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Properties

Label 925.2.bn.b.468.17
Level $925$
Weight $2$
Character 925.468
Analytic conductor $7.386$
Analytic rank $0$
Dimension $204$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [925,2,Mod(18,925)] 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("925.18"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([27, 17])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 925 = 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 925.bn (of order \(36\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [204] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.38616218697\)
Analytic rank: \(0\)
Dimension: \(204\)
Relative dimension: \(17\) over \(\Q(\zeta_{36})\)
Twist minimal: no (minimal twist has level 185)
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 468.17
Character \(\chi\) \(=\) 925.468
Dual form 925.2.bn.b.757.17

$q$-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+(0.469713 - 2.66388i) q^{2} +(1.96091 + 1.37304i) q^{3} +(-4.99622 - 1.81848i) q^{4} +(4.57868 - 4.57868i) q^{6} +(-0.315692 + 3.60837i) q^{7} +(-4.48602 + 7.77001i) q^{8} +(0.933852 + 2.56574i) q^{9} +(1.47450 + 0.851304i) q^{11} +(-7.30028 - 10.4259i) q^{12} +(3.18223 + 1.15824i) q^{13} +(9.46397 + 2.53586i) q^{14} +(10.4453 + 8.76463i) q^{16} +(1.45434 + 3.99576i) q^{17} +(7.27345 - 1.28251i) q^{18} +(1.97777 - 2.82454i) q^{19} +(-5.57349 + 6.64222i) q^{21} +(2.96036 - 3.52802i) q^{22} +(1.74896 + 3.02928i) q^{23} +(-19.4652 + 9.07677i) q^{24} +(4.58014 - 7.93304i) q^{26} +(0.167035 - 0.623382i) q^{27} +(8.13900 - 17.4541i) q^{28} +(-0.442747 - 1.65236i) q^{29} +(-5.32606 - 5.32606i) q^{31} +(14.5082 - 12.1739i) q^{32} +(1.72248 + 3.69388i) q^{33} +(11.3273 - 1.99732i) q^{34} -14.5172i q^{36} +(4.05665 + 4.53250i) q^{37} +(-6.59525 - 6.59525i) q^{38} +(4.64976 + 6.64054i) q^{39} +(-1.37504 + 3.77790i) q^{41} +(15.0761 + 17.9670i) q^{42} -4.73916 q^{43} +(-5.81886 - 6.93465i) q^{44} +(8.89114 - 3.23611i) q^{46} +(2.31610 + 0.620597i) q^{47} +(8.44802 + 31.5284i) q^{48} +(-6.02703 - 1.06273i) q^{49} +(-2.63453 + 9.83218i) q^{51} +(-13.7929 - 11.5736i) q^{52} +(-0.491198 - 5.61442i) q^{53} +(-1.58215 - 0.737770i) q^{54} +(-26.6209 - 18.6401i) q^{56} +(7.75643 - 2.82311i) q^{57} +(-4.60964 + 0.403291i) q^{58} +(-0.132493 - 1.51440i) q^{59} +(2.41726 + 5.18382i) q^{61} +(-16.6897 + 11.6862i) q^{62} +(-9.55295 + 2.55970i) q^{63} +(-11.9796 - 20.7492i) q^{64} +(10.6491 - 2.85342i) q^{66} +(-5.27234 - 0.461270i) q^{67} -22.6084i q^{68} +(-0.729789 + 8.34153i) q^{69} +(-0.493958 - 2.80137i) q^{71} +(-24.1251 - 4.25390i) q^{72} +(5.50833 - 5.50833i) q^{73} +(13.9795 - 8.67744i) q^{74} +(-15.0177 + 10.5155i) q^{76} +(-3.53731 + 5.05180i) q^{77} +(19.8736 - 9.26722i) q^{78} +(7.89820 + 0.691003i) q^{79} +(7.45830 - 6.25825i) q^{81} +(9.41797 + 5.43747i) q^{82} +(4.25349 + 1.98344i) q^{83} +(39.9251 - 23.0508i) q^{84} +(-2.22605 + 12.6245i) q^{86} +(1.40057 - 3.84803i) q^{87} +(-13.2293 + 7.63792i) q^{88} +(-13.9431 + 1.21986i) q^{89} +(-5.18396 + 11.1170i) q^{91} +(-3.22950 - 18.3154i) q^{92} +(-3.13100 - 17.7568i) q^{93} +(2.74110 - 5.87830i) q^{94} +(45.1645 - 3.95138i) q^{96} +(-6.80632 + 3.92963i) q^{97} +(-5.66196 + 15.5561i) q^{98} +(-0.807256 + 4.57818i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 204 q + 12 q^{2} + 6 q^{3} - 24 q^{6} + 12 q^{7} + 24 q^{8} - 36 q^{11} - 36 q^{12} + 12 q^{13} - 24 q^{14} - 24 q^{16} + 30 q^{17} + 90 q^{18} - 24 q^{21} + 24 q^{22} + 6 q^{23} + 36 q^{24} - 12 q^{26}+ \cdots + 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(76\) \(852\)
\(\chi(n)\) \(e\left(\frac{29}{36}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.469713 − 2.66388i 0.332137 − 1.88365i −0.121710 − 0.992566i \(-0.538838\pi\)
0.453848 − 0.891079i \(-0.350051\pi\)
\(3\) 1.96091 + 1.37304i 1.13213 + 0.792726i 0.980657 − 0.195735i \(-0.0627093\pi\)
0.151473 + 0.988461i \(0.451598\pi\)
\(4\) −4.99622 − 1.81848i −2.49811 − 0.909238i
\(5\) 0 0
\(6\) 4.57868 − 4.57868i 1.86924 − 1.86924i
\(7\) −0.315692 + 3.60837i −0.119320 + 1.36384i 0.666557 + 0.745454i \(0.267767\pi\)
−0.785877 + 0.618382i \(0.787788\pi\)
\(8\) −4.48602 + 7.77001i −1.58605 + 2.74711i
\(9\) 0.933852 + 2.56574i 0.311284 + 0.855246i
\(10\) 0 0
\(11\) 1.47450 + 0.851304i 0.444579 + 0.256678i 0.705538 − 0.708672i \(-0.250705\pi\)
−0.260959 + 0.965350i \(0.584039\pi\)
\(12\) −7.30028 − 10.4259i −2.10741 − 3.00969i
\(13\) 3.18223 + 1.15824i 0.882593 + 0.321238i 0.743256 − 0.669007i \(-0.233281\pi\)
0.139337 + 0.990245i \(0.455503\pi\)
\(14\) 9.46397 + 2.53586i 2.52935 + 0.677738i
\(15\) 0 0
\(16\) 10.4453 + 8.76463i 2.61132 + 2.19116i
\(17\) 1.45434 + 3.99576i 0.352729 + 0.969115i 0.981490 + 0.191516i \(0.0613403\pi\)
−0.628761 + 0.777599i \(0.716437\pi\)
\(18\) 7.27345 − 1.28251i 1.71437 − 0.302289i
\(19\) 1.97777 − 2.82454i 0.453731 − 0.647995i −0.525488 − 0.850801i \(-0.676117\pi\)
0.979219 + 0.202806i \(0.0650061\pi\)
\(20\) 0 0
\(21\) −5.57349 + 6.64222i −1.21623 + 1.44945i
\(22\) 2.96036 − 3.52802i 0.631151 − 0.752176i
\(23\) 1.74896 + 3.02928i 0.364682 + 0.631649i 0.988725 − 0.149742i \(-0.0478442\pi\)
−0.624043 + 0.781390i \(0.714511\pi\)
\(24\) −19.4652 + 9.07677i −3.97332 + 1.85279i
\(25\) 0 0
\(26\) 4.58014 − 7.93304i 0.898240 − 1.55580i
\(27\) 0.167035 − 0.623382i 0.0321458 − 0.119970i
\(28\) 8.13900 − 17.4541i 1.53813 − 3.29852i
\(29\) −0.442747 − 1.65236i −0.0822161 − 0.306835i 0.912556 − 0.408951i \(-0.134105\pi\)
−0.994772 + 0.102116i \(0.967439\pi\)
\(30\) 0 0
\(31\) −5.32606 − 5.32606i −0.956588 − 0.956588i 0.0425081 − 0.999096i \(-0.486465\pi\)
−0.999096 + 0.0425081i \(0.986465\pi\)
\(32\) 14.5082 − 12.1739i 2.56472 − 2.15205i
\(33\) 1.72248 + 3.69388i 0.299846 + 0.643022i
\(34\) 11.3273 − 1.99732i 1.94262 − 0.342537i
\(35\) 0 0
\(36\) − 14.5172i − 2.41953i
\(37\) 4.05665 + 4.53250i 0.666909 + 0.745139i
\(38\) −6.59525 − 6.59525i −1.06989 − 1.06989i
\(39\) 4.64976 + 6.64054i 0.744557 + 1.06334i
\(40\) 0 0
\(41\) −1.37504 + 3.77790i −0.214745 + 0.590008i −0.999558 − 0.0297249i \(-0.990537\pi\)
0.784813 + 0.619733i \(0.212759\pi\)
\(42\) 15.0761 + 17.9670i 2.32630 + 2.77237i
\(43\) −4.73916 −0.722716 −0.361358 − 0.932427i \(-0.617687\pi\)
−0.361358 + 0.932427i \(0.617687\pi\)
\(44\) −5.81886 − 6.93465i −0.877226 − 1.04544i
\(45\) 0 0
\(46\) 8.89114 − 3.23611i 1.31093 − 0.477138i
\(47\) 2.31610 + 0.620597i 0.337838 + 0.0905234i 0.423750 − 0.905779i \(-0.360714\pi\)
−0.0859118 + 0.996303i \(0.527380\pi\)
\(48\) 8.44802 + 31.5284i 1.21937 + 4.55074i
\(49\) −6.02703 − 1.06273i −0.861005 − 0.151818i
\(50\) 0 0
\(51\) −2.63453 + 9.83218i −0.368907 + 1.37678i
\(52\) −13.7929 − 11.5736i −1.91273 − 1.60497i
\(53\) −0.491198 − 5.61442i −0.0674712 − 0.771200i −0.952276 − 0.305238i \(-0.901264\pi\)
0.884805 − 0.465962i \(-0.154292\pi\)
\(54\) −1.58215 − 0.737770i −0.215304 − 0.100398i
\(55\) 0 0
\(56\) −26.6209 − 18.6401i −3.55736 − 2.49089i
\(57\) 7.75643 − 2.82311i 1.02736 − 0.373930i
\(58\) −4.60964 + 0.403291i −0.605275 + 0.0529547i
\(59\) −0.132493 − 1.51440i −0.0172491 − 0.197158i −0.999930 − 0.0118511i \(-0.996228\pi\)
0.982681 − 0.185307i \(-0.0593280\pi\)
\(60\) 0 0
\(61\) 2.41726 + 5.18382i 0.309498 + 0.663721i 0.998113 − 0.0614066i \(-0.0195586\pi\)
−0.688615 + 0.725127i \(0.741781\pi\)
\(62\) −16.6897 + 11.6862i −2.11959 + 1.48415i
\(63\) −9.55295 + 2.55970i −1.20356 + 0.322492i
\(64\) −11.9796 − 20.7492i −1.49745 − 2.59366i
\(65\) 0 0
\(66\) 10.6491 − 2.85342i 1.31082 − 0.351232i
\(67\) −5.27234 − 0.461270i −0.644118 − 0.0563531i −0.239584 − 0.970876i \(-0.577011\pi\)
−0.404535 + 0.914523i \(0.632567\pi\)
\(68\) − 22.6084i − 2.74167i
\(69\) −0.729789 + 8.34153i −0.0878563 + 1.00420i
\(70\) 0 0
\(71\) −0.493958 − 2.80137i −0.0586220 − 0.332462i 0.941366 − 0.337387i \(-0.109543\pi\)
−0.999988 + 0.00492555i \(0.998432\pi\)
\(72\) −24.1251 − 4.25390i −2.84317 − 0.501327i
\(73\) 5.50833 − 5.50833i 0.644701 − 0.644701i −0.307006 − 0.951707i \(-0.599327\pi\)
0.951707 + 0.307006i \(0.0993273\pi\)
\(74\) 13.9795 − 8.67744i 1.62508 − 1.00873i
\(75\) 0 0
\(76\) −15.0177 + 10.5155i −1.72265 + 1.20621i
\(77\) −3.53731 + 5.05180i −0.403114 + 0.575706i
\(78\) 19.8736 − 9.26722i 2.25025 − 1.04931i
\(79\) 7.89820 + 0.691003i 0.888617 + 0.0777439i 0.522315 − 0.852752i \(-0.325068\pi\)
0.366301 + 0.930496i \(0.380624\pi\)
\(80\) 0 0
\(81\) 7.45830 − 6.25825i 0.828699 − 0.695361i
\(82\) 9.41797 + 5.43747i 1.04004 + 0.600468i
\(83\) 4.25349 + 1.98344i 0.466881 + 0.217710i 0.641799 − 0.766873i \(-0.278188\pi\)
−0.174918 + 0.984583i \(0.555966\pi\)
\(84\) 39.9251 − 23.0508i 4.35619 − 2.51505i
\(85\) 0 0
\(86\) −2.22605 + 12.6245i −0.240041 + 1.36134i
\(87\) 1.40057 − 3.84803i 0.150157 − 0.412552i
\(88\) −13.2293 + 7.63792i −1.41024 + 0.814205i
\(89\) −13.9431 + 1.21986i −1.47797 + 0.129305i −0.797431 − 0.603410i \(-0.793808\pi\)
−0.680535 + 0.732716i \(0.738253\pi\)
\(90\) 0 0
\(91\) −5.18396 + 11.1170i −0.543427 + 1.16538i
\(92\) −3.22950 − 18.3154i −0.336698 − 1.90951i
\(93\) −3.13100 − 17.7568i −0.324670 − 1.84129i
\(94\) 2.74110 − 5.87830i 0.282722 − 0.606300i
\(95\) 0 0
\(96\) 45.1645 − 3.95138i 4.60958 − 0.403286i
\(97\) −6.80632 + 3.92963i −0.691077 + 0.398993i −0.804015 − 0.594609i \(-0.797307\pi\)
0.112939 + 0.993602i \(0.463974\pi\)
\(98\) −5.66196 + 15.5561i −0.571944 + 1.57140i
\(99\) −0.807256 + 4.57818i −0.0811323 + 0.460124i
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.bn.b.468.17 204
5.2 odd 4 925.2.bq.b.357.17 204
5.3 odd 4 185.2.bc.a.172.1 yes 204
5.4 even 2 185.2.z.a.98.1 yes 204
37.17 odd 36 925.2.bq.b.868.17 204
185.17 even 36 inner 925.2.bn.b.757.17 204
185.54 odd 36 185.2.bc.a.128.1 yes 204
185.128 even 36 185.2.z.a.17.1 ✓ 204
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
185.2.z.a.17.1 ✓ 204 185.128 even 36
185.2.z.a.98.1 yes 204 5.4 even 2
185.2.bc.a.128.1 yes 204 185.54 odd 36
185.2.bc.a.172.1 yes 204 5.3 odd 4
925.2.bn.b.468.17 204 1.1 even 1 trivial
925.2.bn.b.757.17 204 185.17 even 36 inner
925.2.bq.b.357.17 204 5.2 odd 4
925.2.bq.b.868.17 204 37.17 odd 36