Properties

Label 230.6.a.d.1.2
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 1.2
Root \(7.78556\) 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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} -9.47271 q^{3} +16.0000 q^{4} +25.0000 q^{5} -37.8908 q^{6} +37.1792 q^{7} +64.0000 q^{8} -153.268 q^{9} +100.000 q^{10} -642.741 q^{11} -151.563 q^{12} +803.942 q^{13} +148.717 q^{14} -236.818 q^{15} +256.000 q^{16} +285.849 q^{17} -613.071 q^{18} -2353.83 q^{19} +400.000 q^{20} -352.188 q^{21} -2570.96 q^{22} -529.000 q^{23} -606.253 q^{24} +625.000 q^{25} +3215.77 q^{26} +3753.73 q^{27} +594.867 q^{28} -4925.02 q^{29} -947.271 q^{30} -4420.85 q^{31} +1024.00 q^{32} +6088.50 q^{33} +1143.40 q^{34} +929.480 q^{35} -2452.29 q^{36} +10232.8 q^{37} -9415.31 q^{38} -7615.50 q^{39} +1600.00 q^{40} -6340.42 q^{41} -1408.75 q^{42} -19250.1 q^{43} -10283.9 q^{44} -3831.70 q^{45} -2116.00 q^{46} -13293.9 q^{47} -2425.01 q^{48} -15424.7 q^{49} +2500.00 q^{50} -2707.77 q^{51} +12863.1 q^{52} +21717.0 q^{53} +15014.9 q^{54} -16068.5 q^{55} +2379.47 q^{56} +22297.1 q^{57} -19700.1 q^{58} -2645.86 q^{59} -3789.08 q^{60} -52328.4 q^{61} -17683.4 q^{62} -5698.38 q^{63} +4096.00 q^{64} +20098.5 q^{65} +24354.0 q^{66} -70672.3 q^{67} +4573.59 q^{68} +5011.06 q^{69} +3717.92 q^{70} +3524.59 q^{71} -9809.14 q^{72} -87528.6 q^{73} +40931.1 q^{74} -5920.44 q^{75} -37661.2 q^{76} -23896.6 q^{77} -30462.0 q^{78} +75802.8 q^{79} +6400.00 q^{80} +1686.10 q^{81} -25361.7 q^{82} +10147.6 q^{83} -5635.00 q^{84} +7146.23 q^{85} -77000.6 q^{86} +46653.3 q^{87} -41135.4 q^{88} -3051.89 q^{89} -15326.8 q^{90} +29889.9 q^{91} -8464.00 q^{92} +41877.4 q^{93} -53175.4 q^{94} -58845.7 q^{95} -9700.05 q^{96} +105531. q^{97} -61698.8 q^{98} +98511.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 12 q^{2} - 34 q^{3} + 48 q^{4} + 75 q^{5} - 136 q^{6} - 121 q^{7} + 192 q^{8} + 89 q^{9} + 300 q^{10} - 502 q^{11} - 544 q^{12} + 76 q^{13} - 484 q^{14} - 850 q^{15} + 768 q^{16} - 1167 q^{17} + 356 q^{18}+ \cdots + 344510 q^{99}+O(q^{100}) \) 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
\(3\) −9.47271 −0.607674 −0.303837 − 0.952724i \(-0.598268\pi\)
−0.303837 + 0.952724i \(0.598268\pi\)
\(4\) 16.0000 0.500000
\(5\) 25.0000 0.447214
\(6\) −37.8908 −0.429691
\(7\) 37.1792 0.286784 0.143392 − 0.989666i \(-0.454199\pi\)
0.143392 + 0.989666i \(0.454199\pi\)
\(8\) 64.0000 0.353553
\(9\) −153.268 −0.630732
\(10\) 100.000 0.316228
\(11\) −642.741 −1.60160 −0.800800 − 0.598931i \(-0.795592\pi\)
−0.800800 + 0.598931i \(0.795592\pi\)
\(12\) −151.563 −0.303837
\(13\) 803.942 1.31937 0.659684 − 0.751543i \(-0.270690\pi\)
0.659684 + 0.751543i \(0.270690\pi\)
\(14\) 148.717 0.202787
\(15\) −236.818 −0.271760
\(16\) 256.000 0.250000
\(17\) 285.849 0.239892 0.119946 − 0.992780i \(-0.461728\pi\)
0.119946 + 0.992780i \(0.461728\pi\)
\(18\) −613.071 −0.445995
\(19\) −2353.83 −1.49586 −0.747930 − 0.663778i \(-0.768952\pi\)
−0.747930 + 0.663778i \(0.768952\pi\)
\(20\) 400.000 0.223607
\(21\) −352.188 −0.174271
\(22\) −2570.96 −1.13250
\(23\) −529.000 −0.208514
\(24\) −606.253 −0.214845
\(25\) 625.000 0.200000
\(26\) 3215.77 0.932935
\(27\) 3753.73 0.990954
\(28\) 594.867 0.143392
\(29\) −4925.02 −1.08746 −0.543730 − 0.839260i \(-0.682988\pi\)
−0.543730 + 0.839260i \(0.682988\pi\)
\(30\) −947.271 −0.192164
\(31\) −4420.85 −0.826231 −0.413116 − 0.910679i \(-0.635559\pi\)
−0.413116 + 0.910679i \(0.635559\pi\)
\(32\) 1024.00 0.176777
\(33\) 6088.50 0.973252
\(34\) 1143.40 0.169629
\(35\) 929.480 0.128254
\(36\) −2452.29 −0.315366
\(37\) 10232.8 1.22882 0.614412 − 0.788986i \(-0.289393\pi\)
0.614412 + 0.788986i \(0.289393\pi\)
\(38\) −9415.31 −1.05773
\(39\) −7615.50 −0.801747
\(40\) 1600.00 0.158114
\(41\) −6340.42 −0.589059 −0.294529 − 0.955642i \(-0.595163\pi\)
−0.294529 + 0.955642i \(0.595163\pi\)
\(42\) −1408.75 −0.123228
\(43\) −19250.1 −1.58768 −0.793840 − 0.608127i \(-0.791921\pi\)
−0.793840 + 0.608127i \(0.791921\pi\)
\(44\) −10283.9 −0.800800
\(45\) −3831.70 −0.282072
\(46\) −2116.00 −0.147442
\(47\) −13293.9 −0.877821 −0.438911 − 0.898531i \(-0.644636\pi\)
−0.438911 + 0.898531i \(0.644636\pi\)
\(48\) −2425.01 −0.151919
\(49\) −15424.7 −0.917755
\(50\) 2500.00 0.141421
\(51\) −2707.77 −0.145776
\(52\) 12863.1 0.659684
\(53\) 21717.0 1.06197 0.530984 − 0.847382i \(-0.321823\pi\)
0.530984 + 0.847382i \(0.321823\pi\)
\(54\) 15014.9 0.700710
\(55\) −16068.5 −0.716258
\(56\) 2379.47 0.101393
\(57\) 22297.1 0.908995
\(58\) −19700.1 −0.768950
\(59\) −2645.86 −0.0989547 −0.0494774 − 0.998775i \(-0.515756\pi\)
−0.0494774 + 0.998775i \(0.515756\pi\)
\(60\) −3789.08 −0.135880
\(61\) −52328.4 −1.80058 −0.900291 − 0.435289i \(-0.856646\pi\)
−0.900291 + 0.435289i \(0.856646\pi\)
\(62\) −17683.4 −0.584234
\(63\) −5698.38 −0.180884
\(64\) 4096.00 0.125000
\(65\) 20098.5 0.590040
\(66\) 24354.0 0.688193
\(67\) −70672.3 −1.92337 −0.961684 − 0.274161i \(-0.911600\pi\)
−0.961684 + 0.274161i \(0.911600\pi\)
\(68\) 4573.59 0.119946
\(69\) 5011.06 0.126709
\(70\) 3717.92 0.0906891
\(71\) 3524.59 0.0829779 0.0414890 − 0.999139i \(-0.486790\pi\)
0.0414890 + 0.999139i \(0.486790\pi\)
\(72\) −9809.14 −0.222997
\(73\) −87528.6 −1.92240 −0.961198 − 0.275861i \(-0.911037\pi\)
−0.961198 + 0.275861i \(0.911037\pi\)
\(74\) 40931.1 0.868909
\(75\) −5920.44 −0.121535
\(76\) −37661.2 −0.747930
\(77\) −23896.6 −0.459314
\(78\) −30462.0 −0.566921
\(79\) 75802.8 1.36653 0.683263 − 0.730173i \(-0.260560\pi\)
0.683263 + 0.730173i \(0.260560\pi\)
\(80\) 6400.00 0.111803
\(81\) 1686.10 0.0285543
\(82\) −25361.7 −0.416527
\(83\) 10147.6 0.161685 0.0808425 − 0.996727i \(-0.474239\pi\)
0.0808425 + 0.996727i \(0.474239\pi\)
\(84\) −5635.00 −0.0871357
\(85\) 7146.23 0.107283
\(86\) −77000.6 −1.12266
\(87\) 46653.3 0.660822
\(88\) −41135.4 −0.566251
\(89\) −3051.89 −0.0408408 −0.0204204 − 0.999791i \(-0.506500\pi\)
−0.0204204 + 0.999791i \(0.506500\pi\)
\(90\) −15326.8 −0.199455
\(91\) 29889.9 0.378374
\(92\) −8464.00 −0.104257
\(93\) 41877.4 0.502080
\(94\) −53175.4 −0.620713
\(95\) −58845.7 −0.668969
\(96\) −9700.05 −0.107423
\(97\) 105531. 1.13880 0.569402 − 0.822059i \(-0.307175\pi\)
0.569402 + 0.822059i \(0.307175\pi\)
\(98\) −61698.8 −0.648951
\(99\) 98511.5 1.01018
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.1.2 ✓ 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.6.a.d.1.2 ✓ 3 1.1 even 1 trivial