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Properties

Label 39.3.i.b
Level $39$
Weight $3$
Character orbit 39.i
Analytic conductor $1.063$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [39,3,Mod(29,39)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(39, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("39.29");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 39 = 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 39.i (of order \(6\), 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: \(1.06267303101\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 5 x^{10} - 2 x^{9} - 11 x^{8} + 25 x^{7} - 50 x^{6} + 75 x^{5} - 99 x^{4} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( - \beta_{8} - \beta_{7} - \beta_{5} + \cdots - 1) q^{3}+ \cdots + (\beta_{11} - \beta_{9} + 3 \beta_{8} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + ( - \beta_{8} - \beta_{7} - \beta_{5} + \cdots - 1) q^{3}+ \cdots + ( - \beta_{11} - \beta_{10} + \cdots + 26) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{3} + 10 q^{4} - 12 q^{6} - 16 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{3} + 10 q^{4} - 12 q^{6} - 16 q^{7} + 6 q^{9} - 6 q^{10} - 34 q^{13} + 2 q^{15} + 50 q^{16} - 80 q^{18} + 84 q^{19} - 100 q^{21} - 40 q^{22} + 48 q^{24} + 8 q^{25} + 128 q^{27} + 12 q^{28} + 142 q^{30} + 32 q^{31} + 14 q^{33} - 124 q^{34} + 74 q^{36} - 90 q^{37} + 138 q^{39} + 4 q^{40} - 78 q^{42} - 76 q^{43} - 152 q^{45} - 100 q^{46} + 56 q^{48} + 62 q^{49} - 368 q^{51} - 456 q^{52} - 144 q^{54} + 152 q^{55} - 36 q^{57} + 238 q^{58} + 324 q^{60} + 38 q^{61} + 220 q^{63} + 476 q^{64} - 252 q^{66} + 24 q^{67} + 242 q^{69} + 896 q^{70} + 12 q^{72} - 580 q^{73} + 20 q^{75} - 320 q^{76} + 464 q^{78} + 80 q^{79} - 102 q^{81} - 570 q^{82} - 250 q^{84} + 278 q^{85} + 70 q^{87} - 12 q^{88} - 852 q^{90} - 124 q^{91} - 300 q^{93} - 184 q^{94} - 928 q^{96} + 120 q^{97} + 384 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 5 x^{10} - 2 x^{9} - 11 x^{8} + 25 x^{7} - 50 x^{6} + 75 x^{5} - 99 x^{4} + \cdots + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 11 \nu^{11} + 8 \nu^{10} + 49 \nu^{9} + 21 \nu^{8} - 50 \nu^{7} + 211 \nu^{6} - 353 \nu^{5} + \cdots - 6237 ) / 4212 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 26 \nu^{11} - 3 \nu^{10} + 5 \nu^{9} - 164 \nu^{8} - 11 \nu^{7} + 25 \nu^{6} + 1300 \nu^{5} + \cdots - 729 ) / 6318 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 23 \nu^{11} - 57 \nu^{10} + 164 \nu^{9} + 37 \nu^{8} - 233 \nu^{7} + 442 \nu^{6} - 101 \nu^{5} + \cdots - 18954 ) / 6318 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 71 \nu^{11} + 138 \nu^{10} - 131 \nu^{9} + 317 \nu^{8} + 380 \nu^{7} - 709 \nu^{6} + 851 \nu^{5} + \cdots + 13851 ) / 12636 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 14 \nu^{11} - 31 \nu^{10} + \nu^{9} + 54 \nu^{8} - 113 \nu^{7} + 139 \nu^{6} - 272 \nu^{5} + \cdots - 1863 ) / 2106 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 46 \nu^{11} - 15 \nu^{10} - 149 \nu^{9} - 52 \nu^{8} - 25 \nu^{7} - 1033 \nu^{6} + 500 \nu^{5} + \cdots + 24057 ) / 6318 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 50 \nu^{11} - 87 \nu^{10} + 385 \nu^{9} + 188 \nu^{8} - 433 \nu^{7} + 1475 \nu^{6} - 1276 \nu^{5} + \cdots - 55647 ) / 6318 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 16 \nu^{11} - 52 \nu^{10} + 95 \nu^{9} + 20 \nu^{8} - 180 \nu^{7} + 357 \nu^{6} - 582 \nu^{5} + \cdots - 11421 ) / 2106 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 43 \nu^{11} + 96 \nu^{10} - 239 \nu^{9} - 61 \nu^{8} + 410 \nu^{7} - 925 \nu^{6} + 1517 \nu^{5} + \cdots + 29079 ) / 4212 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 203 \nu^{11} - 378 \nu^{10} + 619 \nu^{9} + 965 \nu^{8} - 1372 \nu^{7} + 3101 \nu^{6} - 5725 \nu^{5} + \cdots - 89667 ) / 12636 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 169 \nu^{11} - 266 \nu^{10} + 257 \nu^{9} + 543 \nu^{8} - 1504 \nu^{7} + 2195 \nu^{6} - 3991 \nu^{5} + \cdots - 46737 ) / 4212 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{9} + \beta_{8} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} - \beta_{7} - \beta_{5} + \beta_{2} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - 2\beta_{7} - \beta_{6} + \beta_{4} + 2\beta_{3} - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{11} - 2\beta_{9} - 2\beta_{8} + \beta_{6} + 5\beta_{5} + \beta_{2} + 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{10} - \beta_{8} + 4\beta_{7} + 4\beta_{5} + 5\beta_{4} + \beta_{3} + 6\beta_{2} - 7\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{11} + \beta_{10} + 7\beta_{9} - 8\beta_{6} - 7\beta_{4} + \beta_{3} + 11 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -8\beta_{11} + 21\beta_{9} + 33\beta_{8} - 26\beta_{6} + 12\beta_{5} - 26\beta_{2} - 35\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 27\beta_{10} - 36\beta_{8} - 5\beta_{7} - 5\beta_{5} + 12\beta_{4} + 36\beta_{3} - 5\beta_{2} - 17\beta _1 + 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -6\beta_{11} - 6\beta_{10} - 41\beta_{9} - 21\beta_{7} + 41\beta_{4} + 40\beta_{3} + 64 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( \beta_{11} + 54\beta_{9} - 18\beta_{8} + 53\beta_{6} + 19\beta_{5} + 53\beta_{2} + 431\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 18 \beta_{10} + 175 \beta_{8} - 50 \beta_{7} - 50 \beta_{5} + 169 \beta_{4} - 175 \beta_{3} + 142 \beta_{2} + \cdots - 41 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/39\mathbb{Z}\right)^\times\).

\(n\) \(14\) \(28\)
\(\chi(n)\) \(-1\) \(-\beta_{1}\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
−0.822039 − 1.52455i
−1.71387 + 0.250318i
0.417201 − 1.68105i
1.24724 − 1.20183i
0.640152 + 1.60941i
1.73132 − 0.0503679i
−0.822039 + 1.52455i
−1.71387 − 0.250318i
0.417201 + 1.68105i
1.24724 + 1.20183i
0.640152 − 1.60941i
1.73132 + 0.0503679i
−2.55336 + 1.47418i 2.99371 + 0.194134i 2.34642 − 4.06412i 5.36177i −7.93021 + 3.91758i −4.07426 + 7.05683i 2.04276i 8.92462 + 1.16237i −7.90422 − 13.6905i
29.2 −2.35402 + 1.35909i −1.01926 − 2.82154i 1.69427 − 2.93456i − 5.83712i 6.23410 + 5.25670i 2.52687 − 4.37667i − 1.66205i −6.92222 + 5.75177i 7.93319 + 13.7407i
29.3 −0.830035 + 0.479221i −2.87979 + 0.840707i −1.54070 + 2.66856i 3.19053i 1.98744 − 2.07787i −2.45261 + 4.24805i − 6.78710i 7.58642 − 4.84213i −1.52897 − 2.64825i
29.4 0.830035 − 0.479221i 2.16797 − 2.07362i −1.54070 + 2.66856i − 3.19053i 0.805769 − 2.76011i −2.45261 + 4.24805i 6.78710i 0.400194 − 8.99110i −1.52897 − 2.64825i
29.5 2.35402 − 1.35909i −1.93390 − 2.29348i 1.69427 − 2.93456i 5.83712i −7.66949 − 2.77054i 2.52687 − 4.37667i 1.66205i −1.52007 + 8.87070i 7.93319 + 13.7407i
29.6 2.55336 − 1.47418i −1.32873 + 2.68970i 2.34642 − 4.06412i − 5.36177i 0.572379 + 8.82655i −4.07426 + 7.05683i − 2.04276i −5.46895 − 7.14777i −7.90422 − 13.6905i
35.1 −2.55336 − 1.47418i 2.99371 − 0.194134i 2.34642 + 4.06412i − 5.36177i −7.93021 − 3.91758i −4.07426 − 7.05683i − 2.04276i 8.92462 − 1.16237i −7.90422 + 13.6905i
35.2 −2.35402 − 1.35909i −1.01926 + 2.82154i 1.69427 + 2.93456i 5.83712i 6.23410 − 5.25670i 2.52687 + 4.37667i 1.66205i −6.92222 − 5.75177i 7.93319 − 13.7407i
35.3 −0.830035 − 0.479221i −2.87979 − 0.840707i −1.54070 − 2.66856i − 3.19053i 1.98744 + 2.07787i −2.45261 − 4.24805i 6.78710i 7.58642 + 4.84213i −1.52897 + 2.64825i
35.4 0.830035 + 0.479221i 2.16797 + 2.07362i −1.54070 − 2.66856i 3.19053i 0.805769 + 2.76011i −2.45261 − 4.24805i − 6.78710i 0.400194 + 8.99110i −1.52897 + 2.64825i
35.5 2.35402 + 1.35909i −1.93390 + 2.29348i 1.69427 + 2.93456i − 5.83712i −7.66949 + 2.77054i 2.52687 + 4.37667i − 1.66205i −1.52007 − 8.87070i 7.93319 − 13.7407i
35.6 2.55336 + 1.47418i −1.32873 − 2.68970i 2.34642 + 4.06412i 5.36177i 0.572379 − 8.82655i −4.07426 − 7.05683i 2.04276i −5.46895 + 7.14777i −7.90422 + 13.6905i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.c even 3 1 inner
39.i odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.3.i.b ✓ 12
3.b odd 2 1 inner 39.3.i.b ✓ 12
13.c even 3 1 inner 39.3.i.b ✓ 12
13.c even 3 1 507.3.c.h 6
13.e even 6 1 507.3.c.g 6
13.f odd 12 2 507.3.d.d 12
39.h odd 6 1 507.3.c.g 6
39.i odd 6 1 inner 39.3.i.b ✓ 12
39.i odd 6 1 507.3.c.h 6
39.k even 12 2 507.3.d.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.3.i.b ✓ 12 1.a even 1 1 trivial
39.3.i.b ✓ 12 3.b odd 2 1 inner
39.3.i.b ✓ 12 13.c even 3 1 inner
39.3.i.b ✓ 12 39.i odd 6 1 inner
507.3.c.g 6 13.e even 6 1
507.3.c.g 6 39.h odd 6 1
507.3.c.h 6 13.c even 3 1
507.3.c.h 6 39.i odd 6 1
507.3.d.d 12 13.f odd 12 2
507.3.d.d 12 39.k even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 17T_{2}^{10} + 210T_{2}^{8} - 1225T_{2}^{6} + 5238T_{2}^{4} - 4661T_{2}^{2} + 3481 \) acting on \(S_{3}^{\mathrm{new}}(39, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 17 T^{10} + \cdots + 3481 \) Copy content Toggle raw display
$3$ \( T^{12} + 4 T^{11} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( (T^{6} + 73 T^{4} + \cdots + 9971)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 8 T^{5} + \cdots + 40804)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 133726096 \) Copy content Toggle raw display
$13$ \( (T^{6} + 17 T^{5} + \cdots + 4826809)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 28\!\cdots\!81 \) Copy content Toggle raw display
$19$ \( (T^{6} - 42 T^{5} + \cdots + 636804)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 845380978590096 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 27\!\cdots\!61 \) Copy content Toggle raw display
$31$ \( (T^{3} - 8 T^{2} + \cdots + 23428)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + 45 T^{5} + \cdots + 26569)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 78\!\cdots\!41 \) Copy content Toggle raw display
$43$ \( (T^{6} + 38 T^{5} + \cdots + 12996)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + 11240 T^{4} + \cdots + 7969182156)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 5449 T^{4} + \cdots + 2502730971)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 27\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( (T^{6} - 19 T^{5} + \cdots + 155775361)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 12 T^{5} + \cdots + 33990821956)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 40\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( (T^{3} + 145 T^{2} + \cdots + 93367)^{4} \) Copy content Toggle raw display
$79$ \( (T^{3} - 20 T^{2} + \cdots - 8384)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 12224 T^{4} + \cdots + 4692312716)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 2448210984976)^{2} \) Copy content Toggle raw display
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