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(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 183.20
Character \(\chi\) \(=\) 230.183
Dual form 230.4.e.a.137.20

$q$-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} +(7.67471 + 110.037i) 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} +(440.147 - 30.6989i) 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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\)

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 −0.912639 − 0.408767i \(-0.865959\pi\)
−0.408767 − 0.912639i \(-0.634041\pi\)
\(4\) − 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 −0.988174 − 0.153335i \(-0.950999\pi\)
−0.153335 − 0.988174i \(-0.549001\pi\)
\(8\) −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 0.160467 − 0.987041i \(-0.448700\pi\)
−0.987041 + 0.160467i \(0.948700\pi\)
\(14\) −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 \(-0.243726\pi\)
−0.693034 + 0.720905i \(0.743726\pi\)
\(18\) 95.1626 + 95.1626i 1.24611 + 1.24611i
\(19\) 44.9887 0.543217 0.271609 − 0.962408i \(-0.412444\pi\)
0.271609 + 0.962408i \(0.412444\pi\)
\(20\) 0.982037 − 44.7106i 0.0109795 − 0.499879i
\(21\) 290.318i 3.01679i
\(22\) −23.0115 − 23.0115i −0.223003 − 0.223003i
\(23\) 7.67471 + 110.037i 0.0695777 + 0.997577i
\(24\) 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 \(0.198767\pi\)
−0.811288 + 0.584647i \(0.801233\pi\)
\(30\) −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 \(-0.377152\pi\)
−0.926446 + 0.376428i \(0.877152\pi\)
\(38\) 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 0.0209612 − 0.999780i \(-0.493327\pi\)
0.999780 − 0.0209612i \(-0.00667265\pi\)
\(44\) −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 −0.954048 − 0.299653i \(-0.903129\pi\)
0.299653 + 0.954048i \(0.403129\pi\)
\(48\) 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 −0.0444090 + 0.999013i \(0.514140\pi\)
−0.999013 + 0.0444090i \(0.985860\pi\)
\(54\) − 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 \(-0.843023\pi\)
0.880842 − 0.473409i \(-0.156977\pi\)
\(60\) −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 −0.953829 − 0.300350i \(-0.902896\pi\)
−0.300350 − 0.953829i \(-0.597104\pi\)
\(68\) 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 0.975288 − 0.220937i \(-0.0709114\pi\)
−0.220937 + 0.975288i \(0.570911\pi\)
\(74\) −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 \(-0.432720\pi\)
0.209796 + 0.977745i \(0.432720\pi\)
\(80\) −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 \(-0.824318\pi\)
0.524325 + 0.851518i \(0.324318\pi\)
\(84\) 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 \(-0.489829\pi\)
0.0319481 + 0.999490i \(0.489829\pi\)
\(90\) 1040.33 + 1087.06i 1.21845 + 1.27318i
\(91\) 1638.15i 1.88708i
\(92\) 440.147 − 30.6989i 0.498788 − 0.0347889i
\(93\) −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 −0.905402 − 0.424554i \(-0.860431\pi\)
−0.424554 − 0.905402i \(-0.639569\pi\)
\(98\) 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.183.20 yes 72
5.2 odd 4 inner 230.4.e.a.137.19 ✓ 72
23.22 odd 2 inner 230.4.e.a.183.19 yes 72
115.22 even 4 inner 230.4.e.a.137.20 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.e.a.137.19 ✓ 72 5.2 odd 4 inner
230.4.e.a.137.20 yes 72 115.22 even 4 inner
230.4.e.a.183.19 yes 72 23.22 odd 2 inner
230.4.e.a.183.20 yes 72 1.1 even 1 trivial