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Properties

Label 47.7.d.a
Level $47$
Weight $7$
Character orbit 47.d
Analytic conductor $10.813$
Analytic rank $0$
Dimension $506$
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] = [47,7,Mod(5,47)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(47, base_ring=CyclotomicField(46))
 
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("47.5");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 47 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 47.d (of order \(46\), degree \(22\), 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: \(10.8125419301\)
Analytic rank: \(0\)
Dimension: \(506\)
Relative dimension: \(23\) over \(\Q(\zeta_{46})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{46}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 506 q - 21 q^{2} + 11 q^{3} - 857 q^{4} - 23 q^{5} + 717 q^{6} - 165 q^{7} - 589 q^{8} - 2740 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 506 q - 21 q^{2} + 11 q^{3} - 857 q^{4} - 23 q^{5} + 717 q^{6} - 165 q^{7} - 589 q^{8} - 2740 q^{9} - 23 q^{10} - 23 q^{11} + 5240 q^{12} - 23 q^{13} - 10068 q^{14} - 23 q^{15} - 21465 q^{16} - 165 q^{17} + 130 q^{18} - 23 q^{19} - 23 q^{20} + 35109 q^{21} - 23 q^{22} - 23 q^{23} + 5952 q^{24} + 29978 q^{25} - 23 q^{26} + 60725 q^{27} - 12167 q^{28} - 23 q^{29} - 23 q^{30} - 23 q^{31} - 98770 q^{32} - 23 q^{33} - 219559 q^{34} + 348105 q^{35} - 536757 q^{36} - 288957 q^{37} - 478423 q^{38} - 328463 q^{39} + 278185 q^{40} + 540937 q^{41} + 1868844 q^{42} + 281497 q^{43} + 942057 q^{44} + 678017 q^{45} - 162670 q^{47} - 2364698 q^{48} - 930692 q^{49} - 1417297 q^{50} - 1978915 q^{51} - 1059863 q^{52} - 448429 q^{53} + 2079412 q^{54} + 1564705 q^{55} + 3225697 q^{56} + 2428777 q^{57} + 1424137 q^{58} + 1240667 q^{59} - 2802711 q^{60} - 1875477 q^{61} - 23 q^{62} - 1114025 q^{63} - 599497 q^{64} - 1048607 q^{65} - 23 q^{66} - 23 q^{67} + 149348 q^{68} - 23 q^{69} - 23 q^{70} - 656637 q^{71} - 1995273 q^{72} - 23 q^{73} + 188501 q^{74} + 1628915 q^{75} + 10160503 q^{76} + 3293577 q^{77} - 2738633 q^{78} - 6817245 q^{79} - 13225690 q^{80} - 11391226 q^{81} - 11100398 q^{82} - 2463729 q^{83} + 4770224 q^{84} + 5904169 q^{85} + 9960081 q^{86} + 6334177 q^{87} + 20778637 q^{88} + 5973763 q^{89} + 23339641 q^{90} + 10765357 q^{91} + 5181624 q^{92} - 4090293 q^{94} - 5773526 q^{95} - 23950599 q^{96} - 9210333 q^{97} - 21213872 q^{98} - 22691363 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −5.11465 − 14.3912i −4.71685 − 22.6987i −131.303 + 106.823i −180.384 − 12.3386i −302.538 + 183.977i 141.845 − 200.948i 1373.70 + 835.365i 175.663 − 76.3013i 745.034 + 2659.06i
5.2 −4.86000 − 13.6747i 0.306700 + 1.47592i −113.733 + 92.5285i 141.235 + 9.66075i 18.6923 − 11.3670i −183.568 + 260.057i 1024.45 + 622.981i 666.563 − 289.529i −554.294 − 1978.30i
5.3 −4.12252 − 11.5997i 8.16454 + 39.2899i −67.9116 + 55.2502i 26.0746 + 1.78355i 422.092 − 256.680i −10.7660 + 15.2519i 247.680 + 150.618i −808.391 + 351.134i −86.8046 − 309.810i
5.4 −3.73398 − 10.5064i −9.64912 − 46.4341i −46.7966 + 38.0718i 63.8310 + 4.36616i −451.826 + 274.762i −134.516 + 190.566i −34.9896 − 21.2777i −1394.37 + 605.662i −192.471 − 686.937i
5.5 −3.71774 − 10.4607i 4.52796 + 21.7897i −45.9597 + 37.3910i −181.075 − 12.3859i 211.103 − 128.374i 4.80428 − 6.80611i −45.0705 − 27.4080i 214.357 − 93.1084i 543.626 + 1940.23i
5.6 −3.36415 − 9.46580i −2.69034 − 12.9466i −28.6383 + 23.2990i 125.285 + 8.56975i −113.499 + 69.0205i 348.901 − 494.280i −232.448 − 141.354i 508.270 − 220.773i −340.359 − 1214.76i
5.7 −2.32982 − 6.55549i −4.61263 − 22.1972i 12.0992 − 9.84341i −102.502 − 7.01135i −134.767 + 81.9535i −204.857 + 290.216i −473.156 − 287.733i 197.208 − 85.6597i 192.849 + 688.288i
5.8 −1.57856 − 4.44164i 1.29510 + 6.23237i 32.4092 − 26.3668i −49.3905 − 3.37841i 25.6376 − 15.5906i 20.4348 − 28.9495i −426.037 − 259.079i 631.482 − 274.291i 62.9603 + 224.708i
5.9 −1.37824 − 3.87798i 4.55116 + 21.9014i 36.5063 − 29.7001i 196.998 + 13.4750i 78.6607 − 47.8347i −310.172 + 439.414i −390.544 − 237.495i 209.689 − 91.0806i −219.254 − 782.527i
5.10 −0.981718 − 2.76229i 8.89442 + 42.8023i 42.9791 − 34.9661i 79.9515 + 5.46883i 109.501 − 66.5888i 282.210 − 399.801i −299.085 − 181.878i −1084.28 + 470.970i −63.3833 − 226.218i
5.11 −0.400717 − 1.12751i −9.53739 − 45.8965i 48.5348 − 39.4860i −210.970 − 14.4308i −47.9269 + 29.1450i 357.081 − 505.869i −129.403 − 78.6917i −1346.88 + 585.031i 68.2686 + 243.654i
5.12 0.271106 + 0.762819i −7.36377 − 35.4364i 49.1371 − 39.9760i 202.591 + 13.8576i 25.0352 − 15.2242i −11.0044 + 15.5897i 88.0850 + 53.5657i −532.867 + 231.457i 44.3528 + 158.297i
5.13 0.384047 + 1.08061i −4.68156 − 22.5289i 48.6253 − 39.5596i 64.0738 + 4.38277i 22.5469 − 13.7111i 13.1986 − 18.6981i 124.134 + 75.4877i 183.012 − 79.4933i 19.8713 + 70.9217i
5.14 0.434371 + 1.22220i 8.01246 + 38.5581i 48.3404 − 39.3278i −187.985 − 12.8586i −43.6454 + 26.5414i −166.231 + 235.496i 139.993 + 85.1318i −753.879 + 327.456i −65.9397 − 235.342i
5.15 1.54420 + 4.34496i 1.64064 + 7.89520i 33.1514 − 26.9707i −76.6135 − 5.24050i −31.7708 + 19.3203i 197.591 − 279.922i 420.533 + 255.731i 609.005 − 264.528i −95.5366 − 340.975i
5.16 2.29336 + 6.45290i −3.25869 − 15.6817i 13.2651 − 10.7919i −83.0294 − 5.67937i 93.7191 − 56.9918i −319.154 + 452.139i 474.546 + 288.578i 433.351 − 188.231i −153.768 − 548.806i
5.17 2.62030 + 7.37281i 7.63618 + 36.7473i 2.15317 − 1.75173i 72.2377 + 4.94119i −250.922 + 152.589i −248.010 + 351.350i 446.428 + 271.479i −623.406 + 270.783i 152.854 + 545.542i
5.18 2.89395 + 8.14279i 2.60156 + 12.5194i −8.28455 + 6.73998i 185.699 + 12.7022i −94.4141 + 57.4145i 134.004 − 189.841i 393.698 + 239.413i 518.680 − 225.294i 433.973 + 1548.87i
5.19 3.05950 + 8.60862i −9.45315 − 45.4911i −15.1023 + 12.2866i −42.7527 − 2.92437i 362.693 − 220.559i −127.271 + 180.302i 347.614 + 211.388i −1311.43 + 569.633i −105.627 − 376.989i
5.20 4.20126 + 11.8212i −1.12357 − 5.40689i −72.4451 + 58.9384i −229.877 − 15.7241i 59.1957 − 35.9977i 114.125 − 161.678i −315.056 − 191.590i 640.675 − 278.284i −779.898 − 2783.49i
See next 80 embeddings (of 506 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.23
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
47.d odd 46 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 47.7.d.a ✓ 506
47.d odd 46 1 inner 47.7.d.a ✓ 506
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
47.7.d.a ✓ 506 1.a even 1 1 trivial
47.7.d.a ✓ 506 47.d odd 46 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(47, [\chi])\).