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Properties

Label 1350.3.d.o
Level $1350$
Weight $3$
Character orbit 1350.d
Analytic conductor $36.785$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1350,3,Mod(701,1350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1350.701");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1350.d (of order \(2\), degree \(1\), 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: \(36.7848356886\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.4030726144.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 4x^{6} - 16x^{5} + 18x^{4} - 8x^{3} + 172x^{2} + 184x + 274 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 270)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 2 q^{4} + ( - \beta_{7} - \beta_{3}) q^{7} - 2 \beta_1 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - 2 q^{4} + ( - \beta_{7} - \beta_{3}) q^{7} - 2 \beta_1 q^{8} + ( - \beta_{5} + 2 \beta_{4}) q^{11} + ( - 2 \beta_{7} - \beta_{3}) q^{13} + (\beta_{5} + \beta_{4}) q^{14} + 4 q^{16} + ( - \beta_{2} + 7 \beta_1) q^{17} + (\beta_{6} + 3) q^{19} + (\beta_{7} - 2 \beta_{3}) q^{22} + ( - 5 \beta_{2} - 14 \beta_1) q^{23} + (\beta_{5} + 3 \beta_{4}) q^{26} + (2 \beta_{7} + 2 \beta_{3}) q^{28} - 9 \beta_{4} q^{29} + ( - 5 \beta_{6} - 17) q^{31} + 4 \beta_1 q^{32} + (\beta_{6} - 14) q^{34} + ( - 6 \beta_{7} + 4 \beta_{3}) q^{37} + (2 \beta_{2} + 3 \beta_1) q^{38} + ( - 2 \beta_{5} - \beta_{4}) q^{41} + (5 \beta_{7} - \beta_{3}) q^{43} + (2 \beta_{5} - 4 \beta_{4}) q^{44} + (5 \beta_{6} + 28) q^{46} + ( - 20 \beta_{2} + 14 \beta_1) q^{47} + (22 \beta_{6} + 45) q^{49} + (4 \beta_{7} + 2 \beta_{3}) q^{52} + ( - 17 \beta_{2} + 9 \beta_1) q^{53} + ( - 2 \beta_{5} - 2 \beta_{4}) q^{56} - 9 \beta_{7} q^{58} + ( - 3 \beta_{5} + 4 \beta_{4}) q^{59} + (8 \beta_{6} - 49) q^{61} + ( - 10 \beta_{2} - 17 \beta_1) q^{62} - 8 q^{64} + (20 \beta_{7} + 6 \beta_{3}) q^{67} + (2 \beta_{2} - 14 \beta_1) q^{68} + (7 \beta_{5} - 6 \beta_{4}) q^{71} + (10 \beta_{7} - 3 \beta_{3}) q^{73} + ( - 4 \beta_{5} + 16 \beta_{4}) q^{74} + ( - 2 \beta_{6} - 6) q^{76} + ( - 26 \beta_{2} - 64 \beta_1) q^{77} + (21 \beta_{6} - 45) q^{79} + ( - 3 \beta_{7} - 4 \beta_{3}) q^{82} + ( - 9 \beta_{2} + 58 \beta_1) q^{83} + (\beta_{5} - 11 \beta_{4}) q^{86} + ( - 2 \beta_{7} + 4 \beta_{3}) q^{88} + 29 \beta_{4} q^{89} + (28 \beta_{6} + 114) q^{91} + (10 \beta_{2} + 28 \beta_1) q^{92} + (20 \beta_{6} - 28) q^{94} + ( - 18 \beta_{7} - 4 \beta_{3}) q^{97} + (44 \beta_{2} + 45 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} + 32 q^{16} + 24 q^{19} - 136 q^{31} - 112 q^{34} + 224 q^{46} + 360 q^{49} - 392 q^{61} - 64 q^{64} - 48 q^{76} - 360 q^{79} + 912 q^{91} - 224 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 4x^{6} - 16x^{5} + 18x^{4} - 8x^{3} + 172x^{2} + 184x + 274 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{7} + 11\nu^{6} - 12\nu^{5} + 25\nu^{4} - 86\nu^{3} + 70\nu^{2} - 424\nu - 82 ) / 375 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{7} - 4\nu^{6} + 68\nu^{5} - 135\nu^{4} + 304\nu^{3} - 510\nu^{2} - 204\nu - 2017 ) / 125 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{7} - 24\nu^{6} + 58\nu^{5} - 55\nu^{4} + 74\nu^{3} - 20\nu^{2} + 496\nu - 782 ) / 125 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -9\nu^{7} - 8\nu^{6} + 236\nu^{5} - 505\nu^{4} + 1058\nu^{3} - 1650\nu^{2} - 48\nu - 6574 ) / 375 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -11\nu^{7} + 18\nu^{6} + 94\nu^{5} - 145\nu^{4} + 432\nu^{3} - 600\nu^{2} - 1392\nu - 3896 ) / 125 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -16\nu^{7} + 83\nu^{6} - 136\nu^{5} + 255\nu^{4} - 258\nu^{3} - 100\nu^{2} - 1752\nu - 1076 ) / 125 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{7} - 20\nu^{6} + 30\nu^{5} - 62\nu^{4} + 70\nu^{3} + 54\nu^{2} + 452\nu + 322 ) / 25 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 2\beta_{6} - \beta_{5} + 3\beta_{4} + 2\beta_{3} - 2\beta_{2} + 6 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 4\beta_{7} + 5\beta_{6} + 2\beta_{3} + 9\beta _1 + 6 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 10\beta_{7} + 14\beta_{6} - \beta_{5} - 6\beta_{4} + 5\beta_{3} + 10\beta_{2} + 48 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{7} + 5\beta_{6} - \beta_{5} - 3\beta_{4} + 4\beta_{3} + 5\beta_{2} + 3\beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 58\beta_{7} + 89\beta_{6} - 26\beta_{5} - 21\beta_{4} + 71\beta_{3} + 74\beta_{2} + 225\beta _1 + 528 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 115\beta_{7} + 170\beta_{6} - 54\beta_{5} - 27\beta_{4} + 95\beta_{3} + 135\beta_{2} + 405\beta _1 + 687 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 373\beta_{7} + 536\beta_{6} - 182\beta_{5} - 228\beta_{4} + 281\beta_{3} + 599\beta_{2} + 1260\beta _1 + 1839 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
701.1
3.21154 − 0.330265i
0.487323 − 2.17650i
−0.901537 + 1.17650i
−0.797330 + 1.33026i
3.21154 + 0.330265i
0.487323 + 2.17650i
−0.901537 − 1.17650i
−0.797330 − 1.33026i
− 1.41421i 0 −2.00000 0 0 −13.6872 2.82843i 0 0
701.2 − 1.41421i 0 −2.00000 0 0 −0.813575 2.82843i 0 0
701.3 − 1.41421i 0 −2.00000 0 0 0.813575 2.82843i 0 0
701.4 − 1.41421i 0 −2.00000 0 0 13.6872 2.82843i 0 0
701.5 1.41421i 0 −2.00000 0 0 −13.6872 − 2.82843i 0 0
701.6 1.41421i 0 −2.00000 0 0 −0.813575 − 2.82843i 0 0
701.7 1.41421i 0 −2.00000 0 0 0.813575 − 2.82843i 0 0
701.8 1.41421i 0 −2.00000 0 0 13.6872 − 2.82843i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 701.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.3.d.o 8
3.b odd 2 1 inner 1350.3.d.o 8
5.b even 2 1 inner 1350.3.d.o 8
5.c odd 4 1 270.3.b.a ✓ 4
5.c odd 4 1 270.3.b.d yes 4
15.d odd 2 1 inner 1350.3.d.o 8
15.e even 4 1 270.3.b.a ✓ 4
15.e even 4 1 270.3.b.d yes 4
20.e even 4 1 2160.3.c.g 4
20.e even 4 1 2160.3.c.m 4
45.k odd 12 2 810.3.j.a 8
45.k odd 12 2 810.3.j.f 8
45.l even 12 2 810.3.j.a 8
45.l even 12 2 810.3.j.f 8
60.l odd 4 1 2160.3.c.g 4
60.l odd 4 1 2160.3.c.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.3.b.a ✓ 4 5.c odd 4 1
270.3.b.a ✓ 4 15.e even 4 1
270.3.b.d yes 4 5.c odd 4 1
270.3.b.d yes 4 15.e even 4 1
810.3.j.a 8 45.k odd 12 2
810.3.j.a 8 45.l even 12 2
810.3.j.f 8 45.k odd 12 2
810.3.j.f 8 45.l even 12 2
1350.3.d.o 8 1.a even 1 1 trivial
1350.3.d.o 8 3.b odd 2 1 inner
1350.3.d.o 8 5.b even 2 1 inner
1350.3.d.o 8 15.d odd 2 1 inner
2160.3.c.g 4 20.e even 4 1
2160.3.c.g 4 60.l odd 4 1
2160.3.c.m 4 20.e even 4 1
2160.3.c.m 4 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1350, [\chi])\):

\( T_{7}^{4} - 188T_{7}^{2} + 124 \) Copy content Toggle raw display
\( T_{11}^{4} + 388T_{11}^{2} + 35836 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 188 T^{2} + 124)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 388 T^{2} + 35836)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 324 T^{2} + 10044)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 214 T^{2} + 7921)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 6 T - 9)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1234 T^{2} + 27889)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 2268 T^{2} + 813564)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 34 T - 161)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 5408 T^{2} + 2293504)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 1372 T^{2} + 65596)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 1724 T^{2} + 65596)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 7984 T^{2} + 10291264)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 5526 T^{2} + 5948721)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 3076 T^{2} + 2327356)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 98 T + 1249)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 24464 T^{2} + 131041216)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 15876 T^{2} + 53524476)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 8036 T^{2} + 35836)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 90 T - 5913)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 14914 T^{2} + 35988001)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 23548 T^{2} + 87702844)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 18464 T^{2} + 84193024)^{2} \) Copy content Toggle raw display
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