Properties

Label 425.4.b.c
Level $425$
Weight $4$
Character orbit 425.b
Analytic conductor $25.076$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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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: \(1\)
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 17)
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.

\(f(q)\) \(=\) \( q + 3 i q^{2} - 8 i q^{3} - q^{4} + 24 q^{6} + 28 i q^{7} + 21 i q^{8} - 37 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{2} - 8 i q^{3} - q^{4} + 24 q^{6} + 28 i q^{7} + 21 i q^{8} - 37 q^{9} - 24 q^{11} + 8 i q^{12} - 58 i q^{13} - 84 q^{14} - 71 q^{16} - 17 i q^{17} - 111 i q^{18} - 116 q^{19} + 224 q^{21} - 72 i q^{22} - 60 i q^{23} + 168 q^{24} + 174 q^{26} + 80 i q^{27} - 28 i q^{28} - 30 q^{29} - 172 q^{31} - 45 i q^{32} + 192 i q^{33} + 51 q^{34} + 37 q^{36} + 58 i q^{37} - 348 i q^{38} - 464 q^{39} - 342 q^{41} + 672 i q^{42} - 148 i q^{43} + 24 q^{44} + 180 q^{46} - 288 i q^{47} + 568 i q^{48} - 441 q^{49} - 136 q^{51} + 58 i q^{52} + 318 i q^{53} - 240 q^{54} - 588 q^{56} + 928 i q^{57} - 90 i q^{58} - 252 q^{59} + 110 q^{61} - 516 i q^{62} - 1036 i q^{63} - 433 q^{64} - 576 q^{66} + 484 i q^{67} + 17 i q^{68} - 480 q^{69} - 708 q^{71} - 777 i q^{72} + 362 i q^{73} - 174 q^{74} + 116 q^{76} - 672 i q^{77} - 1392 i q^{78} + 484 q^{79} - 359 q^{81} - 1026 i q^{82} + 756 i q^{83} - 224 q^{84} + 444 q^{86} + 240 i q^{87} - 504 i q^{88} + 774 q^{89} + 1624 q^{91} + 60 i q^{92} + 1376 i q^{93} + 864 q^{94} - 360 q^{96} + 382 i q^{97} - 1323 i q^{98} + 888 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 48 q^{6} - 74 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{4} + 48 q^{6} - 74 q^{9} - 48 q^{11} - 168 q^{14} - 142 q^{16} - 232 q^{19} + 448 q^{21} + 336 q^{24} + 348 q^{26} - 60 q^{29} - 344 q^{31} + 102 q^{34} + 74 q^{36} - 928 q^{39} - 684 q^{41} + 48 q^{44} + 360 q^{46} - 882 q^{49} - 272 q^{51} - 480 q^{54} - 1176 q^{56} - 504 q^{59} + 220 q^{61} - 866 q^{64} - 1152 q^{66} - 960 q^{69} - 1416 q^{71} - 348 q^{74} + 232 q^{76} + 968 q^{79} - 718 q^{81} - 448 q^{84} + 888 q^{86} + 1548 q^{89} + 3248 q^{91} + 1728 q^{94} - 720 q^{96} + 1776 q^{99}+O(q^{100}) \) 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 8.00000i −1.00000 0 24.0000 28.0000i 21.0000i −37.0000 0
324.2 3.00000i 8.00000i −1.00000 0 24.0000 28.0000i 21.0000i −37.0000 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 425.4.b.c 2
5.b even 2 1 inner 425.4.b.c 2
5.c odd 4 1 17.4.a.a 1
5.c odd 4 1 425.4.a.d 1
15.e even 4 1 153.4.a.d 1
20.e even 4 1 272.4.a.d 1
35.f even 4 1 833.4.a.a 1
40.i odd 4 1 1088.4.a.l 1
40.k even 4 1 1088.4.a.a 1
55.e even 4 1 2057.4.a.d 1
60.l odd 4 1 2448.4.a.f 1
85.f odd 4 1 289.4.b.a 2
85.g odd 4 1 289.4.a.a 1
85.i odd 4 1 289.4.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
17.4.a.a 1 5.c odd 4 1
153.4.a.d 1 15.e even 4 1
272.4.a.d 1 20.e even 4 1
289.4.a.a 1 85.g odd 4 1
289.4.b.a 2 85.f odd 4 1
289.4.b.a 2 85.i odd 4 1
425.4.a.d 1 5.c odd 4 1
425.4.b.c 2 1.a even 1 1 trivial
425.4.b.c 2 5.b even 2 1 inner
833.4.a.a 1 35.f even 4 1
1088.4.a.a 1 40.k even 4 1
1088.4.a.l 1 40.i odd 4 1
2057.4.a.d 1 55.e even 4 1
2448.4.a.f 1 60.l 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
\( T_{3}^{2} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 9 \) Copy content Toggle raw display
$3$ \( T^{2} + 64 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 784 \) Copy content Toggle raw display
$11$ \( (T + 24)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 3364 \) Copy content Toggle raw display
$17$ \( T^{2} + 289 \) Copy content Toggle raw display
$19$ \( (T + 116)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 3600 \) Copy content Toggle raw display
$29$ \( (T + 30)^{2} \) Copy content Toggle raw display
$31$ \( (T + 172)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 3364 \) Copy content Toggle raw display
$41$ \( (T + 342)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 21904 \) Copy content Toggle raw display
$47$ \( T^{2} + 82944 \) Copy content Toggle raw display
$53$ \( T^{2} + 101124 \) Copy content Toggle raw display
$59$ \( (T + 252)^{2} \) Copy content Toggle raw display
$61$ \( (T - 110)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 234256 \) Copy content Toggle raw display
$71$ \( (T + 708)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 131044 \) Copy content Toggle raw display
$79$ \( (T - 484)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 571536 \) Copy content Toggle raw display
$89$ \( (T - 774)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 145924 \) Copy content Toggle raw display
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