LMFDB - Embedded newform 4046.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4046,2,Mod(1,4046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4046, 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("4046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4046 = 2 \cdot 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4046.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:	$32.3074726578$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 238)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -3.23607 q^{3} +1.00000 q^{4} +1.23607 q^{5} +3.23607 q^{6} +1.00000 q^{7} -1.00000 q^{8} +7.47214 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -3.23607 q^{3} +1.00000 q^{4} +1.23607 q^{5} +3.23607 q^{6} +1.00000 q^{7} -1.00000 q^{8} +7.47214 q^{9} -1.23607 q^{10} -5.23607 q^{11} -3.23607 q^{12} -2.47214 q^{13} -1.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} -7.47214 q^{18} -8.47214 q^{19} +1.23607 q^{20} -3.23607 q^{21} +5.23607 q^{22} -8.00000 q^{23} +3.23607 q^{24} -3.47214 q^{25} +2.47214 q^{26} -14.4721 q^{27} +1.00000 q^{28} +5.70820 q^{29} +4.00000 q^{30} +1.52786 q^{31} -1.00000 q^{32} +16.9443 q^{33} +1.23607 q^{35} +7.47214 q^{36} +3.23607 q^{37} +8.47214 q^{38} +8.00000 q^{39} -1.23607 q^{40} +3.52786 q^{41} +3.23607 q^{42} +2.47214 q^{43} -5.23607 q^{44} +9.23607 q^{45} +8.00000 q^{46} -2.47214 q^{47} -3.23607 q^{48} +1.00000 q^{49} +3.47214 q^{50} -2.47214 q^{52} -4.47214 q^{53} +14.4721 q^{54} -6.47214 q^{55} -1.00000 q^{56} +27.4164 q^{57} -5.70820 q^{58} -6.00000 q^{59} -4.00000 q^{60} -1.23607 q^{61} -1.52786 q^{62} +7.47214 q^{63} +1.00000 q^{64} -3.05573 q^{65} -16.9443 q^{66} -1.52786 q^{67} +25.8885 q^{69} -1.23607 q^{70} +2.47214 q^{71} -7.47214 q^{72} -13.4164 q^{73} -3.23607 q^{74} +11.2361 q^{75} -8.47214 q^{76} -5.23607 q^{77} -8.00000 q^{78} +10.4721 q^{79} +1.23607 q^{80} +24.4164 q^{81} -3.52786 q^{82} +2.00000 q^{83} -3.23607 q^{84} -2.47214 q^{86} -18.4721 q^{87} +5.23607 q^{88} +2.00000 q^{89} -9.23607 q^{90} -2.47214 q^{91} -8.00000 q^{92} -4.94427 q^{93} +2.47214 q^{94} -10.4721 q^{95} +3.23607 q^{96} +8.47214 q^{97} -1.00000 q^{98} -39.1246 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^7 - 2 * q^8 + 6 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 6 q^{9} + 2 q^{10} - 6 q^{11} - 2 q^{12} + 4 q^{13} - 2 q^{14} - 8 q^{15} + 2 q^{16} - 6 q^{18} - 8 q^{19} - 2 q^{20} - 2 q^{21} + 6 q^{22} - 16 q^{23} + 2 q^{24} + 2 q^{25} - 4 q^{26} - 20 q^{27} + 2 q^{28} - 2 q^{29} + 8 q^{30} + 12 q^{31} - 2 q^{32} + 16 q^{33} - 2 q^{35} + 6 q^{36} + 2 q^{37} + 8 q^{38} + 16 q^{39} + 2 q^{40} + 16 q^{41} + 2 q^{42} - 4 q^{43} - 6 q^{44} + 14 q^{45} + 16 q^{46} + 4 q^{47} - 2 q^{48} + 2 q^{49} - 2 q^{50} + 4 q^{52} + 20 q^{54} - 4 q^{55} - 2 q^{56} + 28 q^{57} + 2 q^{58} - 12 q^{59} - 8 q^{60} + 2 q^{61} - 12 q^{62} + 6 q^{63} + 2 q^{64} - 24 q^{65} - 16 q^{66} - 12 q^{67} + 16 q^{69} + 2 q^{70} - 4 q^{71} - 6 q^{72} - 2 q^{74} + 18 q^{75} - 8 q^{76} - 6 q^{77} - 16 q^{78} + 12 q^{79} - 2 q^{80} + 22 q^{81} - 16 q^{82} + 4 q^{83} - 2 q^{84} + 4 q^{86} - 28 q^{87} + 6 q^{88} + 4 q^{89} - 14 q^{90} + 4 q^{91} - 16 q^{92} + 8 q^{93} - 4 q^{94} - 12 q^{95} + 2 q^{96} + 8 q^{97} - 2 q^{98} - 38 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^7 - 2 * q^8 + 6 * q^9 + 2 * q^10 - 6 * q^11 - 2 * q^12 + 4 * q^13 - 2 * q^14 - 8 * q^15 + 2 * q^16 - 6 * q^18 - 8 * q^19 - 2 * q^20 - 2 * q^21 + 6 * q^22 - 16 * q^23 + 2 * q^24 + 2 * q^25 - 4 * q^26 - 20 * q^27 + 2 * q^28 - 2 * q^29 + 8 * q^30 + 12 * q^31 - 2 * q^32 + 16 * q^33 - 2 * q^35 + 6 * q^36 + 2 * q^37 + 8 * q^38 + 16 * q^39 + 2 * q^40 + 16 * q^41 + 2 * q^42 - 4 * q^43 - 6 * q^44 + 14 * q^45 + 16 * q^46 + 4 * q^47 - 2 * q^48 + 2 * q^49 - 2 * q^50 + 4 * q^52 + 20 * q^54 - 4 * q^55 - 2 * q^56 + 28 * q^57 + 2 * q^58 - 12 * q^59 - 8 * q^60 + 2 * q^61 - 12 * q^62 + 6 * q^63 + 2 * q^64 - 24 * q^65 - 16 * q^66 - 12 * q^67 + 16 * q^69 + 2 * q^70 - 4 * q^71 - 6 * q^72 - 2 * q^74 + 18 * q^75 - 8 * q^76 - 6 * q^77 - 16 * q^78 + 12 * q^79 - 2 * q^80 + 22 * q^81 - 16 * q^82 + 4 * q^83 - 2 * q^84 + 4 * q^86 - 28 * q^87 + 6 * q^88 + 4 * q^89 - 14 * q^90 + 4 * q^91 - 16 * q^92 + 8 * q^93 - 4 * q^94 - 12 * q^95 + 2 * q^96 + 8 * q^97 - 2 * q^98 - 38 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−3.23607	−1.86834	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−3.23607	−1.86834	−0.934172	+	0.356822i	$0.883860\pi$
$4$	1.00000	0.500000
$5$	1.23607	0.552786	0.276393	−	0.961045i	$-0.410861\pi$
$5$	1.23607	0.552786	0.276393	+	0.961045i	$0.410861\pi$
$6$	3.23607	1.32112
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−1.00000	−0.353553
$9$	7.47214	2.49071
$10$	−1.23607	−0.390879
$11$	−5.23607	−1.57873	−0.789367	−	0.613922i	$-0.789591\pi$
$11$	−5.23607	−1.57873	−0.789367	+	0.613922i	$0.789591\pi$
$12$	−3.23607	−0.934172
$13$	−2.47214	−0.685647	−0.342824	−	0.939400i	$-0.611383\pi$
$13$	−2.47214	−0.685647	−0.342824	+	0.939400i	$0.611383\pi$
$14$	−1.00000	−0.267261
$15$	−4.00000	−1.03280
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−7.47214	−1.76120
$19$	−8.47214	−1.94364	−0.971821	−	0.235722i	$-0.924255\pi$
$19$	−8.47214	−1.94364	−0.971821	+	0.235722i	$0.924255\pi$
$20$	1.23607	0.276393
$21$	−3.23607	−0.706168
$22$	5.23607	1.11633
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	3.23607	0.660560
$25$	−3.47214	−0.694427
$26$	2.47214	0.484826
$27$	−14.4721	−2.78516
$28$	1.00000	0.188982
$29$	5.70820	1.05999	0.529993	−	0.848002i	$-0.322194\pi$
$29$	5.70820	1.05999	0.529993	+	0.848002i	$0.322194\pi$
$30$	4.00000	0.730297
$31$	1.52786	0.274412	0.137206	−	0.990543i	$-0.456188\pi$
$31$	1.52786	0.274412	0.137206	+	0.990543i	$0.456188\pi$
$32$	−1.00000	−0.176777
$33$	16.9443	2.94962
$34$	0	0
$35$	1.23607	0.208934
$36$	7.47214	1.24536
$37$	3.23607	0.532006	0.266003	−	0.963972i	$-0.414297\pi$
$37$	3.23607	0.532006	0.266003	+	0.963972i	$0.414297\pi$
$38$	8.47214	1.37436
$39$	8.00000	1.28103
$40$	−1.23607	−0.195440
$41$	3.52786	0.550960	0.275480	−	0.961307i	$-0.411163\pi$
$41$	3.52786	0.550960	0.275480	+	0.961307i	$0.411163\pi$
$42$	3.23607	0.499336
$43$	2.47214	0.376997	0.188499	−	0.982073i	$-0.439638\pi$
$43$	2.47214	0.376997	0.188499	+	0.982073i	$0.439638\pi$
$44$	−5.23607	−0.789367
$45$	9.23607	1.37683
$46$	8.00000	1.17954
$47$	−2.47214	−0.360598	−0.180299	−	0.983612i	$-0.557707\pi$
$47$	−2.47214	−0.360598	−0.180299	+	0.983612i	$0.557707\pi$
$48$	−3.23607	−0.467086
$49$	1.00000	0.142857
$50$	3.47214	0.491034
$51$	0	0
$52$	−2.47214	−0.342824
$53$	−4.47214	−0.614295	−0.307148	−	0.951662i	$-0.599375\pi$
$53$	−4.47214	−0.614295	−0.307148	+	0.951662i	$0.599375\pi$
$54$	14.4721	1.96941
$55$	−6.47214	−0.872703
$56$	−1.00000	−0.133631
$57$	27.4164	3.63139
$58$	−5.70820	−0.749524
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	−4.00000	−0.516398
$61$	−1.23607	−0.158262	−0.0791311	−	0.996864i	$-0.525215\pi$
$61$	−1.23607	−0.158262	−0.0791311	+	0.996864i	$0.525215\pi$
$62$	−1.52786	−0.194039
$63$	7.47214	0.941401
$64$	1.00000	0.125000
$65$	−3.05573	−0.379016
$66$	−16.9443	−2.08570
$67$	−1.52786	−0.186658	−0.0933292	−	0.995635i	$-0.529751\pi$
$67$	−1.52786	−0.186658	−0.0933292	+	0.995635i	$0.529751\pi$
$68$	0	0
$69$	25.8885	3.11661
$70$	−1.23607	−0.147738
$71$	2.47214	0.293389	0.146694	−	0.989182i	$-0.453137\pi$
$71$	2.47214	0.293389	0.146694	+	0.989182i	$0.453137\pi$
$72$	−7.47214	−0.880600
$73$	−13.4164	−1.57027	−0.785136	−	0.619324i	$-0.787407\pi$
$73$	−13.4164	−1.57027	−0.785136	+	0.619324i	$0.787407\pi$
$74$	−3.23607	−0.376185
$75$	11.2361	1.29743
$76$	−8.47214	−0.971821
$77$	−5.23607	−0.596705
$78$	−8.00000	−0.905822
$79$	10.4721	1.17821	0.589104	−	0.808057i	$-0.299481\pi$
$79$	10.4721	1.17821	0.589104	+	0.808057i	$0.299481\pi$
$80$	1.23607	0.138197
$81$	24.4164	2.71293
$82$	−3.52786	−0.389587
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	−3.23607	−0.353084
$85$	0	0
$86$	−2.47214	−0.266577
$87$	−18.4721	−1.98042
$88$	5.23607	0.558167
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	−9.23607	−0.973567
$91$	−2.47214	−0.259150
$92$	−8.00000	−0.834058
$93$	−4.94427	−0.512697
$94$	2.47214	0.254981
$95$	−10.4721	−1.07442
$96$	3.23607	0.330280
$97$	8.47214	0.860215	0.430108	−	0.902778i	$-0.358476\pi$
$97$	8.47214	0.860215	0.430108	+	0.902778i	$0.358476\pi$
$98$	−1.00000	−0.101015
$99$	−39.1246	−3.93217
$100$	−3.47214	−0.347214
$101$	−6.47214	−0.644002	−0.322001	−	0.946739i	$-0.604355\pi$
$101$	−6.47214	−0.644002	−0.322001	+	0.946739i	$0.604355\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	2.47214	0.242413
$105$	−4.00000	−0.390360
$106$	4.47214	0.434372
$107$	1.23607	0.119495	0.0597476	−	0.998214i	$-0.480970\pi$
$107$	1.23607	0.119495	0.0597476	+	0.998214i	$0.480970\pi$
$108$	−14.4721	−1.39258
$109$	−12.1803	−1.16666	−0.583332	−	0.812233i	$-0.698252\pi$
$109$	−12.1803	−1.16666	−0.583332	+	0.812233i	$0.698252\pi$
$110$	6.47214	0.617094
$111$	−10.4721	−0.993971
$112$	1.00000	0.0944911
$113$	−13.4164	−1.26211	−0.631055	−	0.775738i	$-0.717378\pi$
$113$	−13.4164	−1.26211	−0.631055	+	0.775738i	$0.717378\pi$
$114$	−27.4164	−2.56778
$115$	−9.88854	−0.922111
$116$	5.70820	0.529993
$117$	−18.4721	−1.70775
$118$	6.00000	0.552345
$119$	0	0
$120$	4.00000	0.365148
$121$	16.4164	1.49240
$122$	1.23607	0.111908
$123$	−11.4164	−1.02938
$124$	1.52786	0.137206
$125$	−10.4721	−0.936656
$126$	−7.47214	−0.665671
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	−1.00000	−0.0883883
$129$	−8.00000	−0.704361
$130$	3.05573	0.268005
$131$	−1.70820	−0.149246	−0.0746232	−	0.997212i	$-0.523775\pi$
$131$	−1.70820	−0.149246	−0.0746232	+	0.997212i	$0.523775\pi$
$132$	16.9443	1.47481
$133$	−8.47214	−0.734627
$134$	1.52786	0.131987
$135$	−17.8885	−1.53960
$136$	0	0
$137$	3.52786	0.301406	0.150703	−	0.988579i	$-0.451846\pi$
$137$	3.52786	0.301406	0.150703	+	0.988579i	$0.451846\pi$
$138$	−25.8885	−2.20378
$139$	12.1803	1.03312	0.516561	−	0.856250i	$-0.327212\pi$
$139$	12.1803	1.03312	0.516561	+	0.856250i	$0.327212\pi$
$140$	1.23607	0.104467
$141$	8.00000	0.673722
$142$	−2.47214	−0.207457
$143$	12.9443	1.08245
$144$	7.47214	0.622678
$145$	7.05573	0.585946
$146$	13.4164	1.11035
$147$	−3.23607	−0.266906
$148$	3.23607	0.266003
$149$	18.9443	1.55198	0.775988	−	0.630748i	$-0.217252\pi$
$149$	18.9443	1.55198	0.775988	+	0.630748i	$0.217252\pi$
$150$	−11.2361	−0.917421
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	8.47214	0.687181
$153$	0	0
$154$	5.23607	0.421934
$155$	1.88854	0.151691
$156$	8.00000	0.640513
$157$	20.9443	1.67153	0.835767	−	0.549084i	$-0.185023\pi$
$157$	20.9443	1.67153	0.835767	+	0.549084i	$0.185023\pi$
$158$	−10.4721	−0.833118
$159$	14.4721	1.14772
$160$	−1.23607	−0.0977198
$161$	−8.00000	−0.630488
$162$	−24.4164	−1.91833
$163$	0.291796	0.0228552	0.0114276	−	0.999935i	$-0.496362\pi$
$163$	0.291796	0.0228552	0.0114276	+	0.999935i	$0.496362\pi$
$164$	3.52786	0.275480
$165$	20.9443	1.63051
$166$	−2.00000	−0.155230
$167$	14.4721	1.11989	0.559944	−	0.828531i	$-0.310823\pi$
$167$	14.4721	1.11989	0.559944	+	0.828531i	$0.310823\pi$
$168$	3.23607	0.249668
$169$	−6.88854	−0.529888
$170$	0	0
$171$	−63.3050	−4.84105
$172$	2.47214	0.188499
$173$	−20.6525	−1.57018	−0.785089	−	0.619383i	$-0.787383\pi$
$173$	−20.6525	−1.57018	−0.785089	+	0.619383i	$0.787383\pi$
$174$	18.4721	1.40037
$175$	−3.47214	−0.262469
$176$	−5.23607	−0.394683
$177$	19.4164	1.45943
$178$	−2.00000	−0.149906
$179$	−8.94427	−0.668526	−0.334263	−	0.942480i	$-0.608487\pi$
$179$	−8.94427	−0.668526	−0.334263	+	0.942480i	$0.608487\pi$
$180$	9.23607	0.688416
$181$	2.76393	0.205441	0.102721	−	0.994710i	$-0.467245\pi$
$181$	2.76393	0.205441	0.102721	+	0.994710i	$0.467245\pi$
$182$	2.47214	0.183247
$183$	4.00000	0.295689
$184$	8.00000	0.589768
$185$	4.00000	0.294086
$186$	4.94427	0.362532
$187$	0	0
$188$	−2.47214	−0.180299
$189$	−14.4721	−1.05269
$190$	10.4721	0.759729
$191$	12.9443	0.936615	0.468307	−	0.883566i	$-0.344864\pi$
$191$	12.9443	0.936615	0.468307	+	0.883566i	$0.344864\pi$
$192$	−3.23607	−0.233543
$193$	11.8885	0.855756	0.427878	−	0.903836i	$-0.359261\pi$
$193$	11.8885	0.855756	0.427878	+	0.903836i	$0.359261\pi$
$194$	−8.47214	−0.608264
$195$	9.88854	0.708133
$196$	1.00000	0.0714286
$197$	−21.7082	−1.54665	−0.773323	−	0.634013i	$-0.781407\pi$
$197$	−21.7082	−1.54665	−0.773323	+	0.634013i	$0.781407\pi$
$198$	39.1246	2.78047
$199$	−14.4721	−1.02590	−0.512951	−	0.858418i	$-0.671448\pi$
$199$	−14.4721	−1.02590	−0.512951	+	0.858418i	$0.671448\pi$
$200$	3.47214	0.245517
$201$	4.94427	0.348742
$202$	6.47214	0.455378
$203$	5.70820	0.400637
$204$	0	0
$205$	4.36068	0.304563
$206$	4.00000	0.278693
$207$	−59.7771	−4.15479
$208$	−2.47214	−0.171412
$209$	44.3607	3.06849
$210$	4.00000	0.276026
$211$	27.7082	1.90751	0.953756	−	0.300583i	$-0.0971812\pi$
$211$	27.7082	1.90751	0.953756	+	0.300583i	$0.0971812\pi$
$212$	−4.47214	−0.307148
$213$	−8.00000	−0.548151
$214$	−1.23607	−0.0844959
$215$	3.05573	0.208399
$216$	14.4721	0.984704
$217$	1.52786	0.103718
$218$	12.1803	0.824957
$219$	43.4164	2.93381
$220$	−6.47214	−0.436351
$221$	0	0
$222$	10.4721	0.702844
$223$	1.52786	0.102313	0.0511567	−	0.998691i	$-0.483709\pi$
$223$	1.52786	0.102313	0.0511567	+	0.998691i	$0.483709\pi$
$224$	−1.00000	−0.0668153
$225$	−25.9443	−1.72962
$226$	13.4164	0.892446
$227$	−27.5967	−1.83166	−0.915830	−	0.401566i	$-0.868466\pi$
$227$	−27.5967	−1.83166	−0.915830	+	0.401566i	$0.868466\pi$
$228$	27.4164	1.81570
$229$	−4.94427	−0.326727	−0.163363	−	0.986566i	$-0.552234\pi$
$229$	−4.94427	−0.326727	−0.163363	+	0.986566i	$0.552234\pi$
$230$	9.88854	0.652031
$231$	16.9443	1.11485
$232$	−5.70820	−0.374762
$233$	−19.8885	−1.30294	−0.651471	−	0.758674i	$-0.725848\pi$
$233$	−19.8885	−1.30294	−0.651471	+	0.758674i	$0.725848\pi$
$234$	18.4721	1.20756
$235$	−3.05573	−0.199334
$236$	−6.00000	−0.390567
$237$	−33.8885	−2.20130
$238$	0	0
$239$	−4.94427	−0.319818	−0.159909	−	0.987132i	$-0.551120\pi$
$239$	−4.94427	−0.319818	−0.159909	+	0.987132i	$0.551120\pi$
$240$	−4.00000	−0.258199
$241$	−11.8885	−0.765808	−0.382904	−	0.923788i	$-0.625076\pi$
$241$	−11.8885	−0.765808	−0.382904	+	0.923788i	$0.625076\pi$
$242$	−16.4164	−1.05529
$243$	−35.5967	−2.28353
$244$	−1.23607	−0.0791311
$245$	1.23607	0.0789695
$246$	11.4164	0.727884
$247$	20.9443	1.33265
$248$	−1.52786	−0.0970195
$249$	−6.47214	−0.410155
$250$	10.4721	0.662316
$251$	14.0000	0.883672	0.441836	−	0.897096i	$-0.354327\pi$
$251$	14.0000	0.883672	0.441836	+	0.897096i	$0.354327\pi$
$252$	7.47214	0.470700
$253$	41.8885	2.63351
$254$	−12.0000	−0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	8.00000	0.498058
$259$	3.23607	0.201079
$260$	−3.05573	−0.189508
$261$	42.6525	2.64012
$262$	1.70820	0.105533
$263$	7.05573	0.435075	0.217537	−	0.976052i	$-0.430198\pi$
$263$	7.05573	0.435075	0.217537	+	0.976052i	$0.430198\pi$
$264$	−16.9443	−1.04285
$265$	−5.52786	−0.339574
$266$	8.47214	0.519460
$267$	−6.47214	−0.396088
$268$	−1.52786	−0.0933292
$269$	0.291796	0.0177911	0.00889556	−	0.999960i	$-0.497168\pi$
$269$	0.291796	0.0177911	0.00889556	+	0.999960i	$0.497168\pi$
$270$	17.8885	1.08866
$271$	19.4164	1.17946	0.589731	−	0.807599i	$-0.299234\pi$
$271$	19.4164	1.17946	0.589731	+	0.807599i	$0.299234\pi$
$272$	0	0
$273$	8.00000	0.484182
$274$	−3.52786	−0.213126
$275$	18.1803	1.09632
$276$	25.8885	1.55831
$277$	22.6525	1.36106	0.680528	−	0.732722i	$-0.261751\pi$
$277$	22.6525	1.36106	0.680528	+	0.732722i	$0.261751\pi$
$278$	−12.1803	−0.730528
$279$	11.4164	0.683482
$280$	−1.23607	−0.0738692
$281$	3.52786	0.210455	0.105227	−	0.994448i	$-0.466443\pi$
$281$	3.52786	0.210455	0.105227	+	0.994448i	$0.466443\pi$
$282$	−8.00000	−0.476393
$283$	−10.2918	−0.611784	−0.305892	−	0.952066i	$-0.598955\pi$
$283$	−10.2918	−0.611784	−0.305892	+	0.952066i	$0.598955\pi$
$284$	2.47214	0.146694
$285$	33.8885	2.00738
$286$	−12.9443	−0.765411
$287$	3.52786	0.208243
$288$	−7.47214	−0.440300
$289$	0	0
$290$	−7.05573	−0.414327
$291$	−27.4164	−1.60718
$292$	−13.4164	−0.785136
$293$	−28.0000	−1.63578	−0.817889	−	0.575376i	$-0.804856\pi$
$293$	−28.0000	−1.63578	−0.817889	+	0.575376i	$0.804856\pi$
$294$	3.23607	0.188731
$295$	−7.41641	−0.431800
$296$	−3.23607	−0.188093
$297$	75.7771	4.39703
$298$	−18.9443	−1.09741
$299$	19.7771	1.14374
$300$	11.2361	0.648715
$301$	2.47214	0.142492
$302$	−4.00000	−0.230174
$303$	20.9443	1.20322
$304$	−8.47214	−0.485910
$305$	−1.52786	−0.0874852
$306$	0	0
$307$	−3.52786	−0.201346	−0.100673	−	0.994920i	$-0.532100\pi$
$307$	−3.52786	−0.201346	−0.100673	+	0.994920i	$0.532100\pi$
$308$	−5.23607	−0.298353
$309$	12.9443	0.736374
$310$	−1.88854	−0.107262
$311$	20.9443	1.18764	0.593820	−	0.804598i	$-0.297619\pi$
$311$	20.9443	1.18764	0.593820	+	0.804598i	$0.297619\pi$
$312$	−8.00000	−0.452911
$313$	5.41641	0.306153	0.153077	−	0.988214i	$-0.451082\pi$
$313$	5.41641	0.306153	0.153077	+	0.988214i	$0.451082\pi$
$314$	−20.9443	−1.18195
$315$	9.23607	0.520393
$316$	10.4721	0.589104
$317$	28.1803	1.58277	0.791383	−	0.611321i	$-0.209362\pi$
$317$	28.1803	1.58277	0.791383	+	0.611321i	$0.209362\pi$
$318$	−14.4721	−0.811557
$319$	−29.8885	−1.67344
$320$	1.23607	0.0690983
$321$	−4.00000	−0.223258
$322$	8.00000	0.445823
$323$	0	0
$324$	24.4164	1.35647
$325$	8.58359	0.476132
$326$	−0.291796	−0.0161611
$327$	39.4164	2.17973
$328$	−3.52786	−0.194794
$329$	−2.47214	−0.136293
$330$	−20.9443	−1.15294
$331$	−20.3607	−1.11912	−0.559562	−	0.828788i	$-0.689031\pi$
$331$	−20.3607	−1.11912	−0.559562	+	0.828788i	$0.689031\pi$
$332$	2.00000	0.109764
$333$	24.1803	1.32507
$334$	−14.4721	−0.791880
$335$	−1.88854	−0.103182
$336$	−3.23607	−0.176542
$337$	20.4721	1.11519	0.557594	−	0.830114i	$-0.311725\pi$
$337$	20.4721	1.11519	0.557594	+	0.830114i	$0.311725\pi$
$338$	6.88854	0.374687
$339$	43.4164	2.35806
$340$	0	0
$341$	−8.00000	−0.433224
$342$	63.3050	3.42314
$343$	1.00000	0.0539949
$344$	−2.47214	−0.133289
$345$	32.0000	1.72282
$346$	20.6525	1.11028
$347$	−4.65248	−0.249758	−0.124879	−	0.992172i	$-0.539854\pi$
$347$	−4.65248	−0.249758	−0.124879	+	0.992172i	$0.539854\pi$
$348$	−18.4721	−0.990210
$349$	−21.5279	−1.15236	−0.576180	−	0.817323i	$-0.695457\pi$
$349$	−21.5279	−1.15236	−0.576180	+	0.817323i	$0.695457\pi$
$350$	3.47214	0.185593
$351$	35.7771	1.90964
$352$	5.23607	0.279083
$353$	−11.8885	−0.632763	−0.316382	−	0.948632i	$-0.602468\pi$
$353$	−11.8885	−0.632763	−0.316382	+	0.948632i	$0.602468\pi$
$354$	−19.4164	−1.03197
$355$	3.05573	0.162181
$356$	2.00000	0.106000
$357$	0	0
$358$	8.94427	0.472719
$359$	22.8328	1.20507	0.602535	−	0.798092i	$-0.294157\pi$
$359$	22.8328	1.20507	0.602535	+	0.798092i	$0.294157\pi$
$360$	−9.23607	−0.486784
$361$	52.7771	2.77774
$362$	−2.76393	−0.145269
$363$	−53.1246	−2.78832
$364$	−2.47214	−0.129575
$365$	−16.5836	−0.868025
$366$	−4.00000	−0.209083
$367$	28.9443	1.51088	0.755439	−	0.655219i	$-0.227423\pi$
$367$	28.9443	1.51088	0.755439	+	0.655219i	$0.227423\pi$
$368$	−8.00000	−0.417029
$369$	26.3607	1.37228
$370$	−4.00000	−0.207950
$371$	−4.47214	−0.232182
$372$	−4.94427	−0.256349
$373$	13.4164	0.694675	0.347338	−	0.937740i	$-0.387086\pi$
$373$	13.4164	0.694675	0.347338	+	0.937740i	$0.387086\pi$
$374$	0	0
$375$	33.8885	1.75000
$376$	2.47214	0.127491
$377$	−14.1115	−0.726777
$378$	14.4721	0.744366
$379$	17.5967	0.903884	0.451942	−	0.892047i	$-0.350731\pi$
$379$	17.5967	0.903884	0.451942	+	0.892047i	$0.350731\pi$
$380$	−10.4721	−0.537209
$381$	−38.8328	−1.98947
$382$	−12.9443	−0.662287
$383$	−8.94427	−0.457031	−0.228515	−	0.973540i	$-0.573387\pi$
$383$	−8.94427	−0.457031	−0.228515	+	0.973540i	$0.573387\pi$
$384$	3.23607	0.165140
$385$	−6.47214	−0.329851
$386$	−11.8885	−0.605111
$387$	18.4721	0.938991
$388$	8.47214	0.430108
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	−9.88854	−0.500726
$391$	0	0
$392$	−1.00000	−0.0505076
$393$	5.52786	0.278844
$394$	21.7082	1.09364
$395$	12.9443	0.651297
$396$	−39.1246	−1.96609
$397$	−18.1803	−0.912445	−0.456223	−	0.889866i	$-0.650798\pi$
$397$	−18.1803	−0.912445	−0.456223	+	0.889866i	$0.650798\pi$
$398$	14.4721	0.725423
$399$	27.4164	1.37254
$400$	−3.47214	−0.173607
$401$	−2.94427	−0.147030	−0.0735150	−	0.997294i	$-0.523422\pi$
$401$	−2.94427	−0.147030	−0.0735150	+	0.997294i	$0.523422\pi$
$402$	−4.94427	−0.246598
$403$	−3.77709	−0.188150
$404$	−6.47214	−0.322001
$405$	30.1803	1.49967
$406$	−5.70820	−0.283293
$407$	−16.9443	−0.839896
$408$	0	0
$409$	14.9443	0.738947	0.369473	−	0.929241i	$-0.379538\pi$
$409$	14.9443	0.738947	0.369473	+	0.929241i	$0.379538\pi$
$410$	−4.36068	−0.215359
$411$	−11.4164	−0.563130
$412$	−4.00000	−0.197066
$413$	−6.00000	−0.295241
$414$	59.7771	2.93788
$415$	2.47214	0.121352
$416$	2.47214	0.121206
$417$	−39.4164	−1.93023
$418$	−44.3607	−2.16975
$419$	30.0689	1.46896	0.734481	−	0.678630i	$-0.237426\pi$
$419$	30.0689	1.46896	0.734481	+	0.678630i	$0.237426\pi$
$420$	−4.00000	−0.195180
$421$	−15.5279	−0.756782	−0.378391	−	0.925646i	$-0.623522\pi$
$421$	−15.5279	−0.756782	−0.378391	+	0.925646i	$0.623522\pi$
$422$	−27.7082	−1.34881
$423$	−18.4721	−0.898146
$424$	4.47214	0.217186
$425$	0	0
$426$	8.00000	0.387601
$427$	−1.23607	−0.0598175
$428$	1.23607	0.0597476
$429$	−41.8885	−2.02240
$430$	−3.05573	−0.147360
$431$	12.9443	0.623504	0.311752	−	0.950164i	$-0.399084\pi$
$431$	12.9443	0.623504	0.311752	+	0.950164i	$0.399084\pi$
$432$	−14.4721	−0.696291
$433$	10.9443	0.525948	0.262974	−	0.964803i	$-0.415297\pi$
$433$	10.9443	0.525948	0.262974	+	0.964803i	$0.415297\pi$
$434$	−1.52786	−0.0733398
$435$	−22.8328	−1.09475
$436$	−12.1803	−0.583332
$437$	67.7771	3.24222
$438$	−43.4164	−2.07452
$439$	3.05573	0.145842	0.0729210	−	0.997338i	$-0.476768\pi$
$439$	3.05573	0.145842	0.0729210	+	0.997338i	$0.476768\pi$
$440$	6.47214	0.308547
$441$	7.47214	0.355816
$442$	0	0
$443$	−8.36068	−0.397228	−0.198614	−	0.980078i	$-0.563644\pi$
$443$	−8.36068	−0.397228	−0.198614	+	0.980078i	$0.563644\pi$
$444$	−10.4721	−0.496986
$445$	2.47214	0.117190
$446$	−1.52786	−0.0723465
$447$	−61.3050	−2.89962
$448$	1.00000	0.0472456
$449$	−9.41641	−0.444388	−0.222194	−	0.975003i	$-0.571322\pi$
$449$	−9.41641	−0.444388	−0.222194	+	0.975003i	$0.571322\pi$
$450$	25.9443	1.22302
$451$	−18.4721	−0.869819
$452$	−13.4164	−0.631055
$453$	−12.9443	−0.608175
$454$	27.5967	1.29518
$455$	−3.05573	−0.143255
$456$	−27.4164	−1.28389
$457$	15.8885	0.743235	0.371617	−	0.928386i	$-0.378803\pi$
$457$	15.8885	0.743235	0.371617	+	0.928386i	$0.378803\pi$
$458$	4.94427	0.231031
$459$	0	0
$460$	−9.88854	−0.461056
$461$	−5.52786	−0.257458	−0.128729	−	0.991680i	$-0.541090\pi$
$461$	−5.52786	−0.257458	−0.128729	+	0.991680i	$0.541090\pi$
$462$	−16.9443	−0.788319
$463$	32.9443	1.53105	0.765525	−	0.643406i	$-0.222479\pi$
$463$	32.9443	1.53105	0.765525	+	0.643406i	$0.222479\pi$
$464$	5.70820	0.264997
$465$	−6.11146	−0.283412
$466$	19.8885	0.921319
$467$	−10.3607	−0.479435	−0.239718	−	0.970843i	$-0.577055\pi$
$467$	−10.3607	−0.479435	−0.239718	+	0.970843i	$0.577055\pi$
$468$	−18.4721	−0.853875
$469$	−1.52786	−0.0705502
$470$	3.05573	0.140950
$471$	−67.7771	−3.12300
$472$	6.00000	0.276172
$473$	−12.9443	−0.595178
$474$	33.8885	1.55655
$475$	29.4164	1.34972
$476$	0	0
$477$	−33.4164	−1.53003
$478$	4.94427	0.226146
$479$	19.4164	0.887158	0.443579	−	0.896235i	$-0.353709\pi$
$479$	19.4164	0.887158	0.443579	+	0.896235i	$0.353709\pi$
$480$	4.00000	0.182574
$481$	−8.00000	−0.364769
$482$	11.8885	0.541508
$483$	25.8885	1.17797
$484$	16.4164	0.746200
$485$	10.4721	0.475515
$486$	35.5967	1.61470
$487$	−3.05573	−0.138468	−0.0692341	−	0.997600i	$-0.522056\pi$
$487$	−3.05573	−0.138468	−0.0692341	+	0.997600i	$0.522056\pi$
$488$	1.23607	0.0559542
$489$	−0.944272	−0.0427015
$490$	−1.23607	−0.0558399
$491$	−32.9443	−1.48675	−0.743377	−	0.668873i	$-0.766777\pi$
$491$	−32.9443	−1.48675	−0.743377	+	0.668873i	$0.766777\pi$
$492$	−11.4164	−0.514691
$493$	0	0
$494$	−20.9443	−0.942327
$495$	−48.3607	−2.17365
$496$	1.52786	0.0686031
$497$	2.47214	0.110890
$498$	6.47214	0.290023
$499$	40.6525	1.81985	0.909927	−	0.414768i	$-0.136137\pi$
$499$	40.6525	1.81985	0.909927	+	0.414768i	$0.136137\pi$
$500$	−10.4721	−0.468328
$501$	−46.8328	−2.09234
$502$	−14.0000	−0.624851
$503$	−12.9443	−0.577157	−0.288578	−	0.957456i	$-0.593183\pi$
$503$	−12.9443	−0.577157	−0.288578	+	0.957456i	$0.593183\pi$
$504$	−7.47214	−0.332835
$505$	−8.00000	−0.355995
$506$	−41.8885	−1.86217
$507$	22.2918	0.990013
$508$	12.0000	0.532414
$509$	−3.05573	−0.135443	−0.0677214	−	0.997704i	$-0.521573\pi$
$509$	−3.05573	−0.135443	−0.0677214	+	0.997704i	$0.521573\pi$
$510$	0	0
$511$	−13.4164	−0.593507
$512$	−1.00000	−0.0441942
$513$	122.610	5.41336
$514$	2.00000	0.0882162
$515$	−4.94427	−0.217871
$516$	−8.00000	−0.352180
$517$	12.9443	0.569288
$518$	−3.23607	−0.142185
$519$	66.8328	2.93364
$520$	3.05573	0.134003
$521$	1.05573	0.0462523	0.0231261	−	0.999733i	$-0.492638\pi$
$521$	1.05573	0.0462523	0.0231261	+	0.999733i	$0.492638\pi$
$522$	−42.6525	−1.86685
$523$	−42.9443	−1.87782	−0.938911	−	0.344160i	$-0.888164\pi$
$523$	−42.9443	−1.87782	−0.938911	+	0.344160i	$0.888164\pi$
$524$	−1.70820	−0.0746232
$525$	11.2361	0.490382
$526$	−7.05573	−0.307644
$527$	0	0
$528$	16.9443	0.737405
$529$	41.0000	1.78261
$530$	5.52786	0.240115
$531$	−44.8328	−1.94558
$532$	−8.47214	−0.367314
$533$	−8.72136	−0.377764
$534$	6.47214	0.280077
$535$	1.52786	0.0660553
$536$	1.52786	0.0659937
$537$	28.9443	1.24904
$538$	−0.291796	−0.0125802
$539$	−5.23607	−0.225533
$540$	−17.8885	−0.769800
$541$	23.2361	0.998997	0.499498	−	0.866315i	$-0.333518\pi$
$541$	23.2361	0.998997	0.499498	+	0.866315i	$0.333518\pi$
$542$	−19.4164	−0.834006
$543$	−8.94427	−0.383835
$544$	0	0
$545$	−15.0557	−0.644917
$546$	−8.00000	−0.342368
$547$	−41.5967	−1.77855	−0.889274	−	0.457374i	$-0.848790\pi$
$547$	−41.5967	−1.77855	−0.889274	+	0.457374i	$0.848790\pi$
$548$	3.52786	0.150703
$549$	−9.23607	−0.394186
$550$	−18.1803	−0.775212
$551$	−48.3607	−2.06023
$552$	−25.8885	−1.10189
$553$	10.4721	0.445321
$554$	−22.6525	−0.962411
$555$	−12.9443	−0.549454
$556$	12.1803	0.516561
$557$	34.3607	1.45591	0.727954	−	0.685626i	$-0.240471\pi$
$557$	34.3607	1.45591	0.727954	+	0.685626i	$0.240471\pi$
$558$	−11.4164	−0.483295
$559$	−6.11146	−0.258487
$560$	1.23607	0.0522334
$561$	0	0
$562$	−3.52786	−0.148814
$563$	−32.8328	−1.38374	−0.691869	−	0.722023i	$-0.743212\pi$
$563$	−32.8328	−1.38374	−0.691869	+	0.722023i	$0.743212\pi$
$564$	8.00000	0.336861
$565$	−16.5836	−0.697677
$566$	10.2918	0.432596
$567$	24.4164	1.02539
$568$	−2.47214	−0.103729
$569$	15.8885	0.666082	0.333041	−	0.942912i	$-0.391925\pi$
$569$	15.8885	0.666082	0.333041	+	0.942912i	$0.391925\pi$
$570$	−33.8885	−1.41943
$571$	13.2361	0.553912	0.276956	−	0.960883i	$-0.410674\pi$
$571$	13.2361	0.553912	0.276956	+	0.960883i	$0.410674\pi$
$572$	12.9443	0.541227
$573$	−41.8885	−1.74992
$574$	−3.52786	−0.147250
$575$	27.7771	1.15838
$576$	7.47214	0.311339
$577$	30.0000	1.24892	0.624458	−	0.781058i	$-0.285320\pi$
$577$	30.0000	1.24892	0.624458	+	0.781058i	$0.285320\pi$
$578$	0	0
$579$	−38.4721	−1.59885
$580$	7.05573	0.292973
$581$	2.00000	0.0829740
$582$	27.4164	1.13645
$583$	23.4164	0.969809
$584$	13.4164	0.555175
$585$	−22.8328	−0.944021
$586$	28.0000	1.15667
$587$	21.4164	0.883950	0.441975	−	0.897027i	$-0.354278\pi$
$587$	21.4164	0.883950	0.441975	+	0.897027i	$0.354278\pi$
$588$	−3.23607	−0.133453
$589$	−12.9443	−0.533359
$590$	7.41641	0.305329
$591$	70.2492	2.88967
$592$	3.23607	0.133002
$593$	−2.94427	−0.120907	−0.0604534	−	0.998171i	$-0.519255\pi$
$593$	−2.94427	−0.120907	−0.0604534	+	0.998171i	$0.519255\pi$
$594$	−75.7771	−3.10917
$595$	0	0
$596$	18.9443	0.775988
$597$	46.8328	1.91674
$598$	−19.7771	−0.808745
$599$	31.7771	1.29838	0.649188	−	0.760628i	$-0.275109\pi$
$599$	31.7771	1.29838	0.649188	+	0.760628i	$0.275109\pi$
$600$	−11.2361	−0.458711
$601$	−14.0000	−0.571072	−0.285536	−	0.958368i	$-0.592172\pi$
$601$	−14.0000	−0.571072	−0.285536	+	0.958368i	$0.592172\pi$
$602$	−2.47214	−0.100757
$603$	−11.4164	−0.464912
$604$	4.00000	0.162758
$605$	20.2918	0.824979
$606$	−20.9443	−0.850803
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	8.47214	0.343590
$609$	−18.4721	−0.748529
$610$	1.52786	0.0618614
$611$	6.11146	0.247243
$612$	0	0
$613$	−44.8328	−1.81078	−0.905390	−	0.424581i	$-0.860422\pi$
$613$	−44.8328	−1.81078	−0.905390	+	0.424581i	$0.860422\pi$
$614$	3.52786	0.142373
$615$	−14.1115	−0.569029
$616$	5.23607	0.210967
$617$	−10.3607	−0.417105	−0.208553	−	0.978011i	$-0.566875\pi$
$617$	−10.3607	−0.417105	−0.208553	+	0.978011i	$0.566875\pi$
$618$	−12.9443	−0.520695
$619$	−25.7082	−1.03330	−0.516650	−	0.856197i	$-0.672821\pi$
$619$	−25.7082	−1.03330	−0.516650	+	0.856197i	$0.672821\pi$
$620$	1.88854	0.0758457
$621$	115.777	4.64597
$622$	−20.9443	−0.839789
$623$	2.00000	0.0801283
$624$	8.00000	0.320256
$625$	4.41641	0.176656
$626$	−5.41641	−0.216483
$627$	−143.554	−5.73300
$628$	20.9443	0.835767
$629$	0	0
$630$	−9.23607	−0.367974
$631$	3.05573	0.121647	0.0608233	−	0.998149i	$-0.480627\pi$
$631$	3.05573	0.121647	0.0608233	+	0.998149i	$0.480627\pi$
$632$	−10.4721	−0.416559
$633$	−89.6656	−3.56389
$634$	−28.1803	−1.11918
$635$	14.8328	0.588622
$636$	14.4721	0.573858
$637$	−2.47214	−0.0979496
$638$	29.8885	1.18330
$639$	18.4721	0.730746
$640$	−1.23607	−0.0488599
$641$	−9.05573	−0.357680	−0.178840	−	0.983878i	$-0.557234\pi$
$641$	−9.05573	−0.357680	−0.178840	+	0.983878i	$0.557234\pi$
$642$	4.00000	0.157867
$643$	−13.1246	−0.517584	−0.258792	−	0.965933i	$-0.583324\pi$
$643$	−13.1246	−0.517584	−0.258792	+	0.965933i	$0.583324\pi$
$644$	−8.00000	−0.315244
$645$	−9.88854	−0.389361
$646$	0	0
$647$	20.3607	0.800461	0.400230	−	0.916415i	$-0.368930\pi$
$647$	20.3607	0.800461	0.400230	+	0.916415i	$0.368930\pi$
$648$	−24.4164	−0.959167
$649$	31.4164	1.23320
$650$	−8.58359	−0.336676
$651$	−4.94427	−0.193781
$652$	0.291796	0.0114276
$653$	7.59675	0.297284	0.148642	−	0.988891i	$-0.452510\pi$
$653$	7.59675	0.297284	0.148642	+	0.988891i	$0.452510\pi$
$654$	−39.4164	−1.54130
$655$	−2.11146	−0.0825014
$656$	3.52786	0.137740
$657$	−100.249	−3.91109
$658$	2.47214	0.0963739
$659$	−20.9443	−0.815873	−0.407936	−	0.913010i	$-0.633752\pi$
$659$	−20.9443	−0.815873	−0.407936	+	0.913010i	$0.633752\pi$
$660$	20.9443	0.815255
$661$	−3.41641	−0.132883	−0.0664414	−	0.997790i	$-0.521165\pi$
$661$	−3.41641	−0.132883	−0.0664414	+	0.997790i	$0.521165\pi$
$662$	20.3607	0.791340
$663$	0	0
$664$	−2.00000	−0.0776151
$665$	−10.4721	−0.406092
$666$	−24.1803	−0.936969
$667$	−45.6656	−1.76818
$668$	14.4721	0.559944
$669$	−4.94427	−0.191157
$670$	1.88854	0.0729608
$671$	6.47214	0.249854
$672$	3.23607	0.124834
$673$	23.3050	0.898340	0.449170	−	0.893446i	$-0.351720\pi$
$673$	23.3050	0.898340	0.449170	+	0.893446i	$0.351720\pi$
$674$	−20.4721	−0.788557
$675$	50.2492	1.93409
$676$	−6.88854	−0.264944
$677$	−34.1803	−1.31366	−0.656829	−	0.754040i	$-0.728102\pi$
$677$	−34.1803	−1.31366	−0.656829	+	0.754040i	$0.728102\pi$
$678$	−43.4164	−1.66740
$679$	8.47214	0.325131
$680$	0	0
$681$	89.3050	3.42217
$682$	8.00000	0.306336
$683$	−46.1803	−1.76704	−0.883521	−	0.468392i	$-0.844834\pi$
$683$	−46.1803	−1.76704	−0.883521	+	0.468392i	$0.844834\pi$
$684$	−63.3050	−2.42053
$685$	4.36068	0.166613
$686$	−1.00000	−0.0381802
$687$	16.0000	0.610438
$688$	2.47214	0.0942493
$689$	11.0557	0.421190
$690$	−32.0000	−1.21822
$691$	34.6525	1.31824	0.659121	−	0.752037i	$-0.270928\pi$
$691$	34.6525	1.31824	0.659121	+	0.752037i	$0.270928\pi$
$692$	−20.6525	−0.785089
$693$	−39.1246	−1.48622
$694$	4.65248	0.176606
$695$	15.0557	0.571096
$696$	18.4721	0.700185
$697$	0	0
$698$	21.5279	0.814842
$699$	64.3607	2.43434
$700$	−3.47214	−0.131234
$701$	−36.2492	−1.36911	−0.684557	−	0.728959i	$-0.740004\pi$
$701$	−36.2492	−1.36911	−0.684557	+	0.728959i	$0.740004\pi$
$702$	−35.7771	−1.35032
$703$	−27.4164	−1.03403
$704$	−5.23607	−0.197342
$705$	9.88854	0.372424
$706$	11.8885	0.447431
$707$	−6.47214	−0.243410
$708$	19.4164	0.729713
$709$	23.5967	0.886194	0.443097	−	0.896474i	$-0.353880\pi$
$709$	23.5967	0.886194	0.443097	+	0.896474i	$0.353880\pi$
$710$	−3.05573	−0.114679
$711$	78.2492	2.93458
$712$	−2.00000	−0.0749532
$713$	−12.2229	−0.457752
$714$	0	0
$715$	16.0000	0.598366
$716$	−8.94427	−0.334263
$717$	16.0000	0.597531
$718$	−22.8328	−0.852113
$719$	27.4164	1.02246	0.511230	−	0.859444i	$-0.329190\pi$
$719$	27.4164	1.02246	0.511230	+	0.859444i	$0.329190\pi$
$720$	9.23607	0.344208
$721$	−4.00000	−0.148968
$722$	−52.7771	−1.96416
$723$	38.4721	1.43079
$724$	2.76393	0.102721
$725$	−19.8197	−0.736084
$726$	53.1246	1.97164
$727$	−37.5279	−1.39183	−0.695916	−	0.718123i	$-0.745001\pi$
$727$	−37.5279	−1.39183	−0.695916	+	0.718123i	$0.745001\pi$
$728$	2.47214	0.0916235
$729$	41.9443	1.55349
$730$	16.5836	0.613786
$731$	0	0
$732$	4.00000	0.147844
$733$	−4.58359	−0.169299	−0.0846494	−	0.996411i	$-0.526977\pi$
$733$	−4.58359	−0.169299	−0.0846494	+	0.996411i	$0.526977\pi$
$734$	−28.9443	−1.06835
$735$	−4.00000	−0.147542
$736$	8.00000	0.294884
$737$	8.00000	0.294684
$738$	−26.3607	−0.970350
$739$	46.8328	1.72277	0.861386	−	0.507950i	$-0.169597\pi$
$739$	46.8328	1.72277	0.861386	+	0.507950i	$0.169597\pi$
$740$	4.00000	0.147043
$741$	−67.7771	−2.48985
$742$	4.47214	0.164177
$743$	−29.5279	−1.08327	−0.541636	−	0.840613i	$-0.682195\pi$
$743$	−29.5279	−1.08327	−0.541636	+	0.840613i	$0.682195\pi$
$744$	4.94427	0.181266
$745$	23.4164	0.857911
$746$	−13.4164	−0.491210
$747$	14.9443	0.546782
$748$	0	0
$749$	1.23607	0.0451649
$750$	−33.8885	−1.23743
$751$	−28.9443	−1.05619	−0.528096	−	0.849185i	$-0.677094\pi$
$751$	−28.9443	−1.05619	−0.528096	+	0.849185i	$0.677094\pi$
$752$	−2.47214	−0.0901495
$753$	−45.3050	−1.65100
$754$	14.1115	0.513909
$755$	4.94427	0.179940
$756$	−14.4721	−0.526346
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	−17.5967	−0.639143
$759$	−135.554	−4.92030
$760$	10.4721	0.379864
$761$	3.88854	0.140960	0.0704798	−	0.997513i	$-0.477547\pi$
$761$	3.88854	0.140960	0.0704798	+	0.997513i	$0.477547\pi$
$762$	38.8328	1.40676
$763$	−12.1803	−0.440958
$764$	12.9443	0.468307
$765$	0	0
$766$	8.94427	0.323170
$767$	14.8328	0.535582
$768$	−3.23607	−0.116772
$769$	−4.83282	−0.174276	−0.0871379	−	0.996196i	$-0.527772\pi$
$769$	−4.83282	−0.174276	−0.0871379	+	0.996196i	$0.527772\pi$
$770$	6.47214	0.233240
$771$	6.47214	0.233088
$772$	11.8885	0.427878
$773$	4.00000	0.143870	0.0719350	−	0.997409i	$-0.477083\pi$
$773$	4.00000	0.143870	0.0719350	+	0.997409i	$0.477083\pi$
$774$	−18.4721	−0.663967
$775$	−5.30495	−0.190559
$776$	−8.47214	−0.304132
$777$	−10.4721	−0.375686
$778$	−26.0000	−0.932145
$779$	−29.8885	−1.07087
$780$	9.88854	0.354067
$781$	−12.9443	−0.463182
$782$	0	0
$783$	−82.6099	−2.95224
$784$	1.00000	0.0357143
$785$	25.8885	0.924002
$786$	−5.52786	−0.197172
$787$	−9.70820	−0.346060	−0.173030	−	0.984917i	$-0.555356\pi$
$787$	−9.70820	−0.346060	−0.173030	+	0.984917i	$0.555356\pi$
$788$	−21.7082	−0.773323
$789$	−22.8328	−0.812870
$790$	−12.9443	−0.460537
$791$	−13.4164	−0.477033
$792$	39.1246	1.39023
$793$	3.05573	0.108512
$794$	18.1803	0.645196
$795$	17.8885	0.634441
$796$	−14.4721	−0.512951
$797$	39.4164	1.39620	0.698100	−	0.716000i	$-0.254029\pi$
$797$	39.4164	1.39620	0.698100	+	0.716000i	$0.254029\pi$
$798$	−27.4164	−0.970530
$799$	0	0
$800$	3.47214	0.122759
$801$	14.9443	0.528030
$802$	2.94427	0.103966
$803$	70.2492	2.47904
$804$	4.94427	0.174371
$805$	−9.88854	−0.348525
$806$	3.77709	0.133042
$807$	−0.944272	−0.0332399
$808$	6.47214	0.227689
$809$	−1.05573	−0.0371174	−0.0185587	−	0.999828i	$-0.505908\pi$
$809$	−1.05573	−0.0371174	−0.0185587	+	0.999828i	$0.505908\pi$
$810$	−30.1803	−1.06043
$811$	19.8197	0.695962	0.347981	−	0.937502i	$-0.386867\pi$
$811$	19.8197	0.695962	0.347981	+	0.937502i	$0.386867\pi$
$812$	5.70820	0.200319
$813$	−62.8328	−2.20364
$814$	16.9443	0.593896
$815$	0.360680	0.0126341
$816$	0	0
$817$	−20.9443	−0.732747
$818$	−14.9443	−0.522514
$819$	−18.4721	−0.645469
$820$	4.36068	0.152282
$821$	43.0132	1.50117	0.750585	−	0.660774i	$-0.229772\pi$
$821$	43.0132	1.50117	0.750585	+	0.660774i	$0.229772\pi$
$822$	11.4164	0.398193
$823$	2.47214	0.0861732	0.0430866	−	0.999071i	$-0.486281\pi$
$823$	2.47214	0.0861732	0.0430866	+	0.999071i	$0.486281\pi$
$824$	4.00000	0.139347
$825$	−58.8328	−2.04830
$826$	6.00000	0.208767
$827$	37.5967	1.30737	0.653684	−	0.756768i	$-0.273223\pi$
$827$	37.5967	1.30737	0.653684	+	0.756768i	$0.273223\pi$
$828$	−59.7771	−2.07740
$829$	40.9443	1.42205	0.711027	−	0.703165i	$-0.248231\pi$
$829$	40.9443	1.42205	0.711027	+	0.703165i	$0.248231\pi$
$830$	−2.47214	−0.0858091
$831$	−73.3050	−2.54292
$832$	−2.47214	−0.0857059
$833$	0	0
$834$	39.4164	1.36488
$835$	17.8885	0.619059
$836$	44.3607	1.53425
$837$	−22.1115	−0.764284
$838$	−30.0689	−1.03871
$839$	−7.63932	−0.263739	−0.131869	−	0.991267i	$-0.542098\pi$
$839$	−7.63932	−0.263739	−0.131869	+	0.991267i	$0.542098\pi$
$840$	4.00000	0.138013
$841$	3.58359	0.123572
$842$	15.5279	0.535126
$843$	−11.4164	−0.393202
$844$	27.7082	0.953756
$845$	−8.51471	−0.292915
$846$	18.4721	0.635085
$847$	16.4164	0.564074
$848$	−4.47214	−0.153574
$849$	33.3050	1.14302
$850$	0	0
$851$	−25.8885	−0.887448
$852$	−8.00000	−0.274075
$853$	21.2361	0.727109	0.363555	−	0.931573i	$-0.381563\pi$
$853$	21.2361	0.727109	0.363555	+	0.931573i	$0.381563\pi$
$854$	1.23607	0.0422974
$855$	−78.2492	−2.67607
$856$	−1.23607	−0.0422479
$857$	3.52786	0.120510	0.0602548	−	0.998183i	$-0.480809\pi$
$857$	3.52786	0.120510	0.0602548	+	0.998183i	$0.480809\pi$
$858$	41.8885	1.43005
$859$	−35.8885	−1.22450	−0.612251	−	0.790664i	$-0.709736\pi$
$859$	−35.8885	−1.22450	−0.612251	+	0.790664i	$0.709736\pi$
$860$	3.05573	0.104199
$861$	−11.4164	−0.389070
$862$	−12.9443	−0.440884
$863$	16.9443	0.576790	0.288395	−	0.957512i	$-0.406878\pi$
$863$	16.9443	0.576790	0.288395	+	0.957512i	$0.406878\pi$
$864$	14.4721	0.492352
$865$	−25.5279	−0.867973
$866$	−10.9443	−0.371901
$867$	0	0
$868$	1.52786	0.0518591
$869$	−54.8328	−1.86008
$870$	22.8328	0.774105
$871$	3.77709	0.127982
$872$	12.1803	0.412478
$873$	63.3050	2.14255
$874$	−67.7771	−2.29259
$875$	−10.4721	−0.354023
$876$	43.4164	1.46690
$877$	−9.12461	−0.308116	−0.154058	−	0.988062i	$-0.549234\pi$
$877$	−9.12461	−0.308116	−0.154058	+	0.988062i	$0.549234\pi$
$878$	−3.05573	−0.103126
$879$	90.6099	3.05620
$880$	−6.47214	−0.218176
$881$	46.9443	1.58159	0.790796	−	0.612079i	$-0.209667\pi$
$881$	46.9443	1.58159	0.790796	+	0.612079i	$0.209667\pi$
$882$	−7.47214	−0.251600
$883$	3.41641	0.114971	0.0574856	−	0.998346i	$-0.481692\pi$
$883$	3.41641	0.114971	0.0574856	+	0.998346i	$0.481692\pi$
$884$	0	0
$885$	24.0000	0.806751
$886$	8.36068	0.280883
$887$	24.3607	0.817952	0.408976	−	0.912545i	$-0.365886\pi$
$887$	24.3607	0.817952	0.408976	+	0.912545i	$0.365886\pi$
$888$	10.4721	0.351422
$889$	12.0000	0.402467
$890$	−2.47214	−0.0828662
$891$	−127.846	−4.28300
$892$	1.52786	0.0511567
$893$	20.9443	0.700873
$894$	61.3050	2.05034
$895$	−11.0557	−0.369552
$896$	−1.00000	−0.0334077
$897$	−64.0000	−2.13690
$898$	9.41641	0.314230
$899$	8.72136	0.290874
$900$	−25.9443	−0.864809
$901$	0	0
$902$	18.4721	0.615055
$903$	−8.00000	−0.266223
$904$	13.4164	0.446223
$905$	3.41641	0.113565
$906$	12.9443	0.430045
$907$	6.76393	0.224593	0.112296	−	0.993675i	$-0.464179\pi$
$907$	6.76393	0.224593	0.112296	+	0.993675i	$0.464179\pi$
$908$	−27.5967	−0.915830
$909$	−48.3607	−1.60402
$910$	3.05573	0.101296
$911$	−12.3607	−0.409528	−0.204764	−	0.978811i	$-0.565643\pi$
$911$	−12.3607	−0.409528	−0.204764	+	0.978811i	$0.565643\pi$
$912$	27.4164	0.907848
$913$	−10.4721	−0.346577
$914$	−15.8885	−0.525546
$915$	4.94427	0.163453
$916$	−4.94427	−0.163363
$917$	−1.70820	−0.0564099
$918$	0	0
$919$	−27.0557	−0.892486	−0.446243	−	0.894912i	$-0.647238\pi$
$919$	−27.0557	−0.892486	−0.446243	+	0.894912i	$0.647238\pi$
$920$	9.88854	0.326016
$921$	11.4164	0.376183
$922$	5.52786	0.182051
$923$	−6.11146	−0.201161
$924$	16.9443	0.557426
$925$	−11.2361	−0.369440
$926$	−32.9443	−1.08262
$927$	−29.8885	−0.981669
$928$	−5.70820	−0.187381
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	6.11146	0.200403
$931$	−8.47214	−0.277663
$932$	−19.8885	−0.651471
$933$	−67.7771	−2.21892
$934$	10.3607	0.339012
$935$	0	0
$936$	18.4721	0.603781
$937$	20.1115	0.657013	0.328506	−	0.944502i	$-0.393455\pi$
$937$	20.1115	0.657013	0.328506	+	0.944502i	$0.393455\pi$
$938$	1.52786	0.0498865
$939$	−17.5279	−0.572000
$940$	−3.05573	−0.0996669
$941$	−59.1246	−1.92741	−0.963704	−	0.266974i	$-0.913976\pi$
$941$	−59.1246	−1.92741	−0.963704	+	0.266974i	$0.913976\pi$
$942$	67.7771	2.20830
$943$	−28.2229	−0.919064
$944$	−6.00000	−0.195283
$945$	−17.8885	−0.581914
$946$	12.9443	0.420855
$947$	47.1246	1.53134	0.765672	−	0.643231i	$-0.222407\pi$
$947$	47.1246	1.53134	0.765672	+	0.643231i	$0.222407\pi$
$948$	−33.8885	−1.10065
$949$	33.1672	1.07665
$950$	−29.4164	−0.954394
$951$	−91.1935	−2.95715
$952$	0	0
$953$	5.05573	0.163771	0.0818855	−	0.996642i	$-0.473906\pi$
$953$	5.05573	0.163771	0.0818855	+	0.996642i	$0.473906\pi$
$954$	33.4164	1.08190
$955$	16.0000	0.517748
$956$	−4.94427	−0.159909
$957$	96.7214	3.12656
$958$	−19.4164	−0.627316
$959$	3.52786	0.113921
$960$	−4.00000	−0.129099
$961$	−28.6656	−0.924698
$962$	8.00000	0.257930
$963$	9.23607	0.297628
$964$	−11.8885	−0.382904
$965$	14.6950	0.473050
$966$	−25.8885	−0.832950
$967$	44.9443	1.44531	0.722655	−	0.691209i	$-0.242921\pi$
$967$	44.9443	1.44531	0.722655	+	0.691209i	$0.242921\pi$
$968$	−16.4164	−0.527643
$969$	0	0
$970$	−10.4721	−0.336240
$971$	24.8328	0.796923	0.398461	−	0.917185i	$-0.369544\pi$
$971$	24.8328	0.796923	0.398461	+	0.917185i	$0.369544\pi$
$972$	−35.5967	−1.14177
$973$	12.1803	0.390484
$974$	3.05573	0.0979118
$975$	−27.7771	−0.889579
$976$	−1.23607	−0.0395656
$977$	−58.9443	−1.88579	−0.942897	−	0.333084i	$-0.891911\pi$
$977$	−58.9443	−1.88579	−0.942897	+	0.333084i	$0.891911\pi$
$978$	0.944272	0.0301945
$979$	−10.4721	−0.334691
$980$	1.23607	0.0394847
$981$	−91.0132	−2.90583
$982$	32.9443	1.05129
$983$	−54.4721	−1.73739	−0.868696	−	0.495346i	$-0.835041\pi$
$983$	−54.4721	−1.73739	−0.868696	+	0.495346i	$0.835041\pi$
$984$	11.4164	0.363942
$985$	−26.8328	−0.854965
$986$	0	0
$987$	8.00000	0.254643
$988$	20.9443	0.666326
$989$	−19.7771	−0.628875
$990$	48.3607	1.53700
$991$	22.2492	0.706770	0.353385	−	0.935478i	$-0.385031\pi$
$991$	22.2492	0.706770	0.353385	+	0.935478i	$0.385031\pi$
$992$	−1.52786	−0.0485097
$993$	65.8885	2.09091
$994$	−2.47214	−0.0784114
$995$	−17.8885	−0.567105
$996$	−6.47214	−0.205077
$997$	47.4853	1.50387	0.751937	−	0.659235i	$-0.229120\pi$
$997$	47.4853	1.50387	0.751937	+	0.659235i	$0.229120\pi$
$998$	−40.6525	−1.28683
$999$	−46.8328	−1.48172

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4046.2.a.v.1.1		2
17.16	even	2		238.2.a.f.1.2	✓	2
51.50	odd	2		2142.2.a.x.1.2		2
68.67	odd	2		1904.2.a.f.1.1		2
85.84	even	2		5950.2.a.x.1.1		2
119.118	odd	2		1666.2.a.o.1.1		2
136.67	odd	2		7616.2.a.y.1.2		2
136.101	even	2		7616.2.a.n.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
238.2.a.f.1.2	✓	2	17.16	even	2
1666.2.a.o.1.1		2	119.118	odd	2
1904.2.a.f.1.1		2	68.67	odd	2
2142.2.a.x.1.2		2	51.50	odd	2
4046.2.a.v.1.1		2	1.1	even	1	trivial
5950.2.a.x.1.1		2	85.84	even	2
7616.2.a.n.1.1		2	136.101	even	2
7616.2.a.y.1.2		2	136.67	odd	2

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:	\( N \)	\(=\)	\( 4046 = 2 \cdot 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -3.23607 q^{3} +1.00000 q^{4} +1.23607 q^{5} +3.23607 q^{6} +1.00000 q^{7} -1.00000 q^{8} +7.47214 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -3.23607 q^{3} +1.00000 q^{4} +1.23607 q^{5} +3.23607 q^{6} +1.00000 q^{7} -1.00000 q^{8} +7.47214 q^{9} -1.23607 q^{10} -5.23607 q^{11} -3.23607 q^{12} -2.47214 q^{13} -1.00000 q^{14} -4.00000 q^{15} +1.00000 q^{16} -7.47214 q^{18} -8.47214 q^{19} +1.23607 q^{20} -3.23607 q^{21} +5.23607 q^{22} -8.00000 q^{23} +3.23607 q^{24} -3.47214 q^{25} +2.47214 q^{26} -14.4721 q^{27} +1.00000 q^{28} +5.70820 q^{29} +4.00000 q^{30} +1.52786 q^{31} -1.00000 q^{32} +16.9443 q^{33} +1.23607 q^{35} +7.47214 q^{36} +3.23607 q^{37} +8.47214 q^{38} +8.00000 q^{39} -1.23607 q^{40} +3.52786 q^{41} +3.23607 q^{42} +2.47214 q^{43} -5.23607 q^{44} +9.23607 q^{45} +8.00000 q^{46} -2.47214 q^{47} -3.23607 q^{48} +1.00000 q^{49} +3.47214 q^{50} -2.47214 q^{52} -4.47214 q^{53} +14.4721 q^{54} -6.47214 q^{55} -1.00000 q^{56} +27.4164 q^{57} -5.70820 q^{58} -6.00000 q^{59} -4.00000 q^{60} -1.23607 q^{61} -1.52786 q^{62} +7.47214 q^{63} +1.00000 q^{64} -3.05573 q^{65} -16.9443 q^{66} -1.52786 q^{67} +25.8885 q^{69} -1.23607 q^{70} +2.47214 q^{71} -7.47214 q^{72} -13.4164 q^{73} -3.23607 q^{74} +11.2361 q^{75} -8.47214 q^{76} -5.23607 q^{77} -8.00000 q^{78} +10.4721 q^{79} +1.23607 q^{80} +24.4164 q^{81} -3.52786 q^{82} +2.00000 q^{83} -3.23607 q^{84} -2.47214 q^{86} -18.4721 q^{87} +5.23607 q^{88} +2.00000 q^{89} -9.23607 q^{90} -2.47214 q^{91} -8.00000 q^{92} -4.94427 q^{93} +2.47214 q^{94} -10.4721 q^{95} +3.23607 q^{96} +8.47214 q^{97} -1.00000 q^{98} -39.1246 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^7 - 2 * q^8 + 6 * q^9	\( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} + 2 q^{6} + 2 q^{7} - 2 q^{8} + 6 q^{9} + 2 q^{10} - 6 q^{11} - 2 q^{12} + 4 q^{13} - 2 q^{14} - 8 q^{15} + 2 q^{16} - 6 q^{18} - 8 q^{19} - 2 q^{20} - 2 q^{21} + 6 q^{22} - 16 q^{23} + 2 q^{24} + 2 q^{25} - 4 q^{26} - 20 q^{27} + 2 q^{28} - 2 q^{29} + 8 q^{30} + 12 q^{31} - 2 q^{32} + 16 q^{33} - 2 q^{35} + 6 q^{36} + 2 q^{37} + 8 q^{38} + 16 q^{39} + 2 q^{40} + 16 q^{41} + 2 q^{42} - 4 q^{43} - 6 q^{44} + 14 q^{45} + 16 q^{46} + 4 q^{47} - 2 q^{48} + 2 q^{49} - 2 q^{50} + 4 q^{52} + 20 q^{54} - 4 q^{55} - 2 q^{56} + 28 q^{57} + 2 q^{58} - 12 q^{59} - 8 q^{60} + 2 q^{61} - 12 q^{62} + 6 q^{63} + 2 q^{64} - 24 q^{65} - 16 q^{66} - 12 q^{67} + 16 q^{69} + 2 q^{70} - 4 q^{71} - 6 q^{72} - 2 q^{74} + 18 q^{75} - 8 q^{76} - 6 q^{77} - 16 q^{78} + 12 q^{79} - 2 q^{80} + 22 q^{81} - 16 q^{82} + 4 q^{83} - 2 q^{84} + 4 q^{86} - 28 q^{87} + 6 q^{88} + 4 q^{89} - 14 q^{90} + 4 q^{91} - 16 q^{92} + 8 q^{93} - 4 q^{94} - 12 q^{95} + 2 q^{96} + 8 q^{97} - 2 q^{98} - 38 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^5 + 2 * q^6 + 2 * q^7 - 2 * q^8 + 6 * q^9 + 2 * q^10 - 6 * q^11 - 2 * q^12 + 4 * q^13 - 2 * q^14 - 8 * q^15 + 2 * q^16 - 6 * q^18 - 8 * q^19 - 2 * q^20 - 2 * q^21 + 6 * q^22 - 16 * q^23 + 2 * q^24 + 2 * q^25 - 4 * q^26 - 20 * q^27 + 2 * q^28 - 2 * q^29 + 8 * q^30 + 12 * q^31 - 2 * q^32 + 16 * q^33 - 2 * q^35 + 6 * q^36 + 2 * q^37 + 8 * q^38 + 16 * q^39 + 2 * q^40 + 16 * q^41 + 2 * q^42 - 4 * q^43 - 6 * q^44 + 14 * q^45 + 16 * q^46 + 4 * q^47 - 2 * q^48 + 2 * q^49 - 2 * q^50 + 4 * q^52 + 20 * q^54 - 4 * q^55 - 2 * q^56 + 28 * q^57 + 2 * q^58 - 12 * q^59 - 8 * q^60 + 2 * q^61 - 12 * q^62 + 6 * q^63 + 2 * q^64 - 24 * q^65 - 16 * q^66 - 12 * q^67 + 16 * q^69 + 2 * q^70 - 4 * q^71 - 6 * q^72 - 2 * q^74 + 18 * q^75 - 8 * q^76 - 6 * q^77 - 16 * q^78 + 12 * q^79 - 2 * q^80 + 22 * q^81 - 16 * q^82 + 4 * q^83 - 2 * q^84 + 4 * q^86 - 28 * q^87 + 6 * q^88 + 4 * q^89 - 14 * q^90 + 4 * q^91 - 16 * q^92 + 8 * q^93 - 4 * q^94 - 12 * q^95 + 2 * q^96 + 8 * q^97 - 2 * q^98 - 38 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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