LMFDB - Embedded newform 1694.2.a.z.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1694,2,Mod(1,1694)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1694, 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("1694.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1694 = 2 \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1694.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:	$13.5266581024$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.7625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 4x + 16 $ x^4 - x^3 - 9x^2 + 4x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 154)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.96645$ of defining polynomial
Character	$\chi$	$=$	1694.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.96645 q^{3} +1.00000 q^{4} -4.18178 q^{5} +1.96645 q^{6} -1.00000 q^{7} +1.00000 q^{8} +0.866918 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.96645 q^{3} +1.00000 q^{4} -4.18178 q^{5} +1.96645 q^{6} -1.00000 q^{7} +1.00000 q^{8} +0.866918 q^{9} -4.18178 q^{10} +1.96645 q^{12} -4.40270 q^{13} -1.00000 q^{14} -8.22325 q^{15} +1.00000 q^{16} +1.10299 q^{17} +0.866918 q^{18} -1.43625 q^{19} -4.18178 q^{20} -1.96645 q^{21} -1.16663 q^{23} +1.96645 q^{24} +12.4873 q^{25} -4.40270 q^{26} -4.19460 q^{27} -1.00000 q^{28} -7.30550 q^{29} -8.22325 q^{30} -2.83337 q^{31} +1.00000 q^{32} +1.10299 q^{34} +4.18178 q^{35} +0.866918 q^{36} -7.08458 q^{37} -1.43625 q^{38} -8.65769 q^{39} -4.18178 q^{40} -1.13308 q^{41} -1.96645 q^{42} -4.55093 q^{43} -3.62526 q^{45} -1.16663 q^{46} -7.19316 q^{47} +1.96645 q^{48} +1.00000 q^{49} +12.4873 q^{50} +2.16896 q^{51} -4.40270 q^{52} +10.0023 q^{53} -4.19460 q^{54} -1.00000 q^{56} -2.82432 q^{57} -7.30550 q^{58} +7.41439 q^{59} -8.22325 q^{60} -11.0151 q^{61} -2.83337 q^{62} -0.866918 q^{63} +1.00000 q^{64} +18.4111 q^{65} +11.7327 q^{67} +1.10299 q^{68} -2.29413 q^{69} +4.18178 q^{70} +8.47447 q^{71} +0.866918 q^{72} -13.6090 q^{73} -7.08458 q^{74} +24.5556 q^{75} -1.43625 q^{76} -8.65769 q^{78} -7.26403 q^{79} -4.18178 q^{80} -10.8492 q^{81} -1.13308 q^{82} +6.44235 q^{83} -1.96645 q^{84} -4.61244 q^{85} -4.55093 q^{86} -14.3659 q^{87} +7.52083 q^{89} -3.62526 q^{90} +4.40270 q^{91} -1.16663 q^{92} -5.57167 q^{93} -7.19316 q^{94} +6.00610 q^{95} +1.96645 q^{96} +1.40616 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} - 3 q^{3} + 4 q^{4} - 5 q^{5} - 3 q^{6} - 4 q^{7} + 4 q^{8} + 9 q^{9}+O(q^{10}) $ 4 * q + 4 * q^2 - 3 * q^3 + 4 * q^4 - 5 * q^5 - 3 * q^6 - 4 * q^7 + 4 * q^8 + 9 * q^9	$ 4 q + 4 q^{2} - 3 q^{3} + 4 q^{4} - 5 q^{5} - 3 q^{6} - 4 q^{7} + 4 q^{8} + 9 q^{9} - 5 q^{10} - 3 q^{12} - 14 q^{13} - 4 q^{14} - 5 q^{15} + 4 q^{16} + q^{17} + 9 q^{18} - 13 q^{19} - 5 q^{20} + 3 q^{21} - 10 q^{23} - 3 q^{24} + 15 q^{25} - 14 q^{26} - 24 q^{27} - 4 q^{28} - 6 q^{29} - 5 q^{30} - 6 q^{31} + 4 q^{32} + q^{34} + 5 q^{35} + 9 q^{36} + 3 q^{37} - 13 q^{38} - 2 q^{39} - 5 q^{40} + q^{41} + 3 q^{42} + 8 q^{43} - 20 q^{45} - 10 q^{46} - 3 q^{47} - 3 q^{48} + 4 q^{49} + 15 q^{50} - 22 q^{51} - 14 q^{52} + 4 q^{53} - 24 q^{54} - 4 q^{56} + 16 q^{57} - 6 q^{58} - 4 q^{59} - 5 q^{60} - 27 q^{61} - 6 q^{62} - 9 q^{63} + 4 q^{64} - 5 q^{65} + 9 q^{67} + q^{68} + 5 q^{70} - 20 q^{71} + 9 q^{72} - 28 q^{73} + 3 q^{74} - 13 q^{76} - 2 q^{78} - 22 q^{79} - 5 q^{80} + 16 q^{81} + q^{82} - 6 q^{83} + 3 q^{84} - 5 q^{85} + 8 q^{86} + 2 q^{87} + 6 q^{89} - 20 q^{90} + 14 q^{91} - 10 q^{92} + 12 q^{93} - 3 q^{94} - 15 q^{95} - 3 q^{96} + 15 q^{97} + 4 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 - 3 * q^3 + 4 * q^4 - 5 * q^5 - 3 * q^6 - 4 * q^7 + 4 * q^8 + 9 * q^9 - 5 * q^10 - 3 * q^12 - 14 * q^13 - 4 * q^14 - 5 * q^15 + 4 * q^16 + q^17 + 9 * q^18 - 13 * q^19 - 5 * q^20 + 3 * q^21 - 10 * q^23 - 3 * q^24 + 15 * q^25 - 14 * q^26 - 24 * q^27 - 4 * q^28 - 6 * q^29 - 5 * q^30 - 6 * q^31 + 4 * q^32 + q^34 + 5 * q^35 + 9 * q^36 + 3 * q^37 - 13 * q^38 - 2 * q^39 - 5 * q^40 + q^41 + 3 * q^42 + 8 * q^43 - 20 * q^45 - 10 * q^46 - 3 * q^47 - 3 * q^48 + 4 * q^49 + 15 * q^50 - 22 * q^51 - 14 * q^52 + 4 * q^53 - 24 * q^54 - 4 * q^56 + 16 * q^57 - 6 * q^58 - 4 * q^59 - 5 * q^60 - 27 * q^61 - 6 * q^62 - 9 * q^63 + 4 * q^64 - 5 * q^65 + 9 * q^67 + q^68 + 5 * q^70 - 20 * q^71 + 9 * q^72 - 28 * q^73 + 3 * q^74 - 13 * q^76 - 2 * q^78 - 22 * q^79 - 5 * q^80 + 16 * q^81 + q^82 - 6 * q^83 + 3 * q^84 - 5 * q^85 + 8 * q^86 + 2 * q^87 + 6 * q^89 - 20 * q^90 + 14 * q^91 - 10 * q^92 + 12 * q^93 - 3 * q^94 - 15 * q^95 - 3 * q^96 + 15 * q^97 + 4 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.96645	1.13533	0.567665	−	0.823260i	$-0.307847\pi$
$3$	1.96645	1.13533	0.567665	+	0.823260i	$0.307847\pi$
$4$	1.00000	0.500000
$5$	−4.18178	−1.87015	−0.935074	−	0.354452i	$-0.884668\pi$
$5$	−4.18178	−1.87015	−0.935074	+	0.354452i	$0.884668\pi$
$6$	1.96645	0.802799
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0.866918	0.288973
$10$	−4.18178	−1.32239
$11$	0	0
$11$	0	0
$12$	1.96645	0.567665
$13$	−4.40270	−1.22109	−0.610545	−	0.791982i	$-0.709050\pi$
$13$	−4.40270	−1.22109	−0.610545	+	0.791982i	$0.709050\pi$
$14$	−1.00000	−0.267261
$15$	−8.22325	−2.12323
$16$	1.00000	0.250000
$17$	1.10299	0.267513	0.133757	−	0.991014i	$-0.457296\pi$
$17$	1.10299	0.267513	0.133757	+	0.991014i	$0.457296\pi$
$18$	0.866918	0.204334
$19$	−1.43625	−0.329499	−0.164750	−	0.986335i	$-0.552682\pi$
$19$	−1.43625	−0.329499	−0.164750	+	0.986335i	$0.552682\pi$
$20$	−4.18178	−0.935074
$21$	−1.96645	−0.429114
$22$	0	0
$23$	−1.16663	−0.243260	−0.121630	−	0.992576i	$-0.538812\pi$
$23$	−1.16663	−0.243260	−0.121630	+	0.992576i	$0.538812\pi$
$24$	1.96645	0.401400
$25$	12.4873	2.49746
$26$	−4.40270	−0.863441
$27$	−4.19460	−0.807250
$28$	−1.00000	−0.188982
$29$	−7.30550	−1.35660	−0.678299	−	0.734786i	$-0.737282\pi$
$29$	−7.30550	−1.35660	−0.678299	+	0.734786i	$0.737282\pi$
$30$	−8.22325	−1.50135
$31$	−2.83337	−0.508887	−0.254444	−	0.967088i	$-0.581892\pi$
$31$	−2.83337	−0.508887	−0.254444	+	0.967088i	$0.581892\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.10299	0.189160
$35$	4.18178	0.706850
$36$	0.866918	0.144486
$37$	−7.08458	−1.16470	−0.582349	−	0.812939i	$-0.697866\pi$
$37$	−7.08458	−1.16470	−0.582349	+	0.812939i	$0.697866\pi$
$38$	−1.43625	−0.232991
$39$	−8.65769	−1.38634
$40$	−4.18178	−0.661197
$41$	−1.13308	−0.176958	−0.0884789	−	0.996078i	$-0.528201\pi$
$41$	−1.13308	−0.176958	−0.0884789	+	0.996078i	$0.528201\pi$
$42$	−1.96645	−0.303430
$43$	−4.55093	−0.694010	−0.347005	−	0.937863i	$-0.612801\pi$
$43$	−4.55093	−0.694010	−0.347005	+	0.937863i	$0.612801\pi$
$44$	0	0
$45$	−3.62526	−0.540422
$46$	−1.16663	−0.172011
$47$	−7.19316	−1.04923	−0.524615	−	0.851340i	$-0.675791\pi$
$47$	−7.19316	−1.04923	−0.524615	+	0.851340i	$0.675791\pi$
$48$	1.96645	0.283832
$49$	1.00000	0.142857
$50$	12.4873	1.76597
$51$	2.16896	0.303716
$52$	−4.40270	−0.610545
$53$	10.0023	1.37393	0.686963	−	0.726693i	$-0.258944\pi$
$53$	10.0023	1.37393	0.686963	+	0.726693i	$0.258944\pi$
$54$	−4.19460	−0.570812
$55$	0	0
$56$	−1.00000	−0.133631
$57$	−2.82432	−0.374090
$58$	−7.30550	−0.959259
$59$	7.41439	0.965272	0.482636	−	0.875821i	$-0.339679\pi$
$59$	7.41439	0.965272	0.482636	+	0.875821i	$0.339679\pi$
$60$	−8.22325	−1.06162
$61$	−11.0151	−1.41034	−0.705172	−	0.709036i	$-0.749130\pi$
$61$	−11.0151	−1.41034	−0.705172	+	0.709036i	$0.749130\pi$
$62$	−2.83337	−0.359838
$63$	−0.866918	−0.109221
$64$	1.00000	0.125000
$65$	18.4111	2.28362
$66$	0	0
$67$	11.7327	1.43338	0.716689	−	0.697393i	$-0.245657\pi$
$67$	11.7327	1.43338	0.716689	+	0.697393i	$0.245657\pi$
$68$	1.10299	0.133757
$69$	−2.29413	−0.276180
$70$	4.18178	0.499818
$71$	8.47447	1.00573	0.502867	−	0.864364i	$-0.332279\pi$
$71$	8.47447	1.00573	0.502867	+	0.864364i	$0.332279\pi$
$72$	0.866918	0.102167
$73$	−13.6090	−1.59281	−0.796406	−	0.604763i	$-0.793268\pi$
$73$	−13.6090	−1.59281	−0.796406	+	0.604763i	$0.793268\pi$
$74$	−7.08458	−0.823566
$75$	24.5556	2.83544
$76$	−1.43625	−0.164750
$77$	0	0
$78$	−8.65769	−0.980290
$79$	−7.26403	−0.817267	−0.408634	−	0.912699i	$-0.633995\pi$
$79$	−7.26403	−0.817267	−0.408634	+	0.912699i	$0.633995\pi$
$80$	−4.18178	−0.467537
$81$	−10.8492	−1.20547
$82$	−1.13308	−0.125128
$83$	6.44235	0.707140	0.353570	−	0.935408i	$-0.384968\pi$
$83$	6.44235	0.707140	0.353570	+	0.935408i	$0.384968\pi$
$84$	−1.96645	−0.214557
$85$	−4.61244	−0.500290
$86$	−4.55093	−0.490739
$87$	−14.3659	−1.54018
$88$	0	0
$89$	7.52083	0.797207	0.398603	−	0.917123i	$-0.369495\pi$
$89$	7.52083	0.797207	0.398603	+	0.917123i	$0.369495\pi$
$90$	−3.62526	−0.382136
$91$	4.40270	0.461529
$92$	−1.16663	−0.121630
$93$	−5.57167	−0.577755
$94$	−7.19316	−0.741917
$95$	6.00610	0.616213
$96$	1.96645	0.200700
$97$	1.40616	0.142774	0.0713868	−	0.997449i	$-0.477258\pi$
$97$	1.40616	0.142774	0.0713868	+	0.997449i	$0.477258\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	12.4873	1.24873
$101$	−8.40270	−0.836100	−0.418050	−	0.908424i	$-0.637286\pi$
$101$	−8.40270	−0.836100	−0.418050	+	0.908424i	$0.637286\pi$
$102$	2.16896	0.214759
$103$	17.6449	1.73860	0.869300	−	0.494284i	$-0.164570\pi$
$103$	17.6449	1.73860	0.869300	+	0.494284i	$0.164570\pi$
$104$	−4.40270	−0.431720
$105$	8.22325	0.802507
$106$	10.0023	0.971512
$107$	−13.7199	−1.32635	−0.663176	−	0.748463i	$-0.730792\pi$
$107$	−13.7199	−1.32635	−0.663176	+	0.748463i	$0.730792\pi$
$108$	−4.19460	−0.403625
$109$	8.94804	0.857067	0.428533	−	0.903526i	$-0.359030\pi$
$109$	8.94804	0.857067	0.428533	+	0.903526i	$0.359030\pi$
$110$	0	0
$111$	−13.9315	−1.32232
$112$	−1.00000	−0.0944911
$113$	−5.80214	−0.545820	−0.272910	−	0.962040i	$-0.587986\pi$
$113$	−5.80214	−0.545820	−0.272910	+	0.962040i	$0.587986\pi$
$114$	−2.82432	−0.264522
$115$	4.87861	0.454933
$116$	−7.30550	−0.678299
$117$	−3.81678	−0.352861
$118$	7.41439	0.682550
$119$	−1.10299	−0.101111
$120$	−8.22325	−0.750677
$121$	0	0
$122$	−11.0151	−0.997264
$123$	−2.22815	−0.200905
$124$	−2.83337	−0.254444
$125$	−31.3102	−2.80047
$126$	−0.866918	−0.0772312
$127$	10.7233	0.951543	0.475772	−	0.879569i	$-0.342169\pi$
$127$	10.7233	0.951543	0.475772	+	0.879569i	$0.342169\pi$
$128$	1.00000	0.0883883
$129$	−8.94917	−0.787930
$130$	18.4111	1.61476
$131$	9.33001	0.815167	0.407583	−	0.913168i	$-0.366372\pi$
$131$	9.33001	0.815167	0.407583	+	0.913168i	$0.366372\pi$
$132$	0	0
$133$	1.43625	0.124539
$134$	11.7327	1.01355
$135$	17.5409	1.50968
$136$	1.10299	0.0945802
$137$	13.2048	1.12817	0.564083	−	0.825718i	$-0.309230\pi$
$137$	13.2048	1.12817	0.564083	+	0.825718i	$0.309230\pi$
$138$	−2.29413	−0.195289
$139$	−7.19460	−0.610238	−0.305119	−	0.952314i	$-0.598696\pi$
$139$	−7.19460	−0.610238	−0.305119	+	0.952314i	$0.598696\pi$
$140$	4.18178	0.353425
$141$	−14.1450	−1.19122
$142$	8.47447	0.711161
$143$	0	0
$144$	0.866918	0.0722431
$145$	30.5500	2.53704
$146$	−13.6090	−1.12629
$147$	1.96645	0.162190
$148$	−7.08458	−0.582349
$149$	17.4225	1.42731	0.713654	−	0.700499i	$-0.247039\pi$
$149$	17.4225	1.42731	0.713654	+	0.700499i	$0.247039\pi$
$150$	24.5556	2.00496
$151$	12.0718	0.982386	0.491193	−	0.871051i	$-0.336561\pi$
$151$	12.0718	0.982386	0.491193	+	0.871051i	$0.336561\pi$
$152$	−1.43625	−0.116496
$153$	0.956198	0.0773040
$154$	0	0
$155$	11.8485	0.951695
$156$	−8.65769	−0.693170
$157$	−20.5719	−1.64181	−0.820907	−	0.571062i	$-0.806531\pi$
$157$	−20.5719	−1.64181	−0.820907	+	0.571062i	$0.806531\pi$
$158$	−7.26403	−0.577895
$159$	19.6691	1.55986
$160$	−4.18178	−0.330599
$161$	1.16663	0.0919437
$162$	−10.8492	−0.852394
$163$	1.58774	0.124362	0.0621808	−	0.998065i	$-0.480194\pi$
$163$	1.58774	0.124362	0.0621808	+	0.998065i	$0.480194\pi$
$164$	−1.13308	−0.0884789
$165$	0	0
$166$	6.44235	0.500024
$167$	20.9293	1.61956	0.809780	−	0.586734i	$-0.199587\pi$
$167$	20.9293	1.61956	0.809780	+	0.586734i	$0.199587\pi$
$168$	−1.96645	−0.151715
$169$	6.38379	0.491061
$170$	−4.61244	−0.353758
$171$	−1.24511	−0.0952163
$172$	−4.55093	−0.347005
$173$	−1.03934	−0.0790193	−0.0395097	−	0.999219i	$-0.512580\pi$
$173$	−1.03934	−0.0790193	−0.0395097	+	0.999219i	$0.512580\pi$
$174$	−14.3659	−1.08908
$175$	−12.4873	−0.943950
$176$	0	0
$177$	14.5800	1.09590
$178$	7.52083	0.563710
$179$	−0.409927	−0.0306394	−0.0153197	−	0.999883i	$-0.504877\pi$
$179$	−0.409927	−0.0306394	−0.0153197	+	0.999883i	$0.504877\pi$
$180$	−3.62526	−0.270211
$181$	−10.9630	−0.814873	−0.407436	−	0.913234i	$-0.633577\pi$
$181$	−10.9630	−0.814873	−0.407436	+	0.913234i	$0.633577\pi$
$182$	4.40270	0.326350
$183$	−21.6607	−1.60121
$184$	−1.16663	−0.0860054
$185$	29.6261	2.17816
$186$	−5.57167	−0.408534
$187$	0	0
$188$	−7.19316	−0.524615
$189$	4.19460	0.305112
$190$	6.00610	0.435728
$191$	6.89903	0.499196	0.249598	−	0.968350i	$-0.419701\pi$
$191$	6.89903	0.499196	0.249598	+	0.968350i	$0.419701\pi$
$192$	1.96645	0.141916
$193$	−3.41408	−0.245751	−0.122875	−	0.992422i	$-0.539212\pi$
$193$	−3.41408	−0.245751	−0.122875	+	0.992422i	$0.539212\pi$
$194$	1.40616	0.100956
$195$	36.2045	2.59266
$196$	1.00000	0.0714286
$197$	−1.70964	−0.121807	−0.0609035	−	0.998144i	$-0.519398\pi$
$197$	−1.70964	−0.121807	−0.0609035	+	0.998144i	$0.519398\pi$
$198$	0	0
$199$	−14.0415	−0.995374	−0.497687	−	0.867357i	$-0.665817\pi$
$199$	−14.0415	−0.995374	−0.497687	+	0.867357i	$0.665817\pi$
$200$	12.4873	0.882984
$201$	23.0718	1.62736
$202$	−8.40270	−0.591212
$203$	7.30550	0.512746
$204$	2.16896	0.151858
$205$	4.73830	0.330937
$206$	17.6449	1.22938
$207$	−1.01138	−0.0702955
$208$	−4.40270	−0.305272
$209$	0	0
$210$	8.22325	0.567458
$211$	−17.0081	−1.17089	−0.585443	−	0.810713i	$-0.699080\pi$
$211$	−17.0081	−1.17089	−0.585443	+	0.810713i	$0.699080\pi$
$212$	10.0023	0.686963
$213$	16.6646	1.14184
$214$	−13.7199	−0.937873
$215$	19.0310	1.29790
$216$	−4.19460	−0.285406
$217$	2.83337	0.192341
$218$	8.94804	0.606038
$219$	−26.7614	−1.80837
$220$	0	0
$221$	−4.85612	−0.326658
$222$	−13.9315	−0.935018
$223$	9.01980	0.604011	0.302005	−	0.953306i	$-0.402344\pi$
$223$	9.01980	0.604011	0.302005	+	0.953306i	$0.402344\pi$
$224$	−1.00000	−0.0668153
$225$	10.8254	0.721696
$226$	−5.80214	−0.385953
$227$	−12.5419	−0.832434	−0.416217	−	0.909265i	$-0.636644\pi$
$227$	−12.5419	−0.832434	−0.416217	+	0.909265i	$0.636644\pi$
$228$	−2.82432	−0.187045
$229$	−19.4202	−1.28332	−0.641661	−	0.766989i	$-0.721754\pi$
$229$	−19.4202	−1.28332	−0.641661	+	0.766989i	$0.721754\pi$
$230$	4.87861	0.321686
$231$	0	0
$232$	−7.30550	−0.479630
$233$	−14.2721	−0.934999	−0.467500	−	0.883993i	$-0.654845\pi$
$233$	−14.2721	−0.934999	−0.467500	+	0.883993i	$0.654845\pi$
$234$	−3.81678	−0.249511
$235$	30.0802	1.96222
$236$	7.41439	0.482636
$237$	−14.2843	−0.927867
$238$	−1.10299	−0.0714959
$239$	3.80704	0.246257	0.123128	−	0.992391i	$-0.460707\pi$
$239$	3.80704	0.246257	0.123128	+	0.992391i	$0.460707\pi$
$240$	−8.22325	−0.530809
$241$	−5.80125	−0.373692	−0.186846	−	0.982389i	$-0.559826\pi$
$241$	−5.80125	−0.373692	−0.186846	+	0.982389i	$0.559826\pi$
$242$	0	0
$243$	−8.75061	−0.561352
$244$	−11.0151	−0.705172
$245$	−4.18178	−0.267164
$246$	−2.22815	−0.142061
$247$	6.32340	0.402348
$248$	−2.83337	−0.179919
$249$	12.6686	0.802837
$250$	−31.3102	−1.98023
$251$	−11.7269	−0.740197	−0.370098	−	0.928993i	$-0.620676\pi$
$251$	−11.7269	−0.740197	−0.370098	+	0.928993i	$0.620676\pi$
$252$	−0.866918	−0.0546107
$253$	0	0
$254$	10.7233	0.672843
$255$	−9.07013	−0.567994
$256$	1.00000	0.0625000
$257$	−7.94082	−0.495335	−0.247667	−	0.968845i	$-0.579664\pi$
$257$	−7.94082	−0.495335	−0.247667	+	0.968845i	$0.579664\pi$
$258$	−8.94917	−0.557151
$259$	7.08458	0.440214
$260$	18.4111	1.14181
$261$	−6.33327	−0.392019
$262$	9.33001	0.576410
$263$	−5.78832	−0.356923	−0.178461	−	0.983947i	$-0.557112\pi$
$263$	−5.78832	−0.356923	−0.178461	+	0.983947i	$0.557112\pi$
$264$	0	0
$265$	−41.8275	−2.56945
$266$	1.43625	0.0880624
$267$	14.7893	0.905092
$268$	11.7327	0.716689
$269$	9.92680	0.605247	0.302624	−	0.953110i	$-0.402137\pi$
$269$	9.92680	0.605247	0.302624	+	0.953110i	$0.402137\pi$
$270$	17.5409	1.06750
$271$	9.78374	0.594320	0.297160	−	0.954828i	$-0.403961\pi$
$271$	9.78374	0.594320	0.297160	+	0.954828i	$0.403961\pi$
$272$	1.10299	0.0668783
$273$	8.65769	0.523987
$274$	13.2048	0.797734
$275$	0	0
$276$	−2.29413	−0.138090
$277$	−14.0415	−0.843670	−0.421835	−	0.906673i	$-0.638614\pi$
$277$	−14.0415	−0.843670	−0.421835	+	0.906673i	$0.638614\pi$
$278$	−7.19460	−0.431503
$279$	−2.45629	−0.147055
$280$	4.18178	0.249909
$281$	−19.5722	−1.16758	−0.583789	−	0.811905i	$-0.698430\pi$
$281$	−19.5722	−1.16758	−0.583789	+	0.811905i	$0.698430\pi$
$282$	−14.1450	−0.842320
$283$	4.24977	0.252623	0.126311	−	0.991991i	$-0.459686\pi$
$283$	4.24977	0.252623	0.126311	+	0.991991i	$0.459686\pi$
$284$	8.47447	0.502867
$285$	11.8107	0.699604
$286$	0	0
$287$	1.13308	0.0668837
$288$	0.866918	0.0510836
$289$	−15.7834	−0.928437
$290$	30.5500	1.79396
$291$	2.76514	0.162095
$292$	−13.6090	−0.796406
$293$	14.1123	0.824452	0.412226	−	0.911082i	$-0.364751\pi$
$293$	14.1123	0.824452	0.412226	+	0.911082i	$0.364751\pi$
$294$	1.96645	0.114686
$295$	−31.0054	−1.80520
$296$	−7.08458	−0.411783
$297$	0	0
$298$	17.4225	1.00926
$299$	5.13634	0.297042
$300$	24.5556	1.41772
$301$	4.55093	0.262311
$302$	12.0718	0.694652
$303$	−16.5235	−0.949249
$304$	−1.43625	−0.0823748
$305$	46.0629	2.63755
$306$	0.956198	0.0546622
$307$	1.03336	0.0589769	0.0294884	−	0.999565i	$-0.490612\pi$
$307$	1.03336	0.0589769	0.0294884	+	0.999565i	$0.490612\pi$
$308$	0	0
$309$	34.6977	1.97388
$310$	11.8485	0.672950
$311$	−6.12963	−0.347579	−0.173790	−	0.984783i	$-0.555601\pi$
$311$	−6.12963	−0.347579	−0.173790	+	0.984783i	$0.555601\pi$
$312$	−8.65769	−0.490145
$313$	0.225006	0.0127181	0.00635905	−	0.999980i	$-0.497976\pi$
$313$	0.225006	0.0127181	0.00635905	+	0.999980i	$0.497976\pi$
$314$	−20.5719	−1.16094
$315$	3.62526	0.204260
$316$	−7.26403	−0.408634
$317$	−19.9767	−1.12200	−0.561002	−	0.827815i	$-0.689584\pi$
$317$	−19.9767	−1.12200	−0.561002	+	0.827815i	$0.689584\pi$
$318$	19.6691	1.10299
$319$	0	0
$320$	−4.18178	−0.233769
$321$	−26.9795	−1.50585
$322$	1.16663	0.0650140
$323$	−1.58417	−0.0881455
$324$	−10.8492	−0.602734
$325$	−54.9778	−3.04962
$326$	1.58774	0.0879370
$327$	17.5959	0.973053
$328$	−1.13308	−0.0625640
$329$	7.19316	0.396572
$330$	0	0
$331$	−10.1632	−0.558619	−0.279309	−	0.960201i	$-0.590106\pi$
$331$	−10.1632	−0.558619	−0.279309	+	0.960201i	$0.590106\pi$
$332$	6.44235	0.353570
$333$	−6.14175	−0.336566
$334$	20.9293	1.14520
$335$	−49.0636	−2.68063
$336$	−1.96645	−0.107279
$337$	−11.2759	−0.614238	−0.307119	−	0.951671i	$-0.599365\pi$
$337$	−11.2759	−0.614238	−0.307119	+	0.951671i	$0.599365\pi$
$338$	6.38379	0.347232
$339$	−11.4096	−0.619685
$340$	−4.61244	−0.250145
$341$	0	0
$342$	−1.24511	−0.0673281
$343$	−1.00000	−0.0539949
$344$	−4.55093	−0.245370
$345$	9.59353	0.516498
$346$	−1.03934	−0.0558751
$347$	−7.42507	−0.398599	−0.199299	−	0.979939i	$-0.563867\pi$
$347$	−7.42507	−0.398599	−0.199299	+	0.979939i	$0.563867\pi$
$348$	−14.3659	−0.770092
$349$	−24.5929	−1.31643	−0.658214	−	0.752831i	$-0.728688\pi$
$349$	−24.5929	−1.31643	−0.658214	+	0.752831i	$0.728688\pi$
$350$	−12.4873	−0.667473
$351$	18.4676	0.985725
$352$	0	0
$353$	−0.826141	−0.0439710	−0.0219855	−	0.999758i	$-0.506999\pi$
$353$	−0.826141	−0.0439710	−0.0219855	+	0.999758i	$0.506999\pi$
$354$	14.5800	0.774919
$355$	−35.4383	−1.88087
$356$	7.52083	0.398603
$357$	−2.16896	−0.114794
$358$	−0.409927	−0.0216653
$359$	−9.16519	−0.483720	−0.241860	−	0.970311i	$-0.577758\pi$
$359$	−9.16519	−0.483720	−0.241860	+	0.970311i	$0.577758\pi$
$360$	−3.62526	−0.191068
$361$	−16.9372	−0.891430
$362$	−10.9630	−0.576202
$363$	0	0
$364$	4.40270	0.230764
$365$	56.9098	2.97879
$366$	−21.6607	−1.13222
$367$	−15.5377	−0.811063	−0.405532	−	0.914081i	$-0.632914\pi$
$367$	−15.5377	−0.811063	−0.405532	+	0.914081i	$0.632914\pi$
$368$	−1.16663	−0.0608150
$369$	−0.982289	−0.0511359
$370$	29.6261	1.54019
$371$	−10.0023	−0.519295
$372$	−5.57167	−0.288877
$373$	23.3026	1.20656	0.603282	−	0.797528i	$-0.293859\pi$
$373$	23.3026	1.20656	0.603282	+	0.797528i	$0.293859\pi$
$374$	0	0
$375$	−61.5698	−3.17945
$376$	−7.19316	−0.370959
$377$	32.1639	1.65653
$378$	4.19460	0.215747
$379$	−15.0571	−0.773433	−0.386716	−	0.922199i	$-0.626391\pi$
$379$	−15.0571	−0.773433	−0.386716	+	0.922199i	$0.626391\pi$
$380$	6.00610	0.308106
$381$	21.0869	1.08032
$382$	6.89903	0.352985
$383$	−11.8070	−0.603311	−0.301656	−	0.953417i	$-0.597539\pi$
$383$	−11.8070	−0.603311	−0.301656	+	0.953417i	$0.597539\pi$
$384$	1.96645	0.100350
$385$	0	0
$386$	−3.41408	−0.173772
$387$	−3.94528	−0.200550
$388$	1.40616	0.0713868
$389$	−12.6493	−0.641343	−0.320671	−	0.947191i	$-0.603909\pi$
$389$	−12.6493	−0.641343	−0.320671	+	0.947191i	$0.603909\pi$
$390$	36.2045	1.83329
$391$	−1.28678	−0.0650753
$392$	1.00000	0.0505076
$393$	18.3470	0.925482
$394$	−1.70964	−0.0861306
$395$	30.3766	1.52841
$396$	0	0
$397$	4.31688	0.216658	0.108329	−	0.994115i	$-0.465450\pi$
$397$	4.31688	0.216658	0.108329	+	0.994115i	$0.465450\pi$
$398$	−14.0415	−0.703835
$399$	2.82432	0.141393
$400$	12.4873	0.624364
$401$	31.5661	1.57633	0.788167	−	0.615461i	$-0.211030\pi$
$401$	31.5661	1.57633	0.788167	+	0.615461i	$0.211030\pi$
$402$	23.0718	1.15071
$403$	12.4745	0.621397
$404$	−8.40270	−0.418050
$405$	45.3690	2.25440
$406$	7.30550	0.362566
$407$	0	0
$408$	2.16896	0.107380
$409$	−26.7821	−1.32429	−0.662145	−	0.749376i	$-0.730354\pi$
$409$	−26.7821	−1.32429	−0.662145	+	0.749376i	$0.730354\pi$
$410$	4.73830	0.234008
$411$	25.9666	1.28084
$412$	17.6449	0.869300
$413$	−7.41439	−0.364838
$414$	−1.01138	−0.0497064
$415$	−26.9405	−1.32246
$416$	−4.40270	−0.215860
$417$	−14.1478	−0.692821
$418$	0	0
$419$	23.6393	1.15485	0.577427	−	0.816442i	$-0.304057\pi$
$419$	23.6393	1.15485	0.577427	+	0.816442i	$0.304057\pi$
$420$	8.22325	0.401254
$421$	−7.19693	−0.350757	−0.175378	−	0.984501i	$-0.556115\pi$
$421$	−7.19693	−0.350757	−0.175378	+	0.984501i	$0.556115\pi$
$422$	−17.0081	−0.827942
$423$	−6.23587	−0.303199
$424$	10.0023	0.485756
$425$	13.7733	0.668103
$426$	16.6646	0.807402
$427$	11.0151	0.533060
$428$	−13.7199	−0.663176
$429$	0	0
$430$	19.0310	0.917756
$431$	−18.2386	−0.878522	−0.439261	−	0.898359i	$-0.644760\pi$
$431$	−18.2386	−0.878522	−0.439261	+	0.898359i	$0.644760\pi$
$432$	−4.19460	−0.201813
$433$	−3.93868	−0.189281	−0.0946405	−	0.995512i	$-0.530170\pi$
$433$	−3.93868	−0.189281	−0.0946405	+	0.995512i	$0.530170\pi$
$434$	2.83337	0.136006
$435$	60.0750	2.88037
$436$	8.94804	0.428533
$437$	1.67558	0.0801540
$438$	−26.7614	−1.27871
$439$	−21.2519	−1.01430	−0.507149	−	0.861858i	$-0.669301\pi$
$439$	−21.2519	−1.01430	−0.507149	+	0.861858i	$0.669301\pi$
$440$	0	0
$441$	0.866918	0.0412818
$442$	−4.85612	−0.230982
$443$	−8.63846	−0.410425	−0.205213	−	0.978717i	$-0.565789\pi$
$443$	−8.63846	−0.410425	−0.205213	+	0.978717i	$0.565789\pi$
$444$	−13.9315	−0.661158
$445$	−31.4505	−1.49090
$446$	9.01980	0.427100
$447$	34.2605	1.62046
$448$	−1.00000	−0.0472456
$449$	30.3247	1.43111	0.715556	−	0.698555i	$-0.246173\pi$
$449$	30.3247	1.43111	0.715556	+	0.698555i	$0.246173\pi$
$450$	10.8254	0.510316
$451$	0	0
$452$	−5.80214	−0.272910
$453$	23.7385	1.11533
$454$	−12.5419	−0.588620
$455$	−18.4111	−0.863127
$456$	−2.82432	−0.132261
$457$	17.7260	0.829187	0.414593	−	0.910007i	$-0.363924\pi$
$457$	17.7260	0.829187	0.414593	+	0.910007i	$0.363924\pi$
$458$	−19.4202	−0.907445
$459$	−4.62658	−0.215950
$460$	4.87861	0.227466
$461$	9.33956	0.434987	0.217493	−	0.976062i	$-0.430212\pi$
$461$	9.33956	0.434987	0.217493	+	0.976062i	$0.430212\pi$
$462$	0	0
$463$	32.9755	1.53250	0.766250	−	0.642543i	$-0.222121\pi$
$463$	32.9755	1.53250	0.766250	+	0.642543i	$0.222121\pi$
$464$	−7.30550	−0.339149
$465$	23.2995	1.08049
$466$	−14.2721	−0.661144
$467$	−22.0928	−1.02233	−0.511167	−	0.859482i	$-0.670787\pi$
$467$	−22.0928	−1.02233	−0.511167	+	0.859482i	$0.670787\pi$
$468$	−3.81678	−0.176431
$469$	−11.7327	−0.541766
$470$	30.0802	1.38750
$471$	−40.4535	−1.86400
$472$	7.41439	0.341275
$473$	0	0
$474$	−14.2843	−0.656101
$475$	−17.9349	−0.822910
$476$	−1.10299	−0.0505553
$477$	8.67120	0.397027
$478$	3.80704	0.174130
$479$	3.81678	0.174393	0.0871966	−	0.996191i	$-0.472209\pi$
$479$	3.81678	0.174393	0.0871966	+	0.996191i	$0.472209\pi$
$480$	−8.22325	−0.375338
$481$	31.1913	1.42220
$482$	−5.80125	−0.264240
$483$	2.29413	0.104386
$484$	0	0
$485$	−5.88024	−0.267008
$486$	−8.75061	−0.396936
$487$	−12.9840	−0.588363	−0.294182	−	0.955750i	$-0.595047\pi$
$487$	−12.9840	−0.588363	−0.294182	+	0.955750i	$0.595047\pi$
$488$	−11.0151	−0.498632
$489$	3.12221	0.141191
$490$	−4.18178	−0.188914
$491$	−8.76614	−0.395610	−0.197805	−	0.980241i	$-0.563381\pi$
$491$	−8.76614	−0.395610	−0.197805	+	0.980241i	$0.563381\pi$
$492$	−2.22815	−0.100453
$493$	−8.05786	−0.362908
$494$	6.32340	0.284503
$495$	0	0
$496$	−2.83337	−0.127222
$497$	−8.47447	−0.380132
$498$	12.6686	0.567691
$499$	9.07522	0.406263	0.203131	−	0.979152i	$-0.434888\pi$
$499$	9.07522	0.406263	0.203131	+	0.979152i	$0.434888\pi$
$500$	−31.3102	−1.40023
$501$	41.1564	1.83873
$502$	−11.7269	−0.523398
$503$	39.6593	1.76832	0.884160	−	0.467185i	$-0.154732\pi$
$503$	39.6593	1.76832	0.884160	+	0.467185i	$0.154732\pi$
$504$	−0.866918	−0.0386156
$505$	35.1382	1.56363
$506$	0	0
$507$	12.5534	0.557515
$508$	10.7233	0.475772
$509$	2.07464	0.0919569	0.0459785	−	0.998942i	$-0.485359\pi$
$509$	2.07464	0.0919569	0.0459785	+	0.998942i	$0.485359\pi$
$510$	−9.07013	−0.401632
$511$	13.6090	0.602026
$512$	1.00000	0.0441942
$513$	6.02451	0.265988
$514$	−7.94082	−0.350254
$515$	−73.7870	−3.25144
$516$	−8.94917	−0.393965
$517$	0	0
$518$	7.08458	0.311279
$519$	−2.04380	−0.0897130
$520$	18.4111	0.807382
$521$	1.97468	0.0865124	0.0432562	−	0.999064i	$-0.486227\pi$
$521$	1.97468	0.0865124	0.0432562	+	0.999064i	$0.486227\pi$
$522$	−6.33327	−0.277200
$523$	18.7960	0.821891	0.410945	−	0.911660i	$-0.365199\pi$
$523$	18.7960	0.821891	0.410945	+	0.911660i	$0.365199\pi$
$524$	9.33001	0.407583
$525$	−24.5556	−1.07169
$526$	−5.78832	−0.252383
$527$	−3.12516	−0.136134
$528$	0	0
$529$	−21.6390	−0.940825
$530$	−41.8275	−1.81687
$531$	6.42767	0.278937
$532$	1.43625	0.0622695
$533$	4.98862	0.216081
$534$	14.7893	0.639997
$535$	57.3736	2.48048
$536$	11.7327	0.506776
$537$	−0.806100	−0.0347858
$538$	9.92680	0.427975
$539$	0	0
$540$	17.5409	0.754839
$541$	20.2224	0.869431	0.434715	−	0.900568i	$-0.356849\pi$
$541$	20.2224	0.869431	0.434715	+	0.900568i	$0.356849\pi$
$542$	9.78374	0.420247
$543$	−21.5582	−0.925149
$544$	1.10299	0.0472901
$545$	−37.4187	−1.60284
$546$	8.65769	0.370515
$547$	−23.3883	−1.00001	−0.500005	−	0.866023i	$-0.666668\pi$
$547$	−23.3883	−1.00001	−0.500005	+	0.866023i	$0.666668\pi$
$548$	13.2048	0.564083
$549$	−9.54922	−0.407551
$550$	0	0
$551$	10.4926	0.446998
$552$	−2.29413	−0.0976445
$553$	7.26403	0.308898
$554$	−14.0415	−0.596565
$555$	58.2583	2.47293
$556$	−7.19460	−0.305119
$557$	−22.2195	−0.941470	−0.470735	−	0.882275i	$-0.656011\pi$
$557$	−22.2195	−0.941470	−0.470735	+	0.882275i	$0.656011\pi$
$558$	−2.45629	−0.103983
$559$	20.0364	0.847449
$560$	4.18178	0.176712
$561$	0	0
$562$	−19.5722	−0.825602
$563$	37.5171	1.58116	0.790580	−	0.612359i	$-0.209779\pi$
$563$	37.5171	1.58116	0.790580	+	0.612359i	$0.209779\pi$
$564$	−14.1450	−0.595611
$565$	24.2633	1.02076
$566$	4.24977	0.178631
$567$	10.8492	0.455624
$568$	8.47447	0.355581
$569$	−9.19661	−0.385542	−0.192771	−	0.981244i	$-0.561747\pi$
$569$	−9.19661	−0.385542	−0.192771	+	0.981244i	$0.561747\pi$
$570$	11.8107	0.494695
$571$	27.0930	1.13381	0.566904	−	0.823784i	$-0.308141\pi$
$571$	27.0930	1.13381	0.566904	+	0.823784i	$0.308141\pi$
$572$	0	0
$573$	13.5666	0.566752
$574$	1.13308	0.0472939
$575$	−14.5681	−0.607531
$576$	0.866918	0.0361216
$577$	−13.2726	−0.552547	−0.276274	−	0.961079i	$-0.589100\pi$
$577$	−13.2726	−0.552547	−0.276274	+	0.961079i	$0.589100\pi$
$578$	−15.7834	−0.656504
$579$	−6.71361	−0.279008
$580$	30.5500	1.26852
$581$	−6.44235	−0.267274
$582$	2.76514	0.114619
$583$	0	0
$584$	−13.6090	−0.563144
$585$	15.9609	0.659903
$586$	14.1123	0.582976
$587$	−31.9137	−1.31722	−0.658609	−	0.752485i	$-0.728855\pi$
$587$	−31.9137	−1.31722	−0.658609	+	0.752485i	$0.728855\pi$
$588$	1.96645	0.0810949
$589$	4.06943	0.167678
$590$	−31.0054	−1.27647
$591$	−3.36193	−0.138291
$592$	−7.08458	−0.291174
$593$	18.1466	0.745191	0.372596	−	0.927994i	$-0.378468\pi$
$593$	18.1466	0.745191	0.372596	+	0.927994i	$0.378468\pi$
$594$	0	0
$595$	4.61244	0.189092
$596$	17.4225	0.713654
$597$	−27.6118	−1.13008
$598$	5.13634	0.210041
$599$	24.1156	0.985335	0.492668	−	0.870218i	$-0.336022\pi$
$599$	24.1156	0.985335	0.492668	+	0.870218i	$0.336022\pi$
$600$	24.5556	1.00248
$601$	−0.369700	−0.0150804	−0.00754019	−	0.999972i	$-0.502400\pi$
$601$	−0.369700	−0.0150804	−0.00754019	+	0.999972i	$0.502400\pi$
$602$	4.55093	0.185482
$603$	10.1713	0.414207
$604$	12.0718	0.491193
$605$	0	0
$606$	−16.5235	−0.671220
$607$	−40.9761	−1.66317	−0.831584	−	0.555399i	$-0.812565\pi$
$607$	−40.9761	−1.66317	−0.831584	+	0.555399i	$0.812565\pi$
$608$	−1.43625	−0.0582478
$609$	14.3659	0.582135
$610$	46.0629	1.86503
$611$	31.6693	1.28120
$612$	0.956198	0.0386520
$613$	18.0620	0.729518	0.364759	−	0.931102i	$-0.381151\pi$
$613$	18.0620	0.729518	0.364759	+	0.931102i	$0.381151\pi$
$614$	1.03336	0.0417029
$615$	9.31762	0.375723
$616$	0	0
$617$	40.2608	1.62084	0.810419	−	0.585851i	$-0.199240\pi$
$617$	40.2608	1.62084	0.810419	+	0.585851i	$0.199240\pi$
$618$	34.6977	1.39575
$619$	1.01169	0.0406633	0.0203316	−	0.999793i	$-0.493528\pi$
$619$	1.01169	0.0406633	0.0203316	+	0.999793i	$0.493528\pi$
$620$	11.8485	0.475848
$621$	4.89356	0.196372
$622$	−6.12963	−0.245776
$623$	−7.52083	−0.301316
$624$	−8.65769	−0.346585
$625$	68.4958	2.73983
$626$	0.225006	0.00899305
$627$	0	0
$628$	−20.5719	−0.820907
$629$	−7.81419	−0.311572
$630$	3.62526	0.144434
$631$	−9.81364	−0.390675	−0.195337	−	0.980736i	$-0.562580\pi$
$631$	−9.81364	−0.390675	−0.195337	+	0.980736i	$0.562580\pi$
$632$	−7.26403	−0.288948
$633$	−33.4456	−1.32934
$634$	−19.9767	−0.793376
$635$	−44.8427	−1.77953
$636$	19.6691	0.779929
$637$	−4.40270	−0.174441
$638$	0	0
$639$	7.34666	0.290630
$640$	−4.18178	−0.165299
$641$	36.6269	1.44667	0.723337	−	0.690495i	$-0.242607\pi$
$641$	36.6269	1.44667	0.723337	+	0.690495i	$0.242607\pi$
$642$	−26.9795	−1.06479
$643$	2.67485	0.105486	0.0527428	−	0.998608i	$-0.483204\pi$
$643$	2.67485	0.105486	0.0527428	+	0.998608i	$0.483204\pi$
$644$	1.16663	0.0459718
$645$	37.4234	1.47355
$646$	−1.58417	−0.0623283
$647$	19.5046	0.766803	0.383402	−	0.923582i	$-0.374752\pi$
$647$	19.5046	0.766803	0.383402	+	0.923582i	$0.374752\pi$
$648$	−10.8492	−0.426197
$649$	0	0
$650$	−54.9778	−2.15641
$651$	5.57167	0.218371
$652$	1.58774	0.0621808
$653$	21.8705	0.855857	0.427928	−	0.903813i	$-0.359244\pi$
$653$	21.8705	0.855857	0.427928	+	0.903813i	$0.359244\pi$
$654$	17.5959	0.688052
$655$	−39.0160	−1.52448
$656$	−1.13308	−0.0442394
$657$	−11.7979	−0.460279
$658$	7.19316	0.280418
$659$	−37.5352	−1.46216	−0.731081	−	0.682290i	$-0.760984\pi$
$659$	−37.5352	−1.46216	−0.731081	+	0.682290i	$0.760984\pi$
$660$	0	0
$661$	−5.62382	−0.218741	−0.109371	−	0.994001i	$-0.534884\pi$
$661$	−5.62382	−0.218741	−0.109371	+	0.994001i	$0.534884\pi$
$662$	−10.1632	−0.395003
$663$	−9.54930	−0.370864
$664$	6.44235	0.250012
$665$	−6.00610	−0.232907
$666$	−6.14175	−0.237988
$667$	8.52285	0.330006
$668$	20.9293	0.809780
$669$	17.7370	0.685751
$670$	−49.0636	−1.89549
$671$	0	0
$672$	−1.96645	−0.0758574
$673$	−21.5916	−0.832295	−0.416147	−	0.909297i	$-0.636620\pi$
$673$	−21.5916	−0.832295	−0.416147	+	0.909297i	$0.636620\pi$
$674$	−11.2759	−0.434332
$675$	−52.3791	−2.01607
$676$	6.38379	0.245530
$677$	13.2775	0.510297	0.255149	−	0.966902i	$-0.417876\pi$
$677$	13.2775	0.510297	0.255149	+	0.966902i	$0.417876\pi$
$678$	−11.4096	−0.438184
$679$	−1.40616	−0.0539634
$680$	−4.61244	−0.176879
$681$	−24.6630	−0.945087
$682$	0	0
$683$	−31.0217	−1.18701	−0.593507	−	0.804829i	$-0.702257\pi$
$683$	−31.0217	−1.18701	−0.593507	+	0.804829i	$0.702257\pi$
$684$	−1.24511	−0.0476081
$685$	−55.2198	−2.10984
$686$	−1.00000	−0.0381802
$687$	−38.1888	−1.45699
$688$	−4.55093	−0.173503
$689$	−44.0373	−1.67769
$690$	9.59353	0.365219
$691$	−37.1621	−1.41371	−0.706857	−	0.707356i	$-0.749888\pi$
$691$	−37.1621	−1.41371	−0.706857	+	0.707356i	$0.749888\pi$
$692$	−1.03934	−0.0395097
$693$	0	0
$694$	−7.42507	−0.281852
$695$	30.0862	1.14124
$696$	−14.3659	−0.544538
$697$	−1.24977	−0.0473385
$698$	−24.5929	−0.930855
$699$	−28.0654	−1.06153
$700$	−12.4873	−0.471975
$701$	−46.1821	−1.74427	−0.872137	−	0.489261i	$-0.837266\pi$
$701$	−46.1821	−1.74427	−0.872137	+	0.489261i	$0.837266\pi$
$702$	18.4676	0.697013
$703$	10.1753	0.383767
$704$	0	0
$705$	59.1511	2.22776
$706$	−0.826141	−0.0310922
$707$	8.40270	0.316016
$708$	14.5800	0.547951
$709$	24.1245	0.906016	0.453008	−	0.891506i	$-0.350351\pi$
$709$	24.1245	0.906016	0.453008	+	0.891506i	$0.350351\pi$
$710$	−35.4383	−1.32998
$711$	−6.29732	−0.236168
$712$	7.52083	0.281855
$713$	3.30550	0.123792
$714$	−2.16896	−0.0811714
$715$	0	0
$716$	−0.409927	−0.0153197
$717$	7.48634	0.279583
$718$	−9.16519	−0.342042
$719$	−6.15421	−0.229513	−0.114757	−	0.993394i	$-0.536609\pi$
$719$	−6.15421	−0.229513	−0.114757	+	0.993394i	$0.536609\pi$
$720$	−3.62526	−0.135105
$721$	−17.6449	−0.657129
$722$	−16.9372	−0.630336
$723$	−11.4079	−0.424263
$724$	−10.9630	−0.407436
$725$	−91.2259	−3.38804
$726$	0	0
$727$	−48.7522	−1.80812	−0.904060	−	0.427407i	$-0.859427\pi$
$727$	−48.7522	−1.80812	−0.904060	+	0.427407i	$0.859427\pi$
$728$	4.40270	0.163175
$729$	15.3400	0.568148
$730$	56.9098	2.10633
$731$	−5.01961	−0.185657
$732$	−21.6607	−0.800603
$733$	9.16286	0.338438	0.169219	−	0.985578i	$-0.445875\pi$
$733$	9.16286	0.338438	0.169219	+	0.985578i	$0.445875\pi$
$734$	−15.5377	−0.573508
$735$	−8.22325	−0.303319
$736$	−1.16663	−0.0430027
$737$	0	0
$738$	−0.982289	−0.0361586
$739$	−15.2214	−0.559929	−0.279965	−	0.960010i	$-0.590323\pi$
$739$	−15.2214	−0.559929	−0.279965	+	0.960010i	$0.590323\pi$
$740$	29.6261	1.08908
$741$	12.4346	0.456798
$742$	−10.0023	−0.367197
$743$	−22.9769	−0.842940	−0.421470	−	0.906842i	$-0.638486\pi$
$743$	−22.9769	−0.842940	−0.421470	+	0.906842i	$0.638486\pi$
$744$	−5.57167	−0.204267
$745$	−72.8571	−2.66928
$746$	23.3026	0.853170
$747$	5.58499	0.204344
$748$	0	0
$749$	13.7199	0.501314
$750$	−61.5698	−2.24821
$751$	−4.11324	−0.150094	−0.0750471	−	0.997180i	$-0.523911\pi$
$751$	−4.11324	−0.150094	−0.0750471	+	0.997180i	$0.523911\pi$
$752$	−7.19316	−0.262307
$753$	−23.0604	−0.840367
$754$	32.1639	1.17134
$755$	−50.4815	−1.83721
$756$	4.19460	0.152556
$757$	48.8412	1.77517	0.887583	−	0.460649i	$-0.152383\pi$
$757$	48.8412	1.77517	0.887583	+	0.460649i	$0.152383\pi$
$758$	−15.0571	−0.546899
$759$	0	0
$760$	6.00610	0.217864
$761$	−28.8211	−1.04476	−0.522381	−	0.852712i	$-0.674956\pi$
$761$	−28.8211	−1.04476	−0.522381	+	0.852712i	$0.674956\pi$
$762$	21.0869	0.763898
$763$	−8.94804	−0.323941
$764$	6.89903	0.249598
$765$	−3.99861	−0.144570
$766$	−11.8070	−0.426606
$767$	−32.6434	−1.17868
$768$	1.96645	0.0709581
$769$	11.5702	0.417233	0.208617	−	0.977998i	$-0.433104\pi$
$769$	11.5702	0.417233	0.208617	+	0.977998i	$0.433104\pi$
$770$	0	0
$771$	−15.6152	−0.562368
$772$	−3.41408	−0.122875
$773$	0.505306	0.0181746	0.00908730	−	0.999959i	$-0.497107\pi$
$773$	0.505306	0.0181746	0.00908730	+	0.999959i	$0.497107\pi$
$774$	−3.94528	−0.141810
$775$	−35.3810	−1.27092
$776$	1.40616	0.0504781
$777$	13.9315	0.499788
$778$	−12.6493	−0.453498
$779$	1.62739	0.0583075
$780$	36.2045	1.29633
$781$	0	0
$782$	−1.28678	−0.0460152
$783$	30.6436	1.09511
$784$	1.00000	0.0357143
$785$	86.0270	3.07043
$786$	18.3470	0.654415
$787$	32.3153	1.15191	0.575957	−	0.817480i	$-0.304630\pi$
$787$	32.3153	1.15191	0.575957	+	0.817480i	$0.304630\pi$
$788$	−1.70964	−0.0609035
$789$	−11.3824	−0.405225
$790$	30.3766	1.08075
$791$	5.80214	0.206300
$792$	0	0
$793$	48.4964	1.72216
$794$	4.31688	0.153200
$795$	−82.2517	−2.91717
$796$	−14.0415	−0.497687
$797$	−44.8930	−1.59019	−0.795096	−	0.606484i	$-0.792580\pi$
$797$	−44.8930	−1.59019	−0.795096	+	0.606484i	$0.792580\pi$
$798$	2.82432	0.0999798
$799$	−7.93395	−0.280683
$800$	12.4873	0.441492
$801$	6.51994	0.230371
$802$	31.5661	1.11464
$803$	0	0
$804$	23.0718	0.813678
$805$	−4.87861	−0.171948
$806$	12.4745	0.439394
$807$	19.5205	0.687155
$808$	−8.40270	−0.295606
$809$	22.4523	0.789380	0.394690	−	0.918814i	$-0.370852\pi$
$809$	22.4523	0.789380	0.394690	+	0.918814i	$0.370852\pi$
$810$	45.3690	1.59410
$811$	22.7984	0.800559	0.400280	−	0.916393i	$-0.368913\pi$
$811$	22.7984	0.800559	0.400280	+	0.916393i	$0.368913\pi$
$812$	7.30550	0.256373
$813$	19.2392	0.674749
$814$	0	0
$815$	−6.63959	−0.232575
$816$	2.16896	0.0759289
$817$	6.53629	0.228676
$818$	−26.7821	−0.936414
$819$	3.81678	0.133369
$820$	4.73830	0.165469
$821$	35.6299	1.24349	0.621746	−	0.783219i	$-0.286423\pi$
$821$	35.6299	1.24349	0.621746	+	0.783219i	$0.286423\pi$
$822$	25.9666	0.905691
$823$	39.7077	1.38412	0.692061	−	0.721839i	$-0.256703\pi$
$823$	39.7077	1.38412	0.692061	+	0.721839i	$0.256703\pi$
$824$	17.6449	0.614688
$825$	0	0
$826$	−7.41439	−0.257980
$827$	49.5272	1.72223	0.861115	−	0.508411i	$-0.169767\pi$
$827$	49.5272	1.72223	0.861115	+	0.508411i	$0.169767\pi$
$828$	−1.01138	−0.0351477
$829$	−11.7293	−0.407374	−0.203687	−	0.979036i	$-0.565292\pi$
$829$	−11.7293	−0.407374	−0.203687	+	0.979036i	$0.565292\pi$
$830$	−26.9405	−0.935118
$831$	−27.6118	−0.957844
$832$	−4.40270	−0.152636
$833$	1.10299	0.0382162
$834$	−14.1478	−0.489898
$835$	−87.5218	−3.02882
$836$	0	0
$837$	11.8848	0.410800
$838$	23.6393	0.816605
$839$	12.4730	0.430617	0.215308	−	0.976546i	$-0.430924\pi$
$839$	12.4730	0.430617	0.215308	+	0.976546i	$0.430924\pi$
$840$	8.22325	0.283729
$841$	24.3704	0.840357
$842$	−7.19693	−0.248022
$843$	−38.4877	−1.32559
$844$	−17.0081	−0.585443
$845$	−26.6956	−0.918356
$846$	−6.23587	−0.214394
$847$	0	0
$848$	10.0023	0.343481
$849$	8.35696	0.286810
$850$	13.7733	0.472420
$851$	8.26511	0.283324
$852$	16.6646	0.570920
$853$	−36.9885	−1.26646	−0.633230	−	0.773963i	$-0.718271\pi$
$853$	−36.9885	−1.26646	−0.633230	+	0.773963i	$0.718271\pi$
$854$	11.0151	0.376931
$855$	5.20679	0.178069
$856$	−13.7199	−0.468936
$857$	−43.4178	−1.48312	−0.741562	−	0.670884i	$-0.765915\pi$
$857$	−43.4178	−1.48312	−0.741562	+	0.670884i	$0.765915\pi$
$858$	0	0
$859$	29.6963	1.01322	0.506612	−	0.862174i	$-0.330898\pi$
$859$	29.6963	1.01322	0.506612	+	0.862174i	$0.330898\pi$
$860$	19.0310	0.648951
$861$	2.22815	0.0759351
$862$	−18.2386	−0.621209
$863$	−48.5575	−1.65292	−0.826459	−	0.562997i	$-0.809648\pi$
$863$	−48.5575	−1.65292	−0.826459	+	0.562997i	$0.809648\pi$
$864$	−4.19460	−0.142703
$865$	4.34628	0.147778
$866$	−3.93868	−0.133842
$867$	−31.0373	−1.05408
$868$	2.83337	0.0961707
$869$	0	0
$870$	60.0750	2.03673
$871$	−51.6556	−1.75028
$872$	8.94804	0.303019
$873$	1.21902	0.0412577
$874$	1.67558	0.0566775
$875$	31.3102	1.05848
$876$	−26.7614	−0.904183
$877$	1.82626	0.0616684	0.0308342	−	0.999525i	$-0.490184\pi$
$877$	1.82626	0.0616684	0.0308342	+	0.999525i	$0.490184\pi$
$878$	−21.2519	−0.717217
$879$	27.7512	0.936025
$880$	0	0
$881$	−4.27735	−0.144108	−0.0720538	−	0.997401i	$-0.522955\pi$
$881$	−4.27735	−0.144108	−0.0720538	+	0.997401i	$0.522955\pi$
$882$	0.866918	0.0291906
$883$	−4.44029	−0.149428	−0.0747138	−	0.997205i	$-0.523804\pi$
$883$	−4.44029	−0.149428	−0.0747138	+	0.997205i	$0.523804\pi$
$884$	−4.85612	−0.163329
$885$	−60.9704	−2.04950
$886$	−8.63846	−0.290215
$887$	−48.1556	−1.61691	−0.808453	−	0.588560i	$-0.799695\pi$
$887$	−48.1556	−1.61691	−0.808453	+	0.588560i	$0.799695\pi$
$888$	−13.9315	−0.467509
$889$	−10.7233	−0.359650
$890$	−31.4505	−1.05422
$891$	0	0
$892$	9.01980	0.302005
$893$	10.3312	0.345720
$894$	34.2605	1.14584
$895$	1.71422	0.0573002
$896$	−1.00000	−0.0334077
$897$	10.1004	0.337241
$898$	30.3247	1.01195
$899$	20.6992	0.690356
$900$	10.8254	0.360848
$901$	11.0324	0.367543
$902$	0	0
$903$	8.94917	0.297810
$904$	−5.80214	−0.192976
$905$	45.8448	1.52393
$906$	23.7385	0.788659
$907$	12.2063	0.405303	0.202652	−	0.979251i	$-0.435044\pi$
$907$	12.2063	0.405303	0.202652	+	0.979251i	$0.435044\pi$
$908$	−12.5419	−0.416217
$909$	−7.28445	−0.241610
$910$	−18.4111	−0.610323
$911$	23.0040	0.762155	0.381078	−	0.924543i	$-0.375553\pi$
$911$	23.0040	0.762155	0.381078	+	0.924543i	$0.375553\pi$
$912$	−2.82432	−0.0935226
$913$	0	0
$914$	17.7260	0.586324
$915$	90.5803	2.99449
$916$	−19.4202	−0.641661
$917$	−9.33001	−0.308104
$918$	−4.62658	−0.152700
$919$	−17.1900	−0.567046	−0.283523	−	0.958965i	$-0.591503\pi$
$919$	−17.1900	−0.567046	−0.283523	+	0.958965i	$0.591503\pi$
$920$	4.87861	0.160843
$921$	2.03204	0.0669581
$922$	9.33956	0.307582
$923$	−37.3105	−1.22809
$924$	0	0
$925$	−88.4671	−2.90878
$926$	32.9755	1.08364
$927$	15.2966	0.502408
$928$	−7.30550	−0.239815
$929$	−4.33020	−0.142069	−0.0710347	−	0.997474i	$-0.522630\pi$
$929$	−4.33020	−0.142069	−0.0710347	+	0.997474i	$0.522630\pi$
$930$	23.2995	0.764020
$931$	−1.43625	−0.0470713
$932$	−14.2721	−0.467500
$933$	−12.0536	−0.394617
$934$	−22.0928	−0.722899
$935$	0	0
$936$	−3.81678	−0.124755
$937$	37.2144	1.21574	0.607870	−	0.794037i	$-0.292024\pi$
$937$	37.2144	1.21574	0.607870	+	0.794037i	$0.292024\pi$
$938$	−11.7327	−0.383087
$939$	0.442463	0.0144392
$940$	30.0802	0.981108
$941$	38.7116	1.26196	0.630981	−	0.775798i	$-0.282652\pi$
$941$	38.7116	1.26196	0.630981	+	0.775798i	$0.282652\pi$
$942$	−40.4535	−1.31805
$943$	1.32189	0.0430467
$944$	7.41439	0.241318
$945$	−17.5409	−0.570605
$946$	0	0
$947$	−25.2894	−0.821796	−0.410898	−	0.911681i	$-0.634785\pi$
$947$	−25.2894	−0.821796	−0.410898	+	0.911681i	$0.634785\pi$
$948$	−14.2843	−0.463934
$949$	59.9163	1.94497
$950$	−17.9349	−0.581885
$951$	−39.2831	−1.27384
$952$	−1.10299	−0.0357480
$953$	50.5392	1.63713	0.818563	−	0.574417i	$-0.194771\pi$
$953$	50.5392	1.63713	0.818563	+	0.574417i	$0.194771\pi$
$954$	8.67120	0.280740
$955$	−28.8502	−0.933571
$956$	3.80704	0.123128
$957$	0	0
$958$	3.81678	0.123315
$959$	−13.2048	−0.426407
$960$	−8.22325	−0.265404
$961$	−22.9720	−0.741034
$962$	31.1913	1.00565
$963$	−11.8940	−0.383279
$964$	−5.80125	−0.186846
$965$	14.2769	0.459590
$966$	2.29413	0.0738123
$967$	−59.9151	−1.92674	−0.963371	−	0.268174i	$-0.913580\pi$
$967$	−59.9151	−1.92674	−0.963371	+	0.268174i	$0.913580\pi$
$968$	0	0
$969$	−3.11518	−0.100074
$970$	−5.88024	−0.188803
$971$	6.59372	0.211603	0.105801	−	0.994387i	$-0.466259\pi$
$971$	6.59372	0.211603	0.105801	+	0.994387i	$0.466259\pi$
$972$	−8.75061	−0.280676
$973$	7.19460	0.230648
$974$	−12.9840	−0.416036
$975$	−108.111	−3.46232
$976$	−11.0151	−0.352586
$977$	−9.87207	−0.315836	−0.157918	−	0.987452i	$-0.550478\pi$
$977$	−9.87207	−0.315836	−0.157918	+	0.987452i	$0.550478\pi$
$978$	3.12221	0.0998374
$979$	0	0
$980$	−4.18178	−0.133582
$981$	7.75722	0.247669
$982$	−8.76614	−0.279739
$983$	−42.2271	−1.34684	−0.673418	−	0.739262i	$-0.735174\pi$
$983$	−42.2271	−1.34684	−0.673418	+	0.739262i	$0.735174\pi$
$984$	−2.22815	−0.0710307
$985$	7.14935	0.227797
$986$	−8.05786	−0.256615
$987$	14.1450	0.450239
$988$	6.32340	0.201174
$989$	5.30927	0.168825
$990$	0	0
$991$	−14.0297	−0.445667	−0.222833	−	0.974857i	$-0.571531\pi$
$991$	−14.0297	−0.445667	−0.222833	+	0.974857i	$0.571531\pi$
$992$	−2.83337	−0.0899595
$993$	−19.9854	−0.634216
$994$	−8.47447	−0.268794
$995$	58.7183	1.86150
$996$	12.6686	0.401418
$997$	24.6954	0.782111	0.391055	−	0.920367i	$-0.372110\pi$
$997$	24.6954	0.782111	0.391055	+	0.920367i	$0.372110\pi$
$998$	9.07522	0.287271
$999$	29.7169	0.940202

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1694.2.a.z.1.4		4
11.7	odd	10		154.2.f.e.71.1	✓	8
11.8	odd	10		154.2.f.e.141.1	yes	8
11.10	odd	2		1694.2.a.x.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
154.2.f.e.71.1	✓	8	11.7	odd	10
154.2.f.e.141.1	yes	8	11.8	odd	10
1694.2.a.x.1.4		4	11.10	odd	2
1694.2.a.z.1.4		4	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:	\( N \)	\(=\)	\( 1694 = 2 \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1694.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.96645 q^{3} +1.00000 q^{4} -4.18178 q^{5} +1.96645 q^{6} -1.00000 q^{7} +1.00000 q^{8} +0.866918 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.96645 q^{3} +1.00000 q^{4} -4.18178 q^{5} +1.96645 q^{6} -1.00000 q^{7} +1.00000 q^{8} +0.866918 q^{9} -4.18178 q^{10} +1.96645 q^{12} -4.40270 q^{13} -1.00000 q^{14} -8.22325 q^{15} +1.00000 q^{16} +1.10299 q^{17} +0.866918 q^{18} -1.43625 q^{19} -4.18178 q^{20} -1.96645 q^{21} -1.16663 q^{23} +1.96645 q^{24} +12.4873 q^{25} -4.40270 q^{26} -4.19460 q^{27} -1.00000 q^{28} -7.30550 q^{29} -8.22325 q^{30} -2.83337 q^{31} +1.00000 q^{32} +1.10299 q^{34} +4.18178 q^{35} +0.866918 q^{36} -7.08458 q^{37} -1.43625 q^{38} -8.65769 q^{39} -4.18178 q^{40} -1.13308 q^{41} -1.96645 q^{42} -4.55093 q^{43} -3.62526 q^{45} -1.16663 q^{46} -7.19316 q^{47} +1.96645 q^{48} +1.00000 q^{49} +12.4873 q^{50} +2.16896 q^{51} -4.40270 q^{52} +10.0023 q^{53} -4.19460 q^{54} -1.00000 q^{56} -2.82432 q^{57} -7.30550 q^{58} +7.41439 q^{59} -8.22325 q^{60} -11.0151 q^{61} -2.83337 q^{62} -0.866918 q^{63} +1.00000 q^{64} +18.4111 q^{65} +11.7327 q^{67} +1.10299 q^{68} -2.29413 q^{69} +4.18178 q^{70} +8.47447 q^{71} +0.866918 q^{72} -13.6090 q^{73} -7.08458 q^{74} +24.5556 q^{75} -1.43625 q^{76} -8.65769 q^{78} -7.26403 q^{79} -4.18178 q^{80} -10.8492 q^{81} -1.13308 q^{82} +6.44235 q^{83} -1.96645 q^{84} -4.61244 q^{85} -4.55093 q^{86} -14.3659 q^{87} +7.52083 q^{89} -3.62526 q^{90} +4.40270 q^{91} -1.16663 q^{92} -5.57167 q^{93} -7.19316 q^{94} +6.00610 q^{95} +1.96645 q^{96} +1.40616 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} - 3 q^{3} + 4 q^{4} - 5 q^{5} - 3 q^{6} - 4 q^{7} + 4 q^{8} + 9 q^{9}+O(q^{10}) \) 4 * q + 4 * q^2 - 3 * q^3 + 4 * q^4 - 5 * q^5 - 3 * q^6 - 4 * q^7 + 4 * q^8 + 9 * q^9	\( 4 q + 4 q^{2} - 3 q^{3} + 4 q^{4} - 5 q^{5} - 3 q^{6} - 4 q^{7} + 4 q^{8} + 9 q^{9} - 5 q^{10} - 3 q^{12} - 14 q^{13} - 4 q^{14} - 5 q^{15} + 4 q^{16} + q^{17} + 9 q^{18} - 13 q^{19} - 5 q^{20} + 3 q^{21} - 10 q^{23} - 3 q^{24} + 15 q^{25} - 14 q^{26} - 24 q^{27} - 4 q^{28} - 6 q^{29} - 5 q^{30} - 6 q^{31} + 4 q^{32} + q^{34} + 5 q^{35} + 9 q^{36} + 3 q^{37} - 13 q^{38} - 2 q^{39} - 5 q^{40} + q^{41} + 3 q^{42} + 8 q^{43} - 20 q^{45} - 10 q^{46} - 3 q^{47} - 3 q^{48} + 4 q^{49} + 15 q^{50} - 22 q^{51} - 14 q^{52} + 4 q^{53} - 24 q^{54} - 4 q^{56} + 16 q^{57} - 6 q^{58} - 4 q^{59} - 5 q^{60} - 27 q^{61} - 6 q^{62} - 9 q^{63} + 4 q^{64} - 5 q^{65} + 9 q^{67} + q^{68} + 5 q^{70} - 20 q^{71} + 9 q^{72} - 28 q^{73} + 3 q^{74} - 13 q^{76} - 2 q^{78} - 22 q^{79} - 5 q^{80} + 16 q^{81} + q^{82} - 6 q^{83} + 3 q^{84} - 5 q^{85} + 8 q^{86} + 2 q^{87} + 6 q^{89} - 20 q^{90} + 14 q^{91} - 10 q^{92} + 12 q^{93} - 3 q^{94} - 15 q^{95} - 3 q^{96} + 15 q^{97} + 4 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 - 3 * q^3 + 4 * q^4 - 5 * q^5 - 3 * q^6 - 4 * q^7 + 4 * q^8 + 9 * q^9 - 5 * q^10 - 3 * q^12 - 14 * q^13 - 4 * q^14 - 5 * q^15 + 4 * q^16 + q^17 + 9 * q^18 - 13 * q^19 - 5 * q^20 + 3 * q^21 - 10 * q^23 - 3 * q^24 + 15 * q^25 - 14 * q^26 - 24 * q^27 - 4 * q^28 - 6 * q^29 - 5 * q^30 - 6 * q^31 + 4 * q^32 + q^34 + 5 * q^35 + 9 * q^36 + 3 * q^37 - 13 * q^38 - 2 * q^39 - 5 * q^40 + q^41 + 3 * q^42 + 8 * q^43 - 20 * q^45 - 10 * q^46 - 3 * q^47 - 3 * q^48 + 4 * q^49 + 15 * q^50 - 22 * q^51 - 14 * q^52 + 4 * q^53 - 24 * q^54 - 4 * q^56 + 16 * q^57 - 6 * q^58 - 4 * q^59 - 5 * q^60 - 27 * q^61 - 6 * q^62 - 9 * q^63 + 4 * q^64 - 5 * q^65 + 9 * q^67 + q^68 + 5 * q^70 - 20 * q^71 + 9 * q^72 - 28 * q^73 + 3 * q^74 - 13 * q^76 - 2 * q^78 - 22 * q^79 - 5 * q^80 + 16 * q^81 + q^82 - 6 * q^83 + 3 * q^84 - 5 * q^85 + 8 * q^86 + 2 * q^87 + 6 * q^89 - 20 * q^90 + 14 * q^91 - 10 * q^92 + 12 * q^93 - 3 * q^94 - 15 * q^95 - 3 * q^96 + 15 * q^97 + 4 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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