Properties

Label 25.5.c.a
Level $25$
Weight $5$
Character orbit 25.c
Analytic conductor $2.584$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [25,5,Mod(7,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.7");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 25.c (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.58424907710\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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

\(f(q)\) \(=\) \( q + (i + 1) q^{2} + ( - 6 i + 6) q^{3} - 14 i q^{4} + 12 q^{6} + (26 i + 26) q^{7} + ( - 30 i + 30) q^{8} + 9 i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (i + 1) q^{2} + ( - 6 i + 6) q^{3} - 14 i q^{4} + 12 q^{6} + (26 i + 26) q^{7} + ( - 30 i + 30) q^{8} + 9 i q^{9} - 8 q^{11} + ( - 84 i - 84) q^{12} + (139 i - 139) q^{13} + 52 i q^{14} - 164 q^{16} + (i + 1) q^{17} + (9 i - 9) q^{18} + 180 i q^{19} + 312 q^{21} + ( - 8 i - 8) q^{22} + ( - 166 i + 166) q^{23} - 360 i q^{24} - 278 q^{26} + (540 i + 540) q^{27} + ( - 364 i + 364) q^{28} - 480 i q^{29} + 572 q^{31} + ( - 644 i - 644) q^{32} + (48 i - 48) q^{33} + 2 i q^{34} + 126 q^{36} + (251 i + 251) q^{37} + (180 i - 180) q^{38} + 1668 i q^{39} - 1688 q^{41} + (312 i + 312) q^{42} + (1474 i - 1474) q^{43} + 112 i q^{44} + 332 q^{46} + ( - 2474 i - 2474) q^{47} + (984 i - 984) q^{48} - 1049 i q^{49} + 12 q^{51} + (1946 i + 1946) q^{52} + ( - 3331 i + 3331) q^{53} + 1080 i q^{54} + 1560 q^{56} + (1080 i + 1080) q^{57} + ( - 480 i + 480) q^{58} - 3660 i q^{59} + 1592 q^{61} + (572 i + 572) q^{62} + (234 i - 234) q^{63} + 1336 i q^{64} - 96 q^{66} + ( - 874 i - 874) q^{67} + ( - 14 i + 14) q^{68} - 1992 i q^{69} - 6068 q^{71} + (270 i + 270) q^{72} + ( - 791 i + 791) q^{73} + 502 i q^{74} + 2520 q^{76} + ( - 208 i - 208) q^{77} + (1668 i - 1668) q^{78} + 9120 i q^{79} + 5751 q^{81} + ( - 1688 i - 1688) q^{82} + (5654 i - 5654) q^{83} - 4368 i q^{84} - 2948 q^{86} + ( - 2880 i - 2880) q^{87} + (240 i - 240) q^{88} + 2160 i q^{89} - 7228 q^{91} + ( - 2324 i - 2324) q^{92} + ( - 3432 i + 3432) q^{93} - 4948 i q^{94} - 7728 q^{96} + (6551 i + 6551) q^{97} + ( - 1049 i + 1049) q^{98} - 72 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 12 q^{3} + 24 q^{6} + 52 q^{7} + 60 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 12 q^{3} + 24 q^{6} + 52 q^{7} + 60 q^{8} - 16 q^{11} - 168 q^{12} - 278 q^{13} - 328 q^{16} + 2 q^{17} - 18 q^{18} + 624 q^{21} - 16 q^{22} + 332 q^{23} - 556 q^{26} + 1080 q^{27} + 728 q^{28} + 1144 q^{31} - 1288 q^{32} - 96 q^{33} + 252 q^{36} + 502 q^{37} - 360 q^{38} - 3376 q^{41} + 624 q^{42} - 2948 q^{43} + 664 q^{46} - 4948 q^{47} - 1968 q^{48} + 24 q^{51} + 3892 q^{52} + 6662 q^{53} + 3120 q^{56} + 2160 q^{57} + 960 q^{58} + 3184 q^{61} + 1144 q^{62} - 468 q^{63} - 192 q^{66} - 1748 q^{67} + 28 q^{68} - 12136 q^{71} + 540 q^{72} + 1582 q^{73} + 5040 q^{76} - 416 q^{77} - 3336 q^{78} + 11502 q^{81} - 3376 q^{82} - 11308 q^{83} - 5896 q^{86} - 5760 q^{87} - 480 q^{88} - 14456 q^{91} - 4648 q^{92} + 6864 q^{93} - 15456 q^{96} + 13102 q^{97} + 2098 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(i\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
1.00000i
1.00000i
1.00000 + 1.00000i 6.00000 6.00000i 14.0000i 0 12.0000 26.0000 + 26.0000i 30.0000 30.0000i 9.00000i 0
18.1 1.00000 1.00000i 6.00000 + 6.00000i 14.0000i 0 12.0000 26.0000 26.0000i 30.0000 + 30.0000i 9.00000i 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.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 25.5.c.a 2
3.b odd 2 1 225.5.g.b 2
4.b odd 2 1 400.5.p.a 2
5.b even 2 1 5.5.c.a 2
5.c odd 4 1 5.5.c.a 2
5.c odd 4 1 inner 25.5.c.a 2
15.d odd 2 1 45.5.g.b 2
15.e even 4 1 45.5.g.b 2
15.e even 4 1 225.5.g.b 2
20.d odd 2 1 80.5.p.d 2
20.e even 4 1 80.5.p.d 2
20.e even 4 1 400.5.p.a 2
40.e odd 2 1 320.5.p.c 2
40.f even 2 1 320.5.p.h 2
40.i odd 4 1 320.5.p.h 2
40.k even 4 1 320.5.p.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.5.c.a 2 5.b even 2 1
5.5.c.a 2 5.c odd 4 1
25.5.c.a 2 1.a even 1 1 trivial
25.5.c.a 2 5.c odd 4 1 inner
45.5.g.b 2 15.d odd 2 1
45.5.g.b 2 15.e even 4 1
80.5.p.d 2 20.d odd 2 1
80.5.p.d 2 20.e even 4 1
225.5.g.b 2 3.b odd 2 1
225.5.g.b 2 15.e even 4 1
320.5.p.c 2 40.e odd 2 1
320.5.p.c 2 40.k even 4 1
320.5.p.h 2 40.f even 2 1
320.5.p.h 2 40.i odd 4 1
400.5.p.a 2 4.b odd 2 1
400.5.p.a 2 20.e even 4 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$3$ \( T^{2} - 12T + 72 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 52T + 1352 \) Copy content Toggle raw display
$11$ \( (T + 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 278T + 38642 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 32400 \) Copy content Toggle raw display
$23$ \( T^{2} - 332T + 55112 \) Copy content Toggle raw display
$29$ \( T^{2} + 230400 \) Copy content Toggle raw display
$31$ \( (T - 572)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 502T + 126002 \) Copy content Toggle raw display
$41$ \( (T + 1688)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 2948 T + 4345352 \) Copy content Toggle raw display
$47$ \( T^{2} + 4948 T + 12241352 \) Copy content Toggle raw display
$53$ \( T^{2} - 6662 T + 22191122 \) Copy content Toggle raw display
$59$ \( T^{2} + 13395600 \) Copy content Toggle raw display
$61$ \( (T - 1592)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1748 T + 1527752 \) Copy content Toggle raw display
$71$ \( (T + 6068)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 1582 T + 1251362 \) Copy content Toggle raw display
$79$ \( T^{2} + 83174400 \) Copy content Toggle raw display
$83$ \( T^{2} + 11308 T + 63935432 \) Copy content Toggle raw display
$89$ \( T^{2} + 4665600 \) Copy content Toggle raw display
$97$ \( T^{2} - 13102 T + 85831202 \) Copy content Toggle raw display
show more
show less