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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(151,925)] 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("925.151"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([0, 13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.bb (of order \(18\), degree \(6\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [72,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(12\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 176.4
Character \(\chi\) \(=\) 925.176
Dual form 925.2.bb.b.226.4

$q$-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+(-1.67806 + 0.295887i) q^{2} +(-0.576068 + 3.26704i) q^{3} +(0.848938 - 0.308988i) q^{4} -5.65273i q^{6} +(-3.03262 - 2.54467i) q^{7} +(1.61818 - 0.934254i) q^{8} +(-7.52263 - 2.73801i) q^{9} +(-0.0511557 - 0.0886042i) q^{11} +(0.520431 + 2.95151i) q^{12} +(-1.22226 - 3.35813i) q^{13} +(5.84184 + 3.37279i) q^{14} +(-3.82308 + 3.20794i) q^{16} +(-1.83918 + 5.05311i) q^{17} +(13.4335 + 2.36869i) q^{18} +(-1.79373 - 0.316283i) q^{19} +(10.0605 - 8.44180i) q^{21} +(0.112059 + 0.133547i) q^{22} +(1.23928 + 0.715500i) q^{23} +(2.12007 + 5.82484i) q^{24} +(3.04465 + 5.27348i) q^{26} +(8.30258 - 14.3805i) q^{27} +(-3.36078 - 1.22322i) q^{28} +(6.83529 - 3.94636i) q^{29} +5.98103i q^{31} +(3.06404 - 3.65158i) q^{32} +(0.318943 - 0.116086i) q^{33} +(1.59110 - 9.02358i) q^{34} -7.23226 q^{36} +(5.33549 - 2.92105i) q^{37} +3.10357 q^{38} +(11.6753 - 2.05866i) q^{39} +(-2.22936 + 0.811422i) q^{41} +(-14.3843 + 17.1426i) q^{42} +8.74266i q^{43} +(-0.0708056 - 0.0594130i) q^{44} +(-2.29129 - 0.833962i) q^{46} +(2.36106 - 4.08948i) q^{47} +(-8.27813 - 14.3381i) q^{48} +(1.50590 + 8.54040i) q^{49} +(-15.4492 - 8.91961i) q^{51} +(-2.07524 - 2.47318i) q^{52} +(0.334163 - 0.280396i) q^{53} +(-9.67720 + 26.5879i) q^{54} +(-7.28469 - 1.28449i) q^{56} +(2.06662 - 5.67800i) q^{57} +(-10.3023 + 8.64468i) q^{58} +(-2.87715 - 3.42885i) q^{59} +(-0.650273 - 1.78661i) q^{61} +(-1.76971 - 10.0365i) q^{62} +(15.8459 + 27.4460i) q^{63} +(0.929493 - 1.60993i) q^{64} +(-0.500855 + 0.289169i) q^{66} +(3.62636 + 3.04287i) q^{67} +4.85806i q^{68} +(-3.05148 + 3.63661i) q^{69} +(-0.233151 + 1.32227i) q^{71} +(-14.7309 + 2.59746i) q^{72} +9.92546 q^{73} +(-8.08895 + 6.48039i) q^{74} +(-1.62049 + 0.285737i) q^{76} +(-0.0703328 + 0.398877i) q^{77} +(-18.9826 + 6.90910i) q^{78} +(-0.493949 + 0.588665i) q^{79} +(23.8013 + 19.9717i) q^{81} +(3.50091 - 2.02125i) q^{82} +(-5.00073 - 1.82012i) q^{83} +(5.93236 - 10.2751i) q^{84} +(-2.58684 - 14.6707i) q^{86} +(8.95533 + 24.6046i) q^{87} +(-0.165558 - 0.0955848i) q^{88} +(-0.593471 - 0.707272i) q^{89} +(-4.83869 + 13.2942i) q^{91} +(1.27316 + 0.224492i) q^{92} +(-19.5403 - 3.44548i) q^{93} +(-2.75197 + 7.56098i) q^{94} +(10.1648 + 12.1139i) q^{96} +(-6.92255 - 3.99673i) q^{97} +(-5.05398 - 13.8857i) q^{98} +(0.142226 + 0.806601i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 6 q^{2} - 6 q^{4} + 6 q^{7} + 18 q^{8} - 30 q^{12} + 30 q^{13} - 18 q^{14} - 18 q^{16} + 30 q^{18} - 18 q^{19} - 30 q^{22} - 30 q^{24} + 24 q^{27} - 30 q^{28} + 18 q^{29} + 36 q^{32} + 12 q^{33}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{17}{18}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.67806 + 0.295887i −1.18656 + 0.209223i −0.731883 − 0.681431i \(-0.761358\pi\)
−0.454682 + 0.890654i \(0.650247\pi\)
\(3\) −0.576068 + 3.26704i −0.332593 + 1.88623i 0.117222 + 0.993106i \(0.462601\pi\)
−0.449815 + 0.893122i \(0.648510\pi\)
\(4\) 0.848938 − 0.308988i 0.424469 − 0.154494i
\(5\) 0 0
\(6\) − 5.65273i − 2.30772i
\(7\) −3.03262 − 2.54467i −1.14622 − 0.961796i −0.146599 − 0.989196i \(-0.546833\pi\)
−0.999625 + 0.0274005i \(0.991277\pi\)
\(8\) 1.61818 − 0.934254i 0.572112 − 0.330309i
\(9\) −7.52263 − 2.73801i −2.50754 − 0.912671i
\(10\) 0 0
\(11\) −0.0511557 − 0.0886042i −0.0154240 − 0.0267152i 0.858210 − 0.513298i \(-0.171576\pi\)
−0.873634 + 0.486583i \(0.838243\pi\)
\(12\) 0.520431 + 2.95151i 0.150236 + 0.852028i
\(13\) −1.22226 − 3.35813i −0.338994 − 0.931378i −0.985681 − 0.168622i \(-0.946068\pi\)
0.646687 − 0.762756i \(-0.276154\pi\)
\(14\) 5.84184 + 3.37279i 1.56130 + 0.901416i
\(15\) 0 0
\(16\) −3.82308 + 3.20794i −0.955769 + 0.801986i
\(17\) −1.83918 + 5.05311i −0.446067 + 1.22556i 0.489374 + 0.872074i \(0.337225\pi\)
−0.935441 + 0.353484i \(0.884997\pi\)
\(18\) 13.4335 + 2.36869i 3.16631 + 0.558307i
\(19\) −1.79373 − 0.316283i −0.411510 − 0.0725604i −0.0359393 − 0.999354i \(-0.511442\pi\)
−0.375571 + 0.926794i \(0.622553\pi\)
\(20\) 0 0
\(21\) 10.0605 − 8.44180i 2.19539 − 1.84215i
\(22\) 0.112059 + 0.133547i 0.0238910 + 0.0284722i
\(23\) 1.23928 + 0.715500i 0.258408 + 0.149192i 0.623608 − 0.781737i \(-0.285666\pi\)
−0.365200 + 0.930929i \(0.618999\pi\)
\(24\) 2.12007 + 5.82484i 0.432757 + 1.18899i
\(25\) 0 0
\(26\) 3.04465 + 5.27348i 0.597104 + 1.03421i
\(27\) 8.30258 − 14.3805i 1.59783 − 2.76753i
\(28\) −3.36078 − 1.22322i −0.635128 − 0.231168i
\(29\) 6.83529 − 3.94636i 1.26928 − 0.732820i 0.294430 − 0.955673i \(-0.404870\pi\)
0.974852 + 0.222853i \(0.0715368\pi\)
\(30\) 0 0
\(31\) 5.98103i 1.07423i 0.843511 + 0.537113i \(0.180485\pi\)
−0.843511 + 0.537113i \(0.819515\pi\)
\(32\) 3.06404 − 3.65158i 0.541651 − 0.645515i
\(33\) 0.318943 − 0.116086i 0.0555208 − 0.0202079i
\(34\) 1.59110 − 9.02358i 0.272872 − 1.54753i
\(35\) 0 0
\(36\) −7.23226 −1.20538
\(37\) 5.33549 − 2.92105i 0.877149 − 0.480218i
\(38\) 3.10357 0.503465
\(39\) 11.6753 − 2.05866i 1.86954 − 0.329650i
\(40\) 0 0
\(41\) −2.22936 + 0.811422i −0.348168 + 0.126723i −0.510184 − 0.860065i \(-0.670423\pi\)
0.162016 + 0.986788i \(0.448200\pi\)
\(42\) −14.3843 + 17.1426i −2.21955 + 2.64516i
\(43\) 8.74266i 1.33324i 0.745396 + 0.666621i \(0.232260\pi\)
−0.745396 + 0.666621i \(0.767740\pi\)
\(44\) −0.0708056 − 0.0594130i −0.0106743 − 0.00895684i
\(45\) 0 0
\(46\) −2.29129 − 0.833962i −0.337833 − 0.122961i
\(47\) 2.36106 − 4.08948i 0.344396 − 0.596512i −0.640848 − 0.767668i \(-0.721417\pi\)
0.985244 + 0.171156i \(0.0547502\pi\)
\(48\) −8.27813 − 14.3381i −1.19485 − 2.06953i
\(49\) 1.50590 + 8.54040i 0.215129 + 1.22006i
\(50\) 0 0
\(51\) −15.4492 − 8.91961i −2.16332 − 1.24899i
\(52\) −2.07524 − 2.47318i −0.287785 − 0.342968i
\(53\) 0.334163 − 0.280396i 0.0459008 − 0.0385153i −0.619549 − 0.784958i \(-0.712684\pi\)
0.665449 + 0.746443i \(0.268240\pi\)
\(54\) −9.67720 + 26.5879i −1.31690 + 3.61815i
\(55\) 0 0
\(56\) −7.28469 − 1.28449i −0.973457 − 0.171647i
\(57\) 2.06662 − 5.67800i 0.273731 − 0.752069i
\(58\) −10.3023 + 8.64468i −1.35276 + 1.13510i
\(59\) −2.87715 − 3.42885i −0.374573 − 0.446398i 0.545521 − 0.838097i \(-0.316332\pi\)
−0.920093 + 0.391699i \(0.871887\pi\)
\(60\) 0 0
\(61\) −0.650273 − 1.78661i −0.0832589 − 0.228752i 0.891076 − 0.453853i \(-0.149951\pi\)
−0.974335 + 0.225101i \(0.927729\pi\)
\(62\) −1.76971 − 10.0365i −0.224753 − 1.27464i
\(63\) 15.8459 + 27.4460i 1.99640 + 3.45787i
\(64\) 0.929493 − 1.60993i 0.116187 − 0.201241i
\(65\) 0 0
\(66\) −0.500855 + 0.289169i −0.0616510 + 0.0355942i
\(67\) 3.62636 + 3.04287i 0.443030 + 0.371746i 0.836842 − 0.547445i \(-0.184399\pi\)
−0.393812 + 0.919191i \(0.628844\pi\)
\(68\) 4.85806i 0.589126i
\(69\) −3.05148 + 3.63661i −0.367355 + 0.437797i
\(70\) 0 0
\(71\) −0.233151 + 1.32227i −0.0276699 + 0.156924i −0.995512 − 0.0946341i \(-0.969832\pi\)
0.967842 + 0.251558i \(0.0809430\pi\)
\(72\) −14.7309 + 2.59746i −1.73606 + 0.306114i
\(73\) 9.92546 1.16169 0.580844 − 0.814015i \(-0.302723\pi\)
0.580844 + 0.814015i \(0.302723\pi\)
\(74\) −8.08895 + 6.48039i −0.940321 + 0.753330i
\(75\) 0 0
\(76\) −1.62049 + 0.285737i −0.185884 + 0.0327763i
\(77\) −0.0703328 + 0.398877i −0.00801517 + 0.0454563i
\(78\) −18.9826 + 6.90910i −2.14936 + 0.782302i
\(79\) −0.493949 + 0.588665i −0.0555736 + 0.0662300i −0.793115 − 0.609072i \(-0.791542\pi\)
0.737542 + 0.675302i \(0.235987\pi\)
\(80\) 0 0
\(81\) 23.8013 + 19.9717i 2.64459 + 2.21908i
\(82\) 3.50091 − 2.02125i 0.386611 − 0.223210i
\(83\) −5.00073 − 1.82012i −0.548901 − 0.199784i 0.0526569 − 0.998613i \(-0.483231\pi\)
−0.601558 + 0.798829i \(0.705453\pi\)
\(84\) 5.93236 − 10.2751i 0.647274 − 1.12111i
\(85\) 0 0
\(86\) −2.58684 − 14.6707i −0.278946 − 1.58198i
\(87\) 8.95533 + 24.6046i 0.960112 + 2.63789i
\(88\) −0.165558 − 0.0955848i −0.0176485 − 0.0101894i
\(89\) −0.593471 − 0.707272i −0.0629078 − 0.0749706i 0.733673 − 0.679503i \(-0.237804\pi\)
−0.796581 + 0.604532i \(0.793360\pi\)
\(90\) 0 0
\(91\) −4.83869 + 13.2942i −0.507233 + 1.39361i
\(92\) 1.27316 + 0.224492i 0.132736 + 0.0234049i
\(93\) −19.5403 − 3.44548i −2.02623 − 0.357280i
\(94\) −2.75197 + 7.56098i −0.283844 + 0.779856i
\(95\) 0 0
\(96\) 10.1648 + 12.1139i 1.03744 + 1.23637i
\(97\) −6.92255 − 3.99673i −0.702878 − 0.405807i 0.105540 − 0.994415i \(-0.466343\pi\)
−0.808418 + 0.588608i \(0.799676\pi\)
\(98\) −5.05398 − 13.8857i −0.510529 − 1.40267i
\(99\) 0.142226 + 0.806601i 0.0142942 + 0.0810665i
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 925.2.bb.b.176.4 72
5.2 odd 4 925.2.ba.b.324.3 72
5.3 odd 4 925.2.ba.c.324.10 72
5.4 even 2 185.2.w.a.176.9 yes 72
37.4 even 18 inner 925.2.bb.b.226.4 72
185.4 even 18 185.2.w.a.41.9 ✓ 72
185.39 odd 36 6845.2.a.t.1.29 36
185.78 odd 36 925.2.ba.b.374.3 72
185.109 odd 36 6845.2.a.u.1.8 36
185.152 odd 36 925.2.ba.c.374.10 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.41.9 ✓ 72 185.4 even 18
185.2.w.a.176.9 yes 72 5.4 even 2
925.2.ba.b.324.3 72 5.2 odd 4
925.2.ba.b.374.3 72 185.78 odd 36
925.2.ba.c.324.10 72 5.3 odd 4
925.2.ba.c.374.10 72 185.152 odd 36
925.2.bb.b.176.4 72 1.1 even 1 trivial
925.2.bb.b.226.4 72 37.4 even 18 inner
6845.2.a.t.1.29 36 185.39 odd 36
6845.2.a.u.1.8 36 185.109 odd 36