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Properties

Label 465.2.bm.e
Level $465$
Weight $2$
Character orbit 465.bm
Analytic conductor $3.713$
Analytic rank $0$
Dimension $448$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [465,2,Mod(44,465)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(465, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([15, 15, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("465.44");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 465 = 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 465.bm (of order \(30\), degree \(8\), 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: \(3.71304369399\)
Analytic rank: \(0\)
Dimension: \(448\)
Relative dimension: \(56\) over \(\Q(\zeta_{30})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{30}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 448 q - 128 q^{4} - 30 q^{6} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 448 q - 128 q^{4} - 30 q^{6} - 10 q^{9} + 20 q^{10} - 8 q^{16} - 104 q^{19} - 14 q^{21} + 84 q^{24} + 62 q^{25} - 32 q^{31} - 92 q^{34} - 134 q^{36} - 32 q^{39} - 124 q^{40} + 39 q^{45} - 20 q^{46} - 32 q^{49} - 104 q^{51} - 110 q^{54} - 82 q^{55} + 325 q^{60} + 56 q^{64} - 224 q^{66} - 96 q^{69} - 14 q^{70} + 16 q^{75} + 12 q^{76} - 16 q^{79} - 62 q^{81} + 152 q^{84} + 50 q^{85} + 19 q^{90} - 20 q^{91} - 392 q^{94} + 24 q^{96} - 156 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
44.1 −2.09650 + 1.52320i −1.59221 − 0.681807i 1.45716 − 4.48467i 2.04539 + 0.903546i 4.37661 − 0.995845i 2.27270 − 2.04635i 2.17452 + 6.69249i 2.07028 + 2.17116i −5.66444 + 1.22124i
44.2 −2.09650 + 1.52320i 0.558717 − 1.63946i 1.45716 − 4.48467i −0.240199 + 2.22313i 1.32588 + 4.28817i −2.27270 + 2.04635i 2.17452 + 6.69249i −2.37567 − 1.83199i −2.88269 − 5.02667i
44.3 −2.01444 + 1.46358i −1.38659 + 1.03796i 1.29788 − 3.99448i −1.69560 + 1.45771i 1.27407 − 4.12030i −0.164955 + 0.148526i 1.69282 + 5.20997i 0.845266 − 2.87846i 1.28222 − 5.41813i
44.4 −2.01444 + 1.46358i 1.69917 − 0.335905i 1.29788 − 3.99448i 2.11022 − 0.739581i −2.93125 + 3.16353i 0.164955 − 0.148526i 1.69282 + 5.20997i 2.77434 − 1.14152i −3.16848 + 4.57831i
44.5 −1.96936 + 1.43082i −1.21977 − 1.22970i 1.21309 − 3.73351i −1.94659 − 1.10036i 4.16165 + 0.676445i −2.57601 + 2.31945i 1.44852 + 4.45809i −0.0243214 + 2.99990i 5.40796 − 0.618219i
44.6 −1.96936 + 1.43082i −0.0976592 − 1.72930i 1.21309 − 3.73351i 0.0203540 − 2.23598i 2.66664 + 3.26587i 2.57601 − 2.31945i 1.44852 + 4.45809i −2.98093 + 0.337763i 3.15920 + 4.43257i
44.7 −1.80162 + 1.30895i 0.179056 + 1.72277i 0.914441 − 2.81436i 0.801495 + 2.08749i −2.57762 − 2.86940i 3.34690 − 3.01356i 0.660076 + 2.03150i −2.93588 + 0.616946i −4.17641 − 2.71174i
44.8 −1.80162 + 1.30895i 1.16046 + 1.28582i 0.914441 − 2.81436i 1.40707 + 1.73786i −3.77378 − 0.797582i −3.34690 + 3.01356i 0.660076 + 2.03150i −0.306684 + 2.98428i −4.80978 − 1.28917i
44.9 −1.67212 + 1.21487i −1.34920 + 1.08613i 0.702056 − 2.16071i 0.813904 − 2.08268i 0.936523 − 3.45523i 1.37897 − 1.24163i 0.173662 + 0.534475i 0.640663 − 2.93079i 1.16924 + 4.47129i
44.10 −1.67212 + 1.21487i 1.70994 − 0.275889i 0.702056 − 2.16071i −2.21061 − 0.336479i −2.52406 + 2.53867i −1.37897 + 1.24163i 0.173662 + 0.534475i 2.84777 − 0.943505i 4.10518 − 2.12296i
44.11 −1.26575 + 0.919623i −0.367596 + 1.69259i 0.138389 − 0.425916i −1.73175 − 1.41459i −1.09126 − 2.48045i −2.60792 + 2.34818i −0.750432 − 2.30959i −2.72975 − 1.24438i 3.49285 + 0.197965i
44.12 −1.26575 + 0.919623i 1.50381 + 0.859389i 0.138389 − 0.425916i −0.359196 − 2.20703i −2.69377 + 0.295166i 2.60792 − 2.34818i −0.750432 − 2.30959i 1.52290 + 2.58472i 2.48429 + 2.46323i
44.13 −1.23373 + 0.896357i −1.72083 − 0.196856i 0.100599 − 0.309612i −2.23320 − 0.113289i 2.29949 − 1.29961i 2.18043 − 1.96326i −0.789075 − 2.42852i 2.92250 + 0.677511i 2.85671 − 1.86197i
44.14 −1.23373 + 0.896357i 1.00517 − 1.41055i 0.100599 − 0.309612i 1.01849 − 1.99065i 0.0242506 + 2.64122i −2.18043 + 1.96326i −0.789075 − 2.42852i −0.979284 − 2.83567i 0.527794 + 3.36885i
44.15 −1.23115 + 0.894486i −0.893641 − 1.48371i 0.0976011 − 0.300385i 2.23150 + 0.142837i 2.42737 + 1.02733i −1.39935 + 1.25998i −0.791989 − 2.43749i −1.40281 + 2.65181i −2.87509 + 1.82019i
44.16 −1.23115 + 0.894486i −0.504651 − 1.65690i 0.0976011 − 0.300385i −0.992050 + 2.00396i 2.10338 + 1.58850i 1.39935 − 1.25998i −0.791989 − 2.43749i −2.49065 + 1.67232i −0.571143 − 3.35455i
44.17 −1.15375 + 0.838251i −1.30353 + 1.14053i 0.0104483 − 0.0321565i 1.23250 + 1.86573i 0.547909 − 2.40857i −2.16976 + 1.95366i −0.866489 − 2.66678i 0.398400 − 2.97343i −2.98595 − 1.11944i
44.18 −1.15375 + 0.838251i 1.71981 − 0.205553i 0.0104483 − 0.0321565i 0.999516 + 2.00024i −1.81193 + 1.67879i 2.16976 − 1.95366i −0.866489 − 2.66678i 2.91550 − 0.707026i −2.82990 − 1.46994i
44.19 −0.832173 + 0.604609i 0.239294 + 1.71544i −0.291074 + 0.895835i −2.00254 + 0.994896i −1.23631 − 1.28286i 0.828350 − 0.745849i −0.935129 − 2.87803i −2.88548 + 0.820991i 1.06494 − 2.03868i
44.20 −0.832173 + 0.604609i 1.11470 + 1.32568i −0.291074 + 0.895835i 1.86288 − 1.23681i −1.72915 − 0.429240i −0.828350 + 0.745849i −0.935129 − 2.87803i −0.514879 + 2.95549i −0.802451 + 2.15555i
See next 80 embeddings (of 448 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 44.56
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
31.h odd 30 1 inner
93.p even 30 1 inner
155.v odd 30 1 inner
465.bm even 30 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 465.2.bm.e ✓ 448
3.b odd 2 1 inner 465.2.bm.e ✓ 448
5.b even 2 1 inner 465.2.bm.e ✓ 448
15.d odd 2 1 inner 465.2.bm.e ✓ 448
31.h odd 30 1 inner 465.2.bm.e ✓ 448
93.p even 30 1 inner 465.2.bm.e ✓ 448
155.v odd 30 1 inner 465.2.bm.e ✓ 448
465.bm even 30 1 inner 465.2.bm.e ✓ 448
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
465.2.bm.e ✓ 448 1.a even 1 1 trivial
465.2.bm.e ✓ 448 3.b odd 2 1 inner
465.2.bm.e ✓ 448 5.b even 2 1 inner
465.2.bm.e ✓ 448 15.d odd 2 1 inner
465.2.bm.e ✓ 448 31.h odd 30 1 inner
465.2.bm.e ✓ 448 93.p even 30 1 inner
465.2.bm.e ✓ 448 155.v odd 30 1 inner
465.2.bm.e ✓ 448 465.bm even 30 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{224} + 88 T_{2}^{222} + 4036 T_{2}^{220} + 128590 T_{2}^{218} + 3202376 T_{2}^{216} + \cdots + 181063936 \) acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\). Copy content Toggle raw display