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Properties

Label 312.6.q.b
Level $312$
Weight $6$
Character orbit 312.q
Analytic conductor $50.040$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [312,6,Mod(217,312)] 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("312.217"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(312, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 312.q (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [18,0,-81] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(50.0397517816\)
Analytic rank: \(0\)
Dimension: \(18\)
Relative dimension: \(9\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 9 x^{17} - 31561 x^{16} + 110968 x^{15} + 409124362 x^{14} + 1115716662 x^{13} + \cdots + 20\!\cdots\!67 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{40}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (9 \beta_1 - 9) q^{3} + ( - \beta_{2} + 5) q^{5} + (\beta_{14} + \beta_{3} - 5 \beta_1) q^{7} - 81 \beta_1 q^{9} + ( - \beta_{9} - \beta_{7} - \beta_{6} + \cdots - 67) q^{11} + ( - \beta_{4} - \beta_{3} - 30 \beta_1 + 39) q^{13}+ \cdots + (81 \beta_{9} + 81 \beta_{2} + 5427) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 81 q^{3} + 90 q^{5} - 48 q^{7} - 729 q^{9} - 602 q^{11} + 429 q^{13} - 405 q^{15} + 877 q^{17} - 594 q^{19} + 864 q^{21} - 3346 q^{23} + 7412 q^{25} + 13122 q^{27} + 6955 q^{29} - 2804 q^{31} - 5418 q^{33}+ \cdots + 97524 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 9 x^{17} - 31561 x^{16} + 110968 x^{15} + 409124362 x^{14} + 1115716662 x^{13} + \cdots + 20\!\cdots\!67 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 12\!\cdots\!77 \nu^{17} + \cdots + 90\!\cdots\!47 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 12\!\cdots\!77 \nu^{17} + \cdots - 90\!\cdots\!47 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 31\!\cdots\!91 \nu^{17} + \cdots - 41\!\cdots\!41 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 74\!\cdots\!58 \nu^{17} + \cdots - 97\!\cdots\!35 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 57\!\cdots\!53 \nu^{17} + \cdots + 42\!\cdots\!53 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14\!\cdots\!38 \nu^{17} + \cdots + 17\!\cdots\!01 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 32\!\cdots\!92 \nu^{17} + \cdots + 20\!\cdots\!43 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 41\!\cdots\!26 \nu^{17} + \cdots + 97\!\cdots\!45 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 71\!\cdots\!93 \nu^{17} + \cdots + 23\!\cdots\!47 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 82\!\cdots\!75 \nu^{17} + \cdots - 85\!\cdots\!24 ) / 27\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 42\!\cdots\!27 \nu^{17} + \cdots + 28\!\cdots\!49 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 30\!\cdots\!45 \nu^{17} + \cdots + 43\!\cdots\!07 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 13\!\cdots\!99 \nu^{17} + \cdots - 96\!\cdots\!05 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 29\!\cdots\!47 \nu^{17} + \cdots - 15\!\cdots\!48 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 68\!\cdots\!29 \nu^{17} + \cdots - 28\!\cdots\!08 ) / 87\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 16\!\cdots\!18 \nu^{17} + \cdots - 69\!\cdots\!35 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 41\!\cdots\!47 \nu^{17} + \cdots + 25\!\cdots\!76 ) / 11\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + 2\beta_{9} - 2\beta_{7} - 2\beta_{5} - 2\beta_{3} + 10\beta_{2} + \beta _1 + 3511 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - 6 \beta_{17} - 50 \beta_{15} - 6 \beta_{14} + \beta_{13} - 50 \beta_{12} + 11 \beta_{11} + \cdots + 23332 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 40 \beta_{17} + 44 \beta_{16} - 3897 \beta_{15} - 2112 \beta_{14} + 6061 \beta_{13} - 3161 \beta_{12} + \cdots + 20052624 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 83520 \beta_{17} - 14065 \beta_{16} - 466520 \beta_{15} - 20050 \beta_{14} - 18187 \beta_{13} + \cdots + 135719971 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 809994 \beta_{17} + 790161 \beta_{16} - 45957386 \beta_{15} - 24662724 \beta_{14} + 40679594 \beta_{13} + \cdots + 131475654342 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 877849182 \beta_{17} - 250519185 \beta_{16} - 4004052537 \beta_{15} + 50395968 \beta_{14} + \cdots + 398738108135 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 11956588684 \beta_{17} + 9389679438 \beta_{16} - 432760342541 \beta_{15} - 226350063160 \beta_{14} + \cdots + 916667942958217 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 8340608185416 \beta_{17} - 3040307066547 \beta_{16} - 32800979332654 \beta_{15} + \cdots - 19\!\cdots\!59 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 152337696372882 \beta_{17} + 93791131792053 \beta_{16} + \cdots + 65\!\cdots\!93 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 75\!\cdots\!76 \beta_{17} + \cdots - 50\!\cdots\!49 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 17\!\cdots\!08 \beta_{17} + \cdots + 48\!\cdots\!99 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 66\!\cdots\!92 \beta_{17} + \cdots - 63\!\cdots\!46 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 18\!\cdots\!96 \beta_{17} + \cdots + 35\!\cdots\!31 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 57\!\cdots\!10 \beta_{17} + \cdots - 66\!\cdots\!70 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 18\!\cdots\!36 \beta_{17} + \cdots + 26\!\cdots\!02 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 48\!\cdots\!84 \beta_{17} + \cdots - 63\!\cdots\!41 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/312\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(157\) \(209\)
\(\chi(n)\) \(1\) \(-\beta_{1}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
217.1
85.5783 + 0.866025i
78.1965 + 0.866025i
56.9251 + 0.866025i
18.2734 + 0.866025i
−8.14732 + 0.866025i
−33.5119 + 0.866025i
−43.7864 + 0.866025i
−60.6534 + 0.866025i
−88.3742 + 0.866025i
85.5783 − 0.866025i
78.1965 − 0.866025i
56.9251 − 0.866025i
18.2734 − 0.866025i
−8.14732 − 0.866025i
−33.5119 − 0.866025i
−43.7864 − 0.866025i
−60.6534 − 0.866025i
−88.3742 − 0.866025i
0 −4.50000 + 7.79423i 0 −80.0783 0 2.24140 + 3.88221i 0 −40.5000 − 70.1481i 0
217.2 0 −4.50000 + 7.79423i 0 −72.6965 0 −73.6706 − 127.601i 0 −40.5000 − 70.1481i 0
217.3 0 −4.50000 + 7.79423i 0 −51.4251 0 53.9930 + 93.5186i 0 −40.5000 − 70.1481i 0
217.4 0 −4.50000 + 7.79423i 0 −12.7734 0 109.401 + 189.489i 0 −40.5000 − 70.1481i 0
217.5 0 −4.50000 + 7.79423i 0 13.6473 0 22.5118 + 38.9916i 0 −40.5000 − 70.1481i 0
217.6 0 −4.50000 + 7.79423i 0 39.0119 0 −93.9251 − 162.683i 0 −40.5000 − 70.1481i 0
217.7 0 −4.50000 + 7.79423i 0 49.2864 0 −113.416 − 196.443i 0 −40.5000 − 70.1481i 0
217.8 0 −4.50000 + 7.79423i 0 66.1534 0 33.0014 + 57.1601i 0 −40.5000 − 70.1481i 0
217.9 0 −4.50000 + 7.79423i 0 93.8742 0 35.8631 + 62.1168i 0 −40.5000 − 70.1481i 0
289.1 0 −4.50000 − 7.79423i 0 −80.0783 0 2.24140 − 3.88221i 0 −40.5000 + 70.1481i 0
289.2 0 −4.50000 − 7.79423i 0 −72.6965 0 −73.6706 + 127.601i 0 −40.5000 + 70.1481i 0
289.3 0 −4.50000 − 7.79423i 0 −51.4251 0 53.9930 − 93.5186i 0 −40.5000 + 70.1481i 0
289.4 0 −4.50000 − 7.79423i 0 −12.7734 0 109.401 − 189.489i 0 −40.5000 + 70.1481i 0
289.5 0 −4.50000 − 7.79423i 0 13.6473 0 22.5118 − 38.9916i 0 −40.5000 + 70.1481i 0
289.6 0 −4.50000 − 7.79423i 0 39.0119 0 −93.9251 + 162.683i 0 −40.5000 + 70.1481i 0
289.7 0 −4.50000 − 7.79423i 0 49.2864 0 −113.416 + 196.443i 0 −40.5000 + 70.1481i 0
289.8 0 −4.50000 − 7.79423i 0 66.1534 0 33.0014 − 57.1601i 0 −40.5000 + 70.1481i 0
289.9 0 −4.50000 − 7.79423i 0 93.8742 0 35.8631 − 62.1168i 0 −40.5000 + 70.1481i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 217.9
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 312.6.q.b ✓ 18
13.c even 3 1 inner 312.6.q.b ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.6.q.b ✓ 18 1.a even 1 1 trivial
312.6.q.b ✓ 18 13.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{9} - 45 T_{5}^{8} - 14903 T_{5}^{7} + 613467 T_{5}^{6} + 69364623 T_{5}^{5} + \cdots - 623132622132612 \) acting on \(S_{6}^{\mathrm{new}}(312, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9 T + 81)^{9} \) Copy content Toggle raw display
$5$ \( (T^{9} + \cdots - 623132622132612)^{2} \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 20\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 50\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 13\!\cdots\!93 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 44\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 51\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{18} + \cdots + 60\!\cdots\!36 \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots - 18\!\cdots\!36)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 31\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{18} + \cdots + 95\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 97\!\cdots\!44 \) Copy content Toggle raw display
$47$ \( (T^{9} + \cdots - 38\!\cdots\!40)^{2} \) Copy content Toggle raw display
$53$ \( (T^{9} + \cdots - 57\!\cdots\!64)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 37\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( T^{18} + \cdots + 82\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots + 13\!\cdots\!33)^{2} \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots - 37\!\cdots\!16)^{2} \) Copy content Toggle raw display
$83$ \( (T^{9} + \cdots + 75\!\cdots\!20)^{2} \) Copy content Toggle raw display
$89$ \( T^{18} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 85\!\cdots\!84 \) Copy content Toggle raw display
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