↑

Properties

Label 90.12.e.c
Level $90$
Weight $12$
Character orbit 90.e
Analytic conductor $69.151$
Analytic rank $0$
Dimension $22$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [90,12,Mod(31,90)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("90.31"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 90.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,352] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(69.1508862504\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 352 q^{2} + 141 q^{3} - 11264 q^{4} - 34375 q^{5} + 26976 q^{6} - 21316 q^{7} - 720896 q^{8} - 222459 q^{9} - 2200000 q^{10} + 713391 q^{11} + 718848 q^{12} - 355192 q^{13} + 682112 q^{14} + 2193750 q^{15}+ \cdots + 281036208384 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 16.0000 + 27.7128i −420.183 − 24.3513i −512.000 + 886.810i −1562.50 + 2706.33i −6048.09 − 12034.1i −14232.2 − 24650.9i −32768.0 175961. + 20464.0i −100000.
31.2 16.0000 + 27.7128i −304.393 + 290.675i −512.000 + 886.810i −1562.50 + 2706.33i −12925.7 − 3784.79i 29048.3 + 50313.1i −32768.0 8163.25 − 176959.i −100000.
31.3 16.0000 + 27.7128i −299.147 − 296.071i −512.000 + 886.810i −1562.50 + 2706.33i 3418.60 − 13027.3i 26113.8 + 45230.5i −32768.0 1831.09 + 177138.i −100000.
31.4 16.0000 + 27.7128i −243.513 − 343.290i −512.000 + 886.810i −1562.50 + 2706.33i 5617.33 − 12241.1i −37557.5 − 65051.5i −32768.0 −58549.6 + 167192.i −100000.
31.5 16.0000 + 27.7128i −83.2483 + 412.573i −512.000 + 886.810i −1562.50 + 2706.33i −12765.5 + 4294.13i −23853.8 − 41316.0i −32768.0 −163286. − 68692.1i −100000.
31.6 16.0000 + 27.7128i 0.0909322 + 420.888i −512.000 + 886.810i −1562.50 + 2706.33i −11662.5 + 6736.73i −415.494 − 719.656i −32768.0 −177147. + 76.5446i −100000.
31.7 16.0000 + 27.7128i 61.2263 − 416.411i −512.000 + 886.810i −1562.50 + 2706.33i 12519.5 − 4965.83i 10265.4 + 17780.2i −32768.0 −169650. − 50990.7i −100000.
31.8 16.0000 + 27.7128i 198.579 − 371.098i −512.000 + 886.810i −1562.50 + 2706.33i 13461.4 − 434.378i 643.507 + 1114.59i −32768.0 −98279.8 − 147384.i −100000.
31.9 16.0000 + 27.7128i 364.224 − 210.922i −512.000 + 886.810i −1562.50 + 2706.33i 11672.8 + 6718.93i −34208.5 − 59250.9i −32768.0 88171.2 − 153645.i −100000.
31.10 16.0000 + 27.7128i 383.797 + 172.762i −512.000 + 886.810i −1562.50 + 2706.33i 1353.02 + 13400.3i 22839.5 + 39559.2i −32768.0 117453. + 132611.i −100000.
31.11 16.0000 + 27.7128i 413.068 − 80.7590i −512.000 + 886.810i −1562.50 + 2706.33i 8847.14 + 10155.1i 10699.0 + 18531.2i −32768.0 164103. − 66717.9i −100000.
61.1 16.0000 − 27.7128i −420.183 + 24.3513i −512.000 − 886.810i −1562.50 − 2706.33i −6048.09 + 12034.1i −14232.2 + 24650.9i −32768.0 175961. − 20464.0i −100000.
61.2 16.0000 − 27.7128i −304.393 − 290.675i −512.000 − 886.810i −1562.50 − 2706.33i −12925.7 + 3784.79i 29048.3 − 50313.1i −32768.0 8163.25 + 176959.i −100000.
61.3 16.0000 − 27.7128i −299.147 + 296.071i −512.000 − 886.810i −1562.50 − 2706.33i 3418.60 + 13027.3i 26113.8 − 45230.5i −32768.0 1831.09 − 177138.i −100000.
61.4 16.0000 − 27.7128i −243.513 + 343.290i −512.000 − 886.810i −1562.50 − 2706.33i 5617.33 + 12241.1i −37557.5 + 65051.5i −32768.0 −58549.6 − 167192.i −100000.
61.5 16.0000 − 27.7128i −83.2483 − 412.573i −512.000 − 886.810i −1562.50 − 2706.33i −12765.5 − 4294.13i −23853.8 + 41316.0i −32768.0 −163286. + 68692.1i −100000.
61.6 16.0000 − 27.7128i 0.0909322 − 420.888i −512.000 − 886.810i −1562.50 − 2706.33i −11662.5 − 6736.73i −415.494 + 719.656i −32768.0 −177147. − 76.5446i −100000.
61.7 16.0000 − 27.7128i 61.2263 + 416.411i −512.000 − 886.810i −1562.50 − 2706.33i 12519.5 + 4965.83i 10265.4 − 17780.2i −32768.0 −169650. + 50990.7i −100000.
61.8 16.0000 − 27.7128i 198.579 + 371.098i −512.000 − 886.810i −1562.50 − 2706.33i 13461.4 + 434.378i 643.507 − 1114.59i −32768.0 −98279.8 + 147384.i −100000.
61.9 16.0000 − 27.7128i 364.224 + 210.922i −512.000 − 886.810i −1562.50 − 2706.33i 11672.8 − 6718.93i −34208.5 + 59250.9i −32768.0 88171.2 + 153645.i −100000.
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.11
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.12.e.c ✓ 22
3.b odd 2 1 270.12.e.b 22
9.c even 3 1 inner 90.12.e.c ✓ 22
9.d odd 6 1 270.12.e.b 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.12.e.c ✓ 22 1.a even 1 1 trivial
90.12.e.c ✓ 22 9.c even 3 1 inner
270.12.e.b 22 3.b odd 2 1
270.12.e.b 22 9.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{22} + 21316 T_{7}^{21} + 11467509519 T_{7}^{20} - 41054299704104 T_{7}^{19} + \cdots + 20\!\cdots\!36 \) acting on \(S_{12}^{\mathrm{new}}(90, [\chi])\). Copy content Toggle raw display