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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [95,11,Mod(94,95)] 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("95.94"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(95, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 95.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2048,3951] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(60.3589390040\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-19}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 11 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 94.2
Root \(0.500000 + 2.17945i\) of defining polynomial
Character \(\chi\) \(=\) 95.94
Dual form 95.11.d.a.94.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1024.00 q^{4} +(1975.50 + 2421.37i) q^{5} +8486.78i q^{7} -59049.0 q^{9} -203523. q^{11} +1.04858e6 q^{16} -1.85908e6i q^{17} -2.47610e6 q^{19} +(-2.02291e6 - 2.47948e6i) q^{20} +1.18025e7i q^{23} +(-1.96042e6 + 9.56683e6i) q^{25} -8.69046e6i q^{28} +(-2.05496e7 + 1.67656e7i) q^{35} +6.04662e7 q^{36} -2.03243e8i q^{43} +2.08408e8 q^{44} +(-1.16651e8 - 1.42979e8i) q^{45} +4.28715e7i q^{47} +2.10450e8 q^{49} +(-4.02060e8 - 4.92804e8i) q^{55} +1.60684e9 q^{61} -5.01136e8i q^{63} -1.07374e9 q^{64} +1.90370e9i q^{68} -2.70383e9i q^{73} +2.53553e9 q^{76} -1.72725e9i q^{77} +(2.07146e9 + 2.53899e9i) q^{80} +3.48678e9 q^{81} -7.57882e9i q^{83} +(4.50153e9 - 3.67262e9i) q^{85} -1.20857e10i q^{92} +(-4.89153e9 - 5.99555e9i) q^{95} +1.20178e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2048 q^{4} + 3951 q^{5} - 118098 q^{9} - 407046 q^{11} + 2097152 q^{16} - 4952198 q^{19} - 4045824 q^{20} - 3920849 q^{25} - 41099223 q^{35} + 120932352 q^{36} + 416815104 q^{44} - 233302599 q^{45}+ \cdots + 24035659254 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/95\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −1024.00 −1.00000
\(5\) 1975.50 + 2421.37i 0.632160 + 0.774838i
\(6\) 0 0
\(7\) 8486.78i 0.504955i 0.967603 + 0.252477i \(0.0812453\pi\)
−0.967603 + 0.252477i \(0.918755\pi\)
\(8\) 0 0
\(9\) −59049.0 −1.00000
\(10\) 0 0
\(11\) −203523. −1.26372 −0.631859 − 0.775083i \(-0.717708\pi\)
−0.631859 + 0.775083i \(0.717708\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.04858e6 1.00000
\(17\) − 1.85908e6i − 1.30935i −0.755912 − 0.654673i \(-0.772806\pi\)
0.755912 − 0.654673i \(-0.227194\pi\)
\(18\) 0 0
\(19\) −2.47610e6 −1.00000
\(20\) −2.02291e6 − 2.47948e6i −0.632160 − 0.774838i
\(21\) 0 0
\(22\) 0 0
\(23\) 1.18025e7i 1.83372i 0.399206 + 0.916861i \(0.369286\pi\)
−0.399206 + 0.916861i \(0.630714\pi\)
\(24\) 0 0
\(25\) −1.96042e6 + 9.56683e6i −0.200747 + 0.979643i
\(26\) 0 0
\(27\) 0 0
\(28\) − 8.69046e6i − 0.504955i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.05496e7 + 1.67656e7i −0.391258 + 0.319212i
\(36\) 6.04662e7 1.00000
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) − 2.03243e8i − 1.38252i −0.722605 − 0.691262i \(-0.757055\pi\)
0.722605 − 0.691262i \(-0.242945\pi\)
\(44\) 2.08408e8 1.26372
\(45\) −1.16651e8 − 1.42979e8i −0.632160 − 0.774838i
\(46\) 0 0
\(47\) 4.28715e7i 0.186930i 0.995623 + 0.0934651i \(0.0297943\pi\)
−0.995623 + 0.0934651i \(0.970206\pi\)
\(48\) 0 0
\(49\) 2.10450e8 0.745021
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −4.02060e8 − 4.92804e8i −0.798872 − 0.979176i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 1.60684e9 1.90249 0.951246 − 0.308435i \(-0.0998051\pi\)
0.951246 + 0.308435i \(0.0998051\pi\)
\(62\) 0 0
\(63\) − 5.01136e8i − 0.504955i
\(64\) −1.07374e9 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 1.90370e9i 1.30935i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) − 2.70383e9i − 1.30426i −0.758106 − 0.652131i \(-0.773875\pi\)
0.758106 − 0.652131i \(-0.226125\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 2.53553e9 1.00000
\(77\) − 1.72725e9i − 0.638120i
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 2.07146e9 + 2.53899e9i 0.632160 + 0.774838i
\(81\) 3.48678e9 1.00000
\(82\) 0 0
\(83\) − 7.57882e9i − 1.92403i −0.273003 − 0.962013i \(-0.588017\pi\)
0.273003 − 0.962013i \(-0.411983\pi\)
\(84\) 0 0
\(85\) 4.50153e9 − 3.67262e9i 1.01453 − 0.827716i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 1.20857e10i − 1.83372i
\(93\) 0 0
\(94\) 0 0
\(95\) −4.89153e9 − 5.99555e9i −0.632160 − 0.774838i
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 1.20178e10 1.26372
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 95.11.d.a.94.2 yes 2
5.4 even 2 inner 95.11.d.a.94.1 ✓ 2
19.18 odd 2 CM 95.11.d.a.94.2 yes 2
95.94 odd 2 inner 95.11.d.a.94.1 ✓ 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.11.d.a.94.1 ✓ 2 5.4 even 2 inner
95.11.d.a.94.1 ✓ 2 95.94 odd 2 inner
95.11.d.a.94.2 yes 2 1.1 even 1 trivial
95.11.d.a.94.2 yes 2 19.18 odd 2 CM