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Properties

Label 75.9.h.a
Level $75$
Weight $9$
Character orbit 75.h
Analytic conductor $30.553$
Analytic rank $0$
Dimension $312$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,9,Mod(14,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.14");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 75.h (of order \(10\), degree \(4\), 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: \(30.5533957546\)
Analytic rank: \(0\)
Dimension: \(312\)
Relative dimension: \(78\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 312 q - 5 q^{3} - 9734 q^{4} + 509 q^{6} + 7907 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 312 q - 5 q^{3} - 9734 q^{4} + 509 q^{6} + 7907 q^{9} + 14020 q^{10} + 89595 q^{12} - 10 q^{13} + 229305 q^{15} - 984170 q^{16} - 215676 q^{19} + 322110 q^{21} + 1011190 q^{22} - 858332 q^{24} - 524440 q^{25} + 752320 q^{27} + 501750 q^{28} - 3563465 q^{30} - 137826 q^{31} - 3326380 q^{33} - 3352182 q^{34} + 1480071 q^{36} - 18160810 q^{37} + 6122545 q^{39} - 7576300 q^{40} - 19813675 q^{42} - 13693905 q^{45} + 5724198 q^{46} - 1395750 q^{48} - 219341528 q^{49} - 4033356 q^{51} - 45870200 q^{52} - 11569756 q^{54} + 16903430 q^{55} + 44829290 q^{58} - 154497850 q^{60} + 25290294 q^{61} - 19840425 q^{63} - 114049486 q^{64} + 23158870 q^{66} + 126689140 q^{67} + 3877679 q^{69} + 114378120 q^{70} - 180619525 q^{72} + 99769290 q^{73} + 76194020 q^{75} + 213385472 q^{76} + 111822625 q^{78} - 106524466 q^{79} + 147986067 q^{81} - 46036320 q^{84} + 306911390 q^{85} + 209395370 q^{87} + 65557890 q^{88} + 332360180 q^{90} + 49394050 q^{91} + 436958008 q^{94} + 121790792 q^{96} + 459701190 q^{97} + 368603340 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
14.1 −25.6605 + 18.6435i −48.2910 − 65.0306i 231.776 − 713.332i 67.6825 + 621.324i 2451.57 + 768.408i − 1262.78i 4842.33 + 14903.2i −1896.96 + 6280.79i −13320.4 − 14681.7i
14.2 −25.4144 + 18.4646i −59.0839 + 55.4084i 225.839 − 695.062i −560.608 − 276.304i 478.483 − 2499.13i 1022.59i 4609.39 + 14186.2i 420.809 − 6547.49i 19349.3 − 3329.33i
14.3 −23.8636 + 17.3379i 79.8661 + 13.5058i 189.759 − 584.018i 16.9081 − 624.771i −2140.05 + 1062.41i − 3660.43i 3263.86 + 10045.1i 6196.19 + 2157.32i 10428.7 + 15202.4i
14.4 −23.0990 + 16.7824i 9.41452 + 80.4510i 172.806 − 531.842i 624.983 + 4.61001i −1567.63 − 1700.34i 222.249i 2675.25 + 8233.57i −6383.73 + 1514.82i −14513.8 + 10382.2i
14.5 −22.8983 + 16.6366i 72.3223 − 36.4757i 168.448 − 518.428i 624.768 + 17.0121i −1049.23 + 2038.43i 2817.17i 2528.64 + 7782.36i 3900.04 − 5276.02i −14589.2 + 10004.5i
14.6 −22.7133 + 16.5022i 67.8683 + 44.2141i 164.463 − 506.165i −246.775 + 574.219i −2271.14 + 115.726i 2811.00i 2396.35 + 7375.19i 2651.22 + 6001.48i −3870.77 − 17114.7i
14.7 −22.2790 + 16.1866i 39.2288 − 70.8668i 155.238 − 477.774i −615.859 − 106.502i 273.117 + 2213.82i 2155.09i 2096.49 + 6452.33i −3483.21 − 5560.04i 15444.6 − 7595.94i
14.8 −21.7195 + 15.7802i −4.20061 − 80.8910i 143.616 − 442.005i 39.5040 − 623.750i 1367.71 + 1690.63i − 797.376i 1731.83 + 5330.03i −6525.71 + 679.583i 8984.87 + 14170.9i
14.9 −21.4718 + 15.6002i −77.7877 − 22.5849i 138.565 − 426.459i 474.402 − 406.900i 2022.57 − 728.564i 1038.32i 1578.02 + 4856.64i 5540.84 + 3513.65i −3838.57 + 16137.7i
14.10 −21.0004 + 15.2577i 21.6916 + 78.0415i 129.111 − 397.363i −530.378 + 330.642i −1646.26 − 1307.94i − 4080.13i 1297.96 + 3994.72i −5619.95 + 3385.69i 6093.30 − 15035.9i
14.11 −20.1876 + 14.6672i 65.8472 − 47.1715i 113.306 − 348.720i −10.8504 + 624.906i −637.425 + 1918.07i − 2570.63i 853.340 + 2626.31i 2110.70 − 6212.22i −8946.55 − 12774.5i
14.12 −19.5520 + 14.2053i −67.0891 + 45.3878i 101.380 − 312.015i 349.479 + 518.159i 666.976 − 1840.45i − 1921.07i 538.248 + 1656.56i 2440.90 − 6090.05i −14193.7 − 5166.57i
14.13 −18.8000 + 13.6590i −75.0376 − 30.5018i 87.7640 − 270.110i −584.080 − 222.430i 1827.33 − 451.505i − 2583.17i 201.140 + 619.045i 4700.28 + 4577.56i 14018.9 − 3796.27i
14.14 −18.0666 + 13.1261i −80.1960 + 11.3845i 74.9973 − 230.818i −327.176 + 532.523i 1299.43 − 1258.34i 4310.48i −91.8061 − 282.550i 6301.79 − 1825.98i −1079.03 − 13915.4i
14.15 −18.0586 + 13.1203i 53.6805 + 60.6581i 74.8614 − 230.400i −279.152 − 559.195i −1765.25 − 391.094i 3019.69i −94.7984 − 291.759i −797.810 + 6512.31i 12377.9 + 6435.71i
14.16 −17.4091 + 12.6484i −39.1771 + 70.8954i 63.9847 − 196.925i 111.859 − 614.909i −214.678 − 1729.75i 1610.32i −325.442 − 1001.61i −3491.30 − 5554.95i 5830.27 + 12119.8i
14.17 −16.4605 + 11.9592i −22.1397 − 77.9156i 48.8152 − 150.238i 498.223 + 377.357i 1296.24 + 1017.75i 4096.60i −616.352 − 1896.94i −5580.67 + 3450.05i −12713.9 − 253.111i
14.18 −15.9050 + 11.5557i 10.6351 − 80.2988i 40.3279 − 124.116i 624.153 − 32.5283i 758.756 + 1400.05i − 4310.11i −762.412 − 2346.46i −6334.79 − 1707.97i −9551.29 + 7729.88i
14.19 −15.4924 + 11.2559i −33.5962 − 73.7041i 34.2115 − 105.292i −557.929 + 281.674i 1350.09 + 763.699i 172.487i −859.760 − 2646.07i −4303.59 + 4952.36i 5473.18 − 10643.8i
14.20 −13.9097 + 10.1060i 79.4483 + 15.7785i 12.2408 − 37.6733i 623.492 + 43.3919i −1264.56 + 583.432i − 251.140i −1149.68 − 3538.35i 6063.08 + 2507.15i −9111.12 + 5697.45i
See next 80 embeddings (of 312 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 14.78
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.e even 10 1 inner
75.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.9.h.a ✓ 312
3.b odd 2 1 inner 75.9.h.a ✓ 312
25.e even 10 1 inner 75.9.h.a ✓ 312
75.h odd 10 1 inner 75.9.h.a ✓ 312
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.9.h.a ✓ 312 1.a even 1 1 trivial
75.9.h.a ✓ 312 3.b odd 2 1 inner
75.9.h.a ✓ 312 25.e even 10 1 inner
75.9.h.a ✓ 312 75.h odd 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(75, [\chi])\).