Properties

Label 25.38.b.a
Level 25
Weight 38
Character orbit 25.b
Analytic conductor 216.785
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 38 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(216.785095312\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
Defining polynomial: \(x^{4} + 31868761 x^{2} + 253904465984400\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2}\cdot 5^{4} \)
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.

\(f(q)\) \(=\) \( q + ( \beta_{1} - 1944 \beta_{2} ) q^{2} + ( 72 \beta_{1} - 139914 \beta_{2} ) q^{3} + ( -18860134912 + 3888 \beta_{3} ) q^{4} + ( -11253271923648 + 279882 \beta_{3} ) q^{6} + ( 9650004336 \beta_{1} - 34484439534860 \beta_{2} ) q^{7} + ( 99683138560 \beta_{1} + 340440430436352 \beta_{2} ) q^{8} + ( 449473689200884707 + 20147616 \beta_{3} ) q^{9} +O(q^{10})\) \( q +(\beta_{1} - 1944 \beta_{2}) q^{2} +(72 \beta_{1} - 139914 \beta_{2}) q^{3} +(-18860134912 + 3888 \beta_{3}) q^{4} +(-11253271923648 + 279882 \beta_{3}) q^{6} +(9650004336 \beta_{1} - 34484439534860 \beta_{2}) q^{7} +(99683138560 \beta_{1} + 340440430436352 \beta_{2}) q^{8} +(449473689200884707 + 20147616 \beta_{3}) q^{9} +(-13367018177424269028 - 643726129420 \beta_{3}) q^{11} +(-2717893793664 \beta_{1} + 43747747983700992 \beta_{2}) q^{12} +(-724443400725408 \beta_{1} - 5305817416539331783 \beta_{2}) q^{13} +(-\)\(15\!\cdots\!24\)\( + 53244047964044 \beta_{3}) q^{14} +(-\)\(15\!\cdots\!04\)\( + 387706242023424 \beta_{3}) q^{16} +(24633489938571456 \beta_{1} - \)\(89\!\cdots\!27\)\( \beta_{2}) q^{17} +(449375771787124707 \beta_{1} - \)\(87\!\cdots\!64\)\( \beta_{2}) q^{18} +(-\)\(18\!\cdots\!60\)\( - 25859523645743148 \beta_{3}) q^{19} +(-\)\(11\!\cdots\!28\)\( + 3833050353177024 \beta_{3}) q^{21} +(-10238509188443069028 \beta_{1} - \)\(68\!\cdots\!48\)\( \beta_{2}) q^{22} +(-34024622863285905488 \beta_{1} + \)\(26\!\cdots\!28\)\( \beta_{2}) q^{23} +(-\)\(93\!\cdots\!80\)\( - 10564644342933504 \beta_{3}) q^{24} +(\)\(80\!\cdots\!72\)\( + 3897499445529138631 \beta_{3}) q^{26} +(64775499512752949040 \beta_{1} - \)\(12\!\cdots\!12\)\( \beta_{2}) q^{27} +(-\)\(51\!\cdots\!32\)\( \beta_{1} + \)\(61\!\cdots\!32\)\( \beta_{2}) q^{28} +(\)\(63\!\cdots\!10\)\( + 46166393183925012208 \beta_{3}) q^{29} +(\)\(13\!\cdots\!12\)\( + \)\(12\!\cdots\!40\)\( \beta_{3}) q^{31} +(-\)\(37\!\cdots\!84\)\( \beta_{1} + \)\(13\!\cdots\!36\)\( \beta_{2}) q^{32} +(-\)\(73\!\cdots\!16\)\( \beta_{1} - \)\(49\!\cdots\!68\)\( \beta_{2}) q^{33} +(-\)\(79\!\cdots\!04\)\( + \)\(94\!\cdots\!91\)\( \beta_{3}) q^{34} +(-\)\(84\!\cdots\!84\)\( + \)\(17\!\cdots\!24\)\( \beta_{3}) q^{36} +(-\)\(25\!\cdots\!04\)\( \beta_{1} - \)\(68\!\cdots\!81\)\( \beta_{2}) q^{37} +(-\)\(60\!\cdots\!60\)\( \beta_{1} - \)\(34\!\cdots\!92\)\( \beta_{2}) q^{38} +(\)\(58\!\cdots\!84\)\( + \)\(28\!\cdots\!64\)\( \beta_{3}) q^{39} +(-\)\(63\!\cdots\!18\)\( + \)\(11\!\cdots\!40\)\( \beta_{3}) q^{41} +(-\)\(13\!\cdots\!28\)\( \beta_{1} + \)\(78\!\cdots\!48\)\( \beta_{2}) q^{42} +(\)\(42\!\cdots\!52\)\( \beta_{1} + \)\(25\!\cdots\!50\)\( \beta_{2}) q^{43} +(-\)\(66\!\cdots\!64\)\( - \)\(39\!\cdots\!24\)\( \beta_{3}) q^{44} +(\)\(62\!\cdots\!92\)\( - \)\(32\!\cdots\!00\)\( \beta_{3}) q^{46} +(\)\(38\!\cdots\!16\)\( \beta_{1} + \)\(42\!\cdots\!32\)\( \beta_{2}) q^{47} +(-\)\(12\!\cdots\!88\)\( \beta_{1} + \)\(62\!\cdots\!08\)\( \beta_{2}) q^{48} +(\)\(19\!\cdots\!43\)\( + \)\(66\!\cdots\!20\)\( \beta_{3}) q^{49} +(-\)\(57\!\cdots\!88\)\( + \)\(67\!\cdots\!28\)\( \beta_{3}) q^{51} +(-\)\(37\!\cdots\!04\)\( \beta_{1} - \)\(31\!\cdots\!40\)\( \beta_{2}) q^{52} +(\)\(11\!\cdots\!72\)\( \beta_{1} - \)\(15\!\cdots\!39\)\( \beta_{2}) q^{53} +(-\)\(10\!\cdots\!60\)\( + \)\(25\!\cdots\!72\)\( \beta_{3}) q^{54} +(-\)\(11\!\cdots\!40\)\( + \)\(15\!\cdots\!28\)\( \beta_{3}) q^{56} +(-\)\(43\!\cdots\!20\)\( \beta_{1} - \)\(24\!\cdots\!64\)\( \beta_{2}) q^{57} +(\)\(41\!\cdots\!10\)\( \beta_{1} + \)\(55\!\cdots\!32\)\( \beta_{2}) q^{58} +(\)\(11\!\cdots\!20\)\( + \)\(96\!\cdots\!16\)\( \beta_{3}) q^{59} +(\)\(50\!\cdots\!22\)\( + \)\(57\!\cdots\!00\)\( \beta_{3}) q^{61} +(-\)\(49\!\cdots\!88\)\( \beta_{1} + \)\(18\!\cdots\!32\)\( \beta_{2}) q^{62} +(\)\(43\!\cdots\!52\)\( \beta_{1} - \)\(15\!\cdots\!36\)\( \beta_{2}) q^{63} +(-\)\(93\!\cdots\!32\)\( - \)\(88\!\cdots\!04\)\( \beta_{3}) q^{64} +(\)\(84\!\cdots\!44\)\( + \)\(35\!\cdots\!64\)\( \beta_{3}) q^{66} +(\)\(78\!\cdots\!56\)\( \beta_{1} - \)\(10\!\cdots\!62\)\( \beta_{2}) q^{67} +(-\)\(91\!\cdots\!72\)\( \beta_{1} + \)\(30\!\cdots\!76\)\( \beta_{2}) q^{68} +(\)\(45\!\cdots\!24\)\( - \)\(23\!\cdots\!48\)\( \beta_{3}) q^{69} +(-\)\(37\!\cdots\!08\)\( + \)\(34\!\cdots\!00\)\( \beta_{3}) q^{71} +(\)\(44\!\cdots\!20\)\( \beta_{1} + \)\(15\!\cdots\!04\)\( \beta_{2}) q^{72} +(-\)\(11\!\cdots\!88\)\( \beta_{1} - \)\(19\!\cdots\!77\)\( \beta_{2}) q^{73} +(\)\(33\!\cdots\!36\)\( + \)\(19\!\cdots\!05\)\( \beta_{3}) q^{74} +(-\)\(33\!\cdots\!80\)\( - \)\(23\!\cdots\!04\)\( \beta_{3}) q^{76} +(-\)\(73\!\cdots\!08\)\( \beta_{1} - \)\(45\!\cdots\!00\)\( \beta_{2}) q^{77} +(\)\(44\!\cdots\!84\)\( \beta_{1} + \)\(29\!\cdots\!80\)\( \beta_{2}) q^{78} +(-\)\(13\!\cdots\!40\)\( + \)\(81\!\cdots\!48\)\( \beta_{3}) q^{79} +(\)\(20\!\cdots\!21\)\( + \)\(27\!\cdots\!32\)\( \beta_{3}) q^{81} +(-\)\(68\!\cdots\!18\)\( \beta_{1} + \)\(29\!\cdots\!52\)\( \beta_{2}) q^{82} +(\)\(23\!\cdots\!32\)\( \beta_{1} + \)\(47\!\cdots\!34\)\( \beta_{2}) q^{83} +(\)\(76\!\cdots\!36\)\( - \)\(51\!\cdots\!52\)\( \beta_{3}) q^{84} +(-\)\(50\!\cdots\!68\)\( - \)\(17\!\cdots\!62\)\( \beta_{3}) q^{86} +(\)\(29\!\cdots\!20\)\( \beta_{1} + \)\(39\!\cdots\!44\)\( \beta_{2}) q^{87} +(-\)\(18\!\cdots\!80\)\( \beta_{1} - \)\(13\!\cdots\!56\)\( \beta_{2}) q^{88} +(\)\(66\!\cdots\!30\)\( + \)\(46\!\cdots\!04\)\( \beta_{3}) q^{89} +(\)\(56\!\cdots\!92\)\( + \)\(26\!\cdots\!08\)\( \beta_{3}) q^{91} +(\)\(31\!\cdots\!56\)\( \beta_{1} - \)\(24\!\cdots\!32\)\( \beta_{2}) q^{92} +(-\)\(35\!\cdots\!36\)\( \beta_{1} + \)\(13\!\cdots\!52\)\( \beta_{2}) q^{93} +(-\)\(35\!\cdots\!44\)\( - \)\(35\!\cdots\!28\)\( \beta_{3}) q^{94} +(\)\(86\!\cdots\!32\)\( - \)\(10\!\cdots\!68\)\( \beta_{3}) q^{96} +(-\)\(10\!\cdots\!84\)\( \beta_{1} + \)\(60\!\cdots\!77\)\( \beta_{2}) q^{97} +(-\)\(13\!\cdots\!57\)\( \beta_{1} + \)\(94\!\cdots\!88\)\( \beta_{2}) q^{98} +(-\)\(60\!\cdots\!96\)\( - \)\(28\!\cdots\!88\)\( \beta_{3}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 75440539648q^{4} - 45013087694592q^{6} + 1797894756803538828q^{9} + O(q^{10}) \) \( 4q - 75440539648q^{4} - 45013087694592q^{6} + 1797894756803538828q^{9} - 53468072709697076112q^{11} - \)\(63\!\cdots\!96\)\(q^{14} - \)\(62\!\cdots\!16\)\(q^{16} - \)\(74\!\cdots\!40\)\(q^{19} - \)\(45\!\cdots\!12\)\(q^{21} - \)\(37\!\cdots\!20\)\(q^{24} + \)\(32\!\cdots\!88\)\(q^{26} + \)\(25\!\cdots\!40\)\(q^{29} + \)\(52\!\cdots\!48\)\(q^{31} - \)\(31\!\cdots\!16\)\(q^{34} - \)\(33\!\cdots\!36\)\(q^{36} + \)\(23\!\cdots\!36\)\(q^{39} - \)\(25\!\cdots\!72\)\(q^{41} - \)\(26\!\cdots\!56\)\(q^{44} + \)\(25\!\cdots\!68\)\(q^{46} + \)\(76\!\cdots\!72\)\(q^{49} - \)\(22\!\cdots\!52\)\(q^{51} - \)\(40\!\cdots\!40\)\(q^{54} - \)\(44\!\cdots\!60\)\(q^{56} + \)\(47\!\cdots\!80\)\(q^{59} + \)\(20\!\cdots\!88\)\(q^{61} - \)\(37\!\cdots\!28\)\(q^{64} + \)\(33\!\cdots\!76\)\(q^{66} + \)\(18\!\cdots\!96\)\(q^{69} - \)\(14\!\cdots\!32\)\(q^{71} + \)\(13\!\cdots\!44\)\(q^{74} - \)\(13\!\cdots\!20\)\(q^{76} - \)\(54\!\cdots\!60\)\(q^{79} + \)\(80\!\cdots\!84\)\(q^{81} + \)\(30\!\cdots\!44\)\(q^{84} - \)\(20\!\cdots\!72\)\(q^{86} + \)\(26\!\cdots\!20\)\(q^{89} + \)\(22\!\cdots\!68\)\(q^{91} - \)\(14\!\cdots\!76\)\(q^{94} + \)\(34\!\cdots\!28\)\(q^{96} - \)\(24\!\cdots\!84\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 31868761 x^{2} + 253904465984400\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( 4 \nu^{3} + 191212564 \nu \)\()/1327865\)
\(\beta_{2}\)\(=\)\((\)\( 5 \nu^{3} + 79671905 \nu \)\()/1593438\)
\(\beta_{3}\)\(=\)\( 4800 \nu^{2} + 76485026400 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-24 \beta_{2} + 25 \beta_{1}\)\()/2400\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 76485026400\)\()/4800\)
\(\nu^{3}\)\(=\)\((\)\(1147275384 \beta_{2} - 398359525 \beta_{1}\)\()/2400\)

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
3992.29i
3991.29i
3991.29i
3992.29i
480412.i 3.45869e7i −9.33565e10 0 −1.66160e13 5.42222e15i 2.11777e16i 4.49088e17 0
24.2 286012.i 2.05955e7i 5.56362e10 0 −5.89057e12 1.97377e15i 5.52218e16i 4.49860e17 0
24.3 286012.i 2.05955e7i 5.56362e10 0 −5.89057e12 1.97377e15i 5.52218e16i 4.49860e17 0
24.4 480412.i 3.45869e7i −9.33565e10 0 −1.66160e13 5.42222e15i 2.11777e16i 4.49088e17 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 25.38.b.a 4
5.b even 2 1 inner 25.38.b.a 4
5.c odd 4 1 1.38.a.a 2
5.c odd 4 1 25.38.a.a 2
15.e even 4 1 9.38.a.a 2
20.e even 4 1 16.38.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.38.a.a 2 5.c odd 4 1
9.38.a.a 2 15.e even 4 1
16.38.a.b 2 20.e even 4 1
25.38.a.a 2 5.c odd 4 1
25.38.b.a 4 1.a even 1 1 trivial
25.38.b.a 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 312598176768 T_{2}^{2} + \)\(18\!\cdots\!56\)\( \) acting on \(S_{38}^{\mathrm{new}}(25, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 237157637120 T^{2} + \)\(46\!\cdots\!68\)\( T^{4} - \)\(44\!\cdots\!80\)\( T^{6} + \)\(35\!\cdots\!56\)\( T^{8} \)
$3$ \( 1 - 1799515190183764140 T^{2} + \)\(12\!\cdots\!38\)\( T^{4} - \)\(36\!\cdots\!60\)\( T^{6} + \)\(41\!\cdots\!61\)\( T^{8} \)
$5$ 1
$7$ \( 1 - \)\(40\!\cdots\!00\)\( T^{2} + \)\(94\!\cdots\!98\)\( T^{4} - \)\(14\!\cdots\!00\)\( T^{6} + \)\(11\!\cdots\!01\)\( T^{8} \)
$11$ \( ( 1 + 26734036354848538056 T + \)\(70\!\cdots\!26\)\( T^{2} + \)\(90\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} )^{2} \)
$13$ \( 1 - \)\(36\!\cdots\!80\)\( T^{2} + \)\(65\!\cdots\!78\)\( T^{4} - \)\(98\!\cdots\!20\)\( T^{6} + \)\(73\!\cdots\!21\)\( T^{8} \)
$17$ \( 1 - \)\(92\!\cdots\!60\)\( T^{2} + \)\(43\!\cdots\!58\)\( T^{4} - \)\(10\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!41\)\( T^{8} \)
$19$ \( ( 1 + \)\(37\!\cdots\!20\)\( T + \)\(20\!\cdots\!78\)\( T^{2} + \)\(76\!\cdots\!80\)\( T^{3} + \)\(42\!\cdots\!21\)\( T^{4} )^{2} \)
$23$ \( 1 - \)\(28\!\cdots\!20\)\( T^{2} + \)\(20\!\cdots\!18\)\( T^{4} - \)\(16\!\cdots\!80\)\( T^{6} + \)\(34\!\cdots\!81\)\( T^{8} \)
$29$ \( ( 1 - \)\(12\!\cdots\!20\)\( T + \)\(21\!\cdots\!18\)\( T^{2} - \)\(16\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!81\)\( T^{4} )^{2} \)
$31$ \( ( 1 - \)\(26\!\cdots\!24\)\( T + \)\(24\!\cdots\!66\)\( T^{2} - \)\(39\!\cdots\!64\)\( T^{3} + \)\(22\!\cdots\!21\)\( T^{4} )^{2} \)
$37$ \( 1 - \)\(21\!\cdots\!80\)\( T^{2} + \)\(29\!\cdots\!78\)\( T^{4} - \)\(23\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!21\)\( T^{8} \)
$41$ \( ( 1 + \)\(12\!\cdots\!36\)\( T + \)\(12\!\cdots\!86\)\( T^{2} + \)\(59\!\cdots\!16\)\( T^{3} + \)\(22\!\cdots\!61\)\( T^{4} )^{2} \)
$43$ \( 1 - \)\(22\!\cdots\!00\)\( T^{2} - \)\(13\!\cdots\!02\)\( T^{4} - \)\(17\!\cdots\!00\)\( T^{6} + \)\(56\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - \)\(28\!\cdots\!40\)\( T^{2} + \)\(30\!\cdots\!38\)\( T^{4} - \)\(15\!\cdots\!60\)\( T^{6} + \)\(29\!\cdots\!61\)\( T^{8} \)
$53$ \( 1 - \)\(84\!\cdots\!40\)\( T^{2} + \)\(46\!\cdots\!38\)\( T^{4} - \)\(33\!\cdots\!60\)\( T^{6} + \)\(15\!\cdots\!61\)\( T^{8} \)
$59$ \( ( 1 - \)\(23\!\cdots\!40\)\( T + \)\(64\!\cdots\!38\)\( T^{2} - \)\(79\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!61\)\( T^{4} )^{2} \)
$61$ \( ( 1 - \)\(10\!\cdots\!44\)\( T + \)\(10\!\cdots\!26\)\( T^{2} - \)\(11\!\cdots\!24\)\( T^{3} + \)\(13\!\cdots\!41\)\( T^{4} )^{2} \)
$67$ \( 1 - \)\(69\!\cdots\!60\)\( T^{2} + \)\(28\!\cdots\!58\)\( T^{4} - \)\(93\!\cdots\!40\)\( T^{6} + \)\(18\!\cdots\!41\)\( T^{8} \)
$71$ \( ( 1 + \)\(74\!\cdots\!16\)\( T + \)\(58\!\cdots\!46\)\( T^{2} + \)\(23\!\cdots\!56\)\( T^{3} + \)\(98\!\cdots\!81\)\( T^{4} )^{2} \)
$73$ \( 1 - \)\(33\!\cdots\!20\)\( T^{2} + \)\(42\!\cdots\!18\)\( T^{4} - \)\(25\!\cdots\!80\)\( T^{6} + \)\(59\!\cdots\!81\)\( T^{8} \)
$79$ \( ( 1 + \)\(27\!\cdots\!80\)\( T + \)\(50\!\cdots\!18\)\( T^{2} + \)\(44\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!81\)\( T^{4} )^{2} \)
$83$ \( 1 - \)\(27\!\cdots\!60\)\( T^{2} + \)\(38\!\cdots\!58\)\( T^{4} - \)\(28\!\cdots\!40\)\( T^{6} + \)\(10\!\cdots\!41\)\( T^{8} \)
$89$ \( ( 1 - \)\(13\!\cdots\!60\)\( T + \)\(23\!\cdots\!58\)\( T^{2} - \)\(17\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!41\)\( T^{4} )^{2} \)
$97$ \( 1 - \)\(78\!\cdots\!40\)\( T^{2} + \)\(30\!\cdots\!38\)\( T^{4} - \)\(82\!\cdots\!60\)\( T^{6} + \)\(11\!\cdots\!61\)\( T^{8} \)
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