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Properties

Label 175.4.a.h.1.1
Level $175$
Weight $4$
Character 175.1
Self dual yes
Analytic conductor $10.325$
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] = [175,4,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, 4); 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, 4, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4] 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: \(10.3253342510\)
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: \( x^{4} - 32x^{2} - 35x + 120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.87199\) 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)
 
\(f(q)\) \(=\) \(q-4.87199 q^{2} -4.14916 q^{3} +15.7363 q^{4} +20.2147 q^{6} +7.00000 q^{7} -37.6910 q^{8} -9.78444 q^{9} +36.9922 q^{11} -65.2924 q^{12} -61.3165 q^{13} -34.1039 q^{14} +57.7401 q^{16} -44.8345 q^{17} +47.6697 q^{18} -139.701 q^{19} -29.0441 q^{21} -180.226 q^{22} +217.580 q^{23} +156.386 q^{24} +298.733 q^{26} +152.625 q^{27} +110.154 q^{28} -33.8226 q^{29} +124.437 q^{31} +20.2192 q^{32} -153.487 q^{33} +218.433 q^{34} -153.971 q^{36} +237.270 q^{37} +680.620 q^{38} +254.412 q^{39} +195.117 q^{41} +141.503 q^{42} -343.725 q^{43} +582.119 q^{44} -1060.05 q^{46} +16.8224 q^{47} -239.573 q^{48} +49.0000 q^{49} +186.026 q^{51} -964.894 q^{52} +346.965 q^{53} -743.586 q^{54} -263.837 q^{56} +579.641 q^{57} +164.783 q^{58} +135.340 q^{59} +490.414 q^{61} -606.255 q^{62} -68.4911 q^{63} -560.428 q^{64} +747.785 q^{66} +477.969 q^{67} -705.529 q^{68} -902.777 q^{69} +45.2557 q^{71} +368.786 q^{72} +100.781 q^{73} -1155.98 q^{74} -2198.37 q^{76} +258.945 q^{77} -1239.49 q^{78} +880.534 q^{79} -369.085 q^{81} -950.609 q^{82} -1155.03 q^{83} -457.047 q^{84} +1674.62 q^{86} +140.336 q^{87} -1394.27 q^{88} +619.374 q^{89} -429.216 q^{91} +3423.90 q^{92} -516.309 q^{93} -81.9585 q^{94} -83.8929 q^{96} -231.195 q^{97} -238.727 q^{98} -361.948 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 3 q^{3} + 36 q^{4} + q^{6} + 28 q^{7} + 27 q^{8} + 61 q^{9} + 100 q^{11} - 165 q^{12} + 44 q^{13} + 28 q^{14} + 160 q^{16} - 53 q^{17} + 433 q^{18} - 29 q^{19} - 21 q^{21} - 152 q^{22}+ \cdots + 2383 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.87199 −1.72251 −0.861254 − 0.508175i \(-0.830320\pi\)
−0.861254 + 0.508175i \(0.830320\pi\)
\(3\) −4.14916 −0.798507 −0.399253 − 0.916841i \(-0.630731\pi\)
−0.399253 + 0.916841i \(0.630731\pi\)
\(4\) 15.7363 1.96703
\(5\) 0 0
\(6\) 20.2147 1.37543
\(7\) 7.00000 0.377964
\(8\) −37.6910 −1.66572
\(9\) −9.78444 −0.362387
\(10\) 0 0
\(11\) 36.9922 1.01396 0.506980 − 0.861958i \(-0.330762\pi\)
0.506980 + 0.861958i \(0.330762\pi\)
\(12\) −65.2924 −1.57069
\(13\) −61.3165 −1.30817 −0.654083 − 0.756423i \(-0.726945\pi\)
−0.654083 + 0.756423i \(0.726945\pi\)
\(14\) −34.1039 −0.651047
\(15\) 0 0
\(16\) 57.7401 0.902189
\(17\) −44.8345 −0.639646 −0.319823 − 0.947477i \(-0.603623\pi\)
−0.319823 + 0.947477i \(0.603623\pi\)
\(18\) 47.6697 0.624214
\(19\) −139.701 −1.68682 −0.843408 − 0.537273i \(-0.819455\pi\)
−0.843408 + 0.537273i \(0.819455\pi\)
\(20\) 0 0
\(21\) −29.0441 −0.301807
\(22\) −180.226 −1.74656
\(23\) 217.580 1.97255 0.986275 − 0.165110i \(-0.0527979\pi\)
0.986275 + 0.165110i \(0.0527979\pi\)
\(24\) 156.386 1.33009
\(25\) 0 0
\(26\) 298.733 2.25333
\(27\) 152.625 1.08788
\(28\) 110.154 0.743469
\(29\) −33.8226 −0.216576 −0.108288 − 0.994120i \(-0.534537\pi\)
−0.108288 + 0.994120i \(0.534537\pi\)
\(30\) 0 0
\(31\) 124.437 0.720952 0.360476 − 0.932769i \(-0.382614\pi\)
0.360476 + 0.932769i \(0.382614\pi\)
\(32\) 20.2192 0.111697
\(33\) −153.487 −0.809655
\(34\) 218.433 1.10179
\(35\) 0 0
\(36\) −153.971 −0.712827
\(37\) 237.270 1.05424 0.527121 − 0.849790i \(-0.323271\pi\)
0.527121 + 0.849790i \(0.323271\pi\)
\(38\) 680.620 2.90556
\(39\) 254.412 1.04458
\(40\) 0 0
\(41\) 195.117 0.743224 0.371612 − 0.928388i \(-0.378805\pi\)
0.371612 + 0.928388i \(0.378805\pi\)
\(42\) 141.503 0.519865
\(43\) −343.725 −1.21901 −0.609506 − 0.792781i \(-0.708632\pi\)
−0.609506 + 0.792781i \(0.708632\pi\)
\(44\) 582.119 1.99450
\(45\) 0 0
\(46\) −1060.05 −3.39773
\(47\) 16.8224 0.0522085 0.0261042 − 0.999659i \(-0.491690\pi\)
0.0261042 + 0.999659i \(0.491690\pi\)
\(48\) −239.573 −0.720404
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 186.026 0.510761
\(52\) −964.894 −2.57321
\(53\) 346.965 0.899231 0.449616 − 0.893222i \(-0.351561\pi\)
0.449616 + 0.893222i \(0.351561\pi\)
\(54\) −743.586 −1.87387
\(55\) 0 0
\(56\) −263.837 −0.629584
\(57\) 579.641 1.34694
\(58\) 164.783 0.373054
\(59\) 135.340 0.298640 0.149320 − 0.988789i \(-0.452292\pi\)
0.149320 + 0.988789i \(0.452292\pi\)
\(60\) 0 0
\(61\) 490.414 1.02936 0.514681 − 0.857382i \(-0.327911\pi\)
0.514681 + 0.857382i \(0.327911\pi\)
\(62\) −606.255 −1.24185
\(63\) −68.4911 −0.136969
\(64\) −560.428 −1.09459
\(65\) 0 0
\(66\) 747.785 1.39464
\(67\) 477.969 0.871540 0.435770 − 0.900058i \(-0.356476\pi\)
0.435770 + 0.900058i \(0.356476\pi\)
\(68\) −705.529 −1.25820
\(69\) −902.777 −1.57510
\(70\) 0 0
\(71\) 45.2557 0.0756460 0.0378230 − 0.999284i \(-0.487958\pi\)
0.0378230 + 0.999284i \(0.487958\pi\)
\(72\) 368.786 0.603636
\(73\) 100.781 0.161583 0.0807913 − 0.996731i \(-0.474255\pi\)
0.0807913 + 0.996731i \(0.474255\pi\)
\(74\) −1155.98 −1.81594
\(75\) 0 0
\(76\) −2198.37 −3.31803
\(77\) 258.945 0.383241
\(78\) −1239.49 −1.79930
\(79\) 880.534 1.25402 0.627012 − 0.779010i \(-0.284278\pi\)
0.627012 + 0.779010i \(0.284278\pi\)
\(80\) 0 0
\(81\) −369.085 −0.506289
\(82\) −950.609 −1.28021
\(83\) −1155.03 −1.52748 −0.763739 − 0.645525i \(-0.776638\pi\)
−0.763739 + 0.645525i \(0.776638\pi\)
\(84\) −457.047 −0.593665
\(85\) 0 0
\(86\) 1674.62 2.09976
\(87\) 140.336 0.172937
\(88\) −1394.27 −1.68898
\(89\) 619.374 0.737680 0.368840 − 0.929493i \(-0.379755\pi\)
0.368840 + 0.929493i \(0.379755\pi\)
\(90\) 0 0
\(91\) −429.216 −0.494440
\(92\) 3423.90 3.88007
\(93\) −516.309 −0.575685
\(94\) −81.9585 −0.0899295
\(95\) 0 0
\(96\) −83.8929 −0.0891904
\(97\) −231.195 −0.242003 −0.121001 − 0.992652i \(-0.538611\pi\)
−0.121001 + 0.992652i \(0.538611\pi\)
\(98\) −238.727 −0.246073
\(99\) −361.948 −0.367446
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 175.4.a.h.1.1 yes 4
3.2 odd 2 1575.4.a.bg.1.4 4
5.2 odd 4 175.4.b.f.99.1 8
5.3 odd 4 175.4.b.f.99.8 8
5.4 even 2 175.4.a.g.1.4 ✓ 4
7.6 odd 2 1225.4.a.bd.1.1 4
15.14 odd 2 1575.4.a.bl.1.1 4
35.34 odd 2 1225.4.a.z.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.4.a.g.1.4 ✓ 4 5.4 even 2
175.4.a.h.1.1 yes 4 1.1 even 1 trivial
175.4.b.f.99.1 8 5.2 odd 4
175.4.b.f.99.8 8 5.3 odd 4
1225.4.a.z.1.4 4 35.34 odd 2
1225.4.a.bd.1.1 4 7.6 odd 2
1575.4.a.bg.1.4 4 3.2 odd 2
1575.4.a.bl.1.1 4 15.14 odd 2