500i
$636$		0			0
$637$		−	36.5939i		−	1.44990i
$638$		0			0
$639$	−12.3346			−0.487947
$640$		0			0
$641$	−35.4136			−1.39875			−0.699376	−	0.714754i	$-0.746539\pi$
$641$	−35.4136			−1.39875			−0.699376	+	0.714754i	$0.746539\pi$
$642$		0			0
$643$		−	27.6618i		−	1.09088i	−0.838151	−	0.545438i	$-0.816363\pi$
$643$		−	27.6618i		−	1.09088i	0.838151	−	0.545438i	$-0.183637\pi$
$644$		0			0
$645$	−6.54402	−	20.1207i	−0.257670	−	0.792253i
$646$		0			0
$647$			29.0871i			1.14353i	0.820417	+	0.571765i	$0.193741\pi$
$647$			29.0871i			1.14353i	−0.820417	+	0.571765i	$0.806259\pi$
$648$		0			0
$649$	51.9794			2.04037
$650$		0			0
$651$	−1.55903			−0.0611033
$652$		0			0
$653$			22.7798i			0.891444i	0.895171	+	0.445722i	$0.147053\pi$
$653$			22.7798i			0.891444i	−0.895171	+	0.445722i	$0.852947\pi$
$654$		0			0
$655$	8.38515	+	25.7816i	0.327635	+	1.00737i
$656$		0			0
$657$			38.0219i			1.48337i
$658$		0			0
$659$	13.5698			0.528605			0.264302	−	0.964440i	$-0.414858\pi$
$659$	13.5698			0.528605			0.264302	+	0.964440i	$0.414858\pi$
$660$		0			0
$661$	47.4318			1.84489			0.922443	−	0.386134i	$-0.126190\pi$
$661$	47.4318			1.84489			0.922443	+	0.386134i	$0.126190\pi$
$662$		0			0
$663$			106.808i			4.14809i
$664$		0			0
$665$	−0.308405	+	0.100305i	−0.0119594	+	0.00388966i
$666$		0			0
$667$			3.40533i			0.131855i
$668$		0			0
$669$	42.4857			1.64259
$670$		0			0
$671$	46.6972			1.80273
$672$		0			0
$673$			4.71398i			0.181711i	0.995864	+	0.0908553i	$0.0289601\pi$
$673$			4.71398i			0.181711i	−0.995864	+	0.0908553i	$0.971040\pi$
$674$		0			0
$675$	17.6530	+	24.2679i	0.679464	+	0.934072i
$676$		0			0
$677$			42.5524i			1.63542i	0.575629	+	0.817711i	$0.304757\pi$
$677$			42.5524i			1.63542i	−0.575629	+	0.817711i	$0.695243\pi$
$678$		0			0
$679$	0.722747			0.0277365
$680$		0			0
$681$	−38.2757			−1.46673
$682$		0			0
$683$			34.6186i			1.32464i	0.749220	+	0.662321i	$0.230429\pi$
$683$			34.6186i			1.32464i	−0.749220	+	0.662321i	$0.769571\pi$
$684$		0			0
$685$	42.6105	−	13.8585i	1.62806	−	0.529508i
$686$		0			0
$687$			20.9747i			0.800235i
$688$		0			0
$689$	36.5584			1.39276
$690$		0			0
$691$	22.4131			0.852636			0.426318	−	0.904573i	$-0.359811\pi$
$691$	22.4131			0.852636			0.426318	+	0.904573i	$0.359811\pi$
$692$		0			0
$693$		−	4.23576i		−	0.160903i
$694$		0			0
$695$	−7.88393	−	24.2405i	−0.299055	−	0.919495i
$696$		0			0
$697$			41.3140i			1.56488i
$698$		0			0
$699$	12.0846			0.457082
$700$		0			0
$701$	−30.3599			−1.14668			−0.573339	−	0.819318i	$-0.694352\pi$
$701$	−30.3599			−1.14668			−0.573339	+	0.819318i	$0.694352\pi$
$702$		0			0
$703$			5.03553i			0.189919i
$704$		0			0
$705$	−11.5336	−	35.4622i	−0.434382	−	1.33558i
$706$		0			0
$707$			0.776149i			0.0291901i
$708$		0			0
$709$	11.2281			0.421679			0.210839	−	0.977521i	$-0.432380\pi$
$709$	11.2281			0.421679			0.210839	+	0.977521i	$0.432380\pi$
$710$		0			0
$711$	−49.5196			−1.85713
$712$		0			0
$713$		−	2.35101i		−	0.0880459i
$714$		0			0
$715$	−63.7518	+	20.7345i	−2.38418	+	0.775427i
$716$		0			0
$717$			43.6757i			1.63110i
$718$		0			0
$719$	27.4485			1.02366			0.511829	−	0.859088i	$-0.328968\pi$
$719$	27.4485			1.02366			0.511829	+	0.859088i	$0.328968\pi$
$720$		0			0
$721$	0.521392			0.0194177
$722$		0			0
$723$			76.0336i			2.82772i
$724$		0			0
$725$	−22.1097	+	16.0831i	−0.821135	+	0.597312i
$726$		0			0
$727$		−	17.2650i		−	0.640323i	−0.947363	−	0.320162i	$-0.896263\pi$
$727$		−	17.2650i		−	0.640323i	0.947363	−	0.320162i	$-0.103737\pi$
$728$		0			0
$729$	42.2472			1.56471
$730$		0			0
$731$	−23.7726			−0.879261
$732$		0			0
$733$			15.6789i			0.579112i	0.957161	+	0.289556i	$0.0935077\pi$
$733$			15.6789i			0.579112i	−0.957161	+	0.289556i	$0.906492\pi$
$734$		0			0
$735$	−42.2565	+	13.7434i	−1.55866	+	0.506934i
$736$		0			0
$737$		−	78.0946i		−	2.87665i
$738$		0			0
$739$	−47.7761			−1.75747			−0.878737	−	0.477307i	$-0.841613\pi$
$739$	−47.7761			−1.75747			−0.878737	+	0.477307i	$0.841613\pi$
$740$		0			0
$741$	−14.9303			−0.548479
$742$		0			0
$743$		−	37.5789i		−	1.37864i	−0.724459	−	0.689318i	$-0.757910\pi$
$743$		−	37.5789i		−	1.37864i	0.724459	−	0.689318i	$-0.242090\pi$
$744$		0			0
$745$	10.7763	+	33.1337i	0.394814	+	1.21392i
$746$		0			0
$747$		−	11.0899i		−	0.405758i
$748$		0			0
$749$	−2.22654			−0.0813561
$750$		0			0
$751$	50.4411			1.84062			0.920311	−	0.391188i	$-0.127936\pi$
$751$	50.4411			1.84062			0.920311	+	0.391188i	$0.127936\pi$
$752$		0			0
$753$			10.3778i			0.378188i
$754$		0			0
$755$	15.4722	+	47.5719i	0.563090	+	1.73132i
$756$		0			0
$757$			10.8756i			0.395280i	0.980275	+	0.197640i	$0.0633276\pi$
$757$			10.8756i			0.395280i	−0.980275	+	0.197640i	$0.936672\pi$
$758$		0			0
$759$	10.1391			0.368026
$760$		0			0
$761$	10.7675			0.390322			0.195161	−	0.980771i	$-0.437477\pi$
$761$	10.7675			0.390322			0.195161	+	0.980771i	$0.437477\pi$
$762$		0			0
$763$			0.319229i			0.0115569i
$764$		0			0
$765$	77.7002	−	25.2710i	2.80926	−	0.913677i
$766$		0			0
$767$			47.6677i			1.72118i
$768$		0			0
$769$	1.76320			0.0635827			0.0317914	−	0.999495i	$-0.489879\pi$
$769$	1.76320			0.0635827			0.0317914	+	0.999495i	$0.489879\pi$
$770$		0			0
$771$	−27.6514			−0.995841
$772$		0			0
$773$		−	26.5661i		−	0.955517i	−0.878491	−	0.477759i	$-0.841449\pi$
$773$		−	26.5661i		−	0.955517i	0.878491	−	0.477759i	$-0.158551\pi$
$774$		0			0
$775$	15.2643	−	11.1036i	0.548311	−	0.398853i
$776$		0			0
$777$		−	2.07954i		−	0.0746032i
$778$		0			0
$779$	−5.77513			−0.206915
$780$		0			0
$781$	−13.8074			−0.494068
$782$		0			0
$783$		−	32.8189i		−	1.17285i
$784$		0			0
$785$	−49.1304	+	15.9791i	−1.75354	+	0.570317i
$786$		0			0
$787$			34.4557i			1.22821i	0.789223	+	0.614106i	$0.210483\pi$
$787$			34.4557i			1.22821i	−0.789223	+	0.614106i	$0.789517\pi$
$788$		0			0
$789$	−5.18379			−0.184548
$790$		0			0
$791$	1.79282			0.0637453
$792$		0			0
$793$			42.8236i			1.52071i
$794$		0			0
$795$	−13.7301	−	42.2155i	−0.486956	−	1.49723i
$796$		0			0
$797$		−	11.5121i		−	0.407779i	−0.978994	−	0.203890i	$-0.934642\pi$
$797$		−	11.5121i		−	0.407779i	0.978994	−	0.203890i	$-0.0653584\pi$
$798$		0			0
$799$	−41.8985			−1.48226
$800$		0			0
$801$	19.9604			0.705267
$802$		0			0
$803$			42.5621i			1.50198i
$804$		0			0
$805$	0.0624661	+	0.192063i	0.00220164	+	0.00676933i
$806$		0			0
$807$		−	61.6048i		−	2.16859i
$808$		0			0
$809$	−35.7367			−1.25643			−0.628217	−	0.778038i	$-0.716215\pi$
$809$	−35.7367			−1.25643			−0.628217	+	0.778038i	$0.716215\pi$
$810$		0			0
$811$	−47.4425			−1.66593			−0.832966	−	0.553324i	$-0.813359\pi$
$811$	−47.4425			−1.66593			−0.832966	+	0.553324i	$0.813359\pi$
$812$		0			0
$813$		−	21.4820i		−	0.753406i
$814$		0			0
$815$	24.1400	−	7.85124i	0.845588	−	0.275017i
$816$		0			0
$817$		−	3.32308i		−	0.116260i
$818$		0			0
$819$	3.88440			0.135732
$820$		0			0
$821$	−5.70237			−0.199014			−0.0995070	−	0.995037i	$-0.531727\pi$
$821$	−5.70237			−0.199014			−0.0995070	+	0.995037i	$0.531727\pi$
$822$		0			0
$823$			3.37125i			0.117514i	0.998272	+	0.0587572i	$0.0187138\pi$
$823$			3.37125i			0.117514i	−0.998272	+	0.0587572i	$0.981286\pi$
$824$		0			0
$825$	47.8861	+	65.8298i	1.66718	+	2.29190i
$826$		0			0
$827$		−	3.22704i		−	0.112215i	−0.998425	−	0.0561075i	$-0.982131\pi$
$827$		−	3.22704i		−	0.112215i	0.998425	−	0.0561075i	$-0.0178690\pi$
$828$		0			0
$829$	−10.5746			−0.367273			−0.183636	−	0.982994i	$-0.558787\pi$
$829$	−10.5746			−0.367273			−0.183636	+	0.982994i	$0.558787\pi$
$830$		0			0
$831$	35.7519			1.24022
$832$		0			0
$833$			49.9260i			1.72983i
$834$		0			0
$835$	−12.9710	+	4.21865i	−0.448880	+	0.145993i
$836$		0			0
$837$			22.6578i			0.783168i
$838$		0			0
$839$	−51.5214			−1.77872			−0.889359	−	0.457210i	$-0.848849\pi$
$839$	−51.5214			−1.77872			−0.889359	+	0.457210i	$0.848849\pi$
$840$		0			0
$841$	0.900274			0.0310439
$842$		0			0
$843$			63.1977i			2.17665i
$844$		0			0
$845$	−10.0238	−	30.8200i	−0.344830	−	1.06024i
$846$		0			0
$847$		−	3.14617i		−	0.108104i
$848$		0			0
$849$	58.7055			2.01477
$850$		0			0
$851$	3.13593			0.107498
$852$		0			0
$853$			14.3074i			0.489874i	0.969539	+	0.244937i	$0.0787673\pi$
$853$			14.3074i			0.489874i	−0.969539	+	0.244937i	$0.921233\pi$
$854$		0			0
$855$	3.53254	+	10.8614i	0.120810	+	0.371453i
$856$		0			0
$857$			3.55632i			0.121482i	0.998154	+	0.0607408i	$0.0193463\pi$
$857$			3.55632i			0.121482i	−0.998154	+	0.0607408i	$0.980654\pi$
$858$		0			0
$859$	−13.8084			−0.471135			−0.235568	−	0.971858i	$-0.575695\pi$
$859$	−13.8084			−0.471135			−0.235568	+	0.971858i	$0.575695\pi$
$860$		0			0
$861$	2.38498			0.0812799
$862$		0			0
$863$		−	11.3476i		−	0.386277i	−0.981171	−	0.193139i	$-0.938133\pi$
$863$		−	11.3476i		−	0.386277i	0.981171	−	0.193139i	$-0.0618667\pi$
$864$		0			0
$865$	−23.1676	+	7.53498i	−0.787722	+	0.256197i
$866$		0			0
$867$		−	97.3154i		−	3.30500i
$868$		0			0
$869$	−55.4327			−1.88042
$870$		0			0
$871$	71.6166			2.42664
$872$		0			0
$873$		−	25.4537i		−	0.861478i
$874$		0			0
$875$	−0.951981	+	1.31267i	−0.0321828	+	0.0443764i
$876$		0			0
$877$		−	44.6155i		−	1.50656i	−0.657701	−	0.753279i	$-0.728471\pi$
$877$		−	44.6155i		−	1.50656i	0.657701	−	0.753279i	$-0.271529\pi$
$878$		0			0
$879$	63.0479			2.12655
$880$		0			0
$881$	34.3973			1.15887			0.579437	−	0.815017i	$-0.303272\pi$
$881$	34.3973			1.15887			0.579437	+	0.815017i	$0.303272\pi$
$882$		0			0
$883$		−	54.2057i		−	1.82417i	−0.410004	−	0.912084i	$-0.634473\pi$
$883$		−	54.2057i		−	1.82417i	0.410004	−	0.912084i	$-0.365527\pi$
$884$		0			0
$885$	55.0439	−	17.9024i	1.85028	−	0.601781i
$886$		0			0
$887$		−	1.69998i		−	0.0570798i	−0.999593	−	0.0285399i	$-0.990914\pi$
$887$		−	1.69998i		−	0.0570798i	0.999593	−	0.0285399i	$-0.00908577\pi$
$888$		0			0
$889$	2.62377			0.0879984
$890$		0			0
$891$	−10.0997			−0.338353
$892$		0			0
$893$		−	5.85683i		−	0.195991i
$894$		0			0
$895$	−3.08536	−	9.48648i	−0.103132	−	0.317098i
$896$		0			0
$897$			9.29804i			0.310453i
$898$		0			0
$899$	−20.6428			−0.688477
$900$		0			0
$901$	−49.8776			−1.66166
$902$		0			0
$903$			1.37235i			0.0456689i
$904$		0			0
$905$	2.48133	+	7.62928i	0.0824822	+	0.253606i
$906$		0			0
$907$			13.1338i			0.436101i	0.975938	+	0.218050i	$0.0699697\pi$
$907$			13.1338i			0.436101i	−0.975938	+	0.218050i	$0.930030\pi$
$908$		0			0
$909$	27.3344			0.906626
$910$		0			0
$911$	24.7511			0.820040			0.410020	−	0.912076i	$-0.365522\pi$
$911$	24.7511			0.820040			0.410020	+	0.912076i	$0.365522\pi$
$912$		0			0
$913$		−	12.4141i		−	0.410848i
$914$		0			0
$915$	49.4503	−	16.0831i	1.63478	−	0.531691i
$916$		0			0
$917$		−	1.75845i		−	0.0580692i
$918$		0			0
$919$	14.2789			0.471019			0.235510	−	0.971872i	$-0.424324\pi$
$919$	14.2789			0.471019			0.235510	+	0.971872i	$0.424324\pi$
$920$		0			0
$921$	5.12707			0.168943
$922$		0			0
$923$		−	12.6621i		−	0.416777i
$924$		0			0
$925$	14.8108	+	20.3606i	0.486975	+	0.669453i
$926$		0			0
$927$		−	18.3624i		−	0.603100i
$928$		0			0
$929$	−18.6484			−0.611834			−0.305917	−	0.952058i	$-0.598963\pi$
$929$	−18.6484			−0.611834			−0.305917	+	0.952058i	$0.598963\pi$
$930$		0			0
$931$	−6.97897			−0.228726
$932$		0			0
$933$		−	26.3829i		−	0.863738i
$934$		0			0
$935$	86.9784	−	28.2887i	2.84450	−	0.925138i
$936$		0			0
$937$		−	17.9120i		−	0.585159i	−0.956241	−	0.292580i	$-0.905486\pi$
$937$		−	17.9120i		−	0.585159i	0.956241	−	0.292580i	$-0.0945136\pi$
$938$		0			0
$939$	−12.2376			−0.399360
$940$		0			0
$941$	10.7412			0.350152			0.175076	−	0.984555i	$-0.443983\pi$
$941$	10.7412			0.350152			0.175076	+	0.984555i	$0.443983\pi$
$942$		0			0
$943$			3.59653i			0.117119i
$944$		0			0
$945$	−0.602017	−	1.85101i	−0.0195836	−	0.0602132i
$946$		0			0
$947$			58.5220i			1.90171i	0.309636	+	0.950855i	$0.399793\pi$
$947$			58.5220i			1.90171i	−0.309636	+	0.950855i	$0.600207\pi$
$948$		0			0
$949$	−39.0315			−1.26702
$950$		0			0
$951$	13.1285			0.425720
$952$		0			0
$953$		−	2.93656i		−	0.0951244i	−0.998868	−	0.0475622i	$-0.984855\pi$
$953$		−	2.93656i		−	0.0951244i	0.998868	−	0.0475622i	$-0.0151452\pi$
$954$		0			0
$955$	5.53813	+	17.0280i	0.179210	+	0.551012i
$956$		0			0
$957$		−	89.0255i		−	2.87779i
$958$		0			0
$959$	−2.90628			−0.0938486
$960$		0			0
$961$	−16.7484			−0.540271
$962$		0			0
$963$			78.4144i			2.52687i
$964$		0			0
$965$	−3.09695	+	1.00725i	−0.0996944	+	0.0324244i
$966$		0			0
$967$			20.5192i			0.659852i	0.944007	+	0.329926i	$0.107024\pi$
$967$			20.5192i			0.659852i	−0.944007	+	0.329926i	$0.892976\pi$
$968$		0			0
$969$	20.3699			0.654374
$970$		0			0
$971$	−2.37837			−0.0763255			−0.0381628	−	0.999272i	$-0.512151\pi$
$971$	−2.37837			−0.0763255			−0.0381628	+	0.999272i	$0.512151\pi$
$972$		0			0
$973$			1.65334i			0.0530037i
$974$		0			0
$975$	−60.3692	+	43.9139i	−1.93336	+	1.40637i
$976$		0			0
$977$		−	46.2822i		−	1.48070i	−0.672222	−	0.740349i	$-0.734660\pi$
$977$		−	46.2822i		−	1.48070i	0.672222	−	0.740349i	$-0.265340\pi$
$978$		0			0
$979$	22.3439			0.714114
$980$		0			0
$981$	11.2426			0.358948
$982$		0			0
$983$			49.2913i			1.57215i	0.618133	+	0.786074i	$0.287889\pi$
$983$			49.2913i			1.57215i	−0.618133	+	0.786074i	$0.712111\pi$
$984$		0			0
$985$	−56.0335	+	18.2242i	−1.78538	+	0.580672i
$986$		0			0
$987$			2.41872i			0.0769887i
$988$		0			0
$989$	−2.06949			−0.0658059
$990$		0			0
$991$	6.92186			0.219880			0.109940	−	0.993938i	$-0.464934\pi$
$991$	6.92186			0.219880			0.109940	+	0.993938i	$0.464934\pi$
$992$		0			0
$993$		−	2.66054i		−	0.0844296i
$994$		0			0
$995$	−8.41940	−	25.8869i	−0.266913	−	0.820670i
$996$		0			0
$997$		−	28.1533i		−	0.891624i	−0.895127	−	0.445812i	$-0.852915\pi$
$997$		−	28.1533i		−	0.891624i	0.895127	−	0.445812i	$-0.147085\pi$
$998$		0			0
$999$	−30.2225			−0.956199

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	760.2.d.e.609.11	yes	12
4.3	odd	2		1520.2.d.k.609.2		12
5.2	odd	4		3800.2.a.z.1.6		6
5.3	odd	4		3800.2.a.be.1.1		6
5.4	even	2	inner	760.2.d.e.609.2	✓	12
20.3	even	4		7600.2.a.cg.1.6		6
20.7	even	4		7600.2.a.cn.1.1		6
20.19	odd	2		1520.2.d.k.609.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.d.e.609.2	✓	12	5.4	even	2	inner
760.2.d.e.609.11	yes	12	1.1	even	1	trivial
1520.2.d.k.609.2		12	4.3	odd	2
1520.2.d.k.609.11		12	20.19	odd	2
3800.2.a.z.1.6		6	5.2	odd	4
3800.2.a.be.1.1		6	5.3	odd	4
7600.2.a.cg.1.6		6	20.3	even	4
7600.2.a.cn.1.1		6	20.7	even	4

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:	\( N \)	\(=\)	\( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	760.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.84742i q^{3} +(0.691595 + 2.12643i) q^{5} -0.145034i q^{7} -5.10782 q^{9} -5.71774 q^{11} -5.24345i q^{13} +(-6.05484 + 1.96926i) q^{15} +7.15378i q^{17} -1.00000 q^{19} +0.412974 q^{21} +0.622762i q^{23} +(-4.04339 + 2.94125i) q^{25} -6.00186i q^{27} +5.46811 q^{29} -3.77513 q^{31} -16.2808i q^{33} +(0.308405 - 0.100305i) q^{35} -5.03553i q^{37} +14.9303 q^{39} +5.77513 q^{41} +3.32308i q^{43} +(-3.53254 - 10.8614i) q^{45} +5.85683i q^{47} +6.97897 q^{49} -20.3699 q^{51} +6.97219i q^{53} +(-3.95436 - 12.1584i) q^{55} -2.84742i q^{57} -9.09089 q^{59} -8.16707 q^{61} +0.740810i q^{63} +(11.1498 - 3.62634i) q^{65} +13.6583i q^{67} -1.77327 q^{69} +2.41484 q^{71} -7.44385i q^{73} +(-8.37499 - 11.5133i) q^{75} +0.829270i q^{77} +9.69485 q^{79} +1.76638 q^{81} +2.17116i q^{83} +(-15.2120 + 4.94752i) q^{85} +15.5700i q^{87} -3.90782 q^{89} -0.760481 q^{91} -10.7494i q^{93} +(-0.691595 - 2.12643i) q^{95} +4.98328i q^{97} +29.2052 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{5} - 16 q^{9} - 4 q^{11} - 12 q^{15} - 12 q^{19} + 36 q^{21} + 18 q^{25} + 4 q^{29} + 16 q^{31} + 6 q^{35} + 36 q^{39} + 8 q^{41} - 2 q^{45} - 4 q^{49} - 68 q^{51} - 18 q^{55} + 4 q^{59} + 20 q^{61}+ \cdots + 44 q^{99}+O(q^{100}) \) 12 * q + 6 * q^5 - 16 * q^9 - 4 * q^11 - 12 * q^15 - 12 * q^19 + 36 * q^21 + 18 * q^25 + 4 * q^29 + 16 * q^31 + 6 * q^35 + 36 * q^39 + 8 * q^41 - 2 * q^45 - 4 * q^49 - 68 * q^51 - 18 * q^55 + 4 * q^59 + 20 * q^61 + 20 * q^65 - 36 * q^69 - 16 * q^71 - 16 * q^75 + 40 * q^79 + 12 * q^81 + 6 * q^85 - 64 * q^89 + 20 * q^91 - 6 * q^95 + 44 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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