Properties

Label 700.4.e.f
Level $700$
Weight $4$
Character orbit 700.e
Analytic conductor $41.301$
Analytic rank $1$
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] = [700,4,Mod(449,700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(700, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("700.449");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 700.e (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: \(41.3013370040\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
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.

\(f(q)\) \(=\) \( q + 4 i q^{3} - 7 i q^{7} + 11 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 4 i q^{3} - 7 i q^{7} + 11 q^{9} - 12 q^{11} - 82 i q^{13} + 30 i q^{17} - 68 q^{19} + 28 q^{21} + 216 i q^{23} + 152 i q^{27} - 246 q^{29} - 112 q^{31} - 48 i q^{33} - 110 i q^{37} + 328 q^{39} - 246 q^{41} - 172 i q^{43} - 192 i q^{47} - 49 q^{49} - 120 q^{51} + 558 i q^{53} - 272 i q^{57} - 540 q^{59} + 110 q^{61} - 77 i q^{63} - 140 i q^{67} - 864 q^{69} - 840 q^{71} - 550 i q^{73} + 84 i q^{77} + 208 q^{79} - 311 q^{81} + 516 i q^{83} - 984 i q^{87} + 1398 q^{89} - 574 q^{91} - 448 i q^{93} - 1586 i q^{97} - 132 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 22 q^{9} - 24 q^{11} - 136 q^{19} + 56 q^{21} - 492 q^{29} - 224 q^{31} + 656 q^{39} - 492 q^{41} - 98 q^{49} - 240 q^{51} - 1080 q^{59} + 220 q^{61} - 1728 q^{69} - 1680 q^{71} + 416 q^{79} - 622 q^{81} + 2796 q^{89} - 1148 q^{91} - 264 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/700\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(477\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
1.00000i
1.00000i
0 4.00000i 0 0 0 7.00000i 0 11.0000 0
449.2 0 4.00000i 0 0 0 7.00000i 0 11.0000 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 700.4.e.f 2
5.b even 2 1 inner 700.4.e.f 2
5.c odd 4 1 28.4.a.b 1
5.c odd 4 1 700.4.a.e 1
15.e even 4 1 252.4.a.c 1
20.e even 4 1 112.4.a.c 1
35.f even 4 1 196.4.a.b 1
35.k even 12 2 196.4.e.d 2
35.l odd 12 2 196.4.e.c 2
40.i odd 4 1 448.4.a.d 1
40.k even 4 1 448.4.a.m 1
60.l odd 4 1 1008.4.a.f 1
105.k odd 4 1 1764.4.a.k 1
105.w odd 12 2 1764.4.k.e 2
105.x even 12 2 1764.4.k.k 2
140.j odd 4 1 784.4.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.b 1 5.c odd 4 1
112.4.a.c 1 20.e even 4 1
196.4.a.b 1 35.f even 4 1
196.4.e.c 2 35.l odd 12 2
196.4.e.d 2 35.k even 12 2
252.4.a.c 1 15.e even 4 1
448.4.a.d 1 40.i odd 4 1
448.4.a.m 1 40.k even 4 1
700.4.a.e 1 5.c odd 4 1
700.4.e.f 2 1.a even 1 1 trivial
700.4.e.f 2 5.b even 2 1 inner
784.4.a.n 1 140.j odd 4 1
1008.4.a.f 1 60.l odd 4 1
1764.4.a.k 1 105.k odd 4 1
1764.4.k.e 2 105.w odd 12 2
1764.4.k.k 2 105.x even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(700, [\chi])\):

\( T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{11} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T + 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 6724 \) Copy content Toggle raw display
$17$ \( T^{2} + 900 \) Copy content Toggle raw display
$19$ \( (T + 68)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 46656 \) Copy content Toggle raw display
$29$ \( (T + 246)^{2} \) Copy content Toggle raw display
$31$ \( (T + 112)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 12100 \) Copy content Toggle raw display
$41$ \( (T + 246)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 29584 \) Copy content Toggle raw display
$47$ \( T^{2} + 36864 \) Copy content Toggle raw display
$53$ \( T^{2} + 311364 \) Copy content Toggle raw display
$59$ \( (T + 540)^{2} \) Copy content Toggle raw display
$61$ \( (T - 110)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 19600 \) Copy content Toggle raw display
$71$ \( (T + 840)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 302500 \) Copy content Toggle raw display
$79$ \( (T - 208)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 266256 \) Copy content Toggle raw display
$89$ \( (T - 1398)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2515396 \) Copy content Toggle raw display
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