Properties

Label 25.9.c.a
Level $25$
Weight $9$
Character orbit 25.c
Analytic conductor $10.184$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 25.c (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.1844652515\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{141})\)
Defining polynomial: \(x^{4} + 71 x^{2} + 1225\)
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} + \beta_{2} q^{3} -26 \beta_{1} q^{4} + 282 q^{6} -231 \beta_{3} q^{7} + 230 \beta_{2} q^{8} -6279 \beta_{1} q^{9} +O(q^{10})\) \( q -\beta_{3} q^{2} + \beta_{2} q^{3} -26 \beta_{1} q^{4} + 282 q^{6} -231 \beta_{3} q^{7} + 230 \beta_{2} q^{8} -6279 \beta_{1} q^{9} + 12132 q^{11} -26 \beta_{3} q^{12} -204 \beta_{2} q^{13} -65142 \beta_{1} q^{14} + 71516 q^{16} + 6484 \beta_{3} q^{17} -6279 \beta_{2} q^{18} -168380 \beta_{1} q^{19} + 65142 q^{21} -12132 \beta_{3} q^{22} + 18641 \beta_{2} q^{23} + 64860 \beta_{1} q^{24} -57528 q^{26} -12840 \beta_{3} q^{27} -6006 \beta_{2} q^{28} + 666630 \beta_{1} q^{29} -1042808 q^{31} -12636 \beta_{3} q^{32} + 12132 \beta_{2} q^{33} + 1828488 \beta_{1} q^{34} -163254 q^{36} + 174864 \beta_{3} q^{37} -168380 \beta_{2} q^{38} -57528 \beta_{1} q^{39} -1321128 q^{41} -65142 \beta_{3} q^{42} + 234231 \beta_{2} q^{43} -315432 \beta_{1} q^{44} + 5256762 q^{46} -305721 \beta_{3} q^{47} + 71516 \beta_{2} q^{48} -9283001 \beta_{1} q^{49} -1828488 q^{51} + 5304 \beta_{3} q^{52} -264524 \beta_{2} q^{53} -3620880 \beta_{1} q^{54} + 14982660 q^{56} -168380 \beta_{3} q^{57} + 666630 \beta_{2} q^{58} -6498540 \beta_{1} q^{59} -14393968 q^{61} + 1042808 \beta_{3} q^{62} -1450449 \beta_{2} q^{63} + 14744744 \beta_{1} q^{64} + 3421224 q^{66} + 964809 \beta_{3} q^{67} + 168584 \beta_{2} q^{68} + 5256762 \beta_{1} q^{69} -23065488 q^{71} -1444170 \beta_{3} q^{72} + 1468116 \beta_{2} q^{73} + 49311648 \beta_{1} q^{74} -4377880 q^{76} -2802492 \beta_{3} q^{77} -57528 \beta_{2} q^{78} + 2760680 \beta_{1} q^{79} -37575639 q^{81} + 1321128 \beta_{3} q^{82} -970489 \beta_{2} q^{83} -1693692 \beta_{1} q^{84} + 66053142 q^{86} + 666630 \beta_{3} q^{87} + 2790360 \beta_{2} q^{88} -26130510 \beta_{1} q^{89} -13288968 q^{91} -484666 \beta_{3} q^{92} -1042808 \beta_{2} q^{93} -86213322 \beta_{1} q^{94} + 3563352 q^{96} + 6815604 \beta_{3} q^{97} -9283001 \beta_{2} q^{98} -76176828 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 1128q^{6} + O(q^{10}) \) \( 4q + 1128q^{6} + 48528q^{11} + 286064q^{16} + 260568q^{21} - 230112q^{26} - 4171232q^{31} - 653016q^{36} - 5284512q^{41} + 21027048q^{46} - 7313952q^{51} + 59930640q^{56} - 57575872q^{61} + 13684896q^{66} - 92261952q^{71} - 17511520q^{76} - 150302556q^{81} + 264212568q^{86} - 53155872q^{91} + 14253408q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 71 x^{2} + 1225\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 36 \nu \)\()/35\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 70 \nu^{2} + 106 \nu + 2485 \)\()/35\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - 70 \nu^{2} + 106 \nu - 2485 \)\()/35\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + \beta_{2} - 142\)\()/4\)
\(\nu^{3}\)\(=\)\(-9 \beta_{3} - 9 \beta_{2} + 53 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/25\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
6.43717i
5.43717i
6.43717i
5.43717i
−11.8743 11.8743i −11.8743 + 11.8743i 26.0000i 0 282.000 −2742.97 2742.97i −2731.10 + 2731.10i 6279.00i 0
7.2 11.8743 + 11.8743i 11.8743 11.8743i 26.0000i 0 282.000 2742.97 + 2742.97i 2731.10 2731.10i 6279.00i 0
18.1 −11.8743 + 11.8743i −11.8743 11.8743i 26.0000i 0 282.000 −2742.97 + 2742.97i −2731.10 2731.10i 6279.00i 0
18.2 11.8743 11.8743i 11.8743 + 11.8743i 26.0000i 0 282.000 2742.97 2742.97i 2731.10 + 2731.10i 6279.00i 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
5.c odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.9.c.a 4
5.b even 2 1 inner 25.9.c.a 4
5.c odd 4 2 inner 25.9.c.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.9.c.a 4 1.a even 1 1 trivial
25.9.c.a 4 5.b even 2 1 inner
25.9.c.a 4 5.c odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 79524 \) acting on \(S_{9}^{\mathrm{new}}(25, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 79524 + T^{4} \)
$3$ \( 79524 + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 226436345031204 + T^{4} \)
$11$ \( ( -12132 + T )^{4} \)
$13$ \( 137726936146944 + T^{4} \)
$17$ \( \)\(14\!\cdots\!64\)\( + T^{4} \)
$19$ \( ( 28351824400 + T^{2} )^{2} \)
$23$ \( \)\(96\!\cdots\!64\)\( + T^{4} \)
$29$ \( ( 444395556900 + T^{2} )^{2} \)
$31$ \( ( 1042808 + T )^{4} \)
$37$ \( \)\(74\!\cdots\!84\)\( + T^{4} \)
$41$ \( ( 1321128 + T )^{4} \)
$43$ \( \)\(23\!\cdots\!04\)\( + T^{4} \)
$47$ \( \)\(69\!\cdots\!44\)\( + T^{4} \)
$53$ \( \)\(38\!\cdots\!24\)\( + T^{4} \)
$59$ \( ( 42231022131600 + T^{2} )^{2} \)
$61$ \( ( 14393968 + T )^{4} \)
$67$ \( \)\(68\!\cdots\!64\)\( + T^{4} \)
$71$ \( ( 23065488 + T )^{4} \)
$73$ \( \)\(36\!\cdots\!64\)\( + T^{4} \)
$79$ \( ( 7621354062400 + T^{2} )^{2} \)
$83$ \( \)\(70\!\cdots\!84\)\( + T^{4} \)
$89$ \( ( 682803552860100 + T^{2} )^{2} \)
$97$ \( \)\(17\!\cdots\!44\)\( + T^{4} \)
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