LMFDB - Embedded newform 6720.2.a.da.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:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 3360)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.48119$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.61213 q^{11} +6.31265 q^{13} +1.00000 q^{15} +7.92478 q^{17} -2.38787 q^{19} -1.00000 q^{21} -6.70052 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.92478 q^{29} +9.08840 q^{31} +1.61213 q^{33} -1.00000 q^{35} +7.92478 q^{37} -6.31265 q^{39} -1.22425 q^{41} +5.92478 q^{43} -1.00000 q^{45} +6.70052 q^{47} +1.00000 q^{49} -7.92478 q^{51} +3.61213 q^{53} +1.61213 q^{55} +2.38787 q^{57} -8.00000 q^{59} -4.70052 q^{61} +1.00000 q^{63} -6.31265 q^{65} -9.14903 q^{67} +6.70052 q^{69} +8.31265 q^{71} +10.3127 q^{73} -1.00000 q^{75} -1.61213 q^{77} +4.00000 q^{79} +1.00000 q^{81} -0.775746 q^{83} -7.92478 q^{85} +7.92478 q^{87} -14.6253 q^{89} +6.31265 q^{91} -9.08840 q^{93} +2.38787 q^{95} -6.31265 q^{97} -1.61213 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} - 4 q^{11} - 2 q^{13} + 3 q^{15} + 2 q^{17} - 8 q^{19} - 3 q^{21} + 3 q^{25} - 3 q^{27} - 2 q^{29} + 8 q^{31} + 4 q^{33} - 3 q^{35} + 2 q^{37} + 2 q^{39} - 2 q^{41} - 4 q^{43} - 3 q^{45} + 3 q^{49} - 2 q^{51} + 10 q^{53} + 4 q^{55} + 8 q^{57} - 24 q^{59} + 6 q^{61} + 3 q^{63} + 2 q^{65} - 4 q^{67} + 4 q^{71} + 10 q^{73} - 3 q^{75} - 4 q^{77} + 12 q^{79} + 3 q^{81} - 4 q^{83} - 2 q^{85} + 2 q^{87} - 2 q^{89} - 2 q^{91} - 8 q^{93} + 8 q^{95} + 2 q^{97} - 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 - 4 * q^11 - 2 * q^13 + 3 * q^15 + 2 * q^17 - 8 * q^19 - 3 * q^21 + 3 * q^25 - 3 * q^27 - 2 * q^29 + 8 * q^31 + 4 * q^33 - 3 * q^35 + 2 * q^37 + 2 * q^39 - 2 * q^41 - 4 * q^43 - 3 * q^45 + 3 * q^49 - 2 * q^51 + 10 * q^53 + 4 * q^55 + 8 * q^57 - 24 * q^59 + 6 * q^61 + 3 * q^63 + 2 * q^65 - 4 * q^67 + 4 * q^71 + 10 * q^73 - 3 * q^75 - 4 * q^77 + 12 * q^79 + 3 * q^81 - 4 * q^83 - 2 * q^85 + 2 * q^87 - 2 * q^89 - 2 * q^91 - 8 * q^93 + 8 * q^95 + 2 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−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
$11$	−1.61213	−0.486075	−0.243037	−	0.970017i	$-0.578144\pi$
$11$	−1.61213	−0.486075	−0.243037	+	0.970017i	$0.578144\pi$
$12$	0	0
$13$	6.31265	1.75081	0.875407	−	0.483386i	$-0.160593\pi$
$13$	6.31265	1.75081	0.875407	+	0.483386i	$0.160593\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	7.92478	1.92204	0.961020	−	0.276477i	$-0.0891671\pi$
$17$	7.92478	1.92204	0.961020	+	0.276477i	$0.0891671\pi$
$18$	0	0
$19$	−2.38787	−0.547816	−0.273908	−	0.961756i	$-0.588316\pi$
$19$	−2.38787	−0.547816	−0.273908	+	0.961756i	$0.588316\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−6.70052	−1.39716	−0.698578	−	0.715534i	$-0.746183\pi$
$23$	−6.70052	−1.39716	−0.698578	+	0.715534i	$0.746183\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.92478	−1.47159	−0.735797	−	0.677202i	$-0.763192\pi$
$29$	−7.92478	−1.47159	−0.735797	+	0.677202i	$0.763192\pi$
$30$	0	0
$31$	9.08840	1.63232	0.816162	−	0.577823i	$-0.196098\pi$
$31$	9.08840	1.63232	0.816162	+	0.577823i	$0.196098\pi$
$32$	0	0
$33$	1.61213	0.280635
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	7.92478	1.30283	0.651413	−	0.758724i	$-0.274177\pi$
$37$	7.92478	1.30283	0.651413	+	0.758724i	$0.274177\pi$
$38$	0	0
$39$	−6.31265	−1.01083
$40$	0	0
$41$	−1.22425	−0.191196	−0.0955982	−	0.995420i	$-0.530476\pi$
$41$	−1.22425	−0.191196	−0.0955982	+	0.995420i	$0.530476\pi$
$42$	0	0
$43$	5.92478	0.903520	0.451760	−	0.892139i	$-0.350796\pi$
$43$	5.92478	0.903520	0.451760	+	0.892139i	$0.350796\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	6.70052	0.977372	0.488686	−	0.872460i	$-0.337476\pi$
$47$	6.70052	0.977372	0.488686	+	0.872460i	$0.337476\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.92478	−1.10969
$52$	0	0
$53$	3.61213	0.496164	0.248082	−	0.968739i	$-0.420200\pi$
$53$	3.61213	0.496164	0.248082	+	0.968739i	$0.420200\pi$
$54$	0	0
$55$	1.61213	0.217379
$56$	0	0
$57$	2.38787	0.316282
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−4.70052	−0.601840	−0.300920	−	0.953649i	$-0.597294\pi$
$61$	−4.70052	−0.601840	−0.300920	+	0.953649i	$0.597294\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−6.31265	−0.782988
$66$	0	0
$67$	−9.14903	−1.11773	−0.558866	−	0.829258i	$-0.688763\pi$
$67$	−9.14903	−1.11773	−0.558866	+	0.829258i	$0.688763\pi$
$68$	0	0
$69$	6.70052	0.806648
$70$	0	0
$71$	8.31265	0.986530	0.493265	−	0.869879i	$-0.335803\pi$
$71$	8.31265	0.986530	0.493265	+	0.869879i	$0.335803\pi$
$72$	0	0
$73$	10.3127	1.20700	0.603502	−	0.797361i	$-0.293771\pi$
$73$	10.3127	1.20700	0.603502	+	0.797361i	$0.293771\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−1.61213	−0.183719
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.775746	−0.0851492	−0.0425746	−	0.999093i	$-0.513556\pi$
$83$	−0.775746	−0.0851492	−0.0425746	+	0.999093i	$0.513556\pi$
$84$	0	0
$85$	−7.92478	−0.859563
$86$	0	0
$87$	7.92478	0.849625
$88$	0	0
$89$	−14.6253	−1.55028	−0.775139	−	0.631790i	$-0.782320\pi$
$89$	−14.6253	−1.55028	−0.775139	+	0.631790i	$0.782320\pi$
$90$	0	0
$91$	6.31265	0.661746
$92$	0	0
$93$	−9.08840	−0.942423
$94$	0	0
$95$	2.38787	0.244991
$96$	0	0
$97$	−6.31265	−0.640953	−0.320476	−	0.947257i	$-0.603843\pi$
$97$	−6.31265	−0.640953	−0.320476	+	0.947257i	$0.603843\pi$
$98$	0	0
$99$	−1.61213	−0.162025
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	−10.7005	−1.03446	−0.517229	−	0.855847i	$-0.673037\pi$
$107$	−10.7005	−1.03446	−0.517229	+	0.855847i	$0.673037\pi$
$108$	0	0
$109$	−9.84955	−0.943416	−0.471708	−	0.881755i	$-0.656362\pi$
$109$	−9.84955	−0.943416	−0.471708	+	0.881755i	$0.656362\pi$
$110$	0	0
$111$	−7.92478	−0.752187
$112$	0	0
$113$	1.16362	0.109464	0.0547321	−	0.998501i	$-0.482570\pi$
$113$	1.16362	0.109464	0.0547321	+	0.998501i	$0.482570\pi$
$114$	0	0
$115$	6.70052	0.624827
$116$	0	0
$117$	6.31265	0.583605
$118$	0	0
$119$	7.92478	0.726463
$120$	0	0
$121$	−8.40105	−0.763732
$122$	0	0
$123$	1.22425	0.110387
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.62530	−0.765372	−0.382686	−	0.923879i	$-0.625001\pi$
$127$	−8.62530	−0.765372	−0.382686	+	0.923879i	$0.625001\pi$
$128$	0	0
$129$	−5.92478	−0.521648
$130$	0	0
$131$	8.62530	0.753596	0.376798	−	0.926295i	$-0.377025\pi$
$131$	8.62530	0.753596	0.376798	+	0.926295i	$0.377025\pi$
$132$	0	0
$133$	−2.38787	−0.207055
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	8.23743	0.703771	0.351885	−	0.936043i	$-0.385541\pi$
$137$	8.23743	0.703771	0.351885	+	0.936043i	$0.385541\pi$
$138$	0	0
$139$	−10.3879	−0.881088	−0.440544	−	0.897731i	$-0.645214\pi$
$139$	−10.3879	−0.881088	−0.440544	+	0.897731i	$0.645214\pi$
$140$	0	0
$141$	−6.70052	−0.564286
$142$	0	0
$143$	−10.1768	−0.851026
$144$	0	0
$145$	7.92478	0.658117
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	16.7005	1.36816	0.684080	−	0.729407i	$-0.260204\pi$
$149$	16.7005	1.36816	0.684080	+	0.729407i	$0.260204\pi$
$150$	0	0
$151$	7.22425	0.587901	0.293951	−	0.955821i	$-0.405030\pi$
$151$	7.22425	0.587901	0.293951	+	0.955821i	$0.405030\pi$
$152$	0	0
$153$	7.92478	0.640680
$154$	0	0
$155$	−9.08840	−0.729998
$156$	0	0
$157$	−2.31265	−0.184570	−0.0922848	−	0.995733i	$-0.529417\pi$
$157$	−2.31265	−0.184570	−0.0922848	+	0.995733i	$0.529417\pi$
$158$	0	0
$159$	−3.61213	−0.286460
$160$	0	0
$161$	−6.70052	−0.528075
$162$	0	0
$163$	2.07522	0.162544	0.0812720	−	0.996692i	$-0.474102\pi$
$163$	2.07522	0.162544	0.0812720	+	0.996692i	$0.474102\pi$
$164$	0	0
$165$	−1.61213	−0.125504
$166$	0	0
$167$	−17.9248	−1.38706	−0.693530	−	0.720427i	$-0.743946\pi$
$167$	−17.9248	−1.38706	−0.693530	+	0.720427i	$0.743946\pi$
$168$	0	0
$169$	26.8496	2.06535
$170$	0	0
$171$	−2.38787	−0.182605
$172$	0	0
$173$	−1.22425	−0.0930783	−0.0465391	−	0.998916i	$-0.514819\pi$
$173$	−1.22425	−0.0930783	−0.0465391	+	0.998916i	$0.514819\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	8.00000	0.601317
$178$	0	0
$179$	−17.6121	−1.31639	−0.658196	−	0.752847i	$-0.728680\pi$
$179$	−17.6121	−1.31639	−0.658196	+	0.752847i	$0.728680\pi$
$180$	0	0
$181$	8.07522	0.600227	0.300113	−	0.953904i	$-0.402976\pi$
$181$	8.07522	0.600227	0.300113	+	0.953904i	$0.402976\pi$
$182$	0	0
$183$	4.70052	0.347473
$184$	0	0
$185$	−7.92478	−0.582641
$186$	0	0
$187$	−12.7757	−0.934255
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−8.31265	−0.601482	−0.300741	−	0.953706i	$-0.597234\pi$
$191$	−8.31265	−0.601482	−0.300741	+	0.953706i	$0.597234\pi$
$192$	0	0
$193$	−9.84955	−0.708986	−0.354493	−	0.935059i	$-0.615347\pi$
$193$	−9.84955	−0.708986	−0.354493	+	0.935059i	$0.615347\pi$
$194$	0	0
$195$	6.31265	0.452058
$196$	0	0
$197$	20.2374	1.44186	0.720929	−	0.693009i	$-0.243716\pi$
$197$	20.2374	1.44186	0.720929	+	0.693009i	$0.243716\pi$
$198$	0	0
$199$	23.5369	1.66849	0.834243	−	0.551396i	$-0.185905\pi$
$199$	23.5369	1.66849	0.834243	+	0.551396i	$0.185905\pi$
$200$	0	0
$201$	9.14903	0.645323
$202$	0	0
$203$	−7.92478	−0.556210
$204$	0	0
$205$	1.22425	0.0855056
$206$	0	0
$207$	−6.70052	−0.465719
$208$	0	0
$209$	3.84955	0.266279
$210$	0	0
$211$	−9.40105	−0.647195	−0.323597	−	0.946195i	$-0.604892\pi$
$211$	−9.40105	−0.647195	−0.323597	+	0.946195i	$0.604892\pi$
$212$	0	0
$213$	−8.31265	−0.569573
$214$	0	0
$215$	−5.92478	−0.404066
$216$	0	0
$217$	9.08840	0.616961
$218$	0	0
$219$	−10.3127	−0.696864
$220$	0	0
$221$	50.0263	3.36514
$222$	0	0
$223$	28.4749	1.90682	0.953409	−	0.301682i	$-0.0975480\pi$
$223$	28.4749	1.90682	0.953409	+	0.301682i	$0.0975480\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	24.4749	1.62445	0.812227	−	0.583342i	$-0.198255\pi$
$227$	24.4749	1.62445	0.812227	+	0.583342i	$0.198255\pi$
$228$	0	0
$229$	26.9986	1.78412	0.892058	−	0.451920i	$-0.149261\pi$
$229$	26.9986	1.78412	0.892058	+	0.451920i	$0.149261\pi$
$230$	0	0
$231$	1.61213	0.106070
$232$	0	0
$233$	11.4617	0.750880	0.375440	−	0.926847i	$-0.377492\pi$
$233$	11.4617	0.750880	0.375440	+	0.926847i	$0.377492\pi$
$234$	0	0
$235$	−6.70052	−0.437094
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	8.31265	0.537701	0.268850	−	0.963182i	$-0.413356\pi$
$239$	8.31265	0.537701	0.268850	+	0.963182i	$0.413356\pi$
$240$	0	0
$241$	5.22425	0.336524	0.168262	−	0.985742i	$-0.446185\pi$
$241$	5.22425	0.336524	0.168262	+	0.985742i	$0.446185\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−15.0738	−0.959123
$248$	0	0
$249$	0.775746	0.0491609
$250$	0	0
$251$	−11.8496	−0.747937	−0.373969	−	0.927441i	$-0.622003\pi$
$251$	−11.8496	−0.747937	−0.373969	+	0.927441i	$0.622003\pi$
$252$	0	0
$253$	10.8021	0.679122
$254$	0	0
$255$	7.92478	0.496269
$256$	0	0
$257$	0.850969	0.0530820	0.0265410	−	0.999648i	$-0.491551\pi$
$257$	0.850969	0.0530820	0.0265410	+	0.999648i	$0.491551\pi$
$258$	0	0
$259$	7.92478	0.492422
$260$	0	0
$261$	−7.92478	−0.490531
$262$	0	0
$263$	24.8773	1.53400	0.767001	−	0.641646i	$-0.221748\pi$
$263$	24.8773	1.53400	0.767001	+	0.641646i	$0.221748\pi$
$264$	0	0
$265$	−3.61213	−0.221891
$266$	0	0
$267$	14.6253	0.895054
$268$	0	0
$269$	28.2981	1.72536	0.862682	−	0.505747i	$-0.168783\pi$
$269$	28.2981	1.72536	0.862682	+	0.505747i	$0.168783\pi$
$270$	0	0
$271$	−1.08840	−0.0661154	−0.0330577	−	0.999453i	$-0.510525\pi$
$271$	−1.08840	−0.0661154	−0.0330577	+	0.999453i	$0.510525\pi$
$272$	0	0
$273$	−6.31265	−0.382059
$274$	0	0
$275$	−1.61213	−0.0972149
$276$	0	0
$277$	12.0752	0.725530	0.362765	−	0.931881i	$-0.381833\pi$
$277$	12.0752	0.725530	0.362765	+	0.931881i	$0.381833\pi$
$278$	0	0
$279$	9.08840	0.544108
$280$	0	0
$281$	−1.22425	−0.0730329	−0.0365164	−	0.999333i	$-0.511626\pi$
$281$	−1.22425	−0.0730329	−0.0365164	+	0.999333i	$0.511626\pi$
$282$	0	0
$283$	5.55149	0.330002	0.165001	−	0.986293i	$-0.447237\pi$
$283$	5.55149	0.330002	0.165001	+	0.986293i	$0.447237\pi$
$284$	0	0
$285$	−2.38787	−0.141445
$286$	0	0
$287$	−1.22425	−0.0722654
$288$	0	0
$289$	45.8021	2.69424
$290$	0	0
$291$	6.31265	0.370054
$292$	0	0
$293$	1.37470	0.0803108	0.0401554	−	0.999193i	$-0.487215\pi$
$293$	1.37470	0.0803108	0.0401554	+	0.999193i	$0.487215\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	1.61213	0.0935451
$298$	0	0
$299$	−42.2981	−2.44616
$300$	0	0
$301$	5.92478	0.341498
$302$	0	0
$303$	−10.0000	−0.574485
$304$	0	0
$305$	4.70052	0.269151
$306$	0	0
$307$	−11.0738	−0.632016	−0.316008	−	0.948757i	$-0.602343\pi$
$307$	−11.0738	−0.632016	−0.316008	+	0.948757i	$0.602343\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−6.44851	−0.365661	−0.182831	−	0.983144i	$-0.558526\pi$
$311$	−6.44851	−0.365661	−0.182831	+	0.983144i	$0.558526\pi$
$312$	0	0
$313$	23.0884	1.30503	0.652517	−	0.757774i	$-0.273713\pi$
$313$	23.0884	1.30503	0.652517	+	0.757774i	$0.273713\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	−21.6385	−1.21534	−0.607669	−	0.794190i	$-0.707895\pi$
$317$	−21.6385	−1.21534	−0.607669	+	0.794190i	$0.707895\pi$
$318$	0	0
$319$	12.7757	0.715304
$320$	0	0
$321$	10.7005	0.597245
$322$	0	0
$323$	−18.9234	−1.05292
$324$	0	0
$325$	6.31265	0.350163
$326$	0	0
$327$	9.84955	0.544682
$328$	0	0
$329$	6.70052	0.369412
$330$	0	0
$331$	33.4010	1.83589	0.917944	−	0.396710i	$-0.129848\pi$
$331$	33.4010	1.83589	0.917944	+	0.396710i	$0.129848\pi$
$332$	0	0
$333$	7.92478	0.434275
$334$	0	0
$335$	9.14903	0.499865
$336$	0	0
$337$	−3.40105	−0.185267	−0.0926334	−	0.995700i	$-0.529528\pi$
$337$	−3.40105	−0.185267	−0.0926334	+	0.995700i	$0.529528\pi$
$338$	0	0
$339$	−1.16362	−0.0631991
$340$	0	0
$341$	−14.6516	−0.793431
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−6.70052	−0.360744
$346$	0	0
$347$	5.92478	0.318059	0.159029	−	0.987274i	$-0.449164\pi$
$347$	5.92478	0.318059	0.159029	+	0.987274i	$0.449164\pi$
$348$	0	0
$349$	−18.1016	−0.968955	−0.484477	−	0.874804i	$-0.660990\pi$
$349$	−18.1016	−0.968955	−0.484477	+	0.874804i	$0.660990\pi$
$350$	0	0
$351$	−6.31265	−0.336944
$352$	0	0
$353$	−7.02776	−0.374050	−0.187025	−	0.982355i	$-0.559885\pi$
$353$	−7.02776	−0.374050	−0.187025	+	0.982355i	$0.559885\pi$
$354$	0	0
$355$	−8.31265	−0.441190
$356$	0	0
$357$	−7.92478	−0.419424
$358$	0	0
$359$	−13.7137	−0.723781	−0.361891	−	0.932221i	$-0.617869\pi$
$359$	−13.7137	−0.723781	−0.361891	+	0.932221i	$0.617869\pi$
$360$	0	0
$361$	−13.2981	−0.699898
$362$	0	0
$363$	8.40105	0.440941
$364$	0	0
$365$	−10.3127	−0.539789
$366$	0	0
$367$	32.6253	1.70303	0.851513	−	0.524333i	$-0.175685\pi$
$367$	32.6253	1.70303	0.851513	+	0.524333i	$0.175685\pi$
$368$	0	0
$369$	−1.22425	−0.0637321
$370$	0	0
$371$	3.61213	0.187532
$372$	0	0
$373$	−13.5975	−0.704054	−0.352027	−	0.935990i	$-0.614507\pi$
$373$	−13.5975	−0.704054	−0.352027	+	0.935990i	$0.614507\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−50.0263	−2.57649
$378$	0	0
$379$	18.0263	0.925951	0.462976	−	0.886371i	$-0.346782\pi$
$379$	18.0263	0.925951	0.462976	+	0.886371i	$0.346782\pi$
$380$	0	0
$381$	8.62530	0.441888
$382$	0	0
$383$	−24.3733	−1.24542	−0.622708	−	0.782454i	$-0.713968\pi$
$383$	−24.3733	−1.24542	−0.622708	+	0.782454i	$0.713968\pi$
$384$	0	0
$385$	1.61213	0.0821616
$386$	0	0
$387$	5.92478	0.301173
$388$	0	0
$389$	19.2995	0.978522	0.489261	−	0.872137i	$-0.337266\pi$
$389$	19.2995	0.978522	0.489261	+	0.872137i	$0.337266\pi$
$390$	0	0
$391$	−53.1002	−2.68539
$392$	0	0
$393$	−8.62530	−0.435089
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	29.2652	1.46878	0.734389	−	0.678729i	$-0.237469\pi$
$397$	29.2652	1.46878	0.734389	+	0.678729i	$0.237469\pi$
$398$	0	0
$399$	2.38787	0.119543
$400$	0	0
$401$	37.8496	1.89012	0.945058	−	0.326902i	$-0.106005\pi$
$401$	37.8496	1.89012	0.945058	+	0.326902i	$0.106005\pi$
$402$	0	0
$403$	57.3719	2.85790
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−12.7757	−0.633270
$408$	0	0
$409$	−36.6516	−1.81231	−0.906154	−	0.422949i	$-0.860995\pi$
$409$	−36.6516	−1.81231	−0.906154	+	0.422949i	$0.860995\pi$
$410$	0	0
$411$	−8.23743	−0.406322
$412$	0	0
$413$	−8.00000	−0.393654
$414$	0	0
$415$	0.775746	0.0380799
$416$	0	0
$417$	10.3879	0.508696
$418$	0	0
$419$	7.07381	0.345578	0.172789	−	0.984959i	$-0.444722\pi$
$419$	7.07381	0.345578	0.172789	+	0.984959i	$0.444722\pi$
$420$	0	0
$421$	−12.4485	−0.606703	−0.303352	−	0.952879i	$-0.598106\pi$
$421$	−12.4485	−0.606703	−0.303352	+	0.952879i	$0.598106\pi$
$422$	0	0
$423$	6.70052	0.325791
$424$	0	0
$425$	7.92478	0.384408
$426$	0	0
$427$	−4.70052	−0.227474
$428$	0	0
$429$	10.1768	0.491340
$430$	0	0
$431$	30.7612	1.48171	0.740856	−	0.671663i	$-0.234420\pi$
$431$	30.7612	1.48171	0.740856	+	0.671663i	$0.234420\pi$
$432$	0	0
$433$	−34.1622	−1.64173	−0.820865	−	0.571122i	$-0.806508\pi$
$433$	−34.1622	−1.64173	−0.820865	+	0.571122i	$0.806508\pi$
$434$	0	0
$435$	−7.92478	−0.379964
$436$	0	0
$437$	16.0000	0.765384
$438$	0	0
$439$	3.68735	0.175988	0.0879938	−	0.996121i	$-0.471954\pi$
$439$	3.68735	0.175988	0.0879938	+	0.996121i	$0.471954\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	28.3733	1.34806	0.674028	−	0.738706i	$-0.264563\pi$
$443$	28.3733	1.34806	0.674028	+	0.738706i	$0.264563\pi$
$444$	0	0
$445$	14.6253	0.693306
$446$	0	0
$447$	−16.7005	−0.789908
$448$	0	0
$449$	−11.4010	−0.538049	−0.269024	−	0.963133i	$-0.586701\pi$
$449$	−11.4010	−0.538049	−0.269024	+	0.963133i	$0.586701\pi$
$450$	0	0
$451$	1.97365	0.0929357
$452$	0	0
$453$	−7.22425	−0.339425
$454$	0	0
$455$	−6.31265	−0.295942
$456$	0	0
$457$	26.6253	1.24548	0.622740	−	0.782429i	$-0.286020\pi$
$457$	26.6253	1.24548	0.622740	+	0.782429i	$0.286020\pi$
$458$	0	0
$459$	−7.92478	−0.369897
$460$	0	0
$461$	31.4010	1.46249	0.731246	−	0.682114i	$-0.238939\pi$
$461$	31.4010	1.46249	0.731246	+	0.682114i	$0.238939\pi$
$462$	0	0
$463$	−30.0263	−1.39544	−0.697721	−	0.716369i	$-0.745803\pi$
$463$	−30.0263	−1.39544	−0.697721	+	0.716369i	$0.745803\pi$
$464$	0	0
$465$	9.08840	0.421464
$466$	0	0
$467$	−3.37470	−0.156162	−0.0780812	−	0.996947i	$-0.524879\pi$
$467$	−3.37470	−0.156162	−0.0780812	+	0.996947i	$0.524879\pi$
$468$	0	0
$469$	−9.14903	−0.422463
$470$	0	0
$471$	2.31265	0.106561
$472$	0	0
$473$	−9.55149	−0.439178
$474$	0	0
$475$	−2.38787	−0.109563
$476$	0	0
$477$	3.61213	0.165388
$478$	0	0
$479$	−23.6991	−1.08284	−0.541420	−	0.840752i	$-0.682113\pi$
$479$	−23.6991	−1.08284	−0.541420	+	0.840752i	$0.682113\pi$
$480$	0	0
$481$	50.0263	2.28101
$482$	0	0
$483$	6.70052	0.304884
$484$	0	0
$485$	6.31265	0.286643
$486$	0	0
$487$	−12.7757	−0.578924	−0.289462	−	0.957189i	$-0.593476\pi$
$487$	−12.7757	−0.578924	−0.289462	+	0.957189i	$0.593476\pi$
$488$	0	0
$489$	−2.07522	−0.0938448
$490$	0	0
$491$	38.0870	1.71884	0.859421	−	0.511269i	$-0.170824\pi$
$491$	38.0870	1.71884	0.859421	+	0.511269i	$0.170824\pi$
$492$	0	0
$493$	−62.8021	−2.82846
$494$	0	0
$495$	1.61213	0.0724597
$496$	0	0
$497$	8.31265	0.372873
$498$	0	0
$499$	−8.47486	−0.379387	−0.189693	−	0.981843i	$-0.560749\pi$
$499$	−8.47486	−0.379387	−0.189693	+	0.981843i	$0.560749\pi$
$500$	0	0
$501$	17.9248	0.800820
$502$	0	0
$503$	−40.8773	−1.82263	−0.911315	−	0.411710i	$-0.864932\pi$
$503$	−40.8773	−1.82263	−0.911315	+	0.411710i	$0.864932\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	0	0
$507$	−26.8496	−1.19243
$508$	0	0
$509$	−24.2981	−1.07699	−0.538496	−	0.842628i	$-0.681007\pi$
$509$	−24.2981	−1.07699	−0.538496	+	0.842628i	$0.681007\pi$
$510$	0	0
$511$	10.3127	0.456205
$512$	0	0
$513$	2.38787	0.105427
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−10.8021	−0.475076
$518$	0	0
$519$	1.22425	0.0537388
$520$	0	0
$521$	4.17679	0.182989	0.0914943	−	0.995806i	$-0.470836\pi$
$521$	4.17679	0.182989	0.0914943	+	0.995806i	$0.470836\pi$
$522$	0	0
$523$	−12.6253	−0.552066	−0.276033	−	0.961148i	$-0.589020\pi$
$523$	−12.6253	−0.552066	−0.276033	+	0.961148i	$0.589020\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	72.0235	3.13739
$528$	0	0
$529$	21.8970	0.952044
$530$	0	0
$531$	−8.00000	−0.347170
$532$	0	0
$533$	−7.72829	−0.334749
$534$	0	0
$535$	10.7005	0.462624
$536$	0	0
$537$	17.6121	0.760019
$538$	0	0
$539$	−1.61213	−0.0694392
$540$	0	0
$541$	4.29806	0.184788	0.0923941	−	0.995723i	$-0.470548\pi$
$541$	4.29806	0.184788	0.0923941	+	0.995723i	$0.470548\pi$
$542$	0	0
$543$	−8.07522	−0.346541
$544$	0	0
$545$	9.84955	0.421909
$546$	0	0
$547$	−19.9511	−0.853049	−0.426524	−	0.904476i	$-0.640262\pi$
$547$	−19.9511	−0.853049	−0.426524	+	0.904476i	$0.640262\pi$
$548$	0	0
$549$	−4.70052	−0.200613
$550$	0	0
$551$	18.9234	0.806162
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	7.92478	0.336388
$556$	0	0
$557$	10.0606	0.426283	0.213141	−	0.977021i	$-0.431631\pi$
$557$	10.0606	0.426283	0.213141	+	0.977021i	$0.431631\pi$
$558$	0	0
$559$	37.4010	1.58190
$560$	0	0
$561$	12.7757	0.539392
$562$	0	0
$563$	−13.2506	−0.558446	−0.279223	−	0.960226i	$-0.590077\pi$
$563$	−13.2506	−0.558446	−0.279223	+	0.960226i	$0.590077\pi$
$564$	0	0
$565$	−1.16362	−0.0489538
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−16.2981	−0.683250	−0.341625	−	0.939836i	$-0.610977\pi$
$569$	−16.2981	−0.683250	−0.341625	+	0.939836i	$0.610977\pi$
$570$	0	0
$571$	32.7757	1.37162	0.685811	−	0.727780i	$-0.259448\pi$
$571$	32.7757	1.37162	0.685811	+	0.727780i	$0.259448\pi$
$572$	0	0
$573$	8.31265	0.347266
$574$	0	0
$575$	−6.70052	−0.279431
$576$	0	0
$577$	−8.91160	−0.370995	−0.185497	−	0.982645i	$-0.559390\pi$
$577$	−8.91160	−0.370995	−0.185497	+	0.982645i	$0.559390\pi$
$578$	0	0
$579$	9.84955	0.409334
$580$	0	0
$581$	−0.775746	−0.0321834
$582$	0	0
$583$	−5.82321	−0.241173
$584$	0	0
$585$	−6.31265	−0.260996
$586$	0	0
$587$	−5.25060	−0.216716	−0.108358	−	0.994112i	$-0.534559\pi$
$587$	−5.25060	−0.216716	−0.108358	+	0.994112i	$0.534559\pi$
$588$	0	0
$589$	−21.7019	−0.894213
$590$	0	0
$591$	−20.2374	−0.832457
$592$	0	0
$593$	−22.2228	−0.912583	−0.456291	−	0.889830i	$-0.650823\pi$
$593$	−22.2228	−0.912583	−0.456291	+	0.889830i	$0.650823\pi$
$594$	0	0
$595$	−7.92478	−0.324884
$596$	0	0
$597$	−23.5369	−0.963301
$598$	0	0
$599$	32.9380	1.34581	0.672904	−	0.739730i	$-0.265047\pi$
$599$	32.9380	1.34581	0.672904	+	0.739730i	$0.265047\pi$
$600$	0	0
$601$	17.0738	0.696455	0.348228	−	0.937410i	$-0.386784\pi$
$601$	17.0738	0.696455	0.348228	+	0.937410i	$0.386784\pi$
$602$	0	0
$603$	−9.14903	−0.372577
$604$	0	0
$605$	8.40105	0.341551
$606$	0	0
$607$	−6.32724	−0.256815	−0.128407	−	0.991722i	$-0.540986\pi$
$607$	−6.32724	−0.256815	−0.128407	+	0.991722i	$0.540986\pi$
$608$	0	0
$609$	7.92478	0.321128
$610$	0	0
$611$	42.2981	1.71120
$612$	0	0
$613$	−25.3258	−1.02290	−0.511450	−	0.859313i	$-0.670892\pi$
$613$	−25.3258	−1.02290	−0.511450	+	0.859313i	$0.670892\pi$
$614$	0	0
$615$	−1.22425	−0.0493667
$616$	0	0
$617$	−5.78892	−0.233053	−0.116527	−	0.993188i	$-0.537176\pi$
$617$	−5.78892	−0.233053	−0.116527	+	0.993188i	$0.537176\pi$
$618$	0	0
$619$	−23.7889	−0.956157	−0.478079	−	0.878317i	$-0.658667\pi$
$619$	−23.7889	−0.956157	−0.478079	+	0.878317i	$0.658667\pi$
$620$	0	0
$621$	6.70052	0.268883
$622$	0	0
$623$	−14.6253	−0.585950
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−3.84955	−0.153736
$628$	0	0
$629$	62.8021	2.50408
$630$	0	0
$631$	42.6516	1.69794	0.848968	−	0.528445i	$-0.177225\pi$
$631$	42.6516	1.69794	0.848968	+	0.528445i	$0.177225\pi$
$632$	0	0
$633$	9.40105	0.373658
$634$	0	0
$635$	8.62530	0.342285
$636$	0	0
$637$	6.31265	0.250116
$638$	0	0
$639$	8.31265	0.328843
$640$	0	0
$641$	−16.1768	−0.638945	−0.319472	−	0.947596i	$-0.603506\pi$
$641$	−16.1768	−0.638945	−0.319472	+	0.947596i	$0.603506\pi$
$642$	0	0
$643$	42.0263	1.65736	0.828679	−	0.559725i	$-0.189093\pi$
$643$	42.0263	1.65736	0.828679	+	0.559725i	$0.189093\pi$
$644$	0	0
$645$	5.92478	0.233288
$646$	0	0
$647$	−4.52373	−0.177846	−0.0889231	−	0.996038i	$-0.528343\pi$
$647$	−4.52373	−0.177846	−0.0889231	+	0.996038i	$0.528343\pi$
$648$	0	0
$649$	12.8970	0.506252
$650$	0	0
$651$	−9.08840	−0.356202
$652$	0	0
$653$	24.0870	0.942596	0.471298	−	0.881974i	$-0.343786\pi$
$653$	24.0870	0.942596	0.471298	+	0.881974i	$0.343786\pi$
$654$	0	0
$655$	−8.62530	−0.337018
$656$	0	0
$657$	10.3127	0.402335
$658$	0	0
$659$	13.4617	0.524393	0.262196	−	0.965015i	$-0.415553\pi$
$659$	13.4617	0.524393	0.262196	+	0.965015i	$0.415553\pi$
$660$	0	0
$661$	39.6531	1.54233	0.771163	−	0.636638i	$-0.219675\pi$
$661$	39.6531	1.54233	0.771163	+	0.636638i	$0.219675\pi$
$662$	0	0
$663$	−50.0263	−1.94286
$664$	0	0
$665$	2.38787	0.0925977
$666$	0	0
$667$	53.1002	2.05605
$668$	0	0
$669$	−28.4749	−1.10090
$670$	0	0
$671$	7.57784	0.292539
$672$	0	0
$673$	30.4749	1.17472	0.587360	−	0.809326i	$-0.300167\pi$
$673$	30.4749	1.17472	0.587360	+	0.809326i	$0.300167\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	24.0263	0.923408	0.461704	−	0.887034i	$-0.347238\pi$
$677$	24.0263	0.923408	0.461704	+	0.887034i	$0.347238\pi$
$678$	0	0
$679$	−6.31265	−0.242257
$680$	0	0
$681$	−24.4749	−0.937878
$682$	0	0
$683$	36.8773	1.41107	0.705536	−	0.708674i	$-0.250706\pi$
$683$	36.8773	1.41107	0.705536	+	0.708674i	$0.250706\pi$
$684$	0	0
$685$	−8.23743	−0.314736
$686$	0	0
$687$	−26.9986	−1.03006
$688$	0	0
$689$	22.8021	0.868691
$690$	0	0
$691$	37.1900	1.41477	0.707387	−	0.706827i	$-0.249874\pi$
$691$	37.1900	1.41477	0.707387	+	0.706827i	$0.249874\pi$
$692$	0	0
$693$	−1.61213	−0.0612396
$694$	0	0
$695$	10.3879	0.394034
$696$	0	0
$697$	−9.70194	−0.367487
$698$	0	0
$699$	−11.4617	−0.433521
$700$	0	0
$701$	−14.3733	−0.542871	−0.271436	−	0.962457i	$-0.587498\pi$
$701$	−14.3733	−0.542871	−0.271436	+	0.962457i	$0.587498\pi$
$702$	0	0
$703$	−18.9234	−0.713708
$704$	0	0
$705$	6.70052	0.252356
$706$	0	0
$707$	10.0000	0.376089
$708$	0	0
$709$	−18.1504	−0.681654	−0.340827	−	0.940126i	$-0.610707\pi$
$709$	−18.1504	−0.681654	−0.340827	+	0.940126i	$0.610707\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	−60.8970	−2.28061
$714$	0	0
$715$	10.1768	0.380590
$716$	0	0
$717$	−8.31265	−0.310442
$718$	0	0
$719$	−20.7757	−0.774805	−0.387402	−	0.921911i	$-0.626628\pi$
$719$	−20.7757	−0.774805	−0.387402	+	0.921911i	$0.626628\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−5.22425	−0.194292
$724$	0	0
$725$	−7.92478	−0.294319
$726$	0	0
$727$	−23.0738	−0.855760	−0.427880	−	0.903836i	$-0.640739\pi$
$727$	−23.0738	−0.855760	−0.427880	+	0.903836i	$0.640739\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	46.9525	1.73660
$732$	0	0
$733$	−50.9380	−1.88144	−0.940718	−	0.339189i	$-0.889847\pi$
$733$	−50.9380	−1.88144	−0.940718	+	0.339189i	$0.889847\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	14.7494	0.543301
$738$	0	0
$739$	28.6253	1.05300	0.526499	−	0.850176i	$-0.323504\pi$
$739$	28.6253	1.05300	0.526499	+	0.850176i	$0.323504\pi$
$740$	0	0
$741$	15.0738	0.553750
$742$	0	0
$743$	−41.9248	−1.53807	−0.769035	−	0.639206i	$-0.779263\pi$
$743$	−41.9248	−1.53807	−0.769035	+	0.639206i	$0.779263\pi$
$744$	0	0
$745$	−16.7005	−0.611860
$746$	0	0
$747$	−0.775746	−0.0283831
$748$	0	0
$749$	−10.7005	−0.390989
$750$	0	0
$751$	27.6991	1.01075	0.505377	−	0.862898i	$-0.331353\pi$
$751$	27.6991	1.01075	0.505377	+	0.862898i	$0.331353\pi$
$752$	0	0
$753$	11.8496	0.431822
$754$	0	0
$755$	−7.22425	−0.262918
$756$	0	0
$757$	5.74798	0.208914	0.104457	−	0.994529i	$-0.466690\pi$
$757$	5.74798	0.208914	0.104457	+	0.994529i	$0.466690\pi$
$758$	0	0
$759$	−10.8021	−0.392091
$760$	0	0
$761$	3.67276	0.133137	0.0665687	−	0.997782i	$-0.478795\pi$
$761$	3.67276	0.133137	0.0665687	+	0.997782i	$0.478795\pi$
$762$	0	0
$763$	−9.84955	−0.356578
$764$	0	0
$765$	−7.92478	−0.286521
$766$	0	0
$767$	−50.5012	−1.82349
$768$	0	0
$769$	−24.9234	−0.898759	−0.449380	−	0.893341i	$-0.648355\pi$
$769$	−24.9234	−0.898759	−0.449380	+	0.893341i	$0.648355\pi$
$770$	0	0
$771$	−0.850969	−0.0306469
$772$	0	0
$773$	5.72829	0.206032	0.103016	−	0.994680i	$-0.467151\pi$
$773$	5.72829	0.206032	0.103016	+	0.994680i	$0.467151\pi$
$774$	0	0
$775$	9.08840	0.326465
$776$	0	0
$777$	−7.92478	−0.284300
$778$	0	0
$779$	2.92336	0.104740
$780$	0	0
$781$	−13.4010	−0.479527
$782$	0	0
$783$	7.92478	0.283208
$784$	0	0
$785$	2.31265	0.0825420
$786$	0	0
$787$	48.4749	1.72794	0.863971	−	0.503542i	$-0.167970\pi$
$787$	48.4749	1.72794	0.863971	+	0.503542i	$0.167970\pi$
$788$	0	0
$789$	−24.8773	−0.885656
$790$	0	0
$791$	1.16362	0.0413735
$792$	0	0
$793$	−29.6728	−1.05371
$794$	0	0
$795$	3.61213	0.128109
$796$	0	0
$797$	−1.22425	−0.0433653	−0.0216826	−	0.999765i	$-0.506902\pi$
$797$	−1.22425	−0.0433653	−0.0216826	+	0.999765i	$0.506902\pi$
$798$	0	0
$799$	53.1002	1.87855
$800$	0	0
$801$	−14.6253	−0.516760
$802$	0	0
$803$	−16.6253	−0.586694
$804$	0	0
$805$	6.70052	0.236162
$806$	0	0
$807$	−28.2981	−0.996139
$808$	0	0
$809$	5.72829	0.201396	0.100698	−	0.994917i	$-0.467892\pi$
$809$	5.72829	0.201396	0.100698	+	0.994917i	$0.467892\pi$
$810$	0	0
$811$	−29.1900	−1.02500	−0.512499	−	0.858688i	$-0.671280\pi$
$811$	−29.1900	−1.02500	−0.512499	+	0.858688i	$0.671280\pi$
$812$	0	0
$813$	1.08840	0.0381717
$814$	0	0
$815$	−2.07522	−0.0726919
$816$	0	0
$817$	−14.1476	−0.494962
$818$	0	0
$819$	6.31265	0.220582
$820$	0	0
$821$	27.6239	0.964080	0.482040	−	0.876149i	$-0.339896\pi$
$821$	27.6239	0.964080	0.482040	+	0.876149i	$0.339896\pi$
$822$	0	0
$823$	−22.3272	−0.778279	−0.389139	−	0.921179i	$-0.627228\pi$
$823$	−22.3272	−0.778279	−0.389139	+	0.921179i	$0.627228\pi$
$824$	0	0
$825$	1.61213	0.0561271
$826$	0	0
$827$	−16.1016	−0.559906	−0.279953	−	0.960014i	$-0.590319\pi$
$827$	−16.1016	−0.559906	−0.279953	+	0.960014i	$0.590319\pi$
$828$	0	0
$829$	−42.2228	−1.46646	−0.733230	−	0.679981i	$-0.761988\pi$
$829$	−42.2228	−1.46646	−0.733230	+	0.679981i	$0.761988\pi$
$830$	0	0
$831$	−12.0752	−0.418885
$832$	0	0
$833$	7.92478	0.274577
$834$	0	0
$835$	17.9248	0.620312
$836$	0	0
$837$	−9.08840	−0.314141
$838$	0	0
$839$	−28.7757	−0.993449	−0.496725	−	0.867908i	$-0.665464\pi$
$839$	−28.7757	−0.993449	−0.496725	+	0.867908i	$0.665464\pi$
$840$	0	0
$841$	33.8021	1.16559
$842$	0	0
$843$	1.22425	0.0421655
$844$	0	0
$845$	−26.8496	−0.923653
$846$	0	0
$847$	−8.40105	−0.288663
$848$	0	0
$849$	−5.55149	−0.190527
$850$	0	0
$851$	−53.1002	−1.82025
$852$	0	0
$853$	43.7137	1.49673	0.748364	−	0.663288i	$-0.230840\pi$
$853$	43.7137	1.49673	0.748364	+	0.663288i	$0.230840\pi$
$854$	0	0
$855$	2.38787	0.0816635
$856$	0	0
$857$	10.1016	0.345063	0.172532	−	0.985004i	$-0.444805\pi$
$857$	10.1016	0.345063	0.172532	+	0.985004i	$0.444805\pi$
$858$	0	0
$859$	−35.0132	−1.19463	−0.597317	−	0.802005i	$-0.703767\pi$
$859$	−35.0132	−1.19463	−0.597317	+	0.802005i	$0.703767\pi$
$860$	0	0
$861$	1.22425	0.0417225
$862$	0	0
$863$	−32.9986	−1.12328	−0.561642	−	0.827380i	$-0.689830\pi$
$863$	−32.9986	−1.12328	−0.561642	+	0.827380i	$0.689830\pi$
$864$	0	0
$865$	1.22425	0.0416259
$866$	0	0
$867$	−45.8021	−1.55552
$868$	0	0
$869$	−6.44851	−0.218751
$870$	0	0
$871$	−57.7546	−1.95694
$872$	0	0
$873$	−6.31265	−0.213651
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−2.37328	−0.0801401	−0.0400701	−	0.999197i	$-0.512758\pi$
$877$	−2.37328	−0.0801401	−0.0400701	+	0.999197i	$0.512758\pi$
$878$	0	0
$879$	−1.37470	−0.0463675
$880$	0	0
$881$	1.07381	0.0361775	0.0180888	−	0.999836i	$-0.494242\pi$
$881$	1.07381	0.0361775	0.0180888	+	0.999836i	$0.494242\pi$
$882$	0	0
$883$	35.9511	1.20985	0.604926	−	0.796282i	$-0.293203\pi$
$883$	35.9511	1.20985	0.604926	+	0.796282i	$0.293203\pi$
$884$	0	0
$885$	−8.00000	−0.268917
$886$	0	0
$887$	−54.3996	−1.82656	−0.913280	−	0.407331i	$-0.866459\pi$
$887$	−54.3996	−1.82656	−0.913280	+	0.407331i	$0.866459\pi$
$888$	0	0
$889$	−8.62530	−0.289283
$890$	0	0
$891$	−1.61213	−0.0540083
$892$	0	0
$893$	−16.0000	−0.535420
$894$	0	0
$895$	17.6121	0.588708
$896$	0	0
$897$	42.2981	1.41229
$898$	0	0
$899$	−72.0235	−2.40212
$900$	0	0
$901$	28.6253	0.953647
$902$	0	0
$903$	−5.92478	−0.197164
$904$	0	0
$905$	−8.07522	−0.268429
$906$	0	0
$907$	1.44992	0.0481439	0.0240719	−	0.999710i	$-0.492337\pi$
$907$	1.44992	0.0481439	0.0240719	+	0.999710i	$0.492337\pi$
$908$	0	0
$909$	10.0000	0.331679
$910$	0	0
$911$	−49.5633	−1.64210	−0.821052	−	0.570854i	$-0.806612\pi$
$911$	−49.5633	−1.64210	−0.821052	+	0.570854i	$0.806612\pi$
$912$	0	0
$913$	1.25060	0.0413889
$914$	0	0
$915$	−4.70052	−0.155395
$916$	0	0
$917$	8.62530	0.284833
$918$	0	0
$919$	34.4485	1.13635	0.568176	−	0.822907i	$-0.307649\pi$
$919$	34.4485	1.13635	0.568176	+	0.822907i	$0.307649\pi$
$920$	0	0
$921$	11.0738	0.364894
$922$	0	0
$923$	52.4749	1.72723
$924$	0	0
$925$	7.92478	0.260565
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−4.32724	−0.141972	−0.0709860	−	0.997477i	$-0.522615\pi$
$929$	−4.32724	−0.141972	−0.0709860	+	0.997477i	$0.522615\pi$
$930$	0	0
$931$	−2.38787	−0.0782594
$932$	0	0
$933$	6.44851	0.211115
$934$	0	0
$935$	12.7757	0.417812
$936$	0	0
$937$	−49.5369	−1.61830	−0.809150	−	0.587602i	$-0.800072\pi$
$937$	−49.5369	−1.61830	−0.809150	+	0.587602i	$0.800072\pi$
$938$	0	0
$939$	−23.0884	−0.753461
$940$	0	0
$941$	−48.5012	−1.58109	−0.790547	−	0.612401i	$-0.790204\pi$
$941$	−48.5012	−1.58109	−0.790547	+	0.612401i	$0.790204\pi$
$942$	0	0
$943$	8.20314	0.267131
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	2.57925	0.0838145	0.0419073	−	0.999122i	$-0.486657\pi$
$947$	2.57925	0.0838145	0.0419073	+	0.999122i	$0.486657\pi$
$948$	0	0
$949$	65.1002	2.11324
$950$	0	0
$951$	21.6385	0.701676
$952$	0	0
$953$	−17.9394	−0.581113	−0.290557	−	0.956858i	$-0.593840\pi$
$953$	−17.9394	−0.581113	−0.290557	+	0.956858i	$0.593840\pi$
$954$	0	0
$955$	8.31265	0.268991
$956$	0	0
$957$	−12.7757	−0.412981
$958$	0	0
$959$	8.23743	0.266000
$960$	0	0
$961$	51.5990	1.66448
$962$	0	0
$963$	−10.7005	−0.344820
$964$	0	0
$965$	9.84955	0.317068
$966$	0	0
$967$	30.6516	0.985691	0.492845	−	0.870117i	$-0.335957\pi$
$967$	30.6516	0.985691	0.492845	+	0.870117i	$0.335957\pi$
$968$	0	0
$969$	18.9234	0.607906
$970$	0	0
$971$	−28.3536	−0.909910	−0.454955	−	0.890514i	$-0.650345\pi$
$971$	−28.3536	−0.909910	−0.454955	+	0.890514i	$0.650345\pi$
$972$	0	0
$973$	−10.3879	−0.333020
$974$	0	0
$975$	−6.31265	−0.202167
$976$	0	0
$977$	−2.68594	−0.0859307	−0.0429653	−	0.999077i	$-0.513681\pi$
$977$	−2.68594	−0.0859307	−0.0429653	+	0.999077i	$0.513681\pi$
$978$	0	0
$979$	23.5778	0.753551
$980$	0	0
$981$	−9.84955	−0.314472
$982$	0	0
$983$	43.8007	1.39702	0.698512	−	0.715598i	$-0.253846\pi$
$983$	43.8007	1.39702	0.698512	+	0.715598i	$0.253846\pi$
$984$	0	0
$985$	−20.2374	−0.644818
$986$	0	0
$987$	−6.70052	−0.213280
$988$	0	0
$989$	−39.6991	−1.26236
$990$	0	0
$991$	21.5515	0.684606	0.342303	−	0.939590i	$-0.388793\pi$
$991$	21.5515	0.684606	0.342303	+	0.939590i	$0.388793\pi$
$992$	0	0
$993$	−33.4010	−1.05995
$994$	0	0
$995$	−23.5369	−0.746170
$996$	0	0
$997$	26.1622	0.828565	0.414283	−	0.910148i	$-0.364032\pi$
$997$	26.1622	0.828565	0.414283	+	0.910148i	$0.364032\pi$
$998$	0	0
$999$	−7.92478	−0.250729

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6720.2.a.da.1.2		3
4.3	odd	2		6720.2.a.db.1.2		3
8.3	odd	2		3360.2.a.bi.1.2	✓	3
8.5	even	2		3360.2.a.bj.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3360.2.a.bi.1.2	✓	3	8.3	odd	2
3360.2.a.bj.1.2	yes	3	8.5	even	2
6720.2.a.da.1.2		3	1.1	even	1	trivial
6720.2.a.db.1.2		3	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.61213 q^{11} +6.31265 q^{13} +1.00000 q^{15} +7.92478 q^{17} -2.38787 q^{19} -1.00000 q^{21} -6.70052 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.92478 q^{29} +9.08840 q^{31} +1.61213 q^{33} -1.00000 q^{35} +7.92478 q^{37} -6.31265 q^{39} -1.22425 q^{41} +5.92478 q^{43} -1.00000 q^{45} +6.70052 q^{47} +1.00000 q^{49} -7.92478 q^{51} +3.61213 q^{53} +1.61213 q^{55} +2.38787 q^{57} -8.00000 q^{59} -4.70052 q^{61} +1.00000 q^{63} -6.31265 q^{65} -9.14903 q^{67} +6.70052 q^{69} +8.31265 q^{71} +10.3127 q^{73} -1.00000 q^{75} -1.61213 q^{77} +4.00000 q^{79} +1.00000 q^{81} -0.775746 q^{83} -7.92478 q^{85} +7.92478 q^{87} -14.6253 q^{89} +6.31265 q^{91} -9.08840 q^{93} +2.38787 q^{95} -6.31265 q^{97} -1.61213 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} - 4 q^{11} - 2 q^{13} + 3 q^{15} + 2 q^{17} - 8 q^{19} - 3 q^{21} + 3 q^{25} - 3 q^{27} - 2 q^{29} + 8 q^{31} + 4 q^{33} - 3 q^{35} + 2 q^{37} + 2 q^{39} - 2 q^{41} - 4 q^{43} - 3 q^{45} + 3 q^{49} - 2 q^{51} + 10 q^{53} + 4 q^{55} + 8 q^{57} - 24 q^{59} + 6 q^{61} + 3 q^{63} + 2 q^{65} - 4 q^{67} + 4 q^{71} + 10 q^{73} - 3 q^{75} - 4 q^{77} + 12 q^{79} + 3 q^{81} - 4 q^{83} - 2 q^{85} + 2 q^{87} - 2 q^{89} - 2 q^{91} - 8 q^{93} + 8 q^{95} + 2 q^{97} - 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 + 3 * q^7 + 3 * q^9 - 4 * q^11 - 2 * q^13 + 3 * q^15 + 2 * q^17 - 8 * q^19 - 3 * q^21 + 3 * q^25 - 3 * q^27 - 2 * q^29 + 8 * q^31 + 4 * q^33 - 3 * q^35 + 2 * q^37 + 2 * q^39 - 2 * q^41 - 4 * q^43 - 3 * q^45 + 3 * q^49 - 2 * q^51 + 10 * q^53 + 4 * q^55 + 8 * q^57 - 24 * q^59 + 6 * q^61 + 3 * q^63 + 2 * q^65 - 4 * q^67 + 4 * q^71 + 10 * q^73 - 3 * q^75 - 4 * q^77 + 12 * q^79 + 3 * q^81 - 4 * q^83 - 2 * q^85 + 2 * q^87 - 2 * q^89 - 2 * q^91 - 8 * q^93 + 8 * q^95 + 2 * q^97 - 4 * q^99

Introduction

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