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Properties

Label 304.3.z.a
Level $304$
Weight $3$
Character orbit 304.z
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(33,304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(304, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 0, 7]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("304.33");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 304.z (of order \(18\), degree \(6\), not 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: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 24x^{10} + 216x^{8} + 905x^{6} + 1770x^{4} + 1395x^{2} + 361 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{11} + \beta_{9} + \beta_{8} + \cdots - \beta_1) q^{3}+ \cdots + (\beta_{10} - 2 \beta_{9} + \cdots + 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{11} + \beta_{9} + \beta_{8} + \cdots - \beta_1) q^{3}+ \cdots + ( - 7 \beta_{11} + 12 \beta_{10} + \cdots - 40) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{5} - 6 q^{7} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{5} - 6 q^{7} - 24 q^{9} + 18 q^{11} + 21 q^{13} - 63 q^{15} - 3 q^{17} + 24 q^{19} + 30 q^{21} + 102 q^{23} - 156 q^{25} + 27 q^{27} + 147 q^{29} - 99 q^{31} + 84 q^{33} - 96 q^{35} + 108 q^{39} - 144 q^{41} + 27 q^{43} - 3 q^{45} + 99 q^{47} - 24 q^{49} + 42 q^{51} + 111 q^{53} - 162 q^{55} - 168 q^{57} - 3 q^{59} + 150 q^{61} - 234 q^{63} + 126 q^{65} - 135 q^{67} + 72 q^{69} + 168 q^{71} - 90 q^{73} + 246 q^{77} + 75 q^{79} - 159 q^{81} + 156 q^{83} - 300 q^{85} - 69 q^{87} - 558 q^{89} + 453 q^{91} - 57 q^{93} + 69 q^{95} + 465 q^{97} - 462 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 24x^{10} + 216x^{8} + 905x^{6} + 1770x^{4} + 1395x^{2} + 361 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{11} + 2 \nu^{10} - 27 \nu^{9} + 28 \nu^{8} - 277 \nu^{7} + 38 \nu^{6} - 1304 \nu^{5} + \cdots - 1520 ) / 304 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} - 5 \nu^{10} - \nu^{9} - 89 \nu^{8} + 239 \nu^{7} - 437 \nu^{6} + 2078 \nu^{5} + \cdots + 1881 ) / 304 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{11} + 5 \nu^{10} - \nu^{9} + 89 \nu^{8} + 239 \nu^{7} + 437 \nu^{6} + 2078 \nu^{5} + \cdots - 1881 ) / 304 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{11} - 2 \nu^{10} - 27 \nu^{9} - 28 \nu^{8} - 277 \nu^{7} - 38 \nu^{6} - 1304 \nu^{5} + \cdots + 1520 ) / 304 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3 \nu^{11} + 17 \nu^{10} + 55 \nu^{9} + 333 \nu^{8} + 315 \nu^{7} + 2185 \nu^{6} + 530 \nu^{5} + \cdots + 171 ) / 608 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3 \nu^{11} - 17 \nu^{10} + 55 \nu^{9} - 333 \nu^{8} + 315 \nu^{7} - 2185 \nu^{6} + 530 \nu^{5} + \cdots - 171 ) / 608 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{10} + 21\nu^{8} + 153\nu^{6} + 454\nu^{4} + 504\nu^{2} + 8\nu + 171 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{10} - 21\nu^{8} - 153\nu^{6} - 454\nu^{4} - 504\nu^{2} + 8\nu - 171 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{11} + \nu^{10} - 56 \nu^{9} + 14 \nu^{8} - 329 \nu^{7} + 19 \nu^{6} - 549 \nu^{5} + \cdots - 380 ) / 152 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17 \nu^{11} + 17 \nu^{10} + 333 \nu^{9} + 333 \nu^{8} + 2185 \nu^{7} + 2185 \nu^{6} + 5334 \nu^{5} + \cdots + 1083 ) / 608 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 9\nu^{11} + 197\nu^{9} + 1545\nu^{7} + 5238\nu^{5} + 7304\nu^{3} + 2979\nu - 152 ) / 304 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{8} + \beta_{7} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{9} - \beta_{5} + \beta_{4} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -5\beta_{8} - 5\beta_{7} + 2\beta_{6} + 2\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 7 \beta_{10} - 7 \beta_{9} - \beta_{8} + \beta_{7} + 2 \beta_{6} + 5 \beta_{5} - 11 \beta_{4} + \cdots + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{11} - 3 \beta_{10} + 3 \beta_{9} + 32 \beta_{8} + 32 \beta_{7} - 18 \beta_{6} - 15 \beta_{5} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 50 \beta_{10} + 50 \beta_{9} + 14 \beta_{8} - 14 \beta_{7} - 30 \beta_{6} - 20 \beta_{5} + 102 \beta_{4} + \cdots - 153 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 38 \beta_{11} + 44 \beta_{10} - 44 \beta_{9} - 226 \beta_{8} - 226 \beta_{7} + 158 \beta_{6} + \cdots + 25 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 384 \beta_{10} - 384 \beta_{9} - 149 \beta_{8} + 149 \beta_{7} + 334 \beta_{6} + 50 \beta_{5} + \cdots + 1104 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 494 \beta_{11} - 483 \beta_{10} + 483 \beta_{9} + 1688 \beta_{8} + 1688 \beta_{7} - 1400 \beta_{6} + \cdots - 236 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 3088 \beta_{10} + 3088 \beta_{9} + 1433 \beta_{8} - 1433 \beta_{7} - 3332 \beta_{6} + 244 \beta_{5} + \cdots - 8372 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 5420 \beta_{11} + 4765 \beta_{10} - 4765 \beta_{9} - 13049 \beta_{8} - 13049 \beta_{7} + 12374 \beta_{6} + \cdots + 2055 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(\beta_{1} - \beta_{5}\) \(1\) \(1\)

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
− 2.57727i
1.89323i
2.88811i
− 0.918492i
2.57727i
− 1.89323i
− 2.01431i
0.728740i
2.01431i
− 0.728740i
− 2.88811i
0.918492i
0 0.384565 + 0.0678091i 0 1.73199 + 1.45331i 0 −5.72163 − 9.91015i 0 −8.31394 − 3.02603i 0
33.2 0 1.91357 + 0.337414i 0 −2.13959 − 1.79533i 0 1.20796 + 2.09224i 0 −4.90934 − 1.78685i 0
97.1 0 −3.28392 + 3.91363i 0 −3.00117 + 1.09234i 0 −3.87208 − 6.70664i 0 −2.96949 − 16.8408i 0
97.2 0 0.464845 − 0.553981i 0 0.295437 − 0.107530i 0 0.328846 + 0.569578i 0 1.47202 + 8.34824i 0
129.1 0 0.384565 − 0.0678091i 0 1.73199 − 1.45331i 0 −5.72163 + 9.91015i 0 −8.31394 + 3.02603i 0
129.2 0 1.91357 − 0.337414i 0 −2.13959 + 1.79533i 0 1.20796 − 2.09224i 0 −4.90934 + 1.78685i 0
193.1 0 −0.621128 − 1.70654i 0 1.24445 − 7.05761i 0 −0.422527 − 0.731838i 0 4.36793 − 3.66513i 0
193.2 0 1.14207 + 3.13782i 0 −1.13111 + 6.41483i 0 5.47943 + 9.49065i 0 −1.64718 + 1.38215i 0
241.1 0 −0.621128 + 1.70654i 0 1.24445 + 7.05761i 0 −0.422527 + 0.731838i 0 4.36793 + 3.66513i 0
241.2 0 1.14207 − 3.13782i 0 −1.13111 − 6.41483i 0 5.47943 − 9.49065i 0 −1.64718 − 1.38215i 0
257.1 0 −3.28392 − 3.91363i 0 −3.00117 − 1.09234i 0 −3.87208 + 6.70664i 0 −2.96949 + 16.8408i 0
257.2 0 0.464845 + 0.553981i 0 0.295437 + 0.107530i 0 0.328846 − 0.569578i 0 1.47202 − 8.34824i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.f odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.3.z.a 12
4.b odd 2 1 19.3.f.a ✓ 12
12.b even 2 1 171.3.ba.b 12
19.f odd 18 1 inner 304.3.z.a 12
76.d even 2 1 361.3.f.g 12
76.f even 6 1 361.3.f.b 12
76.f even 6 1 361.3.f.c 12
76.g odd 6 1 361.3.f.e 12
76.g odd 6 1 361.3.f.f 12
76.k even 18 1 19.3.f.a ✓ 12
76.k even 18 1 361.3.b.c 12
76.k even 18 1 361.3.d.d 12
76.k even 18 1 361.3.d.f 12
76.k even 18 1 361.3.f.e 12
76.k even 18 1 361.3.f.f 12
76.l odd 18 1 361.3.b.c 12
76.l odd 18 1 361.3.d.d 12
76.l odd 18 1 361.3.d.f 12
76.l odd 18 1 361.3.f.b 12
76.l odd 18 1 361.3.f.c 12
76.l odd 18 1 361.3.f.g 12
228.u odd 18 1 171.3.ba.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.f.a ✓ 12 4.b odd 2 1
19.3.f.a ✓ 12 76.k even 18 1
171.3.ba.b 12 12.b even 2 1
171.3.ba.b 12 228.u odd 18 1
304.3.z.a 12 1.a even 1 1 trivial
304.3.z.a 12 19.f odd 18 1 inner
361.3.b.c 12 76.k even 18 1
361.3.b.c 12 76.l odd 18 1
361.3.d.d 12 76.k even 18 1
361.3.d.d 12 76.l odd 18 1
361.3.d.f 12 76.k even 18 1
361.3.d.f 12 76.l odd 18 1
361.3.f.b 12 76.f even 6 1
361.3.f.b 12 76.l odd 18 1
361.3.f.c 12 76.f even 6 1
361.3.f.c 12 76.l odd 18 1
361.3.f.e 12 76.g odd 6 1
361.3.f.e 12 76.k even 18 1
361.3.f.f 12 76.g odd 6 1
361.3.f.f 12 76.k even 18 1
361.3.f.g 12 76.d even 2 1
361.3.f.g 12 76.l odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 12 T_{3}^{10} - 63 T_{3}^{9} + 375 T_{3}^{8} - 1395 T_{3}^{7} + 2699 T_{3}^{6} - 4905 T_{3}^{5} + \cdots + 289 \) 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
$3$ \( T^{12} + 12 T^{10} + \cdots + 289 \) Copy content Toggle raw display
$5$ \( T^{12} + 6 T^{11} + \cdots + 87616 \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{11} + \cdots + 1700416 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 131774082049 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 3568865052736 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 21556993329 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 22\!\cdots\!61 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 114585206984704 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 484594358358016 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 125789503910464 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 58527273697344 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 17\!\cdots\!69 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 126559282520449 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 51\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 48\!\cdots\!41 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 42\!\cdots\!56 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 17\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 30\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 15\!\cdots\!89 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 14\!\cdots\!89 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
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