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Properties

Label 715.2.z.c
Level $715$
Weight $2$
Character orbit 715.z
Analytic conductor $5.709$
Analytic rank $0$
Dimension $56$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [715,2,Mod(56,715)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(715, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("715.56");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 715 = 5 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 715.z (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: \(5.70930374452\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(28\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 6 q^{3} + 34 q^{4} + 18 q^{6} + 6 q^{7} - 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 56 q - 6 q^{3} + 34 q^{4} + 18 q^{6} + 6 q^{7} - 46 q^{9} + 2 q^{10} - 44 q^{12} - 2 q^{13} + 32 q^{14} - 54 q^{16} + 26 q^{17} - 6 q^{19} - 2 q^{22} + 30 q^{23} + 36 q^{24} - 56 q^{25} - 18 q^{28} - 36 q^{29} - 10 q^{30} - 36 q^{32} - 8 q^{35} + 58 q^{36} - 6 q^{37} - 36 q^{38} + 16 q^{39} + 12 q^{40} - 12 q^{41} - 74 q^{42} + 138 q^{46} - 50 q^{48} + 46 q^{49} - 24 q^{51} + 26 q^{52} + 104 q^{53} - 18 q^{54} - 28 q^{55} + 6 q^{58} + 24 q^{59} - 4 q^{61} + 14 q^{62} - 24 q^{63} - 112 q^{64} + 8 q^{65} - 20 q^{66} + 30 q^{67} - 92 q^{68} - 16 q^{69} + 156 q^{72} - 32 q^{74} + 6 q^{75} - 24 q^{76} - 16 q^{77} + 150 q^{78} - 12 q^{79} + 24 q^{80} - 116 q^{81} - 36 q^{82} + 6 q^{84} + 12 q^{85} - 16 q^{87} + 6 q^{88} - 24 q^{89} - 72 q^{90} + 74 q^{91} + 36 q^{92} + 114 q^{93} - 66 q^{94} - 8 q^{95} + 18 q^{97} + 30 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
56.1 −2.39949 − 1.38535i −0.676788 + 1.17223i 2.83838 + 4.91621i − 1.00000i 3.24790 − 1.87517i −1.45329 + 0.839057i − 10.1872i 0.583915 + 1.01137i −1.38535 + 2.39949i
56.2 −2.25316 − 1.30087i −0.450452 + 0.780205i 2.38450 + 4.13008i 1.00000i 2.02988 − 1.17195i −0.304840 + 0.175999i − 7.20420i 1.09419 + 1.89519i 1.30087 − 2.25316i
56.3 −2.12606 − 1.22748i −1.57545 + 2.72877i 2.01341 + 3.48733i 1.00000i 6.69901 − 3.86768i −1.00897 + 0.582528i − 4.97577i −3.46412 − 6.00003i 1.22748 − 2.12606i
56.4 −1.93564 − 1.11754i −0.193236 + 0.334695i 1.49779 + 2.59425i − 1.00000i 0.748069 − 0.431898i 3.67315 − 2.12069i − 2.22519i 1.42532 + 2.46873i −1.11754 + 1.93564i
56.5 −1.77070 − 1.02231i −1.62819 + 2.82011i 1.09025 + 1.88837i − 1.00000i 5.76608 − 3.32905i −3.46306 + 1.99940i − 0.369070i −3.80201 − 6.58528i −1.02231 + 1.77070i
56.6 −1.72187 − 0.994120i 1.62718 − 2.81836i 0.976547 + 1.69143i 1.00000i −5.60358 + 3.23523i −2.14080 + 1.23599i 0.0932592i −3.79545 − 6.57392i 0.994120 − 1.72187i
56.7 −1.71306 − 0.989037i 1.06706 − 1.84820i 0.956388 + 1.65651i 1.00000i −3.65589 + 2.11073i 2.57195 − 1.48492i 0.172536i −0.777241 − 1.34622i 0.989037 − 1.71306i
56.8 −1.31219 − 0.757592i 1.57755 − 2.73240i 0.147892 + 0.256157i − 1.00000i −4.14009 + 2.39028i 2.98641 − 1.72421i 2.58220i −3.47734 − 6.02292i −0.757592 + 1.31219i
56.9 −1.15203 − 0.665122i −0.758065 + 1.31301i −0.115224 − 0.199574i 1.00000i 1.74662 − 1.00841i −3.46125 + 1.99835i 2.96704i 0.350674 + 0.607386i 0.665122 − 1.15203i
56.10 −0.732105 − 0.422681i −0.267146 + 0.462711i −0.642681 − 1.11316i − 1.00000i 0.391158 − 0.225835i −0.674330 + 0.389325i 2.77732i 1.35727 + 2.35085i −0.422681 + 0.732105i
56.11 −0.568502 − 0.328225i −1.67550 + 2.90206i −0.784537 − 1.35886i 1.00000i 1.90506 − 1.09988i 1.00819 − 0.582078i 2.34292i −4.11463 − 7.12674i 0.328225 − 0.568502i
56.12 −0.568047 − 0.327962i 0.672602 − 1.16498i −0.784882 − 1.35946i 1.00000i −0.764139 + 0.441176i −3.28423 + 1.89615i 2.34149i 0.595213 + 1.03094i 0.327962 − 0.568047i
56.13 −0.514544 − 0.297072i −1.46111 + 2.53072i −0.823496 − 1.42634i − 1.00000i 1.50361 − 0.868111i 3.01624 − 1.74142i 2.16684i −2.76969 − 4.79724i −0.297072 + 0.514544i
56.14 −0.414828 − 0.239501i −0.351830 + 0.609388i −0.885279 − 1.53335i − 1.00000i 0.291898 − 0.168527i −1.35370 + 0.781560i 1.80610i 1.25243 + 2.16927i −0.239501 + 0.414828i
56.15 0.0990239 + 0.0571715i −0.983059 + 1.70271i −0.993463 − 1.72073i 1.00000i −0.194693 + 0.112406i 1.02052 − 0.589196i − 0.455877i −0.432811 − 0.749651i −0.0571715 + 0.0990239i
56.16 0.285336 + 0.164739i 1.07380 − 1.85987i −0.945722 − 1.63804i − 1.00000i 0.612786 − 0.353792i −3.71712 + 2.14608i − 1.28214i −0.806075 − 1.39616i 0.164739 − 0.285336i
56.17 0.412241 + 0.238008i 1.04877 − 1.81652i −0.886705 − 1.53582i − 1.00000i 0.864693 − 0.499231i 3.89305 − 2.24766i − 1.79620i −0.699838 − 1.21215i 0.238008 − 0.412241i
56.18 0.711389 + 0.410720i 1.47581 − 2.55617i −0.662617 − 1.14769i 1.00000i 2.09975 − 1.21229i 3.27635 − 1.89160i − 2.73148i −2.85602 − 4.94677i −0.410720 + 0.711389i
56.19 0.809271 + 0.467233i 0.346641 − 0.600400i −0.563387 − 0.975815i 1.00000i 0.561053 − 0.323924i 0.420118 − 0.242555i − 2.92186i 1.25968 + 2.18183i −0.467233 + 0.809271i
56.20 1.15246 + 0.665374i −1.20045 + 2.07925i −0.114555 − 0.198415i − 1.00000i −2.76695 + 1.59750i 2.38205 − 1.37528i − 2.96638i −1.38218 − 2.39400i 0.665374 − 1.15246i
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 56.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 715.2.z.c ✓ 56
13.e even 6 1 inner 715.2.z.c ✓ 56
13.f odd 12 1 9295.2.a.bj 28
13.f odd 12 1 9295.2.a.bk 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
715.2.z.c ✓ 56 1.a even 1 1 trivial
715.2.z.c ✓ 56 13.e even 6 1 inner
9295.2.a.bj 28 13.f odd 12 1
9295.2.a.bk 28 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{56} - 45 T_{2}^{54} + 1133 T_{2}^{52} + 36 T_{2}^{51} - 19630 T_{2}^{50} - 1410 T_{2}^{49} + \cdots + 531441 \) acting on \(S_{2}^{\mathrm{new}}(715, [\chi])\). Copy content Toggle raw display