LMFDB - Embedded newform 864.2.f.a.431.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(431,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

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

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

chi := DirichletCharacter("864.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	864.f (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:	$6.89907473464$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.23123460096.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 2x^{6} + 6x^{4} + 8x^{2} + 16 $ x^8 + 2x^6 + 6x^4 + 8*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.2
Root			$-0.563016 - 1.29731i$ of defining polynomial
Character	$\chi$	$=$	864.431
Dual form			864.2.f.a.431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.07638 q^{5} +3.99102i q^{7} +O(q^{10})$	$q-3.07638 q^{5} +3.99102i q^{7} -1.89939i q^{11} -1.06939i q^{13} -7.08863i q^{17} -3.73205 q^{19} -0.824313 q^{23} +4.46410 q^{25} +4.50413 q^{29} -5.84325i q^{31} -12.2779i q^{35} -6.91264i q^{37} -3.79879i q^{41} -2.00000 q^{43} -6.97707 q^{47} -8.92820 q^{49} -10.6569 q^{53} +5.84325i q^{55} -1.89939i q^{59} +1.06939i q^{61} +3.28985i q^{65} +9.19615 q^{67} -10.6569 q^{71} +5.92820 q^{73} +7.58051 q^{77} -6.12979i q^{79} +10.3785i q^{83} +21.8073i q^{85} -13.6683i q^{89} +4.26795 q^{91} +11.4812 q^{95} -11.3923 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 16 q^{19} + 8 q^{25} - 16 q^{43} - 16 q^{49} + 32 q^{67} - 8 q^{73} + 48 q^{91} - 8 q^{97}+O(q^{100}) $ 8 * q - 16 * q^19 + 8 * q^25 - 16 * q^43 - 16 * q^49 + 32 * q^67 - 8 * q^73 + 48 * q^91 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.07638	−1.37580	−0.687899	−	0.725806i	$-0.741467\pi$
$5$	−3.07638	−1.37580	−0.687899	+	0.725806i	$0.741467\pi$
$6$	0	0
$7$	3.99102i	1.50846i	0.656609	+	0.754231i	$0.271990\pi$
$7$	3.99102i	1.50846i	−0.656609	+	0.754231i	$0.728010\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1.89939i	− 0.572689i	−0.958127	−	0.286344i	$-0.907560\pi$
$11$	− 1.89939i	− 0.572689i	0.958127	−	0.286344i	$-0.0924401\pi$
$12$	0	0
$13$	− 1.06939i	− 0.296595i	−0.988943	−	0.148298i	$-0.952621\pi$
$13$	− 1.06939i	− 0.296595i	0.988943	−	0.148298i	$-0.0473794\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 7.08863i	− 1.71925i	−0.510929	−	0.859623i	$-0.670698\pi$
$17$	− 7.08863i	− 1.71925i	0.510929	−	0.859623i	$-0.329302\pi$
$18$	0	0
$19$	−3.73205	−0.856191	−0.428096	−	0.903733i	$-0.640815\pi$
$19$	−3.73205	−0.856191	−0.428096	+	0.903733i	$0.640815\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.824313	−0.171881	−0.0859406	−	0.996300i	$-0.527390\pi$
$23$	−0.824313	−0.171881	−0.0859406	+	0.996300i	$0.527390\pi$
$24$	0	0
$25$	4.46410	0.892820
$26$	0	0
$27$	0	0
$28$	0	0
$29$	4.50413	0.836396	0.418198	−	0.908356i	$-0.362662\pi$
$29$	4.50413	0.836396	0.418198	+	0.908356i	$0.362662\pi$
$30$	0	0
$31$	− 5.84325i	− 1.04948i	−0.851263	−	0.524740i	$-0.824163\pi$
$31$	− 5.84325i	− 1.04948i	0.851263	−	0.524740i	$-0.175837\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 12.2779i	− 2.07534i
$36$	0	0
$37$	− 6.91264i	− 1.13643i	−0.822880	−	0.568216i	$-0.807634\pi$
$37$	− 6.91264i	− 1.13643i	0.822880	−	0.568216i	$-0.192366\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 3.79879i	− 0.593271i	−0.954991	−	0.296635i	$-0.904135\pi$
$41$	− 3.79879i	− 0.593271i	0.954991	−	0.296635i	$-0.0958646\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.97707	−1.01771	−0.508855	−	0.860852i	$-0.669931\pi$
$47$	−6.97707	−1.01771	−0.508855	+	0.860852i	$0.669931\pi$
$48$	0	0
$49$	−8.92820	−1.27546
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.6569	−1.46384	−0.731918	−	0.681393i	$-0.761375\pi$
$53$	−10.6569	−1.46384	−0.731918	+	0.681393i	$0.761375\pi$
$54$	0	0
$55$	5.84325i	0.787904i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 1.89939i	− 0.247280i	−0.992327	−	0.123640i	$-0.960543\pi$
$59$	− 1.89939i	− 0.247280i	0.992327	−	0.123640i	$-0.0394568\pi$
$60$	0	0
$61$	1.06939i	0.136921i	0.997654	+	0.0684606i	$0.0218088\pi$
$61$	1.06939i	0.136921i	−0.997654	+	0.0684606i	$0.978191\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.28985i	0.408055i
$66$	0	0
$67$	9.19615	1.12349	0.561744	−	0.827311i	$-0.310130\pi$
$67$	9.19615	1.12349	0.561744	+	0.827311i	$0.310130\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.6569	−1.26474	−0.632370	−	0.774667i	$-0.717918\pi$
$71$	−10.6569	−1.26474	−0.632370	+	0.774667i	$0.717918\pi$
$72$	0	0
$73$	5.92820	0.693844	0.346922	−	0.937894i	$-0.387227\pi$
$73$	5.92820	0.693844	0.346922	+	0.937894i	$0.387227\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7.58051	0.863879
$78$	0	0
$79$	− 6.12979i	− 0.689656i	−0.938666	−	0.344828i	$-0.887937\pi$
$79$	− 6.12979i	− 0.689656i	0.938666	−	0.344828i	$-0.112063\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.3785i	1.13919i	0.821927	+	0.569593i	$0.192899\pi$
$83$	10.3785i	1.13919i	−0.821927	+	0.569593i	$0.807101\pi$
$84$	0	0
$85$	21.8073i	2.36534i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 13.6683i	− 1.44884i	−0.689359	−	0.724420i	$-0.742108\pi$
$89$	− 13.6683i	− 1.44884i	0.689359	−	0.724420i	$-0.257892\pi$
$90$	0	0
$91$	4.26795	0.447403
$92$	0	0
$93$	0	0
$94$	0	0
$95$	11.4812	1.17795
$96$	0	0
$97$	−11.3923	−1.15671	−0.578357	−	0.815784i	$-0.696306\pi$
$97$	−11.3923	−1.15671	−0.578357	+	0.815784i	$0.696306\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−6.15276	−0.612222	−0.306111	−	0.951996i	$-0.599028\pi$
$101$	−6.15276	−0.612222	−0.306111	+	0.951996i	$0.599028\pi$
$102$	0	0
$103$	− 3.99102i	− 0.393246i	−0.980479	−	0.196623i	$-0.937002\pi$
$103$	− 3.99102i	− 0.393246i	0.980479	−	0.196623i	$-0.0629976\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 8.47908i	− 0.819704i	−0.912152	−	0.409852i	$-0.865580\pi$
$107$	− 8.47908i	− 0.819704i	0.912152	−	0.409852i	$-0.134420\pi$
$108$	0	0
$109$	10.1208i	0.969398i	0.874681	+	0.484699i	$0.161071\pi$
$109$	10.1208i	0.969398i	−0.874681	+	0.484699i	$0.838929\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	10.8874i	1.02420i	0.858925	+	0.512101i	$0.171133\pi$
$113$	10.8874i	1.02420i	−0.858925	+	0.512101i	$0.828867\pi$
$114$	0	0
$115$	2.53590	0.236474
$116$	0	0
$117$	0	0
$118$	0	0
$119$	28.2908	2.59342
$120$	0	0
$121$	7.39230	0.672028
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.64863	0.147458
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	10.3785i	0.906772i	0.891314	+	0.453386i	$0.149784\pi$
$131$	10.3785i	0.906772i	−0.891314	+	0.453386i	$0.850216\pi$
$132$	0	0
$133$	− 14.8947i	− 1.29153i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17.4671i	1.49232i	0.665769	+	0.746158i	$0.268104\pi$
$137$	17.4671i	1.49232i	−0.665769	+	0.746158i	$0.731896\pi$
$138$	0	0
$139$	3.19615	0.271094	0.135547	−	0.990771i	$-0.456721\pi$
$139$	3.19615	0.271094	0.135547	+	0.990771i	$0.456721\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.03119	−0.169857
$144$	0	0
$145$	−13.8564	−1.15071
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.15276	0.504053	0.252027	−	0.967720i	$-0.418903\pi$
$149$	6.15276	0.504053	0.252027	+	0.967720i	$0.418903\pi$
$150$	0	0
$151$	− 9.83427i	− 0.800301i	−0.916449	−	0.400151i	$-0.868958\pi$
$151$	− 9.83427i	− 0.800301i	0.916449	−	0.400151i	$-0.131042\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	17.9761i	1.44387i
$156$	0	0
$157$	− 21.8073i	− 1.74041i	−0.492687	−	0.870207i	$-0.663985\pi$
$157$	− 21.8073i	− 1.74041i	0.492687	−	0.870207i	$-0.336015\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 3.28985i	− 0.259276i
$162$	0	0
$163$	−8.80385	−0.689571	−0.344785	−	0.938682i	$-0.612048\pi$
$163$	−8.80385	−0.689571	−0.344785	+	0.938682i	$0.612048\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−22.1381	−1.71310	−0.856548	−	0.516067i	$-0.827396\pi$
$167$	−22.1381	−1.71310	−0.856548	+	0.516067i	$0.827396\pi$
$168$	0	0
$169$	11.8564	0.912031
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.64863	0.125343	0.0626714	−	0.998034i	$-0.480038\pi$
$173$	1.64863	0.125343	0.0626714	+	0.998034i	$0.480038\pi$
$174$	0	0
$175$	17.8163i	1.34679i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.79879i	0.283935i	0.989871	+	0.141967i	$0.0453428\pi$
$179$	3.79879i	0.283935i	−0.989871	+	0.141967i	$0.954657\pi$
$180$	0	0
$181$	− 3.20817i	− 0.238461i	−0.992867	−	0.119231i	$-0.961957\pi$
$181$	− 3.20817i	− 0.238461i	0.992867	−	0.119231i	$-0.0380428\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	21.2659i	1.56350i
$186$	0	0
$187$	−13.4641	−0.984593
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.97707	−0.504843	−0.252421	−	0.967617i	$-0.581227\pi$
$191$	−6.97707	−0.504843	−0.252421	+	0.967617i	$0.581227\pi$
$192$	0	0
$193$	−5.39230	−0.388147	−0.194073	−	0.980987i	$-0.562170\pi$
$193$	−5.39230	−0.388147	−0.194073	+	0.980987i	$0.562170\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.0305	1.21337	0.606687	−	0.794941i	$-0.292498\pi$
$197$	17.0305	1.21337	0.606687	+	0.794941i	$0.292498\pi$
$198$	0	0
$199$	− 15.6775i	− 1.11135i	−0.831400	−	0.555675i	$-0.812460\pi$
$199$	− 15.6775i	− 1.11135i	0.831400	−	0.555675i	$-0.187540\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	17.9761i	1.26167i
$204$	0	0
$205$	11.6865i	0.816221i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	7.08863i	0.490331i
$210$	0	0
$211$	−21.0526	−1.44932	−0.724659	−	0.689108i	$-0.758003\pi$
$211$	−21.0526	−1.44932	−0.724659	+	0.689108i	$0.758003\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.15276	0.419614
$216$	0	0
$217$	23.3205	1.58310
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−7.58051	−0.509920
$222$	0	0
$223$	26.0849i	1.74677i	0.487029	+	0.873386i	$0.338081\pi$
$223$	26.0849i	1.74677i	−0.487029	+	0.873386i	$0.661919\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.79879i	0.252134i	0.992022	+	0.126067i	$0.0402355\pi$
$227$	3.79879i	0.252134i	−0.992022	+	0.126067i	$0.959765\pi$
$228$	0	0
$229$	− 11.6865i	− 0.772266i	−0.922443	−	0.386133i	$-0.873811\pi$
$229$	− 11.6865i	− 0.772266i	0.922443	−	0.386133i	$-0.126189\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	14.1773i	0.928784i	0.885630	+	0.464392i	$0.153727\pi$
$233$	14.1773i	0.928784i	−0.885630	+	0.464392i	$0.846273\pi$
$234$	0	0
$235$	21.4641	1.40016
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.3055	0.795977	0.397989	−	0.917390i	$-0.369708\pi$
$239$	12.3055	0.795977	0.397989	+	0.917390i	$0.369708\pi$
$240$	0	0
$241$	−5.39230	−0.347349	−0.173674	−	0.984803i	$-0.555564\pi$
$241$	−5.39230	−0.347349	−0.173674	+	0.984803i	$0.555564\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	27.4665	1.75477
$246$	0	0
$247$	3.99102i	0.253942i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 14.1773i	− 0.894861i	−0.894319	−	0.447431i	$-0.852339\pi$
$251$	− 14.1773i	− 0.894861i	0.894319	−	0.447431i	$-0.147661\pi$
$252$	0	0
$253$	1.56569i	0.0984344i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 3.79879i	− 0.236962i	−0.992956	−	0.118481i	$-0.962198\pi$
$257$	− 3.79879i	− 0.236962i	0.992956	−	0.118481i	$-0.0378024\pi$
$258$	0	0
$259$	27.5885	1.71426
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3.29725	0.203317	0.101659	−	0.994819i	$-0.467585\pi$
$263$	3.29725	0.203317	0.101659	+	0.994819i	$0.467585\pi$
$264$	0	0
$265$	32.7846	2.01394
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−19.8860	−1.21247	−0.606236	−	0.795285i	$-0.707321\pi$
$269$	−19.8860	−1.21247	−0.606236	+	0.795285i	$0.707321\pi$
$270$	0	0
$271$	11.9730i	0.727311i	0.931534	+	0.363655i	$0.118471\pi$
$271$	11.9730i	0.727311i	−0.931534	+	0.363655i	$0.881529\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 8.47908i	− 0.511308i
$276$	0	0
$277$	21.8073i	1.31027i	0.755510	+	0.655137i	$0.227389\pi$
$277$	21.8073i	1.31027i	−0.755510	+	0.655137i	$0.772611\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 10.3785i	− 0.619128i	−0.950879	−	0.309564i	$-0.899817\pi$
$281$	− 10.3785i	− 0.619128i	0.950879	−	0.309564i	$-0.100183\pi$
$282$	0	0
$283$	−27.8564	−1.65589	−0.827946	−	0.560808i	$-0.810491\pi$
$283$	−27.8564	−1.65589	−0.827946	+	0.560808i	$0.810491\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.1610	0.894926
$288$	0	0
$289$	−33.2487	−1.95581
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.58051	0.442858	0.221429	−	0.975176i	$-0.428928\pi$
$293$	7.58051	0.442858	0.221429	+	0.975176i	$0.428928\pi$
$294$	0	0
$295$	5.84325i	0.340207i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.881512i	0.0509791i
$300$	0	0
$301$	− 7.98203i	− 0.460077i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 3.28985i	− 0.188376i
$306$	0	0
$307$	−20.9282	−1.19444	−0.597218	−	0.802079i	$-0.703727\pi$
$307$	−20.9282	−1.19444	−0.597218	+	0.802079i	$0.703727\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	20.4895	1.16185	0.580925	−	0.813957i	$-0.302691\pi$
$311$	20.4895	1.16185	0.580925	+	0.813957i	$0.302691\pi$
$312$	0	0
$313$	9.39230	0.530884	0.265442	−	0.964127i	$-0.414482\pi$
$313$	9.39230	0.530884	0.265442	+	0.964127i	$0.414482\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−16.8096	−0.944124	−0.472062	−	0.881565i	$-0.656490\pi$
$317$	−16.8096	−0.944124	−0.472062	+	0.881565i	$0.656490\pi$
$318$	0	0
$319$	− 8.55511i	− 0.478994i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	26.4551i	1.47200i
$324$	0	0
$325$	− 4.77386i	− 0.264806i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 27.8456i	− 1.53518i
$330$	0	0
$331$	−1.87564	−0.103095	−0.0515474	−	0.998671i	$-0.516415\pi$
$331$	−1.87564	−0.103095	−0.0515474	+	0.998671i	$0.516415\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−28.2908	−1.54569
$336$	0	0
$337$	−11.3923	−0.620578	−0.310289	−	0.950642i	$-0.600426\pi$
$337$	−11.3923	−0.620578	−0.310289	+	0.950642i	$0.600426\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−11.0986	−0.601025
$342$	0	0
$343$	− 7.69549i	− 0.415517i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 7.59757i	− 0.407859i	−0.978986	−	0.203930i	$-0.934629\pi$
$347$	− 7.59757i	− 0.407859i	0.978986	−	0.203930i	$-0.0653714\pi$
$348$	0	0
$349$	− 2.63508i	− 0.141053i	−0.997510	−	0.0705264i	$-0.977532\pi$
$349$	− 2.63508i	− 0.141053i	0.997510	−	0.0705264i	$-0.0224679\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	7.59757i	0.404378i	0.979347	+	0.202189i	$0.0648055\pi$
$353$	7.59757i	0.404378i	−0.979347	+	0.202189i	$0.935194\pi$
$354$	0	0
$355$	32.7846	1.74003
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−23.7867	−1.25541	−0.627707	−	0.778449i	$-0.716007\pi$
$359$	−23.7867	−1.25541	−0.627707	+	0.778449i	$0.716007\pi$
$360$	0	0
$361$	−5.07180	−0.266937
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−18.2374	−0.954589
$366$	0	0
$367$	− 37.4848i	− 1.95669i	−0.206975	−	0.978346i	$-0.566362\pi$
$367$	− 37.4848i	− 1.95669i	0.206975	−	0.978346i	$-0.433638\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 42.5318i	− 2.20814i
$372$	0	0
$373$	− 18.5991i	− 0.963027i	−0.876439	−	0.481514i	$-0.840087\pi$
$373$	− 18.5991i	− 0.963027i	0.876439	−	0.481514i	$-0.159913\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 4.81667i	− 0.248071i
$378$	0	0
$379$	26.2679	1.34929	0.674647	−	0.738141i	$-0.264296\pi$
$379$	26.2679	1.34929	0.674647	+	0.738141i	$0.264296\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.64863	0.0842409	0.0421204	−	0.999113i	$-0.486589\pi$
$383$	1.64863	0.0842409	0.0421204	+	0.999113i	$0.486589\pi$
$384$	0	0
$385$	−23.3205	−1.18852
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−15.3819	−0.779893	−0.389946	−	0.920838i	$-0.627506\pi$
$389$	−15.3819	−0.779893	−0.389946	+	0.920838i	$0.627506\pi$
$390$	0	0
$391$	5.84325i	0.295506i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	18.8576i	0.948827i
$396$	0	0
$397$	10.1208i	0.507949i	0.967211	+	0.253974i	$0.0817379\pi$
$397$	10.1208i	0.507949i	−0.967211	+	0.253974i	$0.918262\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 15.1951i	− 0.758809i	−0.925231	−	0.379405i	$-0.876129\pi$
$401$	− 15.1951i	− 0.758809i	0.925231	−	0.379405i	$-0.123871\pi$
$402$	0	0
$403$	−6.24871	−0.311270
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−13.1298	−0.650821
$408$	0	0
$409$	−8.60770	−0.425623	−0.212812	−	0.977093i	$-0.568262\pi$
$409$	−8.60770	−0.425623	−0.212812	+	0.977093i	$0.568262\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.58051	0.373012
$414$	0	0
$415$	− 31.9281i	− 1.56729i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 26.4551i	− 1.29242i	−0.763160	−	0.646209i	$-0.776353\pi$
$419$	− 26.4551i	− 1.29242i	0.763160	−	0.646209i	$-0.223647\pi$
$420$	0	0
$421$	4.77386i	0.232664i	0.993210	+	0.116332i	$0.0371136\pi$
$421$	4.77386i	0.232664i	−0.993210	+	0.116332i	$0.962886\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 31.6444i	− 1.53498i
$426$	0	0
$427$	−4.26795	−0.206541
$428$	0	0
$429$	0	0
$430$	0	0
$431$	0.382565	0.0184275	0.00921375	−	0.999958i	$-0.497067\pi$
$431$	0.382565	0.0184275	0.00921375	+	0.999958i	$0.497067\pi$
$432$	0	0
$433$	4.53590	0.217981	0.108991	−	0.994043i	$-0.465238\pi$
$433$	4.53590	0.217981	0.108991	+	0.994043i	$0.465238\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.07638	0.147163
$438$	0	0
$439$	− 14.3984i	− 0.687197i	−0.939117	−	0.343598i	$-0.888354\pi$
$439$	− 14.3984i	− 0.687197i	0.939117	−	0.343598i	$-0.111646\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 27.3366i	− 1.29880i	−0.760446	−	0.649402i	$-0.775019\pi$
$443$	− 27.3366i	− 1.29880i	0.760446	−	0.649402i	$-0.224981\pi$
$444$	0	0
$445$	42.0489i	1.99331i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	24.0468i	1.13484i	0.823429	+	0.567419i	$0.192058\pi$
$449$	24.0468i	1.13484i	−0.823429	+	0.567419i	$0.807942\pi$
$450$	0	0
$451$	−7.21539	−0.339759
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−13.1298	−0.615536
$456$	0	0
$457$	6.39230	0.299019	0.149510	−	0.988760i	$-0.452230\pi$
$457$	6.39230	0.299019	0.149510	+	0.988760i	$0.452230\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.93188	0.276275	0.138138	−	0.990413i	$-0.455888\pi$
$461$	5.93188	0.276275	0.138138	+	0.990413i	$0.455888\pi$
$462$	0	0
$463$	2.42532i	0.112714i	0.998411	+	0.0563571i	$0.0179485\pi$
$463$	2.42532i	0.112714i	−0.998411	+	0.0563571i	$0.982051\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 19.8754i	− 0.919726i	−0.887990	−	0.459863i	$-0.847899\pi$
$467$	− 19.8754i	− 0.919726i	0.887990	−	0.459863i	$-0.152101\pi$
$468$	0	0
$469$	36.7020i	1.69474i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.79879i	0.174668i
$474$	0	0
$475$	−16.6603	−0.764425
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.15276	0.281127	0.140563	−	0.990072i	$-0.455109\pi$
$479$	6.15276	0.281127	0.140563	+	0.990072i	$0.455109\pi$
$480$	0	0
$481$	−7.39230	−0.337060
$482$	0	0
$483$	0	0
$484$	0	0
$485$	35.0470	1.59140
$486$	0	0
$487$	33.2073i	1.50477i	0.658726	+	0.752383i	$0.271096\pi$
$487$	33.2073i	1.50477i	−0.658726	+	0.752383i	$0.728904\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 33.0348i	− 1.49084i	−0.666595	−	0.745420i	$-0.732249\pi$
$491$	− 33.0348i	− 1.49084i	0.666595	−	0.745420i	$-0.267751\pi$
$492$	0	0
$493$	− 31.9281i	− 1.43797i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 42.5318i	− 1.90781i
$498$	0	0
$499$	5.60770	0.251035	0.125517	−	0.992091i	$-0.459941\pi$
$499$	5.60770	0.251035	0.125517	+	0.992091i	$0.459941\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	29.4977	1.31524	0.657619	−	0.753351i	$-0.271564\pi$
$503$	29.4977	1.31524	0.657619	+	0.753351i	$0.271564\pi$
$504$	0	0
$505$	18.9282	0.842294
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−0.220874	−0.00979007	−0.00489503	−	0.999988i	$-0.501558\pi$
$509$	−0.220874	−0.00979007	−0.00489503	+	0.999988i	$0.501558\pi$
$510$	0	0
$511$	23.6595i	1.04664i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	12.2779i	0.541028i
$516$	0	0
$517$	13.2522i	0.582831i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 7.08863i	− 0.310559i	−0.987871	−	0.155279i	$-0.950372\pi$
$521$	− 7.08863i	− 0.310559i	0.987871	−	0.155279i	$-0.0496278\pi$
$522$	0	0
$523$	37.5885	1.64363	0.821814	−	0.569756i	$-0.192962\pi$
$523$	37.5885	1.64363	0.821814	+	0.569756i	$0.192962\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−41.4207	−1.80431
$528$	0	0
$529$	−22.3205	−0.970457
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−4.06238	−0.175961
$534$	0	0
$535$	26.0849i	1.12775i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	16.9582i	0.730440i
$540$	0	0
$541$	− 34.5632i	− 1.48599i	−0.669298	−	0.742994i	$-0.733405\pi$
$541$	− 34.5632i	− 1.48599i	0.669298	−	0.742994i	$-0.266595\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 31.1354i	− 1.33370i
$546$	0	0
$547$	31.5885	1.35062	0.675312	−	0.737532i	$-0.264009\pi$
$547$	31.5885	1.35062	0.675312	+	0.737532i	$0.264009\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−16.8096	−0.716115
$552$	0	0
$553$	24.4641	1.04032
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.9401	0.633034	0.316517	−	0.948587i	$-0.397487\pi$
$557$	14.9401	0.633034	0.316517	+	0.948587i	$0.397487\pi$
$558$	0	0
$559$	2.13878i	0.0904607i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	34.9342i	1.47230i	0.676817	+	0.736151i	$0.263359\pi$
$563$	34.9342i	1.47230i	−0.676817	+	0.736151i	$0.736641\pi$
$564$	0	0
$565$	− 33.4938i	− 1.40910i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6.07075i	0.254499i	0.991871	+	0.127250i	$0.0406149\pi$
$569$	6.07075i	0.254499i	−0.991871	+	0.127250i	$0.959385\pi$
$570$	0	0
$571$	3.19615	0.133755	0.0668774	−	0.997761i	$-0.478696\pi$
$571$	3.19615	0.133755	0.0668774	+	0.997761i	$0.478696\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.67982	−0.153459
$576$	0	0
$577$	11.9282	0.496578	0.248289	−	0.968686i	$-0.420132\pi$
$577$	11.9282	0.496578	0.248289	+	0.968686i	$0.420132\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−41.4207	−1.71842
$582$	0	0
$583$	20.2416i	0.838322i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 33.0348i	− 1.36349i	−0.731588	−	0.681747i	$-0.761221\pi$
$587$	− 33.0348i	− 1.36349i	0.731588	−	0.681747i	$-0.238779\pi$
$588$	0	0
$589$	21.8073i	0.898555i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	38.7330i	1.59057i	0.606233	+	0.795287i	$0.292680\pi$
$593$	38.7330i	1.59057i	−0.606233	+	0.795287i	$0.707320\pi$
$594$	0	0
$595$	−87.0333	−3.56802
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−22.5207	−0.920169	−0.460084	−	0.887875i	$-0.652181\pi$
$599$	−22.5207	−0.920169	−0.460084	+	0.887875i	$0.652181\pi$
$600$	0	0
$601$	30.3923	1.23973	0.619864	−	0.784709i	$-0.287188\pi$
$601$	30.3923	1.23973	0.619864	+	0.784709i	$0.287188\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−22.7415	−0.924574
$606$	0	0
$607$	− 35.3461i	− 1.43465i	−0.696738	−	0.717326i	$-0.745366\pi$
$607$	− 35.3461i	− 1.43465i	0.696738	−	0.717326i	$-0.254634\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	7.46120i	0.301848i
$612$	0	0
$613$	6.91264i	0.279199i	0.990208	+	0.139599i	$0.0445815\pi$
$613$	6.91264i	0.279199i	−0.990208	+	0.139599i	$0.955418\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	28.8635i	1.16200i	0.813904	+	0.581000i	$0.197338\pi$
$617$	28.8635i	1.16200i	−0.813904	+	0.581000i	$0.802662\pi$
$618$	0	0
$619$	−37.1962	−1.49504	−0.747520	−	0.664240i	$-0.768755\pi$
$619$	−37.1962	−1.49504	−0.747520	+	0.664240i	$0.768755\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	54.5505	2.18552
$624$	0	0
$625$	−27.3923	−1.09569
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−49.0012	−1.95380
$630$	0	0
$631$	22.0939i	0.879542i	0.898110	+	0.439771i	$0.144940\pi$
$631$	22.0939i	0.879542i	−0.898110	+	0.439771i	$0.855060\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	9.54773i	0.378295i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 34.9342i	− 1.37982i	−0.723896	−	0.689909i	$-0.757650\pi$
$641$	− 34.9342i	− 1.37982i	0.723896	−	0.689909i	$-0.242350\pi$
$642$	0	0
$643$	18.7846	0.740793	0.370396	−	0.928874i	$-0.379222\pi$
$643$	18.7846	0.740793	0.370396	+	0.928874i	$0.379222\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	39.7720	1.56360	0.781800	−	0.623529i	$-0.214302\pi$
$647$	39.7720	1.56360	0.781800	+	0.623529i	$0.214302\pi$
$648$	0	0
$649$	−3.60770	−0.141614
$650$	0	0
$651$	0	0
$652$	0	0
$653$	32.4124	1.26840	0.634198	−	0.773171i	$-0.281331\pi$
$653$	32.4124	1.26840	0.634198	+	0.773171i	$0.281331\pi$
$654$	0	0
$655$	− 31.9281i	− 1.24753i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	21.7748i	0.848227i	0.905609	+	0.424114i	$0.139414\pi$
$659$	21.7748i	0.848227i	−0.905609	+	0.424114i	$0.860586\pi$
$660$	0	0
$661$	− 45.2571i	− 1.76030i	−0.474698	−	0.880149i	$-0.657443\pi$
$661$	− 45.2571i	− 1.76030i	0.474698	−	0.880149i	$-0.342557\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	45.8216i	1.77689i
$666$	0	0
$667$	−3.71281	−0.143761
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.03119	0.0784133
$672$	0	0
$673$	35.2487	1.35874	0.679369	−	0.733797i	$-0.262254\pi$
$673$	35.2487	1.35874	0.679369	+	0.733797i	$0.262254\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−44.4970	−1.71016	−0.855080	−	0.518496i	$-0.826492\pi$
$677$	−44.4970	−1.71016	−0.855080	+	0.518496i	$0.826492\pi$
$678$	0	0
$679$	− 45.4669i	− 1.74486i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	39.7509i	1.52103i	0.649323	+	0.760513i	$0.275052\pi$
$683$	39.7509i	1.52103i	−0.649323	+	0.760513i	$0.724948\pi$
$684$	0	0
$685$	− 53.7354i	− 2.05313i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	11.3964i	0.434167i
$690$	0	0
$691$	−3.85641	−0.146705	−0.0733523	−	0.997306i	$-0.523370\pi$
$691$	−3.85641	−0.146705	−0.0733523	+	0.997306i	$0.523370\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.83257	−0.372971
$696$	0	0
$697$	−26.9282	−1.01998
$698$	0	0
$699$	0	0
$700$	0	0
$701$	51.8567	1.95860	0.979300	−	0.202415i	$-0.0648790\pi$
$701$	51.8567	1.95860	0.979300	+	0.202415i	$0.0648790\pi$
$702$	0	0
$703$	25.7983i	0.973002i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 24.5557i	− 0.923514i
$708$	0	0
$709$	35.1363i	1.31957i	0.751454	+	0.659786i	$0.229353\pi$
$709$	35.1363i	1.31957i	−0.751454	+	0.659786i	$0.770647\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	4.81667i	0.180386i
$714$	0	0
$715$	6.24871	0.233689
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−7.80138	−0.290942	−0.145471	−	0.989362i	$-0.546470\pi$
$719$	−7.80138	−0.290942	−0.145471	+	0.989362i	$0.546470\pi$
$720$	0	0
$721$	15.9282	0.593197
$722$	0	0
$723$	0	0
$724$	0	0
$725$	20.1069	0.746751
$726$	0	0
$727$	26.0849i	0.967434i	0.875224	+	0.483717i	$0.160714\pi$
$727$	26.0849i	0.967434i	−0.875224	+	0.483717i	$0.839286\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	14.1773i	0.524365i
$732$	0	0
$733$	13.2522i	0.489481i	0.969589	+	0.244741i	$0.0787028\pi$
$733$	13.2522i	0.489481i	−0.969589	+	0.244741i	$0.921297\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 17.4671i	− 0.643409i
$738$	0	0
$739$	20.6410	0.759292	0.379646	−	0.925132i	$-0.376046\pi$
$739$	20.6410	0.759292	0.379646	+	0.925132i	$0.376046\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	51.6950	1.89651	0.948253	−	0.317517i	$-0.102849\pi$
$743$	51.6950	1.89651	0.948253	+	0.317517i	$0.102849\pi$
$744$	0	0
$745$	−18.9282	−0.693476
$746$	0	0
$747$	0	0
$748$	0	0
$749$	33.8402	1.23649
$750$	0	0
$751$	19.9551i	0.728171i	0.931366	+	0.364086i	$0.118618\pi$
$751$	19.9551i	0.728171i	−0.931366	+	0.364086i	$0.881382\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	30.2539i	1.10105i
$756$	0	0
$757$	− 52.0930i	− 1.89335i	−0.322189	−	0.946675i	$-0.604419\pi$
$757$	− 52.0930i	− 1.89335i	0.322189	−	0.946675i	$-0.395581\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 43.0407i	− 1.56023i	−0.625639	−	0.780113i	$-0.715162\pi$
$761$	− 43.0407i	− 1.56023i	0.625639	−	0.780113i	$-0.284838\pi$
$762$	0	0
$763$	−40.3923	−1.46230
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.03119	−0.0733421
$768$	0	0
$769$	4.07180	0.146833	0.0734164	−	0.997301i	$-0.476610\pi$
$769$	4.07180	0.146833	0.0734164	+	0.997301i	$0.476610\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	44.7179	1.60839	0.804196	−	0.594364i	$-0.202596\pi$
$773$	44.7179	1.60839	0.804196	+	0.594364i	$0.202596\pi$
$774$	0	0
$775$	− 26.0849i	− 0.936996i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.1773i	0.507953i
$780$	0	0
$781$	20.2416i	0.724302i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	67.0875i	2.39446i
$786$	0	0
$787$	−7.44486	−0.265381	−0.132690	−	0.991158i	$-0.542362\pi$
$787$	−7.44486	−0.265381	−0.132690	+	0.991158i	$0.542362\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−43.4519	−1.54497
$792$	0	0
$793$	1.14359	0.0406102
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27.4665	0.972914	0.486457	−	0.873704i	$-0.338289\pi$
$797$	27.4665	0.972914	0.486457	+	0.873704i	$0.338289\pi$
$798$	0	0
$799$	49.4579i	1.74969i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 11.2600i	− 0.397356i
$804$	0	0
$805$	10.1208i	0.356712i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	14.1773i	0.498446i	0.968446	+	0.249223i	$0.0801752\pi$
$809$	14.1773i	0.498446i	−0.968446	+	0.249223i	$0.919825\pi$
$810$	0	0
$811$	27.0718	0.950619	0.475310	−	0.879819i	$-0.342336\pi$
$811$	27.0718	0.950619	0.475310	+	0.879819i	$0.342336\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	27.0840	0.948710
$816$	0	0
$817$	7.46410	0.261136
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10.8778	0.379636	0.189818	−	0.981819i	$-0.439210\pi$
$821$	10.8778	0.379636	0.189818	+	0.981819i	$0.439210\pi$
$822$	0	0
$823$	− 24.2326i	− 0.844697i	−0.906434	−	0.422348i	$-0.861206\pi$
$823$	− 24.2326i	− 0.844697i	0.906434	−	0.422348i	$-0.138794\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	40.6324i	1.41293i	0.707749	+	0.706464i	$0.249711\pi$
$827$	40.6324i	1.41293i	−0.707749	+	0.706464i	$0.750289\pi$
$828$	0	0
$829$	34.5632i	1.20043i	0.799839	+	0.600215i	$0.204918\pi$
$829$	34.5632i	1.20043i	−0.799839	+	0.600215i	$0.795082\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	63.2888i	2.19283i
$834$	0	0
$835$	68.1051	2.35687
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−25.0528	−0.864918	−0.432459	−	0.901654i	$-0.642354\pi$
$839$	−25.0528	−0.864918	−0.432459	+	0.901654i	$0.642354\pi$
$840$	0	0
$841$	−8.71281	−0.300442
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−36.4748	−1.25477
$846$	0	0
$847$	29.5028i	1.01373i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5.69818i	0.195331i
$852$	0	0
$853$	14.8947i	0.509984i	0.966943	+	0.254992i	$0.0820728\pi$
$853$	14.8947i	0.509984i	−0.966943	+	0.254992i	$0.917927\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.01788i	0.0347702i	0.999849	+	0.0173851i	$0.00553413\pi$
$857$	1.01788i	0.0347702i	−0.999849	+	0.0173851i	$0.994466\pi$
$858$	0	0
$859$	−11.5885	−0.395393	−0.197697	−	0.980263i	$-0.563346\pi$
$859$	−11.5885	−0.395393	−0.197697	+	0.980263i	$0.563346\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−15.9853	−0.544147	−0.272073	−	0.962276i	$-0.587709\pi$
$863$	−15.9853	−0.544147	−0.272073	+	0.962276i	$0.587709\pi$
$864$	0	0
$865$	−5.07180	−0.172446
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−11.6429	−0.394958
$870$	0	0
$871$	− 9.83427i	− 0.333221i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	6.57969i	0.222434i
$876$	0	0
$877$	19.1722i	0.647400i	0.946160	+	0.323700i	$0.104927\pi$
$877$	19.1722i	0.647400i	−0.946160	+	0.323700i	$0.895073\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	6.07075i	0.204529i	0.994757	+	0.102264i	$0.0326088\pi$
$881$	6.07075i	0.204529i	−0.994757	+	0.102264i	$0.967391\pi$
$882$	0	0
$883$	−44.1244	−1.48490	−0.742451	−	0.669900i	$-0.766337\pi$
$883$	−44.1244	−1.48490	−0.742451	+	0.669900i	$0.766337\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.8179	0.866880	0.433440	−	0.901182i	$-0.357300\pi$
$887$	25.8179	0.866880	0.433440	+	0.901182i	$0.357300\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	26.0388	0.871354
$894$	0	0
$895$	− 11.6865i	− 0.390637i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 26.3188i	− 0.877780i
$900$	0	0
$901$	75.5427i	2.51669i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.86954i	0.328075i
$906$	0	0
$907$	46.3731	1.53979	0.769896	−	0.638169i	$-0.220308\pi$
$907$	46.3731	1.53979	0.769896	+	0.638169i	$0.220308\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	0.441748	0.0146358	0.00731788	−	0.999973i	$-0.497671\pi$
$911$	0.441748	0.0146358	0.00731788	+	0.999973i	$0.497671\pi$
$912$	0	0
$913$	19.7128	0.652399
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−41.4207	−1.36783
$918$	0	0
$919$	− 58.0130i	− 1.91367i	−0.290628	−	0.956836i	$-0.593864\pi$
$919$	− 58.0130i	− 1.91367i	0.290628	−	0.956836i	$-0.406136\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	11.3964i	0.375116i
$924$	0	0
$925$	− 30.8587i	− 1.01463i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	20.7570i	0.681014i	0.940242	+	0.340507i	$0.110599\pi$
$929$	20.7570i	0.681014i	−0.940242	+	0.340507i	$0.889401\pi$
$930$	0	0
$931$	33.3205	1.09204
$932$	0	0
$933$	0	0
$934$	0	0
$935$	41.4207	1.35460
$936$	0	0
$937$	−23.3923	−0.764193	−0.382097	−	0.924122i	$-0.624798\pi$
$937$	−23.3923	−0.764193	−0.382097	+	0.924122i	$0.624798\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−18.6791	−0.608923	−0.304461	−	0.952525i	$-0.598476\pi$
$941$	−18.6791	−0.608923	−0.304461	+	0.952525i	$0.598476\pi$
$942$	0	0
$943$	3.13139i	0.101972i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29.2360i	0.950044i	0.879974	+	0.475022i	$0.157560\pi$
$947$	29.2360i	0.950044i	−0.879974	+	0.475022i	$0.842440\pi$
$948$	0	0
$949$	− 6.33956i	− 0.205791i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	15.7041i	0.508705i	0.967112	+	0.254353i	$0.0818624\pi$
$953$	15.7041i	0.508705i	−0.967112	+	0.254353i	$0.918138\pi$
$954$	0	0
$955$	21.4641	0.694562
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−69.7115	−2.25110
$960$	0	0
$961$	−3.14359	−0.101406
$962$	0	0
$963$	0	0
$964$	0	0
$965$	16.5888	0.534011
$966$	0	0
$967$	9.83427i	0.316249i	0.987419	+	0.158124i	$0.0505447\pi$
$967$	9.83427i	0.316249i	−0.987419	+	0.158124i	$0.949455\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 55.8275i	− 1.79159i	−0.444466	−	0.895796i	$-0.646607\pi$
$971$	− 55.8275i	− 1.79159i	0.444466	−	0.895796i	$-0.353393\pi$
$972$	0	0
$973$	12.7559i	0.408935i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 23.5379i	− 0.753043i	−0.926408	−	0.376521i	$-0.877120\pi$
$977$	− 23.5379i	− 0.753043i	0.926408	−	0.376521i	$-0.122880\pi$
$978$	0	0
$979$	−25.9615	−0.829734
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−55.3156	−1.76429	−0.882147	−	0.470974i	$-0.843903\pi$
$983$	−55.3156	−1.76429	−0.882147	+	0.470974i	$0.843903\pi$
$984$	0	0
$985$	−52.3923	−1.66936
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.64863	0.0524233
$990$	0	0
$991$	− 25.7983i	− 0.819511i	−0.912195	−	0.409755i	$-0.865614\pi$
$991$	− 25.7983i	− 0.819511i	0.912195	−	0.409755i	$-0.134386\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	48.2300i	1.52899i
$996$	0	0
$997$	1.56569i	0.0495860i	0.999693	+	0.0247930i	$0.00789267\pi$
$997$	1.56569i	0.0495860i	−0.999693	+	0.0247930i	$0.992107\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.f.a.431.2		8
3.2	odd	2	inner	864.2.f.a.431.8		8
4.3	odd	2		216.2.f.a.107.5	yes	8
8.3	odd	2	inner	864.2.f.a.431.7		8
8.5	even	2		216.2.f.a.107.3	✓	8
9.2	odd	6		2592.2.p.f.2159.2		16
9.4	even	3		2592.2.p.f.431.7		16
9.5	odd	6		2592.2.p.f.431.1		16
9.7	even	3		2592.2.p.f.2159.8		16
12.11	even	2		216.2.f.a.107.4	yes	8
24.5	odd	2		216.2.f.a.107.6	yes	8
24.11	even	2	inner	864.2.f.a.431.1		8
36.7	odd	6		648.2.l.f.539.1		16
36.11	even	6		648.2.l.f.539.8		16
36.23	even	6		648.2.l.f.107.3		16
36.31	odd	6		648.2.l.f.107.6		16
72.5	odd	6		648.2.l.f.107.1		16
72.11	even	6		2592.2.p.f.2159.7		16
72.13	even	6		648.2.l.f.107.8		16
72.29	odd	6		648.2.l.f.539.6		16
72.43	odd	6		2592.2.p.f.2159.1		16
72.59	even	6		2592.2.p.f.431.8		16
72.61	even	6		648.2.l.f.539.3		16
72.67	odd	6		2592.2.p.f.431.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.f.a.107.3	✓	8	8.5	even	2
216.2.f.a.107.4	yes	8	12.11	even	2
216.2.f.a.107.5	yes	8	4.3	odd	2
216.2.f.a.107.6	yes	8	24.5	odd	2
648.2.l.f.107.1		16	72.5	odd	6
648.2.l.f.107.3		16	36.23	even	6
648.2.l.f.107.6		16	36.31	odd	6
648.2.l.f.107.8		16	72.13	even	6
648.2.l.f.539.1		16	36.7	odd	6
648.2.l.f.539.3		16	72.61	even	6
648.2.l.f.539.6		16	72.29	odd	6
648.2.l.f.539.8		16	36.11	even	6
864.2.f.a.431.1		8	24.11	even	2	inner
864.2.f.a.431.2		8	1.1	even	1	trivial
864.2.f.a.431.7		8	8.3	odd	2	inner
864.2.f.a.431.8		8	3.2	odd	2	inner
2592.2.p.f.431.1		16	9.5	odd	6
2592.2.p.f.431.2		16	72.67	odd	6
2592.2.p.f.431.7		16	9.4	even	3
2592.2.p.f.431.8		16	72.59	even	6
2592.2.p.f.2159.1		16	72.43	odd	6
2592.2.p.f.2159.2		16	9.2	odd	6
2592.2.p.f.2159.7		16	72.11	even	6
2592.2.p.f.2159.8		16	9.7	even	3

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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.07638 q^{5} +3.99102i q^{7} +O(q^{10})\)	\(q-3.07638 q^{5} +3.99102i q^{7} -1.89939i q^{11} -1.06939i q^{13} -7.08863i q^{17} -3.73205 q^{19} -0.824313 q^{23} +4.46410 q^{25} +4.50413 q^{29} -5.84325i q^{31} -12.2779i q^{35} -6.91264i q^{37} -3.79879i q^{41} -2.00000 q^{43} -6.97707 q^{47} -8.92820 q^{49} -10.6569 q^{53} +5.84325i q^{55} -1.89939i q^{59} +1.06939i q^{61} +3.28985i q^{65} +9.19615 q^{67} -10.6569 q^{71} +5.92820 q^{73} +7.58051 q^{77} -6.12979i q^{79} +10.3785i q^{83} +21.8073i q^{85} -13.6683i q^{89} +4.26795 q^{91} +11.4812 q^{95} -11.3923 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 16 q^{19} + 8 q^{25} - 16 q^{43} - 16 q^{49} + 32 q^{67} - 8 q^{73} + 48 q^{91} - 8 q^{97}+O(q^{100}) \) 8 * q - 16 * q^19 + 8 * q^25 - 16 * q^43 - 16 * q^49 + 32 * q^67 - 8 * q^73 + 48 * q^91 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more