LMFDB - Embedded newform 480.4.h.a.191.22

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			191.22
Character	$\chi$	$=$	480.191
Dual form			480.4.h.a.191.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(4.47955 + 2.63318i) q^{3} -5.00000i q^{5} -30.6962i q^{7} +(13.1328 + 23.5909i) q^{9} +8.44415 q^{11} +1.42157 q^{13} +(13.1659 - 22.3978i) q^{15} -53.1406i q^{17} +42.2378i q^{19} +(80.8285 - 137.505i) q^{21} -75.5043 q^{23} -25.0000 q^{25} +(-3.29015 + 140.258i) q^{27} -184.020i q^{29} -313.528i q^{31} +(37.8260 + 22.2349i) q^{33} -153.481 q^{35} -344.423 q^{37} +(6.36799 + 3.74324i) q^{39} -349.174i q^{41} -161.727i q^{43} +(117.955 - 65.6638i) q^{45} +505.194 q^{47} -599.257 q^{49} +(139.929 - 238.046i) q^{51} +215.195i q^{53} -42.2207i q^{55} +(-111.220 + 189.207i) q^{57} +582.343 q^{59} +635.634 q^{61} +(724.151 - 403.126i) q^{63} -7.10785i q^{65} +143.076i q^{67} +(-338.225 - 198.816i) q^{69} +473.307 q^{71} -224.614 q^{73} +(-111.989 - 65.8294i) q^{75} -259.203i q^{77} +1188.82i q^{79} +(-384.061 + 619.627i) q^{81} +425.854 q^{83} -265.703 q^{85} +(484.557 - 824.327i) q^{87} -827.327i q^{89} -43.6368i q^{91} +(825.575 - 1404.46i) q^{93} +211.189 q^{95} -1041.70 q^{97} +(110.895 + 199.205i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	4.47955	+	2.63318i	0.862090	+	0.506755i
$3$	4.47955	+	2.63318i	0.862090	+	0.506755i
$4$		0			0
$5$		−	5.00000i		−	0.447214i
$5$		−	5.00000i		−	0.447214i
$6$		0			0
$7$		−	30.6962i		−	1.65744i	−0.559664	−	0.828720i	$-0.689070\pi$
$7$		−	30.6962i		−	1.65744i	0.559664	−	0.828720i	$-0.310930\pi$
$8$		0			0
$9$	13.1328	+	23.5909i	0.486398	+	0.873737i
$10$		0			0
$11$	8.44415			0.231455			0.115728	−	0.993281i	$-0.463080\pi$
$11$	8.44415			0.231455			0.115728	+	0.993281i	$0.463080\pi$
$12$		0			0
$13$	1.42157			0.0303287			0.0151643	−	0.999885i	$-0.495173\pi$
$13$	1.42157			0.0303287			0.0151643	+	0.999885i	$0.495173\pi$
$14$		0			0
$15$	13.1659	−	22.3978i	0.226628	−	0.385538i
$16$		0			0
$17$		−	53.1406i		−	0.758147i	−0.925367	−	0.379074i	$-0.876243\pi$
$17$		−	53.1406i		−	0.758147i	0.925367	−	0.379074i	$-0.123757\pi$
$18$		0			0
$19$			42.2378i			0.510001i	0.966941	+	0.255001i	$0.0820757\pi$
$19$			42.2378i			0.510001i	−0.966941	+	0.255001i	$0.917924\pi$
$20$		0			0
$21$	80.8285	−	137.505i	0.839916	−	1.42886i
$22$		0			0
$23$	−75.5043			−0.684510			−0.342255	−	0.939607i	$-0.611191\pi$
$23$	−75.5043			−0.684510			−0.342255	+	0.939607i	$0.611191\pi$
$24$		0			0
$25$	−25.0000			−0.200000
$26$		0			0
$27$	−3.29015	+	140.258i	−0.0234515	+	0.999725i
$28$		0			0
$29$		−	184.020i		−	1.17833i	−0.808012	−	0.589167i	$-0.799456\pi$
$29$		−	184.020i		−	1.17833i	0.808012	−	0.589167i	$-0.200544\pi$
$30$		0			0
$31$		−	313.528i		−	1.81649i	−0.418435	−	0.908247i	$-0.637421\pi$
$31$		−	313.528i		−	1.81649i	0.418435	−	0.908247i	$-0.362579\pi$
$32$		0			0
$33$	37.8260	+	22.2349i	0.199535	+	0.117291i
$34$		0			0
$35$	−153.481			−0.741229
$36$		0			0
$37$	−344.423			−1.53035			−0.765174	−	0.643824i	$-0.777347\pi$
$37$	−344.423			−1.53035			−0.765174	+	0.643824i	$0.777347\pi$
$38$		0			0
$39$	6.36799	+	3.74324i	0.0261460	+	0.0153692i
$40$		0			0
$41$		−	349.174i		−	1.33004i	−0.746824	−	0.665022i	$-0.768422\pi$
$41$		−	349.174i		−	1.33004i	0.746824	−	0.665022i	$-0.231578\pi$
$42$		0			0
$43$		−	161.727i		−	0.573562i	−0.957996	−	0.286781i	$-0.907415\pi$
$43$		−	161.727i		−	0.573562i	0.957996	−	0.286781i	$-0.0925852\pi$
$44$		0			0
$45$	117.955	−	65.6638i	0.390747	−	0.217524i
$46$		0			0
$47$	505.194			1.56788			0.783938	−	0.620839i	$-0.213208\pi$
$47$	505.194			1.56788			0.783938	+	0.620839i	$0.213208\pi$
$48$		0			0
$49$	−599.257			−1.74710
$50$		0			0
$51$	139.929	−	238.046i	0.384195	−	0.653591i
$52$		0			0
$53$			215.195i			0.557722i	0.960331	+	0.278861i	$0.0899569\pi$
$53$			215.195i			0.557722i	−0.960331	+	0.278861i	$0.910043\pi$
$54$		0			0
$55$		−	42.2207i		−	0.103510i
$56$		0			0
$57$	−111.220	+	189.207i	−0.258446	+	0.439667i
$58$		0			0
$59$	582.343			1.28499			0.642497	−	0.766288i	$-0.277899\pi$
$59$	582.343			1.28499			0.642497	+	0.766288i	$0.277899\pi$
$60$		0			0
$61$	635.634			1.33417			0.667087	−	0.744980i	$-0.267541\pi$
$61$	635.634			1.33417			0.667087	+	0.744980i	$0.267541\pi$
$62$		0			0
$63$	724.151	−	403.126i	1.44817	−	0.806176i
$64$		0			0
$65$		−	7.10785i		−	0.0135634i
$66$		0			0
$67$			143.076i			0.260889i	0.991456	+	0.130444i	$0.0416404\pi$
$67$			143.076i			0.260889i	−0.991456	+	0.130444i	$0.958360\pi$
$68$		0			0
$69$	−338.225	−	198.816i	−0.590109	−	0.346879i
$70$		0			0
$71$	473.307			0.791144			0.395572	−	0.918435i	$-0.370546\pi$
$71$	473.307			0.791144			0.395572	+	0.918435i	$0.370546\pi$
$72$		0			0
$73$	−224.614			−0.360124			−0.180062	−	0.983655i	$-0.557630\pi$
$73$	−224.614			−0.360124			−0.180062	+	0.983655i	$0.557630\pi$
$74$		0			0
$75$	−111.989	−	65.8294i	−0.172418	−	0.101351i
$76$		0			0
$77$		−	259.203i		−	0.383623i
$78$		0			0
$79$			1188.82i			1.69307i	0.532334	+	0.846534i	$0.321315\pi$
$79$			1188.82i			1.69307i	−0.532334	+	0.846534i	$0.678685\pi$
$80$		0			0
$81$	−384.061	+	619.627i	−0.526833	+	0.849969i
$82$		0			0
$83$	425.854			0.563175			0.281588	−	0.959535i	$-0.409139\pi$
$83$	425.854			0.563175			0.281588	+	0.959535i	$0.409139\pi$
$84$		0			0
$85$	−265.703			−0.339054
$86$		0			0
$87$	484.557	−	824.327i	0.597126	−	1.01583i
$88$		0			0
$89$		−	827.327i		−	0.985354i	−0.870212	−	0.492677i	$-0.836018\pi$
$89$		−	827.327i		−	0.985354i	0.870212	−	0.492677i	$-0.163982\pi$
$90$		0			0
$91$		−	43.6368i		−	0.0502679i
$92$		0			0
$93$	825.575	−	1404.46i	0.920517	−	1.56598i
$94$		0			0
$95$	211.189			0.228079
$96$		0			0
$97$	−1041.70			−1.09040			−0.545199	−	0.838307i	$-0.683546\pi$
$97$	−1041.70			−1.09040			−0.545199	+	0.838307i	$0.683546\pi$
$98$		0			0
$99$	110.895	+	199.205i	0.112579	+	0.202231i
$100$		0			0
$101$		−	1024.94i		−	1.00975i	−0.863192	−	0.504876i	$-0.831538\pi$
$101$		−	1024.94i		−	1.00975i	0.863192	−	0.504876i	$-0.168462\pi$
$102$		0			0
$103$			509.724i			0.487617i	0.969823	+	0.243809i	$0.0783968\pi$
$103$			509.724i			0.487617i	−0.969823	+	0.243809i	$0.921603\pi$
$104$		0			0
$105$	−687.526	−	404.143i	−0.639006	−	0.375622i
$106$		0			0
$107$	−371.393			−0.335550			−0.167775	−	0.985825i	$-0.553658\pi$
$107$	−371.393			−0.335550			−0.167775	+	0.985825i	$0.553658\pi$
$108$		0			0
$109$	96.7636			0.0850300			0.0425150	−	0.999096i	$-0.486463\pi$
$109$	96.7636			0.0850300			0.0425150	+	0.999096i	$0.486463\pi$
$110$		0			0
$111$	−1542.86	−	906.928i	−1.31930	−	0.775511i
$112$		0			0
$113$			1027.85i			0.855683i	0.903854	+	0.427842i	$0.140726\pi$
$113$			1027.85i			0.855683i	−0.903854	+	0.427842i	$0.859274\pi$
$114$		0			0
$115$			377.521i			0.306122i
$116$		0			0
$117$	18.6691	+	33.5361i	0.0147518	+	0.0264993i
$118$		0			0
$119$	−1631.22			−1.25658
$120$		0			0
$121$	−1259.70			−0.946429
$122$		0			0
$123$	919.437	−	1564.14i	0.674007	−	1.14662i
$124$		0			0
$125$			125.000i			0.0894427i
$126$		0			0
$127$		−	1704.60i		−	1.19101i	−0.803350	−	0.595507i	$-0.796951\pi$
$127$		−	1704.60i		−	1.19101i	0.803350	−	0.595507i	$-0.203049\pi$
$128$		0			0
$129$	425.856	−	724.465i	0.290655	−	0.494462i
$130$		0			0
$131$	2830.28			1.88766			0.943828	−	0.330437i	$-0.107196\pi$
$131$	2830.28			1.88766			0.943828	+	0.330437i	$0.107196\pi$
$132$		0			0
$133$	1296.54			0.845296
$134$		0			0
$135$	701.288	+	16.4507i	0.447091	+	0.0104878i
$136$		0			0
$137$			822.413i			0.512872i	0.966561	+	0.256436i	$0.0825483\pi$
$137$			822.413i			0.512872i	−0.966561	+	0.256436i	$0.917452\pi$
$138$		0			0
$139$			939.127i			0.573063i	0.958071	+	0.286531i	$0.0925022\pi$
$139$			939.127i			0.573063i	−0.958071	+	0.286531i	$0.907498\pi$
$140$		0			0
$141$	2263.04	+	1330.27i	1.35165	+	0.794529i
$142$		0			0
$143$	12.0039			0.00701972
$144$		0			0
$145$	−920.100			−0.526967
$146$		0			0
$147$	−2684.40	−	1577.95i	−1.50616	−	0.885354i
$148$		0			0
$149$		−	1598.11i		−	0.878675i	−0.898322	−	0.439338i	$-0.855213\pi$
$149$		−	1598.11i		−	0.878675i	0.898322	−	0.439338i	$-0.144787\pi$
$150$		0			0
$151$			2314.47i			1.24734i	0.781687	+	0.623671i	$0.214360\pi$
$151$			2314.47i			1.24734i	−0.781687	+	0.623671i	$0.785640\pi$
$152$		0			0
$153$	1253.64	−	697.883i	0.662421	−	0.368762i
$154$		0			0
$155$	−1567.64			−0.812360
$156$		0			0
$157$	1902.00			0.966853			0.483426	−	0.875385i	$-0.339392\pi$
$157$	1902.00			0.966853			0.483426	+	0.875385i	$0.339392\pi$
$158$		0			0
$159$	−566.646	+	963.976i	−0.282629	+	0.480807i
$160$		0			0
$161$			2317.69i			1.13453i
$162$		0			0
$163$		−	1448.44i		−	0.696014i	−0.937492	−	0.348007i	$-0.886859\pi$
$163$		−	1448.44i		−	0.696014i	0.937492	−	0.348007i	$-0.113141\pi$
$164$		0			0
$165$	111.175	−	189.130i	0.0524542	−	0.0892348i
$166$		0			0
$167$	−2873.68			−1.33157			−0.665784	−	0.746144i	$-0.731903\pi$
$167$	−2873.68			−1.33157			−0.665784	+	0.746144i	$0.731903\pi$
$168$		0			0
$169$	−2194.98			−0.999080
$170$		0			0
$171$	−996.428	+	554.699i	−0.445607	+	0.248064i
$172$		0			0
$173$		−	84.4452i		−	0.0371113i	−0.999828	−	0.0185556i	$-0.994093\pi$
$173$		−	84.4452i		−	0.0371113i	0.999828	−	0.0185556i	$-0.00590678\pi$
$174$		0			0
$175$			767.405i			0.331488i
$176$		0			0
$177$	2608.64	+	1533.41i	1.10778	+	0.651177i
$178$		0			0
$179$	1333.14			0.556670			0.278335	−	0.960484i	$-0.410217\pi$
$179$	1333.14			0.556670			0.278335	+	0.960484i	$0.410217\pi$
$180$		0			0
$181$	−3560.10			−1.46199			−0.730996	−	0.682382i	$-0.760944\pi$
$181$	−3560.10			−1.46199			−0.730996	+	0.682382i	$0.760944\pi$
$182$		0			0
$183$	2847.35	+	1673.74i	1.15018	+	0.676099i
$184$		0			0
$185$			1722.12i			0.684392i
$186$		0			0
$187$		−	448.728i		−	0.175477i
$188$		0			0
$189$	4305.37	+	100.995i	1.65698	+	0.0388694i
$190$		0			0
$191$	−1156.95			−0.438293			−0.219146	−	0.975692i	$-0.570327\pi$
$191$	−1156.95			−0.438293			−0.219146	+	0.975692i	$0.570327\pi$
$192$		0			0
$193$	947.332			0.353318			0.176659	−	0.984272i	$-0.443471\pi$
$193$	947.332			0.353318			0.176659	+	0.984272i	$0.443471\pi$
$194$		0			0
$195$	18.7162	−	31.8400i	0.00687332	−	0.0116929i
$196$		0			0
$197$		−	1578.72i		−	0.570960i	−0.958385	−	0.285480i	$-0.907847\pi$
$197$		−	1578.72i		−	0.570960i	0.958385	−	0.285480i	$-0.0921530\pi$
$198$		0			0
$199$			3388.77i			1.20715i	0.797305	+	0.603576i	$0.206258\pi$
$199$			3388.77i			1.20715i	−0.797305	+	0.603576i	$0.793742\pi$
$200$		0			0
$201$	−376.745	+	640.918i	−0.132207	+	0.224910i
$202$		0			0
$203$	−5648.72			−1.95302
$204$		0			0
$205$	−1745.87			−0.594814
$206$		0			0
$207$	−991.579	−	1781.21i	−0.332945	−	0.598082i
$208$		0			0
$209$			356.663i			0.118042i
$210$		0			0
$211$			4291.74i			1.40026i	0.714014	+	0.700132i	$0.246875\pi$
$211$			4291.74i			1.40026i	−0.714014	+	0.700132i	$0.753125\pi$
$212$		0			0
$213$	2120.20	+	1246.30i	0.682037	+	0.400916i
$214$		0			0
$215$	−808.636			−0.256505
$216$		0			0
$217$	−9624.12			−3.01073
$218$		0			0
$219$	−1006.17	−	591.447i	−0.310459	−	0.182495i
$220$		0			0
$221$		−	75.5431i		−	0.0229936i
$222$		0			0
$223$			3722.34i			1.11779i	0.829239	+	0.558894i	$0.188774\pi$
$223$			3722.34i			1.11779i	−0.829239	+	0.558894i	$0.811226\pi$
$224$		0			0
$225$	−328.319	−	589.773i	−0.0972797	−	0.174747i
$226$		0			0
$227$	2987.21			0.873428			0.436714	−	0.899600i	$-0.356142\pi$
$227$	2987.21			0.873428			0.436714	+	0.899600i	$0.356142\pi$
$228$		0			0
$229$	5291.96			1.52708			0.763542	−	0.645758i	$-0.223458\pi$
$229$	5291.96			1.52708			0.763542	+	0.645758i	$0.223458\pi$
$230$		0			0
$231$	682.528	−	1161.11i	0.194403	−	0.330717i
$232$		0			0
$233$			6277.11i			1.76492i	0.470385	+	0.882461i	$0.344115\pi$
$233$			6277.11i			1.76492i	−0.470385	+	0.882461i	$0.655885\pi$
$234$		0			0
$235$		−	2525.97i		−	0.701176i
$236$		0			0
$237$	−3130.37	+	5325.37i	−0.857971	+	1.45958i
$238$		0			0
$239$	960.397			0.259929			0.129964	−	0.991519i	$-0.458514\pi$
$239$	960.397			0.259929			0.129964	+	0.991519i	$0.458514\pi$
$240$		0			0
$241$	7032.90			1.87979			0.939894	−	0.341467i	$-0.110924\pi$
$241$	7032.90			1.87979			0.939894	+	0.341467i	$0.110924\pi$
$242$		0			0
$243$	−3352.01	+	1764.35i	−0.884904	+	0.465774i
$244$		0			0
$245$			2996.28i			0.781329i
$246$		0			0
$247$			60.0440i			0.0154676i
$248$		0			0
$249$	1907.64	+	1121.35i	0.485508	+	0.285392i
$250$		0			0
$251$	3405.37			0.856354			0.428177	−	0.903695i	$-0.359156\pi$
$251$	3405.37			0.856354			0.428177	+	0.903695i	$0.359156\pi$
$252$		0			0
$253$	−637.569			−0.158433
$254$		0			0
$255$	−1190.23	−	699.644i	−0.292295	−	0.171817i
$256$		0			0
$257$			3398.21i			0.824804i	0.911002	+	0.412402i	$0.135310\pi$
$257$			3398.21i			0.824804i	−0.911002	+	0.412402i	$0.864690\pi$
$258$		0			0
$259$			10572.5i			2.53646i
$260$		0			0
$261$	4341.20	−	2416.69i	1.02955	−	0.573139i
$262$		0			0
$263$	2025.02			0.474783			0.237391	−	0.971414i	$-0.423708\pi$
$263$	2025.02			0.474783			0.237391	+	0.971414i	$0.423708\pi$
$264$		0			0
$265$	1075.97			0.249421
$266$		0			0
$267$	2178.50	−	3706.05i	0.499333	−	0.849464i
$268$		0			0
$269$			3700.78i			0.838812i	0.907799	+	0.419406i	$0.137762\pi$
$269$			3700.78i			0.838812i	−0.907799	+	0.419406i	$0.862238\pi$
$270$		0			0
$271$		−	5405.03i		−	1.21156i	−0.795633	−	0.605779i	$-0.792861\pi$
$271$		−	5405.03i		−	1.21156i	0.795633	−	0.605779i	$-0.207139\pi$
$272$		0			0
$273$	114.903	−	195.473i	0.0254735	−	0.0433355i
$274$		0			0
$275$	−211.104			−0.0462910
$276$		0			0
$277$	−2927.21			−0.634943			−0.317471	−	0.948268i	$-0.602834\pi$
$277$	−2927.21			−0.634943			−0.317471	+	0.948268i	$0.602834\pi$
$278$		0			0
$279$	7396.41	−	4117.49i	1.58714	−	0.883540i
$280$		0			0
$281$		−	3813.62i		−	0.809613i	−0.914402	−	0.404807i	$-0.867339\pi$
$281$		−	3813.62i		−	0.809613i	0.914402	−	0.404807i	$-0.132661\pi$
$282$		0			0
$283$			7631.39i			1.60297i	0.598018	+	0.801483i	$0.295955\pi$
$283$			7631.39i			1.60297i	−0.598018	+	0.801483i	$0.704045\pi$
$284$		0			0
$285$	946.033	+	556.098i	0.196625	+	0.115580i
$286$		0			0
$287$	−10718.3			−2.20447
$288$		0			0
$289$	2089.07			0.425213
$290$		0			0
$291$	−4666.35	−	2742.98i	−0.940021	−	0.552565i
$292$		0			0
$293$		−	3321.29i		−	0.662226i	−0.943591	−	0.331113i	$-0.892576\pi$
$293$		−	3321.29i		−	0.662226i	0.943591	−	0.331113i	$-0.107424\pi$
$294$		0			0
$295$		−	2911.72i		−	0.574667i
$296$		0			0
$297$	−27.7825	+	1184.36i	−0.00542796	+	0.231391i
$298$		0			0
$299$	−107.335			−0.0207603
$300$		0			0
$301$	−4964.41			−0.950644
$302$		0			0
$303$	2698.84	−	4591.25i	0.511697	−	0.870497i
$304$		0			0
$305$		−	3178.17i		−	0.596661i
$306$		0			0
$307$			309.855i			0.0576037i	0.999585	+	0.0288019i	$0.00916918\pi$
$307$			309.855i			0.0576037i	−0.999585	+	0.0288019i	$0.990831\pi$
$308$		0			0
$309$	−1342.19	+	2283.33i	−0.247102	+	0.420370i
$310$		0			0
$311$	−7712.08			−1.40615			−0.703074	−	0.711117i	$-0.748190\pi$
$311$	−7712.08			−1.40615			−0.703074	+	0.711117i	$0.748190\pi$
$312$		0			0
$313$	−305.245			−0.0551229			−0.0275615	−	0.999620i	$-0.508774\pi$
$313$	−305.245			−0.0551229			−0.0275615	+	0.999620i	$0.508774\pi$
$314$		0			0
$315$	−2015.63	−	3620.76i	−0.360533	−	0.647640i
$316$		0			0
$317$			6944.11i			1.23035i	0.788391	+	0.615174i	$0.210914\pi$
$317$			6944.11i			1.23035i	−0.788391	+	0.615174i	$0.789086\pi$
$318$		0			0
$319$		−	1553.89i		−	0.272731i
$320$		0			0
$321$	−1663.67	−	977.943i	−0.289275	−	0.170042i
$322$		0			0
$323$	2244.55			0.386656
$324$		0			0
$325$	−35.5392			−0.00606573
$326$		0			0
$327$	433.458	+	254.796i	0.0733036	+	0.0430894i
$328$		0			0
$329$		−	15507.5i		−	2.59866i
$330$		0			0
$331$			1753.57i			0.291193i	0.989344	+	0.145596i	$0.0465100\pi$
$331$			1753.57i			0.291193i	−0.989344	+	0.145596i	$0.953490\pi$
$332$		0			0
$333$	−4523.23	−	8125.26i	−0.744359	−	1.33712i
$334$		0			0
$335$	715.381			0.116673
$336$		0			0
$337$	−3893.43			−0.629343			−0.314671	−	0.949201i	$-0.601894\pi$
$337$	−3893.43			−0.629343			−0.314671	+	0.949201i	$0.601894\pi$
$338$		0			0
$339$	−2706.52	+	4604.32i	−0.433622	+	0.737676i
$340$		0			0
$341$		−	2647.48i		−	0.420437i
$342$		0			0
$343$			7866.11i			1.23828i
$344$		0			0
$345$	−994.080	+	1691.13i	−0.155129	+	0.263905i
$346$		0			0
$347$	−3069.04			−0.474798			−0.237399	−	0.971412i	$-0.576295\pi$
$347$	−3069.04			−0.474798			−0.237399	+	0.971412i	$0.576295\pi$
$348$		0			0
$349$	9549.74			1.46472			0.732358	−	0.680919i	$-0.238420\pi$
$349$	9549.74			1.46472			0.732358	+	0.680919i	$0.238420\pi$
$350$		0			0
$351$	−4.67717	+	199.386i	−0.000711251	+	0.0303203i
$352$		0			0
$353$		−	12359.0i		−	1.86346i	−0.363150	−	0.931731i	$-0.618299\pi$
$353$		−	12359.0i		−	1.86346i	0.363150	−	0.931731i	$-0.381701\pi$
$354$		0			0
$355$		−	2366.53i		−	0.353810i
$356$		0			0
$357$	−7307.12	−	4295.28i	−1.08329	−	0.636780i
$358$		0			0
$359$	8035.82			1.18138			0.590689	−	0.806900i	$-0.298856\pi$
$359$	8035.82			1.18138			0.590689	+	0.806900i	$0.298856\pi$
$360$		0			0
$361$	5074.97			0.739899
$362$		0			0
$363$	−5642.87	−	3317.00i	−0.815907	−	0.479607i
$364$		0			0
$365$			1123.07i			0.161052i
$366$		0			0
$367$		−	5634.42i		−	0.801402i	−0.916209	−	0.400701i	$-0.868767\pi$
$367$		−	5634.42i		−	0.801402i	0.916209	−	0.400701i	$-0.131233\pi$
$368$		0			0
$369$	8237.33	−	4585.62i	1.16211	−	0.646931i
$370$		0			0
$371$	6605.66			0.924390
$372$		0			0
$373$	−6939.30			−0.963280			−0.481640	−	0.876369i	$-0.659959\pi$
$373$	−6939.30			−0.963280			−0.481640	+	0.876369i	$0.659959\pi$
$374$		0			0
$375$	−329.147	+	559.944i	−0.0453256	+	0.0771077i
$376$		0			0
$377$		−	261.597i		−	0.0357373i
$378$		0			0
$379$		−	12425.5i		−	1.68404i	−0.539443	−	0.842022i	$-0.681365\pi$
$379$		−	12425.5i		−	1.68404i	0.539443	−	0.842022i	$-0.318635\pi$
$380$		0			0
$381$	4488.51	−	7635.84i	0.603552	−	1.02676i
$382$		0			0
$383$	4037.90			0.538713			0.269356	−	0.963041i	$-0.413189\pi$
$383$	4037.90			0.538713			0.269356	+	0.963041i	$0.413189\pi$
$384$		0			0
$385$	−1296.02			−0.171561
$386$		0			0
$387$	3815.29	−	2123.92i	0.501142	−	0.278980i
$388$		0			0
$389$			9958.38i			1.29797i	0.760802	+	0.648984i	$0.224806\pi$
$389$			9958.38i			1.29797i	−0.760802	+	0.648984i	$0.775194\pi$
$390$		0			0
$391$			4012.35i			0.518959i
$392$		0			0
$393$	12678.4	+	7452.64i	1.62733	+	0.956580i
$394$		0			0
$395$	5944.09			0.757163
$396$		0			0
$397$	−15000.3			−1.89633			−0.948164	−	0.317781i	$-0.897062\pi$
$397$	−15000.3			−1.89633			−0.948164	+	0.317781i	$0.897062\pi$
$398$		0			0
$399$	5807.92	+	3414.02i	0.728721	+	0.428358i
$400$		0			0
$401$			12157.9i			1.51406i	0.653382	+	0.757029i	$0.273350\pi$
$401$			12157.9i			1.51406i	−0.653382	+	0.757029i	$0.726650\pi$
$402$		0			0
$403$		−	445.702i		−	0.0550918i
$404$		0			0
$405$	3098.14	+	1920.31i	0.380118	+	0.235607i
$406$		0			0
$407$	−2908.36			−0.354207
$408$		0			0
$409$	7049.34			0.852243			0.426121	−	0.904666i	$-0.359880\pi$
$409$	7049.34			0.852243			0.426121	+	0.904666i	$0.359880\pi$
$410$		0			0
$411$	−2165.56	+	3684.04i	−0.259901	+	0.442142i
$412$		0			0
$413$		−	17875.7i		−	2.12980i
$414$		0			0
$415$		−	2129.27i		−	0.251860i
$416$		0			0
$417$	−2472.89	+	4206.87i	−0.290402	+	0.494032i
$418$		0			0
$419$	14345.9			1.67265			0.836327	−	0.548231i	$-0.184699\pi$
$419$	14345.9			1.67265			0.836327	+	0.548231i	$0.184699\pi$
$420$		0			0
$421$	−12852.8			−1.48790			−0.743949	−	0.668236i	$-0.767049\pi$
$421$	−12852.8			−1.48790			−0.743949	+	0.668236i	$0.767049\pi$
$422$		0			0
$423$	6634.60	+	11918.0i	0.762613	+	1.36991i
$424$		0			0
$425$			1328.52i			0.151629i
$426$		0			0
$427$		−	19511.5i		−	2.21131i
$428$		0			0
$429$	53.7723	+	31.6085i	0.00605163	+	0.00355728i
$430$		0			0
$431$	2402.92			0.268548			0.134274	−	0.990944i	$-0.457130\pi$
$431$	2402.92			0.268548			0.134274	+	0.990944i	$0.457130\pi$
$432$		0			0
$433$	−494.140			−0.0548426			−0.0274213	−	0.999624i	$-0.508730\pi$
$433$	−494.140			−0.0548426			−0.0274213	+	0.999624i	$0.508730\pi$
$434$		0			0
$435$	−4121.64	−	2422.79i	−0.454293	−	0.267043i
$436$		0			0
$437$		−	3189.14i		−	0.349101i
$438$		0			0
$439$			6222.73i			0.676525i	0.941052	+	0.338263i	$0.109839\pi$
$439$			6222.73i			0.676525i	−0.941052	+	0.338263i	$0.890161\pi$
$440$		0			0
$441$	−7869.89	−	14137.0i	−0.849789	−	1.52651i
$442$		0			0
$443$	15317.1			1.64275			0.821375	−	0.570389i	$-0.193207\pi$
$443$	15317.1			1.64275			0.821375	+	0.570389i	$0.193207\pi$
$444$		0			0
$445$	−4136.63			−0.440664
$446$		0			0
$447$	4208.12	−	7158.83i	0.445273	−	0.757497i
$448$		0			0
$449$		−	3891.31i		−	0.409003i	−0.978866	−	0.204501i	$-0.934443\pi$
$449$		−	3891.31i		−	0.409003i	0.978866	−	0.204501i	$-0.0655573\pi$
$450$		0			0
$451$		−	2948.48i		−	0.307846i
$452$		0			0
$453$	−6094.40	+	10367.8i	−0.632097	+	1.07532i
$454$		0			0
$455$	−218.184			−0.0224805
$456$		0			0
$457$	2951.67			0.302130			0.151065	−	0.988524i	$-0.451730\pi$
$457$	2951.67			0.302130			0.151065	+	0.988524i	$0.451730\pi$
$458$		0			0
$459$	7453.38	+	174.841i	0.757939	+	0.0177796i
$460$		0			0
$461$			18289.8i			1.84781i	0.382622	+	0.923905i	$0.375021\pi$
$461$			18289.8i			1.84781i	−0.382622	+	0.923905i	$0.624979\pi$
$462$		0			0
$463$		−	5620.10i		−	0.564122i	−0.959397	−	0.282061i	$-0.908982\pi$
$463$		−	5620.10i		−	0.564122i	0.959397	−	0.282061i	$-0.0910180\pi$
$464$		0			0
$465$	−7022.32	−	4127.87i	−0.700328	−	0.411668i
$466$		0			0
$467$	−926.584			−0.0918141			−0.0459070	−	0.998946i	$-0.514618\pi$
$467$	−926.584			−0.0918141			−0.0459070	+	0.998946i	$0.514618\pi$
$468$		0			0
$469$	4391.90			0.432407
$470$		0			0
$471$	8520.09	+	5008.29i	0.833514	+	0.489957i
$472$		0			0
$473$		−	1365.65i		−	0.132754i
$474$		0			0
$475$		−	1055.95i		−	0.102000i
$476$		0			0
$477$	−5076.64	+	2826.10i	−0.487302	+	0.271275i
$478$		0			0
$479$	−9333.96			−0.890354			−0.445177	−	0.895443i	$-0.646859\pi$
$479$	−9333.96			−0.890354			−0.445177	+	0.895443i	$0.646859\pi$
$480$		0			0
$481$	−489.622			−0.0464134
$482$		0			0
$483$	−6102.90	+	10382.2i	−0.574931	+	0.978070i
$484$		0			0
$485$			5208.50i			0.487641i
$486$		0			0
$487$		−	10323.9i		−	0.960619i	−0.877099	−	0.480309i	$-0.840524\pi$
$487$		−	10323.9i		−	0.960619i	0.877099	−	0.480309i	$-0.159476\pi$
$488$		0			0
$489$	3813.99	−	6488.34i	0.352709	−	0.600027i
$490$		0			0
$491$	4918.30			0.452056			0.226028	−	0.974121i	$-0.427426\pi$
$491$	4918.30			0.452056			0.226028	+	0.974121i	$0.427426\pi$
$492$		0			0
$493$	−9778.94			−0.893350
$494$		0			0
$495$	996.025	−	554.475i	0.0904404	−	0.0503471i
$496$		0			0
$497$		−	14528.7i		−	1.31127i
$498$		0			0
$499$			4521.90i			0.405667i	0.979213	+	0.202834i	$0.0650151\pi$
$499$			4521.90i			0.405667i	−0.979213	+	0.202834i	$0.934985\pi$
$500$		0			0
$501$	−12872.8	−	7566.90i	−1.14793	−	0.674779i
$502$		0			0
$503$	−3614.99			−0.320446			−0.160223	−	0.987081i	$-0.551221\pi$
$503$	−3614.99			−0.320446			−0.160223	+	0.987081i	$0.551221\pi$
$504$		0			0
$505$	−5124.68			−0.451575
$506$		0			0
$507$	−9832.52	−	5779.77i	−0.861297	−	0.506289i
$508$		0			0
$509$		−	9543.27i		−	0.831037i	−0.909585	−	0.415519i	$-0.863600\pi$
$509$		−	9543.27i		−	0.831037i	0.909585	−	0.415519i	$-0.136400\pi$
$510$		0			0
$511$			6894.78i			0.596883i
$512$		0			0
$513$	−5924.17	−	138.969i	−0.509861	−	0.0119603i
$514$		0			0
$515$	2548.62			0.218069
$516$		0			0
$517$	4265.94			0.362893
$518$		0			0
$519$	222.359	−	378.277i	0.0188063	−	0.0319933i
$520$		0			0
$521$			5298.70i			0.445567i	0.974868	+	0.222783i	$0.0715143\pi$
$521$			5298.70i			0.445567i	−0.974868	+	0.222783i	$0.928486\pi$
$522$		0			0
$523$		−	7441.69i		−	0.622184i	−0.950380	−	0.311092i	$-0.899305\pi$
$523$		−	7441.69i		−	0.622184i	0.950380	−	0.311092i	$-0.100695\pi$
$524$		0			0
$525$	−2020.71	+	3437.63i	−0.167983	+	0.285772i
$526$		0			0
$527$	−16661.1			−1.37717
$528$		0			0
$529$	−6466.11			−0.531446
$530$		0			0
$531$	7647.77	+	13738.0i	0.625019	+	1.12275i
$532$		0			0
$533$		−	496.375i		−	0.0403384i
$534$		0			0
$535$			1856.96i			0.150063i
$536$		0			0
$537$	5971.89	+	3510.40i	0.479899	+	0.282095i
$538$		0			0
$539$	−5060.21			−0.404376
$540$		0			0
$541$	5059.45			0.402075			0.201038	−	0.979583i	$-0.435569\pi$
$541$	5059.45			0.402075			0.201038	+	0.979583i	$0.435569\pi$
$542$		0			0
$543$	−15947.7	−	9374.39i	−1.26037	−	0.740872i
$544$		0			0
$545$		−	483.818i		−	0.0380266i
$546$		0			0
$547$			10615.4i			0.829769i	0.909874	+	0.414884i	$0.136178\pi$
$547$			10615.4i			0.829769i	−0.909874	+	0.414884i	$0.863822\pi$
$548$		0			0
$549$	8347.63	+	14995.2i	0.648940	+	1.16572i
$550$		0			0
$551$	7772.61			0.600951
$552$		0			0
$553$	36492.2			2.80616
$554$		0			0
$555$	−4534.64	+	7714.31i	−0.346819	+	0.590008i
$556$		0			0
$557$			2043.42i			0.155445i	0.996975	+	0.0777223i	$0.0247648\pi$
$557$			2043.42i			0.155445i	−0.996975	+	0.0777223i	$0.975235\pi$
$558$		0			0
$559$		−	229.906i		−	0.0173954i
$560$		0			0
$561$	1181.58	−	2010.10i	0.0889239	−	0.151277i
$562$		0			0
$563$	16858.7			1.26201			0.631004	−	0.775779i	$-0.282643\pi$
$563$	16858.7			1.26201			0.631004	+	0.775779i	$0.282643\pi$
$564$		0			0
$565$	5139.26			0.382673
$566$		0			0
$567$	19020.2	+	11789.2i	1.40877	+	0.873194i
$568$		0			0
$569$		−	21832.7i		−	1.60857i	−0.594244	−	0.804285i	$-0.702549\pi$
$569$		−	21832.7i		−	1.60857i	0.594244	−	0.804285i	$-0.297451\pi$
$570$		0			0
$571$		−	7921.03i		−	0.580534i	−0.956946	−	0.290267i	$-0.906256\pi$
$571$		−	7921.03i		−	0.580534i	0.956946	−	0.290267i	$-0.0937440\pi$
$572$		0			0
$573$	−5182.61	−	3046.45i	−0.377848	−	0.222107i
$574$		0			0
$575$	1887.61			0.136902
$576$		0			0
$577$	12065.5			0.870526			0.435263	−	0.900303i	$-0.356655\pi$
$577$	12065.5			0.870526			0.435263	+	0.900303i	$0.356655\pi$
$578$		0			0
$579$	4243.62	+	2494.49i	0.304592	+	0.179046i
$580$		0			0
$581$		−	13072.1i		−	0.933429i
$582$		0			0
$583$			1817.14i			0.129088i
$584$		0			0
$585$	167.681	−	93.3457i	0.0118508	−	0.00659721i
$586$		0			0
$587$	−19991.9			−1.40571			−0.702855	−	0.711333i	$-0.748092\pi$
$587$	−19991.9			−1.40571			−0.702855	+	0.711333i	$0.748092\pi$
$588$		0			0
$589$	13242.7			0.926414
$590$		0			0
$591$	4157.05	−	7071.95i	0.289337	−	0.492219i
$592$		0			0
$593$			21435.3i			1.48439i	0.670183	+	0.742196i	$0.266215\pi$
$593$			21435.3i			1.48439i	−0.670183	+	0.742196i	$0.733785\pi$
$594$		0			0
$595$			8156.08i			0.561961i
$596$		0			0
$597$	−8923.22	+	15180.2i	−0.611730	+	1.04067i
$598$		0			0
$599$	−13834.2			−0.943655			−0.471828	−	0.881691i	$-0.656406\pi$
$599$	−13834.2			−0.943655			−0.471828	+	0.881691i	$0.656406\pi$
$600$		0			0
$601$	8360.75			0.567458			0.283729	−	0.958905i	$-0.408428\pi$
$601$	8360.75			0.567458			0.283729	+	0.958905i	$0.408428\pi$
$602$		0			0
$603$	−3375.30	+	1878.99i	−0.227948	+	0.126896i
$604$		0			0
$605$			6298.48i			0.423256i
$606$		0			0
$607$		−	21745.8i		−	1.45410i	−0.686587	−	0.727048i	$-0.740892\pi$
$607$		−	21745.8i		−	1.45410i	0.686587	−	0.727048i	$-0.259108\pi$
$608$		0			0
$609$	−25303.7	−	14874.1i	−1.68368	−	0.989701i
$610$		0			0
$611$	718.169			0.0475516
$612$		0			0
$613$	24312.8			1.60193			0.800966	−	0.598709i	$-0.204320\pi$
$613$	24312.8			1.60193			0.800966	+	0.598709i	$0.204320\pi$
$614$		0			0
$615$	−7820.71	−	4597.18i	−0.512783	−	0.301425i
$616$		0			0
$617$			19786.2i			1.29102i	0.763750	+	0.645512i	$0.223356\pi$
$617$			19786.2i			1.29102i	−0.763750	+	0.645512i	$0.776644\pi$
$618$		0			0
$619$		−	3627.94i		−	0.235572i	−0.993039	−	0.117786i	$-0.962420\pi$
$619$		−	3627.94i		−	0.235572i	0.993039	−	0.117786i	$-0.0375798\pi$
$620$		0			0
$621$	248.420	−	10590.0i	0.0160527	−	0.684322i
$622$		0			0
$623$	−25395.8			−1.63316
$624$		0			0
$625$	625.000			0.0400000
$626$		0			0
$627$	−939.155	+	1597.69i	−0.0598186	+	0.101763i
$628$		0			0
$629$			18302.9i			1.16023i
$630$		0			0
$631$			3799.85i			0.239730i	0.992790	+	0.119865i	$0.0382461\pi$
$631$			3799.85i			0.239730i	−0.992790	+	0.119865i	$0.961754\pi$
$632$		0			0
$633$	−11300.9	+	19225.1i	−0.709591	+	1.20715i
$634$		0			0
$635$	−8523.00			−0.532638
$636$		0			0
$637$	−851.885			−0.0529873
$638$		0			0
$639$	6215.83	+	11165.7i	0.384811	+	0.691251i
$640$		0			0
$641$			17703.6i			1.09088i	0.838151	+	0.545438i	$0.183637\pi$
$641$			17703.6i			1.09088i	−0.838151	+	0.545438i	$0.816363\pi$
$642$		0			0
$643$		−	5277.19i		−	0.323658i	−0.986819	−	0.161829i	$-0.948261\pi$
$643$		−	5277.19i		−	0.323658i	0.986819	−	0.161829i	$-0.0517393\pi$
$644$		0			0
$645$	−3622.33	−	2129.28i	−0.221130	−	0.129985i
$646$		0			0
$647$	15962.4			0.969931			0.484966	−	0.874533i	$-0.338832\pi$
$647$	15962.4			0.969931			0.484966	+	0.874533i	$0.338832\pi$
$648$		0			0
$649$	4917.39			0.297418
$650$		0			0
$651$	−43111.7	−	25342.0i	−2.59552	−	1.52570i
$652$		0			0
$653$		−	21347.1i		−	1.27929i	−0.768671	−	0.639645i	$-0.779081\pi$
$653$		−	21347.1i		−	1.27929i	0.768671	−	0.639645i	$-0.220919\pi$
$654$		0			0
$655$		−	14151.4i		−	0.844186i
$656$		0			0
$657$	−2949.80	−	5298.84i	−0.175164	−	0.314653i
$658$		0			0
$659$	−4923.78			−0.291052			−0.145526	−	0.989354i	$-0.546487\pi$
$659$	−4923.78			−0.291052			−0.145526	+	0.989354i	$0.546487\pi$
$660$		0			0
$661$	24819.3			1.46045			0.730226	−	0.683206i	$-0.239415\pi$
$661$	24819.3			1.46045			0.730226	+	0.683206i	$0.239415\pi$
$662$		0			0
$663$	198.918	−	338.399i	0.0116521	−	0.0198225i
$664$		0			0
$665$		−	6482.70i		−	0.378028i
$666$		0			0
$667$			13894.3i			0.806581i
$668$		0			0
$669$	−9801.59	+	16674.4i	−0.566444	+	0.963633i
$670$		0			0
$671$	5367.39			0.308801
$672$		0			0
$673$	1774.38			0.101631			0.0508154	−	0.998708i	$-0.483818\pi$
$673$	1774.38			0.101631			0.0508154	+	0.998708i	$0.483818\pi$
$674$		0			0
$675$	82.2537	−	3506.44i	0.00469029	−	0.199945i
$676$		0			0
$677$		−	24981.1i		−	1.41817i	−0.705122	−	0.709086i	$-0.749108\pi$
$677$		−	24981.1i		−	1.41817i	0.705122	−	0.709086i	$-0.250892\pi$
$678$		0			0
$679$			31976.2i			1.80727i
$680$		0			0
$681$	13381.4	+	7865.85i	0.752973	+	0.442614i
$682$		0			0
$683$	−5711.76			−0.319992			−0.159996	−	0.987118i	$-0.551148\pi$
$683$	−5711.76			−0.319992			−0.159996	+	0.987118i	$0.551148\pi$
$684$		0			0
$685$	4112.06			0.229363
$686$		0			0
$687$	23705.6	+	13934.7i	1.31648	+	0.773858i
$688$		0			0
$689$			305.914i			0.0169150i
$690$		0			0
$691$		−	6656.68i		−	0.366472i	−0.983069	−	0.183236i	$-0.941343\pi$
$691$		−	6656.68i		−	0.366472i	0.983069	−	0.183236i	$-0.0586573\pi$
$692$		0			0
$693$	6114.84	−	3404.05i	0.335185	−	0.186594i
$694$		0			0
$695$	4695.63			0.256281
$696$		0			0
$697$	−18555.3			−1.00837
$698$		0			0
$699$	−16528.7	+	28118.6i	−0.894383	+	1.52152i
$700$		0			0
$701$			7363.29i			0.396730i	0.980128	+	0.198365i	$0.0635631\pi$
$701$			7363.29i			0.396730i	−0.980128	+	0.198365i	$0.936437\pi$
$702$		0			0
$703$		−	14547.7i		−	0.780479i
$704$		0			0
$705$	6651.33	−	11315.2i	0.355324	−	0.604476i
$706$		0			0
$707$	−31461.6			−1.67360
$708$		0			0
$709$	−5408.48			−0.286487			−0.143244	−	0.989687i	$-0.545753\pi$
$709$	−5408.48			−0.286487			−0.143244	+	0.989687i	$0.545753\pi$
$710$		0			0
$711$	−28045.3	+	15612.5i	−1.47930	+	0.823506i
$712$		0			0
$713$			23672.7i			1.24341i
$714$		0			0
$715$		−	60.0197i		−	0.00313932i
$716$		0			0
$717$	4302.15	+	2528.90i	0.224082	+	0.131720i
$718$		0			0
$719$	−33674.5			−1.74666			−0.873329	−	0.487131i	$-0.838043\pi$
$719$	−33674.5			−1.74666			−0.873329	+	0.487131i	$0.838043\pi$
$720$		0			0
$721$	15646.6			0.808196
$722$		0			0
$723$	31504.2	+	18518.9i	1.62055	+	0.952592i
$724$		0			0
$725$			4600.50i			0.235667i
$726$		0			0
$727$		−	355.017i		−	0.0181112i	−0.999959	−	0.00905559i	$-0.997117\pi$
$727$		−	355.017i		−	0.0181112i	0.999959	−	0.00905559i	$-0.00288252\pi$
$728$		0			0
$729$	−19661.3	−	922.936i	−0.998900	−	0.0468900i
$730$		0			0
$731$	−8594.29			−0.434844
$732$		0			0
$733$	−13274.7			−0.668911			−0.334456	−	0.942411i	$-0.608552\pi$
$733$	−13274.7			−0.668911			−0.334456	+	0.942411i	$0.608552\pi$
$734$		0			0
$735$	−7889.75	+	13422.0i	−0.395942	+	0.673576i
$736$		0			0
$737$			1208.16i			0.0603841i
$738$		0			0
$739$			2921.85i			0.145443i	0.997352	+	0.0727213i	$0.0231683\pi$
$739$			2921.85i			0.145443i	−0.997352	+	0.0727213i	$0.976832\pi$
$740$		0			0
$741$	−158.107	+	268.970i	−0.00783831	+	0.0133345i
$742$		0			0
$743$	1539.97			0.0760377			0.0380189	−	0.999277i	$-0.487895\pi$
$743$	1539.97			0.0760377			0.0380189	+	0.999277i	$0.487895\pi$
$744$		0			0
$745$	−7990.57			−0.392955
$746$		0			0
$747$	5592.64	+	10046.3i	0.273928	+	0.492067i
$748$		0			0
$749$			11400.3i			0.556154i
$750$		0			0
$751$		−	28405.8i		−	1.38022i	−0.723707	−	0.690108i	$-0.757563\pi$
$751$		−	28405.8i		−	1.38022i	0.723707	−	0.690108i	$-0.242437\pi$
$752$		0			0
$753$	15254.5	+	8966.94i	0.738254	+	0.433962i
$754$		0			0
$755$	11572.3			0.557829
$756$		0			0
$757$	24017.6			1.15315			0.576575	−	0.817044i	$-0.304389\pi$
$757$	24017.6			1.15315			0.576575	+	0.817044i	$0.304389\pi$
$758$		0			0
$759$	−2856.02	−	1678.83i	−0.136584	−	0.0802869i
$760$		0			0
$761$			27752.8i			1.32200i	0.750388	+	0.660998i	$0.229867\pi$
$761$			27752.8i			1.32200i	−0.750388	+	0.660998i	$0.770133\pi$
$762$		0			0
$763$		−	2970.28i		−	0.140932i
$764$		0			0
$765$	−3489.42	−	6268.18i	−0.164915	−	0.296244i
$766$		0			0
$767$	827.842			0.0389721
$768$		0			0
$769$	−32632.8			−1.53026			−0.765129	−	0.643877i	$-0.777325\pi$
$769$	−32632.8			−1.53026			−0.765129	+	0.643877i	$0.777325\pi$
$770$		0			0
$771$	−8948.09	+	15222.5i	−0.417973	+	0.711055i
$772$		0			0
$773$			1400.10i			0.0651463i	0.999469	+	0.0325731i	$0.0103702\pi$
$773$			1400.10i			0.0651463i	−0.999469	+	0.0325731i	$0.989630\pi$
$774$		0			0
$775$			7838.20i			0.363299i
$776$		0			0
$777$	−27839.2	+	47360.0i	−1.28536	+	2.18666i
$778$		0			0
$779$	14748.3			0.678324
$780$		0			0
$781$	3996.67			0.183114
$782$		0			0
$783$	25810.2	+	605.453i	1.17801	+	0.0276336i
$784$		0			0
$785$		−	9509.98i		−	0.432390i
$786$		0			0
$787$			13989.4i			0.633634i	0.948487	+	0.316817i	$0.102614\pi$
$787$			13989.4i			0.633634i	−0.948487	+	0.316817i	$0.897386\pi$
$788$		0			0
$789$	9071.17	+	5332.23i	0.409305	+	0.240599i
$790$		0			0
$791$	31551.2			1.41824
$792$		0			0
$793$	903.598			0.0404637
$794$		0			0
$795$	4819.88	+	2833.23i	0.215023	+	0.126395i
$796$		0			0
$797$			24386.5i			1.08383i	0.840432	+	0.541917i	$0.182301\pi$
$797$			24386.5i			1.08383i	−0.840432	+	0.541917i	$0.817699\pi$
$798$		0			0
$799$		−	26846.4i		−	1.18868i
$800$		0			0
$801$	19517.4	−	10865.1i	0.860940	−	0.479275i
$802$		0			0
$803$	−1896.67			−0.0833525
$804$		0			0
$805$	11588.5			0.507379
$806$		0			0
$807$	−9744.80	+	16577.8i	−0.425072	+	0.723131i
$808$		0			0
$809$		−	24639.1i		−	1.07079i	−0.844603	−	0.535393i	$-0.820164\pi$
$809$		−	24639.1i		−	1.07079i	0.844603	−	0.535393i	$-0.179836\pi$
$810$		0			0
$811$		−	11558.0i		−	0.500438i	−0.968189	−	0.250219i	$-0.919497\pi$
$811$		−	11558.0i		−	0.500438i	0.968189	−	0.250219i	$-0.0805026\pi$
$812$		0			0
$813$	14232.4	−	24212.1i	0.613963	−	1.04447i
$814$		0			0
$815$	−7242.18			−0.311267
$816$		0			0
$817$	6831.01			0.292517
$818$		0			0
$819$	1029.43	−	573.071i	0.0439209	−	0.0244502i
$820$		0			0
$821$			11931.0i			0.507182i	0.967312	+	0.253591i	$0.0816117\pi$
$821$			11931.0i			0.507182i	−0.967312	+	0.253591i	$0.918388\pi$
$822$		0			0
$823$		−	29996.1i		−	1.27047i	−0.772319	−	0.635235i	$-0.780903\pi$
$823$		−	29996.1i		−	1.27047i	0.772319	−	0.635235i	$-0.219097\pi$
$824$		0			0
$825$	−945.650	−	555.873i	−0.0399070	−	0.0234582i
$826$		0			0
$827$	43041.2			1.80978			0.904891	−	0.425644i	$-0.139952\pi$
$827$	43041.2			1.80978			0.904891	+	0.425644i	$0.139952\pi$
$828$		0			0
$829$	11938.5			0.500172			0.250086	−	0.968224i	$-0.419541\pi$
$829$	11938.5			0.500172			0.250086	+	0.968224i	$0.419541\pi$
$830$		0			0
$831$	−13112.6	−	7707.87i	−0.547378	−	0.321760i
$832$		0			0
$833$			31844.9i			1.32456i
$834$		0			0
$835$			14368.4i			0.595495i
$836$		0			0
$837$	43974.7	+	1031.55i	1.81599	+	0.0425994i
$838$		0			0
$839$	24079.1			0.990825			0.495412	−	0.868658i	$-0.335017\pi$
$839$	24079.1			0.990825			0.495412	+	0.868658i	$0.335017\pi$
$840$		0			0
$841$	−9474.37			−0.388469
$842$		0			0
$843$	10041.9	−	17083.3i	0.410276	−	0.697959i
$844$		0			0
$845$			10974.9i			0.446802i
$846$		0			0
$847$			38667.9i			1.56865i
$848$		0			0
$849$	−20094.8	+	34185.2i	−0.812311	+	1.38190i
$850$		0			0
$851$	26005.4			1.04754
$852$		0			0
$853$	−20697.6			−0.830801			−0.415400	−	0.909639i	$-0.636358\pi$
$853$	−20697.6			−0.830801			−0.415400	+	0.909639i	$0.636358\pi$
$854$		0			0
$855$	2773.50	+	4982.14i	0.110938	+	0.199281i
$856$		0			0
$857$		−	40049.1i		−	1.59633i	−0.602441	−	0.798163i	$-0.705805\pi$
$857$		−	40049.1i		−	1.59633i	0.602441	−	0.798163i	$-0.294195\pi$
$858$		0			0
$859$		−	36206.3i		−	1.43812i	−0.694948	−	0.719060i	$-0.744573\pi$
$859$		−	36206.3i		−	1.43812i	0.694948	−	0.719060i	$-0.255427\pi$
$860$		0			0
$861$	−48013.2	−	28223.2i	−1.90045	−	1.11712i
$862$		0			0
$863$	−16377.6			−0.646002			−0.323001	−	0.946399i	$-0.604692\pi$
$863$	−16377.6			−0.646002			−0.323001	+	0.946399i	$0.604692\pi$
$864$		0			0
$865$	−422.226			−0.0165967
$866$		0			0
$867$	9358.10	+	5500.89i	0.366572	+	0.215479i
$868$		0			0
$869$			10038.6i			0.391869i
$870$		0			0
$871$			203.393i			0.00791241i
$872$		0			0
$873$	−13680.4	−	24574.6i	−0.530368	−	0.952721i
$874$		0			0
$875$	3837.03			0.148246
$876$		0			0
$877$	20486.3			0.788795			0.394397	−	0.918940i	$-0.370953\pi$
$877$	20486.3			0.788795			0.394397	+	0.918940i	$0.370953\pi$
$878$		0			0
$879$	8745.56	−	14877.9i	0.335586	−	0.570898i
$880$		0			0
$881$			5581.92i			0.213461i	0.994288	+	0.106731i	$0.0340383\pi$
$881$			5581.92i			0.213461i	−0.994288	+	0.106731i	$0.965962\pi$
$882$		0			0
$883$			13037.8i			0.496892i	0.968646	+	0.248446i	$0.0799199\pi$
$883$			13037.8i			0.496892i	−0.968646	+	0.248446i	$0.920080\pi$
$884$		0			0
$885$	7667.06	−	13043.2i	0.291215	−	0.495414i
$886$		0			0
$887$	−25693.7			−0.972614			−0.486307	−	0.873788i	$-0.661656\pi$
$887$	−25693.7			−0.972614			−0.486307	+	0.873788i	$0.661656\pi$
$888$		0			0
$889$	−52324.7			−1.97403
$890$		0			0
$891$	−3243.07	+	5232.22i	−0.121938	+	0.196730i
$892$		0			0
$893$			21338.3i			0.799619i
$894$		0			0
$895$		−	6665.72i		−	0.248950i
$896$		0			0
$897$	−480.811	−	282.631i	−0.0178972	−	0.0105204i
$898$		0			0
$899$	−57695.4			−2.14043
$900$		0			0
$901$	11435.6			0.422835
$902$		0			0
$903$	−22238.3	−	13072.2i	−0.819541	−	0.481744i
$904$		0			0
$905$			17800.5i			0.653823i
$906$		0			0
$907$		−	10443.8i		−	0.382339i	−0.981557	−	0.191170i	$-0.938772\pi$
$907$		−	10443.8i		−	0.382339i	0.981557	−	0.191170i	$-0.0612280\pi$
$908$		0			0
$909$	24179.2	−	13460.2i	0.882257	−	0.491142i
$910$		0			0
$911$	29090.8			1.05798			0.528991	−	0.848627i	$-0.322571\pi$
$911$	29090.8			1.05798			0.528991	+	0.848627i	$0.322571\pi$
$912$		0			0
$913$	3595.98			0.130350
$914$		0			0
$915$	8368.68	−	14236.8i	0.302361	−	0.514375i
$916$		0			0
$917$		−	86878.9i		−	3.12868i
$918$		0			0
$919$			8558.40i			0.307199i	0.988133	+	0.153599i	$0.0490865\pi$
$919$			8558.40i			0.307199i	−0.988133	+	0.153599i	$0.950914\pi$
$920$		0			0
$921$	−815.902	+	1388.01i	−0.0291910	+	0.0496596i
$922$		0			0
$923$	672.839			0.0239943
$924$		0			0
$925$	8610.59			0.306070
$926$		0			0
$927$	−12024.8	+	6694.08i	−0.426049	+	0.237176i
$928$		0			0
$929$		−	24717.9i		−	0.872948i	−0.899717	−	0.436474i	$-0.856227\pi$
$929$		−	24717.9i		−	0.872948i	0.899717	−	0.436474i	$-0.143773\pi$
$930$		0			0
$931$		−	25311.3i		−	0.891025i
$932$		0			0
$933$	−34546.6	−	20307.3i	−1.21223	−	0.712572i
$934$		0			0
$935$	−2243.64			−0.0784757
$936$		0			0
$937$	55762.6			1.94417			0.972083	−	0.234638i	$-0.0753905\pi$
$937$	55762.6			1.94417			0.972083	+	0.234638i	$0.0753905\pi$
$938$		0			0
$939$	−1367.36	−	803.765i	−0.0475209	−	0.0279338i
$940$		0			0
$941$		−	24516.0i		−	0.849306i	−0.905356	−	0.424653i	$-0.860396\pi$
$941$		−	24516.0i		−	0.849306i	0.905356	−	0.424653i	$-0.139604\pi$
$942$		0			0
$943$			26364.1i			0.910428i
$944$		0			0
$945$	504.975	−	21526.9i	0.0173829	−	0.741025i
$946$		0			0
$947$	15597.2			0.535207			0.267603	−	0.963529i	$-0.413768\pi$
$947$	15597.2			0.535207			0.267603	+	0.963529i	$0.413768\pi$
$948$		0			0
$949$	−319.304			−0.0109221
$950$		0			0
$951$	−18285.1	+	31106.5i	−0.623485	+	1.06067i
$952$		0			0
$953$		−	10953.5i		−	0.372317i	−0.982520	−	0.186159i	$-0.940396\pi$
$953$		−	10953.5i		−	0.372317i	0.982520	−	0.186159i	$-0.0596038\pi$
$954$		0			0
$955$			5784.75i			0.196010i
$956$		0			0
$957$	4091.67	−	6960.74i	0.138208	−	0.235119i
$958$		0			0
$959$	25244.9			0.850054
$960$		0			0
$961$	−68508.8			−2.29965
$962$		0			0
$963$	−4877.41	−	8761.49i	−0.163211	−	0.293183i
$964$		0			0
$965$		−	4736.66i		−	0.158009i
$966$		0			0
$967$		−	3021.09i		−	0.100467i	−0.998737	−	0.0502336i	$-0.984003\pi$
$967$		−	3021.09i		−	0.100467i	0.998737	−	0.0502336i	$-0.0159966\pi$
$968$		0			0
$969$	10054.6	+	5910.29i	0.333332	+	0.195940i
$970$		0			0
$971$	800.261			0.0264486			0.0132243	−	0.999913i	$-0.495790\pi$
$971$	800.261			0.0264486			0.0132243	+	0.999913i	$0.495790\pi$
$972$		0			0
$973$	28827.6			0.949816
$974$		0			0
$975$	−159.200	−	93.5811i	−0.00522921	−	0.00307384i
$976$		0			0
$977$		−	45423.5i		−	1.48744i	−0.668493	−	0.743719i	$-0.733060\pi$
$977$		−	45423.5i		−	1.48744i	0.668493	−	0.743719i	$-0.266940\pi$
$978$		0			0
$979$		−	6986.07i		−	0.228065i
$980$		0			0
$981$	1270.77	+	2282.74i	0.0413585	+	0.0742939i
$982$		0			0
$983$	−23420.3			−0.759910			−0.379955	−	0.925005i	$-0.624061\pi$
$983$	−23420.3			−0.759910			−0.379955	+	0.925005i	$0.624061\pi$
$984$		0			0
$985$	−7893.59			−0.255341
$986$		0			0
$987$	40834.1	−	69466.9i	1.31688	−	2.24028i
$988$		0			0
$989$			12211.1i			0.392609i
$990$		0			0
$991$			44803.2i			1.43615i	0.695968	+	0.718073i	$0.254976\pi$
$991$			44803.2i			1.43615i	−0.695968	+	0.718073i	$0.745024\pi$
$992$		0			0
$993$	−4617.45	+	7855.19i	−0.147563	+	0.251034i
$994$		0			0
$995$	16943.8			0.539855
$996$		0			0
$997$	−31601.9			−1.00385			−0.501927	−	0.864910i	$-0.667375\pi$
$997$	−31601.9			−1.00385			−0.501927	+	0.864910i	$0.667375\pi$
$998$		0			0
$999$	1133.20	−	48308.0i	0.0358889	−	1.52993i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.h.a.191.22	yes	24
3.2	odd	2		480.4.h.b.191.4	yes	24
4.3	odd	2		480.4.h.b.191.3	yes	24
8.3	odd	2		960.4.h.c.191.22		24
8.5	even	2		960.4.h.e.191.3		24
12.11	even	2	inner	480.4.h.a.191.21	✓	24
24.5	odd	2		960.4.h.c.191.21		24
24.11	even	2		960.4.h.e.191.4		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.h.a.191.21	✓	24	12.11	even	2	inner
480.4.h.a.191.22	yes	24	1.1	even	1	trivial
480.4.h.b.191.3	yes	24	4.3	odd	2
480.4.h.b.191.4	yes	24	3.2	odd	2
960.4.h.c.191.21		24	24.5	odd	2
960.4.h.c.191.22		24	8.3	odd	2
960.4.h.e.191.3		24	8.5	even	2
960.4.h.e.191.4		24	24.11	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)

\(f(q)\)	\(=\)	\(q+(4.47955 + 2.63318i) q^{3} -5.00000i q^{5} -30.6962i q^{7} +(13.1328 + 23.5909i) q^{9} +8.44415 q^{11} +1.42157 q^{13} +(13.1659 - 22.3978i) q^{15} -53.1406i q^{17} +42.2378i q^{19} +(80.8285 - 137.505i) q^{21} -75.5043 q^{23} -25.0000 q^{25} +(-3.29015 + 140.258i) q^{27} -184.020i q^{29} -313.528i q^{31} +(37.8260 + 22.2349i) q^{33} -153.481 q^{35} -344.423 q^{37} +(6.36799 + 3.74324i) q^{39} -349.174i q^{41} -161.727i q^{43} +(117.955 - 65.6638i) q^{45} +505.194 q^{47} -599.257 q^{49} +(139.929 - 238.046i) q^{51} +215.195i q^{53} -42.2207i q^{55} +(-111.220 + 189.207i) q^{57} +582.343 q^{59} +635.634 q^{61} +(724.151 - 403.126i) q^{63} -7.10785i q^{65} +143.076i q^{67} +(-338.225 - 198.816i) q^{69} +473.307 q^{71} -224.614 q^{73} +(-111.989 - 65.8294i) q^{75} -259.203i q^{77} +1188.82i q^{79} +(-384.061 + 619.627i) q^{81} +425.854 q^{83} -265.703 q^{85} +(484.557 - 824.327i) q^{87} -827.327i q^{89} -43.6368i q^{91} +(825.575 - 1404.46i) q^{93} +211.189 q^{95} -1041.70 q^{97} +(110.895 + 199.205i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 20 q^{9} - 72 q^{11} - 72 q^{13} - 20 q^{15} - 68 q^{21} - 96 q^{23} - 600 q^{25} - 168 q^{27} - 80 q^{33} - 504 q^{37} + 456 q^{39} - 220 q^{45} - 432 q^{47} - 816 q^{49} - 1240 q^{51} + 40 q^{57}+ \cdots - 3160 q^{99}+O(q^{100}) \) 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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