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Properties

Label 630.4.k.t
Level $630$
Weight $4$
Character orbit 630.k
Analytic conductor $37.171$
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] = [630,4,Mod(361,630)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(630, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("630.361");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 630 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 630.k (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: \(37.1712033036\)
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: \( x^{10} + 438 x^{8} - 116 x^{7} + 177736 x^{6} - 51040 x^{5} + 6182668 x^{4} - 21638872 x^{3} + \cdots + 657204496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3\cdot 7 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{3} + 2) q^{2} - 4 \beta_{3} q^{4} + ( - 5 \beta_{3} + 5) q^{5} + (\beta_{6} - \beta_{3} + \beta_1 - 1) q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{3} + 2) q^{2} - 4 \beta_{3} q^{4} + ( - 5 \beta_{3} + 5) q^{5} + (\beta_{6} - \beta_{3} + \beta_1 - 1) q^{7} - 8 q^{8} - 10 \beta_{3} q^{10} + (\beta_{8} + \beta_{6} + \cdots + 5 \beta_{3}) q^{11}+ \cdots + (10 \beta_{9} - 10 \beta_{8} + \cdots + 142) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} - 20 q^{4} + 25 q^{5} - 13 q^{7} - 80 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} - 20 q^{4} + 25 q^{5} - 13 q^{7} - 80 q^{8} - 50 q^{10} + 26 q^{11} + 86 q^{13} - 22 q^{14} - 80 q^{16} + 86 q^{17} - 7 q^{19} - 200 q^{20} + 104 q^{22} + 68 q^{23} - 125 q^{25} + 86 q^{26} + 8 q^{28} - 288 q^{29} - 45 q^{31} + 160 q^{32} + 344 q^{34} - 55 q^{35} - 259 q^{37} + 14 q^{38} - 200 q^{40} - 388 q^{41} + 94 q^{43} + 104 q^{44} - 136 q^{46} + 516 q^{47} + 145 q^{49} - 500 q^{50} - 172 q^{52} + 326 q^{53} + 260 q^{55} + 104 q^{56} - 288 q^{58} - 36 q^{59} - 142 q^{61} - 180 q^{62} + 640 q^{64} + 215 q^{65} - 835 q^{67} + 344 q^{68} + 20 q^{70} - 3072 q^{71} - 477 q^{73} + 518 q^{74} + 56 q^{76} + 968 q^{77} - 327 q^{79} + 400 q^{80} - 388 q^{82} + 20 q^{83} + 860 q^{85} + 94 q^{86} - 208 q^{88} + 1178 q^{89} + 399 q^{91} - 544 q^{92} - 1032 q^{94} + 35 q^{95} - 1488 q^{97} + 1864 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 438 x^{8} - 116 x^{7} + 177736 x^{6} - 51040 x^{5} + 6182668 x^{4} - 21638872 x^{3} + \cdots + 657204496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 20\!\cdots\!24 \nu^{9} + \cdots + 35\!\cdots\!04 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 22\!\cdots\!81 \nu^{9} + \cdots + 10\!\cdots\!44 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 24\!\cdots\!83 \nu^{9} + \cdots + 88\!\cdots\!48 ) / 88\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 62\!\cdots\!87 \nu^{9} + \cdots + 22\!\cdots\!88 ) / 82\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 37\!\cdots\!22 \nu^{9} + \cdots - 12\!\cdots\!48 ) / 39\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 39\!\cdots\!31 \nu^{9} + \cdots + 56\!\cdots\!16 ) / 41\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27\!\cdots\!66 \nu^{9} + \cdots + 14\!\cdots\!76 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 83\!\cdots\!77 \nu^{9} + \cdots - 29\!\cdots\!12 ) / 82\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 98\!\cdots\!14 \nu^{9} + \cdots + 71\!\cdots\!04 ) / 48\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -4\beta_{7} + 3\beta_{6} + \beta_{5} + 9\beta _1 - 5 ) / 21 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 63 \beta_{9} - 42 \beta_{8} - \beta_{7} - 99 \beta_{6} - 5 \beta_{5} - 42 \beta_{4} - 3633 \beta_{3} + \cdots + 4 ) / 21 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 672\beta_{8} + 2122\beta_{7} - 3072\beta_{6} + 1622\beta_{5} + 168\beta_{2} - 1740\beta _1 + 1046 ) / 21 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 8680 \beta_{9} - 366 \beta_{7} + 10372 \beta_{6} + 564 \beta_{5} + 6342 \beta_{4} + 450926 \beta_{3} + \cdots - 451856 ) / 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 69930 \beta_{9} - 296772 \beta_{8} - 204950 \beta_{7} + 671730 \beta_{6} - 859186 \beta_{5} + \cdots + 654236 ) / 21 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1111404 \beta_{8} + 163768 \beta_{7} + 888204 \beta_{6} + 59432 \beta_{5} - 1500996 \beta_{2} + \cdots + 77432060 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 28363944 \beta_{9} - 263633332 \beta_{7} + 219995856 \beta_{6} + 82786600 \beta_{5} + 120532356 \beta_{4} + \cdots - 285282128 ) / 21 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1411644108 \beta_{9} - 1046377080 \beta_{8} - 98364180 \beta_{7} - 2546021532 \beta_{6} + \cdots + 54364008 ) / 7 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 48569792040 \beta_{8} + 139605499496 \beta_{7} - 197273126448 \beta_{6} + 106237418992 \beta_{5} + \cdots + 4553434840 ) / 21 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/630\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(281\) \(451\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
361.1
−3.34046 − 5.78584i
−10.0335 − 17.3785i
2.29607 + 3.97692i
1.03675 + 1.79571i
10.0411 + 17.3917i
−3.34046 + 5.78584i
−10.0335 + 17.3785i
2.29607 − 3.97692i
1.03675 − 1.79571i
10.0411 − 17.3917i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i 2.50000 + 4.33013i 0 −14.4029 + 11.6429i −8.00000 0 −5.00000 + 8.66025i
361.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 2.50000 + 4.33013i 0 −14.2082 − 11.8796i −8.00000 0 −5.00000 + 8.66025i
361.3 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 2.50000 + 4.33013i 0 −6.43445 + 17.3666i −8.00000 0 −5.00000 + 8.66025i
361.4 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 2.50000 + 4.33013i 0 10.0536 − 15.5540i −8.00000 0 −5.00000 + 8.66025i
361.5 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 2.50000 + 4.33013i 0 18.4920 + 1.02224i −8.00000 0 −5.00000 + 8.66025i
541.1 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 2.50000 − 4.33013i 0 −14.4029 − 11.6429i −8.00000 0 −5.00000 − 8.66025i
541.2 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 2.50000 − 4.33013i 0 −14.2082 + 11.8796i −8.00000 0 −5.00000 − 8.66025i
541.3 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 2.50000 − 4.33013i 0 −6.43445 − 17.3666i −8.00000 0 −5.00000 − 8.66025i
541.4 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 2.50000 − 4.33013i 0 10.0536 + 15.5540i −8.00000 0 −5.00000 − 8.66025i
541.5 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 2.50000 − 4.33013i 0 18.4920 − 1.02224i −8.00000 0 −5.00000 − 8.66025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 361.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 630.4.k.t yes 10
3.b odd 2 1 630.4.k.s ✓ 10
7.c even 3 1 inner 630.4.k.t yes 10
21.h odd 6 1 630.4.k.s ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.4.k.s ✓ 10 3.b odd 2 1
630.4.k.s ✓ 10 21.h odd 6 1
630.4.k.t yes 10 1.a even 1 1 trivial
630.4.k.t yes 10 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(630, [\chi])\):

\( T_{11}^{10} - 26 T_{11}^{9} + 4109 T_{11}^{8} - 182270 T_{11}^{7} + 16684583 T_{11}^{6} + \cdots + 16532372264004 \) Copy content Toggle raw display
\( T_{13}^{5} - 43T_{13}^{4} - 8714T_{13}^{3} + 260492T_{13}^{2} + 18559981T_{13} - 191424337 \) 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
$5$ \( (T^{2} - 5 T + 25)^{5} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 4747561509943 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 16532372264004 \) Copy content Toggle raw display
$13$ \( (T^{5} - 43 T^{4} + \cdots - 191424337)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 16\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( (T^{5} + 144 T^{4} + \cdots - 138231273600)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 77\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( (T^{5} + 194 T^{4} + \cdots - 37141624170)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} - 47 T^{4} + \cdots - 12346056504)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 13\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 55\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 54\!\cdots\!76 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 24\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots - 16496864942760)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 92632100866416)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 14\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 764944370633536)^{2} \) Copy content Toggle raw display
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