Properties

Label 300.3.l.e
Level 300300
Weight 33
Character orbit 300.l
Analytic conductor 8.1748.174
Analytic rank 00
Dimension 88
CM discriminant -15
Inner twists 1616

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [300,3,Mod(107,300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(300, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 1])) 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("300.107"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: N N == 300=22352 300 = 2^{2} \cdot 3 \cdot 5^{2}
Weight: k k == 3 3
Character orbit: [χ][\chi] == 300.l (of order 44, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 8.174407930818.17440793081
Analytic rank: 00
Dimension: 88
Relative dimension: 44 over Q(i)\Q(i)
Coefficient field: 8.0.3317760000.4
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x8+17x4+256 x^{8} + 17x^{4} + 256 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: U(1)[D4]\mathrm{U}(1)[D_{4}]

qq-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 qq-expansion are expressed in terms of a basis 1,β1,,β71,\beta_1,\ldots,\beta_{7} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+β1q2+3β5q3+(β6+4β4)q4+(3β2+3)q6+(3β7+4β5)q8+9β4q9+(12β3+3β1)q12+(7β25)q16+49β7q98+O(q100) q + \beta_1 q^{2} + 3 \beta_{5} q^{3} + (\beta_{6} + 4 \beta_{4}) q^{4} + (3 \beta_{2} + 3) q^{6} + (3 \beta_{7} + 4 \beta_{5}) q^{8} + 9 \beta_{4} q^{9} + (12 \beta_{3} + 3 \beta_1) q^{12} + (7 \beta_{2} - 5) q^{16}+ \cdots - 49 \beta_{7} q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+12q668q16252q36+136q46+944q61+480q76648q81732q96+O(q100) 8 q + 12 q^{6} - 68 q^{16} - 252 q^{36} + 136 q^{46} + 944 q^{61} + 480 q^{76} - 648 q^{81} - 732 q^{96}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8+17x4+256 x^{8} + 17x^{4} + 256 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν4+5)/7 ( \nu^{4} + 5 ) / 7 Copy content Toggle raw display
β3\beta_{3}== (ν5+5ν)/28 ( \nu^{5} + 5\nu ) / 28 Copy content Toggle raw display
β4\beta_{4}== (ν6+33ν2)/112 ( \nu^{6} + 33\nu^{2} ) / 112 Copy content Toggle raw display
β5\beta_{5}== (3ν7+13ν3)/448 ( -3\nu^{7} + 13\nu^{3} ) / 448 Copy content Toggle raw display
β6\beta_{6}== (ν65ν2)/28 ( -\nu^{6} - 5\nu^{2} ) / 28 Copy content Toggle raw display
β7\beta_{7}== (ν7+33ν3)/112 ( \nu^{7} + 33\nu^{3} ) / 112 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β6+4β4 \beta_{6} + 4\beta_{4} Copy content Toggle raw display
ν3\nu^{3}== 3β7+4β5 3\beta_{7} + 4\beta_{5} Copy content Toggle raw display
ν4\nu^{4}== 7β25 7\beta_{2} - 5 Copy content Toggle raw display
ν5\nu^{5}== 28β35β1 28\beta_{3} - 5\beta_1 Copy content Toggle raw display
ν6\nu^{6}== 33β620β4 -33\beta_{6} - 20\beta_{4} Copy content Toggle raw display
ν7\nu^{7}== 13β7132β5 13\beta_{7} - 132\beta_{5} Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/300Z)×\left(\mathbb{Z}/300\mathbb{Z}\right)^\times.

nn 101101 151151 277277
χ(n)\chi(n) 1-1 1-1 β4-\beta_{4}

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
107.1
−1.72286 + 1.01575i
−1.01575 + 1.72286i
1.01575 1.72286i
1.72286 1.01575i
−1.72286 1.01575i
−1.01575 1.72286i
1.01575 + 1.72286i
1.72286 + 1.01575i
−1.72286 + 1.01575i −2.12132 + 2.12132i 1.93649 3.50000i 0 1.50000 5.80948i 0 0.218832 + 7.99701i 9.00000i 0
107.2 −1.01575 + 1.72286i 2.12132 2.12132i −1.93649 3.50000i 0 1.50000 + 5.80948i 0 7.99701 + 0.218832i 9.00000i 0
107.3 1.01575 1.72286i −2.12132 + 2.12132i −1.93649 3.50000i 0 1.50000 + 5.80948i 0 −7.99701 0.218832i 9.00000i 0
107.4 1.72286 1.01575i 2.12132 2.12132i 1.93649 3.50000i 0 1.50000 5.80948i 0 −0.218832 7.99701i 9.00000i 0
143.1 −1.72286 1.01575i −2.12132 2.12132i 1.93649 + 3.50000i 0 1.50000 + 5.80948i 0 0.218832 7.99701i 9.00000i 0
143.2 −1.01575 1.72286i 2.12132 + 2.12132i −1.93649 + 3.50000i 0 1.50000 5.80948i 0 7.99701 0.218832i 9.00000i 0
143.3 1.01575 + 1.72286i −2.12132 2.12132i −1.93649 + 3.50000i 0 1.50000 5.80948i 0 −7.99701 + 0.218832i 9.00000i 0
143.4 1.72286 + 1.01575i 2.12132 + 2.12132i 1.93649 + 3.50000i 0 1.50000 + 5.80948i 0 −0.218832 + 7.99701i 9.00000i 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by Q(15)\Q(\sqrt{-15})
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
12.b even 2 1 inner
15.e even 4 2 inner
20.d odd 2 1 inner
20.e even 4 2 inner
60.h even 2 1 inner
60.l odd 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.3.l.e 8
3.b odd 2 1 inner 300.3.l.e 8
4.b odd 2 1 inner 300.3.l.e 8
5.b even 2 1 inner 300.3.l.e 8
5.c odd 4 2 inner 300.3.l.e 8
12.b even 2 1 inner 300.3.l.e 8
15.d odd 2 1 CM 300.3.l.e 8
15.e even 4 2 inner 300.3.l.e 8
20.d odd 2 1 inner 300.3.l.e 8
20.e even 4 2 inner 300.3.l.e 8
60.h even 2 1 inner 300.3.l.e 8
60.l odd 4 2 inner 300.3.l.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.3.l.e 8 1.a even 1 1 trivial
300.3.l.e 8 3.b odd 2 1 inner
300.3.l.e 8 4.b odd 2 1 inner
300.3.l.e 8 5.b even 2 1 inner
300.3.l.e 8 5.c odd 4 2 inner
300.3.l.e 8 12.b even 2 1 inner
300.3.l.e 8 15.d odd 2 1 CM
300.3.l.e 8 15.e even 4 2 inner
300.3.l.e 8 20.d odd 2 1 inner
300.3.l.e 8 20.e even 4 2 inner
300.3.l.e 8 60.h even 2 1 inner
300.3.l.e 8 60.l odd 4 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S3new(300,[χ])S_{3}^{\mathrm{new}}(300, [\chi]):

T7 T_{7} Copy content Toggle raw display
T174+921600 T_{17}^{4} + 921600 Copy content Toggle raw display
T192960 T_{19}^{2} - 960 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8+17T4+256 T^{8} + 17T^{4} + 256 Copy content Toggle raw display
33 (T4+81)2 (T^{4} + 81)^{2} Copy content Toggle raw display
55 T8 T^{8} Copy content Toggle raw display
77 T8 T^{8} Copy content Toggle raw display
1111 T8 T^{8} Copy content Toggle raw display
1313 T8 T^{8} Copy content Toggle raw display
1717 (T4+921600)2 (T^{4} + 921600)^{2} Copy content Toggle raw display
1919 (T2960)4 (T^{2} - 960)^{4} Copy content Toggle raw display
2323 (T4+1336336)2 (T^{4} + 1336336)^{2} Copy content Toggle raw display
2929 T8 T^{8} Copy content Toggle raw display
3131 (T2+3840)4 (T^{2} + 3840)^{4} Copy content Toggle raw display
3737 T8 T^{8} Copy content Toggle raw display
4141 T8 T^{8} Copy content Toggle raw display
4343 T8 T^{8} Copy content Toggle raw display
4747 (T4+38416)2 (T^{4} + 38416)^{2} Copy content Toggle raw display
5353 (T4+14745600)2 (T^{4} + 14745600)^{2} Copy content Toggle raw display
5959 T8 T^{8} Copy content Toggle raw display
6161 (T118)8 (T - 118)^{8} Copy content Toggle raw display
6767 T8 T^{8} Copy content Toggle raw display
7171 T8 T^{8} Copy content Toggle raw display
7373 T8 T^{8} Copy content Toggle raw display
7979 (T215360)4 (T^{2} - 15360)^{4} Copy content Toggle raw display
8383 (T4+562448656)2 (T^{4} + 562448656)^{2} Copy content Toggle raw display
8989 T8 T^{8} Copy content Toggle raw display
9797 T8 T^{8} Copy content Toggle raw display
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