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Properties

Label 175.6.a.h.1.4
Level $175$
Weight $6$
Character 175.1
Self dual yes
Analytic conductor $28.067$
Analytic rank $1$
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] = [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 = [4,1] 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: \(1\)
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: \( x^{4} - x^{3} - 56x^{2} + 128x + 120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(6.19902\) 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)
 
\(f(q)\) \(=\) \(q+6.19902 q^{2} -2.67299 q^{3} +6.42787 q^{4} -16.5699 q^{6} +49.0000 q^{7} -158.522 q^{8} -235.855 q^{9} +590.159 q^{11} -17.1816 q^{12} +443.258 q^{13} +303.752 q^{14} -1188.37 q^{16} -1831.75 q^{17} -1462.07 q^{18} -2718.10 q^{19} -130.977 q^{21} +3658.41 q^{22} -5047.42 q^{23} +423.729 q^{24} +2747.77 q^{26} +1279.98 q^{27} +314.966 q^{28} +607.777 q^{29} +521.900 q^{31} -2294.05 q^{32} -1577.49 q^{33} -11355.1 q^{34} -1516.05 q^{36} -3103.51 q^{37} -16849.6 q^{38} -1184.83 q^{39} -11463.7 q^{41} -811.927 q^{42} -4164.16 q^{43} +3793.46 q^{44} -31289.1 q^{46} +26845.9 q^{47} +3176.52 q^{48} +2401.00 q^{49} +4896.26 q^{51} +2849.20 q^{52} +9880.38 q^{53} +7934.60 q^{54} -7767.59 q^{56} +7265.47 q^{57} +3767.62 q^{58} -33380.5 q^{59} -26745.3 q^{61} +3235.27 q^{62} -11556.9 q^{63} +23807.1 q^{64} -9778.90 q^{66} +29097.5 q^{67} -11774.3 q^{68} +13491.7 q^{69} -78389.0 q^{71} +37388.3 q^{72} +63102.9 q^{73} -19238.7 q^{74} -17471.6 q^{76} +28917.8 q^{77} -7344.76 q^{78} -27250.3 q^{79} +53891.4 q^{81} -71064.0 q^{82} +13368.9 q^{83} -841.901 q^{84} -25813.7 q^{86} -1624.58 q^{87} -93553.3 q^{88} +49633.1 q^{89} +21719.6 q^{91} -32444.2 q^{92} -1395.03 q^{93} +166418. q^{94} +6131.97 q^{96} +81372.3 q^{97} +14883.9 q^{98} -139192. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} - 6 q^{3} - 15 q^{4} + 79 q^{6} + 196 q^{7} - 279 q^{8} - 206 q^{9} - 260 q^{11} + 165 q^{12} - 214 q^{13} + 49 q^{14} - 1247 q^{16} - 146 q^{17} + 688 q^{18} + 856 q^{19} - 294 q^{21} + 4267 q^{22}+ \cdots - 100868 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.19902 1.09584 0.547921 − 0.836530i \(-0.315419\pi\)
0.547921 + 0.836530i \(0.315419\pi\)
\(3\) −2.67299 −0.171473 −0.0857363 − 0.996318i \(-0.527324\pi\)
−0.0857363 + 0.996318i \(0.527324\pi\)
\(4\) 6.42787 0.200871
\(5\) 0 0
\(6\) −16.5699 −0.187907
\(7\) 49.0000 0.377964
\(8\) −158.522 −0.875720
\(9\) −235.855 −0.970597
\(10\) 0 0
\(11\) 590.159 1.47058 0.735288 − 0.677755i \(-0.237047\pi\)
0.735288 + 0.677755i \(0.237047\pi\)
\(12\) −17.1816 −0.0344438
\(13\) 443.258 0.727441 0.363721 − 0.931508i \(-0.381506\pi\)
0.363721 + 0.931508i \(0.381506\pi\)
\(14\) 303.752 0.414190
\(15\) 0 0
\(16\) −1188.37 −1.16052
\(17\) −1831.75 −1.53725 −0.768625 − 0.639699i \(-0.779059\pi\)
−0.768625 + 0.639699i \(0.779059\pi\)
\(18\) −1462.07 −1.06362
\(19\) −2718.10 −1.72736 −0.863679 − 0.504043i \(-0.831845\pi\)
−0.863679 + 0.504043i \(0.831845\pi\)
\(20\) 0 0
\(21\) −130.977 −0.0648105
\(22\) 3658.41 1.61152
\(23\) −5047.42 −1.98953 −0.994764 − 0.102196i \(-0.967413\pi\)
−0.994764 + 0.102196i \(0.967413\pi\)
\(24\) 423.729 0.150162
\(25\) 0 0
\(26\) 2747.77 0.797161
\(27\) 1279.98 0.337903
\(28\) 314.966 0.0759221
\(29\) 607.777 0.134199 0.0670995 − 0.997746i \(-0.478625\pi\)
0.0670995 + 0.997746i \(0.478625\pi\)
\(30\) 0 0
\(31\) 521.900 0.0975401 0.0487700 − 0.998810i \(-0.484470\pi\)
0.0487700 + 0.998810i \(0.484470\pi\)
\(32\) −2294.05 −0.396029
\(33\) −1577.49 −0.252163
\(34\) −11355.1 −1.68458
\(35\) 0 0
\(36\) −1516.05 −0.194965
\(37\) −3103.51 −0.372690 −0.186345 − 0.982484i \(-0.559664\pi\)
−0.186345 + 0.982484i \(0.559664\pi\)
\(38\) −16849.6 −1.89291
\(39\) −1184.83 −0.124736
\(40\) 0 0
\(41\) −11463.7 −1.06504 −0.532521 − 0.846417i \(-0.678755\pi\)
−0.532521 + 0.846417i \(0.678755\pi\)
\(42\) −811.927 −0.0710221
\(43\) −4164.16 −0.343444 −0.171722 − 0.985145i \(-0.554933\pi\)
−0.171722 + 0.985145i \(0.554933\pi\)
\(44\) 3793.46 0.295396
\(45\) 0 0
\(46\) −31289.1 −2.18021
\(47\) 26845.9 1.77269 0.886346 − 0.463024i \(-0.153236\pi\)
0.886346 + 0.463024i \(0.153236\pi\)
\(48\) 3176.52 0.198998
\(49\) 2401.00 0.142857
\(50\) 0 0
\(51\) 4896.26 0.263596
\(52\) 2849.20 0.146122
\(53\) 9880.38 0.483152 0.241576 − 0.970382i \(-0.422336\pi\)
0.241576 + 0.970382i \(0.422336\pi\)
\(54\) 7934.60 0.370289
\(55\) 0 0
\(56\) −7767.59 −0.330991
\(57\) 7265.47 0.296194
\(58\) 3767.62 0.147061
\(59\) −33380.5 −1.24843 −0.624213 − 0.781254i \(-0.714580\pi\)
−0.624213 + 0.781254i \(0.714580\pi\)
\(60\) 0 0
\(61\) −26745.3 −0.920287 −0.460144 − 0.887844i \(-0.652202\pi\)
−0.460144 + 0.887844i \(0.652202\pi\)
\(62\) 3235.27 0.106889
\(63\) −11556.9 −0.366851
\(64\) 23807.1 0.726536
\(65\) 0 0
\(66\) −9778.90 −0.276331
\(67\) 29097.5 0.791897 0.395949 − 0.918273i \(-0.370416\pi\)
0.395949 + 0.918273i \(0.370416\pi\)
\(68\) −11774.3 −0.308789
\(69\) 13491.7 0.341150
\(70\) 0 0
\(71\) −78389.0 −1.84548 −0.922740 − 0.385423i \(-0.874056\pi\)
−0.922740 + 0.385423i \(0.874056\pi\)
\(72\) 37388.3 0.849971
\(73\) 63102.9 1.38593 0.692967 − 0.720970i \(-0.256303\pi\)
0.692967 + 0.720970i \(0.256303\pi\)
\(74\) −19238.7 −0.408410
\(75\) 0 0
\(76\) −17471.6 −0.346976
\(77\) 28917.8 0.555825
\(78\) −7344.76 −0.136691
\(79\) −27250.3 −0.491252 −0.245626 − 0.969365i \(-0.578993\pi\)
−0.245626 + 0.969365i \(0.578993\pi\)
\(80\) 0 0
\(81\) 53891.4 0.912656
\(82\) −71064.0 −1.16712
\(83\) 13368.9 0.213010 0.106505 − 0.994312i \(-0.466034\pi\)
0.106505 + 0.994312i \(0.466034\pi\)
\(84\) −841.901 −0.0130185
\(85\) 0 0
\(86\) −25813.7 −0.376360
\(87\) −1624.58 −0.0230115
\(88\) −93553.3 −1.28781
\(89\) 49633.1 0.664196 0.332098 − 0.943245i \(-0.392244\pi\)
0.332098 + 0.943245i \(0.392244\pi\)
\(90\) 0 0
\(91\) 21719.6 0.274947
\(92\) −32444.2 −0.399638
\(93\) −1395.03 −0.0167254
\(94\) 166418. 1.94259
\(95\) 0 0
\(96\) 6131.97 0.0679082
\(97\) 81372.3 0.878107 0.439053 − 0.898461i \(-0.355314\pi\)
0.439053 + 0.898461i \(0.355314\pi\)
\(98\) 14883.9 0.156549
\(99\) −139192. −1.42734
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.h.1.4 yes 4
5.2 odd 4 175.6.b.g.99.7 8
5.3 odd 4 175.6.b.g.99.2 8
5.4 even 2 175.6.a.g.1.1 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.6.a.g.1.1 ✓ 4 5.4 even 2
175.6.a.h.1.4 yes 4 1.1 even 1 trivial
175.6.b.g.99.2 8 5.3 odd 4
175.6.b.g.99.7 8 5.2 odd 4