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Properties

Label 145.6.a.c
Level $145$
Weight $6$
Character orbit 145.a
Self dual yes
Analytic conductor $23.256$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [145,6,Mod(1,145)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("145.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(145, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 145.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [13,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(23.2556538729\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 6 x^{12} - 296 x^{11} + 1238 x^{10} + 33250 x^{9} - 78360 x^{8} - 1708024 x^{7} + \cdots + 251513192 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{2} + \beta_{4} q^{3} + (\beta_{2} + 16) q^{4} - 25 q^{5} + (\beta_{8} + \beta_{7} + 2 \beta_{4} - \beta_1) q^{6} + (\beta_{8} - \beta_{6} + 2 \beta_{4} + \cdots + 1) q^{7} + ( - \beta_{12} + \beta_{8} + 3 \beta_{4} + \cdots - 22) q^{8}+ \cdots + ( - 68 \beta_{12} + 65 \beta_{11} + \cdots - 2734) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q + 7 q^{2} + 2 q^{3} + 213 q^{4} - 325 q^{5} - 3 q^{6} - 18 q^{7} - 399 q^{8} + 1155 q^{9} - 175 q^{10} + 844 q^{11} + 167 q^{12} - 704 q^{13} + 4425 q^{14} - 50 q^{15} + 5805 q^{16} - 210 q^{17} + 7378 q^{18}+ \cdots - 3872 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{13} - 6 x^{12} - 296 x^{11} + 1238 x^{10} + 33250 x^{9} - 78360 x^{8} - 1708024 x^{7} + \cdots + 251513192 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 47 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 17\!\cdots\!52 \nu^{12} + \cdots + 31\!\cdots\!88 ) / 12\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25\!\cdots\!11 \nu^{12} + \cdots + 45\!\cdots\!04 ) / 13\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 57\!\cdots\!73 \nu^{12} + \cdots + 10\!\cdots\!84 ) / 13\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 30\!\cdots\!37 \nu^{12} + \cdots + 55\!\cdots\!52 ) / 52\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15\!\cdots\!93 \nu^{12} + \cdots - 28\!\cdots\!80 ) / 13\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16\!\cdots\!41 \nu^{12} + \cdots + 30\!\cdots\!88 ) / 13\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 43\!\cdots\!77 \nu^{12} + \cdots - 77\!\cdots\!96 ) / 34\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 11\!\cdots\!15 \nu^{12} + \cdots + 20\!\cdots\!24 ) / 68\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 23\!\cdots\!49 \nu^{12} + \cdots - 41\!\cdots\!36 ) / 13\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 43\!\cdots\!83 \nu^{12} + \cdots + 78\!\cdots\!20 ) / 22\!\cdots\!84 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 47 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} - \beta_{8} - 3\beta_{4} - \beta_{3} + 3\beta_{2} + 88\beta _1 + 100 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{12} - \beta_{11} - 6 \beta_{10} - 3 \beta_{9} - \beta_{8} - 7 \beta_{6} + 8 \beta_{5} + \cdots + 4116 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 143 \beta_{12} - 34 \beta_{11} - 26 \beta_{10} - 16 \beta_{9} - 199 \beta_{8} - 6 \beta_{7} + \cdots + 18324 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 693 \beta_{12} - 349 \beta_{11} - 1236 \beta_{10} - 463 \beta_{9} - 663 \beta_{8} - 296 \beta_{7} + \cdots + 431594 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 18025 \beta_{12} - 7556 \beta_{11} - 7248 \beta_{10} - 2438 \beta_{9} - 29785 \beta_{8} - 3622 \beta_{7} + \cdots + 2812652 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 114161 \beta_{12} - 75953 \beta_{11} - 193518 \beta_{10} - 52127 \beta_{9} - 163999 \beta_{8} + \cdots + 50183340 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2227681 \beta_{12} - 1295144 \beta_{11} - 1414102 \beta_{10} - 253658 \beta_{9} - 4236429 \beta_{8} + \cdots + 407537682 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 16826577 \beta_{12} - 13635801 \beta_{11} - 27999368 \beta_{10} - 4934479 \beta_{9} - 31379235 \beta_{8} + \cdots + 6220852362 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 277125797 \beta_{12} - 203336196 \beta_{11} - 239329964 \beta_{10} - 18466038 \beta_{9} + \cdots + 57731633352 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 2369432269 \beta_{12} - 2233390933 \beta_{11} - 3957483850 \beta_{10} - 371779707 \beta_{9} + \cdots + 803047759380 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
11.9547
10.7368
8.18919
6.43969
3.37801
0.805122
−1.39799
−1.39945
−2.22498
−5.63626
−7.41417
−7.79320
−9.63745
−10.9547 13.6840 88.0059 −25.0000 −149.904 −126.162 −613.529 −55.7482 273.868
1.2 −9.73676 −23.9644 62.8045 −25.0000 233.336 34.2066 −299.936 331.293 243.419
1.3 −7.18919 6.65927 19.6844 −25.0000 −47.8747 −243.085 88.5391 −198.654 179.730
1.4 −5.43969 −3.47310 −2.40976 −25.0000 18.8926 183.942 187.178 −230.938 135.992
1.5 −2.37801 22.5809 −26.3451 −25.0000 −53.6977 −54.8749 138.745 266.898 59.4503
1.6 0.194878 −10.1754 −31.9620 −25.0000 −1.98296 −172.604 −12.4648 −139.461 −4.87196
1.7 2.39799 24.0072 −26.2496 −25.0000 57.5690 216.784 −139.682 333.344 −59.9498
1.8 2.39945 −14.4285 −26.2426 −25.0000 −34.6205 4.40159 −139.750 −34.8176 −59.9863
1.9 3.22498 −0.213539 −21.5995 −25.0000 −0.688659 −23.5466 −172.857 −242.954 −80.6245
1.10 6.63626 −30.6482 12.0399 −25.0000 −203.389 −171.700 −132.460 696.311 −165.906
1.11 8.41417 −17.8297 38.7982 −25.0000 −150.022 108.201 57.2014 74.8998 −210.354
1.12 8.79320 27.8996 45.3203 −25.0000 245.326 55.3204 117.128 535.387 −219.830
1.13 10.6375 7.90198 81.1554 −25.0000 84.0569 171.117 522.888 −180.559 −265.936
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(29\) \( -1 \)

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 145.6.a.c ✓ 13
5.b even 2 1 725.6.a.f 13
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.6.a.c ✓ 13 1.a even 1 1 trivial
725.6.a.f 13 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{13} - 7 T_{2}^{12} - 290 T_{2}^{11} + 2128 T_{2}^{10} + 28745 T_{2}^{9} - 226077 T_{2}^{8} + \cdots + 187349760 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(145))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{13} + \cdots + 187349760 \) Copy content Toggle raw display
$3$ \( T^{13} + \cdots + 15528887348544 \) Copy content Toggle raw display
$5$ \( (T + 25)^{13} \) Copy content Toggle raw display
$7$ \( T^{13} + \cdots - 72\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{13} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$13$ \( T^{13} + \cdots + 99\!\cdots\!92 \) Copy content Toggle raw display
$17$ \( T^{13} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{13} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{13} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T - 841)^{13} \) Copy content Toggle raw display
$31$ \( T^{13} + \cdots + 42\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{13} + \cdots - 13\!\cdots\!48 \) Copy content Toggle raw display
$41$ \( T^{13} + \cdots - 29\!\cdots\!68 \) Copy content Toggle raw display
$43$ \( T^{13} + \cdots - 58\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{13} + \cdots - 57\!\cdots\!72 \) Copy content Toggle raw display
$53$ \( T^{13} + \cdots - 11\!\cdots\!48 \) Copy content Toggle raw display
$59$ \( T^{13} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{13} + \cdots - 22\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{13} + \cdots - 96\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{13} + \cdots - 84\!\cdots\!52 \) Copy content Toggle raw display
$73$ \( T^{13} + \cdots + 36\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{13} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{13} + \cdots - 23\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{13} + \cdots - 41\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{13} + \cdots - 65\!\cdots\!72 \) Copy content Toggle raw display
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