LMFDB - Embedded newform 1520.2.g.g.1519.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1519,1520)]

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([1, 0, 1, 1]))

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

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

chi := DirichletCharacter("1520.1519");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.g (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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 20x^{14} + 271x^{12} - 2000x^{10} + 10645x^{8} - 29570x^{6} + 58816x^{4} - 56840x^{2} + 38416 $ x^16 - 20x^14 + 271x^12 - 2000x^10 + 10645x^8 - 29570x^6 + 58816x^4 - 56840*x^2 + 38416
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1519.8
Root			$0.957255 - 0.552672i$ of defining polynomial
Character	$\chi$	$=$	1520.1519
Dual form			1520.2.g.g.1519.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.10534i q^{3} +(0.602114 - 2.15348i) q^{5} +2.26573 q^{7} +1.77822 q^{9} +O(q^{10})$	$q-1.10534i q^{3} +(0.602114 - 2.15348i) q^{5} +2.26573 q^{7} +1.77822 q^{9} -1.56319i q^{11} -6.54594 q^{13} +(-2.38033 - 0.665543i) q^{15} -2.59328i q^{17} +(3.27492 - 2.87662i) q^{19} -2.50441i q^{21} -3.39112 q^{23} +(-4.27492 - 2.59328i) q^{25} -5.28157i q^{27} +0.621972i q^{29} +7.20423 q^{31} -1.72786 q^{33} +(1.36423 - 4.87920i) q^{35} +1.45178 q^{37} +7.23551i q^{39} +7.46927i q^{41} +5.63203 q^{43} +(1.07069 - 3.82935i) q^{45} +3.82902 q^{47} -1.86646 q^{49} -2.86646 q^{51} -5.96830 q^{53} +(-3.36630 - 0.941220i) q^{55} +(-3.17965 - 3.61991i) q^{57} -8.28152 q^{59} -2.70753 q^{61} +4.02896 q^{63} +(-3.94140 + 14.0965i) q^{65} -10.0068i q^{67} +3.74835i q^{69} -7.20423 q^{71} +0.644042i q^{73} +(-2.86646 + 4.72525i) q^{75} -3.54177i q^{77} +11.9649 q^{79} -0.503299 q^{81} +8.56042 q^{83} +(-5.58456 - 1.56145i) q^{85} +0.687492 q^{87} -7.00176i q^{89} -14.8313 q^{91} -7.96315i q^{93} +(-4.22285 - 8.78451i) q^{95} -5.28081 q^{97} -2.77969i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{9}+O(q^{10}) $ 16 * q - 32 * q^9	$ 16 q - 32 q^{9} + 32 q^{15} - 8 q^{19} - 8 q^{25} + 96 q^{31} - 24 q^{45} - 8 q^{49} - 24 q^{51} - 72 q^{59} - 24 q^{61} - 96 q^{71} - 24 q^{75} + 32 q^{79} - 8 q^{81} - 56 q^{91}+O(q^{100}) $ 16 * q - 32 * q^9 + 32 * q^15 - 8 * q^19 - 8 * q^25 + 96 * q^31 - 24 * q^45 - 8 * q^49 - 24 * q^51 - 72 * q^59 - 24 * q^61 - 96 * q^71 - 24 * q^75 + 32 * q^79 - 8 * q^81 - 56 * q^91

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
$3$		−	1.10534i		−	0.638170i	−0.947726	−	0.319085i	$-0.896624\pi$
$3$		−	1.10534i		−	0.638170i	0.947726	−	0.319085i	$-0.103376\pi$
$4$		0			0
$5$	0.602114	−	2.15348i	0.269274	−	0.963064i
$5$	0.602114	−	2.15348i	0.269274	−	0.963064i
$6$		0			0
$7$	2.26573			0.856366			0.428183	−	0.903692i	$-0.359154\pi$
$7$	2.26573			0.856366			0.428183	+	0.903692i	$0.359154\pi$
$8$		0			0
$9$	1.77822			0.592739
$10$		0			0
$11$		−	1.56319i		−	0.471320i	−0.971836	−	0.235660i	$-0.924275\pi$
$11$		−	1.56319i		−	0.471320i	0.971836	−	0.235660i	$-0.0757252\pi$
$12$		0			0
$13$	−6.54594			−1.81552			−0.907759	−	0.419492i	$-0.862208\pi$
$13$	−6.54594			−1.81552			−0.907759	+	0.419492i	$0.862208\pi$
$14$		0			0
$15$	−2.38033	−	0.665543i	−0.614599	−	0.171842i
$16$		0			0
$17$		−	2.59328i		−	0.628962i	−0.949264	−	0.314481i	$-0.898170\pi$
$17$		−	2.59328i		−	0.628962i	0.949264	−	0.314481i	$-0.101830\pi$
$18$		0			0
$19$	3.27492	−	2.87662i	0.751318	−	0.659941i
$19$	3.27492	−	2.87662i	0.751318	−	0.659941i
$20$		0			0
$21$		−	2.50441i		−	0.546507i
$22$		0			0
$23$	−3.39112			−0.707098			−0.353549	−	0.935416i	$-0.615025\pi$
$23$	−3.39112			−0.707098			−0.353549	+	0.935416i	$0.615025\pi$
$24$		0			0
$25$	−4.27492	−	2.59328i	−0.854983	−	0.518655i
$26$		0			0
$27$		−	5.28157i		−	1.01644i
$28$		0			0
$29$			0.621972i			0.115497i	0.998331	+	0.0577486i	$0.0183922\pi$
$29$			0.621972i			0.115497i	−0.998331	+	0.0577486i	$0.981608\pi$
$30$		0			0
$31$	7.20423			1.29392			0.646959	−	0.762525i	$-0.276041\pi$
$31$	7.20423			1.29392			0.646959	+	0.762525i	$0.276041\pi$
$32$		0			0
$33$	−1.72786			−0.300782
$34$		0			0
$35$	1.36423	−	4.87920i	0.230597	−	0.824735i
$36$		0			0
$37$	1.45178			0.238672			0.119336	−	0.992854i	$-0.461923\pi$
$37$	1.45178			0.238672			0.119336	+	0.992854i	$0.461923\pi$
$38$		0			0
$39$			7.23551i			1.15861i
$40$		0			0
$41$			7.46927i			1.16650i	0.812291	+	0.583252i	$0.198220\pi$
$41$			7.46927i			1.16650i	−0.812291	+	0.583252i	$0.801780\pi$
$42$		0			0
$43$	5.63203			0.858876			0.429438	−	0.903096i	$-0.358712\pi$
$43$	5.63203			0.858876			0.429438	+	0.903096i	$0.358712\pi$
$44$		0			0
$45$	1.07069	−	3.82935i	0.159609	−	0.570845i
$46$		0			0
$47$	3.82902			0.558520			0.279260	−	0.960215i	$-0.409911\pi$
$47$	3.82902			0.558520			0.279260	+	0.960215i	$0.409911\pi$
$48$		0			0
$49$	−1.86646			−0.266637
$50$		0			0
$51$	−2.86646			−0.401385
$52$		0			0
$53$	−5.96830			−0.819809			−0.409904	−	0.912128i	$-0.634438\pi$
$53$	−5.96830			−0.819809			−0.409904	+	0.912128i	$0.634438\pi$
$54$		0			0
$55$	−3.36630	−	0.941220i	−0.453911	−	0.126914i
$56$		0			0
$57$	−3.17965	−	3.61991i	−0.421155	−	0.479469i
$58$		0			0
$59$	−8.28152			−1.07816			−0.539081	−	0.842254i	$-0.681228\pi$
$59$	−8.28152			−1.07816			−0.539081	+	0.842254i	$0.681228\pi$
$60$		0			0
$61$	−2.70753			−0.346663			−0.173332	−	0.984864i	$-0.555453\pi$
$61$	−2.70753			−0.346663			−0.173332	+	0.984864i	$0.555453\pi$
$62$		0			0
$63$	4.02896			0.507601
$64$		0			0
$65$	−3.94140	+	14.0965i	−0.488871	+	1.74846i
$66$		0			0
$67$		−	10.0068i		−	1.22253i	−0.791427	−	0.611264i	$-0.790661\pi$
$67$		−	10.0068i		−	1.22253i	0.791427	−	0.611264i	$-0.209339\pi$
$68$		0			0
$69$			3.74835i			0.451249i
$70$		0			0
$71$	−7.20423			−0.854985			−0.427492	−	0.904019i	$-0.640603\pi$
$71$	−7.20423			−0.854985			−0.427492	+	0.904019i	$0.640603\pi$
$72$		0			0
$73$			0.644042i			0.0753794i	0.999289	+	0.0376897i	$0.0119998\pi$
$73$			0.644042i			0.0753794i	−0.999289	+	0.0376897i	$0.988000\pi$
$74$		0			0
$75$	−2.86646	+	4.72525i	−0.330990	+	0.545625i
$76$		0			0
$77$		−	3.54177i		−	0.403622i
$78$		0			0
$79$	11.9649			1.34616			0.673078	−	0.739572i	$-0.264972\pi$
$79$	11.9649			1.34616			0.673078	+	0.739572i	$0.264972\pi$
$80$		0			0
$81$	−0.503299			−0.0559221
$82$		0			0
$83$	8.56042			0.939629			0.469814	−	0.882765i	$-0.344321\pi$
$83$	8.56042			0.939629			0.469814	+	0.882765i	$0.344321\pi$
$84$		0			0
$85$	−5.58456	−	1.56145i	−0.605730	−	0.169363i
$86$		0			0
$87$	0.687492			0.0737069
$88$		0			0
$89$		−	7.00176i		−	0.742185i	−0.928596	−	0.371093i	$-0.878983\pi$
$89$		−	7.00176i		−	0.742185i	0.928596	−	0.371093i	$-0.121017\pi$
$90$		0			0
$91$	−14.8313			−1.55475
$92$		0			0
$93$		−	7.96315i		−	0.825740i
$94$		0			0
$95$	−4.22285	−	8.78451i	−0.433255	−	0.901271i
$96$		0			0
$97$	−5.28081			−0.536185			−0.268092	−	0.963393i	$-0.586393\pi$
$97$	−5.28081			−0.536185			−0.268092	+	0.963393i	$0.586393\pi$
$98$		0			0
$99$		−	2.77969i		−	0.279370i
$100$		0			0
$101$	−13.2574			−1.31916			−0.659578	−	0.751636i	$-0.729265\pi$
$101$	−13.2574			−1.31916			−0.659578	+	0.751636i	$0.729265\pi$
$102$		0			0
$103$			5.50731i			0.542652i	0.962488	+	0.271326i	$0.0874621\pi$
$103$			5.50731i			0.542652i	−0.962488	+	0.271326i	$0.912538\pi$
$104$		0			0
$105$	−5.39319	−	1.50794i	−0.526321	−	0.147160i
$106$		0			0
$107$		−	2.76700i		−	0.267496i	−0.991015	−	0.133748i	$-0.957299\pi$
$107$		−	2.76700i		−	0.267496i	0.991015	−	0.133748i	$-0.0427013\pi$
$108$		0			0
$109$		−	15.0930i		−	1.44565i	−0.691033	−	0.722824i	$-0.742844\pi$
$109$		−	15.0930i		−	1.44565i	0.691033	−	0.722824i	$-0.257156\pi$
$110$		0			0
$111$		−	1.60472i		−	0.152313i
$112$		0			0
$113$	−3.17965			−0.299116			−0.149558	−	0.988753i	$-0.547785\pi$
$113$	−3.17965			−0.299116			−0.149558	+	0.988753i	$0.547785\pi$
$114$		0			0
$115$	−2.04184	+	7.30270i	−0.190403	+	0.680980i
$116$		0			0
$117$	−11.6401			−1.07613
$118$		0			0
$119$		−	5.87567i		−	0.538622i
$120$		0			0
$121$	8.55643			0.777857
$122$		0			0
$123$	8.25610			0.744428
$124$		0			0
$125$	−8.15855	+	7.64448i	−0.729723	+	0.683743i
$126$		0			0
$127$		−	5.17674i		−	0.459361i	−0.973266	−	0.229681i	$-0.926232\pi$
$127$		−	5.17674i		−	0.459361i	0.973266	−	0.229681i	$-0.0737681\pi$
$128$		0			0
$129$		−	6.22532i		−	0.548109i
$130$		0			0
$131$		−	1.54472i		−	0.134963i	−0.997721	−	0.0674814i	$-0.978504\pi$
$131$		−	1.54472i		−	0.134963i	0.997721	−	0.0674814i	$-0.0214963\pi$
$132$		0			0
$133$	7.42008	−	6.51764i	0.643403	−	0.565151i
$134$		0			0
$135$	−11.3737	−	3.18011i	−0.978895	−	0.273700i
$136$		0			0
$137$		−	16.3937i		−	1.40061i	−0.713843	−	0.700306i	$-0.753047\pi$
$137$		−	16.3937i		−	1.40061i	0.713843	−	0.700306i	$-0.246953\pi$
$138$		0			0
$139$			12.7735i			1.08343i	0.840562	+	0.541716i	$0.182225\pi$
$139$			12.7735i			1.08343i	−0.840562	+	0.541716i	$0.817775\pi$
$140$		0			0
$141$		−	4.23238i		−	0.356431i
$142$		0			0
$143$			10.2326i			0.855690i
$144$		0			0
$145$	1.33940	+	0.374498i	0.111231	+	0.0311004i
$146$		0			0
$147$			2.06308i			0.170160i
$148$		0			0
$149$	19.0114			1.55748			0.778738	−	0.627349i	$-0.215860\pi$
$149$	19.0114			1.55748			0.778738	+	0.627349i	$0.215860\pi$
$150$		0			0
$151$	5.73292			0.466539			0.233269	−	0.972412i	$-0.425058\pi$
$151$	5.73292			0.466539			0.233269	+	0.972412i	$0.425058\pi$
$152$		0			0
$153$		−	4.61141i		−	0.372810i
$154$		0			0
$155$	4.33777	−	15.5141i	0.348418	−	1.24613i
$156$		0			0
$157$			18.2218i			1.45426i	0.686500	+	0.727130i	$0.259146\pi$
$157$			18.2218i			1.45426i	−0.686500	+	0.727130i	$0.740854\pi$
$158$		0			0
$159$			6.59702i			0.523178i
$160$		0			0
$161$	−7.68337			−0.605535
$162$		0			0
$163$	2.50545			0.196242			0.0981209	−	0.995175i	$-0.468717\pi$
$163$	2.50545			0.196242			0.0981209	+	0.995175i	$0.468717\pi$
$164$		0			0
$165$	−1.04037	+	3.72091i	−0.0809928	+	0.289673i
$166$		0			0
$167$			21.4923i			1.66313i	0.555430	+	0.831563i	$0.312554\pi$
$167$			21.4923i			1.66313i	−0.555430	+	0.831563i	$0.687446\pi$
$168$		0			0
$169$	29.8494			2.29611
$170$		0			0
$171$	5.82351	−	5.11524i	0.445335	−	0.391173i
$172$		0			0
$173$	−7.23344			−0.549948			−0.274974	−	0.961452i	$-0.588669\pi$
$173$	−7.23344			−0.549948			−0.274974	+	0.961452i	$0.588669\pi$
$174$		0			0
$175$	−9.68581	−	5.87567i	−0.732179	−	0.444159i
$176$		0			0
$177$			9.15392i			0.688051i
$178$		0			0
$179$	8.99340			0.672198			0.336099	−	0.941827i	$-0.390892\pi$
$179$	8.99340			0.672198			0.336099	+	0.941827i	$0.390892\pi$
$180$		0			0
$181$			22.6062i			1.68031i	0.542348	+	0.840154i	$0.317536\pi$
$181$			22.6062i			1.68031i	−0.542348	+	0.840154i	$0.682464\pi$
$182$		0			0
$183$			2.99275i			0.221230i
$184$		0			0
$185$	0.874140	−	3.12638i	0.0642680	−	0.229856i
$186$		0			0
$187$	−4.05379			−0.296442
$188$		0			0
$189$		−	11.9666i		−	0.870443i
$190$		0			0
$191$			16.4759i			1.19215i	0.802927	+	0.596077i	$0.203275\pi$
$191$			16.4759i			1.19215i	−0.802927	+	0.596077i	$0.796725\pi$
$192$		0			0
$193$	10.4848			0.754713			0.377357	−	0.926068i	$-0.376833\pi$
$193$	10.4848			0.754713			0.377357	+	0.926068i	$0.376833\pi$
$194$		0			0
$195$	15.5815	+	4.35661i	1.11581	+	0.311983i
$196$		0			0
$197$		−	2.06366i		−	0.147029i	−0.997294	−	0.0735147i	$-0.976578\pi$
$197$		−	2.06366i		−	0.147029i	0.997294	−	0.0735147i	$-0.0234216\pi$
$198$		0			0
$199$			10.8155i			0.766692i	0.923605	+	0.383346i	$0.125228\pi$
$199$			10.8155i			0.766692i	−0.923605	+	0.383346i	$0.874772\pi$
$200$		0			0
$201$	−11.0610			−0.780181
$202$		0			0
$203$			1.40922i			0.0989079i
$204$		0			0
$205$	16.0849	+	4.49735i	1.12342	+	0.314109i
$206$		0			0
$207$	−6.03015			−0.419124
$208$		0			0
$209$	−4.49670	−	5.11932i	−0.311043	−	0.354111i
$210$		0			0
$211$	10.7251			0.738345			0.369173	−	0.929361i	$-0.379641\pi$
$211$	10.7251			0.738345			0.369173	+	0.929361i	$0.379641\pi$
$212$		0			0
$213$			7.96315i			0.545626i
$214$		0			0
$215$	3.39112	−	12.1284i	0.231273	−	0.827152i
$216$		0			0
$217$	16.3228			1.10807
$218$		0			0
$219$	0.711887			0.0481049
$220$		0			0
$221$			16.9754i			1.14189i
$222$		0			0
$223$		−	2.51456i		−	0.168388i	−0.996449	−	0.0841938i	$-0.973169\pi$
$223$		−	2.51456i		−	0.168388i	0.996449	−	0.0841938i	$-0.0268315\pi$
$224$		0			0
$225$	−7.60173	−	4.61141i	−0.506782	−	0.307427i
$226$		0			0
$227$		−	24.3302i		−	1.61485i	−0.589970	−	0.807425i	$-0.700860\pi$
$227$		−	24.3302i		−	1.61485i	0.589970	−	0.807425i	$-0.299140\pi$
$228$		0			0
$229$	−3.04654			−0.201321			−0.100660	−	0.994921i	$-0.532096\pi$
$229$	−3.04654			−0.201321			−0.100660	+	0.994921i	$0.532096\pi$
$230$		0			0
$231$	−3.91487			−0.257580
$232$		0			0
$233$			14.5134i			0.950804i	0.879769	+	0.475402i	$0.157697\pi$
$233$			14.5134i			0.950804i	−0.879769	+	0.475402i	$0.842303\pi$
$234$		0			0
$235$	2.30551	−	8.24571i	0.150395	−	0.537891i
$236$		0			0
$237$		−	13.2253i		−	0.859076i
$238$		0			0
$239$			3.92426i			0.253839i	0.991913	+	0.126920i	$0.0405090\pi$
$239$			3.92426i			0.253839i	−0.991913	+	0.126920i	$0.959491\pi$
$240$		0			0
$241$		−	16.9589i		−	1.09242i	−0.837648	−	0.546210i	$-0.816070\pi$
$241$		−	16.9589i		−	1.09242i	0.837648	−	0.546210i	$-0.183930\pi$
$242$		0			0
$243$		−	15.2884i		−	0.980751i
$244$		0			0
$245$	−1.12382	+	4.01938i	−0.0717984	+	0.256789i
$246$		0			0
$247$	−21.4374	+	18.8302i	−1.36403	+	1.19813i
$248$		0			0
$249$		−	9.46221i		−	0.599643i
$250$		0			0
$251$		−	28.9375i		−	1.82652i	−0.407381	−	0.913258i	$-0.633558\pi$
$251$		−	28.9375i		−	1.82652i	0.407381	−	0.913258i	$-0.366442\pi$
$252$		0			0
$253$			5.30097i			0.333269i
$254$		0			0
$255$	−1.72594	+	6.17285i	−0.108082	+	0.386559i
$256$		0			0
$257$	16.8492			1.05102			0.525511	−	0.850787i	$-0.323874\pi$
$257$	16.8492			1.05102			0.525511	+	0.850787i	$0.323874\pi$
$258$		0			0
$259$	3.28935			0.204391
$260$		0			0
$261$			1.10600i			0.0684597i
$262$		0			0
$263$	−24.3260			−1.50001			−0.750004	−	0.661433i	$-0.769948\pi$
$263$	−24.3260			−1.50001			−0.750004	+	0.661433i	$0.769948\pi$
$264$		0			0
$265$	−3.59360	+	12.8526i	−0.220753	+	0.789528i
$266$		0			0
$267$	−7.73935			−0.473641
$268$		0			0
$269$		−	27.0596i		−	1.64985i	−0.565239	−	0.824927i	$-0.691216\pi$
$269$		−	27.0596i		−	1.64985i	0.565239	−	0.824927i	$-0.308784\pi$
$270$		0			0
$271$		−	15.7084i		−	0.954220i	−0.878844	−	0.477110i	$-0.841684\pi$
$271$		−	15.7084i		−	0.954220i	0.878844	−	0.477110i	$-0.158316\pi$
$272$		0			0
$273$			16.3937i			0.992194i
$274$		0			0
$275$	−4.05379	+	6.68251i	−0.244453	+	0.402971i
$276$		0			0
$277$			14.3234i			0.860607i	0.902684	+	0.430304i	$0.141594\pi$
$277$			14.3234i			0.860607i	−0.902684	+	0.430304i	$0.858406\pi$
$278$		0			0
$279$	12.8107			0.766955
$280$		0			0
$281$			22.7167i			1.35517i	0.735446	+	0.677583i	$0.236973\pi$
$281$			22.7167i			1.35517i	−0.735446	+	0.677583i	$0.763027\pi$
$282$		0			0
$283$	15.4222			0.916756			0.458378	−	0.888757i	$-0.348431\pi$
$283$	15.4222			0.916756			0.458378	+	0.888757i	$0.348431\pi$
$284$		0			0
$285$	−9.70990	+	4.66770i	−0.575165	+	0.276491i
$286$		0			0
$287$			16.9234i			0.998954i
$288$		0			0
$289$	10.2749			0.604407
$290$		0			0
$291$			5.83710i			0.342177i
$292$		0			0
$293$	6.75029			0.394356			0.197178	−	0.980368i	$-0.436822\pi$
$293$	6.75029			0.394356			0.197178	+	0.980368i	$0.436822\pi$
$294$		0			0
$295$	−4.98642	+	17.8340i	−0.290320	+	1.03834i
$296$		0			0
$297$	−8.25610			−0.479068
$298$		0			0
$299$	22.1981			1.28375
$300$		0			0
$301$	12.7607			0.735512
$302$		0			0
$303$			14.6539i			0.841847i
$304$		0			0
$305$	−1.63024	+	5.83059i	−0.0933473	+	0.333859i
$306$		0			0
$307$			26.8520i			1.53253i	0.642527	+	0.766263i	$0.277886\pi$
$307$			26.8520i			1.53253i	−0.642527	+	0.766263i	$0.722114\pi$
$308$		0			0
$309$	6.08747			0.346304
$310$		0			0
$311$		−	14.0339i		−	0.795791i	−0.917431	−	0.397896i	$-0.869741\pi$
$311$		−	14.0339i		−	0.795791i	0.917431	−	0.397896i	$-0.130259\pi$
$312$		0			0
$313$			28.6638i			1.62018i	0.586309	+	0.810088i	$0.300581\pi$
$313$			28.6638i			1.62018i	−0.586309	+	0.810088i	$0.699419\pi$
$314$		0			0
$315$	2.42589	−	8.67627i	0.136684	−	0.488852i
$316$		0			0
$317$	22.6849			1.27411			0.637054	−	0.770819i	$-0.280153\pi$
$317$	22.6849			1.27411			0.637054	+	0.770819i	$0.280153\pi$
$318$		0			0
$319$	0.972261			0.0544362
$320$		0			0
$321$	−3.05849			−0.170708
$322$		0			0
$323$	−7.45986	−	8.49277i	−0.415078	−	0.472550i
$324$		0			0
$325$	27.9834	+	16.9754i	1.55224	+	0.941628i
$326$		0			0
$327$	−16.6829			−0.922569
$328$		0			0
$329$	8.67553			0.478298
$330$		0			0
$331$	−27.6132			−1.51776			−0.758878	−	0.651233i	$-0.774252\pi$
$331$	−27.6132			−1.51776			−0.758878	+	0.651233i	$0.774252\pi$
$332$		0			0
$333$	2.58159			0.141470
$334$		0			0
$335$	−21.5494	−	6.02525i	−1.17737	−	0.329194i
$336$		0			0
$337$	5.85338			0.318854			0.159427	−	0.987210i	$-0.449035\pi$
$337$	5.85338			0.318854			0.159427	+	0.987210i	$0.449035\pi$
$338$		0			0
$339$			3.51460i			0.190887i
$340$		0			0
$341$		−	11.2616i		−	0.609849i
$342$		0			0
$343$	−20.0890			−1.08471
$344$		0			0
$345$	8.07199	+	2.25694i	0.434581	+	0.121509i
$346$		0			0
$347$	26.8862			1.44333			0.721664	−	0.692244i	$-0.243378\pi$
$347$	26.8862			1.44333			0.721664	+	0.692244i	$0.243378\pi$
$348$		0			0
$349$	11.8103			0.632192			0.316096	−	0.948727i	$-0.397628\pi$
$349$	11.8103			0.632192			0.316096	+	0.948727i	$0.397628\pi$
$350$		0			0
$351$			34.5729i			1.84536i
$352$		0			0
$353$		−	3.11618i		−	0.165858i	−0.996555	−	0.0829288i	$-0.973573\pi$
$353$		−	3.11618i		−	0.165858i	0.996555	−	0.0829288i	$-0.0264274\pi$
$354$		0			0
$355$	−4.33777	+	15.5141i	−0.230225	+	0.823405i
$356$		0			0
$357$	−6.49463			−0.343732
$358$		0			0
$359$		−	4.17810i		−	0.220512i	−0.993903	−	0.110256i	$-0.964833\pi$
$359$		−	4.17810i		−	0.220512i	0.993903	−	0.110256i	$-0.0351670\pi$
$360$		0			0
$361$	2.45017	−	18.8414i	0.128956	−	0.991650i
$362$		0			0
$363$		−	9.45780i		−	0.496406i
$364$		0			0
$365$	1.38693	+	0.387787i	0.0725951	+	0.0202977i
$366$		0			0
$367$	−7.28475			−0.380261			−0.190130	−	0.981759i	$-0.560891\pi$
$367$	−7.28475			−0.380261			−0.190130	+	0.981759i	$0.560891\pi$
$368$		0			0
$369$			13.2820i			0.691432i
$370$		0			0
$371$	−13.5226			−0.702056
$372$		0			0
$373$	29.2485			1.51443			0.757215	−	0.653166i	$-0.226559\pi$
$373$	29.2485			1.51443			0.757215	+	0.653166i	$0.226559\pi$
$374$		0			0
$375$	8.44978	+	9.01799i	0.436345	+	0.465687i
$376$		0			0
$377$		−	4.07139i		−	0.209687i
$378$		0			0
$379$	13.2047			0.678280			0.339140	−	0.940736i	$-0.389864\pi$
$379$	13.2047			0.678280			0.339140	+	0.940736i	$0.389864\pi$
$380$		0			0
$381$	−5.72207			−0.293151
$382$		0			0
$383$			4.60434i			0.235271i	0.993057	+	0.117635i	$0.0375314\pi$
$383$			4.60434i			0.235271i	−0.993057	+	0.117635i	$0.962469\pi$
$384$		0			0
$385$	−7.62712	−	2.13255i	−0.388714	−	0.108685i
$386$		0			0
$387$	10.0150			0.509089
$388$		0			0
$389$	15.4651			0.784110			0.392055	−	0.919942i	$-0.371764\pi$
$389$	15.4651			0.784110			0.392055	+	0.919942i	$0.371764\pi$
$390$		0			0
$391$			8.79412i			0.444738i
$392$		0			0
$393$	−1.70745			−0.0861293
$394$		0			0
$395$	7.20423	−	25.7661i	0.362484	−	1.29643i
$396$		0			0
$397$			21.2303i			1.06552i	0.846267	+	0.532759i	$0.178845\pi$
$397$			21.2303i			1.06552i	−0.846267	+	0.532759i	$0.821155\pi$
$398$		0			0
$399$	−7.20423	−	8.20174i	−0.360663	−	0.410601i
$400$		0			0
$401$			7.00176i			0.349651i	0.984599	+	0.174826i	$0.0559362\pi$
$401$			7.00176i			0.349651i	−0.984599	+	0.174826i	$0.944064\pi$
$402$		0			0
$403$	−47.1585			−2.34913
$404$		0			0
$405$	−0.303043	+	1.08384i	−0.0150583	+	0.0538565i
$406$		0			0
$407$		−	2.26942i		−	0.112491i
$408$		0			0
$409$		−	22.4078i		−	1.10799i	−0.832519	−	0.553997i	$-0.813102\pi$
$409$		−	22.4078i		−	1.10799i	0.832519	−	0.553997i	$-0.186898\pi$
$410$		0			0
$411$	−18.1207			−0.893828
$412$		0			0
$413$	−18.7637			−0.923301
$414$		0			0
$415$	5.15435	−	18.4347i	0.253017	−	0.904922i
$416$		0			0
$417$	14.1191			0.691413
$418$		0			0
$419$			39.5697i			1.93311i	0.256463	+	0.966554i	$0.417443\pi$
$419$			39.5697i			1.93311i	−0.256463	+	0.966554i	$0.582557\pi$
$420$		0			0
$421$		−	22.5897i		−	1.10096i	−0.834850	−	0.550478i	$-0.814446\pi$
$421$		−	22.5897i		−	1.10096i	0.834850	−	0.550478i	$-0.185554\pi$
$422$		0			0
$423$	6.80883			0.331057
$424$		0			0
$425$	−6.72508	+	11.0860i	−0.326214	+	0.537752i
$426$		0			0
$427$	−6.13453			−0.296871
$428$		0			0
$429$	11.3105			0.546076
$430$		0			0
$431$	28.6904			1.38197			0.690985	−	0.722869i	$-0.257177\pi$
$431$	28.6904			1.38197			0.690985	+	0.722869i	$0.257177\pi$
$432$		0			0
$433$	20.8825			1.00355			0.501776	−	0.864998i	$-0.332680\pi$
$433$	20.8825			1.00355			0.501776	+	0.864998i	$0.332680\pi$
$434$		0			0
$435$	0.413949	−	1.48050i	0.0198473	−	0.0709845i
$436$		0			0
$437$	−11.1056	+	9.75496i	−0.531255	+	0.466643i
$438$		0			0
$439$	36.8669			1.75956			0.879781	−	0.475379i	$-0.157689\pi$
$439$	36.8669			1.75956			0.879781	+	0.475379i	$0.157689\pi$
$440$		0			0
$441$	−3.31897			−0.158046
$442$		0			0
$443$	20.1220			0.956024			0.478012	−	0.878353i	$-0.341358\pi$
$443$	20.1220			0.956024			0.478012	+	0.878353i	$0.341358\pi$
$444$		0			0
$445$	−15.0781	−	4.21586i	−0.714772	−	0.199851i
$446$		0			0
$447$		−	21.0142i		−	0.993935i
$448$		0			0
$449$		−	39.4272i		−	1.86068i	−0.366692	−	0.930342i	$-0.619510\pi$
$449$		−	39.4272i		−	1.86068i	0.366692	−	0.930342i	$-0.380490\pi$
$450$		0			0
$451$	11.6759			0.549796
$452$		0			0
$453$		−	6.33685i		−	0.297731i
$454$		0			0
$455$	−8.93016	+	31.9390i	−0.418653	+	1.49732i
$456$		0			0
$457$			7.43353i			0.347726i	0.984770	+	0.173863i	$0.0556249\pi$
$457$			7.43353i			0.347726i	−0.984770	+	0.173863i	$0.944375\pi$
$458$		0			0
$459$	−13.6966			−0.639301
$460$		0			0
$461$	−7.32369			−0.341098			−0.170549	−	0.985349i	$-0.554554\pi$
$461$	−7.32369			−0.341098			−0.170549	+	0.985349i	$0.554554\pi$
$462$		0			0
$463$	26.6564			1.23883			0.619413	−	0.785065i	$-0.287370\pi$
$463$	26.6564			1.23883			0.619413	+	0.785065i	$0.287370\pi$
$464$		0			0
$465$	−17.1484	−	4.79472i	−0.795240	−	0.222350i
$466$		0			0
$467$	40.7992			1.88796			0.943980	−	0.330002i	$-0.107049\pi$
$467$	40.7992			1.88796			0.943980	+	0.330002i	$0.107049\pi$
$468$		0			0
$469$		−	22.6728i		−	1.04693i
$470$		0			0
$471$	20.1414			0.928066
$472$		0			0
$473$		−	8.80394i		−	0.404805i
$474$		0			0
$475$	−21.4599	+	3.80453i	−0.984646	+	0.174564i
$476$		0			0
$477$	−10.6129			−0.485932
$478$		0			0
$479$			19.4329i			0.887913i	0.896048	+	0.443956i	$0.146425\pi$
$479$			19.4329i			0.887913i	−0.896048	+	0.443956i	$0.853575\pi$
$480$		0			0
$481$	−9.50330			−0.433313
$482$		0			0
$483$			8.49277i			0.386434i
$484$		0			0
$485$	−3.17965	+	11.3721i	−0.144380	+	0.516380i
$486$		0			0
$487$			15.1908i			0.688363i	0.938903	+	0.344182i	$0.111844\pi$
$487$			15.1908i			0.688363i	−0.938903	+	0.344182i	$0.888156\pi$
$488$		0			0
$489$		−	2.76938i		−	0.125236i
$490$		0			0
$491$			5.67967i			0.256320i	0.991754	+	0.128160i	$0.0409071\pi$
$491$			5.67967i			0.256320i	−0.991754	+	0.128160i	$0.959093\pi$
$492$		0			0
$493$	1.61294			0.0726434
$494$		0			0
$495$	−5.98600	−	1.67369i	−0.269051	−	0.0752269i
$496$		0			0
$497$	−16.3228			−0.732180
$498$		0			0
$499$			10.0410i			0.449498i	0.974417	+	0.224749i	$0.0721562\pi$
$499$			10.0410i			0.449498i	−0.974417	+	0.224749i	$0.927844\pi$
$500$		0			0
$501$	23.7564			1.06136
$502$		0			0
$503$	−43.1196			−1.92261			−0.961305	−	0.275488i	$-0.911161\pi$
$503$	−43.1196			−1.92261			−0.961305	+	0.275488i	$0.911161\pi$
$504$		0			0
$505$	−7.98244	+	28.5494i	−0.355214	+	1.27043i
$506$		0			0
$507$		−	32.9938i		−	1.46531i
$508$		0			0
$509$			25.0941i			1.11228i	0.831089	+	0.556139i	$0.187718\pi$
$509$			25.0941i			1.11228i	−0.831089	+	0.556139i	$0.812282\pi$
$510$		0			0
$511$			1.45923i			0.0645523i
$512$		0			0
$513$	−15.1930	−	17.2967i	−0.670789	−	0.763668i
$514$		0			0
$515$	11.8599	+	3.31603i	0.522608	+	0.146122i
$516$		0			0
$517$		−	5.98549i		−	0.263242i
$518$		0			0
$519$			7.99543i			0.350960i
$520$		0			0
$521$			23.9332i			1.04853i	0.851554	+	0.524267i	$0.175661\pi$
$521$			23.9332i			1.04853i	−0.851554	+	0.524267i	$0.824339\pi$
$522$		0			0
$523$			5.97820i			0.261408i	0.991421	+	0.130704i	$0.0417238\pi$
$523$			5.97820i			0.261408i	−0.991421	+	0.130704i	$0.958276\pi$
$524$		0			0
$525$	−6.49463	+	10.7062i	−0.283449	+	0.467255i
$526$		0			0
$527$		−	18.6826i		−	0.813825i
$528$		0			0
$529$	−11.5003			−0.500012
$530$		0			0
$531$	−14.7263			−0.639068
$532$		0			0
$533$		−	48.8934i		−	2.11781i
$534$		0			0
$535$	−5.95868	−	1.66605i	−0.257616	−	0.0720297i
$536$		0			0
$537$		−	9.94080i		−	0.428977i
$538$		0			0
$539$			2.91764i			0.125671i
$540$		0			0
$541$	−20.5116			−0.881863			−0.440931	−	0.897541i	$-0.645352\pi$
$541$	−20.5116			−0.881863			−0.440931	+	0.897541i	$0.645352\pi$
$542$		0			0
$543$	24.9877			1.07232
$544$		0			0
$545$	−32.5024	−	9.08771i	−1.39225	−	0.389275i
$546$		0			0
$547$			42.8371i			1.83158i	0.401657	+	0.915790i	$0.368434\pi$
$547$			42.8371i			1.83158i	−0.401657	+	0.915790i	$0.631566\pi$
$548$		0			0
$549$	−4.81457			−0.205481
$550$		0			0
$551$	1.78917	+	2.03691i	0.0762214	+	0.0867751i
$552$		0			0
$553$	27.1092			1.15280
$554$		0			0
$555$	−3.45573	−	0.966225i	−0.146687	−	0.0410140i
$556$		0			0
$557$			10.8960i			0.461679i	0.972992	+	0.230839i	$0.0741472\pi$
$557$			10.8960i			0.461679i	−0.972992	+	0.230839i	$0.925853\pi$
$558$		0			0
$559$	−36.8669			−1.55930
$560$		0			0
$561$			4.48083i			0.189181i
$562$		0			0
$563$			6.48445i			0.273287i	0.990620	+	0.136644i	$0.0436315\pi$
$563$			6.48445i			0.273287i	−0.990620	+	0.136644i	$0.956369\pi$
$564$		0			0
$565$	−1.91451	+	6.84730i	−0.0805441	+	0.288068i
$566$		0			0
$567$	−1.14034			−0.0478898
$568$		0			0
$569$			6.72027i			0.281728i	0.990029	+	0.140864i	$0.0449881\pi$
$569$			6.72027i			0.281728i	−0.990029	+	0.140864i	$0.955012\pi$
$570$		0			0
$571$		−	31.1715i		−	1.30449i	−0.758009	−	0.652244i	$-0.773828\pi$
$571$		−	31.1715i		−	1.30449i	0.758009	−	0.652244i	$-0.226172\pi$
$572$		0			0
$573$	18.2115			0.760798
$574$		0			0
$575$	14.4968	+	8.79412i	0.604557	+	0.366740i
$576$		0			0
$577$			26.0017i			1.08246i	0.840874	+	0.541232i	$0.182042\pi$
$577$			26.0017i			1.08246i	−0.840874	+	0.541232i	$0.817958\pi$
$578$		0			0
$579$		−	11.5893i		−	0.481636i
$580$		0			0
$581$	19.3956			0.804666
$582$		0			0
$583$			9.32959i			0.386392i
$584$		0			0
$585$	−7.00867	+	25.0667i	−0.289773	+	1.03638i
$586$		0			0
$587$	15.6911			0.647643			0.323821	−	0.946118i	$-0.395032\pi$
$587$	15.6911			0.647643			0.323821	+	0.946118i	$0.395032\pi$
$588$		0			0
$589$	23.5933	−	20.7238i	0.972143	−	0.853909i
$590$		0			0
$591$	−2.28105			−0.0938298
$592$		0			0
$593$		−	43.6835i		−	1.79387i	−0.442167	−	0.896933i	$-0.645790\pi$
$593$		−	43.6835i		−	1.79387i	0.442167	−	0.896933i	$-0.354210\pi$
$594$		0			0
$595$	−12.6531	−	3.53782i	−0.518727	−	0.145037i
$596$		0			0
$597$	11.9549			0.489280
$598$		0			0
$599$	−1.47131			−0.0601160			−0.0300580	−	0.999548i	$-0.509569\pi$
$599$	−1.47131			−0.0601160			−0.0300580	+	0.999548i	$0.509569\pi$
$600$		0			0
$601$		−	1.21650i		−	0.0496221i	−0.999692	−	0.0248111i	$-0.992102\pi$
$601$		−	1.21650i		−	0.0496221i	0.999692	−	0.0248111i	$-0.00789841\pi$
$602$		0			0
$603$		−	17.7943i		−	0.724639i
$604$		0			0
$605$	5.15195	−	18.4261i	0.209456	−	0.749126i
$606$		0			0
$607$		−	48.4686i		−	1.96728i	−0.180145	−	0.983640i	$-0.557657\pi$
$607$		−	48.4686i		−	1.96728i	0.180145	−	0.983640i	$-0.442343\pi$
$608$		0			0
$609$	1.55767			0.0631201
$610$		0			0
$611$	−25.0646			−1.01400
$612$		0			0
$613$			35.1789i			1.42086i	0.703766	+	0.710432i	$0.251500\pi$
$613$			35.1789i			1.42086i	−0.703766	+	0.710432i	$0.748500\pi$
$614$		0			0
$615$	4.97112	−	17.7793i	0.200455	−	0.716931i
$616$		0			0
$617$		−	23.4084i		−	0.942386i	−0.882030	−	0.471193i	$-0.843824\pi$
$617$		−	23.4084i		−	0.942386i	0.882030	−	0.471193i	$-0.156176\pi$
$618$		0			0
$619$		−	18.1615i		−	0.729974i	−0.931013	−	0.364987i	$-0.881073\pi$
$619$		−	18.1615i		−	0.729974i	0.931013	−	0.364987i	$-0.118927\pi$
$620$		0			0
$621$			17.9104i			0.718722i
$622$		0			0
$623$		−	15.8641i		−	0.635582i
$624$		0			0
$625$	11.5498	+	22.1721i	0.461993	+	0.886883i
$626$		0			0
$627$	−5.65861	+	4.97040i	−0.225983	+	0.198499i
$628$		0			0
$629$		−	3.76488i		−	0.150116i
$630$		0			0
$631$			27.8830i			1.11000i	0.831849	+	0.555002i	$0.187283\pi$
$631$			27.8830i			1.11000i	−0.831849	+	0.555002i	$0.812717\pi$
$632$		0			0
$633$		−	11.8549i		−	0.471190i
$634$		0			0
$635$	−11.1480	−	3.11699i	−0.442394	−	0.123694i
$636$		0			0
$637$	12.2177			0.484085
$638$		0			0
$639$	−12.8107			−0.506782
$640$		0			0
$641$			2.46044i			0.0971817i	0.998819	+	0.0485909i	$0.0154730\pi$
$641$			2.46044i			0.0971817i	−0.998819	+	0.0485909i	$0.984527\pi$
$642$		0			0
$643$	−16.4483			−0.648658			−0.324329	−	0.945944i	$-0.605138\pi$
$643$	−16.4483			−0.648658			−0.324329	+	0.945944i	$0.605138\pi$
$644$		0			0
$645$	−13.4061	−	3.74835i	−0.527864	−	0.147591i
$646$		0			0
$647$	−26.2980			−1.03388			−0.516941	−	0.856021i	$-0.672929\pi$
$647$	−26.2980			−1.03388			−0.516941	+	0.856021i	$0.672929\pi$
$648$		0			0
$649$			12.9456i			0.508159i
$650$		0			0
$651$		−	18.0423i		−	0.707135i
$652$		0			0
$653$			17.4178i			0.681613i	0.940134	+	0.340806i	$0.110700\pi$
$653$			17.4178i			0.681613i	−0.940134	+	0.340806i	$0.889300\pi$
$654$		0			0
$655$	−3.32652	−	0.930098i	−0.129978	−	0.0363419i
$656$		0			0
$657$			1.14525i			0.0446803i
$658$		0			0
$659$	20.5354			0.799946			0.399973	−	0.916527i	$-0.369020\pi$
$659$	20.5354			0.799946			0.399973	+	0.916527i	$0.369020\pi$
$660$		0			0
$661$			1.39841i			0.0543919i	0.999630	+	0.0271959i	$0.00865780\pi$
$661$			1.39841i			0.0543919i	−0.999630	+	0.0271959i	$0.991342\pi$
$662$		0			0
$663$	18.7637			0.728721
$664$		0			0
$665$	−9.56784	−	19.9033i	−0.371025	−	0.771818i
$666$		0			0
$667$		−	2.10918i		−	0.0816679i
$668$		0			0
$669$	−2.77946			−0.107460
$670$		0			0
$671$			4.23238i			0.163389i
$672$		0			0
$673$	−21.6241			−0.833547			−0.416773	−	0.909010i	$-0.636839\pi$
$673$	−21.6241			−0.833547			−0.416773	+	0.909010i	$0.636839\pi$
$674$		0			0
$675$	−13.6966	+	22.5783i	−0.527181	+	0.869038i
$676$		0			0
$677$	−34.0030			−1.30684			−0.653420	−	0.756995i	$-0.726667\pi$
$677$	−34.0030			−1.30684			−0.653420	+	0.756995i	$0.726667\pi$
$678$		0			0
$679$	−11.9649			−0.459170
$680$		0			0
$681$	−26.8932			−1.03055
$682$		0			0
$683$			2.06308i			0.0789416i	0.999221	+	0.0394708i	$0.0125672\pi$
$683$			2.06308i			0.0789416i	−0.999221	+	0.0394708i	$0.987433\pi$
$684$		0			0
$685$	−35.3035	−	9.87090i	−1.34888	−	0.377148i
$686$		0			0
$687$			3.36747i			0.128477i
$688$		0			0
$689$	39.0681			1.48838
$690$		0			0
$691$		−	21.9908i		−	0.836569i	−0.908316	−	0.418285i	$-0.862631\pi$
$691$		−	21.9908i		−	0.836569i	0.908316	−	0.418285i	$-0.137369\pi$
$692$		0			0
$693$		−	6.29804i		−	0.239243i
$694$		0			0
$695$	27.5073	+	7.69108i	1.04341	+	0.291739i
$696$		0			0
$697$	19.3699			0.733686
$698$		0			0
$699$	16.0423			0.606775
$700$		0			0
$701$	−39.7509			−1.50137			−0.750686	−	0.660659i	$-0.770277\pi$
$701$	−39.7509			−1.50137			−0.750686	+	0.660659i	$0.770277\pi$
$702$		0			0
$703$	4.75447	−	4.17623i	0.179318	−	0.157509i
$704$		0			0
$705$	−9.11434	−	2.54838i	−0.343266	−	0.0959775i
$706$		0			0
$707$	−30.0376			−1.12968
$708$		0			0
$709$	−17.8947			−0.672048			−0.336024	−	0.941853i	$-0.609082\pi$
$709$	−17.8947			−0.672048			−0.336024	+	0.941853i	$0.609082\pi$
$710$		0			0
$711$	21.2762			0.797918
$712$		0			0
$713$	−24.4304			−0.914927
$714$		0			0
$715$	22.0356	+	6.16117i	0.824084	+	0.230415i
$716$		0			0
$717$	4.33765			0.161993
$718$		0			0
$719$		−	38.1437i		−	1.42252i	−0.702929	−	0.711260i	$-0.748125\pi$
$719$		−	38.1437i		−	1.42252i	0.702929	−	0.711260i	$-0.251875\pi$
$720$		0			0
$721$			12.4781i			0.464708i
$722$		0			0
$723$	−18.7454			−0.697150
$724$		0			0
$725$	1.61294	−	2.65888i	0.0599033	−	0.0987483i
$726$		0			0
$727$	36.8366			1.36619			0.683096	−	0.730328i	$-0.260633\pi$
$727$	36.8366			1.36619			0.683096	+	0.730328i	$0.260633\pi$
$728$		0			0
$729$	−18.4088			−0.681808
$730$		0			0
$731$		−	14.6054i		−	0.540200i
$732$		0			0
$733$		−	19.5959i		−	0.723792i	−0.932219	−	0.361896i	$-0.882130\pi$
$733$		−	19.5959i		−	0.723792i	0.932219	−	0.361896i	$-0.117870\pi$
$734$		0			0
$735$	4.44279	+	1.24221i	0.163875	+	0.0458196i
$736$		0			0
$737$	−15.6426			−0.576202
$738$		0			0
$739$		−	36.8500i		−	1.35555i	−0.735269	−	0.677775i	$-0.762944\pi$
$739$		−	36.8500i		−	1.35555i	0.735269	−	0.677775i	$-0.237056\pi$
$740$		0			0
$741$	20.8138	+	23.6957i	0.764614	+	0.870484i
$742$		0			0
$743$		−	22.1001i		−	0.810774i	−0.914145	−	0.405387i	$-0.867137\pi$
$743$		−	22.1001i		−	0.810774i	0.914145	−	0.405387i	$-0.132863\pi$
$744$		0			0
$745$	11.4470	−	40.9406i	0.419387	−	1.49995i
$746$		0			0
$747$	15.2223			0.556954
$748$		0			0
$749$		−	6.26929i		−	0.229075i
$750$		0			0
$751$	15.7058			0.573111			0.286556	−	0.958064i	$-0.407490\pi$
$751$	15.7058			0.573111			0.286556	+	0.958064i	$0.407490\pi$
$752$		0			0
$753$	−31.9858			−1.16563
$754$		0			0
$755$	3.45187	−	12.3457i	0.125627	−	0.449306i
$756$		0			0
$757$		−	13.2641i		−	0.482093i	−0.970514	−	0.241046i	$-0.922509\pi$
$757$		−	13.2641i		−	0.482093i	0.970514	−	0.241046i	$-0.0774906\pi$
$758$		0			0
$759$	5.85940			0.212683
$760$		0			0
$761$	−4.74310			−0.171937			−0.0859687	−	0.996298i	$-0.527399\pi$
$761$	−4.74310			−0.171937			−0.0859687	+	0.996298i	$0.527399\pi$
$762$		0			0
$763$		−	34.1967i		−	1.23800i
$764$		0			0
$765$	−9.93055	−	2.77659i	−0.359040	−	0.100388i
$766$		0			0
$767$	54.2103			1.95742
$768$		0			0
$769$	−11.5045			−0.414864			−0.207432	−	0.978249i	$-0.566511\pi$
$769$	−11.5045			−0.414864			−0.207432	+	0.978249i	$0.566511\pi$
$770$		0			0
$771$		−	18.6241i		−	0.670732i
$772$		0			0
$773$	−20.9366			−0.753036			−0.376518	−	0.926409i	$-0.622879\pi$
$773$	−20.9366			−0.753036			−0.376518	+	0.926409i	$0.622879\pi$
$774$		0			0
$775$	−30.7975	−	18.6826i	−1.10628	−	0.671097i
$776$		0			0
$777$		−	3.63587i		−	0.130436i
$778$		0			0
$779$	21.4862	+	24.4612i	0.769823	+	0.876415i
$780$		0			0
$781$			11.2616i			0.402971i
$782$		0			0
$783$	3.28499			0.117396
$784$		0			0
$785$	39.2403	+	10.9716i	1.40055	+	0.391594i
$786$		0			0
$787$			21.6239i			0.770808i	0.922748	+	0.385404i	$0.125938\pi$
$787$			21.6239i			0.770808i	−0.922748	+	0.385404i	$0.874062\pi$
$788$		0			0
$789$			26.8886i			0.957261i
$790$		0			0
$791$	−7.20423			−0.256153
$792$		0			0
$793$	17.7233			0.629373
$794$		0			0
$795$	14.2065	+	3.97216i	0.503853	+	0.140878i
$796$		0			0
$797$	−21.2844			−0.753932			−0.376966	−	0.926227i	$-0.623033\pi$
$797$	−21.2844			−0.753932			−0.376966	+	0.926227i	$0.623033\pi$
$798$		0			0
$799$		−	9.92971i		−	0.351288i
$800$		0			0
$801$		−	12.4506i		−	0.439922i
$802$		0			0
$803$	1.00676			0.0355278
$804$		0			0
$805$	−4.62627	+	16.5460i	−0.163055	+	0.583168i
$806$		0			0
$807$	−29.9102			−1.05289
$808$		0			0
$809$	19.8028			0.696231			0.348115	−	0.937452i	$-0.386822\pi$
$809$	19.8028			0.696231			0.348115	+	0.937452i	$0.386822\pi$
$810$		0			0
$811$	−25.1696			−0.883824			−0.441912	−	0.897059i	$-0.645700\pi$
$811$	−25.1696			−0.883824			−0.441912	+	0.897059i	$0.645700\pi$
$812$		0			0
$813$	−17.3632			−0.608955
$814$		0			0
$815$	1.50856	−	5.39542i	0.0528427	−	0.188993i
$816$		0			0
$817$	18.4444	−	16.2012i	0.645289	−	0.566807i
$818$		0			0
$819$	−26.3733			−0.921559
$820$		0			0
$821$	−30.7467			−1.07307			−0.536533	−	0.843879i	$-0.680266\pi$
$821$	−30.7467			−1.07307			−0.536533	+	0.843879i	$0.680266\pi$
$822$		0			0
$823$	−12.4242			−0.433079			−0.216539	−	0.976274i	$-0.569477\pi$
$823$	−12.4242			−0.433079			−0.216539	+	0.976274i	$0.569477\pi$
$824$		0			0
$825$	7.38647	+	4.48083i	0.257164	+	0.156002i
$826$		0			0
$827$		−	29.3152i		−	1.01939i	−0.860356	−	0.509694i	$-0.829759\pi$
$827$		−	29.3152i		−	1.01939i	0.860356	−	0.509694i	$-0.170241\pi$
$828$		0			0
$829$			43.8141i			1.52173i	0.648913	+	0.760863i	$0.275224\pi$
$829$			43.8141i			1.52173i	−0.648913	+	0.760863i	$0.724776\pi$
$830$		0			0
$831$	15.8322			0.549214
$832$		0			0
$833$			4.84025i			0.167705i
$834$		0			0
$835$	46.2832	+	12.9408i	1.60170	+	0.447836i
$836$		0			0
$837$		−	38.0496i		−	1.31519i
$838$		0			0
$839$	15.7533			0.543864			0.271932	−	0.962316i	$-0.412337\pi$
$839$	15.7533			0.543864			0.271932	+	0.962316i	$0.412337\pi$
$840$		0			0
$841$	28.6132			0.986660
$842$		0			0
$843$	25.1098			0.864827
$844$		0			0
$845$	17.9727	−	64.2799i	0.618281	−	2.21130i
$846$		0			0
$847$	19.3866			0.666131
$848$		0			0
$849$		−	17.0468i		−	0.585046i
$850$		0			0
$851$	−4.92318			−0.168764
$852$		0			0
$853$			11.0343i			0.377806i	0.981996	+	0.188903i	$0.0604931\pi$
$853$			11.0343i			0.377806i	−0.981996	+	0.188903i	$0.939507\pi$
$854$		0			0
$855$	−7.50914	−	15.6208i	−0.256807	−	0.534218i
$856$		0			0
$857$	−55.8137			−1.90656			−0.953280	−	0.302089i	$-0.902316\pi$
$857$	−55.8137			−1.90656			−0.953280	+	0.302089i	$0.902316\pi$
$858$		0			0
$859$			50.0771i			1.70861i	0.519772	+	0.854305i	$0.326017\pi$
$859$			50.0771i			1.70861i	−0.519772	+	0.854305i	$0.673983\pi$
$860$		0			0
$861$	18.7061			0.637503
$862$		0			0
$863$		−	29.8836i		−	1.01725i	−0.860988	−	0.508625i	$-0.830154\pi$
$863$		−	29.8836i		−	1.01725i	0.860988	−	0.508625i	$-0.169846\pi$
$864$		0			0
$865$	−4.35535	+	15.5770i	−0.148086	+	0.529635i
$866$		0			0
$867$		−	11.3573i		−	0.385715i
$868$		0			0
$869$		−	18.7034i		−	0.634470i
$870$		0			0
$871$			65.5041i			2.21952i
$872$		0			0
$873$	−9.39041			−0.317817
$874$		0			0
$875$	−18.4851	+	17.3203i	−0.624910	+	0.585535i
$876$		0			0
$877$	−37.5097			−1.26661			−0.633306	−	0.773902i	$-0.718302\pi$
$877$	−37.5097			−1.26661			−0.633306	+	0.773902i	$0.718302\pi$
$878$		0			0
$879$		−	7.46139i		−	0.251666i
$880$		0			0
$881$	−21.7377			−0.732363			−0.366182	−	0.930543i	$-0.619335\pi$
$881$	−21.7377			−0.732363			−0.366182	+	0.930543i	$0.619335\pi$
$882$		0			0
$883$	−34.3443			−1.15578			−0.577889	−	0.816115i	$-0.696123\pi$
$883$	−34.3443			−1.15578			−0.577889	+	0.816115i	$0.696123\pi$
$884$		0			0
$885$	19.7127	+	5.51170i	0.662637	+	0.185274i
$886$		0			0
$887$			4.60434i			0.154599i	0.997008	+	0.0772993i	$0.0246297\pi$
$887$			4.60434i			0.154599i	−0.997008	+	0.0772993i	$0.975370\pi$
$888$		0			0
$889$		−	11.7291i		−	0.393381i
$890$		0			0
$891$			0.786753i			0.0263572i
$892$		0			0
$893$	12.5397	−	11.0146i	0.419626	−	0.368590i
$894$		0			0
$895$	5.41505	−	19.3671i	0.181005	−	0.647370i
$896$		0			0
$897$		−	24.5365i		−	0.819250i
$898$		0			0
$899$			4.48083i			0.149444i
$900$		0			0
$901$			15.4774i			0.515629i
$902$		0			0
$903$		−	14.1049i		−	0.469382i
$904$		0			0
$905$	48.6820	+	13.6115i	1.61824	+	0.452463i
$906$		0			0
$907$			19.9622i			0.662834i	0.943484	+	0.331417i	$0.107527\pi$
$907$			19.9622i			0.662834i	−0.943484	+	0.331417i	$0.892473\pi$
$908$		0			0
$909$	−23.5745			−0.781915
$910$		0			0
$911$	−22.2960			−0.738698			−0.369349	−	0.929291i	$-0.620419\pi$
$911$	−22.2960			−0.738698			−0.369349	+	0.929291i	$0.620419\pi$
$912$		0			0
$913$		−	13.3816i		−	0.442866i
$914$		0			0
$915$	6.44481	+	1.80198i	0.213059	+	0.0595715i
$916$		0			0
$917$		−	3.49992i		−	0.115578i
$918$		0			0
$919$			24.5781i			0.810755i	0.914149	+	0.405378i	$0.132860\pi$
$919$			24.5781i			0.810755i	−0.914149	+	0.405378i	$0.867140\pi$
$920$		0			0
$921$	29.6807			0.978013
$922$		0			0
$923$	47.1585			1.55224
$924$		0			0
$925$	−6.20626	−	3.76488i	−0.204061	−	0.123788i
$926$		0			0
$927$			9.79319i			0.321651i
$928$		0			0
$929$	−12.1269			−0.397872			−0.198936	−	0.980012i	$-0.563749\pi$
$929$	−12.1269			−0.397872			−0.198936	+	0.980012i	$0.563749\pi$
$930$		0			0
$931$	−6.11250	+	5.36909i	−0.200329	+	0.175965i
$932$		0			0
$933$	−15.5123			−0.507850
$934$		0			0
$935$	−2.44084	+	8.72973i	−0.0798241	+	0.285493i
$936$		0			0
$937$		−	17.8066i		−	0.581717i	−0.956766	−	0.290859i	$-0.906059\pi$
$937$		−	17.8066i		−	0.581717i	0.956766	−	0.290859i	$-0.0939409\pi$
$938$		0			0
$939$	31.6834			1.03395
$940$		0			0
$941$		−	33.0749i		−	1.07821i	−0.842239	−	0.539105i	$-0.818763\pi$
$941$		−	33.0749i		−	1.07821i	0.842239	−	0.539105i	$-0.181237\pi$
$942$		0			0
$943$		−	25.3292i		−	0.824832i
$944$		0			0
$945$	−25.7698	−	7.20527i	−0.838292	−	0.234387i
$946$		0			0
$947$	−17.4234			−0.566185			−0.283092	−	0.959093i	$-0.591360\pi$
$947$	−17.4234			−0.566185			−0.283092	+	0.959093i	$0.591360\pi$
$948$		0			0
$949$		−	4.21586i		−	0.136853i
$950$		0			0
$951$		−	25.0746i		−	0.813098i
$952$		0			0
$953$	−54.0441			−1.75066			−0.875330	−	0.483526i	$-0.839356\pi$
$953$	−54.0441			−1.75066			−0.875330	+	0.483526i	$0.839356\pi$
$954$		0			0
$955$	35.4805	+	9.92037i	1.14812	+	0.321016i
$956$		0			0
$957$		−	1.07468i		−	0.0347395i
$958$		0			0
$959$		−	37.1438i		−	1.19944i
$960$		0			0
$961$	20.9009			0.674223
$962$		0			0
$963$		−	4.92033i		−	0.158555i
$964$		0			0
$965$	6.31305	−	22.5788i	0.203224	−	0.726837i
$966$		0			0
$967$	44.9325			1.44493			0.722466	−	0.691406i	$-0.243009\pi$
$967$	44.9325			1.44493			0.722466	+	0.691406i	$0.243009\pi$
$968$		0			0
$969$	−9.38742	+	8.24571i	−0.301567	+	0.264890i
$970$		0			0
$971$	−61.2771			−1.96648			−0.983238	−	0.182328i	$-0.941637\pi$
$971$	−61.2771			−1.96648			−0.983238	+	0.182328i	$0.941637\pi$
$972$		0			0
$973$			28.9412i			0.927814i
$974$		0			0
$975$	18.7637	−	30.9312i	0.600919	−	0.990592i
$976$		0			0
$977$	27.9021			0.892666			0.446333	−	0.894867i	$-0.352730\pi$
$977$	27.9021			0.892666			0.446333	+	0.894867i	$0.352730\pi$
$978$		0			0
$979$	−10.9451			−0.349807
$980$		0			0
$981$		−	26.8386i		−	0.856891i
$982$		0			0
$983$			41.1754i			1.31329i	0.754199	+	0.656646i	$0.228025\pi$
$983$			41.1754i			1.31329i	−0.754199	+	0.656646i	$0.771975\pi$
$984$		0			0
$985$	−4.44403	−	1.24256i	−0.141599	−	0.0395911i
$986$		0			0
$987$		−	9.58944i		−	0.305235i
$988$		0			0
$989$	−19.0989			−0.607310
$990$		0			0
$991$	57.8324			1.83711			0.918553	−	0.395297i	$-0.129358\pi$
$991$	57.8324			1.83711			0.918553	+	0.395297i	$0.129358\pi$
$992$		0			0
$993$			30.5220i			0.968587i
$994$		0			0
$995$	23.2910	+	6.51218i	0.738373	+	0.206450i
$996$		0			0
$997$		−	3.26069i		−	0.103267i	−0.998666	−	0.0516335i	$-0.983557\pi$
$997$		−	3.26069i		−	0.103267i	0.998666	−	0.0516335i	$-0.0164428\pi$
$998$		0			0
$999$		−	7.66770i		−	0.242595i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.g.g.1519.8	yes	16
4.3	odd	2		1520.2.g.e.1519.9	yes	16
5.4	even	2	inner	1520.2.g.g.1519.9	yes	16
19.18	odd	2		1520.2.g.e.1519.10	yes	16
20.19	odd	2		1520.2.g.e.1519.8	yes	16
76.75	even	2	inner	1520.2.g.g.1519.7	yes	16
95.94	odd	2		1520.2.g.e.1519.7	✓	16
380.379	even	2	inner	1520.2.g.g.1519.10	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1520.2.g.e.1519.7	✓	16	95.94	odd	2
1520.2.g.e.1519.8	yes	16	20.19	odd	2
1520.2.g.e.1519.9	yes	16	4.3	odd	2
1520.2.g.e.1519.10	yes	16	19.18	odd	2
1520.2.g.g.1519.7	yes	16	76.75	even	2	inner
1520.2.g.g.1519.8	yes	16	1.1	even	1	trivial
1520.2.g.g.1519.9	yes	16	5.4	even	2	inner
1520.2.g.g.1519.10	yes	16	380.379	even	2	inner

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.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.10534i q^{3} +(0.602114 - 2.15348i) q^{5} +2.26573 q^{7} +1.77822 q^{9} +O(q^{10})\)	\(q-1.10534i q^{3} +(0.602114 - 2.15348i) q^{5} +2.26573 q^{7} +1.77822 q^{9} -1.56319i q^{11} -6.54594 q^{13} +(-2.38033 - 0.665543i) q^{15} -2.59328i q^{17} +(3.27492 - 2.87662i) q^{19} -2.50441i q^{21} -3.39112 q^{23} +(-4.27492 - 2.59328i) q^{25} -5.28157i q^{27} +0.621972i q^{29} +7.20423 q^{31} -1.72786 q^{33} +(1.36423 - 4.87920i) q^{35} +1.45178 q^{37} +7.23551i q^{39} +7.46927i q^{41} +5.63203 q^{43} +(1.07069 - 3.82935i) q^{45} +3.82902 q^{47} -1.86646 q^{49} -2.86646 q^{51} -5.96830 q^{53} +(-3.36630 - 0.941220i) q^{55} +(-3.17965 - 3.61991i) q^{57} -8.28152 q^{59} -2.70753 q^{61} +4.02896 q^{63} +(-3.94140 + 14.0965i) q^{65} -10.0068i q^{67} +3.74835i q^{69} -7.20423 q^{71} +0.644042i q^{73} +(-2.86646 + 4.72525i) q^{75} -3.54177i q^{77} +11.9649 q^{79} -0.503299 q^{81} +8.56042 q^{83} +(-5.58456 - 1.56145i) q^{85} +0.687492 q^{87} -7.00176i q^{89} -14.8313 q^{91} -7.96315i q^{93} +(-4.22285 - 8.78451i) q^{95} -5.28081 q^{97} -2.77969i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{9}+O(q^{10}) \) 16 * q - 32 * q^9	\( 16 q - 32 q^{9} + 32 q^{15} - 8 q^{19} - 8 q^{25} + 96 q^{31} - 24 q^{45} - 8 q^{49} - 24 q^{51} - 72 q^{59} - 24 q^{61} - 96 q^{71} - 24 q^{75} + 32 q^{79} - 8 q^{81} - 56 q^{91}+O(q^{100}) \) 16 * q - 32 * q^9 + 32 * q^15 - 8 * q^19 - 8 * q^25 + 96 * q^31 - 24 * q^45 - 8 * q^49 - 24 * q^51 - 72 * q^59 - 24 * q^61 - 96 * q^71 - 24 * q^75 + 32 * q^79 - 8 * q^81 - 56 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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