↑

Properties

Label 605.6.a.b.1.2
Level $605$
Weight $6$
Character 605.1
Self dual yes
Analytic conductor $97.032$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [605,6,Mod(1,605)] 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("605.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(605, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 605.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(97.0322109869\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 605.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.94924 q^{2} +5.84068 q^{3} -7.50499 q^{4} +25.0000 q^{5} +28.9069 q^{6} +1.89401 q^{7} -195.520 q^{8} -208.886 q^{9} +123.731 q^{10} -43.8342 q^{12} +378.388 q^{13} +9.37392 q^{14} +146.017 q^{15} -727.515 q^{16} +1083.22 q^{17} -1033.83 q^{18} +3117.36 q^{19} -187.625 q^{20} +11.0623 q^{21} -3605.50 q^{23} -1141.97 q^{24} +625.000 q^{25} +1872.73 q^{26} -2639.32 q^{27} -14.2145 q^{28} -2834.03 q^{29} +722.673 q^{30} +3702.72 q^{31} +2655.98 q^{32} +5361.12 q^{34} +47.3503 q^{35} +1567.69 q^{36} -1867.45 q^{37} +15428.6 q^{38} +2210.04 q^{39} -4888.00 q^{40} -11772.7 q^{41} +54.7500 q^{42} -18783.0 q^{43} -5222.16 q^{45} -17844.5 q^{46} -19172.7 q^{47} -4249.18 q^{48} -16803.4 q^{49} +3093.28 q^{50} +6326.74 q^{51} -2839.80 q^{52} -36095.8 q^{53} -13062.7 q^{54} -370.317 q^{56} +18207.5 q^{57} -14026.3 q^{58} +32752.8 q^{59} -1095.86 q^{60} -11854.9 q^{61} +18325.7 q^{62} -395.633 q^{63} +36425.6 q^{64} +9459.70 q^{65} -26101.1 q^{67} -8129.56 q^{68} -21058.5 q^{69} +234.348 q^{70} -51549.5 q^{71} +40841.4 q^{72} -37342.3 q^{73} -9242.49 q^{74} +3650.42 q^{75} -23395.8 q^{76} +10938.0 q^{78} -44628.8 q^{79} -18187.9 q^{80} +35344.0 q^{81} -58265.7 q^{82} +41233.4 q^{83} -83.0225 q^{84} +27080.5 q^{85} -92961.8 q^{86} -16552.7 q^{87} +43519.3 q^{89} -25845.8 q^{90} +716.671 q^{91} +27059.2 q^{92} +21626.4 q^{93} -94890.3 q^{94} +77934.0 q^{95} +15512.7 q^{96} -85975.4 q^{97} -83164.2 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 7 q^{2} - 36 q^{3} + 41 q^{4} + 75 q^{5} - 101 q^{6} + 102 q^{7} + 15 q^{8} + 249 q^{9} + 175 q^{10} - 1237 q^{12} + 1646 q^{13} - 963 q^{14} - 900 q^{15} - 2687 q^{16} + 1742 q^{17} + 3076 q^{18}+ \cdots - 209376 q^{98}+O(q^{100}) \) 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 \(-0.440013\pi\)
0.187340 + 0.982295i \(0.440013\pi\)
\(4\) −7.50499 −0.234531
\(5\) 25.0000 0.447214
\(6\) 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
\(10\) 123.731 0.391272
\(11\) 0 0
\(12\) −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
\(15\) 146.017 0.167562
\(16\) −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 \(-0.0438133\pi\)
0.990542 + 0.137209i \(0.0438133\pi\)
\(20\) −187.625 −0.104885
\(21\) 11.0623 0.00547391
\(22\) 0 0
\(23\) −3605.50 −1.42117 −0.710584 − 0.703613i \(-0.751569\pi\)
−0.710584 + 0.703613i \(0.751569\pi\)
\(24\) −1141.97 −0.404693
\(25\) 625.000 0.200000
\(26\) 1872.73 0.543304
\(27\) −2639.32 −0.696760
\(28\) −14.2145 −0.00342640
\(29\) −2834.03 −0.625763 −0.312882 − 0.949792i \(-0.601294\pi\)
−0.312882 + 0.949792i \(0.601294\pi\)
\(30\) 722.673 0.146602
\(31\) 3702.72 0.692018 0.346009 − 0.938231i \(-0.387537\pi\)
0.346009 + 0.938231i \(0.387537\pi\)
\(32\) 2655.98 0.458512
\(33\) 0 0
\(34\) 5361.12 0.795350
\(35\) 47.3503 0.00653360
\(36\) 1567.69 0.201606
\(37\) −1867.45 −0.224257 −0.112128 − 0.993694i \(-0.535767\pi\)
−0.112128 + 0.993694i \(0.535767\pi\)
\(38\) 15428.6 1.73327
\(39\) 2210.04 0.232669
\(40\) −4888.00 −0.483037
\(41\) −11772.7 −1.09374 −0.546871 − 0.837217i \(-0.684181\pi\)
−0.546871 + 0.837217i \(0.684181\pi\)
\(42\) 54.7500 0.00478918
\(43\) −18783.0 −1.54915 −0.774577 − 0.632480i \(-0.782037\pi\)
−0.774577 + 0.632480i \(0.782037\pi\)
\(44\) 0 0
\(45\) −5222.16 −0.384432
\(46\) −17844.5 −1.24340
\(47\) −19172.7 −1.26601 −0.633007 − 0.774146i \(-0.718180\pi\)
−0.633007 + 0.774146i \(0.718180\pi\)
\(48\) −4249.18 −0.266196
\(49\) −16803.4 −0.999787
\(50\) 3093.28 0.174982
\(51\) 6326.74 0.340608
\(52\) −2839.80 −0.145640
\(53\) −36095.8 −1.76509 −0.882545 − 0.470228i \(-0.844172\pi\)
−0.882545 + 0.470228i \(0.844172\pi\)
\(54\) −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 \(0.290174\pi\)
\(60\) −1095.86 −0.0392984
\(61\) −11854.9 −0.407917 −0.203959 − 0.978980i \(-0.565381\pi\)
−0.203959 + 0.978980i \(0.565381\pi\)
\(62\) 18325.7 0.605454
\(63\) −395.633 −0.0125586
\(64\) 36425.6 1.11162
\(65\) 9459.70 0.277712
\(66\) 0 0
\(67\) −26101.1 −0.710348 −0.355174 − 0.934800i \(-0.615578\pi\)
−0.355174 + 0.934800i \(0.615578\pi\)
\(68\) −8129.56 −0.213204
\(69\) −21058.5 −0.532483
\(70\) 234.348 0.00571631
\(71\) −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
\(75\) 3650.42 0.0749359
\(76\) −23395.8 −0.464626
\(77\) 0 0
\(78\) 10938.0 0.203565
\(79\) −44628.8 −0.804540 −0.402270 − 0.915521i \(-0.631779\pi\)
−0.402270 + 0.915521i \(0.631779\pi\)
\(80\) −18187.9 −0.317729
\(81\) 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
\(85\) 27080.5 0.406546
\(86\) −92961.8 −1.35537
\(87\) −16552.7 −0.234461
\(88\) 0 0
\(89\) 43519.3 0.582381 0.291190 − 0.956665i \(-0.405949\pi\)
0.291190 + 0.956665i \(0.405949\pi\)
\(90\) −25845.8 −0.336343
\(91\) 716.671 0.00907228
\(92\) 27059.2 0.333308
\(93\) 21626.4 0.259285
\(94\) −94890.3 −1.10765
\(95\) 77934.0 0.885968
\(96\) 15512.7 0.171795
\(97\) −85975.4 −0.927779 −0.463890 − 0.885893i \(-0.653547\pi\)
−0.463890 + 0.885893i \(0.653547\pi\)
\(98\) −83164.2 −0.874724
\(99\) 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 605.6.a.b.1.2 3
11.10 odd 2 55.6.a.a.1.2 ✓ 3
33.32 even 2 495.6.a.f.1.2 3
44.43 even 2 880.6.a.l.1.1 3
55.32 even 4 275.6.b.c.199.3 6
55.43 even 4 275.6.b.c.199.4 6
55.54 odd 2 275.6.a.c.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.a.1.2 ✓ 3 11.10 odd 2
275.6.a.c.1.2 3 55.54 odd 2
275.6.b.c.199.3 6 55.32 even 4
275.6.b.c.199.4 6 55.43 even 4
495.6.a.f.1.2 3 33.32 even 2
605.6.a.b.1.2 3 1.1 even 1 trivial
880.6.a.l.1.1 3 44.43 even 2