Properties

Label 78.4.e.d
Level $78$
Weight $4$
Character orbit 78.e
Analytic conductor $4.602$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [78,4,Mod(55,78)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(78, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("78.55");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 78.e (of order \(3\), degree \(2\), 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: \(4.60214898045\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 283x^{4} - 1716x^{3} + 80089x^{2} - 242814x + 736164 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{2} + 2) q^{2} + ( - 3 \beta_{2} - 3) q^{3} + 4 \beta_{2} q^{4} + ( - \beta_{3} - 4) q^{5} - 6 \beta_{2} q^{6} + (\beta_{5} - 6 \beta_{2}) q^{7} - 8 q^{8} + 9 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{2} + 2) q^{2} + ( - 3 \beta_{2} - 3) q^{3} + 4 \beta_{2} q^{4} + ( - \beta_{3} - 4) q^{5} - 6 \beta_{2} q^{6} + (\beta_{5} - 6 \beta_{2}) q^{7} - 8 q^{8} + 9 \beta_{2} q^{9} + ( - 2 \beta_{3} - 8 \beta_{2} + 2 \beta_1 - 8) q^{10} + (\beta_{5} + \beta_{4} - 3 \beta_{3} + 13 \beta_{2} + 3 \beta_1 + 13) q^{11} + 12 q^{12} + ( - \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_1 - 5) q^{13} + ( - 2 \beta_{4} + 12) q^{14} + (3 \beta_{3} + 12 \beta_{2} - 3 \beta_1 + 12) q^{15} + ( - 16 \beta_{2} - 16) q^{16} + (\beta_{5} + 5 \beta_{2} - 4 \beta_1) q^{17} - 18 q^{18} + ( - 3 \beta_{5} + 3 \beta_{2} + 5 \beta_1) q^{19} + ( - 16 \beta_{2} + 4 \beta_1) q^{20} + (3 \beta_{4} - 18) q^{21} + (2 \beta_{5} + 26 \beta_{2} + 6 \beta_1) q^{22} + (\beta_{5} + \beta_{4} + 5 \beta_{3} - 99 \beta_{2} - 5 \beta_1 - 99) q^{23} + (24 \beta_{2} + 24) q^{24} + ( - 4 \beta_{4} + 3 \beta_{3} + 81) q^{25} + ( - 2 \beta_{5} - 6 \beta_{3} - 10 \beta_{2} + 4 \beta_1 - 10) q^{26} + 27 q^{27} + ( - 4 \beta_{5} - 4 \beta_{4} + 24 \beta_{2} + 24) q^{28} + (4 \beta_{5} + 4 \beta_{4} + 5 \beta_{3} + 84 \beta_{2} - 5 \beta_1 + 84) q^{29} + (24 \beta_{2} - 6 \beta_1) q^{30} + (2 \beta_{4} + 13 \beta_{3} - 73) q^{31} - 32 \beta_{2} q^{32} + ( - 3 \beta_{5} - 39 \beta_{2} - 9 \beta_1) q^{33} + ( - 2 \beta_{4} - 8 \beta_{3} - 10) q^{34} + ( - 9 \beta_{5} + 47 \beta_{2} - 23 \beta_1) q^{35} + ( - 36 \beta_{2} - 36) q^{36} + (3 \beta_{5} + 3 \beta_{4} - 10 \beta_{3} + 9 \beta_{2} + 10 \beta_1 + 9) q^{37} + (6 \beta_{4} + 10 \beta_{3} - 6) q^{38} + (3 \beta_{5} + 9 \beta_{3} + 15 \beta_{2} - 6 \beta_1 + 15) q^{39} + (8 \beta_{3} + 32) q^{40} + (5 \beta_{5} + 5 \beta_{4} + 2 \beta_{3} + 217 \beta_{2} - 2 \beta_1 + 217) q^{41} + (6 \beta_{5} + 6 \beta_{4} - 36 \beta_{2} - 36) q^{42} + (\beta_{5} + 18 \beta_{2}) q^{43} + ( - 4 \beta_{4} + 12 \beta_{3} - 52) q^{44} + ( - 36 \beta_{2} + 9 \beta_1) q^{45} + (2 \beta_{5} - 198 \beta_{2} - 10 \beta_1) q^{46} + ( - \beta_{4} + 3 \beta_{3} + 47) q^{47} + 48 \beta_{2} q^{48} + ( - 6 \beta_{5} - 6 \beta_{4} - 27 \beta_{3} - 558 \beta_{2} + 27 \beta_1 - 558) q^{49} + ( - 8 \beta_{5} - 8 \beta_{4} + 6 \beta_{3} + 162 \beta_{2} - 6 \beta_1 + 162) q^{50} + (3 \beta_{4} + 12 \beta_{3} + 15) q^{51} + (4 \beta_{4} - 4 \beta_{3} - 20 \beta_{2} + 12 \beta_1) q^{52} + (4 \beta_{4} + \beta_{3} - 370) q^{53} + (54 \beta_{2} + 54) q^{54} + ( - 21 \beta_{5} - 21 \beta_{4} + \beta_{3} + 541 \beta_{2} - \beta_1 + 541) q^{55} + ( - 8 \beta_{5} + 48 \beta_{2}) q^{56} + ( - 9 \beta_{4} - 15 \beta_{3} + 9) q^{57} + (8 \beta_{5} + 168 \beta_{2} - 10 \beta_1) q^{58} + ( - 2 \beta_{5} - 416 \beta_{2} + 24 \beta_1) q^{59} + ( - 12 \beta_{3} - 48) q^{60} + (19 \beta_{5} + 156 \beta_{2} + 15 \beta_1) q^{61} + (4 \beta_{5} + 4 \beta_{4} + 26 \beta_{3} - 146 \beta_{2} - 26 \beta_1 - 146) q^{62} + ( - 9 \beta_{5} - 9 \beta_{4} + 54 \beta_{2} + 54) q^{63} + 64 q^{64} + (13 \beta_{5} + \beta_{4} - 14 \beta_{3} - 213 \beta_{2} + 16 \beta_1 + 377) q^{65} + (6 \beta_{4} - 18 \beta_{3} + 78) q^{66} + (7 \beta_{5} + 7 \beta_{4} - 16 \beta_{3} + 70 \beta_{2} + 16 \beta_1 + 70) q^{67} + ( - 4 \beta_{5} - 4 \beta_{4} - 16 \beta_{3} - 20 \beta_{2} + 16 \beta_1 - 20) q^{68} + ( - 3 \beta_{5} + 297 \beta_{2} + 15 \beta_1) q^{69} + (18 \beta_{4} - 46 \beta_{3} - 94) q^{70} + (15 \beta_{5} - 137 \beta_{2} - 19 \beta_1) q^{71} - 72 \beta_{2} q^{72} + ( - 12 \beta_{4} + 30 \beta_{3} + 17) q^{73} + (6 \beta_{5} + 18 \beta_{2} + 20 \beta_1) q^{74} + (12 \beta_{5} + 12 \beta_{4} - 9 \beta_{3} - 243 \beta_{2} + 9 \beta_1 - 243) q^{75} + (12 \beta_{5} + 12 \beta_{4} + 20 \beta_{3} - 12 \beta_{2} - 20 \beta_1 - 12) q^{76} + ( - 10 \beta_{4} - 96 \beta_{3} - 856) q^{77} + ( - 6 \beta_{4} + 6 \beta_{3} + 30 \beta_{2} - 18 \beta_1) q^{78} + (12 \beta_{4} - 15 \beta_{3} - 373) q^{79} + (16 \beta_{3} + 64 \beta_{2} - 16 \beta_1 + 64) q^{80} + ( - 81 \beta_{2} - 81) q^{81} + (10 \beta_{5} + 434 \beta_{2} - 4 \beta_1) q^{82} + (3 \beta_{4} + 9 \beta_{3} - 231) q^{83} + (12 \beta_{5} - 72 \beta_{2}) q^{84} + (7 \beta_{5} - 757 \beta_{2} - 16 \beta_1) q^{85} + ( - 2 \beta_{4} - 36) q^{86} + ( - 12 \beta_{5} - 252 \beta_{2} + 15 \beta_1) q^{87} + ( - 8 \beta_{5} - 8 \beta_{4} + 24 \beta_{3} - 104 \beta_{2} - 24 \beta_1 - 104) q^{88} + ( - 4 \beta_{5} - 4 \beta_{4} - 22 \beta_{3} + 868 \beta_{2} + 22 \beta_1 + 868) q^{89} + (18 \beta_{3} + 72) q^{90} + ( - 20 \beta_{5} + 7 \beta_{4} + 50 \beta_{3} + 99 \beta_{2} - 69 \beta_1 + 888) q^{91} + ( - 4 \beta_{4} - 20 \beta_{3} + 396) q^{92} + ( - 6 \beta_{5} - 6 \beta_{4} - 39 \beta_{3} + 219 \beta_{2} + 39 \beta_1 + 219) q^{93} + ( - 2 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} + 94 \beta_{2} - 6 \beta_1 + 94) q^{94} + (7 \beta_{5} + 869 \beta_{2} + 59 \beta_1) q^{95} - 96 q^{96} + ( - 22 \beta_{5} - 63 \beta_{2} - 17 \beta_1) q^{97} + ( - 12 \beta_{5} - 1116 \beta_{2} + 54 \beta_1) q^{98} + ( - 9 \beta_{4} + 27 \beta_{3} - 117) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} - 9 q^{3} - 12 q^{4} - 24 q^{5} + 18 q^{6} + 17 q^{7} - 48 q^{8} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} - 9 q^{3} - 12 q^{4} - 24 q^{5} + 18 q^{6} + 17 q^{7} - 48 q^{8} - 27 q^{9} - 24 q^{10} + 40 q^{11} + 72 q^{12} - 31 q^{13} + 68 q^{14} + 36 q^{15} - 48 q^{16} - 16 q^{17} - 108 q^{18} - 6 q^{19} + 48 q^{20} - 102 q^{21} - 80 q^{22} - 296 q^{23} + 72 q^{24} + 478 q^{25} - 28 q^{26} + 162 q^{27} + 68 q^{28} + 256 q^{29} - 72 q^{30} - 434 q^{31} + 96 q^{32} + 120 q^{33} - 64 q^{34} - 132 q^{35} - 108 q^{36} + 30 q^{37} - 24 q^{38} + 42 q^{39} + 192 q^{40} + 656 q^{41} - 102 q^{42} - 55 q^{43} - 320 q^{44} + 108 q^{45} + 592 q^{46} + 280 q^{47} - 144 q^{48} - 1680 q^{49} + 478 q^{50} + 96 q^{51} + 68 q^{52} - 2212 q^{53} + 162 q^{54} + 1602 q^{55} - 136 q^{56} + 36 q^{57} - 512 q^{58} + 1250 q^{59} - 288 q^{60} - 487 q^{61} - 434 q^{62} + 153 q^{63} + 384 q^{64} + 2890 q^{65} + 480 q^{66} + 217 q^{67} - 64 q^{68} - 888 q^{69} - 528 q^{70} + 396 q^{71} + 216 q^{72} + 78 q^{73} - 60 q^{74} - 717 q^{75} - 24 q^{76} - 5156 q^{77} - 102 q^{78} - 2214 q^{79} + 192 q^{80} - 243 q^{81} - 1312 q^{82} - 1380 q^{83} + 204 q^{84} + 2264 q^{85} - 220 q^{86} + 768 q^{87} - 320 q^{88} + 2600 q^{89} + 432 q^{90} + 5065 q^{91} + 2368 q^{92} + 651 q^{93} + 280 q^{94} - 2614 q^{95} - 576 q^{96} + 211 q^{97} + 3360 q^{98} - 720 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 283x^{4} - 1716x^{3} + 80089x^{2} - 242814x + 736164 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 283\nu^{3} + 858\nu^{2} - 80089\nu ) / 242814 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 858 ) / 283 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 5\nu^{3} + 283\nu^{2} - 858\nu + 49480 ) / 1132 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -95\nu^{5} - 429\nu^{4} - 26885\nu^{3} + 81510\nu^{2} - 6633338\nu ) / 485628 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 4\beta_{5} + 4\beta_{4} + 5\beta_{3} - 190\beta_{2} - 5\beta _1 - 190 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -283\beta_{3} + 858 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -1132\beta_{5} + 53770\beta_{2} + 2273\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3432\beta_{5} + 3432\beta_{4} + 84379\beta_{3} - 405834\beta_{2} - 84379\beta _1 - 405834 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/78\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(67\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
7.51525 13.0168i
1.57067 2.72048i
−9.08592 + 15.7373i
7.51525 + 13.0168i
1.57067 + 2.72048i
−9.08592 15.7373i
1.00000 + 1.73205i −1.50000 2.59808i −2.00000 + 3.46410i −19.0305 3.00000 5.19615i 16.8836 29.2432i −8.00000 −4.50000 + 7.79423i −19.0305 32.9618i
55.2 1.00000 + 1.73205i −1.50000 2.59808i −2.00000 + 3.46410i −7.14134 3.00000 5.19615i −17.5532 + 30.4030i −8.00000 −4.50000 + 7.79423i −7.14134 12.3692i
55.3 1.00000 + 1.73205i −1.50000 2.59808i −2.00000 + 3.46410i 14.1718 3.00000 5.19615i 9.16959 15.8822i −8.00000 −4.50000 + 7.79423i 14.1718 + 24.5464i
61.1 1.00000 1.73205i −1.50000 + 2.59808i −2.00000 3.46410i −19.0305 3.00000 + 5.19615i 16.8836 + 29.2432i −8.00000 −4.50000 7.79423i −19.0305 + 32.9618i
61.2 1.00000 1.73205i −1.50000 + 2.59808i −2.00000 3.46410i −7.14134 3.00000 + 5.19615i −17.5532 30.4030i −8.00000 −4.50000 7.79423i −7.14134 + 12.3692i
61.3 1.00000 1.73205i −1.50000 + 2.59808i −2.00000 3.46410i 14.1718 3.00000 + 5.19615i 9.16959 + 15.8822i −8.00000 −4.50000 7.79423i 14.1718 24.5464i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 78.4.e.d 6
3.b odd 2 1 234.4.h.j 6
4.b odd 2 1 624.4.q.h 6
13.c even 3 1 inner 78.4.e.d 6
13.c even 3 1 1014.4.a.u 3
13.e even 6 1 1014.4.a.y 3
13.f odd 12 2 1014.4.b.m 6
39.i odd 6 1 234.4.h.j 6
52.j odd 6 1 624.4.q.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.4.e.d 6 1.a even 1 1 trivial
78.4.e.d 6 13.c even 3 1 inner
234.4.h.j 6 3.b odd 2 1
234.4.h.j 6 39.i odd 6 1
624.4.q.h 6 4.b odd 2 1
624.4.q.h 6 52.j odd 6 1
1014.4.a.u 3 13.c even 3 1
1014.4.a.y 3 13.e even 6 1
1014.4.b.m 6 13.f odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 12T_{5}^{2} - 235T_{5} - 1926 \) acting on \(S_{4}^{\mathrm{new}}(78, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 9)^{3} \) Copy content Toggle raw display
$5$ \( (T^{3} + 12 T^{2} - 235 T - 1926)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} - 17 T^{5} + \cdots + 472627600 \) Copy content Toggle raw display
$11$ \( T^{6} - 40 T^{5} + \cdots + 21778675776 \) Copy content Toggle raw display
$13$ \( T^{6} + 31 T^{5} + \cdots + 10604499373 \) Copy content Toggle raw display
$17$ \( T^{6} + 16 T^{5} + \cdots + 30762353664 \) Copy content Toggle raw display
$19$ \( T^{6} + 6 T^{5} + \cdots + 39394310400 \) Copy content Toggle raw display
$23$ \( T^{6} + 296 T^{5} + \cdots + 193846478400 \) Copy content Toggle raw display
$29$ \( T^{6} - 256 T^{5} + \cdots + 509753160900 \) Copy content Toggle raw display
$31$ \( (T^{3} + 217 T^{2} - 35692 T + 1117424)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} - 30 T^{5} + \cdots + 12134674712196 \) Copy content Toggle raw display
$41$ \( T^{6} - 656 T^{5} + \cdots + 30479011808400 \) Copy content Toggle raw display
$43$ \( T^{6} + 55 T^{5} + 3323 T^{4} + \cdots + 10679824 \) Copy content Toggle raw display
$47$ \( (T^{3} - 140 T^{2} + 2488 T - 8856)^{2} \) Copy content Toggle raw display
$53$ \( (T^{3} + 1106 T^{2} + 386817 T + 41277168)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 343510935266304 \) Copy content Toggle raw display
$61$ \( T^{6} + 487 T^{5} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 947501727258256 \) Copy content Toggle raw display
$71$ \( T^{6} - 396 T^{5} + \cdots + 92\!\cdots\!96 \) Copy content Toggle raw display
$73$ \( (T^{3} - 39 T^{2} - 465345 T - 116327705)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 1107 T^{2} + 145176 T - 16936400)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} + 690 T^{2} + 125748 T + 6135264)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 2600 T^{5} + \cdots + 29\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{6} - 211 T^{5} + \cdots + 54\!\cdots\!24 \) Copy content Toggle raw display
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