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Properties

Label 1150.2.e.b.1057.1
Level $1150$
Weight $2$
Character 1150.1057
Analytic conductor $9.183$
Analytic rank $0$
Dimension $8$
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(643,1150)] 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("1150.643"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1150, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,4,0,0,4,0,0,0,0,0,4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.18279623245\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.110166016.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 10x^{6} + 19x^{4} + 10x^{2} + 1 \) 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: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1057.1
Root \(-0.814115i\) of defining polynomial
Character \(\chi\) \(=\) 1150.1057
Dual form 1150.2.e.b.643.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.707107 - 0.707107i) q^{2} +(-0.575666 + 0.575666i) q^{3} +1.00000i q^{4} +0.814115 q^{6} +(2.09689 - 2.09689i) q^{7} +(0.707107 - 0.707107i) q^{8} +2.33722i q^{9} -1.39990i q^{11} +(-0.575666 - 0.575666i) q^{12} +(-4.24822 + 4.24822i) q^{13} -2.96545 q^{14} -1.00000 q^{16} +(-4.38978 + 4.38978i) q^{17} +(1.65266 - 1.65266i) q^{18} +2.37966 q^{19} +2.41421i q^{21} +(-0.989880 + 0.989880i) q^{22} +(4.74955 + 0.664664i) q^{23} +0.814115i q^{24} +6.00789 q^{26} +(-3.07245 - 3.07245i) q^{27} +(2.09689 + 2.09689i) q^{28} -3.87087i q^{29} -5.74501 q^{31} +(0.707107 + 0.707107i) q^{32} +(0.805875 + 0.805875i) q^{33} +6.20809 q^{34} -2.33722 q^{36} +(-2.27719 + 2.27719i) q^{37} +(-1.68267 - 1.68267i) q^{38} -4.89111i q^{39} +2.49121 q^{41} +(1.70711 - 1.70711i) q^{42} +(-2.85390 - 2.85390i) q^{43} +1.39990 q^{44} +(-2.88845 - 3.82843i) q^{46} +(1.26288 + 1.26288i) q^{47} +(0.575666 - 0.575666i) q^{48} -1.79387i q^{49} -5.05409i q^{51} +(-4.24822 - 4.24822i) q^{52} +(4.13375 + 4.13375i) q^{53} +4.34511i q^{54} -2.96545i q^{56} +(-1.36989 + 1.36989i) q^{57} +(-2.73712 + 2.73712i) q^{58} +5.66801i q^{59} +13.1168i q^{61} +(4.06233 + 4.06233i) q^{62} +(4.90088 + 4.90088i) q^{63} -1.00000i q^{64} -1.13968i q^{66} +(-11.1390 + 11.1390i) q^{67} +(-4.38978 - 4.38978i) q^{68} +(-3.11678 + 2.35153i) q^{69} -9.16188 q^{71} +(1.65266 + 1.65266i) q^{72} +(-10.7875 + 10.7875i) q^{73} +3.22044 q^{74} +2.37966i q^{76} +(-2.93543 - 2.93543i) q^{77} +(-3.45854 + 3.45854i) q^{78} +5.70196 q^{79} -3.47424 q^{81} +(-1.76155 - 1.76155i) q^{82} +(11.6335 + 11.6335i) q^{83} -2.41421 q^{84} +4.03602i q^{86} +(2.22833 + 2.22833i) q^{87} +(-0.989880 - 0.989880i) q^{88} -1.31441 q^{89} +17.8161i q^{91} +(-0.664664 + 4.74955i) q^{92} +(3.30721 - 3.30721i) q^{93} -1.78598i q^{94} -0.814115 q^{96} +(7.84001 - 7.84001i) q^{97} +(-1.26846 + 1.26846i) q^{98} +3.27187 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 4 q^{6} + 4 q^{12} - 4 q^{14} - 8 q^{16} - 24 q^{17} + 8 q^{18} - 12 q^{19} + 12 q^{22} + 16 q^{23} + 12 q^{26} - 8 q^{27} - 4 q^{31} - 20 q^{33} - 4 q^{34} - 4 q^{36} - 4 q^{37} - 8 q^{38}+ \cdots - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(51\) \(277\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.707107 − 0.707107i −0.500000 − 0.500000i
\(3\) −0.575666 + 0.575666i −0.332361 + 0.332361i −0.853482 − 0.521122i \(-0.825514\pi\)
0.521122 + 0.853482i \(0.325514\pi\)
\(4\) 1.00000i 0.500000i
\(5\) 0 0
\(6\) 0.814115 0.332361
\(7\) 2.09689 − 2.09689i 0.792549 − 0.792549i −0.189359 − 0.981908i \(-0.560641\pi\)
0.981908 + 0.189359i \(0.0606410\pi\)
\(8\) 0.707107 − 0.707107i 0.250000 − 0.250000i
\(9\) 2.33722i 0.779072i
\(10\) 0 0
\(11\) − 1.39990i − 0.422086i −0.977477 − 0.211043i \(-0.932314\pi\)
0.977477 − 0.211043i \(-0.0676860\pi\)
\(12\) −0.575666 − 0.575666i −0.166180 − 0.166180i
\(13\) −4.24822 + 4.24822i −1.17824 + 1.17824i −0.198053 + 0.980191i \(0.563462\pi\)
−0.980191 + 0.198053i \(0.936538\pi\)
\(14\) −2.96545 −0.792549
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −4.38978 + 4.38978i −1.06468 + 1.06468i −0.0669198 + 0.997758i \(0.521317\pi\)
−0.997758 + 0.0669198i \(0.978683\pi\)
\(18\) 1.65266 − 1.65266i 0.389536 − 0.389536i
\(19\) 2.37966 0.545931 0.272966 − 0.962024i \(-0.411995\pi\)
0.272966 + 0.962024i \(0.411995\pi\)
\(20\) 0 0
\(21\) 2.41421i 0.526825i
\(22\) −0.989880 + 0.989880i −0.211043 + 0.211043i
\(23\) 4.74955 + 0.664664i 0.990350 + 0.138592i
\(24\) 0.814115i 0.166180i
\(25\) 0 0
\(26\) 6.00789 1.17824
\(27\) −3.07245 − 3.07245i −0.591294 − 0.591294i
\(28\) 2.09689 + 2.09689i 0.396274 + 0.396274i
\(29\) − 3.87087i − 0.718803i −0.933183 − 0.359401i \(-0.882981\pi\)
0.933183 − 0.359401i \(-0.117019\pi\)
\(30\) 0 0
\(31\) −5.74501 −1.03183 −0.515917 − 0.856639i \(-0.672549\pi\)
−0.515917 + 0.856639i \(0.672549\pi\)
\(32\) 0.707107 + 0.707107i 0.125000 + 0.125000i
\(33\) 0.805875 + 0.805875i 0.140285 + 0.140285i
\(34\) 6.20809 1.06468
\(35\) 0 0
\(36\) −2.33722 −0.389536
\(37\) −2.27719 + 2.27719i −0.374368 + 0.374368i −0.869065 − 0.494697i \(-0.835279\pi\)
0.494697 + 0.869065i \(0.335279\pi\)
\(38\) −1.68267 − 1.68267i −0.272966 − 0.272966i
\(39\) − 4.89111i − 0.783205i
\(40\) 0 0
\(41\) 2.49121 0.389062 0.194531 − 0.980896i \(-0.437682\pi\)
0.194531 + 0.980896i \(0.437682\pi\)
\(42\) 1.70711 − 1.70711i 0.263412 − 0.263412i
\(43\) −2.85390 − 2.85390i −0.435215 − 0.435215i 0.455183 − 0.890398i \(-0.349574\pi\)
−0.890398 + 0.455183i \(0.849574\pi\)
\(44\) 1.39990 0.211043
\(45\) 0 0
\(46\) −2.88845 − 3.82843i −0.425879 − 0.564471i
\(47\) 1.26288 + 1.26288i 0.184210 + 0.184210i 0.793188 − 0.608977i \(-0.208420\pi\)
−0.608977 + 0.793188i \(0.708420\pi\)
\(48\) 0.575666 − 0.575666i 0.0830902 − 0.0830902i
\(49\) − 1.79387i − 0.256268i
\(50\) 0 0
\(51\) − 5.05409i − 0.707715i
\(52\) −4.24822 − 4.24822i −0.589122 − 0.589122i
\(53\) 4.13375 + 4.13375i 0.567814 + 0.567814i 0.931516 − 0.363701i \(-0.118487\pi\)
−0.363701 + 0.931516i \(0.618487\pi\)
\(54\) 4.34511i 0.591294i
\(55\) 0 0
\(56\) − 2.96545i − 0.396274i
\(57\) −1.36989 + 1.36989i −0.181446 + 0.181446i
\(58\) −2.73712 + 2.73712i −0.359401 + 0.359401i
\(59\) 5.66801i 0.737912i 0.929447 + 0.368956i \(0.120285\pi\)
−0.929447 + 0.368956i \(0.879715\pi\)
\(60\) 0 0
\(61\) 13.1168i 1.67943i 0.543026 + 0.839716i \(0.317278\pi\)
−0.543026 + 0.839716i \(0.682722\pi\)
\(62\) 4.06233 + 4.06233i 0.515917 + 0.515917i
\(63\) 4.90088 + 4.90088i 0.617453 + 0.617453i
\(64\) − 1.00000i − 0.125000i
\(65\) 0 0
\(66\) − 1.13968i − 0.140285i
\(67\) −11.1390 + 11.1390i −1.36084 + 1.36084i −0.487999 + 0.872844i \(0.662273\pi\)
−0.872844 + 0.487999i \(0.837727\pi\)
\(68\) −4.38978 − 4.38978i −0.532339 − 0.532339i
\(69\) −3.11678 + 2.35153i −0.375216 + 0.283091i
\(70\) 0 0
\(71\) −9.16188 −1.08732 −0.543658 − 0.839307i \(-0.682961\pi\)
−0.543658 + 0.839307i \(0.682961\pi\)
\(72\) 1.65266 + 1.65266i 0.194768 + 0.194768i
\(73\) −10.7875 + 10.7875i −1.26258 + 1.26258i −0.312735 + 0.949840i \(0.601245\pi\)
−0.949840 + 0.312735i \(0.898755\pi\)
\(74\) 3.22044 0.374368
\(75\) 0 0
\(76\) 2.37966i 0.272966i
\(77\) −2.93543 − 2.93543i −0.334524 − 0.334524i
\(78\) −3.45854 + 3.45854i −0.391602 + 0.391602i
\(79\) 5.70196 0.641520 0.320760 − 0.947160i \(-0.396062\pi\)
0.320760 + 0.947160i \(0.396062\pi\)
\(80\) 0 0
\(81\) −3.47424 −0.386026
\(82\) −1.76155 − 1.76155i −0.194531 − 0.194531i
\(83\) 11.6335 + 11.6335i 1.27694 + 1.27694i 0.942373 + 0.334565i \(0.108590\pi\)
0.334565 + 0.942373i \(0.391410\pi\)
\(84\) −2.41421 −0.263412
\(85\) 0 0
\(86\) 4.03602i 0.435215i
\(87\) 2.22833 + 2.22833i 0.238902 + 0.238902i
\(88\) −0.989880 − 0.989880i −0.105522 − 0.105522i
\(89\) −1.31441 −0.139327 −0.0696635 − 0.997571i \(-0.522193\pi\)
−0.0696635 + 0.997571i \(0.522193\pi\)
\(90\) 0 0
\(91\) 17.8161i 1.86763i
\(92\) −0.664664 + 4.74955i −0.0692960 + 0.495175i
\(93\) 3.30721 − 3.30721i 0.342941 − 0.342941i
\(94\) − 1.78598i − 0.184210i
\(95\) 0 0
\(96\) −0.814115 −0.0830902
\(97\) 7.84001 − 7.84001i 0.796033 − 0.796033i −0.186435 − 0.982467i \(-0.559693\pi\)
0.982467 + 0.186435i \(0.0596932\pi\)
\(98\) −1.26846 + 1.26846i −0.128134 + 0.128134i
\(99\) 3.27187 0.328836
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.e.b.1057.1 8
5.2 odd 4 230.2.e.a.183.4 yes 8
5.3 odd 4 1150.2.e.c.643.1 8
5.4 even 2 230.2.e.b.137.4 yes 8
23.22 odd 2 1150.2.e.c.1057.1 8
115.22 even 4 230.2.e.b.183.4 yes 8
115.68 even 4 inner 1150.2.e.b.643.1 8
115.114 odd 2 230.2.e.a.137.4 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.e.a.137.4 ✓ 8 115.114 odd 2
230.2.e.a.183.4 yes 8 5.2 odd 4
230.2.e.b.137.4 yes 8 5.4 even 2
230.2.e.b.183.4 yes 8 115.22 even 4
1150.2.e.b.643.1 8 115.68 even 4 inner
1150.2.e.b.1057.1 8 1.1 even 1 trivial
1150.2.e.c.643.1 8 5.3 odd 4
1150.2.e.c.1057.1 8 23.22 odd 2