Properties

Label 2312.4.a.k
Level $2312$
Weight $4$
Character orbit 2312.a
Self dual yes
Analytic conductor $136.412$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2312 = 2^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2312.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(136.412415933\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 95x^{6} + 756x^{4} - 1780x^{2} + 1152 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: no (minimal twist has level 136)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_{4} q^{5} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{7} + ( - \beta_{5} + 16) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + \beta_{4} q^{5} + ( - \beta_{4} - \beta_{2} - \beta_1) q^{7} + ( - \beta_{5} + 16) q^{9} + (\beta_{7} + \beta_{4} - 2 \beta_{2}) q^{11} + (\beta_{6} + 5) q^{13} + (\beta_{6} + \beta_{3} - 3) q^{15} + ( - \beta_{6} + \beta_{3} - 5) q^{19} + ( - 2 \beta_{6} - 2 \beta_{5} - \beta_{3} + 38) q^{21} + (\beta_{7} - 5 \beta_1) q^{23} + (\beta_{6} - 3 \beta_{5} + 63) q^{25} + (3 \beta_{7} - 3 \beta_{4} - 19 \beta_{2} - 6 \beta_1) q^{27} + (2 \beta_{7} - 5 \beta_{4} - 18 \beta_{2}) q^{29} + ( - 3 \beta_{7} + 4 \beta_{4} - 14 \beta_{2} - 5 \beta_1) q^{31} + (3 \beta_{6} - 5 \beta_{5} + \beta_{3} + 98) q^{33} + (2 \beta_{6} + 8 \beta_{5} - 130) q^{35} + ( - \beta_{4} + 28 \beta_{2} - 4 \beta_1) q^{37} + (3 \beta_{7} + 21 \beta_{4} - 3 \beta_{2} + 10 \beta_1) q^{39} + ( - 6 \beta_{7} - 16 \beta_{4} + 2 \beta_{2} + 4 \beta_1) q^{41} + (\beta_{6} - 4 \beta_{5} + \beta_{3} - 3) q^{43} + (2 \beta_{7} + 43 \beta_{4} + 10 \beta_{2} + 20 \beta_1) q^{45} + ( - 3 \beta_{6} + \beta_{3} + 41) q^{47} + ( - \beta_{6} - 6 \beta_{5} - 4 \beta_{3} + 136) q^{49} + (3 \beta_{6} + 10 \beta_{5} - 3 \beta_{3} - 57) q^{53} + (7 \beta_{6} - 8 \beta_{5} + 3 \beta_{3} + 171) q^{55} + ( - 4 \beta_{7} + 28 \beta_{4} + 8 \beta_{2}) q^{57} + ( - \beta_{6} + 4 \beta_{5} - \beta_{3} + 11) q^{59} + ( - 6 \beta_{7} + 5 \beta_{4} - 74 \beta_{2} - 16 \beta_1) q^{61} + (\beta_{7} - 70 \beta_{4} - 80 \beta_{2} - 15 \beta_1) q^{63} + (10 \beta_{7} + 16 \beta_{4} - 78 \beta_{2} - 20 \beta_1) q^{65} + ( - 9 \beta_{6} + 4 \beta_{5} - 3 \beta_{3} - 73) q^{67} + ( - 3 \beta_{6} - 8 \beta_{5} - 25) q^{69} + (6 \beta_{7} - 27 \beta_{4} + 31 \beta_{2} - 29 \beta_1) q^{71} + ( - 6 \beta_{7} + 26 \beta_{4} + 6 \beta_{2} - 20 \beta_1) q^{73} + (12 \beta_{7} + 12 \beta_{4} - 151 \beta_{2} - 8 \beta_1) q^{75} + ( - 3 \beta_{6} + 2 \beta_{5} - 4 \beta_{3} - 207) q^{77} + ( - 11 \beta_{7} - 16 \beta_{4} - 64 \beta_{2} - 5 \beta_1) q^{79} + ( - 3 \beta_{6} - 7 \beta_{5} - 3 \beta_{3} + 391) q^{81} + ( - 9 \beta_{6} + 16 \beta_{5} - \beta_{3} - 297) q^{83} + ( - \beta_{6} - 24 \beta_{5} - 5 \beta_{3} + 819) q^{87} + (\beta_{6} + 18 \beta_{5} + 4 \beta_{3} + 19) q^{89} + (\beta_{7} + 21 \beta_{4} + 139 \beta_{2} - 20 \beta_1) q^{91} + ( - 7 \beta_{6} - 10 \beta_{5} + 4 \beta_{3} + 505) q^{93} + ( - 8 \beta_{7} - 12 \beta_{4} - 120 \beta_{2} + 32 \beta_1) q^{95} + ( - 16 \beta_{7} + 22 \beta_{4} - 140 \beta_{2} + 24 \beta_1) q^{97} + ( - 4 \beta_{7} + 70 \beta_{4} - 183 \beta_{2} + 10 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 132 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 132 q^{9} + 44 q^{13} - 24 q^{15} - 48 q^{19} + 308 q^{21} + 520 q^{25} + 812 q^{33} - 1064 q^{35} - 8 q^{43} + 312 q^{47} + 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 180 q^{69} - 1660 q^{77} + 3156 q^{81} - 2472 q^{83} + 6664 q^{87} + 68 q^{89} + 4036 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 95x^{6} + 756x^{4} - 1780x^{2} + 1152 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{7} - 284\nu^{5} + 2159\nu^{3} - 3402\nu ) / 172 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 35\nu^{6} - 3127\nu^{4} + 8318\nu^{2} + 12168 ) / 172 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -59\nu^{7} + 5485\nu^{5} - 33588\nu^{3} + 47900\nu ) / 1032 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 41\nu^{6} - 3781\nu^{4} + 20634\nu^{2} - 22844 ) / 172 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -45\nu^{6} + 4217\nu^{4} - 28386\nu^{2} + 35292 ) / 172 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 109\nu^{7} - 9989\nu^{5} + 48702\nu^{3} - 15160\nu ) / 1032 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{6} + 5\beta_{5} - 2\beta_{3} + 190 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{7} - 23\beta_{4} - 33\beta_{2} + 78\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 277\beta_{6} + 450\beta_{5} - 171\beta_{3} + 15027 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -619\beta_{7} - 2075\beta_{4} - 3053\beta_{2} + 6708\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 24035\beta_{6} + 39016\beta_{5} - 14763\beta_{3} + 1294619 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -53561\beta_{7} - 179881\beta_{4} - 265039\beta_{2} + 580024\beta_1 ) / 4 \) 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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.20783
−1.03229
1.60125
−9.30031
9.30031
−1.60125
1.03229
−2.20783
0 −9.26065 0 16.4090 0 −34.5010 0 58.7597 0
1.2 0 −8.52350 0 −18.2701 0 13.8757 0 45.6501 0
1.3 0 −2.95309 0 −4.89575 0 −4.46235 0 −18.2793 0
1.4 0 −2.62097 0 11.5318 0 23.0485 0 −20.1305 0
1.5 0 2.62097 0 −11.5318 0 −23.0485 0 −20.1305 0
1.6 0 2.95309 0 4.89575 0 4.46235 0 −18.2793 0
1.7 0 8.52350 0 18.2701 0 −13.8757 0 45.6501 0
1.8 0 9.26065 0 −16.4090 0 34.5010 0 58.7597 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(17\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2312.4.a.k 8
17.b even 2 1 inner 2312.4.a.k 8
17.c even 4 2 136.4.b.b 8
51.f odd 4 2 1224.4.c.e 8
68.f odd 4 2 272.4.b.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.4.b.b 8 17.c even 4 2
272.4.b.f 8 68.f odd 4 2
1224.4.c.e 8 51.f odd 4 2
2312.4.a.k 8 1.a even 1 1 trivial
2312.4.a.k 8 17.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 174T_{3}^{6} + 8760T_{3}^{4} - 106624T_{3}^{2} + 373248 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2312))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 174 T^{6} + 8760 T^{4} + \cdots + 373248 \) Copy content Toggle raw display
$5$ \( T^{8} - 760 T^{6} + \cdots + 286466048 \) Copy content Toggle raw display
$7$ \( T^{8} - 1934 T^{6} + \cdots + 2424307712 \) Copy content Toggle raw display
$11$ \( T^{8} - 7406 T^{6} + \cdots + 1063158272 \) Copy content Toggle raw display
$13$ \( (T^{4} - 22 T^{3} - 5836 T^{2} + \cdots + 8525216)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 24 T^{3} - 21056 T^{2} + \cdots + 44946176)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 36734 T^{6} + \cdots + 2935871578112 \) Copy content Toggle raw display
$29$ \( T^{8} - 106232 T^{6} + \cdots + 14\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( T^{8} - 155918 T^{6} + \cdots + 32\!\cdots\!52 \) Copy content Toggle raw display
$37$ \( T^{8} - 166072 T^{6} + \cdots + 54\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( T^{8} - 419392 T^{6} + \cdots + 55\!\cdots\!32 \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{3} - 60592 T^{2} + \cdots + 112195072)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 156 T^{3} - 57344 T^{2} + \cdots + 134217728)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 236 T^{3} + \cdots - 14761769616)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 36 T^{3} - 60112 T^{2} + \cdots + 70465536)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} - 1597240 T^{6} + \cdots + 18\!\cdots\!28 \) Copy content Toggle raw display
$67$ \( (T^{4} + 312 T^{3} + \cdots + 30967766784)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 1607086 T^{6} + \cdots + 14\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( T^{8} - 1779648 T^{6} + \cdots + 21\!\cdots\!68 \) Copy content Toggle raw display
$79$ \( T^{8} - 1578638 T^{6} + \cdots + 17\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( (T^{4} + 1236 T^{3} + \cdots - 54225864704)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 34 T^{3} - 1202908 T^{2} + \cdots + 21605388512)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 5185024 T^{6} + \cdots + 13\!\cdots\!92 \) Copy content Toggle raw display
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