↑

Properties

Label 80.10.s.a
Level $80$
Weight $10$
Character orbit 80.s
Analytic conductor $41.203$
Analytic rank $0$
Dimension $212$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [80,10,Mod(3,80)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(80, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 3]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("80.3");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 80 = 2^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 80.s (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: \(41.2028668931\)
Analytic rank: \(0\)
Dimension: \(212\)
Relative dimension: \(106\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 212 q - 2 q^{2} - 4 q^{3} - 684 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} + 712 q^{8} + 1338444 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 212 q - 2 q^{2} - 4 q^{3} - 684 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} + 712 q^{8} + 1338444 q^{9} + 1022 q^{10} - 4 q^{11} - 155360 q^{12} - 78732 q^{15} + 534200 q^{16} - 4 q^{17} - 759866 q^{18} + 480888 q^{19} + 360352 q^{20} - 4 q^{21} - 1659308 q^{22} - 4 q^{23} - 78732 q^{24} + 6874252 q^{26} - 78736 q^{27} + 282588 q^{28} - 8081124 q^{30} - 3431972 q^{32} - 4 q^{33} + 14019176 q^{34} - 15214060 q^{35} - 18999476 q^{36} + 66252864 q^{38} + 38890752 q^{40} - 82908936 q^{42} - 144493960 q^{44} - 3945618 q^{45} + 13958060 q^{46} + 112458264 q^{47} + 269076820 q^{48} - 11251806 q^{50} - 90014332 q^{51} - 87877584 q^{52} - 4 q^{53} + 49163684 q^{54} - 4 q^{55} - 87069868 q^{56} - 78732 q^{57} + 534661688 q^{58} - 144037616 q^{59} - 425720880 q^{60} + 90121100 q^{61} + 26063004 q^{62} - 78732 q^{63} + 317209104 q^{64} - 4 q^{65} + 193062388 q^{66} - 1095329816 q^{68} + 382281444 q^{69} - 884221412 q^{70} - 238101480 q^{71} + 440475536 q^{72} + 241709752 q^{73} - 652460148 q^{74} - 1034969596 q^{75} + 214323196 q^{76} - 161414432 q^{77} - 115110388 q^{78} - 426894316 q^{80} + 7748409772 q^{81} - 647768040 q^{82} - 795523164 q^{83} + 1454815008 q^{84} + 3906248 q^{85} + 1472476340 q^{86} - 2350951628 q^{87} + 296521744 q^{88} - 6912111786 q^{90} + 1393037580 q^{91} + 1769972156 q^{92} - 3013336692 q^{94} - 2812876760 q^{95} + 5969986888 q^{96} - 4 q^{97} - 2904618678 q^{98} - 778491244 q^{99}+O(q^{100}) \) 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 −22.5978 + 1.15663i 35.9411 509.324 − 52.2747i −689.512 − 1215.61i −812.192 + 41.5706i −7555.03 + 7555.03i −11449.2 + 1770.40i −18391.2 16987.5 + 26672.6i
3.2 −22.5314 − 2.08215i 254.588 503.329 + 93.8277i 740.772 + 1185.07i −5736.24 − 530.092i −1753.55 + 1753.55i −11145.4 − 3162.08i 45132.3 −14223.1 − 28243.6i
3.3 −22.5234 + 2.16751i 229.908 502.604 − 97.6391i −33.4403 − 1397.14i −5178.29 + 498.326i 6720.80 − 6720.80i −11108.7 + 3288.56i 33174.5 3781.50 + 31395.9i
3.4 −22.5110 + 2.29204i 81.2054 501.493 − 103.192i −1377.59 + 235.325i −1828.02 + 186.126i 2013.40 − 2013.40i −11052.6 + 3472.41i −13088.7 30471.5 − 8454.91i
3.5 −22.4911 − 2.47971i −137.350 499.702 + 111.543i 1092.93 − 870.992i 3089.17 + 340.590i 1610.35 − 1610.35i −10962.3 − 3747.85i −817.840 −26741.1 + 16879.4i
3.6 −22.2621 + 4.04965i −163.388 479.201 − 180.307i −789.060 + 1153.48i 3637.36 − 661.664i 689.070 − 689.070i −9937.83 + 5954.60i 7012.72 12895.0 − 28874.2i
3.7 −22.0049 + 5.27104i −171.903 456.432 − 231.978i 1177.13 + 753.319i 3782.70 − 906.106i −6622.54 + 6622.54i −8820.99 + 7510.52i 9867.54 −29873.4 − 10372.0i
3.8 −21.9579 − 5.46374i 28.7674 452.295 + 239.944i 1393.02 + 112.281i −631.670 − 157.177i 2170.86 − 2170.86i −8620.44 − 7739.88i −18855.4 −29974.4 − 10076.6i
3.9 −21.9279 + 5.58271i −274.531 449.667 − 244.835i −703.832 − 1207.37i 6019.89 − 1532.63i −645.613 + 645.613i −8493.41 + 7879.07i 55684.2 22174.0 + 22545.8i
3.10 −21.9195 − 5.61581i −129.877 448.925 + 246.191i −1336.10 − 409.839i 2846.83 + 729.362i 7695.48 − 7695.48i −8457.64 − 7917.45i −2815.05 26985.0 + 16486.7i
3.11 −21.8231 − 5.97938i 121.510 440.494 + 260.977i 430.124 + 1329.71i −2651.73 − 726.555i −454.419 + 454.419i −8052.46 − 8329.20i −4918.27 −1435.81 − 31590.2i
3.12 −21.5130 − 7.01367i −40.2909 413.617 + 301.770i −637.187 + 1243.83i 866.777 + 282.587i −6457.63 + 6457.63i −6781.61 − 9392.95i −18059.6 22431.6 − 22289.5i
3.13 −21.3799 + 7.40953i 148.371 402.198 − 316.829i 1347.79 − 369.572i −3172.15 + 1099.36i −4144.58 + 4144.58i −6251.38 + 9753.87i 2330.92 −26077.3 + 17887.9i
3.14 −21.1924 + 7.92985i −20.5791 386.235 − 336.105i 842.754 + 1114.85i 436.120 − 163.189i 8070.25 − 8070.25i −5519.98 + 10185.7i −19259.5 −26700.6 − 16943.4i
3.15 −20.1988 + 10.1985i 198.563 303.982 − 411.994i −1034.37 + 939.790i −4010.73 + 2025.04i 951.975 − 951.975i −1938.36 + 11421.9i 19744.1 11308.6 − 29531.6i
3.16 −20.0411 − 10.5050i −237.515 291.290 + 421.063i 392.185 + 1341.39i 4760.07 + 2495.10i 2682.83 − 2682.83i −1414.52 − 11498.6i 36730.6 6231.43 − 31002.7i
3.17 −20.0199 − 10.5454i 103.099 289.590 + 422.234i 308.191 − 1363.14i −2064.04 − 1087.22i 2060.70 − 2060.70i −1344.93 − 11506.9i −9053.52 −20544.7 + 24039.8i
3.18 −19.7999 − 10.9527i −149.768 272.075 + 433.727i −1158.99 − 780.940i 2965.40 + 1640.37i −1701.60 + 1701.60i −636.585 − 11567.7i 2747.49 14394.5 + 28156.7i
3.19 −19.4293 − 11.5974i 228.197 242.999 + 450.661i −1389.37 − 150.953i −4433.72 − 2646.50i −2921.86 + 2921.86i 505.202 − 11574.2i 32390.9 25243.8 + 19046.0i
3.20 −19.3875 + 11.6672i −76.3175 239.752 − 452.397i 345.726 − 1354.10i 1479.61 − 890.413i 2928.50 − 2928.50i 630.021 + 11568.1i −13858.6 9095.87 + 30286.4i
See next 80 embeddings (of 212 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.106
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 80.10.s.a yes 212
5.c odd 4 1 80.10.j.a ✓ 212
16.f odd 4 1 80.10.j.a ✓ 212
80.s even 4 1 inner 80.10.s.a yes 212
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.10.j.a ✓ 212 5.c odd 4 1
80.10.j.a ✓ 212 16.f odd 4 1
80.10.s.a yes 212 1.a even 1 1 trivial
80.10.s.a yes 212 80.s even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(80, [\chi])\).