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

$q$-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)
 
\(f(q)\) \(=\) \(q+7.82466i q^{2} +16.3044i q^{3} -29.2253 q^{4} -127.576 q^{6} -125.436i q^{7} +21.7111i q^{8} -22.8321 q^{9} -121.000 q^{11} -476.500i q^{12} -532.300i q^{13} +981.491 q^{14} -1105.09 q^{16} -1373.09i q^{17} -178.654i q^{18} +554.639 q^{19} +2045.15 q^{21} -946.784i q^{22} -4250.72i q^{23} -353.986 q^{24} +4165.06 q^{26} +3589.70i q^{27} +3665.89i q^{28} +6973.40 q^{29} +3130.03 q^{31} -7952.21i q^{32} -1972.83i q^{33} +10744.0 q^{34} +667.276 q^{36} +1384.70i q^{37} +4339.86i q^{38} +8678.81 q^{39} +679.385 q^{41} +16002.6i q^{42} +1721.06i q^{43} +3536.26 q^{44} +33260.4 q^{46} -15143.3i q^{47} -18017.8i q^{48} +1072.91 q^{49} +22387.4 q^{51} +15556.6i q^{52} +9544.44i q^{53} -28088.1 q^{54} +2723.35 q^{56} +9043.04i q^{57} +54564.5i q^{58} -27582.7 q^{59} -40527.5 q^{61} +24491.5i q^{62} +2863.96i q^{63} +26860.4 q^{64} +15436.7 q^{66} -58726.5i q^{67} +40129.0i q^{68} +69305.2 q^{69} -42527.8 q^{71} -495.712i q^{72} -23753.0i q^{73} -10834.8 q^{74} -16209.5 q^{76} +15177.7i q^{77} +67908.7i q^{78} +78690.1 q^{79} -64075.9 q^{81} +5315.96i q^{82} -52252.1i q^{83} -59770.0 q^{84} -13466.7 q^{86} +113697. i q^{87} -2627.05i q^{88} -8156.09 q^{89} -66769.3 q^{91} +124228. i q^{92} +51033.2i q^{93} +118491. q^{94} +129656. q^{96} +79010.1i q^{97} +8395.17i q^{98} +2762.69 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 122 q^{4} - 314 q^{6} - 44 q^{9} - 968 q^{11} + 3374 q^{14} - 5342 q^{16} + 6788 q^{19} - 9416 q^{21} - 5442 q^{24} - 7300 q^{26} + 10496 q^{29} + 9464 q^{31} - 19518 q^{34} + 14300 q^{36} + 42160 q^{39}+ \cdots + 5324 q^{99}+O(q^{100}) \) 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 \(0.243098\pi\)
−0.722272 + 0.691609i \(0.756902\pi\)
\(3\) 16.3044i 1.04593i 0.852356 + 0.522963i \(0.175173\pi\)
−0.852356 + 0.522963i \(0.824827\pi\)
\(4\) −29.2253 −0.913290
\(5\) 0 0
\(6\) −127.576 −1.44674
\(7\) − 125.436i − 0.967555i −0.875191 − 0.483778i \(-0.839264\pi\)
0.875191 − 0.483778i \(-0.160736\pi\)
\(8\) 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 \(-0.856117\pi\)
0.899566 − 0.436785i \(-0.143883\pi\)
\(14\) 981.491 1.33834
\(15\) 0 0
\(16\) −1105.09 −1.07919
\(17\) − 1373.09i − 1.15233i −0.817333 − 0.576166i \(-0.804548\pi\)
0.817333 − 0.576166i \(-0.195452\pi\)
\(18\) − 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 \(-0.683873\pi\)
0.546060 − 0.837746i \(-0.316127\pi\)
\(24\) −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 \(0.0264956\pi\)
−0.996538 + 0.0831422i \(0.973504\pi\)
\(38\) 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 \(0.0226105\pi\)
−0.997478 + 0.0709732i \(0.977390\pi\)
\(44\) 3536.26 0.275367
\(45\) 0 0
\(46\) 33260.4 2.31757
\(47\) − 15143.3i − 0.999945i −0.866041 − 0.499972i \(-0.833344\pi\)
0.866041 − 0.499972i \(-0.166656\pi\)
\(48\) − 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 \(0.0749728\pi\)
−0.972390 + 0.233362i \(0.925027\pi\)
\(54\) −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 \(-0.705294\pi\)
0.601159 − 0.799130i \(-0.294706\pi\)
\(68\) 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 \(-0.915999\pi\)
0.965381 − 0.260845i \(-0.0840011\pi\)
\(74\) −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 \(-0.863336\pi\)
0.909240 − 0.416273i \(-0.136664\pi\)
\(84\) −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 \(0.140186\pi\)
−0.904578 + 0.426308i \(0.859814\pi\)
\(98\) 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.199.7 8
5.2 odd 4 275.6.a.d.1.1 4
5.3 odd 4 55.6.a.b.1.4 ✓ 4
5.4 even 2 inner 275.6.b.d.199.2 8
15.8 even 4 495.6.a.g.1.1 4
20.3 even 4 880.6.a.n.1.4 4
55.43 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 5.3 odd 4
275.6.a.d.1.1 4 5.2 odd 4
275.6.b.d.199.2 8 5.4 even 2 inner
275.6.b.d.199.7 8 1.1 even 1 trivial
495.6.a.g.1.1 4 15.8 even 4
605.6.a.c.1.1 4 55.43 even 4
880.6.a.n.1.4 4 20.3 even 4