LMFDB - Embedded newform 6720.2.a.cy.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6720,2,Mod(1,6720)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6720, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6720.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6720.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:	$53.6594701583$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3360)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	6720.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.82843 q^{11} +4.82843 q^{13} +1.00000 q^{15} -3.65685 q^{17} +2.82843 q^{19} +1.00000 q^{21} +4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.65685 q^{29} -2.82843 q^{31} +2.82843 q^{33} +1.00000 q^{35} +0.343146 q^{37} +4.82843 q^{39} +3.65685 q^{41} -9.65685 q^{43} +1.00000 q^{45} +11.3137 q^{47} +1.00000 q^{49} -3.65685 q^{51} +6.48528 q^{53} +2.82843 q^{55} +2.82843 q^{57} +13.3137 q^{61} +1.00000 q^{63} +4.82843 q^{65} -4.00000 q^{67} +4.00000 q^{69} +5.17157 q^{71} +0.828427 q^{73} +1.00000 q^{75} +2.82843 q^{77} -7.31371 q^{79} +1.00000 q^{81} -6.34315 q^{83} -3.65685 q^{85} -3.65685 q^{87} +0.343146 q^{89} +4.82843 q^{91} -2.82843 q^{93} +2.82843 q^{95} -12.8284 q^{97} +2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{13} + 2 q^{15} + 4 q^{17} + 2 q^{21} + 8 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} + 2 q^{35} + 12 q^{37} + 4 q^{39} - 4 q^{41} - 8 q^{43} + 2 q^{45} + 2 q^{49} + 4 q^{51} - 4 q^{53} + 4 q^{61} + 2 q^{63} + 4 q^{65} - 8 q^{67} + 8 q^{69} + 16 q^{71} - 4 q^{73} + 2 q^{75} + 8 q^{79} + 2 q^{81} - 24 q^{83} + 4 q^{85} + 4 q^{87} + 12 q^{89} + 4 q^{91} - 20 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 + 4 * q^13 + 2 * q^15 + 4 * q^17 + 2 * q^21 + 8 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^29 + 2 * q^35 + 12 * q^37 + 4 * q^39 - 4 * q^41 - 8 * q^43 + 2 * q^45 + 2 * q^49 + 4 * q^51 - 4 * q^53 + 4 * q^61 + 2 * q^63 + 4 * q^65 - 8 * q^67 + 8 * q^69 + 16 * q^71 - 4 * q^73 + 2 * q^75 + 8 * q^79 + 2 * q^81 - 24 * q^83 + 4 * q^85 + 4 * q^87 + 12 * q^89 + 4 * q^91 - 20 * q^97

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.82843	0.852803	0.426401	−	0.904534i	$-0.359781\pi$
$11$	2.82843	0.852803	0.426401	+	0.904534i	$0.359781\pi$
$12$	0	0
$13$	4.82843	1.33916	0.669582	−	0.742738i	$-0.266473\pi$
$13$	4.82843	1.33916	0.669582	+	0.742738i	$0.266473\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−3.65685	−0.886917	−0.443459	−	0.896295i	$-0.646249\pi$
$17$	−3.65685	−0.886917	−0.443459	+	0.896295i	$0.646249\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.65685	−0.679061	−0.339530	−	0.940595i	$-0.610268\pi$
$29$	−3.65685	−0.679061	−0.339530	+	0.940595i	$0.610268\pi$
$30$	0	0
$31$	−2.82843	−0.508001	−0.254000	−	0.967204i	$-0.581746\pi$
$31$	−2.82843	−0.508001	−0.254000	+	0.967204i	$0.581746\pi$
$32$	0	0
$33$	2.82843	0.492366
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	0.343146	0.0564128	0.0282064	−	0.999602i	$-0.491020\pi$
$37$	0.343146	0.0564128	0.0282064	+	0.999602i	$0.491020\pi$
$38$	0	0
$39$	4.82843	0.773167
$40$	0	0
$41$	3.65685	0.571105	0.285552	−	0.958363i	$-0.407823\pi$
$41$	3.65685	0.571105	0.285552	+	0.958363i	$0.407823\pi$
$42$	0	0
$43$	−9.65685	−1.47266	−0.736328	−	0.676625i	$-0.763442\pi$
$43$	−9.65685	−1.47266	−0.736328	+	0.676625i	$0.763442\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	11.3137	1.65027	0.825137	−	0.564933i	$-0.191098\pi$
$47$	11.3137	1.65027	0.825137	+	0.564933i	$0.191098\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.65685	−0.512062
$52$	0	0
$53$	6.48528	0.890822	0.445411	−	0.895326i	$-0.353058\pi$
$53$	6.48528	0.890822	0.445411	+	0.895326i	$0.353058\pi$
$54$	0	0
$55$	2.82843	0.381385
$56$	0	0
$57$	2.82843	0.374634
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	13.3137	1.70465	0.852323	−	0.523016i	$-0.175193\pi$
$61$	13.3137	1.70465	0.852323	+	0.523016i	$0.175193\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	4.82843	0.598893
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	4.00000	0.481543
$70$	0	0
$71$	5.17157	0.613753	0.306876	−	0.951749i	$-0.400716\pi$
$71$	5.17157	0.613753	0.306876	+	0.951749i	$0.400716\pi$
$72$	0	0
$73$	0.828427	0.0969601	0.0484800	−	0.998824i	$-0.484562\pi$
$73$	0.828427	0.0969601	0.0484800	+	0.998824i	$0.484562\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	2.82843	0.322329
$78$	0	0
$79$	−7.31371	−0.822856	−0.411428	−	0.911442i	$-0.634970\pi$
$79$	−7.31371	−0.822856	−0.411428	+	0.911442i	$0.634970\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.34315	−0.696251	−0.348125	−	0.937448i	$-0.613182\pi$
$83$	−6.34315	−0.696251	−0.348125	+	0.937448i	$0.613182\pi$
$84$	0	0
$85$	−3.65685	−0.396642
$86$	0	0
$87$	−3.65685	−0.392056
$88$	0	0
$89$	0.343146	0.0363734	0.0181867	−	0.999835i	$-0.494211\pi$
$89$	0.343146	0.0363734	0.0181867	+	0.999835i	$0.494211\pi$
$90$	0	0
$91$	4.82843	0.506157
$92$	0	0
$93$	−2.82843	−0.293294
$94$	0	0
$95$	2.82843	0.290191
$96$	0	0
$97$	−12.8284	−1.30253	−0.651265	−	0.758851i	$-0.725761\pi$
$97$	−12.8284	−1.30253	−0.651265	+	0.758851i	$0.725761\pi$
$98$	0	0
$99$	2.82843	0.284268
$100$	0	0
$101$	−5.31371	−0.528734	−0.264367	−	0.964422i	$-0.585163\pi$
$101$	−5.31371	−0.528734	−0.264367	+	0.964422i	$0.585163\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	−9.31371	−0.892091	−0.446046	−	0.895010i	$-0.647168\pi$
$109$	−9.31371	−0.892091	−0.446046	+	0.895010i	$0.647168\pi$
$110$	0	0
$111$	0.343146	0.0325700
$112$	0	0
$113$	15.1716	1.42722	0.713611	−	0.700542i	$-0.247059\pi$
$113$	15.1716	1.42722	0.713611	+	0.700542i	$0.247059\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	4.82843	0.446388
$118$	0	0
$119$	−3.65685	−0.335223
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	3.65685	0.329727
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−5.65685	−0.501965	−0.250982	−	0.967992i	$-0.580754\pi$
$127$	−5.65685	−0.501965	−0.250982	+	0.967992i	$0.580754\pi$
$128$	0	0
$129$	−9.65685	−0.850239
$130$	0	0
$131$	−5.65685	−0.494242	−0.247121	−	0.968985i	$-0.579484\pi$
$131$	−5.65685	−0.494242	−0.247121	+	0.968985i	$0.579484\pi$
$132$	0	0
$133$	2.82843	0.245256
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	4.82843	0.412520	0.206260	−	0.978497i	$-0.433871\pi$
$137$	4.82843	0.412520	0.206260	+	0.978497i	$0.433871\pi$
$138$	0	0
$139$	−19.7990	−1.67933	−0.839664	−	0.543106i	$-0.817248\pi$
$139$	−19.7990	−1.67933	−0.839664	+	0.543106i	$0.817248\pi$
$140$	0	0
$141$	11.3137	0.952786
$142$	0	0
$143$	13.6569	1.14204
$144$	0	0
$145$	−3.65685	−0.303685
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	5.31371	0.435316	0.217658	−	0.976025i	$-0.430158\pi$
$149$	5.31371	0.435316	0.217658	+	0.976025i	$0.430158\pi$
$150$	0	0
$151$	9.65685	0.785864	0.392932	−	0.919568i	$-0.371461\pi$
$151$	9.65685	0.785864	0.392932	+	0.919568i	$0.371461\pi$
$152$	0	0
$153$	−3.65685	−0.295639
$154$	0	0
$155$	−2.82843	−0.227185
$156$	0	0
$157$	7.17157	0.572354	0.286177	−	0.958177i	$-0.407615\pi$
$157$	7.17157	0.572354	0.286177	+	0.958177i	$0.407615\pi$
$158$	0	0
$159$	6.48528	0.514316
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	−6.34315	−0.496834	−0.248417	−	0.968653i	$-0.579910\pi$
$163$	−6.34315	−0.496834	−0.248417	+	0.968653i	$0.579910\pi$
$164$	0	0
$165$	2.82843	0.220193
$166$	0	0
$167$	−8.97056	−0.694163	−0.347081	−	0.937835i	$-0.612827\pi$
$167$	−8.97056	−0.694163	−0.347081	+	0.937835i	$0.612827\pi$
$168$	0	0
$169$	10.3137	0.793362
$170$	0	0
$171$	2.82843	0.216295
$172$	0	0
$173$	−18.9706	−1.44231	−0.721153	−	0.692776i	$-0.756387\pi$
$173$	−18.9706	−1.44231	−0.721153	+	0.692776i	$0.756387\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.8284	0.809355	0.404677	−	0.914460i	$-0.367384\pi$
$179$	10.8284	0.809355	0.404677	+	0.914460i	$0.367384\pi$
$180$	0	0
$181$	23.6569	1.75840	0.879200	−	0.476453i	$-0.158078\pi$
$181$	23.6569	1.75840	0.879200	+	0.476453i	$0.158078\pi$
$182$	0	0
$183$	13.3137	0.984178
$184$	0	0
$185$	0.343146	0.0252286
$186$	0	0
$187$	−10.3431	−0.756366
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	18.8284	1.36238	0.681189	−	0.732108i	$-0.261463\pi$
$191$	18.8284	1.36238	0.681189	+	0.732108i	$0.261463\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	4.82843	0.345771
$196$	0	0
$197$	−7.17157	−0.510953	−0.255477	−	0.966815i	$-0.582232\pi$
$197$	−7.17157	−0.510953	−0.255477	+	0.966815i	$0.582232\pi$
$198$	0	0
$199$	8.48528	0.601506	0.300753	−	0.953702i	$-0.402762\pi$
$199$	8.48528	0.601506	0.300753	+	0.953702i	$0.402762\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	−3.65685	−0.256661
$204$	0	0
$205$	3.65685	0.255406
$206$	0	0
$207$	4.00000	0.278019
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	0	0
$213$	5.17157	0.354350
$214$	0	0
$215$	−9.65685	−0.658592
$216$	0	0
$217$	−2.82843	−0.192006
$218$	0	0
$219$	0.828427	0.0559799
$220$	0	0
$221$	−17.6569	−1.18773
$222$	0	0
$223$	16.9706	1.13643	0.568216	−	0.822879i	$-0.307634\pi$
$223$	16.9706	1.13643	0.568216	+	0.822879i	$0.307634\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−17.6569	−1.17193	−0.585963	−	0.810338i	$-0.699284\pi$
$227$	−17.6569	−1.17193	−0.585963	+	0.810338i	$0.699284\pi$
$228$	0	0
$229$	−25.3137	−1.67278	−0.836388	−	0.548137i	$-0.815337\pi$
$229$	−25.3137	−1.67278	−0.836388	+	0.548137i	$0.815337\pi$
$230$	0	0
$231$	2.82843	0.186097
$232$	0	0
$233$	23.1716	1.51802	0.759010	−	0.651079i	$-0.225683\pi$
$233$	23.1716	1.51802	0.759010	+	0.651079i	$0.225683\pi$
$234$	0	0
$235$	11.3137	0.738025
$236$	0	0
$237$	−7.31371	−0.475076
$238$	0	0
$239$	−18.8284	−1.21791	−0.608955	−	0.793205i	$-0.708411\pi$
$239$	−18.8284	−1.21791	−0.608955	+	0.793205i	$0.708411\pi$
$240$	0	0
$241$	7.65685	0.493221	0.246611	−	0.969115i	$-0.420683\pi$
$241$	7.65685	0.493221	0.246611	+	0.969115i	$0.420683\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	13.6569	0.868965
$248$	0	0
$249$	−6.34315	−0.401981
$250$	0	0
$251$	−11.3137	−0.714115	−0.357057	−	0.934082i	$-0.616220\pi$
$251$	−11.3137	−0.714115	−0.357057	+	0.934082i	$0.616220\pi$
$252$	0	0
$253$	11.3137	0.711287
$254$	0	0
$255$	−3.65685	−0.229001
$256$	0	0
$257$	29.3137	1.82854	0.914269	−	0.405107i	$-0.132766\pi$
$257$	29.3137	1.82854	0.914269	+	0.405107i	$0.132766\pi$
$258$	0	0
$259$	0.343146	0.0213220
$260$	0	0
$261$	−3.65685	−0.226354
$262$	0	0
$263$	−12.9706	−0.799799	−0.399900	−	0.916559i	$-0.630955\pi$
$263$	−12.9706	−0.799799	−0.399900	+	0.916559i	$0.630955\pi$
$264$	0	0
$265$	6.48528	0.398388
$266$	0	0
$267$	0.343146	0.0210002
$268$	0	0
$269$	17.3137	1.05564	0.527818	−	0.849358i	$-0.323010\pi$
$269$	17.3137	1.05564	0.527818	+	0.849358i	$0.323010\pi$
$270$	0	0
$271$	−5.17157	−0.314151	−0.157075	−	0.987587i	$-0.550207\pi$
$271$	−5.17157	−0.314151	−0.157075	+	0.987587i	$0.550207\pi$
$272$	0	0
$273$	4.82843	0.292230
$274$	0	0
$275$	2.82843	0.170561
$276$	0	0
$277$	30.9706	1.86084	0.930420	−	0.366494i	$-0.119442\pi$
$277$	30.9706	1.86084	0.930420	+	0.366494i	$0.119442\pi$
$278$	0	0
$279$	−2.82843	−0.169334
$280$	0	0
$281$	−14.9706	−0.893069	−0.446534	−	0.894766i	$-0.647342\pi$
$281$	−14.9706	−0.893069	−0.446534	+	0.894766i	$0.647342\pi$
$282$	0	0
$283$	7.31371	0.434755	0.217377	−	0.976088i	$-0.430250\pi$
$283$	7.31371	0.434755	0.217377	+	0.976088i	$0.430250\pi$
$284$	0	0
$285$	2.82843	0.167542
$286$	0	0
$287$	3.65685	0.215857
$288$	0	0
$289$	−3.62742	−0.213377
$290$	0	0
$291$	−12.8284	−0.752016
$292$	0	0
$293$	8.34315	0.487412	0.243706	−	0.969849i	$-0.421637\pi$
$293$	8.34315	0.487412	0.243706	+	0.969849i	$0.421637\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.82843	0.164122
$298$	0	0
$299$	19.3137	1.11694
$300$	0	0
$301$	−9.65685	−0.556612
$302$	0	0
$303$	−5.31371	−0.305265
$304$	0	0
$305$	13.3137	0.762341
$306$	0	0
$307$	−6.34315	−0.362022	−0.181011	−	0.983481i	$-0.557937\pi$
$307$	−6.34315	−0.362022	−0.181011	+	0.983481i	$0.557937\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	−4.82843	−0.272919	−0.136459	−	0.990646i	$-0.543572\pi$
$313$	−4.82843	−0.272919	−0.136459	+	0.990646i	$0.543572\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	−12.8284	−0.720516	−0.360258	−	0.932853i	$-0.617311\pi$
$317$	−12.8284	−0.720516	−0.360258	+	0.932853i	$0.617311\pi$
$318$	0	0
$319$	−10.3431	−0.579105
$320$	0	0
$321$	−8.00000	−0.446516
$322$	0	0
$323$	−10.3431	−0.575508
$324$	0	0
$325$	4.82843	0.267833
$326$	0	0
$327$	−9.31371	−0.515049
$328$	0	0
$329$	11.3137	0.623745
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	0.343146	0.0188043
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	21.3137	1.16103	0.580516	−	0.814249i	$-0.302851\pi$
$337$	21.3137	1.16103	0.580516	+	0.814249i	$0.302851\pi$
$338$	0	0
$339$	15.1716	0.824007
$340$	0	0
$341$	−8.00000	−0.433224
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	4.00000	0.215353
$346$	0	0
$347$	18.3431	0.984712	0.492356	−	0.870394i	$-0.336136\pi$
$347$	18.3431	0.984712	0.492356	+	0.870394i	$0.336136\pi$
$348$	0	0
$349$	29.3137	1.56913	0.784563	−	0.620049i	$-0.212887\pi$
$349$	29.3137	1.56913	0.784563	+	0.620049i	$0.212887\pi$
$350$	0	0
$351$	4.82843	0.257722
$352$	0	0
$353$	26.9706	1.43550	0.717749	−	0.696302i	$-0.245172\pi$
$353$	26.9706	1.43550	0.717749	+	0.696302i	$0.245172\pi$
$354$	0	0
$355$	5.17157	0.274479
$356$	0	0
$357$	−3.65685	−0.193541
$358$	0	0
$359$	17.4558	0.921284	0.460642	−	0.887586i	$-0.347619\pi$
$359$	17.4558	0.921284	0.460642	+	0.887586i	$0.347619\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	−3.00000	−0.157459
$364$	0	0
$365$	0.828427	0.0433619
$366$	0	0
$367$	−18.3431	−0.957504	−0.478752	−	0.877950i	$-0.658911\pi$
$367$	−18.3431	−0.957504	−0.478752	+	0.877950i	$0.658911\pi$
$368$	0	0
$369$	3.65685	0.190368
$370$	0	0
$371$	6.48528	0.336699
$372$	0	0
$373$	−10.0000	−0.517780	−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000	−0.517780	−0.258890	+	0.965907i	$0.583357\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−17.6569	−0.909374
$378$	0	0
$379$	9.65685	0.496039	0.248020	−	0.968755i	$-0.420220\pi$
$379$	9.65685	0.496039	0.248020	+	0.968755i	$0.420220\pi$
$380$	0	0
$381$	−5.65685	−0.289809
$382$	0	0
$383$	−28.2843	−1.44526	−0.722629	−	0.691236i	$-0.757067\pi$
$383$	−28.2843	−1.44526	−0.722629	+	0.691236i	$0.757067\pi$
$384$	0	0
$385$	2.82843	0.144150
$386$	0	0
$387$	−9.65685	−0.490885
$388$	0	0
$389$	13.3137	0.675032	0.337516	−	0.941320i	$-0.390413\pi$
$389$	13.3137	0.675032	0.337516	+	0.941320i	$0.390413\pi$
$390$	0	0
$391$	−14.6274	−0.739740
$392$	0	0
$393$	−5.65685	−0.285351
$394$	0	0
$395$	−7.31371	−0.367993
$396$	0	0
$397$	−14.4853	−0.726995	−0.363498	−	0.931595i	$-0.618418\pi$
$397$	−14.4853	−0.726995	−0.363498	+	0.931595i	$0.618418\pi$
$398$	0	0
$399$	2.82843	0.141598
$400$	0	0
$401$	21.3137	1.06436	0.532178	−	0.846633i	$-0.321374\pi$
$401$	21.3137	1.06436	0.532178	+	0.846633i	$0.321374\pi$
$402$	0	0
$403$	−13.6569	−0.680296
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	0.970563	0.0481090
$408$	0	0
$409$	−9.31371	−0.460533	−0.230267	−	0.973128i	$-0.573960\pi$
$409$	−9.31371	−0.460533	−0.230267	+	0.973128i	$0.573960\pi$
$410$	0	0
$411$	4.82843	0.238169
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−6.34315	−0.311373
$416$	0	0
$417$	−19.7990	−0.969561
$418$	0	0
$419$	16.9706	0.829066	0.414533	−	0.910034i	$-0.363945\pi$
$419$	16.9706	0.829066	0.414533	+	0.910034i	$0.363945\pi$
$420$	0	0
$421$	−1.31371	−0.0640262	−0.0320131	−	0.999487i	$-0.510192\pi$
$421$	−1.31371	−0.0640262	−0.0320131	+	0.999487i	$0.510192\pi$
$422$	0	0
$423$	11.3137	0.550091
$424$	0	0
$425$	−3.65685	−0.177383
$426$	0	0
$427$	13.3137	0.644296
$428$	0	0
$429$	13.6569	0.659359
$430$	0	0
$431$	16.4853	0.794068	0.397034	−	0.917804i	$-0.370039\pi$
$431$	16.4853	0.794068	0.397034	+	0.917804i	$0.370039\pi$
$432$	0	0
$433$	30.4853	1.46503	0.732515	−	0.680751i	$-0.238347\pi$
$433$	30.4853	1.46503	0.732515	+	0.680751i	$0.238347\pi$
$434$	0	0
$435$	−3.65685	−0.175333
$436$	0	0
$437$	11.3137	0.541208
$438$	0	0
$439$	−18.8284	−0.898632	−0.449316	−	0.893373i	$-0.648332\pi$
$439$	−18.8284	−0.898632	−0.449316	+	0.893373i	$0.648332\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−8.97056	−0.426204	−0.213102	−	0.977030i	$-0.568357\pi$
$443$	−8.97056	−0.426204	−0.213102	+	0.977030i	$0.568357\pi$
$444$	0	0
$445$	0.343146	0.0162667
$446$	0	0
$447$	5.31371	0.251330
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	0	0
$451$	10.3431	0.487040
$452$	0	0
$453$	9.65685	0.453719
$454$	0	0
$455$	4.82843	0.226360
$456$	0	0
$457$	−19.6569	−0.919509	−0.459754	−	0.888046i	$-0.652063\pi$
$457$	−19.6569	−0.919509	−0.459754	+	0.888046i	$0.652063\pi$
$458$	0	0
$459$	−3.65685	−0.170687
$460$	0	0
$461$	28.6274	1.33331	0.666656	−	0.745366i	$-0.267725\pi$
$461$	28.6274	1.33331	0.666656	+	0.745366i	$0.267725\pi$
$462$	0	0
$463$	−36.2843	−1.68627	−0.843137	−	0.537700i	$-0.819293\pi$
$463$	−36.2843	−1.68627	−0.843137	+	0.537700i	$0.819293\pi$
$464$	0	0
$465$	−2.82843	−0.131165
$466$	0	0
$467$	−4.97056	−0.230010	−0.115005	−	0.993365i	$-0.536688\pi$
$467$	−4.97056	−0.230010	−0.115005	+	0.993365i	$0.536688\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	7.17157	0.330449
$472$	0	0
$473$	−27.3137	−1.25589
$474$	0	0
$475$	2.82843	0.129777
$476$	0	0
$477$	6.48528	0.296941
$478$	0	0
$479$	−19.3137	−0.882466	−0.441233	−	0.897393i	$-0.645459\pi$
$479$	−19.3137	−0.882466	−0.441233	+	0.897393i	$0.645459\pi$
$480$	0	0
$481$	1.65685	0.0755461
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	−12.8284	−0.582509
$486$	0	0
$487$	5.65685	0.256337	0.128168	−	0.991752i	$-0.459090\pi$
$487$	5.65685	0.256337	0.128168	+	0.991752i	$0.459090\pi$
$488$	0	0
$489$	−6.34315	−0.286847
$490$	0	0
$491$	17.4558	0.787771	0.393886	−	0.919159i	$-0.371131\pi$
$491$	17.4558	0.787771	0.393886	+	0.919159i	$0.371131\pi$
$492$	0	0
$493$	13.3726	0.602271
$494$	0	0
$495$	2.82843	0.127128
$496$	0	0
$497$	5.17157	0.231977
$498$	0	0
$499$	17.6569	0.790429	0.395215	−	0.918589i	$-0.370670\pi$
$499$	17.6569	0.790429	0.395215	+	0.918589i	$0.370670\pi$
$500$	0	0
$501$	−8.97056	−0.400775
$502$	0	0
$503$	39.5980	1.76559	0.882793	−	0.469762i	$-0.155660\pi$
$503$	39.5980	1.76559	0.882793	+	0.469762i	$0.155660\pi$
$504$	0	0
$505$	−5.31371	−0.236457
$506$	0	0
$507$	10.3137	0.458048
$508$	0	0
$509$	1.31371	0.0582291	0.0291146	−	0.999576i	$-0.490731\pi$
$509$	1.31371	0.0582291	0.0291146	+	0.999576i	$0.490731\pi$
$510$	0	0
$511$	0.828427	0.0366475
$512$	0	0
$513$	2.82843	0.124878
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	32.0000	1.40736
$518$	0	0
$519$	−18.9706	−0.832715
$520$	0	0
$521$	−18.9706	−0.831115	−0.415558	−	0.909567i	$-0.636414\pi$
$521$	−18.9706	−0.831115	−0.415558	+	0.909567i	$0.636414\pi$
$522$	0	0
$523$	−33.6569	−1.47171	−0.735856	−	0.677138i	$-0.763220\pi$
$523$	−33.6569	−1.47171	−0.735856	+	0.677138i	$0.763220\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	10.3431	0.450555
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	17.6569	0.764803
$534$	0	0
$535$	−8.00000	−0.345870
$536$	0	0
$537$	10.8284	0.467281
$538$	0	0
$539$	2.82843	0.121829
$540$	0	0
$541$	−44.6274	−1.91868	−0.959341	−	0.282249i	$-0.908920\pi$
$541$	−44.6274	−1.91868	−0.959341	+	0.282249i	$0.908920\pi$
$542$	0	0
$543$	23.6569	1.01521
$544$	0	0
$545$	−9.31371	−0.398955
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	0	0
$549$	13.3137	0.568215
$550$	0	0
$551$	−10.3431	−0.440633
$552$	0	0
$553$	−7.31371	−0.311011
$554$	0	0
$555$	0.343146	0.0145657
$556$	0	0
$557$	−20.8284	−0.882529	−0.441264	−	0.897377i	$-0.645470\pi$
$557$	−20.8284	−0.882529	−0.441264	+	0.897377i	$0.645470\pi$
$558$	0	0
$559$	−46.6274	−1.97213
$560$	0	0
$561$	−10.3431	−0.436688
$562$	0	0
$563$	−23.3137	−0.982556	−0.491278	−	0.871003i	$-0.663470\pi$
$563$	−23.3137	−0.982556	−0.491278	+	0.871003i	$0.663470\pi$
$564$	0	0
$565$	15.1716	0.638273
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	24.6274	1.03244	0.516218	−	0.856458i	$-0.327340\pi$
$569$	24.6274	1.03244	0.516218	+	0.856458i	$0.327340\pi$
$570$	0	0
$571$	−43.5980	−1.82452	−0.912259	−	0.409613i	$-0.865664\pi$
$571$	−43.5980	−1.82452	−0.912259	+	0.409613i	$0.865664\pi$
$572$	0	0
$573$	18.8284	0.786569
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	−4.82843	−0.201010	−0.100505	−	0.994937i	$-0.532046\pi$
$577$	−4.82843	−0.201010	−0.100505	+	0.994937i	$0.532046\pi$
$578$	0	0
$579$	10.0000	0.415586
$580$	0	0
$581$	−6.34315	−0.263158
$582$	0	0
$583$	18.3431	0.759695
$584$	0	0
$585$	4.82843	0.199631
$586$	0	0
$587$	7.31371	0.301869	0.150935	−	0.988544i	$-0.451772\pi$
$587$	7.31371	0.301869	0.150935	+	0.988544i	$0.451772\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	−7.17157	−0.294999
$592$	0	0
$593$	28.3431	1.16391	0.581957	−	0.813220i	$-0.302287\pi$
$593$	28.3431	1.16391	0.581957	+	0.813220i	$0.302287\pi$
$594$	0	0
$595$	−3.65685	−0.149916
$596$	0	0
$597$	8.48528	0.347279
$598$	0	0
$599$	−32.4853	−1.32731	−0.663656	−	0.748038i	$-0.730996\pi$
$599$	−32.4853	−1.32731	−0.663656	+	0.748038i	$0.730996\pi$
$600$	0	0
$601$	−35.6569	−1.45447	−0.727237	−	0.686387i	$-0.759196\pi$
$601$	−35.6569	−1.45447	−0.727237	+	0.686387i	$0.759196\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	−3.00000	−0.121967
$606$	0	0
$607$	−29.6569	−1.20373	−0.601867	−	0.798596i	$-0.705576\pi$
$607$	−29.6569	−1.20373	−0.601867	+	0.798596i	$0.705576\pi$
$608$	0	0
$609$	−3.65685	−0.148183
$610$	0	0
$611$	54.6274	2.20999
$612$	0	0
$613$	16.3431	0.660093	0.330047	−	0.943965i	$-0.392935\pi$
$613$	16.3431	0.660093	0.330047	+	0.943965i	$0.392935\pi$
$614$	0	0
$615$	3.65685	0.147459
$616$	0	0
$617$	34.4853	1.38833	0.694163	−	0.719818i	$-0.255775\pi$
$617$	34.4853	1.38833	0.694163	+	0.719818i	$0.255775\pi$
$618$	0	0
$619$	41.4558	1.66625	0.833126	−	0.553084i	$-0.186549\pi$
$619$	41.4558	1.66625	0.833126	+	0.553084i	$0.186549\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	0.343146	0.0137478
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	8.00000	0.319489
$628$	0	0
$629$	−1.25483	−0.0500335
$630$	0	0
$631$	18.6274	0.741546	0.370773	−	0.928724i	$-0.379093\pi$
$631$	18.6274	0.741546	0.370773	+	0.928724i	$0.379093\pi$
$632$	0	0
$633$	−20.0000	−0.794929
$634$	0	0
$635$	−5.65685	−0.224485
$636$	0	0
$637$	4.82843	0.191309
$638$	0	0
$639$	5.17157	0.204584
$640$	0	0
$641$	−11.6569	−0.460418	−0.230209	−	0.973141i	$-0.573941\pi$
$641$	−11.6569	−0.460418	−0.230209	+	0.973141i	$0.573941\pi$
$642$	0	0
$643$	−22.3431	−0.881128	−0.440564	−	0.897721i	$-0.645221\pi$
$643$	−22.3431	−0.881128	−0.440564	+	0.897721i	$0.645221\pi$
$644$	0	0
$645$	−9.65685	−0.380238
$646$	0	0
$647$	−8.97056	−0.352669	−0.176335	−	0.984330i	$-0.556424\pi$
$647$	−8.97056	−0.352669	−0.176335	+	0.984330i	$0.556424\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−2.82843	−0.110855
$652$	0	0
$653$	−26.4853	−1.03645	−0.518225	−	0.855245i	$-0.673407\pi$
$653$	−26.4853	−1.03645	−0.518225	+	0.855245i	$0.673407\pi$
$654$	0	0
$655$	−5.65685	−0.221032
$656$	0	0
$657$	0.828427	0.0323200
$658$	0	0
$659$	−36.7696	−1.43234	−0.716169	−	0.697927i	$-0.754106\pi$
$659$	−36.7696	−1.43234	−0.716169	+	0.697927i	$0.754106\pi$
$660$	0	0
$661$	−6.00000	−0.233373	−0.116686	−	0.993169i	$-0.537227\pi$
$661$	−6.00000	−0.233373	−0.116686	+	0.993169i	$0.537227\pi$
$662$	0	0
$663$	−17.6569	−0.685735
$664$	0	0
$665$	2.82843	0.109682
$666$	0	0
$667$	−14.6274	−0.566376
$668$	0	0
$669$	16.9706	0.656120
$670$	0	0
$671$	37.6569	1.45373
$672$	0	0
$673$	39.6569	1.52866	0.764330	−	0.644826i	$-0.223070\pi$
$673$	39.6569	1.52866	0.764330	+	0.644826i	$0.223070\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−26.9706	−1.03656	−0.518281	−	0.855210i	$-0.673428\pi$
$677$	−26.9706	−1.03656	−0.518281	+	0.855210i	$0.673428\pi$
$678$	0	0
$679$	−12.8284	−0.492310
$680$	0	0
$681$	−17.6569	−0.676612
$682$	0	0
$683$	21.6569	0.828676	0.414338	−	0.910123i	$-0.364013\pi$
$683$	21.6569	0.828676	0.414338	+	0.910123i	$0.364013\pi$
$684$	0	0
$685$	4.82843	0.184485
$686$	0	0
$687$	−25.3137	−0.965778
$688$	0	0
$689$	31.3137	1.19296
$690$	0	0
$691$	−12.2010	−0.464148	−0.232074	−	0.972698i	$-0.574551\pi$
$691$	−12.2010	−0.464148	−0.232074	+	0.972698i	$0.574551\pi$
$692$	0	0
$693$	2.82843	0.107443
$694$	0	0
$695$	−19.7990	−0.751018
$696$	0	0
$697$	−13.3726	−0.506523
$698$	0	0
$699$	23.1716	0.876429
$700$	0	0
$701$	−46.9706	−1.77405	−0.887027	−	0.461718i	$-0.847233\pi$
$701$	−46.9706	−1.77405	−0.887027	+	0.461718i	$0.847233\pi$
$702$	0	0
$703$	0.970563	0.0366055
$704$	0	0
$705$	11.3137	0.426099
$706$	0	0
$707$	−5.31371	−0.199843
$708$	0	0
$709$	−17.3137	−0.650230	−0.325115	−	0.945674i	$-0.605403\pi$
$709$	−17.3137	−0.650230	−0.325115	+	0.945674i	$0.605403\pi$
$710$	0	0
$711$	−7.31371	−0.274285
$712$	0	0
$713$	−11.3137	−0.423702
$714$	0	0
$715$	13.6569	0.510737
$716$	0	0
$717$	−18.8284	−0.703160
$718$	0	0
$719$	−26.3431	−0.982434	−0.491217	−	0.871037i	$-0.663448\pi$
$719$	−26.3431	−0.982434	−0.491217	+	0.871037i	$0.663448\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	7.65685	0.284761
$724$	0	0
$725$	−3.65685	−0.135812
$726$	0	0
$727$	−26.3431	−0.977013	−0.488507	−	0.872560i	$-0.662458\pi$
$727$	−26.3431	−0.977013	−0.488507	+	0.872560i	$0.662458\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	35.3137	1.30612
$732$	0	0
$733$	30.7696	1.13650	0.568250	−	0.822856i	$-0.307621\pi$
$733$	30.7696	1.13650	0.568250	+	0.822856i	$0.307621\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	−11.3137	−0.416746
$738$	0	0
$739$	14.3431	0.527621	0.263811	−	0.964575i	$-0.415021\pi$
$739$	14.3431	0.527621	0.263811	+	0.964575i	$0.415021\pi$
$740$	0	0
$741$	13.6569	0.501697
$742$	0	0
$743$	25.6569	0.941259	0.470629	−	0.882331i	$-0.344027\pi$
$743$	25.6569	0.941259	0.470629	+	0.882331i	$0.344027\pi$
$744$	0	0
$745$	5.31371	0.194679
$746$	0	0
$747$	−6.34315	−0.232084
$748$	0	0
$749$	−8.00000	−0.292314
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	0	0
$753$	−11.3137	−0.412294
$754$	0	0
$755$	9.65685	0.351449
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	0	0
$759$	11.3137	0.410662
$760$	0	0
$761$	−7.65685	−0.277561	−0.138780	−	0.990323i	$-0.544318\pi$
$761$	−7.65685	−0.277561	−0.138780	+	0.990323i	$0.544318\pi$
$762$	0	0
$763$	−9.31371	−0.337179
$764$	0	0
$765$	−3.65685	−0.132214
$766$	0	0
$767$	0	0
$768$	0	0
$769$	10.9706	0.395609	0.197804	−	0.980242i	$-0.436619\pi$
$769$	10.9706	0.395609	0.197804	+	0.980242i	$0.436619\pi$
$770$	0	0
$771$	29.3137	1.05571
$772$	0	0
$773$	30.9706	1.11393	0.556967	−	0.830535i	$-0.311965\pi$
$773$	30.9706	1.11393	0.556967	+	0.830535i	$0.311965\pi$
$774$	0	0
$775$	−2.82843	−0.101600
$776$	0	0
$777$	0.343146	0.0123103
$778$	0	0
$779$	10.3431	0.370582
$780$	0	0
$781$	14.6274	0.523410
$782$	0	0
$783$	−3.65685	−0.130685
$784$	0	0
$785$	7.17157	0.255964
$786$	0	0
$787$	−41.6569	−1.48491	−0.742453	−	0.669898i	$-0.766338\pi$
$787$	−41.6569	−1.48491	−0.742453	+	0.669898i	$0.766338\pi$
$788$	0	0
$789$	−12.9706	−0.461764
$790$	0	0
$791$	15.1716	0.539439
$792$	0	0
$793$	64.2843	2.28280
$794$	0	0
$795$	6.48528	0.230009
$796$	0	0
$797$	−34.9706	−1.23872	−0.619360	−	0.785107i	$-0.712608\pi$
$797$	−34.9706	−1.23872	−0.619360	+	0.785107i	$0.712608\pi$
$798$	0	0
$799$	−41.3726	−1.46366
$800$	0	0
$801$	0.343146	0.0121245
$802$	0	0
$803$	2.34315	0.0826878
$804$	0	0
$805$	4.00000	0.140981
$806$	0	0
$807$	17.3137	0.609471
$808$	0	0
$809$	−5.02944	−0.176826	−0.0884128	−	0.996084i	$-0.528179\pi$
$809$	−5.02944	−0.176826	−0.0884128	+	0.996084i	$0.528179\pi$
$810$	0	0
$811$	49.4558	1.73663	0.868315	−	0.496014i	$-0.165203\pi$
$811$	49.4558	1.73663	0.868315	+	0.496014i	$0.165203\pi$
$812$	0	0
$813$	−5.17157	−0.181375
$814$	0	0
$815$	−6.34315	−0.222191
$816$	0	0
$817$	−27.3137	−0.955586
$818$	0	0
$819$	4.82843	0.168719
$820$	0	0
$821$	46.2843	1.61533	0.807666	−	0.589640i	$-0.200730\pi$
$821$	46.2843	1.61533	0.807666	+	0.589640i	$0.200730\pi$
$822$	0	0
$823$	−36.2843	−1.26479	−0.632395	−	0.774646i	$-0.717928\pi$
$823$	−36.2843	−1.26479	−0.632395	+	0.774646i	$0.717928\pi$
$824$	0	0
$825$	2.82843	0.0984732
$826$	0	0
$827$	45.2548	1.57366	0.786832	−	0.617167i	$-0.211720\pi$
$827$	45.2548	1.57366	0.786832	+	0.617167i	$0.211720\pi$
$828$	0	0
$829$	22.2843	0.773965	0.386982	−	0.922087i	$-0.373517\pi$
$829$	22.2843	0.773965	0.386982	+	0.922087i	$0.373517\pi$
$830$	0	0
$831$	30.9706	1.07436
$832$	0	0
$833$	−3.65685	−0.126702
$834$	0	0
$835$	−8.97056	−0.310439
$836$	0	0
$837$	−2.82843	−0.0977647
$838$	0	0
$839$	−15.0294	−0.518874	−0.259437	−	0.965760i	$-0.583537\pi$
$839$	−15.0294	−0.518874	−0.259437	+	0.965760i	$0.583537\pi$
$840$	0	0
$841$	−15.6274	−0.538876
$842$	0	0
$843$	−14.9706	−0.515614
$844$	0	0
$845$	10.3137	0.354802
$846$	0	0
$847$	−3.00000	−0.103081
$848$	0	0
$849$	7.31371	0.251006
$850$	0	0
$851$	1.37258	0.0470515
$852$	0	0
$853$	36.8284	1.26098	0.630491	−	0.776197i	$-0.282854\pi$
$853$	36.8284	1.26098	0.630491	+	0.776197i	$0.282854\pi$
$854$	0	0
$855$	2.82843	0.0967302
$856$	0	0
$857$	43.9411	1.50100	0.750500	−	0.660870i	$-0.229813\pi$
$857$	43.9411	1.50100	0.750500	+	0.660870i	$0.229813\pi$
$858$	0	0
$859$	−33.4558	−1.14150	−0.570749	−	0.821124i	$-0.693347\pi$
$859$	−33.4558	−1.14150	−0.570749	+	0.821124i	$0.693347\pi$
$860$	0	0
$861$	3.65685	0.124625
$862$	0	0
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	0	0
$865$	−18.9706	−0.645018
$866$	0	0
$867$	−3.62742	−0.123194
$868$	0	0
$869$	−20.6863	−0.701734
$870$	0	0
$871$	−19.3137	−0.654420
$872$	0	0
$873$	−12.8284	−0.434176
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	2.28427	0.0771344	0.0385672	−	0.999256i	$-0.487721\pi$
$877$	2.28427	0.0771344	0.0385672	+	0.999256i	$0.487721\pi$
$878$	0	0
$879$	8.34315	0.281407
$880$	0	0
$881$	−9.02944	−0.304209	−0.152105	−	0.988364i	$-0.548605\pi$
$881$	−9.02944	−0.304209	−0.152105	+	0.988364i	$0.548605\pi$
$882$	0	0
$883$	−26.6274	−0.896084	−0.448042	−	0.894013i	$-0.647878\pi$
$883$	−26.6274	−0.896084	−0.448042	+	0.894013i	$0.647878\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−35.3137	−1.18572	−0.592859	−	0.805306i	$-0.702001\pi$
$887$	−35.3137	−1.18572	−0.592859	+	0.805306i	$0.702001\pi$
$888$	0	0
$889$	−5.65685	−0.189725
$890$	0	0
$891$	2.82843	0.0947559
$892$	0	0
$893$	32.0000	1.07084
$894$	0	0
$895$	10.8284	0.361954
$896$	0	0
$897$	19.3137	0.644866
$898$	0	0
$899$	10.3431	0.344963
$900$	0	0
$901$	−23.7157	−0.790085
$902$	0	0
$903$	−9.65685	−0.321360
$904$	0	0
$905$	23.6569	0.786380
$906$	0	0
$907$	34.6274	1.14978	0.574892	−	0.818229i	$-0.305044\pi$
$907$	34.6274	1.14978	0.574892	+	0.818229i	$0.305044\pi$
$908$	0	0
$909$	−5.31371	−0.176245
$910$	0	0
$911$	−0.485281	−0.0160781	−0.00803904	−	0.999968i	$-0.502559\pi$
$911$	−0.485281	−0.0160781	−0.00803904	+	0.999968i	$0.502559\pi$
$912$	0	0
$913$	−17.9411	−0.593765
$914$	0	0
$915$	13.3137	0.440138
$916$	0	0
$917$	−5.65685	−0.186806
$918$	0	0
$919$	−0.686292	−0.0226387	−0.0113193	−	0.999936i	$-0.503603\pi$
$919$	−0.686292	−0.0226387	−0.0113193	+	0.999936i	$0.503603\pi$
$920$	0	0
$921$	−6.34315	−0.209014
$922$	0	0
$923$	24.9706	0.821916
$924$	0	0
$925$	0.343146	0.0112826
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	−22.2843	−0.731123	−0.365562	−	0.930787i	$-0.619123\pi$
$929$	−22.2843	−0.731123	−0.365562	+	0.930787i	$0.619123\pi$
$930$	0	0
$931$	2.82843	0.0926980
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	−10.3431	−0.338257
$936$	0	0
$937$	−58.4853	−1.91063	−0.955315	−	0.295588i	$-0.904484\pi$
$937$	−58.4853	−1.91063	−0.955315	+	0.295588i	$0.904484\pi$
$938$	0	0
$939$	−4.82843	−0.157570
$940$	0	0
$941$	−34.0000	−1.10837	−0.554184	−	0.832394i	$-0.686970\pi$
$941$	−34.0000	−1.10837	−0.554184	+	0.832394i	$0.686970\pi$
$942$	0	0
$943$	14.6274	0.476334
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	50.9117	1.65441	0.827204	−	0.561902i	$-0.189930\pi$
$947$	50.9117	1.65441	0.827204	+	0.561902i	$0.189930\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	−12.8284	−0.415990
$952$	0	0
$953$	28.4264	0.920822	0.460411	−	0.887706i	$-0.347702\pi$
$953$	28.4264	0.920822	0.460411	+	0.887706i	$0.347702\pi$
$954$	0	0
$955$	18.8284	0.609274
$956$	0	0
$957$	−10.3431	−0.334346
$958$	0	0
$959$	4.82843	0.155918
$960$	0	0
$961$	−23.0000	−0.741935
$962$	0	0
$963$	−8.00000	−0.257796
$964$	0	0
$965$	10.0000	0.321911
$966$	0	0
$967$	25.9411	0.834210	0.417105	−	0.908858i	$-0.363045\pi$
$967$	25.9411	0.834210	0.417105	+	0.908858i	$0.363045\pi$
$968$	0	0
$969$	−10.3431	−0.332270
$970$	0	0
$971$	−8.00000	−0.256732	−0.128366	−	0.991727i	$-0.540973\pi$
$971$	−8.00000	−0.256732	−0.128366	+	0.991727i	$0.540973\pi$
$972$	0	0
$973$	−19.7990	−0.634726
$974$	0	0
$975$	4.82843	0.154633
$976$	0	0
$977$	−36.1421	−1.15629	−0.578145	−	0.815934i	$-0.696223\pi$
$977$	−36.1421	−1.15629	−0.578145	+	0.815934i	$0.696223\pi$
$978$	0	0
$979$	0.970563	0.0310193
$980$	0	0
$981$	−9.31371	−0.297364
$982$	0	0
$983$	28.6863	0.914951	0.457475	−	0.889222i	$-0.348754\pi$
$983$	28.6863	0.914951	0.457475	+	0.889222i	$0.348754\pi$
$984$	0	0
$985$	−7.17157	−0.228505
$986$	0	0
$987$	11.3137	0.360119
$988$	0	0
$989$	−38.6274	−1.22828
$990$	0	0
$991$	−13.9411	−0.442854	−0.221427	−	0.975177i	$-0.571072\pi$
$991$	−13.9411	−0.442854	−0.221427	+	0.975177i	$0.571072\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	0	0
$995$	8.48528	0.269002
$996$	0	0
$997$	24.1421	0.764589	0.382295	−	0.924041i	$-0.375134\pi$
$997$	24.1421	0.764589	0.382295	+	0.924041i	$0.375134\pi$
$998$	0	0
$999$	0.343146	0.0108567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6720.2.a.cy.1.2		2
4.3	odd	2		6720.2.a.cr.1.1		2
8.3	odd	2		3360.2.a.be.1.2	yes	2
8.5	even	2		3360.2.a.bc.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3360.2.a.bc.1.1	✓	2	8.5	even	2
3360.2.a.be.1.2	yes	2	8.3	odd	2
6720.2.a.cr.1.1		2	4.3	odd	2
6720.2.a.cy.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6720.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.82843 q^{11} +4.82843 q^{13} +1.00000 q^{15} -3.65685 q^{17} +2.82843 q^{19} +1.00000 q^{21} +4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.65685 q^{29} -2.82843 q^{31} +2.82843 q^{33} +1.00000 q^{35} +0.343146 q^{37} +4.82843 q^{39} +3.65685 q^{41} -9.65685 q^{43} +1.00000 q^{45} +11.3137 q^{47} +1.00000 q^{49} -3.65685 q^{51} +6.48528 q^{53} +2.82843 q^{55} +2.82843 q^{57} +13.3137 q^{61} +1.00000 q^{63} +4.82843 q^{65} -4.00000 q^{67} +4.00000 q^{69} +5.17157 q^{71} +0.828427 q^{73} +1.00000 q^{75} +2.82843 q^{77} -7.31371 q^{79} +1.00000 q^{81} -6.34315 q^{83} -3.65685 q^{85} -3.65685 q^{87} +0.343146 q^{89} +4.82843 q^{91} -2.82843 q^{93} +2.82843 q^{95} -12.8284 q^{97} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{13} + 2 q^{15} + 4 q^{17} + 2 q^{21} + 8 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} + 2 q^{35} + 12 q^{37} + 4 q^{39} - 4 q^{41} - 8 q^{43} + 2 q^{45} + 2 q^{49} + 4 q^{51} - 4 q^{53} + 4 q^{61} + 2 q^{63} + 4 q^{65} - 8 q^{67} + 8 q^{69} + 16 q^{71} - 4 q^{73} + 2 q^{75} + 8 q^{79} + 2 q^{81} - 24 q^{83} + 4 q^{85} + 4 q^{87} + 12 q^{89} + 4 q^{91} - 20 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 + 4 * q^13 + 2 * q^15 + 4 * q^17 + 2 * q^21 + 8 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^29 + 2 * q^35 + 12 * q^37 + 4 * q^39 - 4 * q^41 - 8 * q^43 + 2 * q^45 + 2 * q^49 + 4 * q^51 - 4 * q^53 + 4 * q^61 + 2 * q^63 + 4 * q^65 - 8 * q^67 + 8 * q^69 + 16 * q^71 - 4 * q^73 + 2 * q^75 + 8 * q^79 + 2 * q^81 - 24 * q^83 + 4 * q^85 + 4 * q^87 + 12 * q^89 + 4 * q^91 - 20 * q^97

Introduction

L-functions

Modular forms

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Database

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