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Properties

Label 42.7.g.a
Level $42$
Weight $7$
Character orbit 42.g
Analytic conductor $9.662$
Analytic rank $0$
Dimension $8$
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] = [42,7,Mod(19,42)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(42, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("42.19");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 42.g (of order \(6\), 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: \(9.66227151203\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 193x^{6} + 306x^{5} + 29845x^{4} + 16988x^{3} + 1125468x^{2} + 214128x + 35378704 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + (9 \beta_1 - 18) q^{3} - 32 \beta_1 q^{4} + (\beta_{7} - \beta_{6} + \beta_{3} + \cdots + 38) q^{5}+ \cdots + ( - 243 \beta_1 + 243) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + (9 \beta_1 - 18) q^{3} - 32 \beta_1 q^{4} + (\beta_{7} - \beta_{6} + \beta_{3} + \cdots + 38) q^{5}+ \cdots + (3402 \beta_{7} + 2916 \beta_{6} + \cdots + 107163) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 108 q^{3} - 128 q^{4} + 462 q^{5} + 580 q^{7} + 972 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 108 q^{3} - 128 q^{4} + 462 q^{5} + 580 q^{7} + 972 q^{9} + 576 q^{10} + 1806 q^{11} + 3456 q^{12} + 384 q^{14} - 8316 q^{15} - 4096 q^{16} + 9564 q^{17} + 23022 q^{19} - 1350 q^{21} - 17664 q^{22} + 2400 q^{23} + 32762 q^{25} - 31872 q^{26} - 32320 q^{28} - 2484 q^{29} - 5184 q^{30} + 148416 q^{31} - 48762 q^{33} + 11412 q^{35} - 62208 q^{36} - 84046 q^{37} - 76800 q^{38} - 75222 q^{39} - 18432 q^{40} - 50112 q^{42} + 92972 q^{43} + 57792 q^{44} + 112266 q^{45} + 210432 q^{46} - 323124 q^{47} - 8644 q^{49} - 774912 q^{50} - 86076 q^{51} + 267456 q^{52} + 358086 q^{53} + 153600 q^{56} - 414396 q^{57} - 7104 q^{58} + 719382 q^{59} + 133056 q^{60} + 421536 q^{61} - 104490 q^{63} + 262144 q^{64} - 322740 q^{65} + 238464 q^{66} + 267010 q^{67} - 306048 q^{68} + 854016 q^{70} - 464664 q^{71} + 1944486 q^{73} + 1077504 q^{74} - 884574 q^{75} - 1713498 q^{77} + 573696 q^{78} - 685904 q^{79} - 473088 q^{80} - 236196 q^{81} - 587520 q^{82} + 457920 q^{84} - 3876168 q^{85} + 575616 q^{86} + 33534 q^{87} + 282624 q^{88} + 4130604 q^{89} - 484266 q^{91} - 153600 q^{92} - 1335744 q^{93} - 1724544 q^{94} - 2105232 q^{95} - 2102400 q^{98} + 877716 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 193x^{6} + 306x^{5} + 29845x^{4} + 16988x^{3} + 1125468x^{2} + 214128x + 35378704 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1636073 \nu^{7} + 177960445 \nu^{6} - 557057579 \nu^{5} + 27650057549 \nu^{4} + \cdots + 197665427324196 ) / 164920217026180 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1898862927 \nu^{7} + 209877374095 \nu^{6} - 656964989609 \nu^{5} + 52466273535399 \nu^{4} + \cdots + 23\!\cdots\!16 ) / 28\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10482751509 \nu^{7} - 656074423920 \nu^{6} + 3615202460937 \nu^{5} - 99138904513972 \nu^{4} + \cdots - 11\!\cdots\!28 ) / 57\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 375805237 \nu^{7} + 3599844475 \nu^{6} - 160074398171 \nu^{5} + 525986544591 \nu^{4} + \cdots + 834698877937384 ) / 64517830110580 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 49034533 \nu^{7} - 149328930 \nu^{6} + 11302608029 \nu^{5} + 4418887706 \nu^{4} + \cdots + 19165651090004 ) / 5865257282780 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 780769416286 \nu^{7} + 1428965569135 \nu^{6} - 110191784832028 \nu^{5} + \cdots - 21\!\cdots\!28 ) / 57\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 562755681534 \nu^{7} + 830242904705 \nu^{6} - 68594335566832 \nu^{5} + \cdots - 48\!\cdots\!02 ) / 28\!\cdots\!30 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} + 5\beta_{6} + 5\beta_{5} + 2\beta_{4} - 13\beta_{2} + 41\beta_1 ) / 84 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{7} - 3\beta_{6} + 6\beta_{5} + 8\beta_{4} + 405\beta_{3} - 3\beta_{2} + 4013\beta _1 - 4017 ) / 42 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 124\beta_{7} - 150\beta_{6} + 75\beta_{5} + 62\beta_{4} + 489\beta_{3} + 564\beta_{2} + 62\beta _1 - 3195 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 577\beta_{7} - 501\beta_{6} - 501\beta_{5} - 577\beta_{4} + 40959\beta_{2} - 264322\beta_1 ) / 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 74458 \beta_{7} + 73277 \beta_{6} - 146554 \beta_{5} - 148916 \beta_{4} - 919599 \beta_{3} + \cdots + 5793909 ) / 84 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 74560 \beta_{7} + 67246 \beta_{6} - 33623 \beta_{5} - 37280 \beta_{4} - 1936113 \beta_{3} + \cdots + 11692593 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 12451882 \beta_{7} + 11595953 \beta_{6} + 11595953 \beta_{5} + 12451882 \beta_{4} + \cdots + 1226282077 \beta_1 ) / 84 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(31\)
\(\chi(n)\) \(1\) \(1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
6.71087 + 11.6236i
−5.50376 − 9.53280i
3.07067 + 5.31856i
−3.27778 − 5.67728i
6.71087 − 11.6236i
−5.50376 + 9.53280i
3.07067 − 5.31856i
−3.27778 + 5.67728i
−2.82843 + 4.89898i −13.5000 + 7.79423i −16.0000 − 27.7128i −124.321 − 71.7768i − 88.1816i 342.924 − 7.21089i 181.019 121.500 − 210.444i 703.266 − 406.031i
19.2 −2.82843 + 4.89898i −13.5000 + 7.79423i −16.0000 − 27.7128i 214.365 + 123.764i − 88.1816i 14.2078 − 342.706i 181.019 121.500 − 210.444i −1212.63 + 700.114i
19.3 2.82843 − 4.89898i −13.5000 + 7.79423i −16.0000 − 27.7128i 63.2658 + 36.5265i 88.1816i −271.352 + 209.802i −181.019 121.500 − 210.444i 357.886 − 206.625i
19.4 2.82843 − 4.89898i −13.5000 + 7.79423i −16.0000 − 27.7128i 77.6900 + 44.8543i 88.1816i 204.220 − 275.578i −181.019 121.500 − 210.444i 439.481 − 253.734i
31.1 −2.82843 − 4.89898i −13.5000 − 7.79423i −16.0000 + 27.7128i −124.321 + 71.7768i 88.1816i 342.924 + 7.21089i 181.019 121.500 + 210.444i 703.266 + 406.031i
31.2 −2.82843 − 4.89898i −13.5000 − 7.79423i −16.0000 + 27.7128i 214.365 − 123.764i 88.1816i 14.2078 + 342.706i 181.019 121.500 + 210.444i −1212.63 − 700.114i
31.3 2.82843 + 4.89898i −13.5000 − 7.79423i −16.0000 + 27.7128i 63.2658 − 36.5265i − 88.1816i −271.352 − 209.802i −181.019 121.500 + 210.444i 357.886 + 206.625i
31.4 2.82843 + 4.89898i −13.5000 − 7.79423i −16.0000 + 27.7128i 77.6900 − 44.8543i − 88.1816i 204.220 + 275.578i −181.019 121.500 + 210.444i 439.481 + 253.734i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.7.g.a ✓ 8
3.b odd 2 1 126.7.n.a 8
4.b odd 2 1 336.7.bh.f 8
7.b odd 2 1 294.7.g.d 8
7.c even 3 1 294.7.c.b 8
7.c even 3 1 294.7.g.d 8
7.d odd 6 1 inner 42.7.g.a ✓ 8
7.d odd 6 1 294.7.c.b 8
21.g even 6 1 126.7.n.a 8
28.f even 6 1 336.7.bh.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.7.g.a ✓ 8 1.a even 1 1 trivial
42.7.g.a ✓ 8 7.d odd 6 1 inner
126.7.n.a 8 3.b odd 2 1
126.7.n.a 8 21.g even 6 1
294.7.c.b 8 7.c even 3 1
294.7.c.b 8 7.d odd 6 1
294.7.g.d 8 7.b odd 2 1
294.7.g.d 8 7.c even 3 1
336.7.bh.f 8 4.b odd 2 1
336.7.bh.f 8 28.f even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 462 T_{5}^{7} + 59091 T_{5}^{6} + 5570334 T_{5}^{5} - 982673451 T_{5}^{4} + \cdots + 54\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(42, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 27 T + 243)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 32\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 24\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 32\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 23\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 23\!\cdots\!44)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 37\!\cdots\!49 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 14\!\cdots\!12)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 42\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 68\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 16\!\cdots\!64)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 59\!\cdots\!49 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 12\!\cdots\!24 \) Copy content Toggle raw display
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