Properties

Label 90.5.h.a
Level $90$
Weight $5$
Character orbit 90.h
Analytic conductor $9.303$
Analytic rank $0$
Dimension $32$
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] = [90,5,Mod(11,90)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(90, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("90.11");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 90.h (of order \(6\), 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: \(9.30329667755\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{3} + 128 q^{4} - 32 q^{6} + 52 q^{7} + 176 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{3} + 128 q^{4} - 32 q^{6} + 52 q^{7} + 176 q^{9} + 972 q^{11} + 128 q^{12} - 20 q^{13} - 288 q^{14} + 100 q^{15} - 1024 q^{16} + 1408 q^{18} - 200 q^{19} + 1168 q^{21} - 672 q^{22} - 3996 q^{23} - 512 q^{24} + 2000 q^{25} + 1172 q^{27} + 832 q^{28} + 4500 q^{29} + 800 q^{30} + 736 q^{31} + 5628 q^{33} + 192 q^{34} - 160 q^{36} - 4136 q^{37} - 4896 q^{38} - 1184 q^{39} + 864 q^{41} - 1984 q^{42} + 136 q^{43} + 2200 q^{45} - 2112 q^{46} - 12528 q^{47} + 512 q^{48} - 11148 q^{49} + 10620 q^{51} + 160 q^{52} - 9152 q^{54} - 2304 q^{56} - 3704 q^{57} - 16668 q^{59} - 800 q^{60} + 6172 q^{61} + 18328 q^{63} - 16384 q^{64} + 17100 q^{65} + 36288 q^{66} - 20468 q^{67} + 576 q^{68} + 16500 q^{69} - 2400 q^{70} - 512 q^{72} - 41432 q^{73} - 14400 q^{74} - 1000 q^{75} - 800 q^{76} - 21348 q^{77} - 4672 q^{78} + 7024 q^{79} + 20192 q^{81} - 17664 q^{82} + 54468 q^{83} + 18976 q^{84} + 8700 q^{85} + 37152 q^{86} - 37980 q^{87} + 5376 q^{88} + 12800 q^{90} + 69392 q^{91} - 31968 q^{92} - 52748 q^{93} - 672 q^{94} - 12600 q^{95} - 2048 q^{96} - 18332 q^{97} - 24720 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −2.44949 1.41421i −8.23651 3.62767i 4.00000 + 6.92820i 9.68246 5.59017i 15.0449 + 20.5341i 30.9376 53.5855i 22.6274i 54.6801 + 59.7586i −31.6228
11.2 −2.44949 1.41421i −7.00963 5.64492i 4.00000 + 6.92820i −9.68246 + 5.59017i 9.18690 + 23.7403i −10.8913 + 18.8643i 22.6274i 17.2698 + 79.1376i 31.6228
11.3 −2.44949 1.41421i −1.37920 + 8.89369i 4.00000 + 6.92820i −9.68246 + 5.59017i 15.9559 19.8345i −23.8583 + 41.3239i 22.6274i −77.1956 24.5324i 31.6228
11.4 −2.44949 1.41421i −1.32770 + 8.90153i 4.00000 + 6.92820i 9.68246 5.59017i 15.8409 19.9266i 27.8508 48.2391i 22.6274i −77.4744 23.6371i −31.6228
11.5 −2.44949 1.41421i −0.611821 8.97918i 4.00000 + 6.92820i 9.68246 5.59017i −11.1998 + 22.8597i −45.2607 + 78.3939i 22.6274i −80.2513 + 10.9873i −31.6228
11.6 −2.44949 1.41421i 4.21210 7.95350i 4.00000 + 6.92820i −9.68246 + 5.59017i −21.5655 + 13.5252i 12.4476 21.5599i 22.6274i −45.5164 67.0020i 31.6228
11.7 −2.44949 1.41421i 8.01375 4.09632i 4.00000 + 6.92820i 9.68246 5.59017i −25.4227 1.29927i 9.80759 16.9872i 22.6274i 47.4404 65.6537i −31.6228
11.8 −2.44949 1.41421i 8.33901 + 3.38540i 4.00000 + 6.92820i −9.68246 + 5.59017i −15.6386 20.0856i 26.6637 46.1828i 22.6274i 58.0781 + 56.4618i 31.6228
11.9 2.44949 + 1.41421i −8.95263 + 0.922181i 4.00000 + 6.92820i −9.68246 + 5.59017i −23.2335 10.4021i 4.22228 7.31320i 22.6274i 79.2992 16.5119i −31.6228
11.10 2.44949 + 1.41421i −8.93009 1.11963i 4.00000 + 6.92820i 9.68246 5.59017i −20.2908 15.3716i 13.1727 22.8158i 22.6274i 78.4929 + 19.9968i 31.6228
11.11 2.44949 + 1.41421i −4.63058 + 7.71736i 4.00000 + 6.92820i 9.68246 5.59017i −22.2566 + 12.3550i −36.2886 + 62.8537i 22.6274i −38.1154 71.4718i 31.6228
11.12 2.44949 + 1.41421i −4.34936 7.87928i 4.00000 + 6.92820i −9.68246 + 5.59017i 0.489266 25.4511i −24.5976 + 42.6042i 22.6274i −43.1661 + 68.5397i −31.6228
11.13 2.44949 + 1.41421i 5.35414 + 7.23417i 4.00000 + 6.92820i −9.68246 + 5.59017i 2.88424 + 25.2919i −10.7341 + 18.5921i 22.6274i −23.6664 + 77.4655i −31.6228
11.14 2.44949 + 1.41421i 5.78558 6.89399i 4.00000 + 6.92820i −9.68246 + 5.59017i 23.9213 8.70471i 39.7478 68.8452i 22.6274i −14.0542 79.7714i −31.6228
11.15 2.44949 + 1.41421i 8.72295 + 2.21590i 4.00000 + 6.92820i 9.68246 5.59017i 18.2330 + 17.7639i 45.5675 78.9253i 22.6274i 71.1795 + 38.6584i 31.6228
11.16 2.44949 + 1.41421i 9.00000 0.00397022i 4.00000 + 6.92820i 9.68246 5.59017i 22.0510 + 12.7182i −32.7869 + 56.7886i 22.6274i 81.0000 0.0714639i 31.6228
41.1 −2.44949 + 1.41421i −8.23651 + 3.62767i 4.00000 6.92820i 9.68246 + 5.59017i 15.0449 20.5341i 30.9376 + 53.5855i 22.6274i 54.6801 59.7586i −31.6228
41.2 −2.44949 + 1.41421i −7.00963 + 5.64492i 4.00000 6.92820i −9.68246 5.59017i 9.18690 23.7403i −10.8913 18.8643i 22.6274i 17.2698 79.1376i 31.6228
41.3 −2.44949 + 1.41421i −1.37920 8.89369i 4.00000 6.92820i −9.68246 5.59017i 15.9559 + 19.8345i −23.8583 41.3239i 22.6274i −77.1956 + 24.5324i 31.6228
41.4 −2.44949 + 1.41421i −1.32770 8.90153i 4.00000 6.92820i 9.68246 + 5.59017i 15.8409 + 19.9266i 27.8508 + 48.2391i 22.6274i −77.4744 + 23.6371i −31.6228
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.5.h.a 32
3.b odd 2 1 270.5.h.a 32
9.c even 3 1 270.5.h.a 32
9.c even 3 1 810.5.d.c 32
9.d odd 6 1 inner 90.5.h.a 32
9.d odd 6 1 810.5.d.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.5.h.a 32 1.a even 1 1 trivial
90.5.h.a 32 9.d odd 6 1 inner
270.5.h.a 32 3.b odd 2 1
270.5.h.a 32 9.c even 3 1
810.5.d.c 32 9.c even 3 1
810.5.d.c 32 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(90, [\chi])\).