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Properties

Label 1150.2.b.a.599.2
Level $1150$
Weight $2$
Character 1150.599
Analytic conductor $9.183$
Analytic rank $0$
Dimension $2$
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 = [2,0,0,-2,0,-6,0,0,-12,0,6] 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: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.a.599.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} -4.00000i q^{7} -1.00000i q^{8} -6.00000 q^{9} +3.00000 q^{11} -3.00000i q^{12} -6.00000i q^{13} +4.00000 q^{14} +1.00000 q^{16} -5.00000i q^{17} -6.00000i q^{18} +1.00000 q^{19} +12.0000 q^{21} +3.00000i q^{22} +1.00000i q^{23} +3.00000 q^{24} +6.00000 q^{26} -9.00000i q^{27} +4.00000i q^{28} +8.00000 q^{29} -8.00000 q^{31} +1.00000i q^{32} +9.00000i q^{33} +5.00000 q^{34} +6.00000 q^{36} -2.00000i q^{37} +1.00000i q^{38} +18.0000 q^{39} -7.00000 q^{41} +12.0000i q^{42} +4.00000i q^{43} -3.00000 q^{44} -1.00000 q^{46} -10.0000i q^{47} +3.00000i q^{48} -9.00000 q^{49} +15.0000 q^{51} +6.00000i q^{52} -12.0000i q^{53} +9.00000 q^{54} -4.00000 q^{56} +3.00000i q^{57} +8.00000i q^{58} -4.00000 q^{59} -8.00000 q^{61} -8.00000i q^{62} +24.0000i q^{63} -1.00000 q^{64} -9.00000 q^{66} -3.00000i q^{67} +5.00000i q^{68} -3.00000 q^{69} +4.00000 q^{71} +6.00000i q^{72} -7.00000i q^{73} +2.00000 q^{74} -1.00000 q^{76} -12.0000i q^{77} +18.0000i q^{78} +6.00000 q^{79} +9.00000 q^{81} -7.00000i q^{82} +11.0000i q^{83} -12.0000 q^{84} -4.00000 q^{86} +24.0000i q^{87} -3.00000i q^{88} +3.00000 q^{89} -24.0000 q^{91} -1.00000i q^{92} -24.0000i q^{93} +10.0000 q^{94} -3.00000 q^{96} +14.0000i q^{97} -9.00000i q^{98} -18.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 6 q^{6} - 12 q^{9} + 6 q^{11} + 8 q^{14} + 2 q^{16} + 2 q^{19} + 24 q^{21} + 6 q^{24} + 12 q^{26} + 16 q^{29} - 16 q^{31} + 10 q^{34} + 12 q^{36} + 36 q^{39} - 14 q^{41} - 6 q^{44} - 2 q^{46}+ \cdots - 36 q^{99}+O(q^{100}) \) 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
\(3\) 3.00000i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −3.00000 −1.22474
\(7\) − 4.00000i − 1.51186i −0.654654 − 0.755929i \(-0.727186\pi\)
0.654654 − 0.755929i \(-0.272814\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −6.00000 −2.00000
\(10\) 0 0
\(11\) 3.00000 0.904534 0.452267 − 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) − 3.00000i − 0.866025i
\(13\) − 6.00000i − 1.66410i −0.554700 − 0.832050i \(-0.687167\pi\)
0.554700 − 0.832050i \(-0.312833\pi\)
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 5.00000i − 1.21268i −0.795206 − 0.606339i \(-0.792637\pi\)
0.795206 − 0.606339i \(-0.207363\pi\)
\(18\) − 6.00000i − 1.41421i
\(19\) 1.00000 0.229416 0.114708 − 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 12.0000 2.61861
\(22\) 3.00000i 0.639602i
\(23\) 1.00000i 0.208514i
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) 6.00000 1.17670
\(27\) − 9.00000i − 1.73205i
\(28\) 4.00000i 0.755929i
\(29\) 8.00000 1.48556 0.742781 − 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 9.00000i 1.56670i
\(34\) 5.00000 0.857493
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) − 2.00000i − 0.328798i −0.986394 − 0.164399i \(-0.947432\pi\)
0.986394 − 0.164399i \(-0.0525685\pi\)
\(38\) 1.00000i 0.162221i
\(39\) 18.0000 2.88231
\(40\) 0 0
\(41\) −7.00000 −1.09322 −0.546608 − 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 12.0000i 1.85164i
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) −3.00000 −0.452267
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) − 10.0000i − 1.45865i −0.684167 − 0.729325i \(-0.739834\pi\)
0.684167 − 0.729325i \(-0.260166\pi\)
\(48\) 3.00000i 0.433013i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 15.0000 2.10042
\(52\) 6.00000i 0.832050i
\(53\) − 12.0000i − 1.64833i −0.566352 − 0.824163i \(-0.691646\pi\)
0.566352 − 0.824163i \(-0.308354\pi\)
\(54\) 9.00000 1.22474
\(55\) 0 0
\(56\) −4.00000 −0.534522
\(57\) 3.00000i 0.397360i
\(58\) 8.00000i 1.05045i
\(59\) −4.00000 −0.520756 −0.260378 − 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −8.00000 −1.02430 −0.512148 − 0.858898i \(-0.671150\pi\)
−0.512148 + 0.858898i \(0.671150\pi\)
\(62\) − 8.00000i − 1.01600i
\(63\) 24.0000i 3.02372i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −9.00000 −1.10782
\(67\) − 3.00000i − 0.366508i −0.983066 − 0.183254i \(-0.941337\pi\)
0.983066 − 0.183254i \(-0.0586631\pi\)
\(68\) 5.00000i 0.606339i
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 − 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 6.00000i 0.707107i
\(73\) − 7.00000i − 0.819288i −0.912245 − 0.409644i \(-0.865653\pi\)
0.912245 − 0.409644i \(-0.134347\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) − 12.0000i − 1.36753i
\(78\) 18.0000i 2.03810i
\(79\) 6.00000 0.675053 0.337526 − 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) − 7.00000i − 0.773021i
\(83\) 11.0000i 1.20741i 0.797209 + 0.603703i \(0.206309\pi\)
−0.797209 + 0.603703i \(0.793691\pi\)
\(84\) −12.0000 −1.30931
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 24.0000i 2.57307i
\(88\) − 3.00000i − 0.319801i
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) −24.0000 −2.51588
\(92\) − 1.00000i − 0.104257i
\(93\) − 24.0000i − 2.48868i
\(94\) 10.0000 1.03142
\(95\) 0 0
\(96\) −3.00000 −0.306186
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) − 9.00000i − 0.909137i
\(99\) −18.0000 −1.80907
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.a.599.2 2
5.2 odd 4 1150.2.a.d.1.1 ✓ 1
5.3 odd 4 1150.2.a.e.1.1 yes 1
5.4 even 2 inner 1150.2.b.a.599.1 2
20.3 even 4 9200.2.a.bl.1.1 1
20.7 even 4 9200.2.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1150.2.a.d.1.1 ✓ 1 5.2 odd 4
1150.2.a.e.1.1 yes 1 5.3 odd 4
1150.2.b.a.599.1 2 5.4 even 2 inner
1150.2.b.a.599.2 2 1.1 even 1 trivial
9200.2.a.a.1.1 1 20.7 even 4
9200.2.a.bl.1.1 1 20.3 even 4