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Properties

Label 11.9.d.a
Level $11$
Weight $9$
Character orbit 11.d
Analytic conductor $4.481$
Analytic rank $0$
Dimension $28$
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] = [11,9,Mod(2,11)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(11, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.2");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 11.d (of order \(10\), 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: \(4.48116471067\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(7\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 5 q^{2} + 144 q^{3} + 951 q^{4} - 708 q^{5} - 4485 q^{6} + 5470 q^{7} - 3845 q^{8} - 14225 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 5 q^{2} + 144 q^{3} + 951 q^{4} - 708 q^{5} - 4485 q^{6} + 5470 q^{7} - 3845 q^{8} - 14225 q^{9} + 38567 q^{11} + 63954 q^{12} - 12500 q^{13} - 49262 q^{14} - 211056 q^{15} + 65335 q^{16} + 368200 q^{17} - 316350 q^{18} - 442685 q^{19} - 101488 q^{20} + 1686925 q^{22} + 704764 q^{23} - 426415 q^{24} - 960933 q^{25} + 659122 q^{26} - 1523847 q^{27} - 2014380 q^{28} - 795320 q^{29} + 680860 q^{30} + 1317554 q^{31} - 3224101 q^{33} + 897538 q^{34} + 8132260 q^{35} + 8527262 q^{36} + 8163354 q^{37} + 1456780 q^{38} - 14657890 q^{39} - 24701940 q^{40} - 13192700 q^{41} + 3806030 q^{42} - 8229340 q^{44} + 17701416 q^{45} + 38601510 q^{46} + 11954754 q^{47} + 49075784 q^{48} - 11198359 q^{49} - 41838705 q^{50} - 41019825 q^{51} - 81977530 q^{52} - 2537916 q^{53} + 40359452 q^{55} + 96646244 q^{56} + 88596375 q^{57} - 13252700 q^{58} + 49581537 q^{59} - 58912456 q^{60} - 29302910 q^{61} - 207755000 q^{62} - 178685550 q^{63} + 22221799 q^{64} + 119858070 q^{66} + 105502894 q^{67} + 335519000 q^{68} + 32103578 q^{69} + 228302300 q^{70} + 30044916 q^{71} - 375829355 q^{72} - 272936400 q^{73} - 400896650 q^{74} + 30339519 q^{75} + 23326180 q^{77} + 387475480 q^{78} + 319699810 q^{79} + 208513052 q^{80} + 403883776 q^{81} - 197581295 q^{82} - 558510965 q^{83} - 721672470 q^{84} - 360970180 q^{85} + 119645907 q^{86} + 85623835 q^{88} + 317966134 q^{89} + 1020614800 q^{90} + 285678712 q^{91} + 599604784 q^{92} - 125102972 q^{93} - 650361480 q^{94} - 641153840 q^{95} - 523100900 q^{96} + 85897149 q^{97} - 149389979 q^{99}+O(q^{100}) \) 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 −27.1410 − 8.81864i 102.046 − 74.1411i 451.757 + 328.221i −273.027 − 840.292i −3423.47 + 1112.35i −623.359 + 857.980i −5072.51 − 6981.72i 2889.11 − 8891.78i 25214.1i
2.2 −20.7053 − 6.72755i −78.6006 + 57.1067i 176.340 + 128.119i −34.6528 − 106.650i 2011.64 − 653.620i 1699.37 − 2338.98i 486.673 + 669.848i 889.425 − 2737.37i 2441.35i
2.3 −11.5394 − 3.74936i 17.8878 − 12.9963i −88.0094 − 63.9426i 189.885 + 584.407i −255.142 + 82.9005i −1973.26 + 2715.95i 2601.55 + 3580.72i −1876.39 + 5774.93i − 7455.63i
2.4 2.53840 + 0.824777i 83.6636 − 60.7851i −201.345 − 146.286i 23.5313 + 72.4220i 262.506 − 85.2933i 2431.93 − 3347.27i −792.059 − 1090.18i 1277.30 − 3931.12i 203.244i
2.5 6.67744 + 2.16963i −62.7933 + 45.6220i −167.227 − 121.498i −217.294 − 668.763i −518.281 + 168.400i −790.545 + 1088.09i −1909.53 − 2628.24i −165.830 + 510.373i − 4937.07i
2.6 21.1366 + 6.86769i −66.9452 + 48.6385i 192.481 + 139.846i 346.140 + 1065.31i −1749.03 + 568.293i 1612.97 − 2220.06i −236.187 − 325.083i 88.4908 − 272.347i 24894.2i
2.7 23.8701 + 7.75587i 59.7479 − 43.4094i 302.520 + 219.794i −128.848 − 396.552i 1762.87 − 572.790i −1690.62 + 2326.94i 1739.84 + 2394.69i −342.025 + 1052.64i − 10465.1i
6.1 −27.1410 + 8.81864i 102.046 + 74.1411i 451.757 − 328.221i −273.027 + 840.292i −3423.47 − 1112.35i −623.359 − 857.980i −5072.51 + 6981.72i 2889.11 + 8891.78i − 25214.1i
6.2 −20.7053 + 6.72755i −78.6006 − 57.1067i 176.340 − 128.119i −34.6528 + 106.650i 2011.64 + 653.620i 1699.37 + 2338.98i 486.673 − 669.848i 889.425 + 2737.37i − 2441.35i
6.3 −11.5394 + 3.74936i 17.8878 + 12.9963i −88.0094 + 63.9426i 189.885 − 584.407i −255.142 − 82.9005i −1973.26 − 2715.95i 2601.55 − 3580.72i −1876.39 − 5774.93i 7455.63i
6.4 2.53840 − 0.824777i 83.6636 + 60.7851i −201.345 + 146.286i 23.5313 − 72.4220i 262.506 + 85.2933i 2431.93 + 3347.27i −792.059 + 1090.18i 1277.30 + 3931.12i − 203.244i
6.5 6.67744 − 2.16963i −62.7933 − 45.6220i −167.227 + 121.498i −217.294 + 668.763i −518.281 − 168.400i −790.545 − 1088.09i −1909.53 + 2628.24i −165.830 − 510.373i 4937.07i
6.6 21.1366 − 6.86769i −66.9452 − 48.6385i 192.481 − 139.846i 346.140 − 1065.31i −1749.03 − 568.293i 1612.97 + 2220.06i −236.187 + 325.083i 88.4908 + 272.347i − 24894.2i
6.7 23.8701 − 7.75587i 59.7479 + 43.4094i 302.520 − 219.794i −128.848 + 396.552i 1762.87 + 572.790i −1690.62 − 2326.94i 1739.84 − 2394.69i −342.025 − 1052.64i 10465.1i
7.1 −15.5037 + 21.3390i 26.4821 − 81.5035i −135.880 − 418.196i 31.1978 − 22.6665i 1328.63 + 1828.71i 3043.44 − 988.873i 4608.64 + 1497.44i −633.558 − 460.307i 1017.14i
7.2 −11.7272 + 16.1411i −37.1442 + 114.318i −43.8989 − 135.107i 106.382 − 77.2908i −1409.62 − 1940.17i 701.839 − 228.041i −2162.01 − 702.479i −6380.96 − 4636.04i 2623.51i
7.3 −5.67443 + 7.81018i 11.2934 − 34.7576i 50.3085 + 154.834i −363.947 + 264.423i 207.379 + 285.433i −3818.23 + 1240.62i −3845.20 − 1249.38i 4227.41 + 3071.39i − 4342.95i
7.4 1.86291 − 2.56408i 3.27133 − 10.0681i 76.0043 + 233.917i 699.890 − 508.500i −19.7213 − 27.1440i 1515.95 − 492.563i 1513.02 + 491.611i 5217.30 + 3790.59i − 2741.87i
7.5 8.19681 − 11.2819i −32.6782 + 100.573i 19.0138 + 58.5184i −585.798 + 425.607i 866.803 + 1193.05i 1206.27 − 391.939i 4211.31 + 1368.34i −3739.12 − 2716.63i 10097.6i
7.6 9.10817 − 12.5363i 47.5466 − 146.333i 4.90775 + 15.1045i −526.079 + 382.219i −1401.42 − 1928.89i 1608.54 − 522.646i 4006.81 + 1301.89i −13844.8 − 10058.9i 10076.4i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.9.d.a ✓ 28
3.b odd 2 1 99.9.k.a 28
11.c even 5 1 121.9.b.b 28
11.d odd 10 1 inner 11.9.d.a ✓ 28
11.d odd 10 1 121.9.b.b 28
33.f even 10 1 99.9.k.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.9.d.a ✓ 28 1.a even 1 1 trivial
11.9.d.a ✓ 28 11.d odd 10 1 inner
99.9.k.a 28 3.b odd 2 1
99.9.k.a 28 33.f even 10 1
121.9.b.b 28 11.c even 5 1
121.9.b.b 28 11.d odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(11, [\chi])\).