Properties

Label 95.5.d.a.94.1
Level $95$
Weight $5$
Character 95.94
Analytic conductor $9.820$
Analytic rank $0$
Dimension $2$
CM discriminant -19
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [95,5,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, 5); 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, 5, names="a")
 
Level: \( N \) \(=\) \( 95 = 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
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,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.82014649297\)
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: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 94.1
Root \(0.500000 - 2.17945i\) of defining polynomial
Character \(\chi\) \(=\) 95.94
Dual form 95.5.d.a.94.2

$q$-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)
 
\(f(q)\) \(=\) \(q-16.0000 q^{4} +(15.5000 - 19.6150i) q^{5} +65.3835i q^{7} -81.0000 q^{9} +233.000 q^{11} +256.000 q^{16} +457.684i q^{17} +361.000 q^{19} +(-248.000 + 313.841i) q^{20} +1046.14i q^{23} +(-144.500 - 608.066i) q^{25} -1046.14i q^{28} +(1282.50 + 1013.44i) q^{35} +1296.00 q^{36} +1111.52i q^{43} -3728.00 q^{44} +(-1255.50 + 1588.82i) q^{45} -4249.93i q^{47} -1874.00 q^{49} +(3611.50 - 4570.31i) q^{55} -3167.00 q^{61} -5296.06i q^{63} -4096.00 q^{64} -7322.95i q^{68} +3596.09i q^{73} -5776.00 q^{76} +15234.4i q^{77} +(3968.00 - 5021.45i) q^{80} +6561.00 q^{81} +12553.6i q^{83} +(8977.50 + 7094.11i) q^{85} -16738.2i q^{92} +(5595.50 - 7081.03i) q^{95} -18873.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4} + 31 q^{5} - 162 q^{9} + 466 q^{11} + 512 q^{16} + 722 q^{19} - 496 q^{20} - 289 q^{25} + 2565 q^{35} + 2592 q^{36} - 7456 q^{44} - 2511 q^{45} - 3748 q^{49} + 7223 q^{55} - 6334 q^{61} - 8192 q^{64}+ \cdots - 37746 q^{99}+O(q^{100}) \) 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 \(0.5\pi\)
\(4\) −16.0000 −1.00000
\(5\) 15.5000 − 19.6150i 0.620000 − 0.784602i
\(6\) 0 0
\(7\) 65.3835i 1.33436i 0.744898 + 0.667178i \(0.232498\pi\)
−0.744898 + 0.667178i \(0.767502\pi\)
\(8\) 0 0
\(9\) −81.0000 −1.00000
\(10\) 0 0
\(11\) 233.000 1.92562 0.962810 − 0.270180i \(-0.0870831\pi\)
0.962810 + 0.270180i \(0.0870831\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 256.000 1.00000
\(17\) 457.684i 1.58368i 0.610727 + 0.791842i \(0.290877\pi\)
−0.610727 + 0.791842i \(0.709123\pi\)
\(18\) 0 0
\(19\) 361.000 1.00000
\(20\) −248.000 + 313.841i −0.620000 + 0.784602i
\(21\) 0 0
\(22\) 0 0
\(23\) 1046.14i 1.97757i 0.149338 + 0.988786i \(0.452286\pi\)
−0.149338 + 0.988786i \(0.547714\pi\)
\(24\) 0 0
\(25\) −144.500 − 608.066i −0.231200 − 0.972906i
\(26\) 0 0
\(27\) 0 0
\(28\) − 1046.14i − 1.33436i
\(29\) 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
\(35\) 1282.50 + 1013.44i 1.04694 + 0.827301i
\(36\) 1296.00 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
\(43\) 1111.52i 0.601146i 0.953759 + 0.300573i \(0.0971779\pi\)
−0.953759 + 0.300573i \(0.902822\pi\)
\(44\) −3728.00 −1.92562
\(45\) −1255.50 + 1588.82i −0.620000 + 0.784602i
\(46\) 0 0
\(47\) − 4249.93i − 1.92391i −0.273201 − 0.961957i \(-0.588082\pi\)
0.273201 − 0.961957i \(-0.411918\pi\)
\(48\) 0 0
\(49\) −1874.00 −0.780508
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 3611.50 − 4570.31i 1.19388 − 1.51084i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −3167.00 −0.851115 −0.425558 − 0.904931i \(-0.639922\pi\)
−0.425558 + 0.904931i \(0.639922\pi\)
\(62\) 0 0
\(63\) − 5296.06i − 1.33436i
\(64\) −4096.00 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) − 7322.95i − 1.58368i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 3596.09i 0.674815i 0.941359 + 0.337408i \(0.109550\pi\)
−0.941359 + 0.337408i \(0.890450\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −5776.00 −1.00000
\(77\) 15234.4i 2.56946i
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 3968.00 − 5021.45i 0.620000 − 0.784602i
\(81\) 6561.00 1.00000
\(82\) 0 0
\(83\) 12553.6i 1.82227i 0.412106 + 0.911136i \(0.364793\pi\)
−0.412106 + 0.911136i \(0.635207\pi\)
\(84\) 0 0
\(85\) 8977.50 + 7094.11i 1.24256 + 0.981883i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 16738.2i − 1.97757i
\(93\) 0 0
\(94\) 0 0
\(95\) 5595.50 − 7081.03i 0.620000 − 0.784602i
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) −18873.0 −1.92562
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 95.5.d.a.94.1 ✓ 2
5.4 even 2 inner 95.5.d.a.94.2 yes 2
19.18 odd 2 CM 95.5.d.a.94.1 ✓ 2
95.94 odd 2 inner 95.5.d.a.94.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.5.d.a.94.1 ✓ 2 1.1 even 1 trivial
95.5.d.a.94.1 ✓ 2 19.18 odd 2 CM
95.5.d.a.94.2 yes 2 5.4 even 2 inner
95.5.d.a.94.2 yes 2 95.94 odd 2 inner