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Properties

Label 4620.2.h.e
Level $4620$
Weight $2$
Character orbit 4620.h
Analytic conductor $36.891$
Analytic rank $0$
Dimension $14$
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] = [4620,2,Mod(1849,4620)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4620, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4620.1849");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4620.h (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: \(36.8908857338\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} + 12 x^{12} - 32 x^{11} + 86 x^{10} - 220 x^{9} + 585 x^{8} - 1536 x^{7} + \cdots + 78125 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_{3} q^{5} + \beta_{2} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_{3} q^{5} + \beta_{2} q^{7} - q^{9} + q^{11} + ( - \beta_{6} - \beta_{2}) q^{13} - \beta_{9} q^{15} + (\beta_{6} + \beta_{4} - \beta_{2}) q^{17} + ( - \beta_{11} + \beta_{9} + \cdots + \beta_1) q^{19}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 4 q^{5} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 4 q^{5} - 14 q^{9} + 14 q^{11} + 6 q^{19} - 14 q^{21} - 8 q^{25} + 38 q^{29} + 4 q^{31} + 8 q^{39} - 12 q^{41} + 4 q^{45} - 14 q^{49} + 14 q^{51} - 4 q^{55} + 18 q^{59} - 54 q^{61} - 2 q^{65} + 26 q^{69} - 24 q^{71} + 48 q^{79} + 14 q^{81} + 12 q^{85} + 46 q^{89} + 8 q^{91} - 72 q^{95} - 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 4 x^{13} + 12 x^{12} - 32 x^{11} + 86 x^{10} - 220 x^{9} + 585 x^{8} - 1536 x^{7} + \cdots + 78125 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{13} + 55 \nu^{12} - 53 \nu^{11} + 247 \nu^{10} - 807 \nu^{9} + 1425 \nu^{8} - 4872 \nu^{7} + \cdots + 703125 ) / 120000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{13} - 4 \nu^{12} + 12 \nu^{11} - 32 \nu^{10} + 86 \nu^{9} - 220 \nu^{8} + 585 \nu^{7} + \cdots - 62500 ) / 15625 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5 \nu^{13} + 228 \nu^{12} - 317 \nu^{11} + 1156 \nu^{10} - 3511 \nu^{9} + 5688 \nu^{8} + \cdots + 3275000 ) / 300000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 42 \nu^{13} - 233 \nu^{12} + 664 \nu^{11} - 1599 \nu^{10} + 2742 \nu^{9} - 8255 \nu^{8} + \cdots - 1828125 ) / 500000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 317 \nu^{13} - 3237 \nu^{12} + 2191 \nu^{11} - 15941 \nu^{10} + 49973 \nu^{9} - 52515 \nu^{8} + \cdots - 63859375 ) / 3000000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 196 \nu^{13} + 1919 \nu^{12} - 2292 \nu^{11} + 8367 \nu^{10} - 27976 \nu^{9} + 45405 \nu^{8} + \cdots + 27640625 ) / 1500000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 51 \nu^{13} - 41 \nu^{12} + 215 \nu^{11} - 721 \nu^{10} + 1205 \nu^{9} - 4287 \nu^{8} + \cdots + 78125 ) / 120000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9 \nu^{13} + 35 \nu^{12} - 53 \nu^{11} + 235 \nu^{10} - 527 \nu^{9} + 1173 \nu^{8} + \cdots + 414625 ) / 24000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 79 \nu^{13} + 211 \nu^{12} - 478 \nu^{11} + 1693 \nu^{10} - 3459 \nu^{9} + 9875 \nu^{8} + \cdots + 2140625 ) / 125000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 453 \nu^{13} + 1493 \nu^{12} - 3005 \nu^{11} + 10873 \nu^{10} - 23315 \nu^{9} + 60291 \nu^{8} + \cdots + 17246875 ) / 600000 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3139 \nu^{13} - 5421 \nu^{12} + 16003 \nu^{11} - 54953 \nu^{10} + 96509 \nu^{9} - 326595 \nu^{8} + \cdots - 38171875 ) / 3000000 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 673 \nu^{13} - 2043 \nu^{12} + 4345 \nu^{11} - 16583 \nu^{10} + 35615 \nu^{9} - 89301 \nu^{8} + \cdots - 24878125 ) / 600000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + \beta_{11} - \beta_{8} - \beta_{6} - \beta_{4} - \beta_{3} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} + 2\beta_{11} - \beta_{10} - \beta_{8} - \beta_{7} + \beta_{5} + 2\beta_{4} + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{13} - 2 \beta_{12} + \beta_{11} - 6 \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{6} + \cdots - 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3 \beta_{13} + 9 \beta_{12} + 12 \beta_{11} - 6 \beta_{10} - 5 \beta_{9} - 7 \beta_{8} - 4 \beta_{7} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 13 \beta_{13} + 27 \beta_{12} + 14 \beta_{11} + 6 \beta_{10} + 9 \beta_{9} + 9 \beta_{8} - 20 \beta_{7} + \cdots - 37 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5 \beta_{13} + 17 \beta_{12} - 8 \beta_{11} - 10 \beta_{10} + 75 \beta_{9} - 3 \beta_{8} - 24 \beta_{7} + \cdots + 36 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 34 \beta_{13} - \beta_{12} - 47 \beta_{11} + 42 \beta_{10} - 19 \beta_{9} + 54 \beta_{8} - 64 \beta_{7} + \cdots + 477 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 48 \beta_{13} - 83 \beta_{12} - 90 \beta_{11} + 111 \beta_{10} + 87 \beta_{9} + 406 \beta_{8} + \cdots - 162 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 280 \beta_{13} + 177 \beta_{12} + 321 \beta_{11} + 8 \beta_{10} + 392 \beta_{9} - 465 \beta_{8} + \cdots - 401 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 112 \beta_{13} - 312 \beta_{12} + 44 \beta_{11} + 68 \beta_{10} + 532 \beta_{9} + 576 \beta_{8} + \cdots - 20 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 248 \beta_{13} - 1448 \beta_{12} + 168 \beta_{11} - 2760 \beta_{10} + 2472 \beta_{9} + 2080 \beta_{8} + \cdots - 3471 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 5112 \beta_{13} + 4736 \beta_{12} + 2900 \beta_{11} - 1540 \beta_{10} - 5540 \beta_{9} + 1296 \beta_{8} + \cdots - 4396 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4620\mathbb{Z}\right)^\times\).

\(n\) \(661\) \(1541\) \(2311\) \(2521\) \(3697\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
2.21140 − 0.331193i
1.60267 − 1.55931i
1.56500 + 1.59711i
0.151739 + 2.23091i
−0.217835 − 2.22543i
−1.52705 + 1.63344i
−1.78593 − 1.34553i
2.21140 + 0.331193i
1.60267 + 1.55931i
1.56500 − 1.59711i
0.151739 − 2.23091i
−0.217835 + 2.22543i
−1.52705 − 1.63344i
−1.78593 + 1.34553i
0 − 1.00000i 0 −2.21140 − 0.331193i 0 − 1.00000i 0 −1.00000 0
1849.2 0 − 1.00000i 0 −1.60267 − 1.55931i 0 − 1.00000i 0 −1.00000 0
1849.3 0 − 1.00000i 0 −1.56500 + 1.59711i 0 − 1.00000i 0 −1.00000 0
1849.4 0 − 1.00000i 0 −0.151739 + 2.23091i 0 − 1.00000i 0 −1.00000 0
1849.5 0 − 1.00000i 0 0.217835 − 2.22543i 0 − 1.00000i 0 −1.00000 0
1849.6 0 − 1.00000i 0 1.52705 + 1.63344i 0 − 1.00000i 0 −1.00000 0
1849.7 0 − 1.00000i 0 1.78593 − 1.34553i 0 − 1.00000i 0 −1.00000 0
1849.8 0 1.00000i 0 −2.21140 + 0.331193i 0 1.00000i 0 −1.00000 0
1849.9 0 1.00000i 0 −1.60267 + 1.55931i 0 1.00000i 0 −1.00000 0
1849.10 0 1.00000i 0 −1.56500 − 1.59711i 0 1.00000i 0 −1.00000 0
1849.11 0 1.00000i 0 −0.151739 − 2.23091i 0 1.00000i 0 −1.00000 0
1849.12 0 1.00000i 0 0.217835 + 2.22543i 0 1.00000i 0 −1.00000 0
1849.13 0 1.00000i 0 1.52705 − 1.63344i 0 1.00000i 0 −1.00000 0
1849.14 0 1.00000i 0 1.78593 + 1.34553i 0 1.00000i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1849.14
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4620.2.h.e ✓ 14
5.b even 2 1 inner 4620.2.h.e ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4620.2.h.e ✓ 14 1.a even 1 1 trivial
4620.2.h.e ✓ 14 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4620, [\chi])\):

\( T_{13}^{14} + 74T_{13}^{12} + 1829T_{13}^{10} + 17268T_{13}^{8} + 57020T_{13}^{6} + 61812T_{13}^{4} + 17872T_{13}^{2} + 256 \) Copy content Toggle raw display
\( T_{17}^{14} + 95T_{17}^{12} + 3199T_{17}^{10} + 45181T_{17}^{8} + 243228T_{17}^{6} + 482416T_{17}^{4} + 278592T_{17}^{2} + 36864 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{14} + 4 T^{13} + \cdots + 78125 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{7} \) Copy content Toggle raw display
$11$ \( (T - 1)^{14} \) Copy content Toggle raw display
$13$ \( T^{14} + 74 T^{12} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{14} + 95 T^{12} + \cdots + 36864 \) Copy content Toggle raw display
$19$ \( (T^{7} - 3 T^{6} + \cdots + 6128)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 215 T^{12} + \cdots + 589824 \) Copy content Toggle raw display
$29$ \( (T^{7} - 19 T^{6} + \cdots - 98996)^{2} \) Copy content Toggle raw display
$31$ \( (T^{7} - 2 T^{6} + \cdots + 108608)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 1207701504 \) Copy content Toggle raw display
$41$ \( (T^{7} + 6 T^{6} + \cdots + 269056)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 3377004544 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 14086841344 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 13046775169024 \) Copy content Toggle raw display
$59$ \( (T^{7} - 9 T^{6} + \cdots - 2435984)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} + 27 T^{6} + \cdots - 2850688)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 138074669056 \) Copy content Toggle raw display
$71$ \( (T^{7} + 12 T^{6} + \cdots + 147456)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 232013824 \) Copy content Toggle raw display
$79$ \( (T^{7} - 24 T^{6} + \cdots - 7266304)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 2727346176 \) Copy content Toggle raw display
$89$ \( (T^{7} - 23 T^{6} + \cdots + 1295872)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 1416167424 \) Copy content Toggle raw display
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