LMFDB - Embedded newform 6930.2.a.cj.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6930,2,Mod(1,6930)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6930, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6930.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6930.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:	$55.3363286007$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	6930.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +3.75877 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.75877 q^{17} -8.12836 q^{19} -1.00000 q^{20} -1.00000 q^{22} -7.14796 q^{23} +1.00000 q^{25} +3.75877 q^{26} +1.00000 q^{28} -9.51754 q^{29} +6.12836 q^{31} +1.00000 q^{32} +1.75877 q^{34} -1.00000 q^{35} -5.75877 q^{37} -8.12836 q^{38} -1.00000 q^{40} +4.12836 q^{41} -8.12836 q^{43} -1.00000 q^{44} -7.14796 q^{46} -6.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} +3.75877 q^{52} -0.610815 q^{53} +1.00000 q^{55} +1.00000 q^{56} -9.51754 q^{58} -10.1284 q^{59} -4.00000 q^{61} +6.12836 q^{62} +1.00000 q^{64} -3.75877 q^{65} +4.98040 q^{67} +1.75877 q^{68} -1.00000 q^{70} +13.0351 q^{71} -8.12836 q^{73} -5.75877 q^{74} -8.12836 q^{76} -1.00000 q^{77} +15.0351 q^{79} -1.00000 q^{80} +4.12836 q^{82} +8.90673 q^{83} -1.75877 q^{85} -8.12836 q^{86} -1.00000 q^{88} -16.6655 q^{89} +3.75877 q^{91} -7.14796 q^{92} -6.00000 q^{94} +8.12836 q^{95} +15.6459 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 + 3 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8} - 3 q^{10} - 3 q^{11} + 3 q^{14} + 3 q^{16} - 6 q^{17} - 6 q^{19} - 3 q^{20} - 3 q^{22} - 6 q^{23} + 3 q^{25} + 3 q^{28} - 6 q^{29} + 3 q^{32} - 6 q^{34} - 3 q^{35} - 6 q^{37} - 6 q^{38} - 3 q^{40} - 6 q^{41} - 6 q^{43} - 3 q^{44} - 6 q^{46} - 18 q^{47} + 3 q^{49} + 3 q^{50} - 6 q^{53} + 3 q^{55} + 3 q^{56} - 6 q^{58} - 12 q^{59} - 12 q^{61} + 3 q^{64} + 12 q^{67} - 6 q^{68} - 3 q^{70} - 6 q^{71} - 6 q^{73} - 6 q^{74} - 6 q^{76} - 3 q^{77} - 3 q^{80} - 6 q^{82} + 6 q^{85} - 6 q^{86} - 3 q^{88} - 12 q^{89} - 6 q^{92} - 18 q^{94} + 6 q^{95} + 6 q^{97} + 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 + 3 * q^7 + 3 * q^8 - 3 * q^10 - 3 * q^11 + 3 * q^14 + 3 * q^16 - 6 * q^17 - 6 * q^19 - 3 * q^20 - 3 * q^22 - 6 * q^23 + 3 * q^25 + 3 * q^28 - 6 * q^29 + 3 * q^32 - 6 * q^34 - 3 * q^35 - 6 * q^37 - 6 * q^38 - 3 * q^40 - 6 * q^41 - 6 * q^43 - 3 * q^44 - 6 * q^46 - 18 * q^47 + 3 * q^49 + 3 * q^50 - 6 * q^53 + 3 * q^55 + 3 * q^56 - 6 * q^58 - 12 * q^59 - 12 * q^61 + 3 * q^64 + 12 * q^67 - 6 * q^68 - 3 * q^70 - 6 * q^71 - 6 * q^73 - 6 * q^74 - 6 * q^76 - 3 * q^77 - 3 * q^80 - 6 * q^82 + 6 * q^85 - 6 * q^86 - 3 * q^88 - 12 * q^89 - 6 * q^92 - 18 * q^94 + 6 * q^95 + 6 * q^97 + 3 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.75877	1.04250	0.521248	−	0.853405i	$-0.325467\pi$
$13$	3.75877	1.04250	0.521248	+	0.853405i	$0.325467\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	1.75877	0.426564	0.213282	−	0.976991i	$-0.431585\pi$
$17$	1.75877	0.426564	0.213282	+	0.976991i	$0.431585\pi$
$18$	0	0
$19$	−8.12836	−1.86477	−0.932386	−	0.361463i	$-0.882277\pi$
$19$	−8.12836	−1.86477	−0.932386	+	0.361463i	$0.882277\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−1.00000	−0.213201
$23$	−7.14796	−1.49045	−0.745226	−	0.666812i	$-0.767658\pi$
$23$	−7.14796	−1.49045	−0.745226	+	0.666812i	$0.767658\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	3.75877	0.737156
$27$	0	0
$28$	1.00000	0.188982
$29$	−9.51754	−1.76736	−0.883681	−	0.468089i	$-0.844943\pi$
$29$	−9.51754	−1.76736	−0.883681	+	0.468089i	$0.844943\pi$
$30$	0	0
$31$	6.12836	1.10069	0.550343	−	0.834939i	$-0.314497\pi$
$31$	6.12836	1.10069	0.550343	+	0.834939i	$0.314497\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.75877	0.301627
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−5.75877	−0.946736	−0.473368	−	0.880865i	$-0.656962\pi$
$37$	−5.75877	−0.946736	−0.473368	+	0.880865i	$0.656962\pi$
$38$	−8.12836	−1.31859
$39$	0	0
$40$	−1.00000	−0.158114
$41$	4.12836	0.644741	0.322370	−	0.946614i	$-0.395520\pi$
$41$	4.12836	0.644741	0.322370	+	0.946614i	$0.395520\pi$
$42$	0	0
$43$	−8.12836	−1.23956	−0.619781	−	0.784775i	$-0.712779\pi$
$43$	−8.12836	−1.23956	−0.619781	+	0.784775i	$0.712779\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	−7.14796	−1.05391
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	3.75877	0.521248
$53$	−0.610815	−0.0839018	−0.0419509	−	0.999120i	$-0.513357\pi$
$53$	−0.610815	−0.0839018	−0.0419509	+	0.999120i	$0.513357\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	1.00000	0.133631
$57$	0	0
$58$	−9.51754	−1.24971
$59$	−10.1284	−1.31860	−0.659300	−	0.751880i	$-0.729147\pi$
$59$	−10.1284	−1.31860	−0.659300	+	0.751880i	$0.729147\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	6.12836	0.778302
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.75877	−0.466218
$66$	0	0
$67$	4.98040	0.608453	0.304226	−	0.952600i	$-0.401602\pi$
$67$	4.98040	0.608453	0.304226	+	0.952600i	$0.401602\pi$
$68$	1.75877	0.213282
$69$	0	0
$70$	−1.00000	−0.119523
$71$	13.0351	1.54698	0.773490	−	0.633809i	$-0.218509\pi$
$71$	13.0351	1.54698	0.773490	+	0.633809i	$0.218509\pi$
$72$	0	0
$73$	−8.12836	−0.951352	−0.475676	−	0.879621i	$-0.657797\pi$
$73$	−8.12836	−0.951352	−0.475676	+	0.879621i	$0.657797\pi$
$74$	−5.75877	−0.669443
$75$	0	0
$76$	−8.12836	−0.932386
$77$	−1.00000	−0.113961
$78$	0	0
$79$	15.0351	1.69158	0.845789	−	0.533517i	$-0.179130\pi$
$79$	15.0351	1.69158	0.845789	+	0.533517i	$0.179130\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	4.12836	0.455901
$83$	8.90673	0.977640	0.488820	−	0.872385i	$-0.337427\pi$
$83$	8.90673	0.977640	0.488820	+	0.872385i	$0.337427\pi$
$84$	0	0
$85$	−1.75877	−0.190765
$86$	−8.12836	−0.876503
$87$	0	0
$88$	−1.00000	−0.106600
$89$	−16.6655	−1.76654	−0.883270	−	0.468866i	$-0.844663\pi$
$89$	−16.6655	−1.76654	−0.883270	+	0.468866i	$0.844663\pi$
$90$	0	0
$91$	3.75877	0.394026
$92$	−7.14796	−0.745226
$93$	0	0
$94$	−6.00000	−0.618853
$95$	8.12836	0.833952
$96$	0	0
$97$	15.6459	1.58860	0.794300	−	0.607526i	$-0.207838\pi$
$97$	15.6459	1.58860	0.794300	+	0.607526i	$0.207838\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−10.7392	−1.06859	−0.534294	−	0.845299i	$-0.679422\pi$
$101$	−10.7392	−1.06859	−0.534294	+	0.845299i	$0.679422\pi$
$102$	0	0
$103$	6.12836	0.603845	0.301922	−	0.953333i	$-0.402372\pi$
$103$	6.12836	0.603845	0.301922	+	0.953333i	$0.402372\pi$
$104$	3.75877	0.368578
$105$	0	0
$106$	−0.610815	−0.0593276
$107$	−14.8675	−1.43730	−0.718649	−	0.695373i	$-0.755239\pi$
$107$	−14.8675	−1.43730	−0.718649	+	0.695373i	$0.755239\pi$
$108$	0	0
$109$	−15.2763	−1.46321	−0.731603	−	0.681731i	$-0.761227\pi$
$109$	−15.2763	−1.46321	−0.731603	+	0.681731i	$0.761227\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	1.00000	0.0944911
$113$	5.88713	0.553814	0.276907	−	0.960897i	$-0.410691\pi$
$113$	5.88713	0.553814	0.276907	+	0.960897i	$0.410691\pi$
$114$	0	0
$115$	7.14796	0.666550
$116$	−9.51754	−0.883681
$117$	0	0
$118$	−10.1284	−0.932391
$119$	1.75877	0.161226
$120$	0	0
$121$	1.00000	0.0909091
$122$	−4.00000	−0.362143
$123$	0	0
$124$	6.12836	0.550343
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−2.12836	−0.188861	−0.0944305	−	0.995531i	$-0.530103\pi$
$127$	−2.12836	−0.188861	−0.0944305	+	0.995531i	$0.530103\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−3.75877	−0.329666
$131$	3.51754	0.307329	0.153664	−	0.988123i	$-0.450893\pi$
$131$	3.51754	0.307329	0.153664	+	0.988123i	$0.450893\pi$
$132$	0	0
$133$	−8.12836	−0.704818
$134$	4.98040	0.430241
$135$	0	0
$136$	1.75877	0.150813
$137$	−13.1480	−1.12331	−0.561653	−	0.827373i	$-0.689834\pi$
$137$	−13.1480	−1.12331	−0.561653	+	0.827373i	$0.689834\pi$
$138$	0	0
$139$	3.64590	0.309241	0.154620	−	0.987974i	$-0.450585\pi$
$139$	3.64590	0.309241	0.154620	+	0.987974i	$0.450585\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	13.0351	1.09388
$143$	−3.75877	−0.314324
$144$	0	0
$145$	9.51754	0.790389
$146$	−8.12836	−0.672707
$147$	0	0
$148$	−5.75877	−0.473368
$149$	9.51754	0.779707	0.389854	−	0.920877i	$-0.372526\pi$
$149$	9.51754	0.779707	0.389854	+	0.920877i	$0.372526\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−8.12836	−0.659297
$153$	0	0
$154$	−1.00000	−0.0805823
$155$	−6.12836	−0.492241
$156$	0	0
$157$	−16.3851	−1.30767	−0.653835	−	0.756637i	$-0.726841\pi$
$157$	−16.3851	−1.30767	−0.653835	+	0.756637i	$0.726841\pi$
$158$	15.0351	1.19613
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−7.14796	−0.563338
$162$	0	0
$163$	−15.2763	−1.19653	−0.598267	−	0.801297i	$-0.704144\pi$
$163$	−15.2763	−1.19653	−0.598267	+	0.801297i	$0.704144\pi$
$164$	4.12836	0.322370
$165$	0	0
$166$	8.90673	0.691296
$167$	−6.11287	−0.473028	−0.236514	−	0.971628i	$-0.576005\pi$
$167$	−6.11287	−0.473028	−0.236514	+	0.971628i	$0.576005\pi$
$168$	0	0
$169$	1.12836	0.0867966
$170$	−1.75877	−0.134892
$171$	0	0
$172$	−8.12836	−0.619781
$173$	−25.0351	−1.90338	−0.951691	−	0.307057i	$-0.900656\pi$
$173$	−25.0351	−1.90338	−0.951691	+	0.307057i	$0.900656\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	−16.6655	−1.24913
$179$	17.1634	1.28286	0.641428	−	0.767183i	$-0.278342\pi$
$179$	17.1634	1.28286	0.641428	+	0.767183i	$0.278342\pi$
$180$	0	0
$181$	−24.7939	−1.84291	−0.921456	−	0.388482i	$-0.873000\pi$
$181$	−24.7939	−1.84291	−0.921456	+	0.388482i	$0.873000\pi$
$182$	3.75877	0.278619
$183$	0	0
$184$	−7.14796	−0.526954
$185$	5.75877	0.423393
$186$	0	0
$187$	−1.75877	−0.128614
$188$	−6.00000	−0.437595
$189$	0	0
$190$	8.12836	0.589693
$191$	9.51754	0.688665	0.344333	−	0.938848i	$-0.388105\pi$
$191$	9.51754	0.688665	0.344333	+	0.938848i	$0.388105\pi$
$192$	0	0
$193$	12.1284	0.873018	0.436509	−	0.899700i	$-0.356215\pi$
$193$	12.1284	0.873018	0.436509	+	0.899700i	$0.356215\pi$
$194$	15.6459	1.12331
$195$	0	0
$196$	1.00000	0.0714286
$197$	14.9067	1.06206	0.531030	−	0.847353i	$-0.321805\pi$
$197$	14.9067	1.06206	0.531030	+	0.847353i	$0.321805\pi$
$198$	0	0
$199$	21.6459	1.53444	0.767218	−	0.641386i	$-0.221640\pi$
$199$	21.6459	1.53444	0.767218	+	0.641386i	$0.221640\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−10.7392	−0.755605
$203$	−9.51754	−0.668000
$204$	0	0
$205$	−4.12836	−0.288337
$206$	6.12836	0.426983
$207$	0	0
$208$	3.75877	0.260624
$209$	8.12836	0.562250
$210$	0	0
$211$	22.7547	1.56649	0.783247	−	0.621710i	$-0.213562\pi$
$211$	22.7547	1.56649	0.783247	+	0.621710i	$0.213562\pi$
$212$	−0.610815	−0.0419509
$213$	0	0
$214$	−14.8675	−1.01632
$215$	8.12836	0.554349
$216$	0	0
$217$	6.12836	0.416020
$218$	−15.2763	−1.03464
$219$	0	0
$220$	1.00000	0.0674200
$221$	6.61081	0.444692
$222$	0	0
$223$	3.26083	0.218361	0.109181	−	0.994022i	$-0.465177\pi$
$223$	3.26083	0.218361	0.109181	+	0.994022i	$0.465177\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	5.88713	0.391606
$227$	−24.4243	−1.62110	−0.810548	−	0.585673i	$-0.800830\pi$
$227$	−24.4243	−1.62110	−0.810548	+	0.585673i	$0.800830\pi$
$228$	0	0
$229$	−2.24123	−0.148105	−0.0740523	−	0.997254i	$-0.523593\pi$
$229$	−2.24123	−0.148105	−0.0740523	+	0.997254i	$0.523593\pi$
$230$	7.14796	0.471322
$231$	0	0
$232$	−9.51754	−0.624857
$233$	−4.77837	−0.313041	−0.156521	−	0.987675i	$-0.550028\pi$
$233$	−4.77837	−0.313041	−0.156521	+	0.987675i	$0.550028\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	−10.1284	−0.659300
$237$	0	0
$238$	1.75877	0.114004
$239$	−28.0155	−1.81217	−0.906085	−	0.423095i	$-0.860944\pi$
$239$	−28.0155	−1.81217	−0.906085	+	0.423095i	$0.860944\pi$
$240$	0	0
$241$	−18.9067	−1.21789	−0.608945	−	0.793213i	$-0.708407\pi$
$241$	−18.9067	−1.21789	−0.608945	+	0.793213i	$0.708407\pi$
$242$	1.00000	0.0642824
$243$	0	0
$244$	−4.00000	−0.256074
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−30.5526	−1.94402
$248$	6.12836	0.389151
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−4.16756	−0.263054	−0.131527	−	0.991313i	$-0.541988\pi$
$251$	−4.16756	−0.263054	−0.131527	+	0.991313i	$0.541988\pi$
$252$	0	0
$253$	7.14796	0.449388
$254$	−2.12836	−0.133545
$255$	0	0
$256$	1.00000	0.0625000
$257$	−25.6459	−1.59975	−0.799874	−	0.600169i	$-0.795100\pi$
$257$	−25.6459	−1.59975	−0.799874	+	0.600169i	$0.795100\pi$
$258$	0	0
$259$	−5.75877	−0.357833
$260$	−3.75877	−0.233109
$261$	0	0
$262$	3.51754	0.217314
$263$	15.5175	0.956853	0.478426	−	0.878128i	$-0.341207\pi$
$263$	15.5175	0.956853	0.478426	+	0.878128i	$0.341207\pi$
$264$	0	0
$265$	0.610815	0.0375220
$266$	−8.12836	−0.498381
$267$	0	0
$268$	4.98040	0.304226
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	−31.2918	−1.90084	−0.950421	−	0.310968i	$-0.899347\pi$
$271$	−31.2918	−1.90084	−0.950421	+	0.310968i	$0.899347\pi$
$272$	1.75877	0.106641
$273$	0	0
$274$	−13.1480	−0.794297
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−6.86753	−0.412630	−0.206315	−	0.978486i	$-0.566147\pi$
$277$	−6.86753	−0.412630	−0.206315	+	0.978486i	$0.566147\pi$
$278$	3.64590	0.218666
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	29.2763	1.74648	0.873239	−	0.487292i	$-0.162015\pi$
$281$	29.2763	1.74648	0.873239	+	0.487292i	$0.162015\pi$
$282$	0	0
$283$	7.88713	0.468841	0.234420	−	0.972135i	$-0.424681\pi$
$283$	7.88713	0.468841	0.234420	+	0.972135i	$0.424681\pi$
$284$	13.0351	0.773490
$285$	0	0
$286$	−3.75877	−0.222261
$287$	4.12836	0.243689
$288$	0	0
$289$	−13.9067	−0.818043
$290$	9.51754	0.558889
$291$	0	0
$292$	−8.12836	−0.475676
$293$	−18.9959	−1.10975	−0.554876	−	0.831933i	$-0.687234\pi$
$293$	−18.9959	−1.10975	−0.554876	+	0.831933i	$0.687234\pi$
$294$	0	0
$295$	10.1284	0.589696
$296$	−5.75877	−0.334722
$297$	0	0
$298$	9.51754	0.551336
$299$	−26.8675	−1.55379
$300$	0	0
$301$	−8.12836	−0.468511
$302$	8.00000	0.460348
$303$	0	0
$304$	−8.12836	−0.466193
$305$	4.00000	0.229039
$306$	0	0
$307$	28.1438	1.60625	0.803127	−	0.595808i	$-0.203168\pi$
$307$	28.1438	1.60625	0.803127	+	0.595808i	$0.203168\pi$
$308$	−1.00000	−0.0569803
$309$	0	0
$310$	−6.12836	−0.348067
$311$	−16.0547	−0.910378	−0.455189	−	0.890395i	$-0.650428\pi$
$311$	−16.0547	−0.910378	−0.455189	+	0.890395i	$0.650428\pi$
$312$	0	0
$313$	26.9959	1.52590	0.762949	−	0.646459i	$-0.223751\pi$
$313$	26.9959	1.52590	0.762949	+	0.646459i	$0.223751\pi$
$314$	−16.3851	−0.924663
$315$	0	0
$316$	15.0351	0.845789
$317$	−6.65002	−0.373502	−0.186751	−	0.982407i	$-0.559796\pi$
$317$	−6.65002	−0.373502	−0.186751	+	0.982407i	$0.559796\pi$
$318$	0	0
$319$	9.51754	0.532880
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−7.14796	−0.398340
$323$	−14.2959	−0.795446
$324$	0	0
$325$	3.75877	0.208499
$326$	−15.2763	−0.846077
$327$	0	0
$328$	4.12836	0.227950
$329$	−6.00000	−0.330791
$330$	0	0
$331$	−5.22163	−0.287007	−0.143503	−	0.989650i	$-0.545837\pi$
$331$	−5.22163	−0.287007	−0.143503	+	0.989650i	$0.545837\pi$
$332$	8.90673	0.488820
$333$	0	0
$334$	−6.11287	−0.334482
$335$	−4.98040	−0.272108
$336$	0	0
$337$	−14.7392	−0.802894	−0.401447	−	0.915882i	$-0.631493\pi$
$337$	−14.7392	−0.802894	−0.401447	+	0.915882i	$0.631493\pi$
$338$	1.12836	0.0613745
$339$	0	0
$340$	−1.75877	−0.0953827
$341$	−6.12836	−0.331869
$342$	0	0
$343$	1.00000	0.0539949
$344$	−8.12836	−0.438252
$345$	0	0
$346$	−25.0351	−1.34589
$347$	−21.9026	−1.17579	−0.587897	−	0.808936i	$-0.700044\pi$
$347$	−21.9026	−1.17579	−0.587897	+	0.808936i	$0.700044\pi$
$348$	0	0
$349$	−33.8135	−1.80999	−0.904996	−	0.425419i	$-0.860127\pi$
$349$	−33.8135	−1.80999	−0.904996	+	0.425419i	$0.860127\pi$
$350$	1.00000	0.0534522
$351$	0	0
$352$	−1.00000	−0.0533002
$353$	−14.8675	−0.791319	−0.395659	−	0.918397i	$-0.629484\pi$
$353$	−14.8675	−0.791319	−0.395659	+	0.918397i	$0.629484\pi$
$354$	0	0
$355$	−13.0351	−0.691830
$356$	−16.6655	−0.883270
$357$	0	0
$358$	17.1634	0.907116
$359$	17.3155	0.913878	0.456939	−	0.889498i	$-0.348946\pi$
$359$	17.3155	0.913878	0.456939	+	0.889498i	$0.348946\pi$
$360$	0	0
$361$	47.0702	2.47738
$362$	−24.7939	−1.30314
$363$	0	0
$364$	3.75877	0.197013
$365$	8.12836	0.425458
$366$	0	0
$367$	−4.65002	−0.242729	−0.121364	−	0.992608i	$-0.538727\pi$
$367$	−4.65002	−0.242729	−0.121364	+	0.992608i	$0.538727\pi$
$368$	−7.14796	−0.372613
$369$	0	0
$370$	5.75877	0.299384
$371$	−0.610815	−0.0317119
$372$	0	0
$373$	4.90673	0.254061	0.127030	−	0.991899i	$-0.459455\pi$
$373$	4.90673	0.254061	0.127030	+	0.991899i	$0.459455\pi$
$374$	−1.75877	−0.0909439
$375$	0	0
$376$	−6.00000	−0.309426
$377$	−35.7743	−1.84247
$378$	0	0
$379$	5.55674	0.285431	0.142715	−	0.989764i	$-0.454417\pi$
$379$	5.55674	0.285431	0.142715	+	0.989764i	$0.454417\pi$
$380$	8.12836	0.416976
$381$	0	0
$382$	9.51754	0.486960
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	12.1284	0.617317
$387$	0	0
$388$	15.6459	0.794300
$389$	6.42427	0.325723	0.162862	−	0.986649i	$-0.447928\pi$
$389$	6.42427	0.325723	0.162862	+	0.986649i	$0.447928\pi$
$390$	0	0
$391$	−12.5716	−0.635774
$392$	1.00000	0.0505076
$393$	0	0
$394$	14.9067	0.750990
$395$	−15.0351	−0.756497
$396$	0	0
$397$	12.3541	0.620035	0.310017	−	0.950731i	$-0.399665\pi$
$397$	12.3541	0.620035	0.310017	+	0.950731i	$0.399665\pi$
$398$	21.6459	1.08501
$399$	0	0
$400$	1.00000	0.0500000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	23.0351	1.14746
$404$	−10.7392	−0.534294
$405$	0	0
$406$	−9.51754	−0.472348
$407$	5.75877	0.285452
$408$	0	0
$409$	−14.7392	−0.728805	−0.364403	−	0.931242i	$-0.618727\pi$
$409$	−14.7392	−0.728805	−0.364403	+	0.931242i	$0.618727\pi$
$410$	−4.12836	−0.203885
$411$	0	0
$412$	6.12836	0.301922
$413$	−10.1284	−0.498384
$414$	0	0
$415$	−8.90673	−0.437214
$416$	3.75877	0.184289
$417$	0	0
$418$	8.12836	0.397571
$419$	−4.16756	−0.203598	−0.101799	−	0.994805i	$-0.532460\pi$
$419$	−4.16756	−0.203598	−0.101799	+	0.994805i	$0.532460\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	22.7547	1.10768
$423$	0	0
$424$	−0.610815	−0.0296638
$425$	1.75877	0.0853129
$426$	0	0
$427$	−4.00000	−0.193574
$428$	−14.8675	−0.718649
$429$	0	0
$430$	8.12836	0.391984
$431$	−19.7588	−0.951746	−0.475873	−	0.879514i	$-0.657868\pi$
$431$	−19.7588	−0.951746	−0.475873	+	0.879514i	$0.657868\pi$
$432$	0	0
$433$	−22.4243	−1.07764	−0.538821	−	0.842420i	$-0.681130\pi$
$433$	−22.4243	−1.07764	−0.538821	+	0.842420i	$0.681130\pi$
$434$	6.12836	0.294170
$435$	0	0
$436$	−15.2763	−0.731603
$437$	58.1011	2.77935
$438$	0	0
$439$	−10.3851	−0.495652	−0.247826	−	0.968805i	$-0.579716\pi$
$439$	−10.3851	−0.495652	−0.247826	+	0.968805i	$0.579716\pi$
$440$	1.00000	0.0476731
$441$	0	0
$442$	6.61081	0.314444
$443$	23.1242	1.09867	0.549333	−	0.835604i	$-0.314882\pi$
$443$	23.1242	1.09867	0.549333	+	0.835604i	$0.314882\pi$
$444$	0	0
$445$	16.6655	0.790020
$446$	3.26083	0.154405
$447$	0	0
$448$	1.00000	0.0472456
$449$	19.0351	0.898321	0.449161	−	0.893451i	$-0.351723\pi$
$449$	19.0351	0.898321	0.449161	+	0.893451i	$0.351723\pi$
$450$	0	0
$451$	−4.12836	−0.194397
$452$	5.88713	0.276907
$453$	0	0
$454$	−24.4243	−1.14629
$455$	−3.75877	−0.176214
$456$	0	0
$457$	38.1985	1.78685	0.893426	−	0.449211i	$-0.148295\pi$
$457$	38.1985	1.78685	0.893426	+	0.449211i	$0.148295\pi$
$458$	−2.24123	−0.104726
$459$	0	0
$460$	7.14796	0.333275
$461$	12.6108	0.587344	0.293672	−	0.955906i	$-0.405123\pi$
$461$	12.6108	0.587344	0.293672	+	0.955906i	$0.405123\pi$
$462$	0	0
$463$	−2.55262	−0.118630	−0.0593152	−	0.998239i	$-0.518892\pi$
$463$	−2.55262	−0.118630	−0.0593152	+	0.998239i	$0.518892\pi$
$464$	−9.51754	−0.441841
$465$	0	0
$466$	−4.77837	−0.221354
$467$	−13.8716	−0.641903	−0.320952	−	0.947096i	$-0.604003\pi$
$467$	−13.8716	−0.641903	−0.320952	+	0.947096i	$0.604003\pi$
$468$	0	0
$469$	4.98040	0.229973
$470$	6.00000	0.276759
$471$	0	0
$472$	−10.1284	−0.466195
$473$	8.12836	0.373742
$474$	0	0
$475$	−8.12836	−0.372955
$476$	1.75877	0.0806131
$477$	0	0
$478$	−28.0155	−1.28140
$479$	28.5134	1.30281	0.651406	−	0.758730i	$-0.274180\pi$
$479$	28.5134	1.30281	0.651406	+	0.758730i	$0.274180\pi$
$480$	0	0
$481$	−21.6459	−0.986968
$482$	−18.9067	−0.861178
$483$	0	0
$484$	1.00000	0.0454545
$485$	−15.6459	−0.710444
$486$	0	0
$487$	40.2567	1.82421	0.912103	−	0.409961i	$-0.134458\pi$
$487$	40.2567	1.82421	0.912103	+	0.409961i	$0.134458\pi$
$488$	−4.00000	−0.181071
$489$	0	0
$490$	−1.00000	−0.0451754
$491$	0.225748	0.0101879	0.00509393	−	0.999987i	$-0.498379\pi$
$491$	0.225748	0.0101879	0.00509393	+	0.999987i	$0.498379\pi$
$492$	0	0
$493$	−16.7392	−0.753894
$494$	−30.5526	−1.37463
$495$	0	0
$496$	6.12836	0.275171
$497$	13.0351	0.584703
$498$	0	0
$499$	9.07428	0.406221	0.203110	−	0.979156i	$-0.434895\pi$
$499$	9.07428	0.406221	0.203110	+	0.979156i	$0.434895\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−4.16756	−0.186007
$503$	31.9573	1.42491	0.712453	−	0.701720i	$-0.247584\pi$
$503$	31.9573	1.42491	0.712453	+	0.701720i	$0.247584\pi$
$504$	0	0
$505$	10.7392	0.477887
$506$	7.14796	0.317765
$507$	0	0
$508$	−2.12836	−0.0944305
$509$	4.55262	0.201791	0.100896	−	0.994897i	$-0.467829\pi$
$509$	4.55262	0.201791	0.100896	+	0.994897i	$0.467829\pi$
$510$	0	0
$511$	−8.12836	−0.359577
$512$	1.00000	0.0441942
$513$	0	0
$514$	−25.6459	−1.13119
$515$	−6.12836	−0.270048
$516$	0	0
$517$	6.00000	0.263880
$518$	−5.75877	−0.253026
$519$	0	0
$520$	−3.75877	−0.164833
$521$	4.89124	0.214289	0.107145	−	0.994243i	$-0.465829\pi$
$521$	4.89124	0.214289	0.107145	+	0.994243i	$0.465829\pi$
$522$	0	0
$523$	23.4047	1.02341	0.511707	−	0.859160i	$-0.329013\pi$
$523$	23.4047	1.02341	0.511707	+	0.859160i	$0.329013\pi$
$524$	3.51754	0.153664
$525$	0	0
$526$	15.5175	0.676597
$527$	10.7784	0.469513
$528$	0	0
$529$	28.0933	1.22145
$530$	0.610815	0.0265321
$531$	0	0
$532$	−8.12836	−0.352409
$533$	15.5175	0.672139
$534$	0	0
$535$	14.8675	0.642779
$536$	4.98040	0.215120
$537$	0	0
$538$	6.00000	0.258678
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−12.9804	−0.558071	−0.279035	−	0.960281i	$-0.590015\pi$
$541$	−12.9804	−0.558071	−0.279035	+	0.960281i	$0.590015\pi$
$542$	−31.2918	−1.34410
$543$	0	0
$544$	1.75877	0.0754067
$545$	15.2763	0.654365
$546$	0	0
$547$	−21.1242	−0.903207	−0.451604	−	0.892219i	$-0.649148\pi$
$547$	−21.1242	−0.903207	−0.451604	+	0.892219i	$0.649148\pi$
$548$	−13.1480	−0.561653
$549$	0	0
$550$	−1.00000	−0.0426401
$551$	77.3620	3.29573
$552$	0	0
$553$	15.0351	0.639357
$554$	−6.86753	−0.291773
$555$	0	0
$556$	3.64590	0.154620
$557$	23.1634	0.981466	0.490733	−	0.871310i	$-0.336729\pi$
$557$	23.1634	0.981466	0.490733	+	0.871310i	$0.336729\pi$
$558$	0	0
$559$	−30.5526	−1.29224
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	29.2763	1.23495
$563$	37.6459	1.58659	0.793293	−	0.608840i	$-0.208365\pi$
$563$	37.6459	1.58659	0.793293	+	0.608840i	$0.208365\pi$
$564$	0	0
$565$	−5.88713	−0.247673
$566$	7.88713	0.331521
$567$	0	0
$568$	13.0351	0.546940
$569$	−19.7980	−0.829974	−0.414987	−	0.909827i	$-0.636214\pi$
$569$	−19.7980	−0.829974	−0.414987	+	0.909827i	$0.636214\pi$
$570$	0	0
$571$	8.53714	0.357268	0.178634	−	0.983916i	$-0.442832\pi$
$571$	8.53714	0.357268	0.178634	+	0.983916i	$0.442832\pi$
$572$	−3.75877	−0.157162
$573$	0	0
$574$	4.12836	0.172314
$575$	−7.14796	−0.298090
$576$	0	0
$577$	10.4825	0.436390	0.218195	−	0.975905i	$-0.429983\pi$
$577$	10.4825	0.436390	0.218195	+	0.975905i	$0.429983\pi$
$578$	−13.9067	−0.578444
$579$	0	0
$580$	9.51754	0.395194
$581$	8.90673	0.369513
$582$	0	0
$583$	0.610815	0.0252974
$584$	−8.12836	−0.336354
$585$	0	0
$586$	−18.9959	−0.784713
$587$	−32.9067	−1.35821	−0.679103	−	0.734043i	$-0.737631\pi$
$587$	−32.9067	−1.35821	−0.679103	+	0.734043i	$0.737631\pi$
$588$	0	0
$589$	−49.8135	−2.05253
$590$	10.1284	0.416978
$591$	0	0
$592$	−5.75877	−0.236684
$593$	−43.5722	−1.78930	−0.894648	−	0.446771i	$-0.852574\pi$
$593$	−43.5722	−1.78930	−0.894648	+	0.446771i	$0.852574\pi$
$594$	0	0
$595$	−1.75877	−0.0721026
$596$	9.51754	0.389854
$597$	0	0
$598$	−26.8675	−1.09869
$599$	−48.8093	−1.99430	−0.997148	−	0.0754754i	$-0.975953\pi$
$599$	−48.8093	−1.99430	−0.997148	+	0.0754754i	$0.975953\pi$
$600$	0	0
$601$	38.9959	1.59068	0.795338	−	0.606167i	$-0.207294\pi$
$601$	38.9959	1.59068	0.795338	+	0.606167i	$0.207294\pi$
$602$	−8.12836	−0.331287
$603$	0	0
$604$	8.00000	0.325515
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	42.5526	1.72716	0.863579	−	0.504214i	$-0.168218\pi$
$607$	42.5526	1.72716	0.863579	+	0.504214i	$0.168218\pi$
$608$	−8.12836	−0.329648
$609$	0	0
$610$	4.00000	0.161955
$611$	−22.5526	−0.912381
$612$	0	0
$613$	48.9377	1.97657	0.988287	−	0.152605i	$-0.0487661\pi$
$613$	48.9377	1.97657	0.988287	+	0.152605i	$0.0487661\pi$
$614$	28.1438	1.13579
$615$	0	0
$616$	−1.00000	−0.0402911
$617$	−3.59121	−0.144577	−0.0722884	−	0.997384i	$-0.523030\pi$
$617$	−3.59121	−0.144577	−0.0722884	+	0.997384i	$0.523030\pi$
$618$	0	0
$619$	28.1046	1.12962	0.564810	−	0.825221i	$-0.308949\pi$
$619$	28.1046	1.12962	0.564810	+	0.825221i	$0.308949\pi$
$620$	−6.12836	−0.246121
$621$	0	0
$622$	−16.0547	−0.643734
$623$	−16.6655	−0.667689
$624$	0	0
$625$	1.00000	0.0400000
$626$	26.9959	1.07897
$627$	0	0
$628$	−16.3851	−0.653835
$629$	−10.1284	−0.403844
$630$	0	0
$631$	−23.4593	−0.933902	−0.466951	−	0.884283i	$-0.654648\pi$
$631$	−23.4593	−0.933902	−0.466951	+	0.884283i	$0.654648\pi$
$632$	15.0351	0.598063
$633$	0	0
$634$	−6.65002	−0.264106
$635$	2.12836	0.0844612
$636$	0	0
$637$	3.75877	0.148928
$638$	9.51754	0.376803
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−35.5485	−1.40408	−0.702041	−	0.712137i	$-0.747728\pi$
$641$	−35.5485	−1.40408	−0.702041	+	0.712137i	$0.747728\pi$
$642$	0	0
$643$	−13.5567	−0.534626	−0.267313	−	0.963610i	$-0.586136\pi$
$643$	−13.5567	−0.534626	−0.267313	+	0.963610i	$0.586136\pi$
$644$	−7.14796	−0.281669
$645$	0	0
$646$	−14.2959	−0.562465
$647$	11.8135	0.464435	0.232217	−	0.972664i	$-0.425402\pi$
$647$	11.8135	0.464435	0.232217	+	0.972664i	$0.425402\pi$
$648$	0	0
$649$	10.1284	0.397573
$650$	3.75877	0.147431
$651$	0	0
$652$	−15.2763	−0.598267
$653$	−46.9377	−1.83681	−0.918407	−	0.395637i	$-0.870524\pi$
$653$	−46.9377	−1.83681	−0.918407	+	0.395637i	$0.870524\pi$
$654$	0	0
$655$	−3.51754	−0.137442
$656$	4.12836	0.161185
$657$	0	0
$658$	−6.00000	−0.233904
$659$	1.22163	0.0475879	0.0237940	−	0.999717i	$-0.492425\pi$
$659$	1.22163	0.0475879	0.0237940	+	0.999717i	$0.492425\pi$
$660$	0	0
$661$	−8.28043	−0.322071	−0.161036	−	0.986949i	$-0.551483\pi$
$661$	−8.28043	−0.322071	−0.161036	+	0.986949i	$0.551483\pi$
$662$	−5.22163	−0.202944
$663$	0	0
$664$	8.90673	0.345648
$665$	8.12836	0.315204
$666$	0	0
$667$	68.0310	2.63417
$668$	−6.11287	−0.236514
$669$	0	0
$670$	−4.98040	−0.192410
$671$	4.00000	0.154418
$672$	0	0
$673$	−30.1094	−1.16063	−0.580315	−	0.814392i	$-0.697071\pi$
$673$	−30.1094	−1.16063	−0.580315	+	0.814392i	$0.697071\pi$
$674$	−14.7392	−0.567732
$675$	0	0
$676$	1.12836	0.0433983
$677$	11.9608	0.459691	0.229845	−	0.973227i	$-0.426178\pi$
$677$	11.9608	0.459691	0.229845	+	0.973227i	$0.426178\pi$
$678$	0	0
$679$	15.6459	0.600434
$680$	−1.75877	−0.0674458
$681$	0	0
$682$	−6.12836	−0.234667
$683$	18.4635	0.706485	0.353242	−	0.935532i	$-0.385079\pi$
$683$	18.4635	0.706485	0.353242	+	0.935532i	$0.385079\pi$
$684$	0	0
$685$	13.1480	0.502358
$686$	1.00000	0.0381802
$687$	0	0
$688$	−8.12836	−0.309891
$689$	−2.29591	−0.0874673
$690$	0	0
$691$	−15.7006	−0.597278	−0.298639	−	0.954366i	$-0.596533\pi$
$691$	−15.7006	−0.597278	−0.298639	+	0.954366i	$0.596533\pi$
$692$	−25.0351	−0.951691
$693$	0	0
$694$	−21.9026	−0.831412
$695$	−3.64590	−0.138297
$696$	0	0
$697$	7.26083	0.275024
$698$	−33.8135	−1.27986
$699$	0	0
$700$	1.00000	0.0377964
$701$	−26.0310	−0.983176	−0.491588	−	0.870828i	$-0.663583\pi$
$701$	−26.0310	−0.983176	−0.491588	+	0.870828i	$0.663583\pi$
$702$	0	0
$703$	46.8093	1.76545
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−14.8675	−0.559547
$707$	−10.7392	−0.403888
$708$	0	0
$709$	−25.2918	−0.949853	−0.474927	−	0.880025i	$-0.657525\pi$
$709$	−25.2918	−0.949853	−0.474927	+	0.880025i	$0.657525\pi$
$710$	−13.0351	−0.489198
$711$	0	0
$712$	−16.6655	−0.624566
$713$	−43.8052	−1.64052
$714$	0	0
$715$	3.75877	0.140570
$716$	17.1634	0.641428
$717$	0	0
$718$	17.3155	0.646209
$719$	−18.7237	−0.698276	−0.349138	−	0.937071i	$-0.613526\pi$
$719$	−18.7237	−0.698276	−0.349138	+	0.937071i	$0.613526\pi$
$720$	0	0
$721$	6.12836	0.228232
$722$	47.0702	1.75177
$723$	0	0
$724$	−24.7939	−0.921456
$725$	−9.51754	−0.353473
$726$	0	0
$727$	34.2959	1.27196	0.635982	−	0.771703i	$-0.280595\pi$
$727$	34.2959	1.27196	0.635982	+	0.771703i	$0.280595\pi$
$728$	3.75877	0.139309
$729$	0	0
$730$	8.12836	0.300844
$731$	−14.2959	−0.528753
$732$	0	0
$733$	32.3506	1.19490	0.597448	−	0.801907i	$-0.296181\pi$
$733$	32.3506	1.19490	0.597448	+	0.801907i	$0.296181\pi$
$734$	−4.65002	−0.171635
$735$	0	0
$736$	−7.14796	−0.263477
$737$	−4.98040	−0.183455
$738$	0	0
$739$	−27.0114	−0.993629	−0.496815	−	0.867857i	$-0.665497\pi$
$739$	−27.0114	−0.993629	−0.496815	+	0.867857i	$0.665497\pi$
$740$	5.75877	0.211697
$741$	0	0
$742$	−0.610815	−0.0224237
$743$	−7.03508	−0.258092	−0.129046	−	0.991639i	$-0.541192\pi$
$743$	−7.03508	−0.258092	−0.129046	+	0.991639i	$0.541192\pi$
$744$	0	0
$745$	−9.51754	−0.348696
$746$	4.90673	0.179648
$747$	0	0
$748$	−1.75877	−0.0643070
$749$	−14.8675	−0.543248
$750$	0	0
$751$	41.9026	1.52905	0.764524	−	0.644595i	$-0.222974\pi$
$751$	41.9026	1.52905	0.764524	+	0.644595i	$0.222974\pi$
$752$	−6.00000	−0.218797
$753$	0	0
$754$	−35.7743	−1.30282
$755$	−8.00000	−0.291150
$756$	0	0
$757$	43.0898	1.56612	0.783062	−	0.621944i	$-0.213657\pi$
$757$	43.0898	1.56612	0.783062	+	0.621944i	$0.213657\pi$
$758$	5.55674	0.201830
$759$	0	0
$760$	8.12836	0.294846
$761$	6.65002	0.241063	0.120531	−	0.992710i	$-0.461540\pi$
$761$	6.65002	0.241063	0.120531	+	0.992710i	$0.461540\pi$
$762$	0	0
$763$	−15.2763	−0.553040
$764$	9.51754	0.344333
$765$	0	0
$766$	−6.00000	−0.216789
$767$	−38.0702	−1.37463
$768$	0	0
$769$	31.5877	1.13908	0.569541	−	0.821963i	$-0.307121\pi$
$769$	31.5877	1.13908	0.569541	+	0.821963i	$0.307121\pi$
$770$	1.00000	0.0360375
$771$	0	0
$772$	12.1284	0.436509
$773$	19.6459	0.706614	0.353307	−	0.935507i	$-0.385057\pi$
$773$	19.6459	0.706614	0.353307	+	0.935507i	$0.385057\pi$
$774$	0	0
$775$	6.12836	0.220137
$776$	15.6459	0.561655
$777$	0	0
$778$	6.42427	0.230321
$779$	−33.5567	−1.20230
$780$	0	0
$781$	−13.0351	−0.466432
$782$	−12.5716	−0.449560
$783$	0	0
$784$	1.00000	0.0357143
$785$	16.3851	0.584808
$786$	0	0
$787$	−33.4748	−1.19325	−0.596624	−	0.802521i	$-0.703492\pi$
$787$	−33.4748	−1.19325	−0.596624	+	0.802521i	$0.703492\pi$
$788$	14.9067	0.531030
$789$	0	0
$790$	−15.0351	−0.534924
$791$	5.88713	0.209322
$792$	0	0
$793$	−15.0351	−0.533911
$794$	12.3541	0.438431
$795$	0	0
$796$	21.6459	0.767218
$797$	14.6810	0.520027	0.260013	−	0.965605i	$-0.416273\pi$
$797$	14.6810	0.520027	0.260013	+	0.965605i	$0.416273\pi$
$798$	0	0
$799$	−10.5526	−0.373325
$800$	1.00000	0.0353553
$801$	0	0
$802$	0	0
$803$	8.12836	0.286843
$804$	0	0
$805$	7.14796	0.251932
$806$	23.0351	0.811376
$807$	0	0
$808$	−10.7392	−0.377803
$809$	−26.8331	−0.943400	−0.471700	−	0.881759i	$-0.656359\pi$
$809$	−26.8331	−0.943400	−0.471700	+	0.881759i	$0.656359\pi$
$810$	0	0
$811$	28.7202	1.00850	0.504251	−	0.863557i	$-0.331769\pi$
$811$	28.7202	1.00850	0.504251	+	0.863557i	$0.331769\pi$
$812$	−9.51754	−0.334000
$813$	0	0
$814$	5.75877	0.201845
$815$	15.2763	0.535106
$816$	0	0
$817$	66.0702	2.31150
$818$	−14.7392	−0.515343
$819$	0	0
$820$	−4.12836	−0.144168
$821$	42.8485	1.49542	0.747712	−	0.664023i	$-0.231152\pi$
$821$	42.8485	1.49542	0.747712	+	0.664023i	$0.231152\pi$
$822$	0	0
$823$	−53.8444	−1.87690	−0.938449	−	0.345417i	$-0.887737\pi$
$823$	−53.8444	−1.87690	−0.938449	+	0.345417i	$0.887737\pi$
$824$	6.12836	0.213491
$825$	0	0
$826$	−10.1284	−0.352411
$827$	−19.4593	−0.676668	−0.338334	−	0.941026i	$-0.609863\pi$
$827$	−19.4593	−0.676668	−0.338334	+	0.941026i	$0.609863\pi$
$828$	0	0
$829$	−28.4587	−0.988413	−0.494206	−	0.869345i	$-0.664541\pi$
$829$	−28.4587	−0.988413	−0.494206	+	0.869345i	$0.664541\pi$
$830$	−8.90673	−0.309157
$831$	0	0
$832$	3.75877	0.130312
$833$	1.75877	0.0609378
$834$	0	0
$835$	6.11287	0.211545
$836$	8.12836	0.281125
$837$	0	0
$838$	−4.16756	−0.143966
$839$	−44.5681	−1.53866	−0.769331	−	0.638850i	$-0.779410\pi$
$839$	−44.5681	−1.53866	−0.769331	+	0.638850i	$0.779410\pi$
$840$	0	0
$841$	61.5836	2.12357
$842$	14.0000	0.482472
$843$	0	0
$844$	22.7547	0.783247
$845$	−1.12836	−0.0388166
$846$	0	0
$847$	1.00000	0.0343604
$848$	−0.610815	−0.0209755
$849$	0	0
$850$	1.75877	0.0603253
$851$	41.1634	1.41106
$852$	0	0
$853$	−5.71957	−0.195834	−0.0979172	−	0.995195i	$-0.531218\pi$
$853$	−5.71957	−0.195834	−0.0979172	+	0.995195i	$0.531218\pi$
$854$	−4.00000	−0.136877
$855$	0	0
$856$	−14.8675	−0.508162
$857$	17.2763	0.590148	0.295074	−	0.955474i	$-0.404656\pi$
$857$	17.2763	0.590148	0.295074	+	0.955474i	$0.404656\pi$
$858$	0	0
$859$	15.1088	0.515504	0.257752	−	0.966211i	$-0.417018\pi$
$859$	15.1088	0.515504	0.257752	+	0.966211i	$0.417018\pi$
$860$	8.12836	0.277175
$861$	0	0
$862$	−19.7588	−0.672986
$863$	−0.0344725	−0.00117346	−0.000586729	−	1.00000i	$-0.500187\pi$
$863$	−0.0344725	−0.00117346	−0.000586729		1.00000i	$0.500187\pi$
$864$	0	0
$865$	25.0351	0.851218
$866$	−22.4243	−0.762008
$867$	0	0
$868$	6.12836	0.208010
$869$	−15.0351	−0.510030
$870$	0	0
$871$	18.7202	0.634309
$872$	−15.2763	−0.517321
$873$	0	0
$874$	58.1011	1.96530
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	36.9377	1.24730	0.623649	−	0.781705i	$-0.285650\pi$
$877$	36.9377	1.24730	0.623649	+	0.781705i	$0.285650\pi$
$878$	−10.3851	−0.350479
$879$	0	0
$880$	1.00000	0.0337100
$881$	−4.66550	−0.157185	−0.0785923	−	0.996907i	$-0.525043\pi$
$881$	−4.66550	−0.157185	−0.0785923	+	0.996907i	$0.525043\pi$
$882$	0	0
$883$	−0.754652	−0.0253961	−0.0126980	−	0.999919i	$-0.504042\pi$
$883$	−0.754652	−0.0253961	−0.0126980	+	0.999919i	$0.504042\pi$
$884$	6.61081	0.222346
$885$	0	0
$886$	23.1242	0.776874
$887$	−3.44387	−0.115634	−0.0578169	−	0.998327i	$-0.518414\pi$
$887$	−3.44387	−0.115634	−0.0578169	+	0.998327i	$0.518414\pi$
$888$	0	0
$889$	−2.12836	−0.0713828
$890$	16.6655	0.558629
$891$	0	0
$892$	3.26083	0.109181
$893$	48.7701	1.63203
$894$	0	0
$895$	−17.1634	−0.573710
$896$	1.00000	0.0334077
$897$	0	0
$898$	19.0351	0.635209
$899$	−58.3269	−1.94531
$900$	0	0
$901$	−1.07428	−0.0357895
$902$	−4.12836	−0.137459
$903$	0	0
$904$	5.88713	0.195803
$905$	24.7939	0.824176
$906$	0	0
$907$	12.2412	0.406463	0.203232	−	0.979131i	$-0.434856\pi$
$907$	12.2412	0.406463	0.203232	+	0.979131i	$0.434856\pi$
$908$	−24.4243	−0.810548
$909$	0	0
$910$	−3.75877	−0.124602
$911$	50.2567	1.66508	0.832540	−	0.553966i	$-0.186886\pi$
$911$	50.2567	1.66508	0.832540	+	0.553966i	$0.186886\pi$
$912$	0	0
$913$	−8.90673	−0.294770
$914$	38.1985	1.26349
$915$	0	0
$916$	−2.24123	−0.0740523
$917$	3.51754	0.116159
$918$	0	0
$919$	26.9567	0.889219	0.444609	−	0.895725i	$-0.353342\pi$
$919$	26.9567	0.889219	0.444609	+	0.895725i	$0.353342\pi$
$920$	7.14796	0.235661
$921$	0	0
$922$	12.6108	0.415315
$923$	48.9959	1.61272
$924$	0	0
$925$	−5.75877	−0.189347
$926$	−2.55262	−0.0838844
$927$	0	0
$928$	−9.51754	−0.312429
$929$	15.3655	0.504125	0.252062	−	0.967711i	$-0.418891\pi$
$929$	15.3655	0.504125	0.252062	+	0.967711i	$0.418891\pi$
$930$	0	0
$931$	−8.12836	−0.266396
$932$	−4.77837	−0.156521
$933$	0	0
$934$	−13.8716	−0.453894
$935$	1.75877	0.0575179
$936$	0	0
$937$	−34.1985	−1.11722	−0.558608	−	0.829431i	$-0.688665\pi$
$937$	−34.1985	−1.11722	−0.558608	+	0.829431i	$0.688665\pi$
$938$	4.98040	0.162616
$939$	0	0
$940$	6.00000	0.195698
$941$	−4.55262	−0.148411	−0.0742056	−	0.997243i	$-0.523642\pi$
$941$	−4.55262	−0.148411	−0.0742056	+	0.997243i	$0.523642\pi$
$942$	0	0
$943$	−29.5093	−0.960955
$944$	−10.1284	−0.329650
$945$	0	0
$946$	8.12836	0.264276
$947$	19.8324	0.644468	0.322234	−	0.946660i	$-0.395566\pi$
$947$	19.8324	0.644468	0.322234	+	0.946660i	$0.395566\pi$
$948$	0	0
$949$	−30.5526	−0.991780
$950$	−8.12836	−0.263719
$951$	0	0
$952$	1.75877	0.0570021
$953$	53.7743	1.74192	0.870959	−	0.491355i	$-0.163498\pi$
$953$	53.7743	1.74192	0.870959	+	0.491355i	$0.163498\pi$
$954$	0	0
$955$	−9.51754	−0.307980
$956$	−28.0155	−0.906085
$957$	0	0
$958$	28.5134	0.921227
$959$	−13.1480	−0.424570
$960$	0	0
$961$	6.55674	0.211508
$962$	−21.6459	−0.697892
$963$	0	0
$964$	−18.9067	−0.608945
$965$	−12.1284	−0.390426
$966$	0	0
$967$	−2.12836	−0.0684433	−0.0342217	−	0.999414i	$-0.510895\pi$
$967$	−2.12836	−0.0684433	−0.0342217	+	0.999414i	$0.510895\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	−15.6459	−0.502359
$971$	−47.3500	−1.51953	−0.759767	−	0.650196i	$-0.774687\pi$
$971$	−47.3500	−1.51953	−0.759767	+	0.650196i	$0.774687\pi$
$972$	0	0
$973$	3.64590	0.116882
$974$	40.2567	1.28991
$975$	0	0
$976$	−4.00000	−0.128037
$977$	−11.7006	−0.374335	−0.187167	−	0.982328i	$-0.559931\pi$
$977$	−11.7006	−0.374335	−0.187167	+	0.982328i	$0.559931\pi$
$978$	0	0
$979$	16.6655	0.532632
$980$	−1.00000	−0.0319438
$981$	0	0
$982$	0.225748	0.00720391
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	0	0
$985$	−14.9067	−0.474968
$986$	−16.7392	−0.533084
$987$	0	0
$988$	−30.5526	−0.972008
$989$	58.1011	1.84751
$990$	0	0
$991$	56.4243	1.79238	0.896188	−	0.443675i	$-0.146325\pi$
$991$	56.4243	1.79238	0.896188	+	0.443675i	$0.146325\pi$
$992$	6.12836	0.194575
$993$	0	0
$994$	13.0351	0.413448
$995$	−21.6459	−0.686221
$996$	0	0
$997$	20.4979	0.649176	0.324588	−	0.945855i	$-0.394774\pi$
$997$	20.4979	0.649176	0.324588	+	0.945855i	$0.394774\pi$
$998$	9.07428	0.287241
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6930.2.a.cj.1.3	yes	3
3.2	odd	2		6930.2.a.ci.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6930.2.a.ci.1.3	✓	3	3.2	odd	2
6930.2.a.cj.1.3	yes	3	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:	\( N \)	\(=\)	\( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6930.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} -1.00000 q^{11} +3.75877 q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.75877 q^{17} -8.12836 q^{19} -1.00000 q^{20} -1.00000 q^{22} -7.14796 q^{23} +1.00000 q^{25} +3.75877 q^{26} +1.00000 q^{28} -9.51754 q^{29} +6.12836 q^{31} +1.00000 q^{32} +1.75877 q^{34} -1.00000 q^{35} -5.75877 q^{37} -8.12836 q^{38} -1.00000 q^{40} +4.12836 q^{41} -8.12836 q^{43} -1.00000 q^{44} -7.14796 q^{46} -6.00000 q^{47} +1.00000 q^{49} +1.00000 q^{50} +3.75877 q^{52} -0.610815 q^{53} +1.00000 q^{55} +1.00000 q^{56} -9.51754 q^{58} -10.1284 q^{59} -4.00000 q^{61} +6.12836 q^{62} +1.00000 q^{64} -3.75877 q^{65} +4.98040 q^{67} +1.75877 q^{68} -1.00000 q^{70} +13.0351 q^{71} -8.12836 q^{73} -5.75877 q^{74} -8.12836 q^{76} -1.00000 q^{77} +15.0351 q^{79} -1.00000 q^{80} +4.12836 q^{82} +8.90673 q^{83} -1.75877 q^{85} -8.12836 q^{86} -1.00000 q^{88} -16.6655 q^{89} +3.75877 q^{91} -7.14796 q^{92} -6.00000 q^{94} +8.12836 q^{95} +15.6459 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 + 3 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} - 3 q^{5} + 3 q^{7} + 3 q^{8} - 3 q^{10} - 3 q^{11} + 3 q^{14} + 3 q^{16} - 6 q^{17} - 6 q^{19} - 3 q^{20} - 3 q^{22} - 6 q^{23} + 3 q^{25} + 3 q^{28} - 6 q^{29} + 3 q^{32} - 6 q^{34} - 3 q^{35} - 6 q^{37} - 6 q^{38} - 3 q^{40} - 6 q^{41} - 6 q^{43} - 3 q^{44} - 6 q^{46} - 18 q^{47} + 3 q^{49} + 3 q^{50} - 6 q^{53} + 3 q^{55} + 3 q^{56} - 6 q^{58} - 12 q^{59} - 12 q^{61} + 3 q^{64} + 12 q^{67} - 6 q^{68} - 3 q^{70} - 6 q^{71} - 6 q^{73} - 6 q^{74} - 6 q^{76} - 3 q^{77} - 3 q^{80} - 6 q^{82} + 6 q^{85} - 6 q^{86} - 3 q^{88} - 12 q^{89} - 6 q^{92} - 18 q^{94} + 6 q^{95} + 6 q^{97} + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 - 3 * q^5 + 3 * q^7 + 3 * q^8 - 3 * q^10 - 3 * q^11 + 3 * q^14 + 3 * q^16 - 6 * q^17 - 6 * q^19 - 3 * q^20 - 3 * q^22 - 6 * q^23 + 3 * q^25 + 3 * q^28 - 6 * q^29 + 3 * q^32 - 6 * q^34 - 3 * q^35 - 6 * q^37 - 6 * q^38 - 3 * q^40 - 6 * q^41 - 6 * q^43 - 3 * q^44 - 6 * q^46 - 18 * q^47 + 3 * q^49 + 3 * q^50 - 6 * q^53 + 3 * q^55 + 3 * q^56 - 6 * q^58 - 12 * q^59 - 12 * q^61 + 3 * q^64 + 12 * q^67 - 6 * q^68 - 3 * q^70 - 6 * q^71 - 6 * q^73 - 6 * q^74 - 6 * q^76 - 3 * q^77 - 3 * q^80 - 6 * q^82 + 6 * q^85 - 6 * q^86 - 3 * q^88 - 12 * q^89 - 6 * q^92 - 18 * q^94 + 6 * q^95 + 6 * q^97 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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