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(1,275)] 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("275.1"); 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([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 275.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,5,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.1055504486\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 33x^{2} - 8x + 116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.95665\) of defining polynomial
Character \(\chi\) \(=\) 275.1

$q$-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)
 
\(f(q)\) \(=\) \(q-7.82466 q^{2} +16.3044 q^{3} +29.2253 q^{4} -127.576 q^{6} +125.436 q^{7} +21.7111 q^{8} +22.8321 q^{9} -121.000 q^{11} +476.500 q^{12} -532.300 q^{13} -981.491 q^{14} -1105.09 q^{16} +1373.09 q^{17} -178.654 q^{18} -554.639 q^{19} +2045.15 q^{21} +946.784 q^{22} -4250.72 q^{23} +353.986 q^{24} +4165.06 q^{26} -3589.70 q^{27} +3665.89 q^{28} -6973.40 q^{29} +3130.03 q^{31} +7952.21 q^{32} -1972.83 q^{33} -10744.0 q^{34} +667.276 q^{36} -1384.70 q^{37} +4339.86 q^{38} -8678.81 q^{39} +679.385 q^{41} -16002.6 q^{42} +1721.06 q^{43} -3536.26 q^{44} +33260.4 q^{46} +15143.3 q^{47} -18017.8 q^{48} -1072.91 q^{49} +22387.4 q^{51} -15556.6 q^{52} +9544.44 q^{53} +28088.1 q^{54} +2723.35 q^{56} -9043.04 q^{57} +54564.5 q^{58} +27582.7 q^{59} -40527.5 q^{61} -24491.5 q^{62} +2863.96 q^{63} -26860.4 q^{64} +15436.7 q^{66} +58726.5 q^{67} +40129.0 q^{68} -69305.2 q^{69} -42527.8 q^{71} +495.712 q^{72} -23753.0 q^{73} +10834.8 q^{74} -16209.5 q^{76} -15177.7 q^{77} +67908.7 q^{78} -78690.1 q^{79} -64075.9 q^{81} -5315.96 q^{82} -52252.1 q^{83} +59770.0 q^{84} -13466.7 q^{86} -113697. q^{87} -2627.05 q^{88} +8156.09 q^{89} -66769.3 q^{91} -124228. q^{92} +51033.2 q^{93} -118491. q^{94} +129656. q^{96} -79010.1 q^{97} +8395.17 q^{98} -2762.69 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{2} + 61 q^{4} - 157 q^{6} + 90 q^{7} + 135 q^{8} + 22 q^{9} - 484 q^{11} + 795 q^{12} - 820 q^{13} - 1687 q^{14} - 2671 q^{16} + 3800 q^{17} + 1610 q^{18} - 3394 q^{19} - 4708 q^{21} - 605 q^{22}+ \cdots - 2662 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.82466 −1.38322 −0.691609 − 0.722272i \(-0.743098\pi\)
−0.691609 + 0.722272i \(0.743098\pi\)
\(3\) 16.3044 1.04593 0.522963 − 0.852356i \(-0.324827\pi\)
0.522963 + 0.852356i \(0.324827\pi\)
\(4\) 29.2253 0.913290
\(5\) 0 0
\(6\) −127.576 −1.44674
\(7\) 125.436 0.967555 0.483778 − 0.875191i \(-0.339264\pi\)
0.483778 + 0.875191i \(0.339264\pi\)
\(8\) 21.7111 0.119938
\(9\) 22.8321 0.0939594
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 476.500 0.955233
\(13\) −532.300 −0.873570 −0.436785 − 0.899566i \(-0.643883\pi\)
−0.436785 + 0.899566i \(0.643883\pi\)
\(14\) −981.491 −1.33834
\(15\) 0 0
\(16\) −1105.09 −1.07919
\(17\) 1373.09 1.15233 0.576166 − 0.817333i \(-0.304548\pi\)
0.576166 + 0.817333i \(0.304548\pi\)
\(18\) −178.654 −0.129966
\(19\) −554.639 −0.352474 −0.176237 − 0.984348i \(-0.556392\pi\)
−0.176237 + 0.984348i \(0.556392\pi\)
\(20\) 0 0
\(21\) 2045.15 1.01199
\(22\) 946.784 0.417056
\(23\) −4250.72 −1.67549 −0.837746 − 0.546060i \(-0.816127\pi\)
−0.837746 + 0.546060i \(0.816127\pi\)
\(24\) 353.986 0.125446
\(25\) 0 0
\(26\) 4165.06 1.20834
\(27\) −3589.70 −0.947651
\(28\) 3665.89 0.883659
\(29\) −6973.40 −1.53975 −0.769874 − 0.638196i \(-0.779681\pi\)
−0.769874 + 0.638196i \(0.779681\pi\)
\(30\) 0 0
\(31\) 3130.03 0.584985 0.292493 − 0.956268i \(-0.405515\pi\)
0.292493 + 0.956268i \(0.405515\pi\)
\(32\) 7952.21 1.37282
\(33\) −1972.83 −0.315358
\(34\) −10744.0 −1.59392
\(35\) 0 0
\(36\) 667.276 0.0858122
\(37\) −1384.70 −0.166284 −0.0831422 − 0.996538i \(-0.526496\pi\)
−0.0831422 + 0.996538i \(0.526496\pi\)
\(38\) 4339.86 0.487548
\(39\) −8678.81 −0.913689
\(40\) 0 0
\(41\) 679.385 0.0631185 0.0315592 − 0.999502i \(-0.489953\pi\)
0.0315592 + 0.999502i \(0.489953\pi\)
\(42\) −16002.6 −1.39980
\(43\) 1721.06 0.141946 0.0709732 − 0.997478i \(-0.477390\pi\)
0.0709732 + 0.997478i \(0.477390\pi\)
\(44\) −3536.26 −0.275367
\(45\) 0 0
\(46\) 33260.4 2.31757
\(47\) 15143.3 0.999945 0.499972 − 0.866041i \(-0.333344\pi\)
0.499972 + 0.866041i \(0.333344\pi\)
\(48\) −18017.8 −1.12875
\(49\) −1072.91 −0.0638372
\(50\) 0 0
\(51\) 22387.4 1.20525
\(52\) −15556.6 −0.797823
\(53\) 9544.44 0.466724 0.233362 − 0.972390i \(-0.425027\pi\)
0.233362 + 0.972390i \(0.425027\pi\)
\(54\) 28088.1 1.31081
\(55\) 0 0
\(56\) 2723.35 0.116047
\(57\) −9043.04 −0.368661
\(58\) 54564.5 2.12981
\(59\) 27582.7 1.03159 0.515794 − 0.856713i \(-0.327497\pi\)
0.515794 + 0.856713i \(0.327497\pi\)
\(60\) 0 0
\(61\) −40527.5 −1.39452 −0.697261 − 0.716817i \(-0.745598\pi\)
−0.697261 + 0.716817i \(0.745598\pi\)
\(62\) −24491.5 −0.809162
\(63\) 2863.96 0.0909109
\(64\) −26860.4 −0.819714
\(65\) 0 0
\(66\) 15436.7 0.436209
\(67\) 58726.5 1.59826 0.799130 − 0.601159i \(-0.205294\pi\)
0.799130 + 0.601159i \(0.205294\pi\)
\(68\) 40129.0 1.05241
\(69\) −69305.2 −1.75244
\(70\) 0 0
\(71\) −42527.8 −1.00121 −0.500607 − 0.865675i \(-0.666890\pi\)
−0.500607 + 0.865675i \(0.666890\pi\)
\(72\) 495.712 0.0112693
\(73\) −23753.0 −0.521690 −0.260845 − 0.965381i \(-0.584001\pi\)
−0.260845 + 0.965381i \(0.584001\pi\)
\(74\) 10834.8 0.230007
\(75\) 0 0
\(76\) −16209.5 −0.321911
\(77\) −15177.7 −0.291729
\(78\) 67908.7 1.26383
\(79\) −78690.1 −1.41857 −0.709287 − 0.704919i \(-0.750983\pi\)
−0.709287 + 0.704919i \(0.750983\pi\)
\(80\) 0 0
\(81\) −64075.9 −1.08513
\(82\) −5315.96 −0.0873066
\(83\) −52252.1 −0.832546 −0.416273 − 0.909240i \(-0.636664\pi\)
−0.416273 + 0.909240i \(0.636664\pi\)
\(84\) 59770.0 0.924241
\(85\) 0 0
\(86\) −13466.7 −0.196343
\(87\) −113697. −1.61046
\(88\) −2627.05 −0.0361627
\(89\) 8156.09 0.109146 0.0545729 − 0.998510i \(-0.482620\pi\)
0.0545729 + 0.998510i \(0.482620\pi\)
\(90\) 0 0
\(91\) −66769.3 −0.845227
\(92\) −124228. −1.53021
\(93\) 51033.2 0.611851
\(94\) −118491. −1.38314
\(95\) 0 0
\(96\) 129656. 1.43586
\(97\) −79010.1 −0.852615 −0.426308 − 0.904578i \(-0.640186\pi\)
−0.426308 + 0.904578i \(0.640186\pi\)
\(98\) 8395.17 0.0883007
\(99\) −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.a.d.1.1 4
5.2 odd 4 275.6.b.d.199.2 8
5.3 odd 4 275.6.b.d.199.7 8
5.4 even 2 55.6.a.b.1.4 ✓ 4
15.14 odd 2 495.6.a.g.1.1 4
20.19 odd 2 880.6.a.n.1.4 4
55.54 odd 2 605.6.a.c.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.b.1.4 ✓ 4 5.4 even 2
275.6.a.d.1.1 4 1.1 even 1 trivial
275.6.b.d.199.2 8 5.2 odd 4
275.6.b.d.199.7 8 5.3 odd 4
495.6.a.g.1.1 4 15.14 odd 2
605.6.a.c.1.1 4 55.54 odd 2
880.6.a.n.1.4 4 20.19 odd 2