LMFDB - Embedded newform 6422.2.a.bd.1.1

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:	$0$
Dimension:	$6$
Coefficient field:	6.6.316645497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 12x^{4} + 24x^{3} + 13x^{2} - 24x + 5 $ x^6 - 2x^5 - 12x^4 + 24x^3 + 13x^2 - 24*x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 494)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.28714$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -3.28714 q^{3} +1.00000 q^{4} +2.88198 q^{5} -3.28714 q^{6} +3.02408 q^{7} +1.00000 q^{8} +7.80527 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.28714 q^{3} +1.00000 q^{4} +2.88198 q^{5} -3.28714 q^{6} +3.02408 q^{7} +1.00000 q^{8} +7.80527 q^{9} +2.88198 q^{10} +2.88198 q^{11} -3.28714 q^{12} +3.02408 q^{14} -9.47348 q^{15} +1.00000 q^{16} +0.0993160 q^{17} +7.80527 q^{18} -1.00000 q^{19} +2.88198 q^{20} -9.94055 q^{21} +2.88198 q^{22} +7.51133 q^{23} -3.28714 q^{24} +3.30584 q^{25} -15.7956 q^{27} +3.02408 q^{28} +3.34220 q^{29} -9.47348 q^{30} +5.90708 q^{31} +1.00000 q^{32} -9.47348 q^{33} +0.0993160 q^{34} +8.71534 q^{35} +7.80527 q^{36} -6.88543 q^{37} -1.00000 q^{38} +2.88198 q^{40} +6.94736 q^{41} -9.94055 q^{42} +6.08702 q^{43} +2.88198 q^{44} +22.4947 q^{45} +7.51133 q^{46} +5.88198 q^{47} -3.28714 q^{48} +2.14504 q^{49} +3.30584 q^{50} -0.326465 q^{51} +9.17556 q^{53} -15.7956 q^{54} +8.30584 q^{55} +3.02408 q^{56} +3.28714 q^{57} +3.34220 q^{58} +0.856427 q^{59} -9.47348 q^{60} -12.0129 q^{61} +5.90708 q^{62} +23.6037 q^{63} +1.00000 q^{64} -9.47348 q^{66} -11.4641 q^{67} +0.0993160 q^{68} -24.6908 q^{69} +8.71534 q^{70} +8.97303 q^{71} +7.80527 q^{72} -0.646560 q^{73} -6.88543 q^{74} -10.8667 q^{75} -1.00000 q^{76} +8.71534 q^{77} -14.0089 q^{79} +2.88198 q^{80} +28.5064 q^{81} +6.94736 q^{82} +10.0518 q^{83} -9.94055 q^{84} +0.286227 q^{85} +6.08702 q^{86} -10.9863 q^{87} +2.88198 q^{88} -0.951375 q^{89} +22.4947 q^{90} +7.51133 q^{92} -19.4174 q^{93} +5.88198 q^{94} -2.88198 q^{95} -3.28714 q^{96} -16.4426 q^{97} +2.14504 q^{98} +22.4947 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 6 q^{8} + 10 q^{9}+O(q^{10}) $ 6 * q + 6 * q^2 + 2 * q^3 + 6 * q^4 + 2 * q^5 + 2 * q^6 + q^7 + 6 * q^8 + 10 * q^9	$ 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 6 q^{8} + 10 q^{9} + 2 q^{10} + 2 q^{11} + 2 q^{12} + q^{14} - 9 q^{15} + 6 q^{16} - 2 q^{17} + 10 q^{18} - 6 q^{19} + 2 q^{20} + q^{21} + 2 q^{22} + 8 q^{23} + 2 q^{24} + 16 q^{25} - 4 q^{27} + q^{28} + 20 q^{29} - 9 q^{30} + 3 q^{31} + 6 q^{32} - 9 q^{33} - 2 q^{34} + 7 q^{35} + 10 q^{36} - 9 q^{37} - 6 q^{38} + 2 q^{40} + 3 q^{41} + q^{42} + 13 q^{43} + 2 q^{44} + 38 q^{45} + 8 q^{46} + 20 q^{47} + 2 q^{48} - 7 q^{49} + 16 q^{50} + 2 q^{51} + 25 q^{53} - 4 q^{54} + 46 q^{55} + q^{56} - 2 q^{57} + 20 q^{58} - 9 q^{60} + 6 q^{61} + 3 q^{62} + 46 q^{63} + 6 q^{64} - 9 q^{66} - 32 q^{67} - 2 q^{68} - 29 q^{69} + 7 q^{70} + 39 q^{71} + 10 q^{72} + 7 q^{73} - 9 q^{74} - 15 q^{75} - 6 q^{76} + 7 q^{77} + 18 q^{79} + 2 q^{80} + 54 q^{81} + 3 q^{82} - 7 q^{83} + q^{84} - 2 q^{85} + 13 q^{86} + 12 q^{87} + 2 q^{88} + 9 q^{89} + 38 q^{90} + 8 q^{92} - 47 q^{93} + 20 q^{94} - 2 q^{95} + 2 q^{96} - 4 q^{97} - 7 q^{98} + 38 q^{99}+O(q^{100}) $ 6 * q + 6 * q^2 + 2 * q^3 + 6 * q^4 + 2 * q^5 + 2 * q^6 + q^7 + 6 * q^8 + 10 * q^9 + 2 * q^10 + 2 * q^11 + 2 * q^12 + q^14 - 9 * q^15 + 6 * q^16 - 2 * q^17 + 10 * q^18 - 6 * q^19 + 2 * q^20 + q^21 + 2 * q^22 + 8 * q^23 + 2 * q^24 + 16 * q^25 - 4 * q^27 + q^28 + 20 * q^29 - 9 * q^30 + 3 * q^31 + 6 * q^32 - 9 * q^33 - 2 * q^34 + 7 * q^35 + 10 * q^36 - 9 * q^37 - 6 * q^38 + 2 * q^40 + 3 * q^41 + q^42 + 13 * q^43 + 2 * q^44 + 38 * q^45 + 8 * q^46 + 20 * q^47 + 2 * q^48 - 7 * q^49 + 16 * q^50 + 2 * q^51 + 25 * q^53 - 4 * q^54 + 46 * q^55 + q^56 - 2 * q^57 + 20 * q^58 - 9 * q^60 + 6 * q^61 + 3 * q^62 + 46 * q^63 + 6 * q^64 - 9 * q^66 - 32 * q^67 - 2 * q^68 - 29 * q^69 + 7 * q^70 + 39 * q^71 + 10 * q^72 + 7 * q^73 - 9 * q^74 - 15 * q^75 - 6 * q^76 + 7 * q^77 + 18 * q^79 + 2 * q^80 + 54 * q^81 + 3 * q^82 - 7 * q^83 + q^84 - 2 * q^85 + 13 * q^86 + 12 * q^87 + 2 * q^88 + 9 * q^89 + 38 * q^90 + 8 * q^92 - 47 * q^93 + 20 * q^94 - 2 * q^95 + 2 * q^96 - 4 * q^97 - 7 * q^98 + 38 * 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
$3$	−3.28714	−1.89783	−0.948915	−	0.315533i	$-0.897817\pi$
$3$	−3.28714	−1.89783	−0.948915	+	0.315533i	$0.897817\pi$
$4$	1.00000	0.500000
$5$	2.88198	1.28886	0.644431	−	0.764662i	$-0.277094\pi$
$5$	2.88198	1.28886	0.644431	+	0.764662i	$0.277094\pi$
$6$	−3.28714	−1.34197
$7$	3.02408	1.14299	0.571497	−	0.820604i	$-0.306363\pi$
$7$	3.02408	1.14299	0.571497	+	0.820604i	$0.306363\pi$
$8$	1.00000	0.353553
$9$	7.80527	2.60176
$10$	2.88198	0.911364
$11$	2.88198	0.868951	0.434476	−	0.900684i	$-0.356934\pi$
$11$	2.88198	0.868951	0.434476	+	0.900684i	$0.356934\pi$
$12$	−3.28714	−0.948915
$13$	0	0
$13$	0	0
$14$	3.02408	0.808219
$15$	−9.47348	−2.44604
$16$	1.00000	0.250000
$17$	0.0993160	0.0240877	0.0120438	−	0.999927i	$-0.496166\pi$
$17$	0.0993160	0.0240877	0.0120438	+	0.999927i	$0.496166\pi$
$18$	7.80527	1.83972
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	2.88198	0.644431
$21$	−9.94055	−2.16921
$22$	2.88198	0.614441
$23$	7.51133	1.56622	0.783110	−	0.621883i	$-0.213632\pi$
$23$	7.51133	1.56622	0.783110	+	0.621883i	$0.213632\pi$
$24$	−3.28714	−0.670984
$25$	3.30584	0.661167
$26$	0	0
$27$	−15.7956	−3.03986
$28$	3.02408	0.571497
$29$	3.34220	0.620632	0.310316	−	0.950633i	$-0.399565\pi$
$29$	3.34220	0.620632	0.310316	+	0.950633i	$0.399565\pi$
$30$	−9.47348	−1.72961
$31$	5.90708	1.06094	0.530472	−	0.847703i	$-0.322015\pi$
$31$	5.90708	1.06094	0.530472	+	0.847703i	$0.322015\pi$
$32$	1.00000	0.176777
$33$	−9.47348	−1.64912
$34$	0.0993160	0.0170326
$35$	8.71534	1.47316
$36$	7.80527	1.30088
$37$	−6.88543	−1.13196	−0.565979	−	0.824420i	$-0.691502\pi$
$37$	−6.88543	−1.13196	−0.565979	+	0.824420i	$0.691502\pi$
$38$	−1.00000	−0.162221
$39$	0	0
$40$	2.88198	0.455682
$41$	6.94736	1.08500	0.542498	−	0.840057i	$-0.317479\pi$
$41$	6.94736	1.08500	0.542498	+	0.840057i	$0.317479\pi$
$42$	−9.94055	−1.53386
$43$	6.08702	0.928263	0.464131	−	0.885766i	$-0.346367\pi$
$43$	6.08702	0.928263	0.464131	+	0.885766i	$0.346367\pi$
$44$	2.88198	0.434476
$45$	22.4947	3.35331
$46$	7.51133	1.10748
$47$	5.88198	0.857976	0.428988	−	0.903310i	$-0.358870\pi$
$47$	5.88198	0.857976	0.428988	+	0.903310i	$0.358870\pi$
$48$	−3.28714	−0.474457
$49$	2.14504	0.306435
$50$	3.30584	0.467516
$51$	−0.326465	−0.0457143
$52$	0	0
$53$	9.17556	1.26036	0.630180	−	0.776449i	$-0.282981\pi$
$53$	9.17556	1.26036	0.630180	+	0.776449i	$0.282981\pi$
$54$	−15.7956	−2.14950
$55$	8.30584	1.11996
$56$	3.02408	0.404109
$57$	3.28714	0.435392
$58$	3.34220	0.438853
$59$	0.856427	0.111497	0.0557487	−	0.998445i	$-0.482245\pi$
$59$	0.856427	0.111497	0.0557487	+	0.998445i	$0.482245\pi$
$60$	−9.47348	−1.22302
$61$	−12.0129	−1.53809	−0.769047	−	0.639192i	$-0.779269\pi$
$61$	−12.0129	−1.53809	−0.769047	+	0.639192i	$0.779269\pi$
$62$	5.90708	0.750200
$63$	23.6037	2.97379
$64$	1.00000	0.125000
$65$	0	0
$66$	−9.47348	−1.16610
$67$	−11.4641	−1.40056	−0.700280	−	0.713868i	$-0.746942\pi$
$67$	−11.4641	−1.40056	−0.700280	+	0.713868i	$0.746942\pi$
$68$	0.0993160	0.0120438
$69$	−24.6908	−2.97242
$70$	8.71534	1.04168
$71$	8.97303	1.06490	0.532451	−	0.846461i	$-0.321271\pi$
$71$	8.97303	1.06490	0.532451	+	0.846461i	$0.321271\pi$
$72$	7.80527	0.919859
$73$	−0.646560	−0.0756742	−0.0378371	−	0.999284i	$-0.512047\pi$
$73$	−0.646560	−0.0756742	−0.0378371	+	0.999284i	$0.512047\pi$
$74$	−6.88543	−0.800415
$75$	−10.8667	−1.25478
$76$	−1.00000	−0.114708
$77$	8.71534	0.993206
$78$	0	0
$79$	−14.0089	−1.57612	−0.788061	−	0.615597i	$-0.788915\pi$
$79$	−14.0089	−1.57612	−0.788061	+	0.615597i	$0.788915\pi$
$80$	2.88198	0.322216
$81$	28.5064	3.16738
$82$	6.94736	0.767207
$83$	10.0518	1.10332	0.551662	−	0.834068i	$-0.313994\pi$
$83$	10.0518	1.10332	0.551662	+	0.834068i	$0.313994\pi$
$84$	−9.94055	−1.08460
$85$	0.286227	0.0310457
$86$	6.08702	0.656381
$87$	−10.9863	−1.17785
$88$	2.88198	0.307221
$89$	−0.951375	−0.100846	−0.0504228	−	0.998728i	$-0.516057\pi$
$89$	−0.951375	−0.100846	−0.0504228	+	0.998728i	$0.516057\pi$
$90$	22.4947	2.37115
$91$	0	0
$92$	7.51133	0.783110
$93$	−19.4174	−2.01349
$94$	5.88198	0.606680
$95$	−2.88198	−0.295685
$96$	−3.28714	−0.335492
$97$	−16.4426	−1.66949	−0.834746	−	0.550636i	$-0.814385\pi$
$97$	−16.4426	−1.66949	−0.834746	+	0.550636i	$0.814385\pi$
$98$	2.14504	0.216682
$99$	22.4947	2.26080
$100$	3.30584	0.330584
$101$	−9.72510	−0.967684	−0.483842	−	0.875155i	$-0.660759\pi$
$101$	−9.72510	−0.967684	−0.483842	+	0.875155i	$0.660759\pi$
$102$	−0.326465	−0.0323249
$103$	−13.0880	−1.28960	−0.644801	−	0.764351i	$-0.723060\pi$
$103$	−13.0880	−1.28960	−0.644801	+	0.764351i	$0.723060\pi$
$104$	0	0
$105$	−28.6485	−2.79581
$106$	9.17556	0.891210
$107$	8.21201	0.793885	0.396942	−	0.917844i	$-0.370071\pi$
$107$	8.21201	0.793885	0.396942	+	0.917844i	$0.370071\pi$
$108$	−15.7956	−1.51993
$109$	−6.16653	−0.590646	−0.295323	−	0.955397i	$-0.595427\pi$
$109$	−6.16653	−0.590646	−0.295323	+	0.955397i	$0.595427\pi$
$110$	8.30584	0.791930
$111$	22.6334	2.14826
$112$	3.02408	0.285748
$113$	−15.9957	−1.50475	−0.752373	−	0.658737i	$-0.771091\pi$
$113$	−15.9957	−1.50475	−0.752373	+	0.658737i	$0.771091\pi$
$114$	3.28714	0.307869
$115$	21.6475	2.01864
$116$	3.34220	0.310316
$117$	0	0
$118$	0.856427	0.0788405
$119$	0.300339	0.0275321
$120$	−9.47348	−0.864806
$121$	−2.69416	−0.244924
$122$	−12.0129	−1.08760
$123$	−22.8369	−2.05914
$124$	5.90708	0.530472
$125$	−4.88256	−0.436709
$126$	23.6037	2.10279
$127$	1.15297	0.102309	0.0511546	−	0.998691i	$-0.483710\pi$
$127$	1.15297	0.102309	0.0511546	+	0.998691i	$0.483710\pi$
$128$	1.00000	0.0883883
$129$	−20.0089	−1.76168
$130$	0	0
$131$	−9.25910	−0.808971	−0.404486	−	0.914544i	$-0.632549\pi$
$131$	−9.25910	−0.808971	−0.404486	+	0.914544i	$0.632549\pi$
$132$	−9.47348	−0.824560
$133$	−3.02408	−0.262221
$134$	−11.4641	−0.990346
$135$	−45.5226	−3.91796
$136$	0.0993160	0.00851628
$137$	6.17416	0.527494	0.263747	−	0.964592i	$-0.415042\pi$
$137$	6.17416	0.527494	0.263747	+	0.964592i	$0.415042\pi$
$138$	−24.6908	−2.10182
$139$	−6.61882	−0.561401	−0.280700	−	0.959795i	$-0.590567\pi$
$139$	−6.61882	−0.561401	−0.280700	+	0.959795i	$0.590567\pi$
$140$	8.71534	0.736581
$141$	−19.3349	−1.62829
$142$	8.97303	0.753000
$143$	0	0
$144$	7.80527	0.650439
$145$	9.63218	0.799909
$146$	−0.646560	−0.0535097
$147$	−7.05105	−0.581561
$148$	−6.88543	−0.565979
$149$	−2.91053	−0.238440	−0.119220	−	0.992868i	$-0.538039\pi$
$149$	−2.91053	−0.238440	−0.119220	+	0.992868i	$0.538039\pi$
$150$	−10.8667	−0.887265
$151$	−3.38645	−0.275586	−0.137793	−	0.990461i	$-0.544001\pi$
$151$	−3.38645	−0.275586	−0.137793	+	0.990461i	$0.544001\pi$
$152$	−1.00000	−0.0811107
$153$	0.775188	0.0626702
$154$	8.71534	0.702302
$155$	17.0241	1.36741
$156$	0	0
$157$	9.59018	0.765380	0.382690	−	0.923877i	$-0.374998\pi$
$157$	9.59018	0.765380	0.382690	+	0.923877i	$0.374998\pi$
$158$	−14.0089	−1.11449
$159$	−30.1613	−2.39195
$160$	2.88198	0.227841
$161$	22.7148	1.79018
$162$	28.5064	2.23967
$163$	4.35938	0.341453	0.170726	−	0.985318i	$-0.445389\pi$
$163$	4.35938	0.341453	0.170726	+	0.985318i	$0.445389\pi$
$164$	6.94736	0.542498
$165$	−27.3024	−2.12549
$166$	10.0518	0.780168
$167$	19.9405	1.54305	0.771523	−	0.636202i	$-0.219495\pi$
$167$	19.9405	1.54305	0.771523	+	0.636202i	$0.219495\pi$
$168$	−9.94055	−0.766930
$169$	0	0
$170$	0.286227	0.0219526
$171$	−7.80527	−0.596884
$172$	6.08702	0.464131
$173$	−4.85829	−0.369369	−0.184685	−	0.982798i	$-0.559126\pi$
$173$	−4.85829	−0.369369	−0.184685	+	0.982798i	$0.559126\pi$
$174$	−10.9863	−0.832868
$175$	9.99710	0.755710
$176$	2.88198	0.217238
$177$	−2.81519	−0.211603
$178$	−0.951375	−0.0713086
$179$	−12.5379	−0.937126	−0.468563	−	0.883430i	$-0.655228\pi$
$179$	−12.5379	−0.937126	−0.468563	+	0.883430i	$0.655228\pi$
$180$	22.4947	1.67665
$181$	−21.6396	−1.60846	−0.804229	−	0.594319i	$-0.797421\pi$
$181$	−21.6396	−1.60846	−0.804229	+	0.594319i	$0.797421\pi$
$182$	0	0
$183$	39.4880	2.91904
$184$	7.51133	0.553742
$185$	−19.8437	−1.45894
$186$	−19.4174	−1.42375
$187$	0.286227	0.0209310
$188$	5.88198	0.428988
$189$	−47.7670	−3.47454
$190$	−2.88198	−0.209081
$191$	−10.2866	−0.744310	−0.372155	−	0.928171i	$-0.621381\pi$
$191$	−10.2866	−0.744310	−0.372155	+	0.928171i	$0.621381\pi$
$192$	−3.28714	−0.237229
$193$	22.5115	1.62041	0.810206	−	0.586146i	$-0.199355\pi$
$193$	22.5115	1.62041	0.810206	+	0.586146i	$0.199355\pi$
$194$	−16.4426	−1.18051
$195$	0	0
$196$	2.14504	0.153217
$197$	−7.05750	−0.502826	−0.251413	−	0.967880i	$-0.580895\pi$
$197$	−7.05750	−0.502826	−0.251413	+	0.967880i	$0.580895\pi$
$198$	22.4947	1.59863
$199$	26.2412	1.86019	0.930096	−	0.367317i	$-0.119724\pi$
$199$	26.2412	1.86019	0.930096	+	0.367317i	$0.119724\pi$
$200$	3.30584	0.233758
$201$	37.6840	2.65802
$202$	−9.72510	−0.684256
$203$	10.1071	0.709378
$204$	−0.326465	−0.0228571
$205$	20.0222	1.39841
$206$	−13.0880	−0.911886
$207$	58.6279	4.07492
$208$	0	0
$209$	−2.88198	−0.199351
$210$	−28.6485	−1.97694
$211$	11.2897	0.777217	0.388608	−	0.921403i	$-0.372956\pi$
$211$	11.2897	0.777217	0.388608	+	0.921403i	$0.372956\pi$
$212$	9.17556	0.630180
$213$	−29.4956	−2.02100
$214$	8.21201	0.561361
$215$	17.5427	1.19640
$216$	−15.7956	−1.07475
$217$	17.8635	1.21265
$218$	−6.16653	−0.417650
$219$	2.12533	0.143617
$220$	8.30584	0.559979
$221$	0	0
$222$	22.6334	1.51905
$223$	7.80973	0.522979	0.261489	−	0.965206i	$-0.415786\pi$
$223$	7.80973	0.522979	0.261489	+	0.965206i	$0.415786\pi$
$224$	3.02408	0.202055
$225$	25.8029	1.72020
$226$	−15.9957	−1.06402
$227$	−6.31625	−0.419224	−0.209612	−	0.977785i	$-0.567220\pi$
$227$	−6.31625	−0.419224	−0.209612	+	0.977785i	$0.567220\pi$
$228$	3.28714	0.217696
$229$	−6.89920	−0.455912	−0.227956	−	0.973671i	$-0.573204\pi$
$229$	−6.89920	−0.455912	−0.227956	+	0.973671i	$0.573204\pi$
$230$	21.6475	1.42740
$231$	−28.6485	−1.88493
$232$	3.34220	0.219426
$233$	−2.39251	−0.156739	−0.0783693	−	0.996924i	$-0.524971\pi$
$233$	−2.39251	−0.156739	−0.0783693	+	0.996924i	$0.524971\pi$
$234$	0	0
$235$	16.9518	1.10581
$236$	0.856427	0.0557487
$237$	46.0491	2.99121
$238$	0.300339	0.0194681
$239$	21.5847	1.39620	0.698100	−	0.716000i	$-0.254029\pi$
$239$	21.5847	1.39620	0.698100	+	0.716000i	$0.254029\pi$
$240$	−9.47348	−0.611510
$241$	−16.4075	−1.05690	−0.528451	−	0.848964i	$-0.677227\pi$
$241$	−16.4075	−1.05690	−0.528451	+	0.848964i	$0.677227\pi$
$242$	−2.69416	−0.173187
$243$	−46.3177	−2.97128
$244$	−12.0129	−0.769047
$245$	6.18198	0.394952
$246$	−22.8369	−1.45603
$247$	0	0
$248$	5.90708	0.375100
$249$	−33.0415	−2.09392
$250$	−4.88256	−0.308800
$251$	−18.5904	−1.17342	−0.586709	−	0.809798i	$-0.699577\pi$
$251$	−18.5904	−1.17342	−0.586709	+	0.809798i	$0.699577\pi$
$252$	23.6037	1.48690
$253$	21.6475	1.36097
$254$	1.15297	0.0723436
$255$	−0.940868	−0.0589194
$256$	1.00000	0.0625000
$257$	6.96066	0.434194	0.217097	−	0.976150i	$-0.430341\pi$
$257$	6.96066	0.434194	0.217097	+	0.976150i	$0.430341\pi$
$258$	−20.0089	−1.24570
$259$	−20.8221	−1.29382
$260$	0	0
$261$	26.0868	1.61473
$262$	−9.25910	−0.572029
$263$	1.68874	0.104132	0.0520661	−	0.998644i	$-0.483419\pi$
$263$	1.68874	0.104132	0.0520661	+	0.998644i	$0.483419\pi$
$264$	−9.47348	−0.583052
$265$	26.4438	1.62443
$266$	−3.02408	−0.185418
$267$	3.12730	0.191388
$268$	−11.4641	−0.700280
$269$	28.3668	1.72956	0.864778	−	0.502154i	$-0.167459\pi$
$269$	28.3668	1.72956	0.864778	+	0.502154i	$0.167459\pi$
$270$	−45.5226	−2.77042
$271$	−14.0040	−0.850680	−0.425340	−	0.905034i	$-0.639845\pi$
$271$	−14.0040	−0.850680	−0.425340	+	0.905034i	$0.639845\pi$
$272$	0.0993160	0.00602192
$273$	0	0
$274$	6.17416	0.372995
$275$	9.52737	0.574522
$276$	−24.6908	−1.48621
$277$	9.57854	0.575519	0.287759	−	0.957703i	$-0.407090\pi$
$277$	9.57854	0.575519	0.287759	+	0.957703i	$0.407090\pi$
$278$	−6.61882	−0.396970
$279$	46.1064	2.76032
$280$	8.71534	0.520841
$281$	0.150742	0.00899250	0.00449625	−	0.999990i	$-0.498569\pi$
$281$	0.150742	0.00899250	0.00449625	+	0.999990i	$0.498569\pi$
$282$	−19.3349	−1.15138
$283$	−29.5322	−1.75551	−0.877755	−	0.479110i	$-0.840959\pi$
$283$	−29.5322	−1.75551	−0.877755	+	0.479110i	$0.840959\pi$
$284$	8.97303	0.532451
$285$	9.47348	0.561160
$286$	0	0
$287$	21.0094	1.24014
$288$	7.80527	0.459930
$289$	−16.9901	−0.999420
$290$	9.63218	0.565621
$291$	54.0490	3.16841
$292$	−0.646560	−0.0378371
$293$	3.94681	0.230575	0.115287	−	0.993332i	$-0.463221\pi$
$293$	3.94681	0.230575	0.115287	+	0.993332i	$0.463221\pi$
$294$	−7.05105	−0.411226
$295$	2.46821	0.143705
$296$	−6.88543	−0.400208
$297$	−45.5226	−2.64149
$298$	−2.91053	−0.168603
$299$	0	0
$300$	−10.8667	−0.627391
$301$	18.4076	1.06100
$302$	−3.38645	−0.194868
$303$	31.9677	1.83650
$304$	−1.00000	−0.0573539
$305$	−34.6210	−1.98239
$306$	0.775188	0.0443145
$307$	−0.213487	−0.0121844	−0.00609219	−	0.999981i	$-0.501939\pi$
$307$	−0.213487	−0.0121844	−0.00609219	+	0.999981i	$0.501939\pi$
$308$	8.71534	0.496603
$309$	43.0221	2.44744
$310$	17.0241	0.966905
$311$	−30.1703	−1.71080	−0.855399	−	0.517969i	$-0.826688\pi$
$311$	−30.1703	−1.71080	−0.855399	+	0.517969i	$0.826688\pi$
$312$	0	0
$313$	−19.5575	−1.10545	−0.552727	−	0.833362i	$-0.686413\pi$
$313$	−19.5575	−1.10545	−0.552727	+	0.833362i	$0.686413\pi$
$314$	9.59018	0.541205
$315$	68.0256	3.83281
$316$	−14.0089	−0.788061
$317$	−6.31782	−0.354844	−0.177422	−	0.984135i	$-0.556776\pi$
$317$	−6.31782	−0.354844	−0.177422	+	0.984135i	$0.556776\pi$
$318$	−30.1613	−1.69136
$319$	9.63218	0.539299
$320$	2.88198	0.161108
$321$	−26.9940	−1.50666
$322$	22.7148	1.26585
$323$	−0.0993160	−0.00552609
$324$	28.5064	1.58369
$325$	0	0
$326$	4.35938	0.241444
$327$	20.2702	1.12095
$328$	6.94736	0.383604
$329$	17.7876	0.980661
$330$	−27.3024	−1.50295
$331$	7.47193	0.410695	0.205347	−	0.978689i	$-0.434168\pi$
$331$	7.47193	0.410695	0.205347	+	0.978689i	$0.434168\pi$
$332$	10.0518	0.551662
$333$	−53.7426	−2.94508
$334$	19.9405	1.09110
$335$	−33.0393	−1.80513
$336$	−9.94055	−0.542302
$337$	−3.39531	−0.184955	−0.0924773	−	0.995715i	$-0.529479\pi$
$337$	−3.39531	−0.184955	−0.0924773	+	0.995715i	$0.529479\pi$
$338$	0	0
$339$	52.5800	2.85575
$340$	0.286227	0.0155228
$341$	17.0241	0.921908
$342$	−7.80527	−0.422060
$343$	−14.6818	−0.792741
$344$	6.08702	0.328190
$345$	−71.1584	−3.83104
$346$	−4.85829	−0.261183
$347$	1.90912	0.102487	0.0512436	−	0.998686i	$-0.483682\pi$
$347$	1.90912	0.102487	0.0512436	+	0.998686i	$0.483682\pi$
$348$	−10.9863	−0.588927
$349$	−26.1257	−1.39848	−0.699239	−	0.714888i	$-0.746478\pi$
$349$	−26.1257	−1.39848	−0.699239	+	0.714888i	$0.746478\pi$
$350$	9.99710	0.534368
$351$	0	0
$352$	2.88198	0.153610
$353$	−11.5915	−0.616951	−0.308475	−	0.951232i	$-0.599819\pi$
$353$	−11.5915	−0.616951	−0.308475	+	0.951232i	$0.599819\pi$
$354$	−2.81519	−0.149626
$355$	25.8601	1.37251
$356$	−0.951375	−0.0504228
$357$	−0.987256	−0.0522511
$358$	−12.5379	−0.662648
$359$	4.12940	0.217941	0.108971	−	0.994045i	$-0.465245\pi$
$359$	4.12940	0.217941	0.108971	+	0.994045i	$0.465245\pi$
$360$	22.4947	1.18557
$361$	1.00000	0.0526316
$362$	−21.6396	−1.13735
$363$	8.85609	0.464824
$364$	0	0
$365$	−1.86338	−0.0975336
$366$	39.4880	2.06407
$367$	22.0476	1.15087	0.575437	−	0.817846i	$-0.304832\pi$
$367$	22.0476	1.15087	0.575437	+	0.817846i	$0.304832\pi$
$368$	7.51133	0.391555
$369$	54.2260	2.82289
$370$	−19.8437	−1.03163
$371$	27.7476	1.44058
$372$	−19.4174	−1.00674
$373$	22.9777	1.18974	0.594870	−	0.803822i	$-0.297203\pi$
$373$	22.9777	1.18974	0.594870	+	0.803822i	$0.297203\pi$
$374$	0.286227	0.0148005
$375$	16.0496	0.828799
$376$	5.88198	0.303340
$377$	0	0
$378$	−47.7670	−2.45687
$379$	−10.1844	−0.523138	−0.261569	−	0.965185i	$-0.584240\pi$
$379$	−10.1844	−0.523138	−0.261569	+	0.965185i	$0.584240\pi$
$380$	−2.88198	−0.147843
$381$	−3.78996	−0.194166
$382$	−10.2866	−0.526307
$383$	34.1328	1.74410	0.872052	−	0.489413i	$-0.162789\pi$
$383$	34.1328	1.74410	0.872052	+	0.489413i	$0.162789\pi$
$384$	−3.28714	−0.167746
$385$	25.1175	1.28011
$386$	22.5115	1.14580
$387$	47.5108	2.41511
$388$	−16.4426	−0.834746
$389$	−11.1658	−0.566127	−0.283063	−	0.959101i	$-0.591351\pi$
$389$	−11.1658	−0.566127	−0.283063	+	0.959101i	$0.591351\pi$
$390$	0	0
$391$	0.745995	0.0377266
$392$	2.14504	0.108341
$393$	30.4359	1.53529
$394$	−7.05750	−0.355551
$395$	−40.3734	−2.03141
$396$	22.4947	1.13040
$397$	1.89726	0.0952209	0.0476104	−	0.998866i	$-0.484839\pi$
$397$	1.89726	0.0952209	0.0476104	+	0.998866i	$0.484839\pi$
$398$	26.2412	1.31535
$399$	9.94055	0.497650
$400$	3.30584	0.165292
$401$	30.7695	1.53656	0.768278	−	0.640117i	$-0.221114\pi$
$401$	30.7695	1.53656	0.768278	+	0.640117i	$0.221114\pi$
$402$	37.6840	1.87951
$403$	0	0
$404$	−9.72510	−0.483842
$405$	82.1550	4.08231
$406$	10.1071	0.501606
$407$	−19.8437	−0.983616
$408$	−0.326465	−0.0161624
$409$	−13.5808	−0.671525	−0.335763	−	0.941947i	$-0.608994\pi$
$409$	−13.5808	−0.671525	−0.335763	+	0.941947i	$0.608994\pi$
$410$	20.0222	0.988825
$411$	−20.2953	−1.00109
$412$	−13.0880	−0.644801
$413$	2.58990	0.127441
$414$	58.6279	2.88140
$415$	28.9690	1.42203
$416$	0	0
$417$	21.7570	1.06544
$418$	−2.88198	−0.140962
$419$	11.5178	0.562684	0.281342	−	0.959608i	$-0.409221\pi$
$419$	11.5178	0.562684	0.281342	+	0.959608i	$0.409221\pi$
$420$	−28.6485	−1.39790
$421$	−14.7578	−0.719249	−0.359625	−	0.933097i	$-0.617095\pi$
$421$	−14.7578	−0.719249	−0.359625	+	0.933097i	$0.617095\pi$
$422$	11.2897	0.549575
$423$	45.9105	2.23224
$424$	9.17556	0.445605
$425$	0.328322	0.0159260
$426$	−29.4956	−1.42906
$427$	−36.3279	−1.75803
$428$	8.21201	0.396942
$429$	0	0
$430$	17.5427	0.845985
$431$	0.0489741	0.00235900	0.00117950	−	0.999999i	$-0.499625\pi$
$431$	0.0489741	0.00235900	0.00117950	+	0.999999i	$0.499625\pi$
$432$	−15.7956	−0.759964
$433$	15.4388	0.741941	0.370970	−	0.928645i	$-0.379025\pi$
$433$	15.4388	0.741941	0.370970	+	0.928645i	$0.379025\pi$
$434$	17.8635	0.857474
$435$	−31.6623	−1.51809
$436$	−6.16653	−0.295323
$437$	−7.51133	−0.359315
$438$	2.12533	0.101552
$439$	23.4821	1.12074	0.560370	−	0.828243i	$-0.310659\pi$
$439$	23.4821	1.12074	0.560370	+	0.828243i	$0.310659\pi$
$440$	8.30584	0.395965
$441$	16.7426	0.797268
$442$	0	0
$443$	−0.851887	−0.0404744	−0.0202372	−	0.999795i	$-0.506442\pi$
$443$	−0.851887	−0.0404744	−0.0202372	+	0.999795i	$0.506442\pi$
$444$	22.6334	1.07413
$445$	−2.74185	−0.129976
$446$	7.80973	0.369802
$447$	9.56731	0.452518
$448$	3.02408	0.142874
$449$	37.3624	1.76324	0.881620	−	0.471959i	$-0.156453\pi$
$449$	37.3624	1.76324	0.881620	+	0.471959i	$0.156453\pi$
$450$	25.8029	1.21636
$451$	20.0222	0.942808
$452$	−15.9957	−0.752373
$453$	11.1317	0.523014
$454$	−6.31625	−0.296436
$455$	0	0
$456$	3.28714	0.153934
$457$	−18.0321	−0.843505	−0.421753	−	0.906711i	$-0.638585\pi$
$457$	−18.0321	−0.843505	−0.421753	+	0.906711i	$0.638585\pi$
$458$	−6.89920	−0.322379
$459$	−1.56875	−0.0732231
$460$	21.6475	1.00932
$461$	−20.5994	−0.959410	−0.479705	−	0.877430i	$-0.659256\pi$
$461$	−20.5994	−0.959410	−0.479705	+	0.877430i	$0.659256\pi$
$462$	−28.6485	−1.33285
$463$	0.835400	0.0388243	0.0194122	−	0.999812i	$-0.493821\pi$
$463$	0.835400	0.0388243	0.0194122	+	0.999812i	$0.493821\pi$
$464$	3.34220	0.155158
$465$	−55.9606	−2.59511
$466$	−2.39251	−0.110831
$467$	4.94829	0.228979	0.114490	−	0.993424i	$-0.463477\pi$
$467$	4.94829	0.228979	0.114490	+	0.993424i	$0.463477\pi$
$468$	0	0
$469$	−34.6683	−1.60083
$470$	16.9518	0.781928
$471$	−31.5242	−1.45256
$472$	0.856427	0.0394203
$473$	17.5427	0.806615
$474$	46.0491	2.11511
$475$	−3.30584	−0.151682
$476$	0.300339	0.0137660
$477$	71.6177	3.27915
$478$	21.5847	0.987263
$479$	3.05994	0.139812	0.0699062	−	0.997554i	$-0.477730\pi$
$479$	3.05994	0.139812	0.0699062	+	0.997554i	$0.477730\pi$
$480$	−9.47348	−0.432403
$481$	0	0
$482$	−16.4075	−0.747343
$483$	−74.6667	−3.39745
$484$	−2.69416	−0.122462
$485$	−47.3873	−2.15175
$486$	−46.3177	−2.10101
$487$	−12.8479	−0.582192	−0.291096	−	0.956694i	$-0.594020\pi$
$487$	−12.8479	−0.582192	−0.291096	+	0.956694i	$0.594020\pi$
$488$	−12.0129	−0.543798
$489$	−14.3299	−0.648019
$490$	6.18198	0.279274
$491$	12.0690	0.544665	0.272332	−	0.962203i	$-0.412205\pi$
$491$	12.0690	0.544665	0.272332	+	0.962203i	$0.412205\pi$
$492$	−22.8369	−1.02957
$493$	0.331934	0.0149496
$494$	0	0
$495$	64.8293	2.91386
$496$	5.90708	0.265236
$497$	27.1351	1.21718
$498$	−33.0415	−1.48063
$499$	35.1067	1.57159	0.785796	−	0.618485i	$-0.212253\pi$
$499$	35.1067	1.57159	0.785796	+	0.618485i	$0.212253\pi$
$500$	−4.88256	−0.218355
$501$	−65.5473	−2.92844
$502$	−18.5904	−0.829732
$503$	−6.31482	−0.281564	−0.140782	−	0.990041i	$-0.544962\pi$
$503$	−6.31482	−0.281564	−0.140782	+	0.990041i	$0.544962\pi$
$504$	23.6037	1.05139
$505$	−28.0276	−1.24721
$506$	21.6475	0.962350
$507$	0	0
$508$	1.15297	0.0511546
$509$	−30.7506	−1.36300	−0.681498	−	0.731820i	$-0.738671\pi$
$509$	−30.7506	−1.36300	−0.681498	+	0.731820i	$0.738671\pi$
$510$	−0.940868	−0.0416623
$511$	−1.95525	−0.0864951
$512$	1.00000	0.0441942
$513$	15.7956	0.697391
$514$	6.96066	0.307022
$515$	−37.7195	−1.66212
$516$	−20.0089	−0.880842
$517$	16.9518	0.745539
$518$	−20.8221	−0.914870
$519$	15.9699	0.700999
$520$	0	0
$521$	32.8734	1.44021	0.720105	−	0.693865i	$-0.244094\pi$
$521$	32.8734	1.44021	0.720105	+	0.693865i	$0.244094\pi$
$522$	26.0868	1.14179
$523$	39.3400	1.72022	0.860108	−	0.510112i	$-0.170396\pi$
$523$	39.3400	1.72022	0.860108	+	0.510112i	$0.170396\pi$
$524$	−9.25910	−0.404486
$525$	−32.8618	−1.43421
$526$	1.68874	0.0736326
$527$	0.586668	0.0255557
$528$	−9.47348	−0.412280
$529$	33.4200	1.45304
$530$	26.4438	1.14865
$531$	6.68464	0.290089
$532$	−3.02408	−0.131110
$533$	0	0
$534$	3.12730	0.135332
$535$	23.6669	1.02321
$536$	−11.4641	−0.495173
$537$	41.2138	1.77851
$538$	28.3668	1.22298
$539$	6.18198	0.266277
$540$	−45.5226	−1.95898
$541$	19.8357	0.852802	0.426401	−	0.904534i	$-0.359781\pi$
$541$	19.8357	0.852802	0.426401	+	0.904534i	$0.359781\pi$
$542$	−14.0040	−0.601521
$543$	71.1323	3.05258
$544$	0.0993160	0.00425814
$545$	−17.7718	−0.761262
$546$	0	0
$547$	−7.35779	−0.314596	−0.157298	−	0.987551i	$-0.550278\pi$
$547$	−7.35779	−0.314596	−0.157298	+	0.987551i	$0.550278\pi$
$548$	6.17416	0.263747
$549$	−93.7639	−4.00174
$550$	9.52737	0.406248
$551$	−3.34220	−0.142383
$552$	−24.6908	−1.05091
$553$	−42.3639	−1.80150
$554$	9.57854	0.406953
$555$	65.2290	2.76882
$556$	−6.61882	−0.280700
$557$	13.4915	0.571654	0.285827	−	0.958281i	$-0.407732\pi$
$557$	13.4915	0.571654	0.285827	+	0.958281i	$0.407732\pi$
$558$	46.1064	1.95184
$559$	0	0
$560$	8.71534	0.368291
$561$	−0.940868	−0.0397235
$562$	0.150742	0.00635866
$563$	−25.5347	−1.07616	−0.538079	−	0.842894i	$-0.680850\pi$
$563$	−25.5347	−1.07616	−0.538079	+	0.842894i	$0.680850\pi$
$564$	−19.3349	−0.814146
$565$	−46.0993	−1.93941
$566$	−29.5322	−1.24133
$567$	86.2055	3.62029
$568$	8.97303	0.376500
$569$	−3.68679	−0.154558	−0.0772792	−	0.997009i	$-0.524623\pi$
$569$	−3.68679	−0.154558	−0.0772792	+	0.997009i	$0.524623\pi$
$570$	9.47348	0.396800
$571$	−2.85909	−0.119649	−0.0598246	−	0.998209i	$-0.519054\pi$
$571$	−2.85909	−0.119649	−0.0598246	+	0.998209i	$0.519054\pi$
$572$	0	0
$573$	33.8134	1.41257
$574$	21.0094	0.876913
$575$	24.8312	1.03553
$576$	7.80527	0.325219
$577$	−1.98920	−0.0828114	−0.0414057	−	0.999142i	$-0.513184\pi$
$577$	−1.98920	−0.0828114	−0.0414057	+	0.999142i	$0.513184\pi$
$578$	−16.9901	−0.706697
$579$	−73.9983	−3.07526
$580$	9.63218	0.399955
$581$	30.3973	1.26109
$582$	54.0490	2.24040
$583$	26.4438	1.09519
$584$	−0.646560	−0.0267549
$585$	0	0
$586$	3.94681	0.163041
$587$	22.4820	0.927933	0.463967	−	0.885853i	$-0.346426\pi$
$587$	22.4820	0.927933	0.463967	+	0.885853i	$0.346426\pi$
$588$	−7.05105	−0.290780
$589$	−5.90708	−0.243397
$590$	2.46821	0.101615
$591$	23.1989	0.954277
$592$	−6.88543	−0.282990
$593$	−20.4746	−0.840789	−0.420395	−	0.907341i	$-0.638108\pi$
$593$	−20.4746	−0.840789	−0.420395	+	0.907341i	$0.638108\pi$
$594$	−45.5226	−1.86781
$595$	0.865573	0.0354850
$596$	−2.91053	−0.119220
$597$	−86.2585	−3.53033
$598$	0	0
$599$	−7.90709	−0.323075	−0.161538	−	0.986867i	$-0.551645\pi$
$599$	−7.90709	−0.323075	−0.161538	+	0.986867i	$0.551645\pi$
$600$	−10.8667	−0.443633
$601$	−5.32201	−0.217089	−0.108545	−	0.994092i	$-0.534619\pi$
$601$	−5.32201	−0.217089	−0.108545	+	0.994092i	$0.534619\pi$
$602$	18.4076	0.750239
$603$	−89.4802	−3.64392
$604$	−3.38645	−0.137793
$605$	−7.76454	−0.315673
$606$	31.9677	1.29860
$607$	19.0936	0.774985	0.387493	−	0.921873i	$-0.373341\pi$
$607$	19.0936	0.774985	0.387493	+	0.921873i	$0.373341\pi$
$608$	−1.00000	−0.0405554
$609$	−33.2234	−1.34628
$610$	−34.6210	−1.40176
$611$	0	0
$612$	0.775188	0.0313351
$613$	46.5932	1.88188	0.940941	−	0.338571i	$-0.109943\pi$
$613$	46.5932	1.88188	0.940941	+	0.338571i	$0.109943\pi$
$614$	−0.213487	−0.00861565
$615$	−65.8156	−2.65394
$616$	8.71534	0.351151
$617$	37.8493	1.52376	0.761878	−	0.647720i	$-0.224278\pi$
$617$	37.8493	1.52376	0.761878	+	0.647720i	$0.224278\pi$
$618$	43.0221	1.73060
$619$	25.6498	1.03095	0.515477	−	0.856904i	$-0.327615\pi$
$619$	25.6498	1.03095	0.515477	+	0.856904i	$0.327615\pi$
$620$	17.0241	0.683705
$621$	−118.646	−4.76109
$622$	−30.1703	−1.20972
$623$	−2.87703	−0.115266
$624$	0	0
$625$	−30.6006	−1.22403
$626$	−19.5575	−0.781675
$627$	9.47348	0.378334
$628$	9.59018	0.382690
$629$	−0.683834	−0.0272662
$630$	68.0256	2.71020
$631$	−37.2968	−1.48476	−0.742380	−	0.669979i	$-0.766303\pi$
$631$	−37.2968	−1.48476	−0.742380	+	0.669979i	$0.766303\pi$
$632$	−14.0089	−0.557243
$633$	−37.1109	−1.47502
$634$	−6.31782	−0.250913
$635$	3.32283	0.131863
$636$	−30.1613	−1.19597
$637$	0	0
$638$	9.63218	0.381342
$639$	70.0369	2.77062
$640$	2.88198	0.113920
$641$	17.1654	0.677991	0.338995	−	0.940788i	$-0.389913\pi$
$641$	17.1654	0.677991	0.338995	+	0.940788i	$0.389913\pi$
$642$	−26.9940	−1.06537
$643$	−30.6064	−1.20700	−0.603500	−	0.797363i	$-0.706228\pi$
$643$	−30.6064	−1.20700	−0.603500	+	0.797363i	$0.706228\pi$
$644$	22.7148	0.895090
$645$	−57.6653	−2.27057
$646$	−0.0993160	−0.00390754
$647$	−4.87167	−0.191525	−0.0957626	−	0.995404i	$-0.530529\pi$
$647$	−4.87167	−0.191525	−0.0957626	+	0.995404i	$0.530529\pi$
$648$	28.5064	1.11984
$649$	2.46821	0.0968857
$650$	0	0
$651$	−58.7197	−2.30141
$652$	4.35938	0.170726
$653$	−12.0799	−0.472725	−0.236362	−	0.971665i	$-0.575955\pi$
$653$	−12.0799	−0.472725	−0.236362	+	0.971665i	$0.575955\pi$
$654$	20.2702	0.792629
$655$	−26.6846	−1.04265
$656$	6.94736	0.271249
$657$	−5.04658	−0.196886
$658$	17.7876	0.693432
$659$	33.0823	1.28870	0.644351	−	0.764730i	$-0.277128\pi$
$659$	33.0823	1.28870	0.644351	+	0.764730i	$0.277128\pi$
$660$	−27.3024	−1.06275
$661$	−38.7512	−1.50725	−0.753624	−	0.657305i	$-0.771696\pi$
$661$	−38.7512	−1.50725	−0.753624	+	0.657305i	$0.771696\pi$
$662$	7.47193	0.290405
$663$	0	0
$664$	10.0518	0.390084
$665$	−8.71534	−0.337967
$666$	−53.7426	−2.08248
$667$	25.1044	0.972046
$668$	19.9405	0.771523
$669$	−25.6717	−0.992524
$670$	−33.0393	−1.27642
$671$	−34.6210	−1.33653
$672$	−9.94055	−0.383465
$673$	14.3390	0.552729	0.276365	−	0.961053i	$-0.410870\pi$
$673$	14.3390	0.552729	0.276365	+	0.961053i	$0.410870\pi$
$674$	−3.39531	−0.130783
$675$	−52.2175	−2.00985
$676$	0	0
$677$	−6.46495	−0.248468	−0.124234	−	0.992253i	$-0.539647\pi$
$677$	−6.46495	−0.248468	−0.124234	+	0.992253i	$0.539647\pi$
$678$	52.5800	2.01932
$679$	−49.7236	−1.90822
$680$	0.286227	0.0109763
$681$	20.7624	0.795616
$682$	17.0241	0.651887
$683$	17.8309	0.682280	0.341140	−	0.940012i	$-0.389187\pi$
$683$	17.8309	0.682280	0.341140	+	0.940012i	$0.389187\pi$
$684$	−7.80527	−0.298442
$685$	17.7938	0.679867
$686$	−14.6818	−0.560552
$687$	22.6786	0.865244
$688$	6.08702	0.232066
$689$	0	0
$690$	−71.1584	−2.70895
$691$	26.5402	1.00964	0.504819	−	0.863225i	$-0.331559\pi$
$691$	26.5402	1.00964	0.504819	+	0.863225i	$0.331559\pi$
$692$	−4.85829	−0.184685
$693$	68.0256	2.58408
$694$	1.90912	0.0724694
$695$	−19.0753	−0.723568
$696$	−10.9863	−0.416434
$697$	0.689984	0.0261350
$698$	−26.1257	−0.988874
$699$	7.86451	0.297463
$700$	9.99710	0.377855
$701$	−30.9001	−1.16708	−0.583541	−	0.812083i	$-0.698333\pi$
$701$	−30.9001	−1.16708	−0.583541	+	0.812083i	$0.698333\pi$
$702$	0	0
$703$	6.88543	0.259689
$704$	2.88198	0.108619
$705$	−55.7228	−2.09864
$706$	−11.5915	−0.436250
$707$	−29.4095	−1.10606
$708$	−2.81519	−0.105801
$709$	31.0370	1.16562	0.582809	−	0.812609i	$-0.301954\pi$
$709$	31.0370	1.16562	0.582809	+	0.812609i	$0.301954\pi$
$710$	25.8601	0.970513
$711$	−109.343	−4.10068
$712$	−0.951375	−0.0356543
$713$	44.3700	1.66167
$714$	−0.987256	−0.0369471
$715$	0	0
$716$	−12.5379	−0.468563
$717$	−70.9520	−2.64975
$718$	4.12940	0.154108
$719$	−21.3270	−0.795362	−0.397681	−	0.917524i	$-0.630185\pi$
$719$	−21.3270	−0.795362	−0.397681	+	0.917524i	$0.630185\pi$
$720$	22.4947	0.838326
$721$	−39.5792	−1.47401
$722$	1.00000	0.0372161
$723$	53.9338	2.00582
$724$	−21.6396	−0.804229
$725$	11.0488	0.410341
$726$	8.85609	0.328680
$727$	−2.45044	−0.0908817	−0.0454409	−	0.998967i	$-0.514469\pi$
$727$	−2.45044	−0.0908817	−0.0454409	+	0.998967i	$0.514469\pi$
$728$	0	0
$729$	66.7333	2.47161
$730$	−1.86338	−0.0689667
$731$	0.604539	0.0223597
$732$	39.4880	1.45952
$733$	14.7075	0.543235	0.271618	−	0.962405i	$-0.412441\pi$
$733$	14.7075	0.543235	0.271618	+	0.962405i	$0.412441\pi$
$734$	22.0476	0.813791
$735$	−20.3210	−0.749552
$736$	7.51133	0.276871
$737$	−33.0393	−1.21702
$738$	54.2260	1.99609
$739$	−9.86821	−0.363008	−0.181504	−	0.983390i	$-0.558097\pi$
$739$	−9.86821	−0.363008	−0.181504	+	0.983390i	$0.558097\pi$
$740$	−19.8437	−0.729469
$741$	0	0
$742$	27.7476	1.01865
$743$	−38.6841	−1.41918	−0.709591	−	0.704614i	$-0.751120\pi$
$743$	−38.6841	−1.41918	−0.709591	+	0.704614i	$0.751120\pi$
$744$	−19.4174	−0.711876
$745$	−8.38811	−0.307316
$746$	22.9777	0.841273
$747$	78.4567	2.87058
$748$	0.286227	0.0104655
$749$	24.8337	0.907405
$750$	16.0496	0.586049
$751$	−22.6420	−0.826217	−0.413108	−	0.910682i	$-0.635557\pi$
$751$	−22.6420	−0.826217	−0.413108	+	0.910682i	$0.635557\pi$
$752$	5.88198	0.214494
$753$	61.1093	2.22695
$754$	0	0
$755$	−9.75970	−0.355192
$756$	−47.7670	−1.73727
$757$	−13.1303	−0.477229	−0.238614	−	0.971114i	$-0.576693\pi$
$757$	−13.1303	−0.477229	−0.238614	+	0.971114i	$0.576693\pi$
$758$	−10.1844	−0.369915
$759$	−71.1584	−2.58289
$760$	−2.88198	−0.104541
$761$	16.9767	0.615407	0.307703	−	0.951482i	$-0.400440\pi$
$761$	16.9767	0.615407	0.307703	+	0.951482i	$0.400440\pi$
$762$	−3.78996	−0.137296
$763$	−18.6481	−0.675105
$764$	−10.2866	−0.372155
$765$	2.23408	0.0807733
$766$	34.1328	1.23327
$767$	0	0
$768$	−3.28714	−0.118614
$769$	5.94960	0.214548	0.107274	−	0.994230i	$-0.465788\pi$
$769$	5.94960	0.214548	0.107274	+	0.994230i	$0.465788\pi$
$770$	25.1175	0.905171
$771$	−22.8806	−0.824026
$772$	22.5115	0.810206
$773$	−54.6636	−1.96611	−0.983056	−	0.183305i	$-0.941320\pi$
$773$	−54.6636	−1.96611	−0.983056	+	0.183305i	$0.941320\pi$
$774$	47.5108	1.70774
$775$	19.5278	0.701461
$776$	−16.4426	−0.590254
$777$	68.4450	2.45545
$778$	−11.1658	−0.400312
$779$	−6.94736	−0.248915
$780$	0	0
$781$	25.8601	0.925348
$782$	0.745995	0.0266767
$783$	−52.7920	−1.88663
$784$	2.14504	0.0766087
$785$	27.6388	0.986470
$786$	30.4359	1.08561
$787$	−39.4329	−1.40563	−0.702816	−	0.711372i	$-0.748074\pi$
$787$	−39.4329	−1.40563	−0.702816	+	0.711372i	$0.748074\pi$
$788$	−7.05750	−0.251413
$789$	−5.55112	−0.197625
$790$	−40.3734	−1.43642
$791$	−48.3722	−1.71992
$792$	22.4947	0.799313
$793$	0	0
$794$	1.89726	0.0673313
$795$	−86.9245	−3.08289
$796$	26.2412	0.930096
$797$	−39.0593	−1.38355	−0.691776	−	0.722112i	$-0.743171\pi$
$797$	−39.0593	−1.38355	−0.691776	+	0.722112i	$0.743171\pi$
$798$	9.94055	0.351892
$799$	0.584175	0.0206666
$800$	3.30584	0.116879
$801$	−7.42574	−0.262376
$802$	30.7695	1.08651
$803$	−1.86338	−0.0657572
$804$	37.6840	1.32901
$805$	65.4638	2.30730
$806$	0	0
$807$	−93.2456	−3.28240
$808$	−9.72510	−0.342128
$809$	6.72501	0.236439	0.118219	−	0.992988i	$-0.462281\pi$
$809$	6.72501	0.236439	0.118219	+	0.992988i	$0.462281\pi$
$810$	82.1550	2.88663
$811$	−18.0290	−0.633083	−0.316542	−	0.948579i	$-0.602522\pi$
$811$	−18.0290	−0.633083	−0.316542	+	0.948579i	$0.602522\pi$
$812$	10.1071	0.354689
$813$	46.0329	1.61444
$814$	−19.8437	−0.695522
$815$	12.5637	0.440086
$816$	−0.326465	−0.0114286
$817$	−6.08702	−0.212958
$818$	−13.5808	−0.474840
$819$	0	0
$820$	20.0222	0.699205
$821$	−41.6151	−1.45238	−0.726188	−	0.687496i	$-0.758710\pi$
$821$	−41.6151	−1.45238	−0.726188	+	0.687496i	$0.758710\pi$
$822$	−20.2953	−0.707880
$823$	−21.2485	−0.740678	−0.370339	−	0.928897i	$-0.620758\pi$
$823$	−21.2485	−0.740678	−0.370339	+	0.928897i	$0.620758\pi$
$824$	−13.0880	−0.455943
$825$	−31.3178	−1.09034
$826$	2.58990	0.0901142
$827$	18.1628	0.631581	0.315791	−	0.948829i	$-0.397730\pi$
$827$	18.1628	0.631581	0.315791	+	0.948829i	$0.397730\pi$
$828$	58.6279	2.03746
$829$	5.54733	0.192667	0.0963334	−	0.995349i	$-0.469288\pi$
$829$	5.54733	0.192667	0.0963334	+	0.995349i	$0.469288\pi$
$830$	28.9690	1.00553
$831$	−31.4860	−1.09224
$832$	0	0
$833$	0.213037	0.00738130
$834$	21.7570	0.753382
$835$	57.4683	1.98877
$836$	−2.88198	−0.0996755
$837$	−93.3057	−3.22512
$838$	11.5178	0.397877
$839$	−12.1478	−0.419389	−0.209695	−	0.977767i	$-0.567247\pi$
$839$	−12.1478	−0.419389	−0.209695	+	0.977767i	$0.567247\pi$
$840$	−28.6485	−0.988468
$841$	−17.8297	−0.614816
$842$	−14.7578	−0.508586
$843$	−0.495509	−0.0170662
$844$	11.2897	0.388608
$845$	0	0
$846$	45.9105	1.57843
$847$	−8.14736	−0.279947
$848$	9.17556	0.315090
$849$	97.0765	3.33166
$850$	0.328322	0.0112614
$851$	−51.7187	−1.77290
$852$	−29.4956	−1.01050
$853$	17.4434	0.597250	0.298625	−	0.954371i	$-0.403472\pi$
$853$	17.4434	0.597250	0.298625	+	0.954371i	$0.403472\pi$
$854$	−36.3279	−1.24312
$855$	−22.4947	−0.769301
$856$	8.21201	0.280681
$857$	2.87797	0.0983095	0.0491548	−	0.998791i	$-0.484347\pi$
$857$	2.87797	0.0983095	0.0491548	+	0.998791i	$0.484347\pi$
$858$	0	0
$859$	−42.4295	−1.44767	−0.723837	−	0.689971i	$-0.757623\pi$
$859$	−42.4295	−1.44767	−0.723837	+	0.689971i	$0.757623\pi$
$860$	17.5427	0.598201
$861$	−69.0606	−2.35358
$862$	0.0489741	0.00166807
$863$	36.3443	1.23717	0.618587	−	0.785716i	$-0.287706\pi$
$863$	36.3443	1.23717	0.618587	+	0.785716i	$0.287706\pi$
$864$	−15.7956	−0.537376
$865$	−14.0015	−0.476066
$866$	15.4388	0.524631
$867$	55.8489	1.89673
$868$	17.8635	0.606326
$869$	−40.3734	−1.36957
$870$	−31.6623	−1.07345
$871$	0	0
$872$	−6.16653	−0.208825
$873$	−128.339	−4.34361
$874$	−7.51133	−0.254074
$875$	−14.7652	−0.499156
$876$	2.12533	0.0718083
$877$	55.0880	1.86019	0.930094	−	0.367321i	$-0.119725\pi$
$877$	55.0880	1.86019	0.930094	+	0.367321i	$0.119725\pi$
$878$	23.4821	0.792482
$879$	−12.9737	−0.437592
$880$	8.30584	0.279990
$881$	−43.6793	−1.47159	−0.735797	−	0.677202i	$-0.763192\pi$
$881$	−43.6793	−1.47159	−0.735797	+	0.677202i	$0.763192\pi$
$882$	16.7426	0.563754
$883$	−11.2873	−0.379849	−0.189925	−	0.981799i	$-0.560824\pi$
$883$	−11.2873	−0.379849	−0.189925	+	0.981799i	$0.560824\pi$
$884$	0	0
$885$	−8.11335	−0.272727
$886$	−0.851887	−0.0286197
$887$	50.8257	1.70656	0.853280	−	0.521453i	$-0.174610\pi$
$887$	50.8257	1.70656	0.853280	+	0.521453i	$0.174610\pi$
$888$	22.6334	0.759526
$889$	3.48666	0.116939
$890$	−2.74185	−0.0919070
$891$	82.1550	2.75229
$892$	7.80973	0.261489
$893$	−5.88198	−0.196833
$894$	9.56731	0.319979
$895$	−36.1340	−1.20783
$896$	3.02408	0.101027
$897$	0	0
$898$	37.3624	1.24680
$899$	19.7427	0.658455
$900$	25.8029	0.860098
$901$	0.911280	0.0303592
$902$	20.0222	0.666666
$903$	−60.5084	−2.01359
$904$	−15.9957	−0.532008
$905$	−62.3650	−2.07308
$906$	11.1317	0.369827
$907$	7.63723	0.253590	0.126795	−	0.991929i	$-0.459531\pi$
$907$	7.63723	0.253590	0.126795	+	0.991929i	$0.459531\pi$
$908$	−6.31625	−0.209612
$909$	−75.9070	−2.51768
$910$	0	0
$911$	10.2286	0.338889	0.169445	−	0.985540i	$-0.445803\pi$
$911$	10.2286	0.338889	0.169445	+	0.985540i	$0.445803\pi$
$912$	3.28714	0.108848
$913$	28.9690	0.958735
$914$	−18.0321	−0.596448
$915$	113.804	3.76224
$916$	−6.89920	−0.227956
$917$	−28.0002	−0.924649
$918$	−1.56875	−0.0517765
$919$	36.6393	1.20862	0.604310	−	0.796749i	$-0.293449\pi$
$919$	36.6393	1.20862	0.604310	+	0.796749i	$0.293449\pi$
$920$	21.6475	0.713698
$921$	0.701762	0.0231239
$922$	−20.5994	−0.678406
$923$	0	0
$924$	−28.6485	−0.942467
$925$	−22.7621	−0.748414
$926$	0.835400	0.0274529
$927$	−102.156	−3.35523
$928$	3.34220	0.109713
$929$	2.79520	0.0917076	0.0458538	−	0.998948i	$-0.485399\pi$
$929$	2.79520	0.0917076	0.0458538	+	0.998948i	$0.485399\pi$
$930$	−55.9606	−1.83502
$931$	−2.14504	−0.0703010
$932$	−2.39251	−0.0783693
$933$	99.1738	3.24680
$934$	4.94829	0.161913
$935$	0.824902	0.0269772
$936$	0	0
$937$	23.0454	0.752862	0.376431	−	0.926445i	$-0.377151\pi$
$937$	23.0454	0.752862	0.376431	+	0.926445i	$0.377151\pi$
$938$	−34.6683	−1.13196
$939$	64.2882	2.09796
$940$	16.9518	0.552906
$941$	2.39612	0.0781114	0.0390557	−	0.999237i	$-0.487565\pi$
$941$	2.39612	0.0781114	0.0390557	+	0.999237i	$0.487565\pi$
$942$	−31.5242	−1.02712
$943$	52.1839	1.69934
$944$	0.856427	0.0278743
$945$	−137.664	−4.47820
$946$	17.5427	0.570363
$947$	41.5019	1.34863	0.674315	−	0.738444i	$-0.264439\pi$
$947$	41.5019	1.34863	0.674315	+	0.738444i	$0.264439\pi$
$948$	46.0491	1.49561
$949$	0	0
$950$	−3.30584	−0.107255
$951$	20.7675	0.673434
$952$	0.300339	0.00973405
$953$	13.9965	0.453390	0.226695	−	0.973966i	$-0.427208\pi$
$953$	13.9965	0.453390	0.226695	+	0.973966i	$0.427208\pi$
$954$	71.6177	2.31871
$955$	−29.6457	−0.959313
$956$	21.5847	0.698100
$957$	−31.6623	−1.02350
$958$	3.05994	0.0988622
$959$	18.6711	0.602922
$960$	−9.47348	−0.305755
$961$	3.89363	0.125601
$962$	0	0
$963$	64.0969	2.06549
$964$	−16.4075	−0.528451
$965$	64.8777	2.08849
$966$	−74.6667	−2.40236
$967$	5.30132	0.170479	0.0852394	−	0.996360i	$-0.472834\pi$
$967$	5.30132	0.170479	0.0852394	+	0.996360i	$0.472834\pi$
$968$	−2.69416	−0.0865937
$969$	0.326465	0.0104876
$970$	−47.3873	−1.52151
$971$	42.2763	1.35671	0.678355	−	0.734734i	$-0.262693\pi$
$971$	42.2763	1.35671	0.678355	+	0.734734i	$0.262693\pi$
$972$	−46.3177	−1.48564
$973$	−20.0158	−0.641677
$974$	−12.8479	−0.411672
$975$	0	0
$976$	−12.0129	−0.384523
$977$	−15.7624	−0.504282	−0.252141	−	0.967690i	$-0.581135\pi$
$977$	−15.7624	−0.504282	−0.252141	+	0.967690i	$0.581135\pi$
$978$	−14.3299	−0.458219
$979$	−2.74185	−0.0876299
$980$	6.18198	0.197476
$981$	−48.1314	−1.53672
$982$	12.0690	0.385136
$983$	−23.2214	−0.740647	−0.370324	−	0.928903i	$-0.620753\pi$
$983$	−23.2214	−0.740647	−0.370324	+	0.928903i	$0.620753\pi$
$984$	−22.8369	−0.728014
$985$	−20.3396	−0.648073
$986$	0.331934	0.0105709
$987$	−58.4702	−1.86113
$988$	0	0
$989$	45.7216	1.45386
$990$	64.8293	2.06041
$991$	4.63538	0.147248	0.0736239	−	0.997286i	$-0.476544\pi$
$991$	4.63538	0.147248	0.0736239	+	0.997286i	$0.476544\pi$
$992$	5.90708	0.187550
$993$	−24.5613	−0.779428
$994$	27.1351	0.860674
$995$	75.6268	2.39753
$996$	−33.0415	−1.04696
$997$	−11.7982	−0.373652	−0.186826	−	0.982393i	$-0.559820\pi$
$997$	−11.7982	−0.373652	−0.186826	+	0.982393i	$0.559820\pi$
$998$	35.1067	1.11128
$999$	108.759	3.44099

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6422.2.a.bd.1.1		6
13.3	even	3		494.2.g.f.191.6	✓	12
13.9	even	3		494.2.g.f.419.6	yes	12
13.12	even	2		6422.2.a.bc.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
494.2.g.f.191.6	✓	12	13.3	even	3
494.2.g.f.419.6	yes	12	13.9	even	3
6422.2.a.bc.1.1		6	13.12	even	2
6422.2.a.bd.1.1		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6422.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.28714 q^{3} +1.00000 q^{4} +2.88198 q^{5} -3.28714 q^{6} +3.02408 q^{7} +1.00000 q^{8} +7.80527 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.28714 q^{3} +1.00000 q^{4} +2.88198 q^{5} -3.28714 q^{6} +3.02408 q^{7} +1.00000 q^{8} +7.80527 q^{9} +2.88198 q^{10} +2.88198 q^{11} -3.28714 q^{12} +3.02408 q^{14} -9.47348 q^{15} +1.00000 q^{16} +0.0993160 q^{17} +7.80527 q^{18} -1.00000 q^{19} +2.88198 q^{20} -9.94055 q^{21} +2.88198 q^{22} +7.51133 q^{23} -3.28714 q^{24} +3.30584 q^{25} -15.7956 q^{27} +3.02408 q^{28} +3.34220 q^{29} -9.47348 q^{30} +5.90708 q^{31} +1.00000 q^{32} -9.47348 q^{33} +0.0993160 q^{34} +8.71534 q^{35} +7.80527 q^{36} -6.88543 q^{37} -1.00000 q^{38} +2.88198 q^{40} +6.94736 q^{41} -9.94055 q^{42} +6.08702 q^{43} +2.88198 q^{44} +22.4947 q^{45} +7.51133 q^{46} +5.88198 q^{47} -3.28714 q^{48} +2.14504 q^{49} +3.30584 q^{50} -0.326465 q^{51} +9.17556 q^{53} -15.7956 q^{54} +8.30584 q^{55} +3.02408 q^{56} +3.28714 q^{57} +3.34220 q^{58} +0.856427 q^{59} -9.47348 q^{60} -12.0129 q^{61} +5.90708 q^{62} +23.6037 q^{63} +1.00000 q^{64} -9.47348 q^{66} -11.4641 q^{67} +0.0993160 q^{68} -24.6908 q^{69} +8.71534 q^{70} +8.97303 q^{71} +7.80527 q^{72} -0.646560 q^{73} -6.88543 q^{74} -10.8667 q^{75} -1.00000 q^{76} +8.71534 q^{77} -14.0089 q^{79} +2.88198 q^{80} +28.5064 q^{81} +6.94736 q^{82} +10.0518 q^{83} -9.94055 q^{84} +0.286227 q^{85} +6.08702 q^{86} -10.9863 q^{87} +2.88198 q^{88} -0.951375 q^{89} +22.4947 q^{90} +7.51133 q^{92} -19.4174 q^{93} +5.88198 q^{94} -2.88198 q^{95} -3.28714 q^{96} -16.4426 q^{97} +2.14504 q^{98} +22.4947 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 6 q^{8} + 10 q^{9}+O(q^{10}) \) 6 * q + 6 * q^2 + 2 * q^3 + 6 * q^4 + 2 * q^5 + 2 * q^6 + q^7 + 6 * q^8 + 10 * q^9	\( 6 q + 6 q^{2} + 2 q^{3} + 6 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 6 q^{8} + 10 q^{9} + 2 q^{10} + 2 q^{11} + 2 q^{12} + q^{14} - 9 q^{15} + 6 q^{16} - 2 q^{17} + 10 q^{18} - 6 q^{19} + 2 q^{20} + q^{21} + 2 q^{22} + 8 q^{23} + 2 q^{24} + 16 q^{25} - 4 q^{27} + q^{28} + 20 q^{29} - 9 q^{30} + 3 q^{31} + 6 q^{32} - 9 q^{33} - 2 q^{34} + 7 q^{35} + 10 q^{36} - 9 q^{37} - 6 q^{38} + 2 q^{40} + 3 q^{41} + q^{42} + 13 q^{43} + 2 q^{44} + 38 q^{45} + 8 q^{46} + 20 q^{47} + 2 q^{48} - 7 q^{49} + 16 q^{50} + 2 q^{51} + 25 q^{53} - 4 q^{54} + 46 q^{55} + q^{56} - 2 q^{57} + 20 q^{58} - 9 q^{60} + 6 q^{61} + 3 q^{62} + 46 q^{63} + 6 q^{64} - 9 q^{66} - 32 q^{67} - 2 q^{68} - 29 q^{69} + 7 q^{70} + 39 q^{71} + 10 q^{72} + 7 q^{73} - 9 q^{74} - 15 q^{75} - 6 q^{76} + 7 q^{77} + 18 q^{79} + 2 q^{80} + 54 q^{81} + 3 q^{82} - 7 q^{83} + q^{84} - 2 q^{85} + 13 q^{86} + 12 q^{87} + 2 q^{88} + 9 q^{89} + 38 q^{90} + 8 q^{92} - 47 q^{93} + 20 q^{94} - 2 q^{95} + 2 q^{96} - 4 q^{97} - 7 q^{98} + 38 q^{99}+O(q^{100}) \) 6 * q + 6 * q^2 + 2 * q^3 + 6 * q^4 + 2 * q^5 + 2 * q^6 + q^7 + 6 * q^8 + 10 * q^9 + 2 * q^10 + 2 * q^11 + 2 * q^12 + q^14 - 9 * q^15 + 6 * q^16 - 2 * q^17 + 10 * q^18 - 6 * q^19 + 2 * q^20 + q^21 + 2 * q^22 + 8 * q^23 + 2 * q^24 + 16 * q^25 - 4 * q^27 + q^28 + 20 * q^29 - 9 * q^30 + 3 * q^31 + 6 * q^32 - 9 * q^33 - 2 * q^34 + 7 * q^35 + 10 * q^36 - 9 * q^37 - 6 * q^38 + 2 * q^40 + 3 * q^41 + q^42 + 13 * q^43 + 2 * q^44 + 38 * q^45 + 8 * q^46 + 20 * q^47 + 2 * q^48 - 7 * q^49 + 16 * q^50 + 2 * q^51 + 25 * q^53 - 4 * q^54 + 46 * q^55 + q^56 - 2 * q^57 + 20 * q^58 - 9 * q^60 + 6 * q^61 + 3 * q^62 + 46 * q^63 + 6 * q^64 - 9 * q^66 - 32 * q^67 - 2 * q^68 - 29 * q^69 + 7 * q^70 + 39 * q^71 + 10 * q^72 + 7 * q^73 - 9 * q^74 - 15 * q^75 - 6 * q^76 + 7 * q^77 + 18 * q^79 + 2 * q^80 + 54 * q^81 + 3 * q^82 - 7 * q^83 + q^84 - 2 * q^85 + 13 * q^86 + 12 * q^87 + 2 * q^88 + 9 * q^89 + 38 * q^90 + 8 * q^92 - 47 * q^93 + 20 * q^94 - 2 * q^95 + 2 * q^96 - 4 * q^97 - 7 * q^98 + 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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