Properties

Label 78.14.a.g
Level $78$
Weight $14$
Character orbit 78.a
Self dual yes
Analytic conductor $83.640$
Analytic rank $0$
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] = [78,14,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, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("78.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 14 \)
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: \(83.6401245825\)
Analytic rank: \(0\)
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: \( x^{4} - 2x^{3} - 140869910x^{2} - 625019925408x + 47227437332856 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q - 64 q^{2} - 729 q^{3} + 4096 q^{4} + ( - \beta_{2} + 4326) q^{5} + 46656 q^{6} + (\beta_{3} + 4 \beta_{2} + 27692) q^{7} - 262144 q^{8} + 531441 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 64 q^{2} - 729 q^{3} + 4096 q^{4} + ( - \beta_{2} + 4326) q^{5} + 46656 q^{6} + (\beta_{3} + 4 \beta_{2} + 27692) q^{7} - 262144 q^{8} + 531441 q^{9} + (64 \beta_{2} - 276864) q^{10} + (42 \beta_{3} - 61 \beta_{2} + 59 \beta_1 + 1257287) q^{11} - 2985984 q^{12} + 4826809 q^{13} + ( - 64 \beta_{3} - 256 \beta_{2} - 1772288) q^{14} + (729 \beta_{2} - 3153654) q^{15} + 16777216 q^{16} + (504 \beta_{3} - 448 \beta_{2} - 1040 \beta_1 + 44639026) q^{17} - 34012224 q^{18} + ( - 231 \beta_{3} - 436 \beta_{2} + 1610 \beta_1 + 82623298) q^{19} + ( - 4096 \beta_{2} + 17719296) q^{20} + ( - 729 \beta_{3} - 2916 \beta_{2} - 20187468) q^{21} + ( - 2688 \beta_{3} + 3904 \beta_{2} - 3776 \beta_1 - 80466368) q^{22} + (1939 \beta_{3} - 14920 \beta_{2} - 2469 \beta_1 + 120983775) q^{23} + 191102976 q^{24} + (3605 \beta_{3} - 26326 \beta_{2} + 13845 \beta_1 + 486086776) q^{25} - 308915776 q^{26} - 387420489 q^{27} + (4096 \beta_{3} + 16384 \beta_{2} + 113426432) q^{28} + (1267 \beta_{3} - 52144 \beta_{2} + 50161 \beta_1 + 862656595) q^{29} + ( - 46656 \beta_{2} + 201833856) q^{30} + (1981 \beta_{3} + 113498 \beta_{2} - 134594 \beta_1 - 440130462) q^{31} - 1073741824 q^{32} + ( - 30618 \beta_{3} + 44469 \beta_{2} - 43011 \beta_1 - 916562223) q^{33} + ( - 32256 \beta_{3} + 28672 \beta_{2} + \cdots - 2856897664) q^{34}+ \cdots + (22320522 \beta_{3} - 32417901 \beta_{2} + \cdots + 668173860567) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 256 q^{2} - 2916 q^{3} + 16384 q^{4} + 17302 q^{5} + 186624 q^{6} + 110778 q^{7} - 1048576 q^{8} + 2125764 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 256 q^{2} - 2916 q^{3} + 16384 q^{4} + 17302 q^{5} + 186624 q^{6} + 110778 q^{7} - 1048576 q^{8} + 2125764 q^{9} - 1107328 q^{10} + 5029228 q^{11} - 11943936 q^{12} + 19307236 q^{13} - 7089792 q^{14} - 12613158 q^{15} + 67108864 q^{16} + 178554136 q^{17} - 136048896 q^{18} + 330495078 q^{19} + 70868992 q^{20} - 80757162 q^{21} - 321870592 q^{22} + 483904200 q^{23} + 764411904 q^{24} + 1944329352 q^{25} - 1235663104 q^{26} - 1549681956 q^{27} + 453746688 q^{28} + 3450624948 q^{29} + 807242112 q^{30} - 1760560078 q^{31} - 4294967296 q^{32} - 3666307212 q^{33} - 11427464704 q^{34} - 25854753440 q^{35} + 8707129344 q^{36} - 21294929844 q^{37} - 21151684992 q^{38} - 14074975044 q^{39} - 4535615488 q^{40} - 70606057250 q^{41} + 5168458368 q^{42} - 75558364632 q^{43} + 20599717888 q^{44} + 9194992182 q^{45} - 30969868800 q^{46} - 112183555592 q^{47} - 48922361856 q^{48} - 153830912232 q^{49} - 124437078528 q^{50} - 130165965144 q^{51} + 79082438656 q^{52} - 104575883144 q^{53} + 99179645184 q^{54} + 285535404320 q^{55} - 29039788032 q^{56} - 240930911862 q^{57} - 220839996672 q^{58} - 953392600312 q^{59} - 51663495168 q^{60} - 173990512884 q^{61} + 112675844992 q^{62} + 58871971098 q^{63} + 274877906944 q^{64} + 83513449318 q^{65} + 234643661568 q^{66} - 163049862070 q^{67} + 731357741056 q^{68} - 352766161800 q^{69} + 1654704220160 q^{70} + 1141535539628 q^{71} - 557256278016 q^{72} + 2505023734484 q^{73} + 1362875510016 q^{74} - 1417416097608 q^{75} + 1353707839488 q^{76} + 4228634258008 q^{77} + 900798402816 q^{78} + 2114849719736 q^{79} + 290279391232 q^{80} + 1129718145924 q^{81} + 4518787664000 q^{82} + 3245912192784 q^{83} - 330781335552 q^{84} + 7248384065516 q^{85} + 4835735336448 q^{86} - 2515505587092 q^{87} - 1318381944832 q^{88} - 4151539197470 q^{89} - 588479499648 q^{90} + 534704247402 q^{91} + 1982071603200 q^{92} + 1283448296862 q^{93} + 7179747557888 q^{94} - 597992324600 q^{95} + 3131031158784 q^{96} + 7366767632100 q^{97} + 9845178382848 q^{98} + 2672737957548 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 140869910x^{2} - 625019925408x + 47227437332856 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 743\nu^{3} - 5178340\nu^{2} - 71445673330\nu + 16323041686596 ) / 1043493360 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 251\nu^{3} - 1187572\nu^{2} - 26376103114\nu - 34052955813276 ) / 626096016 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -281\nu^{3} - 3659300\nu^{2} + 124555533070\nu + 389464247911668 ) / 3130480080 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_{2} - \beta _1 + 11 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 997\beta_{3} + 8681\beta_{2} - 4762\beta _1 + 422607278 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 61976375\beta_{3} + 217162870\beta_{2} - 97603535\beta _1 + 5627624219221 ) / 12 \) 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
−6116.94
13653.6
−7608.98
74.3167
−64.0000 −729.000 4096.00 −36250.1 46656.0 47833.1 −262144. 531441. 2.32001e6
1.2 −64.0000 −729.000 4096.00 −32895.6 46656.0 397852. −262144. 531441. 2.10532e6
1.3 −64.0000 −729.000 4096.00 24591.2 46656.0 −259838. −262144. 531441. −1.57384e6
1.4 −64.0000 −729.000 4096.00 61856.5 46656.0 −75069.1 −262144. 531441. −3.95881e6
\(n\): 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 78.14.a.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.14.a.g 4 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 17302T_{5}^{3} - 3263891324T_{5}^{2} + 2093196490200T_{5} + 1813895761795020000 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(78))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 64)^{4} \) Copy content Toggle raw display
$3$ \( (T + 729)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 17302 T^{3} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{4} - 110778 T^{3} + \cdots + 37\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{4} - 5029228 T^{3} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( (T - 4826809)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} - 178554136 T^{3} + \cdots - 47\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{4} - 330495078 T^{3} + \cdots - 28\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{4} - 483904200 T^{3} + \cdots - 15\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} - 3450624948 T^{3} + \cdots + 49\!\cdots\!92 \) Copy content Toggle raw display
$31$ \( T^{4} + 1760560078 T^{3} + \cdots + 29\!\cdots\!48 \) Copy content Toggle raw display
$37$ \( T^{4} + 21294929844 T^{3} + \cdots - 27\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{4} + 70606057250 T^{3} + \cdots - 11\!\cdots\!40 \) Copy content Toggle raw display
$43$ \( T^{4} + 75558364632 T^{3} + \cdots - 33\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{4} + 112183555592 T^{3} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{4} + 104575883144 T^{3} + \cdots + 21\!\cdots\!40 \) Copy content Toggle raw display
$59$ \( T^{4} + 953392600312 T^{3} + \cdots - 19\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{4} + 173990512884 T^{3} + \cdots - 15\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{4} + 163049862070 T^{3} + \cdots + 72\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{4} - 1141535539628 T^{3} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{4} - 2505023734484 T^{3} + \cdots - 12\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( T^{4} - 2114849719736 T^{3} + \cdots + 24\!\cdots\!72 \) Copy content Toggle raw display
$83$ \( T^{4} - 3245912192784 T^{3} + \cdots - 32\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{4} + 4151539197470 T^{3} + \cdots - 35\!\cdots\!28 \) Copy content Toggle raw display
$97$ \( T^{4} - 7366767632100 T^{3} + \cdots - 17\!\cdots\!40 \) Copy content Toggle raw display
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