LMFDB - Embedded newform 8010.2.a.bn.1.3

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			1.3
Root			$1.14258$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -0.407283 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -0.407283 q^{7} +1.00000 q^{8} +1.00000 q^{10} -5.17907 q^{11} -5.41054 q^{13} -0.407283 q^{14} +1.00000 q^{16} -5.29339 q^{17} +4.89391 q^{19} +1.00000 q^{20} -5.17907 q^{22} +6.02245 q^{23} +1.00000 q^{25} -5.41054 q^{26} -0.407283 q^{28} +6.80359 q^{29} +6.59597 q^{31} +1.00000 q^{32} -5.29339 q^{34} -0.407283 q^{35} -5.41551 q^{37} +4.89391 q^{38} +1.00000 q^{40} -5.72949 q^{41} +7.23034 q^{43} -5.17907 q^{44} +6.02245 q^{46} -2.28199 q^{47} -6.83412 q^{49} +1.00000 q^{50} -5.41054 q^{52} +6.57529 q^{53} -5.17907 q^{55} -0.407283 q^{56} +6.80359 q^{58} +13.1937 q^{59} +5.18842 q^{61} +6.59597 q^{62} +1.00000 q^{64} -5.41054 q^{65} -2.71810 q^{67} -5.29339 q^{68} -0.407283 q^{70} +3.59571 q^{71} +11.7231 q^{73} -5.41551 q^{74} +4.89391 q^{76} +2.10935 q^{77} +16.4412 q^{79} +1.00000 q^{80} -5.72949 q^{82} +2.68629 q^{83} -5.29339 q^{85} +7.23034 q^{86} -5.17907 q^{88} -1.00000 q^{89} +2.20362 q^{91} +6.02245 q^{92} -2.28199 q^{94} +4.89391 q^{95} +14.0449 q^{97} -6.83412 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−0.407283	−0.153939	−0.0769693	−	0.997033i	$-0.524524\pi$
$7$	−0.407283	−0.153939	−0.0769693	+	0.997033i	$0.524524\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−5.17907	−1.56155	−0.780774	−	0.624814i	$-0.785175\pi$
$11$	−5.17907	−1.56155	−0.780774	+	0.624814i	$0.785175\pi$
$12$	0	0
$13$	−5.41054	−1.50061	−0.750307	−	0.661090i	$-0.770094\pi$
$13$	−5.41054	−1.50061	−0.750307	+	0.661090i	$0.770094\pi$
$14$	−0.407283	−0.108851
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.29339	−1.28383	−0.641917	−	0.766774i	$-0.721861\pi$
$17$	−5.29339	−1.28383	−0.641917	+	0.766774i	$0.721861\pi$
$18$	0	0
$19$	4.89391	1.12274	0.561370	−	0.827565i	$-0.310275\pi$
$19$	4.89391	1.12274	0.561370	+	0.827565i	$0.310275\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−5.17907	−1.10418
$23$	6.02245	1.25577	0.627884	−	0.778307i	$-0.283921\pi$
$23$	6.02245	1.25577	0.627884	+	0.778307i	$0.283921\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−5.41054	−1.06109
$27$	0	0
$28$	−0.407283	−0.0769693
$29$	6.80359	1.26340	0.631698	−	0.775215i	$-0.282358\pi$
$29$	6.80359	1.26340	0.631698	+	0.775215i	$0.282358\pi$
$30$	0	0
$31$	6.59597	1.18467	0.592336	−	0.805691i	$-0.298206\pi$
$31$	6.59597	1.18467	0.592336	+	0.805691i	$0.298206\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−5.29339	−0.907808
$35$	−0.407283	−0.0688434
$36$	0	0
$37$	−5.41551	−0.890304	−0.445152	−	0.895455i	$-0.646850\pi$
$37$	−5.41551	−0.890304	−0.445152	+	0.895455i	$0.646850\pi$
$38$	4.89391	0.793897
$39$	0	0
$40$	1.00000	0.158114
$41$	−5.72949	−0.894796	−0.447398	−	0.894335i	$-0.647649\pi$
$41$	−5.72949	−0.894796	−0.447398	+	0.894335i	$0.647649\pi$
$42$	0	0
$43$	7.23034	1.10262	0.551308	−	0.834301i	$-0.314129\pi$
$43$	7.23034	1.10262	0.551308	+	0.834301i	$0.314129\pi$
$44$	−5.17907	−0.780774
$45$	0	0
$46$	6.02245	0.887962
$47$	−2.28199	−0.332863	−0.166431	−	0.986053i	$-0.553224\pi$
$47$	−2.28199	−0.332863	−0.166431	+	0.986053i	$0.553224\pi$
$48$	0	0
$49$	−6.83412	−0.976303
$50$	1.00000	0.141421
$51$	0	0
$52$	−5.41054	−0.750307
$53$	6.57529	0.903185	0.451593	−	0.892224i	$-0.350856\pi$
$53$	6.57529	0.903185	0.451593	+	0.892224i	$0.350856\pi$
$54$	0	0
$55$	−5.17907	−0.698345
$56$	−0.407283	−0.0544255
$57$	0	0
$58$	6.80359	0.893356
$59$	13.1937	1.71768	0.858838	−	0.512248i	$-0.171187\pi$
$59$	13.1937	1.71768	0.858838	+	0.512248i	$0.171187\pi$
$60$	0	0
$61$	5.18842	0.664309	0.332155	−	0.943225i	$-0.392224\pi$
$61$	5.18842	0.664309	0.332155	+	0.943225i	$0.392224\pi$
$62$	6.59597	0.837689
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.41054	−0.671095
$66$	0	0
$67$	−2.71810	−0.332068	−0.166034	−	0.986120i	$-0.553096\pi$
$67$	−2.71810	−0.332068	−0.166034	+	0.986120i	$0.553096\pi$
$68$	−5.29339	−0.641917
$69$	0	0
$70$	−0.407283	−0.0486796
$71$	3.59571	0.426732	0.213366	−	0.976972i	$-0.431557\pi$
$71$	3.59571	0.426732	0.213366	+	0.976972i	$0.431557\pi$
$72$	0	0
$73$	11.7231	1.37209	0.686044	−	0.727560i	$-0.259346\pi$
$73$	11.7231	1.37209	0.686044	+	0.727560i	$0.259346\pi$
$74$	−5.41551	−0.629540
$75$	0	0
$76$	4.89391	0.561370
$77$	2.10935	0.240382
$78$	0	0
$79$	16.4412	1.84978	0.924891	−	0.380233i	$-0.124156\pi$
$79$	16.4412	1.84978	0.924891	+	0.380233i	$0.124156\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−5.72949	−0.632716
$83$	2.68629	0.294858	0.147429	−	0.989073i	$-0.452900\pi$
$83$	2.68629	0.294858	0.147429	+	0.989073i	$0.452900\pi$
$84$	0	0
$85$	−5.29339	−0.574148
$86$	7.23034	0.779668
$87$	0	0
$88$	−5.17907	−0.552090
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	2.20362	0.231002
$92$	6.02245	0.627884
$93$	0	0
$94$	−2.28199	−0.235369
$95$	4.89391	0.502104
$96$	0	0
$97$	14.0449	1.42604	0.713022	−	0.701141i	$-0.247326\pi$
$97$	14.0449	1.42604	0.713022	+	0.701141i	$0.247326\pi$
$98$	−6.83412	−0.690350
$99$	0	0
$100$	1.00000	0.100000
$101$	7.37391	0.733731	0.366866	−	0.930274i	$-0.380431\pi$
$101$	7.37391	0.733731	0.366866	+	0.930274i	$0.380431\pi$
$102$	0	0
$103$	−2.52160	−0.248461	−0.124230	−	0.992253i	$-0.539646\pi$
$103$	−2.52160	−0.248461	−0.124230	+	0.992253i	$0.539646\pi$
$104$	−5.41054	−0.530547
$105$	0	0
$106$	6.57529	0.638648
$107$	−18.4030	−1.77909	−0.889545	−	0.456848i	$-0.848978\pi$
$107$	−18.4030	−1.77909	−0.889545	+	0.456848i	$0.848978\pi$
$108$	0	0
$109$	3.21886	0.308311	0.154155	−	0.988047i	$-0.450734\pi$
$109$	3.21886	0.308311	0.154155	+	0.988047i	$0.450734\pi$
$110$	−5.17907	−0.493805
$111$	0	0
$112$	−0.407283	−0.0384846
$113$	−0.835582	−0.0786049	−0.0393025	−	0.999227i	$-0.512514\pi$
$113$	−0.835582	−0.0786049	−0.0393025	+	0.999227i	$0.512514\pi$
$114$	0	0
$115$	6.02245	0.561597
$116$	6.80359	0.631698
$117$	0	0
$118$	13.1937	1.21458
$119$	2.15591	0.197632
$120$	0	0
$121$	15.8227	1.43843
$122$	5.18842	0.469738
$123$	0	0
$124$	6.59597	0.592336
$125$	1.00000	0.0894427
$126$	0	0
$127$	8.70377	0.772334	0.386167	−	0.922429i	$-0.373799\pi$
$127$	8.70377	0.772334	0.386167	+	0.922429i	$0.373799\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−5.41054	−0.474536
$131$	9.31551	0.813900	0.406950	−	0.913451i	$-0.366592\pi$
$131$	9.31551	0.813900	0.406950	+	0.913451i	$0.366592\pi$
$132$	0	0
$133$	−1.99321	−0.172833
$134$	−2.71810	−0.234808
$135$	0	0
$136$	−5.29339	−0.453904
$137$	7.19371	0.614600	0.307300	−	0.951613i	$-0.400574\pi$
$137$	7.19371	0.614600	0.307300	+	0.951613i	$0.400574\pi$
$138$	0	0
$139$	−5.54357	−0.470199	−0.235100	−	0.971971i	$-0.575542\pi$
$139$	−5.54357	−0.470199	−0.235100	+	0.971971i	$0.575542\pi$
$140$	−0.407283	−0.0344217
$141$	0	0
$142$	3.59571	0.301745
$143$	28.0215	2.34328
$144$	0	0
$145$	6.80359	0.565008
$146$	11.7231	0.970213
$147$	0	0
$148$	−5.41551	−0.445152
$149$	−14.0290	−1.14930	−0.574649	−	0.818400i	$-0.694861\pi$
$149$	−14.0290	−1.14930	−0.574649	+	0.818400i	$0.694861\pi$
$150$	0	0
$151$	−9.76867	−0.794963	−0.397482	−	0.917610i	$-0.630116\pi$
$151$	−9.76867	−0.794963	−0.397482	+	0.917610i	$0.630116\pi$
$152$	4.89391	0.396948
$153$	0	0
$154$	2.10935	0.169976
$155$	6.59597	0.529801
$156$	0	0
$157$	−2.29981	−0.183544	−0.0917722	−	0.995780i	$-0.529253\pi$
$157$	−2.29981	−0.183544	−0.0917722	+	0.995780i	$0.529253\pi$
$158$	16.4412	1.30799
$159$	0	0
$160$	1.00000	0.0790569
$161$	−2.45284	−0.193311
$162$	0	0
$163$	16.1459	1.26465	0.632324	−	0.774704i	$-0.282101\pi$
$163$	16.1459	1.26465	0.632324	+	0.774704i	$0.282101\pi$
$164$	−5.72949	−0.447398
$165$	0	0
$166$	2.68629	0.208496
$167$	1.85445	0.143502	0.0717508	−	0.997423i	$-0.477141\pi$
$167$	1.85445	0.143502	0.0717508	+	0.997423i	$0.477141\pi$
$168$	0	0
$169$	16.2739	1.25184
$170$	−5.29339	−0.405984
$171$	0	0
$172$	7.23034	0.551308
$173$	−20.6267	−1.56822	−0.784110	−	0.620622i	$-0.786880\pi$
$173$	−20.6267	−1.56822	−0.784110	+	0.620622i	$0.786880\pi$
$174$	0	0
$175$	−0.407283	−0.0307877
$176$	−5.17907	−0.390387
$177$	0	0
$178$	−1.00000	−0.0749532
$179$	20.6072	1.54026	0.770129	−	0.637888i	$-0.220192\pi$
$179$	20.6072	1.54026	0.770129	+	0.637888i	$0.220192\pi$
$180$	0	0
$181$	22.6109	1.68065	0.840327	−	0.542080i	$-0.182363\pi$
$181$	22.6109	1.68065	0.840327	+	0.542080i	$0.182363\pi$
$182$	2.20362	0.163343
$183$	0	0
$184$	6.02245	0.443981
$185$	−5.41551	−0.398156
$186$	0	0
$187$	27.4148	2.00477
$188$	−2.28199	−0.166431
$189$	0	0
$190$	4.89391	0.355041
$191$	−6.42333	−0.464776	−0.232388	−	0.972623i	$-0.574654\pi$
$191$	−6.42333	−0.464776	−0.232388	+	0.972623i	$0.574654\pi$
$192$	0	0
$193$	1.42151	0.102323	0.0511613	−	0.998690i	$-0.483708\pi$
$193$	1.42151	0.102323	0.0511613	+	0.998690i	$0.483708\pi$
$194$	14.0449	1.00837
$195$	0	0
$196$	−6.83412	−0.488151
$197$	−12.0190	−0.856321	−0.428160	−	0.903703i	$-0.640838\pi$
$197$	−12.0190	−0.856321	−0.428160	+	0.903703i	$0.640838\pi$
$198$	0	0
$199$	15.2287	1.07953	0.539767	−	0.841815i	$-0.318512\pi$
$199$	15.2287	1.07953	0.539767	+	0.841815i	$0.318512\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	7.37391	0.518826
$203$	−2.77099	−0.194485
$204$	0	0
$205$	−5.72949	−0.400165
$206$	−2.52160	−0.175688
$207$	0	0
$208$	−5.41054	−0.375153
$209$	−25.3459	−1.75321
$210$	0	0
$211$	28.5967	1.96868	0.984339	−	0.176285i	$-0.0564082\pi$
$211$	28.5967	1.96868	0.984339	+	0.176285i	$0.0564082\pi$
$212$	6.57529	0.451593
$213$	0	0
$214$	−18.4030	−1.25801
$215$	7.23034	0.493105
$216$	0	0
$217$	−2.68643	−0.182367
$218$	3.21886	0.218009
$219$	0	0
$220$	−5.17907	−0.349173
$221$	28.6401	1.92654
$222$	0	0
$223$	−21.8121	−1.46065	−0.730324	−	0.683100i	$-0.760631\pi$
$223$	−21.8121	−1.46065	−0.730324	+	0.683100i	$0.760631\pi$
$224$	−0.407283	−0.0272127
$225$	0	0
$226$	−0.835582	−0.0555821
$227$	−17.8327	−1.18360	−0.591800	−	0.806085i	$-0.701582\pi$
$227$	−17.8327	−1.18360	−0.591800	+	0.806085i	$0.701582\pi$
$228$	0	0
$229$	10.3158	0.681684	0.340842	−	0.940121i	$-0.389288\pi$
$229$	10.3158	0.681684	0.340842	+	0.940121i	$0.389288\pi$
$230$	6.02245	0.397109
$231$	0	0
$232$	6.80359	0.446678
$233$	−21.3566	−1.39912	−0.699558	−	0.714576i	$-0.746620\pi$
$233$	−21.3566	−1.39912	−0.699558	+	0.714576i	$0.746620\pi$
$234$	0	0
$235$	−2.28199	−0.148861
$236$	13.1937	0.858838
$237$	0	0
$238$	2.15591	0.139747
$239$	−9.89792	−0.640243	−0.320122	−	0.947376i	$-0.603724\pi$
$239$	−9.89792	−0.640243	−0.320122	+	0.947376i	$0.603724\pi$
$240$	0	0
$241$	10.8722	0.700341	0.350170	−	0.936686i	$-0.386124\pi$
$241$	10.8722	0.700341	0.350170	+	0.936686i	$0.386124\pi$
$242$	15.8227	1.01712
$243$	0	0
$244$	5.18842	0.332155
$245$	−6.83412	−0.436616
$246$	0	0
$247$	−26.4787	−1.68480
$248$	6.59597	0.418845
$249$	0	0
$250$	1.00000	0.0632456
$251$	−10.7651	−0.679486	−0.339743	−	0.940518i	$-0.610340\pi$
$251$	−10.7651	−0.679486	−0.339743	+	0.940518i	$0.610340\pi$
$252$	0	0
$253$	−31.1907	−1.96094
$254$	8.70377	0.546123
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.44781	0.526960	0.263480	−	0.964665i	$-0.415130\pi$
$257$	8.44781	0.526960	0.263480	+	0.964665i	$0.415130\pi$
$258$	0	0
$259$	2.20565	0.137052
$260$	−5.41054	−0.335547
$261$	0	0
$262$	9.31551	0.575514
$263$	−28.9186	−1.78320	−0.891600	−	0.452824i	$-0.850417\pi$
$263$	−28.9186	−1.78320	−0.891600	+	0.452824i	$0.850417\pi$
$264$	0	0
$265$	6.57529	0.403917
$266$	−1.99321	−0.122211
$267$	0	0
$268$	−2.71810	−0.166034
$269$	−13.8782	−0.846171	−0.423085	−	0.906090i	$-0.639053\pi$
$269$	−13.8782	−0.846171	−0.423085	+	0.906090i	$0.639053\pi$
$270$	0	0
$271$	−31.6948	−1.92532	−0.962661	−	0.270709i	$-0.912742\pi$
$271$	−31.6948	−1.92532	−0.962661	+	0.270709i	$0.912742\pi$
$272$	−5.29339	−0.320959
$273$	0	0
$274$	7.19371	0.434588
$275$	−5.17907	−0.312309
$276$	0	0
$277$	4.71752	0.283448	0.141724	−	0.989906i	$-0.454735\pi$
$277$	4.71752	0.283448	0.141724	+	0.989906i	$0.454735\pi$
$278$	−5.54357	−0.332481
$279$	0	0
$280$	−0.407283	−0.0243398
$281$	−3.49434	−0.208455	−0.104228	−	0.994553i	$-0.533237\pi$
$281$	−3.49434	−0.208455	−0.104228	+	0.994553i	$0.533237\pi$
$282$	0	0
$283$	−9.90353	−0.588704	−0.294352	−	0.955697i	$-0.595104\pi$
$283$	−9.90353	−0.588704	−0.294352	+	0.955697i	$0.595104\pi$
$284$	3.59571	0.213366
$285$	0	0
$286$	28.0215	1.65695
$287$	2.33352	0.137744
$288$	0	0
$289$	11.0199	0.648231
$290$	6.80359	0.399521
$291$	0	0
$292$	11.7231	0.686044
$293$	25.7439	1.50398	0.751989	−	0.659176i	$-0.229095\pi$
$293$	25.7439	1.50398	0.751989	+	0.659176i	$0.229095\pi$
$294$	0	0
$295$	13.1937	0.768168
$296$	−5.41551	−0.314770
$297$	0	0
$298$	−14.0290	−0.812676
$299$	−32.5847	−1.88442
$300$	0	0
$301$	−2.94480	−0.169735
$302$	−9.76867	−0.562124
$303$	0	0
$304$	4.89391	0.280685
$305$	5.18842	0.297088
$306$	0	0
$307$	16.2600	0.928008	0.464004	−	0.885833i	$-0.346412\pi$
$307$	16.2600	0.928008	0.464004	+	0.885833i	$0.346412\pi$
$308$	2.10935	0.120191
$309$	0	0
$310$	6.59597	0.374626
$311$	−3.47459	−0.197026	−0.0985131	−	0.995136i	$-0.531409\pi$
$311$	−3.47459	−0.197026	−0.0985131	+	0.995136i	$0.531409\pi$
$312$	0	0
$313$	6.61345	0.373814	0.186907	−	0.982378i	$-0.440154\pi$
$313$	6.61345	0.373814	0.186907	+	0.982378i	$0.440154\pi$
$314$	−2.29981	−0.129786
$315$	0	0
$316$	16.4412	0.924891
$317$	1.51014	0.0848179	0.0424089	−	0.999100i	$-0.486497\pi$
$317$	1.51014	0.0848179	0.0424089	+	0.999100i	$0.486497\pi$
$318$	0	0
$319$	−35.2363	−1.97285
$320$	1.00000	0.0559017
$321$	0	0
$322$	−2.45284	−0.136692
$323$	−25.9053	−1.44141
$324$	0	0
$325$	−5.41054	−0.300123
$326$	16.1459	0.894241
$327$	0	0
$328$	−5.72949	−0.316358
$329$	0.929417	0.0512404
$330$	0	0
$331$	−7.53706	−0.414274	−0.207137	−	0.978312i	$-0.566415\pi$
$331$	−7.53706	−0.414274	−0.207137	+	0.978312i	$0.566415\pi$
$332$	2.68629	0.147429
$333$	0	0
$334$	1.85445	0.101471
$335$	−2.71810	−0.148505
$336$	0	0
$337$	3.49257	0.190253	0.0951263	−	0.995465i	$-0.469675\pi$
$337$	3.49257	0.190253	0.0951263	+	0.995465i	$0.469675\pi$
$338$	16.2739	0.885185
$339$	0	0
$340$	−5.29339	−0.287074
$341$	−34.1610	−1.84992
$342$	0	0
$343$	5.63440	0.304229
$344$	7.23034	0.389834
$345$	0	0
$346$	−20.6267	−1.10890
$347$	−23.6329	−1.26868	−0.634339	−	0.773055i	$-0.718728\pi$
$347$	−23.6329	−1.26868	−0.634339	+	0.773055i	$0.718728\pi$
$348$	0	0
$349$	−5.98240	−0.320231	−0.160115	−	0.987098i	$-0.551187\pi$
$349$	−5.98240	−0.320231	−0.160115	+	0.987098i	$0.551187\pi$
$350$	−0.407283	−0.0217702
$351$	0	0
$352$	−5.17907	−0.276045
$353$	35.2167	1.87439	0.937197	−	0.348800i	$-0.113411\pi$
$353$	35.2167	1.87439	0.937197	+	0.348800i	$0.113411\pi$
$354$	0	0
$355$	3.59571	0.190840
$356$	−1.00000	−0.0529999
$357$	0	0
$358$	20.6072	1.08913
$359$	0.590467	0.0311637	0.0155818	−	0.999879i	$-0.495040\pi$
$359$	0.590467	0.0311637	0.0155818	+	0.999879i	$0.495040\pi$
$360$	0	0
$361$	4.95034	0.260544
$362$	22.6109	1.18840
$363$	0	0
$364$	2.20362	0.115501
$365$	11.7231	0.613616
$366$	0	0
$367$	−3.94733	−0.206049	−0.103024	−	0.994679i	$-0.532852\pi$
$367$	−3.94733	−0.206049	−0.103024	+	0.994679i	$0.532852\pi$
$368$	6.02245	0.313942
$369$	0	0
$370$	−5.41551	−0.281539
$371$	−2.67800	−0.139035
$372$	0	0
$373$	20.4976	1.06133	0.530663	−	0.847583i	$-0.321943\pi$
$373$	20.4976	1.06133	0.530663	+	0.847583i	$0.321943\pi$
$374$	27.4148	1.41759
$375$	0	0
$376$	−2.28199	−0.117685
$377$	−36.8111	−1.89587
$378$	0	0
$379$	−25.8205	−1.32631	−0.663156	−	0.748481i	$-0.730783\pi$
$379$	−25.8205	−1.32631	−0.663156	+	0.748481i	$0.730783\pi$
$380$	4.89391	0.251052
$381$	0	0
$382$	−6.42333	−0.328646
$383$	−2.12501	−0.108583	−0.0542914	−	0.998525i	$-0.517290\pi$
$383$	−2.12501	−0.108583	−0.0542914	+	0.998525i	$0.517290\pi$
$384$	0	0
$385$	2.10935	0.107502
$386$	1.42151	0.0723530
$387$	0	0
$388$	14.0449	0.713022
$389$	−17.9264	−0.908905	−0.454453	−	0.890771i	$-0.650165\pi$
$389$	−17.9264	−0.908905	−0.454453	+	0.890771i	$0.650165\pi$
$390$	0	0
$391$	−31.8792	−1.61220
$392$	−6.83412	−0.345175
$393$	0	0
$394$	−12.0190	−0.605510
$395$	16.4412	0.827248
$396$	0	0
$397$	9.80916	0.492308	0.246154	−	0.969231i	$-0.420833\pi$
$397$	9.80916	0.492308	0.246154	+	0.969231i	$0.420833\pi$
$398$	15.2287	0.763345
$399$	0	0
$400$	1.00000	0.0500000
$401$	−9.90529	−0.494646	−0.247323	−	0.968933i	$-0.579551\pi$
$401$	−9.90529	−0.494646	−0.247323	+	0.968933i	$0.579551\pi$
$402$	0	0
$403$	−35.6878	−1.77773
$404$	7.37391	0.366866
$405$	0	0
$406$	−2.77099	−0.137522
$407$	28.0473	1.39025
$408$	0	0
$409$	33.4419	1.65359	0.826797	−	0.562501i	$-0.190161\pi$
$409$	33.4419	1.65359	0.826797	+	0.562501i	$0.190161\pi$
$410$	−5.72949	−0.282959
$411$	0	0
$412$	−2.52160	−0.124230
$413$	−5.37358	−0.264416
$414$	0	0
$415$	2.68629	0.131865
$416$	−5.41054	−0.265273
$417$	0	0
$418$	−25.3459	−1.23971
$419$	−7.58019	−0.370317	−0.185158	−	0.982709i	$-0.559280\pi$
$419$	−7.58019	−0.370317	−0.185158	+	0.982709i	$0.559280\pi$
$420$	0	0
$421$	−0.491585	−0.0239584	−0.0119792	−	0.999928i	$-0.503813\pi$
$421$	−0.491585	−0.0239584	−0.0119792	+	0.999928i	$0.503813\pi$
$422$	28.5967	1.39207
$423$	0	0
$424$	6.57529	0.319324
$425$	−5.29339	−0.256767
$426$	0	0
$427$	−2.11316	−0.102263
$428$	−18.4030	−0.889545
$429$	0	0
$430$	7.23034	0.348678
$431$	−13.8339	−0.666358	−0.333179	−	0.942864i	$-0.608121\pi$
$431$	−13.8339	−0.666358	−0.333179	+	0.942864i	$0.608121\pi$
$432$	0	0
$433$	30.9164	1.48575	0.742875	−	0.669431i	$-0.233462\pi$
$433$	30.9164	1.48575	0.742875	+	0.669431i	$0.233462\pi$
$434$	−2.68643	−0.128953
$435$	0	0
$436$	3.21886	0.154155
$437$	29.4733	1.40990
$438$	0	0
$439$	−24.7853	−1.18294	−0.591469	−	0.806328i	$-0.701452\pi$
$439$	−24.7853	−1.18294	−0.591469	+	0.806328i	$0.701452\pi$
$440$	−5.17907	−0.246902
$441$	0	0
$442$	28.6401	1.36227
$443$	−9.49946	−0.451333	−0.225666	−	0.974205i	$-0.572456\pi$
$443$	−9.49946	−0.451333	−0.225666	+	0.974205i	$0.572456\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	−21.8121	−1.03283
$447$	0	0
$448$	−0.407283	−0.0192423
$449$	7.95642	0.375487	0.187743	−	0.982218i	$-0.439883\pi$
$449$	7.95642	0.375487	0.187743	+	0.982218i	$0.439883\pi$
$450$	0	0
$451$	29.6734	1.39727
$452$	−0.835582	−0.0393025
$453$	0	0
$454$	−17.8327	−0.836931
$455$	2.20362	0.103307
$456$	0	0
$457$	3.76143	0.175952	0.0879761	−	0.996123i	$-0.471960\pi$
$457$	3.76143	0.175952	0.0879761	+	0.996123i	$0.471960\pi$
$458$	10.3158	0.482023
$459$	0	0
$460$	6.02245	0.280798
$461$	−27.7844	−1.29405	−0.647024	−	0.762470i	$-0.723986\pi$
$461$	−27.7844	−1.29405	−0.647024	+	0.762470i	$0.723986\pi$
$462$	0	0
$463$	27.1332	1.26099	0.630493	−	0.776195i	$-0.282853\pi$
$463$	27.1332	1.26099	0.630493	+	0.776195i	$0.282853\pi$
$464$	6.80359	0.315849
$465$	0	0
$466$	−21.3566	−0.989324
$467$	35.7083	1.65238	0.826192	−	0.563388i	$-0.190502\pi$
$467$	35.7083	1.65238	0.826192	+	0.563388i	$0.190502\pi$
$468$	0	0
$469$	1.10704	0.0511181
$470$	−2.28199	−0.105260
$471$	0	0
$472$	13.1937	0.607290
$473$	−37.4464	−1.72179
$474$	0	0
$475$	4.89391	0.224548
$476$	2.15591	0.0988158
$477$	0	0
$478$	−9.89792	−0.452720
$479$	13.7666	0.629010	0.314505	−	0.949256i	$-0.398161\pi$
$479$	13.7666	0.629010	0.314505	+	0.949256i	$0.398161\pi$
$480$	0	0
$481$	29.3008	1.33600
$482$	10.8722	0.495216
$483$	0	0
$484$	15.8227	0.719215
$485$	14.0449	0.637746
$486$	0	0
$487$	−2.25061	−0.101985	−0.0509924	−	0.998699i	$-0.516238\pi$
$487$	−2.25061	−0.101985	−0.0509924	+	0.998699i	$0.516238\pi$
$488$	5.18842	0.234869
$489$	0	0
$490$	−6.83412	−0.308734
$491$	30.4684	1.37502	0.687511	−	0.726174i	$-0.258703\pi$
$491$	30.4684	1.37502	0.687511	+	0.726174i	$0.258703\pi$
$492$	0	0
$493$	−36.0140	−1.62199
$494$	−26.4787	−1.19133
$495$	0	0
$496$	6.59597	0.296168
$497$	−1.46447	−0.0656905
$498$	0	0
$499$	41.3846	1.85263	0.926314	−	0.376753i	$-0.122959\pi$
$499$	41.3846	1.85263	0.926314	+	0.376753i	$0.122959\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−10.7651	−0.480469
$503$	32.0212	1.42775	0.713877	−	0.700271i	$-0.246937\pi$
$503$	32.0212	1.42775	0.713877	+	0.700271i	$0.246937\pi$
$504$	0	0
$505$	7.37391	0.328135
$506$	−31.1907	−1.38660
$507$	0	0
$508$	8.70377	0.386167
$509$	43.4494	1.92586	0.962931	−	0.269747i	$-0.0869402\pi$
$509$	43.4494	1.92586	0.962931	+	0.269747i	$0.0869402\pi$
$510$	0	0
$511$	−4.77463	−0.211217
$512$	1.00000	0.0441942
$513$	0	0
$514$	8.44781	0.372617
$515$	−2.52160	−0.111115
$516$	0	0
$517$	11.8186	0.519781
$518$	2.20565	0.0969105
$519$	0	0
$520$	−5.41054	−0.237268
$521$	−31.0235	−1.35917	−0.679583	−	0.733598i	$-0.737839\pi$
$521$	−31.0235	−1.35917	−0.679583	+	0.733598i	$0.737839\pi$
$522$	0	0
$523$	−34.5689	−1.51159	−0.755797	−	0.654806i	$-0.772750\pi$
$523$	−34.5689	−1.51159	−0.755797	+	0.654806i	$0.772750\pi$
$524$	9.31551	0.406950
$525$	0	0
$526$	−28.9186	−1.26091
$527$	−34.9150	−1.52092
$528$	0	0
$529$	13.2700	0.576955
$530$	6.57529	0.285612
$531$	0	0
$532$	−1.99321	−0.0864165
$533$	30.9996	1.34274
$534$	0	0
$535$	−18.4030	−0.795633
$536$	−2.71810	−0.117404
$537$	0	0
$538$	−13.8782	−0.598333
$539$	35.3944	1.52454
$540$	0	0
$541$	−28.5471	−1.22734	−0.613669	−	0.789564i	$-0.710307\pi$
$541$	−28.5471	−1.22734	−0.613669	+	0.789564i	$0.710307\pi$
$542$	−31.6948	−1.36141
$543$	0	0
$544$	−5.29339	−0.226952
$545$	3.21886	0.137881
$546$	0	0
$547$	−41.9543	−1.79384	−0.896918	−	0.442198i	$-0.854199\pi$
$547$	−41.9543	−1.79384	−0.896918	+	0.442198i	$0.854199\pi$
$548$	7.19371	0.307300
$549$	0	0
$550$	−5.17907	−0.220836
$551$	33.2962	1.41846
$552$	0	0
$553$	−6.69623	−0.284753
$554$	4.71752	0.200428
$555$	0	0
$556$	−5.54357	−0.235100
$557$	17.2690	0.731711	0.365855	−	0.930672i	$-0.380777\pi$
$557$	17.2690	0.731711	0.365855	+	0.930672i	$0.380777\pi$
$558$	0	0
$559$	−39.1200	−1.65460
$560$	−0.407283	−0.0172109
$561$	0	0
$562$	−3.49434	−0.147400
$563$	41.3314	1.74191	0.870956	−	0.491361i	$-0.163501\pi$
$563$	41.3314	1.74191	0.870956	+	0.491361i	$0.163501\pi$
$564$	0	0
$565$	−0.835582	−0.0351532
$566$	−9.90353	−0.416276
$567$	0	0
$568$	3.59571	0.150872
$569$	−15.3101	−0.641833	−0.320917	−	0.947107i	$-0.603991\pi$
$569$	−15.3101	−0.641833	−0.320917	+	0.947107i	$0.603991\pi$
$570$	0	0
$571$	−10.0473	−0.420465	−0.210233	−	0.977651i	$-0.567422\pi$
$571$	−10.0473	−0.420465	−0.210233	+	0.977651i	$0.567422\pi$
$572$	28.0215	1.17164
$573$	0	0
$574$	2.33352	0.0973994
$575$	6.02245	0.251154
$576$	0	0
$577$	20.5197	0.854247	0.427124	−	0.904193i	$-0.359527\pi$
$577$	20.5197	0.854247	0.427124	+	0.904193i	$0.359527\pi$
$578$	11.0199	0.458369
$579$	0	0
$580$	6.80359	0.282504
$581$	−1.09408	−0.0453900
$582$	0	0
$583$	−34.0539	−1.41037
$584$	11.7231	0.485106
$585$	0	0
$586$	25.7439	1.06347
$587$	−31.1416	−1.28535	−0.642676	−	0.766138i	$-0.722176\pi$
$587$	−31.1416	−1.28535	−0.642676	+	0.766138i	$0.722176\pi$
$588$	0	0
$589$	32.2801	1.33008
$590$	13.1937	0.543177
$591$	0	0
$592$	−5.41551	−0.222576
$593$	5.71505	0.234689	0.117344	−	0.993091i	$-0.462562\pi$
$593$	5.71505	0.234689	0.117344	+	0.993091i	$0.462562\pi$
$594$	0	0
$595$	2.15591	0.0883835
$596$	−14.0290	−0.574649
$597$	0	0
$598$	−32.5847	−1.33249
$599$	−6.21809	−0.254064	−0.127032	−	0.991899i	$-0.540545\pi$
$599$	−6.21809	−0.254064	−0.127032	+	0.991899i	$0.540545\pi$
$600$	0	0
$601$	−1.94185	−0.0792098	−0.0396049	−	0.999215i	$-0.512610\pi$
$601$	−1.94185	−0.0792098	−0.0396049	+	0.999215i	$0.512610\pi$
$602$	−2.94480	−0.120021
$603$	0	0
$604$	−9.76867	−0.397482
$605$	15.8227	0.643285
$606$	0	0
$607$	18.2929	0.742485	0.371243	−	0.928536i	$-0.378932\pi$
$607$	18.2929	0.742485	0.371243	+	0.928536i	$0.378932\pi$
$608$	4.89391	0.198474
$609$	0	0
$610$	5.18842	0.210073
$611$	12.3468	0.499498
$612$	0	0
$613$	−14.8370	−0.599262	−0.299631	−	0.954055i	$-0.596864\pi$
$613$	−14.8370	−0.599262	−0.299631	+	0.954055i	$0.596864\pi$
$614$	16.2600	0.656201
$615$	0	0
$616$	2.10935	0.0849880
$617$	36.0213	1.45016	0.725081	−	0.688663i	$-0.241802\pi$
$617$	36.0213	1.45016	0.725081	+	0.688663i	$0.241802\pi$
$618$	0	0
$619$	11.8568	0.476565	0.238282	−	0.971196i	$-0.423416\pi$
$619$	11.8568	0.476565	0.238282	+	0.971196i	$0.423416\pi$
$620$	6.59597	0.264901
$621$	0	0
$622$	−3.47459	−0.139318
$623$	0.407283	0.0163175
$624$	0	0
$625$	1.00000	0.0400000
$626$	6.61345	0.264327
$627$	0	0
$628$	−2.29981	−0.0917722
$629$	28.6664	1.14300
$630$	0	0
$631$	21.3267	0.849004	0.424502	−	0.905427i	$-0.360449\pi$
$631$	21.3267	0.849004	0.424502	+	0.905427i	$0.360449\pi$
$632$	16.4412	0.653997
$633$	0	0
$634$	1.51014	0.0599753
$635$	8.70377	0.345398
$636$	0	0
$637$	36.9763	1.46505
$638$	−35.2363	−1.39502
$639$	0	0
$640$	1.00000	0.0395285
$641$	16.2122	0.640343	0.320171	−	0.947360i	$-0.396260\pi$
$641$	16.2122	0.640343	0.320171	+	0.947360i	$0.396260\pi$
$642$	0	0
$643$	−33.6309	−1.32627	−0.663137	−	0.748498i	$-0.730775\pi$
$643$	−33.6309	−1.32627	−0.663137	+	0.748498i	$0.730775\pi$
$644$	−2.45284	−0.0966556
$645$	0	0
$646$	−25.9053	−1.01923
$647$	0.394681	0.0155165	0.00775826	−	0.999970i	$-0.497530\pi$
$647$	0.394681	0.0155165	0.00775826	+	0.999970i	$0.497530\pi$
$648$	0	0
$649$	−68.3311	−2.68223
$650$	−5.41054	−0.212219
$651$	0	0
$652$	16.1459	0.632324
$653$	9.05973	0.354535	0.177267	−	0.984163i	$-0.443274\pi$
$653$	9.05973	0.354535	0.177267	+	0.984163i	$0.443274\pi$
$654$	0	0
$655$	9.31551	0.363987
$656$	−5.72949	−0.223699
$657$	0	0
$658$	0.929417	0.0362324
$659$	44.0529	1.71606	0.858029	−	0.513602i	$-0.171689\pi$
$659$	44.0529	1.71606	0.858029	+	0.513602i	$0.171689\pi$
$660$	0	0
$661$	26.2277	1.02014	0.510070	−	0.860133i	$-0.329619\pi$
$661$	26.2277	1.02014	0.510070	+	0.860133i	$0.329619\pi$
$662$	−7.53706	−0.292936
$663$	0	0
$664$	2.68629	0.104248
$665$	−1.99321	−0.0772932
$666$	0	0
$667$	40.9743	1.58653
$668$	1.85445	0.0717508
$669$	0	0
$670$	−2.71810	−0.105009
$671$	−26.8712	−1.03735
$672$	0	0
$673$	−27.2662	−1.05103	−0.525517	−	0.850783i	$-0.676128\pi$
$673$	−27.2662	−1.05103	−0.525517	+	0.850783i	$0.676128\pi$
$674$	3.49257	0.134529
$675$	0	0
$676$	16.2739	0.625920
$677$	−13.4765	−0.517943	−0.258972	−	0.965885i	$-0.583384\pi$
$677$	−13.4765	−0.517943	−0.258972	+	0.965885i	$0.583384\pi$
$678$	0	0
$679$	−5.72025	−0.219523
$680$	−5.29339	−0.202992
$681$	0	0
$682$	−34.1610	−1.30809
$683$	18.7435	0.717200	0.358600	−	0.933491i	$-0.383254\pi$
$683$	18.7435	0.717200	0.358600	+	0.933491i	$0.383254\pi$
$684$	0	0
$685$	7.19371	0.274858
$686$	5.63440	0.215123
$687$	0	0
$688$	7.23034	0.275654
$689$	−35.5758	−1.35533
$690$	0	0
$691$	−13.3365	−0.507346	−0.253673	−	0.967290i	$-0.581639\pi$
$691$	−13.3365	−0.507346	−0.253673	+	0.967290i	$0.581639\pi$
$692$	−20.6267	−0.784110
$693$	0	0
$694$	−23.6329	−0.897091
$695$	−5.54357	−0.210279
$696$	0	0
$697$	30.3284	1.14877
$698$	−5.98240	−0.226437
$699$	0	0
$700$	−0.407283	−0.0153939
$701$	−1.41696	−0.0535177	−0.0267588	−	0.999642i	$-0.508519\pi$
$701$	−1.41696	−0.0535177	−0.0267588	+	0.999642i	$0.508519\pi$
$702$	0	0
$703$	−26.5030	−0.999580
$704$	−5.17907	−0.195193
$705$	0	0
$706$	35.2167	1.32540
$707$	−3.00327	−0.112950
$708$	0	0
$709$	−1.06234	−0.0398969	−0.0199485	−	0.999801i	$-0.506350\pi$
$709$	−1.06234	−0.0398969	−0.0199485	+	0.999801i	$0.506350\pi$
$710$	3.59571	0.134944
$711$	0	0
$712$	−1.00000	−0.0374766
$713$	39.7239	1.48767
$714$	0	0
$715$	28.0215	1.04795
$716$	20.6072	0.770129
$717$	0	0
$718$	0.590467	0.0220360
$719$	24.2128	0.902986	0.451493	−	0.892275i	$-0.350892\pi$
$719$	24.2128	0.902986	0.451493	+	0.892275i	$0.350892\pi$
$720$	0	0
$721$	1.02701	0.0382477
$722$	4.95034	0.184233
$723$	0	0
$724$	22.6109	0.840327
$725$	6.80359	0.252679
$726$	0	0
$727$	15.3861	0.570638	0.285319	−	0.958433i	$-0.407900\pi$
$727$	15.3861	0.570638	0.285319	+	0.958433i	$0.407900\pi$
$728$	2.20362	0.0816716
$729$	0	0
$730$	11.7231	0.433892
$731$	−38.2730	−1.41558
$732$	0	0
$733$	5.16018	0.190596	0.0952978	−	0.995449i	$-0.469620\pi$
$733$	5.16018	0.190596	0.0952978	+	0.995449i	$0.469620\pi$
$734$	−3.94733	−0.145699
$735$	0	0
$736$	6.02245	0.221991
$737$	14.0772	0.518540
$738$	0	0
$739$	−12.8966	−0.474410	−0.237205	−	0.971460i	$-0.576231\pi$
$739$	−12.8966	−0.474410	−0.237205	+	0.971460i	$0.576231\pi$
$740$	−5.41551	−0.199078
$741$	0	0
$742$	−2.67800	−0.0983126
$743$	11.4830	0.421269	0.210635	−	0.977565i	$-0.432447\pi$
$743$	11.4830	0.421269	0.210635	+	0.977565i	$0.432447\pi$
$744$	0	0
$745$	−14.0290	−0.513981
$746$	20.4976	0.750471
$747$	0	0
$748$	27.4148	1.00238
$749$	7.49525	0.273870
$750$	0	0
$751$	10.0444	0.366525	0.183262	−	0.983064i	$-0.441334\pi$
$751$	10.0444	0.366525	0.183262	+	0.983064i	$0.441334\pi$
$752$	−2.28199	−0.0832157
$753$	0	0
$754$	−36.8111	−1.34058
$755$	−9.76867	−0.355518
$756$	0	0
$757$	31.6550	1.15052	0.575260	−	0.817971i	$-0.304901\pi$
$757$	31.6550	1.15052	0.575260	+	0.817971i	$0.304901\pi$
$758$	−25.8205	−0.937844
$759$	0	0
$760$	4.89391	0.177521
$761$	29.8456	1.08190	0.540950	−	0.841054i	$-0.318065\pi$
$761$	29.8456	1.08190	0.540950	+	0.841054i	$0.318065\pi$
$762$	0	0
$763$	−1.31099	−0.0474609
$764$	−6.42333	−0.232388
$765$	0	0
$766$	−2.12501	−0.0767796
$767$	−71.3851	−2.57757
$768$	0	0
$769$	46.0659	1.66118	0.830589	−	0.556886i	$-0.188004\pi$
$769$	46.0659	1.66118	0.830589	+	0.556886i	$0.188004\pi$
$770$	2.10935	0.0760156
$771$	0	0
$772$	1.42151	0.0511613
$773$	46.8553	1.68527	0.842634	−	0.538486i	$-0.181004\pi$
$773$	46.8553	1.68527	0.842634	+	0.538486i	$0.181004\pi$
$774$	0	0
$775$	6.59597	0.236934
$776$	14.0449	0.504183
$777$	0	0
$778$	−17.9264	−0.642693
$779$	−28.0396	−1.00462
$780$	0	0
$781$	−18.6224	−0.666362
$782$	−31.8792	−1.14000
$783$	0	0
$784$	−6.83412	−0.244076
$785$	−2.29981	−0.0820836
$786$	0	0
$787$	−48.8526	−1.74141	−0.870703	−	0.491809i	$-0.836336\pi$
$787$	−48.8526	−1.74141	−0.870703	+	0.491809i	$0.836336\pi$
$788$	−12.0190	−0.428160
$789$	0	0
$790$	16.4412	0.584952
$791$	0.340318	0.0121003
$792$	0	0
$793$	−28.0722	−0.996872
$794$	9.80916	0.348114
$795$	0	0
$796$	15.2287	0.539767
$797$	−40.9607	−1.45090	−0.725451	−	0.688274i	$-0.758369\pi$
$797$	−40.9607	−1.45090	−0.725451	+	0.688274i	$0.758369\pi$
$798$	0	0
$799$	12.0795	0.427341
$800$	1.00000	0.0353553
$801$	0	0
$802$	−9.90529	−0.349768
$803$	−60.7148	−2.14258
$804$	0	0
$805$	−2.45284	−0.0864514
$806$	−35.6878	−1.25705
$807$	0	0
$808$	7.37391	0.259413
$809$	−16.2655	−0.571865	−0.285933	−	0.958250i	$-0.592303\pi$
$809$	−16.2655	−0.571865	−0.285933	+	0.958250i	$0.592303\pi$
$810$	0	0
$811$	−33.5231	−1.17716	−0.588578	−	0.808440i	$-0.700312\pi$
$811$	−33.5231	−1.17716	−0.588578	+	0.808440i	$0.700312\pi$
$812$	−2.77099	−0.0972427
$813$	0	0
$814$	28.0473	0.983057
$815$	16.1459	0.565568
$816$	0	0
$817$	35.3846	1.23795
$818$	33.4419	1.16927
$819$	0	0
$820$	−5.72949	−0.200082
$821$	−9.39583	−0.327917	−0.163958	−	0.986467i	$-0.552426\pi$
$821$	−9.39583	−0.327917	−0.163958	+	0.986467i	$0.552426\pi$
$822$	0	0
$823$	−16.2379	−0.566018	−0.283009	−	0.959117i	$-0.591333\pi$
$823$	−16.2379	−0.566018	−0.283009	+	0.959117i	$0.591333\pi$
$824$	−2.52160	−0.0878442
$825$	0	0
$826$	−5.37358	−0.186971
$827$	−27.8607	−0.968811	−0.484405	−	0.874844i	$-0.660964\pi$
$827$	−27.8607	−0.968811	−0.484405	+	0.874844i	$0.660964\pi$
$828$	0	0
$829$	−24.2324	−0.841625	−0.420813	−	0.907148i	$-0.638255\pi$
$829$	−24.2324	−0.841625	−0.420813	+	0.907148i	$0.638255\pi$
$830$	2.68629	0.0932423
$831$	0	0
$832$	−5.41054	−0.187577
$833$	36.1756	1.25341
$834$	0	0
$835$	1.85445	0.0641759
$836$	−25.3459	−0.876605
$837$	0	0
$838$	−7.58019	−0.261853
$839$	−8.97915	−0.309995	−0.154997	−	0.987915i	$-0.549537\pi$
$839$	−8.97915	−0.309995	−0.154997	+	0.987915i	$0.549537\pi$
$840$	0	0
$841$	17.2889	0.596169
$842$	−0.491585	−0.0169411
$843$	0	0
$844$	28.5967	0.984339
$845$	16.2739	0.559840
$846$	0	0
$847$	−6.44433	−0.221430
$848$	6.57529	0.225796
$849$	0	0
$850$	−5.29339	−0.181562
$851$	−32.6147	−1.11802
$852$	0	0
$853$	49.1815	1.68394	0.841971	−	0.539523i	$-0.181395\pi$
$853$	49.1815	1.68394	0.841971	+	0.539523i	$0.181395\pi$
$854$	−2.11316	−0.0723107
$855$	0	0
$856$	−18.4030	−0.629003
$857$	8.28995	0.283179	0.141590	−	0.989925i	$-0.454779\pi$
$857$	8.28995	0.283179	0.141590	+	0.989925i	$0.454779\pi$
$858$	0	0
$859$	15.8620	0.541206	0.270603	−	0.962691i	$-0.412777\pi$
$859$	15.8620	0.541206	0.270603	+	0.962691i	$0.412777\pi$
$860$	7.23034	0.246553
$861$	0	0
$862$	−13.8339	−0.471186
$863$	−33.6468	−1.14535	−0.572675	−	0.819782i	$-0.694094\pi$
$863$	−33.6468	−1.14535	−0.572675	+	0.819782i	$0.694094\pi$
$864$	0	0
$865$	−20.6267	−0.701329
$866$	30.9164	1.05058
$867$	0	0
$868$	−2.68643	−0.0911833
$869$	−85.1502	−2.88852
$870$	0	0
$871$	14.7064	0.498306
$872$	3.21886	0.109004
$873$	0	0
$874$	29.4733	0.996951
$875$	−0.407283	−0.0137687
$876$	0	0
$877$	−20.9635	−0.707887	−0.353943	−	0.935267i	$-0.615159\pi$
$877$	−20.9635	−0.707887	−0.353943	+	0.935267i	$0.615159\pi$
$878$	−24.7853	−0.836463
$879$	0	0
$880$	−5.17907	−0.174586
$881$	24.7804	0.834874	0.417437	−	0.908706i	$-0.362929\pi$
$881$	24.7804	0.834874	0.417437	+	0.908706i	$0.362929\pi$
$882$	0	0
$883$	−38.9052	−1.30927	−0.654633	−	0.755947i	$-0.727177\pi$
$883$	−38.9052	−1.30927	−0.654633	+	0.755947i	$0.727177\pi$
$884$	28.6401	0.963269
$885$	0	0
$886$	−9.49946	−0.319141
$887$	22.0542	0.740507	0.370254	−	0.928931i	$-0.379271\pi$
$887$	22.0542	0.740507	0.370254	+	0.928931i	$0.379271\pi$
$888$	0	0
$889$	−3.54490	−0.118892
$890$	−1.00000	−0.0335201
$891$	0	0
$892$	−21.8121	−0.730324
$893$	−11.1679	−0.373718
$894$	0	0
$895$	20.6072	0.688824
$896$	−0.407283	−0.0136064
$897$	0	0
$898$	7.95642	0.265509
$899$	44.8763	1.49671
$900$	0	0
$901$	−34.8055	−1.15954
$902$	29.6734	0.988016
$903$	0	0
$904$	−0.835582	−0.0277910
$905$	22.6109	0.751611
$906$	0	0
$907$	−8.17430	−0.271423	−0.135712	−	0.990748i	$-0.543332\pi$
$907$	−8.17430	−0.271423	−0.135712	+	0.990748i	$0.543332\pi$
$908$	−17.8327	−0.591800
$909$	0	0
$910$	2.20362	0.0730493
$911$	−41.4375	−1.37289	−0.686443	−	0.727184i	$-0.740829\pi$
$911$	−41.4375	−1.37289	−0.686443	+	0.727184i	$0.740829\pi$
$912$	0	0
$913$	−13.9125	−0.460435
$914$	3.76143	0.124417
$915$	0	0
$916$	10.3158	0.340842
$917$	−3.79405	−0.125291
$918$	0	0
$919$	26.8892	0.886994	0.443497	−	0.896276i	$-0.353738\pi$
$919$	26.8892	0.886994	0.443497	+	0.896276i	$0.353738\pi$
$920$	6.02245	0.198554
$921$	0	0
$922$	−27.7844	−0.915029
$923$	−19.4547	−0.640359
$924$	0	0
$925$	−5.41551	−0.178061
$926$	27.1332	0.891652
$927$	0	0
$928$	6.80359	0.223339
$929$	8.33309	0.273400	0.136700	−	0.990613i	$-0.456350\pi$
$929$	8.33309	0.273400	0.136700	+	0.990613i	$0.456350\pi$
$930$	0	0
$931$	−33.4456	−1.09613
$932$	−21.3566	−0.699558
$933$	0	0
$934$	35.7083	1.16841
$935$	27.4148	0.896560
$936$	0	0
$937$	33.6177	1.09824	0.549121	−	0.835743i	$-0.314963\pi$
$937$	33.6177	1.09824	0.549121	+	0.835743i	$0.314963\pi$
$938$	1.10704	0.0361460
$939$	0	0
$940$	−2.28199	−0.0744304
$941$	15.3181	0.499357	0.249679	−	0.968329i	$-0.419675\pi$
$941$	15.3181	0.499357	0.249679	+	0.968329i	$0.419675\pi$
$942$	0	0
$943$	−34.5056	−1.12366
$944$	13.1937	0.429419
$945$	0	0
$946$	−37.4464	−1.21749
$947$	4.05126	0.131648	0.0658242	−	0.997831i	$-0.479032\pi$
$947$	4.05126	0.131648	0.0658242	+	0.997831i	$0.479032\pi$
$948$	0	0
$949$	−63.4284	−2.05897
$950$	4.89391	0.158779
$951$	0	0
$952$	2.15591	0.0698733
$953$	−48.9812	−1.58666	−0.793328	−	0.608794i	$-0.791653\pi$
$953$	−48.9812	−1.58666	−0.793328	+	0.608794i	$0.791653\pi$
$954$	0	0
$955$	−6.42333	−0.207854
$956$	−9.89792	−0.320122
$957$	0	0
$958$	13.7666	0.444778
$959$	−2.92988	−0.0946107
$960$	0	0
$961$	12.5068	0.403446
$962$	29.3008	0.944697
$963$	0	0
$964$	10.8722	0.350170
$965$	1.42151	0.0457600
$966$	0	0
$967$	0.228811	0.00735806	0.00367903	−	0.999993i	$-0.498829\pi$
$967$	0.228811	0.00735806	0.00367903	+	0.999993i	$0.498829\pi$
$968$	15.8227	0.508562
$969$	0	0
$970$	14.0449	0.450955
$971$	19.9419	0.639964	0.319982	−	0.947424i	$-0.396323\pi$
$971$	19.9419	0.639964	0.319982	+	0.947424i	$0.396323\pi$
$972$	0	0
$973$	2.25780	0.0723818
$974$	−2.25061	−0.0721141
$975$	0	0
$976$	5.18842	0.166077
$977$	−25.3376	−0.810623	−0.405311	−	0.914179i	$-0.632837\pi$
$977$	−25.3376	−0.810623	−0.405311	+	0.914179i	$0.632837\pi$
$978$	0	0
$979$	5.17907	0.165524
$980$	−6.83412	−0.218308
$981$	0	0
$982$	30.4684	0.972287
$983$	−38.4085	−1.22504	−0.612521	−	0.790454i	$-0.709845\pi$
$983$	−38.4085	−1.22504	−0.612521	+	0.790454i	$0.709845\pi$
$984$	0	0
$985$	−12.0190	−0.382958
$986$	−36.0140	−1.14692
$987$	0	0
$988$	−26.4787	−0.842399
$989$	43.5444	1.38463
$990$	0	0
$991$	−2.31776	−0.0736262	−0.0368131	−	0.999322i	$-0.511721\pi$
$991$	−2.31776	−0.0736262	−0.0368131	+	0.999322i	$0.511721\pi$
$992$	6.59597	0.209422
$993$	0	0
$994$	−1.46447	−0.0464502
$995$	15.2287	0.482782
$996$	0	0
$997$	8.95160	0.283500	0.141750	−	0.989902i	$-0.454727\pi$
$997$	8.95160	0.283500	0.141750	+	0.989902i	$0.454727\pi$
$998$	41.3846	1.31001
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8010.2.a.bn.1.3		7
3.2	odd	2		2670.2.a.t.1.3	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2670.2.a.t.1.3	✓	7	3.2	odd	2
8010.2.a.bn.1.3		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -0.407283 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -0.407283 q^{7} +1.00000 q^{8} +1.00000 q^{10} -5.17907 q^{11} -5.41054 q^{13} -0.407283 q^{14} +1.00000 q^{16} -5.29339 q^{17} +4.89391 q^{19} +1.00000 q^{20} -5.17907 q^{22} +6.02245 q^{23} +1.00000 q^{25} -5.41054 q^{26} -0.407283 q^{28} +6.80359 q^{29} +6.59597 q^{31} +1.00000 q^{32} -5.29339 q^{34} -0.407283 q^{35} -5.41551 q^{37} +4.89391 q^{38} +1.00000 q^{40} -5.72949 q^{41} +7.23034 q^{43} -5.17907 q^{44} +6.02245 q^{46} -2.28199 q^{47} -6.83412 q^{49} +1.00000 q^{50} -5.41054 q^{52} +6.57529 q^{53} -5.17907 q^{55} -0.407283 q^{56} +6.80359 q^{58} +13.1937 q^{59} +5.18842 q^{61} +6.59597 q^{62} +1.00000 q^{64} -5.41054 q^{65} -2.71810 q^{67} -5.29339 q^{68} -0.407283 q^{70} +3.59571 q^{71} +11.7231 q^{73} -5.41551 q^{74} +4.89391 q^{76} +2.10935 q^{77} +16.4412 q^{79} +1.00000 q^{80} -5.72949 q^{82} +2.68629 q^{83} -5.29339 q^{85} +7.23034 q^{86} -5.17907 q^{88} -1.00000 q^{89} +2.20362 q^{91} +6.02245 q^{92} -2.28199 q^{94} +4.89391 q^{95} +14.0449 q^{97} -6.83412 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + q^{7} + 7 q^{8}+O(q^{10}) \) 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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