Properties

Label 25.6.b.a
Level $25$
Weight $6$
Character orbit 25.b
Analytic conductor $4.010$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.00959549532\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 2 \beta q^{3} + 28 q^{4} - 8 q^{6} + 96 \beta q^{7} + 60 \beta q^{8} + 227 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + 2 \beta q^{3} + 28 q^{4} - 8 q^{6} + 96 \beta q^{7} + 60 \beta q^{8} + 227 q^{9} - 148 q^{11} + 56 \beta q^{12} - 143 \beta q^{13} - 384 q^{14} + 656 q^{16} - 839 \beta q^{17} + 227 \beta q^{18} - 1060 q^{19} - 768 q^{21} - 148 \beta q^{22} - 1488 \beta q^{23} - 480 q^{24} + 572 q^{26} + 940 \beta q^{27} + 2688 \beta q^{28} + 3410 q^{29} - 2448 q^{31} + 2576 \beta q^{32} - 296 \beta q^{33} + 3356 q^{34} + 6356 q^{36} + 91 \beta q^{37} - 1060 \beta q^{38} + 1144 q^{39} - 9398 q^{41} - 768 \beta q^{42} + 622 \beta q^{43} - 4144 q^{44} + 5952 q^{46} - 6044 \beta q^{47} + 1312 \beta q^{48} - 20057 q^{49} + 6712 q^{51} - 4004 \beta q^{52} - 11923 \beta q^{53} - 3760 q^{54} - 23040 q^{56} - 2120 \beta q^{57} + 3410 \beta q^{58} + 20020 q^{59} + 32302 q^{61} - 2448 \beta q^{62} + 21792 \beta q^{63} + 10688 q^{64} + 1184 q^{66} + 30486 \beta q^{67} - 23492 \beta q^{68} + 11904 q^{69} - 32648 q^{71} + 13620 \beta q^{72} + 19387 \beta q^{73} - 364 q^{74} - 29680 q^{76} - 14208 \beta q^{77} + 1144 \beta q^{78} + 33360 q^{79} + 47641 q^{81} - 9398 \beta q^{82} - 8358 \beta q^{83} - 21504 q^{84} - 2488 q^{86} + 6820 \beta q^{87} - 8880 \beta q^{88} - 101370 q^{89} + 54912 q^{91} - 41664 \beta q^{92} - 4896 \beta q^{93} + 24176 q^{94} - 20608 q^{96} - 59519 \beta q^{97} - 20057 \beta q^{98} - 33596 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 56 q^{4} - 16 q^{6} + 454 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 56 q^{4} - 16 q^{6} + 454 q^{9} - 296 q^{11} - 768 q^{14} + 1312 q^{16} - 2120 q^{19} - 1536 q^{21} - 960 q^{24} + 1144 q^{26} + 6820 q^{29} - 4896 q^{31} + 6712 q^{34} + 12712 q^{36} + 2288 q^{39} - 18796 q^{41} - 8288 q^{44} + 11904 q^{46} - 40114 q^{49} + 13424 q^{51} - 7520 q^{54} - 46080 q^{56} + 40040 q^{59} + 64604 q^{61} + 21376 q^{64} + 2368 q^{66} + 23808 q^{69} - 65296 q^{71} - 728 q^{74} - 59360 q^{76} + 66720 q^{79} + 95282 q^{81} - 43008 q^{84} - 4976 q^{86} - 202740 q^{89} + 109824 q^{91} + 48352 q^{94} - 41216 q^{96} - 67192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
1.00000i
1.00000i
2.00000i 4.00000i 28.0000 0 −8.00000 192.000i 120.000i 227.000 0
24.2 2.00000i 4.00000i 28.0000 0 −8.00000 192.000i 120.000i 227.000 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 25.6.b.a 2
3.b odd 2 1 225.6.b.e 2
4.b odd 2 1 400.6.c.j 2
5.b even 2 1 inner 25.6.b.a 2
5.c odd 4 1 5.6.a.a 1
5.c odd 4 1 25.6.a.a 1
15.d odd 2 1 225.6.b.e 2
15.e even 4 1 45.6.a.b 1
15.e even 4 1 225.6.a.f 1
20.d odd 2 1 400.6.c.j 2
20.e even 4 1 80.6.a.e 1
20.e even 4 1 400.6.a.g 1
35.f even 4 1 245.6.a.b 1
40.i odd 4 1 320.6.a.j 1
40.k even 4 1 320.6.a.g 1
55.e even 4 1 605.6.a.a 1
60.l odd 4 1 720.6.a.a 1
65.h odd 4 1 845.6.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.a.a 1 5.c odd 4 1
25.6.a.a 1 5.c odd 4 1
25.6.b.a 2 1.a even 1 1 trivial
25.6.b.a 2 5.b even 2 1 inner
45.6.a.b 1 15.e even 4 1
80.6.a.e 1 20.e even 4 1
225.6.a.f 1 15.e even 4 1
225.6.b.e 2 3.b odd 2 1
225.6.b.e 2 15.d odd 2 1
245.6.a.b 1 35.f even 4 1
320.6.a.g 1 40.k even 4 1
320.6.a.j 1 40.i odd 4 1
400.6.a.g 1 20.e even 4 1
400.6.c.j 2 4.b odd 2 1
400.6.c.j 2 20.d odd 2 1
605.6.a.a 1 55.e even 4 1
720.6.a.a 1 60.l odd 4 1
845.6.a.b 1 65.h odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 4 \) acting on \(S_{6}^{\mathrm{new}}(25, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 36864 \) Copy content Toggle raw display
$11$ \( (T + 148)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 81796 \) Copy content Toggle raw display
$17$ \( T^{2} + 2815684 \) Copy content Toggle raw display
$19$ \( (T + 1060)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 8856576 \) Copy content Toggle raw display
$29$ \( (T - 3410)^{2} \) Copy content Toggle raw display
$31$ \( (T + 2448)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 33124 \) Copy content Toggle raw display
$41$ \( (T + 9398)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 1547536 \) Copy content Toggle raw display
$47$ \( T^{2} + 146119744 \) Copy content Toggle raw display
$53$ \( T^{2} + 568631716 \) Copy content Toggle raw display
$59$ \( (T - 20020)^{2} \) Copy content Toggle raw display
$61$ \( (T - 32302)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 3717584784 \) Copy content Toggle raw display
$71$ \( (T + 32648)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1503423076 \) Copy content Toggle raw display
$79$ \( (T - 33360)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 279424656 \) Copy content Toggle raw display
$89$ \( (T + 101370)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 14170045444 \) Copy content Toggle raw display
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