LMFDB - Embedded newform 7942.2.a.ca.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7942,2,Mod(1,7942)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7942, 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("7942.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7942 = 2 \cdot 11 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7942.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:	$63.4171892853$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 3 x^{14} - 33 x^{13} + 101 x^{12} + 408 x^{11} - 1314 x^{10} - 2271 x^{9} + 8292 x^{8} + \cdots - 3592 $ x^15 - 3x^14 - 33x^13 + 101x^12 + 408x^11 - 1314x^10 - 2271x^9 + 8292x^8 + 4944x^7 - 26254x^6 + 1401x^5 + 39318x^4 - 19069x^3 - 20118x^2 + 18180x - 3592
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.11106$ of defining polynomial
Character	$\chi$	$=$	7942.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.11106 q^{3} +1.00000 q^{4} +2.93101 q^{5} -3.11106 q^{6} +4.25846 q^{7} +1.00000 q^{8} +6.67869 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.11106 q^{3} +1.00000 q^{4} +2.93101 q^{5} -3.11106 q^{6} +4.25846 q^{7} +1.00000 q^{8} +6.67869 q^{9} +2.93101 q^{10} -1.00000 q^{11} -3.11106 q^{12} +4.02336 q^{13} +4.25846 q^{14} -9.11854 q^{15} +1.00000 q^{16} -3.10643 q^{17} +6.67869 q^{18} +2.93101 q^{20} -13.2483 q^{21} -1.00000 q^{22} +8.63048 q^{23} -3.11106 q^{24} +3.59080 q^{25} +4.02336 q^{26} -11.4446 q^{27} +4.25846 q^{28} +1.90985 q^{29} -9.11854 q^{30} -1.66607 q^{31} +1.00000 q^{32} +3.11106 q^{33} -3.10643 q^{34} +12.4816 q^{35} +6.67869 q^{36} -6.62481 q^{37} -12.5169 q^{39} +2.93101 q^{40} +4.69405 q^{41} -13.2483 q^{42} +3.02750 q^{43} -1.00000 q^{44} +19.5753 q^{45} +8.63048 q^{46} +5.87560 q^{47} -3.11106 q^{48} +11.1345 q^{49} +3.59080 q^{50} +9.66427 q^{51} +4.02336 q^{52} +0.147678 q^{53} -11.4446 q^{54} -2.93101 q^{55} +4.25846 q^{56} +1.90985 q^{58} +4.30306 q^{59} -9.11854 q^{60} -7.53100 q^{61} -1.66607 q^{62} +28.4409 q^{63} +1.00000 q^{64} +11.7925 q^{65} +3.11106 q^{66} +2.17528 q^{67} -3.10643 q^{68} -26.8499 q^{69} +12.4816 q^{70} +13.7827 q^{71} +6.67869 q^{72} -10.8014 q^{73} -6.62481 q^{74} -11.1712 q^{75} -4.25846 q^{77} -12.5169 q^{78} -0.921801 q^{79} +2.93101 q^{80} +15.5688 q^{81} +4.69405 q^{82} +0.684858 q^{83} -13.2483 q^{84} -9.10496 q^{85} +3.02750 q^{86} -5.94165 q^{87} -1.00000 q^{88} -13.4881 q^{89} +19.5753 q^{90} +17.1333 q^{91} +8.63048 q^{92} +5.18324 q^{93} +5.87560 q^{94} -3.11106 q^{96} +16.7593 q^{97} +11.1345 q^{98} -6.67869 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 15 q^{2} + 3 q^{3} + 15 q^{4} + 9 q^{5} + 3 q^{6} + 15 q^{8} + 30 q^{9}+O(q^{10}) $ 15 * q + 15 * q^2 + 3 * q^3 + 15 * q^4 + 9 * q^5 + 3 * q^6 + 15 * q^8 + 30 * q^9	$ 15 q + 15 q^{2} + 3 q^{3} + 15 q^{4} + 9 q^{5} + 3 q^{6} + 15 q^{8} + 30 q^{9} + 9 q^{10} - 15 q^{11} + 3 q^{12} + 21 q^{15} + 15 q^{16} + 21 q^{17} + 30 q^{18} + 9 q^{20} - 9 q^{21} - 15 q^{22} + 21 q^{23} + 3 q^{24} + 24 q^{25} + 3 q^{27} - 9 q^{29} + 21 q^{30} + 18 q^{31} + 15 q^{32} - 3 q^{33} + 21 q^{34} + 18 q^{35} + 30 q^{36} - 9 q^{37} + 9 q^{40} + 15 q^{41} - 9 q^{42} + 3 q^{43} - 15 q^{44} + 54 q^{45} + 21 q^{46} + 39 q^{47} + 3 q^{48} + 33 q^{49} + 24 q^{50} + 30 q^{51} + 18 q^{53} + 3 q^{54} - 9 q^{55} - 9 q^{58} + 6 q^{59} + 21 q^{60} - 30 q^{61} + 18 q^{62} + 24 q^{63} + 15 q^{64} + 6 q^{65} - 3 q^{66} + 9 q^{67} + 21 q^{68} - 42 q^{69} + 18 q^{70} + 9 q^{71} + 30 q^{72} + 12 q^{73} - 9 q^{74} + 21 q^{75} + 12 q^{79} + 9 q^{80} + 63 q^{81} + 15 q^{82} + 30 q^{83} - 9 q^{84} - 3 q^{85} + 3 q^{86} + 9 q^{87} - 15 q^{88} - 6 q^{89} + 54 q^{90} + 96 q^{91} + 21 q^{92} + 102 q^{93} + 39 q^{94} + 3 q^{96} + 33 q^{98} - 30 q^{99}+O(q^{100}) $ 15 * q + 15 * q^2 + 3 * q^3 + 15 * q^4 + 9 * q^5 + 3 * q^6 + 15 * q^8 + 30 * q^9 + 9 * q^10 - 15 * q^11 + 3 * q^12 + 21 * q^15 + 15 * q^16 + 21 * q^17 + 30 * q^18 + 9 * q^20 - 9 * q^21 - 15 * q^22 + 21 * q^23 + 3 * q^24 + 24 * q^25 + 3 * q^27 - 9 * q^29 + 21 * q^30 + 18 * q^31 + 15 * q^32 - 3 * q^33 + 21 * q^34 + 18 * q^35 + 30 * q^36 - 9 * q^37 + 9 * q^40 + 15 * q^41 - 9 * q^42 + 3 * q^43 - 15 * q^44 + 54 * q^45 + 21 * q^46 + 39 * q^47 + 3 * q^48 + 33 * q^49 + 24 * q^50 + 30 * q^51 + 18 * q^53 + 3 * q^54 - 9 * q^55 - 9 * q^58 + 6 * q^59 + 21 * q^60 - 30 * q^61 + 18 * q^62 + 24 * q^63 + 15 * q^64 + 6 * q^65 - 3 * q^66 + 9 * q^67 + 21 * q^68 - 42 * q^69 + 18 * q^70 + 9 * q^71 + 30 * q^72 + 12 * q^73 - 9 * q^74 + 21 * q^75 + 12 * q^79 + 9 * q^80 + 63 * q^81 + 15 * q^82 + 30 * q^83 - 9 * q^84 - 3 * q^85 + 3 * q^86 + 9 * q^87 - 15 * q^88 - 6 * q^89 + 54 * q^90 + 96 * q^91 + 21 * q^92 + 102 * q^93 + 39 * q^94 + 3 * q^96 + 33 * q^98 - 30 * 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
$3$	−3.11106	−1.79617	−0.898085	−	0.439821i	$-0.855042\pi$
$3$	−3.11106	−1.79617	−0.898085	+	0.439821i	$0.855042\pi$
$4$	1.00000	0.500000
$5$	2.93101	1.31079	0.655393	−	0.755288i	$-0.272503\pi$
$5$	2.93101	1.31079	0.655393	+	0.755288i	$0.272503\pi$
$6$	−3.11106	−1.27008
$7$	4.25846	1.60955	0.804773	−	0.593582i	$-0.202287\pi$
$7$	4.25846	1.60955	0.804773	+	0.593582i	$0.202287\pi$
$8$	1.00000	0.353553
$9$	6.67869	2.22623
$10$	2.93101	0.926866
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−3.11106	−0.898085
$13$	4.02336	1.11588	0.557940	−	0.829882i	$-0.311592\pi$
$13$	4.02336	1.11588	0.557940	+	0.829882i	$0.311592\pi$
$14$	4.25846	1.13812
$15$	−9.11854	−2.35440
$16$	1.00000	0.250000
$17$	−3.10643	−0.753419	−0.376710	−	0.926331i	$-0.622945\pi$
$17$	−3.10643	−0.753419	−0.376710	+	0.926331i	$0.622945\pi$
$18$	6.67869	1.57418
$19$	0	0
$19$	0	0
$20$	2.93101	0.655393
$21$	−13.2483	−2.89102
$22$	−1.00000	−0.213201
$23$	8.63048	1.79958	0.899790	−	0.436323i	$-0.143720\pi$
$23$	8.63048	1.79958	0.899790	+	0.436323i	$0.143720\pi$
$24$	−3.11106	−0.635042
$25$	3.59080	0.718161
$26$	4.02336	0.789046
$27$	−11.4446	−2.20252
$28$	4.25846	0.804773
$29$	1.90985	0.354650	0.177325	−	0.984152i	$-0.443256\pi$
$29$	1.90985	0.354650	0.177325	+	0.984152i	$0.443256\pi$
$30$	−9.11854	−1.66481
$31$	−1.66607	−0.299235	−0.149618	−	0.988744i	$-0.547804\pi$
$31$	−1.66607	−0.299235	−0.149618	+	0.988744i	$0.547804\pi$
$32$	1.00000	0.176777
$33$	3.11106	0.541566
$34$	−3.10643	−0.532748
$35$	12.4816	2.10977
$36$	6.67869	1.11311
$37$	−6.62481	−1.08911	−0.544556	−	0.838724i	$-0.683302\pi$
$37$	−6.62481	−1.08911	−0.544556	+	0.838724i	$0.683302\pi$
$38$	0	0
$39$	−12.5169	−2.00431
$40$	2.93101	0.463433
$41$	4.69405	0.733087	0.366543	−	0.930401i	$-0.380541\pi$
$41$	4.69405	0.733087	0.366543	+	0.930401i	$0.380541\pi$
$42$	−13.2483	−2.04426
$43$	3.02750	0.461689	0.230845	−	0.972991i	$-0.425851\pi$
$43$	3.02750	0.461689	0.230845	+	0.972991i	$0.425851\pi$
$44$	−1.00000	−0.150756
$45$	19.5753	2.91811
$46$	8.63048	1.27250
$47$	5.87560	0.857044	0.428522	−	0.903531i	$-0.359035\pi$
$47$	5.87560	0.857044	0.428522	+	0.903531i	$0.359035\pi$
$48$	−3.11106	−0.449043
$49$	11.1345	1.59064
$50$	3.59080	0.507816
$51$	9.66427	1.35327
$52$	4.02336	0.557940
$53$	0.147678	0.0202851	0.0101426	−	0.999949i	$-0.496771\pi$
$53$	0.147678	0.0202851	0.0101426	+	0.999949i	$0.496771\pi$
$54$	−11.4446	−1.55741
$55$	−2.93101	−0.395217
$56$	4.25846	0.569061
$57$	0	0
$58$	1.90985	0.250775
$59$	4.30306	0.560210	0.280105	−	0.959969i	$-0.409631\pi$
$59$	4.30306	0.560210	0.280105	+	0.959969i	$0.409631\pi$
$60$	−9.11854	−1.17720
$61$	−7.53100	−0.964246	−0.482123	−	0.876103i	$-0.660134\pi$
$61$	−7.53100	−0.964246	−0.482123	+	0.876103i	$0.660134\pi$
$62$	−1.66607	−0.211591
$63$	28.4409	3.58322
$64$	1.00000	0.125000
$65$	11.7925	1.46268
$66$	3.11106	0.382945
$67$	2.17528	0.265752	0.132876	−	0.991133i	$-0.457579\pi$
$67$	2.17528	0.265752	0.132876	+	0.991133i	$0.457579\pi$
$68$	−3.10643	−0.376710
$69$	−26.8499	−3.23235
$70$	12.4816	1.49183
$71$	13.7827	1.63571	0.817854	−	0.575426i	$-0.195164\pi$
$71$	13.7827	1.63571	0.817854	+	0.575426i	$0.195164\pi$
$72$	6.67869	0.787091
$73$	−10.8014	−1.26421	−0.632106	−	0.774882i	$-0.717809\pi$
$73$	−10.8014	−1.26421	−0.632106	+	0.774882i	$0.717809\pi$
$74$	−6.62481	−0.770119
$75$	−11.1712	−1.28994
$76$	0	0
$77$	−4.25846	−0.485297
$78$	−12.5169	−1.41726
$79$	−0.921801	−0.103711	−0.0518553	−	0.998655i	$-0.516513\pi$
$79$	−0.921801	−0.103711	−0.0518553	+	0.998655i	$0.516513\pi$
$80$	2.93101	0.327697
$81$	15.5688	1.72986
$82$	4.69405	0.518371
$83$	0.684858	0.0751729	0.0375865	−	0.999293i	$-0.488033\pi$
$83$	0.684858	0.0751729	0.0375865	+	0.999293i	$0.488033\pi$
$84$	−13.2483	−1.44551
$85$	−9.10496	−0.987571
$86$	3.02750	0.326464
$87$	−5.94165	−0.637011
$88$	−1.00000	−0.106600
$89$	−13.4881	−1.42974	−0.714868	−	0.699260i	$-0.753513\pi$
$89$	−13.4881	−1.42974	−0.714868	+	0.699260i	$0.753513\pi$
$90$	19.5753	2.06342
$91$	17.1333	1.79606
$92$	8.63048	0.899790
$93$	5.18324	0.537477
$94$	5.87560	0.606021
$95$	0	0
$96$	−3.11106	−0.317521
$97$	16.7593	1.70165	0.850824	−	0.525451i	$-0.176103\pi$
$97$	16.7593	1.70165	0.850824	+	0.525451i	$0.176103\pi$
$98$	11.1345	1.12475
$99$	−6.67869	−0.671233
$100$	3.59080	0.359080
$101$	7.09312	0.705792	0.352896	−	0.935663i	$-0.385197\pi$
$101$	7.09312	0.705792	0.352896	+	0.935663i	$0.385197\pi$
$102$	9.66427	0.956906
$103$	15.3656	1.51402	0.757010	−	0.653404i	$-0.226660\pi$
$103$	15.3656	1.51402	0.757010	+	0.653404i	$0.226660\pi$
$104$	4.02336	0.394523
$105$	−38.8309	−3.78951
$106$	0.147678	0.0143438
$107$	−16.1326	−1.55960	−0.779801	−	0.626028i	$-0.784680\pi$
$107$	−16.1326	−1.55960	−0.779801	+	0.626028i	$0.784680\pi$
$108$	−11.4446	−1.10126
$109$	5.98480	0.573240	0.286620	−	0.958044i	$-0.407468\pi$
$109$	5.98480	0.573240	0.286620	+	0.958044i	$0.407468\pi$
$110$	−2.93101	−0.279461
$111$	20.6102	1.95623
$112$	4.25846	0.402387
$113$	−10.4515	−0.983194	−0.491597	−	0.870823i	$-0.663587\pi$
$113$	−10.4515	−0.983194	−0.491597	+	0.870823i	$0.663587\pi$
$114$	0	0
$115$	25.2960	2.35886
$116$	1.90985	0.177325
$117$	26.8708	2.48420
$118$	4.30306	0.396128
$119$	−13.2286	−1.21266
$120$	−9.11854	−0.832405
$121$	1.00000	0.0909091
$122$	−7.53100	−0.681825
$123$	−14.6035	−1.31675
$124$	−1.66607	−0.149618
$125$	−4.13036	−0.369431
$126$	28.4409	2.53372
$127$	−1.99805	−0.177298	−0.0886491	−	0.996063i	$-0.528255\pi$
$127$	−1.99805	−0.177298	−0.0886491	+	0.996063i	$0.528255\pi$
$128$	1.00000	0.0883883
$129$	−9.41873	−0.829273
$130$	11.7925	1.03427
$131$	3.20339	0.279881	0.139941	−	0.990160i	$-0.455309\pi$
$131$	3.20339	0.279881	0.139941	+	0.990160i	$0.455309\pi$
$132$	3.11106	0.270783
$133$	0	0
$134$	2.17528	0.187915
$135$	−33.5442	−2.88703
$136$	−3.10643	−0.266374
$137$	−20.7455	−1.77241	−0.886206	−	0.463292i	$-0.846668\pi$
$137$	−20.7455	−1.77241	−0.886206	+	0.463292i	$0.846668\pi$
$138$	−26.8499	−2.28562
$139$	20.9110	1.77364	0.886822	−	0.462111i	$-0.152908\pi$
$139$	20.9110	1.77364	0.886822	+	0.462111i	$0.152908\pi$
$140$	12.4816	1.05489
$141$	−18.2793	−1.53940
$142$	13.7827	1.15662
$143$	−4.02336	−0.336450
$144$	6.67869	0.556557
$145$	5.59778	0.464870
$146$	−10.8014	−0.893932
$147$	−34.6400	−2.85706
$148$	−6.62481	−0.544556
$149$	−21.6038	−1.76985	−0.884924	−	0.465735i	$-0.845790\pi$
$149$	−21.6038	−1.76985	−0.884924	+	0.465735i	$0.845790\pi$
$150$	−11.1712	−0.912125
$151$	−15.8472	−1.28963	−0.644813	−	0.764341i	$-0.723065\pi$
$151$	−15.8472	−1.28963	−0.644813	+	0.764341i	$0.723065\pi$
$152$	0	0
$153$	−20.7468	−1.67728
$154$	−4.25846	−0.343157
$155$	−4.88326	−0.392233
$156$	−12.5169	−1.00215
$157$	−2.58688	−0.206456	−0.103228	−	0.994658i	$-0.532917\pi$
$157$	−2.58688	−0.206456	−0.103228	+	0.994658i	$0.532917\pi$
$158$	−0.921801	−0.0733345
$159$	−0.459435	−0.0364356
$160$	2.93101	0.231716
$161$	36.7526	2.89651
$162$	15.5688	1.22320
$163$	8.80855	0.689939	0.344969	−	0.938614i	$-0.387889\pi$
$163$	8.80855	0.689939	0.344969	+	0.938614i	$0.387889\pi$
$164$	4.69405	0.366543
$165$	9.11854	0.709877
$166$	0.684858	0.0531553
$167$	−5.57202	−0.431176	−0.215588	−	0.976484i	$-0.569167\pi$
$167$	−5.57202	−0.431176	−0.215588	+	0.976484i	$0.569167\pi$
$168$	−13.2483	−1.02213
$169$	3.18742	0.245186
$170$	−9.10496	−0.698319
$171$	0	0
$172$	3.02750	0.230845
$173$	−12.4821	−0.948993	−0.474496	−	0.880257i	$-0.657370\pi$
$173$	−12.4821	−0.948993	−0.474496	+	0.880257i	$0.657370\pi$
$174$	−5.94165	−0.450435
$175$	15.2913	1.15591
$176$	−1.00000	−0.0753778
$177$	−13.3871	−1.00623
$178$	−13.4881	−1.01098
$179$	−9.84076	−0.735533	−0.367767	−	0.929918i	$-0.619877\pi$
$179$	−9.84076	−0.735533	−0.367767	+	0.929918i	$0.619877\pi$
$180$	19.5753	1.45905
$181$	−2.32324	−0.172685	−0.0863425	−	0.996266i	$-0.527518\pi$
$181$	−2.32324	−0.172685	−0.0863425	+	0.996266i	$0.527518\pi$
$182$	17.1333	1.27001
$183$	23.4294	1.73195
$184$	8.63048	0.636248
$185$	−19.4174	−1.42759
$186$	5.18324	0.380054
$187$	3.10643	0.227164
$188$	5.87560	0.428522
$189$	−48.7364	−3.54505
$190$	0	0
$191$	−6.59698	−0.477341	−0.238670	−	0.971101i	$-0.576712\pi$
$191$	−6.59698	−0.477341	−0.238670	+	0.971101i	$0.576712\pi$
$192$	−3.11106	−0.224521
$193$	−8.15324	−0.586883	−0.293442	−	0.955977i	$-0.594801\pi$
$193$	−8.15324	−0.586883	−0.293442	+	0.955977i	$0.594801\pi$
$194$	16.7593	1.20325
$195$	−36.6871	−2.62722
$196$	11.1345	0.795321
$197$	−21.6285	−1.54097	−0.770483	−	0.637460i	$-0.779985\pi$
$197$	−21.6285	−1.54097	−0.770483	+	0.637460i	$0.779985\pi$
$198$	−6.67869	−0.474633
$199$	−10.4974	−0.744142	−0.372071	−	0.928204i	$-0.621352\pi$
$199$	−10.4974	−0.744142	−0.372071	+	0.928204i	$0.621352\pi$
$200$	3.59080	0.253908
$201$	−6.76741	−0.477336
$202$	7.09312	0.499070
$203$	8.13301	0.570825
$204$	9.66427	0.676635
$205$	13.7583	0.960920
$206$	15.3656	1.07057
$207$	57.6403	4.00628
$208$	4.02336	0.278970
$209$	0	0
$210$	−38.8309	−2.67959
$211$	−4.99694	−0.344003	−0.172002	−	0.985097i	$-0.555023\pi$
$211$	−4.99694	−0.344003	−0.172002	+	0.985097i	$0.555023\pi$
$212$	0.147678	0.0101426
$213$	−42.8789	−2.93801
$214$	−16.1326	−1.10280
$215$	8.87362	0.605176
$216$	−11.4446	−0.778707
$217$	−7.09490	−0.481633
$218$	5.98480	0.405342
$219$	33.6039	2.27074
$220$	−2.93101	−0.197608
$221$	−12.4983	−0.840725
$222$	20.6102	1.38326
$223$	−2.42462	−0.162364	−0.0811822	−	0.996699i	$-0.525870\pi$
$223$	−2.42462	−0.162364	−0.0811822	+	0.996699i	$0.525870\pi$
$224$	4.25846	0.284530
$225$	23.9819	1.59879
$226$	−10.4515	−0.695223
$227$	−1.20313	−0.0798546	−0.0399273	−	0.999203i	$-0.512713\pi$
$227$	−1.20313	−0.0798546	−0.0399273	+	0.999203i	$0.512713\pi$
$228$	0	0
$229$	−21.9280	−1.44904	−0.724522	−	0.689252i	$-0.757939\pi$
$229$	−21.9280	−1.44904	−0.724522	+	0.689252i	$0.757939\pi$
$230$	25.2960	1.66797
$231$	13.2483	0.871675
$232$	1.90985	0.125388
$233$	21.8999	1.43471	0.717354	−	0.696709i	$-0.245353\pi$
$233$	21.8999	1.43471	0.717354	+	0.696709i	$0.245353\pi$
$234$	26.8708	1.75660
$235$	17.2214	1.12340
$236$	4.30306	0.280105
$237$	2.86778	0.186282
$238$	−13.2286	−0.857482
$239$	6.87525	0.444723	0.222362	−	0.974964i	$-0.428623\pi$
$239$	6.87525	0.444723	0.222362	+	0.974964i	$0.428623\pi$
$240$	−9.11854	−0.588599
$241$	4.08165	0.262922	0.131461	−	0.991321i	$-0.458033\pi$
$241$	4.08165	0.262922	0.131461	+	0.991321i	$0.458033\pi$
$242$	1.00000	0.0642824
$243$	−14.1016	−0.904616
$244$	−7.53100	−0.482123
$245$	32.6353	2.08499
$246$	−14.6035	−0.931082
$247$	0	0
$248$	−1.66607	−0.105796
$249$	−2.13063	−0.135023
$250$	−4.13036	−0.261227
$251$	23.2980	1.47055	0.735277	−	0.677766i	$-0.237052\pi$
$251$	23.2980	1.47055	0.735277	+	0.677766i	$0.237052\pi$
$252$	28.4409	1.79161
$253$	−8.63048	−0.542594
$254$	−1.99805	−0.125369
$255$	28.3261	1.77385
$256$	1.00000	0.0625000
$257$	23.5456	1.46873	0.734366	−	0.678754i	$-0.237480\pi$
$257$	23.5456	1.46873	0.734366	+	0.678754i	$0.237480\pi$
$258$	−9.41873	−0.586384
$259$	−28.2115	−1.75298
$260$	11.7925	0.731340
$261$	12.7553	0.789531
$262$	3.20339	0.197906
$263$	−19.8530	−1.22419	−0.612093	−	0.790786i	$-0.709672\pi$
$263$	−19.8530	−1.22419	−0.612093	+	0.790786i	$0.709672\pi$
$264$	3.11106	0.191472
$265$	0.432846	0.0265895
$266$	0	0
$267$	41.9623	2.56805
$268$	2.17528	0.132876
$269$	8.77117	0.534788	0.267394	−	0.963587i	$-0.413837\pi$
$269$	8.77117	0.534788	0.267394	+	0.963587i	$0.413837\pi$
$270$	−33.5442	−2.04144
$271$	−2.22037	−0.134878	−0.0674389	−	0.997723i	$-0.521483\pi$
$271$	−2.22037	−0.134878	−0.0674389	+	0.997723i	$0.521483\pi$
$272$	−3.10643	−0.188355
$273$	−53.3028	−3.22603
$274$	−20.7455	−1.25328
$275$	−3.59080	−0.216534
$276$	−26.8499	−1.61618
$277$	−16.7586	−1.00692	−0.503462	−	0.864017i	$-0.667941\pi$
$277$	−16.7586	−1.00692	−0.503462	+	0.864017i	$0.667941\pi$
$278$	20.9110	1.25416
$279$	−11.1272	−0.666166
$280$	12.4816	0.745917
$281$	11.6638	0.695803	0.347901	−	0.937531i	$-0.386894\pi$
$281$	11.6638	0.695803	0.347901	+	0.937531i	$0.386894\pi$
$282$	−18.2793	−1.08852
$283$	−28.1627	−1.67410	−0.837048	−	0.547129i	$-0.815721\pi$
$283$	−28.1627	−1.67410	−0.837048	+	0.547129i	$0.815721\pi$
$284$	13.7827	0.817854
$285$	0	0
$286$	−4.02336	−0.237906
$287$	19.9894	1.17994
$288$	6.67869	0.393545
$289$	−7.35011	−0.432360
$290$	5.59778	0.328713
$291$	−52.1391	−3.05645
$292$	−10.8014	−0.632106
$293$	6.84160	0.399690	0.199845	−	0.979827i	$-0.435956\pi$
$293$	6.84160	0.399690	0.199845	+	0.979827i	$0.435956\pi$
$294$	−34.6400	−2.02025
$295$	12.6123	0.734316
$296$	−6.62481	−0.385059
$297$	11.4446	0.664083
$298$	−21.6038	−1.25147
$299$	34.7235	2.00811
$300$	−11.1712	−0.644970
$301$	12.8925	0.743110
$302$	−15.8472	−0.911903
$303$	−22.0671	−1.26772
$304$	0	0
$305$	−22.0734	−1.26392
$306$	−20.7468	−1.18602
$307$	−15.8624	−0.905315	−0.452658	−	0.891684i	$-0.649524\pi$
$307$	−15.8624	−0.905315	−0.452658	+	0.891684i	$0.649524\pi$
$308$	−4.25846	−0.242648
$309$	−47.8033	−2.71944
$310$	−4.88326	−0.277351
$311$	24.2979	1.37781	0.688905	−	0.724852i	$-0.258092\pi$
$311$	24.2979	1.37781	0.688905	+	0.724852i	$0.258092\pi$
$312$	−12.5169	−0.708630
$313$	24.5147	1.38566	0.692828	−	0.721103i	$-0.256365\pi$
$313$	24.5147	1.38566	0.692828	+	0.721103i	$0.256365\pi$
$314$	−2.58688	−0.145986
$315$	83.3605	4.69683
$316$	−0.921801	−0.0518553
$317$	18.7963	1.05571	0.527854	−	0.849335i	$-0.322997\pi$
$317$	18.7963	1.05571	0.527854	+	0.849335i	$0.322997\pi$
$318$	−0.459435	−0.0257638
$319$	−1.90985	−0.106931
$320$	2.93101	0.163848
$321$	50.1896	2.80131
$322$	36.7526	2.04814
$323$	0	0
$324$	15.5688	0.864932
$325$	14.4471	0.801381
$326$	8.80855	0.487860
$327$	−18.6191	−1.02964
$328$	4.69405	0.259185
$329$	25.0210	1.37945
$330$	9.11854	0.501959
$331$	6.59203	0.362331	0.181165	−	0.983453i	$-0.442013\pi$
$331$	6.59203	0.362331	0.181165	+	0.983453i	$0.442013\pi$
$332$	0.684858	0.0375865
$333$	−44.2450	−2.42461
$334$	−5.57202	−0.304887
$335$	6.37575	0.348344
$336$	−13.2483	−0.722755
$337$	11.5919	0.631453	0.315726	−	0.948850i	$-0.397752\pi$
$337$	11.5919	0.631453	0.315726	+	0.948850i	$0.397752\pi$
$338$	3.18742	0.173373
$339$	32.5152	1.76598
$340$	−9.10496	−0.493786
$341$	1.66607	0.0902228
$342$	0	0
$343$	17.6066	0.950664
$344$	3.02750	0.163232
$345$	−78.6974	−4.23692
$346$	−12.4821	−0.671039
$347$	21.1616	1.13601	0.568007	−	0.823024i	$-0.307715\pi$
$347$	21.1616	1.13601	0.568007	+	0.823024i	$0.307715\pi$
$348$	−5.94165	−0.318506
$349$	26.0415	1.39397	0.696984	−	0.717086i	$-0.254525\pi$
$349$	26.0415	1.39397	0.696984	+	0.717086i	$0.254525\pi$
$350$	15.2913	0.817354
$351$	−46.0458	−2.45774
$352$	−1.00000	−0.0533002
$353$	1.56380	0.0832325	0.0416163	−	0.999134i	$-0.486749\pi$
$353$	1.56380	0.0832325	0.0416163	+	0.999134i	$0.486749\pi$
$354$	−13.3871	−0.711514
$355$	40.3973	2.14406
$356$	−13.4881	−0.714868
$357$	41.1549	2.17815
$358$	−9.84076	−0.520101
$359$	28.9364	1.52720	0.763601	−	0.645688i	$-0.223429\pi$
$359$	28.9364	1.52720	0.763601	+	0.645688i	$0.223429\pi$
$360$	19.5753	1.03171
$361$	0	0
$362$	−2.32324	−0.122107
$363$	−3.11106	−0.163288
$364$	17.1333	0.898030
$365$	−31.6591	−1.65711
$366$	23.4294	1.22467
$367$	−5.92852	−0.309466	−0.154733	−	0.987956i	$-0.549452\pi$
$367$	−5.92852	−0.309466	−0.154733	+	0.987956i	$0.549452\pi$
$368$	8.63048	0.449895
$369$	31.3501	1.63202
$370$	−19.4174	−1.00946
$371$	0.628881	0.0326499
$372$	5.18324	0.268739
$373$	12.4999	0.647221	0.323610	−	0.946190i	$-0.395103\pi$
$373$	12.4999	0.647221	0.323610	+	0.946190i	$0.395103\pi$
$374$	3.10643	0.160629
$375$	12.8498	0.663561
$376$	5.87560	0.303011
$377$	7.68400	0.395746
$378$	−48.7364	−2.50673
$379$	15.7294	0.807965	0.403983	−	0.914767i	$-0.367626\pi$
$379$	15.7294	0.807965	0.403983	+	0.914767i	$0.367626\pi$
$380$	0	0
$381$	6.21605	0.318458
$382$	−6.59698	−0.337531
$383$	−20.6193	−1.05360	−0.526798	−	0.849991i	$-0.676607\pi$
$383$	−20.6193	−1.05360	−0.526798	+	0.849991i	$0.676607\pi$
$384$	−3.11106	−0.158761
$385$	−12.4816	−0.636120
$386$	−8.15324	−0.414989
$387$	20.2197	1.02783
$388$	16.7593	0.850824
$389$	4.97188	0.252084	0.126042	−	0.992025i	$-0.459773\pi$
$389$	4.97188	0.252084	0.126042	+	0.992025i	$0.459773\pi$
$390$	−36.6871	−1.85773
$391$	−26.8100	−1.35584
$392$	11.1345	0.562377
$393$	−9.96593	−0.502714
$394$	−21.6285	−1.08963
$395$	−2.70180	−0.135943
$396$	−6.67869	−0.335617
$397$	−19.3533	−0.971316	−0.485658	−	0.874149i	$-0.661420\pi$
$397$	−19.3533	−0.971316	−0.485658	+	0.874149i	$0.661420\pi$
$398$	−10.4974	−0.526188
$399$	0	0
$400$	3.59080	0.179540
$401$	22.7854	1.13785	0.568924	−	0.822390i	$-0.307360\pi$
$401$	22.7854	1.13785	0.568924	+	0.822390i	$0.307360\pi$
$402$	−6.76741	−0.337528
$403$	−6.70320	−0.333910
$404$	7.09312	0.352896
$405$	45.6322	2.26748
$406$	8.13301	0.403634
$407$	6.62481	0.328380
$408$	9.66427	0.478453
$409$	30.6973	1.51789	0.758943	−	0.651157i	$-0.225716\pi$
$409$	30.6973	1.51789	0.758943	+	0.651157i	$0.225716\pi$
$410$	13.7583	0.679473
$411$	64.5406	3.18355
$412$	15.3656	0.757010
$413$	18.3244	0.901685
$414$	57.6403	2.83287
$415$	2.00732	0.0985356
$416$	4.02336	0.197261
$417$	−65.0552	−3.18577
$418$	0	0
$419$	−15.5346	−0.758916	−0.379458	−	0.925209i	$-0.623889\pi$
$419$	−15.5346	−0.758916	−0.379458	+	0.925209i	$0.623889\pi$
$420$	−38.8309	−1.89476
$421$	−29.5170	−1.43857	−0.719286	−	0.694714i	$-0.755531\pi$
$421$	−29.5170	−1.43857	−0.719286	+	0.694714i	$0.755531\pi$
$422$	−4.99694	−0.243247
$423$	39.2412	1.90797
$424$	0.147678	0.00717188
$425$	−11.1546	−0.541076
$426$	−42.8789	−2.07749
$427$	−32.0705	−1.55200
$428$	−16.1326	−0.779801
$429$	12.5169	0.604322
$430$	8.87362	0.427924
$431$	−28.5012	−1.37285	−0.686426	−	0.727199i	$-0.740822\pi$
$431$	−28.5012	−1.37285	−0.686426	+	0.727199i	$0.740822\pi$
$432$	−11.4446	−0.550629
$433$	25.4790	1.22444	0.612221	−	0.790687i	$-0.290276\pi$
$433$	25.4790	1.22444	0.612221	+	0.790687i	$0.290276\pi$
$434$	−7.09490	−0.340566
$435$	−17.4150	−0.834986
$436$	5.98480	0.286620
$437$	0	0
$438$	33.6039	1.60565
$439$	−5.68911	−0.271527	−0.135763	−	0.990741i	$-0.543349\pi$
$439$	−5.68911	−0.271527	−0.135763	+	0.990741i	$0.543349\pi$
$440$	−2.93101	−0.139730
$441$	74.3637	3.54113
$442$	−12.4983	−0.594482
$443$	10.1297	0.481278	0.240639	−	0.970615i	$-0.422643\pi$
$443$	10.1297	0.481278	0.240639	+	0.970615i	$0.422643\pi$
$444$	20.6102	0.978116
$445$	−39.5337	−1.87408
$446$	−2.42462	−0.114809
$447$	67.2105	3.17895
$448$	4.25846	0.201193
$449$	−12.5988	−0.594575	−0.297287	−	0.954788i	$-0.596082\pi$
$449$	−12.5988	−0.594575	−0.297287	+	0.954788i	$0.596082\pi$
$450$	23.9819	1.13052
$451$	−4.69405	−0.221034
$452$	−10.4515	−0.491597
$453$	49.3015	2.31639
$454$	−1.20313	−0.0564658
$455$	50.2179	2.35425
$456$	0	0
$457$	26.2312	1.22704	0.613521	−	0.789678i	$-0.289753\pi$
$457$	26.2312	1.22704	0.613521	+	0.789678i	$0.289753\pi$
$458$	−21.9280	−1.02463
$459$	35.5518	1.65942
$460$	25.2960	1.17943
$461$	−3.60532	−0.167916	−0.0839581	−	0.996469i	$-0.526756\pi$
$461$	−3.60532	−0.167916	−0.0839581	+	0.996469i	$0.526756\pi$
$462$	13.2483	0.616368
$463$	19.6525	0.913328	0.456664	−	0.889639i	$-0.349044\pi$
$463$	19.6525	0.913328	0.456664	+	0.889639i	$0.349044\pi$
$464$	1.90985	0.0886624
$465$	15.1921	0.704518
$466$	21.8999	1.01449
$467$	−18.8897	−0.874111	−0.437056	−	0.899434i	$-0.643979\pi$
$467$	−18.8897	−0.874111	−0.437056	+	0.899434i	$0.643979\pi$
$468$	26.8708	1.24210
$469$	9.26332	0.427741
$470$	17.2214	0.794365
$471$	8.04794	0.370830
$472$	4.30306	0.198064
$473$	−3.02750	−0.139205
$474$	2.86778	0.131721
$475$	0	0
$476$	−13.2286	−0.606332
$477$	0.986295	0.0451594
$478$	6.87525	0.314467
$479$	10.7653	0.491879	0.245939	−	0.969285i	$-0.420904\pi$
$479$	10.7653	0.491879	0.245939	+	0.969285i	$0.420904\pi$
$480$	−9.11854	−0.416202
$481$	−26.6540	−1.21532
$482$	4.08165	0.185914
$483$	−114.339	−5.20262
$484$	1.00000	0.0454545
$485$	49.1216	2.23050
$486$	−14.1016	−0.639660
$487$	−16.2735	−0.737425	−0.368712	−	0.929544i	$-0.620201\pi$
$487$	−16.2735	−0.737425	−0.368712	+	0.929544i	$0.620201\pi$
$488$	−7.53100	−0.340913
$489$	−27.4039	−1.23925
$490$	32.6353	1.47431
$491$	−7.38583	−0.333318	−0.166659	−	0.986015i	$-0.553298\pi$
$491$	−7.38583	−0.333318	−0.166659	+	0.986015i	$0.553298\pi$
$492$	−14.6035	−0.658375
$493$	−5.93280	−0.267200
$494$	0	0
$495$	−19.5753	−0.879843
$496$	−1.66607	−0.0748088
$497$	58.6932	2.63275
$498$	−2.13063	−0.0954759
$499$	13.4720	0.603091	0.301546	−	0.953452i	$-0.402497\pi$
$499$	13.4720	0.603091	0.301546	+	0.953452i	$0.402497\pi$
$500$	−4.13036	−0.184715
$501$	17.3349	0.774465
$502$	23.2980	1.03984
$503$	−23.1330	−1.03145	−0.515724	−	0.856755i	$-0.672477\pi$
$503$	−23.1330	−1.03145	−0.515724	+	0.856755i	$0.672477\pi$
$504$	28.4409	1.26686
$505$	20.7900	0.925143
$506$	−8.63048	−0.383672
$507$	−9.91626	−0.440397
$508$	−1.99805	−0.0886491
$509$	−15.3868	−0.682010	−0.341005	−	0.940062i	$-0.610767\pi$
$509$	−15.3868	−0.682010	−0.341005	+	0.940062i	$0.610767\pi$
$510$	28.3261	1.25430
$511$	−45.9974	−2.03481
$512$	1.00000	0.0441942
$513$	0	0
$514$	23.5456	1.03855
$515$	45.0367	1.98456
$516$	−9.41873	−0.414636
$517$	−5.87560	−0.258408
$518$	−28.2115	−1.23954
$519$	38.8324	1.70455
$520$	11.7925	0.517135
$521$	4.68553	0.205277	0.102638	−	0.994719i	$-0.467272\pi$
$521$	4.68553	0.205277	0.102638	+	0.994719i	$0.467272\pi$
$522$	12.7553	0.558283
$523$	6.40148	0.279917	0.139959	−	0.990157i	$-0.455303\pi$
$523$	6.40148	0.279917	0.139959	+	0.990157i	$0.455303\pi$
$524$	3.20339	0.139941
$525$	−47.5721	−2.07622
$526$	−19.8530	−0.865630
$527$	5.17553	0.225449
$528$	3.11106	0.135391
$529$	51.4852	2.23849
$530$	0.432846	0.0188016
$531$	28.7388	1.24716
$532$	0	0
$533$	18.8858	0.818037
$534$	41.9623	1.81588
$535$	−47.2849	−2.04430
$536$	2.17528	0.0939576
$537$	30.6152	1.32114
$538$	8.77117	0.378152
$539$	−11.1345	−0.479596
$540$	−33.5442	−1.44351
$541$	−37.5757	−1.61551	−0.807754	−	0.589520i	$-0.799317\pi$
$541$	−37.5757	−1.61551	−0.807754	+	0.589520i	$0.799317\pi$
$542$	−2.22037	−0.0953730
$543$	7.22773	0.310172
$544$	−3.10643	−0.133187
$545$	17.5415	0.751395
$546$	−53.3028	−2.28115
$547$	26.8868	1.14960	0.574799	−	0.818295i	$-0.305080\pi$
$547$	26.8868	1.14960	0.574799	+	0.818295i	$0.305080\pi$
$548$	−20.7455	−0.886206
$549$	−50.2972	−2.14663
$550$	−3.59080	−0.153112
$551$	0	0
$552$	−26.8499	−1.14281
$553$	−3.92545	−0.166927
$554$	−16.7586	−0.712003
$555$	60.4086	2.56420
$556$	20.9110	0.886822
$557$	−5.76159	−0.244127	−0.122063	−	0.992522i	$-0.538951\pi$
$557$	−5.76159	−0.244127	−0.122063	+	0.992522i	$0.538951\pi$
$558$	−11.1272	−0.471050
$559$	12.1807	0.515189
$560$	12.4816	0.527443
$561$	−9.66427	−0.408026
$562$	11.6638	0.492007
$563$	−38.4988	−1.62253	−0.811265	−	0.584679i	$-0.801220\pi$
$563$	−38.4988	−1.62253	−0.811265	+	0.584679i	$0.801220\pi$
$564$	−18.2793	−0.769698
$565$	−30.6334	−1.28876
$566$	−28.1627	−1.18377
$567$	66.2990	2.78430
$568$	13.7827	0.578310
$569$	4.86166	0.203811	0.101906	−	0.994794i	$-0.467506\pi$
$569$	4.86166	0.203811	0.101906	+	0.994794i	$0.467506\pi$
$570$	0	0
$571$	−5.30945	−0.222194	−0.111097	−	0.993810i	$-0.535436\pi$
$571$	−5.30945	−0.222194	−0.111097	+	0.993810i	$0.535436\pi$
$572$	−4.02336	−0.168225
$573$	20.5236	0.857385
$574$	19.9894	0.834342
$575$	30.9904	1.29239
$576$	6.67869	0.278279
$577$	−9.59686	−0.399522	−0.199761	−	0.979845i	$-0.564017\pi$
$577$	−9.59686	−0.399522	−0.199761	+	0.979845i	$0.564017\pi$
$578$	−7.35011	−0.305724
$579$	25.3652	1.05414
$580$	5.59778	0.232435
$581$	2.91644	0.120994
$582$	−52.1391	−2.16124
$583$	−0.147678	−0.00611620
$584$	−10.8014	−0.446966
$585$	78.7584	3.25626
$586$	6.84160	0.282624
$587$	27.4748	1.13401	0.567003	−	0.823716i	$-0.308103\pi$
$587$	27.4748	1.13401	0.567003	+	0.823716i	$0.308103\pi$
$588$	−34.6400	−1.42853
$589$	0	0
$590$	12.6123	0.519240
$591$	67.2875	2.76784
$592$	−6.62481	−0.272278
$593$	8.73711	0.358790	0.179395	−	0.983777i	$-0.442586\pi$
$593$	8.73711	0.358790	0.179395	+	0.983777i	$0.442586\pi$
$594$	11.4446	0.469578
$595$	−38.7731	−1.58954
$596$	−21.6038	−0.884924
$597$	32.6581	1.33661
$598$	34.7235	1.41995
$599$	10.9467	0.447269	0.223634	−	0.974673i	$-0.428208\pi$
$599$	10.9467	0.447269	0.223634	+	0.974673i	$0.428208\pi$
$600$	−11.1712	−0.456062
$601$	−10.7923	−0.440225	−0.220113	−	0.975474i	$-0.570642\pi$
$601$	−10.7923	−0.440225	−0.220113	+	0.975474i	$0.570642\pi$
$602$	12.8925	0.525458
$603$	14.5280	0.591625
$604$	−15.8472	−0.644813
$605$	2.93101	0.119162
$606$	−22.0671	−0.896416
$607$	22.8681	0.928186	0.464093	−	0.885786i	$-0.346380\pi$
$607$	22.8681	0.928186	0.464093	+	0.885786i	$0.346380\pi$
$608$	0	0
$609$	−25.3023	−1.02530
$610$	−22.0734	−0.893727
$611$	23.6396	0.956357
$612$	−20.7468	−0.838641
$613$	−21.9097	−0.884925	−0.442463	−	0.896787i	$-0.645895\pi$
$613$	−21.9097	−0.884925	−0.442463	+	0.896787i	$0.645895\pi$
$614$	−15.8624	−0.640155
$615$	−42.8028	−1.72598
$616$	−4.25846	−0.171578
$617$	3.11107	0.125247	0.0626234	−	0.998037i	$-0.480053\pi$
$617$	3.11107	0.125247	0.0626234	+	0.998037i	$0.480053\pi$
$618$	−47.8033	−1.92293
$619$	−2.62989	−0.105704	−0.0528521	−	0.998602i	$-0.516831\pi$
$619$	−2.62989	−0.105704	−0.0528521	+	0.998602i	$0.516831\pi$
$620$	−4.88326	−0.196117
$621$	−98.7725	−3.96360
$622$	24.2979	0.974258
$623$	−57.4385	−2.30123
$624$	−12.5169	−0.501077
$625$	−30.0601	−1.20241
$626$	24.5147	0.979806
$627$	0	0
$628$	−2.58688	−0.103228
$629$	20.5795	0.820558
$630$	83.3605	3.32116
$631$	−23.9282	−0.952568	−0.476284	−	0.879291i	$-0.658017\pi$
$631$	−23.9282	−0.952568	−0.476284	+	0.879291i	$0.658017\pi$
$632$	−0.921801	−0.0366673
$633$	15.5458	0.617889
$634$	18.7963	0.746498
$635$	−5.85630	−0.232400
$636$	−0.459435	−0.0182178
$637$	44.7980	1.77496
$638$	−1.90985	−0.0756116
$639$	92.0505	3.64146
$640$	2.93101	0.115858
$641$	10.2628	0.405355	0.202678	−	0.979246i	$-0.435036\pi$
$641$	10.2628	0.405355	0.202678	+	0.979246i	$0.435036\pi$
$642$	50.1896	1.98083
$643$	−11.3552	−0.447807	−0.223904	−	0.974611i	$-0.571880\pi$
$643$	−11.3552	−0.447807	−0.223904	+	0.974611i	$0.571880\pi$
$644$	36.7526	1.44825
$645$	−27.6064	−1.08700
$646$	0	0
$647$	39.0097	1.53363	0.766816	−	0.641867i	$-0.221840\pi$
$647$	39.0097	1.53363	0.766816	+	0.641867i	$0.221840\pi$
$648$	15.5688	0.611599
$649$	−4.30306	−0.168910
$650$	14.4471	0.566662
$651$	22.0726	0.865095
$652$	8.80855	0.344969
$653$	−2.69528	−0.105474	−0.0527372	−	0.998608i	$-0.516795\pi$
$653$	−2.69528	−0.105474	−0.0527372	+	0.998608i	$0.516795\pi$
$654$	−18.6191	−0.728063
$655$	9.38915	0.366865
$656$	4.69405	0.183272
$657$	−72.1393	−2.81442
$658$	25.0210	0.975420
$659$	−3.06221	−0.119287	−0.0596433	−	0.998220i	$-0.518996\pi$
$659$	−3.06221	−0.119287	−0.0596433	+	0.998220i	$0.518996\pi$
$660$	9.11854	0.354939
$661$	−26.4162	−1.02747	−0.513735	−	0.857949i	$-0.671738\pi$
$661$	−26.4162	−1.02747	−0.513735	+	0.857949i	$0.671738\pi$
$662$	6.59203	0.256206
$663$	38.8829	1.51008
$664$	0.684858	0.0265776
$665$	0	0
$666$	−44.2450	−1.71446
$667$	16.4829	0.638220
$668$	−5.57202	−0.215588
$669$	7.54313	0.291634
$670$	6.37575	0.246317
$671$	7.53100	0.290731
$672$	−13.2483	−0.511065
$673$	−13.3309	−0.513869	−0.256934	−	0.966429i	$-0.582712\pi$
$673$	−13.3309	−0.513869	−0.256934	+	0.966429i	$0.582712\pi$
$674$	11.5919	0.446505
$675$	−41.0953	−1.58176
$676$	3.18742	0.122593
$677$	32.8102	1.26100	0.630498	−	0.776191i	$-0.282851\pi$
$677$	32.8102	1.26100	0.630498	+	0.776191i	$0.282851\pi$
$678$	32.5152	1.24874
$679$	71.3688	2.73888
$680$	−9.10496	−0.349159
$681$	3.74301	0.143433
$682$	1.66607	0.0637971
$683$	9.86506	0.377476	0.188738	−	0.982027i	$-0.439560\pi$
$683$	9.86506	0.377476	0.188738	+	0.982027i	$0.439560\pi$
$684$	0	0
$685$	−60.8053	−2.32325
$686$	17.6066	0.672221
$687$	68.2193	2.60273
$688$	3.02750	0.115422
$689$	0.594162	0.0226358
$690$	−78.6974	−2.99596
$691$	−24.6086	−0.936156	−0.468078	−	0.883687i	$-0.655053\pi$
$691$	−24.6086	−0.936156	−0.468078	+	0.883687i	$0.655053\pi$
$692$	−12.4821	−0.474496
$693$	−28.4409	−1.08038
$694$	21.1616	0.803284
$695$	61.2902	2.32487
$696$	−5.94165	−0.225217
$697$	−14.5817	−0.552322
$698$	26.0415	0.985685
$699$	−68.1318	−2.57698
$700$	15.2913	0.577957
$701$	−5.28347	−0.199554	−0.0997770	−	0.995010i	$-0.531813\pi$
$701$	−5.28347	−0.199554	−0.0997770	+	0.995010i	$0.531813\pi$
$702$	−46.0458	−1.73789
$703$	0	0
$704$	−1.00000	−0.0376889
$705$	−53.5768	−2.01782
$706$	1.56380	0.0588543
$707$	30.2058	1.13601
$708$	−13.3871	−0.503117
$709$	−32.6775	−1.22723	−0.613614	−	0.789606i	$-0.710285\pi$
$709$	−32.6775	−1.22723	−0.613614	+	0.789606i	$0.710285\pi$
$710$	40.3973	1.51608
$711$	−6.15642	−0.230884
$712$	−13.4881	−0.505488
$713$	−14.3790	−0.538497
$714$	41.1549	1.54018
$715$	−11.7925	−0.441014
$716$	−9.84076	−0.367767
$717$	−21.3893	−0.798798
$718$	28.9364	1.07990
$719$	−50.2764	−1.87499	−0.937497	−	0.347994i	$-0.886863\pi$
$719$	−50.2764	−1.87499	−0.937497	+	0.347994i	$0.886863\pi$
$720$	19.5753	0.729527
$721$	65.4339	2.43688
$722$	0	0
$723$	−12.6982	−0.472253
$724$	−2.32324	−0.0863425
$725$	6.85789	0.254696
$726$	−3.11106	−0.115462
$727$	−17.3620	−0.643921	−0.321960	−	0.946753i	$-0.604342\pi$
$727$	−17.3620	−0.643921	−0.321960	+	0.946753i	$0.604342\pi$
$728$	17.1333	0.635003
$729$	−2.83553	−0.105020
$730$	−31.6591	−1.17175
$731$	−9.40470	−0.347845
$732$	23.4294	0.865975
$733$	−17.6506	−0.651939	−0.325970	−	0.945380i	$-0.605691\pi$
$733$	−17.6506	−0.651939	−0.325970	+	0.945380i	$0.605691\pi$
$734$	−5.92852	−0.218826
$735$	−101.530	−3.74500
$736$	8.63048	0.318124
$737$	−2.17528	−0.0801273
$738$	31.3501	1.15401
$739$	31.3603	1.15361	0.576803	−	0.816883i	$-0.304300\pi$
$739$	31.3603	1.15361	0.576803	+	0.816883i	$0.304300\pi$
$740$	−19.4174	−0.713797
$741$	0	0
$742$	0.628881	0.0230870
$743$	−10.6136	−0.389374	−0.194687	−	0.980865i	$-0.562369\pi$
$743$	−10.6136	−0.389374	−0.194687	+	0.980865i	$0.562369\pi$
$744$	5.18324	0.190027
$745$	−63.3208	−2.31989
$746$	12.4999	0.457654
$747$	4.57395	0.167352
$748$	3.10643	0.113582
$749$	−68.7002	−2.51025
$750$	12.8498	0.469208
$751$	21.4131	0.781376	0.390688	−	0.920523i	$-0.372237\pi$
$751$	21.4131	0.781376	0.390688	+	0.920523i	$0.372237\pi$
$752$	5.87560	0.214261
$753$	−72.4813	−2.64137
$754$	7.68400	0.279835
$755$	−46.4482	−1.69042
$756$	−48.7364	−1.77253
$757$	−12.4086	−0.450997	−0.225499	−	0.974244i	$-0.572401\pi$
$757$	−12.4086	−0.450997	−0.225499	+	0.974244i	$0.572401\pi$
$758$	15.7294	0.571318
$759$	26.8499	0.974591
$760$	0	0
$761$	−22.9365	−0.831446	−0.415723	−	0.909491i	$-0.636471\pi$
$761$	−22.9365	−0.831446	−0.415723	+	0.909491i	$0.636471\pi$
$762$	6.21605	0.225184
$763$	25.4860	0.922656
$764$	−6.59698	−0.238670
$765$	−60.8092	−2.19856
$766$	−20.6193	−0.745005
$767$	17.3127	0.625127
$768$	−3.11106	−0.112261
$769$	−13.3853	−0.482686	−0.241343	−	0.970440i	$-0.577588\pi$
$769$	−13.3853	−0.482686	−0.241343	+	0.970440i	$0.577588\pi$
$770$	−12.4816	−0.449805
$771$	−73.2516	−2.63809
$772$	−8.15324	−0.293442
$773$	30.8919	1.11111	0.555553	−	0.831481i	$-0.312507\pi$
$773$	30.8919	1.11111	0.555553	+	0.831481i	$0.312507\pi$
$774$	20.2197	0.726782
$775$	−5.98253	−0.214899
$776$	16.7593	0.601623
$777$	87.7677	3.14865
$778$	4.97188	0.178251
$779$	0	0
$780$	−36.6871	−1.31361
$781$	−13.7827	−0.493185
$782$	−26.8100	−0.958722
$783$	−21.8574	−0.781121
$784$	11.1345	0.397660
$785$	−7.58217	−0.270619
$786$	−9.96593	−0.355473
$787$	46.8608	1.67041	0.835204	−	0.549941i	$-0.185350\pi$
$787$	46.8608	1.67041	0.835204	+	0.549941i	$0.185350\pi$
$788$	−21.6285	−0.770483
$789$	61.7637	2.19885
$790$	−2.70180	−0.0961259
$791$	−44.5073	−1.58250
$792$	−6.67869	−0.237317
$793$	−30.2999	−1.07598
$794$	−19.3533	−0.686824
$795$	−1.34661	−0.0477593
$796$	−10.4974	−0.372071
$797$	−24.5961	−0.871239	−0.435619	−	0.900131i	$-0.643471\pi$
$797$	−24.5961	−0.871239	−0.435619	+	0.900131i	$0.643471\pi$
$798$	0	0
$799$	−18.2521	−0.645713
$800$	3.59080	0.126954
$801$	−90.0827	−3.18292
$802$	22.7854	0.804580
$803$	10.8014	0.381174
$804$	−6.76741	−0.238668
$805$	107.722	3.79670
$806$	−6.70320	−0.236110
$807$	−27.2876	−0.960570
$808$	7.09312	0.249535
$809$	−12.3574	−0.434464	−0.217232	−	0.976120i	$-0.569703\pi$
$809$	−12.3574	−0.434464	−0.217232	+	0.976120i	$0.569703\pi$
$810$	45.6322	1.60335
$811$	11.3975	0.400219	0.200109	−	0.979774i	$-0.435870\pi$
$811$	11.3975	0.400219	0.200109	+	0.979774i	$0.435870\pi$
$812$	8.13301	0.285413
$813$	6.90769	0.242263
$814$	6.62481	0.232200
$815$	25.8179	0.904362
$816$	9.66427	0.338317
$817$	0	0
$818$	30.6973	1.07331
$819$	114.428	3.99844
$820$	13.7583	0.480460
$821$	20.1977	0.704905	0.352452	−	0.935830i	$-0.385348\pi$
$821$	20.1977	0.704905	0.352452	+	0.935830i	$0.385348\pi$
$822$	64.5406	2.25111
$823$	18.1889	0.634027	0.317013	−	0.948421i	$-0.397320\pi$
$823$	18.1889	0.634027	0.317013	+	0.948421i	$0.397320\pi$
$824$	15.3656	0.535287
$825$	11.1712	0.388931
$826$	18.3244	0.637587
$827$	−6.57550	−0.228652	−0.114326	−	0.993443i	$-0.536471\pi$
$827$	−6.57550	−0.228652	−0.114326	+	0.993443i	$0.536471\pi$
$828$	57.6403	2.00314
$829$	−29.0737	−1.00977	−0.504885	−	0.863187i	$-0.668465\pi$
$829$	−29.0737	−1.00977	−0.504885	+	0.863187i	$0.668465\pi$
$830$	2.00732	0.0696752
$831$	52.1369	1.80861
$832$	4.02336	0.139485
$833$	−34.5885	−1.19842
$834$	−65.0552	−2.25268
$835$	−16.3316	−0.565179
$836$	0	0
$837$	19.0675	0.659070
$838$	−15.5346	−0.536634
$839$	34.5337	1.19224	0.596119	−	0.802896i	$-0.296709\pi$
$839$	34.5337	1.19224	0.596119	+	0.802896i	$0.296709\pi$
$840$	−38.8309	−1.33979
$841$	−25.3525	−0.874224
$842$	−29.5170	−1.01722
$843$	−36.2867	−1.24978
$844$	−4.99694	−0.172002
$845$	9.34236	0.321387
$846$	39.2412	1.34914
$847$	4.25846	0.146322
$848$	0.147678	0.00507129
$849$	87.6157	3.00696
$850$	−11.1546	−0.382599
$851$	−57.1753	−1.95995
$852$	−42.8789	−1.46901
$853$	−3.19532	−0.109406	−0.0547028	−	0.998503i	$-0.517421\pi$
$853$	−3.19532	−0.109406	−0.0547028	+	0.998503i	$0.517421\pi$
$854$	−32.0705	−1.09743
$855$	0	0
$856$	−16.1326	−0.551402
$857$	14.2661	0.487322	0.243661	−	0.969861i	$-0.421652\pi$
$857$	14.2661	0.487322	0.243661	+	0.969861i	$0.421652\pi$
$858$	12.5169	0.427320
$859$	48.7818	1.66441	0.832206	−	0.554467i	$-0.187078\pi$
$859$	48.7818	1.66441	0.832206	+	0.554467i	$0.187078\pi$
$860$	8.87362	0.302588
$861$	−62.1882	−2.11937
$862$	−28.5012	−0.970754
$863$	−41.5018	−1.41274	−0.706369	−	0.707844i	$-0.749668\pi$
$863$	−41.5018	−1.41274	−0.706369	+	0.707844i	$0.749668\pi$
$864$	−11.4446	−0.389353
$865$	−36.5850	−1.24393
$866$	25.4790	0.865812
$867$	22.8666	0.776592
$868$	−7.09490	−0.240816
$869$	0.921801	0.0312699
$870$	−17.4150	−0.590424
$871$	8.75191	0.296547
$872$	5.98480	0.202671
$873$	111.930	3.78826
$874$	0	0
$875$	−17.5890	−0.594616
$876$	33.6039	1.13537
$877$	−37.5768	−1.26888	−0.634439	−	0.772973i	$-0.718769\pi$
$877$	−37.5768	−1.26888	−0.634439	+	0.772973i	$0.718769\pi$
$878$	−5.68911	−0.191998
$879$	−21.2846	−0.717912
$880$	−2.93101	−0.0988042
$881$	23.7402	0.799829	0.399914	−	0.916553i	$-0.369040\pi$
$881$	23.7402	0.799829	0.399914	+	0.916553i	$0.369040\pi$
$882$	74.3637	2.50396
$883$	−0.524925	−0.0176651	−0.00883256	−	0.999961i	$-0.502812\pi$
$883$	−0.524925	−0.0176651	−0.00883256	+	0.999961i	$0.502812\pi$
$884$	−12.4983	−0.420362
$885$	−39.2376	−1.31896
$886$	10.1297	0.340315
$887$	−23.9127	−0.802910	−0.401455	−	0.915879i	$-0.631495\pi$
$887$	−23.9127	−0.802910	−0.401455	+	0.915879i	$0.631495\pi$
$888$	20.6102	0.691632
$889$	−8.50861	−0.285370
$890$	−39.5337	−1.32517
$891$	−15.5688	−0.521574
$892$	−2.42462	−0.0811822
$893$	0	0
$894$	67.2105	2.24786
$895$	−28.8434	−0.964127
$896$	4.25846	0.142265
$897$	−108.027	−3.60692
$898$	−12.5988	−0.420428
$899$	−3.18194	−0.106124
$900$	23.9819	0.799395
$901$	−0.458751	−0.0152832
$902$	−4.69405	−0.156295
$903$	−40.1093	−1.33475
$904$	−10.4515	−0.347611
$905$	−6.80943	−0.226353
$906$	49.3015	1.63793
$907$	−27.1398	−0.901163	−0.450582	−	0.892735i	$-0.648783\pi$
$907$	−27.1398	−0.901163	−0.450582	+	0.892735i	$0.648783\pi$
$908$	−1.20313	−0.0399273
$909$	47.3727	1.57125
$910$	50.2179	1.66471
$911$	9.64387	0.319516	0.159758	−	0.987156i	$-0.448929\pi$
$911$	9.64387	0.319516	0.159758	+	0.987156i	$0.448929\pi$
$912$	0	0
$913$	−0.684858	−0.0226655
$914$	26.2312	0.867650
$915$	68.6717	2.27022
$916$	−21.9280	−0.724522
$917$	13.6415	0.450482
$918$	35.5518	1.17339
$919$	−25.5610	−0.843180	−0.421590	−	0.906787i	$-0.638528\pi$
$919$	−25.5610	−0.843180	−0.421590	+	0.906787i	$0.638528\pi$
$920$	25.2960	0.833985
$921$	49.3489	1.62610
$922$	−3.60532	−0.118735
$923$	55.4529	1.82525
$924$	13.2483	0.435838
$925$	−23.7884	−0.782158
$926$	19.6525	0.645820
$927$	102.622	3.37055
$928$	1.90985	0.0626938
$929$	29.3269	0.962183	0.481092	−	0.876670i	$-0.340240\pi$
$929$	29.3269	0.962183	0.481092	+	0.876670i	$0.340240\pi$
$930$	15.1921	0.498169
$931$	0	0
$932$	21.8999	0.717354
$933$	−75.5923	−2.47478
$934$	−18.8897	−0.618090
$935$	9.10496	0.297764
$936$	26.8708	0.878298
$937$	−1.72269	−0.0562778	−0.0281389	−	0.999604i	$-0.508958\pi$
$937$	−1.72269	−0.0562778	−0.0281389	+	0.999604i	$0.508958\pi$
$938$	9.26332	0.302458
$939$	−76.2668	−2.48887
$940$	17.2214	0.561701
$941$	−38.4777	−1.25434	−0.627169	−	0.778883i	$-0.715786\pi$
$941$	−38.4777	−1.25434	−0.627169	+	0.778883i	$0.715786\pi$
$942$	8.04794	0.262216
$943$	40.5119	1.31925
$944$	4.30306	0.140053
$945$	−142.847	−4.64680
$946$	−3.02750	−0.0984325
$947$	−3.08491	−0.100246	−0.0501231	−	0.998743i	$-0.515961\pi$
$947$	−3.08491	−0.100246	−0.0501231	+	0.998743i	$0.515961\pi$
$948$	2.86778	0.0931410
$949$	−43.4580	−1.41071
$950$	0	0
$951$	−58.4765	−1.89623
$952$	−13.2286	−0.428741
$953$	1.06262	0.0344215	0.0172108	−	0.999852i	$-0.494521\pi$
$953$	1.06262	0.0344215	0.0172108	+	0.999852i	$0.494521\pi$
$954$	0.986295	0.0319325
$955$	−19.3358	−0.625691
$956$	6.87525	0.222362
$957$	5.94165	0.192066
$958$	10.7653	0.347811
$959$	−88.3441	−2.85278
$960$	−9.11854	−0.294299
$961$	−28.2242	−0.910458
$962$	−26.6540	−0.859360
$963$	−107.745	−3.47203
$964$	4.08165	0.131461
$965$	−23.8972	−0.769278
$966$	−114.339	−3.67881
$967$	−20.4889	−0.658879	−0.329440	−	0.944177i	$-0.606860\pi$
$967$	−20.4889	−0.658879	−0.329440	+	0.944177i	$0.606860\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	49.1216	1.57720
$971$	1.69698	0.0544587	0.0272294	−	0.999629i	$-0.491332\pi$
$971$	1.69698	0.0544587	0.0272294	+	0.999629i	$0.491332\pi$
$972$	−14.1016	−0.452308
$973$	89.0485	2.85476
$974$	−16.2735	−0.521438
$975$	−44.9458	−1.43942
$976$	−7.53100	−0.241062
$977$	−4.17331	−0.133516	−0.0667581	−	0.997769i	$-0.521266\pi$
$977$	−4.17331	−0.133516	−0.0667581	+	0.997769i	$0.521266\pi$
$978$	−27.4039	−0.876281
$979$	13.4881	0.431081
$980$	32.6353	1.04250
$981$	39.9706	1.27616
$982$	−7.38583	−0.235691
$983$	5.02905	0.160402	0.0802009	−	0.996779i	$-0.474444\pi$
$983$	5.02905	0.160402	0.0802009	+	0.996779i	$0.474444\pi$
$984$	−14.6035	−0.465541
$985$	−63.3933	−2.01988
$986$	−5.93280	−0.188939
$987$	−77.8418	−2.47773
$988$	0	0
$989$	26.1288	0.830847
$990$	−19.5753	−0.622143
$991$	−50.3171	−1.59837	−0.799187	−	0.601082i	$-0.794736\pi$
$991$	−50.3171	−1.59837	−0.799187	+	0.601082i	$0.794736\pi$
$992$	−1.66607	−0.0528978
$993$	−20.5082	−0.650807
$994$	58.6932	1.86163
$995$	−30.7680	−0.975411
$996$	−2.13063	−0.0675117
$997$	28.0631	0.888768	0.444384	−	0.895836i	$-0.353423\pi$
$997$	28.0631	0.888768	0.444384	+	0.895836i	$0.353423\pi$
$998$	13.4720	0.426450
$999$	75.8184	2.39879

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7942.2.a.ca.1.1		15
19.6	even	9		418.2.j.d.397.1	yes	30
19.16	even	9		418.2.j.d.199.1	✓	30
19.18	odd	2		7942.2.a.by.1.15		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.j.d.199.1	✓	30	19.16	even	9
418.2.j.d.397.1	yes	30	19.6	even	9
7942.2.a.by.1.15		15	19.18	odd	2
7942.2.a.ca.1.1		15	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:	\( N \)	\(=\)	\( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7942.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.11106 q^{3} +1.00000 q^{4} +2.93101 q^{5} -3.11106 q^{6} +4.25846 q^{7} +1.00000 q^{8} +6.67869 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.11106 q^{3} +1.00000 q^{4} +2.93101 q^{5} -3.11106 q^{6} +4.25846 q^{7} +1.00000 q^{8} +6.67869 q^{9} +2.93101 q^{10} -1.00000 q^{11} -3.11106 q^{12} +4.02336 q^{13} +4.25846 q^{14} -9.11854 q^{15} +1.00000 q^{16} -3.10643 q^{17} +6.67869 q^{18} +2.93101 q^{20} -13.2483 q^{21} -1.00000 q^{22} +8.63048 q^{23} -3.11106 q^{24} +3.59080 q^{25} +4.02336 q^{26} -11.4446 q^{27} +4.25846 q^{28} +1.90985 q^{29} -9.11854 q^{30} -1.66607 q^{31} +1.00000 q^{32} +3.11106 q^{33} -3.10643 q^{34} +12.4816 q^{35} +6.67869 q^{36} -6.62481 q^{37} -12.5169 q^{39} +2.93101 q^{40} +4.69405 q^{41} -13.2483 q^{42} +3.02750 q^{43} -1.00000 q^{44} +19.5753 q^{45} +8.63048 q^{46} +5.87560 q^{47} -3.11106 q^{48} +11.1345 q^{49} +3.59080 q^{50} +9.66427 q^{51} +4.02336 q^{52} +0.147678 q^{53} -11.4446 q^{54} -2.93101 q^{55} +4.25846 q^{56} +1.90985 q^{58} +4.30306 q^{59} -9.11854 q^{60} -7.53100 q^{61} -1.66607 q^{62} +28.4409 q^{63} +1.00000 q^{64} +11.7925 q^{65} +3.11106 q^{66} +2.17528 q^{67} -3.10643 q^{68} -26.8499 q^{69} +12.4816 q^{70} +13.7827 q^{71} +6.67869 q^{72} -10.8014 q^{73} -6.62481 q^{74} -11.1712 q^{75} -4.25846 q^{77} -12.5169 q^{78} -0.921801 q^{79} +2.93101 q^{80} +15.5688 q^{81} +4.69405 q^{82} +0.684858 q^{83} -13.2483 q^{84} -9.10496 q^{85} +3.02750 q^{86} -5.94165 q^{87} -1.00000 q^{88} -13.4881 q^{89} +19.5753 q^{90} +17.1333 q^{91} +8.63048 q^{92} +5.18324 q^{93} +5.87560 q^{94} -3.11106 q^{96} +16.7593 q^{97} +11.1345 q^{98} -6.67869 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 15 q^{2} + 3 q^{3} + 15 q^{4} + 9 q^{5} + 3 q^{6} + 15 q^{8} + 30 q^{9}+O(q^{10}) \) 15 * q + 15 * q^2 + 3 * q^3 + 15 * q^4 + 9 * q^5 + 3 * q^6 + 15 * q^8 + 30 * q^9	\( 15 q + 15 q^{2} + 3 q^{3} + 15 q^{4} + 9 q^{5} + 3 q^{6} + 15 q^{8} + 30 q^{9} + 9 q^{10} - 15 q^{11} + 3 q^{12} + 21 q^{15} + 15 q^{16} + 21 q^{17} + 30 q^{18} + 9 q^{20} - 9 q^{21} - 15 q^{22} + 21 q^{23} + 3 q^{24} + 24 q^{25} + 3 q^{27} - 9 q^{29} + 21 q^{30} + 18 q^{31} + 15 q^{32} - 3 q^{33} + 21 q^{34} + 18 q^{35} + 30 q^{36} - 9 q^{37} + 9 q^{40} + 15 q^{41} - 9 q^{42} + 3 q^{43} - 15 q^{44} + 54 q^{45} + 21 q^{46} + 39 q^{47} + 3 q^{48} + 33 q^{49} + 24 q^{50} + 30 q^{51} + 18 q^{53} + 3 q^{54} - 9 q^{55} - 9 q^{58} + 6 q^{59} + 21 q^{60} - 30 q^{61} + 18 q^{62} + 24 q^{63} + 15 q^{64} + 6 q^{65} - 3 q^{66} + 9 q^{67} + 21 q^{68} - 42 q^{69} + 18 q^{70} + 9 q^{71} + 30 q^{72} + 12 q^{73} - 9 q^{74} + 21 q^{75} + 12 q^{79} + 9 q^{80} + 63 q^{81} + 15 q^{82} + 30 q^{83} - 9 q^{84} - 3 q^{85} + 3 q^{86} + 9 q^{87} - 15 q^{88} - 6 q^{89} + 54 q^{90} + 96 q^{91} + 21 q^{92} + 102 q^{93} + 39 q^{94} + 3 q^{96} + 33 q^{98} - 30 q^{99}+O(q^{100}) \) 15 * q + 15 * q^2 + 3 * q^3 + 15 * q^4 + 9 * q^5 + 3 * q^6 + 15 * q^8 + 30 * q^9 + 9 * q^10 - 15 * q^11 + 3 * q^12 + 21 * q^15 + 15 * q^16 + 21 * q^17 + 30 * q^18 + 9 * q^20 - 9 * q^21 - 15 * q^22 + 21 * q^23 + 3 * q^24 + 24 * q^25 + 3 * q^27 - 9 * q^29 + 21 * q^30 + 18 * q^31 + 15 * q^32 - 3 * q^33 + 21 * q^34 + 18 * q^35 + 30 * q^36 - 9 * q^37 + 9 * q^40 + 15 * q^41 - 9 * q^42 + 3 * q^43 - 15 * q^44 + 54 * q^45 + 21 * q^46 + 39 * q^47 + 3 * q^48 + 33 * q^49 + 24 * q^50 + 30 * q^51 + 18 * q^53 + 3 * q^54 - 9 * q^55 - 9 * q^58 + 6 * q^59 + 21 * q^60 - 30 * q^61 + 18 * q^62 + 24 * q^63 + 15 * q^64 + 6 * q^65 - 3 * q^66 + 9 * q^67 + 21 * q^68 - 42 * q^69 + 18 * q^70 + 9 * q^71 + 30 * q^72 + 12 * q^73 - 9 * q^74 + 21 * q^75 + 12 * q^79 + 9 * q^80 + 63 * q^81 + 15 * q^82 + 30 * q^83 - 9 * q^84 - 3 * q^85 + 3 * q^86 + 9 * q^87 - 15 * q^88 - 6 * q^89 + 54 * q^90 + 96 * q^91 + 21 * q^92 + 102 * q^93 + 39 * q^94 + 3 * q^96 + 33 * q^98 - 30 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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