Properties

Label 25.34.b.a
Level $25$
Weight $34$
Character orbit 25.b
Analytic conductor $172.457$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(172.457072203\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} + \cdots)\)
Defining polynomial: \(x^{4} + 1178101 x^{2} + 346979902500\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4}\cdot 5^{2} \)
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.

\(f(q)\) \(=\) \( q + ( 6084 \beta_{1} + \beta_{2} ) q^{2} + ( 1895994 \beta_{1} + 312 \beta_{2} ) q^{3} + ( -7326116992 + 12168 \beta_{3} ) q^{4} + ( -4964461096608 + 3794202 \beta_{3} ) q^{6} + ( 3357654003340 \beta_{1} - 801594864 \beta_{2} ) q^{7} + ( -140937529254912 \beta_{1} - 6139193600 \beta_{2} ) q^{8} + ( 4010568477485427 + 1183100256 \beta_{3} ) q^{9} +O(q^{10})\) \( q +(6084 \beta_{1} + \beta_{2}) q^{2} +(1895994 \beta_{1} + 312 \beta_{2}) q^{3} +(-7326116992 + 12168 \beta_{3}) q^{4} +(-4964461096608 + 3794202 \beta_{3}) q^{6} +(3357654003340 \beta_{1} - 801594864 \beta_{2}) q^{7} +(-140937529254912 \beta_{1} - 6139193600 \beta_{2}) q^{8} +(4010568477485427 + 1183100256 \beta_{3}) q^{9} +(66935907720957132 + 134661178300 \beta_{3}) q^{11} +(-60261771666523392 \beta_{1} - 4592794000704 \beta_{2}) q^{12} +(-149080523912972197 \beta_{1} - 3738595861728 \beta_{2}) q^{13} +(7748320631234170176 - 1519249149236 \beta_{3}) q^{14} +(97802989555947175936 - 73766059001856 \beta_{3}) q^{16} +(3968057463058776267 \beta_{1} + 2662090686805056 \beta_{2}) q^{17} +(9949266136427165964 \beta_{1} + 3290770281735027 \beta_{2}) q^{18} +(\)\(68\!\cdots\!00\)\( + 488470315076892 \beta_{3}) q^{19} +(\)\(24\!\cdots\!12\)\( - 472231003532736 \beta_{3}) q^{21} +(-\)\(12\!\cdots\!12\)\( \beta_{1} - 14991953156762868 \beta_{2}) q^{22} +(\)\(13\!\cdots\!72\)\( \beta_{1} + 164629195362887632 \beta_{2}) q^{23} +(\)\(50\!\cdots\!00\)\( - 55612383357970944 \beta_{3}) q^{24} +(\)\(13\!\cdots\!52\)\( - 171826141135725349 \beta_{3}) q^{26} +(\)\(13\!\cdots\!52\)\( \beta_{1} + 2761409163063330000 \beta_{2}) q^{27} +(\)\(94\!\cdots\!88\)\( \beta_{1} + 1786984362586217088 \beta_{2}) q^{28} +(\)\(83\!\cdots\!50\)\( - 608514536018417392 \beta_{3}) q^{29} +(-\)\(31\!\cdots\!08\)\( - 3291649706447157600 \beta_{3}) q^{31} +(\)\(28\!\cdots\!24\)\( \beta_{1} + 89946988381051355136 \beta_{2}) q^{32} +(-\)\(38\!\cdots\!92\)\( \beta_{1} - 4647675400034394816 \beta_{2}) q^{33} +(-\)\(34\!\cdots\!04\)\( + 20164217201580736971 \beta_{3}) q^{34} +(-\)\(11\!\cdots\!84\)\( + 40133066345321525784 \beta_{3}) q^{36} +(\)\(52\!\cdots\!81\)\( \beta_{1} - \)\(20\!\cdots\!04\)\( \beta_{2}) q^{37} +(-\)\(18\!\cdots\!28\)\( \beta_{1} + \)\(38\!\cdots\!00\)\( \beta_{2}) q^{38} +(\)\(42\!\cdots\!24\)\( - 53601478783108443096 \beta_{3}) q^{39} +(\)\(13\!\cdots\!22\)\( + \)\(32\!\cdots\!00\)\( \beta_{3}) q^{41} +(\)\(20\!\cdots\!32\)\( \beta_{1} + \)\(27\!\cdots\!12\)\( \beta_{2}) q^{42} +(\)\(78\!\cdots\!90\)\( \beta_{1} + \)\(88\!\cdots\!52\)\( \beta_{2}) q^{43} +(\)\(15\!\cdots\!56\)\( - \)\(17\!\cdots\!24\)\( \beta_{3}) q^{44} +(-\)\(28\!\cdots\!88\)\( + \)\(23\!\cdots\!60\)\( \beta_{3}) q^{46} +(-\)\(27\!\cdots\!32\)\( \beta_{1} - \)\(49\!\cdots\!84\)\( \beta_{2}) q^{47} +(\)\(46\!\cdots\!32\)\( \beta_{1} + \)\(44\!\cdots\!32\)\( \beta_{2}) q^{48} +(-\)\(12\!\cdots\!57\)\( - \)\(53\!\cdots\!20\)\( \beta_{3}) q^{49} +(-\)\(10\!\cdots\!48\)\( + \)\(62\!\cdots\!68\)\( \beta_{3}) q^{51} +(\)\(16\!\cdots\!60\)\( \beta_{1} + \)\(20\!\cdots\!76\)\( \beta_{2}) q^{52} +(-\)\(13\!\cdots\!61\)\( \beta_{1} + \)\(20\!\cdots\!12\)\( \beta_{2}) q^{53} +(-\)\(42\!\cdots\!00\)\( + \)\(30\!\cdots\!52\)\( \beta_{3}) q^{54} +(-\)\(12\!\cdots\!00\)\( + \)\(92\!\cdots\!68\)\( \beta_{3}) q^{56} +(-\)\(57\!\cdots\!36\)\( \beta_{1} + \)\(11\!\cdots\!00\)\( \beta_{2}) q^{57} +(\)\(12\!\cdots\!28\)\( \beta_{1} + \)\(12\!\cdots\!50\)\( \beta_{2}) q^{58} +(\)\(15\!\cdots\!00\)\( + \)\(31\!\cdots\!36\)\( \beta_{3}) q^{59} +(-\)\(28\!\cdots\!18\)\( + \)\(63\!\cdots\!00\)\( \beta_{3}) q^{61} +(\)\(21\!\cdots\!28\)\( \beta_{1} - \)\(11\!\cdots\!08\)\( \beta_{2}) q^{62} +(\)\(25\!\cdots\!36\)\( \beta_{1} - \)\(36\!\cdots\!28\)\( \beta_{2}) q^{63} +(-\)\(43\!\cdots\!12\)\( + \)\(19\!\cdots\!96\)\( \beta_{3}) q^{64} +(\)\(29\!\cdots\!44\)\( - \)\(41\!\cdots\!36\)\( \beta_{3}) q^{66} +(-\)\(79\!\cdots\!98\)\( \beta_{1} - \)\(28\!\cdots\!44\)\( \beta_{2}) q^{67} +(-\)\(42\!\cdots\!36\)\( \beta_{1} - \)\(24\!\cdots\!52\)\( \beta_{2}) q^{68} +(-\)\(87\!\cdots\!56\)\( + \)\(72\!\cdots\!72\)\( \beta_{3}) q^{69} +(-\)\(13\!\cdots\!88\)\( - \)\(18\!\cdots\!00\)\( \beta_{3}) q^{71} +(-\)\(47\!\cdots\!24\)\( \beta_{1} - \)\(79\!\cdots\!00\)\( \beta_{2}) q^{72} +(\)\(47\!\cdots\!17\)\( \beta_{1} - \)\(59\!\cdots\!68\)\( \beta_{2}) q^{73} +(-\)\(72\!\cdots\!64\)\( + \)\(40\!\cdots\!45\)\( \beta_{3}) q^{74} +(\)\(22\!\cdots\!00\)\( + \)\(46\!\cdots\!36\)\( \beta_{3}) q^{76} +(\)\(15\!\cdots\!80\)\( \beta_{1} - \)\(98\!\cdots\!48\)\( \beta_{2}) q^{77} +(\)\(91\!\cdots\!80\)\( \beta_{1} + \)\(75\!\cdots\!24\)\( \beta_{2}) q^{78} +(\)\(42\!\cdots\!00\)\( - \)\(11\!\cdots\!12\)\( \beta_{3}) q^{79} +(\)\(91\!\cdots\!21\)\( + \)\(16\!\cdots\!12\)\( \beta_{3}) q^{81} +(-\)\(31\!\cdots\!52\)\( \beta_{1} - \)\(58\!\cdots\!78\)\( \beta_{2}) q^{82} +(\)\(14\!\cdots\!46\)\( \beta_{1} + \)\(24\!\cdots\!92\)\( \beta_{2}) q^{83} +(-\)\(24\!\cdots\!04\)\( + \)\(32\!\cdots\!28\)\( \beta_{3}) q^{84} +(-\)\(15\!\cdots\!68\)\( + \)\(13\!\cdots\!58\)\( \beta_{3}) q^{86} +(\)\(39\!\cdots\!36\)\( \beta_{1} + \)\(37\!\cdots\!00\)\( \beta_{2}) q^{87} +(\)\(66\!\cdots\!16\)\( \beta_{1} + \)\(14\!\cdots\!00\)\( \beta_{2}) q^{88} +(-\)\(66\!\cdots\!50\)\( - \)\(38\!\cdots\!16\)\( \beta_{3}) q^{89} +(\)\(13\!\cdots\!72\)\( + \)\(10\!\cdots\!88\)\( \beta_{3}) q^{91} +(-\)\(34\!\cdots\!08\)\( \beta_{1} - \)\(27\!\cdots\!44\)\( \beta_{2}) q^{92} +(\)\(66\!\cdots\!48\)\( \beta_{1} - \)\(34\!\cdots\!96\)\( \beta_{2}) q^{93} +(\)\(22\!\cdots\!56\)\( - \)\(30\!\cdots\!88\)\( \beta_{3}) q^{94} +(-\)\(39\!\cdots\!88\)\( + \)\(25\!\cdots\!72\)\( \beta_{3}) q^{96} +(\)\(18\!\cdots\!03\)\( \beta_{1} - \)\(53\!\cdots\!84\)\( \beta_{2}) q^{97} +(\)\(58\!\cdots\!92\)\( \beta_{1} + \)\(20\!\cdots\!43\)\( \beta_{2}) q^{98} +(\)\(46\!\cdots\!64\)\( + \)\(61\!\cdots\!92\)\( \beta_{3}) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 29304467968q^{4} - 19857844386432q^{6} + 16042273909941708q^{9} + O(q^{10}) \) \( 4q - 29304467968q^{4} - 19857844386432q^{6} + 16042273909941708q^{9} + 267743630883828528q^{11} + 30993282524936680704q^{14} + \)\(39\!\cdots\!44\)\(q^{16} + \)\(27\!\cdots\!00\)\(q^{19} + \)\(96\!\cdots\!48\)\(q^{21} + \)\(20\!\cdots\!00\)\(q^{24} + \)\(54\!\cdots\!08\)\(q^{26} + \)\(33\!\cdots\!00\)\(q^{29} - \)\(12\!\cdots\!32\)\(q^{31} - \)\(13\!\cdots\!16\)\(q^{34} - \)\(47\!\cdots\!36\)\(q^{36} + \)\(17\!\cdots\!96\)\(q^{39} + \)\(55\!\cdots\!88\)\(q^{41} + \)\(60\!\cdots\!24\)\(q^{44} - \)\(11\!\cdots\!52\)\(q^{46} - \)\(49\!\cdots\!28\)\(q^{49} - \)\(43\!\cdots\!92\)\(q^{51} - \)\(16\!\cdots\!00\)\(q^{54} - \)\(51\!\cdots\!00\)\(q^{56} + \)\(61\!\cdots\!00\)\(q^{59} - \)\(11\!\cdots\!72\)\(q^{61} - \)\(17\!\cdots\!48\)\(q^{64} + \)\(11\!\cdots\!76\)\(q^{66} - \)\(35\!\cdots\!24\)\(q^{69} - \)\(53\!\cdots\!52\)\(q^{71} - \)\(29\!\cdots\!56\)\(q^{74} + \)\(91\!\cdots\!00\)\(q^{76} + \)\(17\!\cdots\!00\)\(q^{79} + \)\(36\!\cdots\!84\)\(q^{81} - \)\(98\!\cdots\!16\)\(q^{84} - \)\(62\!\cdots\!72\)\(q^{86} - \)\(26\!\cdots\!00\)\(q^{89} + \)\(53\!\cdots\!88\)\(q^{91} + \)\(90\!\cdots\!24\)\(q^{94} - \)\(15\!\cdots\!52\)\(q^{96} + \)\(18\!\cdots\!56\)\(q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 1178101 x^{2} + 346979902500\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} - 589051 \nu \)\()/58905\)
\(\beta_{2}\)\(=\)\((\)\( 4 \nu^{3} + 7068604 \nu \)\()/32725\)
\(\beta_{3}\)\(=\)\( 1440 \nu^{2} + 848232720 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(5 \beta_{2} + 36 \beta_{1}\)\()/720\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 848232720\)\()/1440\)
\(\nu^{3}\)\(=\)\((\)\(-2945255 \beta_{2} - 63617436 \beta_{1}\)\()/720\)

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
767.996i
766.996i
766.996i
767.996i
171359.i 5.34420e7i −2.07741e10 0 −9.15779e12 5.50153e13i 2.08788e15i 2.70301e15 0
24.2 49679.4i 1.55221e7i 6.12189e9 0 −7.71130e11 1.22168e14i 7.30875e14i 5.31812e15 0
24.3 49679.4i 1.55221e7i 6.12189e9 0 −7.71130e11 1.22168e14i 7.30875e14i 5.31812e15 0
24.4 171359.i 5.34420e7i −2.07741e10 0 −9.15779e12 5.50153e13i 2.08788e15i 2.70301e15 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 25.34.b.a 4
5.b even 2 1 inner 25.34.b.a 4
5.c odd 4 1 1.34.a.a 2
5.c odd 4 1 25.34.a.a 2
15.e even 4 1 9.34.a.b 2
20.e even 4 1 16.34.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.34.a.a 2 5.c odd 4 1
9.34.a.b 2 15.e even 4 1
16.34.a.b 2 20.e even 4 1
25.34.a.a 2 5.c odd 4 1
25.34.b.a 4 1.a even 1 1 trivial
25.34.b.a 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 31832103168 T_{2}^{2} + \)72471856579614867456

'>\(72\!\cdots\!56\)\( \) acting on \(S_{34}^{\mathrm{new}}(25, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2527635200 T^{2} - 31677653929187868672 T^{4} - \)\(18\!\cdots\!00\)\( T^{6} + \)\(54\!\cdots\!96\)\( T^{8} \)
$3$ \( 1 - 19139258088081900 T^{2} + \)\(15\!\cdots\!58\)\( T^{4} - \)\(59\!\cdots\!00\)\( T^{6} + \)\(95\!\cdots\!41\)\( T^{8} \)
$5$ 1
$7$ \( 1 - \)\(12\!\cdots\!00\)\( T^{2} + \)\(12\!\cdots\!98\)\( T^{4} - \)\(77\!\cdots\!00\)\( T^{6} + \)\(35\!\cdots\!01\)\( T^{8} \)
$11$ \( ( 1 - 133871815441914264 T + \)\(28\!\cdots\!86\)\( T^{2} - \)\(31\!\cdots\!84\)\( T^{3} + \)\(53\!\cdots\!61\)\( T^{4} )^{2} \)
$13$ \( 1 - \)\(18\!\cdots\!00\)\( T^{2} + \)\(14\!\cdots\!18\)\( T^{4} - \)\(60\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!81\)\( T^{8} \)
$17$ \( 1 + \)\(15\!\cdots\!00\)\( T^{2} + \)\(27\!\cdots\!38\)\( T^{4} + \)\(24\!\cdots\!00\)\( T^{6} + \)\(26\!\cdots\!61\)\( T^{8} \)
$19$ \( ( 1 - \)\(13\!\cdots\!00\)\( T + \)\(33\!\cdots\!18\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!81\)\( T^{4} )^{2} \)
$23$ \( 1 - \)\(24\!\cdots\!00\)\( T^{2} + \)\(27\!\cdots\!78\)\( T^{4} - \)\(18\!\cdots\!00\)\( T^{6} + \)\(55\!\cdots\!21\)\( T^{8} \)
$29$ \( ( 1 - \)\(16\!\cdots\!00\)\( T + \)\(38\!\cdots\!78\)\( T^{2} - \)\(30\!\cdots\!00\)\( T^{3} + \)\(32\!\cdots\!21\)\( T^{4} )^{2} \)
$31$ \( ( 1 + \)\(62\!\cdots\!16\)\( T + \)\(29\!\cdots\!46\)\( T^{2} + \)\(10\!\cdots\!56\)\( T^{3} + \)\(26\!\cdots\!81\)\( T^{4} )^{2} \)
$37$ \( 1 - \)\(16\!\cdots\!00\)\( T^{2} + \)\(12\!\cdots\!18\)\( T^{4} - \)\(50\!\cdots\!00\)\( T^{6} + \)\(10\!\cdots\!81\)\( T^{8} \)
$41$ \( ( 1 - \)\(27\!\cdots\!44\)\( T + \)\(22\!\cdots\!26\)\( T^{2} - \)\(46\!\cdots\!24\)\( T^{3} + \)\(27\!\cdots\!41\)\( T^{4} )^{2} \)
$43$ \( 1 - \)\(48\!\cdots\!00\)\( T^{2} - \)\(10\!\cdots\!02\)\( T^{4} - \)\(30\!\cdots\!00\)\( T^{6} + \)\(41\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - \)\(45\!\cdots\!00\)\( T^{2} + \)\(95\!\cdots\!58\)\( T^{4} - \)\(10\!\cdots\!00\)\( T^{6} + \)\(52\!\cdots\!41\)\( T^{8} \)
$53$ \( 1 - \)\(18\!\cdots\!00\)\( T^{2} + \)\(17\!\cdots\!58\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{6} + \)\(40\!\cdots\!41\)\( T^{8} \)
$59$ \( ( 1 - \)\(30\!\cdots\!00\)\( T + \)\(77\!\cdots\!58\)\( T^{2} - \)\(83\!\cdots\!00\)\( T^{3} + \)\(75\!\cdots\!41\)\( T^{4} )^{2} \)
$61$ \( ( 1 + \)\(57\!\cdots\!36\)\( T + \)\(16\!\cdots\!86\)\( T^{2} + \)\(47\!\cdots\!16\)\( T^{3} + \)\(67\!\cdots\!61\)\( T^{4} )^{2} \)
$67$ \( 1 - \)\(58\!\cdots\!00\)\( T^{2} + \)\(14\!\cdots\!38\)\( T^{4} - \)\(19\!\cdots\!00\)\( T^{6} + \)\(11\!\cdots\!61\)\( T^{8} \)
$71$ \( ( 1 + \)\(26\!\cdots\!76\)\( T + \)\(22\!\cdots\!66\)\( T^{2} + \)\(32\!\cdots\!36\)\( T^{3} + \)\(15\!\cdots\!21\)\( T^{4} )^{2} \)
$73$ \( 1 - \)\(37\!\cdots\!00\)\( T^{2} + \)\(22\!\cdots\!78\)\( T^{4} - \)\(35\!\cdots\!00\)\( T^{6} + \)\(90\!\cdots\!21\)\( T^{8} \)
$79$ \( ( 1 - \)\(85\!\cdots\!00\)\( T + \)\(85\!\cdots\!78\)\( T^{2} - \)\(35\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!21\)\( T^{4} )^{2} \)
$83$ \( 1 - \)\(66\!\cdots\!00\)\( T^{2} + \)\(19\!\cdots\!38\)\( T^{4} - \)\(30\!\cdots\!00\)\( T^{6} + \)\(20\!\cdots\!61\)\( T^{8} \)
$89$ \( ( 1 + \)\(13\!\cdots\!00\)\( T + \)\(47\!\cdots\!38\)\( T^{2} + \)\(28\!\cdots\!00\)\( T^{3} + \)\(45\!\cdots\!61\)\( T^{4} )^{2} \)
$97$ \( 1 - \)\(70\!\cdots\!00\)\( T^{2} + \)\(34\!\cdots\!58\)\( T^{4} - \)\(94\!\cdots\!00\)\( T^{6} + \)\(17\!\cdots\!41\)\( T^{8} \)
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