Properties

Label 414.8.a.c
Level $414$
Weight $8$
Character orbit 414.a
Self dual yes
Analytic conductor $129.327$
Analytic rank $0$
Dimension $3$
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] = [414,8,Mod(1,414)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(414, 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("414.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 414.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: \(129.327400550\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 684x - 5052 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
Twist minimal: no (minimal twist has level 138)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 q^{2} + 64 q^{4} + ( - \beta_{2} - 31) q^{5} - 286 q^{7} - 512 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 8 q^{2} + 64 q^{4} + ( - \beta_{2} - 31) q^{5} - 286 q^{7} - 512 q^{8} + (8 \beta_{2} + 248) q^{10} + (21 \beta_{2} + 14 \beta_1 + 609) q^{11} + ( - 52 \beta_{2} + 54 \beta_1 - 1268) q^{13} + 2288 q^{14} + 4096 q^{16} + ( - 23 \beta_{2} + 131 \beta_1 - 1106) q^{17} + (196 \beta_{2} + 65 \beta_1 - 4701) q^{19} + ( - 64 \beta_{2} - 1984) q^{20} + ( - 168 \beta_{2} - 112 \beta_1 - 4872) q^{22} + 12167 q^{23} + (27 \beta_{2} + 77 \beta_1 - 56723) q^{25} + (416 \beta_{2} - 432 \beta_1 + 10144) q^{26} - 18304 q^{28} + ( - 545 \beta_{2} + 105 \beta_1 + 57088) q^{29} + ( - 233 \beta_{2} - 51 \beta_1 - 41718) q^{31} - 32768 q^{32} + (184 \beta_{2} - 1048 \beta_1 + 8848) q^{34} + (286 \beta_{2} + 8866) q^{35} + (733 \beta_{2} - 38 \beta_1 - 226101) q^{37} + ( - 1568 \beta_{2} - 520 \beta_1 + 37608) q^{38} + (512 \beta_{2} + 15872) q^{40} + (2369 \beta_{2} + 77 \beta_1 + 111218) q^{41} + ( - 5078 \beta_{2} + 1613 \beta_1 - 50871) q^{43} + (1344 \beta_{2} + 896 \beta_1 + 38976) q^{44} - 97336 q^{46} + ( - 4507 \beta_{2} - 2371 \beta_1 + 607352) q^{47} - 741747 q^{49} + ( - 216 \beta_{2} - 616 \beta_1 + 453784) q^{50} + ( - 3328 \beta_{2} + 3456 \beta_1 - 81152) q^{52} + ( - 285 \beta_{2} + 5072 \beta_1 + 1145697) q^{53} + ( - 2177 \beta_{2} - 2527 \beta_1 - 515942) q^{55} + 146432 q^{56} + (4360 \beta_{2} - 840 \beta_1 - 456704) q^{58} + ( - 3600 \beta_{2} + 2016 \beta_1 + 1047348) q^{59} + ( - 3063 \beta_{2} - 7700 \beta_1 - 1146631) q^{61} + (1864 \beta_{2} + 408 \beta_1 + 333744) q^{62} + 262144 q^{64} + ( - 5312 \beta_{2} + 494 \beta_1 + 840718) q^{65} + (21354 \beta_{2} + 4995 \beta_1 - 301837) q^{67} + ( - 1472 \beta_{2} + 8384 \beta_1 - 70784) q^{68} + ( - 2288 \beta_{2} - 70928) q^{70} + (18009 \beta_{2} - 23895 \beta_1 - 850872) q^{71} + ( - 24711 \beta_{2} + 13283 \beta_1 - 192940) q^{73} + ( - 5864 \beta_{2} + 304 \beta_1 + 1808808) q^{74} + (12544 \beta_{2} + 4160 \beta_1 - 300864) q^{76} + ( - 6006 \beta_{2} - 4004 \beta_1 - 174174) q^{77} + (42994 \beta_{2} - 6150 \beta_1 + 122046) q^{79} + ( - 4096 \beta_{2} - 126976) q^{80} + ( - 18952 \beta_{2} - 616 \beta_1 - 889744) q^{82} + (30495 \beta_{2} + 31190 \beta_1 - 1373337) q^{83} + ( - 14444 \beta_{2} - 6744 \beta_1 - 130004) q^{85} + (40624 \beta_{2} - 12904 \beta_1 + 406968) q^{86} + ( - 10752 \beta_{2} - 7168 \beta_1 - 311808) q^{88} + ( - 30741 \beta_{2} + 11019 \beta_1 - 486312) q^{89} + (14872 \beta_{2} - 15444 \beta_1 + 362648) q^{91} + 778688 q^{92} + (36056 \beta_{2} + 18968 \beta_1 - 4858816) q^{94} + ( - 2185 \beta_{2} - 19317 \beta_1 - 4175500) q^{95} + ( - 16514 \beta_{2} + 14628 \beta_1 - 5040156) q^{97} + 5933976 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 24 q^{2} + 192 q^{4} - 92 q^{5} - 858 q^{7} - 1536 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 24 q^{2} + 192 q^{4} - 92 q^{5} - 858 q^{7} - 1536 q^{8} + 736 q^{10} + 1820 q^{11} - 3698 q^{13} + 6864 q^{14} + 12288 q^{16} - 3164 q^{17} - 14234 q^{19} - 5888 q^{20} - 14560 q^{22} + 36501 q^{23} - 170119 q^{25} + 29584 q^{26} - 54912 q^{28} + 171914 q^{29} - 124972 q^{31} - 98304 q^{32} + 25312 q^{34} + 26312 q^{35} - 679074 q^{37} + 113872 q^{38} + 47104 q^{40} + 331362 q^{41} - 145922 q^{43} + 116480 q^{44} - 292008 q^{46} + 1824192 q^{47} - 2225241 q^{49} + 1360952 q^{50} - 236672 q^{52} + 3442448 q^{53} - 1548176 q^{55} + 439296 q^{56} - 1375312 q^{58} + 3147660 q^{59} - 3444530 q^{61} + 999776 q^{62} + 786432 q^{64} + 2527960 q^{65} - 921870 q^{67} - 202496 q^{68} - 210496 q^{70} - 2594520 q^{71} - 540826 q^{73} + 5432592 q^{74} - 910976 q^{76} - 520520 q^{77} + 316994 q^{79} - 376832 q^{80} - 2650896 q^{82} - 4119316 q^{83} - 382312 q^{85} + 1167376 q^{86} - 931840 q^{88} - 1417176 q^{89} + 1057628 q^{91} + 2336064 q^{92} - 14593536 q^{94} - 12543632 q^{95} - 15089326 q^{97} + 17801928 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 684x - 5052 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{2} + 27\nu + 448 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 3\nu - 456 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 4 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 9\beta_{2} + \beta _1 + 1828 ) / 4 \) 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
29.7267
−20.3930
−8.33366
−8.00000 0 64.0000 −200.248 0 −286.000 −512.000 0 1601.98
1.2 −8.00000 0 64.0000 −41.5271 0 −286.000 −512.000 0 332.217
1.3 −8.00000 0 64.0000 149.775 0 −286.000 −512.000 0 −1198.20
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(23\) \(-1\)

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 414.8.a.c 3
3.b odd 2 1 138.8.a.d 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.8.a.d 3 3.b odd 2 1
414.8.a.c 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 92T_{5}^{2} - 27896T_{5} - 1245480 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(414))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 8)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 92 T^{2} - 27896 T - 1245480 \) Copy content Toggle raw display
$7$ \( (T + 286)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 1820 T^{2} + \cdots - 39808807752 \) Copy content Toggle raw display
$13$ \( T^{3} + 3698 T^{2} + \cdots - 31329885032 \) Copy content Toggle raw display
$17$ \( T^{3} + 3164 T^{2} + \cdots + 8811060158208 \) Copy content Toggle raw display
$19$ \( T^{3} + 14234 T^{2} + \cdots - 27667220902592 \) Copy content Toggle raw display
$23$ \( (T - 12167)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} - 171914 T^{2} + \cdots + 62745468718728 \) Copy content Toggle raw display
$31$ \( T^{3} + 124972 T^{2} + \cdots + 9840206494080 \) Copy content Toggle raw display
$37$ \( T^{3} + 679074 T^{2} + \cdots + 82\!\cdots\!28 \) Copy content Toggle raw display
$41$ \( T^{3} - 331362 T^{2} + \cdots + 19\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{3} + 145922 T^{2} + \cdots - 31\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( T^{3} - 1824192 T^{2} + \cdots + 86\!\cdots\!88 \) Copy content Toggle raw display
$53$ \( T^{3} - 3442448 T^{2} + \cdots + 55\!\cdots\!52 \) Copy content Toggle raw display
$59$ \( T^{3} - 3147660 T^{2} + \cdots - 72\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{3} + 3444530 T^{2} + \cdots - 29\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{3} + 921870 T^{2} + \cdots - 21\!\cdots\!88 \) Copy content Toggle raw display
$71$ \( T^{3} + 2594520 T^{2} + \cdots - 71\!\cdots\!68 \) Copy content Toggle raw display
$73$ \( T^{3} + 540826 T^{2} + \cdots - 39\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{3} - 316994 T^{2} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( T^{3} + 4119316 T^{2} + \cdots - 39\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{3} + 1417176 T^{2} + \cdots - 79\!\cdots\!12 \) Copy content Toggle raw display
$97$ \( T^{3} + 15089326 T^{2} + \cdots + 43\!\cdots\!24 \) Copy content Toggle raw display
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