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Properties

Label 43.8.e.a
Level $43$
Weight $8$
Character orbit 43.e
Analytic conductor $13.433$
Analytic rank $0$
Dimension $144$
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] = [43,8,Mod(4,43)]
 
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([4]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("43.4");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 43.e (of order \(7\), 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: \(13.4325560958\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(24\) over \(\Q(\zeta_{7})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{7}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q + q^{2} - 33 q^{3} - 1531 q^{4} - 253 q^{5} + 484 q^{6} + 3440 q^{7} - 6339 q^{8} - 7461 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 144 q + q^{2} - 33 q^{3} - 1531 q^{4} - 253 q^{5} + 484 q^{6} + 3440 q^{7} - 6339 q^{8} - 7461 q^{9} - 3341 q^{10} - 2960 q^{11} + 13689 q^{12} + 21868 q^{13} + 39897 q^{14} - 29921 q^{15} - 190983 q^{16} - 47792 q^{17} - 8652 q^{18} - 57870 q^{19} - 29763 q^{20} - 5368 q^{21} - 158945 q^{22} - 4594 q^{23} + 615935 q^{24} - 395817 q^{25} - 393069 q^{26} - 12195 q^{27} - 310163 q^{28} - 351549 q^{29} + 446684 q^{30} - 1062532 q^{31} - 18837 q^{32} + 864286 q^{33} - 60405 q^{34} + 12628 q^{35} + 3122128 q^{36} - 699954 q^{37} - 3468417 q^{38} - 353795 q^{39} - 128577 q^{40} + 2170261 q^{41} + 395298 q^{42} - 817514 q^{43} + 7454824 q^{44} - 4456426 q^{45} + 4453814 q^{46} - 748674 q^{47} - 12117143 q^{48} + 10546384 q^{49} + 3042978 q^{50} + 1049484 q^{51} - 718239 q^{52} + 4445244 q^{53} + 6955665 q^{54} - 4115285 q^{55} - 3152781 q^{56} - 9217501 q^{57} + 9825959 q^{58} + 4995358 q^{59} - 2930621 q^{60} - 7764277 q^{61} + 4452543 q^{62} - 16888467 q^{63} + 9029845 q^{64} - 5616137 q^{65} + 11011393 q^{66} - 9868514 q^{67} + 19838976 q^{68} - 21931867 q^{69} + 5753942 q^{70} + 14147202 q^{71} + 7754068 q^{72} + 14623156 q^{73} - 12810086 q^{74} - 8522837 q^{75} - 49153968 q^{76} - 39877497 q^{77} - 24880828 q^{78} - 4675142 q^{79} + 26108098 q^{80} - 10465710 q^{81} - 20492413 q^{82} - 11604907 q^{83} + 63335734 q^{84} + 72214364 q^{85} - 23830431 q^{86} + 86376126 q^{87} - 35967739 q^{88} + 41753866 q^{89} - 47033632 q^{90} + 1637756 q^{91} - 33725352 q^{92} - 79434044 q^{93} - 92694615 q^{94} - 8060431 q^{95} - 98386924 q^{96} + 45598073 q^{97} - 13251727 q^{98} + 111493864 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −13.6930 + 17.1704i 16.1961 + 20.3092i −78.8438 − 345.437i −261.650 − 126.004i −570.491 −1281.81 4478.19 + 2156.58i 336.501 − 1474.31i 5746.31 − 2767.28i
4.2 −11.8404 + 14.8473i −33.9228 − 42.5378i −51.7667 − 226.805i 73.7187 + 35.5010i 1033.23 537.033 1790.33 + 862.177i −172.058 + 753.837i −1399.95 + 674.181i
4.3 −10.9550 + 13.7371i 47.1173 + 59.0832i −40.2142 − 176.190i 310.040 + 149.307i −1327.80 256.844 834.593 + 401.919i −784.134 + 3435.52i −5447.55 + 2623.40i
4.4 −10.6610 + 13.3685i 28.9138 + 36.2567i −36.5770 − 160.254i −268.999 − 129.543i −792.950 1688.99 560.390 + 269.870i 8.10929 − 35.5291i 4599.62 − 2215.06i
4.5 −9.93690 + 12.4605i 1.86370 + 2.33701i −28.0389 − 122.846i 289.176 + 139.260i −47.6396 −811.897 −28.6375 − 13.7911i 484.665 − 2123.46i −4608.76 + 2219.46i
4.6 −7.44470 + 9.33536i −31.2055 − 39.1304i −3.24267 − 14.2071i −407.844 − 196.407i 597.612 −473.219 −1220.24 − 587.638i −70.7554 + 310.000i 4869.81 − 2345.18i
4.7 −5.59101 + 7.01091i 18.1552 + 22.7659i 10.5893 + 46.3946i −162.859 − 78.4289i −261.116 −216.009 −1418.62 − 683.170i 297.979 − 1305.53i 1460.41 − 703.295i
4.8 −4.78449 + 5.99956i −52.2910 − 65.5708i 15.3793 + 67.3812i 273.792 + 131.851i 643.582 −1239.27 −1362.80 − 656.292i −1078.53 + 4725.36i −2101.01 + 1011.79i
4.9 −4.39463 + 5.51070i 50.6548 + 63.5192i 17.4277 + 76.3558i −128.537 − 61.9003i −572.644 −1177.27 −1310.22 − 630.967i −982.117 + 4302.94i 905.989 − 436.301i
4.10 −3.90249 + 4.89357i −1.47000 − 1.84332i 19.7651 + 86.5966i 272.275 + 131.121i 14.7571 920.308 −1222.72 − 588.833i 485.416 − 2126.75i −1704.20 + 820.700i
4.11 −1.71133 + 2.14594i −42.9085 − 53.8056i 26.8063 + 117.446i −41.1887 − 19.8354i 188.894 1550.98 −614.443 − 295.900i −567.247 + 2485.27i 113.053 − 54.4434i
4.12 1.49727 − 1.87752i 39.7176 + 49.8043i 27.1994 + 119.168i 407.717 + 196.346i 152.977 121.661 541.410 + 260.729i −416.327 + 1824.05i 979.108 − 471.514i
4.13 1.73999 − 2.18187i −10.7544 − 13.4856i 26.7497 + 117.198i 123.310 + 59.3829i −48.1364 −1435.32 624.093 + 300.547i 420.449 − 1842.11i 344.124 − 165.721i
4.14 2.60908 − 3.27168i −1.73382 − 2.17414i 24.5861 + 107.719i −285.032 − 137.264i −11.6368 191.171 899.157 + 433.011i 484.933 − 2124.63i −1192.76 + 574.401i
4.15 2.92047 − 3.66215i 41.4965 + 52.0349i 23.6005 + 103.400i −227.345 − 109.484i 311.749 1135.34 987.778 + 475.689i −499.024 + 2186.37i −1064.90 + 512.829i
4.16 4.62525 − 5.79988i −36.3827 − 45.6224i 16.2370 + 71.1389i −224.417 − 108.074i −432.884 −299.806 1343.21 + 646.855i −271.054 + 1187.56i −1664.80 + 801.726i
4.17 6.90154 − 8.65426i −34.3931 − 43.1276i 1.21773 + 5.33523i 421.489 + 202.978i −610.603 685.995 1331.12 + 641.035i −190.451 + 834.420i 4665.55 − 2246.81i
4.18 8.24618 − 10.3404i 33.7479 + 42.3185i −10.4413 − 45.7463i 173.526 + 83.5656i 715.880 −747.001 966.122 + 465.260i −165.282 + 724.148i 2295.02 − 1105.23i
4.19 9.61177 − 12.0528i 6.35304 + 7.96646i −24.4006 − 106.906i 89.1946 + 42.9539i 157.082 751.914 254.795 + 122.703i 463.550 − 2030.94i 1375.03 − 662.180i
4.20 10.0321 − 12.5798i 30.9538 + 38.8148i −29.1265 − 127.612i −433.070 − 208.556i 798.812 −1469.95 −41.9462 − 20.2002i −61.7996 + 270.762i −6968.17 + 3355.69i
See next 80 embeddings (of 144 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.e even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.8.e.a ✓ 144
43.e even 7 1 inner 43.8.e.a ✓ 144
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.8.e.a ✓ 144 1.a even 1 1 trivial
43.8.e.a ✓ 144 43.e even 7 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(43, [\chi])\).