Properties

Label 9200.2.a.de
Level $9200$
Weight $2$
Character orbit 9200.a
Self dual yes
Analytic conductor $73.462$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9200.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(73.4623698596\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 3 x^{7} - 13 x^{6} + 38 x^{5} + 41 x^{4} - 123 x^{3} + 15 x^{2} + 32 x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 920)
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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{3} + ( 1 + \beta_{7} ) q^{7} + ( 2 + \beta_{3} + \beta_{5} + \beta_{7} ) q^{9} +O(q^{10})\) \( q + \beta_{1} q^{3} + ( 1 + \beta_{7} ) q^{7} + ( 2 + \beta_{3} + \beta_{5} + \beta_{7} ) q^{9} + ( -1 - \beta_{3} ) q^{11} + ( 1 + \beta_{2} ) q^{13} + ( -1 + \beta_{4} + \beta_{6} ) q^{17} + ( -1 + \beta_{4} + \beta_{7} ) q^{19} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{21} - q^{23} + ( 1 + \beta_{1} + \beta_{3} + \beta_{6} + \beta_{7} ) q^{27} + ( 3 - \beta_{1} + \beta_{4} + \beta_{6} ) q^{29} + ( -2 + \beta_{1} - \beta_{3} - \beta_{4} - \beta_{7} ) q^{31} + ( -1 - 2 \beta_{1} - 2 \beta_{4} - \beta_{7} ) q^{33} + ( 2 - \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{39} + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{41} + ( 3 + \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{43} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{47} + ( 5 + \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} ) q^{49} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{6} ) q^{51} + ( 3 - 3 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{53} + ( -2 + 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{57} + ( -5 + \beta_{1} - \beta_{4} - \beta_{5} ) q^{59} + ( 3 - \beta_{2} + \beta_{5} + \beta_{7} ) q^{61} + ( 6 + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{63} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{67} -\beta_{1} q^{69} + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{71} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{73} + ( 4 - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 4 \beta_{6} ) q^{77} + ( 2 \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{79} + ( 3 \beta_{1} + \beta_{2} - 2 \beta_{5} + 2 \beta_{6} ) q^{81} + ( 4 + \beta_{4} + 2 \beta_{6} ) q^{83} + ( -3 + 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{87} + ( 4 + \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{89} + ( 4 + 3 \beta_{1} + \beta_{2} - 2 \beta_{6} + 3 \beta_{7} ) q^{91} + ( 6 - 4 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} ) q^{93} + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{97} + ( -5 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{3} + 7 q^{7} + 11 q^{9} + O(q^{10}) \) \( 8 q + 3 q^{3} + 7 q^{7} + 11 q^{9} - 7 q^{11} + 11 q^{13} - 7 q^{17} - 11 q^{19} - 8 q^{23} + 12 q^{27} + 22 q^{29} - 9 q^{31} - 9 q^{33} + 4 q^{37} + 7 q^{41} + 22 q^{43} + 4 q^{47} + 39 q^{49} + 19 q^{51} + 4 q^{53} - 32 q^{59} + 17 q^{61} + 44 q^{63} - 4 q^{67} - 3 q^{69} - 15 q^{71} + 6 q^{73} + 18 q^{77} + 2 q^{79} + 24 q^{81} + 36 q^{83} - 4 q^{87} + 46 q^{89} + 35 q^{91} + 20 q^{93} + 3 q^{97} - 61 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} - 13 x^{6} + 38 x^{5} + 41 x^{4} - 123 x^{3} + 15 x^{2} + 32 x - 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 7 \nu^{2} + 11 \nu + 1 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{6} - 4 \nu^{5} - 5 \nu^{4} + 29 \nu^{3} - 8 \nu^{2} - 21 \nu - 2 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} - 3 \nu^{6} - 11 \nu^{5} + 30 \nu^{4} + 33 \nu^{3} - 69 \nu^{2} - 17 \nu + 12 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 3 \nu^{6} - 13 \nu^{5} + 36 \nu^{4} + 45 \nu^{3} - 107 \nu^{2} - 9 \nu + 18 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 3 \nu^{6} - 13 \nu^{5} + 36 \nu^{4} + 47 \nu^{3} - 109 \nu^{2} - 23 \nu + 26 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} + 17 \nu^{5} - 31 \nu^{4} - 74 \nu^{3} + 117 \nu^{2} + 30 \nu - 26 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} + \beta_{5} + \beta_{3} + 5\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{3} + 7 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(9 \beta_{7} + 2 \beta_{6} + 7 \beta_{5} + 9 \beta_{3} + \beta_{2} + 3 \beta_{1} + 36\)
\(\nu^{5}\)\(=\)\(14 \beta_{7} + 12 \beta_{6} + \beta_{5} + \beta_{4} + 14 \beta_{3} + 3 \beta_{2} + 55 \beta_{1} + 22\)
\(\nu^{6}\)\(=\)\(80 \beta_{7} + 29 \beta_{6} + 47 \beta_{5} + 4 \beta_{4} + 82 \beta_{3} + 17 \beta_{2} + 53 \beta_{1} + 281\)
\(\nu^{7}\)\(=\)\(160 \beta_{7} + 126 \beta_{6} + 11 \beta_{5} + 25 \beta_{4} + 166 \beta_{3} + 54 \beta_{2} + 460 \beta_{1} + 305\)

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
−2.54092
−2.51561
−0.540724
0.296848
0.493532
1.69755
2.89996
3.20935
0 −2.54092 0 0 0 −0.780573 0 3.45626 0
1.2 0 −2.51561 0 0 0 4.64022 0 3.32827 0
1.3 0 −0.540724 0 0 0 1.15693 0 −2.70762 0
1.4 0 0.296848 0 0 0 −3.46037 0 −2.91188 0
1.5 0 0.493532 0 0 0 4.54439 0 −2.75643 0
1.6 0 1.69755 0 0 0 −4.22860 0 −0.118308 0
1.7 0 2.89996 0 0 0 0.580879 0 5.40975 0
1.8 0 3.20935 0 0 0 4.54713 0 7.29996 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
\(2\) \(1\)
\(5\) \(-1\)
\(23\) \(1\)

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 9200.2.a.de 8
4.b odd 2 1 4600.2.a.bj 8
5.b even 2 1 9200.2.a.dd 8
5.c odd 4 2 1840.2.e.h 16
20.d odd 2 1 4600.2.a.bk 8
20.e even 4 2 920.2.e.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.e.c 16 20.e even 4 2
1840.2.e.h 16 5.c odd 4 2
4600.2.a.bj 8 4.b odd 2 1
4600.2.a.bk 8 20.d odd 2 1
9200.2.a.dd 8 5.b even 2 1
9200.2.a.de 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9200))\):

\(T_{3}^{8} - \cdots\)
\(T_{7}^{8} - \cdots\)
\(T_{11}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( -8 + 32 T + 15 T^{2} - 123 T^{3} + 41 T^{4} + 38 T^{5} - 13 T^{6} - 3 T^{7} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( -736 + 1056 T + 1316 T^{2} - 1720 T^{3} + 20 T^{4} + 218 T^{5} - 23 T^{6} - 7 T^{7} + T^{8} \)
$11$ \( 440 - 5456 T + 28 T^{2} + 2364 T^{3} + 162 T^{4} - 248 T^{5} - 29 T^{6} + 7 T^{7} + T^{8} \)
$13$ \( -14020 - 1696 T + 11569 T^{2} - 2883 T^{3} - 1115 T^{4} + 438 T^{5} - 7 T^{6} - 11 T^{7} + T^{8} \)
$17$ \( 176 - 64 T - 2700 T^{2} + 2292 T^{3} + 706 T^{4} - 296 T^{5} - 49 T^{6} + 7 T^{7} + T^{8} \)
$19$ \( 11192 + 23600 T + 16488 T^{2} + 3044 T^{3} - 1116 T^{4} - 432 T^{5} - 7 T^{6} + 11 T^{7} + T^{8} \)
$23$ \( ( 1 + T )^{8} \)
$29$ \( -400 - 4632 T - 344 T^{2} + 5614 T^{3} - 2107 T^{4} - 96 T^{5} + 146 T^{6} - 22 T^{7} + T^{8} \)
$31$ \( -287276 - 183760 T + 5147 T^{2} + 20781 T^{3} + 1671 T^{4} - 766 T^{5} - 79 T^{6} + 9 T^{7} + T^{8} \)
$37$ \( 16096 - 3792 T - 22120 T^{2} - 112 T^{3} + 3340 T^{4} + 212 T^{5} - 106 T^{6} - 4 T^{7} + T^{8} \)
$41$ \( 1197584 + 601624 T - 181313 T^{2} - 44475 T^{3} + 8683 T^{4} + 1026 T^{5} - 163 T^{6} - 7 T^{7} + T^{8} \)
$43$ \( -244448 - 74096 T + 72296 T^{2} + 4616 T^{3} - 6976 T^{4} + 716 T^{5} + 108 T^{6} - 22 T^{7} + T^{8} \)
$47$ \( -73520 - 138952 T - 20048 T^{2} + 27562 T^{3} + 8173 T^{4} - 34 T^{5} - 166 T^{6} - 4 T^{7} + T^{8} \)
$53$ \( 202880 - 254336 T - 228000 T^{2} - 13376 T^{3} + 12488 T^{4} + 568 T^{5} - 206 T^{6} - 4 T^{7} + T^{8} \)
$59$ \( 29696 + 1536 T - 24896 T^{2} - 9024 T^{3} + 2464 T^{4} + 1872 T^{5} + 376 T^{6} + 32 T^{7} + T^{8} \)
$61$ \( 200 + 320 T - 592 T^{2} - 1276 T^{3} - 400 T^{4} + 208 T^{5} + 49 T^{6} - 17 T^{7} + T^{8} \)
$67$ \( 680608 - 115216 T - 135192 T^{2} + 13376 T^{3} + 8428 T^{4} - 388 T^{5} - 174 T^{6} + 4 T^{7} + T^{8} \)
$71$ \( -8900000 - 4219360 T + 126317 T^{2} + 242207 T^{3} + 10847 T^{4} - 3982 T^{5} - 253 T^{6} + 15 T^{7} + T^{8} \)
$73$ \( -557680 - 780576 T - 245128 T^{2} + 28224 T^{3} + 16801 T^{4} + 298 T^{5} - 258 T^{6} - 6 T^{7} + T^{8} \)
$79$ \( -5248 + 34496 T - 13136 T^{2} - 11992 T^{3} + 4108 T^{4} + 984 T^{5} - 206 T^{6} - 2 T^{7} + T^{8} \)
$83$ \( -4183520 - 553168 T + 505448 T^{2} + 25120 T^{3} - 22872 T^{4} + 708 T^{5} + 352 T^{6} - 36 T^{7} + T^{8} \)
$89$ \( 12804160 - 4496608 T - 440608 T^{2} + 349256 T^{3} - 36348 T^{4} - 2600 T^{5} + 710 T^{6} - 46 T^{7} + T^{8} \)
$97$ \( 2960 + 3344 T - 8172 T^{2} + 1216 T^{3} + 1712 T^{4} - 174 T^{5} - 123 T^{6} - 3 T^{7} + T^{8} \)
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