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Properties

Label 1275.4.a.bf
Level $1275$
Weight $4$
Character orbit 1275.a
Self dual yes
Analytic conductor $75.227$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1275,4,Mod(1,1275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1275, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1275.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1275.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: \(75.2274352573\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} - 88 x^{12} + 151 x^{11} + 2969 x^{10} - 4260 x^{9} - 48218 x^{8} + 56779 x^{7} + \cdots - 40000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 255)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 5) q^{4} + 3 \beta_1 q^{6} + ( - \beta_{8} - \beta_1) q^{7} + ( - \beta_{3} - \beta_{2} - 5 \beta_1 - 3) q^{8} + 9 q^{9} + (\beta_{5} + \beta_{2} + 2 \beta_1 + 4) q^{11}+ \cdots + (9 \beta_{5} + 9 \beta_{2} + 18 \beta_1 + 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 2 q^{2} - 42 q^{3} + 68 q^{4} + 6 q^{6} - 8 q^{7} - 51 q^{8} + 126 q^{9} + 58 q^{11} - 204 q^{12} + 72 q^{13} + 257 q^{14} + 404 q^{16} + 238 q^{17} - 18 q^{18} + 282 q^{19} + 24 q^{21} - 361 q^{22}+ \cdots + 522 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} - 88 x^{12} + 151 x^{11} + 2969 x^{10} - 4260 x^{9} - 48218 x^{8} + 56779 x^{7} + \cdots - 40000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - \nu^{2} - 21\nu + 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 176609217 \nu^{13} - 2312044785 \nu^{12} + 6212759975 \nu^{11} + 168156663758 \nu^{10} + \cdots + 416462785924672 ) / 27430161172224 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 127662187 \nu^{13} - 238182179 \nu^{12} - 22051809835 \nu^{11} + 41566314066 \nu^{10} + \cdots + 979449212069952 ) / 13715080586112 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 477488279 \nu^{13} + 2947765039 \nu^{12} + 10097972023 \nu^{11} - 136871632074 \nu^{10} + \cdots - 859299883450816 ) / 13715080586112 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 177958463 \nu^{13} + 1278858587 \nu^{12} - 19745373977 \nu^{11} - 106588520304 \nu^{10} + \cdots - 29789772952608 ) / 3428770146528 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2512401207 \nu^{13} + 2399130631 \nu^{12} + 217717477055 \nu^{11} + \cdots + 205131186074688 ) / 27430161172224 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1786246825 \nu^{13} + 6730871103 \nu^{12} - 178830978633 \nu^{11} - 548905547674 \nu^{10} + \cdots + 916942348159168 ) / 13715080586112 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 999532879 \nu^{13} - 6168640919 \nu^{12} - 74477545111 \nu^{11} + 448885148274 \nu^{10} + \cdots - 125480230128960 ) / 6857540293056 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 312032447 \nu^{13} + 1126621134 \nu^{12} + 24534211506 \nu^{11} - 82353535853 \nu^{10} + \cdots + 65771491451088 ) / 1714385073264 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1895908879 \nu^{13} + 4647347087 \nu^{12} + 165145310199 \nu^{11} + \cdots + 614513882074816 ) / 6857540293056 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3869468765 \nu^{13} + 7005005677 \nu^{12} + 319591479445 \nu^{11} + \cdots + 863704101637312 ) / 13715080586112 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + \beta_{2} + 21\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{12} - \beta_{11} - \beta_{8} + \beta_{5} - \beta_{4} + 29\beta_{2} + 6\beta _1 + 277 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{13} - 2 \beta_{12} - 3 \beta_{11} - 3 \beta_{10} - \beta_{9} + \beta_{8} + \beta_{7} + \cdots + 153 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{13} + 46 \beta_{12} - 41 \beta_{11} + 3 \beta_{10} - 5 \beta_{9} - 63 \beta_{8} + 5 \beta_{7} + \cdots + 6811 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 88 \beta_{13} - 93 \beta_{12} - 138 \beta_{11} - 153 \beta_{10} - 73 \beta_{9} + 16 \beta_{8} + \cdots + 5697 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 132 \beta_{13} + 1606 \beta_{12} - 1326 \beta_{11} + 140 \beta_{10} - 306 \beta_{9} - 2688 \beta_{8} + \cdots + 180183 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2826 \beta_{13} - 3238 \beta_{12} - 4738 \beta_{11} - 5856 \beta_{10} - 3416 \beta_{9} - 482 \beta_{8} + \cdots + 189255 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 5318 \beta_{13} + 50989 \beta_{12} - 39699 \beta_{11} + 4082 \beta_{10} - 13458 \beta_{9} + \cdots + 4975787 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 80384 \beta_{13} - 102142 \beta_{12} - 146209 \beta_{11} - 202653 \beta_{10} - 134191 \beta_{9} + \cdots + 5970699 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 170892 \beta_{13} + 1554134 \beta_{12} - 1148807 \beta_{11} + 83037 \beta_{10} - 520935 \beta_{9} + \cdots + 141065285 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2143278 \beta_{13} - 3089173 \beta_{12} - 4286184 \beta_{11} - 6692943 \beta_{10} - 4831043 \beta_{9} + \cdots + 184145367 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
5.54024
5.18549
4.30657
3.10924
2.88401
1.25969
0.734249
−0.0356257
−1.57341
−2.03748
−3.68670
−4.02588
−4.25159
−5.40881
−5.54024 −3.00000 22.6943 0 16.6207 18.7639 −81.4099 9.00000 0
1.2 −5.18549 −3.00000 18.8893 0 15.5565 −33.1355 −56.4663 9.00000 0
1.3 −4.30657 −3.00000 10.5466 0 12.9197 −16.5875 −10.9670 9.00000 0
1.4 −3.10924 −3.00000 1.66738 0 9.32772 −19.5118 19.6896 9.00000 0
1.5 −2.88401 −3.00000 0.317510 0 8.65203 23.6654 22.1564 9.00000 0
1.6 −1.25969 −3.00000 −6.41317 0 3.77908 −12.7244 18.1562 9.00000 0
1.7 −0.734249 −3.00000 −7.46088 0 2.20275 6.33704 11.3521 9.00000 0
1.8 0.0356257 −3.00000 −7.99873 0 −0.106877 −8.56009 −0.569967 9.00000 0
1.9 1.57341 −3.00000 −5.52439 0 −4.72022 34.0065 −21.2794 9.00000 0
1.10 2.03748 −3.00000 −3.84869 0 −6.11243 −16.5301 −24.1414 9.00000 0
1.11 3.68670 −3.00000 5.59179 0 −11.0601 13.9744 −8.87836 9.00000 0
1.12 4.02588 −3.00000 8.20770 0 −12.0776 5.14700 0.836172 9.00000 0
1.13 4.25159 −3.00000 10.0761 0 −12.7548 −32.4321 8.82654 9.00000 0
1.14 5.40881 −3.00000 21.2553 0 −16.2264 29.5873 71.6953 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)
\(17\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1275.4.a.bf 14
5.b even 2 1 1275.4.a.bg 14
5.c odd 4 2 255.4.b.b ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
255.4.b.b ✓ 28 5.c odd 4 2
1275.4.a.bf 14 1.a even 1 1 trivial
1275.4.a.bg 14 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1275))\):

\( T_{2}^{14} + 2 T_{2}^{13} - 88 T_{2}^{12} - 151 T_{2}^{11} + 2969 T_{2}^{10} + 4260 T_{2}^{9} + \cdots - 40000 \) Copy content Toggle raw display
\( T_{7}^{14} + 8 T_{7}^{13} - 3228 T_{7}^{12} - 25500 T_{7}^{11} + 3908473 T_{7}^{10} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 2 T^{13} + \cdots - 40000 \) Copy content Toggle raw display
$3$ \( (T + 3)^{14} \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots - 12\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots - 61\!\cdots\!68 \) Copy content Toggle raw display
$17$ \( (T - 17)^{14} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 33\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 63\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 52\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 72\!\cdots\!28 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 47\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 23\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 37\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 12\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 70\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 27\!\cdots\!20 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 23\!\cdots\!40 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 53\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 57\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 12\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 13\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 14\!\cdots\!20 \) Copy content Toggle raw display
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