Properties

Label 225.10.a.s
Level 225
Weight 10
Character orbit 225.a
Self dual yes
Analytic conductor 115.883
Analytic rank 1
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(115.883063137\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.49740556.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} + ( 342 - \beta_{3} ) q^{4} + ( 133 \beta_{1} - 49 \beta_{2} ) q^{7} + ( -684 \beta_{1} + 8 \beta_{2} ) q^{8} +O(q^{10})\) \( q -\beta_{1} q^{2} + ( 342 - \beta_{3} ) q^{4} + ( 133 \beta_{1} - 49 \beta_{2} ) q^{7} + ( -684 \beta_{1} + 8 \beta_{2} ) q^{8} + ( -27492 - 20 \beta_{3} ) q^{11} + ( 1918 \beta_{1} + 110 \beta_{2} ) q^{13} + ( -106134 + 133 \beta_{3} ) q^{14} + ( 407816 - 172 \beta_{3} ) q^{16} + ( 7448 \beta_{1} + 1632 \beta_{2} ) q^{17} + ( 159220 + 476 \beta_{3} ) q^{19} + ( 10412 \beta_{1} + 160 \beta_{2} ) q^{22} + ( 17005 \beta_{1} + 5199 \beta_{2} ) q^{23} + ( -1654692 + 1918 \beta_{3} ) q^{26} + ( 151620 \beta_{1} + 24024 \beta_{2} ) q^{28} + ( -882930 + 1976 \beta_{3} ) q^{29} + ( -2646928 + 760 \beta_{3} ) q^{31} + ( -204496 \beta_{1} - 2720 \beta_{2} ) q^{32} + ( -6608656 + 7448 \beta_{3} ) q^{34} + ( -335370 \beta_{1} - 51378 \beta_{2} ) q^{37} + ( 247284 \beta_{1} - 3808 \beta_{2} ) q^{38} + ( 4197138 + 24890 \beta_{3} ) q^{41} + ( -719131 \beta_{1} + 132643 \beta_{2} ) q^{43} + ( 5159736 + 20652 \beta_{3} ) q^{44} + ( -15312518 + 17005 \beta_{3} ) q^{46} + ( 1012111 \beta_{1} - 214259 \beta_{2} ) q^{47} + ( 11730257 + 27930 \beta_{3} ) q^{49} + ( 2310648 \beta_{1} - 71664 \beta_{2} ) q^{52} + ( -1349666 \beta_{1} - 259794 \beta_{2} ) q^{53} + ( -78794520 + 83524 \beta_{3} ) q^{56} + ( 2570434 \beta_{1} - 15808 \beta_{2} ) q^{58} + ( -115207260 + 52972 \beta_{3} ) q^{59} + ( 90122642 + 43150 \beta_{3} ) q^{61} + ( 3295968 \beta_{1} - 6080 \beta_{2} ) q^{62} + ( -33748768 - 116432 \beta_{3} ) q^{64} + ( 6669647 \beta_{1} + 1448953 \beta_{2} ) q^{67} + ( 9155872 \beta_{1} - 895168 \beta_{2} ) q^{68} + ( 11902968 - 91200 \beta_{3} ) q^{71} + ( 1847940 \beta_{1} - 90564 \beta_{2} ) q^{73} + ( 294215436 - 335370 \beta_{3} ) q^{74} + ( -292122360 + 3572 \beta_{3} ) q^{76} + ( -1533756 \beta_{1} + 2162748 \beta_{2} ) q^{77} + ( 182010880 - 267976 \beta_{3} ) q^{79} + ( 17058922 \beta_{1} - 199120 \beta_{2} ) q^{82} + ( -12737657 \beta_{1} + 1180353 \beta_{2} ) q^{83} + ( 593976138 - 719131 \beta_{3} ) q^{86} + ( 7146128 \beta_{1} - 247136 \beta_{2} ) q^{88} + ( -395675190 - 185592 \beta_{3} ) q^{89} + ( 118330632 - 357504 \beta_{3} ) q^{91} + ( 21128228 \beta_{1} - 2797928 \beta_{2} ) q^{92} + ( -831775426 + 1012111 \beta_{3} ) q^{94} + ( 14786464 \beta_{1} - 1954216 \beta_{2} ) q^{97} + ( 12121963 \beta_{1} - 223440 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 1368q^{4} + O(q^{10}) \) \( 4q + 1368q^{4} - 109968q^{11} - 424536q^{14} + 1631264q^{16} + 636880q^{19} - 6618768q^{26} - 3531720q^{29} - 10587712q^{31} - 26434624q^{34} + 16788552q^{41} + 20638944q^{44} - 61250072q^{46} + 46921028q^{49} - 315178080q^{56} - 460829040q^{59} + 360490568q^{61} - 134995072q^{64} + 47611872q^{71} + 1176861744q^{74} - 1168489440q^{76} + 728043520q^{79} + 2375904552q^{86} - 1582700760q^{89} + 473322528q^{91} - 3327101704q^{94} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 45 x^{2} + 304\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( -\nu^{3} + 37 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\( \nu^{3} - 7 \nu \)
\(\beta_{3}\)\(=\)\( 60 \nu^{2} - 1350 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + 2 \beta_{1}\)\()/30\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} + 1350\)\()/60\)
\(\nu^{3}\)\(=\)\((\)\(37 \beta_{2} + 14 \beta_{1}\)\()/30\)

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
2.87724
6.05982
−6.05982
−2.87724
−41.3193 0 1195.29 0 0 5315.22 −28233.0 0 0
1.2 −0.843944 0 −511.288 0 0 −8712.99 863.597 0 0
1.3 0.843944 0 −511.288 0 0 8712.99 −863.597 0 0
1.4 41.3193 0 1195.29 0 0 −5315.22 28233.0 0 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 225.10.a.s 4
3.b odd 2 1 25.10.a.e 4
5.b even 2 1 inner 225.10.a.s 4
5.c odd 4 2 45.10.b.b 4
12.b even 2 1 400.10.a.ba 4
15.d odd 2 1 25.10.a.e 4
15.e even 4 2 5.10.b.a 4
60.h even 2 1 400.10.a.ba 4
60.l odd 4 2 80.10.c.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.10.b.a 4 15.e even 4 2
25.10.a.e 4 3.b odd 2 1
25.10.a.e 4 15.d odd 2 1
45.10.b.b 4 5.c odd 4 2
80.10.c.c 4 60.l odd 4 2
225.10.a.s 4 1.a even 1 1 trivial
225.10.a.s 4 5.b even 2 1 inner
400.10.a.ba 4 12.b even 2 1
400.10.a.ba 4 60.h even 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(225))\):

\( T_{2}^{4} - 1708 T_{2}^{2} + 1216 \)
\( T_{7}^{4} - 104167728 T_{7}^{2} + \)\(21\!\cdots\!96\)\( \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 340 T^{2} - 174912 T^{4} + 89128960 T^{6} + 68719476736 T^{8} \)
$3$ 1
$5$ 1
$7$ \( 1 + 57246700 T^{2} + 3508143545353398 T^{4} + \)\(93\!\cdots\!00\)\( T^{6} + \)\(26\!\cdots\!01\)\( T^{8} \)
$11$ \( ( 1 + 54984 T + 5180465446 T^{2} + 129649395841944 T^{3} + 5559917313492231481 T^{4} )^{2} \)
$13$ \( 1 + 35613791860 T^{2} + \)\(53\!\cdots\!58\)\( T^{4} + \)\(40\!\cdots\!40\)\( T^{6} + \)\(12\!\cdots\!41\)\( T^{8} \)
$17$ \( 1 + 285780369220 T^{2} + \)\(48\!\cdots\!18\)\( T^{4} + \)\(40\!\cdots\!80\)\( T^{6} + \)\(19\!\cdots\!81\)\( T^{8} \)
$19$ \( ( 1 - 318440 T + 505756418358 T^{2} - 102756670480744760 T^{3} + \)\(10\!\cdots\!41\)\( T^{4} )^{2} \)
$23$ \( 1 + 5779790962540 T^{2} + \)\(14\!\cdots\!38\)\( T^{4} + \)\(18\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!61\)\( T^{8} \)
$29$ \( ( 1 + 1765860 T + 26950935551038 T^{2} + 25617588792948032340 T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \)
$31$ \( ( 1 + 5293856 T + 59464921598526 T^{2} + \)\(13\!\cdots\!76\)\( T^{3} + \)\(69\!\cdots\!41\)\( T^{4} )^{2} \)
$37$ \( 1 + 231603274936660 T^{2} + \)\(44\!\cdots\!58\)\( T^{4} + \)\(39\!\cdots\!40\)\( T^{6} + \)\(28\!\cdots\!41\)\( T^{8} \)
$41$ \( ( 1 - 8394276 T + 221313076168966 T^{2} - \)\(27\!\cdots\!36\)\( T^{3} + \)\(10\!\cdots\!21\)\( T^{4} )^{2} \)
$43$ \( 1 + 614109141147100 T^{2} + \)\(57\!\cdots\!98\)\( T^{4} + \)\(15\!\cdots\!00\)\( T^{6} + \)\(63\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 + 1368976020813580 T^{2} + \)\(29\!\cdots\!78\)\( T^{4} + \)\(17\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!21\)\( T^{8} \)
$53$ \( 1 + 7684297973864980 T^{2} + \)\(36\!\cdots\!78\)\( T^{4} + \)\(83\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!21\)\( T^{8} \)
$59$ \( ( 1 + 230414520 T + 28555631923987078 T^{2} + \)\(19\!\cdots\!80\)\( T^{3} + \)\(75\!\cdots\!21\)\( T^{4} )^{2} \)
$61$ \( ( 1 - 180245284 T + 30154717014478446 T^{2} - \)\(21\!\cdots\!44\)\( T^{3} + \)\(13\!\cdots\!81\)\( T^{4} )^{2} \)
$67$ \( 1 - 41160407446058180 T^{2} + \)\(18\!\cdots\!18\)\( T^{4} - \)\(30\!\cdots\!20\)\( T^{6} + \)\(54\!\cdots\!81\)\( T^{8} \)
$71$ \( ( 1 - 23805936 T + 85782754020107086 T^{2} - \)\(10\!\cdots\!16\)\( T^{3} + \)\(21\!\cdots\!61\)\( T^{4} )^{2} \)
$73$ \( 1 + 229489314868712740 T^{2} + \)\(20\!\cdots\!38\)\( T^{4} + \)\(79\!\cdots\!60\)\( T^{6} + \)\(12\!\cdots\!61\)\( T^{8} \)
$79$ \( ( 1 - 364021760 T + 220545463862625438 T^{2} - \)\(43\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!61\)\( T^{4} )^{2} \)
$83$ \( 1 + 434569632367965820 T^{2} + \)\(10\!\cdots\!18\)\( T^{4} + \)\(15\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!81\)\( T^{8} \)
$89$ \( ( 1 + 791350380 T + 832192702699668118 T^{2} + \)\(27\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!81\)\( T^{4} )^{2} \)
$97$ \( 1 + 2561123777205326980 T^{2} + \)\(27\!\cdots\!78\)\( T^{4} + \)\(14\!\cdots\!20\)\( T^{6} + \)\(33\!\cdots\!21\)\( T^{8} \)
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