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Properties

Label 304.3.d.b
Level $304$
Weight $3$
Character orbit 304.d
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(191,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.191");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 304.d (of order \(2\), degree \(1\), 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 58x^{10} + 1185x^{8} + 10500x^{6} + 43184x^{4} + 78272x^{2} + 50176 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{5} q^{5} + ( - \beta_{9} + \beta_{8}) q^{7} + (\beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{5} q^{5} + ( - \beta_{9} + \beta_{8}) q^{7} + (\beta_{2} - 1) q^{9} + ( - \beta_{9} + \beta_{6}) q^{11} + ( - \beta_{11} - \beta_{10} - \beta_{5} - 3) q^{13} + ( - \beta_{9} - \beta_{8} + \cdots + \beta_1) q^{15}+ \cdots + (7 \beta_{9} + 2 \beta_{8} + \cdots + 20 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{5} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{5} - 8 q^{9} - 36 q^{13} - 2 q^{17} - 12 q^{21} + 82 q^{25} + 24 q^{29} - 36 q^{33} - 40 q^{37} + 32 q^{41} + 62 q^{45} - 130 q^{49} + 92 q^{53} - 18 q^{61} - 248 q^{65} + 280 q^{69} + 262 q^{73} - 194 q^{77} - 172 q^{81} - 214 q^{85} - 108 q^{89} + 104 q^{93} - 196 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 58x^{10} + 1185x^{8} + 10500x^{6} + 43184x^{4} + 78272x^{2} + 50176 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{10} - 262\nu^{8} - 4485\nu^{6} - 28336\nu^{4} - 66160\nu^{2} - 46976 ) / 2176 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25\nu^{11} + 1310\nu^{9} + 22289\nu^{7} + 136920\nu^{5} + 286192\nu^{3} + 104320\nu ) / 30464 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{10} - 66\nu^{8} - 1577\nu^{6} - 16180\nu^{4} - 63824\nu^{2} - 67712 ) / 1088 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{11} + 164\nu^{9} + 3031\nu^{7} + 22530\nu^{5} + 72880\nu^{3} + 95968\nu ) / 2176 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 97\nu^{11} + 5654\nu^{9} + 114889\nu^{7} + 967456\nu^{5} + 3200560\nu^{3} + 2936320\nu ) / 60928 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -2\nu^{11} - 115\nu^{9} - 2304\nu^{7} - 19355\nu^{5} - 67808\nu^{3} - 72592\nu ) / 1088 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 9\nu^{11} + 494\nu^{9} + 9153\nu^{7} + 67368\nu^{5} + 201392\nu^{3} + 192384\nu ) / 3584 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 7\nu^{10} + 394\nu^{8} + 7639\nu^{6} + 61240\nu^{4} + 207408\nu^{2} + 223744 ) / 1088 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -3\nu^{10} - 164\nu^{8} - 3014\nu^{6} - 21799\nu^{4} - 63088\nu^{2} - 57072 ) / 272 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} + \beta_{8} + \beta_{7} + 2\beta_{6} - 16\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{10} + 4\beta_{5} + 4\beta_{3} - 25\beta_{2} + 174 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 39\beta_{9} - 23\beta_{8} - 31\beta_{7} - 54\beta_{6} - 20\beta_{4} + 316\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 16\beta_{11} - 54\beta_{10} - 140\beta_{5} - 172\beta_{3} + 563\beta_{2} - 3594 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -1197\beta_{9} + 397\beta_{8} + 757\beta_{7} + 1314\beta_{6} + 876\beta_{4} - 6692\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -800\beta_{11} + 1154\beta_{10} + 3828\beta_{5} + 5540\beta_{3} - 12545\beta_{2} + 78110 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 33407\beta_{9} - 5839\beta_{8} - 17527\beta_{7} - 31430\beta_{6} - 28500\beta_{4} + 145612\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 27568\beta_{11} - 23366\beta_{10} - 97676\beta_{5} - 159116\beta_{3} + 280795\beta_{2} - 1732314 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -885477\beta_{9} + 66533\beta_{8} + 401837\beta_{7} + 748274\beta_{6} + 823148\beta_{4} - 3215428\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
4.85859i
4.35666i
2.54548i
2.28954i
1.47611i
1.23012i
− 1.23012i
− 1.47611i
− 2.28954i
− 2.54548i
− 4.35666i
− 4.85859i
0 − 4.85859i 0 2.61079 0 − 1.78742i 0 −14.6059 0
191.2 0 − 4.35666i 0 −4.07679 0 − 5.65700i 0 −9.98053 0
191.3 0 − 2.54548i 0 −8.62010 0 6.87830i 0 2.52054 0
191.4 0 − 2.28954i 0 3.24364 0 3.98949i 0 3.75802 0
191.5 0 − 1.47611i 0 8.65137 0 10.6065i 0 6.82110 0
191.6 0 − 1.23012i 0 −2.80891 0 − 12.1687i 0 7.48680 0
191.7 0 1.23012i 0 −2.80891 0 12.1687i 0 7.48680 0
191.8 0 1.47611i 0 8.65137 0 − 10.6065i 0 6.82110 0
191.9 0 2.28954i 0 3.24364 0 − 3.98949i 0 3.75802 0
191.10 0 2.54548i 0 −8.62010 0 − 6.87830i 0 2.52054 0
191.11 0 4.35666i 0 −4.07679 0 5.65700i 0 −9.98053 0
191.12 0 4.85859i 0 2.61079 0 1.78742i 0 −14.6059 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.3.d.b ✓ 12
3.b odd 2 1 2736.3.m.c 12
4.b odd 2 1 inner 304.3.d.b ✓ 12
8.b even 2 1 1216.3.d.c 12
8.d odd 2 1 1216.3.d.c 12
12.b even 2 1 2736.3.m.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
304.3.d.b ✓ 12 1.a even 1 1 trivial
304.3.d.b ✓ 12 4.b odd 2 1 inner
1216.3.d.c 12 8.b even 2 1
1216.3.d.c 12 8.d odd 2 1
2736.3.m.c 12 3.b odd 2 1
2736.3.m.c 12 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 58T_{3}^{10} + 1185T_{3}^{8} + 10500T_{3}^{6} + 43184T_{3}^{4} + 78272T_{3}^{2} + 50176 \) 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
$3$ \( T^{12} + 58 T^{10} + \cdots + 50176 \) Copy content Toggle raw display
$5$ \( (T^{6} + T^{5} - 95 T^{4} + \cdots - 7232)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 1282499344 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 4201113856 \) Copy content Toggle raw display
$13$ \( (T^{6} + 18 T^{5} + \cdots - 2903936)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + T^{5} + \cdots + 34282)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 19)^{6} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 35734762356736 \) Copy content Toggle raw display
$29$ \( (T^{6} - 12 T^{5} + \cdots + 10123216)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( (T^{6} + 20 T^{5} + \cdots - 6670035776)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 16 T^{5} + \cdots + 700475392)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 61\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{6} - 46 T^{5} + \cdots - 695782976)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( (T^{6} + 9 T^{5} + \cdots - 53304973688)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$73$ \( (T^{6} - 131 T^{5} + \cdots - 132936187334)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 25\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 34\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( (T^{6} + 54 T^{5} + \cdots - 5266722816)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 98 T^{5} + \cdots + 192261332992)^{2} \) Copy content Toggle raw display
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