LMFDB - Embedded newform 480.4.b.a.431.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,4,Mod(431,480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(480, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("480.431");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	480.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$28.3209168028$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.23
Character	$\chi$	$=$	480.431
Dual form			480.4.b.a.431.24

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.19284 - 0.185395i) q^{3} -5.00000 q^{5} +11.8996i q^{7} +(26.9313 - 1.92545i) q^{9} +O(q^{10})$	\(q+(5.19284 - 0.185395i) q^{3} -5.00000 q^{5} +11.8996i q^{7} +(26.9313 - 1.92545i) q^{9} -19.5939i q^{11} +80.7203i q^{13} +(-25.9642 + 0.926975i) q^{15} +12.9580i q^{17} -81.9017 q^{19} +(2.20613 + 61.7928i) q^{21} +147.701 q^{23} +25.0000 q^{25} +(139.493 - 14.9915i) q^{27} -217.553 q^{29} +139.101i q^{31} +(-3.63262 - 101.748i) q^{33} -59.4981i q^{35} +272.588i q^{37} +(14.9651 + 419.168i) q^{39} +480.713i q^{41} +24.1939 q^{43} +(-134.656 + 9.62727i) q^{45} +370.290 q^{47} +201.399 q^{49} +(2.40235 + 67.2888i) q^{51} -271.318 q^{53} +97.9697i q^{55} +(-425.303 + 15.1842i) q^{57} -323.342i q^{59} +79.6391i q^{61} +(22.9122 + 320.472i) q^{63} -403.602i q^{65} +563.244 q^{67} +(766.991 - 27.3831i) q^{69} -537.552 q^{71} +98.9937 q^{73} +(129.821 - 4.63488i) q^{75} +233.160 q^{77} -292.524i q^{79} +(721.585 - 103.710i) q^{81} +1235.15i q^{83} -64.7900i q^{85} +(-1129.72 + 40.3332i) q^{87} -657.532i q^{89} -960.541 q^{91} +(25.7885 + 722.327i) q^{93} +409.508 q^{95} -574.564 q^{97} +(-37.7272 - 527.689i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 120 q^{5}+O(q^{10}) $ 24 * q - 120 * q^5	$ 24 q - 120 q^{5} - 12 q^{19} + 4 q^{21} + 228 q^{23} + 600 q^{25} - 132 q^{27} + 116 q^{33} - 656 q^{39} - 924 q^{47} - 816 q^{49} + 700 q^{51} - 528 q^{53} - 172 q^{57} + 476 q^{63} - 1632 q^{67} - 980 q^{69} + 216 q^{71} - 216 q^{73} + 152 q^{81} + 252 q^{87} + 1800 q^{91} + 60 q^{95} + 792 q^{97} + 1328 q^{99}+O(q^{100}) $ 24 * q - 120 * q^5 - 12 * q^19 + 4 * q^21 + 228 * q^23 + 600 * q^25 - 132 * q^27 + 116 * q^33 - 656 * q^39 - 924 * q^47 - 816 * q^49 + 700 * q^51 - 528 * q^53 - 172 * q^57 + 476 * q^63 - 1632 * q^67 - 980 * q^69 + 216 * q^71 - 216 * q^73 + 152 * q^81 + 252 * q^87 + 1800 * q^91 + 60 * q^95 + 792 * q^97 + 1328 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$3$	5.19284	−	0.185395i	0.999363	−	0.0356793i
$3$	5.19284	−	0.185395i	0.999363	−	0.0356793i
$4$		0			0
$5$	−5.00000			−0.447214
$5$	−5.00000			−0.447214
$6$		0			0
$7$			11.8996i			0.642519i	0.946991	+	0.321259i	$0.104106\pi$
$7$			11.8996i			0.642519i	−0.946991	+	0.321259i	$0.895894\pi$
$8$		0			0
$9$	26.9313	−	1.92545i	0.997454	−	0.0713131i
$10$		0			0
$11$		−	19.5939i		−	0.537072i	−0.963270	−	0.268536i	$-0.913460\pi$
$11$		−	19.5939i		−	0.537072i	0.963270	−	0.268536i	$-0.0865398\pi$
$12$		0			0
$13$			80.7203i			1.72214i	0.508488	+	0.861069i	$0.330205\pi$
$13$			80.7203i			1.72214i	−0.508488	+	0.861069i	$0.669795\pi$
$14$		0			0
$15$	−25.9642	+	0.926975i	−0.446929	+	0.0159563i
$16$		0			0
$17$			12.9580i			0.184869i	0.995719	+	0.0924345i	$0.0294649\pi$
$17$			12.9580i			0.184869i	−0.995719	+	0.0924345i	$0.970535\pi$
$18$		0			0
$19$	−81.9017			−0.988923			−0.494462	−	0.869200i	$-0.664635\pi$
$19$	−81.9017			−0.988923			−0.494462	+	0.869200i	$0.664635\pi$
$20$		0			0
$21$	2.20613	+	61.7928i	0.0229246	+	0.642110i
$22$		0			0
$23$	147.701			1.33904			0.669519	−	0.742795i	$-0.266500\pi$
$23$	147.701			1.33904			0.669519	+	0.742795i	$0.266500\pi$
$24$		0			0
$25$	25.0000			0.200000
$26$		0			0
$27$	139.493	−	14.9915i	0.994274	−	0.106856i
$28$		0			0
$29$	−217.553			−1.39305			−0.696527	−	0.717531i	$-0.745272\pi$
$29$	−217.553			−1.39305			−0.696527	+	0.717531i	$0.745272\pi$
$30$		0			0
$31$			139.101i			0.805909i	0.915220	+	0.402955i	$0.132017\pi$
$31$			139.101i			0.805909i	−0.915220	+	0.402955i	$0.867983\pi$
$32$		0			0
$33$	−3.63262	−	101.748i	−0.0191624	−	0.536730i
$34$		0			0
$35$		−	59.4981i		−	0.287343i
$36$		0			0
$37$			272.588i			1.21117i	0.795781	+	0.605584i	$0.207060\pi$
$37$			272.588i			1.21117i	−0.795781	+	0.605584i	$0.792940\pi$
$38$		0			0
$39$	14.9651	+	419.168i	0.0614447	+	1.72104i
$40$		0			0
$41$			480.713i			1.83109i	0.402212	+	0.915547i	$0.368242\pi$
$41$			480.713i			1.83109i	−0.402212	+	0.915547i	$0.631758\pi$
$42$		0			0
$43$	24.1939			0.0858033			0.0429016	−	0.999079i	$-0.486340\pi$
$43$	24.1939			0.0858033			0.0429016	+	0.999079i	$0.486340\pi$
$44$		0			0
$45$	−134.656	+	9.62727i	−0.446075	+	0.0318922i
$46$		0			0
$47$	370.290			1.14920			0.574599	−	0.818435i	$-0.305158\pi$
$47$	370.290			1.14920			0.574599	+	0.818435i	$0.305158\pi$
$48$		0			0
$49$	201.399			0.587170
$50$		0			0
$51$	2.40235	+	67.2888i	0.00659600	+	0.184751i
$52$		0			0
$53$	−271.318			−0.703178			−0.351589	−	0.936155i	$-0.614358\pi$
$53$	−271.318			−0.703178			−0.351589	+	0.936155i	$0.614358\pi$
$54$		0			0
$55$			97.9697i			0.240186i
$56$		0			0
$57$	−425.303	+	15.1842i	−0.988293	+	0.0352841i
$58$		0			0
$59$		−	323.342i		−	0.713485i	−0.934203	−	0.356742i	$-0.883887\pi$
$59$		−	323.342i		−	0.713485i	0.934203	−	0.356742i	$-0.116113\pi$
$60$		0			0
$61$			79.6391i			0.167160i	0.996501	+	0.0835799i	$0.0266354\pi$
$61$			79.6391i			0.167160i	−0.996501	+	0.0835799i	$0.973365\pi$
$62$		0			0
$63$	22.9122	+	320.472i	0.0458200	+	0.640883i
$64$		0			0
$65$		−	403.602i		−	0.770164i
$66$		0			0
$67$	563.244			1.02703			0.513517	−	0.858080i	$-0.328342\pi$
$67$	563.244			1.02703			0.513517	+	0.858080i	$0.328342\pi$
$68$		0			0
$69$	766.991	−	27.3831i	1.33819	−	0.0477759i
$70$		0			0
$71$	−537.552			−0.898531			−0.449265	−	0.893398i	$-0.648314\pi$
$71$	−537.552			−0.898531			−0.449265	+	0.893398i	$0.648314\pi$
$72$		0			0
$73$	98.9937			0.158717			0.0793585	−	0.996846i	$-0.474713\pi$
$73$	98.9937			0.158717			0.0793585	+	0.996846i	$0.474713\pi$
$74$		0			0
$75$	129.821	−	4.63488i	0.199873	−	0.00713586i
$76$		0			0
$77$	233.160			0.345079
$78$		0			0
$79$		−	292.524i		−	0.416602i	−0.978065	−	0.208301i	$-0.933207\pi$
$79$		−	292.524i		−	0.416602i	0.978065	−	0.208301i	$-0.0667933\pi$
$80$		0			0
$81$	721.585	−	103.710i	0.989829	−	0.142263i
$82$		0			0
$83$			1235.15i			1.63344i	0.577033	+	0.816721i	$0.304210\pi$
$83$			1235.15i			1.63344i	−0.577033	+	0.816721i	$0.695790\pi$
$84$		0			0
$85$		−	64.7900i		−	0.0826760i
$86$		0			0
$87$	−1129.72	+	40.3332i	−1.39217	+	0.0497032i
$88$		0			0
$89$		−	657.532i		−	0.783126i	−0.920151	−	0.391563i	$-0.871935\pi$
$89$		−	657.532i		−	0.783126i	0.920151	−	0.391563i	$-0.128065\pi$
$90$		0			0
$91$	−960.541			−1.10651
$92$		0			0
$93$	25.7885	+	722.327i	0.0287543	+	0.805396i
$94$		0			0
$95$	409.508			0.442260
$96$		0			0
$97$	−574.564			−0.601424			−0.300712	−	0.953715i	$-0.597224\pi$
$97$	−574.564			−0.601424			−0.300712	+	0.953715i	$0.597224\pi$
$98$		0			0
$99$	−37.7272	−	527.689i	−0.0383003	−	0.535705i
$100$		0			0
$101$	95.4087			0.0939953			0.0469976	−	0.998895i	$-0.485035\pi$
$101$	95.4087			0.0939953			0.0469976	+	0.998895i	$0.485035\pi$
$102$		0			0
$103$			190.207i			0.181957i	0.995853	+	0.0909787i	$0.0289995\pi$
$103$			190.207i			0.181957i	−0.995853	+	0.0909787i	$0.971000\pi$
$104$		0			0
$105$	−11.0306	−	308.964i	−0.0102522	−	0.287160i
$106$		0			0
$107$		−	1394.13i		−	1.25958i	−0.776764	−	0.629792i	$-0.783140\pi$
$107$		−	1394.13i		−	1.25958i	0.776764	−	0.629792i	$-0.216860\pi$
$108$		0			0
$109$			1052.71i			0.925061i	0.886603	+	0.462531i	$0.153058\pi$
$109$			1052.71i			0.925061i	−0.886603	+	0.462531i	$0.846942\pi$
$110$		0			0
$111$	50.5365	+	1415.51i	0.0432136	+	1.21040i
$112$		0			0
$113$			1385.09i			1.15308i	0.817069	+	0.576540i	$0.195598\pi$
$113$			1385.09i			1.15308i	−0.817069	+	0.576540i	$0.804402\pi$
$114$		0			0
$115$	−738.507			−0.598836
$116$		0			0
$117$	155.423	+	2173.90i	0.122811	+	1.71775i
$118$		0			0
$119$	−154.195			−0.118782
$120$		0			0
$121$	947.078			0.711554
$122$		0			0
$123$	89.1219	+	2496.27i	0.0653321	+	1.82993i
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$		−	345.019i		−	0.241067i	−0.992709	−	0.120533i	$-0.961540\pi$
$127$		−	345.019i		−	0.241067i	0.992709	−	0.120533i	$-0.0384605\pi$
$128$		0			0
$129$	125.635	−	4.48544i	0.0857487	−	0.00306140i
$130$		0			0
$131$		−	481.159i		−	0.320909i	−0.987043	−	0.160454i	$-0.948704\pi$
$131$		−	481.159i		−	0.320909i	0.987043	−	0.160454i	$-0.0512960\pi$
$132$		0			0
$133$		−	974.599i		−	0.635402i
$134$		0			0
$135$	−697.464	+	74.9575i	−0.444653	+	0.0477875i
$136$		0			0
$137$		−	1207.42i		−	0.752967i	−0.926423	−	0.376483i	$-0.877133\pi$
$137$		−	1207.42i		−	0.752967i	0.926423	−	0.376483i	$-0.122867\pi$
$138$		0			0
$139$	2439.69			1.48872			0.744359	−	0.667779i	$-0.232755\pi$
$139$	2439.69			1.48872			0.744359	+	0.667779i	$0.232755\pi$
$140$		0			0
$141$	1922.86	−	68.6499i	1.14847	−	0.0410026i
$142$		0			0
$143$	1581.63			0.924912
$144$		0			0
$145$	1087.76			0.622993
$146$		0			0
$147$	1045.83	−	37.3384i	0.586796	−	0.0209498i
$148$		0			0
$149$	−2052.93			−1.12874			−0.564372	−	0.825521i	$-0.690882\pi$
$149$	−2052.93			−1.12874			−0.564372	+	0.825521i	$0.690882\pi$
$150$		0			0
$151$		−	232.344i		−	0.125218i	−0.998038	−	0.0626088i	$-0.980058\pi$
$151$		−	232.344i		−	0.125218i	0.998038	−	0.0626088i	$-0.0199420\pi$
$152$		0			0
$153$	24.9500	+	348.975i	0.0131836	+	0.184398i
$154$		0			0
$155$		−	695.503i		−	0.360414i
$156$		0			0
$157$		−	3057.94i		−	1.55446i	−0.629216	−	0.777231i	$-0.716624\pi$
$157$		−	3057.94i		−	1.55446i	0.629216	−	0.777231i	$-0.283376\pi$
$158$		0			0
$159$	−1408.91	+	50.3010i	−0.702730	+	0.0250889i
$160$		0			0
$161$			1757.59i			0.860358i
$162$		0			0
$163$	−621.974			−0.298876			−0.149438	−	0.988771i	$-0.547746\pi$
$163$	−621.974			−0.298876			−0.149438	+	0.988771i	$0.547746\pi$
$164$		0			0
$165$	18.1631	+	508.741i	0.00856966	+	0.240033i
$166$		0			0
$167$	−1489.97			−0.690402			−0.345201	−	0.938529i	$-0.612189\pi$
$167$	−1489.97			−0.690402			−0.345201	+	0.938529i	$0.612189\pi$
$168$		0			0
$169$	−4318.77			−1.96576
$170$		0			0
$171$	−2205.72	+	157.698i	−0.986405	+	0.0705232i
$172$		0			0
$173$	2319.39			1.01930			0.509652	−	0.860381i	$-0.329774\pi$
$173$	2319.39			1.01930			0.509652	+	0.860381i	$0.329774\pi$
$174$		0			0
$175$			297.490i			0.128504i
$176$		0			0
$177$	−59.9461	−	1679.07i	−0.0254566	−	0.713031i
$178$		0			0
$179$			2525.19i			1.05442i	0.849734	+	0.527212i	$0.176763\pi$
$179$			2525.19i			1.05442i	−0.849734	+	0.527212i	$0.823237\pi$
$180$		0			0
$181$		−	3943.26i		−	1.61934i	−0.586888	−	0.809668i	$-0.699647\pi$
$181$		−	3943.26i		−	1.61934i	0.586888	−	0.809668i	$-0.300353\pi$
$182$		0			0
$183$	14.7647	+	413.554i	0.00596414	+	0.167053i
$184$		0			0
$185$		−	1362.94i		−	0.541651i
$186$		0			0
$187$	253.898			0.0992880
$188$		0			0
$189$	178.393	+	1659.91i	0.0686571	+	0.638840i
$190$		0			0
$191$	1918.68			0.726862			0.363431	−	0.931621i	$-0.381605\pi$
$191$	1918.68			0.726862			0.363431	+	0.931621i	$0.381605\pi$
$192$		0			0
$193$	−1417.65			−0.528731			−0.264365	−	0.964423i	$-0.585162\pi$
$193$	−1417.65			−0.528731			−0.264365	+	0.964423i	$0.585162\pi$
$194$		0			0
$195$	−74.8257	−	2095.84i	−0.0274789	−	0.769673i
$196$		0			0
$197$	−3744.03			−1.35407			−0.677033	−	0.735952i	$-0.736735\pi$
$197$	−3744.03			−1.35407			−0.677033	+	0.735952i	$0.736735\pi$
$198$		0			0
$199$		−	4872.55i		−	1.73571i	−0.496820	−	0.867854i	$-0.665499\pi$
$199$		−	4872.55i		−	1.73571i	0.496820	−	0.867854i	$-0.334501\pi$
$200$		0			0
$201$	2924.84	−	104.423i	1.02638	−	0.0366438i
$202$		0			0
$203$		−	2588.80i		−	0.895063i
$204$		0			0
$205$		−	2403.57i		−	0.818890i
$206$		0			0
$207$	3977.79	−	284.393i	1.33563	−	0.0954911i
$208$		0			0
$209$			1604.78i			0.531123i
$210$		0			0
$211$	−2120.64			−0.691900			−0.345950	−	0.938253i	$-0.612443\pi$
$211$	−2120.64			−0.691900			−0.345950	+	0.938253i	$0.612443\pi$
$212$		0			0
$213$	−2791.42	+	99.6594i	−0.897958	+	0.0320589i
$214$		0			0
$215$	−120.970			−0.0383724
$216$		0			0
$217$	−1655.24			−0.517812
$218$		0			0
$219$	514.059	−	18.3529i	0.158616	−	0.00566291i
$220$		0			0
$221$	−1045.97			−0.318370
$222$		0			0
$223$		−	5310.83i		−	1.59480i	−0.603454	−	0.797398i	$-0.706209\pi$
$223$		−	5310.83i		−	1.59480i	0.603454	−	0.797398i	$-0.293791\pi$
$224$		0			0
$225$	673.281	−	48.1364i	0.199491	−	0.0142626i
$226$		0			0
$227$		−	3911.60i		−	1.14371i	−0.820355	−	0.571854i	$-0.806224\pi$
$227$		−	3911.60i		−	1.14371i	0.820355	−	0.571854i	$-0.193776\pi$
$228$		0			0
$229$		−	5988.71i		−	1.72814i	−0.503368	−	0.864072i	$-0.667906\pi$
$229$		−	5988.71i		−	1.72814i	0.503368	−	0.864072i	$-0.332094\pi$
$230$		0			0
$231$	1210.76	−	43.2268i	0.344859	−	0.0123122i
$232$		0			0
$233$			3640.14i			1.02349i	0.859137	+	0.511745i	$0.171001\pi$
$233$			3640.14i			1.02349i	−0.859137	+	0.511745i	$0.828999\pi$
$234$		0			0
$235$	−1851.45			−0.513937
$236$		0			0
$237$	−54.2325	−	1519.03i	−0.0148640	−	0.416336i
$238$		0			0
$239$	6300.66			1.70525			0.852627	−	0.522521i	$-0.175008\pi$
$239$	6300.66			1.70525			0.852627	+	0.522521i	$0.175008\pi$
$240$		0			0
$241$	−186.979			−0.0499766			−0.0249883	−	0.999688i	$-0.507955\pi$
$241$	−186.979			−0.0499766			−0.0249883	+	0.999688i	$0.507955\pi$
$242$		0			0
$243$	3727.85	−	672.327i	0.984123	−	0.177489i
$244$		0			0
$245$	−1007.00			−0.262590
$246$		0			0
$247$		−	6611.13i		−	1.70306i
$248$		0			0
$249$	228.991	+	6413.96i	0.0582800	+	1.63240i
$250$		0			0
$251$			5717.83i			1.43787i	0.695075	+	0.718937i	$0.255371\pi$
$251$			5717.83i			1.43787i	−0.695075	+	0.718937i	$0.744629\pi$
$252$		0			0
$253$		−	2894.05i		−	0.719160i
$254$		0			0
$255$	−12.0117	−	336.444i	−0.00294982	−	0.0826233i
$256$		0			0
$257$		−	6608.04i		−	1.60388i	−0.597402	−	0.801942i	$-0.703800\pi$
$257$		−	6608.04i		−	1.60388i	0.597402	−	0.801942i	$-0.296200\pi$
$258$		0			0
$259$	−3243.69			−0.778198
$260$		0			0
$261$	−5858.97	+	418.888i	−1.38951	+	0.0993431i
$262$		0			0
$263$	−5341.51			−1.25236			−0.626182	−	0.779677i	$-0.715383\pi$
$263$	−5341.51			−1.25236			−0.626182	+	0.779677i	$0.715383\pi$
$264$		0			0
$265$	1356.59			0.314471
$266$		0			0
$267$	−121.903	−	3414.46i	−0.0279414	−	0.782627i
$268$		0			0
$269$	7051.54			1.59829			0.799144	−	0.601139i	$-0.205286\pi$
$269$	7051.54			1.59829			0.799144	+	0.601139i	$0.205286\pi$
$270$		0			0
$271$		−	4785.65i		−	1.07272i	−0.843989	−	0.536361i	$-0.819799\pi$
$271$		−	4785.65i		−	1.07272i	0.843989	−	0.536361i	$-0.180201\pi$
$272$		0			0
$273$	−4987.94	+	178.080i	−1.10580	+	0.0394794i
$274$		0			0
$275$		−	489.848i		−	0.107414i
$276$		0			0
$277$			4080.47i			0.885097i	0.896745	+	0.442548i	$0.145926\pi$
$277$			4080.47i			0.885097i	−0.896745	+	0.442548i	$0.854074\pi$
$278$		0			0
$279$	267.832	+	3746.15i	0.0574719	+	0.803858i
$280$		0			0
$281$		−	1042.56i		−	0.221331i	−0.993858	−	0.110665i	$-0.964702\pi$
$281$		−	1042.56i		−	0.221331i	0.993858	−	0.110665i	$-0.0352982\pi$
$282$		0			0
$283$	−2816.83			−0.591672			−0.295836	−	0.955239i	$-0.595598\pi$
$283$	−2816.83			−0.591672			−0.295836	+	0.955239i	$0.595598\pi$
$284$		0			0
$285$	2126.51	−	75.9208i	0.441978	−	0.0157795i
$286$		0			0
$287$	−5720.30			−1.17651
$288$		0			0
$289$	4745.09			0.965823
$290$		0			0
$291$	−2983.62	+	106.521i	−0.601041	+	0.0214584i
$292$		0			0
$293$	4373.43			0.872009			0.436005	−	0.899944i	$-0.356393\pi$
$293$	4373.43			0.872009			0.436005	+	0.899944i	$0.356393\pi$
$294$		0			0
$295$			1616.71i			0.319080i
$296$		0			0
$297$	−293.743	−	2733.21i	−0.0573895	−	0.533997i
$298$		0			0
$299$			11922.5i			2.30601i
$300$		0			0
$301$			287.899i			0.0551302i
$302$		0			0
$303$	495.443	−	17.6883i	0.0939354	−	0.00335369i
$304$		0			0
$305$		−	398.196i		−	0.0747561i
$306$		0			0
$307$	4848.29			0.901324			0.450662	−	0.892695i	$-0.351188\pi$
$307$	4848.29			0.901324			0.450662	+	0.892695i	$0.351188\pi$
$308$		0			0
$309$	35.2634	+	987.714i	0.00649211	+	0.181842i
$310$		0			0
$311$	−4111.79			−0.749705			−0.374853	−	0.927084i	$-0.622307\pi$
$311$	−4111.79			−0.749705			−0.374853	+	0.927084i	$0.622307\pi$
$312$		0			0
$313$	7169.64			1.29474			0.647368	−	0.762178i	$-0.275870\pi$
$313$	7169.64			1.29474			0.647368	+	0.762178i	$0.275870\pi$
$314$		0			0
$315$	−114.561	−	1602.36i	−0.0204913	−	0.286612i
$316$		0			0
$317$	4371.23			0.774489			0.387244	−	0.921977i	$-0.373427\pi$
$317$	4371.23			0.774489			0.387244	+	0.921977i	$0.373427\pi$
$318$		0			0
$319$			4262.72i			0.748170i
$320$		0			0
$321$	−258.465	−	7239.49i	−0.0449411	−	1.25878i
$322$		0			0
$323$		−	1061.28i		−	0.182821i
$324$		0			0
$325$			2018.01i			0.344428i
$326$		0			0
$327$	195.168	+	5466.58i	0.0330055	+	0.924472i
$328$		0			0
$329$			4406.31i			0.738381i
$330$		0			0
$331$	5590.60			0.928360			0.464180	−	0.885741i	$-0.346349\pi$
$331$	5590.60			0.928360			0.464180	+	0.885741i	$0.346349\pi$
$332$		0			0
$333$	524.856	+	7341.14i	0.0863722	+	1.20808i
$334$		0			0
$335$	−2816.22			−0.459303
$336$		0			0
$337$	−7898.02			−1.27665			−0.638327	−	0.769765i	$-0.720373\pi$
$337$	−7898.02			−1.27665			−0.638327	+	0.769765i	$0.720373\pi$
$338$		0			0
$339$	256.788	+	7192.54i	0.0411411	+	1.15235i
$340$		0			0
$341$	2725.53			0.432831
$342$		0			0
$343$			6478.14i			1.01979i
$344$		0			0
$345$	−3834.95	+	136.916i	−0.598455	+	0.0213661i
$346$		0			0
$347$		−	10863.8i		−	1.68069i	−0.542048	−	0.840347i	$-0.682351\pi$
$347$		−	10863.8i		−	1.68069i	0.542048	−	0.840347i	$-0.317649\pi$
$348$		0			0
$349$			4377.57i			0.671421i	0.941965	+	0.335711i	$0.108976\pi$
$349$			4377.57i			0.671421i	−0.941965	+	0.335711i	$0.891024\pi$
$350$		0			0
$351$	1210.12	+	11259.9i	0.184021	+	1.71228i
$352$		0			0
$353$			1333.66i			0.201086i	0.994933	+	0.100543i	$0.0320580\pi$
$353$			1333.66i			0.201086i	−0.994933	+	0.100543i	$0.967942\pi$
$354$		0			0
$355$	2687.76			0.401835
$356$		0			0
$357$	−800.711	+	28.5870i	−0.118706	+	0.00423805i
$358$		0			0
$359$	−3527.91			−0.518651			−0.259326	−	0.965790i	$-0.583500\pi$
$359$	−3527.91			−0.518651			−0.259326	+	0.965790i	$0.583500\pi$
$360$		0			0
$361$	−151.112			−0.0220312
$362$		0			0
$363$	4918.03	−	175.584i	0.711101	−	0.0253877i
$364$		0			0
$365$	−494.969			−0.0709804
$366$		0			0
$367$			4297.18i			0.611202i	0.952160	+	0.305601i	$0.0988574\pi$
$367$			4297.18i			0.611202i	−0.952160	+	0.305601i	$0.901143\pi$
$368$		0			0
$369$	925.592	+	12946.2i	0.130581	+	1.82643i
$370$		0			0
$371$		−	3228.58i		−	0.451805i
$372$		0			0
$373$		−	1702.18i		−	0.236288i	−0.992996	−	0.118144i	$-0.962306\pi$
$373$		−	1702.18i		−	0.236288i	0.992996	−	0.118144i	$-0.0376944\pi$
$374$		0			0
$375$	−649.105	+	23.1744i	−0.0893858	+	0.00319125i
$376$		0			0
$377$		−	17560.9i		−	2.39903i
$378$		0			0
$379$	−1377.13			−0.186645			−0.0933223	−	0.995636i	$-0.529749\pi$
$379$	−1377.13			−0.186645			−0.0933223	+	0.995636i	$0.529749\pi$
$380$		0			0
$381$	−63.9648	−	1791.63i	−0.00860109	−	0.240913i
$382$		0			0
$383$	−1957.46			−0.261152			−0.130576	−	0.991438i	$-0.541683\pi$
$383$	−1957.46			−0.261152			−0.130576	+	0.991438i	$0.541683\pi$
$384$		0			0
$385$	−1165.80			−0.154324
$386$		0			0
$387$	651.573	−	46.5843i	0.0855848	−	0.00611890i
$388$		0			0
$389$	−7121.90			−0.928264			−0.464132	−	0.885766i	$-0.653634\pi$
$389$	−7121.90			−0.928264			−0.464132	+	0.885766i	$0.653634\pi$
$390$		0			0
$391$			1913.91i			0.247547i
$392$		0			0
$393$	−89.2045	−	2498.58i	−0.0114498	−	0.320705i
$394$		0			0
$395$			1462.62i			0.186310i
$396$		0			0
$397$			3576.18i			0.452099i	0.974116	+	0.226050i	$0.0725811\pi$
$397$			3576.18i			0.452099i	−0.974116	+	0.226050i	$0.927419\pi$
$398$		0			0
$399$	−180.686	−	5060.94i	−0.0226707	−	0.634997i
$400$		0			0
$401$			4642.92i			0.578196i	0.957300	+	0.289098i	$0.0933553\pi$
$401$			4642.92i			0.578196i	−0.957300	+	0.289098i	$0.906645\pi$
$402$		0			0
$403$	−11228.2			−1.38789
$404$		0			0
$405$	−3607.93	+	518.549i	−0.442665	+	0.0636220i
$406$		0			0
$407$	5341.07			0.650485
$408$		0			0
$409$	−2880.41			−0.348233			−0.174116	−	0.984725i	$-0.555707\pi$
$409$	−2880.41			−0.348233			−0.174116	+	0.984725i	$0.555707\pi$
$410$		0			0
$411$	−223.849	−	6269.92i	−0.0268653	−	0.752487i
$412$		0			0
$413$	3847.65			0.458427
$414$		0			0
$415$		−	6175.77i		−	0.730497i
$416$		0			0
$417$	12668.9	−	452.307i	1.48777	−	0.0531164i
$418$		0			0
$419$			11503.6i			1.34126i	0.741792	+	0.670630i	$0.233976\pi$
$419$			11503.6i			1.34126i	−0.741792	+	0.670630i	$0.766024\pi$
$420$		0			0
$421$			12380.8i			1.43327i	0.697450	+	0.716633i	$0.254318\pi$
$421$			12380.8i			1.43327i	−0.697450	+	0.716633i	$0.745682\pi$
$422$		0			0
$423$	9972.37	−	712.976i	1.14627	−	0.0819529i
$424$		0			0
$425$			323.950i			0.0369738i
$426$		0			0
$427$	−947.675			−0.107403
$428$		0			0
$429$	8213.15	−	293.226i	0.924323	−	0.0330002i
$430$		0			0
$431$	−1329.74			−0.148611			−0.0743053	−	0.997236i	$-0.523674\pi$
$431$	−1329.74			−0.148611			−0.0743053	+	0.997236i	$0.523674\pi$
$432$		0			0
$433$	7568.50			0.839997			0.419999	−	0.907525i	$-0.362031\pi$
$433$	7568.50			0.839997			0.419999	+	0.907525i	$0.362031\pi$
$434$		0			0
$435$	5648.59	−	201.666i	0.622596	−	0.0222279i
$436$		0			0
$437$	−12097.0			−1.32421
$438$		0			0
$439$			6198.59i			0.673901i	0.941522	+	0.336951i	$0.109396\pi$
$439$			6198.59i			0.673901i	−0.941522	+	0.336951i	$0.890604\pi$
$440$		0			0
$441$	5423.93	−	387.785i	0.585675	−	0.0418729i
$442$		0			0
$443$			4385.56i			0.470349i	0.971953	+	0.235174i	$0.0755661\pi$
$443$			4385.56i			0.470349i	−0.971953	+	0.235174i	$0.924434\pi$
$444$		0			0
$445$			3287.66i			0.350225i
$446$		0			0
$447$	−10660.6	+	380.604i	−1.12803	+	0.0402728i
$448$		0			0
$449$			10366.2i			1.08955i	0.838581	+	0.544777i	$0.183386\pi$
$449$			10366.2i			1.08955i	−0.838581	+	0.544777i	$0.816614\pi$
$450$		0			0
$451$	9419.07			0.983429
$452$		0			0
$453$	−43.0754	−	1206.52i	−0.00446768	−	0.125138i
$454$		0			0
$455$	4802.71			0.494845
$456$		0			0
$457$	8503.91			0.870451			0.435226	−	0.900321i	$-0.356669\pi$
$457$	8503.91			0.870451			0.435226	+	0.900321i	$0.356669\pi$
$458$		0			0
$459$	194.260	+	1807.55i	0.0197544	+	0.183811i
$460$		0			0
$461$	14344.7			1.44924			0.724618	−	0.689151i	$-0.242016\pi$
$461$	14344.7			1.44924			0.724618	+	0.689151i	$0.242016\pi$
$462$		0			0
$463$			14521.6i			1.45762i	0.684717	+	0.728809i	$0.259926\pi$
$463$			14521.6i			1.45762i	−0.684717	+	0.728809i	$0.740074\pi$
$464$		0			0
$465$	−128.943	−	3611.64i	−0.0128593	−	0.360184i
$466$		0			0
$467$		−	15334.2i		−	1.51945i	−0.650247	−	0.759723i	$-0.725335\pi$
$467$		−	15334.2i		−	1.51945i	0.650247	−	0.759723i	$-0.274665\pi$
$468$		0			0
$469$			6702.39i			0.659888i
$470$		0			0
$471$	−566.928	−	15879.4i	−0.0554621	−	1.55347i
$472$		0			0
$473$		−	474.054i		−	0.0460825i
$474$		0			0
$475$	−2047.54			−0.197785
$476$		0			0
$477$	−7306.94	+	522.411i	−0.701387	+	0.0501458i
$478$		0			0
$479$	2635.56			0.251403			0.125701	−	0.992068i	$-0.459882\pi$
$479$	2635.56			0.251403			0.125701	+	0.992068i	$0.459882\pi$
$480$		0			0
$481$	−22003.4			−2.08580
$482$		0			0
$483$	325.849	+	9126.90i	0.0306969	+	0.859810i
$484$		0			0
$485$	2872.82			0.268965
$486$		0			0
$487$		−	627.098i		−	0.0583502i	−0.999574	−	0.0291751i	$-0.990712\pi$
$487$		−	627.098i		−	0.0583502i	0.999574	−	0.0291751i	$-0.00928803\pi$
$488$		0			0
$489$	−3229.82	+	115.311i	−0.298686	+	0.0106637i
$490$		0			0
$491$		−	106.671i		−	0.00980444i	−0.999988	−	0.00490222i	$-0.998440\pi$
$491$		−	106.671i		−	0.00980444i	0.999988	−	0.00490222i	$-0.00156043\pi$
$492$		0			0
$493$		−	2819.05i		−	0.257533i
$494$		0			0
$495$	188.636	+	2638.45i	0.0171284	+	0.239574i
$496$		0			0
$497$		−	6396.66i		−	0.577323i
$498$		0			0
$499$	−3347.75			−0.300332			−0.150166	−	0.988661i	$-0.547981\pi$
$499$	−3347.75			−0.300332			−0.150166	+	0.988661i	$0.547981\pi$
$500$		0			0
$501$	−7737.16	+	276.232i	−0.689962	+	0.0246330i
$502$		0			0
$503$	574.905			0.0509617			0.0254809	−	0.999675i	$-0.491888\pi$
$503$	574.905			0.0509617			0.0254809	+	0.999675i	$0.491888\pi$
$504$		0			0
$505$	−477.044			−0.0420360
$506$		0			0
$507$	−22426.7	+	800.679i	−1.96451	+	0.0701369i
$508$		0			0
$509$	846.131			0.0736819			0.0368410	−	0.999321i	$-0.488271\pi$
$509$	846.131			0.0736819			0.0368410	+	0.999321i	$0.488271\pi$
$510$		0			0
$511$			1177.99i			0.101979i
$512$		0			0
$513$	−11424.7	+	1227.83i	−0.983261	+	0.105673i
$514$		0			0
$515$		−	951.033i		−	0.0813739i
$516$		0			0
$517$		−	7255.43i		−	0.617202i
$518$		0			0
$519$	12044.2	−	430.003i	1.01866	−	0.0363680i
$520$		0			0
$521$			12770.5i			1.07387i	0.843624	+	0.536935i	$0.180418\pi$
$521$			12770.5i			1.07387i	−0.843624	+	0.536935i	$0.819582\pi$
$522$		0			0
$523$	5813.64			0.486067			0.243033	−	0.970018i	$-0.421858\pi$
$523$	5813.64			0.486067			0.243033	+	0.970018i	$0.421858\pi$
$524$		0			0
$525$	55.1532	+	1544.82i	0.00458492	+	0.128422i
$526$		0			0
$527$	−1802.46			−0.148988
$528$		0			0
$529$	9648.73			0.793025
$530$		0			0
$531$	−622.581	−	8708.02i	−0.0508808	−	0.711668i
$532$		0			0
$533$	−38803.4			−3.15340
$534$		0			0
$535$			6970.64i			0.563303i
$536$		0			0
$537$	468.158	+	13112.9i	0.0376211	+	1.05375i
$538$		0			0
$539$		−	3946.20i		−	0.315352i
$540$		0			0
$541$			4442.49i			0.353045i	0.984297	+	0.176523i	$0.0564849\pi$
$541$			4442.49i			0.353045i	−0.984297	+	0.176523i	$0.943515\pi$
$542$		0			0
$543$	−731.060	−	20476.7i	−0.0577768	−	1.61831i
$544$		0			0
$545$		−	5263.57i		−	0.413700i
$546$		0			0
$547$	2460.07			0.192295			0.0961473	−	0.995367i	$-0.469348\pi$
$547$	2460.07			0.192295			0.0961473	+	0.995367i	$0.469348\pi$
$548$		0			0
$549$	153.342	+	2144.78i	0.0119207	+	0.166734i
$550$		0			0
$551$	17818.0			1.37762
$552$		0			0
$553$	3480.92			0.267674
$554$		0			0
$555$	−252.682	−	7077.54i	−0.0193257	−	0.541306i
$556$		0			0
$557$	20307.0			1.54477			0.772385	−	0.635155i	$-0.219064\pi$
$557$	20307.0			1.54477			0.772385	+	0.635155i	$0.219064\pi$
$558$		0			0
$559$			1952.94i			0.147765i
$560$		0			0
$561$	1318.45	−	47.0714i	0.0992248	−	0.00354253i
$562$		0			0
$563$			1927.79i			0.144310i	0.997393	+	0.0721550i	$0.0229876\pi$
$563$			1927.79i			0.144310i	−0.997393	+	0.0721550i	$0.977012\pi$
$564$		0			0
$565$		−	6925.44i		−	0.515673i
$566$		0			0
$567$	1234.11	+	8586.59i	0.0914068	+	0.635984i
$568$		0			0
$569$			5528.65i			0.407334i	0.979040	+	0.203667i	$0.0652861\pi$
$569$			5528.65i			0.407334i	−0.979040	+	0.203667i	$0.934714\pi$
$570$		0			0
$571$	22158.5			1.62400			0.812001	−	0.583656i	$-0.198379\pi$
$571$	22158.5			1.62400			0.812001	+	0.583656i	$0.198379\pi$
$572$		0			0
$573$	9963.39	−	355.713i	0.726399	−	0.0259339i
$574$		0			0
$575$	3692.54			0.267808
$576$		0			0
$577$	16291.1			1.17541			0.587703	−	0.809077i	$-0.300032\pi$
$577$	16291.1			1.17541			0.587703	+	0.809077i	$0.300032\pi$
$578$		0			0
$579$	−7361.66	+	262.826i	−0.528394	+	0.0188647i
$580$		0			0
$581$	−14697.8			−1.04952
$582$		0			0
$583$			5316.19i			0.377657i
$584$		0			0
$585$	−777.117	−	10869.5i	−0.0549228	−	0.768203i
$586$		0			0
$587$		−	14991.4i		−	1.05411i	−0.849831	−	0.527055i	$-0.823296\pi$
$587$		−	14991.4i		−	1.05411i	0.849831	−	0.527055i	$-0.176704\pi$
$588$		0			0
$589$		−	11392.6i		−	0.796982i
$590$		0			0
$591$	−19442.2	+	694.125i	−1.35320	+	0.0483121i
$592$		0			0
$593$		−	11604.3i		−	0.803591i	−0.915729	−	0.401796i	$-0.868386\pi$
$593$		−	11604.3i		−	0.803591i	0.915729	−	0.401796i	$-0.131614\pi$
$594$		0			0
$595$	770.976			0.0531209
$596$		0			0
$597$	−903.346	−	25302.4i	−0.0619288	−	1.73460i
$598$		0			0
$599$	2109.64			0.143902			0.0719511	−	0.997408i	$-0.477077\pi$
$599$	2109.64			0.143902			0.0719511	+	0.997408i	$0.477077\pi$
$600$		0			0
$601$	−17029.9			−1.15585			−0.577923	−	0.816091i	$-0.696137\pi$
$601$	−17029.9			−1.15585			−0.577923	+	0.816091i	$0.696137\pi$
$602$		0			0
$603$	15168.9	−	1084.50i	1.02442	−	0.0732410i
$604$		0			0
$605$	−4735.39			−0.318216
$606$		0			0
$607$			18341.5i			1.22646i	0.789905	+	0.613229i	$0.210130\pi$
$607$			18341.5i			1.22646i	−0.789905	+	0.613229i	$0.789870\pi$
$608$		0			0
$609$	−479.950	−	13443.2i	−0.0319352	−	0.894494i
$610$		0			0
$611$			29889.9i			1.97908i
$612$		0			0
$613$			9663.26i			0.636697i	0.947974	+	0.318349i	$0.103128\pi$
$613$			9663.26i			0.636697i	−0.947974	+	0.318349i	$0.896872\pi$
$614$		0			0
$615$	−445.609	−	12481.3i	−0.0292174	−	0.818368i
$616$		0			0
$617$			2815.64i			0.183717i	0.995772	+	0.0918586i	$0.0292808\pi$
$617$			2815.64i			0.183717i	−0.995772	+	0.0918586i	$0.970719\pi$
$618$		0			0
$619$	14824.5			0.962596			0.481298	−	0.876557i	$-0.340166\pi$
$619$	14824.5			0.962596			0.481298	+	0.876557i	$0.340166\pi$
$620$		0			0
$621$	20603.3	−	2214.27i	1.33137	−	0.143085i
$622$		0			0
$623$	7824.37			0.503173
$624$		0			0
$625$	625.000			0.0400000
$626$		0			0
$627$	297.518	+	8333.35i	0.0189501	+	0.530785i
$628$		0			0
$629$	−3532.20			−0.223908
$630$		0			0
$631$		−	2273.51i		−	0.143434i	−0.997425	−	0.0717171i	$-0.977152\pi$
$631$		−	2273.51i		−	0.143434i	0.997425	−	0.0717171i	$-0.0228479\pi$
$632$		0			0
$633$	−11012.2	+	393.156i	−0.691460	+	0.0246865i
$634$		0			0
$635$			1725.09i			0.107808i
$636$		0			0
$637$			16257.0i			1.01119i
$638$		0			0
$639$	−14476.9	+	1035.03i	−0.896243	+	0.0640770i
$640$		0			0
$641$			19455.2i			1.19880i	0.800448	+	0.599402i	$0.204595\pi$
$641$			19455.2i			1.19880i	−0.800448	+	0.599402i	$0.795405\pi$
$642$		0			0
$643$	−15696.0			−0.962657			−0.481328	−	0.876540i	$-0.659846\pi$
$643$	−15696.0			−0.962657			−0.481328	+	0.876540i	$0.659846\pi$
$644$		0			0
$645$	−628.177	+	22.4272i	−0.0383480	+	0.00136910i
$646$		0			0
$647$	−10884.6			−0.661389			−0.330694	−	0.943738i	$-0.607283\pi$
$647$	−10884.6			−0.661389			−0.330694	+	0.943738i	$0.607283\pi$
$648$		0			0
$649$	−6335.55			−0.383193
$650$		0			0
$651$	−8595.42	+	306.874i	−0.517482	+	0.0184752i
$652$		0			0
$653$	27481.9			1.64694			0.823469	−	0.567361i	$-0.192036\pi$
$653$	27481.9			1.64694			0.823469	+	0.567361i	$0.192036\pi$
$654$		0			0
$655$			2405.80i			0.143515i
$656$		0			0
$657$	2666.03	−	190.608i	0.158313	−	0.0113186i
$658$		0			0
$659$		−	31077.7i		−	1.83705i	−0.395367	−	0.918523i	$-0.629383\pi$
$659$		−	31077.7i		−	1.83705i	0.395367	−	0.918523i	$-0.370617\pi$
$660$		0			0
$661$			7446.83i			0.438197i	0.975703	+	0.219098i	$0.0703116\pi$
$661$			7446.83i			0.438197i	−0.975703	+	0.219098i	$0.929688\pi$
$662$		0			0
$663$	−5431.58	+	193.918i	−0.318167	+	0.0113592i
$664$		0			0
$665$			4872.99i			0.284160i
$666$		0			0
$667$	−32132.9			−1.86535
$668$		0			0
$669$	−984.602	−	27578.3i	−0.0569012	−	1.59378i
$670$		0			0
$671$	1560.44			0.0897768
$672$		0			0
$673$	11727.3			0.671700			0.335850	−	0.941915i	$-0.390976\pi$
$673$	11727.3			0.671700			0.335850	+	0.941915i	$0.390976\pi$
$674$		0			0
$675$	3487.32	−	374.788i	0.198855	−	0.0213712i
$676$		0			0
$677$	−16194.0			−0.919331			−0.459666	−	0.888092i	$-0.652031\pi$
$677$	−16194.0			−0.919331			−0.459666	+	0.888092i	$0.652031\pi$
$678$		0			0
$679$		−	6837.09i		−	0.386426i
$680$		0			0
$681$	−725.190	−	20312.3i	−0.0408067	−	1.14298i
$682$		0			0
$683$			22713.8i			1.27250i	0.771481	+	0.636252i	$0.219516\pi$
$683$			22713.8i			1.27250i	−0.771481	+	0.636252i	$0.780484\pi$
$684$		0			0
$685$			6037.08i			0.336737i
$686$		0			0
$687$	−1110.28	−	31098.4i	−0.0616590	−	1.72704i
$688$		0			0
$689$		−	21900.9i		−	1.21097i
$690$		0			0
$691$	10207.3			0.561945			0.280972	−	0.959716i	$-0.409343\pi$
$691$	10207.3			0.561945			0.280972	+	0.959716i	$0.409343\pi$
$692$		0			0
$693$	6279.30	−	448.940i	0.344200	−	0.0246087i
$694$		0			0
$695$	−12198.5			−0.665775
$696$		0			0
$697$	−6229.08			−0.338513
$698$		0			0
$699$	674.863	+	18902.7i	0.0365174	+	1.02284i
$700$		0			0
$701$	2137.78			0.115182			0.0575911	−	0.998340i	$-0.481658\pi$
$701$	2137.78			0.115182			0.0575911	+	0.998340i	$0.481658\pi$
$702$		0			0
$703$		−	22325.4i		−	1.19775i
$704$		0			0
$705$	−9614.28	+	343.249i	−0.513610	+	0.0183369i
$706$		0			0
$707$			1135.33i			0.0603937i
$708$		0			0
$709$			7932.80i			0.420201i	0.977680	+	0.210101i	$0.0673792\pi$
$709$			7932.80i			0.420201i	−0.977680	+	0.210101i	$0.932621\pi$
$710$		0			0
$711$	−563.242	−	7878.04i	−0.0297092	−	0.415541i
$712$		0			0
$713$			20545.4i			1.07914i
$714$		0			0
$715$	−7908.15			−0.413633
$716$		0			0
$717$	32718.3	−	1168.11i	1.70417	−	0.0608422i
$718$		0			0
$719$	−24949.4			−1.29410			−0.647048	−	0.762449i	$-0.723997\pi$
$719$	−24949.4			−1.29410			−0.647048	+	0.762449i	$0.723997\pi$
$720$		0			0
$721$	−2263.39			−0.116911
$722$		0			0
$723$	−970.951	+	34.6649i	−0.0499447	+	0.00178313i
$724$		0			0
$725$	−5438.82			−0.278611
$726$		0			0
$727$		−	13284.0i		−	0.677685i	−0.940843	−	0.338843i	$-0.889965\pi$
$727$		−	13284.0i		−	0.677685i	0.940843	−	0.338843i	$-0.110035\pi$
$728$		0			0
$729$	19233.5	−	4182.42i	0.977164	−	0.212489i
$730$		0			0
$731$			313.505i			0.0158624i
$732$		0			0
$733$		−	16860.0i		−	0.849575i	−0.905293	−	0.424788i	$-0.860349\pi$
$733$		−	16860.0i		−	0.849575i	0.905293	−	0.424788i	$-0.139651\pi$
$734$		0			0
$735$	−5229.17	+	186.692i	−0.262423	+	0.00936903i
$736$		0			0
$737$		−	11036.2i		−	0.551591i
$738$		0			0
$739$	−21665.0			−1.07843			−0.539216	−	0.842168i	$-0.681279\pi$
$739$	−21665.0			−1.07843			−0.539216	+	0.842168i	$0.681279\pi$
$740$		0			0
$741$	−1225.67	−	34330.6i	−0.0607640	−	1.70198i
$742$		0			0
$743$	23700.7			1.17025			0.585123	−	0.810944i	$-0.301046\pi$
$743$	23700.7			1.17025			0.585123	+	0.810944i	$0.301046\pi$
$744$		0			0
$745$	10264.7			0.504790
$746$		0			0
$747$	2378.23	+	33264.2i	0.116486	+	1.62928i
$748$		0			0
$749$	16589.6			0.809307
$750$		0			0
$751$		−	38822.5i		−	1.88636i	−0.332286	−	0.943179i	$-0.607820\pi$
$751$		−	38822.5i		−	1.88636i	0.332286	−	0.943179i	$-0.392180\pi$
$752$		0			0
$753$	1060.06	+	29691.8i	0.0513023	+	1.43696i
$754$		0			0
$755$			1161.72i			0.0559990i
$756$		0			0
$757$		−	21198.9i		−	1.01782i	−0.860821	−	0.508908i	$-0.830049\pi$
$757$		−	21198.9i		−	1.01782i	0.860821	−	0.508908i	$-0.169951\pi$
$758$		0			0
$759$	−536.543	−	15028.4i	−0.0256591	−	0.718702i
$760$		0			0
$761$			2028.32i			0.0966185i	0.998832	+	0.0483092i	$0.0153833\pi$
$761$			2028.32i			0.0966185i	−0.998832	+	0.0483092i	$0.984617\pi$
$762$		0			0
$763$	−12526.9			−0.594369
$764$		0			0
$765$	−124.750	−	1744.88i	−0.00589588	−	0.0824655i
$766$		0			0
$767$	26100.3			1.22872
$768$		0			0
$769$	−257.053			−0.0120541			−0.00602703	−	0.999982i	$-0.501918\pi$
$769$	−257.053			−0.0120541			−0.00602703	+	0.999982i	$0.501918\pi$
$770$		0			0
$771$	−1225.10	−	34314.5i	−0.0572254	−	1.60286i
$772$		0			0
$773$	19191.4			0.892971			0.446486	−	0.894791i	$-0.352675\pi$
$773$	19191.4			0.892971			0.446486	+	0.894791i	$0.352675\pi$
$774$		0			0
$775$			3477.51i			0.161182i
$776$		0			0
$777$	−16844.0	+	601.365i	−0.777703	+	0.0277656i
$778$		0			0
$779$		−	39371.2i		−	1.81081i
$780$		0			0
$781$			10532.8i			0.482576i
$782$		0			0
$783$	−30347.1	+	3261.45i	−1.38508	+	0.148856i
$784$		0			0
$785$			15289.7i			0.695176i
$786$		0			0
$787$	41816.6			1.89403			0.947015	−	0.321191i	$-0.104083\pi$
$787$	41816.6			1.89403			0.947015	+	0.321191i	$0.104083\pi$
$788$		0			0
$789$	−27737.6	+	990.289i	−1.25157	+	0.0446834i
$790$		0			0
$791$	−16482.0			−0.740876
$792$		0			0
$793$	−6428.50			−0.287872
$794$		0			0
$795$	7044.57	−	251.505i	0.314270	−	0.0112201i
$796$		0			0
$797$	−25232.5			−1.12143			−0.560717	−	0.828008i	$-0.689474\pi$
$797$	−25232.5			−1.12143			−0.560717	+	0.828008i	$0.689474\pi$
$798$		0			0
$799$			4798.21i			0.212451i
$800$		0			0
$801$	−1266.05	−	17708.2i	−0.0558472	−	0.781132i
$802$		0			0
$803$		−	1939.68i		−	0.0852425i
$804$		0			0
$805$		−	8787.96i		−	0.384764i
$806$		0			0
$807$	36617.5	−	1307.32i	1.59727	−	0.0570258i
$808$		0			0
$809$		−	35880.6i		−	1.55933i	−0.626199	−	0.779663i	$-0.715390\pi$
$809$		−	35880.6i		−	1.55933i	0.626199	−	0.779663i	$-0.284610\pi$
$810$		0			0
$811$	−24028.6			−1.04039			−0.520196	−	0.854047i	$-0.674141\pi$
$811$	−24028.6			−1.04039			−0.520196	+	0.854047i	$0.674141\pi$
$812$		0			0
$813$	−887.235	−	24851.1i	−0.0382739	−	1.07204i
$814$		0			0
$815$	3109.87			0.133661
$816$		0			0
$817$	−1981.52			−0.0848528
$818$		0			0
$819$	−25868.6	+	1849.48i	−1.10369	+	0.0789084i
$820$		0			0
$821$	−9237.39			−0.392676			−0.196338	−	0.980536i	$-0.562905\pi$
$821$	−9237.39			−0.392676			−0.196338	+	0.980536i	$0.562905\pi$
$822$		0			0
$823$			2873.15i			0.121691i	0.998147	+	0.0608454i	$0.0193797\pi$
$823$			2873.15i			0.121691i	−0.998147	+	0.0608454i	$0.980620\pi$
$824$		0			0
$825$	−90.8154	−	2543.71i	−0.00383247	−	0.107346i
$826$		0			0
$827$			15113.0i			0.635465i	0.948180	+	0.317733i	$0.102921\pi$
$827$			15113.0i			0.635465i	−0.948180	+	0.317733i	$0.897079\pi$
$828$		0			0
$829$			10373.3i			0.434595i	0.976105	+	0.217298i	$0.0697242\pi$
$829$			10373.3i			0.434595i	−0.976105	+	0.217298i	$0.930276\pi$
$830$		0			0
$831$	756.499	+	21189.3i	0.0315796	+	0.884533i
$832$		0			0
$833$			2609.73i			0.108549i
$834$		0			0
$835$	7449.83			0.308757
$836$		0			0
$837$	2085.33	+	19403.5i	0.0861164	+	0.801295i
$838$		0			0
$839$	14010.2			0.576502			0.288251	−	0.957555i	$-0.406926\pi$
$839$	14010.2			0.576502			0.288251	+	0.957555i	$0.406926\pi$
$840$		0			0
$841$	22940.3			0.940599
$842$		0			0
$843$	−193.285	−	5413.85i	−0.00789692	−	0.221190i
$844$		0			0
$845$	21593.9			0.879114
$846$		0			0
$847$			11269.9i			0.457187i
$848$		0			0
$849$	−14627.4	+	522.227i	−0.591296	+	0.0211104i
$850$		0			0
$851$			40261.7i			1.62180i
$852$		0			0
$853$			11697.1i			0.469520i	0.972053	+	0.234760i	$0.0754304\pi$
$853$			11697.1i			0.469520i	−0.972053	+	0.234760i	$0.924570\pi$
$854$		0			0
$855$	11028.6	−	788.490i	0.441134	−	0.0315389i
$856$		0			0
$857$		−	4954.60i		−	0.197486i	−0.995113	−	0.0987432i	$-0.968518\pi$
$857$		−	4954.60i		−	0.197486i	0.995113	−	0.0987432i	$-0.0314822\pi$
$858$		0			0
$859$	31596.1			1.25500			0.627499	−	0.778617i	$-0.284079\pi$
$859$	31596.1			1.25500			0.627499	+	0.778617i	$0.284079\pi$
$860$		0			0
$861$	−29704.7	+	1060.52i	−1.17576	+	0.0419771i
$862$		0			0
$863$	−6703.04			−0.264396			−0.132198	−	0.991223i	$-0.542204\pi$
$863$	−6703.04			−0.264396			−0.132198	+	0.991223i	$0.542204\pi$
$864$		0			0
$865$	−11596.9			−0.455847
$866$		0			0
$867$	24640.5	−	879.716i	0.965208	−	0.0344599i
$868$		0			0
$869$	−5731.70			−0.223745
$870$		0			0
$871$			45465.3i			1.76869i
$872$		0			0
$873$	−15473.7	+	1106.30i	−0.599893	+	0.0428894i
$874$		0			0
$875$		−	1487.45i		−	0.0574686i
$876$		0			0
$877$			2640.85i			0.101682i	0.998707	+	0.0508411i	$0.0161902\pi$
$877$			2640.85i			0.101682i	−0.998707	+	0.0508411i	$0.983810\pi$
$878$		0			0
$879$	22710.6	−	810.813i	0.871454	−	0.0311127i
$880$		0			0
$881$		−	948.785i		−	0.0362831i	−0.999835	−	0.0181415i	$-0.994225\pi$
$881$		−	948.785i		−	0.0362831i	0.999835	−	0.0181415i	$-0.00577495\pi$
$882$		0			0
$883$	7226.54			0.275416			0.137708	−	0.990473i	$-0.456026\pi$
$883$	7226.54			0.275416			0.137708	+	0.990473i	$0.456026\pi$
$884$		0			0
$885$	299.730	+	8395.34i	0.0113846	+	0.318877i
$886$		0			0
$887$	−48118.6			−1.82149			−0.910746	−	0.412966i	$-0.864493\pi$
$887$	−48118.6			−1.82149			−0.910746	+	0.412966i	$0.864493\pi$
$888$		0			0
$889$	4105.59			0.154890
$890$		0			0
$891$	−2032.08	−	14138.7i	−0.0764056	−	0.531609i
$892$		0			0
$893$	−30327.4			−1.13647
$894$		0			0
$895$		−	12626.0i		−	0.471553i
$896$		0			0
$897$	2210.37	+	61911.8i	0.0822768	+	2.30454i
$898$		0			0
$899$		−	30261.7i		−	1.12268i
$900$		0			0
$901$		−	3515.74i		−	0.129996i
$902$		0			0
$903$	53.3750	+	1495.01i	0.00196701	+	0.0550951i
$904$		0			0
$905$			19716.3i			0.724189i
$906$		0			0
$907$	−3998.99			−0.146399			−0.0731997	−	0.997317i	$-0.523321\pi$
$907$	−3998.99			−0.146399			−0.0731997	+	0.997317i	$0.523321\pi$
$908$		0			0
$909$	2569.48	−	183.705i	0.0937560	−	0.00670310i
$910$		0			0
$911$	44298.6			1.61106			0.805532	−	0.592552i	$-0.201880\pi$
$911$	44298.6			1.61106			0.805532	+	0.592552i	$0.201880\pi$
$912$		0			0
$913$	24201.5			0.877276
$914$		0			0
$915$	−73.8235	−	2067.77i	−0.00266725	−	0.0747085i
$916$		0			0
$917$	5725.61			0.206190
$918$		0			0
$919$			2554.93i			0.0917078i	0.998948	+	0.0458539i	$0.0146009\pi$
$919$			2554.93i			0.0917078i	−0.998948	+	0.0458539i	$0.985399\pi$
$920$		0			0
$921$	25176.4	−	898.849i	0.900751	−	0.0321586i
$922$		0			0
$923$		−	43391.4i		−	1.54739i
$924$		0			0
$925$			6814.70i			0.242234i
$926$		0			0
$927$	366.234	+	5122.51i	0.0129760	+	0.181494i
$928$		0			0
$929$		−	11706.4i		−	0.413428i	−0.978401	−	0.206714i	$-0.933723\pi$
$929$		−	11706.4i		−	0.413428i	0.978401	−	0.206714i	$-0.0662770\pi$
$930$		0			0
$931$	−16494.9			−0.580665
$932$		0			0
$933$	−21351.9	+	762.306i	−0.749228	+	0.0267490i
$934$		0			0
$935$	−1269.49			−0.0444030
$936$		0			0
$937$	−41893.6			−1.46062			−0.730311	−	0.683114i	$-0.760625\pi$
$937$	−41893.6			−1.46062			−0.730311	+	0.683114i	$0.760625\pi$
$938$		0			0
$939$	37230.8	−	1329.22i	1.29391	−	0.0461952i
$940$		0			0
$941$	3148.26			0.109065			0.0545326	−	0.998512i	$-0.482633\pi$
$941$	3148.26			0.109065			0.0545326	+	0.998512i	$0.482633\pi$
$942$		0			0
$943$			71002.1i			2.45190i
$944$		0			0
$945$	−891.966	−	8299.56i	−0.0307044	−	0.285698i
$946$		0			0
$947$			24539.5i			0.842055i	0.907048	+	0.421028i	$0.138330\pi$
$947$			24539.5i			0.842055i	−0.907048	+	0.421028i	$0.861670\pi$
$948$		0			0
$949$			7990.81i			0.273333i
$950$		0			0
$951$	22699.1	−	810.405i	0.773996	−	0.0276332i
$952$		0			0
$953$		−	34706.3i		−	1.17969i	−0.807515	−	0.589847i	$-0.799188\pi$
$953$		−	34706.3i		−	1.17969i	0.807515	−	0.589847i	$-0.200812\pi$
$954$		0			0
$955$	−9593.39			−0.325063
$956$		0			0
$957$	790.287	+	22135.6i	0.0266942	+	0.747694i
$958$		0			0
$959$	14367.8			0.483795
$960$		0			0
$961$	10442.0			0.350510
$962$		0			0
$963$	−2684.33	−	37545.6i	−0.0898249	−	1.25638i
$964$		0			0
$965$	7088.27			0.236455
$966$		0			0
$967$		−	8683.49i		−	0.288772i	−0.989521	−	0.144386i	$-0.953879\pi$
$967$		−	8683.49i		−	0.288772i	0.989521	−	0.144386i	$-0.0461207\pi$
$968$		0			0
$969$	−196.756	−	5511.07i	−0.00652293	−	0.182705i
$970$		0			0
$971$			35296.8i			1.16656i	0.812272	+	0.583279i	$0.198231\pi$
$971$			35296.8i			1.16656i	−0.812272	+	0.583279i	$0.801769\pi$
$972$		0			0
$973$			29031.4i			0.956530i
$974$		0			0
$975$	374.129	+	10479.2i	0.0122889	+	0.344208i
$976$		0			0
$977$			22443.3i			0.734927i	0.930038	+	0.367463i	$0.119774\pi$
$977$			22443.3i			0.734927i	−0.930038	+	0.367463i	$0.880226\pi$
$978$		0			0
$979$	−12883.6			−0.420595
$980$		0			0
$981$	2026.95	+	28350.9i	0.0659690	+	0.922706i
$982$		0			0
$983$	−47336.6			−1.53591			−0.767957	−	0.640502i	$-0.778726\pi$
$983$	−47336.6			−1.53591			−0.767957	+	0.640502i	$0.778726\pi$
$984$		0			0
$985$	18720.2			0.605557
$986$		0			0
$987$	816.907	+	22881.3i	0.0263449	+	0.737911i
$988$		0			0
$989$	3573.48			0.114894
$990$		0			0
$991$			34234.4i			1.09737i	0.836030	+	0.548683i	$0.184871\pi$
$991$			34234.4i			1.09737i	−0.836030	+	0.548683i	$0.815129\pi$
$992$		0			0
$993$	29031.1	−	1036.47i	0.927769	−	0.0331232i
$994$		0			0
$995$			24362.7i			0.776232i
$996$		0			0
$997$			50553.3i			1.60586i	0.596076	+	0.802928i	$0.296726\pi$
$997$			50553.3i			1.60586i	−0.596076	+	0.802928i	$0.703274\pi$
$998$		0			0
$999$	4086.51	+	38024.1i	0.129421	+	1.20423i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.b.a.431.23		24
3.2	odd	2		480.4.b.b.431.24		24
4.3	odd	2		120.4.b.b.11.13	yes	24
8.3	odd	2		480.4.b.b.431.23		24
8.5	even	2		120.4.b.a.11.11	✓	24
12.11	even	2		120.4.b.a.11.12	yes	24
24.5	odd	2		120.4.b.b.11.14	yes	24
24.11	even	2	inner	480.4.b.a.431.24		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.b.a.11.11	✓	24	8.5	even	2
120.4.b.a.11.12	yes	24	12.11	even	2
120.4.b.b.11.13	yes	24	4.3	odd	2
120.4.b.b.11.14	yes	24	24.5	odd	2
480.4.b.a.431.23		24	1.1	even	1	trivial
480.4.b.a.431.24		24	24.11	even	2	inner
480.4.b.b.431.23		24	8.3	odd	2
480.4.b.b.431.24		24	3.2	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:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	480.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(5.19284 - 0.185395i) q^{3} -5.00000 q^{5} +11.8996i q^{7} +(26.9313 - 1.92545i) q^{9} +O(q^{10})\)	\(q+(5.19284 - 0.185395i) q^{3} -5.00000 q^{5} +11.8996i q^{7} +(26.9313 - 1.92545i) q^{9} -19.5939i q^{11} +80.7203i q^{13} +(-25.9642 + 0.926975i) q^{15} +12.9580i q^{17} -81.9017 q^{19} +(2.20613 + 61.7928i) q^{21} +147.701 q^{23} +25.0000 q^{25} +(139.493 - 14.9915i) q^{27} -217.553 q^{29} +139.101i q^{31} +(-3.63262 - 101.748i) q^{33} -59.4981i q^{35} +272.588i q^{37} +(14.9651 + 419.168i) q^{39} +480.713i q^{41} +24.1939 q^{43} +(-134.656 + 9.62727i) q^{45} +370.290 q^{47} +201.399 q^{49} +(2.40235 + 67.2888i) q^{51} -271.318 q^{53} +97.9697i q^{55} +(-425.303 + 15.1842i) q^{57} -323.342i q^{59} +79.6391i q^{61} +(22.9122 + 320.472i) q^{63} -403.602i q^{65} +563.244 q^{67} +(766.991 - 27.3831i) q^{69} -537.552 q^{71} +98.9937 q^{73} +(129.821 - 4.63488i) q^{75} +233.160 q^{77} -292.524i q^{79} +(721.585 - 103.710i) q^{81} +1235.15i q^{83} -64.7900i q^{85} +(-1129.72 + 40.3332i) q^{87} -657.532i q^{89} -960.541 q^{91} +(25.7885 + 722.327i) q^{93} +409.508 q^{95} -574.564 q^{97} +(-37.7272 - 527.689i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 120 q^{5}+O(q^{10}) \) 24 * q - 120 * q^5	\( 24 q - 120 q^{5} - 12 q^{19} + 4 q^{21} + 228 q^{23} + 600 q^{25} - 132 q^{27} + 116 q^{33} - 656 q^{39} - 924 q^{47} - 816 q^{49} + 700 q^{51} - 528 q^{53} - 172 q^{57} + 476 q^{63} - 1632 q^{67} - 980 q^{69} + 216 q^{71} - 216 q^{73} + 152 q^{81} + 252 q^{87} + 1800 q^{91} + 60 q^{95} + 792 q^{97} + 1328 q^{99}+O(q^{100}) \) 24 * q - 120 * q^5 - 12 * q^19 + 4 * q^21 + 228 * q^23 + 600 * q^25 - 132 * q^27 + 116 * q^33 - 656 * q^39 - 924 * q^47 - 816 * q^49 + 700 * q^51 - 528 * q^53 - 172 * q^57 + 476 * q^63 - 1632 * q^67 - 980 * q^69 + 216 * q^71 - 216 * q^73 + 152 * q^81 + 252 * q^87 + 1800 * q^91 + 60 * q^95 + 792 * q^97 + 1328 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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