LMFDB - Embedded newform 1520.2.d.k.609.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(609,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, 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("1520.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.d (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:	$12.1372611072$
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:	no (minimal twist has level 760)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.4
Root			$1.62784 - 0.591735i$ of defining polynomial
Character	$\chi$	$=$	1520.609
Dual form			1520.2.d.k.609.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.59277i q^{3} +(-1.59909 - 1.56299i) q^{5} -1.66290i q^{7} +0.463073 q^{9} +O(q^{10})$	$q-1.59277i q^{3} +(-1.59909 - 1.56299i) q^{5} -1.66290i q^{7} +0.463073 q^{9} -5.56511 q^{11} -6.31979i q^{13} +(-2.48948 + 2.54698i) q^{15} +4.12142i q^{17} +1.00000 q^{19} -2.64862 q^{21} -1.82315i q^{23} +(0.114152 + 4.99870i) q^{25} -5.51589i q^{27} +4.08942 q^{29} -6.61202 q^{31} +8.86396i q^{33} +(-2.59909 + 2.65912i) q^{35} +9.66787i q^{37} -10.0660 q^{39} -4.61202 q^{41} -3.75231i q^{43} +(-0.740494 - 0.723777i) q^{45} +3.85299i q^{47} +4.23477 q^{49} +6.56449 q^{51} -5.24594i q^{53} +(8.89910 + 8.69819i) q^{55} -1.59277i q^{57} -11.5291 q^{59} +8.02587 q^{61} -0.770043i q^{63} +(-9.87775 + 10.1059i) q^{65} -0.155284i q^{67} -2.90387 q^{69} +12.1645 q^{71} -0.795627i q^{73} +(7.96179 - 0.181818i) q^{75} +9.25422i q^{77} -7.18555 q^{79} -7.39634 q^{81} +5.88500i q^{83} +(6.44173 - 6.59051i) q^{85} -6.51351i q^{87} -18.5476 q^{89} -10.5092 q^{91} +10.5315i q^{93} +(-1.59909 - 1.56299i) q^{95} +2.77634i q^{97} -2.57705 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{5} - 16 q^{9}+O(q^{10}) $ 12 * q + 6 * q^5 - 16 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$1$	$1$	$-1$

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	$-0.816843\pi$
$11$	−5.56511			−1.67794			−0.838972	+	0.544174i	$0.816843\pi$
$12$		0			0
$13$		−	6.31979i		−	1.75280i	−0.481588	−	0.876398i	$-0.659940\pi$
$13$		−	6.31979i		−	1.75280i	0.481588	−	0.876398i	$-0.340060\pi$
$14$		0			0
$15$	−2.48948	+	2.54698i	−0.642781	+	0.657628i
$16$		0			0
$17$			4.12142i			0.999592i	0.866143	+	0.499796i	$0.166592\pi$
$17$			4.12142i			0.999592i	−0.866143	+	0.499796i	$0.833408\pi$
$18$		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	$-0.702364\pi$
$31$	−6.61202			−1.18755			−0.593777	+	0.804630i	$0.702364\pi$
$32$		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	$0.292369\pi$
$37$			9.66787i			1.58939i	−0.607010	+	0.794694i	$0.707631\pi$
$38$		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	$-0.882677\pi$
$53$		−	5.24594i		−	0.720585i	0.932839	−	0.360293i	$-0.117323\pi$
$54$		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	$-0.770179\pi$
$59$	−11.5291			−1.50097			−0.750483	+	0.660890i	$0.770179\pi$
$60$		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	$-0.243301\pi$
$71$	12.1645			1.44366			0.721831	+	0.692069i	$0.243301\pi$
$72$		0			0
$73$		−	0.795627i		−	0.0931211i	−0.998915	−	0.0465606i	$-0.985174\pi$
$73$		−	0.795627i		−	0.0931211i	0.998915	−	0.0465606i	$-0.0148261\pi$
$74$		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	$-0.632456\pi$
$79$	−7.18555			−0.808437			−0.404219	+	0.914662i	$0.632456\pi$
$80$		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	$0.0450148\pi$
$97$			2.77634i			0.281894i	−0.990017	+	0.140947i	$0.954985\pi$
$98$		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	$-0.880118\pi$
$113$		−	7.81913i		−	0.735562i	0.929912	−	0.367781i	$-0.119882\pi$
$114$		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	$-0.657533\pi$
$131$	−10.8720			−0.949895			−0.474947	+	0.880014i	$0.657533\pi$
$132$		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	$0.104366\pi$
$137$			7.53861i			0.644067i	−0.946728	+	0.322033i	$0.895634\pi$
$138$		0			0
$139$	−0.885954			−0.0751457			−0.0375728	−	0.999294i	$-0.511963\pi$
$139$	−0.885954			−0.0751457			−0.0375728	+	0.999294i	$0.511963\pi$
$140$		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	$-0.461681\pi$
$151$	2.95140			0.240181			0.120091	+	0.992763i	$0.461681\pi$
$152$		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	$-0.782935\pi$
$157$		−	15.7951i		−	1.26058i	0.776358	−	0.630292i	$-0.217065\pi$
$158$		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	$-0.735579\pi$
$173$		−	19.4244i		−	1.47681i	0.674356	−	0.738406i	$-0.264421\pi$
$174$		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	$-0.586852\pi$
$179$	−7.21080			−0.538960			−0.269480	+	0.963006i	$0.586852\pi$
$180$		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	$-0.155641\pi$
$191$	24.4017			1.76564			0.882821	+	0.469709i	$0.155641\pi$
$192$		0			0
$193$		−	19.1845i		−	1.38093i	−0.723366	−	0.690465i	$-0.757406\pi$
$193$		−	19.1845i		−	1.38093i	0.723366	−	0.690465i	$-0.242594\pi$
$194$		0			0
$195$	16.0964	+	15.7330i	1.15269	+	1.12666i
$196$		0			0
$197$			11.9453i			0.851066i	0.904943	+	0.425533i	$0.139913\pi$
$197$			11.9453i			0.851066i	−0.904943	+	0.425533i	$0.860087\pi$
$198$		0			0
$199$	9.20530			0.652546			0.326273	−	0.945276i	$-0.394207\pi$
$199$	9.20530			0.652546			0.326273	+	0.945276i	$0.394207\pi$
$200$		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	$-0.716867\pi$
$211$	−18.2971			−1.25962			−0.629811	+	0.776749i	$0.716867\pi$
$212$		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	$-0.989356\pi$
$233$		−	1.02066i		−	0.0668658i	0.999441	−	0.0334329i	$-0.0106440\pi$
$234$		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	$-0.462152\pi$
$239$	3.66778			0.237249			0.118624	+	0.992939i	$0.462152\pi$
$240$		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	$-0.115420\pi$
$251$	29.6256			1.86995			0.934977	+	0.354709i	$0.115420\pi$
$252$		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	$-0.856422\pi$
$257$		−	13.9768i		−	0.871848i	0.899984	−	0.435924i	$-0.143578\pi$
$258$		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	$-0.654242\pi$
$271$	−15.3369			−0.931649			−0.465824	+	0.884877i	$0.654242\pi$
$272$		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	$-0.999478\pi$
$277$		−	0.0546196i		−	0.00328177i	0.999999	−	0.00164089i	$-0.000522311\pi$
$278$		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	$-0.872002\pi$
$293$		−	13.3983i		−	0.782736i	0.920234	−	0.391368i	$-0.127998\pi$
$294$		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	$-0.856123\pi$
$311$	−31.7283			−1.79915			−0.899573	+	0.436770i	$0.856123\pi$
$312$		0			0
$313$		−	26.2396i		−	1.48315i	−0.670870	−	0.741575i	$-0.734079\pi$
$313$		−	26.2396i		−	1.48315i	0.670870	−	0.741575i	$-0.265921\pi$
$314$		0			0
$315$	−1.20357	+	1.23137i	−0.0678133	+	0.0693796i
$316$		0			0
$317$			0.269210i			0.0151203i	0.999971	+	0.00756016i	$0.00240650\pi$
$317$			0.269210i			0.0151203i	−0.999971	+	0.00756016i	$0.997594\pi$
$318$		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	$-0.414154\pi$
$331$	9.69472			0.532870			0.266435	+	0.963853i	$0.414154\pi$
$332$		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	$-0.832689\pi$
$337$		−	18.4219i		−	1.00350i	0.865012	−	0.501751i	$-0.167311\pi$
$338$		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	$0.248216\pi$
$353$			26.4213i			1.40627i	−0.711058	+	0.703133i	$0.751784\pi$
$354$		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	$-0.492402\pi$
$359$	0.904490			0.0477372			0.0238686	+	0.999715i	$0.492402\pi$
$360$		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	$-0.788773\pi$
$373$		−	23.7919i		−	1.23190i	0.787786	−	0.615949i	$-0.211227\pi$
$374$		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	$-0.244931\pi$
$379$	27.9667			1.43655			0.718276	+	0.695758i	$0.244931\pi$
$380$		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	$-0.825357\pi$
$397$		−	20.7833i		−	1.04308i	0.853226	−	0.521542i	$-0.174643\pi$
$398$		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	$-0.459811\pi$
$419$	5.15507			0.251842			0.125921	+	0.992040i	$0.459811\pi$
$420$		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	$-0.328726\pi$
$431$	21.2788			1.02496			0.512481	+	0.858699i	$0.328726\pi$
$432$		0			0
$433$			6.44963i			0.309949i	0.987918	+	0.154975i	$0.0495296\pi$
$433$			6.44963i			0.309949i	−0.987918	+	0.154975i	$0.950470\pi$
$434$		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	$-0.321986\pi$
$439$	22.2324			1.06110			0.530548	+	0.847655i	$0.321986\pi$
$440$		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	$0.247502\pi$
$457$			29.9942i			1.40307i	−0.712634	+	0.701536i	$0.752498\pi$
$458$		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	$-0.572475\pi$
$479$	−9.88045			−0.451450			−0.225725	+	0.974191i	$0.572475\pi$
$480$		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	$-0.347965\pi$
$491$	20.3715			0.919353			0.459676	+	0.888086i	$0.347965\pi$
$492$		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	$-0.600995\pi$
$499$	−13.9386			−0.623979			−0.311990	+	0.950086i	$0.600995\pi$
$500$		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	$0.0628582\pi$
$557$			9.26068i			0.392388i	−0.980565	+	0.196194i	$0.937142\pi$
$558$		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	$-0.462309\pi$
$571$	5.64571			0.236266			0.118133	+	0.992998i	$0.462309\pi$
$572$		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	$0.245992\pi$
$577$			33.5402i			1.39630i	−0.715953	+	0.698148i	$0.754008\pi$
$578$		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	$0.167299\pi$
$593$			24.4353i			1.00344i	−0.865031	+	0.501719i	$0.832701\pi$
$594$		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	$-0.522880\pi$
$599$	−3.51535			−0.143633			−0.0718166	+	0.997418i	$0.522880\pi$
$600$		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	$-0.937379\pi$
$613$		−	9.67880i		−	0.390923i	0.980711	−	0.195462i	$-0.0626205\pi$
$614$		0			0
$615$	11.4815	−	11.7467i	0.462981	−	0.473674i
$616$		0			0
$617$		−	10.4077i		−	0.418999i	−0.977809	−	0.209500i	$-0.932816\pi$
$617$		−	10.4077i		−	0.418999i	0.977809	−	0.209500i	$-0.0671835\pi$
$618$		0			0
$619$	−32.7144			−1.31490			−0.657452	−	0.753496i	$-0.728366\pi$
$619$	−32.7144			−1.31490			−0.657452	+	0.753496i	$0.728366\pi$
$620$		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	$-0.234323\pi$
$631$	37.2305			1.48212			0.741061	+	0.671438i	$0.234323\pi$
$632$		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	$0.211673\pi$
$653$			31.5361i			1.23410i	−0.786924	+	0.617051i	$0.788327\pi$
$654$		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	$-0.636520\pi$
$659$	−21.3511			−0.831722			−0.415861	+	0.909428i	$0.636520\pi$
$660$		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	$-0.974436\pi$
$673$		−	4.16240i		−	0.160449i	0.996777	−	0.0802243i	$-0.0255637\pi$
$674$		0			0
$675$	27.5723	−	0.629648i	1.06126	−	0.0242352i
$676$		0			0
$677$		−	35.5734i		−	1.36720i	−0.729857	−	0.683599i	$-0.760414\pi$
$677$		−	35.5734i		−	1.36720i	0.729857	−	0.683599i	$-0.239586\pi$
$678$		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	$-0.708345\pi$
$691$	−32.0064			−1.21758			−0.608791	+	0.793331i	$0.708345\pi$
$692$		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	$-0.416688\pi$
$719$	13.8765			0.517506			0.258753	+	0.965943i	$0.416688\pi$
$720$		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	$0.336856\pi$
$733$			47.1903i			1.74301i	−0.490384	+	0.871506i	$0.663144\pi$
$734$		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	$-0.544716\pi$
$739$	−7.61271			−0.280038			−0.140019	+	0.990149i	$0.544716\pi$
$740$		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	$-0.726194\pi$
$751$	−35.7516			−1.30459			−0.652296	+	0.757964i	$0.726194\pi$
$752$		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	$-0.925192\pi$
$757$		−	12.8136i		−	0.465717i	0.972511	−	0.232859i	$-0.0748079\pi$
$758$		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	$-0.942964\pi$
$773$		−	9.91039i		−	0.356452i	0.983990	−	0.178226i	$-0.0570358\pi$
$774$		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
$797$		−	54.8059i		−	1.94132i	−0.240448	−	0.970662i	$-0.577294\pi$
$797$		−	54.8059i		−	1.94132i	0.240448	−	0.970662i	$-0.422706\pi$
$798$		0			0
$799$	−15.8798			−0.561787
$800$		0			0
$801$	−8.58890			−0.303474
$802$		0			0
$803$			4.42776i			0.156252i
$804$		0			0
$805$	4.84798	+	4.73853i	0.170869	+	0.167011i
$806$		0			0
$807$			25.6221i			0.901939i
$808$		0			0
$809$	−25.6862			−0.903080			−0.451540	−	0.892251i	$-0.649125\pi$
$809$	−25.6862			−0.903080			−0.451540	+	0.892251i	$0.649125\pi$
$810$		0			0
$811$	−8.01224			−0.281348			−0.140674	−	0.990056i	$-0.544927\pi$
$811$	−8.01224			−0.281348			−0.140674	+	0.990056i	$0.544927\pi$
$812$		0			0
$813$			24.4282i			0.856733i
$814$		0			0
$815$	−31.9566	+	32.6947i	−1.11939	+	1.14524i
$816$		0			0
$817$		−	3.75231i		−	0.131277i
$818$		0			0
$819$	−4.86652			−0.170050
$820$		0			0
$821$	−48.8946			−1.70643			−0.853217	−	0.521555i	$-0.825352\pi$
$821$	−48.8946			−1.70643			−0.853217	+	0.521555i	$0.825352\pi$
$822$		0			0
$823$			39.8314i			1.38843i	0.719766	+	0.694217i	$0.244249\pi$
$823$			39.8314i			1.38843i	−0.719766	+	0.694217i	$0.755751\pi$
$824$		0			0
$825$	−44.3083	+	1.01184i	−1.54262	+	0.0352276i
$826$		0			0
$827$		−	28.8758i		−	1.00411i	−0.864835	−	0.502056i	$-0.832577\pi$
$827$		−	28.8758i		−	1.00411i	0.864835	−	0.502056i	$-0.167423\pi$
$828$		0			0
$829$	2.35519			0.0817991			0.0408995	−	0.999163i	$-0.486978\pi$
$829$	2.35519			0.0817991			0.0408995	+	0.999163i	$0.486978\pi$
$830$		0			0
$831$	−0.0869966			−0.00301788
$832$		0			0
$833$			17.4533i			0.604720i
$834$		0			0
$835$	−22.8326	+	23.3600i	−0.790155	+	0.808406i
$836$		0			0
$837$			36.4712i			1.26063i
$838$		0			0
$839$	26.0230			0.898415			0.449207	−	0.893428i	$-0.351706\pi$
$839$	26.0230			0.898415			0.449207	+	0.893428i	$0.351706\pi$
$840$		0			0
$841$	−12.2767			−0.423334
$842$		0			0
$843$		−	32.3835i		−	1.11535i
$844$		0			0
$845$	43.0791	+	42.1065i	1.48196	+	1.44851i
$846$		0			0
$847$		−	33.2089i		−	1.14107i
$848$		0			0
$849$	−35.6636			−1.22397
$850$		0			0
$851$	17.6260			0.604211
$852$		0			0
$853$		−	37.7907i		−	1.29393i	−0.762520	−	0.646964i	$-0.776038\pi$
$853$		−	37.7907i		−	1.29393i	0.762520	−	0.646964i	$-0.223962\pi$
$854$		0			0
$855$	−0.740494	−	0.723777i	−0.0253244	−	0.0247526i
$856$		0			0
$857$		−	19.7796i		−	0.675657i	−0.941208	−	0.337828i	$-0.890308\pi$
$857$		−	19.7796i		−	0.675657i	0.941208	−	0.337828i	$-0.109692\pi$
$858$		0			0
$859$	22.0641			0.752816			0.376408	−	0.926454i	$-0.377159\pi$
$859$	22.0641			0.752816			0.376408	+	0.926454i	$0.377159\pi$
$860$		0			0
$861$	12.2155			0.416303
$862$		0			0
$863$		−	32.3505i		−	1.10122i	−0.834761	−	0.550612i	$-0.814394\pi$
$863$		−	32.3505i		−	1.10122i	0.834761	−	0.550612i	$-0.185606\pi$
$864$		0			0
$865$	−30.3601	+	31.0613i	−1.03227	+	1.05612i
$866$		0			0
$867$		−	0.0220864i		−	0.000750094i
$868$		0			0
$869$	39.9884			1.35651
$870$		0			0
$871$	−0.981363			−0.0332522
$872$		0			0
$873$			1.28565i			0.0435126i
$874$		0			0
$875$	−13.5888	−	12.6885i	−0.459386	−	0.428950i
$876$		0			0
$877$		−	22.5876i		−	0.762728i	−0.924425	−	0.381364i	$-0.875454\pi$
$877$		−	22.5876i		−	0.762728i	0.924425	−	0.381364i	$-0.124546\pi$
$878$		0			0
$879$	−21.3404			−0.719794
$880$		0			0
$881$	31.2658			1.05337			0.526686	−	0.850060i	$-0.323434\pi$
$881$	31.2658			1.05337			0.526686	+	0.850060i	$0.323434\pi$
$882$		0			0
$883$			22.4584i			0.755786i	0.925849	+	0.377893i	$0.123351\pi$
$883$			22.4584i			0.755786i	−0.925849	+	0.377893i	$0.876649\pi$
$884$		0			0
$885$	28.7016	−	29.3645i	0.964793	−	0.987077i
$886$		0			0
$887$		−	14.4883i		−	0.486470i	−0.969967	−	0.243235i	$-0.921791\pi$
$887$		−	14.4883i		−	0.486470i	0.969967	−	0.243235i	$-0.0782086\pi$
$888$		0			0
$889$	−8.43117			−0.282772
$890$		0			0
$891$	41.1615			1.37896
$892$		0			0
$893$			3.85299i			0.128935i
$894$		0			0
$895$	11.5307	+	11.2704i	0.385428	+	0.376727i
$896$		0			0
$897$			18.3519i			0.612751i
$898$		0			0
$899$	−27.0393			−0.901811
$900$		0			0
$901$	21.6207			0.720291
$902$		0			0
$903$			9.93845i			0.330731i
$904$		0			0
$905$	16.1887	+	15.8232i	0.538129	+	0.525981i
$906$		0			0
$907$			9.58465i			0.318253i	0.987258	+	0.159127i	$0.0508678\pi$
$907$			9.58465i			0.318253i	−0.987258	+	0.159127i	$0.949132\pi$
$908$		0			0
$909$	−6.34370			−0.210407
$910$		0			0
$911$	−29.8122			−0.987721			−0.493860	−	0.869541i	$-0.664415\pi$
$911$	−29.8122			−0.987721			−0.493860	+	0.869541i	$0.664415\pi$
$912$		0			0
$913$		−	32.7507i		−	1.08389i
$914$		0			0
$915$	−19.9803	+	20.4417i	−0.660527	+	0.675783i
$916$		0			0
$917$			18.0791i			0.597025i
$918$		0			0
$919$	−22.0075			−0.725961			−0.362981	−	0.931797i	$-0.618241\pi$
$919$	−22.0075			−0.725961			−0.362981	+	0.931797i	$0.618241\pi$
$920$		0			0
$921$	−12.1115			−0.399088
$922$		0			0
$923$		−	76.8772i		−	2.53044i
$924$		0			0
$925$	−48.3267	+	1.10360i	−1.58897	+	0.0362863i
$926$		0			0
$927$			3.43729i			0.112896i
$928$		0			0
$929$	22.7036			0.744880			0.372440	−	0.928056i	$-0.378521\pi$
$929$	22.7036			0.744880			0.372440	+	0.928056i	$0.378521\pi$
$930$		0			0
$931$	4.23477			0.138789
$932$		0			0
$933$			50.5360i			1.65447i
$934$		0			0
$935$	−35.8489	+	36.6769i	−1.17239	+	1.19946i
$936$		0			0
$937$			10.5567i			0.344871i	0.985021	+	0.172436i	$0.0551637\pi$
$937$			10.5567i			0.344871i	−0.985021	+	0.172436i	$0.944836\pi$
$938$		0			0
$939$	−41.7937			−1.36389
$940$		0			0
$941$	−24.7961			−0.808331			−0.404166	−	0.914686i	$-0.632438\pi$
$941$	−24.7961			−0.808331			−0.404166	+	0.914686i	$0.632438\pi$
$942$		0			0
$943$			8.40842i			0.273816i
$944$		0			0
$945$	14.6674	+	14.3363i	0.477131	+	0.466359i
$946$		0			0
$947$			5.83000i			0.189449i	0.995504	+	0.0947247i	$0.0301971\pi$
$947$			5.83000i			0.189449i	−0.995504	+	0.0947247i	$0.969803\pi$
$948$		0			0
$949$	−5.02820			−0.163222
$950$		0			0
$951$	0.428790			0.0139045
$952$		0			0
$953$		−	7.15932i		−	0.231913i	−0.993254	−	0.115957i	$-0.963007\pi$
$953$		−	7.15932i		−	0.231913i	0.993254	−	0.115957i	$-0.0369933\pi$
$954$		0			0
$955$	−39.0204	−	38.1394i	−1.26267	−	1.23416i
$956$		0			0
$957$			36.2484i			1.17175i
$958$		0			0
$959$	12.5359			0.404806
$960$		0			0
$961$	12.7188			0.410285
$962$		0			0
$963$			6.39088i			0.205943i
$964$		0			0
$965$	−29.9851	+	30.6777i	−0.965254	+	0.987549i
$966$		0			0
$967$			10.3120i			0.331612i	0.986158	+	0.165806i	$0.0530226\pi$
$967$			10.3120i			0.331612i	−0.986158	+	0.165806i	$0.946977\pi$
$968$		0			0
$969$	6.56449			0.210882
$970$		0			0
$971$	−38.2430			−1.22728			−0.613639	−	0.789587i	$-0.710295\pi$
$971$	−38.2430			−1.22728			−0.613639	+	0.789587i	$0.710295\pi$
$972$		0			0
$973$			1.47325i			0.0472303i
$974$		0			0
$975$	−1.14905	−	50.3169i	−0.0367991	−	1.61143i
$976$		0			0
$977$		−	18.5114i		−	0.592232i	−0.955152	−	0.296116i	$-0.904308\pi$
$977$		−	18.5114i		−	0.592232i	0.955152	−	0.296116i	$-0.0956915\pi$
$978$		0			0
$979$	103.220			3.29891
$980$		0			0
$981$	5.90615			0.188569
$982$		0			0
$983$		−	10.9590i		−	0.349539i	−0.984609	−	0.174769i	$-0.944082\pi$
$983$		−	10.9590i		−	0.349539i	0.984609	−	0.174769i	$-0.0559180\pi$
$984$		0			0
$985$	18.6703	−	19.1015i	0.594885	−	0.608625i
$986$		0			0
$987$		−	10.2051i		−	0.324832i
$988$		0			0
$989$	−6.84104			−0.217532
$990$		0			0
$991$	36.8766			1.17142			0.585712	−	0.810519i	$-0.300815\pi$
$991$	36.8766			1.17142			0.585712	+	0.810519i	$0.300815\pi$
$992$		0			0
$993$		−	15.4415i		−	0.490021i
$994$		0			0
$995$	−14.7201	−	14.3877i	−0.466657	−	0.456122i
$996$		0			0
$997$			36.4988i			1.15593i	0.816062	+	0.577964i	$0.196153\pi$
$997$			36.4988i			1.15593i	−0.816062	+	0.577964i	$0.803847\pi$
$998$		0			0
$999$	53.3269			1.68719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.d.k.609.4		12
4.3	odd	2		760.2.d.e.609.9	yes	12
5.2	odd	4		7600.2.a.cn.1.2		6
5.3	odd	4		7600.2.a.cg.1.5		6
5.4	even	2	inner	1520.2.d.k.609.9		12
20.3	even	4		3800.2.a.be.1.2		6
20.7	even	4		3800.2.a.z.1.5		6
20.19	odd	2		760.2.d.e.609.4	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.d.e.609.4	✓	12	20.19	odd	2
760.2.d.e.609.9	yes	12	4.3	odd	2
1520.2.d.k.609.4		12	1.1	even	1	trivial
1520.2.d.k.609.9		12	5.4	even	2	inner
3800.2.a.z.1.5		6	20.7	even	4
3800.2.a.be.1.2		6	20.3	even	4
7600.2.a.cg.1.5		6	5.3	odd	4
7600.2.a.cn.1.2		6	5.2	odd	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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.59277i q^{3} +(-1.59909 - 1.56299i) q^{5} -1.66290i q^{7} +0.463073 q^{9} +O(q^{10})\)	\(q-1.59277i q^{3} +(-1.59909 - 1.56299i) q^{5} -1.66290i q^{7} +0.463073 q^{9} -5.56511 q^{11} -6.31979i q^{13} +(-2.48948 + 2.54698i) q^{15} +4.12142i q^{17} +1.00000 q^{19} -2.64862 q^{21} -1.82315i q^{23} +(0.114152 + 4.99870i) q^{25} -5.51589i q^{27} +4.08942 q^{29} -6.61202 q^{31} +8.86396i q^{33} +(-2.59909 + 2.65912i) q^{35} +9.66787i q^{37} -10.0660 q^{39} -4.61202 q^{41} -3.75231i q^{43} +(-0.740494 - 0.723777i) q^{45} +3.85299i q^{47} +4.23477 q^{49} +6.56449 q^{51} -5.24594i q^{53} +(8.89910 + 8.69819i) q^{55} -1.59277i q^{57} -11.5291 q^{59} +8.02587 q^{61} -0.770043i q^{63} +(-9.87775 + 10.1059i) q^{65} -0.155284i q^{67} -2.90387 q^{69} +12.1645 q^{71} -0.795627i q^{73} +(7.96179 - 0.181818i) q^{75} +9.25422i q^{77} -7.18555 q^{79} -7.39634 q^{81} +5.88500i q^{83} +(6.44173 - 6.59051i) q^{85} -6.51351i q^{87} -18.5476 q^{89} -10.5092 q^{91} +10.5315i q^{93} +(-1.59909 - 1.56299i) q^{95} +2.77634i q^{97} -2.57705 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{5} - 16 q^{9}+O(q^{10}) \) 12 * q + 6 * q^5 - 16 * q^9	\( 12 q + 6 q^{5} - 16 q^{9} + 4 q^{11} + 12 q^{15} + 12 q^{19} + 36 q^{21} + 18 q^{25} + 4 q^{29} - 16 q^{31} - 6 q^{35} - 36 q^{39} + 8 q^{41} - 2 q^{45} - 4 q^{49} + 68 q^{51} + 18 q^{55} - 4 q^{59} + 20 q^{61} + 20 q^{65} - 36 q^{69} + 16 q^{71} + 16 q^{75} - 40 q^{79} + 12 q^{81} + 6 q^{85} - 64 q^{89} - 20 q^{91} + 6 q^{95} - 44 q^{99}+O(q^{100}) \) 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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