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Properties

Label 91.10.e.b
Level $91$
Weight $10$
Character orbit 91.e
Analytic conductor $46.868$
Analytic rank $0$
Dimension $74$
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] = [91,10,Mod(53,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.53");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.e (of order \(3\), degree \(2\), 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: \(46.8682610909\)
Analytic rank: \(0\)
Dimension: \(74\)
Relative dimension: \(37\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 74 q - 31 q^{2} - 10069 q^{4} - 950 q^{5} + 2098 q^{6} + 14708 q^{7} + 40476 q^{8} - 222999 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 74 q - 31 q^{2} - 10069 q^{4} - 950 q^{5} + 2098 q^{6} + 14708 q^{7} + 40476 q^{8} - 222999 q^{9} + 16725 q^{10} - 175574 q^{11} - 23825 q^{12} + 2113514 q^{13} - 292904 q^{14} + 911274 q^{15} - 3286989 q^{16} - 486316 q^{17} + 144656 q^{18} + 27986 q^{19} + 3775310 q^{20} - 1809055 q^{21} + 5239236 q^{22} - 6116470 q^{23} - 4806522 q^{24} - 16260673 q^{25} - 885391 q^{26} + 23318082 q^{27} - 8088817 q^{28} - 10931388 q^{29} - 42923236 q^{30} - 2802773 q^{31} - 4751377 q^{32} - 3572238 q^{33} + 87356180 q^{34} + 14122538 q^{35} + 51389732 q^{36} - 46304940 q^{37} + 27230026 q^{38} + 23295512 q^{40} + 99808744 q^{41} - 7075896 q^{42} + 52036284 q^{43} - 198463932 q^{44} - 11938243 q^{45} - 20401011 q^{46} - 3627742 q^{47} + 231854622 q^{48} + 6764228 q^{49} + 327622404 q^{50} - 219825785 q^{51} - 287580709 q^{52} - 120210032 q^{53} + 289077612 q^{54} + 125782264 q^{55} - 419349423 q^{56} + 193882328 q^{57} - 128094865 q^{58} - 137363570 q^{59} - 341425250 q^{60} + 50702546 q^{61} - 115183288 q^{62} - 128206045 q^{63} + 2041940080 q^{64} - 27132950 q^{65} - 127254421 q^{66} - 465048728 q^{67} + 801011066 q^{68} - 2004165456 q^{69} - 50818418 q^{70} + 1973360460 q^{71} + 143931886 q^{72} + 61649244 q^{73} - 327082833 q^{74} - 178745790 q^{75} - 620930172 q^{76} - 708179008 q^{77} + 59920978 q^{78} + 383420594 q^{79} - 1514677411 q^{80} - 1680713753 q^{81} + 2234687846 q^{82} + 915242052 q^{83} - 1902598826 q^{84} + 3271040878 q^{85} - 829481488 q^{86} + 356584192 q^{87} - 1144409331 q^{88} - 655489346 q^{89} - 452855218 q^{90} + 420075188 q^{91} + 8905154190 q^{92} - 2341464256 q^{93} - 853193898 q^{94} + 313415190 q^{95} + 1851737061 q^{96} - 3054101548 q^{97} - 1509842425 q^{98} + 13258700904 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −22.1923 − 38.4382i 4.88280 − 8.45725i −728.995 + 1262.66i −883.973 − 1531.09i −433.442 5156.53 − 3709.96i 41987.4 9793.82 + 16963.4i −39234.8 + 67956.6i
53.2 −21.1972 − 36.7147i 101.475 − 175.759i −642.646 + 1113.10i 308.076 + 533.604i −8603.92 −6128.19 + 1672.99i 32783.3 −10752.7 − 18624.2i 13060.7 − 22621.8i
53.3 −20.7022 − 35.8572i 10.6263 − 18.4052i −601.161 + 1041.24i 1366.51 + 2366.87i −879.947 6347.04 + 262.189i 28582.4 9615.67 + 16654.8i 56579.6 − 97998.8i
53.4 −20.1416 − 34.8863i −116.865 + 202.416i −555.370 + 961.929i −1062.37 − 1840.07i 9415.40 −5968.54 − 2174.88i 24119.2 −17473.4 − 30264.8i −42795.6 + 74124.2i
53.5 −19.4067 − 33.6134i −114.121 + 197.663i −497.241 + 861.247i 296.326 + 513.252i 8858.86 3226.98 + 5471.76i 18726.8 −16205.7 − 28069.1i 11501.4 − 19921.1i
53.6 −16.6999 − 28.9251i 35.4919 − 61.4738i −301.774 + 522.687i 279.241 + 483.660i −2370.85 −4077.16 − 4871.38i 3057.67 7322.15 + 12682.3i 9326.60 − 16154.1i
53.7 −16.2740 − 28.1873i 18.8760 − 32.6941i −273.683 + 474.033i −1028.12 − 1780.75i −1228.75 −3975.25 + 4954.89i 1151.09 9128.90 + 15811.7i −33463.0 + 57959.6i
53.8 −15.0637 − 26.0910i −43.9456 + 76.1160i −197.828 + 342.649i 628.597 + 1088.76i 2647.93 −6095.25 − 1789.28i −3505.11 5979.07 + 10356.0i 18938.0 − 32801.5i
53.9 −14.4794 − 25.0791i −56.0520 + 97.0849i −163.306 + 282.855i −348.476 − 603.578i 3246.40 2711.07 − 5744.88i −5368.60 3557.84 + 6162.36i −10091.4 + 17478.9i
53.10 −13.0597 − 22.6201i 137.175 − 237.593i −85.1112 + 147.417i 1051.96 + 1822.06i −7165.84 1176.32 − 6242.59i −8927.02 −27792.3 − 48137.6i 27476.7 − 47591.0i
53.11 −11.4525 − 19.8363i 70.4885 − 122.090i −6.31964 + 10.9459i −77.1296 − 133.592i −3229.08 −201.220 + 6349.26i −11437.9 −95.7443 − 165.834i −1766.65 + 3059.93i
53.12 −11.2945 − 19.5627i 97.1363 − 168.245i 0.867948 − 1.50333i −998.354 − 1729.20i −4388.43 6130.71 − 1663.74i −11604.8 −9029.41 − 15639.4i −22551.8 + 39060.9i
53.13 −11.1700 − 19.3471i −28.6697 + 49.6574i 6.46093 − 11.1907i 515.147 + 892.262i 1280.97 6171.64 + 1504.80i −11726.8 8197.60 + 14198.7i 11508.4 − 19933.2i
53.14 −5.45177 − 9.44275i −119.544 + 207.056i 196.556 − 340.446i 23.5616 + 40.8099i 2606.90 −2571.17 − 5808.85i −9868.94 −18739.9 − 32458.4i 256.905 − 444.972i
53.15 −5.43693 − 9.41703i −100.952 + 174.854i 196.880 − 341.006i −1193.41 − 2067.05i 2195.47 4807.09 + 4152.77i −9849.09 −10541.1 − 18257.8i −12977.0 + 22476.8i
53.16 −4.43108 − 7.67486i 81.5279 − 141.210i 216.731 − 375.389i −394.495 − 683.286i −1445.03 −1617.61 − 6143.04i −8378.84 −3452.08 − 5979.18i −3496.08 + 6055.39i
53.17 −3.64361 − 6.31093i −54.7738 + 94.8710i 229.448 − 397.416i −507.566 − 879.130i 798.299 −3368.02 + 5386.10i −7075.14 3841.16 + 6653.08i −3698.75 + 6406.42i
53.18 −1.69451 − 2.93498i 94.4566 − 163.604i 250.257 − 433.458i 1073.60 + 1859.53i −640.230 2825.00 + 5689.73i −3431.43 −8002.59 − 13860.9i 3638.46 − 6301.99i
53.19 −0.961851 − 1.66597i 0.279794 − 0.484618i 254.150 − 440.200i 1027.43 + 1779.55i −1.07648 2479.92 − 5848.38i −1962.75 9841.34 + 17045.7i 1976.46 − 3423.33i
53.20 −0.469964 − 0.814001i 27.9128 − 48.3464i 255.558 − 442.640i 387.691 + 671.501i −52.4720 −6160.71 + 1548.96i −961.655 8283.25 + 14347.0i 364.402 − 631.162i
See all 74 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.37
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.10.e.b ✓ 74
7.c even 3 1 inner 91.10.e.b ✓ 74
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.10.e.b ✓ 74 1.a even 1 1 trivial
91.10.e.b ✓ 74 7.c even 3 1 inner