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Properties

Label 31.3.f.a
Level $31$
Weight $3$
Character orbit 31.f
Analytic conductor $0.845$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [31,3,Mod(15,31)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([7])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("31.15"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 31.f (of order \(10\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.844688819517\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} + 20 x^{18} - 33 x^{17} + 250 x^{16} - 510 x^{15} + 2908 x^{14} - 6447 x^{13} + \cdots + 731025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{10} q^{2} - \beta_{13} q^{3} + (\beta_{19} + 3 \beta_{11}) q^{4} + ( - \beta_{12} + \beta_{8} - \beta_{6} + \cdots - 2) q^{5} + ( - \beta_{19} + 2 \beta_{18} + \cdots + 1) q^{6} + ( - \beta_{19} + \beta_{15} + \cdots - \beta_{7}) q^{7}+ \cdots + (3 \beta_{19} + 14 \beta_{18} + \cdots - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 3 q^{2} - 5 q^{3} - 11 q^{4} - 14 q^{5} - q^{7} - 19 q^{8} + 2 q^{9} + 12 q^{10} - 10 q^{11} + 90 q^{12} + 10 q^{13} - 85 q^{15} - 103 q^{16} + 35 q^{17} + 6 q^{18} + 47 q^{19} - 125 q^{20} - 125 q^{21}+ \cdots - 1000 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 20 x^{18} - 33 x^{17} + 250 x^{16} - 510 x^{15} + 2908 x^{14} - 6447 x^{13} + \cdots + 731025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 44\!\cdots\!91 \nu^{19} + \cdots + 79\!\cdots\!35 ) / 11\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 53\!\cdots\!64 \nu^{19} + \cdots + 84\!\cdots\!50 ) / 11\!\cdots\!05 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 38\!\cdots\!28 \nu^{19} + \cdots + 43\!\cdots\!50 ) / 10\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11\!\cdots\!83 \nu^{19} + \cdots + 58\!\cdots\!50 ) / 21\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 31\!\cdots\!66 \nu^{19} + \cdots - 58\!\cdots\!50 ) / 58\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 33\!\cdots\!70 \nu^{19} + \cdots + 34\!\cdots\!50 ) / 58\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 37\!\cdots\!82 \nu^{19} + \cdots + 32\!\cdots\!80 ) / 58\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15\!\cdots\!77 \nu^{19} + \cdots - 17\!\cdots\!25 ) / 21\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 78\!\cdots\!61 \nu^{19} + \cdots - 47\!\cdots\!50 ) / 10\!\cdots\!45 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 62\!\cdots\!98 \nu^{19} + \cdots - 10\!\cdots\!90 ) / 58\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 80\!\cdots\!48 \nu^{19} + \cdots - 27\!\cdots\!50 ) / 64\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 32\!\cdots\!83 \nu^{19} + \cdots + 67\!\cdots\!75 ) / 21\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 94\!\cdots\!13 \nu^{19} + \cdots - 12\!\cdots\!75 ) / 58\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 22\!\cdots\!58 \nu^{19} + \cdots - 37\!\cdots\!00 ) / 58\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 23\!\cdots\!90 \nu^{19} + \cdots + 23\!\cdots\!50 ) / 58\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 24\!\cdots\!27 \nu^{19} + \cdots + 21\!\cdots\!35 ) / 58\!\cdots\!65 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 59\!\cdots\!49 \nu^{19} + \cdots + 86\!\cdots\!40 ) / 12\!\cdots\!70 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 42\!\cdots\!56 \nu^{19} + \cdots + 69\!\cdots\!30 ) / 58\!\cdots\!65 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{16} - 7\beta_{7} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2 \beta_{18} + \beta_{17} + 2 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{10} + 2 \beta_{5} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12 \beta_{19} - \beta_{17} - \beta_{16} + 2 \beta_{15} + 2 \beta_{13} + 2 \beta_{12} + 69 \beta_{11} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 19 \beta_{19} - 19 \beta_{17} - 32 \beta_{15} + 32 \beta_{14} + 36 \beta_{13} + 2 \beta_{12} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 6 \beta_{18} + 164 \beta_{17} + 22 \beta_{16} - 6 \beta_{15} - 32 \beta_{14} - 34 \beta_{13} + \cdots + 184 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 280 \beta_{19} - 444 \beta_{18} + 64 \beta_{16} - 50 \beta_{14} - 134 \beta_{13} - 444 \beta_{12} + \cdots + 190 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1719 \beta_{19} + 570 \beta_{18} - 2109 \beta_{17} - 1719 \beta_{16} + 158 \beta_{15} - 570 \beta_{14} + \cdots - 11833 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1008 \beta_{19} + 6924 \beta_{18} + 3841 \beta_{17} + 3841 \beta_{16} + 5998 \beta_{15} + 1190 \beta_{13} + \cdots + 6494 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 6263 \beta_{19} + 6263 \beta_{17} - 8016 \beta_{15} + 8016 \beta_{14} + 16328 \beta_{13} - 4938 \beta_{12} + \cdots + 156379 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 95984 \beta_{18} - 37104 \beta_{17} - 51187 \beta_{16} - 95984 \beta_{15} + 15308 \beta_{14} + \cdots - 145498 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 95140 \beta_{19} - 110600 \beta_{18} + 266540 \beta_{16} + 56836 \beta_{14} - 130108 \beta_{13} + \cdots - 871692 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 184856 \beta_{19} + 1085696 \beta_{18} + 487544 \beta_{17} - 184856 \beta_{16} + 1325416 \beta_{15} + \cdots + 227080 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 3381373 \beta_{19} + 888180 \beta_{18} - 1398220 \beta_{17} - 1398220 \beta_{16} + 1519456 \beta_{15} + \cdots + 1910816 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 8761393 \beta_{19} - 8761393 \beta_{17} - 14631402 \beta_{15} + 14631402 \beta_{14} + 18020750 \beta_{13} + \cdots + 3398490 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 14172340 \beta_{18} + 63366093 \beta_{17} + 20130213 \beta_{16} - 14172340 \beta_{15} + \cdots + 146888192 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( 113596275 \beta_{19} - 197466448 \beta_{18} + 28815675 \beta_{16} - 54361786 \beta_{14} + \cdots + 35190172 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 556031402 \beta_{19} + 287901398 \beta_{18} - 841992700 \beta_{17} - 556031402 \beta_{16} + \cdots - 4994265487 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 348933632 \beta_{19} + 3469253478 \beta_{18} + 1468094680 \beta_{17} + 1468094680 \beta_{16} + \cdots + 2873523836 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(\beta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1
1.15230 + 3.54642i
0.651822 + 2.00610i
0.276623 + 0.851359i
−0.453363 − 1.39531i
−1.12738 − 3.46973i
2.39225 − 1.73807i
2.18177 − 1.58515i
0.339254 − 0.246482i
−1.55178 + 1.12743i
−2.86150 + 2.07900i
2.39225 + 1.73807i
2.18177 + 1.58515i
0.339254 + 0.246482i
−1.55178 − 1.12743i
−2.86150 − 2.07900i
1.15230 − 3.54642i
0.651822 − 2.00610i
0.276623 − 0.851359i
−0.453363 + 1.39531i
−1.12738 + 3.46973i
−3.01677 + 2.19181i −1.12307 + 1.54577i 3.06079 − 9.42013i −7.37547 − 7.12479i −1.10519 + 3.40142i 6.80424 + 20.9413i 1.65302 + 5.08748i 22.2501 − 16.1656i
15.2 −1.70649 + 1.23984i 3.05635 − 4.20670i 0.138848 − 0.427329i 4.35092 10.9681i −2.01919 + 6.21441i −2.31441 − 7.12303i −5.57392 − 17.1548i −7.42481 + 5.39444i
15.3 −0.724209 + 0.526168i −2.01819 + 2.77780i −0.988443 + 3.04211i 4.26245 − 3.07362i 1.06330 − 3.27251i −1.99132 − 6.12865i −0.861933 − 2.65276i −3.08690 + 2.24277i
15.4 1.18692 − 0.862348i 1.08425 − 1.49234i −0.570933 + 1.75715i −4.14776 − 2.70629i 1.06697 − 3.28380i 2.65108 + 8.15917i 1.72966 + 5.32336i −4.92306 + 3.57681i
15.5 2.95153 − 2.14441i −2.80836 + 3.86537i 2.87696 − 8.85438i −3.94423 17.4310i −0.932951 + 2.87133i −5.98647 − 18.4245i −4.27307 − 13.1512i −11.6415 + 8.45807i
23.1 −0.913758 + 2.81226i 5.20635 − 1.69165i −3.83778 − 2.78831i −5.21585 16.1874i −5.08304 − 3.69304i 1.77924 − 1.29269i 16.9633 − 12.3246i 4.76602 − 14.6683i
23.2 −0.833363 + 2.56483i −3.19530 + 1.03822i −2.64778 − 1.92372i −1.06496 − 9.06060i 10.2892 + 7.47554i −1.58651 + 1.15266i 1.85090 − 1.34475i 0.887497 − 2.73143i
23.3 −0.129583 + 0.398816i −0.358441 + 0.116465i 3.09381 + 2.24778i 3.98636 − 0.158044i −9.49164 − 6.89608i −2.65437 + 1.92851i −7.16624 + 5.20658i −0.516566 + 1.58983i
23.4 0.592727 − 1.82423i 1.04418 − 0.339275i 0.259595 + 0.188607i −6.67828 − 2.10592i 3.63065 + 2.63783i 6.70505 − 4.87150i −6.30595 + 4.58154i −3.95840 + 12.1827i
23.5 1.09299 − 3.36389i −3.38778 + 1.10076i −6.88507 − 5.00229i 8.82683 12.5992i 2.08188 + 1.51257i −12.9065 + 9.37714i 2.98422 − 2.16816i 9.64768 − 29.6925i
27.1 −0.913758 − 2.81226i 5.20635 + 1.69165i −3.83778 + 2.78831i −5.21585 − 16.1874i −5.08304 + 3.69304i 1.77924 + 1.29269i 16.9633 + 12.3246i 4.76602 + 14.6683i
27.2 −0.833363 − 2.56483i −3.19530 − 1.03822i −2.64778 + 1.92372i −1.06496 9.06060i 10.2892 − 7.47554i −1.58651 − 1.15266i 1.85090 + 1.34475i 0.887497 + 2.73143i
27.3 −0.129583 − 0.398816i −0.358441 − 0.116465i 3.09381 − 2.24778i 3.98636 0.158044i −9.49164 + 6.89608i −2.65437 − 1.92851i −7.16624 − 5.20658i −0.516566 − 1.58983i
27.4 0.592727 + 1.82423i 1.04418 + 0.339275i 0.259595 − 0.188607i −6.67828 2.10592i 3.63065 − 2.63783i 6.70505 + 4.87150i −6.30595 − 4.58154i −3.95840 − 12.1827i
27.5 1.09299 + 3.36389i −3.38778 − 1.10076i −6.88507 + 5.00229i 8.82683 − 12.5992i 2.08188 − 1.51257i −12.9065 − 9.37714i 2.98422 + 2.16816i 9.64768 + 29.6925i
29.1 −3.01677 − 2.19181i −1.12307 − 1.54577i 3.06079 + 9.42013i −7.37547 7.12479i −1.10519 − 3.40142i 6.80424 − 20.9413i 1.65302 − 5.08748i 22.2501 + 16.1656i
29.2 −1.70649 − 1.23984i 3.05635 + 4.20670i 0.138848 + 0.427329i 4.35092 − 10.9681i −2.01919 − 6.21441i −2.31441 + 7.12303i −5.57392 + 17.1548i −7.42481 − 5.39444i
29.3 −0.724209 − 0.526168i −2.01819 − 2.77780i −0.988443 − 3.04211i 4.26245 3.07362i 1.06330 + 3.27251i −1.99132 + 6.12865i −0.861933 + 2.65276i −3.08690 − 2.24277i
29.4 1.18692 + 0.862348i 1.08425 + 1.49234i −0.570933 − 1.75715i −4.14776 2.70629i 1.06697 + 3.28380i 2.65108 − 8.15917i 1.72966 − 5.32336i −4.92306 − 3.57681i
29.5 2.95153 + 2.14441i −2.80836 − 3.86537i 2.87696 + 8.85438i −3.94423 − 17.4310i −0.932951 − 2.87133i −5.98647 + 18.4245i −4.27307 + 13.1512i −11.6415 − 8.45807i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.f odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.3.f.a ✓ 20
3.b odd 2 1 279.3.v.a 20
31.f odd 10 1 inner 31.3.f.a ✓ 20
93.k even 10 1 279.3.v.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.3.f.a ✓ 20 1.a even 1 1 trivial
31.3.f.a ✓ 20 31.f odd 10 1 inner
279.3.v.a 20 3.b odd 2 1
279.3.v.a 20 93.k even 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(31, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + 3 T^{19} + \cdots + 731025 \) Copy content Toggle raw display
$3$ \( T^{20} + 5 T^{19} + \cdots + 66430125 \) Copy content Toggle raw display
$5$ \( (T^{10} + 7 T^{9} + \cdots + 2920851)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 82363701948025 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 98\!\cdots\!25 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 48\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 67\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 27\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 36\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 19\!\cdots\!21 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 70\!\cdots\!25)^{2} \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 14\!\cdots\!01 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 48\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 53\!\cdots\!25 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 69\!\cdots\!25 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 61\!\cdots\!25 \) Copy content Toggle raw display
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