LMFDB - Embedded newform 6720.2.a.cp.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{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			$-0.618034$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -4.47214 q^{13} +1.00000 q^{15} +4.47214 q^{17} -1.00000 q^{21} -2.47214 q^{23} +1.00000 q^{25} -1.00000 q^{27} +0.472136 q^{29} +2.47214 q^{31} -1.00000 q^{35} +0.472136 q^{37} +4.47214 q^{39} -6.94427 q^{41} -1.52786 q^{43} -1.00000 q^{45} +6.47214 q^{47} +1.00000 q^{49} -4.47214 q^{51} -2.00000 q^{53} +4.00000 q^{59} -3.52786 q^{61} +1.00000 q^{63} +4.47214 q^{65} +6.47214 q^{67} +2.47214 q^{69} -11.4164 q^{71} -0.472136 q^{73} -1.00000 q^{75} -4.94427 q^{79} +1.00000 q^{81} -0.944272 q^{83} -4.47214 q^{85} -0.472136 q^{87} +10.9443 q^{89} -4.47214 q^{91} -2.47214 q^{93} +4.47214 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{15} - 2 q^{21} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 8 q^{29} - 4 q^{31} - 2 q^{35} - 8 q^{37} + 4 q^{41} - 12 q^{43} - 2 q^{45} + 4 q^{47} + 2 q^{49} - 4 q^{53} + 8 q^{59} - 16 q^{61} + 2 q^{63} + 4 q^{67} - 4 q^{69} + 4 q^{71} + 8 q^{73} - 2 q^{75} + 8 q^{79} + 2 q^{81} + 16 q^{83} + 8 q^{87} + 4 q^{89} + 4 q^{93}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 + 2 * q^15 - 2 * q^21 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 8 * q^29 - 4 * q^31 - 2 * q^35 - 8 * q^37 + 4 * q^41 - 12 * q^43 - 2 * q^45 + 4 * q^47 + 2 * q^49 - 4 * q^53 + 8 * q^59 - 16 * q^61 + 2 * q^63 + 4 * q^67 - 4 * q^69 + 4 * q^71 + 8 * q^73 - 2 * q^75 + 8 * q^79 + 2 * q^81 + 16 * q^83 + 8 * q^87 + 4 * q^89 + 4 * q^93

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
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−2.47214	−0.515476	−0.257738	−	0.966215i	$-0.582977\pi$
$23$	−2.47214	−0.515476	−0.257738	+	0.966215i	$0.582977\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0.472136	0.0876734	0.0438367	−	0.999039i	$-0.486042\pi$
$29$	0.472136	0.0876734	0.0438367	+	0.999039i	$0.486042\pi$
$30$	0	0
$31$	2.47214	0.444009	0.222004	−	0.975046i	$-0.428740\pi$
$31$	2.47214	0.444009	0.222004	+	0.975046i	$0.428740\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	0.472136	0.0776187	0.0388093	−	0.999247i	$-0.487644\pi$
$37$	0.472136	0.0776187	0.0388093	+	0.999247i	$0.487644\pi$
$38$	0	0
$39$	4.47214	0.716115
$40$	0	0
$41$	−6.94427	−1.08451	−0.542257	−	0.840213i	$-0.682430\pi$
$41$	−6.94427	−1.08451	−0.542257	+	0.840213i	$0.682430\pi$
$42$	0	0
$43$	−1.52786	−0.232997	−0.116499	−	0.993191i	$-0.537167\pi$
$43$	−1.52786	−0.232997	−0.116499	+	0.993191i	$0.537167\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	6.47214	0.944058	0.472029	−	0.881583i	$-0.343522\pi$
$47$	6.47214	0.944058	0.472029	+	0.881583i	$0.343522\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.47214	−0.626224
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	−3.52786	−0.451697	−0.225848	−	0.974162i	$-0.572515\pi$
$61$	−3.52786	−0.451697	−0.225848	+	0.974162i	$0.572515\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	4.47214	0.554700
$66$	0	0
$67$	6.47214	0.790697	0.395349	−	0.918531i	$-0.370624\pi$
$67$	6.47214	0.790697	0.395349	+	0.918531i	$0.370624\pi$
$68$	0	0
$69$	2.47214	0.297610
$70$	0	0
$71$	−11.4164	−1.35488	−0.677439	−	0.735579i	$-0.736910\pi$
$71$	−11.4164	−1.35488	−0.677439	+	0.735579i	$0.736910\pi$
$72$	0	0
$73$	−0.472136	−0.0552593	−0.0276297	−	0.999618i	$-0.508796\pi$
$73$	−0.472136	−0.0552593	−0.0276297	+	0.999618i	$0.508796\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.94427	−0.556274	−0.278137	−	0.960541i	$-0.589717\pi$
$79$	−4.94427	−0.556274	−0.278137	+	0.960541i	$0.589717\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.944272	−0.103647	−0.0518237	−	0.998656i	$-0.516503\pi$
$83$	−0.944272	−0.103647	−0.0518237	+	0.998656i	$0.516503\pi$
$84$	0	0
$85$	−4.47214	−0.485071
$86$	0	0
$87$	−0.472136	−0.0506183
$88$	0	0
$89$	10.9443	1.16009	0.580045	−	0.814584i	$-0.303035\pi$
$89$	10.9443	1.16009	0.580045	+	0.814584i	$0.303035\pi$
$90$	0	0
$91$	−4.47214	−0.468807
$92$	0	0
$93$	−2.47214	−0.256349
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.47214	0.454077	0.227038	−	0.973886i	$-0.427096\pi$
$97$	4.47214	0.454077	0.227038	+	0.973886i	$0.427096\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.9443	−1.08900	−0.544498	−	0.838762i	$-0.683280\pi$
$101$	−10.9443	−1.08900	−0.544498	+	0.838762i	$0.683280\pi$
$102$	0	0
$103$	17.8885	1.76261	0.881305	−	0.472547i	$-0.156665\pi$
$103$	17.8885	1.76261	0.881305	+	0.472547i	$0.156665\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	−2.47214	−0.238990	−0.119495	−	0.992835i	$-0.538128\pi$
$107$	−2.47214	−0.238990	−0.119495	+	0.992835i	$0.538128\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	0	0
$111$	−0.472136	−0.0448132
$112$	0	0
$113$	−2.00000	−0.188144	−0.0940721	−	0.995565i	$-0.529988\pi$
$113$	−2.00000	−0.188144	−0.0940721	+	0.995565i	$0.529988\pi$
$114$	0	0
$115$	2.47214	0.230528
$116$	0	0
$117$	−4.47214	−0.413449
$118$	0	0
$119$	4.47214	0.409960
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	6.94427	0.626144
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	1.52786	0.134521
$130$	0	0
$131$	16.9443	1.48043	0.740214	−	0.672371i	$-0.234724\pi$
$131$	16.9443	1.48043	0.740214	+	0.672371i	$0.234724\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−6.94427	−0.593289	−0.296645	−	0.954988i	$-0.595868\pi$
$137$	−6.94427	−0.593289	−0.296645	+	0.954988i	$0.595868\pi$
$138$	0	0
$139$	0	0		−	1.00000i	$-0.5\pi$
$139$	0	0			1.00000i	$0.5\pi$
$140$	0	0
$141$	−6.47214	−0.545052
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.472136	−0.0392088
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−12.4721	−1.02176	−0.510879	−	0.859653i	$-0.670680\pi$
$149$	−12.4721	−1.02176	−0.510879	+	0.859653i	$0.670680\pi$
$150$	0	0
$151$	−3.05573	−0.248672	−0.124336	−	0.992240i	$-0.539680\pi$
$151$	−3.05573	−0.248672	−0.124336	+	0.992240i	$0.539680\pi$
$152$	0	0
$153$	4.47214	0.361551
$154$	0	0
$155$	−2.47214	−0.198567
$156$	0	0
$157$	−7.52786	−0.600789	−0.300394	−	0.953815i	$-0.597118\pi$
$157$	−7.52786	−0.600789	−0.300394	+	0.953815i	$0.597118\pi$
$158$	0	0
$159$	2.00000	0.158610
$160$	0	0
$161$	−2.47214	−0.194832
$162$	0	0
$163$	−6.47214	−0.506937	−0.253468	−	0.967344i	$-0.581571\pi$
$163$	−6.47214	−0.506937	−0.253468	+	0.967344i	$0.581571\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.52786	−0.118230	−0.0591148	−	0.998251i	$-0.518828\pi$
$167$	−1.52786	−0.118230	−0.0591148	+	0.998251i	$0.518828\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	−13.4164	−0.997234	−0.498617	−	0.866822i	$-0.666159\pi$
$181$	−13.4164	−0.997234	−0.498617	+	0.866822i	$0.666159\pi$
$182$	0	0
$183$	3.52786	0.260787
$184$	0	0
$185$	−0.472136	−0.0347121
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	3.41641	0.247203	0.123601	−	0.992332i	$-0.460556\pi$
$191$	3.41641	0.247203	0.123601	+	0.992332i	$0.460556\pi$
$192$	0	0
$193$	−1.05573	−0.0759930	−0.0379965	−	0.999278i	$-0.512098\pi$
$193$	−1.05573	−0.0759930	−0.0379965	+	0.999278i	$0.512098\pi$
$194$	0	0
$195$	−4.47214	−0.320256
$196$	0	0
$197$	2.94427	0.209771	0.104885	−	0.994484i	$-0.466552\pi$
$197$	2.94427	0.209771	0.104885	+	0.994484i	$0.466552\pi$
$198$	0	0
$199$	−15.4164	−1.09284	−0.546420	−	0.837511i	$-0.684010\pi$
$199$	−15.4164	−1.09284	−0.546420	+	0.837511i	$0.684010\pi$
$200$	0	0
$201$	−6.47214	−0.456509
$202$	0	0
$203$	0.472136	0.0331374
$204$	0	0
$205$	6.94427	0.485009
$206$	0	0
$207$	−2.47214	−0.171825
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−8.00000	−0.550743	−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	0	0
$213$	11.4164	0.782239
$214$	0	0
$215$	1.52786	0.104199
$216$	0	0
$217$	2.47214	0.167820
$218$	0	0
$219$	0.472136	0.0319040
$220$	0	0
$221$	−20.0000	−1.34535
$222$	0	0
$223$	−9.88854	−0.662186	−0.331093	−	0.943598i	$-0.607417\pi$
$223$	−9.88854	−0.662186	−0.331093	+	0.943598i	$0.607417\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	16.9443	1.12463	0.562315	−	0.826923i	$-0.309911\pi$
$227$	16.9443	1.12463	0.562315	+	0.826923i	$0.309911\pi$
$228$	0	0
$229$	−11.5279	−0.761783	−0.380891	−	0.924620i	$-0.624383\pi$
$229$	−11.5279	−0.761783	−0.380891	+	0.924620i	$0.624383\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.9443	−0.979032	−0.489516	−	0.871994i	$-0.662826\pi$
$233$	−14.9443	−0.979032	−0.489516	+	0.871994i	$0.662826\pi$
$234$	0	0
$235$	−6.47214	−0.422196
$236$	0	0
$237$	4.94427	0.321165
$238$	0	0
$239$	24.3607	1.57576	0.787881	−	0.615828i	$-0.211178\pi$
$239$	24.3607	1.57576	0.787881	+	0.615828i	$0.211178\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0.944272	0.0598408
$250$	0	0
$251$	24.9443	1.57447	0.787234	−	0.616654i	$-0.211512\pi$
$251$	24.9443	1.57447	0.787234	+	0.616654i	$0.211512\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	4.47214	0.280056
$256$	0	0
$257$	−0.472136	−0.0294510	−0.0147255	−	0.999892i	$-0.504687\pi$
$257$	−0.472136	−0.0294510	−0.0147255	+	0.999892i	$0.504687\pi$
$258$	0	0
$259$	0.472136	0.0293371
$260$	0	0
$261$	0.472136	0.0292245
$262$	0	0
$263$	−18.4721	−1.13904	−0.569520	−	0.821977i	$-0.692871\pi$
$263$	−18.4721	−1.13904	−0.569520	+	0.821977i	$0.692871\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	−10.9443	−0.669779
$268$	0	0
$269$	−22.0000	−1.34136	−0.670682	−	0.741745i	$-0.733998\pi$
$269$	−22.0000	−1.34136	−0.670682	+	0.741745i	$0.733998\pi$
$270$	0	0
$271$	−20.3607	−1.23682	−0.618412	−	0.785854i	$-0.712224\pi$
$271$	−20.3607	−1.23682	−0.618412	+	0.785854i	$0.712224\pi$
$272$	0	0
$273$	4.47214	0.270666
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−4.47214	−0.268705	−0.134352	−	0.990934i	$-0.542895\pi$
$277$	−4.47214	−0.268705	−0.134352	+	0.990934i	$0.542895\pi$
$278$	0	0
$279$	2.47214	0.148003
$280$	0	0
$281$	3.88854	0.231971	0.115986	−	0.993251i	$-0.462997\pi$
$281$	3.88854	0.231971	0.115986	+	0.993251i	$0.462997\pi$
$282$	0	0
$283$	−29.8885	−1.77669	−0.888345	−	0.459177i	$-0.848144\pi$
$283$	−29.8885	−1.77669	−0.888345	+	0.459177i	$0.848144\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.94427	−0.409907
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	−4.47214	−0.262161
$292$	0	0
$293$	22.9443	1.34042	0.670209	−	0.742172i	$-0.266204\pi$
$293$	22.9443	1.34042	0.670209	+	0.742172i	$0.266204\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	0	0
$298$	0	0
$299$	11.0557	0.639369
$300$	0	0
$301$	−1.52786	−0.0880646
$302$	0	0
$303$	10.9443	0.628732
$304$	0	0
$305$	3.52786	0.202005
$306$	0	0
$307$	7.05573	0.402692	0.201346	−	0.979520i	$-0.435468\pi$
$307$	7.05573	0.402692	0.201346	+	0.979520i	$0.435468\pi$
$308$	0	0
$309$	−17.8885	−1.01764
$310$	0	0
$311$	1.88854	0.107089	0.0535447	−	0.998565i	$-0.482948\pi$
$311$	1.88854	0.107089	0.0535447	+	0.998565i	$0.482948\pi$
$312$	0	0
$313$	−8.47214	−0.478873	−0.239437	−	0.970912i	$-0.576963\pi$
$313$	−8.47214	−0.478873	−0.239437	+	0.970912i	$0.576963\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	−19.8885	−1.11705	−0.558526	−	0.829487i	$-0.688633\pi$
$317$	−19.8885	−1.11705	−0.558526	+	0.829487i	$0.688633\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	2.47214	0.137981
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−4.47214	−0.248069
$326$	0	0
$327$	14.0000	0.774202
$328$	0	0
$329$	6.47214	0.356820
$330$	0	0
$331$	−32.0000	−1.75888	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−32.0000	−1.75888	−0.879440	+	0.476011i	$0.842082\pi$
$332$	0	0
$333$	0.472136	0.0258729
$334$	0	0
$335$	−6.47214	−0.353611
$336$	0	0
$337$	−10.9443	−0.596172	−0.298086	−	0.954539i	$-0.596348\pi$
$337$	−10.9443	−0.596172	−0.298086	+	0.954539i	$0.596348\pi$
$338$	0	0
$339$	2.00000	0.108625
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−2.47214	−0.133095
$346$	0	0
$347$	18.4721	0.991636	0.495818	−	0.868426i	$-0.334868\pi$
$347$	18.4721	0.991636	0.495818	+	0.868426i	$0.334868\pi$
$348$	0	0
$349$	−24.4721	−1.30996	−0.654982	−	0.755645i	$-0.727324\pi$
$349$	−24.4721	−1.30996	−0.654982	+	0.755645i	$0.727324\pi$
$350$	0	0
$351$	4.47214	0.238705
$352$	0	0
$353$	9.41641	0.501185	0.250592	−	0.968093i	$-0.419375\pi$
$353$	9.41641	0.501185	0.250592	+	0.968093i	$0.419375\pi$
$354$	0	0
$355$	11.4164	0.605920
$356$	0	0
$357$	−4.47214	−0.236691
$358$	0	0
$359$	−11.4164	−0.602535	−0.301267	−	0.953540i	$-0.597410\pi$
$359$	−11.4164	−0.602535	−0.301267	+	0.953540i	$0.597410\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0	0
$363$	11.0000	0.577350
$364$	0	0
$365$	0.472136	0.0247127
$366$	0	0
$367$	−3.05573	−0.159508	−0.0797539	−	0.996815i	$-0.525413\pi$
$367$	−3.05573	−0.159508	−0.0797539	+	0.996815i	$0.525413\pi$
$368$	0	0
$369$	−6.94427	−0.361504
$370$	0	0
$371$	−2.00000	−0.103835
$372$	0	0
$373$	29.4164	1.52312	0.761562	−	0.648092i	$-0.224433\pi$
$373$	29.4164	1.52312	0.761562	+	0.648092i	$0.224433\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−2.11146	−0.108746
$378$	0	0
$379$	−36.9443	−1.89770	−0.948850	−	0.315728i	$-0.897751\pi$
$379$	−36.9443	−1.89770	−0.948850	+	0.315728i	$0.897751\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.41641	−0.174570	−0.0872851	−	0.996183i	$-0.527819\pi$
$383$	−3.41641	−0.174570	−0.0872851	+	0.996183i	$0.527819\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.52786	−0.0776657
$388$	0	0
$389$	−15.5279	−0.787294	−0.393647	−	0.919262i	$-0.628787\pi$
$389$	−15.5279	−0.787294	−0.393647	+	0.919262i	$0.628787\pi$
$390$	0	0
$391$	−11.0557	−0.559112
$392$	0	0
$393$	−16.9443	−0.854725
$394$	0	0
$395$	4.94427	0.248773
$396$	0	0
$397$	−25.4164	−1.27561	−0.637806	−	0.770197i	$-0.720158\pi$
$397$	−25.4164	−1.27561	−0.637806	+	0.770197i	$0.720158\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9.05573	−0.452221	−0.226111	−	0.974102i	$-0.572601\pi$
$401$	−9.05573	−0.452221	−0.226111	+	0.974102i	$0.572601\pi$
$402$	0	0
$403$	−11.0557	−0.550725
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	22.9443	1.13452	0.567261	−	0.823538i	$-0.308003\pi$
$409$	22.9443	1.13452	0.567261	+	0.823538i	$0.308003\pi$
$410$	0	0
$411$	6.94427	0.342536
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	0.944272	0.0463525
$416$	0	0
$417$	0	0
$418$	0	0
$419$	32.9443	1.60943	0.804717	−	0.593659i	$-0.202317\pi$
$419$	32.9443	1.60943	0.804717	+	0.593659i	$0.202317\pi$
$420$	0	0
$421$	11.8885	0.579412	0.289706	−	0.957116i	$-0.406442\pi$
$421$	11.8885	0.579412	0.289706	+	0.957116i	$0.406442\pi$
$422$	0	0
$423$	6.47214	0.314686
$424$	0	0
$425$	4.47214	0.216930
$426$	0	0
$427$	−3.52786	−0.170725
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−25.5279	−1.22963	−0.614817	−	0.788670i	$-0.710770\pi$
$431$	−25.5279	−1.22963	−0.614817	+	0.788670i	$0.710770\pi$
$432$	0	0
$433$	−26.3607	−1.26681	−0.633407	−	0.773819i	$-0.718344\pi$
$433$	−26.3607	−1.26681	−0.633407	+	0.773819i	$0.718344\pi$
$434$	0	0
$435$	0.472136	0.0226372
$436$	0	0
$437$	0	0
$438$	0	0
$439$	21.5279	1.02747	0.513734	−	0.857949i	$-0.328262\pi$
$439$	21.5279	1.02747	0.513734	+	0.857949i	$0.328262\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	10.4721	0.497546	0.248773	−	0.968562i	$-0.419973\pi$
$443$	10.4721	0.497546	0.248773	+	0.968562i	$0.419973\pi$
$444$	0	0
$445$	−10.9443	−0.518808
$446$	0	0
$447$	12.4721	0.589912
$448$	0	0
$449$	−20.8328	−0.983161	−0.491581	−	0.870832i	$-0.663581\pi$
$449$	−20.8328	−0.983161	−0.491581	+	0.870832i	$0.663581\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	3.05573	0.143571
$454$	0	0
$455$	4.47214	0.209657
$456$	0	0
$457$	−17.0557	−0.797833	−0.398917	−	0.916987i	$-0.630614\pi$
$457$	−17.0557	−0.797833	−0.398917	+	0.916987i	$0.630614\pi$
$458$	0	0
$459$	−4.47214	−0.208741
$460$	0	0
$461$	24.8328	1.15658	0.578290	−	0.815831i	$-0.303720\pi$
$461$	24.8328	1.15658	0.578290	+	0.815831i	$0.303720\pi$
$462$	0	0
$463$	20.9443	0.973363	0.486681	−	0.873580i	$-0.338207\pi$
$463$	20.9443	0.973363	0.486681	+	0.873580i	$0.338207\pi$
$464$	0	0
$465$	2.47214	0.114643
$466$	0	0
$467$	−5.88854	−0.272489	−0.136245	−	0.990675i	$-0.543503\pi$
$467$	−5.88854	−0.272489	−0.136245	+	0.990675i	$0.543503\pi$
$468$	0	0
$469$	6.47214	0.298855
$470$	0	0
$471$	7.52786	0.346866
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−25.8885	−1.18288	−0.591439	−	0.806350i	$-0.701440\pi$
$479$	−25.8885	−1.18288	−0.591439	+	0.806350i	$0.701440\pi$
$480$	0	0
$481$	−2.11146	−0.0962741
$482$	0	0
$483$	2.47214	0.112486
$484$	0	0
$485$	−4.47214	−0.203069
$486$	0	0
$487$	−36.9443	−1.67410	−0.837052	−	0.547123i	$-0.815723\pi$
$487$	−36.9443	−1.67410	−0.837052	+	0.547123i	$0.815723\pi$
$488$	0	0
$489$	6.47214	0.292680
$490$	0	0
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	0	0
$493$	2.11146	0.0950952
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−11.4164	−0.512096
$498$	0	0
$499$	−28.9443	−1.29572	−0.647862	−	0.761758i	$-0.724337\pi$
$499$	−28.9443	−1.29572	−0.647862	+	0.761758i	$0.724337\pi$
$500$	0	0
$501$	1.52786	0.0682599
$502$	0	0
$503$	−30.4721	−1.35869	−0.679343	−	0.733821i	$-0.737735\pi$
$503$	−30.4721	−1.35869	−0.679343	+	0.733821i	$0.737735\pi$
$504$	0	0
$505$	10.9443	0.487014
$506$	0	0
$507$	−7.00000	−0.310881
$508$	0	0
$509$	−22.0000	−0.975133	−0.487566	−	0.873086i	$-0.662115\pi$
$509$	−22.0000	−0.975133	−0.487566	+	0.873086i	$0.662115\pi$
$510$	0	0
$511$	−0.472136	−0.0208861
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−17.8885	−0.788263
$516$	0	0
$517$	0	0
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	−34.0000	−1.48957	−0.744784	−	0.667306i	$-0.767447\pi$
$521$	−34.0000	−1.48957	−0.744784	+	0.667306i	$0.767447\pi$
$522$	0	0
$523$	−7.05573	−0.308525	−0.154263	−	0.988030i	$-0.549300\pi$
$523$	−7.05573	−0.308525	−0.154263	+	0.988030i	$0.549300\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	11.0557	0.481595
$528$	0	0
$529$	−16.8885	−0.734285
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	31.0557	1.34517
$534$	0	0
$535$	2.47214	0.106880
$536$	0	0
$537$	−16.0000	−0.690451
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−2.94427	−0.126584	−0.0632921	−	0.997995i	$-0.520160\pi$
$541$	−2.94427	−0.126584	−0.0632921	+	0.997995i	$0.520160\pi$
$542$	0	0
$543$	13.4164	0.575753
$544$	0	0
$545$	14.0000	0.599694
$546$	0	0
$547$	−11.4164	−0.488130	−0.244065	−	0.969759i	$-0.578481\pi$
$547$	−11.4164	−0.488130	−0.244065	+	0.969759i	$0.578481\pi$
$548$	0	0
$549$	−3.52786	−0.150566
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−4.94427	−0.210252
$554$	0	0
$555$	0.472136	0.0200411
$556$	0	0
$557$	−10.0000	−0.423714	−0.211857	−	0.977301i	$-0.567951\pi$
$557$	−10.0000	−0.423714	−0.211857	+	0.977301i	$0.567951\pi$
$558$	0	0
$559$	6.83282	0.288997
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	0	0
$565$	2.00000	0.0841406
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−20.8328	−0.873357	−0.436679	−	0.899618i	$-0.643845\pi$
$569$	−20.8328	−0.873357	−0.436679	+	0.899618i	$0.643845\pi$
$570$	0	0
$571$	−12.9443	−0.541701	−0.270850	−	0.962621i	$-0.587305\pi$
$571$	−12.9443	−0.541701	−0.270850	+	0.962621i	$0.587305\pi$
$572$	0	0
$573$	−3.41641	−0.142722
$574$	0	0
$575$	−2.47214	−0.103095
$576$	0	0
$577$	−2.36068	−0.0982764	−0.0491382	−	0.998792i	$-0.515647\pi$
$577$	−2.36068	−0.0982764	−0.0491382	+	0.998792i	$0.515647\pi$
$578$	0	0
$579$	1.05573	0.0438746
$580$	0	0
$581$	−0.944272	−0.0391750
$582$	0	0
$583$	0	0
$584$	0	0
$585$	4.47214	0.184900
$586$	0	0
$587$	29.8885	1.23363	0.616816	−	0.787107i	$-0.288422\pi$
$587$	29.8885	1.23363	0.616816	+	0.787107i	$0.288422\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−2.94427	−0.121111
$592$	0	0
$593$	−3.52786	−0.144872	−0.0724360	−	0.997373i	$-0.523077\pi$
$593$	−3.52786	−0.144872	−0.0724360	+	0.997373i	$0.523077\pi$
$594$	0	0
$595$	−4.47214	−0.183340
$596$	0	0
$597$	15.4164	0.630952
$598$	0	0
$599$	−6.47214	−0.264444	−0.132222	−	0.991220i	$-0.542211\pi$
$599$	−6.47214	−0.264444	−0.132222	+	0.991220i	$0.542211\pi$
$600$	0	0
$601$	−26.9443	−1.09908	−0.549540	−	0.835467i	$-0.685197\pi$
$601$	−26.9443	−1.09908	−0.549540	+	0.835467i	$0.685197\pi$
$602$	0	0
$603$	6.47214	0.263566
$604$	0	0
$605$	11.0000	0.447214
$606$	0	0
$607$	25.8885	1.05078	0.525392	−	0.850860i	$-0.323919\pi$
$607$	25.8885	1.05078	0.525392	+	0.850860i	$0.323919\pi$
$608$	0	0
$609$	−0.472136	−0.0191319
$610$	0	0
$611$	−28.9443	−1.17096
$612$	0	0
$613$	32.4721	1.31154	0.655769	−	0.754962i	$-0.272345\pi$
$613$	32.4721	1.31154	0.655769	+	0.754962i	$0.272345\pi$
$614$	0	0
$615$	−6.94427	−0.280020
$616$	0	0
$617$	−21.0557	−0.847672	−0.423836	−	0.905739i	$-0.639317\pi$
$617$	−21.0557	−0.847672	−0.423836	+	0.905739i	$0.639317\pi$
$618$	0	0
$619$	−27.0557	−1.08746	−0.543731	−	0.839260i	$-0.682989\pi$
$619$	−27.0557	−1.08746	−0.543731	+	0.839260i	$0.682989\pi$
$620$	0	0
$621$	2.47214	0.0992034
$622$	0	0
$623$	10.9443	0.438473
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.11146	0.0841893
$630$	0	0
$631$	−1.88854	−0.0751817	−0.0375909	−	0.999293i	$-0.511968\pi$
$631$	−1.88854	−0.0751817	−0.0375909	+	0.999293i	$0.511968\pi$
$632$	0	0
$633$	8.00000	0.317971
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4.47214	−0.177192
$638$	0	0
$639$	−11.4164	−0.451626
$640$	0	0
$641$	−10.9443	−0.432273	−0.216136	−	0.976363i	$-0.569346\pi$
$641$	−10.9443	−0.432273	−0.216136	+	0.976363i	$0.569346\pi$
$642$	0	0
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	0	0
$645$	−1.52786	−0.0601596
$646$	0	0
$647$	−4.58359	−0.180200	−0.0900998	−	0.995933i	$-0.528719\pi$
$647$	−4.58359	−0.180200	−0.0900998	+	0.995933i	$0.528719\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−2.47214	−0.0968906
$652$	0	0
$653$	4.11146	0.160894	0.0804469	−	0.996759i	$-0.474365\pi$
$653$	4.11146	0.160894	0.0804469	+	0.996759i	$0.474365\pi$
$654$	0	0
$655$	−16.9443	−0.662067
$656$	0	0
$657$	−0.472136	−0.0184198
$658$	0	0
$659$	30.8328	1.20108	0.600538	−	0.799596i	$-0.294953\pi$
$659$	30.8328	1.20108	0.600538	+	0.799596i	$0.294953\pi$
$660$	0	0
$661$	46.3607	1.80322	0.901611	−	0.432548i	$-0.142386\pi$
$661$	46.3607	1.80322	0.901611	+	0.432548i	$0.142386\pi$
$662$	0	0
$663$	20.0000	0.776736
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.16718	−0.0451936
$668$	0	0
$669$	9.88854	0.382313
$670$	0	0
$671$	0	0
$672$	0	0
$673$	45.7771	1.76458	0.882289	−	0.470709i	$-0.156002\pi$
$673$	45.7771	1.76458	0.882289	+	0.470709i	$0.156002\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	11.8885	0.456914	0.228457	−	0.973554i	$-0.426632\pi$
$677$	11.8885	0.456914	0.228457	+	0.973554i	$0.426632\pi$
$678$	0	0
$679$	4.47214	0.171625
$680$	0	0
$681$	−16.9443	−0.649306
$682$	0	0
$683$	−12.3607	−0.472968	−0.236484	−	0.971635i	$-0.575995\pi$
$683$	−12.3607	−0.472968	−0.236484	+	0.971635i	$0.575995\pi$
$684$	0	0
$685$	6.94427	0.265327
$686$	0	0
$687$	11.5279	0.439815
$688$	0	0
$689$	8.94427	0.340750
$690$	0	0
$691$	−25.8885	−0.984847	−0.492423	−	0.870356i	$-0.663889\pi$
$691$	−25.8885	−0.984847	−0.492423	+	0.870356i	$0.663889\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−31.0557	−1.17632
$698$	0	0
$699$	14.9443	0.565244
$700$	0	0
$701$	−9.41641	−0.355653	−0.177826	−	0.984062i	$-0.556907\pi$
$701$	−9.41641	−0.355653	−0.177826	+	0.984062i	$0.556907\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	6.47214	0.243755
$706$	0	0
$707$	−10.9443	−0.411602
$708$	0	0
$709$	−12.8328	−0.481947	−0.240973	−	0.970532i	$-0.577467\pi$
$709$	−12.8328	−0.481947	−0.240973	+	0.970532i	$0.577467\pi$
$710$	0	0
$711$	−4.94427	−0.185425
$712$	0	0
$713$	−6.11146	−0.228876
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−24.3607	−0.909766
$718$	0	0
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	0	0
$721$	17.8885	0.666204
$722$	0	0
$723$	6.00000	0.223142
$724$	0	0
$725$	0.472136	0.0175347
$726$	0	0
$727$	−28.9443	−1.07348	−0.536742	−	0.843747i	$-0.680345\pi$
$727$	−28.9443	−1.07348	−0.536742	+	0.843747i	$0.680345\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.83282	−0.252721
$732$	0	0
$733$	29.4164	1.08652	0.543260	−	0.839565i	$-0.317190\pi$
$733$	29.4164	1.08652	0.543260	+	0.839565i	$0.317190\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.41641	0.272082	0.136041	−	0.990703i	$-0.456562\pi$
$743$	7.41641	0.272082	0.136041	+	0.990703i	$0.456562\pi$
$744$	0	0
$745$	12.4721	0.456944
$746$	0	0
$747$	−0.944272	−0.0345491
$748$	0	0
$749$	−2.47214	−0.0903299
$750$	0	0
$751$	11.0557	0.403429	0.201715	−	0.979444i	$-0.435349\pi$
$751$	11.0557	0.403429	0.201715	+	0.979444i	$0.435349\pi$
$752$	0	0
$753$	−24.9443	−0.909020
$754$	0	0
$755$	3.05573	0.111209
$756$	0	0
$757$	11.5279	0.418987	0.209494	−	0.977810i	$-0.432818\pi$
$757$	11.5279	0.418987	0.209494	+	0.977810i	$0.432818\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−35.8885	−1.30096	−0.650479	−	0.759524i	$-0.725432\pi$
$761$	−35.8885	−1.30096	−0.650479	+	0.759524i	$0.725432\pi$
$762$	0	0
$763$	−14.0000	−0.506834
$764$	0	0
$765$	−4.47214	−0.161690
$766$	0	0
$767$	−17.8885	−0.645918
$768$	0	0
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	0.472136	0.0170036
$772$	0	0
$773$	38.9443	1.40073	0.700364	−	0.713786i	$-0.253021\pi$
$773$	38.9443	1.40073	0.700364	+	0.713786i	$0.253021\pi$
$774$	0	0
$775$	2.47214	0.0888017
$776$	0	0
$777$	−0.472136	−0.0169378
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−0.472136	−0.0168728
$784$	0	0
$785$	7.52786	0.268681
$786$	0	0
$787$	47.7771	1.70307	0.851535	−	0.524298i	$-0.175672\pi$
$787$	47.7771	1.70307	0.851535	+	0.524298i	$0.175672\pi$
$788$	0	0
$789$	18.4721	0.657625
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	15.7771	0.560261
$794$	0	0
$795$	−2.00000	−0.0709327
$796$	0	0
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	28.9443	1.02397
$800$	0	0
$801$	10.9443	0.386697
$802$	0	0
$803$	0	0
$804$	0	0
$805$	2.47214	0.0871313
$806$	0	0
$807$	22.0000	0.774437
$808$	0	0
$809$	40.8328	1.43561	0.717803	−	0.696247i	$-0.245148\pi$
$809$	40.8328	1.43561	0.717803	+	0.696247i	$0.245148\pi$
$810$	0	0
$811$	43.7771	1.53722	0.768611	−	0.639717i	$-0.220948\pi$
$811$	43.7771	1.53722	0.768611	+	0.639717i	$0.220948\pi$
$812$	0	0
$813$	20.3607	0.714080
$814$	0	0
$815$	6.47214	0.226709
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−4.47214	−0.156269
$820$	0	0
$821$	31.3050	1.09255	0.546275	−	0.837606i	$-0.316045\pi$
$821$	31.3050	1.09255	0.546275	+	0.837606i	$0.316045\pi$
$822$	0	0
$823$	−20.9443	−0.730071	−0.365036	−	0.930994i	$-0.618943\pi$
$823$	−20.9443	−0.730071	−0.365036	+	0.930994i	$0.618943\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−15.4164	−0.536081	−0.268041	−	0.963408i	$-0.586376\pi$
$827$	−15.4164	−0.536081	−0.268041	+	0.963408i	$0.586376\pi$
$828$	0	0
$829$	−6.58359	−0.228658	−0.114329	−	0.993443i	$-0.536472\pi$
$829$	−6.58359	−0.228658	−0.114329	+	0.993443i	$0.536472\pi$
$830$	0	0
$831$	4.47214	0.155137
$832$	0	0
$833$	4.47214	0.154950
$834$	0	0
$835$	1.52786	0.0528739
$836$	0	0
$837$	−2.47214	−0.0854495
$838$	0	0
$839$	−14.1115	−0.487182	−0.243591	−	0.969878i	$-0.578325\pi$
$839$	−14.1115	−0.487182	−0.243591	+	0.969878i	$0.578325\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	0	0
$843$	−3.88854	−0.133929
$844$	0	0
$845$	−7.00000	−0.240807
$846$	0	0
$847$	−11.0000	−0.377964
$848$	0	0
$849$	29.8885	1.02577
$850$	0	0
$851$	−1.16718	−0.0400106
$852$	0	0
$853$	40.4721	1.38574	0.692870	−	0.721063i	$-0.256346\pi$
$853$	40.4721	1.38574	0.692870	+	0.721063i	$0.256346\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−26.3607	−0.900464	−0.450232	−	0.892912i	$-0.648659\pi$
$857$	−26.3607	−0.900464	−0.450232	+	0.892912i	$0.648659\pi$
$858$	0	0
$859$	−12.9443	−0.441653	−0.220826	−	0.975313i	$-0.570875\pi$
$859$	−12.9443	−0.441653	−0.220826	+	0.975313i	$0.570875\pi$
$860$	0	0
$861$	6.94427	0.236660
$862$	0	0
$863$	−3.63932	−0.123884	−0.0619420	−	0.998080i	$-0.519729\pi$
$863$	−3.63932	−0.123884	−0.0619420	+	0.998080i	$0.519729\pi$
$864$	0	0
$865$	6.00000	0.204006
$866$	0	0
$867$	−3.00000	−0.101885
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−28.9443	−0.980739
$872$	0	0
$873$	4.47214	0.151359
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	15.3050	0.516811	0.258406	−	0.966036i	$-0.416803\pi$
$877$	15.3050	0.516811	0.258406	+	0.966036i	$0.416803\pi$
$878$	0	0
$879$	−22.9443	−0.773891
$880$	0	0
$881$	20.8328	0.701875	0.350938	−	0.936399i	$-0.385863\pi$
$881$	20.8328	0.701875	0.350938	+	0.936399i	$0.385863\pi$
$882$	0	0
$883$	37.3050	1.25541	0.627706	−	0.778451i	$-0.283994\pi$
$883$	37.3050	1.25541	0.627706	+	0.778451i	$0.283994\pi$
$884$	0	0
$885$	4.00000	0.134459
$886$	0	0
$887$	51.4164	1.72639	0.863197	−	0.504867i	$-0.168459\pi$
$887$	51.4164	1.72639	0.863197	+	0.504867i	$0.168459\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−16.0000	−0.534821
$896$	0	0
$897$	−11.0557	−0.369140
$898$	0	0
$899$	1.16718	0.0389278
$900$	0	0
$901$	−8.94427	−0.297977
$902$	0	0
$903$	1.52786	0.0508441
$904$	0	0
$905$	13.4164	0.445976
$906$	0	0
$907$	−8.36068	−0.277612	−0.138806	−	0.990320i	$-0.544326\pi$
$907$	−8.36068	−0.277612	−0.138806	+	0.990320i	$0.544326\pi$
$908$	0	0
$909$	−10.9443	−0.362999
$910$	0	0
$911$	27.4164	0.908346	0.454173	−	0.890913i	$-0.349935\pi$
$911$	27.4164	0.908346	0.454173	+	0.890913i	$0.349935\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−3.52786	−0.116628
$916$	0	0
$917$	16.9443	0.559549
$918$	0	0
$919$	32.7214	1.07938	0.539689	−	0.841864i	$-0.318542\pi$
$919$	32.7214	1.07938	0.539689	+	0.841864i	$0.318542\pi$
$920$	0	0
$921$	−7.05573	−0.232494
$922$	0	0
$923$	51.0557	1.68052
$924$	0	0
$925$	0.472136	0.0155237
$926$	0	0
$927$	17.8885	0.587537
$928$	0	0
$929$	39.8885	1.30870	0.654350	−	0.756192i	$-0.272942\pi$
$929$	39.8885	1.30870	0.654350	+	0.756192i	$0.272942\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−1.88854	−0.0618281
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−31.3050	−1.02269	−0.511344	−	0.859376i	$-0.670852\pi$
$937$	−31.3050	−1.02269	−0.511344	+	0.859376i	$0.670852\pi$
$938$	0	0
$939$	8.47214	0.276478
$940$	0	0
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	0	0
$943$	17.1672	0.559040
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	−11.6393	−0.378227	−0.189114	−	0.981955i	$-0.560561\pi$
$947$	−11.6393	−0.378227	−0.189114	+	0.981955i	$0.560561\pi$
$948$	0	0
$949$	2.11146	0.0685408
$950$	0	0
$951$	19.8885	0.644930
$952$	0	0
$953$	−29.7771	−0.964574	−0.482287	−	0.876013i	$-0.660194\pi$
$953$	−29.7771	−0.964574	−0.482287	+	0.876013i	$0.660194\pi$
$954$	0	0
$955$	−3.41641	−0.110552
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.94427	−0.224242
$960$	0	0
$961$	−24.8885	−0.802856
$962$	0	0
$963$	−2.47214	−0.0796635
$964$	0	0
$965$	1.05573	0.0339851
$966$	0	0
$967$	−20.9443	−0.673522	−0.336761	−	0.941590i	$-0.609332\pi$
$967$	−20.9443	−0.673522	−0.336761	+	0.941590i	$0.609332\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	4.47214	0.143223
$976$	0	0
$977$	28.8328	0.922443	0.461222	−	0.887285i	$-0.347411\pi$
$977$	28.8328	0.922443	0.461222	+	0.887285i	$0.347411\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−14.0000	−0.446986
$982$	0	0
$983$	−50.2492	−1.60270	−0.801351	−	0.598195i	$-0.795885\pi$
$983$	−50.2492	−1.60270	−0.801351	+	0.598195i	$0.795885\pi$
$984$	0	0
$985$	−2.94427	−0.0938123
$986$	0	0
$987$	−6.47214	−0.206010
$988$	0	0
$989$	3.77709	0.120104
$990$	0	0
$991$	−19.0557	−0.605325	−0.302663	−	0.953098i	$-0.597876\pi$
$991$	−19.0557	−0.605325	−0.302663	+	0.953098i	$0.597876\pi$
$992$	0	0
$993$	32.0000	1.01549
$994$	0	0
$995$	15.4164	0.488733
$996$	0	0
$997$	−9.41641	−0.298221	−0.149110	−	0.988821i	$-0.547641\pi$
$997$	−9.41641	−0.298221	−0.149110	+	0.988821i	$0.547641\pi$
$998$	0	0
$999$	−0.472136	−0.0149377

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6720.2.a.cp.1.1		2
4.3	odd	2		6720.2.a.cv.1.1		2
8.3	odd	2		3360.2.a.bd.1.2	✓	2
8.5	even	2		3360.2.a.bh.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3360.2.a.bd.1.2	✓	2	8.3	odd	2
3360.2.a.bh.1.2	yes	2	8.5	even	2
6720.2.a.cp.1.1		2	1.1	even	1	trivial
6720.2.a.cv.1.1		2	4.3	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:	\( N \)	\(=\)	\( 6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6720.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -4.47214 q^{13} +1.00000 q^{15} +4.47214 q^{17} -1.00000 q^{21} -2.47214 q^{23} +1.00000 q^{25} -1.00000 q^{27} +0.472136 q^{29} +2.47214 q^{31} -1.00000 q^{35} +0.472136 q^{37} +4.47214 q^{39} -6.94427 q^{41} -1.52786 q^{43} -1.00000 q^{45} +6.47214 q^{47} +1.00000 q^{49} -4.47214 q^{51} -2.00000 q^{53} +4.00000 q^{59} -3.52786 q^{61} +1.00000 q^{63} +4.47214 q^{65} +6.47214 q^{67} +2.47214 q^{69} -11.4164 q^{71} -0.472136 q^{73} -1.00000 q^{75} -4.94427 q^{79} +1.00000 q^{81} -0.944272 q^{83} -4.47214 q^{85} -0.472136 q^{87} +10.9443 q^{89} -4.47214 q^{91} -2.47214 q^{93} +4.47214 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{15} - 2 q^{21} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 8 q^{29} - 4 q^{31} - 2 q^{35} - 8 q^{37} + 4 q^{41} - 12 q^{43} - 2 q^{45} + 4 q^{47} + 2 q^{49} - 4 q^{53} + 8 q^{59} - 16 q^{61} + 2 q^{63} + 4 q^{67} - 4 q^{69} + 4 q^{71} + 8 q^{73} - 2 q^{75} + 8 q^{79} + 2 q^{81} + 16 q^{83} + 8 q^{87} + 4 q^{89} + 4 q^{93}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 + 2 * q^15 - 2 * q^21 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 8 * q^29 - 4 * q^31 - 2 * q^35 - 8 * q^37 + 4 * q^41 - 12 * q^43 - 2 * q^45 + 4 * q^47 + 2 * q^49 - 4 * q^53 + 8 * q^59 - 16 * q^61 + 2 * q^63 + 4 * q^67 - 4 * q^69 + 4 * q^71 + 8 * q^73 - 2 * q^75 + 8 * q^79 + 2 * q^81 + 16 * q^83 + 8 * q^87 + 4 * q^89 + 4 * q^93

Introduction

L-functions

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