Properties

Label 25.34.a.a
Level $25$
Weight $34$
Character orbit 25.a
Self dual yes
Analytic conductor $172.457$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(172.457072203\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 589050\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 72\sqrt{2356201}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 60840 - \beta ) q^{2} + ( -18959940 + 312 \beta ) q^{3} + ( 7326116992 - 121680 \beta ) q^{4} + ( -4964461096608 + 37942020 \beta ) q^{6} + ( 33576540033400 + 801594864 \beta ) q^{7} + ( 1409375292549120 - 6139193600 \beta ) q^{8} + ( -4010568477485427 - 11831002560 \beta ) q^{9} +O(q^{10})\) \( q +(60840 - \beta) q^{2} +(-18959940 + 312 \beta) q^{3} +(7326116992 - 121680 \beta) q^{4} +(-4964461096608 + 37942020 \beta) q^{6} +(33576540033400 + 801594864 \beta) q^{7} +(1409375292549120 - 6139193600 \beta) q^{8} +(-4010568477485427 - 11831002560 \beta) q^{9} +(66935907720957132 + 1346611783000 \beta) q^{11} +(-602617716665233920 + 4592794000704 \beta) q^{12} +(1490805239129721970 - 3738595861728 \beta) q^{13} +(-7748320631234170176 + 15192491492360 \beta) q^{14} +(97802989555947175936 - 737660590018560 \beta) q^{16} +(39680574630587762670 - 2662090686805056 \beta) q^{17} +(-99492661364271659640 + 3290770281735027 \beta) q^{18} +(-\)\(68\!\cdots\!00\)\( - 4884703150768920 \beta) q^{19} +(\)\(24\!\cdots\!12\)\( - 4722310035327360 \beta) q^{21} +(-\)\(12\!\cdots\!20\)\( + 14991953156762868 \beta) q^{22} +(-\)\(13\!\cdots\!20\)\( + 164629195362887632 \beta) q^{23} +(-\)\(50\!\cdots\!00\)\( + 556123833579709440 \beta) q^{24} +(\)\(13\!\cdots\!52\)\( - 1718261411357253490 \beta) q^{26} +(\)\(13\!\cdots\!20\)\( - 2761409163063330000 \beta) q^{27} +(-\)\(94\!\cdots\!80\)\( + 1786984362586217088 \beta) q^{28} +(-\)\(83\!\cdots\!50\)\( + 6085145360184173920 \beta) q^{29} +(-\)\(31\!\cdots\!08\)\( - 32916497064471576000 \beta) q^{31} +(\)\(28\!\cdots\!40\)\( - 89946988381051355136 \beta) q^{32} +(\)\(38\!\cdots\!20\)\( - 4647675400034394816 \beta) q^{33} +(\)\(34\!\cdots\!04\)\( - \)\(20\!\cdots\!10\)\( \beta) q^{34} +(-\)\(11\!\cdots\!84\)\( + \)\(40\!\cdots\!40\)\( \beta) q^{36} +(\)\(52\!\cdots\!10\)\( + \)\(20\!\cdots\!04\)\( \beta) q^{37} +(\)\(18\!\cdots\!80\)\( + \)\(38\!\cdots\!00\)\( \beta) q^{38} +(-\)\(42\!\cdots\!24\)\( + \)\(53\!\cdots\!60\)\( \beta) q^{39} +(\)\(13\!\cdots\!22\)\( + \)\(32\!\cdots\!00\)\( \beta) q^{41} +(\)\(20\!\cdots\!20\)\( - \)\(27\!\cdots\!12\)\( \beta) q^{42} +(-\)\(78\!\cdots\!00\)\( + \)\(88\!\cdots\!52\)\( \beta) q^{43} +(-\)\(15\!\cdots\!56\)\( + \)\(17\!\cdots\!40\)\( \beta) q^{44} +(-\)\(28\!\cdots\!88\)\( + \)\(23\!\cdots\!00\)\( \beta) q^{46} +(-\)\(27\!\cdots\!20\)\( + \)\(49\!\cdots\!84\)\( \beta) q^{47} +(-\)\(46\!\cdots\!20\)\( + \)\(44\!\cdots\!32\)\( \beta) q^{48} +(\)\(12\!\cdots\!57\)\( + \)\(53\!\cdots\!00\)\( \beta) q^{49} +(-\)\(10\!\cdots\!48\)\( + \)\(62\!\cdots\!80\)\( \beta) q^{51} +(\)\(16\!\cdots\!00\)\( - \)\(20\!\cdots\!76\)\( \beta) q^{52} +(\)\(13\!\cdots\!10\)\( + \)\(20\!\cdots\!12\)\( \beta) q^{53} +(\)\(42\!\cdots\!00\)\( - \)\(30\!\cdots\!20\)\( \beta) q^{54} +(-\)\(12\!\cdots\!00\)\( + \)\(92\!\cdots\!80\)\( \beta) q^{56} +(-\)\(57\!\cdots\!60\)\( - \)\(11\!\cdots\!00\)\( \beta) q^{57} +(-\)\(12\!\cdots\!80\)\( + \)\(12\!\cdots\!50\)\( \beta) q^{58} +(-\)\(15\!\cdots\!00\)\( - \)\(31\!\cdots\!60\)\( \beta) q^{59} +(-\)\(28\!\cdots\!18\)\( + \)\(63\!\cdots\!00\)\( \beta) q^{61} +(\)\(21\!\cdots\!80\)\( + \)\(11\!\cdots\!08\)\( \beta) q^{62} +(-\)\(25\!\cdots\!60\)\( - \)\(36\!\cdots\!28\)\( \beta) q^{63} +(\)\(43\!\cdots\!12\)\( - \)\(19\!\cdots\!60\)\( \beta) q^{64} +(\)\(29\!\cdots\!44\)\( - \)\(41\!\cdots\!60\)\( \beta) q^{66} +(-\)\(79\!\cdots\!80\)\( + \)\(28\!\cdots\!44\)\( \beta) q^{67} +(\)\(42\!\cdots\!60\)\( - \)\(24\!\cdots\!52\)\( \beta) q^{68} +(\)\(87\!\cdots\!56\)\( - \)\(72\!\cdots\!20\)\( \beta) q^{69} +(-\)\(13\!\cdots\!88\)\( - \)\(18\!\cdots\!00\)\( \beta) q^{71} +(-\)\(47\!\cdots\!40\)\( + \)\(79\!\cdots\!00\)\( \beta) q^{72} +(-\)\(47\!\cdots\!70\)\( - \)\(59\!\cdots\!68\)\( \beta) q^{73} +(\)\(72\!\cdots\!64\)\( - \)\(40\!\cdots\!50\)\( \beta) q^{74} +(\)\(22\!\cdots\!00\)\( + \)\(46\!\cdots\!60\)\( \beta) q^{76} +(\)\(15\!\cdots\!00\)\( + \)\(98\!\cdots\!48\)\( \beta) q^{77} +(-\)\(91\!\cdots\!00\)\( + \)\(75\!\cdots\!24\)\( \beta) q^{78} +(-\)\(42\!\cdots\!00\)\( + \)\(11\!\cdots\!20\)\( \beta) q^{79} +(\)\(91\!\cdots\!21\)\( + \)\(16\!\cdots\!20\)\( \beta) q^{81} +(-\)\(31\!\cdots\!20\)\( + \)\(58\!\cdots\!78\)\( \beta) q^{82} +(-\)\(14\!\cdots\!60\)\( + \)\(24\!\cdots\!92\)\( \beta) q^{83} +(\)\(24\!\cdots\!04\)\( - \)\(32\!\cdots\!80\)\( \beta) q^{84} +(-\)\(15\!\cdots\!68\)\( + \)\(13\!\cdots\!80\)\( \beta) q^{86} +(\)\(39\!\cdots\!60\)\( - \)\(37\!\cdots\!00\)\( \beta) q^{87} +(-\)\(66\!\cdots\!60\)\( + \)\(14\!\cdots\!00\)\( \beta) q^{88} +(\)\(66\!\cdots\!50\)\( + \)\(38\!\cdots\!60\)\( \beta) q^{89} +(\)\(13\!\cdots\!72\)\( + \)\(10\!\cdots\!80\)\( \beta) q^{91} +(-\)\(34\!\cdots\!80\)\( + \)\(27\!\cdots\!44\)\( \beta) q^{92} +(-\)\(66\!\cdots\!80\)\( - \)\(34\!\cdots\!96\)\( \beta) q^{93} +(-\)\(22\!\cdots\!56\)\( + \)\(30\!\cdots\!80\)\( \beta) q^{94} +(-\)\(39\!\cdots\!88\)\( + \)\(25\!\cdots\!20\)\( \beta) q^{96} +(\)\(18\!\cdots\!30\)\( + \)\(53\!\cdots\!84\)\( \beta) q^{97} +(-\)\(58\!\cdots\!20\)\( + \)\(20\!\cdots\!43\)\( \beta) q^{98} +(-\)\(46\!\cdots\!64\)\( - \)\(61\!\cdots\!20\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 121680q^{2} - 37919880q^{3} + 14652233984q^{4} - 9928922193216q^{6} + 67153080066800q^{7} + 2818750585098240q^{8} - 8021136954970854q^{9} + O(q^{10}) \) \( 2q + 121680q^{2} - 37919880q^{3} + 14652233984q^{4} - 9928922193216q^{6} + 67153080066800q^{7} + 2818750585098240q^{8} - 8021136954970854q^{9} + 133871815441914264q^{11} - 1205235433330467840q^{12} + 2981610478259443940q^{13} - 15496641262468340352q^{14} + \)\(19\!\cdots\!72\)\(q^{16} + 79361149261175525340q^{17} - \)\(19\!\cdots\!80\)\(q^{18} - \)\(13\!\cdots\!00\)\(q^{19} + \)\(48\!\cdots\!24\)\(q^{21} - \)\(24\!\cdots\!40\)\(q^{22} - \)\(26\!\cdots\!40\)\(q^{23} - \)\(10\!\cdots\!00\)\(q^{24} + \)\(27\!\cdots\!04\)\(q^{26} + \)\(27\!\cdots\!40\)\(q^{27} - \)\(18\!\cdots\!60\)\(q^{28} - \)\(16\!\cdots\!00\)\(q^{29} - \)\(62\!\cdots\!16\)\(q^{31} + \)\(57\!\cdots\!80\)\(q^{32} + \)\(77\!\cdots\!40\)\(q^{33} + \)\(69\!\cdots\!08\)\(q^{34} - \)\(23\!\cdots\!68\)\(q^{36} + \)\(10\!\cdots\!20\)\(q^{37} + \)\(36\!\cdots\!60\)\(q^{38} - \)\(85\!\cdots\!48\)\(q^{39} + \)\(27\!\cdots\!44\)\(q^{41} + \)\(40\!\cdots\!40\)\(q^{42} - \)\(15\!\cdots\!00\)\(q^{43} - \)\(30\!\cdots\!12\)\(q^{44} - \)\(56\!\cdots\!76\)\(q^{46} - \)\(54\!\cdots\!40\)\(q^{47} - \)\(93\!\cdots\!40\)\(q^{48} + \)\(24\!\cdots\!14\)\(q^{49} - \)\(21\!\cdots\!96\)\(q^{51} + \)\(32\!\cdots\!00\)\(q^{52} + \)\(26\!\cdots\!20\)\(q^{53} + \)\(84\!\cdots\!00\)\(q^{54} - \)\(25\!\cdots\!00\)\(q^{56} - \)\(11\!\cdots\!20\)\(q^{57} - \)\(25\!\cdots\!60\)\(q^{58} - \)\(30\!\cdots\!00\)\(q^{59} - \)\(57\!\cdots\!36\)\(q^{61} + \)\(42\!\cdots\!60\)\(q^{62} - \)\(50\!\cdots\!20\)\(q^{63} + \)\(86\!\cdots\!24\)\(q^{64} + \)\(58\!\cdots\!88\)\(q^{66} - \)\(15\!\cdots\!60\)\(q^{67} + \)\(84\!\cdots\!20\)\(q^{68} + \)\(17\!\cdots\!12\)\(q^{69} - \)\(26\!\cdots\!76\)\(q^{71} - \)\(95\!\cdots\!80\)\(q^{72} - \)\(94\!\cdots\!40\)\(q^{73} + \)\(14\!\cdots\!28\)\(q^{74} + \)\(45\!\cdots\!00\)\(q^{76} + \)\(30\!\cdots\!00\)\(q^{77} - \)\(18\!\cdots\!00\)\(q^{78} - \)\(85\!\cdots\!00\)\(q^{79} + \)\(18\!\cdots\!42\)\(q^{81} - \)\(62\!\cdots\!40\)\(q^{82} - \)\(29\!\cdots\!20\)\(q^{83} + \)\(49\!\cdots\!08\)\(q^{84} - \)\(31\!\cdots\!36\)\(q^{86} + \)\(78\!\cdots\!20\)\(q^{87} - \)\(13\!\cdots\!20\)\(q^{88} + \)\(13\!\cdots\!00\)\(q^{89} + \)\(26\!\cdots\!44\)\(q^{91} - \)\(68\!\cdots\!60\)\(q^{92} - \)\(13\!\cdots\!60\)\(q^{93} - \)\(45\!\cdots\!12\)\(q^{94} - \)\(79\!\cdots\!76\)\(q^{96} + \)\(36\!\cdots\!60\)\(q^{97} - \)\(11\!\cdots\!40\)\(q^{98} - \)\(92\!\cdots\!28\)\(q^{99} + O(q^{100}) \)

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
767.996
−766.996
−49679.4 1.55221e7 −6.12189e9 0 −7.71130e11 1.22168e14 7.30875e14 −5.31812e15 0
1.2 171359. −5.34420e7 2.07741e10 0 −9.15779e12 −5.50153e13 2.08788e15 −2.70301e15 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(1\)

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 25.34.a.a 2
5.b even 2 1 1.34.a.a 2
5.c odd 4 2 25.34.b.a 4
15.d odd 2 1 9.34.a.b 2
20.d odd 2 1 16.34.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.34.a.a 2 5.b even 2 1
9.34.a.b 2 15.d odd 2 1
16.34.a.b 2 20.d odd 2 1
25.34.a.a 2 1.a even 1 1 trivial
25.34.b.a 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 121680 T_{2} - 8513040384 \) acting on \(S_{34}^{\mathrm{new}}(\Gamma_0(25))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 121680 T + 8666828800 T^{2} - 1045223241154560 T^{3} + 73786976294838206464 T^{4} \)
$3$ \( 1 + 37919880 T + 10288587693648150 T^{2} + \)\(21\!\cdots\!40\)\( T^{3} + \)\(30\!\cdots\!29\)\( T^{4} \)
$5$ 1
$7$ \( 1 - 67153080066800 T + \)\(87\!\cdots\!50\)\( T^{2} - \)\(51\!\cdots\!00\)\( T^{3} + \)\(59\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 - 133871815441914264 T + \)\(28\!\cdots\!86\)\( T^{2} - \)\(31\!\cdots\!84\)\( T^{3} + \)\(53\!\cdots\!61\)\( T^{4} \)
$13$ \( 1 - 2981610478259443940 T + \)\(13\!\cdots\!50\)\( T^{2} - \)\(17\!\cdots\!20\)\( T^{3} + \)\(33\!\cdots\!09\)\( T^{4} \)
$17$ \( 1 - 79361149261175525340 T - \)\(44\!\cdots\!50\)\( T^{2} - \)\(31\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!69\)\( T^{4} \)
$19$ \( 1 + \)\(13\!\cdots\!00\)\( T + \)\(33\!\cdots\!18\)\( T^{2} + \)\(21\!\cdots\!00\)\( T^{3} + \)\(24\!\cdots\!81\)\( T^{4} \)
$23$ \( 1 + \)\(26\!\cdots\!40\)\( T + \)\(15\!\cdots\!50\)\( T^{2} + \)\(22\!\cdots\!20\)\( T^{3} + \)\(74\!\cdots\!89\)\( T^{4} \)
$29$ \( 1 + \)\(16\!\cdots\!00\)\( T + \)\(38\!\cdots\!78\)\( T^{2} + \)\(30\!\cdots\!00\)\( T^{3} + \)\(32\!\cdots\!21\)\( T^{4} \)
$31$ \( 1 + \)\(62\!\cdots\!16\)\( T + \)\(29\!\cdots\!46\)\( T^{2} + \)\(10\!\cdots\!56\)\( T^{3} + \)\(26\!\cdots\!81\)\( T^{4} \)
$37$ \( 1 - \)\(10\!\cdots\!20\)\( T + \)\(13\!\cdots\!50\)\( T^{2} - \)\(59\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!09\)\( T^{4} \)
$41$ \( 1 - \)\(27\!\cdots\!44\)\( T + \)\(22\!\cdots\!26\)\( T^{2} - \)\(46\!\cdots\!24\)\( T^{3} + \)\(27\!\cdots\!41\)\( T^{4} \)
$43$ \( 1 + \)\(15\!\cdots\!00\)\( T + \)\(12\!\cdots\!50\)\( T^{2} + \)\(12\!\cdots\!00\)\( T^{3} + \)\(64\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 + \)\(54\!\cdots\!40\)\( T + \)\(37\!\cdots\!50\)\( T^{2} + \)\(81\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!29\)\( T^{4} \)
$53$ \( 1 - \)\(26\!\cdots\!20\)\( T + \)\(12\!\cdots\!50\)\( T^{2} - \)\(21\!\cdots\!60\)\( T^{3} + \)\(63\!\cdots\!29\)\( T^{4} \)
$59$ \( 1 + \)\(30\!\cdots\!00\)\( T + \)\(77\!\cdots\!58\)\( T^{2} + \)\(83\!\cdots\!00\)\( T^{3} + \)\(75\!\cdots\!41\)\( T^{4} \)
$61$ \( 1 + \)\(57\!\cdots\!36\)\( T + \)\(16\!\cdots\!86\)\( T^{2} + \)\(47\!\cdots\!16\)\( T^{3} + \)\(67\!\cdots\!61\)\( T^{4} \)
$67$ \( 1 + \)\(15\!\cdots\!60\)\( T + \)\(41\!\cdots\!50\)\( T^{2} + \)\(29\!\cdots\!20\)\( T^{3} + \)\(33\!\cdots\!69\)\( T^{4} \)
$71$ \( 1 + \)\(26\!\cdots\!76\)\( T + \)\(22\!\cdots\!66\)\( T^{2} + \)\(32\!\cdots\!36\)\( T^{3} + \)\(15\!\cdots\!21\)\( T^{4} \)
$73$ \( 1 + \)\(94\!\cdots\!40\)\( T + \)\(19\!\cdots\!50\)\( T^{2} + \)\(29\!\cdots\!20\)\( T^{3} + \)\(95\!\cdots\!89\)\( T^{4} \)
$79$ \( 1 + \)\(85\!\cdots\!00\)\( T + \)\(85\!\cdots\!78\)\( T^{2} + \)\(35\!\cdots\!00\)\( T^{3} + \)\(17\!\cdots\!21\)\( T^{4} \)
$83$ \( 1 + \)\(29\!\cdots\!20\)\( T + \)\(37\!\cdots\!50\)\( T^{2} + \)\(62\!\cdots\!60\)\( T^{3} + \)\(45\!\cdots\!69\)\( T^{4} \)
$89$ \( 1 - \)\(13\!\cdots\!00\)\( T + \)\(47\!\cdots\!38\)\( T^{2} - \)\(28\!\cdots\!00\)\( T^{3} + \)\(45\!\cdots\!61\)\( T^{4} \)
$97$ \( 1 - \)\(36\!\cdots\!60\)\( T + \)\(42\!\cdots\!50\)\( T^{2} - \)\(13\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!29\)\( T^{4} \)
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