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Properties

Label 525.4.a.v.1.2
Level $525$
Weight $4$
Character 525.1
Self dual yes
Analytic conductor $30.976$
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] = [525,4,Mod(1,525)] 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("525.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,6,12,16,0,18,-28,93,36,0,57] 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: \(30.9760027530\)
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: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.21734\) of defining polynomial
Character \(\chi\) \(=\) 525.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.217342 q^{2} +3.00000 q^{3} -7.95276 q^{4} -0.652027 q^{6} -7.00000 q^{7} +3.46721 q^{8} +9.00000 q^{9} -30.6085 q^{11} -23.8583 q^{12} -25.3178 q^{13} +1.52140 q^{14} +62.8685 q^{16} -72.8676 q^{17} -1.95608 q^{18} +122.711 q^{19} -21.0000 q^{21} +6.65253 q^{22} +194.258 q^{23} +10.4016 q^{24} +5.50264 q^{26} +27.0000 q^{27} +55.6693 q^{28} +48.6103 q^{29} -288.907 q^{31} -41.4017 q^{32} -91.8255 q^{33} +15.8372 q^{34} -71.5749 q^{36} -15.8251 q^{37} -26.6702 q^{38} -75.9535 q^{39} +452.905 q^{41} +4.56419 q^{42} +152.574 q^{43} +243.422 q^{44} -42.2205 q^{46} +164.435 q^{47} +188.606 q^{48} +49.0000 q^{49} -218.603 q^{51} +201.347 q^{52} +591.600 q^{53} -5.86824 q^{54} -24.2705 q^{56} +368.132 q^{57} -10.5651 q^{58} -180.823 q^{59} +115.773 q^{61} +62.7918 q^{62} -63.0000 q^{63} -493.950 q^{64} +19.9576 q^{66} +605.264 q^{67} +579.499 q^{68} +582.774 q^{69} -990.917 q^{71} +31.2049 q^{72} +863.756 q^{73} +3.43947 q^{74} -975.889 q^{76} +214.260 q^{77} +16.5079 q^{78} +965.930 q^{79} +81.0000 q^{81} -98.4355 q^{82} +160.924 q^{83} +167.008 q^{84} -33.1608 q^{86} +145.831 q^{87} -106.126 q^{88} -51.6227 q^{89} +177.225 q^{91} -1544.89 q^{92} -866.722 q^{93} -35.7387 q^{94} -124.205 q^{96} +1497.31 q^{97} -10.6498 q^{98} -275.477 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{2} + 12 q^{3} + 16 q^{4} + 18 q^{6} - 28 q^{7} + 93 q^{8} + 36 q^{9} + 57 q^{11} + 48 q^{12} + 43 q^{13} - 42 q^{14} + 216 q^{16} + 99 q^{17} + 54 q^{18} - 12 q^{19} - 84 q^{21} - 41 q^{22} + 156 q^{23}+ \cdots + 513 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.217342 −0.0768421 −0.0384211 − 0.999262i \(-0.512233\pi\)
−0.0384211 + 0.999262i \(0.512233\pi\)
\(3\) 3.00000 0.577350
\(4\) −7.95276 −0.994095
\(5\) 0 0
\(6\) −0.652027 −0.0443648
\(7\) −7.00000 −0.377964
\(8\) 3.46721 0.153231
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −30.6085 −0.838983 −0.419492 − 0.907759i \(-0.637792\pi\)
−0.419492 + 0.907759i \(0.637792\pi\)
\(12\) −23.8583 −0.573941
\(13\) −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 \(-0.673991\pi\)
−0.519794 + 0.854292i \(0.673991\pi\)
\(18\) −1.95608 −0.0256140
\(19\) 122.711 1.48167 0.740836 − 0.671686i \(-0.234430\pi\)
0.740836 + 0.671686i \(0.234430\pi\)
\(20\) 0 0
\(21\) −21.0000 −0.218218
\(22\) 6.65253 0.0644692
\(23\) 194.258 1.76111 0.880556 − 0.473943i \(-0.157170\pi\)
0.880556 + 0.473943i \(0.157170\pi\)
\(24\) 10.4016 0.0884677
\(25\) 0 0
\(26\) 5.50264 0.0415060
\(27\) 27.0000 0.192450
\(28\) 55.6693 0.375733
\(29\) 48.6103 0.311266 0.155633 − 0.987815i \(-0.450258\pi\)
0.155633 + 0.987815i \(0.450258\pi\)
\(30\) 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
\(33\) −91.8255 −0.484387
\(34\) 15.8372 0.0798841
\(35\) 0 0
\(36\) −71.5749 −0.331365
\(37\) −15.8251 −0.0703144 −0.0351572 − 0.999382i \(-0.511193\pi\)
−0.0351572 + 0.999382i \(0.511193\pi\)
\(38\) −26.6702 −0.113855
\(39\) −75.9535 −0.311854
\(40\) 0 0
\(41\) 452.905 1.72517 0.862584 − 0.505914i \(-0.168845\pi\)
0.862584 + 0.505914i \(0.168845\pi\)
\(42\) 4.56419 0.0167683
\(43\) 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 \(-0.417871\pi\)
0.255163 + 0.966898i \(0.417871\pi\)
\(48\) 188.606 0.567143
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −218.603 −0.600206
\(52\) 201.347 0.536957
\(53\) 591.600 1.53326 0.766628 − 0.642092i \(-0.221933\pi\)
0.766628 + 0.642092i \(0.221933\pi\)
\(54\) −5.86824 −0.0147883
\(55\) 0 0
\(56\) −24.2705 −0.0579157
\(57\) 368.132 0.855443
\(58\) −10.5651 −0.0239183
\(59\) −180.823 −0.399003 −0.199501 − 0.979898i \(-0.563932\pi\)
−0.199501 + 0.979898i \(0.563932\pi\)
\(60\) 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
\(63\) −63.0000 −0.125988
\(64\) −493.950 −0.964746
\(65\) 0 0
\(66\) 19.9576 0.0372213
\(67\) 605.264 1.10365 0.551827 − 0.833959i \(-0.313931\pi\)
0.551827 + 0.833959i \(0.313931\pi\)
\(68\) 579.499 1.03345
\(69\) 582.774 1.01678
\(70\) 0 0
\(71\) −990.917 −1.65634 −0.828170 − 0.560477i \(-0.810618\pi\)
−0.828170 + 0.560477i \(0.810618\pi\)
\(72\) 31.2049 0.0510768
\(73\) 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
\(78\) 16.5079 0.0239635
\(79\) 965.930 1.37564 0.687821 − 0.725881i \(-0.258568\pi\)
0.687821 + 0.725881i \(0.258568\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −98.4355 −0.132566
\(83\) 160.924 0.212816 0.106408 − 0.994323i \(-0.466065\pi\)
0.106408 + 0.994323i \(0.466065\pi\)
\(84\) 167.008 0.216929
\(85\) 0 0
\(86\) −33.1608 −0.0415793
\(87\) 145.831 0.179709
\(88\) −106.126 −0.128558
\(89\) −51.6227 −0.0614831 −0.0307415 − 0.999527i \(-0.509787\pi\)
−0.0307415 + 0.999527i \(0.509787\pi\)
\(90\) 0 0
\(91\) 177.225 0.204156
\(92\) −1544.89 −1.75071
\(93\) −866.722 −0.966396
\(94\) −35.7387 −0.0392145
\(95\) 0 0
\(96\) −124.205 −0.132048
\(97\) 1497.31 1.56731 0.783656 − 0.621195i \(-0.213353\pi\)
0.783656 + 0.621195i \(0.213353\pi\)
\(98\) −10.6498 −0.0109774
\(99\) −275.477 −0.279661
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 525.4.a.v.1.2 yes 4
3.2 odd 2 1575.4.a.bf.1.3 4
5.2 odd 4 525.4.d.o.274.4 8
5.3 odd 4 525.4.d.o.274.5 8
5.4 even 2 525.4.a.s.1.3 ✓ 4
15.14 odd 2 1575.4.a.bm.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
525.4.a.s.1.3 ✓ 4 5.4 even 2
525.4.a.v.1.2 yes 4 1.1 even 1 trivial
525.4.d.o.274.4 8 5.2 odd 4
525.4.d.o.274.5 8 5.3 odd 4
1575.4.a.bf.1.3 4 3.2 odd 2
1575.4.a.bm.1.2 4 15.14 odd 2