LMFDB - Embedded newform 8470.2.a.cy.1.5

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:	6.6.13298000.1
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			1.5
Root			$0.0677009$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +2.44508 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.44508 q^{6} -1.00000 q^{7} -1.00000 q^{8} +2.97843 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +2.44508 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.44508 q^{6} -1.00000 q^{7} -1.00000 q^{8} +2.97843 q^{9} -1.00000 q^{10} +2.44508 q^{12} -1.00429 q^{13} +1.00000 q^{14} +2.44508 q^{15} +1.00000 q^{16} -0.974141 q^{17} -2.97843 q^{18} -5.13082 q^{19} +1.00000 q^{20} -2.44508 q^{21} -4.97385 q^{23} -2.44508 q^{24} +1.00000 q^{25} +1.00429 q^{26} -0.0527323 q^{27} -1.00000 q^{28} -2.89329 q^{29} -2.44508 q^{30} -6.46901 q^{31} -1.00000 q^{32} +0.974141 q^{34} -1.00000 q^{35} +2.97843 q^{36} +9.79242 q^{37} +5.13082 q^{38} -2.45558 q^{39} -1.00000 q^{40} -3.10213 q^{41} +2.44508 q^{42} +9.46992 q^{43} +2.97843 q^{45} +4.97385 q^{46} +7.29309 q^{47} +2.44508 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.38186 q^{51} -1.00429 q^{52} +3.98903 q^{53} +0.0527323 q^{54} +1.00000 q^{56} -12.5453 q^{57} +2.89329 q^{58} -0.902167 q^{59} +2.44508 q^{60} -7.11156 q^{61} +6.46901 q^{62} -2.97843 q^{63} +1.00000 q^{64} -1.00429 q^{65} -1.88571 q^{67} -0.974141 q^{68} -12.1615 q^{69} +1.00000 q^{70} +2.21005 q^{71} -2.97843 q^{72} +4.41995 q^{73} -9.79242 q^{74} +2.44508 q^{75} -5.13082 q^{76} +2.45558 q^{78} -12.7528 q^{79} +1.00000 q^{80} -9.06423 q^{81} +3.10213 q^{82} -8.95284 q^{83} -2.44508 q^{84} -0.974141 q^{85} -9.46992 q^{86} -7.07434 q^{87} -5.28764 q^{89} -2.97843 q^{90} +1.00429 q^{91} -4.97385 q^{92} -15.8173 q^{93} -7.29309 q^{94} -5.13082 q^{95} -2.44508 q^{96} -9.41520 q^{97} -1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.44508	1.41167	0.705835	−	0.708376i	$-0.250572\pi$
$3$	2.44508	1.41167	0.705835	+	0.708376i	$0.250572\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−2.44508	−0.998201
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	2.97843	0.992811
$10$	−1.00000	−0.316228
$11$	0	0
$11$	0	0
$12$	2.44508	0.705835
$13$	−1.00429	−0.278541	−0.139270	−	0.990254i	$-0.544476\pi$
$13$	−1.00429	−0.278541	−0.139270	+	0.990254i	$0.544476\pi$
$14$	1.00000	0.267261
$15$	2.44508	0.631318
$16$	1.00000	0.250000
$17$	−0.974141	−0.236264	−0.118132	−	0.992998i	$-0.537691\pi$
$17$	−0.974141	−0.236264	−0.118132	+	0.992998i	$0.537691\pi$
$18$	−2.97843	−0.702023
$19$	−5.13082	−1.17709	−0.588545	−	0.808464i	$-0.700299\pi$
$19$	−5.13082	−1.17709	−0.588545	+	0.808464i	$0.700299\pi$
$20$	1.00000	0.223607
$21$	−2.44508	−0.533561
$22$	0	0
$23$	−4.97385	−1.03712	−0.518560	−	0.855041i	$-0.673532\pi$
$23$	−4.97385	−1.03712	−0.518560	+	0.855041i	$0.673532\pi$
$24$	−2.44508	−0.499101
$25$	1.00000	0.200000
$26$	1.00429	0.196958
$27$	−0.0527323	−0.0101483
$28$	−1.00000	−0.188982
$29$	−2.89329	−0.537271	−0.268635	−	0.963242i	$-0.586573\pi$
$29$	−2.89329	−0.537271	−0.268635	+	0.963242i	$0.586573\pi$
$30$	−2.44508	−0.446409
$31$	−6.46901	−1.16187	−0.580934	−	0.813950i	$-0.697313\pi$
$31$	−6.46901	−1.16187	−0.580934	+	0.813950i	$0.697313\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	0.974141	0.167064
$35$	−1.00000	−0.169031
$36$	2.97843	0.496406
$37$	9.79242	1.60986	0.804932	−	0.593367i	$-0.202202\pi$
$37$	9.79242	1.60986	0.804932	+	0.593367i	$0.202202\pi$
$38$	5.13082	0.832329
$39$	−2.45558	−0.393207
$40$	−1.00000	−0.158114
$41$	−3.10213	−0.484471	−0.242235	−	0.970218i	$-0.577881\pi$
$41$	−3.10213	−0.484471	−0.242235	+	0.970218i	$0.577881\pi$
$42$	2.44508	0.377285
$43$	9.46992	1.44415	0.722074	−	0.691816i	$-0.243189\pi$
$43$	9.46992	1.44415	0.722074	+	0.691816i	$0.243189\pi$
$44$	0	0
$45$	2.97843	0.443999
$46$	4.97385	0.733354
$47$	7.29309	1.06381	0.531903	−	0.846805i	$-0.321477\pi$
$47$	7.29309	1.06381	0.531903	+	0.846805i	$0.321477\pi$
$48$	2.44508	0.352917
$49$	1.00000	0.142857
$50$	−1.00000	−0.141421
$51$	−2.38186	−0.333526
$52$	−1.00429	−0.139270
$53$	3.98903	0.547936	0.273968	−	0.961739i	$-0.411664\pi$
$53$	3.98903	0.547936	0.273968	+	0.961739i	$0.411664\pi$
$54$	0.0527323	0.00717595
$55$	0	0
$56$	1.00000	0.133631
$57$	−12.5453	−1.66166
$58$	2.89329	0.379908
$59$	−0.902167	−0.117452	−0.0587260	−	0.998274i	$-0.518704\pi$
$59$	−0.902167	−0.117452	−0.0587260	+	0.998274i	$0.518704\pi$
$60$	2.44508	0.315659
$61$	−7.11156	−0.910542	−0.455271	−	0.890353i	$-0.650458\pi$
$61$	−7.11156	−0.910542	−0.455271	+	0.890353i	$0.650458\pi$
$62$	6.46901	0.821565
$63$	−2.97843	−0.375247
$64$	1.00000	0.125000
$65$	−1.00429	−0.124567
$66$	0	0
$67$	−1.88571	−0.230377	−0.115188	−	0.993344i	$-0.536747\pi$
$67$	−1.88571	−0.230377	−0.115188	+	0.993344i	$0.536747\pi$
$68$	−0.974141	−0.118132
$69$	−12.1615	−1.46407
$70$	1.00000	0.119523
$71$	2.21005	0.262285	0.131142	−	0.991364i	$-0.458136\pi$
$71$	2.21005	0.262285	0.131142	+	0.991364i	$0.458136\pi$
$72$	−2.97843	−0.351012
$73$	4.41995	0.517316	0.258658	−	0.965969i	$-0.416720\pi$
$73$	4.41995	0.517316	0.258658	+	0.965969i	$0.416720\pi$
$74$	−9.79242	−1.13835
$75$	2.44508	0.282334
$76$	−5.13082	−0.588545
$77$	0	0
$78$	2.45558	0.278040
$79$	−12.7528	−1.43480	−0.717399	−	0.696663i	$-0.754667\pi$
$79$	−12.7528	−1.43480	−0.717399	+	0.696663i	$0.754667\pi$
$80$	1.00000	0.111803
$81$	−9.06423	−1.00714
$82$	3.10213	0.342573
$83$	−8.95284	−0.982702	−0.491351	−	0.870962i	$-0.663497\pi$
$83$	−8.95284	−0.982702	−0.491351	+	0.870962i	$0.663497\pi$
$84$	−2.44508	−0.266780
$85$	−0.974141	−0.105660
$86$	−9.46992	−1.02117
$87$	−7.07434	−0.758448
$88$	0	0
$89$	−5.28764	−0.560489	−0.280245	−	0.959929i	$-0.590416\pi$
$89$	−5.28764	−0.560489	−0.280245	+	0.959929i	$0.590416\pi$
$90$	−2.97843	−0.313954
$91$	1.00429	0.105278
$92$	−4.97385	−0.518560
$93$	−15.8173	−1.64018
$94$	−7.29309	−0.752225
$95$	−5.13082	−0.526411
$96$	−2.44508	−0.249550
$97$	−9.41520	−0.955969	−0.477985	−	0.878368i	$-0.658633\pi$
$97$	−9.41520	−0.955969	−0.477985	+	0.878368i	$0.658633\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	1.00000	0.100000
$101$	−6.43796	−0.640601	−0.320300	−	0.947316i	$-0.603784\pi$
$101$	−6.43796	−0.640601	−0.320300	+	0.947316i	$0.603784\pi$
$102$	2.38186	0.235839
$103$	6.53919	0.644325	0.322163	−	0.946684i	$-0.395590\pi$
$103$	6.53919	0.644325	0.322163	+	0.946684i	$0.395590\pi$
$104$	1.00429	0.0984790
$105$	−2.44508	−0.238616
$106$	−3.98903	−0.387449
$107$	−13.6151	−1.31622	−0.658110	−	0.752922i	$-0.728644\pi$
$107$	−13.6151	−1.31622	−0.658110	+	0.752922i	$0.728644\pi$
$108$	−0.0527323	−0.00507417
$109$	4.46766	0.427925	0.213962	−	0.976842i	$-0.431363\pi$
$109$	4.46766	0.427925	0.213962	+	0.976842i	$0.431363\pi$
$110$	0	0
$111$	23.9433	2.27260
$112$	−1.00000	−0.0944911
$113$	−17.2962	−1.62709	−0.813544	−	0.581504i	$-0.802465\pi$
$113$	−17.2962	−1.62709	−0.813544	+	0.581504i	$0.802465\pi$
$114$	12.5453	1.17497
$115$	−4.97385	−0.463814
$116$	−2.89329	−0.268635
$117$	−2.99122	−0.276538
$118$	0.902167	0.0830512
$119$	0.974141	0.0892993
$120$	−2.44508	−0.223205
$121$	0	0
$122$	7.11156	0.643850
$123$	−7.58496	−0.683913
$124$	−6.46901	−0.580934
$125$	1.00000	0.0894427
$126$	2.97843	0.265340
$127$	−20.1769	−1.79041	−0.895207	−	0.445651i	$-0.852972\pi$
$127$	−20.1769	−1.79041	−0.895207	+	0.445651i	$0.852972\pi$
$128$	−1.00000	−0.0883883
$129$	23.1547	2.03866
$130$	1.00429	0.0880823
$131$	11.0785	0.967935	0.483967	−	0.875086i	$-0.339195\pi$
$131$	11.0785	0.967935	0.483967	+	0.875086i	$0.339195\pi$
$132$	0	0
$133$	5.13082	0.444898
$134$	1.88571	0.162901
$135$	−0.0527323	−0.00453847
$136$	0.974141	0.0835319
$137$	−19.6702	−1.68054	−0.840269	−	0.542170i	$-0.817603\pi$
$137$	−19.6702	−1.68054	−0.840269	+	0.542170i	$0.817603\pi$
$138$	12.1615	1.03525
$139$	9.53624	0.808854	0.404427	−	0.914570i	$-0.367471\pi$
$139$	9.53624	0.808854	0.404427	+	0.914570i	$0.367471\pi$
$140$	−1.00000	−0.0845154
$141$	17.8322	1.50174
$142$	−2.21005	−0.185463
$143$	0	0
$144$	2.97843	0.248203
$145$	−2.89329	−0.240275
$146$	−4.41995	−0.365797
$147$	2.44508	0.201667
$148$	9.79242	0.804932
$149$	9.27662	0.759970	0.379985	−	0.924993i	$-0.375929\pi$
$149$	9.27662	0.759970	0.379985	+	0.924993i	$0.375929\pi$
$150$	−2.44508	−0.199640
$151$	−0.800899	−0.0651763	−0.0325881	−	0.999469i	$-0.510375\pi$
$151$	−0.800899	−0.0651763	−0.0325881	+	0.999469i	$0.510375\pi$
$152$	5.13082	0.416164
$153$	−2.90141	−0.234565
$154$	0	0
$155$	−6.46901	−0.519604
$156$	−2.45558	−0.196604
$157$	−18.0983	−1.44440	−0.722199	−	0.691685i	$-0.756868\pi$
$157$	−18.0983	−1.44440	−0.722199	+	0.691685i	$0.756868\pi$
$158$	12.7528	1.01456
$159$	9.75352	0.773504
$160$	−1.00000	−0.0790569
$161$	4.97385	0.391994
$162$	9.06423	0.712154
$163$	8.78949	0.688446	0.344223	−	0.938888i	$-0.388142\pi$
$163$	8.78949	0.688446	0.344223	+	0.938888i	$0.388142\pi$
$164$	−3.10213	−0.242235
$165$	0	0
$166$	8.95284	0.694875
$167$	11.5772	0.895871	0.447935	−	0.894066i	$-0.352159\pi$
$167$	11.5772	0.895871	0.447935	+	0.894066i	$0.352159\pi$
$168$	2.44508	0.188642
$169$	−11.9914	−0.922415
$170$	0.974141	0.0747132
$171$	−15.2818	−1.16863
$172$	9.46992	0.722074
$173$	−5.23233	−0.397807	−0.198903	−	0.980019i	$-0.563738\pi$
$173$	−5.23233	−0.397807	−0.198903	+	0.980019i	$0.563738\pi$
$174$	7.07434	0.536304
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−2.20587	−0.165804
$178$	5.28764	0.396326
$179$	9.84737	0.736027	0.368014	−	0.929820i	$-0.380038\pi$
$179$	9.84737	0.736027	0.368014	+	0.929820i	$0.380038\pi$
$180$	2.97843	0.221999
$181$	−20.0533	−1.49055	−0.745275	−	0.666757i	$-0.767682\pi$
$181$	−20.0533	−1.49055	−0.745275	+	0.666757i	$0.767682\pi$
$182$	−1.00429	−0.0744431
$183$	−17.3884	−1.28538
$184$	4.97385	0.366677
$185$	9.79242	0.719953
$186$	15.8173	1.15978
$187$	0	0
$188$	7.29309	0.531903
$189$	0.0527323	0.00383571
$190$	5.13082	0.372229
$191$	−17.4033	−1.25926	−0.629629	−	0.776896i	$-0.716793\pi$
$191$	−17.4033	−1.25926	−0.629629	+	0.776896i	$0.716793\pi$
$192$	2.44508	0.176459
$193$	−25.2939	−1.82069	−0.910346	−	0.413847i	$-0.864185\pi$
$193$	−25.2939	−1.82069	−0.910346	+	0.413847i	$0.864185\pi$
$194$	9.41520	0.675972
$195$	−2.45558	−0.175848
$196$	1.00000	0.0714286
$197$	9.89470	0.704968	0.352484	−	0.935818i	$-0.385337\pi$
$197$	9.89470	0.704968	0.352484	+	0.935818i	$0.385337\pi$
$198$	0	0
$199$	−1.57633	−0.111743	−0.0558717	−	0.998438i	$-0.517794\pi$
$199$	−1.57633	−0.111743	−0.0558717	+	0.998438i	$0.517794\pi$
$200$	−1.00000	−0.0707107
$201$	−4.61073	−0.325216
$202$	6.43796	0.452973
$203$	2.89329	0.203069
$204$	−2.38186	−0.166763
$205$	−3.10213	−0.216662
$206$	−6.53919	−0.455607
$207$	−14.8143	−1.02966
$208$	−1.00429	−0.0696352
$209$	0	0
$210$	2.44508	0.168727
$211$	10.2108	0.702938	0.351469	−	0.936199i	$-0.385682\pi$
$211$	10.2108	0.702938	0.351469	+	0.936199i	$0.385682\pi$
$212$	3.98903	0.273968
$213$	5.40375	0.370259
$214$	13.6151	0.930708
$215$	9.46992	0.645843
$216$	0.0527323	0.00358798
$217$	6.46901	0.439145
$218$	−4.46766	−0.302588
$219$	10.8071	0.730279
$220$	0	0
$221$	0.978322	0.0658091
$222$	−23.9433	−1.60697
$223$	−9.38022	−0.628146	−0.314073	−	0.949399i	$-0.601694\pi$
$223$	−9.38022	−0.628146	−0.314073	+	0.949399i	$0.601694\pi$
$224$	1.00000	0.0668153
$225$	2.97843	0.198562
$226$	17.2962	1.15052
$227$	−13.6115	−0.903430	−0.451715	−	0.892162i	$-0.649188\pi$
$227$	−13.6115	−0.903430	−0.451715	+	0.892162i	$0.649188\pi$
$228$	−12.5453	−0.830831
$229$	−16.2404	−1.07319	−0.536597	−	0.843839i	$-0.680290\pi$
$229$	−16.2404	−1.07319	−0.536597	+	0.843839i	$0.680290\pi$
$230$	4.97385	0.327966
$231$	0	0
$232$	2.89329	0.189954
$233$	−8.03950	−0.526685	−0.263343	−	0.964702i	$-0.584825\pi$
$233$	−8.03950	−0.526685	−0.263343	+	0.964702i	$0.584825\pi$
$234$	2.99122	0.195542
$235$	7.29309	0.475749
$236$	−0.902167	−0.0587260
$237$	−31.1816	−2.02546
$238$	−0.974141	−0.0631442
$239$	2.23140	0.144337	0.0721686	−	0.997392i	$-0.477008\pi$
$239$	2.23140	0.144337	0.0721686	+	0.997392i	$0.477008\pi$
$240$	2.44508	0.157829
$241$	−3.19243	−0.205643	−0.102821	−	0.994700i	$-0.532787\pi$
$241$	−3.19243	−0.205643	−0.102821	+	0.994700i	$0.532787\pi$
$242$	0	0
$243$	−22.0046	−1.41160
$244$	−7.11156	−0.455271
$245$	1.00000	0.0638877
$246$	7.58496	0.483599
$247$	5.15284	0.327868
$248$	6.46901	0.410783
$249$	−21.8904	−1.38725
$250$	−1.00000	−0.0632456
$251$	18.3437	1.15785	0.578923	−	0.815383i	$-0.303473\pi$
$251$	18.3437	1.15785	0.578923	+	0.815383i	$0.303473\pi$
$252$	−2.97843	−0.187624
$253$	0	0
$254$	20.1769	1.26601
$255$	−2.38186	−0.149158
$256$	1.00000	0.0625000
$257$	17.7717	1.10857	0.554285	−	0.832327i	$-0.312992\pi$
$257$	17.7717	1.10857	0.554285	+	0.832327i	$0.312992\pi$
$258$	−23.1547	−1.44155
$259$	−9.79242	−0.608471
$260$	−1.00429	−0.0622836
$261$	−8.61747	−0.533408
$262$	−11.0785	−0.684433
$263$	16.3390	1.00751	0.503754	−	0.863847i	$-0.331952\pi$
$263$	16.3390	1.00751	0.503754	+	0.863847i	$0.331952\pi$
$264$	0	0
$265$	3.98903	0.245044
$266$	−5.13082	−0.314591
$267$	−12.9287	−0.791226
$268$	−1.88571	−0.115188
$269$	22.6073	1.37839	0.689197	−	0.724574i	$-0.257964\pi$
$269$	22.6073	1.37839	0.689197	+	0.724574i	$0.257964\pi$
$270$	0.0527323	0.00320918
$271$	0.691377	0.0419981	0.0209991	−	0.999779i	$-0.493315\pi$
$271$	0.691377	0.0419981	0.0209991	+	0.999779i	$0.493315\pi$
$272$	−0.974141	−0.0590659
$273$	2.45558	0.148618
$274$	19.6702	1.18832
$275$	0	0
$276$	−12.1615	−0.732035
$277$	7.69613	0.462415	0.231208	−	0.972904i	$-0.425732\pi$
$277$	7.69613	0.462415	0.231208	+	0.972904i	$0.425732\pi$
$278$	−9.53624	−0.571946
$279$	−19.2675	−1.15352
$280$	1.00000	0.0597614
$281$	−12.7976	−0.763443	−0.381722	−	0.924277i	$-0.624669\pi$
$281$	−12.7976	−0.763443	−0.381722	+	0.924277i	$0.624669\pi$
$282$	−17.8322	−1.06189
$283$	16.7641	0.996522	0.498261	−	0.867027i	$-0.333972\pi$
$283$	16.7641	0.996522	0.498261	+	0.867027i	$0.333972\pi$
$284$	2.21005	0.131142
$285$	−12.5453	−0.743118
$286$	0	0
$287$	3.10213	0.183113
$288$	−2.97843	−0.175506
$289$	−16.0511	−0.944179
$290$	2.89329	0.169900
$291$	−23.0210	−1.34951
$292$	4.41995	0.258658
$293$	−7.67994	−0.448667	−0.224333	−	0.974512i	$-0.572020\pi$
$293$	−7.67994	−0.448667	−0.224333	+	0.974512i	$0.572020\pi$
$294$	−2.44508	−0.142600
$295$	−0.902167	−0.0525262
$296$	−9.79242	−0.569173
$297$	0	0
$298$	−9.27662	−0.537380
$299$	4.99520	0.288880
$300$	2.44508	0.141167
$301$	−9.46992	−0.545837
$302$	0.800899	0.0460866
$303$	−15.7413	−0.904317
$304$	−5.13082	−0.294273
$305$	−7.11156	−0.407207
$306$	2.90141	0.165863
$307$	18.3456	1.04704	0.523518	−	0.852015i	$-0.324619\pi$
$307$	18.3456	1.04704	0.523518	+	0.852015i	$0.324619\pi$
$308$	0	0
$309$	15.9889	0.909574
$310$	6.46901	0.367415
$311$	−17.8031	−1.00952	−0.504761	−	0.863259i	$-0.668420\pi$
$311$	−17.8031	−1.00952	−0.504761	+	0.863259i	$0.668420\pi$
$312$	2.45558	0.139020
$313$	26.7803	1.51371	0.756857	−	0.653581i	$-0.226734\pi$
$313$	26.7803	1.51371	0.756857	+	0.653581i	$0.226734\pi$
$314$	18.0983	1.02134
$315$	−2.97843	−0.167816
$316$	−12.7528	−0.717399
$317$	−35.0441	−1.96827	−0.984137	−	0.177411i	$-0.943228\pi$
$317$	−35.0441	−1.96827	−0.984137	+	0.177411i	$0.943228\pi$
$318$	−9.75352	−0.546950
$319$	0	0
$320$	1.00000	0.0559017
$321$	−33.2900	−1.85807
$322$	−4.97385	−0.277182
$323$	4.99814	0.278104
$324$	−9.06423	−0.503569
$325$	−1.00429	−0.0557081
$326$	−8.78949	−0.486805
$327$	10.9238	0.604088
$328$	3.10213	0.171286
$329$	−7.29309	−0.402081
$330$	0	0
$331$	17.7187	0.973908	0.486954	−	0.873428i	$-0.338108\pi$
$331$	17.7187	0.973908	0.486954	+	0.873428i	$0.338108\pi$
$332$	−8.95284	−0.491351
$333$	29.1661	1.59829
$334$	−11.5772	−0.633476
$335$	−1.88571	−0.103028
$336$	−2.44508	−0.133390
$337$	33.1859	1.80775	0.903875	−	0.427796i	$-0.140710\pi$
$337$	33.1859	1.80775	0.903875	+	0.427796i	$0.140710\pi$
$338$	11.9914	0.652246
$339$	−42.2906	−2.29691
$340$	−0.974141	−0.0528302
$341$	0	0
$342$	15.2818	0.826345
$343$	−1.00000	−0.0539949
$344$	−9.46992	−0.510584
$345$	−12.1615	−0.654752
$346$	5.23233	0.281292
$347$	−32.1471	−1.72575	−0.862874	−	0.505420i	$-0.831338\pi$
$347$	−32.1471	−1.72575	−0.862874	+	0.505420i	$0.831338\pi$
$348$	−7.07434	−0.379224
$349$	2.51940	0.134861	0.0674303	−	0.997724i	$-0.478520\pi$
$349$	2.51940	0.134861	0.0674303	+	0.997724i	$0.478520\pi$
$350$	1.00000	0.0534522
$351$	0.0529586	0.00282672
$352$	0	0
$353$	−5.22226	−0.277953	−0.138976	−	0.990296i	$-0.544381\pi$
$353$	−5.22226	−0.277953	−0.138976	+	0.990296i	$0.544381\pi$
$354$	2.20587	0.117241
$355$	2.21005	0.117297
$356$	−5.28764	−0.280245
$357$	2.38186	0.126061
$358$	−9.84737	−0.520450
$359$	−9.49040	−0.500884	−0.250442	−	0.968132i	$-0.580576\pi$
$359$	−9.49040	−0.500884	−0.250442	+	0.968132i	$0.580576\pi$
$360$	−2.97843	−0.156977
$361$	7.32530	0.385542
$362$	20.0533	1.05398
$363$	0	0
$364$	1.00429	0.0526392
$365$	4.41995	0.231351
$366$	17.3884	0.908904
$367$	−19.6449	−1.02545	−0.512727	−	0.858552i	$-0.671365\pi$
$367$	−19.6449	−1.02545	−0.512727	+	0.858552i	$0.671365\pi$
$368$	−4.97385	−0.259280
$369$	−9.23948	−0.480988
$370$	−9.79242	−0.509084
$371$	−3.98903	−0.207100
$372$	−15.8173	−0.820088
$373$	−20.5117	−1.06206	−0.531029	−	0.847354i	$-0.678194\pi$
$373$	−20.5117	−1.06206	−0.531029	+	0.847354i	$0.678194\pi$
$374$	0	0
$375$	2.44508	0.126264
$376$	−7.29309	−0.376113
$377$	2.90571	0.149652
$378$	−0.0527323	−0.00271226
$379$	32.1657	1.65224	0.826120	−	0.563494i	$-0.190543\pi$
$379$	32.1657	1.65224	0.826120	+	0.563494i	$0.190543\pi$
$380$	−5.13082	−0.263205
$381$	−49.3343	−2.52747
$382$	17.4033	0.890430
$383$	6.01133	0.307165	0.153582	−	0.988136i	$-0.450919\pi$
$383$	6.01133	0.307165	0.153582	+	0.988136i	$0.450919\pi$
$384$	−2.44508	−0.124775
$385$	0	0
$386$	25.2939	1.28742
$387$	28.2055	1.43377
$388$	−9.41520	−0.477985
$389$	18.5721	0.941644	0.470822	−	0.882228i	$-0.343957\pi$
$389$	18.5721	0.941644	0.470822	+	0.882228i	$0.343957\pi$
$390$	2.45558	0.124343
$391$	4.84523	0.245034
$392$	−1.00000	−0.0505076
$393$	27.0879	1.36640
$394$	−9.89470	−0.498488
$395$	−12.7528	−0.641661
$396$	0	0
$397$	5.21019	0.261492	0.130746	−	0.991416i	$-0.458263\pi$
$397$	5.21019	0.261492	0.130746	+	0.991416i	$0.458263\pi$
$398$	1.57633	0.0790145
$399$	12.5453	0.628050
$400$	1.00000	0.0500000
$401$	−22.0522	−1.10124	−0.550618	−	0.834758i	$-0.685608\pi$
$401$	−22.0522	−1.10124	−0.550618	+	0.834758i	$0.685608\pi$
$402$	4.61073	0.229962
$403$	6.49678	0.323628
$404$	−6.43796	−0.320300
$405$	−9.06423	−0.450405
$406$	−2.89329	−0.143592
$407$	0	0
$408$	2.38186	0.117919
$409$	35.6323	1.76190	0.880952	−	0.473205i	$-0.156903\pi$
$409$	35.6323	1.76190	0.880952	+	0.473205i	$0.156903\pi$
$410$	3.10213	0.153203
$411$	−48.0953	−2.37236
$412$	6.53919	0.322163
$413$	0.902167	0.0443927
$414$	14.8143	0.728082
$415$	−8.95284	−0.439478
$416$	1.00429	0.0492395
$417$	23.3169	1.14183
$418$	0	0
$419$	2.71308	0.132543	0.0662713	−	0.997802i	$-0.478890\pi$
$419$	2.71308	0.132543	0.0662713	+	0.997802i	$0.478890\pi$
$420$	−2.44508	−0.119308
$421$	−19.3933	−0.945169	−0.472585	−	0.881285i	$-0.656679\pi$
$421$	−19.3933	−0.945169	−0.472585	+	0.881285i	$0.656679\pi$
$422$	−10.2108	−0.497053
$423$	21.7220	1.05616
$424$	−3.98903	−0.193725
$425$	−0.974141	−0.0472528
$426$	−5.40375	−0.261813
$427$	7.11156	0.344152
$428$	−13.6151	−0.658110
$429$	0	0
$430$	−9.46992	−0.456680
$431$	−27.1661	−1.30854	−0.654271	−	0.756260i	$-0.727025\pi$
$431$	−27.1661	−1.30854	−0.654271	+	0.756260i	$0.727025\pi$
$432$	−0.0527323	−0.00253708
$433$	6.77403	0.325539	0.162770	−	0.986664i	$-0.447957\pi$
$433$	6.77403	0.325539	0.162770	+	0.986664i	$0.447957\pi$
$434$	−6.46901	−0.310523
$435$	−7.07434	−0.339188
$436$	4.46766	0.213962
$437$	25.5199	1.22078
$438$	−10.8071	−0.516385
$439$	36.5942	1.74654	0.873272	−	0.487232i	$-0.161993\pi$
$439$	36.5942	1.74654	0.873272	+	0.487232i	$0.161993\pi$
$440$	0	0
$441$	2.97843	0.141830
$442$	−0.978322	−0.0465340
$443$	−13.8146	−0.656352	−0.328176	−	0.944617i	$-0.606434\pi$
$443$	−13.8146	−0.656352	−0.328176	+	0.944617i	$0.606434\pi$
$444$	23.9433	1.13630
$445$	−5.28764	−0.250658
$446$	9.38022	0.444166
$447$	22.6821	1.07283
$448$	−1.00000	−0.0472456
$449$	24.7466	1.16787	0.583933	−	0.811802i	$-0.301513\pi$
$449$	24.7466	1.16787	0.583933	+	0.811802i	$0.301513\pi$
$450$	−2.97843	−0.140405
$451$	0	0
$452$	−17.2962	−0.813544
$453$	−1.95827	−0.0920074
$454$	13.6115	0.638821
$455$	1.00429	0.0470820
$456$	12.5453	0.587487
$457$	−11.0626	−0.517485	−0.258743	−	0.965946i	$-0.583308\pi$
$457$	−11.0626	−0.517485	−0.258743	+	0.965946i	$0.583308\pi$
$458$	16.2404	0.758862
$459$	0.0513687	0.00239768
$460$	−4.97385	−0.231907
$461$	17.8851	0.832990	0.416495	−	0.909138i	$-0.363258\pi$
$461$	17.8851	0.832990	0.416495	+	0.909138i	$0.363258\pi$
$462$	0	0
$463$	−8.47374	−0.393808	−0.196904	−	0.980423i	$-0.563089\pi$
$463$	−8.47374	−0.393808	−0.196904	+	0.980423i	$0.563089\pi$
$464$	−2.89329	−0.134318
$465$	−15.8173	−0.733509
$466$	8.03950	0.372423
$467$	14.7272	0.681495	0.340747	−	0.940155i	$-0.389320\pi$
$467$	14.7272	0.681495	0.340747	+	0.940155i	$0.389320\pi$
$468$	−2.99122	−0.138269
$469$	1.88571	0.0870742
$470$	−7.29309	−0.336405
$471$	−44.2517	−2.03901
$472$	0.902167	0.0415256
$473$	0	0
$474$	31.1816	1.43222
$475$	−5.13082	−0.235418
$476$	0.974141	0.0446497
$477$	11.8811	0.543997
$478$	−2.23140	−0.102062
$479$	34.1872	1.56205	0.781026	−	0.624498i	$-0.214696\pi$
$479$	34.1872	1.56205	0.781026	+	0.624498i	$0.214696\pi$
$480$	−2.44508	−0.111602
$481$	−9.83445	−0.448413
$482$	3.19243	0.145411
$483$	12.1615	0.553366
$484$	0	0
$485$	−9.41520	−0.427522
$486$	22.0046	0.998150
$487$	−32.1518	−1.45694	−0.728468	−	0.685080i	$-0.759767\pi$
$487$	−32.1518	−1.45694	−0.728468	+	0.685080i	$0.759767\pi$
$488$	7.11156	0.321925
$489$	21.4910	0.971858
$490$	−1.00000	−0.0451754
$491$	16.6210	0.750097	0.375048	−	0.927005i	$-0.377626\pi$
$491$	16.6210	0.750097	0.375048	+	0.927005i	$0.377626\pi$
$492$	−7.58496	−0.341956
$493$	2.81847	0.126938
$494$	−5.15284	−0.231837
$495$	0	0
$496$	−6.46901	−0.290467
$497$	−2.21005	−0.0991342
$498$	21.8904	0.980934
$499$	26.8144	1.20038	0.600189	−	0.799858i	$-0.295092\pi$
$499$	26.8144	1.20038	0.600189	+	0.799858i	$0.295092\pi$
$500$	1.00000	0.0447214
$501$	28.3072	1.26467
$502$	−18.3437	−0.818720
$503$	1.76608	0.0787458	0.0393729	−	0.999225i	$-0.487464\pi$
$503$	1.76608	0.0787458	0.0393729	+	0.999225i	$0.487464\pi$
$504$	2.97843	0.132670
$505$	−6.43796	−0.286485
$506$	0	0
$507$	−29.3200	−1.30215
$508$	−20.1769	−0.895207
$509$	26.5080	1.17495	0.587474	−	0.809243i	$-0.300122\pi$
$509$	26.5080	1.17495	0.587474	+	0.809243i	$0.300122\pi$
$510$	2.38186	0.105470
$511$	−4.41995	−0.195527
$512$	−1.00000	−0.0441942
$513$	0.270560	0.0119455
$514$	−17.7717	−0.783878
$515$	6.53919	0.288151
$516$	23.1547	1.01933
$517$	0	0
$518$	9.79242	0.430254
$519$	−12.7935	−0.561572
$520$	1.00429	0.0440412
$521$	−15.0004	−0.657181	−0.328590	−	0.944473i	$-0.606574\pi$
$521$	−15.0004	−0.657181	−0.328590	+	0.944473i	$0.606574\pi$
$522$	8.61747	0.377177
$523$	28.8065	1.25962	0.629809	−	0.776750i	$-0.283133\pi$
$523$	28.8065	1.25962	0.629809	+	0.776750i	$0.283133\pi$
$524$	11.0785	0.483967
$525$	−2.44508	−0.106712
$526$	−16.3390	−0.712416
$527$	6.30173	0.274508
$528$	0	0
$529$	1.73918	0.0756167
$530$	−3.98903	−0.173273
$531$	−2.68704	−0.116608
$532$	5.13082	0.222449
$533$	3.11544	0.134945
$534$	12.9287	0.559481
$535$	−13.6151	−0.588632
$536$	1.88571	0.0814505
$537$	24.0776	1.03903
$538$	−22.6073	−0.974671
$539$	0	0
$540$	−0.0527323	−0.00226924
$541$	26.0652	1.12063	0.560315	−	0.828280i	$-0.310680\pi$
$541$	26.0652	1.12063	0.560315	+	0.828280i	$0.310680\pi$
$542$	−0.691377	−0.0296972
$543$	−49.0320	−2.10417
$544$	0.974141	0.0417659
$545$	4.46766	0.191374
$546$	−2.45558	−0.105089
$547$	23.8493	1.01972	0.509860	−	0.860257i	$-0.329697\pi$
$547$	23.8493	1.01972	0.509860	+	0.860257i	$0.329697\pi$
$548$	−19.6702	−0.840269
$549$	−21.1813	−0.903996
$550$	0	0
$551$	14.8449	0.632416
$552$	12.1615	0.517627
$553$	12.7528	0.542302
$554$	−7.69613	−0.326977
$555$	23.9433	1.01634
$556$	9.53624	0.404427
$557$	−28.5937	−1.21155	−0.605776	−	0.795635i	$-0.707137\pi$
$557$	−28.5937	−1.21155	−0.605776	+	0.795635i	$0.707137\pi$
$558$	19.2675	0.815659
$559$	−9.51057	−0.402254
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	12.7976	0.539836
$563$	−46.9795	−1.97995	−0.989974	−	0.141249i	$-0.954888\pi$
$563$	−46.9795	−1.97995	−0.989974	+	0.141249i	$0.954888\pi$
$564$	17.8322	0.750872
$565$	−17.2962	−0.727656
$566$	−16.7641	−0.704648
$567$	9.06423	0.380662
$568$	−2.21005	−0.0927316
$569$	10.8453	0.454657	0.227329	−	0.973818i	$-0.427001\pi$
$569$	10.8453	0.454657	0.227329	+	0.973818i	$0.427001\pi$
$570$	12.5453	0.525464
$571$	−12.5385	−0.524718	−0.262359	−	0.964970i	$-0.584500\pi$
$571$	−12.5385	−0.524718	−0.262359	+	0.964970i	$0.584500\pi$
$572$	0	0
$573$	−42.5525	−1.77766
$574$	−3.10213	−0.129480
$575$	−4.97385	−0.207424
$576$	2.97843	0.124101
$577$	−16.5167	−0.687601	−0.343800	−	0.939043i	$-0.611714\pi$
$577$	−16.5167	−0.687601	−0.343800	+	0.939043i	$0.611714\pi$
$578$	16.0511	0.667636
$579$	−61.8456	−2.57022
$580$	−2.89329	−0.120137
$581$	8.95284	0.371426
$582$	23.0210	0.954249
$583$	0	0
$584$	−4.41995	−0.182899
$585$	−2.99122	−0.123672
$586$	7.67994	0.317255
$587$	−0.475698	−0.0196341	−0.00981707	−	0.999952i	$-0.503125\pi$
$587$	−0.475698	−0.0196341	−0.00981707	+	0.999952i	$0.503125\pi$
$588$	2.44508	0.100834
$589$	33.1913	1.36762
$590$	0.902167	0.0371416
$591$	24.1934	0.995183
$592$	9.79242	0.402466
$593$	−27.6428	−1.13515	−0.567576	−	0.823321i	$-0.692119\pi$
$593$	−27.6428	−1.13515	−0.567576	+	0.823321i	$0.692119\pi$
$594$	0	0
$595$	0.974141	0.0399359
$596$	9.27662	0.379985
$597$	−3.85427	−0.157745
$598$	−4.99520	−0.204269
$599$	−4.39806	−0.179700	−0.0898500	−	0.995955i	$-0.528639\pi$
$599$	−4.39806	−0.179700	−0.0898500	+	0.995955i	$0.528639\pi$
$600$	−2.44508	−0.0998201
$601$	−32.1891	−1.31302	−0.656511	−	0.754316i	$-0.727968\pi$
$601$	−32.1891	−1.31302	−0.656511	+	0.754316i	$0.727968\pi$
$602$	9.46992	0.385965
$603$	−5.61647	−0.228721
$604$	−0.800899	−0.0325881
$605$	0	0
$606$	15.7413	0.639448
$607$	−15.0219	−0.609721	−0.304861	−	0.952397i	$-0.598610\pi$
$607$	−15.0219	−0.609721	−0.304861	+	0.952397i	$0.598610\pi$
$608$	5.13082	0.208082
$609$	7.07434	0.286667
$610$	7.11156	0.287939
$611$	−7.32440	−0.296314
$612$	−2.90141	−0.117283
$613$	1.12624	0.0454885	0.0227443	−	0.999741i	$-0.492760\pi$
$613$	1.12624	0.0454885	0.0227443	+	0.999741i	$0.492760\pi$
$614$	−18.3456	−0.740366
$615$	−7.58496	−0.305855
$616$	0	0
$617$	−6.79925	−0.273728	−0.136864	−	0.990590i	$-0.543702\pi$
$617$	−6.79925	−0.273728	−0.136864	+	0.990590i	$0.543702\pi$
$618$	−15.9889	−0.643166
$619$	36.5101	1.46746	0.733732	−	0.679439i	$-0.237777\pi$
$619$	36.5101	1.46746	0.733732	+	0.679439i	$0.237777\pi$
$620$	−6.46901	−0.259802
$621$	0.262282	0.0105250
$622$	17.8031	0.713841
$623$	5.28764	0.211845
$624$	−2.45558	−0.0983019
$625$	1.00000	0.0400000
$626$	−26.7803	−1.07036
$627$	0	0
$628$	−18.0983	−0.722199
$629$	−9.53919	−0.380352
$630$	2.97843	0.118664
$631$	0.487747	0.0194169	0.00970845	−	0.999953i	$-0.496910\pi$
$631$	0.487747	0.0194169	0.00970845	+	0.999953i	$0.496910\pi$
$632$	12.7528	0.507278
$633$	24.9662	0.992317
$634$	35.0441	1.39178
$635$	−20.1769	−0.800697
$636$	9.75352	0.386752
$637$	−1.00429	−0.0397915
$638$	0	0
$639$	6.58248	0.260399
$640$	−1.00000	−0.0395285
$641$	29.9446	1.18274	0.591371	−	0.806400i	$-0.298587\pi$
$641$	29.9446	1.18274	0.591371	+	0.806400i	$0.298587\pi$
$642$	33.2900	1.31385
$643$	36.7794	1.45044	0.725220	−	0.688518i	$-0.241738\pi$
$643$	36.7794	1.45044	0.725220	+	0.688518i	$0.241738\pi$
$644$	4.97385	0.195997
$645$	23.1547	0.911717
$646$	−4.99814	−0.196649
$647$	−24.8585	−0.977287	−0.488643	−	0.872484i	$-0.662508\pi$
$647$	−24.8585	−0.977287	−0.488643	+	0.872484i	$0.662508\pi$
$648$	9.06423	0.356077
$649$	0	0
$650$	1.00429	0.0393916
$651$	15.8173	0.619928
$652$	8.78949	0.344223
$653$	−25.7286	−1.00684	−0.503420	−	0.864042i	$-0.667925\pi$
$653$	−25.7286	−1.00684	−0.503420	+	0.864042i	$0.667925\pi$
$654$	−10.9238	−0.427155
$655$	11.0785	0.432874
$656$	−3.10213	−0.121118
$657$	13.1645	0.513597
$658$	7.29309	0.284314
$659$	45.1117	1.75730	0.878652	−	0.477463i	$-0.158443\pi$
$659$	45.1117	1.75730	0.878652	+	0.477463i	$0.158443\pi$
$660$	0	0
$661$	−41.3111	−1.60682	−0.803409	−	0.595428i	$-0.796983\pi$
$661$	−41.3111	−1.60682	−0.803409	+	0.595428i	$0.796983\pi$
$662$	−17.7187	−0.688657
$663$	2.39208	0.0929007
$664$	8.95284	0.347437
$665$	5.13082	0.198965
$666$	−29.1661	−1.13016
$667$	14.3908	0.557214
$668$	11.5772	0.447935
$669$	−22.9354	−0.886735
$670$	1.88571	0.0728515
$671$	0	0
$672$	2.44508	0.0943211
$673$	35.7108	1.37655	0.688275	−	0.725450i	$-0.258368\pi$
$673$	35.7108	1.37655	0.688275	+	0.725450i	$0.258368\pi$
$674$	−33.1859	−1.27827
$675$	−0.0527323	−0.00202967
$676$	−11.9914	−0.461208
$677$	44.0214	1.69188	0.845940	−	0.533278i	$-0.179040\pi$
$677$	44.0214	1.69188	0.845940	+	0.533278i	$0.179040\pi$
$678$	42.2906	1.62416
$679$	9.41520	0.361322
$680$	0.974141	0.0373566
$681$	−33.2814	−1.27534
$682$	0	0
$683$	48.6683	1.86224	0.931120	−	0.364712i	$-0.118833\pi$
$683$	48.6683	1.86224	0.931120	+	0.364712i	$0.118833\pi$
$684$	−15.2818	−0.584314
$685$	−19.6702	−0.751559
$686$	1.00000	0.0381802
$687$	−39.7091	−1.51499
$688$	9.46992	0.361037
$689$	−4.00616	−0.152622
$690$	12.1615	0.462980
$691$	2.67864	0.101900	0.0509501	−	0.998701i	$-0.483775\pi$
$691$	2.67864	0.101900	0.0509501	+	0.998701i	$0.483775\pi$
$692$	−5.23233	−0.198903
$693$	0	0
$694$	32.1471	1.22029
$695$	9.53624	0.361730
$696$	7.07434	0.268152
$697$	3.02191	0.114463
$698$	−2.51940	−0.0953608
$699$	−19.6572	−0.743505
$700$	−1.00000	−0.0377964
$701$	3.34202	0.126226	0.0631131	−	0.998006i	$-0.479897\pi$
$701$	3.34202	0.126226	0.0631131	+	0.998006i	$0.479897\pi$
$702$	−0.0529586	−0.00199880
$703$	−50.2431	−1.89496
$704$	0	0
$705$	17.8322	0.671600
$706$	5.22226	0.196542
$707$	6.43796	0.242124
$708$	−2.20587	−0.0829018
$709$	−14.6950	−0.551882	−0.275941	−	0.961175i	$-0.588989\pi$
$709$	−14.6950	−0.551882	−0.275941	+	0.961175i	$0.588989\pi$
$710$	−2.21005	−0.0829416
$711$	−37.9832	−1.42448
$712$	5.28764	0.198163
$713$	32.1759	1.20500
$714$	−2.38186	−0.0891387
$715$	0	0
$716$	9.84737	0.368014
$717$	5.45596	0.203757
$718$	9.49040	0.354178
$719$	−22.0502	−0.822332	−0.411166	−	0.911560i	$-0.634878\pi$
$719$	−22.0502	−0.822332	−0.411166	+	0.911560i	$0.634878\pi$
$720$	2.97843	0.111000
$721$	−6.53919	−0.243532
$722$	−7.32530	−0.272619
$723$	−7.80576	−0.290299
$724$	−20.0533	−0.745275
$725$	−2.89329	−0.107454
$726$	0	0
$727$	51.5714	1.91268	0.956339	−	0.292259i	$-0.0944068\pi$
$727$	51.5714	1.91268	0.956339	+	0.292259i	$0.0944068\pi$
$728$	−1.00429	−0.0372216
$729$	−26.6104	−0.985571
$730$	−4.41995	−0.163590
$731$	−9.22503	−0.341200
$732$	−17.3884	−0.642692
$733$	15.0644	0.556417	0.278209	−	0.960521i	$-0.410259\pi$
$733$	15.0644	0.556417	0.278209	+	0.960521i	$0.410259\pi$
$734$	19.6449	0.725106
$735$	2.44508	0.0901883
$736$	4.97385	0.183339
$737$	0	0
$738$	9.23948	0.340110
$739$	−2.91374	−0.107184	−0.0535918	−	0.998563i	$-0.517067\pi$
$739$	−2.91374	−0.107184	−0.0535918	+	0.998563i	$0.517067\pi$
$740$	9.79242	0.359976
$741$	12.5991	0.462841
$742$	3.98903	0.146442
$743$	19.3058	0.708261	0.354131	−	0.935196i	$-0.384777\pi$
$743$	19.3058	0.708261	0.354131	+	0.935196i	$0.384777\pi$
$744$	15.8173	0.579889
$745$	9.27662	0.339869
$746$	20.5117	0.750988
$747$	−26.6654	−0.975637
$748$	0	0
$749$	13.6151	0.497485
$750$	−2.44508	−0.0892818
$751$	−2.00896	−0.0733080	−0.0366540	−	0.999328i	$-0.511670\pi$
$751$	−2.00896	−0.0733080	−0.0366540	+	0.999328i	$0.511670\pi$
$752$	7.29309	0.265952
$753$	44.8519	1.63449
$754$	−2.90571	−0.105820
$755$	−0.800899	−0.0291477
$756$	0.0527323	0.00191785
$757$	−12.9220	−0.469658	−0.234829	−	0.972037i	$-0.575453\pi$
$757$	−12.9220	−0.469658	−0.234829	+	0.972037i	$0.575453\pi$
$758$	−32.1657	−1.16831
$759$	0	0
$760$	5.13082	0.186114
$761$	28.5718	1.03573	0.517864	−	0.855463i	$-0.326727\pi$
$761$	28.5718	1.03573	0.517864	+	0.855463i	$0.326727\pi$
$762$	49.3343	1.78719
$763$	−4.46766	−0.161740
$764$	−17.4033	−0.629629
$765$	−2.90141	−0.104901
$766$	−6.01133	−0.217198
$767$	0.906039	0.0327152
$768$	2.44508	0.0882294
$769$	−28.4564	−1.02616	−0.513082	−	0.858340i	$-0.671496\pi$
$769$	−28.4564	−1.02616	−0.513082	+	0.858340i	$0.671496\pi$
$770$	0	0
$771$	43.4534	1.56494
$772$	−25.2939	−0.910346
$773$	−17.7792	−0.639472	−0.319736	−	0.947507i	$-0.603594\pi$
$773$	−17.7792	−0.639472	−0.319736	+	0.947507i	$0.603594\pi$
$774$	−28.2055	−1.01383
$775$	−6.46901	−0.232374
$776$	9.41520	0.337986
$777$	−23.9433	−0.858960
$778$	−18.5721	−0.665843
$779$	15.9164	0.570266
$780$	−2.45558	−0.0879239
$781$	0	0
$782$	−4.84523	−0.173265
$783$	0.152570	0.00545240
$784$	1.00000	0.0357143
$785$	−18.0983	−0.645954
$786$	−27.0879	−0.966193
$787$	36.4440	1.29909	0.649544	−	0.760324i	$-0.274960\pi$
$787$	36.4440	1.29909	0.649544	+	0.760324i	$0.274960\pi$
$788$	9.89470	0.352484
$789$	39.9503	1.42227
$790$	12.7528	0.453723
$791$	17.2962	0.614981
$792$	0	0
$793$	7.14209	0.253623
$794$	−5.21019	−0.184903
$795$	9.75352	0.345922
$796$	−1.57633	−0.0558717
$797$	45.6253	1.61613	0.808066	−	0.589092i	$-0.200514\pi$
$797$	45.6253	1.61613	0.808066	+	0.589092i	$0.200514\pi$
$798$	−12.5453	−0.444098
$799$	−7.10450	−0.251339
$800$	−1.00000	−0.0353553
$801$	−15.7489	−0.556460
$802$	22.0522	0.778691
$803$	0	0
$804$	−4.61073	−0.162608
$805$	4.97385	0.175305
$806$	−6.49678	−0.228839
$807$	55.2768	1.94584
$808$	6.43796	0.226487
$809$	−29.3137	−1.03062	−0.515308	−	0.857005i	$-0.672322\pi$
$809$	−29.3137	−1.03062	−0.515308	+	0.857005i	$0.672322\pi$
$810$	9.06423	0.318485
$811$	−22.8889	−0.803739	−0.401870	−	0.915697i	$-0.631640\pi$
$811$	−22.8889	−0.803739	−0.401870	+	0.915697i	$0.631640\pi$
$812$	2.89329	0.101535
$813$	1.69047	0.0592875
$814$	0	0
$815$	8.78949	0.307882
$816$	−2.38186	−0.0833816
$817$	−48.5884	−1.69989
$818$	−35.6323	−1.24585
$819$	2.99122	0.104522
$820$	−3.10213	−0.108331
$821$	−17.6975	−0.617648	−0.308824	−	0.951119i	$-0.599935\pi$
$821$	−17.6975	−0.617648	−0.308824	+	0.951119i	$0.599935\pi$
$822$	48.0953	1.67751
$823$	−18.5622	−0.647037	−0.323518	−	0.946222i	$-0.604866\pi$
$823$	−18.5622	−0.647037	−0.323518	+	0.946222i	$0.604866\pi$
$824$	−6.53919	−0.227803
$825$	0	0
$826$	−0.902167	−0.0313904
$827$	−26.0843	−0.907041	−0.453521	−	0.891246i	$-0.649832\pi$
$827$	−26.0843	−0.907041	−0.453521	+	0.891246i	$0.649832\pi$
$828$	−14.8143	−0.514832
$829$	42.9069	1.49022	0.745109	−	0.666943i	$-0.232398\pi$
$829$	42.9069	1.49022	0.745109	+	0.666943i	$0.232398\pi$
$830$	8.95284	0.310758
$831$	18.8177	0.652778
$832$	−1.00429	−0.0348176
$833$	−0.974141	−0.0337520
$834$	−23.3169	−0.807399
$835$	11.5772	0.400645
$836$	0	0
$837$	0.341126	0.0117910
$838$	−2.71308	−0.0937217
$839$	−16.6107	−0.573464	−0.286732	−	0.958011i	$-0.592569\pi$
$839$	−16.6107	−0.573464	−0.286732	+	0.958011i	$0.592569\pi$
$840$	2.44508	0.0843634
$841$	−20.6289	−0.711340
$842$	19.3933	0.668336
$843$	−31.2913	−1.07773
$844$	10.2108	0.351469
$845$	−11.9914	−0.412517
$846$	−21.7220	−0.746817
$847$	0	0
$848$	3.98903	0.136984
$849$	40.9896	1.40676
$850$	0.974141	0.0334127
$851$	−48.7060	−1.66962
$852$	5.40375	0.185130
$853$	−44.6067	−1.52730	−0.763652	−	0.645628i	$-0.776596\pi$
$853$	−44.6067	−1.52730	−0.763652	+	0.645628i	$0.776596\pi$
$854$	−7.11156	−0.243352
$855$	−15.2818	−0.522627
$856$	13.6151	0.465354
$857$	30.6150	1.04579	0.522895	−	0.852397i	$-0.324852\pi$
$857$	30.6150	1.04579	0.522895	+	0.852397i	$0.324852\pi$
$858$	0	0
$859$	−11.1817	−0.381515	−0.190757	−	0.981637i	$-0.561094\pi$
$859$	−11.1817	−0.381515	−0.190757	+	0.981637i	$0.561094\pi$
$860$	9.46992	0.322921
$861$	7.58496	0.258495
$862$	27.1661	0.925279
$863$	−44.5350	−1.51599	−0.757995	−	0.652260i	$-0.773821\pi$
$863$	−44.5350	−1.51599	−0.757995	+	0.652260i	$0.773821\pi$
$864$	0.0527323	0.00179399
$865$	−5.23233	−0.177905
$866$	−6.77403	−0.230191
$867$	−39.2462	−1.33287
$868$	6.46901	0.219573
$869$	0	0
$870$	7.07434	0.239842
$871$	1.89381	0.0641693
$872$	−4.46766	−0.151294
$873$	−28.0426	−0.949097
$874$	−25.5199	−0.863224
$875$	−1.00000	−0.0338062
$876$	10.8071	0.365139
$877$	−16.9159	−0.571209	−0.285604	−	0.958348i	$-0.592194\pi$
$877$	−16.9159	−0.571209	−0.285604	+	0.958348i	$0.592194\pi$
$878$	−36.5942	−1.23499
$879$	−18.7781	−0.633369
$880$	0	0
$881$	−9.06950	−0.305559	−0.152780	−	0.988260i	$-0.548822\pi$
$881$	−9.06950	−0.305559	−0.152780	+	0.988260i	$0.548822\pi$
$882$	−2.97843	−0.100289
$883$	25.8923	0.871344	0.435672	−	0.900105i	$-0.356511\pi$
$883$	25.8923	0.871344	0.435672	+	0.900105i	$0.356511\pi$
$884$	0.978322	0.0329045
$885$	−2.20587	−0.0741496
$886$	13.8146	0.464111
$887$	25.9193	0.870284	0.435142	−	0.900362i	$-0.356698\pi$
$887$	25.9193	0.870284	0.435142	+	0.900362i	$0.356698\pi$
$888$	−23.9433	−0.803484
$889$	20.1769	0.676713
$890$	5.28764	0.177242
$891$	0	0
$892$	−9.38022	−0.314073
$893$	−37.4195	−1.25220
$894$	−22.6821	−0.758603
$895$	9.84737	0.329161
$896$	1.00000	0.0334077
$897$	12.2137	0.407803
$898$	−24.7466	−0.825806
$899$	18.7167	0.624238
$900$	2.97843	0.0992811
$901$	−3.88588	−0.129457
$902$	0	0
$903$	−23.1547	−0.770541
$904$	17.2962	0.575262
$905$	−20.0533	−0.666595
$906$	1.95827	0.0650590
$907$	12.0946	0.401596	0.200798	−	0.979633i	$-0.435647\pi$
$907$	12.0946	0.401596	0.200798	+	0.979633i	$0.435647\pi$
$908$	−13.6115	−0.451715
$909$	−19.1750	−0.635996
$910$	−1.00429	−0.0332920
$911$	12.1158	0.401416	0.200708	−	0.979651i	$-0.435676\pi$
$911$	12.1158	0.401416	0.200708	+	0.979651i	$0.435676\pi$
$912$	−12.5453	−0.415416
$913$	0	0
$914$	11.0626	0.365917
$915$	−17.3884	−0.574841
$916$	−16.2404	−0.536597
$917$	−11.0785	−0.365845
$918$	−0.0513687	−0.00169542
$919$	53.3173	1.75877	0.879387	−	0.476107i	$-0.157953\pi$
$919$	53.3173	1.75877	0.879387	+	0.476107i	$0.157953\pi$
$920$	4.97385	0.163983
$921$	44.8564	1.47807
$922$	−17.8851	−0.589013
$923$	−2.21954	−0.0730569
$924$	0	0
$925$	9.79242	0.321973
$926$	8.47374	0.278464
$927$	19.4765	0.639693
$928$	2.89329	0.0949769
$929$	−33.2434	−1.09068	−0.545340	−	0.838215i	$-0.683600\pi$
$929$	−33.2434	−1.09068	−0.545340	+	0.838215i	$0.683600\pi$
$930$	15.8173	0.518669
$931$	−5.13082	−0.168156
$932$	−8.03950	−0.263343
$933$	−43.5301	−1.42511
$934$	−14.7272	−0.481889
$935$	0	0
$936$	2.99122	0.0977711
$937$	41.0718	1.34176	0.670879	−	0.741567i	$-0.265917\pi$
$937$	41.0718	1.34176	0.670879	+	0.741567i	$0.265917\pi$
$938$	−1.88571	−0.0615708
$939$	65.4801	2.13686
$940$	7.29309	0.237874
$941$	13.1369	0.428251	0.214125	−	0.976806i	$-0.431310\pi$
$941$	13.1369	0.428251	0.214125	+	0.976806i	$0.431310\pi$
$942$	44.2517	1.44180
$943$	15.4295	0.502454
$944$	−0.902167	−0.0293630
$945$	0.0527323	0.00171538
$946$	0	0
$947$	−49.2785	−1.60133	−0.800667	−	0.599109i	$-0.795522\pi$
$947$	−49.2785	−1.60133	−0.800667	+	0.599109i	$0.795522\pi$
$948$	−31.1816	−1.01273
$949$	−4.43892	−0.144093
$950$	5.13082	0.166466
$951$	−85.6858	−2.77855
$952$	−0.974141	−0.0315721
$953$	42.4198	1.37411	0.687056	−	0.726604i	$-0.258903\pi$
$953$	42.4198	1.37411	0.687056	+	0.726604i	$0.258903\pi$
$954$	−11.8811	−0.384664
$955$	−17.4033	−0.563158
$956$	2.23140	0.0721686
$957$	0	0
$958$	−34.1872	−1.10454
$959$	19.6702	0.635184
$960$	2.44508	0.0789147
$961$	10.8481	0.349939
$962$	9.83445	0.317076
$963$	−40.5516	−1.30676
$964$	−3.19243	−0.102821
$965$	−25.2939	−0.814238
$966$	−12.1615	−0.391289
$967$	26.3349	0.846874	0.423437	−	0.905926i	$-0.360823\pi$
$967$	26.3349	0.846874	0.423437	+	0.905926i	$0.360823\pi$
$968$	0	0
$969$	12.2209	0.392591
$970$	9.41520	0.302304
$971$	18.1663	0.582983	0.291491	−	0.956573i	$-0.405848\pi$
$971$	18.1663	0.582983	0.291491	+	0.956573i	$0.405848\pi$
$972$	−22.0046	−0.705798
$973$	−9.53624	−0.305718
$974$	32.1518	1.03021
$975$	−2.45558	−0.0786415
$976$	−7.11156	−0.227635
$977$	−10.9394	−0.349983	−0.174992	−	0.984570i	$-0.555990\pi$
$977$	−10.9394	−0.349983	−0.174992	+	0.984570i	$0.555990\pi$
$978$	−21.4910	−0.687207
$979$	0	0
$980$	1.00000	0.0319438
$981$	13.3066	0.424848
$982$	−16.6210	−0.530398
$983$	−38.4069	−1.22499	−0.612496	−	0.790474i	$-0.709834\pi$
$983$	−38.4069	−1.22499	−0.612496	+	0.790474i	$0.709834\pi$
$984$	7.58496	0.241800
$985$	9.89470	0.315271
$986$	−2.81847	−0.0897584
$987$	−17.8322	−0.567606
$988$	5.15284	0.163934
$989$	−47.1019	−1.49775
$990$	0	0
$991$	55.6207	1.76685	0.883425	−	0.468572i	$-0.155232\pi$
$991$	55.6207	1.76685	0.883425	+	0.468572i	$0.155232\pi$
$992$	6.46901	0.205391
$993$	43.3237	1.37484
$994$	2.21005	0.0700985
$995$	−1.57633	−0.0499731
$996$	−21.8904	−0.693625
$997$	30.2402	0.957718	0.478859	−	0.877892i	$-0.341051\pi$
$997$	30.2402	0.957718	0.478859	+	0.877892i	$0.341051\pi$
$998$	−26.8144	−0.848796
$999$	−0.516377	−0.0163374

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8470.2.a.cy.1.5		6
11.7	odd	10		770.2.n.i.71.1	✓	12
11.8	odd	10		770.2.n.i.141.1	yes	12
11.10	odd	2		8470.2.a.de.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.i.71.1	✓	12	11.7	odd	10
770.2.n.i.141.1	yes	12	11.8	odd	10
8470.2.a.cy.1.5		6	1.1	even	1	trivial
8470.2.a.de.1.5		6	11.10	odd	2

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +2.44508 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.44508 q^{6} -1.00000 q^{7} -1.00000 q^{8} +2.97843 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +2.44508 q^{3} +1.00000 q^{4} +1.00000 q^{5} -2.44508 q^{6} -1.00000 q^{7} -1.00000 q^{8} +2.97843 q^{9} -1.00000 q^{10} +2.44508 q^{12} -1.00429 q^{13} +1.00000 q^{14} +2.44508 q^{15} +1.00000 q^{16} -0.974141 q^{17} -2.97843 q^{18} -5.13082 q^{19} +1.00000 q^{20} -2.44508 q^{21} -4.97385 q^{23} -2.44508 q^{24} +1.00000 q^{25} +1.00429 q^{26} -0.0527323 q^{27} -1.00000 q^{28} -2.89329 q^{29} -2.44508 q^{30} -6.46901 q^{31} -1.00000 q^{32} +0.974141 q^{34} -1.00000 q^{35} +2.97843 q^{36} +9.79242 q^{37} +5.13082 q^{38} -2.45558 q^{39} -1.00000 q^{40} -3.10213 q^{41} +2.44508 q^{42} +9.46992 q^{43} +2.97843 q^{45} +4.97385 q^{46} +7.29309 q^{47} +2.44508 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.38186 q^{51} -1.00429 q^{52} +3.98903 q^{53} +0.0527323 q^{54} +1.00000 q^{56} -12.5453 q^{57} +2.89329 q^{58} -0.902167 q^{59} +2.44508 q^{60} -7.11156 q^{61} +6.46901 q^{62} -2.97843 q^{63} +1.00000 q^{64} -1.00429 q^{65} -1.88571 q^{67} -0.974141 q^{68} -12.1615 q^{69} +1.00000 q^{70} +2.21005 q^{71} -2.97843 q^{72} +4.41995 q^{73} -9.79242 q^{74} +2.44508 q^{75} -5.13082 q^{76} +2.45558 q^{78} -12.7528 q^{79} +1.00000 q^{80} -9.06423 q^{81} +3.10213 q^{82} -8.95284 q^{83} -2.44508 q^{84} -0.974141 q^{85} -9.46992 q^{86} -7.07434 q^{87} -5.28764 q^{89} -2.97843 q^{90} +1.00429 q^{91} -4.97385 q^{92} -15.8173 q^{93} -7.29309 q^{94} -5.13082 q^{95} -2.44508 q^{96} -9.41520 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + q^{3} + 6 q^{4} + 6 q^{5} - q^{6} - 6 q^{7} - 6 q^{8} + 15 q^{9}+O(q^{10}) \) 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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