Properties

Label 64.8.a.h
Level $64$
Weight $8$
Character orbit 64.a
Self dual yes
Analytic conductor $19.993$
Analytic rank $0$
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] = [64,8,Mod(1,64)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(64, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 64.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: \(19.9926416310\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 32)
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 \(\beta = 16\sqrt{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 8) q^{3} + ( - 8 \beta + 90) q^{5} + (14 \beta - 624) q^{7} + ( - 16 \beta + 437) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 8) q^{3} + ( - 8 \beta + 90) q^{5} + (14 \beta - 624) q^{7} + ( - 16 \beta + 437) q^{9} + (75 \beta + 4520) q^{11} + (120 \beta + 1250) q^{13} + (154 \beta - 21200) q^{15} + (240 \beta + 8610) q^{17} + ( - 155 \beta + 37080) q^{19} + ( - 736 \beta + 40832) q^{21} + ( - 1302 \beta - 9552) q^{23} + ( - 1440 \beta + 93815) q^{25} + ( - 1622 \beta - 26960) q^{27} + (2200 \beta + 122514) q^{29} + (2200 \beta + 125760) q^{31} + (3920 \beta + 155840) q^{33} + (6252 \beta - 342880) q^{35} + ( - 2280 \beta + 265370) q^{37} + (290 \beta + 297200) q^{39} + ( - 1696 \beta + 363450) q^{41} + ( - 6109 \beta + 22248) q^{43} + ( - 4936 \beta + 367010) q^{45} + (6228 \beta - 247456) q^{47} + ( - 17472 \beta + 67593) q^{49} + (6690 \beta + 545520) q^{51} + (18200 \beta - 659670) q^{53} + ( - 29410 \beta - 1129200) q^{55} + (38320 \beta - 693440) q^{57} + ( - 27945 \beta + 1309000) q^{59} + ( - 19272 \beta - 418350) q^{61} + (16102 \beta - 846128) q^{63} + (800 \beta - 2345100) q^{65} + (1505 \beta + 1187064) q^{67} + (864 \beta - 3256704) q^{69} + (22110 \beta + 1418000) q^{71} + ( - 79440 \beta - 85590) q^{73} + (105335 \beta - 4436920) q^{75} + (16480 \beta - 132480) q^{77} + (1900 \beta + 1249440) q^{79} + (21008 \beta - 4892359) q^{81} + ( - 67803 \beta - 4765992) q^{83} + ( - 47280 \beta - 4140300) q^{85} + (104914 \beta + 4651888) q^{87} + (103600 \beta + 3659034) q^{89} + ( - 57380 \beta + 3520800) q^{91} + (108160 \beta + 4625920) q^{93} + ( - 310590 \beta + 6511600) q^{95} + (127920 \beta - 4658030) q^{97} + ( - 39545 \beta - 1096760) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{3} + 180 q^{5} - 1248 q^{7} + 874 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 16 q^{3} + 180 q^{5} - 1248 q^{7} + 874 q^{9} + 9040 q^{11} + 2500 q^{13} - 42400 q^{15} + 17220 q^{17} + 74160 q^{19} + 81664 q^{21} - 19104 q^{23} + 187630 q^{25} - 53920 q^{27} + 245028 q^{29} + 251520 q^{31} + 311680 q^{33} - 685760 q^{35} + 530740 q^{37} + 594400 q^{39} + 726900 q^{41} + 44496 q^{43} + 734020 q^{45} - 494912 q^{47} + 135186 q^{49} + 1091040 q^{51} - 1319340 q^{53} - 2258400 q^{55} - 1386880 q^{57} + 2618000 q^{59} - 836700 q^{61} - 1692256 q^{63} - 4690200 q^{65} + 2374128 q^{67} - 6513408 q^{69} + 2836000 q^{71} - 171180 q^{73} - 8873840 q^{75} - 264960 q^{77} + 2498880 q^{79} - 9784718 q^{81} - 9531984 q^{83} - 8280600 q^{85} + 9303776 q^{87} + 7318068 q^{89} + 7041600 q^{91} + 9251840 q^{93} + 13023200 q^{95} - 9316060 q^{97} - 2193520 q^{99}+O(q^{100}) \) 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
−3.16228
3.16228
0 −58.5964 0 494.772 0 −1332.35 0 1246.54 0
1.2 0 42.5964 0 −314.772 0 84.3502 0 −372.543 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

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 64.8.a.h 2
3.b odd 2 1 576.8.a.be 2
4.b odd 2 1 64.8.a.j 2
8.b even 2 1 32.8.a.d yes 2
8.d odd 2 1 32.8.a.b 2
12.b even 2 1 576.8.a.bf 2
16.e even 4 2 256.8.b.j 4
16.f odd 4 2 256.8.b.h 4
24.f even 2 1 288.8.a.o 2
24.h odd 2 1 288.8.a.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.8.a.b 2 8.d odd 2 1
32.8.a.d yes 2 8.b even 2 1
64.8.a.h 2 1.a even 1 1 trivial
64.8.a.j 2 4.b odd 2 1
256.8.b.h 4 16.f odd 4 2
256.8.b.j 4 16.e even 4 2
288.8.a.n 2 24.h odd 2 1
288.8.a.o 2 24.f even 2 1
576.8.a.be 2 3.b odd 2 1
576.8.a.bf 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 16T_{3} - 2496 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(64))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 16T - 2496 \) Copy content Toggle raw display
$5$ \( T^{2} - 180T - 155740 \) Copy content Toggle raw display
$7$ \( T^{2} + 1248 T - 112384 \) Copy content Toggle raw display
$11$ \( T^{2} - 9040 T + 6030400 \) Copy content Toggle raw display
$13$ \( T^{2} - 2500 T - 35301500 \) Copy content Toggle raw display
$17$ \( T^{2} - 17220 T - 73323900 \) Copy content Toggle raw display
$19$ \( T^{2} - 74160 T + 1313422400 \) Copy content Toggle raw display
$23$ \( T^{2} + 19104 T - 4248481536 \) Copy content Toggle raw display
$29$ \( T^{2} - 245028 T + 2619280196 \) Copy content Toggle raw display
$31$ \( T^{2} - 251520 T + 3425177600 \) Copy content Toggle raw display
$37$ \( T^{2} - 530740 T + 57113332900 \) Copy content Toggle raw display
$41$ \( T^{2} - 726900 T + 124732277540 \) Copy content Toggle raw display
$43$ \( T^{2} - 44496 T - 95043921856 \) Copy content Toggle raw display
$47$ \( T^{2} + 494912 T - 38062767104 \) Copy content Toggle raw display
$53$ \( T^{2} + 1319340 T - 412809891100 \) Copy content Toggle raw display
$59$ \( T^{2} - 2618000 T - 285681944000 \) Copy content Toggle raw display
$61$ \( T^{2} + 836700 T - 775792836540 \) Copy content Toggle raw display
$67$ \( T^{2} - 2374128 T + 1403322476096 \) Copy content Toggle raw display
$71$ \( T^{2} - 2836000 T + 759262624000 \) Copy content Toggle raw display
$73$ \( T^{2} + 171180 T - 16148101167900 \) Copy content Toggle raw display
$79$ \( T^{2} - 2498880 T + 1551858713600 \) Copy content Toggle raw display
$83$ \( T^{2} + 9531984 T + 10945727913024 \) Copy content Toggle raw display
$89$ \( T^{2} - 7318068 T - 14087847786844 \) Copy content Toggle raw display
$97$ \( T^{2} + 9316060 T - 20193384103100 \) Copy content Toggle raw display
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