LMFDB - Embedded newform 480.4.h.b.191.10

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			191.10
Character	$\chi$	$=$	480.191
Dual form			480.4.h.b.191.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.67023 + 4.45756i) q^{3} -5.00000i q^{5} -28.8929i q^{7} +(-12.7397 - 23.8055i) q^{9} -26.6844 q^{11} -63.0266 q^{13} +(22.2878 + 13.3512i) q^{15} +4.23575i q^{17} +158.425i q^{19} +(128.792 + 77.1507i) q^{21} +193.058 q^{23} -25.0000 q^{25} +(140.132 + 6.77825i) q^{27} -225.068i q^{29} +176.599i q^{31} +(71.2536 - 118.947i) q^{33} -144.464 q^{35} -36.2616 q^{37} +(168.296 - 280.945i) q^{39} +299.719i q^{41} +509.850i q^{43} +(-119.027 + 63.6985i) q^{45} +211.187 q^{47} -491.797 q^{49} +(-18.8811 - 11.3104i) q^{51} +701.674i q^{53} +133.422i q^{55} +(-706.189 - 423.032i) q^{57} +299.376 q^{59} +133.906 q^{61} +(-687.808 + 368.086i) q^{63} +315.133i q^{65} -292.983i q^{67} +(-515.510 + 860.568i) q^{69} -630.368 q^{71} -268.336 q^{73} +(66.7559 - 111.439i) q^{75} +770.989i q^{77} -549.289i q^{79} +(-404.401 + 606.549i) q^{81} +192.312 q^{83} +21.1787 q^{85} +(1003.25 + 600.983i) q^{87} -1202.98i q^{89} +1821.02i q^{91} +(-787.200 - 471.560i) q^{93} +792.125 q^{95} +831.831 q^{97} +(339.951 + 635.235i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.67023	+	4.45756i	−0.513887	+	0.857858i
$3$	−2.67023	+	4.45756i	−0.513887	+	0.857858i
$4$		0			0
$5$		−	5.00000i		−	0.447214i
$5$		−	5.00000i		−	0.447214i
$6$		0			0
$7$		−	28.8929i		−	1.56007i	−0.625737	−	0.780034i	$-0.715202\pi$
$7$		−	28.8929i		−	1.56007i	0.625737	−	0.780034i	$-0.284798\pi$
$8$		0			0
$9$	−12.7397	−	23.8055i	−0.471840	−	0.881684i
$10$		0			0
$11$	−26.6844			−0.731423			−0.365711	−	0.930728i	$-0.619174\pi$
$11$	−26.6844			−0.731423			−0.365711	+	0.930728i	$0.619174\pi$
$12$		0			0
$13$	−63.0266			−1.34465			−0.672325	−	0.740256i	$-0.734704\pi$
$13$	−63.0266			−1.34465			−0.672325	+	0.740256i	$0.734704\pi$
$14$		0			0
$15$	22.2878	+	13.3512i	0.383646	+	0.229817i
$16$		0			0
$17$			4.23575i			0.0604306i	0.999543	+	0.0302153i	$0.00961929\pi$
$17$			4.23575i			0.0604306i	−0.999543	+	0.0302153i	$0.990381\pi$
$18$		0			0
$19$			158.425i			1.91291i	0.291887	+	0.956453i	$0.405717\pi$
$19$			158.425i			1.91291i	−0.291887	+	0.956453i	$0.594283\pi$
$20$		0			0
$21$	128.792	+	77.1507i	1.33832	+	0.801698i
$22$		0			0
$23$	193.058			1.75023			0.875117	−	0.483911i	$-0.160784\pi$
$23$	193.058			1.75023			0.875117	+	0.483911i	$0.160784\pi$
$24$		0			0
$25$	−25.0000			−0.200000
$26$		0			0
$27$	140.132	+	6.77825i	0.998832	+	0.0483139i
$28$		0			0
$29$		−	225.068i		−	1.44117i	−0.693365	−	0.720586i	$-0.743873\pi$
$29$		−	225.068i		−	1.44117i	0.693365	−	0.720586i	$-0.256127\pi$
$30$		0			0
$31$			176.599i			1.02316i	0.859235	+	0.511582i	$0.170940\pi$
$31$			176.599i			1.02316i	−0.859235	+	0.511582i	$0.829060\pi$
$32$		0			0
$33$	71.2536	−	118.947i	0.375869	−	0.627457i
$34$		0			0
$35$	−144.464			−0.697683
$36$		0			0
$37$	−36.2616			−0.161118			−0.0805592	−	0.996750i	$-0.525671\pi$
$37$	−36.2616			−0.161118			−0.0805592	+	0.996750i	$0.525671\pi$
$38$		0			0
$39$	168.296	−	280.945i	0.690998	−	1.15352i
$40$		0			0
$41$			299.719i			1.14167i	0.821066	+	0.570833i	$0.193379\pi$
$41$			299.719i			1.14167i	−0.821066	+	0.570833i	$0.806621\pi$
$42$		0			0
$43$			509.850i			1.80817i	0.427349	+	0.904087i	$0.359448\pi$
$43$			509.850i			1.80817i	−0.427349	+	0.904087i	$0.640552\pi$
$44$		0			0
$45$	−119.027	+	63.6985i	−0.394301	+	0.211013i
$46$		0			0
$47$	211.187			0.655420			0.327710	−	0.944778i	$-0.393723\pi$
$47$	211.187			0.655420			0.327710	+	0.944778i	$0.393723\pi$
$48$		0			0
$49$	−491.797			−1.43381
$50$		0			0
$51$	−18.8811	−	11.3104i	−0.0518408	−	0.0310545i
$52$		0			0
$53$			701.674i			1.81854i	0.416212	+	0.909268i	$0.363357\pi$
$53$			701.674i			1.81854i	−0.416212	+	0.909268i	$0.636643\pi$
$54$		0			0
$55$			133.422i			0.327102i
$56$		0			0
$57$	−706.189	−	423.032i	−1.64100	−	0.983017i
$58$		0			0
$59$	299.376			0.660600			0.330300	−	0.943876i	$-0.392850\pi$
$59$	299.376			0.660600			0.330300	+	0.943876i	$0.392850\pi$
$60$		0			0
$61$	133.906			0.281064			0.140532	−	0.990076i	$-0.455119\pi$
$61$	133.906			0.281064			0.140532	+	0.990076i	$0.455119\pi$
$62$		0			0
$63$	−687.808	+	368.086i	−1.37549	+	0.736103i
$64$		0			0
$65$			315.133i			0.601346i
$66$		0			0
$67$		−	292.983i		−	0.534233i	−0.963664	−	0.267116i	$-0.913929\pi$
$67$		−	292.983i		−	0.534233i	0.963664	−	0.267116i	$-0.0860708\pi$
$68$		0			0
$69$	−515.510	+	860.568i	−0.899423	+	1.50145i
$70$		0			0
$71$	−630.368			−1.05368			−0.526838	−	0.849966i	$-0.676622\pi$
$71$	−630.368			−1.05368			−0.526838	+	0.849966i	$0.676622\pi$
$72$		0			0
$73$	−268.336			−0.430225			−0.215112	−	0.976589i	$-0.569012\pi$
$73$	−268.336			−0.430225			−0.215112	+	0.976589i	$0.569012\pi$
$74$		0			0
$75$	66.7559	−	111.439i	0.102777	−	0.171572i
$76$		0			0
$77$			770.989i			1.14107i
$78$		0			0
$79$		−	549.289i		−	0.782276i	−0.920332	−	0.391138i	$-0.872082\pi$
$79$		−	549.289i		−	0.782276i	0.920332	−	0.391138i	$-0.127918\pi$
$80$		0			0
$81$	−404.401	+	606.549i	−0.554733	+	0.832028i
$82$		0			0
$83$	192.312			0.254325			0.127162	−	0.991882i	$-0.459413\pi$
$83$	192.312			0.254325			0.127162	+	0.991882i	$0.459413\pi$
$84$		0			0
$85$	21.1787			0.0270254
$86$		0			0
$87$	1003.25	+	600.983i	1.23632	+	0.740600i
$88$		0			0
$89$		−	1202.98i		−	1.43276i	−0.697712	−	0.716379i	$-0.745798\pi$
$89$		−	1202.98i		−	1.43276i	0.697712	−	0.716379i	$-0.254202\pi$
$90$		0			0
$91$			1821.02i			2.09774i
$92$		0			0
$93$	−787.200	−	471.560i	−0.877729	−	0.525791i
$94$		0			0
$95$	792.125			0.855477
$96$		0			0
$97$	831.831			0.870718			0.435359	−	0.900257i	$-0.356621\pi$
$97$	831.831			0.870718			0.435359	+	0.900257i	$0.356621\pi$
$98$		0			0
$99$	339.951	+	635.235i	0.345115	+	0.644884i
$100$		0			0
$101$		−	161.237i		−	0.158848i	−0.996841	−	0.0794240i	$-0.974692\pi$
$101$		−	161.237i		−	0.158848i	0.996841	−	0.0794240i	$-0.0253081\pi$
$102$		0			0
$103$			775.670i			0.742029i	0.928627	+	0.371015i	$0.120990\pi$
$103$			775.670i			0.742029i	−0.928627	+	0.371015i	$0.879010\pi$
$104$		0			0
$105$	385.754	−	643.958i	0.358530	−	0.598513i
$106$		0			0
$107$	−634.062			−0.572870			−0.286435	−	0.958100i	$-0.592470\pi$
$107$	−634.062			−0.572870			−0.286435	+	0.958100i	$0.592470\pi$
$108$		0			0
$109$	−987.416			−0.867682			−0.433841	−	0.900989i	$-0.642842\pi$
$109$	−987.416			−0.867682			−0.433841	+	0.900989i	$0.642842\pi$
$110$		0			0
$111$	96.8271	−	161.638i	0.0827966	−	0.138217i
$112$		0			0
$113$		−	121.165i		−	0.100870i	−0.998727	−	0.0504348i	$-0.983939\pi$
$113$		−	121.165i		−	0.100870i	0.998727	−	0.0504348i	$-0.0160607\pi$
$114$		0			0
$115$		−	965.290i		−	0.782729i
$116$		0			0
$117$	802.940	+	1500.38i	0.634460	+	1.18556i
$118$		0			0
$119$	122.383			0.0942758
$120$		0			0
$121$	−618.942			−0.465021
$122$		0			0
$123$	−1336.02	−	800.321i	−0.979387	−	0.586687i
$124$		0			0
$125$			125.000i			0.0894427i
$126$		0			0
$127$			2114.71i			1.47756i	0.673946	+	0.738781i	$0.264598\pi$
$127$			2114.71i			1.47756i	−0.673946	+	0.738781i	$0.735402\pi$
$128$		0			0
$129$	−2272.69	−	1361.42i	−1.55116	−	0.929197i
$130$		0			0
$131$	854.166			0.569685			0.284843	−	0.958574i	$-0.408059\pi$
$131$	854.166			0.569685			0.284843	+	0.958574i	$0.408059\pi$
$132$		0			0
$133$	4577.35			2.98426
$134$		0			0
$135$	33.8912	−	700.661i	0.0216066	−	0.446691i
$136$		0			0
$137$			361.652i			0.225533i	0.993622	+	0.112766i	$0.0359712\pi$
$137$			361.652i			0.225533i	−0.993622	+	0.112766i	$0.964029\pi$
$138$		0			0
$139$		−	455.917i		−	0.278204i	−0.990278	−	0.139102i	$-0.955578\pi$
$139$		−	455.917i		−	0.278204i	0.990278	−	0.139102i	$-0.0444216\pi$
$140$		0			0
$141$	−563.918	+	941.377i	−0.336812	+	0.562257i
$142$		0			0
$143$	1681.83			0.983507
$144$		0			0
$145$	−1125.34			−0.644512
$146$		0			0
$147$	1313.21	−	2192.22i	0.736817	−	1.23001i
$148$		0			0
$149$			534.101i			0.293660i	0.989162	+	0.146830i	$0.0469070\pi$
$149$			534.101i			0.293660i	−0.989162	+	0.146830i	$0.953093\pi$
$150$		0			0
$151$		−	376.313i		−	0.202807i	−0.994845	−	0.101404i	$-0.967667\pi$
$151$		−	376.313i		−	0.202807i	0.994845	−	0.101404i	$-0.0323334\pi$
$152$		0			0
$153$	100.834	−	53.9621i	0.0532807	−	0.0285136i
$154$		0			0
$155$	882.994			0.457573
$156$		0			0
$157$	1376.01			0.699475			0.349737	−	0.936848i	$-0.386271\pi$
$157$	1376.01			0.699475			0.349737	+	0.936848i	$0.386271\pi$
$158$		0			0
$159$	−3127.76	−	1873.64i	−1.56004	−	0.934522i
$160$		0			0
$161$		−	5578.00i		−	2.73048i
$162$		0			0
$163$			1699.68i			0.816742i	0.912816	+	0.408371i	$0.133903\pi$
$163$			1699.68i			0.816742i	−0.912816	+	0.408371i	$0.866097\pi$
$164$		0			0
$165$	−594.737	−	356.268i	−0.280607	−	0.168094i
$166$		0			0
$167$	−519.812			−0.240864			−0.120432	−	0.992722i	$-0.538428\pi$
$167$	−519.812			−0.240864			−0.120432	+	0.992722i	$0.538428\pi$
$168$		0			0
$169$	1775.36			0.808083
$170$		0			0
$171$	3771.38	−	2018.29i	1.68658	−	0.902586i
$172$		0			0
$173$			2232.08i			0.980937i	0.871459	+	0.490469i	$0.163174\pi$
$173$			2232.08i			0.980937i	−0.871459	+	0.490469i	$0.836826\pi$
$174$		0			0
$175$			722.321i			0.312014i
$176$		0			0
$177$	−799.403	+	1334.49i	−0.339474	+	0.566701i
$178$		0			0
$179$	−1289.04			−0.538254			−0.269127	−	0.963105i	$-0.586735\pi$
$179$	−1289.04			−0.538254			−0.269127	+	0.963105i	$0.586735\pi$
$180$		0			0
$181$	2499.79			1.02657			0.513283	−	0.858220i	$-0.328429\pi$
$181$	2499.79			1.02657			0.513283	+	0.858220i	$0.328429\pi$
$182$		0			0
$183$	−357.561	+	596.895i	−0.144435	+	0.241113i
$184$		0			0
$185$			181.308i			0.0720543i
$186$		0			0
$187$		−	113.028i		−	0.0442003i
$188$		0			0
$189$	195.843	−	4048.82i	0.0753729	−	1.55825i
$190$		0			0
$191$	−1663.96			−0.630364			−0.315182	−	0.949031i	$-0.602066\pi$
$191$	−1663.96			−0.630364			−0.315182	+	0.949031i	$0.602066\pi$
$192$		0			0
$193$	−1442.16			−0.537870			−0.268935	−	0.963158i	$-0.586672\pi$
$193$	−1442.16			−0.537870			−0.268935	+	0.963158i	$0.586672\pi$
$194$		0			0
$195$	−1404.73	−	841.480i	−0.515869	−	0.309024i
$196$		0			0
$197$			2405.09i			0.869824i	0.900473	+	0.434912i	$0.143221\pi$
$197$			2405.09i			0.869824i	−0.900473	+	0.434912i	$0.856779\pi$
$198$		0			0
$199$			3959.74i			1.41055i	0.708936	+	0.705273i	$0.249176\pi$
$199$			3959.74i			1.41055i	−0.708936	+	0.705273i	$0.750824\pi$
$200$		0			0
$201$	1305.99	+	782.334i	0.458296	+	0.274535i
$202$		0			0
$203$	−6502.85			−2.24833
$204$		0			0
$205$	1498.60			0.510568
$206$		0			0
$207$	−2459.50	−	4595.84i	−0.825831	−	1.54315i
$208$		0			0
$209$		−	4227.48i		−	1.39914i
$210$		0			0
$211$		−	1010.91i		−	0.329828i	−0.986308	−	0.164914i	$-0.947265\pi$
$211$		−	1010.91i		−	0.329828i	0.986308	−	0.164914i	$-0.0527347\pi$
$212$		0			0
$213$	1683.23	−	2809.91i	0.541470	−	0.903904i
$214$		0			0
$215$	2549.25			0.808640
$216$		0			0
$217$	5102.44			1.59620
$218$		0			0
$219$	716.521	−	1196.13i	0.221087	−	0.369072i
$220$		0			0
$221$		−	266.965i		−	0.0812579i
$222$		0			0
$223$			1163.13i			0.349278i	0.984633	+	0.174639i	$0.0558758\pi$
$223$			1163.13i			0.349278i	−0.984633	+	0.174639i	$0.944124\pi$
$224$		0			0
$225$	318.492	+	595.137i	0.0943681	+	0.176337i
$226$		0			0
$227$	3253.44			0.951271			0.475636	−	0.879642i	$-0.342218\pi$
$227$	3253.44			0.951271			0.475636	+	0.879642i	$0.342218\pi$
$228$		0			0
$229$	−818.571			−0.236212			−0.118106	−	0.993001i	$-0.537682\pi$
$229$	−818.571			−0.236212			−0.118106	+	0.993001i	$0.537682\pi$
$230$		0			0
$231$	−3436.73	−	2058.72i	−0.978875	−	0.586380i
$232$		0			0
$233$		−	1180.72i		−	0.331981i	−0.986127	−	0.165990i	$-0.946918\pi$
$233$		−	1180.72i		−	0.331981i	0.986127	−	0.165990i	$-0.0530821\pi$
$234$		0			0
$235$		−	1055.93i		−	0.293113i
$236$		0			0
$237$	2448.49	+	1466.73i	0.671082	+	0.402001i
$238$		0			0
$239$	−5108.43			−1.38258			−0.691291	−	0.722576i	$-0.742958\pi$
$239$	−5108.43			−1.38258			−0.691291	+	0.722576i	$0.742958\pi$
$240$		0			0
$241$	451.204			0.120600			0.0603000	−	0.998180i	$-0.480794\pi$
$241$	451.204			0.120600			0.0603000	+	0.998180i	$0.480794\pi$
$242$		0			0
$243$	−1623.88	−	3422.27i	−0.428692	−	0.903451i
$244$		0			0
$245$			2458.99i			0.641220i
$246$		0			0
$247$		−	9985.00i		−	2.57219i
$248$		0			0
$249$	−513.518	+	857.241i	−0.130694	+	0.218175i
$250$		0			0
$251$	−6881.30			−1.73045			−0.865226	−	0.501382i	$-0.832825\pi$
$251$	−6881.30			−1.73045			−0.865226	+	0.501382i	$0.832825\pi$
$252$		0			0
$253$	−5151.64			−1.28016
$254$		0			0
$255$	−56.5522	+	94.4055i	−0.0138880	+	0.0231839i
$256$		0			0
$257$			4051.17i			0.983288i	0.870796	+	0.491644i	$0.163604\pi$
$257$			4051.17i			0.983288i	−0.870796	+	0.491644i	$0.836396\pi$
$258$		0			0
$259$			1047.70i			0.251355i
$260$		0			0
$261$	−5357.84	+	2867.29i	−1.27066	+	0.680003i
$262$		0			0
$263$	3242.34			0.760195			0.380098	−	0.924946i	$-0.375890\pi$
$263$	3242.34			0.760195			0.380098	+	0.924946i	$0.375890\pi$
$264$		0			0
$265$	3508.37			0.813274
$266$		0			0
$267$	5362.35	+	3212.23i	1.22910	+	0.736275i
$268$		0			0
$269$			3239.40i			0.734236i	0.930174	+	0.367118i	$0.119656\pi$
$269$			3239.40i			0.734236i	−0.930174	+	0.367118i	$0.880344\pi$
$270$		0			0
$271$		−	1311.45i		−	0.293965i	−0.989139	−	0.146983i	$-0.953044\pi$
$271$		−	1311.45i		−	0.293965i	0.989139	−	0.146983i	$-0.0469562\pi$
$272$		0			0
$273$	−8117.31	−	4862.55i	−1.79957	−	1.07800i
$274$		0			0
$275$	667.110			0.146285
$276$		0			0
$277$	3169.82			0.687566			0.343783	−	0.939049i	$-0.388292\pi$
$277$	3169.82			0.687566			0.343783	+	0.939049i	$0.388292\pi$
$278$		0			0
$279$	4204.02	−	2249.81i	0.902107	−	0.482770i
$280$		0			0
$281$			4444.46i			0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$281$			4444.46i			0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$282$		0			0
$283$		−	1059.45i		−	0.222537i	−0.993790	−	0.111269i	$-0.964509\pi$
$283$		−	1059.45i		−	0.222537i	0.993790	−	0.111269i	$-0.0354914\pi$
$284$		0			0
$285$	−2115.16	+	3530.95i	−0.439619	+	0.733878i
$286$		0			0
$287$	8659.74			1.78107
$288$		0			0
$289$	4895.06			0.996348
$290$		0			0
$291$	−2221.18	+	3707.94i	−0.447451	+	0.746952i
$292$		0			0
$293$		−	9336.99i		−	1.86168i	−0.365425	−	0.930841i	$-0.619076\pi$
$293$		−	9336.99i		−	1.86168i	0.365425	−	0.930841i	$-0.380924\pi$
$294$		0			0
$295$		−	1496.88i		−	0.295429i
$296$		0			0
$297$	−3739.35	−	180.874i	−0.730569	−	0.0353379i
$298$		0			0
$299$	−12167.8			−2.35345
$300$		0			0
$301$	14731.0			2.82087
$302$		0			0
$303$	718.722	+	430.540i	0.136269	+	0.0816299i
$304$		0			0
$305$		−	669.531i		−	0.125696i
$306$		0			0
$307$			1882.91i			0.350043i	0.984565	+	0.175021i	$0.0559995\pi$
$307$			1882.91i			0.350043i	−0.984565	+	0.175021i	$0.944001\pi$
$308$		0			0
$309$	−3457.59	−	2071.22i	−0.636555	−	0.381319i
$310$		0			0
$311$	3394.17			0.618860			0.309430	−	0.950922i	$-0.399862\pi$
$311$	3394.17			0.618860			0.309430	+	0.950922i	$0.399862\pi$
$312$		0			0
$313$	5075.88			0.916632			0.458316	−	0.888789i	$-0.348453\pi$
$313$	5075.88			0.916632			0.458316	+	0.888789i	$0.348453\pi$
$314$		0			0
$315$	1840.43	+	3439.04i	0.329195	+	0.615136i
$316$		0			0
$317$		−	7198.07i		−	1.27534i	−0.770308	−	0.637671i	$-0.779898\pi$
$317$		−	7198.07i		−	1.27534i	0.770308	−	0.637671i	$-0.220102\pi$
$318$		0			0
$319$			6005.80i			1.05411i
$320$		0			0
$321$	1693.10	−	2826.37i	0.294391	−	0.491441i
$322$		0			0
$323$	−671.049			−0.115598
$324$		0			0
$325$	1575.67			0.268930
$326$		0			0
$327$	2636.63	−	4401.47i	0.445890	−	0.744348i
$328$		0			0
$329$		−	6101.79i		−	1.02250i
$330$		0			0
$331$			6817.22i			1.13205i	0.824388	+	0.566025i	$0.191519\pi$
$331$			6817.22i			1.13205i	−0.824388	+	0.566025i	$0.808481\pi$
$332$		0			0
$333$	461.962	+	863.225i	0.0760221	+	0.142055i
$334$		0			0
$335$	−1464.92			−0.238916
$336$		0			0
$337$	3161.73			0.511070			0.255535	−	0.966800i	$-0.417748\pi$
$337$	3161.73			0.511070			0.255535	+	0.966800i	$0.417748\pi$
$338$		0			0
$339$	540.101	+	323.539i	0.0865318	+	0.0518356i
$340$		0			0
$341$		−	4712.43i		−	0.748365i
$342$		0			0
$343$			4299.17i			0.676774i
$344$		0			0
$345$	4302.84	+	2577.55i	0.671470	+	0.402234i
$346$		0			0
$347$	6583.03			1.01843			0.509215	−	0.860639i	$-0.329936\pi$
$347$	6583.03			1.01843			0.509215	+	0.860639i	$0.329936\pi$
$348$		0			0
$349$	−10406.5			−1.59613			−0.798064	−	0.602573i	$-0.794142\pi$
$349$	−10406.5			−1.59613			−0.798064	+	0.602573i	$0.794142\pi$
$350$		0			0
$351$	−8832.07	−	427.210i	−1.34308	−	0.0649652i
$352$		0			0
$353$			9283.14i			1.39969i	0.714293	+	0.699846i	$0.246748\pi$
$353$			9283.14i			1.39969i	−0.714293	+	0.699846i	$0.753252\pi$
$354$		0			0
$355$			3151.84i			0.471218i
$356$		0			0
$357$	−326.791	+	545.529i	−0.0484471	+	0.0808752i
$358$		0			0
$359$	2998.98			0.440891			0.220446	−	0.975399i	$-0.429249\pi$
$359$	2998.98			0.440891			0.220446	+	0.975399i	$0.429249\pi$
$360$		0			0
$361$	−18239.5			−2.65921
$362$		0			0
$363$	1652.72	−	2758.97i	0.238968	−	0.398922i
$364$		0			0
$365$			1341.68i			0.192402i
$366$		0			0
$367$			7161.60i			1.01862i	0.860584	+	0.509309i	$0.170099\pi$
$367$			7161.60i			1.01862i	−0.860584	+	0.509309i	$0.829901\pi$
$368$		0			0
$369$	7134.96	−	3818.33i	1.00659	−	0.538684i
$370$		0			0
$371$	20273.4			2.83704
$372$		0			0
$373$	−1535.63			−0.213169			−0.106584	−	0.994304i	$-0.533991\pi$
$373$	−1535.63			−0.213169			−0.106584	+	0.994304i	$0.533991\pi$
$374$		0			0
$375$	−557.195	−	333.779i	−0.0767291	−	0.0459634i
$376$		0			0
$377$			14185.3i			1.93787i
$378$		0			0
$379$			11691.9i			1.58462i	0.610118	+	0.792311i	$0.291122\pi$
$379$			11691.9i			1.58462i	−0.610118	+	0.792311i	$0.708878\pi$
$380$		0			0
$381$	−9426.45	−	5646.78i	−1.26754	−	0.759300i
$382$		0			0
$383$	3893.22			0.519411			0.259705	−	0.965688i	$-0.416375\pi$
$383$	3893.22			0.519411			0.259705	+	0.965688i	$0.416375\pi$
$384$		0			0
$385$	3854.94			0.510302
$386$		0			0
$387$	12137.2	−	6495.34i	1.59424	−	0.853169i
$388$		0			0
$389$		−	9888.33i		−	1.28884i	−0.764672	−	0.644419i	$-0.777099\pi$
$389$		−	9888.33i		−	1.28884i	0.764672	−	0.644419i	$-0.222901\pi$
$390$		0			0
$391$			817.745i			0.105768i
$392$		0			0
$393$	−2280.82	+	3807.49i	−0.292754	+	0.488709i
$394$		0			0
$395$	−2746.44			−0.349844
$396$		0			0
$397$	−8095.45			−1.02342			−0.511712	−	0.859157i	$-0.670988\pi$
$397$	−8095.45			−1.02342			−0.511712	+	0.859157i	$0.670988\pi$
$398$		0			0
$399$	−12222.6	+	20403.8i	−1.53357	+	2.56007i
$400$		0			0
$401$			9999.28i			1.24524i	0.782525	+	0.622619i	$0.213931\pi$
$401$			9999.28i			1.24524i	−0.782525	+	0.622619i	$0.786069\pi$
$402$		0			0
$403$		−	11130.4i		−	1.37580i
$404$		0			0
$405$	3032.74	+	2022.00i	0.372094	+	0.248084i
$406$		0			0
$407$	967.620			0.117846
$408$		0			0
$409$	2043.61			0.247066			0.123533	−	0.992340i	$-0.460578\pi$
$409$	2043.61			0.247066			0.123533	+	0.992340i	$0.460578\pi$
$410$		0			0
$411$	−1612.09	−	965.696i	−0.193475	−	0.115898i
$412$		0			0
$413$		−	8649.82i		−	1.03058i
$414$		0			0
$415$		−	961.559i		−	0.113737i
$416$		0			0
$417$	2032.28	+	1217.41i	0.238660	+	0.142966i
$418$		0			0
$419$	−11593.8			−1.35177			−0.675887	−	0.737005i	$-0.736239\pi$
$419$	−11593.8			−1.35177			−0.675887	+	0.737005i	$0.736239\pi$
$420$		0			0
$421$	−12434.0			−1.43942			−0.719711	−	0.694274i	$-0.755726\pi$
$421$	−12434.0			−1.43942			−0.719711	+	0.694274i	$0.755726\pi$
$422$		0			0
$423$	−2690.45	−	5027.40i	−0.309254	−	0.577873i
$424$		0			0
$425$		−	105.894i		−	0.0120861i
$426$		0			0
$427$		−	3868.93i		−	0.438479i
$428$		0			0
$429$	−4490.88	+	7496.85i	−0.505412	+	0.843710i
$430$		0			0
$431$	3677.25			0.410967			0.205484	−	0.978661i	$-0.434123\pi$
$431$	3677.25			0.410967			0.205484	+	0.978661i	$0.434123\pi$
$432$		0			0
$433$	−17733.6			−1.96818			−0.984090	−	0.177672i	$-0.943143\pi$
$433$	−17733.6			−1.96818			−0.984090	+	0.177672i	$0.943143\pi$
$434$		0			0
$435$	3004.92	−	5016.26i	0.331206	−	0.552900i
$436$		0			0
$437$			30585.2i			3.34803i
$438$		0			0
$439$			5085.24i			0.552859i	0.961034	+	0.276430i	$0.0891513\pi$
$439$			5085.24i			0.552859i	−0.961034	+	0.276430i	$0.910849\pi$
$440$		0			0
$441$	6265.34	+	11707.5i	0.676530	+	1.26417i
$442$		0			0
$443$	−14786.8			−1.58587			−0.792935	−	0.609306i	$-0.791448\pi$
$443$	−14786.8			−1.58587			−0.792935	+	0.609306i	$0.791448\pi$
$444$		0			0
$445$	−6014.89			−0.640749
$446$		0			0
$447$	−2380.79	−	1426.18i	−0.251918	−	0.150908i
$448$		0			0
$449$			4836.54i			0.508353i	0.967158	+	0.254176i	$0.0818044\pi$
$449$			4836.54i			0.508353i	−0.967158	+	0.254176i	$0.918196\pi$
$450$		0			0
$451$		−	7997.83i		−	0.835040i
$452$		0			0
$453$	1677.44	+	1004.84i	0.173980	+	0.104220i
$454$		0			0
$455$	9105.10			0.938140
$456$		0			0
$457$	1610.66			0.164866			0.0824329	−	0.996597i	$-0.473731\pi$
$457$	1610.66			0.164866			0.0824329	+	0.996597i	$0.473731\pi$
$458$		0			0
$459$	−28.7110	+	593.565i	−0.00291964	+	0.0603600i
$460$		0			0
$461$		−	14397.3i		−	1.45455i	−0.686346	−	0.727275i	$-0.740786\pi$
$461$		−	14397.3i		−	1.45455i	0.686346	−	0.727275i	$-0.259214\pi$
$462$		0			0
$463$			4605.81i			0.462311i	0.972917	+	0.231156i	$0.0742506\pi$
$463$			4605.81i			0.462311i	−0.972917	+	0.231156i	$0.925749\pi$
$464$		0			0
$465$	−2357.80	+	3936.00i	−0.235141	+	0.392532i
$466$		0			0
$467$	−2603.62			−0.257989			−0.128995	−	0.991645i	$-0.541175\pi$
$467$	−2603.62			−0.257989			−0.128995	+	0.991645i	$0.541175\pi$
$468$		0			0
$469$	−8465.12			−0.833439
$470$		0			0
$471$	−3674.27	+	6133.65i	−0.359451	+	0.600050i
$472$		0			0
$473$		−	13605.1i		−	1.32254i
$474$		0			0
$475$		−	3960.63i		−	0.382581i
$476$		0			0
$477$	16703.7	−	8939.11i	1.60337	−	0.858058i
$478$		0			0
$479$	−904.077			−0.0862387			−0.0431193	−	0.999070i	$-0.513730\pi$
$479$	−904.077			−0.0862387			−0.0431193	+	0.999070i	$0.513730\pi$
$480$		0			0
$481$	2285.45			0.216648
$482$		0			0
$483$	24864.3	+	14894.6i	2.34237	+	1.40316i
$484$		0			0
$485$		−	4159.15i		−	0.389397i
$486$		0			0
$487$		−	2787.27i		−	0.259350i	−0.991557	−	0.129675i	$-0.958607\pi$
$487$		−	2787.27i		−	0.259350i	0.991557	−	0.129675i	$-0.0413934\pi$
$488$		0			0
$489$	−7576.41	−	4538.53i	−0.700648	−	0.419713i
$490$		0			0
$491$	−6234.26			−0.573011			−0.286506	−	0.958079i	$-0.592494\pi$
$491$	−6234.26			−0.573011			−0.286506	+	0.958079i	$0.592494\pi$
$492$		0			0
$493$	953.330			0.0870909
$494$		0			0
$495$	3176.17	−	1699.76i	0.288401	−	0.154340i
$496$		0			0
$497$			18213.1i			1.64381i
$498$		0			0
$499$			8631.50i			0.774347i	0.922007	+	0.387174i	$0.126549\pi$
$499$			8631.50i			0.774347i	−0.922007	+	0.387174i	$0.873451\pi$
$500$		0			0
$501$	1388.02	−	2317.09i	0.123777	−	0.206627i
$502$		0			0
$503$	−3401.02			−0.301479			−0.150740	−	0.988574i	$-0.548165\pi$
$503$	−3401.02			−0.301479			−0.150740	+	0.988574i	$0.548165\pi$
$504$		0			0
$505$	−806.183			−0.0710390
$506$		0			0
$507$	−4740.62	+	7913.76i	−0.415263	+	0.693220i
$508$		0			0
$509$			9241.80i			0.804785i	0.915467	+	0.402392i	$0.131821\pi$
$509$			9241.80i			0.804785i	−0.915467	+	0.402392i	$0.868179\pi$
$510$		0			0
$511$			7753.00i			0.671179i
$512$		0			0
$513$	−1073.84	+	22200.5i	−0.0924199	+	1.91067i
$514$		0			0
$515$	3878.35			0.331845
$516$		0			0
$517$	−5635.39			−0.479389
$518$		0			0
$519$	−9949.65	−	5960.19i	−0.841505	−	0.504091i
$520$		0			0
$521$		−	13462.9i		−	1.13209i	−0.824374	−	0.566046i	$-0.808473\pi$
$521$		−	13462.9i		−	1.13209i	0.824374	−	0.566046i	$-0.191527\pi$
$522$		0			0
$523$			16904.0i			1.41331i	0.707560	+	0.706654i	$0.249796\pi$
$523$			16904.0i			1.41331i	−0.707560	+	0.706654i	$0.750204\pi$
$524$		0			0
$525$	−3219.79	−	1928.77i	−0.267663	−	0.160340i
$526$		0			0
$527$	−748.028			−0.0618304
$528$		0			0
$529$	25104.4			2.06332
$530$		0			0
$531$	−3813.95	−	7126.78i	−0.311698	−	0.582440i
$532$		0			0
$533$		−	18890.3i		−	1.53514i
$534$		0			0
$535$			3170.31i			0.256195i
$536$		0			0
$537$	3442.04	−	5745.98i	0.276602	−	0.461746i
$538$		0			0
$539$	13123.3			1.04872
$540$		0			0
$541$	1830.55			0.145474			0.0727372	−	0.997351i	$-0.476827\pi$
$541$	1830.55			0.145474			0.0727372	+	0.997351i	$0.476827\pi$
$542$		0			0
$543$	−6675.04	+	11143.0i	−0.527538	+	0.880647i
$544$		0			0
$545$			4937.08i			0.388039i
$546$		0			0
$547$		−	13415.3i		−	1.04862i	−0.851527	−	0.524312i	$-0.824323\pi$
$547$		−	13415.3i		−	1.04862i	0.851527	−	0.524312i	$-0.175677\pi$
$548$		0			0
$549$	−1705.92	−	3187.70i	−0.132618	−	0.247810i
$550$		0			0
$551$	35656.4			2.75683
$552$		0			0
$553$	−15870.5			−1.22040
$554$		0			0
$555$	−808.192	−	484.136i	−0.0618124	−	0.0370278i
$556$		0			0
$557$			11587.0i			0.881430i	0.897647	+	0.440715i	$0.145275\pi$
$557$			11587.0i			0.881430i	−0.897647	+	0.440715i	$0.854725\pi$
$558$		0			0
$559$		−	32134.2i		−	2.43136i
$560$		0			0
$561$	503.831	+	301.812i	0.0379176	+	0.0227140i
$562$		0			0
$563$	−7284.20			−0.545280			−0.272640	−	0.962116i	$-0.587897\pi$
$563$	−7284.20			−0.545280			−0.272640	+	0.962116i	$0.587897\pi$
$564$		0			0
$565$	−605.826			−0.0451102
$566$		0			0
$567$	17524.9	+	11684.3i	1.29802	+	0.865421i
$568$		0			0
$569$			7036.03i			0.518393i	0.965825	+	0.259196i	$0.0834577\pi$
$569$			7036.03i			0.518393i	−0.965825	+	0.259196i	$0.916542\pi$
$570$		0			0
$571$		−	9718.85i		−	0.712296i	−0.934430	−	0.356148i	$-0.884090\pi$
$571$		−	9718.85i		−	0.712296i	0.934430	−	0.356148i	$-0.115910\pi$
$572$		0			0
$573$	4443.15	−	7417.18i	0.323936	−	0.540763i
$574$		0			0
$575$	−4826.45			−0.350047
$576$		0			0
$577$	8860.76			0.639304			0.319652	−	0.947535i	$-0.396434\pi$
$577$	8860.76			0.639304			0.319652	+	0.947535i	$0.396434\pi$
$578$		0			0
$579$	3850.91	−	6428.52i	0.276405	−	0.461416i
$580$		0			0
$581$		−	5556.44i		−	0.396764i
$582$		0			0
$583$		−	18723.8i		−	1.33012i
$584$		0			0
$585$	7501.89	−	4014.70i	0.530197	−	0.283739i
$586$		0			0
$587$	−27951.9			−1.96542			−0.982708	−	0.185162i	$-0.940719\pi$
$587$	−27951.9			−1.96542			−0.982708	+	0.185162i	$0.940719\pi$
$588$		0			0
$589$	−27977.7			−1.95722
$590$		0			0
$591$	−10720.8	−	6422.15i	−0.746186	−	0.446991i
$592$		0			0
$593$			2873.12i			0.198962i	0.995039	+	0.0994812i	$0.0317183\pi$
$593$			2873.12i			0.198962i	−0.995039	+	0.0994812i	$0.968282\pi$
$594$		0			0
$595$		−	611.914i		−	0.0421614i
$596$		0			0
$597$	−17650.8	−	10573.4i	−1.21005	−	0.724861i
$598$		0			0
$599$	−4288.34			−0.292515			−0.146258	−	0.989247i	$-0.546723\pi$
$599$	−4288.34			−0.292515			−0.146258	+	0.989247i	$0.546723\pi$
$600$		0			0
$601$	−3043.25			−0.206551			−0.103275	−	0.994653i	$-0.532932\pi$
$601$	−3043.25			−0.206551			−0.103275	+	0.994653i	$0.532932\pi$
$602$		0			0
$603$	−6974.60	+	3732.52i	−0.471024	+	0.252073i
$604$		0			0
$605$			3094.71i			0.207964i
$606$		0			0
$607$			1658.77i			0.110918i	0.998461	+	0.0554591i	$0.0176622\pi$
$607$			1658.77i			0.110918i	−0.998461	+	0.0554591i	$0.982338\pi$
$608$		0			0
$609$	17364.1	−	28986.8i	1.15539	−	1.92874i
$610$		0			0
$611$	−13310.4			−0.881310
$612$		0			0
$613$	28493.9			1.87742			0.938710	−	0.344709i	$-0.112022\pi$
$613$	28493.9			1.87742			0.938710	+	0.344709i	$0.112022\pi$
$614$		0			0
$615$	−4001.60	+	6680.08i	−0.262374	+	0.437995i
$616$		0			0
$617$		−	20481.2i		−	1.33637i	−0.743994	−	0.668186i	$-0.767071\pi$
$617$		−	20481.2i		−	1.33637i	0.743994	−	0.668186i	$-0.232929\pi$
$618$		0			0
$619$			1943.40i			0.126190i	0.998008	+	0.0630952i	$0.0200972\pi$
$619$			1943.40i			0.126190i	−0.998008	+	0.0630952i	$0.979903\pi$
$620$		0			0
$621$	27053.7	+	1308.60i	1.74819	+	0.0845606i
$622$		0			0
$623$	−34757.5			−2.23520
$624$		0			0
$625$	625.000			0.0400000
$626$		0			0
$627$	18844.2	+	11288.4i	1.20027	+	0.719001i
$628$		0			0
$629$		−	153.595i		−	0.00973647i
$630$		0			0
$631$		−	15964.7i		−	1.00720i	−0.863936	−	0.503602i	$-0.832008\pi$
$631$		−	15964.7i		−	1.00720i	0.863936	−	0.503602i	$-0.167992\pi$
$632$		0			0
$633$	4506.18	+	2699.36i	0.282946	+	0.169494i
$634$		0			0
$635$	10573.6			0.660786
$636$		0			0
$637$	30996.3			1.92797
$638$		0			0
$639$	8030.70	+	15006.2i	0.497167	+	0.929009i
$640$		0			0
$641$		−	4330.89i		−	0.266864i	−0.991058	−	0.133432i	$-0.957400\pi$
$641$		−	4330.89i		−	0.266864i	0.991058	−	0.133432i	$-0.0425998\pi$
$642$		0			0
$643$		−	3889.04i		−	0.238520i	−0.992863	−	0.119260i	$-0.961948\pi$
$643$		−	3889.04i		−	0.238520i	0.992863	−	0.119260i	$-0.0380523\pi$
$644$		0			0
$645$	−6807.10	+	11363.4i	−0.415549	+	0.693698i
$646$		0			0
$647$	21741.6			1.32110			0.660548	−	0.750784i	$-0.270324\pi$
$647$	21741.6			1.32110			0.660548	+	0.750784i	$0.270324\pi$
$648$		0			0
$649$	−7988.66			−0.483178
$650$		0			0
$651$	−13624.7	+	22744.4i	−0.820269	+	1.36932i
$652$		0			0
$653$			15086.0i			0.904075i	0.891999	+	0.452037i	$0.149303\pi$
$653$			15086.0i			0.904075i	−0.891999	+	0.452037i	$0.850697\pi$
$654$		0			0
$655$		−	4270.83i		−	0.254771i
$656$		0			0
$657$	3418.52	+	6387.87i	0.202997	+	0.379322i
$658$		0			0
$659$	4047.71			0.239266			0.119633	−	0.992818i	$-0.461828\pi$
$659$	4047.71			0.239266			0.119633	+	0.992818i	$0.461828\pi$
$660$		0			0
$661$	−6148.64			−0.361807			−0.180904	−	0.983501i	$-0.557902\pi$
$661$	−6148.64			−0.361807			−0.180904	+	0.983501i	$0.557902\pi$
$662$		0			0
$663$	1190.01	+	712.859i	0.0697078	+	0.0417574i
$664$		0			0
$665$		−	22886.8i		−	1.33460i
$666$		0			0
$667$		−	43451.1i		−	2.52239i
$668$		0			0
$669$	−5184.72	−	3105.83i	−0.299631	−	0.179489i
$670$		0			0
$671$	−3573.21			−0.205577
$672$		0			0
$673$	−5643.29			−0.323229			−0.161614	−	0.986854i	$-0.551670\pi$
$673$	−5643.29			−0.323229			−0.161614	+	0.986854i	$0.551670\pi$
$674$		0			0
$675$	−3503.31	−	169.456i	−0.199766	−	0.00966278i
$676$		0			0
$677$		−	21993.0i		−	1.24854i	−0.781209	−	0.624269i	$-0.785397\pi$
$677$		−	21993.0i		−	1.24854i	0.781209	−	0.624269i	$-0.214603\pi$
$678$		0			0
$679$		−	24034.0i		−	1.35838i
$680$		0			0
$681$	−8687.45	+	14502.4i	−0.488846	+	0.816055i
$682$		0			0
$683$	26585.8			1.48943			0.744713	−	0.667384i	$-0.232586\pi$
$683$	26585.8			1.48943			0.744713	+	0.667384i	$0.232586\pi$
$684$		0			0
$685$	1808.26			0.100861
$686$		0			0
$687$	2185.78	−	3648.83i	0.121387	−	0.202637i
$688$		0			0
$689$		−	44224.2i		−	2.44529i
$690$		0			0
$691$			4363.60i			0.240231i	0.992760	+	0.120115i	$0.0383264\pi$
$691$			4363.60i			0.240231i	−0.992760	+	0.120115i	$0.961674\pi$
$692$		0			0
$693$	18353.7	−	9822.16i	1.00606	−	0.538402i
$694$		0			0
$695$	−2279.59			−0.124417
$696$		0			0
$697$	−1269.53			−0.0689915
$698$		0			0
$699$	5263.13	+	3152.80i	0.284792	+	0.170601i
$700$		0			0
$701$			15497.9i			0.835016i	0.908673	+	0.417508i	$0.137096\pi$
$701$			15497.9i			0.835016i	−0.908673	+	0.417508i	$0.862904\pi$
$702$		0			0
$703$		−	5744.75i		−	0.308204i
$704$		0			0
$705$	4706.89	+	2819.59i	0.251449	+	0.150627i
$706$		0			0
$707$	−4658.59			−0.247814
$708$		0			0
$709$	−24070.2			−1.27500			−0.637500	−	0.770450i	$-0.720032\pi$
$709$	−24070.2			−1.27500			−0.637500	+	0.770450i	$0.720032\pi$
$710$		0			0
$711$	−13076.1	+	6997.77i	−0.689720	+	0.369109i
$712$		0			0
$713$			34093.8i			1.79078i
$714$		0			0
$715$		−	8409.14i		−	0.439838i
$716$		0			0
$717$	13640.7	−	22771.2i	0.710491	−	1.18606i
$718$		0			0
$719$	−9769.54			−0.506735			−0.253367	−	0.967370i	$-0.581538\pi$
$719$	−9769.54			−0.506735			−0.253367	+	0.967370i	$0.581538\pi$
$720$		0			0
$721$	22411.3			1.15762
$722$		0			0
$723$	−1204.82	+	2011.27i	−0.0619747	+	0.103458i
$724$		0			0
$725$			5626.69i			0.288235i
$726$		0			0
$727$			10716.6i			0.546707i	0.961914	+	0.273354i	$0.0881329\pi$
$727$			10716.6i			0.546707i	−0.961914	+	0.273354i	$0.911867\pi$
$728$		0			0
$729$	19591.1	+	1899.70i	0.995332	+	0.0965149i
$730$		0			0
$731$	−2159.60			−0.109269
$732$		0			0
$733$	22492.7			1.13341			0.566703	−	0.823922i	$-0.308219\pi$
$733$	22492.7			1.13341			0.566703	+	0.823922i	$0.308219\pi$
$734$		0			0
$735$	−10961.1	−	6566.07i	−0.550075	−	0.329514i
$736$		0			0
$737$			7818.08i			0.390750i
$738$		0			0
$739$		−	25323.1i		−	1.26052i	−0.776385	−	0.630260i	$-0.782948\pi$
$739$		−	25323.1i		−	1.26052i	0.776385	−	0.630260i	$-0.217052\pi$
$740$		0			0
$741$	44508.7	+	26662.3i	2.20657	+	1.32181i
$742$		0			0
$743$	17744.8			0.876171			0.438086	−	0.898933i	$-0.355657\pi$
$743$	17744.8			0.876171			0.438086	+	0.898933i	$0.355657\pi$
$744$		0			0
$745$	2670.51			0.131329
$746$		0			0
$747$	−2449.99	−	4578.07i	−0.120001	−	0.224234i
$748$		0			0
$749$			18319.9i			0.893716i
$750$		0			0
$751$			8439.26i			0.410057i	0.978756	+	0.205029i	$0.0657287\pi$
$751$			8439.26i			0.410057i	−0.978756	+	0.205029i	$0.934271\pi$
$752$		0			0
$753$	18374.7	−	30673.8i	0.889257	−	1.48448i
$754$		0			0
$755$	−1881.56			−0.0906982
$756$		0			0
$757$	−10486.3			−0.503475			−0.251738	−	0.967796i	$-0.581002\pi$
$757$	−10486.3			−0.503475			−0.251738	+	0.967796i	$0.581002\pi$
$758$		0			0
$759$	13756.1	−	22963.7i	0.657858	−	1.09820i
$760$		0			0
$761$			136.290i			0.00649213i	0.999995	+	0.00324607i	$0.00103326\pi$
$761$			136.290i			0.00649213i	−0.999995	+	0.00324607i	$0.998967\pi$
$762$		0			0
$763$			28529.3i			1.35364i
$764$		0			0
$765$	−269.811	−	504.170i	−0.0127517	−	0.0238278i
$766$		0			0
$767$	−18868.6			−0.888275
$768$		0			0
$769$	−28694.9			−1.34560			−0.672798	−	0.739826i	$-0.734908\pi$
$769$	−28694.9			−1.34560			−0.672798	+	0.739826i	$0.734908\pi$
$770$		0			0
$771$	−18058.3	−	10817.6i	−0.843522	−	0.505299i
$772$		0			0
$773$			1959.31i			0.0911663i	0.998961	+	0.0455831i	$0.0145146\pi$
$773$			1959.31i			0.0911663i	−0.998961	+	0.0455831i	$0.985485\pi$
$774$		0			0
$775$		−	4414.97i		−	0.204633i
$776$		0			0
$777$	−4670.20	−	2797.61i	−0.215627	−	0.129168i
$778$		0			0
$779$	−47483.0			−2.18390
$780$		0			0
$781$	16821.0			0.770682
$782$		0			0
$783$	1525.56	−	31539.2i	0.0696286	−	1.43949i
$784$		0			0
$785$		−	6880.05i		−	0.312815i
$786$		0			0
$787$			7819.69i			0.354183i	0.984194	+	0.177091i	$0.0566688\pi$
$787$			7819.69i			0.354183i	−0.984194	+	0.177091i	$0.943331\pi$
$788$		0			0
$789$	−8657.81	+	14452.9i	−0.390654	+	0.652140i
$790$		0			0
$791$	−3500.81			−0.157363
$792$		0			0
$793$	−8439.66			−0.377933
$794$		0			0
$795$	−9368.18	+	15638.8i	−0.417931	+	0.697673i
$796$		0			0
$797$			12070.6i			0.536466i	0.963354	+	0.268233i	$0.0864397\pi$
$797$			12070.6i			0.536466i	−0.963354	+	0.268233i	$0.913560\pi$
$798$		0			0
$799$			894.533i			0.0396074i
$800$		0			0
$801$	−28637.5	+	15325.6i	−1.26324	+	0.676033i
$802$		0			0
$803$	7160.40			0.314676
$804$		0			0
$805$	−27890.0			−1.22111
$806$		0			0
$807$	−14439.8	−	8649.95i	−0.629871	−	0.377315i
$808$		0			0
$809$			12118.8i			0.526666i	0.964705	+	0.263333i	$0.0848218\pi$
$809$			12118.8i			0.526666i	−0.964705	+	0.263333i	$0.915178\pi$
$810$		0			0
$811$			26889.8i			1.16428i	0.813090	+	0.582138i	$0.197784\pi$
$811$			26889.8i			1.16428i	−0.813090	+	0.582138i	$0.802216\pi$
$812$		0			0
$813$	5845.85	+	3501.87i	0.252181	+	0.151065i
$814$		0			0
$815$	8498.38			0.365258
$816$		0			0
$817$	−80773.1			−3.45887
$818$		0			0
$819$	43350.2	−	23199.2i	1.84955	−	0.989800i
$820$		0			0
$821$		−	23410.7i		−	0.995176i	−0.867414	−	0.497588i	$-0.834219\pi$
$821$		−	23410.7i		−	0.995176i	0.867414	−	0.497588i	$-0.165781\pi$
$822$		0			0
$823$		−	33677.4i		−	1.42639i	−0.700965	−	0.713196i	$-0.747247\pi$
$823$		−	33677.4i		−	1.42639i	0.700965	−	0.713196i	$-0.252753\pi$
$824$		0			0
$825$	−1781.34	+	2973.68i	−0.0751737	+	0.125491i
$826$		0			0
$827$	−38868.3			−1.63432			−0.817161	−	0.576409i	$-0.804453\pi$
$827$	−38868.3			−1.63432			−0.817161	+	0.576409i	$0.804453\pi$
$828$		0			0
$829$	17670.7			0.740325			0.370162	−	0.928967i	$-0.379302\pi$
$829$	17670.7			0.740325			0.370162	+	0.928967i	$0.379302\pi$
$830$		0			0
$831$	−8464.15	+	14129.6i	−0.353331	+	0.589834i
$832$		0			0
$833$		−	2083.13i		−	0.0866460i
$834$		0			0
$835$			2599.06i			0.107718i
$836$		0			0
$837$	−1197.03	+	24747.2i	−0.0494330	+	1.02197i
$838$		0			0
$839$	20023.0			0.823924			0.411962	−	0.911201i	$-0.364844\pi$
$839$	20023.0			0.823924			0.411962	+	0.911201i	$0.364844\pi$
$840$		0			0
$841$	−26266.4			−1.07698
$842$		0			0
$843$	−19811.4	−	11867.7i	−0.809421	−	0.484872i
$844$		0			0
$845$		−	8876.79i		−	0.361386i
$846$		0			0
$847$			17883.0i			0.725464i
$848$		0			0
$849$	4722.58	+	2828.99i	0.190905	+	0.114359i
$850$		0			0
$851$	−7000.60			−0.281995
$852$		0			0
$853$	18904.0			0.758807			0.379403	−	0.925231i	$-0.376129\pi$
$853$	18904.0			0.758807			0.379403	+	0.925231i	$0.376129\pi$
$854$		0			0
$855$	−10091.4	−	18856.9i	−0.403649	−	0.754261i
$856$		0			0
$857$		−	26919.1i		−	1.07297i	−0.843908	−	0.536487i	$-0.819751\pi$
$857$		−	26919.1i		−	1.07297i	0.843908	−	0.536487i	$-0.180249\pi$
$858$		0			0
$859$		−	20880.5i		−	0.829377i	−0.909964	−	0.414688i	$-0.863891\pi$
$859$		−	20880.5i		−	0.829377i	0.909964	−	0.414688i	$-0.136109\pi$
$860$		0			0
$861$	−23123.5	+	38601.3i	−0.915271	+	1.52791i
$862$		0			0
$863$	−14655.5			−0.578077			−0.289038	−	0.957318i	$-0.593336\pi$
$863$	−14655.5			−0.578077			−0.289038	+	0.957318i	$0.593336\pi$
$864$		0			0
$865$	11160.4			0.438689
$866$		0			0
$867$	−13071.0	+	21820.0i	−0.512010	+	0.854725i
$868$		0			0
$869$			14657.4i			0.572175i
$870$		0			0
$871$			18465.7i			0.718356i
$872$		0			0
$873$	−10597.3	−	19802.1i	−0.410840	−	0.767698i
$874$		0			0
$875$	3611.61			0.139537
$876$		0			0
$877$	−16056.0			−0.618214			−0.309107	−	0.951027i	$-0.600030\pi$
$877$	−16056.0			−0.618214			−0.309107	+	0.951027i	$0.600030\pi$
$878$		0			0
$879$	41620.2	+	24932.0i	1.59706	+	0.956694i
$880$		0			0
$881$		−	37335.5i		−	1.42777i	−0.700264	−	0.713884i	$-0.746934\pi$
$881$		−	37335.5i		−	1.42777i	0.700264	−	0.713884i	$-0.253066\pi$
$882$		0			0
$883$		−	29969.1i		−	1.14217i	−0.820890	−	0.571087i	$-0.806522\pi$
$883$		−	29969.1i		−	1.14217i	0.820890	−	0.571087i	$-0.193478\pi$
$884$		0			0
$885$	6672.43	+	3997.02i	0.253436	+	0.151817i
$886$		0			0
$887$	−15847.3			−0.599887			−0.299944	−	0.953957i	$-0.596968\pi$
$887$	−15847.3			−0.599887			−0.299944	+	0.953957i	$0.596968\pi$
$888$		0			0
$889$	61100.1			2.30510
$890$		0			0
$891$	10791.2	−	16185.4i	0.405745	−	0.608564i
$892$		0			0
$893$			33457.3i			1.25376i
$894$		0			0
$895$			6445.21i			0.240715i
$896$		0			0
$897$	32490.9	−	54238.7i	1.20941	−	2.01893i
$898$		0			0
$899$	39746.7			1.47456
$900$		0			0
$901$	−2972.11			−0.109895
$902$		0			0
$903$	−39335.3	+	65664.5i	−1.44961	+	2.41991i
$904$		0			0
$905$		−	12499.0i		−	0.459094i
$906$		0			0
$907$			41184.7i			1.50773i	0.657027	+	0.753867i	$0.271814\pi$
$907$			41184.7i			1.50773i	−0.657027	+	0.753867i	$0.728186\pi$
$908$		0			0
$909$	−3838.31	+	2054.10i	−0.140054	+	0.0749509i
$910$		0			0
$911$	36382.4			1.32316			0.661582	−	0.749873i	$-0.269885\pi$
$911$	36382.4			1.32316			0.661582	+	0.749873i	$0.269885\pi$
$912$		0			0
$913$	−5131.72			−0.186019
$914$		0			0
$915$	2984.47	+	1787.80i	0.107829	+	0.0645934i
$916$		0			0
$917$		−	24679.3i		−	0.888748i
$918$		0			0
$919$			21252.5i			0.762844i	0.924401	+	0.381422i	$0.124566\pi$
$919$			21252.5i			0.762844i	−0.924401	+	0.381422i	$0.875434\pi$
$920$		0			0
$921$	−8393.17	−	5027.80i	−0.300287	−	0.179882i
$922$		0			0
$923$	39730.0			1.41682
$924$		0			0
$925$	906.541			0.0322237
$926$		0			0
$927$	18465.2	−	9881.79i	0.654235	−	0.350119i
$928$		0			0
$929$			21790.5i			0.769563i	0.923008	+	0.384781i	$0.125723\pi$
$929$			21790.5i			0.769563i	−0.923008	+	0.384781i	$0.874277\pi$
$930$		0			0
$931$		−	77913.0i		−	2.74274i
$932$		0			0
$933$	−9063.23	+	15129.7i	−0.318024	+	0.530894i
$934$		0			0
$935$	−565.142			−0.0197670
$936$		0			0
$937$	8598.71			0.299795			0.149897	−	0.988702i	$-0.452106\pi$
$937$	8598.71			0.299795			0.149897	+	0.988702i	$0.452106\pi$
$938$		0			0
$939$	−13553.8	+	22626.1i	−0.471045	+	0.786340i
$940$		0			0
$941$		−	49480.0i		−	1.71414i	−0.515203	−	0.857068i	$-0.672283\pi$
$941$		−	49480.0i		−	1.71414i	0.515203	−	0.857068i	$-0.327717\pi$
$942$		0			0
$943$			57863.2i			1.99818i
$944$		0			0
$945$	−20244.1	−	979.215i	−0.696869	−	0.0337078i
$946$		0			0
$947$	24727.0			0.848489			0.424245	−	0.905548i	$-0.360540\pi$
$947$	24727.0			0.848489			0.424245	+	0.905548i	$0.360540\pi$
$948$		0			0
$949$	16912.3			0.578501
$950$		0			0
$951$	32085.8	+	19220.5i	1.09406	+	0.655382i
$952$		0			0
$953$			52173.5i			1.77341i	0.462331	+	0.886707i	$0.347013\pi$
$953$			52173.5i			1.77341i	−0.462331	+	0.886707i	$0.652987\pi$
$954$		0			0
$955$			8319.78i			0.281907i
$956$		0			0
$957$	−26771.2	−	16036.9i	−0.904274	−	0.541692i
$958$		0			0
$959$	10449.2			0.351847
$960$		0			0
$961$	−1396.13			−0.0468642
$962$		0			0
$963$	8077.76	+	15094.1i	0.270303	+	0.505091i
$964$		0			0
$965$			7210.80i			0.240543i
$966$		0			0
$967$			5925.67i			0.197060i	0.995134	+	0.0985298i	$0.0314140\pi$
$967$			5925.67i			0.197060i	−0.995134	+	0.0985298i	$0.968586\pi$
$968$		0			0
$969$	1791.86	−	2991.24i	0.0594043	−	0.0991666i
$970$		0			0
$971$	56824.1			1.87803			0.939017	−	0.343869i	$-0.111738\pi$
$971$	56824.1			1.87803			0.939017	+	0.343869i	$0.111738\pi$
$972$		0			0
$973$	−13172.7			−0.434017
$974$		0			0
$975$	−4207.40	+	7023.63i	−0.138200	+	0.230704i
$976$		0			0
$977$			59000.6i			1.93203i	0.258477	+	0.966017i	$0.416779\pi$
$977$			59000.6i			1.93203i	−0.258477	+	0.966017i	$0.583221\pi$
$978$		0			0
$979$			32100.8i			1.04795i
$980$		0			0
$981$	12579.4	+	23505.9i	0.409407	+	0.765021i
$982$		0			0
$983$	30068.3			0.975615			0.487807	−	0.872951i	$-0.337797\pi$
$983$	30068.3			0.975615			0.487807	+	0.872951i	$0.337797\pi$
$984$		0			0
$985$	12025.4			0.388997
$986$		0			0
$987$	27199.1	+	16293.2i	0.877159	+	0.525449i
$988$		0			0
$989$			98430.7i			3.16473i
$990$		0			0
$991$		−	30736.5i		−	0.985243i	−0.870244	−	0.492622i	$-0.836039\pi$
$991$		−	30736.5i		−	0.985243i	0.870244	−	0.492622i	$-0.163961\pi$
$992$		0			0
$993$	−30388.2	−	18203.6i	−0.971137	−	0.581745i
$994$		0			0
$995$	19798.7			0.630816
$996$		0			0
$997$	34289.3			1.08922			0.544610	−	0.838689i	$-0.316678\pi$
$997$	34289.3			1.08922			0.544610	+	0.838689i	$0.316678\pi$
$998$		0			0
$999$	−5081.43	−	245.790i	−0.160930	−	0.00778425i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.h.b.191.10	yes	24
3.2	odd	2		480.4.h.a.191.16	yes	24
4.3	odd	2		480.4.h.a.191.15	✓	24
8.3	odd	2		960.4.h.e.191.10		24
8.5	even	2		960.4.h.c.191.15		24
12.11	even	2	inner	480.4.h.b.191.9	yes	24
24.5	odd	2		960.4.h.e.191.9		24
24.11	even	2		960.4.h.c.191.16		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.h.a.191.15	✓	24	4.3	odd	2
480.4.h.a.191.16	yes	24	3.2	odd	2
480.4.h.b.191.9	yes	24	12.11	even	2	inner
480.4.h.b.191.10	yes	24	1.1	even	1	trivial
960.4.h.c.191.15		24	8.5	even	2
960.4.h.c.191.16		24	24.11	even	2
960.4.h.e.191.9		24	24.5	odd	2
960.4.h.e.191.10		24	8.3	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)

\(f(q)\)	\(=\)	\(q+(-2.67023 + 4.45756i) q^{3} -5.00000i q^{5} -28.8929i q^{7} +(-12.7397 - 23.8055i) q^{9} -26.6844 q^{11} -63.0266 q^{13} +(22.2878 + 13.3512i) q^{15} +4.23575i q^{17} +158.425i q^{19} +(128.792 + 77.1507i) q^{21} +193.058 q^{23} -25.0000 q^{25} +(140.132 + 6.77825i) q^{27} -225.068i q^{29} +176.599i q^{31} +(71.2536 - 118.947i) q^{33} -144.464 q^{35} -36.2616 q^{37} +(168.296 - 280.945i) q^{39} +299.719i q^{41} +509.850i q^{43} +(-119.027 + 63.6985i) q^{45} +211.187 q^{47} -491.797 q^{49} +(-18.8811 - 11.3104i) q^{51} +701.674i q^{53} +133.422i q^{55} +(-706.189 - 423.032i) q^{57} +299.376 q^{59} +133.906 q^{61} +(-687.808 + 368.086i) q^{63} +315.133i q^{65} -292.983i q^{67} +(-515.510 + 860.568i) q^{69} -630.368 q^{71} -268.336 q^{73} +(66.7559 - 111.439i) q^{75} +770.989i q^{77} -549.289i q^{79} +(-404.401 + 606.549i) q^{81} +192.312 q^{83} +21.1787 q^{85} +(1003.25 + 600.983i) q^{87} -1202.98i q^{89} +1821.02i q^{91} +(-787.200 - 471.560i) q^{93} +792.125 q^{95} +831.831 q^{97} +(339.951 + 635.235i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 20 q^{9} + 72 q^{11} - 72 q^{13} + 20 q^{15} - 68 q^{21} + 96 q^{23} - 600 q^{25} + 168 q^{27} - 80 q^{33} - 504 q^{37} - 456 q^{39} - 220 q^{45} + 432 q^{47} - 816 q^{49} + 1240 q^{51} + 40 q^{57}+ \cdots + 3160 q^{99}+O(q^{100}) \) 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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Database

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