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Properties

Label 315.2.be.c
Level $315$
Weight $2$
Character orbit 315.be
Analytic conductor $2.515$
Analytic rank $0$
Dimension $32$
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] = [315,2,Mod(236,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("315.236");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.be (of order \(6\), 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: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + q^{3} + 16 q^{4} - 32 q^{5} + 2 q^{6} + q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + q^{3} + 16 q^{4} - 32 q^{5} + 2 q^{6} + q^{7} + 7 q^{9} + 15 q^{12} - 6 q^{13} - 6 q^{14} - q^{15} - 16 q^{16} + 3 q^{17} + 41 q^{18} - 16 q^{20} - 17 q^{21} - 21 q^{22} - 26 q^{24} + 32 q^{25} - 12 q^{26} - 23 q^{27} - 31 q^{28} + 18 q^{29} - 2 q^{30} + 24 q^{31} - 19 q^{33} + 30 q^{34} - q^{35} + 18 q^{36} - q^{37} - 60 q^{38} - 36 q^{39} - 6 q^{41} + 44 q^{42} - 19 q^{43} - 21 q^{44} - 7 q^{45} + 6 q^{46} - 15 q^{47} + 35 q^{48} + 23 q^{49} - 9 q^{51} + 24 q^{53} - 58 q^{54} + 33 q^{56} + 27 q^{57} + 15 q^{59} - 15 q^{60} - 9 q^{61} - 11 q^{63} + 76 q^{64} + 6 q^{65} + 22 q^{66} + 25 q^{67} - 6 q^{68} + 50 q^{69} + 6 q^{70} + 61 q^{72} + 12 q^{73} + q^{75} - 54 q^{76} - 27 q^{77} - 42 q^{78} - 2 q^{79} + 16 q^{80} + 43 q^{81} - 24 q^{82} + 42 q^{83} - 36 q^{84} - 3 q^{85} - 55 q^{87} - 84 q^{88} - 30 q^{89} - 41 q^{90} - 57 q^{91} + 6 q^{92} - 48 q^{93} + 24 q^{94} - 9 q^{96} + 42 q^{97} - 6 q^{98} - q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
236.1 −2.27381 + 1.31279i −0.933716 + 1.45883i 2.44681 − 4.23800i −1.00000 0.207969 − 4.54286i −2.64220 − 0.136993i 7.59741i −1.25635 − 2.72426i 2.27381 − 1.31279i
236.2 −2.03337 + 1.17397i 1.49299 + 0.878053i 1.75639 − 3.04216i −1.00000 −4.06660 − 0.0326841i 0.135526 − 2.64228i 3.55190i 1.45805 + 2.62185i 2.03337 − 1.17397i
236.3 −1.91788 + 1.10729i 0.302150 − 1.70549i 1.45217 − 2.51523i −1.00000 1.30898 + 3.60549i 2.53521 − 0.756781i 2.00271i −2.81741 − 1.03063i 1.91788 − 1.10729i
236.4 −1.67628 + 0.967799i −1.71487 + 0.243339i 0.873269 − 1.51255i −1.00000 2.63910 − 2.06755i 2.28906 + 1.32673i − 0.490599i 2.88157 − 0.834589i 1.67628 − 0.967799i
236.5 −1.07088 + 0.618275i 1.48704 − 0.888095i −0.235471 + 0.407847i −1.00000 −1.04336 + 1.87045i −2.10494 − 1.60288i − 3.05545i 1.42257 − 2.64127i 1.07088 − 0.618275i
236.6 −0.948743 + 0.547757i −0.237931 + 1.71563i −0.399925 + 0.692690i −1.00000 −0.714013 − 1.75802i 1.38314 + 2.25542i − 3.06727i −2.88678 − 0.816404i 0.948743 − 0.547757i
236.7 −0.454062 + 0.262153i 0.322156 − 1.70183i −0.862552 + 1.49398i −1.00000 0.299860 + 0.857190i −1.09499 + 2.40853i − 1.95309i −2.79243 − 1.09651i 0.454062 − 0.262153i
236.8 −0.0900587 + 0.0519954i −1.13092 − 1.31188i −0.994593 + 1.72269i −1.00000 0.170061 + 0.0593436i 2.34312 − 1.22873i − 0.414839i −0.442054 + 2.96725i 0.0900587 − 0.0519954i
236.9 0.343770 − 0.198476i −1.65197 + 0.520559i −0.921215 + 1.59559i −1.00000 −0.464581 + 0.506829i −0.324456 − 2.62578i 1.52526i 2.45804 − 1.71990i −0.343770 + 0.198476i
236.10 0.403396 − 0.232901i 0.625194 + 1.61528i −0.891514 + 1.54415i −1.00000 0.628402 + 0.505990i −2.59662 − 0.507526i 1.76214i −2.21827 + 2.01973i −0.403396 + 0.232901i
236.11 0.559417 − 0.322980i 1.73059 + 0.0712304i −0.791368 + 1.37069i −1.00000 0.991125 − 0.519096i 1.24717 + 2.33336i 2.31430i 2.98985 + 0.246540i −0.559417 + 0.322980i
236.12 1.36556 − 0.788404i −1.67710 + 0.432827i 0.243162 − 0.421168i −1.00000 −1.94893 + 1.91328i 0.569408 + 2.58375i 2.38678i 2.62532 − 1.45179i −1.36556 + 0.788404i
236.13 1.48159 − 0.855399i 1.02526 − 1.39601i 0.463416 − 0.802660i −1.00000 0.324870 − 2.94533i 1.20122 − 2.35734i 1.83597i −0.897694 − 2.86254i −1.48159 + 0.855399i
236.14 2.01305 − 1.16223i 0.708464 + 1.58053i 1.70157 − 2.94720i −1.00000 3.26312 + 2.35828i 2.64011 − 0.172663i − 3.26155i −1.99616 + 2.23950i −2.01305 + 1.16223i
236.15 2.11687 − 1.22217i 1.73076 + 0.0669518i 1.98742 − 3.44231i −1.00000 3.74561 − 1.97356i −2.50358 + 0.855621i − 4.82720i 2.99103 + 0.231754i −2.11687 + 1.22217i
236.16 2.18143 − 1.25945i −1.57808 − 0.713898i 2.17243 − 3.76276i −1.00000 −4.34160 + 0.430199i −2.57718 − 0.598455i − 5.90648i 1.98070 + 2.25318i −2.18143 + 1.25945i
311.1 −2.27381 − 1.31279i −0.933716 − 1.45883i 2.44681 + 4.23800i −1.00000 0.207969 + 4.54286i −2.64220 + 0.136993i − 7.59741i −1.25635 + 2.72426i 2.27381 + 1.31279i
311.2 −2.03337 − 1.17397i 1.49299 − 0.878053i 1.75639 + 3.04216i −1.00000 −4.06660 + 0.0326841i 0.135526 + 2.64228i − 3.55190i 1.45805 − 2.62185i 2.03337 + 1.17397i
311.3 −1.91788 − 1.10729i 0.302150 + 1.70549i 1.45217 + 2.51523i −1.00000 1.30898 − 3.60549i 2.53521 + 0.756781i − 2.00271i −2.81741 + 1.03063i 1.91788 + 1.10729i
311.4 −1.67628 − 0.967799i −1.71487 − 0.243339i 0.873269 + 1.51255i −1.00000 2.63910 + 2.06755i 2.28906 − 1.32673i 0.490599i 2.88157 + 0.834589i 1.67628 + 0.967799i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 236.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.s even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.be.c yes 32
3.b odd 2 1 945.2.be.c 32
7.d odd 6 1 315.2.t.c ✓ 32
9.c even 3 1 945.2.t.c 32
9.d odd 6 1 315.2.t.c ✓ 32
21.g even 6 1 945.2.t.c 32
63.k odd 6 1 945.2.be.c 32
63.s even 6 1 inner 315.2.be.c yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.t.c ✓ 32 7.d odd 6 1
315.2.t.c ✓ 32 9.d odd 6 1
315.2.be.c yes 32 1.a even 1 1 trivial
315.2.be.c yes 32 63.s even 6 1 inner
945.2.t.c 32 9.c even 3 1
945.2.t.c 32 21.g even 6 1
945.2.be.c 32 3.b odd 2 1
945.2.be.c 32 63.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{32} - 24 T_{2}^{30} + 348 T_{2}^{28} - 3298 T_{2}^{26} - 6 T_{2}^{25} + 23178 T_{2}^{24} + \cdots + 81 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display