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Properties

Label 83.7.d.a
Level $83$
Weight $7$
Character orbit 83.d
Analytic conductor $19.094$
Analytic rank $0$
Dimension $1640$
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] = [83,7,Mod(2,83)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(83, base_ring=CyclotomicField(82))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("83.2");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 83 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 83.d (of order \(82\), degree \(40\), 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: \(19.0944889404\)
Analytic rank: \(0\)
Dimension: \(1640\)
Relative dimension: \(41\) over \(\Q(\zeta_{82})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{82}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1640 q - 41 q^{2} - 59 q^{3} + 1109 q^{4} - 41 q^{5} - 41 q^{6} - 303 q^{7} - 41 q^{8} - 11296 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1640 q - 41 q^{2} - 59 q^{3} + 1109 q^{4} - 41 q^{5} - 41 q^{6} - 303 q^{7} - 41 q^{8} - 11296 q^{9} - 2099 q^{10} + 957 q^{11} - 2583 q^{12} - 41 q^{13} - 41 q^{14} - 41 q^{15} - 16723 q^{16} - 12235 q^{17} - 41 q^{18} - 41 q^{19} - 41 q^{20} + 34123 q^{21} - 41 q^{22} - 26615 q^{23} - 41 q^{24} + 132650 q^{25} - 69339 q^{26} - 51701 q^{27} - 13989 q^{28} + 35377 q^{29} - 15241 q^{30} + 81705 q^{31} - 41 q^{32} + 130747 q^{33} - 41 q^{34} - 41 q^{35} + 201031 q^{36} + 27609 q^{37} - 60519 q^{38} - 41 q^{39} + 73393 q^{40} + 149821 q^{41} - 41 q^{42} - 41 q^{43} - 165199 q^{44} - 41 q^{45} - 41 q^{46} - 41 q^{47} + 388893 q^{48} - 744260 q^{49} - 41 q^{50} - 121205 q^{51} - 41 q^{52} - 41 q^{53} - 41 q^{54} - 41 q^{55} - 41 q^{56} - 41 q^{57} - 41 q^{58} - 668403 q^{59} - 41 q^{60} + 849969 q^{61} - 41 q^{62} - 495771 q^{63} - 732127 q^{64} - 1043689 q^{65} + 7229079 q^{66} + 2332039 q^{67} - 2268889 q^{68} - 5928949 q^{69} - 6014153 q^{70} - 2074641 q^{71} - 13316841 q^{72} - 1222169 q^{73} + 1036439 q^{74} + 9891933 q^{75} + 7557079 q^{76} + 5147115 q^{77} + 21212475 q^{78} + 5889199 q^{79} + 14707479 q^{80} - 1199158 q^{81} - 3011046 q^{83} - 20863344 q^{84} - 6898865 q^{85} - 11130831 q^{86} - 12312637 q^{87} - 29165801 q^{88} - 7507961 q^{89} - 18733139 q^{90} - 2125481 q^{91} + 4164545 q^{92} + 12801403 q^{93} + 12718003 q^{94} + 12649855 q^{95} + 52807959 q^{96} + 5195479 q^{97} + 15754455 q^{98} + 8768937 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −15.3167 − 0.587101i −44.9709 + 18.1249i 170.443 + 13.0857i 43.3890 + 72.9834i 699.445 − 251.210i 491.385 + 155.508i −1628.42 − 187.994i 1168.61 − 1124.67i −621.726 − 1143.34i
2.2 −14.9277 − 0.572191i 12.9522 − 5.22021i 158.696 + 12.1838i −85.3525 − 143.569i −196.334 + 70.5145i 214.884 + 68.0041i −1412.23 − 163.035i −384.751 + 370.286i 1191.97 + 2191.99i
2.3 −14.6029 − 0.559742i 1.93913 − 0.781538i 149.119 + 11.4486i 60.2088 + 101.275i −28.7543 + 10.3273i −323.468 − 102.368i −1242.06 − 143.390i −522.111 + 502.481i −822.535 − 1512.62i
2.4 −13.6619 − 0.523674i 41.7860 − 16.8413i 122.562 + 9.40965i 68.6462 + 115.468i −579.698 + 208.202i −178.288 − 56.4228i −800.277 − 92.3880i 937.183 − 901.948i −877.373 − 1613.46i
2.5 −12.6779 − 0.485957i −35.9510 + 14.4896i 96.6819 + 7.42271i −106.188 − 178.615i 462.827 − 166.227i −374.891 − 118.642i −415.494 − 47.9668i 557.269 − 536.318i 1259.44 + 2316.08i
2.6 −12.1952 − 0.467452i 38.1785 − 15.3873i 84.6917 + 6.50216i −31.6221 − 53.1907i −472.786 + 169.804i 232.350 + 73.5318i −253.880 − 29.3092i 695.567 − 669.416i 360.773 + 663.451i
2.7 −12.1701 − 0.466492i −20.5988 + 8.30208i 84.0827 + 6.45541i −20.5680 − 34.5968i 254.564 − 91.4283i −7.16794 − 2.26843i −245.969 − 28.3959i −169.873 + 163.486i 234.176 + 430.643i
2.8 −10.8741 − 0.416815i −12.6069 + 5.08102i 54.2610 + 4.16587i 59.6849 + 100.394i 139.207 − 49.9970i 376.808 + 119.248i 103.555 + 11.9549i −392.144 + 377.401i −607.177 − 1116.58i
2.9 −9.76443 − 0.374279i 22.0514 − 8.88751i 31.3918 + 2.41009i 104.675 + 176.071i −218.646 + 78.5281i 425.283 + 134.589i 315.635 + 36.4386i −117.984 + 113.548i −956.195 − 1758.41i
2.10 −9.29306 − 0.356211i −40.0220 + 16.1303i 22.4218 + 1.72142i 101.756 + 171.160i 377.672 − 135.643i −487.620 − 154.317i 383.512 + 44.2745i 816.310 − 785.620i −884.652 − 1626.85i
2.11 −8.99731 − 0.344874i 20.3689 − 8.20941i 17.0204 + 1.30673i −60.5827 − 101.904i −186.097 + 66.8378i −541.159 − 171.260i 419.762 + 48.4594i −177.762 + 171.079i 509.937 + 937.758i
2.12 −7.31922 − 0.280552i 31.7744 − 12.8062i −10.3200 − 0.792314i −17.3859 − 29.2443i −236.156 + 84.8172i 80.7056 + 25.5408i 540.993 + 62.4550i 320.352 − 308.308i 119.046 + 218.923i
2.13 −6.89424 − 0.264262i −3.64108 + 1.46749i −16.3515 − 1.25538i −88.6322 − 149.086i 25.4903 − 9.15501i 536.570 + 169.808i 551.041 + 63.6150i −514.156 + 494.826i 571.654 + 1051.25i
2.14 −6.21684 − 0.238297i −32.9241 + 13.2696i −25.2199 − 1.93624i −20.4449 − 34.3897i 207.846 − 74.6494i 10.7334 + 3.39679i 551.870 + 63.7106i 382.656 − 368.269i 118.907 + 218.667i
2.15 −6.00030 − 0.229997i 3.46851 − 1.39794i −27.8616 − 2.13906i 65.3032 + 109.845i −21.1336 + 7.59028i −283.124 − 89.6002i 548.451 + 63.3160i −515.184 + 495.815i −366.575 − 674.120i
2.16 −3.22490 − 0.123613i −47.4038 + 19.1054i −53.4275 − 4.10187i −25.3618 − 42.6603i 155.234 − 55.7534i 305.333 + 96.6285i 376.973 + 43.5197i 1356.84 − 1305.83i 76.5158 + 140.710i
2.17 −2.12012 − 0.0812662i −6.63643 + 2.67472i −59.3239 − 4.55456i 30.9138 + 51.9991i 14.2874 − 5.13142i −206.590 − 65.3795i 260.296 + 30.0499i −488.372 + 470.011i −61.3152 − 112.757i
2.18 −2.01706 − 0.0773157i 49.2662 − 19.8561i −59.7497 − 4.58725i 88.9810 + 149.672i −100.908 + 36.2418i −469.266 − 148.508i 248.498 + 28.6879i 1507.63 − 1450.95i −167.908 − 308.778i
2.19 −1.79653 − 0.0688624i 25.1198 − 10.1242i −60.5894 − 4.65173i 56.5031 + 95.0423i −45.8255 + 16.4586i 177.445 + 56.1559i 222.833 + 25.7250i 3.24473 − 3.12274i −94.9645 − 174.637i
2.20 −1.48144 − 0.0567850i 46.0598 − 18.5638i −61.6208 − 4.73090i −109.073 − 183.468i −69.2891 + 24.8856i 103.456 + 32.7406i 185.275 + 21.3891i 1251.63 − 1204.58i 151.166 + 277.990i
See next 80 embeddings (of 1640 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.41
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
83.d odd 82 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 83.7.d.a ✓ 1640
83.d odd 82 1 inner 83.7.d.a ✓ 1640
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
83.7.d.a ✓ 1640 1.a even 1 1 trivial
83.7.d.a ✓ 1640 83.d odd 82 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(83, [\chi])\).