Properties

Label 425.4.b.b
Level $425$
Weight $4$
Character orbit 425.b
Analytic conductor $25.076$
Analytic rank $0$
Dimension $2$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [425,4,Mod(324,425)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(425, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.324");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.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: \(25.0758117524\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 85)
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.

\(f(q)\) \(=\) \( q + 3 i q^{2} + 5 i q^{3} - q^{4} - 15 q^{6} - 22 i q^{7} + 21 i q^{8} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{2} + 5 i q^{3} - q^{4} - 15 q^{6} - 22 i q^{7} + 21 i q^{8} + 2 q^{9} + 60 q^{11} - 5 i q^{12} + 31 i q^{13} + 66 q^{14} - 71 q^{16} + 17 i q^{17} + 6 i q^{18} + 61 q^{19} + 110 q^{21} + 180 i q^{22} + 78 i q^{23} - 105 q^{24} - 93 q^{26} + 145 i q^{27} + 22 i q^{28} - 69 q^{29} - 31 q^{31} - 45 i q^{32} + 300 i q^{33} - 51 q^{34} - 2 q^{36} + 56 i q^{37} + 183 i q^{38} - 155 q^{39} - 6 q^{41} + 330 i q^{42} + 538 i q^{43} - 60 q^{44} - 234 q^{46} - 465 i q^{47} - 355 i q^{48} - 141 q^{49} - 85 q^{51} - 31 i q^{52} - 723 i q^{53} - 435 q^{54} + 462 q^{56} + 305 i q^{57} - 207 i q^{58} + 753 q^{59} + 35 q^{61} - 93 i q^{62} - 44 i q^{63} - 433 q^{64} - 900 q^{66} - 322 i q^{67} - 17 i q^{68} - 390 q^{69} - 99 q^{71} + 42 i q^{72} + 1123 i q^{73} - 168 q^{74} - 61 q^{76} - 1320 i q^{77} - 465 i q^{78} - 488 q^{79} - 671 q^{81} - 18 i q^{82} + 852 i q^{83} - 110 q^{84} - 1614 q^{86} - 345 i q^{87} + 1260 i q^{88} - 1215 q^{89} + 682 q^{91} - 78 i q^{92} - 155 i q^{93} + 1395 q^{94} + 225 q^{96} - 601 i q^{97} - 423 i q^{98} + 120 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 30 q^{6} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} - 30 q^{6} + 4 q^{9} + 120 q^{11} + 132 q^{14} - 142 q^{16} + 122 q^{19} + 220 q^{21} - 210 q^{24} - 186 q^{26} - 138 q^{29} - 62 q^{31} - 102 q^{34} - 4 q^{36} - 310 q^{39} - 12 q^{41} - 120 q^{44} - 468 q^{46} - 282 q^{49} - 170 q^{51} - 870 q^{54} + 924 q^{56} + 1506 q^{59} + 70 q^{61} - 866 q^{64} - 1800 q^{66} - 780 q^{69} - 198 q^{71} - 336 q^{74} - 122 q^{76} - 976 q^{79} - 1342 q^{81} - 220 q^{84} - 3228 q^{86} - 2430 q^{89} + 1364 q^{91} + 2790 q^{94} + 450 q^{96} + 240 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
324.1
1.00000i
1.00000i
3.00000i 5.00000i −1.00000 0 −15.0000 22.0000i 21.0000i 2.00000 0
324.2 3.00000i 5.00000i −1.00000 0 −15.0000 22.0000i 21.0000i 2.00000 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 425.4.b.b 2
5.b even 2 1 inner 425.4.b.b 2
5.c odd 4 1 85.4.a.b 1
5.c odd 4 1 425.4.a.b 1
15.e even 4 1 765.4.a.c 1
20.e even 4 1 1360.4.a.g 1
85.g odd 4 1 1445.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.4.a.b 1 5.c odd 4 1
425.4.a.b 1 5.c odd 4 1
425.4.b.b 2 1.a even 1 1 trivial
425.4.b.b 2 5.b even 2 1 inner
765.4.a.c 1 15.e even 4 1
1360.4.a.g 1 20.e even 4 1
1445.4.a.g 1 85.g odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(425, [\chi])\):

\( T_{2}^{2} + 9 \) Copy content Toggle raw display
\( T_{3}^{2} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{2} + 25 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 484 \) Copy content Toggle raw display
$11$ \( (T - 60)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 961 \) Copy content Toggle raw display
$17$ \( T^{2} + 289 \) Copy content Toggle raw display
$19$ \( (T - 61)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6084 \) Copy content Toggle raw display
$29$ \( (T + 69)^{2} \) Copy content Toggle raw display
$31$ \( (T + 31)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 3136 \) Copy content Toggle raw display
$41$ \( (T + 6)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 289444 \) Copy content Toggle raw display
$47$ \( T^{2} + 216225 \) Copy content Toggle raw display
$53$ \( T^{2} + 522729 \) Copy content Toggle raw display
$59$ \( (T - 753)^{2} \) Copy content Toggle raw display
$61$ \( (T - 35)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 103684 \) Copy content Toggle raw display
$71$ \( (T + 99)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 1261129 \) Copy content Toggle raw display
$79$ \( (T + 488)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 725904 \) Copy content Toggle raw display
$89$ \( (T + 1215)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 361201 \) Copy content Toggle raw display
show more
show less