Properties

Label 912.6.a.p
Level $912$
Weight $6$
Character orbit 912.a
Self dual yes
Analytic conductor $146.270$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,6,Mod(1,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 912.a (trivial)

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: yes
Analytic conductor: \(146.270043669\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 3227x^{2} + 17265x + 1197450 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 228)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 q^{3} + (\beta_{2} - 21) q^{5} + (\beta_{2} - \beta_1 - 13) q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} + (\beta_{2} - 21) q^{5} + (\beta_{2} - \beta_1 - 13) q^{7} + 81 q^{9} + ( - \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 90) q^{11} + ( - 3 \beta_{3} - 7 \beta_{2} - 4 \beta_1 + 15) q^{13} + (9 \beta_{2} - 189) q^{15} + (4 \beta_{3} - 5 \beta_{2} + 5 \beta_1 + 507) q^{17} - 361 q^{19} + (9 \beta_{2} - 9 \beta_1 - 117) q^{21} + (8 \beta_{3} - 40 \beta_{2} - 11 \beta_1 - 210) q^{23} + (9 \beta_{3} - 36 \beta_{2} + 42 \beta_1 + 979) q^{25} + 729 q^{27} + (21 \beta_{3} - 23 \beta_{2} - 39 \beta_1 - 1779) q^{29} + (30 \beta_{3} - 12 \beta_{2} - 41 \beta_1 - 54) q^{31} + ( - 9 \beta_{3} + 18 \beta_{2} + 36 \beta_1 - 810) q^{33} + (9 \beta_{3} - 98 \beta_{2} + 78 \beta_1 + 4926) q^{35} + (24 \beta_{3} - 66 \beta_{2} - 47 \beta_1 - 2570) q^{37} + ( - 27 \beta_{3} - 63 \beta_{2} - 36 \beta_1 + 135) q^{39} + ( - 104 \beta_{3} + 134 \beta_{2} - 4 \beta_1 - 6996) q^{41} + ( - 54 \beta_{3} + 15 \beta_{2} + 187 \beta_1 + 4581) q^{43} + (81 \beta_{2} - 1701) q^{45} + ( - 67 \beta_{3} - 54 \beta_{2} - 71 \beta_1 + 4740) q^{47} + (33 \beta_{3} - 224 \beta_{2} + 72 \beta_1 - 4539) q^{49} + (36 \beta_{3} - 45 \beta_{2} + 45 \beta_1 + 4563) q^{51} + (62 \beta_{3} - 330 \beta_{2} + 106 \beta_1 - 10764) q^{53} + (30 \beta_{3} + 93 \beta_{2} - 87 \beta_1 + 3483) q^{55} - 3249 q^{57} + (29 \beta_{3} - 119 \beta_{2} - 203 \beta_1 + 17151) q^{59} + ( - 147 \beta_{3} + 16 \beta_{2} + 12 \beta_1 - 16846) q^{61} + (81 \beta_{2} - 81 \beta_1 - 1053) q^{63} + ( - 27 \beta_{3} - 361 \beta_{2} - 231 \beta_1 - 27315) q^{65} + (237 \beta_{3} - 31 \beta_{2} + 259 \beta_1 + 20211) q^{67} + (72 \beta_{3} - 360 \beta_{2} - 99 \beta_1 - 1890) q^{69} + ( - 275 \beta_{3} + 65 \beta_{2} + 77 \beta_1 + 15627) q^{71} + (159 \beta_{3} - 228 \beta_{2} + 526 \beta_1 - 20834) q^{73} + (81 \beta_{3} - 324 \beta_{2} + 378 \beta_1 + 8811) q^{75} + ( - 116 \beta_{3} + 297 \beta_{2} + 245 \beta_1 - 20409) q^{77} + ( - 483 \beta_{3} + 53 \beta_{2} + 290 \beta_1 - 2095) q^{79} + 6561 q^{81} + (208 \beta_{3} - 356 \beta_{2} - 676 \beta_1 + 14004) q^{83} + ( - 93 \beta_{3} + 1200 \beta_{2} - 282 \beta_1 - 26820) q^{85} + (189 \beta_{3} - 207 \beta_{2} - 351 \beta_1 - 16011) q^{87} + ( - 152 \beta_{3} + 60 \beta_{2} - 820 \beta_1 - 48894) q^{89} + ( - 81 \beta_{3} + 33 \beta_{2} - 259 \beta_1 - 6285) q^{91} + (270 \beta_{3} - 108 \beta_{2} - 369 \beta_1 - 486) q^{93} + ( - 361 \beta_{2} + 7581) q^{95} + ( - 237 \beta_{3} + 1647 \beta_{2} + 501 \beta_1 - 46189) q^{97} + ( - 81 \beta_{3} + 162 \beta_{2} + 324 \beta_1 - 7290) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 36 q^{3} - 84 q^{5} - 54 q^{7} + 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 36 q^{3} - 84 q^{5} - 54 q^{7} + 324 q^{9} - 354 q^{11} + 46 q^{13} - 756 q^{15} + 2046 q^{17} - 1444 q^{19} - 486 q^{21} - 846 q^{23} + 4018 q^{25} + 2916 q^{27} - 7152 q^{29} - 238 q^{31} - 3186 q^{33} + 19878 q^{35} - 10326 q^{37} + 414 q^{39} - 28200 q^{41} + 18590 q^{43} - 6804 q^{45} + 18684 q^{47} - 17946 q^{49} + 18414 q^{51} - 42720 q^{53} + 13818 q^{55} - 12996 q^{57} + 68256 q^{59} - 67654 q^{61} - 4374 q^{63} - 109776 q^{65} + 81836 q^{67} - 7614 q^{69} + 62112 q^{71} - 81966 q^{73} + 36162 q^{75} - 81378 q^{77} - 8766 q^{79} + 26244 q^{81} + 55080 q^{83} - 108030 q^{85} - 64368 q^{87} - 197520 q^{89} - 25820 q^{91} - 2142 q^{93} + 30324 q^{95} - 184228 q^{97} - 28674 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 3227x^{2} + 17265x + 1197450 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} - 34\nu^{2} + 2037\nu + 43830 ) / 680 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{3} + 17\nu^{2} + 5179\nu - 49530 ) / 510 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 12\beta_{3} - 32\beta_{2} - 13\beta _1 + 3228 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -408\beta_{3} - 272\beta_{2} + 2479\beta _1 - 22092 ) / 2 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
49.3415
−55.6309
−17.5574
24.8468
0 9.00000 0 −107.122 0 −197.805 0 81.0000 0
1.2 0 9.00000 0 −24.7456 0 94.5162 0 81.0000 0
1.3 0 9.00000 0 −16.5926 0 26.5221 0 81.0000 0
1.4 0 9.00000 0 64.4605 0 22.7670 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(19\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.6.a.p 4
4.b odd 2 1 228.6.a.b 4
12.b even 2 1 684.6.a.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.6.a.b 4 4.b odd 2 1
684.6.a.d 4 12.b even 2 1
912.6.a.p 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 84T_{5}^{3} - 4731T_{5}^{2} - 267930T_{5} - 2835216 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(912))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 9)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 84 T^{3} - 4731 T^{2} + \cdots - 2835216 \) Copy content Toggle raw display
$7$ \( T^{4} + 54 T^{3} - 23183 T^{2} + \cdots - 11289056 \) Copy content Toggle raw display
$11$ \( T^{4} + 354 T^{3} + \cdots - 285160800 \) Copy content Toggle raw display
$13$ \( T^{4} - 46 T^{3} + \cdots - 24059151936 \) Copy content Toggle raw display
$17$ \( T^{4} - 2046 T^{3} + \cdots - 340390896708 \) Copy content Toggle raw display
$19$ \( (T + 361)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 846 T^{3} + \cdots + 30031102634496 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots - 312735730784112 \) Copy content Toggle raw display
$31$ \( T^{4} + 238 T^{3} + \cdots + 44788434085728 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 371485366814720 \) Copy content Toggle raw display
$41$ \( T^{4} + 28200 T^{3} + \cdots - 57\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{4} - 18590 T^{3} + \cdots - 47\!\cdots\!60 \) Copy content Toggle raw display
$47$ \( T^{4} - 18684 T^{3} + \cdots - 18\!\cdots\!68 \) Copy content Toggle raw display
$53$ \( T^{4} + 42720 T^{3} + \cdots - 33\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{4} - 68256 T^{3} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + 67654 T^{3} + \cdots - 40\!\cdots\!60 \) Copy content Toggle raw display
$67$ \( T^{4} - 81836 T^{3} + \cdots - 26\!\cdots\!12 \) Copy content Toggle raw display
$71$ \( T^{4} - 62112 T^{3} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{4} + 81966 T^{3} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + 8766 T^{3} + \cdots + 56\!\cdots\!04 \) Copy content Toggle raw display
$83$ \( T^{4} - 55080 T^{3} + \cdots - 82\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{4} + 197520 T^{3} + \cdots + 58\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( T^{4} + 184228 T^{3} + \cdots + 22\!\cdots\!68 \) Copy content Toggle raw display
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