LMFDB - Embedded newform 6720.2.a.cw.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6720.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$53.6594701583$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3360)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.12311 q^{11} +3.12311 q^{13} +1.00000 q^{15} +2.00000 q^{17} -1.12311 q^{19} -1.00000 q^{21} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{29} -5.12311 q^{31} +5.12311 q^{33} -1.00000 q^{35} +2.00000 q^{37} +3.12311 q^{39} +2.00000 q^{41} +10.2462 q^{43} +1.00000 q^{45} +1.00000 q^{49} +2.00000 q^{51} -13.3693 q^{53} +5.12311 q^{55} -1.12311 q^{57} -4.00000 q^{59} +8.24621 q^{61} -1.00000 q^{63} +3.12311 q^{65} -2.24621 q^{67} +5.12311 q^{71} +15.1231 q^{73} +1.00000 q^{75} -5.12311 q^{77} +2.24621 q^{79} +1.00000 q^{81} -4.00000 q^{83} +2.00000 q^{85} +2.00000 q^{87} -0.246211 q^{89} -3.12311 q^{91} -5.12311 q^{93} -1.12311 q^{95} +4.87689 q^{97} +5.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{11} - 2 q^{13} + 2 q^{15} + 4 q^{17} + 6 q^{19} - 2 q^{21} + 2 q^{25} + 2 q^{27} + 4 q^{29} - 2 q^{31} + 2 q^{33} - 2 q^{35} + 4 q^{37} - 2 q^{39} + 4 q^{41} + 4 q^{43} + 2 q^{45} + 2 q^{49} + 4 q^{51} - 2 q^{53} + 2 q^{55} + 6 q^{57} - 8 q^{59} - 2 q^{63} - 2 q^{65} + 12 q^{67} + 2 q^{71} + 22 q^{73} + 2 q^{75} - 2 q^{77} - 12 q^{79} + 2 q^{81} - 8 q^{83} + 4 q^{85} + 4 q^{87} + 16 q^{89} + 2 q^{91} - 2 q^{93} + 6 q^{95} + 18 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 - 2 * q^13 + 2 * q^15 + 4 * q^17 + 6 * q^19 - 2 * q^21 + 2 * q^25 + 2 * q^27 + 4 * q^29 - 2 * q^31 + 2 * q^33 - 2 * q^35 + 4 * q^37 - 2 * q^39 + 4 * q^41 + 4 * q^43 + 2 * q^45 + 2 * q^49 + 4 * q^51 - 2 * q^53 + 2 * q^55 + 6 * q^57 - 8 * q^59 - 2 * q^63 - 2 * q^65 + 12 * q^67 + 2 * q^71 + 22 * q^73 + 2 * q^75 - 2 * q^77 - 12 * q^79 + 2 * q^81 - 8 * q^83 + 4 * q^85 + 4 * q^87 + 16 * q^89 + 2 * q^91 - 2 * q^93 + 6 * q^95 + 18 * q^97 + 2 * 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
$11$	5.12311	1.54467	0.772337	−	0.635213i	$-0.219088\pi$
$11$	5.12311	1.54467	0.772337	+	0.635213i	$0.219088\pi$
$12$	0	0
$13$	3.12311	0.866194	0.433097	−	0.901347i	$-0.357421\pi$
$13$	3.12311	0.866194	0.433097	+	0.901347i	$0.357421\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−1.12311	−0.257658	−0.128829	−	0.991667i	$-0.541122\pi$
$19$	−1.12311	−0.257658	−0.128829	+	0.991667i	$0.541122\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−5.12311	−0.920137	−0.460068	−	0.887883i	$-0.652175\pi$
$31$	−5.12311	−0.920137	−0.460068	+	0.887883i	$0.652175\pi$
$32$	0	0
$33$	5.12311	0.891818
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	3.12311	0.500097
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	10.2462	1.56253	0.781266	−	0.624198i	$-0.214574\pi$
$43$	10.2462	1.56253	0.781266	+	0.624198i	$0.214574\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	−13.3693	−1.83642	−0.918208	−	0.396098i	$-0.870364\pi$
$53$	−13.3693	−1.83642	−0.918208	+	0.396098i	$0.870364\pi$
$54$	0	0
$55$	5.12311	0.690799
$56$	0	0
$57$	−1.12311	−0.148759
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	8.24621	1.05582	0.527910	−	0.849301i	$-0.322976\pi$
$61$	8.24621	1.05582	0.527910	+	0.849301i	$0.322976\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	3.12311	0.387374
$66$	0	0
$67$	−2.24621	−0.274418	−0.137209	−	0.990542i	$-0.543813\pi$
$67$	−2.24621	−0.274418	−0.137209	+	0.990542i	$0.543813\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.12311	0.608001	0.304000	−	0.952672i	$-0.401678\pi$
$71$	5.12311	0.608001	0.304000	+	0.952672i	$0.401678\pi$
$72$	0	0
$73$	15.1231	1.77003	0.885013	−	0.465567i	$-0.154149\pi$
$73$	15.1231	1.77003	0.885013	+	0.465567i	$0.154149\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−5.12311	−0.583832
$78$	0	0
$79$	2.24621	0.252719	0.126359	−	0.991985i	$-0.459671\pi$
$79$	2.24621	0.252719	0.126359	+	0.991985i	$0.459671\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	2.00000	0.216930
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	−0.246211	−0.0260983	−0.0130492	−	0.999915i	$-0.504154\pi$
$89$	−0.246211	−0.0260983	−0.0130492	+	0.999915i	$0.504154\pi$
$90$	0	0
$91$	−3.12311	−0.327390
$92$	0	0
$93$	−5.12311	−0.531241
$94$	0	0
$95$	−1.12311	−0.115228
$96$	0	0
$97$	4.87689	0.495174	0.247587	−	0.968866i	$-0.420362\pi$
$97$	4.87689	0.495174	0.247587	+	0.968866i	$0.420362\pi$
$98$	0	0
$99$	5.12311	0.514891
$100$	0	0
$101$	0.246211	0.0244989	0.0122495	−	0.999925i	$-0.496101\pi$
$101$	0.246211	0.0244989	0.0122495	+	0.999925i	$0.496101\pi$
$102$	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
$107$	−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
$109$	−8.24621	−0.789844	−0.394922	−	0.918715i	$-0.629228\pi$
$109$	−8.24621	−0.789844	−0.394922	+	0.918715i	$0.629228\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	−5.36932	−0.505103	−0.252551	−	0.967583i	$-0.581270\pi$
$113$	−5.36932	−0.505103	−0.252551	+	0.967583i	$0.581270\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.12311	0.288731
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	15.2462	1.38602
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−20.4924	−1.81841	−0.909204	−	0.416350i	$-0.863309\pi$
$127$	−20.4924	−1.81841	−0.909204	+	0.416350i	$0.863309\pi$
$128$	0	0
$129$	10.2462	0.902129
$130$	0	0
$131$	1.75379	0.153229	0.0766146	−	0.997061i	$-0.475589\pi$
$131$	1.75379	0.153229	0.0766146	+	0.997061i	$0.475589\pi$
$132$	0	0
$133$	1.12311	0.0973856
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	12.8769	1.10015	0.550074	−	0.835116i	$-0.314600\pi$
$137$	12.8769	1.10015	0.550074	+	0.835116i	$0.314600\pi$
$138$	0	0
$139$	−9.12311	−0.773812	−0.386906	−	0.922119i	$-0.626456\pi$
$139$	−9.12311	−0.773812	−0.386906	+	0.922119i	$0.626456\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	16.0000	1.33799
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	12.2462	1.00325	0.501624	−	0.865086i	$-0.332736\pi$
$149$	12.2462	1.00325	0.501624	+	0.865086i	$0.332736\pi$
$150$	0	0
$151$	−20.4924	−1.66765	−0.833825	−	0.552029i	$-0.813854\pi$
$151$	−20.4924	−1.66765	−0.833825	+	0.552029i	$0.813854\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	−5.12311	−0.411498
$156$	0	0
$157$	−7.12311	−0.568486	−0.284243	−	0.958752i	$-0.591742\pi$
$157$	−7.12311	−0.568486	−0.284243	+	0.958752i	$0.591742\pi$
$158$	0	0
$159$	−13.3693	−1.06026
$160$	0	0
$161$	0	0
$162$	0	0
$163$	10.2462	0.802545	0.401273	−	0.915959i	$-0.368568\pi$
$163$	10.2462	0.802545	0.401273	+	0.915959i	$0.368568\pi$
$164$	0	0
$165$	5.12311	0.398833
$166$	0	0
$167$	16.0000	1.23812	0.619059	−	0.785345i	$-0.287514\pi$
$167$	16.0000	1.23812	0.619059	+	0.785345i	$0.287514\pi$
$168$	0	0
$169$	−3.24621	−0.249709
$170$	0	0
$171$	−1.12311	−0.0858860
$172$	0	0
$173$	0.246211	0.0187191	0.00935955	−	0.999956i	$-0.497021\pi$
$173$	0.246211	0.0187191	0.00935955	+	0.999956i	$0.497021\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−2.87689	−0.215029	−0.107515	−	0.994204i	$-0.534289\pi$
$179$	−2.87689	−0.215029	−0.107515	+	0.994204i	$0.534289\pi$
$180$	0	0
$181$	−22.4924	−1.67185	−0.835924	−	0.548845i	$-0.815068\pi$
$181$	−22.4924	−1.67185	−0.835924	+	0.548845i	$0.815068\pi$
$182$	0	0
$183$	8.24621	0.609577
$184$	0	0
$185$	2.00000	0.147043
$186$	0	0
$187$	10.2462	0.749277
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−5.12311	−0.370695	−0.185347	−	0.982673i	$-0.559341\pi$
$191$	−5.12311	−0.370695	−0.185347	+	0.982673i	$0.559341\pi$
$192$	0	0
$193$	18.0000	1.29567	0.647834	−	0.761781i	$-0.275675\pi$
$193$	18.0000	1.29567	0.647834	+	0.761781i	$0.275675\pi$
$194$	0	0
$195$	3.12311	0.223650
$196$	0	0
$197$	−23.6155	−1.68254	−0.841268	−	0.540618i	$-0.818191\pi$
$197$	−23.6155	−1.68254	−0.841268	+	0.540618i	$0.818191\pi$
$198$	0	0
$199$	−5.12311	−0.363167	−0.181584	−	0.983375i	$-0.558122\pi$
$199$	−5.12311	−0.363167	−0.181584	+	0.983375i	$0.558122\pi$
$200$	0	0
$201$	−2.24621	−0.158436
$202$	0	0
$203$	−2.00000	−0.140372
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.75379	−0.397998
$210$	0	0
$211$	5.75379	0.396107	0.198054	−	0.980191i	$-0.436538\pi$
$211$	5.75379	0.396107	0.198054	+	0.980191i	$0.436538\pi$
$212$	0	0
$213$	5.12311	0.351029
$214$	0	0
$215$	10.2462	0.698786
$216$	0	0
$217$	5.12311	0.347779
$218$	0	0
$219$	15.1231	1.02192
$220$	0	0
$221$	6.24621	0.420166
$222$	0	0
$223$	18.2462	1.22186	0.610928	−	0.791686i	$-0.290796\pi$
$223$	18.2462	1.22186	0.610928	+	0.791686i	$0.290796\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	3.75379	0.248057	0.124029	−	0.992279i	$-0.460419\pi$
$229$	3.75379	0.248057	0.124029	+	0.992279i	$0.460419\pi$
$230$	0	0
$231$	−5.12311	−0.337076
$232$	0	0
$233$	23.1231	1.51485	0.757423	−	0.652925i	$-0.226458\pi$
$233$	23.1231	1.51485	0.757423	+	0.652925i	$0.226458\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	2.24621	0.145907
$238$	0	0
$239$	13.1231	0.848863	0.424432	−	0.905460i	$-0.360474\pi$
$239$	13.1231	0.848863	0.424432	+	0.905460i	$0.360474\pi$
$240$	0	0
$241$	20.2462	1.30417	0.652087	−	0.758145i	$-0.273894\pi$
$241$	20.2462	1.30417	0.652087	+	0.758145i	$0.273894\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−3.50758	−0.223182
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	2.00000	0.125245
$256$	0	0
$257$	20.2462	1.26292	0.631462	−	0.775407i	$-0.282455\pi$
$257$	20.2462	1.26292	0.631462	+	0.775407i	$0.282455\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	2.00000	0.123797
$262$	0	0
$263$	18.2462	1.12511	0.562555	−	0.826760i	$-0.309819\pi$
$263$	18.2462	1.12511	0.562555	+	0.826760i	$0.309819\pi$
$264$	0	0
$265$	−13.3693	−0.821271
$266$	0	0
$267$	−0.246211	−0.0150679
$268$	0	0
$269$	16.2462	0.990549	0.495274	−	0.868737i	$-0.335067\pi$
$269$	16.2462	0.990549	0.495274	+	0.868737i	$0.335067\pi$
$270$	0	0
$271$	17.6155	1.07007	0.535034	−	0.844831i	$-0.320299\pi$
$271$	17.6155	1.07007	0.535034	+	0.844831i	$0.320299\pi$
$272$	0	0
$273$	−3.12311	−0.189019
$274$	0	0
$275$	5.12311	0.308935
$276$	0	0
$277$	14.4924	0.870765	0.435383	−	0.900245i	$-0.356613\pi$
$277$	14.4924	0.870765	0.435383	+	0.900245i	$0.356613\pi$
$278$	0	0
$279$	−5.12311	−0.306712
$280$	0	0
$281$	24.7386	1.47578	0.737892	−	0.674919i	$-0.235822\pi$
$281$	24.7386	1.47578	0.737892	+	0.674919i	$0.235822\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	0	0
$285$	−1.12311	−0.0665270
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	4.87689	0.285889
$292$	0	0
$293$	16.2462	0.949114	0.474557	−	0.880225i	$-0.342608\pi$
$293$	16.2462	0.949114	0.474557	+	0.880225i	$0.342608\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	5.12311	0.297273
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−10.2462	−0.590582
$302$	0	0
$303$	0.246211	0.0141445
$304$	0	0
$305$	8.24621	0.472177
$306$	0	0
$307$	−9.75379	−0.556678	−0.278339	−	0.960483i	$-0.589784\pi$
$307$	−9.75379	−0.556678	−0.278339	+	0.960483i	$0.589784\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	33.3693	1.88615	0.943073	−	0.332587i	$-0.107921\pi$
$313$	33.3693	1.88615	0.943073	+	0.332587i	$0.107921\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	15.1231	0.849398	0.424699	−	0.905335i	$-0.360380\pi$
$317$	15.1231	0.849398	0.424699	+	0.905335i	$0.360380\pi$
$318$	0	0
$319$	10.2462	0.573678
$320$	0	0
$321$	−8.00000	−0.446516
$322$	0	0
$323$	−2.24621	−0.124983
$324$	0	0
$325$	3.12311	0.173239
$326$	0	0
$327$	−8.24621	−0.456017
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.24621	0.123463	0.0617315	−	0.998093i	$-0.480338\pi$
$331$	2.24621	0.123463	0.0617315	+	0.998093i	$0.480338\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	−2.24621	−0.122724
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	−5.36932	−0.291621
$340$	0	0
$341$	−26.2462	−1.42131
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−26.2462	−1.40897	−0.704485	−	0.709719i	$-0.748822\pi$
$347$	−26.2462	−1.40897	−0.704485	+	0.709719i	$0.748822\pi$
$348$	0	0
$349$	11.7538	0.629166	0.314583	−	0.949230i	$-0.398135\pi$
$349$	11.7538	0.629166	0.314583	+	0.949230i	$0.398135\pi$
$350$	0	0
$351$	3.12311	0.166699
$352$	0	0
$353$	−10.4924	−0.558455	−0.279228	−	0.960225i	$-0.590078\pi$
$353$	−10.4924	−0.558455	−0.279228	+	0.960225i	$0.590078\pi$
$354$	0	0
$355$	5.12311	0.271906
$356$	0	0
$357$	−2.00000	−0.105851
$358$	0	0
$359$	−33.6155	−1.77416	−0.887080	−	0.461616i	$-0.847270\pi$
$359$	−33.6155	−1.77416	−0.887080	+	0.461616i	$0.847270\pi$
$360$	0	0
$361$	−17.7386	−0.933612
$362$	0	0
$363$	15.2462	0.800219
$364$	0	0
$365$	15.1231	0.791580
$366$	0	0
$367$	−13.7538	−0.717942	−0.358971	−	0.933349i	$-0.616872\pi$
$367$	−13.7538	−0.717942	−0.358971	+	0.933349i	$0.616872\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	13.3693	0.694100
$372$	0	0
$373$	20.2462	1.04831	0.524155	−	0.851623i	$-0.324381\pi$
$373$	20.2462	1.04831	0.524155	+	0.851623i	$0.324381\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	6.24621	0.321696
$378$	0	0
$379$	12.4924	0.641693	0.320846	−	0.947131i	$-0.396033\pi$
$379$	12.4924	0.641693	0.320846	+	0.947131i	$0.396033\pi$
$380$	0	0
$381$	−20.4924	−1.04986
$382$	0	0
$383$	−4.49242	−0.229552	−0.114776	−	0.993391i	$-0.536615\pi$
$383$	−4.49242	−0.229552	−0.114776	+	0.993391i	$0.536615\pi$
$384$	0	0
$385$	−5.12311	−0.261098
$386$	0	0
$387$	10.2462	0.520844
$388$	0	0
$389$	20.2462	1.02652	0.513262	−	0.858232i	$-0.328437\pi$
$389$	20.2462	1.02652	0.513262	+	0.858232i	$0.328437\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	1.75379	0.0884669
$394$	0	0
$395$	2.24621	0.113019
$396$	0	0
$397$	19.1231	0.959761	0.479881	−	0.877334i	$-0.340680\pi$
$397$	19.1231	0.959761	0.479881	+	0.877334i	$0.340680\pi$
$398$	0	0
$399$	1.12311	0.0562256
$400$	0	0
$401$	−38.9848	−1.94681	−0.973405	−	0.229090i	$-0.926425\pi$
$401$	−38.9848	−1.94681	−0.973405	+	0.229090i	$0.926425\pi$
$402$	0	0
$403$	−16.0000	−0.797017
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	10.2462	0.507886
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	12.8769	0.635170
$412$	0	0
$413$	4.00000	0.196827
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	−9.12311	−0.446760
$418$	0	0
$419$	−17.7538	−0.867329	−0.433665	−	0.901074i	$-0.642780\pi$
$419$	−17.7538	−0.867329	−0.433665	+	0.901074i	$0.642780\pi$
$420$	0	0
$421$	6.49242	0.316421	0.158211	−	0.987405i	$-0.449427\pi$
$421$	6.49242	0.316421	0.158211	+	0.987405i	$0.449427\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	−8.24621	−0.399062
$428$	0	0
$429$	16.0000	0.772487
$430$	0	0
$431$	33.6155	1.61920	0.809602	−	0.586980i	$-0.199683\pi$
$431$	33.6155	1.61920	0.809602	+	0.586980i	$0.199683\pi$
$432$	0	0
$433$	4.87689	0.234369	0.117184	−	0.993110i	$-0.462613\pi$
$433$	4.87689	0.234369	0.117184	+	0.993110i	$0.462613\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−33.6155	−1.60438	−0.802191	−	0.597068i	$-0.796332\pi$
$439$	−33.6155	−1.60438	−0.802191	+	0.597068i	$0.796332\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−22.7386	−1.08035	−0.540173	−	0.841554i	$-0.681641\pi$
$443$	−22.7386	−1.08035	−0.540173	+	0.841554i	$0.681641\pi$
$444$	0	0
$445$	−0.246211	−0.0116715
$446$	0	0
$447$	12.2462	0.579226
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	10.2462	0.482475
$452$	0	0
$453$	−20.4924	−0.962818
$454$	0	0
$455$	−3.12311	−0.146413
$456$	0	0
$457$	4.24621	0.198629	0.0993147	−	0.995056i	$-0.468335\pi$
$457$	4.24621	0.198629	0.0993147	+	0.995056i	$0.468335\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	−14.4924	−0.674979	−0.337490	−	0.941329i	$-0.609578\pi$
$461$	−14.4924	−0.674979	−0.337490	+	0.941329i	$0.609578\pi$
$462$	0	0
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	0	0
$465$	−5.12311	−0.237578
$466$	0	0
$467$	36.9848	1.71145	0.855727	−	0.517427i	$-0.173110\pi$
$467$	36.9848	1.71145	0.855727	+	0.517427i	$0.173110\pi$
$468$	0	0
$469$	2.24621	0.103720
$470$	0	0
$471$	−7.12311	−0.328215
$472$	0	0
$473$	52.4924	2.41360
$474$	0	0
$475$	−1.12311	−0.0515316
$476$	0	0
$477$	−13.3693	−0.612139
$478$	0	0
$479$	−8.00000	−0.365529	−0.182765	−	0.983157i	$-0.558505\pi$
$479$	−8.00000	−0.365529	−0.182765	+	0.983157i	$0.558505\pi$
$480$	0	0
$481$	6.24621	0.284803
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.87689	0.221448
$486$	0	0
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	0	0
$489$	10.2462	0.463350
$490$	0	0
$491$	13.1231	0.592237	0.296119	−	0.955151i	$-0.404308\pi$
$491$	13.1231	0.592237	0.296119	+	0.955151i	$0.404308\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	5.12311	0.230266
$496$	0	0
$497$	−5.12311	−0.229803
$498$	0	0
$499$	−20.4924	−0.917367	−0.458683	−	0.888600i	$-0.651679\pi$
$499$	−20.4924	−0.917367	−0.458683	+	0.888600i	$0.651679\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	0	0
$503$	3.50758	0.156395	0.0781976	−	0.996938i	$-0.475084\pi$
$503$	3.50758	0.156395	0.0781976	+	0.996938i	$0.475084\pi$
$504$	0	0
$505$	0.246211	0.0109563
$506$	0	0
$507$	−3.24621	−0.144169
$508$	0	0
$509$	11.7538	0.520978	0.260489	−	0.965477i	$-0.416116\pi$
$509$	11.7538	0.520978	0.260489	+	0.965477i	$0.416116\pi$
$510$	0	0
$511$	−15.1231	−0.669007
$512$	0	0
$513$	−1.12311	−0.0495863
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0.246211	0.0108075
$520$	0	0
$521$	22.4924	0.985411	0.492705	−	0.870196i	$-0.336008\pi$
$521$	22.4924	0.985411	0.492705	+	0.870196i	$0.336008\pi$
$522$	0	0
$523$	−34.7386	−1.51901	−0.759507	−	0.650499i	$-0.774560\pi$
$523$	−34.7386	−1.51901	−0.759507	+	0.650499i	$0.774560\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	−10.2462	−0.446332
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	6.24621	0.270553
$534$	0	0
$535$	−8.00000	−0.345870
$536$	0	0
$537$	−2.87689	−0.124147
$538$	0	0
$539$	5.12311	0.220668
$540$	0	0
$541$	14.4924	0.623078	0.311539	−	0.950233i	$-0.399156\pi$
$541$	14.4924	0.623078	0.311539	+	0.950233i	$0.399156\pi$
$542$	0	0
$543$	−22.4924	−0.965242
$544$	0	0
$545$	−8.24621	−0.353229
$546$	0	0
$547$	21.7538	0.930125	0.465062	−	0.885278i	$-0.346032\pi$
$547$	21.7538	0.930125	0.465062	+	0.885278i	$0.346032\pi$
$548$	0	0
$549$	8.24621	0.351940
$550$	0	0
$551$	−2.24621	−0.0956918
$552$	0	0
$553$	−2.24621	−0.0955186
$554$	0	0
$555$	2.00000	0.0848953
$556$	0	0
$557$	15.1231	0.640787	0.320393	−	0.947285i	$-0.396185\pi$
$557$	15.1231	0.640787	0.320393	+	0.947285i	$0.396185\pi$
$558$	0	0
$559$	32.0000	1.35346
$560$	0	0
$561$	10.2462	0.432595
$562$	0	0
$563$	−16.4924	−0.695073	−0.347536	−	0.937667i	$-0.612982\pi$
$563$	−16.4924	−0.695073	−0.347536	+	0.937667i	$0.612982\pi$
$564$	0	0
$565$	−5.36932	−0.225889
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−3.50758	−0.146788	−0.0733938	−	0.997303i	$-0.523383\pi$
$571$	−3.50758	−0.146788	−0.0733938	+	0.997303i	$0.523383\pi$
$572$	0	0
$573$	−5.12311	−0.214021
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−23.6155	−0.983127	−0.491564	−	0.870842i	$-0.663574\pi$
$577$	−23.6155	−0.983127	−0.491564	+	0.870842i	$0.663574\pi$
$578$	0	0
$579$	18.0000	0.748054
$580$	0	0
$581$	4.00000	0.165948
$582$	0	0
$583$	−68.4924	−2.83667
$584$	0	0
$585$	3.12311	0.129125
$586$	0	0
$587$	−32.4924	−1.34111	−0.670553	−	0.741862i	$-0.733943\pi$
$587$	−32.4924	−1.34111	−0.670553	+	0.741862i	$0.733943\pi$
$588$	0	0
$589$	5.75379	0.237081
$590$	0	0
$591$	−23.6155	−0.971413
$592$	0	0
$593$	30.4924	1.25217	0.626087	−	0.779753i	$-0.284656\pi$
$593$	30.4924	1.25217	0.626087	+	0.779753i	$0.284656\pi$
$594$	0	0
$595$	−2.00000	−0.0819920
$596$	0	0
$597$	−5.12311	−0.209675
$598$	0	0
$599$	−5.12311	−0.209324	−0.104662	−	0.994508i	$-0.533376\pi$
$599$	−5.12311	−0.209324	−0.104662	+	0.994508i	$0.533376\pi$
$600$	0	0
$601$	−16.2462	−0.662697	−0.331348	−	0.943508i	$-0.607504\pi$
$601$	−16.2462	−0.662697	−0.331348	+	0.943508i	$0.607504\pi$
$602$	0	0
$603$	−2.24621	−0.0914728
$604$	0	0
$605$	15.2462	0.619847
$606$	0	0
$607$	−13.7538	−0.558249	−0.279125	−	0.960255i	$-0.590044\pi$
$607$	−13.7538	−0.558249	−0.279125	+	0.960255i	$0.590044\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−22.9848	−0.928349	−0.464175	−	0.885744i	$-0.653649\pi$
$613$	−22.9848	−0.928349	−0.464175	+	0.885744i	$0.653649\pi$
$614$	0	0
$615$	2.00000	0.0806478
$616$	0	0
$617$	−13.3693	−0.538228	−0.269114	−	0.963108i	$-0.586731\pi$
$617$	−13.3693	−0.538228	−0.269114	+	0.963108i	$0.586731\pi$
$618$	0	0
$619$	−17.1231	−0.688236	−0.344118	−	0.938926i	$-0.611822\pi$
$619$	−17.1231	−0.688236	−0.344118	+	0.938926i	$0.611822\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.246211	0.00986425
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.75379	−0.229784
$628$	0	0
$629$	4.00000	0.159490
$630$	0	0
$631$	2.24621	0.0894203	0.0447101	−	0.999000i	$-0.485764\pi$
$631$	2.24621	0.0894203	0.0447101	+	0.999000i	$0.485764\pi$
$632$	0	0
$633$	5.75379	0.228693
$634$	0	0
$635$	−20.4924	−0.813217
$636$	0	0
$637$	3.12311	0.123742
$638$	0	0
$639$	5.12311	0.202667
$640$	0	0
$641$	−8.24621	−0.325706	−0.162853	−	0.986650i	$-0.552070\pi$
$641$	−8.24621	−0.325706	−0.162853	+	0.986650i	$0.552070\pi$
$642$	0	0
$643$	−22.2462	−0.877305	−0.438652	−	0.898657i	$-0.644544\pi$
$643$	−22.2462	−0.877305	−0.438652	+	0.898657i	$0.644544\pi$
$644$	0	0
$645$	10.2462	0.403444
$646$	0	0
$647$	40.9848	1.61128	0.805640	−	0.592405i	$-0.201821\pi$
$647$	40.9848	1.61128	0.805640	+	0.592405i	$0.201821\pi$
$648$	0	0
$649$	−20.4924	−0.804398
$650$	0	0
$651$	5.12311	0.200790
$652$	0	0
$653$	−7.61553	−0.298019	−0.149009	−	0.988836i	$-0.547608\pi$
$653$	−7.61553	−0.298019	−0.149009	+	0.988836i	$0.547608\pi$
$654$	0	0
$655$	1.75379	0.0685262
$656$	0	0
$657$	15.1231	0.590009
$658$	0	0
$659$	−9.61553	−0.374568	−0.187284	−	0.982306i	$-0.559968\pi$
$659$	−9.61553	−0.374568	−0.187284	+	0.982306i	$0.559968\pi$
$660$	0	0
$661$	−31.7538	−1.23508	−0.617540	−	0.786540i	$-0.711870\pi$
$661$	−31.7538	−1.23508	−0.617540	+	0.786540i	$0.711870\pi$
$662$	0	0
$663$	6.24621	0.242583
$664$	0	0
$665$	1.12311	0.0435522
$666$	0	0
$667$	0	0
$668$	0	0
$669$	18.2462	0.705439
$670$	0	0
$671$	42.2462	1.63090
$672$	0	0
$673$	37.2311	1.43515	0.717576	−	0.696480i	$-0.245252\pi$
$673$	37.2311	1.43515	0.717576	+	0.696480i	$0.245252\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−7.75379	−0.298002	−0.149001	−	0.988837i	$-0.547606\pi$
$677$	−7.75379	−0.298002	−0.149001	+	0.988837i	$0.547606\pi$
$678$	0	0
$679$	−4.87689	−0.187158
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	−13.7538	−0.526274	−0.263137	−	0.964758i	$-0.584757\pi$
$683$	−13.7538	−0.526274	−0.263137	+	0.964758i	$0.584757\pi$
$684$	0	0
$685$	12.8769	0.492001
$686$	0	0
$687$	3.75379	0.143216
$688$	0	0
$689$	−41.7538	−1.59069
$690$	0	0
$691$	20.6307	0.784828	0.392414	−	0.919789i	$-0.371640\pi$
$691$	20.6307	0.784828	0.392414	+	0.919789i	$0.371640\pi$
$692$	0	0
$693$	−5.12311	−0.194611
$694$	0	0
$695$	−9.12311	−0.346059
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	23.1231	0.874596
$700$	0	0
$701$	38.4924	1.45384	0.726919	−	0.686723i	$-0.240951\pi$
$701$	38.4924	1.45384	0.726919	+	0.686723i	$0.240951\pi$
$702$	0	0
$703$	−2.24621	−0.0847175
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−0.246211	−0.00925973
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	2.24621	0.0842395
$712$	0	0
$713$	0	0
$714$	0	0
$715$	16.0000	0.598366
$716$	0	0
$717$	13.1231	0.490091
$718$	0	0
$719$	43.2311	1.61225	0.806123	−	0.591748i	$-0.201562\pi$
$719$	43.2311	1.61225	0.806123	+	0.591748i	$0.201562\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	20.2462	0.752965
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	−30.7386	−1.14003	−0.570016	−	0.821633i	$-0.693063\pi$
$727$	−30.7386	−1.14003	−0.570016	+	0.821633i	$0.693063\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	20.4924	0.757940
$732$	0	0
$733$	47.6155	1.75872	0.879360	−	0.476158i	$-0.157971\pi$
$733$	47.6155	1.75872	0.879360	+	0.476158i	$0.157971\pi$
$734$	0	0
$735$	1.00000	0.0368856
$736$	0	0
$737$	−11.5076	−0.423887
$738$	0	0
$739$	−40.0000	−1.47142	−0.735712	−	0.677295i	$-0.763152\pi$
$739$	−40.0000	−1.47142	−0.735712	+	0.677295i	$0.763152\pi$
$740$	0	0
$741$	−3.50758	−0.128854
$742$	0	0
$743$	−14.7386	−0.540708	−0.270354	−	0.962761i	$-0.587141\pi$
$743$	−14.7386	−0.540708	−0.270354	+	0.962761i	$0.587141\pi$
$744$	0	0
$745$	12.2462	0.448666
$746$	0	0
$747$	−4.00000	−0.146352
$748$	0	0
$749$	8.00000	0.292314
$750$	0	0
$751$	−10.2462	−0.373890	−0.186945	−	0.982370i	$-0.559859\pi$
$751$	−10.2462	−0.373890	−0.186945	+	0.982370i	$0.559859\pi$
$752$	0	0
$753$	−12.0000	−0.437304
$754$	0	0
$755$	−20.4924	−0.745796
$756$	0	0
$757$	−35.7538	−1.29949	−0.649747	−	0.760151i	$-0.725125\pi$
$757$	−35.7538	−1.29949	−0.649747	+	0.760151i	$0.725125\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−28.7386	−1.04177	−0.520887	−	0.853625i	$-0.674399\pi$
$761$	−28.7386	−1.04177	−0.520887	+	0.853625i	$0.674399\pi$
$762$	0	0
$763$	8.24621	0.298533
$764$	0	0
$765$	2.00000	0.0723102
$766$	0	0
$767$	−12.4924	−0.451075
$768$	0	0
$769$	−32.2462	−1.16283	−0.581414	−	0.813608i	$-0.697500\pi$
$769$	−32.2462	−1.16283	−0.581414	+	0.813608i	$0.697500\pi$
$770$	0	0
$771$	20.2462	0.729149
$772$	0	0
$773$	16.2462	0.584336	0.292168	−	0.956367i	$-0.405623\pi$
$773$	16.2462	0.584336	0.292168	+	0.956367i	$0.405623\pi$
$774$	0	0
$775$	−5.12311	−0.184027
$776$	0	0
$777$	−2.00000	−0.0717496
$778$	0	0
$779$	−2.24621	−0.0804789
$780$	0	0
$781$	26.2462	0.939163
$782$	0	0
$783$	2.00000	0.0714742
$784$	0	0
$785$	−7.12311	−0.254235
$786$	0	0
$787$	31.2311	1.11327	0.556633	−	0.830758i	$-0.312093\pi$
$787$	31.2311	1.11327	0.556633	+	0.830758i	$0.312093\pi$
$788$	0	0
$789$	18.2462	0.649582
$790$	0	0
$791$	5.36932	0.190911
$792$	0	0
$793$	25.7538	0.914544
$794$	0	0
$795$	−13.3693	−0.474161
$796$	0	0
$797$	36.7386	1.30135	0.650675	−	0.759357i	$-0.274486\pi$
$797$	36.7386	1.30135	0.650675	+	0.759357i	$0.274486\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−0.246211	−0.00869945
$802$	0	0
$803$	77.4773	2.73411
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16.2462	0.571894
$808$	0	0
$809$	20.2462	0.711819	0.355909	−	0.934520i	$-0.384171\pi$
$809$	20.2462	0.711819	0.355909	+	0.934520i	$0.384171\pi$
$810$	0	0
$811$	−53.6155	−1.88270	−0.941348	−	0.337438i	$-0.890440\pi$
$811$	−53.6155	−1.88270	−0.941348	+	0.337438i	$0.890440\pi$
$812$	0	0
$813$	17.6155	0.617804
$814$	0	0
$815$	10.2462	0.358909
$816$	0	0
$817$	−11.5076	−0.402599
$818$	0	0
$819$	−3.12311	−0.109130
$820$	0	0
$821$	−54.9848	−1.91898	−0.959492	−	0.281735i	$-0.909090\pi$
$821$	−54.9848	−1.91898	−0.959492	+	0.281735i	$0.909090\pi$
$822$	0	0
$823$	12.4924	0.435458	0.217729	−	0.976009i	$-0.430135\pi$
$823$	12.4924	0.435458	0.217729	+	0.976009i	$0.430135\pi$
$824$	0	0
$825$	5.12311	0.178364
$826$	0	0
$827$	48.9848	1.70337	0.851685	−	0.524054i	$-0.175581\pi$
$827$	48.9848	1.70337	0.851685	+	0.524054i	$0.175581\pi$
$828$	0	0
$829$	−6.49242	−0.225491	−0.112746	−	0.993624i	$-0.535965\pi$
$829$	−6.49242	−0.225491	−0.112746	+	0.993624i	$0.535965\pi$
$830$	0	0
$831$	14.4924	0.502737
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	16.0000	0.553703
$836$	0	0
$837$	−5.12311	−0.177080
$838$	0	0
$839$	38.7386	1.33741	0.668703	−	0.743530i	$-0.266850\pi$
$839$	38.7386	1.33741	0.668703	+	0.743530i	$0.266850\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	24.7386	0.852044
$844$	0	0
$845$	−3.24621	−0.111673
$846$	0	0
$847$	−15.2462	−0.523866
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−0.384472	−0.0131641	−0.00658203	−	0.999978i	$-0.502095\pi$
$853$	−0.384472	−0.0131641	−0.00658203	+	0.999978i	$0.502095\pi$
$854$	0	0
$855$	−1.12311	−0.0384094
$856$	0	0
$857$	−48.2462	−1.64806	−0.824030	−	0.566547i	$-0.808279\pi$
$857$	−48.2462	−1.64806	−0.824030	+	0.566547i	$0.808279\pi$
$858$	0	0
$859$	21.6155	0.737512	0.368756	−	0.929526i	$-0.379784\pi$
$859$	21.6155	0.737512	0.368756	+	0.929526i	$0.379784\pi$
$860$	0	0
$861$	−2.00000	−0.0681598
$862$	0	0
$863$	−40.0000	−1.36162	−0.680808	−	0.732462i	$-0.738371\pi$
$863$	−40.0000	−1.36162	−0.680808	+	0.732462i	$0.738371\pi$
$864$	0	0
$865$	0.246211	0.00837143
$866$	0	0
$867$	−13.0000	−0.441503
$868$	0	0
$869$	11.5076	0.390368
$870$	0	0
$871$	−7.01515	−0.237699
$872$	0	0
$873$	4.87689	0.165058
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−10.4924	−0.354304	−0.177152	−	0.984184i	$-0.556688\pi$
$877$	−10.4924	−0.354304	−0.177152	+	0.984184i	$0.556688\pi$
$878$	0	0
$879$	16.2462	0.547971
$880$	0	0
$881$	−40.2462	−1.35593	−0.677965	−	0.735095i	$-0.737138\pi$
$881$	−40.2462	−1.35593	−0.677965	+	0.735095i	$0.737138\pi$
$882$	0	0
$883$	18.2462	0.614034	0.307017	−	0.951704i	$-0.400669\pi$
$883$	18.2462	0.614034	0.307017	+	0.951704i	$0.400669\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	0	0
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	0	0
$889$	20.4924	0.687294
$890$	0	0
$891$	5.12311	0.171630
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−2.87689	−0.0961640
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−10.2462	−0.341730
$900$	0	0
$901$	−26.7386	−0.890793
$902$	0	0
$903$	−10.2462	−0.340973
$904$	0	0
$905$	−22.4924	−0.747673
$906$	0	0
$907$	−38.7386	−1.28630	−0.643148	−	0.765742i	$-0.722372\pi$
$907$	−38.7386	−1.28630	−0.643148	+	0.765742i	$0.722372\pi$
$908$	0	0
$909$	0.246211	0.00816631
$910$	0	0
$911$	−1.61553	−0.0535248	−0.0267624	−	0.999642i	$-0.508520\pi$
$911$	−1.61553	−0.0535248	−0.0267624	+	0.999642i	$0.508520\pi$
$912$	0	0
$913$	−20.4924	−0.678200
$914$	0	0
$915$	8.24621	0.272611
$916$	0	0
$917$	−1.75379	−0.0579152
$918$	0	0
$919$	18.2462	0.601887	0.300943	−	0.953642i	$-0.402698\pi$
$919$	18.2462	0.601887	0.300943	+	0.953642i	$0.402698\pi$
$920$	0	0
$921$	−9.75379	−0.321398
$922$	0	0
$923$	16.0000	0.526646
$924$	0	0
$925$	2.00000	0.0657596
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−11.7538	−0.385629	−0.192815	−	0.981235i	$-0.561762\pi$
$929$	−11.7538	−0.385629	−0.192815	+	0.981235i	$0.561762\pi$
$930$	0	0
$931$	−1.12311	−0.0368083
$932$	0	0
$933$	16.0000	0.523816
$934$	0	0
$935$	10.2462	0.335087
$936$	0	0
$937$	40.1080	1.31027	0.655135	−	0.755512i	$-0.272612\pi$
$937$	40.1080	1.31027	0.655135	+	0.755512i	$0.272612\pi$
$938$	0	0
$939$	33.3693	1.08897
$940$	0	0
$941$	−42.9848	−1.40127	−0.700633	−	0.713522i	$-0.747099\pi$
$941$	−42.9848	−1.40127	−0.700633	+	0.713522i	$0.747099\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−1.00000	−0.0325300
$946$	0	0
$947$	−51.2311	−1.66479	−0.832393	−	0.554186i	$-0.813030\pi$
$947$	−51.2311	−1.66479	−0.832393	+	0.554186i	$0.813030\pi$
$948$	0	0
$949$	47.2311	1.53318
$950$	0	0
$951$	15.1231	0.490400
$952$	0	0
$953$	2.63068	0.0852162	0.0426081	−	0.999092i	$-0.486433\pi$
$953$	2.63068	0.0852162	0.0426081	+	0.999092i	$0.486433\pi$
$954$	0	0
$955$	−5.12311	−0.165780
$956$	0	0
$957$	10.2462	0.331213
$958$	0	0
$959$	−12.8769	−0.415817
$960$	0	0
$961$	−4.75379	−0.153348
$962$	0	0
$963$	−8.00000	−0.257796
$964$	0	0
$965$	18.0000	0.579441
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	0	0
$969$	−2.24621	−0.0721587
$970$	0	0
$971$	44.0000	1.41203	0.706014	−	0.708198i	$-0.250492\pi$
$971$	44.0000	1.41203	0.706014	+	0.708198i	$0.250492\pi$
$972$	0	0
$973$	9.12311	0.292473
$974$	0	0
$975$	3.12311	0.100019
$976$	0	0
$977$	10.6307	0.340106	0.170053	−	0.985435i	$-0.445606\pi$
$977$	10.6307	0.340106	0.170053	+	0.985435i	$0.445606\pi$
$978$	0	0
$979$	−1.26137	−0.0403134
$980$	0	0
$981$	−8.24621	−0.263281
$982$	0	0
$983$	11.5076	0.367035	0.183517	−	0.983016i	$-0.441252\pi$
$983$	11.5076	0.367035	0.183517	+	0.983016i	$0.441252\pi$
$984$	0	0
$985$	−23.6155	−0.752453
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	9.26137	0.294197	0.147098	−	0.989122i	$-0.453007\pi$
$991$	9.26137	0.294197	0.147098	+	0.989122i	$0.453007\pi$
$992$	0	0
$993$	2.24621	0.0712814
$994$	0	0
$995$	−5.12311	−0.162413
$996$	0	0
$997$	−37.8617	−1.19909	−0.599547	−	0.800340i	$-0.704652\pi$
$997$	−37.8617	−1.19909	−0.599547	+	0.800340i	$0.704652\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6720.2.a.cw.1.2		2
4.3	odd	2		6720.2.a.ct.1.1		2
8.3	odd	2		3360.2.a.bg.1.2	yes	2
8.5	even	2		3360.2.a.ba.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3360.2.a.ba.1.1	✓	2	8.5	even	2
3360.2.a.bg.1.2	yes	2	8.3	odd	2
6720.2.a.ct.1.1		2	4.3	odd	2
6720.2.a.cw.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.12311 q^{11} +3.12311 q^{13} +1.00000 q^{15} +2.00000 q^{17} -1.12311 q^{19} -1.00000 q^{21} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{29} -5.12311 q^{31} +5.12311 q^{33} -1.00000 q^{35} +2.00000 q^{37} +3.12311 q^{39} +2.00000 q^{41} +10.2462 q^{43} +1.00000 q^{45} +1.00000 q^{49} +2.00000 q^{51} -13.3693 q^{53} +5.12311 q^{55} -1.12311 q^{57} -4.00000 q^{59} +8.24621 q^{61} -1.00000 q^{63} +3.12311 q^{65} -2.24621 q^{67} +5.12311 q^{71} +15.1231 q^{73} +1.00000 q^{75} -5.12311 q^{77} +2.24621 q^{79} +1.00000 q^{81} -4.00000 q^{83} +2.00000 q^{85} +2.00000 q^{87} -0.246211 q^{89} -3.12311 q^{91} -5.12311 q^{93} -1.12311 q^{95} +4.87689 q^{97} +5.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} - 2 q^{13} + 2 q^{15} + 4 q^{17} + 6 q^{19} - 2 q^{21} + 2 q^{25} + 2 q^{27} + 4 q^{29} - 2 q^{31} + 2 q^{33} - 2 q^{35} + 4 q^{37} - 2 q^{39} + 4 q^{41} + 4 q^{43} + 2 q^{45} + 2 q^{49} + 4 q^{51} - 2 q^{53} + 2 q^{55} + 6 q^{57} - 8 q^{59} - 2 q^{63} - 2 q^{65} + 12 q^{67} + 2 q^{71} + 22 q^{73} + 2 q^{75} - 2 q^{77} - 12 q^{79} + 2 q^{81} - 8 q^{83} + 4 q^{85} + 4 q^{87} + 16 q^{89} + 2 q^{91} - 2 q^{93} + 6 q^{95} + 18 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 2 * q^11 - 2 * q^13 + 2 * q^15 + 4 * q^17 + 6 * q^19 - 2 * q^21 + 2 * q^25 + 2 * q^27 + 4 * q^29 - 2 * q^31 + 2 * q^33 - 2 * q^35 + 4 * q^37 - 2 * q^39 + 4 * q^41 + 4 * q^43 + 2 * q^45 + 2 * q^49 + 4 * q^51 - 2 * q^53 + 2 * q^55 + 6 * q^57 - 8 * q^59 - 2 * q^63 - 2 * q^65 + 12 * q^67 + 2 * q^71 + 22 * q^73 + 2 * q^75 - 2 * q^77 - 12 * q^79 + 2 * q^81 - 8 * q^83 + 4 * q^85 + 4 * q^87 + 16 * q^89 + 2 * q^91 - 2 * q^93 + 6 * q^95 + 18 * q^97 + 2 * q^99

Introduction

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