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Properties

Label 368.3.f.d
Level $368$
Weight $3$
Character orbit 368.f
Analytic conductor $10.027$
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] = [368,3,Mod(321,368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("368.321");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 368.f (of order \(2\), degree \(1\), 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: \(10.0272737285\)
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: \( x^{12} + 28x^{10} + 274x^{8} + 1108x^{6} + 1649x^{4} + 816x^{2} + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{14}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 184)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{6} q^{5} - \beta_{9} q^{7} + (\beta_{10} - \beta_1 + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{6} q^{5} - \beta_{9} q^{7} + (\beta_{10} - \beta_1 + 4) q^{9} - \beta_{4} q^{11} - \beta_{8} q^{13} + (\beta_{7} + \beta_{6} + \beta_{5}) q^{15} + ( - \beta_{11} - \beta_{9} + \beta_{4}) q^{17} + (\beta_{11} - \beta_{7} - \beta_{6}) q^{19} + (2 \beta_{9} + \beta_{7} + \cdots + 2 \beta_{4}) q^{21}+ \cdots + (3 \beta_{11} + 4 \beta_{9} + \cdots - 4 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 52 q^{9} + 4 q^{23} - 28 q^{25} - 96 q^{27} - 72 q^{29} + 8 q^{31} - 152 q^{35} + 8 q^{39} + 88 q^{41} + 216 q^{47} - 140 q^{49} + 56 q^{55} + 72 q^{59} - 280 q^{69} + 64 q^{71} - 144 q^{73} + 136 q^{75} + 216 q^{77} - 12 q^{81} - 312 q^{85} + 304 q^{87} + 552 q^{93} - 128 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 28x^{10} + 274x^{8} + 1108x^{6} + 1649x^{4} + 816x^{2} + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{8} - 58\nu^{6} - 285\nu^{4} - 176\nu^{2} + 64 ) / 16 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{8} - 20\nu^{6} - 107\nu^{4} - 114\nu^{2} - 16 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{10} + 47\nu^{8} + 352\nu^{6} + 925\nu^{4} + 520\nu^{2} - 80 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{11} - 45\nu^{9} - 308\nu^{7} - 639\nu^{5} + 72\nu^{3} + 592\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} + 21\nu^{9} + 113\nu^{7} - 55\nu^{5} - 1280\nu^{3} - 1248\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} + 26\nu^{9} + 231\nu^{7} + 836\nu^{5} + 1136\nu^{3} + 592\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{11} - 25\nu^{9} - 217\nu^{7} - 845\nu^{5} - 1600\nu^{3} - 952\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{10} - 24\nu^{8} - 187\nu^{6} - 538\nu^{4} - 424\nu^{2} + 24 ) / 4 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2\nu^{11} + 53\nu^{9} + 476\nu^{7} + 1655\nu^{5} + 1696\nu^{3} + 448\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 5\nu^{10} + 120\nu^{8} + 935\nu^{6} + 2702\nu^{4} + 2304\nu^{2} + 272 ) / 16 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -7\nu^{11} - 190\nu^{9} - 1769\nu^{7} - 6532\nu^{5} - 7720\nu^{3} - 2160\nu ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + 6\beta_{7} + 14\beta_{6} - 9\beta_{5} - 7\beta_{4} ) / 112 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{10} + 8\beta_{8} + 6\beta_{3} - 3\beta_{2} + 6\beta _1 - 134 ) / 28 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 13\beta_{11} + 84\beta_{9} - 62\beta_{7} - 126\beta_{6} + 79\beta_{5} + 77\beta_{4} ) / 112 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -12\beta_{10} - 38\beta_{8} - 46\beta_{3} + 23\beta_{2} - 46\beta _1 + 570 ) / 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -229\beta_{11} - 1400\beta_{9} + 698\beta_{7} + 1778\beta_{6} - 683\beta_{5} - 749\beta_{4} ) / 112 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 100\beta_{10} + 704\beta_{8} + 1158\beta_{3} - 663\beta_{2} + 1382\beta _1 - 10966 ) / 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 3107\beta_{11} + 19012\beta_{9} - 8042\beta_{7} - 23450\beta_{6} + 6169\beta_{5} + 7539\beta_{4} ) / 112 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4\beta_{10} - 245\beta_{8} - 500\beta_{3} + 308\beta_{2} - 660\beta _1 + 4006 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -39167\beta_{11} - 239792\beta_{9} + 93158\beta_{7} + 292670\beta_{6} - 59097\beta_{5} - 79079\beta_{4} ) / 112 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -10172\beta_{10} + 70376\beta_{8} + 166406\beta_{3} - 106471\beta_{2} + 232038\beta _1 - 1197222 ) / 28 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 476709 \beta_{11} + 2917796 \beta_{9} - 1081054 \beta_{7} - 3542238 \beta_{6} + \cdots + 857381 \beta_{4} ) / 112 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/368\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(97\) \(277\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
1.16242i
− 1.16242i
3.40823i
− 3.40823i
− 2.63612i
2.63612i
0.984364i
− 0.984364i
2.65929i
− 2.65929i
0.206917i
− 0.206917i
0 −5.34058 0 − 6.94967i 0 − 12.4507i 0 19.5218 0
321.2 0 −5.34058 0 6.94967i 0 12.4507i 0 19.5218 0
321.3 0 −3.71974 0 − 4.26358i 0 7.57283i 0 4.83645 0
321.4 0 −3.71974 0 4.26358i 0 − 7.57283i 0 4.83645 0
321.5 0 −0.512226 0 − 3.89376i 0 5.33774i 0 −8.73762 0
321.6 0 −0.512226 0 3.89376i 0 − 5.33774i 0 −8.73762 0
321.7 0 1.06705 0 − 5.41904i 0 1.81644i 0 −7.86140 0
321.8 0 1.06705 0 5.41904i 0 − 1.81644i 0 −7.86140 0
321.9 0 4.06690 0 − 1.82844i 0 − 9.79270i 0 7.53967 0
321.10 0 4.06690 0 1.82844i 0 9.79270i 0 7.53967 0
321.11 0 4.43859 0 − 7.04652i 0 − 4.89337i 0 10.7011 0
321.12 0 4.43859 0 7.04652i 0 4.89337i 0 10.7011 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 321.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 368.3.f.d 12
4.b odd 2 1 184.3.f.a ✓ 12
8.b even 2 1 1472.3.f.h 12
8.d odd 2 1 1472.3.f.g 12
12.b even 2 1 1656.3.c.a 12
23.b odd 2 1 inner 368.3.f.d 12
92.b even 2 1 184.3.f.a ✓ 12
184.e odd 2 1 1472.3.f.h 12
184.h even 2 1 1472.3.f.g 12
276.h odd 2 1 1656.3.c.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.3.f.a ✓ 12 4.b odd 2 1
184.3.f.a ✓ 12 92.b even 2 1
368.3.f.d 12 1.a even 1 1 trivial
368.3.f.d 12 23.b odd 2 1 inner
1472.3.f.g 12 8.d odd 2 1
1472.3.f.g 12 184.h even 2 1
1472.3.f.h 12 8.b even 2 1
1472.3.f.h 12 184.e odd 2 1
1656.3.c.a 12 12.b even 2 1
1656.3.c.a 12 276.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 40T_{3}^{4} + 16T_{3}^{3} + 383T_{3}^{2} - 196T_{3} - 196 \) acting on \(S_{3}^{\mathrm{new}}(368, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} - 40 T^{4} + \cdots - 196)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + 164 T^{10} + \cdots + 64888832 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 1919025152 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 222108459008 \) Copy content Toggle raw display
$13$ \( (T^{6} - 502 T^{4} + \cdots - 10012)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 41361040474112 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 102936684265472 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 21\!\cdots\!21 \) Copy content Toggle raw display
$29$ \( (T^{6} + 36 T^{5} + \cdots - 52328476)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 4 T^{5} + \cdots - 18906948)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 49994880131072 \) Copy content Toggle raw display
$41$ \( (T^{6} - 44 T^{5} + \cdots + 1028809732)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 29\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( (T^{6} - 108 T^{5} + \cdots - 511001188)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 12\!\cdots\!08 \) Copy content Toggle raw display
$59$ \( (T^{6} - 36 T^{5} + \cdots - 4486769216)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 10\!\cdots\!48 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 17\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( (T^{6} - 32 T^{5} + \cdots - 210441316)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 72 T^{5} + \cdots + 5153462692)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 66\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 22\!\cdots\!12 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 22\!\cdots\!28 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 35\!\cdots\!88 \) Copy content Toggle raw display
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