LMFDB - Embedded newform 480.4.h.a.191.15

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			191.15
Character	$\chi$	$=$	480.191
Dual form			480.4.h.a.191.16

$q$-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	$0.284798\pi$
$7$			28.8929i			1.56007i	−0.625737	+	0.780034i	$0.715202\pi$
$8$		0			0
$9$	−12.7397	−	23.8055i	−0.471840	−	0.881684i
$10$		0			0
$11$	26.6844			0.731423			0.365711	−	0.930728i	$-0.380826\pi$
$11$	26.6844			0.731423			0.365711	+	0.930728i	$0.380826\pi$
$12$		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	$-0.594283\pi$
$19$		−	158.425i		−	1.91291i	0.291887	−	0.956453i	$-0.405717\pi$
$20$		0			0
$21$	128.792	+	77.1507i	1.33832	+	0.801698i
$22$		0			0
$23$	−193.058			−1.75023			−0.875117	−	0.483911i	$-0.839216\pi$
$23$	−193.058			−1.75023			−0.875117	+	0.483911i	$0.839216\pi$
$24$		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	$-0.829060\pi$
$31$		−	176.599i		−	1.02316i	0.859235	−	0.511582i	$-0.170940\pi$
$32$		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	$-0.640552\pi$
$43$		−	509.850i		−	1.80817i	0.427349	−	0.904087i	$-0.359448\pi$
$44$		0			0
$45$	−119.027	+	63.6985i	−0.394301	+	0.211013i
$46$		0			0
$47$	−211.187			−0.655420			−0.327710	−	0.944778i	$-0.606277\pi$
$47$	−211.187			−0.655420			−0.327710	+	0.944778i	$0.606277\pi$
$48$		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	$-0.607150\pi$
$59$	−299.376			−0.660600			−0.330300	+	0.943876i	$0.607150\pi$
$60$		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	$0.0860708\pi$
$67$			292.983i			0.534233i	−0.963664	+	0.267116i	$0.913929\pi$
$68$		0			0
$69$	−515.510	+	860.568i	−0.899423	+	1.50145i
$70$		0			0
$71$	630.368			1.05368			0.526838	−	0.849966i	$-0.323378\pi$
$71$	630.368			1.05368			0.526838	+	0.849966i	$0.323378\pi$
$72$		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	$0.127918\pi$
$79$			549.289i			0.782276i	−0.920332	+	0.391138i	$0.872082\pi$
$80$		0			0
$81$	−404.401	+	606.549i	−0.554733	+	0.832028i
$82$		0			0
$83$	−192.312			−0.254325			−0.127162	−	0.991882i	$-0.540587\pi$
$83$	−192.312			−0.254325			−0.127162	+	0.991882i	$0.540587\pi$
$84$		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	$-0.879010\pi$
$103$		−	775.670i		−	0.742029i	0.928627	−	0.371015i	$-0.120990\pi$
$104$		0			0
$105$	385.754	−	643.958i	0.358530	−	0.598513i
$106$		0			0
$107$	634.062			0.572870			0.286435	−	0.958100i	$-0.407530\pi$
$107$	634.062			0.572870			0.286435	+	0.958100i	$0.407530\pi$
$108$		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	$-0.735402\pi$
$127$		−	2114.71i		−	1.47756i	0.673946	−	0.738781i	$-0.264598\pi$
$128$		0			0
$129$	−2272.69	−	1361.42i	−1.55116	−	0.929197i
$130$		0			0
$131$	−854.166			−0.569685			−0.284843	−	0.958574i	$-0.591941\pi$
$131$	−854.166			−0.569685			−0.284843	+	0.958574i	$0.591941\pi$
$132$		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	$0.0444216\pi$
$139$			455.917i			0.278204i	−0.990278	+	0.139102i	$0.955578\pi$
$140$		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	$0.0323334\pi$
$151$			376.313i			0.202807i	−0.994845	+	0.101404i	$0.967667\pi$
$152$		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	$-0.866097\pi$
$163$		−	1699.68i		−	0.816742i	0.912816	−	0.408371i	$-0.133903\pi$
$164$		0			0
$165$	−594.737	−	356.268i	−0.280607	−	0.168094i
$166$		0			0
$167$	519.812			0.240864			0.120432	−	0.992722i	$-0.461572\pi$
$167$	519.812			0.240864			0.120432	+	0.992722i	$0.461572\pi$
$168$		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	$-0.413265\pi$
$179$	1289.04			0.538254			0.269127	+	0.963105i	$0.413265\pi$
$180$		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	$-0.397934\pi$
$191$	1663.96			0.630364			0.315182	+	0.949031i	$0.397934\pi$
$192$		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	$-0.750824\pi$
$199$		−	3959.74i		−	1.41055i	0.708936	−	0.705273i	$-0.249176\pi$
$200$		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	$0.0527347\pi$
$211$			1010.91i			0.329828i	−0.986308	+	0.164914i	$0.947265\pi$
$212$		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	$-0.944124\pi$
$223$		−	1163.13i		−	0.349278i	0.984633	−	0.174639i	$-0.0558758\pi$
$224$		0			0
$225$	318.492	+	595.137i	0.0943681	+	0.176337i
$226$		0			0
$227$	−3253.44			−0.951271			−0.475636	−	0.879642i	$-0.657782\pi$
$227$	−3253.44			−0.951271			−0.475636	+	0.879642i	$0.657782\pi$
$228$		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	$-0.257042\pi$
$239$	5108.43			1.38258			0.691291	+	0.722576i	$0.257042\pi$
$240$		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	$-0.167175\pi$
$251$	6881.30			1.73045			0.865226	+	0.501382i	$0.167175\pi$
$252$		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	$-0.624110\pi$
$263$	−3242.34			−0.760195			−0.380098	+	0.924946i	$0.624110\pi$
$264$		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	$0.0469562\pi$
$271$			1311.45i			0.293965i	−0.989139	+	0.146983i	$0.953044\pi$
$272$		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	$0.0354914\pi$
$283$			1059.45i			0.222537i	−0.993790	+	0.111269i	$0.964509\pi$
$284$		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	$-0.944001\pi$
$307$		−	1882.91i		−	0.350043i	0.984565	−	0.175021i	$-0.0559995\pi$
$308$		0			0
$309$	−3457.59	−	2071.22i	−0.636555	−	0.381319i
$310$		0			0
$311$	−3394.17			−0.618860			−0.309430	−	0.950922i	$-0.600138\pi$
$311$	−3394.17			−0.618860			−0.309430	+	0.950922i	$0.600138\pi$
$312$		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	$-0.808481\pi$
$331$		−	6817.22i		−	1.13205i	0.824388	−	0.566025i	$-0.191519\pi$
$332$		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	$-0.670064\pi$
$347$	−6583.03			−1.01843			−0.509215	+	0.860639i	$0.670064\pi$
$348$		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	$-0.570751\pi$
$359$	−2998.98			−0.440891			−0.220446	+	0.975399i	$0.570751\pi$
$360$		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	$-0.829901\pi$
$367$		−	7161.60i		−	1.01862i	0.860584	−	0.509309i	$-0.170099\pi$
$368$		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	$-0.708878\pi$
$379$		−	11691.9i		−	1.58462i	0.610118	−	0.792311i	$-0.291122\pi$
$380$		0			0
$381$	−9426.45	−	5646.78i	−1.26754	−	0.759300i
$382$		0			0
$383$	−3893.22			−0.519411			−0.259705	−	0.965688i	$-0.583625\pi$
$383$	−3893.22			−0.519411			−0.259705	+	0.965688i	$0.583625\pi$
$384$		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	$-0.263761\pi$
$419$	11593.8			1.35177			0.675887	+	0.737005i	$0.263761\pi$
$420$		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	$-0.565877\pi$
$431$	−3677.25			−0.410967			−0.205484	+	0.978661i	$0.565877\pi$
$432$		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	$-0.910849\pi$
$439$		−	5085.24i		−	0.552859i	0.961034	−	0.276430i	$-0.0891513\pi$
$440$		0			0
$441$	6265.34	+	11707.5i	0.676530	+	1.26417i
$442$		0			0
$443$	14786.8			1.58587			0.792935	−	0.609306i	$-0.208552\pi$
$443$	14786.8			1.58587			0.792935	+	0.609306i	$0.208552\pi$
$444$		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	$-0.925749\pi$
$463$		−	4605.81i		−	0.462311i	0.972917	−	0.231156i	$-0.0742506\pi$
$464$		0			0
$465$	−2357.80	+	3936.00i	−0.235141	+	0.392532i
$466$		0			0
$467$	2603.62			0.257989			0.128995	−	0.991645i	$-0.458825\pi$
$467$	2603.62			0.257989			0.128995	+	0.991645i	$0.458825\pi$
$468$		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	$-0.486270\pi$
$479$	904.077			0.0862387			0.0431193	+	0.999070i	$0.486270\pi$
$480$		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	$0.0413934\pi$
$487$			2787.27i			0.259350i	−0.991557	+	0.129675i	$0.958607\pi$
$488$		0			0
$489$	−7576.41	−	4538.53i	−0.700648	−	0.419713i
$490$		0			0
$491$	6234.26			0.573011			0.286506	−	0.958079i	$-0.407506\pi$
$491$	6234.26			0.573011			0.286506	+	0.958079i	$0.407506\pi$
$492$		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	$-0.873451\pi$
$499$		−	8631.50i		−	0.774347i	0.922007	−	0.387174i	$-0.126549\pi$
$500$		0			0
$501$	1388.02	−	2317.09i	0.123777	−	0.206627i
$502$		0			0
$503$	3401.02			0.301479			0.150740	−	0.988574i	$-0.451835\pi$
$503$	3401.02			0.301479			0.150740	+	0.988574i	$0.451835\pi$
$504$		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	$-0.750204\pi$
$523$		−	16904.0i		−	1.41331i	0.707560	−	0.706654i	$-0.249796\pi$
$524$		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	$0.175677\pi$
$547$			13415.3i			1.04862i	−0.851527	+	0.524312i	$0.824323\pi$
$548$		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	$-0.412103\pi$
$563$	7284.20			0.545280			0.272640	+	0.962116i	$0.412103\pi$
$564$		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	$0.115910\pi$
$571$			9718.85i			0.712296i	−0.934430	+	0.356148i	$0.884090\pi$
$572$		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	$-0.0592811\pi$
$587$	27951.9			1.96542			0.982708	+	0.185162i	$0.0592811\pi$
$588$		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	$-0.453277\pi$
$599$	4288.34			0.292515			0.146258	+	0.989247i	$0.453277\pi$
$600$		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	$-0.982338\pi$
$607$		−	1658.77i		−	0.110918i	0.998461	−	0.0554591i	$-0.0176622\pi$
$608$		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	$-0.979903\pi$
$619$		−	1943.40i		−	0.126190i	0.998008	−	0.0630952i	$-0.0200972\pi$
$620$		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	$0.167992\pi$
$631$			15964.7i			1.00720i	−0.863936	+	0.503602i	$0.832008\pi$
$632$		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	$0.0380523\pi$
$643$			3889.04i			0.238520i	−0.992863	+	0.119260i	$0.961948\pi$
$644$		0			0
$645$	−6807.10	+	11363.4i	−0.415549	+	0.693698i
$646$		0			0
$647$	−21741.6			−1.32110			−0.660548	−	0.750784i	$-0.729676\pi$
$647$	−21741.6			−1.32110			−0.660548	+	0.750784i	$0.729676\pi$
$648$		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	$-0.538172\pi$
$659$	−4047.71			−0.239266			−0.119633	+	0.992818i	$0.538172\pi$
$660$		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	$-0.767414\pi$
$683$	−26585.8			−1.48943			−0.744713	+	0.667384i	$0.767414\pi$
$684$		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	$-0.961674\pi$
$691$		−	4363.60i		−	0.240231i	0.992760	−	0.120115i	$-0.0383264\pi$
$692$		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	$-0.418462\pi$
$719$	9769.54			0.506735			0.253367	+	0.967370i	$0.418462\pi$
$720$		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	$-0.911867\pi$
$727$		−	10716.6i		−	0.546707i	0.961914	−	0.273354i	$-0.0881329\pi$
$728$		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	$0.217052\pi$
$739$			25323.1i			1.26052i	−0.776385	+	0.630260i	$0.782948\pi$
$740$		0			0
$741$	44508.7	+	26662.3i	2.20657	+	1.32181i
$742$		0			0
$743$	−17744.8			−0.876171			−0.438086	−	0.898933i	$-0.644343\pi$
$743$	−17744.8			−0.876171			−0.438086	+	0.898933i	$0.644343\pi$
$744$		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	$-0.934271\pi$
$751$		−	8439.26i		−	0.410057i	0.978756	−	0.205029i	$-0.0657287\pi$
$752$		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	$-0.943331\pi$
$787$		−	7819.69i		−	0.354183i	0.984194	−	0.177091i	$-0.0566688\pi$
$788$		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	$-0.802216\pi$
$811$		−	26889.8i		−	1.16428i	0.813090	−	0.582138i	$-0.197784\pi$
$812$		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	$0.252753\pi$
$823$			33677.4i			1.42639i	−0.700965	+	0.713196i	$0.747247\pi$
$824$		0			0
$825$	−1781.34	+	2973.68i	−0.0751737	+	0.125491i
$826$		0			0
$827$	38868.3			1.63432			0.817161	−	0.576409i	$-0.195547\pi$
$827$	38868.3			1.63432			0.817161	+	0.576409i	$0.195547\pi$
$828$		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	$-0.635156\pi$
$839$	−20023.0			−0.823924			−0.411962	+	0.911201i	$0.635156\pi$
$840$		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	$0.136109\pi$
$859$			20880.5i			0.829377i	−0.909964	+	0.414688i	$0.863891\pi$
$860$		0			0
$861$	−23123.5	+	38601.3i	−0.915271	+	1.52791i
$862$		0			0
$863$	14655.5			0.578077			0.289038	−	0.957318i	$-0.406664\pi$
$863$	14655.5			0.578077			0.289038	+	0.957318i	$0.406664\pi$
$864$		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	$0.193478\pi$
$883$			29969.1i			1.14217i	−0.820890	+	0.571087i	$0.806522\pi$
$884$		0			0
$885$	6672.43	+	3997.02i	0.253436	+	0.151817i
$886$		0			0
$887$	15847.3			0.599887			0.299944	−	0.953957i	$-0.403032\pi$
$887$	15847.3			0.599887			0.299944	+	0.953957i	$0.403032\pi$
$888$		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	$-0.728186\pi$
$907$		−	41184.7i		−	1.50773i	0.657027	−	0.753867i	$-0.271814\pi$
$908$		0			0
$909$	−3838.31	+	2054.10i	−0.140054	+	0.0749509i
$910$		0			0
$911$	−36382.4			−1.32316			−0.661582	−	0.749873i	$-0.730115\pi$
$911$	−36382.4			−1.32316			−0.661582	+	0.749873i	$0.730115\pi$
$912$		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	$-0.875434\pi$
$919$		−	21252.5i		−	0.762844i	0.924401	−	0.381422i	$-0.124566\pi$
$920$		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	$-0.639460\pi$
$947$	−24727.0			−0.848489			−0.424245	+	0.905548i	$0.639460\pi$
$948$		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	$-0.968586\pi$
$967$		−	5925.67i		−	0.197060i	0.995134	−	0.0985298i	$-0.0314140\pi$
$968$		0			0
$969$	1791.86	−	2991.24i	0.0594043	−	0.0991666i
$970$		0			0
$971$	−56824.1			−1.87803			−0.939017	−	0.343869i	$-0.888262\pi$
$971$	−56824.1			−1.87803			−0.939017	+	0.343869i	$0.888262\pi$
$972$		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	$-0.662203\pi$
$983$	−30068.3			−0.975615			−0.487807	+	0.872951i	$0.662203\pi$
$984$		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	$0.163961\pi$
$991$			30736.5i			0.985243i	−0.870244	+	0.492622i	$0.836039\pi$
$992$		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.a.191.15	✓	24
3.2	odd	2		480.4.h.b.191.9	yes	24
4.3	odd	2		480.4.h.b.191.10	yes	24
8.3	odd	2		960.4.h.c.191.15		24
8.5	even	2		960.4.h.e.191.10		24
12.11	even	2	inner	480.4.h.a.191.16	yes	24
24.5	odd	2		960.4.h.c.191.16		24
24.11	even	2		960.4.h.e.191.9		24

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

Fields

Representations

Groups

Database

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