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Properties

Label 1150.2.b.j.599.1
Level $1150$
Weight $2$
Character 1150.599
Analytic conductor $9.183$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1150,2,Mod(599,1150)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1150.599"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-6,0,2,0,0,-20,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.18279623245\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.77580864.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 19x^{4} + 105x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.1
Root \(-3.11903i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.j.599.6

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000i q^{2} -3.11903i q^{3} -1.00000 q^{4} -3.11903 q^{6} -4.50973i q^{7} +1.00000i q^{8} -6.72833 q^{9} +4.33763 q^{11} +3.11903i q^{12} -3.72833i q^{13} -4.50973 q^{14} +1.00000 q^{16} -1.11903i q^{17} +6.72833i q^{18} -4.50973 q^{19} -14.0660 q^{21} -4.33763i q^{22} -1.00000i q^{23} +3.11903 q^{24} -3.72833 q^{26} +11.6288i q^{27} +4.50973i q^{28} +8.23805 q^{29} +1.72833 q^{31} -1.00000i q^{32} -13.5292i q^{33} -1.11903 q^{34} +6.72833 q^{36} +0.781399i q^{37} +4.50973i q^{38} -11.6288 q^{39} +3.90043 q^{41} +14.0660i q^{42} +8.00000i q^{43} -4.33763 q^{44} -1.00000 q^{46} +11.4567i q^{47} -3.11903i q^{48} -13.3376 q^{49} -3.49027 q^{51} +3.72833i q^{52} -6.00000i q^{53} +11.6288 q^{54} +4.50973 q^{56} +14.0660i q^{57} -8.23805i q^{58} +2.23805 q^{59} +3.55623 q^{61} -1.72833i q^{62} +30.3429i q^{63} -1.00000 q^{64} -13.5292 q^{66} -2.43720i q^{67} +1.11903i q^{68} -3.11903 q^{69} +7.11903 q^{71} -6.72833i q^{72} -9.45665i q^{73} +0.781399 q^{74} +4.50973 q^{76} -19.5615i q^{77} +11.6288i q^{78} +14.9133 q^{79} +16.0854 q^{81} -3.90043i q^{82} +2.78140i q^{83} +14.0660 q^{84} +8.00000 q^{86} -25.6947i q^{87} +4.33763i q^{88} +7.69471 q^{89} -16.8137 q^{91} +1.00000i q^{92} -5.39070i q^{93} +11.4567 q^{94} -3.11903 q^{96} +0.642920i q^{97} +13.3376i q^{98} -29.1850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} + 2 q^{6} - 20 q^{9} + 6 q^{11} - 6 q^{14} + 6 q^{16} - 6 q^{19} - 44 q^{21} - 2 q^{24} - 2 q^{26} + 8 q^{29} - 10 q^{31} + 14 q^{34} + 20 q^{36} - 28 q^{39} + 2 q^{41} - 6 q^{44} - 6 q^{46}+ \cdots - 114 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) − 3.11903i − 1.80077i −0.435093 − 0.900385i \(-0.643285\pi\)
0.435093 − 0.900385i \(-0.356715\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −3.11903 −1.27334
\(7\) − 4.50973i − 1.70452i −0.523122 − 0.852258i \(-0.675233\pi\)
0.523122 − 0.852258i \(-0.324767\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −6.72833 −2.24278
\(10\) 0 0
\(11\) 4.33763 1.30784 0.653922 − 0.756562i \(-0.273122\pi\)
0.653922 + 0.756562i \(0.273122\pi\)
\(12\) 3.11903i 0.900385i
\(13\) − 3.72833i − 1.03405i −0.855970 − 0.517026i \(-0.827039\pi\)
0.855970 − 0.517026i \(-0.172961\pi\)
\(14\) −4.50973 −1.20527
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 1.11903i − 0.271404i −0.990750 − 0.135702i \(-0.956671\pi\)
0.990750 − 0.135702i \(-0.0433289\pi\)
\(18\) 6.72833i 1.58588i
\(19\) −4.50973 −1.03460 −0.517301 − 0.855803i \(-0.673063\pi\)
−0.517301 + 0.855803i \(0.673063\pi\)
\(20\) 0 0
\(21\) −14.0660 −3.06944
\(22\) − 4.33763i − 0.924785i
\(23\) − 1.00000i − 0.208514i
\(24\) 3.11903 0.636669
\(25\) 0 0
\(26\) −3.72833 −0.731185
\(27\) 11.6288i 2.23795i
\(28\) 4.50973i 0.852258i
\(29\) 8.23805 1.52977 0.764884 − 0.644168i \(-0.222796\pi\)
0.764884 + 0.644168i \(0.222796\pi\)
\(30\) 0 0
\(31\) 1.72833 0.310417 0.155208 − 0.987882i \(-0.450395\pi\)
0.155208 + 0.987882i \(0.450395\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 13.5292i − 2.35513i
\(34\) −1.11903 −0.191911
\(35\) 0 0
\(36\) 6.72833 1.12139
\(37\) 0.781399i 0.128461i 0.997935 + 0.0642306i \(0.0204593\pi\)
−0.997935 + 0.0642306i \(0.979541\pi\)
\(38\) 4.50973i 0.731574i
\(39\) −11.6288 −1.86209
\(40\) 0 0
\(41\) 3.90043 0.609144 0.304572 − 0.952489i \(-0.401487\pi\)
0.304572 + 0.952489i \(0.401487\pi\)
\(42\) 14.0660i 2.17042i
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) −4.33763 −0.653922
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) 11.4567i 1.67112i 0.549396 + 0.835562i \(0.314858\pi\)
−0.549396 + 0.835562i \(0.685142\pi\)
\(48\) − 3.11903i − 0.450193i
\(49\) −13.3376 −1.90538
\(50\) 0 0
\(51\) −3.49027 −0.488736
\(52\) 3.72833i 0.517026i
\(53\) − 6.00000i − 0.824163i −0.911147 − 0.412082i \(-0.864802\pi\)
0.911147 − 0.412082i \(-0.135198\pi\)
\(54\) 11.6288 1.58247
\(55\) 0 0
\(56\) 4.50973 0.602637
\(57\) 14.0660i 1.86308i
\(58\) − 8.23805i − 1.08171i
\(59\) 2.23805 0.291370 0.145685 − 0.989331i \(-0.453461\pi\)
0.145685 + 0.989331i \(0.453461\pi\)
\(60\) 0 0
\(61\) 3.55623 0.455329 0.227664 − 0.973740i \(-0.426891\pi\)
0.227664 + 0.973740i \(0.426891\pi\)
\(62\) − 1.72833i − 0.219498i
\(63\) 30.3429i 3.82285i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −13.5292 −1.66533
\(67\) − 2.43720i − 0.297752i −0.988856 − 0.148876i \(-0.952435\pi\)
0.988856 − 0.148876i \(-0.0475655\pi\)
\(68\) 1.11903i 0.135702i
\(69\) −3.11903 −0.375487
\(70\) 0 0
\(71\) 7.11903 0.844873 0.422437 − 0.906393i \(-0.361175\pi\)
0.422437 + 0.906393i \(0.361175\pi\)
\(72\) − 6.72833i − 0.792941i
\(73\) − 9.45665i − 1.10682i −0.832910 − 0.553409i \(-0.813327\pi\)
0.832910 − 0.553409i \(-0.186673\pi\)
\(74\) 0.781399 0.0908357
\(75\) 0 0
\(76\) 4.50973 0.517301
\(77\) − 19.5615i − 2.22924i
\(78\) 11.6288i 1.31670i
\(79\) 14.9133 1.67788 0.838939 − 0.544225i \(-0.183176\pi\)
0.838939 + 0.544225i \(0.183176\pi\)
\(80\) 0 0
\(81\) 16.0854 1.78727
\(82\) − 3.90043i − 0.430730i
\(83\) 2.78140i 0.305298i 0.988280 + 0.152649i \(0.0487804\pi\)
−0.988280 + 0.152649i \(0.951220\pi\)
\(84\) 14.0660 1.53472
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) − 25.6947i − 2.75476i
\(88\) 4.33763i 0.462393i
\(89\) 7.69471 0.815637 0.407819 − 0.913063i \(-0.366290\pi\)
0.407819 + 0.913063i \(0.366290\pi\)
\(90\) 0 0
\(91\) −16.8137 −1.76256
\(92\) 1.00000i 0.104257i
\(93\) − 5.39070i − 0.558989i
\(94\) 11.4567 1.18166
\(95\) 0 0
\(96\) −3.11903 −0.318334
\(97\) 0.642920i 0.0652786i 0.999467 + 0.0326393i \(0.0103913\pi\)
−0.999467 + 0.0326393i \(0.989609\pi\)
\(98\) 13.3376i 1.34730i
\(99\) −29.1850 −2.93320
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1150.2.b.j.599.1 6
5.2 odd 4 230.2.a.d.1.1 ✓ 3
5.3 odd 4 1150.2.a.q.1.3 3
5.4 even 2 inner 1150.2.b.j.599.6 6
15.2 even 4 2070.2.a.z.1.3 3
20.3 even 4 9200.2.a.cf.1.1 3
20.7 even 4 1840.2.a.r.1.3 3
40.27 even 4 7360.2.a.ce.1.1 3
40.37 odd 4 7360.2.a.bz.1.3 3
115.22 even 4 5290.2.a.r.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.1 ✓ 3 5.2 odd 4
1150.2.a.q.1.3 3 5.3 odd 4
1150.2.b.j.599.1 6 1.1 even 1 trivial
1150.2.b.j.599.6 6 5.4 even 2 inner
1840.2.a.r.1.3 3 20.7 even 4
2070.2.a.z.1.3 3 15.2 even 4
5290.2.a.r.1.1 3 115.22 even 4
7360.2.a.bz.1.3 3 40.37 odd 4
7360.2.a.ce.1.1 3 40.27 even 4
9200.2.a.cf.1.1 3 20.3 even 4