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Properties

Label 624.4.d.b
Level $624$
Weight $4$
Character orbit 624.d
Analytic conductor $36.817$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,4,Mod(287,624)]
 
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([1, 0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.287");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.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: \(36.8171918436\)
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: \( x^{12} + 120x^{10} + 5196x^{8} + 96803x^{6} + 702900x^{4} + 976752x^{2} + 254016 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{10} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} - \beta_{5} q^{5} - \beta_{7} q^{7} + (\beta_{4} + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} - \beta_{5} q^{5} - \beta_{7} q^{7} + (\beta_{4} + 4) q^{9} + (\beta_{10} + \beta_{9} - \beta_{2}) q^{11} - 13 q^{13} + ( - \beta_{10} - 2 \beta_{8} + \cdots + \beta_{2}) q^{15}+ \cdots + ( - 3 \beta_{10} - 15 \beta_{9} + \cdots - 9 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 54 q^{9} - 156 q^{13} - 54 q^{21} - 408 q^{25} - 360 q^{33} + 636 q^{37} - 810 q^{45} + 336 q^{49} - 1260 q^{57} + 960 q^{61} - 252 q^{69} + 3216 q^{73} - 2538 q^{81} + 2196 q^{85} - 1116 q^{93} + 4800 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 120x^{10} + 5196x^{8} + 96803x^{6} + 702900x^{4} + 976752x^{2} + 254016 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -335\nu^{10} - 22548\nu^{8} - 532644\nu^{6} - 9478141\nu^{4} - 112501200\nu^{2} - 208515312 ) / 2222688 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 243823 \nu^{11} + 543494 \nu^{10} + 28334004 \nu^{9} + 54476520 \nu^{8} + 1177630788 \nu^{7} + \cdots + 114851937312 ) / 14376345984 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 6845 \nu^{11} + 1806 \nu^{10} - 708252 \nu^{9} + 678888 \nu^{8} - 26600460 \nu^{7} + \cdots + 5516717472 ) / 186705792 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 6845 \nu^{11} + 1806 \nu^{10} + 708252 \nu^{9} + 678888 \nu^{8} + 26600460 \nu^{7} + \cdots + 5703423264 ) / 186705792 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1355 \nu^{11} - 168396 \nu^{9} - 7577172 \nu^{7} - 147021553 \nu^{5} - 1099988520 \nu^{3} - 1369886448 \nu ) / 31117632 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 243823 \nu^{11} - 543494 \nu^{10} + 28334004 \nu^{9} - 54476520 \nu^{8} + 1177630788 \nu^{7} + \cdots - 114851937312 ) / 4792115328 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 42337 \nu^{11} - 5150244 \nu^{9} - 223427388 \nu^{7} - 4111524179 \nu^{5} + \cdots - 39817168272 \nu ) / 653470272 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 985711 \nu^{11} - 543494 \nu^{10} - 108387684 \nu^{9} - 54476520 \nu^{8} + \cdots - 114851937312 ) / 14376345984 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1324297 \nu^{11} + 1921318 \nu^{10} - 153347532 \nu^{9} + 174755784 \nu^{8} + \cdots - 252319384800 ) / 14376345984 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 196015 \nu^{11} - 4011 \nu^{10} + 22710192 \nu^{9} + 3533544 \nu^{8} + 948958068 \nu^{7} + \cdots + 63207372816 ) / 1797043248 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -412\nu^{11} - 49629\nu^{9} - 2138736\nu^{7} - 38905616\nu^{5} - 261099639\nu^{3} - 190007244\nu ) / 1667016 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} + 3\beta_{8} + 3\beta_{7} + 3\beta_{6} - 5\beta_{5} + \beta_{4} - \beta_{3} + 12\beta_{2} - 1 ) / 36 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 9 \beta_{10} - 15 \beta_{9} + 3 \beta_{8} + 3 \beta_{7} - 7 \beta_{6} + 3 \beta_{4} + 3 \beta_{3} + \cdots - 720 ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 25 \beta_{11} - 39 \beta_{8} - 51 \beta_{7} - 72 \beta_{6} + 89 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} + \cdots + 4 ) / 18 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 561 \beta_{10} + 543 \beta_{9} + 9 \beta_{8} + 9 \beta_{7} + 119 \beta_{6} - 213 \beta_{4} + \cdots + 24048 ) / 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2293 \beta_{11} + 2469 \beta_{8} + 3177 \beta_{7} + 6033 \beta_{6} - 6749 \beta_{5} - 107 \beta_{4} + \cdots + 107 ) / 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 13128 \beta_{10} - 10110 \beta_{9} - 1509 \beta_{8} - 1509 \beta_{7} - 560 \beta_{6} + 5310 \beta_{4} + \cdots - 443547 ) / 18 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 98123 \beta_{11} - 88809 \beta_{8} - 106845 \beta_{7} - 249057 \beta_{6} + 259747 \beta_{5} + \cdots - 12253 ) / 36 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1132191 \beta_{10} + 778809 \beta_{9} + 176691 \beta_{8} + 176691 \beta_{7} - 35623 \beta_{6} + \cdots + 34326432 ) / 36 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2038607 \beta_{11} + 1715181 \beta_{8} + 1916985 \beta_{7} + 5098968 \beta_{6} - 5097847 \beta_{5} + \cdots + 331504 ) / 18 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 47308263 \beta_{10} - 30595929 \beta_{9} - 8356167 \beta_{8} - 8356167 \beta_{7} + 3162383 \beta_{6} + \cdots - 1360964520 ) / 36 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 167219603 \beta_{11} - 137302539 \beta_{8} - 143941383 \beta_{7} - 415377303 \beta_{6} + \cdots - 30971317 ) / 36 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
4.65383i
− 4.65383i
− 6.39800i
6.39800i
1.16673i
− 1.16673i
− 6.34801i
6.34801i
− 3.92558i
3.92558i
− 0.582192i
0.582192i
0 −4.84460 − 1.87879i 0 18.9647i 0 − 27.0499i 0 19.9403 + 18.2040i 0
287.2 0 −4.84460 + 1.87879i 0 − 18.9647i 0 27.0499i 0 19.9403 − 18.2040i 0
287.3 0 −4.31028 − 2.90198i 0 − 9.74037i 0 − 7.39870i 0 10.1570 + 25.0167i 0
287.4 0 −4.31028 + 2.90198i 0 9.74037i 0 7.39870i 0 10.1570 − 25.0167i 0
287.5 0 −2.28065 − 4.66890i 0 4.73978i 0 12.5922i 0 −16.5973 + 21.2962i 0
287.6 0 −2.28065 + 4.66890i 0 − 4.73978i 0 − 12.5922i 0 −16.5973 − 21.2962i 0
287.7 0 2.28065 − 4.66890i 0 − 4.73978i 0 12.5922i 0 −16.5973 − 21.2962i 0
287.8 0 2.28065 + 4.66890i 0 4.73978i 0 − 12.5922i 0 −16.5973 + 21.2962i 0
287.9 0 4.31028 − 2.90198i 0 9.74037i 0 − 7.39870i 0 10.1570 − 25.0167i 0
287.10 0 4.31028 + 2.90198i 0 − 9.74037i 0 7.39870i 0 10.1570 + 25.0167i 0
287.11 0 4.84460 − 1.87879i 0 − 18.9647i 0 − 27.0499i 0 19.9403 − 18.2040i 0
287.12 0 4.84460 + 1.87879i 0 18.9647i 0 27.0499i 0 19.9403 + 18.2040i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.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.d.b ✓ 12
3.b odd 2 1 inner 624.4.d.b ✓ 12
4.b odd 2 1 inner 624.4.d.b ✓ 12
12.b even 2 1 inner 624.4.d.b ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.4.d.b ✓ 12 1.a even 1 1 trivial
624.4.d.b ✓ 12 3.b odd 2 1 inner
624.4.d.b ✓ 12 4.b odd 2 1 inner
624.4.d.b ✓ 12 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 477T_{5}^{4} + 44334T_{5}^{2} + 766584 \) acting on \(S_{4}^{\mathrm{new}}(624, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 387420489 \) Copy content Toggle raw display
$5$ \( (T^{6} + 477 T^{4} + \cdots + 766584)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 945 T^{4} + \cdots + 6351048)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 4896 T^{4} + \cdots - 455196672)^{2} \) Copy content Toggle raw display
$13$ \( (T + 13)^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} + 5931 T^{4} + \cdots + 12265344)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 5562 T^{4} + \cdots + 1223343648)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 49392 T^{4} + \cdots - 265323921408)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 70272 T^{4} + \cdots + 449255890944)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 16616597875488)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 159 T^{2} + \cdots + 20472076)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 147159152439264)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 259700619110688)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 3793562044272)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 3344080287744)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 122300245847808)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 240 T^{2} + \cdots - 23804144)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 116304797191968)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 25122002130288)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 804 T^{2} + \cdots + 130948496)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 15\!\cdots\!92)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 94\!\cdots\!68)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 15\!\cdots\!36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 1200 T^{2} + \cdots + 40416992)^{4} \) Copy content Toggle raw display
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