↑

Properties

Label 1011.6.a.b
Level $1011$
Weight $6$
Character orbit 1011.a
Self dual yes
Analytic conductor $162.148$
Analytic rank $1$
Dimension $66$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1011,6,Mod(1,1011)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1011, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1011.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1011 = 3 \cdot 337 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1011.a (trivial)

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: yes
Analytic conductor: \(162.148041831\)
Analytic rank: \(1\)
Dimension: \(66\)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 17 q^{2} - 594 q^{3} + 971 q^{4} - 222 q^{5} + 153 q^{6} + 549 q^{7} - 711 q^{8} + 5346 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q - 17 q^{2} - 594 q^{3} + 971 q^{4} - 222 q^{5} + 153 q^{6} + 549 q^{7} - 711 q^{8} + 5346 q^{9} + 492 q^{10} - 1182 q^{11} - 8739 q^{12} - 1793 q^{13} - 1785 q^{14} + 1998 q^{15} + 13075 q^{16} - 2162 q^{17} - 1377 q^{18} + 2288 q^{19} - 6947 q^{20} - 4941 q^{21} - 4621 q^{22} - 6517 q^{23} + 6399 q^{24} + 32240 q^{25} - 12158 q^{26} - 48114 q^{27} + 20016 q^{28} - 37522 q^{29} - 4428 q^{30} + 8332 q^{31} - 32949 q^{32} + 10638 q^{33} - 19528 q^{34} - 27262 q^{35} + 78651 q^{36} - 16628 q^{37} - 34042 q^{38} + 16137 q^{39} + 10811 q^{40} - 44288 q^{41} + 16065 q^{42} + 4211 q^{43} - 39572 q^{44} - 17982 q^{45} - 64951 q^{46} - 20820 q^{47} - 117675 q^{48} + 117857 q^{49} - 66591 q^{50} + 19458 q^{51} - 19892 q^{52} - 107763 q^{53} + 12393 q^{54} + 99895 q^{55} - 95751 q^{56} - 20592 q^{57} - 23890 q^{58} - 98902 q^{59} + 62523 q^{60} - 69412 q^{61} + 1216 q^{62} + 44469 q^{63} + 67555 q^{64} - 68617 q^{65} + 41589 q^{66} - 7629 q^{67} - 91991 q^{68} + 58653 q^{69} - 10510 q^{70} - 165158 q^{71} - 57591 q^{72} + 15099 q^{73} - 101693 q^{74} - 290160 q^{75} + 107873 q^{76} - 298539 q^{77} + 109422 q^{78} - 45288 q^{79} - 81423 q^{80} + 433026 q^{81} + 241529 q^{82} - 88964 q^{83} - 180144 q^{84} - 42632 q^{85} + 220761 q^{86} + 337698 q^{87} + 134050 q^{88} - 153981 q^{89} + 39852 q^{90} + 22229 q^{91} + 441817 q^{92} - 74988 q^{93} + 157032 q^{94} + 80326 q^{95} + 296541 q^{96} + 289294 q^{97} + 805107 q^{98} - 95742 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −11.0715 −9.00000 90.5772 −68.9950 99.6431 49.4404 −648.535 81.0000 763.875
1.2 −10.7011 −9.00000 82.5135 56.3945 96.3099 210.094 −540.549 81.0000 −603.483
1.3 −10.6223 −9.00000 80.8336 −62.0035 95.6009 30.9331 −518.726 81.0000 658.621
1.4 −10.3439 −9.00000 74.9972 74.1790 93.0955 30.5015 −444.761 81.0000 −767.304
1.5 −10.0176 −9.00000 68.3523 13.9065 90.1584 −165.733 −364.162 81.0000 −139.310
1.6 −9.83277 −9.00000 64.6833 −72.2614 88.4949 −159.171 −321.367 81.0000 710.530
1.7 −9.74085 −9.00000 62.8842 −27.1041 87.6677 181.529 −300.839 81.0000 264.017
1.8 −9.37687 −9.00000 55.9257 −100.627 84.3918 166.393 −224.348 81.0000 943.566
1.9 −9.14176 −9.00000 51.5717 36.8612 82.2758 29.0778 −178.920 81.0000 −336.976
1.10 −9.12299 −9.00000 51.2290 107.496 82.1069 −5.15199 −175.426 81.0000 −980.683
1.11 −8.67828 −9.00000 43.3126 79.8802 78.1045 −4.15985 −98.1735 81.0000 −693.223
1.12 −8.33712 −9.00000 37.5076 −1.89296 75.0341 −171.280 −45.9179 81.0000 15.7818
1.13 −7.27821 −9.00000 20.9724 32.7498 65.5039 −150.973 80.2616 81.0000 −238.360
1.14 −7.01426 −9.00000 17.1998 −24.6293 63.1283 −145.032 103.812 81.0000 172.756
1.15 −6.88573 −9.00000 15.4133 13.2523 61.9716 63.3109 114.212 81.0000 −91.2519
1.16 −6.85591 −9.00000 15.0035 −103.192 61.7032 130.556 116.526 81.0000 707.478
1.17 −6.73488 −9.00000 13.3587 −39.3332 60.6140 120.120 125.547 81.0000 264.904
1.18 −6.25365 −9.00000 7.10819 −71.0028 56.2829 122.650 155.665 81.0000 444.027
1.19 −6.01504 −9.00000 4.18066 −6.69474 54.1353 −59.4917 167.334 81.0000 40.2691
1.20 −5.77404 −9.00000 1.33957 −66.6440 51.9664 −129.926 177.035 81.0000 384.805
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.66
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(337\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1011.6.a.b ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1011.6.a.b ✓ 66 1.a even 1 1 trivial