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Properties

Label 63.7.m.d
Level $63$
Weight $7$
Character orbit 63.m
Analytic conductor $14.493$
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] = [63,7,Mod(10,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.10");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 63.m (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: \(14.4934072681\)
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: \( x^{8} - x^{7} + 212x^{6} + 473x^{5} + 39800x^{4} + 36821x^{3} + 985651x^{2} - 601290x + 21068100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 7 \)
Twist minimal: no (minimal twist has level 21)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 + 1) q^{2} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 1) q^{4}+ \cdots + ( - 8 \beta_{7} - 12 \beta_{6} + \cdots - 392) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1 + 1) q^{2} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 1) q^{4}+ \cdots + ( - 14371 \beta_{7} - 4121 \beta_{6} + \cdots - 150010) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 5 q^{2} - 173 q^{4} + 294 q^{5} - 656 q^{7} - 3326 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 5 q^{2} - 173 q^{4} + 294 q^{5} - 656 q^{7} - 3326 q^{8} - 3411 q^{10} + 314 q^{11} + 5360 q^{14} - 12721 q^{16} + 5532 q^{17} - 18234 q^{19} + 86106 q^{22} - 3928 q^{23} - 17038 q^{25} - 12366 q^{26} + 85037 q^{28} + 8300 q^{29} - 89508 q^{31} + 186207 q^{32} + 25860 q^{35} + 64706 q^{37} + 77136 q^{38} + 221823 q^{40} + 45740 q^{43} - 92529 q^{44} - 111504 q^{46} - 483276 q^{47} - 310684 q^{49} - 967216 q^{50} - 1673988 q^{52} + 540974 q^{53} + 241885 q^{56} + 539799 q^{58} + 181770 q^{59} + 418224 q^{61} + 2378626 q^{64} + 414204 q^{65} - 1158902 q^{67} + 821250 q^{68} + 1087917 q^{70} - 1442344 q^{71} - 378666 q^{73} + 432940 q^{74} - 1065994 q^{77} + 611452 q^{79} + 2094945 q^{80} - 1561266 q^{82} - 275112 q^{85} - 816224 q^{86} - 366441 q^{88} + 989196 q^{89} + 304446 q^{91} - 678720 q^{92} - 716148 q^{94} + 591792 q^{95} - 3509629 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 212x^{6} + 473x^{5} + 39800x^{4} + 36821x^{3} + 985651x^{2} - 601290x + 21068100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 405518177 \nu^{7} - 12595814413 \nu^{6} - 65484675874 \nu^{5} - 2673339327091 \nu^{4} + \cdots - 10\!\cdots\!70 ) / 11\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8497603 \nu^{7} - 13389005 \nu^{6} + 1621914529 \nu^{5} + 3085757533 \nu^{4} + \cdots - 5583985297290 ) / 7542500798721 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 30359760961 \nu^{7} - 17272473736 \nu^{6} + 12542253415247 \nu^{5} - 45647117796292 \nu^{4} + \cdots - 88\!\cdots\!20 ) / 23\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 36613259519 \nu^{7} + 484838322556 \nu^{6} + 240706081393 \nu^{5} + 69967211645572 \nu^{4} + \cdots - 10\!\cdots\!80 ) / 23\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20960695316 \nu^{7} - 201659710511 \nu^{6} + 3951590320162 \nu^{5} - 21226630078907 \nu^{4} + \cdots - 12\!\cdots\!60 ) / 32\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 14593340681 \nu^{7} + 40700920519 \nu^{6} + 3013487903677 \nu^{5} + 21592891249843 \nu^{4} + \cdots + 66\!\cdots\!80 ) / 13\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{3} - 106\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -8\beta_{7} - 12\beta_{6} - 10\beta_{5} - 6\beta_{4} + 165\beta_{3} - 4\beta_{2} - 161\beta _1 - 204 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 213\beta_{7} - 217\beta_{6} - 8\beta_{5} - 205\beta_{4} + 4\beta_{3} + 17990\beta_{2} - 718\beta _1 - 17986 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2462\beta_{7} + 930\beta_{6} + 766\beta_{5} + 2544\beta_{4} - 30357\beta_{3} + 77878\beta_{2} - 848\beta _1 - 1614 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4432 \beta_{7} + 6648 \beta_{6} + 45813 \beta_{5} + 43597 \beta_{4} - 195138 \beta_{3} + 2216 \beta_{2} + \cdots + 3368601 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 221730 \beta_{7} + 385038 \beta_{6} + 326616 \beta_{5} - 104886 \beta_{4} - 163308 \beta_{3} + \cdots + 20878866 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(1 - \beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
−6.30797 − 10.9257i
−2.75320 − 4.76869i
2.26350 + 3.92050i
7.29767 + 12.6399i
−6.30797 + 10.9257i
−2.75320 + 4.76869i
2.26350 − 3.92050i
7.29767 − 12.6399i
−5.80797 − 10.0597i 0 −35.4650 + 61.4271i 165.302 − 95.4373i 0 −103.254 + 327.089i 80.4975 0 −1920.14 − 1108.59i
10.2 −2.25320 − 3.90266i 0 21.8461 − 37.8386i −53.9244 + 31.1333i 0 218.833 − 264.124i −485.305 0 243.005 + 140.299i
10.3 2.76350 + 4.78652i 0 16.7261 − 28.9705i 57.9943 − 33.4830i 0 −240.457 + 244.600i 538.619 0 320.535 + 185.061i
10.4 7.79767 + 13.5060i 0 −89.6073 + 155.204i −22.3721 + 12.9165i 0 −203.121 − 276.389i −1796.81 0 −348.901 − 201.438i
19.1 −5.80797 + 10.0597i 0 −35.4650 − 61.4271i 165.302 + 95.4373i 0 −103.254 − 327.089i 80.4975 0 −1920.14 + 1108.59i
19.2 −2.25320 + 3.90266i 0 21.8461 + 37.8386i −53.9244 − 31.1333i 0 218.833 + 264.124i −485.305 0 243.005 − 140.299i
19.3 2.76350 − 4.78652i 0 16.7261 + 28.9705i 57.9943 + 33.4830i 0 −240.457 − 244.600i 538.619 0 320.535 − 185.061i
19.4 7.79767 − 13.5060i 0 −89.6073 − 155.204i −22.3721 − 12.9165i 0 −203.121 + 276.389i −1796.81 0 −348.901 + 201.438i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.4
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 63.7.m.d 8
3.b odd 2 1 21.7.f.a ✓ 8
7.c even 3 1 441.7.d.c 8
7.d odd 6 1 inner 63.7.m.d 8
7.d odd 6 1 441.7.d.c 8
12.b even 2 1 336.7.bh.d 8
21.c even 2 1 147.7.f.d 8
21.g even 6 1 21.7.f.a ✓ 8
21.g even 6 1 147.7.d.b 8
21.h odd 6 1 147.7.d.b 8
21.h odd 6 1 147.7.f.d 8
84.j odd 6 1 336.7.bh.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.7.f.a ✓ 8 3.b odd 2 1
21.7.f.a ✓ 8 21.g even 6 1
63.7.m.d 8 1.a even 1 1 trivial
63.7.m.d 8 7.d odd 6 1 inner
147.7.d.b 8 21.g even 6 1
147.7.d.b 8 21.h odd 6 1
147.7.f.d 8 21.c even 2 1
147.7.f.d 8 21.h odd 6 1
336.7.bh.d 8 12.b even 2 1
336.7.bh.d 8 84.j odd 6 1
441.7.d.c 8 7.c even 3 1
441.7.d.c 8 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 5T_{2}^{7} + 227T_{2}^{6} + 442T_{2}^{5} + 37712T_{2}^{4} - 12248T_{2}^{3} + 992080T_{2}^{2} + 1281408T_{2} + 20358144 \) acting on \(S_{7}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 5 T^{7} + \cdots + 20358144 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 422734160250000 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 40\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 13\!\cdots\!80)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 23\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 36\!\cdots\!44 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots - 29\!\cdots\!96)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 99\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 73\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 29\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 19\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 77\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 99\!\cdots\!01 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 52\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 87\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 90\!\cdots\!76 \) Copy content Toggle raw display
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