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Properties

Label 145.2.n.a
Level $145$
Weight $2$
Character orbit 145.n
Analytic conductor $1.158$
Analytic rank $0$
Dimension $84$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [145,2,Mod(24,145)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(145, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("145.24");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 145 = 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 145.n (of order \(14\), degree \(6\), 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: \(1.15783082931\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(14\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84 q + 4 q^{4} - 7 q^{5} - 32 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q + 4 q^{4} - 7 q^{5} - 32 q^{6} - 4 q^{9} - 9 q^{10} - 6 q^{11} - 22 q^{14} - 19 q^{15} - 32 q^{16} - 2 q^{19} - 25 q^{20} - 42 q^{21} + 84 q^{24} + 33 q^{25} - 38 q^{26} - 6 q^{29} - 6 q^{30} - 20 q^{31} + 42 q^{34} + 27 q^{35} - 4 q^{36} - 36 q^{39} - 83 q^{40} - 32 q^{41} + 42 q^{44} - 16 q^{45} + 12 q^{46} + 50 q^{49} + 56 q^{50} + 4 q^{51} + 6 q^{54} + 75 q^{55} - 86 q^{56} - 156 q^{59} + 142 q^{60} + 22 q^{61} + 12 q^{64} - 18 q^{65} - 102 q^{66} + 10 q^{69} + 22 q^{70} + 12 q^{71} + 38 q^{74} - 78 q^{75} + 118 q^{76} + 26 q^{79} - 146 q^{80} + 74 q^{81} - 82 q^{84} + 34 q^{85} + 288 q^{86} - 26 q^{89} - 32 q^{90} - 28 q^{91} - 90 q^{94} - 3 q^{95} + 266 q^{96} + 152 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
24.1 −2.04092 + 1.62758i 2.56463 − 0.585361i 1.07130 − 4.69368i −1.89560 + 1.18604i −4.28149 + 5.36882i 2.05651 − 0.469385i 3.18765 + 6.61922i 3.53179 − 1.70082i 1.93840 − 5.50586i
24.2 −1.84695 + 1.47289i −1.13127 + 0.258204i 0.796768 − 3.49087i 2.15047 − 0.612769i 1.70908 − 2.14312i 3.09934 − 0.707404i 1.62013 + 3.36423i −1.48981 + 0.717456i −3.06927 + 4.29916i
24.3 −1.64816 + 1.31436i 0.294280 − 0.0671676i 0.543838 − 2.38271i −0.651398 − 2.13908i −0.396739 + 0.497495i −4.74780 + 1.08365i 0.406096 + 0.843267i −2.62082 + 1.26212i 3.88514 + 2.66938i
24.4 −1.13529 + 0.905360i −0.671128 + 0.153181i 0.0241547 − 0.105828i −1.17325 + 1.90354i 0.623238 − 0.781516i 1.94424 − 0.443761i −1.19168 − 2.47455i −2.27596 + 1.09604i −0.391416 − 3.22328i
24.5 −0.829215 + 0.661277i 2.14210 − 0.488921i −0.194731 + 0.853174i 1.56923 + 1.59296i −1.45295 + 1.82194i −1.60527 + 0.366393i −1.32307 − 2.74738i 1.64665 − 0.792984i −2.35462 − 0.283215i
24.6 −0.523263 + 0.417288i −3.18534 + 0.727033i −0.345367 + 1.51315i −0.222530 − 2.22497i 1.36339 − 1.70963i 1.32294 − 0.301953i −1.03148 − 2.14189i 6.91491 − 3.33004i 1.04489 + 1.07138i
24.7 −0.145382 + 0.115939i 1.00475 − 0.229327i −0.437348 + 1.91614i 0.776499 − 2.09691i −0.119484 + 0.149829i 2.55822 − 0.583896i −0.319935 − 0.664351i −1.74598 + 0.840822i 0.130224 + 0.394881i
24.8 0.145382 − 0.115939i −1.00475 + 0.229327i −0.437348 + 1.91614i −2.21713 + 0.290423i −0.119484 + 0.149829i −2.55822 + 0.583896i 0.319935 + 0.664351i −1.74598 + 0.840822i −0.288660 + 0.299273i
24.9 0.523263 − 0.417288i 3.18534 − 0.727033i −0.345367 + 1.51315i −2.11967 − 0.712053i 1.36339 − 1.70963i −1.32294 + 0.301953i 1.03148 + 2.14189i 6.91491 − 3.33004i −1.40627 + 0.511920i
24.10 0.829215 − 0.661277i −2.14210 + 0.488921i −0.194731 + 0.853174i 1.20384 + 1.88435i −1.45295 + 1.82194i 1.60527 − 0.366393i 1.32307 + 2.74738i 1.64665 − 0.792984i 2.24432 + 0.766461i
24.11 1.13529 − 0.905360i 0.671128 − 0.153181i 0.0241547 − 0.105828i 2.11689 − 0.720258i 0.623238 − 0.781516i −1.94424 + 0.443761i 1.19168 + 2.47455i −2.27596 + 1.09604i 1.75118 − 2.73425i
24.12 1.64816 − 1.31436i −0.294280 + 0.0671676i 0.543838 − 2.38271i −1.94050 − 1.11106i −0.396739 + 0.497495i 4.74780 − 1.08365i −0.406096 − 0.843267i −2.62082 + 1.26212i −4.65859 + 0.719326i
24.13 1.84695 − 1.47289i 1.13127 − 0.258204i 0.796768 − 3.49087i −1.07593 + 1.96020i 1.70908 − 2.14312i −3.09934 + 0.707404i −1.62013 − 3.36423i −1.48981 + 0.717456i 0.899975 + 5.20512i
24.14 2.04092 − 1.62758i −2.56463 + 0.585361i 1.07130 − 4.69368i 1.57812 − 1.58416i −4.28149 + 5.36882i −2.05651 + 0.469385i −3.18765 − 6.61922i 3.53179 − 1.70082i 0.642471 − 5.80165i
49.1 −1.10463 − 2.29379i 2.31121 − 1.84313i −2.79427 + 3.50391i 1.75990 − 1.37941i −6.78077 − 3.26545i −2.08897 + 1.66590i 6.15969 + 1.40591i 1.27700 − 5.59492i −5.10810 − 2.51309i
49.2 −1.04951 − 2.17932i −1.68851 + 1.34655i −2.40100 + 3.01076i 1.93738 + 1.11650i 4.70666 + 2.26661i 1.96329 − 1.56567i 4.36483 + 0.996243i 0.370334 − 1.62254i 0.399931 − 5.39394i
49.3 −1.02776 − 2.13417i −0.235394 + 0.187720i −2.25142 + 2.82319i −2.20400 + 0.377329i 0.642556 + 0.309439i −2.13630 + 1.70365i 3.72038 + 0.849153i −0.647392 + 2.83641i 3.07048 + 4.31592i
49.4 −0.696602 − 1.44651i 1.76606 − 1.40839i −0.360153 + 0.451618i −0.897062 + 2.04824i −3.26749 − 1.57354i 3.55346 − 2.83379i −2.22635 − 0.508149i 0.467858 − 2.04982i 3.58769 − 0.129198i
49.5 −0.380152 − 0.789393i 0.364055 − 0.290324i 0.768353 − 0.963484i 2.19264 + 0.438567i −0.367576 − 0.177015i −0.279069 + 0.222550i −2.76105 − 0.630191i −0.619315 + 2.71340i −0.487333 − 1.89758i
49.6 −0.301823 − 0.626741i −2.11369 + 1.68561i 0.945272 − 1.18533i −2.13615 − 0.660949i 1.69440 + 0.815981i 3.58861 − 2.86182i −2.38458 − 0.544265i 0.958837 − 4.20094i 0.230495 + 1.53830i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 24.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
29.d even 7 1 inner
145.n even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 145.2.n.a ✓ 84
5.b even 2 1 inner 145.2.n.a ✓ 84
5.c odd 4 2 725.2.l.h 84
29.d even 7 1 inner 145.2.n.a ✓ 84
145.n even 14 1 inner 145.2.n.a ✓ 84
145.p odd 28 2 725.2.l.h 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.n.a ✓ 84 1.a even 1 1 trivial
145.2.n.a ✓ 84 5.b even 2 1 inner
145.2.n.a ✓ 84 29.d even 7 1 inner
145.2.n.a ✓ 84 145.n even 14 1 inner
725.2.l.h 84 5.c odd 4 2
725.2.l.h 84 145.p odd 28 2

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(145, [\chi])\).