Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [275,6,Mod(199,275)] 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("275.199"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 275.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-122] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(44.1055504486\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 70x^{6} + 1541x^{4} + 10660x^{2} + 5476 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(4.95665i\) of defining polynomial
Character \(\chi\) \(=\) 275.199
Dual form 275.6.b.d.199.7

$q$-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)
 
\(f(q)\) \(=\) \(q-7.82466i q^{2} -16.3044i q^{3} -29.2253 q^{4} -127.576 q^{6} +125.436i q^{7} -21.7111i q^{8} -22.8321 q^{9} -121.000 q^{11} +476.500i q^{12} +532.300i q^{13} +981.491 q^{14} -1105.09 q^{16} +1373.09i q^{17} +178.654i q^{18} +554.639 q^{19} +2045.15 q^{21} +946.784i q^{22} +4250.72i q^{23} -353.986 q^{24} +4165.06 q^{26} -3589.70i q^{27} -3665.89i q^{28} +6973.40 q^{29} +3130.03 q^{31} +7952.21i q^{32} +1972.83i q^{33} +10744.0 q^{34} +667.276 q^{36} -1384.70i q^{37} -4339.86i q^{38} +8678.81 q^{39} +679.385 q^{41} -16002.6i q^{42} -1721.06i q^{43} +3536.26 q^{44} +33260.4 q^{46} +15143.3i q^{47} +18017.8i q^{48} +1072.91 q^{49} +22387.4 q^{51} -15556.6i q^{52} -9544.44i q^{53} -28088.1 q^{54} +2723.35 q^{56} -9043.04i q^{57} -54564.5i q^{58} -27582.7 q^{59} -40527.5 q^{61} -24491.5i q^{62} -2863.96i q^{63} +26860.4 q^{64} +15436.7 q^{66} +58726.5i q^{67} -40129.0i q^{68} +69305.2 q^{69} -42527.8 q^{71} +495.712i q^{72} +23753.0i q^{73} -10834.8 q^{74} -16209.5 q^{76} -15177.7i q^{77} -67908.7i q^{78} +78690.1 q^{79} -64075.9 q^{81} -5315.96i q^{82} +52252.1i q^{83} -59770.0 q^{84} -13466.7 q^{86} -113697. i q^{87} +2627.05i q^{88} -8156.09 q^{89} -66769.3 q^{91} -124228. i q^{92} -51033.2i q^{93} +118491. q^{94} +129656. q^{96} -79010.1i q^{97} -8395.17i q^{98} +2762.69 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 122 q^{4} - 314 q^{6} - 44 q^{9} - 968 q^{11} + 3374 q^{14} - 5342 q^{16} + 6788 q^{19} - 9416 q^{21} - 5442 q^{24} - 7300 q^{26} + 10496 q^{29} + 9464 q^{31} - 19518 q^{34} + 14300 q^{36} + 42160 q^{39}+ \cdots + 5324 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 7.82466i − 1.38322i −0.722272 − 0.691609i \(-0.756902\pi\)
0.722272 − 0.691609i \(-0.243098\pi\)
\(3\) − 16.3044i − 1.04593i −0.852356 − 0.522963i \(-0.824827\pi\)
0.852356 − 0.522963i \(-0.175173\pi\)
\(4\) −29.2253 −0.913290
\(5\) 0 0
\(6\) −127.576 −1.44674
\(7\) 125.436i 0.967555i 0.875191 + 0.483778i \(0.160736\pi\)
−0.875191 + 0.483778i \(0.839264\pi\)
\(8\) − 21.7111i − 0.119938i
\(9\) −22.8321 −0.0939594
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 476.500i 0.955233i
\(13\) 532.300i 0.873570i 0.899566 + 0.436785i \(0.143883\pi\)
−0.899566 + 0.436785i \(0.856117\pi\)
\(14\) 981.491 1.33834
\(15\) 0 0
\(16\) −1105.09 −1.07919
\(17\) 1373.09i 1.15233i 0.817333 + 0.576166i \(0.195452\pi\)
−0.817333 + 0.576166i \(0.804548\pi\)
\(18\) 178.654i 0.129966i
\(19\) 554.639 0.352474 0.176237 − 0.984348i \(-0.443608\pi\)
0.176237 + 0.984348i \(0.443608\pi\)
\(20\) 0 0
\(21\) 2045.15 1.01199
\(22\) 946.784i 0.417056i
\(23\) 4250.72i 1.67549i 0.546060 + 0.837746i \(0.316127\pi\)
−0.546060 + 0.837746i \(0.683873\pi\)
\(24\) −353.986 −0.125446
\(25\) 0 0
\(26\) 4165.06 1.20834
\(27\) − 3589.70i − 0.947651i
\(28\) − 3665.89i − 0.883659i
\(29\) 6973.40 1.53975 0.769874 − 0.638196i \(-0.220319\pi\)
0.769874 + 0.638196i \(0.220319\pi\)
\(30\) 0 0
\(31\) 3130.03 0.584985 0.292493 − 0.956268i \(-0.405515\pi\)
0.292493 + 0.956268i \(0.405515\pi\)
\(32\) 7952.21i 1.37282i
\(33\) 1972.83i 0.315358i
\(34\) 10744.0 1.59392
\(35\) 0 0
\(36\) 667.276 0.0858122
\(37\) − 1384.70i − 0.166284i −0.996538 − 0.0831422i \(-0.973504\pi\)
0.996538 − 0.0831422i \(-0.0264956\pi\)
\(38\) − 4339.86i − 0.487548i
\(39\) 8678.81 0.913689
\(40\) 0 0
\(41\) 679.385 0.0631185 0.0315592 − 0.999502i \(-0.489953\pi\)
0.0315592 + 0.999502i \(0.489953\pi\)
\(42\) − 16002.6i − 1.39980i
\(43\) − 1721.06i − 0.141946i −0.997478 − 0.0709732i \(-0.977390\pi\)
0.997478 − 0.0709732i \(-0.0226105\pi\)
\(44\) 3536.26 0.275367
\(45\) 0 0
\(46\) 33260.4 2.31757
\(47\) 15143.3i 0.999945i 0.866041 + 0.499972i \(0.166656\pi\)
−0.866041 + 0.499972i \(0.833344\pi\)
\(48\) 18017.8i 1.12875i
\(49\) 1072.91 0.0638372
\(50\) 0 0
\(51\) 22387.4 1.20525
\(52\) − 15556.6i − 0.797823i
\(53\) − 9544.44i − 0.466724i −0.972390 − 0.233362i \(-0.925027\pi\)
0.972390 − 0.233362i \(-0.0749728\pi\)
\(54\) −28088.1 −1.31081
\(55\) 0 0
\(56\) 2723.35 0.116047
\(57\) − 9043.04i − 0.368661i
\(58\) − 54564.5i − 2.12981i
\(59\) −27582.7 −1.03159 −0.515794 − 0.856713i \(-0.672503\pi\)
−0.515794 + 0.856713i \(0.672503\pi\)
\(60\) 0 0
\(61\) −40527.5 −1.39452 −0.697261 − 0.716817i \(-0.745598\pi\)
−0.697261 + 0.716817i \(0.745598\pi\)
\(62\) − 24491.5i − 0.809162i
\(63\) − 2863.96i − 0.0909109i
\(64\) 26860.4 0.819714
\(65\) 0 0
\(66\) 15436.7 0.436209
\(67\) 58726.5i 1.59826i 0.601159 + 0.799130i \(0.294706\pi\)
−0.601159 + 0.799130i \(0.705294\pi\)
\(68\) − 40129.0i − 1.05241i
\(69\) 69305.2 1.75244
\(70\) 0 0
\(71\) −42527.8 −1.00121 −0.500607 − 0.865675i \(-0.666890\pi\)
−0.500607 + 0.865675i \(0.666890\pi\)
\(72\) 495.712i 0.0112693i
\(73\) 23753.0i 0.521690i 0.965381 + 0.260845i \(0.0840011\pi\)
−0.965381 + 0.260845i \(0.915999\pi\)
\(74\) −10834.8 −0.230007
\(75\) 0 0
\(76\) −16209.5 −0.321911
\(77\) − 15177.7i − 0.291729i
\(78\) − 67908.7i − 1.26383i
\(79\) 78690.1 1.41857 0.709287 − 0.704919i \(-0.249017\pi\)
0.709287 + 0.704919i \(0.249017\pi\)
\(80\) 0 0
\(81\) −64075.9 −1.08513
\(82\) − 5315.96i − 0.0873066i
\(83\) 52252.1i 0.832546i 0.909240 + 0.416273i \(0.136664\pi\)
−0.909240 + 0.416273i \(0.863336\pi\)
\(84\) −59770.0 −0.924241
\(85\) 0 0
\(86\) −13466.7 −0.196343
\(87\) − 113697.i − 1.61046i
\(88\) 2627.05i 0.0361627i
\(89\) −8156.09 −0.109146 −0.0545729 − 0.998510i \(-0.517380\pi\)
−0.0545729 + 0.998510i \(0.517380\pi\)
\(90\) 0 0
\(91\) −66769.3 −0.845227
\(92\) − 124228.i − 1.53021i
\(93\) − 51033.2i − 0.611851i
\(94\) 118491. 1.38314
\(95\) 0 0
\(96\) 129656. 1.43586
\(97\) − 79010.1i − 0.852615i −0.904578 − 0.426308i \(-0.859814\pi\)
0.904578 − 0.426308i \(-0.140186\pi\)
\(98\) − 8395.17i − 0.0883007i
\(99\) 2762.69 0.0283298
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 275.6.b.d.199.2 8
5.2 odd 4 55.6.a.b.1.4 ✓ 4
5.3 odd 4 275.6.a.d.1.1 4
5.4 even 2 inner 275.6.b.d.199.7 8
15.2 even 4 495.6.a.g.1.1 4
20.7 even 4 880.6.a.n.1.4 4
55.32 even 4 605.6.a.c.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.b.1.4 ✓ 4 5.2 odd 4
275.6.a.d.1.1 4 5.3 odd 4
275.6.b.d.199.2 8 1.1 even 1 trivial
275.6.b.d.199.7 8 5.4 even 2 inner
495.6.a.g.1.1 4 15.2 even 4
605.6.a.c.1.1 4 55.32 even 4
880.6.a.n.1.4 4 20.7 even 4