Properties

Label 25.4.b.b
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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [25,4,Mod(24,25)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(25, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("25.24");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 25.b (of order \(2\), degree \(1\), not 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: \(1.47504775014\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + i q^{2} - 7 i q^{3} + 7 q^{4} + 7 q^{6} + 6 i q^{7} + 15 i q^{8} - 22 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} - 7 i q^{3} + 7 q^{4} + 7 q^{6} + 6 i q^{7} + 15 i q^{8} - 22 q^{9} - 43 q^{11} - 49 i q^{12} + 28 i q^{13} - 6 q^{14} + 41 q^{16} + 91 i q^{17} - 22 i q^{18} + 35 q^{19} + 42 q^{21} - 43 i q^{22} - 162 i q^{23} + 105 q^{24} - 28 q^{26} - 35 i q^{27} + 42 i q^{28} - 160 q^{29} + 42 q^{31} + 161 i q^{32} + 301 i q^{33} - 91 q^{34} - 154 q^{36} - 314 i q^{37} + 35 i q^{38} + 196 q^{39} - 203 q^{41} + 42 i q^{42} - 92 i q^{43} - 301 q^{44} + 162 q^{46} + 196 i q^{47} - 287 i q^{48} + 307 q^{49} + 637 q^{51} + 196 i q^{52} - 82 i q^{53} + 35 q^{54} - 90 q^{56} - 245 i q^{57} - 160 i q^{58} + 280 q^{59} - 518 q^{61} + 42 i q^{62} - 132 i q^{63} + 167 q^{64} - 301 q^{66} + 141 i q^{67} + 637 i q^{68} - 1134 q^{69} + 412 q^{71} - 330 i q^{72} + 763 i q^{73} + 314 q^{74} + 245 q^{76} - 258 i q^{77} + 196 i q^{78} - 510 q^{79} - 839 q^{81} - 203 i q^{82} - 777 i q^{83} + 294 q^{84} + 92 q^{86} + 1120 i q^{87} - 645 i q^{88} + 945 q^{89} - 168 q^{91} - 1134 i q^{92} - 294 i q^{93} - 196 q^{94} + 1127 q^{96} + 1246 i q^{97} + 307 i q^{98} + 946 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{4} + 14 q^{6} - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 14 q^{4} + 14 q^{6} - 44 q^{9} - 86 q^{11} - 12 q^{14} + 82 q^{16} + 70 q^{19} + 84 q^{21} + 210 q^{24} - 56 q^{26} - 320 q^{29} + 84 q^{31} - 182 q^{34} - 308 q^{36} + 392 q^{39} - 406 q^{41} - 602 q^{44} + 324 q^{46} + 614 q^{49} + 1274 q^{51} + 70 q^{54} - 180 q^{56} + 560 q^{59} - 1036 q^{61} + 334 q^{64} - 602 q^{66} - 2268 q^{69} + 824 q^{71} + 628 q^{74} + 490 q^{76} - 1020 q^{79} - 1678 q^{81} + 588 q^{84} + 184 q^{86} + 1890 q^{89} - 336 q^{91} - 392 q^{94} + 2254 q^{96} + 1892 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1
1.00000i
1.00000i
1.00000i 7.00000i 7.00000 0 7.00000 6.00000i 15.0000i −22.0000 0
24.2 1.00000i 7.00000i 7.00000 0 7.00000 6.00000i 15.0000i −22.0000 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.b 2
3.b odd 2 1 225.4.b.f 2
4.b odd 2 1 400.4.c.e 2
5.b even 2 1 inner 25.4.b.b 2
5.c odd 4 1 25.4.a.a 1
5.c odd 4 1 25.4.a.b yes 1
15.d odd 2 1 225.4.b.f 2
15.e even 4 1 225.4.a.c 1
15.e even 4 1 225.4.a.e 1
20.d odd 2 1 400.4.c.e 2
20.e even 4 1 400.4.a.c 1
20.e even 4 1 400.4.a.s 1
35.f even 4 1 1225.4.a.h 1
35.f even 4 1 1225.4.a.i 1
40.i odd 4 1 1600.4.a.i 1
40.i odd 4 1 1600.4.a.bt 1
40.k even 4 1 1600.4.a.h 1
40.k even 4 1 1600.4.a.bs 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.a.a 1 5.c odd 4 1
25.4.a.b yes 1 5.c odd 4 1
25.4.b.b 2 1.a even 1 1 trivial
25.4.b.b 2 5.b even 2 1 inner
225.4.a.c 1 15.e even 4 1
225.4.a.e 1 15.e even 4 1
225.4.b.f 2 3.b odd 2 1
225.4.b.f 2 15.d odd 2 1
400.4.a.c 1 20.e even 4 1
400.4.a.s 1 20.e even 4 1
400.4.c.e 2 4.b odd 2 1
400.4.c.e 2 20.d odd 2 1
1225.4.a.h 1 35.f even 4 1
1225.4.a.i 1 35.f even 4 1
1600.4.a.h 1 40.k even 4 1
1600.4.a.i 1 40.i odd 4 1
1600.4.a.bs 1 40.k even 4 1
1600.4.a.bt 1 40.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 1 \) acting on \(S_{4}^{\mathrm{new}}(25, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 49 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 36 \) Copy content Toggle raw display
$11$ \( (T + 43)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 784 \) Copy content Toggle raw display
$17$ \( T^{2} + 8281 \) Copy content Toggle raw display
$19$ \( (T - 35)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 26244 \) Copy content Toggle raw display
$29$ \( (T + 160)^{2} \) Copy content Toggle raw display
$31$ \( (T - 42)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 98596 \) Copy content Toggle raw display
$41$ \( (T + 203)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 8464 \) Copy content Toggle raw display
$47$ \( T^{2} + 38416 \) Copy content Toggle raw display
$53$ \( T^{2} + 6724 \) Copy content Toggle raw display
$59$ \( (T - 280)^{2} \) Copy content Toggle raw display
$61$ \( (T + 518)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 19881 \) Copy content Toggle raw display
$71$ \( (T - 412)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 582169 \) Copy content Toggle raw display
$79$ \( (T + 510)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 603729 \) Copy content Toggle raw display
$89$ \( (T - 945)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1552516 \) Copy content Toggle raw display
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