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 = [3,7,36] 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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.21865.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 30x + 40 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-5.61356\) 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)
 
\(f(q)\) \(=\) \(q+4.94924 q^{2} -5.84068 q^{3} -7.50499 q^{4} -28.9069 q^{6} +1.89401 q^{7} -195.520 q^{8} -208.886 q^{9} +121.000 q^{11} +43.8342 q^{12} +378.388 q^{13} +9.37392 q^{14} -727.515 q^{16} +1083.22 q^{17} -1033.83 q^{18} -3117.36 q^{19} -11.0623 q^{21} +598.858 q^{22} +3605.50 q^{23} +1141.97 q^{24} +1872.73 q^{26} +2639.32 q^{27} -14.2145 q^{28} +2834.03 q^{29} +3702.72 q^{31} +2655.98 q^{32} -706.722 q^{33} +5361.12 q^{34} +1567.69 q^{36} +1867.45 q^{37} -15428.6 q^{38} -2210.04 q^{39} +11772.7 q^{41} -54.7500 q^{42} -18783.0 q^{43} -908.104 q^{44} +17844.5 q^{46} +19172.7 q^{47} +4249.18 q^{48} -16803.4 q^{49} -6326.74 q^{51} -2839.80 q^{52} +36095.8 q^{53} +13062.7 q^{54} -370.317 q^{56} +18207.5 q^{57} +14026.3 q^{58} +32752.8 q^{59} +11854.9 q^{61} +18325.7 q^{62} -395.633 q^{63} +36425.6 q^{64} -3497.74 q^{66} +26101.1 q^{67} -8129.56 q^{68} -21058.5 q^{69} -51549.5 q^{71} +40841.4 q^{72} -37342.3 q^{73} +9242.49 q^{74} +23395.8 q^{76} +229.175 q^{77} -10938.0 q^{78} +44628.8 q^{79} +35344.0 q^{81} +58265.7 q^{82} +41233.4 q^{83} +83.0225 q^{84} -92961.8 q^{86} -16552.7 q^{87} -23657.9 q^{88} +43519.3 q^{89} +716.671 q^{91} -27059.2 q^{92} -21626.4 q^{93} +94890.3 q^{94} -15512.7 q^{96} +85975.4 q^{97} -83164.2 q^{98} -25275.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 7 q^{2} + 36 q^{3} + 41 q^{4} + 101 q^{6} + 102 q^{7} + 15 q^{8} + 249 q^{9} + 363 q^{11} + 1237 q^{12} + 1646 q^{13} - 963 q^{14} - 2687 q^{16} + 1742 q^{17} + 3076 q^{18} - 10 q^{19} + 1236 q^{21}+ \cdots + 30129 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.94924 0.874911 0.437455 − 0.899240i \(-0.355880\pi\)
0.437455 + 0.899240i \(0.355880\pi\)
\(3\) −5.84068 −0.374680 −0.187340 − 0.982295i \(-0.559987\pi\)
−0.187340 + 0.982295i \(0.559987\pi\)
\(4\) −7.50499 −0.234531
\(5\) 0 0
\(6\) −28.9069 −0.327811
\(7\) 1.89401 0.0146096 0.00730478 − 0.999973i \(-0.497675\pi\)
0.00730478 + 0.999973i \(0.497675\pi\)
\(8\) −195.520 −1.08010
\(9\) −208.886 −0.859615
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 43.8342 0.0878740
\(13\) 378.388 0.620982 0.310491 − 0.950576i \(-0.399507\pi\)
0.310491 + 0.950576i \(0.399507\pi\)
\(14\) 9.37392 0.0127821
\(15\) 0 0
\(16\) −727.515 −0.710464
\(17\) 1083.22 0.909064 0.454532 − 0.890730i \(-0.349807\pi\)
0.454532 + 0.890730i \(0.349807\pi\)
\(18\) −1033.83 −0.752087
\(19\) −3117.36 −1.98108 −0.990542 − 0.137209i \(-0.956187\pi\)
−0.990542 + 0.137209i \(0.956187\pi\)
\(20\) 0 0
\(21\) −11.0623 −0.00547391
\(22\) 598.858 0.263796
\(23\) 3605.50 1.42117 0.710584 − 0.703613i \(-0.248431\pi\)
0.710584 + 0.703613i \(0.248431\pi\)
\(24\) 1141.97 0.404693
\(25\) 0 0
\(26\) 1872.73 0.543304
\(27\) 2639.32 0.696760
\(28\) −14.2145 −0.00342640
\(29\) 2834.03 0.625763 0.312882 − 0.949792i \(-0.398706\pi\)
0.312882 + 0.949792i \(0.398706\pi\)
\(30\) 0 0
\(31\) 3702.72 0.692018 0.346009 − 0.938231i \(-0.387537\pi\)
0.346009 + 0.938231i \(0.387537\pi\)
\(32\) 2655.98 0.458512
\(33\) −706.722 −0.112970
\(34\) 5361.12 0.795350
\(35\) 0 0
\(36\) 1567.69 0.201606
\(37\) 1867.45 0.224257 0.112128 − 0.993694i \(-0.464233\pi\)
0.112128 + 0.993694i \(0.464233\pi\)
\(38\) −15428.6 −1.73327
\(39\) −2210.04 −0.232669
\(40\) 0 0
\(41\) 11772.7 1.09374 0.546871 − 0.837217i \(-0.315819\pi\)
0.546871 + 0.837217i \(0.315819\pi\)
\(42\) −54.7500 −0.00478918
\(43\) −18783.0 −1.54915 −0.774577 − 0.632480i \(-0.782037\pi\)
−0.774577 + 0.632480i \(0.782037\pi\)
\(44\) −908.104 −0.0707138
\(45\) 0 0
\(46\) 17844.5 1.24340
\(47\) 19172.7 1.26601 0.633007 − 0.774146i \(-0.281820\pi\)
0.633007 + 0.774146i \(0.281820\pi\)
\(48\) 4249.18 0.266196
\(49\) −16803.4 −0.999787
\(50\) 0 0
\(51\) −6326.74 −0.340608
\(52\) −2839.80 −0.145640
\(53\) 36095.8 1.76509 0.882545 − 0.470228i \(-0.155828\pi\)
0.882545 + 0.470228i \(0.155828\pi\)
\(54\) 13062.7 0.609603
\(55\) 0 0
\(56\) −370.317 −0.0157799
\(57\) 18207.5 0.742272
\(58\) 14026.3 0.547487
\(59\) 32752.8 1.22495 0.612475 − 0.790490i \(-0.290174\pi\)
0.612475 + 0.790490i \(0.290174\pi\)
\(60\) 0 0
\(61\) 11854.9 0.407917 0.203959 − 0.978980i \(-0.434619\pi\)
0.203959 + 0.978980i \(0.434619\pi\)
\(62\) 18325.7 0.605454
\(63\) −395.633 −0.0125586
\(64\) 36425.6 1.11162
\(65\) 0 0
\(66\) −3497.74 −0.0988388
\(67\) 26101.1 0.710348 0.355174 − 0.934800i \(-0.384422\pi\)
0.355174 + 0.934800i \(0.384422\pi\)
\(68\) −8129.56 −0.213204
\(69\) −21058.5 −0.532483
\(70\) 0 0
\(71\) −51549.5 −1.21361 −0.606805 − 0.794851i \(-0.707549\pi\)
−0.606805 + 0.794851i \(0.707549\pi\)
\(72\) 40841.4 0.928474
\(73\) −37342.3 −0.820151 −0.410075 − 0.912052i \(-0.634498\pi\)
−0.410075 + 0.912052i \(0.634498\pi\)
\(74\) 9242.49 0.196205
\(75\) 0 0
\(76\) 23395.8 0.464626
\(77\) 229.175 0.00440495
\(78\) −10938.0 −0.203565
\(79\) 44628.8 0.804540 0.402270 − 0.915521i \(-0.368221\pi\)
0.402270 + 0.915521i \(0.368221\pi\)
\(80\) 0 0
\(81\) 35344.0 0.598553
\(82\) 58265.7 0.956926
\(83\) 41233.4 0.656983 0.328491 − 0.944507i \(-0.393460\pi\)
0.328491 + 0.944507i \(0.393460\pi\)
\(84\) 83.0225 0.00128380
\(85\) 0 0
\(86\) −92961.8 −1.35537
\(87\) −16552.7 −0.234461
\(88\) −23657.9 −0.325664
\(89\) 43519.3 0.582381 0.291190 − 0.956665i \(-0.405949\pi\)
0.291190 + 0.956665i \(0.405949\pi\)
\(90\) 0 0
\(91\) 716.671 0.00907228
\(92\) −27059.2 −0.333308
\(93\) −21626.4 −0.259285
\(94\) 94890.3 1.10765
\(95\) 0 0
\(96\) −15512.7 −0.171795
\(97\) 85975.4 0.927779 0.463890 − 0.885893i \(-0.346453\pi\)
0.463890 + 0.885893i \(0.346453\pi\)
\(98\) −83164.2 −0.874724
\(99\) −25275.3 −0.259184
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.c.1.2 3
5.2 odd 4 275.6.b.c.199.4 6
5.3 odd 4 275.6.b.c.199.3 6
5.4 even 2 55.6.a.a.1.2 ✓ 3
15.14 odd 2 495.6.a.f.1.2 3
20.19 odd 2 880.6.a.l.1.1 3
55.54 odd 2 605.6.a.b.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.a.1.2 ✓ 3 5.4 even 2
275.6.a.c.1.2 3 1.1 even 1 trivial
275.6.b.c.199.3 6 5.3 odd 4
275.6.b.c.199.4 6 5.2 odd 4
495.6.a.f.1.2 3 15.14 odd 2
605.6.a.b.1.2 3 55.54 odd 2
880.6.a.l.1.1 3 20.19 odd 2