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Properties

Label 3552.2.f.b.1777.30
Level $3552$
Weight $2$
Character 3552.1777
Analytic conductor $28.363$
Analytic rank $0$
Dimension $44$
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] = [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 1777.30
Character \(\chi\) \(=\) 3552.1777
Dual form 3552.2.f.b.1777.15

$q$-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)
 
\(f(q)\) \(=\) \(q+1.00000i q^{3} -2.04572i q^{5} -2.83247 q^{7} -1.00000 q^{9} +0.148585i q^{11} +5.95266i q^{13} +2.04572 q^{15} +5.00506 q^{17} -1.84992i q^{19} -2.83247i q^{21} -9.18677 q^{23} +0.815031 q^{25} -1.00000i q^{27} -7.39851i q^{29} +7.74902 q^{31} -0.148585 q^{33} +5.79444i q^{35} -1.00000i q^{37} -5.95266 q^{39} -0.418741 q^{41} +12.4090i q^{43} +2.04572i q^{45} -8.07530 q^{47} +1.02288 q^{49} +5.00506i q^{51} -4.21989i q^{53} +0.303963 q^{55} +1.84992 q^{57} -5.45357i q^{59} -9.80288i q^{61} +2.83247 q^{63} +12.1775 q^{65} -15.3029i q^{67} -9.18677i q^{69} -6.84982 q^{71} -3.32175 q^{73} +0.815031i q^{75} -0.420861i q^{77} +10.8437 q^{79} +1.00000 q^{81} -1.66981i q^{83} -10.2389i q^{85} +7.39851 q^{87} -12.5465 q^{89} -16.8607i q^{91} +7.74902i q^{93} -3.78443 q^{95} -7.92559 q^{97} -0.148585i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q - 44 q^{9} + 4 q^{15} + 20 q^{17} - 76 q^{25} + 4 q^{31} - 8 q^{39} - 56 q^{41} - 24 q^{47} + 92 q^{49} - 28 q^{57} + 72 q^{65} + 32 q^{71} - 48 q^{73} + 28 q^{79} + 44 q^{81} - 12 q^{87} - 76 q^{89}+ \cdots + 32 q^{97}+O(q^{100}) \) 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
\(5\) − 2.04572i − 0.914874i −0.889242 − 0.457437i \(-0.848768\pi\)
0.889242 − 0.457437i \(-0.151232\pi\)
\(6\) 0 0
\(7\) −2.83247 −1.07057 −0.535286 − 0.844671i \(-0.679796\pi\)
−0.535286 + 0.844671i \(0.679796\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0.148585i 0.0448000i 0.999749 + 0.0224000i \(0.00713073\pi\)
−0.999749 + 0.0224000i \(0.992869\pi\)
\(12\) 0 0
\(13\) 5.95266i 1.65097i 0.564423 + 0.825486i \(0.309099\pi\)
−0.564423 + 0.825486i \(0.690901\pi\)
\(14\) 0 0
\(15\) 2.04572 0.528203
\(16\) 0 0
\(17\) 5.00506 1.21390 0.606952 − 0.794738i \(-0.292392\pi\)
0.606952 + 0.794738i \(0.292392\pi\)
\(18\) 0 0
\(19\) − 1.84992i − 0.424402i −0.977226 − 0.212201i \(-0.931937\pi\)
0.977226 − 0.212201i \(-0.0680631\pi\)
\(20\) 0 0
\(21\) − 2.83247i − 0.618095i
\(22\) 0 0
\(23\) −9.18677 −1.91557 −0.957787 − 0.287479i \(-0.907183\pi\)
−0.957787 + 0.287479i \(0.907183\pi\)
\(24\) 0 0
\(25\) 0.815031 0.163006
\(26\) 0 0
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) − 7.39851i − 1.37387i −0.726719 − 0.686935i \(-0.758956\pi\)
0.726719 − 0.686935i \(-0.241044\pi\)
\(30\) 0 0
\(31\) 7.74902 1.39177 0.695883 − 0.718156i \(-0.255013\pi\)
0.695883 + 0.718156i \(0.255013\pi\)
\(32\) 0 0
\(33\) −0.148585 −0.0258653
\(34\) 0 0
\(35\) 5.79444i 0.979439i
\(36\) 0 0
\(37\) − 1.00000i − 0.164399i
\(38\) 0 0
\(39\) −5.95266 −0.953189
\(40\) 0 0
\(41\) −0.418741 −0.0653964 −0.0326982 − 0.999465i \(-0.510410\pi\)
−0.0326982 + 0.999465i \(0.510410\pi\)
\(42\) 0 0
\(43\) 12.4090i 1.89236i 0.323646 + 0.946178i \(0.395091\pi\)
−0.323646 + 0.946178i \(0.604909\pi\)
\(44\) 0 0
\(45\) 2.04572i 0.304958i
\(46\) 0 0
\(47\) −8.07530 −1.17790 −0.588952 − 0.808168i \(-0.700459\pi\)
−0.588952 + 0.808168i \(0.700459\pi\)
\(48\) 0 0
\(49\) 1.02288 0.146126
\(50\) 0 0
\(51\) 5.00506i 0.700848i
\(52\) 0 0
\(53\) − 4.21989i − 0.579647i −0.957080 − 0.289823i \(-0.906403\pi\)
0.957080 − 0.289823i \(-0.0935966\pi\)
\(54\) 0 0
\(55\) 0.303963 0.0409863
\(56\) 0 0
\(57\) 1.84992 0.245029
\(58\) 0 0
\(59\) − 5.45357i − 0.709994i −0.934867 − 0.354997i \(-0.884482\pi\)
0.934867 − 0.354997i \(-0.115518\pi\)
\(60\) 0 0
\(61\) − 9.80288i − 1.25513i −0.778564 − 0.627565i \(-0.784052\pi\)
0.778564 − 0.627565i \(-0.215948\pi\)
\(62\) 0 0
\(63\) 2.83247 0.356858
\(64\) 0 0
\(65\) 12.1775 1.51043
\(66\) 0 0
\(67\) − 15.3029i − 1.86954i −0.355250 − 0.934771i \(-0.615604\pi\)
0.355250 − 0.934771i \(-0.384396\pi\)
\(68\) 0 0
\(69\) − 9.18677i − 1.10596i
\(70\) 0 0
\(71\) −6.84982 −0.812924 −0.406462 − 0.913668i \(-0.633238\pi\)
−0.406462 + 0.913668i \(0.633238\pi\)
\(72\) 0 0
\(73\) −3.32175 −0.388782 −0.194391 − 0.980924i \(-0.562273\pi\)
−0.194391 + 0.980924i \(0.562273\pi\)
\(74\) 0 0
\(75\) 0.815031i 0.0941117i
\(76\) 0 0
\(77\) − 0.420861i − 0.0479616i
\(78\) 0 0
\(79\) 10.8437 1.22001 0.610005 − 0.792398i \(-0.291167\pi\)
0.610005 + 0.792398i \(0.291167\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 1.66981i − 0.183286i −0.995792 − 0.0916429i \(-0.970788\pi\)
0.995792 − 0.0916429i \(-0.0292118\pi\)
\(84\) 0 0
\(85\) − 10.2389i − 1.11057i
\(86\) 0 0
\(87\) 7.39851 0.793204
\(88\) 0 0
\(89\) −12.5465 −1.32993 −0.664964 − 0.746875i \(-0.731553\pi\)
−0.664964 + 0.746875i \(0.731553\pi\)
\(90\) 0 0
\(91\) − 16.8607i − 1.76748i
\(92\) 0 0
\(93\) 7.74902i 0.803536i
\(94\) 0 0
\(95\) −3.78443 −0.388274
\(96\) 0 0
\(97\) −7.92559 −0.804722 −0.402361 − 0.915481i \(-0.631810\pi\)
−0.402361 + 0.915481i \(0.631810\pi\)
\(98\) 0 0
\(99\) − 0.148585i − 0.0149333i
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.1777.30 44
4.3 odd 2 888.2.f.b.445.14 yes 44
8.3 odd 2 888.2.f.b.445.13 ✓ 44
8.5 even 2 inner 3552.2.f.b.1777.15 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
888.2.f.b.445.13 ✓ 44 8.3 odd 2
888.2.f.b.445.14 yes 44 4.3 odd 2
3552.2.f.b.1777.15 44 8.5 even 2 inner
3552.2.f.b.1777.30 44 1.1 even 1 trivial