LMFDB - Embedded newform 8470.2.a.de.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:	$0$
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			1.5
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	$-0.455524\pi$
$13$	1.00429	0.278541	0.139270	+	0.990254i	$0.455524\pi$
$14$	1.00000	0.267261
$15$	2.44508	0.631318
$16$	1.00000	0.250000
$17$	0.974141	0.236264	0.118132	−	0.992998i	$-0.462309\pi$
$17$	0.974141	0.236264	0.118132	+	0.992998i	$0.462309\pi$
$18$	2.97843	0.702023
$19$	5.13082	1.17709	0.588545	−	0.808464i	$-0.299701\pi$
$19$	5.13082	1.17709	0.588545	+	0.808464i	$0.299701\pi$
$20$	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	$-0.413427\pi$
$29$	2.89329	0.537271	0.268635	+	0.963242i	$0.413427\pi$
$30$	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	$-0.422119\pi$
$41$	3.10213	0.484471	0.242235	+	0.970218i	$0.422119\pi$
$42$	2.44508	0.377285
$43$	−9.46992	−1.44415	−0.722074	−	0.691816i	$-0.756811\pi$
$43$	−9.46992	−1.44415	−0.722074	+	0.691816i	$0.756811\pi$
$44$	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	$-0.349542\pi$
$61$	7.11156	0.910542	0.455271	+	0.890353i	$0.349542\pi$
$62$	−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	$-0.583280\pi$
$73$	−4.41995	−0.517316	−0.258658	+	0.965969i	$0.583280\pi$
$74$	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	$-0.245333\pi$
$79$	12.7528	1.43480	0.717399	+	0.696663i	$0.245333\pi$
$80$	1.00000	0.111803
$81$	−9.06423	−1.00714
$82$	3.10213	0.342573
$83$	8.95284	0.982702	0.491351	−	0.870962i	$-0.336503\pi$
$83$	8.95284	0.982702	0.491351	+	0.870962i	$0.336503\pi$
$84$	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	$-0.396216\pi$
$101$	6.43796	0.640601	0.320300	+	0.947316i	$0.396216\pi$
$102$	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	$-0.271356\pi$
$107$	13.6151	1.31622	0.658110	+	0.752922i	$0.271356\pi$
$108$	−0.0527323	−0.00507417
$109$	−4.46766	−0.427925	−0.213962	−	0.976842i	$-0.568637\pi$
$109$	−4.46766	−0.427925	−0.213962	+	0.976842i	$0.568637\pi$
$110$	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	$-0.147028\pi$
$127$	20.1769	1.79041	0.895207	+	0.445651i	$0.147028\pi$
$128$	1.00000	0.0883883
$129$	−23.1547	−2.03866
$130$	1.00429	0.0880823
$131$	−11.0785	−0.967935	−0.483967	−	0.875086i	$-0.660805\pi$
$131$	−11.0785	−0.967935	−0.483967	+	0.875086i	$0.660805\pi$
$132$	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	$-0.632529\pi$
$139$	−9.53624	−0.808854	−0.404427	+	0.914570i	$0.632529\pi$
$140$	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	$-0.624071\pi$
$149$	−9.27662	−0.759970	−0.379985	+	0.924993i	$0.624071\pi$
$150$	2.44508	0.199640
$151$	0.800899	0.0651763	0.0325881	−	0.999469i	$-0.489625\pi$
$151$	0.800899	0.0651763	0.0325881	+	0.999469i	$0.489625\pi$
$152$	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	$-0.647841\pi$
$167$	−11.5772	−0.895871	−0.447935	+	0.894066i	$0.647841\pi$
$168$	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	$-0.436262\pi$
$173$	5.23233	0.397807	0.198903	+	0.980019i	$0.436262\pi$
$174$	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	$-0.135815\pi$
$193$	25.2939	1.82069	0.910346	+	0.413847i	$0.135815\pi$
$194$	−9.41520	−0.675972
$195$	2.45558	0.175848
$196$	1.00000	0.0714286
$197$	−9.89470	−0.704968	−0.352484	−	0.935818i	$-0.614663\pi$
$197$	−9.89470	−0.704968	−0.352484	+	0.935818i	$0.614663\pi$
$198$	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	$-0.614318\pi$
$211$	−10.2108	−0.702938	−0.351469	+	0.936199i	$0.614318\pi$
$212$	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	$-0.350812\pi$
$227$	13.6115	0.903430	0.451715	+	0.892162i	$0.350812\pi$
$228$	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	$-0.415175\pi$
$233$	8.03950	0.526685	0.263343	+	0.964702i	$0.415175\pi$
$234$	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	$-0.522992\pi$
$239$	−2.23140	−0.144337	−0.0721686	+	0.997392i	$0.522992\pi$
$240$	2.44508	0.157829
$241$	3.19243	0.205643	0.102821	−	0.994700i	$-0.467213\pi$
$241$	3.19243	0.205643	0.102821	+	0.994700i	$0.467213\pi$
$242$	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	$-0.668048\pi$
$263$	−16.3390	−1.00751	−0.503754	+	0.863847i	$0.668048\pi$
$264$	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	$-0.506685\pi$
$271$	−0.691377	−0.0419981	−0.0209991	+	0.999779i	$0.506685\pi$
$272$	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	$-0.574268\pi$
$277$	−7.69613	−0.462415	−0.231208	+	0.972904i	$0.574268\pi$
$278$	−9.53624	−0.571946
$279$	−19.2675	−1.15352
$280$	1.00000	0.0597614
$281$	12.7976	0.763443	0.381722	−	0.924277i	$-0.375331\pi$
$281$	12.7976	0.763443	0.381722	+	0.924277i	$0.375331\pi$
$282$	17.8322	1.06189
$283$	−16.7641	−0.996522	−0.498261	−	0.867027i	$-0.666028\pi$
$283$	−16.7641	−0.996522	−0.498261	+	0.867027i	$0.666028\pi$
$284$	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	$-0.427980\pi$
$293$	7.67994	0.448667	0.224333	+	0.974512i	$0.427980\pi$
$294$	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	$-0.675381\pi$
$307$	−18.3456	−1.04704	−0.523518	+	0.852015i	$0.675381\pi$
$308$	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	$-0.859290\pi$
$337$	−33.1859	−1.80775	−0.903875	+	0.427796i	$0.859290\pi$
$338$	−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	$-0.168662\pi$
$347$	32.1471	1.72575	0.862874	+	0.505420i	$0.168662\pi$
$348$	7.07434	0.379224
$349$	−2.51940	−0.134861	−0.0674303	−	0.997724i	$-0.521480\pi$
$349$	−2.51940	−0.134861	−0.0674303	+	0.997724i	$0.521480\pi$
$350$	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	$-0.419424\pi$
$359$	9.49040	0.500884	0.250442	+	0.968132i	$0.419424\pi$
$360$	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	$-0.321806\pi$
$373$	20.5117	1.06206	0.531029	+	0.847354i	$0.321806\pi$
$374$	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	$-0.843097\pi$
$409$	−35.6323	−1.76190	−0.880952	+	0.473205i	$0.843097\pi$
$410$	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	$-0.272975\pi$
$431$	27.1661	1.30854	0.654271	+	0.756260i	$0.272975\pi$
$432$	−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	$-0.838007\pi$
$439$	−36.5942	−1.74654	−0.873272	+	0.487232i	$0.838007\pi$
$440$	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	$-0.416692\pi$
$457$	11.0626	0.517485	0.258743	+	0.965946i	$0.416692\pi$
$458$	−16.2404	−0.758862
$459$	−0.0513687	−0.00239768
$460$	−4.97385	−0.231907
$461$	−17.8851	−0.832990	−0.416495	−	0.909138i	$-0.636742\pi$
$461$	−17.8851	−0.832990	−0.416495	+	0.909138i	$0.636742\pi$
$462$	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	$-0.785304\pi$
$479$	−34.1872	−1.56205	−0.781026	+	0.624498i	$0.785304\pi$
$480$	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	$-0.622374\pi$
$491$	−16.6210	−0.750097	−0.375048	+	0.927005i	$0.622374\pi$
$492$	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	$-0.512536\pi$
$503$	−1.76608	−0.0787458	−0.0393729	+	0.999225i	$0.512536\pi$
$504$	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	$-0.716867\pi$
$523$	−28.8065	−1.25962	−0.629809	+	0.776750i	$0.716867\pi$
$524$	−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	$-0.689320\pi$
$541$	−26.0652	−1.12063	−0.560315	+	0.828280i	$0.689320\pi$
$542$	−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	$-0.670303\pi$
$547$	−23.8493	−1.01972	−0.509860	+	0.860257i	$0.670303\pi$
$548$	−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	$-0.292863\pi$
$557$	28.5937	1.21155	0.605776	+	0.795635i	$0.292863\pi$
$558$	−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	$-0.0451119\pi$
$563$	46.9795	1.97995	0.989974	+	0.141249i	$0.0451119\pi$
$564$	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	$-0.572999\pi$
$569$	−10.8453	−0.454657	−0.227329	+	0.973818i	$0.572999\pi$
$570$	12.5453	0.525464
$571$	12.5385	0.524718	0.262359	−	0.964970i	$-0.415500\pi$
$571$	12.5385	0.524718	0.262359	+	0.964970i	$0.415500\pi$
$572$	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	$-0.307881\pi$
$593$	27.6428	1.13515	0.567576	+	0.823321i	$0.307881\pi$
$594$	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	$-0.272032\pi$
$601$	32.1891	1.31302	0.656511	+	0.754316i	$0.272032\pi$
$602$	−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	$-0.401390\pi$
$607$	15.0219	0.609721	0.304861	+	0.952397i	$0.401390\pi$
$608$	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	$-0.507240\pi$
$613$	−1.12624	−0.0454885	−0.0227443	+	0.999741i	$0.507240\pi$
$614$	−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	$-0.841557\pi$
$659$	−45.1117	−1.75730	−0.878652	+	0.477463i	$0.841557\pi$
$660$	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	$-0.741632\pi$
$673$	−35.7108	−1.37655	−0.688275	+	0.725450i	$0.741632\pi$
$674$	−33.1859	−1.27827
$675$	−0.0527323	−0.00202967
$676$	−11.9914	−0.461208
$677$	−44.0214	−1.69188	−0.845940	−	0.533278i	$-0.820960\pi$
$677$	−44.0214	−1.69188	−0.845940	+	0.533278i	$0.820960\pi$
$678$	−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	$-0.520103\pi$
$701$	−3.34202	−0.126226	−0.0631131	+	0.998006i	$0.520103\pi$
$702$	−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	$-0.589741\pi$
$733$	−15.0644	−0.556417	−0.278209	+	0.960521i	$0.589741\pi$
$734$	−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	$-0.482933\pi$
$739$	2.91374	0.107184	0.0535918	+	0.998563i	$0.482933\pi$
$740$	9.79242	0.359976
$741$	12.5991	0.462841
$742$	3.98903	0.146442
$743$	−19.3058	−0.708261	−0.354131	−	0.935196i	$-0.615223\pi$
$743$	−19.3058	−0.708261	−0.354131	+	0.935196i	$0.615223\pi$
$744$	−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	$-0.673273\pi$
$761$	−28.5718	−1.03573	−0.517864	+	0.855463i	$0.673273\pi$
$762$	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	$-0.328504\pi$
$769$	28.4564	1.02616	0.513082	+	0.858340i	$0.328504\pi$
$770$	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	$-0.725040\pi$
$787$	−36.4440	−1.29909	−0.649544	+	0.760324i	$0.725040\pi$
$788$	−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	$-0.327678\pi$
$809$	29.3137	1.03062	0.515308	+	0.857005i	$0.327678\pi$
$810$	−9.06423	−0.318485
$811$	22.8889	0.803739	0.401870	−	0.915697i	$-0.368360\pi$
$811$	22.8889	0.803739	0.401870	+	0.915697i	$0.368360\pi$
$812$	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	$-0.400065\pi$
$821$	17.6975	0.617648	0.308824	+	0.951119i	$0.400065\pi$
$822$	−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	$-0.350168\pi$
$827$	26.0843	0.907041	0.453521	+	0.891246i	$0.350168\pi$
$828$	−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	$-0.223404\pi$
$853$	44.6067	1.52730	0.763652	+	0.645628i	$0.223404\pi$
$854$	7.11156	0.243352
$855$	15.2818	0.522627
$856$	13.6151	0.465354
$857$	−30.6150	−1.04579	−0.522895	−	0.852397i	$-0.675148\pi$
$857$	−30.6150	−1.04579	−0.522895	+	0.852397i	$0.675148\pi$
$858$	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	$-0.407806\pi$
$877$	16.9159	0.571209	0.285604	+	0.958348i	$0.407806\pi$
$878$	−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	$-0.643302\pi$
$887$	−25.9193	−0.870284	−0.435142	+	0.900362i	$0.643302\pi$
$888$	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	$-0.842047\pi$
$919$	−53.3173	−1.75877	−0.879387	+	0.476107i	$0.842047\pi$
$920$	−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	$-0.734083\pi$
$937$	−41.0718	−1.34176	−0.670879	+	0.741567i	$0.734083\pi$
$938$	−1.88571	−0.0615708
$939$	65.4801	2.13686
$940$	7.29309	0.237874
$941$	−13.1369	−0.428251	−0.214125	−	0.976806i	$-0.568690\pi$
$941$	−13.1369	−0.428251	−0.214125	+	0.976806i	$0.568690\pi$
$942$	−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	$-0.741097\pi$
$953$	−42.4198	−1.37411	−0.687056	+	0.726604i	$0.741097\pi$
$954$	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	$-0.639177\pi$
$967$	−26.3349	−0.846874	−0.423437	+	0.905926i	$0.639177\pi$
$968$	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	$-0.658949\pi$
$997$	−30.2402	−0.957718	−0.478859	+	0.877892i	$0.658949\pi$
$998$	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.de.1.5		6
11.3	even	5		770.2.n.i.141.1	yes	12
11.4	even	5		770.2.n.i.71.1	✓	12
11.10	odd	2		8470.2.a.cy.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.n.i.71.1	✓	12	11.4	even	5
770.2.n.i.141.1	yes	12	11.3	even	5
8470.2.a.cy.1.5		6	11.10	odd	2
8470.2.a.de.1.5		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)

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

Fields

Representations

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Database

Properties

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