Properties

Label 78.16.a.f
Level $78$
Weight $16$
Character orbit 78.a
Self dual yes
Analytic conductor $111.301$
Analytic rank $1$
Dimension $4$
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] = [78,16,Mod(1,78)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(78, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("78.1");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 78.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: \(111.300933978\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6327x^{2} - 123410x + 3269236 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{6}\cdot 5\cdot 7\cdot 23 \)
Twist minimal: yes
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 128 q^{2} - 2187 q^{3} + 16384 q^{4} + (\beta_{2} + 47 \beta_1 + 4795) q^{5} - 279936 q^{6} + ( - \beta_{3} - 10 \beta_{2} + 871 \beta_1 + 627745) q^{7} + 2097152 q^{8} + 4782969 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 128 q^{2} - 2187 q^{3} + 16384 q^{4} + (\beta_{2} + 47 \beta_1 + 4795) q^{5} - 279936 q^{6} + ( - \beta_{3} - 10 \beta_{2} + 871 \beta_1 + 627745) q^{7} + 2097152 q^{8} + 4782969 q^{9} + (128 \beta_{2} + 6016 \beta_1 + 613760) q^{10} + (8 \beta_{3} + 153 \beta_{2} - 27708 \beta_1 - 10277230) q^{11} - 35831808 q^{12} - 62748517 q^{13} + ( - 128 \beta_{3} - 1280 \beta_{2} + 111488 \beta_1 + 80351360) q^{14} + ( - 2187 \beta_{2} - 102789 \beta_1 - 10486665) q^{15} + 268435456 q^{16} + (2472 \beta_{3} - 6236 \beta_{2} - 432898 \beta_1 - 621877080) q^{17} + 612220032 q^{18} + ( - 3321 \beta_{3} - 870 \beta_{2} - 374675 \beta_1 - 2837823033) q^{19} + (16384 \beta_{2} + 770048 \beta_1 + 78561280) q^{20} + (2187 \beta_{3} + 21870 \beta_{2} - 1904877 \beta_1 - 1372878315) q^{21} + (1024 \beta_{3} + 19584 \beta_{2} - 3546624 \beta_1 - 1315485440) q^{22} + ( - 20527 \beta_{3} + 39546 \beta_{2} + \cdots - 1684632872) q^{23}+ \cdots + (38263752 \beta_{3} + 731794257 \beta_{2} + \cdots - 49155672495870) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 512 q^{2} - 8748 q^{3} + 65536 q^{4} + 19180 q^{5} - 1119744 q^{6} + 2510980 q^{7} + 8388608 q^{8} + 19131876 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 512 q^{2} - 8748 q^{3} + 65536 q^{4} + 19180 q^{5} - 1119744 q^{6} + 2510980 q^{7} + 8388608 q^{8} + 19131876 q^{9} + 2455040 q^{10} - 41108920 q^{11} - 143327232 q^{12} - 250994068 q^{13} + 321405440 q^{14} - 41946660 q^{15} + 1073741824 q^{16} - 2487508320 q^{17} + 2448880128 q^{18} - 11351292132 q^{19} + 314245120 q^{20} - 5491513260 q^{21} - 5261941760 q^{22} - 6738531488 q^{23} - 18345885696 q^{24} + 76363379660 q^{25} - 32127240704 q^{26} - 41841412812 q^{27} + 41139896320 q^{28} + 205829454856 q^{29} - 5369172480 q^{30} + 245508625876 q^{31} + 137438953472 q^{32} + 89905208040 q^{33} - 318401064960 q^{34} - 792057763760 q^{35} + 313456656384 q^{36} - 1174927152528 q^{37} - 1452965392896 q^{38} + 548924026716 q^{39} + 40223375360 q^{40} + 1786321433340 q^{41} - 702913697280 q^{42} - 2478301019624 q^{43} - 673528545280 q^{44} + 91737345420 q^{45} - 862532030464 q^{46} - 4173552490816 q^{47} - 2348273369088 q^{48} + 12442477653636 q^{49} + 9774512596480 q^{50} + 5440180695840 q^{51} - 4112286810112 q^{52} - 5700182621112 q^{53} - 5355700839936 q^{54} + 251218497560 q^{55} + 5265906728960 q^{56} + 24825275892684 q^{57} + 26346170221568 q^{58} - 19103066746688 q^{59} - 687254077440 q^{60} - 35026221391592 q^{61} + 31425104112128 q^{62} + 12009939499620 q^{63} + 17592186044416 q^{64} - 1203516556060 q^{65} + 11507866629120 q^{66} - 146670006946988 q^{67} - 40755336314880 q^{68} + 14737168364256 q^{69} - 101383393761280 q^{70} + 54469169758856 q^{71} + 40122452017152 q^{72} - 280227940264048 q^{73} - 150390675523584 q^{74} - 167006711316420 q^{75} - 185979570290688 q^{76} - 705615415653880 q^{77} + 70262275419648 q^{78} - 76381735869712 q^{79} + 5148592046080 q^{80} + 91507169819844 q^{81} + 228649143467520 q^{82} - 899056113684240 q^{83} - 89972953251840 q^{84} - 14\!\cdots\!40 q^{85}+ \cdots - 196622689983480 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 6327x^{2} - 123410x + 3269236 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{3} - 93\nu^{2} + 20238\nu + 571878 ) / 230 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 543\nu^{3} - 28707\nu^{2} - 2358978\nu + 40555872 ) / 115 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1893\nu^{3} - 90357\nu^{2} - 6684378\nu + 110633022 ) / 115 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 11\beta_{3} - 36\beta_{2} + 850\beta_1 ) / 173880 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{3} - 14\beta_{2} - 1282\beta _1 + 5238756 ) / 1656 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 64441\beta_{3} - 197286\beta_{2} - 3423790\beta _1 + 16093898100 ) / 173880 \) 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
85.5728
−54.1820
15.1502
−46.5411
128.000 −2187.00 16384.0 −319623. −279936. 1.82306e6 2.09715e6 4.78297e6 −4.09117e7
1.2 128.000 −2187.00 16384.0 −80479.8 −279936. 425425. 2.09715e6 4.78297e6 −1.03014e7
1.3 128.000 −2187.00 16384.0 178827. −279936. 3.86580e6 2.09715e6 4.78297e6 2.28898e7
1.4 128.000 −2187.00 16384.0 240456. −279936. −3.60330e6 2.09715e6 4.78297e6 3.07783e7
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(13\) \(1\)

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 78.16.a.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.16.a.f 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 19180T_{5}^{3} - 99032909880T_{5}^{2} + 6419103628128800T_{5} + 1106094956596202320000 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(78))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 128)^{4} \) Copy content Toggle raw display
$3$ \( (T + 2187)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 19180 T^{3} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{4} - 2510980 T^{3} + \cdots - 10\!\cdots\!68 \) Copy content Toggle raw display
$11$ \( T^{4} + 41108920 T^{3} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T + 62748517)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 2487508320 T^{3} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{4} + 11351292132 T^{3} + \cdots - 24\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} + 6738531488 T^{3} + \cdots + 61\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} - 205829454856 T^{3} + \cdots - 17\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} - 245508625876 T^{3} + \cdots - 11\!\cdots\!32 \) Copy content Toggle raw display
$37$ \( T^{4} + 1174927152528 T^{3} + \cdots - 98\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{4} - 1786321433340 T^{3} + \cdots - 16\!\cdots\!88 \) Copy content Toggle raw display
$43$ \( T^{4} + 2478301019624 T^{3} + \cdots - 61\!\cdots\!72 \) Copy content Toggle raw display
$47$ \( T^{4} + 4173552490816 T^{3} + \cdots - 20\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{4} + 5700182621112 T^{3} + \cdots - 31\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{4} + 19103066746688 T^{3} + \cdots - 35\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + 35026221391592 T^{3} + \cdots - 47\!\cdots\!72 \) Copy content Toggle raw display
$67$ \( T^{4} + 146670006946988 T^{3} + \cdots - 39\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{4} - 54469169758856 T^{3} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{4} + 280227940264048 T^{3} + \cdots - 22\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{4} + 76381735869712 T^{3} + \cdots - 12\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{4} + 899056113684240 T^{3} + \cdots + 62\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{4} - 191408718717108 T^{3} + \cdots + 76\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 93\!\cdots\!28 \) Copy content Toggle raw display
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