Properties

Label 25.32.a.a
Level $25$
Weight $32$
Character orbit 25.a
Self dual yes
Analytic conductor $152.193$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 32 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(152.192832048\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
Defining polynomial: \(x^{2} - x - 4573872\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
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 \(\beta = 12\sqrt{18295489}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -19980 - \beta ) q^{2} + ( -8681580 - 432 \beta ) q^{3} + ( 886267168 + 39960 \beta ) q^{4} + ( 1311583748112 + 17312940 \beta ) q^{6} + ( -15128763788600 + 71928864 \beta ) q^{7} + ( -80077529352960 + 462815680 \beta ) q^{8} + ( -50633228151963 + 7500885120 \beta ) q^{9} +O(q^{10})\) \( q +(-19980 - \beta) q^{2} +(-8681580 - 432 \beta) q^{3} +(886267168 + 39960 \beta) q^{4} +(1311583748112 + 17312940 \beta) q^{6} +(-15128763788600 + 71928864 \beta) q^{7} +(-80077529352960 + 462815680 \beta) q^{8} +(-50633228151963 + 7500885120 \beta) q^{9} +(-3891176872559388 + 29000909200 \beta) q^{11} +(-53173705477656960 - 729783353376 \beta) q^{12} +(-37354476525130310 + 4661016429888 \beta) q^{13} +(112772481922620576 + 13691625085880 \beta) q^{14} +(-1522606456842450944 - 14982974507520 \beta) q^{16} +(-8612303914493544690 - 45085348093056 \beta) q^{17} +(-18749808114787989180 - 99234456545637 \beta) q^{18} +(-6185281664011082020 - 157805792764560 \beta) q^{19} +(49477478708035580832 + 5911169769550080 \beta) q^{21} +(1341356516498345040 + 3311738706743388 \beta) q^{22} +(-\)\(94\!\cdots\!40\)\( + 18557618179251808 \beta) q^{23} +(\)\(16\!\cdots\!40\)\( + 30575521329304320 \beta) q^{24} +(-\)\(11\!\cdots\!08\)\( - 55772631744031930 \beta) q^{26} +(-\)\(27\!\cdots\!40\)\( + 223588927516223520 \beta) q^{27} +(-\)\(58\!\cdots\!60\)\( - 540797210397718848 \beta) q^{28} +(\)\(64\!\cdots\!70\)\( + 234192357384448960 \beta) q^{29} +(\)\(62\!\cdots\!32\)\( + 2412621171020361600 \beta) q^{31} +(\)\(24\!\cdots\!20\)\( + 828077182664699904 \beta) q^{32} +(\)\(77\!\cdots\!40\)\( + 1429214695653119616 \beta) q^{33} +(\)\(29\!\cdots\!96\)\( + 9513109169392803570 \beta) q^{34} +(\)\(74\!\cdots\!16\)\( + 4624484415843298680 \beta) q^{36} +(\)\(41\!\cdots\!30\)\( - 32224244113578511296 \beta) q^{37} +(\)\(53\!\cdots\!60\)\( + 9338241403446990820 \beta) q^{38} +(-\)\(49\!\cdots\!56\)\( - 24327853158530769120 \beta) q^{39} +(\)\(43\!\cdots\!42\)\( + \)\(18\!\cdots\!00\)\( \beta) q^{41} +(-\)\(16\!\cdots\!40\)\( - \)\(16\!\cdots\!32\)\( \beta) q^{42} +(\)\(91\!\cdots\!00\)\( + \)\(34\!\cdots\!48\)\( \beta) q^{43} +(-\)\(39\!\cdots\!84\)\( - \)\(12\!\cdots\!80\)\( \beta) q^{44} +(-\)\(29\!\cdots\!28\)\( + \)\(57\!\cdots\!00\)\( \beta) q^{46} +(-\)\(47\!\cdots\!60\)\( - \)\(51\!\cdots\!16\)\( \beta) q^{47} +(\)\(30\!\cdots\!60\)\( + \)\(78\!\cdots\!08\)\( \beta) q^{48} +(\)\(84\!\cdots\!93\)\( - \)\(21\!\cdots\!00\)\( \beta) q^{49} +(\)\(12\!\cdots\!72\)\( + \)\(41\!\cdots\!60\)\( \beta) q^{51} +(\)\(45\!\cdots\!00\)\( + \)\(26\!\cdots\!84\)\( \beta) q^{52} +(-\)\(97\!\cdots\!30\)\( + \)\(12\!\cdots\!68\)\( \beta) q^{53} +(-\)\(53\!\cdots\!20\)\( - \)\(17\!\cdots\!60\)\( \beta) q^{54} +(\)\(12\!\cdots\!20\)\( - \)\(12\!\cdots\!40\)\( \beta) q^{56} +(\)\(23\!\cdots\!20\)\( + \)\(40\!\cdots\!40\)\( \beta) q^{57} +(-\)\(19\!\cdots\!60\)\( - \)\(68\!\cdots\!70\)\( \beta) q^{58} +(-\)\(99\!\cdots\!60\)\( - \)\(44\!\cdots\!80\)\( \beta) q^{59} +(-\)\(60\!\cdots\!38\)\( + \)\(11\!\cdots\!00\)\( \beta) q^{61} +(-\)\(76\!\cdots\!60\)\( - \)\(11\!\cdots\!32\)\( \beta) q^{62} +(\)\(21\!\cdots\!80\)\( - \)\(11\!\cdots\!32\)\( \beta) q^{63} +(-\)\(37\!\cdots\!52\)\( - \)\(22\!\cdots\!80\)\( \beta) q^{64} +(-\)\(37\!\cdots\!56\)\( - \)\(29\!\cdots\!20\)\( \beta) q^{66} +(\)\(48\!\cdots\!60\)\( + \)\(88\!\cdots\!44\)\( \beta) q^{67} +(-\)\(12\!\cdots\!80\)\( - \)\(38\!\cdots\!08\)\( \beta) q^{68} +(-\)\(12\!\cdots\!96\)\( + \)\(24\!\cdots\!40\)\( \beta) q^{69} +(\)\(27\!\cdots\!72\)\( - \)\(95\!\cdots\!00\)\( \beta) q^{71} +(\)\(13\!\cdots\!80\)\( - \)\(62\!\cdots\!40\)\( \beta) q^{72} +(-\)\(31\!\cdots\!90\)\( - \)\(15\!\cdots\!92\)\( \beta) q^{73} +(\)\(76\!\cdots\!36\)\( + \)\(22\!\cdots\!50\)\( \beta) q^{74} +(-\)\(22\!\cdots\!60\)\( - \)\(38\!\cdots\!80\)\( \beta) q^{76} +(\)\(64\!\cdots\!00\)\( - \)\(71\!\cdots\!32\)\( \beta) q^{77} +(\)\(16\!\cdots\!00\)\( + \)\(54\!\cdots\!56\)\( \beta) q^{78} +(-\)\(59\!\cdots\!80\)\( + \)\(46\!\cdots\!60\)\( \beta) q^{79} +(-\)\(19\!\cdots\!79\)\( - \)\(53\!\cdots\!60\)\( \beta) q^{81} +(-\)\(57\!\cdots\!60\)\( - \)\(80\!\cdots\!42\)\( \beta) q^{82} +(\)\(13\!\cdots\!80\)\( + \)\(62\!\cdots\!28\)\( \beta) q^{83} +(\)\(66\!\cdots\!76\)\( + \)\(72\!\cdots\!60\)\( \beta) q^{84} +(-\)\(10\!\cdots\!68\)\( - \)\(16\!\cdots\!40\)\( \beta) q^{86} +(-\)\(82\!\cdots\!20\)\( - \)\(29\!\cdots\!40\)\( \beta) q^{87} +(\)\(34\!\cdots\!80\)\( - \)\(41\!\cdots\!40\)\( \beta) q^{88} +(-\)\(10\!\cdots\!90\)\( + \)\(26\!\cdots\!80\)\( \beta) q^{89} +(\)\(14\!\cdots\!12\)\( - \)\(73\!\cdots\!40\)\( \beta) q^{91} +(\)\(11\!\cdots\!60\)\( - \)\(21\!\cdots\!56\)\( \beta) q^{92} +(-\)\(32\!\cdots\!60\)\( - \)\(48\!\cdots\!24\)\( \beta) q^{93} +(\)\(23\!\cdots\!56\)\( + \)\(57\!\cdots\!40\)\( \beta) q^{94} +(-\)\(30\!\cdots\!48\)\( - \)\(11\!\cdots\!60\)\( \beta) q^{96} +(\)\(45\!\cdots\!90\)\( + \)\(16\!\cdots\!84\)\( \beta) q^{97} +(\)\(40\!\cdots\!60\)\( - \)\(41\!\cdots\!93\)\( \beta) q^{98} +(\)\(77\!\cdots\!44\)\( - \)\(30\!\cdots\!60\)\( \beta) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 39960q^{2} - 17363160q^{3} + 1772534336q^{4} + 2623167496224q^{6} - 30257527577200q^{7} - 160155058705920q^{8} - 101266456303926q^{9} + O(q^{10}) \) \( 2q - 39960q^{2} - 17363160q^{3} + 1772534336q^{4} + 2623167496224q^{6} - 30257527577200q^{7} - 160155058705920q^{8} - 101266456303926q^{9} - 7782353745118776q^{11} - 106347410955313920q^{12} - 74708953050260620q^{13} + 225544963845241152q^{14} - 3045212913684901888q^{16} - 17224607828987089380q^{17} - 37499616229575978360q^{18} - 12370563328022164040q^{19} + 98954957416071161664q^{21} + 2682713032996690080q^{22} - \)\(18\!\cdots\!80\)\(q^{23} + \)\(33\!\cdots\!80\)\(q^{24} - \)\(23\!\cdots\!16\)\(q^{26} - \)\(54\!\cdots\!80\)\(q^{27} - \)\(11\!\cdots\!20\)\(q^{28} + \)\(12\!\cdots\!40\)\(q^{29} + \)\(12\!\cdots\!64\)\(q^{31} + \)\(48\!\cdots\!40\)\(q^{32} + \)\(15\!\cdots\!80\)\(q^{33} + \)\(58\!\cdots\!92\)\(q^{34} + \)\(14\!\cdots\!32\)\(q^{36} + \)\(83\!\cdots\!60\)\(q^{37} + \)\(10\!\cdots\!20\)\(q^{38} - \)\(99\!\cdots\!12\)\(q^{39} + \)\(87\!\cdots\!84\)\(q^{41} - \)\(33\!\cdots\!80\)\(q^{42} + \)\(18\!\cdots\!00\)\(q^{43} - \)\(79\!\cdots\!68\)\(q^{44} - \)\(59\!\cdots\!56\)\(q^{46} - \)\(95\!\cdots\!20\)\(q^{47} + \)\(60\!\cdots\!20\)\(q^{48} + \)\(16\!\cdots\!86\)\(q^{49} + \)\(25\!\cdots\!44\)\(q^{51} + \)\(91\!\cdots\!00\)\(q^{52} - \)\(19\!\cdots\!60\)\(q^{53} - \)\(10\!\cdots\!40\)\(q^{54} + \)\(25\!\cdots\!40\)\(q^{56} + \)\(46\!\cdots\!40\)\(q^{57} - \)\(38\!\cdots\!20\)\(q^{58} - \)\(19\!\cdots\!20\)\(q^{59} - \)\(12\!\cdots\!76\)\(q^{61} - \)\(15\!\cdots\!20\)\(q^{62} + \)\(43\!\cdots\!60\)\(q^{63} - \)\(74\!\cdots\!04\)\(q^{64} - \)\(75\!\cdots\!12\)\(q^{66} + \)\(96\!\cdots\!20\)\(q^{67} - \)\(24\!\cdots\!60\)\(q^{68} - \)\(25\!\cdots\!92\)\(q^{69} + \)\(55\!\cdots\!44\)\(q^{71} + \)\(26\!\cdots\!60\)\(q^{72} - \)\(62\!\cdots\!80\)\(q^{73} + \)\(15\!\cdots\!72\)\(q^{74} - \)\(44\!\cdots\!20\)\(q^{76} + \)\(12\!\cdots\!00\)\(q^{77} + \)\(32\!\cdots\!00\)\(q^{78} - \)\(11\!\cdots\!60\)\(q^{79} - \)\(39\!\cdots\!58\)\(q^{81} - \)\(11\!\cdots\!20\)\(q^{82} + \)\(26\!\cdots\!60\)\(q^{83} + \)\(13\!\cdots\!52\)\(q^{84} - \)\(21\!\cdots\!36\)\(q^{86} - \)\(16\!\cdots\!40\)\(q^{87} + \)\(69\!\cdots\!60\)\(q^{88} - \)\(21\!\cdots\!80\)\(q^{89} + \)\(28\!\cdots\!24\)\(q^{91} + \)\(22\!\cdots\!20\)\(q^{92} - \)\(65\!\cdots\!20\)\(q^{93} + \)\(46\!\cdots\!12\)\(q^{94} - \)\(60\!\cdots\!96\)\(q^{96} + \)\(90\!\cdots\!80\)\(q^{97} + \)\(80\!\cdots\!20\)\(q^{98} + \)\(15\!\cdots\!88\)\(q^{99} + O(q^{100}) \)

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
2139.16
−2138.16
−71307.9 −3.08552e7 2.93733e9 0 2.20022e12 −1.14368e13 −5.63222e13 3.34371e14 0
1.2 31347.9 1.34921e7 −1.16479e9 0 4.22947e11 −1.88207e13 −1.03833e14 −4.35638e14 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 25.32.a.a 2
5.b even 2 1 1.32.a.a 2
5.c odd 4 2 25.32.b.a 4
15.d odd 2 1 9.32.a.a 2
20.d odd 2 1 16.32.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.32.a.a 2 5.b even 2 1
9.32.a.a 2 15.d odd 2 1
16.32.a.b 2 20.d odd 2 1
25.32.a.a 2 1.a even 1 1 trivial
25.32.b.a 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 39960 T_{2} - 2235350016 \) acting on \(S_{32}^{\mathrm{new}}(\Gamma_0(25))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -2235350016 + 39960 T + T^{2} \)
$3$ \( -416300505539184 + 17363160 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( \)\(21\!\cdots\!64\)\( + 30257527577200 T + T^{2} \)
$11$ \( \)\(12\!\cdots\!44\)\( + 7782353745118776 T + T^{2} \)
$13$ \( -\)\(55\!\cdots\!04\)\( + 74708953050260620 T + T^{2} \)
$17$ \( \)\(68\!\cdots\!24\)\( + 17224607828987089380 T + T^{2} \)
$19$ \( -\)\(27\!\cdots\!00\)\( + 12370563328022164040 T + T^{2} \)
$23$ \( -\)\(73\!\cdots\!24\)\( + \)\(18\!\cdots\!80\)\( T + T^{2} \)
$29$ \( \)\(39\!\cdots\!00\)\( - \)\(12\!\cdots\!40\)\( T + T^{2} \)
$31$ \( -\)\(11\!\cdots\!76\)\( - \)\(12\!\cdots\!64\)\( T + T^{2} \)
$37$ \( -\)\(25\!\cdots\!56\)\( - \)\(83\!\cdots\!60\)\( T + T^{2} \)
$41$ \( -\)\(70\!\cdots\!36\)\( - \)\(87\!\cdots\!84\)\( T + T^{2} \)
$43$ \( -\)\(22\!\cdots\!64\)\( - \)\(18\!\cdots\!00\)\( T + T^{2} \)
$47$ \( \)\(15\!\cdots\!04\)\( + \)\(95\!\cdots\!20\)\( T + T^{2} \)
$53$ \( \)\(56\!\cdots\!16\)\( + \)\(19\!\cdots\!60\)\( T + T^{2} \)
$59$ \( -\)\(51\!\cdots\!00\)\( + \)\(19\!\cdots\!20\)\( T + T^{2} \)
$61$ \( \)\(36\!\cdots\!44\)\( + \)\(12\!\cdots\!76\)\( T + T^{2} \)
$67$ \( \)\(26\!\cdots\!24\)\( - \)\(96\!\cdots\!20\)\( T + T^{2} \)
$71$ \( -\)\(16\!\cdots\!16\)\( - \)\(55\!\cdots\!44\)\( T + T^{2} \)
$73$ \( \)\(90\!\cdots\!76\)\( + \)\(62\!\cdots\!80\)\( T + T^{2} \)
$79$ \( -\)\(53\!\cdots\!00\)\( + \)\(11\!\cdots\!60\)\( T + T^{2} \)
$83$ \( -\)\(86\!\cdots\!44\)\( - \)\(26\!\cdots\!60\)\( T + T^{2} \)
$89$ \( -\)\(62\!\cdots\!00\)\( + \)\(21\!\cdots\!80\)\( T + T^{2} \)
$97$ \( \)\(19\!\cdots\!04\)\( - \)\(90\!\cdots\!80\)\( T + T^{2} \)
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