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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1665,2,Mod(334,1665)] 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("1665.334"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1665, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1665 = 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1665.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(13.2950919365\)
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: no (minimal twist has level 555)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 334.1
Root \(1.16407 + 1.16407i\) of defining polynomial
Character \(\chi\) \(=\) 1665.334
Dual form 1665.2.c.d.334.7

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1665\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(-0.806144\pi\)
0.820211 − 0.572061i \(-0.193856\pi\)
\(3\) 0 0
\(4\) −0.618034 −0.309017
\(5\) −2.14896 + 0.618034i −0.961045 + 0.276393i
\(6\) 0 0
\(7\) − 3.32813i − 1.25792i −0.777440 − 0.628958i \(-0.783482\pi\)
0.777440 − 0.628958i \(-0.216518\pi\)
\(8\) − 2.23607i − 0.790569i
\(9\) 0 0
\(10\) 1.00000 + 3.47709i 0.316228 + 1.09955i
\(11\) −1.14896 −0.346425 −0.173212 − 0.984884i \(-0.555415\pi\)
−0.173212 + 0.984884i \(0.555415\pi\)
\(12\) 0 0
\(13\) − 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 0
\(16\) −4.85410 −1.21353
\(17\) 2.67989i 0.649968i 0.945719 + 0.324984i \(0.105359\pi\)
−0.945719 + 0.324984i \(0.894641\pi\)
\(18\) 0 0
\(19\) 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
\(21\) 0 0
\(22\) 1.85906i 0.396353i
\(23\) 0.420194i 0.0876165i 0.999040 + 0.0438083i \(0.0139491\pi\)
−0.999040 + 0.0438083i \(0.986051\pi\)
\(24\) 0 0
\(25\) 4.23607 − 2.65626i 0.847214 − 0.531252i
\(26\) −3.00802 −0.589921
\(27\) 0 0
\(28\) 2.05690i 0.388717i
\(29\) −10.0413 −1.86462 −0.932310 − 0.361659i \(-0.882210\pi\)
−0.932310 + 0.361659i \(0.882210\pi\)
\(30\) 0 0
\(31\) −6.00306 −1.07818 −0.539091 − 0.842248i \(-0.681232\pi\)
−0.539091 + 0.842248i \(0.681232\pi\)
\(32\) 3.38197i 0.597853i
\(33\) 0 0
\(34\) 4.33615 0.743644
\(35\) 2.05690 + 7.15202i 0.347679 + 1.20891i
\(36\) 0 0
\(37\) − 1.00000i − 0.164399i
\(38\) − 2.94617i − 0.477931i
\(39\) 0 0
\(40\) 1.38197 + 4.80522i 0.218508 + 0.759772i
\(41\) 4.14094 0.646706 0.323353 − 0.946278i \(-0.395190\pi\)
0.323353 + 0.946278i \(0.395190\pi\)
\(42\) 0 0
\(43\) 8.24409i 1.25721i 0.777724 + 0.628606i \(0.216374\pi\)
−0.777724 + 0.628606i \(0.783626\pi\)
\(44\) 0.710097 0.107051
\(45\) 0 0
\(46\) 0.679888 0.100244
\(47\) − 11.7212i − 1.70971i −0.518867 − 0.854855i \(-0.673646\pi\)
0.518867 − 0.854855i \(-0.326354\pi\)
\(48\) 0 0
\(49\) −4.07646 −0.582351
\(50\) −4.29792 − 6.85410i −0.607818 − 0.969316i
\(51\) 0 0
\(52\) 1.14896i 0.159332i
\(53\) 1.64166i 0.225499i 0.993623 + 0.112750i \(0.0359658\pi\)
−0.993623 + 0.112750i \(0.964034\pi\)
\(54\) 0 0
\(55\) 2.46907 − 0.710097i 0.332930 − 0.0957495i
\(56\) −7.44193 −0.994469
\(57\) 0 0
\(58\) 16.2472i 2.13336i
\(59\) −7.07150 −0.920631 −0.460315 − 0.887755i \(-0.652264\pi\)
−0.460315 + 0.887755i \(0.652264\pi\)
\(60\) 0 0
\(61\) −0.294859 −0.0377528 −0.0188764 − 0.999822i \(-0.506009\pi\)
−0.0188764 + 0.999822i \(0.506009\pi\)
\(62\) 9.71316i 1.23357i
\(63\) 0 0
\(64\) −4.23607 −0.529508
\(65\) 1.14896 + 3.99504i 0.142511 + 0.495524i
\(66\) 0 0
\(67\) 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 0
\(70\) 11.5722 − 3.32813i 1.38314 − 0.397788i
\(71\) 10.3518 1.22853 0.614264 − 0.789101i \(-0.289453\pi\)
0.614264 + 0.789101i \(0.289453\pi\)
\(72\) 0 0
\(73\) 9.07150i 1.06174i 0.847454 + 0.530869i \(0.178135\pi\)
−0.847454 + 0.530869i \(0.821865\pi\)
\(74\) −1.61803 −0.188093
\(75\) 0 0
\(76\) −1.12533 −0.129085
\(77\) 3.82389i 0.435773i
\(78\) 0 0
\(79\) −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
\(81\) 0 0
\(82\) − 6.70018i − 0.739912i
\(83\) − 0.115689i − 0.0126985i −0.999980 − 0.00634927i \(-0.997979\pi\)
0.999980 − 0.00634927i \(-0.00202105\pi\)
\(84\) 0 0
\(85\) −1.65626 − 5.75898i −0.179647 − 0.624649i
\(86\) 13.3392 1.43840
\(87\) 0 0
\(88\) 2.56916i 0.273873i
\(89\) −5.51226 −0.584298 −0.292149 − 0.956373i \(-0.594370\pi\)
−0.292149 + 0.956373i \(0.594370\pi\)
\(90\) 0 0
\(91\) −6.18719 −0.648594
\(92\) − 0.259694i − 0.0270750i
\(93\) 0 0
\(94\) −18.9653 −1.95612
\(95\) −3.91289 + 1.12533i −0.401454 + 0.115457i
\(96\) 0 0
\(97\) 1.38692i 0.140821i 0.997518 + 0.0704103i \(0.0224309\pi\)
−0.997518 + 0.0704103i \(0.977569\pi\)
\(98\) 6.59584i 0.666281i
\(99\) 0 0
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 1665.2.c.d.334.1 8
3.2 odd 2 555.2.c.b.334.8 yes 8
5.2 odd 4 8325.2.a.bw.1.4 4
5.3 odd 4 8325.2.a.bt.1.1 4
5.4 even 2 inner 1665.2.c.d.334.7 8
15.2 even 4 2775.2.a.w.1.2 4
15.8 even 4 2775.2.a.y.1.3 4
15.14 odd 2 555.2.c.b.334.2 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
555.2.c.b.334.2 ✓ 8 15.14 odd 2
555.2.c.b.334.8 yes 8 3.2 odd 2
1665.2.c.d.334.1 8 1.1 even 1 trivial
1665.2.c.d.334.7 8 5.4 even 2 inner
2775.2.a.w.1.2 4 15.2 even 4
2775.2.a.y.1.3 4 15.8 even 4
8325.2.a.bt.1.1 4 5.3 odd 4
8325.2.a.bw.1.4 4 5.2 odd 4