LMFDB - Embedded newform 8010.2.a.bn.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8010,2,Mod(1,8010)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8010, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8010.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8010 = 2 \cdot 3^{2} \cdot 5 \cdot 89 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8010.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:	$63.9601720190$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 24x^{5} + 34x^{4} + 111x^{3} - 127x^{2} - 20x + 16 $ x^7 - 3x^6 - 24x^5 + 34x^4 + 111x^3 - 127x^2 - 20x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2670)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.21696$ of defining polynomial
Character	$\chi$	$=$	8010.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.69061 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.69061 q^{7} +1.00000 q^{8} +1.00000 q^{10} +3.22841 q^{11} +1.66654 q^{13} +2.69061 q^{14} +1.00000 q^{16} +0.666496 q^{17} -5.66234 q^{19} +1.00000 q^{20} +3.22841 q^{22} +3.67557 q^{23} +1.00000 q^{25} +1.66654 q^{26} +2.69061 q^{28} -9.13315 q^{29} +5.71468 q^{31} +1.00000 q^{32} +0.666496 q^{34} +2.69061 q^{35} +5.79103 q^{37} -5.66234 q^{38} +1.00000 q^{40} -8.92810 q^{41} +8.73237 q^{43} +3.22841 q^{44} +3.67557 q^{46} +13.0045 q^{47} +0.239400 q^{49} +1.00000 q^{50} +1.66654 q^{52} +6.74336 q^{53} +3.22841 q^{55} +2.69061 q^{56} -9.13315 q^{58} +10.1336 q^{59} -11.4993 q^{61} +5.71468 q^{62} +1.00000 q^{64} +1.66654 q^{65} +3.40986 q^{67} +0.666496 q^{68} +2.69061 q^{70} -16.1900 q^{71} +0.318464 q^{73} +5.79103 q^{74} -5.66234 q^{76} +8.68641 q^{77} -1.09139 q^{79} +1.00000 q^{80} -8.92810 q^{82} +7.18550 q^{83} +0.666496 q^{85} +8.73237 q^{86} +3.22841 q^{88} -1.00000 q^{89} +4.48402 q^{91} +3.67557 q^{92} +13.0045 q^{94} -5.66234 q^{95} +9.35115 q^{97} +0.239400 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + q^{7} + 7 q^{8}+O(q^{10}) $ 7 * q + 7 * q^2 + 7 * q^4 + 7 * q^5 + q^7 + 7 * q^8	$ 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + q^{7} + 7 q^{8} + 7 q^{10} - q^{11} + q^{14} + 7 q^{16} + 9 q^{17} + 9 q^{19} + 7 q^{20} - q^{22} + 8 q^{23} + 7 q^{25} + q^{28} + 4 q^{29} + 16 q^{31} + 7 q^{32} + 9 q^{34} + q^{35} + 2 q^{37} + 9 q^{38} + 7 q^{40} + q^{41} - 10 q^{43} - q^{44} + 8 q^{46} + 13 q^{47} + 26 q^{49} + 7 q^{50} + 24 q^{53} - q^{55} + q^{56} + 4 q^{58} + 6 q^{59} + 23 q^{61} + 16 q^{62} + 7 q^{64} + 5 q^{67} + 9 q^{68} + q^{70} + 8 q^{71} - 2 q^{73} + 2 q^{74} + 9 q^{76} + 6 q^{77} + 7 q^{79} + 7 q^{80} + q^{82} + 7 q^{83} + 9 q^{85} - 10 q^{86} - q^{88} - 7 q^{89} + 48 q^{91} + 8 q^{92} + 13 q^{94} + 9 q^{95} + 30 q^{97} + 26 q^{98}+O(q^{100}) $ 7 * q + 7 * q^2 + 7 * q^4 + 7 * q^5 + q^7 + 7 * q^8 + 7 * q^10 - q^11 + q^14 + 7 * q^16 + 9 * q^17 + 9 * q^19 + 7 * q^20 - q^22 + 8 * q^23 + 7 * q^25 + q^28 + 4 * q^29 + 16 * q^31 + 7 * q^32 + 9 * q^34 + q^35 + 2 * q^37 + 9 * q^38 + 7 * q^40 + q^41 - 10 * q^43 - q^44 + 8 * q^46 + 13 * q^47 + 26 * q^49 + 7 * q^50 + 24 * q^53 - q^55 + q^56 + 4 * q^58 + 6 * q^59 + 23 * q^61 + 16 * q^62 + 7 * q^64 + 5 * q^67 + 9 * q^68 + q^70 + 8 * q^71 - 2 * q^73 + 2 * q^74 + 9 * q^76 + 6 * q^77 + 7 * q^79 + 7 * q^80 + q^82 + 7 * q^83 + 9 * q^85 - 10 * q^86 - q^88 - 7 * q^89 + 48 * q^91 + 8 * q^92 + 13 * q^94 + 9 * q^95 + 30 * q^97 + 26 * 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
$7$	2.69061	1.01696	0.508478	−	0.861075i	$-0.330208\pi$
$7$	2.69061	1.01696	0.508478	+	0.861075i	$0.330208\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	3.22841	0.973403	0.486701	−	0.873568i	$-0.338200\pi$
$11$	3.22841	0.973403	0.486701	+	0.873568i	$0.338200\pi$
$12$	0	0
$13$	1.66654	0.462216	0.231108	−	0.972928i	$-0.425765\pi$
$13$	1.66654	0.462216	0.231108	+	0.972928i	$0.425765\pi$
$14$	2.69061	0.719097
$15$	0	0
$16$	1.00000	0.250000
$17$	0.666496	0.161649	0.0808245	−	0.996728i	$-0.474245\pi$
$17$	0.666496	0.161649	0.0808245	+	0.996728i	$0.474245\pi$
$18$	0	0
$19$	−5.66234	−1.29903	−0.649515	−	0.760349i	$-0.725028\pi$
$19$	−5.66234	−1.29903	−0.649515	+	0.760349i	$0.725028\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	3.22841	0.688300
$23$	3.67557	0.766410	0.383205	−	0.923663i	$-0.374820\pi$
$23$	3.67557	0.766410	0.383205	+	0.923663i	$0.374820\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.66654	0.326836
$27$	0	0
$28$	2.69061	0.508478
$29$	−9.13315	−1.69598	−0.847992	−	0.530009i	$-0.822188\pi$
$29$	−9.13315	−1.69598	−0.847992	+	0.530009i	$0.822188\pi$
$30$	0	0
$31$	5.71468	1.02639	0.513194	−	0.858273i	$-0.328462\pi$
$31$	5.71468	1.02639	0.513194	+	0.858273i	$0.328462\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.666496	0.114303
$35$	2.69061	0.454797
$36$	0	0
$37$	5.79103	0.952040	0.476020	−	0.879434i	$-0.342079\pi$
$37$	5.79103	0.952040	0.476020	+	0.879434i	$0.342079\pi$
$38$	−5.66234	−0.918552
$39$	0	0
$40$	1.00000	0.158114
$41$	−8.92810	−1.39434	−0.697168	−	0.716908i	$-0.745557\pi$
$41$	−8.92810	−1.39434	−0.697168	+	0.716908i	$0.745557\pi$
$42$	0	0
$43$	8.73237	1.33167	0.665837	−	0.746097i	$-0.268075\pi$
$43$	8.73237	1.33167	0.665837	+	0.746097i	$0.268075\pi$
$44$	3.22841	0.486701
$45$	0	0
$46$	3.67557	0.541934
$47$	13.0045	1.89689	0.948447	−	0.316935i	$-0.102654\pi$
$47$	13.0045	1.89689	0.948447	+	0.316935i	$0.102654\pi$
$48$	0	0
$49$	0.239400	0.0342000
$50$	1.00000	0.141421
$51$	0	0
$52$	1.66654	0.231108
$53$	6.74336	0.926272	0.463136	−	0.886287i	$-0.346724\pi$
$53$	6.74336	0.926272	0.463136	+	0.886287i	$0.346724\pi$
$54$	0	0
$55$	3.22841	0.435319
$56$	2.69061	0.359548
$57$	0	0
$58$	−9.13315	−1.19924
$59$	10.1336	1.31928	0.659642	−	0.751580i	$-0.270708\pi$
$59$	10.1336	1.31928	0.659642	+	0.751580i	$0.270708\pi$
$60$	0	0
$61$	−11.4993	−1.47234	−0.736170	−	0.676797i	$-0.763367\pi$
$61$	−11.4993	−1.47234	−0.736170	+	0.676797i	$0.763367\pi$
$62$	5.71468	0.725765
$63$	0	0
$64$	1.00000	0.125000
$65$	1.66654	0.206709
$66$	0	0
$67$	3.40986	0.416580	0.208290	−	0.978067i	$-0.433210\pi$
$67$	3.40986	0.416580	0.208290	+	0.978067i	$0.433210\pi$
$68$	0.666496	0.0808245
$69$	0	0
$70$	2.69061	0.321590
$71$	−16.1900	−1.92139	−0.960697	−	0.277600i	$-0.910461\pi$
$71$	−16.1900	−1.92139	−0.960697	+	0.277600i	$0.910461\pi$
$72$	0	0
$73$	0.318464	0.0372734	0.0186367	−	0.999826i	$-0.494067\pi$
$73$	0.318464	0.0372734	0.0186367	+	0.999826i	$0.494067\pi$
$74$	5.79103	0.673194
$75$	0	0
$76$	−5.66234	−0.649515
$77$	8.68641	0.989908
$78$	0	0
$79$	−1.09139	−0.122791	−0.0613956	−	0.998114i	$-0.519555\pi$
$79$	−1.09139	−0.122791	−0.0613956	+	0.998114i	$0.519555\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−8.92810	−0.985944
$83$	7.18550	0.788711	0.394355	−	0.918958i	$-0.370968\pi$
$83$	7.18550	0.788711	0.394355	+	0.918958i	$0.370968\pi$
$84$	0	0
$85$	0.666496	0.0722916
$86$	8.73237	0.941636
$87$	0	0
$88$	3.22841	0.344150
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	4.48402	0.470053
$92$	3.67557	0.383205
$93$	0	0
$94$	13.0045	1.34131
$95$	−5.66234	−0.580944
$96$	0	0
$97$	9.35115	0.949465	0.474732	−	0.880130i	$-0.342545\pi$
$97$	9.35115	0.949465	0.474732	+	0.880130i	$0.342545\pi$
$98$	0.239400	0.0241830
$99$	0	0
$100$	1.00000	0.100000
$101$	−4.26530	−0.424413	−0.212207	−	0.977225i	$-0.568065\pi$
$101$	−4.26530	−0.424413	−0.212207	+	0.977225i	$0.568065\pi$
$102$	0	0
$103$	−1.87130	−0.184385	−0.0921925	−	0.995741i	$-0.529388\pi$
$103$	−1.87130	−0.184385	−0.0921925	+	0.995741i	$0.529388\pi$
$104$	1.66654	0.163418
$105$	0	0
$106$	6.74336	0.654973
$107$	3.10568	0.300237	0.150119	−	0.988668i	$-0.452034\pi$
$107$	3.10568	0.300237	0.150119	+	0.988668i	$0.452034\pi$
$108$	0	0
$109$	16.8087	1.60998	0.804992	−	0.593286i	$-0.202170\pi$
$109$	16.8087	1.60998	0.804992	+	0.593286i	$0.202170\pi$
$110$	3.22841	0.307817
$111$	0	0
$112$	2.69061	0.254239
$113$	−14.5904	−1.37255	−0.686277	−	0.727341i	$-0.740756\pi$
$113$	−14.5904	−1.37255	−0.686277	+	0.727341i	$0.740756\pi$
$114$	0	0
$115$	3.67557	0.342749
$116$	−9.13315	−0.847992
$117$	0	0
$118$	10.1336	0.932875
$119$	1.79328	0.164390
$120$	0	0
$121$	−0.577358	−0.0524871
$122$	−11.4993	−1.04110
$123$	0	0
$124$	5.71468	0.513194
$125$	1.00000	0.0894427
$126$	0	0
$127$	14.9856	1.32975	0.664877	−	0.746953i	$-0.268484\pi$
$127$	14.9856	1.32975	0.664877	+	0.746953i	$0.268484\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.66654	0.146166
$131$	−19.9913	−1.74665	−0.873323	−	0.487142i	$-0.838039\pi$
$131$	−19.9913	−1.74665	−0.873323	+	0.487142i	$0.838039\pi$
$132$	0	0
$133$	−15.2352	−1.32106
$134$	3.40986	0.294567
$135$	0	0
$136$	0.666496	0.0571515
$137$	4.13362	0.353159	0.176579	−	0.984286i	$-0.443497\pi$
$137$	4.13362	0.353159	0.176579	+	0.984286i	$0.443497\pi$
$138$	0	0
$139$	5.07560	0.430506	0.215253	−	0.976558i	$-0.430942\pi$
$139$	5.07560	0.430506	0.215253	+	0.976558i	$0.430942\pi$
$140$	2.69061	0.227398
$141$	0	0
$142$	−16.1900	−1.35863
$143$	5.38029	0.449922
$144$	0	0
$145$	−9.13315	−0.758467
$146$	0.318464	0.0263563
$147$	0	0
$148$	5.79103	0.476020
$149$	−3.72371	−0.305058	−0.152529	−	0.988299i	$-0.548742\pi$
$149$	−3.72371	−0.305058	−0.152529	+	0.988299i	$0.548742\pi$
$150$	0	0
$151$	14.1234	1.14934	0.574672	−	0.818384i	$-0.305130\pi$
$151$	14.1234	1.14934	0.574672	+	0.818384i	$0.305130\pi$
$152$	−5.66234	−0.459276
$153$	0	0
$154$	8.68641	0.699971
$155$	5.71468	0.459014
$156$	0	0
$157$	−9.79595	−0.781802	−0.390901	−	0.920433i	$-0.627837\pi$
$157$	−9.79595	−0.781802	−0.390901	+	0.920433i	$0.627837\pi$
$158$	−1.09139	−0.0868265
$159$	0	0
$160$	1.00000	0.0790569
$161$	9.88955	0.779405
$162$	0	0
$163$	−21.7815	−1.70606	−0.853029	−	0.521863i	$-0.825237\pi$
$163$	−21.7815	−1.70606	−0.853029	+	0.521863i	$0.825237\pi$
$164$	−8.92810	−0.697168
$165$	0	0
$166$	7.18550	0.557703
$167$	−3.75840	−0.290834	−0.145417	−	0.989370i	$-0.546452\pi$
$167$	−3.75840	−0.290834	−0.145417	+	0.989370i	$0.546452\pi$
$168$	0	0
$169$	−10.2226	−0.786356
$170$	0.666496	0.0511179
$171$	0	0
$172$	8.73237	0.665837
$173$	−8.14265	−0.619074	−0.309537	−	0.950887i	$-0.600174\pi$
$173$	−8.14265	−0.619074	−0.309537	+	0.950887i	$0.600174\pi$
$174$	0	0
$175$	2.69061	0.203391
$176$	3.22841	0.243351
$177$	0	0
$178$	−1.00000	−0.0749532
$179$	−12.4135	−0.927828	−0.463914	−	0.885880i	$-0.653555\pi$
$179$	−12.4135	−0.927828	−0.463914	+	0.885880i	$0.653555\pi$
$180$	0	0
$181$	6.09831	0.453284	0.226642	−	0.973978i	$-0.427225\pi$
$181$	6.09831	0.453284	0.226642	+	0.973978i	$0.427225\pi$
$182$	4.48402	0.332378
$183$	0	0
$184$	3.67557	0.270967
$185$	5.79103	0.425765
$186$	0	0
$187$	2.15172	0.157349
$188$	13.0045	0.948447
$189$	0	0
$190$	−5.66234	−0.410789
$191$	−27.4055	−1.98300	−0.991498	−	0.130123i	$-0.958463\pi$
$191$	−27.4055	−1.98300	−0.991498	+	0.130123i	$0.958463\pi$
$192$	0	0
$193$	4.08538	0.294072	0.147036	−	0.989131i	$-0.453027\pi$
$193$	4.08538	0.294072	0.147036	+	0.989131i	$0.453027\pi$
$194$	9.35115	0.671373
$195$	0	0
$196$	0.239400	0.0171000
$197$	−13.2156	−0.941572	−0.470786	−	0.882247i	$-0.656030\pi$
$197$	−13.2156	−0.941572	−0.470786	+	0.882247i	$0.656030\pi$
$198$	0	0
$199$	18.9766	1.34522	0.672609	−	0.739998i	$-0.265173\pi$
$199$	18.9766	1.34522	0.672609	+	0.739998i	$0.265173\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−4.26530	−0.300105
$203$	−24.5738	−1.72474
$204$	0	0
$205$	−8.92810	−0.623566
$206$	−1.87130	−0.130380
$207$	0	0
$208$	1.66654	0.115554
$209$	−18.2804	−1.26448
$210$	0	0
$211$	5.17513	0.356270	0.178135	−	0.984006i	$-0.442994\pi$
$211$	5.17513	0.356270	0.178135	+	0.984006i	$0.442994\pi$
$212$	6.74336	0.463136
$213$	0	0
$214$	3.10568	0.212300
$215$	8.73237	0.595543
$216$	0	0
$217$	15.3760	1.04379
$218$	16.8087	1.13843
$219$	0	0
$220$	3.22841	0.217659
$221$	1.11074	0.0747167
$222$	0	0
$223$	−15.5239	−1.03956	−0.519778	−	0.854302i	$-0.673985\pi$
$223$	−15.5239	−1.03956	−0.519778	+	0.854302i	$0.673985\pi$
$224$	2.69061	0.179774
$225$	0	0
$226$	−14.5904	−0.970542
$227$	7.97353	0.529222	0.264611	−	0.964355i	$-0.414756\pi$
$227$	7.97353	0.529222	0.264611	+	0.964355i	$0.414756\pi$
$228$	0	0
$229$	23.4234	1.54786	0.773931	−	0.633271i	$-0.218288\pi$
$229$	23.4234	1.54786	0.773931	+	0.633271i	$0.218288\pi$
$230$	3.67557	0.242360
$231$	0	0
$232$	−9.13315	−0.599621
$233$	14.6269	0.958238	0.479119	−	0.877750i	$-0.340956\pi$
$233$	14.6269	0.958238	0.479119	+	0.877750i	$0.340956\pi$
$234$	0	0
$235$	13.0045	0.848317
$236$	10.1336	0.659642
$237$	0	0
$238$	1.79328	0.116241
$239$	−21.8888	−1.41587	−0.707936	−	0.706277i	$-0.750373\pi$
$239$	−21.8888	−1.41587	−0.707936	+	0.706277i	$0.750373\pi$
$240$	0	0
$241$	29.1892	1.88024	0.940121	−	0.340842i	$-0.110712\pi$
$241$	29.1892	1.88024	0.940121	+	0.340842i	$0.110712\pi$
$242$	−0.577358	−0.0371140
$243$	0	0
$244$	−11.4993	−0.736170
$245$	0.239400	0.0152947
$246$	0	0
$247$	−9.43653	−0.600432
$248$	5.71468	0.362883
$249$	0	0
$250$	1.00000	0.0632456
$251$	30.1478	1.90291	0.951456	−	0.307786i	$-0.0995882\pi$
$251$	30.1478	1.90291	0.951456	+	0.307786i	$0.0995882\pi$
$252$	0	0
$253$	11.8663	0.746026
$254$	14.9856	0.940278
$255$	0	0
$256$	1.00000	0.0625000
$257$	4.80697	0.299850	0.149925	−	0.988697i	$-0.452097\pi$
$257$	4.80697	0.299850	0.149925	+	0.988697i	$0.452097\pi$
$258$	0	0
$259$	15.5814	0.968183
$260$	1.66654	0.103355
$261$	0	0
$262$	−19.9913	−1.23506
$263$	7.11079	0.438470	0.219235	−	0.975672i	$-0.429644\pi$
$263$	7.11079	0.438470	0.219235	+	0.975672i	$0.429644\pi$
$264$	0	0
$265$	6.74336	0.414241
$266$	−15.2352	−0.934128
$267$	0	0
$268$	3.40986	0.208290
$269$	−16.6149	−1.01303	−0.506513	−	0.862232i	$-0.669066\pi$
$269$	−16.6149	−1.01303	−0.506513	+	0.862232i	$0.669066\pi$
$270$	0	0
$271$	13.7950	0.837988	0.418994	−	0.907989i	$-0.362383\pi$
$271$	13.7950	0.837988	0.418994	+	0.907989i	$0.362383\pi$
$272$	0.666496	0.0404122
$273$	0	0
$274$	4.13362	0.249721
$275$	3.22841	0.194681
$276$	0	0
$277$	12.3820	0.743961	0.371981	−	0.928241i	$-0.378679\pi$
$277$	12.3820	0.743961	0.371981	+	0.928241i	$0.378679\pi$
$278$	5.07560	0.304414
$279$	0	0
$280$	2.69061	0.160795
$281$	−9.75613	−0.582002	−0.291001	−	0.956723i	$-0.593988\pi$
$281$	−9.75613	−0.582002	−0.291001	+	0.956723i	$0.593988\pi$
$282$	0	0
$283$	−9.97137	−0.592736	−0.296368	−	0.955074i	$-0.595776\pi$
$283$	−9.97137	−0.592736	−0.296368	+	0.955074i	$0.595776\pi$
$284$	−16.1900	−0.960697
$285$	0	0
$286$	5.38029	0.318143
$287$	−24.0221	−1.41798
$288$	0	0
$289$	−16.5558	−0.973870
$290$	−9.13315	−0.536317
$291$	0	0
$292$	0.318464	0.0186367
$293$	−9.27171	−0.541659	−0.270830	−	0.962627i	$-0.587298\pi$
$293$	−9.27171	−0.541659	−0.270830	+	0.962627i	$0.587298\pi$
$294$	0	0
$295$	10.1336	0.590002
$296$	5.79103	0.336597
$297$	0	0
$298$	−3.72371	−0.215709
$299$	6.12550	0.354247
$300$	0	0
$301$	23.4954	1.35425
$302$	14.1234	0.812708
$303$	0	0
$304$	−5.66234	−0.324757
$305$	−11.4993	−0.658450
$306$	0	0
$307$	1.75882	0.100381	0.0501905	−	0.998740i	$-0.484017\pi$
$307$	1.75882	0.100381	0.0501905	+	0.998740i	$0.484017\pi$
$308$	8.68641	0.494954
$309$	0	0
$310$	5.71468	0.324572
$311$	5.51671	0.312824	0.156412	−	0.987692i	$-0.450007\pi$
$311$	5.51671	0.312824	0.156412	+	0.987692i	$0.450007\pi$
$312$	0	0
$313$	7.51475	0.424759	0.212379	−	0.977187i	$-0.431879\pi$
$313$	7.51475	0.424759	0.212379	+	0.977187i	$0.431879\pi$
$314$	−9.79595	−0.552818
$315$	0	0
$316$	−1.09139	−0.0613956
$317$	−16.5031	−0.926906	−0.463453	−	0.886121i	$-0.653390\pi$
$317$	−16.5031	−0.926906	−0.463453	+	0.886121i	$0.653390\pi$
$318$	0	0
$319$	−29.4856	−1.65088
$320$	1.00000	0.0559017
$321$	0	0
$322$	9.88955	0.551123
$323$	−3.77392	−0.209987
$324$	0	0
$325$	1.66654	0.0924432
$326$	−21.7815	−1.20637
$327$	0	0
$328$	−8.92810	−0.492972
$329$	34.9900	1.92906
$330$	0	0
$331$	−4.87626	−0.268024	−0.134012	−	0.990980i	$-0.542786\pi$
$331$	−4.87626	−0.268024	−0.134012	+	0.990980i	$0.542786\pi$
$332$	7.18550	0.394355
$333$	0	0
$334$	−3.75840	−0.205651
$335$	3.40986	0.186300
$336$	0	0
$337$	11.0519	0.602034	0.301017	−	0.953619i	$-0.402674\pi$
$337$	11.0519	0.602034	0.301017	+	0.953619i	$0.402674\pi$
$338$	−10.2226	−0.556038
$339$	0	0
$340$	0.666496	0.0361458
$341$	18.4493	0.999088
$342$	0	0
$343$	−18.1902	−0.982176
$344$	8.73237	0.470818
$345$	0	0
$346$	−8.14265	−0.437751
$347$	34.1826	1.83502	0.917508	−	0.397717i	$-0.130197\pi$
$347$	34.1826	1.83502	0.917508	+	0.397717i	$0.130197\pi$
$348$	0	0
$349$	−2.61684	−0.140076	−0.0700381	−	0.997544i	$-0.522312\pi$
$349$	−2.61684	−0.140076	−0.0700381	+	0.997544i	$0.522312\pi$
$350$	2.69061	0.143819
$351$	0	0
$352$	3.22841	0.172075
$353$	7.98091	0.424781	0.212391	−	0.977185i	$-0.431875\pi$
$353$	7.98091	0.424781	0.212391	+	0.977185i	$0.431875\pi$
$354$	0	0
$355$	−16.1900	−0.859273
$356$	−1.00000	−0.0529999
$357$	0	0
$358$	−12.4135	−0.656073
$359$	−33.9701	−1.79287	−0.896436	−	0.443172i	$-0.853853\pi$
$359$	−33.9701	−1.79287	−0.896436	+	0.443172i	$0.853853\pi$
$360$	0	0
$361$	13.0621	0.687477
$362$	6.09831	0.320520
$363$	0	0
$364$	4.48402	0.235027
$365$	0.318464	0.0166692
$366$	0	0
$367$	24.6949	1.28906	0.644532	−	0.764577i	$-0.277052\pi$
$367$	24.6949	1.28906	0.644532	+	0.764577i	$0.277052\pi$
$368$	3.67557	0.191602
$369$	0	0
$370$	5.79103	0.301062
$371$	18.1438	0.941978
$372$	0	0
$373$	−13.8166	−0.715396	−0.357698	−	0.933837i	$-0.616438\pi$
$373$	−13.8166	−0.715396	−0.357698	+	0.933837i	$0.616438\pi$
$374$	2.15172	0.111263
$375$	0	0
$376$	13.0045	0.670654
$377$	−15.2208	−0.783911
$378$	0	0
$379$	−11.8594	−0.609174	−0.304587	−	0.952484i	$-0.598519\pi$
$379$	−11.8594	−0.609174	−0.304587	+	0.952484i	$0.598519\pi$
$380$	−5.66234	−0.290472
$381$	0	0
$382$	−27.4055	−1.40219
$383$	−19.5971	−1.00137	−0.500684	−	0.865630i	$-0.666918\pi$
$383$	−19.5971	−1.00137	−0.500684	+	0.865630i	$0.666918\pi$
$384$	0	0
$385$	8.68641	0.442700
$386$	4.08538	0.207940
$387$	0	0
$388$	9.35115	0.474732
$389$	12.1979	0.618456	0.309228	−	0.950988i	$-0.399929\pi$
$389$	12.1979	0.618456	0.309228	+	0.950988i	$0.399929\pi$
$390$	0	0
$391$	2.44975	0.123889
$392$	0.239400	0.0120915
$393$	0	0
$394$	−13.2156	−0.665792
$395$	−1.09139	−0.0549139
$396$	0	0
$397$	29.6473	1.48796	0.743979	−	0.668203i	$-0.232936\pi$
$397$	29.6473	1.48796	0.743979	+	0.668203i	$0.232936\pi$
$398$	18.9766	0.951213
$399$	0	0
$400$	1.00000	0.0500000
$401$	−5.43272	−0.271297	−0.135649	−	0.990757i	$-0.543312\pi$
$401$	−5.43272	−0.271297	−0.135649	+	0.990757i	$0.543312\pi$
$402$	0	0
$403$	9.52377	0.474413
$404$	−4.26530	−0.212207
$405$	0	0
$406$	−24.5738	−1.21958
$407$	18.6958	0.926719
$408$	0	0
$409$	0.969581	0.0479427	0.0239714	−	0.999713i	$-0.492369\pi$
$409$	0.969581	0.0479427	0.0239714	+	0.999713i	$0.492369\pi$
$410$	−8.92810	−0.440928
$411$	0	0
$412$	−1.87130	−0.0921925
$413$	27.2656	1.34165
$414$	0	0
$415$	7.18550	0.352722
$416$	1.66654	0.0817090
$417$	0	0
$418$	−18.2804	−0.894121
$419$	−1.52316	−0.0744112	−0.0372056	−	0.999308i	$-0.511846\pi$
$419$	−1.52316	−0.0744112	−0.0372056	+	0.999308i	$0.511846\pi$
$420$	0	0
$421$	14.3391	0.698846	0.349423	−	0.936965i	$-0.386378\pi$
$421$	14.3391	0.698846	0.349423	+	0.936965i	$0.386378\pi$
$422$	5.17513	0.251921
$423$	0	0
$424$	6.74336	0.327487
$425$	0.666496	0.0323298
$426$	0	0
$427$	−30.9403	−1.49730
$428$	3.10568	0.150119
$429$	0	0
$430$	8.73237	0.421112
$431$	33.5583	1.61645	0.808224	−	0.588875i	$-0.200429\pi$
$431$	33.5583	1.61645	0.808224	+	0.588875i	$0.200429\pi$
$432$	0	0
$433$	−12.5847	−0.604782	−0.302391	−	0.953184i	$-0.597785\pi$
$433$	−12.5847	−0.604782	−0.302391	+	0.953184i	$0.597785\pi$
$434$	15.3760	0.738072
$435$	0	0
$436$	16.8087	0.804992
$437$	−20.8123	−0.995589
$438$	0	0
$439$	14.8883	0.710580	0.355290	−	0.934756i	$-0.384382\pi$
$439$	14.8883	0.710580	0.355290	+	0.934756i	$0.384382\pi$
$440$	3.22841	0.153908
$441$	0	0
$442$	1.11074	0.0528327
$443$	10.9299	0.519293	0.259647	−	0.965704i	$-0.416394\pi$
$443$	10.9299	0.519293	0.259647	+	0.965704i	$0.416394\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−15.5239	−0.735077
$447$	0	0
$448$	2.69061	0.127120
$449$	35.9550	1.69682	0.848411	−	0.529338i	$-0.177560\pi$
$449$	35.9550	1.69682	0.848411	+	0.529338i	$0.177560\pi$
$450$	0	0
$451$	−28.8236	−1.35725
$452$	−14.5904	−0.686277
$453$	0	0
$454$	7.97353	0.374216
$455$	4.48402	0.210214
$456$	0	0
$457$	10.3680	0.484997	0.242498	−	0.970152i	$-0.422033\pi$
$457$	10.3680	0.484997	0.242498	+	0.970152i	$0.422033\pi$
$458$	23.4234	1.09450
$459$	0	0
$460$	3.67557	0.171374
$461$	38.8513	1.80949	0.904743	−	0.425958i	$-0.140063\pi$
$461$	38.8513	1.80949	0.904743	+	0.425958i	$0.140063\pi$
$462$	0	0
$463$	−26.3837	−1.22616	−0.613078	−	0.790023i	$-0.710069\pi$
$463$	−26.3837	−1.22616	−0.613078	+	0.790023i	$0.710069\pi$
$464$	−9.13315	−0.423996
$465$	0	0
$466$	14.6269	0.677576
$467$	−31.6933	−1.46659	−0.733296	−	0.679909i	$-0.762019\pi$
$467$	−31.6933	−1.46659	−0.733296	+	0.679909i	$0.762019\pi$
$468$	0	0
$469$	9.17460	0.423644
$470$	13.0045	0.599851
$471$	0	0
$472$	10.1336	0.466438
$473$	28.1917	1.29626
$474$	0	0
$475$	−5.66234	−0.259806
$476$	1.79328	0.0821949
$477$	0	0
$478$	−21.8888	−1.00117
$479$	−29.8280	−1.36287	−0.681437	−	0.731877i	$-0.738645\pi$
$479$	−29.8280	−1.36287	−0.681437	+	0.731877i	$0.738645\pi$
$480$	0	0
$481$	9.65101	0.440048
$482$	29.1892	1.32953
$483$	0	0
$484$	−0.577358	−0.0262436
$485$	9.35115	0.424614
$486$	0	0
$487$	30.2539	1.37094	0.685468	−	0.728103i	$-0.259598\pi$
$487$	30.2539	1.37094	0.685468	+	0.728103i	$0.259598\pi$
$488$	−11.4993	−0.520551
$489$	0	0
$490$	0.239400	0.0108150
$491$	−13.5921	−0.613404	−0.306702	−	0.951806i	$-0.599226\pi$
$491$	−13.5921	−0.613404	−0.306702	+	0.951806i	$0.599226\pi$
$492$	0	0
$493$	−6.08720	−0.274154
$494$	−9.43653	−0.424570
$495$	0	0
$496$	5.71468	0.256597
$497$	−43.5609	−1.95397
$498$	0	0
$499$	−4.29674	−0.192348	−0.0961742	−	0.995365i	$-0.530661\pi$
$499$	−4.29674	−0.192348	−0.0961742	+	0.995365i	$0.530661\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	30.1478	1.34556
$503$	−11.6201	−0.518113	−0.259056	−	0.965862i	$-0.583412\pi$
$503$	−11.6201	−0.518113	−0.259056	+	0.965862i	$0.583412\pi$
$504$	0	0
$505$	−4.26530	−0.189803
$506$	11.8663	0.527520
$507$	0	0
$508$	14.9856	0.664877
$509$	14.5653	0.645595	0.322798	−	0.946468i	$-0.395377\pi$
$509$	14.5653	0.645595	0.322798	+	0.946468i	$0.395377\pi$
$510$	0	0
$511$	0.856864	0.0379054
$512$	1.00000	0.0441942
$513$	0	0
$514$	4.80697	0.212026
$515$	−1.87130	−0.0824595
$516$	0	0
$517$	41.9837	1.84644
$518$	15.5814	0.684609
$519$	0	0
$520$	1.66654	0.0730828
$521$	−43.5094	−1.90618	−0.953090	−	0.302687i	$-0.902116\pi$
$521$	−43.5094	−1.90618	−0.953090	+	0.302687i	$0.902116\pi$
$522$	0	0
$523$	−20.1337	−0.880386	−0.440193	−	0.897903i	$-0.645090\pi$
$523$	−20.1337	−0.880386	−0.440193	+	0.897903i	$0.645090\pi$
$524$	−19.9913	−0.873323
$525$	0	0
$526$	7.11079	0.310045
$527$	3.80881	0.165914
$528$	0	0
$529$	−9.49016	−0.412616
$530$	6.74336	0.292913
$531$	0	0
$532$	−15.2352	−0.660528
$533$	−14.8791	−0.644484
$534$	0	0
$535$	3.10568	0.134270
$536$	3.40986	0.147283
$537$	0	0
$538$	−16.6149	−0.716318
$539$	0.772882	0.0332904
$540$	0	0
$541$	−31.7059	−1.36314	−0.681572	−	0.731751i	$-0.738703\pi$
$541$	−31.7059	−1.36314	−0.681572	+	0.731751i	$0.738703\pi$
$542$	13.7950	0.592547
$543$	0	0
$544$	0.666496	0.0285758
$545$	16.8087	0.720007
$546$	0	0
$547$	25.7168	1.09957	0.549786	−	0.835306i	$-0.314709\pi$
$547$	25.7168	1.09957	0.549786	+	0.835306i	$0.314709\pi$
$548$	4.13362	0.176579
$549$	0	0
$550$	3.22841	0.137660
$551$	51.7150	2.20313
$552$	0	0
$553$	−2.93651	−0.124873
$554$	12.3820	0.526060
$555$	0	0
$556$	5.07560	0.215253
$557$	−46.1284	−1.95452	−0.977261	−	0.212041i	$-0.931989\pi$
$557$	−46.1284	−1.95452	−0.977261	+	0.212041i	$0.931989\pi$
$558$	0	0
$559$	14.5529	0.615521
$560$	2.69061	0.113699
$561$	0	0
$562$	−9.75613	−0.411538
$563$	22.0696	0.930122	0.465061	−	0.885279i	$-0.346032\pi$
$563$	22.0696	0.930122	0.465061	+	0.885279i	$0.346032\pi$
$564$	0	0
$565$	−14.5904	−0.613824
$566$	−9.97137	−0.419128
$567$	0	0
$568$	−16.1900	−0.679315
$569$	−17.9780	−0.753678	−0.376839	−	0.926279i	$-0.622989\pi$
$569$	−17.9780	−0.753678	−0.376839	+	0.926279i	$0.622989\pi$
$570$	0	0
$571$	−37.2926	−1.56065	−0.780324	−	0.625375i	$-0.784946\pi$
$571$	−37.2926	−1.56065	−0.780324	+	0.625375i	$0.784946\pi$
$572$	5.38029	0.224961
$573$	0	0
$574$	−24.0221	−1.00266
$575$	3.67557	0.153282
$576$	0	0
$577$	33.3977	1.39037	0.695183	−	0.718833i	$-0.255323\pi$
$577$	33.3977	1.39037	0.695183	+	0.718833i	$0.255323\pi$
$578$	−16.5558	−0.688630
$579$	0	0
$580$	−9.13315	−0.379233
$581$	19.3334	0.802084
$582$	0	0
$583$	21.7703	0.901636
$584$	0.318464	0.0131781
$585$	0	0
$586$	−9.27171	−0.383011
$587$	17.5960	0.726266	0.363133	−	0.931737i	$-0.381707\pi$
$587$	17.5960	0.726266	0.363133	+	0.931737i	$0.381707\pi$
$588$	0	0
$589$	−32.3585	−1.33331
$590$	10.1336	0.417194
$591$	0	0
$592$	5.79103	0.238010
$593$	−23.5674	−0.967799	−0.483900	−	0.875124i	$-0.660780\pi$
$593$	−23.5674	−0.967799	−0.483900	+	0.875124i	$0.660780\pi$
$594$	0	0
$595$	1.79328	0.0735174
$596$	−3.72371	−0.152529
$597$	0	0
$598$	6.12550	0.250490
$599$	−6.89867	−0.281872	−0.140936	−	0.990019i	$-0.545011\pi$
$599$	−6.89867	−0.281872	−0.140936	+	0.990019i	$0.545011\pi$
$600$	0	0
$601$	16.5758	0.676143	0.338071	−	0.941120i	$-0.390225\pi$
$601$	16.5758	0.676143	0.338071	+	0.941120i	$0.390225\pi$
$602$	23.4954	0.957603
$603$	0	0
$604$	14.1234	0.574672
$605$	−0.577358	−0.0234730
$606$	0	0
$607$	−42.3181	−1.71764	−0.858818	−	0.512280i	$-0.828801\pi$
$607$	−42.3181	−1.71764	−0.858818	+	0.512280i	$0.828801\pi$
$608$	−5.66234	−0.229638
$609$	0	0
$610$	−11.4993	−0.465595
$611$	21.6725	0.876775
$612$	0	0
$613$	−20.1081	−0.812160	−0.406080	−	0.913838i	$-0.633105\pi$
$613$	−20.1081	−0.812160	−0.406080	+	0.913838i	$0.633105\pi$
$614$	1.75882	0.0709801
$615$	0	0
$616$	8.68641	0.349985
$617$	10.8487	0.436751	0.218375	−	0.975865i	$-0.429924\pi$
$617$	10.8487	0.436751	0.218375	+	0.975865i	$0.429924\pi$
$618$	0	0
$619$	−10.8836	−0.437447	−0.218724	−	0.975787i	$-0.570189\pi$
$619$	−10.8836	−0.437447	−0.218724	+	0.975787i	$0.570189\pi$
$620$	5.71468	0.229507
$621$	0	0
$622$	5.51671	0.221200
$623$	−2.69061	−0.107797
$624$	0	0
$625$	1.00000	0.0400000
$626$	7.51475	0.300350
$627$	0	0
$628$	−9.79595	−0.390901
$629$	3.85970	0.153896
$630$	0	0
$631$	0.910782	0.0362577	0.0181288	−	0.999836i	$-0.494229\pi$
$631$	0.910782	0.0362577	0.0181288	+	0.999836i	$0.494229\pi$
$632$	−1.09139	−0.0434133
$633$	0	0
$634$	−16.5031	−0.655422
$635$	14.9856	0.594684
$636$	0	0
$637$	0.398970	0.0158078
$638$	−29.4856	−1.16734
$639$	0	0
$640$	1.00000	0.0395285
$641$	−35.2520	−1.39237	−0.696185	−	0.717862i	$-0.745121\pi$
$641$	−35.2520	−1.39237	−0.696185	+	0.717862i	$0.745121\pi$
$642$	0	0
$643$	−30.3741	−1.19784	−0.598918	−	0.800810i	$-0.704403\pi$
$643$	−30.3741	−1.19784	−0.598918	+	0.800810i	$0.704403\pi$
$644$	9.88955	0.389703
$645$	0	0
$646$	−3.77392	−0.148483
$647$	49.4393	1.94366	0.971829	−	0.235686i	$-0.0757337\pi$
$647$	49.4393	1.94366	0.971829	+	0.235686i	$0.0757337\pi$
$648$	0	0
$649$	32.7155	1.28420
$650$	1.66654	0.0653672
$651$	0	0
$652$	−21.7815	−0.853029
$653$	−33.7934	−1.32244	−0.661218	−	0.750194i	$-0.729960\pi$
$653$	−33.7934	−1.32244	−0.661218	+	0.750194i	$0.729960\pi$
$654$	0	0
$655$	−19.9913	−0.781123
$656$	−8.92810	−0.348584
$657$	0	0
$658$	34.9900	1.36405
$659$	2.01527	0.0785040	0.0392520	−	0.999229i	$-0.487502\pi$
$659$	2.01527	0.0785040	0.0392520	+	0.999229i	$0.487502\pi$
$660$	0	0
$661$	−46.0530	−1.79126	−0.895628	−	0.444804i	$-0.853273\pi$
$661$	−46.0530	−1.79126	−0.895628	+	0.444804i	$0.853273\pi$
$662$	−4.87626	−0.189521
$663$	0	0
$664$	7.18550	0.278851
$665$	−15.2352	−0.590794
$666$	0	0
$667$	−33.5696	−1.29982
$668$	−3.75840	−0.145417
$669$	0	0
$670$	3.40986	0.131734
$671$	−37.1246	−1.43318
$672$	0	0
$673$	32.5664	1.25534	0.627671	−	0.778479i	$-0.284008\pi$
$673$	32.5664	1.25534	0.627671	+	0.778479i	$0.284008\pi$
$674$	11.0519	0.425703
$675$	0	0
$676$	−10.2226	−0.393178
$677$	−24.8992	−0.956953	−0.478476	−	0.878100i	$-0.658811\pi$
$677$	−24.8992	−0.956953	−0.478476	+	0.878100i	$0.658811\pi$
$678$	0	0
$679$	25.1603	0.965564
$680$	0.666496	0.0255589
$681$	0	0
$682$	18.4493	0.706462
$683$	38.9863	1.49177	0.745884	−	0.666076i	$-0.232027\pi$
$683$	38.9863	1.49177	0.745884	+	0.666076i	$0.232027\pi$
$684$	0	0
$685$	4.13362	0.157937
$686$	−18.1902	−0.694504
$687$	0	0
$688$	8.73237	0.332919
$689$	11.2381	0.428138
$690$	0	0
$691$	−7.22330	−0.274787	−0.137394	−	0.990517i	$-0.543873\pi$
$691$	−7.22330	−0.274787	−0.137394	+	0.990517i	$0.543873\pi$
$692$	−8.14265	−0.309537
$693$	0	0
$694$	34.1826	1.29755
$695$	5.07560	0.192528
$696$	0	0
$697$	−5.95054	−0.225393
$698$	−2.61684	−0.0990488
$699$	0	0
$700$	2.69061	0.101696
$701$	−1.85518	−0.0700691	−0.0350346	−	0.999386i	$-0.511154\pi$
$701$	−1.85518	−0.0700691	−0.0350346	+	0.999386i	$0.511154\pi$
$702$	0	0
$703$	−32.7908	−1.23673
$704$	3.22841	0.121675
$705$	0	0
$706$	7.98091	0.300366
$707$	−11.4763	−0.431610
$708$	0	0
$709$	0.701602	0.0263492	0.0131746	−	0.999913i	$-0.495806\pi$
$709$	0.701602	0.0263492	0.0131746	+	0.999913i	$0.495806\pi$
$710$	−16.1900	−0.607598
$711$	0	0
$712$	−1.00000	−0.0374766
$713$	21.0047	0.786633
$714$	0	0
$715$	5.38029	0.201211
$716$	−12.4135	−0.463914
$717$	0	0
$718$	−33.9701	−1.26775
$719$	24.4449	0.911642	0.455821	−	0.890072i	$-0.349346\pi$
$719$	24.4449	0.911642	0.455821	+	0.890072i	$0.349346\pi$
$720$	0	0
$721$	−5.03495	−0.187511
$722$	13.0621	0.486120
$723$	0	0
$724$	6.09831	0.226642
$725$	−9.13315	−0.339197
$726$	0	0
$727$	−11.3411	−0.420618	−0.210309	−	0.977635i	$-0.567447\pi$
$727$	−11.3411	−0.420618	−0.210309	+	0.977635i	$0.567447\pi$
$728$	4.48402	0.166189
$729$	0	0
$730$	0.318464	0.0117869
$731$	5.82009	0.215264
$732$	0	0
$733$	−37.5764	−1.38792	−0.693958	−	0.720016i	$-0.744135\pi$
$733$	−37.5764	−1.38792	−0.693958	+	0.720016i	$0.744135\pi$
$734$	24.6949	0.911506
$735$	0	0
$736$	3.67557	0.135483
$737$	11.0084	0.405500
$738$	0	0
$739$	−44.3779	−1.63247	−0.816235	−	0.577721i	$-0.803942\pi$
$739$	−44.3779	−1.63247	−0.816235	+	0.577721i	$0.803942\pi$
$740$	5.79103	0.212883
$741$	0	0
$742$	18.1438	0.666079
$743$	11.7044	0.429393	0.214697	−	0.976681i	$-0.431124\pi$
$743$	11.7044	0.429393	0.214697	+	0.976681i	$0.431124\pi$
$744$	0	0
$745$	−3.72371	−0.136426
$746$	−13.8166	−0.505862
$747$	0	0
$748$	2.15172	0.0786747
$749$	8.35618	0.305328
$750$	0	0
$751$	1.48418	0.0541587	0.0270793	−	0.999633i	$-0.491379\pi$
$751$	1.48418	0.0541587	0.0270793	+	0.999633i	$0.491379\pi$
$752$	13.0045	0.474224
$753$	0	0
$754$	−15.2208	−0.554309
$755$	14.1234	0.514002
$756$	0	0
$757$	−49.5073	−1.79938	−0.899688	−	0.436535i	$-0.856206\pi$
$757$	−49.5073	−1.79938	−0.899688	+	0.436535i	$0.856206\pi$
$758$	−11.8594	−0.430751
$759$	0	0
$760$	−5.66234	−0.205395
$761$	30.9514	1.12199	0.560993	−	0.827820i	$-0.310419\pi$
$761$	30.9514	1.12199	0.560993	+	0.827820i	$0.310419\pi$
$762$	0	0
$763$	45.2258	1.63728
$764$	−27.4055	−0.991498
$765$	0	0
$766$	−19.5971	−0.708073
$767$	16.8881	0.609794
$768$	0	0
$769$	32.5233	1.17282	0.586410	−	0.810015i	$-0.300541\pi$
$769$	32.5233	1.17282	0.586410	+	0.810015i	$0.300541\pi$
$770$	8.68641	0.313036
$771$	0	0
$772$	4.08538	0.147036
$773$	−23.6140	−0.849335	−0.424668	−	0.905349i	$-0.639609\pi$
$773$	−23.6140	−0.849335	−0.424668	+	0.905349i	$0.639609\pi$
$774$	0	0
$775$	5.71468	0.205277
$776$	9.35115	0.335687
$777$	0	0
$778$	12.1979	0.437314
$779$	50.5539	1.81128
$780$	0	0
$781$	−52.2678	−1.87029
$782$	2.44975	0.0876030
$783$	0	0
$784$	0.239400	0.00855000
$785$	−9.79595	−0.349633
$786$	0	0
$787$	53.0834	1.89222	0.946109	−	0.323849i	$-0.104977\pi$
$787$	53.0834	1.89222	0.946109	+	0.323849i	$0.104977\pi$
$788$	−13.2156	−0.470786
$789$	0	0
$790$	−1.09139	−0.0388300
$791$	−39.2572	−1.39583
$792$	0	0
$793$	−19.1641	−0.680539
$794$	29.6473	1.05214
$795$	0	0
$796$	18.9766	0.672609
$797$	45.9913	1.62910	0.814549	−	0.580095i	$-0.196985\pi$
$797$	45.9913	1.62910	0.814549	+	0.580095i	$0.196985\pi$
$798$	0	0
$799$	8.66741	0.306631
$800$	1.00000	0.0353553
$801$	0	0
$802$	−5.43272	−0.191836
$803$	1.02813	0.0362820
$804$	0	0
$805$	9.88955	0.348561
$806$	9.52377	0.335460
$807$	0	0
$808$	−4.26530	−0.150053
$809$	28.6274	1.00648	0.503242	−	0.864145i	$-0.332140\pi$
$809$	28.6274	1.00648	0.503242	+	0.864145i	$0.332140\pi$
$810$	0	0
$811$	−56.4748	−1.98310	−0.991549	−	0.129736i	$-0.958587\pi$
$811$	−56.4748	−1.98310	−0.991549	+	0.129736i	$0.958587\pi$
$812$	−24.5738	−0.862371
$813$	0	0
$814$	18.6958	0.655289
$815$	−21.7815	−0.762973
$816$	0	0
$817$	−49.4456	−1.72988
$818$	0.969581	0.0339006
$819$	0	0
$820$	−8.92810	−0.311783
$821$	−34.7588	−1.21309	−0.606544	−	0.795050i	$-0.707445\pi$
$821$	−34.7588	−1.21309	−0.606544	+	0.795050i	$0.707445\pi$
$822$	0	0
$823$	29.1258	1.01526	0.507631	−	0.861574i	$-0.330521\pi$
$823$	29.1258	1.01526	0.507631	+	0.861574i	$0.330521\pi$
$824$	−1.87130	−0.0651899
$825$	0	0
$826$	27.2656	0.948693
$827$	−19.5730	−0.680621	−0.340311	−	0.940313i	$-0.610532\pi$
$827$	−19.5730	−0.680621	−0.340311	+	0.940313i	$0.610532\pi$
$828$	0	0
$829$	−43.3413	−1.50530	−0.752652	−	0.658418i	$-0.771226\pi$
$829$	−43.3413	−1.50530	−0.752652	+	0.658418i	$0.771226\pi$
$830$	7.18550	0.249412
$831$	0	0
$832$	1.66654	0.0577770
$833$	0.159559	0.00552839
$834$	0	0
$835$	−3.75840	−0.130065
$836$	−18.2804	−0.632239
$837$	0	0
$838$	−1.52316	−0.0526167
$839$	−9.59277	−0.331179	−0.165590	−	0.986195i	$-0.552953\pi$
$839$	−9.59277	−0.331179	−0.165590	+	0.986195i	$0.552953\pi$
$840$	0	0
$841$	54.4145	1.87636
$842$	14.3391	0.494159
$843$	0	0
$844$	5.17513	0.178135
$845$	−10.2226	−0.351669
$846$	0	0
$847$	−1.55345	−0.0533771
$848$	6.74336	0.231568
$849$	0	0
$850$	0.666496	0.0228606
$851$	21.2854	0.729653
$852$	0	0
$853$	−4.91394	−0.168250	−0.0841250	−	0.996455i	$-0.526810\pi$
$853$	−4.91394	−0.168250	−0.0841250	+	0.996455i	$0.526810\pi$
$854$	−30.9403	−1.05875
$855$	0	0
$856$	3.10568	0.106150
$857$	−23.4386	−0.800648	−0.400324	−	0.916374i	$-0.631102\pi$
$857$	−23.4386	−0.800648	−0.400324	+	0.916374i	$0.631102\pi$
$858$	0	0
$859$	3.99343	0.136254	0.0681270	−	0.997677i	$-0.478298\pi$
$859$	3.99343	0.136254	0.0681270	+	0.997677i	$0.478298\pi$
$860$	8.73237	0.297771
$861$	0	0
$862$	33.5583	1.14300
$863$	19.7910	0.673694	0.336847	−	0.941559i	$-0.390640\pi$
$863$	19.7910	0.673694	0.336847	+	0.941559i	$0.390640\pi$
$864$	0	0
$865$	−8.14265	−0.276858
$866$	−12.5847	−0.427646
$867$	0	0
$868$	15.3760	0.521896
$869$	−3.52346	−0.119525
$870$	0	0
$871$	5.68267	0.192550
$872$	16.8087	0.569215
$873$	0	0
$874$	−20.8123	−0.703988
$875$	2.69061	0.0909593
$876$	0	0
$877$	49.2862	1.66428	0.832138	−	0.554568i	$-0.187117\pi$
$877$	49.2862	1.66428	0.832138	+	0.554568i	$0.187117\pi$
$878$	14.8883	0.502456
$879$	0	0
$880$	3.22841	0.108830
$881$	7.70492	0.259585	0.129793	−	0.991541i	$-0.458569\pi$
$881$	7.70492	0.259585	0.129793	+	0.991541i	$0.458569\pi$
$882$	0	0
$883$	−22.4931	−0.756954	−0.378477	−	0.925611i	$-0.623552\pi$
$883$	−22.4931	−0.756954	−0.378477	+	0.925611i	$0.623552\pi$
$884$	1.11074	0.0373584
$885$	0	0
$886$	10.9299	0.367196
$887$	−12.7700	−0.428775	−0.214387	−	0.976749i	$-0.568775\pi$
$887$	−12.7700	−0.428775	−0.214387	+	0.976749i	$0.568775\pi$
$888$	0	0
$889$	40.3203	1.35230
$890$	−1.00000	−0.0335201
$891$	0	0
$892$	−15.5239	−0.519778
$893$	−73.6356	−2.46412
$894$	0	0
$895$	−12.4135	−0.414937
$896$	2.69061	0.0898871
$897$	0	0
$898$	35.9550	1.19983
$899$	−52.1931	−1.74074
$900$	0	0
$901$	4.49442	0.149731
$902$	−28.8236	−0.959721
$903$	0	0
$904$	−14.5904	−0.485271
$905$	6.09831	0.202715
$906$	0	0
$907$	15.9351	0.529117	0.264559	−	0.964370i	$-0.414774\pi$
$907$	15.9351	0.529117	0.264559	+	0.964370i	$0.414774\pi$
$908$	7.97353	0.264611
$909$	0	0
$910$	4.48402	0.148644
$911$	51.0935	1.69280	0.846402	−	0.532544i	$-0.178764\pi$
$911$	51.0935	1.69280	0.846402	+	0.532544i	$0.178764\pi$
$912$	0	0
$913$	23.1977	0.767733
$914$	10.3680	0.342945
$915$	0	0
$916$	23.4234	0.773931
$917$	−53.7888	−1.77626
$918$	0	0
$919$	−10.8471	−0.357812	−0.178906	−	0.983866i	$-0.557256\pi$
$919$	−10.8471	−0.357812	−0.178906	+	0.983866i	$0.557256\pi$
$920$	3.67557	0.121180
$921$	0	0
$922$	38.8513	1.27950
$923$	−26.9813	−0.888099
$924$	0	0
$925$	5.79103	0.190408
$926$	−26.3837	−0.867023
$927$	0	0
$928$	−9.13315	−0.299810
$929$	−26.7917	−0.879008	−0.439504	−	0.898241i	$-0.644846\pi$
$929$	−26.7917	−0.879008	−0.439504	+	0.898241i	$0.644846\pi$
$930$	0	0
$931$	−1.35556	−0.0444268
$932$	14.6269	0.479119
$933$	0	0
$934$	−31.6933	−1.03704
$935$	2.15172	0.0703688
$936$	0	0
$937$	31.2940	1.02233	0.511165	−	0.859483i	$-0.329214\pi$
$937$	31.2940	1.02233	0.511165	+	0.859483i	$0.329214\pi$
$938$	9.17460	0.299561
$939$	0	0
$940$	13.0045	0.424159
$941$	−23.7913	−0.775573	−0.387787	−	0.921749i	$-0.626760\pi$
$941$	−23.7913	−0.775573	−0.387787	+	0.921749i	$0.626760\pi$
$942$	0	0
$943$	−32.8159	−1.06863
$944$	10.1336	0.329821
$945$	0	0
$946$	28.1917	0.916591
$947$	−31.8630	−1.03541	−0.517704	−	0.855560i	$-0.673213\pi$
$947$	−31.8630	−1.03541	−0.517704	+	0.855560i	$0.673213\pi$
$948$	0	0
$949$	0.530734	0.0172284
$950$	−5.66234	−0.183710
$951$	0	0
$952$	1.79328	0.0581206
$953$	55.8289	1.80848	0.904238	−	0.427028i	$-0.140439\pi$
$953$	55.8289	1.80848	0.904238	+	0.427028i	$0.140439\pi$
$954$	0	0
$955$	−27.4055	−0.886823
$956$	−21.8888	−0.707936
$957$	0	0
$958$	−29.8280	−0.963698
$959$	11.1220	0.359147
$960$	0	0
$961$	1.65760	0.0534710
$962$	9.65101	0.311161
$963$	0	0
$964$	29.1892	0.940121
$965$	4.08538	0.131513
$966$	0	0
$967$	−21.8711	−0.703327	−0.351663	−	0.936127i	$-0.614384\pi$
$967$	−21.8711	−0.703327	−0.351663	+	0.936127i	$0.614384\pi$
$968$	−0.577358	−0.0185570
$969$	0	0
$970$	9.35115	0.300247
$971$	1.42416	0.0457034	0.0228517	−	0.999739i	$-0.492725\pi$
$971$	1.42416	0.0457034	0.0228517	+	0.999739i	$0.492725\pi$
$972$	0	0
$973$	13.6565	0.437806
$974$	30.2539	0.969397
$975$	0	0
$976$	−11.4993	−0.368085
$977$	27.3160	0.873917	0.436959	−	0.899482i	$-0.356056\pi$
$977$	27.3160	0.873917	0.436959	+	0.899482i	$0.356056\pi$
$978$	0	0
$979$	−3.22841	−0.103180
$980$	0.239400	0.00764735
$981$	0	0
$982$	−13.5921	−0.433742
$983$	−4.75534	−0.151672	−0.0758358	−	0.997120i	$-0.524162\pi$
$983$	−4.75534	−0.151672	−0.0758358	+	0.997120i	$0.524162\pi$
$984$	0	0
$985$	−13.2156	−0.421084
$986$	−6.08720	−0.193856
$987$	0	0
$988$	−9.43653	−0.300216
$989$	32.0965	1.02061
$990$	0	0
$991$	33.2730	1.05695	0.528476	−	0.848948i	$-0.322764\pi$
$991$	33.2730	1.05695	0.528476	+	0.848948i	$0.322764\pi$
$992$	5.71468	0.181441
$993$	0	0
$994$	−43.5609	−1.38167
$995$	18.9766	0.601600
$996$	0	0
$997$	15.0932	0.478005	0.239003	−	0.971019i	$-0.423180\pi$
$997$	15.0932	0.478005	0.239003	+	0.971019i	$0.423180\pi$
$998$	−4.29674	−0.136011
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8010.2.a.bn.1.6		7
3.2	odd	2		2670.2.a.t.1.6	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2670.2.a.t.1.6	✓	7	3.2	odd	2
8010.2.a.bn.1.6		7	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:	\( N \)	\(=\)	\( 8010 = 2 \cdot 3^{2} \cdot 5 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8010.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.69061 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.69061 q^{7} +1.00000 q^{8} +1.00000 q^{10} +3.22841 q^{11} +1.66654 q^{13} +2.69061 q^{14} +1.00000 q^{16} +0.666496 q^{17} -5.66234 q^{19} +1.00000 q^{20} +3.22841 q^{22} +3.67557 q^{23} +1.00000 q^{25} +1.66654 q^{26} +2.69061 q^{28} -9.13315 q^{29} +5.71468 q^{31} +1.00000 q^{32} +0.666496 q^{34} +2.69061 q^{35} +5.79103 q^{37} -5.66234 q^{38} +1.00000 q^{40} -8.92810 q^{41} +8.73237 q^{43} +3.22841 q^{44} +3.67557 q^{46} +13.0045 q^{47} +0.239400 q^{49} +1.00000 q^{50} +1.66654 q^{52} +6.74336 q^{53} +3.22841 q^{55} +2.69061 q^{56} -9.13315 q^{58} +10.1336 q^{59} -11.4993 q^{61} +5.71468 q^{62} +1.00000 q^{64} +1.66654 q^{65} +3.40986 q^{67} +0.666496 q^{68} +2.69061 q^{70} -16.1900 q^{71} +0.318464 q^{73} +5.79103 q^{74} -5.66234 q^{76} +8.68641 q^{77} -1.09139 q^{79} +1.00000 q^{80} -8.92810 q^{82} +7.18550 q^{83} +0.666496 q^{85} +8.73237 q^{86} +3.22841 q^{88} -1.00000 q^{89} +4.48402 q^{91} +3.67557 q^{92} +13.0045 q^{94} -5.66234 q^{95} +9.35115 q^{97} +0.239400 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + q^{7} + 7 q^{8}+O(q^{10}) \) 7 * q + 7 * q^2 + 7 * q^4 + 7 * q^5 + q^7 + 7 * q^8	\( 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + q^{7} + 7 q^{8} + 7 q^{10} - q^{11} + q^{14} + 7 q^{16} + 9 q^{17} + 9 q^{19} + 7 q^{20} - q^{22} + 8 q^{23} + 7 q^{25} + q^{28} + 4 q^{29} + 16 q^{31} + 7 q^{32} + 9 q^{34} + q^{35} + 2 q^{37} + 9 q^{38} + 7 q^{40} + q^{41} - 10 q^{43} - q^{44} + 8 q^{46} + 13 q^{47} + 26 q^{49} + 7 q^{50} + 24 q^{53} - q^{55} + q^{56} + 4 q^{58} + 6 q^{59} + 23 q^{61} + 16 q^{62} + 7 q^{64} + 5 q^{67} + 9 q^{68} + q^{70} + 8 q^{71} - 2 q^{73} + 2 q^{74} + 9 q^{76} + 6 q^{77} + 7 q^{79} + 7 q^{80} + q^{82} + 7 q^{83} + 9 q^{85} - 10 q^{86} - q^{88} - 7 q^{89} + 48 q^{91} + 8 q^{92} + 13 q^{94} + 9 q^{95} + 30 q^{97} + 26 q^{98}+O(q^{100}) \) 7 * q + 7 * q^2 + 7 * q^4 + 7 * q^5 + q^7 + 7 * q^8 + 7 * q^10 - q^11 + q^14 + 7 * q^16 + 9 * q^17 + 9 * q^19 + 7 * q^20 - q^22 + 8 * q^23 + 7 * q^25 + q^28 + 4 * q^29 + 16 * q^31 + 7 * q^32 + 9 * q^34 + q^35 + 2 * q^37 + 9 * q^38 + 7 * q^40 + q^41 - 10 * q^43 - q^44 + 8 * q^46 + 13 * q^47 + 26 * q^49 + 7 * q^50 + 24 * q^53 - q^55 + q^56 + 4 * q^58 + 6 * q^59 + 23 * q^61 + 16 * q^62 + 7 * q^64 + 5 * q^67 + 9 * q^68 + q^70 + 8 * q^71 - 2 * q^73 + 2 * q^74 + 9 * q^76 + 6 * q^77 + 7 * q^79 + 7 * q^80 + q^82 + 7 * q^83 + 9 * q^85 - 10 * q^86 - q^88 - 7 * q^89 + 48 * q^91 + 8 * q^92 + 13 * q^94 + 9 * q^95 + 30 * q^97 + 26 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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