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Properties

Label 1575.4.a.bl.1.2
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
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] = [1575,4,Mod(1,1575)] 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("1575.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1575, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4,0,36,0,0,-28,27,0,0,-100] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(92.9280082590\)
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: \( x^{4} - 32x^{2} - 35x + 120 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 175)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.50478\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.504784 q^{2} -7.74519 q^{4} -7.00000 q^{7} +7.94792 q^{8} -54.8800 q^{11} -16.0073 q^{13} +3.53349 q^{14} +57.9496 q^{16} -0.422056 q^{17} +127.501 q^{19} +27.7025 q^{22} +51.1101 q^{23} +8.08024 q^{26} +54.2164 q^{28} -41.4750 q^{29} +192.354 q^{31} -92.8353 q^{32} +0.213047 q^{34} +189.232 q^{37} -64.3605 q^{38} +76.3187 q^{41} -294.499 q^{43} +425.056 q^{44} -25.7996 q^{46} +540.297 q^{47} +49.0000 q^{49} +123.980 q^{52} -661.316 q^{53} -55.6354 q^{56} +20.9359 q^{58} -410.312 q^{59} +46.0495 q^{61} -97.0974 q^{62} -416.735 q^{64} -10.4074 q^{67} +3.26890 q^{68} +491.117 q^{71} +814.540 q^{73} -95.5215 q^{74} -987.522 q^{76} +384.160 q^{77} -858.725 q^{79} -38.5244 q^{82} -1055.80 q^{83} +148.658 q^{86} -436.182 q^{88} -341.567 q^{89} +112.051 q^{91} -395.858 q^{92} -272.733 q^{94} +1417.21 q^{97} -24.7344 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 36 q^{4} - 28 q^{7} + 27 q^{8} - 100 q^{11} - 44 q^{13} - 28 q^{14} + 160 q^{16} - 53 q^{17} - 29 q^{19} + 152 q^{22} + 295 q^{23} - 700 q^{26} - 252 q^{28} - 129 q^{29} + 114 q^{31} - 310 q^{32}+ \cdots + 196 q^{98}+O(q^{100}) \) 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 \(0.528442\pi\)
\(3\) 0 0
\(4\) −7.74519 −0.968149
\(5\) 0 0
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) 7.94792 0.351252
\(9\) 0 0
\(10\) 0 0
\(11\) −54.8800 −1.50427 −0.752134 − 0.659010i \(-0.770975\pi\)
−0.752134 + 0.659010i \(0.770975\pi\)
\(12\) 0 0
\(13\) −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 \(-0.500958\pi\)
−0.00301069 + 0.999995i \(0.500958\pi\)
\(18\) 0 0
\(19\) 127.501 1.53952 0.769758 − 0.638336i \(-0.220377\pi\)
0.769758 + 0.638336i \(0.220377\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 27.7025 0.268464
\(23\) 51.1101 0.463357 0.231678 − 0.972792i \(-0.425578\pi\)
0.231678 + 0.972792i \(0.425578\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 8.08024 0.0609487
\(27\) 0 0
\(28\) 54.2164 0.365926
\(29\) −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 \(0.311866\pi\)
\(32\) −92.8353 −0.512848
\(33\) 0 0
\(34\) 0.213047 0.00107462
\(35\) 0 0
\(36\) 0 0
\(37\) 189.232 0.840801 0.420400 − 0.907339i \(-0.361890\pi\)
0.420400 + 0.907339i \(0.361890\pi\)
\(38\) −64.3605 −0.274754
\(39\) 0 0
\(40\) 0 0
\(41\) 76.3187 0.290707 0.145353 − 0.989380i \(-0.453568\pi\)
0.145353 + 0.989380i \(0.453568\pi\)
\(42\) 0 0
\(43\) −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 \(-0.183487\pi\)
0.838408 + 0.545043i \(0.183487\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 123.980 0.330633
\(53\) −661.316 −1.71394 −0.856969 − 0.515368i \(-0.827655\pi\)
−0.856969 + 0.515368i \(0.827655\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −55.6354 −0.132761
\(57\) 0 0
\(58\) 20.9359 0.0473969
\(59\) −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 \(0.484611\pi\)
\(62\) −97.0974 −0.198893
\(63\) 0 0
\(64\) −416.735 −0.813935
\(65\) 0 0
\(66\) 0 0
\(67\) −10.4074 −0.0189771 −0.00948854 − 0.999955i \(-0.503020\pi\)
−0.00948854 + 0.999955i \(0.503020\pi\)
\(68\) 3.26890 0.00582960
\(69\) 0 0
\(70\) 0 0
\(71\) 491.117 0.820913 0.410456 − 0.911880i \(-0.365369\pi\)
0.410456 + 0.911880i \(0.365369\pi\)
\(72\) 0 0
\(73\) 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
\(78\) 0 0
\(79\) −858.725 −1.22296 −0.611482 − 0.791258i \(-0.709426\pi\)
−0.611482 + 0.791258i \(0.709426\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −38.5244 −0.0518818
\(83\) −1055.80 −1.39626 −0.698129 − 0.715972i \(-0.745984\pi\)
−0.698129 + 0.715972i \(0.745984\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 148.658 0.186398
\(87\) 0 0
\(88\) −436.182 −0.528376
\(89\) −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
\(92\) −395.858 −0.448598
\(93\) 0 0
\(94\) −272.733 −0.299258
\(95\) 0 0
\(96\) 0 0
\(97\) 1417.21 1.48346 0.741731 − 0.670697i \(-0.234005\pi\)
0.741731 + 0.670697i \(0.234005\pi\)
\(98\) −24.7344 −0.0254954
\(99\) 0 0
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 1575.4.a.bl.1.2 4
3.2 odd 2 175.4.a.g.1.3 ✓ 4
5.4 even 2 1575.4.a.bg.1.3 4
15.2 even 4 175.4.b.f.99.5 8
15.8 even 4 175.4.b.f.99.4 8
15.14 odd 2 175.4.a.h.1.2 yes 4
21.20 even 2 1225.4.a.z.1.3 4
105.104 even 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 3.2 odd 2
175.4.a.h.1.2 yes 4 15.14 odd 2
175.4.b.f.99.4 8 15.8 even 4
175.4.b.f.99.5 8 15.2 even 4
1225.4.a.z.1.3 4 21.20 even 2
1225.4.a.bd.1.2 4 105.104 even 2
1575.4.a.bg.1.3 4 5.4 even 2
1575.4.a.bl.1.2 4 1.1 even 1 trivial