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Properties

Label 624.4.c.g
Level $624$
Weight $4$
Character orbit 624.c
Analytic conductor $36.817$
Analytic rank $0$
Dimension $10$
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] = [624,4,Mod(337,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.337");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.c (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: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 171x^{8} + 10475x^{6} + 270297x^{4} + 2501800x^{2} + 1354896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{25} \)
Twist minimal: no (minimal twist has level 312)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} + \beta_1 q^{5} - \beta_{3} q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} + \beta_1 q^{5} - \beta_{3} q^{7} + 9 q^{9} + (\beta_{9} + \beta_{6}) q^{11} + ( - \beta_{7} + \beta_{6} + \beta_{3} + \cdots - 1) q^{13}+ \cdots + (9 \beta_{9} + 9 \beta_{6}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 30 q^{3} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 30 q^{3} + 90 q^{9} - 6 q^{13} + 92 q^{17} + 168 q^{23} - 118 q^{25} - 270 q^{27} + 412 q^{29} - 504 q^{35} + 18 q^{39} - 616 q^{43} + 110 q^{49} - 276 q^{51} + 308 q^{53} - 592 q^{55} + 436 q^{61} + 216 q^{65} - 504 q^{69} + 354 q^{75} + 368 q^{77} + 1312 q^{79} + 810 q^{81} - 1236 q^{87} + 2184 q^{91} - 1896 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 171x^{8} + 10475x^{6} + 270297x^{4} + 2501800x^{2} + 1354896 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 31\nu^{8} + 970\nu^{6} - 75569\nu^{4} - 291260\nu^{2} + 50382912 ) / 849108 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -467\nu^{9} + 12778\nu^{7} + 5977413\nu^{5} + 232469056\nu^{3} + 1407290312\nu ) / 164726952 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -441\nu^{8} - 59450\nu^{6} - 2650089\nu^{4} - 40877576\nu^{2} - 73086300 ) / 849108 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1427\nu^{8} + 172474\nu^{6} + 6272435\nu^{4} + 63742780\nu^{2} - 28790928 ) / 849108 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 79\nu^{9} + 9144\nu^{7} + 282191\nu^{5} - 92946\nu^{3} - 66888440\nu ) / 3167826 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6031 \nu^{9} - 3007 \nu^{8} + 805000 \nu^{7} - 94090 \nu^{6} + 39266747 \nu^{5} + \cdots + 6314290272 ) / 164726952 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 6031 \nu^{9} - 3007 \nu^{8} - 805000 \nu^{7} - 94090 \nu^{6} - 39266747 \nu^{5} + \cdots + 6314290272 ) / 164726952 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12229\nu^{9} + 2016954\nu^{7} + 112492445\nu^{5} + 2402342640\nu^{3} + 15884420296\nu ) / 164726952 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} + \beta_{7} + \beta_{2} - 136 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} - 4\beta_{6} - 9\beta_{3} - 52\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -72\beta_{8} - 72\beta_{7} - 5\beta_{5} - 14\beta_{4} - 41\beta_{2} + 6578 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -93\beta_{9} - 30\beta_{8} + 30\beta_{7} + 273\beta_{6} + 741\beta_{3} + 2864\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4889\beta_{8} + 4889\beta_{7} + 408\beta_{5} + 1068\beta_{4} + 1301\beta_{2} - 346244 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7169\beta_{9} + 3432\beta_{8} - 3432\beta_{7} - 16790\beta_{6} - 48609\beta_{3} - 162332\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -319098\beta_{8} - 319098\beta_{7} - 24955\beta_{5} - 67546\beta_{4} - 21697\beta_{2} + 19090534 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -496413\beta_{9} - 290082\beta_{8} + 290082\beta_{7} + 1043715\beta_{6} + 2968869\beta_{3} + 9344608\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
− 7.80054i
− 7.38657i
− 5.87408i
− 4.52865i
− 0.759412i
0.759412i
4.52865i
5.87408i
7.38657i
7.80054i
0 −3.00000 0 − 15.6011i 0 5.33637i 0 9.00000 0
337.2 0 −3.00000 0 − 14.7731i 0 13.6208i 0 9.00000 0
337.3 0 −3.00000 0 − 11.7482i 0 − 24.4020i 0 9.00000 0
337.4 0 −3.00000 0 − 9.05729i 0 − 28.5653i 0 9.00000 0
337.5 0 −3.00000 0 − 1.51882i 0 5.87887i 0 9.00000 0
337.6 0 −3.00000 0 1.51882i 0 − 5.87887i 0 9.00000 0
337.7 0 −3.00000 0 9.05729i 0 28.5653i 0 9.00000 0
337.8 0 −3.00000 0 11.7482i 0 24.4020i 0 9.00000 0
337.9 0 −3.00000 0 14.7731i 0 − 13.6208i 0 9.00000 0
337.10 0 −3.00000 0 15.6011i 0 − 5.33637i 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 337.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.4.c.g 10
4.b odd 2 1 312.4.c.a ✓ 10
12.b even 2 1 936.4.c.b 10
13.b even 2 1 inner 624.4.c.g 10
52.b odd 2 1 312.4.c.a ✓ 10
156.h even 2 1 936.4.c.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.4.c.a ✓ 10 4.b odd 2 1
312.4.c.a ✓ 10 52.b odd 2 1
624.4.c.g 10 1.a even 1 1 trivial
624.4.c.g 10 13.b even 2 1 inner
936.4.c.b 10 12.b even 2 1
936.4.c.b 10 156.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 684T_{5}^{8} + 167600T_{5}^{6} + 17299008T_{5}^{4} + 640460800T_{5}^{2} + 1387413504 \) acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T + 3)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 1387413504 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 88718196736 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 68807688601600 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 51\!\cdots\!57 \) Copy content Toggle raw display
$17$ \( (T^{5} - 46 T^{4} + \cdots + 31342880)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{5} - 84 T^{4} + \cdots - 7538884608)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} - 206 T^{4} + \cdots + 3429287200)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 50\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 186488083972096 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{5} + 308 T^{4} + \cdots - 466635771904)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots + 1173012629984)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 53\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 45015746615200)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 97\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 22\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 41830265651200)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
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