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Properties

Label 304.3.q.b
Level $304$
Weight $3$
Character orbit 304.q
Analytic conductor $8.283$
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] = [304,3,Mod(159,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.159");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 304.q (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: \(8.28340003655\)
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: \( x^{12} - 6 x^{11} + 85 x^{10} - 370 x^{9} + 2400 x^{8} - 7446 x^{7} + 26325 x^{6} - 54402 x^{5} + \cdots + 4699 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{8} - \beta_{7} - \beta_1 + 1) q^{3} + \beta_{3} q^{5} + (3 \beta_{8} - \beta_{6} - \beta_{4} + \cdots - 2) q^{7}+ \cdots + (\beta_{11} - 4 \beta_{8} - \beta_{7} + \cdots + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{8} - \beta_{7} - \beta_1 + 1) q^{3} + \beta_{3} q^{5} + (3 \beta_{8} - \beta_{6} - \beta_{4} + \cdots - 2) q^{7}+ \cdots + (\beta_{10} - 6 \beta_{8} + 24 \beta_{7} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{3} + q^{5} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{3} + q^{5} + 16 q^{9} - 3 q^{13} + 9 q^{15} + 25 q^{17} + 21 q^{19} + 42 q^{21} - 21 q^{23} + 19 q^{25} - 15 q^{29} - 21 q^{33} + 72 q^{35} + 20 q^{37} - 28 q^{41} - 141 q^{43} + 212 q^{45} - 99 q^{47} - 52 q^{49} + 33 q^{51} - 43 q^{53} - 66 q^{55} + 189 q^{57} + 12 q^{59} - 27 q^{61} + 468 q^{63} + 94 q^{65} - 120 q^{67} - 374 q^{69} + 33 q^{71} - 62 q^{73} + 4 q^{77} + 123 q^{79} - 250 q^{81} - 19 q^{85} - 123 q^{89} - 420 q^{91} - 4 q^{93} - 63 q^{95} - 4 q^{97} - 258 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 85 x^{10} - 370 x^{9} + 2400 x^{8} - 7446 x^{7} + 26325 x^{6} - 54402 x^{5} + \cdots + 4699 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 169 \nu^{10} + 845 \nu^{9} - 13451 \nu^{8} + 48734 \nu^{7} - 351615 \nu^{6} + 887825 \nu^{5} + \cdots - 2230631 ) / 2200 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 18355 \nu^{11} + 205479 \nu^{10} - 2030785 \nu^{9} + 14348986 \nu^{8} - 71001389 \nu^{7} + \cdots + 1472411636 ) / 2721400 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 8034 \nu^{11} + 88719 \nu^{10} - 881601 \nu^{9} + 6176600 \nu^{8} - 30674859 \nu^{7} + \cdots + 634302201 ) / 680350 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 32136 \nu^{11} + 207673 \nu^{10} - 2790389 \nu^{9} + 13003143 \nu^{8} - 80302498 \nu^{7} + \cdots + 646799427 ) / 2721400 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 8034 \nu^{11} - 345 \nu^{10} - 436281 \nu^{9} - 908936 \nu^{8} - 5004635 \nu^{7} + \cdots - 536286980 ) / 680350 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 5098 \nu^{11} + 26802 \nu^{10} - 412507 \nu^{9} + 1579191 \nu^{8} - 11009562 \nu^{7} + \cdots + 19270743 ) / 340175 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 10196 \nu^{11} - 56078 \nu^{10} + 837384 \nu^{9} - 3347643 \nu^{8} + 22701948 \nu^{7} + \cdots - 67181747 ) / 340175 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 250913 \nu^{11} + 1607011 \nu^{10} - 21745637 \nu^{9} + 100528032 \nu^{8} - 624602171 \nu^{7} + \cdots + 4718721694 ) / 2721400 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 250913 \nu^{11} - 1153032 \nu^{10} + 19475742 \nu^{9} - 64267851 \nu^{8} + 493180817 \nu^{7} + \cdots + 1360159787 ) / 2721400 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 962309 \nu^{11} - 5366301 \nu^{10} + 79427931 \nu^{9} - 321926038 \nu^{8} + 2165265711 \nu^{7} + \cdots - 7322600864 ) / 2721400 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{5} + \beta_{2} + \beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + 2\beta_{10} - 2\beta_{9} + 2\beta_{5} + \beta_{4} - 3\beta_{3} - 21\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 2 \beta_{11} + 6 \beta_{10} - 2 \beta_{9} - 5 \beta_{8} - 10 \beta_{7} + 30 \beta_{6} - 22 \beta_{5} + \cdots + 264 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 28 \beta_{11} - 60 \beta_{10} + 70 \beta_{9} + 53 \beta_{8} - 25 \beta_{7} + 38 \beta_{6} - 74 \beta_{5} + \cdots + 374 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 89 \beta_{11} - 282 \beta_{10} + 128 \beta_{9} + 381 \beta_{8} + 369 \beta_{7} - 752 \beta_{6} + \cdots - 6102 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 659 \beta_{11} + 1370 \beta_{10} - 1944 \beta_{9} - 1760 \beta_{8} + 1379 \beta_{7} - 1899 \beta_{6} + \cdots - 14516 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3056 \beta_{11} + 9648 \beta_{10} - 5540 \beta_{9} - 15515 \beta_{8} - 9600 \beta_{7} + 17364 \beta_{6} + \cdots + 139883 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 14220 \beta_{11} - 25996 \beta_{10} + 47968 \beta_{9} + 36029 \beta_{8} - 51579 \beta_{7} + \cdots + 495420 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 94653 \beta_{11} - 288748 \beta_{10} + 197892 \beta_{9} + 494685 \beta_{8} + 207709 \beta_{7} + \cdots - 3086292 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 276038 \beta_{11} + 370950 \beta_{10} - 1078492 \beta_{9} - 441136 \beta_{8} + 1630651 \beta_{7} + \cdots - 15585293 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-1 + \beta_{8}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
159.1
0.500000 − 5.05406i
0.500000 − 3.67154i
0.500000 − 0.328955i
0.500000 + 0.320428i
0.500000 + 1.95978i
0.500000 + 5.04229i
0.500000 + 5.05406i
0.500000 + 3.67154i
0.500000 + 0.328955i
0.500000 − 0.320428i
0.500000 − 1.95978i
0.500000 − 5.04229i
0 −3.62694 + 2.09402i 0 0.0357555 + 0.0619303i 0 5.49673i 0 4.26981 − 7.39553i 0
159.2 0 −2.42964 + 1.40275i 0 1.20757 + 2.09157i 0 − 6.42125i 0 −0.564557 + 0.977842i 0
159.3 0 0.465117 − 0.268535i 0 −3.57045 − 6.18420i 0 9.72821i 0 −4.35578 + 7.54443i 0
159.4 0 1.02750 − 0.593227i 0 3.26338 + 5.65234i 0 − 3.01538i 0 −3.79616 + 6.57515i 0
159.5 0 2.44722 − 1.41290i 0 −2.19266 − 3.79780i 0 − 5.89021i 0 −0.507419 + 0.878875i 0
159.6 0 5.11675 − 2.95416i 0 1.75641 + 3.04219i 0 10.4942i 0 12.9541 − 22.4372i 0
239.1 0 −3.62694 − 2.09402i 0 0.0357555 − 0.0619303i 0 − 5.49673i 0 4.26981 + 7.39553i 0
239.2 0 −2.42964 − 1.40275i 0 1.20757 − 2.09157i 0 6.42125i 0 −0.564557 − 0.977842i 0
239.3 0 0.465117 + 0.268535i 0 −3.57045 + 6.18420i 0 − 9.72821i 0 −4.35578 − 7.54443i 0
239.4 0 1.02750 + 0.593227i 0 3.26338 − 5.65234i 0 3.01538i 0 −3.79616 − 6.57515i 0
239.5 0 2.44722 + 1.41290i 0 −2.19266 + 3.79780i 0 5.89021i 0 −0.507419 − 0.878875i 0
239.6 0 5.11675 + 2.95416i 0 1.75641 − 3.04219i 0 − 10.4942i 0 12.9541 + 22.4372i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 159.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.3.q.b yes 12
4.b odd 2 1 304.3.q.a ✓ 12
19.c even 3 1 304.3.q.a ✓ 12
76.g odd 6 1 inner 304.3.q.b yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.3.q.a ✓ 12 4.b odd 2 1
304.3.q.a ✓ 12 19.c even 3 1
304.3.q.b yes 12 1.a even 1 1 trivial
304.3.q.b yes 12 76.g odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 6 T_{3}^{11} - 17 T_{3}^{10} + 174 T_{3}^{9} + 450 T_{3}^{8} - 2910 T_{3}^{7} + \cdots + 15625 \) acting on \(S_{3}^{\mathrm{new}}(304, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 6 T^{11} + \cdots + 15625 \) Copy content Toggle raw display
$5$ \( T^{12} - T^{11} + \cdots + 15376 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 4096000000 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 12630163456 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 5601025600 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 128477691040000 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 41271373518400 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 808330461184 \) Copy content Toggle raw display
$37$ \( (T^{6} - 10 T^{5} + \cdots + 686464000)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 25\!\cdots\!09 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 72\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 28\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 28\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 31\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 86\!\cdots\!25 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 84\!\cdots\!25 \) Copy content Toggle raw display
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