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Properties

Label 83.8.c.a
Level $83$
Weight $8$
Character orbit 83.c
Analytic conductor $25.928$
Analytic rank $0$
Dimension $1920$
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] = [83,8,Mod(3,83)]
 
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([72]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("83.3");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 83 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 83.c (of order \(41\), 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: \(25.9279571152\)
Analytic rank: \(0\)
Dimension: \(1920\)
Relative dimension: \(48\) over \(\Q(\zeta_{41})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{41}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1920 q - 47 q^{2} - 13 q^{3} - 2995 q^{4} - 287 q^{5} + 133 q^{6} + 1703 q^{7} - 1319 q^{8} - 32875 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1920 q - 47 q^{2} - 13 q^{3} - 2995 q^{4} - 287 q^{5} + 133 q^{6} + 1703 q^{7} - 1319 q^{8} - 32875 q^{9} - 10291 q^{10} + 1051 q^{11} + 15689 q^{12} + 353 q^{13} - 9137 q^{14} - 26829 q^{15} - 159251 q^{16} - 32747 q^{17} + 27647 q^{18} + 46541 q^{19} + 56713 q^{20} - 107002 q^{21} - 56865 q^{22} + 77524 q^{23} + 90555 q^{24} - 684473 q^{25} + 11757 q^{26} + 166724 q^{27} + 490395 q^{28} - 9267 q^{29} - 463493 q^{30} - 241993 q^{31} - 348281 q^{32} + 277688 q^{33} - 1016211 q^{34} - 388501 q^{35} - 2680957 q^{36} - 81263 q^{37} - 103347 q^{38} - 923271 q^{39} - 2352295 q^{40} + 1730128 q^{41} + 1224589 q^{42} - 810005 q^{43} - 743999 q^{44} - 1983365 q^{45} + 3457535 q^{46} - 892253 q^{47} + 1765793 q^{48} - 5422351 q^{49} - 2983631 q^{50} + 4220528 q^{51} + 3387269 q^{52} - 3354163 q^{53} + 710861 q^{54} + 694021 q^{55} - 6071253 q^{56} - 2213157 q^{57} - 822197 q^{58} + 4406167 q^{59} - 9406931 q^{60} + 6450229 q^{61} - 477893 q^{62} + 3791232 q^{63} - 16475263 q^{64} + 7595367 q^{65} - 60109381 q^{66} - 10951832 q^{67} + 53286979 q^{68} + 70863440 q^{69} + 72026511 q^{70} - 65974 q^{71} - 27628649 q^{72} - 54170424 q^{73} - 47432779 q^{74} - 129609148 q^{75} - 53284287 q^{76} - 32057153 q^{77} - 14580233 q^{78} + 14803814 q^{79} + 178600077 q^{80} + 46266739 q^{81} + 92096374 q^{82} + 68727322 q^{83} + 217717964 q^{84} + 43226808 q^{85} + 40133945 q^{86} + 4742423 q^{87} - 186777517 q^{88} - 65450416 q^{89} - 311071143 q^{90} - 92729208 q^{91} - 146526967 q^{92} - 120126893 q^{93} + 16819659 q^{94} + 70502112 q^{95} + 399699865 q^{96} + 46800790 q^{97} + 297957145 q^{98} + 145299428 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −20.7225 + 8.35191i −10.7239 − 8.84396i 267.440 − 257.385i 211.034 − 276.850i 296.090 + 93.7035i −880.112 − 67.5702i −2222.51 + 4957.72i −379.598 − 1957.31i −2060.91 + 7499.56i
3.2 −19.6224 + 7.90852i −52.7686 − 43.5179i 230.265 − 221.608i −305.713 + 401.057i 1379.61 + 436.603i −164.556 − 12.6337i −1658.00 + 3698.48i 474.327 + 2445.76i 2827.03 − 10287.4i
3.3 −18.4375 + 7.43097i 51.7147 + 42.6488i 192.494 − 185.257i 208.923 − 274.082i −1270.41 − 402.045i 480.991 + 36.9279i −1131.60 + 2524.25i 439.105 + 2264.14i −1815.32 + 6605.87i
3.4 −18.4107 + 7.42019i 34.7925 + 28.6931i 191.669 − 184.463i −157.064 + 206.049i −853.463 − 270.095i 1325.37 + 101.755i −1120.66 + 2499.85i −29.1654 − 150.385i 1362.74 − 4958.95i
3.5 −18.3863 + 7.41033i −35.3069 − 29.1174i 190.915 − 183.737i 56.0093 − 73.4773i 864.931 + 273.724i 1255.72 + 96.4077i −1110.69 + 2477.60i −17.6301 − 90.9057i −485.311 + 1766.02i
3.6 −17.6241 + 7.10317i 49.7247 + 41.0077i 167.929 − 161.615i −171.744 + 225.307i −1167.64 − 369.522i −1476.57 − 113.363i −816.676 + 1821.75i 374.534 + 1931.20i 1426.45 − 5190.77i
3.7 −17.5645 + 7.07912i −5.73046 − 4.72588i 166.170 − 159.922i −112.623 + 147.748i 134.107 + 42.4409i −183.469 − 14.0857i −794.999 + 1773.39i −405.881 − 2092.83i 932.244 − 3392.38i
3.8 −16.3158 + 6.57586i −66.7079 − 55.0136i 130.737 − 125.821i 247.712 − 324.967i 1450.16 + 458.930i −576.011 − 44.2230i −384.606 + 857.934i 1007.06 + 5192.70i −1904.68 + 6931.02i
3.9 −14.8823 + 5.99809i −9.36785 − 7.72561i 93.2778 − 89.7709i 77.6836 − 101.911i 185.754 + 58.7853i −624.571 − 47.9511i −9.57439 + 21.3575i −388.314 − 2002.25i −544.834 + 1982.62i
3.10 −12.8832 + 5.19240i −47.7816 − 39.4052i 46.7895 − 45.0303i 21.1210 − 27.7081i 820.188 + 259.564i 1344.58 + 103.230i 358.320 − 799.299i 313.925 + 1618.68i −128.234 + 466.638i
3.11 −12.6787 + 5.10999i −53.7092 − 44.2936i 42.4116 − 40.8171i −126.725 + 166.247i 907.305 + 287.134i −1330.03 − 102.113i 386.609 − 862.402i 506.365 + 2610.96i 757.188 − 2755.37i
3.12 −12.0163 + 4.84301i 41.5140 + 34.2363i 28.7101 − 27.6307i 186.992 − 245.311i −664.652 − 210.342i −569.562 − 43.7279i 467.189 − 1042.15i 134.899 + 695.578i −1058.91 + 3853.33i
3.13 −11.2880 + 4.54949i −0.409123 − 0.337401i 14.4953 − 13.9503i −260.718 + 342.030i 6.15320 + 1.94730i −303.562 − 23.3059i 537.093 − 1198.09i −416.332 − 2146.72i 1386.94 − 5046.99i
3.14 −10.9701 + 4.42134i 11.9103 + 9.82238i 8.56782 − 8.24570i 213.868 − 280.569i −174.086 − 55.0927i 976.242 + 74.9505i 561.768 − 1253.13i −371.008 − 1913.02i −1105.66 + 4023.45i
3.15 −9.99723 + 4.02925i 60.9522 + 50.2669i −8.51700 + 8.19679i −135.785 + 178.133i −811.890 − 256.938i 962.862 + 73.9233i 616.498 − 1375.21i 772.022 + 3980.76i 639.732 − 2327.95i
3.16 −8.89220 + 3.58388i 16.6131 + 13.7007i −25.9997 + 25.0222i −203.876 + 267.460i −196.828 − 62.2901i 800.750 + 61.4772i 643.514 − 1435.48i −328.101 − 1691.78i 854.362 − 3108.98i
3.17 −8.56895 + 3.45360i 67.6803 + 55.8155i −30.7272 + 29.5720i −83.7983 + 109.933i −772.713 − 244.540i −90.0085 − 6.91036i 644.918 − 1438.61i 1048.86 + 5408.23i 338.399 − 1231.42i
3.18 −6.09375 + 2.45600i −33.9808 − 28.0237i −61.1249 + 58.8268i 321.041 − 421.166i 275.897 + 87.3129i 127.960 + 9.82404i 572.015 − 1275.99i −47.0222 − 242.459i −921.959 + 3354.96i
3.19 −6.05331 + 2.43970i −53.8462 − 44.4066i −61.5363 + 59.2228i −163.894 + 215.008i 434.287 + 137.438i 448.008 + 34.3956i 569.743 − 1270.92i 511.079 + 2635.26i 467.543 − 1701.36i
3.20 −5.26289 + 2.12114i −35.6981 − 29.4400i −69.0280 + 66.4328i 112.718 − 147.872i 250.321 + 79.2189i −1556.04 − 119.464i 519.482 − 1158.80i −8.74592 − 45.0964i −279.565 + 1017.32i
See next 80 embeddings (of 1920 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
83.c even 41 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 83.8.c.a ✓ 1920
83.c even 41 1 inner 83.8.c.a ✓ 1920
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
83.8.c.a ✓ 1920 1.a even 1 1 trivial
83.8.c.a ✓ 1920 83.c even 41 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(83, [\chi])\).