Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [555,2,Mod(334,555)] 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("555.334"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(555, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 555 = 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 555.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] 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: \(4.43169731218\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.309760000.3
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 8x^{6} - 2x^{5} - x^{4} - 2x^{3} + 18x^{2} + 6x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 334.8
Root \(-0.164066 + 0.164066i\) of defining polynomial
Character \(\chi\) \(=\) 555.334
Dual form 555.2.c.b.334.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.61803i q^{2} -1.00000i q^{3} -0.618034 q^{4} +(2.14896 - 0.618034i) q^{5} +1.61803 q^{6} -3.32813i q^{7} +2.23607i q^{8} -1.00000 q^{9} +(1.00000 + 3.47709i) q^{10} +1.14896 q^{11} +0.618034i q^{12} -1.85906i q^{13} +5.38503 q^{14} +(-0.618034 - 2.14896i) q^{15} -4.85410 q^{16} -2.67989i q^{17} -1.61803i q^{18} +1.82083 q^{19} +(-1.32813 + 0.381966i) q^{20} -3.32813 q^{21} +1.85906i q^{22} -0.420194i q^{23} +2.23607 q^{24} +(4.23607 - 2.65626i) q^{25} +3.00802 q^{26} +1.00000i q^{27} +2.05690i q^{28} +10.0413 q^{29} +(3.47709 - 1.00000i) q^{30} -6.00306 q^{31} -3.38197i q^{32} -1.14896i q^{33} +4.33615 q^{34} +(-2.05690 - 7.15202i) q^{35} +0.618034 q^{36} -1.00000i q^{37} +2.94617i q^{38} -1.85906 q^{39} +(1.38197 + 4.80522i) q^{40} -4.14094 q^{41} -5.38503i q^{42} +8.24409i q^{43} -0.710097 q^{44} +(-2.14896 + 0.618034i) q^{45} +0.679888 q^{46} +11.7212i q^{47} +4.85410i q^{48} -4.07646 q^{49} +(4.29792 + 6.85410i) q^{50} -2.67989 q^{51} +1.14896i q^{52} -1.64166i q^{53} -1.61803 q^{54} +(2.46907 - 0.710097i) q^{55} +7.44193 q^{56} -1.82083i q^{57} +16.2472i q^{58} +7.07150 q^{59} +(0.381966 + 1.32813i) q^{60} -0.294859 q^{61} -9.71316i q^{62} +3.32813i q^{63} -4.23607 q^{64} +(-1.14896 - 3.99504i) q^{65} +1.85906 q^{66} +4.44688i q^{67} +1.65626i q^{68} -0.420194 q^{69} +(11.5722 - 3.32813i) q^{70} -10.3518 q^{71} -2.23607i q^{72} +9.07150i q^{73} +1.61803 q^{74} +(-2.65626 - 4.23607i) q^{75} -1.12533 q^{76} -3.82389i q^{77} -3.00802i q^{78} -9.63977 q^{79} +(-10.4313 + 3.00000i) q^{80} +1.00000 q^{81} -6.70018i q^{82} +0.115689i q^{83} +2.05690 q^{84} +(-1.65626 - 5.75898i) q^{85} -13.3392 q^{86} -10.0413i q^{87} +2.56916i q^{88} +5.51226 q^{89} +(-1.00000 - 3.47709i) q^{90} -6.18719 q^{91} +0.259694i q^{92} +6.00306i q^{93} -18.9653 q^{94} +(3.91289 - 1.12533i) q^{95} -3.38197 q^{96} +1.38692i q^{97} -6.59584i q^{98} -1.14896 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4} + 4 q^{6} - 8 q^{9} + 8 q^{10} - 8 q^{11} + 8 q^{14} + 4 q^{15} - 12 q^{16} + 8 q^{19} - 16 q^{21} + 16 q^{25} - 12 q^{26} + 24 q^{29} - 4 q^{31} - 12 q^{34} + 8 q^{35} - 4 q^{36} + 4 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/555\mathbb{Z}\right)^\times\).

\(n\) \(76\) \(112\) \(371\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.61803i 1.14412i 0.820211 + 0.572061i \(0.193856\pi\)
−0.820211 + 0.572061i \(0.806144\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −0.618034 −0.309017
\(5\) 2.14896 − 0.618034i 0.961045 − 0.276393i
\(6\) 1.61803 0.660560
\(7\) − 3.32813i − 1.25792i −0.777440 − 0.628958i \(-0.783482\pi\)
0.777440 − 0.628958i \(-0.216518\pi\)
\(8\) 2.23607i 0.790569i
\(9\) −1.00000 −0.333333
\(10\) 1.00000 + 3.47709i 0.316228 + 1.09955i
\(11\) 1.14896 0.346425 0.173212 − 0.984884i \(-0.444585\pi\)
0.173212 + 0.984884i \(0.444585\pi\)
\(12\) 0.618034i 0.178411i
\(13\) − 1.85906i − 0.515610i −0.966197 − 0.257805i \(-0.917001\pi\)
0.966197 − 0.257805i \(-0.0829992\pi\)
\(14\) 5.38503 1.43921
\(15\) −0.618034 − 2.14896i −0.159576 − 0.554859i
\(16\) −4.85410 −1.21353
\(17\) − 2.67989i − 0.649968i −0.945719 − 0.324984i \(-0.894641\pi\)
0.945719 − 0.324984i \(-0.105359\pi\)
\(18\) − 1.61803i − 0.381374i
\(19\) 1.82083 0.417727 0.208864 − 0.977945i \(-0.433024\pi\)
0.208864 + 0.977945i \(0.433024\pi\)
\(20\) −1.32813 + 0.381966i −0.296979 + 0.0854102i
\(21\) −3.32813 −0.726258
\(22\) 1.85906i 0.396353i
\(23\) − 0.420194i − 0.0876165i −0.999040 − 0.0438083i \(-0.986051\pi\)
0.999040 − 0.0438083i \(-0.0139491\pi\)
\(24\) 2.23607 0.456435
\(25\) 4.23607 − 2.65626i 0.847214 − 0.531252i
\(26\) 3.00802 0.589921
\(27\) 1.00000i 0.192450i
\(28\) 2.05690i 0.388717i
\(29\) 10.0413 1.86462 0.932310 − 0.361659i \(-0.117790\pi\)
0.932310 + 0.361659i \(0.117790\pi\)
\(30\) 3.47709 − 1.00000i 0.634827 − 0.182574i
\(31\) −6.00306 −1.07818 −0.539091 − 0.842248i \(-0.681232\pi\)
−0.539091 + 0.842248i \(0.681232\pi\)
\(32\) − 3.38197i − 0.597853i
\(33\) − 1.14896i − 0.200008i
\(34\) 4.33615 0.743644
\(35\) −2.05690 − 7.15202i −0.347679 − 1.20891i
\(36\) 0.618034 0.103006
\(37\) − 1.00000i − 0.164399i
\(38\) 2.94617i 0.477931i
\(39\) −1.85906 −0.297688
\(40\) 1.38197 + 4.80522i 0.218508 + 0.759772i
\(41\) −4.14094 −0.646706 −0.323353 − 0.946278i \(-0.604810\pi\)
−0.323353 + 0.946278i \(0.604810\pi\)
\(42\) − 5.38503i − 0.830928i
\(43\) 8.24409i 1.25721i 0.777724 + 0.628606i \(0.216374\pi\)
−0.777724 + 0.628606i \(0.783626\pi\)
\(44\) −0.710097 −0.107051
\(45\) −2.14896 + 0.618034i −0.320348 + 0.0921311i
\(46\) 0.679888 0.100244
\(47\) 11.7212i 1.70971i 0.518867 + 0.854855i \(0.326354\pi\)
−0.518867 + 0.854855i \(0.673646\pi\)
\(48\) 4.85410i 0.700629i
\(49\) −4.07646 −0.582351
\(50\) 4.29792 + 6.85410i 0.607818 + 0.969316i
\(51\) −2.67989 −0.375259
\(52\) 1.14896i 0.159332i
\(53\) − 1.64166i − 0.225499i −0.993623 − 0.112750i \(-0.964034\pi\)
0.993623 − 0.112750i \(-0.0359658\pi\)
\(54\) −1.61803 −0.220187
\(55\) 2.46907 − 0.710097i 0.332930 − 0.0957495i
\(56\) 7.44193 0.994469
\(57\) − 1.82083i − 0.241175i
\(58\) 16.2472i 2.13336i
\(59\) 7.07150 0.920631 0.460315 − 0.887755i \(-0.347736\pi\)
0.460315 + 0.887755i \(0.347736\pi\)
\(60\) 0.381966 + 1.32813i 0.0493116 + 0.171461i
\(61\) −0.294859 −0.0377528 −0.0188764 − 0.999822i \(-0.506009\pi\)
−0.0188764 + 0.999822i \(0.506009\pi\)
\(62\) − 9.71316i − 1.23357i
\(63\) 3.32813i 0.419305i
\(64\) −4.23607 −0.529508
\(65\) −1.14896 − 3.99504i −0.142511 − 0.495524i
\(66\) 1.85906 0.228834
\(67\) 4.44688i 0.543273i 0.962400 + 0.271637i \(0.0875649\pi\)
−0.962400 + 0.271637i \(0.912435\pi\)
\(68\) 1.65626i 0.200851i
\(69\) −0.420194 −0.0505854
\(70\) 11.5722 − 3.32813i 1.38314 − 0.397788i
\(71\) −10.3518 −1.22853 −0.614264 − 0.789101i \(-0.710547\pi\)
−0.614264 + 0.789101i \(0.710547\pi\)
\(72\) − 2.23607i − 0.263523i
\(73\) 9.07150i 1.06174i 0.847454 + 0.530869i \(0.178135\pi\)
−0.847454 + 0.530869i \(0.821865\pi\)
\(74\) 1.61803 0.188093
\(75\) −2.65626 − 4.23607i −0.306719 − 0.489139i
\(76\) −1.12533 −0.129085
\(77\) − 3.82389i − 0.435773i
\(78\) − 3.00802i − 0.340591i
\(79\) −9.63977 −1.08456 −0.542279 − 0.840198i \(-0.682439\pi\)
−0.542279 + 0.840198i \(0.682439\pi\)
\(80\) −10.4313 + 3.00000i −1.16625 + 0.335410i
\(81\) 1.00000 0.111111
\(82\) − 6.70018i − 0.739912i
\(83\) 0.115689i 0.0126985i 0.999980 + 0.00634927i \(0.00202105\pi\)
−0.999980 + 0.00634927i \(0.997979\pi\)
\(84\) 2.05690 0.224426
\(85\) −1.65626 − 5.75898i −0.179647 − 0.624649i
\(86\) −13.3392 −1.43840
\(87\) − 10.0413i − 1.07654i
\(88\) 2.56916i 0.273873i
\(89\) 5.51226 0.584298 0.292149 − 0.956373i \(-0.405630\pi\)
0.292149 + 0.956373i \(0.405630\pi\)
\(90\) −1.00000 − 3.47709i −0.105409 − 0.366518i
\(91\) −6.18719 −0.648594
\(92\) 0.259694i 0.0270750i
\(93\) 6.00306i 0.622489i
\(94\) −18.9653 −1.95612
\(95\) 3.91289 − 1.12533i 0.401454 − 0.115457i
\(96\) −3.38197 −0.345170
\(97\) 1.38692i 0.140821i 0.997518 + 0.0704103i \(0.0224309\pi\)
−0.997518 + 0.0704103i \(0.977569\pi\)
\(98\) − 6.59584i − 0.666281i
\(99\) −1.14896 −0.115475
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 555.2.c.b.334.8 yes 8
3.2 odd 2 1665.2.c.d.334.1 8
5.2 odd 4 2775.2.a.w.1.2 4
5.3 odd 4 2775.2.a.y.1.3 4
5.4 even 2 inner 555.2.c.b.334.2 ✓ 8
15.2 even 4 8325.2.a.bw.1.4 4
15.8 even 4 8325.2.a.bt.1.1 4
15.14 odd 2 1665.2.c.d.334.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
555.2.c.b.334.2 ✓ 8 5.4 even 2 inner
555.2.c.b.334.8 yes 8 1.1 even 1 trivial
1665.2.c.d.334.1 8 3.2 odd 2
1665.2.c.d.334.7 8 15.14 odd 2
2775.2.a.w.1.2 4 5.2 odd 4
2775.2.a.y.1.3 4 5.3 odd 4
8325.2.a.bt.1.1 4 15.8 even 4
8325.2.a.bw.1.4 4 15.2 even 4