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Properties

Label 175.4.a.h.1.2
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 1.2
Root \(1.50478\) 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)
 
\(f(q)\) \(=\) \(q-0.504784 q^{2} +4.26379 q^{3} -7.74519 q^{4} -2.15229 q^{6} +7.00000 q^{7} +7.94792 q^{8} -8.82008 q^{9} +54.8800 q^{11} -33.0239 q^{12} +16.0073 q^{13} -3.53349 q^{14} +57.9496 q^{16} -0.422056 q^{17} +4.45223 q^{18} +127.501 q^{19} +29.8465 q^{21} -27.7025 q^{22} +51.1101 q^{23} +33.8883 q^{24} -8.08024 q^{26} -152.729 q^{27} -54.2164 q^{28} +41.4750 q^{29} +192.354 q^{31} -92.8353 q^{32} +233.997 q^{33} +0.213047 q^{34} +68.3132 q^{36} -189.232 q^{37} -64.3605 q^{38} +68.2519 q^{39} -76.3187 q^{41} -15.0660 q^{42} +294.499 q^{43} -425.056 q^{44} -25.7996 q^{46} +540.297 q^{47} +247.085 q^{48} +49.0000 q^{49} -1.79956 q^{51} -123.980 q^{52} -661.316 q^{53} +77.0953 q^{54} +55.6354 q^{56} +543.639 q^{57} -20.9359 q^{58} +410.312 q^{59} +46.0495 q^{61} -97.0974 q^{62} -61.7405 q^{63} -416.735 q^{64} -118.118 q^{66} +10.4074 q^{67} +3.26890 q^{68} +217.923 q^{69} -491.117 q^{71} -70.1012 q^{72} -814.540 q^{73} +95.5215 q^{74} -987.522 q^{76} +384.160 q^{77} -34.4525 q^{78} -858.725 q^{79} -413.064 q^{81} +38.5244 q^{82} -1055.80 q^{83} -231.167 q^{84} -148.658 q^{86} +176.841 q^{87} +436.182 q^{88} +341.567 q^{89} +112.051 q^{91} -395.858 q^{92} +820.159 q^{93} -272.733 q^{94} -395.831 q^{96} -1417.21 q^{97} -24.7344 q^{98} -484.046 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 3 q^{3} + 36 q^{4} + q^{6} + 28 q^{7} + 27 q^{8} + 61 q^{9} + 100 q^{11} - 165 q^{12} + 44 q^{13} + 28 q^{14} + 160 q^{16} - 53 q^{17} + 433 q^{18} - 29 q^{19} - 21 q^{21} - 152 q^{22}+ \cdots + 2383 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.504784 −0.178468 −0.0892340 − 0.996011i \(-0.528442\pi\)
−0.0892340 + 0.996011i \(0.528442\pi\)
\(3\) 4.26379 0.820567 0.410284 − 0.911958i \(-0.365430\pi\)
0.410284 + 0.911958i \(0.365430\pi\)
\(4\) −7.74519 −0.968149
\(5\) 0 0
\(6\) −2.15229 −0.146445
\(7\) 7.00000 0.377964
\(8\) 7.94792 0.351252
\(9\) −8.82008 −0.326670
\(10\) 0 0
\(11\) 54.8800 1.50427 0.752134 − 0.659010i \(-0.229025\pi\)
0.752134 + 0.659010i \(0.229025\pi\)
\(12\) −33.0239 −0.794431
\(13\) 16.0073 0.341510 0.170755 − 0.985313i \(-0.445379\pi\)
0.170755 + 0.985313i \(0.445379\pi\)
\(14\) −3.53349 −0.0674546
\(15\) 0 0
\(16\) 57.9496 0.905462
\(17\) −0.422056 −0.00602139 −0.00301069 − 0.999995i \(-0.500958\pi\)
−0.00301069 + 0.999995i \(0.500958\pi\)
\(18\) 4.45223 0.0583000
\(19\) 127.501 1.53952 0.769758 − 0.638336i \(-0.220377\pi\)
0.769758 + 0.638336i \(0.220377\pi\)
\(20\) 0 0
\(21\) 29.8465 0.310145
\(22\) −27.7025 −0.268464
\(23\) 51.1101 0.463357 0.231678 − 0.972792i \(-0.425578\pi\)
0.231678 + 0.972792i \(0.425578\pi\)
\(24\) 33.8883 0.288226
\(25\) 0 0
\(26\) −8.08024 −0.0609487
\(27\) −152.729 −1.08862
\(28\) −54.2164 −0.365926
\(29\) 41.4750 0.265576 0.132788 − 0.991144i \(-0.457607\pi\)
0.132788 + 0.991144i \(0.457607\pi\)
\(30\) 0 0
\(31\) 192.354 1.11445 0.557224 − 0.830362i \(-0.311866\pi\)
0.557224 + 0.830362i \(0.311866\pi\)
\(32\) −92.8353 −0.512848
\(33\) 233.997 1.23435
\(34\) 0.213047 0.00107462
\(35\) 0 0
\(36\) 68.3132 0.316265
\(37\) −189.232 −0.840801 −0.420400 − 0.907339i \(-0.638110\pi\)
−0.420400 + 0.907339i \(0.638110\pi\)
\(38\) −64.3605 −0.274754
\(39\) 68.2519 0.280232
\(40\) 0 0
\(41\) −76.3187 −0.290707 −0.145353 − 0.989380i \(-0.546432\pi\)
−0.145353 + 0.989380i \(0.546432\pi\)
\(42\) −15.0660 −0.0553510
\(43\) 294.499 1.04443 0.522216 − 0.852813i \(-0.325105\pi\)
0.522216 + 0.852813i \(0.325105\pi\)
\(44\) −425.056 −1.45636
\(45\) 0 0
\(46\) −25.7996 −0.0826943
\(47\) 540.297 1.67682 0.838408 − 0.545043i \(-0.183487\pi\)
0.838408 + 0.545043i \(0.183487\pi\)
\(48\) 247.085 0.742992
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −1.79956 −0.00494095
\(52\) −123.980 −0.330633
\(53\) −661.316 −1.71394 −0.856969 − 0.515368i \(-0.827655\pi\)
−0.856969 + 0.515368i \(0.827655\pi\)
\(54\) 77.0953 0.194284
\(55\) 0 0
\(56\) 55.6354 0.132761
\(57\) 543.639 1.26328
\(58\) −20.9359 −0.0473969
\(59\) 410.312 0.905390 0.452695 − 0.891665i \(-0.350463\pi\)
0.452695 + 0.891665i \(0.350463\pi\)
\(60\) 0 0
\(61\) 46.0495 0.0966563 0.0483281 − 0.998832i \(-0.484611\pi\)
0.0483281 + 0.998832i \(0.484611\pi\)
\(62\) −97.0974 −0.198893
\(63\) −61.7405 −0.123469
\(64\) −416.735 −0.813935
\(65\) 0 0
\(66\) −118.118 −0.220292
\(67\) 10.4074 0.0189771 0.00948854 − 0.999955i \(-0.496980\pi\)
0.00948854 + 0.999955i \(0.496980\pi\)
\(68\) 3.26890 0.00582960
\(69\) 217.923 0.380215
\(70\) 0 0
\(71\) −491.117 −0.820913 −0.410456 − 0.911880i \(-0.634631\pi\)
−0.410456 + 0.911880i \(0.634631\pi\)
\(72\) −70.1012 −0.114743
\(73\) −814.540 −1.30595 −0.652977 − 0.757378i \(-0.726480\pi\)
−0.652977 + 0.757378i \(0.726480\pi\)
\(74\) 95.5215 0.150056
\(75\) 0 0
\(76\) −987.522 −1.49048
\(77\) 384.160 0.568560
\(78\) −34.4525 −0.0500125
\(79\) −858.725 −1.22296 −0.611482 − 0.791258i \(-0.709426\pi\)
−0.611482 + 0.791258i \(0.709426\pi\)
\(80\) 0 0
\(81\) −413.064 −0.566617
\(82\) 38.5244 0.0518818
\(83\) −1055.80 −1.39626 −0.698129 − 0.715972i \(-0.745984\pi\)
−0.698129 + 0.715972i \(0.745984\pi\)
\(84\) −231.167 −0.300267
\(85\) 0 0
\(86\) −148.658 −0.186398
\(87\) 176.841 0.217923
\(88\) 436.182 0.528376
\(89\) 341.567 0.406809 0.203405 − 0.979095i \(-0.434799\pi\)
0.203405 + 0.979095i \(0.434799\pi\)
\(90\) 0 0
\(91\) 112.051 0.129079
\(92\) −395.858 −0.448598
\(93\) 820.159 0.914479
\(94\) −272.733 −0.299258
\(95\) 0 0
\(96\) −395.831 −0.420826
\(97\) −1417.21 −1.48346 −0.741731 − 0.670697i \(-0.765995\pi\)
−0.741731 + 0.670697i \(0.765995\pi\)
\(98\) −24.7344 −0.0254954
\(99\) −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.a.h.1.2 yes 4
3.2 odd 2 1575.4.a.bg.1.3 4
5.2 odd 4 175.4.b.f.99.4 8
5.3 odd 4 175.4.b.f.99.5 8
5.4 even 2 175.4.a.g.1.3 ✓ 4
7.6 odd 2 1225.4.a.bd.1.2 4
15.14 odd 2 1575.4.a.bl.1.2 4
35.34 odd 2 1225.4.a.z.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.4.a.g.1.3 ✓ 4 5.4 even 2
175.4.a.h.1.2 yes 4 1.1 even 1 trivial
175.4.b.f.99.4 8 5.2 odd 4
175.4.b.f.99.5 8 5.3 odd 4
1225.4.a.z.1.3 4 35.34 odd 2
1225.4.a.bd.1.2 4 7.6 odd 2
1575.4.a.bg.1.3 4 3.2 odd 2
1575.4.a.bl.1.2 4 15.14 odd 2