↑

Properties

Label 675.3.j.e
Level $675$
Weight $3$
Character orbit 675.j
Analytic conductor $18.392$
Analytic rank $0$
Dimension $20$
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [675,3,Mod(251,675)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("675.251"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 675.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [20,0,0,18,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.3924178443\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 3 x^{18} - 19 x^{16} - 66 x^{14} + 109 x^{12} + 813 x^{10} + 981 x^{8} - 5346 x^{6} + \cdots + 59049 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{12} \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} + ( - \beta_{9} + 2 \beta_{6} - \beta_{3} + 2) q^{4} - \beta_{15} q^{7} + (\beta_{19} + \beta_{18} + \cdots - \beta_1) q^{8} + (\beta_{11} + \beta_{6} + 2) q^{11} + (\beta_{19} - \beta_{15} + \cdots - \beta_1) q^{13}+ \cdots + (13 \beta_{19} + 13 \beta_{18} + \cdots - 10 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 18 q^{4} + 24 q^{11} + 30 q^{14} - 26 q^{16} + 8 q^{19} - 114 q^{29} + 28 q^{31} + 4 q^{34} - 102 q^{41} + 116 q^{46} + 40 q^{49} + 618 q^{56} + 120 q^{59} - 50 q^{61} - 140 q^{64} + 504 q^{74} - 96 q^{76}+ \cdots - 218 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 3 x^{18} - 19 x^{16} - 66 x^{14} + 109 x^{12} + 813 x^{10} + 981 x^{8} - 5346 x^{6} + \cdots + 59049 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 29 \nu^{19} + 609 \nu^{17} + 286 \nu^{15} - 8916 \nu^{13} - 19627 \nu^{11} + 25071 \nu^{9} + \cdots - 2407887 \nu ) / 393660 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 23 \nu^{18} + 327 \nu^{16} + 1058 \nu^{14} - 3333 \nu^{12} - 13496 \nu^{10} - 8097 \nu^{8} + \cdots - 1154736 ) / 65610 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{19} + 3 \nu^{17} - 19 \nu^{15} - 66 \nu^{13} + 109 \nu^{11} + 813 \nu^{9} + 981 \nu^{7} + \cdots + 19683 \nu ) / 6561 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 11 \nu^{18} + 33 \nu^{16} - 128 \nu^{14} - 483 \nu^{12} - 340 \nu^{10} + 3597 \nu^{8} + \cdots - 20412 \nu^{2} ) / 13122 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 6 \nu^{18} + 4 \nu^{16} + 126 \nu^{14} + 59 \nu^{12} - 837 \nu^{10} - 2804 \nu^{8} + 1497 \nu^{6} + \cdots - 96957 ) / 7290 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 107 \nu^{19} + 357 \nu^{17} - 2492 \nu^{15} - 16008 \nu^{13} - 8371 \nu^{11} + 102093 \nu^{9} + \cdots - 3287061 \nu ) / 393660 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 47 \nu^{18} - 87 \nu^{16} + 488 \nu^{14} + 1104 \nu^{12} - 101 \nu^{10} - 15315 \nu^{8} + \cdots - 111537 ) / 26244 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 14 \nu^{18} - 42 \nu^{16} + 185 \nu^{14} + 681 \nu^{12} + 13 \nu^{10} - 6036 \nu^{8} + \cdots - 45927 ) / 6561 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 101 \nu^{18} - 324 \nu^{16} - 3476 \nu^{14} + 3546 \nu^{12} + 37487 \nu^{10} + 49734 \nu^{8} + \cdots + 5196312 ) / 43740 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 344 \nu^{18} - 1509 \nu^{16} + 7454 \nu^{14} + 22776 \nu^{12} - 26588 \nu^{10} - 208221 \nu^{8} + \cdots - 6055803 ) / 131220 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 103 \nu^{19} - 132 \nu^{17} - 2578 \nu^{15} + 528 \nu^{13} + 22621 \nu^{11} + 47172 \nu^{9} + \cdots + 2948076 \nu ) / 131220 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 89 \nu^{19} - 168 \nu^{17} + 530 \nu^{15} + 3264 \nu^{13} + 7093 \nu^{11} - 28032 \nu^{9} + \cdots + 1285956 \nu ) / 78732 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 223 \nu^{18} + 63 \nu^{16} - 4678 \nu^{14} - 4662 \nu^{12} + 30121 \nu^{10} + 106497 \nu^{8} + \cdots + 3855681 ) / 43740 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 703 \nu^{19} + 42 \nu^{17} + 14788 \nu^{15} - 8808 \nu^{13} - 140671 \nu^{11} + \cdots - 24511896 \nu ) / 393660 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 146 \nu^{19} + 267 \nu^{17} - 2234 \nu^{15} - 4686 \nu^{13} + 9380 \nu^{11} + 58263 \nu^{9} + \cdots + 1883007 \nu ) / 78732 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 509 \nu^{18} + 69 \nu^{16} - 7889 \nu^{14} - 546 \nu^{12} + 54428 \nu^{10} + 156966 \nu^{8} + \cdots + 8975448 ) / 65610 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 41 \nu^{19} - 39 \nu^{17} - 671 \nu^{15} + 696 \nu^{13} + 6062 \nu^{11} + 10734 \nu^{9} + \cdots + 931662 \nu ) / 21870 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 6 \nu^{19} + 4 \nu^{17} + 126 \nu^{15} + 59 \nu^{13} - 837 \nu^{11} - 2804 \nu^{9} + \cdots - 89667 \nu ) / 2430 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - 2\beta_{6} + \beta_{5} + \beta_{3} - 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{19} + \beta_{16} + \beta_{15} + \beta_{13} - \beta_{12} - 2\beta_{7} - \beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{14} - \beta_{11} - \beta_{10} + 3\beta_{9} - 2\beta_{8} + 4\beta_{6} - 2\beta_{3} + 15 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3\beta_{19} + 4\beta_{18} - 3\beta_{16} + 3\beta_{15} + 12\beta_{12} + 2\beta_{4} + 15\beta_{2} + 5\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{17} + 3\beta_{14} + 3\beta_{11} + 3\beta_{10} - 6\beta_{8} + 13\beta_{6} + 3\beta_{5} + 4\beta_{3} + 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - \beta_{19} - 8 \beta_{18} + 11 \beta_{16} + 8 \beta_{15} - \beta_{13} + 16 \beta_{12} - 19 \beta_{7} + \cdots + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3 \beta_{17} - 3 \beta_{14} - 4 \beta_{11} + 29 \beta_{9} - 23 \beta_{8} - 75 \beta_{6} - 5 \beta_{5} + \cdots + 38 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 10 \beta_{19} + 17 \beta_{18} - 18 \beta_{16} + 21 \beta_{15} + 9 \beta_{13} + 24 \beta_{12} + \cdots + 42 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( \beta_{17} + 48 \beta_{14} + 3 \beta_{11} - 9 \beta_{10} + 62 \beta_{9} - 27 \beta_{8} + 147 \beta_{6} + \cdots - 358 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 69 \beta_{19} + 49 \beta_{18} + 55 \beta_{16} + 124 \beta_{15} + 25 \beta_{13} + 296 \beta_{12} + \cdots - 182 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 57 \beta_{17} - 13 \beta_{14} + 6 \beta_{11} + 82 \beta_{10} - 73 \beta_{9} - 309 \beta_{8} + \cdots + 410 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 131 \beta_{19} - 59 \beta_{18} - 267 \beta_{16} - 99 \beta_{15} - 243 \beta_{13} + 561 \beta_{12} + \cdots + 208 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 48 \beta_{17} + 99 \beta_{14} + 261 \beta_{11} + 69 \beta_{10} - 28 \beta_{9} + 204 \beta_{8} + \cdots - 3049 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 119 \beta_{19} - 150 \beta_{18} + 917 \beta_{16} + 476 \beta_{15} + 197 \beta_{13} + \cdots - 1353 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 63 \beta_{17} - 469 \beta_{14} - 764 \beta_{11} + 28 \beta_{10} + 573 \beta_{9} - 1105 \beta_{8} + \cdots - 10023 ) / 3 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 1014 \beta_{19} + 1103 \beta_{18} - 2886 \beta_{16} - 1515 \beta_{15} - 567 \beta_{13} + \cdots - 2633 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 1253 \beta_{17} + 2040 \beta_{14} + 2283 \beta_{11} - 669 \beta_{10} - 7515 \beta_{9} + 3633 \beta_{8} + \cdots - 8064 ) / 3 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 7120 \beta_{19} + 17 \beta_{18} + 4120 \beta_{16} - 1493 \beta_{15} - 5804 \beta_{13} + \cdots - 12322 \beta_1 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(1 + \beta_{6}\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−0.961330 + 1.44078i
1.69702 − 0.346576i
0.105167 − 1.72886i
0.185238 + 1.72212i
−1.70311 − 0.315300i
1.70311 + 0.315300i
−0.185238 − 1.72212i
−0.105167 + 1.72886i
−1.69702 + 0.346576i
0.961330 − 1.44078i
−0.961330 − 1.44078i
1.69702 + 0.346576i
0.105167 + 1.72886i
0.185238 − 1.72212i
−1.70311 + 0.315300i
1.70311 − 0.315300i
−0.185238 + 1.72212i
−0.105167 − 1.72886i
−1.69702 − 0.346576i
0.961330 + 1.44078i
−3.19328 − 1.84364i 0 4.79800 + 8.31039i 0 0 −1.28370 + 2.22343i − 20.6340i 0 0
251.2 −2.46092 − 1.42081i 0 2.03740 + 3.52889i 0 0 −4.86464 + 8.42581i − 0.212574i 0 0
251.3 −1.98614 − 1.14670i 0 0.629835 + 1.09091i 0 0 3.49042 − 6.04559i 6.28466i 0 0
251.4 −1.15863 − 0.668935i 0 −1.10505 − 1.91401i 0 0 4.10376 − 7.10792i 8.30831i 0 0
251.5 −0.457947 − 0.264396i 0 −1.86019 − 3.22194i 0 0 1.38329 − 2.39593i 4.08247i 0 0
251.6 0.457947 + 0.264396i 0 −1.86019 − 3.22194i 0 0 −1.38329 + 2.39593i − 4.08247i 0 0
251.7 1.15863 + 0.668935i 0 −1.10505 − 1.91401i 0 0 −4.10376 + 7.10792i − 8.30831i 0 0
251.8 1.98614 + 1.14670i 0 0.629835 + 1.09091i 0 0 −3.49042 + 6.04559i − 6.28466i 0 0
251.9 2.46092 + 1.42081i 0 2.03740 + 3.52889i 0 0 4.86464 − 8.42581i 0.212574i 0 0
251.10 3.19328 + 1.84364i 0 4.79800 + 8.31039i 0 0 1.28370 − 2.22343i 20.6340i 0 0
476.1 −3.19328 + 1.84364i 0 4.79800 − 8.31039i 0 0 −1.28370 − 2.22343i 20.6340i 0 0
476.2 −2.46092 + 1.42081i 0 2.03740 − 3.52889i 0 0 −4.86464 − 8.42581i 0.212574i 0 0
476.3 −1.98614 + 1.14670i 0 0.629835 − 1.09091i 0 0 3.49042 + 6.04559i − 6.28466i 0 0
476.4 −1.15863 + 0.668935i 0 −1.10505 + 1.91401i 0 0 4.10376 + 7.10792i − 8.30831i 0 0
476.5 −0.457947 + 0.264396i 0 −1.86019 + 3.22194i 0 0 1.38329 + 2.39593i − 4.08247i 0 0
476.6 0.457947 − 0.264396i 0 −1.86019 + 3.22194i 0 0 −1.38329 − 2.39593i 4.08247i 0 0
476.7 1.15863 − 0.668935i 0 −1.10505 + 1.91401i 0 0 −4.10376 − 7.10792i 8.30831i 0 0
476.8 1.98614 − 1.14670i 0 0.629835 − 1.09091i 0 0 −3.49042 − 6.04559i 6.28466i 0 0
476.9 2.46092 − 1.42081i 0 2.03740 − 3.52889i 0 0 4.86464 + 8.42581i − 0.212574i 0 0
476.10 3.19328 − 1.84364i 0 4.79800 − 8.31039i 0 0 1.28370 + 2.22343i − 20.6340i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.d odd 6 1 inner
45.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.3.j.e 20
3.b odd 2 1 225.3.j.e 20
5.b even 2 1 inner 675.3.j.e 20
5.c odd 4 2 135.3.h.a 20
9.c even 3 1 225.3.j.e 20
9.d odd 6 1 inner 675.3.j.e 20
15.d odd 2 1 225.3.j.e 20
15.e even 4 2 45.3.h.a ✓ 20
45.h odd 6 1 inner 675.3.j.e 20
45.j even 6 1 225.3.j.e 20
45.k odd 12 2 45.3.h.a ✓ 20
45.k odd 12 2 405.3.d.a 20
45.l even 12 2 135.3.h.a 20
45.l even 12 2 405.3.d.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.3.h.a ✓ 20 15.e even 4 2
45.3.h.a ✓ 20 45.k odd 12 2
135.3.h.a 20 5.c odd 4 2
135.3.h.a 20 45.l even 12 2
225.3.j.e 20 3.b odd 2 1
225.3.j.e 20 9.c even 3 1
225.3.j.e 20 15.d odd 2 1
225.3.j.e 20 45.j even 6 1
405.3.d.a 20 45.k odd 12 2
405.3.d.a 20 45.l even 12 2
675.3.j.e 20 1.a even 1 1 trivial
675.3.j.e 20 5.b even 2 1 inner
675.3.j.e 20 9.d odd 6 1 inner
675.3.j.e 20 45.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 29 T_{2}^{18} + 561 T_{2}^{16} - 6012 T_{2}^{14} + 46527 T_{2}^{12} - 219603 T_{2}^{10} + \cdots + 83521 \) acting on \(S_{3}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 29 T^{18} + \cdots + 83521 \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 245780494502544 \) Copy content Toggle raw display
$11$ \( (T^{10} - 12 T^{9} + \cdots + 21579372)^{2} \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( (T^{10} + 704 T^{8} + \cdots + 1532096164)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} - 2 T^{4} + \cdots + 7946)^{4} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 15352201216 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 19167316554672)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 116686747082896)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots - 2777948395200)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 957338655721083)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 70779040697088)^{2} \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 457796433007684)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 42\!\cdots\!09 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 18\!\cdots\!36)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 42\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
show more
show less