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Properties

Label 315.4.m.a
Level $315$
Weight $4$
Character orbit 315.m
Analytic conductor $18.586$
Analytic rank $0$
Dimension $36$
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] = [315,4,Mod(8,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.8");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 315.m (of order \(4\), 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: \(18.5856016518\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(18\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 8 q^{5} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 8 q^{5} - 24 q^{8} - 144 q^{10} + 108 q^{13} - 56 q^{14} - 120 q^{16} + 88 q^{17} + 492 q^{20} - 576 q^{22} - 128 q^{23} + 72 q^{25} + 32 q^{29} + 960 q^{31} - 192 q^{32} + 112 q^{35} - 612 q^{37} - 1584 q^{38} - 1008 q^{40} - 240 q^{43} + 1024 q^{44} - 672 q^{46} - 456 q^{47} + 2704 q^{50} + 192 q^{52} - 3200 q^{53} + 384 q^{55} + 816 q^{58} + 2480 q^{59} + 1824 q^{61} - 544 q^{62} + 1248 q^{65} + 384 q^{67} - 4472 q^{68} + 1008 q^{70} + 420 q^{73} + 2944 q^{74} - 2832 q^{76} + 2748 q^{80} + 3000 q^{82} - 3504 q^{83} + 2400 q^{85} - 3000 q^{88} + 5936 q^{89} - 8072 q^{92} - 3248 q^{95} + 1284 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −3.56629 − 3.56629i 0 17.4368i −3.17571 − 10.7198i 0 4.94975 − 4.94975i 33.6545 − 33.6545i 0 −26.9045 + 49.5555i
8.2 −3.52448 − 3.52448i 0 16.8439i 4.03728 − 10.4259i 0 −4.94975 + 4.94975i 31.1701 − 31.1701i 0 −50.9753 + 22.5167i
8.3 −2.94732 − 2.94732i 0 9.37337i 5.48437 + 9.74277i 0 4.94975 − 4.94975i 4.04776 − 4.04776i 0 12.5509 − 44.8793i
8.4 −2.35931 − 2.35931i 0 3.13271i 8.93400 + 6.72187i 0 −4.94975 + 4.94975i −11.4835 + 11.4835i 0 −5.21911 − 36.9371i
8.5 −2.02826 − 2.02826i 0 0.227694i −9.94964 + 5.09948i 0 −4.94975 + 4.94975i −15.7643 + 15.7643i 0 30.5236 + 9.83739i
8.6 −1.90638 − 1.90638i 0 − 0.731412i 9.20141 − 6.35092i 0 4.94975 − 4.94975i −16.6454 + 16.6454i 0 −29.6487 − 5.43413i
8.7 −1.64649 − 1.64649i 0 − 2.57816i −7.80929 + 8.00094i 0 4.94975 − 4.94975i −17.4168 + 17.4168i 0 26.0313 − 0.315554i
8.8 −0.889917 − 0.889917i 0 − 6.41610i 3.64096 − 10.5709i 0 −4.94975 + 4.94975i −12.8291 + 12.8291i 0 −12.6474 + 6.16705i
8.9 −0.194219 − 0.194219i 0 − 7.92456i −10.4148 + 4.06585i 0 4.94975 − 4.94975i −3.09286 + 3.09286i 0 2.81243 + 1.23310i
8.10 0.494985 + 0.494985i 0 − 7.50998i 9.62910 + 5.68159i 0 −4.94975 + 4.94975i 7.67721 − 7.67721i 0 1.95396 + 7.57857i
8.11 0.720585 + 0.720585i 0 − 6.96151i 0.174705 − 11.1790i 0 4.94975 − 4.94975i 10.7810 − 10.7810i 0 8.18130 − 7.92952i
8.12 1.32823 + 1.32823i 0 − 4.47160i −7.54015 − 8.25506i 0 −4.94975 + 4.94975i 16.5652 − 16.5652i 0 0.949568 − 20.9797i
8.13 1.51016 + 1.51016i 0 − 3.43886i −6.17420 + 9.32091i 0 −4.94975 + 4.94975i 17.2745 − 17.2745i 0 −23.4000 + 4.75202i
8.14 2.13530 + 2.13530i 0 1.11901i 2.59134 + 10.8759i 0 4.94975 − 4.94975i 14.6930 − 14.6930i 0 −17.6900 + 28.7566i
8.15 2.68742 + 2.68742i 0 6.44443i −11.1546 − 0.757676i 0 4.94975 − 4.94975i 4.18046 − 4.18046i 0 −27.9410 − 32.0134i
8.16 3.18686 + 3.18686i 0 12.3122i 8.71878 − 6.99877i 0 −4.94975 + 4.94975i −13.7424 + 13.7424i 0 50.0897 + 5.48143i
8.17 3.30318 + 3.30318i 0 13.8220i 10.9813 + 2.10013i 0 4.94975 − 4.94975i −19.2311 + 19.2311i 0 29.3362 + 43.2104i
8.18 3.69595 + 3.69595i 0 19.3200i −11.1748 − 0.351359i 0 −4.94975 + 4.94975i −41.8383 + 41.8383i 0 −40.0029 − 42.6001i
197.1 −3.56629 + 3.56629i 0 − 17.4368i −3.17571 + 10.7198i 0 4.94975 + 4.94975i 33.6545 + 33.6545i 0 −26.9045 − 49.5555i
197.2 −3.52448 + 3.52448i 0 − 16.8439i 4.03728 + 10.4259i 0 −4.94975 − 4.94975i 31.1701 + 31.1701i 0 −50.9753 − 22.5167i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.4.m.a ✓ 36
3.b odd 2 1 315.4.m.b yes 36
5.c odd 4 1 315.4.m.b yes 36
15.e even 4 1 inner 315.4.m.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.4.m.a ✓ 36 1.a even 1 1 trivial
315.4.m.a ✓ 36 15.e even 4 1 inner
315.4.m.b yes 36 3.b odd 2 1
315.4.m.b yes 36 5.c odd 4 1