↑

Properties

Label 275.2.a.f.1.2
Level $275$
Weight $2$
Character 275.1
Self dual yes
Analytic conductor $2.196$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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(1,275)] 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("275.1"); 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([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 275.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.30278 q^{2} -1.30278 q^{3} +3.30278 q^{4} -3.00000 q^{6} +4.30278 q^{7} +3.00000 q^{8} -1.30278 q^{9} -1.00000 q^{11} -4.30278 q^{12} +5.00000 q^{13} +9.90833 q^{14} +0.302776 q^{16} -3.90833 q^{17} -3.00000 q^{18} -1.00000 q^{19} -5.60555 q^{21} -2.30278 q^{22} -3.69722 q^{23} -3.90833 q^{24} +11.5139 q^{26} +5.60555 q^{27} +14.2111 q^{28} -9.90833 q^{29} -4.21110 q^{31} -5.30278 q^{32} +1.30278 q^{33} -9.00000 q^{34} -4.30278 q^{36} +9.60555 q^{37} -2.30278 q^{38} -6.51388 q^{39} +1.60555 q^{41} -12.9083 q^{42} -7.21110 q^{43} -3.30278 q^{44} -8.51388 q^{46} -3.00000 q^{47} -0.394449 q^{48} +11.5139 q^{49} +5.09167 q^{51} +16.5139 q^{52} +2.30278 q^{53} +12.9083 q^{54} +12.9083 q^{56} +1.30278 q^{57} -22.8167 q^{58} +0.211103 q^{59} +2.90833 q^{61} -9.69722 q^{62} -5.60555 q^{63} -12.8167 q^{64} +3.00000 q^{66} -4.00000 q^{67} -12.9083 q^{68} +4.81665 q^{69} +4.60555 q^{71} -3.90833 q^{72} +2.90833 q^{73} +22.1194 q^{74} -3.30278 q^{76} -4.30278 q^{77} -15.0000 q^{78} -0.0916731 q^{79} -3.39445 q^{81} +3.69722 q^{82} +14.5139 q^{83} -18.5139 q^{84} -16.6056 q^{86} +12.9083 q^{87} -3.00000 q^{88} +5.30278 q^{89} +21.5139 q^{91} -12.2111 q^{92} +5.48612 q^{93} -6.90833 q^{94} +6.90833 q^{96} +11.6972 q^{97} +26.5139 q^{98} +1.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + q^{3} + 3 q^{4} - 6 q^{6} + 5 q^{7} + 6 q^{8} + q^{9} - 2 q^{11} - 5 q^{12} + 10 q^{13} + 9 q^{14} - 3 q^{16} + 3 q^{17} - 6 q^{18} - 2 q^{19} - 4 q^{21} - q^{22} - 11 q^{23} + 3 q^{24}+ \cdots - q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.30278 1.62831 0.814154 − 0.580649i \(-0.197201\pi\)
0.814154 + 0.580649i \(0.197201\pi\)
\(3\) −1.30278 −0.752158 −0.376079 − 0.926588i \(-0.622728\pi\)
−0.376079 + 0.926588i \(0.622728\pi\)
\(4\) 3.30278 1.65139
\(5\) 0 0
\(6\) −3.00000 −1.22474
\(7\) 4.30278 1.62630 0.813148 − 0.582057i \(-0.197752\pi\)
0.813148 + 0.582057i \(0.197752\pi\)
\(8\) 3.00000 1.06066
\(9\) −1.30278 −0.434259
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −4.30278 −1.24210
\(13\) 5.00000 1.38675 0.693375 − 0.720577i \(-0.256123\pi\)
0.693375 + 0.720577i \(0.256123\pi\)
\(14\) 9.90833 2.64811
\(15\) 0 0
\(16\) 0.302776 0.0756939
\(17\) −3.90833 −0.947909 −0.473954 − 0.880549i \(-0.657174\pi\)
−0.473954 + 0.880549i \(0.657174\pi\)
\(18\) −3.00000 −0.707107
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −5.60555 −1.22323
\(22\) −2.30278 −0.490953
\(23\) −3.69722 −0.770925 −0.385462 − 0.922724i \(-0.625958\pi\)
−0.385462 + 0.922724i \(0.625958\pi\)
\(24\) −3.90833 −0.797784
\(25\) 0 0
\(26\) 11.5139 2.25806
\(27\) 5.60555 1.07879
\(28\) 14.2111 2.68565
\(29\) −9.90833 −1.83993 −0.919965 − 0.392000i \(-0.871783\pi\)
−0.919965 + 0.392000i \(0.871783\pi\)
\(30\) 0 0
\(31\) −4.21110 −0.756336 −0.378168 − 0.925737i \(-0.623446\pi\)
−0.378168 + 0.925737i \(0.623446\pi\)
\(32\) −5.30278 −0.937407
\(33\) 1.30278 0.226784
\(34\) −9.00000 −1.54349
\(35\) 0 0
\(36\) −4.30278 −0.717129
\(37\) 9.60555 1.57914 0.789571 − 0.613659i \(-0.210303\pi\)
0.789571 + 0.613659i \(0.210303\pi\)
\(38\) −2.30278 −0.373560
\(39\) −6.51388 −1.04306
\(40\) 0 0
\(41\) 1.60555 0.250745 0.125372 − 0.992110i \(-0.459987\pi\)
0.125372 + 0.992110i \(0.459987\pi\)
\(42\) −12.9083 −1.99180
\(43\) −7.21110 −1.09968 −0.549841 − 0.835269i \(-0.685312\pi\)
−0.549841 + 0.835269i \(0.685312\pi\)
\(44\) −3.30278 −0.497912
\(45\) 0 0
\(46\) −8.51388 −1.25530
\(47\) −3.00000 −0.437595 −0.218797 − 0.975770i \(-0.570213\pi\)
−0.218797 + 0.975770i \(0.570213\pi\)
\(48\) −0.394449 −0.0569338
\(49\) 11.5139 1.64484
\(50\) 0 0
\(51\) 5.09167 0.712977
\(52\) 16.5139 2.29006
\(53\) 2.30278 0.316311 0.158155 − 0.987414i \(-0.449445\pi\)
0.158155 + 0.987414i \(0.449445\pi\)
\(54\) 12.9083 1.75660
\(55\) 0 0
\(56\) 12.9083 1.72495
\(57\) 1.30278 0.172557
\(58\) −22.8167 −2.99597
\(59\) 0.211103 0.0274832 0.0137416 − 0.999906i \(-0.495626\pi\)
0.0137416 + 0.999906i \(0.495626\pi\)
\(60\) 0 0
\(61\) 2.90833 0.372373 0.186187 − 0.982514i \(-0.440387\pi\)
0.186187 + 0.982514i \(0.440387\pi\)
\(62\) −9.69722 −1.23155
\(63\) −5.60555 −0.706233
\(64\) −12.8167 −1.60208
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −12.9083 −1.56536
\(69\) 4.81665 0.579857
\(70\) 0 0
\(71\) 4.60555 0.546578 0.273289 − 0.961932i \(-0.411888\pi\)
0.273289 + 0.961932i \(0.411888\pi\)
\(72\) −3.90833 −0.460601
\(73\) 2.90833 0.340394 0.170197 − 0.985410i \(-0.445560\pi\)
0.170197 + 0.985410i \(0.445560\pi\)
\(74\) 22.1194 2.57133
\(75\) 0 0
\(76\) −3.30278 −0.378854
\(77\) −4.30278 −0.490347
\(78\) −15.0000 −1.69842
\(79\) −0.0916731 −0.0103140 −0.00515701 − 0.999987i \(-0.501642\pi\)
−0.00515701 + 0.999987i \(0.501642\pi\)
\(80\) 0 0
\(81\) −3.39445 −0.377161
\(82\) 3.69722 0.408290
\(83\) 14.5139 1.59311 0.796553 − 0.604569i \(-0.206655\pi\)
0.796553 + 0.604569i \(0.206655\pi\)
\(84\) −18.5139 −2.02003
\(85\) 0 0
\(86\) −16.6056 −1.79062
\(87\) 12.9083 1.38392
\(88\) −3.00000 −0.319801
\(89\) 5.30278 0.562093 0.281047 − 0.959694i \(-0.409318\pi\)
0.281047 + 0.959694i \(0.409318\pi\)
\(90\) 0 0
\(91\) 21.5139 2.25527
\(92\) −12.2111 −1.27310
\(93\) 5.48612 0.568884
\(94\) −6.90833 −0.712540
\(95\) 0 0
\(96\) 6.90833 0.705078
\(97\) 11.6972 1.18767 0.593837 − 0.804586i \(-0.297613\pi\)
0.593837 + 0.804586i \(0.297613\pi\)
\(98\) 26.5139 2.67831
\(99\) 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.a.f.1.2 yes 2
3.2 odd 2 2475.2.a.o.1.1 2
4.3 odd 2 4400.2.a.bh.1.2 2
5.2 odd 4 275.2.b.c.199.4 4
5.3 odd 4 275.2.b.c.199.1 4
5.4 even 2 275.2.a.e.1.1 ✓ 2
11.10 odd 2 3025.2.a.h.1.1 2
15.2 even 4 2475.2.c.k.199.1 4
15.8 even 4 2475.2.c.k.199.4 4
15.14 odd 2 2475.2.a.t.1.2 2
20.3 even 4 4400.2.b.y.4049.3 4
20.7 even 4 4400.2.b.y.4049.2 4
20.19 odd 2 4400.2.a.bs.1.1 2
55.54 odd 2 3025.2.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.1 ✓ 2 5.4 even 2
275.2.a.f.1.2 yes 2 1.1 even 1 trivial
275.2.b.c.199.1 4 5.3 odd 4
275.2.b.c.199.4 4 5.2 odd 4
2475.2.a.o.1.1 2 3.2 odd 2
2475.2.a.t.1.2 2 15.14 odd 2
2475.2.c.k.199.1 4 15.2 even 4
2475.2.c.k.199.4 4 15.8 even 4
3025.2.a.h.1.1 2 11.10 odd 2
3025.2.a.n.1.2 2 55.54 odd 2
4400.2.a.bh.1.2 2 4.3 odd 2
4400.2.a.bs.1.1 2 20.19 odd 2
4400.2.b.y.4049.2 4 20.7 even 4
4400.2.b.y.4049.3 4 20.3 even 4