Properties

Label 11.10.a.a
Level $11$
Weight $10$
Character orbit 11.a
Self dual yes
Analytic conductor $5.665$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [11,10,Mod(1,11)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(11, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("11.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 11.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: \(5.66539419780\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.2659452.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 306x - 836 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{2} + 4 \beta_1 - 62) q^{3} + (8 \beta_{2} + 12 \beta_1 + 304) q^{4} + ( - 3 \beta_{2} + 34 \beta_1 - 608) q^{5} + ( - 44 \beta_{2} + 47 \beta_1 - 3652) q^{6} + (22 \beta_{2} - 122 \beta_1 - 2420) q^{7} + ( - 200 \beta_1 - 6688) q^{8} + (315 \beta_{2} - 666 \beta_1 + 4851) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{2} + 4 \beta_1 - 62) q^{3} + (8 \beta_{2} + 12 \beta_1 + 304) q^{4} + ( - 3 \beta_{2} + 34 \beta_1 - 608) q^{5} + ( - 44 \beta_{2} + 47 \beta_1 - 3652) q^{6} + (22 \beta_{2} - 122 \beta_1 - 2420) q^{7} + ( - 200 \beta_1 - 6688) q^{8} + (315 \beta_{2} - 666 \beta_1 + 4851) q^{9} + ( - 308 \beta_{2} + 299 \beta_1 - 28908) q^{10} - 14641 q^{11} + ( - 392 \beta_{2} + 2492 \beta_1 - 23680) q^{12} + ( - 726 \beta_{2} + 538 \beta_1 - 31086) q^{13} + (1240 \beta_{2} + 3158 \beta_1 + 108088) q^{14} + (2335 \beta_{2} - 4720 \beta_1 + 180110) q^{15} + ( - 2496 \beta_{2} + 2944 \beta_1 + 7552) q^{16} + ( - 2156 \beta_{2} - 9154 \beta_1 + 6226) q^{17} + (9108 \beta_{2} - 7254 \beta_1 + 665676) q^{18} + ( - 7876 \beta_{2} - 13262 \beta_1 - 342452) q^{19} + ( - 4552 \beta_{2} + 18076 \beta_1 - 52192) q^{20} + ( - 4642 \beta_{2} + 1114 \beta_1 - 429308) q^{21} + 14641 \beta_1 q^{22} + (23639 \beta_{2} + 39292 \beta_1 + 558230) q^{23} + ( - 2112 \beta_{2} - 17352 \beta_1 - 315744) q^{24} + (15047 \beta_{2} - 35086 \beta_1 - 520243) q^{25} + ( - 13016 \beta_{2} + 48588 \beta_1 - 720696) q^{26} + ( - 38727 \beta_{2} + 44424 \beta_1 - 3428478) q^{27} + ( - 21648 \beta_{2} - 124440 \beta_1 - 856768) q^{28} + (15422 \beta_{2} + 56616 \beta_1 - 897886) q^{29} + (65780 \beta_{2} - 200525 \beta_1 + 4757500) q^{30} + (4231 \beta_{2} + 37548 \beta_1 - 1508434) q^{31} + ( - 53504 \beta_{2} + 141888 \beta_1 + 53504) q^{32} + (14641 \beta_{2} - 58564 \beta_1 + 907742) q^{33} + (47360 \beta_{2} + 174770 \beta_1 + 6633136) q^{34} + ( - 50490 \beta_{2} - 14650 \beta_1 - 2644620) q^{35} + (6048 \beta_{2} - 538200 \beta_1 + 6969456) q^{36} + ( - 73373 \beta_{2} + 181182 \beta_1 - 2940068) q^{37} + (11584 \beta_{2} + 761504 \beta_1 + 7765904) q^{38} + (110660 \beta_{2} - 316610 \beta_1 + 8307640) q^{39} + ( - 41536 \beta_{2} - 167592 \beta_1 - 1715296) q^{40} + ( - 13046 \beta_{2} + 147690 \beta_1 - 3257034) q^{41} + ( - 64616 \beta_{2} + 569126 \beta_1 - 2710120) q^{42} + (409750 \beta_{2} - 61556 \beta_1 + 6265248) q^{43} + ( - 117128 \beta_{2} - 175692 \beta_1 - 4450864) q^{44} + ( - 508536 \beta_{2} + 810108 \beta_1 - 30638466) q^{45} + ( - 30668 \beta_{2} - 1809821 \beta_1 - 22890340) q^{46} + ( - 45640 \beta_{2} - 1380980 \beta_1 - 10385272) q^{47} + (314176 \beta_{2} - 682240 \beta_1 + 25463936) q^{48} + (61036 \beta_{2} + 544512 \beta_1 - 18076559) q^{49} + (461252 \beta_{2} + 444724 \beta_1 + 34468412) q^{50} + ( - 242990 \beta_{2} - 40738 \beta_1 - 20703628) q^{51} + ( - 173184 \beta_{2} + 291712 \beta_1 - 28781984) q^{52} + (450204 \beta_{2} + 1561532 \beta_1 + 15833374) q^{53} + ( - 820116 \beta_{2} + 4173381 \beta_1 - 51276060) q^{54} + (43923 \beta_{2} - 497794 \beta_1 + 8901728) q^{55} + (100864 \beta_{2} + 1447536 \beta_1 + 37802560) q^{56} + (365376 \beta_{2} - 2557974 \beta_1 + 20701032) q^{57} + ( - 267864 \beta_{2} - 290432 \beta_1 - 40214920) q^{58} + ( - 264531 \beta_{2} + \cdots + 110712790) q^{59}+ \cdots + ( - 4611915 \beta_{2} + \cdots - 71023491) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 186 q^{3} + 912 q^{4} - 1824 q^{5} - 10956 q^{6} - 7260 q^{7} - 20064 q^{8} + 14553 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 186 q^{3} + 912 q^{4} - 1824 q^{5} - 10956 q^{6} - 7260 q^{7} - 20064 q^{8} + 14553 q^{9} - 86724 q^{10} - 43923 q^{11} - 71040 q^{12} - 93258 q^{13} + 324264 q^{14} + 540330 q^{15} + 22656 q^{16} + 18678 q^{17} + 1997028 q^{18} - 1027356 q^{19} - 156576 q^{20} - 1287924 q^{21} + 1674690 q^{23} - 947232 q^{24} - 1560729 q^{25} - 2162088 q^{26} - 10285434 q^{27} - 2570304 q^{28} - 2693658 q^{29} + 14272500 q^{30} - 4525302 q^{31} + 160512 q^{32} + 2723226 q^{33} + 19899408 q^{34} - 7933860 q^{35} + 20908368 q^{36} - 8820204 q^{37} + 23297712 q^{38} + 24922920 q^{39} - 5145888 q^{40} - 9771102 q^{41} - 8130360 q^{42} + 18795744 q^{43} - 13352592 q^{44} - 91915398 q^{45} - 68671020 q^{46} - 31155816 q^{47} + 76391808 q^{48} - 54229677 q^{49} + 103405236 q^{50} - 62110884 q^{51} - 86345952 q^{52} + 47500122 q^{53} - 153828180 q^{54} + 26705184 q^{55} + 113407680 q^{56} + 62103096 q^{57} - 120644760 q^{58} + 332138370 q^{59} + 290804160 q^{60} - 49031730 q^{61} - 86992620 q^{62} + 319652784 q^{63} - 421220352 q^{64} + 161689572 q^{65} + 160406796 q^{66} + 330560082 q^{67} - 382273056 q^{68} - 104664822 q^{69} - 22907160 q^{70} - 57835050 q^{71} + 302075136 q^{72} - 458816886 q^{73} - 528939708 q^{74} - 562184580 q^{75} - 1324671744 q^{76} + 106293660 q^{77} + 903869520 q^{78} - 798908748 q^{79} + 442084224 q^{80} + 1544572395 q^{81} - 376730664 q^{82} + 1239784920 q^{83} - 809016384 q^{84} - 632001744 q^{85} + 627638088 q^{86} + 505901880 q^{87} + 293757024 q^{88} - 699523368 q^{89} - 2575080288 q^{90} - 268926960 q^{91} + 3537302976 q^{92} + 614745786 q^{93} + 3327514080 q^{94} + 107303856 q^{95} + 2520807168 q^{96} - 2207436012 q^{97} - 1261919472 q^{98} - 213070473 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 306x - 836 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 6\nu - 204 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} + 3\beta _1 + 204 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
18.7255
−2.80408
−15.9214
−37.4510 70.6582 890.580 613.897 −2646.22 −6611.82 −14178.2 −14690.4 −22991.1
1.2 5.60816 5.22371 −480.549 −529.708 29.2954 −3708.24 −5566.37 −19655.7 −2970.69
1.3 31.8429 −261.882 501.969 −1908.19 −8339.07 3060.06 −319.425 48899.1 −60762.2
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \( +1 \)

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 11.10.a.a 3
3.b odd 2 1 99.10.a.b 3
4.b odd 2 1 176.10.a.g 3
5.b even 2 1 275.10.a.a 3
11.b odd 2 1 121.10.a.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.10.a.a 3 1.a even 1 1 trivial
99.10.a.b 3 3.b odd 2 1
121.10.a.b 3 11.b odd 2 1
176.10.a.g 3 4.b odd 2 1
275.10.a.a 3 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 1224T_{2} + 6688 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(11))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - 1224T + 6688 \) Copy content Toggle raw display
$3$ \( T^{3} + 186 T^{2} + \cdots + 96660 \) Copy content Toggle raw display
$5$ \( T^{3} + 1824 T^{2} + \cdots - 620517350 \) Copy content Toggle raw display
$7$ \( T^{3} + \cdots - 75027235360 \) Copy content Toggle raw display
$11$ \( (T + 14641)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} + \cdots - 73087940648800 \) Copy content Toggle raw display
$17$ \( T^{3} + \cdots + 13\!\cdots\!12 \) Copy content Toggle raw display
$19$ \( T^{3} + \cdots - 20\!\cdots\!76 \) Copy content Toggle raw display
$23$ \( T^{3} + \cdots + 44\!\cdots\!32 \) Copy content Toggle raw display
$29$ \( T^{3} + \cdots - 61\!\cdots\!96 \) Copy content Toggle raw display
$31$ \( T^{3} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{3} + \cdots + 95\!\cdots\!26 \) Copy content Toggle raw display
$41$ \( T^{3} + \cdots - 53\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{3} + \cdots + 12\!\cdots\!88 \) Copy content Toggle raw display
$47$ \( T^{3} + \cdots + 27\!\cdots\!48 \) Copy content Toggle raw display
$53$ \( T^{3} + \cdots - 34\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{3} + \cdots - 11\!\cdots\!88 \) Copy content Toggle raw display
$61$ \( T^{3} + \cdots - 99\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{3} + \cdots + 92\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{3} + \cdots - 24\!\cdots\!84 \) Copy content Toggle raw display
$73$ \( T^{3} + \cdots - 66\!\cdots\!08 \) Copy content Toggle raw display
$79$ \( T^{3} + \cdots - 10\!\cdots\!48 \) Copy content Toggle raw display
$83$ \( T^{3} + \cdots - 48\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{3} + \cdots - 39\!\cdots\!30 \) Copy content Toggle raw display
$97$ \( T^{3} + \cdots + 24\!\cdots\!50 \) Copy content Toggle raw display
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