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Properties

Label 45.10.e.a
Level $45$
Weight $10$
Character orbit 45.e
Analytic conductor $23.177$
Analytic rank $0$
Dimension $34$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [45,10,Mod(16,45)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(45, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("45.16");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 45.e (of order \(3\), 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: \(23.1766126274\)
Analytic rank: \(0\)
Dimension: \(34\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 34 q - 16 q^{2} - 285 q^{3} - 3840 q^{4} + 10625 q^{5} + 1635 q^{6} + 9946 q^{7} + 38772 q^{8} + 7959 q^{9} - 20000 q^{10} - 32881 q^{11} + 595719 q^{12} + 100272 q^{13} - 333129 q^{14} - 157500 q^{15}+ \cdots - 3426367056 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −20.9424 + 36.2733i −70.5260 + 121.281i −621.169 − 1075.90i 312.500 + 541.266i −2922.28 − 5098.13i 1139.06 − 1972.90i 30590.0 −9735.16 − 17106.9i −26178.0
16.2 −18.6441 + 32.2925i −106.488 − 91.3412i −439.206 − 760.727i 312.500 + 541.266i 4935.02 − 1735.80i −5448.66 + 9437.36i 13662.8 2996.55 + 19453.6i −23305.1
16.3 −18.2909 + 31.6808i 111.356 + 85.3390i −413.115 − 715.537i 312.500 + 541.266i −4740.42 + 1966.93i −40.7021 + 70.4981i 11495.1 5117.53 + 19006.1i −22863.6
16.4 −16.0901 + 27.8688i 111.444 − 85.2244i −261.781 − 453.418i 312.500 + 541.266i 581.961 + 4477.08i 1094.17 − 1895.16i 372.064 5156.59 − 18995.5i −20112.6
16.5 −11.7369 + 20.3289i −136.620 + 31.9052i −19.5089 − 33.7904i 312.500 + 541.266i 954.898 − 3151.80i −1177.74 + 2039.90i −11102.7 17647.1 − 8717.78i −14671.1
16.6 −9.23989 + 16.0040i −91.4361 − 106.407i 85.2489 + 147.655i 312.500 + 541.266i 2547.79 − 480.152i 5895.83 − 10211.9i −12612.4 −2961.87 + 19458.9i −11549.9
16.7 −6.24767 + 10.8213i 29.0629 + 137.253i 177.933 + 308.190i 312.500 + 541.266i −1666.83 − 543.012i −634.215 + 1098.49i −10844.3 −17993.7 + 7977.93i −7809.58
16.8 −4.48496 + 7.76817i 31.8778 − 136.627i 215.770 + 373.725i 312.500 + 541.266i 918.368 + 860.396i −2964.42 + 5134.52i −8463.48 −17650.6 − 8710.70i −5606.20
16.9 −1.56935 + 2.71819i 139.810 + 11.6685i 251.074 + 434.873i 312.500 + 541.266i −251.128 + 361.719i 5780.42 − 10012.0i −3183.11 19410.7 + 3262.73i −1961.69
16.10 4.56079 − 7.89952i −123.844 + 65.9219i 214.398 + 371.349i 312.500 + 541.266i −44.0741 + 1278.96i 3087.01 − 5346.86i 8581.55 10991.6 − 16328.1i 5700.99
16.11 5.53551 − 9.58779i 136.385 − 32.8940i 194.716 + 337.258i 312.500 + 541.266i 439.582 − 1489.72i −2339.14 + 4051.50i 9979.78 17519.0 − 8972.53i 6919.39
16.12 9.49113 − 16.4391i −50.4183 + 130.924i 75.8370 + 131.354i 312.500 + 541.266i 1673.74 + 2071.44i −5510.81 + 9545.00i 12598.0 −14599.0 − 13201.9i 11863.9
16.13 10.9609 − 18.9848i −90.8749 − 106.887i 15.7180 + 27.2244i 312.500 + 541.266i −3025.29 + 553.672i −784.227 + 1358.32i 11913.1 −3166.49 + 19426.6i 13701.1
16.14 13.2029 − 22.8680i 21.4862 − 138.641i −92.6316 − 160.443i 312.500 + 541.266i −2886.77 − 2321.81i 4067.52 − 7045.16i 8627.73 −18759.7 − 5957.73i 16503.6
16.15 16.1149 − 27.9119i 120.125 + 72.4778i −263.383 − 456.193i 312.500 + 541.266i 3958.80 − 2184.94i −761.321 + 1318.65i −475.920 9176.94 + 17412.8i 20143.7
16.16 19.1635 − 33.1922i −136.859 − 30.8636i −478.483 − 828.757i 312.500 + 541.266i −3647.14 + 3951.21i −984.642 + 1705.45i −17054.3 17777.9 + 8447.94i 23954.4
16.17 20.2166 − 35.0161i −36.9808 + 135.334i −561.419 − 972.406i 312.500 + 541.266i 3991.26 + 4030.92i 4554.85 − 7889.24i −24698.1 −16947.8 − 10009.6i 25270.7
31.1 −20.9424 − 36.2733i −70.5260 − 121.281i −621.169 + 1075.90i 312.500 − 541.266i −2922.28 + 5098.13i 1139.06 + 1972.90i 30590.0 −9735.16 + 17106.9i −26178.0
31.2 −18.6441 − 32.2925i −106.488 + 91.3412i −439.206 + 760.727i 312.500 − 541.266i 4935.02 + 1735.80i −5448.66 − 9437.36i 13662.8 2996.55 − 19453.6i −23305.1
31.3 −18.2909 − 31.6808i 111.356 − 85.3390i −413.115 + 715.537i 312.500 − 541.266i −4740.42 − 1966.93i −40.7021 − 70.4981i 11495.1 5117.53 − 19006.1i −22863.6
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.17
Significant digits:
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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 45.10.e.a ✓ 34
3.b odd 2 1 135.10.e.a 34
9.c even 3 1 inner 45.10.e.a ✓ 34
9.d odd 6 1 135.10.e.a 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.10.e.a ✓ 34 1.a even 1 1 trivial
45.10.e.a ✓ 34 9.c even 3 1 inner
135.10.e.a 34 3.b odd 2 1
135.10.e.a 34 9.d odd 6 1