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Properties

Label 1148.4.a.d
Level $1148$
Weight $4$
Character orbit 1148.a
Self dual yes
Analytic conductor $67.734$
Analytic rank $0$
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: \(0\)
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: \( x^{15} - 3 x^{14} - 238 x^{13} + 602 x^{12} + 21013 x^{11} - 44923 x^{10} - 876344 x^{9} + \cdots - 45134496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + \beta_{4} q^{5} - 7 q^{7} + (\beta_{2} - \beta_1 + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + \beta_{4} q^{5} - 7 q^{7} + (\beta_{2} - \beta_1 + 6) q^{9} + ( - \beta_{9} + \beta_{4} + \beta_1 - 1) q^{11} + (\beta_{14} - \beta_{13} - \beta_{9} + \cdots + 2) q^{13}+ \cdots + ( - 6 \beta_{14} + 14 \beta_{13} + \cdots + 187) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 12 q^{3} - 4 q^{5} - 105 q^{7} + 89 q^{9} - 14 q^{11} + 34 q^{13} - 160 q^{15} - 100 q^{17} + 26 q^{19} - 84 q^{21} + 158 q^{23} + 441 q^{25} + 450 q^{27} - 156 q^{29} + 252 q^{31} - 668 q^{33} + 28 q^{35} + 182 q^{37} + 370 q^{39} + 615 q^{41} + 894 q^{43} - 158 q^{45} + 1728 q^{47} + 735 q^{49} + 630 q^{51} + 1034 q^{53} + 1944 q^{55} + 54 q^{57} + 262 q^{59} + 322 q^{61} - 623 q^{63} + 188 q^{65} + 1808 q^{67} - 168 q^{69} + 584 q^{71} - 1290 q^{73} + 5188 q^{75} + 98 q^{77} + 3726 q^{79} + 3043 q^{81} + 2484 q^{83} + 3404 q^{85} + 5448 q^{87} + 876 q^{89} - 238 q^{91} + 6174 q^{93} + 5714 q^{95} - 154 q^{97} + 2854 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 3 x^{14} - 238 x^{13} + 602 x^{12} + 21013 x^{11} - 44923 x^{10} - 876344 x^{9} + \cdots - 45134496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 49\!\cdots\!35 \nu^{14} + \cdots + 76\!\cdots\!16 ) / 10\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 42\!\cdots\!77 \nu^{14} + \cdots - 39\!\cdots\!44 ) / 34\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11\!\cdots\!43 \nu^{14} + \cdots + 17\!\cdots\!80 ) / 78\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14\!\cdots\!32 \nu^{14} + \cdots + 42\!\cdots\!28 ) / 78\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 25\!\cdots\!27 \nu^{14} + \cdots - 83\!\cdots\!08 ) / 10\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 32\!\cdots\!19 \nu^{14} + \cdots + 61\!\cdots\!96 ) / 10\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 84\!\cdots\!71 \nu^{14} + \cdots - 29\!\cdots\!24 ) / 15\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 52\!\cdots\!45 \nu^{14} + \cdots + 49\!\cdots\!56 ) / 78\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 92\!\cdots\!85 \nu^{14} + \cdots + 11\!\cdots\!92 ) / 78\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 93\!\cdots\!65 \nu^{14} + \cdots + 58\!\cdots\!64 ) / 78\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14\!\cdots\!91 \nu^{14} + \cdots + 61\!\cdots\!44 ) / 10\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 13\!\cdots\!96 \nu^{14} + \cdots - 35\!\cdots\!60 ) / 78\!\cdots\!16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{12} - 2 \beta_{11} - 2 \beta_{9} + \beta_{8} + 3 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 11 \beta_{14} + \beta_{13} - 3 \beta_{12} - 10 \beta_{11} + \beta_{10} - 10 \beta_{9} + 12 \beta_{8} + \cdots + 2005 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 137 \beta_{14} + 31 \beta_{13} + 56 \beta_{12} - 220 \beta_{11} - 30 \beta_{10} - 289 \beta_{9} + \cdots + 2281 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1784 \beta_{14} + 120 \beta_{13} - 615 \beta_{12} - 1340 \beta_{11} - 201 \beta_{10} - 1689 \beta_{9} + \cdots + 157335 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 15790 \beta_{14} + 5813 \beta_{13} + 1813 \beta_{12} - 22537 \beta_{11} - 5876 \beta_{10} + \cdots + 350971 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 216635 \beta_{14} + 18217 \beta_{13} - 81610 \beta_{12} - 150126 \beta_{11} - 51113 \beta_{10} + \cdots + 13685802 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1749858 \beta_{14} + 763936 \beta_{13} - 98860 \beta_{12} - 2292550 \beta_{11} - 798242 \beta_{10} + \cdots + 47468914 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 24071836 \beta_{14} + 2647022 \beta_{13} - 9369183 \beta_{12} - 16349800 \beta_{11} - 7572660 \beta_{10} + \cdots + 1262666052 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 191893719 \beta_{14} + 87980250 \beta_{13} - 30987995 \beta_{12} - 234053130 \beta_{11} + \cdots + 5927548838 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2584683043 \beta_{14} + 353612254 \beta_{13} - 1011951483 \beta_{12} - 1775727673 \beta_{11} + \cdots + 121068757941 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 20960422129 \beta_{14} + 9534993422 \beta_{13} - 4757748033 \beta_{12} - 24041884630 \beta_{11} + \cdots + 703513887888 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 273562524585 \beta_{14} + 44377572483 \beta_{13} - 106367945258 \beta_{12} - 192839943462 \beta_{11} + \cdots + 11927897543931 \) 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
10.3108
7.69288
5.89624
5.79784
3.33715
3.24851
1.93335
−0.0293365
−1.63762
−2.79210
−2.86399
−4.83487
−5.65179
−8.38889
−9.01814
0 −9.31076 0 −4.97499 0 −7.00000 0 59.6903 0
1.2 0 −6.69288 0 15.9636 0 −7.00000 0 17.7947 0
1.3 0 −4.89624 0 −3.91686 0 −7.00000 0 −3.02684 0
1.4 0 −4.79784 0 12.6199 0 −7.00000 0 −3.98075 0
1.5 0 −2.33715 0 8.20734 0 −7.00000 0 −21.5377 0
1.6 0 −2.24851 0 −12.4858 0 −7.00000 0 −21.9442 0
1.7 0 −0.933352 0 1.08946 0 −7.00000 0 −26.1289 0
1.8 0 1.02934 0 −17.2720 0 −7.00000 0 −25.9405 0
1.9 0 2.63762 0 −9.55517 0 −7.00000 0 −20.0430 0
1.10 0 3.79210 0 13.9605 0 −7.00000 0 −12.6200 0
1.11 0 3.86399 0 −6.90167 0 −7.00000 0 −12.0696 0
1.12 0 5.83487 0 16.1201 0 −7.00000 0 7.04574 0
1.13 0 6.65179 0 −10.2494 0 −7.00000 0 17.2463 0
1.14 0 9.38889 0 14.2830 0 −7.00000 0 61.1513 0
1.15 0 10.0181 0 −20.8879 0 −7.00000 0 73.3631 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.d ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.4.a.d ✓ 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{15} - 12 T_{3}^{14} - 175 T_{3}^{13} + 2310 T_{3}^{12} + 9946 T_{3}^{11} - 157884 T_{3}^{10} + \cdots + 1042773904 \) 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
$3$ \( T^{15} + \cdots + 1042773904 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots - 343528737493248 \) Copy content Toggle raw display
$7$ \( (T + 7)^{15} \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots - 10\!\cdots\!28 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots + 32\!\cdots\!12 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 97\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 56\!\cdots\!92 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 25\!\cdots\!92 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 10\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( (T - 41)^{15} \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 33\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots - 81\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 30\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 19\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 36\!\cdots\!04 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 20\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 23\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 43\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 58\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots + 88\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 54\!\cdots\!96 \) Copy content Toggle raw display
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