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Properties

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

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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(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: \(1\)
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 \(-1.30278\) 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)
 
\(f(q)\) \(=\) \(q+1.30278 q^{2} -2.30278 q^{3} -0.302776 q^{4} -3.00000 q^{6} -0.697224 q^{7} -3.00000 q^{8} +2.30278 q^{9} -1.00000 q^{11} +0.697224 q^{12} -5.00000 q^{13} -0.908327 q^{14} -3.30278 q^{16} -6.90833 q^{17} +3.00000 q^{18} -1.00000 q^{19} +1.60555 q^{21} -1.30278 q^{22} +7.30278 q^{23} +6.90833 q^{24} -6.51388 q^{26} +1.60555 q^{27} +0.211103 q^{28} +0.908327 q^{29} +10.2111 q^{31} +1.69722 q^{32} +2.30278 q^{33} -9.00000 q^{34} -0.697224 q^{36} -2.39445 q^{37} -1.30278 q^{38} +11.5139 q^{39} -5.60555 q^{41} +2.09167 q^{42} -7.21110 q^{43} +0.302776 q^{44} +9.51388 q^{46} +3.00000 q^{47} +7.60555 q^{48} -6.51388 q^{49} +15.9083 q^{51} +1.51388 q^{52} +1.30278 q^{53} +2.09167 q^{54} +2.09167 q^{56} +2.30278 q^{57} +1.18335 q^{58} -14.2111 q^{59} -7.90833 q^{61} +13.3028 q^{62} -1.60555 q^{63} +8.81665 q^{64} +3.00000 q^{66} +4.00000 q^{67} +2.09167 q^{68} -16.8167 q^{69} -2.60555 q^{71} -6.90833 q^{72} +7.90833 q^{73} -3.11943 q^{74} +0.302776 q^{76} +0.697224 q^{77} +15.0000 q^{78} -10.9083 q^{79} -10.6056 q^{81} -7.30278 q^{82} +3.51388 q^{83} -0.486122 q^{84} -9.39445 q^{86} -2.09167 q^{87} +3.00000 q^{88} +1.69722 q^{89} +3.48612 q^{91} -2.21110 q^{92} -23.5139 q^{93} +3.90833 q^{94} -3.90833 q^{96} -15.3028 q^{97} -8.48612 q^{98} -2.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{3} + 3 q^{4} - 6 q^{6} - 5 q^{7} - 6 q^{8} + q^{9} - 2 q^{11} + 5 q^{12} - 10 q^{13} + 9 q^{14} - 3 q^{16} - 3 q^{17} + 6 q^{18} - 2 q^{19} - 4 q^{21} + q^{22} + 11 q^{23} + 3 q^{24}+ \cdots - q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.30278 0.921201 0.460601 − 0.887607i \(-0.347634\pi\)
0.460601 + 0.887607i \(0.347634\pi\)
\(3\) −2.30278 −1.32951 −0.664754 − 0.747062i \(-0.731464\pi\)
−0.664754 + 0.747062i \(0.731464\pi\)
\(4\) −0.302776 −0.151388
\(5\) 0 0
\(6\) −3.00000 −1.22474
\(7\) −0.697224 −0.263526 −0.131763 − 0.991281i \(-0.542064\pi\)
−0.131763 + 0.991281i \(0.542064\pi\)
\(8\) −3.00000 −1.06066
\(9\) 2.30278 0.767592
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0.697224 0.201271
\(13\) −5.00000 −1.38675 −0.693375 − 0.720577i \(-0.743877\pi\)
−0.693375 + 0.720577i \(0.743877\pi\)
\(14\) −0.908327 −0.242761
\(15\) 0 0
\(16\) −3.30278 −0.825694
\(17\) −6.90833 −1.67552 −0.837758 − 0.546042i \(-0.816134\pi\)
−0.837758 + 0.546042i \(0.816134\pi\)
\(18\) 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
\(21\) 1.60555 0.350360
\(22\) −1.30278 −0.277753
\(23\) 7.30278 1.52273 0.761367 − 0.648321i \(-0.224529\pi\)
0.761367 + 0.648321i \(0.224529\pi\)
\(24\) 6.90833 1.41016
\(25\) 0 0
\(26\) −6.51388 −1.27748
\(27\) 1.60555 0.308988
\(28\) 0.211103 0.0398946
\(29\) 0.908327 0.168672 0.0843360 − 0.996437i \(-0.473123\pi\)
0.0843360 + 0.996437i \(0.473123\pi\)
\(30\) 0 0
\(31\) 10.2111 1.83397 0.916984 − 0.398924i \(-0.130616\pi\)
0.916984 + 0.398924i \(0.130616\pi\)
\(32\) 1.69722 0.300030
\(33\) 2.30278 0.400862
\(34\) −9.00000 −1.54349
\(35\) 0 0
\(36\) −0.697224 −0.116204
\(37\) −2.39445 −0.393645 −0.196822 − 0.980439i \(-0.563062\pi\)
−0.196822 + 0.980439i \(0.563062\pi\)
\(38\) −1.30278 −0.211338
\(39\) 11.5139 1.84370
\(40\) 0 0
\(41\) −5.60555 −0.875440 −0.437720 − 0.899111i \(-0.644214\pi\)
−0.437720 + 0.899111i \(0.644214\pi\)
\(42\) 2.09167 0.322752
\(43\) −7.21110 −1.09968 −0.549841 − 0.835269i \(-0.685312\pi\)
−0.549841 + 0.835269i \(0.685312\pi\)
\(44\) 0.302776 0.0456451
\(45\) 0 0
\(46\) 9.51388 1.40274
\(47\) 3.00000 0.437595 0.218797 − 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 7.60555 1.09777
\(49\) −6.51388 −0.930554
\(50\) 0 0
\(51\) 15.9083 2.22761
\(52\) 1.51388 0.209937
\(53\) 1.30278 0.178950 0.0894750 − 0.995989i \(-0.471481\pi\)
0.0894750 + 0.995989i \(0.471481\pi\)
\(54\) 2.09167 0.284641
\(55\) 0 0
\(56\) 2.09167 0.279512
\(57\) 2.30278 0.305010
\(58\) 1.18335 0.155381
\(59\) −14.2111 −1.85013 −0.925064 − 0.379811i \(-0.875989\pi\)
−0.925064 + 0.379811i \(0.875989\pi\)
\(60\) 0 0
\(61\) −7.90833 −1.01256 −0.506279 − 0.862370i \(-0.668979\pi\)
−0.506279 + 0.862370i \(0.668979\pi\)
\(62\) 13.3028 1.68945
\(63\) −1.60555 −0.202280
\(64\) 8.81665 1.10208
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) 4.00000 0.488678 0.244339 − 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 2.09167 0.253653
\(69\) −16.8167 −2.02449
\(70\) 0 0
\(71\) −2.60555 −0.309222 −0.154611 − 0.987975i \(-0.549412\pi\)
−0.154611 + 0.987975i \(0.549412\pi\)
\(72\) −6.90833 −0.814154
\(73\) 7.90833 0.925600 0.462800 − 0.886463i \(-0.346845\pi\)
0.462800 + 0.886463i \(0.346845\pi\)
\(74\) −3.11943 −0.362626
\(75\) 0 0
\(76\) 0.302776 0.0347307
\(77\) 0.697224 0.0794561
\(78\) 15.0000 1.69842
\(79\) −10.9083 −1.22728 −0.613641 − 0.789585i \(-0.710296\pi\)
−0.613641 + 0.789585i \(0.710296\pi\)
\(80\) 0 0
\(81\) −10.6056 −1.17839
\(82\) −7.30278 −0.806457
\(83\) 3.51388 0.385698 0.192849 − 0.981228i \(-0.438227\pi\)
0.192849 + 0.981228i \(0.438227\pi\)
\(84\) −0.486122 −0.0530402
\(85\) 0 0
\(86\) −9.39445 −1.01303
\(87\) −2.09167 −0.224251
\(88\) 3.00000 0.319801
\(89\) 1.69722 0.179905 0.0899527 − 0.995946i \(-0.471328\pi\)
0.0899527 + 0.995946i \(0.471328\pi\)
\(90\) 0 0
\(91\) 3.48612 0.365445
\(92\) −2.21110 −0.230523
\(93\) −23.5139 −2.43828
\(94\) 3.90833 0.403113
\(95\) 0 0
\(96\) −3.90833 −0.398892
\(97\) −15.3028 −1.55376 −0.776881 − 0.629648i \(-0.783199\pi\)
−0.776881 + 0.629648i \(0.783199\pi\)
\(98\) −8.48612 −0.857228
\(99\) −2.30278 −0.231438
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.e.1.2 ✓ 2
3.2 odd 2 2475.2.a.t.1.1 2
4.3 odd 2 4400.2.a.bs.1.2 2
5.2 odd 4 275.2.b.c.199.3 4
5.3 odd 4 275.2.b.c.199.2 4
5.4 even 2 275.2.a.f.1.1 yes 2
11.10 odd 2 3025.2.a.n.1.1 2
15.2 even 4 2475.2.c.k.199.2 4
15.8 even 4 2475.2.c.k.199.3 4
15.14 odd 2 2475.2.a.o.1.2 2
20.3 even 4 4400.2.b.y.4049.4 4
20.7 even 4 4400.2.b.y.4049.1 4
20.19 odd 2 4400.2.a.bh.1.1 2
55.54 odd 2 3025.2.a.h.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.2 ✓ 2 1.1 even 1 trivial
275.2.a.f.1.1 yes 2 5.4 even 2
275.2.b.c.199.2 4 5.3 odd 4
275.2.b.c.199.3 4 5.2 odd 4
2475.2.a.o.1.2 2 15.14 odd 2
2475.2.a.t.1.1 2 3.2 odd 2
2475.2.c.k.199.2 4 15.2 even 4
2475.2.c.k.199.3 4 15.8 even 4
3025.2.a.h.1.2 2 55.54 odd 2
3025.2.a.n.1.1 2 11.10 odd 2
4400.2.a.bh.1.1 2 20.19 odd 2
4400.2.a.bs.1.2 2 4.3 odd 2
4400.2.b.y.4049.1 4 20.7 even 4
4400.2.b.y.4049.4 4 20.3 even 4