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Properties

Label 31.4.c.a
Level $31$
Weight $4$
Character orbit 31.c
Analytic conductor $1.829$
Analytic rank $0$
Dimension $14$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [31,4,Mod(5,31)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(31, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("31.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 31.c (of order \(3\), degree \(2\), minimal)

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: no
Analytic conductor: \(1.82905921018\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 2 x^{13} + 38 x^{12} - 56 x^{11} + 1015 x^{10} - 1484 x^{9} + 11598 x^{8} - 12482 x^{7} + \cdots + 123904 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_1) q^{2} + ( - \beta_{10} - \beta_{8} + 2 \beta_{3} - 2) q^{3} + ( - \beta_{2} + 2) q^{4} + ( - \beta_{6} - \beta_{3}) q^{5} + (\beta_{11} + \beta_{7} - 3 \beta_{4} + \cdots - 3) q^{6}+ \cdots + (\beta_{9} + \beta_{8} + \cdots + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} - \beta_1) q^{2} + ( - \beta_{10} - \beta_{8} + 2 \beta_{3} - 2) q^{3} + ( - \beta_{2} + 2) q^{4} + ( - \beta_{6} - \beta_{3}) q^{5} + (\beta_{11} + \beta_{7} - 3 \beta_{4} + \cdots - 3) q^{6}+ \cdots + ( - 30 \beta_{13} + 7 \beta_{12} + \cdots + 209) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 4 q^{2} - 11 q^{3} + 32 q^{4} - 5 q^{5} - 12 q^{6} + 31 q^{7} + 12 q^{8} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 4 q^{2} - 11 q^{3} + 32 q^{4} - 5 q^{5} - 12 q^{6} + 31 q^{7} + 12 q^{8} - 14 q^{9} - 18 q^{10} - 35 q^{11} - 94 q^{12} + 15 q^{13} - 60 q^{14} + 78 q^{15} + 72 q^{16} - 81 q^{17} - 160 q^{18} + 73 q^{19} - 260 q^{20} + 157 q^{21} - 28 q^{22} + 264 q^{23} + 56 q^{24} - 284 q^{25} + 334 q^{26} - 314 q^{27} + 430 q^{28} + 692 q^{29} + 438 q^{31} - 508 q^{32} + 66 q^{33} - 838 q^{34} + 2 q^{35} + 492 q^{36} + 311 q^{37} + 256 q^{38} + 450 q^{39} - 402 q^{40} - 397 q^{41} + 122 q^{42} - 783 q^{43} - 1722 q^{44} - 234 q^{45} + 1288 q^{46} - 816 q^{47} - 1114 q^{48} - 1406 q^{49} + 440 q^{50} - 471 q^{51} + 772 q^{52} + 1183 q^{53} + 1424 q^{54} - 211 q^{55} - 1848 q^{56} - 19 q^{57} + 440 q^{58} + 1147 q^{59} + 4996 q^{60} - 1340 q^{61} + 1148 q^{62} - 2768 q^{63} - 2192 q^{64} - 403 q^{65} + 3404 q^{66} + 1589 q^{67} + 1572 q^{68} - 48 q^{69} + 5536 q^{70} - 1053 q^{71} - 856 q^{72} + 1799 q^{73} - 2486 q^{74} - 3152 q^{75} - 1562 q^{76} + 634 q^{77} - 9248 q^{78} - 3633 q^{79} - 2328 q^{80} + 3165 q^{81} - 1306 q^{82} - 1301 q^{83} + 3300 q^{84} + 6062 q^{85} + 1828 q^{86} - 3164 q^{87} - 1936 q^{88} + 2724 q^{89} + 816 q^{90} + 4934 q^{91} - 2664 q^{92} + 5205 q^{93} - 8568 q^{94} - 2146 q^{95} - 884 q^{96} + 2620 q^{97} + 7168 q^{98} + 1688 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 2 x^{13} + 38 x^{12} - 56 x^{11} + 1015 x^{10} - 1484 x^{9} + 11598 x^{8} - 12482 x^{7} + \cdots + 123904 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 46\!\cdots\!36 \nu^{13} + \cdots + 11\!\cdots\!28 ) / 11\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 18\!\cdots\!26 \nu^{13} + \cdots + 44\!\cdots\!20 ) / 42\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20\!\cdots\!61 \nu^{13} + \cdots - 20\!\cdots\!64 ) / 38\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 14\!\cdots\!98 \nu^{13} + \cdots - 29\!\cdots\!36 ) / 21\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 14\!\cdots\!82 \nu^{13} + \cdots + 50\!\cdots\!76 ) / 10\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 68\!\cdots\!18 \nu^{13} + \cdots - 67\!\cdots\!92 ) / 42\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 10\!\cdots\!38 \nu^{13} + \cdots - 61\!\cdots\!52 ) / 42\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 82\!\cdots\!94 \nu^{13} + \cdots - 20\!\cdots\!08 ) / 21\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 19\!\cdots\!89 \nu^{13} + \cdots + 27\!\cdots\!40 ) / 19\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 31\!\cdots\!69 \nu^{13} + \cdots + 35\!\cdots\!56 ) / 19\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 13\!\cdots\!07 \nu^{13} + \cdots + 13\!\cdots\!20 ) / 76\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 14\!\cdots\!73 \nu^{13} + \cdots + 80\!\cdots\!00 ) / 84\!\cdots\!68 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 10\beta_{3} + \beta_{2} - 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + \beta_{12} - \beta_{11} - \beta_{10} + \beta_{6} + 19\beta_{4} + \beta_{2} - 19\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -23\beta_{9} + 7\beta_{8} - 2\beta_{7} - \beta_{6} - 3\beta_{5} - 185\beta_{3} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 22 \beta_{13} - 32 \beta_{12} + 32 \beta_{11} + 24 \beta_{10} - 28 \beta_{9} + 24 \beta_{8} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 36 \beta_{13} - 112 \beta_{12} - 66 \beta_{11} + 240 \beta_{10} + 36 \beta_{6} - 206 \beta_{4} + \cdots + 3892 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 735\beta_{9} - 597\beta_{8} - 815\beta_{7} - 473\beta_{6} + 861\beta_{5} + 36\beta_{3} + 9029\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 941 \beta_{13} + 3187 \beta_{12} + 1632 \beta_{11} - 6447 \beta_{10} + 11913 \beta_{9} - 6447 \beta_{8} + \cdots - 85911 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 10358 \beta_{13} + 21724 \beta_{12} - 19378 \beta_{11} - 15196 \beta_{10} + 10358 \beta_{6} + \cdots - 14858 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 276381 \beta_{9} + 160432 \beta_{8} - 36556 \beta_{7} - 22016 \beta_{6} - 82920 \beta_{5} + \cdots - 191140 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 230017 \beta_{13} - 532617 \beta_{12} + 449021 \beta_{11} + 386393 \beta_{10} - 491869 \beta_{9} + \cdots + 682010 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 489769 \beta_{13} - 2078707 \beta_{12} - 782414 \beta_{11} + 3874791 \beta_{10} + 489769 \beta_{6} + \cdots + 44504309 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 12583968 \beta_{9} - 9754256 \beta_{8} - 10301348 \beta_{7} - 5154374 \beta_{6} + 12883912 \beta_{5} + \cdots + 110180241 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
2.45227 + 4.24746i
1.72284 + 2.98404i
0.676171 + 1.17116i
0.551662 + 0.955506i
−0.469231 − 0.812733i
−1.57992 − 2.73650i
−2.35379 − 4.07688i
2.45227 − 4.24746i
1.72284 − 2.98404i
0.676171 − 1.17116i
0.551662 − 0.955506i
−0.469231 + 0.812733i
−1.57992 + 2.73650i
−2.35379 + 4.07688i
−4.90455 −1.91417 + 3.31543i 16.0546 −4.95460 − 8.58162i 9.38812 − 16.2607i 17.6619 − 30.5913i −39.5040 6.17193 + 10.6901i 24.3001 + 42.0889i
5.2 −3.44567 0.460799 − 0.798127i 3.87266 8.49030 + 14.7056i −1.58776 + 2.75008i −11.0327 + 19.1091i 14.2215 13.0753 + 22.6471i −29.2548 − 50.6708i
5.3 −1.35234 3.54046 − 6.13226i −6.17117 −3.22870 − 5.59227i −4.78792 + 8.29292i 9.25891 − 16.0369i 19.1643 −11.5697 − 20.0394i 4.36631 + 7.56267i
5.4 −1.10332 −2.77097 + 4.79946i −6.78268 −7.29176 − 12.6297i 3.05727 − 5.29535i −13.8631 + 24.0116i 16.3101 −1.85651 − 3.21558i 8.04517 + 13.9346i
5.5 0.938463 −3.60837 + 6.24987i −7.11929 9.47653 + 16.4138i −3.38632 + 5.86527i 14.4457 − 25.0206i −14.1889 −12.5406 − 21.7210i 8.89338 + 15.4038i
5.6 3.15984 1.93687 − 3.35476i 1.98461 1.19593 + 2.07141i 6.12022 − 10.6005i −4.42400 + 7.66259i −19.0077 5.99704 + 10.3872i 3.77894 + 6.54532i
5.7 4.70758 −3.14463 + 5.44667i 14.1613 −6.18770 − 10.7174i −14.8036 + 25.6406i 3.45332 − 5.98133i 29.0048 −6.27745 − 10.8729i −29.1291 − 50.4530i
25.1 −4.90455 −1.91417 − 3.31543i 16.0546 −4.95460 + 8.58162i 9.38812 + 16.2607i 17.6619 + 30.5913i −39.5040 6.17193 − 10.6901i 24.3001 − 42.0889i
25.2 −3.44567 0.460799 + 0.798127i 3.87266 8.49030 − 14.7056i −1.58776 − 2.75008i −11.0327 − 19.1091i 14.2215 13.0753 − 22.6471i −29.2548 + 50.6708i
25.3 −1.35234 3.54046 + 6.13226i −6.17117 −3.22870 + 5.59227i −4.78792 − 8.29292i 9.25891 + 16.0369i 19.1643 −11.5697 + 20.0394i 4.36631 − 7.56267i
25.4 −1.10332 −2.77097 − 4.79946i −6.78268 −7.29176 + 12.6297i 3.05727 + 5.29535i −13.8631 − 24.0116i 16.3101 −1.85651 + 3.21558i 8.04517 − 13.9346i
25.5 0.938463 −3.60837 − 6.24987i −7.11929 9.47653 − 16.4138i −3.38632 − 5.86527i 14.4457 + 25.0206i −14.1889 −12.5406 + 21.7210i 8.89338 − 15.4038i
25.6 3.15984 1.93687 + 3.35476i 1.98461 1.19593 − 2.07141i 6.12022 + 10.6005i −4.42400 − 7.66259i −19.0077 5.99704 − 10.3872i 3.77894 − 6.54532i
25.7 4.70758 −3.14463 − 5.44667i 14.1613 −6.18770 + 10.7174i −14.8036 − 25.6406i 3.45332 + 5.98133i 29.0048 −6.27745 + 10.8729i −29.1291 + 50.4530i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.4.c.a ✓ 14
3.b odd 2 1 279.4.h.a 14
4.b odd 2 1 496.4.i.f 14
31.c even 3 1 inner 31.4.c.a ✓ 14
31.c even 3 1 961.4.a.g 7
31.e odd 6 1 961.4.a.f 7
93.h odd 6 1 279.4.h.a 14
124.i odd 6 1 496.4.i.f 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.4.c.a ✓ 14 1.a even 1 1 trivial
31.4.c.a ✓ 14 31.c even 3 1 inner
279.4.h.a 14 3.b odd 2 1
279.4.h.a 14 93.h odd 6 1
496.4.i.f 14 4.b odd 2 1
496.4.i.f 14 124.i odd 6 1
961.4.a.f 7 31.e odd 6 1
961.4.a.g 7 31.c even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(31, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{7} + 2 T^{6} + \cdots - 352)^{2} \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 592581649 \) Copy content Toggle raw display
$5$ \( T^{14} + \cdots + 79025930460409 \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 49\!\cdots\!89 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 90\!\cdots\!29 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 21\!\cdots\!09 \) Copy content Toggle raw display
$17$ \( T^{14} + \cdots + 14\!\cdots\!21 \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 11\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots + 20701839761408)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots - 240223668944768)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 20\!\cdots\!31 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 38\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 10\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 13\!\cdots\!81 \) Copy content Toggle raw display
$47$ \( (T^{7} + \cdots - 23\!\cdots\!36)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 84\!\cdots\!89 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 11\!\cdots\!81 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 39\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 84\!\cdots\!89 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 25\!\cdots\!21 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 61\!\cdots\!49 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 27\!\cdots\!29 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 26\!\cdots\!81 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots - 25\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots - 35\!\cdots\!16)^{2} \) Copy content Toggle raw display
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