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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 151.2
Character \(\chi\) \(=\) 925.151
Dual form 925.2.bb.b.876.2

$q$-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+(-1.29158 + 1.53924i) q^{2} +(-0.583938 + 0.489982i) q^{3} +(-0.353796 - 2.00647i) q^{4} -1.53167i q^{6} +(1.63114 + 0.593686i) q^{7} +(0.0651292 + 0.0376024i) q^{8} +(-0.420044 + 2.38219i) q^{9} +(-0.381991 + 0.661628i) q^{11} +(1.18973 + 0.998303i) q^{12} +(0.0386391 - 0.00681312i) q^{13} +(-3.02057 + 1.74393i) q^{14} +(3.68711 - 1.34200i) q^{16} +(4.30304 + 0.758742i) q^{17} +(-3.12424 - 3.72332i) q^{18} +(2.89648 + 3.45189i) q^{19} +(-1.24338 + 0.452553i) q^{21} +(-0.525034 - 1.44252i) q^{22} +(-1.58218 + 0.913472i) q^{23} +(-0.0564559 + 0.00995470i) q^{24} +(-0.0394183 + 0.0682745i) q^{26} +(-2.06536 - 3.57731i) q^{27} +(0.614127 - 3.48288i) q^{28} +(-4.07808 - 2.35448i) q^{29} +6.00718i q^{31} +(-2.74797 + 7.54998i) q^{32} +(-0.101127 - 0.573518i) q^{33} +(-6.72559 + 5.64344i) q^{34} +4.92840 q^{36} +(1.93907 + 5.76541i) q^{37} -9.05431 q^{38} +(-0.0192245 + 0.0229109i) q^{39} +(1.01550 + 5.75920i) q^{41} +(0.909331 - 2.49837i) q^{42} -3.43382i q^{43} +(1.46269 + 0.532374i) q^{44} +(0.637453 - 3.61517i) q^{46} +(1.81560 + 3.14471i) q^{47} +(-1.49549 + 2.59026i) q^{48} +(-3.05416 - 2.56274i) q^{49} +(-2.88448 + 1.66535i) q^{51} +(-0.0273407 - 0.0751179i) q^{52} +(-11.2190 + 4.08338i) q^{53} +(8.17392 + 1.44128i) q^{54} +(0.0839109 + 0.100001i) q^{56} +(-3.38273 - 0.596466i) q^{57} +(8.89125 - 3.23615i) q^{58} +(-2.59304 - 7.12432i) q^{59} +(-5.71230 + 1.00723i) q^{61} +(-9.24649 - 7.75873i) q^{62} +(-2.09942 + 3.63630i) q^{63} +(-4.14828 - 7.18504i) q^{64} +(1.01339 + 0.585084i) q^{66} +(-0.268567 - 0.0977503i) q^{67} -8.90238i q^{68} +(0.476310 - 1.30865i) q^{69} +(-4.33845 + 3.64039i) q^{71} +(-0.116933 + 0.139355i) q^{72} +4.13245 q^{73} +(-11.3788 - 4.46177i) q^{74} +(5.90137 - 7.03298i) q^{76} +(-1.01588 + 0.852425i) q^{77} +(-0.0104354 - 0.0591823i) q^{78} +(-0.799712 + 2.19719i) q^{79} +(-3.86030 - 1.40504i) q^{81} +(-10.1764 - 5.87534i) q^{82} +(-0.460892 + 2.61385i) q^{83} +(1.34794 + 2.33470i) q^{84} +(5.28548 + 4.43504i) q^{86} +(3.53499 - 0.623315i) q^{87} +(-0.0497576 + 0.0287276i) q^{88} +(-5.87216 - 16.1336i) q^{89} +(0.0670707 + 0.0118264i) q^{91} +(2.39263 + 2.85142i) q^{92} +(-2.94341 - 3.50782i) q^{93} +(-7.18544 - 1.26699i) q^{94} +(-2.09471 - 5.75518i) q^{96} +(12.8808 - 7.43673i) q^{97} +(7.88935 - 1.39111i) q^{98} +(-1.41567 - 1.18789i) 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 \(\left(\mathbb{Z}/925\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{13}{18}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.29158 + 1.53924i −0.913282 + 1.08841i 0.0824941 + 0.996592i \(0.473711\pi\)
−0.995776 + 0.0918155i \(0.970733\pi\)
\(3\) −0.583938 + 0.489982i −0.337137 + 0.282891i −0.795600 − 0.605822i \(-0.792844\pi\)
0.458464 + 0.888713i \(0.348400\pi\)
\(4\) −0.353796 − 2.00647i −0.176898 − 1.00324i
\(5\) 0 0
\(6\) − 1.53167i − 0.625301i
\(7\) 1.63114 + 0.593686i 0.616513 + 0.224392i 0.631351 − 0.775498i \(-0.282501\pi\)
−0.0148376 + 0.999890i \(0.504723\pi\)
\(8\) 0.0651292 + 0.0376024i 0.0230267 + 0.0132945i
\(9\) −0.420044 + 2.38219i −0.140015 + 0.794062i
\(10\) 0 0
\(11\) −0.381991 + 0.661628i −0.115175 + 0.199488i −0.917850 − 0.396928i \(-0.870076\pi\)
0.802675 + 0.596417i \(0.203409\pi\)
\(12\) 1.18973 + 0.998303i 0.343446 + 0.288185i
\(13\) 0.0386391 − 0.00681312i 0.0107166 − 0.00188962i −0.168287 − 0.985738i \(-0.553824\pi\)
0.179004 + 0.983848i \(0.442713\pi\)
\(14\) −3.02057 + 1.74393i −0.807280 + 0.466084i
\(15\) 0 0
\(16\) 3.68711 − 1.34200i 0.921778 − 0.335500i
\(17\) 4.30304 + 0.758742i 1.04364 + 0.184022i 0.669088 − 0.743184i \(-0.266685\pi\)
0.374553 + 0.927206i \(0.377796\pi\)
\(18\) −3.12424 − 3.72332i −0.736390 − 0.877595i
\(19\) 2.89648 + 3.45189i 0.664498 + 0.791918i 0.988024 − 0.154301i \(-0.0493127\pi\)
−0.323526 + 0.946219i \(0.604868\pi\)
\(20\) 0 0
\(21\) −1.24338 + 0.452553i −0.271328 + 0.0987552i
\(22\) −0.525034 − 1.44252i −0.111938 − 0.307546i
\(23\) −1.58218 + 0.913472i −0.329907 + 0.190472i −0.655800 − 0.754935i \(-0.727669\pi\)
0.325893 + 0.945407i \(0.394335\pi\)
\(24\) −0.0564559 + 0.00995470i −0.0115240 + 0.00203199i
\(25\) 0 0
\(26\) −0.0394183 + 0.0682745i −0.00773057 + 0.0133897i
\(27\) −2.06536 − 3.57731i −0.397479 − 0.688454i
\(28\) 0.614127 − 3.48288i 0.116059 − 0.658203i
\(29\) −4.07808 − 2.35448i −0.757280 − 0.437216i 0.0710385 − 0.997474i \(-0.477369\pi\)
−0.828318 + 0.560258i \(0.810702\pi\)
\(30\) 0 0
\(31\) 6.00718i 1.07892i 0.842011 + 0.539461i \(0.181372\pi\)
−0.842011 + 0.539461i \(0.818628\pi\)
\(32\) −2.74797 + 7.54998i −0.485777 + 1.33466i
\(33\) −0.101127 − 0.573518i −0.0176039 − 0.0998367i
\(34\) −6.72559 + 5.64344i −1.15343 + 0.967842i
\(35\) 0 0
\(36\) 4.92840 0.821401
\(37\) 1.93907 + 5.76541i 0.318782 + 0.947828i
\(38\) −9.05431 −1.46880
\(39\) −0.0192245 + 0.0229109i −0.00307839 + 0.00366868i
\(40\) 0 0
\(41\) 1.01550 + 5.75920i 0.158595 + 0.899436i 0.955425 + 0.295233i \(0.0953973\pi\)
−0.796830 + 0.604203i \(0.793492\pi\)
\(42\) 0.909331 − 2.49837i 0.140313 − 0.385506i
\(43\) − 3.43382i − 0.523653i −0.965115 − 0.261827i \(-0.915675\pi\)
0.965115 − 0.261827i \(-0.0843248\pi\)
\(44\) 1.46269 + 0.532374i 0.220508 + 0.0802584i
\(45\) 0 0
\(46\) 0.637453 − 3.61517i 0.0939873 − 0.533028i
\(47\) 1.81560 + 3.14471i 0.264832 + 0.458703i 0.967520 − 0.252796i \(-0.0813501\pi\)
−0.702688 + 0.711499i \(0.748017\pi\)
\(48\) −1.49549 + 2.59026i −0.215855 + 0.373872i
\(49\) −3.05416 − 2.56274i −0.436308 − 0.366106i
\(50\) 0 0
\(51\) −2.88448 + 1.66535i −0.403907 + 0.233196i
\(52\) −0.0273407 − 0.0751179i −0.00379147 − 0.0104170i
\(53\) −11.2190 + 4.08338i −1.54105 + 0.560895i −0.966296 − 0.257434i \(-0.917123\pi\)
−0.574750 + 0.818329i \(0.694901\pi\)
\(54\) 8.17392 + 1.44128i 1.11233 + 0.196134i
\(55\) 0 0
\(56\) 0.0839109 + 0.100001i 0.0112131 + 0.0133632i
\(57\) −3.38273 − 0.596466i −0.448053 − 0.0790039i
\(58\) 8.89125 − 3.23615i 1.16748 − 0.424927i
\(59\) −2.59304 − 7.12432i −0.337585 − 0.927507i −0.986078 − 0.166286i \(-0.946822\pi\)
0.648492 − 0.761221i \(-0.275400\pi\)
\(60\) 0 0
\(61\) −5.71230 + 1.00723i −0.731385 + 0.128963i −0.526925 − 0.849912i \(-0.676655\pi\)
−0.204460 + 0.978875i \(0.565544\pi\)
\(62\) −9.24649 − 7.75873i −1.17431 − 0.985359i
\(63\) −2.09942 + 3.63630i −0.264502 + 0.458131i
\(64\) −4.14828 − 7.18504i −0.518535 − 0.898130i
\(65\) 0 0
\(66\) 1.01339 + 0.585084i 0.124740 + 0.0720188i
\(67\) −0.268567 − 0.0977503i −0.0328106 − 0.0119421i 0.325563 − 0.945520i \(-0.394446\pi\)
−0.358373 + 0.933578i \(0.616669\pi\)
\(68\) − 8.90238i − 1.07957i
\(69\) 0.476310 − 1.30865i 0.0573410 − 0.157543i
\(70\) 0 0
\(71\) −4.33845 + 3.64039i −0.514879 + 0.432035i −0.862842 − 0.505473i \(-0.831318\pi\)
0.347963 + 0.937508i \(0.386874\pi\)
\(72\) −0.116933 + 0.139355i −0.0137807 + 0.0164232i
\(73\) 4.13245 0.483667 0.241834 − 0.970318i \(-0.422251\pi\)
0.241834 + 0.970318i \(0.422251\pi\)
\(74\) −11.3788 − 4.46177i −1.32276 − 0.518670i
\(75\) 0 0
\(76\) 5.90137 − 7.03298i 0.676933 − 0.806738i
\(77\) −1.01588 + 0.852425i −0.115770 + 0.0971428i
\(78\) −0.0104354 − 0.0591823i −0.00118158 − 0.00670108i
\(79\) −0.799712 + 2.19719i −0.0899746 + 0.247203i −0.976516 − 0.215447i \(-0.930879\pi\)
0.886541 + 0.462650i \(0.153101\pi\)
\(80\) 0 0
\(81\) −3.86030 − 1.40504i −0.428923 − 0.156115i
\(82\) −10.1764 − 5.87534i −1.12379 − 0.648823i
\(83\) −0.460892 + 2.61385i −0.0505895 + 0.286907i −0.999598 − 0.0283424i \(-0.990977\pi\)
0.949009 + 0.315250i \(0.102088\pi\)
\(84\) 1.34794 + 2.33470i 0.147072 + 0.254736i
\(85\) 0 0
\(86\) 5.28548 + 4.43504i 0.569948 + 0.478243i
\(87\) 3.53499 − 0.623315i 0.378991 − 0.0668264i
\(88\) −0.0497576 + 0.0287276i −0.00530418 + 0.00306237i
\(89\) −5.87216 − 16.1336i −0.622448 − 1.71016i −0.700916 − 0.713244i \(-0.747225\pi\)
0.0784679 − 0.996917i \(-0.474997\pi\)
\(90\) 0 0
\(91\) 0.0670707 + 0.0118264i 0.00703092 + 0.00123974i
\(92\) 2.39263 + 2.85142i 0.249449 + 0.297281i
\(93\) −2.94341 − 3.50782i −0.305217 − 0.363744i
\(94\) −7.18544 − 1.26699i −0.741122 − 0.130680i
\(95\) 0 0
\(96\) −2.09471 − 5.75518i −0.213791 − 0.587385i
\(97\) 12.8808 − 7.43673i 1.30785 − 0.755086i 0.326111 − 0.945332i \(-0.394262\pi\)
0.981736 + 0.190246i \(0.0609285\pi\)
\(98\) 7.88935 − 1.39111i 0.796945 − 0.140523i
\(99\) −1.41567 − 1.18789i −0.142280 − 0.119387i
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.151.2 72
5.2 odd 4 925.2.ba.b.299.2 72
5.3 odd 4 925.2.ba.c.299.11 72
5.4 even 2 185.2.w.a.151.11 yes 72
37.25 even 18 inner 925.2.bb.b.876.2 72
185.62 odd 36 925.2.ba.c.99.11 72
185.69 odd 36 6845.2.a.t.1.32 36
185.79 odd 36 6845.2.a.u.1.5 36
185.99 even 18 185.2.w.a.136.11 ✓ 72
185.173 odd 36 925.2.ba.b.99.2 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.w.a.136.11 ✓ 72 185.99 even 18
185.2.w.a.151.11 yes 72 5.4 even 2
925.2.ba.b.99.2 72 185.173 odd 36
925.2.ba.b.299.2 72 5.2 odd 4
925.2.ba.c.99.11 72 185.62 odd 36
925.2.ba.c.299.11 72 5.3 odd 4
925.2.bb.b.151.2 72 1.1 even 1 trivial
925.2.bb.b.876.2 72 37.25 even 18 inner
6845.2.a.t.1.32 36 185.69 odd 36
6845.2.a.u.1.5 36 185.79 odd 36