Properties

Label 25.38.a.a
Level $25$
Weight $38$
Character orbit 25.a
Self dual yes
Analytic conductor $216.785$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 38 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(216.785095312\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 15934380\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
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 \(\beta = 48\sqrt{63737521}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 97200 - \beta ) q^{2} + ( -6995700 + 72 \beta ) q^{3} + ( 18860134912 - 194400 \beta ) q^{4} + ( -11253271923648 + 13994100 \beta ) q^{6} + ( 1724221976743000 - 9650004336 \beta ) q^{7} + ( 17022021521817600 + 99683138560 \beta ) q^{8} + ( -449473689200884707 - 1007380800 \beta ) q^{9} +O(q^{10})\) \( q +(97200 - \beta) q^{2} +(-6995700 + 72 \beta) q^{3} +(18860134912 - 194400 \beta) q^{4} +(-11253271923648 + 13994100 \beta) q^{6} +(1724221976743000 - 9650004336 \beta) q^{7} +(17022021521817600 + 99683138560 \beta) q^{8} +(-449473689200884707 - 1007380800 \beta) q^{9} +(-13367018177424269028 - 32186306471000 \beta) q^{11} +(-2187387399185049600 + 2717893793664 \beta) q^{12} +(-\)\(26\!\cdots\!50\)\( - 724443400725408 \beta) q^{13} +(\)\(15\!\cdots\!24\)\( - 2662202398202200 \beta) q^{14} +(-\)\(15\!\cdots\!04\)\( + 19385312101171200 \beta) q^{16} +(\)\(44\!\cdots\!50\)\( - 24633489938571456 \beta) q^{17} +(-\)\(43\!\cdots\!00\)\( + 449375771787124707 \beta) q^{18} +(\)\(18\!\cdots\!60\)\( + 1292976182287157400 \beta) q^{19} +(-\)\(11\!\cdots\!28\)\( + 191652517658851200 \beta) q^{21} +(\)\(34\!\cdots\!00\)\( + 10238509188443069028 \beta) q^{22} +(\)\(13\!\cdots\!00\)\( - 34024622863285905488 \beta) q^{23} +(\)\(93\!\cdots\!80\)\( + 528232217146675200 \beta) q^{24} +(\)\(80\!\cdots\!72\)\( + \)\(19\!\cdots\!50\)\( \beta) q^{26} +(\)\(62\!\cdots\!00\)\( - 64775499512752949040 \beta) q^{27} +(\)\(30\!\cdots\!00\)\( - \)\(51\!\cdots\!32\)\( \beta) q^{28} +(-\)\(63\!\cdots\!10\)\( - \)\(23\!\cdots\!00\)\( \beta) q^{29} +(\)\(13\!\cdots\!12\)\( + \)\(64\!\cdots\!00\)\( \beta) q^{31} +(-\)\(67\!\cdots\!00\)\( + \)\(37\!\cdots\!84\)\( \beta) q^{32} +(-\)\(24\!\cdots\!00\)\( - \)\(73\!\cdots\!16\)\( \beta) q^{33} +(\)\(79\!\cdots\!04\)\( - \)\(47\!\cdots\!50\)\( \beta) q^{34} +(-\)\(84\!\cdots\!84\)\( + \)\(87\!\cdots\!00\)\( \beta) q^{36} +(\)\(34\!\cdots\!50\)\( + \)\(25\!\cdots\!04\)\( \beta) q^{37} +(-\)\(17\!\cdots\!00\)\( - \)\(60\!\cdots\!60\)\( \beta) q^{38} +(-\)\(58\!\cdots\!84\)\( - \)\(14\!\cdots\!00\)\( \beta) q^{39} +(-\)\(63\!\cdots\!18\)\( + \)\(58\!\cdots\!00\)\( \beta) q^{41} +(-\)\(39\!\cdots\!00\)\( + \)\(13\!\cdots\!28\)\( \beta) q^{42} +(\)\(12\!\cdots\!00\)\( + \)\(42\!\cdots\!52\)\( \beta) q^{43} +(\)\(66\!\cdots\!64\)\( + \)\(19\!\cdots\!00\)\( \beta) q^{44} +(\)\(62\!\cdots\!92\)\( - \)\(16\!\cdots\!00\)\( \beta) q^{46} +(-\)\(21\!\cdots\!00\)\( - \)\(38\!\cdots\!16\)\( \beta) q^{47} +(\)\(31\!\cdots\!00\)\( - \)\(12\!\cdots\!88\)\( \beta) q^{48} +(-\)\(19\!\cdots\!43\)\( - \)\(33\!\cdots\!00\)\( \beta) q^{49} +(-\)\(57\!\cdots\!88\)\( + \)\(33\!\cdots\!00\)\( \beta) q^{51} +(\)\(15\!\cdots\!00\)\( + \)\(37\!\cdots\!04\)\( \beta) q^{52} +(-\)\(79\!\cdots\!50\)\( + \)\(11\!\cdots\!72\)\( \beta) q^{53} +(\)\(10\!\cdots\!60\)\( - \)\(12\!\cdots\!00\)\( \beta) q^{54} +(-\)\(11\!\cdots\!40\)\( + \)\(76\!\cdots\!00\)\( \beta) q^{56} +(\)\(12\!\cdots\!00\)\( + \)\(43\!\cdots\!20\)\( \beta) q^{57} +(\)\(27\!\cdots\!00\)\( + \)\(41\!\cdots\!10\)\( \beta) q^{58} +(-\)\(11\!\cdots\!20\)\( - \)\(48\!\cdots\!00\)\( \beta) q^{59} +(\)\(50\!\cdots\!22\)\( + \)\(28\!\cdots\!00\)\( \beta) q^{61} +(-\)\(93\!\cdots\!00\)\( + \)\(49\!\cdots\!88\)\( \beta) q^{62} +(-\)\(77\!\cdots\!00\)\( + \)\(43\!\cdots\!52\)\( \beta) q^{63} +(\)\(93\!\cdots\!32\)\( + \)\(44\!\cdots\!00\)\( \beta) q^{64} +(\)\(84\!\cdots\!44\)\( + \)\(17\!\cdots\!00\)\( \beta) q^{66} +(\)\(54\!\cdots\!00\)\( - \)\(78\!\cdots\!56\)\( \beta) q^{67} +(\)\(15\!\cdots\!00\)\( - \)\(91\!\cdots\!72\)\( \beta) q^{68} +(-\)\(45\!\cdots\!24\)\( + \)\(11\!\cdots\!00\)\( \beta) q^{69} +(-\)\(37\!\cdots\!08\)\( + \)\(17\!\cdots\!00\)\( \beta) q^{71} +(-\)\(76\!\cdots\!00\)\( - \)\(44\!\cdots\!20\)\( \beta) q^{72} +(-\)\(98\!\cdots\!50\)\( - \)\(11\!\cdots\!88\)\( \beta) q^{73} +(-\)\(33\!\cdots\!36\)\( - \)\(96\!\cdots\!50\)\( \beta) q^{74} +(-\)\(33\!\cdots\!80\)\( - \)\(11\!\cdots\!00\)\( \beta) q^{76} +(\)\(22\!\cdots\!00\)\( + \)\(73\!\cdots\!08\)\( \beta) q^{77} +(\)\(14\!\cdots\!00\)\( + \)\(44\!\cdots\!84\)\( \beta) q^{78} +(\)\(13\!\cdots\!40\)\( - \)\(40\!\cdots\!00\)\( \beta) q^{79} +(\)\(20\!\cdots\!21\)\( + \)\(13\!\cdots\!00\)\( \beta) q^{81} +(-\)\(14\!\cdots\!00\)\( + \)\(68\!\cdots\!18\)\( \beta) q^{82} +(\)\(23\!\cdots\!00\)\( + \)\(23\!\cdots\!32\)\( \beta) q^{83} +(-\)\(76\!\cdots\!36\)\( + \)\(25\!\cdots\!00\)\( \beta) q^{84} +(-\)\(50\!\cdots\!68\)\( - \)\(86\!\cdots\!00\)\( \beta) q^{86} +(-\)\(19\!\cdots\!00\)\( - \)\(29\!\cdots\!20\)\( \beta) q^{87} +(-\)\(69\!\cdots\!00\)\( - \)\(18\!\cdots\!80\)\( \beta) q^{88} +(-\)\(66\!\cdots\!30\)\( - \)\(23\!\cdots\!00\)\( \beta) q^{89} +(\)\(56\!\cdots\!92\)\( + \)\(13\!\cdots\!00\)\( \beta) q^{91} +(\)\(12\!\cdots\!00\)\( - \)\(31\!\cdots\!56\)\( \beta) q^{92} +(\)\(67\!\cdots\!00\)\( - \)\(35\!\cdots\!36\)\( \beta) q^{93} +(\)\(35\!\cdots\!44\)\( + \)\(17\!\cdots\!00\)\( \beta) q^{94} +(\)\(86\!\cdots\!32\)\( - \)\(50\!\cdots\!00\)\( \beta) q^{96} +(-\)\(30\!\cdots\!50\)\( + \)\(10\!\cdots\!84\)\( \beta) q^{97} +(\)\(47\!\cdots\!00\)\( - \)\(13\!\cdots\!57\)\( \beta) q^{98} +(\)\(60\!\cdots\!96\)\( + \)\(14\!\cdots\!00\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 194400q^{2} - 13991400q^{3} + 37720269824q^{4} - 22506543847296q^{6} + 3448443953486000q^{7} + 34044043043635200q^{8} - 898947378401769414q^{9} + O(q^{10}) \) \( 2q + 194400q^{2} - 13991400q^{3} + 37720269824q^{4} - 22506543847296q^{6} + 3448443953486000q^{7} + 34044043043635200q^{8} - 898947378401769414q^{9} - 26734036354848538056q^{11} - 4374774798370099200q^{12} - \)\(53\!\cdots\!00\)\(q^{13} + \)\(31\!\cdots\!48\)\(q^{14} - \)\(31\!\cdots\!08\)\(q^{16} + \)\(89\!\cdots\!00\)\(q^{17} - \)\(87\!\cdots\!00\)\(q^{18} + \)\(37\!\cdots\!20\)\(q^{19} - \)\(22\!\cdots\!56\)\(q^{21} + \)\(68\!\cdots\!00\)\(q^{22} + \)\(26\!\cdots\!00\)\(q^{23} + \)\(18\!\cdots\!60\)\(q^{24} + \)\(16\!\cdots\!44\)\(q^{26} + \)\(12\!\cdots\!00\)\(q^{27} + \)\(61\!\cdots\!00\)\(q^{28} - \)\(12\!\cdots\!20\)\(q^{29} + \)\(26\!\cdots\!24\)\(q^{31} - \)\(13\!\cdots\!00\)\(q^{32} - \)\(49\!\cdots\!00\)\(q^{33} + \)\(15\!\cdots\!08\)\(q^{34} - \)\(16\!\cdots\!68\)\(q^{36} + \)\(68\!\cdots\!00\)\(q^{37} - \)\(34\!\cdots\!00\)\(q^{38} - \)\(11\!\cdots\!68\)\(q^{39} - \)\(12\!\cdots\!36\)\(q^{41} - \)\(78\!\cdots\!00\)\(q^{42} + \)\(25\!\cdots\!00\)\(q^{43} + \)\(13\!\cdots\!28\)\(q^{44} + \)\(12\!\cdots\!84\)\(q^{46} - \)\(42\!\cdots\!00\)\(q^{47} + \)\(62\!\cdots\!00\)\(q^{48} - \)\(38\!\cdots\!86\)\(q^{49} - \)\(11\!\cdots\!76\)\(q^{51} + \)\(31\!\cdots\!00\)\(q^{52} - \)\(15\!\cdots\!00\)\(q^{53} + \)\(20\!\cdots\!20\)\(q^{54} - \)\(22\!\cdots\!80\)\(q^{56} + \)\(24\!\cdots\!00\)\(q^{57} + \)\(55\!\cdots\!00\)\(q^{58} - \)\(23\!\cdots\!40\)\(q^{59} + \)\(10\!\cdots\!44\)\(q^{61} - \)\(18\!\cdots\!00\)\(q^{62} - \)\(15\!\cdots\!00\)\(q^{63} + \)\(18\!\cdots\!64\)\(q^{64} + \)\(16\!\cdots\!88\)\(q^{66} + \)\(10\!\cdots\!00\)\(q^{67} + \)\(30\!\cdots\!00\)\(q^{68} - \)\(90\!\cdots\!48\)\(q^{69} - \)\(74\!\cdots\!16\)\(q^{71} - \)\(15\!\cdots\!00\)\(q^{72} - \)\(19\!\cdots\!00\)\(q^{73} - \)\(67\!\cdots\!72\)\(q^{74} - \)\(66\!\cdots\!60\)\(q^{76} + \)\(45\!\cdots\!00\)\(q^{77} + \)\(29\!\cdots\!00\)\(q^{78} + \)\(27\!\cdots\!80\)\(q^{79} + \)\(40\!\cdots\!42\)\(q^{81} - \)\(29\!\cdots\!00\)\(q^{82} + \)\(47\!\cdots\!00\)\(q^{83} - \)\(15\!\cdots\!72\)\(q^{84} - \)\(10\!\cdots\!36\)\(q^{86} - \)\(39\!\cdots\!00\)\(q^{87} - \)\(13\!\cdots\!00\)\(q^{88} - \)\(13\!\cdots\!60\)\(q^{89} + \)\(11\!\cdots\!84\)\(q^{91} + \)\(24\!\cdots\!00\)\(q^{92} + \)\(13\!\cdots\!00\)\(q^{93} + \)\(70\!\cdots\!88\)\(q^{94} + \)\(17\!\cdots\!64\)\(q^{96} - \)\(60\!\cdots\!00\)\(q^{97} + \)\(94\!\cdots\!00\)\(q^{98} + \)\(12\!\cdots\!92\)\(q^{99} + O(q^{100}) \)

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
3992.29
−3991.29
−286012. 2.05955e7 −5.56362e10 0 −5.89057e12 −1.97377e15 5.52218e16 −4.49860e17 0
1.2 480412. −3.45869e7 9.33565e10 0 −1.66160e13 5.42222e15 −2.11777e16 −4.49088e17 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 25.38.a.a 2
5.b even 2 1 1.38.a.a 2
5.c odd 4 2 25.38.b.a 4
15.d odd 2 1 9.38.a.a 2
20.d odd 2 1 16.38.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.38.a.a 2 5.b even 2 1
9.38.a.a 2 15.d odd 2 1
16.38.a.b 2 20.d odd 2 1
25.38.a.a 2 1.a even 1 1 trivial
25.38.b.a 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 194400 T_{2} - 137403408384 \) acting on \(S_{38}^{\mathrm{new}}(\Gamma_0(25))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 194400 T + 137474498560 T^{2} - 26718132554956800 T^{3} + \)\(18\!\cdots\!84\)\( T^{4} \)
$3$ \( 1 + 13991400 T + 899855474728862070 T^{2} + \)\(63\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!69\)\( T^{4} \)
$5$ 1
$7$ \( 1 - 3448443953486000 T + \)\(26\!\cdots\!50\)\( T^{2} - \)\(64\!\cdots\!00\)\( T^{3} + \)\(34\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 + 26734036354848538056 T + \)\(70\!\cdots\!26\)\( T^{2} + \)\(90\!\cdots\!76\)\( T^{3} + \)\(11\!\cdots\!41\)\( T^{4} \)
$13$ \( 1 + \)\(53\!\cdots\!00\)\( T + \)\(32\!\cdots\!90\)\( T^{2} + \)\(87\!\cdots\!00\)\( T^{3} + \)\(27\!\cdots\!89\)\( T^{4} \)
$17$ \( 1 - \)\(89\!\cdots\!00\)\( T + \)\(86\!\cdots\!30\)\( T^{2} - \)\(30\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!29\)\( T^{4} \)
$19$ \( 1 - \)\(37\!\cdots\!20\)\( T + \)\(20\!\cdots\!78\)\( T^{2} - \)\(76\!\cdots\!80\)\( T^{3} + \)\(42\!\cdots\!21\)\( T^{4} \)
$23$ \( 1 - \)\(26\!\cdots\!00\)\( T + \)\(48\!\cdots\!10\)\( T^{2} - \)\(63\!\cdots\!00\)\( T^{3} + \)\(58\!\cdots\!09\)\( T^{4} \)
$29$ \( 1 + \)\(12\!\cdots\!20\)\( T + \)\(21\!\cdots\!18\)\( T^{2} + \)\(16\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!81\)\( T^{4} \)
$31$ \( 1 - \)\(26\!\cdots\!24\)\( T + \)\(24\!\cdots\!66\)\( T^{2} - \)\(39\!\cdots\!64\)\( T^{3} + \)\(22\!\cdots\!21\)\( T^{4} \)
$37$ \( 1 - \)\(68\!\cdots\!00\)\( T + \)\(13\!\cdots\!90\)\( T^{2} - \)\(71\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \)
$41$ \( 1 + \)\(12\!\cdots\!36\)\( T + \)\(12\!\cdots\!86\)\( T^{2} + \)\(59\!\cdots\!16\)\( T^{3} + \)\(22\!\cdots\!61\)\( T^{4} \)
$43$ \( 1 - \)\(25\!\cdots\!00\)\( T + \)\(44\!\cdots\!50\)\( T^{2} - \)\(70\!\cdots\!00\)\( T^{3} + \)\(75\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 + \)\(42\!\cdots\!00\)\( T + \)\(14\!\cdots\!70\)\( T^{2} + \)\(31\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!69\)\( T^{4} \)
$53$ \( 1 + \)\(15\!\cdots\!00\)\( T + \)\(16\!\cdots\!70\)\( T^{2} + \)\(10\!\cdots\!00\)\( T^{3} + \)\(39\!\cdots\!69\)\( T^{4} \)
$59$ \( 1 + \)\(23\!\cdots\!40\)\( T + \)\(64\!\cdots\!38\)\( T^{2} + \)\(79\!\cdots\!60\)\( T^{3} + \)\(11\!\cdots\!61\)\( T^{4} \)
$61$ \( 1 - \)\(10\!\cdots\!44\)\( T + \)\(10\!\cdots\!26\)\( T^{2} - \)\(11\!\cdots\!24\)\( T^{3} + \)\(13\!\cdots\!41\)\( T^{4} \)
$67$ \( 1 - \)\(10\!\cdots\!00\)\( T + \)\(94\!\cdots\!30\)\( T^{2} - \)\(39\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!29\)\( T^{4} \)
$71$ \( 1 + \)\(74\!\cdots\!16\)\( T + \)\(58\!\cdots\!46\)\( T^{2} + \)\(23\!\cdots\!56\)\( T^{3} + \)\(98\!\cdots\!81\)\( T^{4} \)
$73$ \( 1 + \)\(19\!\cdots\!00\)\( T + \)\(18\!\cdots\!10\)\( T^{2} + \)\(17\!\cdots\!00\)\( T^{3} + \)\(76\!\cdots\!09\)\( T^{4} \)
$79$ \( 1 - \)\(27\!\cdots\!80\)\( T + \)\(50\!\cdots\!18\)\( T^{2} - \)\(44\!\cdots\!20\)\( T^{3} + \)\(26\!\cdots\!81\)\( T^{4} \)
$83$ \( 1 - \)\(47\!\cdots\!00\)\( T + \)\(24\!\cdots\!30\)\( T^{2} - \)\(47\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!29\)\( T^{4} \)
$89$ \( 1 + \)\(13\!\cdots\!60\)\( T + \)\(23\!\cdots\!58\)\( T^{2} + \)\(17\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!41\)\( T^{4} \)
$97$ \( 1 + \)\(60\!\cdots\!00\)\( T + \)\(57\!\cdots\!70\)\( T^{2} + \)\(19\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!69\)\( T^{4} \)
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