LMFDB - Embedded newform 380.3.g.c.189.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [380,3,Mod(189,380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(380, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("380.189");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 380 = 2^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	380.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:	$10.3542500457$
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} + 20x^{10} + 44x^{8} - 270x^{6} + 36676x^{4} - 71664x^{2} + 687241 $ x^12 + 20x^10 + 44x^8 - 270x^6 + 36676x^4 - 71664*x^2 + 687241
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{19} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			189.12
Root			$-2.80718 - 1.46321i$ of defining polynomial
Character	$\chi$	$=$	380.189
Dual form			380.3.g.c.189.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.61436 q^{3} +(1.48938 + 4.77302i) q^{5} +7.03410i q^{7} +22.5210 q^{9} +O(q^{10})$	$q+5.61436 q^{3} +(1.48938 + 4.77302i) q^{5} +7.03410i q^{7} +22.5210 q^{9} -1.02125 q^{11} -11.3480 q^{13} +(8.36189 + 26.7975i) q^{15} -16.5801i q^{17} +(-9.54227 - 16.4300i) q^{19} +39.4919i q^{21} +35.1705i q^{23} +(-20.5635 + 14.2177i) q^{25} +75.9119 q^{27} -6.63193i q^{29} -25.3890i q^{31} -5.73365 q^{33} +(-33.5739 + 10.4764i) q^{35} +47.8401 q^{37} -63.7118 q^{39} -12.1251i q^{41} -50.0927i q^{43} +(33.5423 + 107.493i) q^{45} -78.3252i q^{47} -0.478526 q^{49} -93.0869i q^{51} +45.0342 q^{53} +(-1.52102 - 4.87444i) q^{55} +(-53.5737 - 92.2439i) q^{57} +9.44897i q^{59} -43.1487 q^{61} +158.415i q^{63} +(-16.9015 - 54.1643i) q^{65} +112.523 q^{67} +197.460i q^{69} -40.3311i q^{71} +45.4211i q^{73} +(-115.451 + 79.8230i) q^{75} -7.18356i q^{77} +144.704i q^{79} +223.507 q^{81} +56.8706i q^{83} +(79.1374 - 24.6941i) q^{85} -37.2341i q^{87} -81.8009i q^{89} -79.8230i q^{91} -142.543i q^{93} +(64.2088 - 70.0159i) q^{95} -103.506 q^{97} -22.9995 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{5} + 76 q^{9}+O(q^{10}) $ 12 * q + 12 * q^5 + 76 * q^9	$ 12 q + 12 q^{5} + 76 q^{9} - 24 q^{11} + 68 q^{19} - 76 q^{25} - 32 q^{35} - 264 q^{39} + 220 q^{45} + 212 q^{49} + 176 q^{55} - 600 q^{61} + 492 q^{81} + 408 q^{85} + 124 q^{95} + 136 q^{99}+O(q^{100}) $ 12 * q + 12 * q^5 + 76 * q^9 - 24 * q^11 + 68 * q^19 - 76 * q^25 - 32 * q^35 - 264 * q^39 + 220 * q^45 + 212 * q^49 + 176 * q^55 - 600 * q^61 + 492 * q^81 + 408 * q^85 + 124 * q^95 + 136 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/380\mathbb{Z}\right)^\times$.

$n$	$21$	$77$	$191$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	5.61436			1.87145			0.935726	−	0.352727i	$-0.114745\pi$
$3$	5.61436			1.87145			0.935726	+	0.352727i	$0.114745\pi$
$4$		0			0
$5$	1.48938	+	4.77302i	0.297875	+	0.954605i
$5$	1.48938	+	4.77302i	0.297875	+	0.954605i
$6$		0			0
$7$			7.03410i			1.00487i	0.864615	+	0.502436i	$0.167563\pi$
$7$			7.03410i			1.00487i	−0.864615	+	0.502436i	$0.832437\pi$
$8$		0			0
$9$	22.5210			2.50234
$10$		0			0
$11$	−1.02125			−0.0928407			−0.0464204	−	0.998922i	$-0.514781\pi$
$11$	−1.02125			−0.0928407			−0.0464204	+	0.998922i	$0.514781\pi$
$12$		0			0
$13$	−11.3480			−0.872924			−0.436462	−	0.899723i	$-0.643769\pi$
$13$	−11.3480			−0.872924			−0.436462	+	0.899723i	$0.643769\pi$
$14$		0			0
$15$	8.36189	+	26.7975i	0.557459	+	1.78650i
$16$		0			0
$17$		−	16.5801i		−	0.975303i	−0.873038	−	0.487651i	$-0.837854\pi$
$17$		−	16.5801i		−	0.975303i	0.873038	−	0.487651i	$-0.162146\pi$
$18$		0			0
$19$	−9.54227	−	16.4300i	−0.502225	−	0.864737i
$19$	−9.54227	−	16.4300i	−0.502225	−	0.864737i
$20$		0			0
$21$			39.4919i			1.88057i
$22$		0			0
$23$			35.1705i			1.52915i	0.644534	+	0.764576i	$0.277051\pi$
$23$			35.1705i			1.52915i	−0.644534	+	0.764576i	$0.722949\pi$
$24$		0			0
$25$	−20.5635	+	14.2177i	−0.822541	+	0.568706i
$26$		0			0
$27$	75.9119			2.81155
$28$		0			0
$29$		−	6.63193i		−	0.228687i	−0.993441	−	0.114344i	$-0.963523\pi$
$29$		−	6.63193i		−	0.228687i	0.993441	−	0.114344i	$-0.0364765\pi$
$30$		0			0
$31$		−	25.3890i		−	0.818998i	−0.912310	−	0.409499i	$-0.865703\pi$
$31$		−	25.3890i		−	0.818998i	0.912310	−	0.409499i	$-0.134297\pi$
$32$		0			0
$33$	−5.73365			−0.173747
$34$		0			0
$35$	−33.5739	+	10.4764i	−0.959255	+	0.299326i
$36$		0			0
$37$	47.8401			1.29297			0.646487	−	0.762925i	$-0.276237\pi$
$37$	47.8401			1.29297			0.646487	+	0.762925i	$0.276237\pi$
$38$		0			0
$39$	−63.7118			−1.63364
$40$		0			0
$41$		−	12.1251i		−	0.295734i	−0.989007	−	0.147867i	$-0.952759\pi$
$41$		−	12.1251i		−	0.295734i	0.989007	−	0.147867i	$-0.0472407\pi$
$42$		0			0
$43$		−	50.0927i		−	1.16495i	−0.812850	−	0.582474i	$-0.802085\pi$
$43$		−	50.0927i		−	1.16495i	0.812850	−	0.582474i	$-0.197915\pi$
$44$		0			0
$45$	33.5423	+	107.493i	0.745384	+	2.38874i
$46$		0			0
$47$		−	78.3252i		−	1.66649i	−0.552901	−	0.833247i	$-0.686479\pi$
$47$		−	78.3252i		−	1.66649i	0.552901	−	0.833247i	$-0.313521\pi$
$48$		0			0
$49$	−0.478526			−0.00976584
$50$		0			0
$51$		−	93.0869i		−	1.82523i
$52$		0			0
$53$	45.0342			0.849701			0.424851	−	0.905263i	$-0.360327\pi$
$53$	45.0342			0.849701			0.424851	+	0.905263i	$0.360327\pi$
$54$		0			0
$55$	−1.52102	−	4.87444i	−0.0276549	−	0.0886262i
$56$		0			0
$57$	−53.5737	−	92.2439i	−0.939890	−	1.61831i
$58$		0			0
$59$			9.44897i			0.160152i	0.996789	+	0.0800760i	$0.0255163\pi$
$59$			9.44897i			0.160152i	−0.996789	+	0.0800760i	$0.974484\pi$
$60$		0			0
$61$	−43.1487			−0.707356			−0.353678	−	0.935367i	$-0.615069\pi$
$61$	−43.1487			−0.707356			−0.353678	+	0.935367i	$0.615069\pi$
$62$		0			0
$63$			158.415i			2.51452i
$64$		0			0
$65$	−16.9015	−	54.1643i	−0.260022	−	0.833297i
$66$		0			0
$67$	112.523			1.67945			0.839726	−	0.543011i	$-0.182716\pi$
$67$	112.523			1.67945			0.839726	+	0.543011i	$0.182716\pi$
$68$		0			0
$69$			197.460i			2.86174i
$70$		0			0
$71$		−	40.3311i		−	0.568043i	−0.958818	−	0.284022i	$-0.908331\pi$
$71$		−	40.3311i		−	0.568043i	0.958818	−	0.284022i	$-0.0916687\pi$
$72$		0			0
$73$			45.4211i			0.622207i	0.950376	+	0.311104i	$0.100699\pi$
$73$			45.4211i			0.622207i	−0.950376	+	0.311104i	$0.899301\pi$
$74$		0			0
$75$	−115.451	+	79.8230i	−1.53935	+	1.06431i
$76$		0			0
$77$		−	7.18356i		−	0.0932930i
$78$		0			0
$79$			144.704i			1.83170i	0.401526	+	0.915848i	$0.368480\pi$
$79$			144.704i			1.83170i	−0.401526	+	0.915848i	$0.631520\pi$
$80$		0			0
$81$	223.507			2.75935
$82$		0			0
$83$			56.8706i			0.685188i	0.939484	+	0.342594i	$0.111306\pi$
$83$			56.8706i			0.685188i	−0.939484	+	0.342594i	$0.888694\pi$
$84$		0			0
$85$	79.1374	−	24.6941i	0.931029	−	0.290518i
$86$		0			0
$87$		−	37.2341i		−	0.427978i
$88$		0			0
$89$		−	81.8009i		−	0.919111i	−0.888149	−	0.459556i	$-0.848009\pi$
$89$		−	81.8009i		−	0.919111i	0.888149	−	0.459556i	$-0.151991\pi$
$90$		0			0
$91$		−	79.8230i		−	0.877176i
$92$		0			0
$93$		−	142.543i		−	1.53272i
$94$		0			0
$95$	64.2088	−	70.0159i	0.675882	−	0.737010i
$96$		0			0
$97$	−103.506			−1.06708			−0.533539	−	0.845776i	$-0.679138\pi$
$97$	−103.506			−1.06708			−0.533539	+	0.845776i	$0.679138\pi$
$98$		0			0
$99$	−22.9995			−0.232319
$100$		0			0
$101$	−19.5967			−0.194027			−0.0970136	−	0.995283i	$-0.530929\pi$
$101$	−19.5967			−0.194027			−0.0970136	+	0.995283i	$0.530929\pi$
$102$		0			0
$103$	−110.433			−1.07217			−0.536083	−	0.844165i	$-0.680097\pi$
$103$	−110.433			−1.07217			−0.536083	+	0.844165i	$0.680097\pi$
$104$		0			0
$105$	−188.496	+	58.8184i	−1.79520	+	0.560175i
$106$		0			0
$107$	−38.8233			−0.362835			−0.181417	−	0.983406i	$-0.558069\pi$
$107$	−38.8233			−0.362835			−0.181417	+	0.983406i	$0.558069\pi$
$108$		0			0
$109$		−	98.8797i		−	0.907153i	−0.891217	−	0.453577i	$-0.850148\pi$
$109$		−	98.8797i		−	0.907153i	0.891217	−	0.453577i	$-0.149852\pi$
$110$		0			0
$111$	268.591			2.41974
$112$		0			0
$113$	−74.5996			−0.660174			−0.330087	−	0.943951i	$-0.607078\pi$
$113$	−74.5996			−0.660174			−0.330087	+	0.943951i	$0.607078\pi$
$114$		0			0
$115$	−167.870	+	52.3821i	−1.45974	+	0.455496i
$116$		0			0
$117$	−255.569			−2.18435
$118$		0			0
$119$	116.626			0.980053
$120$		0			0
$121$	−119.957			−0.991381
$122$		0			0
$123$		−	68.0746i		−	0.553452i
$124$		0			0
$125$	−98.4880	−	76.9747i	−0.787904	−	0.615798i
$126$		0			0
$127$	193.575			1.52421			0.762106	−	0.647453i	$-0.224166\pi$
$127$	193.575			1.52421			0.762106	+	0.647453i	$0.224166\pi$
$128$		0			0
$129$		−	281.239i		−	2.18014i
$130$		0			0
$131$	−32.7455			−0.249965			−0.124983	−	0.992159i	$-0.539888\pi$
$131$	−32.7455			−0.249965			−0.124983	+	0.992159i	$0.539888\pi$
$132$		0			0
$133$	115.570	−	67.1213i	0.868949	−	0.504671i
$134$		0			0
$135$	113.061	+	362.329i	0.837491	+	2.68392i
$136$		0			0
$137$			16.9858i			0.123984i	0.998077	+	0.0619920i	$0.0197453\pi$
$137$			16.9858i			0.123984i	−0.998077	+	0.0619920i	$0.980255\pi$
$138$		0			0
$139$	−114.212			−0.821666			−0.410833	−	0.911711i	$-0.634762\pi$
$139$	−114.212			−0.821666			−0.410833	+	0.911711i	$0.634762\pi$
$140$		0			0
$141$		−	439.746i		−	3.11876i
$142$		0			0
$143$	11.5891			0.0810429
$144$		0			0
$145$	31.6544	−	9.87744i	0.218306	−	0.0681203i
$146$		0			0
$147$	−2.68662			−0.0182763
$148$		0			0
$149$	−179.871			−1.20719			−0.603593	−	0.797293i	$-0.706265\pi$
$149$	−179.871			−1.20719			−0.603593	+	0.797293i	$0.706265\pi$
$150$		0			0
$151$			203.393i			1.34698i	0.739198	+	0.673488i	$0.235205\pi$
$151$			203.393i			1.34698i	−0.739198	+	0.673488i	$0.764795\pi$
$152$		0			0
$153$		−	373.402i		−	2.44053i
$154$		0			0
$155$	121.182	−	37.8137i	0.781820	−	0.243959i
$156$		0			0
$157$			128.869i			0.820825i	0.911900	+	0.410412i	$0.134615\pi$
$157$			128.869i			0.820825i	−0.911900	+	0.410412i	$0.865385\pi$
$158$		0			0
$159$	252.838			1.59018
$160$		0			0
$161$	−247.393			−1.53660
$162$		0			0
$163$			266.033i			1.63210i	0.577979	+	0.816052i	$0.303842\pi$
$163$			266.033i			1.63210i	−0.577979	+	0.816052i	$0.696158\pi$
$164$		0			0
$165$	−8.53956	−	27.3669i	−0.0517549	−	0.165860i
$166$		0			0
$167$	−296.182			−1.77354			−0.886771	−	0.462209i	$-0.847057\pi$
$167$	−296.182			−1.77354			−0.886771	+	0.462209i	$0.847057\pi$
$168$		0			0
$169$	−40.2226			−0.238004
$170$		0			0
$171$	−214.902	−	370.021i	−1.25673	−	2.16386i
$172$		0			0
$173$	−289.021			−1.67064			−0.835322	−	0.549761i	$-0.814719\pi$
$173$	−289.021			−1.67064			−0.835322	+	0.549761i	$0.814719\pi$
$174$		0			0
$175$	−100.008	−	144.646i	−0.571476	−	0.826547i
$176$		0			0
$177$			53.0499i			0.299717i
$178$		0			0
$179$		−	209.726i		−	1.17165i	−0.810437	−	0.585826i	$-0.800770\pi$
$179$		−	209.726i		−	1.17165i	0.810437	−	0.585826i	$-0.199230\pi$
$180$		0			0
$181$		−	177.564i		−	0.981016i	−0.871437	−	0.490508i	$-0.836811\pi$
$181$		−	177.564i		−	0.981016i	0.871437	−	0.490508i	$-0.163189\pi$
$182$		0			0
$183$	−242.252			−1.32378
$184$		0			0
$185$	71.2519	+	228.342i	0.385145	+	1.23428i
$186$		0			0
$187$			16.9324i			0.0905478i
$188$		0			0
$189$			533.971i			2.82525i
$190$		0			0
$191$	318.581			1.66796			0.833982	−	0.551792i	$-0.186056\pi$
$191$	318.581			1.66796			0.833982	+	0.551792i	$0.186056\pi$
$192$		0			0
$193$	278.981			1.44550			0.722748	−	0.691112i	$-0.242879\pi$
$193$	278.981			1.44550			0.722748	+	0.691112i	$0.242879\pi$
$194$		0			0
$195$	−94.8908	−	304.098i	−0.486620	−	1.55948i
$196$		0			0
$197$		−	297.749i		−	1.51141i	−0.654909	−	0.755707i	$-0.727293\pi$
$197$		−	297.749i		−	1.51141i	0.654909	−	0.755707i	$-0.272707\pi$
$198$		0			0
$199$	44.1173			0.221695			0.110848	−	0.993837i	$-0.464643\pi$
$199$	44.1173			0.221695			0.110848	+	0.993837i	$0.464643\pi$
$200$		0			0
$201$	631.746			3.14301
$202$		0			0
$203$	46.6497			0.229801
$204$		0			0
$205$	57.8733	−	18.0588i	0.282309	−	0.0880918i
$206$		0			0
$207$			792.075i			3.82645i
$208$		0			0
$209$	9.74502	+	16.7791i	0.0466269	+	0.0802828i
$210$		0			0
$211$			306.768i			1.45388i	0.686702	+	0.726939i	$0.259058\pi$
$211$			306.768i			1.45388i	−0.686702	+	0.726939i	$0.740942\pi$
$212$		0			0
$213$		−	226.433i		−	1.06307i
$214$		0			0
$215$	239.094	−	74.6069i	1.11206	−	0.347009i
$216$		0			0
$217$	178.588			0.822988
$218$		0			0
$219$			255.010i			1.16443i
$220$		0			0
$221$			188.152i			0.851365i
$222$		0			0
$223$	157.741			0.707360			0.353680	−	0.935366i	$-0.384930\pi$
$223$	157.741			0.707360			0.353680	+	0.935366i	$0.384930\pi$
$224$		0			0
$225$	−463.111	+	320.196i	−2.05827	+	1.42309i
$226$		0			0
$227$	−27.8332			−0.122613			−0.0613066	−	0.998119i	$-0.519527\pi$
$227$	−27.8332			−0.122613			−0.0613066	+	0.998119i	$0.519527\pi$
$228$		0			0
$229$	−164.314			−0.717529			−0.358765	−	0.933428i	$-0.616802\pi$
$229$	−164.314			−0.717529			−0.358765	+	0.933428i	$0.616802\pi$
$230$		0			0
$231$		−	40.3311i		−	0.174593i
$232$		0			0
$233$			150.324i			0.645168i	0.946541	+	0.322584i	$0.104551\pi$
$233$			150.324i			0.645168i	−0.946541	+	0.322584i	$0.895449\pi$
$234$		0			0
$235$	373.848	−	116.656i	1.59084	−	0.496407i
$236$		0			0
$237$			812.420i			3.42793i
$238$		0			0
$239$	−240.092			−1.00457			−0.502285	−	0.864702i	$-0.667507\pi$
$239$	−240.092			−1.00457			−0.502285	+	0.864702i	$0.667507\pi$
$240$		0			0
$241$		−	332.556i		−	1.37990i	−0.723857	−	0.689950i	$-0.757632\pi$
$241$		−	332.556i		−	1.37990i	0.723857	−	0.689950i	$-0.242368\pi$
$242$		0			0
$243$	571.643			2.35244
$244$		0			0
$245$	−0.712706	−	2.28402i	−0.00290900	−	0.00932252i
$246$		0			0
$247$	108.286	+	186.448i	0.438404	+	0.754850i
$248$		0			0
$249$			319.292i			1.28230i
$250$		0			0
$251$	275.443			1.09738			0.548692	−	0.836024i	$-0.315126\pi$
$251$	275.443			1.09738			0.548692	+	0.836024i	$0.315126\pi$
$252$		0			0
$253$		−	35.9178i		−	0.141968i
$254$		0			0
$255$	444.306	−	138.641i	1.74238	−	0.543692i
$256$		0			0
$257$	−307.172			−1.19522			−0.597610	−	0.801787i	$-0.703883\pi$
$257$	−307.172			−1.19522			−0.597610	+	0.801787i	$0.703883\pi$
$258$		0			0
$259$			336.512i			1.29927i
$260$		0			0
$261$		−	149.358i		−	0.572253i
$262$		0			0
$263$			56.8141i			0.216023i	0.994150	+	0.108012i	$0.0344483\pi$
$263$			56.8141i			0.216023i	−0.994150	+	0.108012i	$0.965552\pi$
$264$		0			0
$265$	67.0728	+	214.949i	0.253105	+	0.811129i
$266$		0			0
$267$		−	459.260i		−	1.72007i
$268$		0			0
$269$			488.147i			1.81467i	0.420404	+	0.907337i	$0.361888\pi$
$269$			488.147i			1.81467i	−0.420404	+	0.907337i	$0.638112\pi$
$270$		0			0
$271$	123.352			0.455175			0.227587	−	0.973758i	$-0.426916\pi$
$271$	123.352			0.455175			0.227587	+	0.973758i	$0.426916\pi$
$272$		0			0
$273$		−	448.155i		−	1.64159i
$274$		0			0
$275$	21.0005	−	14.5198i	0.0763653	−	0.0527991i
$276$		0			0
$277$			66.0782i			0.238549i	0.992861	+	0.119275i	$0.0380569\pi$
$277$			66.0782i			0.238549i	−0.992861	+	0.119275i	$0.961943\pi$
$278$		0			0
$279$		−	571.785i		−	2.04941i
$280$		0			0
$281$		−	412.679i		−	1.46861i	−0.678821	−	0.734304i	$-0.737509\pi$
$281$		−	412.679i		−	1.46861i	0.678821	−	0.734304i	$-0.262491\pi$
$282$		0			0
$283$			402.243i			1.42135i	0.703519	+	0.710676i	$0.251611\pi$
$283$			402.243i			1.42135i	−0.703519	+	0.710676i	$0.748389\pi$
$284$		0			0
$285$	360.491	−	393.095i	1.26488	−	1.37928i
$286$		0			0
$287$	85.2890			0.297174
$288$		0			0
$289$	14.0988			0.0487847
$290$		0			0
$291$	−581.123			−1.99698
$292$		0			0
$293$	9.43424			0.0321988			0.0160994	−	0.999870i	$-0.494875\pi$
$293$	9.43424			0.0321988			0.0160994	+	0.999870i	$0.494875\pi$
$294$		0			0
$295$	−45.1001	+	14.0731i	−0.152882	+	0.0477053i
$296$		0			0
$297$	−77.5248			−0.261026
$298$		0			0
$299$		−	399.115i		−	1.33483i
$300$		0			0
$301$	352.357			1.17062
$302$		0			0
$303$	−110.023			−0.363113
$304$		0			0
$305$	−64.2647	−	205.950i	−0.210704	−	0.675246i
$306$		0			0
$307$	266.792			0.869031			0.434515	−	0.900664i	$-0.356920\pi$
$307$	266.792			0.869031			0.434515	+	0.900664i	$0.356920\pi$
$308$		0			0
$309$	−620.011			−2.00651
$310$		0			0
$311$	−436.217			−1.40263			−0.701314	−	0.712853i	$-0.747403\pi$
$311$	−436.217			−1.40263			−0.701314	+	0.712853i	$0.747403\pi$
$312$		0			0
$313$			325.754i			1.04075i	0.853939	+	0.520374i	$0.174207\pi$
$313$			325.754i			1.04075i	−0.853939	+	0.520374i	$0.825793\pi$
$314$		0			0
$315$	−756.119	+	235.940i	−2.40038	+	0.749015i
$316$		0			0
$317$	−91.1420			−0.287514			−0.143757	−	0.989613i	$-0.545918\pi$
$317$	−91.1420			−0.287514			−0.143757	+	0.989613i	$0.545918\pi$
$318$		0			0
$319$			6.77285i			0.0212315i
$320$		0			0
$321$	−217.968			−0.679028
$322$		0			0
$323$	−272.412	+	158.212i	−0.843380	+	0.489821i
$324$		0			0
$325$	233.355	−	161.342i	0.718015	−	0.496437i
$326$		0			0
$327$		−	555.146i		−	1.69769i
$328$		0			0
$329$	550.947			1.67461
$330$		0			0
$331$			72.8914i			0.220216i	0.993920	+	0.110108i	$0.0351196\pi$
$331$			72.8914i			0.220216i	−0.993920	+	0.110108i	$0.964880\pi$
$332$		0			0
$333$	1077.41			3.23546
$334$		0			0
$335$	167.589	+	537.076i	0.500267	+	1.60321i
$336$		0			0
$337$	76.9804			0.228429			0.114214	−	0.993456i	$-0.463565\pi$
$337$	76.9804			0.228429			0.114214	+	0.993456i	$0.463565\pi$
$338$		0			0
$339$	−418.829			−1.23548
$340$		0			0
$341$			25.9284i			0.0760364i
$342$		0			0
$343$			341.305i			0.995058i
$344$		0			0
$345$	−942.480	+	294.092i	−2.73183	+	0.852440i
$346$		0			0
$347$		−	261.448i		−	0.753452i	−0.926325	−	0.376726i	$-0.877050\pi$
$347$		−	261.448i		−	0.753452i	0.926325	−	0.376726i	$-0.122950\pi$
$348$		0			0
$349$	220.895			0.632937			0.316469	−	0.948603i	$-0.397503\pi$
$349$	220.895			0.632937			0.316469	+	0.948603i	$0.397503\pi$
$350$		0			0
$351$	−861.449			−2.45427
$352$		0			0
$353$		−	264.271i		−	0.748644i	−0.927299	−	0.374322i	$-0.877875\pi$
$353$		−	264.271i		−	0.748644i	0.927299	−	0.374322i	$-0.122125\pi$
$354$		0			0
$355$	192.501	−	60.0681i	0.542257	−	0.169206i
$356$		0			0
$357$	654.782			1.83412
$358$		0			0
$359$	347.108			0.966874			0.483437	−	0.875379i	$-0.339388\pi$
$359$	347.108			0.966874			0.483437	+	0.875379i	$0.339388\pi$
$360$		0			0
$361$	−178.890	+	313.559i	−0.495541	+	0.868585i
$362$		0			0
$363$	−673.482			−1.85532
$364$		0			0
$365$	−216.796	+	67.6491i	−0.593962	+	0.185340i
$366$		0			0
$367$			17.2357i			0.0469637i	0.999724	+	0.0234818i	$0.00747519\pi$
$367$			17.2357i			0.0469637i	−0.999724	+	0.0234818i	$0.992525\pi$
$368$		0			0
$369$		−	273.069i		−	0.740025i
$370$		0			0
$371$			316.775i			0.853840i
$372$		0			0
$373$	116.587			0.312565			0.156283	−	0.987712i	$-0.450049\pi$
$373$	116.587			0.312565			0.156283	+	0.987712i	$0.450049\pi$
$374$		0			0
$375$	−552.947	−	432.164i	−1.47453	−	1.15244i
$376$		0			0
$377$			75.2593i			0.199627i
$378$		0			0
$379$		−	325.385i		−	0.858534i	−0.903178	−	0.429267i	$-0.858772\pi$
$379$		−	325.385i		−	0.858534i	0.903178	−	0.429267i	$-0.141228\pi$
$380$		0			0
$381$	1086.80			2.85249
$382$		0			0
$383$	−300.829			−0.785454			−0.392727	−	0.919655i	$-0.628468\pi$
$383$	−300.829			−0.785454			−0.392727	+	0.919655i	$0.628468\pi$
$384$		0			0
$385$	34.2873	−	10.6990i	0.0890579	−	0.0277897i
$386$		0			0
$387$		−	1128.14i		−	2.91509i
$388$		0			0
$389$	−48.0199			−0.123444			−0.0617222	−	0.998093i	$-0.519659\pi$
$389$	−48.0199			−0.123444			−0.0617222	+	0.998093i	$0.519659\pi$
$390$		0			0
$391$	583.132			1.49139
$392$		0			0
$393$	−183.845			−0.467799
$394$		0			0
$395$	−690.675	+	215.519i	−1.74854	+	0.545617i
$396$		0			0
$397$			125.169i			0.315288i	0.987496	+	0.157644i	$0.0503898\pi$
$397$			125.169i			0.315288i	−0.987496	+	0.157644i	$0.949610\pi$
$398$		0			0
$399$	648.853	−	376.843i	1.62620	−	0.944468i
$400$		0			0
$401$			350.633i			0.874395i	0.899365	+	0.437198i	$0.144029\pi$
$401$			350.633i			0.874395i	−0.899365	+	0.437198i	$0.855971\pi$
$402$		0			0
$403$			288.114i			0.714923i
$404$		0			0
$405$	332.886	+	1066.81i	0.821942	+	2.63409i
$406$		0			0
$407$	−48.8566			−0.120041
$408$		0			0
$409$		−	317.356i		−	0.775932i	−0.921674	−	0.387966i	$-0.873178\pi$
$409$		−	317.356i		−	0.775932i	0.921674	−	0.387966i	$-0.126822\pi$
$410$		0			0
$411$			95.3645i			0.232030i
$412$		0			0
$413$	−66.4650			−0.160932
$414$		0			0
$415$	−271.445	+	84.7017i	−0.654084	+	0.204101i
$416$		0			0
$417$	−641.225			−1.53771
$418$		0			0
$419$	240.313			0.573539			0.286770	−	0.958000i	$-0.407419\pi$
$419$	240.313			0.573539			0.286770	+	0.958000i	$0.407419\pi$
$420$		0			0
$421$			188.532i			0.447820i	0.974610	+	0.223910i	$0.0718822\pi$
$421$			188.532i			0.447820i	−0.974610	+	0.223910i	$0.928118\pi$
$422$		0			0
$423$		−	1763.96i		−	4.17013i
$424$		0			0
$425$	235.731	+	340.946i	0.554661	+	0.802226i
$426$		0			0
$427$		−	303.512i		−	0.710802i
$428$		0			0
$429$	65.0655			0.151668
$430$		0			0
$431$		−	341.606i		−	0.792590i	−0.918123	−	0.396295i	$-0.870296\pi$
$431$		−	341.606i		−	0.792590i	0.918123	−	0.396295i	$-0.129704\pi$
$432$		0			0
$433$	−487.775			−1.12650			−0.563251	−	0.826286i	$-0.690450\pi$
$433$	−487.775			−1.12650			−0.563251	+	0.826286i	$0.690450\pi$
$434$		0			0
$435$	177.719	−	55.4555i	0.408550	−	0.127484i
$436$		0			0
$437$	577.851	−	335.606i	1.32231	−	0.767978i
$438$		0			0
$439$			509.200i			1.15991i	0.814649	+	0.579954i	$0.196930\pi$
$439$			509.200i			1.15991i	−0.814649	+	0.579954i	$0.803070\pi$
$440$		0			0
$441$	−10.7769			−0.0244374
$442$		0			0
$443$			772.768i			1.74440i	0.489152	+	0.872198i	$0.337306\pi$
$443$			772.768i			1.74440i	−0.489152	+	0.872198i	$0.662694\pi$
$444$		0			0
$445$	390.438	−	121.832i	0.877388	−	0.273781i
$446$		0			0
$447$	−1009.86			−2.25919
$448$		0			0
$449$		−	669.385i		−	1.49084i	−0.666598	−	0.745418i	$-0.732250\pi$
$449$		−	669.385i		−	1.49084i	0.666598	−	0.745418i	$-0.267750\pi$
$450$		0			0
$451$			12.3827i			0.0274561i
$452$		0			0
$453$			1141.92i			2.52080i
$454$		0			0
$455$	380.997	−	118.886i	0.837356	−	0.261289i
$456$		0			0
$457$		−	677.202i		−	1.48184i	−0.671591	−	0.740922i	$-0.734389\pi$
$457$		−	677.202i		−	1.48184i	0.671591	−	0.740922i	$-0.265611\pi$
$458$		0			0
$459$		−	1258.63i		−	2.74211i
$460$		0			0
$461$	77.2391			0.167547			0.0837735	−	0.996485i	$-0.473303\pi$
$461$	77.2391			0.167547			0.0837735	+	0.996485i	$0.473303\pi$
$462$		0			0
$463$			494.756i			1.06859i	0.845299	+	0.534293i	$0.179422\pi$
$463$			494.756i			1.06859i	−0.845299	+	0.534293i	$0.820578\pi$
$464$		0			0
$465$	680.360	−	212.300i	1.46314	−	0.456558i
$466$		0			0
$467$			79.7450i			0.170760i	0.996348	+	0.0853801i	$0.0272105\pi$
$467$			79.7450i			0.170760i	−0.996348	+	0.0853801i	$0.972790\pi$
$468$		0			0
$469$			791.499i			1.68763i
$470$		0			0
$471$			723.520i			1.53613i
$472$		0			0
$473$			51.1571i			0.108155i
$474$		0			0
$475$	429.819	+	202.190i	0.904882	+	0.425663i
$476$		0			0
$477$	1014.22			2.12624
$478$		0			0
$479$	522.458			1.09073			0.545364	−	0.838200i	$-0.316392\pi$
$479$	522.458			1.09073			0.545364	+	0.838200i	$0.316392\pi$
$480$		0			0
$481$	−542.890			−1.12867
$482$		0			0
$483$	−1388.95			−2.87567
$484$		0			0
$485$	−154.160	−	494.039i	−0.317856	−	1.01864i
$486$		0			0
$487$	362.955			0.745287			0.372644	−	0.927975i	$-0.378451\pi$
$487$	362.955			0.745287			0.372644	+	0.927975i	$0.378451\pi$
$488$		0			0
$489$			1493.60i			3.05440i
$490$		0			0
$491$	858.606			1.74869			0.874344	−	0.485307i	$-0.161292\pi$
$491$	858.606			1.74869			0.874344	+	0.485307i	$0.161292\pi$
$492$		0			0
$493$	−109.958			−0.223039
$494$		0			0
$495$	−34.2550	−	109.777i	−0.0692020	−	0.221773i
$496$		0			0
$497$	283.693			0.570810
$498$		0			0
$499$	467.505			0.936883			0.468441	−	0.883495i	$-0.344816\pi$
$499$	467.505			0.936883			0.468441	+	0.883495i	$0.344816\pi$
$500$		0			0
$501$	−1662.87			−3.31910
$502$		0			0
$503$		−	222.767i		−	0.442877i	−0.975174	−	0.221438i	$-0.928925\pi$
$503$		−	222.767i		−	0.442877i	0.975174	−	0.221438i	$-0.0710752\pi$
$504$		0			0
$505$	−29.1869	−	93.5357i	−0.0577959	−	0.185219i
$506$		0			0
$507$	−225.824			−0.445413
$508$		0			0
$509$			305.647i			0.600486i	0.953863	+	0.300243i	$0.0970678\pi$
$509$			305.647i			0.600486i	−0.953863	+	0.300243i	$0.902932\pi$
$510$		0			0
$511$	−319.497			−0.625238
$512$		0			0
$513$	−724.372	−	1247.23i	−1.41203	−	2.43125i
$514$		0			0
$515$	−164.476	−	527.100i	−0.319372	−	1.02349i
$516$		0			0
$517$			79.9895i			0.154718i
$518$		0			0
$519$	−1622.67			−3.12653
$520$		0			0
$521$		−	254.171i		−	0.487853i	−0.969794	−	0.243926i	$-0.921564\pi$
$521$		−	254.171i		−	0.487853i	0.969794	−	0.243926i	$-0.0784355\pi$
$522$		0			0
$523$	498.490			0.953136			0.476568	−	0.879138i	$-0.341881\pi$
$523$	498.490			0.953136			0.476568	+	0.879138i	$0.341881\pi$
$524$		0			0
$525$	−561.483	−	812.093i	−1.06949	−	1.54684i
$526$		0			0
$527$	−420.953			−0.798771
$528$		0			0
$529$	−707.963			−1.33830
$530$		0			0
$531$			212.800i			0.400754i
$532$		0			0
$533$			137.596i			0.258153i
$534$		0			0
$535$	−57.8225	−	185.305i	−0.108080	−	0.346364i
$536$		0			0
$537$		−	1177.48i		−	2.19269i
$538$		0			0
$539$	0.488694			0.000906668
$540$		0			0
$541$	−106.934			−0.197659			−0.0988295	−	0.995104i	$-0.531510\pi$
$541$	−106.934			−0.197659			−0.0988295	+	0.995104i	$0.531510\pi$
$542$		0			0
$543$		−	996.908i		−	1.83593i
$544$		0			0
$545$	471.955	−	147.269i	0.865973	−	0.270218i
$546$		0			0
$547$	−243.162			−0.444538			−0.222269	−	0.974985i	$-0.571346\pi$
$547$	−243.162			−0.444538			−0.222269	+	0.974985i	$0.571346\pi$
$548$		0			0
$549$	−971.754			−1.77004
$550$		0			0
$551$	−108.963	+	63.2837i	−0.197754	+	0.114852i
$552$		0			0
$553$	−1017.86			−1.84062
$554$		0			0
$555$	400.033	+	1281.99i	0.720781	+	2.30990i
$556$		0			0
$557$			335.753i			0.602787i	0.953500	+	0.301394i	$0.0974518\pi$
$557$			335.753i			0.602787i	−0.953500	+	0.301394i	$0.902548\pi$
$558$		0			0
$559$			568.453i			1.01691i
$560$		0			0
$561$			95.0648i			0.169456i
$562$		0			0
$563$	−271.689			−0.482573			−0.241286	−	0.970454i	$-0.577569\pi$
$563$	−271.689			−0.482573			−0.241286	+	0.970454i	$0.577569\pi$
$564$		0			0
$565$	−111.107	−	356.066i	−0.196649	−	0.630205i
$566$		0			0
$567$			1572.17i			2.77279i
$568$		0			0
$569$		−	636.906i		−	1.11934i	−0.828715	−	0.559671i	$-0.810927\pi$
$569$		−	636.906i		−	1.11934i	0.828715	−	0.559671i	$-0.189073\pi$
$570$		0			0
$571$	15.3268			0.0268421			0.0134210	−	0.999910i	$-0.495728\pi$
$571$	15.3268			0.0268421			0.0134210	+	0.999910i	$0.495728\pi$
$572$		0			0
$573$	1788.63			3.12152
$574$		0			0
$575$	−500.042	−	723.229i	−0.869638	−	1.25779i
$576$		0			0
$577$			62.8264i			0.108885i	0.998517	+	0.0544423i	$0.0173381\pi$
$577$			62.8264i			0.108885i	−0.998517	+	0.0544423i	$0.982662\pi$
$578$		0			0
$579$	1566.30			2.70518
$580$		0			0
$581$	−400.033			−0.688526
$582$		0			0
$583$	−45.9910			−0.0788869
$584$		0			0
$585$	−380.638	−	1219.84i	−0.650663	−	2.08519i
$586$		0			0
$587$		−	718.521i		−	1.22406i	−0.790836	−	0.612028i	$-0.790354\pi$
$587$		−	718.521i		−	1.22406i	0.790836	−	0.612028i	$-0.209646\pi$
$588$		0			0
$589$	−417.141	+	242.268i	−0.708218	+	0.411321i
$590$		0			0
$591$		−	1671.67i		−	2.82854i
$592$		0			0
$593$			80.6940i			0.136078i	0.997683	+	0.0680388i	$0.0216742\pi$
$593$			80.6940i			0.136078i	−0.997683	+	0.0680388i	$0.978326\pi$
$594$		0			0
$595$	173.701	+	556.660i	0.291934	+	0.935564i
$596$		0			0
$597$	247.690			0.414892
$598$		0			0
$599$		−	350.033i		−	0.584363i	−0.956363	−	0.292181i	$-0.905619\pi$
$599$		−	350.033i		−	0.584363i	0.956363	−	0.292181i	$-0.0943811\pi$
$600$		0			0
$601$			255.825i			0.425666i	0.977089	+	0.212833i	$0.0682691\pi$
$601$			255.825i			0.425666i	−0.977089	+	0.212833i	$0.931731\pi$
$602$		0			0
$603$	2534.14			4.20255
$604$		0			0
$605$	−178.661	−	572.558i	−0.295308	−	0.946377i
$606$		0			0
$607$	760.876			1.25350			0.626751	−	0.779219i	$-0.284384\pi$
$607$	760.876			1.25350			0.626751	+	0.779219i	$0.284384\pi$
$608$		0			0
$609$	261.908			0.430062
$610$		0			0
$611$			888.835i			1.45472i
$612$		0			0
$613$		−	65.5926i		−	0.107003i	−0.998568	−	0.0535013i	$-0.982962\pi$
$613$		−	65.5926i		−	0.107003i	0.998568	−	0.0535013i	$-0.0170381\pi$
$614$		0			0
$615$	324.922	−	101.389i	0.528328	−	0.164860i
$616$		0			0
$617$			386.387i			0.626236i	0.949714	+	0.313118i	$0.101373\pi$
$617$			386.387i			0.626236i	−0.949714	+	0.313118i	$0.898627\pi$
$618$		0			0
$619$	107.682			0.173961			0.0869804	−	0.996210i	$-0.472278\pi$
$619$	107.682			0.173961			0.0869804	+	0.996210i	$0.472278\pi$
$620$		0			0
$621$			2669.86i			4.29929i
$622$		0			0
$623$	575.396			0.923589
$624$		0			0
$625$	220.717	−	584.730i	0.353146	−	0.935568i
$626$		0			0
$627$	54.7121	+	94.2039i	0.0872601	+	0.150246i
$628$		0			0
$629$		−	793.195i		−	1.26104i
$630$		0			0
$631$	−617.627			−0.978807			−0.489403	−	0.872058i	$-0.662785\pi$
$631$	−617.627			−0.978807			−0.489403	+	0.872058i	$0.662785\pi$
$632$		0			0
$633$			1722.31i			2.72087i
$634$		0			0
$635$	288.306	+	923.937i	0.454025	+	1.45502i
$636$		0			0
$637$	5.43032			0.00852484
$638$		0			0
$639$		−	908.297i		−	1.42143i
$640$		0			0
$641$			452.435i			0.705826i	0.935656	+	0.352913i	$0.114809\pi$
$641$			452.435i			0.705826i	−0.935656	+	0.352913i	$0.885191\pi$
$642$		0			0
$643$			235.639i			0.366468i	0.983069	+	0.183234i	$0.0586565\pi$
$643$			235.639i			0.366468i	−0.983069	+	0.183234i	$0.941343\pi$
$644$		0			0
$645$	1342.36	−	418.870i	2.08118	−	0.649411i
$646$		0			0
$647$			563.621i			0.871130i	0.900157	+	0.435565i	$0.143451\pi$
$647$			563.621i			0.871130i	−0.900157	+	0.435565i	$0.856549\pi$
$648$		0			0
$649$		−	9.64974i		−	0.0148686i
$650$		0			0
$651$	1002.66			1.54018
$652$		0			0
$653$			498.748i			0.763780i	0.924208	+	0.381890i	$0.124727\pi$
$653$			498.748i			0.763780i	−0.924208	+	0.381890i	$0.875273\pi$
$654$		0			0
$655$	−48.7703	−	156.295i	−0.0744585	−	0.238618i
$656$		0			0
$657$			1022.93i			1.55697i
$658$		0			0
$659$		−	36.4563i		−	0.0553207i	−0.999617	−	0.0276603i	$-0.991194\pi$
$659$		−	36.4563i		−	0.0553207i	0.999617	−	0.0276603i	$-0.00880568\pi$
$660$		0			0
$661$		−	260.521i		−	0.394132i	−0.980390	−	0.197066i	$-0.936859\pi$
$661$		−	260.521i		−	0.394132i	0.980390	−	0.197066i	$-0.0631414\pi$
$662$		0			0
$663$			1056.35i			1.59329i
$664$		0			0
$665$	492.499	+	451.651i	0.740600	+	0.679174i
$666$		0			0
$667$	233.248			0.349698
$668$		0			0
$669$	885.617			1.32379
$670$		0			0
$671$	44.0656			0.0656715
$672$		0			0
$673$	−448.262			−0.666065			−0.333032	−	0.942915i	$-0.608072\pi$
$673$	−448.262			−0.666065			−0.333032	+	0.942915i	$0.608072\pi$
$674$		0			0
$675$	−1561.02	+	1079.29i	−2.31261	+	1.59895i
$676$		0			0
$677$	−106.193			−0.156858			−0.0784292	−	0.996920i	$-0.524990\pi$
$677$	−106.193			−0.156858			−0.0784292	+	0.996920i	$0.524990\pi$
$678$		0			0
$679$		−	728.075i		−	1.07227i
$680$		0			0
$681$	−156.266			−0.229465
$682$		0			0
$683$	−604.017			−0.884359			−0.442179	−	0.896927i	$-0.645795\pi$
$683$	−604.017			−0.884359			−0.442179	+	0.896927i	$0.645795\pi$
$684$		0			0
$685$	−81.0737	+	25.2983i	−0.118356	+	0.0369318i
$686$		0			0
$687$	−922.519			−1.34282
$688$		0			0
$689$	−511.048			−0.741724
$690$		0			0
$691$	531.425			0.769067			0.384534	−	0.923111i	$-0.374362\pi$
$691$	531.425			0.769067			0.384534	+	0.923111i	$0.374362\pi$
$692$		0			0
$693$		−	161.781i		−	0.233450i
$694$		0			0
$695$	−170.104	−	545.135i	−0.244754	−	0.784366i
$696$		0			0
$697$	−201.036			−0.288430
$698$		0			0
$699$			843.973i			1.20740i
$700$		0			0
$701$	−868.092			−1.23836			−0.619181	−	0.785248i	$-0.712535\pi$
$701$	−868.092			−1.23836			−0.619181	+	0.785248i	$0.712535\pi$
$702$		0			0
$703$	−456.503	−	786.013i	−0.649364	−	1.11808i
$704$		0			0
$705$	2098.92	−	654.947i	2.97719	−	0.929003i
$706$		0			0
$707$		−	137.845i		−	0.194972i
$708$		0			0
$709$	285.032			0.402020			0.201010	−	0.979589i	$-0.435578\pi$
$709$	285.032			0.402020			0.201010	+	0.979589i	$0.435578\pi$
$710$		0			0
$711$			3258.88i			4.58352i
$712$		0			0
$713$	892.942			1.25237
$714$		0			0
$715$	17.2606	+	55.3152i	0.0241407	+	0.0773639i
$716$		0			0
$717$	−1347.97			−1.88001
$718$		0			0
$719$	450.375			0.626390			0.313195	−	0.949689i	$-0.398601\pi$
$719$	450.375			0.626390			0.313195	+	0.949689i	$0.398601\pi$
$720$		0			0
$721$		−	776.797i		−	1.07739i
$722$		0			0
$723$		−	1867.09i		−	2.58242i
$724$		0			0
$725$	94.2906	+	136.376i	0.130056	+	0.188105i
$726$		0			0
$727$			190.836i			0.262499i	0.991349	+	0.131249i	$0.0418988\pi$
$727$			190.836i			0.262499i	−0.991349	+	0.131249i	$0.958101\pi$
$728$		0			0
$729$	1197.84			1.64313
$730$		0			0
$731$	−830.545			−1.13618
$732$		0			0
$733$			729.335i			0.995001i	0.867464	+	0.497500i	$0.165749\pi$
$733$			729.335i			0.995001i	−0.867464	+	0.497500i	$0.834251\pi$
$734$		0			0
$735$	−4.00138	−	12.8233i	−0.00544406	−	0.0174467i
$736$		0			0
$737$	−114.914			−0.155921
$738$		0			0
$739$	1321.24			1.78787			0.893936	−	0.448194i	$-0.147933\pi$
$739$	1321.24			1.78787			0.893936	+	0.448194i	$0.147933\pi$
$740$		0			0
$741$	607.955	+	1046.79i	0.820452	+	1.41267i
$742$		0			0
$743$	76.4830			0.102938			0.0514690	−	0.998675i	$-0.483610\pi$
$743$	76.4830			0.102938			0.0514690	+	0.998675i	$0.483610\pi$
$744$		0			0
$745$	−267.895	−	858.527i	−0.359591	−	1.15239i
$746$		0			0
$747$			1280.78i			1.71457i
$748$		0			0
$749$		−	273.087i		−	0.364602i
$750$		0			0
$751$			11.5258i			0.0153472i	0.999971	+	0.00767361i	$0.00244261\pi$
$751$			11.5258i			0.0153472i	−0.999971	+	0.00767361i	$0.997557\pi$
$752$		0			0
$753$	1546.44			2.05370
$754$		0			0
$755$	−970.802	+	302.929i	−1.28583	+	0.401231i
$756$		0			0
$757$			516.354i			0.682106i	0.940044	+	0.341053i	$0.110784\pi$
$757$			516.354i			0.682106i	−0.940044	+	0.341053i	$0.889216\pi$
$758$		0			0
$759$		−	201.655i		−	0.265686i
$760$		0			0
$761$	−437.830			−0.575335			−0.287667	−	0.957730i	$-0.592880\pi$
$761$	−437.830			−0.575335			−0.287667	+	0.957730i	$0.592880\pi$
$762$		0			0
$763$	695.529			0.911572
$764$		0			0
$765$	1782.26	−	556.136i	2.32975	−	0.726975i
$766$		0			0
$767$		−	107.227i		−	0.139800i
$768$		0			0
$769$	−514.298			−0.668788			−0.334394	−	0.942433i	$-0.608532\pi$
$769$	−514.298			−0.668788			−0.334394	+	0.942433i	$0.608532\pi$
$770$		0			0
$771$	−1724.57			−2.23680
$772$		0			0
$773$	−899.495			−1.16364			−0.581821	−	0.813317i	$-0.697660\pi$
$773$	−899.495			−1.16364			−0.581821	+	0.813317i	$0.697660\pi$
$774$		0			0
$775$	360.971	+	522.086i	0.465770	+	0.673660i
$776$		0			0
$777$			1889.30i			2.43153i
$778$		0			0
$779$	−199.215	+	115.701i	−0.255732	+	0.148525i
$780$		0			0
$781$			41.1880i			0.0527375i
$782$		0			0
$783$		−	503.443i		−	0.642966i
$784$		0			0
$785$	−615.097	+	191.935i	−0.783563	+	0.244503i
$786$		0			0
$787$	84.0809			0.106837			0.0534186	−	0.998572i	$-0.482988\pi$
$787$	84.0809			0.106837			0.0534186	+	0.998572i	$0.482988\pi$
$788$		0			0
$789$			318.975i			0.404277i
$790$		0			0
$791$		−	524.741i		−	0.663390i
$792$		0			0
$793$	489.652			0.617468
$794$		0			0
$795$	376.571	+	1206.80i	0.473674	+	1.51799i
$796$		0			0
$797$	826.262			1.03672			0.518358	−	0.855164i	$-0.326544\pi$
$797$	826.262			1.03672			0.518358	+	0.855164i	$0.326544\pi$
$798$		0			0
$799$	−1298.64			−1.62534
$800$		0			0
$801$		−	1842.24i		−	2.29993i
$802$		0			0
$803$		−	46.3862i		−	0.0577662i
$804$		0			0
$805$	−368.461	−	1180.81i	−0.457715	−	1.46685i
$806$		0			0
$807$			2740.63i			3.39608i
$808$		0			0
$809$	−26.0920			−0.0322522			−0.0161261	−	0.999870i	$-0.505133\pi$
$809$	−26.0920			−0.0322522			−0.0161261	+	0.999870i	$0.505133\pi$
$810$		0			0
$811$		−	1359.80i		−	1.67670i	−0.545131	−	0.838351i	$-0.683520\pi$
$811$		−	1359.80i		−	1.67670i	0.545131	−	0.838351i	$-0.316480\pi$
$812$		0			0
$813$	692.545			0.851838
$814$		0			0
$815$	−1269.78	+	396.223i	−1.55801	+	0.486163i
$816$		0			0
$817$	−823.024	+	477.998i	−1.00737	+	0.585065i
$818$		0			0
$819$		−	1797.70i		−	2.19499i
$820$		0			0
$821$	−164.334			−0.200163			−0.100081	−	0.994979i	$-0.531910\pi$
$821$	−164.334			−0.200163			−0.100081	+	0.994979i	$0.531910\pi$
$822$		0			0
$823$		−	1040.58i		−	1.26438i	−0.774815	−	0.632188i	$-0.782157\pi$
$823$		−	1040.58i		−	1.26438i	0.774815	−	0.632188i	$-0.217843\pi$
$824$		0			0
$825$	117.904	−	81.5191i	0.142914	−	0.0988110i
$826$		0			0
$827$	1418.49			1.71522			0.857611	−	0.514299i	$-0.171948\pi$
$827$	1418.49			1.71522			0.857611	+	0.514299i	$0.171948\pi$
$828$		0			0
$829$		−	683.294i		−	0.824239i	−0.911130	−	0.412119i	$-0.864789\pi$
$829$		−	683.294i		−	0.824239i	0.911130	−	0.412119i	$-0.135211\pi$
$830$		0			0
$831$			370.987i			0.446434i
$832$		0			0
$833$			7.93404i			0.00952465i
$834$		0			0
$835$	−441.126	−	1413.68i	−0.528294	−	1.69303i
$836$		0			0
$837$		−	1927.32i		−	2.30266i
$838$		0			0
$839$		−	1206.13i		−	1.43758i	−0.695226	−	0.718791i	$-0.744696\pi$
$839$		−	1206.13i		−	1.43758i	0.695226	−	0.718791i	$-0.255304\pi$
$840$		0			0
$841$	797.017			0.947702
$842$		0			0
$843$		−	2316.93i		−	2.74843i
$844$		0			0
$845$	−59.9066	−	191.984i	−0.0708954	−	0.227200i
$846$		0			0
$847$		−	843.790i		−	0.996210i
$848$		0			0
$849$			2258.34i			2.65999i
$850$		0			0
$851$			1682.56i			1.97715i
$852$		0			0
$853$			484.057i			0.567476i	0.958902	+	0.283738i	$0.0915746\pi$
$853$			484.057i			0.567476i	−0.958902	+	0.283738i	$0.908425\pi$
$854$		0			0
$855$	1446.05	−	1576.83i	1.69128	−	1.84425i
$856$		0			0
$857$	−142.789			−0.166615			−0.0833076	−	0.996524i	$-0.526548\pi$
$857$	−142.789			−0.166615			−0.0833076	+	0.996524i	$0.526548\pi$
$858$		0			0
$859$	−737.199			−0.858206			−0.429103	−	0.903256i	$-0.641170\pi$
$859$	−737.199			−0.858206			−0.429103	+	0.903256i	$0.641170\pi$
$860$		0			0
$861$	478.843			0.556148
$862$		0			0
$863$	−1101.44			−1.27629			−0.638145	−	0.769916i	$-0.720298\pi$
$863$	−1101.44			−1.27629			−0.638145	+	0.769916i	$0.720298\pi$
$864$		0			0
$865$	−430.462	−	1379.51i	−0.497643	−	1.59480i
$866$		0			0
$867$	79.1556			0.0912982
$868$		0			0
$869$		−	147.779i		−	0.170056i
$870$		0			0
$871$	−1276.91			−1.46603
$872$		0			0
$873$	−2331.07			−2.67019
$874$		0			0
$875$	541.448	−	692.774i	0.618797	−	0.791742i
$876$		0			0
$877$	−1668.88			−1.90295			−0.951473	−	0.307733i	$-0.900430\pi$
$877$	−1668.88			−1.90295			−0.951473	+	0.307733i	$0.900430\pi$
$878$		0			0
$879$	52.9672			0.0602585
$880$		0			0
$881$	1534.98			1.74231			0.871156	−	0.491007i	$-0.163371\pi$
$881$	1534.98			1.74231			0.871156	+	0.491007i	$0.163371\pi$
$882$		0			0
$883$			1192.75i			1.35079i	0.737457	+	0.675394i	$0.236026\pi$
$883$			1192.75i			1.35079i	−0.737457	+	0.675394i	$0.763974\pi$
$884$		0			0
$885$	−253.208	+	79.0112i	−0.286111	+	0.0892782i
$886$		0			0
$887$	90.7193			0.102277			0.0511383	−	0.998692i	$-0.483715\pi$
$887$	90.7193			0.102277			0.0511383	+	0.998692i	$0.483715\pi$
$888$		0			0
$889$			1361.62i			1.53164i
$890$		0			0
$891$	−228.256			−0.256180
$892$		0			0
$893$	−1286.88	+	747.400i	−1.44108	+	0.836954i
$894$		0			0
$895$	1001.03	−	312.360i	1.11846	−	0.349006i
$896$		0			0
$897$		−	2240.78i		−	2.49808i
$898$		0			0
$899$	−168.378			−0.187295
$900$		0			0
$901$		−	746.673i		−	0.828716i
$902$		0			0
$903$	1978.26			2.19076
$904$		0			0
$905$	847.517	−	264.459i	0.936483	−	0.292220i
$906$		0			0
$907$	−756.388			−0.833944			−0.416972	−	0.908919i	$-0.636909\pi$
$907$	−756.388			−0.833944			−0.416972	+	0.908919i	$0.636909\pi$
$908$		0			0
$909$	−441.339			−0.485521
$910$		0			0
$911$		−	1422.27i		−	1.56122i	−0.625017	−	0.780611i	$-0.714908\pi$
$911$		−	1422.27i		−	1.56122i	0.625017	−	0.780611i	$-0.285092\pi$
$912$		0			0
$913$		−	58.0790i		−	0.0636134i
$914$		0			0
$915$	−360.805	−	1156.28i	−0.394322	−	1.26369i
$916$		0			0
$917$		−	230.335i		−	0.251183i
$918$		0			0
$919$	−802.847			−0.873609			−0.436804	−	0.899556i	$-0.643890\pi$
$919$	−802.847			−0.873609			−0.436804	+	0.899556i	$0.643890\pi$
$920$		0			0
$921$	1497.87			1.62635
$922$		0			0
$923$			457.677i			0.495859i
$924$		0			0
$925$	−983.760	+	680.174i	−1.06352	+	0.735323i
$926$		0			0
$927$	−2487.07			−2.68292
$928$		0			0
$929$	−1818.21			−1.95717			−0.978584	−	0.205849i	$-0.934004\pi$
$929$	−1818.21			−1.95717			−0.978584	+	0.205849i	$0.934004\pi$
$930$		0			0
$931$	4.56623	+	7.86219i	0.00490465	+	0.00844489i
$932$		0			0
$933$	−2449.08			−2.62495
$934$		0			0
$935$	−80.8189	+	25.2188i	−0.0864374	+	0.0269719i
$936$		0			0
$937$		−	1262.41i		−	1.34729i	−0.739053	−	0.673647i	$-0.764727\pi$
$937$		−	1262.41i		−	1.34729i	0.739053	−	0.673647i	$-0.235273\pi$
$938$		0			0
$939$			1828.90i			1.94771i
$940$		0			0
$941$		−	65.8851i		−	0.0700161i	−0.999387	−	0.0350080i	$-0.988854\pi$
$941$		−	65.8851i		−	0.0700161i	0.999387	−	0.0350080i	$-0.0111457\pi$
$942$		0			0
$943$	426.445			0.452222
$944$		0			0
$945$	−2548.66	+	795.284i	−2.69699	+	0.841571i
$946$		0			0
$947$			354.752i			0.374606i	0.982302	+	0.187303i	$0.0599746\pi$
$947$			354.752i			0.374606i	−0.982302	+	0.187303i	$0.940025\pi$
$948$		0			0
$949$		−	515.439i		−	0.543140i
$950$		0			0
$951$	−511.704			−0.538069
$952$		0			0
$953$	−198.204			−0.207979			−0.103990	−	0.994578i	$-0.533161\pi$
$953$	−198.204			−0.207979			−0.103990	+	0.994578i	$0.533161\pi$
$954$		0			0
$955$	474.487	+	1520.60i	0.496845	+	1.59225i
$956$		0			0
$957$			38.0252i			0.0397338i
$958$		0			0
$959$	−119.480			−0.124588
$960$		0			0
$961$	316.401			0.329242
$962$		0			0
$963$	−874.341			−0.907935
$964$		0			0
$965$	415.507	+	1331.58i	0.430577	+	1.37988i
$966$		0			0
$967$			736.898i			0.762045i	0.924566	+	0.381023i	$0.124428\pi$
$967$			736.898i			0.762045i	−0.924566	+	0.381023i	$0.875572\pi$
$968$		0			0
$969$	−1529.42	+	888.260i	−1.57835	+	0.916677i
$970$		0			0
$971$			1371.95i			1.41292i	0.707752	+	0.706461i	$0.249710\pi$
$971$			1371.95i			1.41292i	−0.707752	+	0.706461i	$0.750290\pi$
$972$		0			0
$973$		−	803.375i		−	0.825668i
$974$		0			0
$975$	1310.14	−	905.832i	1.34373	−	0.929059i
$976$		0			0
$977$	1562.15			1.59893			0.799464	−	0.600713i	$-0.205117\pi$
$977$	1562.15			1.59893			0.799464	+	0.600713i	$0.205117\pi$
$978$		0			0
$979$			83.5390i			0.0853310i
$980$		0			0
$981$		−	2226.87i		−	2.27000i
$982$		0			0
$983$	1552.98			1.57983			0.789916	−	0.613215i	$-0.210124\pi$
$983$	1552.98			1.57983			0.789916	+	0.613215i	$0.210124\pi$
$984$		0			0
$985$	1421.16	−	443.460i	1.44280	−	0.450213i
$986$		0			0
$987$	3093.21			3.13396
$988$		0			0
$989$	1761.79			1.78138
$990$		0			0
$991$			26.8694i			0.0271134i	0.999908	+	0.0135567i	$0.00431537\pi$
$991$			26.8694i			0.0271134i	−0.999908	+	0.0135567i	$0.995685\pi$
$992$		0			0
$993$			409.239i			0.412123i
$994$		0			0
$995$	65.7073	+	210.573i	0.0660375	+	0.211631i
$996$		0			0
$997$		−	1303.14i		−	1.30706i	−0.756902	−	0.653529i	$-0.773288\pi$
$997$		−	1303.14i		−	1.30706i	0.756902	−	0.653529i	$-0.226712\pi$
$998$		0			0
$999$	3631.63			3.63526

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	380.3.g.c.189.12	yes	12
3.2	odd	2		3420.3.h.e.2089.5		12
5.2	odd	4		1900.3.e.e.1101.1		12
5.3	odd	4		1900.3.e.e.1101.12		12
5.4	even	2	inner	380.3.g.c.189.1	✓	12
15.14	odd	2		3420.3.h.e.2089.8		12
19.18	odd	2	inner	380.3.g.c.189.2	yes	12
57.56	even	2		3420.3.h.e.2089.6		12
95.18	even	4		1900.3.e.e.1101.2		12
95.37	even	4		1900.3.e.e.1101.11		12
95.94	odd	2	inner	380.3.g.c.189.11	yes	12
285.284	even	2		3420.3.h.e.2089.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.3.g.c.189.1	✓	12	5.4	even	2	inner
380.3.g.c.189.2	yes	12	19.18	odd	2	inner
380.3.g.c.189.11	yes	12	95.94	odd	2	inner
380.3.g.c.189.12	yes	12	1.1	even	1	trivial
1900.3.e.e.1101.1		12	5.2	odd	4
1900.3.e.e.1101.2		12	95.18	even	4
1900.3.e.e.1101.11		12	95.37	even	4
1900.3.e.e.1101.12		12	5.3	odd	4
3420.3.h.e.2089.5		12	3.2	odd	2
3420.3.h.e.2089.6		12	57.56	even	2
3420.3.h.e.2089.7		12	285.284	even	2
3420.3.h.e.2089.8		12	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	380.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+5.61436 q^{3} +(1.48938 + 4.77302i) q^{5} +7.03410i q^{7} +22.5210 q^{9} +O(q^{10})\)	\(q+5.61436 q^{3} +(1.48938 + 4.77302i) q^{5} +7.03410i q^{7} +22.5210 q^{9} -1.02125 q^{11} -11.3480 q^{13} +(8.36189 + 26.7975i) q^{15} -16.5801i q^{17} +(-9.54227 - 16.4300i) q^{19} +39.4919i q^{21} +35.1705i q^{23} +(-20.5635 + 14.2177i) q^{25} +75.9119 q^{27} -6.63193i q^{29} -25.3890i q^{31} -5.73365 q^{33} +(-33.5739 + 10.4764i) q^{35} +47.8401 q^{37} -63.7118 q^{39} -12.1251i q^{41} -50.0927i q^{43} +(33.5423 + 107.493i) q^{45} -78.3252i q^{47} -0.478526 q^{49} -93.0869i q^{51} +45.0342 q^{53} +(-1.52102 - 4.87444i) q^{55} +(-53.5737 - 92.2439i) q^{57} +9.44897i q^{59} -43.1487 q^{61} +158.415i q^{63} +(-16.9015 - 54.1643i) q^{65} +112.523 q^{67} +197.460i q^{69} -40.3311i q^{71} +45.4211i q^{73} +(-115.451 + 79.8230i) q^{75} -7.18356i q^{77} +144.704i q^{79} +223.507 q^{81} +56.8706i q^{83} +(79.1374 - 24.6941i) q^{85} -37.2341i q^{87} -81.8009i q^{89} -79.8230i q^{91} -142.543i q^{93} +(64.2088 - 70.0159i) q^{95} -103.506 q^{97} -22.9995 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{5} + 76 q^{9}+O(q^{10}) \) 12 * q + 12 * q^5 + 76 * q^9	\( 12 q + 12 q^{5} + 76 q^{9} - 24 q^{11} + 68 q^{19} - 76 q^{25} - 32 q^{35} - 264 q^{39} + 220 q^{45} + 212 q^{49} + 176 q^{55} - 600 q^{61} + 492 q^{81} + 408 q^{85} + 124 q^{95} + 136 q^{99}+O(q^{100}) \) 12 * q + 12 * q^5 + 76 * q^9 - 24 * q^11 + 68 * q^19 - 76 * q^25 - 32 * q^35 - 264 * q^39 + 220 * q^45 + 212 * q^49 + 176 * q^55 - 600 * q^61 + 492 * q^81 + 408 * q^85 + 124 * q^95 + 136 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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