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Properties

Label 84.9.g.a
Level $84$
Weight $9$
Character orbit 84.g
Analytic conductor $34.220$
Analytic rank $0$
Dimension $48$
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] = [84,9,Mod(43,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.43");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 84.g (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: \(34.2198032451\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{2} + 290 q^{4} + 672 q^{5} - 2268 q^{6} + 4230 q^{8} - 104976 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 6 q^{2} + 290 q^{4} + 672 q^{5} - 2268 q^{6} + 4230 q^{8} - 104976 q^{9} - 41972 q^{10} + 22680 q^{12} + 57120 q^{13} - 14406 q^{14} - 207950 q^{16} + 386400 q^{17} + 13122 q^{18} + 562380 q^{20} + 1186136 q^{22} + 74844 q^{24} + 5520720 q^{25} + 1682100 q^{26} + 148862 q^{28} - 1994592 q^{29} - 2054808 q^{30} + 7754814 q^{32} + 2395820 q^{34} - 634230 q^{36} - 5931232 q^{37} - 4284000 q^{38} - 16076956 q^{40} + 10936800 q^{41} + 12531912 q^{44} - 1469664 q^{45} + 9456072 q^{46} - 12365136 q^{48} - 39530064 q^{49} - 6199074 q^{50} - 6937420 q^{52} - 7150560 q^{53} + 4960116 q^{54} + 14218722 q^{56} + 518452 q^{58} + 3633984 q^{60} - 42829024 q^{61} - 93291072 q^{62} - 69128614 q^{64} + 21653184 q^{65} + 45582264 q^{66} + 82618620 q^{68} + 17273088 q^{69} - 11611236 q^{70} - 9251010 q^{72} - 49489440 q^{73} - 126365628 q^{74} - 223942320 q^{76} + 94964352 q^{77} + 136267920 q^{78} + 238226604 q^{80} + 229582512 q^{81} + 269696252 q^{82} + 301906816 q^{85} - 188004816 q^{86} - 116561288 q^{88} - 315415968 q^{89} + 91792764 q^{90} + 487386984 q^{92} + 37454400 q^{93} + 148324176 q^{94} - 320488812 q^{96} - 3460128 q^{97} + 4941258 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1 −15.9320 − 1.47308i − 46.7654i 251.660 + 46.9385i 959.139 −68.8894 + 745.068i − 907.493i −3940.31 − 1118.54i −2187.00 −15281.0 − 1412.89i
43.2 −15.9320 + 1.47308i 46.7654i 251.660 − 46.9385i 959.139 −68.8894 − 745.068i 907.493i −3940.31 + 1118.54i −2187.00 −15281.0 + 1412.89i
43.3 −15.1205 − 5.23171i 46.7654i 201.258 + 158.212i 237.168 244.663 − 707.115i 907.493i −2215.40 − 3445.17i −2187.00 −3586.10 − 1240.80i
43.4 −15.1205 + 5.23171i − 46.7654i 201.258 − 158.212i 237.168 244.663 + 707.115i − 907.493i −2215.40 + 3445.17i −2187.00 −3586.10 + 1240.80i
43.5 −14.8262 − 6.01528i − 46.7654i 183.633 + 178.368i −242.776 −281.307 + 693.353i 907.493i −1649.65 − 3749.12i −2187.00 3599.45 + 1460.37i
43.6 −14.8262 + 6.01528i 46.7654i 183.633 − 178.368i −242.776 −281.307 − 693.353i − 907.493i −1649.65 + 3749.12i −2187.00 3599.45 − 1460.37i
43.7 −13.7684 − 8.15059i − 46.7654i 123.136 + 224.441i 1187.65 −381.165 + 643.883i 907.493i 133.944 − 4093.81i −2187.00 −16352.1 − 9680.09i
43.8 −13.7684 + 8.15059i 46.7654i 123.136 − 224.441i 1187.65 −381.165 − 643.883i − 907.493i 133.944 + 4093.81i −2187.00 −16352.1 + 9680.09i
43.9 −13.3322 − 8.84604i − 46.7654i 99.4951 + 235.874i −172.633 −413.688 + 623.485i − 907.493i 760.065 − 4024.86i −2187.00 2301.58 + 1527.12i
43.10 −13.3322 + 8.84604i 46.7654i 99.4951 − 235.874i −172.633 −413.688 − 623.485i 907.493i 760.065 + 4024.86i −2187.00 2301.58 − 1527.12i
43.11 −11.8878 − 10.7089i 46.7654i 26.6398 + 254.610i −971.506 500.805 − 555.938i − 907.493i 2409.90 − 3312.04i −2187.00 11549.1 + 10403.7i
43.12 −11.8878 + 10.7089i − 46.7654i 26.6398 − 254.610i −971.506 500.805 + 555.938i 907.493i 2409.90 + 3312.04i −2187.00 11549.1 − 10403.7i
43.13 −11.4743 − 11.1509i 46.7654i 7.31688 + 255.895i 743.303 521.474 − 536.598i − 907.493i 2769.50 − 3017.80i −2187.00 −8528.84 − 8288.46i
43.14 −11.4743 + 11.1509i − 46.7654i 7.31688 − 255.895i 743.303 521.474 + 536.598i 907.493i 2769.50 + 3017.80i −2187.00 −8528.84 + 8288.46i
43.15 −11.3279 − 11.2995i 46.7654i 0.641301 + 255.999i −681.653 528.427 − 529.752i 907.493i 2885.41 − 2907.17i −2187.00 7721.67 + 7702.35i
43.16 −11.3279 + 11.2995i − 46.7654i 0.641301 − 255.999i −681.653 528.427 + 529.752i − 907.493i 2885.41 + 2907.17i −2187.00 7721.67 − 7702.35i
43.17 −7.95920 − 13.8799i − 46.7654i −129.302 + 220.946i −823.528 −649.098 + 372.215i 907.493i 4095.84 + 36.1503i −2187.00 6554.62 + 11430.5i
43.18 −7.95920 + 13.8799i 46.7654i −129.302 − 220.946i −823.528 −649.098 − 372.215i − 907.493i 4095.84 − 36.1503i −2187.00 6554.62 − 11430.5i
43.19 −7.12094 − 14.3280i − 46.7654i −154.584 + 204.058i 99.4009 −670.055 + 333.013i − 907.493i 4024.53 + 761.804i −2187.00 −707.828 − 1424.22i
43.20 −7.12094 + 14.3280i 46.7654i −154.584 − 204.058i 99.4009 −670.055 − 333.013i 907.493i 4024.53 − 761.804i −2187.00 −707.828 + 1424.22i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.48
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 84.9.g.a ✓ 48
4.b odd 2 1 inner 84.9.g.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.9.g.a ✓ 48 1.a even 1 1 trivial
84.9.g.a ✓ 48 4.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(84, [\chi])\).