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Properties

Label 230.2.a.b.1.1
Level $230$
Weight $2$
Character 230.1
Self dual yes
Analytic conductor $1.837$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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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(1,230)] 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("230.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-2,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.83655924649\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 230.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.302776 q^{3} +1.00000 q^{4} +1.00000 q^{5} +0.302776 q^{6} +3.30278 q^{7} -1.00000 q^{8} -2.90833 q^{9} -1.00000 q^{10} -1.69722 q^{11} -0.302776 q^{12} +3.30278 q^{13} -3.30278 q^{14} -0.302776 q^{15} +1.00000 q^{16} +6.90833 q^{17} +2.90833 q^{18} +5.90833 q^{19} +1.00000 q^{20} -1.00000 q^{21} +1.69722 q^{22} -1.00000 q^{23} +0.302776 q^{24} +1.00000 q^{25} -3.30278 q^{26} +1.78890 q^{27} +3.30278 q^{28} -2.60555 q^{29} +0.302776 q^{30} -7.90833 q^{31} -1.00000 q^{32} +0.513878 q^{33} -6.90833 q^{34} +3.30278 q^{35} -2.90833 q^{36} +8.00000 q^{37} -5.90833 q^{38} -1.00000 q^{39} -1.00000 q^{40} +0.908327 q^{41} +1.00000 q^{42} -9.21110 q^{43} -1.69722 q^{44} -2.90833 q^{45} +1.00000 q^{46} -2.60555 q^{47} -0.302776 q^{48} +3.90833 q^{49} -1.00000 q^{50} -2.09167 q^{51} +3.30278 q^{52} -11.2111 q^{53} -1.78890 q^{54} -1.69722 q^{55} -3.30278 q^{56} -1.78890 q^{57} +2.60555 q^{58} -3.39445 q^{59} -0.302776 q^{60} +11.5139 q^{61} +7.90833 q^{62} -9.60555 q^{63} +1.00000 q^{64} +3.30278 q^{65} -0.513878 q^{66} -4.00000 q^{67} +6.90833 q^{68} +0.302776 q^{69} -3.30278 q^{70} -16.3028 q^{71} +2.90833 q^{72} -5.81665 q^{73} -8.00000 q^{74} -0.302776 q^{75} +5.90833 q^{76} -5.60555 q^{77} +1.00000 q^{78} -14.4222 q^{79} +1.00000 q^{80} +8.18335 q^{81} -0.908327 q^{82} +11.2111 q^{83} -1.00000 q^{84} +6.90833 q^{85} +9.21110 q^{86} +0.788897 q^{87} +1.69722 q^{88} +2.90833 q^{90} +10.9083 q^{91} -1.00000 q^{92} +2.39445 q^{93} +2.60555 q^{94} +5.90833 q^{95} +0.302776 q^{96} +6.30278 q^{97} -3.90833 q^{98} +4.93608 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} + 2 q^{5} - 3 q^{6} + 3 q^{7} - 2 q^{8} + 5 q^{9} - 2 q^{10} - 7 q^{11} + 3 q^{12} + 3 q^{13} - 3 q^{14} + 3 q^{15} + 2 q^{16} + 3 q^{17} - 5 q^{18} + q^{19} + 2 q^{20}+ \cdots - 37 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −0.302776 −0.174808 −0.0874038 − 0.996173i \(-0.527857\pi\)
−0.0874038 + 0.996173i \(0.527857\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0.302776 0.123608
\(7\) 3.30278 1.24833 0.624166 − 0.781292i \(-0.285439\pi\)
0.624166 + 0.781292i \(0.285439\pi\)
\(8\) −1.00000 −0.353553
\(9\) −2.90833 −0.969442
\(10\) −1.00000 −0.316228
\(11\) −1.69722 −0.511732 −0.255866 − 0.966712i \(-0.582361\pi\)
−0.255866 + 0.966712i \(0.582361\pi\)
\(12\) −0.302776 −0.0874038
\(13\) 3.30278 0.916025 0.458013 − 0.888946i \(-0.348561\pi\)
0.458013 + 0.888946i \(0.348561\pi\)
\(14\) −3.30278 −0.882704
\(15\) −0.302776 −0.0781763
\(16\) 1.00000 0.250000
\(17\) 6.90833 1.67552 0.837758 − 0.546042i \(-0.183866\pi\)
0.837758 + 0.546042i \(0.183866\pi\)
\(18\) 2.90833 0.685499
\(19\) 5.90833 1.35546 0.677732 − 0.735309i \(-0.262963\pi\)
0.677732 + 0.735309i \(0.262963\pi\)
\(20\) 1.00000 0.223607
\(21\) −1.00000 −0.218218
\(22\) 1.69722 0.361849
\(23\) −1.00000 −0.208514
\(24\) 0.302776 0.0618038
\(25\) 1.00000 0.200000
\(26\) −3.30278 −0.647728
\(27\) 1.78890 0.344273
\(28\) 3.30278 0.624166
\(29\) −2.60555 −0.483839 −0.241919 − 0.970296i \(-0.577777\pi\)
−0.241919 + 0.970296i \(0.577777\pi\)
\(30\) 0.302776 0.0552790
\(31\) −7.90833 −1.42038 −0.710189 − 0.704011i \(-0.751390\pi\)
−0.710189 + 0.704011i \(0.751390\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.513878 0.0894547
\(34\) −6.90833 −1.18477
\(35\) 3.30278 0.558271
\(36\) −2.90833 −0.484721
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) −5.90833 −0.958457
\(39\) −1.00000 −0.160128
\(40\) −1.00000 −0.158114
\(41\) 0.908327 0.141857 0.0709284 − 0.997481i \(-0.477404\pi\)
0.0709284 + 0.997481i \(0.477404\pi\)
\(42\) 1.00000 0.154303
\(43\) −9.21110 −1.40468 −0.702340 − 0.711842i \(-0.747861\pi\)
−0.702340 + 0.711842i \(0.747861\pi\)
\(44\) −1.69722 −0.255866
\(45\) −2.90833 −0.433548
\(46\) 1.00000 0.147442
\(47\) −2.60555 −0.380059 −0.190029 − 0.981778i \(-0.560858\pi\)
−0.190029 + 0.981778i \(0.560858\pi\)
\(48\) −0.302776 −0.0437019
\(49\) 3.90833 0.558332
\(50\) −1.00000 −0.141421
\(51\) −2.09167 −0.292893
\(52\) 3.30278 0.458013
\(53\) −11.2111 −1.53996 −0.769982 − 0.638066i \(-0.779735\pi\)
−0.769982 + 0.638066i \(0.779735\pi\)
\(54\) −1.78890 −0.243438
\(55\) −1.69722 −0.228854
\(56\) −3.30278 −0.441352
\(57\) −1.78890 −0.236945
\(58\) 2.60555 0.342126
\(59\) −3.39445 −0.441920 −0.220960 − 0.975283i \(-0.570919\pi\)
−0.220960 + 0.975283i \(0.570919\pi\)
\(60\) −0.302776 −0.0390882
\(61\) 11.5139 1.47420 0.737101 − 0.675783i \(-0.236194\pi\)
0.737101 + 0.675783i \(0.236194\pi\)
\(62\) 7.90833 1.00436
\(63\) −9.60555 −1.21019
\(64\) 1.00000 0.125000
\(65\) 3.30278 0.409659
\(66\) −0.513878 −0.0632540
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 6.90833 0.837758
\(69\) 0.302776 0.0364499
\(70\) −3.30278 −0.394757
\(71\) −16.3028 −1.93478 −0.967392 − 0.253285i \(-0.918489\pi\)
−0.967392 + 0.253285i \(0.918489\pi\)
\(72\) 2.90833 0.342750
\(73\) −5.81665 −0.680788 −0.340394 − 0.940283i \(-0.610560\pi\)
−0.340394 + 0.940283i \(0.610560\pi\)
\(74\) −8.00000 −0.929981
\(75\) −0.302776 −0.0349615
\(76\) 5.90833 0.677732
\(77\) −5.60555 −0.638812
\(78\) 1.00000 0.113228
\(79\) −14.4222 −1.62262 −0.811312 − 0.584613i \(-0.801246\pi\)
−0.811312 + 0.584613i \(0.801246\pi\)
\(80\) 1.00000 0.111803
\(81\) 8.18335 0.909261
\(82\) −0.908327 −0.100308
\(83\) 11.2111 1.23058 0.615289 − 0.788301i \(-0.289039\pi\)
0.615289 + 0.788301i \(0.289039\pi\)
\(84\) −1.00000 −0.109109
\(85\) 6.90833 0.749313
\(86\) 9.21110 0.993259
\(87\) 0.788897 0.0845787
\(88\) 1.69722 0.180925
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 2.90833 0.306565
\(91\) 10.9083 1.14350
\(92\) −1.00000 −0.104257
\(93\) 2.39445 0.248293
\(94\) 2.60555 0.268742
\(95\) 5.90833 0.606182
\(96\) 0.302776 0.0309019
\(97\) 6.30278 0.639950 0.319975 − 0.947426i \(-0.396325\pi\)
0.319975 + 0.947426i \(0.396325\pi\)
\(98\) −3.90833 −0.394801
\(99\) 4.93608 0.496095
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.a.b.1.1 ✓ 2
3.2 odd 2 2070.2.a.w.1.2 2
4.3 odd 2 1840.2.a.j.1.2 2
5.2 odd 4 1150.2.b.f.599.2 4
5.3 odd 4 1150.2.b.f.599.3 4
5.4 even 2 1150.2.a.m.1.2 2
8.3 odd 2 7360.2.a.bu.1.1 2
8.5 even 2 7360.2.a.bc.1.2 2
20.19 odd 2 9200.2.a.ca.1.1 2
23.22 odd 2 5290.2.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.b.1.1 ✓ 2 1.1 even 1 trivial
1150.2.a.m.1.2 2 5.4 even 2
1150.2.b.f.599.2 4 5.2 odd 4
1150.2.b.f.599.3 4 5.3 odd 4
1840.2.a.j.1.2 2 4.3 odd 2
2070.2.a.w.1.2 2 3.2 odd 2
5290.2.a.j.1.1 2 23.22 odd 2
7360.2.a.bc.1.2 2 8.5 even 2
7360.2.a.bu.1.1 2 8.3 odd 2
9200.2.a.ca.1.1 2 20.19 odd 2