LMFDB - Embedded newform 480.4.h.b.191.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,4,Mod(191,480)]

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, 0, 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("480.191");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	480.h (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:	$28.3209168028$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			191.13
Character	$\chi$	$=$	480.191
Dual form			480.4.h.b.191.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.991849 - 5.10061i) q^{3} +5.00000i q^{5} +7.56208i q^{7} +(-25.0325 - 10.1181i) q^{9} -3.60156 q^{11} +0.331659 q^{13} +(25.5031 + 4.95925i) q^{15} +2.26149i q^{17} +106.718i q^{19} +(38.5712 + 7.50045i) q^{21} +163.714 q^{23} -25.0000 q^{25} +(-76.4368 + 117.645i) q^{27} -39.3759i q^{29} +211.404i q^{31} +(-3.57220 + 18.3702i) q^{33} -37.8104 q^{35} +30.8049 q^{37} +(0.328956 - 1.69166i) q^{39} +268.456i q^{41} +202.814i q^{43} +(50.5904 - 125.162i) q^{45} +13.8352 q^{47} +285.815 q^{49} +(11.5350 + 2.24306i) q^{51} -222.334i q^{53} -18.0078i q^{55} +(544.329 + 105.849i) q^{57} -442.239 q^{59} +601.295 q^{61} +(76.5137 - 189.298i) q^{63} +1.65829i q^{65} +347.079i q^{67} +(162.379 - 835.040i) q^{69} +634.275 q^{71} +620.653 q^{73} +(-24.7962 + 127.515i) q^{75} -27.2353i q^{77} +946.140i q^{79} +(524.249 + 506.561i) q^{81} +655.062 q^{83} -11.3075 q^{85} +(-200.841 - 39.0550i) q^{87} -920.102i q^{89} +2.50803i q^{91} +(1078.29 + 209.681i) q^{93} -533.592 q^{95} +264.948 q^{97} +(90.1559 + 36.4409i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 20 q^{9} + 72 q^{11} - 72 q^{13} + 20 q^{15} - 68 q^{21} + 96 q^{23} - 600 q^{25} + 168 q^{27} - 80 q^{33} - 504 q^{37} - 456 q^{39} - 220 q^{45} + 432 q^{47} - 816 q^{49} + 1240 q^{51} + 40 q^{57}+ \cdots + 3160 q^{99}+O(q^{100}) $ 24 * q + 20 * q^9 + 72 * q^11 - 72 * q^13 + 20 * q^15 - 68 * q^21 + 96 * q^23 - 600 * q^25 + 168 * q^27 - 80 * q^33 - 504 * q^37 - 456 * q^39 - 220 * q^45 + 432 * q^47 - 816 * q^49 + 1240 * q^51 + 40 * q^57 - 2184 * q^59 - 1080 * q^61 - 1720 * q^63 - 1548 * q^69 + 816 * q^71 - 1440 * q^73 - 2304 * q^81 - 2016 * q^83 - 2144 * q^87 - 112 * q^93 + 1800 * q^95 + 720 * q^97 + 3160 * q^99

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
$3$	0.991849	−	5.10061i	0.190882	−	0.981613i
$3$	0.991849	−	5.10061i	0.190882	−	0.981613i
$4$		0			0
$5$			5.00000i			0.447214i
$5$			5.00000i			0.447214i
$6$		0			0
$7$			7.56208i			0.408314i	0.978938	+	0.204157i	$0.0654453\pi$
$7$			7.56208i			0.408314i	−0.978938	+	0.204157i	$0.934555\pi$
$8$		0			0
$9$	−25.0325	−	10.1181i	−0.927128	−	0.374744i
$10$		0			0
$11$	−3.60156			−0.0987192			−0.0493596	−	0.998781i	$-0.515718\pi$
$11$	−3.60156			−0.0987192			−0.0493596	+	0.998781i	$0.515718\pi$
$12$		0			0
$13$	0.331659			0.00707582			0.00353791	−	0.999994i	$-0.498874\pi$
$13$	0.331659			0.00707582			0.00353791	+	0.999994i	$0.498874\pi$
$14$		0			0
$15$	25.5031	+	4.95925i	0.438991	+	0.0853648i
$16$		0			0
$17$			2.26149i			0.0322643i	0.999870	+	0.0161321i	$0.00513524\pi$
$17$			2.26149i			0.0322643i	−0.999870	+	0.0161321i	$0.994865\pi$
$18$		0			0
$19$			106.718i			1.28857i	0.764784	+	0.644286i	$0.222845\pi$
$19$			106.718i			1.28857i	−0.764784	+	0.644286i	$0.777155\pi$
$20$		0			0
$21$	38.5712	+	7.50045i	0.400806	+	0.0779396i
$22$		0			0
$23$	163.714			1.48420			0.742101	−	0.670288i	$-0.233829\pi$
$23$	163.714			1.48420			0.742101	+	0.670288i	$0.233829\pi$
$24$		0			0
$25$	−25.0000			−0.200000
$26$		0			0
$27$	−76.4368	+	117.645i	−0.544825	+	0.838550i
$28$		0			0
$29$		−	39.3759i		−	0.252135i	−0.992022	−	0.126068i	$-0.959764\pi$
$29$		−	39.3759i		−	0.252135i	0.992022	−	0.126068i	$-0.0402356\pi$
$30$		0			0
$31$			211.404i			1.22482i	0.790542	+	0.612408i	$0.209799\pi$
$31$			211.404i			1.22482i	−0.790542	+	0.612408i	$0.790201\pi$
$32$		0			0
$33$	−3.57220	+	18.3702i	−0.0188437	+	0.0969040i
$34$		0			0
$35$	−37.8104			−0.182604
$36$		0			0
$37$	30.8049			0.136873			0.0684365	−	0.997655i	$-0.478199\pi$
$37$	30.8049			0.136873			0.0684365	+	0.997655i	$0.478199\pi$
$38$		0			0
$39$	0.328956	−	1.69166i	0.00135064	−	0.00694571i
$40$		0			0
$41$			268.456i			1.02258i	0.859408	+	0.511291i	$0.170832\pi$
$41$			268.456i			1.02258i	−0.859408	+	0.511291i	$0.829168\pi$
$42$		0			0
$43$			202.814i			0.719275i	0.933092	+	0.359637i	$0.117100\pi$
$43$			202.814i			0.719275i	−0.933092	+	0.359637i	$0.882900\pi$
$44$		0			0
$45$	50.5904	−	125.162i	0.167590	−	0.414624i
$46$		0			0
$47$	13.8352			0.0429377			0.0214689	−	0.999770i	$-0.493166\pi$
$47$	13.8352			0.0429377			0.0214689	+	0.999770i	$0.493166\pi$
$48$		0			0
$49$	285.815			0.833280
$50$		0			0
$51$	11.5350	+	2.24306i	0.0316711	+	0.00615866i
$52$		0			0
$53$		−	222.334i		−	0.576225i	−0.957597	−	0.288112i	$-0.906972\pi$
$53$		−	222.334i		−	0.576225i	0.957597	−	0.288112i	$-0.0930277\pi$
$54$		0			0
$55$		−	18.0078i		−	0.0441486i
$56$		0			0
$57$	544.329	+	105.849i	1.26488	+	0.245965i
$58$		0			0
$59$	−442.239			−0.975842			−0.487921	−	0.872888i	$-0.662245\pi$
$59$	−442.239			−0.975842			−0.487921	+	0.872888i	$0.662245\pi$
$60$		0			0
$61$	601.295			1.26210			0.631049	−	0.775743i	$-0.282625\pi$
$61$	601.295			1.26210			0.631049	+	0.775743i	$0.282625\pi$
$62$		0			0
$63$	76.5137	−	189.298i	0.153013	−	0.378560i
$64$		0			0
$65$			1.65829i			0.00316440i
$66$		0			0
$67$			347.079i			0.632872i	0.948614	+	0.316436i	$0.102486\pi$
$67$			347.079i			0.632872i	−0.948614	+	0.316436i	$0.897514\pi$
$68$		0			0
$69$	162.379	−	835.040i	0.283307	−	1.45691i
$70$		0			0
$71$	634.275			1.06021			0.530103	−	0.847934i	$-0.322153\pi$
$71$	634.275			1.06021			0.530103	+	0.847934i	$0.322153\pi$
$72$		0			0
$73$	620.653			0.995096			0.497548	−	0.867437i	$-0.334234\pi$
$73$	620.653			0.995096			0.497548	+	0.867437i	$0.334234\pi$
$74$		0			0
$75$	−24.7962	+	127.515i	−0.0381763	+	0.196323i
$76$		0			0
$77$		−	27.2353i		−	0.0403084i
$78$		0			0
$79$			946.140i			1.34746i	0.738979	+	0.673728i	$0.235308\pi$
$79$			946.140i			1.34746i	−0.738979	+	0.673728i	$0.764692\pi$
$80$		0			0
$81$	524.249	+	506.561i	0.719134	+	0.694871i
$82$		0			0
$83$	655.062			0.866294			0.433147	−	0.901323i	$-0.357403\pi$
$83$	655.062			0.866294			0.433147	+	0.901323i	$0.357403\pi$
$84$		0			0
$85$	−11.3075			−0.0144290
$86$		0			0
$87$	−200.841	−	39.0550i	−0.247499	−	0.0481279i
$88$		0			0
$89$		−	920.102i		−	1.09585i	−0.836528	−	0.547925i	$-0.815418\pi$
$89$		−	920.102i		−	1.09585i	0.836528	−	0.547925i	$-0.184582\pi$
$90$		0			0
$91$			2.50803i			0.00288915i
$92$		0			0
$93$	1078.29	+	209.681i	1.20229	+	0.233795i
$94$		0			0
$95$	−533.592			−0.576267
$96$		0			0
$97$	264.948			0.277334			0.138667	−	0.990339i	$-0.455718\pi$
$97$	264.948			0.277334			0.138667	+	0.990339i	$0.455718\pi$
$98$		0			0
$99$	90.1559	+	36.4409i	0.0915254	+	0.0369944i
$100$		0			0
$101$		−	1339.31i		−	1.31947i	−0.751500	−	0.659733i	$-0.770669\pi$
$101$		−	1339.31i		−	1.31947i	0.751500	−	0.659733i	$-0.229331\pi$
$102$		0			0
$103$			78.5803i			0.0751723i	0.999293	+	0.0375861i	$0.0119669\pi$
$103$			78.5803i			0.0751723i	−0.999293	+	0.0375861i	$0.988033\pi$
$104$		0			0
$105$	−37.5022	+	192.856i	−0.0348556	+	0.179246i
$106$		0			0
$107$	−535.098			−0.483457			−0.241728	−	0.970344i	$-0.577714\pi$
$107$	−535.098			−0.483457			−0.241728	+	0.970344i	$0.577714\pi$
$108$		0			0
$109$	−1412.38			−1.24111			−0.620556	−	0.784162i	$-0.713093\pi$
$109$	−1412.38			−1.24111			−0.620556	+	0.784162i	$0.713093\pi$
$110$		0			0
$111$	30.5539	−	157.124i	0.0261265	−	0.134356i
$112$		0			0
$113$			2249.99i			1.87311i	0.350523	+	0.936554i	$0.386004\pi$
$113$			2249.99i			1.87311i	−0.350523	+	0.936554i	$0.613996\pi$
$114$		0			0
$115$			818.569i			0.663756i
$116$		0			0
$117$	−8.30224	−	3.35575i	−0.00656019	−	0.00265162i
$118$		0			0
$119$	−17.1016			−0.0131740
$120$		0			0
$121$	−1318.03			−0.990255
$122$		0			0
$123$	1369.29	+	266.268i	1.00378	+	0.195192i
$124$		0			0
$125$		−	125.000i		−	0.0894427i
$126$		0			0
$127$			1311.90i			0.916634i	0.888789	+	0.458317i	$0.151547\pi$
$127$			1311.90i			0.916634i	−0.888789	+	0.458317i	$0.848453\pi$
$128$		0			0
$129$	1034.47	+	201.161i	0.706050	+	0.137296i
$130$		0			0
$131$	331.540			0.221120			0.110560	−	0.993869i	$-0.464736\pi$
$131$	331.540			0.221120			0.110560	+	0.993869i	$0.464736\pi$
$132$		0			0
$133$	−807.013			−0.526142
$134$		0			0
$135$	−588.226	−	382.184i	−0.375011	−	0.243653i
$136$		0			0
$137$			2390.13i			1.49053i	0.666767	+	0.745266i	$0.267678\pi$
$137$			2390.13i			1.49053i	−0.666767	+	0.745266i	$0.732322\pi$
$138$		0			0
$139$		−	83.4868i		−	0.0509443i	−0.999676	−	0.0254721i	$-0.991891\pi$
$139$		−	83.4868i		−	0.0509443i	0.999676	−	0.0254721i	$-0.00810891\pi$
$140$		0			0
$141$	13.7225	−	70.5681i	0.00819602	−	0.0421482i
$142$		0			0
$143$	−1.19449			−0.000698519
$144$		0			0
$145$	196.879			0.112758
$146$		0			0
$147$	283.485	−	1457.83i	0.159058	−	0.817958i
$148$		0			0
$149$		−	3584.43i		−	1.97079i	−0.170280	−	0.985396i	$-0.554467\pi$
$149$		−	3584.43i		−	1.97079i	0.170280	−	0.985396i	$-0.445533\pi$
$150$		0			0
$151$			1708.54i			0.920788i	0.887715	+	0.460394i	$0.152292\pi$
$151$			1708.54i			0.920788i	−0.887715	+	0.460394i	$0.847708\pi$
$152$		0			0
$153$	22.8820	−	56.6108i	0.0120908	−	0.0299131i
$154$		0			0
$155$	−1057.02			−0.547754
$156$		0			0
$157$	−698.665			−0.355157			−0.177578	−	0.984107i	$-0.556826\pi$
$157$	−698.665			−0.355157			−0.177578	+	0.984107i	$0.556826\pi$
$158$		0			0
$159$	−1134.04	−	220.522i	−0.565630	−	0.109991i
$160$		0			0
$161$			1238.02i			0.606021i
$162$		0			0
$163$			1342.24i			0.644986i	0.946572	+	0.322493i	$0.104521\pi$
$163$			1342.24i			0.644986i	−0.946572	+	0.322493i	$0.895479\pi$
$164$		0			0
$165$	−91.8508	−	17.8610i	−0.0433368	−	0.00842714i
$166$		0			0
$167$	−309.196			−0.143271			−0.0716356	−	0.997431i	$-0.522822\pi$
$167$	−309.196			−0.143271			−0.0716356	+	0.997431i	$0.522822\pi$
$168$		0			0
$169$	−2196.89			−0.999950
$170$		0			0
$171$	1079.78	−	2671.42i	0.482884	−	1.19467i
$172$		0			0
$173$		−	699.135i		−	0.307250i	−0.988129	−	0.153625i	$-0.950905\pi$
$173$		−	699.135i		−	0.307250i	0.988129	−	0.153625i	$-0.0490948\pi$
$174$		0			0
$175$		−	189.052i		−	0.0816628i
$176$		0			0
$177$	−438.635	+	2255.69i	−0.186270	+	0.957899i
$178$		0			0
$179$	1734.56			0.724286			0.362143	−	0.932123i	$-0.382045\pi$
$179$	1734.56			0.724286			0.362143	+	0.932123i	$0.382045\pi$
$180$		0			0
$181$	−1557.96			−0.639792			−0.319896	−	0.947453i	$-0.603648\pi$
$181$	−1557.96			−0.639792			−0.319896	+	0.947453i	$0.603648\pi$
$182$		0			0
$183$	596.394	−	3066.97i	0.240911	−	1.23889i
$184$		0			0
$185$			154.025i			0.0612115i
$186$		0			0
$187$		−	8.14491i		−	0.00318510i
$188$		0			0
$189$	−889.643	−	578.021i	−0.342392	−	0.222460i
$190$		0			0
$191$	92.4394			0.0350193			0.0175096	−	0.999847i	$-0.494426\pi$
$191$	92.4394			0.0350193			0.0175096	+	0.999847i	$0.494426\pi$
$192$		0			0
$193$	−3015.83			−1.12479			−0.562394	−	0.826869i	$-0.690119\pi$
$193$	−3015.83			−1.12479			−0.562394	+	0.826869i	$0.690119\pi$
$194$		0			0
$195$	8.45831	+	1.64478i	0.00310622	+	0.000604026i
$196$		0			0
$197$			1482.37i			0.536115i	0.963403	+	0.268057i	$0.0863817\pi$
$197$			1482.37i			0.536115i	−0.963403	+	0.268057i	$0.913618\pi$
$198$		0			0
$199$		−	527.079i		−	0.187757i	−0.995584	−	0.0938785i	$-0.970073\pi$
$199$		−	527.079i		−	0.187757i	0.995584	−	0.0938785i	$-0.0299265\pi$
$200$		0			0
$201$	1770.32	+	344.250i	0.621236	+	0.120804i
$202$		0			0
$203$	297.764			0.102950
$204$		0			0
$205$	−1342.28			−0.457312
$206$		0			0
$207$	−4098.16	−	1656.47i	−1.37605	−	0.556196i
$208$		0			0
$209$		−	384.353i		−	0.127207i
$210$		0			0
$211$		−	3224.65i		−	1.05211i	−0.850452	−	0.526053i	$-0.823671\pi$
$211$		−	3224.65i		−	1.05211i	0.850452	−	0.526053i	$-0.176329\pi$
$212$		0			0
$213$	629.105	−	3235.19i	0.202374	−	1.04071i
$214$		0			0
$215$	−1014.07			−0.321669
$216$		0			0
$217$	−1598.65			−0.500109
$218$		0			0
$219$	615.595	−	3165.71i	0.189945	−	0.976799i
$220$		0			0
$221$			0.750045i			0.000228296i
$222$		0			0
$223$		−	5592.65i		−	1.67942i	−0.543032	−	0.839712i	$-0.682724\pi$
$223$		−	5592.65i		−	1.67942i	0.543032	−	0.839712i	$-0.317276\pi$
$224$		0			0
$225$	625.812	+	252.952i	0.185426	+	0.0749487i
$226$		0			0
$227$	−4878.86			−1.42653			−0.713263	−	0.700896i	$-0.752784\pi$
$227$	−4878.86			−1.42653			−0.713263	+	0.700896i	$0.752784\pi$
$228$		0			0
$229$	−3451.84			−0.996087			−0.498044	−	0.867152i	$-0.665948\pi$
$229$	−3451.84			−0.996087			−0.498044	+	0.867152i	$0.665948\pi$
$230$		0			0
$231$	−138.917	−	27.0133i	−0.0395673	−	0.00769413i
$232$		0			0
$233$		−	1819.14i		−	0.511485i	−0.966745	−	0.255742i	$-0.917680\pi$
$233$		−	1819.14i		−	0.511485i	0.966745	−	0.255742i	$-0.0823199\pi$
$234$		0			0
$235$			69.1761i			0.0192023i
$236$		0			0
$237$	4825.89	+	938.429i	1.32268	+	0.257205i
$238$		0			0
$239$	5962.04			1.61361			0.806803	−	0.590820i	$-0.201196\pi$
$239$	5962.04			1.61361			0.806803	+	0.590820i	$0.201196\pi$
$240$		0			0
$241$	−1303.84			−0.348497			−0.174249	−	0.984702i	$-0.555750\pi$
$241$	−1303.84			−0.348497			−0.174249	+	0.984702i	$0.555750\pi$
$242$		0			0
$243$	3103.75	−	2171.56i	0.819364	−	0.573274i
$244$		0			0
$245$			1429.07i			0.372654i
$246$		0			0
$247$			35.3941i			0.00911770i
$248$		0			0
$249$	649.723	−	3341.22i	0.165359	−	0.850365i
$250$		0			0
$251$	−6591.43			−1.65756			−0.828780	−	0.559574i	$-0.810965\pi$
$251$	−6591.43			−1.65756			−0.828780	+	0.559574i	$0.810965\pi$
$252$		0			0
$253$	−589.625			−0.146519
$254$		0			0
$255$	−11.2153	+	57.6750i	−0.00275424	+	0.0141637i
$256$		0			0
$257$			6088.10i			1.47769i	0.673878	+	0.738843i	$0.264627\pi$
$257$			6088.10i			1.47769i	−0.673878	+	0.738843i	$0.735373\pi$
$258$		0			0
$259$			232.949i			0.0558872i
$260$		0			0
$261$	−398.408	+	985.676i	−0.0944860	+	0.233762i
$262$		0			0
$263$	3905.07			0.915577			0.457788	−	0.889061i	$-0.348642\pi$
$263$	3905.07			0.915577			0.457788	+	0.889061i	$0.348642\pi$
$264$		0			0
$265$	1111.67			0.257696
$266$		0			0
$267$	−4693.08	−	912.603i	−1.07570	−	0.209177i
$268$		0			0
$269$		−	1436.08i		−	0.325499i	−0.986667	−	0.162750i	$-0.947964\pi$
$269$		−	1436.08i		−	0.325499i	0.986667	−	0.162750i	$-0.0520363\pi$
$270$		0			0
$271$			3055.89i			0.684988i	0.939520	+	0.342494i	$0.111272\pi$
$271$			3055.89i			0.684988i	−0.939520	+	0.342494i	$0.888728\pi$
$272$		0			0
$273$	12.7925	+	2.48759i	0.00283603	+	0.000551486i
$274$		0			0
$275$	90.0390			0.0197438
$276$		0			0
$277$	−3584.60			−0.777538			−0.388769	−	0.921335i	$-0.627100\pi$
$277$	−3584.60			−0.777538			−0.388769	+	0.921335i	$0.627100\pi$
$278$		0			0
$279$	2139.00	−	5291.96i	0.458992	−	1.13556i
$280$		0			0
$281$		−	4437.97i		−	0.942161i	−0.882090	−	0.471081i	$-0.843864\pi$
$281$		−	4437.97i		−	0.942161i	0.882090	−	0.471081i	$-0.156136\pi$
$282$		0			0
$283$			6626.00i			1.39178i	0.718146	+	0.695892i	$0.244991\pi$
$283$			6626.00i			1.39178i	−0.718146	+	0.695892i	$0.755009\pi$
$284$		0			0
$285$	−529.243	+	2721.65i	−0.109999	+	0.565671i
$286$		0			0
$287$	−2030.09			−0.417534
$288$		0			0
$289$	4907.89			0.998959
$290$		0			0
$291$	262.789	−	1351.40i	0.0529380	−	0.272235i
$292$		0			0
$293$		−	2675.75i		−	0.533512i	−0.963764	−	0.266756i	$-0.914048\pi$
$293$		−	2675.75i		−	0.533512i	0.963764	−	0.266756i	$-0.0859517\pi$
$294$		0			0
$295$		−	2211.20i		−	0.436410i
$296$		0			0
$297$	275.292	−	423.706i	0.0537847	−	0.0827809i
$298$		0			0
$299$	54.2971			0.0105019
$300$		0			0
$301$	−1533.69			−0.293690
$302$		0			0
$303$	−6831.29	−	1328.39i	−1.29521	−	0.251862i
$304$		0			0
$305$			3006.48i			0.564427i
$306$		0			0
$307$			627.027i			0.116568i	0.998300	+	0.0582839i	$0.0185629\pi$
$307$			627.027i			0.116568i	−0.998300	+	0.0582839i	$0.981437\pi$
$308$		0			0
$309$	400.807	+	77.9398i	0.0737901	+	0.0143490i
$310$		0			0
$311$	7966.93			1.45261			0.726307	−	0.687370i	$-0.241235\pi$
$311$	7966.93			1.45261			0.726307	+	0.687370i	$0.241235\pi$
$312$		0			0
$313$	−9332.71			−1.68536			−0.842678	−	0.538418i	$-0.819022\pi$
$313$	−9332.71			−1.68536			−0.842678	+	0.538418i	$0.819022\pi$
$314$		0			0
$315$	946.488	+	382.569i	0.169297	+	0.0684295i
$316$		0			0
$317$			7948.93i			1.40838i	0.710012	+	0.704190i	$0.248690\pi$
$317$			7948.93i			1.40838i	−0.710012	+	0.704190i	$0.751310\pi$
$318$		0			0
$319$			141.815i			0.0248906i
$320$		0			0
$321$	−530.737	+	2729.33i	−0.0922830	+	0.474568i
$322$		0			0
$323$	−241.343			−0.0415749
$324$		0			0
$325$	−8.29147			−0.00141516
$326$		0			0
$327$	−1400.87	+	7203.99i	−0.236905	+	1.21829i
$328$		0			0
$329$			104.623i			0.0175321i
$330$		0			0
$331$			3174.68i			0.527179i	0.964635	+	0.263589i	$0.0849064\pi$
$331$			3174.68i			0.527179i	−0.964635	+	0.263589i	$0.915094\pi$
$332$		0			0
$333$	−771.124	−	311.687i	−0.126899	−	0.0512923i
$334$		0			0
$335$	−1735.40			−0.283029
$336$		0			0
$337$	−596.381			−0.0964004			−0.0482002	−	0.998838i	$-0.515349\pi$
$337$	−596.381			−0.0964004			−0.0482002	+	0.998838i	$0.515349\pi$
$338$		0			0
$339$	11476.3	+	2231.65i	1.83867	+	0.357542i
$340$		0			0
$341$		−	761.384i		−	0.120913i
$342$		0			0
$343$			4755.15i			0.748554i
$344$		0			0
$345$	4175.20	+	811.897i	0.651551	+	0.126699i
$346$		0			0
$347$	−9910.10			−1.53315			−0.766573	−	0.642157i	$-0.778040\pi$
$347$	−9910.10			−1.53315			−0.766573	+	0.642157i	$0.778040\pi$
$348$		0			0
$349$	−7302.75			−1.12008			−0.560039	−	0.828466i	$-0.689214\pi$
$349$	−7302.75			−1.12008			−0.560039	+	0.828466i	$0.689214\pi$
$350$		0			0
$351$	−25.3509	+	39.0181i	−0.00385508	+	0.00593342i
$352$		0			0
$353$		−	11157.8i		−	1.68235i	−0.540766	−	0.841173i	$-0.681866\pi$
$353$		−	11157.8i		−	1.68235i	0.540766	−	0.841173i	$-0.318134\pi$
$354$		0			0
$355$			3171.37i			0.474138i
$356$		0			0
$357$	−16.9622	+	87.2286i	−0.00251467	+	0.0129317i
$358$		0			0
$359$	13045.4			1.91785			0.958927	−	0.283652i	$-0.0915461\pi$
$359$	13045.4			1.91785			0.958927	+	0.283652i	$0.0915461\pi$
$360$		0			0
$361$	−4529.81			−0.660419
$362$		0			0
$363$	−1307.29	+	6722.75i	−0.189021	+	0.972047i
$364$		0			0
$365$			3103.27i			0.445020i
$366$		0			0
$367$			8284.26i			1.17830i	0.808025	+	0.589148i	$0.200537\pi$
$367$			8284.26i			1.17830i	−0.808025	+	0.589148i	$0.799463\pi$
$368$		0			0
$369$	2716.26	−	6720.12i	0.383206	−	0.948064i
$370$		0			0
$371$	1681.31			0.235281
$372$		0			0
$373$	13725.9			1.90537			0.952684	−	0.303962i	$-0.0983097\pi$
$373$	13725.9			1.90537			0.952684	+	0.303962i	$0.0983097\pi$
$374$		0			0
$375$	−637.576	−	123.981i	−0.0877981	−	0.0170730i
$376$		0			0
$377$		−	13.0594i		−	0.00178406i
$378$		0			0
$379$		−	492.460i		−	0.0667440i	−0.999443	−	0.0333720i	$-0.989375\pi$
$379$		−	492.460i		−	0.0667440i	0.999443	−	0.0333720i	$-0.0106246\pi$
$380$		0			0
$381$	6691.50	+	1301.21i	0.899780	+	0.174968i
$382$		0			0
$383$	−2848.35			−0.380010			−0.190005	−	0.981783i	$-0.560850\pi$
$383$	−2848.35			−0.380010			−0.190005	+	0.981783i	$0.560850\pi$
$384$		0			0
$385$	136.176			0.0180265
$386$		0			0
$387$	2052.09	−	5076.93i	0.269544	−	0.666860i
$388$		0			0
$389$			7422.19i			0.967404i	0.875233	+	0.483702i	$0.160708\pi$
$389$			7422.19i			0.967404i	−0.875233	+	0.483702i	$0.839292\pi$
$390$		0			0
$391$			370.238i			0.0478868i
$392$		0			0
$393$	328.838	−	1691.06i	0.0422078	−	0.217055i
$394$		0			0
$395$	−4730.70			−0.602601
$396$		0			0
$397$	10609.7			1.34127			0.670637	−	0.741785i	$-0.266021\pi$
$397$	10609.7			1.34127			0.670637	+	0.741785i	$0.266021\pi$
$398$		0			0
$399$	−800.436	+	4116.26i	−0.100431	+	0.516468i
$400$		0			0
$401$		−	12084.4i		−	1.50490i	−0.658650	−	0.752449i	$-0.728872\pi$
$401$		−	12084.4i		−	1.50490i	0.658650	−	0.752449i	$-0.271128\pi$
$402$		0			0
$403$			70.1140i			0.00866657i
$404$		0			0
$405$	−2532.80	+	2621.25i	−0.310756	+	0.321607i
$406$		0			0
$407$	−110.946			−0.0135120
$408$		0			0
$409$	11303.0			1.36650			0.683251	−	0.730184i	$-0.260566\pi$
$409$	11303.0			1.36650			0.683251	+	0.730184i	$0.260566\pi$
$410$		0			0
$411$	12191.1	+	2370.65i	1.46313	+	0.284515i
$412$		0			0
$413$		−	3344.25i		−	0.398450i
$414$		0			0
$415$			3275.31i			0.387418i
$416$		0			0
$417$	−425.834	−	82.8063i	−0.0500076	−	0.00972433i
$418$		0			0
$419$	13156.2			1.53395			0.766974	−	0.641678i	$-0.221762\pi$
$419$	13156.2			1.53395			0.766974	+	0.641678i	$0.221762\pi$
$420$		0			0
$421$	10659.2			1.23396			0.616979	−	0.786980i	$-0.288356\pi$
$421$	10659.2			1.23396			0.616979	+	0.786980i	$0.288356\pi$
$422$		0			0
$423$	−346.330	−	139.986i	−0.0398088	−	0.0160906i
$424$		0			0
$425$		−	56.5374i		−	0.00645286i
$426$		0			0
$427$			4547.04i			0.515332i
$428$		0			0
$429$	−1.18475	+	6.09262i	−0.000133334	+	0.000685675i
$430$		0			0
$431$	−13912.5			−1.55485			−0.777426	−	0.628975i	$-0.783475\pi$
$431$	−13912.5			−1.55485			−0.777426	+	0.628975i	$0.783475\pi$
$432$		0			0
$433$	−675.722			−0.0749956			−0.0374978	−	0.999297i	$-0.511939\pi$
$433$	−675.722			−0.0749956			−0.0374978	+	0.999297i	$0.511939\pi$
$434$		0			0
$435$	195.275	−	1004.21i	0.0215235	−	0.110685i
$436$		0			0
$437$			17471.3i			1.91250i
$438$		0			0
$439$			445.601i			0.0484451i	0.999707	+	0.0242225i	$0.00771102\pi$
$439$			445.601i			0.0484451i	−0.999707	+	0.0242225i	$0.992289\pi$
$440$		0			0
$441$	−7154.65	−	2891.90i	−0.772557	−	0.312266i
$442$		0			0
$443$	−2942.79			−0.315612			−0.157806	−	0.987470i	$-0.550442\pi$
$443$	−2942.79			−0.315612			−0.157806	+	0.987470i	$0.550442\pi$
$444$		0			0
$445$	4600.51			0.490079
$446$		0			0
$447$	−18282.8	−	3555.21i	−1.93455	−	0.376188i
$448$		0			0
$449$		−	17229.3i		−	1.81091i	−0.424438	−	0.905457i	$-0.639528\pi$
$449$		−	17229.3i		−	1.81091i	0.424438	−	0.905457i	$-0.360472\pi$
$450$		0			0
$451$		−	966.861i		−	0.100948i
$452$		0			0
$453$	8714.60	+	1694.62i	0.903858	+	0.175762i
$454$		0			0
$455$	−12.5402			−0.00129207
$456$		0			0
$457$	13952.0			1.42811			0.714055	−	0.700090i	$-0.246857\pi$
$457$	13952.0			1.42811			0.714055	+	0.700090i	$0.246857\pi$
$458$		0			0
$459$	−266.054	−	172.861i	−0.0270552	−	0.0175784i
$460$		0			0
$461$		−	9609.93i		−	0.970887i	−0.874268	−	0.485444i	$-0.838658\pi$
$461$		−	9609.93i		−	0.970887i	0.874268	−	0.485444i	$-0.161342\pi$
$462$		0			0
$463$			1619.02i			0.162510i	0.996693	+	0.0812550i	$0.0258928\pi$
$463$			1619.02i			0.162510i	−0.996693	+	0.0812550i	$0.974107\pi$
$464$		0			0
$465$	−1048.40	+	5391.45i	−0.104556	+	0.537683i
$466$		0			0
$467$	2089.28			0.207024			0.103512	−	0.994628i	$-0.466992\pi$
$467$	2089.28			0.207024			0.103512	+	0.994628i	$0.466992\pi$
$468$		0			0
$469$	−2624.64			−0.258411
$470$		0			0
$471$	−692.971	+	3563.62i	−0.0677928	+	0.348626i
$472$		0			0
$473$		−	730.446i		−	0.0710062i
$474$		0			0
$475$		−	2667.96i		−	0.257715i
$476$		0			0
$477$	−2249.59	+	5565.57i	−0.215937	+	0.534234i
$478$		0			0
$479$	5299.72			0.505533			0.252767	−	0.967527i	$-0.418660\pi$
$479$	5299.72			0.505533			0.252767	+	0.967527i	$0.418660\pi$
$480$		0			0
$481$	10.2167			0.000968488
$482$		0			0
$483$	6314.64	+	1227.93i	0.594878	+	0.115678i
$484$		0			0
$485$			1324.74i			0.124028i
$486$		0			0
$487$		−	4111.11i		−	0.382530i	−0.981538	−	0.191265i	$-0.938741\pi$
$487$		−	4111.11i		−	0.382530i	0.981538	−	0.191265i	$-0.0612590\pi$
$488$		0			0
$489$	6846.27	+	1331.30i	0.633126	+	0.123116i
$490$		0			0
$491$	−19369.9			−1.78035			−0.890176	−	0.455617i	$-0.849419\pi$
$491$	−19369.9			−1.78035			−0.890176	+	0.455617i	$0.849419\pi$
$492$		0			0
$493$	89.0484			0.00813496
$494$		0			0
$495$	−182.204	+	450.780i	−0.0165444	+	0.0409314i
$496$		0			0
$497$			4796.44i			0.432897i
$498$		0			0
$499$		−	11765.0i		−	1.05546i	−0.849412	−	0.527731i	$-0.823043\pi$
$499$		−	11765.0i		−	1.05546i	0.849412	−	0.527731i	$-0.176957\pi$
$500$		0			0
$501$	−306.676	+	1577.09i	−0.0273478	+	0.140637i
$502$		0			0
$503$	15301.1			1.35635			0.678175	−	0.734900i	$-0.262771\pi$
$503$	15301.1			1.35635			0.678175	+	0.734900i	$0.262771\pi$
$504$		0			0
$505$	6696.54			0.590083
$506$		0			0
$507$	−2178.98	+	11205.5i	−0.190872	+	0.981564i
$508$		0			0
$509$			3104.14i			0.270312i	0.990824	+	0.135156i	$0.0431535\pi$
$509$			3104.14i			0.270312i	−0.990824	+	0.135156i	$0.956846\pi$
$510$		0			0
$511$			4693.43i			0.406312i
$512$		0			0
$513$	−12554.9	−	8157.21i	−1.08053	−	0.702046i
$514$		0			0
$515$	−392.901			−0.0336181
$516$		0			0
$517$	−49.8283			−0.00423878
$518$		0			0
$519$	−3566.01	−	693.436i	−0.301601	−	0.0586483i
$520$		0			0
$521$			9588.86i			0.806326i	0.915128	+	0.403163i	$0.132089\pi$
$521$			9588.86i			0.806326i	−0.915128	+	0.403163i	$0.867911\pi$
$522$		0			0
$523$		−	10769.0i		−	0.900374i	−0.892934	−	0.450187i	$-0.851357\pi$
$523$		−	10769.0i		−	0.900374i	0.892934	−	0.450187i	$-0.148643\pi$
$524$		0			0
$525$	−964.281	−	187.511i	−0.0801613	−	0.0155879i
$526$		0			0
$527$	−478.089			−0.0395178
$528$		0			0
$529$	14635.2			1.20286
$530$		0			0
$531$	11070.3	+	4474.61i	0.904730	+	0.365690i
$532$		0			0
$533$			89.0359i			0.00723560i
$534$		0			0
$535$		−	2675.49i		−	0.216208i
$536$		0			0
$537$	1720.42	−	8847.32i	0.138253	−	0.710968i
$538$		0			0
$539$	−1029.38			−0.0822607
$540$		0			0
$541$	14203.8			1.12878			0.564391	−	0.825508i	$-0.309111\pi$
$541$	14203.8			1.12878			0.564391	+	0.825508i	$0.309111\pi$
$542$		0			0
$543$	−1545.26	+	7946.55i	−0.122124	+	0.628028i
$544$		0			0
$545$		−	7061.89i		−	0.555043i
$546$		0			0
$547$			7545.49i			0.589803i	0.955528	+	0.294901i	$0.0952868\pi$
$547$			7545.49i			0.589803i	−0.955528	+	0.294901i	$0.904713\pi$
$548$		0			0
$549$	−15051.9	−	6083.95i	−1.17013	−	0.472963i
$550$		0			0
$551$	4202.13			0.324894
$552$		0			0
$553$	−7154.79			−0.550185
$554$		0			0
$555$	785.620	+	152.769i	0.0600860	+	0.0116841i
$556$		0			0
$557$			1999.80i			0.152126i	0.997103	+	0.0760631i	$0.0242350\pi$
$557$			1999.80i			0.152126i	−0.997103	+	0.0760631i	$0.975765\pi$
$558$		0			0
$559$			67.2650i			0.00508946i
$560$		0			0
$561$	−41.5440	−	8.07852i	−0.00312654	−	0.000607978i
$562$		0			0
$563$	7507.55			0.561999			0.281000	−	0.959708i	$-0.409334\pi$
$563$	7507.55			0.561999			0.281000	+	0.959708i	$0.409334\pi$
$564$		0			0
$565$	−11250.0			−0.837680
$566$		0			0
$567$	−3830.65	+	3964.41i	−0.283726	+	0.293633i
$568$		0			0
$569$		−	8616.82i		−	0.634861i	−0.948282	−	0.317430i	$-0.897180\pi$
$569$		−	8616.82i		−	0.634861i	0.948282	−	0.317430i	$-0.102820\pi$
$570$		0			0
$571$		−	7168.20i		−	0.525359i	−0.964883	−	0.262680i	$-0.915394\pi$
$571$		−	7168.20i		−	0.525359i	0.964883	−	0.262680i	$-0.0846062\pi$
$572$		0			0
$573$	91.6860	−	471.498i	0.00668453	−	0.0343754i
$574$		0			0
$575$	−4092.84			−0.296841
$576$		0			0
$577$	−11297.9			−0.815144			−0.407572	−	0.913173i	$-0.633624\pi$
$577$	−11297.9			−0.815144			−0.407572	+	0.913173i	$0.633624\pi$
$578$		0			0
$579$	−2991.25	+	15382.6i	−0.214701	+	1.10411i
$580$		0			0
$581$			4953.63i			0.353720i
$582$		0			0
$583$			800.749i			0.0568844i
$584$		0			0
$585$	16.7787	−	41.5112i	0.00118584	−	0.00293381i
$586$		0			0
$587$	−6935.92			−0.487694			−0.243847	−	0.969814i	$-0.578409\pi$
$587$	−6935.92			−0.487694			−0.243847	+	0.969814i	$0.578409\pi$
$588$		0			0
$589$	−22560.7			−1.57826
$590$		0			0
$591$	7561.00	+	1470.29i	0.526257	+	0.102334i
$592$		0			0
$593$			7039.12i			0.487457i	0.969843	+	0.243729i	$0.0783706\pi$
$593$			7039.12i			0.487457i	−0.969843	+	0.243729i	$0.921629\pi$
$594$		0			0
$595$		−	85.5080i		−	0.00589158i
$596$		0			0
$597$	−2688.43	−	522.783i	−0.184305	−	0.0358393i
$598$		0			0
$599$	10574.2			0.721283			0.360641	−	0.932705i	$-0.382558\pi$
$599$	10574.2			0.721283			0.360641	+	0.932705i	$0.382558\pi$
$600$		0			0
$601$	−12740.8			−0.864741			−0.432370	−	0.901696i	$-0.642323\pi$
$601$	−12740.8			−0.864741			−0.432370	+	0.901696i	$0.642323\pi$
$602$		0			0
$603$	3511.77	−	8688.25i	0.237165	−	0.586754i
$604$		0			0
$605$		−	6590.14i		−	0.442855i
$606$		0			0
$607$		−	25748.3i		−	1.72173i	−0.508831	−	0.860867i	$-0.669922\pi$
$607$		−	25748.3i		−	1.72173i	0.508831	−	0.860867i	$-0.330078\pi$
$608$		0			0
$609$	295.337	−	1518.78i	0.0196513	−	0.101057i
$610$		0			0
$611$	4.58857			0.000303820
$612$		0			0
$613$	16632.3			1.09587			0.547937	−	0.836520i	$-0.315413\pi$
$613$	16632.3			1.09587			0.547937	+	0.836520i	$0.315413\pi$
$614$		0			0
$615$	−1331.34	+	6846.46i	−0.0872925	+	0.448904i
$616$		0			0
$617$			24761.8i			1.61568i	0.589405	+	0.807838i	$0.299362\pi$
$617$			24761.8i			1.61568i	−0.589405	+	0.807838i	$0.700638\pi$
$618$		0			0
$619$			3258.09i			0.211557i	0.994390	+	0.105779i	$0.0337335\pi$
$619$			3258.09i			0.211557i	−0.994390	+	0.105779i	$0.966267\pi$
$620$		0			0
$621$	−12513.8	+	19260.1i	−0.808631	+	1.24458i
$622$		0			0
$623$	6957.89			0.447451
$624$		0			0
$625$	625.000			0.0400000
$626$		0			0
$627$	−1960.43	−	381.220i	−0.124868	−	0.0242814i
$628$		0			0
$629$			69.6652i			0.00441611i
$630$		0			0
$631$		−	23435.1i		−	1.47850i	−0.673429	−	0.739251i	$-0.735179\pi$
$631$		−	23435.1i		−	1.47850i	0.673429	−	0.739251i	$-0.264821\pi$
$632$		0			0
$633$	−16447.7	−	3198.37i	−1.03276	−	0.200827i
$634$		0			0
$635$	−6559.51			−0.409931
$636$		0			0
$637$	94.7931			0.00589613
$638$		0			0
$639$	−15877.5	−	6417.64i	−0.982946	−	0.397305i
$640$		0			0
$641$		−	5791.54i		−	0.356867i	−0.983952	−	0.178434i	$-0.942897\pi$
$641$		−	5791.54i		−	0.356867i	0.983952	−	0.178434i	$-0.0571030\pi$
$642$		0			0
$643$		−	28184.8i		−	1.72862i	−0.502963	−	0.864308i	$-0.667757\pi$
$643$		−	28184.8i		−	1.72862i	0.502963	−	0.864308i	$-0.332243\pi$
$644$		0			0
$645$	−1005.80	+	5172.37i	−0.0614008	+	0.315755i
$646$		0			0
$647$	18326.1			1.11356			0.556780	−	0.830660i	$-0.312037\pi$
$647$	18326.1			1.11356			0.556780	+	0.830660i	$0.312037\pi$
$648$		0			0
$649$	1592.75			0.0963343
$650$		0			0
$651$	−1585.62	+	8154.11i	−0.0954616	+	0.490914i
$652$		0			0
$653$		−	3131.15i		−	0.187644i	−0.995589	−	0.0938220i	$-0.970092\pi$
$653$		−	3131.15i		−	0.187644i	0.995589	−	0.0938220i	$-0.0299084\pi$
$654$		0			0
$655$			1657.70i			0.0988880i
$656$		0			0
$657$	−15536.5	−	6279.82i	−0.922582	−	0.372906i
$658$		0			0
$659$	−10369.6			−0.612961			−0.306480	−	0.951877i	$-0.599151\pi$
$659$	−10369.6			−0.612961			−0.306480	+	0.951877i	$0.599151\pi$
$660$		0			0
$661$	2158.66			0.127023			0.0635116	−	0.997981i	$-0.479770\pi$
$661$	2158.66			0.127023			0.0635116	+	0.997981i	$0.479770\pi$
$662$		0			0
$663$	3.82569	+	0.743931i	0.000224099	+	4.35775e-5i
$664$		0			0
$665$		−	4035.07i		−	0.235298i
$666$		0			0
$667$		−	6446.37i		−	0.374220i
$668$		0			0
$669$	−28525.9	−	5547.07i	−1.64854	−	0.320571i
$670$		0			0
$671$	−2165.60			−0.124593
$672$		0			0
$673$	−12912.3			−0.739574			−0.369787	−	0.929117i	$-0.620569\pi$
$673$	−12912.3			−0.739574			−0.369787	+	0.929117i	$0.620569\pi$
$674$		0			0
$675$	1910.92	−	2941.13i	0.108965	−	0.167710i
$676$		0			0
$677$			12719.5i			0.722081i	0.932550	+	0.361041i	$0.117578\pi$
$677$			12719.5i			0.722081i	−0.932550	+	0.361041i	$0.882422\pi$
$678$		0			0
$679$			2003.56i			0.113239i
$680$		0			0
$681$	−4839.10	+	24885.2i	−0.272298	+	1.40030i
$682$		0			0
$683$	−16158.9			−0.905273			−0.452637	−	0.891695i	$-0.649517\pi$
$683$	−16158.9			−0.905273			−0.452637	+	0.891695i	$0.649517\pi$
$684$		0			0
$685$	−11950.7			−0.666586
$686$		0			0
$687$	−3423.70	+	17606.5i	−0.190135	+	0.977772i
$688$		0			0
$689$		−	73.7390i		−	0.00407726i
$690$		0			0
$691$			16564.6i			0.911934i	0.889997	+	0.455967i	$0.150707\pi$
$691$			16564.6i			0.911934i	−0.889997	+	0.455967i	$0.849293\pi$
$692$		0			0
$693$	−275.569	+	681.766i	−0.0151053	+	0.0373711i
$694$		0			0
$695$	417.434			0.0227830
$696$		0			0
$697$	−607.113			−0.0329929
$698$		0			0
$699$	−9278.74	−	1804.32i	−0.502080	−	0.0976330i
$700$		0			0
$701$		−	18242.4i		−	0.982893i	−0.870908	−	0.491446i	$-0.836468\pi$
$701$		−	18242.4i		−	0.982893i	0.870908	−	0.491446i	$-0.163532\pi$
$702$		0			0
$703$			3287.45i			0.176371i
$704$		0			0
$705$	352.840	+	68.6123i	0.0188493	+	0.00366537i
$706$		0			0
$707$	10128.0			0.538757
$708$		0			0
$709$	−2575.81			−0.136441			−0.0682203	−	0.997670i	$-0.521732\pi$
$709$	−2575.81			−0.136441			−0.0682203	+	0.997670i	$0.521732\pi$
$710$		0			0
$711$	9573.12	−	23684.2i	0.504951	−	1.24927i
$712$		0			0
$713$			34609.7i			1.81787i
$714$		0			0
$715$		−	5.97244i		−	0.000312387i
$716$		0			0
$717$	5913.44	−	30410.0i	0.308008	−	1.58394i
$718$		0			0
$719$	−22087.5			−1.14565			−0.572827	−	0.819677i	$-0.694153\pi$
$719$	−22087.5			−1.14565			−0.572827	+	0.819677i	$0.694153\pi$
$720$		0			0
$721$	−594.230			−0.0306939
$722$		0			0
$723$	−1293.22	+	6650.39i	−0.0665217	+	0.342090i
$724$		0			0
$725$			984.397i			0.0504270i
$726$		0			0
$727$		−	20412.7i		−	1.04136i	−0.853753	−	0.520678i	$-0.825679\pi$
$727$		−	20412.7i		−	1.04136i	0.853753	−	0.520678i	$-0.174321\pi$
$728$		0			0
$729$	−7997.83	−	17984.9i	−0.406332	−	0.913726i
$730$		0			0
$731$	−458.662			−0.0232069
$732$		0			0
$733$	21094.4			1.06294			0.531472	−	0.847076i	$-0.321639\pi$
$733$	21094.4			1.06294			0.531472	+	0.847076i	$0.321639\pi$
$734$		0			0
$735$	7289.15	+	1417.43i	0.365802	+	0.0711328i
$736$		0			0
$737$		−	1250.03i		−	0.0624766i
$738$		0			0
$739$		−	23587.2i		−	1.17411i	−0.809546	−	0.587056i	$-0.800287\pi$
$739$		−	23587.2i		−	1.17411i	0.809546	−	0.587056i	$-0.199713\pi$
$740$		0			0
$741$	180.532	+	35.1056i	0.00895006	+	0.00174040i
$742$		0			0
$743$	21493.1			1.06125			0.530624	−	0.847608i	$-0.321958\pi$
$743$	21493.1			1.06125			0.530624	+	0.847608i	$0.321958\pi$
$744$		0			0
$745$	17922.1			0.881365
$746$		0			0
$747$	−16397.8	−	6627.97i	−0.803165	−	0.324638i
$748$		0			0
$749$		−	4046.46i		−	0.197402i
$750$		0			0
$751$		−	28360.3i		−	1.37801i	−0.724759	−	0.689003i	$-0.758049\pi$
$751$		−	28360.3i		−	1.37801i	0.724759	−	0.689003i	$-0.241951\pi$
$752$		0			0
$753$	−6537.71	+	33620.3i	−0.316398	+	1.62708i
$754$		0			0
$755$	−8542.70			−0.411789
$756$		0			0
$757$	−2336.95			−0.112203			−0.0561017	−	0.998425i	$-0.517867\pi$
$757$	−2336.95			−0.112203			−0.0561017	+	0.998425i	$0.517867\pi$
$758$		0			0
$759$	−584.819	+	3007.45i	−0.0279678	+	0.143825i
$760$		0			0
$761$		−	15498.7i		−	0.738274i	−0.929375	−	0.369137i	$-0.879653\pi$
$761$		−	15498.7i		−	0.738274i	0.929375	−	0.369137i	$-0.120347\pi$
$762$		0			0
$763$		−	10680.5i		−	0.506764i
$764$		0			0
$765$	283.054	+	114.410i	0.0133776	+	0.00540719i
$766$		0			0
$767$	−146.673			−0.00690488
$768$		0			0
$769$	34398.0			1.61304			0.806518	−	0.591210i	$-0.201349\pi$
$769$	34398.0			1.61304			0.806518	+	0.591210i	$0.201349\pi$
$770$		0			0
$771$	31053.0	+	6038.48i	1.45052	+	0.282063i
$772$		0			0
$773$			8017.03i			0.373030i	0.982452	+	0.186515i	$0.0597194\pi$
$773$			8017.03i			0.373030i	−0.982452	+	0.186515i	$0.940281\pi$
$774$		0			0
$775$		−	5285.10i		−	0.244963i
$776$		0			0
$777$	1188.18	+	231.051i	0.0548596	+	0.0106678i
$778$		0			0
$779$	−28649.2			−1.31767
$780$		0			0
$781$	−2284.38			−0.104663
$782$		0			0
$783$	4632.39	+	3009.77i	0.211428	+	0.137370i
$784$		0			0
$785$		−	3493.33i		−	0.158831i
$786$		0			0
$787$		−	4787.96i		−	0.216864i	−0.994104	−	0.108432i	$-0.965417\pi$
$787$		−	4787.96i		−	0.216864i	0.994104	−	0.108432i	$-0.0345830\pi$
$788$		0			0
$789$	3873.24	−	19918.2i	0.174767	−	0.898742i
$790$		0			0
$791$	−17014.6			−0.764816
$792$		0			0
$793$	199.425			0.00893037
$794$		0			0
$795$	1102.61	−	5670.19i	0.0491893	−	0.252957i
$796$		0			0
$797$		−	42855.5i		−	1.90467i	−0.305059	−	0.952334i	$-0.598676\pi$
$797$		−	42855.5i		−	1.90467i	0.305059	−	0.952334i	$-0.401324\pi$
$798$		0			0
$799$			31.2883i			0.00138536i
$800$		0			0
$801$	−9309.67	+	23032.4i	−0.410663	+	1.01599i
$802$		0			0
$803$	−2235.32			−0.0982350
$804$		0			0
$805$	−6190.08			−0.271021
$806$		0			0
$807$	−7324.89	−	1424.38i	−0.319514	−	0.0621318i
$808$		0			0
$809$			39261.4i			1.70625i	0.521705	+	0.853126i	$0.325296\pi$
$809$			39261.4i			1.70625i	−0.521705	+	0.853126i	$0.674704\pi$
$810$		0			0
$811$		−	18358.6i		−	0.794890i	−0.917626	−	0.397445i	$-0.869897\pi$
$811$		−	18358.6i		−	0.794890i	0.917626	−	0.397445i	$-0.130103\pi$
$812$		0			0
$813$	15586.9	+	3030.98i	0.672394	+	0.130752i
$814$		0			0
$815$	−6711.22			−0.288446
$816$		0			0
$817$	−21644.0			−0.926838
$818$		0			0
$819$	25.3765	−	62.7822i	0.00108269	−	0.00267862i
$820$		0			0
$821$		−	17109.3i		−	0.727307i	−0.931534	−	0.363653i	$-0.881529\pi$
$821$		−	17109.3i		−	0.727307i	0.931534	−	0.363653i	$-0.118471\pi$
$822$		0			0
$823$			2383.56i			0.100955i	0.998725	+	0.0504773i	$0.0160742\pi$
$823$			2383.56i			0.100955i	−0.998725	+	0.0504773i	$0.983926\pi$
$824$		0			0
$825$	89.3051	−	459.254i	0.00376873	−	0.0193808i
$826$		0			0
$827$	42820.0			1.80048			0.900241	−	0.435392i	$-0.143390\pi$
$827$	42820.0			1.80048			0.900241	+	0.435392i	$0.143390\pi$
$828$		0			0
$829$	−29767.6			−1.24713			−0.623565	−	0.781772i	$-0.714316\pi$
$829$	−29767.6			−1.24713			−0.623565	+	0.781772i	$0.714316\pi$
$830$		0			0
$831$	−3555.39	+	18283.7i	−0.148418	+	0.763241i
$832$		0			0
$833$			646.369i			0.0268852i
$834$		0			0
$835$		−	1545.98i		−	0.0640728i
$836$		0			0
$837$	−24870.7	−	16159.0i	−1.02707	−	0.667310i
$838$		0			0
$839$	−4587.05			−0.188752			−0.0943758	−	0.995537i	$-0.530086\pi$
$839$	−4587.05			−0.188752			−0.0943758	+	0.995537i	$0.530086\pi$
$840$		0			0
$841$	22838.5			0.936428
$842$		0			0
$843$	−22636.4	−	4401.80i	−0.924838	−	0.179841i
$844$		0			0
$845$		−	10984.5i		−	0.447191i
$846$		0			0
$847$		−	9967.04i		−	0.404335i
$848$		0			0
$849$	33796.7	+	6572.00i	1.36619	+	0.265666i
$850$		0			0
$851$	5043.19			0.203147
$852$		0			0
$853$	13351.4			0.535924			0.267962	−	0.963429i	$-0.413650\pi$
$853$	13351.4			0.535924			0.267962	+	0.963429i	$0.413650\pi$
$854$		0			0
$855$	13357.1	+	5398.92i	0.534274	+	0.215952i
$856$		0			0
$857$		−	7865.61i		−	0.313517i	−0.987637	−	0.156759i	$-0.949896\pi$
$857$		−	7865.61i		−	0.313517i	0.987637	−	0.156759i	$-0.0501044\pi$
$858$		0			0
$859$			16368.8i			0.650171i	0.945685	+	0.325085i	$0.105393\pi$
$859$			16368.8i			0.650171i	−0.945685	+	0.325085i	$0.894607\pi$
$860$		0			0
$861$	−2013.54	+	10354.7i	−0.0796996	+	0.409857i
$862$		0			0
$863$	−15437.7			−0.608930			−0.304465	−	0.952524i	$-0.598478\pi$
$863$	−15437.7			−0.608930			−0.304465	+	0.952524i	$0.598478\pi$
$864$		0			0
$865$	3495.67			0.137406
$866$		0			0
$867$	4867.88	−	25033.2i	0.190683	−	0.980591i
$868$		0			0
$869$		−	3407.58i		−	0.133020i
$870$		0			0
$871$			115.112i			0.00447809i
$872$		0			0
$873$	−6632.31	−	2680.77i	−0.257125	−	0.103929i
$874$		0			0
$875$	945.260			0.0365207
$876$		0			0
$877$	49168.8			1.89317			0.946586	−	0.322451i	$-0.104507\pi$
$877$	49168.8			1.89317			0.946586	+	0.322451i	$0.104507\pi$
$878$		0			0
$879$	−13648.0	−	2653.94i	−0.523702	−	0.101837i
$880$		0			0
$881$			681.297i			0.0260539i	0.999915	+	0.0130270i	$0.00414672\pi$
$881$			681.297i			0.0260539i	−0.999915	+	0.0130270i	$0.995853\pi$
$882$		0			0
$883$		−	4787.17i		−	0.182447i	−0.995830	−	0.0912237i	$-0.970922\pi$
$883$		−	4787.17i		−	0.182447i	0.995830	−	0.0912237i	$-0.0290778\pi$
$884$		0			0
$885$	−11278.5	−	2193.17i	−0.428385	−	0.0833025i
$886$		0			0
$887$	34679.9			1.31278			0.656391	−	0.754421i	$-0.272082\pi$
$887$	34679.9			1.31278			0.656391	+	0.754421i	$0.272082\pi$
$888$		0			0
$889$	−9920.71			−0.374274
$890$		0			0
$891$	−1888.11	−	1824.41i	−0.0709924	−	0.0685971i
$892$		0			0
$893$			1476.47i			0.0553284i
$894$		0			0
$895$			8672.80i			0.323910i
$896$		0			0
$897$	53.8546	−	276.948i	0.00200463	−	0.0103088i
$898$		0			0
$899$	8324.22			0.308819
$900$		0			0
$901$	502.807			0.0185915
$902$		0			0
$903$	−1521.19	+	7822.78i	−0.0560600	+	0.288290i
$904$		0			0
$905$		−	7789.81i		−	0.286124i
$906$		0			0
$907$			48652.6i			1.78113i	0.454859	+	0.890564i	$0.349690\pi$
$907$			48652.6i			1.78113i	−0.454859	+	0.890564i	$0.650310\pi$
$908$		0			0
$909$	−13551.2	+	33526.2i	−0.494462	+	1.22332i
$910$		0			0
$911$	−27930.6			−1.01579			−0.507894	−	0.861420i	$-0.669576\pi$
$911$	−27930.6			−1.01579			−0.507894	+	0.861420i	$0.669576\pi$
$912$		0			0
$913$	−2359.24			−0.0855198
$914$		0			0
$915$	15334.9	+	2981.97i	0.554049	+	0.107739i
$916$		0			0
$917$			2507.13i			0.0902865i
$918$		0			0
$919$			13528.1i			0.485584i	0.970078	+	0.242792i	$0.0780633\pi$
$919$			13528.1i			0.485584i	−0.970078	+	0.242792i	$0.921937\pi$
$920$		0			0
$921$	3198.22	+	621.917i	0.114425	+	0.0222507i
$922$		0			0
$923$	210.363			0.00750182
$924$		0			0
$925$	−770.124			−0.0273746
$926$		0			0
$927$	795.081	−	1967.06i	0.0281703	−	0.0696944i
$928$		0			0
$929$			19568.9i			0.691104i	0.938400	+	0.345552i	$0.112308\pi$
$929$			19568.9i			0.691104i	−0.938400	+	0.345552i	$0.887692\pi$
$930$		0			0
$931$			30501.7i			1.07374i
$932$		0			0
$933$	7902.00	−	40636.2i	0.277277	−	1.42591i
$934$		0			0
$935$	40.7245			0.00142442
$936$		0			0
$937$	24127.4			0.841203			0.420602	−	0.907245i	$-0.361819\pi$
$937$	24127.4			0.841203			0.420602	+	0.907245i	$0.361819\pi$
$938$		0			0
$939$	−9256.65	+	47602.5i	−0.321703	+	1.65437i
$940$		0			0
$941$		−	16631.5i		−	0.576166i	−0.957605	−	0.288083i	$-0.906982\pi$
$941$		−	16631.5i		−	0.576166i	0.957605	−	0.288083i	$-0.0930180\pi$
$942$		0			0
$943$			43950.0i			1.51772i
$944$		0			0
$945$	2890.11	−	4448.22i	0.0994870	−	0.153122i
$946$		0			0
$947$	53121.4			1.82282			0.911411	−	0.411496i	$-0.134994\pi$
$947$	53121.4			1.82282			0.911411	+	0.411496i	$0.134994\pi$
$948$		0			0
$949$	205.845			0.00704111
$950$		0			0
$951$	40544.4	+	7884.14i	1.38248	+	0.268834i
$952$		0			0
$953$			4330.26i			0.147189i	0.997288	+	0.0735944i	$0.0234470\pi$
$953$			4330.26i			0.147189i	−0.997288	+	0.0735944i	$0.976553\pi$
$954$		0			0
$955$			462.197i			0.0156611i
$956$		0			0
$957$	723.341	+	140.659i	0.0244329	+	0.00475115i
$958$		0			0
$959$	−18074.4			−0.608605
$960$		0			0
$961$	−14900.6			−0.500173
$962$		0			0
$963$	13394.8	+	5414.16i	0.448227	+	0.181172i
$964$		0			0
$965$		−	15079.1i		−	0.503020i
$966$		0			0
$967$			20993.9i			0.698157i	0.937093	+	0.349078i	$0.113505\pi$
$967$			20993.9i			0.698157i	−0.937093	+	0.349078i	$0.886495\pi$
$968$		0			0
$969$	−239.376	+	1231.00i	−0.00793588	+	0.0408104i
$970$		0			0
$971$	−21909.0			−0.724093			−0.362047	−	0.932160i	$-0.617922\pi$
$971$	−21909.0			−0.724093			−0.362047	+	0.932160i	$0.617922\pi$
$972$		0			0
$973$	631.334			0.0208013
$974$		0			0
$975$	−8.22389	+	42.2916i	−0.000270129	+	0.00138914i
$976$		0			0
$977$			25168.2i			0.824158i	0.911148	+	0.412079i	$0.135197\pi$
$977$			25168.2i			0.824158i	−0.911148	+	0.412079i	$0.864803\pi$
$978$		0			0
$979$			3313.80i			0.108181i
$980$		0			0
$981$	35355.3	+	14290.6i	1.15067	+	0.465099i
$982$		0			0
$983$	−43478.1			−1.41072			−0.705360	−	0.708850i	$-0.749214\pi$
$983$	−43478.1			−1.41072			−0.705360	+	0.708850i	$0.749214\pi$
$984$		0			0
$985$	−7411.86			−0.239758
$986$		0			0
$987$	533.641	+	103.770i	0.0172097	+	0.00334655i
$988$		0			0
$989$			33203.4i			1.06755i
$990$		0			0
$991$			54030.2i			1.73191i	0.500121	+	0.865956i	$0.333289\pi$
$991$			54030.2i			1.73191i	−0.500121	+	0.865956i	$0.666711\pi$
$992$		0			0
$993$	16192.8	+	3148.80i	0.517485	+	0.100629i
$994$		0			0
$995$	2635.40			0.0839675
$996$		0			0
$997$	−46940.4			−1.49109			−0.745545	−	0.666455i	$-0.767811\pi$
$997$	−46940.4			−1.49109			−0.745545	+	0.666455i	$0.767811\pi$
$998$		0			0
$999$	−2354.63	+	3624.06i	−0.0745718	+	0.114775i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.h.b.191.13	yes	24
3.2	odd	2		480.4.h.a.191.11	✓	24
4.3	odd	2		480.4.h.a.191.12	yes	24
8.3	odd	2		960.4.h.e.191.13		24
8.5	even	2		960.4.h.c.191.12		24
12.11	even	2	inner	480.4.h.b.191.14	yes	24
24.5	odd	2		960.4.h.e.191.14		24
24.11	even	2		960.4.h.c.191.11		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.h.a.191.11	✓	24	3.2	odd	2
480.4.h.a.191.12	yes	24	4.3	odd	2
480.4.h.b.191.13	yes	24	1.1	even	1	trivial
480.4.h.b.191.14	yes	24	12.11	even	2	inner
960.4.h.c.191.11		24	24.11	even	2
960.4.h.c.191.12		24	8.5	even	2
960.4.h.e.191.13		24	8.3	odd	2
960.4.h.e.191.14		24	24.5	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.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.991849 - 5.10061i) q^{3} +5.00000i q^{5} +7.56208i q^{7} +(-25.0325 - 10.1181i) q^{9} -3.60156 q^{11} +0.331659 q^{13} +(25.5031 + 4.95925i) q^{15} +2.26149i q^{17} +106.718i q^{19} +(38.5712 + 7.50045i) q^{21} +163.714 q^{23} -25.0000 q^{25} +(-76.4368 + 117.645i) q^{27} -39.3759i q^{29} +211.404i q^{31} +(-3.57220 + 18.3702i) q^{33} -37.8104 q^{35} +30.8049 q^{37} +(0.328956 - 1.69166i) q^{39} +268.456i q^{41} +202.814i q^{43} +(50.5904 - 125.162i) q^{45} +13.8352 q^{47} +285.815 q^{49} +(11.5350 + 2.24306i) q^{51} -222.334i q^{53} -18.0078i q^{55} +(544.329 + 105.849i) q^{57} -442.239 q^{59} +601.295 q^{61} +(76.5137 - 189.298i) q^{63} +1.65829i q^{65} +347.079i q^{67} +(162.379 - 835.040i) q^{69} +634.275 q^{71} +620.653 q^{73} +(-24.7962 + 127.515i) q^{75} -27.2353i q^{77} +946.140i q^{79} +(524.249 + 506.561i) q^{81} +655.062 q^{83} -11.3075 q^{85} +(-200.841 - 39.0550i) q^{87} -920.102i q^{89} +2.50803i q^{91} +(1078.29 + 209.681i) q^{93} -533.592 q^{95} +264.948 q^{97} +(90.1559 + 36.4409i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 20 q^{9} + 72 q^{11} - 72 q^{13} + 20 q^{15} - 68 q^{21} + 96 q^{23} - 600 q^{25} + 168 q^{27} - 80 q^{33} - 504 q^{37} - 456 q^{39} - 220 q^{45} + 432 q^{47} - 816 q^{49} + 1240 q^{51} + 40 q^{57}+ \cdots + 3160 q^{99}+O(q^{100}) \) 24 * q + 20 * q^9 + 72 * q^11 - 72 * q^13 + 20 * q^15 - 68 * q^21 + 96 * q^23 - 600 * q^25 + 168 * q^27 - 80 * q^33 - 504 * q^37 - 456 * q^39 - 220 * q^45 + 432 * q^47 - 816 * q^49 + 1240 * q^51 + 40 * q^57 - 2184 * q^59 - 1080 * q^61 - 1720 * q^63 - 1548 * q^69 + 816 * q^71 - 1440 * q^73 - 2304 * q^81 - 2016 * q^83 - 2144 * q^87 - 112 * q^93 + 1800 * q^95 + 720 * q^97 + 3160 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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