LMFDB - Embedded newform 760.2.d.e.609.4

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			609.4
Root			$1.62784 - 0.591735i$ of defining polynomial
Character	$\chi$	$=$	760.609
Dual form			760.2.d.e.609.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.59277i q^{3} +(-1.59909 + 1.56299i) q^{5} -1.66290i q^{7} +0.463073 q^{9} +5.56511 q^{11} +6.31979i q^{13} +(2.48948 + 2.54698i) q^{15} -4.12142i q^{17} -1.00000 q^{19} -2.64862 q^{21} -1.82315i q^{23} +(0.114152 - 4.99870i) q^{25} -5.51589i q^{27} +4.08942 q^{29} +6.61202 q^{31} -8.86396i q^{33} +(2.59909 + 2.65912i) q^{35} -9.66787i q^{37} +10.0660 q^{39} -4.61202 q^{41} -3.75231i q^{43} +(-0.740494 + 0.723777i) q^{45} +3.85299i q^{47} +4.23477 q^{49} -6.56449 q^{51} +5.24594i q^{53} +(-8.89910 + 8.69819i) q^{55} +1.59277i q^{57} +11.5291 q^{59} +8.02587 q^{61} -0.770043i q^{63} +(-9.87775 - 10.1059i) q^{65} -0.155284i q^{67} -2.90387 q^{69} -12.1645 q^{71} +0.795627i q^{73} +(-7.96179 - 0.181818i) q^{75} -9.25422i q^{77} +7.18555 q^{79} -7.39634 q^{81} +5.88500i q^{83} +(6.44173 + 6.59051i) q^{85} -6.51351i q^{87} -18.5476 q^{89} +10.5092 q^{91} -10.5315i q^{93} +(1.59909 - 1.56299i) q^{95} -2.77634i q^{97} +2.57705 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$		−	1.59277i		−	0.919588i	−0.888026	−	0.459794i	$-0.847923\pi$
$3$		−	1.59277i		−	0.919588i	0.888026	−	0.459794i	$-0.152077\pi$
$4$		0			0
$5$	−1.59909	+	1.56299i	−0.715133	+	0.698988i
$5$	−1.59909	+	1.56299i	−0.715133	+	0.698988i
$6$		0			0
$7$		−	1.66290i		−	0.628516i	−0.949338	−	0.314258i	$-0.898244\pi$
$7$		−	1.66290i		−	0.628516i	0.949338	−	0.314258i	$-0.101756\pi$
$8$		0			0
$9$	0.463073			0.154358
$10$		0			0
$11$	5.56511			1.67794			0.838972	−	0.544174i	$-0.183157\pi$
$11$	5.56511			1.67794			0.838972	+	0.544174i	$0.183157\pi$
$12$		0			0
$13$			6.31979i			1.75280i	0.481588	+	0.876398i	$0.340060\pi$
$13$			6.31979i			1.75280i	−0.481588	+	0.876398i	$0.659940\pi$
$14$		0			0
$15$	2.48948	+	2.54698i	0.642781	+	0.657628i
$16$		0			0
$17$		−	4.12142i		−	0.999592i	−0.866143	−	0.499796i	$-0.833408\pi$
$17$		−	4.12142i		−	0.999592i	0.866143	−	0.499796i	$-0.166592\pi$
$18$		0			0
$19$	−1.00000			−0.229416
$19$	−1.00000			−0.229416
$20$		0			0
$21$	−2.64862			−0.577976
$22$		0			0
$23$		−	1.82315i		−	0.380154i	−0.981769	−	0.190077i	$-0.939126\pi$
$23$		−	1.82315i		−	0.380154i	0.981769	−	0.190077i	$-0.0608737\pi$
$24$		0			0
$25$	0.114152	−	4.99870i	0.0228303	−	0.999739i
$26$		0			0
$27$		−	5.51589i		−	1.06153i
$28$		0			0
$29$	4.08942			0.759385			0.379693	−	0.925113i	$-0.376030\pi$
$29$	4.08942			0.759385			0.379693	+	0.925113i	$0.376030\pi$
$30$		0			0
$31$	6.61202			1.18755			0.593777	−	0.804630i	$-0.297636\pi$
$31$	6.61202			1.18755			0.593777	+	0.804630i	$0.297636\pi$
$32$		0			0
$33$		−	8.86396i		−	1.54302i
$34$		0			0
$35$	2.59909	+	2.65912i	0.439326	+	0.449473i
$36$		0			0
$37$		−	9.66787i		−	1.58939i	−0.607010	−	0.794694i	$-0.707631\pi$
$37$		−	9.66787i		−	1.58939i	0.607010	−	0.794694i	$-0.292369\pi$
$38$		0			0
$39$	10.0660			1.61185
$40$		0			0
$41$	−4.61202			−0.720277			−0.360138	−	0.932899i	$-0.617271\pi$
$41$	−4.61202			−0.720277			−0.360138	+	0.932899i	$0.617271\pi$
$42$		0			0
$43$		−	3.75231i		−	0.572222i	−0.958196	−	0.286111i	$-0.907637\pi$
$43$		−	3.75231i		−	0.572222i	0.958196	−	0.286111i	$-0.0923627\pi$
$44$		0			0
$45$	−0.740494	+	0.723777i	−0.110386	+	0.107894i
$46$		0			0
$47$			3.85299i			0.562017i	0.959705	+	0.281008i	$0.0906689\pi$
$47$			3.85299i			0.562017i	−0.959705	+	0.281008i	$0.909331\pi$
$48$		0			0
$49$	4.23477			0.604967
$50$		0			0
$51$	−6.56449			−0.919213
$52$		0			0
$53$			5.24594i			0.720585i	0.932839	+	0.360293i	$0.117323\pi$
$53$			5.24594i			0.720585i	−0.932839	+	0.360293i	$0.882677\pi$
$54$		0			0
$55$	−8.89910	+	8.69819i	−1.19995	+	1.17286i
$56$		0			0
$57$			1.59277i			0.210968i
$58$		0			0
$59$	11.5291			1.50097			0.750483	−	0.660890i	$-0.229821\pi$
$59$	11.5291			1.50097			0.750483	+	0.660890i	$0.229821\pi$
$60$		0			0
$61$	8.02587			1.02761			0.513804	−	0.857908i	$-0.328236\pi$
$61$	8.02587			1.02761			0.513804	+	0.857908i	$0.328236\pi$
$62$		0			0
$63$		−	0.770043i		−	0.0970164i
$64$		0			0
$65$	−9.87775	−	10.1059i	−1.22518	−	1.25348i
$66$		0			0
$67$		−	0.155284i		−	0.0189710i	−0.999955	−	0.00948548i	$-0.996981\pi$
$67$		−	0.155284i		−	0.0189710i	0.999955	−	0.00948548i	$-0.00301937\pi$
$68$		0			0
$69$	−2.90387			−0.349585
$70$		0			0
$71$	−12.1645			−1.44366			−0.721831	−	0.692069i	$-0.756699\pi$
$71$	−12.1645			−1.44366			−0.721831	+	0.692069i	$0.756699\pi$
$72$		0			0
$73$			0.795627i			0.0931211i	0.998915	+	0.0465606i	$0.0148261\pi$
$73$			0.795627i			0.0931211i	−0.998915	+	0.0465606i	$0.985174\pi$
$74$		0			0
$75$	−7.96179	−	0.181818i	−0.919348	−	0.0209945i
$76$		0			0
$77$		−	9.25422i		−	1.05462i
$78$		0			0
$79$	7.18555			0.808437			0.404219	−	0.914662i	$-0.367544\pi$
$79$	7.18555			0.808437			0.404219	+	0.914662i	$0.367544\pi$
$80$		0			0
$81$	−7.39634			−0.821816
$82$		0			0
$83$			5.88500i			0.645963i	0.946405	+	0.322981i	$0.104685\pi$
$83$			5.88500i			0.645963i	−0.946405	+	0.322981i	$0.895315\pi$
$84$		0			0
$85$	6.44173	+	6.59051i	0.698703	+	0.714841i
$86$		0			0
$87$		−	6.51351i		−	0.698322i
$88$		0			0
$89$	−18.5476			−1.96604			−0.983021	−	0.183492i	$-0.941260\pi$
$89$	−18.5476			−1.96604			−0.983021	+	0.183492i	$0.941260\pi$
$90$		0			0
$91$	10.5092			1.10166
$92$		0			0
$93$		−	10.5315i		−	1.09206i
$94$		0			0
$95$	1.59909	−	1.56299i	0.164063	−	0.160359i
$96$		0			0
$97$		−	2.77634i		−	0.281894i	−0.990017	−	0.140947i	$-0.954985\pi$
$97$		−	2.77634i		−	0.281894i	0.990017	−	0.140947i	$-0.0450148\pi$
$98$		0			0
$99$	2.57705			0.259004
$100$		0			0
$101$	−13.6991			−1.36311			−0.681557	−	0.731765i	$-0.738697\pi$
$101$	−13.6991			−1.36311			−0.681557	+	0.731765i	$0.738697\pi$
$102$		0			0
$103$			7.42279i			0.731389i	0.930735	+	0.365695i	$0.119169\pi$
$103$			7.42279i			0.731389i	−0.930735	+	0.365695i	$0.880831\pi$
$104$		0			0
$105$	4.23537	−	4.13975i	0.413330	−	0.403999i
$106$		0			0
$107$			13.8010i			1.33419i	0.744971	+	0.667097i	$0.232463\pi$
$107$			13.8010i			1.33419i	−0.744971	+	0.667097i	$0.767537\pi$
$108$		0			0
$109$	12.7543			1.22164			0.610818	−	0.791771i	$-0.290841\pi$
$109$	12.7543			1.22164			0.610818	+	0.791771i	$0.290841\pi$
$110$		0			0
$111$	−15.3987			−1.46158
$112$		0			0
$113$			7.81913i			0.735562i	0.929912	+	0.367781i	$0.119882\pi$
$113$			7.81913i			0.735562i	−0.929912	+	0.367781i	$0.880118\pi$
$114$		0			0
$115$	2.84956	+	2.91538i	0.265723	+	0.271860i
$116$		0			0
$117$			2.92653i			0.270558i
$118$		0			0
$119$	−6.85351			−0.628260
$120$		0			0
$121$	19.9705			1.81550
$122$		0			0
$123$			7.34591i			0.662358i
$124$		0			0
$125$	7.63035	+	8.17176i	0.682480	+	0.730905i
$126$		0			0
$127$		−	5.07017i		−	0.449905i	−0.974370	−	0.224952i	$-0.927777\pi$
$127$		−	5.07017i		−	0.449905i	0.974370	−	0.224952i	$-0.0722226\pi$
$128$		0			0
$129$	−5.97658			−0.526209
$130$		0			0
$131$	10.8720			0.949895			0.474947	−	0.880014i	$-0.342467\pi$
$131$	10.8720			0.949895			0.474947	+	0.880014i	$0.342467\pi$
$132$		0			0
$133$			1.66290i			0.144192i
$134$		0			0
$135$	8.62126	+	8.82038i	0.742000	+	0.759138i
$136$		0			0
$137$		−	7.53861i		−	0.644067i	−0.946728	−	0.322033i	$-0.895634\pi$
$137$		−	7.53861i		−	0.644067i	0.946728	−	0.322033i	$-0.104366\pi$
$138$		0			0
$139$	0.885954			0.0751457			0.0375728	−	0.999294i	$-0.488037\pi$
$139$	0.885954			0.0751457			0.0375728	+	0.999294i	$0.488037\pi$
$140$		0			0
$141$	6.13694			0.516824
$142$		0			0
$143$			35.1704i			2.94109i
$144$		0			0
$145$	−6.53933	+	6.39170i	−0.543062	+	0.530802i
$146$		0			0
$147$		−	6.74503i		−	0.556321i
$148$		0			0
$149$	−12.2594			−1.00433			−0.502166	−	0.864771i	$-0.667463\pi$
$149$	−12.2594			−1.00433			−0.502166	+	0.864771i	$0.667463\pi$
$150$		0			0
$151$	−2.95140			−0.240181			−0.120091	−	0.992763i	$-0.538319\pi$
$151$	−2.95140			−0.240181			−0.120091	+	0.992763i	$0.538319\pi$
$152$		0			0
$153$		−	1.90852i		−	0.154295i
$154$		0			0
$155$	−10.5732	+	10.3345i	−0.849259	+	0.830087i
$156$		0			0
$157$			15.7951i			1.26058i	0.776358	+	0.630292i	$0.217065\pi$
$157$			15.7951i			1.26058i	−0.776358	+	0.630292i	$0.782935\pi$
$158$		0			0
$159$	8.35559			0.662642
$160$		0			0
$161$	−3.03172			−0.238933
$162$		0			0
$163$		−	20.4459i		−	1.60144i	−0.599037	−	0.800721i	$-0.704450\pi$
$163$		−	20.4459i		−	1.60144i	0.599037	−	0.800721i	$-0.295550\pi$
$164$		0			0
$165$	13.8542	+	14.1742i	1.07855	+	1.10346i
$166$		0			0
$167$		−	14.6083i		−	1.13043i	−0.824945	−	0.565213i	$-0.808794\pi$
$167$		−	14.6083i		−	1.13043i	0.824945	−	0.565213i	$-0.191206\pi$
$168$		0			0
$169$	−26.9398			−2.07229
$170$		0			0
$171$	−0.463073			−0.0354121
$172$		0			0
$173$			19.4244i			1.47681i	0.674356	+	0.738406i	$0.264421\pi$
$173$			19.4244i			1.47681i	−0.674356	+	0.738406i	$0.735579\pi$
$174$		0			0
$175$	−8.31232	−	0.189823i	−0.628353	−	0.0143492i
$176$		0			0
$177$		−	18.3633i		−	1.38027i
$178$		0			0
$179$	7.21080			0.538960			0.269480	−	0.963006i	$-0.413148\pi$
$179$	7.21080			0.538960			0.269480	+	0.963006i	$0.413148\pi$
$180$		0			0
$181$	−10.1237			−0.752488			−0.376244	−	0.926521i	$-0.622785\pi$
$181$	−10.1237			−0.752488			−0.376244	+	0.926521i	$0.622785\pi$
$182$		0			0
$183$		−	12.7834i		−	0.944976i
$184$		0			0
$185$	15.1107	+	15.4598i	1.11096	+	1.13662i
$186$		0			0
$187$		−	22.9362i		−	1.67726i
$188$		0			0
$189$	−9.17236			−0.667191
$190$		0			0
$191$	−24.4017			−1.76564			−0.882821	−	0.469709i	$-0.844359\pi$
$191$	−24.4017			−1.76564			−0.882821	+	0.469709i	$0.844359\pi$
$192$		0			0
$193$			19.1845i			1.38093i	0.723366	+	0.690465i	$0.242594\pi$
$193$			19.1845i			1.38093i	−0.723366	+	0.690465i	$0.757406\pi$
$194$		0			0
$195$	−16.0964	+	15.7330i	−1.15269	+	1.12666i
$196$		0			0
$197$		−	11.9453i		−	0.851066i	−0.904943	−	0.425533i	$-0.860087\pi$
$197$		−	11.9453i		−	0.851066i	0.904943	−	0.425533i	$-0.139913\pi$
$198$		0			0
$199$	−9.20530			−0.652546			−0.326273	−	0.945276i	$-0.605793\pi$
$199$	−9.20530			−0.652546			−0.326273	+	0.945276i	$0.605793\pi$
$200$		0			0
$201$	−0.247332			−0.0174455
$202$		0			0
$203$		−	6.80028i		−	0.477286i
$204$		0			0
$205$	7.37502	−	7.20852i	0.515094	−	0.503465i
$206$		0			0
$207$		−	0.844253i		−	0.0586796i
$208$		0			0
$209$	−5.56511			−0.384947
$210$		0			0
$211$	18.2971			1.25962			0.629811	−	0.776749i	$-0.283133\pi$
$211$	18.2971			1.25962			0.629811	+	0.776749i	$0.283133\pi$
$212$		0			0
$213$			19.3753i			1.32757i
$214$		0			0
$215$	5.86481	+	6.00027i	0.399977	+	0.409215i
$216$		0			0
$217$		−	10.9951i		−	0.746397i
$218$		0			0
$219$	1.26725			0.0856331
$220$		0			0
$221$	26.0466			1.75208
$222$		0			0
$223$		−	1.57405i		−	0.105406i	−0.998610	−	0.0527032i	$-0.983216\pi$
$223$		−	1.57405i		−	0.105406i	0.998610	−	0.0527032i	$-0.0167837\pi$
$224$		0			0
$225$	0.0528606	−	2.31476i	0.00352404	−	0.154317i
$226$		0			0
$227$			14.6034i			0.969260i	0.874719	+	0.484630i	$0.161046\pi$
$227$			14.6034i			0.969260i	−0.874719	+	0.484630i	$0.838954\pi$
$228$		0			0
$229$	−9.33328			−0.616761			−0.308380	−	0.951263i	$-0.599787\pi$
$229$	−9.33328			−0.616761			−0.308380	+	0.951263i	$0.599787\pi$
$230$		0			0
$231$	−14.7399			−0.969812
$232$		0			0
$233$			1.02066i			0.0668658i	0.999441	+	0.0334329i	$0.0106440\pi$
$233$			1.02066i			0.0668658i	−0.999441	+	0.0334329i	$0.989356\pi$
$234$		0			0
$235$	−6.02217	−	6.16127i	−0.392843	−	0.401917i
$236$		0			0
$237$		−	11.4449i		−	0.743429i
$238$		0			0
$239$	−3.66778			−0.237249			−0.118624	−	0.992939i	$-0.537848\pi$
$239$	−3.66778			−0.237249			−0.118624	+	0.992939i	$0.537848\pi$
$240$		0			0
$241$	−23.1700			−1.49251			−0.746254	−	0.665661i	$-0.768150\pi$
$241$	−23.1700			−1.49251			−0.746254	+	0.665661i	$0.768150\pi$
$242$		0			0
$243$		−	4.76697i		−	0.305801i
$244$		0			0
$245$	−6.77176	+	6.61888i	−0.432632	+	0.422865i
$246$		0			0
$247$		−	6.31979i		−	0.402119i
$248$		0			0
$249$	9.37347			0.594020
$250$		0			0
$251$	−29.6256			−1.86995			−0.934977	−	0.354709i	$-0.884580\pi$
$251$	−29.6256			−1.86995			−0.934977	+	0.354709i	$0.884580\pi$
$252$		0			0
$253$		−	10.1461i		−	0.637877i
$254$		0			0
$255$	10.4972	−	10.2602i	0.657360	−	0.642519i
$256$		0			0
$257$			13.9768i			0.871848i	0.899984	+	0.435924i	$0.143578\pi$
$257$			13.9768i			0.871848i	−0.899984	+	0.435924i	$0.856422\pi$
$258$		0			0
$259$	−16.0767			−0.998956
$260$		0			0
$261$	1.89370			0.117217
$262$		0			0
$263$		−	11.4699i		−	0.707264i	−0.935385	−	0.353632i	$-0.884947\pi$
$263$		−	11.4699i		−	0.707264i	0.935385	−	0.353632i	$-0.115053\pi$
$264$		0			0
$265$	−8.19933	−	8.38871i	−0.503681	−	0.515314i
$266$		0			0
$267$			29.5421i			1.80795i
$268$		0			0
$269$	−16.0864			−0.980808			−0.490404	−	0.871495i	$-0.663151\pi$
$269$	−16.0864			−0.980808			−0.490404	+	0.871495i	$0.663151\pi$
$270$		0			0
$271$	15.3369			0.931649			0.465824	−	0.884877i	$-0.345758\pi$
$271$	15.3369			0.931649			0.465824	+	0.884877i	$0.345758\pi$
$272$		0			0
$273$		−	16.7387i		−	1.01307i
$274$		0			0
$275$	0.635267	−	27.8183i	0.0383081	−	1.67751i
$276$		0			0
$277$			0.0546196i			0.00328177i	0.999999	+	0.00164089i	$0.000522311\pi$
$277$			0.0546196i			0.00328177i	−0.999999	+	0.00164089i	$0.999478\pi$
$278$		0			0
$279$	3.06185			0.183308
$280$		0			0
$281$	20.3315			1.21288			0.606438	−	0.795130i	$-0.292598\pi$
$281$	20.3315			1.21288			0.606438	+	0.795130i	$0.292598\pi$
$282$		0			0
$283$		−	22.3909i		−	1.33100i	−0.746397	−	0.665501i	$-0.768218\pi$
$283$		−	22.3909i		−	1.33100i	0.746397	−	0.665501i	$-0.231782\pi$
$284$		0			0
$285$	−2.48948	−	2.54698i	−0.147464	−	0.150870i
$286$		0			0
$287$			7.66932i			0.452706i
$288$		0			0
$289$	0.0138666			0.000815685
$290$		0			0
$291$	−4.42208			−0.259227
$292$		0			0
$293$			13.3983i			0.782736i	0.920234	+	0.391368i	$0.127998\pi$
$293$			13.3983i			0.782736i	−0.920234	+	0.391368i	$0.872002\pi$
$294$		0			0
$295$	−18.4361	+	18.0199i	−1.07339	+	1.04916i
$296$		0			0
$297$		−	30.6966i		−	1.78119i
$298$		0			0
$299$	11.5219			0.666332
$300$		0			0
$301$	−6.23971			−0.359651
$302$		0			0
$303$			21.8196i			1.25350i
$304$		0			0
$305$	−12.8341	+	12.5443i	−0.734876	+	0.718286i
$306$		0			0
$307$		−	7.60404i		−	0.433985i	−0.976173	−	0.216993i	$-0.930375\pi$
$307$		−	7.60404i		−	0.433985i	0.976173	−	0.216993i	$-0.0696248\pi$
$308$		0			0
$309$	11.8228			0.672577
$310$		0			0
$311$	31.7283			1.79915			0.899573	−	0.436770i	$-0.143877\pi$
$311$	31.7283			1.79915			0.899573	+	0.436770i	$0.143877\pi$
$312$		0			0
$313$			26.2396i			1.48315i	0.670870	+	0.741575i	$0.265921\pi$
$313$			26.2396i			1.48315i	−0.670870	+	0.741575i	$0.734079\pi$
$314$		0			0
$315$	1.20357	+	1.23137i	0.0678133	+	0.0693796i
$316$		0			0
$317$		−	0.269210i		−	0.0151203i	−0.999971	−	0.00756016i	$-0.997594\pi$
$317$		−	0.269210i		−	0.0151203i	0.999971	−	0.00756016i	$-0.00240650\pi$
$318$		0			0
$319$	22.7581			1.27421
$320$		0			0
$321$	21.9819			1.22691
$322$		0			0
$323$			4.12142i			0.229322i
$324$		0			0
$325$	31.5907	+	0.721415i	1.75234	+	0.0400169i
$326$		0			0
$327$		−	20.3146i		−	1.12340i
$328$		0			0
$329$	6.40713			0.353237
$330$		0			0
$331$	−9.69472			−0.532870			−0.266435	−	0.963853i	$-0.585846\pi$
$331$	−9.69472			−0.532870			−0.266435	+	0.963853i	$0.585846\pi$
$332$		0			0
$333$		−	4.47693i		−	0.245334i
$334$		0			0
$335$	0.242707	+	0.248313i	0.0132605	+	0.0135668i
$336$		0			0
$337$			18.4219i			1.00350i	0.865012	+	0.501751i	$0.167311\pi$
$337$			18.4219i			1.00350i	−0.865012	+	0.501751i	$0.832689\pi$
$338$		0			0
$339$	12.4541			0.676414
$340$		0			0
$341$	36.7967			1.99265
$342$		0			0
$343$		−	18.6823i		−	1.00875i
$344$		0			0
$345$	4.64354	−	4.53870i	0.250000	−	0.244356i
$346$		0			0
$347$			8.85913i			0.475583i	0.971316	+	0.237791i	$0.0764235\pi$
$347$			8.85913i			0.475583i	−0.971316	+	0.237791i	$0.923577\pi$
$348$		0			0
$349$	6.36382			0.340647			0.170324	−	0.985388i	$-0.445519\pi$
$349$	6.36382			0.340647			0.170324	+	0.985388i	$0.445519\pi$
$350$		0			0
$351$	34.8593			1.86065
$352$		0			0
$353$		−	26.4213i		−	1.40627i	−0.711058	−	0.703133i	$-0.751784\pi$
$353$		−	26.4213i		−	1.40627i	0.711058	−	0.703133i	$-0.248216\pi$
$354$		0			0
$355$	19.4521	−	19.0130i	1.03241	−	1.00910i
$356$		0			0
$357$			10.9161i			0.577740i
$358$		0			0
$359$	−0.904490			−0.0477372			−0.0238686	−	0.999715i	$-0.507598\pi$
$359$	−0.904490			−0.0477372			−0.0238686	+	0.999715i	$0.507598\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$		−	31.8085i		−	1.66951i
$364$		0			0
$365$	−1.24355	−	1.27228i	−0.0650906	−	0.0665940i
$366$		0			0
$367$		−	20.6657i		−	1.07874i	−0.842069	−	0.539370i	$-0.818662\pi$
$367$		−	20.6657i		−	1.07874i	0.842069	−	0.539370i	$-0.181338\pi$
$368$		0			0
$369$	−2.13570			−0.111180
$370$		0			0
$371$	8.72347			0.452900
$372$		0			0
$373$			23.7919i			1.23190i	0.787786	+	0.615949i	$0.211227\pi$
$373$			23.7919i			1.23190i	−0.787786	+	0.615949i	$0.788773\pi$
$374$		0			0
$375$	13.0158	−	12.1534i	0.672131	−	0.627600i
$376$		0			0
$377$			25.8443i			1.33105i
$378$		0			0
$379$	−27.9667			−1.43655			−0.718276	−	0.695758i	$-0.755069\pi$
$379$	−27.9667			−1.43655			−0.718276	+	0.695758i	$0.755069\pi$
$380$		0			0
$381$	−8.07563			−0.413727
$382$		0			0
$383$			12.1083i			0.618707i	0.950947	+	0.309353i	$0.100113\pi$
$383$			12.1083i			0.618707i	−0.950947	+	0.309353i	$0.899887\pi$
$384$		0			0
$385$	14.4642	+	14.7983i	0.737164	+	0.754191i
$386$		0			0
$387$		−	1.73760i		−	0.0883269i
$388$		0			0
$389$	−0.871634			−0.0441936			−0.0220968	−	0.999756i	$-0.507034\pi$
$389$	−0.871634			−0.0441936			−0.0220968	+	0.999756i	$0.507034\pi$
$390$		0			0
$391$	−7.51398			−0.379998
$392$		0			0
$393$		−	17.3167i		−	0.873512i
$394$		0			0
$395$	−11.4903	+	11.2309i	−0.578140	+	0.565088i
$396$		0			0
$397$			20.7833i			1.04308i	0.853226	+	0.521542i	$0.174643\pi$
$397$			20.7833i			1.04308i	−0.853226	+	0.521542i	$0.825357\pi$
$398$		0			0
$399$	2.64862			0.132597
$400$		0			0
$401$	−16.0237			−0.800184			−0.400092	−	0.916475i	$-0.631022\pi$
$401$	−16.0237			−0.800184			−0.400092	+	0.916475i	$0.631022\pi$
$402$		0			0
$403$			41.7866i			2.08154i
$404$		0			0
$405$	11.8274	−	11.5604i	0.587708	−	0.574440i
$406$		0			0
$407$		−	53.8028i		−	2.66691i
$408$		0			0
$409$	−20.1554			−0.996621			−0.498311	−	0.866999i	$-0.666046\pi$
$409$	−20.1554			−0.996621			−0.498311	+	0.866999i	$0.666046\pi$
$410$		0			0
$411$	−12.0073			−0.592276
$412$		0			0
$413$		−	19.1718i		−	0.943382i
$414$		0			0
$415$	−9.19817	−	9.41062i	−0.451520	−	0.461949i
$416$		0			0
$417$		−	1.41112i		−	0.0691031i
$418$		0			0
$419$	−5.15507			−0.251842			−0.125921	−	0.992040i	$-0.540189\pi$
$419$	−5.15507			−0.251842			−0.125921	+	0.992040i	$0.540189\pi$
$420$		0			0
$421$	26.5417			1.29356			0.646781	−	0.762676i	$-0.276115\pi$
$421$	26.5417			1.29356			0.646781	+	0.762676i	$0.276115\pi$
$422$		0			0
$423$			1.78422i			0.0867516i
$424$		0			0
$425$	−20.6017	−	0.470468i	−0.999332	−	0.0228210i
$426$		0			0
$427$		−	13.3462i		−	0.645868i
$428$		0			0
$429$	56.0184			2.70460
$430$		0			0
$431$	−21.2788			−1.02496			−0.512481	−	0.858699i	$-0.671274\pi$
$431$	−21.2788			−1.02496			−0.512481	+	0.858699i	$0.671274\pi$
$432$		0			0
$433$		−	6.44963i		−	0.309949i	−0.987918	−	0.154975i	$-0.950470\pi$
$433$		−	6.44963i		−	0.309949i	0.987918	−	0.154975i	$-0.0495296\pi$
$434$		0			0
$435$	10.1805	+	10.4157i	0.488119	+	0.499393i
$436$		0			0
$437$			1.82315i			0.0872132i
$438$		0			0
$439$	−22.2324			−1.06110			−0.530548	−	0.847655i	$-0.678014\pi$
$439$	−22.2324			−1.06110			−0.530548	+	0.847655i	$0.678014\pi$
$440$		0			0
$441$	1.96101			0.0933813
$442$		0			0
$443$			7.56611i			0.359477i	0.983714	+	0.179738i	$0.0575251\pi$
$443$			7.56611i			0.359477i	−0.983714	+	0.179738i	$0.942475\pi$
$444$		0			0
$445$	29.6592	−	28.9896i	1.40598	−	1.37424i
$446$		0			0
$447$			19.5265i			0.923571i
$448$		0			0
$449$	−22.5411			−1.06378			−0.531891	−	0.846813i	$-0.678518\pi$
$449$	−22.5411			−1.06378			−0.531891	+	0.846813i	$0.678518\pi$
$450$		0			0
$451$	−25.6664			−1.20858
$452$		0			0
$453$			4.70091i			0.220868i
$454$		0			0
$455$	−16.8051	+	16.4257i	−0.787834	+	0.770048i
$456$		0			0
$457$		−	29.9942i		−	1.40307i	−0.712634	−	0.701536i	$-0.752498\pi$
$457$		−	29.9942i		−	1.40307i	0.712634	−	0.701536i	$-0.247502\pi$
$458$		0			0
$459$	−22.7333			−1.06110
$460$		0			0
$461$	−35.7493			−1.66501			−0.832506	−	0.554016i	$-0.813095\pi$
$461$	−35.7493			−1.66501			−0.832506	+	0.554016i	$0.813095\pi$
$462$		0			0
$463$		−	9.81829i		−	0.456295i	−0.973627	−	0.228147i	$-0.926733\pi$
$463$		−	9.81829i		−	0.456295i	0.973627	−	0.228147i	$-0.0732668\pi$
$464$		0			0
$465$	16.4605	+	16.8407i	0.763338	+	0.780969i
$466$		0			0
$467$		−	1.96046i		−	0.0907194i	−0.998971	−	0.0453597i	$-0.985557\pi$
$467$		−	1.96046i		−	0.0907194i	0.998971	−	0.0453597i	$-0.0144434\pi$
$468$		0			0
$469$	−0.258222			−0.0119236
$470$		0			0
$471$	25.1580			1.15922
$472$		0			0
$473$		−	20.8821i		−	0.960158i
$474$		0			0
$475$	−0.114152	+	4.99870i	−0.00523764	+	0.229356i
$476$		0			0
$477$			2.42925i			0.111228i
$478$		0			0
$479$	9.88045			0.451450			0.225725	−	0.974191i	$-0.427525\pi$
$479$	9.88045			0.451450			0.225725	+	0.974191i	$0.427525\pi$
$480$		0			0
$481$	61.0989			2.78587
$482$		0			0
$483$			4.82884i			0.219720i
$484$		0			0
$485$	4.33937	+	4.43960i	0.197041	+	0.201592i
$486$		0			0
$487$		−	9.03706i		−	0.409508i	−0.978813	−	0.204754i	$-0.934361\pi$
$487$		−	9.03706i		−	0.409508i	0.978813	−	0.204754i	$-0.0656395\pi$
$488$		0			0
$489$	−32.5656			−1.47267
$490$		0			0
$491$	−20.3715			−0.919353			−0.459676	−	0.888086i	$-0.652035\pi$
$491$	−20.3715			−0.919353			−0.459676	+	0.888086i	$0.652035\pi$
$492$		0			0
$493$		−	16.8542i		−	0.759076i
$494$		0			0
$495$	−4.12093	+	4.02790i	−0.185222	+	0.181041i
$496$		0			0
$497$			20.2283i			0.907365i
$498$		0			0
$499$	13.9386			0.623979			0.311990	−	0.950086i	$-0.399005\pi$
$499$	13.9386			0.623979			0.311990	+	0.950086i	$0.399005\pi$
$500$		0			0
$501$	−23.2678			−1.03953
$502$		0			0
$503$		−	39.7299i		−	1.77147i	−0.464192	−	0.885735i	$-0.653655\pi$
$503$		−	39.7299i		−	1.77147i	0.464192	−	0.885735i	$-0.346345\pi$
$504$		0			0
$505$	21.9061	−	21.4115i	0.974807	−	0.952801i
$506$		0			0
$507$			42.9090i			1.90566i
$508$		0			0
$509$	12.5200			0.554941			0.277471	−	0.960734i	$-0.410504\pi$
$509$	12.5200			0.554941			0.277471	+	0.960734i	$0.410504\pi$
$510$		0			0
$511$	1.32305			0.0585282
$512$		0			0
$513$			5.51589i			0.243533i
$514$		0			0
$515$	−11.6017	−	11.8697i	−0.511232	−	0.523040i
$516$		0			0
$517$			21.4423i			0.943033i
$518$		0			0
$519$	30.9387			1.35806
$520$		0			0
$521$	−0.0426487			−0.00186847			−0.000934237	−	1.00000i	$-0.500297\pi$
$521$	−0.0426487			−0.00186847			−0.000934237		1.00000i	$0.500297\pi$
$522$		0			0
$523$			29.4459i			1.28758i	0.765203	+	0.643789i	$0.222639\pi$
$523$			29.4459i			1.28758i	−0.765203	+	0.643789i	$0.777361\pi$
$524$		0			0
$525$	−0.302344	+	13.2396i	−0.0131954	+	0.577826i
$526$		0			0
$527$		−	27.2509i		−	1.18707i
$528$		0			0
$529$	19.6761			0.855483
$530$		0			0
$531$	5.33883			0.231686
$532$		0			0
$533$		−	29.1470i		−	1.26250i
$534$		0			0
$535$	−21.5708	−	22.0690i	−0.932586	−	0.954126i
$536$		0			0
$537$		−	11.4852i		−	0.495621i
$538$		0			0
$539$	23.5670			1.01510
$540$		0			0
$541$	−20.0558			−0.862264			−0.431132	−	0.902289i	$-0.641886\pi$
$541$	−20.0558			−0.862264			−0.431132	+	0.902289i	$0.641886\pi$
$542$		0			0
$543$			16.1248i			0.691979i
$544$		0			0
$545$	−20.3951	+	19.9347i	−0.873632	+	0.853909i
$546$		0			0
$547$		−	14.8952i		−	0.636874i	−0.947944	−	0.318437i	$-0.896842\pi$
$547$		−	14.8952i		−	0.636874i	0.947944	−	0.318437i	$-0.103158\pi$
$548$		0			0
$549$	3.71657			0.158619
$550$		0			0
$551$	−4.08942			−0.174215
$552$		0			0
$553$		−	11.9488i		−	0.508116i
$554$		0			0
$555$	24.6239	−	24.0680i	1.04523	−	1.02163i
$556$		0			0
$557$		−	9.26068i		−	0.392388i	−0.980565	−	0.196194i	$-0.937142\pi$
$557$		−	9.26068i		−	0.392388i	0.980565	−	0.196194i	$-0.0628582\pi$
$558$		0			0
$559$	23.7138			1.00299
$560$		0			0
$561$	−36.5322			−1.54239
$562$		0			0
$563$			12.9472i			0.545661i	0.962062	+	0.272830i	$0.0879598\pi$
$563$			12.9472i			0.545661i	−0.962062	+	0.272830i	$0.912040\pi$
$564$		0			0
$565$	−12.2212	−	12.5035i	−0.514149	−	0.526025i
$566$		0			0
$567$			12.2994i			0.516525i
$568$		0			0
$569$	−20.0593			−0.840929			−0.420464	−	0.907309i	$-0.638133\pi$
$569$	−20.0593			−0.840929			−0.420464	+	0.907309i	$0.638133\pi$
$570$		0			0
$571$	−5.64571			−0.236266			−0.118133	−	0.992998i	$-0.537691\pi$
$571$	−5.64571			−0.236266			−0.118133	+	0.992998i	$0.537691\pi$
$572$		0			0
$573$			38.8663i			1.62366i
$574$		0			0
$575$	−9.11339	−	0.208116i	−0.380054	−	0.00867904i
$576$		0			0
$577$		−	33.5402i		−	1.39630i	−0.715953	−	0.698148i	$-0.754008\pi$
$577$		−	33.5402i		−	1.39630i	0.715953	−	0.698148i	$-0.245992\pi$
$578$		0			0
$579$	30.5566			1.26989
$580$		0			0
$581$	9.78616			0.405998
$582$		0			0
$583$			29.1943i			1.20910i
$584$		0			0
$585$	−4.57412	−	4.67977i	−0.189117	−	0.193485i
$586$		0			0
$587$			27.5517i			1.13718i	0.822621	+	0.568591i	$0.192511\pi$
$587$			27.5517i			1.13718i	−0.822621	+	0.568591i	$0.807489\pi$
$588$		0			0
$589$	−6.61202			−0.272444
$590$		0			0
$591$	−19.0261			−0.782630
$592$		0			0
$593$		−	24.4353i		−	1.00344i	−0.865031	−	0.501719i	$-0.832701\pi$
$593$		−	24.4353i		−	1.00344i	0.865031	−	0.501719i	$-0.167299\pi$
$594$		0			0
$595$	10.9593	−	10.7119i	0.449289	−	0.439146i
$596$		0			0
$597$			14.6620i			0.600074i
$598$		0			0
$599$	3.51535			0.143633			0.0718166	−	0.997418i	$-0.477120\pi$
$599$	3.51535			0.143633			0.0718166	+	0.997418i	$0.477120\pi$
$600$		0			0
$601$	−1.21824			−0.0496932			−0.0248466	−	0.999691i	$-0.507910\pi$
$601$	−1.21824			−0.0496932			−0.0248466	+	0.999691i	$0.507910\pi$
$602$		0			0
$603$		−	0.0719079i		−	0.00292832i
$604$		0			0
$605$	−31.9345	+	31.2136i	−1.29832	+	1.26901i
$606$		0			0
$607$		−	13.1872i		−	0.535251i	−0.963523	−	0.267625i	$-0.913761\pi$
$607$		−	13.1872i		−	0.535251i	0.963523	−	0.267625i	$-0.0862390\pi$
$608$		0			0
$609$	−10.8313			−0.438907
$610$		0			0
$611$	−24.3501			−0.985100
$612$		0			0
$613$			9.67880i			0.390923i	0.980711	+	0.195462i	$0.0626205\pi$
$613$			9.67880i			0.390923i	−0.980711	+	0.195462i	$0.937379\pi$
$614$		0			0
$615$	−11.4815	−	11.7467i	−0.462981	−	0.473674i
$616$		0			0
$617$			10.4077i			0.418999i	0.977809	+	0.209500i	$0.0671835\pi$
$617$			10.4077i			0.418999i	−0.977809	+	0.209500i	$0.932816\pi$
$618$		0			0
$619$	32.7144			1.31490			0.657452	−	0.753496i	$-0.271634\pi$
$619$	32.7144			1.31490			0.657452	+	0.753496i	$0.271634\pi$
$620$		0			0
$621$	−10.0563			−0.403546
$622$		0			0
$623$			30.8428i			1.23569i
$624$		0			0
$625$	−24.9739	−	1.14122i	−0.998958	−	0.0456488i
$626$		0			0
$627$			8.86396i			0.353993i
$628$		0			0
$629$	−39.8454			−1.58874
$630$		0			0
$631$	−37.2305			−1.48212			−0.741061	−	0.671438i	$-0.765677\pi$
$631$	−37.2305			−1.48212			−0.741061	+	0.671438i	$0.765677\pi$
$632$		0			0
$633$		−	29.1431i		−	1.15833i
$634$		0			0
$635$	7.92460	+	8.10763i	0.314478	+	0.321742i
$636$		0			0
$637$			26.7629i			1.06038i
$638$		0			0
$639$	−5.63306			−0.222840
$640$		0			0
$641$	23.3803			0.923464			0.461732	−	0.887019i	$-0.347228\pi$
$641$	23.3803			0.923464			0.461732	+	0.887019i	$0.347228\pi$
$642$		0			0
$643$			24.6705i			0.972911i	0.873705	+	0.486455i	$0.161710\pi$
$643$			24.6705i			0.972911i	−0.873705	+	0.486455i	$0.838290\pi$
$644$		0			0
$645$	9.55707	−	9.34132i	0.376309	−	0.367814i
$646$		0			0
$647$			47.2591i			1.85795i	0.370147	+	0.928973i	$0.379307\pi$
$647$			47.2591i			1.85795i	−0.370147	+	0.928973i	$0.620693\pi$
$648$		0			0
$649$	64.1610			2.51854
$650$		0			0
$651$	−17.5127			−0.686378
$652$		0			0
$653$		−	31.5361i		−	1.23410i	−0.786924	−	0.617051i	$-0.788327\pi$
$653$		−	31.5361i		−	1.23410i	0.786924	−	0.617051i	$-0.211673\pi$
$654$		0			0
$655$	−17.3853	+	16.9928i	−0.679301	+	0.663966i
$656$		0			0
$657$			0.368434i			0.0143740i
$658$		0			0
$659$	21.3511			0.831722			0.415861	−	0.909428i	$-0.363480\pi$
$659$	21.3511			0.831722			0.415861	+	0.909428i	$0.363480\pi$
$660$		0			0
$661$	−31.2161			−1.21416			−0.607082	−	0.794639i	$-0.707660\pi$
$661$	−31.2161			−1.21416			−0.607082	+	0.794639i	$0.707660\pi$
$662$		0			0
$663$		−	41.4863i		−	1.61119i
$664$		0			0
$665$	−2.59909	−	2.65912i	−0.100788	−	0.103116i
$666$		0			0
$667$		−	7.45563i		−	0.288683i
$668$		0			0
$669$	−2.50711			−0.0969304
$670$		0			0
$671$	44.6649			1.72427
$672$		0			0
$673$			4.16240i			0.160449i	0.996777	+	0.0802243i	$0.0255637\pi$
$673$			4.16240i			0.160449i	−0.996777	+	0.0802243i	$0.974436\pi$
$674$		0			0
$675$	−27.5723	−	0.629648i	−1.06126	−	0.0242352i
$676$		0			0
$677$			35.5734i			1.36720i	0.729857	+	0.683599i	$0.239586\pi$
$677$			35.5734i			1.36720i	−0.729857	+	0.683599i	$0.760414\pi$
$678$		0			0
$679$	−4.61677			−0.177175
$680$		0			0
$681$	23.2599			0.891320
$682$		0			0
$683$		−	26.8628i		−	1.02788i	−0.857827	−	0.513938i	$-0.828186\pi$
$683$		−	26.8628i		−	1.02788i	0.857827	−	0.513938i	$-0.171814\pi$
$684$		0			0
$685$	11.7827	+	12.0549i	0.450195	+	0.460593i
$686$		0			0
$687$			14.8658i			0.567166i
$688$		0			0
$689$	−33.1533			−1.26304
$690$		0			0
$691$	32.0064			1.21758			0.608791	−	0.793331i	$-0.291655\pi$
$691$	32.0064			1.21758			0.608791	+	0.793331i	$0.291655\pi$
$692$		0			0
$693$		−	4.28538i		−	0.162788i
$694$		0			0
$695$	−1.41672	+	1.38473i	−0.0537391	+	0.0525259i
$696$		0			0
$697$			19.0081i			0.719983i
$698$		0			0
$699$	1.62568			0.0614890
$700$		0			0
$701$	−0.822942			−0.0310821			−0.0155410	−	0.999879i	$-0.504947\pi$
$701$	−0.822942			−0.0310821			−0.0155410	+	0.999879i	$0.504947\pi$
$702$		0			0
$703$			9.66787i			0.364631i
$704$		0			0
$705$	−9.81350	+	9.59195i	−0.369598	+	0.361254i
$706$		0			0
$707$			22.7802i			0.856739i
$708$		0			0
$709$	−24.8057			−0.931599			−0.465800	−	0.884890i	$-0.654233\pi$
$709$	−24.8057			−0.931599			−0.465800	+	0.884890i	$0.654233\pi$
$710$		0			0
$711$	3.32743			0.124789
$712$		0			0
$713$		−	12.0547i		−	0.451453i
$714$		0			0
$715$	−54.9708	−	56.2405i	−2.05579	−	2.10327i
$716$		0			0
$717$			5.84194i			0.218171i
$718$		0			0
$719$	−13.8765			−0.517506			−0.258753	−	0.965943i	$-0.583312\pi$
$719$	−13.8765			−0.517506			−0.258753	+	0.965943i	$0.583312\pi$
$720$		0			0
$721$	12.3433			0.459690
$722$		0			0
$723$			36.9045i			1.37249i
$724$		0			0
$725$	0.466814	−	20.4417i	0.0173370	−	0.759187i
$726$		0			0
$727$			28.7158i			1.06501i	0.846426	+	0.532506i	$0.178749\pi$
$727$			28.7158i			1.06501i	−0.846426	+	0.532506i	$0.821251\pi$
$728$		0			0
$729$	−29.7817			−1.10303
$730$		0			0
$731$	−15.4649			−0.571989
$732$		0			0
$733$		−	47.1903i		−	1.74301i	−0.490384	−	0.871506i	$-0.663144\pi$
$733$		−	47.1903i		−	1.74301i	0.490384	−	0.871506i	$-0.336856\pi$
$734$		0			0
$735$	10.5424	+	10.7859i	0.388862	+	0.397843i
$736$		0			0
$737$		−	0.864174i		−	0.0318322i
$738$		0			0
$739$	7.61271			0.280038			0.140019	−	0.990149i	$-0.455284\pi$
$739$	7.61271			0.280038			0.140019	+	0.990149i	$0.455284\pi$
$740$		0			0
$741$	−10.0660			−0.369784
$742$		0			0
$743$			15.8127i			0.580111i	0.957010	+	0.290055i	$0.0936738\pi$
$743$			15.8127i			0.580111i	−0.957010	+	0.290055i	$0.906326\pi$
$744$		0			0
$745$	19.6039	−	19.1613i	0.718230	−	0.702016i
$746$		0			0
$747$			2.72519i			0.0997093i
$748$		0			0
$749$	22.9497			0.838563
$750$		0			0
$751$	35.7516			1.30459			0.652296	−	0.757964i	$-0.273806\pi$
$751$	35.7516			1.30459			0.652296	+	0.757964i	$0.273806\pi$
$752$		0			0
$753$			47.1869i			1.71959i
$754$		0			0
$755$	4.71954	−	4.61299i	0.171762	−	0.167884i
$756$		0			0
$757$			12.8136i			0.465717i	0.972511	+	0.232859i	$0.0748079\pi$
$757$			12.8136i			0.465717i	−0.972511	+	0.232859i	$0.925192\pi$
$758$		0			0
$759$	−16.1604			−0.586584
$760$		0			0
$761$	18.8830			0.684508			0.342254	−	0.939608i	$-0.388810\pi$
$761$	18.8830			0.684508			0.342254	+	0.939608i	$0.388810\pi$
$762$		0			0
$763$		−	21.2090i		−	0.767818i
$764$		0			0
$765$	2.98299	+	3.05189i	0.107850	+	0.110341i
$766$		0			0
$767$			72.8618i			2.63089i
$768$		0			0
$769$	16.0228			0.577798			0.288899	−	0.957360i	$-0.406711\pi$
$769$	16.0228			0.577798			0.288899	+	0.957360i	$0.406711\pi$
$770$		0			0
$771$	22.2618			0.801741
$772$		0			0
$773$			9.91039i			0.356452i	0.983990	+	0.178226i	$0.0570358\pi$
$773$			9.91039i			0.356452i	−0.983990	+	0.178226i	$0.942964\pi$
$774$		0			0
$775$	0.754774	−	33.0515i	0.0271123	−	1.18724i
$776$		0			0
$777$			25.6065i			0.918628i
$778$		0			0
$779$	4.61202			0.165243
$780$		0			0
$781$	−67.6969			−2.42238
$782$		0			0
$783$		−	22.5568i		−	0.806113i
$784$		0			0
$785$	−24.6875	−	25.2577i	−0.881133	−	0.901485i
$786$		0			0
$787$			30.9671i			1.10386i	0.833891	+	0.551929i	$0.186108\pi$
$787$			30.9671i			1.10386i	−0.833891	+	0.551929i	$0.813892\pi$
$788$		0			0
$789$	−18.2689			−0.650392
$790$		0			0
$791$	13.0024			0.462313
$792$		0			0
$793$			50.7219i			1.80119i
$794$		0			0
$795$	−13.3613	+	13.0597i	−0.473877	+	0.463179i
$796$		0			0
$797$

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)

\(f(q)\)	\(=\)	\(q-1.59277i q^{3} +(-1.59909 + 1.56299i) q^{5} -1.66290i q^{7} +0.463073 q^{9} +5.56511 q^{11} +6.31979i q^{13} +(2.48948 + 2.54698i) q^{15} -4.12142i q^{17} -1.00000 q^{19} -2.64862 q^{21} -1.82315i q^{23} +(0.114152 - 4.99870i) q^{25} -5.51589i q^{27} +4.08942 q^{29} +6.61202 q^{31} -8.86396i q^{33} +(2.59909 + 2.65912i) q^{35} -9.66787i q^{37} +10.0660 q^{39} -4.61202 q^{41} -3.75231i q^{43} +(-0.740494 + 0.723777i) q^{45} +3.85299i q^{47} +4.23477 q^{49} -6.56449 q^{51} +5.24594i q^{53} +(-8.89910 + 8.69819i) q^{55} +1.59277i q^{57} +11.5291 q^{59} +8.02587 q^{61} -0.770043i q^{63} +(-9.87775 - 10.1059i) q^{65} -0.155284i q^{67} -2.90387 q^{69} -12.1645 q^{71} +0.795627i q^{73} +(-7.96179 - 0.181818i) q^{75} -9.25422i q^{77} +7.18555 q^{79} -7.39634 q^{81} +5.88500i q^{83} +(6.44173 + 6.59051i) q^{85} -6.51351i q^{87} -18.5476 q^{89} +10.5092 q^{91} -10.5315i q^{93} +(1.59909 - 1.56299i) q^{95} -2.77634i q^{97} +2.57705 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{5} - 16 q^{9} - 4 q^{11} - 12 q^{15} - 12 q^{19} + 36 q^{21} + 18 q^{25} + 4 q^{29} + 16 q^{31} + 6 q^{35} + 36 q^{39} + 8 q^{41} - 2 q^{45} - 4 q^{49} - 68 q^{51} - 18 q^{55} + 4 q^{59} + 20 q^{61}+ \cdots + 44 q^{99}+O(q^{100}) \) 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

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