↑

Properties

Label 175.4.b.f.99.5
Level $175$
Weight $4$
Character 175.99
Analytic conductor $10.325$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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] = [175,4,Mod(99,175)] 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("175.99"); 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([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-72,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.3253342510\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 64x^{6} + 1264x^{4} + 8905x^{2} + 14400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.5
Root \(1.50478i\) of defining polynomial
Character \(\chi\) \(=\) 175.99
Dual form 175.4.b.f.99.4

$q$-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)
 
\(f(q)\) \(=\) \(q+0.504784i q^{2} +4.26379i q^{3} +7.74519 q^{4} -2.15229 q^{6} -7.00000i q^{7} +7.94792i q^{8} +8.82008 q^{9} +54.8800 q^{11} +33.0239i q^{12} +16.0073i q^{13} +3.53349 q^{14} +57.9496 q^{16} +0.422056i q^{17} +4.45223i q^{18} -127.501 q^{19} +29.8465 q^{21} +27.7025i q^{22} +51.1101i q^{23} -33.8883 q^{24} -8.08024 q^{26} +152.729i q^{27} -54.2164i q^{28} -41.4750 q^{29} +192.354 q^{31} +92.8353i q^{32} +233.997i q^{33} -0.213047 q^{34} +68.3132 q^{36} +189.232i q^{37} -64.3605i q^{38} -68.2519 q^{39} -76.3187 q^{41} +15.0660i q^{42} +294.499i q^{43} +425.056 q^{44} -25.7996 q^{46} -540.297i q^{47} +247.085i q^{48} -49.0000 q^{49} -1.79956 q^{51} +123.980i q^{52} -661.316i q^{53} -77.0953 q^{54} +55.6354 q^{56} -543.639i q^{57} -20.9359i q^{58} -410.312 q^{59} +46.0495 q^{61} +97.0974i q^{62} -61.7405i q^{63} +416.735 q^{64} -118.118 q^{66} -10.4074i q^{67} +3.26890i q^{68} -217.923 q^{69} -491.117 q^{71} +70.1012i q^{72} -814.540i q^{73} -95.5215 q^{74} -987.522 q^{76} -384.160i q^{77} -34.4525i q^{78} +858.725 q^{79} -413.064 q^{81} -38.5244i q^{82} -1055.80i q^{83} +231.167 q^{84} -148.658 q^{86} -176.841i q^{87} +436.182i q^{88} -341.567 q^{89} +112.051 q^{91} +395.858i q^{92} +820.159i q^{93} +272.733 q^{94} -395.831 q^{96} +1417.21i q^{97} -24.7344i q^{98} +484.046 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 72 q^{4} + 2 q^{6} - 122 q^{9} + 200 q^{11} - 56 q^{14} + 320 q^{16} + 58 q^{19} - 42 q^{21} + 42 q^{24} + 1400 q^{26} - 258 q^{29} + 228 q^{31} - 406 q^{34} + 2202 q^{36} - 1348 q^{39} + 1342 q^{41}+ \cdots - 4766 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.504784i 0.178468i 0.996011 + 0.0892340i \(0.0284419\pi\)
−0.996011 + 0.0892340i \(0.971558\pi\)
\(3\) 4.26379i 0.820567i 0.911958 + 0.410284i \(0.134570\pi\)
−0.911958 + 0.410284i \(0.865430\pi\)
\(4\) 7.74519 0.968149
\(5\) 0 0
\(6\) −2.15229 −0.146445
\(7\) − 7.00000i − 0.377964i
\(8\) 7.94792i 0.351252i
\(9\) 8.82008 0.326670
\(10\) 0 0
\(11\) 54.8800 1.50427 0.752134 − 0.659010i \(-0.229025\pi\)
0.752134 + 0.659010i \(0.229025\pi\)
\(12\) 33.0239i 0.794431i
\(13\) 16.0073i 0.341510i 0.985313 + 0.170755i \(0.0546207\pi\)
−0.985313 + 0.170755i \(0.945379\pi\)
\(14\) 3.53349 0.0674546
\(15\) 0 0
\(16\) 57.9496 0.905462
\(17\) 0.422056i 0.00602139i 0.999995 + 0.00301069i \(0.000958335\pi\)
−0.999995 + 0.00301069i \(0.999042\pi\)
\(18\) 4.45223i 0.0583000i
\(19\) −127.501 −1.53952 −0.769758 − 0.638336i \(-0.779623\pi\)
−0.769758 + 0.638336i \(0.779623\pi\)
\(20\) 0 0
\(21\) 29.8465 0.310145
\(22\) 27.7025i 0.268464i
\(23\) 51.1101i 0.463357i 0.972792 + 0.231678i \(0.0744217\pi\)
−0.972792 + 0.231678i \(0.925578\pi\)
\(24\) −33.8883 −0.288226
\(25\) 0 0
\(26\) −8.08024 −0.0609487
\(27\) 152.729i 1.08862i
\(28\) − 54.2164i − 0.365926i
\(29\) −41.4750 −0.265576 −0.132788 − 0.991144i \(-0.542393\pi\)
−0.132788 + 0.991144i \(0.542393\pi\)
\(30\) 0 0
\(31\) 192.354 1.11445 0.557224 − 0.830362i \(-0.311866\pi\)
0.557224 + 0.830362i \(0.311866\pi\)
\(32\) 92.8353i 0.512848i
\(33\) 233.997i 1.23435i
\(34\) −0.213047 −0.00107462
\(35\) 0 0
\(36\) 68.3132 0.316265
\(37\) 189.232i 0.840801i 0.907339 + 0.420400i \(0.138110\pi\)
−0.907339 + 0.420400i \(0.861890\pi\)
\(38\) − 64.3605i − 0.274754i
\(39\) −68.2519 −0.280232
\(40\) 0 0
\(41\) −76.3187 −0.290707 −0.145353 − 0.989380i \(-0.546432\pi\)
−0.145353 + 0.989380i \(0.546432\pi\)
\(42\) 15.0660i 0.0553510i
\(43\) 294.499i 1.04443i 0.852813 + 0.522216i \(0.174895\pi\)
−0.852813 + 0.522216i \(0.825105\pi\)
\(44\) 425.056 1.45636
\(45\) 0 0
\(46\) −25.7996 −0.0826943
\(47\) − 540.297i − 1.67682i −0.545043 − 0.838408i \(-0.683487\pi\)
0.545043 − 0.838408i \(-0.316513\pi\)
\(48\) 247.085i 0.742992i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) −1.79956 −0.00494095
\(52\) 123.980i 0.330633i
\(53\) − 661.316i − 1.71394i −0.515368 − 0.856969i \(-0.672345\pi\)
0.515368 − 0.856969i \(-0.327655\pi\)
\(54\) −77.0953 −0.194284
\(55\) 0 0
\(56\) 55.6354 0.132761
\(57\) − 543.639i − 1.26328i
\(58\) − 20.9359i − 0.0473969i
\(59\) −410.312 −0.905390 −0.452695 − 0.891665i \(-0.649537\pi\)
−0.452695 + 0.891665i \(0.649537\pi\)
\(60\) 0 0
\(61\) 46.0495 0.0966563 0.0483281 − 0.998832i \(-0.484611\pi\)
0.0483281 + 0.998832i \(0.484611\pi\)
\(62\) 97.0974i 0.198893i
\(63\) − 61.7405i − 0.123469i
\(64\) 416.735 0.813935
\(65\) 0 0
\(66\) −118.118 −0.220292
\(67\) − 10.4074i − 0.0189771i −0.999955 − 0.00948854i \(-0.996980\pi\)
0.999955 − 0.00948854i \(-0.00302034\pi\)
\(68\) 3.26890i 0.00582960i
\(69\) −217.923 −0.380215
\(70\) 0 0
\(71\) −491.117 −0.820913 −0.410456 − 0.911880i \(-0.634631\pi\)
−0.410456 + 0.911880i \(0.634631\pi\)
\(72\) 70.1012i 0.114743i
\(73\) − 814.540i − 1.30595i −0.757378 − 0.652977i \(-0.773520\pi\)
0.757378 − 0.652977i \(-0.226480\pi\)
\(74\) −95.5215 −0.150056
\(75\) 0 0
\(76\) −987.522 −1.49048
\(77\) − 384.160i − 0.568560i
\(78\) − 34.4525i − 0.0500125i
\(79\) 858.725 1.22296 0.611482 − 0.791258i \(-0.290574\pi\)
0.611482 + 0.791258i \(0.290574\pi\)
\(80\) 0 0
\(81\) −413.064 −0.566617
\(82\) − 38.5244i − 0.0518818i
\(83\) − 1055.80i − 1.39626i −0.715972 − 0.698129i \(-0.754016\pi\)
0.715972 − 0.698129i \(-0.245984\pi\)
\(84\) 231.167 0.300267
\(85\) 0 0
\(86\) −148.658 −0.186398
\(87\) − 176.841i − 0.217923i
\(88\) 436.182i 0.528376i
\(89\) −341.567 −0.406809 −0.203405 − 0.979095i \(-0.565201\pi\)
−0.203405 + 0.979095i \(0.565201\pi\)
\(90\) 0 0
\(91\) 112.051 0.129079
\(92\) 395.858i 0.448598i
\(93\) 820.159i 0.914479i
\(94\) 272.733 0.299258
\(95\) 0 0
\(96\) −395.831 −0.420826
\(97\) 1417.21i 1.48346i 0.670697 + 0.741731i \(0.265995\pi\)
−0.670697 + 0.741731i \(0.734005\pi\)
\(98\) − 24.7344i − 0.0254954i
\(99\) 484.046 0.491398
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.b.f.99.5 8
5.2 odd 4 175.4.a.h.1.2 yes 4
5.3 odd 4 175.4.a.g.1.3 ✓ 4
5.4 even 2 inner 175.4.b.f.99.4 8
15.2 even 4 1575.4.a.bg.1.3 4
15.8 even 4 1575.4.a.bl.1.2 4
35.13 even 4 1225.4.a.z.1.3 4
35.27 even 4 1225.4.a.bd.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.4.a.g.1.3 ✓ 4 5.3 odd 4
175.4.a.h.1.2 yes 4 5.2 odd 4
175.4.b.f.99.4 8 5.4 even 2 inner
175.4.b.f.99.5 8 1.1 even 1 trivial
1225.4.a.z.1.3 4 35.13 even 4
1225.4.a.bd.1.2 4 35.27 even 4
1575.4.a.bg.1.3 4 15.2 even 4
1575.4.a.bl.1.2 4 15.8 even 4