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Properties

Label 605.6.a.c.1.3
Level $605$
Weight $6$
Character 605.1
Self dual yes
Analytic conductor $97.032$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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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 = [4,5] 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: \(0\)
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 1.3
Root \(-2.50110\) 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)
 
\(f(q)\) \(=\) \(q+6.96278 q^{2} +22.6701 q^{3} +16.4803 q^{4} -25.0000 q^{5} +157.847 q^{6} +169.118 q^{7} -108.060 q^{8} +270.932 q^{9} -174.070 q^{10} +373.610 q^{12} -25.3182 q^{13} +1177.53 q^{14} -566.752 q^{15} -1279.77 q^{16} +2016.26 q^{17} +1886.44 q^{18} +773.486 q^{19} -412.008 q^{20} +3833.91 q^{21} -541.643 q^{23} -2449.73 q^{24} +625.000 q^{25} -176.285 q^{26} +633.231 q^{27} +2787.11 q^{28} +5882.28 q^{29} -3946.17 q^{30} -915.584 q^{31} -5452.83 q^{32} +14038.8 q^{34} -4227.94 q^{35} +4465.06 q^{36} +11360.3 q^{37} +5385.62 q^{38} -573.964 q^{39} +2701.50 q^{40} +15477.7 q^{41} +26694.7 q^{42} -6097.77 q^{43} -6773.31 q^{45} -3771.34 q^{46} +15131.8 q^{47} -29012.5 q^{48} +11793.8 q^{49} +4351.74 q^{50} +45708.8 q^{51} -417.252 q^{52} +10443.0 q^{53} +4409.05 q^{54} -18274.9 q^{56} +17535.0 q^{57} +40957.0 q^{58} -50295.3 q^{59} -9340.26 q^{60} -45523.0 q^{61} -6375.01 q^{62} +45819.4 q^{63} +2985.73 q^{64} +632.954 q^{65} +11285.0 q^{67} +33228.7 q^{68} -12279.1 q^{69} -29438.2 q^{70} +64741.3 q^{71} -29277.0 q^{72} -77769.4 q^{73} +79099.6 q^{74} +14168.8 q^{75} +12747.3 q^{76} -3996.39 q^{78} +87890.2 q^{79} +31994.2 q^{80} -51481.2 q^{81} +107768. q^{82} -18403.3 q^{83} +63184.1 q^{84} -50406.5 q^{85} -42457.4 q^{86} +133352. q^{87} +52660.1 q^{89} -47161.1 q^{90} -4281.74 q^{91} -8926.45 q^{92} -20756.4 q^{93} +105359. q^{94} -19337.2 q^{95} -123616. q^{96} -38745.0 q^{97} +82117.3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{2} + 61 q^{4} - 100 q^{5} + 157 q^{6} + 90 q^{7} + 135 q^{8} + 22 q^{9} - 125 q^{10} - 795 q^{12} - 820 q^{13} - 1687 q^{14} - 2671 q^{16} + 3800 q^{17} + 1610 q^{18} + 3394 q^{19} - 1525 q^{20}+ \cdots + 393590 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.96278 1.23086 0.615429 − 0.788192i \(-0.288983\pi\)
0.615429 + 0.788192i \(0.288983\pi\)
\(3\) 22.6701 1.45429 0.727143 − 0.686486i \(-0.240848\pi\)
0.727143 + 0.686486i \(0.240848\pi\)
\(4\) 16.4803 0.515010
\(5\) −25.0000 −0.447214
\(6\) 157.847 1.79002
\(7\) 169.118 1.30450 0.652249 − 0.758004i \(-0.273826\pi\)
0.652249 + 0.758004i \(0.273826\pi\)
\(8\) −108.060 −0.596953
\(9\) 270.932 1.11495
\(10\) −174.070 −0.550456
\(11\) 0 0
\(12\) 373.610 0.748973
\(13\) −25.3182 −0.0415502 −0.0207751 − 0.999784i \(-0.506613\pi\)
−0.0207751 + 0.999784i \(0.506613\pi\)
\(14\) 1177.53 1.60565
\(15\) −566.752 −0.650377
\(16\) −1279.77 −1.24977
\(17\) 2016.26 1.69209 0.846047 − 0.533108i \(-0.178976\pi\)
0.846047 + 0.533108i \(0.178976\pi\)
\(18\) 1886.44 1.37234
\(19\) 773.486 0.491551 0.245775 − 0.969327i \(-0.420957\pi\)
0.245775 + 0.969327i \(0.420957\pi\)
\(20\) −412.008 −0.230320
\(21\) 3833.91 1.89711
\(22\) 0 0
\(23\) −541.643 −0.213498 −0.106749 − 0.994286i \(-0.534044\pi\)
−0.106749 + 0.994286i \(0.534044\pi\)
\(24\) −2449.73 −0.868141
\(25\) 625.000 0.200000
\(26\) −176.285 −0.0511424
\(27\) 633.231 0.167168
\(28\) 2787.11 0.671830
\(29\) 5882.28 1.29882 0.649412 − 0.760437i \(-0.275015\pi\)
0.649412 + 0.760437i \(0.275015\pi\)
\(30\) −3946.17 −0.800521
\(31\) −915.584 −0.171117 −0.0855587 − 0.996333i \(-0.527268\pi\)
−0.0855587 + 0.996333i \(0.527268\pi\)
\(32\) −5452.83 −0.941342
\(33\) 0 0
\(34\) 14038.8 2.08273
\(35\) −4227.94 −0.583390
\(36\) 4465.06 0.574210
\(37\) 11360.3 1.36423 0.682115 − 0.731245i \(-0.261061\pi\)
0.682115 + 0.731245i \(0.261061\pi\)
\(38\) 5385.62 0.605029
\(39\) −573.964 −0.0604259
\(40\) 2701.50 0.266966
\(41\) 15477.7 1.43796 0.718979 − 0.695031i \(-0.244610\pi\)
0.718979 + 0.695031i \(0.244610\pi\)
\(42\) 26694.7 2.33508
\(43\) −6097.77 −0.502921 −0.251460 − 0.967868i \(-0.580911\pi\)
−0.251460 + 0.967868i \(0.580911\pi\)
\(44\) 0 0
\(45\) −6773.31 −0.498620
\(46\) −3771.34 −0.262785
\(47\) 15131.8 0.999185 0.499593 − 0.866260i \(-0.333483\pi\)
0.499593 + 0.866260i \(0.333483\pi\)
\(48\) −29012.5 −1.81753
\(49\) 11793.8 0.701717
\(50\) 4351.74 0.246172
\(51\) 45708.8 2.46079
\(52\) −417.252 −0.0213988
\(53\) 10443.0 0.510666 0.255333 − 0.966853i \(-0.417815\pi\)
0.255333 + 0.966853i \(0.417815\pi\)
\(54\) 4409.05 0.205760
\(55\) 0 0
\(56\) −18274.9 −0.778724
\(57\) 17535.0 0.714856
\(58\) 40957.0 1.59867
\(59\) −50295.3 −1.88104 −0.940519 − 0.339741i \(-0.889661\pi\)
−0.940519 + 0.339741i \(0.889661\pi\)
\(60\) −9340.26 −0.334951
\(61\) −45523.0 −1.56641 −0.783206 − 0.621762i \(-0.786417\pi\)
−0.783206 + 0.621762i \(0.786417\pi\)
\(62\) −6375.01 −0.210621
\(63\) 45819.4 1.45445
\(64\) 2985.73 0.0911172
\(65\) 632.954 0.0185818
\(66\) 0 0
\(67\) 11285.0 0.307124 0.153562 − 0.988139i \(-0.450926\pi\)
0.153562 + 0.988139i \(0.450926\pi\)
\(68\) 33228.7 0.871446
\(69\) −12279.1 −0.310487
\(70\) −29438.2 −0.718069
\(71\) 64741.3 1.52418 0.762089 − 0.647473i \(-0.224174\pi\)
0.762089 + 0.647473i \(0.224174\pi\)
\(72\) −29277.0 −0.665572
\(73\) −77769.4 −1.70805 −0.854027 − 0.520229i \(-0.825847\pi\)
−0.854027 + 0.520229i \(0.825847\pi\)
\(74\) 79099.6 1.67917
\(75\) 14168.8 0.290857
\(76\) 12747.3 0.253154
\(77\) 0 0
\(78\) −3996.39 −0.0743757
\(79\) 87890.2 1.58443 0.792214 − 0.610243i \(-0.208928\pi\)
0.792214 + 0.610243i \(0.208928\pi\)
\(80\) 31994.2 0.558916
\(81\) −51481.2 −0.871838
\(82\) 107768. 1.76992
\(83\) −18403.3 −0.293225 −0.146613 − 0.989194i \(-0.546837\pi\)
−0.146613 + 0.989194i \(0.546837\pi\)
\(84\) 63184.1 0.977034
\(85\) −50406.5 −0.756728
\(86\) −42457.4 −0.619024
\(87\) 133352. 1.88886
\(88\) 0 0
\(89\) 52660.1 0.704704 0.352352 − 0.935868i \(-0.385382\pi\)
0.352352 + 0.935868i \(0.385382\pi\)
\(90\) −47161.1 −0.613730
\(91\) −4281.74 −0.0542022
\(92\) −8926.45 −0.109954
\(93\) −20756.4 −0.248854
\(94\) 105359. 1.22985
\(95\) −19337.2 −0.219828
\(96\) −123616. −1.36898
\(97\) −38745.0 −0.418106 −0.209053 − 0.977904i \(-0.567038\pi\)
−0.209053 + 0.977904i \(0.567038\pi\)
\(98\) 82117.3 0.863714
\(99\) 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.c.1.3 4
11.10 odd 2 55.6.a.b.1.2 ✓ 4
33.32 even 2 495.6.a.g.1.3 4
44.43 even 2 880.6.a.n.1.1 4
55.32 even 4 275.6.b.d.199.3 8
55.43 even 4 275.6.b.d.199.6 8
55.54 odd 2 275.6.a.d.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.6.a.b.1.2 ✓ 4 11.10 odd 2
275.6.a.d.1.3 4 55.54 odd 2
275.6.b.d.199.3 8 55.32 even 4
275.6.b.d.199.6 8 55.43 even 4
495.6.a.g.1.3 4 33.32 even 2
605.6.a.c.1.3 4 1.1 even 1 trivial
880.6.a.n.1.1 4 44.43 even 2