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Properties

Label 1148.4.a.c
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: \( x^{15} - 5 x^{14} - 270 x^{13} + 1158 x^{12} + 28413 x^{11} - 102669 x^{10} - 1445580 x^{9} + \cdots + 1052740152 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + \beta_{3} q^{5} + 7 q^{7} + (\beta_{2} - \beta_1 + 11) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + \beta_{3} q^{5} + 7 q^{7} + (\beta_{2} - \beta_1 + 11) q^{9} + (\beta_{5} - \beta_1 + 4) q^{11} + (\beta_{12} + 10) q^{13} + ( - \beta_{8} + \beta_{2} - \beta_1 + 10) q^{15} + ( - \beta_{6} + \beta_{3} + 8) q^{17} + (\beta_{9} + \beta_{2} + \beta_1 + 17) q^{19} + ( - 7 \beta_1 + 7) q^{21} + ( - \beta_{13} + \beta_{12} - \beta_{9} + \cdots - 2) q^{23}+ \cdots + (6 \beta_{14} + \beta_{13} + \cdots + 211) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + 10 q^{3} + 6 q^{5} + 105 q^{7} + 165 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q + 10 q^{3} + 6 q^{5} + 105 q^{7} + 165 q^{9} + 62 q^{11} + 148 q^{13} + 152 q^{15} + 132 q^{17} + 260 q^{19} + 70 q^{21} - 26 q^{23} + 453 q^{25} + 454 q^{27} + 12 q^{29} + 144 q^{31} + 336 q^{33} + 42 q^{35} + 274 q^{37} + 218 q^{39} - 615 q^{41} + 986 q^{43} + 648 q^{45} + 44 q^{47} + 735 q^{49} + 202 q^{51} - 366 q^{53} + 928 q^{55} - 294 q^{57} - 8 q^{59} + 1396 q^{61} + 1155 q^{63} + 156 q^{65} - 24 q^{67} + 892 q^{69} + 1464 q^{71} + 1174 q^{73} + 318 q^{75} + 434 q^{77} + 1590 q^{79} + 1959 q^{81} + 1970 q^{83} + 2348 q^{85} + 2544 q^{87} + 1972 q^{89} + 1036 q^{91} + 4034 q^{93} + 1070 q^{95} + 954 q^{97} + 3398 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 5 x^{14} - 270 x^{13} + 1158 x^{12} + 28413 x^{11} - 102669 x^{10} - 1445580 x^{9} + \cdots + 1052740152 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 37 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13\!\cdots\!56 \nu^{14} + \cdots + 44\!\cdots\!62 ) / 88\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 27\!\cdots\!92 \nu^{14} + \cdots + 73\!\cdots\!64 ) / 79\!\cdots\!18 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28\!\cdots\!83 \nu^{14} + \cdots + 72\!\cdots\!88 ) / 26\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 75\!\cdots\!98 \nu^{14} + \cdots + 50\!\cdots\!72 ) / 53\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 28\!\cdots\!43 \nu^{14} + \cdots + 46\!\cdots\!00 ) / 17\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18\!\cdots\!02 \nu^{14} + \cdots - 53\!\cdots\!04 ) / 88\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 76\!\cdots\!73 \nu^{14} + \cdots + 12\!\cdots\!92 ) / 26\!\cdots\!06 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 15\!\cdots\!44 \nu^{14} + \cdots - 81\!\cdots\!24 ) / 53\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 54\!\cdots\!33 \nu^{14} + \cdots - 82\!\cdots\!72 ) / 15\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 84\!\cdots\!59 \nu^{14} + \cdots + 43\!\cdots\!36 ) / 15\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 10\!\cdots\!70 \nu^{14} + \cdots + 49\!\cdots\!52 ) / 15\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10\!\cdots\!81 \nu^{14} + \cdots - 16\!\cdots\!88 ) / 15\!\cdots\!36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 37 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{12} + \beta_{5} - 3\beta_{3} + 2\beta_{2} + 64\beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} + 2 \beta_{13} + 3 \beta_{12} + 2 \beta_{10} + \beta_{9} - \beta_{8} - 2 \beta_{7} + \cdots + 2371 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 96 \beta_{14} + 57 \beta_{13} + 129 \beta_{12} + 24 \beta_{11} - 6 \beta_{10} + 6 \beta_{9} + \cdots + 4448 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 203 \beta_{14} + 534 \beta_{13} + 617 \beta_{12} + 66 \beta_{11} + 102 \beta_{10} + 42 \beta_{9} + \cdots + 170164 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8138 \beta_{14} + 8590 \beta_{13} + 13203 \beta_{12} + 3471 \beta_{11} - 1157 \beta_{10} + 1307 \beta_{9} + \cdots + 512078 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 27315 \beta_{14} + 75897 \beta_{13} + 80931 \beta_{12} + 10860 \beta_{11} - 3879 \beta_{10} + \cdots + 12936427 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 680029 \beta_{14} + 988257 \beta_{13} + 1263148 \beta_{12} + 371307 \beta_{11} - 175146 \beta_{10} + \cdots + 52120169 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3097633 \beta_{14} + 8902403 \beta_{13} + 9052827 \beta_{12} + 1338333 \beta_{11} - 1605673 \beta_{10} + \cdots + 1021983574 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 57534978 \beta_{14} + 103820640 \beta_{13} + 118485540 \beta_{12} + 35724015 \beta_{11} + \cdots + 5035634255 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 323291870 \beta_{14} + 957161151 \beta_{13} + 940395788 \beta_{12} + 147283227 \beta_{11} + \cdots + 83043419032 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4959995696 \beta_{14} + 10463816590 \beta_{13} + 11064367536 \beta_{12} + 3277689021 \beta_{11} + \cdots + 473947045916 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 32292513570 \beta_{14} + 98150024571 \beta_{13} + 93897733002 \beta_{12} + 15246859215 \beta_{11} + \cdots + 6896492585512 \) 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
9.62561
9.07011
8.44021
6.52747
2.98071
2.89042
1.92676
0.707687
−1.04602
−1.16028
−5.16749
−5.92647
−7.49607
−7.80263
−8.57002
0 −8.62561 0 −15.0933 0 7.00000 0 47.4011 0
1.2 0 −8.07011 0 −2.73391 0 7.00000 0 38.1266 0
1.3 0 −7.44021 0 16.9428 0 7.00000 0 28.3567 0
1.4 0 −5.52747 0 −5.01451 0 7.00000 0 3.55296 0
1.5 0 −1.98071 0 5.23598 0 7.00000 0 −23.0768 0
1.6 0 −1.89042 0 18.4916 0 7.00000 0 −23.4263 0
1.7 0 −0.926761 0 −7.25280 0 7.00000 0 −26.1411 0
1.8 0 0.292313 0 −17.4774 0 7.00000 0 −26.9146 0
1.9 0 2.04602 0 7.01684 0 7.00000 0 −22.8138 0
1.10 0 2.16028 0 −14.3655 0 7.00000 0 −22.3332 0
1.11 0 6.16749 0 16.2801 0 7.00000 0 11.0379 0
1.12 0 6.92647 0 −13.5350 0 7.00000 0 20.9760 0
1.13 0 8.49607 0 1.41643 0 7.00000 0 45.1831 0
1.14 0 8.80263 0 18.4805 0 7.00000 0 50.4862 0
1.15 0 9.57002 0 −2.39168 0 7.00000 0 64.5852 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.c ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1148.4.a.c ✓ 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{15} - 10 T_{3}^{14} - 235 T_{3}^{13} + 2352 T_{3}^{12} + 20794 T_{3}^{11} - 207080 T_{3}^{10} + \cdots + 392425976 \) 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
$3$ \( T^{15} + \cdots + 392425976 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots - 59831496108288 \) Copy content Toggle raw display
$7$ \( (T - 7)^{15} \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 41\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots - 38\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 54\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 28\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 40\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 31\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 34\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T + 41)^{15} \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 25\!\cdots\!68 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 11\!\cdots\!28 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 72\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 27\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 69\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 12\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 21\!\cdots\!08 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 77\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 11\!\cdots\!12 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 32\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 19\!\cdots\!28 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 35\!\cdots\!56 \) Copy content Toggle raw display
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