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Properties

Label 71.7.h.a
Level $71$
Weight $7$
Character orbit 71.h
Analytic conductor $16.334$
Analytic rank $0$
Dimension $840$
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] = [71,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("71.7");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(16.3338399370\)
Analytic rank: \(0\)
Dimension: \(840\)
Relative dimension: \(35\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 840 q - 10 q^{2} + 55 q^{3} + 1198 q^{4} + 126 q^{5} - 918 q^{6} - 23 q^{7} + 131 q^{8} + 7546 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 840 q - 10 q^{2} + 55 q^{3} + 1198 q^{4} + 126 q^{5} - 918 q^{6} - 23 q^{7} + 131 q^{8} + 7546 q^{9} - 2738 q^{10} - 1463 q^{11} + 3287 q^{12} - 23 q^{13} - 18290 q^{14} + 5228 q^{15} + 68462 q^{16} + 2570 q^{17} - 26745 q^{18} - 6745 q^{19} - 52864 q^{20} + 15937 q^{21} - 19928 q^{22} - 28 q^{23} - 35893 q^{24} - 571092 q^{25} + 66997 q^{26} + 198565 q^{27} + 128154 q^{28} - 40961 q^{29} + 407435 q^{30} - 128519 q^{31} + 401320 q^{32} + 75353 q^{33} - 476 q^{34} + 131968 q^{35} - 200068 q^{36} - 21044 q^{37} + 145205 q^{38} + 123396 q^{39} - 149980 q^{40} - 34188 q^{41} + 509224 q^{42} + 516743 q^{43} - 350483 q^{44} - 1812421 q^{45} + 839500 q^{46} - 397055 q^{47} + 132073 q^{48} - 516660 q^{49} - 275587 q^{50} - 883071 q^{51} + 391433 q^{52} + 251317 q^{53} + 404299 q^{54} - 435352 q^{55} + 2535081 q^{56} + 1392540 q^{57} - 3191058 q^{58} - 1813895 q^{59} - 7905336 q^{60} + 934381 q^{61} - 239618 q^{62} - 1259119 q^{63} + 5909363 q^{64} + 910632 q^{65} + 12964425 q^{66} + 2624713 q^{67} + 2922405 q^{68} + 2751764 q^{69} - 300728 q^{71} - 7000676 q^{72} - 4735993 q^{73} - 2407708 q^{74} - 5888351 q^{75} - 8042728 q^{76} - 7609131 q^{77} - 868580 q^{78} - 602041 q^{79} + 1791625 q^{80} + 5757840 q^{81} + 2634308 q^{82} + 3040503 q^{83} + 20965752 q^{84} + 700170 q^{85} + 5182335 q^{86} - 9283491 q^{87} - 15723405 q^{88} + 5631815 q^{89} + 10615935 q^{90} + 8939142 q^{91} - 12987536 q^{92} - 4837027 q^{93} + 4628120 q^{94} - 5586244 q^{95} + 6473670 q^{96} - 5779648 q^{97} + 8628336 q^{98} + 2652449 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −15.2739 − 4.21533i −1.95932 − 4.58405i 160.582 + 95.9433i −40.0905 − 123.386i 10.6031 + 78.2755i 544.702 + 24.4626i −1347.49 − 1409.37i 486.610 − 508.954i 92.2262 + 2053.58i
7.2 −15.1256 − 4.17440i 18.4728 + 43.2192i 156.418 + 93.4553i 22.7232 + 69.9349i −98.9976 − 730.829i −243.576 − 10.9390i −1281.81 − 1340.67i −1022.87 + 1069.84i −51.7662 − 1152.66i
7.3 −13.0375 − 3.59812i −13.3025 − 31.1228i 102.089 + 60.9956i 38.2454 + 117.707i 61.4482 + 453.628i 44.2938 + 1.98924i −513.342 − 536.914i −287.889 + 301.108i −75.0995 − 1672.22i
7.4 −12.7499 − 3.51876i 0.748793 + 1.75189i 95.2385 + 56.9023i 9.12516 + 28.0843i −3.38259 − 24.9713i −640.489 − 28.7644i −429.073 − 448.776i 501.276 − 524.294i −17.5231 − 390.183i
7.5 −11.7883 − 3.25337i −2.38020 − 5.56876i 73.4391 + 43.8778i −61.5994 − 189.584i 9.94135 + 73.3900i −317.670 − 14.2666i −182.108 − 190.470i 478.439 − 500.408i 109.368 + 2435.27i
7.6 −11.1626 − 3.08068i 8.43097 + 19.7252i 60.1720 + 35.9511i 11.4959 + 35.3807i −33.3443 − 246.158i 185.499 + 8.33077i −48.7661 − 51.0054i 185.781 − 194.312i −19.3273 − 430.355i
7.7 −10.7285 − 2.96088i 8.58642 + 20.0889i 51.3936 + 30.7062i 57.6814 + 177.525i −32.6386 − 240.948i 419.972 + 18.8610i 31.7804 + 33.2397i 173.947 − 181.934i −93.2050 − 2075.37i
7.8 −10.0937 − 2.78569i −17.9722 − 42.0480i 39.1824 + 23.4104i −47.2973 − 145.566i 64.2734 + 474.486i 69.0572 + 3.10136i 132.832 + 138.932i −941.250 + 984.470i 71.9036 + 1601.06i
7.9 −9.90452 − 2.73348i 16.4768 + 38.5493i 35.6870 + 21.3220i −50.4989 − 155.420i −57.8209 − 426.851i 97.4996 + 4.37871i 159.254 + 166.567i −710.779 + 743.417i 75.3319 + 1677.39i
7.10 −6.28533 − 1.73464i −12.2770 − 28.7235i −18.4443 − 11.0200i 14.2323 + 43.8025i 27.3401 + 201.833i −186.361 − 8.36950i 385.193 + 402.880i −170.528 + 178.359i −13.4731 − 300.001i
7.11 −5.56073 − 1.53466i −3.55799 − 8.32432i −26.3742 − 15.7578i 67.7243 + 208.434i 7.00996 + 51.7496i −106.490 − 4.78248i 377.611 + 394.950i 447.150 − 467.682i −56.7205 − 1262.98i
7.12 −5.51078 − 1.52088i −3.98586 − 9.32538i −26.8851 − 16.0631i −20.0486 − 61.7032i 7.78243 + 57.4522i 448.974 + 20.1634i 376.570 + 393.862i 432.709 − 452.578i 16.6403 + 370.525i
7.13 −5.03164 − 1.38864i 8.03612 + 18.8014i −31.5516 − 18.8512i −9.96867 − 30.6804i −14.3264 − 105.761i −396.116 − 17.7896i 363.438 + 380.126i 214.870 − 224.736i 7.55458 + 168.216i
7.14 −4.94555 − 1.36489i 18.2494 + 42.6966i −32.3451 − 19.3253i 49.7071 + 152.983i −31.9774 − 236.067i −399.783 − 17.9543i 360.497 + 377.050i −986.173 + 1031.46i −37.0251 − 824.428i
7.15 −0.973541 − 0.268680i −16.8932 − 39.5235i −54.0651 − 32.3024i −6.63702 − 20.4266i 5.82700 + 43.0167i −586.915 − 26.3584i 88.6231 + 92.6925i −772.946 + 808.439i 0.973174 + 21.6694i
7.16 −0.797558 − 0.220112i 2.13348 + 4.99154i −54.3531 − 32.4744i −70.9212 − 218.273i −0.602881 − 4.45064i −296.563 − 13.3187i 72.7948 + 76.1374i 483.421 − 505.619i 8.51926 + 189.696i
7.17 −0.449020 − 0.123922i 15.3782 + 35.9791i −54.7545 − 32.7143i −26.5640 − 81.7555i −2.44653 − 18.0610i 249.103 + 11.1872i 41.1335 + 43.0223i −554.219 + 579.668i 1.79648 + 40.0017i
7.18 −0.345531 − 0.0953604i −20.5465 − 48.0709i −54.8304 − 32.7597i 61.7134 + 189.934i 2.51538 + 18.5693i 620.967 + 27.8877i 31.6751 + 33.1295i −1384.87 + 1448.46i −3.21166 − 71.5131i
7.19 1.36837 + 0.377645i 12.6133 + 29.5104i −53.2109 − 31.7920i 36.0848 + 111.058i 6.11521 + 45.1443i 528.286 + 23.7253i −123.588 − 129.263i −207.980 + 217.530i 7.43686 + 165.595i
7.20 2.59612 + 0.716483i 3.23565 + 7.57018i −48.7143 − 29.1054i 38.6534 + 118.963i 2.97622 + 21.9714i −263.508 − 11.8342i −224.728 − 235.047i 456.946 − 477.929i 15.1138 + 336.536i
See next 80 embeddings (of 840 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.35
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 71.7.h.a ✓ 840
71.h odd 70 1 inner 71.7.h.a ✓ 840
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.7.h.a ✓ 840 1.a even 1 1 trivial
71.7.h.a ✓ 840 71.h odd 70 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(71, [\chi])\).