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Properties

Label 37.10.f.a
Level $37$
Weight $10$
Character orbit 37.f
Analytic conductor $19.056$
Analytic rank $0$
Dimension $162$
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] = [37,10,Mod(7,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([16]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.7");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 37.f (of order \(9\), 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: \(19.0563259381\)
Analytic rank: \(0\)
Dimension: \(162\)
Relative dimension: \(27\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 162 q - 57 q^{2} - 228 q^{3} + 249 q^{4} - 3420 q^{5} - 12 q^{6} + 5808 q^{7} + 61698 q^{8} - 30198 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 162 q - 57 q^{2} - 228 q^{3} + 249 q^{4} - 3420 q^{5} - 12 q^{6} + 5808 q^{7} + 61698 q^{8} - 30198 q^{9} + 59997 q^{10} + 175689 q^{11} - 147411 q^{12} + 713586 q^{13} - 96867 q^{14} - 143316 q^{15} + 556665 q^{16} - 119334 q^{17} - 711948 q^{18} + 1752348 q^{19} + 4774248 q^{20} + 57972 q^{21} + 1413495 q^{22} - 5098719 q^{23} + 3891975 q^{24} + 10869768 q^{25} + 5518221 q^{26} + 13281609 q^{27} + 9017757 q^{28} - 2782023 q^{29} - 10231020 q^{30} + 8407836 q^{31} + 6197838 q^{32} - 8252229 q^{33} - 49656453 q^{34} - 23563824 q^{35} + 151165428 q^{36} - 23634336 q^{37} - 12277422 q^{38} + 100476834 q^{39} + 165006327 q^{40} + 74094072 q^{41} - 3662109 q^{42} - 90803088 q^{43} - 350087604 q^{44} - 184820799 q^{45} - 114643110 q^{46} + 158694333 q^{47} + 359750805 q^{48} + 36118224 q^{49} - 259200273 q^{50} + 176035647 q^{51} - 203883429 q^{52} - 31488120 q^{53} - 109796865 q^{54} + 23137227 q^{55} + 37371291 q^{56} - 12449808 q^{57} - 840994983 q^{58} - 77893794 q^{59} + 244178580 q^{60} - 980597586 q^{61} - 586588080 q^{62} - 48634623 q^{63} - 917012376 q^{64} + 1432770144 q^{65} - 177475497 q^{66} - 302695398 q^{67} + 1615796082 q^{68} + 699674787 q^{69} + 2092804794 q^{70} - 774435768 q^{71} - 2899807611 q^{72} - 2505680652 q^{73} + 1161236205 q^{74} - 2302474524 q^{75} + 1445475621 q^{76} - 1374645213 q^{77} + 2942410749 q^{78} - 29533980 q^{79} + 9154934064 q^{80} - 2232253221 q^{81} + 707552205 q^{82} - 392438496 q^{83} - 1388366664 q^{84} - 556053567 q^{85} + 852383496 q^{86} - 6650019951 q^{87} + 3660093510 q^{88} + 100313124 q^{89} + 12897388398 q^{90} + 684844038 q^{91} + 621259308 q^{92} + 2212599060 q^{93} + 3136220628 q^{94} + 1257937926 q^{95} - 6811500039 q^{96} + 70258497 q^{97} - 8202052869 q^{98} - 2679000153 q^{99}+O(q^{100}) \) 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 −40.6791 + 14.8060i 175.800 + 63.9860i 1043.36 − 875.479i −20.2507 − 114.847i −8098.75 813.249 + 4612.17i −18398.2 + 31866.7i 11733.4 + 9845.47i 2524.20 + 4372.05i
7.2 −38.2858 + 13.9349i −178.096 − 64.8216i 879.406 − 737.909i −21.3928 − 121.324i 7721.82 −389.844 − 2210.92i −12955.9 + 22440.3i 12438.2 + 10436.9i 2509.68 + 4346.90i
7.3 −34.4384 + 12.5346i −33.2455 − 12.1004i 636.676 − 534.235i 217.968 + 1236.16i 1296.60 860.947 + 4882.67i −5847.68 + 10128.5i −14119.2 − 11847.4i −23001.2 − 39839.2i
7.4 −33.3678 + 12.1449i 12.1843 + 4.43471i 573.697 − 481.389i −434.250 − 2462.75i −460.421 −1475.57 − 8368.36i −4206.21 + 7285.37i −14949.3 − 12543.9i 44399.8 + 76902.7i
7.5 −29.8419 + 10.8616i 141.024 + 51.3286i 380.353 − 319.154i 229.976 + 1304.26i −4765.95 −1104.54 − 6264.16i 245.880 − 425.877i 2175.16 + 1825.18i −21029.2 − 36423.6i
7.6 −25.6827 + 9.34774i −25.0373 − 9.11281i 180.007 − 151.044i −170.101 − 964.688i 728.209 2070.73 + 11743.7i 3785.58 − 6556.82i −14534.2 − 12195.7i 13386.3 + 23185.8i
7.7 −23.1708 + 8.43347i −142.761 − 51.9607i 73.5461 − 61.7125i 397.153 + 2252.37i 3746.09 −641.640 − 3638.92i 5128.73 − 8883.22i 2602.72 + 2183.94i −28197.6 − 48839.7i
7.8 −20.2342 + 7.36464i 235.992 + 85.8942i −37.0304 + 31.0722i −303.054 − 1718.71i −5407.69 961.861 + 5454.99i 6032.83 − 10449.2i 33236.5 + 27888.8i 18789.7 + 32544.7i
7.9 −19.0907 + 6.94843i −243.959 − 88.7940i −76.0425 + 63.8072i −175.069 − 992.868i 5274.32 314.940 + 1786.11i 6209.20 − 10754.6i 36553.8 + 30672.3i 10241.1 + 17738.0i
7.10 −16.5869 + 6.03712i 162.181 + 59.0289i −153.538 + 128.833i −2.06884 − 11.7330i −3046.43 −571.552 − 3241.43i 6287.68 − 10890.6i 7740.06 + 6494.68i 105.149 + 182.124i
7.11 −12.0936 + 4.40169i −70.1607 − 25.5364i −265.336 + 222.643i −232.994 − 1321.38i 960.895 −66.0058 − 374.338i 5523.49 − 9566.96i −10807.6 − 9068.69i 8634.03 + 14954.6i
7.12 −7.45759 + 2.71434i −52.8450 − 19.2340i −343.967 + 288.622i 69.2798 + 392.905i 446.304 −1588.46 − 9008.63i 3813.41 − 6605.02i −12655.4 − 10619.1i −1583.14 − 2742.08i
7.13 −6.84315 + 2.49070i 103.862 + 37.8028i −351.590 + 295.019i 367.923 + 2086.59i −804.900 1644.65 + 9327.29i 3535.45 − 6123.58i −5719.73 − 4799.43i −7714.83 − 13362.5i
7.14 4.43325 − 1.61357i 83.4023 + 30.3559i −375.165 + 314.801i −96.6824 − 548.313i 418.724 −278.101 − 1577.19i −2362.99 + 4092.82i −9043.60 − 7588.48i −1313.36 − 2274.80i
7.15 5.13813 − 1.87013i −174.502 − 63.5134i −369.312 + 309.889i 226.866 + 1286.62i −1015.39 674.322 + 3824.27i −2717.82 + 4707.40i 11338.8 + 9514.40i 3571.81 + 6186.56i
7.16 8.62864 − 3.14057i 125.460 + 45.6637i −327.624 + 274.910i −365.091 − 2070.54i 1225.96 119.762 + 679.206i −4314.28 + 7472.55i −1423.02 − 1194.05i −9652.90 − 16719.3i
7.17 11.2989 − 4.11245i 246.120 + 89.5804i −281.463 + 236.175i 241.261 + 1368.26i 3149.27 −938.900 − 5324.76i −5287.09 + 9157.51i 37472.4 + 31443.1i 8352.87 + 14467.6i
7.18 14.8129 − 5.39147i −119.764 − 43.5906i −201.860 + 169.380i −288.707 − 1637.34i −2009.08 1626.30 + 9223.21i −6112.41 + 10587.0i −2634.73 − 2210.80i −13104.3 − 22697.2i
7.19 15.9696 − 5.81245i −215.150 − 78.3084i −170.972 + 143.463i −327.821 − 1859.16i −3891.02 −2140.89 − 12141.6i −6247.06 + 10820.2i 25079.5 + 21044.2i −16041.4 − 27784.6i
7.20 21.1586 − 7.70109i −29.5242 − 10.7459i −3.83595 + 3.21875i 289.494 + 1641.80i −707.445 −1798.85 − 10201.8i −5820.60 + 10081.6i −14321.9 − 12017.5i 18768.9 + 32508.7i
See next 80 embeddings (of 162 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.27
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.f even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 37.10.f.a ✓ 162
37.f even 9 1 inner 37.10.f.a ✓ 162
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.10.f.a ✓ 162 1.a even 1 1 trivial
37.10.f.a ✓ 162 37.f even 9 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(37, [\chi])\).