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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Newspace parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 43.e (of order \(7\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.4325560958\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(24\) over \(\Q(\zeta_{7})\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 144q + q^{2} - 33q^{3} - 1531q^{4} - 253q^{5} + 484q^{6} + 3440q^{7} - 6339q^{8} - 7461q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 144q + q^{2} - 33q^{3} - 1531q^{4} - 253q^{5} + 484q^{6} + 3440q^{7} - 6339q^{8} - 7461q^{9} - 3341q^{10} - 2960q^{11} + 13689q^{12} + 21868q^{13} + 39897q^{14} - 29921q^{15} - 190983q^{16} - 47792q^{17} - 8652q^{18} - 57870q^{19} - 29763q^{20} - 5368q^{21} - 158945q^{22} - 4594q^{23} + 615935q^{24} - 395817q^{25} - 393069q^{26} - 12195q^{27} - 310163q^{28} - 351549q^{29} + 446684q^{30} - 1062532q^{31} - 18837q^{32} + 864286q^{33} - 60405q^{34} + 12628q^{35} + 3122128q^{36} - 699954q^{37} - 3468417q^{38} - 353795q^{39} - 128577q^{40} + 2170261q^{41} + 395298q^{42} - 817514q^{43} + 7454824q^{44} - 4456426q^{45} + 4453814q^{46} - 748674q^{47} - 12117143q^{48} + 10546384q^{49} + 3042978q^{50} + 1049484q^{51} - 718239q^{52} + 4445244q^{53} + 6955665q^{54} - 4115285q^{55} - 3152781q^{56} - 9217501q^{57} + 9825959q^{58} + 4995358q^{59} - 2930621q^{60} - 7764277q^{61} + 4452543q^{62} - 16888467q^{63} + 9029845q^{64} - 5616137q^{65} + 11011393q^{66} - 9868514q^{67} + 19838976q^{68} - 21931867q^{69} + 5753942q^{70} + 14147202q^{71} + 7754068q^{72} + 14623156q^{73} - 12810086q^{74} - 8522837q^{75} - 49153968q^{76} - 39877497q^{77} - 24880828q^{78} - 4675142q^{79} + 26108098q^{80} - 10465710q^{81} - 20492413q^{82} - 11604907q^{83} + 63335734q^{84} + 72214364q^{85} - 23830431q^{86} + 86376126q^{87} - 35967739q^{88} + 41753866q^{89} - 47033632q^{90} + 1637756q^{91} - 33725352q^{92} - 79434044q^{93} - 92694615q^{94} - 8060431q^{95} - 98386924q^{96} + 45598073q^{97} - 13251727q^{98} + 111493864q^{99} + O(q^{100}) \)

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.

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 41.24
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 \(S_{8}^{\mathrm{new}}(43, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database