Properties

Label 25.4.b.a
Level $25$
Weight $4$
Character orbit 25.b
Analytic conductor $1.475$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.47504775014\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + 4 i q^{2} + 2 i q^{3} -8 q^{4} -8 q^{6} -6 i q^{7} + 23 q^{9} +O(q^{10})\) \( q + 4 i q^{2} + 2 i q^{3} -8 q^{4} -8 q^{6} -6 i q^{7} + 23 q^{9} + 32 q^{11} -16 i q^{12} -38 i q^{13} + 24 q^{14} -64 q^{16} -26 i q^{17} + 92 i q^{18} -100 q^{19} + 12 q^{21} + 128 i q^{22} -78 i q^{23} + 152 q^{26} + 100 i q^{27} + 48 i q^{28} + 50 q^{29} -108 q^{31} -256 i q^{32} + 64 i q^{33} + 104 q^{34} -184 q^{36} -266 i q^{37} -400 i q^{38} + 76 q^{39} + 22 q^{41} + 48 i q^{42} + 442 i q^{43} -256 q^{44} + 312 q^{46} + 514 i q^{47} -128 i q^{48} + 307 q^{49} + 52 q^{51} + 304 i q^{52} + 2 i q^{53} -400 q^{54} -200 i q^{57} + 200 i q^{58} -500 q^{59} -518 q^{61} -432 i q^{62} -138 i q^{63} + 512 q^{64} -256 q^{66} -126 i q^{67} + 208 i q^{68} + 156 q^{69} + 412 q^{71} -878 i q^{73} + 1064 q^{74} + 800 q^{76} -192 i q^{77} + 304 i q^{78} -600 q^{79} + 421 q^{81} + 88 i q^{82} + 282 i q^{83} -96 q^{84} -1768 q^{86} + 100 i q^{87} + 150 q^{89} -228 q^{91} + 624 i q^{92} -216 i q^{93} -2056 q^{94} + 512 q^{96} -386 i q^{97} + 1228 i q^{98} + 736 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{4} - 16 q^{6} + 46 q^{9} + O(q^{10}) \) \( 2 q - 16 q^{4} - 16 q^{6} + 46 q^{9} + 64 q^{11} + 48 q^{14} - 128 q^{16} - 200 q^{19} + 24 q^{21} + 304 q^{26} + 100 q^{29} - 216 q^{31} + 208 q^{34} - 368 q^{36} + 152 q^{39} + 44 q^{41} - 512 q^{44} + 624 q^{46} + 614 q^{49} + 104 q^{51} - 800 q^{54} - 1000 q^{59} - 1036 q^{61} + 1024 q^{64} - 512 q^{66} + 312 q^{69} + 824 q^{71} + 2128 q^{74} + 1600 q^{76} - 1200 q^{79} + 842 q^{81} - 192 q^{84} - 3536 q^{86} + 300 q^{89} - 456 q^{91} - 4112 q^{94} + 1024 q^{96} + 1472 q^{99} + O(q^{100}) \)

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
4.00000i 2.00000i −8.00000 0 −8.00000 6.00000i 0 23.0000 0
24.2 4.00000i 2.00000i −8.00000 0 −8.00000 6.00000i 0 23.0000 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 25.4.b.a 2
3.b odd 2 1 225.4.b.c 2
4.b odd 2 1 400.4.c.k 2
5.b even 2 1 inner 25.4.b.a 2
5.c odd 4 1 5.4.a.a 1
5.c odd 4 1 25.4.a.c 1
15.d odd 2 1 225.4.b.c 2
15.e even 4 1 45.4.a.d 1
15.e even 4 1 225.4.a.b 1
20.d odd 2 1 400.4.c.k 2
20.e even 4 1 80.4.a.d 1
20.e even 4 1 400.4.a.m 1
35.f even 4 1 245.4.a.a 1
35.f even 4 1 1225.4.a.k 1
35.k even 12 2 245.4.e.g 2
35.l odd 12 2 245.4.e.f 2
40.i odd 4 1 320.4.a.g 1
40.i odd 4 1 1600.4.a.bi 1
40.k even 4 1 320.4.a.h 1
40.k even 4 1 1600.4.a.s 1
45.k odd 12 2 405.4.e.l 2
45.l even 12 2 405.4.e.c 2
55.e even 4 1 605.4.a.d 1
60.l odd 4 1 720.4.a.u 1
65.h odd 4 1 845.4.a.b 1
80.i odd 4 1 1280.4.d.e 2
80.j even 4 1 1280.4.d.l 2
80.s even 4 1 1280.4.d.l 2
80.t odd 4 1 1280.4.d.e 2
85.g odd 4 1 1445.4.a.a 1
95.g even 4 1 1805.4.a.h 1
105.k odd 4 1 2205.4.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.4.a.a 1 5.c odd 4 1
25.4.a.c 1 5.c odd 4 1
25.4.b.a 2 1.a even 1 1 trivial
25.4.b.a 2 5.b even 2 1 inner
45.4.a.d 1 15.e even 4 1
80.4.a.d 1 20.e even 4 1
225.4.a.b 1 15.e even 4 1
225.4.b.c 2 3.b odd 2 1
225.4.b.c 2 15.d odd 2 1
245.4.a.a 1 35.f even 4 1
245.4.e.f 2 35.l odd 12 2
245.4.e.g 2 35.k even 12 2
320.4.a.g 1 40.i odd 4 1
320.4.a.h 1 40.k even 4 1
400.4.a.m 1 20.e even 4 1
400.4.c.k 2 4.b odd 2 1
400.4.c.k 2 20.d odd 2 1
405.4.e.c 2 45.l even 12 2
405.4.e.l 2 45.k odd 12 2
605.4.a.d 1 55.e even 4 1
720.4.a.u 1 60.l odd 4 1
845.4.a.b 1 65.h odd 4 1
1225.4.a.k 1 35.f even 4 1
1280.4.d.e 2 80.i odd 4 1
1280.4.d.e 2 80.t odd 4 1
1280.4.d.l 2 80.j even 4 1
1280.4.d.l 2 80.s even 4 1
1445.4.a.a 1 85.g odd 4 1
1600.4.a.s 1 40.k even 4 1
1600.4.a.bi 1 40.i odd 4 1
1805.4.a.h 1 95.g even 4 1
2205.4.a.q 1 105.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 16 \) acting on \(S_{4}^{\mathrm{new}}(25, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 16 + T^{2} \)
$3$ \( 4 + T^{2} \)
$5$ \( T^{2} \)
$7$ \( 36 + T^{2} \)
$11$ \( ( -32 + T )^{2} \)
$13$ \( 1444 + T^{2} \)
$17$ \( 676 + T^{2} \)
$19$ \( ( 100 + T )^{2} \)
$23$ \( 6084 + T^{2} \)
$29$ \( ( -50 + T )^{2} \)
$31$ \( ( 108 + T )^{2} \)
$37$ \( 70756 + T^{2} \)
$41$ \( ( -22 + T )^{2} \)
$43$ \( 195364 + T^{2} \)
$47$ \( 264196 + T^{2} \)
$53$ \( 4 + T^{2} \)
$59$ \( ( 500 + T )^{2} \)
$61$ \( ( 518 + T )^{2} \)
$67$ \( 15876 + T^{2} \)
$71$ \( ( -412 + T )^{2} \)
$73$ \( 770884 + T^{2} \)
$79$ \( ( 600 + T )^{2} \)
$83$ \( 79524 + T^{2} \)
$89$ \( ( -150 + T )^{2} \)
$97$ \( 148996 + T^{2} \)
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