Properties

Label 704.4.a.p
Level $704$
Weight $4$
Character orbit 704.a
Self dual yes
Analytic conductor $41.537$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [704,4,Mod(1,704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(704, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("704.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 704.a (trivial)

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: yes
Analytic conductor: \(41.5373446440\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 11)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 4\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta + 1) q^{3} + ( - 2 \beta - 1) q^{5} + ( - \beta + 10) q^{7} + (2 \beta + 22) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta + 1) q^{3} + ( - 2 \beta - 1) q^{5} + ( - \beta + 10) q^{7} + (2 \beta + 22) q^{9} + 11 q^{11} + (5 \beta - 40) q^{13} + ( - 3 \beta - 97) q^{15} + (3 \beta - 62) q^{17} + ( - 15 \beta - 36) q^{19} + (9 \beta - 38) q^{21} + ( - 9 \beta - 49) q^{23} + (4 \beta + 68) q^{25} + ( - 3 \beta + 91) q^{27} + (14 \beta - 72) q^{29} + (7 \beta - 17) q^{31} + (11 \beta + 11) q^{33} + ( - 19 \beta + 86) q^{35} + (2 \beta - 27) q^{37} + ( - 35 \beta + 200) q^{39} + ( - \beta + 268) q^{41} + (4 \beta + 30) q^{43} + ( - 46 \beta - 214) q^{45} + ( - 30 \beta - 136) q^{47} + ( - 20 \beta - 195) q^{49} + ( - 59 \beta + 82) q^{51} + (14 \beta + 246) q^{53} + ( - 22 \beta - 11) q^{55} + ( - 51 \beta - 756) q^{57} + (33 \beta - 317) q^{59} + ( - 46 \beta - 420) q^{61} + ( - 2 \beta + 124) q^{63} + (75 \beta - 440) q^{65} + (5 \beta - 377) q^{67} + ( - 58 \beta - 481) q^{69} + (19 \beta - 339) q^{71} + ( - 117 \beta - 200) q^{73} + (72 \beta + 260) q^{75} + ( - 11 \beta + 110) q^{77} + (164 \beta + 158) q^{79} + (34 \beta - 647) q^{81} + ( - 30 \beta - 234) q^{83} + (121 \beta - 226) q^{85} + ( - 58 \beta + 600) q^{87} + ( - 82 \beta - 921) q^{89} + (90 \beta - 640) q^{91} + ( - 10 \beta + 319) q^{93} + (87 \beta + 1476) q^{95} + (36 \beta + 1097) q^{97} + (22 \beta + 242) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 20 q^{7} + 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 20 q^{7} + 44 q^{9} + 22 q^{11} - 80 q^{13} - 194 q^{15} - 124 q^{17} - 72 q^{19} - 76 q^{21} - 98 q^{23} + 136 q^{25} + 182 q^{27} - 144 q^{29} - 34 q^{31} + 22 q^{33} + 172 q^{35} - 54 q^{37} + 400 q^{39} + 536 q^{41} + 60 q^{43} - 428 q^{45} - 272 q^{47} - 390 q^{49} + 164 q^{51} + 492 q^{53} - 22 q^{55} - 1512 q^{57} - 634 q^{59} - 840 q^{61} + 248 q^{63} - 880 q^{65} - 754 q^{67} - 962 q^{69} - 678 q^{71} - 400 q^{73} + 520 q^{75} + 220 q^{77} + 316 q^{79} - 1294 q^{81} - 468 q^{83} - 452 q^{85} + 1200 q^{87} - 1842 q^{89} - 1280 q^{91} + 638 q^{93} + 2952 q^{95} + 2194 q^{97} + 484 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.73205
1.73205
0 −5.92820 0 12.8564 0 16.9282 0 8.14359 0
1.2 0 7.92820 0 −14.8564 0 3.07180 0 35.8564 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(11\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 704.4.a.p 2
4.b odd 2 1 704.4.a.n 2
8.b even 2 1 11.4.a.a 2
8.d odd 2 1 176.4.a.i 2
24.f even 2 1 1584.4.a.bc 2
24.h odd 2 1 99.4.a.c 2
40.f even 2 1 275.4.a.b 2
40.i odd 4 2 275.4.b.c 4
56.h odd 2 1 539.4.a.e 2
88.b odd 2 1 121.4.a.c 2
88.g even 2 1 1936.4.a.w 2
88.o even 10 4 121.4.c.c 8
88.p odd 10 4 121.4.c.f 8
104.e even 2 1 1859.4.a.a 2
120.i odd 2 1 2475.4.a.q 2
264.m even 2 1 1089.4.a.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.4.a.a 2 8.b even 2 1
99.4.a.c 2 24.h odd 2 1
121.4.a.c 2 88.b odd 2 1
121.4.c.c 8 88.o even 10 4
121.4.c.f 8 88.p odd 10 4
176.4.a.i 2 8.d odd 2 1
275.4.a.b 2 40.f even 2 1
275.4.b.c 4 40.i odd 4 2
539.4.a.e 2 56.h odd 2 1
704.4.a.n 2 4.b odd 2 1
704.4.a.p 2 1.a even 1 1 trivial
1089.4.a.v 2 264.m even 2 1
1584.4.a.bc 2 24.f even 2 1
1859.4.a.a 2 104.e even 2 1
1936.4.a.w 2 88.g even 2 1
2475.4.a.q 2 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(704))\):

\( T_{3}^{2} - 2T_{3} - 47 \) Copy content Toggle raw display
\( T_{5}^{2} + 2T_{5} - 191 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T - 47 \) Copy content Toggle raw display
$5$ \( T^{2} + 2T - 191 \) Copy content Toggle raw display
$7$ \( T^{2} - 20T + 52 \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 80T + 400 \) Copy content Toggle raw display
$17$ \( T^{2} + 124T + 3412 \) Copy content Toggle raw display
$19$ \( T^{2} + 72T - 9504 \) Copy content Toggle raw display
$23$ \( T^{2} + 98T - 1487 \) Copy content Toggle raw display
$29$ \( T^{2} + 144T - 4224 \) Copy content Toggle raw display
$31$ \( T^{2} + 34T - 2063 \) Copy content Toggle raw display
$37$ \( T^{2} + 54T + 537 \) Copy content Toggle raw display
$41$ \( T^{2} - 536T + 71776 \) Copy content Toggle raw display
$43$ \( T^{2} - 60T + 132 \) Copy content Toggle raw display
$47$ \( T^{2} + 272T - 24704 \) Copy content Toggle raw display
$53$ \( T^{2} - 492T + 51108 \) Copy content Toggle raw display
$59$ \( T^{2} + 634T + 48217 \) Copy content Toggle raw display
$61$ \( T^{2} + 840T + 74832 \) Copy content Toggle raw display
$67$ \( T^{2} + 754T + 140929 \) Copy content Toggle raw display
$71$ \( T^{2} + 678T + 97593 \) Copy content Toggle raw display
$73$ \( T^{2} + 400T - 617072 \) Copy content Toggle raw display
$79$ \( T^{2} - 316 T - 1266044 \) Copy content Toggle raw display
$83$ \( T^{2} + 468T + 11556 \) Copy content Toggle raw display
$89$ \( T^{2} + 1842 T + 525489 \) Copy content Toggle raw display
$97$ \( T^{2} - 2194 T + 1141201 \) Copy content Toggle raw display
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