↑

Properties

Label 175.4.a.g.1.3
Level $175$
Weight $4$
Character 175.1
Self dual yes
Analytic conductor $10.325$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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(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 1.3
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 \(-0.471558\pi\)
0.0892340 + 0.996011i \(0.471558\pi\)
\(3\) −4.26379 −0.820567 −0.410284 − 0.911958i \(-0.634570\pi\)
−0.410284 + 0.911958i \(0.634570\pi\)
\(4\) −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 \(-0.554621\pi\)
−0.170755 + 0.985313i \(0.554621\pi\)
\(14\) −3.53349 −0.0674546
\(15\) 0 0
\(16\) 57.9496 0.905462
\(17\) 0.422056 0.00602139 0.00301069 − 0.999995i \(-0.499042\pi\)
0.00301069 + 0.999995i \(0.499042\pi\)
\(18\) −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 \(-0.574422\pi\)
−0.231678 + 0.972792i \(0.574422\pi\)
\(24\) 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 \(-0.361890\pi\)
0.420400 + 0.907339i \(0.361890\pi\)
\(38\) 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 \(-0.674895\pi\)
−0.522216 + 0.852813i \(0.674895\pi\)
\(44\) −425.056 −1.45636
\(45\) 0 0
\(46\) −25.7996 −0.0826943
\(47\) −540.297 −1.67682 −0.838408 − 0.545043i \(-0.816513\pi\)
−0.838408 + 0.545043i \(0.816513\pi\)
\(48\) −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 \(-0.172345\pi\)
0.856969 + 0.515368i \(0.172345\pi\)
\(54\) 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 \(-0.503020\pi\)
−0.00948854 + 0.999955i \(0.503020\pi\)
\(68\) −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 \(-0.273520\pi\)
0.652977 + 0.757378i \(0.273520\pi\)
\(74\) 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 \(-0.254016\pi\)
0.698129 + 0.715972i \(0.254016\pi\)
\(84\) −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 \(-0.234005\pi\)
0.741731 + 0.670697i \(0.234005\pi\)
\(98\) 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.g.1.3 ✓ 4
3.2 odd 2 1575.4.a.bl.1.2 4
5.2 odd 4 175.4.b.f.99.5 8
5.3 odd 4 175.4.b.f.99.4 8
5.4 even 2 175.4.a.h.1.2 yes 4
7.6 odd 2 1225.4.a.z.1.3 4
15.14 odd 2 1575.4.a.bg.1.3 4
35.34 odd 2 1225.4.a.bd.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
175.4.a.g.1.3 ✓ 4 1.1 even 1 trivial
175.4.a.h.1.2 yes 4 5.4 even 2
175.4.b.f.99.4 8 5.3 odd 4
175.4.b.f.99.5 8 5.2 odd 4
1225.4.a.z.1.3 4 7.6 odd 2
1225.4.a.bd.1.2 4 35.34 odd 2
1575.4.a.bg.1.3 4 15.14 odd 2
1575.4.a.bl.1.2 4 3.2 odd 2