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Properties

Label 1445.2.d.j.866.6
Level $1445$
Weight $2$
Character 1445.866
Analytic conductor $11.538$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1445,2,Mod(866,1445)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1445.866"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1445.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,8,0,24,0,0,0,24,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(11.5383830921\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 866.6
Character \(\chi\) \(=\) 1445.866
Dual form 1445.2.d.j.866.5

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.43840 q^{2} +0.109907i q^{3} +0.0689897 q^{4} +1.00000i q^{5} -0.158090i q^{6} +0.695085i q^{7} +2.77756 q^{8} +2.98792 q^{9} -1.43840i q^{10} +4.85089i q^{11} +0.00758244i q^{12} +5.63906 q^{13} -0.999809i q^{14} -0.109907 q^{15} -4.13322 q^{16} -4.29782 q^{18} +2.32272 q^{19} +0.0689897i q^{20} -0.0763945 q^{21} -6.97750i q^{22} +4.63686i q^{23} +0.305273i q^{24} -1.00000 q^{25} -8.11121 q^{26} +0.658113i q^{27} +0.0479537i q^{28} -6.50618i q^{29} +0.158090 q^{30} -6.63194i q^{31} +0.390093 q^{32} -0.533145 q^{33} -0.695085 q^{35} +0.206136 q^{36} -0.118625i q^{37} -3.34100 q^{38} +0.619770i q^{39} +2.77756i q^{40} +1.07877i q^{41} +0.109886 q^{42} +0.641108 q^{43} +0.334661i q^{44} +2.98792i q^{45} -6.66965i q^{46} +4.93703 q^{47} -0.454269i q^{48} +6.51686 q^{49} +1.43840 q^{50} +0.389037 q^{52} -11.9864 q^{53} -0.946629i q^{54} -4.85089 q^{55} +1.93064i q^{56} +0.255283i q^{57} +9.35848i q^{58} +9.91829 q^{59} -0.00758244 q^{60} +1.60292i q^{61} +9.53937i q^{62} +2.07686i q^{63} +7.70533 q^{64} +5.63906i q^{65} +0.766875 q^{66} -2.99411 q^{67} -0.509622 q^{69} +0.999809 q^{70} +4.68852i q^{71} +8.29913 q^{72} -5.49911i q^{73} +0.170630i q^{74} -0.109907i q^{75} +0.160244 q^{76} -3.37178 q^{77} -0.891477i q^{78} +14.8439i q^{79} -4.13322i q^{80} +8.89143 q^{81} -1.55170i q^{82} -5.03506 q^{83} -0.00527044 q^{84} -0.922169 q^{86} +0.715073 q^{87} +13.4736i q^{88} -2.35657 q^{89} -4.29782i q^{90} +3.91962i q^{91} +0.319896i q^{92} +0.728895 q^{93} -7.10141 q^{94} +2.32272i q^{95} +0.0428738i q^{96} +2.70080i q^{97} -9.37384 q^{98} +14.4941i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 8 q^{2} + 24 q^{4} + 24 q^{8} - 24 q^{9} - 16 q^{13} + 16 q^{15} + 24 q^{16} + 8 q^{18} + 32 q^{21} - 24 q^{25} - 32 q^{26} - 16 q^{30} + 56 q^{32} - 32 q^{35} - 24 q^{36} - 48 q^{38} + 32 q^{43}+ \cdots - 120 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1445\mathbb{Z}\right)^\times\).

\(n\) \(581\) \(1157\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.43840 −1.01710 −0.508551 − 0.861032i \(-0.669819\pi\)
−0.508551 + 0.861032i \(0.669819\pi\)
\(3\) 0.109907i 0.0634547i 0.999497 + 0.0317274i \(0.0101008\pi\)
−0.999497 + 0.0317274i \(0.989899\pi\)
\(4\) 0.0689897 0.0344949
\(5\) 1.00000i 0.447214i
\(6\) − 0.158090i − 0.0645399i
\(7\) 0.695085i 0.262717i 0.991335 + 0.131359i \(0.0419339\pi\)
−0.991335 + 0.131359i \(0.958066\pi\)
\(8\) 2.77756 0.982016
\(9\) 2.98792 0.995974
\(10\) − 1.43840i − 0.454861i
\(11\) 4.85089i 1.46260i 0.682058 + 0.731298i \(0.261085\pi\)
−0.682058 + 0.731298i \(0.738915\pi\)
\(12\) 0.00758244i 0.00218886i
\(13\) 5.63906 1.56399 0.781996 − 0.623283i \(-0.214202\pi\)
0.781996 + 0.623283i \(0.214202\pi\)
\(14\) − 0.999809i − 0.267210i
\(15\) −0.109907 −0.0283778
\(16\) −4.13322 −1.03330
\(17\) 0 0
\(18\) −4.29782 −1.01301
\(19\) 2.32272 0.532869 0.266434 − 0.963853i \(-0.414154\pi\)
0.266434 + 0.963853i \(0.414154\pi\)
\(20\) 0.0689897i 0.0154266i
\(21\) −0.0763945 −0.0166706
\(22\) − 6.97750i − 1.48761i
\(23\) 4.63686i 0.966852i 0.875385 + 0.483426i \(0.160608\pi\)
−0.875385 + 0.483426i \(0.839392\pi\)
\(24\) 0.305273i 0.0623136i
\(25\) −1.00000 −0.200000
\(26\) −8.11121 −1.59074
\(27\) 0.658113i 0.126654i
\(28\) 0.0479537i 0.00906240i
\(29\) − 6.50618i − 1.20817i −0.796921 − 0.604084i \(-0.793539\pi\)
0.796921 − 0.604084i \(-0.206461\pi\)
\(30\) 0.158090 0.0288631
\(31\) − 6.63194i − 1.19113i −0.803307 − 0.595565i \(-0.796928\pi\)
0.803307 − 0.595565i \(-0.203072\pi\)
\(32\) 0.390093 0.0689593
\(33\) −0.533145 −0.0928087
\(34\) 0 0
\(35\) −0.695085 −0.117491
\(36\) 0.206136 0.0343560
\(37\) − 0.118625i − 0.0195018i −0.999952 − 0.00975091i \(-0.996896\pi\)
0.999952 − 0.00975091i \(-0.00310386\pi\)
\(38\) −3.34100 −0.541981
\(39\) 0.619770i 0.0992427i
\(40\) 2.77756i 0.439171i
\(41\) 1.07877i 0.168475i 0.996446 + 0.0842375i \(0.0268454\pi\)
−0.996446 + 0.0842375i \(0.973155\pi\)
\(42\) 0.109886 0.0169557
\(43\) 0.641108 0.0977681 0.0488840 − 0.998804i \(-0.484434\pi\)
0.0488840 + 0.998804i \(0.484434\pi\)
\(44\) 0.334661i 0.0504521i
\(45\) 2.98792i 0.445413i
\(46\) − 6.66965i − 0.983386i
\(47\) 4.93703 0.720139 0.360070 − 0.932925i \(-0.382753\pi\)
0.360070 + 0.932925i \(0.382753\pi\)
\(48\) − 0.454269i − 0.0655681i
\(49\) 6.51686 0.930980
\(50\) 1.43840 0.203420
\(51\) 0 0
\(52\) 0.389037 0.0539497
\(53\) −11.9864 −1.64646 −0.823228 − 0.567711i \(-0.807829\pi\)
−0.823228 + 0.567711i \(0.807829\pi\)
\(54\) − 0.946629i − 0.128820i
\(55\) −4.85089 −0.654093
\(56\) 1.93064i 0.257993i
\(57\) 0.255283i 0.0338130i
\(58\) 9.35848i 1.22883i
\(59\) 9.91829 1.29125 0.645626 − 0.763654i \(-0.276597\pi\)
0.645626 + 0.763654i \(0.276597\pi\)
\(60\) −0.00758244 −0.000978888 0
\(61\) 1.60292i 0.205233i 0.994721 + 0.102617i \(0.0327215\pi\)
−0.994721 + 0.102617i \(0.967278\pi\)
\(62\) 9.53937i 1.21150i
\(63\) 2.07686i 0.261659i
\(64\) 7.70533 0.963166
\(65\) 5.63906i 0.699439i
\(66\) 0.766875 0.0943958
\(67\) −2.99411 −0.365789 −0.182894 − 0.983133i \(-0.558547\pi\)
−0.182894 + 0.983133i \(0.558547\pi\)
\(68\) 0 0
\(69\) −0.509622 −0.0613513
\(70\) 0.999809 0.119500
\(71\) 4.68852i 0.556425i 0.960520 + 0.278213i \(0.0897420\pi\)
−0.960520 + 0.278213i \(0.910258\pi\)
\(72\) 8.29913 0.978062
\(73\) − 5.49911i − 0.643622i −0.946804 − 0.321811i \(-0.895708\pi\)
0.946804 − 0.321811i \(-0.104292\pi\)
\(74\) 0.170630i 0.0198353i
\(75\) − 0.109907i − 0.0126909i
\(76\) 0.160244 0.0183812
\(77\) −3.37178 −0.384250
\(78\) − 0.891477i − 0.100940i
\(79\) 14.8439i 1.67007i 0.550193 + 0.835037i \(0.314554\pi\)
−0.550193 + 0.835037i \(0.685446\pi\)
\(80\) − 4.13322i − 0.462108i
\(81\) 8.89143 0.987937
\(82\) − 1.55170i − 0.171356i
\(83\) −5.03506 −0.552670 −0.276335 − 0.961061i \(-0.589120\pi\)
−0.276335 + 0.961061i \(0.589120\pi\)
\(84\) −0.00527044 −0.000575052 0
\(85\) 0 0
\(86\) −0.922169 −0.0994400
\(87\) 0.715073 0.0766639
\(88\) 13.4736i 1.43629i
\(89\) −2.35657 −0.249796 −0.124898 − 0.992170i \(-0.539860\pi\)
−0.124898 + 0.992170i \(0.539860\pi\)
\(90\) − 4.29782i − 0.453030i
\(91\) 3.91962i 0.410888i
\(92\) 0.319896i 0.0333514i
\(93\) 0.728895 0.0755829
\(94\) −7.10141 −0.732454
\(95\) 2.32272i 0.238306i
\(96\) 0.0428738i 0.00437579i
\(97\) 2.70080i 0.274225i 0.990555 + 0.137113i \(0.0437822\pi\)
−0.990555 + 0.137113i \(0.956218\pi\)
\(98\) −9.37384 −0.946900
\(99\) 14.4941i 1.45671i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1445.2.d.j.866.6 24
17.4 even 4 1445.2.a.p.1.10 12
17.5 odd 16 85.2.l.a.26.5 ✓ 24
17.10 odd 16 85.2.l.a.36.5 yes 24
17.13 even 4 1445.2.a.q.1.10 12
17.16 even 2 inner 1445.2.d.j.866.5 24
51.5 even 16 765.2.be.b.451.2 24
51.44 even 16 765.2.be.b.631.2 24
85.4 even 4 7225.2.a.bs.1.3 12
85.22 even 16 425.2.n.f.349.5 24
85.27 even 16 425.2.n.c.274.2 24
85.39 odd 16 425.2.m.b.26.2 24
85.44 odd 16 425.2.m.b.376.2 24
85.64 even 4 7225.2.a.bq.1.3 12
85.73 even 16 425.2.n.c.349.2 24
85.78 even 16 425.2.n.f.274.5 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.l.a.26.5 ✓ 24 17.5 odd 16
85.2.l.a.36.5 yes 24 17.10 odd 16
425.2.m.b.26.2 24 85.39 odd 16
425.2.m.b.376.2 24 85.44 odd 16
425.2.n.c.274.2 24 85.27 even 16
425.2.n.c.349.2 24 85.73 even 16
425.2.n.f.274.5 24 85.78 even 16
425.2.n.f.349.5 24 85.22 even 16
765.2.be.b.451.2 24 51.5 even 16
765.2.be.b.631.2 24 51.44 even 16
1445.2.a.p.1.10 12 17.4 even 4
1445.2.a.q.1.10 12 17.13 even 4
1445.2.d.j.866.5 24 17.16 even 2 inner
1445.2.d.j.866.6 24 1.1 even 1 trivial
7225.2.a.bq.1.3 12 85.64 even 4
7225.2.a.bs.1.3 12 85.4 even 4