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Properties

Label 618.2.e.f
Level $618$
Weight $2$
Character orbit 618.e
Analytic conductor $4.935$
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] = [618,2,Mod(355,618)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(618, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("618.355");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 618 = 2 \cdot 3 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 618.e (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: \(4.93475484492\)
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: \( x^{10} - x^{9} + 17x^{8} - 26x^{7} + 265x^{6} - 324x^{5} + 621x^{4} - 132x^{3} + 396x^{2} - 144x + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{2} + q^{3} + ( - \beta_{5} - 1) q^{4} - \beta_1 q^{5} - \beta_{5} q^{6} + (\beta_{6} + \beta_{5} + 1) q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{2} + q^{3} + ( - \beta_{5} - 1) q^{4} - \beta_1 q^{5} - \beta_{5} q^{6} + (\beta_{6} + \beta_{5} + 1) q^{7} - q^{8} + q^{9} + ( - \beta_{4} - \beta_1) q^{10} + ( - \beta_{5} - \beta_{3} + \beta_1 - 1) q^{11} + ( - \beta_{5} - 1) q^{12} + ( - \beta_{8} + \beta_{7} + \beta_{3}) q^{13} + (\beta_{7} + 1) q^{14} - \beta_1 q^{15} + \beta_{5} q^{16} + ( - \beta_{6} - 2 \beta_1) q^{17} - \beta_{5} q^{18} + ( - \beta_{9} - 2 \beta_{5} + \beta_{2}) q^{19} - \beta_{4} q^{20} + (\beta_{6} + \beta_{5} + 1) q^{21} + (\beta_{8} + \beta_{4} - \beta_{3} + \cdots - 1) q^{22}+ \cdots + ( - \beta_{5} - \beta_{3} + \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 5 q^{2} + 10 q^{3} - 5 q^{4} - q^{5} + 5 q^{6} + 3 q^{7} - 10 q^{8} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 5 q^{2} + 10 q^{3} - 5 q^{4} - q^{5} + 5 q^{6} + 3 q^{7} - 10 q^{8} + 10 q^{9} - 2 q^{10} - 3 q^{11} - 5 q^{12} - 6 q^{13} + 6 q^{14} - q^{15} - 5 q^{16} + 5 q^{18} + 11 q^{19} - q^{20} + 3 q^{21} - 6 q^{22} - 4 q^{23} - 10 q^{24} - 8 q^{25} - 3 q^{26} + 10 q^{27} + 3 q^{28} - 5 q^{29} - 2 q^{30} + 4 q^{31} + 5 q^{32} - 3 q^{33} - 13 q^{35} - 5 q^{36} + 8 q^{37} - 11 q^{38} - 6 q^{39} + q^{40} + 10 q^{41} + 6 q^{42} + 13 q^{43} - 3 q^{44} - q^{45} - 2 q^{46} - 12 q^{47} - 5 q^{48} - 8 q^{49} + 8 q^{50} + 3 q^{52} - 25 q^{53} + 5 q^{54} + 17 q^{55} - 3 q^{56} + 11 q^{57} + 5 q^{58} + 13 q^{59} - q^{60} + 2 q^{61} + 2 q^{62} + 3 q^{63} + 10 q^{64} - q^{65} - 6 q^{66} + 8 q^{67} - 4 q^{69} + 13 q^{70} + 22 q^{71} - 10 q^{72} + 4 q^{74} - 8 q^{75} - 22 q^{76} - 4 q^{77} - 3 q^{78} + 20 q^{79} + 2 q^{80} + 10 q^{81} - 10 q^{82} + 3 q^{84} - 52 q^{85} - 13 q^{86} - 5 q^{87} + 3 q^{88} + 44 q^{89} - 2 q^{90} + 20 q^{91} + 2 q^{92} + 4 q^{93} - 24 q^{94} - 32 q^{95} + 5 q^{96} - 41 q^{97} + 8 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 17x^{8} - 26x^{7} + 265x^{6} - 324x^{5} + 621x^{4} - 132x^{3} + 396x^{2} - 144x + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 80303 \nu^{9} + 587675 \nu^{8} - 6357575 \nu^{7} + 9150518 \nu^{6} - 101026675 \nu^{5} + \cdots - 1981053576 ) / 344272944 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1096657 \nu^{9} + 4293350 \nu^{8} + 17452904 \nu^{7} + 54782887 \nu^{6} + 240634273 \nu^{5} + \cdots + 376367364 ) / 1032818832 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 189565 \nu^{9} + 314413 \nu^{8} - 3401377 \nu^{7} + 6862678 \nu^{6} - 54050453 \nu^{5} + \cdots + 40558992 ) / 86068236 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 844979 \nu^{9} - 276284 \nu^{8} + 13421404 \nu^{7} - 11765323 \nu^{6} + 203331401 \nu^{5} + \cdots - 51705792 ) / 258204708 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1380353 \nu^{9} - 2824550 \nu^{8} - 22910136 \nu^{7} - 30111223 \nu^{6} - 315781617 \nu^{5} + \cdots - 413644356 ) / 344272944 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11650727 \nu^{9} - 18186883 \nu^{8} + 196749007 \nu^{7} - 408684726 \nu^{6} + 3126490523 \nu^{5} + \cdots - 2722638072 ) / 688545888 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 36074105 \nu^{9} - 46244615 \nu^{8} + 628080943 \nu^{7} - 1098264712 \nu^{6} + \cdots - 6294131568 ) / 2065637664 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 10912861 \nu^{9} + 152626 \nu^{8} - 175091672 \nu^{7} + 100514117 \nu^{6} - 2608936909 \nu^{5} + \cdots - 2746485684 ) / 516409416 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{8} + \beta_{7} - \beta_{6} + 6\beta_{5} + \beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{8} - 15\beta_{4} + 2\beta_{3} + \beta_{2} - 15\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -15\beta_{9} + 18\beta_{6} - 80\beta_{5} + 16\beta_{3} + 22\beta _1 - 80 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 22\beta_{9} + 27\beta_{8} + 3\beta_{7} - 3\beta_{6} + 130\beta_{5} + 227\beta_{4} - 22\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 229\beta_{8} - 273\beta_{7} - 416\beta_{4} - 229\beta_{3} + 227\beta_{2} - 416\beta _1 + 1204 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -416\beta_{9} + 141\beta_{6} - 2412\beta_{5} - 313\beta_{3} + 3486\beta _1 - 2412 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3486\beta_{9} - 3242\beta_{8} + 4074\beta_{7} - 4074\beta_{6} + 18550\beta_{5} + 7493\beta_{4} - 3486\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -3309\beta_{8} - 3663\beta_{7} - 54031\beta_{4} + 3309\beta_{3} + 7493\beta_{2} - 54031\beta _1 + 42806 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/618\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(413\)
\(\chi(n)\) \(\beta_{5}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
355.1
1.81593 − 3.14529i
0.787881 − 1.36465i
0.319293 − 0.553032i
−0.407204 + 0.705298i
−2.01590 + 3.49165i
1.81593 + 3.14529i
0.787881 + 1.36465i
0.319293 + 0.553032i
−0.407204 − 0.705298i
−2.01590 − 3.49165i
0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −1.81593 + 3.14529i 0.500000 + 0.866025i −1.02581 + 1.77675i −1.00000 1.00000 −3.63187
355.2 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.787881 + 1.36465i 0.500000 + 0.866025i 1.40524 − 2.43395i −1.00000 1.00000 −1.57576
355.3 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i −0.319293 + 0.553032i 0.500000 + 0.866025i −1.61441 + 2.79624i −1.00000 1.00000 −0.638586
355.4 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 0.407204 − 0.705298i 0.500000 + 0.866025i 2.19714 − 3.80557i −1.00000 1.00000 0.814408
355.5 0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 2.01590 − 3.49165i 0.500000 + 0.866025i 0.537829 − 0.931548i −1.00000 1.00000 4.03181
571.1 0.500000 − 0.866025i 1.00000 −0.500000 − 0.866025i −1.81593 − 3.14529i 0.500000 − 0.866025i −1.02581 − 1.77675i −1.00000 1.00000 −3.63187
571.2 0.500000 − 0.866025i 1.00000 −0.500000 − 0.866025i −0.787881 − 1.36465i 0.500000 − 0.866025i 1.40524 + 2.43395i −1.00000 1.00000 −1.57576
571.3 0.500000 − 0.866025i 1.00000 −0.500000 − 0.866025i −0.319293 − 0.553032i 0.500000 − 0.866025i −1.61441 − 2.79624i −1.00000 1.00000 −0.638586
571.4 0.500000 − 0.866025i 1.00000 −0.500000 − 0.866025i 0.407204 + 0.705298i 0.500000 − 0.866025i 2.19714 + 3.80557i −1.00000 1.00000 0.814408
571.5 0.500000 − 0.866025i 1.00000 −0.500000 − 0.866025i 2.01590 + 3.49165i 0.500000 − 0.866025i 0.537829 + 0.931548i −1.00000 1.00000 4.03181
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 355.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 618.2.e.f ✓ 10
3.b odd 2 1 1854.2.f.k 10
103.c even 3 1 inner 618.2.e.f ✓ 10
309.h odd 6 1 1854.2.f.k 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
618.2.e.f ✓ 10 1.a even 1 1 trivial
618.2.e.f ✓ 10 103.c even 3 1 inner
1854.2.f.k 10 3.b odd 2 1
1854.2.f.k 10 309.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + T_{5}^{9} + 17 T_{5}^{8} + 26 T_{5}^{7} + 265 T_{5}^{6} + 324 T_{5}^{5} + 621 T_{5}^{4} + \cdots + 144 \) acting on \(S_{2}^{\mathrm{new}}(618, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{5} \) Copy content Toggle raw display
$3$ \( (T - 1)^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + T^{9} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( T^{10} - 3 T^{9} + \cdots + 7744 \) Copy content Toggle raw display
$11$ \( T^{10} + 3 T^{9} + \cdots + 580644 \) Copy content Toggle raw display
$13$ \( (T^{5} + 3 T^{4} - 35 T^{3} + \cdots - 8)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + 59 T^{8} + \cdots + 756900 \) Copy content Toggle raw display
$19$ \( T^{10} - 11 T^{9} + \cdots + 23658496 \) Copy content Toggle raw display
$23$ \( (T^{5} + 2 T^{4} - 19 T^{3} + \cdots - 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + 5 T^{9} + \cdots + 3283344 \) Copy content Toggle raw display
$31$ \( (T^{5} - 2 T^{4} + \cdots - 608)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} - 4 T^{4} + \cdots - 26752)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 10 T^{9} + \cdots + 352836 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 147865600 \) Copy content Toggle raw display
$47$ \( T^{10} + 12 T^{9} + \cdots + 900 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 1722914064 \) Copy content Toggle raw display
$59$ \( T^{10} - 13 T^{9} + \cdots + 36 \) Copy content Toggle raw display
$61$ \( (T^{5} - T^{4} - 77 T^{3} + \cdots - 320)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} - 8 T^{9} + \cdots + 44729344 \) Copy content Toggle raw display
$71$ \( T^{10} - 22 T^{9} + \cdots + 116964 \) Copy content Toggle raw display
$73$ \( (T^{5} - 126 T^{3} + \cdots + 9603)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 10 T^{4} + \cdots - 64795)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 241864704 \) Copy content Toggle raw display
$89$ \( (T^{5} - 22 T^{4} + \cdots + 8628)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + 41 T^{9} + \cdots + 707281 \) Copy content Toggle raw display
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