LMFDB - Embedded newform 6096.2.a.bk.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6096,2,Mod(1,6096)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6096, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6096.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6096 = 2^{4} \cdot 3 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6096.a (trivial)

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:	yes
Analytic conductor:	$48.6768050722$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 14x^{7} + 26x^{6} + 59x^{5} - 99x^{4} - 66x^{3} + 102x^{2} - 24x + 1 $ x^9 - 2x^8 - 14x^7 + 26x^6 + 59x^5 - 99x^4 - 66x^3 + 102x^2 - 24x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 381)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.247918$ of defining polynomial
Character	$\chi$	$=$	6096.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.730098 q^{5} +3.26267 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.730098 q^{5} +3.26267 q^{7} +1.00000 q^{9} -5.67568 q^{11} +6.18996 q^{13} -0.730098 q^{15} +2.63121 q^{17} -4.63121 q^{19} -3.26267 q^{21} +3.17572 q^{23} -4.46696 q^{25} -1.00000 q^{27} -9.22063 q^{29} -6.09141 q^{31} +5.67568 q^{33} +2.38207 q^{35} -4.24058 q^{37} -6.18996 q^{39} +3.75631 q^{41} -4.26770 q^{43} +0.730098 q^{45} +1.13802 q^{47} +3.64501 q^{49} -2.63121 q^{51} -12.3963 q^{53} -4.14380 q^{55} +4.63121 q^{57} -4.95155 q^{59} +7.49582 q^{61} +3.26267 q^{63} +4.51928 q^{65} -2.08323 q^{67} -3.17572 q^{69} -8.05669 q^{71} -0.449676 q^{73} +4.46696 q^{75} -18.5179 q^{77} +6.95350 q^{79} +1.00000 q^{81} -5.09676 q^{83} +1.92104 q^{85} +9.22063 q^{87} -2.02997 q^{89} +20.1958 q^{91} +6.09141 q^{93} -3.38124 q^{95} -4.40474 q^{97} -5.67568 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{3} - 4 q^{5} - 10 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q - 9 * q^3 - 4 * q^5 - 10 * q^7 + 9 * q^9	$ 9 q - 9 q^{3} - 4 q^{5} - 10 q^{7} + 9 q^{9} - 8 q^{11} + 14 q^{13} + 4 q^{15} - 6 q^{17} - 12 q^{19} + 10 q^{21} + 4 q^{23} + 21 q^{25} - 9 q^{27} - 8 q^{29} - 4 q^{31} + 8 q^{33} - 6 q^{35} + 22 q^{37} - 14 q^{39} - 2 q^{41} - 6 q^{43} - 4 q^{45} + 2 q^{47} + 23 q^{49} + 6 q^{51} - 12 q^{53} + 22 q^{55} + 12 q^{57} + 6 q^{59} + 2 q^{61} - 10 q^{63} + 4 q^{65} - 18 q^{67} - 4 q^{69} - 24 q^{71} + 14 q^{73} - 21 q^{75} - 18 q^{77} - 12 q^{79} + 9 q^{81} + 20 q^{83} - 24 q^{85} + 8 q^{87} - 30 q^{89} - 14 q^{91} + 4 q^{93} + 32 q^{95} + 12 q^{97} - 8 q^{99}+O(q^{100}) $ 9 * q - 9 * q^3 - 4 * q^5 - 10 * q^7 + 9 * q^9 - 8 * q^11 + 14 * q^13 + 4 * q^15 - 6 * q^17 - 12 * q^19 + 10 * q^21 + 4 * q^23 + 21 * q^25 - 9 * q^27 - 8 * q^29 - 4 * q^31 + 8 * q^33 - 6 * q^35 + 22 * q^37 - 14 * q^39 - 2 * q^41 - 6 * q^43 - 4 * q^45 + 2 * q^47 + 23 * q^49 + 6 * q^51 - 12 * q^53 + 22 * q^55 + 12 * q^57 + 6 * q^59 + 2 * q^61 - 10 * q^63 + 4 * q^65 - 18 * q^67 - 4 * q^69 - 24 * q^71 + 14 * q^73 - 21 * q^75 - 18 * q^77 - 12 * q^79 + 9 * q^81 + 20 * q^83 - 24 * q^85 + 8 * q^87 - 30 * q^89 - 14 * q^91 + 4 * q^93 + 32 * q^95 + 12 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0.730098	0.326510	0.163255	−	0.986584i	$-0.447801\pi$
$5$	0.730098	0.326510	0.163255	+	0.986584i	$0.447801\pi$
$6$	0	0
$7$	3.26267	1.23317	0.616587	−	0.787287i	$-0.288515\pi$
$7$	3.26267	1.23317	0.616587	+	0.787287i	$0.288515\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.67568	−1.71128	−0.855641	−	0.517571i	$-0.826836\pi$
$11$	−5.67568	−1.71128	−0.855641	+	0.517571i	$0.826836\pi$
$12$	0	0
$13$	6.18996	1.71679	0.858393	−	0.512993i	$-0.171463\pi$
$13$	6.18996	1.71679	0.858393	+	0.512993i	$0.171463\pi$
$14$	0	0
$15$	−0.730098	−0.188510
$16$	0	0
$17$	2.63121	0.638162	0.319081	−	0.947727i	$-0.396626\pi$
$17$	2.63121	0.638162	0.319081	+	0.947727i	$0.396626\pi$
$18$	0	0
$19$	−4.63121	−1.06247	−0.531236	−	0.847224i	$-0.678272\pi$
$19$	−4.63121	−1.06247	−0.531236	+	0.847224i	$0.678272\pi$
$20$	0	0
$21$	−3.26267	−0.711973
$22$	0	0
$23$	3.17572	0.662184	0.331092	−	0.943598i	$-0.392583\pi$
$23$	3.17572	0.662184	0.331092	+	0.943598i	$0.392583\pi$
$24$	0	0
$25$	−4.46696	−0.893391
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−9.22063	−1.71223	−0.856114	−	0.516788i	$-0.827128\pi$
$29$	−9.22063	−1.71223	−0.856114	+	0.516788i	$0.827128\pi$
$30$	0	0
$31$	−6.09141	−1.09405	−0.547024	−	0.837117i	$-0.684240\pi$
$31$	−6.09141	−1.09405	−0.547024	+	0.837117i	$0.684240\pi$
$32$	0	0
$33$	5.67568	0.988009
$34$	0	0
$35$	2.38207	0.402643
$36$	0	0
$37$	−4.24058	−0.697147	−0.348574	−	0.937281i	$-0.613334\pi$
$37$	−4.24058	−0.697147	−0.348574	+	0.937281i	$0.613334\pi$
$38$	0	0
$39$	−6.18996	−0.991187
$40$	0	0
$41$	3.75631	0.586637	0.293319	−	0.956015i	$-0.405240\pi$
$41$	3.75631	0.586637	0.293319	+	0.956015i	$0.405240\pi$
$42$	0	0
$43$	−4.26770	−0.650819	−0.325409	−	0.945573i	$-0.605502\pi$
$43$	−4.26770	−0.650819	−0.325409	+	0.945573i	$0.605502\pi$
$44$	0	0
$45$	0.730098	0.108837
$46$	0	0
$47$	1.13802	0.165997	0.0829986	−	0.996550i	$-0.473550\pi$
$47$	1.13802	0.165997	0.0829986	+	0.996550i	$0.473550\pi$
$48$	0	0
$49$	3.64501	0.520716
$50$	0	0
$51$	−2.63121	−0.368443
$52$	0	0
$53$	−12.3963	−1.70277	−0.851385	−	0.524542i	$-0.824237\pi$
$53$	−12.3963	−1.70277	−0.851385	+	0.524542i	$0.824237\pi$
$54$	0	0
$55$	−4.14380	−0.558750
$56$	0	0
$57$	4.63121	0.613419
$58$	0	0
$59$	−4.95155	−0.644638	−0.322319	−	0.946631i	$-0.604462\pi$
$59$	−4.95155	−0.644638	−0.322319	+	0.946631i	$0.604462\pi$
$60$	0	0
$61$	7.49582	0.959741	0.479871	−	0.877339i	$-0.340684\pi$
$61$	7.49582	0.959741	0.479871	+	0.877339i	$0.340684\pi$
$62$	0	0
$63$	3.26267	0.411058
$64$	0	0
$65$	4.51928	0.560547
$66$	0	0
$67$	−2.08323	−0.254507	−0.127253	−	0.991870i	$-0.540616\pi$
$67$	−2.08323	−0.254507	−0.127253	+	0.991870i	$0.540616\pi$
$68$	0	0
$69$	−3.17572	−0.382312
$70$	0	0
$71$	−8.05669	−0.956153	−0.478076	−	0.878318i	$-0.658666\pi$
$71$	−8.05669	−0.956153	−0.478076	+	0.878318i	$0.658666\pi$
$72$	0	0
$73$	−0.449676	−0.0526306	−0.0263153	−	0.999654i	$-0.508377\pi$
$73$	−0.449676	−0.0526306	−0.0263153	+	0.999654i	$0.508377\pi$
$74$	0	0
$75$	4.46696	0.515800
$76$	0	0
$77$	−18.5179	−2.11031
$78$	0	0
$79$	6.95350	0.782330	0.391165	−	0.920321i	$-0.372072\pi$
$79$	6.95350	0.782330	0.391165	+	0.920321i	$0.372072\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−5.09676	−0.559442	−0.279721	−	0.960081i	$-0.590242\pi$
$83$	−5.09676	−0.559442	−0.279721	+	0.960081i	$0.590242\pi$
$84$	0	0
$85$	1.92104	0.208366
$86$	0	0
$87$	9.22063	0.988555
$88$	0	0
$89$	−2.02997	−0.215177	−0.107588	−	0.994196i	$-0.534313\pi$
$89$	−2.02997	−0.215177	−0.107588	+	0.994196i	$0.534313\pi$
$90$	0	0
$91$	20.1958	2.11709
$92$	0	0
$93$	6.09141	0.631649
$94$	0	0
$95$	−3.38124	−0.346908
$96$	0	0
$97$	−4.40474	−0.447234	−0.223617	−	0.974677i	$-0.571786\pi$
$97$	−4.40474	−0.447234	−0.223617	+	0.974677i	$0.571786\pi$
$98$	0	0
$99$	−5.67568	−0.570427
$100$	0	0
$101$	−10.3799	−1.03284	−0.516420	−	0.856335i	$-0.672736\pi$
$101$	−10.3799	−1.03284	−0.516420	+	0.856335i	$0.672736\pi$
$102$	0	0
$103$	−13.4120	−1.32152	−0.660760	−	0.750598i	$-0.729766\pi$
$103$	−13.4120	−1.32152	−0.660760	+	0.750598i	$0.729766\pi$
$104$	0	0
$105$	−2.38207	−0.232466
$106$	0	0
$107$	10.7425	1.03852	0.519260	−	0.854616i	$-0.326207\pi$
$107$	10.7425	1.03852	0.519260	+	0.854616i	$0.326207\pi$
$108$	0	0
$109$	6.58912	0.631123	0.315562	−	0.948905i	$-0.397807\pi$
$109$	6.58912	0.631123	0.315562	+	0.948905i	$0.397807\pi$
$110$	0	0
$111$	4.24058	0.402498
$112$	0	0
$113$	−8.68884	−0.817377	−0.408689	−	0.912674i	$-0.634014\pi$
$113$	−8.68884	−0.817377	−0.408689	+	0.912674i	$0.634014\pi$
$114$	0	0
$115$	2.31859	0.216209
$116$	0	0
$117$	6.18996	0.572262
$118$	0	0
$119$	8.58477	0.786965
$120$	0	0
$121$	21.2133	1.92848
$122$	0	0
$123$	−3.75631	−0.338695
$124$	0	0
$125$	−6.91180	−0.618211
$126$	0	0
$127$	1.00000	0.0887357
$127$	1.00000	0.0887357
$128$	0	0
$129$	4.26770	0.375750
$130$	0	0
$131$	13.8370	1.20895	0.604474	−	0.796625i	$-0.293383\pi$
$131$	13.8370	1.20895	0.604474	+	0.796625i	$0.293383\pi$
$132$	0	0
$133$	−15.1101	−1.31021
$134$	0	0
$135$	−0.730098	−0.0628368
$136$	0	0
$137$	2.93954	0.251142	0.125571	−	0.992085i	$-0.459924\pi$
$137$	2.93954	0.251142	0.125571	+	0.992085i	$0.459924\pi$
$138$	0	0
$139$	9.45281	0.801776	0.400888	−	0.916127i	$-0.368702\pi$
$139$	9.45281	0.801776	0.400888	+	0.916127i	$0.368702\pi$
$140$	0	0
$141$	−1.13802	−0.0958386
$142$	0	0
$143$	−35.1322	−2.93790
$144$	0	0
$145$	−6.73196	−0.559059
$146$	0	0
$147$	−3.64501	−0.300635
$148$	0	0
$149$	−19.6394	−1.60892	−0.804460	−	0.594007i	$-0.797545\pi$
$149$	−19.6394	−1.60892	−0.804460	+	0.594007i	$0.797545\pi$
$150$	0	0
$151$	12.8407	1.04496	0.522482	−	0.852650i	$-0.325006\pi$
$151$	12.8407	1.04496	0.522482	+	0.852650i	$0.325006\pi$
$152$	0	0
$153$	2.63121	0.212721
$154$	0	0
$155$	−4.44732	−0.357217
$156$	0	0
$157$	9.51482	0.759365	0.379683	−	0.925117i	$-0.376033\pi$
$157$	9.51482	0.759365	0.379683	+	0.925117i	$0.376033\pi$
$158$	0	0
$159$	12.3963	0.983094
$160$	0	0
$161$	10.3613	0.816587
$162$	0	0
$163$	23.6001	1.84850	0.924252	−	0.381782i	$-0.124690\pi$
$163$	23.6001	1.84850	0.924252	+	0.381782i	$0.124690\pi$
$164$	0	0
$165$	4.14380	0.322594
$166$	0	0
$167$	−23.8515	−1.84569	−0.922844	−	0.385174i	$-0.874141\pi$
$167$	−23.8515	−1.84569	−0.922844	+	0.385174i	$0.874141\pi$
$168$	0	0
$169$	25.3156	1.94735
$170$	0	0
$171$	−4.63121	−0.354158
$172$	0	0
$173$	−4.23447	−0.321941	−0.160970	−	0.986959i	$-0.551462\pi$
$173$	−4.23447	−0.321941	−0.160970	+	0.986959i	$0.551462\pi$
$174$	0	0
$175$	−14.5742	−1.10171
$176$	0	0
$177$	4.95155	0.372182
$178$	0	0
$179$	−3.07916	−0.230148	−0.115074	−	0.993357i	$-0.536710\pi$
$179$	−3.07916	−0.230148	−0.115074	+	0.993357i	$0.536710\pi$
$180$	0	0
$181$	−18.9050	−1.40520	−0.702601	−	0.711584i	$-0.747978\pi$
$181$	−18.9050	−1.40520	−0.702601	+	0.711584i	$0.747978\pi$
$182$	0	0
$183$	−7.49582	−0.554107
$184$	0	0
$185$	−3.09604	−0.227625
$186$	0	0
$187$	−14.9339	−1.09208
$188$	0	0
$189$	−3.26267	−0.237324
$190$	0	0
$191$	23.0731	1.66951	0.834756	−	0.550620i	$-0.185608\pi$
$191$	23.0731	1.66951	0.834756	+	0.550620i	$0.185608\pi$
$192$	0	0
$193$	9.91938	0.714012	0.357006	−	0.934102i	$-0.383797\pi$
$193$	9.91938	0.714012	0.357006	+	0.934102i	$0.383797\pi$
$194$	0	0
$195$	−4.51928	−0.323632
$196$	0	0
$197$	−8.02848	−0.572006	−0.286003	−	0.958229i	$-0.592327\pi$
$197$	−8.02848	−0.572006	−0.286003	+	0.958229i	$0.592327\pi$
$198$	0	0
$199$	−22.3790	−1.58640	−0.793202	−	0.608958i	$-0.791588\pi$
$199$	−22.3790	−1.58640	−0.793202	+	0.608958i	$0.791588\pi$
$200$	0	0
$201$	2.08323	0.146940
$202$	0	0
$203$	−30.0839	−2.11147
$204$	0	0
$205$	2.74248	0.191543
$206$	0	0
$207$	3.17572	0.220728
$208$	0	0
$209$	26.2853	1.81819
$210$	0	0
$211$	−27.3334	−1.88171	−0.940855	−	0.338809i	$-0.889976\pi$
$211$	−27.3334	−1.88171	−0.940855	+	0.338809i	$0.889976\pi$
$212$	0	0
$213$	8.05669	0.552035
$214$	0	0
$215$	−3.11584	−0.212499
$216$	0	0
$217$	−19.8742	−1.34915
$218$	0	0
$219$	0.449676	0.0303863
$220$	0	0
$221$	16.2871	1.09559
$222$	0	0
$223$	28.1512	1.88514	0.942572	−	0.334004i	$-0.108400\pi$
$223$	28.1512	1.88514	0.942572	+	0.334004i	$0.108400\pi$
$224$	0	0
$225$	−4.46696	−0.297797
$226$	0	0
$227$	−6.11576	−0.405917	−0.202959	−	0.979187i	$-0.565056\pi$
$227$	−6.11576	−0.405917	−0.202959	+	0.979187i	$0.565056\pi$
$228$	0	0
$229$	20.8170	1.37562	0.687812	−	0.725889i	$-0.258571\pi$
$229$	20.8170	1.37562	0.687812	+	0.725889i	$0.258571\pi$
$230$	0	0
$231$	18.5179	1.21839
$232$	0	0
$233$	2.31590	0.151720	0.0758598	−	0.997118i	$-0.475830\pi$
$233$	2.31590	0.151720	0.0758598	+	0.997118i	$0.475830\pi$
$234$	0	0
$235$	0.830866	0.0541997
$236$	0	0
$237$	−6.95350	−0.451678
$238$	0	0
$239$	−3.32324	−0.214962	−0.107481	−	0.994207i	$-0.534279\pi$
$239$	−3.32324	−0.214962	−0.107481	+	0.994207i	$0.534279\pi$
$240$	0	0
$241$	−12.5640	−0.809316	−0.404658	−	0.914468i	$-0.632609\pi$
$241$	−12.5640	−0.809316	−0.404658	+	0.914468i	$0.632609\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	2.66121	0.170019
$246$	0	0
$247$	−28.6670	−1.82404
$248$	0	0
$249$	5.09676	0.322994
$250$	0	0
$251$	−9.57205	−0.604183	−0.302091	−	0.953279i	$-0.597685\pi$
$251$	−9.57205	−0.604183	−0.302091	+	0.953279i	$0.597685\pi$
$252$	0	0
$253$	−18.0244	−1.13318
$254$	0	0
$255$	−1.92104	−0.120300
$256$	0	0
$257$	−29.0040	−1.80922	−0.904611	−	0.426238i	$-0.859839\pi$
$257$	−29.0040	−1.80922	−0.904611	+	0.426238i	$0.859839\pi$
$258$	0	0
$259$	−13.8356	−0.859703
$260$	0	0
$261$	−9.22063	−0.570742
$262$	0	0
$263$	28.1825	1.73781	0.868904	−	0.494980i	$-0.164825\pi$
$263$	28.1825	1.73781	0.868904	+	0.494980i	$0.164825\pi$
$264$	0	0
$265$	−9.05055	−0.555971
$266$	0	0
$267$	2.02997	0.124232
$268$	0	0
$269$	23.6254	1.44047	0.720233	−	0.693732i	$-0.244035\pi$
$269$	23.6254	1.44047	0.720233	+	0.693732i	$0.244035\pi$
$270$	0	0
$271$	13.0876	0.795013	0.397507	−	0.917599i	$-0.369876\pi$
$271$	13.0876	0.795013	0.397507	+	0.917599i	$0.369876\pi$
$272$	0	0
$273$	−20.1958	−1.22230
$274$	0	0
$275$	25.3530	1.52884
$276$	0	0
$277$	−13.0813	−0.785981	−0.392991	−	0.919542i	$-0.628560\pi$
$277$	−13.0813	−0.785981	−0.392991	+	0.919542i	$0.628560\pi$
$278$	0	0
$279$	−6.09141	−0.364683
$280$	0	0
$281$	−11.0939	−0.661806	−0.330903	−	0.943665i	$-0.607353\pi$
$281$	−11.0939	−0.661806	−0.330903	+	0.943665i	$0.607353\pi$
$282$	0	0
$283$	4.43449	0.263603	0.131802	−	0.991276i	$-0.457924\pi$
$283$	4.43449	0.263603	0.131802	+	0.991276i	$0.457924\pi$
$284$	0	0
$285$	3.38124	0.200287
$286$	0	0
$287$	12.2556	0.723426
$288$	0	0
$289$	−10.0767	−0.592749
$290$	0	0
$291$	4.40474	0.258210
$292$	0	0
$293$	−33.2796	−1.94422	−0.972108	−	0.234532i	$-0.924644\pi$
$293$	−33.2796	−1.94422	−0.972108	+	0.234532i	$0.924644\pi$
$294$	0	0
$295$	−3.61512	−0.210480
$296$	0	0
$297$	5.67568	0.329336
$298$	0	0
$299$	19.6576	1.13683
$300$	0	0
$301$	−13.9241	−0.802572
$302$	0	0
$303$	10.3799	0.596311
$304$	0	0
$305$	5.47268	0.313365
$306$	0	0
$307$	−0.813465	−0.0464269	−0.0232134	−	0.999731i	$-0.507390\pi$
$307$	−0.813465	−0.0464269	−0.0232134	+	0.999731i	$0.507390\pi$
$308$	0	0
$309$	13.4120	0.762979
$310$	0	0
$311$	−27.4584	−1.55702	−0.778511	−	0.627631i	$-0.784025\pi$
$311$	−27.4584	−1.55702	−0.778511	+	0.627631i	$0.784025\pi$
$312$	0	0
$313$	0.786774	0.0444711	0.0222355	−	0.999753i	$-0.492922\pi$
$313$	0.786774	0.0444711	0.0222355	+	0.999753i	$0.492922\pi$
$314$	0	0
$315$	2.38207	0.134214
$316$	0	0
$317$	−7.59789	−0.426740	−0.213370	−	0.976971i	$-0.568444\pi$
$317$	−7.59789	−0.426740	−0.213370	+	0.976971i	$0.568444\pi$
$318$	0	0
$319$	52.3333	2.93010
$320$	0	0
$321$	−10.7425	−0.599590
$322$	0	0
$323$	−12.1857	−0.678030
$324$	0	0
$325$	−27.6503	−1.53376
$326$	0	0
$327$	−6.58912	−0.364379
$328$	0	0
$329$	3.71298	0.204703
$330$	0	0
$331$	10.0661	0.553285	0.276643	−	0.960973i	$-0.410778\pi$
$331$	10.0661	0.553285	0.276643	+	0.960973i	$0.410778\pi$
$332$	0	0
$333$	−4.24058	−0.232382
$334$	0	0
$335$	−1.52096	−0.0830990
$336$	0	0
$337$	14.2971	0.778814	0.389407	−	0.921066i	$-0.372680\pi$
$337$	14.2971	0.778814	0.389407	+	0.921066i	$0.372680\pi$
$338$	0	0
$339$	8.68884	0.471913
$340$	0	0
$341$	34.5729	1.87222
$342$	0	0
$343$	−10.9462	−0.591040
$344$	0	0
$345$	−2.31859	−0.124829
$346$	0	0
$347$	12.2511	0.657673	0.328836	−	0.944387i	$-0.393344\pi$
$347$	12.2511	0.657673	0.328836	+	0.944387i	$0.393344\pi$
$348$	0	0
$349$	10.7555	0.575730	0.287865	−	0.957671i	$-0.407055\pi$
$349$	10.7555	0.575730	0.287865	+	0.957671i	$0.407055\pi$
$350$	0	0
$351$	−6.18996	−0.330396
$352$	0	0
$353$	8.30567	0.442066	0.221033	−	0.975266i	$-0.429057\pi$
$353$	8.30567	0.442066	0.221033	+	0.975266i	$0.429057\pi$
$354$	0	0
$355$	−5.88217	−0.312193
$356$	0	0
$357$	−8.58477	−0.454354
$358$	0	0
$359$	12.3173	0.650083	0.325041	−	0.945700i	$-0.394622\pi$
$359$	12.3173	0.650083	0.325041	+	0.945700i	$0.394622\pi$
$360$	0	0
$361$	2.44811	0.128848
$362$	0	0
$363$	−21.2133	−1.11341
$364$	0	0
$365$	−0.328307	−0.0171844
$366$	0	0
$367$	2.02127	0.105510	0.0527548	−	0.998607i	$-0.483200\pi$
$367$	2.02127	0.105510	0.0527548	+	0.998607i	$0.483200\pi$
$368$	0	0
$369$	3.75631	0.195546
$370$	0	0
$371$	−40.4452	−2.09981
$372$	0	0
$373$	−16.5782	−0.858388	−0.429194	−	0.903212i	$-0.641202\pi$
$373$	−16.5782	−0.858388	−0.429194	+	0.903212i	$0.641202\pi$
$374$	0	0
$375$	6.91180	0.356924
$376$	0	0
$377$	−57.0753	−2.93953
$378$	0	0
$379$	−20.7107	−1.06384	−0.531919	−	0.846795i	$-0.678529\pi$
$379$	−20.7107	−1.06384	−0.531919	+	0.846795i	$0.678529\pi$
$380$	0	0
$381$	−1.00000	−0.0512316
$382$	0	0
$383$	−25.4844	−1.30219	−0.651095	−	0.758996i	$-0.725690\pi$
$383$	−25.4844	−1.30219	−0.651095	+	0.758996i	$0.725690\pi$
$384$	0	0
$385$	−13.5198	−0.689035
$386$	0	0
$387$	−4.26770	−0.216940
$388$	0	0
$389$	16.8643	0.855053	0.427526	−	0.904003i	$-0.359385\pi$
$389$	16.8643	0.855053	0.427526	+	0.904003i	$0.359385\pi$
$390$	0	0
$391$	8.35599	0.422581
$392$	0	0
$393$	−13.8370	−0.697987
$394$	0	0
$395$	5.07673	0.255438
$396$	0	0
$397$	15.6857	0.787241	0.393620	−	0.919273i	$-0.371222\pi$
$397$	15.6857	0.787241	0.393620	+	0.919273i	$0.371222\pi$
$398$	0	0
$399$	15.1101	0.756452
$400$	0	0
$401$	5.92962	0.296111	0.148055	−	0.988979i	$-0.452699\pi$
$401$	5.92962	0.296111	0.148055	+	0.988979i	$0.452699\pi$
$402$	0	0
$403$	−37.7056	−1.87825
$404$	0	0
$405$	0.730098	0.0362788
$406$	0	0
$407$	24.0682	1.19301
$408$	0	0
$409$	0.136675	0.00675814	0.00337907	−	0.999994i	$-0.498924\pi$
$409$	0.136675	0.00675814	0.00337907	+	0.999994i	$0.498924\pi$
$410$	0	0
$411$	−2.93954	−0.144997
$412$	0	0
$413$	−16.1553	−0.794950
$414$	0	0
$415$	−3.72114	−0.182663
$416$	0	0
$417$	−9.45281	−0.462906
$418$	0	0
$419$	6.48482	0.316804	0.158402	−	0.987375i	$-0.449366\pi$
$419$	6.48482	0.316804	0.158402	+	0.987375i	$0.449366\pi$
$420$	0	0
$421$	21.4386	1.04485	0.522426	−	0.852685i	$-0.325027\pi$
$421$	21.4386	1.04485	0.522426	+	0.852685i	$0.325027\pi$
$422$	0	0
$423$	1.13802	0.0553324
$424$	0	0
$425$	−11.7535	−0.570129
$426$	0	0
$427$	24.4564	1.18353
$428$	0	0
$429$	35.1322	1.69620
$430$	0	0
$431$	35.1794	1.69453	0.847267	−	0.531168i	$-0.178247\pi$
$431$	35.1794	1.69453	0.847267	+	0.531168i	$0.178247\pi$
$432$	0	0
$433$	−12.0399	−0.578602	−0.289301	−	0.957238i	$-0.593423\pi$
$433$	−12.0399	−0.578602	−0.289301	+	0.957238i	$0.593423\pi$
$434$	0	0
$435$	6.73196	0.322773
$436$	0	0
$437$	−14.7074	−0.703552
$438$	0	0
$439$	−23.2467	−1.10950	−0.554751	−	0.832016i	$-0.687187\pi$
$439$	−23.2467	−1.10950	−0.554751	+	0.832016i	$0.687187\pi$
$440$	0	0
$441$	3.64501	0.173572
$442$	0	0
$443$	14.5282	0.690253	0.345127	−	0.938556i	$-0.387836\pi$
$443$	14.5282	0.690253	0.345127	+	0.938556i	$0.387836\pi$
$444$	0	0
$445$	−1.48208	−0.0702573
$446$	0	0
$447$	19.6394	0.928910
$448$	0	0
$449$	16.7413	0.790069	0.395034	−	0.918666i	$-0.370733\pi$
$449$	16.7413	0.790069	0.395034	+	0.918666i	$0.370733\pi$
$450$	0	0
$451$	−21.3196	−1.00390
$452$	0	0
$453$	−12.8407	−0.603310
$454$	0	0
$455$	14.7449	0.691252
$456$	0	0
$457$	−17.5598	−0.821413	−0.410706	−	0.911768i	$-0.634718\pi$
$457$	−17.5598	−0.821413	−0.410706	+	0.911768i	$0.634718\pi$
$458$	0	0
$459$	−2.63121	−0.122814
$460$	0	0
$461$	5.17795	0.241161	0.120581	−	0.992704i	$-0.461524\pi$
$461$	5.17795	0.241161	0.120581	+	0.992704i	$0.461524\pi$
$462$	0	0
$463$	−8.06771	−0.374938	−0.187469	−	0.982271i	$-0.560028\pi$
$463$	−8.06771	−0.374938	−0.187469	+	0.982271i	$0.560028\pi$
$464$	0	0
$465$	4.44732	0.206240
$466$	0	0
$467$	−24.1977	−1.11974	−0.559869	−	0.828581i	$-0.689148\pi$
$467$	−24.1977	−1.11974	−0.559869	+	0.828581i	$0.689148\pi$
$468$	0	0
$469$	−6.79689	−0.313851
$470$	0	0
$471$	−9.51482	−0.438420
$472$	0	0
$473$	24.2221	1.11373
$474$	0	0
$475$	20.6874	0.949204
$476$	0	0
$477$	−12.3963	−0.567590
$478$	0	0
$479$	−16.8571	−0.770222	−0.385111	−	0.922870i	$-0.625837\pi$
$479$	−16.8571	−0.770222	−0.385111	+	0.922870i	$0.625837\pi$
$480$	0	0
$481$	−26.2490	−1.19685
$482$	0	0
$483$	−10.3613	−0.471457
$484$	0	0
$485$	−3.21589	−0.146026
$486$	0	0
$487$	−36.1603	−1.63858	−0.819290	−	0.573379i	$-0.805632\pi$
$487$	−36.1603	−1.63858	−0.819290	+	0.573379i	$0.805632\pi$
$488$	0	0
$489$	−23.6001	−1.06723
$490$	0	0
$491$	−21.4712	−0.968983	−0.484491	−	0.874796i	$-0.660995\pi$
$491$	−21.4712	−0.968983	−0.484491	+	0.874796i	$0.660995\pi$
$492$	0	0
$493$	−24.2614	−1.09268
$494$	0	0
$495$	−4.14380	−0.186250
$496$	0	0
$497$	−26.2863	−1.17910
$498$	0	0
$499$	−23.1230	−1.03513	−0.517563	−	0.855645i	$-0.673161\pi$
$499$	−23.1230	−1.03513	−0.517563	+	0.855645i	$0.673161\pi$
$500$	0	0
$501$	23.8515	1.06561
$502$	0	0
$503$	−11.0323	−0.491906	−0.245953	−	0.969282i	$-0.579101\pi$
$503$	−11.0323	−0.491906	−0.245953	+	0.969282i	$0.579101\pi$
$504$	0	0
$505$	−7.57836	−0.337232
$506$	0	0
$507$	−25.3156	−1.12431
$508$	0	0
$509$	−31.5235	−1.39726	−0.698628	−	0.715485i	$-0.746206\pi$
$509$	−31.5235	−1.39726	−0.698628	+	0.715485i	$0.746206\pi$
$510$	0	0
$511$	−1.46714	−0.0649026
$512$	0	0
$513$	4.63121	0.204473
$514$	0	0
$515$	−9.79204	−0.431489
$516$	0	0
$517$	−6.45903	−0.284068
$518$	0	0
$519$	4.23447	0.185873
$520$	0	0
$521$	−8.07076	−0.353586	−0.176793	−	0.984248i	$-0.556572\pi$
$521$	−8.07076	−0.353586	−0.176793	+	0.984248i	$0.556572\pi$
$522$	0	0
$523$	4.78967	0.209438	0.104719	−	0.994502i	$-0.466606\pi$
$523$	4.78967	0.209438	0.104719	+	0.994502i	$0.466606\pi$
$524$	0	0
$525$	14.5742	0.636070
$526$	0	0
$527$	−16.0278	−0.698181
$528$	0	0
$529$	−12.9148	−0.561513
$530$	0	0
$531$	−4.95155	−0.214879
$532$	0	0
$533$	23.2514	1.00713
$534$	0	0
$535$	7.84311	0.339087
$536$	0	0
$537$	3.07916	0.132876
$538$	0	0
$539$	−20.6879	−0.891091
$540$	0	0
$541$	−16.2940	−0.700532	−0.350266	−	0.936650i	$-0.613909\pi$
$541$	−16.2940	−0.700532	−0.350266	+	0.936650i	$0.613909\pi$
$542$	0	0
$543$	18.9050	0.811293
$544$	0	0
$545$	4.81070	0.206068
$546$	0	0
$547$	5.42893	0.232124	0.116062	−	0.993242i	$-0.462973\pi$
$547$	5.42893	0.232124	0.116062	+	0.993242i	$0.462973\pi$
$548$	0	0
$549$	7.49582	0.319914
$550$	0	0
$551$	42.7027	1.81919
$552$	0	0
$553$	22.6870	0.964748
$554$	0	0
$555$	3.09604	0.131419
$556$	0	0
$557$	−26.3767	−1.11762	−0.558808	−	0.829297i	$-0.688741\pi$
$557$	−26.3767	−1.11762	−0.558808	+	0.829297i	$0.688741\pi$
$558$	0	0
$559$	−26.4169	−1.11732
$560$	0	0
$561$	14.9339	0.630510
$562$	0	0
$563$	−8.73665	−0.368206	−0.184103	−	0.982907i	$-0.558938\pi$
$563$	−8.73665	−0.368206	−0.184103	+	0.982907i	$0.558938\pi$
$564$	0	0
$565$	−6.34370	−0.266882
$566$	0	0
$567$	3.26267	0.137019
$568$	0	0
$569$	−15.4535	−0.647843	−0.323921	−	0.946084i	$-0.605001\pi$
$569$	−15.4535	−0.647843	−0.323921	+	0.946084i	$0.605001\pi$
$570$	0	0
$571$	−30.7959	−1.28877	−0.644384	−	0.764702i	$-0.722886\pi$
$571$	−30.7959	−1.28877	−0.644384	+	0.764702i	$0.722886\pi$
$572$	0	0
$573$	−23.0731	−0.963894
$574$	0	0
$575$	−14.1858	−0.591589
$576$	0	0
$577$	35.3969	1.47359	0.736796	−	0.676116i	$-0.236338\pi$
$577$	35.3969	1.47359	0.736796	+	0.676116i	$0.236338\pi$
$578$	0	0
$579$	−9.91938	−0.412235
$580$	0	0
$581$	−16.6291	−0.689889
$582$	0	0
$583$	70.3577	2.91392
$584$	0	0
$585$	4.51928	0.186849
$586$	0	0
$587$	22.1999	0.916286	0.458143	−	0.888879i	$-0.348515\pi$
$587$	22.1999	0.916286	0.458143	+	0.888879i	$0.348515\pi$
$588$	0	0
$589$	28.2106	1.16240
$590$	0	0
$591$	8.02848	0.330248
$592$	0	0
$593$	−26.2104	−1.07633	−0.538165	−	0.842839i	$-0.680882\pi$
$593$	−26.2104	−1.07633	−0.538165	+	0.842839i	$0.680882\pi$
$594$	0	0
$595$	6.26772	0.256951
$596$	0	0
$597$	22.3790	0.915911
$598$	0	0
$599$	−41.2984	−1.68740	−0.843702	−	0.536811i	$-0.819629\pi$
$599$	−41.2984	−1.68740	−0.843702	+	0.536811i	$0.819629\pi$
$600$	0	0
$601$	33.3376	1.35987	0.679934	−	0.733274i	$-0.262009\pi$
$601$	33.3376	1.35987	0.679934	+	0.733274i	$0.262009\pi$
$602$	0	0
$603$	−2.08323	−0.0848356
$604$	0	0
$605$	15.4878	0.629668
$606$	0	0
$607$	−10.8056	−0.438584	−0.219292	−	0.975659i	$-0.570375\pi$
$607$	−10.8056	−0.438584	−0.219292	+	0.975659i	$0.570375\pi$
$608$	0	0
$609$	30.0839	1.21906
$610$	0	0
$611$	7.04430	0.284982
$612$	0	0
$613$	18.4102	0.743582	0.371791	−	0.928316i	$-0.378744\pi$
$613$	18.4102	0.743582	0.371791	+	0.928316i	$0.378744\pi$
$614$	0	0
$615$	−2.74248	−0.110587
$616$	0	0
$617$	−25.9688	−1.04546	−0.522731	−	0.852497i	$-0.675087\pi$
$617$	−25.9688	−1.04546	−0.522731	+	0.852497i	$0.675087\pi$
$618$	0	0
$619$	−20.2261	−0.812957	−0.406479	−	0.913660i	$-0.633243\pi$
$619$	−20.2261	−0.812957	−0.406479	+	0.913660i	$0.633243\pi$
$620$	0	0
$621$	−3.17572	−0.127437
$622$	0	0
$623$	−6.62313	−0.265350
$624$	0	0
$625$	17.2885	0.691540
$626$	0	0
$627$	−26.2853	−1.04973
$628$	0	0
$629$	−11.1579	−0.444893
$630$	0	0
$631$	−4.91491	−0.195659	−0.0978297	−	0.995203i	$-0.531190\pi$
$631$	−4.91491	−0.195659	−0.0978297	+	0.995203i	$0.531190\pi$
$632$	0	0
$633$	27.3334	1.08641
$634$	0	0
$635$	0.730098	0.0289730
$636$	0	0
$637$	22.5625	0.893958
$638$	0	0
$639$	−8.05669	−0.318718
$640$	0	0
$641$	35.8440	1.41575	0.707876	−	0.706337i	$-0.249653\pi$
$641$	35.8440	1.41575	0.707876	+	0.706337i	$0.249653\pi$
$642$	0	0
$643$	−19.9921	−0.788413	−0.394206	−	0.919022i	$-0.628980\pi$
$643$	−19.9921	−0.788413	−0.394206	+	0.919022i	$0.628980\pi$
$644$	0	0
$645$	3.11584	0.122686
$646$	0	0
$647$	30.8851	1.21422	0.607109	−	0.794618i	$-0.292329\pi$
$647$	30.8851	1.21422	0.607109	+	0.794618i	$0.292329\pi$
$648$	0	0
$649$	28.1034	1.10316
$650$	0	0
$651$	19.8742	0.778933
$652$	0	0
$653$	28.1627	1.10209	0.551046	−	0.834475i	$-0.314229\pi$
$653$	28.1627	1.10209	0.551046	+	0.834475i	$0.314229\pi$
$654$	0	0
$655$	10.1024	0.394733
$656$	0	0
$657$	−0.449676	−0.0175435
$658$	0	0
$659$	1.17991	0.0459629	0.0229814	−	0.999736i	$-0.492684\pi$
$659$	1.17991	0.0459629	0.0229814	+	0.999736i	$0.492684\pi$
$660$	0	0
$661$	−3.56827	−0.138789	−0.0693947	−	0.997589i	$-0.522107\pi$
$661$	−3.56827	−0.138789	−0.0693947	+	0.997589i	$0.522107\pi$
$662$	0	0
$663$	−16.2871	−0.632538
$664$	0	0
$665$	−11.0319	−0.427797
$666$	0	0
$667$	−29.2821	−1.13381
$668$	0	0
$669$	−28.1512	−1.08839
$670$	0	0
$671$	−42.5438	−1.64239
$672$	0	0
$673$	−23.6668	−0.912288	−0.456144	−	0.889906i	$-0.650770\pi$
$673$	−23.6668	−0.912288	−0.456144	+	0.889906i	$0.650770\pi$
$674$	0	0
$675$	4.46696	0.171933
$676$	0	0
$677$	−41.6260	−1.59982	−0.799908	−	0.600123i	$-0.795118\pi$
$677$	−41.6260	−1.59982	−0.799908	+	0.600123i	$0.795118\pi$
$678$	0	0
$679$	−14.3712	−0.551516
$680$	0	0
$681$	6.11576	0.234356
$682$	0	0
$683$	46.9772	1.79753	0.898767	−	0.438428i	$-0.144464\pi$
$683$	46.9772	1.79753	0.898767	+	0.438428i	$0.144464\pi$
$684$	0	0
$685$	2.14615	0.0820002
$686$	0	0
$687$	−20.8170	−0.794217
$688$	0	0
$689$	−76.7329	−2.92329
$690$	0	0
$691$	29.8965	1.13732	0.568659	−	0.822573i	$-0.307462\pi$
$691$	29.8965	1.13732	0.568659	+	0.822573i	$0.307462\pi$
$692$	0	0
$693$	−18.5179	−0.703435
$694$	0	0
$695$	6.90147	0.261788
$696$	0	0
$697$	9.88365	0.374370
$698$	0	0
$699$	−2.31590	−0.0875953
$700$	0	0
$701$	11.6543	0.440176	0.220088	−	0.975480i	$-0.429366\pi$
$701$	11.6543	0.440176	0.220088	+	0.975480i	$0.429366\pi$
$702$	0	0
$703$	19.6390	0.740700
$704$	0	0
$705$	−0.830866	−0.0312922
$706$	0	0
$707$	−33.8662	−1.27367
$708$	0	0
$709$	−14.4932	−0.544303	−0.272151	−	0.962254i	$-0.587735\pi$
$709$	−14.4932	−0.544303	−0.272151	+	0.962254i	$0.587735\pi$
$710$	0	0
$711$	6.95350	0.260777
$712$	0	0
$713$	−19.3446	−0.724461
$714$	0	0
$715$	−25.6499	−0.959254
$716$	0	0
$717$	3.32324	0.124109
$718$	0	0
$719$	14.5782	0.543676	0.271838	−	0.962343i	$-0.412369\pi$
$719$	14.5782	0.543676	0.271838	+	0.962343i	$0.412369\pi$
$720$	0	0
$721$	−43.7588	−1.62966
$722$	0	0
$723$	12.5640	0.467259
$724$	0	0
$725$	41.1881	1.52969
$726$	0	0
$727$	2.11293	0.0783642	0.0391821	−	0.999232i	$-0.487525\pi$
$727$	2.11293	0.0783642	0.0391821	+	0.999232i	$0.487525\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−11.2292	−0.415328
$732$	0	0
$733$	−44.0321	−1.62636	−0.813182	−	0.582010i	$-0.802266\pi$
$733$	−44.0321	−1.62636	−0.813182	+	0.582010i	$0.802266\pi$
$734$	0	0
$735$	−2.66121	−0.0981604
$736$	0	0
$737$	11.8237	0.435533
$738$	0	0
$739$	30.1933	1.11068	0.555340	−	0.831624i	$-0.312588\pi$
$739$	30.1933	1.11068	0.555340	+	0.831624i	$0.312588\pi$
$740$	0	0
$741$	28.6670	1.05311
$742$	0	0
$743$	15.9501	0.585153	0.292576	−	0.956242i	$-0.405487\pi$
$743$	15.9501	0.585153	0.292576	+	0.956242i	$0.405487\pi$
$744$	0	0
$745$	−14.3387	−0.525328
$746$	0	0
$747$	−5.09676	−0.186481
$748$	0	0
$749$	35.0494	1.28068
$750$	0	0
$751$	21.3528	0.779176	0.389588	−	0.920989i	$-0.372617\pi$
$751$	21.3528	0.779176	0.389588	+	0.920989i	$0.372617\pi$
$752$	0	0
$753$	9.57205	0.348825
$754$	0	0
$755$	9.37499	0.341191
$756$	0	0
$757$	−27.3116	−0.992656	−0.496328	−	0.868135i	$-0.665319\pi$
$757$	−27.3116	−0.992656	−0.496328	+	0.868135i	$0.665319\pi$
$758$	0	0
$759$	18.0244	0.654243
$760$	0	0
$761$	−25.3898	−0.920379	−0.460189	−	0.887821i	$-0.652218\pi$
$761$	−25.3898	−0.920379	−0.460189	+	0.887821i	$0.652218\pi$
$762$	0	0
$763$	21.4981	0.778284
$764$	0	0
$765$	1.92104	0.0694554
$766$	0	0
$767$	−30.6499	−1.10670
$768$	0	0
$769$	24.2388	0.874073	0.437037	−	0.899444i	$-0.356028\pi$
$769$	24.2388	0.874073	0.437037	+	0.899444i	$0.356028\pi$
$770$	0	0
$771$	29.0040	1.04455
$772$	0	0
$773$	−18.0085	−0.647720	−0.323860	−	0.946105i	$-0.604981\pi$
$773$	−18.0085	−0.647720	−0.323860	+	0.946105i	$0.604981\pi$
$774$	0	0
$775$	27.2101	0.977414
$776$	0	0
$777$	13.8356	0.496350
$778$	0	0
$779$	−17.3963	−0.623286
$780$	0	0
$781$	45.7272	1.63625
$782$	0	0
$783$	9.22063	0.329518
$784$	0	0
$785$	6.94675	0.247940
$786$	0	0
$787$	16.5498	0.589938	0.294969	−	0.955507i	$-0.404691\pi$
$787$	16.5498	0.589938	0.294969	+	0.955507i	$0.404691\pi$
$788$	0	0
$789$	−28.1825	−1.00332
$790$	0	0
$791$	−28.3488	−1.00797
$792$	0	0
$793$	46.3988	1.64767
$794$	0	0
$795$	9.05055	0.320990
$796$	0	0
$797$	36.4091	1.28968	0.644838	−	0.764319i	$-0.276925\pi$
$797$	36.4091	1.28968	0.644838	+	0.764319i	$0.276925\pi$
$798$	0	0
$799$	2.99437	0.105933
$800$	0	0
$801$	−2.02997	−0.0717256
$802$	0	0
$803$	2.55221	0.0900657
$804$	0	0
$805$	7.56478	0.266624
$806$	0	0
$807$	−23.6254	−0.831654
$808$	0	0
$809$	34.5832	1.21588	0.607940	−	0.793983i	$-0.291996\pi$
$809$	34.5832	1.21588	0.607940	+	0.793983i	$0.291996\pi$
$810$	0	0
$811$	21.2642	0.746687	0.373344	−	0.927693i	$-0.378211\pi$
$811$	21.2642	0.746687	0.373344	+	0.927693i	$0.378211\pi$
$812$	0	0
$813$	−13.0876	−0.459001
$814$	0	0
$815$	17.2304	0.603554
$816$	0	0
$817$	19.7646	0.691477
$818$	0	0
$819$	20.1958	0.705698
$820$	0	0
$821$	32.6487	1.13945	0.569724	−	0.821836i	$-0.307050\pi$
$821$	32.6487	1.13945	0.569724	+	0.821836i	$0.307050\pi$
$822$	0	0
$823$	40.7342	1.41990	0.709952	−	0.704250i	$-0.248717\pi$
$823$	40.7342	1.41990	0.709952	+	0.704250i	$0.248717\pi$
$824$	0	0
$825$	−25.3530	−0.882678
$826$	0	0
$827$	−39.8118	−1.38439	−0.692196	−	0.721710i	$-0.743356\pi$
$827$	−39.8118	−1.38439	−0.692196	+	0.721710i	$0.743356\pi$
$828$	0	0
$829$	43.3664	1.50618	0.753090	−	0.657918i	$-0.228563\pi$
$829$	43.3664	1.50618	0.753090	+	0.657918i	$0.228563\pi$
$830$	0	0
$831$	13.0813	0.453787
$832$	0	0
$833$	9.59079	0.332301
$834$	0	0
$835$	−17.4140	−0.602635
$836$	0	0
$837$	6.09141	0.210550
$838$	0	0
$839$	−43.2174	−1.49203	−0.746015	−	0.665930i	$-0.768035\pi$
$839$	−43.2174	−1.49203	−0.746015	+	0.665930i	$0.768035\pi$
$840$	0	0
$841$	56.0199	1.93172
$842$	0	0
$843$	11.0939	0.382094
$844$	0	0
$845$	18.4829	0.635830
$846$	0	0
$847$	69.2120	2.37815
$848$	0	0
$849$	−4.43449	−0.152191
$850$	0	0
$851$	−13.4669	−0.461640
$852$	0	0
$853$	41.3459	1.41566	0.707828	−	0.706385i	$-0.249675\pi$
$853$	41.3459	1.41566	0.707828	+	0.706385i	$0.249675\pi$
$854$	0	0
$855$	−3.38124	−0.115636
$856$	0	0
$857$	−20.6726	−0.706162	−0.353081	−	0.935593i	$-0.614866\pi$
$857$	−20.6726	−0.706162	−0.353081	+	0.935593i	$0.614866\pi$
$858$	0	0
$859$	28.0249	0.956196	0.478098	−	0.878307i	$-0.341326\pi$
$859$	28.0249	0.956196	0.478098	+	0.878307i	$0.341326\pi$
$860$	0	0
$861$	−12.2556	−0.417670
$862$	0	0
$863$	−30.5578	−1.04020	−0.520100	−	0.854106i	$-0.674105\pi$
$863$	−30.5578	−1.04020	−0.520100	+	0.854106i	$0.674105\pi$
$864$	0	0
$865$	−3.09158	−0.105117
$866$	0	0
$867$	10.0767	0.342224
$868$	0	0
$869$	−39.4658	−1.33879
$870$	0	0
$871$	−12.8951	−0.436934
$872$	0	0
$873$	−4.40474	−0.149078
$874$	0	0
$875$	−22.5509	−0.762361
$876$	0	0
$877$	−17.7657	−0.599904	−0.299952	−	0.953954i	$-0.596971\pi$
$877$	−17.7657	−0.599904	−0.299952	+	0.953954i	$0.596971\pi$
$878$	0	0
$879$	33.2796	1.12249
$880$	0	0
$881$	5.46089	0.183982	0.0919911	−	0.995760i	$-0.470677\pi$
$881$	5.46089	0.183982	0.0919911	+	0.995760i	$0.470677\pi$
$882$	0	0
$883$	−25.3867	−0.854330	−0.427165	−	0.904174i	$-0.640488\pi$
$883$	−25.3867	−0.854330	−0.427165	+	0.904174i	$0.640488\pi$
$884$	0	0
$885$	3.61512	0.121521
$886$	0	0
$887$	19.8564	0.666712	0.333356	−	0.942801i	$-0.391819\pi$
$887$	19.8564	0.666712	0.333356	+	0.942801i	$0.391819\pi$
$888$	0	0
$889$	3.26267	0.109426
$890$	0	0
$891$	−5.67568	−0.190142
$892$	0	0
$893$	−5.27041	−0.176368
$894$	0	0
$895$	−2.24809	−0.0751454
$896$	0	0
$897$	−19.6576	−0.656348
$898$	0	0
$899$	56.1666	1.87326
$900$	0	0
$901$	−32.6174	−1.08664
$902$	0	0
$903$	13.9241	0.463365
$904$	0	0
$905$	−13.8025	−0.458812
$906$	0	0
$907$	13.3616	0.443664	0.221832	−	0.975085i	$-0.428796\pi$
$907$	13.3616	0.443664	0.221832	+	0.975085i	$0.428796\pi$
$908$	0	0
$909$	−10.3799	−0.344280
$910$	0	0
$911$	−2.68407	−0.0889273	−0.0444636	−	0.999011i	$-0.514158\pi$
$911$	−2.68407	−0.0889273	−0.0444636	+	0.999011i	$0.514158\pi$
$912$	0	0
$913$	28.9276	0.957363
$914$	0	0
$915$	−5.47268	−0.180921
$916$	0	0
$917$	45.1457	1.49084
$918$	0	0
$919$	48.1917	1.58970	0.794849	−	0.606808i	$-0.207550\pi$
$919$	48.1917	1.58970	0.794849	+	0.606808i	$0.207550\pi$
$920$	0	0
$921$	0.813465	0.0268046
$922$	0	0
$923$	−49.8706	−1.64151
$924$	0	0
$925$	18.9425	0.622825
$926$	0	0
$927$	−13.4120	−0.440506
$928$	0	0
$929$	44.5292	1.46096	0.730478	−	0.682937i	$-0.239297\pi$
$929$	44.5292	1.46096	0.730478	+	0.682937i	$0.239297\pi$
$930$	0	0
$931$	−16.8808	−0.553246
$932$	0	0
$933$	27.4584	0.898947
$934$	0	0
$935$	−10.9032	−0.356573
$936$	0	0
$937$	42.6300	1.39266	0.696331	−	0.717720i	$-0.254814\pi$
$937$	42.6300	1.39266	0.696331	+	0.717720i	$0.254814\pi$
$938$	0	0
$939$	−0.786774	−0.0256754
$940$	0	0
$941$	37.6984	1.22893	0.614466	−	0.788943i	$-0.289371\pi$
$941$	37.6984	1.22893	0.614466	+	0.788943i	$0.289371\pi$
$942$	0	0
$943$	11.9290	0.388462
$944$	0	0
$945$	−2.38207	−0.0774887
$946$	0	0
$947$	50.0805	1.62740	0.813698	−	0.581287i	$-0.197451\pi$
$947$	50.0805	1.62740	0.813698	+	0.581287i	$0.197451\pi$
$948$	0	0
$949$	−2.78347	−0.0903554
$950$	0	0
$951$	7.59789	0.246379
$952$	0	0
$953$	−23.2341	−0.752626	−0.376313	−	0.926493i	$-0.622808\pi$
$953$	−23.2341	−0.752626	−0.376313	+	0.926493i	$0.622808\pi$
$954$	0	0
$955$	16.8456	0.545112
$956$	0	0
$957$	−52.3333	−1.69170
$958$	0	0
$959$	9.59074	0.309701
$960$	0	0
$961$	6.10523	0.196943
$962$	0	0
$963$	10.7425	0.346174
$964$	0	0
$965$	7.24212	0.233132
$966$	0	0
$967$	−4.12518	−0.132657	−0.0663284	−	0.997798i	$-0.521129\pi$
$967$	−4.12518	−0.132657	−0.0663284	+	0.997798i	$0.521129\pi$
$968$	0	0
$969$	12.1857	0.391461
$970$	0	0
$971$	−33.2746	−1.06783	−0.533917	−	0.845537i	$-0.679281\pi$
$971$	−33.2746	−1.06783	−0.533917	+	0.845537i	$0.679281\pi$
$972$	0	0
$973$	30.8414	0.988729
$974$	0	0
$975$	27.6503	0.885518
$976$	0	0
$977$	17.7539	0.567996	0.283998	−	0.958825i	$-0.408339\pi$
$977$	17.7539	0.567996	0.283998	+	0.958825i	$0.408339\pi$
$978$	0	0
$979$	11.5215	0.368228
$980$	0	0
$981$	6.58912	0.210374
$982$	0	0
$983$	38.8198	1.23816	0.619079	−	0.785329i	$-0.287506\pi$
$983$	38.8198	1.23816	0.619079	+	0.785329i	$0.287506\pi$
$984$	0	0
$985$	−5.86158	−0.186765
$986$	0	0
$987$	−3.71298	−0.118186
$988$	0	0
$989$	−13.5530	−0.430962
$990$	0	0
$991$	23.1476	0.735307	0.367653	−	0.929963i	$-0.380161\pi$
$991$	23.1476	0.735307	0.367653	+	0.929963i	$0.380161\pi$
$992$	0	0
$993$	−10.0661	−0.319439
$994$	0	0
$995$	−16.3388	−0.517976
$996$	0	0
$997$	−35.7648	−1.13268	−0.566341	−	0.824171i	$-0.691642\pi$
$997$	−35.7648	−1.13268	−0.566341	+	0.824171i	$0.691642\pi$
$998$	0	0
$999$	4.24058	0.134166

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6096.2.a.bk.1.6		9
4.3	odd	2		381.2.a.e.1.5	✓	9
12.11	even	2		1143.2.a.j.1.5		9
20.19	odd	2		9525.2.a.p.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
381.2.a.e.1.5	✓	9	4.3	odd	2
1143.2.a.j.1.5		9	12.11	even	2
6096.2.a.bk.1.6		9	1.1	even	1	trivial
9525.2.a.p.1.5		9	20.19	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 6096 = 2^{4} \cdot 3 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6096.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.730098 q^{5} +3.26267 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.730098 q^{5} +3.26267 q^{7} +1.00000 q^{9} -5.67568 q^{11} +6.18996 q^{13} -0.730098 q^{15} +2.63121 q^{17} -4.63121 q^{19} -3.26267 q^{21} +3.17572 q^{23} -4.46696 q^{25} -1.00000 q^{27} -9.22063 q^{29} -6.09141 q^{31} +5.67568 q^{33} +2.38207 q^{35} -4.24058 q^{37} -6.18996 q^{39} +3.75631 q^{41} -4.26770 q^{43} +0.730098 q^{45} +1.13802 q^{47} +3.64501 q^{49} -2.63121 q^{51} -12.3963 q^{53} -4.14380 q^{55} +4.63121 q^{57} -4.95155 q^{59} +7.49582 q^{61} +3.26267 q^{63} +4.51928 q^{65} -2.08323 q^{67} -3.17572 q^{69} -8.05669 q^{71} -0.449676 q^{73} +4.46696 q^{75} -18.5179 q^{77} +6.95350 q^{79} +1.00000 q^{81} -5.09676 q^{83} +1.92104 q^{85} +9.22063 q^{87} -2.02997 q^{89} +20.1958 q^{91} +6.09141 q^{93} -3.38124 q^{95} -4.40474 q^{97} -5.67568 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{3} - 4 q^{5} - 10 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q - 9 * q^3 - 4 * q^5 - 10 * q^7 + 9 * q^9	\( 9 q - 9 q^{3} - 4 q^{5} - 10 q^{7} + 9 q^{9} - 8 q^{11} + 14 q^{13} + 4 q^{15} - 6 q^{17} - 12 q^{19} + 10 q^{21} + 4 q^{23} + 21 q^{25} - 9 q^{27} - 8 q^{29} - 4 q^{31} + 8 q^{33} - 6 q^{35} + 22 q^{37} - 14 q^{39} - 2 q^{41} - 6 q^{43} - 4 q^{45} + 2 q^{47} + 23 q^{49} + 6 q^{51} - 12 q^{53} + 22 q^{55} + 12 q^{57} + 6 q^{59} + 2 q^{61} - 10 q^{63} + 4 q^{65} - 18 q^{67} - 4 q^{69} - 24 q^{71} + 14 q^{73} - 21 q^{75} - 18 q^{77} - 12 q^{79} + 9 q^{81} + 20 q^{83} - 24 q^{85} + 8 q^{87} - 30 q^{89} - 14 q^{91} + 4 q^{93} + 32 q^{95} + 12 q^{97} - 8 q^{99}+O(q^{100}) \) 9 * q - 9 * q^3 - 4 * q^5 - 10 * q^7 + 9 * q^9 - 8 * q^11 + 14 * q^13 + 4 * q^15 - 6 * q^17 - 12 * q^19 + 10 * q^21 + 4 * q^23 + 21 * q^25 - 9 * q^27 - 8 * q^29 - 4 * q^31 + 8 * q^33 - 6 * q^35 + 22 * q^37 - 14 * q^39 - 2 * q^41 - 6 * q^43 - 4 * q^45 + 2 * q^47 + 23 * q^49 + 6 * q^51 - 12 * q^53 + 22 * q^55 + 12 * q^57 + 6 * q^59 + 2 * q^61 - 10 * q^63 + 4 * q^65 - 18 * q^67 - 4 * q^69 - 24 * q^71 + 14 * q^73 - 21 * q^75 - 18 * q^77 - 12 * q^79 + 9 * q^81 + 20 * q^83 - 24 * q^85 + 8 * q^87 - 30 * q^89 - 14 * q^91 + 4 * q^93 + 32 * q^95 + 12 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more