Properties

Label 425.4.d.a
Level $425$
Weight $4$
Character orbit 425.d
Analytic conductor $25.076$
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] = [425,4,Mod(101,425)]
 
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([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("425.101");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.d (of order \(2\), degree \(1\), 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = 2i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - 4 \beta q^{3} - 7 q^{4} + 4 \beta q^{6} + 7 \beta q^{7} + 15 q^{8} - 37 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - 4 \beta q^{3} - 7 q^{4} + 4 \beta q^{6} + 7 \beta q^{7} + 15 q^{8} - 37 q^{9} + 10 \beta q^{11} + 28 \beta q^{12} + 58 q^{13} - 7 \beta q^{14} + 41 q^{16} + ( - 34 \beta + 17) q^{17} + 37 q^{18} + 80 q^{19} + 112 q^{21} - 10 \beta q^{22} - 59 \beta q^{23} - 60 \beta q^{24} - 58 q^{26} + 40 \beta q^{27} - 49 \beta q^{28} + 63 \beta q^{29} + 35 \beta q^{31} - 161 q^{32} + 160 q^{33} + (34 \beta - 17) q^{34} + 259 q^{36} + 67 \beta q^{37} - 80 q^{38} - 232 \beta q^{39} - 50 \beta q^{41} - 112 q^{42} - 272 q^{43} - 70 \beta q^{44} + 59 \beta q^{46} + 464 q^{47} - 164 \beta q^{48} + 147 q^{49} + ( - 68 \beta - 544) q^{51} - 406 q^{52} - 642 q^{53} - 40 \beta q^{54} + 105 \beta q^{56} - 320 \beta q^{57} - 63 \beta q^{58} - 180 q^{59} + 55 \beta q^{61} - 35 \beta q^{62} - 259 \beta q^{63} - 167 q^{64} - 160 q^{66} + 924 q^{67} + (238 \beta - 119) q^{68} - 944 q^{69} + 45 \beta q^{71} - 555 q^{72} - 414 \beta q^{73} - 67 \beta q^{74} - 560 q^{76} - 280 q^{77} + 232 \beta q^{78} - 667 \beta q^{79} - 359 q^{81} + 50 \beta q^{82} - 552 q^{83} - 784 q^{84} + 272 q^{86} + 1008 q^{87} + 150 \beta q^{88} + 1490 q^{89} + 406 \beta q^{91} + 413 \beta q^{92} + 560 q^{93} - 464 q^{94} + 644 \beta q^{96} - 688 \beta q^{97} - 147 q^{98} - 370 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 14 q^{4} + 30 q^{8} - 74 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 14 q^{4} + 30 q^{8} - 74 q^{9} + 116 q^{13} + 82 q^{16} + 34 q^{17} + 74 q^{18} + 160 q^{19} + 224 q^{21} - 116 q^{26} - 322 q^{32} + 320 q^{33} - 34 q^{34} + 518 q^{36} - 160 q^{38} - 224 q^{42} - 544 q^{43} + 928 q^{47} + 294 q^{49} - 1088 q^{51} - 812 q^{52} - 1284 q^{53} - 360 q^{59} - 334 q^{64} - 320 q^{66} + 1848 q^{67} - 238 q^{68} - 1888 q^{69} - 1110 q^{72} - 1120 q^{76} - 560 q^{77} - 718 q^{81} - 1104 q^{83} - 1568 q^{84} + 544 q^{86} + 2016 q^{87} + 2980 q^{89} + 1120 q^{93} - 928 q^{94} - 294 q^{98}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
1.00000i
1.00000i
−1.00000 8.00000i −7.00000 0 8.00000i 14.0000i 15.0000 −37.0000 0
101.2 −1.00000 8.00000i −7.00000 0 8.00000i 14.0000i 15.0000 −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
17.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.d.a 2
5.b even 2 1 85.4.d.a 2
5.c odd 4 1 425.4.c.a 2
5.c odd 4 1 425.4.c.b 2
15.d odd 2 1 765.4.g.a 2
17.b even 2 1 inner 425.4.d.a 2
85.c even 2 1 85.4.d.a 2
85.g odd 4 1 425.4.c.a 2
85.g odd 4 1 425.4.c.b 2
85.j even 4 1 1445.4.a.d 1
85.j even 4 1 1445.4.a.e 1
255.h odd 2 1 765.4.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
85.4.d.a 2 5.b even 2 1
85.4.d.a 2 85.c even 2 1
425.4.c.a 2 5.c odd 4 1
425.4.c.a 2 85.g odd 4 1
425.4.c.b 2 5.c odd 4 1
425.4.c.b 2 85.g odd 4 1
425.4.d.a 2 1.a even 1 1 trivial
425.4.d.a 2 17.b even 2 1 inner
765.4.g.a 2 15.d odd 2 1
765.4.g.a 2 255.h odd 2 1
1445.4.a.d 1 85.j even 4 1
1445.4.a.e 1 85.j even 4 1

Hecke kernels

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

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 64 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 196 \) Copy content Toggle raw display
$11$ \( T^{2} + 400 \) Copy content Toggle raw display
$13$ \( (T - 58)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 34T + 4913 \) Copy content Toggle raw display
$19$ \( (T - 80)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 13924 \) Copy content Toggle raw display
$29$ \( T^{2} + 15876 \) Copy content Toggle raw display
$31$ \( T^{2} + 4900 \) Copy content Toggle raw display
$37$ \( T^{2} + 17956 \) Copy content Toggle raw display
$41$ \( T^{2} + 10000 \) Copy content Toggle raw display
$43$ \( (T + 272)^{2} \) Copy content Toggle raw display
$47$ \( (T - 464)^{2} \) Copy content Toggle raw display
$53$ \( (T + 642)^{2} \) Copy content Toggle raw display
$59$ \( (T + 180)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 12100 \) Copy content Toggle raw display
$67$ \( (T - 924)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 8100 \) Copy content Toggle raw display
$73$ \( T^{2} + 685584 \) Copy content Toggle raw display
$79$ \( T^{2} + 1779556 \) Copy content Toggle raw display
$83$ \( (T + 552)^{2} \) Copy content Toggle raw display
$89$ \( (T - 1490)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1893376 \) Copy content Toggle raw display
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