Properties

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

Related objects

Downloads

Learn more about

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: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
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.

\(f(q)\) \(=\) \( q + 3 \beta_{1} q^{2} + 7 \beta_{3} q^{3} + 11 \beta_{2} q^{4} -63 q^{6} + 28 \beta_{1} q^{7} -15 \beta_{3} q^{8} -66 \beta_{2} q^{9} +O(q^{10})\) \( q + 3 \beta_{1} q^{2} + 7 \beta_{3} q^{3} + 11 \beta_{2} q^{4} -63 q^{6} + 28 \beta_{1} q^{7} -15 \beta_{3} q^{8} -66 \beta_{2} q^{9} + 117 q^{11} -77 \beta_{1} q^{12} + 42 \beta_{3} q^{13} + 252 \beta_{2} q^{14} + 311 q^{16} -147 \beta_{1} q^{17} -198 \beta_{3} q^{18} -595 \beta_{2} q^{19} -588 q^{21} + 351 \beta_{1} q^{22} + 102 \beta_{3} q^{23} + 315 \beta_{2} q^{24} -378 q^{26} -105 \beta_{1} q^{27} + 308 \beta_{3} q^{28} + 1170 \beta_{2} q^{29} + 322 q^{31} + 693 \beta_{1} q^{32} + 819 \beta_{3} q^{33} -1323 \beta_{2} q^{34} + 726 q^{36} -472 \beta_{1} q^{37} -1785 \beta_{3} q^{38} -882 \beta_{2} q^{39} -63 q^{41} -1764 \beta_{1} q^{42} -1028 \beta_{3} q^{43} + 1287 \beta_{2} q^{44} -918 q^{46} + 378 \beta_{1} q^{47} + 2177 \beta_{3} q^{48} -49 \beta_{2} q^{49} + 3087 q^{51} -462 \beta_{1} q^{52} -1218 \beta_{3} q^{53} -945 \beta_{2} q^{54} + 1260 q^{56} + 4165 \beta_{1} q^{57} + 3510 \beta_{3} q^{58} + 1890 \beta_{2} q^{59} -5908 q^{61} + 966 \beta_{1} q^{62} -1848 \beta_{3} q^{63} + 1261 \beta_{2} q^{64} -7371 q^{66} -3897 \beta_{1} q^{67} -1617 \beta_{3} q^{68} -2142 \beta_{2} q^{69} + 2682 q^{71} -990 \beta_{1} q^{72} + 1477 \beta_{3} q^{73} -4248 \beta_{2} q^{74} + 6545 q^{76} + 3276 \beta_{1} q^{77} -2646 \beta_{3} q^{78} + 6520 \beta_{2} q^{79} + 7551 q^{81} -189 \beta_{1} q^{82} + 2037 \beta_{3} q^{83} -6468 \beta_{2} q^{84} + 9252 q^{86} -8190 \beta_{1} q^{87} -1755 \beta_{3} q^{88} + 5985 \beta_{2} q^{89} -3528 q^{91} -1122 \beta_{1} q^{92} + 2254 \beta_{3} q^{93} + 3402 \beta_{2} q^{94} -14553 q^{96} + 2828 \beta_{1} q^{97} -147 \beta_{3} q^{98} -7722 \beta_{2} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 252q^{6} + O(q^{10}) \) \( 4q - 252q^{6} + 468q^{11} + 1244q^{16} - 2352q^{21} - 1512q^{26} + 1288q^{31} + 2904q^{36} - 252q^{41} - 3672q^{46} + 12348q^{51} + 5040q^{56} - 23632q^{61} - 29484q^{66} + 10728q^{71} + 26180q^{76} + 30204q^{81} + 37008q^{86} - 14112q^{91} - 58212q^{96} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)

Character values

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

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

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
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
−3.67423 3.67423i 8.57321 8.57321i 11.0000i 0 −63.0000 −34.2929 34.2929i −18.3712 + 18.3712i 66.0000i 0
7.2 3.67423 + 3.67423i −8.57321 + 8.57321i 11.0000i 0 −63.0000 34.2929 + 34.2929i 18.3712 18.3712i 66.0000i 0
18.1 −3.67423 + 3.67423i 8.57321 + 8.57321i 11.0000i 0 −63.0000 −34.2929 + 34.2929i −18.3712 18.3712i 66.0000i 0
18.2 3.67423 3.67423i −8.57321 8.57321i 11.0000i 0 −63.0000 34.2929 34.2929i 18.3712 + 18.3712i 66.0000i 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.b 4
3.b odd 2 1 225.5.g.f 4
4.b odd 2 1 400.5.p.j 4
5.b even 2 1 inner 25.5.c.b 4
5.c odd 4 2 inner 25.5.c.b 4
15.d odd 2 1 225.5.g.f 4
15.e even 4 2 225.5.g.f 4
20.d odd 2 1 400.5.p.j 4
20.e even 4 2 400.5.p.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.5.c.b 4 1.a even 1 1 trivial
25.5.c.b 4 5.b even 2 1 inner
25.5.c.b 4 5.c odd 4 2 inner
225.5.g.f 4 3.b odd 2 1
225.5.g.f 4 15.d odd 2 1
225.5.g.f 4 15.e even 4 2
400.5.p.j 4 4.b odd 2 1
400.5.p.j 4 20.d odd 2 1
400.5.p.j 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 729 \) acting on \(S_{5}^{\mathrm{new}}(25, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 487 T^{4} + 65536 T^{8} \)
$3$ \( 1 - 12897 T^{4} + 43046721 T^{8} \)
$5$ 1
$7$ \( 1 - 5527102 T^{4} + 33232930569601 T^{8} \)
$11$ \( ( 1 - 117 T + 14641 T^{2} )^{4} \)
$13$ \( 1 + 1054887458 T^{4} + 665416609183179841 T^{8} \)
$17$ \( 1 - 3503608657 T^{4} + 48661191875666868481 T^{8} \)
$19$ \( ( 1 + 93383 T^{2} + 16983563041 T^{4} )^{2} \)
$23$ \( 1 + 122658570338 T^{4} + \)\(61\!\cdots\!61\)\( T^{8} \)
$29$ \( ( 1 - 45662 T^{2} + 500246412961 T^{4} )^{2} \)
$31$ \( ( 1 - 322 T + 923521 T^{2} )^{4} \)
$37$ \( 1 + 2461256293058 T^{4} + \)\(12\!\cdots\!41\)\( T^{8} \)
$41$ \( ( 1 + 63 T + 2825761 T^{2} )^{4} \)
$43$ \( 1 - 9927677992702 T^{4} + \)\(13\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 + 39439575780578 T^{4} + \)\(56\!\cdots\!21\)\( T^{8} \)
$53$ \( 1 + 3858356729378 T^{4} + \)\(38\!\cdots\!21\)\( T^{8} \)
$59$ \( ( 1 - 20662622 T^{2} + 146830437604321 T^{4} )^{2} \)
$61$ \( ( 1 + 5908 T + 13845841 T^{2} )^{4} \)
$67$ \( 1 - 784493155081057 T^{4} + \)\(16\!\cdots\!81\)\( T^{8} \)
$71$ \( ( 1 - 2682 T + 25411681 T^{2} )^{4} \)
$73$ \( 1 + 912332767302863 T^{4} + \)\(65\!\cdots\!61\)\( T^{8} \)
$79$ \( ( 1 - 35389762 T^{2} + 1517108809906561 T^{4} )^{2} \)
$83$ \( 1 + 2296474800768143 T^{4} + \)\(50\!\cdots\!81\)\( T^{8} \)
$89$ \( ( 1 - 89664257 T^{2} + 3936588805702081 T^{4} )^{2} \)
$97$ \( 1 + 7754275002202178 T^{4} + \)\(61\!\cdots\!21\)\( T^{8} \)
show more
show less