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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 230.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.5704393013\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 137.12
Character \(\chi\) \(=\) 230.137
Dual form 230.4.e.a.183.12

$q$-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+(-1.41421 - 1.41421i) q^{2} +(0.933603 - 0.933603i) q^{3} +4.00000i q^{4} +(4.48554 + 10.2411i) q^{5} -2.64063 q^{6} +(3.80706 - 3.80706i) q^{7} +(5.65685 - 5.65685i) q^{8} +25.2568i q^{9} +(8.13958 - 20.8266i) q^{10} -27.1483i q^{11} +(3.73441 + 3.73441i) q^{12} +(-22.6583 + 22.6583i) q^{13} -10.7680 q^{14} +(13.7488 + 5.37340i) q^{15} -16.0000 q^{16} +(-67.3825 + 67.3825i) q^{17} +(35.7185 - 35.7185i) q^{18} +1.14225 q^{19} +(-40.9644 + 17.9422i) q^{20} -7.10856i q^{21} +(-38.3934 + 38.3934i) q^{22} +(-105.690 - 31.5680i) q^{23} -10.5625i q^{24} +(-84.7598 + 91.8737i) q^{25} +64.0875 q^{26} +(48.7871 + 48.7871i) q^{27} +(15.2282 + 15.2282i) q^{28} +147.654i q^{29} +(-11.8446 - 27.0429i) q^{30} +109.302 q^{31} +(22.6274 + 22.6274i) q^{32} +(-25.3457 - 25.3457i) q^{33} +190.586 q^{34} +(56.0651 + 21.9117i) q^{35} -101.027 q^{36} +(-119.900 + 119.900i) q^{37} +(-1.61538 - 1.61538i) q^{38} +42.3078i q^{39} +(83.3064 + 32.5583i) q^{40} +264.541 q^{41} +(-10.0530 + 10.0530i) q^{42} +(-40.5062 - 40.5062i) q^{43} +108.593 q^{44} +(-258.657 + 113.290i) q^{45} +(104.825 + 194.113i) q^{46} +(249.160 + 249.160i) q^{47} +(-14.9376 + 14.9376i) q^{48} +314.013i q^{49} +(249.797 - 10.0604i) q^{50} +125.817i q^{51} +(-90.6334 - 90.6334i) q^{52} +(-15.7902 - 15.7902i) q^{53} -137.991i q^{54} +(278.028 - 121.775i) q^{55} -43.0719i q^{56} +(1.06640 - 1.06640i) q^{57} +(208.814 - 208.814i) q^{58} +416.071i q^{59} +(-21.4936 + 54.9953i) q^{60} +152.167i q^{61} +(-154.576 - 154.576i) q^{62} +(96.1540 + 96.1540i) q^{63} -64.0000i q^{64} +(-333.681 - 130.411i) q^{65} +71.6885i q^{66} +(-29.3613 + 29.3613i) q^{67} +(-269.530 - 269.530i) q^{68} +(-128.145 + 69.2009i) q^{69} +(-48.3002 - 110.276i) q^{70} -711.508 q^{71} +(142.874 + 142.874i) q^{72} +(146.668 - 146.668i) q^{73} +339.130 q^{74} +(6.64147 + 164.906i) q^{75} +4.56898i q^{76} +(-103.355 - 103.355i) q^{77} +(59.8322 - 59.8322i) q^{78} +392.019 q^{79} +(-71.7687 - 163.857i) q^{80} -590.837 q^{81} +(-374.117 - 374.117i) q^{82} +(-697.537 - 697.537i) q^{83} +28.4342 q^{84} +(-992.317 - 387.823i) q^{85} +114.569i q^{86} +(137.850 + 137.850i) q^{87} +(-153.574 - 153.574i) q^{88} +674.229 q^{89} +(526.013 + 205.579i) q^{90} +172.523i q^{91} +(126.272 - 422.762i) q^{92} +(102.045 - 102.045i) q^{93} -704.730i q^{94} +(5.12359 + 11.6978i) q^{95} +42.2500 q^{96} +(705.114 - 705.114i) q^{97} +(444.081 - 444.081i) q^{98} +685.678 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 16 q^{3} - 16 q^{6} - 64 q^{12} + 192 q^{13} - 1152 q^{16} + 32 q^{18} + 276 q^{23} + 880 q^{25} + 304 q^{26} + 728 q^{27} + 608 q^{31} + 688 q^{35} + 2816 q^{36} - 2208 q^{41} - 256 q^{46} + 144 q^{47}+ \cdots - 448 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 − 1.41421i −0.500000 − 0.500000i
\(3\) 0.933603 − 0.933603i 0.179672 − 0.179672i −0.611541 − 0.791213i \(-0.709450\pi\)
0.791213 + 0.611541i \(0.209450\pi\)
\(4\) 4.00000i 0.500000i
\(5\) 4.48554 + 10.2411i 0.401199 + 0.915991i
\(6\) −2.64063 −0.179672
\(7\) 3.80706 − 3.80706i 0.205562 − 0.205562i −0.596816 − 0.802378i \(-0.703568\pi\)
0.802378 + 0.596816i \(0.203568\pi\)
\(8\) 5.65685 − 5.65685i 0.250000 − 0.250000i
\(9\) 25.2568i 0.935436i
\(10\) 8.13958 − 20.8266i 0.257396 − 0.658595i
\(11\) − 27.1483i − 0.744137i −0.928205 − 0.372069i \(-0.878649\pi\)
0.928205 − 0.372069i \(-0.121351\pi\)
\(12\) 3.73441 + 3.73441i 0.0898360 + 0.0898360i
\(13\) −22.6583 + 22.6583i −0.483407 + 0.483407i −0.906218 − 0.422811i \(-0.861043\pi\)
0.422811 + 0.906218i \(0.361043\pi\)
\(14\) −10.7680 −0.205562
\(15\) 13.7488 + 5.37340i 0.236662 + 0.0924937i
\(16\) −16.0000 −0.250000
\(17\) −67.3825 + 67.3825i −0.961333 + 0.961333i −0.999280 − 0.0379472i \(-0.987918\pi\)
0.0379472 + 0.999280i \(0.487918\pi\)
\(18\) 35.7185 − 35.7185i 0.467718 − 0.467718i
\(19\) 1.14225 0.0137921 0.00689603 − 0.999976i \(-0.497805\pi\)
0.00689603 + 0.999976i \(0.497805\pi\)
\(20\) −40.9644 + 17.9422i −0.457995 + 0.200599i
\(21\) − 7.10856i − 0.0738674i
\(22\) −38.3934 + 38.3934i −0.372069 + 0.372069i
\(23\) −105.690 − 31.5680i −0.958173 − 0.286190i
\(24\) − 10.5625i − 0.0898360i
\(25\) −84.7598 + 91.8737i −0.678079 + 0.734989i
\(26\) 64.0875 0.483407
\(27\) 48.7871 + 48.7871i 0.347744 + 0.347744i
\(28\) 15.2282 + 15.2282i 0.102781 + 0.102781i
\(29\) 147.654i 0.945471i 0.881204 + 0.472736i \(0.156733\pi\)
−0.881204 + 0.472736i \(0.843267\pi\)
\(30\) −11.8446 − 27.0429i −0.0720842 − 0.164578i
\(31\) 109.302 0.633266 0.316633 − 0.948548i \(-0.397448\pi\)
0.316633 + 0.948548i \(0.397448\pi\)
\(32\) 22.6274 + 22.6274i 0.125000 + 0.125000i
\(33\) −25.3457 − 25.3457i −0.133701 − 0.133701i
\(34\) 190.586 0.961333
\(35\) 56.0651 + 21.9117i 0.270764 + 0.105822i
\(36\) −101.027 −0.467718
\(37\) −119.900 + 119.900i −0.532744 + 0.532744i −0.921388 − 0.388644i \(-0.872943\pi\)
0.388644 + 0.921388i \(0.372943\pi\)
\(38\) −1.61538 − 1.61538i −0.00689603 − 0.00689603i
\(39\) 42.3078i 0.173709i
\(40\) 83.3064 + 32.5583i 0.329297 + 0.128698i
\(41\) 264.541 1.00767 0.503833 − 0.863801i \(-0.331923\pi\)
0.503833 + 0.863801i \(0.331923\pi\)
\(42\) −10.0530 + 10.0530i −0.0369337 + 0.0369337i
\(43\) −40.5062 − 40.5062i −0.143654 − 0.143654i 0.631622 − 0.775276i \(-0.282389\pi\)
−0.775276 + 0.631622i \(0.782389\pi\)
\(44\) 108.593 0.372069
\(45\) −258.657 + 113.290i −0.856851 + 0.375296i
\(46\) 104.825 + 194.113i 0.335991 + 0.622181i
\(47\) 249.160 + 249.160i 0.773269 + 0.773269i 0.978677 − 0.205407i \(-0.0658519\pi\)
−0.205407 + 0.978677i \(0.565852\pi\)
\(48\) −14.9376 + 14.9376i −0.0449180 + 0.0449180i
\(49\) 314.013i 0.915489i
\(50\) 249.797 − 10.0604i 0.706534 − 0.0284552i
\(51\) 125.817i 0.345449i
\(52\) −90.6334 − 90.6334i −0.241704 − 0.241704i
\(53\) −15.7902 − 15.7902i −0.0409237 − 0.0409237i 0.686349 − 0.727273i \(-0.259212\pi\)
−0.727273 + 0.686349i \(0.759212\pi\)
\(54\) − 137.991i − 0.347744i
\(55\) 278.028 − 121.775i 0.681623 − 0.298547i
\(56\) − 43.0719i − 0.102781i
\(57\) 1.06640 − 1.06640i 0.00247804 − 0.00247804i
\(58\) 208.814 − 208.814i 0.472736 − 0.472736i
\(59\) 416.071i 0.918099i 0.888411 + 0.459049i \(0.151810\pi\)
−0.888411 + 0.459049i \(0.848190\pi\)
\(60\) −21.4936 + 54.9953i −0.0462468 + 0.118331i
\(61\) 152.167i 0.319392i 0.987166 + 0.159696i \(0.0510515\pi\)
−0.987166 + 0.159696i \(0.948949\pi\)
\(62\) −154.576 − 154.576i −0.316633 − 0.316633i
\(63\) 96.1540 + 96.1540i 0.192290 + 0.192290i
\(64\) − 64.0000i − 0.125000i
\(65\) −333.681 − 130.411i −0.636739 − 0.248854i
\(66\) 71.6885i 0.133701i
\(67\) −29.3613 + 29.3613i −0.0535381 + 0.0535381i −0.733369 − 0.679831i \(-0.762053\pi\)
0.679831 + 0.733369i \(0.262053\pi\)
\(68\) −269.530 − 269.530i −0.480666 − 0.480666i
\(69\) −128.145 + 69.2009i −0.223577 + 0.120736i
\(70\) −48.3002 − 110.276i −0.0824712 − 0.188293i
\(71\) −711.508 −1.18930 −0.594651 − 0.803984i \(-0.702710\pi\)
−0.594651 + 0.803984i \(0.702710\pi\)
\(72\) 142.874 + 142.874i 0.233859 + 0.233859i
\(73\) 146.668 − 146.668i 0.235153 − 0.235153i −0.579687 − 0.814840i \(-0.696825\pi\)
0.814840 + 0.579687i \(0.196825\pi\)
\(74\) 339.130 0.532744
\(75\) 6.64147 + 164.906i 0.0102252 + 0.253889i
\(76\) 4.56898i 0.00689603i
\(77\) −103.355 − 103.355i −0.152966 − 0.152966i
\(78\) 59.8322 − 59.8322i 0.0868547 − 0.0868547i
\(79\) 392.019 0.558298 0.279149 − 0.960248i \(-0.409948\pi\)
0.279149 + 0.960248i \(0.409948\pi\)
\(80\) −71.7687 − 163.857i −0.100300 − 0.228998i
\(81\) −590.837 −0.810476
\(82\) −374.117 − 374.117i −0.503833 − 0.503833i
\(83\) −697.537 − 697.537i −0.922466 − 0.922466i 0.0747373 − 0.997203i \(-0.476188\pi\)
−0.997203 + 0.0747373i \(0.976188\pi\)
\(84\) 28.4342 0.0369337
\(85\) −992.317 − 387.823i −1.26626 − 0.494886i
\(86\) 114.569i 0.143654i
\(87\) 137.850 + 137.850i 0.169875 + 0.169875i
\(88\) −153.574 − 153.574i −0.186034 − 0.186034i
\(89\) 674.229 0.803013 0.401506 − 0.915856i \(-0.368487\pi\)
0.401506 + 0.915856i \(0.368487\pi\)
\(90\) 526.013 + 205.579i 0.616073 + 0.240777i
\(91\) 172.523i 0.198740i
\(92\) 126.272 − 422.762i 0.143095 − 0.479086i
\(93\) 102.045 − 102.045i 0.113780 − 0.113780i
\(94\) − 704.730i − 0.773269i
\(95\) 5.12359 + 11.6978i 0.00553336 + 0.0126334i
\(96\) 42.2500 0.0449180
\(97\) 705.114 − 705.114i 0.738077 − 0.738077i −0.234129 − 0.972206i \(-0.575224\pi\)
0.972206 + 0.234129i \(0.0752237\pi\)
\(98\) 444.081 − 444.081i 0.457744 − 0.457744i
\(99\) 685.678 0.696093
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.e.a.137.12 yes 72
5.3 odd 4 inner 230.4.e.a.183.11 yes 72
23.22 odd 2 inner 230.4.e.a.137.11 ✓ 72
115.68 even 4 inner 230.4.e.a.183.12 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.e.a.137.11 ✓ 72 23.22 odd 2 inner
230.4.e.a.137.12 yes 72 1.1 even 1 trivial
230.4.e.a.183.11 yes 72 5.3 odd 4 inner
230.4.e.a.183.12 yes 72 115.68 even 4 inner