LMFDB - Embedded newform 480.2.k.b.241.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [480,2,Mod(241,480)]

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([0, 1, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("480.241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$480 = 2^{5} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	480.k (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:	$3.83281929702$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.399424.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8$ x^6 - 2x^5 + 3x^4 - 6x^3 + 6x^2 - 8*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{6}$
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.2
Root			$0.264658 + 1.38923i$ of defining polynomial
Character	$\chi$	$=$	480.241
Dual form			480.2.k.b.241.5

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +1.00000i q^{5} -0.941367 q^{7} -1.00000 q^{9} -4.49828i q^{11} -5.55691i q^{13} +1.00000 q^{15} +7.55691 q^{17} -1.05863i q^{19} +0.941367i q^{21} +1.05863 q^{23} -1.00000 q^{25} +1.00000i q^{27} -2.00000i q^{29} -3.55691 q^{31} -4.49828 q^{33} -0.941367i q^{35} -7.43965i q^{37} -5.55691 q^{39} -3.88273 q^{41} +1.88273i q^{43} -1.00000i q^{45} +10.0552 q^{47} -6.11383 q^{49} -7.55691i q^{51} +2.00000i q^{53} +4.49828 q^{55} -1.05863 q^{57} +8.49828i q^{59} +8.99656i q^{61} +0.941367 q^{63} +5.55691 q^{65} +4.00000i q^{67} -1.05863i q^{69} +12.9966 q^{71} -6.00000 q^{73} +1.00000i q^{75} +4.23453i q^{77} -11.5569 q^{79} +1.00000 q^{81} -5.88273i q^{83} +7.55691i q^{85} -2.00000 q^{87} -4.11727 q^{89} +5.23109i q^{91} +3.55691i q^{93} +1.05863 q^{95} +17.1138 q^{97} +4.49828i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 4 q^{7} - 6 q^{9} + 6 q^{15} + 12 q^{17} + 8 q^{23} - 6 q^{25} + 12 q^{31} + 8 q^{33} - 20 q^{41} - 8 q^{47} + 30 q^{49} - 8 q^{55} - 8 q^{57} + 4 q^{63} + 8 q^{71} - 36 q^{73} - 36 q^{79} + 6 q^{81}+ \cdots + 36 q^{97}+O(q^{100})$ 6 * q - 4 * q^7 - 6 * q^9 + 6 * q^15 + 12 * q^17 + 8 * q^23 - 6 * q^25 + 12 * q^31 + 8 * q^33 - 20 * q^41 - 8 * q^47 + 30 * q^49 - 8 * q^55 - 8 * q^57 + 4 * q^63 + 8 * q^71 - 36 * q^73 - 36 * q^79 + 6 * q^81 - 12 * q^87 - 28 * q^89 + 8 * q^95 + 36 * q^97

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
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	−0.941367	−0.355803	−0.177902	−	0.984048i	$-0.556931\pi$
$7$	−0.941367	−0.355803	−0.177902	+	0.984048i	$0.556931\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	− 4.49828i	− 1.35628i	−0.734931	−	0.678141i	$-0.762786\pi$
$11$	− 4.49828i	− 1.35628i	0.734931	−	0.678141i	$-0.237214\pi$
$12$	0	0
$13$	− 5.55691i	− 1.54121i	−0.637313	−	0.770605i	$-0.719954\pi$
$13$	− 5.55691i	− 1.54121i	0.637313	−	0.770605i	$-0.280046\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	7.55691	1.83282	0.916410	−	0.400240i	$-0.131073\pi$
$17$	7.55691	1.83282	0.916410	+	0.400240i	$0.131073\pi$
$18$	0	0
$19$	− 1.05863i	− 0.242867i	−0.992600	−	0.121434i	$-0.961251\pi$
$19$	− 1.05863i	− 0.242867i	0.992600	−	0.121434i	$-0.0387491\pi$
$20$	0	0
$21$	0.941367i	0.205423i
$22$	0	0
$23$	1.05863	0.220740	0.110370	−	0.993891i	$-0.464796\pi$
$23$	1.05863	0.220740	0.110370	+	0.993891i	$0.464796\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	− 2.00000i	− 0.371391i	−0.982607	−	0.185695i	$-0.940546\pi$
$29$	− 2.00000i	− 0.371391i	0.982607	−	0.185695i	$-0.0594537\pi$
$30$	0	0
$31$	−3.55691	−0.638841	−0.319420	−	0.947613i	$-0.603488\pi$
$31$	−3.55691	−0.638841	−0.319420	+	0.947613i	$0.603488\pi$
$32$	0	0
$33$	−4.49828	−0.783050
$34$	0	0
$35$	− 0.941367i	− 0.159120i
$36$	0	0
$37$	− 7.43965i	− 1.22307i	−0.791217	−	0.611535i	$-0.790552\pi$
$37$	− 7.43965i	− 1.22307i	0.791217	−	0.611535i	$-0.209448\pi$
$38$	0	0
$39$	−5.55691	−0.889818
$40$	0	0
$41$	−3.88273	−0.606381	−0.303191	−	0.952930i	$-0.598052\pi$
$41$	−3.88273	−0.606381	−0.303191	+	0.952930i	$0.598052\pi$
$42$	0	0
$43$	1.88273i	0.287114i	0.989642	+	0.143557i	$0.0458541\pi$
$43$	1.88273i	0.287114i	−0.989642	+	0.143557i	$0.954146\pi$
$44$	0	0
$45$	− 1.00000i	− 0.149071i
$46$	0	0
$47$	10.0552	1.46670	0.733350	−	0.679851i	$-0.237955\pi$
$47$	10.0552	1.46670	0.733350	+	0.679851i	$0.237955\pi$
$48$	0	0
$49$	−6.11383	−0.873404
$50$	0	0
$51$	− 7.55691i	− 1.05818i
$52$	0	0
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	4.49828	0.606548
$56$	0	0
$57$	−1.05863	−0.140219
$58$	0	0
$59$	8.49828i	1.10638i	0.833054	+	0.553191i	$0.186590\pi$
$59$	8.49828i	1.10638i	−0.833054	+	0.553191i	$0.813410\pi$
$60$	0	0
$61$	8.99656i	1.15189i	0.817488	+	0.575946i	$0.195366\pi$
$61$	8.99656i	1.15189i	−0.817488	+	0.575946i	$0.804634\pi$
$62$	0	0
$63$	0.941367	0.118601
$64$	0	0
$65$	5.55691	0.689250
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	0	0
$69$	− 1.05863i	− 0.127444i
$70$	0	0
$71$	12.9966	1.54241	0.771204	−	0.636588i	$-0.219655\pi$
$71$	12.9966	1.54241	0.771204	+	0.636588i	$0.219655\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	1.00000i	0.115470i
$76$	0	0
$77$	4.23453i	0.482570i
$78$	0	0
$79$	−11.5569	−1.30025	−0.650127	−	0.759825i	$-0.725284\pi$
$79$	−11.5569	−1.30025	−0.650127	+	0.759825i	$0.725284\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 5.88273i	− 0.645714i	−0.946448	−	0.322857i	$-0.895357\pi$
$83$	− 5.88273i	− 0.645714i	0.946448	−	0.322857i	$-0.104643\pi$
$84$	0	0
$85$	7.55691i	0.819662i
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−4.11727	−0.436429	−0.218215	−	0.975901i	$-0.570023\pi$
$89$	−4.11727	−0.436429	−0.218215	+	0.975901i	$0.570023\pi$
$90$	0	0
$91$	5.23109i	0.548368i
$92$	0	0
$93$	3.55691i	0.368835i
$94$	0	0
$95$	1.05863	0.108613
$96$	0	0
$97$	17.1138	1.73765	0.868823	−	0.495123i	$-0.164877\pi$
$97$	17.1138	1.73765	0.868823	+	0.495123i	$0.164877\pi$
$98$	0	0
$99$	4.49828i	0.452094i
$100$	0	0
$101$	2.00000i	0.199007i	0.995037	+	0.0995037i	$0.0317255\pi$
$101$	2.00000i	0.199007i	−0.995037	+	0.0995037i	$0.968274\pi$
$102$	0	0
$103$	−10.1725	−1.00232	−0.501161	−	0.865354i	$-0.667094\pi$
$103$	−10.1725	−1.00232	−0.501161	+	0.865354i	$0.667094\pi$
$104$	0	0
$105$	−0.941367	−0.0918680
$106$	0	0
$107$	17.2311i	1.66579i	0.553429	+	0.832896i	$0.313319\pi$
$107$	17.2311i	1.66579i	−0.553429	+	0.832896i	$0.686681\pi$
$108$	0	0
$109$	1.88273i	0.180333i	0.995927	+	0.0901666i	$0.0287399\pi$
$109$	1.88273i	0.180333i	−0.995927	+	0.0901666i	$0.971260\pi$
$110$	0	0
$111$	−7.43965	−0.706140
$112$	0	0
$113$	15.3224	1.44141	0.720704	−	0.693243i	$-0.243819\pi$
$113$	15.3224	1.44141	0.720704	+	0.693243i	$0.243819\pi$
$114$	0	0
$115$	1.05863i	0.0987181i
$116$	0	0
$117$	5.55691i	0.513737i
$118$	0	0
$119$	−7.11383	−0.652124
$120$	0	0
$121$	−9.23453	−0.839503
$122$	0	0
$123$	3.88273i	0.350094i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	18.1725	1.61255	0.806273	−	0.591544i	$-0.201481\pi$
$127$	18.1725	1.61255	0.806273	+	0.591544i	$0.201481\pi$
$128$	0	0
$129$	1.88273	0.165765
$130$	0	0
$131$	6.38101i	0.557512i	0.960362	+	0.278756i	$0.0899220\pi$
$131$	6.38101i	0.557512i	−0.960362	+	0.278756i	$0.910078\pi$
$132$	0	0
$133$	0.996562i	0.0864129i
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−4.44309	−0.379598	−0.189799	−	0.981823i	$-0.560784\pi$
$137$	−4.44309	−0.379598	−0.189799	+	0.981823i	$0.560784\pi$
$138$	0	0
$139$	− 20.1725i	− 1.71101i	−0.517798	−	0.855503i	$-0.673248\pi$
$139$	− 20.1725i	− 1.71101i	0.517798	−	0.855503i	$-0.326752\pi$
$140$	0	0
$141$	− 10.0552i	− 0.846800i
$142$	0	0
$143$	−24.9966	−2.09032
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	6.11383i	0.504260i
$148$	0	0
$149$	− 2.00000i	− 0.163846i	−0.996639	−	0.0819232i	$-0.973894\pi$
$149$	− 2.00000i	− 0.163846i	0.996639	−	0.0819232i	$-0.0261062\pi$
$150$	0	0
$151$	−9.67418	−0.787274	−0.393637	−	0.919266i	$-0.628783\pi$
$151$	−9.67418	−0.787274	−0.393637	+	0.919266i	$0.628783\pi$
$152$	0	0
$153$	−7.55691	−0.610940
$154$	0	0
$155$	− 3.55691i	− 0.285698i
$156$	0	0
$157$	4.32582i	0.345238i	0.984989	+	0.172619i	$0.0552229\pi$
$157$	4.32582i	0.345238i	−0.984989	+	0.172619i	$0.944777\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	−0.996562	−0.0785401
$162$	0	0
$163$	6.11727i	0.479141i	0.970879	+	0.239571i	$0.0770067\pi$
$163$	6.11727i	0.479141i	−0.970879	+	0.239571i	$0.922993\pi$
$164$	0	0
$165$	− 4.49828i	− 0.350191i
$166$	0	0
$167$	6.05520	0.468565	0.234283	−	0.972169i	$-0.424726\pi$
$167$	6.05520	0.468565	0.234283	+	0.972169i	$0.424726\pi$
$168$	0	0
$169$	−17.8793	−1.37533
$170$	0	0
$171$	1.05863i	0.0809557i
$172$	0	0
$173$	16.8793i	1.28331i	0.766994	+	0.641655i	$0.221752\pi$
$173$	16.8793i	1.28331i	−0.766994	+	0.641655i	$0.778248\pi$
$174$	0	0
$175$	0.941367	0.0711606
$176$	0	0
$177$	8.49828	0.638770
$178$	0	0
$179$	− 10.6155i	− 0.793443i	−0.917939	−	0.396722i	$-0.870148\pi$
$179$	− 10.6155i	− 0.793443i	0.917939	−	0.396722i	$-0.129852\pi$
$180$	0	0
$181$	14.1173i	1.04933i	0.851309	+	0.524664i	$0.175809\pi$
$181$	14.1173i	1.04933i	−0.851309	+	0.524664i	$0.824191\pi$
$182$	0	0
$183$	8.99656	0.665045
$184$	0	0
$185$	7.43965	0.546974
$186$	0	0
$187$	− 33.9931i	− 2.48582i
$188$	0	0
$189$	− 0.941367i	− 0.0684744i
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	−4.87930	−0.351219	−0.175610	−	0.984460i	$-0.556190\pi$
$193$	−4.87930	−0.351219	−0.175610	+	0.984460i	$0.556190\pi$
$194$	0	0
$195$	− 5.55691i	− 0.397939i
$196$	0	0
$197$	− 2.88617i	− 0.205631i	−0.994700	−	0.102816i	$-0.967215\pi$
$197$	− 2.88617i	− 0.205631i	0.994700	−	0.102816i	$-0.0327852\pi$
$198$	0	0
$199$	17.6742	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$199$	17.6742	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	1.88273i	0.132142i
$204$	0	0
$205$	− 3.88273i	− 0.271182i
$206$	0	0
$207$	−1.05863	−0.0735801
$208$	0	0
$209$	−4.76203	−0.329396
$210$	0	0
$211$	23.9379i	1.64795i	0.566623	+	0.823977i	$0.308250\pi$
$211$	23.9379i	1.64795i	−0.566623	+	0.823977i	$0.691750\pi$
$212$	0	0
$213$	− 12.9966i	− 0.890510i
$214$	0	0
$215$	−1.88273	−0.128401
$216$	0	0
$217$	3.34836	0.227302
$218$	0	0
$219$	6.00000i	0.405442i
$220$	0	0
$221$	− 41.9931i	− 2.82476i
$222$	0	0
$223$	24.0552	1.61086	0.805428	−	0.592694i	$-0.201936\pi$
$223$	24.0552	1.61086	0.805428	+	0.592694i	$0.201936\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	− 11.1138i	− 0.737651i	−0.929499	−	0.368825i	$-0.879760\pi$
$227$	− 11.1138i	− 0.737651i	0.929499	−	0.368825i	$-0.120240\pi$
$228$	0	0
$229$	− 17.2311i	− 1.13866i	−0.822108	−	0.569331i	$-0.807202\pi$
$229$	− 17.2311i	− 1.13866i	0.822108	−	0.569331i	$-0.192798\pi$
$230$	0	0
$231$	4.23453	0.278612
$232$	0	0
$233$	−8.44309	−0.553125	−0.276562	−	0.960996i	$-0.589195\pi$
$233$	−8.44309	−0.553125	−0.276562	+	0.960996i	$0.589195\pi$
$234$	0	0
$235$	10.0552i	0.655929i
$236$	0	0
$237$	11.5569i	0.750702i
$238$	0	0
$239$	−10.1173	−0.654432	−0.327216	−	0.944950i	$-0.606110\pi$
$239$	−10.1173	−0.654432	−0.327216	+	0.944950i	$0.606110\pi$
$240$	0	0
$241$	16.8793	1.08729	0.543646	−	0.839315i	$-0.317044\pi$
$241$	16.8793	1.08729	0.543646	+	0.839315i	$0.317044\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	− 6.11383i	− 0.390598i
$246$	0	0
$247$	−5.88273	−0.374309
$248$	0	0
$249$	−5.88273	−0.372803
$250$	0	0
$251$	− 11.8466i	− 0.747753i	−0.927478	−	0.373877i	$-0.878028\pi$
$251$	− 11.8466i	− 0.747753i	0.927478	−	0.373877i	$-0.121972\pi$
$252$	0	0
$253$	− 4.76203i	− 0.299386i
$254$	0	0
$255$	7.55691	0.473232
$256$	0	0
$257$	−10.6707	−0.665623	−0.332811	−	0.942993i	$-0.607997\pi$
$257$	−10.6707	−0.665623	−0.332811	+	0.942993i	$0.607997\pi$
$258$	0	0
$259$	7.00344i	0.435172i
$260$	0	0
$261$	2.00000i	0.123797i
$262$	0	0
$263$	1.94480	0.119922	0.0599609	−	0.998201i	$-0.480902\pi$
$263$	1.94480	0.119922	0.0599609	+	0.998201i	$0.480902\pi$
$264$	0	0
$265$	−2.00000	−0.122859
$266$	0	0
$267$	4.11727i	0.251973i
$268$	0	0
$269$	9.76547i	0.595411i	0.954658	+	0.297706i	$0.0962214\pi$
$269$	9.76547i	0.595411i	−0.954658	+	0.297706i	$0.903779\pi$
$270$	0	0
$271$	−3.44652	−0.209361	−0.104681	−	0.994506i	$-0.533382\pi$
$271$	−3.44652	−0.209361	−0.104681	+	0.994506i	$0.533382\pi$
$272$	0	0
$273$	5.23109	0.316600
$274$	0	0
$275$	4.49828i	0.271257i
$276$	0	0
$277$	− 18.7880i	− 1.12886i	−0.825480	−	0.564431i	$-0.809096\pi$
$277$	− 18.7880i	− 1.12886i	0.825480	−	0.564431i	$-0.190904\pi$
$278$	0	0
$279$	3.55691	0.212947
$280$	0	0
$281$	−16.8793	−1.00693	−0.503467	−	0.864014i	$-0.667943\pi$
$281$	−16.8793	−1.00693	−0.503467	+	0.864014i	$0.667943\pi$
$282$	0	0
$283$	− 20.0000i	− 1.18888i	−0.804141	−	0.594438i	$-0.797374\pi$
$283$	− 20.0000i	− 1.18888i	0.804141	−	0.594438i	$-0.202626\pi$
$284$	0	0
$285$	− 1.05863i	− 0.0627080i
$286$	0	0
$287$	3.65508	0.215752
$288$	0	0
$289$	40.1070	2.35923
$290$	0	0
$291$	− 17.1138i	− 1.00323i
$292$	0	0
$293$	− 20.2277i	− 1.18171i	−0.806777	−	0.590856i	$-0.798790\pi$
$293$	− 20.2277i	− 1.18171i	0.806777	−	0.590856i	$-0.201210\pi$
$294$	0	0
$295$	−8.49828	−0.494789
$296$	0	0
$297$	4.49828	0.261017
$298$	0	0
$299$	− 5.88273i	− 0.340207i
$300$	0	0
$301$	− 1.77234i	− 0.102156i
$302$	0	0
$303$	2.00000	0.114897
$304$	0	0
$305$	−8.99656	−0.515142
$306$	0	0
$307$	− 8.11039i	− 0.462884i	−0.972849	−	0.231442i	$-0.925656\pi$
$307$	− 8.11039i	− 0.462884i	0.972849	−	0.231442i	$-0.0743444\pi$
$308$	0	0
$309$	10.1725i	0.578691i
$310$	0	0
$311$	−31.8759	−1.80751	−0.903757	−	0.428046i	$-0.859202\pi$
$311$	−31.8759	−1.80751	−0.903757	+	0.428046i	$0.859202\pi$
$312$	0	0
$313$	−5.11383	−0.289051	−0.144525	−	0.989501i	$-0.546166\pi$
$313$	−5.11383	−0.289051	−0.144525	+	0.989501i	$0.546166\pi$
$314$	0	0
$315$	0.941367i	0.0530400i
$316$	0	0
$317$	24.6448i	1.38419i	0.721807	+	0.692094i	$0.243312\pi$
$317$	24.6448i	1.38419i	−0.721807	+	0.692094i	$0.756688\pi$
$318$	0	0
$319$	−8.99656	−0.503711
$320$	0	0
$321$	17.2311	0.961746
$322$	0	0
$323$	− 8.00000i	− 0.445132i
$324$	0	0
$325$	5.55691i	0.308242i
$326$	0	0
$327$	1.88273	0.104115
$328$	0	0
$329$	−9.46563	−0.521857
$330$	0	0
$331$	11.0518i	0.607460i	0.952758	+	0.303730i	$0.0982320\pi$
$331$	11.0518i	0.607460i	−0.952758	+	0.303730i	$0.901768\pi$
$332$	0	0
$333$	7.43965i	0.407690i
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−19.9931	−1.08909	−0.544547	−	0.838730i	$-0.683299\pi$
$337$	−19.9931	−1.08909	−0.544547	+	0.838730i	$0.683299\pi$
$338$	0	0
$339$	− 15.3224i	− 0.832198i
$340$	0	0
$341$	16.0000i	0.866449i
$342$	0	0
$343$	12.3449	0.666563
$344$	0	0
$345$	1.05863	0.0569949
$346$	0	0
$347$	6.87930i	0.369300i	0.982804	+	0.184650i	$0.0591151\pi$
$347$	6.87930i	0.369300i	−0.982804	+	0.184650i	$0.940885\pi$
$348$	0	0
$349$	− 4.76203i	− 0.254906i	−0.991845	−	0.127453i	$-0.959320\pi$
$349$	− 4.76203i	− 0.254906i	0.991845	−	0.127453i	$-0.0406801\pi$
$350$	0	0
$351$	5.55691	0.296606
$352$	0	0
$353$	−3.79145	−0.201798	−0.100899	−	0.994897i	$-0.532172\pi$
$353$	−3.79145	−0.201798	−0.100899	+	0.994897i	$0.532172\pi$
$354$	0	0
$355$	12.9966i	0.689786i
$356$	0	0
$357$	7.11383i	0.376504i
$358$	0	0
$359$	12.9966	0.685932	0.342966	−	0.939348i	$-0.388568\pi$
$359$	12.9966	0.685932	0.342966	+	0.939348i	$0.388568\pi$
$360$	0	0
$361$	17.8793	0.941016
$362$	0	0
$363$	9.23453i	0.484687i
$364$	0	0
$365$	− 6.00000i	− 0.314054i
$366$	0	0
$367$	−22.9345	−1.19717	−0.598585	−	0.801059i	$-0.704270\pi$
$367$	−22.9345	−1.19717	−0.598585	+	0.801059i	$0.704270\pi$
$368$	0	0
$369$	3.88273	0.202127
$370$	0	0
$371$	− 1.88273i	− 0.0977467i
$372$	0	0
$373$	15.4396i	0.799435i	0.916638	+	0.399717i	$0.130892\pi$
$373$	15.4396i	0.799435i	−0.916638	+	0.399717i	$0.869108\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−11.1138	−0.572391
$378$	0	0
$379$	6.28973i	0.323082i	0.986866	+	0.161541i	$0.0516463\pi$
$379$	6.28973i	0.323082i	−0.986866	+	0.161541i	$0.948354\pi$
$380$	0	0
$381$	− 18.1725i	− 0.931003i
$382$	0	0
$383$	−2.94137	−0.150297	−0.0751484	−	0.997172i	$-0.523943\pi$
$383$	−2.94137	−0.150297	−0.0751484	+	0.997172i	$0.523943\pi$
$384$	0	0
$385$	−4.23453	−0.215812
$386$	0	0
$387$	− 1.88273i	− 0.0957047i
$388$	0	0
$389$	12.2277i	0.619967i	0.950742	+	0.309983i	$0.100324\pi$
$389$	12.2277i	0.619967i	−0.950742	+	0.309983i	$0.899676\pi$
$390$	0	0
$391$	8.00000	0.404577
$392$	0	0
$393$	6.38101	0.321880
$394$	0	0
$395$	− 11.5569i	− 0.581491i
$396$	0	0
$397$	− 5.32238i	− 0.267123i	−0.991041	−	0.133561i	$-0.957359\pi$
$397$	− 5.32238i	− 0.267123i	0.991041	−	0.133561i	$-0.0426413\pi$
$398$	0	0
$399$	0.996562	0.0498905
$400$	0	0
$401$	6.99656	0.349392	0.174696	−	0.984622i	$-0.444106\pi$
$401$	6.99656	0.349392	0.174696	+	0.984622i	$0.444106\pi$
$402$	0	0
$403$	19.7655i	0.984588i
$404$	0	0
$405$	1.00000i	0.0496904i
$406$	0	0
$407$	−33.4656	−1.65883
$408$	0	0
$409$	16.2277	0.802406	0.401203	−	0.915989i	$-0.368592\pi$
$409$	16.2277	0.802406	0.401203	+	0.915989i	$0.368592\pi$
$410$	0	0
$411$	4.44309i	0.219161i
$412$	0	0
$413$	− 8.00000i	− 0.393654i
$414$	0	0
$415$	5.88273	0.288772
$416$	0	0
$417$	−20.1725	−0.987850
$418$	0	0
$419$	15.6121i	0.762701i	0.924430	+	0.381351i	$0.124541\pi$
$419$	15.6121i	0.762701i	−0.924430	+	0.381351i	$0.875459\pi$
$420$	0	0
$421$	33.2311i	1.61958i	0.586717	+	0.809792i	$0.300420\pi$
$421$	33.2311i	1.61958i	−0.586717	+	0.809792i	$0.699580\pi$
$422$	0	0
$423$	−10.0552	−0.488900
$424$	0	0
$425$	−7.55691	−0.366564
$426$	0	0
$427$	− 8.46907i	− 0.409847i
$428$	0	0
$429$	24.9966i	1.20685i
$430$	0	0
$431$	12.9966	0.626022	0.313011	−	0.949749i	$-0.398662\pi$
$431$	12.9966	0.626022	0.313011	+	0.949749i	$0.398662\pi$
$432$	0	0
$433$	20.2277	0.972079	0.486040	−	0.873937i	$-0.338441\pi$
$433$	20.2277	0.972079	0.486040	+	0.873937i	$0.338441\pi$
$434$	0	0
$435$	− 2.00000i	− 0.0958927i
$436$	0	0
$437$	− 1.12070i	− 0.0536106i
$438$	0	0
$439$	5.43965	0.259620	0.129810	−	0.991539i	$-0.458563\pi$
$439$	5.43965	0.259620	0.129810	+	0.991539i	$0.458563\pi$
$440$	0	0
$441$	6.11383	0.291135
$442$	0	0
$443$	15.3484i	0.729223i	0.931160	+	0.364611i	$0.118798\pi$
$443$	15.3484i	0.729223i	−0.931160	+	0.364611i	$0.881202\pi$
$444$	0	0
$445$	− 4.11727i	− 0.195177i
$446$	0	0
$447$	−2.00000	−0.0945968
$448$	0	0
$449$	4.22766	0.199515	0.0997577	−	0.995012i	$-0.468193\pi$
$449$	4.22766	0.199515	0.0997577	+	0.995012i	$0.468193\pi$
$450$	0	0
$451$	17.4656i	0.822424i
$452$	0	0
$453$	9.67418i	0.454533i
$454$	0	0
$455$	−5.23109	−0.245238
$456$	0	0
$457$	2.65164	0.124038	0.0620192	−	0.998075i	$-0.480246\pi$
$457$	2.65164	0.124038	0.0620192	+	0.998075i	$0.480246\pi$
$458$	0	0
$459$	7.55691i	0.352727i
$460$	0	0
$461$	− 10.2345i	− 0.476670i	−0.971183	−	0.238335i	$-0.923398\pi$
$461$	− 10.2345i	− 0.476670i	0.971183	−	0.238335i	$-0.0766016\pi$
$462$	0	0
$463$	−19.0586	−0.885730	−0.442865	−	0.896588i	$-0.646038\pi$
$463$	−19.0586	−0.885730	−0.442865	+	0.896588i	$0.646038\pi$
$464$	0	0
$465$	−3.55691	−0.164948
$466$	0	0
$467$	4.11039i	0.190206i	0.995467	+	0.0951031i	$0.0303181\pi$
$467$	4.11039i	0.190206i	−0.995467	+	0.0951031i	$0.969682\pi$
$468$	0	0
$469$	− 3.76547i	− 0.173873i
$470$	0	0
$471$	4.32582	0.199323
$472$	0	0
$473$	8.46907	0.389408
$474$	0	0
$475$	1.05863i	0.0485734i
$476$	0	0
$477$	− 2.00000i	− 0.0915737i
$478$	0	0
$479$	25.2311	1.15284	0.576419	−	0.817154i	$-0.304450\pi$
$479$	25.2311	1.15284	0.576419	+	0.817154i	$0.304450\pi$
$480$	0	0
$481$	−41.3415	−1.88501
$482$	0	0
$483$	0.996562i	0.0453451i
$484$	0	0
$485$	17.1138i	0.777099i
$486$	0	0
$487$	21.9379	0.994102	0.497051	−	0.867721i	$-0.334416\pi$
$487$	21.9379	0.994102	0.497051	+	0.867721i	$0.334416\pi$
$488$	0	0
$489$	6.11727	0.276632
$490$	0	0
$491$	7.50172i	0.338548i	0.985569	+	0.169274i	$0.0541423\pi$
$491$	7.50172i	0.338548i	−0.985569	+	0.169274i	$0.945858\pi$
$492$	0	0
$493$	− 15.1138i	− 0.680693i
$494$	0	0
$495$	−4.49828	−0.202183
$496$	0	0
$497$	−12.2345	−0.548794
$498$	0	0
$499$	29.1690i	1.30578i	0.757451	+	0.652892i	$0.226445\pi$
$499$	29.1690i	1.30578i	−0.757451	+	0.652892i	$0.773555\pi$
$500$	0	0
$501$	− 6.05520i	− 0.270526i
$502$	0	0
$503$	23.9379	1.06734	0.533670	−	0.845693i	$-0.320813\pi$
$503$	23.9379	1.06734	0.533670	+	0.845693i	$0.320813\pi$
$504$	0	0
$505$	−2.00000	−0.0889988
$506$	0	0
$507$	17.8793i	0.794047i
$508$	0	0
$509$	28.6967i	1.27196i	0.771706	+	0.635980i	$0.219404\pi$
$509$	28.6967i	1.27196i	−0.771706	+	0.635980i	$0.780596\pi$
$510$	0	0
$511$	5.64820	0.249862
$512$	0	0
$513$	1.05863	0.0467398
$514$	0	0
$515$	− 10.1725i	− 0.448252i
$516$	0	0
$517$	− 45.2311i	− 1.98926i
$518$	0	0
$519$	16.8793	0.740919
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	− 25.7586i	− 1.12634i	−0.826340	−	0.563172i	$-0.809581\pi$
$523$	− 25.7586i	− 1.12634i	0.826340	−	0.563172i	$-0.190419\pi$
$524$	0	0
$525$	− 0.941367i	− 0.0410846i
$526$	0	0
$527$	−26.8793	−1.17088
$528$	0	0
$529$	−21.8793	−0.951274
$530$	0	0
$531$	− 8.49828i	− 0.368794i
$532$	0	0
$533$	21.5760i	0.934561i
$534$	0	0
$535$	−17.2311	−0.744965
$536$	0	0
$537$	−10.6155	−0.458095
$538$	0	0
$539$	27.5017i	1.18458i
$540$	0	0
$541$	12.3449i	0.530750i	0.964145	+	0.265375i	$0.0854957\pi$
$541$	12.3449i	0.530750i	−0.964145	+	0.265375i	$0.914504\pi$
$542$	0	0
$543$	14.1173	0.605830
$544$	0	0
$545$	−1.88273	−0.0806475
$546$	0	0
$547$	19.8759i	0.849830i	0.905233	+	0.424915i	$0.139696\pi$
$547$	19.8759i	0.849830i	−0.905233	+	0.424915i	$0.860304\pi$
$548$	0	0
$549$	− 8.99656i	− 0.383964i
$550$	0	0
$551$	−2.11727	−0.0901986
$552$	0	0
$553$	10.8793	0.462635
$554$	0	0
$555$	− 7.43965i	− 0.315795i
$556$	0	0
$557$	3.12070i	0.132228i	0.997812	+	0.0661142i	$0.0210602\pi$
$557$	3.12070i	0.132228i	−0.997812	+	0.0661142i	$0.978940\pi$
$558$	0	0
$559$	10.4622	0.442503
$560$	0	0
$561$	−33.9931	−1.43519
$562$	0	0
$563$	0.651639i	0.0274633i	0.999906	+	0.0137317i	$0.00437106\pi$
$563$	0.651639i	0.0274633i	−0.999906	+	0.0137317i	$0.995629\pi$
$564$	0	0
$565$	15.3224i	0.644617i
$566$	0	0
$567$	−0.941367	−0.0395337
$568$	0	0
$569$	26.9966	1.13175	0.565877	−	0.824489i	$-0.308538\pi$
$569$	26.9966	1.13175	0.565877	+	0.824489i	$0.308538\pi$
$570$	0	0
$571$	− 14.9414i	− 0.625277i	−0.949872	−	0.312638i	$-0.898787\pi$
$571$	− 14.9414i	− 0.625277i	0.949872	−	0.312638i	$-0.101213\pi$
$572$	0	0
$573$	− 8.00000i	− 0.334205i
$574$	0	0
$575$	−1.05863	−0.0441481
$576$	0	0
$577$	−8.87930	−0.369650	−0.184825	−	0.982771i	$-0.559172\pi$
$577$	−8.87930	−0.369650	−0.184825	+	0.982771i	$0.559172\pi$
$578$	0	0
$579$	4.87930i	0.202777i
$580$	0	0
$581$	5.53781i	0.229747i
$582$	0	0
$583$	8.99656	0.372600
$584$	0	0
$585$	−5.55691	−0.229750
$586$	0	0
$587$	− 1.23109i	− 0.0508127i	−0.999677	−	0.0254064i	$-0.991912\pi$
$587$	− 1.23109i	− 0.0508127i	0.999677	−	0.0254064i	$-0.00808797\pi$
$588$	0	0
$589$	3.76547i	0.155153i
$590$	0	0
$591$	−2.88617	−0.118721
$592$	0	0
$593$	3.55691	0.146065	0.0730325	−	0.997330i	$-0.476732\pi$
$593$	3.55691	0.146065	0.0730325	+	0.997330i	$0.476732\pi$
$594$	0	0
$595$	− 7.11383i	− 0.291639i
$596$	0	0
$597$	− 17.6742i	− 0.723356i
$598$	0	0
$599$	19.2242	0.785480	0.392740	−	0.919649i	$-0.371527\pi$
$599$	19.2242	0.785480	0.392740	+	0.919649i	$0.371527\pi$
$600$	0	0
$601$	−27.7586	−1.13230	−0.566148	−	0.824303i	$-0.691567\pi$
$601$	−27.7586	−1.13230	−0.566148	+	0.824303i	$0.691567\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	0	0
$605$	− 9.23453i	− 0.375437i
$606$	0	0
$607$	−7.16902	−0.290982	−0.145491	−	0.989360i	$-0.546476\pi$
$607$	−7.16902	−0.290982	−0.145491	+	0.989360i	$0.546476\pi$
$608$	0	0
$609$	1.88273	0.0762922
$610$	0	0
$611$	− 55.8759i	− 2.26050i
$612$	0	0
$613$	9.55691i	0.386000i	0.981199	+	0.193000i	$0.0618218\pi$
$613$	9.55691i	0.386000i	−0.981199	+	0.193000i	$0.938178\pi$
$614$	0	0
$615$	−3.88273	−0.156567
$616$	0	0
$617$	1.32926	0.0535139	0.0267569	−	0.999642i	$-0.491482\pi$
$617$	1.32926	0.0535139	0.0267569	+	0.999642i	$0.491482\pi$
$618$	0	0
$619$	− 28.1725i	− 1.13235i	−0.824286	−	0.566173i	$-0.808423\pi$
$619$	− 28.1725i	− 1.13235i	0.824286	−	0.566173i	$-0.191577\pi$
$620$	0	0
$621$	1.05863i	0.0424815i
$622$	0	0
$623$	3.87586	0.155283
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.76203i	0.190177i
$628$	0	0
$629$	− 56.2208i	− 2.24167i
$630$	0	0
$631$	23.3224	0.928449	0.464225	−	0.885717i	$-0.346333\pi$
$631$	23.3224	0.928449	0.464225	+	0.885717i	$0.346333\pi$
$632$	0	0
$633$	23.9379	0.951447
$634$	0	0
$635$	18.1725i	0.721152i
$636$	0	0
$637$	33.9740i	1.34610i
$638$	0	0
$639$	−12.9966	−0.514136
$640$	0	0
$641$	27.1070	1.07066	0.535330	−	0.844643i	$-0.320187\pi$
$641$	27.1070	1.07066	0.535330	+	0.844643i	$0.320187\pi$
$642$	0	0
$643$	− 20.3449i	− 0.802325i	−0.916007	−	0.401163i	$-0.868606\pi$
$643$	− 20.3449i	− 0.802325i	0.916007	−	0.401163i	$-0.131394\pi$
$644$	0	0
$645$	1.88273i	0.0741326i
$646$	0	0
$647$	−37.6965	−1.48200	−0.741002	−	0.671503i	$-0.765649\pi$
$647$	−37.6965	−1.48200	−0.741002	+	0.671503i	$0.765649\pi$
$648$	0	0
$649$	38.2277	1.50057
$650$	0	0
$651$	− 3.34836i	− 0.131233i
$652$	0	0
$653$	− 8.64476i	− 0.338296i	−0.985591	−	0.169148i	$-0.945898\pi$
$653$	− 8.64476i	− 0.338296i	0.985591	−	0.169148i	$-0.0541015\pi$
$654$	0	0
$655$	−6.38101	−0.249327
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	− 29.2603i	− 1.13982i	−0.821707	−	0.569910i	$-0.806978\pi$
$659$	− 29.2603i	− 1.13982i	0.821707	−	0.569910i	$-0.193022\pi$
$660$	0	0
$661$	28.7620i	1.11871i	0.828927	+	0.559357i	$0.188952\pi$
$661$	28.7620i	1.11871i	−0.828927	+	0.559357i	$0.811048\pi$
$662$	0	0
$663$	−41.9931	−1.63088
$664$	0	0
$665$	−0.996562	−0.0386450
$666$	0	0
$667$	− 2.11727i	− 0.0819809i
$668$	0	0
$669$	− 24.0552i	− 0.930028i
$670$	0	0
$671$	40.4691	1.56229
$672$	0	0
$673$	−18.0000	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$674$	0	0
$675$	− 1.00000i	− 0.0384900i
$676$	0	0
$677$	− 42.8724i	− 1.64772i	−0.566793	−	0.823860i	$-0.691816\pi$
$677$	− 42.8724i	− 1.64772i	0.566793	−	0.823860i	$-0.308184\pi$
$678$	0	0
$679$	−16.1104	−0.618260
$680$	0	0
$681$	−11.1138	−0.425883
$682$	0	0
$683$	26.1173i	0.999349i	0.866213	+	0.499675i	$0.166547\pi$
$683$	26.1173i	0.999349i	−0.866213	+	0.499675i	$0.833453\pi$
$684$	0	0
$685$	− 4.44309i	− 0.169762i
$686$	0	0
$687$	−17.2311	−0.657407
$688$	0	0
$689$	11.1138	0.423403
$690$	0	0
$691$	5.29317i	0.201362i	0.994919	+	0.100681i	$0.0321021\pi$
$691$	5.29317i	0.201362i	−0.994919	+	0.100681i	$0.967898\pi$
$692$	0	0
$693$	− 4.23453i	− 0.160857i
$694$	0	0
$695$	20.1725	0.765185
$696$	0	0
$697$	−29.3415	−1.11139
$698$	0	0
$699$	8.44309i	0.319347i
$700$	0	0
$701$	7.99312i	0.301896i	0.988542	+	0.150948i	$0.0482326\pi$
$701$	7.99312i	0.301896i	−0.988542	+	0.150948i	$0.951767\pi$
$702$	0	0
$703$	−7.87586	−0.297044
$704$	0	0
$705$	10.0552	0.378701
$706$	0	0
$707$	− 1.88273i	− 0.0708075i
$708$	0	0
$709$	− 28.9966i	− 1.08899i	−0.838764	−	0.544494i	$-0.816722\pi$
$709$	− 28.9966i	− 1.08899i	0.838764	−	0.544494i	$-0.183278\pi$
$710$	0	0
$711$	11.5569	0.433418
$712$	0	0
$713$	−3.76547	−0.141018
$714$	0	0
$715$	− 24.9966i	− 0.934818i
$716$	0	0
$717$	10.1173i	0.377836i
$718$	0	0
$719$	−26.8793	−1.00243	−0.501214	−	0.865323i	$-0.667113\pi$
$719$	−26.8793	−1.00243	−0.501214	+	0.865323i	$0.667113\pi$
$720$	0	0
$721$	9.57602	0.356630
$722$	0	0
$723$	− 16.8793i	− 0.627748i
$724$	0	0
$725$	2.00000i	0.0742781i
$726$	0	0
$727$	−41.8138	−1.55079	−0.775394	−	0.631478i	$-0.782449\pi$
$727$	−41.8138	−1.55079	−0.775394	+	0.631478i	$0.782449\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	14.2277i	0.526229i
$732$	0	0
$733$	30.0844i	1.11119i	0.831452	+	0.555597i	$0.187510\pi$
$733$	30.0844i	1.11119i	−0.831452	+	0.555597i	$0.812490\pi$
$734$	0	0
$735$	−6.11383	−0.225512
$736$	0	0
$737$	17.9931	0.662785
$738$	0	0
$739$	− 29.0449i	− 1.06843i	−0.845348	−	0.534217i	$-0.820607\pi$
$739$	− 29.0449i	− 1.06843i	0.845348	−	0.534217i	$-0.179393\pi$
$740$	0	0
$741$	5.88273i	0.216108i
$742$	0	0
$743$	43.2863	1.58802	0.794010	−	0.607905i	$-0.207990\pi$
$743$	43.2863	1.58802	0.794010	+	0.607905i	$0.207990\pi$
$744$	0	0
$745$	2.00000	0.0732743
$746$	0	0
$747$	5.88273i	0.215238i
$748$	0	0
$749$	− 16.2208i	− 0.592694i
$750$	0	0
$751$	−41.7846	−1.52474	−0.762370	−	0.647141i	$-0.775964\pi$
$751$	−41.7846	−1.52474	−0.762370	+	0.647141i	$0.775964\pi$
$752$	0	0
$753$	−11.8466	−0.431716
$754$	0	0
$755$	− 9.67418i	− 0.352079i
$756$	0	0
$757$	− 16.3258i	− 0.593372i	−0.954975	−	0.296686i	$-0.904119\pi$
$757$	− 16.3258i	− 0.593372i	0.954975	−	0.296686i	$-0.0958815\pi$
$758$	0	0
$759$	−4.76203	−0.172851
$760$	0	0
$761$	50.2208	1.82050	0.910251	−	0.414057i	$-0.135889\pi$
$761$	50.2208	1.82050	0.910251	+	0.414057i	$0.135889\pi$
$762$	0	0
$763$	− 1.77234i	− 0.0641631i
$764$	0	0
$765$	− 7.55691i	− 0.273221i
$766$	0	0
$767$	47.2242	1.70517
$768$	0	0
$769$	−31.3415	−1.13020	−0.565101	−	0.825021i	$-0.691163\pi$
$769$	−31.3415	−1.13020	−0.565101	+	0.825021i	$0.691163\pi$
$770$	0	0
$771$	10.6707i	0.384297i
$772$	0	0
$773$	− 9.11383i	− 0.327802i	−0.986477	−	0.163901i	$-0.947592\pi$
$773$	− 9.11383i	− 0.327802i	0.986477	−	0.163901i	$-0.0524077\pi$
$774$	0	0
$775$	3.55691	0.127768
$776$	0	0
$777$	7.00344	0.251247
$778$	0	0
$779$	4.11039i	0.147270i
$780$	0	0
$781$	− 58.4622i	− 2.09194i
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	−4.32582	−0.154395
$786$	0	0
$787$	− 36.2208i	− 1.29113i	−0.763705	−	0.645566i	$-0.776622\pi$
$787$	− 36.2208i	− 1.29113i	0.763705	−	0.645566i	$-0.223378\pi$
$788$	0	0
$789$	− 1.94480i	− 0.0692369i
$790$	0	0
$791$	−14.4240	−0.512858
$792$	0	0
$793$	49.9931	1.77531
$794$	0	0
$795$	2.00000i	0.0709327i
$796$	0	0
$797$	10.0000i	0.354218i	0.984191	+	0.177109i	$0.0566745\pi$
$797$	10.0000i	0.354218i	−0.984191	+	0.177109i	$0.943325\pi$
$798$	0	0
$799$	75.9862	2.68820
$800$	0	0
$801$	4.11727	0.145476
$802$	0	0
$803$	26.9897i	0.952445i
$804$	0	0
$805$	− 0.996562i	− 0.0351242i
$806$	0	0
$807$	9.76547	0.343761
$808$	0	0
$809$	47.5760	1.67268	0.836342	−	0.548208i	$-0.184690\pi$
$809$	47.5760	1.67268	0.836342	+	0.548208i	$0.184690\pi$
$810$	0	0
$811$	− 20.5174i	− 0.720463i	−0.932863	−	0.360231i	$-0.882698\pi$
$811$	− 20.5174i	− 0.720463i	0.932863	−	0.360231i	$-0.117302\pi$
$812$	0	0
$813$	3.44652i	0.120875i
$814$	0	0
$815$	−6.11727	−0.214278
$816$	0	0
$817$	1.99312	0.0697306
$818$	0	0
$819$	− 5.23109i	− 0.182789i
$820$	0	0
$821$	44.4622i	1.55174i	0.630892	+	0.775871i	$0.282689\pi$
$821$	44.4622i	1.55174i	−0.630892	+	0.775871i	$0.717311\pi$
$822$	0	0
$823$	−32.1656	−1.12122	−0.560611	−	0.828079i	$-0.689434\pi$
$823$	−32.1656	−1.12122	−0.560611	+	0.828079i	$0.689434\pi$
$824$	0	0
$825$	4.49828	0.156610
$826$	0	0
$827$	20.0000i	0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$827$	20.0000i	0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$828$	0	0
$829$	33.8827i	1.17680i	0.808571	+	0.588398i	$0.200241\pi$
$829$	33.8827i	1.17680i	−0.808571	+	0.588398i	$0.799759\pi$
$830$	0	0
$831$	−18.7880	−0.651749
$832$	0	0
$833$	−46.2017	−1.60079
$834$	0	0
$835$	6.05520i	0.209549i
$836$	0	0
$837$	− 3.55691i	− 0.122945i
$838$	0	0
$839$	−4.52750	−0.156307	−0.0781533	−	0.996941i	$-0.524902\pi$
$839$	−4.52750	−0.156307	−0.0781533	+	0.996941i	$0.524902\pi$
$840$	0	0
$841$	25.0000	0.862069
$842$	0	0
$843$	16.8793i	0.581354i
$844$	0	0
$845$	− 17.8793i	− 0.615066i
$846$	0	0
$847$	8.69308	0.298698
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	− 7.87586i	− 0.269981i
$852$	0	0
$853$	− 50.4293i	− 1.72667i	−0.504633	−	0.863334i	$-0.668372\pi$
$853$	− 50.4293i	− 1.72667i	0.504633	−	0.863334i	$-0.331628\pi$
$854$	0	0
$855$	−1.05863	−0.0362045
$856$	0	0
$857$	26.4362	0.903044	0.451522	−	0.892260i	$-0.350881\pi$
$857$	26.4362	0.903044	0.451522	+	0.892260i	$0.350881\pi$
$858$	0	0
$859$	− 0.406994i	− 0.0138865i	−0.999976	−	0.00694323i	$-0.997790\pi$
$859$	− 0.406994i	− 0.0138865i	0.999976	−	0.00694323i	$-0.00221012\pi$
$860$	0	0
$861$	− 3.65508i	− 0.124565i
$862$	0	0
$863$	29.9311	1.01886	0.509432	−	0.860511i	$-0.329855\pi$
$863$	29.9311	1.01886	0.509432	+	0.860511i	$0.329855\pi$
$864$	0	0
$865$	−16.8793	−0.573913
$866$	0	0
$867$	− 40.1070i	− 1.36210i
$868$	0	0
$869$	51.9862i	1.76351i
$870$	0	0
$871$	22.2277	0.753155
$872$	0	0
$873$	−17.1138	−0.579215
$874$	0	0
$875$	0.941367i	0.0318240i
$876$	0	0
$877$	− 11.2051i	− 0.378370i	−0.981941	−	0.189185i	$-0.939415\pi$
$877$	− 11.2051i	− 0.378370i	0.981941	−	0.189185i	$-0.0605846\pi$
$878$	0	0
$879$	−20.2277	−0.682262
$880$	0	0
$881$	−48.3380	−1.62855	−0.814275	−	0.580479i	$-0.802865\pi$
$881$	−48.3380	−1.62855	−0.814275	+	0.580479i	$0.802865\pi$
$882$	0	0
$883$	50.5726i	1.70190i	0.525244	+	0.850951i	$0.323974\pi$
$883$	50.5726i	1.70190i	−0.525244	+	0.850951i	$0.676026\pi$
$884$	0	0
$885$	8.49828i	0.285667i
$886$	0	0
$887$	48.0483	1.61330	0.806652	−	0.591026i	$-0.201277\pi$
$887$	48.0483	1.61330	0.806652	+	0.591026i	$0.201277\pi$
$888$	0	0
$889$	−17.1070	−0.573749
$890$	0	0
$891$	− 4.49828i	− 0.150698i
$892$	0	0
$893$	− 10.6448i	− 0.356213i
$894$	0	0
$895$	10.6155	0.354839
$896$	0	0
$897$	−5.88273	−0.196419
$898$	0	0
$899$	7.11383i	0.237259i
$900$	0	0
$901$	15.1138i	0.503515i
$902$	0	0
$903$	−1.77234	−0.0589799
$904$	0	0
$905$	−14.1173	−0.469274
$906$	0	0
$907$	6.46219i	0.214573i	0.994228	+	0.107287i	$0.0342163\pi$
$907$	6.46219i	0.214573i	−0.994228	+	0.107287i	$0.965784\pi$
$908$	0	0
$909$	− 2.00000i	− 0.0663358i
$910$	0	0
$911$	−50.3380	−1.66777	−0.833887	−	0.551935i	$-0.813890\pi$
$911$	−50.3380	−1.66777	−0.833887	+	0.551935i	$0.813890\pi$
$912$	0	0
$913$	−26.4622	−0.875771
$914$	0	0
$915$	8.99656i	0.297417i
$916$	0	0
$917$	− 6.00688i	− 0.198365i
$918$	0	0
$919$	−46.4362	−1.53179	−0.765895	−	0.642966i	$-0.777704\pi$
$919$	−46.4362	−1.53179	−0.765895	+	0.642966i	$0.777704\pi$
$920$	0	0
$921$	−8.11039	−0.267246
$922$	0	0
$923$	− 72.2208i	− 2.37718i
$924$	0	0
$925$	7.43965i	0.244614i
$926$	0	0
$927$	10.1725	0.334107
$928$	0	0
$929$	−35.9931	−1.18090	−0.590448	−	0.807076i	$-0.701049\pi$
$929$	−35.9931	−1.18090	−0.590448	+	0.807076i	$0.701049\pi$
$930$	0	0
$931$	6.47230i	0.212121i
$932$	0	0
$933$	31.8759i	1.04357i
$934$	0	0
$935$	33.9931	1.11169
$936$	0	0
$937$	2.70360	0.0883227	0.0441613	−	0.999024i	$-0.485938\pi$
$937$	2.70360	0.0883227	0.0441613	+	0.999024i	$0.485938\pi$
$938$	0	0
$939$	5.11383i	0.166883i
$940$	0	0
$941$	− 17.7655i	− 0.579138i	−0.957157	−	0.289569i	$-0.906488\pi$
$941$	− 17.7655i	− 0.579138i	0.957157	−	0.289569i	$-0.0935119\pi$
$942$	0	0
$943$	−4.11039	−0.133853
$944$	0	0
$945$	0.941367	0.0306227
$946$	0	0
$947$	26.2277i	0.852284i	0.904656	+	0.426142i	$0.140128\pi$
$947$	26.2277i	0.852284i	−0.904656	+	0.426142i	$0.859872\pi$
$948$	0	0
$949$	33.3415i	1.08231i
$950$	0	0
$951$	24.6448	0.799161
$952$	0	0
$953$	9.09472	0.294607	0.147304	−	0.989091i	$-0.452941\pi$
$953$	9.09472	0.294607	0.147304	+	0.989091i	$0.452941\pi$
$954$	0	0
$955$	8.00000i	0.258874i
$956$	0	0
$957$	8.99656i	0.290818i
$958$	0	0
$959$	4.18257	0.135062
$960$	0	0
$961$	−18.3484	−0.591883
$962$	0	0
$963$	− 17.2311i	− 0.555264i
$964$	0	0
$965$	− 4.87930i	− 0.157070i
$966$	0	0
$967$	7.47574	0.240404	0.120202	−	0.992749i	$-0.461646\pi$
$967$	7.47574	0.240404	0.120202	+	0.992749i	$0.461646\pi$
$968$	0	0
$969$	−8.00000	−0.256997
$970$	0	0
$971$	41.0777i	1.31825i	0.752035	+	0.659124i	$0.229073\pi$
$971$	41.0777i	1.31825i	−0.752035	+	0.659124i	$0.770927\pi$
$972$	0	0
$973$	18.9897i	0.608781i
$974$	0	0
$975$	5.55691	0.177964
$976$	0	0
$977$	−4.20855	−0.134644	−0.0673218	−	0.997731i	$-0.521445\pi$
$977$	−4.20855	−0.134644	−0.0673218	+	0.997731i	$0.521445\pi$
$978$	0	0
$979$	18.5206i	0.591922i
$980$	0	0
$981$	− 1.88273i	− 0.0601111i
$982$	0	0
$983$	8.35504	0.266484	0.133242	−	0.991084i	$-0.457461\pi$
$983$	8.35504	0.266484	0.133242	+	0.991084i	$0.457461\pi$
$984$	0	0
$985$	2.88617	0.0919611
$986$	0	0
$987$	9.46563i	0.301294i
$988$	0	0
$989$	1.99312i	0.0633777i
$990$	0	0
$991$	13.9087	0.441825	0.220912	−	0.975294i	$-0.429097\pi$
$991$	13.9087	0.441825	0.220912	+	0.975294i	$0.429097\pi$
$992$	0	0
$993$	11.0518	0.350717
$994$	0	0
$995$	17.6742i	0.560309i
$996$	0	0
$997$	34.8984i	1.10524i	0.833432	+	0.552622i	$0.186373\pi$
$997$	34.8984i	1.10524i	−0.833432	+	0.552622i	$0.813627\pi$
$998$	0	0
$999$	7.43965	0.235380

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.2.k.b.241.2		6
3.2	odd	2		1440.2.k.f.721.2		6
4.3	odd	2		120.2.k.b.61.4	yes	6
5.2	odd	4		2400.2.d.e.49.3		6
5.3	odd	4		2400.2.d.f.49.4		6
5.4	even	2		2400.2.k.c.1201.5		6
8.3	odd	2		120.2.k.b.61.3	✓	6
8.5	even	2	inner	480.2.k.b.241.5		6
12.11	even	2		360.2.k.f.181.3		6
15.2	even	4		7200.2.d.r.2449.3		6
15.8	even	4		7200.2.d.q.2449.4		6
15.14	odd	2		7200.2.k.p.3601.4		6
16.3	odd	4		3840.2.a.bp.1.2		3
16.5	even	4		3840.2.a.bo.1.2		3
16.11	odd	4		3840.2.a.bq.1.2		3
16.13	even	4		3840.2.a.br.1.2		3
20.3	even	4		600.2.d.e.349.5		6
20.7	even	4		600.2.d.f.349.2		6
20.19	odd	2		600.2.k.c.301.3		6
24.5	odd	2		1440.2.k.f.721.5		6
24.11	even	2		360.2.k.f.181.4		6
40.3	even	4		600.2.d.f.349.1		6
40.13	odd	4		2400.2.d.e.49.4		6
40.19	odd	2		600.2.k.c.301.4		6
40.27	even	4		600.2.d.e.349.6		6
40.29	even	2		2400.2.k.c.1201.2		6
40.37	odd	4		2400.2.d.f.49.3		6
60.23	odd	4		1800.2.d.q.1549.2		6
60.47	odd	4		1800.2.d.r.1549.5		6
60.59	even	2		1800.2.k.p.901.4		6
120.29	odd	2		7200.2.k.p.3601.3		6
120.53	even	4		7200.2.d.r.2449.4		6
120.59	even	2		1800.2.k.p.901.3		6
120.77	even	4		7200.2.d.q.2449.3		6
120.83	odd	4		1800.2.d.r.1549.6		6
120.107	odd	4		1800.2.d.q.1549.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.2.k.b.61.3	✓	6	8.3	odd	2
120.2.k.b.61.4	yes	6	4.3	odd	2
360.2.k.f.181.3		6	12.11	even	2
360.2.k.f.181.4		6	24.11	even	2
480.2.k.b.241.2		6	1.1	even	1	trivial
480.2.k.b.241.5		6	8.5	even	2	inner
600.2.d.e.349.5		6	20.3	even	4
600.2.d.e.349.6		6	40.27	even	4
600.2.d.f.349.1		6	40.3	even	4
600.2.d.f.349.2		6	20.7	even	4
600.2.k.c.301.3		6	20.19	odd	2
600.2.k.c.301.4		6	40.19	odd	2
1440.2.k.f.721.2		6	3.2	odd	2
1440.2.k.f.721.5		6	24.5	odd	2
1800.2.d.q.1549.1		6	120.107	odd	4
1800.2.d.q.1549.2		6	60.23	odd	4
1800.2.d.r.1549.5		6	60.47	odd	4
1800.2.d.r.1549.6		6	120.83	odd	4
1800.2.k.p.901.3		6	120.59	even	2
1800.2.k.p.901.4		6	60.59	even	2
2400.2.d.e.49.3		6	5.2	odd	4
2400.2.d.e.49.4		6	40.13	odd	4
2400.2.d.f.49.3		6	40.37	odd	4
2400.2.d.f.49.4		6	5.3	odd	4
2400.2.k.c.1201.2		6	40.29	even	2
2400.2.k.c.1201.5		6	5.4	even	2
3840.2.a.bo.1.2		3	16.5	even	4
3840.2.a.bp.1.2		3	16.3	odd	4
3840.2.a.bq.1.2		3	16.11	odd	4
3840.2.a.br.1.2		3	16.13	even	4
7200.2.d.q.2449.3		6	120.77	even	4
7200.2.d.q.2449.4		6	15.8	even	4
7200.2.d.r.2449.3		6	15.2	even	4
7200.2.d.r.2449.4		6	120.53	even	4
7200.2.k.p.3601.3		6	120.29	odd	2
7200.2.k.p.3601.4		6	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	480.2.k.b.241.2

Level	$480$
Weight	$2$
Character	480.241
Analytic conductor	$3.833$
Analytic rank	$0$
Dimension	$6$
Inner twists	$2$