LMFDB - Embedded newform 8470.2.a.de.1.1

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:	$0$
Dimension:	$6$
Coefficient field:	6.6.13298000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 3x^{3} + 26x^{2} + 13x - 1 $ x^6 - x^5 - 10x^4 + 3x^3 + 26x^2 + 13x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
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			1.1
Root			$-2.27063$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -2.80837 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.80837 q^{6} +1.00000 q^{7} +1.00000 q^{8} +4.88695 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -2.80837 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.80837 q^{6} +1.00000 q^{7} +1.00000 q^{8} +4.88695 q^{9} +1.00000 q^{10} -2.80837 q^{12} -3.05764 q^{13} +1.00000 q^{14} -2.80837 q^{15} +1.00000 q^{16} +6.94459 q^{17} +4.88695 q^{18} -4.69704 q^{19} +1.00000 q^{20} -2.80837 q^{21} -1.73118 q^{23} -2.80837 q^{24} +1.00000 q^{25} -3.05764 q^{26} -5.29926 q^{27} +1.00000 q^{28} -6.74554 q^{29} -2.80837 q^{30} +3.34334 q^{31} +1.00000 q^{32} +6.94459 q^{34} +1.00000 q^{35} +4.88695 q^{36} -7.32736 q^{37} -4.69704 q^{38} +8.58700 q^{39} +1.00000 q^{40} +7.58977 q^{41} -2.80837 q^{42} +7.27592 q^{43} +4.88695 q^{45} -1.73118 q^{46} +4.00555 q^{47} -2.80837 q^{48} +1.00000 q^{49} +1.00000 q^{50} -19.5030 q^{51} -3.05764 q^{52} -1.70116 q^{53} -5.29926 q^{54} +1.00000 q^{56} +13.1910 q^{57} -6.74554 q^{58} +7.64741 q^{59} -2.80837 q^{60} +8.93050 q^{61} +3.34334 q^{62} +4.88695 q^{63} +1.00000 q^{64} -3.05764 q^{65} +10.5718 q^{67} +6.94459 q^{68} +4.86179 q^{69} +1.00000 q^{70} -2.30414 q^{71} +4.88695 q^{72} -13.1113 q^{73} -7.32736 q^{74} -2.80837 q^{75} -4.69704 q^{76} +8.58700 q^{78} +4.25829 q^{79} +1.00000 q^{80} +0.221440 q^{81} +7.58977 q^{82} +3.86073 q^{83} -2.80837 q^{84} +6.94459 q^{85} +7.27592 q^{86} +18.9440 q^{87} -12.3140 q^{89} +4.88695 q^{90} -3.05764 q^{91} -1.73118 q^{92} -9.38933 q^{93} +4.00555 q^{94} -4.69704 q^{95} -2.80837 q^{96} -6.94282 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} + 6 q^{8} + 15 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 + 15 * q^9	$ 6 q + 6 q^{2} + q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} + 6 q^{8} + 15 q^{9} + 6 q^{10} + q^{12} + 2 q^{13} + 6 q^{14} + q^{15} + 6 q^{16} + 7 q^{17} + 15 q^{18} + 11 q^{19} + 6 q^{20} + q^{21} - 6 q^{23} + q^{24} + 6 q^{25} + 2 q^{26} + 4 q^{27} + 6 q^{28} + 2 q^{29} + q^{30} + 6 q^{32} + 7 q^{34} + 6 q^{35} + 15 q^{36} - 14 q^{37} + 11 q^{38} + 20 q^{39} + 6 q^{40} + 13 q^{41} + q^{42} + 19 q^{43} + 15 q^{45} - 6 q^{46} + 22 q^{47} + q^{48} + 6 q^{49} + 6 q^{50} - 14 q^{51} + 2 q^{52} - 10 q^{53} + 4 q^{54} + 6 q^{56} + 32 q^{57} + 2 q^{58} - 7 q^{59} + q^{60} + 22 q^{61} + 15 q^{63} + 6 q^{64} + 2 q^{65} + 5 q^{67} + 7 q^{68} - 36 q^{69} + 6 q^{70} + 8 q^{71} + 15 q^{72} + 13 q^{73} - 14 q^{74} + q^{75} + 11 q^{76} + 20 q^{78} - 16 q^{79} + 6 q^{80} + 18 q^{81} + 13 q^{82} - 5 q^{83} + q^{84} + 7 q^{85} + 19 q^{86} + 14 q^{87} + q^{89} + 15 q^{90} + 2 q^{91} - 6 q^{92} - 42 q^{93} + 22 q^{94} + 11 q^{95} + q^{96} - 3 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 + 15 * q^9 + 6 * q^10 + q^12 + 2 * q^13 + 6 * q^14 + q^15 + 6 * q^16 + 7 * q^17 + 15 * q^18 + 11 * q^19 + 6 * q^20 + q^21 - 6 * q^23 + q^24 + 6 * q^25 + 2 * q^26 + 4 * q^27 + 6 * q^28 + 2 * q^29 + q^30 + 6 * q^32 + 7 * q^34 + 6 * q^35 + 15 * q^36 - 14 * q^37 + 11 * q^38 + 20 * q^39 + 6 * q^40 + 13 * q^41 + q^42 + 19 * q^43 + 15 * q^45 - 6 * q^46 + 22 * q^47 + q^48 + 6 * q^49 + 6 * q^50 - 14 * q^51 + 2 * q^52 - 10 * q^53 + 4 * q^54 + 6 * q^56 + 32 * q^57 + 2 * q^58 - 7 * q^59 + q^60 + 22 * q^61 + 15 * q^63 + 6 * q^64 + 2 * q^65 + 5 * q^67 + 7 * q^68 - 36 * q^69 + 6 * q^70 + 8 * q^71 + 15 * q^72 + 13 * q^73 - 14 * q^74 + q^75 + 11 * q^76 + 20 * q^78 - 16 * q^79 + 6 * q^80 + 18 * q^81 + 13 * q^82 - 5 * q^83 + q^84 + 7 * q^85 + 19 * q^86 + 14 * q^87 + q^89 + 15 * q^90 + 2 * q^91 - 6 * q^92 - 42 * q^93 + 22 * q^94 + 11 * q^95 + q^96 - 3 * 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
$3$	−2.80837	−1.62141	−0.810707	−	0.585452i	$-0.800917\pi$
$3$	−2.80837	−1.62141	−0.810707	+	0.585452i	$0.800917\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.80837	−1.14651
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	4.88695	1.62898
$10$	1.00000	0.316228
$11$	0	0
$11$	0	0
$12$	−2.80837	−0.810707
$13$	−3.05764	−0.848037	−0.424019	−	0.905653i	$-0.639381\pi$
$13$	−3.05764	−0.848037	−0.424019	+	0.905653i	$0.639381\pi$
$14$	1.00000	0.267261
$15$	−2.80837	−0.725118
$16$	1.00000	0.250000
$17$	6.94459	1.68431	0.842156	−	0.539234i	$-0.181286\pi$
$17$	6.94459	1.68431	0.842156	+	0.539234i	$0.181286\pi$
$18$	4.88695	1.15187
$19$	−4.69704	−1.07757	−0.538787	−	0.842442i	$-0.681117\pi$
$19$	−4.69704	−1.07757	−0.538787	+	0.842442i	$0.681117\pi$
$20$	1.00000	0.223607
$21$	−2.80837	−0.612837
$22$	0	0
$23$	−1.73118	−0.360976	−0.180488	−	0.983577i	$-0.557768\pi$
$23$	−1.73118	−0.360976	−0.180488	+	0.983577i	$0.557768\pi$
$24$	−2.80837	−0.573256
$25$	1.00000	0.200000
$26$	−3.05764	−0.599653
$27$	−5.29926	−1.01984
$28$	1.00000	0.188982
$29$	−6.74554	−1.25262	−0.626308	−	0.779576i	$-0.715435\pi$
$29$	−6.74554	−1.25262	−0.626308	+	0.779576i	$0.715435\pi$
$30$	−2.80837	−0.512736
$31$	3.34334	0.600481	0.300240	−	0.953864i	$-0.402933\pi$
$31$	3.34334	0.600481	0.300240	+	0.953864i	$0.402933\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.94459	1.19099
$35$	1.00000	0.169031
$36$	4.88695	0.814492
$37$	−7.32736	−1.20461	−0.602305	−	0.798266i	$-0.705751\pi$
$37$	−7.32736	−1.20461	−0.602305	+	0.798266i	$0.705751\pi$
$38$	−4.69704	−0.761960
$39$	8.58700	1.37502
$40$	1.00000	0.158114
$41$	7.58977	1.18532	0.592662	−	0.805452i	$-0.298077\pi$
$41$	7.58977	1.18532	0.592662	+	0.805452i	$0.298077\pi$
$42$	−2.80837	−0.433341
$43$	7.27592	1.10957	0.554784	−	0.831995i	$-0.312801\pi$
$43$	7.27592	1.10957	0.554784	+	0.831995i	$0.312801\pi$
$44$	0	0
$45$	4.88695	0.728504
$46$	−1.73118	−0.255248
$47$	4.00555	0.584270	0.292135	−	0.956377i	$-0.405634\pi$
$47$	4.00555	0.584270	0.292135	+	0.956377i	$0.405634\pi$
$48$	−2.80837	−0.405354
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	−19.5030	−2.73097
$52$	−3.05764	−0.424019
$53$	−1.70116	−0.233673	−0.116836	−	0.993151i	$-0.537275\pi$
$53$	−1.70116	−0.233673	−0.116836	+	0.993151i	$0.537275\pi$
$54$	−5.29926	−0.721138
$55$	0	0
$56$	1.00000	0.133631
$57$	13.1910	1.74719
$58$	−6.74554	−0.885733
$59$	7.64741	0.995608	0.497804	−	0.867289i	$-0.334140\pi$
$59$	7.64741	0.995608	0.497804	+	0.867289i	$0.334140\pi$
$60$	−2.80837	−0.362559
$61$	8.93050	1.14343	0.571717	−	0.820451i	$-0.306278\pi$
$61$	8.93050	1.14343	0.571717	+	0.820451i	$0.306278\pi$
$62$	3.34334	0.424604
$63$	4.88695	0.615698
$64$	1.00000	0.125000
$65$	−3.05764	−0.379254
$66$	0	0
$67$	10.5718	1.29155	0.645773	−	0.763529i	$-0.276535\pi$
$67$	10.5718	1.29155	0.645773	+	0.763529i	$0.276535\pi$
$68$	6.94459	0.842156
$69$	4.86179	0.585291
$70$	1.00000	0.119523
$71$	−2.30414	−0.273451	−0.136725	−	0.990609i	$-0.543658\pi$
$71$	−2.30414	−0.273451	−0.136725	+	0.990609i	$0.543658\pi$
$72$	4.88695	0.575933
$73$	−13.1113	−1.53456	−0.767278	−	0.641314i	$-0.778389\pi$
$73$	−13.1113	−1.53456	−0.767278	+	0.641314i	$0.778389\pi$
$74$	−7.32736	−0.851788
$75$	−2.80837	−0.324283
$76$	−4.69704	−0.538787
$77$	0	0
$78$	8.58700	0.972286
$79$	4.25829	0.479095	0.239547	−	0.970885i	$-0.423001\pi$
$79$	4.25829	0.479095	0.239547	+	0.970885i	$0.423001\pi$
$80$	1.00000	0.111803
$81$	0.221440	0.0246044
$82$	7.58977	0.838150
$83$	3.86073	0.423770	0.211885	−	0.977295i	$-0.432040\pi$
$83$	3.86073	0.423770	0.211885	+	0.977295i	$0.432040\pi$
$84$	−2.80837	−0.306418
$85$	6.94459	0.753247
$86$	7.27592	0.784583
$87$	18.9440	2.03101
$88$	0	0
$89$	−12.3140	−1.30529	−0.652643	−	0.757665i	$-0.726340\pi$
$89$	−12.3140	−1.30529	−0.652643	+	0.757665i	$0.726340\pi$
$90$	4.88695	0.515130
$91$	−3.05764	−0.320528
$92$	−1.73118	−0.180488
$93$	−9.38933	−0.973628
$94$	4.00555	0.413141
$95$	−4.69704	−0.481906
$96$	−2.80837	−0.286628
$97$	−6.94282	−0.704936	−0.352468	−	0.935824i	$-0.614658\pi$
$97$	−6.94282	−0.704936	−0.352468	+	0.935824i	$0.614658\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	13.5915	1.35240	0.676202	−	0.736717i	$-0.263625\pi$
$101$	13.5915	1.35240	0.676202	+	0.736717i	$0.263625\pi$
$102$	−19.5030	−1.93108
$103$	15.8591	1.56264	0.781320	−	0.624131i	$-0.214547\pi$
$103$	15.8591	1.56264	0.781320	+	0.624131i	$0.214547\pi$
$104$	−3.05764	−0.299826
$105$	−2.80837	−0.274069
$106$	−1.70116	−0.165232
$107$	−4.39685	−0.425060	−0.212530	−	0.977155i	$-0.568170\pi$
$107$	−4.39685	−0.425060	−0.212530	+	0.977155i	$0.568170\pi$
$108$	−5.29926	−0.509922
$109$	−14.4931	−1.38819	−0.694095	−	0.719884i	$-0.744195\pi$
$109$	−14.4931	−1.38819	−0.694095	+	0.719884i	$0.744195\pi$
$110$	0	0
$111$	20.5779	1.95317
$112$	1.00000	0.0944911
$113$	−19.3326	−1.81866	−0.909328	−	0.416081i	$-0.863403\pi$
$113$	−19.3326	−1.81866	−0.909328	+	0.416081i	$0.863403\pi$
$114$	13.1910	1.23545
$115$	−1.73118	−0.161433
$116$	−6.74554	−0.626308
$117$	−14.9425	−1.38144
$118$	7.64741	0.704001
$119$	6.94459	0.636610
$120$	−2.80837	−0.256368
$121$	0	0
$122$	8.93050	0.808529
$123$	−21.3149	−1.92190
$124$	3.34334	0.300240
$125$	1.00000	0.0894427
$126$	4.88695	0.435364
$127$	21.5848	1.91534	0.957669	−	0.287870i	$-0.0929472\pi$
$127$	21.5848	1.91534	0.957669	+	0.287870i	$0.0929472\pi$
$128$	1.00000	0.0883883
$129$	−20.4335	−1.79907
$130$	−3.05764	−0.268173
$131$	5.23468	0.457356	0.228678	−	0.973502i	$-0.426560\pi$
$131$	5.23468	0.457356	0.228678	+	0.973502i	$0.426560\pi$
$132$	0	0
$133$	−4.69704	−0.407285
$134$	10.5718	0.913261
$135$	−5.29926	−0.456088
$136$	6.94459	0.595494
$137$	5.00039	0.427212	0.213606	−	0.976920i	$-0.431479\pi$
$137$	5.00039	0.427212	0.213606	+	0.976920i	$0.431479\pi$
$138$	4.86179	0.413863
$139$	−3.08024	−0.261262	−0.130631	−	0.991431i	$-0.541700\pi$
$139$	−3.08024	−0.261262	−0.130631	+	0.991431i	$0.541700\pi$
$140$	1.00000	0.0845154
$141$	−11.2491	−0.947343
$142$	−2.30414	−0.193359
$143$	0	0
$144$	4.88695	0.407246
$145$	−6.74554	−0.560187
$146$	−13.1113	−1.08510
$147$	−2.80837	−0.231631
$148$	−7.32736	−0.602305
$149$	−15.8858	−1.30141	−0.650706	−	0.759330i	$-0.725527\pi$
$149$	−15.8858	−1.30141	−0.650706	+	0.759330i	$0.725527\pi$
$150$	−2.80837	−0.229303
$151$	−5.45916	−0.444260	−0.222130	−	0.975017i	$-0.571301\pi$
$151$	−5.45916	−0.444260	−0.222130	+	0.975017i	$0.571301\pi$
$152$	−4.69704	−0.380980
$153$	33.9379	2.74372
$154$	0	0
$155$	3.34334	0.268543
$156$	8.58700	0.687510
$157$	8.44070	0.673641	0.336820	−	0.941569i	$-0.390648\pi$
$157$	8.44070	0.673641	0.336820	+	0.941569i	$0.390648\pi$
$158$	4.25829	0.338771
$159$	4.77750	0.378880
$160$	1.00000	0.0790569
$161$	−1.73118	−0.136436
$162$	0.221440	0.0173980
$163$	9.74355	0.763174	0.381587	−	0.924333i	$-0.375378\pi$
$163$	9.74355	0.763174	0.381587	+	0.924333i	$0.375378\pi$
$164$	7.58977	0.592662
$165$	0	0
$166$	3.86073	0.299650
$167$	20.2081	1.56375	0.781873	−	0.623437i	$-0.214264\pi$
$167$	20.2081	1.56375	0.781873	+	0.623437i	$0.214264\pi$
$168$	−2.80837	−0.216671
$169$	−3.65082	−0.280833
$170$	6.94459	0.532626
$171$	−22.9542	−1.75535
$172$	7.27592	0.554784
$173$	1.06417	0.0809073	0.0404537	−	0.999181i	$-0.487120\pi$
$173$	1.06417	0.0809073	0.0404537	+	0.999181i	$0.487120\pi$
$174$	18.9440	1.43614
$175$	1.00000	0.0755929
$176$	0	0
$177$	−21.4768	−1.61429
$178$	−12.3140	−0.922977
$179$	−5.73780	−0.428863	−0.214432	−	0.976739i	$-0.568790\pi$
$179$	−5.73780	−0.428863	−0.214432	+	0.976739i	$0.568790\pi$
$180$	4.88695	0.364252
$181$	3.77619	0.280682	0.140341	−	0.990103i	$-0.455180\pi$
$181$	3.77619	0.280682	0.140341	+	0.990103i	$0.455180\pi$
$182$	−3.05764	−0.226648
$183$	−25.0802	−1.85398
$184$	−1.73118	−0.127624
$185$	−7.32736	−0.538718
$186$	−9.38933	−0.688459
$187$	0	0
$188$	4.00555	0.292135
$189$	−5.29926	−0.385465
$190$	−4.69704	−0.340759
$191$	11.7298	0.848736	0.424368	−	0.905490i	$-0.360496\pi$
$191$	11.7298	0.848736	0.424368	+	0.905490i	$0.360496\pi$
$192$	−2.80837	−0.202677
$193$	18.9879	1.36678	0.683391	−	0.730053i	$-0.260505\pi$
$193$	18.9879	1.36678	0.683391	+	0.730053i	$0.260505\pi$
$194$	−6.94282	−0.498465
$195$	8.58700	0.614928
$196$	1.00000	0.0714286
$197$	−3.27508	−0.233340	−0.116670	−	0.993171i	$-0.537222\pi$
$197$	−3.27508	−0.233340	−0.116670	+	0.993171i	$0.537222\pi$
$198$	0	0
$199$	14.7438	1.04516	0.522580	−	0.852590i	$-0.324970\pi$
$199$	14.7438	1.04516	0.522580	+	0.852590i	$0.324970\pi$
$200$	1.00000	0.0707107
$201$	−29.6894	−2.09413
$202$	13.5915	0.956294
$203$	−6.74554	−0.473444
$204$	−19.5030	−1.36548
$205$	7.58977	0.530093
$206$	15.8591	1.10495
$207$	−8.46019	−0.588024
$208$	−3.05764	−0.212009
$209$	0	0
$210$	−2.80837	−0.193796
$211$	−13.6709	−0.941143	−0.470572	−	0.882362i	$-0.655952\pi$
$211$	−13.6709	−0.941143	−0.470572	+	0.882362i	$0.655952\pi$
$212$	−1.70116	−0.116836
$213$	6.47087	0.443377
$214$	−4.39685	−0.300563
$215$	7.27592	0.496214
$216$	−5.29926	−0.360569
$217$	3.34334	0.226960
$218$	−14.4931	−0.981598
$219$	36.8213	2.48815
$220$	0	0
$221$	−21.2341	−1.42836
$222$	20.5779	1.38110
$223$	−5.23577	−0.350613	−0.175307	−	0.984514i	$-0.556092\pi$
$223$	−5.23577	−0.350613	−0.175307	+	0.984514i	$0.556092\pi$
$224$	1.00000	0.0668153
$225$	4.88695	0.325797
$226$	−19.3326	−1.28598
$227$	−22.5392	−1.49598	−0.747991	−	0.663709i	$-0.768981\pi$
$227$	−22.5392	−1.49598	−0.747991	+	0.663709i	$0.768981\pi$
$228$	13.1910	0.873597
$229$	1.54850	0.102328	0.0511639	−	0.998690i	$-0.483707\pi$
$229$	1.54850	0.102328	0.0511639	+	0.998690i	$0.483707\pi$
$230$	−1.73118	−0.114151
$231$	0	0
$232$	−6.74554	−0.442867
$233$	−6.59831	−0.432270	−0.216135	−	0.976364i	$-0.569345\pi$
$233$	−6.59831	−0.432270	−0.216135	+	0.976364i	$0.569345\pi$
$234$	−14.9425	−0.976825
$235$	4.00555	0.261293
$236$	7.64741	0.497804
$237$	−11.9589	−0.776811
$238$	6.94459	0.450151
$239$	9.32864	0.603420	0.301710	−	0.953400i	$-0.402443\pi$
$239$	9.32864	0.603420	0.301710	+	0.953400i	$0.402443\pi$
$240$	−2.80837	−0.181280
$241$	−4.69741	−0.302587	−0.151293	−	0.988489i	$-0.548344\pi$
$241$	−4.69741	−0.302587	−0.151293	+	0.988489i	$0.548344\pi$
$242$	0	0
$243$	15.2759	0.979949
$244$	8.93050	0.571717
$245$	1.00000	0.0638877
$246$	−21.3149	−1.35899
$247$	14.3619	0.913823
$248$	3.34334	0.212302
$249$	−10.8424	−0.687106
$250$	1.00000	0.0632456
$251$	25.1086	1.58484	0.792420	−	0.609976i	$-0.208821\pi$
$251$	25.1086	1.58484	0.792420	+	0.609976i	$0.208821\pi$
$252$	4.88695	0.307849
$253$	0	0
$254$	21.5848	1.35435
$255$	−19.5030	−1.22133
$256$	1.00000	0.0625000
$257$	22.2671	1.38898	0.694491	−	0.719502i	$-0.255630\pi$
$257$	22.2671	1.38898	0.694491	+	0.719502i	$0.255630\pi$
$258$	−20.4335	−1.27213
$259$	−7.32736	−0.455300
$260$	−3.05764	−0.189627
$261$	−32.9651	−2.04049
$262$	5.23468	0.323400
$263$	−12.5549	−0.774170	−0.387085	−	0.922044i	$-0.626518\pi$
$263$	−12.5549	−0.774170	−0.387085	+	0.922044i	$0.626518\pi$
$264$	0	0
$265$	−1.70116	−0.104502
$266$	−4.69704	−0.287994
$267$	34.5824	2.11641
$268$	10.5718	0.645773
$269$	−7.54481	−0.460015	−0.230007	−	0.973189i	$-0.573875\pi$
$269$	−7.54481	−0.460015	−0.230007	+	0.973189i	$0.573875\pi$
$270$	−5.29926	−0.322503
$271$	27.7828	1.68768	0.843842	−	0.536592i	$-0.180289\pi$
$271$	27.7828	1.68768	0.843842	+	0.536592i	$0.180289\pi$
$272$	6.94459	0.421078
$273$	8.58700	0.519709
$274$	5.00039	0.302085
$275$	0	0
$276$	4.86179	0.292646
$277$	18.8691	1.13374	0.566868	−	0.823809i	$-0.308155\pi$
$277$	18.8691	1.13374	0.566868	+	0.823809i	$0.308155\pi$
$278$	−3.08024	−0.184740
$279$	16.3387	0.978173
$280$	1.00000	0.0597614
$281$	6.86665	0.409630	0.204815	−	0.978801i	$-0.434341\pi$
$281$	6.86665	0.409630	0.204815	+	0.978801i	$0.434341\pi$
$282$	−11.2491	−0.669873
$283$	−25.1097	−1.49261	−0.746307	−	0.665601i	$-0.768175\pi$
$283$	−25.1097	−1.49261	−0.746307	+	0.665601i	$0.768175\pi$
$284$	−2.30414	−0.136725
$285$	13.1910	0.781369
$286$	0	0
$287$	7.58977	0.448010
$288$	4.88695	0.287966
$289$	31.2274	1.83690
$290$	−6.74554	−0.396112
$291$	19.4980	1.14299
$292$	−13.1113	−0.767278
$293$	16.5950	0.969491	0.484746	−	0.874655i	$-0.338912\pi$
$293$	16.5950	0.969491	0.484746	+	0.874655i	$0.338912\pi$
$294$	−2.80837	−0.163788
$295$	7.64741	0.445250
$296$	−7.32736	−0.425894
$297$	0	0
$298$	−15.8858	−0.920237
$299$	5.29332	0.306121
$300$	−2.80837	−0.162141
$301$	7.27592	0.419377
$302$	−5.45916	−0.314139
$303$	−38.1699	−2.19281
$304$	−4.69704	−0.269394
$305$	8.93050	0.511359
$306$	33.9379	1.94010
$307$	−31.2886	−1.78573	−0.892867	−	0.450321i	$-0.851309\pi$
$307$	−31.2886	−1.78573	−0.892867	+	0.450321i	$0.851309\pi$
$308$	0	0
$309$	−44.5381	−2.53368
$310$	3.34334	0.189889
$311$	−5.69208	−0.322768	−0.161384	−	0.986892i	$-0.551596\pi$
$311$	−5.69208	−0.322768	−0.161384	+	0.986892i	$0.551596\pi$
$312$	8.58700	0.486143
$313$	17.4320	0.985315	0.492658	−	0.870223i	$-0.336026\pi$
$313$	17.4320	0.985315	0.492658	+	0.870223i	$0.336026\pi$
$314$	8.44070	0.476336
$315$	4.88695	0.275349
$316$	4.25829	0.239547
$317$	21.2720	1.19475	0.597376	−	0.801961i	$-0.296210\pi$
$317$	21.2720	1.19475	0.597376	+	0.801961i	$0.296210\pi$
$318$	4.77750	0.267909
$319$	0	0
$320$	1.00000	0.0559017
$321$	12.3480	0.689198
$322$	−1.73118	−0.0964748
$323$	−32.6190	−1.81497
$324$	0.221440	0.0123022
$325$	−3.05764	−0.169607
$326$	9.74355	0.539645
$327$	40.7021	2.25083
$328$	7.58977	0.419075
$329$	4.00555	0.220833
$330$	0	0
$331$	0.712600	0.0391680	0.0195840	−	0.999808i	$-0.493766\pi$
$331$	0.712600	0.0391680	0.0195840	+	0.999808i	$0.493766\pi$
$332$	3.86073	0.211885
$333$	−35.8084	−1.96229
$334$	20.2081	1.10574
$335$	10.5718	0.577597
$336$	−2.80837	−0.153209
$337$	28.7577	1.56653	0.783267	−	0.621686i	$-0.213552\pi$
$337$	28.7577	1.56653	0.783267	+	0.621686i	$0.213552\pi$
$338$	−3.65082	−0.198579
$339$	54.2930	2.94879
$340$	6.94459	0.376623
$341$	0	0
$342$	−22.9542	−1.24122
$343$	1.00000	0.0539949
$344$	7.27592	0.392291
$345$	4.86179	0.261750
$346$	1.06417	0.0572101
$347$	34.5981	1.85732	0.928662	−	0.370927i	$-0.120960\pi$
$347$	34.5981	1.85732	0.928662	+	0.370927i	$0.120960\pi$
$348$	18.9440	1.01550
$349$	15.4593	0.827518	0.413759	−	0.910386i	$-0.364216\pi$
$349$	15.4593	0.827518	0.413759	+	0.910386i	$0.364216\pi$
$350$	1.00000	0.0534522
$351$	16.2032	0.864865
$352$	0	0
$353$	2.31479	0.123204	0.0616019	−	0.998101i	$-0.480379\pi$
$353$	2.31479	0.123204	0.0616019	+	0.998101i	$0.480379\pi$
$354$	−21.4768	−1.14148
$355$	−2.30414	−0.122291
$356$	−12.3140	−0.652643
$357$	−19.5030	−1.03221
$358$	−5.73780	−0.303252
$359$	5.56161	0.293530	0.146765	−	0.989171i	$-0.453114\pi$
$359$	5.56161	0.293530	0.146765	+	0.989171i	$0.453114\pi$
$360$	4.88695	0.257565
$361$	3.06217	0.161167
$362$	3.77619	0.198472
$363$	0	0
$364$	−3.05764	−0.160264
$365$	−13.1113	−0.686274
$366$	−25.0802	−1.31096
$367$	26.0861	1.36168	0.680842	−	0.732430i	$-0.261614\pi$
$367$	26.0861	1.36168	0.680842	+	0.732430i	$0.261614\pi$
$368$	−1.73118	−0.0902439
$369$	37.0908	1.93087
$370$	−7.32736	−0.380931
$371$	−1.70116	−0.0883200
$372$	−9.38933	−0.486814
$373$	28.3183	1.46627	0.733134	−	0.680085i	$-0.238057\pi$
$373$	28.3183	1.46627	0.733134	+	0.680085i	$0.238057\pi$
$374$	0	0
$375$	−2.80837	−0.145024
$376$	4.00555	0.206571
$377$	20.6255	1.06227
$378$	−5.29926	−0.272565
$379$	−32.5648	−1.67274	−0.836371	−	0.548164i	$-0.815327\pi$
$379$	−32.5648	−1.67274	−0.836371	+	0.548164i	$0.815327\pi$
$380$	−4.69704	−0.240953
$381$	−60.6181	−3.10556
$382$	11.7298	0.600147
$383$	−12.3943	−0.633319	−0.316660	−	0.948539i	$-0.602561\pi$
$383$	−12.3943	−0.633319	−0.316660	+	0.948539i	$0.602561\pi$
$384$	−2.80837	−0.143314
$385$	0	0
$386$	18.9879	0.966461
$387$	35.5571	1.80747
$388$	−6.94282	−0.352468
$389$	−19.8793	−1.00792	−0.503960	−	0.863727i	$-0.668124\pi$
$389$	−19.8793	−1.00792	−0.503960	+	0.863727i	$0.668124\pi$
$390$	8.58700	0.434819
$391$	−12.0223	−0.607995
$392$	1.00000	0.0505076
$393$	−14.7009	−0.741564
$394$	−3.27508	−0.164996
$395$	4.25829	0.214258
$396$	0	0
$397$	27.6288	1.38665	0.693325	−	0.720625i	$-0.256145\pi$
$397$	27.6288	1.38665	0.693325	+	0.720625i	$0.256145\pi$
$398$	14.7438	0.739040
$399$	13.1910	0.660377
$400$	1.00000	0.0500000
$401$	30.1562	1.50593	0.752965	−	0.658061i	$-0.228623\pi$
$401$	30.1562	1.50593	0.752965	+	0.658061i	$0.228623\pi$
$402$	−29.6894	−1.48077
$403$	−10.2227	−0.509230
$404$	13.5915	0.676202
$405$	0.221440	0.0110034
$406$	−6.74554	−0.334776
$407$	0	0
$408$	−19.5030	−0.965542
$409$	18.8749	0.933302	0.466651	−	0.884442i	$-0.345460\pi$
$409$	18.8749	0.933302	0.466651	+	0.884442i	$0.345460\pi$
$410$	7.58977	0.374832
$411$	−14.0430	−0.692688
$412$	15.8591	0.781320
$413$	7.64741	0.376305
$414$	−8.46019	−0.415795
$415$	3.86073	0.189516
$416$	−3.05764	−0.149913
$417$	8.65045	0.423614
$418$	0	0
$419$	8.51801	0.416132	0.208066	−	0.978115i	$-0.433283\pi$
$419$	8.51801	0.416132	0.208066	+	0.978115i	$0.433283\pi$
$420$	−2.80837	−0.137035
$421$	−13.4733	−0.656647	−0.328323	−	0.944565i	$-0.606484\pi$
$421$	−13.4733	−0.656647	−0.328323	+	0.944565i	$0.606484\pi$
$422$	−13.6709	−0.665489
$423$	19.5749	0.951766
$424$	−1.70116	−0.0826158
$425$	6.94459	0.336862
$426$	6.47087	0.313515
$427$	8.93050	0.432177
$428$	−4.39685	−0.212530
$429$	0	0
$430$	7.27592	0.350876
$431$	−27.8105	−1.33959	−0.669793	−	0.742548i	$-0.733617\pi$
$431$	−27.8105	−1.33959	−0.669793	+	0.742548i	$0.733617\pi$
$432$	−5.29926	−0.254961
$433$	−19.9578	−0.959111	−0.479555	−	0.877512i	$-0.659202\pi$
$433$	−19.9578	−0.959111	−0.479555	+	0.877512i	$0.659202\pi$
$434$	3.34334	0.160485
$435$	18.9440	0.908295
$436$	−14.4931	−0.694095
$437$	8.13141	0.388978
$438$	36.8213	1.75939
$439$	−10.7367	−0.512436	−0.256218	−	0.966619i	$-0.582476\pi$
$439$	−10.7367	−0.512436	−0.256218	+	0.966619i	$0.582476\pi$
$440$	0	0
$441$	4.88695	0.232712
$442$	−21.2341	−1.01000
$443$	−10.0787	−0.478854	−0.239427	−	0.970914i	$-0.576960\pi$
$443$	−10.0787	−0.478854	−0.239427	+	0.970914i	$0.576960\pi$
$444$	20.5779	0.976586
$445$	−12.3140	−0.583742
$446$	−5.23577	−0.247921
$447$	44.6131	2.11013
$448$	1.00000	0.0472456
$449$	−31.1076	−1.46806	−0.734030	−	0.679117i	$-0.762363\pi$
$449$	−31.1076	−1.46806	−0.734030	+	0.679117i	$0.762363\pi$
$450$	4.88695	0.230373
$451$	0	0
$452$	−19.3326	−0.909328
$453$	15.3313	0.720330
$454$	−22.5392	−1.05782
$455$	−3.05764	−0.143344
$456$	13.1910	0.617727
$457$	26.2062	1.22587	0.612936	−	0.790132i	$-0.289988\pi$
$457$	26.2062	1.22587	0.612936	+	0.790132i	$0.289988\pi$
$458$	1.54850	0.0723566
$459$	−36.8012	−1.71773
$460$	−1.73118	−0.0807166
$461$	−5.33572	−0.248509	−0.124255	−	0.992250i	$-0.539654\pi$
$461$	−5.33572	−0.248509	−0.124255	+	0.992250i	$0.539654\pi$
$462$	0	0
$463$	28.8817	1.34224	0.671122	−	0.741347i	$-0.265813\pi$
$463$	28.8817	1.34224	0.671122	+	0.741347i	$0.265813\pi$
$464$	−6.74554	−0.313154
$465$	−9.38933	−0.435420
$466$	−6.59831	−0.305661
$467$	−37.6575	−1.74258	−0.871291	−	0.490766i	$-0.836717\pi$
$467$	−37.6575	−1.74258	−0.871291	+	0.490766i	$0.836717\pi$
$468$	−14.9425	−0.690720
$469$	10.5718	0.488159
$470$	4.00555	0.184762
$471$	−23.7046	−1.09225
$472$	7.64741	0.352001
$473$	0	0
$474$	−11.9589	−0.549288
$475$	−4.69704	−0.215515
$476$	6.94459	0.318305
$477$	−8.31350	−0.380649
$478$	9.32864	0.426682
$479$	25.5852	1.16902	0.584509	−	0.811387i	$-0.301287\pi$
$479$	25.5852	1.16902	0.584509	+	0.811387i	$0.301287\pi$
$480$	−2.80837	−0.128184
$481$	22.4044	1.02155
$482$	−4.69741	−0.213961
$483$	4.86179	0.221219
$484$	0	0
$485$	−6.94282	−0.315257
$486$	15.2759	0.692929
$487$	20.1951	0.915125	0.457563	−	0.889177i	$-0.348723\pi$
$487$	20.1951	0.915125	0.457563	+	0.889177i	$0.348723\pi$
$488$	8.93050	0.404265
$489$	−27.3635	−1.23742
$490$	1.00000	0.0451754
$491$	18.2166	0.822104	0.411052	−	0.911612i	$-0.365161\pi$
$491$	18.2166	0.822104	0.411052	+	0.911612i	$0.365161\pi$
$492$	−21.3149	−0.960950
$493$	−46.8451	−2.10980
$494$	14.3619	0.646171
$495$	0	0
$496$	3.34334	0.150120
$497$	−2.30414	−0.103355
$498$	−10.8424	−0.485857
$499$	−21.4887	−0.961966	−0.480983	−	0.876730i	$-0.659720\pi$
$499$	−21.4887	−0.961966	−0.480983	+	0.876730i	$0.659720\pi$
$500$	1.00000	0.0447214
$501$	−56.7518	−2.53548
$502$	25.1086	1.12065
$503$	−5.56623	−0.248186	−0.124093	−	0.992271i	$-0.539602\pi$
$503$	−5.56623	−0.248186	−0.124093	+	0.992271i	$0.539602\pi$
$504$	4.88695	0.217682
$505$	13.5915	0.604813
$506$	0	0
$507$	10.2529	0.455346
$508$	21.5848	0.957669
$509$	−7.22200	−0.320110	−0.160055	−	0.987108i	$-0.551167\pi$
$509$	−7.22200	−0.320110	−0.160055	+	0.987108i	$0.551167\pi$
$510$	−19.5030	−0.863607
$511$	−13.1113	−0.580008
$512$	1.00000	0.0441942
$513$	24.8908	1.09896
$514$	22.2671	0.982158
$515$	15.8591	0.698833
$516$	−20.4335	−0.899534
$517$	0	0
$518$	−7.32736	−0.321946
$519$	−2.98858	−0.131184
$520$	−3.05764	−0.134086
$521$	−6.64589	−0.291162	−0.145581	−	0.989346i	$-0.546505\pi$
$521$	−6.64589	−0.291162	−0.145581	+	0.989346i	$0.546505\pi$
$522$	−32.9651	−1.44285
$523$	24.8014	1.08449	0.542246	−	0.840220i	$-0.317574\pi$
$523$	24.8014	1.08449	0.542246	+	0.840220i	$0.317574\pi$
$524$	5.23468	0.228678
$525$	−2.80837	−0.122567
$526$	−12.5549	−0.547421
$527$	23.2181	1.01140
$528$	0	0
$529$	−20.0030	−0.869697
$530$	−1.70116	−0.0738938
$531$	37.3725	1.62183
$532$	−4.69704	−0.203642
$533$	−23.2068	−1.00520
$534$	34.5824	1.49653
$535$	−4.39685	−0.190092
$536$	10.5718	0.456631
$537$	16.1139	0.695365
$538$	−7.54481	−0.325280
$539$	0	0
$540$	−5.29926	−0.228044
$541$	−34.8421	−1.49798	−0.748990	−	0.662581i	$-0.769461\pi$
$541$	−34.8421	−1.49798	−0.748990	+	0.662581i	$0.769461\pi$
$542$	27.7828	1.19337
$543$	−10.6049	−0.455101
$544$	6.94459	0.297747
$545$	−14.4931	−0.620817
$546$	8.58700	0.367489
$547$	22.5413	0.963798	0.481899	−	0.876227i	$-0.339947\pi$
$547$	22.5413	0.963798	0.481899	+	0.876227i	$0.339947\pi$
$548$	5.00039	0.213606
$549$	43.6429	1.86263
$550$	0	0
$551$	31.6841	1.34979
$552$	4.86179	0.206932
$553$	4.25829	0.181081
$554$	18.8691	0.801672
$555$	20.5779	0.873485
$556$	−3.08024	−0.130631
$557$	0.716288	0.0303501	0.0151750	−	0.999885i	$-0.495169\pi$
$557$	0.716288	0.0303501	0.0151750	+	0.999885i	$0.495169\pi$
$558$	16.3387	0.691673
$559$	−22.2472	−0.940955
$560$	1.00000	0.0422577
$561$	0	0
$562$	6.86665	0.289652
$563$	44.5640	1.87815	0.939075	−	0.343713i	$-0.111685\pi$
$563$	44.5640	1.87815	0.939075	+	0.343713i	$0.111685\pi$
$564$	−11.2491	−0.473672
$565$	−19.3326	−0.813327
$566$	−25.1097	−1.05544
$567$	0.221440	0.00929961
$568$	−2.30414	−0.0966795
$569$	9.43478	0.395526	0.197763	−	0.980250i	$-0.436632\pi$
$569$	9.43478	0.395526	0.197763	+	0.980250i	$0.436632\pi$
$570$	13.1910	0.552511
$571$	31.2693	1.30858	0.654289	−	0.756244i	$-0.272968\pi$
$571$	31.2693	1.30858	0.654289	+	0.756244i	$0.272968\pi$
$572$	0	0
$573$	−32.9415	−1.37615
$574$	7.58977	0.316791
$575$	−1.73118	−0.0721951
$576$	4.88695	0.203623
$577$	33.7440	1.40478	0.702390	−	0.711792i	$-0.252116\pi$
$577$	33.7440	1.40478	0.702390	+	0.711792i	$0.252116\pi$
$578$	31.2274	1.29889
$579$	−53.3252	−2.21612
$580$	−6.74554	−0.280093
$581$	3.86073	0.160170
$582$	19.4980	0.808219
$583$	0	0
$584$	−13.1113	−0.542548
$585$	−14.9425	−0.617798
$586$	16.5950	0.685534
$587$	26.2246	1.08240	0.541202	−	0.840892i	$-0.317969\pi$
$587$	26.2246	1.08240	0.541202	+	0.840892i	$0.317969\pi$
$588$	−2.80837	−0.115815
$589$	−15.7038	−0.647063
$590$	7.64741	0.314839
$591$	9.19763	0.378340
$592$	−7.32736	−0.301153
$593$	23.6431	0.970906	0.485453	−	0.874263i	$-0.338655\pi$
$593$	23.6431	0.970906	0.485453	+	0.874263i	$0.338655\pi$
$594$	0	0
$595$	6.94459	0.284701
$596$	−15.8858	−0.650706
$597$	−41.4060	−1.69464
$598$	5.29332	0.216460
$599$	−35.9599	−1.46928	−0.734640	−	0.678457i	$-0.762649\pi$
$599$	−35.9599	−1.46928	−0.734640	+	0.678457i	$0.762649\pi$
$600$	−2.80837	−0.114651
$601$	−31.4056	−1.28106	−0.640530	−	0.767933i	$-0.721285\pi$
$601$	−31.4056	−1.28106	−0.640530	+	0.767933i	$0.721285\pi$
$602$	7.27592	0.296544
$603$	51.6637	2.10391
$604$	−5.45916	−0.222130
$605$	0	0
$606$	−38.1699	−1.55055
$607$	35.9288	1.45830	0.729152	−	0.684352i	$-0.239915\pi$
$607$	35.9288	1.45830	0.729152	+	0.684352i	$0.239915\pi$
$608$	−4.69704	−0.190490
$609$	18.9440	0.767649
$610$	8.93050	0.361585
$611$	−12.2475	−0.495482
$612$	33.9379	1.37186
$613$	47.6086	1.92289	0.961446	−	0.274992i	$-0.0886753\pi$
$613$	47.6086	1.92289	0.961446	+	0.274992i	$0.0886753\pi$
$614$	−31.2886	−1.26270
$615$	−21.3149	−0.859500
$616$	0	0
$617$	−5.44411	−0.219172	−0.109586	−	0.993977i	$-0.534952\pi$
$617$	−5.44411	−0.219172	−0.109586	+	0.993977i	$0.534952\pi$
$618$	−44.5381	−1.79159
$619$	24.6723	0.991665	0.495833	−	0.868418i	$-0.334863\pi$
$619$	24.6723	0.991665	0.495833	+	0.868418i	$0.334863\pi$
$620$	3.34334	0.134272
$621$	9.17397	0.368139
$622$	−5.69208	−0.228231
$623$	−12.3140	−0.493352
$624$	8.58700	0.343755
$625$	1.00000	0.0400000
$626$	17.4320	0.696723
$627$	0	0
$628$	8.44070	0.336820
$629$	−50.8855	−2.02894
$630$	4.88695	0.194701
$631$	−15.8756	−0.631997	−0.315998	−	0.948760i	$-0.602339\pi$
$631$	−15.8756	−0.631997	−0.315998	+	0.948760i	$0.602339\pi$
$632$	4.25829	0.169386
$633$	38.3930	1.52598
$634$	21.2720	0.844818
$635$	21.5848	0.856566
$636$	4.77750	0.189440
$637$	−3.05764	−0.121148
$638$	0	0
$639$	−11.2602	−0.445447
$640$	1.00000	0.0395285
$641$	−3.37605	−0.133346	−0.0666730	−	0.997775i	$-0.521238\pi$
$641$	−3.37605	−0.133346	−0.0666730	+	0.997775i	$0.521238\pi$
$642$	12.3480	0.487336
$643$	20.7120	0.816803	0.408402	−	0.912802i	$-0.366086\pi$
$643$	20.7120	0.816803	0.408402	+	0.912802i	$0.366086\pi$
$644$	−1.73118	−0.0682180
$645$	−20.4335	−0.804568
$646$	−32.6190	−1.28338
$647$	−6.82833	−0.268449	−0.134225	−	0.990951i	$-0.542854\pi$
$647$	−6.82833	−0.268449	−0.134225	+	0.990951i	$0.542854\pi$
$648$	0.221440	0.00869898
$649$	0	0
$650$	−3.05764	−0.119931
$651$	−9.38933	−0.367997
$652$	9.74355	0.381587
$653$	30.9046	1.20939	0.604696	−	0.796456i	$-0.293295\pi$
$653$	30.9046	1.20939	0.604696	+	0.796456i	$0.293295\pi$
$654$	40.7021	1.59158
$655$	5.23468	0.204536
$656$	7.58977	0.296331
$657$	−64.0741	−2.49977
$658$	4.00555	0.156153
$659$	17.6185	0.686318	0.343159	−	0.939277i	$-0.388503\pi$
$659$	17.6185	0.686318	0.343159	+	0.939277i	$0.388503\pi$
$660$	0	0
$661$	11.7396	0.456617	0.228309	−	0.973589i	$-0.426680\pi$
$661$	11.7396	0.456617	0.228309	+	0.973589i	$0.426680\pi$
$662$	0.712600	0.0276960
$663$	59.6332	2.31596
$664$	3.86073	0.149825
$665$	−4.69704	−0.182143
$666$	−35.8084	−1.38755
$667$	11.6777	0.452164
$668$	20.2081	0.781873
$669$	14.7040	0.568489
$670$	10.5718	0.408423
$671$	0	0
$672$	−2.80837	−0.108335
$673$	17.0012	0.655347	0.327673	−	0.944791i	$-0.393735\pi$
$673$	17.0012	0.655347	0.327673	+	0.944791i	$0.393735\pi$
$674$	28.7577	1.10771
$675$	−5.29926	−0.203969
$676$	−3.65082	−0.140416
$677$	−3.84431	−0.147749	−0.0738744	−	0.997268i	$-0.523536\pi$
$677$	−3.84431	−0.147749	−0.0738744	+	0.997268i	$0.523536\pi$
$678$	54.2930	2.08511
$679$	−6.94282	−0.266441
$680$	6.94459	0.266313
$681$	63.2986	2.42561
$682$	0	0
$683$	−2.04787	−0.0783595	−0.0391798	−	0.999232i	$-0.512474\pi$
$683$	−2.04787	−0.0783595	−0.0391798	+	0.999232i	$0.512474\pi$
$684$	−22.9542	−0.877676
$685$	5.00039	0.191055
$686$	1.00000	0.0381802
$687$	−4.34876	−0.165916
$688$	7.27592	0.277392
$689$	5.20155	0.198163
$690$	4.86179	0.185085
$691$	−24.0616	−0.915348	−0.457674	−	0.889120i	$-0.651317\pi$
$691$	−24.0616	−0.915348	−0.457674	+	0.889120i	$0.651317\pi$
$692$	1.06417	0.0404537
$693$	0	0
$694$	34.5981	1.31333
$695$	−3.08024	−0.116840
$696$	18.9440	0.718070
$697$	52.7079	1.99645
$698$	15.4593	0.585144
$699$	18.5305	0.700888
$700$	1.00000	0.0377964
$701$	7.99724	0.302052	0.151026	−	0.988530i	$-0.451742\pi$
$701$	7.99724	0.302052	0.151026	+	0.988530i	$0.451742\pi$
$702$	16.2032	0.611552
$703$	34.4169	1.29806
$704$	0	0
$705$	−11.2491	−0.423665
$706$	2.31479	0.0871183
$707$	13.5915	0.511160
$708$	−21.4768	−0.807147
$709$	4.28038	0.160753	0.0803765	−	0.996765i	$-0.474388\pi$
$709$	4.28038	0.160753	0.0803765	+	0.996765i	$0.474388\pi$
$710$	−2.30414	−0.0864727
$711$	20.8100	0.780438
$712$	−12.3140	−0.461488
$713$	−5.78791	−0.216759
$714$	−19.5030	−0.729881
$715$	0	0
$716$	−5.73780	−0.214432
$717$	−26.1983	−0.978393
$718$	5.56161	0.207557
$719$	36.8044	1.37257	0.686286	−	0.727332i	$-0.259240\pi$
$719$	36.8044	1.37257	0.686286	+	0.727332i	$0.259240\pi$
$720$	4.88695	0.182126
$721$	15.8591	0.590622
$722$	3.06217	0.113962
$723$	13.1921	0.490618
$724$	3.77619	0.140341
$725$	−6.74554	−0.250523
$726$	0	0
$727$	−46.5408	−1.72610	−0.863051	−	0.505116i	$-0.831450\pi$
$727$	−46.5408	−1.72610	−0.863051	+	0.505116i	$0.831450\pi$
$728$	−3.05764	−0.113324
$729$	−43.5647	−1.61351
$730$	−13.1113	−0.485269
$731$	50.5283	1.86886
$732$	−25.0802	−0.926989
$733$	15.6659	0.578633	0.289316	−	0.957234i	$-0.406572\pi$
$733$	15.6659	0.578633	0.289316	+	0.957234i	$0.406572\pi$
$734$	26.0861	0.962857
$735$	−2.80837	−0.103588
$736$	−1.73118	−0.0638121
$737$	0	0
$738$	37.0908	1.36533
$739$	6.15005	0.226233	0.113117	−	0.993582i	$-0.463917\pi$
$739$	6.15005	0.226233	0.113117	+	0.993582i	$0.463917\pi$
$740$	−7.32736	−0.269359
$741$	−40.3334	−1.48169
$742$	−1.70116	−0.0624517
$743$	27.2721	1.00052	0.500259	−	0.865876i	$-0.333238\pi$
$743$	27.2721	1.00052	0.500259	+	0.865876i	$0.333238\pi$
$744$	−9.38933	−0.344229
$745$	−15.8858	−0.582009
$746$	28.3183	1.03681
$747$	18.8672	0.690314
$748$	0	0
$749$	−4.39685	−0.160657
$750$	−2.80837	−0.102547
$751$	7.95258	0.290194	0.145097	−	0.989417i	$-0.453651\pi$
$751$	7.95258	0.290194	0.145097	+	0.989417i	$0.453651\pi$
$752$	4.00555	0.146067
$753$	−70.5142	−2.56968
$754$	20.6255	0.751135
$755$	−5.45916	−0.198679
$756$	−5.29926	−0.192732
$757$	24.8369	0.902713	0.451356	−	0.892344i	$-0.350940\pi$
$757$	24.8369	0.902713	0.451356	+	0.892344i	$0.350940\pi$
$758$	−32.5648	−1.18281
$759$	0	0
$760$	−4.69704	−0.170379
$761$	34.0673	1.23494	0.617469	−	0.786595i	$-0.288158\pi$
$761$	34.0673	1.23494	0.617469	+	0.786595i	$0.288158\pi$
$762$	−60.6181	−2.19596
$763$	−14.4931	−0.524686
$764$	11.7298	0.424368
$765$	33.9379	1.22703
$766$	−12.3943	−0.447824
$767$	−23.3831	−0.844313
$768$	−2.80837	−0.101338
$769$	10.7430	0.387404	0.193702	−	0.981060i	$-0.437950\pi$
$769$	10.7430	0.387404	0.193702	+	0.981060i	$0.437950\pi$
$770$	0	0
$771$	−62.5342	−2.25211
$772$	18.9879	0.683391
$773$	−35.4511	−1.27509	−0.637544	−	0.770414i	$-0.720050\pi$
$773$	−35.4511	−1.27509	−0.637544	+	0.770414i	$0.720050\pi$
$774$	35.5571	1.27807
$775$	3.34334	0.120096
$776$	−6.94282	−0.249233
$777$	20.5779	0.738230
$778$	−19.8793	−0.712708
$779$	−35.6494	−1.27727
$780$	8.58700	0.307464
$781$	0	0
$782$	−12.0223	−0.429918
$783$	35.7464	1.27747
$784$	1.00000	0.0357143
$785$	8.44070	0.301261
$786$	−14.7009	−0.524365
$787$	−22.9148	−0.816825	−0.408413	−	0.912797i	$-0.633918\pi$
$787$	−22.9148	−0.816825	−0.408413	+	0.912797i	$0.633918\pi$
$788$	−3.27508	−0.116670
$789$	35.2589	1.25525
$790$	4.25829	0.151503
$791$	−19.3326	−0.687387
$792$	0	0
$793$	−27.3063	−0.969674
$794$	27.6288	0.980510
$795$	4.77750	0.169440
$796$	14.7438	0.522580
$797$	−42.9156	−1.52015	−0.760074	−	0.649836i	$-0.774837\pi$
$797$	−42.9156	−1.52015	−0.760074	+	0.649836i	$0.774837\pi$
$798$	13.1910	0.466957
$799$	27.8169	0.984092
$800$	1.00000	0.0353553
$801$	−60.1782	−2.12629
$802$	30.1562	1.06485
$803$	0	0
$804$	−29.6894	−1.04707
$805$	−1.73118	−0.0610160
$806$	−10.2227	−0.360080
$807$	21.1886	0.745875
$808$	13.5915	0.478147
$809$	16.9974	0.597598	0.298799	−	0.954316i	$-0.403414\pi$
$809$	16.9974	0.597598	0.298799	+	0.954316i	$0.403414\pi$
$810$	0.221440	0.00778061
$811$	−10.6859	−0.375231	−0.187616	−	0.982243i	$-0.560076\pi$
$811$	−10.6859	−0.375231	−0.187616	+	0.982243i	$0.560076\pi$
$812$	−6.74554	−0.236722
$813$	−78.0243	−2.73643
$814$	0	0
$815$	9.74355	0.341302
$816$	−19.5030	−0.682742
$817$	−34.1753	−1.19564
$818$	18.8749	0.659944
$819$	−14.9425	−0.522135
$820$	7.58977	0.265046
$821$	−20.3259	−0.709379	−0.354689	−	0.934984i	$-0.615413\pi$
$821$	−20.3259	−0.709379	−0.354689	+	0.934984i	$0.615413\pi$
$822$	−14.0430	−0.489805
$823$	−29.8555	−1.04070	−0.520348	−	0.853954i	$-0.674198\pi$
$823$	−29.8555	−1.04070	−0.520348	+	0.853954i	$0.674198\pi$
$824$	15.8591	0.552476
$825$	0	0
$826$	7.64741	0.266088
$827$	−4.53541	−0.157712	−0.0788558	−	0.996886i	$-0.525127\pi$
$827$	−4.53541	−0.157712	−0.0788558	+	0.996886i	$0.525127\pi$
$828$	−8.46019	−0.294012
$829$	−29.0225	−1.00799	−0.503997	−	0.863705i	$-0.668138\pi$
$829$	−29.0225	−1.00799	−0.503997	+	0.863705i	$0.668138\pi$
$830$	3.86073	0.134008
$831$	−52.9915	−1.83826
$832$	−3.05764	−0.106005
$833$	6.94459	0.240616
$834$	8.65045	0.299540
$835$	20.2081	0.699329
$836$	0	0
$837$	−17.7172	−0.612396
$838$	8.51801	0.294250
$839$	−3.85780	−0.133186	−0.0665930	−	0.997780i	$-0.521213\pi$
$839$	−3.85780	−0.133186	−0.0665930	+	0.997780i	$0.521213\pi$
$840$	−2.80837	−0.0968980
$841$	16.5024	0.569047
$842$	−13.4733	−0.464319
$843$	−19.2841	−0.664180
$844$	−13.6709	−0.470572
$845$	−3.65082	−0.125592
$846$	19.5749	0.673000
$847$	0	0
$848$	−1.70116	−0.0584182
$849$	70.5173	2.42015
$850$	6.94459	0.238198
$851$	12.6850	0.434835
$852$	6.47087	0.221688
$853$	−14.8560	−0.508659	−0.254329	−	0.967118i	$-0.581855\pi$
$853$	−14.8560	−0.508659	−0.254329	+	0.967118i	$0.581855\pi$
$854$	8.93050	0.305595
$855$	−22.9542	−0.785017
$856$	−4.39685	−0.150281
$857$	−21.4567	−0.732948	−0.366474	−	0.930428i	$-0.619435\pi$
$857$	−21.4567	−0.732948	−0.366474	+	0.930428i	$0.619435\pi$
$858$	0	0
$859$	−50.7945	−1.73309	−0.866543	−	0.499103i	$-0.833663\pi$
$859$	−50.7945	−1.73309	−0.866543	+	0.499103i	$0.833663\pi$
$860$	7.27592	0.248107
$861$	−21.3149	−0.726410
$862$	−27.8105	−0.947231
$863$	−13.6152	−0.463466	−0.231733	−	0.972779i	$-0.574440\pi$
$863$	−13.6152	−0.463466	−0.231733	+	0.972779i	$0.574440\pi$
$864$	−5.29926	−0.180285
$865$	1.06417	0.0361829
$866$	−19.9578	−0.678194
$867$	−87.6981	−2.97838
$868$	3.34334	0.113480
$869$	0	0
$870$	18.9440	0.642262
$871$	−32.3247	−1.09528
$872$	−14.4931	−0.490799
$873$	−33.9292	−1.14833
$874$	8.13141	0.275049
$875$	1.00000	0.0338062
$876$	36.8213	1.24408
$877$	8.90340	0.300647	0.150323	−	0.988637i	$-0.451969\pi$
$877$	8.90340	0.300647	0.150323	+	0.988637i	$0.451969\pi$
$878$	−10.7367	−0.362347
$879$	−46.6050	−1.57195
$880$	0	0
$881$	−2.11899	−0.0713904	−0.0356952	−	0.999363i	$-0.511365\pi$
$881$	−2.11899	−0.0713904	−0.0356952	+	0.999363i	$0.511365\pi$
$882$	4.88695	0.164552
$883$	−10.2827	−0.346041	−0.173021	−	0.984918i	$-0.555353\pi$
$883$	−10.2827	−0.346041	−0.173021	+	0.984918i	$0.555353\pi$
$884$	−21.2341	−0.714179
$885$	−21.4768	−0.721934
$886$	−10.0787	−0.338601
$887$	−5.63590	−0.189235	−0.0946175	−	0.995514i	$-0.530163\pi$
$887$	−5.63590	−0.189235	−0.0946175	+	0.995514i	$0.530163\pi$
$888$	20.5779	0.690551
$889$	21.5848	0.723930
$890$	−12.3140	−0.412768
$891$	0	0
$892$	−5.23577	−0.175307
$893$	−18.8142	−0.629594
$894$	44.6131	1.49209
$895$	−5.73780	−0.191793
$896$	1.00000	0.0334077
$897$	−14.8656	−0.496349
$898$	−31.1076	−1.03808
$899$	−22.5526	−0.752172
$900$	4.88695	0.162898
$901$	−11.8139	−0.393578
$902$	0	0
$903$	−20.4335	−0.679984
$904$	−19.3326	−0.642992
$905$	3.77619	0.125525
$906$	15.3313	0.509350
$907$	−39.0697	−1.29729	−0.648643	−	0.761092i	$-0.724663\pi$
$907$	−39.0697	−1.29729	−0.648643	+	0.761092i	$0.724663\pi$
$908$	−22.5392	−0.747991
$909$	66.4209	2.20304
$910$	−3.05764	−0.101360
$911$	22.8604	0.757398	0.378699	−	0.925520i	$-0.376372\pi$
$911$	22.8604	0.757398	0.378699	+	0.925520i	$0.376372\pi$
$912$	13.1910	0.436799
$913$	0	0
$914$	26.2062	0.866823
$915$	−25.0802	−0.829124
$916$	1.54850	0.0511639
$917$	5.23468	0.172864
$918$	−36.8012	−1.21462
$919$	−13.0335	−0.429935	−0.214968	−	0.976621i	$-0.568965\pi$
$919$	−13.0335	−0.429935	−0.214968	+	0.976621i	$0.568965\pi$
$920$	−1.73118	−0.0570753
$921$	87.8699	2.89541
$922$	−5.33572	−0.175723
$923$	7.04523	0.231896
$924$	0	0
$925$	−7.32736	−0.240922
$926$	28.8817	0.949110
$927$	77.5024	2.54551
$928$	−6.74554	−0.221433
$929$	27.3966	0.898853	0.449427	−	0.893317i	$-0.351628\pi$
$929$	27.3966	0.898853	0.449427	+	0.893317i	$0.351628\pi$
$930$	−9.38933	−0.307888
$931$	−4.69704	−0.153939
$932$	−6.59831	−0.216135
$933$	15.9855	0.523341
$934$	−37.6575	−1.23219
$935$	0	0
$936$	−14.9425	−0.488412
$937$	22.1523	0.723684	0.361842	−	0.932240i	$-0.382148\pi$
$937$	22.1523	0.723684	0.361842	+	0.932240i	$0.382148\pi$
$938$	10.5718	0.345180
$939$	−48.9556	−1.59760
$940$	4.00555	0.130647
$941$	−12.0202	−0.391847	−0.195924	−	0.980619i	$-0.562771\pi$
$941$	−12.0202	−0.391847	−0.195924	+	0.980619i	$0.562771\pi$
$942$	−23.7046	−0.772338
$943$	−13.1392	−0.427873
$944$	7.64741	0.248902
$945$	−5.29926	−0.172385
$946$	0	0
$947$	−29.5380	−0.959857	−0.479929	−	0.877307i	$-0.659337\pi$
$947$	−29.5380	−0.959857	−0.479929	+	0.877307i	$0.659337\pi$
$948$	−11.9589	−0.388406
$949$	40.0895	1.30136
$950$	−4.69704	−0.152392
$951$	−59.7396	−1.93719
$952$	6.94459	0.225076
$953$	38.2985	1.24061	0.620306	−	0.784360i	$-0.287009\pi$
$953$	38.2985	1.24061	0.620306	+	0.784360i	$0.287009\pi$
$954$	−8.31350	−0.269160
$955$	11.7298	0.379566
$956$	9.32864	0.301710
$957$	0	0
$958$	25.5852	0.826621
$959$	5.00039	0.161471
$960$	−2.80837	−0.0906398
$961$	−19.8221	−0.639423
$962$	22.4044	0.722348
$963$	−21.4872	−0.692415
$964$	−4.69741	−0.151293
$965$	18.9879	0.611243
$966$	4.86179	0.156426
$967$	2.04155	0.0656517	0.0328259	−	0.999461i	$-0.489549\pi$
$967$	2.04155	0.0656517	0.0328259	+	0.999461i	$0.489549\pi$
$968$	0	0
$969$	91.6063	2.94282
$970$	−6.94282	−0.222920
$971$	−5.03293	−0.161514	−0.0807572	−	0.996734i	$-0.525734\pi$
$971$	−5.03293	−0.161514	−0.0807572	+	0.996734i	$0.525734\pi$
$972$	15.2759	0.489975
$973$	−3.08024	−0.0987478
$974$	20.1951	0.647091
$975$	8.58700	0.275004
$976$	8.93050	0.285858
$977$	−10.1256	−0.323945	−0.161973	−	0.986795i	$-0.551786\pi$
$977$	−10.1256	−0.323945	−0.161973	+	0.986795i	$0.551786\pi$
$978$	−27.3635	−0.874989
$979$	0	0
$980$	1.00000	0.0319438
$981$	−70.8272	−2.26134
$982$	18.2166	0.581315
$983$	33.8126	1.07846	0.539228	−	0.842160i	$-0.318716\pi$
$983$	33.8126	1.07846	0.539228	+	0.842160i	$0.318716\pi$
$984$	−21.3149	−0.679494
$985$	−3.27508	−0.104353
$986$	−46.8451	−1.49185
$987$	−11.2491	−0.358062
$988$	14.3619	0.456912
$989$	−12.5959	−0.400527
$990$	0	0
$991$	13.8867	0.441127	0.220563	−	0.975373i	$-0.429210\pi$
$991$	13.8867	0.441127	0.220563	+	0.975373i	$0.429210\pi$
$992$	3.34334	0.106151
$993$	−2.00124	−0.0635076
$994$	−2.30414	−0.0730828
$995$	14.7438	0.467410
$996$	−10.8424	−0.343553
$997$	5.72382	0.181275	0.0906375	−	0.995884i	$-0.471110\pi$
$997$	5.72382	0.181275	0.0906375	+	0.995884i	$0.471110\pi$
$998$	−21.4887	−0.680213
$999$	38.8296	1.22851

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.de.1.1		6
11.5	even	5		770.2.n.i.421.1	✓	12
11.9	even	5		770.2.n.i.631.1	yes	12
11.10	odd	2		8470.2.a.cy.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.i.421.1	✓	12	11.5	even	5
770.2.n.i.631.1	yes	12	11.9	even	5
8470.2.a.cy.1.1		6	11.10	odd	2
8470.2.a.de.1.1		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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -2.80837 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.80837 q^{6} +1.00000 q^{7} +1.00000 q^{8} +4.88695 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -2.80837 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.80837 q^{6} +1.00000 q^{7} +1.00000 q^{8} +4.88695 q^{9} +1.00000 q^{10} -2.80837 q^{12} -3.05764 q^{13} +1.00000 q^{14} -2.80837 q^{15} +1.00000 q^{16} +6.94459 q^{17} +4.88695 q^{18} -4.69704 q^{19} +1.00000 q^{20} -2.80837 q^{21} -1.73118 q^{23} -2.80837 q^{24} +1.00000 q^{25} -3.05764 q^{26} -5.29926 q^{27} +1.00000 q^{28} -6.74554 q^{29} -2.80837 q^{30} +3.34334 q^{31} +1.00000 q^{32} +6.94459 q^{34} +1.00000 q^{35} +4.88695 q^{36} -7.32736 q^{37} -4.69704 q^{38} +8.58700 q^{39} +1.00000 q^{40} +7.58977 q^{41} -2.80837 q^{42} +7.27592 q^{43} +4.88695 q^{45} -1.73118 q^{46} +4.00555 q^{47} -2.80837 q^{48} +1.00000 q^{49} +1.00000 q^{50} -19.5030 q^{51} -3.05764 q^{52} -1.70116 q^{53} -5.29926 q^{54} +1.00000 q^{56} +13.1910 q^{57} -6.74554 q^{58} +7.64741 q^{59} -2.80837 q^{60} +8.93050 q^{61} +3.34334 q^{62} +4.88695 q^{63} +1.00000 q^{64} -3.05764 q^{65} +10.5718 q^{67} +6.94459 q^{68} +4.86179 q^{69} +1.00000 q^{70} -2.30414 q^{71} +4.88695 q^{72} -13.1113 q^{73} -7.32736 q^{74} -2.80837 q^{75} -4.69704 q^{76} +8.58700 q^{78} +4.25829 q^{79} +1.00000 q^{80} +0.221440 q^{81} +7.58977 q^{82} +3.86073 q^{83} -2.80837 q^{84} +6.94459 q^{85} +7.27592 q^{86} +18.9440 q^{87} -12.3140 q^{89} +4.88695 q^{90} -3.05764 q^{91} -1.73118 q^{92} -9.38933 q^{93} +4.00555 q^{94} -4.69704 q^{95} -2.80837 q^{96} -6.94282 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} + 6 q^{8} + 15 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 + q^3 + 6 * q^4 + 6 * q^5 + q^6 + 6 * q^7 + 6 * q^8 + 15 * q^9	\( 6 q + 6 q^{2} + q^{3} + 6 q^{4} + 6 q^{5} + q^{6} + 6 q^{7} + 6 q^{8} + 15 q^{9} + 6 q^{10} + q^{12} + 2 q^{13} + 6 q^{14} + q^{15} + 6 q^{16} + 7 q^{17} + 15 q^{18} + 11 q^{19} + 6 q^{20} + q^{21} - 6 q^{23} + q^{24} + 6 q^{25} + 2 q^{26} + 4 q^{27} + 6 q^{28} + 2 q^{29} + q^{30} + 6 q^{32} + 7 q^{34} + 6 q^{35} + 15 q^{36} - 14 q^{37} + 11 q^{38} + 20 q^{39} + 6 q^{40} + 13 q^{41} + q^{42} + 19 q^{43} + 15 q^{45} - 6 q^{46} + 22 q^{47} + q^{48} + 6 q^{49} + 6 q^{50} - 14 q^{51} + 2 q^{52} - 10 q^{53} + 4 q^{54} + 6 q^{56} + 32 q^{57} + 2 q^{58} - 7 q^{59} + q^{60} + 22 q^{61} + 15 q^{63} + 6 q^{64} + 2 q^{65} + 5 q^{67} + 7 q^{68} - 36 q^{69} + 6 q^{70} + 8 q^{71} + 15 q^{72} + 13 q^{73} - 14 q^{74} + q^{75} + 11 q^{76} + 20 q^{78} - 16 q^{79} + 6 q^{80} + 18 q^{81} + 13 q^{82} - 5 q^{83} + q^{84} + 7 q^{85} + 19 q^{86} + 14 q^{87} + q^{89} + 15 q^{90} + 2 q^{91} - 6 q^{92} - 42 q^{93} + 22 q^{94} + 11 q^{95} + q^{96} - 3 q^{97} + 6 q^{98}+O(q^{100}) \) 6 * q + 6 * q^2 + q^3 + 6 * q^4 + 6 * q^5 + q^6 + 6 * q^7 + 6 * q^8 + 15 * q^9 + 6 * q^10 + q^12 + 2 * q^13 + 6 * q^14 + q^15 + 6 * q^16 + 7 * q^17 + 15 * q^18 + 11 * q^19 + 6 * q^20 + q^21 - 6 * q^23 + q^24 + 6 * q^25 + 2 * q^26 + 4 * q^27 + 6 * q^28 + 2 * q^29 + q^30 + 6 * q^32 + 7 * q^34 + 6 * q^35 + 15 * q^36 - 14 * q^37 + 11 * q^38 + 20 * q^39 + 6 * q^40 + 13 * q^41 + q^42 + 19 * q^43 + 15 * q^45 - 6 * q^46 + 22 * q^47 + q^48 + 6 * q^49 + 6 * q^50 - 14 * q^51 + 2 * q^52 - 10 * q^53 + 4 * q^54 + 6 * q^56 + 32 * q^57 + 2 * q^58 - 7 * q^59 + q^60 + 22 * q^61 + 15 * q^63 + 6 * q^64 + 2 * q^65 + 5 * q^67 + 7 * q^68 - 36 * q^69 + 6 * q^70 + 8 * q^71 + 15 * q^72 + 13 * q^73 - 14 * q^74 + q^75 + 11 * q^76 + 20 * q^78 - 16 * q^79 + 6 * q^80 + 18 * q^81 + 13 * q^82 - 5 * q^83 + q^84 + 7 * q^85 + 19 * q^86 + 14 * q^87 + q^89 + 15 * q^90 + 2 * q^91 - 6 * q^92 - 42 * q^93 + 22 * q^94 + 11 * q^95 + q^96 - 3 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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