Properties

Label 950.2.l.h
Level $950$
Weight $2$
Character orbit 950.l
Analytic conductor $7.586$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [950,2,Mod(101,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 14]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.l (of order \(9\), degree \(6\), minimal)

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: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{9})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 12x^{10} + 105x^{8} + 394x^{6} + 1077x^{4} + 1443x^{2} + 1369 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{3} q^{3} - \beta_{7} q^{4} + (\beta_{6} - \beta_{4} + \beta_1) q^{6} + ( - \beta_{9} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_1 - 1) q^{7} + \beta_{5} q^{8} + ( - 2 \beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + \beta_{3} q^{3} - \beta_{7} q^{4} + (\beta_{6} - \beta_{4} + \beta_1) q^{6} + ( - \beta_{9} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_1 - 1) q^{7} + \beta_{5} q^{8} + ( - 2 \beta_{10} - \beta_{8} + \beta_{7} - \beta_{5} + \beta_{2} + 1) q^{9} + (\beta_{11} - \beta_{10} + \beta_{9} + \beta_{7} - \beta_{5} + \beta_{3} + \beta_{2} + \beta_1) q^{11} + (\beta_{6} + \beta_1) q^{12} + ( - \beta_{10} - \beta_{6} + \beta_{5} - \beta_{3} - \beta_1) q^{13} + ( - \beta_{11} - \beta_{10} - \beta_{4} - \beta_{3} + \beta_1) q^{14} + (\beta_{10} - \beta_{2}) q^{16} + ( - \beta_{11} + 2 \beta_{9} - \beta_{8} - \beta_{7} - 2 \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{2} + \cdots + 2) q^{17}+ \cdots + (\beta_{11} - 3 \beta_{10} + 2 \beta_{9} + 4 \beta_{7} - 3 \beta_{6} - 4 \beta_{5} + 2 \beta_{4} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{7} - 6 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{7} - 6 q^{8} + 18 q^{9} + 6 q^{11} - 6 q^{13} + 18 q^{17} + 12 q^{18} + 12 q^{21} + 6 q^{22} + 30 q^{23} - 6 q^{29} + 6 q^{31} + 24 q^{33} + 18 q^{36} - 36 q^{37} - 18 q^{38} - 36 q^{39} - 6 q^{41} + 30 q^{42} + 6 q^{44} - 12 q^{46} + 6 q^{47} - 18 q^{49} + 12 q^{52} + 12 q^{53} + 12 q^{56} + 18 q^{57} + 36 q^{58} - 24 q^{59} - 30 q^{61} + 6 q^{62} - 18 q^{63} - 6 q^{64} + 24 q^{66} - 12 q^{67} - 12 q^{68} + 6 q^{69} - 42 q^{71} - 6 q^{73} + 6 q^{74} + 18 q^{76} + 24 q^{77} - 48 q^{78} + 60 q^{79} + 18 q^{81} - 6 q^{82} - 24 q^{83} - 24 q^{84} - 36 q^{86} - 54 q^{87} + 6 q^{88} - 12 q^{89} + 24 q^{91} - 24 q^{92} - 6 q^{93} + 60 q^{94} - 30 q^{97} - 36 q^{98} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 12x^{10} + 105x^{8} + 394x^{6} + 1077x^{4} + 1443x^{2} + 1369 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -450\nu^{10} - 16352\nu^{8} - 143080\nu^{6} - 950193\nu^{4} - 1861309\nu^{2} - 3541196 ) / 1181151 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 450\nu^{11} + 16352\nu^{9} + 143080\nu^{7} + 950193\nu^{5} + 1861309\nu^{3} + 3541196\nu ) / 1181151 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -811\nu^{11} - 889\nu^{9} + 25031\nu^{7} + 578974\nu^{5} + 2184085\nu^{3} + 4191175\nu ) / 1181151 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -455\nu^{10} - 4868\nu^{8} - 42595\nu^{6} - 133945\nu^{4} - 436903\nu^{2} - 585377 ) / 393717 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 455\nu^{11} + 4868\nu^{9} + 42595\nu^{7} + 133945\nu^{5} + 436903\nu^{3} + 191660\nu ) / 393717 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2176\nu^{10} + 15493\nu^{8} + 102754\nu^{6} - 177139\nu^{4} - 873376\nu^{2} - 2435044 ) / 1181151 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -954\nu^{10} - 13668\nu^{8} - 119595\nu^{6} - 513035\nu^{4} - 1095464\nu^{2} - 1118621 ) / 393717 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -954\nu^{11} - 13668\nu^{9} - 119595\nu^{7} - 513035\nu^{5} - 1095464\nu^{3} - 1118621\nu ) / 393717 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5224\nu^{10} + 49257\nu^{8} + 398189\nu^{6} + 841289\nu^{4} + 1763226\nu^{2} + 50209 ) / 1181151 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5224\nu^{11} + 49257\nu^{9} + 398189\nu^{7} + 841289\nu^{5} + 1763226\nu^{3} + 50209\nu ) / 1181151 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} + \beta_{8} + \beta_{7} - 4\beta_{5} - \beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + \beta_{9} + 5\beta_{6} - \beta_{4} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 19\beta_{10} + 11\beta_{8} - 19\beta_{7} + 22\beta_{5} - 8\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 19\beta_{11} + 11\beta_{9} - 41\beta_{6} + 19\beta_{4} + 8\beta_{3} - 41\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -93\beta_{10} - 150\beta_{8} + 57\beta_{7} + 150\beta_{2} + 145 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -93\beta_{11} - 150\beta_{9} + 57\beta_{6} - 57\beta_{4} - 150\beta_{3} + 202\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -409\beta_{10} + 409\beta_{8} + 724\beta_{7} - 1030\beta_{5} - 724\beta_{2} - 1030 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -409\beta_{11} + 409\beta_{9} + 1754\beta_{6} - 724\beta_{4} + 724\beta_{3} + 724\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 8449\beta_{10} + 5468\beta_{8} - 8449\beta_{7} + 7519\beta_{5} - 2981\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8449\beta_{11} + 5468\beta_{9} - 15968\beta_{6} + 8449\beta_{4} + 2981\beta_{3} - 15968\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-\beta_{7}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
1.36120 + 2.35767i
−1.36120 2.35767i
−0.665830 1.15325i
0.665830 + 1.15325i
1.36120 2.35767i
−1.36120 + 2.35767i
−0.838929 1.45307i
0.838929 + 1.45307i
−0.665830 + 1.15325i
0.665830 1.15325i
−0.838929 + 1.45307i
0.838929 1.45307i
0.766044 0.642788i −2.55822 0.931116i 0.173648 0.984808i 0 −2.55822 + 0.931116i −2.33394 + 4.04250i −0.500000 0.866025i 3.37939 + 2.83564i 0
101.2 0.766044 0.642788i 2.55822 + 0.931116i 0.173648 0.984808i 0 2.55822 0.931116i 1.33394 2.31045i −0.500000 0.866025i 3.37939 + 2.83564i 0
251.1 −0.939693 0.342020i −0.231240 1.31143i 0.766044 + 0.642788i 0 −0.231240 + 1.31143i 1.18594 2.05411i −0.500000 0.866025i 1.15270 0.419550i 0
251.2 −0.939693 0.342020i 0.231240 + 1.31143i 0.766044 + 0.642788i 0 0.231240 1.31143i −2.18594 + 3.78616i −0.500000 0.866025i 1.15270 0.419550i 0
301.1 0.766044 + 0.642788i −2.55822 + 0.931116i 0.173648 + 0.984808i 0 −2.55822 0.931116i −2.33394 4.04250i −0.500000 + 0.866025i 3.37939 2.83564i 0
301.2 0.766044 + 0.642788i 2.55822 0.931116i 0.173648 + 0.984808i 0 2.55822 + 0.931116i 1.33394 + 2.31045i −0.500000 + 0.866025i 3.37939 2.83564i 0
351.1 0.173648 + 0.984808i −1.28531 + 1.07851i −0.939693 + 0.342020i 0 −1.28531 1.07851i −1.23774 + 2.14383i −0.500000 0.866025i −0.0320889 + 0.181985i 0
351.2 0.173648 + 0.984808i 1.28531 1.07851i −0.939693 + 0.342020i 0 1.28531 + 1.07851i 0.237742 0.411781i −0.500000 0.866025i −0.0320889 + 0.181985i 0
651.1 −0.939693 + 0.342020i −0.231240 + 1.31143i 0.766044 0.642788i 0 −0.231240 1.31143i 1.18594 + 2.05411i −0.500000 + 0.866025i 1.15270 + 0.419550i 0
651.2 −0.939693 + 0.342020i 0.231240 1.31143i 0.766044 0.642788i 0 0.231240 + 1.31143i −2.18594 3.78616i −0.500000 + 0.866025i 1.15270 + 0.419550i 0
701.1 0.173648 0.984808i −1.28531 1.07851i −0.939693 0.342020i 0 −1.28531 + 1.07851i −1.23774 2.14383i −0.500000 + 0.866025i −0.0320889 0.181985i 0
701.2 0.173648 0.984808i 1.28531 + 1.07851i −0.939693 0.342020i 0 1.28531 1.07851i 0.237742 + 0.411781i −0.500000 + 0.866025i −0.0320889 0.181985i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.l.h 12
5.b even 2 1 190.2.k.b 12
5.c odd 4 2 950.2.u.e 24
19.e even 9 1 inner 950.2.l.h 12
95.o odd 18 1 3610.2.a.be 6
95.p even 18 1 190.2.k.b 12
95.p even 18 1 3610.2.a.bc 6
95.q odd 36 2 950.2.u.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.k.b 12 5.b even 2 1
190.2.k.b 12 95.p even 18 1
950.2.l.h 12 1.a even 1 1 trivial
950.2.l.h 12 19.e even 9 1 inner
950.2.u.e 24 5.c odd 4 2
950.2.u.e 24 95.q odd 36 2
3610.2.a.bc 6 95.p even 18 1
3610.2.a.be 6 95.o odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 9T_{3}^{10} + 36T_{3}^{8} + 64T_{3}^{6} + 189T_{3}^{4} + 999T_{3}^{2} + 1369 \) acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{3} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} - 9 T^{10} + 36 T^{8} + \cdots + 1369 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{11} + 48 T^{10} + \cdots + 23104 \) Copy content Toggle raw display
$11$ \( T^{12} - 6 T^{11} + 54 T^{10} - 84 T^{9} + \cdots + 729 \) Copy content Toggle raw display
$13$ \( T^{12} + 6 T^{11} + 6 T^{10} + \cdots + 46656 \) Copy content Toggle raw display
$17$ \( T^{12} - 18 T^{11} + 108 T^{10} + \cdots + 12766329 \) Copy content Toggle raw display
$19$ \( T^{12} - 6 T^{10} + 162 T^{9} + \cdots + 47045881 \) Copy content Toggle raw display
$23$ \( T^{12} - 30 T^{11} + 462 T^{10} + \cdots + 1871424 \) Copy content Toggle raw display
$29$ \( T^{12} + 6 T^{11} + 60 T^{10} + \cdots + 5184 \) Copy content Toggle raw display
$31$ \( T^{12} - 6 T^{11} + 108 T^{10} + \cdots + 23104 \) Copy content Toggle raw display
$37$ \( (T^{6} + 18 T^{5} + 48 T^{4} - 560 T^{3} + \cdots - 2168)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 6 T^{11} + 93 T^{10} + \cdots + 114896961 \) Copy content Toggle raw display
$43$ \( T^{12} - 30 T^{10} + 340 T^{9} + \cdots + 12341169 \) Copy content Toggle raw display
$47$ \( T^{12} - 6 T^{11} + 108 T^{10} + \cdots + 16842816 \) Copy content Toggle raw display
$53$ \( T^{12} - 12 T^{11} + 90 T^{10} + \cdots + 46656 \) Copy content Toggle raw display
$59$ \( T^{12} + 24 T^{11} + \cdots + 3234310641 \) Copy content Toggle raw display
$61$ \( T^{12} + 30 T^{11} + \cdots + 224041024 \) Copy content Toggle raw display
$67$ \( T^{12} + 12 T^{11} - 135 T^{10} + \cdots + 2859481 \) Copy content Toggle raw display
$71$ \( T^{12} + 42 T^{11} + \cdots + 6701714496 \) Copy content Toggle raw display
$73$ \( T^{12} + 6 T^{11} + 6 T^{10} + \cdots + 136258929 \) Copy content Toggle raw display
$79$ \( T^{12} - 60 T^{11} + \cdots + 1036324864 \) Copy content Toggle raw display
$83$ \( T^{12} + 24 T^{11} + \cdots + 51075548001 \) Copy content Toggle raw display
$89$ \( T^{12} + 12 T^{11} - 120 T^{10} + \cdots + 2653641 \) Copy content Toggle raw display
$97$ \( T^{12} + 30 T^{11} + \cdots + 2521747089 \) Copy content Toggle raw display
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