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Properties

Label 175.6.a.i.1.5
Level $175$
Weight $6$
Character 175.1
Self dual yes
Analytic conductor $28.067$
Analytic rank $0$
Dimension $6$
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] = [175,6,Mod(1,175)] 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("175.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 175.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-10] 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: \(28.0671684673\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 158x^{4} + 131x^{3} + 6470x^{2} + 700x - 36400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(7.99461\) of defining polynomial
Character \(\chi\) \(=\) 175.1

$q$-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)
 
\(f(q)\) \(=\) \(q+5.99461 q^{2} -22.9537 q^{3} +3.93531 q^{4} -137.599 q^{6} -49.0000 q^{7} -168.237 q^{8} +283.874 q^{9} -231.728 q^{11} -90.3300 q^{12} +123.094 q^{13} -293.736 q^{14} -1134.44 q^{16} -1409.45 q^{17} +1701.71 q^{18} +2708.15 q^{19} +1124.73 q^{21} -1389.12 q^{22} +2504.52 q^{23} +3861.66 q^{24} +737.903 q^{26} -938.209 q^{27} -192.830 q^{28} +5089.72 q^{29} -1960.33 q^{31} -1416.96 q^{32} +5319.03 q^{33} -8449.13 q^{34} +1117.13 q^{36} +4793.23 q^{37} +16234.3 q^{38} -2825.48 q^{39} +9776.81 q^{41} +6742.33 q^{42} -10219.2 q^{43} -911.922 q^{44} +15013.6 q^{46} +12372.9 q^{47} +26039.7 q^{48} +2401.00 q^{49} +32352.2 q^{51} +484.414 q^{52} -23667.6 q^{53} -5624.19 q^{54} +8243.60 q^{56} -62162.0 q^{57} +30510.9 q^{58} +32894.0 q^{59} +49850.8 q^{61} -11751.4 q^{62} -13909.8 q^{63} +27808.0 q^{64} +31885.5 q^{66} -40528.1 q^{67} -5546.64 q^{68} -57488.0 q^{69} -40565.8 q^{71} -47758.0 q^{72} -60363.0 q^{73} +28733.5 q^{74} +10657.4 q^{76} +11354.7 q^{77} -16937.6 q^{78} -71242.1 q^{79} -47446.0 q^{81} +58608.1 q^{82} -28208.3 q^{83} +4426.17 q^{84} -61259.8 q^{86} -116828. q^{87} +38985.2 q^{88} +107437. q^{89} -6031.63 q^{91} +9856.05 q^{92} +44997.0 q^{93} +74170.7 q^{94} +32524.6 q^{96} +118741. q^{97} +14393.1 q^{98} -65781.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{2} - 16 q^{3} + 144 q^{4} + 223 q^{6} - 294 q^{7} - 741 q^{8} + 840 q^{9} + 784 q^{11} + 351 q^{12} + 12 q^{13} + 490 q^{14} + 2640 q^{16} + 562 q^{17} - 635 q^{18} + 2256 q^{19} + 784 q^{21}+ \cdots + 56764 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.99461 1.05971 0.529853 − 0.848089i \(-0.322247\pi\)
0.529853 + 0.848089i \(0.322247\pi\)
\(3\) −22.9537 −1.47248 −0.736241 − 0.676719i \(-0.763401\pi\)
−0.736241 + 0.676719i \(0.763401\pi\)
\(4\) 3.93531 0.122978
\(5\) 0 0
\(6\) −137.599 −1.56040
\(7\) −49.0000 −0.377964
\(8\) −168.237 −0.929386
\(9\) 283.874 1.16821
\(10\) 0 0
\(11\) −231.728 −0.577427 −0.288714 − 0.957416i \(-0.593228\pi\)
−0.288714 + 0.957416i \(0.593228\pi\)
\(12\) −90.3300 −0.181083
\(13\) 123.094 0.202013 0.101007 − 0.994886i \(-0.467794\pi\)
0.101007 + 0.994886i \(0.467794\pi\)
\(14\) −293.736 −0.400531
\(15\) 0 0
\(16\) −1134.44 −1.10785
\(17\) −1409.45 −1.18285 −0.591424 − 0.806361i \(-0.701434\pi\)
−0.591424 + 0.806361i \(0.701434\pi\)
\(18\) 1701.71 1.23796
\(19\) 2708.15 1.72103 0.860514 − 0.509427i \(-0.170143\pi\)
0.860514 + 0.509427i \(0.170143\pi\)
\(20\) 0 0
\(21\) 1124.73 0.556546
\(22\) −1389.12 −0.611903
\(23\) 2504.52 0.987199 0.493599 − 0.869689i \(-0.335681\pi\)
0.493599 + 0.869689i \(0.335681\pi\)
\(24\) 3861.66 1.36850
\(25\) 0 0
\(26\) 737.903 0.214075
\(27\) −938.209 −0.247680
\(28\) −192.830 −0.0464814
\(29\) 5089.72 1.12383 0.561913 − 0.827197i \(-0.310066\pi\)
0.561913 + 0.827197i \(0.310066\pi\)
\(30\) 0 0
\(31\) −1960.33 −0.366375 −0.183187 − 0.983078i \(-0.558642\pi\)
−0.183187 + 0.983078i \(0.558642\pi\)
\(32\) −1416.96 −0.244615
\(33\) 5319.03 0.850251
\(34\) −8449.13 −1.25347
\(35\) 0 0
\(36\) 1117.13 0.143664
\(37\) 4793.23 0.575604 0.287802 − 0.957690i \(-0.407076\pi\)
0.287802 + 0.957690i \(0.407076\pi\)
\(38\) 16234.3 1.82378
\(39\) −2825.48 −0.297461
\(40\) 0 0
\(41\) 9776.81 0.908317 0.454159 − 0.890921i \(-0.349940\pi\)
0.454159 + 0.890921i \(0.349940\pi\)
\(42\) 6742.33 0.589776
\(43\) −10219.2 −0.842838 −0.421419 − 0.906866i \(-0.638468\pi\)
−0.421419 + 0.906866i \(0.638468\pi\)
\(44\) −911.922 −0.0710110
\(45\) 0 0
\(46\) 15013.6 1.04614
\(47\) 12372.9 0.817009 0.408505 − 0.912756i \(-0.366050\pi\)
0.408505 + 0.912756i \(0.366050\pi\)
\(48\) 26039.7 1.63130
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) 32352.2 1.74172
\(52\) 484.414 0.0248433
\(53\) −23667.6 −1.15735 −0.578674 − 0.815559i \(-0.696430\pi\)
−0.578674 + 0.815559i \(0.696430\pi\)
\(54\) −5624.19 −0.262468
\(55\) 0 0
\(56\) 8243.60 0.351275
\(57\) −62162.0 −2.53418
\(58\) 30510.9 1.19093
\(59\) 32894.0 1.23023 0.615115 − 0.788437i \(-0.289109\pi\)
0.615115 + 0.788437i \(0.289109\pi\)
\(60\) 0 0
\(61\) 49850.8 1.71533 0.857665 − 0.514209i \(-0.171914\pi\)
0.857665 + 0.514209i \(0.171914\pi\)
\(62\) −11751.4 −0.388250
\(63\) −13909.8 −0.441540
\(64\) 27808.0 0.848634
\(65\) 0 0
\(66\) 31885.5 0.901017
\(67\) −40528.1 −1.10298 −0.551492 − 0.834180i \(-0.685941\pi\)
−0.551492 + 0.834180i \(0.685941\pi\)
\(68\) −5546.64 −0.145465
\(69\) −57488.0 −1.45363
\(70\) 0 0
\(71\) −40565.8 −0.955023 −0.477512 − 0.878625i \(-0.658461\pi\)
−0.477512 + 0.878625i \(0.658461\pi\)
\(72\) −47758.0 −1.08571
\(73\) −60363.0 −1.32576 −0.662879 − 0.748727i \(-0.730665\pi\)
−0.662879 + 0.748727i \(0.730665\pi\)
\(74\) 28733.5 0.609971
\(75\) 0 0
\(76\) 10657.4 0.211649
\(77\) 11354.7 0.218247
\(78\) −16937.6 −0.315222
\(79\) −71242.1 −1.28431 −0.642154 − 0.766576i \(-0.721959\pi\)
−0.642154 + 0.766576i \(0.721959\pi\)
\(80\) 0 0
\(81\) −47446.0 −0.803502
\(82\) 58608.1 0.962550
\(83\) −28208.3 −0.449451 −0.224725 − 0.974422i \(-0.572148\pi\)
−0.224725 + 0.974422i \(0.572148\pi\)
\(84\) 4426.17 0.0684431
\(85\) 0 0
\(86\) −61259.8 −0.893161
\(87\) −116828. −1.65481
\(88\) 38985.2 0.536653
\(89\) 107437. 1.43774 0.718868 − 0.695146i \(-0.244660\pi\)
0.718868 + 0.695146i \(0.244660\pi\)
\(90\) 0 0
\(91\) −6031.63 −0.0763538
\(92\) 9856.05 0.121404
\(93\) 44997.0 0.539481
\(94\) 74170.7 0.865790
\(95\) 0 0
\(96\) 32524.6 0.360192
\(97\) 118741. 1.28136 0.640681 − 0.767807i \(-0.278652\pi\)
0.640681 + 0.767807i \(0.278652\pi\)
\(98\) 14393.1 0.151387
\(99\) −65781.6 −0.674553
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 175.6.a.i.1.5 ✓ 6
5.2 odd 4 175.6.b.h.99.9 12
5.3 odd 4 175.6.b.h.99.4 12
5.4 even 2 175.6.a.j.1.2 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.6.a.i.1.5 ✓ 6 1.1 even 1 trivial
175.6.a.j.1.2 yes 6 5.4 even 2
175.6.b.h.99.4 12 5.3 odd 4
175.6.b.h.99.9 12 5.2 odd 4