Properties

Label 25.32.b.a
Level $25$
Weight $32$
Character orbit 25.b
Analytic conductor $152.193$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 32 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(152.192832048\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
Defining polynomial: \(x^{4} + 9147745 x^{2} + 20920305072384\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 1)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1998 \beta_{1} + \beta_{2} ) q^{2} + ( -868158 \beta_{1} - 432 \beta_{2} ) q^{3} + ( -886267168 + 3996 \beta_{3} ) q^{4} + ( 1311583748112 - 1731294 \beta_{3} ) q^{6} + ( 1512876378860 \beta_{1} - 71928864 \beta_{2} ) q^{7} + ( -8007752935296 \beta_{1} + 462815680 \beta_{2} ) q^{8} + ( 50633228151963 + 750088512 \beta_{3} ) q^{9} +O(q^{10})\) \( q +(1998 \beta_{1} + \beta_{2}) q^{2} +(-868158 \beta_{1} - 432 \beta_{2}) q^{3} +(-886267168 + 3996 \beta_{3}) q^{4} +(1311583748112 - 1731294 \beta_{3}) q^{6} +(1512876378860 \beta_{1} - 71928864 \beta_{2}) q^{7} +(-8007752935296 \beta_{1} + 462815680 \beta_{2}) q^{8} +(50633228151963 + 750088512 \beta_{3}) q^{9} +(-3891176872559388 - 2900090920 \beta_{3}) q^{11} +(5317370547765696 \beta_{1} + 729783353376 \beta_{2}) q^{12} +(-3735447652513031 \beta_{1} + 4661016429888 \beta_{2}) q^{13} +(-112772481922620576 + 1369162508588 \beta_{3}) q^{14} +(-1522606456842450944 + 1498297450752 \beta_{3}) q^{16} +(861230391449354469 \beta_{1} + 45085348093056 \beta_{2}) q^{17} +(-1874980811478798918 \beta_{1} - 99234456545637 \beta_{2}) q^{18} +(6185281664011082020 - 15780579276456 \beta_{3}) q^{19} +(49477478708035580832 - 591116976955008 \beta_{3}) q^{21} +(-134135651649834504 \beta_{1} - 3311738706743388 \beta_{2}) q^{22} +(-94867242099495560964 \beta_{1} + 18557618179251808 \beta_{2}) q^{23} +(-\)\(16\!\cdots\!40\)\( + 3057552132930432 \beta_{3}) q^{24} +(-\)\(11\!\cdots\!08\)\( + 5577263174403193 \beta_{3}) q^{26} +(\)\(27\!\cdots\!64\)\( \beta_{1} - 223588927516223520 \beta_{2}) q^{27} +(-\)\(58\!\cdots\!76\)\( \beta_{1} - 540797210397718848 \beta_{2}) q^{28} +(-\)\(64\!\cdots\!70\)\( + 23419235738444896 \beta_{3}) q^{29} +(\)\(62\!\cdots\!32\)\( - 241262117102036160 \beta_{3}) q^{31} +(-\)\(24\!\cdots\!52\)\( \beta_{1} - 828077182664699904 \beta_{2}) q^{32} +(77488091766721752264 \beta_{1} + 1429214695653119616 \beta_{2}) q^{33} +(-\)\(29\!\cdots\!96\)\( + 951310916939280357 \beta_{3}) q^{34} +(\)\(74\!\cdots\!16\)\( - 462448441584329868 \beta_{3}) q^{36} +(-\)\(41\!\cdots\!53\)\( \beta_{1} + 32224244113578511296 \beta_{2}) q^{37} +(\)\(53\!\cdots\!56\)\( \beta_{1} + 9338241403446990820 \beta_{2}) q^{38} +(\)\(49\!\cdots\!56\)\( - 2432785315853076912 \beta_{3}) q^{39} +(\)\(43\!\cdots\!42\)\( - 18425545314484703360 \beta_{3}) q^{41} +(\)\(16\!\cdots\!64\)\( \beta_{1} + \)\(16\!\cdots\!32\)\( \beta_{2}) q^{42} +(\)\(91\!\cdots\!30\)\( \beta_{1} + \)\(34\!\cdots\!48\)\( \beta_{2}) q^{43} +(\)\(39\!\cdots\!84\)\( - 12978887416136399888 \beta_{3}) q^{44} +(-\)\(29\!\cdots\!28\)\( - 57789120977350448580 \beta_{3}) q^{46} +(\)\(47\!\cdots\!16\)\( \beta_{1} + \)\(51\!\cdots\!16\)\( \beta_{2}) q^{47} +(\)\(30\!\cdots\!76\)\( \beta_{1} + \)\(78\!\cdots\!08\)\( \beta_{2}) q^{48} +(-\)\(84\!\cdots\!93\)\( - \)\(21\!\cdots\!80\)\( \beta_{3}) q^{49} +(\)\(12\!\cdots\!72\)\( - \)\(41\!\cdots\!56\)\( \beta_{3}) q^{51} +(-\)\(45\!\cdots\!60\)\( \beta_{1} - \)\(26\!\cdots\!84\)\( \beta_{2}) q^{52} +(-\)\(97\!\cdots\!43\)\( \beta_{1} + \)\(12\!\cdots\!68\)\( \beta_{2}) q^{53} +(\)\(53\!\cdots\!20\)\( - \)\(17\!\cdots\!96\)\( \beta_{3}) q^{54} +(\)\(12\!\cdots\!20\)\( + \)\(12\!\cdots\!44\)\( \beta_{3}) q^{56} +(-\)\(23\!\cdots\!32\)\( \beta_{1} - \)\(40\!\cdots\!40\)\( \beta_{2}) q^{57} +(-\)\(19\!\cdots\!96\)\( \beta_{1} - \)\(68\!\cdots\!70\)\( \beta_{2}) q^{58} +(\)\(99\!\cdots\!60\)\( - \)\(44\!\cdots\!48\)\( \beta_{3}) q^{59} +(-\)\(60\!\cdots\!38\)\( - \)\(11\!\cdots\!00\)\( \beta_{3}) q^{61} +(\)\(76\!\cdots\!96\)\( \beta_{1} + \)\(11\!\cdots\!32\)\( \beta_{2}) q^{62} +(\)\(21\!\cdots\!68\)\( \beta_{1} - \)\(11\!\cdots\!32\)\( \beta_{2}) q^{63} +(\)\(37\!\cdots\!52\)\( - \)\(22\!\cdots\!48\)\( \beta_{3}) q^{64} +(-\)\(37\!\cdots\!56\)\( + \)\(29\!\cdots\!32\)\( \beta_{3}) q^{66} +(-\)\(48\!\cdots\!86\)\( \beta_{1} - \)\(88\!\cdots\!44\)\( \beta_{2}) q^{67} +(-\)\(12\!\cdots\!08\)\( \beta_{1} - \)\(38\!\cdots\!08\)\( \beta_{2}) q^{68} +(\)\(12\!\cdots\!96\)\( + \)\(24\!\cdots\!84\)\( \beta_{3}) q^{69} +(\)\(27\!\cdots\!72\)\( + \)\(95\!\cdots\!00\)\( \beta_{3}) q^{71} +(-\)\(13\!\cdots\!08\)\( \beta_{1} + \)\(62\!\cdots\!40\)\( \beta_{2}) q^{72} +(-\)\(31\!\cdots\!69\)\( \beta_{1} - \)\(15\!\cdots\!92\)\( \beta_{2}) q^{73} +(-\)\(76\!\cdots\!36\)\( + \)\(22\!\cdots\!55\)\( \beta_{3}) q^{74} +(-\)\(22\!\cdots\!60\)\( + \)\(38\!\cdots\!28\)\( \beta_{3}) q^{76} +(-\)\(64\!\cdots\!60\)\( \beta_{1} + \)\(71\!\cdots\!32\)\( \beta_{2}) q^{77} +(\)\(16\!\cdots\!80\)\( \beta_{1} + \)\(54\!\cdots\!56\)\( \beta_{2}) q^{78} +(\)\(59\!\cdots\!80\)\( + \)\(46\!\cdots\!96\)\( \beta_{3}) q^{79} +(-\)\(19\!\cdots\!79\)\( + \)\(53\!\cdots\!76\)\( \beta_{3}) q^{81} +(\)\(57\!\cdots\!76\)\( \beta_{1} + \)\(80\!\cdots\!42\)\( \beta_{2}) q^{82} +(\)\(13\!\cdots\!18\)\( \beta_{1} + \)\(62\!\cdots\!28\)\( \beta_{2}) q^{83} +(-\)\(66\!\cdots\!76\)\( + \)\(72\!\cdots\!16\)\( \beta_{3}) q^{84} +(-\)\(10\!\cdots\!68\)\( + \)\(16\!\cdots\!34\)\( \beta_{3}) q^{86} +(\)\(82\!\cdots\!12\)\( \beta_{1} + \)\(29\!\cdots\!40\)\( \beta_{2}) q^{87} +(\)\(34\!\cdots\!48\)\( \beta_{1} - \)\(41\!\cdots\!40\)\( \beta_{2}) q^{88} +(\)\(10\!\cdots\!90\)\( + \)\(26\!\cdots\!88\)\( \beta_{3}) q^{89} +(\)\(14\!\cdots\!12\)\( + \)\(73\!\cdots\!64\)\( \beta_{3}) q^{91} +(-\)\(11\!\cdots\!36\)\( \beta_{1} + \)\(21\!\cdots\!56\)\( \beta_{2}) q^{92} +(-\)\(32\!\cdots\!76\)\( \beta_{1} - \)\(48\!\cdots\!24\)\( \beta_{2}) q^{93} +(-\)\(23\!\cdots\!56\)\( + \)\(57\!\cdots\!84\)\( \beta_{3}) q^{94} +(-\)\(30\!\cdots\!48\)\( + \)\(11\!\cdots\!96\)\( \beta_{3}) q^{96} +(-\)\(45\!\cdots\!59\)\( \beta_{1} - \)\(16\!\cdots\!84\)\( \beta_{2}) q^{97} +(\)\(40\!\cdots\!66\)\( \beta_{1} - \)\(41\!\cdots\!93\)\( \beta_{2}) q^{98} +(-\)\(77\!\cdots\!44\)\( - \)\(30\!\cdots\!16\)\( \beta_{3}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 3545068672q^{4} + 5246334992448q^{6} + 202532912607852q^{9} + O(q^{10}) \) \( 4q - 3545068672q^{4} + 5246334992448q^{6} + 202532912607852q^{9} - 15564707490237552q^{11} - 451089927690482304q^{14} - 6090425827369803776q^{16} + 24741126656044328080q^{19} + \)\(19\!\cdots\!28\)\(q^{21} - \)\(67\!\cdots\!60\)\(q^{24} - \)\(46\!\cdots\!32\)\(q^{26} - \)\(25\!\cdots\!80\)\(q^{29} + \)\(25\!\cdots\!28\)\(q^{31} - \)\(11\!\cdots\!84\)\(q^{34} + \)\(29\!\cdots\!64\)\(q^{36} + \)\(19\!\cdots\!24\)\(q^{39} + \)\(17\!\cdots\!68\)\(q^{41} + \)\(15\!\cdots\!36\)\(q^{44} - \)\(11\!\cdots\!12\)\(q^{46} - \)\(33\!\cdots\!72\)\(q^{49} + \)\(50\!\cdots\!88\)\(q^{51} + \)\(21\!\cdots\!80\)\(q^{54} + \)\(51\!\cdots\!80\)\(q^{56} + \)\(39\!\cdots\!40\)\(q^{59} - \)\(24\!\cdots\!52\)\(q^{61} + \)\(14\!\cdots\!08\)\(q^{64} - \)\(15\!\cdots\!24\)\(q^{66} + \)\(51\!\cdots\!84\)\(q^{69} + \)\(11\!\cdots\!88\)\(q^{71} - \)\(30\!\cdots\!44\)\(q^{74} - \)\(88\!\cdots\!40\)\(q^{76} + \)\(23\!\cdots\!20\)\(q^{79} - \)\(79\!\cdots\!16\)\(q^{81} - \)\(26\!\cdots\!04\)\(q^{84} - \)\(43\!\cdots\!72\)\(q^{86} + \)\(43\!\cdots\!60\)\(q^{89} + \)\(57\!\cdots\!48\)\(q^{91} - \)\(92\!\cdots\!24\)\(q^{94} - \)\(12\!\cdots\!92\)\(q^{96} - \)\(30\!\cdots\!76\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9147745 x^{2} + 20920305072384\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -5 \nu^{3} - 22869365 \nu \)\()/2286936\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 13721617 \nu \)\()/381156\)
\(\beta_{3}\)\(=\)\( 240 \nu^{2} + 1097729400 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(5 \beta_{2} + 6 \beta_{1}\)\()/120\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 1097729400\)\()/240\)
\(\nu^{3}\)\(=\)\((\)\(-22869365 \beta_{2} - 82329702 \beta_{1}\)\()/120\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
2139.16i
2138.16i
2138.16i
2139.16i
71307.9i 3.08552e7i −2.93733e9 0 2.20022e12 1.14368e13i 5.63222e13i −3.34371e14 0
24.2 31347.9i 1.34921e7i 1.16479e9 0 4.22947e11 1.88207e13i 1.03833e14i 4.35638e14 0
24.3 31347.9i 1.34921e7i 1.16479e9 0 4.22947e11 1.88207e13i 1.03833e14i 4.35638e14 0
24.4 71307.9i 3.08552e7i −2.93733e9 0 2.20022e12 1.14368e13i 5.63222e13i −3.34371e14 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.32.b.a 4
5.b even 2 1 inner 25.32.b.a 4
5.c odd 4 1 1.32.a.a 2
5.c odd 4 1 25.32.a.a 2
15.e even 4 1 9.32.a.a 2
20.e even 4 1 16.32.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.32.a.a 2 5.c odd 4 1
9.32.a.a 2 15.e even 4 1
16.32.a.b 2 20.e even 4 1
25.32.a.a 2 5.c odd 4 1
25.32.b.a 4 1.a even 1 1 trivial
25.32.b.a 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 6067501632 T_{2}^{2} + \)\(49\!\cdots\!56\)\( \) acting on \(S_{32}^{\mathrm{new}}(25, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 4996789694031200256 + 6067501632 T^{2} + T^{4} \)
$3$ \( \)\(17\!\cdots\!56\)\( + 1134080336263968 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( \)\(46\!\cdots\!96\)\( + \)\(48\!\cdots\!72\)\( T^{2} + T^{4} \)
$11$ \( ( \)\(12\!\cdots\!44\)\( + 7782353745118776 T + T^{2} )^{2} \)
$13$ \( \)\(31\!\cdots\!16\)\( + \)\(11\!\cdots\!08\)\( T^{2} + T^{4} \)
$17$ \( \)\(47\!\cdots\!76\)\( + \)\(15\!\cdots\!52\)\( T^{2} + T^{4} \)
$19$ \( ( -\)\(27\!\cdots\!00\)\( - 12370563328022164040 T + T^{2} )^{2} \)
$23$ \( \)\(53\!\cdots\!76\)\( + \)\(36\!\cdots\!48\)\( T^{2} + T^{4} \)
$29$ \( ( \)\(39\!\cdots\!00\)\( + \)\(12\!\cdots\!40\)\( T + T^{2} )^{2} \)
$31$ \( ( -\)\(11\!\cdots\!76\)\( - \)\(12\!\cdots\!64\)\( T + T^{2} )^{2} \)
$37$ \( \)\(65\!\cdots\!36\)\( + \)\(58\!\cdots\!12\)\( T^{2} + T^{4} \)
$41$ \( ( -\)\(70\!\cdots\!36\)\( - \)\(87\!\cdots\!84\)\( T + T^{2} )^{2} \)
$43$ \( \)\(50\!\cdots\!96\)\( + \)\(78\!\cdots\!28\)\( T^{2} + T^{4} \)
$47$ \( \)\(25\!\cdots\!16\)\( + \)\(59\!\cdots\!92\)\( T^{2} + T^{4} \)
$53$ \( \)\(32\!\cdots\!56\)\( + \)\(26\!\cdots\!68\)\( T^{2} + T^{4} \)
$59$ \( ( -\)\(51\!\cdots\!00\)\( - \)\(19\!\cdots\!20\)\( T + T^{2} )^{2} \)
$61$ \( ( \)\(36\!\cdots\!44\)\( + \)\(12\!\cdots\!76\)\( T + T^{2} )^{2} \)
$67$ \( \)\(68\!\cdots\!76\)\( + \)\(88\!\cdots\!52\)\( T^{2} + T^{4} \)
$71$ \( ( -\)\(16\!\cdots\!16\)\( - \)\(55\!\cdots\!44\)\( T + T^{2} )^{2} \)
$73$ \( \)\(81\!\cdots\!76\)\( + \)\(20\!\cdots\!48\)\( T^{2} + T^{4} \)
$79$ \( ( -\)\(53\!\cdots\!00\)\( - \)\(11\!\cdots\!60\)\( T + T^{2} )^{2} \)
$83$ \( \)\(74\!\cdots\!36\)\( + \)\(24\!\cdots\!88\)\( T^{2} + T^{4} \)
$89$ \( ( -\)\(62\!\cdots\!00\)\( - \)\(21\!\cdots\!80\)\( T + T^{2} )^{2} \)
$97$ \( \)\(39\!\cdots\!16\)\( + \)\(42\!\cdots\!92\)\( T^{2} + T^{4} \)
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