Properties

Label 25.5.c.c
Level 25
Weight 5
Character orbit 25.c
Analytic conductor 2.584
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.58424907710\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{21})\)
Defining polynomial: \(x^{4} + 11 x^{2} + 25\)
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.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} + \beta_{2} q^{3} -26 \beta_{1} q^{4} + 42 q^{6} + 9 \beta_{3} q^{7} -10 \beta_{2} q^{8} -39 \beta_{1} q^{9} +O(q^{10})\) \( q -\beta_{3} q^{2} + \beta_{2} q^{3} -26 \beta_{1} q^{4} + 42 q^{6} + 9 \beta_{3} q^{7} -10 \beta_{2} q^{8} -39 \beta_{1} q^{9} -108 q^{11} -26 \beta_{3} q^{12} + 36 \beta_{2} q^{13} + 378 \beta_{1} q^{14} -4 q^{16} + 4 \beta_{3} q^{17} -39 \beta_{2} q^{18} -140 \beta_{1} q^{19} -378 q^{21} + 108 \beta_{3} q^{22} -79 \beta_{2} q^{23} -420 \beta_{1} q^{24} + 1512 q^{26} -120 \beta_{3} q^{27} + 234 \beta_{2} q^{28} -810 \beta_{1} q^{29} -728 q^{31} -156 \beta_{3} q^{32} -108 \beta_{2} q^{33} + 168 \beta_{1} q^{34} -1014 q^{36} + 144 \beta_{3} q^{37} -140 \beta_{2} q^{38} + 1512 \beta_{1} q^{39} + 1512 q^{41} + 378 \beta_{3} q^{42} -9 \beta_{2} q^{43} + 2808 \beta_{1} q^{44} -3318 q^{46} + 39 \beta_{3} q^{47} -4 \beta_{2} q^{48} -1001 \beta_{1} q^{49} -168 q^{51} -936 \beta_{3} q^{52} + 676 \beta_{2} q^{53} -5040 \beta_{1} q^{54} + 3780 q^{56} -140 \beta_{3} q^{57} -810 \beta_{2} q^{58} + 3780 \beta_{1} q^{59} + 4592 q^{61} + 728 \beta_{3} q^{62} + 351 \beta_{2} q^{63} -6616 \beta_{1} q^{64} -4536 q^{66} + 729 \beta_{3} q^{67} + 104 \beta_{2} q^{68} -3318 \beta_{1} q^{69} + 432 q^{71} + 390 \beta_{3} q^{72} -1404 \beta_{2} q^{73} + 6048 \beta_{1} q^{74} -3640 q^{76} -972 \beta_{3} q^{77} + 1512 \beta_{2} q^{78} + 8840 \beta_{1} q^{79} + 1881 q^{81} -1512 \beta_{3} q^{82} -169 \beta_{2} q^{83} + 9828 \beta_{1} q^{84} -378 q^{86} -810 \beta_{3} q^{87} + 1080 \beta_{2} q^{88} -13230 \beta_{1} q^{89} -13608 q^{91} + 2054 \beta_{3} q^{92} -728 \beta_{2} q^{93} + 1638 \beta_{1} q^{94} + 6552 q^{96} + 1764 \beta_{3} q^{97} -1001 \beta_{2} q^{98} + 4212 \beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 168q^{6} + O(q^{10}) \) \( 4q + 168q^{6} - 432q^{11} - 16q^{16} - 1512q^{21} + 6048q^{26} - 2912q^{31} - 4056q^{36} + 6048q^{41} - 13272q^{46} - 672q^{51} + 15120q^{56} + 18368q^{61} - 18144q^{66} + 1728q^{71} - 14560q^{76} + 7524q^{81} - 1512q^{86} - 54432q^{91} + 26208q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 11 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 6 \nu \)\()/5\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 10 \nu^{2} + 16 \nu + 55 \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} - 10 \nu^{2} + 16 \nu - 55 \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2} - 2 \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + \beta_{2} - 22\)\()/4\)
\(\nu^{3}\)\(=\)\((\)\(-3 \beta_{3} - 3 \beta_{2} + 16 \beta_{1}\)\()/2\)

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
2.79129i
1.79129i
2.79129i
1.79129i
−4.58258 4.58258i −4.58258 + 4.58258i 26.0000i 0 42.0000 41.2432 + 41.2432i 45.8258 45.8258i 39.0000i 0
7.2 4.58258 + 4.58258i 4.58258 4.58258i 26.0000i 0 42.0000 −41.2432 41.2432i −45.8258 + 45.8258i 39.0000i 0
18.1 −4.58258 + 4.58258i −4.58258 4.58258i 26.0000i 0 42.0000 41.2432 41.2432i 45.8258 + 45.8258i 39.0000i 0
18.2 4.58258 4.58258i 4.58258 + 4.58258i 26.0000i 0 42.0000 −41.2432 + 41.2432i −45.8258 45.8258i 39.0000i 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 25.5.c.c 4
3.b odd 2 1 225.5.g.j 4
4.b odd 2 1 400.5.p.l 4
5.b even 2 1 inner 25.5.c.c 4
5.c odd 4 2 inner 25.5.c.c 4
15.d odd 2 1 225.5.g.j 4
15.e even 4 2 225.5.g.j 4
20.d odd 2 1 400.5.p.l 4
20.e even 4 2 400.5.p.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.5.c.c 4 1.a even 1 1 trivial
25.5.c.c 4 5.b even 2 1 inner
25.5.c.c 4 5.c odd 4 2 inner
225.5.g.j 4 3.b odd 2 1
225.5.g.j 4 15.d odd 2 1
225.5.g.j 4 15.e even 4 2
400.5.p.l 4 4.b odd 2 1
400.5.p.l 4 20.d odd 2 1
400.5.p.l 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 1764 \) acting on \(S_{5}^{\mathrm{new}}(25, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 412 T^{4} + 65536 T^{8} \)
$3$ \( 1 + 1278 T^{4} + 43046721 T^{8} \)
$5$ 1
$7$ \( 1 - 9569602 T^{4} + 33232930569601 T^{8} \)
$11$ \( ( 1 + 108 T + 14641 T^{2} )^{4} \)
$13$ \( 1 - 1624225342 T^{4} + 665416609183179841 T^{8} \)
$17$ \( 1 + 13727462018 T^{4} + 48661191875666868481 T^{8} \)
$19$ \( ( 1 - 241042 T^{2} + 16983563041 T^{4} )^{2} \)
$23$ \( 1 - 68080016962 T^{4} + \)\(61\!\cdots\!61\)\( T^{8} \)
$29$ \( ( 1 - 758462 T^{2} + 500246412961 T^{4} )^{2} \)
$31$ \( ( 1 + 728 T + 923521 T^{2} )^{4} \)
$37$ \( 1 + 1254529400258 T^{4} + \)\(12\!\cdots\!41\)\( T^{8} \)
$41$ \( ( 1 - 1512 T + 2825761 T^{2} )^{4} \)
$43$ \( 1 + 23329889084798 T^{4} + \)\(13\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 + 46379759106878 T^{4} + \)\(56\!\cdots\!21\)\( T^{8} \)
$53$ \( 1 - 112877432101822 T^{4} + \)\(38\!\cdots\!21\)\( T^{8} \)
$59$ \( ( 1 - 9946322 T^{2} + 146830437604321 T^{4} )^{2} \)
$61$ \( ( 1 - 4592 T + 13845841 T^{2} )^{4} \)
$67$ \( 1 - 488793100954882 T^{4} + \)\(16\!\cdots\!81\)\( T^{8} \)
$71$ \( ( 1 - 432 T + 25411681 T^{2} )^{4} \)
$73$ \( 1 - 937201474520062 T^{4} + \)\(65\!\cdots\!61\)\( T^{8} \)
$79$ \( ( 1 + 245438 T^{2} + 1517108809906561 T^{4} )^{2} \)
$83$ \( 1 + 4278306619448318 T^{4} + \)\(50\!\cdots\!81\)\( T^{8} \)
$89$ \( ( 1 + 49548418 T^{2} + 3936588805702081 T^{4} )^{2} \)
$97$ \( 1 - 13524937897425022 T^{4} + \)\(61\!\cdots\!21\)\( T^{8} \)
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