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Properties

Label 69.7.h.a
Level $69$
Weight $7$
Character orbit 69.h
Analytic conductor $15.874$
Analytic rank $0$
Dimension $460$
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] = [69,7,Mod(2,69)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(69, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 2]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("69.2");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 69.h (of order \(22\), degree \(10\), 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: \(15.8737317698\)
Analytic rank: \(0\)
Dimension: \(460\)
Relative dimension: \(46\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 460 q + 23 q^{3} + 1258 q^{4} - 106 q^{6} - 18 q^{7} - 921 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 460 q + 23 q^{3} + 1258 q^{4} - 106 q^{6} - 18 q^{7} - 921 q^{9} - 1774 q^{10} - 630 q^{12} - 1842 q^{13} - 18179 q^{15} - 44990 q^{16} - 25536 q^{18} - 7794 q^{19} - 49959 q^{21} + 7720 q^{22} + 191290 q^{24} + 97900 q^{25} + 85007 q^{27} + 62574 q^{28} - 256521 q^{30} + 1662 q^{31} + 208277 q^{33} - 367948 q^{34} + 123802 q^{36} + 364110 q^{37} - 106127 q^{39} - 869392 q^{40} - 63343 q^{42} - 464490 q^{43} - 455494 q^{45} + 542498 q^{46} + 386726 q^{48} + 1394576 q^{49} + 218821 q^{51} + 592482 q^{52} + 604509 q^{54} - 1673998 q^{55} - 1160439 q^{57} - 1805128 q^{58} + 445865 q^{60} + 422670 q^{61} + 1245641 q^{63} + 2762020 q^{64} + 2925772 q^{66} + 255486 q^{67} - 770451 q^{69} - 1997900 q^{70} - 6119375 q^{72} - 2072562 q^{73} - 6740449 q^{75} + 1407414 q^{76} + 10410184 q^{78} + 1925886 q^{79} + 5228775 q^{81} + 3802304 q^{82} - 6586407 q^{84} + 229034 q^{85} - 6285451 q^{87} + 2567206 q^{88} - 6481942 q^{90} + 3137676 q^{91} + 10359558 q^{93} - 6984096 q^{94} + 18158553 q^{96} - 535890 q^{97} + 5359789 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −8.32833 + 12.9591i −26.9990 − 0.234738i −71.9916 − 157.640i −22.6897 + 77.2739i 227.898 − 347.929i 42.6553 − 49.2269i 1666.59 + 239.619i 728.890 + 12.6754i −812.437 − 937.602i
2.2 −8.20003 + 12.7595i 2.39070 − 26.8939i −68.9776 − 151.040i 54.2696 − 184.825i 323.549 + 251.035i 122.578 − 141.462i 1531.99 + 220.266i −717.569 − 128.591i 1913.26 + 2208.02i
2.3 −7.93618 + 12.3489i 9.25924 + 25.3627i −62.9267 − 137.790i −30.4794 + 103.803i −386.685 − 86.9411i −356.362 + 411.263i 1271.06 + 182.750i −557.533 + 469.679i −1039.97 − 1200.19i
2.4 −7.89511 + 12.2850i 23.0414 − 14.0746i −62.0027 − 135.767i −55.1270 + 187.745i −9.00771 + 394.185i 221.214 − 255.295i 1232.32 + 177.181i 332.812 − 648.596i −1871.22 − 2159.51i
2.5 −7.12881 + 11.0926i −12.5742 + 23.8933i −45.6402 − 99.9381i 59.5858 − 202.930i −175.400 − 309.812i −90.1777 + 104.071i 598.632 + 86.0703i −412.777 − 600.880i 1826.26 + 2107.61i
2.6 −6.88172 + 10.7082i 18.5992 + 19.5722i −40.7202 − 89.1647i 16.4113 − 55.8918i −337.576 + 64.4730i 370.080 − 427.095i 428.662 + 61.6324i −37.1400 + 728.053i 485.560 + 560.366i
2.7 −6.67610 + 10.3882i −9.18523 − 25.3896i −36.7581 − 80.4889i −23.3451 + 79.5062i 325.074 + 74.0853i −342.298 + 395.033i 299.277 + 43.0296i −560.263 + 466.419i −670.073 − 773.306i
2.8 −6.66382 + 10.3691i 26.9817 + 0.992725i −36.5252 − 79.9791i 19.3246 − 65.8136i −190.095 + 273.161i −98.7161 + 113.924i 291.888 + 41.9672i 727.029 + 53.5709i 553.652 + 638.948i
2.9 −6.14428 + 9.56068i −26.2848 − 6.17310i −27.0679 − 59.2704i 20.5398 − 69.9520i 220.520 − 213.372i 122.043 − 140.846i 13.0344 + 1.87406i 652.786 + 324.518i 542.587 + 626.178i
2.10 −5.60625 + 8.72349i −16.9633 + 21.0059i −18.0827 − 39.5956i −33.0589 + 112.588i −88.1443 − 265.743i 124.109 − 143.230i −210.113 − 30.2097i −153.494 − 712.657i −796.827 − 919.587i
2.11 −5.28963 + 8.23082i 16.8611 − 21.0880i −13.1797 − 28.8596i 16.1556 − 55.0211i 84.3823 + 250.328i −182.035 + 210.080i −312.548 − 44.9377i −160.405 − 711.134i 367.412 + 424.016i
2.12 −5.07966 + 7.90410i −10.3745 − 24.9273i −10.0853 − 22.0838i −42.8023 + 145.771i 249.727 + 44.6206i 332.415 − 383.627i −369.417 − 53.1141i −513.738 + 517.218i −934.769 − 1078.78i
2.13 −3.90570 + 6.07739i −27.0000 − 0.0264852i 4.90642 + 10.7436i 22.1683 − 75.4981i 105.615 − 163.986i −370.224 + 427.262i −542.099 − 77.9420i 728.999 + 1.43020i 372.249 + 429.598i
2.14 −3.82614 + 5.95359i 11.8375 + 24.2667i 5.78070 + 12.6580i −31.2078 + 106.284i −189.766 − 22.3724i 23.3861 − 26.9890i −545.799 − 78.4740i −448.748 + 574.514i −513.366 − 592.456i
2.15 −3.42615 + 5.33119i −15.7384 − 21.9386i 9.90347 + 21.6856i 55.1132 − 187.698i 170.881 − 8.73922i 168.523 − 194.486i −550.993 − 79.2208i −233.608 + 690.557i 811.829 + 936.900i
2.16 −3.34719 + 5.20832i 26.9538 − 1.57802i 10.6636 + 23.3500i −60.8684 + 207.299i −82.0007 + 145.666i −164.718 + 190.095i −549.508 − 79.0073i 724.020 − 85.0674i −875.940 − 1010.89i
2.17 −3.31555 + 5.15910i 17.5977 − 20.4773i 10.9632 + 24.0060i 13.6633 − 46.5328i 47.2982 + 158.682i 292.486 − 337.547i −548.691 − 78.8899i −109.640 − 720.708i 194.766 + 224.772i
2.18 −2.83580 + 4.41259i 24.1950 + 11.9833i 15.1574 + 33.1900i 48.8691 − 166.433i −121.490 + 72.7805i −201.582 + 232.638i −521.717 − 75.0115i 441.800 + 579.874i 595.817 + 687.610i
2.19 −2.10499 + 3.27543i 1.61376 + 26.9517i 20.2891 + 44.4269i 49.4821 − 168.521i −91.6755 − 51.4474i 211.874 − 244.516i −434.874 − 62.5255i −723.792 + 86.9871i 447.818 + 516.810i
2.20 −1.80399 + 2.80706i −10.6616 + 24.8058i 21.9613 + 48.0886i 1.89783 − 6.46341i −50.3981 − 74.6773i −302.480 + 349.080i −385.985 − 55.4962i −501.660 − 528.941i 14.7195 + 16.9872i
See next 80 embeddings (of 460 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.46
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.c even 11 1 inner
69.h odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.7.h.a ✓ 460
3.b odd 2 1 inner 69.7.h.a ✓ 460
23.c even 11 1 inner 69.7.h.a ✓ 460
69.h odd 22 1 inner 69.7.h.a ✓ 460
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.7.h.a ✓ 460 1.a even 1 1 trivial
69.7.h.a ✓ 460 3.b odd 2 1 inner
69.7.h.a ✓ 460 23.c even 11 1 inner
69.7.h.a ✓ 460 69.h odd 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(69, [\chi])\).