Properties

Label 20.12.e.b
Level $20$
Weight $12$
Character orbit 20.e
Analytic conductor $15.367$
Analytic rank $0$
Dimension $60$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [20,12,Mod(3,20)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(20, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.3");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 20.e (of order \(4\), 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: \(15.3668636112\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(30\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 66 q^{2} + 5280 q^{5} - 22360 q^{6} + 205908 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - 66 q^{2} + 5280 q^{5} - 22360 q^{6} + 205908 q^{8} + 724790 q^{10} - 822800 q^{12} + 3369956 q^{13} - 1562440 q^{16} + 18698308 q^{17} - 3837102 q^{18} + 45575420 q^{20} + 25132080 q^{21} + 11921800 q^{22} - 15719900 q^{25} + 337854580 q^{26} - 111299120 q^{28} - 488561080 q^{30} + 1162636424 q^{32} + 357391760 q^{33} - 2647348100 q^{36} - 1194640372 q^{37} + 1225929280 q^{38} + 1982902060 q^{40} - 2839065680 q^{41} - 4639041880 q^{42} - 1316578100 q^{45} + 7692828040 q^{46} - 8859314720 q^{48} - 6461950150 q^{50} + 24561645068 q^{52} + 6757133836 q^{53} - 23336652160 q^{56} - 1062279360 q^{57} + 28094273496 q^{58} + 17596714000 q^{60} - 9850089280 q^{61} - 15717288920 q^{62} + 34535629820 q^{65} - 1233051600 q^{66} + 3762316676 q^{68} - 5648378720 q^{70} - 45494381436 q^{72} - 26772166644 q^{73} + 29079600800 q^{76} + 28136790000 q^{77} - 28110803160 q^{78} - 109632407120 q^{80} + 5759242740 q^{81} + 79710432568 q^{82} - 80654329820 q^{85} - 192079958360 q^{86} + 98741644960 q^{88} + 154170121110 q^{90} - 212561751280 q^{92} - 103736158480 q^{93} + 334276984640 q^{96} - 89886397812 q^{97} - 75369431322 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −45.1846 + 2.52028i −18.0064 + 18.0064i 2035.30 227.755i 6954.24 683.153i 768.231 858.993i −18984.5 18984.5i −91390.1 + 15420.5i 176499.i −312503. + 48394.6i
3.2 −44.9401 5.32801i 359.350 359.350i 1991.22 + 478.883i −6959.92 622.606i −18063.8 + 14234.6i −2235.42 2235.42i −86934.3 32130.3i 81117.8i 309462. + 65062.5i
3.3 −43.4480 12.6598i −513.533 + 513.533i 1727.46 + 1100.09i −3227.70 6197.58i 28813.2 15810.8i −25537.2 25537.2i −61127.8 69665.8i 350285.i 61777.1 + 310135.i
3.4 −42.2496 + 16.2165i −165.660 + 165.660i 1522.05 1370.28i −4210.13 + 5577.00i 4312.64 9685.50i 9765.23 + 9765.23i −42084.8 + 82576.1i 122260.i 87436.6 303899.i
3.5 −37.1216 + 25.8841i 544.922 544.922i 708.031 1921.72i 5461.94 + 4358.36i −6123.60 + 34333.2i 41303.5 + 41303.5i 23458.6 + 89664.0i 416734.i −315568. 20412.0i
3.6 −36.3945 26.8968i −266.570 + 266.570i 601.119 + 1957.79i 205.209 + 6984.70i 16871.6 2531.79i 18557.1 + 18557.1i 30781.1 87421.2i 35028.2i 180398. 259724.i
3.7 −35.4618 28.1151i 228.106 228.106i 467.078 + 1994.03i 2265.99 6610.10i −14502.3 + 1675.81i 43448.3 + 43448.3i 39498.9 83843.7i 73082.7i −266200. + 170697.i
3.8 −34.0099 + 29.8551i −257.870 + 257.870i 265.350 2030.74i −1405.66 6844.87i 1071.41 16468.9i 55896.5 + 55896.5i 51603.3 + 76987.2i 44152.8i 252161. + 190827.i
3.9 −29.8551 + 34.0099i 257.870 257.870i −265.350 2030.74i −1405.66 6844.87i 1071.41 + 16468.9i −55896.5 55896.5i 76987.2 + 51603.3i 44152.8i 274760. + 156548.i
3.10 −29.1822 34.5890i 418.357 418.357i −344.796 + 2018.77i 4148.35 + 5623.11i −26679.1 2261.96i −45215.7 45215.7i 79889.0 46986.0i 172898.i 73439.5 307582.i
3.11 −25.8841 + 37.1216i −544.922 + 544.922i −708.031 1921.72i 5461.94 + 4358.36i −6123.60 34333.2i −41303.5 41303.5i 89664.0 + 23458.6i 416734.i −303167. + 89944.3i
3.12 −16.2165 + 42.2496i 165.660 165.660i −1522.05 1370.28i −4210.13 + 5577.00i 4312.64 + 9685.50i −9765.23 9765.23i 82576.1 42084.8i 122260.i −167352. 268316.i
3.13 −16.1888 42.2602i −59.6583 + 59.6583i −1523.84 + 1368.29i −6580.51 2350.54i 3486.97 + 1555.37i −33756.3 33756.3i 82493.3 + 42246.9i 170029.i 7196.45 + 316146.i
3.14 −9.80346 44.1802i −358.943 + 358.943i −1855.78 + 866.239i 6343.79 2929.93i 19377.1 + 12339.3i 14614.2 + 14614.2i 56463.7 + 73496.8i 80532.8i −191636. 251547.i
3.15 −2.52028 + 45.1846i 18.0064 18.0064i −2035.30 227.755i 6954.24 683.153i 768.231 + 858.993i 18984.5 + 18984.5i 15420.5 91390.1i 176499.i 13341.4 + 315946.i
3.16 1.55298 45.2282i 281.443 281.443i −2043.18 140.477i −4194.88 + 5588.48i −12292.1 13166.2i 51844.9 + 51844.9i −9526.55 + 92191.0i 18726.5i 246242. + 198406.i
3.17 5.32801 + 44.9401i −359.350 + 359.350i −1991.22 + 478.883i −6959.92 622.606i −18063.8 14234.6i 2235.42 + 2235.42i −32130.3 86934.3i 81117.8i −9102.56 316097.i
3.18 9.12236 44.3259i 433.162 433.162i −1881.57 808.713i 4261.88 5537.55i −15248.8 23151.7i −8187.22 8187.22i −53011.2 + 76024.6i 198112.i −206578. 239427.i
3.19 12.6598 + 43.4480i 513.533 513.533i −1727.46 + 1100.09i −3227.70 6197.58i 28813.2 + 15810.8i 25537.2 + 25537.2i −69665.8 61127.8i 350285.i 228411. 218697.i
3.20 19.1088 41.0226i −116.548 + 116.548i −1317.71 1567.78i 4797.95 + 5080.14i 2554.02 + 7008.21i −39201.4 39201.4i −89494.4 + 24097.3i 149980.i 300084. 99748.9i
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.12.e.b 60
4.b odd 2 1 inner 20.12.e.b 60
5.b even 2 1 100.12.e.e 60
5.c odd 4 1 inner 20.12.e.b 60
5.c odd 4 1 100.12.e.e 60
20.d odd 2 1 100.12.e.e 60
20.e even 4 1 inner 20.12.e.b 60
20.e even 4 1 100.12.e.e 60
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.12.e.b 60 1.a even 1 1 trivial
20.12.e.b 60 4.b odd 2 1 inner
20.12.e.b 60 5.c odd 4 1 inner
20.12.e.b 60 20.e even 4 1 inner
100.12.e.e 60 5.b even 2 1
100.12.e.e 60 5.c odd 4 1
100.12.e.e 60 20.d odd 2 1
100.12.e.e 60 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{60} + 1410707871720 T_{3}^{56} + \cdots + 10\!\cdots\!00 \) acting on \(S_{12}^{\mathrm{new}}(20, [\chi])\). Copy content Toggle raw display