LMFDB - Embedded newform 6422.2.a.bk.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6422,2,Mod(1,6422)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6422, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6422.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6422 = 2 \cdot 13^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6422.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:	$51.2799281781$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 12x^{7} + 14x^{6} + 54x^{5} - 11x^{4} - 84x^{3} - 48x^{2} - 4x + 1 $ x^9 - 2x^8 - 12x^7 + 14x^6 + 54x^5 - 11x^4 - 84x^3 - 48x^2 - 4x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.69813$ of defining polynomial
Character	$\chi$	$=$	6422.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +0.162557 q^{3} +1.00000 q^{4} +2.16373 q^{5} +0.162557 q^{6} -0.548851 q^{7} +1.00000 q^{8} -2.97358 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +0.162557 q^{3} +1.00000 q^{4} +2.16373 q^{5} +0.162557 q^{6} -0.548851 q^{7} +1.00000 q^{8} -2.97358 q^{9} +2.16373 q^{10} +1.31187 q^{11} +0.162557 q^{12} -0.548851 q^{14} +0.351730 q^{15} +1.00000 q^{16} -6.35302 q^{17} -2.97358 q^{18} +1.00000 q^{19} +2.16373 q^{20} -0.0892196 q^{21} +1.31187 q^{22} -7.53345 q^{23} +0.162557 q^{24} -0.318266 q^{25} -0.971047 q^{27} -0.548851 q^{28} -6.85680 q^{29} +0.351730 q^{30} +7.23514 q^{31} +1.00000 q^{32} +0.213254 q^{33} -6.35302 q^{34} -1.18757 q^{35} -2.97358 q^{36} -6.48452 q^{37} +1.00000 q^{38} +2.16373 q^{40} -10.3417 q^{41} -0.0892196 q^{42} -3.40642 q^{43} +1.31187 q^{44} -6.43402 q^{45} -7.53345 q^{46} +5.17082 q^{47} +0.162557 q^{48} -6.69876 q^{49} -0.318266 q^{50} -1.03273 q^{51} -0.300248 q^{53} -0.971047 q^{54} +2.83854 q^{55} -0.548851 q^{56} +0.162557 q^{57} -6.85680 q^{58} +4.85098 q^{59} +0.351730 q^{60} +7.73312 q^{61} +7.23514 q^{62} +1.63205 q^{63} +1.00000 q^{64} +0.213254 q^{66} -4.46635 q^{67} -6.35302 q^{68} -1.22462 q^{69} -1.18757 q^{70} -11.1952 q^{71} -2.97358 q^{72} -15.4358 q^{73} -6.48452 q^{74} -0.0517365 q^{75} +1.00000 q^{76} -0.720021 q^{77} +0.567305 q^{79} +2.16373 q^{80} +8.76287 q^{81} -10.3417 q^{82} +17.2714 q^{83} -0.0892196 q^{84} -13.7462 q^{85} -3.40642 q^{86} -1.11462 q^{87} +1.31187 q^{88} +3.95909 q^{89} -6.43402 q^{90} -7.53345 q^{92} +1.17612 q^{93} +5.17082 q^{94} +2.16373 q^{95} +0.162557 q^{96} +7.29665 q^{97} -6.69876 q^{98} -3.90095 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} - 5 q^{3} + 9 q^{4} + q^{5} - 5 q^{6} - 13 q^{7} + 9 q^{8} + 10 q^{9}+O(q^{10}) $ 9 * q + 9 * q^2 - 5 * q^3 + 9 * q^4 + q^5 - 5 * q^6 - 13 * q^7 + 9 * q^8 + 10 * q^9	\( 9 q + 9 q^{2} - 5 q^{3} + 9 q^{4} + q^{5} - 5 q^{6} - 13 q^{7} + 9 q^{8} + 10 q^{9} + q^{10} - 13 q^{11} - 5 q^{12} - 13 q^{14} - q^{15} + 9 q^{16} - 12 q^{17} + 10 q^{18} + 9 q^{19} + q^{20} + 18 q^{21} - 13 q^{22} - 22 q^{23} - 5 q^{24} + 4 q^{25} - 26 q^{27} - 13 q^{28} + 12 q^{29} - q^{30} + q^{31} + 9 q^{32} - 28 q^{33} - 12 q^{34} - 18 q^{35} + 10 q^{36} - 25 q^{37} + 9 q^{38} + q^{40} + 11 q^{41} + 18 q^{42} - 10 q^{43} - 13 q^{44} - q^{45} - 22 q^{46} - 12 q^{47} - 5 q^{48} + 2 q^{49} + 4 q^{50} + 35 q^{51} + 9 q^{53} - 26 q^{54} + 18 q^{55} - 13 q^{56} - 5 q^{57} + 12 q^{58} + 10 q^{59} - q^{60} + 32 q^{61} + q^{62} - 63 q^{63} + 9 q^{64} - 28 q^{66} - 73 q^{67} - 12 q^{68} + 2 q^{69} - 18 q^{70} - 51 q^{71} + 10 q^{72} - 14 q^{73} - 25 q^{74} - 49 q^{75} + 9 q^{76} + 18 q^{77} - 28 q^{79} + q^{80} + 29 q^{81} + 11 q^{82} - 22 q^{83} + 18 q^{84} - 51 q^{85} - 10 q^{86} - 20 q^{87} - 13 q^{88} + 3 q^{89} - q^{90} - 22 q^{92} - 59 q^{93} - 12 q^{94} + q^{95} - 5 q^{96} + 2 q^{98} + 25 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 5 * q^3 + 9 * q^4 + q^5 - 5 * q^6 - 13 * q^7 + 9 * q^8 + 10 * q^9 + q^10 - 13 * q^11 - 5 * q^12 - 13 * q^14 - q^15 + 9 * q^16 - 12 * q^17 + 10 * q^18 + 9 * q^19 + q^20 + 18 * q^21 - 13 * q^22 - 22 * q^23 - 5 * q^24 + 4 * q^25 - 26 * q^27 - 13 * q^28 + 12 * q^29 - q^30 + q^31 + 9 * q^32 - 28 * q^33 - 12 * q^34 - 18 * q^35 + 10 * q^36 - 25 * q^37 + 9 * q^38 + q^40 + 11 * q^41 + 18 * q^42 - 10 * q^43 - 13 * q^44 - q^45 - 22 * q^46 - 12 * q^47 - 5 * q^48 + 2 * q^49 + 4 * q^50 + 35 * q^51 + 9 * q^53 - 26 * q^54 + 18 * q^55 - 13 * q^56 - 5 * q^57 + 12 * q^58 + 10 * q^59 - q^60 + 32 * q^61 + q^62 - 63 * q^63 + 9 * q^64 - 28 * q^66 - 73 * q^67 - 12 * q^68 + 2 * q^69 - 18 * q^70 - 51 * q^71 + 10 * q^72 - 14 * q^73 - 25 * q^74 - 49 * q^75 + 9 * q^76 + 18 * q^77 - 28 * q^79 + q^80 + 29 * q^81 + 11 * q^82 - 22 * q^83 + 18 * q^84 - 51 * q^85 - 10 * q^86 - 20 * q^87 - 13 * q^88 + 3 * q^89 - q^90 - 22 * q^92 - 59 * q^93 - 12 * q^94 + q^95 - 5 * q^96 + 2 * q^98 + 25 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0.162557	0.0938524	0.0469262	−	0.998898i	$-0.485057\pi$
$3$	0.162557	0.0938524	0.0469262	+	0.998898i	$0.485057\pi$
$4$	1.00000	0.500000
$5$	2.16373	0.967650	0.483825	−	0.875165i	$-0.339247\pi$
$5$	2.16373	0.967650	0.483825	+	0.875165i	$0.339247\pi$
$6$	0.162557	0.0663637
$7$	−0.548851	−0.207446	−0.103723	−	0.994606i	$-0.533076\pi$
$7$	−0.548851	−0.207446	−0.103723	+	0.994606i	$0.533076\pi$
$8$	1.00000	0.353553
$9$	−2.97358	−0.991192
$10$	2.16373	0.684232
$11$	1.31187	0.395544	0.197772	−	0.980248i	$-0.436629\pi$
$11$	1.31187	0.395544	0.197772	+	0.980248i	$0.436629\pi$
$12$	0.162557	0.0469262
$13$	0	0
$13$	0	0
$14$	−0.548851	−0.146686
$15$	0.351730	0.0908163
$16$	1.00000	0.250000
$17$	−6.35302	−1.54083	−0.770417	−	0.637541i	$-0.779952\pi$
$17$	−6.35302	−1.54083	−0.770417	+	0.637541i	$0.779952\pi$
$18$	−2.97358	−0.700878
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	2.16373	0.483825
$21$	−0.0892196	−0.0194693
$22$	1.31187	0.279692
$23$	−7.53345	−1.57083	−0.785416	−	0.618968i	$-0.787551\pi$
$23$	−7.53345	−1.57083	−0.785416	+	0.618968i	$0.787551\pi$
$24$	0.162557	0.0331818
$25$	−0.318266	−0.0636532
$26$	0	0
$27$	−0.971047	−0.186878
$28$	−0.548851	−0.103723
$29$	−6.85680	−1.27328	−0.636638	−	0.771163i	$-0.719676\pi$
$29$	−6.85680	−1.27328	−0.636638	+	0.771163i	$0.719676\pi$
$30$	0.351730	0.0642168
$31$	7.23514	1.29947	0.649734	−	0.760161i	$-0.274880\pi$
$31$	7.23514	1.29947	0.649734	+	0.760161i	$0.274880\pi$
$32$	1.00000	0.176777
$33$	0.213254	0.0371228
$34$	−6.35302	−1.08953
$35$	−1.18757	−0.200735
$36$	−2.97358	−0.495596
$37$	−6.48452	−1.06605	−0.533024	−	0.846100i	$-0.678945\pi$
$37$	−6.48452	−1.06605	−0.533024	+	0.846100i	$0.678945\pi$
$38$	1.00000	0.162221
$39$	0	0
$40$	2.16373	0.342116
$41$	−10.3417	−1.61511	−0.807553	−	0.589794i	$-0.799209\pi$
$41$	−10.3417	−1.61511	−0.807553	+	0.589794i	$0.799209\pi$
$42$	−0.0892196	−0.0137669
$43$	−3.40642	−0.519475	−0.259737	−	0.965679i	$-0.583636\pi$
$43$	−3.40642	−0.519475	−0.259737	+	0.965679i	$0.583636\pi$
$44$	1.31187	0.197772
$45$	−6.43402	−0.959127
$46$	−7.53345	−1.11075
$47$	5.17082	0.754242	0.377121	−	0.926164i	$-0.376914\pi$
$47$	5.17082	0.754242	0.377121	+	0.926164i	$0.376914\pi$
$48$	0.162557	0.0234631
$49$	−6.69876	−0.956966
$50$	−0.318266	−0.0450096
$51$	−1.03273	−0.144611
$52$	0	0
$53$	−0.300248	−0.0412423	−0.0206211	−	0.999787i	$-0.506564\pi$
$53$	−0.300248	−0.0412423	−0.0206211	+	0.999787i	$0.506564\pi$
$54$	−0.971047	−0.132143
$55$	2.83854	0.382748
$56$	−0.548851	−0.0733432
$57$	0.162557	0.0215312
$58$	−6.85680	−0.900342
$59$	4.85098	0.631543	0.315772	−	0.948835i	$-0.397737\pi$
$59$	4.85098	0.631543	0.315772	+	0.948835i	$0.397737\pi$
$60$	0.351730	0.0454082
$61$	7.73312	0.990124	0.495062	−	0.868858i	$-0.335145\pi$
$61$	7.73312	0.990124	0.495062	+	0.868858i	$0.335145\pi$
$62$	7.23514	0.918863
$63$	1.63205	0.205619
$64$	1.00000	0.125000
$65$	0	0
$66$	0.213254	0.0262498
$67$	−4.46635	−0.545652	−0.272826	−	0.962063i	$-0.587958\pi$
$67$	−4.46635	−0.545652	−0.272826	+	0.962063i	$0.587958\pi$
$68$	−6.35302	−0.770417
$69$	−1.22462	−0.147426
$70$	−1.18757	−0.141941
$71$	−11.1952	−1.32863	−0.664314	−	0.747454i	$-0.731276\pi$
$71$	−11.1952	−1.32863	−0.664314	+	0.747454i	$0.731276\pi$
$72$	−2.97358	−0.350439
$73$	−15.4358	−1.80662	−0.903312	−	0.428984i	$-0.858872\pi$
$73$	−15.4358	−1.80662	−0.903312	+	0.428984i	$0.858872\pi$
$74$	−6.48452	−0.753810
$75$	−0.0517365	−0.00597401
$76$	1.00000	0.114708
$77$	−0.720021	−0.0820540
$78$	0	0
$79$	0.567305	0.0638268	0.0319134	−	0.999491i	$-0.489840\pi$
$79$	0.567305	0.0638268	0.0319134	+	0.999491i	$0.489840\pi$
$80$	2.16373	0.241913
$81$	8.76287	0.973653
$82$	−10.3417	−1.14205
$83$	17.2714	1.89578	0.947890	−	0.318598i	$-0.103212\pi$
$83$	17.2714	1.89578	0.947890	+	0.318598i	$0.103212\pi$
$84$	−0.0892196	−0.00973466
$85$	−13.7462	−1.49099
$86$	−3.40642	−0.367324
$87$	−1.11462	−0.119500
$88$	1.31187	0.139846
$89$	3.95909	0.419662	0.209831	−	0.977738i	$-0.432709\pi$
$89$	3.95909	0.419662	0.209831	+	0.977738i	$0.432709\pi$
$90$	−6.43402	−0.678205
$91$	0	0
$92$	−7.53345	−0.785416
$93$	1.17612	0.121958
$94$	5.17082	0.533329
$95$	2.16373	0.221994
$96$	0.162557	0.0165909
$97$	7.29665	0.740863	0.370431	−	0.928860i	$-0.379210\pi$
$97$	7.29665	0.740863	0.370431	+	0.928860i	$0.379210\pi$
$98$	−6.69876	−0.676677
$99$	−3.90095	−0.392060
$100$	−0.318266	−0.0318266
$101$	−4.10621	−0.408583	−0.204292	−	0.978910i	$-0.565489\pi$
$101$	−4.10621	−0.408583	−0.204292	+	0.978910i	$0.565489\pi$
$102$	−1.03273	−0.102255
$103$	−6.81407	−0.671411	−0.335705	−	0.941967i	$-0.608975\pi$
$103$	−6.81407	−0.671411	−0.335705	+	0.941967i	$0.608975\pi$
$104$	0	0
$105$	−0.193047	−0.0188395
$106$	−0.300248	−0.0291627
$107$	9.85187	0.952416	0.476208	−	0.879333i	$-0.342011\pi$
$107$	9.85187	0.952416	0.476208	+	0.879333i	$0.342011\pi$
$108$	−0.971047	−0.0934391
$109$	5.85926	0.561215	0.280608	−	0.959823i	$-0.409464\pi$
$109$	5.85926	0.561215	0.280608	+	0.959823i	$0.409464\pi$
$110$	2.83854	0.270644
$111$	−1.05411	−0.100051
$112$	−0.548851	−0.0518615
$113$	1.86214	0.175175	0.0875876	−	0.996157i	$-0.472084\pi$
$113$	1.86214	0.175175	0.0875876	+	0.996157i	$0.472084\pi$
$114$	0.162557	0.0152249
$115$	−16.3004	−1.52002
$116$	−6.85680	−0.636638
$117$	0	0
$118$	4.85098	0.446569
$119$	3.48686	0.319640
$120$	0.351730	0.0321084
$121$	−9.27899	−0.843545
$122$	7.73312	0.700124
$123$	−1.68112	−0.151582
$124$	7.23514	0.649734
$125$	−11.5073	−1.02924
$126$	1.63205	0.145394
$127$	16.8264	1.49310	0.746549	−	0.665331i	$-0.231709\pi$
$127$	16.8264	1.49310	0.746549	+	0.665331i	$0.231709\pi$
$128$	1.00000	0.0883883
$129$	−0.553739	−0.0487540
$130$	0	0
$131$	0.576353	0.0503562	0.0251781	−	0.999683i	$-0.491985\pi$
$131$	0.576353	0.0503562	0.0251781	+	0.999683i	$0.491985\pi$
$132$	0.213254	0.0185614
$133$	−0.548851	−0.0475914
$134$	−4.46635	−0.385834
$135$	−2.10109	−0.180833
$136$	−6.35302	−0.544767
$137$	12.0950	1.03334	0.516672	−	0.856183i	$-0.327171\pi$
$137$	12.0950	1.03334	0.516672	+	0.856183i	$0.327171\pi$
$138$	−1.22462	−0.104246
$139$	−17.7106	−1.50220	−0.751098	−	0.660191i	$-0.770475\pi$
$139$	−17.7106	−1.50220	−0.751098	+	0.660191i	$0.770475\pi$
$140$	−1.18757	−0.100368
$141$	0.840554	0.0707874
$142$	−11.1952	−0.939481
$143$	0	0
$144$	−2.97358	−0.247798
$145$	−14.8363	−1.23209
$146$	−15.4358	−1.27748
$147$	−1.08893	−0.0898136
$148$	−6.48452	−0.533024
$149$	16.2820	1.33387	0.666937	−	0.745114i	$-0.267605\pi$
$149$	16.2820	1.33387	0.666937	+	0.745114i	$0.267605\pi$
$150$	−0.0517365	−0.00422426
$151$	2.48553	0.202270	0.101135	−	0.994873i	$-0.467753\pi$
$151$	2.48553	0.202270	0.101135	+	0.994873i	$0.467753\pi$
$152$	1.00000	0.0811107
$153$	18.8912	1.52726
$154$	−0.720021	−0.0580210
$155$	15.6549	1.25743
$156$	0	0
$157$	−17.3964	−1.38839	−0.694193	−	0.719789i	$-0.744239\pi$
$157$	−17.3964	−1.38839	−0.694193	+	0.719789i	$0.744239\pi$
$158$	0.567305	0.0451324
$159$	−0.0488075	−0.00387069
$160$	2.16373	0.171058
$161$	4.13474	0.325863
$162$	8.76287	0.688476
$163$	−17.4200	−1.36444	−0.682219	−	0.731148i	$-0.738985\pi$
$163$	−17.4200	−1.36444	−0.682219	+	0.731148i	$0.738985\pi$
$164$	−10.3417	−0.807553
$165$	0.461424	0.0359218
$166$	17.2714	1.34052
$167$	−24.0810	−1.86344	−0.931721	−	0.363174i	$-0.881693\pi$
$167$	−24.0810	−1.86344	−0.931721	+	0.363174i	$0.881693\pi$
$168$	−0.0892196	−0.00688344
$169$	0	0
$170$	−13.7462	−1.05429
$171$	−2.97358	−0.227395
$172$	−3.40642	−0.259737
$173$	−13.0175	−0.989706	−0.494853	−	0.868977i	$-0.664778\pi$
$173$	−13.0175	−0.989706	−0.494853	+	0.868977i	$0.664778\pi$
$174$	−1.11462	−0.0844993
$175$	0.174681	0.0132046
$176$	1.31187	0.0988860
$177$	0.788561	0.0592719
$178$	3.95909	0.296746
$179$	6.83699	0.511021	0.255510	−	0.966806i	$-0.417757\pi$
$179$	6.83699	0.511021	0.255510	+	0.966806i	$0.417757\pi$
$180$	−6.43402	−0.479563
$181$	14.3069	1.06342	0.531712	−	0.846925i	$-0.321549\pi$
$181$	14.3069	1.06342	0.531712	+	0.846925i	$0.321549\pi$
$182$	0	0
$183$	1.25707	0.0929256
$184$	−7.53345	−0.555373
$185$	−14.0308	−1.03156
$186$	1.17612	0.0862375
$187$	−8.33434	−0.609467
$188$	5.17082	0.377121
$189$	0.532960	0.0387671
$190$	2.16373	0.156974
$191$	−1.71990	−0.124448	−0.0622239	−	0.998062i	$-0.519819\pi$
$191$	−1.71990	−0.124448	−0.0622239	+	0.998062i	$0.519819\pi$
$192$	0.162557	0.0117316
$193$	4.55498	0.327875	0.163937	−	0.986471i	$-0.447581\pi$
$193$	4.55498	0.327875	0.163937	+	0.986471i	$0.447581\pi$
$194$	7.29665	0.523869
$195$	0	0
$196$	−6.69876	−0.478483
$197$	24.1762	1.72248	0.861241	−	0.508197i	$-0.169688\pi$
$197$	24.1762	1.72248	0.861241	+	0.508197i	$0.169688\pi$
$198$	−3.90095	−0.277228
$199$	4.17741	0.296129	0.148064	−	0.988978i	$-0.452696\pi$
$199$	4.17741	0.296129	0.148064	+	0.988978i	$0.452696\pi$
$200$	−0.318266	−0.0225048
$201$	−0.726038	−0.0512108
$202$	−4.10621	−0.288912
$203$	3.76336	0.264136
$204$	−1.03273	−0.0723055
$205$	−22.3767	−1.56286
$206$	−6.81407	−0.474759
$207$	22.4013	1.55700
$208$	0	0
$209$	1.31187	0.0907440
$210$	−0.193047	−0.0133215
$211$	15.2795	1.05189	0.525944	−	0.850519i	$-0.323712\pi$
$211$	15.2795	1.05189	0.525944	+	0.850519i	$0.323712\pi$
$212$	−0.300248	−0.0206211
$213$	−1.81986	−0.124695
$214$	9.85187	0.673460
$215$	−7.37059	−0.502670
$216$	−0.971047	−0.0660714
$217$	−3.97101	−0.269570
$218$	5.85926	0.396839
$219$	−2.50920	−0.169556
$220$	2.83854	0.191374
$221$	0	0
$222$	−1.05411	−0.0707469
$223$	−29.0794	−1.94730	−0.973652	−	0.228040i	$-0.926768\pi$
$223$	−29.0794	−1.94730	−0.973652	+	0.228040i	$0.926768\pi$
$224$	−0.548851	−0.0366716
$225$	0.946388	0.0630926
$226$	1.86214	0.123868
$227$	0.448232	0.0297502	0.0148751	−	0.999889i	$-0.495265\pi$
$227$	0.448232	0.0297502	0.0148751	+	0.999889i	$0.495265\pi$
$228$	0.162557	0.0107656
$229$	−3.11071	−0.205562	−0.102781	−	0.994704i	$-0.532774\pi$
$229$	−3.11071	−0.205562	−0.102781	+	0.994704i	$0.532774\pi$
$230$	−16.3004	−1.07481
$231$	−0.117045	−0.00770097
$232$	−6.85680	−0.450171
$233$	25.9915	1.70276	0.851381	−	0.524547i	$-0.175765\pi$
$233$	25.9915	1.70276	0.851381	+	0.524547i	$0.175765\pi$
$234$	0	0
$235$	11.1883	0.729842
$236$	4.85098	0.315772
$237$	0.0922195	0.00599030
$238$	3.48686	0.226019
$239$	20.4329	1.32170	0.660848	−	0.750520i	$-0.270197\pi$
$239$	20.4329	1.32170	0.660848	+	0.750520i	$0.270197\pi$
$240$	0.351730	0.0227041
$241$	18.1133	1.16678	0.583390	−	0.812192i	$-0.301726\pi$
$241$	18.1133	1.16678	0.583390	+	0.812192i	$0.301726\pi$
$242$	−9.27899	−0.596476
$243$	4.33761	0.278258
$244$	7.73312	0.495062
$245$	−14.4943	−0.926008
$246$	−1.68112	−0.107184
$247$	0	0
$248$	7.23514	0.459432
$249$	2.80759	0.177924
$250$	−11.5073	−0.727786
$251$	−26.8253	−1.69320	−0.846598	−	0.532233i	$-0.821353\pi$
$251$	−26.8253	−1.69320	−0.846598	+	0.532233i	$0.821353\pi$
$252$	1.63205	0.102809
$253$	−9.88291	−0.621333
$254$	16.8264	1.05578
$255$	−2.23455	−0.139933
$256$	1.00000	0.0625000
$257$	24.1659	1.50743	0.753715	−	0.657201i	$-0.228260\pi$
$257$	24.1659	1.50743	0.753715	+	0.657201i	$0.228260\pi$
$258$	−0.553739	−0.0344743
$259$	3.55903	0.221148
$260$	0	0
$261$	20.3892	1.26206
$262$	0.576353	0.0356072
$263$	−9.76631	−0.602217	−0.301108	−	0.953590i	$-0.597357\pi$
$263$	−9.76631	−0.602217	−0.301108	+	0.953590i	$0.597357\pi$
$264$	0.213254	0.0131249
$265$	−0.649656	−0.0399081
$266$	−0.548851	−0.0336522
$267$	0.643578	0.0393863
$268$	−4.46635	−0.272826
$269$	8.31097	0.506729	0.253364	−	0.967371i	$-0.418463\pi$
$269$	8.31097	0.506729	0.253364	+	0.967371i	$0.418463\pi$
$270$	−2.10109	−0.127868
$271$	−1.36058	−0.0826491	−0.0413245	−	0.999146i	$-0.513158\pi$
$271$	−1.36058	−0.0826491	−0.0413245	+	0.999146i	$0.513158\pi$
$272$	−6.35302	−0.385208
$273$	0	0
$274$	12.0950	0.730684
$275$	−0.417524	−0.0251777
$276$	−1.22462	−0.0737132
$277$	0.112845	0.00678020	0.00339010	−	0.999994i	$-0.498921\pi$
$277$	0.112845	0.00678020	0.00339010	+	0.999994i	$0.498921\pi$
$278$	−17.7106	−1.06221
$279$	−21.5142	−1.28802
$280$	−1.18757	−0.0709706
$281$	−25.6082	−1.52766	−0.763830	−	0.645418i	$-0.776683\pi$
$281$	−25.6082	−1.52766	−0.763830	+	0.645418i	$0.776683\pi$
$282$	0.840554	0.0500543
$283$	−9.58160	−0.569567	−0.284784	−	0.958592i	$-0.591922\pi$
$283$	−9.58160	−0.569567	−0.284784	+	0.958592i	$0.591922\pi$
$284$	−11.1952	−0.664314
$285$	0.351730	0.0208347
$286$	0	0
$287$	5.67606	0.335047
$288$	−2.97358	−0.175220
$289$	23.3608	1.37417
$290$	−14.8363	−0.871216
$291$	1.18612	0.0695318
$292$	−15.4358	−0.903312
$293$	1.29251	0.0755095	0.0377547	−	0.999287i	$-0.487979\pi$
$293$	1.29251	0.0755095	0.0377547	+	0.999287i	$0.487979\pi$
$294$	−1.08893	−0.0635078
$295$	10.4962	0.611113
$296$	−6.48452	−0.376905
$297$	−1.27389	−0.0739185
$298$	16.2820	0.943191
$299$	0	0
$300$	−0.0517365	−0.00298701
$301$	1.86962	0.107763
$302$	2.48553	0.143026
$303$	−0.667494	−0.0383465
$304$	1.00000	0.0573539
$305$	16.7324	0.958094
$306$	18.8912	1.07994
$307$	−6.22672	−0.355378	−0.177689	−	0.984087i	$-0.556862\pi$
$307$	−6.22672	−0.355378	−0.177689	+	0.984087i	$0.556862\pi$
$308$	−0.720021	−0.0410270
$309$	−1.10768	−0.0630135
$310$	15.6549	0.889138
$311$	−24.3976	−1.38346	−0.691732	−	0.722155i	$-0.743152\pi$
$311$	−24.3976	−1.38346	−0.691732	+	0.722155i	$0.743152\pi$
$312$	0	0
$313$	−24.1468	−1.36486	−0.682428	−	0.730953i	$-0.739076\pi$
$313$	−24.1468	−1.36486	−0.682428	+	0.730953i	$0.739076\pi$
$314$	−17.3964	−0.981738
$315$	3.53131	0.198967
$316$	0.567305	0.0319134
$317$	0.544955	0.0306077	0.0153039	−	0.999883i	$-0.495128\pi$
$317$	0.544955	0.0306077	0.0153039	+	0.999883i	$0.495128\pi$
$318$	−0.0488075	−0.00273699
$319$	−8.99524	−0.503637
$320$	2.16373	0.120956
$321$	1.60149	0.0893866
$322$	4.13474	0.230420
$323$	−6.35302	−0.353491
$324$	8.76287	0.486826
$325$	0	0
$326$	−17.4200	−0.964803
$327$	0.952464	0.0526714
$328$	−10.3417	−0.571026
$329$	−2.83801	−0.156464
$330$	0.461424	0.0254006
$331$	−20.0298	−1.10094	−0.550470	−	0.834855i	$-0.685551\pi$
$331$	−20.0298	−1.10094	−0.550470	+	0.834855i	$0.685551\pi$
$332$	17.2714	0.947890
$333$	19.2822	1.05666
$334$	−24.0810	−1.31765
$335$	−9.66399	−0.528000
$336$	−0.0892196	−0.00486733
$337$	0.990093	0.0539338	0.0269669	−	0.999636i	$-0.491415\pi$
$337$	0.990093	0.0539338	0.0269669	+	0.999636i	$0.491415\pi$
$338$	0	0
$339$	0.302704	0.0164406
$340$	−13.7462	−0.745494
$341$	9.49156	0.513997
$342$	−2.97358	−0.160793
$343$	7.51857	0.405965
$344$	−3.40642	−0.183662
$345$	−2.64974	−0.142657
$346$	−13.0175	−0.699828
$347$	15.8651	0.851685	0.425843	−	0.904797i	$-0.359978\pi$
$347$	15.8651	0.851685	0.425843	+	0.904797i	$0.359978\pi$
$348$	−1.11462	−0.0597500
$349$	−8.18795	−0.438291	−0.219146	−	0.975692i	$-0.570327\pi$
$349$	−8.18795	−0.438291	−0.219146	+	0.975692i	$0.570327\pi$
$350$	0.174681	0.00933707
$351$	0	0
$352$	1.31187	0.0699230
$353$	−4.64477	−0.247216	−0.123608	−	0.992331i	$-0.539447\pi$
$353$	−4.64477	−0.247216	−0.123608	+	0.992331i	$0.539447\pi$
$354$	0.788561	0.0419115
$355$	−24.2234	−1.28565
$356$	3.95909	0.209831
$357$	0.566814	0.0299990
$358$	6.83699	0.361346
$359$	5.37310	0.283582	0.141791	−	0.989897i	$-0.454714\pi$
$359$	5.37310	0.283582	0.141791	+	0.989897i	$0.454714\pi$
$360$	−6.43402	−0.339103
$361$	1.00000	0.0526316
$362$	14.3069	0.751954
$363$	−1.50837	−0.0791687
$364$	0	0
$365$	−33.3989	−1.74818
$366$	1.25707	0.0657083
$367$	−16.0979	−0.840303	−0.420152	−	0.907454i	$-0.638023\pi$
$367$	−16.0979	−0.840303	−0.420152	+	0.907454i	$0.638023\pi$
$368$	−7.53345	−0.392708
$369$	30.7519	1.60088
$370$	−14.0308	−0.729424
$371$	0.164791	0.00855554
$372$	1.17612	0.0609791
$373$	−24.4583	−1.26640	−0.633201	−	0.773988i	$-0.718259\pi$
$373$	−24.4583	−1.26640	−0.633201	+	0.773988i	$0.718259\pi$
$374$	−8.33434	−0.430958
$375$	−1.87059	−0.0965971
$376$	5.17082	0.266665
$377$	0	0
$378$	0.532960	0.0274125
$379$	8.28671	0.425660	0.212830	−	0.977089i	$-0.431732\pi$
$379$	8.28671	0.425660	0.212830	+	0.977089i	$0.431732\pi$
$380$	2.16373	0.110997
$381$	2.73525	0.140131
$382$	−1.71990	−0.0879979
$383$	5.62436	0.287391	0.143696	−	0.989622i	$-0.454101\pi$
$383$	5.62436	0.287391	0.143696	+	0.989622i	$0.454101\pi$
$384$	0.162557	0.00829546
$385$	−1.55793	−0.0793996
$386$	4.55498	0.231842
$387$	10.1293	0.514899
$388$	7.29665	0.370431
$389$	10.2852	0.521482	0.260741	−	0.965409i	$-0.416033\pi$
$389$	10.2852	0.521482	0.260741	+	0.965409i	$0.416033\pi$
$390$	0	0
$391$	47.8601	2.42039
$392$	−6.69876	−0.338339
$393$	0.0936903	0.00472605
$394$	24.1762	1.21798
$395$	1.22750	0.0617620
$396$	−3.90095	−0.196030
$397$	−20.5442	−1.03108	−0.515541	−	0.856865i	$-0.672409\pi$
$397$	−20.5442	−1.03108	−0.515541	+	0.856865i	$0.672409\pi$
$398$	4.17741	0.209395
$399$	−0.0892196	−0.00446657
$400$	−0.318266	−0.0159133
$401$	−21.8726	−1.09226	−0.546132	−	0.837699i	$-0.683900\pi$
$401$	−21.8726	−1.09226	−0.546132	+	0.837699i	$0.683900\pi$
$402$	−0.726038	−0.0362115
$403$	0	0
$404$	−4.10621	−0.204292
$405$	18.9605	0.942155
$406$	3.76336	0.186772
$407$	−8.50685	−0.421669
$408$	−1.03273	−0.0511277
$409$	−1.43173	−0.0707946	−0.0353973	−	0.999373i	$-0.511270\pi$
$409$	−1.43173	−0.0707946	−0.0353973	+	0.999373i	$0.511270\pi$
$410$	−22.3767	−1.10511
$411$	1.96613	0.0969818
$412$	−6.81407	−0.335705
$413$	−2.66246	−0.131011
$414$	22.4013	1.10096
$415$	37.3706	1.83445
$416$	0	0
$417$	−2.87899	−0.140985
$418$	1.31187	0.0641657
$419$	35.6625	1.74223	0.871114	−	0.491081i	$-0.163398\pi$
$419$	35.6625	1.74223	0.871114	+	0.491081i	$0.163398\pi$
$420$	−0.193047	−0.00941974
$421$	17.5139	0.853576	0.426788	−	0.904352i	$-0.359645\pi$
$421$	17.5139	0.853576	0.426788	+	0.904352i	$0.359645\pi$
$422$	15.2795	0.743796
$423$	−15.3758	−0.747598
$424$	−0.300248	−0.0145813
$425$	2.02195	0.0980790
$426$	−1.81986	−0.0881726
$427$	−4.24433	−0.205397
$428$	9.85187	0.476208
$429$	0	0
$430$	−7.37059	−0.355441
$431$	−17.4842	−0.842185	−0.421092	−	0.907018i	$-0.638353\pi$
$431$	−17.4842	−0.842185	−0.421092	+	0.907018i	$0.638353\pi$
$432$	−0.971047	−0.0467195
$433$	13.0367	0.626504	0.313252	−	0.949670i	$-0.398582\pi$
$433$	13.0367	0.626504	0.313252	+	0.949670i	$0.398582\pi$
$434$	−3.97101	−0.190615
$435$	−2.41174	−0.115634
$436$	5.85926	0.280608
$437$	−7.53345	−0.360374
$438$	−2.50920	−0.119894
$439$	27.5580	1.31527	0.657637	−	0.753335i	$-0.271556\pi$
$439$	27.5580	1.31527	0.657637	+	0.753335i	$0.271556\pi$
$440$	2.83854	0.135322
$441$	19.9193	0.948537
$442$	0	0
$443$	−32.1705	−1.52847	−0.764234	−	0.644939i	$-0.776883\pi$
$443$	−32.1705	−1.52847	−0.764234	+	0.644939i	$0.776883\pi$
$444$	−1.05411	−0.0500256
$445$	8.56640	0.406086
$446$	−29.0794	−1.37695
$447$	2.64676	0.125187
$448$	−0.548851	−0.0259308
$449$	36.6871	1.73137	0.865686	−	0.500587i	$-0.166882\pi$
$449$	36.6871	1.73137	0.865686	+	0.500587i	$0.166882\pi$
$450$	0.946388	0.0446132
$451$	−13.5670	−0.638846
$452$	1.86214	0.0875876
$453$	0.404041	0.0189835
$454$	0.448232	0.0210366
$455$	0	0
$456$	0.162557	0.00761244
$457$	8.52880	0.398960	0.199480	−	0.979902i	$-0.436075\pi$
$457$	8.52880	0.398960	0.199480	+	0.979902i	$0.436075\pi$
$458$	−3.11071	−0.145354
$459$	6.16908	0.287948
$460$	−16.3004	−0.760008
$461$	−19.9532	−0.929315	−0.464657	−	0.885491i	$-0.653822\pi$
$461$	−19.9532	−0.929315	−0.464657	+	0.885491i	$0.653822\pi$
$462$	−0.117045	−0.00544541
$463$	−1.16008	−0.0539133	−0.0269567	−	0.999637i	$-0.508582\pi$
$463$	−1.16008	−0.0539133	−0.0269567	+	0.999637i	$0.508582\pi$
$464$	−6.85680	−0.318319
$465$	2.54481	0.118013
$466$	25.9915	1.20404
$467$	−10.3312	−0.478070	−0.239035	−	0.971011i	$-0.576831\pi$
$467$	−10.3312	−0.478070	−0.239035	+	0.971011i	$0.576831\pi$
$468$	0	0
$469$	2.45136	0.113193
$470$	11.1883	0.516076
$471$	−2.82792	−0.130303
$472$	4.85098	0.223284
$473$	−4.46879	−0.205475
$474$	0.0922195	0.00423578
$475$	−0.318266	−0.0146031
$476$	3.48686	0.159820
$477$	0.892811	0.0408790
$478$	20.4329	0.934581
$479$	1.09845	0.0501895	0.0250947	−	0.999685i	$-0.492011\pi$
$479$	1.09845	0.0501895	0.0250947	+	0.999685i	$0.492011\pi$
$480$	0.351730	0.0160542
$481$	0	0
$482$	18.1133	0.825038
$483$	0.672131	0.0305830
$484$	−9.27899	−0.421772
$485$	15.7880	0.716896
$486$	4.33761	0.196758
$487$	23.6068	1.06972	0.534862	−	0.844939i	$-0.320364\pi$
$487$	23.6068	1.06972	0.534862	+	0.844939i	$0.320364\pi$
$488$	7.73312	0.350062
$489$	−2.83174	−0.128056
$490$	−14.4943	−0.654787
$491$	−0.936015	−0.0422418	−0.0211209	−	0.999777i	$-0.506723\pi$
$491$	−0.936015	−0.0422418	−0.0211209	+	0.999777i	$0.506723\pi$
$492$	−1.68112	−0.0757908
$493$	43.5614	1.96191
$494$	0	0
$495$	−8.44060	−0.379377
$496$	7.23514	0.324867
$497$	6.14450	0.275618
$498$	2.80759	0.125811
$499$	−35.4033	−1.58487	−0.792434	−	0.609957i	$-0.791187\pi$
$499$	−35.4033	−1.58487	−0.792434	+	0.609957i	$0.791187\pi$
$500$	−11.5073	−0.514622
$501$	−3.91454	−0.174889
$502$	−26.8253	−1.19727
$503$	−16.5408	−0.737519	−0.368760	−	0.929525i	$-0.620217\pi$
$503$	−16.5408	−0.737519	−0.368760	+	0.929525i	$0.620217\pi$
$504$	1.63205	0.0726972
$505$	−8.88473	−0.395366
$506$	−9.88291	−0.439349
$507$	0	0
$508$	16.8264	0.746549
$509$	−13.0801	−0.579764	−0.289882	−	0.957062i	$-0.593616\pi$
$509$	−13.0801	−0.579764	−0.289882	+	0.957062i	$0.593616\pi$
$510$	−2.23455	−0.0989474
$511$	8.47195	0.374777
$512$	1.00000	0.0441942
$513$	−0.971047	−0.0428728
$514$	24.1659	1.06591
$515$	−14.7438	−0.649690
$516$	−0.553739	−0.0243770
$517$	6.78345	0.298336
$518$	3.55903	0.156375
$519$	−2.11610	−0.0928863
$520$	0	0
$521$	5.13946	0.225164	0.112582	−	0.993642i	$-0.464088\pi$
$521$	5.13946	0.225164	0.112582	+	0.993642i	$0.464088\pi$
$522$	20.3892	0.892412
$523$	6.42529	0.280958	0.140479	−	0.990084i	$-0.455136\pi$
$523$	6.42529	0.280958	0.140479	+	0.990084i	$0.455136\pi$
$524$	0.576353	0.0251781
$525$	0.0283956	0.00123928
$526$	−9.76631	−0.425831
$527$	−45.9649	−2.00226
$528$	0.213254	0.00928069
$529$	33.7528	1.46751
$530$	−0.649656	−0.0282193
$531$	−14.4247	−0.625981
$532$	−0.548851	−0.0237957
$533$	0	0
$534$	0.643578	0.0278503
$535$	21.3168	0.921606
$536$	−4.46635	−0.192917
$537$	1.11140	0.0479605
$538$	8.31097	0.358311
$539$	−8.78791	−0.378522
$540$	−2.10109	−0.0904163
$541$	−1.94088	−0.0834449	−0.0417224	−	0.999129i	$-0.513285\pi$
$541$	−1.94088	−0.0834449	−0.0417224	+	0.999129i	$0.513285\pi$
$542$	−1.36058	−0.0584417
$543$	2.32569	0.0998049
$544$	−6.35302	−0.272383
$545$	12.6779	0.543060
$546$	0	0
$547$	21.4687	0.917933	0.458967	−	0.888453i	$-0.348220\pi$
$547$	21.4687	0.917933	0.458967	+	0.888453i	$0.348220\pi$
$548$	12.0950	0.516672
$549$	−22.9950	−0.981403
$550$	−0.417524	−0.0178033
$551$	−6.85680	−0.292110
$552$	−1.22462	−0.0521231
$553$	−0.311366	−0.0132406
$554$	0.112845	0.00479432
$555$	−2.28080	−0.0968146
$556$	−17.7106	−0.751098
$557$	−24.7927	−1.05050	−0.525250	−	0.850948i	$-0.676028\pi$
$557$	−24.7927	−1.05050	−0.525250	+	0.850948i	$0.676028\pi$
$558$	−21.5142	−0.910770
$559$	0	0
$560$	−1.18757	−0.0501838
$561$	−1.35481	−0.0572000
$562$	−25.6082	−1.08022
$563$	−21.6765	−0.913555	−0.456778	−	0.889581i	$-0.650996\pi$
$563$	−21.6765	−0.913555	−0.456778	+	0.889581i	$0.650996\pi$
$564$	0.840554	0.0353937
$565$	4.02917	0.169508
$566$	−9.58160	−0.402745
$567$	−4.80951	−0.201980
$568$	−11.1952	−0.469741
$569$	−29.0025	−1.21585	−0.607924	−	0.793996i	$-0.707997\pi$
$569$	−29.0025	−1.21585	−0.607924	+	0.793996i	$0.707997\pi$
$570$	0.351730	0.0147324
$571$	−39.1246	−1.63731	−0.818657	−	0.574282i	$-0.805281\pi$
$571$	−39.1246	−1.63731	−0.818657	+	0.574282i	$0.805281\pi$
$572$	0	0
$573$	−0.279582	−0.0116797
$574$	5.67606	0.236914
$575$	2.39764	0.0999886
$576$	−2.97358	−0.123899
$577$	−8.61003	−0.358440	−0.179220	−	0.983809i	$-0.557357\pi$
$577$	−8.61003	−0.358440	−0.179220	+	0.983809i	$0.557357\pi$
$578$	23.3608	0.971682
$579$	0.740445	0.0307718
$580$	−14.8363	−0.616043
$581$	−9.47940	−0.393272
$582$	1.18612	0.0491664
$583$	−0.393887	−0.0163131
$584$	−15.4358	−0.638738
$585$	0	0
$586$	1.29251	0.0533933
$587$	33.4674	1.38135	0.690673	−	0.723167i	$-0.257314\pi$
$587$	33.4674	1.38135	0.690673	+	0.723167i	$0.257314\pi$
$588$	−1.08893	−0.0449068
$589$	7.23514	0.298119
$590$	10.4962	0.432122
$591$	3.93001	0.161659
$592$	−6.48452	−0.266512
$593$	1.07899	0.0443090	0.0221545	−	0.999755i	$-0.492947\pi$
$593$	1.07899	0.0443090	0.0221545	+	0.999755i	$0.492947\pi$
$594$	−1.27389	−0.0522683
$595$	7.54462	0.309299
$596$	16.2820	0.666937
$597$	0.679068	0.0277924
$598$	0	0
$599$	11.7201	0.478871	0.239435	−	0.970912i	$-0.423038\pi$
$599$	11.7201	0.478871	0.239435	+	0.970912i	$0.423038\pi$
$600$	−0.0517365	−0.00211213
$601$	31.5113	1.28537	0.642685	−	0.766130i	$-0.277820\pi$
$601$	31.5113	1.28537	0.642685	+	0.766130i	$0.277820\pi$
$602$	1.86962	0.0761999
$603$	13.2810	0.540846
$604$	2.48553	0.101135
$605$	−20.0773	−0.816256
$606$	−0.667494	−0.0271151
$607$	−32.7663	−1.32994	−0.664971	−	0.746869i	$-0.731556\pi$
$607$	−32.7663	−1.32994	−0.664971	+	0.746869i	$0.731556\pi$
$608$	1.00000	0.0405554
$609$	0.611761	0.0247898
$610$	16.7324	0.677475
$611$	0	0
$612$	18.8912	0.763630
$613$	−3.25701	−0.131549	−0.0657746	−	0.997835i	$-0.520952\pi$
$613$	−3.25701	−0.131549	−0.0657746	+	0.997835i	$0.520952\pi$
$614$	−6.22672	−0.251290
$615$	−3.63750	−0.146678
$616$	−0.720021	−0.0290105
$617$	−33.8873	−1.36425	−0.682125	−	0.731235i	$-0.738944\pi$
$617$	−33.8873	−1.36425	−0.682125	+	0.731235i	$0.738944\pi$
$618$	−1.10768	−0.0445573
$619$	11.6176	0.466949	0.233475	−	0.972363i	$-0.424990\pi$
$619$	11.6176	0.466949	0.233475	+	0.972363i	$0.424990\pi$
$620$	15.6549	0.628716
$621$	7.31534	0.293554
$622$	−24.3976	−0.978256
$623$	−2.17295	−0.0870573
$624$	0	0
$625$	−23.3074	−0.932295
$626$	−24.1468	−0.965099
$627$	0.213254	0.00851655
$628$	−17.3964	−0.694193
$629$	41.1963	1.64260
$630$	3.53131	0.140691
$631$	2.38365	0.0948917	0.0474459	−	0.998874i	$-0.484892\pi$
$631$	2.38365	0.0948917	0.0474459	+	0.998874i	$0.484892\pi$
$632$	0.567305	0.0225662
$633$	2.48380	0.0987222
$634$	0.544955	0.0216429
$635$	36.4077	1.44480
$636$	−0.0488075	−0.00193534
$637$	0	0
$638$	−8.99524	−0.356125
$639$	33.2898	1.31692
$640$	2.16373	0.0855290
$641$	33.0723	1.30628	0.653138	−	0.757239i	$-0.273452\pi$
$641$	33.0723	1.30628	0.653138	+	0.757239i	$0.273452\pi$
$642$	1.60149	0.0632058
$643$	−4.46359	−0.176027	−0.0880133	−	0.996119i	$-0.528052\pi$
$643$	−4.46359	−0.176027	−0.0880133	+	0.996119i	$0.528052\pi$
$644$	4.13474	0.162931
$645$	−1.19814	−0.0471768
$646$	−6.35302	−0.249956
$647$	20.0786	0.789372	0.394686	−	0.918816i	$-0.370853\pi$
$647$	20.0786	0.789372	0.394686	+	0.918816i	$0.370853\pi$
$648$	8.76287	0.344238
$649$	6.36386	0.249803
$650$	0	0
$651$	−0.645516	−0.0252998
$652$	−17.4200	−0.682219
$653$	26.8627	1.05122	0.525609	−	0.850726i	$-0.323837\pi$
$653$	26.8627	1.05122	0.525609	+	0.850726i	$0.323837\pi$
$654$	0.952464	0.0372443
$655$	1.24707	0.0487272
$656$	−10.3417	−0.403777
$657$	45.8995	1.79071
$658$	−2.83801	−0.110637
$659$	−7.63965	−0.297598	−0.148799	−	0.988867i	$-0.547541\pi$
$659$	−7.63965	−0.297598	−0.148799	+	0.988867i	$0.547541\pi$
$660$	0.461424	0.0179609
$661$	8.04886	0.313064	0.156532	−	0.987673i	$-0.449969\pi$
$661$	8.04886	0.313064	0.156532	+	0.987673i	$0.449969\pi$
$662$	−20.0298	−0.778482
$663$	0	0
$664$	17.2714	0.670259
$665$	−1.18757	−0.0460518
$666$	19.2822	0.747170
$667$	51.6554	2.00010
$668$	−24.0810	−0.931721
$669$	−4.72707	−0.182759
$670$	−9.66399	−0.373352
$671$	10.1449	0.391638
$672$	−0.0892196	−0.00344172
$673$	−46.3430	−1.78639	−0.893196	−	0.449668i	$-0.851542\pi$
$673$	−46.3430	−1.78639	−0.893196	+	0.449668i	$0.851542\pi$
$674$	0.990093	0.0381370
$675$	0.309052	0.0118954
$676$	0	0
$677$	2.34511	0.0901299	0.0450649	−	0.998984i	$-0.485651\pi$
$677$	2.34511	0.0901299	0.0450649	+	0.998984i	$0.485651\pi$
$678$	0.302704	0.0116253
$679$	−4.00477	−0.153689
$680$	−13.7462	−0.527144
$681$	0.0728634	0.00279213
$682$	9.49156	0.363451
$683$	0.463294	0.0177275	0.00886373	−	0.999961i	$-0.497179\pi$
$683$	0.463294	0.0177275	0.00886373	+	0.999961i	$0.497179\pi$
$684$	−2.97358	−0.113697
$685$	26.1703	0.999915
$686$	7.51857	0.287061
$687$	−0.505669	−0.0192925
$688$	−3.40642	−0.129869
$689$	0	0
$690$	−2.64974	−0.100874
$691$	33.5326	1.27564	0.637821	−	0.770185i	$-0.279836\pi$
$691$	33.5326	1.27564	0.637821	+	0.770185i	$0.279836\pi$
$692$	−13.0175	−0.494853
$693$	2.14104	0.0813313
$694$	15.8651	0.602233
$695$	−38.3210	−1.45360
$696$	−1.11462	−0.0422497
$697$	65.7012	2.48861
$698$	−8.18795	−0.309919
$699$	4.22511	0.159808
$700$	0.174681	0.00660231
$701$	−46.8335	−1.76888	−0.884438	−	0.466658i	$-0.845458\pi$
$701$	−46.8335	−1.76888	−0.884438	+	0.466658i	$0.845458\pi$
$702$	0	0
$703$	−6.48452	−0.244568
$704$	1.31187	0.0494430
$705$	1.81873	0.0684975
$706$	−4.64477	−0.174808
$707$	2.25370	0.0847589
$708$	0.788561	0.0296359
$709$	−43.9520	−1.65065	−0.825326	−	0.564656i	$-0.809009\pi$
$709$	−43.9520	−1.65065	−0.825326	+	0.564656i	$0.809009\pi$
$710$	−24.2234	−0.909089
$711$	−1.68692	−0.0632646
$712$	3.95909	0.148373
$713$	−54.5055	−2.04125
$714$	0.566814	0.0212125
$715$	0	0
$716$	6.83699	0.255510
$717$	3.32152	0.124044
$718$	5.37310	0.200522
$719$	0.278039	0.0103691	0.00518456	−	0.999987i	$-0.498350\pi$
$719$	0.278039	0.0103691	0.00518456	+	0.999987i	$0.498350\pi$
$720$	−6.43402	−0.239782
$721$	3.73991	0.139281
$722$	1.00000	0.0372161
$723$	2.94445	0.109505
$724$	14.3069	0.531712
$725$	2.18229	0.0810482
$726$	−1.50837	−0.0559808
$727$	1.93331	0.0717024	0.0358512	−	0.999357i	$-0.488586\pi$
$727$	1.93331	0.0717024	0.0358512	+	0.999357i	$0.488586\pi$
$728$	0	0
$729$	−25.5835	−0.947538
$730$	−33.3989	−1.23615
$731$	21.6411	0.800424
$732$	1.25707	0.0464628
$733$	−3.70812	−0.136963	−0.0684813	−	0.997652i	$-0.521815\pi$
$733$	−3.70812	−0.136963	−0.0684813	+	0.997652i	$0.521815\pi$
$734$	−16.0979	−0.594184
$735$	−2.35616	−0.0869081
$736$	−7.53345	−0.277687
$737$	−5.85928	−0.215829
$738$	30.7519	1.13199
$739$	23.0440	0.847686	0.423843	−	0.905736i	$-0.360681\pi$
$739$	23.0440	0.847686	0.423843	+	0.905736i	$0.360681\pi$
$740$	−14.0308	−0.515781
$741$	0	0
$742$	0.164791	0.00604968
$743$	−10.3271	−0.378863	−0.189431	−	0.981894i	$-0.560664\pi$
$743$	−10.3271	−0.378863	−0.189431	+	0.981894i	$0.560664\pi$
$744$	1.17612	0.0431188
$745$	35.2299	1.29072
$746$	−24.4583	−0.895481
$747$	−51.3577	−1.87908
$748$	−8.33434	−0.304734
$749$	−5.40720	−0.197575
$750$	−1.87059	−0.0683044
$751$	−27.7799	−1.01370	−0.506851	−	0.862034i	$-0.669191\pi$
$751$	−27.7799	−1.01370	−0.506851	+	0.862034i	$0.669191\pi$
$752$	5.17082	0.188560
$753$	−4.36064	−0.158911
$754$	0	0
$755$	5.37803	0.195726
$756$	0.532960	0.0193836
$757$	6.06412	0.220404	0.110202	−	0.993909i	$-0.464850\pi$
$757$	6.06412	0.220404	0.110202	+	0.993909i	$0.464850\pi$
$758$	8.28671	0.300987
$759$	−1.60654	−0.0583136
$760$	2.16373	0.0784868
$761$	25.1643	0.912205	0.456103	−	0.889927i	$-0.349245\pi$
$761$	25.1643	0.912205	0.456103	+	0.889927i	$0.349245\pi$
$762$	2.73525	0.0990875
$763$	−3.21586	−0.116422
$764$	−1.71990	−0.0622239
$765$	40.8754	1.47785
$766$	5.62436	0.203216
$767$	0	0
$768$	0.162557	0.00586578
$769$	15.7253	0.567069	0.283535	−	0.958962i	$-0.408493\pi$
$769$	15.7253	0.567069	0.283535	+	0.958962i	$0.408493\pi$
$770$	−1.55793	−0.0561440
$771$	3.92835	0.141476
$772$	4.55498	0.163937
$773$	−5.38475	−0.193676	−0.0968380	−	0.995300i	$-0.530873\pi$
$773$	−5.38475	−0.193676	−0.0968380	+	0.995300i	$0.530873\pi$
$774$	10.1293	0.364089
$775$	−2.30270	−0.0827154
$776$	7.29665	0.261934
$777$	0.578546	0.0207552
$778$	10.2852	0.368744
$779$	−10.3417	−0.370531
$780$	0	0
$781$	−14.6867	−0.525530
$782$	47.8601	1.71147
$783$	6.65828	0.237948
$784$	−6.69876	−0.239242
$785$	−37.6412	−1.34347
$786$	0.0936903	0.00334182
$787$	−34.7067	−1.23716	−0.618580	−	0.785722i	$-0.712292\pi$
$787$	−34.7067	−1.23716	−0.618580	+	0.785722i	$0.712292\pi$
$788$	24.1762	0.861241
$789$	−1.58758	−0.0565195
$790$	1.22750	0.0436724
$791$	−1.02204	−0.0363394
$792$	−3.90095	−0.138614
$793$	0	0
$794$	−20.5442	−0.729085
$795$	−0.105606	−0.00374547
$796$	4.17741	0.148064
$797$	9.26148	0.328058	0.164029	−	0.986455i	$-0.447551\pi$
$797$	9.26148	0.328058	0.164029	+	0.986455i	$0.447551\pi$
$798$	−0.0892196	−0.00315834
$799$	−32.8503	−1.16216
$800$	−0.318266	−0.0112524
$801$	−11.7726	−0.415966
$802$	−21.8726	−0.772348
$803$	−20.2498	−0.714599
$804$	−0.726038	−0.0256054
$805$	8.94646	0.315321
$806$	0	0
$807$	1.35101	0.0475577
$808$	−4.10621	−0.144456
$809$	7.67498	0.269838	0.134919	−	0.990857i	$-0.456923\pi$
$809$	7.67498	0.269838	0.134919	+	0.990857i	$0.456923\pi$
$810$	18.9605	0.666204
$811$	12.8408	0.450903	0.225452	−	0.974254i	$-0.427614\pi$
$811$	12.8408	0.450903	0.225452	+	0.974254i	$0.427614\pi$
$812$	3.76336	0.132068
$813$	−0.221171	−0.00775681
$814$	−8.50685	−0.298165
$815$	−37.6921	−1.32030
$816$	−1.03273	−0.0361527
$817$	−3.40642	−0.119176
$818$	−1.43173	−0.0500594
$819$	0	0
$820$	−22.3767	−0.781429
$821$	−33.1851	−1.15817	−0.579085	−	0.815267i	$-0.696590\pi$
$821$	−33.1851	−1.15817	−0.579085	+	0.815267i	$0.696590\pi$
$822$	1.96613	0.0685765
$823$	42.2966	1.47437	0.737183	−	0.675693i	$-0.236156\pi$
$823$	42.2966	1.47437	0.737183	+	0.675693i	$0.236156\pi$
$824$	−6.81407	−0.237379
$825$	−0.0678715	−0.00236298
$826$	−2.66246	−0.0926389
$827$	−50.4573	−1.75457	−0.877287	−	0.479967i	$-0.840649\pi$
$827$	−50.4573	−1.75457	−0.877287	+	0.479967i	$0.840649\pi$
$828$	22.4013	0.778498
$829$	5.91594	0.205469	0.102735	−	0.994709i	$-0.467241\pi$
$829$	5.91594	0.205469	0.102735	+	0.994709i	$0.467241\pi$
$830$	37.3706	1.29715
$831$	0.0183438	0.000636338 0
$832$	0	0
$833$	42.5574	1.47453
$834$	−2.87899	−0.0996912
$835$	−52.1048	−1.80316
$836$	1.31187	0.0453720
$837$	−7.02566	−0.242842
$838$	35.6625	1.23194
$839$	−24.4566	−0.844335	−0.422168	−	0.906518i	$-0.638731\pi$
$839$	−24.4566	−0.844335	−0.422168	+	0.906518i	$0.638731\pi$
$840$	−0.193047	−0.00666076
$841$	18.0157	0.621233
$842$	17.5139	0.603569
$843$	−4.16280	−0.143375
$844$	15.2795	0.525944
$845$	0	0
$846$	−15.3758	−0.528632
$847$	5.09278	0.174990
$848$	−0.300248	−0.0103106
$849$	−1.55756	−0.0534553
$850$	2.02195	0.0693523
$851$	48.8508	1.67458
$852$	−1.81986	−0.0623474
$853$	−27.0068	−0.924696	−0.462348	−	0.886698i	$-0.652993\pi$
$853$	−27.0068	−0.924696	−0.462348	+	0.886698i	$0.652993\pi$
$854$	−4.24433	−0.145238
$855$	−6.43402	−0.220039
$856$	9.85187	0.336730
$857$	−19.0368	−0.650286	−0.325143	−	0.945665i	$-0.605412\pi$
$857$	−19.0368	−0.650286	−0.325143	+	0.945665i	$0.605412\pi$
$858$	0	0
$859$	27.0388	0.922550	0.461275	−	0.887257i	$-0.347392\pi$
$859$	27.0388	0.922550	0.461275	+	0.887257i	$0.347392\pi$
$860$	−7.37059	−0.251335
$861$	0.922685	0.0314450
$862$	−17.4842	−0.595515
$863$	10.4043	0.354167	0.177083	−	0.984196i	$-0.443334\pi$
$863$	10.4043	0.354167	0.177083	+	0.984196i	$0.443334\pi$
$864$	−0.971047	−0.0330357
$865$	−28.1665	−0.957689
$866$	13.0367	0.443005
$867$	3.79747	0.128969
$868$	−3.97101	−0.134785
$869$	0.744231	0.0252463
$870$	−2.41174	−0.0817658
$871$	0	0
$872$	5.85926	0.198419
$873$	−21.6971	−0.734337
$874$	−7.53345	−0.254823
$875$	6.31579	0.213513
$876$	−2.50920	−0.0847780
$877$	16.0102	0.540625	0.270313	−	0.962773i	$-0.412873\pi$
$877$	16.0102	0.540625	0.270313	+	0.962773i	$0.412873\pi$
$878$	27.5580	0.930039
$879$	0.210107	0.00708675
$880$	2.83854	0.0956870
$881$	28.3112	0.953828	0.476914	−	0.878950i	$-0.341755\pi$
$881$	28.3112	0.953828	0.476914	+	0.878950i	$0.341755\pi$
$882$	19.9193	0.670717
$883$	−48.8284	−1.64320	−0.821602	−	0.570061i	$-0.806920\pi$
$883$	−48.8284	−1.64320	−0.821602	+	0.570061i	$0.806920\pi$
$884$	0	0
$885$	1.70623	0.0573544
$886$	−32.1705	−1.08079
$887$	54.4966	1.82982	0.914908	−	0.403662i	$-0.132263\pi$
$887$	54.4966	1.82982	0.914908	+	0.403662i	$0.132263\pi$
$888$	−1.05411	−0.0353735
$889$	−9.23516	−0.309737
$890$	8.56640	0.287146
$891$	11.4958	0.385122
$892$	−29.0794	−0.973652
$893$	5.17082	0.173035
$894$	2.64676	0.0885208
$895$	14.7934	0.494489
$896$	−0.548851	−0.0183358
$897$	0	0
$898$	36.6871	1.22427
$899$	−49.6099	−1.65458
$900$	0.946388	0.0315463
$901$	1.90748	0.0635474
$902$	−13.5670	−0.451732
$903$	0.303920	0.0101138
$904$	1.86214	0.0619338
$905$	30.9563	1.02902
$906$	0.404041	0.0134234
$907$	−54.9629	−1.82501	−0.912507	−	0.409060i	$-0.865857\pi$
$907$	−54.9629	−1.82501	−0.912507	+	0.409060i	$0.865857\pi$
$908$	0.448232	0.0148751
$909$	12.2101	0.404984
$910$	0	0
$911$	−37.3100	−1.23614	−0.618068	−	0.786125i	$-0.712084\pi$
$911$	−37.3100	−1.23614	−0.618068	+	0.786125i	$0.712084\pi$
$912$	0.162557	0.00538281
$913$	22.6578	0.749864
$914$	8.52880	0.282108
$915$	2.71997	0.0899194
$916$	−3.11071	−0.102781
$917$	−0.316332	−0.0104462
$918$	6.16908	0.203610
$919$	14.3143	0.472187	0.236093	−	0.971730i	$-0.424133\pi$
$919$	14.3143	0.472187	0.236093	+	0.971730i	$0.424133\pi$
$920$	−16.3004	−0.537407
$921$	−1.01220	−0.0333531
$922$	−19.9532	−0.657125
$923$	0	0
$924$	−0.117045	−0.00385048
$925$	2.06380	0.0678574
$926$	−1.16008	−0.0381225
$927$	20.2622	0.665497
$928$	−6.85680	−0.225086
$929$	31.5417	1.03485	0.517425	−	0.855728i	$-0.326891\pi$
$929$	31.5417	1.03485	0.517425	+	0.855728i	$0.326891\pi$
$930$	2.54481	0.0834478
$931$	−6.69876	−0.219543
$932$	25.9915	0.851381
$933$	−3.96601	−0.129841
$934$	−10.3312	−0.338047
$935$	−18.0333	−0.589751
$936$	0	0
$937$	−4.29696	−0.140375	−0.0701877	−	0.997534i	$-0.522360\pi$
$937$	−4.29696	−0.140375	−0.0701877	+	0.997534i	$0.522360\pi$
$938$	2.45136	0.0800398
$939$	−3.92523	−0.128095
$940$	11.1883	0.364921
$941$	−9.56109	−0.311683	−0.155841	−	0.987782i	$-0.549809\pi$
$941$	−9.56109	−0.311683	−0.155841	+	0.987782i	$0.549809\pi$
$942$	−2.82792	−0.0921385
$943$	77.9089	2.53706
$944$	4.85098	0.157886
$945$	1.15318	0.0375130
$946$	−4.46879	−0.145293
$947$	−11.0171	−0.358006	−0.179003	−	0.983848i	$-0.557287\pi$
$947$	−11.0171	−0.358006	−0.179003	+	0.983848i	$0.557287\pi$
$948$	0.0922195	0.00299515
$949$	0	0
$950$	−0.318266	−0.0103259
$951$	0.0885863	0.00287261
$952$	3.48686	0.113010
$953$	−42.7267	−1.38405	−0.692027	−	0.721871i	$-0.743282\pi$
$953$	−42.7267	−1.38405	−0.692027	+	0.721871i	$0.743282\pi$
$954$	0.892811	0.0289058
$955$	−3.72141	−0.120422
$956$	20.4329	0.660848
$957$	−1.46224	−0.0472675
$958$	1.09845	0.0354893
$959$	−6.63834	−0.214363
$960$	0.351730	0.0113520
$961$	21.3472	0.688619
$962$	0	0
$963$	−29.2953	−0.944027
$964$	18.1133	0.583390
$965$	9.85576	0.317268
$966$	0.672131	0.0216255
$967$	−9.54961	−0.307095	−0.153547	−	0.988141i	$-0.549070\pi$
$967$	−9.54961	−0.307095	−0.153547	+	0.988141i	$0.549070\pi$
$968$	−9.27899	−0.298238
$969$	−1.03273	−0.0331760
$970$	15.7880	0.506922
$971$	−11.6753	−0.374678	−0.187339	−	0.982295i	$-0.559986\pi$
$971$	−11.6753	−0.374678	−0.187339	+	0.982295i	$0.559986\pi$
$972$	4.33761	0.139129
$973$	9.72048	0.311624
$974$	23.6068	0.756409
$975$	0	0
$976$	7.73312	0.247531
$977$	41.8831	1.33996	0.669980	−	0.742379i	$-0.266303\pi$
$977$	41.8831	1.33996	0.669980	+	0.742379i	$0.266303\pi$
$978$	−2.83174	−0.0905491
$979$	5.19381	0.165995
$980$	−14.4943	−0.463004
$981$	−17.4229	−0.556272
$982$	−0.936015	−0.0298694
$983$	6.48960	0.206986	0.103493	−	0.994630i	$-0.466998\pi$
$983$	6.48960	0.206986	0.103493	+	0.994630i	$0.466998\pi$
$984$	−1.68112	−0.0535922
$985$	52.3108	1.66676
$986$	43.5614	1.38728
$987$	−0.461339	−0.0146846
$988$	0	0
$989$	25.6621	0.816008
$990$	−8.44060	−0.268260
$991$	6.97312	0.221509	0.110754	−	0.993848i	$-0.464673\pi$
$991$	6.97312	0.221509	0.110754	+	0.993848i	$0.464673\pi$
$992$	7.23514	0.229716
$993$	−3.25599	−0.103326
$994$	6.14450	0.194892
$995$	9.03879	0.286549
$996$	2.80759	0.0889618
$997$	28.0043	0.886905	0.443452	−	0.896298i	$-0.353754\pi$
$997$	28.0043	0.886905	0.443452	+	0.896298i	$0.353754\pi$
$998$	−35.4033	−1.12067
$999$	6.29678	0.199221

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bk.1.5	yes	9
13.12	even	2		6422.2.a.bi.1.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6422.2.a.bi.1.5	✓	9	13.12	even	2
6422.2.a.bk.1.5	yes	9	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:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +0.162557 q^{3} +1.00000 q^{4} +2.16373 q^{5} +0.162557 q^{6} -0.548851 q^{7} +1.00000 q^{8} -2.97358 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +0.162557 q^{3} +1.00000 q^{4} +2.16373 q^{5} +0.162557 q^{6} -0.548851 q^{7} +1.00000 q^{8} -2.97358 q^{9} +2.16373 q^{10} +1.31187 q^{11} +0.162557 q^{12} -0.548851 q^{14} +0.351730 q^{15} +1.00000 q^{16} -6.35302 q^{17} -2.97358 q^{18} +1.00000 q^{19} +2.16373 q^{20} -0.0892196 q^{21} +1.31187 q^{22} -7.53345 q^{23} +0.162557 q^{24} -0.318266 q^{25} -0.971047 q^{27} -0.548851 q^{28} -6.85680 q^{29} +0.351730 q^{30} +7.23514 q^{31} +1.00000 q^{32} +0.213254 q^{33} -6.35302 q^{34} -1.18757 q^{35} -2.97358 q^{36} -6.48452 q^{37} +1.00000 q^{38} +2.16373 q^{40} -10.3417 q^{41} -0.0892196 q^{42} -3.40642 q^{43} +1.31187 q^{44} -6.43402 q^{45} -7.53345 q^{46} +5.17082 q^{47} +0.162557 q^{48} -6.69876 q^{49} -0.318266 q^{50} -1.03273 q^{51} -0.300248 q^{53} -0.971047 q^{54} +2.83854 q^{55} -0.548851 q^{56} +0.162557 q^{57} -6.85680 q^{58} +4.85098 q^{59} +0.351730 q^{60} +7.73312 q^{61} +7.23514 q^{62} +1.63205 q^{63} +1.00000 q^{64} +0.213254 q^{66} -4.46635 q^{67} -6.35302 q^{68} -1.22462 q^{69} -1.18757 q^{70} -11.1952 q^{71} -2.97358 q^{72} -15.4358 q^{73} -6.48452 q^{74} -0.0517365 q^{75} +1.00000 q^{76} -0.720021 q^{77} +0.567305 q^{79} +2.16373 q^{80} +8.76287 q^{81} -10.3417 q^{82} +17.2714 q^{83} -0.0892196 q^{84} -13.7462 q^{85} -3.40642 q^{86} -1.11462 q^{87} +1.31187 q^{88} +3.95909 q^{89} -6.43402 q^{90} -7.53345 q^{92} +1.17612 q^{93} +5.17082 q^{94} +2.16373 q^{95} +0.162557 q^{96} +7.29665 q^{97} -6.69876 q^{98} -3.90095 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} - 5 q^{3} + 9 q^{4} + q^{5} - 5 q^{6} - 13 q^{7} + 9 q^{8} + 10 q^{9}+O(q^{10}) \) 9 * q + 9 * q^2 - 5 * q^3 + 9 * q^4 + q^5 - 5 * q^6 - 13 * q^7 + 9 * q^8 + 10 * q^9	\( 9 q + 9 q^{2} - 5 q^{3} + 9 q^{4} + q^{5} - 5 q^{6} - 13 q^{7} + 9 q^{8} + 10 q^{9} + q^{10} - 13 q^{11} - 5 q^{12} - 13 q^{14} - q^{15} + 9 q^{16} - 12 q^{17} + 10 q^{18} + 9 q^{19} + q^{20} + 18 q^{21} - 13 q^{22} - 22 q^{23} - 5 q^{24} + 4 q^{25} - 26 q^{27} - 13 q^{28} + 12 q^{29} - q^{30} + q^{31} + 9 q^{32} - 28 q^{33} - 12 q^{34} - 18 q^{35} + 10 q^{36} - 25 q^{37} + 9 q^{38} + q^{40} + 11 q^{41} + 18 q^{42} - 10 q^{43} - 13 q^{44} - q^{45} - 22 q^{46} - 12 q^{47} - 5 q^{48} + 2 q^{49} + 4 q^{50} + 35 q^{51} + 9 q^{53} - 26 q^{54} + 18 q^{55} - 13 q^{56} - 5 q^{57} + 12 q^{58} + 10 q^{59} - q^{60} + 32 q^{61} + q^{62} - 63 q^{63} + 9 q^{64} - 28 q^{66} - 73 q^{67} - 12 q^{68} + 2 q^{69} - 18 q^{70} - 51 q^{71} + 10 q^{72} - 14 q^{73} - 25 q^{74} - 49 q^{75} + 9 q^{76} + 18 q^{77} - 28 q^{79} + q^{80} + 29 q^{81} + 11 q^{82} - 22 q^{83} + 18 q^{84} - 51 q^{85} - 10 q^{86} - 20 q^{87} - 13 q^{88} + 3 q^{89} - q^{90} - 22 q^{92} - 59 q^{93} - 12 q^{94} + q^{95} - 5 q^{96} + 2 q^{98} + 25 q^{99}+O(q^{100}) \) 9 * q + 9 * q^2 - 5 * q^3 + 9 * q^4 + q^5 - 5 * q^6 - 13 * q^7 + 9 * q^8 + 10 * q^9 + q^10 - 13 * q^11 - 5 * q^12 - 13 * q^14 - q^15 + 9 * q^16 - 12 * q^17 + 10 * q^18 + 9 * q^19 + q^20 + 18 * q^21 - 13 * q^22 - 22 * q^23 - 5 * q^24 + 4 * q^25 - 26 * q^27 - 13 * q^28 + 12 * q^29 - q^30 + q^31 + 9 * q^32 - 28 * q^33 - 12 * q^34 - 18 * q^35 + 10 * q^36 - 25 * q^37 + 9 * q^38 + q^40 + 11 * q^41 + 18 * q^42 - 10 * q^43 - 13 * q^44 - q^45 - 22 * q^46 - 12 * q^47 - 5 * q^48 + 2 * q^49 + 4 * q^50 + 35 * q^51 + 9 * q^53 - 26 * q^54 + 18 * q^55 - 13 * q^56 - 5 * q^57 + 12 * q^58 + 10 * q^59 - q^60 + 32 * q^61 + q^62 - 63 * q^63 + 9 * q^64 - 28 * q^66 - 73 * q^67 - 12 * q^68 + 2 * q^69 - 18 * q^70 - 51 * q^71 + 10 * q^72 - 14 * q^73 - 25 * q^74 - 49 * q^75 + 9 * q^76 + 18 * q^77 - 28 * q^79 + q^80 + 29 * q^81 + 11 * q^82 - 22 * q^83 + 18 * q^84 - 51 * q^85 - 10 * q^86 - 20 * q^87 - 13 * q^88 + 3 * q^89 - q^90 - 22 * q^92 - 59 * q^93 - 12 * q^94 + q^95 - 5 * q^96 + 2 * q^98 + 25 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more