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Properties

Label 230.2.e.a.183.4
Level $230$
Weight $2$
Character 230.183
Analytic conductor $1.837$
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] = [230,2,Mod(137,230)] 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("230.137"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([1, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.83655924649\)
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: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 183.4
Root \(1.22833i\) of defining polynomial
Character \(\chi\) \(=\) 230.183
Dual form 230.2.e.a.137.4

$q$-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} +(-0.185885 + 2.22833i) q^{5} +0.814115 q^{6} +(2.09689 + 2.09689i) q^{7} +(-0.707107 - 0.707107i) q^{8} -2.33722i q^{9} +(1.44423 + 1.70711i) q^{10} -1.39990i q^{11} +(0.575666 - 0.575666i) q^{12} +(4.24822 + 4.24822i) q^{13} +2.96545 q^{14} +(-1.38978 + 1.17576i) q^{15} -1.00000 q^{16} +(-4.38978 - 4.38978i) q^{17} +(-1.65266 - 1.65266i) q^{18} -2.37966 q^{19} +(2.22833 + 0.185885i) q^{20} +2.41421i q^{21} +(-0.989880 - 0.989880i) q^{22} +(0.664664 - 4.74955i) q^{23} -0.814115i q^{24} +(-4.93089 - 0.828427i) q^{25} +6.00789 q^{26} +(3.07245 - 3.07245i) q^{27} +(2.09689 - 2.09689i) q^{28} +3.87087i q^{29} +(-0.151332 + 1.81411i) q^{30} -5.74501 q^{31} +(-0.707107 + 0.707107i) q^{32} +(0.805875 - 0.805875i) q^{33} -6.20809 q^{34} +(-5.06233 + 4.28277i) q^{35} -2.33722 q^{36} +(-2.27719 - 2.27719i) q^{37} +(-1.68267 + 1.68267i) q^{38} +4.89111i q^{39} +(1.70711 - 1.44423i) q^{40} +2.49121 q^{41} +(1.70711 + 1.70711i) q^{42} +(-2.85390 + 2.85390i) q^{43} -1.39990 q^{44} +(5.20809 + 0.434454i) q^{45} +(-2.88845 - 3.82843i) q^{46} +(-1.26288 + 1.26288i) q^{47} +(-0.575666 - 0.575666i) q^{48} +1.79387i q^{49} +(-4.07245 + 2.90088i) q^{50} -5.05409i q^{51} +(4.24822 - 4.24822i) q^{52} +(4.13375 - 4.13375i) q^{53} -4.34511i q^{54} +(3.11944 + 0.260221i) q^{55} -2.96545i q^{56} +(-1.36989 - 1.36989i) q^{57} +(2.73712 + 2.73712i) q^{58} -5.66801i q^{59} +(1.17576 + 1.38978i) q^{60} +13.1168i q^{61} +(-4.06233 + 4.06233i) q^{62} +(4.90088 - 4.90088i) q^{63} +1.00000i q^{64} +(-10.2561 + 8.67675i) q^{65} -1.13968i q^{66} +(-11.1390 - 11.1390i) q^{67} +(-4.38978 + 4.38978i) q^{68} +(3.11678 - 2.35153i) q^{69} +(-0.551233 + 6.60799i) q^{70} -9.16188 q^{71} +(-1.65266 + 1.65266i) q^{72} +(10.7875 + 10.7875i) q^{73} -3.22044 q^{74} +(-2.36165 - 3.31544i) q^{75} +2.37966i q^{76} +(2.93543 - 2.93543i) q^{77} +(3.45854 + 3.45854i) q^{78} -5.70196 q^{79} +(0.185885 - 2.22833i) q^{80} -3.47424 q^{81} +(1.76155 - 1.76155i) q^{82} +(11.6335 - 11.6335i) q^{83} +2.41421 q^{84} +(10.5979 - 8.96588i) q^{85} +4.03602i q^{86} +(-2.22833 + 2.22833i) q^{87} +(-0.989880 + 0.989880i) q^{88} +1.31441 q^{89} +(3.98988 - 3.37547i) q^{90} +17.8161i q^{91} +(-4.74955 - 0.664664i) q^{92} +(-3.30721 - 3.30721i) q^{93} +1.78598i q^{94} +(0.442344 - 5.30266i) q^{95} -0.814115 q^{96} +(7.84001 + 7.84001i) q^{97} +(1.26846 + 1.26846i) q^{98} -3.27187 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} - 4 q^{5} + 4 q^{6} - 4 q^{12} + 4 q^{14} - 8 q^{16} - 24 q^{17} - 8 q^{18} + 12 q^{19} + 4 q^{20} + 12 q^{22} + 16 q^{23} + 12 q^{26} + 8 q^{27} + 16 q^{30} - 4 q^{31} - 20 q^{33} + 4 q^{34}+ \cdots + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.707107 − 0.707107i 0.500000 − 0.500000i
\(3\) 0.575666 + 0.575666i 0.332361 + 0.332361i 0.853482 − 0.521122i \(-0.174486\pi\)
−0.521122 + 0.853482i \(0.674486\pi\)
\(4\) − 1.00000i − 0.500000i
\(5\) −0.185885 + 2.22833i −0.0831305 + 0.996539i
\(6\) 0.814115 0.332361
\(7\) 2.09689 + 2.09689i 0.792549 + 0.792549i 0.981908 − 0.189359i \(-0.0606410\pi\)
−0.189359 + 0.981908i \(0.560641\pi\)
\(8\) −0.707107 − 0.707107i −0.250000 − 0.250000i
\(9\) − 2.33722i − 0.779072i
\(10\) 1.44423 + 1.70711i 0.456704 + 0.539835i
\(11\) − 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 0.980191 + 0.198053i \(0.0634617\pi\)
0.198053 + 0.980191i \(0.436538\pi\)
\(14\) 2.96545 0.792549
\(15\) −1.38978 + 1.17576i −0.358840 + 0.303581i
\(16\) −1.00000 −0.250000
\(17\) −4.38978 − 4.38978i −1.06468 − 1.06468i −0.997758 − 0.0669198i \(-0.978683\pi\)
−0.0669198 − 0.997758i \(-0.521317\pi\)
\(18\) −1.65266 − 1.65266i −0.389536 − 0.389536i
\(19\) −2.37966 −0.545931 −0.272966 − 0.962024i \(-0.588005\pi\)
−0.272966 + 0.962024i \(0.588005\pi\)
\(20\) 2.22833 + 0.185885i 0.498269 + 0.0415652i
\(21\) 2.41421i 0.526825i
\(22\) −0.989880 − 0.989880i −0.211043 − 0.211043i
\(23\) 0.664664 − 4.74955i 0.138592 − 0.990350i
\(24\) − 0.814115i − 0.166180i
\(25\) −4.93089 − 0.828427i −0.986179 − 0.165685i
\(26\) 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 \(0.117019\pi\)
−0.933183 + 0.359401i \(0.882981\pi\)
\(30\) −0.151332 + 1.81411i −0.0276293 + 0.331211i
\(31\) −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
\(35\) −5.06233 + 4.28277i −0.855691 + 0.723921i
\(36\) −2.33722 −0.389536
\(37\) −2.27719 − 2.27719i −0.374368 − 0.374368i 0.494697 − 0.869065i \(-0.335279\pi\)
−0.869065 + 0.494697i \(0.835279\pi\)
\(38\) −1.68267 + 1.68267i −0.272966 + 0.272966i
\(39\) 4.89111i 0.783205i
\(40\) 1.70711 − 1.44423i 0.269917 − 0.228352i
\(41\) 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 −0.890398 − 0.455183i \(-0.849574\pi\)
0.455183 + 0.890398i \(0.349574\pi\)
\(44\) −1.39990 −0.211043
\(45\) 5.20809 + 0.434454i 0.776376 + 0.0647646i
\(46\) −2.88845 − 3.82843i −0.425879 − 0.564471i
\(47\) −1.26288 + 1.26288i −0.184210 + 0.184210i −0.793188 − 0.608977i \(-0.791580\pi\)
0.608977 + 0.793188i \(0.291580\pi\)
\(48\) −0.575666 − 0.575666i −0.0830902 − 0.0830902i
\(49\) 1.79387i 0.256268i
\(50\) −4.07245 + 2.90088i −0.575932 + 0.410247i
\(51\) − 5.05409i − 0.707715i
\(52\) 4.24822 − 4.24822i 0.589122 − 0.589122i
\(53\) 4.13375 − 4.13375i 0.567814 − 0.567814i −0.363701 − 0.931516i \(-0.618487\pi\)
0.931516 + 0.363701i \(0.118487\pi\)
\(54\) − 4.34511i − 0.591294i
\(55\) 3.11944 + 0.260221i 0.420625 + 0.0350882i
\(56\) − 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 \(-0.879715\pi\)
0.929447 − 0.368956i \(-0.120285\pi\)
\(60\) 1.17576 + 1.38978i 0.151791 + 0.179420i
\(61\) 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
\(65\) −10.2561 + 8.67675i −1.27211 + 1.07622i
\(66\) − 1.13968i − 0.140285i
\(67\) −11.1390 − 11.1390i −1.36084 − 1.36084i −0.872844 − 0.487999i \(-0.837727\pi\)
−0.487999 − 0.872844i \(-0.662273\pi\)
\(68\) −4.38978 + 4.38978i −0.532339 + 0.532339i
\(69\) 3.11678 − 2.35153i 0.375216 − 0.283091i
\(70\) −0.551233 + 6.60799i −0.0658850 + 0.789806i
\(71\) −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 0.949840 + 0.312735i \(0.101245\pi\)
0.312735 + 0.949840i \(0.398755\pi\)
\(74\) −3.22044 −0.374368
\(75\) −2.36165 − 3.31544i −0.272700 − 0.382835i
\(76\) 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 \(-0.603938\pi\)
−0.320760 + 0.947160i \(0.603938\pi\)
\(80\) 0.185885 − 2.22833i 0.0207826 − 0.249135i
\(81\) −3.47424 −0.386026
\(82\) 1.76155 − 1.76155i 0.194531 − 0.194531i
\(83\) 11.6335 − 11.6335i 1.27694 − 1.27694i 0.334565 − 0.942373i \(-0.391410\pi\)
0.942373 − 0.334565i \(-0.108590\pi\)
\(84\) 2.41421 0.263412
\(85\) 10.5979 − 8.96588i 1.14950 − 0.972486i
\(86\) 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 \(-0.477807\pi\)
0.0696635 + 0.997571i \(0.477807\pi\)
\(90\) 3.98988 − 3.37547i 0.420570 − 0.355806i
\(91\) 17.8161i 1.86763i
\(92\) −4.74955 − 0.664664i −0.495175 − 0.0692960i
\(93\) −3.30721 − 3.30721i −0.342941 − 0.342941i
\(94\) 1.78598i 0.184210i
\(95\) 0.442344 − 5.30266i 0.0453835 − 0.544042i
\(96\) −0.814115 −0.0830902
\(97\) 7.84001 + 7.84001i 0.796033 + 0.796033i 0.982467 − 0.186435i \(-0.0596932\pi\)
−0.186435 + 0.982467i \(0.559693\pi\)
\(98\) 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 230.2.e.a.183.4 yes 8
5.2 odd 4 230.2.e.b.137.4 yes 8
5.3 odd 4 1150.2.e.b.1057.1 8
5.4 even 2 1150.2.e.c.643.1 8
23.22 odd 2 230.2.e.b.183.4 yes 8
115.22 even 4 inner 230.2.e.a.137.4 ✓ 8
115.68 even 4 1150.2.e.c.1057.1 8
115.114 odd 2 1150.2.e.b.643.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.e.a.137.4 ✓ 8 115.22 even 4 inner
230.2.e.a.183.4 yes 8 1.1 even 1 trivial
230.2.e.b.137.4 yes 8 5.2 odd 4
230.2.e.b.183.4 yes 8 23.22 odd 2
1150.2.e.b.643.1 8 115.114 odd 2
1150.2.e.b.1057.1 8 5.3 odd 4
1150.2.e.c.643.1 8 5.4 even 2
1150.2.e.c.1057.1 8 115.68 even 4