Properties

Label 2775.2.a.u.1.2
Level $2775$
Weight $2$
Character 2775.1
Self dual yes
Analytic conductor $22.158$
Analytic rank $0$
Dimension $3$
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] = [2775,2,Mod(1,2775)] 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("2775.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2775, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2775 = 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2775.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,-3,4,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(22.1584865609\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 3 \) 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 \(-1.91223\) of defining polynomial
Character \(\chi\) \(=\) 2775.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.656620 q^{2} -1.00000 q^{3} -1.56885 q^{4} +0.656620 q^{6} +1.34338 q^{7} +2.34338 q^{8} +1.00000 q^{9} +4.16784 q^{11} +1.56885 q^{12} +3.56885 q^{13} -0.882090 q^{14} +1.59899 q^{16} +0.744391 q^{17} -0.656620 q^{18} -1.74439 q^{19} -1.34338 q^{21} -2.73669 q^{22} +4.25561 q^{23} -2.34338 q^{24} -2.34338 q^{26} -1.00000 q^{27} -2.10756 q^{28} +1.68676 q^{29} +8.13770 q^{31} -5.73669 q^{32} -4.16784 q^{33} -0.488783 q^{34} -1.56885 q^{36} +1.00000 q^{37} +1.14540 q^{38} -3.56885 q^{39} -11.6791 q^{41} +0.882090 q^{42} +8.39331 q^{43} -6.53871 q^{44} -2.79432 q^{46} +8.79432 q^{47} -1.59899 q^{48} -5.19533 q^{49} -0.744391 q^{51} -5.59899 q^{52} +1.48108 q^{53} +0.656620 q^{54} +3.14805 q^{56} +1.74439 q^{57} -1.10756 q^{58} -11.0499 q^{59} -5.59899 q^{61} -5.34338 q^{62} +1.34338 q^{63} +0.568850 q^{64} +2.73669 q^{66} -4.28310 q^{67} -1.16784 q^{68} -4.25561 q^{69} -1.70655 q^{71} +2.34338 q^{72} -4.87439 q^{73} -0.656620 q^{74} +2.73669 q^{76} +5.59899 q^{77} +2.34338 q^{78} -7.96986 q^{79} +1.00000 q^{81} +7.66871 q^{82} +7.56115 q^{83} +2.10756 q^{84} -5.51122 q^{86} -1.68676 q^{87} +9.76683 q^{88} +5.37087 q^{89} +4.79432 q^{91} -6.67641 q^{92} -8.13770 q^{93} -5.77453 q^{94} +5.73669 q^{96} -1.13770 q^{97} +3.41136 q^{98} +4.16784 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 4 q^{4} + 6 q^{7} + 9 q^{8} + 3 q^{9} + q^{11} - 4 q^{12} + 2 q^{13} + 10 q^{14} + 2 q^{16} + 7 q^{17} - 10 q^{19} - 6 q^{21} + 12 q^{22} + 8 q^{23} - 9 q^{24} - 9 q^{26} - 3 q^{27} + 17 q^{28}+ \cdots + q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.656620 −0.464301 −0.232150 − 0.972680i \(-0.574576\pi\)
−0.232150 + 0.972680i \(0.574576\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.56885 −0.784425
\(5\) 0 0
\(6\) 0.656620 0.268064
\(7\) 1.34338 0.507750 0.253875 − 0.967237i \(-0.418295\pi\)
0.253875 + 0.967237i \(0.418295\pi\)
\(8\) 2.34338 0.828510
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.16784 1.25665 0.628325 − 0.777951i \(-0.283741\pi\)
0.628325 + 0.777951i \(0.283741\pi\)
\(12\) 1.56885 0.452888
\(13\) 3.56885 0.989821 0.494910 − 0.868944i \(-0.335201\pi\)
0.494910 + 0.868944i \(0.335201\pi\)
\(14\) −0.882090 −0.235749
\(15\) 0 0
\(16\) 1.59899 0.399747
\(17\) 0.744391 0.180541 0.0902707 − 0.995917i \(-0.471227\pi\)
0.0902707 + 0.995917i \(0.471227\pi\)
\(18\) −0.656620 −0.154767
\(19\) −1.74439 −0.400191 −0.200095 − 0.979776i \(-0.564125\pi\)
−0.200095 + 0.979776i \(0.564125\pi\)
\(20\) 0 0
\(21\) −1.34338 −0.293149
\(22\) −2.73669 −0.583464
\(23\) 4.25561 0.887356 0.443678 − 0.896186i \(-0.353673\pi\)
0.443678 + 0.896186i \(0.353673\pi\)
\(24\) −2.34338 −0.478340
\(25\) 0 0
\(26\) −2.34338 −0.459575
\(27\) −1.00000 −0.192450
\(28\) −2.10756 −0.398291
\(29\) 1.68676 0.313223 0.156612 − 0.987660i \(-0.449943\pi\)
0.156612 + 0.987660i \(0.449943\pi\)
\(30\) 0 0
\(31\) 8.13770 1.46157 0.730787 − 0.682606i \(-0.239153\pi\)
0.730787 + 0.682606i \(0.239153\pi\)
\(32\) −5.73669 −1.01411
\(33\) −4.16784 −0.725527
\(34\) −0.488783 −0.0838255
\(35\) 0 0
\(36\) −1.56885 −0.261475
\(37\) 1.00000 0.164399
\(38\) 1.14540 0.185809
\(39\) −3.56885 −0.571473
\(40\) 0 0
\(41\) −11.6791 −1.82396 −0.911981 − 0.410232i \(-0.865448\pi\)
−0.911981 + 0.410232i \(0.865448\pi\)
\(42\) 0.882090 0.136110
\(43\) 8.39331 1.27997 0.639984 − 0.768388i \(-0.278941\pi\)
0.639984 + 0.768388i \(0.278941\pi\)
\(44\) −6.53871 −0.985748
\(45\) 0 0
\(46\) −2.79432 −0.412000
\(47\) 8.79432 1.28278 0.641392 − 0.767214i \(-0.278357\pi\)
0.641392 + 0.767214i \(0.278357\pi\)
\(48\) −1.59899 −0.230794
\(49\) −5.19533 −0.742190
\(50\) 0 0
\(51\) −0.744391 −0.104236
\(52\) −5.59899 −0.776440
\(53\) 1.48108 0.203442 0.101721 − 0.994813i \(-0.467565\pi\)
0.101721 + 0.994813i \(0.467565\pi\)
\(54\) 0.656620 0.0893547
\(55\) 0 0
\(56\) 3.14805 0.420676
\(57\) 1.74439 0.231050
\(58\) −1.10756 −0.145430
\(59\) −11.0499 −1.43858 −0.719289 − 0.694711i \(-0.755532\pi\)
−0.719289 + 0.694711i \(0.755532\pi\)
\(60\) 0 0
\(61\) −5.59899 −0.716877 −0.358438 − 0.933553i \(-0.616691\pi\)
−0.358438 + 0.933553i \(0.616691\pi\)
\(62\) −5.34338 −0.678610
\(63\) 1.34338 0.169250
\(64\) 0.568850 0.0711062
\(65\) 0 0
\(66\) 2.73669 0.336863
\(67\) −4.28310 −0.523264 −0.261632 − 0.965168i \(-0.584261\pi\)
−0.261632 + 0.965168i \(0.584261\pi\)
\(68\) −1.16784 −0.141621
\(69\) −4.25561 −0.512315
\(70\) 0 0
\(71\) −1.70655 −0.202530 −0.101265 − 0.994859i \(-0.532289\pi\)
−0.101265 + 0.994859i \(0.532289\pi\)
\(72\) 2.34338 0.276170
\(73\) −4.87439 −0.570504 −0.285252 − 0.958453i \(-0.592077\pi\)
−0.285252 + 0.958453i \(0.592077\pi\)
\(74\) −0.656620 −0.0763306
\(75\) 0 0
\(76\) 2.73669 0.313920
\(77\) 5.59899 0.638064
\(78\) 2.34338 0.265335
\(79\) −7.96986 −0.896679 −0.448340 − 0.893863i \(-0.647985\pi\)
−0.448340 + 0.893863i \(0.647985\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 7.66871 0.846867
\(83\) 7.56115 0.829944 0.414972 − 0.909834i \(-0.363791\pi\)
0.414972 + 0.909834i \(0.363791\pi\)
\(84\) 2.10756 0.229954
\(85\) 0 0
\(86\) −5.51122 −0.594290
\(87\) −1.68676 −0.180840
\(88\) 9.76683 1.04115
\(89\) 5.37087 0.569311 0.284656 − 0.958630i \(-0.408121\pi\)
0.284656 + 0.958630i \(0.408121\pi\)
\(90\) 0 0
\(91\) 4.79432 0.502581
\(92\) −6.67641 −0.696064
\(93\) −8.13770 −0.843840
\(94\) −5.77453 −0.595597
\(95\) 0 0
\(96\) 5.73669 0.585498
\(97\) −1.13770 −0.115516 −0.0577579 − 0.998331i \(-0.518395\pi\)
−0.0577579 + 0.998331i \(0.518395\pi\)
\(98\) 3.41136 0.344599
\(99\) 4.16784 0.418883
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 2775.2.a.u.1.2 ✓ 3
3.2 odd 2 8325.2.a.bp.1.2 3
5.4 even 2 2775.2.a.v.1.2 yes 3
15.14 odd 2 8325.2.a.bo.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2775.2.a.u.1.2 ✓ 3 1.1 even 1 trivial
2775.2.a.v.1.2 yes 3 5.4 even 2
8325.2.a.bo.1.2 3 15.14 odd 2
8325.2.a.bp.1.2 3 3.2 odd 2