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Properties

Label 666.4.f.d
Level $666$
Weight $4$
Character orbit 666.f
Analytic conductor $39.295$
Analytic rank $0$
Dimension $10$
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] = [666,4,Mod(343,666)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(666, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("666.343");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 666.f (of order \(3\), 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: \(39.2952720638\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 84 x^{8} - 140 x^{7} + 6309 x^{6} - 5214 x^{5} + 67648 x^{4} + 164178 x^{3} + 511389 x^{2} + \cdots + 443556 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: no (minimal twist has level 74)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{3} + 2) q^{2} + 4 \beta_{3} q^{4} + ( - \beta_{6} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{9} + \beta_{6}) q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{3} + 2) q^{2} + 4 \beta_{3} q^{4} + ( - \beta_{6} + \beta_{2} - \beta_1) q^{5} + ( - \beta_{9} + \beta_{6}) q^{7} - 8 q^{8} + (2 \beta_{4} + 2 \beta_{2}) q^{10} + (\beta_{7} + 3 \beta_{5} - 2 \beta_{2} + 8) q^{11} + ( - 3 \beta_{9} - \beta_{8} + \cdots + 4 \beta_1) q^{13}+ \cdots + (6 \beta_{9} + 24 \beta_{8} + \cdots - 26 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} - 20 q^{4} + q^{5} - q^{7} - 80 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} - 20 q^{4} + q^{5} - q^{7} - 80 q^{8} + 4 q^{10} + 80 q^{11} + 73 q^{13} - 4 q^{14} - 80 q^{16} - 69 q^{17} + 33 q^{19} + 4 q^{20} + 80 q^{22} + 524 q^{23} + 132 q^{25} + 292 q^{26} - 4 q^{28} - 296 q^{29} - 784 q^{31} + 160 q^{32} + 138 q^{34} + 263 q^{35} + 24 q^{37} + 132 q^{38} - 8 q^{40} - 345 q^{41} - 492 q^{43} - 160 q^{44} + 524 q^{46} + 32 q^{47} + 150 q^{49} - 264 q^{50} + 292 q^{52} + 19 q^{53} + 340 q^{55} + 8 q^{56} - 296 q^{58} + 105 q^{59} + 219 q^{61} - 784 q^{62} + 640 q^{64} - 1191 q^{65} + 773 q^{67} + 552 q^{68} - 526 q^{70} - 555 q^{71} + 348 q^{73} - 102 q^{74} + 132 q^{76} - 884 q^{77} - 727 q^{79} - 32 q^{80} - 1380 q^{82} - 2229 q^{83} - 3042 q^{85} - 492 q^{86} - 640 q^{88} - 901 q^{89} - 1405 q^{91} - 1048 q^{92} + 32 q^{94} + 3337 q^{95} - 4596 q^{97} - 300 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 84 x^{8} - 140 x^{7} + 6309 x^{6} - 5214 x^{5} + 67648 x^{4} + 164178 x^{3} + 511389 x^{2} + \cdots + 443556 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2923369 \nu^{9} + 1291648158 \nu^{8} - 11318780244 \nu^{7} + 97444452524 \nu^{6} + \cdots - 620934447646350 ) / 703951748722773 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 103592667275 \nu^{9} + 216329306 \nu^{8} - 8797366014792 \nu^{7} + 15340563156556 \nu^{6} + \cdots - 56\!\cdots\!98 ) / 52\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 82964449469 \nu^{9} - 780820708113 \nu^{8} + 6842372630934 \nu^{7} - 79298402920351 \nu^{6} + \cdots - 15\!\cdots\!62 ) / 21\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 276538800347 \nu^{9} + 3364748327259 \nu^{8} - 29485465261362 \nu^{7} + \cdots + 26\!\cdots\!16 ) / 21\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 14502998162881 \nu^{9} - 622808271963 \nu^{8} + \cdots + 62\!\cdots\!06 ) / 78\!\cdots\!30 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 378746420581 \nu^{9} - 4253543126982 \nu^{8} + 37274020494276 \nu^{7} + \cdots - 79\!\cdots\!98 ) / 10\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 21187262572329 \nu^{9} + 33668105489074 \nu^{8} + \cdots - 80\!\cdots\!32 ) / 26\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 94253847831539 \nu^{9} + 20662805213298 \nu^{8} + \cdots + 33\!\cdots\!34 ) / 39\!\cdots\!15 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 2\beta_{8} - 2\beta_{6} - 2\beta_{5} + 34\beta_{3} + 3\beta_{2} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{7} + 7\beta_{5} + 5\beta_{4} - 68\beta_{2} + 41 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -75\beta_{9} - 153\beta_{8} - 75\beta_{7} + 171\beta_{6} + 171\beta_{4} - 2259\beta_{3} + 334\beta _1 - 2259 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 238\beta_{9} + 728\beta_{8} + 280\beta_{6} - 728\beta_{5} + 6490\beta_{3} + 5175\beta_{2} - 5175\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5693\beta_{7} + 11848\beta_{5} - 12520\beta_{4} - 31241\beta_{2} + 167228 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 24414 \beta_{9} - 67965 \beta_{8} - 24414 \beta_{7} - 6483 \beta_{6} - 6483 \beta_{4} - 695307 \beta_{3} + \cdots - 695307 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 440179 \beta_{9} + 936563 \beta_{8} - 901013 \beta_{6} - 936563 \beta_{5} + 12841285 \beta_{3} + \cdots - 2782284 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2321450\beta_{7} + 6096502\beta_{5} - 654874\beta_{4} - 32923277\beta_{2} + 66768212 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/666\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
343.1
−4.57142 + 7.91792i
3.92323 − 6.79524i
−0.632114 + 1.09485i
−0.858393 + 1.48678i
2.13869 − 3.70432i
−4.57142 − 7.91792i
3.92323 + 6.79524i
−0.632114 − 1.09485i
−0.858393 − 1.48678i
2.13869 + 3.70432i
1.00000 − 1.73205i 0 −2.00000 − 3.46410i −3.82762 − 6.62963i 0 −2.78651 − 4.82638i −8.00000 0 −15.3105
343.2 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −2.94299 − 5.09740i 0 8.40192 + 14.5526i −8.00000 0 −11.7719
343.3 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −2.10613 − 3.64793i 0 −10.8960 − 18.8724i −8.00000 0 −8.42453
343.4 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −0.388081 − 0.672176i 0 11.9492 + 20.6965i −8.00000 0 −1.55232
343.5 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 9.76482 + 16.9132i 0 −7.16860 − 12.4164i −8.00000 0 39.0593
433.1 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −3.82762 + 6.62963i 0 −2.78651 + 4.82638i −8.00000 0 −15.3105
433.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −2.94299 + 5.09740i 0 8.40192 − 14.5526i −8.00000 0 −11.7719
433.3 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −2.10613 + 3.64793i 0 −10.8960 + 18.8724i −8.00000 0 −8.42453
433.4 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −0.388081 + 0.672176i 0 11.9492 − 20.6965i −8.00000 0 −1.55232
433.5 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 9.76482 − 16.9132i 0 −7.16860 + 12.4164i −8.00000 0 39.0593
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 343.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.4.f.d 10
3.b odd 2 1 74.4.c.b ✓ 10
37.c even 3 1 inner 666.4.f.d 10
111.i odd 6 1 74.4.c.b ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.4.c.b ✓ 10 3.b odd 2 1
74.4.c.b ✓ 10 111.i odd 6 1
666.4.f.d 10 1.a even 1 1 trivial
666.4.f.d 10 37.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} - T_{5}^{9} + 247 T_{5}^{8} + 4234 T_{5}^{7} + 63629 T_{5}^{6} + 477433 T_{5}^{5} + \cdots + 8277129 \) acting on \(S_{4}^{\mathrm{new}}(666, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - T^{9} + \cdots + 8277129 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 488936577600 \) Copy content Toggle raw display
$11$ \( (T^{5} - 40 T^{4} + \cdots - 147890304)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 43\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 472303425246849 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{5} - 262 T^{4} + \cdots + 8739280512)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + 148 T^{4} + \cdots + 103287670938)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 392 T^{4} + \cdots - 102244293760)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 33\!\cdots\!93 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 21\!\cdots\!21 \) Copy content Toggle raw display
$43$ \( (T^{5} + 246 T^{4} + \cdots + 838671946912)^{2} \) Copy content Toggle raw display
$47$ \( (T^{5} - 16 T^{4} + \cdots + 25825580928)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 29\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 11\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} - 174 T^{4} + \cdots - 551579403872)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 61\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 20\!\cdots\!49 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 15\!\cdots\!26)^{2} \) Copy content Toggle raw display
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