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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 137.20
Character \(\chi\) \(=\) 230.137
Dual form 230.4.e.a.183.20

$q$-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 \(\left(\mathbb{Z}/230\mathbb{Z}\right)^\times\).

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

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 \(0.549001\pi\)
−0.988174 + 0.153335i \(0.950999\pi\)
\(8\) −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 \(0.0715859\pi\)
−0.974818 + 0.223003i \(0.928414\pi\)
\(12\) −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 −0.693034 − 0.720905i \(-0.743726\pi\)
0.720905 + 0.693034i \(0.243726\pi\)
\(18\) 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 \(-0.801233\pi\)
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 −0.926446 − 0.376428i \(-0.877152\pi\)
0.376428 + 0.926446i \(0.377152\pi\)
\(38\) 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 \(0.00667265\pi\)
0.0209612 + 0.999780i \(0.493327\pi\)
\(44\) −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 \(-0.985860\pi\)
−0.0444090 − 0.999013i \(-0.514140\pi\)
\(54\) 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 \(0.313739\pi\)
−0.552329 + 0.833626i \(0.686261\pi\)
\(62\) 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 \(0.597104\pi\)
−0.953829 + 0.300350i \(0.902896\pi\)
\(68\) 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 \(-0.432720\pi\)
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 0.524325 − 0.851518i \(-0.324318\pi\)
−0.851518 + 0.524325i \(0.824318\pi\)
\(84\) 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 −0.424554 + 0.905402i \(0.639569\pi\)
−0.905402 + 0.424554i \(0.860431\pi\)
\(98\) 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.137.20 yes 72
5.3 odd 4 inner 230.4.e.a.183.19 yes 72
23.22 odd 2 inner 230.4.e.a.137.19 ✓ 72
115.68 even 4 inner 230.4.e.a.183.20 yes 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.4.e.a.137.19 ✓ 72 23.22 odd 2 inner
230.4.e.a.137.20 yes 72 1.1 even 1 trivial
230.4.e.a.183.19 yes 72 5.3 odd 4 inner
230.4.e.a.183.20 yes 72 115.68 even 4 inner