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Properties

Label 1011.2.a.e
Level $1011$
Weight $2$
Character orbit 1011.a
Self dual yes
Analytic conductor $8.073$
Analytic rank $0$
Dimension $17$
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] = [1011,2,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, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1011.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1011 = 3 \cdot 337 \)
Weight: \( k \) \(=\) \( 2 \)
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: \(8.07287564435\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - 3 x^{16} - 24 x^{15} + 72 x^{14} + 231 x^{13} - 687 x^{12} - 1148 x^{11} + 3335 x^{10} + \cdots - 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{11} + 1) q^{5} - \beta_1 q^{6} + ( - \beta_{13} - 1) q^{7} + (\beta_{3} + \beta_1 + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + (\beta_{11} + 1) q^{5} - \beta_1 q^{6} + ( - \beta_{13} - 1) q^{7} + (\beta_{3} + \beta_1 + 1) q^{8} + q^{9} + (\beta_{11} - \beta_{9} - \beta_{8} + \cdots + 1) q^{10}+ \cdots + ( - \beta_{15} - \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 3 q^{2} - 17 q^{3} + 23 q^{4} + 13 q^{5} - 3 q^{6} - 12 q^{7} + 15 q^{8} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 17 q + 3 q^{2} - 17 q^{3} + 23 q^{4} + 13 q^{5} - 3 q^{6} - 12 q^{7} + 15 q^{8} + 17 q^{9} - 3 q^{10} + 6 q^{11} - 23 q^{12} + 14 q^{13} + 16 q^{14} - 13 q^{15} + 35 q^{16} + 17 q^{17} + 3 q^{18} - 12 q^{19} + 26 q^{20} + 12 q^{21} + 3 q^{22} + 15 q^{23} - 15 q^{24} + 32 q^{25} + 37 q^{26} - 17 q^{27} - 19 q^{28} + 41 q^{29} + 3 q^{30} - 18 q^{31} + 33 q^{32} - 6 q^{33} + 19 q^{34} + 5 q^{35} + 23 q^{36} + 15 q^{37} + 10 q^{38} - 14 q^{39} - 20 q^{40} + 18 q^{41} - 16 q^{42} - 2 q^{43} + 30 q^{44} + 13 q^{45} + 11 q^{46} + 2 q^{47} - 35 q^{48} + 41 q^{49} + 20 q^{50} - 17 q^{51} + q^{52} + 70 q^{53} - 3 q^{54} - 29 q^{55} + 58 q^{56} + 12 q^{57} - 11 q^{58} + 11 q^{59} - 26 q^{60} + 10 q^{61} + 4 q^{62} - 12 q^{63} + 35 q^{64} + 30 q^{65} - 3 q^{66} - 11 q^{67} + 36 q^{68} - 15 q^{69} + 7 q^{70} + 37 q^{71} + 15 q^{72} + 5 q^{73} + 36 q^{74} - 32 q^{75} + q^{76} + 35 q^{77} - 37 q^{78} + q^{79} + 64 q^{80} + 17 q^{81} - 63 q^{82} + 25 q^{83} + 19 q^{84} + 15 q^{85} + 56 q^{86} - 41 q^{87} + 16 q^{88} + 40 q^{89} - 3 q^{90} + 10 q^{91} + 37 q^{92} + 18 q^{93} - 38 q^{94} + 62 q^{95} - 33 q^{96} - 6 q^{97} - 24 q^{98} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - 3 x^{16} - 24 x^{15} + 72 x^{14} + 231 x^{13} - 687 x^{12} - 1148 x^{11} + 3335 x^{10} + \cdots - 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 89 \nu^{16} + 8005 \nu^{15} - 9652 \nu^{14} - 196016 \nu^{13} + 162191 \nu^{12} + 1897449 \nu^{11} + \cdots + 450688 ) / 162592 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1729 \nu^{16} - 14615 \nu^{15} - 34394 \nu^{14} + 356026 \nu^{13} + 269895 \nu^{12} - 3418435 \nu^{11} + \cdots - 540560 ) / 162592 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1676 \nu^{16} - 314 \nu^{15} - 43775 \nu^{14} + 4903 \nu^{13} + 451908 \nu^{12} - 36106 \nu^{11} + \cdots + 107688 ) / 81296 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4874 \nu^{16} - 3775 \nu^{15} - 117935 \nu^{14} + 70774 \nu^{13} + 1104994 \nu^{12} - 469741 \nu^{11} + \cdots - 130224 ) / 162592 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11565 \nu^{16} + 4133 \nu^{15} - 295336 \nu^{14} - 160152 \nu^{13} + 2959955 \nu^{12} + \cdots - 391008 ) / 325184 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 5865 \nu^{16} - 18373 \nu^{15} - 135606 \nu^{14} + 426842 \nu^{13} + 1249191 \nu^{12} + \cdots + 437840 ) / 162592 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12235 \nu^{16} + 6967 \nu^{15} + 319014 \nu^{14} - 118480 \nu^{13} - 3341037 \nu^{12} + \cdots - 1343008 ) / 325184 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 12441 \nu^{16} + 5223 \nu^{15} + 323200 \nu^{14} - 94096 \nu^{13} - 3341479 \nu^{12} + \cdots - 1432544 ) / 325184 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 6455 \nu^{16} - 1125 \nu^{15} + 173444 \nu^{14} + 44444 \nu^{13} - 1858993 \nu^{12} + \cdots - 136256 ) / 162592 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 4095 \nu^{16} - 5733 \nu^{15} - 108469 \nu^{14} + 133484 \nu^{13} + 1167649 \nu^{12} - 1221433 \nu^{11} + \cdots + 610928 ) / 81296 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 9339 \nu^{16} + 15107 \nu^{15} + 221886 \nu^{14} - 325520 \nu^{13} - 2068421 \nu^{12} + \cdots + 723936 ) / 162592 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 20453 \nu^{16} + 2213 \nu^{15} + 541158 \nu^{14} + 19216 \nu^{13} - 5718227 \nu^{12} + \cdots - 614880 ) / 325184 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 30011 \nu^{16} + 29821 \nu^{15} + 759848 \nu^{14} - 623916 \nu^{13} - 7678789 \nu^{12} + \cdots - 1891456 ) / 325184 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} + \beta_{13} + \beta_{12} + \beta_{9} + \beta_{8} + \beta_{6} - \beta_{5} + \beta_{3} + 7\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{16} - \beta_{14} + 2 \beta_{13} + 2 \beta_{12} + \beta_{10} + 2 \beta_{8} - \beta_{7} + \beta_{6} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12 \beta_{16} + 13 \beta_{13} + 11 \beta_{12} + 2 \beta_{11} + 2 \beta_{10} + 12 \beta_{9} + 14 \beta_{8} + \cdots + 86 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12 \beta_{16} - 14 \beta_{14} + 28 \beta_{13} + 26 \beta_{12} + \beta_{11} + 16 \beta_{10} + 25 \beta_{8} + \cdots + 53 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 106 \beta_{16} + 3 \beta_{15} - 2 \beta_{14} + 126 \beta_{13} + 96 \beta_{12} + 30 \beta_{11} + \cdots + 527 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 109 \beta_{16} - \beta_{15} - 143 \beta_{14} + 282 \beta_{13} + 248 \beta_{12} + 19 \beta_{11} + \cdots + 337 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 845 \beta_{16} + 52 \beta_{15} - 41 \beta_{14} + 1094 \beta_{13} + 774 \beta_{12} + 322 \beta_{11} + \cdots + 3372 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 901 \beta_{16} - 12 \beta_{15} - 1294 \beta_{14} + 2505 \beta_{13} + 2103 \beta_{12} + 241 \beta_{11} + \cdots + 2146 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 6462 \beta_{16} + 603 \beta_{15} - 553 \beta_{14} + 8995 \beta_{13} + 6005 \beta_{12} + 2996 \beta_{11} + \cdots + 22262 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 7161 \beta_{16} - 59 \beta_{15} - 11016 \beta_{14} + 20910 \beta_{13} + 16814 \beta_{12} + 2535 \beta_{11} + \cdots + 13905 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 48567 \beta_{16} + 5920 \beta_{15} - 6200 \beta_{14} + 71694 \beta_{13} + 45566 \beta_{12} + \cdots + 150532 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 55968 \beta_{16} + 303 \beta_{15} - 90547 \beta_{14} + 168496 \beta_{13} + 129968 \beta_{12} + \cdots + 92228 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 362607 \beta_{16} + 53311 \beta_{15} - 62644 \beta_{14} + 560518 \beta_{13} + 340896 \beta_{12} + \cdots + 1037157 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−2.69204
−2.26936
−2.16322
−1.71009
−1.41389
−1.24874
−0.594068
−0.178559
0.240963
0.662303
0.884176
1.26648
2.02273
2.38125
2.39552
2.66648
2.75006
−2.69204 −1.00000 5.24709 3.74216 2.69204 −4.21931 −8.74130 1.00000 −10.0741
1.2 −2.26936 −1.00000 3.15001 −1.52959 2.26936 −3.76576 −2.60978 1.00000 3.47118
1.3 −2.16322 −1.00000 2.67953 2.92625 2.16322 0.577233 −1.46997 1.00000 −6.33013
1.4 −1.71009 −1.00000 0.924414 −1.42715 1.71009 −2.46218 1.83935 1.00000 2.44056
1.5 −1.41389 −1.00000 −0.000917035 0 0.435921 1.41389 3.71148 2.82908 1.00000 −0.616344
1.6 −1.24874 −1.00000 −0.440658 −0.244044 1.24874 1.31583 3.04774 1.00000 0.304747
1.7 −0.594068 −1.00000 −1.64708 4.39269 0.594068 0.435778 2.16662 1.00000 −2.60956
1.8 −0.178559 −1.00000 −1.96812 0.537657 0.178559 −3.37052 0.708545 1.00000 −0.0960038
1.9 0.240963 −1.00000 −1.94194 −3.73777 −0.240963 −4.65940 −0.949860 1.00000 −0.900664
1.10 0.662303 −1.00000 −1.56135 1.48054 −0.662303 4.62040 −2.35870 1.00000 0.980568
1.11 0.884176 −1.00000 −1.21823 3.51366 −0.884176 −4.67973 −2.84548 1.00000 3.10669
1.12 1.26648 −1.00000 −0.396024 −0.897741 −1.26648 −1.60913 −3.03452 1.00000 −1.13697
1.13 2.02273 −1.00000 2.09144 3.57243 −2.02273 0.575771 0.184959 1.00000 7.22606
1.14 2.38125 −1.00000 3.67037 2.42904 −2.38125 3.22155 3.97758 1.00000 5.78416
1.15 2.39552 −1.00000 3.73854 −4.08189 −2.39552 0.358243 4.16471 1.00000 −9.77826
1.16 2.66648 −1.00000 5.11011 −0.434771 −2.66648 1.62787 8.29303 1.00000 −1.15931
1.17 2.75006 −1.00000 5.56283 2.32261 −2.75006 −3.67812 9.79800 1.00000 6.38732
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
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 1011.2.a.e ✓ 17
3.b odd 2 1 3033.2.a.h 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1011.2.a.e ✓ 17 1.a even 1 1 trivial
3033.2.a.h 17 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} - 3 T_{2}^{16} - 24 T_{2}^{15} + 72 T_{2}^{14} + 231 T_{2}^{13} - 687 T_{2}^{12} - 1148 T_{2}^{11} + \cdots - 64 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1011))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} - 3 T^{16} + \cdots - 64 \) Copy content Toggle raw display
$3$ \( (T + 1)^{17} \) Copy content Toggle raw display
$5$ \( T^{17} - 13 T^{16} + \cdots + 3750 \) Copy content Toggle raw display
$7$ \( T^{17} + 12 T^{16} + \cdots - 104480 \) Copy content Toggle raw display
$11$ \( T^{17} - 6 T^{16} + \cdots + 70851584 \) Copy content Toggle raw display
$13$ \( T^{17} - 14 T^{16} + \cdots - 7874 \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots - 5358693666 \) Copy content Toggle raw display
$19$ \( T^{17} + 12 T^{16} + \cdots - 38805504 \) Copy content Toggle raw display
$23$ \( T^{17} - 15 T^{16} + \cdots - 81920 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots + 7239404226 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots + 5520238592 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots - 1258860270 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots - 7143078912 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots - 114065535196 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots - 10535464960 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots + 9063786687918 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots - 738027507824 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots + 2351298248704 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots - 37304139776 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots + 4784608116364 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots + 260589019136 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 407664672885872 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots - 55758424079376 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots - 61\!\cdots\!82 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots + 31391908940800 \) Copy content Toggle raw display
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