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 $2$

Related objects

Downloads

Learn more

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: \(x^{8} - 95 x^{6} + 756 x^{4} - 1780 x^{2} + 1152\)
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.

\(f(q)\) \(=\) \( q -\beta_{2} q^{3} + \beta_{4} q^{5} + ( -\beta_{1} - \beta_{2} - \beta_{4} ) q^{7} + ( 16 - \beta_{5} ) q^{9} +O(q^{10})\) \( q -\beta_{2} q^{3} + \beta_{4} q^{5} + ( -\beta_{1} - \beta_{2} - \beta_{4} ) q^{7} + ( 16 - \beta_{5} ) q^{9} + ( -2 \beta_{2} + \beta_{4} + \beta_{7} ) q^{11} + ( 5 + \beta_{6} ) q^{13} + ( -3 + \beta_{3} + \beta_{6} ) q^{15} + ( -5 + \beta_{3} - \beta_{6} ) q^{19} + ( 38 - \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{21} + ( -5 \beta_{1} + \beta_{7} ) q^{23} + ( 63 - 3 \beta_{5} + \beta_{6} ) q^{25} + ( -6 \beta_{1} - 19 \beta_{2} - 3 \beta_{4} + 3 \beta_{7} ) q^{27} + ( -18 \beta_{2} - 5 \beta_{4} + 2 \beta_{7} ) q^{29} + ( -5 \beta_{1} - 14 \beta_{2} + 4 \beta_{4} - 3 \beta_{7} ) q^{31} + ( 98 + \beta_{3} - 5 \beta_{5} + 3 \beta_{6} ) q^{33} + ( -130 + 8 \beta_{5} + 2 \beta_{6} ) q^{35} + ( -4 \beta_{1} + 28 \beta_{2} - \beta_{4} ) q^{37} + ( 10 \beta_{1} - 3 \beta_{2} + 21 \beta_{4} + 3 \beta_{7} ) q^{39} + ( 4 \beta_{1} + 2 \beta_{2} - 16 \beta_{4} - 6 \beta_{7} ) q^{41} + ( -3 + \beta_{3} - 4 \beta_{5} + \beta_{6} ) q^{43} + ( 20 \beta_{1} + 10 \beta_{2} + 43 \beta_{4} + 2 \beta_{7} ) q^{45} + ( 41 + \beta_{3} - 3 \beta_{6} ) q^{47} + ( 136 - 4 \beta_{3} - 6 \beta_{5} - \beta_{6} ) q^{49} + ( -57 - 3 \beta_{3} + 10 \beta_{5} + 3 \beta_{6} ) q^{53} + ( 171 + 3 \beta_{3} - 8 \beta_{5} + 7 \beta_{6} ) q^{55} + ( 8 \beta_{2} + 28 \beta_{4} - 4 \beta_{7} ) q^{57} + ( 11 - \beta_{3} + 4 \beta_{5} - \beta_{6} ) q^{59} + ( -16 \beta_{1} - 74 \beta_{2} + 5 \beta_{4} - 6 \beta_{7} ) q^{61} + ( -15 \beta_{1} - 80 \beta_{2} - 70 \beta_{4} + \beta_{7} ) q^{63} + ( -20 \beta_{1} - 78 \beta_{2} + 16 \beta_{4} + 10 \beta_{7} ) q^{65} + ( -73 - 3 \beta_{3} + 4 \beta_{5} - 9 \beta_{6} ) q^{67} + ( -25 - 8 \beta_{5} - 3 \beta_{6} ) q^{69} + ( -29 \beta_{1} + 31 \beta_{2} - 27 \beta_{4} + 6 \beta_{7} ) q^{71} + ( -20 \beta_{1} + 6 \beta_{2} + 26 \beta_{4} - 6 \beta_{7} ) q^{73} + ( -8 \beta_{1} - 151 \beta_{2} + 12 \beta_{4} + 12 \beta_{7} ) q^{75} + ( -207 - 4 \beta_{3} + 2 \beta_{5} - 3 \beta_{6} ) q^{77} + ( -5 \beta_{1} - 64 \beta_{2} - 16 \beta_{4} - 11 \beta_{7} ) q^{79} + ( 391 - 3 \beta_{3} - 7 \beta_{5} - 3 \beta_{6} ) q^{81} + ( -297 - \beta_{3} + 16 \beta_{5} - 9 \beta_{6} ) q^{83} + ( 819 - 5 \beta_{3} - 24 \beta_{5} - \beta_{6} ) q^{87} + ( 19 + 4 \beta_{3} + 18 \beta_{5} + \beta_{6} ) q^{89} + ( -20 \beta_{1} + 139 \beta_{2} + 21 \beta_{4} + \beta_{7} ) q^{91} + ( 505 + 4 \beta_{3} - 10 \beta_{5} - 7 \beta_{6} ) q^{93} + ( 32 \beta_{1} - 120 \beta_{2} - 12 \beta_{4} - 8 \beta_{7} ) q^{95} + ( 24 \beta_{1} - 140 \beta_{2} + 22 \beta_{4} - 16 \beta_{7} ) q^{97} + ( 10 \beta_{1} - 183 \beta_{2} + 70 \beta_{4} - 4 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 132 q^{9} + O(q^{10}) \) \( 8 q + 132 q^{9} + 44 q^{13} - 24 q^{15} - 48 q^{19} + 308 q^{21} + 520 q^{25} + 812 q^{33} - 1064 q^{35} - 8 q^{43} + 312 q^{47} + 1124 q^{49} - 472 q^{53} + 1416 q^{55} + 72 q^{59} - 624 q^{67} - 180 q^{69} - 1660 q^{77} + 3156 q^{81} - 2472 q^{83} + 6664 q^{87} + 68 q^{89} + 4036 q^{93} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 4 \nu \)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{7} - 284 \nu^{5} + 2159 \nu^{3} - 3402 \nu \)\()/172\)
\(\beta_{3}\)\(=\)\((\)\( 35 \nu^{6} - 3127 \nu^{4} + 8318 \nu^{2} + 12168 \)\()/172\)
\(\beta_{4}\)\(=\)\((\)\( -59 \nu^{7} + 5485 \nu^{5} - 33588 \nu^{3} + 47900 \nu \)\()/1032\)
\(\beta_{5}\)\(=\)\((\)\( 41 \nu^{6} - 3781 \nu^{4} + 20634 \nu^{2} - 22844 \)\()/172\)
\(\beta_{6}\)\(=\)\((\)\( -45 \nu^{6} + 4217 \nu^{4} - 28386 \nu^{2} + 35292 \)\()/172\)
\(\beta_{7}\)\(=\)\((\)\( 109 \nu^{7} - 9989 \nu^{5} + 48702 \nu^{3} - 15160 \nu \)\()/1032\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)\(/4\)
\(\nu^{2}\)\(=\)\((\)\(3 \beta_{6} + 5 \beta_{5} - 2 \beta_{3} + 190\)\()/8\)
\(\nu^{3}\)\(=\)\((\)\(-7 \beta_{7} - 23 \beta_{4} - 33 \beta_{2} + 78 \beta_{1}\)\()/4\)
\(\nu^{4}\)\(=\)\((\)\(277 \beta_{6} + 450 \beta_{5} - 171 \beta_{3} + 15027\)\()/8\)
\(\nu^{5}\)\(=\)\((\)\(-619 \beta_{7} - 2075 \beta_{4} - 3053 \beta_{2} + 6708 \beta_{1}\)\()/4\)
\(\nu^{6}\)\(=\)\((\)\(24035 \beta_{6} + 39016 \beta_{5} - 14763 \beta_{3} + 1294619\)\()/8\)
\(\nu^{7}\)\(=\)\((\)\(-53561 \beta_{7} - 179881 \beta_{4} - 265039 \beta_{2} + 580024 \beta_{1}\)\()/4\)

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 \( T_{3}^{8} - 174 T_{3}^{6} + 8760 T_{3}^{4} - 106624 T_{3}^{2} + 373248 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2312))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( 373248 - 106624 T^{2} + 8760 T^{4} - 174 T^{6} + T^{8} \)
$5$ \( 286466048 - 16028160 T^{2} + 187712 T^{4} - 760 T^{6} + T^{8} \)
$7$ \( 2424307712 - 140939264 T^{2} + 1001912 T^{4} - 1934 T^{6} + T^{8} \)
$11$ \( 1063158272 - 764940416 T^{2} + 13918712 T^{4} - 7406 T^{6} + T^{8} \)
$13$ \( ( 8525216 + 70680 T - 5836 T^{2} - 22 T^{3} + T^{4} )^{2} \)
$17$ \( T^{8} \)
$19$ \( ( 44946176 - 382336 T - 21056 T^{2} + 24 T^{3} + T^{4} )^{2} \)
$23$ \( 2935871578112 - 126066546688 T^{2} + 134935032 T^{4} - 36734 T^{6} + T^{8} \)
$29$ \( 147297439325233152 - 36222945874432 T^{2} + 3126815040 T^{4} - 106232 T^{6} + T^{8} \)
$31$ \( 321310420423933952 - 83917512114176 T^{2} + 6678611960 T^{4} - 155918 T^{6} + T^{8} \)
$37$ \( 548861080203763712 - 175406389281280 T^{2} + 9074395456 T^{4} - 166072 T^{6} + T^{8} \)
$41$ \( 55409237072210296832 - 3227605845606400 T^{2} + 58679507968 T^{4} - 419392 T^{6} + T^{8} \)
$43$ \( ( 112195072 - 4732224 T - 60592 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$47$ \( ( 134217728 + 3407872 T - 57344 T^{2} - 156 T^{3} + T^{4} )^{2} \)
$53$ \( ( -14761769616 - 165833136 T - 400992 T^{2} + 236 T^{3} + T^{4} )^{2} \)
$59$ \( ( 70465536 + 5698880 T - 60112 T^{2} - 36 T^{3} + T^{4} )^{2} \)
$61$ \( \)\(18\!\cdots\!28\)\( - 216217434642342400 T^{2} + 907789046080 T^{4} - 1597240 T^{6} + T^{8} \)
$67$ \( ( 30967766784 + 70501504 T - 663552 T^{2} + 312 T^{3} + T^{4} )^{2} \)
$71$ \( \)\(14\!\cdots\!28\)\( - 195987755820328960 T^{2} + 887074814648 T^{4} - 1607086 T^{6} + T^{8} \)
$73$ \( \)\(21\!\cdots\!68\)\( - 84967502970355712 T^{2} + 772622417920 T^{4} - 1779648 T^{6} + T^{8} \)
$79$ \( 17195215357908099072 - 8819538685493248 T^{2} + 628706103544 T^{4} - 1578638 T^{6} + T^{8} \)
$83$ \( ( -54225864704 - 540066368 T - 567888 T^{2} + 1236 T^{3} + T^{4} )^{2} \)
$89$ \( ( 21605388512 + 368883272 T - 1202908 T^{2} - 34 T^{3} + T^{4} )^{2} \)
$97$ \( \)\(13\!\cdots\!92\)\( - 6103257558949298176 T^{2} + 9052262153216 T^{4} - 5185024 T^{6} + T^{8} \)
show more
show less