LMFDB - Embedded newform 4011.2.a.k.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4011,2,Mod(1,4011)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4011, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4011.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4011 = 3 \cdot 7 \cdot 191 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4011.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:	$32.0279962507$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	4011.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.768548 q^{2} +1.00000 q^{3} -1.40933 q^{4} +3.25090 q^{5} -0.768548 q^{6} +1.00000 q^{7} +2.62024 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.768548 q^{2} +1.00000 q^{3} -1.40933 q^{4} +3.25090 q^{5} -0.768548 q^{6} +1.00000 q^{7} +2.62024 q^{8} +1.00000 q^{9} -2.49847 q^{10} +1.06849 q^{11} -1.40933 q^{12} -6.23354 q^{13} -0.768548 q^{14} +3.25090 q^{15} +0.804894 q^{16} -2.53511 q^{17} -0.768548 q^{18} +1.54739 q^{19} -4.58160 q^{20} +1.00000 q^{21} -0.821188 q^{22} +4.87110 q^{23} +2.62024 q^{24} +5.56832 q^{25} +4.79077 q^{26} +1.00000 q^{27} -1.40933 q^{28} -2.96297 q^{29} -2.49847 q^{30} +0.605187 q^{31} -5.85907 q^{32} +1.06849 q^{33} +1.94835 q^{34} +3.25090 q^{35} -1.40933 q^{36} +5.49339 q^{37} -1.18924 q^{38} -6.23354 q^{39} +8.51811 q^{40} -11.4299 q^{41} -0.768548 q^{42} +11.9837 q^{43} -1.50586 q^{44} +3.25090 q^{45} -3.74367 q^{46} +12.5422 q^{47} +0.804894 q^{48} +1.00000 q^{49} -4.27952 q^{50} -2.53511 q^{51} +8.78515 q^{52} +6.96574 q^{53} -0.768548 q^{54} +3.47356 q^{55} +2.62024 q^{56} +1.54739 q^{57} +2.27718 q^{58} -11.9246 q^{59} -4.58160 q^{60} +9.46484 q^{61} -0.465115 q^{62} +1.00000 q^{63} +2.89319 q^{64} -20.2646 q^{65} -0.821188 q^{66} +1.33111 q^{67} +3.57282 q^{68} +4.87110 q^{69} -2.49847 q^{70} +4.53921 q^{71} +2.62024 q^{72} +3.57156 q^{73} -4.22193 q^{74} +5.56832 q^{75} -2.18079 q^{76} +1.06849 q^{77} +4.79077 q^{78} +0.524648 q^{79} +2.61663 q^{80} +1.00000 q^{81} +8.78444 q^{82} +16.0480 q^{83} -1.40933 q^{84} -8.24138 q^{85} -9.21003 q^{86} -2.96297 q^{87} +2.79970 q^{88} +6.80046 q^{89} -2.49847 q^{90} -6.23354 q^{91} -6.86501 q^{92} +0.605187 q^{93} -9.63930 q^{94} +5.03039 q^{95} -5.85907 q^{96} +14.0536 q^{97} -0.768548 q^{98} +1.06849 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q + 9 q^{2} + 27 q^{3} + 31 q^{4} + 23 q^{5} + 9 q^{6} + 27 q^{7} + 18 q^{8} + 27 q^{9}+O(q^{10}) $ 27 * q + 9 * q^2 + 27 * q^3 + 31 * q^4 + 23 * q^5 + 9 * q^6 + 27 * q^7 + 18 * q^8 + 27 * q^9	\( 27 q + 9 q^{2} + 27 q^{3} + 31 q^{4} + 23 q^{5} + 9 q^{6} + 27 q^{7} + 18 q^{8} + 27 q^{9} + 10 q^{10} + 22 q^{11} + 31 q^{12} + 15 q^{13} + 9 q^{14} + 23 q^{15} + 39 q^{16} + 22 q^{17} + 9 q^{18} + 24 q^{19} + 46 q^{20} + 27 q^{21} - 19 q^{22} + 36 q^{23} + 18 q^{24} + 38 q^{25} + 19 q^{26} + 27 q^{27} + 31 q^{28} + 32 q^{29} + 10 q^{30} + 11 q^{31} + 15 q^{32} + 22 q^{33} - 4 q^{34} + 23 q^{35} + 31 q^{36} + 6 q^{37} + 8 q^{38} + 15 q^{39} + 16 q^{41} + 9 q^{42} - 11 q^{43} + 22 q^{44} + 23 q^{45} - 56 q^{46} + 39 q^{47} + 39 q^{48} + 27 q^{49} - 7 q^{50} + 22 q^{51} + 10 q^{52} + 24 q^{53} + 9 q^{54} + 16 q^{55} + 18 q^{56} + 24 q^{57} - 31 q^{58} + 31 q^{59} + 46 q^{60} + 28 q^{61} - 4 q^{62} + 27 q^{63} + 40 q^{64} + 33 q^{65} - 19 q^{66} - 7 q^{67} + 25 q^{68} + 36 q^{69} + 10 q^{70} + 49 q^{71} + 18 q^{72} - 13 q^{73} + 7 q^{74} + 38 q^{75} + 15 q^{76} + 22 q^{77} + 19 q^{78} - 18 q^{79} + 78 q^{80} + 27 q^{81} + 31 q^{82} + 59 q^{83} + 31 q^{84} - 20 q^{85} + 35 q^{86} + 32 q^{87} - 49 q^{88} + 49 q^{89} + 10 q^{90} + 15 q^{91} + 52 q^{92} + 11 q^{93} + 17 q^{94} + 38 q^{95} + 15 q^{96} - 7 q^{97} + 9 q^{98} + 22 q^{99}+O(q^{100}) \) 27 * q + 9 * q^2 + 27 * q^3 + 31 * q^4 + 23 * q^5 + 9 * q^6 + 27 * q^7 + 18 * q^8 + 27 * q^9 + 10 * q^10 + 22 * q^11 + 31 * q^12 + 15 * q^13 + 9 * q^14 + 23 * q^15 + 39 * q^16 + 22 * q^17 + 9 * q^18 + 24 * q^19 + 46 * q^20 + 27 * q^21 - 19 * q^22 + 36 * q^23 + 18 * q^24 + 38 * q^25 + 19 * q^26 + 27 * q^27 + 31 * q^28 + 32 * q^29 + 10 * q^30 + 11 * q^31 + 15 * q^32 + 22 * q^33 - 4 * q^34 + 23 * q^35 + 31 * q^36 + 6 * q^37 + 8 * q^38 + 15 * q^39 + 16 * q^41 + 9 * q^42 - 11 * q^43 + 22 * q^44 + 23 * q^45 - 56 * q^46 + 39 * q^47 + 39 * q^48 + 27 * q^49 - 7 * q^50 + 22 * q^51 + 10 * q^52 + 24 * q^53 + 9 * q^54 + 16 * q^55 + 18 * q^56 + 24 * q^57 - 31 * q^58 + 31 * q^59 + 46 * q^60 + 28 * q^61 - 4 * q^62 + 27 * q^63 + 40 * q^64 + 33 * q^65 - 19 * q^66 - 7 * q^67 + 25 * q^68 + 36 * q^69 + 10 * q^70 + 49 * q^71 + 18 * q^72 - 13 * q^73 + 7 * q^74 + 38 * q^75 + 15 * q^76 + 22 * q^77 + 19 * q^78 - 18 * q^79 + 78 * q^80 + 27 * q^81 + 31 * q^82 + 59 * q^83 + 31 * q^84 - 20 * q^85 + 35 * q^86 + 32 * q^87 - 49 * q^88 + 49 * q^89 + 10 * q^90 + 15 * q^91 + 52 * q^92 + 11 * q^93 + 17 * q^94 + 38 * q^95 + 15 * q^96 - 7 * q^97 + 9 * q^98 + 22 * 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 $

$2$	−0.768548	−0.543445	−0.271723	−	0.962376i	$-0.587593\pi$
$2$	−0.768548	−0.543445	−0.271723	+	0.962376i	$0.587593\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.40933	−0.704667
$5$	3.25090	1.45384	0.726922	−	0.686720i	$-0.240950\pi$
$5$	3.25090	1.45384	0.726922	+	0.686720i	$0.240950\pi$
$6$	−0.768548	−0.313758
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.62024	0.926393
$9$	1.00000	0.333333
$10$	−2.49847	−0.790085
$11$	1.06849	0.322163	0.161081	−	0.986941i	$-0.448502\pi$
$11$	1.06849	0.322163	0.161081	+	0.986941i	$0.448502\pi$
$12$	−1.40933	−0.406840
$13$	−6.23354	−1.72887	−0.864437	−	0.502741i	$-0.832325\pi$
$13$	−6.23354	−1.72887	−0.864437	+	0.502741i	$0.832325\pi$
$14$	−0.768548	−0.205403
$15$	3.25090	0.839378
$16$	0.804894	0.201223
$17$	−2.53511	−0.614854	−0.307427	−	0.951572i	$-0.599468\pi$
$17$	−2.53511	−0.614854	−0.307427	+	0.951572i	$0.599468\pi$
$18$	−0.768548	−0.181148
$19$	1.54739	0.354995	0.177497	−	0.984121i	$-0.443200\pi$
$19$	1.54739	0.354995	0.177497	+	0.984121i	$0.443200\pi$
$20$	−4.58160	−1.02448
$21$	1.00000	0.218218
$22$	−0.821188	−0.175078
$23$	4.87110	1.01569	0.507847	−	0.861447i	$-0.330441\pi$
$23$	4.87110	1.01569	0.507847	+	0.861447i	$0.330441\pi$
$24$	2.62024	0.534853
$25$	5.56832	1.11366
$26$	4.79077	0.939548
$27$	1.00000	0.192450
$28$	−1.40933	−0.266339
$29$	−2.96297	−0.550210	−0.275105	−	0.961414i	$-0.588713\pi$
$29$	−2.96297	−0.550210	−0.275105	+	0.961414i	$0.588713\pi$
$30$	−2.49847	−0.456156
$31$	0.605187	0.108695	0.0543474	−	0.998522i	$-0.482692\pi$
$31$	0.605187	0.108695	0.0543474	+	0.998522i	$0.482692\pi$
$32$	−5.85907	−1.03575
$33$	1.06849	0.186001
$34$	1.94835	0.334140
$35$	3.25090	0.549502
$36$	−1.40933	−0.234889
$37$	5.49339	0.903108	0.451554	−	0.892244i	$-0.350870\pi$
$37$	5.49339	0.903108	0.451554	+	0.892244i	$0.350870\pi$
$38$	−1.18924	−0.192920
$39$	−6.23354	−0.998166
$40$	8.51811	1.34683
$41$	−11.4299	−1.78505	−0.892527	−	0.450995i	$-0.851069\pi$
$41$	−11.4299	−1.78505	−0.892527	+	0.450995i	$0.851069\pi$
$42$	−0.768548	−0.118589
$43$	11.9837	1.82750	0.913748	−	0.406282i	$-0.133175\pi$
$43$	11.9837	1.82750	0.913748	+	0.406282i	$0.133175\pi$
$44$	−1.50586	−0.227018
$45$	3.25090	0.484615
$46$	−3.74367	−0.551974
$47$	12.5422	1.82947	0.914736	−	0.404051i	$-0.132398\pi$
$47$	12.5422	1.82947	0.914736	+	0.404051i	$0.132398\pi$
$48$	0.804894	0.116176
$49$	1.00000	0.142857
$50$	−4.27952	−0.605216
$51$	−2.53511	−0.354986
$52$	8.78515	1.21828
$53$	6.96574	0.956818	0.478409	−	0.878137i	$-0.341214\pi$
$53$	6.96574	0.956818	0.478409	+	0.878137i	$0.341214\pi$
$54$	−0.768548	−0.104586
$55$	3.47356	0.468375
$56$	2.62024	0.350144
$57$	1.54739	0.204956
$58$	2.27718	0.299009
$59$	−11.9246	−1.55245	−0.776223	−	0.630459i	$-0.782867\pi$
$59$	−11.9246	−1.55245	−0.776223	+	0.630459i	$0.782867\pi$
$60$	−4.58160	−0.591482
$61$	9.46484	1.21185	0.605924	−	0.795522i	$-0.292803\pi$
$61$	9.46484	1.21185	0.605924	+	0.795522i	$0.292803\pi$
$62$	−0.465115	−0.0590697
$63$	1.00000	0.125988
$64$	2.89319	0.361648
$65$	−20.2646	−2.51351
$66$	−0.821188	−0.101081
$67$	1.33111	0.162621	0.0813104	−	0.996689i	$-0.474089\pi$
$67$	1.33111	0.162621	0.0813104	+	0.996689i	$0.474089\pi$
$68$	3.57282	0.433268
$69$	4.87110	0.586411
$70$	−2.49847	−0.298624
$71$	4.53921	0.538705	0.269353	−	0.963042i	$-0.413190\pi$
$71$	4.53921	0.538705	0.269353	+	0.963042i	$0.413190\pi$
$72$	2.62024	0.308798
$73$	3.57156	0.418019	0.209009	−	0.977914i	$-0.432976\pi$
$73$	3.57156	0.418019	0.209009	+	0.977914i	$0.432976\pi$
$74$	−4.22193	−0.490790
$75$	5.56832	0.642974
$76$	−2.18079	−0.250153
$77$	1.06849	0.121766
$78$	4.79077	0.542448
$79$	0.524648	0.0590275	0.0295138	−	0.999564i	$-0.490604\pi$
$79$	0.524648	0.0590275	0.0295138	+	0.999564i	$0.490604\pi$
$80$	2.61663	0.292548
$81$	1.00000	0.111111
$82$	8.78444	0.970079
$83$	16.0480	1.76150	0.880750	−	0.473582i	$-0.157039\pi$
$83$	16.0480	1.76150	0.880750	+	0.473582i	$0.157039\pi$
$84$	−1.40933	−0.153771
$85$	−8.24138	−0.893903
$86$	−9.21003	−0.993143
$87$	−2.96297	−0.317664
$88$	2.79970	0.298449
$89$	6.80046	0.720847	0.360423	−	0.932789i	$-0.382632\pi$
$89$	6.80046	0.720847	0.360423	+	0.932789i	$0.382632\pi$
$90$	−2.49847	−0.263362
$91$	−6.23354	−0.653453
$92$	−6.86501	−0.715727
$93$	0.605187	0.0627550
$94$	−9.63930	−0.994218
$95$	5.03039	0.516107
$96$	−5.85907	−0.597989
$97$	14.0536	1.42693	0.713465	−	0.700691i	$-0.247125\pi$
$97$	14.0536	1.42693	0.713465	+	0.700691i	$0.247125\pi$
$98$	−0.768548	−0.0776350
$99$	1.06849	0.107388
$100$	−7.84763	−0.784763
$101$	3.02767	0.301264	0.150632	−	0.988590i	$-0.451869\pi$
$101$	3.02767	0.301264	0.150632	+	0.988590i	$0.451869\pi$
$102$	1.94835	0.192916
$103$	13.3948	1.31982	0.659912	−	0.751343i	$-0.270593\pi$
$103$	13.3948	1.31982	0.659912	+	0.751343i	$0.270593\pi$
$104$	−16.3334	−1.60162
$105$	3.25090	0.317255
$106$	−5.35350	−0.519978
$107$	11.8784	1.14833	0.574166	−	0.818739i	$-0.305326\pi$
$107$	11.8784	1.14833	0.574166	+	0.818739i	$0.305326\pi$
$108$	−1.40933	−0.135613
$109$	−10.0687	−0.964411	−0.482205	−	0.876058i	$-0.660164\pi$
$109$	−10.0687	−0.964411	−0.482205	+	0.876058i	$0.660164\pi$
$110$	−2.66960	−0.254536
$111$	5.49339	0.521410
$112$	0.804894	0.0760553
$113$	−6.49082	−0.610605	−0.305303	−	0.952255i	$-0.598758\pi$
$113$	−6.49082	−0.610605	−0.305303	+	0.952255i	$0.598758\pi$
$114$	−1.18924	−0.111383
$115$	15.8354	1.47666
$116$	4.17582	0.387715
$117$	−6.23354	−0.576291
$118$	9.16459	0.843669
$119$	−2.53511	−0.232393
$120$	8.51811	0.777594
$121$	−9.85832	−0.896211
$122$	−7.27418	−0.658573
$123$	−11.4299	−1.03060
$124$	−0.852912	−0.0765937
$125$	1.84756	0.165250
$126$	−0.768548	−0.0684677
$127$	−22.4657	−1.99351	−0.996754	−	0.0805031i	$-0.974347\pi$
$127$	−22.4657	−1.99351	−0.996754	+	0.0805031i	$0.974347\pi$
$128$	9.49459	0.839211
$129$	11.9837	1.05510
$130$	15.5743	1.36596
$131$	6.69996	0.585378	0.292689	−	0.956208i	$-0.405450\pi$
$131$	6.69996	0.585378	0.292689	+	0.956208i	$0.405450\pi$
$132$	−1.50586	−0.131069
$133$	1.54739	0.134175
$134$	−1.02302	−0.0883755
$135$	3.25090	0.279793
$136$	−6.64259	−0.569597
$137$	0.837074	0.0715161	0.0357581	−	0.999360i	$-0.488615\pi$
$137$	0.837074	0.0715161	0.0357581	+	0.999360i	$0.488615\pi$
$138$	−3.74367	−0.318682
$139$	−8.35674	−0.708809	−0.354405	−	0.935092i	$-0.615316\pi$
$139$	−8.35674	−0.708809	−0.354405	+	0.935092i	$0.615316\pi$
$140$	−4.58160	−0.387216
$141$	12.5422	1.05625
$142$	−3.48860	−0.292757
$143$	−6.66050	−0.556979
$144$	0.804894	0.0670745
$145$	−9.63231	−0.799920
$146$	−2.74491	−0.227170
$147$	1.00000	0.0824786
$148$	−7.74203	−0.636391
$149$	−11.1798	−0.915885	−0.457943	−	0.888982i	$-0.651413\pi$
$149$	−11.1798	−0.915885	−0.457943	+	0.888982i	$0.651413\pi$
$150$	−4.27952	−0.349421
$151$	−17.5879	−1.43128	−0.715642	−	0.698467i	$-0.753866\pi$
$151$	−17.5879	−1.43128	−0.715642	+	0.698467i	$0.753866\pi$
$152$	4.05452	0.328865
$153$	−2.53511	−0.204951
$154$	−0.821188	−0.0661732
$155$	1.96740	0.158025
$156$	8.78515	0.703375
$157$	2.75108	0.219560	0.109780	−	0.993956i	$-0.464985\pi$
$157$	2.75108	0.219560	0.109780	+	0.993956i	$0.464985\pi$
$158$	−0.403217	−0.0320782
$159$	6.96574	0.552419
$160$	−19.0472	−1.50582
$161$	4.87110	0.383896
$162$	−0.768548	−0.0603828
$163$	−0.409528	−0.0320767	−0.0160384	−	0.999871i	$-0.505105\pi$
$163$	−0.409528	−0.0320767	−0.0160384	+	0.999871i	$0.505105\pi$
$164$	16.1086	1.25787
$165$	3.47356	0.270416
$166$	−12.3337	−0.957278
$167$	0.559619	0.0433046	0.0216523	−	0.999766i	$-0.493107\pi$
$167$	0.559619	0.0433046	0.0216523	+	0.999766i	$0.493107\pi$
$168$	2.62024	0.202156
$169$	25.8571	1.98900
$170$	6.33389	0.485787
$171$	1.54739	0.118332
$172$	−16.8890	−1.28778
$173$	−19.3505	−1.47119	−0.735595	−	0.677422i	$-0.763097\pi$
$173$	−19.3505	−1.47119	−0.735595	+	0.677422i	$0.763097\pi$
$174$	2.27718	0.172633
$175$	5.56832	0.420926
$176$	0.860023	0.0648267
$177$	−11.9246	−0.896305
$178$	−5.22647	−0.391741
$179$	−5.28479	−0.395003	−0.197502	−	0.980303i	$-0.563283\pi$
$179$	−5.28479	−0.395003	−0.197502	+	0.980303i	$0.563283\pi$
$180$	−4.58160	−0.341492
$181$	−4.61266	−0.342856	−0.171428	−	0.985197i	$-0.554838\pi$
$181$	−4.61266	−0.342856	−0.171428	+	0.985197i	$0.554838\pi$
$182$	4.79077	0.355116
$183$	9.46484	0.699661
$184$	12.7634	0.940932
$185$	17.8584	1.31298
$186$	−0.465115	−0.0341039
$187$	−2.70875	−0.198083
$188$	−17.6762	−1.28917
$189$	1.00000	0.0727393
$190$	−3.86609	−0.280476
$191$	1.00000	0.0723575
$191$	1.00000	0.0723575
$192$	2.89319	0.208798
$193$	3.29663	0.237297	0.118648	−	0.992936i	$-0.462144\pi$
$193$	3.29663	0.237297	0.118648	+	0.992936i	$0.462144\pi$
$194$	−10.8009	−0.775458
$195$	−20.2646	−1.45118
$196$	−1.40933	−0.100667
$197$	−13.8992	−0.990278	−0.495139	−	0.868814i	$-0.664883\pi$
$197$	−13.8992	−0.990278	−0.495139	+	0.868814i	$0.664883\pi$
$198$	−0.821188	−0.0583593
$199$	8.10315	0.574417	0.287209	−	0.957868i	$-0.407273\pi$
$199$	8.10315	0.574417	0.287209	+	0.957868i	$0.407273\pi$
$200$	14.5903	1.03169
$201$	1.33111	0.0938892
$202$	−2.32690	−0.163720
$203$	−2.96297	−0.207960
$204$	3.57282	0.250147
$205$	−37.1575	−2.59519
$206$	−10.2945	−0.717252
$207$	4.87110	0.338565
$208$	−5.01734	−0.347890
$209$	1.65337	0.114366
$210$	−2.49847	−0.172411
$211$	13.1427	0.904778	0.452389	−	0.891821i	$-0.350572\pi$
$211$	13.1427	0.904778	0.452389	+	0.891821i	$0.350572\pi$
$212$	−9.81706	−0.674238
$213$	4.53921	0.311022
$214$	−9.12914	−0.624055
$215$	38.9577	2.65689
$216$	2.62024	0.178284
$217$	0.605187	0.0410828
$218$	7.73831	0.524104
$219$	3.57156	0.241343
$220$	−4.89541	−0.330048
$221$	15.8027	1.06301
$222$	−4.22193	−0.283358
$223$	3.74967	0.251096	0.125548	−	0.992088i	$-0.459931\pi$
$223$	3.74967	0.251096	0.125548	+	0.992088i	$0.459931\pi$
$224$	−5.85907	−0.391476
$225$	5.56832	0.371221
$226$	4.98850	0.331830
$227$	8.50905	0.564765	0.282383	−	0.959302i	$-0.408875\pi$
$227$	8.50905	0.564765	0.282383	+	0.959302i	$0.408875\pi$
$228$	−2.18079	−0.144426
$229$	18.1401	1.19873	0.599367	−	0.800474i	$-0.295419\pi$
$229$	18.1401	1.19873	0.599367	+	0.800474i	$0.295419\pi$
$230$	−12.1703	−0.802485
$231$	1.06849	0.0703017
$232$	−7.76369	−0.509711
$233$	−6.85842	−0.449310	−0.224655	−	0.974438i	$-0.572125\pi$
$233$	−6.85842	−0.449310	−0.224655	+	0.974438i	$0.572125\pi$
$234$	4.79077	0.313183
$235$	40.7735	2.65977
$236$	16.8057	1.09396
$237$	0.524648	0.0340796
$238$	1.94835	0.126293
$239$	3.15820	0.204287	0.102143	−	0.994770i	$-0.467430\pi$
$239$	3.15820	0.204287	0.102143	+	0.994770i	$0.467430\pi$
$240$	2.61663	0.168902
$241$	22.8494	1.47186	0.735929	−	0.677059i	$-0.236746\pi$
$241$	22.8494	1.47186	0.735929	+	0.677059i	$0.236746\pi$
$242$	7.57659	0.487042
$243$	1.00000	0.0641500
$244$	−13.3391	−0.853950
$245$	3.25090	0.207692
$246$	8.78444	0.560075
$247$	−9.64570	−0.613741
$248$	1.58573	0.100694
$249$	16.0480	1.01700
$250$	−1.41993	−0.0898046
$251$	−0.695418	−0.0438944	−0.0219472	−	0.999759i	$-0.506987\pi$
$251$	−0.695418	−0.0438944	−0.0219472	+	0.999759i	$0.506987\pi$
$252$	−1.40933	−0.0887797
$253$	5.20474	0.327219
$254$	17.2660	1.08336
$255$	−8.24138	−0.516095
$256$	−13.0834	−0.817714
$257$	19.3610	1.20771	0.603853	−	0.797095i	$-0.293631\pi$
$257$	19.3610	1.20771	0.603853	+	0.797095i	$0.293631\pi$
$258$	−9.21003	−0.573392
$259$	5.49339	0.341343
$260$	28.5596	1.77119
$261$	−2.96297	−0.183403
$262$	−5.14924	−0.318121
$263$	19.0497	1.17466	0.587328	−	0.809349i	$-0.300180\pi$
$263$	19.0497	1.17466	0.587328	+	0.809349i	$0.300180\pi$
$264$	2.79970	0.172310
$265$	22.6449	1.39106
$266$	−1.18924	−0.0729170
$267$	6.80046	0.416181
$268$	−1.87598	−0.114594
$269$	−1.15964	−0.0707042	−0.0353521	−	0.999375i	$-0.511255\pi$
$269$	−1.15964	−0.0707042	−0.0353521	+	0.999375i	$0.511255\pi$
$270$	−2.49847	−0.152052
$271$	11.8300	0.718621	0.359310	−	0.933218i	$-0.383012\pi$
$271$	11.8300	0.718621	0.359310	+	0.933218i	$0.383012\pi$
$272$	−2.04049	−0.123723
$273$	−6.23354	−0.377271
$274$	−0.643332	−0.0388651
$275$	5.94971	0.358781
$276$	−6.86501	−0.413225
$277$	−3.15641	−0.189650	−0.0948252	−	0.995494i	$-0.530229\pi$
$277$	−3.15641	−0.189650	−0.0948252	+	0.995494i	$0.530229\pi$
$278$	6.42255	0.385199
$279$	0.605187	0.0362316
$280$	8.51811	0.509055
$281$	−7.56086	−0.451043	−0.225522	−	0.974238i	$-0.572409\pi$
$281$	−7.56086	−0.451043	−0.225522	+	0.974238i	$0.572409\pi$
$282$	−9.63930	−0.574012
$283$	0.195204	0.0116037	0.00580185	−	0.999983i	$-0.498153\pi$
$283$	0.195204	0.0116037	0.00580185	+	0.999983i	$0.498153\pi$
$284$	−6.39727	−0.379608
$285$	5.03039	0.297975
$286$	5.11891	0.302687
$287$	−11.4299	−0.674687
$288$	−5.85907	−0.345249
$289$	−10.5732	−0.621954
$290$	7.40289	0.434713
$291$	14.0536	0.823838
$292$	−5.03352	−0.294564
$293$	−1.32091	−0.0771684	−0.0385842	−	0.999255i	$-0.512285\pi$
$293$	−1.32091	−0.0771684	−0.0385842	+	0.999255i	$0.512285\pi$
$294$	−0.768548	−0.0448226
$295$	−38.7655	−2.25702
$296$	14.3940	0.836633
$297$	1.06849	0.0620003
$298$	8.59221	0.497733
$299$	−30.3642	−1.75601
$300$	−7.84763	−0.453083
$301$	11.9837	0.690728
$302$	13.5171	0.777824
$303$	3.02767	0.173935
$304$	1.24548	0.0714333
$305$	30.7692	1.76184
$306$	1.94835	0.111380
$307$	−11.1993	−0.639180	−0.319590	−	0.947556i	$-0.603545\pi$
$307$	−11.1993	−0.639180	−0.319590	+	0.947556i	$0.603545\pi$
$308$	−1.50586	−0.0858046
$309$	13.3948	0.762001
$310$	−1.51204	−0.0858782
$311$	29.7318	1.68594	0.842968	−	0.537963i	$-0.180806\pi$
$311$	29.7318	1.68594	0.842968	+	0.537963i	$0.180806\pi$
$312$	−16.3334	−0.924694
$313$	−6.14632	−0.347410	−0.173705	−	0.984798i	$-0.555574\pi$
$313$	−6.14632	−0.347410	−0.173705	+	0.984798i	$0.555574\pi$
$314$	−2.11434	−0.119319
$315$	3.25090	0.183167
$316$	−0.739405	−0.0415948
$317$	29.8328	1.67558	0.837789	−	0.545995i	$-0.183848\pi$
$317$	29.8328	1.67558	0.837789	+	0.545995i	$0.183848\pi$
$318$	−5.35350	−0.300209
$319$	−3.16591	−0.177257
$320$	9.40545	0.525780
$321$	11.8784	0.662990
$322$	−3.74367	−0.208627
$323$	−3.92279	−0.218270
$324$	−1.40933	−0.0782964
$325$	−34.7104	−1.92539
$326$	0.314742	0.0174319
$327$	−10.0687	−0.556803
$328$	−29.9491	−1.65366
$329$	12.5422	0.691476
$330$	−2.66960	−0.146956
$331$	−8.93475	−0.491098	−0.245549	−	0.969384i	$-0.578968\pi$
$331$	−8.93475	−0.491098	−0.245549	+	0.969384i	$0.578968\pi$
$332$	−22.6170	−1.24127
$333$	5.49339	0.301036
$334$	−0.430093	−0.0235337
$335$	4.32730	0.236425
$336$	0.804894	0.0439106
$337$	13.5956	0.740601	0.370300	−	0.928912i	$-0.379255\pi$
$337$	13.5956	0.740601	0.370300	+	0.928912i	$0.379255\pi$
$338$	−19.8724	−1.08091
$339$	−6.49082	−0.352533
$340$	11.6149	0.629904
$341$	0.646638	0.0350174
$342$	−1.18924	−0.0643067
$343$	1.00000	0.0539949
$344$	31.4001	1.69298
$345$	15.8354	0.852551
$346$	14.8718	0.799511
$347$	−7.32689	−0.393328	−0.196664	−	0.980471i	$-0.563011\pi$
$347$	−7.32689	−0.393328	−0.196664	+	0.980471i	$0.563011\pi$
$348$	4.17582	0.223847
$349$	−18.9708	−1.01548	−0.507741	−	0.861510i	$-0.669519\pi$
$349$	−18.9708	−1.01548	−0.507741	+	0.861510i	$0.669519\pi$
$350$	−4.27952	−0.228750
$351$	−6.23354	−0.332722
$352$	−6.26038	−0.333679
$353$	−10.7448	−0.571890	−0.285945	−	0.958246i	$-0.592307\pi$
$353$	−10.7448	−0.571890	−0.285945	+	0.958246i	$0.592307\pi$
$354$	9.16459	0.487093
$355$	14.7565	0.783194
$356$	−9.58412	−0.507957
$357$	−2.53511	−0.134172
$358$	4.06161	0.214663
$359$	14.2866	0.754019	0.377010	−	0.926209i	$-0.376952\pi$
$359$	14.2866	0.754019	0.377010	+	0.926209i	$0.376952\pi$
$360$	8.51811	0.448944
$361$	−16.6056	−0.873979
$362$	3.54505	0.186323
$363$	−9.85832	−0.517428
$364$	8.78515	0.460467
$365$	11.6108	0.607735
$366$	−7.27418	−0.380228
$367$	−19.8421	−1.03575	−0.517874	−	0.855457i	$-0.673277\pi$
$367$	−19.8421	−1.03575	−0.517874	+	0.855457i	$0.673277\pi$
$368$	3.92072	0.204382
$369$	−11.4299	−0.595018
$370$	−13.7251	−0.713532
$371$	6.96574	0.361643
$372$	−0.852912	−0.0442214
$373$	3.93886	0.203946	0.101973	−	0.994787i	$-0.467484\pi$
$373$	3.93886	0.203946	0.101973	+	0.994787i	$0.467484\pi$
$374$	2.08180	0.107647
$375$	1.84756	0.0954074
$376$	32.8636	1.69481
$377$	18.4698	0.951244
$378$	−0.768548	−0.0395298
$379$	12.4943	0.641790	0.320895	−	0.947115i	$-0.396016\pi$
$379$	12.4943	0.641790	0.320895	+	0.947115i	$0.396016\pi$
$380$	−7.08950	−0.363684
$381$	−22.4657	−1.15095
$382$	−0.768548	−0.0393223
$383$	−11.6028	−0.592876	−0.296438	−	0.955052i	$-0.595799\pi$
$383$	−11.6028	−0.592876	−0.296438	+	0.955052i	$0.595799\pi$
$384$	9.49459	0.484519
$385$	3.47356	0.177029
$386$	−2.53362	−0.128958
$387$	11.9837	0.609165
$388$	−19.8063	−1.00551
$389$	−8.59082	−0.435572	−0.217786	−	0.975997i	$-0.569884\pi$
$389$	−8.59082	−0.435572	−0.217786	+	0.975997i	$0.569884\pi$
$390$	15.5743	0.788636
$391$	−12.3488	−0.624504
$392$	2.62024	0.132342
$393$	6.69996	0.337968
$394$	10.6822	0.538162
$395$	1.70558	0.0858169
$396$	−1.50586	−0.0756725
$397$	−20.2404	−1.01584	−0.507919	−	0.861405i	$-0.669585\pi$
$397$	−20.2404	−1.01584	−0.507919	+	0.861405i	$0.669585\pi$
$398$	−6.22766	−0.312164
$399$	1.54739	0.0774662
$400$	4.48191	0.224095
$401$	−14.9575	−0.746943	−0.373472	−	0.927642i	$-0.621833\pi$
$401$	−14.9575	−0.746943	−0.373472	+	0.927642i	$0.621833\pi$
$402$	−1.02302	−0.0510236
$403$	−3.77246	−0.187920
$404$	−4.26699	−0.212291
$405$	3.25090	0.161538
$406$	2.27718	0.113015
$407$	5.86965	0.290948
$408$	−6.64259	−0.328857
$409$	19.2801	0.953341	0.476671	−	0.879082i	$-0.341843\pi$
$409$	19.2801	0.953341	0.476671	+	0.879082i	$0.341843\pi$
$410$	28.5573	1.41034
$411$	0.837074	0.0412898
$412$	−18.8777	−0.930037
$413$	−11.9246	−0.586769
$414$	−3.74367	−0.183991
$415$	52.1705	2.56095
$416$	36.5228	1.79068
$417$	−8.35674	−0.409231
$418$	−1.27069	−0.0621517
$419$	22.4989	1.09914	0.549571	−	0.835447i	$-0.314791\pi$
$419$	22.4989	1.09914	0.549571	+	0.835447i	$0.314791\pi$
$420$	−4.58160	−0.223559
$421$	−1.20293	−0.0586272	−0.0293136	−	0.999570i	$-0.509332\pi$
$421$	−1.20293	−0.0586272	−0.0293136	+	0.999570i	$0.509332\pi$
$422$	−10.1008	−0.491697
$423$	12.5422	0.609824
$424$	18.2519	0.886389
$425$	−14.1163	−0.684742
$426$	−3.48860	−0.169023
$427$	9.46484	0.458036
$428$	−16.7407	−0.809192
$429$	−6.66050	−0.321572
$430$	−29.9409	−1.44388
$431$	15.4176	0.742641	0.371320	−	0.928505i	$-0.378905\pi$
$431$	15.4176	0.742641	0.371320	+	0.928505i	$0.378905\pi$
$432$	0.804894	0.0387255
$433$	−7.71927	−0.370965	−0.185482	−	0.982648i	$-0.559385\pi$
$433$	−7.71927	−0.370965	−0.185482	+	0.982648i	$0.559385\pi$
$434$	−0.465115	−0.0223262
$435$	−9.63231	−0.461834
$436$	14.1902	0.679589
$437$	7.53747	0.360566
$438$	−2.74491	−0.131157
$439$	−15.6564	−0.747238	−0.373619	−	0.927582i	$-0.621883\pi$
$439$	−15.6564	−0.747238	−0.373619	+	0.927582i	$0.621883\pi$
$440$	9.10154	0.433899
$441$	1.00000	0.0476190
$442$	−12.1451	−0.577685
$443$	−10.3511	−0.491796	−0.245898	−	0.969296i	$-0.579083\pi$
$443$	−10.3511	−0.491796	−0.245898	+	0.969296i	$0.579083\pi$
$444$	−7.74203	−0.367420
$445$	22.1076	1.04800
$446$	−2.88180	−0.136457
$447$	−11.1798	−0.528786
$448$	2.89319	0.136690
$449$	20.7621	0.979826	0.489913	−	0.871771i	$-0.337029\pi$
$449$	20.7621	0.979826	0.489913	+	0.871771i	$0.337029\pi$
$450$	−4.27952	−0.201739
$451$	−12.2128	−0.575078
$452$	9.14774	0.430273
$453$	−17.5879	−0.826352
$454$	−6.53961	−0.306919
$455$	−20.2646	−0.950019
$456$	4.05452	0.189870
$457$	−29.3239	−1.37172	−0.685858	−	0.727735i	$-0.740573\pi$
$457$	−29.3239	−1.37172	−0.685858	+	0.727735i	$0.740573\pi$
$458$	−13.9416	−0.651446
$459$	−2.53511	−0.118329
$460$	−22.3174	−1.04056
$461$	0.716452	0.0333685	0.0166842	−	0.999861i	$-0.494689\pi$
$461$	0.716452	0.0333685	0.0166842	+	0.999861i	$0.494689\pi$
$462$	−0.821188	−0.0382051
$463$	15.9390	0.740748	0.370374	−	0.928883i	$-0.379230\pi$
$463$	15.9390	0.740748	0.370374	+	0.928883i	$0.379230\pi$
$464$	−2.38488	−0.110715
$465$	1.96740	0.0912360
$466$	5.27102	0.244175
$467$	29.5612	1.36793	0.683966	−	0.729514i	$-0.260254\pi$
$467$	29.5612	1.36793	0.683966	+	0.729514i	$0.260254\pi$
$468$	8.78515	0.406094
$469$	1.33111	0.0614649
$470$	−31.3364	−1.44544
$471$	2.75108	0.126763
$472$	−31.2452	−1.43818
$473$	12.8045	0.588751
$474$	−0.403217	−0.0185204
$475$	8.61634	0.395345
$476$	3.57282	0.163760
$477$	6.96574	0.318939
$478$	−2.42722	−0.111019
$479$	−23.6771	−1.08183	−0.540917	−	0.841076i	$-0.681923\pi$
$479$	−23.6771	−1.08183	−0.540917	+	0.841076i	$0.681923\pi$
$480$	−19.0472	−0.869383
$481$	−34.2433	−1.56136
$482$	−17.5608	−0.799874
$483$	4.87110	0.221643
$484$	13.8937	0.631531
$485$	45.6869	2.07453
$486$	−0.768548	−0.0348620
$487$	−33.8554	−1.53414	−0.767068	−	0.641566i	$-0.778285\pi$
$487$	−33.8554	−1.53414	−0.767068	+	0.641566i	$0.778285\pi$
$488$	24.8001	1.12265
$489$	−0.409528	−0.0185195
$490$	−2.49847	−0.112869
$491$	−13.1933	−0.595405	−0.297702	−	0.954659i	$-0.596220\pi$
$491$	−13.1933	−0.595405	−0.297702	+	0.954659i	$0.596220\pi$
$492$	16.1086	0.726231
$493$	7.51146	0.338299
$494$	7.41318	0.333535
$495$	3.47356	0.156125
$496$	0.487112	0.0218720
$497$	4.53921	0.203611
$498$	−12.3337	−0.552685
$499$	−24.1572	−1.08143	−0.540713	−	0.841207i	$-0.681845\pi$
$499$	−24.1572	−1.08143	−0.540713	+	0.841207i	$0.681845\pi$
$500$	−2.60383	−0.116447
$501$	0.559619	0.0250019
$502$	0.534462	0.0238542
$503$	31.2977	1.39550	0.697748	−	0.716343i	$-0.254185\pi$
$503$	31.2977	1.39550	0.697748	+	0.716343i	$0.254185\pi$
$504$	2.62024	0.116715
$505$	9.84262	0.437991
$506$	−4.00009	−0.177826
$507$	25.8571	1.14835
$508$	31.6617	1.40476
$509$	−3.40035	−0.150718	−0.0753588	−	0.997156i	$-0.524010\pi$
$509$	−3.40035	−0.150718	−0.0753588	+	0.997156i	$0.524010\pi$
$510$	6.33389	0.280469
$511$	3.57156	0.157996
$512$	−8.93395	−0.394829
$513$	1.54739	0.0683188
$514$	−14.8799	−0.656322
$515$	43.5449	1.91882
$516$	−16.8890	−0.743498
$517$	13.4013	0.589388
$518$	−4.22193	−0.185501
$519$	−19.3505	−0.849392
$520$	−53.0980	−2.32850
$521$	−16.3624	−0.716850	−0.358425	−	0.933558i	$-0.616686\pi$
$521$	−16.3624	−0.716850	−0.358425	+	0.933558i	$0.616686\pi$
$522$	2.27718	0.0996697
$523$	14.1333	0.618007	0.309004	−	0.951061i	$-0.400004\pi$
$523$	14.1333	0.618007	0.309004	+	0.951061i	$0.400004\pi$
$524$	−9.44249	−0.412497
$525$	5.56832	0.243022
$526$	−14.6406	−0.638361
$527$	−1.53422	−0.0668315
$528$	0.860023	0.0374277
$529$	0.727611	0.0316352
$530$	−17.4037	−0.755967
$531$	−11.9246	−0.517482
$532$	−2.18079	−0.0945490
$533$	71.2489	3.08613
$534$	−5.22647	−0.226172
$535$	38.6155	1.66950
$536$	3.48782	0.150651
$537$	−5.28479	−0.228055
$538$	0.891235	0.0384239
$539$	1.06849	0.0460233
$540$	−4.58160	−0.197161
$541$	27.4713	1.18108	0.590542	−	0.807007i	$-0.298914\pi$
$541$	27.4713	1.18108	0.590542	+	0.807007i	$0.298914\pi$
$542$	−9.09191	−0.390531
$543$	−4.61266	−0.197948
$544$	14.8534	0.636834
$545$	−32.7324	−1.40210
$546$	4.79077	0.205026
$547$	−39.0413	−1.66928	−0.834642	−	0.550793i	$-0.814325\pi$
$547$	−39.0413	−1.66928	−0.834642	+	0.550793i	$0.814325\pi$
$548$	−1.17972	−0.0503951
$549$	9.46484	0.403950
$550$	−4.57264	−0.194978
$551$	−4.58486	−0.195322
$552$	12.7634	0.543248
$553$	0.524648	0.0223103
$554$	2.42585	0.103065
$555$	17.8584	0.758049
$556$	11.7774	0.499475
$557$	7.91297	0.335283	0.167642	−	0.985848i	$-0.446385\pi$
$557$	7.91297	0.335283	0.167642	+	0.985848i	$0.446385\pi$
$558$	−0.465115	−0.0196899
$559$	−74.7008	−3.15951
$560$	2.61663	0.110573
$561$	−2.70875	−0.114363
$562$	5.81088	0.245117
$563$	1.25050	0.0527021	0.0263511	−	0.999653i	$-0.491611\pi$
$563$	1.25050	0.0527021	0.0263511	+	0.999653i	$0.491611\pi$
$564$	−17.6762	−0.744302
$565$	−21.1010	−0.887725
$566$	−0.150024	−0.00630597
$567$	1.00000	0.0419961
$568$	11.8938	0.499053
$569$	−12.0450	−0.504953	−0.252477	−	0.967603i	$-0.581245\pi$
$569$	−12.0450	−0.504953	−0.252477	+	0.967603i	$0.581245\pi$
$570$	−3.86609	−0.161933
$571$	−2.91239	−0.121880	−0.0609398	−	0.998141i	$-0.519410\pi$
$571$	−2.91239	−0.121880	−0.0609398	+	0.998141i	$0.519410\pi$
$572$	9.38687	0.392485
$573$	1.00000	0.0417756
$574$	8.78444	0.366655
$575$	27.1239	1.13114
$576$	2.89319	0.120549
$577$	−6.06362	−0.252432	−0.126216	−	0.992003i	$-0.540283\pi$
$577$	−6.06362	−0.252432	−0.126216	+	0.992003i	$0.540283\pi$
$578$	8.12602	0.337998
$579$	3.29663	0.137003
$580$	13.5752	0.563678
$581$	16.0480	0.665784
$582$	−10.8009	−0.447711
$583$	7.44284	0.308251
$584$	9.35832	0.387250
$585$	−20.2646	−0.837838
$586$	1.01518	0.0419368
$587$	−17.9370	−0.740338	−0.370169	−	0.928964i	$-0.620700\pi$
$587$	−17.9370	−0.740338	−0.370169	+	0.928964i	$0.620700\pi$
$588$	−1.40933	−0.0581200
$589$	0.936459	0.0385861
$590$	29.7931	1.22656
$591$	−13.8992	−0.571737
$592$	4.42160	0.181727
$593$	−0.156989	−0.00644678	−0.00322339	−	0.999995i	$-0.501026\pi$
$593$	−0.156989	−0.00644678	−0.00322339	+	0.999995i	$0.501026\pi$
$594$	−0.821188	−0.0336937
$595$	−8.24138	−0.337864
$596$	15.7561	0.645394
$597$	8.10315	0.331640
$598$	23.3363	0.954294
$599$	−35.7426	−1.46040	−0.730202	−	0.683231i	$-0.760574\pi$
$599$	−35.7426	−1.46040	−0.730202	+	0.683231i	$0.760574\pi$
$600$	14.5903	0.595647
$601$	24.5704	1.00225	0.501124	−	0.865376i	$-0.332920\pi$
$601$	24.5704	1.00225	0.501124	+	0.865376i	$0.332920\pi$
$602$	−9.21003	−0.375373
$603$	1.33111	0.0542070
$604$	24.7873	1.00858
$605$	−32.0484	−1.30295
$606$	−2.32690	−0.0945240
$607$	−47.4270	−1.92500	−0.962501	−	0.271277i	$-0.912554\pi$
$607$	−47.4270	−1.92500	−0.962501	+	0.271277i	$0.912554\pi$
$608$	−9.06624	−0.367685
$609$	−2.96297	−0.120066
$610$	−23.6476	−0.957464
$611$	−78.1825	−3.16293
$612$	3.57282	0.144423
$613$	−12.4805	−0.504084	−0.252042	−	0.967716i	$-0.581102\pi$
$613$	−12.4805	−0.504084	−0.252042	+	0.967716i	$0.581102\pi$
$614$	8.60722	0.347359
$615$	−37.1575	−1.49833
$616$	2.79970	0.112803
$617$	27.1578	1.09333	0.546666	−	0.837351i	$-0.315897\pi$
$617$	27.1578	1.09333	0.546666	+	0.837351i	$0.315897\pi$
$618$	−10.2945	−0.414106
$619$	12.1557	0.488577	0.244289	−	0.969703i	$-0.421446\pi$
$619$	12.1557	0.488577	0.244289	+	0.969703i	$0.421446\pi$
$620$	−2.77273	−0.111355
$621$	4.87110	0.195470
$622$	−22.8503	−0.916214
$623$	6.80046	0.272455
$624$	−5.01734	−0.200854
$625$	−21.8354	−0.873416
$626$	4.72374	0.188798
$627$	1.65337	0.0660293
$628$	−3.87720	−0.154717
$629$	−13.9264	−0.555280
$630$	−2.49847	−0.0995413
$631$	−42.5703	−1.69470	−0.847349	−	0.531037i	$-0.821803\pi$
$631$	−42.5703	−1.69470	−0.847349	+	0.531037i	$0.821803\pi$
$632$	1.37470	0.0546827
$633$	13.1427	0.522374
$634$	−22.9279	−0.910584
$635$	−73.0337	−2.89825
$636$	−9.81706	−0.389272
$637$	−6.23354	−0.246982
$638$	2.43316	0.0963296
$639$	4.53921	0.179568
$640$	30.8659	1.22008
$641$	37.1018	1.46543	0.732716	−	0.680534i	$-0.238252\pi$
$641$	37.1018	1.46543	0.732716	+	0.680534i	$0.238252\pi$
$642$	−9.12914	−0.360298
$643$	19.4086	0.765399	0.382699	−	0.923873i	$-0.374995\pi$
$643$	19.4086	0.765399	0.382699	+	0.923873i	$0.374995\pi$
$644$	−6.86501	−0.270519
$645$	38.9577	1.53396
$646$	3.01485	0.118618
$647$	−34.9184	−1.37278	−0.686392	−	0.727232i	$-0.740807\pi$
$647$	−34.9184	−1.37278	−0.686392	+	0.727232i	$0.740807\pi$
$648$	2.62024	0.102933
$649$	−12.7413	−0.500140
$650$	26.6766	1.04634
$651$	0.605187	0.0237192
$652$	0.577162	0.0226034
$653$	−37.7043	−1.47548	−0.737742	−	0.675082i	$-0.764108\pi$
$653$	−37.7043	−1.47548	−0.737742	+	0.675082i	$0.764108\pi$
$654$	7.73831	0.302592
$655$	21.7809	0.851049
$656$	−9.19987	−0.359195
$657$	3.57156	0.139340
$658$	−9.63930	−0.375779
$659$	24.4696	0.953201	0.476600	−	0.879120i	$-0.341869\pi$
$659$	24.4696	0.953201	0.476600	+	0.879120i	$0.341869\pi$
$660$	−4.89541	−0.190553
$661$	11.9941	0.466517	0.233259	−	0.972415i	$-0.425061\pi$
$661$	11.9941	0.466517	0.233259	+	0.972415i	$0.425061\pi$
$662$	6.86678	0.266885
$663$	15.8027	0.613727
$664$	42.0496	1.63184
$665$	5.03039	0.195070
$666$	−4.22193	−0.163597
$667$	−14.4329	−0.558845
$668$	−0.788690	−0.0305153
$669$	3.74967	0.144970
$670$	−3.32573	−0.128484
$671$	10.1131	0.390413
$672$	−5.85907	−0.226019
$673$	−17.0262	−0.656312	−0.328156	−	0.944623i	$-0.606427\pi$
$673$	−17.0262	−0.656312	−0.328156	+	0.944623i	$0.606427\pi$
$674$	−10.4489	−0.402476
$675$	5.56832	0.214325
$676$	−36.4412	−1.40159
$677$	−12.6238	−0.485172	−0.242586	−	0.970130i	$-0.577996\pi$
$677$	−12.6238	−0.485172	−0.242586	+	0.970130i	$0.577996\pi$
$678$	4.98850	0.191582
$679$	14.0536	0.539329
$680$	−21.5944	−0.828106
$681$	8.50905	0.326067
$682$	−0.496972	−0.0190301
$683$	−19.8537	−0.759679	−0.379839	−	0.925052i	$-0.624021\pi$
$683$	−19.8537	−0.759679	−0.379839	+	0.925052i	$0.624021\pi$
$684$	−2.18079	−0.0833844
$685$	2.72124	0.103973
$686$	−0.768548	−0.0293433
$687$	18.1401	0.692089
$688$	9.64559	0.367735
$689$	−43.4212	−1.65422
$690$	−12.1703	−0.463315
$691$	−7.69107	−0.292582	−0.146291	−	0.989242i	$-0.546734\pi$
$691$	−7.69107	−0.292582	−0.146291	+	0.989242i	$0.546734\pi$
$692$	27.2713	1.03670
$693$	1.06849	0.0405887
$694$	5.63106	0.213752
$695$	−27.1669	−1.03050
$696$	−7.76369	−0.294282
$697$	28.9761	1.09755
$698$	14.5799	0.551858
$699$	−6.85842	−0.259409
$700$	−7.84763	−0.296613
$701$	32.5719	1.23022	0.615111	−	0.788440i	$-0.289111\pi$
$701$	32.5719	1.23022	0.615111	+	0.788440i	$0.289111\pi$
$702$	4.79077	0.180816
$703$	8.50040	0.320599
$704$	3.09135	0.116510
$705$	40.7735	1.53562
$706$	8.25792	0.310791
$707$	3.02767	0.113867
$708$	16.8057	0.631597
$709$	29.2674	1.09916	0.549580	−	0.835441i	$-0.314788\pi$
$709$	29.2674	1.09916	0.549580	+	0.835441i	$0.314788\pi$
$710$	−11.3411	−0.425623
$711$	0.524648	0.0196758
$712$	17.8188	0.667788
$713$	2.94793	0.110401
$714$	1.94835	0.0729153
$715$	−21.6526	−0.809761
$716$	7.44803	0.278346
$717$	3.15820	0.117945
$718$	−10.9800	−0.409768
$719$	−29.1827	−1.08833	−0.544166	−	0.838978i	$-0.683154\pi$
$719$	−29.1827	−1.08833	−0.544166	+	0.838978i	$0.683154\pi$
$720$	2.61663	0.0975159
$721$	13.3948	0.498847
$722$	12.7622	0.474960
$723$	22.8494	0.849777
$724$	6.50078	0.241599
$725$	−16.4988	−0.612749
$726$	7.57659	0.281194
$727$	−1.61164	−0.0597724	−0.0298862	−	0.999553i	$-0.509514\pi$
$727$	−1.61164	−0.0597724	−0.0298862	+	0.999553i	$0.509514\pi$
$728$	−16.3334	−0.605354
$729$	1.00000	0.0370370
$730$	−8.92342	−0.330270
$731$	−30.3800	−1.12364
$732$	−13.3391	−0.493029
$733$	−36.3980	−1.34439	−0.672196	−	0.740374i	$-0.734649\pi$
$733$	−36.3980	−1.34439	−0.672196	+	0.740374i	$0.734649\pi$
$734$	15.2496	0.562873
$735$	3.25090	0.119911
$736$	−28.5401	−1.05200
$737$	1.42228	0.0523904
$738$	8.78444	0.323360
$739$	25.5258	0.938980	0.469490	−	0.882938i	$-0.344438\pi$
$739$	25.5258	0.938980	0.469490	+	0.882938i	$0.344438\pi$
$740$	−25.1685	−0.925214
$741$	−9.64570	−0.354344
$742$	−5.35350	−0.196533
$743$	−25.3357	−0.929477	−0.464739	−	0.885448i	$-0.653852\pi$
$743$	−25.3357	−0.929477	−0.464739	+	0.885448i	$0.653852\pi$
$744$	1.58573	0.0581358
$745$	−36.3444	−1.33155
$746$	−3.02720	−0.110834
$747$	16.0480	0.587167
$748$	3.81753	0.139583
$749$	11.8784	0.434029
$750$	−1.41993	−0.0518487
$751$	−8.98646	−0.327921	−0.163960	−	0.986467i	$-0.552427\pi$
$751$	−8.98646	−0.327921	−0.163960	+	0.986467i	$0.552427\pi$
$752$	10.0952	0.368133
$753$	−0.695418	−0.0253424
$754$	−14.1949	−0.516949
$755$	−57.1765	−2.08087
$756$	−1.40933	−0.0512570
$757$	−6.60900	−0.240208	−0.120104	−	0.992761i	$-0.538323\pi$
$757$	−6.60900	−0.240208	−0.120104	+	0.992761i	$0.538323\pi$
$758$	−9.60247	−0.348778
$759$	5.20474	0.188920
$760$	13.1808	0.478118
$761$	26.1877	0.949303	0.474652	−	0.880174i	$-0.342574\pi$
$761$	26.1877	0.949303	0.474652	+	0.880174i	$0.342574\pi$
$762$	17.2660	0.625480
$763$	−10.0687	−0.364513
$764$	−1.40933	−0.0509879
$765$	−8.24138	−0.297968
$766$	8.91731	0.322196
$767$	74.3323	2.68398
$768$	−13.0834	−0.472107
$769$	32.3192	1.16546	0.582730	−	0.812666i	$-0.301985\pi$
$769$	32.3192	1.16546	0.582730	+	0.812666i	$0.301985\pi$
$770$	−2.66960	−0.0962055
$771$	19.3610	0.697270
$772$	−4.64606	−0.167215
$773$	23.9652	0.861969	0.430985	−	0.902359i	$-0.358166\pi$
$773$	23.9652	0.861969	0.430985	+	0.902359i	$0.358166\pi$
$774$	−9.21003	−0.331048
$775$	3.36988	0.121050
$776$	36.8238	1.32190
$777$	5.49339	0.197074
$778$	6.60246	0.236710
$779$	−17.6865	−0.633685
$780$	28.5596	1.02260
$781$	4.85011	0.173551
$782$	9.49062	0.339384
$783$	−2.96297	−0.105888
$784$	0.804894	0.0287462
$785$	8.94349	0.319207
$786$	−5.14924	−0.183667
$787$	24.7512	0.882285	0.441142	−	0.897437i	$-0.354573\pi$
$787$	24.7512	0.882285	0.441142	+	0.897437i	$0.354573\pi$
$788$	19.5886	0.697816
$789$	19.0497	0.678188
$790$	−1.31082	−0.0466368
$791$	−6.49082	−0.230787
$792$	2.79970	0.0994831
$793$	−58.9995	−2.09513
$794$	15.5557	0.552052
$795$	22.6449	0.803131
$796$	−11.4201	−0.404773
$797$	−24.8337	−0.879655	−0.439828	−	0.898082i	$-0.644960\pi$
$797$	−24.8337	−0.879655	−0.439828	+	0.898082i	$0.644960\pi$
$798$	−1.18924	−0.0420986
$799$	−31.7959	−1.12486
$800$	−32.6252	−1.15347
$801$	6.80046	0.240282
$802$	11.4956	0.405923
$803$	3.81618	0.134670
$804$	−1.87598	−0.0661606
$805$	15.8354	0.558126
$806$	2.89932	0.102124
$807$	−1.15964	−0.0408211
$808$	7.93320	0.279089
$809$	−37.1859	−1.30739	−0.653693	−	0.756760i	$-0.726781\pi$
$809$	−37.1859	−1.30739	−0.653693	+	0.756760i	$0.726781\pi$
$810$	−2.49847	−0.0877872
$811$	17.5690	0.616932	0.308466	−	0.951235i	$-0.400184\pi$
$811$	17.5690	0.616932	0.308466	+	0.951235i	$0.400184\pi$
$812$	4.17582	0.146543
$813$	11.8300	0.414896
$814$	−4.51111	−0.158114
$815$	−1.33133	−0.0466346
$816$	−2.04049	−0.0714316
$817$	18.5434	0.648751
$818$	−14.8177	−0.518089
$819$	−6.23354	−0.217818
$820$	52.3673	1.82875
$821$	−20.3379	−0.709799	−0.354899	−	0.934905i	$-0.615485\pi$
$821$	−20.3379	−0.709799	−0.354899	+	0.934905i	$0.615485\pi$
$822$	−0.643332	−0.0224388
$823$	21.1224	0.736280	0.368140	−	0.929770i	$-0.379995\pi$
$823$	21.1224	0.736280	0.368140	+	0.929770i	$0.379995\pi$
$824$	35.0974	1.22268
$825$	5.94971	0.207142
$826$	9.16459	0.318877
$827$	−13.7241	−0.477234	−0.238617	−	0.971114i	$-0.576694\pi$
$827$	−13.7241	−0.477234	−0.238617	+	0.971114i	$0.576694\pi$
$828$	−6.86501	−0.238576
$829$	9.45028	0.328222	0.164111	−	0.986442i	$-0.447525\pi$
$829$	9.45028	0.328222	0.164111	+	0.986442i	$0.447525\pi$
$830$	−40.0955	−1.39173
$831$	−3.15641	−0.109495
$832$	−18.0348	−0.625244
$833$	−2.53511	−0.0878363
$834$	6.42255	0.222395
$835$	1.81926	0.0629581
$836$	−2.33015	−0.0805900
$837$	0.605187	0.0209183
$838$	−17.2915	−0.597323
$839$	26.6821	0.921167	0.460583	−	0.887616i	$-0.347640\pi$
$839$	26.6821	0.921167	0.460583	+	0.887616i	$0.347640\pi$
$840$	8.51811	0.293903
$841$	−20.2208	−0.697269
$842$	0.924509	0.0318607
$843$	−7.56086	−0.260410
$844$	−18.5224	−0.637568
$845$	84.0586	2.89170
$846$	−9.63930	−0.331406
$847$	−9.85832	−0.338736
$848$	5.60668	0.192534
$849$	0.195204	0.00669940
$850$	10.8491	0.372119
$851$	26.7589	0.917282
$852$	−6.39727	−0.219167
$853$	15.4560	0.529205	0.264603	−	0.964358i	$-0.414759\pi$
$853$	15.4560	0.529205	0.264603	+	0.964358i	$0.414759\pi$
$854$	−7.27418	−0.248917
$855$	5.03039	0.172036
$856$	31.1243	1.06381
$857$	−18.7575	−0.640746	−0.320373	−	0.947292i	$-0.603808\pi$
$857$	−18.7575	−0.640746	−0.320373	+	0.947292i	$0.603808\pi$
$858$	5.11891	0.174757
$859$	−0.262737	−0.00896447	−0.00448224	−	0.999990i	$-0.501427\pi$
$859$	−0.262737	−0.00896447	−0.00448224	+	0.999990i	$0.501427\pi$
$860$	−54.9045	−1.87223
$861$	−11.4299	−0.389531
$862$	−11.8492	−0.403585
$863$	49.4517	1.68335	0.841677	−	0.539981i	$-0.181569\pi$
$863$	49.4517	1.68335	0.841677	+	0.539981i	$0.181569\pi$
$864$	−5.85907	−0.199330
$865$	−62.9064	−2.13888
$866$	5.93263	0.201599
$867$	−10.5732	−0.359085
$868$	−0.852912	−0.0289497
$869$	0.560583	0.0190165
$870$	7.40289	0.250981
$871$	−8.29752	−0.281151
$872$	−26.3825	−0.893424
$873$	14.0536	0.475643
$874$	−5.79291	−0.195948
$875$	1.84756	0.0624588
$876$	−5.03352	−0.170067
$877$	−49.9338	−1.68614	−0.843072	−	0.537800i	$-0.819256\pi$
$877$	−49.9338	−1.68614	−0.843072	+	0.537800i	$0.819256\pi$
$878$	12.0327	0.406083
$879$	−1.32091	−0.0445532
$880$	2.79585	0.0942480
$881$	52.2219	1.75940	0.879699	−	0.475530i	$-0.157744\pi$
$881$	52.2219	1.75940	0.879699	+	0.475530i	$0.157744\pi$
$882$	−0.768548	−0.0258783
$883$	−25.8673	−0.870504	−0.435252	−	0.900309i	$-0.643341\pi$
$883$	−25.8673	−0.870504	−0.435252	+	0.900309i	$0.643341\pi$
$884$	−22.2713	−0.749065
$885$	−38.7655	−1.30309
$886$	7.95532	0.267264
$887$	35.7854	1.20156	0.600778	−	0.799416i	$-0.294857\pi$
$887$	35.7854	1.20156	0.600778	+	0.799416i	$0.294857\pi$
$888$	14.3940	0.483031
$889$	−22.4657	−0.753475
$890$	−16.9907	−0.569530
$891$	1.06849	0.0357959
$892$	−5.28453	−0.176939
$893$	19.4077	0.649453
$894$	8.59221	0.287366
$895$	−17.1803	−0.574274
$896$	9.49459	0.317192
$897$	−30.3642	−1.01383
$898$	−15.9567	−0.532482
$899$	−1.79315	−0.0598050
$900$	−7.84763	−0.261588
$901$	−17.6589	−0.588304
$902$	9.38611	0.312523
$903$	11.9837	0.398792
$904$	−17.0075	−0.565660
$905$	−14.9953	−0.498459
$906$	13.5171	0.449077
$907$	−47.6695	−1.58284	−0.791420	−	0.611272i	$-0.790658\pi$
$907$	−47.6695	−1.58284	−0.791420	+	0.611272i	$0.790658\pi$
$908$	−11.9921	−0.397972
$909$	3.02767	0.100421
$910$	15.5743	0.516283
$911$	−29.5162	−0.977915	−0.488957	−	0.872308i	$-0.662623\pi$
$911$	−29.5162	−0.977915	−0.488957	+	0.872308i	$0.662623\pi$
$912$	1.24548	0.0412420
$913$	17.1472	0.567490
$914$	22.5368	0.745453
$915$	30.7692	1.01720
$916$	−25.5655	−0.844709
$917$	6.69996	0.221252
$918$	1.94835	0.0643052
$919$	−37.7765	−1.24613	−0.623066	−	0.782169i	$-0.714113\pi$
$919$	−37.7765	−1.24613	−0.623066	+	0.782169i	$0.714113\pi$
$920$	41.4926	1.36797
$921$	−11.1993	−0.369031
$922$	−0.550627	−0.0181339
$923$	−28.2954	−0.931353
$924$	−1.50586	−0.0495393
$925$	30.5890	1.00576
$926$	−12.2499	−0.402556
$927$	13.3948	0.439941
$928$	17.3603	0.569879
$929$	29.0930	0.954510	0.477255	−	0.878765i	$-0.341632\pi$
$929$	29.0930	0.954510	0.477255	+	0.878765i	$0.341632\pi$
$930$	−1.51204	−0.0495818
$931$	1.54739	0.0507135
$932$	9.66580	0.316614
$933$	29.7318	0.973376
$934$	−22.7192	−0.743396
$935$	−8.80585	−0.287982
$936$	−16.3334	−0.533872
$937$	−29.9569	−0.978649	−0.489325	−	0.872102i	$-0.662757\pi$
$937$	−29.9569	−0.978649	−0.489325	+	0.872102i	$0.662757\pi$
$938$	−1.02302	−0.0334028
$939$	−6.14632	−0.200577
$940$	−57.4635	−1.87425
$941$	−32.8456	−1.07074	−0.535368	−	0.844619i	$-0.679827\pi$
$941$	−32.8456	−1.07074	−0.535368	+	0.844619i	$0.679827\pi$
$942$	−2.11434	−0.0688889
$943$	−55.6763	−1.81307
$944$	−9.59801	−0.312388
$945$	3.25090	0.105752
$946$	−9.84086	−0.319954
$947$	−45.5633	−1.48061	−0.740304	−	0.672273i	$-0.765318\pi$
$947$	−45.5633	−1.48061	−0.740304	+	0.672273i	$0.765318\pi$
$948$	−0.739405	−0.0240148
$949$	−22.2634	−0.722702
$950$	−6.62207	−0.214848
$951$	29.8328	0.967395
$952$	−6.64259	−0.215287
$953$	17.9329	0.580903	0.290452	−	0.956890i	$-0.406194\pi$
$953$	17.9329	0.580903	0.290452	+	0.956890i	$0.406194\pi$
$954$	−5.35350	−0.173326
$955$	3.25090	0.105197
$956$	−4.45095	−0.143954
$957$	−3.16591	−0.102339
$958$	18.1970	0.587918
$959$	0.837074	0.0270305
$960$	9.40545	0.303560
$961$	−30.6337	−0.988185
$962$	26.3176	0.848514
$963$	11.8784	0.382777
$964$	−32.2024	−1.03717
$965$	10.7170	0.344993
$966$	−3.74367	−0.120451
$967$	37.5147	1.20639	0.603196	−	0.797593i	$-0.293894\pi$
$967$	37.5147	1.20639	0.603196	+	0.797593i	$0.293894\pi$
$968$	−25.8311	−0.830244
$969$	−3.92279	−0.126018
$970$	−35.1125	−1.12740
$971$	−18.5132	−0.594116	−0.297058	−	0.954859i	$-0.596006\pi$
$971$	−18.5132	−0.594116	−0.297058	+	0.954859i	$0.596006\pi$
$972$	−1.40933	−0.0452044
$973$	−8.35674	−0.267905
$974$	26.0195	0.833719
$975$	−34.7104	−1.11162
$976$	7.61819	0.243852
$977$	37.9008	1.21256	0.606278	−	0.795253i	$-0.292662\pi$
$977$	37.9008	1.21256	0.606278	+	0.795253i	$0.292662\pi$
$978$	0.314742	0.0100643
$979$	7.26624	0.232230
$980$	−4.58160	−0.146354
$981$	−10.0687	−0.321470
$982$	10.1397	0.323570
$983$	−47.9061	−1.52797	−0.763983	−	0.645237i	$-0.776759\pi$
$983$	−47.9061	−1.52797	−0.763983	+	0.645237i	$0.776759\pi$
$984$	−29.9491	−0.954742
$985$	−45.1849	−1.43971
$986$	−5.77291	−0.183847
$987$	12.5422	0.399224
$988$	13.5940	0.432483
$989$	58.3737	1.85618
$990$	−2.66960	−0.0848453
$991$	10.9146	0.346715	0.173357	−	0.984859i	$-0.444538\pi$
$991$	10.9146	0.346715	0.173357	+	0.984859i	$0.444538\pi$
$992$	−3.54584	−0.112580
$993$	−8.93475	−0.283536
$994$	−3.48860	−0.110652
$995$	26.3425	0.835114
$996$	−22.6170	−0.716648
$997$	44.9238	1.42275	0.711376	−	0.702812i	$-0.248072\pi$
$997$	44.9238	1.42275	0.711376	+	0.702812i	$0.248072\pi$
$998$	18.5660	0.587695
$999$	5.49339	0.173803

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4011.2.a.k.1.9	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4011.2.a.k.1.9	✓	27	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:	\( N \)	\(=\)	\( 4011 = 3 \cdot 7 \cdot 191 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4011.a (trivial)

\(f(q)\)	\(=\)	\(q-0.768548 q^{2} +1.00000 q^{3} -1.40933 q^{4} +3.25090 q^{5} -0.768548 q^{6} +1.00000 q^{7} +2.62024 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.768548 q^{2} +1.00000 q^{3} -1.40933 q^{4} +3.25090 q^{5} -0.768548 q^{6} +1.00000 q^{7} +2.62024 q^{8} +1.00000 q^{9} -2.49847 q^{10} +1.06849 q^{11} -1.40933 q^{12} -6.23354 q^{13} -0.768548 q^{14} +3.25090 q^{15} +0.804894 q^{16} -2.53511 q^{17} -0.768548 q^{18} +1.54739 q^{19} -4.58160 q^{20} +1.00000 q^{21} -0.821188 q^{22} +4.87110 q^{23} +2.62024 q^{24} +5.56832 q^{25} +4.79077 q^{26} +1.00000 q^{27} -1.40933 q^{28} -2.96297 q^{29} -2.49847 q^{30} +0.605187 q^{31} -5.85907 q^{32} +1.06849 q^{33} +1.94835 q^{34} +3.25090 q^{35} -1.40933 q^{36} +5.49339 q^{37} -1.18924 q^{38} -6.23354 q^{39} +8.51811 q^{40} -11.4299 q^{41} -0.768548 q^{42} +11.9837 q^{43} -1.50586 q^{44} +3.25090 q^{45} -3.74367 q^{46} +12.5422 q^{47} +0.804894 q^{48} +1.00000 q^{49} -4.27952 q^{50} -2.53511 q^{51} +8.78515 q^{52} +6.96574 q^{53} -0.768548 q^{54} +3.47356 q^{55} +2.62024 q^{56} +1.54739 q^{57} +2.27718 q^{58} -11.9246 q^{59} -4.58160 q^{60} +9.46484 q^{61} -0.465115 q^{62} +1.00000 q^{63} +2.89319 q^{64} -20.2646 q^{65} -0.821188 q^{66} +1.33111 q^{67} +3.57282 q^{68} +4.87110 q^{69} -2.49847 q^{70} +4.53921 q^{71} +2.62024 q^{72} +3.57156 q^{73} -4.22193 q^{74} +5.56832 q^{75} -2.18079 q^{76} +1.06849 q^{77} +4.79077 q^{78} +0.524648 q^{79} +2.61663 q^{80} +1.00000 q^{81} +8.78444 q^{82} +16.0480 q^{83} -1.40933 q^{84} -8.24138 q^{85} -9.21003 q^{86} -2.96297 q^{87} +2.79970 q^{88} +6.80046 q^{89} -2.49847 q^{90} -6.23354 q^{91} -6.86501 q^{92} +0.605187 q^{93} -9.63930 q^{94} +5.03039 q^{95} -5.85907 q^{96} +14.0536 q^{97} -0.768548 q^{98} +1.06849 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 9 q^{2} + 27 q^{3} + 31 q^{4} + 23 q^{5} + 9 q^{6} + 27 q^{7} + 18 q^{8} + 27 q^{9}+O(q^{10}) \) 27 * q + 9 * q^2 + 27 * q^3 + 31 * q^4 + 23 * q^5 + 9 * q^6 + 27 * q^7 + 18 * q^8 + 27 * q^9	\( 27 q + 9 q^{2} + 27 q^{3} + 31 q^{4} + 23 q^{5} + 9 q^{6} + 27 q^{7} + 18 q^{8} + 27 q^{9} + 10 q^{10} + 22 q^{11} + 31 q^{12} + 15 q^{13} + 9 q^{14} + 23 q^{15} + 39 q^{16} + 22 q^{17} + 9 q^{18} + 24 q^{19} + 46 q^{20} + 27 q^{21} - 19 q^{22} + 36 q^{23} + 18 q^{24} + 38 q^{25} + 19 q^{26} + 27 q^{27} + 31 q^{28} + 32 q^{29} + 10 q^{30} + 11 q^{31} + 15 q^{32} + 22 q^{33} - 4 q^{34} + 23 q^{35} + 31 q^{36} + 6 q^{37} + 8 q^{38} + 15 q^{39} + 16 q^{41} + 9 q^{42} - 11 q^{43} + 22 q^{44} + 23 q^{45} - 56 q^{46} + 39 q^{47} + 39 q^{48} + 27 q^{49} - 7 q^{50} + 22 q^{51} + 10 q^{52} + 24 q^{53} + 9 q^{54} + 16 q^{55} + 18 q^{56} + 24 q^{57} - 31 q^{58} + 31 q^{59} + 46 q^{60} + 28 q^{61} - 4 q^{62} + 27 q^{63} + 40 q^{64} + 33 q^{65} - 19 q^{66} - 7 q^{67} + 25 q^{68} + 36 q^{69} + 10 q^{70} + 49 q^{71} + 18 q^{72} - 13 q^{73} + 7 q^{74} + 38 q^{75} + 15 q^{76} + 22 q^{77} + 19 q^{78} - 18 q^{79} + 78 q^{80} + 27 q^{81} + 31 q^{82} + 59 q^{83} + 31 q^{84} - 20 q^{85} + 35 q^{86} + 32 q^{87} - 49 q^{88} + 49 q^{89} + 10 q^{90} + 15 q^{91} + 52 q^{92} + 11 q^{93} + 17 q^{94} + 38 q^{95} + 15 q^{96} - 7 q^{97} + 9 q^{98} + 22 q^{99}+O(q^{100}) \) 27 * q + 9 * q^2 + 27 * q^3 + 31 * q^4 + 23 * q^5 + 9 * q^6 + 27 * q^7 + 18 * q^8 + 27 * q^9 + 10 * q^10 + 22 * q^11 + 31 * q^12 + 15 * q^13 + 9 * q^14 + 23 * q^15 + 39 * q^16 + 22 * q^17 + 9 * q^18 + 24 * q^19 + 46 * q^20 + 27 * q^21 - 19 * q^22 + 36 * q^23 + 18 * q^24 + 38 * q^25 + 19 * q^26 + 27 * q^27 + 31 * q^28 + 32 * q^29 + 10 * q^30 + 11 * q^31 + 15 * q^32 + 22 * q^33 - 4 * q^34 + 23 * q^35 + 31 * q^36 + 6 * q^37 + 8 * q^38 + 15 * q^39 + 16 * q^41 + 9 * q^42 - 11 * q^43 + 22 * q^44 + 23 * q^45 - 56 * q^46 + 39 * q^47 + 39 * q^48 + 27 * q^49 - 7 * q^50 + 22 * q^51 + 10 * q^52 + 24 * q^53 + 9 * q^54 + 16 * q^55 + 18 * q^56 + 24 * q^57 - 31 * q^58 + 31 * q^59 + 46 * q^60 + 28 * q^61 - 4 * q^62 + 27 * q^63 + 40 * q^64 + 33 * q^65 - 19 * q^66 - 7 * q^67 + 25 * q^68 + 36 * q^69 + 10 * q^70 + 49 * q^71 + 18 * q^72 - 13 * q^73 + 7 * q^74 + 38 * q^75 + 15 * q^76 + 22 * q^77 + 19 * q^78 - 18 * q^79 + 78 * q^80 + 27 * q^81 + 31 * q^82 + 59 * q^83 + 31 * q^84 - 20 * q^85 + 35 * q^86 + 32 * q^87 - 49 * q^88 + 49 * q^89 + 10 * q^90 + 15 * q^91 + 52 * q^92 + 11 * q^93 + 17 * q^94 + 38 * q^95 + 15 * q^96 - 7 * q^97 + 9 * q^98 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more