LMFDB - Embedded newform 1280.4.d.g.641.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1280,4,Mod(641,1280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1280.641");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1280 = 2^{8} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1280.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$75.5224448073$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			641.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	1280.641
Dual form			1280.4.d.g.641.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.00000i q^{3} +5.00000i q^{5} -4.00000 q^{7} -37.0000 q^{9} +O(q^{10})$	$q-8.00000i q^{3} +5.00000i q^{5} -4.00000 q^{7} -37.0000 q^{9} -12.0000i q^{11} +58.0000i q^{13} +40.0000 q^{15} +66.0000 q^{17} -100.000i q^{19} +32.0000i q^{21} +132.000 q^{23} -25.0000 q^{25} +80.0000i q^{27} +90.0000i q^{29} -152.000 q^{31} -96.0000 q^{33} -20.0000i q^{35} -34.0000i q^{37} +464.000 q^{39} +438.000 q^{41} -32.0000i q^{43} -185.000i q^{45} +204.000 q^{47} -327.000 q^{49} -528.000i q^{51} +222.000i q^{53} +60.0000 q^{55} -800.000 q^{57} -420.000i q^{59} -902.000i q^{61} +148.000 q^{63} -290.000 q^{65} -1024.00i q^{67} -1056.00i q^{69} +432.000 q^{71} -362.000 q^{73} +200.000i q^{75} +48.0000i q^{77} +160.000 q^{79} -359.000 q^{81} +72.0000i q^{83} +330.000i q^{85} +720.000 q^{87} -810.000 q^{89} -232.000i q^{91} +1216.00i q^{93} +500.000 q^{95} +1106.00 q^{97} +444.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{7} - 74 q^{9}+O(q^{10}) $ 2 * q - 8 * q^7 - 74 * q^9	$ 2 q - 8 q^{7} - 74 q^{9} + 80 q^{15} + 132 q^{17} + 264 q^{23} - 50 q^{25} - 304 q^{31} - 192 q^{33} + 928 q^{39} + 876 q^{41} + 408 q^{47} - 654 q^{49} + 120 q^{55} - 1600 q^{57} + 296 q^{63} - 580 q^{65} + 864 q^{71} - 724 q^{73} + 320 q^{79} - 718 q^{81} + 1440 q^{87} - 1620 q^{89} + 1000 q^{95} + 2212 q^{97}+O(q^{100}) $ 2 * q - 8 * q^7 - 74 * q^9 + 80 * q^15 + 132 * q^17 + 264 * q^23 - 50 * q^25 - 304 * q^31 - 192 * q^33 + 928 * q^39 + 876 * q^41 + 408 * q^47 - 654 * q^49 + 120 * q^55 - 1600 * q^57 + 296 * q^63 - 580 * q^65 + 864 * q^71 - 724 * q^73 + 320 * q^79 - 718 * q^81 + 1440 * q^87 - 1620 * q^89 + 1000 * q^95 + 2212 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1280\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	− 8.00000i	− 1.53960i	−0.638285	−	0.769800i	$-0.720356\pi$
$3$	− 8.00000i	− 1.53960i	0.638285	−	0.769800i	$-0.279644\pi$
$4$	0	0
$5$	5.00000i	0.447214i
$5$	5.00000i	0.447214i
$6$	0	0
$7$	−4.00000	−0.215980	−0.107990	−	0.994152i	$-0.534441\pi$
$7$	−4.00000	−0.215980	−0.107990	+	0.994152i	$0.534441\pi$
$8$	0	0
$9$	−37.0000	−1.37037
$10$	0	0
$11$	− 12.0000i	− 0.328921i	−0.986384	−	0.164461i	$-0.947412\pi$
$11$	− 12.0000i	− 0.328921i	0.986384	−	0.164461i	$-0.0525884\pi$
$12$	0	0
$13$	58.0000i	1.23741i	0.785624	+	0.618704i	$0.212342\pi$
$13$	58.0000i	1.23741i	−0.785624	+	0.618704i	$0.787658\pi$
$14$	0	0
$15$	40.0000	0.688530
$16$	0	0
$17$	66.0000	0.941609	0.470804	−	0.882238i	$-0.343964\pi$
$17$	66.0000	0.941609	0.470804	+	0.882238i	$0.343964\pi$
$18$	0	0
$19$	− 100.000i	− 1.20745i	−0.797192	−	0.603726i	$-0.793682\pi$
$19$	− 100.000i	− 1.20745i	0.797192	−	0.603726i	$-0.206318\pi$
$20$	0	0
$21$	32.0000i	0.332522i
$22$	0	0
$23$	132.000	1.19669	0.598346	−	0.801238i	$-0.295825\pi$
$23$	132.000	1.19669	0.598346	+	0.801238i	$0.295825\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	80.0000i	0.570222i
$28$	0	0
$29$	90.0000i	0.576296i	0.957586	+	0.288148i	$0.0930395\pi$
$29$	90.0000i	0.576296i	−0.957586	+	0.288148i	$0.906961\pi$
$30$	0	0
$31$	−152.000	−0.880645	−0.440323	−	0.897840i	$-0.645136\pi$
$31$	−152.000	−0.880645	−0.440323	+	0.897840i	$0.645136\pi$
$32$	0	0
$33$	−96.0000	−0.506408
$34$	0	0
$35$	− 20.0000i	− 0.0965891i
$36$	0	0
$37$	− 34.0000i	− 0.151069i	−0.997143	−	0.0755347i	$-0.975934\pi$
$37$	− 34.0000i	− 0.151069i	0.997143	−	0.0755347i	$-0.0240664\pi$
$38$	0	0
$39$	464.000	1.90511
$40$	0	0
$41$	438.000	1.66839	0.834196	−	0.551467i	$-0.185932\pi$
$41$	438.000	1.66839	0.834196	+	0.551467i	$0.185932\pi$
$42$	0	0
$43$	− 32.0000i	− 0.113487i	−0.998389	−	0.0567437i	$-0.981928\pi$
$43$	− 32.0000i	− 0.113487i	0.998389	−	0.0567437i	$-0.0180718\pi$
$44$	0	0
$45$	− 185.000i	− 0.612848i
$46$	0	0
$47$	204.000	0.633116	0.316558	−	0.948573i	$-0.397473\pi$
$47$	204.000	0.633116	0.316558	+	0.948573i	$0.397473\pi$
$48$	0	0
$49$	−327.000	−0.953353
$50$	0	0
$51$	− 528.000i	− 1.44970i
$52$	0	0
$53$	222.000i	0.575359i	0.957727	+	0.287680i	$0.0928838\pi$
$53$	222.000i	0.575359i	−0.957727	+	0.287680i	$0.907116\pi$
$54$	0	0
$55$	60.0000	0.147098
$56$	0	0
$57$	−800.000	−1.85899
$58$	0	0
$59$	− 420.000i	− 0.926769i	−0.886157	−	0.463384i	$-0.846635\pi$
$59$	− 420.000i	− 0.926769i	0.886157	−	0.463384i	$-0.153365\pi$
$60$	0	0
$61$	− 902.000i	− 1.89327i	−0.322312	−	0.946633i	$-0.604460\pi$
$61$	− 902.000i	− 1.89327i	0.322312	−	0.946633i	$-0.395540\pi$
$62$	0	0
$63$	148.000	0.295972
$64$	0	0
$65$	−290.000	−0.553386
$66$	0	0
$67$	− 1024.00i	− 1.86719i	−0.358334	−	0.933593i	$-0.616655\pi$
$67$	− 1024.00i	− 1.86719i	0.358334	−	0.933593i	$-0.383345\pi$
$68$	0	0
$69$	− 1056.00i	− 1.84243i
$70$	0	0
$71$	432.000	0.722098	0.361049	−	0.932547i	$-0.382419\pi$
$71$	432.000	0.722098	0.361049	+	0.932547i	$0.382419\pi$
$72$	0	0
$73$	−362.000	−0.580396	−0.290198	−	0.956967i	$-0.593721\pi$
$73$	−362.000	−0.580396	−0.290198	+	0.956967i	$0.593721\pi$
$74$	0	0
$75$	200.000i	0.307920i
$76$	0	0
$77$	48.0000i	0.0710404i
$78$	0	0
$79$	160.000	0.227866	0.113933	−	0.993488i	$-0.463655\pi$
$79$	160.000	0.227866	0.113933	+	0.993488i	$0.463655\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	72.0000i	0.0952172i	0.998866	+	0.0476086i	$0.0151600\pi$
$83$	72.0000i	0.0952172i	−0.998866	+	0.0476086i	$0.984840\pi$
$84$	0	0
$85$	330.000i	0.421100i
$86$	0	0
$87$	720.000	0.887266
$88$	0	0
$89$	−810.000	−0.964717	−0.482359	−	0.875974i	$-0.660220\pi$
$89$	−810.000	−0.964717	−0.482359	+	0.875974i	$0.660220\pi$
$90$	0	0
$91$	− 232.000i	− 0.267255i
$92$	0	0
$93$	1216.00i	1.35584i
$94$	0	0
$95$	500.000	0.539989
$96$	0	0
$97$	1106.00	1.15770	0.578852	−	0.815433i	$-0.303501\pi$
$97$	1106.00	1.15770	0.578852	+	0.815433i	$0.303501\pi$
$98$	0	0
$99$	444.000i	0.450744i
$100$	0	0
$101$	− 258.000i	− 0.254178i	−0.991891	−	0.127089i	$-0.959437\pi$
$101$	− 258.000i	− 0.254178i	0.991891	−	0.127089i	$-0.0405634\pi$
$102$	0	0
$103$	−988.000	−0.945151	−0.472575	−	0.881290i	$-0.656676\pi$
$103$	−988.000	−0.945151	−0.472575	+	0.881290i	$0.656676\pi$
$104$	0	0
$105$	−160.000	−0.148709
$106$	0	0
$107$	24.0000i	0.0216838i	0.999941	+	0.0108419i	$0.00345115\pi$
$107$	24.0000i	0.0216838i	−0.999941	+	0.0108419i	$0.996549\pi$
$108$	0	0
$109$	− 950.000i	− 0.834803i	−0.908722	−	0.417401i	$-0.862941\pi$
$109$	− 950.000i	− 0.834803i	0.908722	−	0.417401i	$-0.137059\pi$
$110$	0	0
$111$	−272.000	−0.232586
$112$	0	0
$113$	−1038.00	−0.864131	−0.432066	−	0.901842i	$-0.642215\pi$
$113$	−1038.00	−0.864131	−0.432066	+	0.901842i	$0.642215\pi$
$114$	0	0
$115$	660.000i	0.535177i
$116$	0	0
$117$	− 2146.00i	− 1.69571i
$118$	0	0
$119$	−264.000	−0.203368
$120$	0	0
$121$	1187.00	0.891811
$122$	0	0
$123$	− 3504.00i	− 2.56866i
$124$	0	0
$125$	− 125.000i	− 0.0894427i
$126$	0	0
$127$	124.000	0.0866395	0.0433198	−	0.999061i	$-0.486207\pi$
$127$	124.000	0.0866395	0.0433198	+	0.999061i	$0.486207\pi$
$128$	0	0
$129$	−256.000	−0.174725
$130$	0	0
$131$	132.000i	0.0880374i	0.999031	+	0.0440187i	$0.0140161\pi$
$131$	132.000i	0.0880374i	−0.999031	+	0.0440187i	$0.985984\pi$
$132$	0	0
$133$	400.000i	0.260785i
$134$	0	0
$135$	−400.000	−0.255011
$136$	0	0
$137$	1254.00	0.782018	0.391009	−	0.920387i	$-0.372126\pi$
$137$	1254.00	0.782018	0.391009	+	0.920387i	$0.372126\pi$
$138$	0	0
$139$	2860.00i	1.74519i	0.488440	+	0.872597i	$0.337566\pi$
$139$	2860.00i	1.74519i	−0.488440	+	0.872597i	$0.662434\pi$
$140$	0	0
$141$	− 1632.00i	− 0.974746i
$142$	0	0
$143$	696.000	0.407010
$144$	0	0
$145$	−450.000	−0.257727
$146$	0	0
$147$	2616.00i	1.46778i
$148$	0	0
$149$	750.000i	0.412365i	0.978514	+	0.206183i	$0.0661041\pi$
$149$	750.000i	0.412365i	−0.978514	+	0.206183i	$0.933896\pi$
$150$	0	0
$151$	−448.000	−0.241442	−0.120721	−	0.992686i	$-0.538521\pi$
$151$	−448.000	−0.241442	−0.120721	+	0.992686i	$0.538521\pi$
$152$	0	0
$153$	−2442.00	−1.29035
$154$	0	0
$155$	− 760.000i	− 0.393837i
$156$	0	0
$157$	− 2246.00i	− 1.14172i	−0.821047	−	0.570861i	$-0.806610\pi$
$157$	− 2246.00i	− 1.14172i	0.821047	−	0.570861i	$-0.193390\pi$
$158$	0	0
$159$	1776.00	0.885824
$160$	0	0
$161$	−528.000	−0.258461
$162$	0	0
$163$	− 568.000i	− 0.272940i	−0.990644	−	0.136470i	$-0.956424\pi$
$163$	− 568.000i	− 0.272940i	0.990644	−	0.136470i	$-0.0435757\pi$
$164$	0	0
$165$	− 480.000i	− 0.226472i
$166$	0	0
$167$	−1524.00	−0.706172	−0.353086	−	0.935591i	$-0.614868\pi$
$167$	−1524.00	−0.706172	−0.353086	+	0.935591i	$0.614868\pi$
$168$	0	0
$169$	−1167.00	−0.531179
$170$	0	0
$171$	3700.00i	1.65466i
$172$	0	0
$173$	− 3702.00i	− 1.62692i	−0.581618	−	0.813462i	$-0.697580\pi$
$173$	− 3702.00i	− 1.62692i	0.581618	−	0.813462i	$-0.302420\pi$
$174$	0	0
$175$	100.000	0.0431959
$176$	0	0
$177$	−3360.00	−1.42685
$178$	0	0
$179$	3180.00i	1.32785i	0.747801	+	0.663923i	$0.231110\pi$
$179$	3180.00i	1.32785i	−0.747801	+	0.663923i	$0.768890\pi$
$180$	0	0
$181$	− 2098.00i	− 0.861564i	−0.902456	−	0.430782i	$-0.858238\pi$
$181$	− 2098.00i	− 0.861564i	0.902456	−	0.430782i	$-0.141762\pi$
$182$	0	0
$183$	−7216.00	−2.91487
$184$	0	0
$185$	170.000	0.0675603
$186$	0	0
$187$	− 792.000i	− 0.309715i
$188$	0	0
$189$	− 320.000i	− 0.123156i
$190$	0	0
$191$	−4392.00	−1.66384	−0.831921	−	0.554894i	$-0.812759\pi$
$191$	−4392.00	−1.66384	−0.831921	+	0.554894i	$0.812759\pi$
$192$	0	0
$193$	−2158.00	−0.804851	−0.402425	−	0.915453i	$-0.631833\pi$
$193$	−2158.00	−0.804851	−0.402425	+	0.915453i	$0.631833\pi$
$194$	0	0
$195$	2320.00i	0.851993i
$196$	0	0
$197$	− 1074.00i	− 0.388423i	−0.980960	−	0.194212i	$-0.937785\pi$
$197$	− 1074.00i	− 0.388423i	0.980960	−	0.194212i	$-0.0622148\pi$
$198$	0	0
$199$	2840.00	1.01167	0.505835	−	0.862630i	$-0.331185\pi$
$199$	2840.00	1.01167	0.505835	+	0.862630i	$0.331185\pi$
$200$	0	0
$201$	−8192.00	−2.87472
$202$	0	0
$203$	− 360.000i	− 0.124468i
$204$	0	0
$205$	2190.00i	0.746128i
$206$	0	0
$207$	−4884.00	−1.63991
$208$	0	0
$209$	−1200.00	−0.397157
$210$	0	0
$211$	− 2668.00i	− 0.870487i	−0.900313	−	0.435243i	$-0.856662\pi$
$211$	− 2668.00i	− 0.870487i	0.900313	−	0.435243i	$-0.143338\pi$
$212$	0	0
$213$	− 3456.00i	− 1.11174i
$214$	0	0
$215$	160.000	0.0507531
$216$	0	0
$217$	608.000	0.190202
$218$	0	0
$219$	2896.00i	0.893578i
$220$	0	0
$221$	3828.00i	1.16515i
$222$	0	0
$223$	−1772.00	−0.532116	−0.266058	−	0.963957i	$-0.585721\pi$
$223$	−1772.00	−0.532116	−0.266058	+	0.963957i	$0.585721\pi$
$224$	0	0
$225$	925.000	0.274074
$226$	0	0
$227$	− 2784.00i	− 0.814011i	−0.913426	−	0.407006i	$-0.866573\pi$
$227$	− 2784.00i	− 0.814011i	0.913426	−	0.407006i	$-0.133427\pi$
$228$	0	0
$229$	350.000i	0.100998i	0.998724	+	0.0504992i	$0.0160812\pi$
$229$	350.000i	0.100998i	−0.998724	+	0.0504992i	$0.983919\pi$
$230$	0	0
$231$	384.000	0.109374
$232$	0	0
$233$	−1962.00	−0.551652	−0.275826	−	0.961208i	$-0.588951\pi$
$233$	−1962.00	−0.551652	−0.275826	+	0.961208i	$0.588951\pi$
$234$	0	0
$235$	1020.00i	0.283138i
$236$	0	0
$237$	− 1280.00i	− 0.350823i
$238$	0	0
$239$	4320.00	1.16919	0.584597	−	0.811324i	$-0.301252\pi$
$239$	4320.00	1.16919	0.584597	+	0.811324i	$0.301252\pi$
$240$	0	0
$241$	−478.000	−0.127762	−0.0638811	−	0.997958i	$-0.520348\pi$
$241$	−478.000	−0.127762	−0.0638811	+	0.997958i	$0.520348\pi$
$242$	0	0
$243$	5032.00i	1.32841i
$244$	0	0
$245$	− 1635.00i	− 0.426352i
$246$	0	0
$247$	5800.00	1.49411
$248$	0	0
$249$	576.000	0.146596
$250$	0	0
$251$	− 2652.00i	− 0.666903i	−0.942767	−	0.333452i	$-0.891787\pi$
$251$	− 2652.00i	− 0.666903i	0.942767	−	0.333452i	$-0.108213\pi$
$252$	0	0
$253$	− 1584.00i	− 0.393617i
$254$	0	0
$255$	2640.00	0.648326
$256$	0	0
$257$	−2334.00	−0.566502	−0.283251	−	0.959046i	$-0.591413\pi$
$257$	−2334.00	−0.566502	−0.283251	+	0.959046i	$0.591413\pi$
$258$	0	0
$259$	136.000i	0.0326279i
$260$	0	0
$261$	− 3330.00i	− 0.789739i
$262$	0	0
$263$	−3948.00	−0.925643	−0.462822	−	0.886451i	$-0.653163\pi$
$263$	−3948.00	−0.925643	−0.462822	+	0.886451i	$0.653163\pi$
$264$	0	0
$265$	−1110.00	−0.257309
$266$	0	0
$267$	6480.00i	1.48528i
$268$	0	0
$269$	− 1590.00i	− 0.360387i	−0.983631	−	0.180193i	$-0.942328\pi$
$269$	− 1590.00i	− 0.360387i	0.983631	−	0.180193i	$-0.0576723\pi$
$270$	0	0
$271$	−4952.00	−1.11001	−0.555005	−	0.831847i	$-0.687284\pi$
$271$	−4952.00	−1.11001	−0.555005	+	0.831847i	$0.687284\pi$
$272$	0	0
$273$	−1856.00	−0.411466
$274$	0	0
$275$	300.000i	0.0657843i
$276$	0	0
$277$	1646.00i	0.357034i	0.983937	+	0.178517i	$0.0571300\pi$
$277$	1646.00i	0.357034i	−0.983937	+	0.178517i	$0.942870\pi$
$278$	0	0
$279$	5624.00	1.20681
$280$	0	0
$281$	1158.00	0.245838	0.122919	−	0.992417i	$-0.460774\pi$
$281$	1158.00	0.245838	0.122919	+	0.992417i	$0.460774\pi$
$282$	0	0
$283$	− 6992.00i	− 1.46866i	−0.678792	−	0.734331i	$-0.737496\pi$
$283$	− 6992.00i	− 1.46866i	0.678792	−	0.734331i	$-0.262504\pi$
$284$	0	0
$285$	− 4000.00i	− 0.831367i
$286$	0	0
$287$	−1752.00	−0.360339
$288$	0	0
$289$	−557.000	−0.113373
$290$	0	0
$291$	− 8848.00i	− 1.78240i
$292$	0	0
$293$	− 258.000i	− 0.0514421i	−0.999669	−	0.0257210i	$-0.991812\pi$
$293$	− 258.000i	− 0.0514421i	0.999669	−	0.0257210i	$-0.00818816\pi$
$294$	0	0
$295$	2100.00	0.414463
$296$	0	0
$297$	960.000	0.187558
$298$	0	0
$299$	7656.00i	1.48080i
$300$	0	0
$301$	128.000i	0.0245110i
$302$	0	0
$303$	−2064.00	−0.391332
$304$	0	0
$305$	4510.00	0.846695
$306$	0	0
$307$	− 8944.00i	− 1.66274i	−0.555720	−	0.831370i	$-0.687557\pi$
$307$	− 8944.00i	− 1.66274i	0.555720	−	0.831370i	$-0.312443\pi$
$308$	0	0
$309$	7904.00i	1.45515i
$310$	0	0
$311$	1392.00	0.253804	0.126902	−	0.991915i	$-0.459497\pi$
$311$	1392.00	0.253804	0.126902	+	0.991915i	$0.459497\pi$
$312$	0	0
$313$	5878.00	1.06148	0.530742	−	0.847534i	$-0.321913\pi$
$313$	5878.00	1.06148	0.530742	+	0.847534i	$0.321913\pi$
$314$	0	0
$315$	740.000i	0.132363i
$316$	0	0
$317$	− 10326.0i	− 1.82955i	−0.403969	−	0.914773i	$-0.632370\pi$
$317$	− 10326.0i	− 1.82955i	0.403969	−	0.914773i	$-0.367630\pi$
$318$	0	0
$319$	1080.00	0.189556
$320$	0	0
$321$	192.000	0.0333844
$322$	0	0
$323$	− 6600.00i	− 1.13695i
$324$	0	0
$325$	− 1450.00i	− 0.247482i
$326$	0	0
$327$	−7600.00	−1.28526
$328$	0	0
$329$	−816.000	−0.136740
$330$	0	0
$331$	4228.00i	0.702090i	0.936359	+	0.351045i	$0.114174\pi$
$331$	4228.00i	0.702090i	−0.936359	+	0.351045i	$0.885826\pi$
$332$	0	0
$333$	1258.00i	0.207021i
$334$	0	0
$335$	5120.00	0.835031
$336$	0	0
$337$	1106.00	0.178776	0.0893882	−	0.995997i	$-0.471509\pi$
$337$	1106.00	0.178776	0.0893882	+	0.995997i	$0.471509\pi$
$338$	0	0
$339$	8304.00i	1.33042i
$340$	0	0
$341$	1824.00i	0.289663i
$342$	0	0
$343$	2680.00	0.421885
$344$	0	0
$345$	5280.00	0.823958
$346$	0	0
$347$	− 9336.00i	− 1.44433i	−0.691720	−	0.722165i	$-0.743147\pi$
$347$	− 9336.00i	− 1.44433i	0.691720	−	0.722165i	$-0.256853\pi$
$348$	0	0
$349$	11770.0i	1.80525i	0.430424	+	0.902627i	$0.358364\pi$
$349$	11770.0i	1.80525i	−0.430424	+	0.902627i	$0.641636\pi$
$350$	0	0
$351$	−4640.00	−0.705598
$352$	0	0
$353$	8322.00	1.25477	0.627387	−	0.778707i	$-0.284124\pi$
$353$	8322.00	1.25477	0.627387	+	0.778707i	$0.284124\pi$
$354$	0	0
$355$	2160.00i	0.322932i
$356$	0	0
$357$	2112.00i	0.313106i
$358$	0	0
$359$	10680.0	1.57011	0.785054	−	0.619427i	$-0.212635\pi$
$359$	10680.0	1.57011	0.785054	+	0.619427i	$0.212635\pi$
$360$	0	0
$361$	−3141.00	−0.457938
$362$	0	0
$363$	− 9496.00i	− 1.37303i
$364$	0	0
$365$	− 1810.00i	− 0.259561i
$366$	0	0
$367$	5884.00	0.836900	0.418450	−	0.908240i	$-0.362574\pi$
$367$	5884.00	0.836900	0.418450	+	0.908240i	$0.362574\pi$
$368$	0	0
$369$	−16206.0	−2.28632
$370$	0	0
$371$	− 888.000i	− 0.124266i
$372$	0	0
$373$	− 2098.00i	− 0.291234i	−0.989341	−	0.145617i	$-0.953483\pi$
$373$	− 2098.00i	− 0.291234i	0.989341	−	0.145617i	$-0.0465167\pi$
$374$	0	0
$375$	−1000.00	−0.137706
$376$	0	0
$377$	−5220.00	−0.713113
$378$	0	0
$379$	− 3860.00i	− 0.523153i	−0.965183	−	0.261576i	$-0.915758\pi$
$379$	− 3860.00i	− 0.523153i	0.965183	−	0.261576i	$-0.0842423\pi$
$380$	0	0
$381$	− 992.000i	− 0.133390i
$382$	0	0
$383$	9588.00	1.27917	0.639587	−	0.768718i	$-0.279105\pi$
$383$	9588.00	1.27917	0.639587	+	0.768718i	$0.279105\pi$
$384$	0	0
$385$	−240.000	−0.0317702
$386$	0	0
$387$	1184.00i	0.155520i
$388$	0	0
$389$	− 13410.0i	− 1.74785i	−0.486060	−	0.873925i	$-0.661566\pi$
$389$	− 13410.0i	− 1.74785i	0.486060	−	0.873925i	$-0.338434\pi$
$390$	0	0
$391$	8712.00	1.12682
$392$	0	0
$393$	1056.00	0.135542
$394$	0	0
$395$	800.000i	0.101905i
$396$	0	0
$397$	13114.0i	1.65787i	0.559348	+	0.828933i	$0.311052\pi$
$397$	13114.0i	1.65787i	−0.559348	+	0.828933i	$0.688948\pi$
$398$	0	0
$399$	3200.00	0.401505
$400$	0	0
$401$	−5838.00	−0.727022	−0.363511	−	0.931590i	$-0.618422\pi$
$401$	−5838.00	−0.727022	−0.363511	+	0.931590i	$0.618422\pi$
$402$	0	0
$403$	− 8816.00i	− 1.08972i
$404$	0	0
$405$	− 1795.00i	− 0.220233i
$406$	0	0
$407$	−408.000	−0.0496899
$408$	0	0
$409$	−9530.00	−1.15215	−0.576074	−	0.817398i	$-0.695416\pi$
$409$	−9530.00	−1.15215	−0.576074	+	0.817398i	$0.695416\pi$
$410$	0	0
$411$	− 10032.0i	− 1.20400i
$412$	0	0
$413$	1680.00i	0.200163i
$414$	0	0
$415$	−360.000	−0.0425824
$416$	0	0
$417$	22880.0	2.68690
$418$	0	0
$419$	7260.00i	0.846478i	0.906018	+	0.423239i	$0.139107\pi$
$419$	7260.00i	0.846478i	−0.906018	+	0.423239i	$0.860893\pi$
$420$	0	0
$421$	12062.0i	1.39636i	0.715924	+	0.698178i	$0.246006\pi$
$421$	12062.0i	1.39636i	−0.715924	+	0.698178i	$0.753994\pi$
$422$	0	0
$423$	−7548.00	−0.867604
$424$	0	0
$425$	−1650.00	−0.188322
$426$	0	0
$427$	3608.00i	0.408907i
$428$	0	0
$429$	− 5568.00i	− 0.626633i
$430$	0	0
$431$	13608.0	1.52082	0.760411	−	0.649442i	$-0.224998\pi$
$431$	13608.0	1.52082	0.760411	+	0.649442i	$0.224998\pi$
$432$	0	0
$433$	−3838.00	−0.425964	−0.212982	−	0.977056i	$-0.568318\pi$
$433$	−3838.00	−0.425964	−0.212982	+	0.977056i	$0.568318\pi$
$434$	0	0
$435$	3600.00i	0.396797i
$436$	0	0
$437$	− 13200.0i	− 1.44495i
$438$	0	0
$439$	7400.00	0.804516	0.402258	−	0.915526i	$-0.368225\pi$
$439$	7400.00	0.804516	0.402258	+	0.915526i	$0.368225\pi$
$440$	0	0
$441$	12099.0	1.30645
$442$	0	0
$443$	− 8352.00i	− 0.895746i	−0.894097	−	0.447873i	$-0.852182\pi$
$443$	− 8352.00i	− 0.895746i	0.894097	−	0.447873i	$-0.147818\pi$
$444$	0	0
$445$	− 4050.00i	− 0.431435i
$446$	0	0
$447$	6000.00	0.634878
$448$	0	0
$449$	10770.0	1.13200	0.566000	−	0.824405i	$-0.308490\pi$
$449$	10770.0	1.13200	0.566000	+	0.824405i	$0.308490\pi$
$450$	0	0
$451$	− 5256.00i	− 0.548770i
$452$	0	0
$453$	3584.00i	0.371724i
$454$	0	0
$455$	1160.00	0.119520
$456$	0	0
$457$	6694.00	0.685191	0.342595	−	0.939483i	$-0.388694\pi$
$457$	6694.00	0.685191	0.342595	+	0.939483i	$0.388694\pi$
$458$	0	0
$459$	5280.00i	0.536927i
$460$	0	0
$461$	3018.00i	0.304907i	0.988311	+	0.152454i	$0.0487175\pi$
$461$	3018.00i	0.304907i	−0.988311	+	0.152454i	$0.951283\pi$
$462$	0	0
$463$	−14492.0	−1.45464	−0.727322	−	0.686296i	$-0.759235\pi$
$463$	−14492.0	−1.45464	−0.727322	+	0.686296i	$0.759235\pi$
$464$	0	0
$465$	−6080.00	−0.606351
$466$	0	0
$467$	7776.00i	0.770515i	0.922809	+	0.385257i	$0.125887\pi$
$467$	7776.00i	0.770515i	−0.922809	+	0.385257i	$0.874113\pi$
$468$	0	0
$469$	4096.00i	0.403274i
$470$	0	0
$471$	−17968.0	−1.75780
$472$	0	0
$473$	−384.000	−0.0373284
$474$	0	0
$475$	2500.00i	0.241490i
$476$	0	0
$477$	− 8214.00i	− 0.788455i
$478$	0	0
$479$	13680.0	1.30492	0.652458	−	0.757825i	$-0.273738\pi$
$479$	13680.0	1.30492	0.652458	+	0.757825i	$0.273738\pi$
$480$	0	0
$481$	1972.00	0.186934
$482$	0	0
$483$	4224.00i	0.397927i
$484$	0	0
$485$	5530.00i	0.517741i
$486$	0	0
$487$	7916.00	0.736567	0.368284	−	0.929714i	$-0.379946\pi$
$487$	7916.00	0.736567	0.368284	+	0.929714i	$0.379946\pi$
$488$	0	0
$489$	−4544.00	−0.420218
$490$	0	0
$491$	− 13932.0i	− 1.28053i	−0.768152	−	0.640267i	$-0.778824\pi$
$491$	− 13932.0i	− 1.28053i	0.768152	−	0.640267i	$-0.221176\pi$
$492$	0	0
$493$	5940.00i	0.542645i
$494$	0	0
$495$	−2220.00	−0.201579
$496$	0	0
$497$	−1728.00	−0.155959
$498$	0	0
$499$	− 8260.00i	− 0.741019i	−0.928829	−	0.370509i	$-0.879183\pi$
$499$	− 8260.00i	− 0.741019i	0.928829	−	0.370509i	$-0.120817\pi$
$500$	0	0
$501$	12192.0i	1.08722i
$502$	0	0
$503$	−11148.0	−0.988200	−0.494100	−	0.869405i	$-0.664502\pi$
$503$	−11148.0	−0.988200	−0.494100	+	0.869405i	$0.664502\pi$
$504$	0	0
$505$	1290.00	0.113672
$506$	0	0
$507$	9336.00i	0.817803i
$508$	0	0
$509$	9690.00i	0.843815i	0.906639	+	0.421907i	$0.138639\pi$
$509$	9690.00i	0.843815i	−0.906639	+	0.421907i	$0.861361\pi$
$510$	0	0
$511$	1448.00	0.125354
$512$	0	0
$513$	8000.00	0.688516
$514$	0	0
$515$	− 4940.00i	− 0.422684i
$516$	0	0
$517$	− 2448.00i	− 0.208245i
$518$	0	0
$519$	−29616.0	−2.50481
$520$	0	0
$521$	16038.0	1.34863	0.674316	−	0.738443i	$-0.264438\pi$
$521$	16038.0	1.34863	0.674316	+	0.738443i	$0.264438\pi$
$522$	0	0
$523$	− 992.000i	− 0.0829391i	−0.999140	−	0.0414695i	$-0.986796\pi$
$523$	− 992.000i	− 0.0829391i	0.999140	−	0.0414695i	$-0.0132039\pi$
$524$	0	0
$525$	− 800.000i	− 0.0665045i
$526$	0	0
$527$	−10032.0	−0.829223
$528$	0	0
$529$	5257.00	0.432070
$530$	0	0
$531$	15540.0i	1.27002i
$532$	0	0
$533$	25404.0i	2.06448i
$534$	0	0
$535$	−120.000	−0.00969729
$536$	0	0
$537$	25440.0	2.04435
$538$	0	0
$539$	3924.00i	0.313578i
$540$	0	0
$541$	− 7142.00i	− 0.567576i	−0.958887	−	0.283788i	$-0.908409\pi$
$541$	− 7142.00i	− 0.567576i	0.958887	−	0.283788i	$-0.0915912\pi$
$542$	0	0
$543$	−16784.0	−1.32646
$544$	0	0
$545$	4750.00	0.373335
$546$	0	0
$547$	7616.00i	0.595314i	0.954673	+	0.297657i	$0.0962051\pi$
$547$	7616.00i	0.595314i	−0.954673	+	0.297657i	$0.903795\pi$
$548$	0	0
$549$	33374.0i	2.59448i
$550$	0	0
$551$	9000.00	0.695849
$552$	0	0
$553$	−640.000	−0.0492144
$554$	0	0
$555$	− 1360.00i	− 0.104016i
$556$	0	0
$557$	10314.0i	0.784593i	0.919839	+	0.392296i	$0.128319\pi$
$557$	10314.0i	0.784593i	−0.919839	+	0.392296i	$0.871681\pi$
$558$	0	0
$559$	1856.00	0.140430
$560$	0	0
$561$	−6336.00	−0.476838
$562$	0	0
$563$	− 7128.00i	− 0.533587i	−0.963754	−	0.266793i	$-0.914036\pi$
$563$	− 7128.00i	− 0.533587i	0.963754	−	0.266793i	$-0.0859641\pi$
$564$	0	0
$565$	− 5190.00i	− 0.386451i
$566$	0	0
$567$	1436.00	0.106360
$568$	0	0
$569$	−2010.00	−0.148091	−0.0740453	−	0.997255i	$-0.523591\pi$
$569$	−2010.00	−0.148091	−0.0740453	+	0.997255i	$0.523591\pi$
$570$	0	0
$571$	23188.0i	1.69945i	0.527224	+	0.849726i	$0.323233\pi$
$571$	23188.0i	1.69945i	−0.527224	+	0.849726i	$0.676767\pi$
$572$	0	0
$573$	35136.0i	2.56165i
$574$	0	0
$575$	−3300.00	−0.239338
$576$	0	0
$577$	22466.0	1.62092	0.810461	−	0.585793i	$-0.199217\pi$
$577$	22466.0	1.62092	0.810461	+	0.585793i	$0.199217\pi$
$578$	0	0
$579$	17264.0i	1.23915i
$580$	0	0
$581$	− 288.000i	− 0.0205650i
$582$	0	0
$583$	2664.00	0.189248
$584$	0	0
$585$	10730.0	0.758343
$586$	0	0
$587$	− 22776.0i	− 1.60148i	−0.599015	−	0.800738i	$-0.704441\pi$
$587$	− 22776.0i	− 1.60148i	0.599015	−	0.800738i	$-0.295559\pi$
$588$	0	0
$589$	15200.0i	1.06334i
$590$	0	0
$591$	−8592.00	−0.598016
$592$	0	0
$593$	−21198.0	−1.46796	−0.733978	−	0.679174i	$-0.762338\pi$
$593$	−21198.0	−1.46796	−0.733978	+	0.679174i	$0.762338\pi$
$594$	0	0
$595$	− 1320.00i	− 0.0909491i
$596$	0	0
$597$	− 22720.0i	− 1.55757i
$598$	0	0
$599$	15960.0	1.08866	0.544330	−	0.838871i	$-0.316784\pi$
$599$	15960.0	1.08866	0.544330	+	0.838871i	$0.316784\pi$
$600$	0	0
$601$	−5882.00	−0.399221	−0.199610	−	0.979875i	$-0.563968\pi$
$601$	−5882.00	−0.399221	−0.199610	+	0.979875i	$0.563968\pi$
$602$	0	0
$603$	37888.0i	2.55874i
$604$	0	0
$605$	5935.00i	0.398830i
$606$	0	0
$607$	−8516.00	−0.569446	−0.284723	−	0.958610i	$-0.591902\pi$
$607$	−8516.00	−0.569446	−0.284723	+	0.958610i	$0.591902\pi$
$608$	0	0
$609$	−2880.00	−0.191631
$610$	0	0
$611$	11832.0i	0.783423i
$612$	0	0
$613$	8462.00i	0.557548i	0.960357	+	0.278774i	$0.0899280\pi$
$613$	8462.00i	0.557548i	−0.960357	+	0.278774i	$0.910072\pi$
$614$	0	0
$615$	17520.0	1.14874
$616$	0	0
$617$	11094.0	0.723870	0.361935	−	0.932203i	$-0.382116\pi$
$617$	11094.0	0.723870	0.361935	+	0.932203i	$0.382116\pi$
$618$	0	0
$619$	− 2180.00i	− 0.141553i	−0.997492	−	0.0707767i	$-0.977452\pi$
$619$	− 2180.00i	− 0.141553i	0.997492	−	0.0707767i	$-0.0225478\pi$
$620$	0	0
$621$	10560.0i	0.682380i
$622$	0	0
$623$	3240.00	0.208359
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	9600.00i	0.611463i
$628$	0	0
$629$	− 2244.00i	− 0.142248i
$630$	0	0
$631$	−26848.0	−1.69382	−0.846911	−	0.531734i	$-0.821541\pi$
$631$	−26848.0	−1.69382	−0.846911	+	0.531734i	$0.821541\pi$
$632$	0	0
$633$	−21344.0	−1.34020
$634$	0	0
$635$	620.000i	0.0387464i
$636$	0	0
$637$	− 18966.0i	− 1.17969i
$638$	0	0
$639$	−15984.0	−0.989542
$640$	0	0
$641$	26322.0	1.62193	0.810965	−	0.585095i	$-0.198943\pi$
$641$	26322.0	1.62193	0.810965	+	0.585095i	$0.198943\pi$
$642$	0	0
$643$	− 10168.0i	− 0.623619i	−0.950145	−	0.311809i	$-0.899065\pi$
$643$	− 10168.0i	− 0.623619i	0.950145	−	0.311809i	$-0.100935\pi$
$644$	0	0
$645$	− 1280.00i	− 0.0781395i
$646$	0	0
$647$	−23604.0	−1.43426	−0.717132	−	0.696937i	$-0.754546\pi$
$647$	−23604.0	−1.43426	−0.717132	+	0.696937i	$0.754546\pi$
$648$	0	0
$649$	−5040.00	−0.304834
$650$	0	0
$651$	− 4864.00i	− 0.292834i
$652$	0	0
$653$	− 16422.0i	− 0.984139i	−0.870556	−	0.492069i	$-0.836241\pi$
$653$	− 16422.0i	− 0.984139i	0.870556	−	0.492069i	$-0.163759\pi$
$654$	0	0
$655$	−660.000	−0.0393715
$656$	0	0
$657$	13394.0	0.795357
$658$	0	0
$659$	− 26100.0i	− 1.54281i	−0.636345	−	0.771405i	$-0.719554\pi$
$659$	− 26100.0i	− 1.54281i	0.636345	−	0.771405i	$-0.280446\pi$
$660$	0	0
$661$	− 3058.00i	− 0.179943i	−0.995944	−	0.0899716i	$-0.971322\pi$
$661$	− 3058.00i	− 0.179943i	0.995944	−	0.0899716i	$-0.0286776\pi$
$662$	0	0
$663$	30624.0	1.79387
$664$	0	0
$665$	−2000.00	−0.116627
$666$	0	0
$667$	11880.0i	0.689648i
$668$	0	0
$669$	14176.0i	0.819246i
$670$	0	0
$671$	−10824.0	−0.622736
$672$	0	0
$673$	10802.0	0.618702	0.309351	−	0.950948i	$-0.399888\pi$
$673$	10802.0	0.618702	0.309351	+	0.950948i	$0.399888\pi$
$674$	0	0
$675$	− 2000.00i	− 0.114044i
$676$	0	0
$677$	− 10674.0i	− 0.605960i	−0.952997	−	0.302980i	$-0.902018\pi$
$677$	− 10674.0i	− 0.605960i	0.952997	−	0.302980i	$-0.0979816\pi$
$678$	0	0
$679$	−4424.00	−0.250041
$680$	0	0
$681$	−22272.0	−1.25325
$682$	0	0
$683$	28608.0i	1.60272i	0.598185	+	0.801358i	$0.295889\pi$
$683$	28608.0i	1.60272i	−0.598185	+	0.801358i	$0.704111\pi$
$684$	0	0
$685$	6270.00i	0.349729i
$686$	0	0
$687$	2800.00	0.155497
$688$	0	0
$689$	−12876.0	−0.711954
$690$	0	0
$691$	− 2428.00i	− 0.133669i	−0.997764	−	0.0668346i	$-0.978710\pi$
$691$	− 2428.00i	− 0.133669i	0.997764	−	0.0668346i	$-0.0212900\pi$
$692$	0	0
$693$	− 1776.00i	− 0.0973516i
$694$	0	0
$695$	−14300.0	−0.780475
$696$	0	0
$697$	28908.0	1.57097
$698$	0	0
$699$	15696.0i	0.849324i
$700$	0	0
$701$	6618.00i	0.356574i	0.983979	+	0.178287i	$0.0570556\pi$
$701$	6618.00i	0.356574i	−0.983979	+	0.178287i	$0.942944\pi$
$702$	0	0
$703$	−3400.00	−0.182409
$704$	0	0
$705$	8160.00	0.435920
$706$	0	0
$707$	1032.00i	0.0548972i
$708$	0	0
$709$	20510.0i	1.08642i	0.839598	+	0.543208i	$0.182791\pi$
$709$	20510.0i	1.08642i	−0.839598	+	0.543208i	$0.817209\pi$
$710$	0	0
$711$	−5920.00	−0.312261
$712$	0	0
$713$	−20064.0	−1.05386
$714$	0	0
$715$	3480.00i	0.182020i
$716$	0	0
$717$	− 34560.0i	− 1.80009i
$718$	0	0
$719$	−31680.0	−1.64321	−0.821603	−	0.570061i	$-0.806920\pi$
$719$	−31680.0	−1.64321	−0.821603	+	0.570061i	$0.806920\pi$
$720$	0	0
$721$	3952.00	0.204133
$722$	0	0
$723$	3824.00i	0.196703i
$724$	0	0
$725$	− 2250.00i	− 0.115259i
$726$	0	0
$727$	13196.0	0.673195	0.336597	−	0.941649i	$-0.390724\pi$
$727$	13196.0	0.673195	0.336597	+	0.941649i	$0.390724\pi$
$728$	0	0
$729$	30563.0	1.55276
$730$	0	0
$731$	− 2112.00i	− 0.106861i
$732$	0	0
$733$	− 8102.00i	− 0.408259i	−0.978944	−	0.204130i	$-0.934564\pi$
$733$	− 8102.00i	− 0.408259i	0.978944	−	0.204130i	$-0.0654364\pi$
$734$	0	0
$735$	−13080.0	−0.656412
$736$	0	0
$737$	−12288.0	−0.614158
$738$	0	0
$739$	− 12580.0i	− 0.626201i	−0.949720	−	0.313101i	$-0.898632\pi$
$739$	− 12580.0i	− 0.626201i	0.949720	−	0.313101i	$-0.101368\pi$
$740$	0	0
$741$	− 46400.0i	− 2.30033i
$742$	0	0
$743$	29892.0	1.47595	0.737975	−	0.674828i	$-0.235782\pi$
$743$	29892.0	1.47595	0.737975	+	0.674828i	$0.235782\pi$
$744$	0	0
$745$	−3750.00	−0.184415
$746$	0	0
$747$	− 2664.00i	− 0.130483i
$748$	0	0
$749$	− 96.0000i	− 0.00468326i
$750$	0	0
$751$	40408.0	1.96339	0.981697	−	0.190450i	$-0.0609946\pi$
$751$	40408.0	1.96339	0.981697	+	0.190450i	$0.0609946\pi$
$752$	0	0
$753$	−21216.0	−1.02676
$754$	0	0
$755$	− 2240.00i	− 0.107976i
$756$	0	0
$757$	32366.0i	1.55398i	0.629513	+	0.776990i	$0.283254\pi$
$757$	32366.0i	1.55398i	−0.629513	+	0.776990i	$0.716746\pi$
$758$	0	0
$759$	−12672.0	−0.606014
$760$	0	0
$761$	17238.0	0.821126	0.410563	−	0.911832i	$-0.365332\pi$
$761$	17238.0	0.821126	0.410563	+	0.911832i	$0.365332\pi$
$762$	0	0
$763$	3800.00i	0.180300i
$764$	0	0
$765$	− 12210.0i	− 0.577063i
$766$	0	0
$767$	24360.0	1.14679
$768$	0	0
$769$	10850.0	0.508792	0.254396	−	0.967100i	$-0.418123\pi$
$769$	10850.0	0.508792	0.254396	+	0.967100i	$0.418123\pi$
$770$	0	0
$771$	18672.0i	0.872186i
$772$	0	0
$773$	9102.00i	0.423514i	0.977322	+	0.211757i	$0.0679185\pi$
$773$	9102.00i	0.423514i	−0.977322	+	0.211757i	$0.932081\pi$
$774$	0	0
$775$	3800.00	0.176129
$776$	0	0
$777$	1088.00	0.0502340
$778$	0	0
$779$	− 43800.0i	− 2.01450i
$780$	0	0
$781$	− 5184.00i	− 0.237514i
$782$	0	0
$783$	−7200.00	−0.328617
$784$	0	0
$785$	11230.0	0.510593
$786$	0	0
$787$	− 25504.0i	− 1.15517i	−0.816330	−	0.577585i	$-0.803995\pi$
$787$	− 25504.0i	− 1.15517i	0.816330	−	0.577585i	$-0.196005\pi$
$788$	0	0
$789$	31584.0i	1.42512i
$790$	0	0
$791$	4152.00	0.186635
$792$	0	0
$793$	52316.0	2.34274
$794$	0	0
$795$	8880.00i	0.396152i
$796$	0	0
$797$	− 14166.0i	− 0.629593i	−0.949159	−	0.314796i	$-0.898064\pi$
$797$	− 14166.0i	− 0.629593i	0.949159	−	0.314796i	$-0.101936\pi$
$798$	0	0
$799$	13464.0	0.596148
$800$	0	0
$801$	29970.0	1.32202
$802$	0	0
$803$	4344.00i	0.190905i
$804$	0	0
$805$	− 2640.00i	− 0.115587i
$806$	0	0
$807$	−12720.0	−0.554852
$808$	0	0
$809$	−33210.0	−1.44327	−0.721633	−	0.692276i	$-0.756608\pi$
$809$	−33210.0	−1.44327	−0.721633	+	0.692276i	$0.756608\pi$
$810$	0	0
$811$	− 39212.0i	− 1.69780i	−0.528550	−	0.848902i	$-0.677264\pi$
$811$	− 39212.0i	− 1.69780i	0.528550	−	0.848902i	$-0.322736\pi$
$812$	0	0
$813$	39616.0i	1.70897i
$814$	0	0
$815$	2840.00	0.122062
$816$	0	0
$817$	−3200.00	−0.137030
$818$	0	0
$819$	8584.00i	0.366238i
$820$	0	0
$821$	6222.00i	0.264494i	0.991217	+	0.132247i	$0.0422192\pi$
$821$	6222.00i	0.264494i	−0.991217	+	0.132247i	$0.957781\pi$
$822$	0	0
$823$	31172.0	1.32028	0.660138	−	0.751144i	$-0.270498\pi$
$823$	31172.0	1.32028	0.660138	+	0.751144i	$0.270498\pi$
$824$	0	0
$825$	2400.00	0.101282
$826$	0	0
$827$	264.000i	0.0111006i	0.999985	+	0.00555029i	$0.00176672\pi$
$827$	264.000i	0.0111006i	−0.999985	+	0.00555029i	$0.998233\pi$
$828$	0	0
$829$	29050.0i	1.21707i	0.793528	+	0.608533i	$0.208242\pi$
$829$	29050.0i	1.21707i	−0.793528	+	0.608533i	$0.791758\pi$
$830$	0	0
$831$	13168.0	0.549691
$832$	0	0
$833$	−21582.0	−0.897685
$834$	0	0
$835$	− 7620.00i	− 0.315810i
$836$	0	0
$837$	− 12160.0i	− 0.502164i
$838$	0	0
$839$	−21720.0	−0.893752	−0.446876	−	0.894596i	$-0.647463\pi$
$839$	−21720.0	−0.893752	−0.446876	+	0.894596i	$0.647463\pi$
$840$	0	0
$841$	16289.0	0.667883
$842$	0	0
$843$	− 9264.00i	− 0.378492i
$844$	0	0
$845$	− 5835.00i	− 0.237550i
$846$	0	0
$847$	−4748.00	−0.192613
$848$	0	0
$849$	−55936.0	−2.26115
$850$	0	0
$851$	− 4488.00i	− 0.180783i
$852$	0	0
$853$	− 6658.00i	− 0.267252i	−0.991032	−	0.133626i	$-0.957338\pi$
$853$	− 6658.00i	− 0.267252i	0.991032	−	0.133626i	$-0.0426620\pi$
$854$	0	0
$855$	−18500.0	−0.739984
$856$	0	0
$857$	13974.0	0.556993	0.278496	−	0.960437i	$-0.410164\pi$
$857$	13974.0	0.556993	0.278496	+	0.960437i	$0.410164\pi$
$858$	0	0
$859$	− 23780.0i	− 0.944544i	−0.881453	−	0.472272i	$-0.843434\pi$
$859$	− 23780.0i	− 0.944544i	0.881453	−	0.472272i	$-0.156566\pi$
$860$	0	0
$861$	14016.0i	0.554778i
$862$	0	0
$863$	12228.0	0.482324	0.241162	−	0.970485i	$-0.422471\pi$
$863$	12228.0	0.482324	0.241162	+	0.970485i	$0.422471\pi$
$864$	0	0
$865$	18510.0	0.727583
$866$	0	0
$867$	4456.00i	0.174549i
$868$	0	0
$869$	− 1920.00i	− 0.0749500i
$870$	0	0
$871$	59392.0	2.31047
$872$	0	0
$873$	−40922.0	−1.58648
$874$	0	0
$875$	500.000i	0.0193178i
$876$	0	0
$877$	− 11606.0i	− 0.446872i	−0.974719	−	0.223436i	$-0.928273\pi$
$877$	− 11606.0i	− 0.446872i	0.974719	−	0.223436i	$-0.0717274\pi$
$878$	0	0
$879$	−2064.00	−0.0792002
$880$	0	0
$881$	−32958.0	−1.26037	−0.630183	−	0.776446i	$-0.717020\pi$
$881$	−32958.0	−1.26037	−0.630183	+	0.776446i	$0.717020\pi$
$882$	0	0
$883$	8072.00i	0.307638i	0.988099	+	0.153819i	$0.0491573\pi$
$883$	8072.00i	0.307638i	−0.988099	+	0.153819i	$0.950843\pi$
$884$	0	0
$885$	− 16800.0i	− 0.638108i
$886$	0	0
$887$	15756.0	0.596431	0.298216	−	0.954498i	$-0.403609\pi$
$887$	15756.0	0.596431	0.298216	+	0.954498i	$0.403609\pi$
$888$	0	0
$889$	−496.000	−0.0187124
$890$	0	0
$891$	4308.00i	0.161979i
$892$	0	0
$893$	− 20400.0i	− 0.764457i
$894$	0	0
$895$	−15900.0	−0.593831
$896$	0	0
$897$	61248.0	2.27983
$898$	0	0
$899$	− 13680.0i	− 0.507512i
$900$	0	0
$901$	14652.0i	0.541763i
$902$	0	0
$903$	1024.00	0.0377371
$904$	0	0
$905$	10490.0	0.385303
$906$	0	0
$907$	− 18776.0i	− 0.687372i	−0.939085	−	0.343686i	$-0.888324\pi$
$907$	− 18776.0i	− 0.687372i	0.939085	−	0.343686i	$-0.111676\pi$
$908$	0	0
$909$	9546.00i	0.348318i
$910$	0	0
$911$	20568.0	0.748022	0.374011	−	0.927424i	$-0.377982\pi$
$911$	20568.0	0.748022	0.374011	+	0.927424i	$0.377982\pi$
$912$	0	0
$913$	864.000	0.0313190
$914$	0	0
$915$	− 36080.0i	− 1.30357i
$916$	0	0
$917$	− 528.000i	− 0.0190143i
$918$	0	0
$919$	−6280.00	−0.225417	−0.112708	−	0.993628i	$-0.535953\pi$
$919$	−6280.00	−0.225417	−0.112708	+	0.993628i	$0.535953\pi$
$920$	0	0
$921$	−71552.0	−2.55996
$922$	0	0
$923$	25056.0i	0.893530i
$924$	0	0
$925$	850.000i	0.0302139i
$926$	0	0
$927$	36556.0	1.29521
$928$	0	0
$929$	−20430.0	−0.721514	−0.360757	−	0.932660i	$-0.617482\pi$
$929$	−20430.0	−0.721514	−0.360757	+	0.932660i	$0.617482\pi$
$930$	0	0
$931$	32700.0i	1.15113i
$932$	0	0
$933$	− 11136.0i	− 0.390757i
$934$	0	0
$935$	3960.00	0.138509
$936$	0	0
$937$	−8906.00	−0.310508	−0.155254	−	0.987875i	$-0.549620\pi$
$937$	−8906.00	−0.310508	−0.155254	+	0.987875i	$0.549620\pi$
$938$	0	0
$939$	− 47024.0i	− 1.63426i
$940$	0	0
$941$	17418.0i	0.603412i	0.953401	+	0.301706i	$0.0975561\pi$
$941$	17418.0i	0.603412i	−0.953401	+	0.301706i	$0.902444\pi$
$942$	0	0
$943$	57816.0	1.99655
$944$	0	0
$945$	1600.00	0.0550773
$946$	0	0
$947$	− 2544.00i	− 0.0872956i	−0.999047	−	0.0436478i	$-0.986102\pi$
$947$	− 2544.00i	− 0.0872956i	0.999047	−	0.0436478i	$-0.0138979\pi$
$948$	0	0
$949$	− 20996.0i	− 0.718187i
$950$	0	0
$951$	−82608.0	−2.81677
$952$	0	0
$953$	−15402.0	−0.523525	−0.261763	−	0.965132i	$-0.584304\pi$
$953$	−15402.0	−0.523525	−0.261763	+	0.965132i	$0.584304\pi$
$954$	0	0
$955$	− 21960.0i	− 0.744093i
$956$	0	0
$957$	− 8640.00i	− 0.291841i
$958$	0	0
$959$	−5016.00	−0.168900
$960$	0	0
$961$	−6687.00	−0.224464
$962$	0	0
$963$	− 888.000i	− 0.0297148i
$964$	0	0
$965$	− 10790.0i	− 0.359940i
$966$	0	0
$967$	−49444.0	−1.64427	−0.822136	−	0.569291i	$-0.807218\pi$
$967$	−49444.0	−1.64427	−0.822136	+	0.569291i	$0.807218\pi$
$968$	0	0
$969$	−52800.0	−1.75044
$970$	0	0
$971$	25188.0i	0.832463i	0.909259	+	0.416231i	$0.136649\pi$
$971$	25188.0i	0.832463i	−0.909259	+	0.416231i	$0.863351\pi$
$972$	0	0
$973$	− 11440.0i	− 0.376927i
$974$	0	0
$975$	−11600.0	−0.381023
$976$	0	0
$977$	2946.00	0.0964697	0.0482348	−	0.998836i	$-0.484640\pi$
$977$	2946.00	0.0964697	0.0482348	+	0.998836i	$0.484640\pi$
$978$	0	0
$979$	9720.00i	0.317316i
$980$	0	0
$981$	35150.0i	1.14399i
$982$	0	0
$983$	15012.0	0.487089	0.243544	−	0.969890i	$-0.421690\pi$
$983$	15012.0	0.487089	0.243544	+	0.969890i	$0.421690\pi$
$984$	0	0
$985$	5370.00	0.173708
$986$	0	0
$987$	6528.00i	0.210525i
$988$	0	0
$989$	− 4224.00i	− 0.135809i
$990$	0	0
$991$	5128.00	0.164376	0.0821878	−	0.996617i	$-0.473809\pi$
$991$	5128.00	0.164376	0.0821878	+	0.996617i	$0.473809\pi$
$992$	0	0
$993$	33824.0	1.08094
$994$	0	0
$995$	14200.0i	0.452432i
$996$	0	0
$997$	− 49714.0i	− 1.57920i	−0.613625	−	0.789598i	$-0.710289\pi$
$997$	− 49714.0i	− 1.57920i	0.613625	−	0.789598i	$-0.289711\pi$
$998$	0	0
$999$	2720.00	0.0861431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1280.4.d.g.641.1		2
4.3	odd	2		1280.4.d.j.641.2		2
8.3	odd	2		1280.4.d.j.641.1		2
8.5	even	2	inner	1280.4.d.g.641.2		2
16.3	odd	4		10.4.a.a.1.1	✓	1
16.5	even	4		320.4.a.b.1.1		1
16.11	odd	4		320.4.a.m.1.1		1
16.13	even	4		80.4.a.f.1.1		1
48.29	odd	4		720.4.a.j.1.1		1
48.35	even	4		90.4.a.a.1.1		1
80.3	even	4		50.4.b.a.49.1		2
80.13	odd	4		400.4.c.c.49.2		2
80.19	odd	4		50.4.a.c.1.1		1
80.29	even	4		400.4.a.b.1.1		1
80.59	odd	4		1600.4.a.d.1.1		1
80.67	even	4		50.4.b.a.49.2		2
80.69	even	4		1600.4.a.bx.1.1		1
80.77	odd	4		400.4.c.c.49.1		2
112.3	even	12		490.4.e.a.471.1		2
112.19	even	12		490.4.e.a.361.1		2
112.51	odd	12		490.4.e.i.361.1		2
112.67	odd	12		490.4.e.i.471.1		2
112.83	even	4		490.4.a.o.1.1		1
144.67	odd	12		810.4.e.c.541.1		2
144.83	even	12		810.4.e.w.271.1		2
144.115	odd	12		810.4.e.c.271.1		2
144.131	even	12		810.4.e.w.541.1		2
176.131	even	4		1210.4.a.b.1.1		1
208.51	odd	4		1690.4.a.a.1.1		1
240.83	odd	4		450.4.c.d.199.2		2
240.179	even	4		450.4.a.q.1.1		1
240.227	odd	4		450.4.c.d.199.1		2
560.419	even	4		2450.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.4.a.a.1.1	✓	1	16.3	odd	4
50.4.a.c.1.1		1	80.19	odd	4
50.4.b.a.49.1		2	80.3	even	4
50.4.b.a.49.2		2	80.67	even	4
80.4.a.f.1.1		1	16.13	even	4
90.4.a.a.1.1		1	48.35	even	4
320.4.a.b.1.1		1	16.5	even	4
320.4.a.m.1.1		1	16.11	odd	4
400.4.a.b.1.1		1	80.29	even	4
400.4.c.c.49.1		2	80.77	odd	4
400.4.c.c.49.2		2	80.13	odd	4
450.4.a.q.1.1		1	240.179	even	4
450.4.c.d.199.1		2	240.227	odd	4
450.4.c.d.199.2		2	240.83	odd	4
490.4.a.o.1.1		1	112.83	even	4
490.4.e.a.361.1		2	112.19	even	12
490.4.e.a.471.1		2	112.3	even	12
490.4.e.i.361.1		2	112.51	odd	12
490.4.e.i.471.1		2	112.67	odd	12
720.4.a.j.1.1		1	48.29	odd	4
810.4.e.c.271.1		2	144.115	odd	12
810.4.e.c.541.1		2	144.67	odd	12
810.4.e.w.271.1		2	144.83	even	12
810.4.e.w.541.1		2	144.131	even	12
1210.4.a.b.1.1		1	176.131	even	4
1280.4.d.g.641.1		2	1.1	even	1	trivial
1280.4.d.g.641.2		2	8.5	even	2	inner
1280.4.d.j.641.1		2	8.3	odd	2
1280.4.d.j.641.2		2	4.3	odd	2
1600.4.a.d.1.1		1	80.59	odd	4
1600.4.a.bx.1.1		1	80.69	even	4
1690.4.a.a.1.1		1	208.51	odd	4
2450.4.a.b.1.1		1	560.419	even	4

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:	\( N \)	\(=\)	\( 1280 = 2^{8} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1280.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-8.00000i q^{3} +5.00000i q^{5} -4.00000 q^{7} -37.0000 q^{9} +O(q^{10})\)	\(q-8.00000i q^{3} +5.00000i q^{5} -4.00000 q^{7} -37.0000 q^{9} -12.0000i q^{11} +58.0000i q^{13} +40.0000 q^{15} +66.0000 q^{17} -100.000i q^{19} +32.0000i q^{21} +132.000 q^{23} -25.0000 q^{25} +80.0000i q^{27} +90.0000i q^{29} -152.000 q^{31} -96.0000 q^{33} -20.0000i q^{35} -34.0000i q^{37} +464.000 q^{39} +438.000 q^{41} -32.0000i q^{43} -185.000i q^{45} +204.000 q^{47} -327.000 q^{49} -528.000i q^{51} +222.000i q^{53} +60.0000 q^{55} -800.000 q^{57} -420.000i q^{59} -902.000i q^{61} +148.000 q^{63} -290.000 q^{65} -1024.00i q^{67} -1056.00i q^{69} +432.000 q^{71} -362.000 q^{73} +200.000i q^{75} +48.0000i q^{77} +160.000 q^{79} -359.000 q^{81} +72.0000i q^{83} +330.000i q^{85} +720.000 q^{87} -810.000 q^{89} -232.000i q^{91} +1216.00i q^{93} +500.000 q^{95} +1106.00 q^{97} +444.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{7} - 74 q^{9}+O(q^{10}) \) 2 * q - 8 * q^7 - 74 * q^9	\( 2 q - 8 q^{7} - 74 q^{9} + 80 q^{15} + 132 q^{17} + 264 q^{23} - 50 q^{25} - 304 q^{31} - 192 q^{33} + 928 q^{39} + 876 q^{41} + 408 q^{47} - 654 q^{49} + 120 q^{55} - 1600 q^{57} + 296 q^{63} - 580 q^{65} + 864 q^{71} - 724 q^{73} + 320 q^{79} - 718 q^{81} + 1440 q^{87} - 1620 q^{89} + 1000 q^{95} + 2212 q^{97}+O(q^{100}) \) 2 * q - 8 * q^7 - 74 * q^9 + 80 * q^15 + 132 * q^17 + 264 * q^23 - 50 * q^25 - 304 * q^31 - 192 * q^33 + 928 * q^39 + 876 * q^41 + 408 * q^47 - 654 * q^49 + 120 * q^55 - 1600 * q^57 + 296 * q^63 - 580 * q^65 + 864 * q^71 - 724 * q^73 + 320 * q^79 - 718 * q^81 + 1440 * q^87 - 1620 * q^89 + 1000 * q^95 + 2212 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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