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Properties

Label 75.18.b.b.49.2
Level $75$
Weight $18$
Character 75.49
Analytic conductor $137.417$
Analytic rank $0$
Dimension $4$
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] = [75,18,Mod(49,75)] 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("75.49"); S:= CuspForms(chi, 18); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 18, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-702752] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(137.416565508\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{849})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 425x^{2} + 44944 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.2
Root \(-14.0688i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.18.b.b.49.3

$q$-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)
 
\(f(q)\) \(=\) \(q-346.477i q^{2} +6561.00i q^{3} +11025.8 q^{4} +2.27323e6 q^{6} +1.80797e7i q^{7} -4.92336e7i q^{8} -4.30467e7 q^{9} -1.45296e8 q^{11} +7.23401e7i q^{12} -3.26155e9i q^{13} +6.26421e9 q^{14} -1.56131e10 q^{16} -7.81194e9i q^{17} +1.49147e10i q^{18} +7.18160e10 q^{19} -1.18621e11 q^{21} +5.03416e10i q^{22} +1.67169e11i q^{23} +3.23022e11 q^{24} -1.13005e12 q^{26} -2.82430e11i q^{27} +1.99343e11i q^{28} -1.63211e11 q^{29} -5.38766e12 q^{31} -1.04356e12i q^{32} -9.53285e11i q^{33} -2.70666e12 q^{34} -4.74623e11 q^{36} +3.68759e13i q^{37} -2.48826e13i q^{38} +2.13991e13 q^{39} -7.52018e12 q^{41} +4.10995e13i q^{42} -5.73935e13i q^{43} -1.60200e12 q^{44} +5.79201e13 q^{46} -1.42516e14i q^{47} -1.02438e14i q^{48} -9.42466e13 q^{49} +5.12542e13 q^{51} -3.59611e13i q^{52} +8.01583e14i q^{53} -9.78553e13 q^{54} +8.90131e14 q^{56} +4.71185e14i q^{57} +5.65490e13i q^{58} -9.64437e14 q^{59} -1.29792e15 q^{61} +1.86670e15i q^{62} -7.78274e14i q^{63} -2.40801e15 q^{64} -3.30291e14 q^{66} -4.71635e15i q^{67} -8.61327e13i q^{68} -1.09679e15 q^{69} +9.29884e15 q^{71} +2.11934e15i q^{72} -6.86407e15i q^{73} +1.27767e16 q^{74} +7.91827e14 q^{76} -2.62691e15i q^{77} -7.41428e15i q^{78} -1.21446e16 q^{79} +1.85302e15 q^{81} +2.60557e15i q^{82} -3.23986e16i q^{83} -1.30789e15 q^{84} -1.98855e16 q^{86} -1.07083e15i q^{87} +7.15343e15i q^{88} +4.91787e16 q^{89} +5.89681e16 q^{91} +1.84316e15i q^{92} -3.53484e16i q^{93} -4.93784e16 q^{94} +6.84677e15 q^{96} -1.26124e17i q^{97} +3.26543e16i q^{98} +6.25450e15 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 702752 q^{4} - 4671432 q^{6} - 172186884 q^{9} - 2263259824 q^{11} + 8770444032 q^{14} + 102082634240 q^{16} + 388752254432 q^{19} - 272341231344 q^{21} + 2778418267776 q^{24} - 6214885009072 q^{26}+ \cdots + 97\!\cdots\!04 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 346.477i − 0.957016i −0.878083 − 0.478508i \(-0.841178\pi\)
0.878083 − 0.478508i \(-0.158822\pi\)
\(3\) 6561.00i 0.577350i
\(4\) 11025.8 0.0841199
\(5\) 0 0
\(6\) 2.27323e6 0.552534
\(7\) 1.80797e7i 1.18538i 0.805429 + 0.592692i \(0.201935\pi\)
−0.805429 + 0.592692i \(0.798065\pi\)
\(8\) − 4.92336e7i − 1.03752i
\(9\) −4.30467e7 −0.333333
\(10\) 0 0
\(11\) −1.45296e8 −0.204369 −0.102184 − 0.994765i \(-0.532583\pi\)
−0.102184 + 0.994765i \(0.532583\pi\)
\(12\) 7.23401e7i 0.0485667i
\(13\) − 3.26155e9i − 1.10894i −0.832205 − 0.554468i \(-0.812922\pi\)
0.832205 − 0.554468i \(-0.187078\pi\)
\(14\) 6.26421e9 1.13443
\(15\) 0 0
\(16\) −1.56131e10 −0.908804
\(17\) − 7.81194e9i − 0.271608i −0.990736 − 0.135804i \(-0.956638\pi\)
0.990736 − 0.135804i \(-0.0433618\pi\)
\(18\) 1.49147e10i 0.319005i
\(19\) 7.18160e10 0.970098 0.485049 − 0.874487i \(-0.338802\pi\)
0.485049 + 0.874487i \(0.338802\pi\)
\(20\) 0 0
\(21\) −1.18621e11 −0.684382
\(22\) 5.03416e10i 0.195584i
\(23\) 1.67169e11i 0.445111i 0.974920 + 0.222555i \(0.0714398\pi\)
−0.974920 + 0.222555i \(0.928560\pi\)
\(24\) 3.23022e11 0.599013
\(25\) 0 0
\(26\) −1.13005e12 −1.06127
\(27\) − 2.82430e11i − 0.192450i
\(28\) 1.99343e11i 0.0997144i
\(29\) −1.63211e11 −0.0605854 −0.0302927 − 0.999541i \(-0.509644\pi\)
−0.0302927 + 0.999541i \(0.509644\pi\)
\(30\) 0 0
\(31\) −5.38766e12 −1.13456 −0.567278 − 0.823527i \(-0.692003\pi\)
−0.567278 + 0.823527i \(0.692003\pi\)
\(32\) − 1.04356e12i − 0.167780i
\(33\) − 9.53285e11i − 0.117992i
\(34\) −2.70666e12 −0.259934
\(35\) 0 0
\(36\) −4.74623e11 −0.0280400
\(37\) 3.68759e13i 1.72595i 0.505246 + 0.862976i \(0.331402\pi\)
−0.505246 + 0.862976i \(0.668598\pi\)
\(38\) − 2.48826e13i − 0.928400i
\(39\) 2.13991e13 0.640244
\(40\) 0 0
\(41\) −7.52018e12 −0.147084 −0.0735420 − 0.997292i \(-0.523430\pi\)
−0.0735420 + 0.997292i \(0.523430\pi\)
\(42\) 4.10995e13i 0.654964i
\(43\) − 5.73935e13i − 0.748825i −0.927262 − 0.374413i \(-0.877844\pi\)
0.927262 − 0.374413i \(-0.122156\pi\)
\(44\) −1.60200e12 −0.0171915
\(45\) 0 0
\(46\) 5.79201e13 0.425978
\(47\) − 1.42516e14i − 0.873034i −0.899696 − 0.436517i \(-0.856212\pi\)
0.899696 − 0.436517i \(-0.143788\pi\)
\(48\) − 1.02438e14i − 0.524698i
\(49\) −9.42466e13 −0.405134
\(50\) 0 0
\(51\) 5.12542e13 0.156813
\(52\) − 3.59611e13i − 0.0932836i
\(53\) 8.01583e14i 1.76849i 0.467020 + 0.884247i \(0.345328\pi\)
−0.467020 + 0.884247i \(0.654672\pi\)
\(54\) −9.78553e13 −0.184178
\(55\) 0 0
\(56\) 8.90131e14 1.22986
\(57\) 4.71185e14i 0.560087i
\(58\) 5.65490e13i 0.0579812i
\(59\) −9.64437e14 −0.855129 −0.427565 − 0.903985i \(-0.640628\pi\)
−0.427565 + 0.903985i \(0.640628\pi\)
\(60\) 0 0
\(61\) −1.29792e15 −0.866849 −0.433424 − 0.901190i \(-0.642695\pi\)
−0.433424 + 0.901190i \(0.642695\pi\)
\(62\) 1.86670e15i 1.08579i
\(63\) − 7.78274e14i − 0.395128i
\(64\) −2.40801e15 −1.06937
\(65\) 0 0
\(66\) −3.30291e14 −0.112921
\(67\) − 4.71635e15i − 1.41896i −0.704726 − 0.709480i \(-0.748930\pi\)
0.704726 − 0.709480i \(-0.251070\pi\)
\(68\) − 8.61327e13i − 0.0228477i
\(69\) −1.09679e15 −0.256985
\(70\) 0 0
\(71\) 9.29884e15 1.70896 0.854482 − 0.519482i \(-0.173875\pi\)
0.854482 + 0.519482i \(0.173875\pi\)
\(72\) 2.11934e15i 0.345840i
\(73\) − 6.86407e15i − 0.996180i −0.867125 − 0.498090i \(-0.834035\pi\)
0.867125 − 0.498090i \(-0.165965\pi\)
\(74\) 1.27767e16 1.65176
\(75\) 0 0
\(76\) 7.91827e14 0.0816046
\(77\) − 2.62691e15i − 0.242256i
\(78\) − 7.41428e15i − 0.612724i
\(79\) −1.21446e16 −0.900646 −0.450323 − 0.892866i \(-0.648691\pi\)
−0.450323 + 0.892866i \(0.648691\pi\)
\(80\) 0 0
\(81\) 1.85302e15 0.111111
\(82\) 2.60557e15i 0.140762i
\(83\) − 3.23986e16i − 1.57893i −0.613796 − 0.789465i \(-0.710358\pi\)
0.613796 − 0.789465i \(-0.289642\pi\)
\(84\) −1.30789e15 −0.0575701
\(85\) 0 0
\(86\) −1.98855e16 −0.716638
\(87\) − 1.07083e15i − 0.0349790i
\(88\) 7.15343e15i 0.212037i
\(89\) 4.91787e16 1.32423 0.662113 − 0.749404i \(-0.269660\pi\)
0.662113 + 0.749404i \(0.269660\pi\)
\(90\) 0 0
\(91\) 5.89681e16 1.31451
\(92\) 1.84316e15i 0.0374427i
\(93\) − 3.53484e16i − 0.655036i
\(94\) −4.93784e16 −0.835507
\(95\) 0 0
\(96\) 6.84677e15 0.0968680
\(97\) − 1.26124e17i − 1.63394i −0.576678 − 0.816972i \(-0.695651\pi\)
0.576678 − 0.816972i \(-0.304349\pi\)
\(98\) 3.26543e16i 0.387720i
\(99\) 6.25450e15 0.0681230
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 75.18.b.b.49.2 4
5.2 odd 4 15.18.a.a.1.2 ✓ 2
5.3 odd 4 75.18.a.c.1.1 2
5.4 even 2 inner 75.18.b.b.49.3 4
15.2 even 4 45.18.a.b.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.18.a.a.1.2 ✓ 2 5.2 odd 4
45.18.a.b.1.1 2 15.2 even 4
75.18.a.c.1.1 2 5.3 odd 4
75.18.b.b.49.2 4 1.1 even 1 trivial
75.18.b.b.49.3 4 5.4 even 2 inner