Properties

Label 230.6.a.d.1.1
Level $230$
Weight $6$
Character 230.1
Self dual yes
Analytic conductor $36.888$
Analytic rank $1$
Dimension $3$
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] = [230,6,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, 6); 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, 6, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 230.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,12,-34] 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: \(36.8882785570\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.27980.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 47x - 106 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-5.13326\) 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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} -26.8834 q^{3} +16.0000 q^{4} +25.0000 q^{5} -107.534 q^{6} -148.733 q^{7} +64.0000 q^{8} +479.717 q^{9} +100.000 q^{10} +389.613 q^{11} -430.134 q^{12} +170.844 q^{13} -594.932 q^{14} -672.085 q^{15} +256.000 q^{16} -275.341 q^{17} +1918.87 q^{18} +543.366 q^{19} +400.000 q^{20} +3998.44 q^{21} +1558.45 q^{22} -529.000 q^{23} -1720.54 q^{24} +625.000 q^{25} +683.376 q^{26} -6363.75 q^{27} -2379.73 q^{28} -2840.78 q^{29} -2688.34 q^{30} -4608.21 q^{31} +1024.00 q^{32} -10474.1 q^{33} -1101.36 q^{34} -3718.32 q^{35} +7675.47 q^{36} -3341.86 q^{37} +2173.46 q^{38} -4592.87 q^{39} +1600.00 q^{40} +12310.0 q^{41} +15993.8 q^{42} -19460.7 q^{43} +6233.81 q^{44} +11992.9 q^{45} -2116.00 q^{46} -3595.03 q^{47} -6882.15 q^{48} +5314.47 q^{49} +2500.00 q^{50} +7402.09 q^{51} +2733.51 q^{52} -31873.2 q^{53} -25455.0 q^{54} +9740.34 q^{55} -9518.91 q^{56} -14607.5 q^{57} -11363.1 q^{58} -26083.7 q^{59} -10753.4 q^{60} +2100.05 q^{61} -18432.8 q^{62} -71349.6 q^{63} +4096.00 q^{64} +4271.10 q^{65} -41896.5 q^{66} -9605.18 q^{67} -4405.45 q^{68} +14221.3 q^{69} -14873.3 q^{70} -78484.3 q^{71} +30701.9 q^{72} -8680.81 q^{73} -13367.4 q^{74} -16802.1 q^{75} +8693.86 q^{76} -57948.3 q^{77} -18371.5 q^{78} -19929.4 q^{79} +6400.00 q^{80} +54507.9 q^{81} +49239.8 q^{82} -34625.0 q^{83} +63975.1 q^{84} -6883.52 q^{85} -77842.9 q^{86} +76369.8 q^{87} +24935.3 q^{88} +60541.8 q^{89} +47971.7 q^{90} -25410.1 q^{91} -8464.00 q^{92} +123884. q^{93} -14380.1 q^{94} +13584.1 q^{95} -27528.6 q^{96} +114656. q^{97} +21257.9 q^{98} +186904. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 12 q^{2} - 34 q^{3} + 48 q^{4} + 75 q^{5} - 136 q^{6} - 121 q^{7} + 192 q^{8} + 89 q^{9} + 300 q^{10} - 502 q^{11} - 544 q^{12} + 76 q^{13} - 484 q^{14} - 850 q^{15} + 768 q^{16} - 1167 q^{17} + 356 q^{18}+ \cdots + 344510 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) −26.8834 −1.72457 −0.862285 − 0.506423i \(-0.830967\pi\)
−0.862285 + 0.506423i \(0.830967\pi\)
\(4\) 16.0000 0.500000
\(5\) 25.0000 0.447214
\(6\) −107.534 −1.21946
\(7\) −148.733 −1.14726 −0.573630 − 0.819114i \(-0.694465\pi\)
−0.573630 + 0.819114i \(0.694465\pi\)
\(8\) 64.0000 0.353553
\(9\) 479.717 1.97414
\(10\) 100.000 0.316228
\(11\) 389.613 0.970850 0.485425 − 0.874278i \(-0.338665\pi\)
0.485425 + 0.874278i \(0.338665\pi\)
\(12\) −430.134 −0.862285
\(13\) 170.844 0.280377 0.140188 − 0.990125i \(-0.455229\pi\)
0.140188 + 0.990125i \(0.455229\pi\)
\(14\) −594.932 −0.811235
\(15\) −672.085 −0.771251
\(16\) 256.000 0.250000
\(17\) −275.341 −0.231072 −0.115536 − 0.993303i \(-0.536859\pi\)
−0.115536 + 0.993303i \(0.536859\pi\)
\(18\) 1918.87 1.39593
\(19\) 543.366 0.345309 0.172655 − 0.984982i \(-0.444766\pi\)
0.172655 + 0.984982i \(0.444766\pi\)
\(20\) 400.000 0.223607
\(21\) 3998.44 1.97853
\(22\) 1558.45 0.686495
\(23\) −529.000 −0.208514
\(24\) −1720.54 −0.609728
\(25\) 625.000 0.200000
\(26\) 683.376 0.198256
\(27\) −6363.75 −1.67998
\(28\) −2379.73 −0.573630
\(29\) −2840.78 −0.627253 −0.313627 − 0.949546i \(-0.601544\pi\)
−0.313627 + 0.949546i \(0.601544\pi\)
\(30\) −2688.34 −0.545357
\(31\) −4608.21 −0.861247 −0.430623 − 0.902532i \(-0.641706\pi\)
−0.430623 + 0.902532i \(0.641706\pi\)
\(32\) 1024.00 0.176777
\(33\) −10474.1 −1.67430
\(34\) −1101.36 −0.163393
\(35\) −3718.32 −0.513070
\(36\) 7675.47 0.987071
\(37\) −3341.86 −0.401313 −0.200657 − 0.979662i \(-0.564308\pi\)
−0.200657 + 0.979662i \(0.564308\pi\)
\(38\) 2173.46 0.244171
\(39\) −4592.87 −0.483529
\(40\) 1600.00 0.158114
\(41\) 12310.0 1.14366 0.571830 − 0.820372i \(-0.306234\pi\)
0.571830 + 0.820372i \(0.306234\pi\)
\(42\) 15993.8 1.39903
\(43\) −19460.7 −1.60505 −0.802523 − 0.596621i \(-0.796510\pi\)
−0.802523 + 0.596621i \(0.796510\pi\)
\(44\) 6233.81 0.485425
\(45\) 11992.9 0.882863
\(46\) −2116.00 −0.147442
\(47\) −3595.03 −0.237387 −0.118694 − 0.992931i \(-0.537871\pi\)
−0.118694 + 0.992931i \(0.537871\pi\)
\(48\) −6882.15 −0.431143
\(49\) 5314.47 0.316206
\(50\) 2500.00 0.141421
\(51\) 7402.09 0.398501
\(52\) 2733.51 0.140188
\(53\) −31873.2 −1.55861 −0.779303 − 0.626647i \(-0.784427\pi\)
−0.779303 + 0.626647i \(0.784427\pi\)
\(54\) −25455.0 −1.18792
\(55\) 9740.34 0.434177
\(56\) −9518.91 −0.405618
\(57\) −14607.5 −0.595510
\(58\) −11363.1 −0.443535
\(59\) −26083.7 −0.975525 −0.487763 − 0.872976i \(-0.662187\pi\)
−0.487763 + 0.872976i \(0.662187\pi\)
\(60\) −10753.4 −0.385626
\(61\) 2100.05 0.0722613 0.0361306 − 0.999347i \(-0.488497\pi\)
0.0361306 + 0.999347i \(0.488497\pi\)
\(62\) −18432.8 −0.608993
\(63\) −71349.6 −2.26486
\(64\) 4096.00 0.125000
\(65\) 4271.10 0.125388
\(66\) −41896.5 −1.18391
\(67\) −9605.18 −0.261408 −0.130704 − 0.991421i \(-0.541724\pi\)
−0.130704 + 0.991421i \(0.541724\pi\)
\(68\) −4405.45 −0.115536
\(69\) 14221.3 0.359598
\(70\) −14873.3 −0.362796
\(71\) −78484.3 −1.84772 −0.923862 − 0.382727i \(-0.874985\pi\)
−0.923862 + 0.382727i \(0.874985\pi\)
\(72\) 30701.9 0.697965
\(73\) −8680.81 −0.190657 −0.0953286 − 0.995446i \(-0.530390\pi\)
−0.0953286 + 0.995446i \(0.530390\pi\)
\(74\) −13367.4 −0.283771
\(75\) −16802.1 −0.344914
\(76\) 8693.86 0.172655
\(77\) −57948.3 −1.11382
\(78\) −18371.5 −0.341907
\(79\) −19929.4 −0.359275 −0.179637 − 0.983733i \(-0.557492\pi\)
−0.179637 + 0.983733i \(0.557492\pi\)
\(80\) 6400.00 0.111803
\(81\) 54507.9 0.923096
\(82\) 49239.8 0.808690
\(83\) −34625.0 −0.551689 −0.275844 − 0.961202i \(-0.588957\pi\)
−0.275844 + 0.961202i \(0.588957\pi\)
\(84\) 63975.1 0.989265
\(85\) −6883.52 −0.103339
\(86\) −77842.9 −1.13494
\(87\) 76369.8 1.08174
\(88\) 24935.3 0.343247
\(89\) 60541.8 0.810178 0.405089 − 0.914277i \(-0.367241\pi\)
0.405089 + 0.914277i \(0.367241\pi\)
\(90\) 47971.7 0.624279
\(91\) −25410.1 −0.321665
\(92\) −8464.00 −0.104257
\(93\) 123884. 1.48528
\(94\) −14380.1 −0.167858
\(95\) 13584.1 0.154427
\(96\) −27528.6 −0.304864
\(97\) 114656. 1.23728 0.618640 − 0.785675i \(-0.287684\pi\)
0.618640 + 0.785675i \(0.287684\pi\)
\(98\) 21257.9 0.223591
\(99\) 186904. 1.91660
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 230.6.a.d.1.1 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.6.a.d.1.1 ✓ 3 1.1 even 1 trivial