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Properties

Label 84.12.o.a
Level $84$
Weight $12$
Character orbit 84.o
Analytic conductor $64.541$
Analytic rank $0$
Dimension $88$
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(19,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.19");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 84.o (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: \(64.5408271670\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(44\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 23 q^{2} - 10692 q^{3} - 1541 q^{4} - 11178 q^{6} - 134 q^{7} + 187124 q^{8} - 2598156 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 23 q^{2} - 10692 q^{3} - 1541 q^{4} - 11178 q^{6} - 134 q^{7} + 187124 q^{8} - 2598156 q^{9} + 1091381 q^{10} - 1100874 q^{11} - 374463 q^{12} - 169166 q^{14} - 622105 q^{16} + 1358127 q^{18} - 3491526 q^{19} - 12591974 q^{20} + 65124 q^{21} + 1182 q^{22} + 49581963 q^{24} + 417716902 q^{25} - 149581956 q^{26} + 1262703816 q^{27} - 341697307 q^{28} - 155346416 q^{29} - 63898551 q^{30} - 11193378 q^{31} - 756565887 q^{32} + 267512382 q^{33} + 168109924 q^{34} - 387281676 q^{35} + 181989018 q^{36} - 613457142 q^{37} - 1875742376 q^{38} - 562739886 q^{39} + 1302916259 q^{40} + 714412953 q^{42} + 668738483 q^{44} - 599884140 q^{46} + 896349652 q^{47} + 925136478 q^{48} + 223261868 q^{49} + 2558563726 q^{50} + 66951116 q^{52} - 2193508508 q^{53} + 330024861 q^{54} - 3571854184 q^{55} + 13823445667 q^{56} + 1696881636 q^{57} + 9435091031 q^{58} - 5397050654 q^{59} + 12455812341 q^{60} + 4027823988 q^{61} - 2829599472 q^{62} - 7912566 q^{63} - 33161077550 q^{64} - 2812412500 q^{65} - 10427221611 q^{66} + 40579222026 q^{67} + 33203133214 q^{68} - 68100128295 q^{70} - 23097902085 q^{72} + 38153881674 q^{73} - 11540760116 q^{74} + 101505207186 q^{75} + 54383830612 q^{76} + 2093349112 q^{77} + 31703695326 q^{78} - 49328748606 q^{79} + 91868665515 q^{80} - 153418513644 q^{81} - 96603369146 q^{82} - 18951518972 q^{83} + 39113842302 q^{84} - 85518048736 q^{85} + 180636888210 q^{86} + 18874589544 q^{87} + 188445676651 q^{88} - 48917608776 q^{90} + 260902920238 q^{91} - 256473551516 q^{92} - 2719990854 q^{93} - 248650106008 q^{94} - 72511638648 q^{95} + 191155318929 q^{96} + 120739994155 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −45.0197 − 4.60726i −121.500 − 210.444i 2005.55 + 414.835i −7668.28 − 4427.28i 4500.32 + 10033.9i 42350.4 − 13556.2i −88377.8 − 27915.8i −29524.5 + 51137.9i 324826. + 234645.i
19.2 −44.7333 + 6.85093i −121.500 − 210.444i 1954.13 − 612.929i 8715.53 + 5031.91i 6876.83 + 8581.47i −42922.4 + 11618.9i −83215.5 + 40805.9i −29524.5 + 51137.9i −424347. − 165384.i
19.3 −44.7331 − 6.85206i −121.500 − 210.444i 1954.10 + 613.028i 2978.76 + 1719.79i 3993.09 + 10246.3i 8102.07 + 43722.8i −83212.4 − 40812.2i −29524.5 + 51137.9i −121465. − 97342.0i
19.4 −43.9003 + 10.9891i −121.500 − 210.444i 1806.48 − 964.853i −6001.65 − 3465.06i 7646.49 + 7903.39i −32276.8 − 30586.6i −68702.1 + 62209.0i −29524.5 + 51137.9i 301552. + 86164.1i
19.5 −42.7332 + 14.8956i −121.500 − 210.444i 1604.24 − 1273.07i 3085.96 + 1781.68i 8326.76 + 7183.13i 44227.7 − 4608.41i −49591.4 + 78298.3i −29524.5 + 51137.9i −158412. − 30169.7i
19.6 −41.1758 − 18.7764i −121.500 − 210.444i 1342.90 + 1546.26i −7948.07 − 4588.82i 1051.49 + 10946.5i −43043.0 + 11163.5i −26261.7 − 88883.4i −29524.5 + 51137.9i 241107. + 338184.i
19.7 −39.6075 − 21.8916i −121.500 − 210.444i 1089.52 + 1734.15i 5695.15 + 3288.10i 205.358 + 10995.0i −20608.7 − 39403.2i −5189.85 − 92536.5i −29524.5 + 51137.9i −153589. − 254910.i
19.8 −38.4436 + 23.8765i −121.500 − 210.444i 907.827 − 1835.80i 4167.79 + 2406.27i 9695.57 + 5189.25i 15124.1 − 41816.1i 8932.21 + 92250.5i −29524.5 + 51137.9i −217678. + 7006.20i
19.9 −37.9759 + 24.6137i −121.500 − 210.444i 836.335 − 1869.45i −6590.23 − 3804.87i 9793.87 + 5001.24i −31714.9 + 31168.7i 14253.5 + 91579.3i −29524.5 + 51137.9i 343922. − 17716.4i
19.10 −34.5343 − 29.2470i −121.500 − 210.444i 337.231 + 2020.04i 1104.42 + 637.637i −1958.94 + 10821.0i 18300.6 + 40526.7i 47434.1 − 79623.7i −29524.5 + 51137.9i −19491.4 − 54321.3i
19.11 −31.8222 + 32.1768i −121.500 − 210.444i −22.6905 − 2047.87i −3523.86 − 2034.50i 10637.8 + 2787.33i 24142.3 + 37342.7i 66616.1 + 64437.8i −29524.5 + 51137.9i 177601. − 48644.1i
19.12 −30.5326 − 33.4030i −121.500 − 210.444i −183.524 + 2039.76i −2421.57 − 1398.09i −3319.76 + 10483.9i 20060.6 − 39685.0i 73737.6 − 56148.9i −29524.5 + 51137.9i 27236.2 + 123575.i
19.13 −25.7304 + 37.2283i −121.500 − 210.444i −723.896 − 1915.80i −10891.3 − 6288.10i 10960.7 + 891.565i −1847.65 − 44428.7i 89948.0 + 22344.7i −29524.5 + 51137.9i 514333. − 243670.i
19.14 −23.3719 + 38.7525i −121.500 − 210.444i −955.507 − 1811.44i 10043.6 + 5798.70i 10994.9 + 210.060i 30567.5 + 32294.8i 92529.7 + 5308.57i −29524.5 + 51137.9i −459453. + 253689.i
19.15 −22.1647 − 39.4554i −121.500 − 210.444i −1065.45 + 1749.03i 11752.8 + 6785.51i −5610.14 + 9458.26i 42201.9 + 14011.6i 92624.2 + 3271.05i −29524.5 + 51137.9i 7226.51 − 614112.i
19.16 −17.9468 − 41.5441i −121.500 − 210.444i −1403.82 + 1491.17i −6980.82 − 4030.38i −6562.17 + 8824.41i −44410.2 − 2249.55i 87143.4 + 31558.8i −29524.5 + 51137.9i −42154.8 + 362344.i
19.17 −16.7655 + 42.0347i −121.500 − 210.444i −1485.84 − 1409.47i −1083.64 − 625.639i 10883.0 − 1579.02i −34278.7 − 28324.9i 84157.3 − 38826.4i −29524.5 + 51137.9i 44466.3 − 35061.3i
19.18 −16.2271 − 42.2455i −121.500 − 210.444i −1521.36 + 1371.04i 2278.61 + 1315.56i −6918.73 + 8547.72i −32326.0 + 30534.5i 82607.6 + 42022.9i −29524.5 + 51137.9i 18601.2 − 117609.i
19.19 −11.6257 − 43.7361i −121.500 − 210.444i −1777.69 + 1016.92i −846.887 − 488.950i −7791.48 + 7760.49i 28932.5 − 33767.4i 65143.0 + 65926.7i −29524.5 + 51137.9i −11539.1 + 42723.9i
19.20 −7.19330 + 44.6795i −121.500 − 210.444i −1944.51 − 642.786i 6372.32 + 3679.06i 10276.5 − 3914.77i 23647.2 − 37658.1i 42706.8 − 82256.1i −29524.5 + 51137.9i −210217. + 258247.i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.12.o.a ✓ 88
4.b odd 2 1 84.12.o.b yes 88
7.d odd 6 1 84.12.o.b yes 88
28.f even 6 1 inner 84.12.o.a ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.12.o.a ✓ 88 1.a even 1 1 trivial
84.12.o.a ✓ 88 28.f even 6 1 inner
84.12.o.b yes 88 4.b odd 2 1
84.12.o.b yes 88 7.d odd 6 1