Properties

Label 700.6.e.b
Level $700$
Weight $6$
Character orbit 700.e
Analytic conductor $112.269$
Analytic rank $0$
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] = [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.

\(f(q)\) \(=\) \( q + 26 i q^{3} + 49 i q^{7} - 433 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 26 i q^{3} + 49 i q^{7} - 433 q^{9} + 8 q^{11} + 684 i q^{13} + 2218 i q^{17} + 2698 q^{19} - 1274 q^{21} + 3344 i q^{23} - 4940 i q^{27} + 3254 q^{29} + 4788 q^{31} + 208 i q^{33} + 11470 i q^{37} - 17784 q^{39} + 13350 q^{41} - 928 i q^{43} - 1212 i q^{47} - 2401 q^{49} - 57668 q^{51} + 13110 i q^{53} + 70148 i q^{57} - 34702 q^{59} - 1032 q^{61} - 21217 i q^{63} - 10108 i q^{67} - 86944 q^{69} + 62720 q^{71} - 18926 i q^{73} + 392 i q^{77} - 11400 q^{79} + 23221 q^{81} + 88958 i q^{83} + 84604 i q^{87} - 19722 q^{89} - 33516 q^{91} + 124488 i q^{93} - 17062 i q^{97} - 3464 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 866 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 866 q^{9} + 16 q^{11} + 5396 q^{19} - 2548 q^{21} + 6508 q^{29} + 9576 q^{31} - 35568 q^{39} + 26700 q^{41} - 4802 q^{49} - 115336 q^{51} - 69404 q^{59} - 2064 q^{61} - 173888 q^{69} + 125440 q^{71} - 22800 q^{79} + 46442 q^{81} - 39444 q^{89} - 67032 q^{91} - 6928 q^{99}+O(q^{100}) \) 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 26.0000i 0 0 0 49.0000i 0 −433.000 0
449.2 0 26.0000i 0 0 0 49.0000i 0 −433.000 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.b 2
5.b even 2 1 inner 700.6.e.b 2
5.c odd 4 1 28.6.a.b 1
5.c odd 4 1 700.6.a.b 1
15.e even 4 1 252.6.a.a 1
20.e even 4 1 112.6.a.b 1
35.f even 4 1 196.6.a.a 1
35.k even 12 2 196.6.e.i 2
35.l odd 12 2 196.6.e.a 2
40.i odd 4 1 448.6.a.b 1
40.k even 4 1 448.6.a.o 1
60.l odd 4 1 1008.6.a.l 1
140.j odd 4 1 784.6.a.m 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.6.a.b 1 5.c odd 4 1
112.6.a.b 1 20.e even 4 1
196.6.a.a 1 35.f even 4 1
196.6.e.a 2 35.l odd 12 2
196.6.e.i 2 35.k even 12 2
252.6.a.a 1 15.e even 4 1
448.6.a.b 1 40.i odd 4 1
448.6.a.o 1 40.k even 4 1
700.6.a.b 1 5.c odd 4 1
700.6.e.b 2 1.a even 1 1 trivial
700.6.e.b 2 5.b even 2 1 inner
784.6.a.m 1 140.j odd 4 1
1008.6.a.l 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 676 \) 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
$3$ \( T^{2} + 676 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2401 \) Copy content Toggle raw display
$11$ \( (T - 8)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 467856 \) Copy content Toggle raw display
$17$ \( T^{2} + 4919524 \) Copy content Toggle raw display
$19$ \( (T - 2698)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 11182336 \) Copy content Toggle raw display
$29$ \( (T - 3254)^{2} \) Copy content Toggle raw display
$31$ \( (T - 4788)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 131560900 \) Copy content Toggle raw display
$41$ \( (T - 13350)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 861184 \) Copy content Toggle raw display
$47$ \( T^{2} + 1468944 \) Copy content Toggle raw display
$53$ \( T^{2} + 171872100 \) Copy content Toggle raw display
$59$ \( (T + 34702)^{2} \) Copy content Toggle raw display
$61$ \( (T + 1032)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 102171664 \) Copy content Toggle raw display
$71$ \( (T - 62720)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 358193476 \) Copy content Toggle raw display
$79$ \( (T + 11400)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 7913525764 \) Copy content Toggle raw display
$89$ \( (T + 19722)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 291111844 \) Copy content Toggle raw display
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