Properties

Label 61.7.j.a
Level $61$
Weight $7$
Character orbit 61.j
Analytic conductor $14.033$
Analytic rank $0$
Dimension $240$
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] = [61,7,Mod(8,61)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(61, base_ring=CyclotomicField(20))
 
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("61.8");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 61.j (of order \(20\), degree \(8\), 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: \(240\)
Relative dimension: \(30\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 16 q^{2} - 10 q^{3} - 10 q^{4} - 230 q^{5} + 1384 q^{6} + 248 q^{7} + 632 q^{8} + 13602 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 16 q^{2} - 10 q^{3} - 10 q^{4} - 230 q^{5} + 1384 q^{6} + 248 q^{7} + 632 q^{8} + 13602 q^{9} + 1256 q^{10} - 1128 q^{11} - 2922 q^{12} - 2500 q^{13} + 18726 q^{15} + 70002 q^{16} - 4010 q^{17} + 51414 q^{18} - 10 q^{19} - 69254 q^{20} - 47962 q^{21} - 6 q^{22} + 1888 q^{23} - 177422 q^{24} + 228644 q^{25} + 117514 q^{26} - 62410 q^{27} - 60166 q^{28} + 154546 q^{29} - 132584 q^{30} - 17928 q^{31} - 428258 q^{32} + 44454 q^{33} - 167910 q^{34} - 101034 q^{35} + 641510 q^{36} + 116574 q^{37} - 262448 q^{38} - 329300 q^{39} - 524818 q^{40} + 519110 q^{41} - 82532 q^{42} + 745560 q^{43} + 1335810 q^{44} - 120410 q^{45} - 10 q^{46} + 588192 q^{47} + 5710 q^{49} + 97068 q^{50} + 297094 q^{51} - 1682120 q^{52} - 1503774 q^{53} - 1126786 q^{54} + 564038 q^{55} + 1572190 q^{56} + 555510 q^{57} + 937506 q^{58} - 48960 q^{59} + 42668 q^{61} - 2163580 q^{62} + 2141384 q^{63} - 2703370 q^{64} - 836260 q^{65} - 6744010 q^{66} - 1823136 q^{67} - 787264 q^{68} - 810946 q^{69} + 2027862 q^{70} + 4427536 q^{71} + 2755822 q^{72} + 455594 q^{73} + 1925380 q^{74} - 1199542 q^{76} - 3436182 q^{77} + 2415212 q^{78} - 2921256 q^{79} - 5044870 q^{80} - 7581422 q^{81} + 480560 q^{82} + 2764914 q^{83} + 5311480 q^{84} + 7048220 q^{85} - 2426786 q^{86} + 5291070 q^{87} - 3660970 q^{88} - 2695354 q^{89} + 8317796 q^{90} - 681472 q^{91} - 3392292 q^{92} + 4964430 q^{93} + 2757602 q^{94} + 5554934 q^{95} + 14100214 q^{96} + 2233430 q^{97} + 7768418 q^{98} + 7983712 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −14.9359 2.36562i 24.3471 + 33.5109i 156.618 + 50.8882i 15.7864 + 5.12931i −284.372 558.112i −57.0555 360.234i −1356.52 691.182i −304.927 + 938.468i −223.650 113.955i
8.2 −14.5200 2.29974i 2.94531 + 4.05387i 144.673 + 47.0071i −140.596 45.6824i −33.4430 65.6355i 102.471 + 646.975i −1154.23 588.108i 217.514 669.440i 1936.39 + 986.640i
8.3 −14.3157 2.26738i −17.1536 23.6099i 138.930 + 45.1412i 124.471 + 40.4431i 192.033 + 376.886i −3.66476 23.1384i −1060.01 540.102i −37.9086 + 116.671i −1690.19 861.194i
8.4 −12.0990 1.91629i 10.2537 + 14.1131i 81.8450 + 26.5930i 10.5714 + 3.43485i −97.0148 190.402i −33.3080 210.298i −240.743 122.664i 131.234 403.897i −121.321 61.8159i
8.5 −11.5786 1.83388i −13.3265 18.3423i 69.8344 + 22.6906i −47.6522 15.4831i 120.665 + 236.818i −61.4872 388.215i −98.4799 50.1780i 66.4280 204.444i 523.354 + 266.662i
8.6 −11.0664 1.75275i −26.9919 37.1511i 58.5256 + 19.0161i −159.846 51.9370i 233.587 + 458.439i 12.0727 + 76.2243i 24.5840 + 12.5261i −426.372 + 1312.24i 1677.89 + 854.925i
8.7 −10.4150 1.64958i 11.0227 + 15.1715i 44.8844 + 14.5838i 234.100 + 76.0636i −89.7755 176.194i 60.1035 + 379.479i 157.899 + 80.4537i 116.600 358.859i −2312.69 1178.37i
8.8 −8.70749 1.37913i 11.9365 + 16.4292i 13.0507 + 4.24042i −207.954 67.5685i −81.2790 159.519i −31.8422 201.044i 394.938 + 201.231i 97.8350 301.105i 1717.57 + 875.147i
8.9 −8.37352 1.32624i 29.2938 + 40.3194i 7.48937 + 2.43344i −17.1595 5.57546i −191.819 376.466i 46.4125 + 293.037i 423.962 + 216.020i −542.257 + 1668.90i 136.291 + 69.4437i
8.10 −6.48483 1.02710i −4.26947 5.87641i −19.8695 6.45601i 59.1716 + 19.2260i 21.6511 + 42.4927i −22.3687 141.230i 496.623 + 253.042i 208.969 643.142i −363.971 185.452i
8.11 −5.80061 0.918727i −14.6267 20.1319i −28.0646 9.11873i −3.89918 1.26692i 66.3479 + 130.215i 92.8806 + 586.425i 489.314 + 249.318i 33.9202 104.396i 21.4537 + 10.9312i
8.12 −5.34019 0.845804i −30.9690 42.6251i −33.0653 10.7436i 186.226 + 60.5084i 129.328 + 253.820i −13.0661 82.4960i 475.806 + 242.435i −632.550 + 1946.79i −943.303 480.637i
8.13 −2.46414 0.390282i 19.7627 + 27.2011i −54.9479 17.8537i 152.454 + 49.5352i −38.0821 74.7403i −103.627 654.278i 270.699 + 137.928i −124.060 + 381.816i −356.335 181.562i
8.14 −0.798753 0.126510i 12.4887 + 17.1893i −60.2456 19.5750i −105.947 34.4242i −7.80080 15.3099i 36.6098 + 231.145i 91.7612 + 46.7547i 85.7711 263.976i 80.2704 + 40.8998i
8.15 −0.794529 0.125841i −12.5523 17.2768i −60.2522 19.5771i −219.917 71.4552i 7.79905 + 15.3065i −40.0746 253.021i 91.2809 + 46.5099i 84.3471 259.594i 165.738 + 84.4478i
8.16 0.120641 + 0.0191077i 16.5235 + 22.7426i −60.8534 19.7725i 22.1092 + 7.18371i 1.55885 + 3.05942i 21.9570 + 138.631i −13.9288 7.09710i −18.9281 + 58.2546i 2.53001 + 1.28910i
8.17 0.244988 + 0.0388022i −17.8282 24.5385i −60.8091 19.7581i −7.75245 2.51892i −3.41555 6.70339i −75.3402 475.679i −28.2752 14.4070i −59.0165 + 181.634i −1.80151 0.917917i
8.18 3.55204 + 0.562587i 0.0740963 + 0.101985i −48.5672 15.7804i 141.256 + 45.8967i 0.205817 + 0.403939i 51.7148 + 326.514i −368.712 187.868i 225.268 693.305i 475.924 + 242.495i
8.19 4.37160 + 0.692394i −24.1162 33.1931i −42.2361 13.7233i −99.3178 32.2703i −82.4438 161.805i 50.4996 + 318.842i −427.533 217.839i −294.918 + 907.664i −411.834 209.840i
8.20 4.61063 + 0.730252i −13.3088 18.3180i −40.1430 13.0432i 200.376 + 65.1061i −47.9851 94.1760i 4.63036 + 29.2349i −441.755 225.086i 66.8494 205.741i 876.316 + 446.505i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
61.j odd 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 61.7.j.a 240
61.j odd 20 1 inner 61.7.j.a 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
61.7.j.a 240 1.a even 1 1 trivial
61.7.j.a 240 61.j odd 20 1 inner

Hecke kernels

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