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Properties

Label 361.4.a.m
Level $361$
Weight $4$
Character orbit 361.a
Self dual yes
Analytic conductor $21.300$
Analytic rank $1$
Dimension $12$
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] = [361,4,Mod(1,361)] 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("361.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(361, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 361.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-6] 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: \(21.2996895121\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} - 48 x^{10} + 287 x^{9} + 846 x^{8} - 4893 x^{7} - 7080 x^{6} + 35973 x^{5} + \cdots + 87552 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + (\beta_{4} - 2) q^{3} + (\beta_{6} - \beta_{5} - 2 \beta_{3} + \cdots + 4) q^{4} + (\beta_{11} - \beta_{10} - 2 \beta_{8} + \cdots + 2) q^{5} + ( - \beta_{9} - \beta_{7} + \beta_{6} + \cdots + 4) q^{6}+ \cdots + ( - 4 \beta_{11} + 20 \beta_{10} + \cdots + 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{2} - 18 q^{3} + 36 q^{4} + 12 q^{5} + 24 q^{6} - 3 q^{7} - 75 q^{8} + 54 q^{9} - 84 q^{10} - 39 q^{11} - 219 q^{12} - 6 q^{13} - 108 q^{14} - 261 q^{15} + 48 q^{16} + 75 q^{17} - 132 q^{18}+ \cdots + 1320 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} - 48 x^{10} + 287 x^{9} + 846 x^{8} - 4893 x^{7} - 7080 x^{6} + 35973 x^{5} + \cdots + 87552 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 212033 \nu^{11} - 12864862 \nu^{10} + 82978908 \nu^{9} + 616055573 \nu^{8} + \cdots + 237015718272 ) / 21569580672 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 222909 \nu^{11} + 1855394 \nu^{10} + 6361212 \nu^{9} - 58153479 \nu^{8} + \cdots + 95708745600 ) / 21569580672 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 753111 \nu^{11} - 13608342 \nu^{10} + 64091944 \nu^{9} + 753716847 \nu^{8} + \cdots - 146621003328 ) / 43139161344 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 126766 \nu^{11} - 2595459 \nu^{10} + 17699073 \nu^{9} + 138051241 \nu^{8} + \cdots + 89764057824 ) / 5392395168 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 740849 \nu^{11} + 6193814 \nu^{10} + 539808 \nu^{9} - 180157567 \nu^{8} + \cdots + 76192616832 ) / 21569580672 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3844815 \nu^{11} - 25003186 \nu^{10} - 111365632 \nu^{9} + 1047242049 \nu^{8} + \cdots + 1114228452288 ) / 43139161344 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 8155217 \nu^{11} + 48953054 \nu^{10} + 362009904 \nu^{9} - 2187311887 \nu^{8} + \cdots - 86442063168 ) / 43139161344 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 754321 \nu^{11} - 2100041 \nu^{10} - 40005445 \nu^{9} + 73042508 \nu^{8} + 732787322 \nu^{7} + \cdots - 50344270512 ) / 2696197584 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27691097 \nu^{11} - 218626278 \nu^{10} - 907325432 \nu^{9} + 9917300815 \nu^{8} + \cdots + 2436608461248 ) / 43139161344 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 18829589 \nu^{11} + 95799266 \nu^{10} + 953656020 \nu^{9} - 4342501039 \nu^{8} + \cdots - 480116687232 ) / 21569580672 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} - 2\beta_{3} + \beta_{2} + \beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} - 7 \beta_{8} + 3 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + \beta_{4} - 5 \beta_{3} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{11} - 5 \beta_{9} - 11 \beta_{8} + 11 \beta_{7} + 42 \beta_{6} - 38 \beta_{5} - 88 \beta_{3} + \cdots + 212 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 45 \beta_{11} - 33 \beta_{10} - 25 \beta_{9} - 310 \beta_{8} + 120 \beta_{7} + 200 \beta_{6} + \cdots + 470 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 155 \beta_{11} - 27 \beta_{10} - 256 \beta_{9} - 913 \beta_{8} + 553 \beta_{7} + 1523 \beta_{6} + \cdots + 5275 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 1669 \beta_{11} - 975 \beta_{10} - 1435 \beta_{9} - 11314 \beta_{8} + 4298 \beta_{7} + 7909 \beta_{6} + \cdots + 18475 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 7396 \beta_{11} - 1882 \beta_{10} - 10374 \beta_{9} - 45260 \beta_{8} + 22098 \beta_{7} + 53334 \beta_{6} + \cdots + 152344 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 59150 \beta_{11} - 29124 \beta_{10} - 60358 \beta_{9} - 394462 \beta_{8} + 150400 \beta_{7} + \cdots + 682375 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 298435 \beta_{11} - 89445 \beta_{10} - 387724 \beta_{9} - 1866427 \beta_{8} + 820733 \beta_{7} + \cdots + 4802740 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 2067388 \beta_{11} - 904060 \beta_{10} - 2282115 \beta_{9} - 13616849 \beta_{8} + 5232831 \beta_{7} + \cdots + 24518434 \) 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
−4.36282
−3.75063
−2.89915
−1.74966
−1.30832
−0.802785
1.61628
1.94318
3.28129
3.95274
4.17548
5.90441
−5.36282 −5.44082 20.7599 −9.98217 29.1782 18.8614 −68.4289 2.60253 53.5326
1.2 −4.75063 −7.51924 14.5685 14.9244 35.7211 −7.29197 −31.2044 29.5390 −70.9004
1.3 −3.89915 −3.96313 7.20340 14.9069 15.4529 −10.7029 3.10605 −11.2936 −58.1243
1.4 −2.74966 6.03443 −0.439354 −5.43841 −16.5927 −23.4784 23.2054 9.41438 14.9538
1.5 −2.30832 5.35815 −2.67165 8.20900 −12.3683 2.89172 24.6336 1.70975 −18.9490
1.6 −1.80279 2.75000 −4.74996 −18.0568 −4.95766 28.2555 22.9855 −19.4375 32.5526
1.7 0.616280 −9.41688 −7.62020 12.8984 −5.80343 −12.1139 −9.62642 61.6775 7.94906
1.8 0.943179 −2.03721 −7.11041 3.78557 −1.92145 23.4166 −14.2518 −22.8498 3.57047
1.9 2.28129 4.26039 −2.79573 7.33557 9.71917 −26.2038 −24.6282 −8.84910 16.7345
1.10 2.95274 −9.39455 0.718680 −0.532952 −27.7397 23.7897 −21.4999 61.2575 −1.57367
1.11 3.17548 1.96746 2.08367 −8.65737 6.24763 8.72625 −18.7872 −23.1291 −27.4913
1.12 4.90441 −0.598597 16.0532 −7.39221 −2.93576 −29.1502 39.4963 −26.6417 −36.2544
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(19\) \( -1 \)

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 361.4.a.m 12
19.b odd 2 1 361.4.a.n 12
19.f odd 18 2 19.4.e.a ✓ 24
57.j even 18 2 171.4.u.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.e.a ✓ 24 19.f odd 18 2
171.4.u.b 24 57.j even 18 2
361.4.a.m 12 1.a even 1 1 trivial
361.4.a.n 12 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 6 T_{2}^{11} - 48 T_{2}^{10} - 303 T_{2}^{9} + 774 T_{2}^{8} + 5259 T_{2}^{7} + \cdots + 69312 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(361))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 6 T^{11} + \cdots + 69312 \) Copy content Toggle raw display
$3$ \( T^{12} + 18 T^{11} + \cdots - 13036717 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 21870774936 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 127021689025944 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 17\!\cdots\!41 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots - 89\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 11\!\cdots\!83 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 13\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots - 20\!\cdots\!84 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 14\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 48\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 13\!\cdots\!23 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 30\!\cdots\!79 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots - 24\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 96\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 83\!\cdots\!07 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots - 55\!\cdots\!88 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 24\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 26\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 37\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 32\!\cdots\!68 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 38\!\cdots\!11 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 65\!\cdots\!03 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 15\!\cdots\!49 \) Copy content Toggle raw display
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