LMFDB - Embedded newform 3360.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3360,2,Mod(1,3360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3360, 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("3360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3360.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:	$26.8297350792$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	3360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.46410 q^{11} +3.46410 q^{13} -1.00000 q^{15} +2.00000 q^{17} +5.46410 q^{19} -1.00000 q^{21} -6.92820 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} -5.46410 q^{31} +5.46410 q^{33} +1.00000 q^{35} +2.00000 q^{37} +3.46410 q^{39} -4.92820 q^{41} +4.00000 q^{43} -1.00000 q^{45} +10.9282 q^{47} +1.00000 q^{49} +2.00000 q^{51} -0.535898 q^{53} -5.46410 q^{55} +5.46410 q^{57} -13.8564 q^{59} -4.92820 q^{61} -1.00000 q^{63} -3.46410 q^{65} -6.92820 q^{67} -6.92820 q^{69} +16.3923 q^{71} +0.535898 q^{73} +1.00000 q^{75} -5.46410 q^{77} +4.00000 q^{79} +1.00000 q^{81} +14.9282 q^{83} -2.00000 q^{85} -2.00000 q^{87} -12.9282 q^{89} -3.46410 q^{91} -5.46410 q^{93} -5.46410 q^{95} +19.4641 q^{97} +5.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{15} + 4 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 4 q^{31} + 4 q^{33} + 2 q^{35} + 4 q^{37} + 4 q^{41} + 8 q^{43} - 2 q^{45} + 8 q^{47} + 2 q^{49} + 4 q^{51} - 8 q^{53} - 4 q^{55} + 4 q^{57} + 4 q^{61} - 2 q^{63} + 12 q^{71} + 8 q^{73} + 2 q^{75} - 4 q^{77} + 8 q^{79} + 2 q^{81} + 16 q^{83} - 4 q^{85} - 4 q^{87} - 12 q^{89} - 4 q^{93} - 4 q^{95} + 32 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 4 * q^11 - 2 * q^15 + 4 * q^17 + 4 * q^19 - 2 * q^21 + 2 * q^25 + 2 * q^27 - 4 * q^29 - 4 * q^31 + 4 * q^33 + 2 * q^35 + 4 * q^37 + 4 * q^41 + 8 * q^43 - 2 * q^45 + 8 * q^47 + 2 * q^49 + 4 * q^51 - 8 * q^53 - 4 * q^55 + 4 * q^57 + 4 * q^61 - 2 * q^63 + 12 * q^71 + 8 * q^73 + 2 * q^75 - 4 * q^77 + 8 * q^79 + 2 * q^81 + 16 * q^83 - 4 * q^85 - 4 * q^87 - 12 * q^89 - 4 * q^93 - 4 * q^95 + 32 * 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
$11$	5.46410	1.64749	0.823744	−	0.566961i	$-0.191881\pi$
$11$	5.46410	1.64749	0.823744	+	0.566961i	$0.191881\pi$
$12$	0	0
$13$	3.46410	0.960769	0.480384	−	0.877058i	$-0.340497\pi$
$13$	3.46410	0.960769	0.480384	+	0.877058i	$0.340497\pi$
$14$	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
$19$	5.46410	1.25355	0.626775	−	0.779200i	$-0.284374\pi$
$19$	5.46410	1.25355	0.626775	+	0.779200i	$0.284374\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−6.92820	−1.44463	−0.722315	−	0.691564i	$-0.756922\pi$
$23$	−6.92820	−1.44463	−0.722315	+	0.691564i	$0.756922\pi$
$24$	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	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−5.46410	−0.981382	−0.490691	−	0.871334i	$-0.663256\pi$
$31$	−5.46410	−0.981382	−0.490691	+	0.871334i	$0.663256\pi$
$32$	0	0
$33$	5.46410	0.951178
$34$	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
$39$	3.46410	0.554700
$40$	0	0
$41$	−4.92820	−0.769656	−0.384828	−	0.922988i	$-0.625739\pi$
$41$	−4.92820	−0.769656	−0.384828	+	0.922988i	$0.625739\pi$
$42$	0	0
$43$	4.00000	0.609994	0.304997	−	0.952353i	$-0.401344\pi$
$43$	4.00000	0.609994	0.304997	+	0.952353i	$0.401344\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	10.9282	1.59404	0.797021	−	0.603951i	$-0.206408\pi$
$47$	10.9282	1.59404	0.797021	+	0.603951i	$0.206408\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.00000	0.280056
$52$	0	0
$53$	−0.535898	−0.0736113	−0.0368057	−	0.999322i	$-0.511718\pi$
$53$	−0.535898	−0.0736113	−0.0368057	+	0.999322i	$0.511718\pi$
$54$	0	0
$55$	−5.46410	−0.736779
$56$	0	0
$57$	5.46410	0.723738
$58$	0	0
$59$	−13.8564	−1.80395	−0.901975	−	0.431788i	$-0.857883\pi$
$59$	−13.8564	−1.80395	−0.901975	+	0.431788i	$0.857883\pi$
$60$	0	0
$61$	−4.92820	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$61$	−4.92820	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	−3.46410	−0.429669
$66$	0	0
$67$	−6.92820	−0.846415	−0.423207	−	0.906033i	$-0.639096\pi$
$67$	−6.92820	−0.846415	−0.423207	+	0.906033i	$0.639096\pi$
$68$	0	0
$69$	−6.92820	−0.834058
$70$	0	0
$71$	16.3923	1.94541	0.972704	−	0.232048i	$-0.0745426\pi$
$71$	16.3923	1.94541	0.972704	+	0.232048i	$0.0745426\pi$
$72$	0	0
$73$	0.535898	0.0627222	0.0313611	−	0.999508i	$-0.490016\pi$
$73$	0.535898	0.0627222	0.0313611	+	0.999508i	$0.490016\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−5.46410	−0.622692
$78$	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
$83$	14.9282	1.63858	0.819292	−	0.573377i	$-0.194367\pi$
$83$	14.9282	1.63858	0.819292	+	0.573377i	$0.194367\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−2.00000	−0.214423
$88$	0	0
$89$	−12.9282	−1.37039	−0.685193	−	0.728361i	$-0.740282\pi$
$89$	−12.9282	−1.37039	−0.685193	+	0.728361i	$0.740282\pi$
$90$	0	0
$91$	−3.46410	−0.363137
$92$	0	0
$93$	−5.46410	−0.566601
$94$	0	0
$95$	−5.46410	−0.560605
$96$	0	0
$97$	19.4641	1.97628	0.988140	−	0.153555i	$-0.0490723\pi$
$97$	19.4641	1.97628	0.988140	+	0.153555i	$0.0490723\pi$
$98$	0	0
$99$	5.46410	0.549163
$100$	0	0
$101$	15.8564	1.57777	0.788886	−	0.614540i	$-0.210658\pi$
$101$	15.8564	1.57777	0.788886	+	0.614540i	$0.210658\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	18.9282	1.82986	0.914929	−	0.403614i	$-0.132246\pi$
$107$	18.9282	1.82986	0.914929	+	0.403614i	$0.132246\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	10.3923	0.977626	0.488813	−	0.872389i	$-0.337430\pi$
$113$	10.3923	0.977626	0.488813	+	0.872389i	$0.337430\pi$
$114$	0	0
$115$	6.92820	0.646058
$116$	0	0
$117$	3.46410	0.320256
$118$	0	0
$119$	−2.00000	−0.183340
$120$	0	0
$121$	18.8564	1.71422
$122$	0	0
$123$	−4.92820	−0.444361
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	5.07180	0.450049	0.225025	−	0.974353i	$-0.427754\pi$
$127$	5.07180	0.450049	0.225025	+	0.974353i	$0.427754\pi$
$128$	0	0
$129$	4.00000	0.352180
$130$	0	0
$131$	−2.92820	−0.255838	−0.127919	−	0.991785i	$-0.540830\pi$
$131$	−2.92820	−0.255838	−0.127919	+	0.991785i	$0.540830\pi$
$132$	0	0
$133$	−5.46410	−0.473798
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−0.535898	−0.0457849	−0.0228924	−	0.999738i	$-0.507288\pi$
$137$	−0.535898	−0.0457849	−0.0228924	+	0.999738i	$0.507288\pi$
$138$	0	0
$139$	5.46410	0.463459	0.231730	−	0.972780i	$-0.425562\pi$
$139$	5.46410	0.463459	0.231730	+	0.972780i	$0.425562\pi$
$140$	0	0
$141$	10.9282	0.920321
$142$	0	0
$143$	18.9282	1.58286
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−20.9282	−1.71451	−0.857253	−	0.514896i	$-0.827831\pi$
$149$	−20.9282	−1.71451	−0.857253	+	0.514896i	$0.827831\pi$
$150$	0	0
$151$	14.9282	1.21484	0.607420	−	0.794381i	$-0.292205\pi$
$151$	14.9282	1.21484	0.607420	+	0.794381i	$0.292205\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	5.46410	0.438887
$156$	0	0
$157$	0.535898	0.0427693	0.0213847	−	0.999771i	$-0.493193\pi$
$157$	0.535898	0.0427693	0.0213847	+	0.999771i	$0.493193\pi$
$158$	0	0
$159$	−0.535898	−0.0424995
$160$	0	0
$161$	6.92820	0.546019
$162$	0	0
$163$	12.0000	0.939913	0.469956	−	0.882690i	$-0.344270\pi$
$163$	12.0000	0.939913	0.469956	+	0.882690i	$0.344270\pi$
$164$	0	0
$165$	−5.46410	−0.425380
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	5.46410	0.417850
$172$	0	0
$173$	−24.9282	−1.89526	−0.947628	−	0.319376i	$-0.896527\pi$
$173$	−24.9282	−1.89526	−0.947628	+	0.319376i	$0.896527\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−13.8564	−1.04151
$178$	0	0
$179$	−2.53590	−0.189542	−0.0947710	−	0.995499i	$-0.530212\pi$
$179$	−2.53590	−0.189542	−0.0947710	+	0.995499i	$0.530212\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	−4.92820	−0.364303
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	10.9282	0.799149
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−8.39230	−0.607246	−0.303623	−	0.952792i	$-0.598196\pi$
$191$	−8.39230	−0.607246	−0.303623	+	0.952792i	$0.598196\pi$
$192$	0	0
$193$	−11.8564	−0.853443	−0.426721	−	0.904383i	$-0.640332\pi$
$193$	−11.8564	−0.853443	−0.426721	+	0.904383i	$0.640332\pi$
$194$	0	0
$195$	−3.46410	−0.248069
$196$	0	0
$197$	−19.4641	−1.38676	−0.693380	−	0.720572i	$-0.743879\pi$
$197$	−19.4641	−1.38676	−0.693380	+	0.720572i	$0.743879\pi$
$198$	0	0
$199$	0.392305	0.0278098	0.0139049	−	0.999903i	$-0.495574\pi$
$199$	0.392305	0.0278098	0.0139049	+	0.999903i	$0.495574\pi$
$200$	0	0
$201$	−6.92820	−0.488678
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	4.92820	0.344201
$206$	0	0
$207$	−6.92820	−0.481543
$208$	0	0
$209$	29.8564	2.06521
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	16.3923	1.12318
$214$	0	0
$215$	−4.00000	−0.272798
$216$	0	0
$217$	5.46410	0.370927
$218$	0	0
$219$	0.535898	0.0362127
$220$	0	0
$221$	6.92820	0.466041
$222$	0	0
$223$	26.9282	1.80325	0.901623	−	0.432523i	$-0.142377\pi$
$223$	26.9282	1.80325	0.901623	+	0.432523i	$0.142377\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−6.92820	−0.459841	−0.229920	−	0.973209i	$-0.573847\pi$
$227$	−6.92820	−0.459841	−0.229920	+	0.973209i	$0.573847\pi$
$228$	0	0
$229$	22.7846	1.50565	0.752825	−	0.658221i	$-0.228691\pi$
$229$	22.7846	1.50565	0.752825	+	0.658221i	$0.228691\pi$
$230$	0	0
$231$	−5.46410	−0.359511
$232$	0	0
$233$	−19.4641	−1.27514	−0.637568	−	0.770394i	$-0.720059\pi$
$233$	−19.4641	−1.27514	−0.637568	+	0.770394i	$0.720059\pi$
$234$	0	0
$235$	−10.9282	−0.712877
$236$	0	0
$237$	4.00000	0.259828
$238$	0	0
$239$	−19.3205	−1.24974	−0.624870	−	0.780729i	$-0.714848\pi$
$239$	−19.3205	−1.24974	−0.624870	+	0.780729i	$0.714848\pi$
$240$	0	0
$241$	−14.7846	−0.952360	−0.476180	−	0.879348i	$-0.657979\pi$
$241$	−14.7846	−0.952360	−0.476180	+	0.879348i	$0.657979\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	18.9282	1.20437
$248$	0	0
$249$	14.9282	0.946036
$250$	0	0
$251$	2.14359	0.135302	0.0676512	−	0.997709i	$-0.478449\pi$
$251$	2.14359	0.135302	0.0676512	+	0.997709i	$0.478449\pi$
$252$	0	0
$253$	−37.8564	−2.38001
$254$	0	0
$255$	−2.00000	−0.125245
$256$	0	0
$257$	−3.07180	−0.191613	−0.0958067	−	0.995400i	$-0.530543\pi$
$257$	−3.07180	−0.191613	−0.0958067	+	0.995400i	$0.530543\pi$
$258$	0	0
$259$	−2.00000	−0.124274
$260$	0	0
$261$	−2.00000	−0.123797
$262$	0	0
$263$	−17.8564	−1.10107	−0.550537	−	0.834811i	$-0.685577\pi$
$263$	−17.8564	−1.10107	−0.550537	+	0.834811i	$0.685577\pi$
$264$	0	0
$265$	0.535898	0.0329200
$266$	0	0
$267$	−12.9282	−0.791193
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	13.4641	0.817886	0.408943	−	0.912560i	$-0.365897\pi$
$271$	13.4641	0.817886	0.408943	+	0.912560i	$0.365897\pi$
$272$	0	0
$273$	−3.46410	−0.209657
$274$	0	0
$275$	5.46410	0.329498
$276$	0	0
$277$	−11.8564	−0.712382	−0.356191	−	0.934413i	$-0.615925\pi$
$277$	−11.8564	−0.712382	−0.356191	+	0.934413i	$0.615925\pi$
$278$	0	0
$279$	−5.46410	−0.327127
$280$	0	0
$281$	−30.7846	−1.83646	−0.918228	−	0.396052i	$-0.870380\pi$
$281$	−30.7846	−1.83646	−0.918228	+	0.396052i	$0.870380\pi$
$282$	0	0
$283$	17.8564	1.06145	0.530727	−	0.847543i	$-0.321919\pi$
$283$	17.8564	1.06145	0.530727	+	0.847543i	$0.321919\pi$
$284$	0	0
$285$	−5.46410	−0.323665
$286$	0	0
$287$	4.92820	0.290903
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	19.4641	1.14101
$292$	0	0
$293$	−24.9282	−1.45632	−0.728161	−	0.685407i	$-0.759625\pi$
$293$	−24.9282	−1.45632	−0.728161	+	0.685407i	$0.759625\pi$
$294$	0	0
$295$	13.8564	0.806751
$296$	0	0
$297$	5.46410	0.317059
$298$	0	0
$299$	−24.0000	−1.38796
$300$	0	0
$301$	−4.00000	−0.230556
$302$	0	0
$303$	15.8564	0.910927
$304$	0	0
$305$	4.92820	0.282188
$306$	0	0
$307$	−22.9282	−1.30858	−0.654291	−	0.756243i	$-0.727033\pi$
$307$	−22.9282	−1.30858	−0.654291	+	0.756243i	$0.727033\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	17.3205	0.979013	0.489506	−	0.872000i	$-0.337177\pi$
$313$	17.3205	0.979013	0.489506	+	0.872000i	$0.337177\pi$
$314$	0	0
$315$	1.00000	0.0563436
$316$	0	0
$317$	13.3205	0.748154	0.374077	−	0.927398i	$-0.377960\pi$
$317$	13.3205	0.748154	0.374077	+	0.927398i	$0.377960\pi$
$318$	0	0
$319$	−10.9282	−0.611862
$320$	0	0
$321$	18.9282	1.05647
$322$	0	0
$323$	10.9282	0.608061
$324$	0	0
$325$	3.46410	0.192154
$326$	0	0
$327$	−2.00000	−0.110600
$328$	0	0
$329$	−10.9282	−0.602491
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	0	0
$333$	2.00000	0.109599
$334$	0	0
$335$	6.92820	0.378528
$336$	0	0
$337$	10.0000	0.544735	0.272367	−	0.962193i	$-0.412193\pi$
$337$	10.0000	0.544735	0.272367	+	0.962193i	$0.412193\pi$
$338$	0	0
$339$	10.3923	0.564433
$340$	0	0
$341$	−29.8564	−1.61682
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	6.92820	0.373002
$346$	0	0
$347$	13.8564	0.743851	0.371925	−	0.928263i	$-0.378698\pi$
$347$	13.8564	0.743851	0.371925	+	0.928263i	$0.378698\pi$
$348$	0	0
$349$	16.9282	0.906146	0.453073	−	0.891473i	$-0.350328\pi$
$349$	16.9282	0.906146	0.453073	+	0.891473i	$0.350328\pi$
$350$	0	0
$351$	3.46410	0.184900
$352$	0	0
$353$	4.14359	0.220541	0.110271	−	0.993902i	$-0.464828\pi$
$353$	4.14359	0.220541	0.110271	+	0.993902i	$0.464828\pi$
$354$	0	0
$355$	−16.3923	−0.870013
$356$	0	0
$357$	−2.00000	−0.105851
$358$	0	0
$359$	−10.5359	−0.556063	−0.278032	−	0.960572i	$-0.589682\pi$
$359$	−10.5359	−0.556063	−0.278032	+	0.960572i	$0.589682\pi$
$360$	0	0
$361$	10.8564	0.571390
$362$	0	0
$363$	18.8564	0.989705
$364$	0	0
$365$	−0.535898	−0.0280502
$366$	0	0
$367$	−8.78461	−0.458553	−0.229276	−	0.973361i	$-0.573636\pi$
$367$	−8.78461	−0.458553	−0.229276	+	0.973361i	$0.573636\pi$
$368$	0	0
$369$	−4.92820	−0.256552
$370$	0	0
$371$	0.535898	0.0278225
$372$	0	0
$373$	−19.0718	−0.987500	−0.493750	−	0.869604i	$-0.664374\pi$
$373$	−19.0718	−0.987500	−0.493750	+	0.869604i	$0.664374\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−6.92820	−0.356821
$378$	0	0
$379$	−30.9282	−1.58868	−0.794338	−	0.607477i	$-0.792182\pi$
$379$	−30.9282	−1.58868	−0.794338	+	0.607477i	$0.792182\pi$
$380$	0	0
$381$	5.07180	0.259836
$382$	0	0
$383$	−21.8564	−1.11681	−0.558405	−	0.829568i	$-0.688587\pi$
$383$	−21.8564	−1.11681	−0.558405	+	0.829568i	$0.688587\pi$
$384$	0	0
$385$	5.46410	0.278476
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	−18.7846	−0.952418	−0.476209	−	0.879332i	$-0.657989\pi$
$389$	−18.7846	−0.952418	−0.476209	+	0.879332i	$0.657989\pi$
$390$	0	0
$391$	−13.8564	−0.700749
$392$	0	0
$393$	−2.92820	−0.147708
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	11.4641	0.575367	0.287683	−	0.957726i	$-0.407115\pi$
$397$	11.4641	0.575367	0.287683	+	0.957726i	$0.407115\pi$
$398$	0	0
$399$	−5.46410	−0.273547
$400$	0	0
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	−18.9282	−0.942881
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	10.9282	0.541691
$408$	0	0
$409$	2.00000	0.0988936	0.0494468	−	0.998777i	$-0.484254\pi$
$409$	2.00000	0.0988936	0.0494468	+	0.998777i	$0.484254\pi$
$410$	0	0
$411$	−0.535898	−0.0264339
$412$	0	0
$413$	13.8564	0.681829
$414$	0	0
$415$	−14.9282	−0.732797
$416$	0	0
$417$	5.46410	0.267578
$418$	0	0
$419$	8.78461	0.429156	0.214578	−	0.976707i	$-0.431162\pi$
$419$	8.78461	0.429156	0.214578	+	0.976707i	$0.431162\pi$
$420$	0	0
$421$	−15.8564	−0.772794	−0.386397	−	0.922333i	$-0.626281\pi$
$421$	−15.8564	−0.772794	−0.386397	+	0.922333i	$0.626281\pi$
$422$	0	0
$423$	10.9282	0.531347
$424$	0	0
$425$	2.00000	0.0970143
$426$	0	0
$427$	4.92820	0.238492
$428$	0	0
$429$	18.9282	0.913862
$430$	0	0
$431$	22.2487	1.07168	0.535841	−	0.844319i	$-0.319995\pi$
$431$	22.2487	1.07168	0.535841	+	0.844319i	$0.319995\pi$
$432$	0	0
$433$	35.4641	1.70430	0.852148	−	0.523301i	$-0.175300\pi$
$433$	35.4641	1.70430	0.852148	+	0.523301i	$0.175300\pi$
$434$	0	0
$435$	2.00000	0.0958927
$436$	0	0
$437$	−37.8564	−1.81092
$438$	0	0
$439$	−27.3205	−1.30394	−0.651968	−	0.758246i	$-0.726057\pi$
$439$	−27.3205	−1.30394	−0.651968	+	0.758246i	$0.726057\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	24.0000	1.14027	0.570137	−	0.821549i	$-0.306890\pi$
$443$	24.0000	1.14027	0.570137	+	0.821549i	$0.306890\pi$
$444$	0	0
$445$	12.9282	0.612856
$446$	0	0
$447$	−20.9282	−0.989870
$448$	0	0
$449$	23.8564	1.12585	0.562927	−	0.826507i	$-0.309675\pi$
$449$	23.8564	1.12585	0.562927	+	0.826507i	$0.309675\pi$
$450$	0	0
$451$	−26.9282	−1.26800
$452$	0	0
$453$	14.9282	0.701388
$454$	0	0
$455$	3.46410	0.162400
$456$	0	0
$457$	−14.7846	−0.691595	−0.345797	−	0.938309i	$-0.612392\pi$
$457$	−14.7846	−0.691595	−0.345797	+	0.938309i	$0.612392\pi$
$458$	0	0
$459$	2.00000	0.0933520
$460$	0	0
$461$	4.14359	0.192986	0.0964932	−	0.995334i	$-0.469237\pi$
$461$	4.14359	0.192986	0.0964932	+	0.995334i	$0.469237\pi$
$462$	0	0
$463$	34.9282	1.62325	0.811626	−	0.584178i	$-0.198583\pi$
$463$	34.9282	1.62325	0.811626	+	0.584178i	$0.198583\pi$
$464$	0	0
$465$	5.46410	0.253392
$466$	0	0
$467$	−30.9282	−1.43119	−0.715593	−	0.698517i	$-0.753844\pi$
$467$	−30.9282	−1.43119	−0.715593	+	0.698517i	$0.753844\pi$
$468$	0	0
$469$	6.92820	0.319915
$470$	0	0
$471$	0.535898	0.0246929
$472$	0	0
$473$	21.8564	1.00496
$474$	0	0
$475$	5.46410	0.250710
$476$	0	0
$477$	−0.535898	−0.0245371
$478$	0	0
$479$	13.8564	0.633115	0.316558	−	0.948573i	$-0.397473\pi$
$479$	13.8564	0.633115	0.316558	+	0.948573i	$0.397473\pi$
$480$	0	0
$481$	6.92820	0.315899
$482$	0	0
$483$	6.92820	0.315244
$484$	0	0
$485$	−19.4641	−0.883819
$486$	0	0
$487$	16.7846	0.760583	0.380292	−	0.924867i	$-0.375824\pi$
$487$	16.7846	0.760583	0.380292	+	0.924867i	$0.375824\pi$
$488$	0	0
$489$	12.0000	0.542659
$490$	0	0
$491$	13.4641	0.607626	0.303813	−	0.952732i	$-0.401740\pi$
$491$	13.4641	0.607626	0.303813	+	0.952732i	$0.401740\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	0	0
$495$	−5.46410	−0.245593
$496$	0	0
$497$	−16.3923	−0.735295
$498$	0	0
$499$	−22.9282	−1.02641	−0.513204	−	0.858267i	$-0.671541\pi$
$499$	−22.9282	−1.02641	−0.513204	+	0.858267i	$0.671541\pi$
$500$	0	0
$501$	8.00000	0.357414
$502$	0	0
$503$	16.0000	0.713405	0.356702	−	0.934218i	$-0.383901\pi$
$503$	16.0000	0.713405	0.356702	+	0.934218i	$0.383901\pi$
$504$	0	0
$505$	−15.8564	−0.705601
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	21.7128	0.962404	0.481202	−	0.876610i	$-0.340200\pi$
$509$	21.7128	0.962404	0.481202	+	0.876610i	$0.340200\pi$
$510$	0	0
$511$	−0.535898	−0.0237067
$512$	0	0
$513$	5.46410	0.241246
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	59.7128	2.62617
$518$	0	0
$519$	−24.9282	−1.09423
$520$	0	0
$521$	28.6410	1.25479	0.627393	−	0.778703i	$-0.284122\pi$
$521$	28.6410	1.25479	0.627393	+	0.778703i	$0.284122\pi$
$522$	0	0
$523$	−28.7846	−1.25866	−0.629332	−	0.777137i	$-0.716671\pi$
$523$	−28.7846	−1.25866	−0.629332	+	0.777137i	$0.716671\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	−10.9282	−0.476040
$528$	0	0
$529$	25.0000	1.08696
$530$	0	0
$531$	−13.8564	−0.601317
$532$	0	0
$533$	−17.0718	−0.739462
$534$	0	0
$535$	−18.9282	−0.818338
$536$	0	0
$537$	−2.53590	−0.109432
$538$	0	0
$539$	5.46410	0.235356
$540$	0	0
$541$	0.143594	0.00617357	0.00308678	−	0.999995i	$-0.499017\pi$
$541$	0.143594	0.00617357	0.00308678	+	0.999995i	$0.499017\pi$
$542$	0	0
$543$	6.00000	0.257485
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	−38.9282	−1.66445	−0.832225	−	0.554438i	$-0.812933\pi$
$547$	−38.9282	−1.66445	−0.832225	+	0.554438i	$0.812933\pi$
$548$	0	0
$549$	−4.92820	−0.210331
$550$	0	0
$551$	−10.9282	−0.465557
$552$	0	0
$553$	−4.00000	−0.170097
$554$	0	0
$555$	−2.00000	−0.0848953
$556$	0	0
$557$	9.60770	0.407091	0.203546	−	0.979065i	$-0.434753\pi$
$557$	9.60770	0.407091	0.203546	+	0.979065i	$0.434753\pi$
$558$	0	0
$559$	13.8564	0.586064
$560$	0	0
$561$	10.9282	0.461389
$562$	0	0
$563$	−1.85641	−0.0782382	−0.0391191	−	0.999235i	$-0.512455\pi$
$563$	−1.85641	−0.0782382	−0.0391191	+	0.999235i	$0.512455\pi$
$564$	0	0
$565$	−10.3923	−0.437208
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	7.85641	0.329358	0.164679	−	0.986347i	$-0.447341\pi$
$569$	7.85641	0.329358	0.164679	+	0.986347i	$0.447341\pi$
$570$	0	0
$571$	36.7846	1.53939	0.769694	−	0.638413i	$-0.220409\pi$
$571$	36.7846	1.53939	0.769694	+	0.638413i	$0.220409\pi$
$572$	0	0
$573$	−8.39230	−0.350594
$574$	0	0
$575$	−6.92820	−0.288926
$576$	0	0
$577$	33.3205	1.38715	0.693575	−	0.720384i	$-0.256034\pi$
$577$	33.3205	1.38715	0.693575	+	0.720384i	$0.256034\pi$
$578$	0	0
$579$	−11.8564	−0.492735
$580$	0	0
$581$	−14.9282	−0.619326
$582$	0	0
$583$	−2.92820	−0.121274
$584$	0	0
$585$	−3.46410	−0.143223
$586$	0	0
$587$	−9.85641	−0.406817	−0.203409	−	0.979094i	$-0.565202\pi$
$587$	−9.85641	−0.406817	−0.203409	+	0.979094i	$0.565202\pi$
$588$	0	0
$589$	−29.8564	−1.23021
$590$	0	0
$591$	−19.4641	−0.800646
$592$	0	0
$593$	−9.71281	−0.398857	−0.199429	−	0.979912i	$-0.563909\pi$
$593$	−9.71281	−0.398857	−0.199429	+	0.979912i	$0.563909\pi$
$594$	0	0
$595$	2.00000	0.0819920
$596$	0	0
$597$	0.392305	0.0160560
$598$	0	0
$599$	−26.5359	−1.08423	−0.542114	−	0.840305i	$-0.682376\pi$
$599$	−26.5359	−1.08423	−0.542114	+	0.840305i	$0.682376\pi$
$600$	0	0
$601$	28.9282	1.18001	0.590003	−	0.807401i	$-0.299127\pi$
$601$	28.9282	1.18001	0.590003	+	0.807401i	$0.299127\pi$
$602$	0	0
$603$	−6.92820	−0.282138
$604$	0	0
$605$	−18.8564	−0.766622
$606$	0	0
$607$	34.9282	1.41769	0.708846	−	0.705363i	$-0.249216\pi$
$607$	34.9282	1.41769	0.708846	+	0.705363i	$0.249216\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	37.8564	1.53151
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	4.92820	0.198724
$616$	0	0
$617$	−13.6077	−0.547825	−0.273913	−	0.961755i	$-0.588318\pi$
$617$	−13.6077	−0.547825	−0.273913	+	0.961755i	$0.588318\pi$
$618$	0	0
$619$	−32.3923	−1.30196	−0.650978	−	0.759096i	$-0.725641\pi$
$619$	−32.3923	−1.30196	−0.650978	+	0.759096i	$0.725641\pi$
$620$	0	0
$621$	−6.92820	−0.278019
$622$	0	0
$623$	12.9282	0.517958
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	29.8564	1.19235
$628$	0	0
$629$	4.00000	0.159490
$630$	0	0
$631$	−15.7128	−0.625517	−0.312759	−	0.949833i	$-0.601253\pi$
$631$	−15.7128	−0.625517	−0.312759	+	0.949833i	$0.601253\pi$
$632$	0	0
$633$	4.00000	0.158986
$634$	0	0
$635$	−5.07180	−0.201268
$636$	0	0
$637$	3.46410	0.137253
$638$	0	0
$639$	16.3923	0.648470
$640$	0	0
$641$	−38.7846	−1.53190	−0.765950	−	0.642900i	$-0.777731\pi$
$641$	−38.7846	−1.53190	−0.765950	+	0.642900i	$0.777731\pi$
$642$	0	0
$643$	−44.7846	−1.76613	−0.883066	−	0.469248i	$-0.844525\pi$
$643$	−44.7846	−1.76613	−0.883066	+	0.469248i	$0.844525\pi$
$644$	0	0
$645$	−4.00000	−0.157500
$646$	0	0
$647$	−13.8564	−0.544752	−0.272376	−	0.962191i	$-0.587809\pi$
$647$	−13.8564	−0.544752	−0.272376	+	0.962191i	$0.587809\pi$
$648$	0	0
$649$	−75.7128	−2.97199
$650$	0	0
$651$	5.46410	0.214155
$652$	0	0
$653$	−27.4641	−1.07475	−0.537377	−	0.843342i	$-0.680585\pi$
$653$	−27.4641	−1.07475	−0.537377	+	0.843342i	$0.680585\pi$
$654$	0	0
$655$	2.92820	0.114414
$656$	0	0
$657$	0.535898	0.0209074
$658$	0	0
$659$	−37.4641	−1.45939	−0.729697	−	0.683771i	$-0.760339\pi$
$659$	−37.4641	−1.45939	−0.729697	+	0.683771i	$0.760339\pi$
$660$	0	0
$661$	−7.07180	−0.275061	−0.137531	−	0.990498i	$-0.543917\pi$
$661$	−7.07180	−0.275061	−0.137531	+	0.990498i	$0.543917\pi$
$662$	0	0
$663$	6.92820	0.269069
$664$	0	0
$665$	5.46410	0.211889
$666$	0	0
$667$	13.8564	0.536522
$668$	0	0
$669$	26.9282	1.04110
$670$	0	0
$671$	−26.9282	−1.03955
$672$	0	0
$673$	28.9282	1.11510	0.557550	−	0.830143i	$-0.311741\pi$
$673$	28.9282	1.11510	0.557550	+	0.830143i	$0.311741\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	10.7846	0.414486	0.207243	−	0.978289i	$-0.433551\pi$
$677$	10.7846	0.414486	0.207243	+	0.978289i	$0.433551\pi$
$678$	0	0
$679$	−19.4641	−0.746964
$680$	0	0
$681$	−6.92820	−0.265489
$682$	0	0
$683$	−5.85641	−0.224089	−0.112045	−	0.993703i	$-0.535740\pi$
$683$	−5.85641	−0.224089	−0.112045	+	0.993703i	$0.535740\pi$
$684$	0	0
$685$	0.535898	0.0204756
$686$	0	0
$687$	22.7846	0.869287
$688$	0	0
$689$	−1.85641	−0.0707235
$690$	0	0
$691$	6.24871	0.237712	0.118856	−	0.992911i	$-0.462077\pi$
$691$	6.24871	0.237712	0.118856	+	0.992911i	$0.462077\pi$
$692$	0	0
$693$	−5.46410	−0.207564
$694$	0	0
$695$	−5.46410	−0.207265
$696$	0	0
$697$	−9.85641	−0.373338
$698$	0	0
$699$	−19.4641	−0.736200
$700$	0	0
$701$	19.8564	0.749966	0.374983	−	0.927032i	$-0.377649\pi$
$701$	19.8564	0.749966	0.374983	+	0.927032i	$0.377649\pi$
$702$	0	0
$703$	10.9282	0.412165
$704$	0	0
$705$	−10.9282	−0.411580
$706$	0	0
$707$	−15.8564	−0.596342
$708$	0	0
$709$	−37.7128	−1.41633	−0.708167	−	0.706045i	$-0.750478\pi$
$709$	−37.7128	−1.41633	−0.708167	+	0.706045i	$0.750478\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	37.8564	1.41773
$714$	0	0
$715$	−18.9282	−0.707875
$716$	0	0
$717$	−19.3205	−0.721538
$718$	0	0
$719$	16.7846	0.625960	0.312980	−	0.949760i	$-0.398673\pi$
$719$	16.7846	0.625960	0.312980	+	0.949760i	$0.398673\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	−14.7846	−0.549846
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	10.9282	0.405305	0.202652	−	0.979251i	$-0.435044\pi$
$727$	10.9282	0.405305	0.202652	+	0.979251i	$0.435044\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	−10.3923	−0.383849	−0.191924	−	0.981410i	$-0.561473\pi$
$733$	−10.3923	−0.383849	−0.191924	+	0.981410i	$0.561473\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	−37.8564	−1.39446
$738$	0	0
$739$	12.7846	0.470289	0.235145	−	0.971960i	$-0.424444\pi$
$739$	12.7846	0.470289	0.235145	+	0.971960i	$0.424444\pi$
$740$	0	0
$741$	18.9282	0.695345
$742$	0	0
$743$	25.8564	0.948580	0.474290	−	0.880369i	$-0.342705\pi$
$743$	25.8564	0.948580	0.474290	+	0.880369i	$0.342705\pi$
$744$	0	0
$745$	20.9282	0.766750
$746$	0	0
$747$	14.9282	0.546194
$748$	0	0
$749$	−18.9282	−0.691621
$750$	0	0
$751$	45.5692	1.66284	0.831422	−	0.555641i	$-0.187527\pi$
$751$	45.5692	1.66284	0.831422	+	0.555641i	$0.187527\pi$
$752$	0	0
$753$	2.14359	0.0781169
$754$	0	0
$755$	−14.9282	−0.543293
$756$	0	0
$757$	−19.0718	−0.693176	−0.346588	−	0.938017i	$-0.612660\pi$
$757$	−19.0718	−0.693176	−0.346588	+	0.938017i	$0.612660\pi$
$758$	0	0
$759$	−37.8564	−1.37410
$760$	0	0
$761$	−31.0718	−1.12635	−0.563176	−	0.826337i	$-0.690421\pi$
$761$	−31.0718	−1.12635	−0.563176	+	0.826337i	$0.690421\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	0	0
$765$	−2.00000	−0.0723102
$766$	0	0
$767$	−48.0000	−1.73318
$768$	0	0
$769$	−28.6410	−1.03282	−0.516411	−	0.856341i	$-0.672732\pi$
$769$	−28.6410	−1.03282	−0.516411	+	0.856341i	$0.672732\pi$
$770$	0	0
$771$	−3.07180	−0.110628
$772$	0	0
$773$	−8.92820	−0.321125	−0.160563	−	0.987026i	$-0.551331\pi$
$773$	−8.92820	−0.321125	−0.160563	+	0.987026i	$0.551331\pi$
$774$	0	0
$775$	−5.46410	−0.196276
$776$	0	0
$777$	−2.00000	−0.0717496
$778$	0	0
$779$	−26.9282	−0.964803
$780$	0	0
$781$	89.5692	3.20504
$782$	0	0
$783$	−2.00000	−0.0714742
$784$	0	0
$785$	−0.535898	−0.0191270
$786$	0	0
$787$	12.7846	0.455722	0.227861	−	0.973694i	$-0.426827\pi$
$787$	12.7846	0.455722	0.227861	+	0.973694i	$0.426827\pi$
$788$	0	0
$789$	−17.8564	−0.635705
$790$	0	0
$791$	−10.3923	−0.369508
$792$	0	0
$793$	−17.0718	−0.606237
$794$	0	0
$795$	0.535898	0.0190064
$796$	0	0
$797$	23.0718	0.817245	0.408622	−	0.912703i	$-0.366009\pi$
$797$	23.0718	0.817245	0.408622	+	0.912703i	$0.366009\pi$
$798$	0	0
$799$	21.8564	0.773224
$800$	0	0
$801$	−12.9282	−0.456796
$802$	0	0
$803$	2.92820	0.103334
$804$	0	0
$805$	−6.92820	−0.244187
$806$	0	0
$807$	−6.00000	−0.211210
$808$	0	0
$809$	−38.7846	−1.36359	−0.681797	−	0.731541i	$-0.738801\pi$
$809$	−38.7846	−1.36359	−0.681797	+	0.731541i	$0.738801\pi$
$810$	0	0
$811$	41.1769	1.44592	0.722959	−	0.690891i	$-0.242782\pi$
$811$	41.1769	1.44592	0.722959	+	0.690891i	$0.242782\pi$
$812$	0	0
$813$	13.4641	0.472207
$814$	0	0
$815$	−12.0000	−0.420342
$816$	0	0
$817$	21.8564	0.764659
$818$	0	0
$819$	−3.46410	−0.121046
$820$	0	0
$821$	25.7128	0.897383	0.448692	−	0.893687i	$-0.351890\pi$
$821$	25.7128	0.897383	0.448692	+	0.893687i	$0.351890\pi$
$822$	0	0
$823$	−8.78461	−0.306212	−0.153106	−	0.988210i	$-0.548928\pi$
$823$	−8.78461	−0.306212	−0.153106	+	0.988210i	$0.548928\pi$
$824$	0	0
$825$	5.46410	0.190236
$826$	0	0
$827$	−10.9282	−0.380011	−0.190005	−	0.981783i	$-0.560851\pi$
$827$	−10.9282	−0.380011	−0.190005	+	0.981783i	$0.560851\pi$
$828$	0	0
$829$	3.85641	0.133939	0.0669693	−	0.997755i	$-0.478667\pi$
$829$	3.85641	0.133939	0.0669693	+	0.997755i	$0.478667\pi$
$830$	0	0
$831$	−11.8564	−0.411294
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	−5.46410	−0.188867
$838$	0	0
$839$	−32.7846	−1.13185	−0.565925	−	0.824457i	$-0.691481\pi$
$839$	−32.7846	−1.13185	−0.565925	+	0.824457i	$0.691481\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−30.7846	−1.06028
$844$	0	0
$845$	1.00000	0.0344010
$846$	0	0
$847$	−18.8564	−0.647914
$848$	0	0
$849$	17.8564	0.612830
$850$	0	0
$851$	−13.8564	−0.474991
$852$	0	0
$853$	−18.3923	−0.629741	−0.314870	−	0.949135i	$-0.601961\pi$
$853$	−18.3923	−0.629741	−0.314870	+	0.949135i	$0.601961\pi$
$854$	0	0
$855$	−5.46410	−0.186868
$856$	0	0
$857$	−11.0718	−0.378205	−0.189103	−	0.981957i	$-0.560558\pi$
$857$	−11.0718	−0.378205	−0.189103	+	0.981957i	$0.560558\pi$
$858$	0	0
$859$	−23.6077	−0.805484	−0.402742	−	0.915314i	$-0.631943\pi$
$859$	−23.6077	−0.805484	−0.402742	+	0.915314i	$0.631943\pi$
$860$	0	0
$861$	4.92820	0.167953
$862$	0	0
$863$	−33.0718	−1.12578	−0.562889	−	0.826533i	$-0.690310\pi$
$863$	−33.0718	−1.12578	−0.562889	+	0.826533i	$0.690310\pi$
$864$	0	0
$865$	24.9282	0.847584
$866$	0	0
$867$	−13.0000	−0.441503
$868$	0	0
$869$	21.8564	0.741428
$870$	0	0
$871$	−24.0000	−0.813209
$872$	0	0
$873$	19.4641	0.658760
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	34.0000	1.14810	0.574049	−	0.818821i	$-0.305372\pi$
$877$	34.0000	1.14810	0.574049	+	0.818821i	$0.305372\pi$
$878$	0	0
$879$	−24.9282	−0.840807
$880$	0	0
$881$	46.7846	1.57621	0.788107	−	0.615539i	$-0.211061\pi$
$881$	46.7846	1.57621	0.788107	+	0.615539i	$0.211061\pi$
$882$	0	0
$883$	26.6410	0.896542	0.448271	−	0.893898i	$-0.352040\pi$
$883$	26.6410	0.896542	0.448271	+	0.893898i	$0.352040\pi$
$884$	0	0
$885$	13.8564	0.465778
$886$	0	0
$887$	−2.92820	−0.0983194	−0.0491597	−	0.998791i	$-0.515654\pi$
$887$	−2.92820	−0.0983194	−0.0491597	+	0.998791i	$0.515654\pi$
$888$	0	0
$889$	−5.07180	−0.170103
$890$	0	0
$891$	5.46410	0.183054
$892$	0	0
$893$	59.7128	1.99821
$894$	0	0
$895$	2.53590	0.0847657
$896$	0	0
$897$	−24.0000	−0.801337
$898$	0	0
$899$	10.9282	0.364476
$900$	0	0
$901$	−1.07180	−0.0357067
$902$	0	0
$903$	−4.00000	−0.133112
$904$	0	0
$905$	−6.00000	−0.199447
$906$	0	0
$907$	−12.7846	−0.424506	−0.212253	−	0.977215i	$-0.568080\pi$
$907$	−12.7846	−0.424506	−0.212253	+	0.977215i	$0.568080\pi$
$908$	0	0
$909$	15.8564	0.525924
$910$	0	0
$911$	−22.2487	−0.737133	−0.368566	−	0.929601i	$-0.620151\pi$
$911$	−22.2487	−0.737133	−0.368566	+	0.929601i	$0.620151\pi$
$912$	0	0
$913$	81.5692	2.69955
$914$	0	0
$915$	4.92820	0.162921
$916$	0	0
$917$	2.92820	0.0966978
$918$	0	0
$919$	−28.0000	−0.923635	−0.461817	−	0.886975i	$-0.652802\pi$
$919$	−28.0000	−0.923635	−0.461817	+	0.886975i	$0.652802\pi$
$920$	0	0
$921$	−22.9282	−0.755510
$922$	0	0
$923$	56.7846	1.86909
$924$	0	0
$925$	2.00000	0.0657596
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	52.6410	1.72710	0.863548	−	0.504267i	$-0.168237\pi$
$929$	52.6410	1.72710	0.863548	+	0.504267i	$0.168237\pi$
$930$	0	0
$931$	5.46410	0.179079
$932$	0	0
$933$	−8.00000	−0.261908
$934$	0	0
$935$	−10.9282	−0.357390
$936$	0	0
$937$	−27.1769	−0.887831	−0.443916	−	0.896069i	$-0.646411\pi$
$937$	−27.1769	−0.887831	−0.443916	+	0.896069i	$0.646411\pi$
$938$	0	0
$939$	17.3205	0.565233
$940$	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
$943$	34.1436	1.11187
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	16.0000	0.519930	0.259965	−	0.965618i	$-0.416289\pi$
$947$	16.0000	0.519930	0.259965	+	0.965618i	$0.416289\pi$
$948$	0	0
$949$	1.85641	0.0602615
$950$	0	0
$951$	13.3205	0.431947
$952$	0	0
$953$	−49.3205	−1.59765	−0.798824	−	0.601565i	$-0.794544\pi$
$953$	−49.3205	−1.59765	−0.798824	+	0.601565i	$0.794544\pi$
$954$	0	0
$955$	8.39230	0.271569
$956$	0	0
$957$	−10.9282	−0.353259
$958$	0	0
$959$	0.535898	0.0173051
$960$	0	0
$961$	−1.14359	−0.0368901
$962$	0	0
$963$	18.9282	0.609953
$964$	0	0
$965$	11.8564	0.381671
$966$	0	0
$967$	−8.00000	−0.257263	−0.128631	−	0.991692i	$-0.541058\pi$
$967$	−8.00000	−0.257263	−0.128631	+	0.991692i	$0.541058\pi$
$968$	0	0
$969$	10.9282	0.351064
$970$	0	0
$971$	−10.1436	−0.325523	−0.162762	−	0.986665i	$-0.552040\pi$
$971$	−10.1436	−0.325523	−0.162762	+	0.986665i	$0.552040\pi$
$972$	0	0
$973$	−5.46410	−0.175171
$974$	0	0
$975$	3.46410	0.110940
$976$	0	0
$977$	−13.6077	−0.435349	−0.217674	−	0.976021i	$-0.569847\pi$
$977$	−13.6077	−0.435349	−0.217674	+	0.976021i	$0.569847\pi$
$978$	0	0
$979$	−70.6410	−2.25770
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	40.7846	1.30083	0.650414	−	0.759580i	$-0.274596\pi$
$983$	40.7846	1.30083	0.650414	+	0.759580i	$0.274596\pi$
$984$	0	0
$985$	19.4641	0.620178
$986$	0	0
$987$	−10.9282	−0.347849
$988$	0	0
$989$	−27.7128	−0.881216
$990$	0	0
$991$	−51.4256	−1.63359	−0.816794	−	0.576929i	$-0.804251\pi$
$991$	−51.4256	−1.63359	−0.816794	+	0.576929i	$0.804251\pi$
$992$	0	0
$993$	4.00000	0.126936
$994$	0	0
$995$	−0.392305	−0.0124369
$996$	0	0
$997$	−0.248711	−0.00787677	−0.00393838	−	0.999992i	$-0.501254\pi$
$997$	−0.248711	−0.00787677	−0.00393838	+	0.999992i	$0.501254\pi$
$998$	0	0
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3360.2.a.bf.1.2	yes	2
4.3	odd	2		3360.2.a.bb.1.1	✓	2
8.3	odd	2		6720.2.a.cz.1.2		2
8.5	even	2		6720.2.a.cq.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3360.2.a.bb.1.1	✓	2	4.3	odd	2
3360.2.a.bf.1.2	yes	2	1.1	even	1	trivial
6720.2.a.cq.1.1		2	8.5	even	2
6720.2.a.cz.1.2		2	8.3	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:	\( N \)	\(=\)	\( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3360.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.46410 q^{11} +3.46410 q^{13} -1.00000 q^{15} +2.00000 q^{17} +5.46410 q^{19} -1.00000 q^{21} -6.92820 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{29} -5.46410 q^{31} +5.46410 q^{33} +1.00000 q^{35} +2.00000 q^{37} +3.46410 q^{39} -4.92820 q^{41} +4.00000 q^{43} -1.00000 q^{45} +10.9282 q^{47} +1.00000 q^{49} +2.00000 q^{51} -0.535898 q^{53} -5.46410 q^{55} +5.46410 q^{57} -13.8564 q^{59} -4.92820 q^{61} -1.00000 q^{63} -3.46410 q^{65} -6.92820 q^{67} -6.92820 q^{69} +16.3923 q^{71} +0.535898 q^{73} +1.00000 q^{75} -5.46410 q^{77} +4.00000 q^{79} +1.00000 q^{81} +14.9282 q^{83} -2.00000 q^{85} -2.00000 q^{87} -12.9282 q^{89} -3.46410 q^{91} -5.46410 q^{93} -5.46410 q^{95} +19.4641 q^{97} +5.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{11} - 2 q^{15} + 4 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 4 q^{31} + 4 q^{33} + 2 q^{35} + 4 q^{37} + 4 q^{41} + 8 q^{43} - 2 q^{45} + 8 q^{47} + 2 q^{49} + 4 q^{51} - 8 q^{53} - 4 q^{55} + 4 q^{57} + 4 q^{61} - 2 q^{63} + 12 q^{71} + 8 q^{73} + 2 q^{75} - 4 q^{77} + 8 q^{79} + 2 q^{81} + 16 q^{83} - 4 q^{85} - 4 q^{87} - 12 q^{89} - 4 q^{93} - 4 q^{95} + 32 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 4 * q^11 - 2 * q^15 + 4 * q^17 + 4 * q^19 - 2 * q^21 + 2 * q^25 + 2 * q^27 - 4 * q^29 - 4 * q^31 + 4 * q^33 + 2 * q^35 + 4 * q^37 + 4 * q^41 + 8 * q^43 - 2 * q^45 + 8 * q^47 + 2 * q^49 + 4 * q^51 - 8 * q^53 - 4 * q^55 + 4 * q^57 + 4 * q^61 - 2 * q^63 + 12 * q^71 + 8 * q^73 + 2 * q^75 - 4 * q^77 + 8 * q^79 + 2 * q^81 + 16 * q^83 - 4 * q^85 - 4 * q^87 - 12 * q^89 - 4 * q^93 - 4 * q^95 + 32 * q^97 + 4 * q^99

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