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Properties

Label 104.3.v.d
Level $104$
Weight $3$
Character orbit 104.v
Analytic conductor $2.834$
Analytic rank $0$
Dimension $12$
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] = [104,3,Mod(33,104)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(104, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 0, 11]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("104.33");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 104 = 2^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 104.v (of order \(12\), degree \(4\), 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: \(2.83379474935\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 64x^{10} + 1482x^{8} + 15000x^{6} + 62513x^{4} + 82376x^{2} + 27556 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{3} + (\beta_{10} + 2 \beta_{5} + 2 \beta_{4} + \cdots - 1) q^{5}+ \cdots + (\beta_{11} + \beta_{10} + 2 \beta_{6} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{3} + (\beta_{10} + 2 \beta_{5} + 2 \beta_{4} + \cdots - 1) q^{5}+ \cdots + ( - 9 \beta_{11} + 3 \beta_{10} + \cdots - 21) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{5} + 6 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{5} + 6 q^{7} - 10 q^{9} + 2 q^{11} - 34 q^{13} + 50 q^{15} + 72 q^{17} + 4 q^{19} - 28 q^{21} - 12 q^{23} - 156 q^{27} - 10 q^{29} + 64 q^{31} + 68 q^{33} + 44 q^{35} - 152 q^{37} - 84 q^{39} + 88 q^{41} + 18 q^{43} - 94 q^{45} - 172 q^{47} - 120 q^{49} - 144 q^{53} + 180 q^{55} + 56 q^{57} + 262 q^{59} + 116 q^{61} - 164 q^{63} + 96 q^{65} + 272 q^{67} + 792 q^{69} + 296 q^{71} - 234 q^{73} - 30 q^{75} - 832 q^{79} + 110 q^{81} - 136 q^{83} + 116 q^{85} + 164 q^{87} - 614 q^{89} - 354 q^{91} - 20 q^{93} + 582 q^{95} - 450 q^{97} - 300 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 64x^{10} + 1482x^{8} + 15000x^{6} + 62513x^{4} + 82376x^{2} + 27556 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{10} - 87\nu^{8} - 3483\nu^{6} - 60581\nu^{4} - 350980\nu^{2} + 69056\nu - 248004 ) / 138112 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 27\nu^{11} + 1559\nu^{9} + 31785\nu^{7} + 280509\nu^{5} + 1099076\nu^{3} + 1970004\nu ) / 1054432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9\nu^{11} + 575\nu^{9} + 13251\nu^{7} + 131517\nu^{5} + 502036\nu^{3} + 390404\nu + 69056 ) / 138112 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 702 \nu^{11} - 4233 \nu^{10} + 40534 \nu^{9} - 251739 \nu^{8} + 826410 \nu^{7} - 5334659 \nu^{6} + \cdots - 127402676 ) / 54830464 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 4233 \nu^{11} + 4394 \nu^{10} - 251739 \nu^{9} + 213954 \nu^{8} - 5334659 \nu^{7} + \cdots + 19344312 ) / 54830464 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 4233 \nu^{11} - 4394 \nu^{10} - 251739 \nu^{9} - 213954 \nu^{8} - 5334659 \nu^{7} + \cdots - 19344312 ) / 54830464 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6852 \nu^{11} - 161 \nu^{10} - 419636 \nu^{9} + 37785 \nu^{8} - 9011716 \nu^{7} + \cdots + 258842140 ) / 27415232 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17937 \nu^{11} - 4072 \nu^{10} - 1091011 \nu^{9} - 289524 \nu^{8} - 23358091 \nu^{7} + \cdots - 537028592 ) / 54830464 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 24178 \nu^{11} + 27315 \nu^{10} - 1527742 \nu^{9} + 1720373 \nu^{8} - 34398470 \nu^{7} + \cdots + 833134412 ) / 54830464 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 5237 \nu^{11} - 13620 \nu^{10} - 344235 \nu^{9} - 805132 \nu^{8} - 8187519 \nu^{7} + \cdots - 108496272 ) / 27415232 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{9} + \beta_{8} - \beta_{7} + 2\beta_{6} + 4\beta_{5} - 2\beta_{3} + \beta _1 - 11 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{7} + 3\beta_{6} + 2\beta_{4} + 13\beta_{3} - 16\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{11} - 3 \beta_{10} + 22 \beta_{9} - 19 \beta_{8} + 23 \beta_{7} - 45 \beta_{6} - 130 \beta_{5} + \cdots + 194 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 15 \beta_{11} - 15 \beta_{10} - \beta_{9} + 14 \beta_{8} - 97 \beta_{7} - 98 \beta_{6} - 184 \beta_{4} + \cdots + 92 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 96 \beta_{11} + 96 \beta_{10} - 475 \beta_{9} + 379 \beta_{8} - 525 \beta_{7} + 1000 \beta_{6} + \cdots - 3861 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 654 \beta_{11} + 654 \beta_{10} + 110 \beta_{9} - 544 \beta_{8} + 2521 \beta_{7} + 2631 \beta_{6} + \cdots - 3609 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2411 \beta_{11} - 2411 \beta_{10} + 10386 \beta_{9} - 7975 \beta_{8} + 12643 \beta_{7} - 23029 \beta_{6} + \cdots + 81642 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 21087 \beta_{11} - 21087 \beta_{10} - 4593 \beta_{9} + 16494 \beta_{8} - 61725 \beta_{7} + \cdots + 111232 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 57132 \beta_{11} + 57132 \beta_{10} - 230959 \beta_{9} + 173827 \beta_{8} - 313749 \beta_{7} + \cdots - 1794929 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 603514 \beta_{11} + 603514 \beta_{10} + 146098 \beta_{9} - 457416 \beta_{8} + 1481905 \beta_{7} + \cdots - 3089125 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/104\mathbb{Z}\right)^\times\).

\(n\) \(41\) \(53\) \(79\)
\(\chi(n)\) \(\beta_{3} - \beta_{5}\) \(1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
− 4.20641i
− 0.712068i
4.91848i
4.20641i
0.712068i
− 4.91848i
− 2.51448i
− 1.20485i
3.71933i
2.51448i
1.20485i
− 3.71933i
0 −2.10320 + 3.64286i 0 −5.99353 − 5.99353i 0 −0.737178 − 0.197526i 0 −4.34693 − 7.52911i 0
33.2 0 −0.356034 + 0.616669i 0 2.85532 + 2.85532i 0 1.00999 + 0.270626i 0 4.24648 + 7.35512i 0
33.3 0 2.45924 − 4.25952i 0 −1.69192 − 1.69192i 0 3.82526 + 1.02498i 0 −7.59570 − 13.1561i 0
41.1 0 −2.10320 − 3.64286i 0 −5.99353 + 5.99353i 0 −0.737178 + 0.197526i 0 −4.34693 + 7.52911i 0
41.2 0 −0.356034 − 0.616669i 0 2.85532 − 2.85532i 0 1.00999 − 0.270626i 0 4.24648 − 7.35512i 0
41.3 0 2.45924 + 4.25952i 0 −1.69192 + 1.69192i 0 3.82526 − 1.02498i 0 −7.59570 + 13.1561i 0
89.1 0 −1.25724 − 2.17761i 0 6.94439 + 6.94439i 0 −1.62327 − 6.05812i 0 1.33868 − 2.31867i 0
89.2 0 −0.602423 − 1.04343i 0 −3.28748 − 3.28748i 0 −0.968448 − 3.61430i 0 3.77417 − 6.53706i 0
89.3 0 1.85966 + 3.22103i 0 0.173217 + 0.173217i 0 1.49364 + 5.57434i 0 −2.41671 + 4.18586i 0
97.1 0 −1.25724 + 2.17761i 0 6.94439 − 6.94439i 0 −1.62327 + 6.05812i 0 1.33868 + 2.31867i 0
97.2 0 −0.602423 + 1.04343i 0 −3.28748 + 3.28748i 0 −0.968448 + 3.61430i 0 3.77417 + 6.53706i 0
97.3 0 1.85966 − 3.22103i 0 0.173217 − 0.173217i 0 1.49364 − 5.57434i 0 −2.41671 − 4.18586i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 104.3.v.d ✓ 12
4.b odd 2 1 208.3.bd.h 12
13.f odd 12 1 inner 104.3.v.d ✓ 12
52.l even 12 1 208.3.bd.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.3.v.d ✓ 12 1.a even 1 1 trivial
104.3.v.d ✓ 12 13.f odd 12 1 inner
208.3.bd.h 12 4.b odd 2 1
208.3.bd.h 12 52.l even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 32 T_{3}^{10} + 52 T_{3}^{9} + 795 T_{3}^{8} + 1230 T_{3}^{7} + 8336 T_{3}^{6} + \cdots + 27556 \) acting on \(S_{3}^{\mathrm{new}}(104, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 32 T^{10} + \cdots + 27556 \) Copy content Toggle raw display
$5$ \( T^{12} + 2 T^{11} + \cdots + 839056 \) Copy content Toggle raw display
$7$ \( T^{12} - 6 T^{11} + \cdots + 183184 \) Copy content Toggle raw display
$11$ \( T^{12} - 2 T^{11} + \cdots + 23116864 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 23298085122481 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 19705702129 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 61405831204 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 47\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{12} + 10 T^{11} + \cdots + 81234169 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 75\!\cdots\!81 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 504084176408761 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 32\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( (T^{6} + 72 T^{5} + \cdots + 1126898656)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 14\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 18\!\cdots\!21 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 68\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 71\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 30\!\cdots\!44 \) Copy content Toggle raw display
$79$ \( (T^{6} + 416 T^{5} + \cdots - 229104261632)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 77\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 15\!\cdots\!84 \) Copy content Toggle raw display
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