↑

Properties

Label 43.7.f.a
Level $43$
Weight $7$
Character orbit 43.f
Analytic conductor $9.892$
Analytic rank $0$
Dimension $126$
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] = [43,7,Mod(2,43)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(43, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([9]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("43.2");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 43.f (of order \(14\), degree \(6\), 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: \(9.89232559565\)
Analytic rank: \(0\)
Dimension: \(126\)
Relative dimension: \(21\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 126 q - 7 q^{2} - 7 q^{3} + 725 q^{4} - 7 q^{5} + 244 q^{6} + 2233 q^{8} + 6312 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 126 q - 7 q^{2} - 7 q^{3} + 725 q^{4} - 7 q^{5} + 244 q^{6} + 2233 q^{8} + 6312 q^{9} + 1955 q^{10} - 985 q^{11} - 455 q^{12} - 4485 q^{13} - 6771 q^{14} + 5017 q^{15} - 41935 q^{16} - 10409 q^{17} - 5110 q^{18} - 33397 q^{19} - 24507 q^{20} - 5448 q^{21} + 25193 q^{22} - 29017 q^{23} - 1885 q^{24} - 2080 q^{25} + 34713 q^{26} - 7 q^{27} + 28665 q^{28} + 17353 q^{29} - 290052 q^{30} + 13547 q^{31} + 458283 q^{32} - 94801 q^{33} - 191527 q^{34} + 183008 q^{35} + 956352 q^{36} + 304607 q^{38} - 381031 q^{39} + 66419 q^{40} + 313447 q^{41} - 323228 q^{43} + 281588 q^{44} - 1237551 q^{45} + 80948 q^{46} - 180133 q^{47} - 762055 q^{48} - 1861754 q^{49} + 999215 q^{51} + 471797 q^{52} + 1030011 q^{53} - 92099 q^{54} + 324233 q^{55} - 842713 q^{56} + 331037 q^{57} + 849461 q^{58} - 369609 q^{59} + 553323 q^{60} - 594727 q^{61} - 886711 q^{62} + 1914136 q^{63} - 3250219 q^{64} + 1473521 q^{65} - 427559 q^{66} + 122535 q^{67} - 602346 q^{68} + 4320057 q^{69} - 1694994 q^{70} - 770707 q^{71} - 391202 q^{72} + 476945 q^{73} - 3943812 q^{74} + 2061367 q^{75} - 2044462 q^{76} + 3016216 q^{77} + 239350 q^{78} + 5077268 q^{79} + 2388946 q^{81} + 5820437 q^{82} - 1324235 q^{83} + 3759618 q^{84} - 4967947 q^{86} - 3537894 q^{87} - 7289555 q^{88} - 4331495 q^{89} - 12825206 q^{90} + 1802976 q^{91} - 12038260 q^{92} + 6249341 q^{94} - 5313259 q^{95} + 7069598 q^{96} + 788975 q^{97} + 4724783 q^{98} - 2216971 q^{99}+O(q^{100}) \) 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 −6.68851 − 13.8888i 4.36630 − 9.06671i −108.260 + 135.754i 1.12402 + 0.256549i −155.130 114.871i 1647.71 + 376.078i 391.383 + 490.779i −3.95482 − 17.3272i
2.2 −5.62328 − 11.6769i −22.1545 + 46.0043i −64.8243 + 81.2871i 43.4728 + 9.92237i 661.767 306.680i 505.038 + 115.272i −1171.05 − 1468.45i −128.597 − 563.421i
2.3 −5.10721 − 10.6052i −7.80492 + 16.2071i −46.4840 + 58.2891i −223.373 − 50.9834i 211.741 − 193.213i 121.123 + 27.6454i 252.771 + 316.965i 600.121 + 2629.30i
2.4 −5.02991 − 10.4447i 20.6257 − 42.8298i −43.8887 + 55.0347i 127.013 + 28.9898i −551.091 − 619.099i 72.2435 + 16.4891i −954.446 − 1196.84i −336.072 − 1472.43i
2.5 −4.79242 − 9.95156i −6.97408 + 14.4818i −36.1629 + 45.3469i 117.045 + 26.7147i 177.540 − 466.480i −64.6015 − 14.7449i 293.438 + 367.960i −295.075 − 1292.81i
2.6 −4.35509 − 9.04344i 15.2129 − 31.5900i −22.9137 + 28.7328i −93.5256 − 21.3466i −351.936 345.453i −266.657 − 60.8626i −311.968 − 391.196i 214.266 + 938.759i
2.7 −3.64714 − 7.57336i 2.80632 − 5.82738i −4.15080 + 5.20494i 190.781 + 43.5445i −54.3678 593.422i −469.925 − 107.257i 428.441 + 537.248i −366.026 − 1603.66i
2.8 −2.72088 − 5.64997i −8.85233 + 18.3821i 15.3844 − 19.2915i −52.8314 − 12.0584i 127.944 − 8.48034i −542.136 − 123.739i 194.988 + 244.507i 75.6182 + 331.305i
2.9 −1.12993 − 2.34633i 12.1637 − 25.2581i 35.6748 − 44.7348i −135.516 − 30.9307i −73.0081 − 261.823i −307.765 − 70.2454i −35.4943 − 44.5085i 80.5506 + 352.915i
2.10 −0.716329 − 1.48747i 3.40007 − 7.06031i 38.2039 − 47.9062i 69.5466 + 15.8736i −12.9376 − 168.130i −201.638 − 46.0227i 416.237 + 521.944i −26.2068 − 114.819i
2.11 −0.546272 − 1.13435i −16.4415 + 34.1411i 38.9150 − 48.7979i −89.4697 − 20.4209i 47.7093 356.575i −155.169 − 35.4164i −440.768 − 552.706i 25.7105 + 112.645i
2.12 −0.0322332 − 0.0669329i −17.7426 + 36.8430i 39.8999 − 50.0329i 157.975 + 36.0568i 3.03791 − 203.317i −9.27029 − 2.11588i −588.078 − 737.427i −2.67865 − 11.7360i
2.13 1.11483 + 2.31496i 18.6979 − 38.8265i 35.7871 − 44.8756i 137.550 + 31.3949i 110.727 167.161i 304.102 + 69.4092i −703.365 − 881.992i 80.6666 + 353.424i
2.14 2.61426 + 5.42856i 1.01637 − 2.11052i 17.2684 − 21.6539i −102.842 − 23.4731i 14.1142 486.006i 538.641 + 122.941i 451.103 + 565.665i −141.431 − 619.650i
2.15 2.79658 + 5.80715i −16.1345 + 33.5037i 14.0012 − 17.5569i −177.830 − 40.5886i −239.682 − 522.371i 543.278 + 124.000i −407.649 − 511.176i −261.612 − 1146.20i
2.16 3.53371 + 7.33783i −9.42884 + 19.5792i −1.45324 + 1.82231i 197.653 + 45.1129i −176.987 222.322i 489.664 + 111.763i 160.083 + 200.737i 367.416 + 1609.76i
2.17 3.58164 + 7.43735i 3.11543 − 6.46926i −2.58264 + 3.23853i 50.0201 + 11.4168i 59.2725 − 578.950i 481.727 + 109.951i 422.379 + 529.646i 94.2435 + 412.908i
2.18 4.32423 + 8.97936i 20.4277 − 42.4185i −22.0266 + 27.6205i −183.723 − 41.9336i 469.225 − 28.9921i 278.591 + 63.5867i −927.514 − 1163.07i −417.925 − 1831.05i
2.19 5.62961 + 11.6900i −16.3881 + 34.0303i −65.0604 + 81.5832i 15.2899 + 3.48982i −490.074 124.671i −510.398 − 116.495i −434.969 − 545.434i 45.2802 + 198.386i
2.20 5.84606 + 12.1395i 12.7263 − 26.4263i −73.2871 + 91.8991i 119.457 + 27.2654i 395.200 91.2949i −703.346 − 160.534i −81.8698 − 102.661i 367.368 + 1609.55i
See next 80 embeddings (of 126 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.21
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.f odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.7.f.a ✓ 126
43.f odd 14 1 inner 43.7.f.a ✓ 126
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.7.f.a ✓ 126 1.a even 1 1 trivial
43.7.f.a ✓ 126 43.f odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(43, [\chi])\).