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Properties

Label 7146.2.a.be
Level $7146$
Weight $2$
Character orbit 7146.a
Self dual yes
Analytic conductor $57.061$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7146,2,Mod(1,7146)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7146, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7146.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7146 = 2 \cdot 3^{2} \cdot 397 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7146.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: \(57.0610972844\)
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: \( x^{14} - x^{13} - 32 x^{12} + 33 x^{11} + 399 x^{10} - 423 x^{9} - 2413 x^{8} + 2601 x^{7} + 7136 x^{6} + \cdots - 1498 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 794)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta_{8} q^{5} - \beta_{5} q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + \beta_{8} q^{5} - \beta_{5} q^{7} - q^{8} - \beta_{8} q^{10} - \beta_{6} q^{11} + ( - \beta_{11} + \beta_{9} + \beta_{8} + \cdots + 1) q^{13}+ \cdots + ( - \beta_{13} - \beta_{12} - 2 \beta_{11} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{2} + 14 q^{4} - 7 q^{5} + 7 q^{7} - 14 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{2} + 14 q^{4} - 7 q^{5} + 7 q^{7} - 14 q^{8} + 7 q^{10} - 3 q^{11} + 7 q^{13} - 7 q^{14} + 14 q^{16} - 10 q^{17} + 7 q^{19} - 7 q^{20} + 3 q^{22} - 14 q^{23} + 27 q^{25} - 7 q^{26} + 7 q^{28} - 20 q^{29} + 6 q^{31} - 14 q^{32} + 10 q^{34} + 16 q^{35} + 20 q^{37} - 7 q^{38} + 7 q^{40} - 4 q^{41} + 3 q^{43} - 3 q^{44} + 14 q^{46} + 23 q^{49} - 27 q^{50} + 7 q^{52} - 21 q^{53} - 14 q^{55} - 7 q^{56} + 20 q^{58} + 30 q^{59} + 19 q^{61} - 6 q^{62} + 14 q^{64} - 12 q^{65} - 21 q^{67} - 10 q^{68} - 16 q^{70} + 11 q^{71} + 9 q^{73} - 20 q^{74} + 7 q^{76} + 26 q^{77} + 22 q^{79} - 7 q^{80} + 4 q^{82} + 51 q^{83} - 10 q^{85} - 3 q^{86} + 3 q^{88} + 6 q^{89} - 8 q^{91} - 14 q^{92} + 40 q^{95} - 15 q^{97} - 23 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} - 32 x^{12} + 33 x^{11} + 399 x^{10} - 423 x^{9} - 2413 x^{8} + 2601 x^{7} + 7136 x^{6} + \cdots - 1498 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1023 \nu^{13} + 635968 \nu^{12} - 2370406 \nu^{11} - 16247954 \nu^{10} + 57874659 \nu^{9} + \cdots + 496090857 ) / 8655791 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1023 \nu^{13} - 635968 \nu^{12} + 2370406 \nu^{11} + 16247954 \nu^{10} - 57874659 \nu^{9} + \cdots - 504746648 ) / 8655791 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 19557 \nu^{13} + 381482 \nu^{12} + 310731 \nu^{11} - 8834949 \nu^{10} - 293705 \nu^{9} + \cdots - 68448835 ) / 8655791 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 102943 \nu^{13} + 164623 \nu^{12} + 1634280 \nu^{11} - 3508753 \nu^{10} - 1363639 \nu^{9} + \cdots + 101575778 ) / 8655791 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 132943 \nu^{13} - 298363 \nu^{12} - 3170502 \nu^{11} + 7768242 \nu^{10} + 26147053 \nu^{9} + \cdots - 254010565 ) / 8655791 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 158379 \nu^{13} + 688742 \nu^{12} + 2651835 \nu^{11} - 16733616 \nu^{10} - 3666465 \nu^{9} + \cdots + 123277680 ) / 8655791 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 213110 \nu^{13} - 607998 \nu^{12} + 7848659 \nu^{11} + 14662680 \nu^{10} - 111151991 \nu^{9} + \cdots - 340821080 ) / 8655791 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 437448 \nu^{13} + 11246 \nu^{12} + 11778188 \nu^{11} - 252420 \nu^{10} - 121387906 \nu^{9} + \cdots - 303633016 ) / 8655791 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 438208 \nu^{13} + 1053329 \nu^{12} + 11263135 \nu^{11} - 26316159 \nu^{10} - 106712314 \nu^{9} + \cdots + 27613763 ) / 8655791 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 465546 \nu^{13} - 794347 \nu^{12} - 12029439 \nu^{11} + 19896721 \nu^{10} + 115201523 \nu^{9} + \cdots + 79551825 ) / 8655791 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 599512 \nu^{13} - 456742 \nu^{12} - 17570347 \nu^{11} + 11417009 \nu^{10} + 199223235 \nu^{9} + \cdots + 408190027 ) / 8655791 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{4} - \beta_{3} + 7\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} - \beta_{12} - \beta_{7} - \beta_{3} + 9\beta_{2} + 35 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{12} - \beta_{11} - \beta_{10} - \beta_{8} + \beta_{6} - 9\beta_{4} - 9\beta_{3} + 2\beta_{2} + 53\beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 14 \beta_{13} - 15 \beta_{12} - 3 \beta_{11} + \beta_{10} + \beta_{9} + 2 \beta_{8} - 15 \beta_{7} + \cdots + 264 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 31 \beta_{12} - 16 \beta_{11} - 17 \beta_{10} + 3 \beta_{9} - 12 \beta_{8} + 2 \beta_{7} + 15 \beta_{6} + \cdots - 78 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 147 \beta_{13} - 163 \beta_{12} - 53 \beta_{11} + 15 \beta_{10} + 22 \beta_{9} + 40 \beta_{8} + \cdots + 2037 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 15 \beta_{13} - 344 \beta_{12} - 198 \beta_{11} - 207 \beta_{10} + 58 \beta_{9} - 100 \beta_{8} + \cdots - 553 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1377 \beta_{13} - 1568 \beta_{12} - 669 \beta_{11} + 156 \beta_{10} + 315 \beta_{9} + 547 \beta_{8} + \cdots + 15897 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 343 \beta_{13} - 3373 \beta_{12} - 2193 \beta_{11} - 2209 \beta_{10} + 768 \beta_{9} - 679 \beta_{8} + \cdots - 3650 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 12168 \beta_{13} - 14222 \beta_{12} - 7358 \beta_{11} + 1380 \beta_{10} + 3760 \beta_{9} + 6337 \beta_{8} + \cdots + 125024 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 5044 \beta_{13} - 31201 \beta_{12} - 22724 \beta_{11} - 21981 \beta_{10} + 8686 \beta_{9} + \cdots - 22152 \) 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
−2.62091
1.17281
2.73552
−2.78621
−1.10726
1.35228
−0.447625
1.01338
2.78952
2.88769
−2.85469
2.25617
−0.665954
−2.72473
−1.00000 0 1.00000 −4.21796 0 −0.971822 −1.00000 0 4.21796
1.2 −1.00000 0 1.00000 −3.56462 0 −0.174202 −1.00000 0 3.56462
1.3 −1.00000 0 1.00000 −3.48460 0 −4.07117 −1.00000 0 3.48460
1.4 −1.00000 0 1.00000 −2.10264 0 4.18682 −1.00000 0 2.10264
1.5 −1.00000 0 1.00000 −1.98339 0 −2.32664 −1.00000 0 1.98339
1.6 −1.00000 0 1.00000 −1.87424 0 2.79571 −1.00000 0 1.87424
1.7 −1.00000 0 1.00000 −1.34910 0 4.95731 −1.00000 0 1.34910
1.8 −1.00000 0 1.00000 −1.05384 0 0.315326 −1.00000 0 1.05384
1.9 −1.00000 0 1.00000 −0.323225 0 1.15584 −1.00000 0 0.323225
1.10 −1.00000 0 1.00000 0.963329 0 −1.11291 −1.00000 0 −0.963329
1.11 −1.00000 0 1.00000 1.87995 0 −4.90079 −1.00000 0 −1.87995
1.12 −1.00000 0 1.00000 2.85538 0 3.60685 −1.00000 0 −2.85538
1.13 −1.00000 0 1.00000 3.11545 0 0.732634 −1.00000 0 −3.11545
1.14 −1.00000 0 1.00000 4.13951 0 2.80704 −1.00000 0 −4.13951
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(397\) \(1\)

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 7146.2.a.be 14
3.b odd 2 1 794.2.a.h ✓ 14
12.b even 2 1 6352.2.a.t 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
794.2.a.h ✓ 14 3.b odd 2 1
6352.2.a.t 14 12.b even 2 1
7146.2.a.be 14 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7146))\):

\( T_{5}^{14} + 7 T_{5}^{13} - 24 T_{5}^{12} - 263 T_{5}^{11} - 63 T_{5}^{10} + 3293 T_{5}^{9} + \cdots - 12550 \) Copy content Toggle raw display
\( T_{7}^{14} - 7 T_{7}^{13} - 36 T_{7}^{12} + 337 T_{7}^{11} + 131 T_{7}^{10} - 5025 T_{7}^{9} + \cdots + 1372 \) Copy content Toggle raw display
\( T_{11}^{14} + 3 T_{11}^{13} - 81 T_{11}^{12} - 240 T_{11}^{11} + 2194 T_{11}^{10} + 6001 T_{11}^{9} + \cdots - 22144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{14} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} + 7 T^{13} + \cdots - 12550 \) Copy content Toggle raw display
$7$ \( T^{14} - 7 T^{13} + \cdots + 1372 \) Copy content Toggle raw display
$11$ \( T^{14} + 3 T^{13} + \cdots - 22144 \) Copy content Toggle raw display
$13$ \( T^{14} - 7 T^{13} + \cdots - 624 \) Copy content Toggle raw display
$17$ \( T^{14} + 10 T^{13} + \cdots - 57903104 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots - 875981852 \) Copy content Toggle raw display
$23$ \( T^{14} + 14 T^{13} + \cdots + 46149632 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 2291924992 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 138684416 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 1978729472 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 563525632 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 271795072 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 9420361728 \) Copy content Toggle raw display
$53$ \( T^{14} + 21 T^{13} + \cdots - 98128 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 27507064832 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 99299268768 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 40107641600 \) Copy content Toggle raw display
$71$ \( T^{14} - 11 T^{13} + \cdots - 29460939 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots - 22806666828 \) Copy content Toggle raw display
$79$ \( T^{14} - 22 T^{13} + \cdots + 75169792 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 132093993012 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 66139874394112 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 71069373091 \) Copy content Toggle raw display
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