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Properties

Label 1150.2.b.j.599.4
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 599.4
Root \(-2.68740i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.j.599.3

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -2.68740i q^{3} -1.00000 q^{4} +2.68740 q^{6} -4.59692i q^{7} -1.00000i q^{8} -4.22212 q^{9} +5.13163 q^{11} +2.68740i q^{12} +1.22212i q^{13} +4.59692 q^{14} +1.00000 q^{16} -4.68740i q^{17} -4.22212i q^{18} +4.59692 q^{19} -12.3537 q^{21} +5.13163i q^{22} +1.00000i q^{23} -2.68740 q^{24} -1.22212 q^{26} +3.28432i q^{27} +4.59692i q^{28} -3.37480 q^{29} -0.777884 q^{31} +1.00000i q^{32} -13.7907i q^{33} +4.68740 q^{34} +4.22212 q^{36} +5.81903i q^{37} +4.59692i q^{38} +3.28432 q^{39} -8.50643 q^{41} -12.3537i q^{42} -8.00000i q^{43} -5.13163 q^{44} -1.00000 q^{46} -6.44423i q^{47} -2.68740i q^{48} -14.1316 q^{49} -12.5969 q^{51} -1.22212i q^{52} +6.00000i q^{53} -3.28432 q^{54} -4.59692 q^{56} -12.3537i q^{57} -3.37480i q^{58} -9.37480 q^{59} +10.9507 q^{61} -0.777884i q^{62} +19.4087i q^{63} -1.00000 q^{64} +13.7907 q^{66} +15.6381i q^{67} +4.68740i q^{68} +2.68740 q^{69} +1.31260 q^{71} +4.22212i q^{72} +4.44423i q^{73} -5.81903 q^{74} -4.59692 q^{76} -23.5897i q^{77} +3.28432i q^{78} +4.88847 q^{79} -3.84008 q^{81} -8.50643i q^{82} +3.81903i q^{83} +12.3537 q^{84} +8.00000 q^{86} +9.06943i q^{87} -5.13163i q^{88} -8.93057 q^{89} +5.61797 q^{91} -1.00000i q^{92} +2.09048i q^{93} +6.44423 q^{94} +2.68740 q^{96} -18.0622i q^{97} -14.1316i q^{98} -21.6663 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{4} + 2 q^{6} - 20 q^{9} + 6 q^{11} - 6 q^{14} + 6 q^{16} - 6 q^{19} - 44 q^{21} - 2 q^{24} - 2 q^{26} + 8 q^{29} - 10 q^{31} + 14 q^{34} + 20 q^{36} - 28 q^{39} + 2 q^{41} - 6 q^{44} - 6 q^{46}+ \cdots - 114 q^{99}+O(q^{100}) \) 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
\(3\) − 2.68740i − 1.55157i −0.630997 − 0.775785i \(-0.717354\pi\)
0.630997 − 0.775785i \(-0.282646\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 2.68740 1.09713
\(7\) − 4.59692i − 1.73747i −0.495277 − 0.868735i \(-0.664933\pi\)
0.495277 − 0.868735i \(-0.335067\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −4.22212 −1.40737
\(10\) 0 0
\(11\) 5.13163 1.54725 0.773623 − 0.633647i \(-0.218443\pi\)
0.773623 + 0.633647i \(0.218443\pi\)
\(12\) 2.68740i 0.775785i
\(13\) 1.22212i 0.338954i 0.985534 + 0.169477i \(0.0542079\pi\)
−0.985534 + 0.169477i \(0.945792\pi\)
\(14\) 4.59692 1.22858
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 4.68740i − 1.13686i −0.822731 − 0.568431i \(-0.807551\pi\)
0.822731 − 0.568431i \(-0.192449\pi\)
\(18\) − 4.22212i − 0.995162i
\(19\) 4.59692 1.05460 0.527302 − 0.849678i \(-0.323204\pi\)
0.527302 + 0.849678i \(0.323204\pi\)
\(20\) 0 0
\(21\) −12.3537 −2.69581
\(22\) 5.13163i 1.09407i
\(23\) 1.00000i 0.208514i
\(24\) −2.68740 −0.548563
\(25\) 0 0
\(26\) −1.22212 −0.239677
\(27\) 3.28432i 0.632067i
\(28\) 4.59692i 0.868735i
\(29\) −3.37480 −0.626684 −0.313342 − 0.949640i \(-0.601449\pi\)
−0.313342 + 0.949640i \(0.601449\pi\)
\(30\) 0 0
\(31\) −0.777884 −0.139712 −0.0698560 − 0.997557i \(-0.522254\pi\)
−0.0698560 + 0.997557i \(0.522254\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 13.7907i − 2.40066i
\(34\) 4.68740 0.803882
\(35\) 0 0
\(36\) 4.22212 0.703686
\(37\) 5.81903i 0.956643i 0.878185 + 0.478321i \(0.158755\pi\)
−0.878185 + 0.478321i \(0.841245\pi\)
\(38\) 4.59692i 0.745718i
\(39\) 3.28432 0.525911
\(40\) 0 0
\(41\) −8.50643 −1.32848 −0.664241 − 0.747519i \(-0.731245\pi\)
−0.664241 + 0.747519i \(0.731245\pi\)
\(42\) − 12.3537i − 1.90622i
\(43\) − 8.00000i − 1.21999i −0.792406 − 0.609994i \(-0.791172\pi\)
0.792406 − 0.609994i \(-0.208828\pi\)
\(44\) −5.13163 −0.773623
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) − 6.44423i − 0.939988i −0.882670 − 0.469994i \(-0.844256\pi\)
0.882670 − 0.469994i \(-0.155744\pi\)
\(48\) − 2.68740i − 0.387893i
\(49\) −14.1316 −2.01880
\(50\) 0 0
\(51\) −12.5969 −1.76392
\(52\) − 1.22212i − 0.169477i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) −3.28432 −0.446939
\(55\) 0 0
\(56\) −4.59692 −0.614289
\(57\) − 12.3537i − 1.63629i
\(58\) − 3.37480i − 0.443133i
\(59\) −9.37480 −1.22049 −0.610247 − 0.792211i \(-0.708930\pi\)
−0.610247 + 0.792211i \(0.708930\pi\)
\(60\) 0 0
\(61\) 10.9507 1.40209 0.701044 − 0.713118i \(-0.252717\pi\)
0.701044 + 0.713118i \(0.252717\pi\)
\(62\) − 0.777884i − 0.0987913i
\(63\) 19.4087i 2.44527i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 13.7907 1.69752
\(67\) 15.6381i 1.91049i 0.295810 + 0.955247i \(0.404410\pi\)
−0.295810 + 0.955247i \(0.595590\pi\)
\(68\) 4.68740i 0.568431i
\(69\) 2.68740 0.323525
\(70\) 0 0
\(71\) 1.31260 0.155777 0.0778885 − 0.996962i \(-0.475182\pi\)
0.0778885 + 0.996962i \(0.475182\pi\)
\(72\) 4.22212i 0.497581i
\(73\) 4.44423i 0.520158i 0.965587 + 0.260079i \(0.0837486\pi\)
−0.965587 + 0.260079i \(0.916251\pi\)
\(74\) −5.81903 −0.676449
\(75\) 0 0
\(76\) −4.59692 −0.527302
\(77\) − 23.5897i − 2.68829i
\(78\) 3.28432i 0.371875i
\(79\) 4.88847 0.549995 0.274998 − 0.961445i \(-0.411323\pi\)
0.274998 + 0.961445i \(0.411323\pi\)
\(80\) 0 0
\(81\) −3.84008 −0.426676
\(82\) − 8.50643i − 0.939378i
\(83\) 3.81903i 0.419193i 0.977788 + 0.209597i \(0.0672151\pi\)
−0.977788 + 0.209597i \(0.932785\pi\)
\(84\) 12.3537 1.34790
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 9.06943i 0.972345i
\(88\) − 5.13163i − 0.547034i
\(89\) −8.93057 −0.946638 −0.473319 − 0.880891i \(-0.656944\pi\)
−0.473319 + 0.880891i \(0.656944\pi\)
\(90\) 0 0
\(91\) 5.61797 0.588923
\(92\) − 1.00000i − 0.104257i
\(93\) 2.09048i 0.216773i
\(94\) 6.44423 0.664672
\(95\) 0 0
\(96\) 2.68740 0.274282
\(97\) − 18.0622i − 1.83394i −0.398958 − 0.916969i \(-0.630628\pi\)
0.398958 − 0.916969i \(-0.369372\pi\)
\(98\) − 14.1316i − 1.42751i
\(99\) −21.6663 −2.17755
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.599.4 6
5.2 odd 4 1150.2.a.q.1.1 3
5.3 odd 4 230.2.a.d.1.3 ✓ 3
5.4 even 2 inner 1150.2.b.j.599.3 6
15.8 even 4 2070.2.a.z.1.1 3
20.3 even 4 1840.2.a.r.1.1 3
20.7 even 4 9200.2.a.cf.1.3 3
40.3 even 4 7360.2.a.ce.1.3 3
40.13 odd 4 7360.2.a.bz.1.1 3
115.68 even 4 5290.2.a.r.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.3 ✓ 3 5.3 odd 4
1150.2.a.q.1.1 3 5.2 odd 4
1150.2.b.j.599.3 6 5.4 even 2 inner
1150.2.b.j.599.4 6 1.1 even 1 trivial
1840.2.a.r.1.1 3 20.3 even 4
2070.2.a.z.1.1 3 15.8 even 4
5290.2.a.r.1.3 3 115.68 even 4
7360.2.a.bz.1.1 3 40.13 odd 4
7360.2.a.ce.1.3 3 40.3 even 4
9200.2.a.cf.1.3 3 20.7 even 4