Properties

Label 912.6.a.w
Level $912$
Weight $6$
Character orbit 912.a
Self dual yes
Analytic conductor $146.270$
Analytic rank $0$
Dimension $5$
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: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 4090x^{3} + 112411x^{2} - 577224x - 966924 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: no (minimal twist has level 456)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 9 q^{3} + (\beta_1 + 13) q^{5} - \beta_{2} q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 9 q^{3} + (\beta_1 + 13) q^{5} - \beta_{2} q^{7} + 81 q^{9} + ( - \beta_{4} + \beta_{3} - 2 \beta_1 + 64) q^{11} + (\beta_{4} + \beta_{3} + \beta_1 + 223) q^{13} + ( - 9 \beta_1 - 117) q^{15} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 + 460) q^{17} + 361 q^{19} + 9 \beta_{2} q^{21} + (2 \beta_{4} + 5 \beta_{3} - 8 \beta_{2} - 3 \beta_1 - 177) q^{23} + ( - 5 \beta_{4} - 5 \beta_{3} + 30 \beta_1 + 1391) q^{25} - 729 q^{27} + ( - 7 \beta_{4} + 15 \beta_{3} - 11 \beta_{2} + 2 \beta_1 + 1186) q^{29} + (13 \beta_{3} - 20 \beta_{2} - 37 \beta_1 - 831) q^{31} + (9 \beta_{4} - 9 \beta_{3} + 18 \beta_1 - 576) q^{33} + (7 \beta_{4} - 34 \beta_{3} - 37 \beta_{2} + 30 \beta_1 + 148) q^{35} + (10 \beta_{3} - 59 \beta_{2} + 35 \beta_1 + 1003) q^{37} + ( - 9 \beta_{4} - 9 \beta_{3} - 9 \beta_1 - 2007) q^{39} + ( - 4 \beta_{4} - 8 \beta_{3} - 28 \beta_{2} + 66 \beta_1 + 5396) q^{41} + ( - 14 \beta_{4} + 21 \beta_{3} + 32 \beta_{2} - 12 \beta_1 - 3552) q^{43} + (81 \beta_1 + 1053) q^{45} + (3 \beta_{4} + 15 \beta_{3} + 13 \beta_{2} + 161 \beta_1 - 6719) q^{47} + ( - 19 \beta_{4} + 24 \beta_{3} - 19 \beta_{2} + 154 \beta_1 + 4445) q^{49} + ( - 9 \beta_{3} - 18 \beta_{2} - 18 \beta_1 - 4140) q^{51} + (42 \beta_{4} - 50 \beta_{3} - 18 \beta_{2} - 40 \beta_1 + 2982) q^{53} + ( - 8 \beta_{4} + 25 \beta_{3} + 106 \beta_{2} + 128 \beta_1 - 5564) q^{55} - 3249 q^{57} + (\beta_{4} - 4 \beta_{3} - 4 \beta_{2} + 392 \beta_1 - 3174) q^{59} + (19 \beta_{4} - 20 \beta_{3} - 75 \beta_{2} - 42 \beta_1 - 182) q^{61} - 81 \beta_{2} q^{63} + ( - 17 \beta_{4} - 28 \beta_{3} + 144 \beta_{2} - 86 \beta_1 + 8212) q^{65} + ( - 29 \beta_{4} - 113 \beta_{3} + 39 \beta_{2} + 312 \beta_1 - 966) q^{67} + ( - 18 \beta_{4} - 45 \beta_{3} + 72 \beta_{2} + 27 \beta_1 + 1593) q^{69} + (91 \beta_{4} - 53 \beta_{3} - 219 \beta_{2} + 246 \beta_1 - 9864) q^{71} + (5 \beta_{4} - 13 \beta_{3} + 58 \beta_{2} + 78 \beta_1 + 12054) q^{73} + (45 \beta_{4} + 45 \beta_{3} - 270 \beta_1 - 12519) q^{75} + (70 \beta_{4} + 67 \beta_{3} - 220 \beta_{2} - 564 \beta_1 - 876) q^{77} + (37 \beta_{4} + 26 \beta_{3} + 299 \beta_{2} + 179 \beta_1 - 33309) q^{79} + 6561 q^{81} + ( - 10 \beta_{4} + 141 \beta_{3} - 105 \beta_{2} - 14 \beta_1 - 21370) q^{83} + ( - 39 \beta_{4} + 54 \beta_{3} + 199 \beta_{2} + 320 \beta_1 + 16010) q^{85} + (63 \beta_{4} - 135 \beta_{3} + 99 \beta_{2} - 18 \beta_1 - 10674) q^{87} + ( - 66 \beta_{4} + 267 \beta_{3} + 265 \beta_{2} - 242 \beta_1 - 2272) q^{89} + ( - 79 \beta_{4} + 43 \beta_{3} - 197 \beta_{2} - 622 \beta_1 - 14496) q^{91} + ( - 117 \beta_{3} + 180 \beta_{2} + 333 \beta_1 + 7479) q^{93} + (361 \beta_1 + 4693) q^{95} + ( - 137 \beta_{4} + 83 \beta_{3} + 219 \beta_{2} + 360 \beta_1 + 36536) q^{97} + ( - 81 \beta_{4} + 81 \beta_{3} - 162 \beta_1 + 5184) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 45 q^{3} + 66 q^{5} + 2 q^{7} + 405 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 45 q^{3} + 66 q^{5} + 2 q^{7} + 405 q^{9} + 322 q^{11} + 1116 q^{13} - 594 q^{15} + 2300 q^{17} + 1805 q^{19} - 18 q^{21} - 866 q^{23} + 6985 q^{25} - 3645 q^{27} + 5998 q^{29} - 4126 q^{31} - 2898 q^{33} + 762 q^{35} + 5188 q^{37} - 10044 q^{39} + 27094 q^{41} - 17766 q^{43} + 5346 q^{45} - 33436 q^{47} + 22503 q^{49} - 20700 q^{51} + 14722 q^{53} - 27838 q^{55} - 16245 q^{57} - 15480 q^{59} - 880 q^{61} + 162 q^{63} + 40664 q^{65} - 4764 q^{67} + 7794 q^{69} - 48924 q^{71} + 60196 q^{73} - 62865 q^{75} - 4510 q^{77} - 166986 q^{79} + 32805 q^{81} - 106352 q^{83} + 80158 q^{85} - 53982 q^{87} - 11466 q^{89} - 72464 q^{91} + 37134 q^{93} + 23826 q^{95} + 183042 q^{97} + 26082 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 2x^{4} - 4090x^{3} + 112411x^{2} - 577224x - 966924 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 467\nu^{4} - 7028\nu^{3} - 2305268\nu^{2} + 67458811\nu + 45298336 ) / 5546458 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2110\nu^{4} - 27630\nu^{3} + 7963109\nu^{2} - 128131267\nu + 29222502 ) / 2773229 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2577\nu^{4} - 20602\nu^{3} + 10268377\nu^{2} - 173404246\nu - 21622292 ) / 2773229 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2712\nu^{4} + 17060\nu^{3} + 11297024\nu^{2} - 339945708\nu + 1657498922 ) / 2773229 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_{2} + 2\beta _1 + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -20\beta_{4} - 17\beta_{3} + 25\beta_{2} - 194\beta _1 + 13142 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 826\beta_{4} + 3677\beta_{3} - 4381\beta_{2} + 10586\beta _1 - 505306 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -86296\beta_{4} - 173033\beta_{3} + 201929\beta_{2} - 992226\beta _1 + 56203934 ) / 8 \) 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
−74.5074
43.9219
−1.31969
24.9212
8.98400
0 −9.00000 0 −73.4639 0 −66.8123 0 81.0000 0
1.2 0 −9.00000 0 −40.4531 0 155.140 0 81.0000 0
1.3 0 −9.00000 0 4.39560 0 −76.5325 0 81.0000 0
1.4 0 −9.00000 0 79.0031 0 −194.767 0 81.0000 0
1.5 0 −9.00000 0 96.5183 0 184.972 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.w 5
4.b odd 2 1 456.6.a.e 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.6.a.e 5 4.b odd 2 1
912.6.a.w 5 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{5} - 66T_{5}^{4} - 9127T_{5}^{3} + 388332T_{5}^{2} + 21135676T_{5} - 99608816 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(912))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( (T + 9)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - 66 T^{4} - 9127 T^{3} + \cdots - 99608816 \) Copy content Toggle raw display
$7$ \( T^{5} - 2 T^{4} + \cdots + 28578898944 \) Copy content Toggle raw display
$11$ \( T^{5} - 322 T^{4} + \cdots - 5635869502464 \) Copy content Toggle raw display
$13$ \( T^{5} - 1116 T^{4} + \cdots - 13044480271936 \) Copy content Toggle raw display
$17$ \( T^{5} - 2300 T^{4} + \cdots + 451692086632 \) Copy content Toggle raw display
$19$ \( (T - 361)^{5} \) Copy content Toggle raw display
$23$ \( T^{5} + 866 T^{4} + \cdots + 67\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{5} - 5998 T^{4} + \cdots - 14\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( T^{5} + 4126 T^{4} + \cdots - 43\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{5} - 5188 T^{4} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{5} - 27094 T^{4} + \cdots + 48\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{5} + 17766 T^{4} + \cdots - 41\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{5} + 33436 T^{4} + \cdots - 51\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( T^{5} - 14722 T^{4} + \cdots - 59\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{5} + 15480 T^{4} + \cdots + 37\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{5} + 880 T^{4} + \cdots - 13\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( T^{5} + 4764 T^{4} + \cdots - 96\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{5} + 48924 T^{4} + \cdots + 66\!\cdots\!72 \) Copy content Toggle raw display
$73$ \( T^{5} - 60196 T^{4} + \cdots + 30\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{5} + 166986 T^{4} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{5} + 106352 T^{4} + \cdots - 11\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{5} + 11466 T^{4} + \cdots + 27\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{5} - 183042 T^{4} + \cdots + 10\!\cdots\!24 \) Copy content Toggle raw display
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