Properties

Label 1151.1.b.a
Level $1151$
Weight $1$
Character orbit 1151.b
Self dual yes
Analytic conductor $0.574$
Analytic rank $0$
Dimension $20$
Projective image $D_{41}$
CM discriminant -1151
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1151 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1151.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.574423829541\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{82})^+\)
Defining polynomial: \(x^{20} - x^{19} - 19 x^{18} + 18 x^{17} + 153 x^{16} - 136 x^{15} - 680 x^{14} + 560 x^{13} + 1820 x^{12} - 1365 x^{11} - 3003 x^{10} + 2002 x^{9} + 3003 x^{8} - 1716 x^{7} - 1716 x^{6} + 792 x^{5} + 495 x^{4} - 165 x^{3} - 55 x^{2} + 10 x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{41}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{41} - \cdots)\)

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + \beta_{16} q^{3} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{4} + \beta_{2} q^{5} + ( \beta_{6} - \beta_{15} ) q^{6} -\beta_{3} q^{7} + ( \beta_{10} - \beta_{11} ) q^{8} + ( 1 - \beta_{9} ) q^{9} +O(q^{10})\) \( q + \beta_{10} q^{2} + \beta_{16} q^{3} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{4} + \beta_{2} q^{5} + ( \beta_{6} - \beta_{15} ) q^{6} -\beta_{3} q^{7} + ( \beta_{10} - \beta_{11} ) q^{8} + ( 1 - \beta_{9} ) q^{9} + ( \beta_{8} + \beta_{12} ) q^{10} -\beta_{17} q^{11} + ( \beta_{4} - \beta_{5} + \beta_{16} ) q^{12} + ( -\beta_{7} - \beta_{13} ) q^{14} + ( \beta_{14} + \beta_{18} ) q^{15} + ( -\beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{16} + ( -\beta_{1} + \beta_{10} - \beta_{19} ) q^{18} + ( \beta_{2} + \beta_{18} - \beta_{19} ) q^{20} + ( -\beta_{13} - \beta_{19} ) q^{21} + ( -\beta_{7} + \beta_{14} ) q^{22} + ( -\beta_{5} + \beta_{6} + \beta_{14} - \beta_{15} ) q^{24} + ( 1 + \beta_{4} ) q^{25} + ( -\beta_{7} + \beta_{16} ) q^{27} + ( -\beta_{3} - \beta_{17} + \beta_{18} ) q^{28} + \beta_{12} q^{29} + ( \beta_{4} + \beta_{8} - \beta_{13} - \beta_{17} ) q^{30} + ( -\beta_{9} + \beta_{10} - \beta_{11} ) q^{32} + ( -\beta_{1} + \beta_{8} ) q^{33} + ( -\beta_{1} - \beta_{5} ) q^{35} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{36} -\beta_{5} q^{37} + ( \beta_{8} - \beta_{9} + \beta_{12} - \beta_{13} ) q^{40} + ( -\beta_{3} - \beta_{9} + \beta_{12} + \beta_{18} ) q^{42} -\beta_{11} q^{43} + ( -\beta_{3} + \beta_{4} - \beta_{17} ) q^{44} + ( \beta_{2} - \beta_{7} - \beta_{11} ) q^{45} + \beta_{8} q^{47} + ( \beta_{4} - \beta_{5} - \beta_{15} + \beta_{16} - \beta_{17} ) q^{48} + ( 1 + \beta_{6} ) q^{49} + ( \beta_{6} + \beta_{10} + \beta_{14} ) q^{50} -\beta_{15} q^{53} + ( -\beta_{3} + \beta_{6} - \beta_{15} - \beta_{17} ) q^{54} + ( -\beta_{15} - \beta_{19} ) q^{55} + ( -\beta_{7} + \beta_{8} - \beta_{13} + \beta_{14} ) q^{56} + ( \beta_{2} - \beta_{19} ) q^{58} + \beta_{4} q^{59} + ( \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} + \beta_{14} + \beta_{18} ) q^{60} + ( -\beta_{3} + \beta_{6} + \beta_{12} ) q^{63} + ( -\beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{64} + ( \beta_{2} - \beta_{9} - \beta_{11} + \beta_{18} ) q^{66} -\beta_{13} q^{67} + ( -\beta_{5} - \beta_{9} - \beta_{11} - \beta_{15} ) q^{70} + ( -1 + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} ) q^{72} + ( -\beta_{5} - \beta_{15} ) q^{74} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{75} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} + \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{77} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{18} - \beta_{19} ) q^{80} + ( 1 - \beta_{9} + \beta_{18} ) q^{81} + \beta_{6} q^{83} + ( -\beta_{1} + \beta_{2} - \beta_{7} + \beta_{8} - \beta_{13} - \beta_{19} ) q^{84} + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{86} + ( \beta_{4} - \beta_{13} ) q^{87} + ( \beta_{6} - \beta_{7} - \beta_{13} + \beta_{14} ) q^{88} + ( -1 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{18} + \beta_{19} ) q^{90} + ( \beta_{2} + \beta_{18} ) q^{94} + ( -\beta_{5} + \beta_{6} - \beta_{7} + \beta_{14} - \beta_{15} + \beta_{16} ) q^{96} + ( \beta_{4} + \beta_{10} + \beta_{16} ) q^{98} + ( \beta_{8} - \beta_{15} - \beta_{17} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - q^{2} - q^{3} + 19q^{4} - q^{5} - 2q^{6} - q^{7} - 2q^{8} + 19q^{9} + O(q^{10}) \) \( 20q - q^{2} - q^{3} + 19q^{4} - q^{5} - 2q^{6} - q^{7} - 2q^{8} + 19q^{9} - 2q^{10} - q^{11} - 3q^{12} - 2q^{14} - 2q^{15} + 18q^{16} - 3q^{18} - 3q^{20} - 2q^{21} - 2q^{22} - 4q^{24} + 19q^{25} - 2q^{27} - 3q^{28} - q^{29} - 4q^{30} - 3q^{32} - 2q^{33} - 2q^{35} + 16q^{36} - q^{37} - 4q^{40} - 4q^{42} - q^{43} - 3q^{44} - 3q^{45} - q^{47} - 5q^{48} + 19q^{49} - 3q^{50} - q^{53} - 4q^{54} - 2q^{55} - 4q^{56} - 2q^{58} - q^{59} - 6q^{60} - 3q^{63} + 17q^{64} - 4q^{66} - q^{67} - 4q^{70} - 6q^{72} - 2q^{74} - 3q^{75} - 2q^{77} - 5q^{80} + 18q^{81} - q^{83} - 6q^{84} - 2q^{86} - 2q^{87} - 4q^{88} - 6q^{90} - 2q^{94} - 6q^{96} - 3q^{98} - 3q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1151\mathbb{Z}\right)^\times\).

\(n\) \(17\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1150.1
−1.90679
−1.21245
−0.0766055
1.08714
1.85500
1.94739
1.33065
0.229367
−0.955440
−1.79233
−1.97656
−1.44104
−0.380782
0.818137
1.71914
1.99413
1.54298
0.529963
−0.676034
−1.63586
−1.99413 0.380782 2.97656 1.63586 −0.759330 1.21245 −3.94152 −0.855005 −3.26212
1150.2 −1.94739 −1.08714 2.79233 −0.529963 2.11708 −1.85500 −3.49037 0.181863 1.03205
1150.3 −1.85500 1.63586 2.44104 −1.99413 −3.03453 −0.229367 −2.67314 1.67603 3.69912
1150.4 −1.71914 −1.94739 1.95544 −0.818137 3.34784 1.97656 −1.64253 2.79233 1.40649
1150.5 −1.54298 1.97656 1.38078 1.44104 −3.04979 −0.818137 −0.587539 2.90679 −2.22350
1150.6 −1.33065 −1.71914 0.770633 1.79233 2.28758 −1.54298 0.305207 1.95544 −2.38497
1150.7 −1.08714 1.21245 0.181863 −0.229367 −1.31810 1.63586 0.889426 0.470037 0.249353
1150.8 −0.818137 −0.529963 −0.330651 −1.94739 0.433582 0.676034 1.08866 −0.719139 1.59323
1150.9 −0.529963 −0.229367 −0.719139 −1.08714 0.121556 −1.99413 0.911080 −0.947391 0.576141
1150.10 −0.229367 0.955440 −0.947391 1.21245 −0.219146 0.380782 0.446667 −0.0871351 −0.278096
1150.11 0.0766055 −1.54298 −0.994132 1.90679 −0.118201 1.79233 −0.152761 1.38078 0.146071
1150.12 0.380782 1.90679 −0.855005 0.0766055 0.726073 −1.33065 −0.706353 2.63586 0.0291700
1150.13 0.676034 −1.99413 −0.542978 −1.85500 −1.34810 −1.08714 −1.04311 2.97656 −1.25405
1150.14 0.955440 1.79233 −0.0871351 −1.33065 1.71246 1.90679 −1.03869 2.21245 −1.27136
1150.15 1.21245 −1.33065 0.470037 0.955440 −1.61335 0.0766055 −0.642554 0.770633 1.15842
1150.16 1.44104 0.676034 1.07661 1.97656 0.974194 −1.94739 0.110392 −0.542978 2.84831
1150.17 1.63586 0.0766055 1.67603 0.380782 0.125316 0.955440 1.10590 −0.994132 0.622906
1150.18 1.79233 −0.818137 2.21245 −1.71914 −1.46637 1.44104 2.17311 −0.330651 −3.08127
1150.19 1.90679 1.44104 2.63586 −1.54298 2.74777 −1.71914 3.11924 1.07661 −2.94214
1150.20 1.97656 −1.85500 2.90679 0.676034 −3.66653 −0.529963 3.76889 2.44104 1.33622
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1150.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
1151.b odd 2 1 CM by \(\Q(\sqrt{-1151}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1151.1.b.a 20
1151.b odd 2 1 CM 1151.1.b.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1151.1.b.a 20 1.a even 1 1 trivial
1151.1.b.a 20 1151.b odd 2 1 CM

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1151, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 10 T - 55 T^{2} + 165 T^{3} + 495 T^{4} - 792 T^{5} - 1716 T^{6} + 1716 T^{7} + 3003 T^{8} - 2002 T^{9} - 3003 T^{10} + 1365 T^{11} + 1820 T^{12} - 560 T^{13} - 680 T^{14} + 136 T^{15} + 153 T^{16} - 18 T^{17} - 19 T^{18} + T^{19} + T^{20} \)
$3$ \( 1 - 10 T - 55 T^{2} + 165 T^{3} + 495 T^{4} - 792 T^{5} - 1716 T^{6} + 1716 T^{7} + 3003 T^{8} - 2002 T^{9} - 3003 T^{10} + 1365 T^{11} + 1820 T^{12} - 560 T^{13} - 680 T^{14} + 136 T^{15} + 153 T^{16} - 18 T^{17} - 19 T^{18} + T^{19} + T^{20} \)
$5$ \( 1 - 10 T - 55 T^{2} + 165 T^{3} + 495 T^{4} - 792 T^{5} - 1716 T^{6} + 1716 T^{7} + 3003 T^{8} - 2002 T^{9} - 3003 T^{10} + 1365 T^{11} + 1820 T^{12} - 560 T^{13} - 680 T^{14} + 136 T^{15} + 153 T^{16} - 18 T^{17} - 19 T^{18} + T^{19} + T^{20} \)
$7$ \( 1 - 10 T - 55 T^{2} + 165 T^{3} + 495 T^{4} - 792 T^{5} - 1716 T^{6} + 1716 T^{7} + 3003 T^{8} - 2002 T^{9} - 3003 T^{10} + 1365 T^{11} + 1820 T^{12} - 560 T^{13} - 680 T^{14} + 136 T^{15} + 153 T^{16} - 18 T^{17} - 19 T^{18} + T^{19} + T^{20} \)
$11$ \( 1 - 10 T - 55 T^{2} + 165 T^{3} + 495 T^{4} - 792 T^{5} - 1716 T^{6} + 1716 T^{7} + 3003 T^{8} - 2002 T^{9} - 3003 T^{10} + 1365 T^{11} + 1820 T^{12} - 560 T^{13} - 680 T^{14} + 136 T^{15} + 153 T^{16} - 18 T^{17} - 19 T^{18} + T^{19} + T^{20} \)
$13$ \( T^{20} \)
$17$ \( T^{20} \)
$19$ \( T^{20} \)
$23$ \( T^{20} \)
$29$ \( 1 - 10 T - 55 T^{2} + 165 T^{3} + 495 T^{4} - 792 T^{5} - 1716 T^{6} + 1716 T^{7} + 3003 T^{8} - 2002 T^{9} - 3003 T^{10} + 1365 T^{11} + 1820 T^{12} - 560 T^{13} - 680 T^{14} + 136 T^{15} + 153 T^{16} - 18 T^{17} - 19 T^{18} + T^{19} + T^{20} \)
$31$ \( T^{20} \)
$37$ \( 1 - 10 T - 55 T^{2} + 165 T^{3} + 495 T^{4} - 792 T^{5} - 1716 T^{6} + 1716 T^{7} + 3003 T^{8} - 2002 T^{9} - 3003 T^{10} + 1365 T^{11} + 1820 T^{12} - 560 T^{13} - 680 T^{14} + 136 T^{15} + 153 T^{16} - 18 T^{17} - 19 T^{18} + T^{19} + T^{20} \)
$41$ \( T^{20} \)
$43$ \( 1 - 10 T - 55 T^{2} + 165 T^{3} + 495 T^{4} - 792 T^{5} - 1716 T^{6} + 1716 T^{7} + 3003 T^{8} - 2002 T^{9} - 3003 T^{10} + 1365 T^{11} + 1820 T^{12} - 560 T^{13} - 680 T^{14} + 136 T^{15} + 153 T^{16} - 18 T^{17} - 19 T^{18} + T^{19} + T^{20} \)
$47$ \( 1 - 10 T - 55 T^{2} + 165 T^{3} + 495 T^{4} - 792 T^{5} - 1716 T^{6} + 1716 T^{7} + 3003 T^{8} - 2002 T^{9} - 3003 T^{10} + 1365 T^{11} + 1820 T^{12} - 560 T^{13} - 680 T^{14} + 136 T^{15} + 153 T^{16} - 18 T^{17} - 19 T^{18} + T^{19} + T^{20} \)
$53$ \( 1 - 10 T - 55 T^{2} + 165 T^{3} + 495 T^{4} - 792 T^{5} - 1716 T^{6} + 1716 T^{7} + 3003 T^{8} - 2002 T^{9} - 3003 T^{10} + 1365 T^{11} + 1820 T^{12} - 560 T^{13} - 680 T^{14} + 136 T^{15} + 153 T^{16} - 18 T^{17} - 19 T^{18} + T^{19} + T^{20} \)
$59$ \( 1 - 10 T - 55 T^{2} + 165 T^{3} + 495 T^{4} - 792 T^{5} - 1716 T^{6} + 1716 T^{7} + 3003 T^{8} - 2002 T^{9} - 3003 T^{10} + 1365 T^{11} + 1820 T^{12} - 560 T^{13} - 680 T^{14} + 136 T^{15} + 153 T^{16} - 18 T^{17} - 19 T^{18} + T^{19} + T^{20} \)
$61$ \( T^{20} \)
$67$ \( 1 - 10 T - 55 T^{2} + 165 T^{3} + 495 T^{4} - 792 T^{5} - 1716 T^{6} + 1716 T^{7} + 3003 T^{8} - 2002 T^{9} - 3003 T^{10} + 1365 T^{11} + 1820 T^{12} - 560 T^{13} - 680 T^{14} + 136 T^{15} + 153 T^{16} - 18 T^{17} - 19 T^{18} + T^{19} + T^{20} \)
$71$ \( T^{20} \)
$73$ \( T^{20} \)
$79$ \( T^{20} \)
$83$ \( 1 - 10 T - 55 T^{2} + 165 T^{3} + 495 T^{4} - 792 T^{5} - 1716 T^{6} + 1716 T^{7} + 3003 T^{8} - 2002 T^{9} - 3003 T^{10} + 1365 T^{11} + 1820 T^{12} - 560 T^{13} - 680 T^{14} + 136 T^{15} + 153 T^{16} - 18 T^{17} - 19 T^{18} + T^{19} + T^{20} \)
$89$ \( T^{20} \)
$97$ \( T^{20} \)
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