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Properties

Label 84.12.b.b
Level $84$
Weight $12$
Character orbit 84.b
Analytic conductor $64.541$
Analytic rank $0$
Dimension $44$
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] = [84,12,Mod(55,84)]
 
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, 1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.55");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 84.b (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: \(64.5408271670\)
Analytic rank: \(0\)
Dimension: \(44\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 23 q^{2} + 10692 q^{3} + 1541 q^{4} - 5589 q^{6} + 134 q^{7} - 18695 q^{8} + 2598156 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 23 q^{2} + 10692 q^{3} + 1541 q^{4} - 5589 q^{6} + 134 q^{7} - 18695 q^{8} + 2598156 q^{9} + 828424 q^{10} + 374463 q^{12} - 2939971 q^{14} - 4412759 q^{16} - 1358127 q^{18} - 6983052 q^{19} + 44417108 q^{20} + 32562 q^{21} - 17991462 q^{22} - 4542885 q^{24} - 453628696 q^{25} + 98243700 q^{26} + 631351908 q^{27} - 48862043 q^{28} - 77673208 q^{29} + 201307032 q^{30} - 363408768 q^{31} - 193132863 q^{32} - 244149820 q^{34} + 387281676 q^{35} + 90994509 q^{36} + 341371884 q^{37} + 414182420 q^{38} + 770410876 q^{40} - 714412953 q^{42} - 880133666 q^{44} - 1775052978 q^{46} + 1792699304 q^{47} - 1072300437 q^{48} - 819617000 q^{49} + 2158233149 q^{50} + 4514138716 q^{52} + 1914807152 q^{53} - 330024861 q^{54} - 1785927092 q^{55} - 1038409567 q^{56} - 1696881636 q^{57} + 5084833750 q^{58} - 3729062416 q^{59} + 10793357244 q^{60} + 18353275776 q^{62} + 7912566 q^{63} - 12067108399 q^{64} - 5624825000 q^{65} - 4371925266 q^{66} - 885261520 q^{68} + 6938553312 q^{70} - 1103921055 q^{72} - 40922634778 q^{74} - 110231773128 q^{75} - 82420472236 q^{76} - 2093349112 q^{77} + 23873219100 q^{78} + 62176412340 q^{80} + 153418513644 q^{81} + 5083269068 q^{82} - 34205636344 q^{83} - 11873476449 q^{84} + 105152127628 q^{85} + 242964863598 q^{86} - 18874589544 q^{87} + 218894674838 q^{88} + 48917608776 q^{90} + 6678724880 q^{91} + 101339859530 q^{92} - 88308330624 q^{93} - 220283048096 q^{94} - 46931285709 q^{96} - 142087425367 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1 −45.2221 − 1.72065i 243.000 2042.08 + 155.623i − 479.067i −10989.0 − 418.118i −18444.0 − 40461.6i −92079.3 − 10551.3i 59049.0 −824.307 + 21664.4i
55.2 −45.2221 + 1.72065i 243.000 2042.08 − 155.623i 479.067i −10989.0 + 418.118i −18444.0 + 40461.6i −92079.3 + 10551.3i 59049.0 −824.307 − 21664.4i
55.3 −44.4594 − 8.44758i 243.000 1905.28 + 751.149i − 11354.1i −10803.6 − 2052.76i 41246.7 − 16614.4i −78362.1 − 49490.6i 59049.0 −95914.4 + 504795.i
55.4 −44.4594 + 8.44758i 243.000 1905.28 − 751.149i 11354.1i −10803.6 + 2052.76i 41246.7 + 16614.4i −78362.1 + 49490.6i 59049.0 −95914.4 − 504795.i
55.5 −40.8700 − 19.4330i 243.000 1292.72 + 1588.45i 1211.18i −9931.41 − 4722.22i 25032.4 + 36752.0i −21964.9 − 90041.5i 59049.0 23536.9 − 49501.0i
55.6 −40.8700 + 19.4330i 243.000 1292.72 − 1588.45i − 1211.18i −9931.41 + 4722.22i 25032.4 − 36752.0i −21964.9 + 90041.5i 59049.0 23536.9 + 49501.0i
55.7 −40.1866 − 20.8095i 243.000 1181.93 + 1672.53i 11675.3i −9765.35 − 5056.70i 983.024 − 44456.3i −12693.5 − 91808.6i 59049.0 242957. − 469190.i
55.8 −40.1866 + 20.8095i 243.000 1181.93 − 1672.53i − 11675.3i −9765.35 + 5056.70i 983.024 + 44456.3i −12693.5 + 91808.6i 59049.0 242957. + 469190.i
55.9 −39.5815 − 21.9387i 243.000 1085.38 + 1736.73i − 10289.8i −9618.30 − 5331.11i −43786.1 + 7752.48i −4859.36 − 92554.4i 59049.0 −225745. + 407286.i
55.10 −39.5815 + 21.9387i 243.000 1085.38 − 1736.73i 10289.8i −9618.30 + 5331.11i −43786.1 − 7752.48i −4859.36 + 92554.4i 59049.0 −225745. − 407286.i
55.11 −32.5648 − 31.4250i 243.000 72.9330 + 2046.70i 4483.20i −7913.25 − 7636.29i −44244.4 + 4445.20i 61942.6 − 68942.3i 59049.0 140885. − 145994.i
55.12 −32.5648 + 31.4250i 243.000 72.9330 − 2046.70i − 4483.20i −7913.25 + 7636.29i −44244.4 − 4445.20i 61942.6 + 68942.3i 59049.0 140885. + 145994.i
55.13 −25.9452 − 37.0790i 243.000 −701.697 + 1924.04i − 5436.30i −6304.67 − 9010.19i 42645.9 − 12595.7i 89547.0 − 23901.3i 59049.0 −201572. + 141046.i
55.14 −25.9452 + 37.0790i 243.000 −701.697 − 1924.04i 5436.30i −6304.67 + 9010.19i 42645.9 + 12595.7i 89547.0 + 23901.3i 59049.0 −201572. − 141046.i
55.15 −23.3606 − 38.7593i 243.000 −956.568 + 1810.88i 7022.22i −5676.62 − 9418.51i 17991.9 + 40664.7i 92534.4 − 5227.27i 59049.0 272177. − 164043.i
55.16 −23.3606 + 38.7593i 243.000 −956.568 − 1810.88i − 7022.22i −5676.62 + 9418.51i 17991.9 − 40664.7i 92534.4 + 5227.27i 59049.0 272177. + 164043.i
55.17 −20.9262 − 40.1260i 243.000 −1172.19 + 1679.37i − 6779.01i −5085.06 − 9750.61i −9419.93 − 43457.9i 91915.7 + 11892.5i 59049.0 −272015. + 141859.i
55.18 −20.9262 + 40.1260i 243.000 −1172.19 − 1679.37i 6779.01i −5085.06 + 9750.61i −9419.93 + 43457.9i 91915.7 − 11892.5i 59049.0 −272015. − 141859.i
55.19 −8.52362 − 44.4449i 243.000 −1902.70 + 757.663i 11942.9i −2071.24 − 10800.1i 30635.0 − 32230.8i 49892.1 + 78107.1i 59049.0 530802. − 101797.i
55.20 −8.52362 + 44.4449i 243.000 −1902.70 − 757.663i − 11942.9i −2071.24 + 10800.1i 30635.0 + 32230.8i 49892.1 − 78107.1i 59049.0 530802. + 101797.i
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 55.44
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.12.b.b yes 44
4.b odd 2 1 84.12.b.a ✓ 44
7.b odd 2 1 84.12.b.a ✓ 44
28.d even 2 1 inner 84.12.b.b yes 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.12.b.a ✓ 44 4.b odd 2 1
84.12.b.a ✓ 44 7.b odd 2 1
84.12.b.b yes 44 1.a even 1 1 trivial
84.12.b.b yes 44 28.d even 2 1 inner