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Properties

Label 28.9.c.a
Level $28$
Weight $9$
Character orbit 28.c
Analytic conductor $11.407$
Analytic rank $0$
Dimension $24$
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] = [28,9,Mod(15,28)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(28, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("28.15");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 28 = 2^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 28.c (of order \(2\), degree \(1\), 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: \(11.4066010817\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{2} - 383 q^{4} - 336 q^{5} + 5502 q^{6} - 12069 q^{8} - 56296 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 3 q^{2} - 383 q^{4} - 336 q^{5} + 5502 q^{6} - 12069 q^{8} - 56296 q^{9} + 41972 q^{10} - 31290 q^{12} - 28560 q^{13} + 7203 q^{14} + 166193 q^{16} - 165648 q^{17} - 11609 q^{18} - 71568 q^{20} - 955748 q^{22} + 133322 q^{24} + 989640 q^{25} - 1066632 q^{26} - 372155 q^{28} + 997296 q^{29} + 3246560 q^{30} - 2265357 q^{32} - 5521600 q^{33} + 1423618 q^{34} - 649451 q^{36} + 5387056 q^{37} + 3945690 q^{38} + 11452504 q^{40} - 12346320 q^{41} + 5642350 q^{42} - 2412372 q^{44} + 17486448 q^{45} + 537288 q^{46} + 20895070 q^{48} - 19765032 q^{49} - 28121691 q^{50} - 12804764 q^{52} + 10812720 q^{53} - 42610652 q^{54} - 7109361 q^{56} + 6748288 q^{57} + 36144266 q^{58} - 98388944 q^{60} + 33644464 q^{61} + 12180756 q^{62} - 1102631 q^{64} - 9084960 q^{65} + 83635608 q^{66} + 106943886 q^{68} - 90944448 q^{69} + 11611236 q^{70} - 48881753 q^{72} - 7651728 q^{73} - 66897318 q^{74} + 64861986 q^{76} - 47482176 q^{77} - 41838760 q^{78} + 154560168 q^{80} + 299710744 q^{81} - 13781390 q^{82} + 571438 q^{84} - 280618528 q^{85} - 139907676 q^{86} - 65086060 q^{88} + 215250672 q^{89} + 40480132 q^{90} + 111569640 q^{92} + 181272192 q^{93} + 171294732 q^{94} - 684707590 q^{96} + 114327024 q^{97} - 2470629 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
15.1 −15.6645 − 3.25957i 76.0548i 234.750 + 102.119i 501.992 247.906 − 1191.36i − 907.493i −3344.37 − 2364.82i 776.667 −7863.43 − 1636.28i
15.2 −15.6645 + 3.25957i − 76.0548i 234.750 − 102.119i 501.992 247.906 + 1191.36i 907.493i −3344.37 + 2364.82i 776.667 −7863.43 + 1636.28i
15.3 −15.6254 − 3.44201i − 125.943i 232.305 + 107.565i −1134.70 −433.497 + 1967.91i − 907.493i −3259.61 − 2480.35i −9300.62 17730.1 + 3905.64i
15.4 −15.6254 + 3.44201i 125.943i 232.305 − 107.565i −1134.70 −433.497 − 1967.91i 907.493i −3259.61 + 2480.35i −9300.62 17730.1 − 3905.64i
15.5 −13.2355 − 8.99008i 18.5217i 94.3570 + 237.976i −197.408 166.511 − 245.144i 907.493i 890.564 − 3998.01i 6217.95 2612.80 + 1774.72i
15.6 −13.2355 + 8.99008i − 18.5217i 94.3570 − 237.976i −197.408 166.511 + 245.144i − 907.493i 890.564 + 3998.01i 6217.95 2612.80 − 1774.72i
15.7 −6.31929 − 14.6992i 161.873i −176.133 + 185.777i 682.531 2379.41 − 1022.92i 907.493i 3843.81 + 1415.04i −19642.0 −4313.12 − 10032.7i
15.8 −6.31929 + 14.6992i − 161.873i −176.133 − 185.777i 682.531 2379.41 + 1022.92i − 907.493i 3843.81 − 1415.04i −19642.0 −4313.12 + 10032.7i
15.9 −6.04849 − 14.8127i 43.6557i −182.831 + 179.189i −708.025 646.659 − 264.051i − 907.493i 3760.12 + 1624.40i 4655.18 4282.48 + 10487.7i
15.10 −6.04849 + 14.8127i − 43.6557i −182.831 − 179.189i −708.025 646.659 + 264.051i 907.493i 3760.12 − 1624.40i 4655.18 4282.48 − 10487.7i
15.11 0.480763 − 15.9928i − 113.792i −255.538 − 15.3775i −421.707 −1819.85 − 54.7071i 907.493i −368.781 + 4079.36i −6387.68 −202.741 + 6744.26i
15.12 0.480763 + 15.9928i 113.792i −255.538 + 15.3775i −421.707 −1819.85 + 54.7071i − 907.493i −368.781 − 4079.36i −6387.68 −202.741 − 6744.26i
15.13 2.44338 − 15.8123i 4.01293i −244.060 − 77.2712i 1160.46 63.4538 + 9.80514i − 907.493i −1818.17 + 3670.35i 6544.90 2835.44 − 18349.5i
15.14 2.44338 + 15.8123i − 4.01293i −244.060 + 77.2712i 1160.46 63.4538 − 9.80514i 907.493i −1818.17 − 3670.35i 6544.90 2835.44 + 18349.5i
15.15 4.26375 − 15.4214i 36.4689i −219.641 − 131.506i −275.039 562.402 + 155.494i 907.493i −2964.51 + 2826.46i 5231.02 −1172.70 + 4241.49i
15.16 4.26375 + 15.4214i − 36.4689i −219.641 + 131.506i −275.039 562.402 − 155.494i − 907.493i −2964.51 − 2826.46i 5231.02 −1172.70 − 4241.49i
15.17 10.1564 − 12.3632i 137.158i −49.6953 − 251.130i −657.296 1695.70 + 1393.03i − 907.493i −3609.49 − 1936.19i −12251.2 −6675.76 + 8126.25i
15.18 10.1564 + 12.3632i − 137.158i −49.6953 + 251.130i −657.296 1695.70 − 1393.03i 907.493i −3609.49 + 1936.19i −12251.2 −6675.76 − 8126.25i
15.19 11.5309 − 11.0923i − 94.2293i 9.92223 − 255.808i −179.424 −1045.22 − 1086.55i − 907.493i −2723.08 − 3059.75i −2318.16 −2068.91 + 1990.22i
15.20 11.5309 + 11.0923i 94.2293i 9.92223 + 255.808i −179.424 −1045.22 + 1086.55i 907.493i −2723.08 + 3059.75i −2318.16 −2068.91 − 1990.22i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 15.24
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 28.9.c.a ✓ 24
4.b odd 2 1 inner 28.9.c.a ✓ 24
8.b even 2 1 448.9.d.d 24
8.d odd 2 1 448.9.d.d 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.9.c.a ✓ 24 1.a even 1 1 trivial
28.9.c.a ✓ 24 4.b odd 2 1 inner
448.9.d.d 24 8.b even 2 1
448.9.d.d 24 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(28, [\chi])\).