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Properties

Label 715.2.i.e
Level $715$
Weight $2$
Character orbit 715.i
Analytic conductor $5.709$
Analytic rank $0$
Dimension $28$
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(276,715)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("715.276");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 715 = 5 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 715.i (of order \(3\), 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: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + q^{2} - q^{3} - 17 q^{4} - 28 q^{5} + 2 q^{6} + 5 q^{7} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + q^{2} - q^{3} - 17 q^{4} - 28 q^{5} + 2 q^{6} + 5 q^{7} - 27 q^{9} - q^{10} - 14 q^{11} - 18 q^{12} + q^{13} + 24 q^{14} + q^{15} - 15 q^{16} - 17 q^{17} - 46 q^{18} + 3 q^{19} + 17 q^{20} + 32 q^{21} + q^{22} - 4 q^{23} - 13 q^{24} + 28 q^{25} - 4 q^{26} + 20 q^{27} + 27 q^{28} - 21 q^{29} - 2 q^{30} + 18 q^{31} + 26 q^{32} - q^{33} + 112 q^{34} - 5 q^{35} - 51 q^{36} + 8 q^{37} + 10 q^{38} - 44 q^{39} + q^{41} + 42 q^{42} - 23 q^{43} + 34 q^{44} + 27 q^{45} - q^{46} + 36 q^{47} + 14 q^{48} - 41 q^{49} + q^{50} - 20 q^{52} + 36 q^{53} + 8 q^{54} + 14 q^{55} - 80 q^{56} - 88 q^{57} + 22 q^{58} + 8 q^{59} + 18 q^{60} - 16 q^{61} + 5 q^{62} + 50 q^{63} + 20 q^{64} - q^{65} - 4 q^{66} + q^{67} + 8 q^{68} - 34 q^{69} - 24 q^{70} - 20 q^{71} + 117 q^{72} + 30 q^{73} - 49 q^{74} - q^{75} + 10 q^{76} - 10 q^{77} - 98 q^{78} + 10 q^{79} + 15 q^{80} - 82 q^{81} - 32 q^{82} + 36 q^{83} - 50 q^{84} + 17 q^{85} + 148 q^{86} - 2 q^{87} - 16 q^{89} + 46 q^{90} - q^{91} + 14 q^{92} + 33 q^{93} - 34 q^{94} - 3 q^{95} - 112 q^{96} - 5 q^{97} + 34 q^{98} + 54 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
276.1 −1.29688 − 2.24626i 0.660167 + 1.14344i −2.36378 + 4.09419i −1.00000 1.71231 − 2.96581i −0.577994 + 1.00112i 7.07462 0.628358 − 1.08835i 1.29688 + 2.24626i
276.2 −1.16596 − 2.01950i −0.849640 − 1.47162i −1.71892 + 2.97726i −1.00000 −1.98129 + 3.43170i −1.89350 + 3.27963i 3.35292 0.0562230 − 0.0973812i 1.16596 + 2.01950i
276.3 −1.07492 − 1.86182i 0.540971 + 0.936989i −1.31092 + 2.27058i −1.00000 1.16300 − 2.01438i 1.37555 − 2.38252i 1.33685 0.914701 − 1.58431i 1.07492 + 1.86182i
276.4 −0.818605 − 1.41787i −1.53190 − 2.65333i −0.340229 + 0.589293i −1.00000 −2.50804 + 4.34405i 2.47765 − 4.29141i −2.16037 −3.19343 + 5.53118i 0.818605 + 1.41787i
276.5 −0.718130 − 1.24384i 1.69843 + 2.94176i −0.0314213 + 0.0544233i −1.00000 2.43938 − 4.22513i −0.0161814 + 0.0280271i −2.78226 −4.26930 + 7.39464i 0.718130 + 1.24384i
276.6 −0.239074 − 0.414089i −0.513092 − 0.888701i 0.885687 − 1.53406i −1.00000 −0.245334 + 0.424931i −1.03329 + 1.78972i −1.80328 0.973474 − 1.68611i 0.239074 + 0.414089i
276.7 −0.00698820 − 0.0121039i 0.942737 + 1.63287i 0.999902 − 1.73188i −1.00000 0.0131761 − 0.0228216i 1.24367 − 2.15410i −0.0559029 −0.277507 + 0.480656i 0.00698820 + 0.0121039i
276.8 0.270610 + 0.468710i −0.0383605 − 0.0664423i 0.853541 − 1.47838i −1.00000 0.0207614 − 0.0359599i −1.88734 + 3.26897i 2.00634 1.49706 − 2.59298i −0.270610 − 0.468710i
276.9 0.491162 + 0.850717i −1.07827 − 1.86761i 0.517520 − 0.896371i −1.00000 1.05921 − 1.83460i 1.19144 − 2.06364i 2.98139 −0.825323 + 1.42950i −0.491162 − 0.850717i
276.10 0.591230 + 1.02404i −1.68524 − 2.91891i 0.300893 − 0.521162i −1.00000 1.99272 − 3.45150i −1.85371 + 3.21072i 3.07651 −4.18003 + 7.24003i −0.591230 − 1.02404i
276.11 0.889753 + 1.54110i 0.234124 + 0.405514i −0.583321 + 1.01034i −1.00000 −0.416625 + 0.721616i 1.76184 − 3.05159i 1.48296 1.39037 − 2.40819i −0.889753 − 1.54110i
276.12 0.951047 + 1.64726i 0.907726 + 1.57223i −0.808980 + 1.40119i −1.00000 −1.72658 + 2.99053i −1.42394 + 2.46634i 0.726677 −0.147934 + 0.256230i −0.951047 − 1.64726i
276.13 1.30754 + 2.26472i 1.60797 + 2.78508i −2.41931 + 4.19037i −1.00000 −4.20496 + 7.28320i 2.49496 − 4.32140i −7.42320 −3.67112 + 6.35857i −1.30754 − 2.26472i
276.14 1.31922 + 2.28495i −1.39562 − 2.41729i −2.48067 + 4.29664i −1.00000 3.68226 − 6.37787i 0.640856 − 1.11000i −7.81327 −2.39554 + 4.14919i −1.31922 − 2.28495i
386.1 −1.29688 + 2.24626i 0.660167 − 1.14344i −2.36378 − 4.09419i −1.00000 1.71231 + 2.96581i −0.577994 − 1.00112i 7.07462 0.628358 + 1.08835i 1.29688 − 2.24626i
386.2 −1.16596 + 2.01950i −0.849640 + 1.47162i −1.71892 − 2.97726i −1.00000 −1.98129 − 3.43170i −1.89350 − 3.27963i 3.35292 0.0562230 + 0.0973812i 1.16596 − 2.01950i
386.3 −1.07492 + 1.86182i 0.540971 − 0.936989i −1.31092 − 2.27058i −1.00000 1.16300 + 2.01438i 1.37555 + 2.38252i 1.33685 0.914701 + 1.58431i 1.07492 − 1.86182i
386.4 −0.818605 + 1.41787i −1.53190 + 2.65333i −0.340229 − 0.589293i −1.00000 −2.50804 − 4.34405i 2.47765 + 4.29141i −2.16037 −3.19343 − 5.53118i 0.818605 − 1.41787i
386.5 −0.718130 + 1.24384i 1.69843 − 2.94176i −0.0314213 − 0.0544233i −1.00000 2.43938 + 4.22513i −0.0161814 − 0.0280271i −2.78226 −4.26930 − 7.39464i 0.718130 − 1.24384i
386.6 −0.239074 + 0.414089i −0.513092 + 0.888701i 0.885687 + 1.53406i −1.00000 −0.245334 − 0.424931i −1.03329 − 1.78972i −1.80328 0.973474 + 1.68611i 0.239074 − 0.414089i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 276.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 715.2.i.e ✓ 28
13.c even 3 1 inner 715.2.i.e ✓ 28
13.c even 3 1 9295.2.a.ba 14
13.e even 6 1 9295.2.a.bb 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
715.2.i.e ✓ 28 1.a even 1 1 trivial
715.2.i.e ✓ 28 13.c even 3 1 inner
9295.2.a.ba 14 13.c even 3 1
9295.2.a.bb 14 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} - T_{2}^{27} + 23 T_{2}^{26} - 20 T_{2}^{25} + 319 T_{2}^{24} - 262 T_{2}^{23} + 2851 T_{2}^{22} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(715, [\chi])\). Copy content Toggle raw display