LMFDB - Embedded newform 8010.2.a.bn.1.4

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			1.4
Root			$-2.69745$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.15694 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.15694 q^{7} +1.00000 q^{8} +1.00000 q^{10} +6.34399 q^{11} -5.23358 q^{13} +1.15694 q^{14} +1.00000 q^{16} +6.50216 q^{17} +1.05092 q^{19} +1.00000 q^{20} +6.34399 q^{22} -0.797994 q^{23} +1.00000 q^{25} -5.23358 q^{26} +1.15694 q^{28} +7.76739 q^{29} +9.54746 q^{31} +1.00000 q^{32} +6.50216 q^{34} +1.15694 q^{35} +0.264192 q^{37} +1.05092 q^{38} +1.00000 q^{40} -0.200768 q^{41} -3.28211 q^{43} +6.34399 q^{44} -0.797994 q^{46} -5.08250 q^{47} -5.66149 q^{49} +1.00000 q^{50} -5.23358 q^{52} -14.2876 q^{53} +6.34399 q^{55} +1.15694 q^{56} +7.76739 q^{58} -11.5381 q^{59} +11.4084 q^{61} +9.54746 q^{62} +1.00000 q^{64} -5.23358 q^{65} -11.7854 q^{67} +6.50216 q^{68} +1.15694 q^{70} +8.25150 q^{71} -8.45709 q^{73} +0.264192 q^{74} +1.05092 q^{76} +7.33960 q^{77} +5.32834 q^{79} +1.00000 q^{80} -0.200768 q^{82} +0.830997 q^{83} +6.50216 q^{85} -3.28211 q^{86} +6.34399 q^{88} -1.00000 q^{89} -6.05493 q^{91} -0.797994 q^{92} -5.08250 q^{94} +1.05092 q^{95} +0.404012 q^{97} -5.66149 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	1.15694	0.437282	0.218641	−	0.975805i	$-0.429838\pi$
$7$	1.15694	0.437282	0.218641	+	0.975805i	$0.429838\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	6.34399	1.91278	0.956392	−	0.292087i	$-0.0943496\pi$
$11$	6.34399	1.91278	0.956392	+	0.292087i	$0.0943496\pi$
$12$	0	0
$13$	−5.23358	−1.45153	−0.725767	−	0.687940i	$-0.758515\pi$
$13$	−5.23358	−1.45153	−0.725767	+	0.687940i	$0.758515\pi$
$14$	1.15694	0.309205
$15$	0	0
$16$	1.00000	0.250000
$17$	6.50216	1.57701	0.788503	−	0.615031i	$-0.210857\pi$
$17$	6.50216	1.57701	0.788503	+	0.615031i	$0.210857\pi$
$18$	0	0
$19$	1.05092	0.241098	0.120549	−	0.992707i	$-0.461535\pi$
$19$	1.05092	0.241098	0.120549	+	0.992707i	$0.461535\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	6.34399	1.35254
$23$	−0.797994	−0.166393	−0.0831966	−	0.996533i	$-0.526513\pi$
$23$	−0.797994	−0.166393	−0.0831966	+	0.996533i	$0.526513\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−5.23358	−1.02639
$27$	0	0
$28$	1.15694	0.218641
$29$	7.76739	1.44237	0.721184	−	0.692744i	$-0.243598\pi$
$29$	7.76739	1.44237	0.721184	+	0.692744i	$0.243598\pi$
$30$	0	0
$31$	9.54746	1.71477	0.857387	−	0.514672i	$-0.172086\pi$
$31$	9.54746	1.71477	0.857387	+	0.514672i	$0.172086\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.50216	1.11511
$35$	1.15694	0.195558
$36$	0	0
$37$	0.264192	0.0434329	0.0217165	−	0.999764i	$-0.493087\pi$
$37$	0.264192	0.0434329	0.0217165	+	0.999764i	$0.493087\pi$
$38$	1.05092	0.170482
$39$	0	0
$40$	1.00000	0.158114
$41$	−0.200768	−0.0313547	−0.0156774	−	0.999877i	$-0.504990\pi$
$41$	−0.200768	−0.0313547	−0.0156774	+	0.999877i	$0.504990\pi$
$42$	0	0
$43$	−3.28211	−0.500517	−0.250259	−	0.968179i	$-0.580516\pi$
$43$	−3.28211	−0.500517	−0.250259	+	0.968179i	$0.580516\pi$
$44$	6.34399	0.956392
$45$	0	0
$46$	−0.797994	−0.117658
$47$	−5.08250	−0.741359	−0.370679	−	0.928761i	$-0.620875\pi$
$47$	−5.08250	−0.741359	−0.370679	+	0.928761i	$0.620875\pi$
$48$	0	0
$49$	−5.66149	−0.808785
$50$	1.00000	0.141421
$51$	0	0
$52$	−5.23358	−0.725767
$53$	−14.2876	−1.96255	−0.981276	−	0.192608i	$-0.938305\pi$
$53$	−14.2876	−1.96255	−0.981276	+	0.192608i	$0.938305\pi$
$54$	0	0
$55$	6.34399	0.855423
$56$	1.15694	0.154602
$57$	0	0
$58$	7.76739	1.01991
$59$	−11.5381	−1.50214	−0.751068	−	0.660225i	$-0.770461\pi$
$59$	−11.5381	−1.50214	−0.751068	+	0.660225i	$0.770461\pi$
$60$	0	0
$61$	11.4084	1.46070	0.730351	−	0.683072i	$-0.239357\pi$
$61$	11.4084	1.46070	0.730351	+	0.683072i	$0.239357\pi$
$62$	9.54746	1.21253
$63$	0	0
$64$	1.00000	0.125000
$65$	−5.23358	−0.649146
$66$	0	0
$67$	−11.7854	−1.43982	−0.719910	−	0.694068i	$-0.755817\pi$
$67$	−11.7854	−1.43982	−0.719910	+	0.694068i	$0.755817\pi$
$68$	6.50216	0.788503
$69$	0	0
$70$	1.15694	0.138281
$71$	8.25150	0.979273	0.489637	−	0.871927i	$-0.337129\pi$
$71$	8.25150	0.979273	0.489637	+	0.871927i	$0.337129\pi$
$72$	0	0
$73$	−8.45709	−0.989828	−0.494914	−	0.868942i	$-0.664800\pi$
$73$	−8.45709	−0.989828	−0.494914	+	0.868942i	$0.664800\pi$
$74$	0.264192	0.0307117
$75$	0	0
$76$	1.05092	0.120549
$77$	7.33960	0.836425
$78$	0	0
$79$	5.32834	0.599485	0.299742	−	0.954020i	$-0.403099\pi$
$79$	5.32834	0.599485	0.299742	+	0.954020i	$0.403099\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−0.200768	−0.0221711
$83$	0.830997	0.0912137	0.0456069	−	0.998959i	$-0.485478\pi$
$83$	0.830997	0.0912137	0.0456069	+	0.998959i	$0.485478\pi$
$84$	0	0
$85$	6.50216	0.705258
$86$	−3.28211	−0.353919
$87$	0	0
$88$	6.34399	0.676271
$89$	−1.00000	−0.106000
$89$	−1.00000	−0.106000
$90$	0	0
$91$	−6.05493	−0.634729
$92$	−0.797994	−0.0831966
$93$	0	0
$94$	−5.08250	−0.524220
$95$	1.05092	0.107822
$96$	0	0
$97$	0.404012	0.0410212	0.0205106	−	0.999790i	$-0.493471\pi$
$97$	0.404012	0.0410212	0.0205106	+	0.999790i	$0.493471\pi$
$98$	−5.66149	−0.571897
$99$	0	0
$100$	1.00000	0.100000
$101$	−7.02243	−0.698758	−0.349379	−	0.936982i	$-0.613607\pi$
$101$	−7.02243	−0.698758	−0.349379	+	0.936982i	$0.613607\pi$
$102$	0	0
$103$	−0.684886	−0.0674838	−0.0337419	−	0.999431i	$-0.510742\pi$
$103$	−0.684886	−0.0674838	−0.0337419	+	0.999431i	$0.510742\pi$
$104$	−5.23358	−0.513195
$105$	0	0
$106$	−14.2876	−1.38773
$107$	18.2840	1.76758	0.883789	−	0.467887i	$-0.154984\pi$
$107$	18.2840	1.76758	0.883789	+	0.467887i	$0.154984\pi$
$108$	0	0
$109$	−4.56538	−0.437284	−0.218642	−	0.975805i	$-0.570163\pi$
$109$	−4.56538	−0.437284	−0.218642	+	0.975805i	$0.570163\pi$
$110$	6.34399	0.604875
$111$	0	0
$112$	1.15694	0.109320
$113$	0.850154	0.0799757	0.0399879	−	0.999200i	$-0.487268\pi$
$113$	0.850154	0.0799757	0.0399879	+	0.999200i	$0.487268\pi$
$114$	0	0
$115$	−0.797994	−0.0744133
$116$	7.76739	0.721184
$117$	0	0
$118$	−11.5381	−1.06217
$119$	7.52260	0.689596
$120$	0	0
$121$	29.2462	2.65874
$122$	11.4084	1.03287
$123$	0	0
$124$	9.54746	0.857387
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.53078	0.490777	0.245389	−	0.969425i	$-0.421084\pi$
$127$	5.53078	0.490777	0.245389	+	0.969425i	$0.421084\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−5.23358	−0.459016
$131$	13.0712	1.14203	0.571017	−	0.820938i	$-0.306549\pi$
$131$	13.0712	1.14203	0.571017	+	0.820938i	$0.306549\pi$
$132$	0	0
$133$	1.21585	0.105428
$134$	−11.7854	−1.01811
$135$	0	0
$136$	6.50216	0.557556
$137$	−17.5381	−1.49838	−0.749192	−	0.662353i	$-0.769558\pi$
$137$	−17.5381	−1.49838	−0.749192	+	0.662353i	$0.769558\pi$
$138$	0	0
$139$	14.3741	1.21919	0.609597	−	0.792711i	$-0.291331\pi$
$139$	14.3741	1.21919	0.609597	+	0.792711i	$0.291331\pi$
$140$	1.15694	0.0977791
$141$	0	0
$142$	8.25150	0.692451
$143$	−33.2018	−2.77647
$144$	0	0
$145$	7.76739	0.645046
$146$	−8.45709	−0.699914
$147$	0	0
$148$	0.264192	0.0217165
$149$	−9.98305	−0.817843	−0.408922	−	0.912569i	$-0.634095\pi$
$149$	−9.98305	−0.817843	−0.408922	+	0.912569i	$0.634095\pi$
$150$	0	0
$151$	13.4544	1.09490	0.547451	−	0.836838i	$-0.315598\pi$
$151$	13.4544	1.09490	0.547451	+	0.836838i	$0.315598\pi$
$152$	1.05092	0.0852410
$153$	0	0
$154$	7.33960	0.591442
$155$	9.54746	0.766870
$156$	0	0
$157$	18.5890	1.48357	0.741784	−	0.670639i	$-0.233980\pi$
$157$	18.5890	1.48357	0.741784	+	0.670639i	$0.233980\pi$
$158$	5.32834	0.423900
$159$	0	0
$160$	1.00000	0.0790569
$161$	−0.923230	−0.0727607
$162$	0	0
$163$	−14.5861	−1.14247	−0.571237	−	0.820785i	$-0.693536\pi$
$163$	−14.5861	−1.14247	−0.571237	+	0.820785i	$0.693536\pi$
$164$	−0.200768	−0.0156774
$165$	0	0
$166$	0.830997	0.0644979
$167$	14.3327	1.10909	0.554547	−	0.832152i	$-0.312891\pi$
$167$	14.3327	1.10909	0.554547	+	0.832152i	$0.312891\pi$
$168$	0	0
$169$	14.3904	1.10695
$170$	6.50216	0.498693
$171$	0	0
$172$	−3.28211	−0.250259
$173$	11.1025	0.844110	0.422055	−	0.906570i	$-0.361309\pi$
$173$	11.1025	0.844110	0.422055	+	0.906570i	$0.361309\pi$
$174$	0	0
$175$	1.15694	0.0874563
$176$	6.34399	0.478196
$177$	0	0
$178$	−1.00000	−0.0749532
$179$	−1.20998	−0.0904379	−0.0452190	−	0.998977i	$-0.514399\pi$
$179$	−1.20998	−0.0904379	−0.0452190	+	0.998977i	$0.514399\pi$
$180$	0	0
$181$	5.97667	0.444242	0.222121	−	0.975019i	$-0.428702\pi$
$181$	5.97667	0.444242	0.222121	+	0.975019i	$0.428702\pi$
$182$	−6.05493	−0.448822
$183$	0	0
$184$	−0.797994	−0.0588289
$185$	0.264192	0.0194238
$186$	0	0
$187$	41.2496	3.01647
$188$	−5.08250	−0.370679
$189$	0	0
$190$	1.05092	0.0762419
$191$	−11.8046	−0.854148	−0.427074	−	0.904217i	$-0.640456\pi$
$191$	−11.8046	−0.854148	−0.427074	+	0.904217i	$0.640456\pi$
$192$	0	0
$193$	−2.84768	−0.204980	−0.102490	−	0.994734i	$-0.532681\pi$
$193$	−2.84768	−0.204980	−0.102490	+	0.994734i	$0.532681\pi$
$194$	0.404012	0.0290064
$195$	0	0
$196$	−5.66149	−0.404392
$197$	−22.0561	−1.57143	−0.785715	−	0.618589i	$-0.787705\pi$
$197$	−22.0561	−1.57143	−0.785715	+	0.618589i	$0.787705\pi$
$198$	0	0
$199$	−9.06110	−0.642324	−0.321162	−	0.947024i	$-0.604073\pi$
$199$	−9.06110	−0.642324	−0.321162	+	0.947024i	$0.604073\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−7.02243	−0.494096
$203$	8.98639	0.630721
$204$	0	0
$205$	−0.200768	−0.0140223
$206$	−0.684886	−0.0477182
$207$	0	0
$208$	−5.23358	−0.362884
$209$	6.66703	0.461168
$210$	0	0
$211$	−9.07734	−0.624910	−0.312455	−	0.949933i	$-0.601151\pi$
$211$	−9.07734	−0.624910	−0.312455	+	0.949933i	$0.601151\pi$
$212$	−14.2876	−0.981276
$213$	0	0
$214$	18.2840	1.24987
$215$	−3.28211	−0.223838
$216$	0	0
$217$	11.0458	0.749839
$218$	−4.56538	−0.309207
$219$	0	0
$220$	6.34399	0.427711
$221$	−34.0296	−2.28908
$222$	0	0
$223$	6.78866	0.454602	0.227301	−	0.973825i	$-0.427010\pi$
$223$	6.78866	0.454602	0.227301	+	0.973825i	$0.427010\pi$
$224$	1.15694	0.0773012
$225$	0	0
$226$	0.850154	0.0565514
$227$	3.49414	0.231915	0.115957	−	0.993254i	$-0.463006\pi$
$227$	3.49414	0.231915	0.115957	+	0.993254i	$0.463006\pi$
$228$	0	0
$229$	−20.1681	−1.33274	−0.666372	−	0.745619i	$-0.732154\pi$
$229$	−20.1681	−1.33274	−0.666372	+	0.745619i	$0.732154\pi$
$230$	−0.797994	−0.0526182
$231$	0	0
$232$	7.76739	0.509954
$233$	3.59903	0.235780	0.117890	−	0.993027i	$-0.462387\pi$
$233$	3.59903	0.235780	0.117890	+	0.993027i	$0.462387\pi$
$234$	0	0
$235$	−5.08250	−0.331546
$236$	−11.5381	−0.751068
$237$	0	0
$238$	7.52260	0.487618
$239$	−16.9984	−1.09953	−0.549767	−	0.835318i	$-0.685284\pi$
$239$	−16.9984	−1.09953	−0.549767	+	0.835318i	$0.685284\pi$
$240$	0	0
$241$	23.4059	1.50770	0.753852	−	0.657044i	$-0.228194\pi$
$241$	23.4059	1.50770	0.753852	+	0.657044i	$0.228194\pi$
$242$	29.2462	1.88001
$243$	0	0
$244$	11.4084	0.730351
$245$	−5.66149	−0.361700
$246$	0	0
$247$	−5.50009	−0.349962
$248$	9.54746	0.606264
$249$	0	0
$250$	1.00000	0.0632456
$251$	−8.52642	−0.538183	−0.269091	−	0.963115i	$-0.586723\pi$
$251$	−8.52642	−0.538183	−0.269091	+	0.963115i	$0.586723\pi$
$252$	0	0
$253$	−5.06246	−0.318274
$254$	5.53078	0.347032
$255$	0	0
$256$	1.00000	0.0625000
$257$	−30.0031	−1.87154	−0.935772	−	0.352606i	$-0.885296\pi$
$257$	−30.0031	−1.87154	−0.935772	+	0.352606i	$0.885296\pi$
$258$	0	0
$259$	0.305654	0.0189924
$260$	−5.23358	−0.324573
$261$	0	0
$262$	13.0712	0.807540
$263$	11.0156	0.679250	0.339625	−	0.940561i	$-0.389700\pi$
$263$	11.0156	0.679250	0.339625	+	0.940561i	$0.389700\pi$
$264$	0	0
$265$	−14.2876	−0.877680
$266$	1.21585	0.0745487
$267$	0	0
$268$	−11.7854	−0.719910
$269$	−17.9259	−1.09296	−0.546481	−	0.837471i	$-0.684033\pi$
$269$	−17.9259	−1.09296	−0.546481	+	0.837471i	$0.684033\pi$
$270$	0	0
$271$	−5.04757	−0.306618	−0.153309	−	0.988178i	$-0.548993\pi$
$271$	−5.04757	−0.306618	−0.153309	+	0.988178i	$0.548993\pi$
$272$	6.50216	0.394251
$273$	0	0
$274$	−17.5381	−1.05952
$275$	6.34399	0.382557
$276$	0	0
$277$	−9.34349	−0.561396	−0.280698	−	0.959796i	$-0.590566\pi$
$277$	−9.34349	−0.561396	−0.280698	+	0.959796i	$0.590566\pi$
$278$	14.3741	0.862101
$279$	0	0
$280$	1.15694	0.0691403
$281$	4.26392	0.254364	0.127182	−	0.991879i	$-0.459407\pi$
$281$	4.26392	0.254364	0.127182	+	0.991879i	$0.459407\pi$
$282$	0	0
$283$	−22.0993	−1.31367	−0.656834	−	0.754035i	$-0.728105\pi$
$283$	−22.0993	−1.31367	−0.656834	+	0.754035i	$0.728105\pi$
$284$	8.25150	0.489637
$285$	0	0
$286$	−33.2018	−1.96326
$287$	−0.232276	−0.0137108
$288$	0	0
$289$	25.2781	1.48695
$290$	7.76739	0.456117
$291$	0	0
$292$	−8.45709	−0.494914
$293$	17.5011	1.02243	0.511214	−	0.859454i	$-0.329196\pi$
$293$	17.5011	1.02243	0.511214	+	0.859454i	$0.329196\pi$
$294$	0	0
$295$	−11.5381	−0.671776
$296$	0.264192	0.0153559
$297$	0	0
$298$	−9.98305	−0.578303
$299$	4.17637	0.241526
$300$	0	0
$301$	−3.79720	−0.218867
$302$	13.4544	0.774213
$303$	0	0
$304$	1.05092	0.0602745
$305$	11.4084	0.653245
$306$	0	0
$307$	−8.36765	−0.477567	−0.238784	−	0.971073i	$-0.576749\pi$
$307$	−8.36765	−0.477567	−0.238784	+	0.971073i	$0.576749\pi$
$308$	7.33960	0.418213
$309$	0	0
$310$	9.54746	0.542259
$311$	−5.19383	−0.294515	−0.147257	−	0.989098i	$-0.547045\pi$
$311$	−5.19383	−0.294515	−0.147257	+	0.989098i	$0.547045\pi$
$312$	0	0
$313$	8.24724	0.466162	0.233081	−	0.972457i	$-0.425119\pi$
$313$	8.24724	0.466162	0.233081	+	0.972457i	$0.425119\pi$
$314$	18.5890	1.04904
$315$	0	0
$316$	5.32834	0.299742
$317$	−7.49484	−0.420952	−0.210476	−	0.977599i	$-0.567501\pi$
$317$	−7.49484	−0.420952	−0.210476	+	0.977599i	$0.567501\pi$
$318$	0	0
$319$	49.2762	2.75894
$320$	1.00000	0.0559017
$321$	0	0
$322$	−0.923230	−0.0514496
$323$	6.83326	0.380213
$324$	0	0
$325$	−5.23358	−0.290307
$326$	−14.5861	−0.807851
$327$	0	0
$328$	−0.200768	−0.0110856
$329$	−5.88014	−0.324183
$330$	0	0
$331$	15.1551	0.833002	0.416501	−	0.909135i	$-0.363256\pi$
$331$	15.1551	0.833002	0.416501	+	0.909135i	$0.363256\pi$
$332$	0.830997	0.0456069
$333$	0	0
$334$	14.3327	0.784248
$335$	−11.7854	−0.643907
$336$	0	0
$337$	26.3691	1.43642	0.718209	−	0.695828i	$-0.244962\pi$
$337$	26.3691	1.43642	0.718209	+	0.695828i	$0.244962\pi$
$338$	14.3904	0.782734
$339$	0	0
$340$	6.50216	0.352629
$341$	60.5690	3.27999
$342$	0	0
$343$	−14.6486	−0.790948
$344$	−3.28211	−0.176960
$345$	0	0
$346$	11.1025	0.596876
$347$	20.1580	1.08214	0.541069	−	0.840978i	$-0.318020\pi$
$347$	20.1580	1.08214	0.541069	+	0.840978i	$0.318020\pi$
$348$	0	0
$349$	−1.80005	−0.0963544	−0.0481772	−	0.998839i	$-0.515341\pi$
$349$	−1.80005	−0.0963544	−0.0481772	+	0.998839i	$0.515341\pi$
$350$	1.15694	0.0618410
$351$	0	0
$352$	6.34399	0.338136
$353$	5.02841	0.267635	0.133818	−	0.991006i	$-0.457276\pi$
$353$	5.02841	0.267635	0.133818	+	0.991006i	$0.457276\pi$
$354$	0	0
$355$	8.25150	0.437944
$356$	−1.00000	−0.0529999
$357$	0	0
$358$	−1.20998	−0.0639493
$359$	12.4533	0.657261	0.328631	−	0.944459i	$-0.393413\pi$
$359$	12.4533	0.657261	0.328631	+	0.944459i	$0.393413\pi$
$360$	0	0
$361$	−17.8956	−0.941872
$362$	5.97667	0.314127
$363$	0	0
$364$	−6.05493	−0.317365
$365$	−8.45709	−0.442665
$366$	0	0
$367$	17.2175	0.898747	0.449373	−	0.893344i	$-0.351647\pi$
$367$	17.2175	0.898747	0.449373	+	0.893344i	$0.351647\pi$
$368$	−0.797994	−0.0415983
$369$	0	0
$370$	0.264192	0.0137347
$371$	−16.5299	−0.858188
$372$	0	0
$373$	−6.98439	−0.361638	−0.180819	−	0.983516i	$-0.557875\pi$
$373$	−6.98439	−0.361638	−0.180819	+	0.983516i	$0.557875\pi$
$374$	41.2496	2.13297
$375$	0	0
$376$	−5.08250	−0.262110
$377$	−40.6513	−2.09365
$378$	0	0
$379$	25.1405	1.29138	0.645690	−	0.763599i	$-0.276570\pi$
$379$	25.1405	1.29138	0.645690	+	0.763599i	$0.276570\pi$
$380$	1.05092	0.0539112
$381$	0	0
$382$	−11.8046	−0.603974
$383$	2.67425	0.136648	0.0683238	−	0.997663i	$-0.478235\pi$
$383$	2.67425	0.136648	0.0683238	+	0.997663i	$0.478235\pi$
$384$	0	0
$385$	7.33960	0.374061
$386$	−2.84768	−0.144943
$387$	0	0
$388$	0.404012	0.0205106
$389$	−11.8593	−0.601290	−0.300645	−	0.953736i	$-0.597202\pi$
$389$	−11.8593	−0.601290	−0.300645	+	0.953736i	$0.597202\pi$
$390$	0	0
$391$	−5.18869	−0.262403
$392$	−5.66149	−0.285949
$393$	0	0
$394$	−22.0561	−1.11117
$395$	5.32834	0.268098
$396$	0	0
$397$	−18.8058	−0.943834	−0.471917	−	0.881643i	$-0.656438\pi$
$397$	−18.8058	−0.943834	−0.471917	+	0.881643i	$0.656438\pi$
$398$	−9.06110	−0.454192
$399$	0	0
$400$	1.00000	0.0500000
$401$	11.6112	0.579836	0.289918	−	0.957052i	$-0.406372\pi$
$401$	11.6112	0.579836	0.289918	+	0.957052i	$0.406372\pi$
$402$	0	0
$403$	−49.9674	−2.48906
$404$	−7.02243	−0.349379
$405$	0	0
$406$	8.98639	0.445987
$407$	1.67603	0.0830778
$408$	0	0
$409$	21.7125	1.07361	0.536807	−	0.843705i	$-0.319630\pi$
$409$	21.7125	1.07361	0.536807	+	0.843705i	$0.319630\pi$
$410$	−0.200768	−0.00991523
$411$	0	0
$412$	−0.684886	−0.0337419
$413$	−13.3489	−0.656856
$414$	0	0
$415$	0.830997	0.0407920
$416$	−5.23358	−0.256598
$417$	0	0
$418$	6.66703	0.326095
$419$	−1.88192	−0.0919377	−0.0459689	−	0.998943i	$-0.514638\pi$
$419$	−1.88192	−0.0919377	−0.0459689	+	0.998943i	$0.514638\pi$
$420$	0	0
$421$	10.3674	0.505275	0.252638	−	0.967561i	$-0.418702\pi$
$421$	10.3674	0.505275	0.252638	+	0.967561i	$0.418702\pi$
$422$	−9.07734	−0.441878
$423$	0	0
$424$	−14.2876	−0.693867
$425$	6.50216	0.315401
$426$	0	0
$427$	13.1989	0.638738
$428$	18.2840	0.883789
$429$	0	0
$430$	−3.28211	−0.158277
$431$	−26.2335	−1.26362	−0.631811	−	0.775123i	$-0.717688\pi$
$431$	−26.2335	−1.26362	−0.631811	+	0.775123i	$0.717688\pi$
$432$	0	0
$433$	−13.3883	−0.643400	−0.321700	−	0.946842i	$-0.604254\pi$
$433$	−13.3883	−0.643400	−0.321700	+	0.946842i	$0.604254\pi$
$434$	11.0458	0.530216
$435$	0	0
$436$	−4.56538	−0.218642
$437$	−0.838630	−0.0401171
$438$	0	0
$439$	24.9135	1.18906	0.594528	−	0.804075i	$-0.297339\pi$
$439$	24.9135	1.18906	0.594528	+	0.804075i	$0.297339\pi$
$440$	6.34399	0.302438
$441$	0	0
$442$	−34.0296	−1.61862
$443$	15.6385	0.743008	0.371504	−	0.928431i	$-0.378842\pi$
$443$	15.6385	0.743008	0.371504	+	0.928431i	$0.378842\pi$
$444$	0	0
$445$	−1.00000	−0.0474045
$446$	6.78866	0.321452
$447$	0	0
$448$	1.15694	0.0546602
$449$	−0.672509	−0.0317377	−0.0158688	−	0.999874i	$-0.505051\pi$
$449$	−0.672509	−0.0317377	−0.0158688	+	0.999874i	$0.505051\pi$
$450$	0	0
$451$	−1.27367	−0.0599748
$452$	0.850154	0.0399879
$453$	0	0
$454$	3.49414	0.163988
$455$	−6.05493	−0.283860
$456$	0	0
$457$	18.3894	0.860218	0.430109	−	0.902777i	$-0.358475\pi$
$457$	18.3894	0.860218	0.430109	+	0.902777i	$0.358475\pi$
$458$	−20.1681	−0.942393
$459$	0	0
$460$	−0.797994	−0.0372067
$461$	13.2258	0.615989	0.307994	−	0.951388i	$-0.400342\pi$
$461$	13.2258	0.615989	0.307994	+	0.951388i	$0.400342\pi$
$462$	0	0
$463$	19.9137	0.925466	0.462733	−	0.886498i	$-0.346869\pi$
$463$	19.9137	0.925466	0.462733	+	0.886498i	$0.346869\pi$
$464$	7.76739	0.360592
$465$	0	0
$466$	3.59903	0.166722
$467$	−26.9449	−1.24686	−0.623430	−	0.781879i	$-0.714261\pi$
$467$	−26.9449	−1.24686	−0.623430	+	0.781879i	$0.714261\pi$
$468$	0	0
$469$	−13.6350	−0.629606
$470$	−5.08250	−0.234438
$471$	0	0
$472$	−11.5381	−0.531085
$473$	−20.8217	−0.957381
$474$	0	0
$475$	1.05092	0.0482196
$476$	7.52260	0.344798
$477$	0	0
$478$	−16.9984	−0.777488
$479$	43.5500	1.98985	0.994925	−	0.100621i	$-0.0320829\pi$
$479$	43.5500	1.98985	0.994925	+	0.100621i	$0.0320829\pi$
$480$	0	0
$481$	−1.38267	−0.0630444
$482$	23.4059	1.06611
$483$	0	0
$484$	29.2462	1.32937
$485$	0.404012	0.0183452
$486$	0	0
$487$	17.3139	0.784570	0.392285	−	0.919844i	$-0.371685\pi$
$487$	17.3139	0.784570	0.392285	+	0.919844i	$0.371685\pi$
$488$	11.4084	0.516436
$489$	0	0
$490$	−5.66149	−0.255760
$491$	−15.0081	−0.677308	−0.338654	−	0.940911i	$-0.609972\pi$
$491$	−15.0081	−0.677308	−0.338654	+	0.940911i	$0.609972\pi$
$492$	0	0
$493$	50.5048	2.27462
$494$	−5.50009	−0.247461
$495$	0	0
$496$	9.54746	0.428694
$497$	9.54648	0.428218
$498$	0	0
$499$	−27.7202	−1.24093	−0.620464	−	0.784235i	$-0.713056\pi$
$499$	−27.7202	−1.24093	−0.620464	+	0.784235i	$0.713056\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−8.52642	−0.380553
$503$	−6.83045	−0.304555	−0.152277	−	0.988338i	$-0.548661\pi$
$503$	−6.83045	−0.304555	−0.152277	+	0.988338i	$0.548661\pi$
$504$	0	0
$505$	−7.02243	−0.312494
$506$	−5.06246	−0.225054
$507$	0	0
$508$	5.53078	0.245389
$509$	25.1436	1.11447	0.557236	−	0.830354i	$-0.311862\pi$
$509$	25.1436	1.11447	0.557236	+	0.830354i	$0.311862\pi$
$510$	0	0
$511$	−9.78434	−0.432834
$512$	1.00000	0.0441942
$513$	0	0
$514$	−30.0031	−1.32338
$515$	−0.684886	−0.0301797
$516$	0	0
$517$	−32.2433	−1.41806
$518$	0.305654	0.0134297
$519$	0	0
$520$	−5.23358	−0.229508
$521$	1.58630	0.0694970	0.0347485	−	0.999396i	$-0.488937\pi$
$521$	1.58630	0.0694970	0.0347485	+	0.999396i	$0.488937\pi$
$522$	0	0
$523$	0.945454	0.0413418	0.0206709	−	0.999786i	$-0.493420\pi$
$523$	0.945454	0.0413418	0.0206709	+	0.999786i	$0.493420\pi$
$524$	13.0712	0.571017
$525$	0	0
$526$	11.0156	0.480302
$527$	62.0791	2.70421
$528$	0	0
$529$	−22.3632	−0.972313
$530$	−14.2876	−0.620613
$531$	0	0
$532$	1.21585	0.0527139
$533$	1.05074	0.0455125
$534$	0	0
$535$	18.2840	0.790485
$536$	−11.7854	−0.509053
$537$	0	0
$538$	−17.9259	−0.772841
$539$	−35.9164	−1.54703
$540$	0	0
$541$	15.4324	0.663489	0.331745	−	0.943369i	$-0.392363\pi$
$541$	15.4324	0.663489	0.331745	+	0.943369i	$0.392363\pi$
$542$	−5.04757	−0.216812
$543$	0	0
$544$	6.50216	0.278778
$545$	−4.56538	−0.195559
$546$	0	0
$547$	−34.3808	−1.47002	−0.735009	−	0.678057i	$-0.762822\pi$
$547$	−34.3808	−1.47002	−0.735009	+	0.678057i	$0.762822\pi$
$548$	−17.5381	−0.749192
$549$	0	0
$550$	6.34399	0.270508
$551$	8.16292	0.347752
$552$	0	0
$553$	6.16456	0.262144
$554$	−9.34349	−0.396967
$555$	0	0
$556$	14.3741	0.609597
$557$	−7.52918	−0.319022	−0.159511	−	0.987196i	$-0.550992\pi$
$557$	−7.52918	−0.319022	−0.159511	+	0.987196i	$0.550992\pi$
$558$	0	0
$559$	17.1772	0.726518
$560$	1.15694	0.0488896
$561$	0	0
$562$	4.26392	0.179863
$563$	42.7985	1.80374	0.901871	−	0.432005i	$-0.142194\pi$
$563$	42.7985	1.80374	0.901871	+	0.432005i	$0.142194\pi$
$564$	0	0
$565$	0.850154	0.0357662
$566$	−22.0993	−0.928903
$567$	0	0
$568$	8.25150	0.346225
$569$	−36.0110	−1.50966	−0.754829	−	0.655922i	$-0.772280\pi$
$569$	−36.0110	−1.50966	−0.754829	+	0.655922i	$0.772280\pi$
$570$	0	0
$571$	−4.78702	−0.200330	−0.100165	−	0.994971i	$-0.531937\pi$
$571$	−4.78702	−0.200330	−0.100165	+	0.994971i	$0.531937\pi$
$572$	−33.2018	−1.38824
$573$	0	0
$574$	−0.232276	−0.00969503
$575$	−0.797994	−0.0332787
$576$	0	0
$577$	−8.05913	−0.335506	−0.167753	−	0.985829i	$-0.553651\pi$
$577$	−8.05913	−0.335506	−0.167753	+	0.985829i	$0.553651\pi$
$578$	25.2781	1.05143
$579$	0	0
$580$	7.76739	0.322523
$581$	0.961412	0.0398861
$582$	0	0
$583$	−90.6403	−3.75394
$584$	−8.45709	−0.349957
$585$	0	0
$586$	17.5011	0.722965
$587$	−5.29582	−0.218582	−0.109291	−	0.994010i	$-0.534858\pi$
$587$	−5.29582	−0.218582	−0.109291	+	0.994010i	$0.534858\pi$
$588$	0	0
$589$	10.0336	0.413429
$590$	−11.5381	−0.475017
$591$	0	0
$592$	0.264192	0.0108582
$593$	43.9422	1.80449	0.902245	−	0.431223i	$-0.141918\pi$
$593$	43.9422	1.80449	0.902245	+	0.431223i	$0.141918\pi$
$594$	0	0
$595$	7.52260	0.308397
$596$	−9.98305	−0.408922
$597$	0	0
$598$	4.17637	0.170784
$599$	−41.6932	−1.70354	−0.851769	−	0.523918i	$-0.824470\pi$
$599$	−41.6932	−1.70354	−0.851769	+	0.523918i	$0.824470\pi$
$600$	0	0
$601$	−27.8348	−1.13540	−0.567702	−	0.823234i	$-0.692168\pi$
$601$	−27.8348	−1.13540	−0.567702	+	0.823234i	$0.692168\pi$
$602$	−3.79720	−0.154762
$603$	0	0
$604$	13.4544	0.547451
$605$	29.2462	1.18903
$606$	0	0
$607$	−20.6503	−0.838169	−0.419085	−	0.907947i	$-0.637649\pi$
$607$	−20.6503	−0.838169	−0.419085	+	0.907947i	$0.637649\pi$
$608$	1.05092	0.0426205
$609$	0	0
$610$	11.4084	0.461914
$611$	26.5997	1.07611
$612$	0	0
$613$	46.9888	1.89786	0.948929	−	0.315489i	$-0.102169\pi$
$613$	46.9888	1.89786	0.948929	+	0.315489i	$0.102169\pi$
$614$	−8.36765	−0.337691
$615$	0	0
$616$	7.33960	0.295721
$617$	13.7101	0.551947	0.275974	−	0.961165i	$-0.411000\pi$
$617$	13.7101	0.551947	0.275974	+	0.961165i	$0.411000\pi$
$618$	0	0
$619$	−17.4661	−0.702021	−0.351010	−	0.936372i	$-0.614162\pi$
$619$	−17.4661	−0.702021	−0.351010	+	0.936372i	$0.614162\pi$
$620$	9.54746	0.383435
$621$	0	0
$622$	−5.19383	−0.208253
$623$	−1.15694	−0.0463518
$624$	0	0
$625$	1.00000	0.0400000
$626$	8.24724	0.329626
$627$	0	0
$628$	18.5890	0.741784
$629$	1.71782	0.0684940
$630$	0	0
$631$	28.7311	1.14377	0.571883	−	0.820335i	$-0.306213\pi$
$631$	28.7311	1.14377	0.571883	+	0.820335i	$0.306213\pi$
$632$	5.32834	0.211950
$633$	0	0
$634$	−7.49484	−0.297658
$635$	5.53078	0.219482
$636$	0	0
$637$	29.6299	1.17398
$638$	49.2762	1.95086
$639$	0	0
$640$	1.00000	0.0395285
$641$	6.25163	0.246925	0.123462	−	0.992349i	$-0.460600\pi$
$641$	6.25163	0.246925	0.123462	+	0.992349i	$0.460600\pi$
$642$	0	0
$643$	8.48465	0.334602	0.167301	−	0.985906i	$-0.446495\pi$
$643$	8.48465	0.334602	0.167301	+	0.985906i	$0.446495\pi$
$644$	−0.923230	−0.0363804
$645$	0	0
$646$	6.83326	0.268851
$647$	−15.6273	−0.614372	−0.307186	−	0.951650i	$-0.599387\pi$
$647$	−15.6273	−0.614372	−0.307186	+	0.951650i	$0.599387\pi$
$648$	0	0
$649$	−73.1977	−2.87326
$650$	−5.23358	−0.205278
$651$	0	0
$652$	−14.5861	−0.571237
$653$	−24.4506	−0.956827	−0.478414	−	0.878135i	$-0.658788\pi$
$653$	−24.4506	−0.956827	−0.478414	+	0.878135i	$0.658788\pi$
$654$	0	0
$655$	13.0712	0.510733
$656$	−0.200768	−0.00783868
$657$	0	0
$658$	−5.88014	−0.229232
$659$	−41.2531	−1.60699	−0.803496	−	0.595310i	$-0.797029\pi$
$659$	−41.2531	−1.60699	−0.803496	+	0.595310i	$0.797029\pi$
$660$	0	0
$661$	−39.5552	−1.53852	−0.769260	−	0.638936i	$-0.779375\pi$
$661$	−39.5552	−1.53852	−0.769260	+	0.638936i	$0.779375\pi$
$662$	15.1551	0.589021
$663$	0	0
$664$	0.830997	0.0322489
$665$	1.21585	0.0471487
$666$	0	0
$667$	−6.19833	−0.240000
$668$	14.3327	0.554547
$669$	0	0
$670$	−11.7854	−0.455311
$671$	72.3750	2.79401
$672$	0	0
$673$	9.42310	0.363234	0.181617	−	0.983369i	$-0.441867\pi$
$673$	9.42310	0.363234	0.181617	+	0.983369i	$0.441867\pi$
$674$	26.3691	1.01570
$675$	0	0
$676$	14.3904	0.553477
$677$	−4.17878	−0.160604	−0.0803018	−	0.996771i	$-0.525588\pi$
$677$	−4.17878	−0.160604	−0.0803018	+	0.996771i	$0.525588\pi$
$678$	0	0
$679$	0.467417	0.0179378
$680$	6.50216	0.249347
$681$	0	0
$682$	60.5690	2.31931
$683$	−2.45500	−0.0939380	−0.0469690	−	0.998896i	$-0.514956\pi$
$683$	−2.45500	−0.0939380	−0.0469690	+	0.998896i	$0.514956\pi$
$684$	0	0
$685$	−17.5381	−0.670097
$686$	−14.6486	−0.559285
$687$	0	0
$688$	−3.28211	−0.125129
$689$	74.7753	2.84871
$690$	0	0
$691$	−21.6124	−0.822173	−0.411086	−	0.911596i	$-0.634851\pi$
$691$	−21.6124	−0.822173	−0.411086	+	0.911596i	$0.634851\pi$
$692$	11.1025	0.422055
$693$	0	0
$694$	20.1580	0.765187
$695$	14.3741	0.545241
$696$	0	0
$697$	−1.30543	−0.0494466
$698$	−1.80005	−0.0681329
$699$	0	0
$700$	1.15694	0.0437282
$701$	50.2850	1.89924	0.949619	−	0.313408i	$-0.101471\pi$
$701$	50.2850	1.89924	0.949619	+	0.313408i	$0.101471\pi$
$702$	0	0
$703$	0.277646	0.0104716
$704$	6.34399	0.239098
$705$	0	0
$706$	5.02841	0.189247
$707$	−8.12452	−0.305554
$708$	0	0
$709$	−9.84854	−0.369870	−0.184935	−	0.982751i	$-0.559207\pi$
$709$	−9.84854	−0.369870	−0.184935	+	0.982751i	$0.559207\pi$
$710$	8.25150	0.309673
$711$	0	0
$712$	−1.00000	−0.0374766
$713$	−7.61882	−0.285327
$714$	0	0
$715$	−33.2018	−1.24168
$716$	−1.20998	−0.0452190
$717$	0	0
$718$	12.4533	0.464754
$719$	−11.1493	−0.415801	−0.207900	−	0.978150i	$-0.566663\pi$
$719$	−11.1493	−0.415801	−0.207900	+	0.978150i	$0.566663\pi$
$720$	0	0
$721$	−0.792370	−0.0295094
$722$	−17.8956	−0.666004
$723$	0	0
$724$	5.97667	0.222121
$725$	7.76739	0.288473
$726$	0	0
$727$	27.5465	1.02164	0.510821	−	0.859687i	$-0.329342\pi$
$727$	27.5465	1.02164	0.510821	+	0.859687i	$0.329342\pi$
$728$	−6.05493	−0.224411
$729$	0	0
$730$	−8.45709	−0.313011
$731$	−21.3408	−0.789319
$732$	0	0
$733$	26.6775	0.985356	0.492678	−	0.870212i	$-0.336018\pi$
$733$	26.6775	0.985356	0.492678	+	0.870212i	$0.336018\pi$
$734$	17.2175	0.635510
$735$	0	0
$736$	−0.797994	−0.0294145
$737$	−74.7666	−2.75406
$738$	0	0
$739$	−6.93806	−0.255221	−0.127610	−	0.991824i	$-0.540731\pi$
$739$	−6.93806	−0.255221	−0.127610	+	0.991824i	$0.540731\pi$
$740$	0.264192	0.00971190
$741$	0	0
$742$	−16.5299	−0.606830
$743$	1.88236	0.0690572	0.0345286	−	0.999404i	$-0.489007\pi$
$743$	1.88236	0.0690572	0.0345286	+	0.999404i	$0.489007\pi$
$744$	0	0
$745$	−9.98305	−0.365751
$746$	−6.98439	−0.255717
$747$	0	0
$748$	41.2496	1.50824
$749$	21.1534	0.772929
$750$	0	0
$751$	−37.9470	−1.38470	−0.692352	−	0.721560i	$-0.743426\pi$
$751$	−37.9470	−1.38470	−0.692352	+	0.721560i	$0.743426\pi$
$752$	−5.08250	−0.185340
$753$	0	0
$754$	−40.6513	−1.48043
$755$	13.4544	0.489655
$756$	0	0
$757$	3.41535	0.124133	0.0620665	−	0.998072i	$-0.480231\pi$
$757$	3.41535	0.124133	0.0620665	+	0.998072i	$0.480231\pi$
$758$	25.1405	0.913144
$759$	0	0
$760$	1.05092	0.0381209
$761$	−9.66792	−0.350462	−0.175231	−	0.984527i	$-0.556067\pi$
$761$	−9.66792	−0.350462	−0.175231	+	0.984527i	$0.556067\pi$
$762$	0	0
$763$	−5.28186	−0.191216
$764$	−11.8046	−0.427074
$765$	0	0
$766$	2.67425	0.0966244
$767$	60.3857	2.18040
$768$	0	0
$769$	−51.9267	−1.87252	−0.936262	−	0.351302i	$-0.885739\pi$
$769$	−51.9267	−1.87252	−0.936262	+	0.351302i	$0.885739\pi$
$770$	7.33960	0.264501
$771$	0	0
$772$	−2.84768	−0.102490
$773$	16.2763	0.585417	0.292708	−	0.956202i	$-0.405444\pi$
$773$	16.2763	0.585417	0.292708	+	0.956202i	$0.405444\pi$
$774$	0	0
$775$	9.54746	0.342955
$776$	0.404012	0.0145032
$777$	0	0
$778$	−11.8593	−0.425177
$779$	−0.210992	−0.00755956
$780$	0	0
$781$	52.3474	1.87314
$782$	−5.18869	−0.185547
$783$	0	0
$784$	−5.66149	−0.202196
$785$	18.5890	0.663472
$786$	0	0
$787$	34.5652	1.23212	0.616059	−	0.787700i	$-0.288728\pi$
$787$	34.5652	1.23212	0.616059	+	0.787700i	$0.288728\pi$
$788$	−22.0561	−0.785715
$789$	0	0
$790$	5.32834	0.189574
$791$	0.983576	0.0349719
$792$	0	0
$793$	−59.7070	−2.12026
$794$	−18.8058	−0.667392
$795$	0	0
$796$	−9.06110	−0.321162
$797$	−30.8359	−1.09226	−0.546131	−	0.837700i	$-0.683900\pi$
$797$	−30.8359	−1.09226	−0.546131	+	0.837700i	$0.683900\pi$
$798$	0	0
$799$	−33.0472	−1.16913
$800$	1.00000	0.0353553
$801$	0	0
$802$	11.6112	0.410006
$803$	−53.6517	−1.89333
$804$	0	0
$805$	−0.923230	−0.0325396
$806$	−49.9674	−1.76003
$807$	0	0
$808$	−7.02243	−0.247048
$809$	19.8073	0.696387	0.348193	−	0.937423i	$-0.386795\pi$
$809$	19.8073	0.696387	0.348193	+	0.937423i	$0.386795\pi$
$810$	0	0
$811$	−40.9023	−1.43627	−0.718137	−	0.695901i	$-0.755005\pi$
$811$	−40.9023	−1.43627	−0.718137	+	0.695901i	$0.755005\pi$
$812$	8.98639	0.315360
$813$	0	0
$814$	1.67603	0.0587449
$815$	−14.5861	−0.510930
$816$	0	0
$817$	−3.44924	−0.120674
$818$	21.7125	0.759160
$819$	0	0
$820$	−0.200768	−0.00701113
$821$	−47.7676	−1.66710	−0.833551	−	0.552443i	$-0.813696\pi$
$821$	−47.7676	−1.66710	−0.833551	+	0.552443i	$0.813696\pi$
$822$	0	0
$823$	−49.4308	−1.72305	−0.861524	−	0.507717i	$-0.830490\pi$
$823$	−49.4308	−1.72305	−0.861524	+	0.507717i	$0.830490\pi$
$824$	−0.684886	−0.0238591
$825$	0	0
$826$	−13.3489	−0.464468
$827$	−42.6789	−1.48409	−0.742045	−	0.670350i	$-0.766144\pi$
$827$	−42.6789	−1.48409	−0.742045	+	0.670350i	$0.766144\pi$
$828$	0	0
$829$	18.6192	0.646672	0.323336	−	0.946284i	$-0.395196\pi$
$829$	18.6192	0.646672	0.323336	+	0.946284i	$0.395196\pi$
$830$	0.830997	0.0288443
$831$	0	0
$832$	−5.23358	−0.181442
$833$	−36.8119	−1.27546
$834$	0	0
$835$	14.3327	0.496002
$836$	6.66703	0.230584
$837$	0	0
$838$	−1.88192	−0.0650098
$839$	−3.40953	−0.117710	−0.0588550	−	0.998267i	$-0.518745\pi$
$839$	−3.40953	−0.117710	−0.0588550	+	0.998267i	$0.518745\pi$
$840$	0	0
$841$	31.3323	1.08042
$842$	10.3674	0.357283
$843$	0	0
$844$	−9.07734	−0.312455
$845$	14.3904	0.495045
$846$	0	0
$847$	33.8360	1.16262
$848$	−14.2876	−0.490638
$849$	0	0
$850$	6.50216	0.223022
$851$	−0.210824	−0.00722695
$852$	0	0
$853$	−3.48932	−0.119472	−0.0597360	−	0.998214i	$-0.519026\pi$
$853$	−3.48932	−0.119472	−0.0597360	+	0.998214i	$0.519026\pi$
$854$	13.1989	0.451656
$855$	0	0
$856$	18.2840	0.624933
$857$	−46.2751	−1.58073	−0.790363	−	0.612639i	$-0.790108\pi$
$857$	−46.2751	−1.58073	−0.790363	+	0.612639i	$0.790108\pi$
$858$	0	0
$859$	−28.8050	−0.982813	−0.491406	−	0.870930i	$-0.663517\pi$
$859$	−28.8050	−0.982813	−0.491406	+	0.870930i	$0.663517\pi$
$860$	−3.28211	−0.111919
$861$	0	0
$862$	−26.2335	−0.893515
$863$	−49.8079	−1.69548	−0.847740	−	0.530412i	$-0.822037\pi$
$863$	−49.8079	−1.69548	−0.847740	+	0.530412i	$0.822037\pi$
$864$	0	0
$865$	11.1025	0.377498
$866$	−13.3883	−0.454952
$867$	0	0
$868$	11.0458	0.374920
$869$	33.8029	1.14668
$870$	0	0
$871$	61.6800	2.08995
$872$	−4.56538	−0.154603
$873$	0	0
$874$	−0.838630	−0.0283671
$875$	1.15694	0.0391117
$876$	0	0
$877$	12.8829	0.435025	0.217512	−	0.976058i	$-0.430206\pi$
$877$	12.8829	0.435025	0.217512	+	0.976058i	$0.430206\pi$
$878$	24.9135	0.840789
$879$	0	0
$880$	6.34399	0.213856
$881$	−2.87516	−0.0968667	−0.0484334	−	0.998826i	$-0.515423\pi$
$881$	−2.87516	−0.0968667	−0.0484334	+	0.998826i	$0.515423\pi$
$882$	0	0
$883$	12.5127	0.421084	0.210542	−	0.977585i	$-0.432477\pi$
$883$	12.5127	0.421084	0.210542	+	0.977585i	$0.432477\pi$
$884$	−34.0296	−1.14454
$885$	0	0
$886$	15.6385	0.525386
$887$	56.2689	1.88933	0.944663	−	0.328042i	$-0.106389\pi$
$887$	56.2689	1.88933	0.944663	+	0.328042i	$0.106389\pi$
$888$	0	0
$889$	6.39877	0.214608
$890$	−1.00000	−0.0335201
$891$	0	0
$892$	6.78866	0.227301
$893$	−5.34131	−0.178740
$894$	0	0
$895$	−1.20998	−0.0404451
$896$	1.15694	0.0386506
$897$	0	0
$898$	−0.672509	−0.0224419
$899$	74.1588	2.47333
$900$	0	0
$901$	−92.9002	−3.09496
$902$	−1.27367	−0.0424086
$903$	0	0
$904$	0.850154	0.0282757
$905$	5.97667	0.198671
$906$	0	0
$907$	−19.5219	−0.648214	−0.324107	−	0.946020i	$-0.605064\pi$
$907$	−19.5219	−0.648214	−0.324107	+	0.946020i	$0.605064\pi$
$908$	3.49414	0.115957
$909$	0	0
$910$	−6.05493	−0.200719
$911$	−53.6591	−1.77780	−0.888902	−	0.458098i	$-0.848531\pi$
$911$	−53.6591	−1.77780	−0.888902	+	0.458098i	$0.848531\pi$
$912$	0	0
$913$	5.27183	0.174472
$914$	18.3894	0.608266
$915$	0	0
$916$	−20.1681	−0.666372
$917$	15.1225	0.499390
$918$	0	0
$919$	−47.0818	−1.55309	−0.776543	−	0.630064i	$-0.783029\pi$
$919$	−47.0818	−1.55309	−0.776543	+	0.630064i	$0.783029\pi$
$920$	−0.797994	−0.0263091
$921$	0	0
$922$	13.2258	0.435570
$923$	−43.1849	−1.42145
$924$	0	0
$925$	0.264192	0.00868659
$926$	19.9137	0.654403
$927$	0	0
$928$	7.76739	0.254977
$929$	−29.2380	−0.959268	−0.479634	−	0.877469i	$-0.659231\pi$
$929$	−29.2380	−0.959268	−0.479634	+	0.877469i	$0.659231\pi$
$930$	0	0
$931$	−5.94979	−0.194996
$932$	3.59903	0.117890
$933$	0	0
$934$	−26.9449	−0.881663
$935$	41.2496	1.34901
$936$	0	0
$937$	21.1311	0.690324	0.345162	−	0.938543i	$-0.387824\pi$
$937$	21.1311	0.690324	0.345162	+	0.938543i	$0.387824\pi$
$938$	−13.6350	−0.445199
$939$	0	0
$940$	−5.08250	−0.165773
$941$	−42.9525	−1.40021	−0.700106	−	0.714039i	$-0.746864\pi$
$941$	−42.9525	−1.40021	−0.700106	+	0.714039i	$0.746864\pi$
$942$	0	0
$943$	0.160212	0.00521721
$944$	−11.5381	−0.375534
$945$	0	0
$946$	−20.8217	−0.676971
$947$	−42.0424	−1.36619	−0.683097	−	0.730327i	$-0.739368\pi$
$947$	−42.0424	−1.36619	−0.683097	+	0.730327i	$0.739368\pi$
$948$	0	0
$949$	44.2609	1.43677
$950$	1.05092	0.0340964
$951$	0	0
$952$	7.52260	0.243809
$953$	−43.7077	−1.41583	−0.707915	−	0.706298i	$-0.750364\pi$
$953$	−43.7077	−1.41583	−0.707915	+	0.706298i	$0.750364\pi$
$954$	0	0
$955$	−11.8046	−0.381987
$956$	−16.9984	−0.549767
$957$	0	0
$958$	43.5500	1.40704
$959$	−20.2905	−0.655215
$960$	0	0
$961$	60.1540	1.94045
$962$	−1.38267	−0.0445792
$963$	0	0
$964$	23.4059	0.753852
$965$	−2.84768	−0.0916700
$966$	0	0
$967$	−19.4995	−0.627063	−0.313532	−	0.949578i	$-0.601512\pi$
$967$	−19.4995	−0.627063	−0.313532	+	0.949578i	$0.601512\pi$
$968$	29.2462	0.940007
$969$	0	0
$970$	0.404012	0.0129720
$971$	45.8348	1.47091	0.735454	−	0.677575i	$-0.236969\pi$
$971$	45.8348	1.47091	0.735454	+	0.677575i	$0.236969\pi$
$972$	0	0
$973$	16.6299	0.533131
$974$	17.3139	0.554775
$975$	0	0
$976$	11.4084	0.365175
$977$	−44.6445	−1.42830	−0.714152	−	0.699990i	$-0.753188\pi$
$977$	−44.6445	−1.42830	−0.714152	+	0.699990i	$0.753188\pi$
$978$	0	0
$979$	−6.34399	−0.202755
$980$	−5.66149	−0.180850
$981$	0	0
$982$	−15.0081	−0.478929
$983$	54.2941	1.73171	0.865856	−	0.500293i	$-0.166774\pi$
$983$	54.2941	1.73171	0.865856	+	0.500293i	$0.166774\pi$
$984$	0	0
$985$	−22.0561	−0.702765
$986$	50.5048	1.60840
$987$	0	0
$988$	−5.50009	−0.174981
$989$	2.61911	0.0832827
$990$	0	0
$991$	−7.35886	−0.233762	−0.116881	−	0.993146i	$-0.537290\pi$
$991$	−7.35886	−0.233762	−0.116881	+	0.993146i	$0.537290\pi$
$992$	9.54746	0.303132
$993$	0	0
$994$	9.54648	0.302796
$995$	−9.06110	−0.287256
$996$	0	0
$997$	60.1173	1.90393	0.951967	−	0.306199i	$-0.0990573\pi$
$997$	60.1173	1.90393	0.951967	+	0.306199i	$0.0990573\pi$
$998$	−27.7202	−0.877469
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8010.2.a.bn.1.4		7
3.2	odd	2		2670.2.a.t.1.4	✓	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2670.2.a.t.1.4	✓	7	3.2	odd	2
8010.2.a.bn.1.4		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.15694 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.15694 q^{7} +1.00000 q^{8} +1.00000 q^{10} +6.34399 q^{11} -5.23358 q^{13} +1.15694 q^{14} +1.00000 q^{16} +6.50216 q^{17} +1.05092 q^{19} +1.00000 q^{20} +6.34399 q^{22} -0.797994 q^{23} +1.00000 q^{25} -5.23358 q^{26} +1.15694 q^{28} +7.76739 q^{29} +9.54746 q^{31} +1.00000 q^{32} +6.50216 q^{34} +1.15694 q^{35} +0.264192 q^{37} +1.05092 q^{38} +1.00000 q^{40} -0.200768 q^{41} -3.28211 q^{43} +6.34399 q^{44} -0.797994 q^{46} -5.08250 q^{47} -5.66149 q^{49} +1.00000 q^{50} -5.23358 q^{52} -14.2876 q^{53} +6.34399 q^{55} +1.15694 q^{56} +7.76739 q^{58} -11.5381 q^{59} +11.4084 q^{61} +9.54746 q^{62} +1.00000 q^{64} -5.23358 q^{65} -11.7854 q^{67} +6.50216 q^{68} +1.15694 q^{70} +8.25150 q^{71} -8.45709 q^{73} +0.264192 q^{74} +1.05092 q^{76} +7.33960 q^{77} +5.32834 q^{79} +1.00000 q^{80} -0.200768 q^{82} +0.830997 q^{83} +6.50216 q^{85} -3.28211 q^{86} +6.34399 q^{88} -1.00000 q^{89} -6.05493 q^{91} -0.797994 q^{92} -5.08250 q^{94} +1.05092 q^{95} +0.404012 q^{97} -5.66149 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + q^{7} + 7 q^{8}+O(q^{10}) \) 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

Related objects

Downloads

Learn more