LMFDB - Embedded newform 6720.2.a.cy.1.1

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			1.1
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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.82843 q^{11} -0.828427 q^{13} +1.00000 q^{15} +7.65685 q^{17} -2.82843 q^{19} +1.00000 q^{21} +4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +7.65685 q^{29} +2.82843 q^{31} -2.82843 q^{33} +1.00000 q^{35} +11.6569 q^{37} -0.828427 q^{39} -7.65685 q^{41} +1.65685 q^{43} +1.00000 q^{45} -11.3137 q^{47} +1.00000 q^{49} +7.65685 q^{51} -10.4853 q^{53} -2.82843 q^{55} -2.82843 q^{57} -9.31371 q^{61} +1.00000 q^{63} -0.828427 q^{65} -4.00000 q^{67} +4.00000 q^{69} +10.8284 q^{71} -4.82843 q^{73} +1.00000 q^{75} -2.82843 q^{77} +15.3137 q^{79} +1.00000 q^{81} -17.6569 q^{83} +7.65685 q^{85} +7.65685 q^{87} +11.6569 q^{89} -0.828427 q^{91} +2.82843 q^{93} -2.82843 q^{95} -7.17157 q^{97} -2.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	0	0
$13$	−0.828427	−0.229764	−0.114882	−	0.993379i	$-0.536649\pi$
$13$	−0.828427	−0.229764	−0.114882	+	0.993379i	$0.536649\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	7.65685	1.85706	0.928530	−	0.371257i	$-0.121073\pi$
$17$	7.65685	1.85706	0.928530	+	0.371257i	$0.121073\pi$
$18$	0	0
$19$	−2.82843	−0.648886	−0.324443	−	0.945905i	$-0.605177\pi$
$19$	−2.82843	−0.648886	−0.324443	+	0.945905i	$0.605177\pi$
$20$	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
$29$	7.65685	1.42184	0.710921	−	0.703272i	$-0.248278\pi$
$29$	7.65685	1.42184	0.710921	+	0.703272i	$0.248278\pi$
$30$	0	0
$31$	2.82843	0.508001	0.254000	−	0.967204i	$-0.418254\pi$
$31$	2.82843	0.508001	0.254000	+	0.967204i	$0.418254\pi$
$32$	0	0
$33$	−2.82843	−0.492366
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	11.6569	1.91638	0.958188	−	0.286141i	$-0.0923726\pi$
$37$	11.6569	1.91638	0.958188	+	0.286141i	$0.0923726\pi$
$38$	0	0
$39$	−0.828427	−0.132655
$40$	0	0
$41$	−7.65685	−1.19580	−0.597900	−	0.801571i	$-0.703998\pi$
$41$	−7.65685	−1.19580	−0.597900	+	0.801571i	$0.703998\pi$
$42$	0	0
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−11.3137	−1.65027	−0.825137	−	0.564933i	$-0.808902\pi$
$47$	−11.3137	−1.65027	−0.825137	+	0.564933i	$0.808902\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	7.65685	1.07217
$52$	0	0
$53$	−10.4853	−1.44026	−0.720132	−	0.693837i	$-0.755919\pi$
$53$	−10.4853	−1.44026	−0.720132	+	0.693837i	$0.755919\pi$
$54$	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
$61$	−9.31371	−1.19250	−0.596249	−	0.802799i	$-0.703343\pi$
$61$	−9.31371	−1.19250	−0.596249	+	0.802799i	$0.703343\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−0.828427	−0.102754
$66$	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
$71$	10.8284	1.28510	0.642549	−	0.766245i	$-0.277877\pi$
$71$	10.8284	1.28510	0.642549	+	0.766245i	$0.277877\pi$
$72$	0	0
$73$	−4.82843	−0.565125	−0.282562	−	0.959249i	$-0.591184\pi$
$73$	−4.82843	−0.565125	−0.282562	+	0.959249i	$0.591184\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−2.82843	−0.322329
$78$	0	0
$79$	15.3137	1.72293	0.861463	−	0.507820i	$-0.169548\pi$
$79$	15.3137	1.72293	0.861463	+	0.507820i	$0.169548\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−17.6569	−1.93809	−0.969046	−	0.246881i	$-0.920594\pi$
$83$	−17.6569	−1.93809	−0.969046	+	0.246881i	$0.920594\pi$
$84$	0	0
$85$	7.65685	0.830502
$86$	0	0
$87$	7.65685	0.820901
$88$	0	0
$89$	11.6569	1.23562	0.617812	−	0.786326i	$-0.288019\pi$
$89$	11.6569	1.23562	0.617812	+	0.786326i	$0.288019\pi$
$90$	0	0
$91$	−0.828427	−0.0868428
$92$	0	0
$93$	2.82843	0.293294
$94$	0	0
$95$	−2.82843	−0.290191
$96$	0	0
$97$	−7.17157	−0.728163	−0.364081	−	0.931367i	$-0.618617\pi$
$97$	−7.17157	−0.728163	−0.364081	+	0.931367i	$0.618617\pi$
$98$	0	0
$99$	−2.82843	−0.284268
$100$	0	0
$101$	17.3137	1.72278	0.861389	−	0.507946i	$-0.169595\pi$
$101$	17.3137	1.72278	0.861389	+	0.507946i	$0.169595\pi$
$102$	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
$109$	13.3137	1.27522	0.637611	−	0.770358i	$-0.279923\pi$
$109$	13.3137	1.27522	0.637611	+	0.770358i	$0.279923\pi$
$110$	0	0
$111$	11.6569	1.10642
$112$	0	0
$113$	20.8284	1.95937	0.979687	−	0.200534i	$-0.0642676\pi$
$113$	20.8284	1.95937	0.979687	+	0.200534i	$0.0642676\pi$
$114$	0	0
$115$	4.00000	0.373002
$116$	0	0
$117$	−0.828427	−0.0765881
$118$	0	0
$119$	7.65685	0.701903
$120$	0	0
$121$	−3.00000	−0.272727
$122$	0	0
$123$	−7.65685	−0.690395
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.65685	0.501965	0.250982	−	0.967992i	$-0.419246\pi$
$127$	5.65685	0.501965	0.250982	+	0.967992i	$0.419246\pi$
$128$	0	0
$129$	1.65685	0.145878
$130$	0	0
$131$	5.65685	0.494242	0.247121	−	0.968985i	$-0.420516\pi$
$131$	5.65685	0.494242	0.247121	+	0.968985i	$0.420516\pi$
$132$	0	0
$133$	−2.82843	−0.245256
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−0.828427	−0.0707773	−0.0353887	−	0.999374i	$-0.511267\pi$
$137$	−0.828427	−0.0707773	−0.0353887	+	0.999374i	$0.511267\pi$
$138$	0	0
$139$	19.7990	1.67933	0.839664	−	0.543106i	$-0.182752\pi$
$139$	19.7990	1.67933	0.839664	+	0.543106i	$0.182752\pi$
$140$	0	0
$141$	−11.3137	−0.952786
$142$	0	0
$143$	2.34315	0.195944
$144$	0	0
$145$	7.65685	0.635867
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−17.3137	−1.41839	−0.709197	−	0.705010i	$-0.750942\pi$
$149$	−17.3137	−1.41839	−0.709197	+	0.705010i	$0.750942\pi$
$150$	0	0
$151$	−1.65685	−0.134833	−0.0674164	−	0.997725i	$-0.521476\pi$
$151$	−1.65685	−0.134833	−0.0674164	+	0.997725i	$0.521476\pi$
$152$	0	0
$153$	7.65685	0.619020
$154$	0	0
$155$	2.82843	0.227185
$156$	0	0
$157$	12.8284	1.02382	0.511910	−	0.859039i	$-0.328938\pi$
$157$	12.8284	1.02382	0.511910	+	0.859039i	$0.328938\pi$
$158$	0	0
$159$	−10.4853	−0.831537
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	−17.6569	−1.38299	−0.691496	−	0.722380i	$-0.743048\pi$
$163$	−17.6569	−1.38299	−0.691496	+	0.722380i	$0.743048\pi$
$164$	0	0
$165$	−2.82843	−0.220193
$166$	0	0
$167$	24.9706	1.93228	0.966140	−	0.258018i	$-0.0830694\pi$
$167$	24.9706	1.93228	0.966140	+	0.258018i	$0.0830694\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	0	0
$171$	−2.82843	−0.216295
$172$	0	0
$173$	14.9706	1.13819	0.569095	−	0.822272i	$-0.307294\pi$
$173$	14.9706	1.13819	0.569095	+	0.822272i	$0.307294\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.17157	0.386542	0.193271	−	0.981145i	$-0.438090\pi$
$179$	5.17157	0.386542	0.193271	+	0.981145i	$0.438090\pi$
$180$	0	0
$181$	12.3431	0.917459	0.458729	−	0.888576i	$-0.348305\pi$
$181$	12.3431	0.917459	0.458729	+	0.888576i	$0.348305\pi$
$182$	0	0
$183$	−9.31371	−0.688489
$184$	0	0
$185$	11.6569	0.857029
$186$	0	0
$187$	−21.6569	−1.58371
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	13.1716	0.953062	0.476531	−	0.879158i	$-0.341894\pi$
$191$	13.1716	0.953062	0.476531	+	0.879158i	$0.341894\pi$
$192$	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
$195$	−0.828427	−0.0593249
$196$	0	0
$197$	−12.8284	−0.913988	−0.456994	−	0.889470i	$-0.651074\pi$
$197$	−12.8284	−0.913988	−0.456994	+	0.889470i	$0.651074\pi$
$198$	0	0
$199$	−8.48528	−0.601506	−0.300753	−	0.953702i	$-0.597238\pi$
$199$	−8.48528	−0.601506	−0.300753	+	0.953702i	$0.597238\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	7.65685	0.537406
$204$	0	0
$205$	−7.65685	−0.534778
$206$	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
$213$	10.8284	0.741952
$214$	0	0
$215$	1.65685	0.112997
$216$	0	0
$217$	2.82843	0.192006
$218$	0	0
$219$	−4.82843	−0.326275
$220$	0	0
$221$	−6.34315	−0.426686
$222$	0	0
$223$	−16.9706	−1.13643	−0.568216	−	0.822879i	$-0.692366\pi$
$223$	−16.9706	−1.13643	−0.568216	+	0.822879i	$0.692366\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−6.34315	−0.421009	−0.210505	−	0.977593i	$-0.567511\pi$
$227$	−6.34315	−0.421009	−0.210505	+	0.977593i	$0.567511\pi$
$228$	0	0
$229$	−2.68629	−0.177515	−0.0887576	−	0.996053i	$-0.528290\pi$
$229$	−2.68629	−0.177515	−0.0887576	+	0.996053i	$0.528290\pi$
$230$	0	0
$231$	−2.82843	−0.186097
$232$	0	0
$233$	28.8284	1.88861	0.944307	−	0.329067i	$-0.106734\pi$
$233$	28.8284	1.88861	0.944307	+	0.329067i	$0.106734\pi$
$234$	0	0
$235$	−11.3137	−0.738025
$236$	0	0
$237$	15.3137	0.994732
$238$	0	0
$239$	−13.1716	−0.851998	−0.425999	−	0.904724i	$-0.640077\pi$
$239$	−13.1716	−0.851998	−0.425999	+	0.904724i	$0.640077\pi$
$240$	0	0
$241$	−3.65685	−0.235559	−0.117779	−	0.993040i	$-0.537578\pi$
$241$	−3.65685	−0.235559	−0.117779	+	0.993040i	$0.537578\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	2.34315	0.149091
$248$	0	0
$249$	−17.6569	−1.11896
$250$	0	0
$251$	11.3137	0.714115	0.357057	−	0.934082i	$-0.383780\pi$
$251$	11.3137	0.714115	0.357057	+	0.934082i	$0.383780\pi$
$252$	0	0
$253$	−11.3137	−0.711287
$254$	0	0
$255$	7.65685	0.479491
$256$	0	0
$257$	6.68629	0.417079	0.208540	−	0.978014i	$-0.433129\pi$
$257$	6.68629	0.417079	0.208540	+	0.978014i	$0.433129\pi$
$258$	0	0
$259$	11.6569	0.724322
$260$	0	0
$261$	7.65685	0.473947
$262$	0	0
$263$	20.9706	1.29310	0.646550	−	0.762871i	$-0.276211\pi$
$263$	20.9706	1.29310	0.646550	+	0.762871i	$0.276211\pi$
$264$	0	0
$265$	−10.4853	−0.644106
$266$	0	0
$267$	11.6569	0.713388
$268$	0	0
$269$	−5.31371	−0.323983	−0.161991	−	0.986792i	$-0.551792\pi$
$269$	−5.31371	−0.323983	−0.161991	+	0.986792i	$0.551792\pi$
$270$	0	0
$271$	−10.8284	−0.657780	−0.328890	−	0.944368i	$-0.606675\pi$
$271$	−10.8284	−0.657780	−0.328890	+	0.944368i	$0.606675\pi$
$272$	0	0
$273$	−0.828427	−0.0501387
$274$	0	0
$275$	−2.82843	−0.170561
$276$	0	0
$277$	−2.97056	−0.178484	−0.0892419	−	0.996010i	$-0.528444\pi$
$277$	−2.97056	−0.178484	−0.0892419	+	0.996010i	$0.528444\pi$
$278$	0	0
$279$	2.82843	0.169334
$280$	0	0
$281$	18.9706	1.13169	0.565844	−	0.824512i	$-0.308550\pi$
$281$	18.9706	1.13169	0.565844	+	0.824512i	$0.308550\pi$
$282$	0	0
$283$	−15.3137	−0.910305	−0.455153	−	0.890413i	$-0.650415\pi$
$283$	−15.3137	−0.910305	−0.455153	+	0.890413i	$0.650415\pi$
$284$	0	0
$285$	−2.82843	−0.167542
$286$	0	0
$287$	−7.65685	−0.451970
$288$	0	0
$289$	41.6274	2.44867
$290$	0	0
$291$	−7.17157	−0.420405
$292$	0	0
$293$	19.6569	1.14837	0.574183	−	0.818727i	$-0.305320\pi$
$293$	19.6569	1.14837	0.574183	+	0.818727i	$0.305320\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.82843	−0.164122
$298$	0	0
$299$	−3.31371	−0.191637
$300$	0	0
$301$	1.65685	0.0954995
$302$	0	0
$303$	17.3137	0.994647
$304$	0	0
$305$	−9.31371	−0.533301
$306$	0	0
$307$	−17.6569	−1.00773	−0.503865	−	0.863782i	$-0.668089\pi$
$307$	−17.6569	−1.00773	−0.503865	+	0.863782i	$0.668089\pi$
$308$	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
$313$	0.828427	0.0468255	0.0234127	−	0.999726i	$-0.492547\pi$
$313$	0.828427	0.0468255	0.0234127	+	0.999726i	$0.492547\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	−7.17157	−0.402796	−0.201398	−	0.979510i	$-0.564548\pi$
$317$	−7.17157	−0.402796	−0.201398	+	0.979510i	$0.564548\pi$
$318$	0	0
$319$	−21.6569	−1.21255
$320$	0	0
$321$	−8.00000	−0.446516
$322$	0	0
$323$	−21.6569	−1.20502
$324$	0	0
$325$	−0.828427	−0.0459529
$326$	0	0
$327$	13.3137	0.736250
$328$	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
$333$	11.6569	0.638792
$334$	0	0
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−1.31371	−0.0715623	−0.0357811	−	0.999360i	$-0.511392\pi$
$337$	−1.31371	−0.0715623	−0.0357811	+	0.999360i	$0.511392\pi$
$338$	0	0
$339$	20.8284	1.13124
$340$	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
$347$	29.6569	1.59206	0.796032	−	0.605255i	$-0.206929\pi$
$347$	29.6569	1.59206	0.796032	+	0.605255i	$0.206929\pi$
$348$	0	0
$349$	6.68629	0.357909	0.178954	−	0.983857i	$-0.442729\pi$
$349$	6.68629	0.357909	0.178954	+	0.983857i	$0.442729\pi$
$350$	0	0
$351$	−0.828427	−0.0442182
$352$	0	0
$353$	−6.97056	−0.371006	−0.185503	−	0.982644i	$-0.559391\pi$
$353$	−6.97056	−0.371006	−0.185503	+	0.982644i	$0.559391\pi$
$354$	0	0
$355$	10.8284	0.574713
$356$	0	0
$357$	7.65685	0.405244
$358$	0	0
$359$	−33.4558	−1.76573	−0.882866	−	0.469625i	$-0.844389\pi$
$359$	−33.4558	−1.76573	−0.882866	+	0.469625i	$0.844389\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	−3.00000	−0.157459
$364$	0	0
$365$	−4.82843	−0.252731
$366$	0	0
$367$	−29.6569	−1.54808	−0.774038	−	0.633140i	$-0.781766\pi$
$367$	−29.6569	−1.54808	−0.774038	+	0.633140i	$0.781766\pi$
$368$	0	0
$369$	−7.65685	−0.398600
$370$	0	0
$371$	−10.4853	−0.544369
$372$	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
$377$	−6.34315	−0.326689
$378$	0	0
$379$	−1.65685	−0.0851069	−0.0425534	−	0.999094i	$-0.513549\pi$
$379$	−1.65685	−0.0851069	−0.0425534	+	0.999094i	$0.513549\pi$
$380$	0	0
$381$	5.65685	0.289809
$382$	0	0
$383$	28.2843	1.44526	0.722629	−	0.691236i	$-0.242933\pi$
$383$	28.2843	1.44526	0.722629	+	0.691236i	$0.242933\pi$
$384$	0	0
$385$	−2.82843	−0.144150
$386$	0	0
$387$	1.65685	0.0842226
$388$	0	0
$389$	−9.31371	−0.472224	−0.236112	−	0.971726i	$-0.575873\pi$
$389$	−9.31371	−0.472224	−0.236112	+	0.971726i	$0.575873\pi$
$390$	0	0
$391$	30.6274	1.54890
$392$	0	0
$393$	5.65685	0.285351
$394$	0	0
$395$	15.3137	0.770516
$396$	0	0
$397$	2.48528	0.124733	0.0623663	−	0.998053i	$-0.480135\pi$
$397$	2.48528	0.124733	0.0623663	+	0.998053i	$0.480135\pi$
$398$	0	0
$399$	−2.82843	−0.141598
$400$	0	0
$401$	−1.31371	−0.0656035	−0.0328017	−	0.999462i	$-0.510443\pi$
$401$	−1.31371	−0.0656035	−0.0328017	+	0.999462i	$0.510443\pi$
$402$	0	0
$403$	−2.34315	−0.116720
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−32.9706	−1.63429
$408$	0	0
$409$	13.3137	0.658321	0.329160	−	0.944274i	$-0.393234\pi$
$409$	13.3137	0.658321	0.329160	+	0.944274i	$0.393234\pi$
$410$	0	0
$411$	−0.828427	−0.0408633
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−17.6569	−0.866741
$416$	0	0
$417$	19.7990	0.969561
$418$	0	0
$419$	−16.9706	−0.829066	−0.414533	−	0.910034i	$-0.636055\pi$
$419$	−16.9706	−0.829066	−0.414533	+	0.910034i	$0.636055\pi$
$420$	0	0
$421$	21.3137	1.03877	0.519383	−	0.854541i	$-0.326162\pi$
$421$	21.3137	1.03877	0.519383	+	0.854541i	$0.326162\pi$
$422$	0	0
$423$	−11.3137	−0.550091
$424$	0	0
$425$	7.65685	0.371412
$426$	0	0
$427$	−9.31371	−0.450722
$428$	0	0
$429$	2.34315	0.113128
$430$	0	0
$431$	−0.485281	−0.0233752	−0.0116876	−	0.999932i	$-0.503720\pi$
$431$	−0.485281	−0.0233752	−0.0116876	+	0.999932i	$0.503720\pi$
$432$	0	0
$433$	13.5147	0.649476	0.324738	−	0.945804i	$-0.394724\pi$
$433$	13.5147	0.649476	0.324738	+	0.945804i	$0.394724\pi$
$434$	0	0
$435$	7.65685	0.367118
$436$	0	0
$437$	−11.3137	−0.541208
$438$	0	0
$439$	−13.1716	−0.628645	−0.314322	−	0.949316i	$-0.601777\pi$
$439$	−13.1716	−0.628645	−0.314322	+	0.949316i	$0.601777\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	24.9706	1.18639	0.593194	−	0.805060i	$-0.297867\pi$
$443$	24.9706	1.18639	0.593194	+	0.805060i	$0.297867\pi$
$444$	0	0
$445$	11.6569	0.552588
$446$	0	0
$447$	−17.3137	−0.818910
$448$	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
$451$	21.6569	1.01978
$452$	0	0
$453$	−1.65685	−0.0778458
$454$	0	0
$455$	−0.828427	−0.0388373
$456$	0	0
$457$	−8.34315	−0.390276	−0.195138	−	0.980776i	$-0.562515\pi$
$457$	−8.34315	−0.390276	−0.195138	+	0.980776i	$0.562515\pi$
$458$	0	0
$459$	7.65685	0.357391
$460$	0	0
$461$	−16.6274	−0.774416	−0.387208	−	0.921992i	$-0.626560\pi$
$461$	−16.6274	−0.774416	−0.387208	+	0.921992i	$0.626560\pi$
$462$	0	0
$463$	20.2843	0.942690	0.471345	−	0.881949i	$-0.343769\pi$
$463$	20.2843	0.942690	0.471345	+	0.881949i	$0.343769\pi$
$464$	0	0
$465$	2.82843	0.131165
$466$	0	0
$467$	28.9706	1.34060	0.670299	−	0.742091i	$-0.266166\pi$
$467$	28.9706	1.34060	0.670299	+	0.742091i	$0.266166\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	12.8284	0.591103
$472$	0	0
$473$	−4.68629	−0.215476
$474$	0	0
$475$	−2.82843	−0.129777
$476$	0	0
$477$	−10.4853	−0.480088
$478$	0	0
$479$	3.31371	0.151407	0.0757036	−	0.997130i	$-0.475880\pi$
$479$	3.31371	0.151407	0.0757036	+	0.997130i	$0.475880\pi$
$480$	0	0
$481$	−9.65685	−0.440315
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	−7.17157	−0.325644
$486$	0	0
$487$	−5.65685	−0.256337	−0.128168	−	0.991752i	$-0.540910\pi$
$487$	−5.65685	−0.256337	−0.128168	+	0.991752i	$0.540910\pi$
$488$	0	0
$489$	−17.6569	−0.798471
$490$	0	0
$491$	−33.4558	−1.50984	−0.754921	−	0.655816i	$-0.772325\pi$
$491$	−33.4558	−1.50984	−0.754921	+	0.655816i	$0.772325\pi$
$492$	0	0
$493$	58.6274	2.64045
$494$	0	0
$495$	−2.82843	−0.127128
$496$	0	0
$497$	10.8284	0.485721
$498$	0	0
$499$	6.34315	0.283958	0.141979	−	0.989870i	$-0.454653\pi$
$499$	6.34315	0.283958	0.141979	+	0.989870i	$0.454653\pi$
$500$	0	0
$501$	24.9706	1.11560
$502$	0	0
$503$	−39.5980	−1.76559	−0.882793	−	0.469762i	$-0.844340\pi$
$503$	−39.5980	−1.76559	−0.882793	+	0.469762i	$0.844340\pi$
$504$	0	0
$505$	17.3137	0.770450
$506$	0	0
$507$	−12.3137	−0.546871
$508$	0	0
$509$	−21.3137	−0.944714	−0.472357	−	0.881407i	$-0.656597\pi$
$509$	−21.3137	−0.944714	−0.472357	+	0.881407i	$0.656597\pi$
$510$	0	0
$511$	−4.82843	−0.213597
$512$	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
$519$	14.9706	0.657135
$520$	0	0
$521$	14.9706	0.655872	0.327936	−	0.944700i	$-0.393647\pi$
$521$	14.9706	0.655872	0.327936	+	0.944700i	$0.393647\pi$
$522$	0	0
$523$	−22.3431	−0.976998	−0.488499	−	0.872565i	$-0.662455\pi$
$523$	−22.3431	−0.976998	−0.488499	+	0.872565i	$0.662455\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	21.6569	0.943387
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	6.34315	0.274752
$534$	0	0
$535$	−8.00000	−0.345870
$536$	0	0
$537$	5.17157	0.223170
$538$	0	0
$539$	−2.82843	−0.121829
$540$	0	0
$541$	0.627417	0.0269748	0.0134874	−	0.999909i	$-0.495707\pi$
$541$	0.627417	0.0269748	0.0134874	+	0.999909i	$0.495707\pi$
$542$	0	0
$543$	12.3431	0.529695
$544$	0	0
$545$	13.3137	0.570297
$546$	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
$549$	−9.31371	−0.397499
$550$	0	0
$551$	−21.6569	−0.922613
$552$	0	0
$553$	15.3137	0.651205
$554$	0	0
$555$	11.6569	0.494806
$556$	0	0
$557$	−15.1716	−0.642840	−0.321420	−	0.946937i	$-0.604160\pi$
$557$	−15.1716	−0.642840	−0.321420	+	0.946937i	$0.604160\pi$
$558$	0	0
$559$	−1.37258	−0.0580541
$560$	0	0
$561$	−21.6569	−0.914353
$562$	0	0
$563$	−0.686292	−0.0289237	−0.0144619	−	0.999895i	$-0.504604\pi$
$563$	−0.686292	−0.0289237	−0.0144619	+	0.999895i	$0.504604\pi$
$564$	0	0
$565$	20.8284	0.876259
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−20.6274	−0.864746	−0.432373	−	0.901695i	$-0.642324\pi$
$569$	−20.6274	−0.864746	−0.432373	+	0.901695i	$0.642324\pi$
$570$	0	0
$571$	35.5980	1.48973	0.744865	−	0.667216i	$-0.232514\pi$
$571$	35.5980	1.48973	0.744865	+	0.667216i	$0.232514\pi$
$572$	0	0
$573$	13.1716	0.550250
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	0.828427	0.0344879	0.0172439	−	0.999851i	$-0.494511\pi$
$577$	0.828427	0.0344879	0.0172439	+	0.999851i	$0.494511\pi$
$578$	0	0
$579$	10.0000	0.415586
$580$	0	0
$581$	−17.6569	−0.732530
$582$	0	0
$583$	29.6569	1.22826
$584$	0	0
$585$	−0.828427	−0.0342512
$586$	0	0
$587$	−15.3137	−0.632064	−0.316032	−	0.948748i	$-0.602351\pi$
$587$	−15.3137	−0.632064	−0.316032	+	0.948748i	$0.602351\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	−12.8284	−0.527691
$592$	0	0
$593$	39.6569	1.62851	0.814256	−	0.580506i	$-0.197145\pi$
$593$	39.6569	1.62851	0.814256	+	0.580506i	$0.197145\pi$
$594$	0	0
$595$	7.65685	0.313900
$596$	0	0
$597$	−8.48528	−0.347279
$598$	0	0
$599$	−15.5147	−0.633914	−0.316957	−	0.948440i	$-0.602661\pi$
$599$	−15.5147	−0.633914	−0.316957	+	0.948440i	$0.602661\pi$
$600$	0	0
$601$	−24.3431	−0.992978	−0.496489	−	0.868043i	$-0.665378\pi$
$601$	−24.3431	−0.992978	−0.496489	+	0.868043i	$0.665378\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	−3.00000	−0.121967
$606$	0	0
$607$	−18.3431	−0.744525	−0.372263	−	0.928127i	$-0.621418\pi$
$607$	−18.3431	−0.744525	−0.372263	+	0.928127i	$0.621418\pi$
$608$	0	0
$609$	7.65685	0.310271
$610$	0	0
$611$	9.37258	0.379174
$612$	0	0
$613$	27.6569	1.11705	0.558525	−	0.829488i	$-0.311368\pi$
$613$	27.6569	1.11705	0.558525	+	0.829488i	$0.311368\pi$
$614$	0	0
$615$	−7.65685	−0.308754
$616$	0	0
$617$	17.5147	0.705116	0.352558	−	0.935790i	$-0.385312\pi$
$617$	17.5147	0.705116	0.352558	+	0.935790i	$0.385312\pi$
$618$	0	0
$619$	−9.45584	−0.380062	−0.190031	−	0.981778i	$-0.560859\pi$
$619$	−9.45584	−0.380062	−0.190031	+	0.981778i	$0.560859\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	11.6569	0.467022
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	8.00000	0.319489
$628$	0	0
$629$	89.2548	3.55882
$630$	0	0
$631$	−26.6274	−1.06002	−0.530010	−	0.847991i	$-0.677812\pi$
$631$	−26.6274	−1.06002	−0.530010	+	0.847991i	$0.677812\pi$
$632$	0	0
$633$	−20.0000	−0.794929
$634$	0	0
$635$	5.65685	0.224485
$636$	0	0
$637$	−0.828427	−0.0328235
$638$	0	0
$639$	10.8284	0.428366
$640$	0	0
$641$	−0.343146	−0.0135534	−0.00677672	−	0.999977i	$-0.502157\pi$
$641$	−0.343146	−0.0135534	−0.00677672	+	0.999977i	$0.502157\pi$
$642$	0	0
$643$	−33.6569	−1.32730	−0.663648	−	0.748045i	$-0.730993\pi$
$643$	−33.6569	−1.32730	−0.663648	+	0.748045i	$0.730993\pi$
$644$	0	0
$645$	1.65685	0.0652386
$646$	0	0
$647$	24.9706	0.981694	0.490847	−	0.871246i	$-0.336687\pi$
$647$	24.9706	0.981694	0.490847	+	0.871246i	$0.336687\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	2.82843	0.110855
$652$	0	0
$653$	−9.51472	−0.372340	−0.186170	−	0.982518i	$-0.559607\pi$
$653$	−9.51472	−0.372340	−0.186170	+	0.982518i	$0.559607\pi$
$654$	0	0
$655$	5.65685	0.221032
$656$	0	0
$657$	−4.82843	−0.188375
$658$	0	0
$659$	36.7696	1.43234	0.716169	−	0.697927i	$-0.245894\pi$
$659$	36.7696	1.43234	0.716169	+	0.697927i	$0.245894\pi$
$660$	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
$663$	−6.34315	−0.246347
$664$	0	0
$665$	−2.82843	−0.109682
$666$	0	0
$667$	30.6274	1.18590
$668$	0	0
$669$	−16.9706	−0.656120
$670$	0	0
$671$	26.3431	1.01697
$672$	0	0
$673$	28.3431	1.09255	0.546274	−	0.837607i	$-0.316046\pi$
$673$	28.3431	1.09255	0.546274	+	0.837607i	$0.316046\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	6.97056	0.267900	0.133950	−	0.990988i	$-0.457234\pi$
$677$	6.97056	0.267900	0.133950	+	0.990988i	$0.457234\pi$
$678$	0	0
$679$	−7.17157	−0.275220
$680$	0	0
$681$	−6.34315	−0.243070
$682$	0	0
$683$	10.3431	0.395769	0.197885	−	0.980225i	$-0.436593\pi$
$683$	10.3431	0.395769	0.197885	+	0.980225i	$0.436593\pi$
$684$	0	0
$685$	−0.828427	−0.0316526
$686$	0	0
$687$	−2.68629	−0.102488
$688$	0	0
$689$	8.68629	0.330921
$690$	0	0
$691$	−51.7990	−1.97053	−0.985263	−	0.171045i	$-0.945286\pi$
$691$	−51.7990	−1.97053	−0.985263	+	0.171045i	$0.945286\pi$
$692$	0	0
$693$	−2.82843	−0.107443
$694$	0	0
$695$	19.7990	0.751018
$696$	0	0
$697$	−58.6274	−2.22067
$698$	0	0
$699$	28.8284	1.09039
$700$	0	0
$701$	−13.0294	−0.492115	−0.246058	−	0.969255i	$-0.579135\pi$
$701$	−13.0294	−0.492115	−0.246058	+	0.969255i	$0.579135\pi$
$702$	0	0
$703$	−32.9706	−1.24351
$704$	0	0
$705$	−11.3137	−0.426099
$706$	0	0
$707$	17.3137	0.651149
$708$	0	0
$709$	5.31371	0.199561	0.0997803	−	0.995009i	$-0.468186\pi$
$709$	5.31371	0.199561	0.0997803	+	0.995009i	$0.468186\pi$
$710$	0	0
$711$	15.3137	0.574309
$712$	0	0
$713$	11.3137	0.423702
$714$	0	0
$715$	2.34315	0.0876287
$716$	0	0
$717$	−13.1716	−0.491901
$718$	0	0
$719$	−37.6569	−1.40436	−0.702182	−	0.711998i	$-0.747791\pi$
$719$	−37.6569	−1.40436	−0.702182	+	0.711998i	$0.747791\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	−3.65685	−0.136000
$724$	0	0
$725$	7.65685	0.284368
$726$	0	0
$727$	−37.6569	−1.39662	−0.698308	−	0.715798i	$-0.746063\pi$
$727$	−37.6569	−1.39662	−0.698308	+	0.715798i	$0.746063\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.6863	0.469219
$732$	0	0
$733$	−42.7696	−1.57973	−0.789865	−	0.613281i	$-0.789849\pi$
$733$	−42.7696	−1.57973	−0.789865	+	0.613281i	$0.789849\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	11.3137	0.416746
$738$	0	0
$739$	25.6569	0.943803	0.471901	−	0.881651i	$-0.343568\pi$
$739$	25.6569	0.943803	0.471901	+	0.881651i	$0.343568\pi$
$740$	0	0
$741$	2.34315	0.0860776
$742$	0	0
$743$	14.3431	0.526199	0.263099	−	0.964769i	$-0.415255\pi$
$743$	14.3431	0.526199	0.263099	+	0.964769i	$0.415255\pi$
$744$	0	0
$745$	−17.3137	−0.634325
$746$	0	0
$747$	−17.6569	−0.646031
$748$	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
$755$	−1.65685	−0.0602991
$756$	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
$761$	3.65685	0.132561	0.0662804	−	0.997801i	$-0.478887\pi$
$761$	3.65685	0.132561	0.0662804	+	0.997801i	$0.478887\pi$
$762$	0	0
$763$	13.3137	0.481989
$764$	0	0
$765$	7.65685	0.276834
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−22.9706	−0.828340	−0.414170	−	0.910200i	$-0.635928\pi$
$769$	−22.9706	−0.828340	−0.414170	+	0.910200i	$0.635928\pi$
$770$	0	0
$771$	6.68629	0.240801
$772$	0	0
$773$	−2.97056	−0.106844	−0.0534219	−	0.998572i	$-0.517013\pi$
$773$	−2.97056	−0.106844	−0.0534219	+	0.998572i	$0.517013\pi$
$774$	0	0
$775$	2.82843	0.101600
$776$	0	0
$777$	11.6569	0.418187
$778$	0	0
$779$	21.6569	0.775937
$780$	0	0
$781$	−30.6274	−1.09594
$782$	0	0
$783$	7.65685	0.273634
$784$	0	0
$785$	12.8284	0.457866
$786$	0	0
$787$	−30.3431	−1.08162	−0.540808	−	0.841146i	$-0.681882\pi$
$787$	−30.3431	−1.08162	−0.540808	+	0.841146i	$0.681882\pi$
$788$	0	0
$789$	20.9706	0.746572
$790$	0	0
$791$	20.8284	0.740574
$792$	0	0
$793$	7.71573	0.273994
$794$	0	0
$795$	−10.4853	−0.371875
$796$	0	0
$797$	−1.02944	−0.0364645	−0.0182323	−	0.999834i	$-0.505804\pi$
$797$	−1.02944	−0.0364645	−0.0182323	+	0.999834i	$0.505804\pi$
$798$	0	0
$799$	−86.6274	−3.06466
$800$	0	0
$801$	11.6569	0.411875
$802$	0	0
$803$	13.6569	0.481940
$804$	0	0
$805$	4.00000	0.140981
$806$	0	0
$807$	−5.31371	−0.187051
$808$	0	0
$809$	−38.9706	−1.37013	−0.685066	−	0.728481i	$-0.740227\pi$
$809$	−38.9706	−1.37013	−0.685066	+	0.728481i	$0.740227\pi$
$810$	0	0
$811$	−1.45584	−0.0511216	−0.0255608	−	0.999673i	$-0.508137\pi$
$811$	−1.45584	−0.0511216	−0.0255608	+	0.999673i	$0.508137\pi$
$812$	0	0
$813$	−10.8284	−0.379770
$814$	0	0
$815$	−17.6569	−0.618493
$816$	0	0
$817$	−4.68629	−0.163953
$818$	0	0
$819$	−0.828427	−0.0289476
$820$	0	0
$821$	−10.2843	−0.358924	−0.179462	−	0.983765i	$-0.557436\pi$
$821$	−10.2843	−0.358924	−0.179462	+	0.983765i	$0.557436\pi$
$822$	0	0
$823$	20.2843	0.707065	0.353533	−	0.935422i	$-0.384980\pi$
$823$	20.2843	0.707065	0.353533	+	0.935422i	$0.384980\pi$
$824$	0	0
$825$	−2.82843	−0.0984732
$826$	0	0
$827$	−45.2548	−1.57366	−0.786832	−	0.617167i	$-0.788280\pi$
$827$	−45.2548	−1.57366	−0.786832	+	0.617167i	$0.788280\pi$
$828$	0	0
$829$	−34.2843	−1.19074	−0.595371	−	0.803451i	$-0.702995\pi$
$829$	−34.2843	−1.19074	−0.595371	+	0.803451i	$0.702995\pi$
$830$	0	0
$831$	−2.97056	−0.103048
$832$	0	0
$833$	7.65685	0.265294
$834$	0	0
$835$	24.9706	0.864142
$836$	0	0
$837$	2.82843	0.0977647
$838$	0	0
$839$	−48.9706	−1.69065	−0.845326	−	0.534251i	$-0.820594\pi$
$839$	−48.9706	−1.69065	−0.845326	+	0.534251i	$0.820594\pi$
$840$	0	0
$841$	29.6274	1.02164
$842$	0	0
$843$	18.9706	0.653381
$844$	0	0
$845$	−12.3137	−0.423604
$846$	0	0
$847$	−3.00000	−0.103081
$848$	0	0
$849$	−15.3137	−0.525565
$850$	0	0
$851$	46.6274	1.59837
$852$	0	0
$853$	31.1716	1.06729	0.533647	−	0.845707i	$-0.320821\pi$
$853$	31.1716	1.06729	0.533647	+	0.845707i	$0.320821\pi$
$854$	0	0
$855$	−2.82843	−0.0967302
$856$	0	0
$857$	−23.9411	−0.817813	−0.408907	−	0.912576i	$-0.634090\pi$
$857$	−23.9411	−0.817813	−0.408907	+	0.912576i	$0.634090\pi$
$858$	0	0
$859$	17.4558	0.595586	0.297793	−	0.954631i	$-0.403750\pi$
$859$	17.4558	0.595586	0.297793	+	0.954631i	$0.403750\pi$
$860$	0	0
$861$	−7.65685	−0.260945
$862$	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
$865$	14.9706	0.509014
$866$	0	0
$867$	41.6274	1.41374
$868$	0	0
$869$	−43.3137	−1.46932
$870$	0	0
$871$	3.31371	0.112281
$872$	0	0
$873$	−7.17157	−0.242721
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−54.2843	−1.83305	−0.916525	−	0.399978i	$-0.869018\pi$
$877$	−54.2843	−1.83305	−0.916525	+	0.399978i	$0.869018\pi$
$878$	0	0
$879$	19.6569	0.663009
$880$	0	0
$881$	−42.9706	−1.44772	−0.723858	−	0.689949i	$-0.757633\pi$
$881$	−42.9706	−1.44772	−0.723858	+	0.689949i	$0.757633\pi$
$882$	0	0
$883$	18.6274	0.626862	0.313431	−	0.949611i	$-0.398521\pi$
$883$	18.6274	0.626862	0.313431	+	0.949611i	$0.398521\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−12.6863	−0.425964	−0.212982	−	0.977056i	$-0.568318\pi$
$887$	−12.6863	−0.425964	−0.212982	+	0.977056i	$0.568318\pi$
$888$	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
$895$	5.17157	0.172867
$896$	0	0
$897$	−3.31371	−0.110642
$898$	0	0
$899$	21.6569	0.722297
$900$	0	0
$901$	−80.2843	−2.67466
$902$	0	0
$903$	1.65685	0.0551367
$904$	0	0
$905$	12.3431	0.410300
$906$	0	0
$907$	−10.6274	−0.352878	−0.176439	−	0.984312i	$-0.556458\pi$
$907$	−10.6274	−0.352878	−0.176439	+	0.984312i	$0.556458\pi$
$908$	0	0
$909$	17.3137	0.574259
$910$	0	0
$911$	16.4853	0.546182	0.273091	−	0.961988i	$-0.411954\pi$
$911$	16.4853	0.546182	0.273091	+	0.961988i	$0.411954\pi$
$912$	0	0
$913$	49.9411	1.65281
$914$	0	0
$915$	−9.31371	−0.307902
$916$	0	0
$917$	5.65685	0.186806
$918$	0	0
$919$	−23.3137	−0.769048	−0.384524	−	0.923115i	$-0.625634\pi$
$919$	−23.3137	−0.769048	−0.384524	+	0.923115i	$0.625634\pi$
$920$	0	0
$921$	−17.6569	−0.581813
$922$	0	0
$923$	−8.97056	−0.295270
$924$	0	0
$925$	11.6569	0.383275
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	34.2843	1.12483	0.562415	−	0.826855i	$-0.309872\pi$
$929$	34.2843	1.12483	0.562415	+	0.826855i	$0.309872\pi$
$930$	0	0
$931$	−2.82843	−0.0926980
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	−21.6569	−0.708255
$936$	0	0
$937$	−41.5147	−1.35623	−0.678113	−	0.734957i	$-0.737202\pi$
$937$	−41.5147	−1.35623	−0.678113	+	0.734957i	$0.737202\pi$
$938$	0	0
$939$	0.828427	0.0270347
$940$	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
$943$	−30.6274	−0.997366
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	−50.9117	−1.65441	−0.827204	−	0.561902i	$-0.810070\pi$
$947$	−50.9117	−1.65441	−0.827204	+	0.561902i	$0.810070\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	−7.17157	−0.232554
$952$	0	0
$953$	−56.4264	−1.82783	−0.913915	−	0.405905i	$-0.866956\pi$
$953$	−56.4264	−1.82783	−0.913915	+	0.405905i	$0.866956\pi$
$954$	0	0
$955$	13.1716	0.426222
$956$	0	0
$957$	−21.6569	−0.700067
$958$	0	0
$959$	−0.828427	−0.0267513
$960$	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
$967$	−41.9411	−1.34874	−0.674368	−	0.738396i	$-0.735584\pi$
$967$	−41.9411	−1.34874	−0.674368	+	0.738396i	$0.735584\pi$
$968$	0	0
$969$	−21.6569	−0.695718
$970$	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
$975$	−0.828427	−0.0265309
$976$	0	0
$977$	−7.85786	−0.251395	−0.125698	−	0.992069i	$-0.540117\pi$
$977$	−7.85786	−0.251395	−0.125698	+	0.992069i	$0.540117\pi$
$978$	0	0
$979$	−32.9706	−1.05374
$980$	0	0
$981$	13.3137	0.425074
$982$	0	0
$983$	51.3137	1.63665	0.818327	−	0.574754i	$-0.194902\pi$
$983$	51.3137	1.63665	0.818327	+	0.574754i	$0.194902\pi$
$984$	0	0
$985$	−12.8284	−0.408748
$986$	0	0
$987$	−11.3137	−0.360119
$988$	0	0
$989$	6.62742	0.210740
$990$	0	0
$991$	53.9411	1.71350	0.856748	−	0.515735i	$-0.172481\pi$
$991$	53.9411	1.71350	0.856748	+	0.515735i	$0.172481\pi$
$992$	0	0
$993$	−4.00000	−0.126936
$994$	0	0
$995$	−8.48528	−0.269002
$996$	0	0
$997$	−4.14214	−0.131183	−0.0655914	−	0.997847i	$-0.520893\pi$
$997$	−4.14214	−0.131183	−0.0655914	+	0.997847i	$0.520893\pi$
$998$	0	0
$999$	11.6569	0.368807

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.82843 q^{11} -0.828427 q^{13} +1.00000 q^{15} +7.65685 q^{17} -2.82843 q^{19} +1.00000 q^{21} +4.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +7.65685 q^{29} +2.82843 q^{31} -2.82843 q^{33} +1.00000 q^{35} +11.6569 q^{37} -0.828427 q^{39} -7.65685 q^{41} +1.65685 q^{43} +1.00000 q^{45} -11.3137 q^{47} +1.00000 q^{49} +7.65685 q^{51} -10.4853 q^{53} -2.82843 q^{55} -2.82843 q^{57} -9.31371 q^{61} +1.00000 q^{63} -0.828427 q^{65} -4.00000 q^{67} +4.00000 q^{69} +10.8284 q^{71} -4.82843 q^{73} +1.00000 q^{75} -2.82843 q^{77} +15.3137 q^{79} +1.00000 q^{81} -17.6569 q^{83} +7.65685 q^{85} +7.65685 q^{87} +11.6569 q^{89} -0.828427 q^{91} +2.82843 q^{93} -2.82843 q^{95} -7.17157 q^{97} -2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 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

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