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Properties

Label 275.2.b.c.199.4
Level $275$
Weight $2$
Character 275.199
Analytic conductor $2.196$
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] = [275,2,Mod(199,275)] 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("275.199"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,-6] 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: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.4
Root \(2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 275.199
Dual form 275.2.b.c.199.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.30278i q^{2} +1.30278i q^{3} -3.30278 q^{4} -3.00000 q^{6} +4.30278i q^{7} -3.00000i q^{8} +1.30278 q^{9} -1.00000 q^{11} -4.30278i q^{12} -5.00000i q^{13} -9.90833 q^{14} +0.302776 q^{16} -3.90833i q^{17} +3.00000i q^{18} +1.00000 q^{19} -5.60555 q^{21} -2.30278i q^{22} +3.69722i q^{23} +3.90833 q^{24} +11.5139 q^{26} +5.60555i q^{27} -14.2111i q^{28} +9.90833 q^{29} -4.21110 q^{31} -5.30278i q^{32} -1.30278i q^{33} +9.00000 q^{34} -4.30278 q^{36} +9.60555i q^{37} +2.30278i q^{38} +6.51388 q^{39} +1.60555 q^{41} -12.9083i q^{42} +7.21110i q^{43} +3.30278 q^{44} -8.51388 q^{46} -3.00000i q^{47} +0.394449i q^{48} -11.5139 q^{49} +5.09167 q^{51} +16.5139i q^{52} -2.30278i q^{53} -12.9083 q^{54} +12.9083 q^{56} +1.30278i q^{57} +22.8167i q^{58} -0.211103 q^{59} +2.90833 q^{61} -9.69722i q^{62} +5.60555i q^{63} +12.8167 q^{64} +3.00000 q^{66} -4.00000i q^{67} +12.9083i q^{68} -4.81665 q^{69} +4.60555 q^{71} -3.90833i q^{72} -2.90833i q^{73} -22.1194 q^{74} -3.30278 q^{76} -4.30278i q^{77} +15.0000i q^{78} +0.0916731 q^{79} -3.39445 q^{81} +3.69722i q^{82} -14.5139i q^{83} +18.5139 q^{84} -16.6056 q^{86} +12.9083i q^{87} +3.00000i q^{88} -5.30278 q^{89} +21.5139 q^{91} -12.2111i q^{92} -5.48612i q^{93} +6.90833 q^{94} +6.90833 q^{96} +11.6972i q^{97} -26.5139i q^{98} -1.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{4} - 12 q^{6} - 2 q^{9} - 4 q^{11} - 18 q^{14} - 6 q^{16} + 4 q^{19} - 8 q^{21} - 6 q^{24} + 10 q^{26} + 18 q^{29} + 12 q^{31} + 36 q^{34} - 10 q^{36} - 10 q^{39} - 8 q^{41} + 6 q^{44} + 2 q^{46}+ \cdots + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.30278i 1.62831i 0.580649 + 0.814154i \(0.302799\pi\)
−0.580649 + 0.814154i \(0.697201\pi\)
\(3\) 1.30278i 0.752158i 0.926588 + 0.376079i \(0.122728\pi\)
−0.926588 + 0.376079i \(0.877272\pi\)
\(4\) −3.30278 −1.65139
\(5\) 0 0
\(6\) −3.00000 −1.22474
\(7\) 4.30278i 1.62630i 0.582057 + 0.813148i \(0.302248\pi\)
−0.582057 + 0.813148i \(0.697752\pi\)
\(8\) − 3.00000i − 1.06066i
\(9\) 1.30278 0.434259
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) − 4.30278i − 1.24210i
\(13\) − 5.00000i − 1.38675i −0.720577 − 0.693375i \(-0.756123\pi\)
0.720577 − 0.693375i \(-0.243877\pi\)
\(14\) −9.90833 −2.64811
\(15\) 0 0
\(16\) 0.302776 0.0756939
\(17\) − 3.90833i − 0.947909i −0.880549 − 0.473954i \(-0.842826\pi\)
0.880549 − 0.473954i \(-0.157174\pi\)
\(18\) 3.00000i 0.707107i
\(19\) 1.00000 0.229416 0.114708 − 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) −5.60555 −1.22323
\(22\) − 2.30278i − 0.490953i
\(23\) 3.69722i 0.770925i 0.922724 + 0.385462i \(0.125958\pi\)
−0.922724 + 0.385462i \(0.874042\pi\)
\(24\) 3.90833 0.797784
\(25\) 0 0
\(26\) 11.5139 2.25806
\(27\) 5.60555i 1.07879i
\(28\) − 14.2111i − 2.68565i
\(29\) 9.90833 1.83993 0.919965 − 0.392000i \(-0.128217\pi\)
0.919965 + 0.392000i \(0.128217\pi\)
\(30\) 0 0
\(31\) −4.21110 −0.756336 −0.378168 − 0.925737i \(-0.623446\pi\)
−0.378168 + 0.925737i \(0.623446\pi\)
\(32\) − 5.30278i − 0.937407i
\(33\) − 1.30278i − 0.226784i
\(34\) 9.00000 1.54349
\(35\) 0 0
\(36\) −4.30278 −0.717129
\(37\) 9.60555i 1.57914i 0.613659 + 0.789571i \(0.289697\pi\)
−0.613659 + 0.789571i \(0.710303\pi\)
\(38\) 2.30278i 0.373560i
\(39\) 6.51388 1.04306
\(40\) 0 0
\(41\) 1.60555 0.250745 0.125372 − 0.992110i \(-0.459987\pi\)
0.125372 + 0.992110i \(0.459987\pi\)
\(42\) − 12.9083i − 1.99180i
\(43\) 7.21110i 1.09968i 0.835269 + 0.549841i \(0.185312\pi\)
−0.835269 + 0.549841i \(0.814688\pi\)
\(44\) 3.30278 0.497912
\(45\) 0 0
\(46\) −8.51388 −1.25530
\(47\) − 3.00000i − 0.437595i −0.975770 − 0.218797i \(-0.929787\pi\)
0.975770 − 0.218797i \(-0.0702134\pi\)
\(48\) 0.394449i 0.0569338i
\(49\) −11.5139 −1.64484
\(50\) 0 0
\(51\) 5.09167 0.712977
\(52\) 16.5139i 2.29006i
\(53\) − 2.30278i − 0.316311i −0.987414 − 0.158155i \(-0.949445\pi\)
0.987414 − 0.158155i \(-0.0505547\pi\)
\(54\) −12.9083 −1.75660
\(55\) 0 0
\(56\) 12.9083 1.72495
\(57\) 1.30278i 0.172557i
\(58\) 22.8167i 2.99597i
\(59\) −0.211103 −0.0274832 −0.0137416 − 0.999906i \(-0.504374\pi\)
−0.0137416 + 0.999906i \(0.504374\pi\)
\(60\) 0 0
\(61\) 2.90833 0.372373 0.186187 − 0.982514i \(-0.440387\pi\)
0.186187 + 0.982514i \(0.440387\pi\)
\(62\) − 9.69722i − 1.23155i
\(63\) 5.60555i 0.706233i
\(64\) 12.8167 1.60208
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) 12.9083i 1.56536i
\(69\) −4.81665 −0.579857
\(70\) 0 0
\(71\) 4.60555 0.546578 0.273289 − 0.961932i \(-0.411888\pi\)
0.273289 + 0.961932i \(0.411888\pi\)
\(72\) − 3.90833i − 0.460601i
\(73\) − 2.90833i − 0.340394i −0.985410 − 0.170197i \(-0.945560\pi\)
0.985410 − 0.170197i \(-0.0544404\pi\)
\(74\) −22.1194 −2.57133
\(75\) 0 0
\(76\) −3.30278 −0.378854
\(77\) − 4.30278i − 0.490347i
\(78\) 15.0000i 1.69842i
\(79\) 0.0916731 0.0103140 0.00515701 − 0.999987i \(-0.498358\pi\)
0.00515701 + 0.999987i \(0.498358\pi\)
\(80\) 0 0
\(81\) −3.39445 −0.377161
\(82\) 3.69722i 0.408290i
\(83\) − 14.5139i − 1.59311i −0.604569 − 0.796553i \(-0.706655\pi\)
0.604569 − 0.796553i \(-0.293345\pi\)
\(84\) 18.5139 2.02003
\(85\) 0 0
\(86\) −16.6056 −1.79062
\(87\) 12.9083i 1.38392i
\(88\) 3.00000i 0.319801i
\(89\) −5.30278 −0.562093 −0.281047 − 0.959694i \(-0.590682\pi\)
−0.281047 + 0.959694i \(0.590682\pi\)
\(90\) 0 0
\(91\) 21.5139 2.25527
\(92\) − 12.2111i − 1.27310i
\(93\) − 5.48612i − 0.568884i
\(94\) 6.90833 0.712540
\(95\) 0 0
\(96\) 6.90833 0.705078
\(97\) 11.6972i 1.18767i 0.804586 + 0.593837i \(0.202387\pi\)
−0.804586 + 0.593837i \(0.797613\pi\)
\(98\) − 26.5139i − 2.67831i
\(99\) −1.30278 −0.130934
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 275.2.b.c.199.4 4
3.2 odd 2 2475.2.c.k.199.1 4
4.3 odd 2 4400.2.b.y.4049.2 4
5.2 odd 4 275.2.a.e.1.1 ✓ 2
5.3 odd 4 275.2.a.f.1.2 yes 2
5.4 even 2 inner 275.2.b.c.199.1 4
15.2 even 4 2475.2.a.t.1.2 2
15.8 even 4 2475.2.a.o.1.1 2
15.14 odd 2 2475.2.c.k.199.4 4
20.3 even 4 4400.2.a.bh.1.2 2
20.7 even 4 4400.2.a.bs.1.1 2
20.19 odd 2 4400.2.b.y.4049.3 4
55.32 even 4 3025.2.a.n.1.2 2
55.43 even 4 3025.2.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.1 ✓ 2 5.2 odd 4
275.2.a.f.1.2 yes 2 5.3 odd 4
275.2.b.c.199.1 4 5.4 even 2 inner
275.2.b.c.199.4 4 1.1 even 1 trivial
2475.2.a.o.1.1 2 15.8 even 4
2475.2.a.t.1.2 2 15.2 even 4
2475.2.c.k.199.1 4 3.2 odd 2
2475.2.c.k.199.4 4 15.14 odd 2
3025.2.a.h.1.1 2 55.43 even 4
3025.2.a.n.1.2 2 55.32 even 4
4400.2.a.bh.1.2 2 20.3 even 4
4400.2.a.bs.1.1 2 20.7 even 4
4400.2.b.y.4049.2 4 4.3 odd 2
4400.2.b.y.4049.3 4 20.19 odd 2