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Properties

Label 71.4.d.a
Level $71$
Weight $4$
Character orbit 71.d
Analytic conductor $4.189$
Analytic rank $0$
Dimension $102$
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] = [71,4,Mod(20,71)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(71, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([8]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("71.20");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 71 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 71.d (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: \(4.18913561041\)
Analytic rank: \(0\)
Dimension: \(102\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 102 q - 7 q^{2} + 5 q^{3} - 59 q^{4} - 4 q^{5} - 19 q^{6} - 19 q^{7} - 7 q^{8} - 252 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 102 q - 7 q^{2} + 5 q^{3} - 59 q^{4} - 4 q^{5} - 19 q^{6} - 19 q^{7} - 7 q^{8} - 252 q^{9} - 102 q^{10} - 37 q^{11} + 330 q^{12} - 79 q^{13} + 282 q^{14} - 136 q^{15} - 799 q^{16} + 312 q^{17} - 597 q^{18} - q^{19} - 44 q^{20} - 23 q^{21} + 47 q^{22} + 169 q^{23} + 210 q^{24} + 1818 q^{25} - 886 q^{26} - 1075 q^{27} + 499 q^{28} + 771 q^{29} - 35 q^{30} + 439 q^{31} + 1977 q^{32} - 235 q^{33} - 295 q^{34} - 516 q^{35} + 2513 q^{36} - 21 q^{37} - 12 q^{38} + 1245 q^{39} - 3510 q^{40} - 453 q^{41} + 342 q^{42} - 119 q^{43} - 1005 q^{44} + 1770 q^{45} + 4018 q^{46} - 267 q^{47} - 4575 q^{48} + 1044 q^{49} + 62 q^{50} - 1390 q^{51} - 2679 q^{52} - 819 q^{53} + 230 q^{54} - 3352 q^{55} + 2519 q^{56} - 10634 q^{57} - 3812 q^{58} + 869 q^{59} + 6816 q^{60} + 229 q^{61} - 6276 q^{62} + 2965 q^{63} + 3597 q^{64} - 3906 q^{65} + 7964 q^{66} - 313 q^{67} + 2199 q^{68} + 1753 q^{69} - 748 q^{70} + 1498 q^{71} + 6468 q^{72} + 3227 q^{73} + 2953 q^{74} + 995 q^{75} - 5134 q^{76} + 4285 q^{77} - 1429 q^{78} - 2721 q^{79} - 1174 q^{80} + 4824 q^{81} + 3640 q^{82} + 1219 q^{83} - 13970 q^{84} - 5432 q^{85} - 2025 q^{86} + 1201 q^{87} + 200 q^{88} + 2707 q^{89} - 4197 q^{90} - 3727 q^{91} + 4035 q^{92} + 1329 q^{93} - 4171 q^{94} + 1198 q^{95} + 634 q^{96} + 5965 q^{97} - 13363 q^{98} - 3809 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
20.1 −1.14299 − 5.00776i −1.15595 + 0.556675i −16.5635 + 7.97655i −3.53400 4.10893 + 5.15244i −1.24817 + 5.46858i 33.2558 + 41.7015i −15.8079 + 19.8225i 4.03932 + 17.6974i
20.2 −1.01556 − 4.44947i 7.51618 − 3.61960i −11.5587 + 5.56637i 10.3571 −23.7385 − 29.7671i 1.92096 − 8.41626i 13.7417 + 17.2315i 26.5572 − 33.3017i −10.5183 − 46.0837i
20.3 −0.885062 − 3.87771i −9.00440 + 4.33629i −7.04554 + 3.39295i 10.7775 24.7843 + 31.0786i −3.94038 + 17.2639i −0.446491 − 0.559883i 45.4416 − 56.9820i −9.53873 − 41.7919i
20.4 −0.736288 − 3.22589i 5.78327 − 2.78508i −2.65649 + 1.27930i −20.5360 −13.2425 − 16.6056i 0.898418 − 3.93623i −10.4215 − 13.0681i 8.85535 − 11.1043i 15.1204 + 66.2469i
20.5 −0.703898 − 3.08398i −1.67883 + 0.808484i −1.80771 + 0.870545i 10.7155 3.67508 + 4.60840i 6.49475 − 28.4553i −11.8210 − 14.8231i −14.6694 + 18.3948i −7.54265 − 33.0465i
20.6 −0.492591 − 2.15818i −3.49106 + 1.68121i 2.79265 − 1.34487i −11.2903 5.34801 + 6.70620i −2.61886 + 11.4740i −15.3198 − 19.2104i −7.47317 + 9.37106i 5.56149 + 24.3665i
20.7 −0.405123 − 1.77496i 2.80251 − 1.34962i 4.22139 − 2.03291i 16.8573 −3.53088 − 4.42758i −6.68995 + 29.3106i −14.3996 − 18.0565i −10.8016 + 13.5448i −6.82929 − 29.9211i
20.8 −0.127705 − 0.559514i 6.47117 − 3.11635i 6.91100 − 3.32816i 0.822056 −2.57005 − 3.22273i −1.40447 + 6.15337i −5.60731 − 7.03134i 15.3301 − 19.2234i −0.104981 − 0.459952i
20.9 0.0394326 + 0.172766i −7.58443 + 3.65247i 7.17946 − 3.45744i −8.50493 −0.930095 − 1.16630i 4.16834 − 18.2627i 1.76433 + 2.21240i 27.3488 − 34.2943i −0.335372 − 1.46936i
20.10 0.130707 + 0.572666i 2.86950 − 1.38188i 6.89689 − 3.32137i −2.21170 1.16642 + 1.46264i 4.93751 − 21.6327i 5.73338 + 7.18943i −10.5098 + 13.1789i −0.289086 − 1.26657i
20.11 0.336793 + 1.47559i −4.26137 + 2.05217i 5.14383 − 2.47714i 12.7107 −4.46335 − 5.59686i −0.940880 + 4.12227i 12.9370 + 16.2225i −2.88637 + 3.61940i 4.28088 + 18.7558i
20.12 0.552070 + 2.41877i −1.29344 + 0.622886i 1.66206 − 0.800406i −9.83933 −2.22069 − 2.78465i −7.66836 + 33.5973i 15.2285 + 19.0959i −15.5492 + 19.4981i −5.43200 − 23.7991i
20.13 0.747864 + 3.27661i 8.05716 − 3.88012i −2.96911 + 1.42985i −6.14805 18.7393 + 23.4983i −1.93302 + 8.46911i 9.85822 + 12.3618i 33.0282 − 41.4161i −4.59791 − 20.1448i
20.14 0.840876 + 3.68412i 2.88498 − 1.38933i −5.65791 + 2.72471i 15.3140 7.54437 + 9.46034i 3.36676 − 14.7507i 4.05292 + 5.08220i −10.4414 + 13.0931i 12.8772 + 56.4186i
20.15 0.907276 + 3.97503i −1.28982 + 0.621144i −7.76999 + 3.74183i −20.8618 −3.63929 − 4.56352i 5.47441 − 23.9850i −1.58640 − 1.98928i −15.5564 + 19.5071i −18.9274 − 82.9262i
20.16 1.00486 + 4.40256i −7.45545 + 3.59036i −11.1651 + 5.37680i 3.35159 −23.2984 − 29.2153i −0.468515 + 2.05270i −12.3666 − 15.5073i 25.8589 − 32.4260i 3.36787 + 14.7556i
20.17 1.25128 + 5.48223i 2.45349 − 1.18154i −21.2814 + 10.2486i 1.80042 9.54745 + 11.9721i −3.56288 + 15.6100i −54.7658 − 68.6742i −12.2107 + 15.3117i 2.25284 + 9.87032i
30.1 −4.89137 − 2.35556i 3.88966 + 4.87748i 13.3890 + 16.7892i 4.93212 −7.53659 − 33.0200i −28.6558 + 13.7999i −16.2778 − 71.3177i −2.65230 + 11.6205i −24.1249 − 11.6179i
30.2 −4.42023 − 2.12867i −3.54639 − 4.44704i 10.0193 + 12.5638i 15.1452 6.20960 + 27.2060i 27.7632 − 13.3700i −8.80969 − 38.5978i −1.19117 + 5.21885i −66.9455 − 32.2392i
30.3 −4.13503 − 1.99133i −3.35472 − 4.20668i 8.14521 + 10.2138i −12.7192 5.49499 + 24.0751i −12.6606 + 6.09702i −5.17160 − 22.6583i −0.433990 + 1.90143i 52.5942 + 25.3280i
See next 80 embeddings (of 102 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 20.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
71.d even 7 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 71.4.d.a ✓ 102
71.d even 7 1 inner 71.4.d.a ✓ 102
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
71.4.d.a ✓ 102 1.a even 1 1 trivial
71.4.d.a ✓ 102 71.d even 7 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(71, [\chi])\).