Properties

Label 71.9.h.a
Level $71$
Weight $9$
Character orbit 71.h
Analytic conductor $28.924$
Analytic rank $0$
Dimension $1128$
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] = [71,9,Mod(7,71)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(71, base_ring=CyclotomicField(70))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("71.7");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 71.h (of order \(70\), degree \(24\), 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: \(28.9238813143\)
Analytic rank: \(0\)
Dimension: \(1128\)
Relative dimension: \(47\) over \(\Q(\zeta_{70})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{70}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1128 q - 34 q^{2} - 200 q^{3} + 6142 q^{4} - 312 q^{5} + 4314 q^{6} - 23 q^{7} + 4203 q^{8} + 131119 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 1128 q - 34 q^{2} - 200 q^{3} + 6142 q^{4} - 312 q^{5} + 4314 q^{6} - 23 q^{7} + 4203 q^{8} + 131119 q^{9} - 13378 q^{10} - 28493 q^{11} + 106751 q^{12} - 23 q^{13} + 270270 q^{14} + 326894 q^{15} + 320590 q^{16} - 267225 q^{17} + 465063 q^{18} + 164461 q^{19} - 922112 q^{20} - 1058528 q^{21} - 834160 q^{22} - 28 q^{23} - 1595005 q^{24} - 20357418 q^{25} - 5859763 q^{26} + 559582 q^{27} - 1862590 q^{28} + 1148435 q^{29} - 2981293 q^{30} + 2896955 q^{31} - 3113784 q^{32} + 1507532 q^{33} - 1820 q^{34} + 9926960 q^{35} + 27634172 q^{36} + 4992354 q^{37} - 8073651 q^{38} - 6224400 q^{39} + 30598140 q^{40} + 1813805 q^{41} - 30687440 q^{42} - 14049509 q^{43} - 9423227 q^{44} + 64080683 q^{45} - 19044420 q^{46} + 28275112 q^{47} + 75578641 q^{48} - 37937904 q^{49} + 7051413 q^{50} - 9816807 q^{51} - 8786583 q^{52} - 75873248 q^{53} - 18140909 q^{54} + 15219664 q^{55} - 9157911 q^{56} - 52659636 q^{57} - 293681306 q^{58} + 11183788 q^{59} + 303103944 q^{60} + 54556526 q^{61} + 179011702 q^{62} + 295436189 q^{63} + 15051683 q^{64} - 54857203 q^{65} - 760341255 q^{66} - 225494338 q^{67} - 149279003 q^{68} - 298627003 q^{69} + 109325943 q^{71} + 206829820 q^{72} + 220277704 q^{73} + 466648628 q^{74} + 324120817 q^{75} + 280966232 q^{76} + 15005443 q^{77} - 750508604 q^{78} - 340742879 q^{79} - 864278607 q^{80} - 459821136 q^{81} + 31487532 q^{82} - 140115094 q^{83} + 1846686864 q^{84} + 898151760 q^{85} - 417280977 q^{86} + 308106669 q^{87} - 1204914205 q^{88} - 484041601 q^{89} + 99059031 q^{90} - 201768974 q^{91} + 371909104 q^{92} - 139362073 q^{93} - 1305385032 q^{94} - 58691132 q^{95} + 764939286 q^{96} + 630394177 q^{97} - 186880160 q^{98} + 1668019676 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −29.8117 8.22750i −6.23568 14.5891i 601.281 + 359.249i 300.369 + 924.441i 65.8643 + 486.230i 831.034 + 37.3218i −9498.26 9934.40i 4360.10 4560.31i −1348.67 30030.4i
7.2 −27.9049 7.70125i 30.2488 + 70.7707i 499.609 + 298.502i −201.133 619.023i −299.067 2207.80i −1735.55 77.9435i −6521.40 6820.85i 440.568 460.798i 845.329 + 18822.7i
7.3 −27.5284 7.59736i −32.1065 75.1170i 480.332 + 286.985i −39.5634 121.764i 313.152 + 2311.78i −2612.63 117.333i −5990.27 6265.33i −77.6695 + 81.2359i 164.036 + 3652.54i
7.4 −26.3814 7.28081i −53.2959 124.692i 423.207 + 252.855i −71.1650 219.023i 498.164 + 3677.59i 2315.48 + 103.988i −4482.15 4687.97i −8173.58 + 8548.90i 282.767 + 6296.29i
7.5 −25.5518 7.05184i 27.9094 + 65.2972i 383.402 + 229.072i −57.6539 177.440i −252.668 1865.27i 3990.36 + 179.207i −3491.80 3652.14i 1049.27 1097.45i 221.877 + 4940.48i
7.6 −24.9077 6.87410i 55.6598 + 130.223i 353.379 + 211.134i 176.810 + 544.167i −491.197 3626.16i 593.892 + 26.6717i −2779.31 2906.93i −9325.84 + 9754.06i −663.293 14769.4i
7.7 −24.6515 6.80337i −30.6649 71.7441i 341.646 + 204.124i −287.767 885.657i 267.833 + 1977.22i 1192.92 + 53.5740i −2509.15 2624.37i 327.181 342.204i 1068.43 + 23790.5i
7.8 −20.7036 5.71382i −0.572781 1.34009i 176.227 + 105.291i 86.7721 + 267.057i 4.20159 + 31.0174i −743.577 33.3941i 752.725 + 787.288i 4532.59 4740.72i −270.575 6024.83i
7.9 −20.3971 5.62924i 29.4906 + 68.9966i 164.590 + 98.3379i 294.687 + 906.955i −213.123 1573.34i −3342.08 150.093i 939.804 + 982.958i 643.219 672.755i −905.301 20158.1i
7.10 −20.0821 5.54230i −60.7634 142.163i 152.809 + 91.2992i 335.668 + 1033.08i 432.345 + 3191.70i −3094.13 138.958i 1122.86 + 1174.42i −11984.1 + 12534.4i −1015.27 22606.8i
7.11 −18.9677 5.23475i −22.9381 53.6663i 112.608 + 67.2802i 235.044 + 723.392i 154.153 + 1138.00i 4231.00 + 190.014i 1697.34 + 1775.28i 2180.15 2280.26i −671.471 14951.5i
7.12 −17.0539 4.70658i 1.56231 + 3.65520i 48.9215 + 29.2292i −355.084 1092.84i −9.43996 69.6886i 2393.30 + 107.483i 2433.10 + 2544.82i 4523.14 4730.84i 912.050 + 20308.4i
7.13 −16.2933 4.49666i −10.8948 25.4898i 25.4877 + 15.2282i −10.5522 32.4764i 62.8940 + 464.302i −1827.73 82.0833i 2643.43 + 2764.81i 4003.03 4186.84i 25.8949 + 576.596i
7.14 −15.9670 4.40661i 51.3872 + 120.226i 15.7637 + 9.41834i −267.295 822.651i −290.709 2146.10i −3970.61 178.320i 2720.16 + 2845.06i −7279.68 + 7613.94i 642.803 + 14313.1i
7.15 −14.3352 3.95627i −43.7785 102.425i −29.9160 17.8740i 78.9573 + 243.005i 222.354 + 1641.48i 1308.82 + 58.7790i 2989.02 + 3126.27i −4040.23 + 4225.75i −170.475 3795.92i
7.16 −13.6164 3.75789i 55.4877 + 129.820i −48.4779 28.9642i −100.486 309.264i −267.694 1976.20i 2624.71 + 117.876i 3050.21 + 3190.27i −9240.25 + 9664.54i 206.078 + 4588.69i
7.17 −12.8627 3.54987i −41.6820 97.5198i −66.9162 39.9806i −275.639 848.330i 189.959 + 1402.33i −4784.63 214.878i 3079.43 + 3220.83i −3238.67 + 3387.38i 533.993 + 11890.3i
7.18 −11.4259 3.15335i 24.4083 + 57.1060i −99.1551 59.2424i −100.071 307.987i −98.8116 729.457i 342.125 + 15.3648i 3043.07 + 3182.81i 1868.73 1954.53i 172.212 + 3834.60i
7.19 −7.16801 1.97824i 39.8175 + 93.1578i −172.296 102.942i 178.026 + 547.908i −101.124 746.525i −656.248 29.4721i 2346.89 + 2454.65i −2558.87 + 2676.37i −192.197 4279.59i
7.20 −6.32703 1.74615i −53.6449 125.509i −182.781 109.206i −184.357 567.392i 120.256 + 887.768i 1980.49 + 88.9441i 2126.94 + 2224.61i −8340.55 + 8723.53i 175.680 + 3911.82i
See next 80 embeddings (of 1128 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.47
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.h odd 70 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 71.9.h.a 1128
71.h odd 70 1 inner 71.9.h.a 1128
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.9.h.a 1128 1.a even 1 1 trivial
71.9.h.a 1128 71.h odd 70 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(71, [\chi])\).