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Properties

Label 425.6.a.d.1.4
Level $425$
Weight $6$
Character 425.1
Self dual yes
Analytic conductor $68.163$
Analytic rank $1$
Dimension $5$
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] = [425,6,Mod(1,425)] 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("425.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 425.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [5,7] 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: \(68.1631234205\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 95x^{3} + 220x^{2} + 1668x - 4640 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(3.95319\) of defining polynomial
Character \(\chi\) \(=\) 425.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.95319 q^{2} +9.94571 q^{3} -7.46587 q^{4} +49.2631 q^{6} -13.5050 q^{7} -195.482 q^{8} -144.083 q^{9} +742.882 q^{11} -74.2534 q^{12} +179.248 q^{13} -66.8928 q^{14} -729.353 q^{16} -289.000 q^{17} -713.670 q^{18} -1855.97 q^{19} -134.317 q^{21} +3679.64 q^{22} +2557.44 q^{23} -1944.21 q^{24} +887.850 q^{26} -3849.81 q^{27} +100.826 q^{28} -1108.67 q^{29} -8774.81 q^{31} +2642.80 q^{32} +7388.50 q^{33} -1431.47 q^{34} +1075.70 q^{36} +5237.19 q^{37} -9192.99 q^{38} +1782.75 q^{39} -16598.4 q^{41} -665.297 q^{42} -14484.7 q^{43} -5546.26 q^{44} +12667.5 q^{46} -16088.0 q^{47} -7253.94 q^{48} -16624.6 q^{49} -2874.31 q^{51} -1338.24 q^{52} -11804.4 q^{53} -19068.9 q^{54} +2639.98 q^{56} -18459.0 q^{57} -5491.46 q^{58} -29390.1 q^{59} +29333.5 q^{61} -43463.3 q^{62} +1945.84 q^{63} +36429.6 q^{64} +36596.7 q^{66} -4122.11 q^{67} +2157.64 q^{68} +25435.6 q^{69} -13649.5 q^{71} +28165.6 q^{72} +33582.1 q^{73} +25940.8 q^{74} +13856.4 q^{76} -10032.6 q^{77} +8830.30 q^{78} -86281.4 q^{79} -3277.05 q^{81} -82215.3 q^{82} +73620.8 q^{83} +1002.79 q^{84} -71745.7 q^{86} -11026.5 q^{87} -145220. q^{88} -100199. q^{89} -2420.74 q^{91} -19093.5 q^{92} -87271.8 q^{93} -79687.2 q^{94} +26284.5 q^{96} -4407.67 q^{97} -82344.9 q^{98} -107037. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 7 q^{2} + 36 q^{3} + 43 q^{4} - 105 q^{6} + 204 q^{7} + 63 q^{8} + 531 q^{9} - 792 q^{11} - 785 q^{12} - 88 q^{13} + 860 q^{14} - 2365 q^{16} - 1445 q^{17} + 2052 q^{18} - 5160 q^{19} - 6428 q^{21}+ \cdots - 535112 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.95319 0.875609 0.437805 − 0.899070i \(-0.355756\pi\)
0.437805 + 0.899070i \(0.355756\pi\)
\(3\) 9.94571 0.638018 0.319009 − 0.947752i \(-0.396650\pi\)
0.319009 + 0.947752i \(0.396650\pi\)
\(4\) −7.46587 −0.233308
\(5\) 0 0
\(6\) 49.2631 0.558654
\(7\) −13.5050 −0.104172 −0.0520858 − 0.998643i \(-0.516587\pi\)
−0.0520858 + 0.998643i \(0.516587\pi\)
\(8\) −195.482 −1.07990
\(9\) −144.083 −0.592933
\(10\) 0 0
\(11\) 742.882 1.85114 0.925568 − 0.378581i \(-0.123588\pi\)
0.925568 + 0.378581i \(0.123588\pi\)
\(12\) −74.2534 −0.148855
\(13\) 179.248 0.294168 0.147084 − 0.989124i \(-0.453011\pi\)
0.147084 + 0.989124i \(0.453011\pi\)
\(14\) −66.8928 −0.0912135
\(15\) 0 0
\(16\) −729.353 −0.712259
\(17\) −289.000 −0.242536
\(18\) −713.670 −0.519178
\(19\) −1855.97 −1.17947 −0.589735 − 0.807596i \(-0.700768\pi\)
−0.589735 + 0.807596i \(0.700768\pi\)
\(20\) 0 0
\(21\) −134.317 −0.0664633
\(22\) 3679.64 1.62087
\(23\) 2557.44 1.00806 0.504029 − 0.863687i \(-0.331850\pi\)
0.504029 + 0.863687i \(0.331850\pi\)
\(24\) −1944.21 −0.688993
\(25\) 0 0
\(26\) 887.850 0.257577
\(27\) −3849.81 −1.01632
\(28\) 100.826 0.0243041
\(29\) −1108.67 −0.244798 −0.122399 − 0.992481i \(-0.539059\pi\)
−0.122399 + 0.992481i \(0.539059\pi\)
\(30\) 0 0
\(31\) −8774.81 −1.63996 −0.819980 − 0.572391i \(-0.806016\pi\)
−0.819980 + 0.572391i \(0.806016\pi\)
\(32\) 2642.80 0.456236
\(33\) 7388.50 1.18106
\(34\) −1431.47 −0.212366
\(35\) 0 0
\(36\) 1075.70 0.138336
\(37\) 5237.19 0.628919 0.314459 − 0.949271i \(-0.398177\pi\)
0.314459 + 0.949271i \(0.398177\pi\)
\(38\) −9192.99 −1.03276
\(39\) 1782.75 0.187685
\(40\) 0 0
\(41\) −16598.4 −1.54208 −0.771042 − 0.636785i \(-0.780264\pi\)
−0.771042 + 0.636785i \(0.780264\pi\)
\(42\) −665.297 −0.0581959
\(43\) −14484.7 −1.19465 −0.597324 − 0.802000i \(-0.703769\pi\)
−0.597324 + 0.802000i \(0.703769\pi\)
\(44\) −5546.26 −0.431886
\(45\) 0 0
\(46\) 12667.5 0.882665
\(47\) −16088.0 −1.06233 −0.531164 − 0.847269i \(-0.678245\pi\)
−0.531164 + 0.847269i \(0.678245\pi\)
\(48\) −7253.94 −0.454434
\(49\) −16624.6 −0.989148
\(50\) 0 0
\(51\) −2874.31 −0.154742
\(52\) −1338.24 −0.0686319
\(53\) −11804.4 −0.577237 −0.288618 − 0.957444i \(-0.593196\pi\)
−0.288618 + 0.957444i \(0.593196\pi\)
\(54\) −19068.9 −0.889899
\(55\) 0 0
\(56\) 2639.98 0.112494
\(57\) −18459.0 −0.752524
\(58\) −5491.46 −0.214347
\(59\) −29390.1 −1.09919 −0.549593 − 0.835433i \(-0.685217\pi\)
−0.549593 + 0.835433i \(0.685217\pi\)
\(60\) 0 0
\(61\) 29333.5 1.00934 0.504671 − 0.863311i \(-0.331614\pi\)
0.504671 + 0.863311i \(0.331614\pi\)
\(62\) −43463.3 −1.43597
\(63\) 1945.84 0.0617667
\(64\) 36429.6 1.11174
\(65\) 0 0
\(66\) 36596.7 1.03415
\(67\) −4122.11 −0.112184 −0.0560922 − 0.998426i \(-0.517864\pi\)
−0.0560922 + 0.998426i \(0.517864\pi\)
\(68\) 2157.64 0.0565856
\(69\) 25435.6 0.643159
\(70\) 0 0
\(71\) −13649.5 −0.321343 −0.160672 − 0.987008i \(-0.551366\pi\)
−0.160672 + 0.987008i \(0.551366\pi\)
\(72\) 28165.6 0.640306
\(73\) 33582.1 0.737566 0.368783 − 0.929516i \(-0.379775\pi\)
0.368783 + 0.929516i \(0.379775\pi\)
\(74\) 25940.8 0.550687
\(75\) 0 0
\(76\) 13856.4 0.275180
\(77\) −10032.6 −0.192836
\(78\) 8830.30 0.164338
\(79\) −86281.4 −1.55543 −0.777713 − 0.628619i \(-0.783621\pi\)
−0.777713 + 0.628619i \(0.783621\pi\)
\(80\) 0 0
\(81\) −3277.05 −0.0554971
\(82\) −82215.3 −1.35026
\(83\) 73620.8 1.17302 0.586510 − 0.809942i \(-0.300501\pi\)
0.586510 + 0.809942i \(0.300501\pi\)
\(84\) 1002.79 0.0155064
\(85\) 0 0
\(86\) −71745.7 −1.04604
\(87\) −11026.5 −0.156185
\(88\) −145220. −1.99903
\(89\) −100199. −1.34087 −0.670435 − 0.741968i \(-0.733893\pi\)
−0.670435 + 0.741968i \(0.733893\pi\)
\(90\) 0 0
\(91\) −2420.74 −0.0306440
\(92\) −19093.5 −0.235188
\(93\) −87271.8 −1.04632
\(94\) −79687.2 −0.930184
\(95\) 0 0
\(96\) 26284.5 0.291087
\(97\) −4407.67 −0.0475642 −0.0237821 − 0.999717i \(-0.507571\pi\)
−0.0237821 + 0.999717i \(0.507571\pi\)
\(98\) −82344.9 −0.866107
\(99\) −107037. −1.09760
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 425.6.a.d.1.4 5
5.4 even 2 85.6.a.a.1.2 ✓ 5
15.14 odd 2 765.6.a.g.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.6.a.a.1.2 ✓ 5 5.4 even 2
425.6.a.d.1.4 5 1.1 even 1 trivial
765.6.a.g.1.4 5 15.14 odd 2