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Properties

Label 354.6.a.h
Level $354$
Weight $6$
Character orbit 354.a
Self dual yes
Analytic conductor $56.776$
Analytic rank $0$
Dimension $8$
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] = [354,6,Mod(1,354)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(354, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("354.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 354 = 2 \cdot 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 354.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: \(56.7758722138\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 17196 x^{6} - 154000 x^{5} + 98085975 x^{4} + 1816612536 x^{3} - 184506058580 x^{2} + \cdots - 7060184373200 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 4 q^{2} + 9 q^{3} + 16 q^{4} + (\beta_1 + 5) q^{5} - 36 q^{6} + ( - \beta_{3} + \beta_1 + 23) q^{7} - 64 q^{8} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 4 q^{2} + 9 q^{3} + 16 q^{4} + (\beta_1 + 5) q^{5} - 36 q^{6} + ( - \beta_{3} + \beta_1 + 23) q^{7} - 64 q^{8} + 81 q^{9} + ( - 4 \beta_1 - 20) q^{10} + ( - \beta_{7} - \beta_{4} + \beta_{2} + \cdots - 44) q^{11}+ \cdots + ( - 81 \beta_{7} - 81 \beta_{4} + \cdots - 3564) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{2} + 72 q^{3} + 128 q^{4} + 40 q^{5} - 288 q^{6} + 181 q^{7} - 512 q^{8} + 648 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{2} + 72 q^{3} + 128 q^{4} + 40 q^{5} - 288 q^{6} + 181 q^{7} - 512 q^{8} + 648 q^{9} - 160 q^{10} - 349 q^{11} + 1152 q^{12} + 121 q^{13} - 724 q^{14} + 360 q^{15} + 2048 q^{16} + 437 q^{17} - 2592 q^{18} + 1314 q^{19} + 640 q^{20} + 1629 q^{21} + 1396 q^{22} + 1224 q^{23} - 4608 q^{24} + 9592 q^{25} - 484 q^{26} + 5832 q^{27} + 2896 q^{28} + 5276 q^{29} - 1440 q^{30} + 18332 q^{31} - 8192 q^{32} - 3141 q^{33} - 1748 q^{34} + 19518 q^{35} + 10368 q^{36} + 30331 q^{37} - 5256 q^{38} + 1089 q^{39} - 2560 q^{40} + 8323 q^{41} - 6516 q^{42} + 30851 q^{43} - 5584 q^{44} + 3240 q^{45} - 4896 q^{46} - 5730 q^{47} + 18432 q^{48} + 32295 q^{49} - 38368 q^{50} + 3933 q^{51} + 1936 q^{52} - 33524 q^{53} - 23328 q^{54} + 23660 q^{55} - 11584 q^{56} + 11826 q^{57} - 21104 q^{58} - 27848 q^{59} + 5760 q^{60} + 2692 q^{61} - 73328 q^{62} + 14661 q^{63} + 32768 q^{64} - 59892 q^{65} + 12564 q^{66} + 56244 q^{67} + 6992 q^{68} + 11016 q^{69} - 78072 q^{70} - 48473 q^{71} - 41472 q^{72} - 30796 q^{73} - 121324 q^{74} + 86328 q^{75} + 21024 q^{76} + 59683 q^{77} - 4356 q^{78} + 135513 q^{79} + 10240 q^{80} + 52488 q^{81} - 33292 q^{82} - 88111 q^{83} + 26064 q^{84} + 114418 q^{85} - 123404 q^{86} + 47484 q^{87} + 22336 q^{88} - 112196 q^{89} - 12960 q^{90} + 377433 q^{91} + 19584 q^{92} + 164988 q^{93} + 22920 q^{94} + 328146 q^{95} - 73728 q^{96} + 551378 q^{97} - 129180 q^{98} - 28269 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 17196 x^{6} - 154000 x^{5} + 98085975 x^{4} + 1816612536 x^{3} - 184506058580 x^{2} + \cdots - 7060184373200 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3531716616097 \nu^{7} - 639912128181572 \nu^{6} + \cdots + 37\!\cdots\!00 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4331671669311 \nu^{7} + 339562400775494 \nu^{6} + \cdots + 24\!\cdots\!00 ) / 83\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 18941466661253 \nu^{7} + \cdots - 12\!\cdots\!00 ) / 25\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7432150279107 \nu^{7} + 249623166066313 \nu^{6} + \cdots + 96\!\cdots\!00 ) / 83\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4967625572903 \nu^{7} - 294881418090673 \nu^{6} + \cdots - 28\!\cdots\!00 ) / 50\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 108056909469313 \nu^{7} + \cdots - 78\!\cdots\!00 ) / 25\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 2\beta_{6} - 3\beta_{5} - \beta_{4} - 2\beta_{3} - 13\beta_{2} + 14\beta _1 + 4306 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -7\beta_{7} + 45\beta_{6} - 118\beta_{5} + 196\beta_{4} - 198\beta_{3} + 320\beta_{2} + 5937\beta _1 + 57594 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7588 \beta_{7} + 13622 \beta_{6} - 32590 \beta_{5} - 10207 \beta_{4} + 5106 \beta_{3} - 86263 \beta_{2} + \cdots + 24921557 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 98944 \beta_{7} + 664035 \beta_{6} - 1607631 \beta_{5} + 1790372 \beta_{4} - 1061246 \beta_{3} + \cdots + 517981613 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 54289545 \beta_{7} + 92992622 \beta_{6} - 295325547 \beta_{5} - 49185113 \beta_{4} + 91231074 \beta_{3} + \cdots + 153680491878 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 670933991 \beta_{7} + 6788260045 \beta_{6} - 16409305734 \beta_{5} + 13578877668 \beta_{4} + \cdots + 3952224957842 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−79.5001
−74.0889
−46.8085
−36.8040
−1.38959
75.7365
77.0538
85.8007
−4.00000 9.00000 16.0000 −74.5001 −36.0000 −124.731 −64.0000 81.0000 298.000
1.2 −4.00000 9.00000 16.0000 −69.0889 −36.0000 −83.5010 −64.0000 81.0000 276.356
1.3 −4.00000 9.00000 16.0000 −41.8085 −36.0000 175.859 −64.0000 81.0000 167.234
1.4 −4.00000 9.00000 16.0000 −31.8040 −36.0000 191.018 −64.0000 81.0000 127.216
1.5 −4.00000 9.00000 16.0000 3.61041 −36.0000 −202.768 −64.0000 81.0000 −14.4417
1.6 −4.00000 9.00000 16.0000 80.7365 −36.0000 18.8441 −64.0000 81.0000 −322.946
1.7 −4.00000 9.00000 16.0000 82.0538 −36.0000 186.988 −64.0000 81.0000 −328.215
1.8 −4.00000 9.00000 16.0000 90.8007 −36.0000 19.2910 −64.0000 81.0000 −363.203
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(59\) \(1\)

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 354.6.a.h ✓ 8
3.b odd 2 1 1062.6.a.m 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
354.6.a.h ✓ 8 1.a even 1 1 trivial
1062.6.a.m 8 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 40 T_{5}^{7} - 16496 T_{5}^{6} + 354880 T_{5}^{5} + 95531225 T_{5}^{4} + \cdots + 14863676640480 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(354))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 4)^{8} \) Copy content Toggle raw display
$3$ \( (T - 9)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 14863676640480 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots - 48\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots - 49\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 27\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots - 43\!\cdots\!40 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots - 99\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 79\!\cdots\!20 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots - 83\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots - 13\!\cdots\!12 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots - 51\!\cdots\!08 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 84\!\cdots\!80 \) Copy content Toggle raw display
$59$ \( (T + 3481)^{8} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 36\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 24\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 64\!\cdots\!40 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 22\!\cdots\!04 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots - 74\!\cdots\!92 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots - 82\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 20\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 24\!\cdots\!20 \) Copy content Toggle raw display
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