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Properties

Label 1148.4.a.b
Level $1148$
Weight $4$
Character orbit 1148.a
Self dual yes
Analytic conductor $67.734$
Analytic rank $1$
Dimension $15$
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] = [1148,4,Mod(1,1148)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1148, 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("1148.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1148.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: \(67.7341926866\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 247 x^{13} - 6 x^{12} + 23870 x^{11} + 940 x^{10} - 1147074 x^{9} - 8966 x^{8} + \cdots + 1720288256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{4} q^{5} - 7 q^{7} + (\beta_{2} + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{4} q^{5} - 7 q^{7} + (\beta_{2} + 6) q^{9} + ( - \beta_{10} - \beta_1 - 1) q^{11} + (\beta_{11} - \beta_{5} + \beta_{4} + \cdots - 4) q^{13}+ \cdots + ( - \beta_{14} + 7 \beta_{13} + \cdots - 170) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9} - 20 q^{11} - 70 q^{13} - 20 q^{15} + 160 q^{17} - 6 q^{19} - 118 q^{23} + 569 q^{25} + 18 q^{27} - 162 q^{29} - 164 q^{31} - 292 q^{33} - 42 q^{35} - 410 q^{37} - 206 q^{39} - 615 q^{41} - 1022 q^{43} + 196 q^{45} - 628 q^{47} + 735 q^{49} - 1994 q^{51} - 512 q^{53} - 1128 q^{55} - 266 q^{57} - 144 q^{59} - 256 q^{61} - 623 q^{63} - 1000 q^{65} - 2670 q^{67} + 108 q^{69} - 1048 q^{71} - 606 q^{73} - 3796 q^{75} + 140 q^{77} - 1386 q^{79} - 2541 q^{81} - 2022 q^{83} - 2848 q^{85} - 3700 q^{87} - 500 q^{89} + 490 q^{91} - 2194 q^{93} - 5230 q^{95} + 1326 q^{97} - 2732 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 247 x^{13} - 6 x^{12} + 23870 x^{11} + 940 x^{10} - 1147074 x^{9} - 8966 x^{8} + \cdots + 1720288256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 33 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 83\!\cdots\!60 \nu^{14} + \cdots + 15\!\cdots\!28 ) / 10\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 37\!\cdots\!21 \nu^{14} + \cdots - 81\!\cdots\!68 ) / 36\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 69\!\cdots\!12 \nu^{14} + \cdots + 29\!\cdots\!72 ) / 10\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 87\!\cdots\!51 \nu^{14} + \cdots - 10\!\cdots\!40 ) / 10\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 55\!\cdots\!93 \nu^{14} + \cdots - 32\!\cdots\!96 ) / 36\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13\!\cdots\!56 \nu^{14} + \cdots - 35\!\cdots\!88 ) / 54\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 52\!\cdots\!21 \nu^{14} + \cdots + 46\!\cdots\!44 ) / 18\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 38\!\cdots\!67 \nu^{14} + \cdots + 34\!\cdots\!84 ) / 10\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 21\!\cdots\!11 \nu^{14} + \cdots + 13\!\cdots\!72 ) / 54\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 16\!\cdots\!27 \nu^{14} + \cdots - 13\!\cdots\!36 ) / 36\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 57\!\cdots\!61 \nu^{14} + \cdots + 30\!\cdots\!88 ) / 10\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 75\!\cdots\!56 \nu^{14} + \cdots - 33\!\cdots\!84 ) / 10\!\cdots\!32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 33 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} - 2 \beta_{11} + 3 \beta_{10} - 2 \beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{4} - \beta_{3} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} + 3 \beta_{13} - 3 \beta_{12} - 4 \beta_{11} - 3 \beta_{10} - 2 \beta_{9} - \beta_{8} + \cdots + 1767 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{14} - 23 \beta_{13} + 77 \beta_{12} - 200 \beta_{11} + 301 \beta_{10} + 12 \beta_{9} - 8 \beta_{8} + \cdots + 152 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 132 \beta_{14} + 390 \beta_{13} - 403 \beta_{12} - 491 \beta_{11} - 481 \beta_{10} - 258 \beta_{9} + \cdots + 109286 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 435 \beta_{14} - 3155 \beta_{13} + 5631 \beta_{12} - 16485 \beta_{11} + 25653 \beta_{10} + 1624 \beta_{9} + \cdots - 2953 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 12397 \beta_{14} + 39155 \beta_{13} - 39153 \beta_{12} - 43924 \beta_{11} - 51825 \beta_{10} + \cdots + 7242668 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 68864 \beta_{14} - 320140 \beta_{13} + 424771 \beta_{12} - 1275761 \beta_{11} + 2079091 \beta_{10} + \cdots - 2047171 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1079644 \beta_{14} + 3559990 \beta_{13} - 3403036 \beta_{12} - 3497110 \beta_{11} - 4933121 \beta_{10} + \cdots + 500170051 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8011871 \beta_{14} - 29142841 \beta_{13} + 33035199 \beta_{12} - 96159552 \beta_{11} + 164981344 \beta_{10} + \cdots - 312920853 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 91776972 \beta_{14} + 307799968 \beta_{13} - 281473479 \beta_{12} - 262194221 \beta_{11} + \cdots + 35469171917 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 805144479 \beta_{14} - 2518570859 \beta_{13} + 2619411653 \beta_{12} - 7164448201 \beta_{11} + \cdots - 36102820273 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 7697298900 \beta_{14} + 25843216766 \beta_{13} - 22710514600 \beta_{12} - 18918705922 \beta_{11} + \cdots + 2561078158948 \) 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
−8.88753
−7.72571
−6.19235
−5.58314
−5.32866
−2.65747
−1.99072
0.418169
2.44633
2.66351
3.30160
5.95988
6.88828
8.23201
8.45579
0 −8.88753 0 13.8011 0 −7.00000 0 51.9882 0
1.2 0 −7.72571 0 13.6642 0 −7.00000 0 32.6866 0
1.3 0 −6.19235 0 −17.0650 0 −7.00000 0 11.3451 0
1.4 0 −5.58314 0 −3.35315 0 −7.00000 0 4.17142 0
1.5 0 −5.32866 0 −18.8353 0 −7.00000 0 1.39457 0
1.6 0 −2.65747 0 −8.90352 0 −7.00000 0 −19.9379 0
1.7 0 −1.99072 0 8.14355 0 −7.00000 0 −23.0370 0
1.8 0 0.418169 0 21.2346 0 −7.00000 0 −26.8251 0
1.9 0 2.44633 0 15.1363 0 −7.00000 0 −21.0155 0
1.10 0 2.66351 0 2.69284 0 −7.00000 0 −19.9057 0
1.11 0 3.30160 0 −19.8552 0 −7.00000 0 −16.0994 0
1.12 0 5.95988 0 6.64417 0 −7.00000 0 8.52015 0
1.13 0 6.88828 0 −5.64409 0 −7.00000 0 20.4484 0
1.14 0 8.23201 0 −8.08041 0 −7.00000 0 40.7659 0
1.15 0 8.45579 0 6.41990 0 −7.00000 0 44.5003 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(41\) \(1\)

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 1148.4.a.b ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.4.a.b ✓ 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{15} - 247 T_{3}^{13} - 6 T_{3}^{12} + 23870 T_{3}^{11} + 940 T_{3}^{10} - 1147074 T_{3}^{9} + \cdots + 1720288256 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1148))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots + 1720288256 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots + 492662864498688 \) Copy content Toggle raw display
$7$ \( (T + 7)^{15} \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots - 49\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots - 57\!\cdots\!92 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots - 25\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 17\!\cdots\!28 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 25\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 59\!\cdots\!88 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 43\!\cdots\!04 \) Copy content Toggle raw display
$41$ \( (T + 41)^{15} \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 10\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 37\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 19\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 11\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 26\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 36\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 88\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 57\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 18\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 16\!\cdots\!48 \) Copy content Toggle raw display
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