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Properties

Label 624.2.cq.f
Level $624$
Weight $2$
Character orbit 624.cq
Analytic conductor $4.983$
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] = [624,2,Mod(175,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.175");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.cq (of order \(12\), degree \(4\), 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: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(3\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 2 x^{10} + 38 x^{9} + 201 x^{8} - 4 x^{7} + 328 x^{6} + 3856 x^{5} + 14736 x^{4} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{8} + \beta_{5}) q^{3} + ( - \beta_{8} + \beta_{7} + \cdots + \beta_{2}) q^{5}+ \cdots - \beta_{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{8} + \beta_{5}) q^{3} + ( - \beta_{8} + \beta_{7} + \cdots + \beta_{2}) q^{5}+ \cdots + (\beta_{10} - \beta_{3} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{5} + 4 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{5} + 4 q^{7} + 6 q^{9} + 2 q^{11} + 6 q^{13} - 4 q^{15} - 6 q^{17} - 4 q^{19} - 2 q^{21} - 6 q^{23} - 4 q^{29} + 6 q^{31} + 2 q^{33} + 30 q^{35} - 8 q^{37} - 10 q^{39} + 10 q^{41} + 12 q^{43} + 4 q^{45} + 4 q^{47} - 42 q^{49} - 4 q^{51} + 56 q^{53} - 30 q^{55} - 8 q^{57} + 36 q^{59} + 34 q^{61} + 2 q^{63} + 18 q^{65} - 14 q^{67} - 6 q^{69} - 10 q^{71} + 26 q^{73} - 18 q^{75} - 6 q^{81} + 28 q^{83} + 22 q^{85} - 12 q^{87} - 44 q^{89} - 82 q^{91} - 6 q^{93} + 30 q^{95} - 2 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 2 x^{10} + 38 x^{9} + 201 x^{8} - 4 x^{7} + 328 x^{6} + 3856 x^{5} + 14736 x^{4} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 245319796 \nu^{11} - 155170535 \nu^{10} + 2671353736 \nu^{9} - 17819480466 \nu^{8} + \cdots + 100006595328 ) / 2571211719648 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1558 \nu^{11} + 6773 \nu^{10} - 12060 \nu^{9} - 57922 \nu^{8} - 130652 \nu^{7} + 533365 \nu^{6} + \cdots + 974624 ) / 9842976 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1558 \nu^{11} + 6773 \nu^{10} - 12060 \nu^{9} - 57922 \nu^{8} - 130652 \nu^{7} + 533365 \nu^{6} + \cdots + 974624 ) / 9842976 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 92438296 \nu^{11} + 250337451 \nu^{10} - 363601398 \nu^{9} - 3234236934 \nu^{8} + \cdots - 51408329376 ) / 428535286608 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1562603052 \nu^{11} - 2879886308 \nu^{10} + 3280376639 \nu^{9} + 56707562240 \nu^{8} + \cdots + 3702977015600 ) / 2571211719648 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 6446623577 \nu^{11} - 9488287406 \nu^{10} + 1716366478 \nu^{9} + 271797960498 \nu^{8} + \cdots + 26452053741024 ) / 5142423439296 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 30457 \nu^{11} - 57798 \nu^{10} + 47368 \nu^{9} + 1181486 \nu^{8} + 6237701 \nu^{7} + \cdots + 10462880 ) / 19685952 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1606510293 \nu^{11} + 3397897178 \nu^{10} - 3713695488 \nu^{9} - 60320188338 \nu^{8} + \cdots - 1010840249184 ) / 857070573216 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4346720405 \nu^{11} - 9179015352 \nu^{10} + 10154908506 \nu^{9} + 160585799390 \nu^{8} + \cdots - 4501853643488 ) / 1714141146432 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 642379115 \nu^{11} - 1387336691 \nu^{10} + 1629048358 \nu^{9} + 23707956054 \nu^{8} + \cdots + 61248038352 ) / 116873259984 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 40714662875 \nu^{11} + 86659299064 \nu^{10} - 95396608014 \nu^{9} - 1527134581826 \nu^{8} + \cdots - 18646799228000 ) / 5142423439296 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{11} + \beta_{10} - \beta_{9} + 6\beta_{8} - 2\beta_{4} - \beta_{3} + 2\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{11} + \beta_{10} - \beta_{9} + 6 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \cdots - 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 11 \beta_{11} - 11 \beta_{10} - 13 \beta_{9} - 8 \beta_{8} + 2 \beta_{6} - 16 \beta_{5} - 18 \beta_{4} + \cdots - 70 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 19 \beta_{11} - 33 \beta_{10} - 19 \beta_{9} - 178 \beta_{8} + 74 \beta_{7} - 74 \beta_{5} + 57 \beta_{4} + \cdots - 104 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 201 \beta_{11} - 201 \beta_{10} + 125 \beta_{9} - 996 \beta_{8} + 476 \beta_{7} - 76 \beta_{6} + \cdots + 238 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 691 \beta_{11} - 351 \beta_{10} + 691 \beta_{9} - 1618 \beta_{8} + 1758 \beta_{7} - 702 \beta_{6} + \cdots + 3376 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1609 \beta_{11} + 1609 \beta_{10} + 3431 \beta_{9} + 5488 \beta_{8} - 1822 \beta_{6} + 10976 \beta_{5} + \cdots + 15778 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 6593 \beta_{11} + 13011 \beta_{10} + 6593 \beta_{9} + 62030 \beta_{8} - 36766 \beta_{7} + \cdots + 25264 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 61187 \beta_{11} + 61187 \beta_{10} - 23319 \beta_{9} + 266004 \beta_{8} - 227908 \beta_{7} + \cdots - 113954 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 238393 \beta_{11} + 124325 \beta_{10} - 238393 \beta_{9} + 406382 \beta_{8} - 728410 \beta_{7} + \cdots - 1134792 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(-1\) \(\beta_{5}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
175.1
−1.94897 + 1.94897i
−0.802676 + 0.802676i
2.38562 − 2.38562i
−1.61939 + 1.61939i
−0.0544315 + 0.0544315i
3.03984 − 3.03984i
−1.94897 − 1.94897i
−0.802676 − 0.802676i
2.38562 + 2.38562i
−1.61939 − 1.61939i
−0.0544315 − 0.0544315i
3.03984 + 3.03984i
0 −0.866025 − 0.500000i 0 −1.58295 + 1.58295i 0 2.77862 + 0.744529i 0 0.500000 + 0.866025i 0
175.2 0 −0.866025 − 0.500000i 0 −0.436651 + 0.436651i 0 −2.52282 − 0.675989i 0 0.500000 + 0.866025i 0
175.3 0 −0.866025 − 0.500000i 0 2.75165 − 2.75165i 0 1.61023 + 0.431460i 0 0.500000 + 0.866025i 0
223.1 0 0.866025 − 0.500000i 0 −2.98541 + 2.98541i 0 −1.26422 − 4.71815i 0 0.500000 − 0.866025i 0
223.2 0 0.866025 − 0.500000i 0 −1.42046 + 1.42046i 0 1.04985 + 3.91810i 0 0.500000 − 0.866025i 0
223.3 0 0.866025 − 0.500000i 0 1.67382 − 1.67382i 0 0.348346 + 1.30004i 0 0.500000 − 0.866025i 0
271.1 0 −0.866025 + 0.500000i 0 −1.58295 − 1.58295i 0 2.77862 − 0.744529i 0 0.500000 − 0.866025i 0
271.2 0 −0.866025 + 0.500000i 0 −0.436651 − 0.436651i 0 −2.52282 + 0.675989i 0 0.500000 − 0.866025i 0
271.3 0 −0.866025 + 0.500000i 0 2.75165 + 2.75165i 0 1.61023 − 0.431460i 0 0.500000 − 0.866025i 0
319.1 0 0.866025 + 0.500000i 0 −2.98541 − 2.98541i 0 −1.26422 + 4.71815i 0 0.500000 + 0.866025i 0
319.2 0 0.866025 + 0.500000i 0 −1.42046 − 1.42046i 0 1.04985 − 3.91810i 0 0.500000 + 0.866025i 0
319.3 0 0.866025 + 0.500000i 0 1.67382 + 1.67382i 0 0.348346 − 1.30004i 0 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 175.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
52.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.cq.f yes 12
4.b odd 2 1 624.2.cq.e ✓ 12
13.f odd 12 1 624.2.cq.e ✓ 12
52.l even 12 1 inner 624.2.cq.f yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.2.cq.e ✓ 12 4.b odd 2 1
624.2.cq.e ✓ 12 13.f odd 12 1
624.2.cq.f yes 12 1.a even 1 1 trivial
624.2.cq.f yes 12 52.l even 12 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\):

\( T_{5}^{12} + 4 T_{5}^{11} + 8 T_{5}^{10} + 278 T_{5}^{8} + 1032 T_{5}^{7} + 1904 T_{5}^{6} + \cdots + 11664 \) Copy content Toggle raw display
\( T_{7}^{12} - 4 T_{7}^{11} + 29 T_{7}^{10} - 122 T_{7}^{9} + 267 T_{7}^{8} - 302 T_{7}^{7} + \cdots + 111556 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} + 4 T^{11} + \cdots + 11664 \) Copy content Toggle raw display
$7$ \( T^{12} - 4 T^{11} + \cdots + 111556 \) Copy content Toggle raw display
$11$ \( T^{12} - 2 T^{11} + \cdots + 5184 \) Copy content Toggle raw display
$13$ \( T^{12} - 6 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} + 6 T^{11} + \cdots + 5184 \) Copy content Toggle raw display
$19$ \( T^{12} + 4 T^{11} + \cdots + 59536 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 106007616 \) Copy content Toggle raw display
$29$ \( T^{12} + 4 T^{11} + \cdots + 81649296 \) Copy content Toggle raw display
$31$ \( T^{12} - 6 T^{11} + \cdots + 952576 \) Copy content Toggle raw display
$37$ \( T^{12} + 8 T^{11} + \cdots + 114244 \) Copy content Toggle raw display
$41$ \( T^{12} - 10 T^{11} + \cdots + 21233664 \) Copy content Toggle raw display
$43$ \( T^{12} - 12 T^{11} + \cdots + 1503076 \) Copy content Toggle raw display
$47$ \( T^{12} - 4 T^{11} + \cdots + 876096 \) Copy content Toggle raw display
$53$ \( (T^{6} - 28 T^{5} + \cdots - 5256)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 278682633216 \) Copy content Toggle raw display
$61$ \( T^{12} - 34 T^{11} + \cdots + 37417689 \) Copy content Toggle raw display
$67$ \( T^{12} + 14 T^{11} + \cdots + 484 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 74975201856 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 520959150625 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 168671204416 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 707940497664 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 346195731456 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 5196679744 \) Copy content Toggle raw display
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