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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 137.19
Character \(\chi\) \(=\) 230.137
Dual form 230.4.e.a.183.19

$q$-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} +(-6.86623 + 6.86623i) q^{3} +4.00000i q^{4} +(-11.1776 + 0.245509i) q^{5} -19.4206 q^{6} +(21.1410 - 21.1410i) q^{7} +(-5.65685 + 5.65685i) q^{8} -67.2901i q^{9} +(-16.1548 - 15.4604i) q^{10} -16.2716i q^{11} +(-27.4649 - 27.4649i) q^{12} +(-38.7433 + 38.7433i) q^{13} +59.7959 q^{14} +(75.0625 - 78.4339i) q^{15} -16.0000 q^{16} +(-1.95359 + 1.95359i) q^{17} +(95.1626 - 95.1626i) q^{18} -44.9887 q^{19} +(-0.982037 - 44.7106i) q^{20} +290.318i q^{21} +(23.0115 - 23.0115i) q^{22} +(110.037 - 7.67471i) q^{23} -77.6825i q^{24} +(124.879 - 5.48843i) q^{25} -109.583 q^{26} +(276.641 + 276.641i) q^{27} +(84.5642 + 84.5642i) q^{28} -182.608i q^{29} +(217.077 - 4.76794i) q^{30} +36.1499 q^{31} +(-22.6274 - 22.6274i) q^{32} +(111.724 + 111.724i) q^{33} -5.52559 q^{34} +(-231.117 + 241.497i) q^{35} +269.160 q^{36} +(123.788 - 123.788i) q^{37} +(-63.6237 - 63.6237i) q^{38} -532.040i q^{39} +(61.8415 - 64.6191i) q^{40} -77.5118 q^{41} +(-410.572 + 410.572i) q^{42} +(-287.818 - 287.818i) q^{43} +65.0862 q^{44} +(16.5203 + 752.145i) q^{45} +(166.469 + 144.762i) q^{46} +(-210.857 - 210.857i) q^{47} +(109.860 - 109.860i) q^{48} -550.888i q^{49} +(184.368 + 168.844i) q^{50} -26.8276i q^{51} +(-154.973 - 154.973i) q^{52} +(402.600 + 402.600i) q^{53} +782.458i q^{54} +(3.99482 + 181.878i) q^{55} +239.184i q^{56} +(308.903 - 308.903i) q^{57} +(258.247 - 258.247i) q^{58} +429.087i q^{59} +(313.736 + 300.250i) q^{60} -794.321i q^{61} +(51.1237 + 51.1237i) q^{62} +(-1422.58 - 1422.58i) q^{63} -64.0000i q^{64} +(423.547 - 442.571i) q^{65} +316.004i q^{66} +(687.815 - 687.815i) q^{67} +(-7.81436 - 7.81436i) q^{68} +(-702.841 + 808.234i) q^{69} +(-668.377 + 14.6804i) q^{70} -734.032 q^{71} +(380.650 + 380.650i) q^{72} +(470.498 - 470.498i) q^{73} +350.126 q^{74} +(-819.766 + 895.135i) q^{75} -179.955i q^{76} +(-343.998 - 343.998i) q^{77} +(752.419 - 752.419i) q^{78} -294.624 q^{79} +(178.842 - 3.92815i) q^{80} -1982.12 q^{81} +(-109.618 - 109.618i) q^{82} +(247.412 + 247.412i) q^{83} -1161.27 q^{84} +(21.3569 - 22.3162i) q^{85} -814.073i q^{86} +(1253.83 + 1253.83i) q^{87} +(92.0458 + 92.0458i) q^{88} -53.6489 q^{89} +(-1040.33 + 1087.06i) q^{90} +1638.15i q^{91} +(30.6989 + 440.147i) q^{92} +(-248.214 + 248.214i) q^{93} -596.392i q^{94} +(502.868 - 11.0452i) q^{95} +310.730 q^{96} +(1270.56 - 1270.56i) q^{97} +(779.073 - 779.073i) q^{98} -1094.91 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
\(3\) −6.86623 + 6.86623i −1.32141 + 1.32141i −0.408767 + 0.912639i \(0.634041\pi\)
−0.912639 + 0.408767i \(0.865959\pi\)
\(4\) 4.00000i 0.500000i
\(5\) −11.1776 + 0.245509i −0.999759 + 0.0219590i
\(6\) −19.4206 −1.32141
\(7\) 21.1410 − 21.1410i 1.14151 − 1.14151i 0.153335 − 0.988174i \(-0.450999\pi\)
0.988174 − 0.153335i \(-0.0490013\pi\)
\(8\) −5.65685 + 5.65685i −0.250000 + 0.250000i
\(9\) − 67.2901i − 2.49223i
\(10\) −16.1548 − 15.4604i −0.510859 − 0.488900i
\(11\) − 16.2716i − 0.446005i −0.974818 − 0.223003i \(-0.928414\pi\)
0.974818 − 0.223003i \(-0.0715859\pi\)
\(12\) −27.4649 − 27.4649i −0.660703 − 0.660703i
\(13\) −38.7433 + 38.7433i −0.826574 + 0.826574i −0.987041 − 0.160467i \(-0.948700\pi\)
0.160467 + 0.987041i \(0.448700\pi\)
\(14\) 59.7959 1.14151
\(15\) 75.0625 − 78.4339i 1.29207 − 1.35010i
\(16\) −16.0000 −0.250000
\(17\) −1.95359 + 1.95359i −0.0278715 + 0.0278715i −0.720905 − 0.693034i \(-0.756274\pi\)
0.693034 + 0.720905i \(0.256274\pi\)
\(18\) 95.1626 − 95.1626i 1.24611 − 1.24611i
\(19\) −44.9887 −0.543217 −0.271609 − 0.962408i \(-0.587556\pi\)
−0.271609 + 0.962408i \(0.587556\pi\)
\(20\) −0.982037 − 44.7106i −0.0109795 − 0.499879i
\(21\) 290.318i 3.01679i
\(22\) 23.0115 − 23.0115i 0.223003 − 0.223003i
\(23\) 110.037 − 7.67471i 0.997577 − 0.0695777i
\(24\) − 77.6825i − 0.660703i
\(25\) 124.879 − 5.48843i 0.999036 − 0.0439074i
\(26\) −109.583 −0.826574
\(27\) 276.641 + 276.641i 1.97184 + 1.97184i
\(28\) 84.5642 + 84.5642i 0.570755 + 0.570755i
\(29\) − 182.608i − 1.16929i −0.811288 − 0.584647i \(-0.801233\pi\)
0.811288 − 0.584647i \(-0.198767\pi\)
\(30\) 217.077 − 4.76794i 1.32109 − 0.0290168i
\(31\) 36.1499 0.209443 0.104721 − 0.994502i \(-0.466605\pi\)
0.104721 + 0.994502i \(0.466605\pi\)
\(32\) −22.6274 − 22.6274i −0.125000 − 0.125000i
\(33\) 111.724 + 111.724i 0.589354 + 0.589354i
\(34\) −5.52559 −0.0278715
\(35\) −231.117 + 241.497i −1.11617 + 1.16630i
\(36\) 269.160 1.24611
\(37\) 123.788 − 123.788i 0.550017 − 0.550017i −0.376428 − 0.926446i \(-0.622848\pi\)
0.926446 + 0.376428i \(0.122848\pi\)
\(38\) −63.6237 − 63.6237i −0.271609 − 0.271609i
\(39\) − 532.040i − 2.18448i
\(40\) 61.8415 − 64.6191i 0.244450 − 0.255429i
\(41\) −77.5118 −0.295251 −0.147626 − 0.989043i \(-0.547163\pi\)
−0.147626 + 0.989043i \(0.547163\pi\)
\(42\) −410.572 + 410.572i −1.50840 + 1.50840i
\(43\) −287.818 − 287.818i −1.02074 − 1.02074i −0.999780 − 0.0209612i \(-0.993327\pi\)
−0.0209612 − 0.999780i \(-0.506673\pi\)
\(44\) 65.0862 0.223003
\(45\) 16.5203 + 752.145i 0.0547268 + 2.49162i
\(46\) 166.469 + 144.762i 0.533577 + 0.463999i
\(47\) −210.857 − 210.857i −0.654396 − 0.654396i 0.299653 − 0.954048i \(-0.403129\pi\)
−0.954048 + 0.299653i \(0.903129\pi\)
\(48\) 109.860 − 109.860i 0.330351 − 0.330351i
\(49\) − 550.888i − 1.60609i
\(50\) 184.368 + 168.844i 0.521472 + 0.477564i
\(51\) − 26.8276i − 0.0736591i
\(52\) −154.973 − 154.973i −0.413287 − 0.413287i
\(53\) 402.600 + 402.600i 1.04342 + 1.04342i 0.999013 + 0.0444090i \(0.0141405\pi\)
0.0444090 + 0.999013i \(0.485860\pi\)
\(54\) 782.458i 1.97184i
\(55\) 3.99482 + 181.878i 0.00979384 + 0.445898i
\(56\) 239.184i 0.570755i
\(57\) 308.903 − 308.903i 0.717810 − 0.717810i
\(58\) 258.247 − 258.247i 0.584647 − 0.584647i
\(59\) 429.087i 0.946819i 0.880842 + 0.473409i \(0.156977\pi\)
−0.880842 + 0.473409i \(0.843023\pi\)
\(60\) 313.736 + 300.250i 0.675052 + 0.646035i
\(61\) − 794.321i − 1.66725i −0.552329 − 0.833626i \(-0.686261\pi\)
0.552329 − 0.833626i \(-0.313739\pi\)
\(62\) 51.1237 + 51.1237i 0.104721 + 0.104721i
\(63\) −1422.58 − 1422.58i −2.84490 − 2.84490i
\(64\) − 64.0000i − 0.125000i
\(65\) 423.547 − 442.571i 0.808224 − 0.844525i
\(66\) 316.004i 0.589354i
\(67\) 687.815 − 687.815i 1.25418 − 1.25418i 0.300350 − 0.953829i \(-0.402896\pi\)
0.953829 − 0.300350i \(-0.0971036\pi\)
\(68\) −7.81436 − 7.81436i −0.0139357 − 0.0139357i
\(69\) −702.841 + 808.234i −1.22626 + 1.41014i
\(70\) −668.377 + 14.6804i −1.14123 + 0.0250664i
\(71\) −734.032 −1.22695 −0.613476 − 0.789713i \(-0.710229\pi\)
−0.613476 + 0.789713i \(0.710229\pi\)
\(72\) 380.650 + 380.650i 0.623056 + 0.623056i
\(73\) 470.498 − 470.498i 0.754352 − 0.754352i −0.220937 − 0.975288i \(-0.570911\pi\)
0.975288 + 0.220937i \(0.0709114\pi\)
\(74\) 350.126 0.550017
\(75\) −819.766 + 895.135i −1.26211 + 1.37815i
\(76\) − 179.955i − 0.271609i
\(77\) −343.998 − 343.998i −0.509119 − 0.509119i
\(78\) 752.419 − 752.419i 1.09224 − 1.09224i
\(79\) −294.624 −0.419592 −0.209796 − 0.977745i \(-0.567280\pi\)
−0.209796 + 0.977745i \(0.567280\pi\)
\(80\) 178.842 − 3.92815i 0.249940 − 0.00548975i
\(81\) −1982.12 −2.71896
\(82\) −109.618 − 109.618i −0.147626 − 0.147626i
\(83\) 247.412 + 247.412i 0.327193 + 0.327193i 0.851518 − 0.524325i \(-0.175682\pi\)
−0.524325 + 0.851518i \(0.675682\pi\)
\(84\) −1161.27 −1.50840
\(85\) 21.3569 − 22.3162i 0.0272527 − 0.0284768i
\(86\) − 814.073i − 1.02074i
\(87\) 1253.83 + 1253.83i 1.54511 + 1.54511i
\(88\) 92.0458 + 92.0458i 0.111501 + 0.111501i
\(89\) −53.6489 −0.0638963 −0.0319481 − 0.999490i \(-0.510171\pi\)
−0.0319481 + 0.999490i \(0.510171\pi\)
\(90\) −1040.33 + 1087.06i −1.21845 + 1.27318i
\(91\) 1638.15i 1.88708i
\(92\) 30.6989 + 440.147i 0.0347889 + 0.498788i
\(93\) −248.214 + 248.214i −0.276759 + 0.276759i
\(94\) − 596.392i − 0.654396i
\(95\) 502.868 − 11.0452i 0.543086 − 0.0119285i
\(96\) 310.730 0.330351
\(97\) 1270.56 − 1270.56i 1.32996 − 1.32996i 0.424554 − 0.905402i \(-0.360431\pi\)
0.905402 − 0.424554i \(-0.139569\pi\)
\(98\) 779.073 − 779.073i 0.803043 − 0.803043i
\(99\) −1094.91 −1.11155
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.137.19 ✓ 72
5.3 odd 4 inner 230.4.e.a.183.20 yes 72
23.22 odd 2 inner 230.4.e.a.137.20 yes 72
115.68 even 4 inner 230.4.e.a.183.19 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.e.a.137.19 ✓ 72 1.1 even 1 trivial
230.4.e.a.137.20 yes 72 23.22 odd 2 inner
230.4.e.a.183.19 yes 72 115.68 even 4 inner
230.4.e.a.183.20 yes 72 5.3 odd 4 inner