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Properties

Label 123.6.a.b
Level $123$
Weight $6$
Character orbit 123.a
Self dual yes
Analytic conductor $19.727$
Analytic rank $1$
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] = [123,6,Mod(1,123)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(123, 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("123.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 123 = 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 123.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: \(19.7272098370\)
Analytic rank: \(1\)
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: \( x^{8} - 4x^{7} - 187x^{6} + 490x^{5} + 10739x^{4} - 10232x^{3} - 208249x^{2} - 155342x + 197472 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} - 9 q^{3} + (\beta_{2} + 17) q^{4} + ( - \beta_{5} + \beta_{4} - 9) q^{5} + ( - 9 \beta_1 + 9) q^{6} + (2 \beta_{7} - \beta_{4} - 2 \beta_{2} + \cdots - 26) q^{7}+ \cdots + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{2} - 9 q^{3} + (\beta_{2} + 17) q^{4} + ( - \beta_{5} + \beta_{4} - 9) q^{5} + ( - 9 \beta_1 + 9) q^{6} + (2 \beta_{7} - \beta_{4} - 2 \beta_{2} + \cdots - 26) q^{7}+ \cdots + ( - 243 \beta_{6} - 243 \beta_{5} + \cdots + 8262) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 72 q^{3} + 134 q^{4} - 72 q^{5} + 36 q^{6} - 196 q^{7} - 72 q^{8} + 648 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} - 72 q^{3} + 134 q^{4} - 72 q^{5} + 36 q^{6} - 196 q^{7} - 72 q^{8} + 648 q^{9} + 25 q^{10} + 712 q^{11} - 1206 q^{12} - 1084 q^{13} + 208 q^{14} + 648 q^{15} + 1098 q^{16} - 994 q^{17} - 324 q^{18} - 3482 q^{19} - 2349 q^{20} + 1764 q^{21} - 10502 q^{22} + 1588 q^{23} + 648 q^{24} - 2802 q^{25} + 2863 q^{26} - 5832 q^{27} - 28588 q^{28} + 1436 q^{29} - 225 q^{30} - 3992 q^{31} - 23368 q^{32} - 6408 q^{33} - 60841 q^{34} - 18640 q^{35} + 10854 q^{36} - 34356 q^{37} - 57967 q^{38} + 9756 q^{39} - 66005 q^{40} - 13448 q^{41} - 1872 q^{42} - 312 q^{43} + 5634 q^{44} - 5832 q^{45} - 4626 q^{46} - 6420 q^{47} - 9882 q^{48} + 8596 q^{49} + 2655 q^{50} + 8946 q^{51} - 51511 q^{52} + 8696 q^{53} + 2916 q^{54} - 29592 q^{55} + 55040 q^{56} + 31338 q^{57} - 39336 q^{58} - 16214 q^{59} + 21141 q^{60} - 50844 q^{61} + 101909 q^{62} - 15876 q^{63} + 96706 q^{64} + 8258 q^{65} + 94518 q^{66} - 56610 q^{67} + 1773 q^{68} - 14292 q^{69} + 97482 q^{70} + 62292 q^{71} - 5832 q^{72} - 58046 q^{73} + 174812 q^{74} + 25218 q^{75} - 10565 q^{76} + 16736 q^{77} - 25767 q^{78} - 36676 q^{79} + 163703 q^{80} + 52488 q^{81} + 6724 q^{82} + 65986 q^{83} + 257292 q^{84} - 145364 q^{85} - 15540 q^{86} - 12924 q^{87} - 149754 q^{88} - 218506 q^{89} + 2025 q^{90} - 113248 q^{91} + 373754 q^{92} + 35928 q^{93} + 193202 q^{94} + 517444 q^{95} + 210312 q^{96} - 375260 q^{97} + 333196 q^{98} + 57672 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 187x^{6} + 490x^{5} + 10739x^{4} - 10232x^{3} - 208249x^{2} - 155342x + 197472 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1693 \nu^{7} + 6043 \nu^{6} - 274230 \nu^{5} - 1210688 \nu^{4} + 9269543 \nu^{3} + 49661669 \nu^{2} + \cdots - 12776896 ) / 817696 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2521 \nu^{7} + 7791 \nu^{6} - 395670 \nu^{5} - 1683384 \nu^{4} + 12229091 \nu^{3} + 73857945 \nu^{2} + \cdots - 47873056 ) / 817696 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7251 \nu^{7} - 21233 \nu^{6} + 1176802 \nu^{5} + 4531328 \nu^{4} - 40949321 \nu^{3} + \cdots + 142766624 ) / 1635392 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 16531 \nu^{7} - 74401 \nu^{6} + 2686002 \nu^{5} + 14417360 \nu^{4} - 89471673 \nu^{3} + \cdots + 485725856 ) / 1635392 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22797 \nu^{7} - 74791 \nu^{6} + 3674638 \nu^{5} + 15620960 \nu^{4} - 124366455 \nu^{3} + \cdots + 588842144 ) / 1635392 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 48 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} + \beta_{6} + 6\beta_{5} - 3\beta_{4} + 2\beta_{3} - \beta_{2} + 77\beta _1 + 67 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{7} + 5\beta_{6} + 20\beta_{5} - 7\beta_{4} + 44\beta_{3} + 104\beta_{2} + 246\beta _1 + 3839 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -448\beta_{7} + 162\beta_{6} + 966\beta_{5} - 394\beta_{4} + 430\beta_{3} - 72\beta_{2} + 7051\beta _1 + 9058 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 1288 \beta_{7} + 892 \beta_{6} + 4300 \beta_{5} - 1596 \beta_{4} + 7268 \beta_{3} + 10757 \beta_{2} + \cdots + 359908 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 55119 \beta_{7} + 21157 \beta_{6} + 122574 \beta_{5} - 46703 \beta_{4} + 64706 \beta_{3} + \cdots + 1160567 \) 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
−9.77256
−6.07184
−4.96299
−1.65981
0.667803
7.44057
7.46010
10.8987
−10.7726 −9.00000 84.0482 34.8822 96.9531 −162.539 −560.692 81.0000 −375.771
1.2 −7.07184 −9.00000 18.0109 −15.4083 63.6465 −95.8793 98.9289 81.0000 108.965
1.3 −5.96299 −9.00000 3.55730 −100.801 53.6669 87.8819 169.604 81.0000 601.073
1.4 −2.65981 −9.00000 −24.9254 −15.4120 23.9383 75.9627 151.411 81.0000 40.9931
1.5 −0.332197 −9.00000 −31.8896 44.1817 2.98977 44.8447 21.2239 81.0000 −14.6770
1.6 6.44057 −9.00000 9.48090 64.5494 −57.9651 −178.332 −145.036 81.0000 415.735
1.7 6.46010 −9.00000 9.73284 −23.2942 −58.1409 188.685 −143.848 81.0000 −150.483
1.8 9.89874 −9.00000 65.9850 −60.6982 −89.0886 −156.625 336.409 81.0000 −600.835
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(41\) \(1\)

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 123.6.a.b ✓ 8
3.b odd 2 1 369.6.a.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
123.6.a.b ✓ 8 1.a even 1 1 trivial
369.6.a.c 8 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 4T_{2}^{7} - 187T_{2}^{6} - 660T_{2}^{5} + 10314T_{2}^{4} + 33800T_{2}^{3} - 172472T_{2}^{2} - 558272T_{2} - 165312 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(123))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{7} + \cdots - 165312 \) Copy content Toggle raw display
$3$ \( (T + 9)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots - 3366978515488 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 24\!\cdots\!92 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 13\!\cdots\!28 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 40\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 16\!\cdots\!60 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 16\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 48\!\cdots\!92 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 42\!\cdots\!48 \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 58\!\cdots\!72 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots - 91\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( (T + 1681)^{8} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots - 91\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots - 15\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots - 17\!\cdots\!12 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots - 29\!\cdots\!48 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 16\!\cdots\!88 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots - 31\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 94\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 26\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots - 29\!\cdots\!52 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots - 11\!\cdots\!68 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots - 79\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots - 66\!\cdots\!44 \) Copy content Toggle raw display
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