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Properties

Label 1575.4.a.bf.1.3
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
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] = [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,-6,0,16,0,0,-28,-93,0,0,-57] 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: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{39 +2 \sqrt{185}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 18x^{2} + 19x + 44 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 525)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.21734\) 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)
 
\(f(q)\) \(=\) \(q+0.217342 q^{2} -7.95276 q^{4} -7.00000 q^{7} -3.46721 q^{8} +30.6085 q^{11} -25.3178 q^{13} -1.52140 q^{14} +62.8685 q^{16} +72.8676 q^{17} +122.711 q^{19} +6.65253 q^{22} -194.258 q^{23} -5.50264 q^{26} +55.6693 q^{28} -48.6103 q^{29} -288.907 q^{31} +41.4017 q^{32} +15.8372 q^{34} -15.8251 q^{37} +26.6702 q^{38} -452.905 q^{41} +152.574 q^{43} -243.422 q^{44} -42.2205 q^{46} -164.435 q^{47} +49.0000 q^{49} +201.347 q^{52} -591.600 q^{53} +24.2705 q^{56} -10.5651 q^{58} +180.823 q^{59} +115.773 q^{61} -62.7918 q^{62} -493.950 q^{64} +605.264 q^{67} -579.499 q^{68} +990.917 q^{71} +863.756 q^{73} -3.43947 q^{74} -975.889 q^{76} -214.260 q^{77} +965.930 q^{79} -98.4355 q^{82} -160.924 q^{83} +33.1608 q^{86} -106.126 q^{88} +51.6227 q^{89} +177.225 q^{91} +1544.89 q^{92} -35.7387 q^{94} +1497.31 q^{97} +10.6498 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{2} + 16 q^{4} - 28 q^{7} - 93 q^{8} - 57 q^{11} + 43 q^{13} + 42 q^{14} + 216 q^{16} - 99 q^{17} - 12 q^{19} - 41 q^{22} - 156 q^{23} + 81 q^{26} - 112 q^{28} - 378 q^{29} - 93 q^{31} - 690 q^{32}+ \cdots - 294 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.217342 0.0768421 0.0384211 − 0.999262i \(-0.487767\pi\)
0.0384211 + 0.999262i \(0.487767\pi\)
\(3\) 0 0
\(4\) −7.95276 −0.994095
\(5\) 0 0
\(6\) 0 0
\(7\) −7.00000 −0.377964
\(8\) −3.46721 −0.153231
\(9\) 0 0
\(10\) 0 0
\(11\) 30.6085 0.838983 0.419492 − 0.907759i \(-0.362208\pi\)
0.419492 + 0.907759i \(0.362208\pi\)
\(12\) 0 0
\(13\) −25.3178 −0.540146 −0.270073 − 0.962840i \(-0.587048\pi\)
−0.270073 + 0.962840i \(0.587048\pi\)
\(14\) −1.52140 −0.0290436
\(15\) 0 0
\(16\) 62.8685 0.982321
\(17\) 72.8676 1.03959 0.519794 − 0.854292i \(-0.326009\pi\)
0.519794 + 0.854292i \(0.326009\pi\)
\(18\) 0 0
\(19\) 122.711 1.48167 0.740836 − 0.671686i \(-0.234430\pi\)
0.740836 + 0.671686i \(0.234430\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.65253 0.0644692
\(23\) −194.258 −1.76111 −0.880556 − 0.473943i \(-0.842830\pi\)
−0.880556 + 0.473943i \(0.842830\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −5.50264 −0.0415060
\(27\) 0 0
\(28\) 55.6693 0.375733
\(29\) −48.6103 −0.311266 −0.155633 − 0.987815i \(-0.549742\pi\)
−0.155633 + 0.987815i \(0.549742\pi\)
\(30\) 0 0
\(31\) −288.907 −1.67385 −0.836924 − 0.547320i \(-0.815648\pi\)
−0.836924 + 0.547320i \(0.815648\pi\)
\(32\) 41.4017 0.228714
\(33\) 0 0
\(34\) 15.8372 0.0798841
\(35\) 0 0
\(36\) 0 0
\(37\) −15.8251 −0.0703144 −0.0351572 − 0.999382i \(-0.511193\pi\)
−0.0351572 + 0.999382i \(0.511193\pi\)
\(38\) 26.6702 0.113855
\(39\) 0 0
\(40\) 0 0
\(41\) −452.905 −1.72517 −0.862584 − 0.505914i \(-0.831155\pi\)
−0.862584 + 0.505914i \(0.831155\pi\)
\(42\) 0 0
\(43\) 152.574 0.541101 0.270550 − 0.962706i \(-0.412794\pi\)
0.270550 + 0.962706i \(0.412794\pi\)
\(44\) −243.422 −0.834029
\(45\) 0 0
\(46\) −42.2205 −0.135328
\(47\) −164.435 −0.510325 −0.255163 − 0.966898i \(-0.582129\pi\)
−0.255163 + 0.966898i \(0.582129\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 201.347 0.536957
\(53\) −591.600 −1.53326 −0.766628 − 0.642092i \(-0.778067\pi\)
−0.766628 + 0.642092i \(0.778067\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 24.2705 0.0579157
\(57\) 0 0
\(58\) −10.5651 −0.0239183
\(59\) 180.823 0.399003 0.199501 − 0.979898i \(-0.436068\pi\)
0.199501 + 0.979898i \(0.436068\pi\)
\(60\) 0 0
\(61\) 115.773 0.243004 0.121502 − 0.992591i \(-0.461229\pi\)
0.121502 + 0.992591i \(0.461229\pi\)
\(62\) −62.7918 −0.128622
\(63\) 0 0
\(64\) −493.950 −0.964746
\(65\) 0 0
\(66\) 0 0
\(67\) 605.264 1.10365 0.551827 − 0.833959i \(-0.313931\pi\)
0.551827 + 0.833959i \(0.313931\pi\)
\(68\) −579.499 −1.03345
\(69\) 0 0
\(70\) 0 0
\(71\) 990.917 1.65634 0.828170 − 0.560477i \(-0.189382\pi\)
0.828170 + 0.560477i \(0.189382\pi\)
\(72\) 0 0
\(73\) 863.756 1.38486 0.692431 − 0.721484i \(-0.256540\pi\)
0.692431 + 0.721484i \(0.256540\pi\)
\(74\) −3.43947 −0.00540311
\(75\) 0 0
\(76\) −975.889 −1.47292
\(77\) −214.260 −0.317106
\(78\) 0 0
\(79\) 965.930 1.37564 0.687821 − 0.725881i \(-0.258568\pi\)
0.687821 + 0.725881i \(0.258568\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −98.4355 −0.132566
\(83\) −160.924 −0.212816 −0.106408 − 0.994323i \(-0.533935\pi\)
−0.106408 + 0.994323i \(0.533935\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 33.1608 0.0415793
\(87\) 0 0
\(88\) −106.126 −0.128558
\(89\) 51.6227 0.0614831 0.0307415 − 0.999527i \(-0.490213\pi\)
0.0307415 + 0.999527i \(0.490213\pi\)
\(90\) 0 0
\(91\) 177.225 0.204156
\(92\) 1544.89 1.75071
\(93\) 0 0
\(94\) −35.7387 −0.0392145
\(95\) 0 0
\(96\) 0 0
\(97\) 1497.31 1.56731 0.783656 − 0.621195i \(-0.213353\pi\)
0.783656 + 0.621195i \(0.213353\pi\)
\(98\) 10.6498 0.0109774
\(99\) 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.bf.1.3 4
3.2 odd 2 525.4.a.v.1.2 yes 4
5.4 even 2 1575.4.a.bm.1.2 4
15.2 even 4 525.4.d.o.274.4 8
15.8 even 4 525.4.d.o.274.5 8
15.14 odd 2 525.4.a.s.1.3 ✓ 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.4.a.s.1.3 ✓ 4 15.14 odd 2
525.4.a.v.1.2 yes 4 3.2 odd 2
525.4.d.o.274.4 8 15.2 even 4
525.4.d.o.274.5 8 15.8 even 4
1575.4.a.bf.1.3 4 1.1 even 1 trivial
1575.4.a.bm.1.2 4 5.4 even 2