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Properties

Label 115.2.j.a.9.1
Level $115$
Weight $2$
Character 115.9
Analytic conductor $0.918$
Analytic rank $0$
Dimension $100$
Inner twists $4$

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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] = [115,2,Mod(4,115)] 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("115.4"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(115, base_ring=CyclotomicField(22)) chi = DirichletCharacter(H, H._module([11, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 115.j (of order \(22\), degree \(10\), 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: \(0.918279623245\)
Analytic rank: \(0\)
Dimension: \(100\)
Relative dimension: \(10\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

Embedding invariants

Embedding label 9.1
Character \(\chi\) \(=\) 115.9
Dual form 115.2.j.a.64.1

$q$-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+(-0.743795 - 2.53313i) q^{2} +(-1.34601 + 1.16633i) q^{3} +(-4.18102 + 2.68698i) q^{4} +(-2.23607 + 0.00123367i) q^{5} +(3.95561 + 2.54212i) q^{6} +(1.87277 + 0.855264i) q^{7} +(5.92582 + 5.13475i) q^{8} +(0.0244873 - 0.170313i) q^{9} +(1.66630 + 5.66334i) q^{10} +(-5.55748 - 1.63182i) q^{11} +(2.49381 - 8.49314i) q^{12} +(-2.87256 + 1.31185i) q^{13} +(0.773542 - 5.38010i) q^{14} +(3.00833 - 2.60964i) q^{15} +(4.47020 - 9.78837i) q^{16} +(-1.10511 + 1.71958i) q^{17} +(-0.449639 + 0.0646484i) q^{18} +(-1.77165 + 1.13857i) q^{19} +(9.34573 - 6.01342i) q^{20} +(-3.51828 + 1.03306i) q^{21} +15.2916i q^{22} +(-1.21456 - 4.63949i) q^{23} -13.9650 q^{24} +(5.00000 - 0.00551712i) q^{25} +(5.45969 + 6.30081i) q^{26} +(-2.72301 - 4.23709i) q^{27} +(-10.1281 + 1.45621i) q^{28} +(1.74171 + 1.11933i) q^{29} +(-8.84816 - 5.67947i) q^{30} +(-1.04153 + 1.20200i) q^{31} +(-12.5978 - 1.81129i) q^{32} +(9.38367 - 4.28538i) q^{33} +(5.17791 + 1.52037i) q^{34} +(-4.18869 - 1.91012i) q^{35} +(0.355246 + 0.777880i) q^{36} +(0.620561 + 0.0892232i) q^{37} +(4.20189 + 3.64095i) q^{38} +(2.33645 - 5.11610i) q^{39} +(-13.2569 - 11.4743i) q^{40} +(0.289709 + 2.01497i) q^{41} +(5.23376 + 8.14389i) q^{42} +(-3.14193 + 2.72250i) q^{43} +(27.6206 - 8.11014i) q^{44} +(-0.0545453 + 0.380862i) q^{45} +(-10.8491 + 6.52746i) q^{46} +0.403164i q^{47} +(5.39949 + 18.3890i) q^{48} +(-1.80825 - 2.08683i) q^{49} +(-3.73295 - 12.6615i) q^{50} +(-0.518105 - 3.60350i) q^{51} +(8.48529 - 13.2034i) q^{52} +(7.50209 + 3.42609i) q^{53} +(-8.70774 + 10.0493i) q^{54} +(12.4289 + 3.64201i) q^{55} +(6.70611 + 14.6843i) q^{56} +(1.05672 - 3.59885i) q^{57} +(1.53994 - 5.24454i) q^{58} +(3.73431 + 8.17700i) q^{59} +(-5.56585 + 18.9943i) q^{60} +(0.892000 - 1.02942i) q^{61} +(3.81950 + 1.74431i) q^{62} +(0.191522 - 0.298014i) q^{63} +(1.71909 + 11.9565i) q^{64} +(6.42161 - 2.93693i) q^{65} +(-17.8350 - 20.5826i) q^{66} +(-1.89660 - 6.45923i) q^{67} -10.1590i q^{68} +(7.04596 + 4.82824i) q^{69} +(-1.72306 + 12.0312i) q^{70} +(-13.3319 + 3.91459i) q^{71} +(1.01962 - 0.883509i) q^{72} +(5.58084 + 8.68396i) q^{73} +(-0.235556 - 1.63833i) q^{74} +(-6.72362 + 5.83905i) q^{75} +(4.34798 - 9.52075i) q^{76} +(-9.01222 - 7.80913i) q^{77} +(-14.6976 - 2.11320i) q^{78} +(-3.54206 - 7.75603i) q^{79} +(-9.98359 + 21.8930i) q^{80} +(9.10231 + 2.67268i) q^{81} +(4.88870 - 2.23259i) q^{82} +(-8.15053 - 1.17187i) q^{83} +(11.9342 - 13.7728i) q^{84} +(2.46898 - 3.84647i) q^{85} +(9.23341 + 5.93395i) q^{86} +(-3.64987 + 0.524772i) q^{87} +(-24.5536 - 38.2062i) q^{88} +(2.09382 + 2.41639i) q^{89} +(1.00534 - 0.145113i) q^{90} -6.50160 q^{91} +(17.5443 + 16.1343i) q^{92} -2.83267i q^{93} +(1.02127 - 0.299871i) q^{94} +(3.96012 - 2.54810i) q^{95} +(19.0693 - 12.2551i) q^{96} +(-9.91281 + 1.42525i) q^{97} +(-3.94125 + 6.13270i) q^{98} +(-0.414009 + 0.906553i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 100 q - 14 q^{4} - 9 q^{5} - 18 q^{6} - 12 q^{9} - 13 q^{10} - 26 q^{11} - 26 q^{14} - 10 q^{15} - 18 q^{16} - 14 q^{19} + 49 q^{20} - 22 q^{21} - 68 q^{24} + 21 q^{25} - 42 q^{26} - 24 q^{29} + 19 q^{30}+ \cdots - 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{11}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.743795 − 2.53313i −0.525942 − 1.79119i −0.607256 − 0.794506i \(-0.707730\pi\)
0.0813139 − 0.996689i \(-0.474088\pi\)
\(3\) −1.34601 + 1.16633i −0.777120 + 0.673379i −0.950239 − 0.311522i \(-0.899161\pi\)
0.173118 + 0.984901i \(0.444616\pi\)
\(4\) −4.18102 + 2.68698i −2.09051 + 1.34349i
\(5\) −2.23607 + 0.00123367i −1.00000 + 0.000551712i
\(6\) 3.95561 + 2.54212i 1.61487 + 1.03782i
\(7\) 1.87277 + 0.855264i 0.707839 + 0.323259i 0.736601 − 0.676327i \(-0.236429\pi\)
−0.0287625 + 0.999586i \(0.509157\pi\)
\(8\) 5.92582 + 5.13475i 2.09509 + 1.81541i
\(9\) 0.0244873 − 0.170313i 0.00816245 − 0.0567711i
\(10\) 1.66630 + 5.66334i 0.526930 + 1.79090i
\(11\) −5.55748 − 1.63182i −1.67564 − 0.492013i −0.700511 − 0.713641i \(-0.747045\pi\)
−0.975131 + 0.221628i \(0.928863\pi\)
\(12\) 2.49381 − 8.49314i 0.719901 − 2.45176i
\(13\) −2.87256 + 1.31185i −0.796703 + 0.363842i −0.771814 − 0.635848i \(-0.780651\pi\)
−0.0248891 + 0.999690i \(0.507923\pi\)
\(14\) 0.773542 − 5.38010i 0.206738 − 1.43789i
\(15\) 3.00833 − 2.60964i 0.776749 − 0.673807i
\(16\) 4.47020 − 9.78837i 1.11755 − 2.44709i
\(17\) −1.10511 + 1.71958i −0.268029 + 0.417060i −0.949013 − 0.315238i \(-0.897916\pi\)
0.680984 + 0.732298i \(0.261552\pi\)
\(18\) −0.449639 + 0.0646484i −0.105981 + 0.0152378i
\(19\) −1.77165 + 1.13857i −0.406444 + 0.261206i −0.727851 − 0.685736i \(-0.759481\pi\)
0.321407 + 0.946941i \(0.395844\pi\)
\(20\) 9.34573 − 6.01342i 2.08977 − 1.34464i
\(21\) −3.51828 + 1.03306i −0.767752 + 0.225432i
\(22\) 15.2916i 3.26017i
\(23\) −1.21456 − 4.63949i −0.253253 − 0.967400i
\(24\) −13.9650 −2.85060
\(25\) 5.00000 − 0.00551712i 0.999999 − 0.00110342i
\(26\) 5.45969 + 6.30081i 1.07073 + 1.23569i
\(27\) −2.72301 − 4.23709i −0.524044 − 0.815428i
\(28\) −10.1281 + 1.45621i −1.91404 + 0.275197i
\(29\) 1.74171 + 1.11933i 0.323428 + 0.207855i 0.692272 − 0.721637i \(-0.256610\pi\)
−0.368844 + 0.929491i \(0.620246\pi\)
\(30\) −8.84816 − 5.67947i −1.61544 − 1.03692i
\(31\) −1.04153 + 1.20200i −0.187065 + 0.215885i −0.841534 − 0.540204i \(-0.818347\pi\)
0.654469 + 0.756089i \(0.272892\pi\)
\(32\) −12.5978 − 1.81129i −2.22699 − 0.320193i
\(33\) 9.38367 − 4.28538i 1.63349 − 0.745989i
\(34\) 5.17791 + 1.52037i 0.888004 + 0.260742i
\(35\) −4.18869 − 1.91012i −0.708017 − 0.322869i
\(36\) 0.355246 + 0.777880i 0.0592076 + 0.129647i
\(37\) 0.620561 + 0.0892232i 0.102020 + 0.0146682i 0.193136 − 0.981172i \(-0.438134\pi\)
−0.0911159 + 0.995840i \(0.529043\pi\)
\(38\) 4.20189 + 3.64095i 0.681636 + 0.590641i
\(39\) 2.33645 − 5.11610i 0.374131 − 0.819232i
\(40\) −13.2569 − 11.4743i −2.09609 − 1.81425i
\(41\) 0.289709 + 2.01497i 0.0452449 + 0.314685i 0.999858 + 0.0168374i \(0.00535976\pi\)
−0.954613 + 0.297848i \(0.903731\pi\)
\(42\) 5.23376 + 8.14389i 0.807586 + 1.25663i
\(43\) −3.14193 + 2.72250i −0.479141 + 0.415178i −0.860658 − 0.509184i \(-0.829947\pi\)
0.381517 + 0.924362i \(0.375402\pi\)
\(44\) 27.6206 − 8.11014i 4.16396 − 1.22265i
\(45\) −0.0545453 + 0.380862i −0.00813113 + 0.0567756i
\(46\) −10.8491 + 6.52746i −1.59961 + 0.962421i
\(47\) 0.403164i 0.0588075i 0.999568 + 0.0294037i \(0.00936085\pi\)
−0.999568 + 0.0294037i \(0.990639\pi\)
\(48\) 5.39949 + 18.3890i 0.779349 + 2.65422i
\(49\) −1.80825 − 2.08683i −0.258321 − 0.298119i
\(50\) −3.73295 − 12.6615i −0.527918 − 1.79061i
\(51\) −0.518105 − 3.60350i −0.0725492 − 0.504591i
\(52\) 8.48529 − 13.2034i 1.17670 − 1.83098i
\(53\) 7.50209 + 3.42609i 1.03049 + 0.470610i 0.857593 − 0.514330i \(-0.171959\pi\)
0.172899 + 0.984940i \(0.444687\pi\)
\(54\) −8.70774 + 10.0493i −1.18497 + 1.36753i
\(55\) 12.4289 + 3.64201i 1.67591 + 0.491089i
\(56\) 6.70611 + 14.6843i 0.896141 + 1.96228i
\(57\) 1.05672 − 3.59885i 0.139965 − 0.476679i
\(58\) 1.53994 − 5.24454i 0.202203 − 0.688642i
\(59\) 3.73431 + 8.17700i 0.486166 + 1.06455i 0.980722 + 0.195409i \(0.0626033\pi\)
−0.494556 + 0.869146i \(0.664669\pi\)
\(60\) −5.56585 + 18.9943i −0.718548 + 2.45215i
\(61\) 0.892000 − 1.02942i 0.114209 − 0.131804i −0.695766 − 0.718268i \(-0.744935\pi\)
0.809975 + 0.586464i \(0.199481\pi\)
\(62\) 3.81950 + 1.74431i 0.485077 + 0.221527i
\(63\) 0.191522 − 0.298014i 0.0241295 − 0.0375462i
\(64\) 1.71909 + 11.9565i 0.214886 + 1.49457i
\(65\) 6.42161 − 2.93693i 0.796503 − 0.364282i
\(66\) −17.8350 − 20.5826i −2.19533 − 2.53355i
\(67\) −1.89660 − 6.45923i −0.231707 − 0.789121i −0.990467 − 0.137751i \(-0.956013\pi\)
0.758760 − 0.651370i \(-0.225805\pi\)
\(68\) − 10.1590i − 1.23196i
\(69\) 7.04596 + 4.82824i 0.848234 + 0.581251i
\(70\) −1.72306 + 12.0312i −0.205944 + 1.43801i
\(71\) −13.3319 + 3.91459i −1.58220 + 0.464577i −0.950524 − 0.310653i \(-0.899452\pi\)
−0.631680 + 0.775229i \(0.717634\pi\)
\(72\) 1.01962 − 0.883509i 0.120164 − 0.104123i
\(73\) 5.58084 + 8.68396i 0.653188 + 1.01638i 0.997005 + 0.0773434i \(0.0246438\pi\)
−0.343817 + 0.939037i \(0.611720\pi\)
\(74\) −0.235556 − 1.63833i −0.0273828 − 0.190452i
\(75\) −6.72362 + 5.83905i −0.776377 + 0.674236i
\(76\) 4.34798 − 9.52075i 0.498748 − 1.09211i
\(77\) −9.01222 − 7.80913i −1.02704 − 0.889933i
\(78\) −14.6976 − 2.11320i −1.66418 − 0.239272i
\(79\) −3.54206 − 7.75603i −0.398513 − 0.872621i −0.997419 − 0.0718020i \(-0.977125\pi\)
0.598906 − 0.800819i \(-0.295602\pi\)
\(80\) −9.98359 + 21.8930i −1.11620 + 2.44771i
\(81\) 9.10231 + 2.67268i 1.01137 + 0.296965i
\(82\) 4.88870 − 2.23259i 0.539867 − 0.246549i
\(83\) −8.15053 − 1.17187i −0.894637 − 0.128629i −0.320365 − 0.947294i \(-0.603806\pi\)
−0.574272 + 0.818665i \(0.694715\pi\)
\(84\) 11.9342 − 13.7728i 1.30213 − 1.50273i
\(85\) 2.46898 − 3.84647i 0.267798 − 0.417208i
\(86\) 9.23341 + 5.93395i 0.995664 + 0.639875i
\(87\) −3.64987 + 0.524772i −0.391307 + 0.0562615i
\(88\) −24.5536 − 38.2062i −2.61742 − 4.07279i
\(89\) 2.09382 + 2.41639i 0.221944 + 0.256137i 0.855791 − 0.517321i \(-0.173071\pi\)
−0.633847 + 0.773458i \(0.718525\pi\)
\(90\) 1.00534 − 0.145113i 0.105973 − 0.0152962i
\(91\) −6.50160 −0.681553
\(92\) 17.5443 + 16.1343i 1.82912 + 1.68212i
\(93\) − 2.83267i − 0.293734i
\(94\) 1.02127 − 0.299871i 0.105336 − 0.0309293i
\(95\) 3.96012 − 2.54810i 0.406300 − 0.261430i
\(96\) 19.0693 − 12.2551i 1.94625 − 1.25078i
\(97\) −9.91281 + 1.42525i −1.00649 + 0.144712i −0.625795 − 0.779987i \(-0.715225\pi\)
−0.380698 + 0.924699i \(0.624316\pi\)
\(98\) −3.94125 + 6.13270i −0.398126 + 0.619497i
\(99\) −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 115.2.j.a.9.1 ✓ 100
5.2 odd 4 575.2.k.g.101.10 100
5.3 odd 4 575.2.k.g.101.1 100
5.4 even 2 inner 115.2.j.a.9.10 yes 100
23.18 even 11 inner 115.2.j.a.64.10 yes 100
115.18 odd 44 575.2.k.g.501.1 100
115.64 even 22 inner 115.2.j.a.64.1 yes 100
115.87 odd 44 575.2.k.g.501.10 100
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.2.j.a.9.1 ✓ 100 1.1 even 1 trivial
115.2.j.a.9.10 yes 100 5.4 even 2 inner
115.2.j.a.64.1 yes 100 115.64 even 22 inner
115.2.j.a.64.10 yes 100 23.18 even 11 inner
575.2.k.g.101.1 100 5.3 odd 4
575.2.k.g.101.10 100 5.2 odd 4
575.2.k.g.501.1 100 115.18 odd 44
575.2.k.g.501.10 100 115.87 odd 44