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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [575,2,Mod(26,575)] 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("575.26"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(575, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([0, 16])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 575 = 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 575.k (of order \(11\), degree \(10\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [100,0] 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: \(4.59139811622\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(10\) over \(\Q(\zeta_{11})\)
Twist minimal: no (minimal twist has level 115)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

Embedding invariants

Embedding label 501.1
Character \(\chi\) \(=\) 575.501
Dual form 575.2.k.g.101.1

$q$-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+(-2.53313 - 0.743795i) q^{2} +(-1.16633 + 1.34601i) q^{3} +(4.18102 + 2.68698i) q^{4} +(3.95561 - 2.54212i) q^{6} +(0.855264 + 1.87277i) q^{7} +(-5.13475 - 5.92582i) q^{8} +(-0.0244873 - 0.170313i) q^{9} +(-5.55748 + 1.63182i) q^{11} +(-8.49314 + 2.49381i) q^{12} +(-1.31185 + 2.87256i) q^{13} +(-0.773542 - 5.38010i) q^{14} +(4.47020 + 9.78837i) q^{16} +(1.71958 - 1.10511i) q^{17} +(-0.0646484 + 0.449639i) q^{18} +(1.77165 + 1.13857i) q^{19} +(-3.51828 - 1.03306i) q^{21} +15.2916 q^{22} +(4.63949 + 1.21456i) q^{23} +13.9650 q^{24} +(5.45969 - 6.30081i) q^{26} +(-4.23709 - 2.72301i) q^{27} +(-1.45621 + 10.1281i) q^{28} +(-1.74171 + 1.11933i) q^{29} +(-1.04153 - 1.20200i) q^{31} +(-1.81129 - 12.5978i) q^{32} +(4.28538 - 9.38367i) q^{33} +(-5.17791 + 1.52037i) q^{34} +(0.355246 - 0.777880i) q^{36} +(0.0892232 + 0.620561i) q^{37} +(-3.64095 - 4.20189i) q^{38} +(-2.33645 - 5.11610i) q^{39} +(0.289709 - 2.01497i) q^{41} +(8.14389 + 5.23376i) q^{42} +(-2.72250 + 3.14193i) q^{43} +(-27.6206 - 8.11014i) q^{44} +(-10.8491 - 6.52746i) q^{46} +0.403164 q^{47} +(-18.3890 - 5.39949i) q^{48} +(1.80825 - 2.08683i) q^{49} +(-0.518105 + 3.60350i) q^{51} +(-13.2034 + 8.48529i) q^{52} +(-3.42609 - 7.50209i) q^{53} +(8.70774 + 10.0493i) q^{54} +(6.70611 - 14.6843i) q^{56} +(-3.59885 + 1.05672i) q^{57} +(5.24454 - 1.53994i) q^{58} +(-3.73431 + 8.17700i) q^{59} +(0.892000 + 1.02942i) q^{61} +(1.74431 + 3.81950i) q^{62} +(0.298014 - 0.191522i) q^{63} +(-1.71909 + 11.9565i) q^{64} +(-17.8350 + 20.5826i) q^{66} +(-6.45923 - 1.89660i) q^{67} +10.1590 q^{68} +(-7.04596 + 4.82824i) q^{69} +(-13.3319 - 3.91459i) q^{71} +(-0.883509 + 1.01962i) q^{72} +(-8.68396 - 5.58084i) q^{73} +(0.235556 - 1.63833i) q^{74} +(4.34798 + 9.52075i) q^{76} +(-7.80913 - 9.01222i) q^{77} +(2.11320 + 14.6976i) q^{78} +(3.54206 - 7.75603i) q^{79} +(9.10231 - 2.67268i) q^{81} +(-2.23259 + 4.88870i) q^{82} +(1.17187 + 8.15053i) q^{83} +(-11.9342 - 13.7728i) q^{84} +(9.23341 - 5.93395i) q^{86} +(0.524772 - 3.64987i) q^{87} +(38.2062 + 24.5536i) q^{88} +(-2.09382 + 2.41639i) q^{89} -6.50160 q^{91} +(16.1343 + 17.5443i) q^{92} +2.83267 q^{93} +(-1.02127 - 0.299871i) q^{94} +(19.0693 + 12.2551i) q^{96} +(1.42525 - 9.91281i) q^{97} +(-6.13270 + 3.94125i) q^{98} +(0.414009 + 0.906553i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 100 q + 14 q^{4} - 18 q^{6} + 12 q^{9} - 26 q^{11} + 26 q^{14} - 18 q^{16} + 14 q^{19} - 22 q^{21} + 68 q^{24} - 42 q^{26} + 24 q^{29} - 12 q^{31} - 8 q^{34} - 10 q^{36} - 14 q^{39} + 8 q^{41} - 166 q^{44}+ \cdots + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/575\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(e\left(\frac{6}{11}\right)\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.53313 − 0.743795i −1.79119 − 0.525942i −0.794506 − 0.607256i \(-0.792270\pi\)
−0.996689 + 0.0813139i \(0.974088\pi\)
\(3\) −1.16633 + 1.34601i −0.673379 + 0.777120i −0.984901 − 0.173118i \(-0.944616\pi\)
0.311522 + 0.950239i \(0.399161\pi\)
\(4\) 4.18102 + 2.68698i 2.09051 + 1.34349i
\(5\) 0 0
\(6\) 3.95561 − 2.54212i 1.61487 − 1.03782i
\(7\) 0.855264 + 1.87277i 0.323259 + 0.707839i 0.999586 − 0.0287625i \(-0.00915665\pi\)
−0.676327 + 0.736601i \(0.736429\pi\)
\(8\) −5.13475 − 5.92582i −1.81541 − 2.09509i
\(9\) −0.0244873 − 0.170313i −0.00816245 − 0.0567711i
\(10\) 0 0
\(11\) −5.55748 + 1.63182i −1.67564 + 0.492013i −0.975131 − 0.221628i \(-0.928863\pi\)
−0.700511 + 0.713641i \(0.747045\pi\)
\(12\) −8.49314 + 2.49381i −2.45176 + 0.719901i
\(13\) −1.31185 + 2.87256i −0.363842 + 0.796703i 0.635848 + 0.771814i \(0.280651\pi\)
−0.999690 + 0.0248891i \(0.992077\pi\)
\(14\) −0.773542 − 5.38010i −0.206738 − 1.43789i
\(15\) 0 0
\(16\) 4.47020 + 9.78837i 1.11755 + 2.44709i
\(17\) 1.71958 − 1.10511i 0.417060 − 0.268029i −0.315238 − 0.949013i \(-0.602084\pi\)
0.732298 + 0.680984i \(0.238448\pi\)
\(18\) −0.0646484 + 0.449639i −0.0152378 + 0.105981i
\(19\) 1.77165 + 1.13857i 0.406444 + 0.261206i 0.727851 − 0.685736i \(-0.240519\pi\)
−0.321407 + 0.946941i \(0.604156\pi\)
\(20\) 0 0
\(21\) −3.51828 − 1.03306i −0.767752 − 0.225432i
\(22\) 15.2916 3.26017
\(23\) 4.63949 + 1.21456i 0.967400 + 0.253253i
\(24\) 13.9650 2.85060
\(25\) 0 0
\(26\) 5.45969 − 6.30081i 1.07073 − 1.23569i
\(27\) −4.23709 − 2.72301i −0.815428 − 0.524044i
\(28\) −1.45621 + 10.1281i −0.275197 + 1.91404i
\(29\) −1.74171 + 1.11933i −0.323428 + 0.207855i −0.692272 − 0.721637i \(-0.743390\pi\)
0.368844 + 0.929491i \(0.379754\pi\)
\(30\) 0 0
\(31\) −1.04153 − 1.20200i −0.187065 − 0.215885i 0.654469 − 0.756089i \(-0.272892\pi\)
−0.841534 + 0.540204i \(0.818347\pi\)
\(32\) −1.81129 − 12.5978i −0.320193 − 2.22699i
\(33\) 4.28538 − 9.38367i 0.745989 − 1.63349i
\(34\) −5.17791 + 1.52037i −0.888004 + 0.260742i
\(35\) 0 0
\(36\) 0.355246 − 0.777880i 0.0592076 − 0.129647i
\(37\) 0.0892232 + 0.620561i 0.0146682 + 0.102020i 0.995840 − 0.0911159i \(-0.0290434\pi\)
−0.981172 + 0.193136i \(0.938134\pi\)
\(38\) −3.64095 − 4.20189i −0.590641 − 0.681636i
\(39\) −2.33645 − 5.11610i −0.374131 − 0.819232i
\(40\) 0 0
\(41\) 0.289709 − 2.01497i 0.0452449 − 0.314685i −0.954613 − 0.297848i \(-0.903731\pi\)
0.999858 − 0.0168374i \(-0.00535976\pi\)
\(42\) 8.14389 + 5.23376i 1.25663 + 0.807586i
\(43\) −2.72250 + 3.14193i −0.415178 + 0.479141i −0.924362 − 0.381517i \(-0.875402\pi\)
0.509184 + 0.860658i \(0.329947\pi\)
\(44\) −27.6206 − 8.11014i −4.16396 − 1.22265i
\(45\) 0 0
\(46\) −10.8491 − 6.52746i −1.59961 − 0.962421i
\(47\) 0.403164 0.0588075 0.0294037 − 0.999568i \(-0.490639\pi\)
0.0294037 + 0.999568i \(0.490639\pi\)
\(48\) −18.3890 − 5.39949i −2.65422 − 0.779349i
\(49\) 1.80825 − 2.08683i 0.258321 − 0.298119i
\(50\) 0 0
\(51\) −0.518105 + 3.60350i −0.0725492 + 0.504591i
\(52\) −13.2034 + 8.48529i −1.83098 + 1.17670i
\(53\) −3.42609 − 7.50209i −0.470610 − 1.03049i −0.984940 − 0.172899i \(-0.944687\pi\)
0.514330 − 0.857593i \(-0.328041\pi\)
\(54\) 8.70774 + 10.0493i 1.18497 + 1.36753i
\(55\) 0 0
\(56\) 6.70611 − 14.6843i 0.896141 − 1.96228i
\(57\) −3.59885 + 1.05672i −0.476679 + 0.139965i
\(58\) 5.24454 − 1.53994i 0.688642 − 0.202203i
\(59\) −3.73431 + 8.17700i −0.486166 + 1.06455i 0.494556 + 0.869146i \(0.335331\pi\)
−0.980722 + 0.195409i \(0.937397\pi\)
\(60\) 0 0
\(61\) 0.892000 + 1.02942i 0.114209 + 0.131804i 0.809975 − 0.586464i \(-0.199481\pi\)
−0.695766 + 0.718268i \(0.744935\pi\)
\(62\) 1.74431 + 3.81950i 0.221527 + 0.485077i
\(63\) 0.298014 − 0.191522i 0.0375462 − 0.0241295i
\(64\) −1.71909 + 11.9565i −0.214886 + 1.49457i
\(65\) 0 0
\(66\) −17.8350 + 20.5826i −2.19533 + 2.53355i
\(67\) −6.45923 − 1.89660i −0.789121 − 0.231707i −0.137751 − 0.990467i \(-0.543987\pi\)
−0.651370 + 0.758760i \(0.725805\pi\)
\(68\) 10.1590 1.23196
\(69\) −7.04596 + 4.82824i −0.848234 + 0.581251i
\(70\) 0 0
\(71\) −13.3319 − 3.91459i −1.58220 − 0.464577i −0.631680 − 0.775229i \(-0.717634\pi\)
−0.950524 + 0.310653i \(0.899452\pi\)
\(72\) −0.883509 + 1.01962i −0.104123 + 0.120164i
\(73\) −8.68396 − 5.58084i −1.01638 − 0.653188i −0.0773434 − 0.997005i \(-0.524644\pi\)
−0.939037 + 0.343817i \(0.888280\pi\)
\(74\) 0.235556 − 1.63833i 0.0273828 − 0.190452i
\(75\) 0 0
\(76\) 4.34798 + 9.52075i 0.498748 + 1.09211i
\(77\) −7.80913 − 9.01222i −0.889933 − 1.02704i
\(78\) 2.11320 + 14.6976i 0.239272 + 1.66418i
\(79\) 3.54206 − 7.75603i 0.398513 − 0.872621i −0.598906 − 0.800819i \(-0.704398\pi\)
0.997419 − 0.0718020i \(-0.0228750\pi\)
\(80\) 0 0
\(81\) 9.10231 − 2.67268i 1.01137 − 0.296965i
\(82\) −2.23259 + 4.88870i −0.246549 + 0.539867i
\(83\) 1.17187 + 8.15053i 0.128629 + 0.894637i 0.947294 + 0.320365i \(0.103806\pi\)
−0.818665 + 0.574272i \(0.805285\pi\)
\(84\) −11.9342 − 13.7728i −1.30213 − 1.50273i
\(85\) 0 0
\(86\) 9.23341 − 5.93395i 0.995664 − 0.639875i
\(87\) 0.524772 − 3.64987i 0.0562615 − 0.391307i
\(88\) 38.2062 + 24.5536i 4.07279 + 2.61742i
\(89\) −2.09382 + 2.41639i −0.221944 + 0.256137i −0.855791 − 0.517321i \(-0.826929\pi\)
0.633847 + 0.773458i \(0.281475\pi\)
\(90\) 0 0
\(91\) −6.50160 −0.681553
\(92\) 16.1343 + 17.5443i 1.68212 + 1.82912i
\(93\) 2.83267 0.293734
\(94\) −1.02127 − 0.299871i −0.105336 − 0.0309293i
\(95\) 0 0
\(96\) 19.0693 + 12.2551i 1.94625 + 1.25078i
\(97\) 1.42525 − 9.91281i 0.144712 − 1.00649i −0.779987 − 0.625795i \(-0.784775\pi\)
0.924699 − 0.380698i \(-0.124316\pi\)
\(98\) −6.13270 + 3.94125i −0.619497 + 0.398126i
\(99\) 0.414009 + 0.906553i 0.0416095 + 0.0911120i
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 575.2.k.g.501.1 100
5.2 odd 4 115.2.j.a.64.10 yes 100
5.3 odd 4 115.2.j.a.64.1 yes 100
5.4 even 2 inner 575.2.k.g.501.10 100
23.9 even 11 inner 575.2.k.g.101.1 100
115.9 even 22 inner 575.2.k.g.101.10 100
115.32 odd 44 115.2.j.a.9.1 ✓ 100
115.78 odd 44 115.2.j.a.9.10 yes 100
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.2.j.a.9.1 ✓ 100 115.32 odd 44
115.2.j.a.9.10 yes 100 115.78 odd 44
115.2.j.a.64.1 yes 100 5.3 odd 4
115.2.j.a.64.10 yes 100 5.2 odd 4
575.2.k.g.101.1 100 23.9 even 11 inner
575.2.k.g.101.10 100 115.9 even 22 inner
575.2.k.g.501.1 100 1.1 even 1 trivial
575.2.k.g.501.10 100 5.4 even 2 inner