↑

Properties

Label 61.7.h.a
Level $61$
Weight $7$
Character orbit 61.h
Analytic conductor $14.033$
Analytic rank $0$
Dimension $120$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [61,7,Mod(21,61)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(61, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([11]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("61.21");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 61.h (of order \(12\), 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: \(14.0332991008\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 12 q^{2} - 6 q^{4} + 126 q^{5} - 520 q^{6} + 1644 q^{7} - 1668 q^{8} - 27224 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 120 q - 12 q^{2} - 6 q^{4} + 126 q^{5} - 520 q^{6} + 1644 q^{7} - 1668 q^{8} - 27224 q^{9} - 600 q^{10} - 1124 q^{11} + 1456 q^{12} - 2486 q^{13} + 14844 q^{14} - 12976 q^{15} + 45166 q^{16} - 5686 q^{17} + 4658 q^{18} - 6 q^{19} + 2424 q^{20} - 24144 q^{21} - 2 q^{22} + 17532 q^{23} + 35010 q^{24} + 174468 q^{25} - 50754 q^{26} + 182648 q^{28} + 20108 q^{29} - 102218 q^{30} + 209236 q^{31} - 53184 q^{32} + 48832 q^{33} + 122488 q^{34} + 50330 q^{35} - 375024 q^{36} - 782 q^{37} + 49716 q^{38} - 81312 q^{39} - 306318 q^{40} - 85444 q^{42} - 301172 q^{43} - 590386 q^{44} - 177522 q^{45} - 161754 q^{46} - 110786 q^{47} + 1116816 q^{48} + 257274 q^{49} + 159026 q^{50} - 485728 q^{51} + 228290 q^{53} + 688808 q^{54} - 97686 q^{55} - 60596 q^{56} + 608064 q^{57} - 324536 q^{58} - 358980 q^{59} - 609800 q^{61} - 3011084 q^{62} - 2391588 q^{63} - 503070 q^{65} - 2140122 q^{66} + 922324 q^{67} - 1612038 q^{68} + 2678616 q^{69} + 5804124 q^{70} - 1579684 q^{71} + 3177656 q^{72} + 910278 q^{73} + 2334764 q^{74} + 1030800 q^{75} - 1232306 q^{76} - 602306 q^{77} - 357690 q^{78} + 1734700 q^{79} + 3819720 q^{80} + 5374632 q^{81} - 3367866 q^{82} - 1129162 q^{83} + 6077418 q^{84} - 4767858 q^{85} - 1052452 q^{86} + 1682200 q^{87} - 1609350 q^{89} - 1963536 q^{90} + 194062 q^{91} - 6594884 q^{92} + 109416 q^{93} + 2079718 q^{94} - 4221708 q^{95} - 4588810 q^{96} + 5036610 q^{97} - 3915984 q^{98} - 2911156 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −14.7263 + 3.94589i 47.8343i 145.867 − 84.2165i 62.3808 + 36.0156i −188.749 − 704.421i 420.527 − 112.680i −1125.83 + 1125.83i −1559.12 −1060.75 − 284.227i
21.2 −13.6267 + 3.65126i − 23.5259i 116.930 − 67.5093i −85.0284 − 49.0912i 85.8993 + 320.580i 608.311 − 162.996i −708.441 + 708.441i 175.532 1337.90 + 358.490i
21.3 −13.4567 + 3.60571i 11.8673i 112.656 − 65.0418i −96.9857 − 55.9947i −42.7900 − 159.694i −288.036 + 77.1791i −650.988 + 650.988i 588.167 1507.01 + 403.801i
21.4 −13.1952 + 3.53565i − 21.0635i 106.187 − 61.3072i 212.224 + 122.527i 74.4732 + 277.938i −21.1275 + 5.66110i −566.190 + 566.190i 285.328 −3233.55 − 866.427i
21.5 −12.4908 + 3.34690i − 49.7593i 89.3926 − 51.6108i −65.9349 − 38.0675i 166.539 + 621.534i −478.748 + 128.280i −358.639 + 358.639i −1746.99 950.987 + 254.816i
21.6 −11.6383 + 3.11848i 17.3168i 70.3004 − 40.5880i 41.6062 + 24.0213i −54.0023 − 201.539i −323.078 + 86.5685i −146.337 + 146.337i 429.127 −559.137 − 149.820i
21.7 −8.30883 + 2.22634i 23.8951i 8.65446 − 4.99665i 112.433 + 64.9133i −53.1987 − 198.540i 163.522 − 43.8155i 328.495 − 328.495i 158.025 −1078.71 − 289.039i
21.8 −8.11762 + 2.17511i 28.8782i 5.73904 − 3.31344i −197.939 − 114.280i −62.8133 − 234.422i 383.921 − 102.871i 340.940 − 340.940i −104.951 1855.37 + 497.145i
21.9 −7.99203 + 2.14146i − 20.2061i 3.86103 − 2.22917i −14.4649 − 8.35132i 43.2704 + 161.487i 115.210 − 30.8704i 348.353 − 348.353i 320.715 133.488 + 35.7680i
21.10 −7.04173 + 1.88683i − 30.1012i −9.39977 + 5.42696i 5.00084 + 2.88724i 56.7957 + 211.964i 143.514 − 38.4544i 385.865 − 385.865i −177.080 −40.6623 − 10.8954i
21.11 −6.88158 + 1.84391i 51.5888i −11.4695 + 6.62195i 5.29326 + 3.05607i −95.1252 − 355.012i −375.415 + 100.592i 389.129 − 389.129i −1932.40 −42.0611 − 11.2702i
21.12 −3.95903 + 1.06082i − 9.70786i −40.8770 + 23.6004i −145.611 − 84.0684i 10.2983 + 38.4338i −451.956 + 121.101i 322.283 − 322.283i 634.757 665.659 + 178.363i
21.13 −3.41292 + 0.914490i − 44.4336i −44.6139 + 25.7578i 136.033 + 78.5387i 40.6340 + 151.648i −185.934 + 49.8208i 288.608 − 288.608i −1245.34 −536.093 − 143.646i
21.14 −1.92223 + 0.515061i 17.1247i −51.9959 + 30.0199i 85.0592 + 49.1090i −8.82025 − 32.9176i 529.612 − 141.909i 174.545 − 174.545i 435.745 −188.798 − 50.5882i
21.15 −0.897396 + 0.240456i 10.6535i −54.6781 + 31.5684i 170.258 + 98.2987i −2.56170 − 9.56039i −597.572 + 160.119i 83.5212 − 83.5212i 615.503 −176.426 − 47.2731i
21.16 0.431709 − 0.115676i − 50.3463i −55.2526 + 31.9001i −211.576 − 122.153i −5.82385 − 21.7349i 415.339 − 111.290i −40.3890 + 40.3890i −1805.75 −105.469 − 28.2604i
21.17 1.68644 − 0.451879i − 1.22020i −52.7858 + 30.4759i −56.7024 − 32.7372i −0.551384 − 2.05779i −98.9740 + 26.5200i −154.260 + 154.260i 727.511 −110.418 − 29.5865i
21.18 1.77844 − 0.476532i 36.6190i −52.4899 + 30.3050i −75.6475 − 43.6751i 17.4501 + 65.1247i 9.74841 − 2.61208i −162.231 + 162.231i −611.948 −155.347 − 41.6252i
21.19 2.31982 − 0.621593i − 28.6557i −50.4305 + 29.1160i 98.5331 + 56.8881i −17.8122 − 66.4760i 341.908 − 91.6139i −207.577 + 207.577i −92.1507 263.940 + 70.7225i
21.20 6.59107 − 1.76607i − 12.3202i −15.1025 + 8.71941i −102.755 − 59.3254i −21.7583 − 81.2031i 132.998 − 35.6368i −392.942 + 392.942i 577.213 −782.035 − 209.546i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.h odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.7.h.a ✓ 120
61.h odd 12 1 inner 61.7.h.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.7.h.a ✓ 120 1.a even 1 1 trivial
61.7.h.a ✓ 120 61.h odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(61, [\chi])\).