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Properties

Label 3552.2.f.b.1777.1
Level $3552$
Weight $2$
Character 3552.1777
Analytic conductor $28.363$
Analytic rank $0$
Dimension $44$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3552,2,Mod(1777,3552)] 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("3552.1777"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3552, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3552 = 2^{5} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3552.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [44] 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: \(28.3628627980\)
Analytic rank: \(0\)
Dimension: \(44\)
Twist minimal: no (minimal twist has level 888)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1777.1
Character \(\chi\) \(=\) 3552.1777
Dual form 3552.2.f.b.1777.44

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -4.40147i q^{5} +1.69571 q^{7} -1.00000 q^{9} +2.08645i q^{11} -0.687546i q^{13} -4.40147 q^{15} +3.68856 q^{17} -2.86790i q^{19} -1.69571i q^{21} +3.21526 q^{23} -14.3730 q^{25} +1.00000i q^{27} -6.84958i q^{29} +2.02754 q^{31} +2.08645 q^{33} -7.46360i q^{35} +1.00000i q^{37} -0.687546 q^{39} -6.69225 q^{41} -7.41574i q^{43} +4.40147i q^{45} -1.07133 q^{47} -4.12458 q^{49} -3.68856i q^{51} +2.99631i q^{53} +9.18345 q^{55} -2.86790 q^{57} -6.10312i q^{59} -13.1839i q^{61} -1.69571 q^{63} -3.02621 q^{65} -6.43114i q^{67} -3.21526i q^{69} +7.80505 q^{71} -13.8509 q^{73} +14.3730i q^{75} +3.53801i q^{77} +7.93106 q^{79} +1.00000 q^{81} +14.2158i q^{83} -16.2351i q^{85} -6.84958 q^{87} -1.81697 q^{89} -1.16588i q^{91} -2.02754i q^{93} -12.6230 q^{95} -3.61017 q^{97} -2.08645i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q - 44 q^{9} + 4 q^{15} + 20 q^{17} - 76 q^{25} + 4 q^{31} - 8 q^{39} - 56 q^{41} - 24 q^{47} + 92 q^{49} - 28 q^{57} + 72 q^{65} + 32 q^{71} - 48 q^{73} + 28 q^{79} + 44 q^{81} - 12 q^{87} - 76 q^{89}+ \cdots + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3552\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(2369\) \(3073\) \(3109\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) − 4.40147i − 1.96840i −0.177066 − 0.984199i \(-0.556661\pi\)
0.177066 − 0.984199i \(-0.443339\pi\)
\(6\) 0 0
\(7\) 1.69571 0.640917 0.320458 − 0.947263i \(-0.396163\pi\)
0.320458 + 0.947263i \(0.396163\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 2.08645i 0.629088i 0.949243 + 0.314544i \(0.101852\pi\)
−0.949243 + 0.314544i \(0.898148\pi\)
\(12\) 0 0
\(13\) − 0.687546i − 0.190691i −0.995444 − 0.0953455i \(-0.969604\pi\)
0.995444 − 0.0953455i \(-0.0303956\pi\)
\(14\) 0 0
\(15\) −4.40147 −1.13646
\(16\) 0 0
\(17\) 3.68856 0.894607 0.447303 − 0.894382i \(-0.352384\pi\)
0.447303 + 0.894382i \(0.352384\pi\)
\(18\) 0 0
\(19\) − 2.86790i − 0.657940i −0.944340 − 0.328970i \(-0.893298\pi\)
0.944340 − 0.328970i \(-0.106702\pi\)
\(20\) 0 0
\(21\) − 1.69571i − 0.370033i
\(22\) 0 0
\(23\) 3.21526 0.670427 0.335214 − 0.942142i \(-0.391191\pi\)
0.335214 + 0.942142i \(0.391191\pi\)
\(24\) 0 0
\(25\) −14.3730 −2.87459
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) − 6.84958i − 1.27193i −0.771716 − 0.635967i \(-0.780601\pi\)
0.771716 − 0.635967i \(-0.219399\pi\)
\(30\) 0 0
\(31\) 2.02754 0.364157 0.182078 − 0.983284i \(-0.441718\pi\)
0.182078 + 0.983284i \(0.441718\pi\)
\(32\) 0 0
\(33\) 2.08645 0.363204
\(34\) 0 0
\(35\) − 7.46360i − 1.26158i
\(36\) 0 0
\(37\) 1.00000i 0.164399i
\(38\) 0 0
\(39\) −0.687546 −0.110095
\(40\) 0 0
\(41\) −6.69225 −1.04515 −0.522577 − 0.852592i \(-0.675029\pi\)
−0.522577 + 0.852592i \(0.675029\pi\)
\(42\) 0 0
\(43\) − 7.41574i − 1.13089i −0.824786 − 0.565445i \(-0.808705\pi\)
0.824786 − 0.565445i \(-0.191295\pi\)
\(44\) 0 0
\(45\) 4.40147i 0.656133i
\(46\) 0 0
\(47\) −1.07133 −0.156269 −0.0781345 − 0.996943i \(-0.524896\pi\)
−0.0781345 + 0.996943i \(0.524896\pi\)
\(48\) 0 0
\(49\) −4.12458 −0.589226
\(50\) 0 0
\(51\) − 3.68856i − 0.516502i
\(52\) 0 0
\(53\) 2.99631i 0.411575i 0.978597 + 0.205788i \(0.0659756\pi\)
−0.978597 + 0.205788i \(0.934024\pi\)
\(54\) 0 0
\(55\) 9.18345 1.23830
\(56\) 0 0
\(57\) −2.86790 −0.379862
\(58\) 0 0
\(59\) − 6.10312i − 0.794559i −0.917698 − 0.397279i \(-0.869954\pi\)
0.917698 − 0.397279i \(-0.130046\pi\)
\(60\) 0 0
\(61\) − 13.1839i − 1.68802i −0.536326 − 0.844011i \(-0.680188\pi\)
0.536326 − 0.844011i \(-0.319812\pi\)
\(62\) 0 0
\(63\) −1.69571 −0.213639
\(64\) 0 0
\(65\) −3.02621 −0.375356
\(66\) 0 0
\(67\) − 6.43114i − 0.785689i −0.919605 − 0.392844i \(-0.871491\pi\)
0.919605 − 0.392844i \(-0.128509\pi\)
\(68\) 0 0
\(69\) − 3.21526i − 0.387071i
\(70\) 0 0
\(71\) 7.80505 0.926289 0.463144 − 0.886283i \(-0.346721\pi\)
0.463144 + 0.886283i \(0.346721\pi\)
\(72\) 0 0
\(73\) −13.8509 −1.62113 −0.810563 − 0.585652i \(-0.800839\pi\)
−0.810563 + 0.585652i \(0.800839\pi\)
\(74\) 0 0
\(75\) 14.3730i 1.65965i
\(76\) 0 0
\(77\) 3.53801i 0.403193i
\(78\) 0 0
\(79\) 7.93106 0.892313 0.446157 − 0.894955i \(-0.352792\pi\)
0.446157 + 0.894955i \(0.352792\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 14.2158i 1.56038i 0.625542 + 0.780191i \(0.284878\pi\)
−0.625542 + 0.780191i \(0.715122\pi\)
\(84\) 0 0
\(85\) − 16.2351i − 1.76094i
\(86\) 0 0
\(87\) −6.84958 −0.734352
\(88\) 0 0
\(89\) −1.81697 −0.192599 −0.0962993 − 0.995352i \(-0.530701\pi\)
−0.0962993 + 0.995352i \(0.530701\pi\)
\(90\) 0 0
\(91\) − 1.16588i − 0.122217i
\(92\) 0 0
\(93\) − 2.02754i − 0.210246i
\(94\) 0 0
\(95\) −12.6230 −1.29509
\(96\) 0 0
\(97\) −3.61017 −0.366557 −0.183278 − 0.983061i \(-0.558671\pi\)
−0.183278 + 0.983061i \(0.558671\pi\)
\(98\) 0 0
\(99\) − 2.08645i − 0.209696i
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 3552.2.f.b.1777.1 44
4.3 odd 2 888.2.f.b.445.2 yes 44
8.3 odd 2 888.2.f.b.445.1 ✓ 44
8.5 even 2 inner 3552.2.f.b.1777.44 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.f.b.445.1 ✓ 44 8.3 odd 2
888.2.f.b.445.2 yes 44 4.3 odd 2
3552.2.f.b.1777.1 44 1.1 even 1 trivial
3552.2.f.b.1777.44 44 8.5 even 2 inner