Properties

Label 700.6.e.d
Level $700$
Weight $6$
Character orbit 700.e
Analytic conductor $112.269$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [700,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("700.449");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(112.268673869\)
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: \( 1 \)
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.

\(f(q)\) \(=\) \( q + 2 i q^{3} + 49 i q^{7} + 239 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 i q^{3} + 49 i q^{7} + 239 q^{9} - 720 q^{11} - 572 i q^{13} + 1254 i q^{17} + 94 q^{19} - 98 q^{21} - 96 i q^{23} + 964 i q^{27} + 4374 q^{29} - 6244 q^{31} - 1440 i q^{33} - 10798 i q^{37} + 1144 q^{39} + 12006 q^{41} + 9160 i q^{43} - 25836 i q^{47} - 2401 q^{49} - 2508 q^{51} - 1014 i q^{53} + 188 i q^{57} - 1242 q^{59} + 7592 q^{61} + 11711 i q^{63} + 41132 i q^{67} + 192 q^{69} - 37632 q^{71} + 13438 i q^{73} - 35280 i q^{77} - 6248 q^{79} + 56149 q^{81} + 25254 i q^{83} + 8748 i q^{87} + 45126 q^{89} + 28028 q^{91} - 12488 i q^{93} + 107222 i q^{97} - 172080 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 478 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 478 q^{9} - 1440 q^{11} + 188 q^{19} - 196 q^{21} + 8748 q^{29} - 12488 q^{31} + 2288 q^{39} + 24012 q^{41} - 4802 q^{49} - 5016 q^{51} - 2484 q^{59} + 15184 q^{61} + 384 q^{69} - 75264 q^{71} - 12496 q^{79} + 112298 q^{81} + 90252 q^{89} + 56056 q^{91} - 344160 q^{99}+O(q^{100}) \) 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 2.00000i 0 0 0 49.0000i 0 239.000 0
449.2 0 2.00000i 0 0 0 49.0000i 0 239.000 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 700.6.e.d 2
5.b even 2 1 inner 700.6.e.d 2
5.c odd 4 1 28.6.a.a 1
5.c odd 4 1 700.6.a.d 1
15.e even 4 1 252.6.a.d 1
20.e even 4 1 112.6.a.e 1
35.f even 4 1 196.6.a.d 1
35.k even 12 2 196.6.e.e 2
35.l odd 12 2 196.6.e.f 2
40.i odd 4 1 448.6.a.i 1
40.k even 4 1 448.6.a.h 1
60.l odd 4 1 1008.6.a.bb 1
140.j odd 4 1 784.6.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.6.a.a 1 5.c odd 4 1
112.6.a.e 1 20.e even 4 1
196.6.a.d 1 35.f even 4 1
196.6.e.e 2 35.k even 12 2
196.6.e.f 2 35.l odd 12 2
252.6.a.d 1 15.e even 4 1
448.6.a.h 1 40.k even 4 1
448.6.a.i 1 40.i odd 4 1
700.6.a.d 1 5.c odd 4 1
700.6.e.d 2 1.a even 1 1 trivial
700.6.e.d 2 5.b even 2 1 inner
784.6.a.f 1 140.j odd 4 1
1008.6.a.bb 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 4 \) acting on \(S_{6}^{\mathrm{new}}(700, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T + 720)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 327184 \) Copy content Toggle raw display
$17$ \( T^{2} + 1572516 \) Copy content Toggle raw display
$19$ \( (T - 94)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 9216 \) Copy content Toggle raw display
$29$ \( (T - 4374)^{2} \) Copy content Toggle raw display
$31$ \( (T + 6244)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 116596804 \) Copy content Toggle raw display
$41$ \( (T - 12006)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 83905600 \) Copy content Toggle raw display
$47$ \( T^{2} + 667498896 \) Copy content Toggle raw display
$53$ \( T^{2} + 1028196 \) Copy content Toggle raw display
$59$ \( (T + 1242)^{2} \) Copy content Toggle raw display
$61$ \( (T - 7592)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1691841424 \) Copy content Toggle raw display
$71$ \( (T + 37632)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 180579844 \) Copy content Toggle raw display
$79$ \( (T + 6248)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 637764516 \) Copy content Toggle raw display
$89$ \( (T - 45126)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 11496557284 \) Copy content Toggle raw display
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