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Properties

Label 230.4.a.f.1.2
Level $230$
Weight $4$
Character 230.1
Self dual yes
Analytic conductor $13.570$
Analytic rank $1$
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] = [230,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-4,-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: \(13.5704393013\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{73}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 18 \) 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 1.2
Root \(-3.77200\) 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)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.77200 q^{3} +4.00000 q^{4} +5.00000 q^{5} -5.54400 q^{6} -4.22800 q^{7} -8.00000 q^{8} -19.3160 q^{9} -10.0000 q^{10} -33.7200 q^{11} +11.0880 q^{12} -42.9480 q^{13} +8.45600 q^{14} +13.8600 q^{15} +16.0000 q^{16} +7.49202 q^{17} +38.6320 q^{18} -25.7200 q^{19} +20.0000 q^{20} -11.7200 q^{21} +67.4400 q^{22} -23.0000 q^{23} -22.1760 q^{24} +25.0000 q^{25} +85.8960 q^{26} -128.388 q^{27} -16.9120 q^{28} -47.4560 q^{29} -27.7200 q^{30} +65.0720 q^{31} -32.0000 q^{32} -93.4720 q^{33} -14.9840 q^{34} -21.1400 q^{35} -77.2640 q^{36} -215.596 q^{37} +51.4400 q^{38} -119.052 q^{39} -40.0000 q^{40} +150.128 q^{41} +23.4400 q^{42} -83.0160 q^{43} -134.880 q^{44} -96.5800 q^{45} +46.0000 q^{46} -278.140 q^{47} +44.3520 q^{48} -325.124 q^{49} -50.0000 q^{50} +20.7679 q^{51} -171.792 q^{52} -182.404 q^{53} +256.776 q^{54} -168.600 q^{55} +33.8240 q^{56} -71.2959 q^{57} +94.9120 q^{58} -270.060 q^{59} +55.4400 q^{60} -137.264 q^{61} -130.144 q^{62} +81.6680 q^{63} +64.0000 q^{64} -214.740 q^{65} +186.944 q^{66} -440.652 q^{67} +29.9681 q^{68} -63.7560 q^{69} +42.2800 q^{70} +1071.29 q^{71} +154.528 q^{72} -269.756 q^{73} +431.192 q^{74} +69.3000 q^{75} -102.880 q^{76} +142.568 q^{77} +238.104 q^{78} +195.232 q^{79} +80.0000 q^{80} +165.640 q^{81} -300.256 q^{82} +136.676 q^{83} -46.8801 q^{84} +37.4601 q^{85} +166.032 q^{86} -131.548 q^{87} +269.760 q^{88} +629.896 q^{89} +193.160 q^{90} +181.584 q^{91} -92.0000 q^{92} +180.380 q^{93} +556.280 q^{94} -128.600 q^{95} -88.7041 q^{96} -419.392 q^{97} +650.248 q^{98} +651.336 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 3 q^{3} + 8 q^{4} + 10 q^{5} + 6 q^{6} - 17 q^{7} - 16 q^{8} - 13 q^{9} - 20 q^{10} + 18 q^{11} - 12 q^{12} - 9 q^{13} + 34 q^{14} - 15 q^{15} + 32 q^{16} - 79 q^{17} + 26 q^{18} + 34 q^{19}+ \cdots + 978 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −0.707107
\(3\) 2.77200 0.533472 0.266736 − 0.963770i \(-0.414055\pi\)
0.266736 + 0.963770i \(0.414055\pi\)
\(4\) 4.00000 0.500000
\(5\) 5.00000 0.447214
\(6\) −5.54400 −0.377222
\(7\) −4.22800 −0.228290 −0.114145 − 0.993464i \(-0.536413\pi\)
−0.114145 + 0.993464i \(0.536413\pi\)
\(8\) −8.00000 −0.353553
\(9\) −19.3160 −0.715408
\(10\) −10.0000 −0.316228
\(11\) −33.7200 −0.924270 −0.462135 − 0.886810i \(-0.652916\pi\)
−0.462135 + 0.886810i \(0.652916\pi\)
\(12\) 11.0880 0.266736
\(13\) −42.9480 −0.916280 −0.458140 − 0.888880i \(-0.651484\pi\)
−0.458140 + 0.888880i \(0.651484\pi\)
\(14\) 8.45600 0.161426
\(15\) 13.8600 0.238576
\(16\) 16.0000 0.250000
\(17\) 7.49202 0.106887 0.0534436 − 0.998571i \(-0.482980\pi\)
0.0534436 + 0.998571i \(0.482980\pi\)
\(18\) 38.6320 0.505870
\(19\) −25.7200 −0.310557 −0.155278 − 0.987871i \(-0.549627\pi\)
−0.155278 + 0.987871i \(0.549627\pi\)
\(20\) 20.0000 0.223607
\(21\) −11.7200 −0.121787
\(22\) 67.4400 0.653557
\(23\) −23.0000 −0.208514
\(24\) −22.1760 −0.188611
\(25\) 25.0000 0.200000
\(26\) 85.8960 0.647908
\(27\) −128.388 −0.915122
\(28\) −16.9120 −0.114145
\(29\) −47.4560 −0.303874 −0.151937 − 0.988390i \(-0.548551\pi\)
−0.151937 + 0.988390i \(0.548551\pi\)
\(30\) −27.7200 −0.168699
\(31\) 65.0720 0.377009 0.188505 − 0.982072i \(-0.439636\pi\)
0.188505 + 0.982072i \(0.439636\pi\)
\(32\) −32.0000 −0.176777
\(33\) −93.4720 −0.493072
\(34\) −14.9840 −0.0755806
\(35\) −21.1400 −0.102095
\(36\) −77.2640 −0.357704
\(37\) −215.596 −0.957940 −0.478970 − 0.877831i \(-0.658990\pi\)
−0.478970 + 0.877831i \(0.658990\pi\)
\(38\) 51.4400 0.219597
\(39\) −119.052 −0.488810
\(40\) −40.0000 −0.158114
\(41\) 150.128 0.571856 0.285928 − 0.958251i \(-0.407698\pi\)
0.285928 + 0.958251i \(0.407698\pi\)
\(42\) 23.4400 0.0861161
\(43\) −83.0160 −0.294414 −0.147207 − 0.989106i \(-0.547028\pi\)
−0.147207 + 0.989106i \(0.547028\pi\)
\(44\) −134.880 −0.462135
\(45\) −96.5800 −0.319940
\(46\) 46.0000 0.147442
\(47\) −278.140 −0.863210 −0.431605 − 0.902063i \(-0.642053\pi\)
−0.431605 + 0.902063i \(0.642053\pi\)
\(48\) 44.3520 0.133368
\(49\) −325.124 −0.947883
\(50\) −50.0000 −0.141421
\(51\) 20.7679 0.0570213
\(52\) −171.792 −0.458140
\(53\) −182.404 −0.472738 −0.236369 − 0.971663i \(-0.575957\pi\)
−0.236369 + 0.971663i \(0.575957\pi\)
\(54\) 256.776 0.647089
\(55\) −168.600 −0.413346
\(56\) 33.8240 0.0807129
\(57\) −71.2959 −0.165673
\(58\) 94.9120 0.214872
\(59\) −270.060 −0.595913 −0.297956 − 0.954580i \(-0.596305\pi\)
−0.297956 + 0.954580i \(0.596305\pi\)
\(60\) 55.4400 0.119288
\(61\) −137.264 −0.288112 −0.144056 − 0.989570i \(-0.546015\pi\)
−0.144056 + 0.989570i \(0.546015\pi\)
\(62\) −130.144 −0.266586
\(63\) 81.6680 0.163321
\(64\) 64.0000 0.125000
\(65\) −214.740 −0.409773
\(66\) 186.944 0.348655
\(67\) −440.652 −0.803496 −0.401748 − 0.915750i \(-0.631597\pi\)
−0.401748 + 0.915750i \(0.631597\pi\)
\(68\) 29.9681 0.0534436
\(69\) −63.7560 −0.111237
\(70\) 42.2800 0.0721918
\(71\) 1071.29 1.79068 0.895342 − 0.445380i \(-0.146931\pi\)
0.895342 + 0.445380i \(0.146931\pi\)
\(72\) 154.528 0.252935
\(73\) −269.756 −0.432501 −0.216250 − 0.976338i \(-0.569383\pi\)
−0.216250 + 0.976338i \(0.569383\pi\)
\(74\) 431.192 0.677366
\(75\) 69.3000 0.106694
\(76\) −102.880 −0.155278
\(77\) 142.568 0.211002
\(78\) 238.104 0.345641
\(79\) 195.232 0.278042 0.139021 − 0.990289i \(-0.455604\pi\)
0.139021 + 0.990289i \(0.455604\pi\)
\(80\) 80.0000 0.111803
\(81\) 165.640 0.227216
\(82\) −300.256 −0.404363
\(83\) 136.676 0.180748 0.0903742 − 0.995908i \(-0.471194\pi\)
0.0903742 + 0.995908i \(0.471194\pi\)
\(84\) −46.8801 −0.0608933
\(85\) 37.4601 0.0478014
\(86\) 166.032 0.208182
\(87\) −131.548 −0.162108
\(88\) 269.760 0.326779
\(89\) 629.896 0.750212 0.375106 − 0.926982i \(-0.377606\pi\)
0.375106 + 0.926982i \(0.377606\pi\)
\(90\) 193.160 0.226232
\(91\) 181.584 0.209178
\(92\) −92.0000 −0.104257
\(93\) 180.380 0.201124
\(94\) 556.280 0.610382
\(95\) −128.600 −0.138885
\(96\) −88.7041 −0.0943054
\(97\) −419.392 −0.438998 −0.219499 − 0.975613i \(-0.570442\pi\)
−0.219499 + 0.975613i \(0.570442\pi\)
\(98\) 650.248 0.670255
\(99\) 651.336 0.661230
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 230.4.a.f.1.2 ✓ 2
3.2 odd 2 2070.4.a.s.1.2 2
4.3 odd 2 1840.4.a.i.1.1 2
5.2 odd 4 1150.4.b.k.599.1 4
5.3 odd 4 1150.4.b.k.599.4 4
5.4 even 2 1150.4.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.a.f.1.2 ✓ 2 1.1 even 1 trivial
1150.4.a.l.1.1 2 5.4 even 2
1150.4.b.k.599.1 4 5.2 odd 4
1150.4.b.k.599.4 4 5.3 odd 4
1840.4.a.i.1.1 2 4.3 odd 2
2070.4.a.s.1.2 2 3.2 odd 2