Properties

Label 288.7.b.d
Level $288$
Weight $7$
Character orbit 288.b
Analytic conductor $66.256$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [288,7,Mod(271,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.271");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 288.b (of order \(2\), degree \(1\), not 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: \(66.2555760825\)
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: \( x^{12} - 2 x^{11} + 31 x^{10} - 1286 x^{9} + 7702 x^{8} - 174032 x^{7} + 1952056 x^{6} - 6345392 x^{5} + 7695616 x^{4} - 6850848 x^{3} - 19274256 x^{2} + \cdots + 767595744 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{64}\cdot 3^{11} \)
Twist minimal: no (minimal twist has level 24)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{5} + (\beta_{4} + \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{5} + (\beta_{4} + \beta_{2}) q^{7} + ( - \beta_{6} + 227) q^{11} + ( - 3 \beta_{4} - \beta_{3} - 3 \beta_{2}) q^{13} + ( - \beta_{8} + \beta_{6} - 2 \beta_{5} - 7 \beta_1 + 407) q^{17} + ( - \beta_{10} - \beta_{8} + \beta_{5} + 11 \beta_1 - 328) q^{19} + ( - 2 \beta_{9} + 2 \beta_{7} + 2 \beta_{4} - 44 \beta_{2}) q^{23} + ( - \beta_{10} + 3 \beta_{8} - 3 \beta_{6} - 2 \beta_{5} - 91 \beta_1 - 2266) q^{25} + ( - \beta_{11} - 5 \beta_{9} - 3 \beta_{7} - 11 \beta_{4} + 6 \beta_{3} - 31 \beta_{2}) q^{29} + (6 \beta_{11} + \beta_{9} + 6 \beta_{7} + 22 \beta_{4} + 3 \beta_{3} - 11 \beta_{2}) q^{31} + (6 \beta_{10} - 10 \beta_{8} - 3 \beta_{6} + 2 \beta_{5} + 77 \beta_1 + 13529) q^{35} + (8 \beta_{11} + 2 \beta_{9} - 10 \beta_{7} - 35 \beta_{4} + 7 \beta_{3} + 41 \beta_{2}) q^{37} + (6 \beta_{10} + 5 \beta_{8} - 5 \beta_{6} - 30 \beta_{5} + 123 \beta_1 + 4525) q^{41} + ( - 3 \beta_{10} + 13 \beta_{8} - 30 \beta_{6} + 27 \beta_{5} - 151 \beta_1 + 4162) q^{43} + ( - 19 \beta_{11} + 19 \beta_{9} - 8 \beta_{7} + 140 \beta_{4} + \cdots - 171 \beta_{2}) q^{47}+ \cdots + (90 \beta_{10} - 60 \beta_{8} - 36 \beta_{6} - 360 \beta_{5} - 5252 \beta_1 - 97126) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2720 q^{11} + 4888 q^{17} - 3936 q^{19} - 27204 q^{25} + 162336 q^{35} + 54280 q^{41} + 49824 q^{43} - 304644 q^{49} - 886144 q^{59} - 473376 q^{65} - 1565952 q^{67} + 555480 q^{73} + 2497760 q^{83} - 367400 q^{89} + 4475808 q^{91} - 1165656 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 31 x^{10} - 1286 x^{9} + 7702 x^{8} - 174032 x^{7} + 1952056 x^{6} - 6345392 x^{5} + 7695616 x^{4} - 6850848 x^{3} - 19274256 x^{2} + \cdots + 767595744 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 78\!\cdots\!93 \nu^{11} + \cdots - 48\!\cdots\!44 ) / 78\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 69\!\cdots\!69 \nu^{11} + \cdots + 11\!\cdots\!92 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\!\cdots\!80 \nu^{11} + \cdots + 16\!\cdots\!20 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 17\!\cdots\!33 \nu^{11} + \cdots + 29\!\cdots\!76 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 60\!\cdots\!69 \nu^{11} + \cdots + 41\!\cdots\!68 ) / 23\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 10\!\cdots\!87 \nu^{11} + \cdots - 60\!\cdots\!36 ) / 39\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 30\!\cdots\!83 \nu^{11} + \cdots - 46\!\cdots\!76 ) / 94\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 27\!\cdots\!27 \nu^{11} + \cdots + 11\!\cdots\!00 ) / 78\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13\!\cdots\!29 \nu^{11} + \cdots - 23\!\cdots\!56 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 27\!\cdots\!07 \nu^{11} + \cdots + 17\!\cdots\!08 ) / 19\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 36\!\cdots\!22 \nu^{11} + \cdots + 55\!\cdots\!04 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( - 7 \beta_{11} + 6 \beta_{10} - 8 \beta_{9} - 6 \beta_{8} - 4 \beta_{7} + 18 \beta_{6} + 9 \beta_{4} - 4 \beta_{3} - 11 \beta_{2} - 86 \beta _1 + 1530 ) / 9216 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 11 \beta_{11} - 6 \beta_{10} + 118 \beta_{9} - 90 \beta_{8} - 46 \beta_{7} - 90 \beta_{6} + 108 \beta_{5} + 783 \beta_{4} + 32 \beta_{3} - 1067 \beta_{2} - 982 \beta _1 - 44514 ) / 9216 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 57 \beta_{11} - 114 \beta_{10} - 189 \beta_{9} - 78 \beta_{8} + 519 \beta_{7} - 642 \beta_{6} + 90 \beta_{5} - 4173 \beta_{4} + 18 \beta_{3} + 7548 \beta_{2} + 14744 \beta _1 + 942294 ) / 3072 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 337 \beta_{11} + 12066 \beta_{10} + 3688 \beta_{9} + 3198 \beta_{8} - 1132 \beta_{7} + 62550 \beta_{6} - 3672 \beta_{5} + 24975 \beta_{4} + 116 \beta_{3} + 6499 \beta_{2} - 235802 \beta _1 - 14672754 ) / 9216 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 35923 \beta_{11} - 93066 \beta_{10} + 88789 \beta_{9} - 8118 \beta_{8} - 171799 \beta_{7} - 483282 \beta_{6} + 8694 \beta_{5} + 1534851 \beta_{4} - 25690 \beta_{3} + \cdots + 482412870 ) / 9216 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 198165 \beta_{11} + 51882 \beta_{10} - 896052 \beta_{9} - 292362 \beta_{8} + 493416 \beta_{7} + 269790 \beta_{6} + 321120 \beta_{5} - 6196989 \beta_{4} - 412788 \beta_{3} + \cdots - 1110565194 ) / 3072 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 11046191 \beta_{11} + 9862386 \beta_{10} + 37769887 \beta_{9} - 1840434 \beta_{8} - 3611077 \beta_{7} + 51254010 \beta_{6} + 2026242 \beta_{5} + 157548465 \beta_{4} + \cdots - 38877402366 ) / 9216 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 33855589 \beta_{11} - 206530998 \beta_{10} - 135937424 \beta_{9} + 95490774 \beta_{8} + 75269516 \beta_{7} - 1073096082 \beta_{6} - 104843592 \beta_{5} + \cdots + 1300209250950 ) / 9216 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 130422483 \beta_{11} + 320139798 \beta_{10} - 335384919 \beta_{9} - 37938518 \beta_{8} - 8192451 \beta_{7} + 1663537214 \beta_{6} + 41661774 \beta_{5} + \cdots - 1390347762026 ) / 1024 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 12860137265 \beta_{11} - 18204604194 \beta_{10} + 71160229012 \beta_{9} - 7785963006 \beta_{8} - 35903808088 \beta_{7} - 94592446470 \beta_{6} + \cdots + 41367648684546 ) / 9216 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 103524216217 \beta_{11} - 179379311550 \beta_{10} - 842532471449 \beta_{9} + 16749006462 \beta_{8} + 628560440675 \beta_{7} + \cdots + 806078260932498 ) / 9216 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
8.16014 + 0.886446i
3.47372 1.02840i
0.630233 2.92643i
−9.37784 + 8.67520i
−1.87190 + 1.33932i
−0.0143493 10.5849i
−0.0143493 + 10.5849i
−1.87190 1.33932i
−9.37784 8.67520i
0.630233 + 2.92643i
3.47372 + 1.02840i
8.16014 0.886446i
0 0 0 232.265i 0 483.645i 0 0 0
271.2 0 0 0 128.353i 0 534.624i 0 0 0
271.3 0 0 0 111.403i 0 106.838i 0 0 0
271.4 0 0 0 100.822i 0 277.765i 0 0 0
271.5 0 0 0 87.0704i 0 355.505i 0 0 0
271.6 0 0 0 82.3007i 0 351.467i 0 0 0
271.7 0 0 0 82.3007i 0 351.467i 0 0 0
271.8 0 0 0 87.0704i 0 355.505i 0 0 0
271.9 0 0 0 100.822i 0 277.765i 0 0 0
271.10 0 0 0 111.403i 0 106.838i 0 0 0
271.11 0 0 0 128.353i 0 534.624i 0 0 0
271.12 0 0 0 232.265i 0 483.645i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.7.b.d 12
3.b odd 2 1 96.7.b.a 12
4.b odd 2 1 72.7.b.c 12
8.b even 2 1 72.7.b.c 12
8.d odd 2 1 inner 288.7.b.d 12
12.b even 2 1 24.7.b.a 12
24.f even 2 1 96.7.b.a 12
24.h odd 2 1 24.7.b.a 12
48.i odd 4 2 768.7.g.l 24
48.k even 4 2 768.7.g.l 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.7.b.a 12 12.b even 2 1
24.7.b.a 12 24.h odd 2 1
72.7.b.c 12 4.b odd 2 1
72.7.b.c 12 8.b even 2 1
96.7.b.a 12 3.b odd 2 1
96.7.b.a 12 24.f even 2 1
288.7.b.d 12 1.a even 1 1 trivial
288.7.b.d 12 8.d odd 2 1 inner
768.7.g.l 24 48.i odd 4 2
768.7.g.l 24 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 107352 T_{5}^{10} + 3991016688 T_{5}^{8} + 71113492512000 T_{5}^{6} + \cdots + 57\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(288, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 107352 T^{10} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + 858216 T^{10} + \cdots + 91\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( (T^{6} - 1360 T^{5} + \cdots + 23\!\cdots\!12)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 26982528 T^{10} + \cdots + 36\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( (T^{6} - 2444 T^{5} + \cdots - 15\!\cdots\!24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 1968 T^{5} + \cdots + 24\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 879376800 T^{10} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{12} + 4040599128 T^{10} + \cdots + 28\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{12} + 7044987816 T^{10} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{12} + 20697207840 T^{10} + \cdots + 84\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{6} - 27140 T^{5} + \cdots - 11\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 24912 T^{5} + \cdots - 19\!\cdots\!32)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 86670856608 T^{10} + \cdots + 41\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{12} + 172945812888 T^{10} + \cdots + 33\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{6} + 443072 T^{5} + \cdots + 12\!\cdots\!28)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + 348717987744 T^{10} + \cdots + 34\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{6} + 782976 T^{5} + \cdots - 54\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + 671412159648 T^{10} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{6} - 277740 T^{5} + \cdots + 11\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 1736387849448 T^{10} + \cdots + 65\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( (T^{6} - 1248880 T^{5} + \cdots + 53\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 183700 T^{5} + \cdots + 62\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 582828 T^{5} + \cdots - 50\!\cdots\!12)^{2} \) Copy content Toggle raw display
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