Properties

Label 891.2.a.q
Level 891
Weight 2
Character orbit 891.a
Self dual yes
Analytic conductor 7.115
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.22545.1
Defining polynomial: \(x^{4} - x^{3} - 6 x^{2} + 5 x + 4\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 99)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( 3 + \beta_{3} ) q^{4} + ( 1 - \beta_{1} - \beta_{3} ) q^{5} + \beta_{1} q^{7} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{2} q^{2} + ( 3 + \beta_{3} ) q^{4} + ( 1 - \beta_{1} - \beta_{3} ) q^{5} + \beta_{1} q^{7} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{8} + ( 1 - 3 \beta_{1} ) q^{10} + q^{11} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{13} + ( 2 \beta_{1} + \beta_{3} ) q^{14} + ( 5 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{16} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{17} + ( 2 + 2 \beta_{1} + \beta_{3} ) q^{19} + ( -2 - 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{20} + \beta_{2} q^{22} + ( 4 - \beta_{1} + \beta_{3} ) q^{23} + ( 3 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{25} + ( 6 + \beta_{1} + 3 \beta_{3} ) q^{26} + ( -1 + 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{28} + ( -1 - \beta_{2} + \beta_{3} ) q^{29} + ( -\beta_{2} + \beta_{3} ) q^{31} + ( -10 + 2 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{32} + ( 5 - 4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{34} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{35} + ( 1 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{37} + ( -1 + 5 \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{38} + ( 4 - 3 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{40} + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{41} + ( -5 - \beta_{2} ) q^{43} + ( 3 + \beta_{3} ) q^{44} + ( -1 - \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{46} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{47} + ( -4 + \beta_{2} ) q^{49} + ( 7 - 4 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{50} + ( -5 + 3 \beta_{1} + 7 \beta_{2} ) q^{52} + ( -1 - 3 \beta_{1} - \beta_{3} ) q^{53} + ( 1 - \beta_{1} - \beta_{3} ) q^{55} + ( 4 + 3 \beta_{1} + \beta_{3} ) q^{56} + ( -6 + \beta_{1} - 2 \beta_{3} ) q^{58} + ( 1 - \beta_{1} + \beta_{3} ) q^{59} + ( 4 - \beta_{1} ) q^{61} + ( -6 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{62} + ( 6 + \beta_{1} - 7 \beta_{2} + 2 \beta_{3} ) q^{64} + ( 5 - 2 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{65} + ( 4 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{67} + ( -2 - 5 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{68} + ( -9 + \beta_{1} - 3 \beta_{2} ) q^{70} + ( -1 + 3 \beta_{1} - 2 \beta_{2} ) q^{71} + ( 6 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{73} + ( -8 - 4 \beta_{1} - \beta_{2} - \beta_{3} ) q^{74} + ( 10 + 7 \beta_{1} + 5 \beta_{3} ) q^{76} + \beta_{1} q^{77} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{79} + ( -9 - 2 \beta_{3} ) q^{80} + ( -5 + 6 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{82} + ( 5 + 2 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{83} + ( 4 - 4 \beta_{1} + 4 \beta_{2} + 3 \beta_{3} ) q^{85} + ( -5 - 5 \beta_{2} - \beta_{3} ) q^{86} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{88} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 4 + 3 \beta_{1} ) q^{91} + ( 19 - 2 \beta_{1} - 3 \beta_{2} + 4 \beta_{3} ) q^{92} + ( -5 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{94} + ( -7 - \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{95} + ( 7 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{97} + ( 5 - 4 \beta_{2} + \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{2} + 11q^{4} + 4q^{5} + q^{7} + O(q^{10}) \) \( 4q + q^{2} + 11q^{4} + 4q^{5} + q^{7} + q^{10} + 4q^{11} + 7q^{13} + q^{14} + 17q^{16} - 5q^{17} + 9q^{19} - 10q^{20} + q^{22} + 14q^{23} + 14q^{25} + 22q^{26} - q^{28} - 6q^{29} - 2q^{31} - 34q^{32} + 16q^{34} - 8q^{35} + 3q^{37} + 3q^{38} + 12q^{40} - 2q^{41} - 21q^{43} + 11q^{44} + 2q^{46} - 7q^{47} - 15q^{49} + 23q^{50} - 10q^{52} - 6q^{53} + 4q^{55} + 18q^{56} - 21q^{58} + 2q^{59} + 15q^{61} - 20q^{62} + 16q^{64} + 19q^{65} + 14q^{67} - 7q^{68} - 38q^{70} - 3q^{71} + 22q^{73} - 36q^{74} + 42q^{76} + q^{77} + 11q^{79} - 34q^{80} - 17q^{82} + 18q^{83} + 13q^{85} - 24q^{86} - 6q^{89} + 19q^{91} + 67q^{92} - 19q^{94} - 30q^{95} + 26q^{97} + 15q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 6 x^{2} + 5 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 5 \beta_{1}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.519120
1.45106
−2.27060
2.33866
−2.73051 0 5.45571 −0.936586 0 −0.519120 −9.43585 0 2.55736
1.2 −0.894434 0 −1.19999 3.74893 0 1.45106 2.86218 0 −3.35317
1.3 2.15561 0 2.64667 3.62393 0 −2.27060 1.39396 0 7.81179
1.4 2.46934 0 4.09762 −2.43628 0 2.33866 5.17972 0 −6.01598
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 891.2.a.q 4
3.b odd 2 1 891.2.a.p 4
9.c even 3 2 99.2.e.e 8
9.d odd 6 2 297.2.e.e 8
11.b odd 2 1 9801.2.a.bi 4
33.d even 2 1 9801.2.a.bl 4
99.h odd 6 2 1089.2.e.i 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.e.e 8 9.c even 3 2
297.2.e.e 8 9.d odd 6 2
891.2.a.p 4 3.b odd 2 1
891.2.a.q 4 1.a even 1 1 trivial
1089.2.e.i 8 99.h odd 6 2
9801.2.a.bi 4 11.b odd 2 1
9801.2.a.bl 4 33.d even 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(891))\):

\( T_{2}^{4} - T_{2}^{3} - 9 T_{2}^{2} + 8 T_{2} + 13 \)
\( T_{5}^{4} - 4 T_{5}^{3} - 9 T_{5}^{2} + 29 T_{5} + 31 \)
\( T_{7}^{4} - T_{7}^{3} - 6 T_{7}^{2} + 5 T_{7} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T - T^{2} + 2 T^{3} + T^{4} + 4 T^{5} - 4 T^{6} - 8 T^{7} + 16 T^{8} \)
$3$ 1
$5$ \( 1 - 4 T + 11 T^{2} - 31 T^{3} + 91 T^{4} - 155 T^{5} + 275 T^{6} - 500 T^{7} + 625 T^{8} \)
$7$ \( 1 - T + 22 T^{2} - 16 T^{3} + 214 T^{4} - 112 T^{5} + 1078 T^{6} - 343 T^{7} + 2401 T^{8} \)
$11$ \( ( 1 - T )^{4} \)
$13$ \( 1 - 7 T + 37 T^{2} - 118 T^{3} + 466 T^{4} - 1534 T^{5} + 6253 T^{6} - 15379 T^{7} + 28561 T^{8} \)
$17$ \( 1 + 5 T + 44 T^{2} + 86 T^{3} + 682 T^{4} + 1462 T^{5} + 12716 T^{6} + 24565 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 9 T + 76 T^{2} - 432 T^{3} + 2112 T^{4} - 8208 T^{5} + 27436 T^{6} - 61731 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 14 T + 143 T^{2} - 953 T^{3} + 5332 T^{4} - 21919 T^{5} + 75647 T^{6} - 170338 T^{7} + 279841 T^{8} \)
$29$ \( 1 + 6 T + 107 T^{2} + 417 T^{3} + 4374 T^{4} + 12093 T^{5} + 89987 T^{6} + 146334 T^{7} + 707281 T^{8} \)
$31$ \( 1 + 2 T + 103 T^{2} + 113 T^{3} + 4405 T^{4} + 3503 T^{5} + 98983 T^{6} + 59582 T^{7} + 923521 T^{8} \)
$37$ \( 1 - 3 T + 67 T^{2} - 189 T^{3} + 2277 T^{4} - 6993 T^{5} + 91723 T^{6} - 151959 T^{7} + 1874161 T^{8} \)
$41$ \( 1 + 2 T + 101 T^{2} + 407 T^{3} + 4894 T^{4} + 16687 T^{5} + 169781 T^{6} + 137842 T^{7} + 2825761 T^{8} \)
$43$ \( 1 + 21 T + 328 T^{2} + 3186 T^{3} + 25008 T^{4} + 136998 T^{5} + 606472 T^{6} + 1669647 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 7 T + 173 T^{2} + 985 T^{3} + 11845 T^{4} + 46295 T^{5} + 382157 T^{6} + 726761 T^{7} + 4879681 T^{8} \)
$53$ \( 1 + 6 T + 167 T^{2} + 789 T^{3} + 11961 T^{4} + 41817 T^{5} + 469103 T^{6} + 893262 T^{7} + 7890481 T^{8} \)
$59$ \( 1 - 2 T + 215 T^{2} - 305 T^{3} + 18421 T^{4} - 17995 T^{5} + 748415 T^{6} - 410758 T^{7} + 12117361 T^{8} \)
$61$ \( 1 - 15 T + 322 T^{2} - 2910 T^{3} + 31962 T^{4} - 177510 T^{5} + 1198162 T^{6} - 3404715 T^{7} + 13845841 T^{8} \)
$67$ \( 1 - 14 T + 247 T^{2} - 1979 T^{3} + 21949 T^{4} - 132593 T^{5} + 1108783 T^{6} - 4210682 T^{7} + 20151121 T^{8} \)
$71$ \( 1 + 3 T + 197 T^{2} + 909 T^{3} + 17697 T^{4} + 64539 T^{5} + 993077 T^{6} + 1073733 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 22 T + 421 T^{2} - 4807 T^{3} + 49780 T^{4} - 350911 T^{5} + 2243509 T^{6} - 8558374 T^{7} + 28398241 T^{8} \)
$79$ \( 1 - 11 T + 292 T^{2} - 2582 T^{3} + 33694 T^{4} - 203978 T^{5} + 1822372 T^{6} - 5423429 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 18 T + 329 T^{2} - 3771 T^{3} + 42384 T^{4} - 312993 T^{5} + 2266481 T^{6} - 10292166 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 6 T + 224 T^{2} + 1554 T^{3} + 26094 T^{4} + 138306 T^{5} + 1774304 T^{6} + 4229814 T^{7} + 62742241 T^{8} \)
$97$ \( 1 - 26 T + 544 T^{2} - 7460 T^{3} + 85489 T^{4} - 723620 T^{5} + 5118496 T^{6} - 23729498 T^{7} + 88529281 T^{8} \)
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