LMFDB - Embedded newform 8470.2.a.db.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8470,2,Mod(1,8470)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8470, 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("8470.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8470.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:	$67.6332905120$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.4642000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 8x^{4} + 5x^{3} + 14x^{2} - 9x - 1 $ x^6 - x^5 - 8x^4 + 5x^3 + 14x^2 - 9x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.32077$ of defining polynomial
Character	$\chi$	$=$	8470.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.32077 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.32077 q^{6} -1.00000 q^{7} +1.00000 q^{8} -1.25558 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.32077 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.32077 q^{6} -1.00000 q^{7} +1.00000 q^{8} -1.25558 q^{9} +1.00000 q^{10} -1.32077 q^{12} +5.34554 q^{13} -1.00000 q^{14} -1.32077 q^{15} +1.00000 q^{16} -6.44843 q^{17} -1.25558 q^{18} -6.42752 q^{19} +1.00000 q^{20} +1.32077 q^{21} +5.51016 q^{23} -1.32077 q^{24} +1.00000 q^{25} +5.34554 q^{26} +5.62062 q^{27} -1.00000 q^{28} -8.19519 q^{29} -1.32077 q^{30} +1.77459 q^{31} +1.00000 q^{32} -6.44843 q^{34} -1.00000 q^{35} -1.25558 q^{36} +10.9821 q^{37} -6.42752 q^{38} -7.06020 q^{39} +1.00000 q^{40} +2.08350 q^{41} +1.32077 q^{42} -2.71912 q^{43} -1.25558 q^{45} +5.51016 q^{46} +4.35747 q^{47} -1.32077 q^{48} +1.00000 q^{49} +1.00000 q^{50} +8.51686 q^{51} +5.34554 q^{52} -8.59127 q^{53} +5.62062 q^{54} -1.00000 q^{56} +8.48925 q^{57} -8.19519 q^{58} -5.44657 q^{59} -1.32077 q^{60} +13.3987 q^{61} +1.77459 q^{62} +1.25558 q^{63} +1.00000 q^{64} +5.34554 q^{65} -10.8106 q^{67} -6.44843 q^{68} -7.27763 q^{69} -1.00000 q^{70} -11.8156 q^{71} -1.25558 q^{72} -5.54852 q^{73} +10.9821 q^{74} -1.32077 q^{75} -6.42752 q^{76} -7.06020 q^{78} -7.35859 q^{79} +1.00000 q^{80} -3.65680 q^{81} +2.08350 q^{82} -7.19204 q^{83} +1.32077 q^{84} -6.44843 q^{85} -2.71912 q^{86} +10.8239 q^{87} -5.39995 q^{89} -1.25558 q^{90} -5.34554 q^{91} +5.51016 q^{92} -2.34381 q^{93} +4.35747 q^{94} -6.42752 q^{95} -1.32077 q^{96} -0.548361 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} - q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 + 6 * q^8 - q^9	$ 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} - q^{9} + 6 q^{10} - q^{12} - 6 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} - 21 q^{17} - q^{18} + 3 q^{19} + 6 q^{20} + q^{21} - 10 q^{23} - q^{24} + 6 q^{25} - 6 q^{26} - 4 q^{27} - 6 q^{28} - 10 q^{29} - q^{30} - 4 q^{31} + 6 q^{32} - 21 q^{34} - 6 q^{35} - q^{36} - 2 q^{37} + 3 q^{38} - 26 q^{39} + 6 q^{40} - 7 q^{41} + q^{42} - 19 q^{43} - q^{45} - 10 q^{46} + 10 q^{47} - q^{48} + 6 q^{49} + 6 q^{50} + 4 q^{51} - 6 q^{52} - 16 q^{53} - 4 q^{54} - 6 q^{56} - 16 q^{57} - 10 q^{58} - 3 q^{59} - q^{60} + 8 q^{61} - 4 q^{62} + q^{63} + 6 q^{64} - 6 q^{65} - 27 q^{67} - 21 q^{68} + 4 q^{69} - 6 q^{70} + 4 q^{71} - q^{72} - 13 q^{73} - 2 q^{74} - q^{75} + 3 q^{76} - 26 q^{78} - 14 q^{79} + 6 q^{80} - 14 q^{81} - 7 q^{82} - 51 q^{83} + q^{84} - 21 q^{85} - 19 q^{86} - 8 q^{87} + q^{89} - q^{90} + 6 q^{91} - 10 q^{92} + 4 q^{93} + 10 q^{94} + 3 q^{95} - q^{96} + 7 q^{97} + 6 q^{98}+O(q^{100}) $ 6 * q + 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 + 6 * q^8 - q^9 + 6 * q^10 - q^12 - 6 * q^13 - 6 * q^14 - q^15 + 6 * q^16 - 21 * q^17 - q^18 + 3 * q^19 + 6 * q^20 + q^21 - 10 * q^23 - q^24 + 6 * q^25 - 6 * q^26 - 4 * q^27 - 6 * q^28 - 10 * q^29 - q^30 - 4 * q^31 + 6 * q^32 - 21 * q^34 - 6 * q^35 - q^36 - 2 * q^37 + 3 * q^38 - 26 * q^39 + 6 * q^40 - 7 * q^41 + q^42 - 19 * q^43 - q^45 - 10 * q^46 + 10 * q^47 - q^48 + 6 * q^49 + 6 * q^50 + 4 * q^51 - 6 * q^52 - 16 * q^53 - 4 * q^54 - 6 * q^56 - 16 * q^57 - 10 * q^58 - 3 * q^59 - q^60 + 8 * q^61 - 4 * q^62 + q^63 + 6 * q^64 - 6 * q^65 - 27 * q^67 - 21 * q^68 + 4 * q^69 - 6 * q^70 + 4 * q^71 - q^72 - 13 * q^73 - 2 * q^74 - q^75 + 3 * q^76 - 26 * q^78 - 14 * q^79 + 6 * q^80 - 14 * q^81 - 7 * q^82 - 51 * q^83 + q^84 - 21 * q^85 - 19 * q^86 - 8 * q^87 + q^89 - q^90 + 6 * q^91 - 10 * q^92 + 4 * q^93 + 10 * q^94 + 3 * q^95 - q^96 + 7 * q^97 + 6 * 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
$3$	−1.32077	−0.762545	−0.381272	−	0.924463i	$-0.624514\pi$
$3$	−1.32077	−0.762545	−0.381272	+	0.924463i	$0.624514\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.32077	−0.539201
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	−1.25558	−0.418526
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	−1.32077	−0.381272
$13$	5.34554	1.48259	0.741293	−	0.671182i	$-0.234213\pi$
$13$	5.34554	1.48259	0.741293	+	0.671182i	$0.234213\pi$
$14$	−1.00000	−0.267261
$15$	−1.32077	−0.341020
$16$	1.00000	0.250000
$17$	−6.44843	−1.56397	−0.781987	−	0.623295i	$-0.785794\pi$
$17$	−6.44843	−1.56397	−0.781987	+	0.623295i	$0.785794\pi$
$18$	−1.25558	−0.295942
$19$	−6.42752	−1.47457	−0.737287	−	0.675580i	$-0.763893\pi$
$19$	−6.42752	−1.47457	−0.737287	+	0.675580i	$0.763893\pi$
$20$	1.00000	0.223607
$21$	1.32077	0.288215
$22$	0	0
$23$	5.51016	1.14895	0.574474	−	0.818523i	$-0.305207\pi$
$23$	5.51016	1.14895	0.574474	+	0.818523i	$0.305207\pi$
$24$	−1.32077	−0.269600
$25$	1.00000	0.200000
$26$	5.34554	1.04835
$27$	5.62062	1.08169
$28$	−1.00000	−0.188982
$29$	−8.19519	−1.52181	−0.760904	−	0.648864i	$-0.775244\pi$
$29$	−8.19519	−1.52181	−0.760904	+	0.648864i	$0.775244\pi$
$30$	−1.32077	−0.241138
$31$	1.77459	0.318725	0.159363	−	0.987220i	$-0.449056\pi$
$31$	1.77459	0.318725	0.159363	+	0.987220i	$0.449056\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.44843	−1.10590
$35$	−1.00000	−0.169031
$36$	−1.25558	−0.209263
$37$	10.9821	1.80544	0.902722	−	0.430225i	$-0.141566\pi$
$37$	10.9821	1.80544	0.902722	+	0.430225i	$0.141566\pi$
$38$	−6.42752	−1.04268
$39$	−7.06020	−1.13054
$40$	1.00000	0.158114
$41$	2.08350	0.325389	0.162694	−	0.986677i	$-0.447982\pi$
$41$	2.08350	0.325389	0.162694	+	0.986677i	$0.447982\pi$
$42$	1.32077	0.203799
$43$	−2.71912	−0.414661	−0.207331	−	0.978271i	$-0.566478\pi$
$43$	−2.71912	−0.414661	−0.207331	+	0.978271i	$0.566478\pi$
$44$	0	0
$45$	−1.25558	−0.187170
$46$	5.51016	0.812428
$47$	4.35747	0.635603	0.317801	−	0.948157i	$-0.397056\pi$
$47$	4.35747	0.635603	0.317801	+	0.948157i	$0.397056\pi$
$48$	−1.32077	−0.190636
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	8.51686	1.19260
$52$	5.34554	0.741293
$53$	−8.59127	−1.18010	−0.590051	−	0.807366i	$-0.700892\pi$
$53$	−8.59127	−1.18010	−0.590051	+	0.807366i	$0.700892\pi$
$54$	5.62062	0.764870
$55$	0	0
$56$	−1.00000	−0.133631
$57$	8.48925	1.12443
$58$	−8.19519	−1.07608
$59$	−5.44657	−0.709083	−0.354541	−	0.935040i	$-0.615363\pi$
$59$	−5.44657	−0.709083	−0.354541	+	0.935040i	$0.615363\pi$
$60$	−1.32077	−0.170510
$61$	13.3987	1.71553	0.857766	−	0.514040i	$-0.171852\pi$
$61$	13.3987	1.71553	0.857766	+	0.514040i	$0.171852\pi$
$62$	1.77459	0.225373
$63$	1.25558	0.158188
$64$	1.00000	0.125000
$65$	5.34554	0.663032
$66$	0	0
$67$	−10.8106	−1.32073	−0.660363	−	0.750947i	$-0.729597\pi$
$67$	−10.8106	−1.32073	−0.660363	+	0.750947i	$0.729597\pi$
$68$	−6.44843	−0.781987
$69$	−7.27763	−0.876124
$70$	−1.00000	−0.119523
$71$	−11.8156	−1.40225	−0.701127	−	0.713036i	$-0.747319\pi$
$71$	−11.8156	−1.40225	−0.701127	+	0.713036i	$0.747319\pi$
$72$	−1.25558	−0.147971
$73$	−5.54852	−0.649405	−0.324702	−	0.945816i	$-0.605264\pi$
$73$	−5.54852	−0.649405	−0.324702	+	0.945816i	$0.605264\pi$
$74$	10.9821	1.27664
$75$	−1.32077	−0.152509
$76$	−6.42752	−0.737287
$77$	0	0
$78$	−7.06020	−0.799411
$79$	−7.35859	−0.827906	−0.413953	−	0.910298i	$-0.635852\pi$
$79$	−7.35859	−0.827906	−0.413953	+	0.910298i	$0.635852\pi$
$80$	1.00000	0.111803
$81$	−3.65680	−0.406311
$82$	2.08350	0.230085
$83$	−7.19204	−0.789429	−0.394714	−	0.918804i	$-0.629156\pi$
$83$	−7.19204	−0.789429	−0.394714	+	0.918804i	$0.629156\pi$
$84$	1.32077	0.144107
$85$	−6.44843	−0.699430
$86$	−2.71912	−0.293210
$87$	10.8239	1.16045
$88$	0	0
$89$	−5.39995	−0.572393	−0.286197	−	0.958171i	$-0.592391\pi$
$89$	−5.39995	−0.572393	−0.286197	+	0.958171i	$0.592391\pi$
$90$	−1.25558	−0.132349
$91$	−5.34554	−0.560364
$92$	5.51016	0.574474
$93$	−2.34381	−0.243042
$94$	4.35747	0.449439
$95$	−6.42752	−0.659449
$96$	−1.32077	−0.134800
$97$	−0.548361	−0.0556776	−0.0278388	−	0.999612i	$-0.508863\pi$
$97$	−0.548361	−0.0556776	−0.0278388	+	0.999612i	$0.508863\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	2.44229	0.243017	0.121509	−	0.992590i	$-0.461227\pi$
$101$	2.44229	0.243017	0.121509	+	0.992590i	$0.461227\pi$
$102$	8.51686	0.843295
$103$	0.630289	0.0621042	0.0310521	−	0.999518i	$-0.490114\pi$
$103$	0.630289	0.0621042	0.0310521	+	0.999518i	$0.490114\pi$
$104$	5.34554	0.524173
$105$	1.32077	0.128894
$106$	−8.59127	−0.834458
$107$	11.4352	1.10549	0.552743	−	0.833352i	$-0.313581\pi$
$107$	11.4352	1.10549	0.552743	+	0.833352i	$0.313581\pi$
$108$	5.62062	0.540845
$109$	−8.31582	−0.796512	−0.398256	−	0.917274i	$-0.630384\pi$
$109$	−8.31582	−0.796512	−0.398256	+	0.917274i	$0.630384\pi$
$110$	0	0
$111$	−14.5048	−1.37673
$112$	−1.00000	−0.0944911
$113$	−20.1474	−1.89531	−0.947656	−	0.319294i	$-0.896554\pi$
$113$	−20.1474	−1.89531	−0.947656	+	0.319294i	$0.896554\pi$
$114$	8.48925	0.795091
$115$	5.51016	0.513825
$116$	−8.19519	−0.760904
$117$	−6.71173	−0.620500
$118$	−5.44657	−0.501397
$119$	6.44843	0.591126
$120$	−1.32077	−0.120569
$121$	0	0
$122$	13.3987	1.21306
$123$	−2.75182	−0.248123
$124$	1.77459	0.159363
$125$	1.00000	0.0894427
$126$	1.25558	0.111856
$127$	−11.0010	−0.976180	−0.488090	−	0.872793i	$-0.662306\pi$
$127$	−11.0010	−0.976180	−0.488090	+	0.872793i	$0.662306\pi$
$128$	1.00000	0.0883883
$129$	3.59132	0.316198
$130$	5.34554	0.468835
$131$	−18.2978	−1.59868	−0.799342	−	0.600876i	$-0.794818\pi$
$131$	−18.2978	−1.59868	−0.799342	+	0.600876i	$0.794818\pi$
$132$	0	0
$133$	6.42752	0.557336
$134$	−10.8106	−0.933894
$135$	5.62062	0.483746
$136$	−6.44843	−0.552948
$137$	5.46086	0.466552	0.233276	−	0.972411i	$-0.425055\pi$
$137$	5.46086	0.466552	0.233276	+	0.972411i	$0.425055\pi$
$138$	−7.27763	−0.619513
$139$	−5.25491	−0.445716	−0.222858	−	0.974851i	$-0.571539\pi$
$139$	−5.25491	−0.445716	−0.222858	+	0.974851i	$0.571539\pi$
$140$	−1.00000	−0.0845154
$141$	−5.75520	−0.484675
$142$	−11.8156	−0.991543
$143$	0	0
$144$	−1.25558	−0.104631
$145$	−8.19519	−0.680574
$146$	−5.54852	−0.459199
$147$	−1.32077	−0.108935
$148$	10.9821	0.902722
$149$	9.21250	0.754718	0.377359	−	0.926067i	$-0.376832\pi$
$149$	9.21250	0.754718	0.377359	+	0.926067i	$0.376832\pi$
$150$	−1.32077	−0.107840
$151$	14.2773	1.16187	0.580936	−	0.813949i	$-0.302686\pi$
$151$	14.2773	1.16187	0.580936	+	0.813949i	$0.302686\pi$
$152$	−6.42752	−0.521340
$153$	8.09650	0.654563
$154$	0	0
$155$	1.77459	0.142538
$156$	−7.06020	−0.565269
$157$	14.9439	1.19265	0.596325	−	0.802743i	$-0.296627\pi$
$157$	14.9439	1.19265	0.596325	+	0.802743i	$0.296627\pi$
$158$	−7.35859	−0.585418
$159$	11.3471	0.899880
$160$	1.00000	0.0790569
$161$	−5.51016	−0.434261
$162$	−3.65680	−0.287305
$163$	16.7682	1.31339	0.656695	−	0.754157i	$-0.271954\pi$
$163$	16.7682	1.31339	0.656695	+	0.754157i	$0.271954\pi$
$164$	2.08350	0.162694
$165$	0	0
$166$	−7.19204	−0.558210
$167$	6.20304	0.480006	0.240003	−	0.970772i	$-0.422852\pi$
$167$	6.20304	0.480006	0.240003	+	0.970772i	$0.422852\pi$
$168$	1.32077	0.101899
$169$	15.5748	1.19806
$170$	−6.44843	−0.494572
$171$	8.07024	0.617147
$172$	−2.71912	−0.207331
$173$	−7.35006	−0.558815	−0.279407	−	0.960173i	$-0.590138\pi$
$173$	−7.35006	−0.558815	−0.279407	+	0.960173i	$0.590138\pi$
$174$	10.8239	0.820560
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	7.19364	0.540707
$178$	−5.39995	−0.404743
$179$	−4.40431	−0.329194	−0.164597	−	0.986361i	$-0.552632\pi$
$179$	−4.40431	−0.329194	−0.164597	+	0.986361i	$0.552632\pi$
$180$	−1.25558	−0.0935852
$181$	−11.0910	−0.824387	−0.412193	−	0.911096i	$-0.635237\pi$
$181$	−11.0910	−0.824387	−0.412193	+	0.911096i	$0.635237\pi$
$182$	−5.34554	−0.396238
$183$	−17.6966	−1.30817
$184$	5.51016	0.406214
$185$	10.9821	0.807419
$186$	−2.34381	−0.171857
$187$	0	0
$188$	4.35747	0.317801
$189$	−5.62062	−0.408840
$190$	−6.42752	−0.466301
$191$	−14.3562	−1.03878	−0.519388	−	0.854538i	$-0.673840\pi$
$191$	−14.3562	−1.03878	−0.519388	+	0.854538i	$0.673840\pi$
$192$	−1.32077	−0.0953181
$193$	8.48187	0.610538	0.305269	−	0.952266i	$-0.401254\pi$
$193$	8.48187	0.610538	0.305269	+	0.952266i	$0.401254\pi$
$194$	−0.548361	−0.0393700
$195$	−7.06020	−0.505592
$196$	1.00000	0.0714286
$197$	−4.97357	−0.354352	−0.177176	−	0.984179i	$-0.556696\pi$
$197$	−4.97357	−0.354352	−0.177176	+	0.984179i	$0.556696\pi$
$198$	0	0
$199$	−22.6336	−1.60445	−0.802227	−	0.597018i	$-0.796352\pi$
$199$	−22.6336	−1.60445	−0.802227	+	0.597018i	$0.796352\pi$
$200$	1.00000	0.0707107
$201$	14.2783	1.00711
$202$	2.44229	0.171839
$203$	8.19519	0.575190
$204$	8.51686	0.596300
$205$	2.08350	0.145518
$206$	0.630289	0.0439143
$207$	−6.91843	−0.480864
$208$	5.34554	0.370646
$209$	0	0
$210$	1.32077	0.0911415
$211$	−3.78425	−0.260519	−0.130259	−	0.991480i	$-0.541581\pi$
$211$	−3.78425	−0.260519	−0.130259	+	0.991480i	$0.541581\pi$
$212$	−8.59127	−0.590051
$213$	15.6056	1.06928
$214$	11.4352	0.781697
$215$	−2.71912	−0.185442
$216$	5.62062	0.382435
$217$	−1.77459	−0.120467
$218$	−8.31582	−0.563219
$219$	7.32830	0.495200
$220$	0	0
$221$	−34.4703	−2.31872
$222$	−14.5048	−0.973496
$223$	22.5140	1.50765	0.753824	−	0.657077i	$-0.228207\pi$
$223$	22.5140	1.50765	0.753824	+	0.657077i	$0.228207\pi$
$224$	−1.00000	−0.0668153
$225$	−1.25558	−0.0837051
$226$	−20.1474	−1.34019
$227$	−10.8997	−0.723439	−0.361719	−	0.932287i	$-0.617810\pi$
$227$	−10.8997	−0.723439	−0.361719	+	0.932287i	$0.617810\pi$
$228$	8.48925	0.562214
$229$	−3.17445	−0.209773	−0.104887	−	0.994484i	$-0.533448\pi$
$229$	−3.17445	−0.209773	−0.104887	+	0.994484i	$0.533448\pi$
$230$	5.51016	0.363329
$231$	0	0
$232$	−8.19519	−0.538041
$233$	−25.6323	−1.67923	−0.839614	−	0.543184i	$-0.817219\pi$
$233$	−25.6323	−1.67923	−0.839614	+	0.543184i	$0.817219\pi$
$234$	−6.71173	−0.438760
$235$	4.35747	0.284250
$236$	−5.44657	−0.354541
$237$	9.71897	0.631315
$238$	6.44843	0.417989
$239$	−17.5664	−1.13628	−0.568139	−	0.822932i	$-0.692337\pi$
$239$	−17.5664	−1.13628	−0.568139	+	0.822932i	$0.692337\pi$
$240$	−1.32077	−0.0852551
$241$	5.63187	0.362781	0.181390	−	0.983411i	$-0.441940\pi$
$241$	5.63187	0.362781	0.181390	+	0.983411i	$0.441940\pi$
$242$	0	0
$243$	−12.0321	−0.771859
$244$	13.3987	0.857766
$245$	1.00000	0.0638877
$246$	−2.75182	−0.175450
$247$	−34.3585	−2.18618
$248$	1.77459	0.112686
$249$	9.49900	0.601975
$250$	1.00000	0.0632456
$251$	14.3083	0.903130	0.451565	−	0.892238i	$-0.350866\pi$
$251$	14.3083	0.903130	0.451565	+	0.892238i	$0.350866\pi$
$252$	1.25558	0.0790939
$253$	0	0
$254$	−11.0010	−0.690263
$255$	8.51686	0.533347
$256$	1.00000	0.0625000
$257$	−7.69294	−0.479872	−0.239936	−	0.970789i	$-0.577126\pi$
$257$	−7.69294	−0.479872	−0.239936	+	0.970789i	$0.577126\pi$
$258$	3.59132	0.223586
$259$	−10.9821	−0.682393
$260$	5.34554	0.331516
$261$	10.2897	0.636916
$262$	−18.2978	−1.13044
$263$	−6.81613	−0.420300	−0.210150	−	0.977669i	$-0.567395\pi$
$263$	−6.81613	−0.420300	−0.210150	+	0.977669i	$0.567395\pi$
$264$	0	0
$265$	−8.59127	−0.527758
$266$	6.42752	0.394096
$267$	7.13207	0.436475
$268$	−10.8106	−0.660363
$269$	3.79822	0.231581	0.115791	−	0.993274i	$-0.463060\pi$
$269$	3.79822	0.231581	0.115791	+	0.993274i	$0.463060\pi$
$270$	5.62062	0.342060
$271$	2.35406	0.142999	0.0714996	−	0.997441i	$-0.477222\pi$
$271$	2.35406	0.142999	0.0714996	+	0.997441i	$0.477222\pi$
$272$	−6.44843	−0.390993
$273$	7.06020	0.427303
$274$	5.46086	0.329902
$275$	0	0
$276$	−7.27763	−0.438062
$277$	−17.4409	−1.04792	−0.523962	−	0.851742i	$-0.675547\pi$
$277$	−17.4409	−1.04792	−0.523962	+	0.851742i	$0.675547\pi$
$278$	−5.25491	−0.315169
$279$	−2.22813	−0.133395
$280$	−1.00000	−0.0597614
$281$	−22.6310	−1.35005	−0.675027	−	0.737793i	$-0.735868\pi$
$281$	−22.6310	−1.35005	−0.675027	+	0.737793i	$0.735868\pi$
$282$	−5.75520	−0.342717
$283$	−27.0596	−1.60852	−0.804262	−	0.594274i	$-0.797439\pi$
$283$	−27.0596	−1.60852	−0.804262	+	0.594274i	$0.797439\pi$
$284$	−11.8156	−0.701127
$285$	8.48925	0.502860
$286$	0	0
$287$	−2.08350	−0.122985
$288$	−1.25558	−0.0739856
$289$	24.5822	1.44601
$290$	−8.19519	−0.481238
$291$	0.724257	0.0424567
$292$	−5.54852	−0.324702
$293$	−20.1958	−1.17985	−0.589926	−	0.807458i	$-0.700843\pi$
$293$	−20.1958	−1.17985	−0.589926	+	0.807458i	$0.700843\pi$
$294$	−1.32077	−0.0770286
$295$	−5.44657	−0.317111
$296$	10.9821	0.638321
$297$	0	0
$298$	9.21250	0.533666
$299$	29.4547	1.70341
$300$	−1.32077	−0.0762545
$301$	2.71912	0.156727
$302$	14.2773	0.821568
$303$	−3.22570	−0.185311
$304$	−6.42752	−0.368643
$305$	13.3987	0.767209
$306$	8.09650	0.462846
$307$	24.4128	1.39331	0.696656	−	0.717405i	$-0.254670\pi$
$307$	24.4128	1.39331	0.696656	+	0.717405i	$0.254670\pi$
$308$	0	0
$309$	−0.832464	−0.0473572
$310$	1.77459	0.100790
$311$	25.2667	1.43274	0.716371	−	0.697720i	$-0.245802\pi$
$311$	25.2667	1.43274	0.716371	+	0.697720i	$0.245802\pi$
$312$	−7.06020	−0.399705
$313$	−18.1747	−1.02729	−0.513647	−	0.858002i	$-0.671706\pi$
$313$	−18.1747	−1.02729	−0.513647	+	0.858002i	$0.671706\pi$
$314$	14.9439	0.843331
$315$	1.25558	0.0707437
$316$	−7.35859	−0.413953
$317$	10.6739	0.599506	0.299753	−	0.954017i	$-0.403096\pi$
$317$	10.6739	0.599506	0.299753	+	0.954017i	$0.403096\pi$
$318$	11.3471	0.636312
$319$	0	0
$320$	1.00000	0.0559017
$321$	−15.1033	−0.842983
$322$	−5.51016	−0.307069
$323$	41.4474	2.30619
$324$	−3.65680	−0.203155
$325$	5.34554	0.296517
$326$	16.7682	0.928707
$327$	10.9833	0.607376
$328$	2.08350	0.115042
$329$	−4.35747	−0.240235
$330$	0	0
$331$	−9.75660	−0.536271	−0.268136	−	0.963381i	$-0.586408\pi$
$331$	−9.75660	−0.536271	−0.268136	+	0.963381i	$0.586408\pi$
$332$	−7.19204	−0.394714
$333$	−13.7888	−0.755624
$334$	6.20304	0.339415
$335$	−10.8106	−0.590646
$336$	1.32077	0.0720537
$337$	10.1462	0.552699	0.276350	−	0.961057i	$-0.410875\pi$
$337$	10.1462	0.552699	0.276350	+	0.961057i	$0.410875\pi$
$338$	15.5748	0.847155
$339$	26.6101	1.44526
$340$	−6.44843	−0.349715
$341$	0	0
$342$	8.07024	0.436389
$343$	−1.00000	−0.0539949
$344$	−2.71912	−0.146605
$345$	−7.27763	−0.391814
$346$	−7.35006	−0.395142
$347$	−21.1756	−1.13677	−0.568383	−	0.822764i	$-0.692431\pi$
$347$	−21.1756	−1.13677	−0.568383	+	0.822764i	$0.692431\pi$
$348$	10.8239	0.580224
$349$	−4.58943	−0.245667	−0.122833	−	0.992427i	$-0.539198\pi$
$349$	−4.58943	−0.245667	−0.122833	+	0.992427i	$0.539198\pi$
$350$	−1.00000	−0.0534522
$351$	30.0452	1.60370
$352$	0	0
$353$	32.2094	1.71433	0.857166	−	0.515040i	$-0.172223\pi$
$353$	32.2094	1.71433	0.857166	+	0.515040i	$0.172223\pi$
$354$	7.19364	0.382338
$355$	−11.8156	−0.627107
$356$	−5.39995	−0.286197
$357$	−8.51686	−0.450760
$358$	−4.40431	−0.232775
$359$	−6.54031	−0.345184	−0.172592	−	0.984993i	$-0.555214\pi$
$359$	−6.54031	−0.345184	−0.172592	+	0.984993i	$0.555214\pi$
$360$	−1.25558	−0.0661747
$361$	22.3130	1.17437
$362$	−11.0910	−0.582930
$363$	0	0
$364$	−5.34554	−0.280182
$365$	−5.54852	−0.290423
$366$	−17.6966	−0.925016
$367$	−2.98556	−0.155845	−0.0779225	−	0.996959i	$-0.524829\pi$
$367$	−2.98556	−0.155845	−0.0779225	+	0.996959i	$0.524829\pi$
$368$	5.51016	0.287237
$369$	−2.61600	−0.136183
$370$	10.9821	0.570931
$371$	8.59127	0.446037
$372$	−2.34381	−0.121521
$373$	−16.8471	−0.872311	−0.436155	−	0.899871i	$-0.643660\pi$
$373$	−16.8471	−0.872311	−0.436155	+	0.899871i	$0.643660\pi$
$374$	0	0
$375$	−1.32077	−0.0682041
$376$	4.35747	0.224719
$377$	−43.8077	−2.25621
$378$	−5.62062	−0.289094
$379$	−27.4781	−1.41145	−0.705727	−	0.708484i	$-0.749380\pi$
$379$	−27.4781	−1.41145	−0.705727	+	0.708484i	$0.749380\pi$
$380$	−6.42752	−0.329725
$381$	14.5297	0.744381
$382$	−14.3562	−0.734526
$383$	−31.3930	−1.60411	−0.802055	−	0.597251i	$-0.796260\pi$
$383$	−31.3930	−1.60411	−0.802055	+	0.597251i	$0.796260\pi$
$384$	−1.32077	−0.0674001
$385$	0	0
$386$	8.48187	0.431716
$387$	3.41406	0.173546
$388$	−0.548361	−0.0278388
$389$	−0.115199	−0.00584080	−0.00292040	−	0.999996i	$-0.500930\pi$
$389$	−0.115199	−0.00584080	−0.00292040	+	0.999996i	$0.500930\pi$
$390$	−7.06020	−0.357507
$391$	−35.5318	−1.79692
$392$	1.00000	0.0505076
$393$	24.1671	1.21907
$394$	−4.97357	−0.250565
$395$	−7.35859	−0.370251
$396$	0	0
$397$	−18.2744	−0.917165	−0.458583	−	0.888652i	$-0.651643\pi$
$397$	−18.2744	−0.917165	−0.458583	+	0.888652i	$0.651643\pi$
$398$	−22.6336	−1.13452
$399$	−8.48925	−0.424994
$400$	1.00000	0.0500000
$401$	30.0646	1.50136	0.750678	−	0.660668i	$-0.229727\pi$
$401$	30.0646	1.50136	0.750678	+	0.660668i	$0.229727\pi$
$402$	14.2783	0.712136
$403$	9.48612	0.472537
$404$	2.44229	0.121509
$405$	−3.65680	−0.181708
$406$	8.19519	0.406720
$407$	0	0
$408$	8.51686	0.421648
$409$	29.2637	1.44700	0.723499	−	0.690326i	$-0.242533\pi$
$409$	29.2637	1.44700	0.723499	+	0.690326i	$0.242533\pi$
$410$	2.08350	0.102897
$411$	−7.21252	−0.355767
$412$	0.630289	0.0310521
$413$	5.44657	0.268008
$414$	−6.91843	−0.340022
$415$	−7.19204	−0.353043
$416$	5.34554	0.262086
$417$	6.94051	0.339878
$418$	0	0
$419$	−17.3655	−0.848362	−0.424181	−	0.905577i	$-0.639438\pi$
$419$	−17.3655	−0.848362	−0.424181	+	0.905577i	$0.639438\pi$
$420$	1.32077	0.0644468
$421$	8.29303	0.404177	0.202089	−	0.979367i	$-0.435227\pi$
$421$	8.29303	0.404177	0.202089	+	0.979367i	$0.435227\pi$
$422$	−3.78425	−0.184214
$423$	−5.47114	−0.266016
$424$	−8.59127	−0.417229
$425$	−6.44843	−0.312795
$426$	15.6056	0.756096
$427$	−13.3987	−0.648410
$428$	11.4352	0.552743
$429$	0	0
$430$	−2.71912	−0.131127
$431$	13.3393	0.642530	0.321265	−	0.946989i	$-0.395892\pi$
$431$	13.3393	0.642530	0.321265	+	0.946989i	$0.395892\pi$
$432$	5.62062	0.270422
$433$	−14.8300	−0.712685	−0.356342	−	0.934355i	$-0.615976\pi$
$433$	−14.8300	−0.712685	−0.356342	+	0.934355i	$0.615976\pi$
$434$	−1.77459	−0.0851829
$435$	10.8239	0.518968
$436$	−8.31582	−0.398256
$437$	−35.4166	−1.69421
$438$	7.32830	0.350160
$439$	−6.08156	−0.290257	−0.145128	−	0.989413i	$-0.546360\pi$
$439$	−6.08156	−0.290257	−0.145128	+	0.989413i	$0.546360\pi$
$440$	0	0
$441$	−1.25558	−0.0597894
$442$	−34.4703	−1.63959
$443$	−19.8590	−0.943530	−0.471765	−	0.881724i	$-0.656383\pi$
$443$	−19.8590	−0.943530	−0.471765	+	0.881724i	$0.656383\pi$
$444$	−14.5048	−0.688366
$445$	−5.39995	−0.255982
$446$	22.5140	1.06607
$447$	−12.1676	−0.575506
$448$	−1.00000	−0.0472456
$449$	0.806539	0.0380629	0.0190315	−	0.999819i	$-0.493942\pi$
$449$	0.806539	0.0380629	0.0190315	+	0.999819i	$0.493942\pi$
$450$	−1.25558	−0.0591885
$451$	0	0
$452$	−20.1474	−0.947656
$453$	−18.8570	−0.885980
$454$	−10.8997	−0.511548
$455$	−5.34554	−0.250603
$456$	8.48925	0.397545
$457$	−29.9349	−1.40030	−0.700149	−	0.713997i	$-0.746883\pi$
$457$	−29.9349	−1.40030	−0.700149	+	0.713997i	$0.746883\pi$
$458$	−3.17445	−0.148332
$459$	−36.2442	−1.69173
$460$	5.51016	0.256912
$461$	33.4831	1.55946	0.779732	−	0.626114i	$-0.215355\pi$
$461$	33.4831	1.55946	0.779732	+	0.626114i	$0.215355\pi$
$462$	0	0
$463$	−29.7511	−1.38265	−0.691324	−	0.722545i	$-0.742972\pi$
$463$	−29.7511	−1.38265	−0.691324	+	0.722545i	$0.742972\pi$
$464$	−8.19519	−0.380452
$465$	−2.34381	−0.108692
$466$	−25.6323	−1.18739
$467$	−6.65053	−0.307750	−0.153875	−	0.988090i	$-0.549175\pi$
$467$	−6.65053	−0.307750	−0.153875	+	0.988090i	$0.549175\pi$
$468$	−6.71173	−0.310250
$469$	10.8106	0.499187
$470$	4.35747	0.200995
$471$	−19.7373	−0.909449
$472$	−5.44657	−0.250699
$473$	0	0
$474$	9.71897	0.446407
$475$	−6.42752	−0.294915
$476$	6.44843	0.295563
$477$	10.7870	0.493903
$478$	−17.5664	−0.803470
$479$	−17.9277	−0.819137	−0.409569	−	0.912279i	$-0.634321\pi$
$479$	−17.9277	−0.819137	−0.409569	+	0.912279i	$0.634321\pi$
$480$	−1.32077	−0.0602845
$481$	58.7051	2.67672
$482$	5.63187	0.256525
$483$	7.27763	0.331144
$484$	0	0
$485$	−0.548361	−0.0248998
$486$	−12.0321	−0.545787
$487$	37.2239	1.68677	0.843387	−	0.537307i	$-0.180558\pi$
$487$	37.2239	1.68677	0.843387	+	0.537307i	$0.180558\pi$
$488$	13.3987	0.606532
$489$	−22.1469	−1.00152
$490$	1.00000	0.0451754
$491$	−21.7165	−0.980051	−0.490026	−	0.871708i	$-0.663013\pi$
$491$	−21.7165	−0.980051	−0.490026	+	0.871708i	$0.663013\pi$
$492$	−2.75182	−0.124062
$493$	52.8461	2.38007
$494$	−34.3585	−1.54586
$495$	0	0
$496$	1.77459	0.0796813
$497$	11.8156	0.530002
$498$	9.49900	0.425660
$499$	−15.7823	−0.706515	−0.353257	−	0.935526i	$-0.614926\pi$
$499$	−15.7823	−0.706515	−0.353257	+	0.935526i	$0.614926\pi$
$500$	1.00000	0.0447214
$501$	−8.19277	−0.366026
$502$	14.3083	0.638610
$503$	−8.96024	−0.399517	−0.199759	−	0.979845i	$-0.564016\pi$
$503$	−8.96024	−0.399517	−0.199759	+	0.979845i	$0.564016\pi$
$504$	1.25558	0.0559278
$505$	2.44229	0.108681
$506$	0	0
$507$	−20.5706	−0.913573
$508$	−11.0010	−0.488090
$509$	43.5620	1.93085	0.965426	−	0.260678i	$-0.0839461\pi$
$509$	43.5620	1.93085	0.965426	+	0.260678i	$0.0839461\pi$
$510$	8.51686	0.377133
$511$	5.54852	0.245452
$512$	1.00000	0.0441942
$513$	−36.1266	−1.59503
$514$	−7.69294	−0.339321
$515$	0.630289	0.0277738
$516$	3.59132	0.158099
$517$	0	0
$518$	−10.9821	−0.482525
$519$	9.70771	0.426121
$520$	5.34554	0.234417
$521$	0.872633	0.0382307	0.0191154	−	0.999817i	$-0.493915\pi$
$521$	0.872633	0.0382307	0.0191154	+	0.999817i	$0.493915\pi$
$522$	10.2897	0.450368
$523$	−22.4315	−0.980860	−0.490430	−	0.871481i	$-0.663160\pi$
$523$	−22.4315	−0.980860	−0.490430	+	0.871481i	$0.663160\pi$
$524$	−18.2978	−0.799342
$525$	1.32077	0.0576430
$526$	−6.81613	−0.297197
$527$	−11.4433	−0.498478
$528$	0	0
$529$	7.36183	0.320080
$530$	−8.59127	−0.373181
$531$	6.83858	0.296769
$532$	6.42752	0.278668
$533$	11.1374	0.482416
$534$	7.13207	0.308635
$535$	11.4352	0.494389
$536$	−10.8106	−0.466947
$537$	5.81707	0.251025
$538$	3.79822	0.163753
$539$	0	0
$540$	5.62062	0.241873
$541$	−10.0432	−0.431793	−0.215896	−	0.976416i	$-0.569267\pi$
$541$	−10.0432	−0.431793	−0.215896	+	0.976416i	$0.569267\pi$
$542$	2.35406	0.101116
$543$	14.6486	0.628632
$544$	−6.44843	−0.276474
$545$	−8.31582	−0.356211
$546$	7.06020	0.302149
$547$	−15.3645	−0.656937	−0.328469	−	0.944515i	$-0.606533\pi$
$547$	−15.3645	−0.656937	−0.328469	+	0.944515i	$0.606533\pi$
$548$	5.46086	0.233276
$549$	−16.8231	−0.717994
$550$	0	0
$551$	52.6747	2.24402
$552$	−7.27763	−0.309756
$553$	7.35859	0.312919
$554$	−17.4409	−0.740994
$555$	−14.5048	−0.615693
$556$	−5.25491	−0.222858
$557$	−45.8170	−1.94133	−0.970665	−	0.240437i	$-0.922709\pi$
$557$	−45.8170	−1.94133	−0.970665	+	0.240437i	$0.922709\pi$
$558$	−2.22813	−0.0943243
$559$	−14.5351	−0.614770
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−22.6310	−0.954632
$563$	−17.1178	−0.721429	−0.360714	−	0.932676i	$-0.617467\pi$
$563$	−17.1178	−0.721429	−0.360714	+	0.932676i	$0.617467\pi$
$564$	−5.75520	−0.242338
$565$	−20.1474	−0.847609
$566$	−27.0596	−1.13740
$567$	3.65680	0.153571
$568$	−11.8156	−0.495772
$569$	18.0666	0.757389	0.378695	−	0.925522i	$-0.376373\pi$
$569$	18.0666	0.757389	0.378695	+	0.925522i	$0.376373\pi$
$570$	8.48925	0.355575
$571$	13.8533	0.579743	0.289872	−	0.957066i	$-0.406387\pi$
$571$	13.8533	0.579743	0.289872	+	0.957066i	$0.406387\pi$
$572$	0	0
$573$	18.9612	0.792114
$574$	−2.08350	−0.0869638
$575$	5.51016	0.229789
$576$	−1.25558	−0.0523157
$577$	39.1469	1.62971	0.814854	−	0.579667i	$-0.196817\pi$
$577$	39.1469	1.62971	0.814854	+	0.579667i	$0.196817\pi$
$578$	24.5822	1.02249
$579$	−11.2026	−0.465563
$580$	−8.19519	−0.340287
$581$	7.19204	0.298376
$582$	0.724257	0.0300214
$583$	0	0
$584$	−5.54852	−0.229599
$585$	−6.71173	−0.277496
$586$	−20.1958	−0.834281
$587$	−25.3734	−1.04727	−0.523637	−	0.851942i	$-0.675425\pi$
$587$	−25.3734	−1.04727	−0.523637	+	0.851942i	$0.675425\pi$
$588$	−1.32077	−0.0544675
$589$	−11.4062	−0.469984
$590$	−5.44657	−0.224232
$591$	6.56892	0.270209
$592$	10.9821	0.451361
$593$	24.5601	1.00856	0.504280	−	0.863540i	$-0.331758\pi$
$593$	24.5601	1.00856	0.504280	+	0.863540i	$0.331758\pi$
$594$	0	0
$595$	6.44843	0.264360
$596$	9.21250	0.377359
$597$	29.8937	1.22347
$598$	29.4547	1.20449
$599$	−35.4719	−1.44934	−0.724672	−	0.689094i	$-0.758009\pi$
$599$	−35.4719	−1.44934	−0.724672	+	0.689094i	$0.758009\pi$
$600$	−1.32077	−0.0539201
$601$	−14.7392	−0.601223	−0.300611	−	0.953747i	$-0.597191\pi$
$601$	−14.7392	−0.601223	−0.300611	+	0.953747i	$0.597191\pi$
$602$	2.71912	0.110823
$603$	13.5735	0.552757
$604$	14.2773	0.580936
$605$	0	0
$606$	−3.22570	−0.131035
$607$	6.09924	0.247560	0.123780	−	0.992310i	$-0.460498\pi$
$607$	6.09924	0.247560	0.123780	+	0.992310i	$0.460498\pi$
$608$	−6.42752	−0.260670
$609$	−10.8239	−0.438608
$610$	13.3987	0.542499
$611$	23.2930	0.942335
$612$	8.09650	0.327281
$613$	18.0889	0.730604	0.365302	−	0.930889i	$-0.380966\pi$
$613$	18.0889	0.730604	0.365302	+	0.930889i	$0.380966\pi$
$614$	24.4128	0.985221
$615$	−2.75182	−0.110964
$616$	0	0
$617$	15.9814	0.643386	0.321693	−	0.946844i	$-0.395748\pi$
$617$	15.9814	0.643386	0.321693	+	0.946844i	$0.395748\pi$
$618$	−0.832464	−0.0334866
$619$	5.20453	0.209188	0.104594	−	0.994515i	$-0.466646\pi$
$619$	5.20453	0.209188	0.104594	+	0.994515i	$0.466646\pi$
$620$	1.77459	0.0712691
$621$	30.9705	1.24280
$622$	25.2667	1.01310
$623$	5.39995	0.216344
$624$	−7.06020	−0.282634
$625$	1.00000	0.0400000
$626$	−18.1747	−0.726407
$627$	0	0
$628$	14.9439	0.596325
$629$	−70.8172	−2.82367
$630$	1.25558	0.0500234
$631$	−34.2712	−1.36432	−0.682158	−	0.731204i	$-0.738959\pi$
$631$	−34.2712	−1.36432	−0.682158	+	0.731204i	$0.738959\pi$
$632$	−7.35859	−0.292709
$633$	4.99811	0.198657
$634$	10.6739	0.423915
$635$	−11.0010	−0.436561
$636$	11.3471	0.449940
$637$	5.34554	0.211798
$638$	0	0
$639$	14.8354	0.586879
$640$	1.00000	0.0395285
$641$	38.3894	1.51629	0.758145	−	0.652086i	$-0.226106\pi$
$641$	38.3894	1.51629	0.758145	+	0.652086i	$0.226106\pi$
$642$	−15.1033	−0.596079
$643$	4.86445	0.191835	0.0959177	−	0.995389i	$-0.469421\pi$
$643$	4.86445	0.191835	0.0959177	+	0.995389i	$0.469421\pi$
$644$	−5.51016	−0.217131
$645$	3.59132	0.141408
$646$	41.4474	1.63073
$647$	13.6415	0.536303	0.268151	−	0.963377i	$-0.413587\pi$
$647$	13.6415	0.536303	0.268151	+	0.963377i	$0.413587\pi$
$648$	−3.65680	−0.143653
$649$	0	0
$650$	5.34554	0.209669
$651$	2.34381	0.0918613
$652$	16.7682	0.656695
$653$	21.3235	0.834453	0.417226	−	0.908803i	$-0.363002\pi$
$653$	21.3235	0.834453	0.417226	+	0.908803i	$0.363002\pi$
$654$	10.9833	0.429479
$655$	−18.2978	−0.714953
$656$	2.08350	0.0813472
$657$	6.96659	0.271793
$658$	−4.35747	−0.169872
$659$	−12.6150	−0.491409	−0.245704	−	0.969345i	$-0.579019\pi$
$659$	−12.6150	−0.491409	−0.245704	+	0.969345i	$0.579019\pi$
$660$	0	0
$661$	32.2901	1.25594	0.627969	−	0.778238i	$-0.283886\pi$
$661$	32.2901	1.25594	0.627969	+	0.778238i	$0.283886\pi$
$662$	−9.75660	−0.379201
$663$	45.5272	1.76813
$664$	−7.19204	−0.279105
$665$	6.42752	0.249248
$666$	−13.7888	−0.534307
$667$	−45.1568	−1.74848
$668$	6.20304	0.240003
$669$	−29.7357	−1.14965
$670$	−10.8106	−0.417650
$671$	0	0
$672$	1.32077	0.0509497
$673$	7.77486	0.299699	0.149849	−	0.988709i	$-0.452121\pi$
$673$	7.77486	0.299699	0.149849	+	0.988709i	$0.452121\pi$
$674$	10.1462	0.390817
$675$	5.62062	0.216338
$676$	15.5748	0.599029
$677$	11.1286	0.427708	0.213854	−	0.976866i	$-0.431398\pi$
$677$	11.1286	0.427708	0.213854	+	0.976866i	$0.431398\pi$
$678$	26.6101	1.02195
$679$	0.548361	0.0210442
$680$	−6.44843	−0.247286
$681$	14.3960	0.551654
$682$	0	0
$683$	−6.91240	−0.264496	−0.132248	−	0.991217i	$-0.542219\pi$
$683$	−6.91240	−0.264496	−0.132248	+	0.991217i	$0.542219\pi$
$684$	8.07024	0.308573
$685$	5.46086	0.208649
$686$	−1.00000	−0.0381802
$687$	4.19270	0.159962
$688$	−2.71912	−0.103665
$689$	−45.9250	−1.74960
$690$	−7.27763	−0.277055
$691$	31.3199	1.19146	0.595732	−	0.803183i	$-0.296862\pi$
$691$	31.3199	1.19146	0.595732	+	0.803183i	$0.296862\pi$
$692$	−7.35006	−0.279407
$693$	0	0
$694$	−21.1756	−0.803815
$695$	−5.25491	−0.199330
$696$	10.8239	0.410280
$697$	−13.4353	−0.508899
$698$	−4.58943	−0.173713
$699$	33.8543	1.28049
$700$	−1.00000	−0.0377964
$701$	−12.1542	−0.459058	−0.229529	−	0.973302i	$-0.573719\pi$
$701$	−12.1542	−0.459058	−0.229529	+	0.973302i	$0.573719\pi$
$702$	30.0452	1.13398
$703$	−70.5875	−2.66226
$704$	0	0
$705$	−5.75520	−0.216753
$706$	32.2094	1.21222
$707$	−2.44229	−0.0918518
$708$	7.19364	0.270354
$709$	−32.3970	−1.21669	−0.608347	−	0.793671i	$-0.708167\pi$
$709$	−32.3970	−1.21669	−0.608347	+	0.793671i	$0.708167\pi$
$710$	−11.8156	−0.443432
$711$	9.23927	0.346500
$712$	−5.39995	−0.202372
$713$	9.77825	0.366199
$714$	−8.51686	−0.318736
$715$	0	0
$716$	−4.40431	−0.164597
$717$	23.2012	0.866463
$718$	−6.54031	−0.244082
$719$	12.3255	0.459663	0.229832	−	0.973230i	$-0.426182\pi$
$719$	12.3255	0.459663	0.229832	+	0.973230i	$0.426182\pi$
$720$	−1.25558	−0.0467926
$721$	−0.630289	−0.0234732
$722$	22.3130	0.830403
$723$	−7.43839	−0.276637
$724$	−11.0910	−0.412193
$725$	−8.19519	−0.304362
$726$	0	0
$727$	2.42914	0.0900918	0.0450459	−	0.998985i	$-0.485657\pi$
$727$	2.42914	0.0900918	0.0450459	+	0.998985i	$0.485657\pi$
$728$	−5.34554	−0.198119
$729$	26.8620	0.994888
$730$	−5.54852	−0.205360
$731$	17.5340	0.648519
$732$	−17.6966	−0.654085
$733$	46.1729	1.70543	0.852717	−	0.522373i	$-0.174953\pi$
$733$	46.1729	1.70543	0.852717	+	0.522373i	$0.174953\pi$
$734$	−2.98556	−0.110199
$735$	−1.32077	−0.0487172
$736$	5.51016	0.203107
$737$	0	0
$738$	−2.61600	−0.0962963
$739$	48.8700	1.79771	0.898856	−	0.438244i	$-0.144399\pi$
$739$	48.8700	1.79771	0.898856	+	0.438244i	$0.144399\pi$
$740$	10.9821	0.403709
$741$	45.3796	1.66706
$742$	8.59127	0.315395
$743$	19.7114	0.723141	0.361571	−	0.932345i	$-0.382241\pi$
$743$	19.7114	0.723141	0.361571	+	0.932345i	$0.382241\pi$
$744$	−2.34381	−0.0859284
$745$	9.21250	0.337520
$746$	−16.8471	−0.616817
$747$	9.03015	0.330396
$748$	0	0
$749$	−11.4352	−0.417835
$750$	−1.32077	−0.0482276
$751$	−52.1667	−1.90359	−0.951795	−	0.306735i	$-0.900763\pi$
$751$	−52.1667	−1.90359	−0.951795	+	0.306735i	$0.900763\pi$
$752$	4.35747	0.158901
$753$	−18.8979	−0.688677
$754$	−43.8077	−1.59538
$755$	14.2773	0.519605
$756$	−5.62062	−0.204420
$757$	−32.7016	−1.18856	−0.594280	−	0.804258i	$-0.702563\pi$
$757$	−32.7016	−1.18856	−0.594280	+	0.804258i	$0.702563\pi$
$758$	−27.4781	−0.998049
$759$	0	0
$760$	−6.42752	−0.233151
$761$	24.3007	0.880899	0.440450	−	0.897777i	$-0.354819\pi$
$761$	24.3007	0.880899	0.440450	+	0.897777i	$0.354819\pi$
$762$	14.5297	0.526357
$763$	8.31582	0.301053
$764$	−14.3562	−0.519388
$765$	8.09650	0.292729
$766$	−31.3930	−1.13428
$767$	−29.1148	−1.05128
$768$	−1.32077	−0.0476590
$769$	34.3945	1.24030	0.620148	−	0.784485i	$-0.287073\pi$
$769$	34.3945	1.24030	0.620148	+	0.784485i	$0.287073\pi$
$770$	0	0
$771$	10.1606	0.365924
$772$	8.48187	0.305269
$773$	11.7248	0.421712	0.210856	−	0.977517i	$-0.432375\pi$
$773$	11.7248	0.421712	0.210856	+	0.977517i	$0.432375\pi$
$774$	3.41406	0.122716
$775$	1.77459	0.0637451
$776$	−0.548361	−0.0196850
$777$	14.5048	0.520355
$778$	−0.115199	−0.00413007
$779$	−13.3918	−0.479810
$780$	−7.06020	−0.252796
$781$	0	0
$782$	−35.5318	−1.27062
$783$	−46.0621	−1.64612
$784$	1.00000	0.0357143
$785$	14.9439	0.533369
$786$	24.1671	0.862011
$787$	11.1529	0.397558	0.198779	−	0.980044i	$-0.436302\pi$
$787$	11.1529	0.397558	0.198779	+	0.980044i	$0.436302\pi$
$788$	−4.97357	−0.177176
$789$	9.00251	0.320498
$790$	−7.35859	−0.261807
$791$	20.1474	0.716360
$792$	0	0
$793$	71.6234	2.54342
$794$	−18.2744	−0.648534
$795$	11.3471	0.402439
$796$	−22.6336	−0.802227
$797$	−14.0990	−0.499411	−0.249706	−	0.968322i	$-0.580334\pi$
$797$	−14.0990	−0.499411	−0.249706	+	0.968322i	$0.580334\pi$
$798$	−8.48925	−0.300516
$799$	−28.0988	−0.994065
$800$	1.00000	0.0353553
$801$	6.78005	0.239561
$802$	30.0646	1.06162
$803$	0	0
$804$	14.2783	0.503556
$805$	−5.51016	−0.194208
$806$	9.48612	0.334134
$807$	−5.01655	−0.176591
$808$	2.44229	0.0859195
$809$	44.7742	1.57418	0.787088	−	0.616840i	$-0.211588\pi$
$809$	44.7742	1.57418	0.787088	+	0.616840i	$0.211588\pi$
$810$	−3.65680	−0.128487
$811$	33.4692	1.17526	0.587631	−	0.809129i	$-0.300061\pi$
$811$	33.4692	1.17526	0.587631	+	0.809129i	$0.300061\pi$
$812$	8.19519	0.287595
$813$	−3.10917	−0.109043
$814$	0	0
$815$	16.7682	0.587366
$816$	8.51686	0.298150
$817$	17.4772	0.611448
$818$	29.2637	1.02318
$819$	6.71173	0.234527
$820$	2.08350	0.0727591
$821$	−44.5883	−1.55614	−0.778072	−	0.628175i	$-0.783802\pi$
$821$	−44.5883	−1.55614	−0.778072	+	0.628175i	$0.783802\pi$
$822$	−7.21252	−0.251565
$823$	−41.0470	−1.43081	−0.715405	−	0.698710i	$-0.753758\pi$
$823$	−41.0470	−1.43081	−0.715405	+	0.698710i	$0.753758\pi$
$824$	0.630289	0.0219571
$825$	0	0
$826$	5.44657	0.189510
$827$	34.2851	1.19221	0.596105	−	0.802907i	$-0.296714\pi$
$827$	34.2851	1.19221	0.596105	+	0.802907i	$0.296714\pi$
$828$	−6.91843	−0.240432
$829$	15.5238	0.539162	0.269581	−	0.962978i	$-0.413115\pi$
$829$	15.5238	0.539162	0.269581	+	0.962978i	$0.413115\pi$
$830$	−7.19204	−0.249639
$831$	23.0354	0.799089
$832$	5.34554	0.185323
$833$	−6.44843	−0.223425
$834$	6.94051	0.240330
$835$	6.20304	0.214665
$836$	0	0
$837$	9.97428	0.344762
$838$	−17.3655	−0.599883
$839$	12.0295	0.415304	0.207652	−	0.978203i	$-0.433418\pi$
$839$	12.0295	0.415304	0.207652	+	0.978203i	$0.433418\pi$
$840$	1.32077	0.0455708
$841$	38.1611	1.31590
$842$	8.29303	0.285797
$843$	29.8903	1.02948
$844$	−3.78425	−0.130259
$845$	15.5748	0.535788
$846$	−5.47114	−0.188102
$847$	0	0
$848$	−8.59127	−0.295025
$849$	35.7394	1.22657
$850$	−6.44843	−0.221179
$851$	60.5130	2.07436
$852$	15.6056	0.534641
$853$	−35.5225	−1.21627	−0.608134	−	0.793834i	$-0.708082\pi$
$853$	−35.5225	−1.21627	−0.608134	+	0.793834i	$0.708082\pi$
$854$	−13.3987	−0.458495
$855$	8.07024	0.275996
$856$	11.4352	0.390849
$857$	−48.4891	−1.65636	−0.828178	−	0.560465i	$-0.810622\pi$
$857$	−48.4891	−1.65636	−0.828178	+	0.560465i	$0.810622\pi$
$858$	0	0
$859$	−1.93679	−0.0660823	−0.0330411	−	0.999454i	$-0.510519\pi$
$859$	−1.93679	−0.0660823	−0.0330411	+	0.999454i	$0.510519\pi$
$860$	−2.71912	−0.0927211
$861$	2.75182	0.0937818
$862$	13.3393	0.454338
$863$	21.2807	0.724405	0.362202	−	0.932099i	$-0.382025\pi$
$863$	21.2807	0.724405	0.362202	+	0.932099i	$0.382025\pi$
$864$	5.62062	0.191217
$865$	−7.35006	−0.249910
$866$	−14.8300	−0.503944
$867$	−32.4674	−1.10265
$868$	−1.77459	−0.0602334
$869$	0	0
$870$	10.8239	0.366966
$871$	−57.7885	−1.95809
$872$	−8.31582	−0.281609
$873$	0.688510	0.0233025
$874$	−35.4166	−1.19799
$875$	−1.00000	−0.0338062
$876$	7.32830	0.247600
$877$	28.4478	0.960614	0.480307	−	0.877100i	$-0.340525\pi$
$877$	28.4478	0.960614	0.480307	+	0.877100i	$0.340525\pi$
$878$	−6.08156	−0.205243
$879$	26.6739	0.899689
$880$	0	0
$881$	−19.2969	−0.650127	−0.325064	−	0.945692i	$-0.605386\pi$
$881$	−19.2969	−0.650127	−0.325064	+	0.945692i	$0.605386\pi$
$882$	−1.25558	−0.0422775
$883$	0.696897	0.0234524	0.0117262	−	0.999931i	$-0.496267\pi$
$883$	0.696897	0.0234524	0.0117262	+	0.999931i	$0.496267\pi$
$884$	−34.4703	−1.15936
$885$	7.19364	0.241812
$886$	−19.8590	−0.667176
$887$	10.1179	0.339727	0.169864	−	0.985468i	$-0.445667\pi$
$887$	10.1179	0.339727	0.169864	+	0.985468i	$0.445667\pi$
$888$	−14.5048	−0.486748
$889$	11.0010	0.368961
$890$	−5.39995	−0.181007
$891$	0	0
$892$	22.5140	0.753824
$893$	−28.0077	−0.937243
$894$	−12.1676	−0.406944
$895$	−4.40431	−0.147220
$896$	−1.00000	−0.0334077
$897$	−38.9028	−1.29893
$898$	0.806539	0.0269145
$899$	−14.5431	−0.485039
$900$	−1.25558	−0.0418526
$901$	55.4002	1.84565
$902$	0	0
$903$	−3.59132	−0.119511
$904$	−20.1474	−0.670094
$905$	−11.0910	−0.368677
$906$	−18.8570	−0.626482
$907$	−26.1814	−0.869341	−0.434670	−	0.900590i	$-0.643135\pi$
$907$	−26.1814	−0.869341	−0.434670	+	0.900590i	$0.643135\pi$
$908$	−10.8997	−0.361719
$909$	−3.06648	−0.101709
$910$	−5.34554	−0.177203
$911$	−34.0567	−1.12835	−0.564175	−	0.825655i	$-0.690806\pi$
$911$	−34.0567	−1.12835	−0.564175	+	0.825655i	$0.690806\pi$
$912$	8.48925	0.281107
$913$	0	0
$914$	−29.9349	−0.990160
$915$	−17.6966	−0.585031
$916$	−3.17445	−0.104887
$917$	18.2978	0.604246
$918$	−36.2442	−1.19624
$919$	37.6890	1.24325	0.621623	−	0.783316i	$-0.286473\pi$
$919$	37.6890	1.24325	0.621623	+	0.783316i	$0.286473\pi$
$920$	5.51016	0.181664
$921$	−32.2436	−1.06246
$922$	33.4831	1.10271
$923$	−63.1607	−2.07896
$924$	0	0
$925$	10.9821	0.361089
$926$	−29.7511	−0.977680
$927$	−0.791376	−0.0259922
$928$	−8.19519	−0.269020
$929$	24.5323	0.804879	0.402440	−	0.915446i	$-0.368162\pi$
$929$	24.5323	0.804879	0.402440	+	0.915446i	$0.368162\pi$
$930$	−2.34381	−0.0768567
$931$	−6.42752	−0.210653
$932$	−25.6323	−0.839614
$933$	−33.3714	−1.09253
$934$	−6.65053	−0.217612
$935$	0	0
$936$	−6.71173	−0.219380
$937$	30.8744	1.00862	0.504311	−	0.863522i	$-0.331746\pi$
$937$	30.8744	1.00862	0.504311	+	0.863522i	$0.331746\pi$
$938$	10.8106	0.352979
$939$	24.0045	0.783358
$940$	4.35747	0.142125
$941$	−46.4060	−1.51279	−0.756396	−	0.654114i	$-0.773042\pi$
$941$	−46.4060	−1.51279	−0.756396	+	0.654114i	$0.773042\pi$
$942$	−19.7373	−0.643078
$943$	11.4804	0.373854
$944$	−5.44657	−0.177271
$945$	−5.62062	−0.182839
$946$	0	0
$947$	8.46285	0.275006	0.137503	−	0.990501i	$-0.456092\pi$
$947$	8.46285	0.275006	0.137503	+	0.990501i	$0.456092\pi$
$948$	9.71897	0.315658
$949$	−29.6598	−0.962798
$950$	−6.42752	−0.208536
$951$	−14.0977	−0.457150
$952$	6.44843	0.208995
$953$	21.1245	0.684289	0.342145	−	0.939647i	$-0.388847\pi$
$953$	21.1245	0.684289	0.342145	+	0.939647i	$0.388847\pi$
$954$	10.7870	0.349242
$955$	−14.3562	−0.464555
$956$	−17.5664	−0.568139
$957$	0	0
$958$	−17.9277	−0.579217
$959$	−5.46086	−0.176340
$960$	−1.32077	−0.0426275
$961$	−27.8508	−0.898414
$962$	58.7051	1.89273
$963$	−14.3578	−0.462674
$964$	5.63187	0.181390
$965$	8.48187	0.273041
$966$	7.27763	0.234154
$967$	−5.05793	−0.162652	−0.0813260	−	0.996688i	$-0.525915\pi$
$967$	−5.05793	−0.162652	−0.0813260	+	0.996688i	$0.525915\pi$
$968$	0	0
$969$	−54.7423	−1.75858
$970$	−0.548361	−0.0176068
$971$	14.8010	0.474988	0.237494	−	0.971389i	$-0.423674\pi$
$971$	14.8010	0.474988	0.237494	+	0.971389i	$0.423674\pi$
$972$	−12.0321	−0.385930
$973$	5.25491	0.168465
$974$	37.2239	1.19273
$975$	−7.06020	−0.226107
$976$	13.3987	0.428883
$977$	−7.38146	−0.236154	−0.118077	−	0.993004i	$-0.537673\pi$
$977$	−7.38146	−0.236154	−0.118077	+	0.993004i	$0.537673\pi$
$978$	−22.1469	−0.708180
$979$	0	0
$980$	1.00000	0.0319438
$981$	10.4412	0.333360
$982$	−21.7165	−0.693001
$983$	1.45695	0.0464694	0.0232347	−	0.999730i	$-0.492603\pi$
$983$	1.45695	0.0464694	0.0232347	+	0.999730i	$0.492603\pi$
$984$	−2.75182	−0.0877249
$985$	−4.97357	−0.158471
$986$	52.8461	1.68296
$987$	5.75520	0.183190
$988$	−34.3585	−1.09309
$989$	−14.9828	−0.476424
$990$	0	0
$991$	34.1561	1.08501	0.542503	−	0.840054i	$-0.317477\pi$
$991$	34.1561	1.08501	0.542503	+	0.840054i	$0.317477\pi$
$992$	1.77459	0.0563432
$993$	12.8862	0.408931
$994$	11.8156	0.374768
$995$	−22.6336	−0.717534
$996$	9.49900	0.300987
$997$	−56.9485	−1.80358	−0.901789	−	0.432177i	$-0.857746\pi$
$997$	−56.9485	−1.80358	−0.901789	+	0.432177i	$0.857746\pi$
$998$	−15.7823	−0.499582
$999$	61.7261	1.95293

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.db.1.2		6
11.3	even	5		770.2.n.g.141.3	yes	12
11.4	even	5		770.2.n.g.71.3	✓	12
11.10	odd	2		8470.2.a.cv.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.g.71.3	✓	12	11.4	even	5
770.2.n.g.141.3	yes	12	11.3	even	5
8470.2.a.cv.1.2		6	11.10	odd	2
8470.2.a.db.1.2		6	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:	\( N \)	\(=\)	\( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8470.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.32077 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.32077 q^{6} -1.00000 q^{7} +1.00000 q^{8} -1.25558 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.32077 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.32077 q^{6} -1.00000 q^{7} +1.00000 q^{8} -1.25558 q^{9} +1.00000 q^{10} -1.32077 q^{12} +5.34554 q^{13} -1.00000 q^{14} -1.32077 q^{15} +1.00000 q^{16} -6.44843 q^{17} -1.25558 q^{18} -6.42752 q^{19} +1.00000 q^{20} +1.32077 q^{21} +5.51016 q^{23} -1.32077 q^{24} +1.00000 q^{25} +5.34554 q^{26} +5.62062 q^{27} -1.00000 q^{28} -8.19519 q^{29} -1.32077 q^{30} +1.77459 q^{31} +1.00000 q^{32} -6.44843 q^{34} -1.00000 q^{35} -1.25558 q^{36} +10.9821 q^{37} -6.42752 q^{38} -7.06020 q^{39} +1.00000 q^{40} +2.08350 q^{41} +1.32077 q^{42} -2.71912 q^{43} -1.25558 q^{45} +5.51016 q^{46} +4.35747 q^{47} -1.32077 q^{48} +1.00000 q^{49} +1.00000 q^{50} +8.51686 q^{51} +5.34554 q^{52} -8.59127 q^{53} +5.62062 q^{54} -1.00000 q^{56} +8.48925 q^{57} -8.19519 q^{58} -5.44657 q^{59} -1.32077 q^{60} +13.3987 q^{61} +1.77459 q^{62} +1.25558 q^{63} +1.00000 q^{64} +5.34554 q^{65} -10.8106 q^{67} -6.44843 q^{68} -7.27763 q^{69} -1.00000 q^{70} -11.8156 q^{71} -1.25558 q^{72} -5.54852 q^{73} +10.9821 q^{74} -1.32077 q^{75} -6.42752 q^{76} -7.06020 q^{78} -7.35859 q^{79} +1.00000 q^{80} -3.65680 q^{81} +2.08350 q^{82} -7.19204 q^{83} +1.32077 q^{84} -6.44843 q^{85} -2.71912 q^{86} +10.8239 q^{87} -5.39995 q^{89} -1.25558 q^{90} -5.34554 q^{91} +5.51016 q^{92} -2.34381 q^{93} +4.35747 q^{94} -6.42752 q^{95} -1.32077 q^{96} -0.548361 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} - q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 + 6 * q^8 - q^9	\( 6 q + 6 q^{2} - q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} + 6 q^{8} - q^{9} + 6 q^{10} - q^{12} - 6 q^{13} - 6 q^{14} - q^{15} + 6 q^{16} - 21 q^{17} - q^{18} + 3 q^{19} + 6 q^{20} + q^{21} - 10 q^{23} - q^{24} + 6 q^{25} - 6 q^{26} - 4 q^{27} - 6 q^{28} - 10 q^{29} - q^{30} - 4 q^{31} + 6 q^{32} - 21 q^{34} - 6 q^{35} - q^{36} - 2 q^{37} + 3 q^{38} - 26 q^{39} + 6 q^{40} - 7 q^{41} + q^{42} - 19 q^{43} - q^{45} - 10 q^{46} + 10 q^{47} - q^{48} + 6 q^{49} + 6 q^{50} + 4 q^{51} - 6 q^{52} - 16 q^{53} - 4 q^{54} - 6 q^{56} - 16 q^{57} - 10 q^{58} - 3 q^{59} - q^{60} + 8 q^{61} - 4 q^{62} + q^{63} + 6 q^{64} - 6 q^{65} - 27 q^{67} - 21 q^{68} + 4 q^{69} - 6 q^{70} + 4 q^{71} - q^{72} - 13 q^{73} - 2 q^{74} - q^{75} + 3 q^{76} - 26 q^{78} - 14 q^{79} + 6 q^{80} - 14 q^{81} - 7 q^{82} - 51 q^{83} + q^{84} - 21 q^{85} - 19 q^{86} - 8 q^{87} + q^{89} - q^{90} + 6 q^{91} - 10 q^{92} + 4 q^{93} + 10 q^{94} + 3 q^{95} - q^{96} + 7 q^{97} + 6 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 - q^3 + 6 * q^4 + 6 * q^5 - q^6 - 6 * q^7 + 6 * q^8 - q^9 + 6 * q^10 - q^12 - 6 * q^13 - 6 * q^14 - q^15 + 6 * q^16 - 21 * q^17 - q^18 + 3 * q^19 + 6 * q^20 + q^21 - 10 * q^23 - q^24 + 6 * q^25 - 6 * q^26 - 4 * q^27 - 6 * q^28 - 10 * q^29 - q^30 - 4 * q^31 + 6 * q^32 - 21 * q^34 - 6 * q^35 - q^36 - 2 * q^37 + 3 * q^38 - 26 * q^39 + 6 * q^40 - 7 * q^41 + q^42 - 19 * q^43 - q^45 - 10 * q^46 + 10 * q^47 - q^48 + 6 * q^49 + 6 * q^50 + 4 * q^51 - 6 * q^52 - 16 * q^53 - 4 * q^54 - 6 * q^56 - 16 * q^57 - 10 * q^58 - 3 * q^59 - q^60 + 8 * q^61 - 4 * q^62 + q^63 + 6 * q^64 - 6 * q^65 - 27 * q^67 - 21 * q^68 + 4 * q^69 - 6 * q^70 + 4 * q^71 - q^72 - 13 * q^73 - 2 * q^74 - q^75 + 3 * q^76 - 26 * q^78 - 14 * q^79 + 6 * q^80 - 14 * q^81 - 7 * q^82 - 51 * q^83 + q^84 - 21 * q^85 - 19 * q^86 - 8 * q^87 + q^89 - q^90 + 6 * q^91 - 10 * q^92 + 4 * q^93 + 10 * q^94 + 3 * q^95 - q^96 + 7 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more