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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [425,4,Mod(324,425)] 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("425.324"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,-20,0,72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.0758117524\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.27206656.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 4x^{3} + 25x^{2} - 30x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 324.3
Root \(1.90022 + 1.90022i\) of defining polynomial
Character \(\chi\) \(=\) 425.324
Dual form 425.4.b.g.324.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.22168i q^{2} -1.15753i q^{3} +6.50750 q^{4} -1.41413 q^{6} -14.6743i q^{7} -17.7235i q^{8} +25.6601 q^{9} -71.3743 q^{11} -7.53262i q^{12} -28.4035i q^{13} -17.9273 q^{14} +30.4076 q^{16} -17.0000i q^{17} -31.3484i q^{18} -85.9592 q^{19} -16.9860 q^{21} +87.1964i q^{22} -7.17872i q^{23} -20.5154 q^{24} -34.7000 q^{26} -60.9556i q^{27} -95.4934i q^{28} +27.1610 q^{29} -15.6170 q^{31} -178.936i q^{32} +82.6178i q^{33} -20.7685 q^{34} +166.983 q^{36} +67.0036i q^{37} +105.014i q^{38} -32.8779 q^{39} -38.5701 q^{41} +20.7514i q^{42} -251.495i q^{43} -464.469 q^{44} -8.77008 q^{46} -57.9121i q^{47} -35.1977i q^{48} +127.663 q^{49} -19.6780 q^{51} -184.836i q^{52} -677.807i q^{53} -74.4680 q^{54} -260.081 q^{56} +99.5002i q^{57} -33.1819i q^{58} -598.645 q^{59} +346.063 q^{61} +19.0789i q^{62} -376.546i q^{63} +24.6588 q^{64} +100.932 q^{66} +849.923i q^{67} -110.628i q^{68} -8.30957 q^{69} -911.836 q^{71} -454.787i q^{72} +704.352i q^{73} +81.8568 q^{74} -559.380 q^{76} +1047.37i q^{77} +40.1662i q^{78} -55.8414 q^{79} +622.266 q^{81} +47.1202i q^{82} -1235.19i q^{83} -110.536 q^{84} -307.245 q^{86} -31.4396i q^{87} +1265.00i q^{88} +72.8069 q^{89} -416.803 q^{91} -46.7155i q^{92} +18.0771i q^{93} -70.7499 q^{94} -207.124 q^{96} -972.939i q^{97} -155.964i q^{98} -1831.47 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 20 q^{4} + 72 q^{6} - 78 q^{9} - 236 q^{11} + 136 q^{14} + 484 q^{16} + 320 q^{19} - 944 q^{21} - 1608 q^{24} - 88 q^{26} - 748 q^{29} + 140 q^{31} + 204 q^{34} + 2516 q^{36} - 496 q^{39} - 708 q^{41}+ \cdots - 1540 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.22168i − 0.431928i −0.976401 − 0.215964i \(-0.930711\pi\)
0.976401 − 0.215964i \(-0.0692894\pi\)
\(3\) − 1.15753i − 0.222766i −0.993778 − 0.111383i \(-0.964472\pi\)
0.993778 − 0.111383i \(-0.0355281\pi\)
\(4\) 6.50750 0.813438
\(5\) 0 0
\(6\) −1.41413 −0.0962191
\(7\) − 14.6743i − 0.792340i −0.918177 − 0.396170i \(-0.870339\pi\)
0.918177 − 0.396170i \(-0.129661\pi\)
\(8\) − 17.7235i − 0.783275i
\(9\) 25.6601 0.950375
\(10\) 0 0
\(11\) −71.3743 −1.95638 −0.978189 − 0.207716i \(-0.933397\pi\)
−0.978189 + 0.207716i \(0.933397\pi\)
\(12\) − 7.53262i − 0.181207i
\(13\) − 28.4035i − 0.605979i −0.952994 − 0.302989i \(-0.902015\pi\)
0.952994 − 0.302989i \(-0.0979847\pi\)
\(14\) −17.9273 −0.342234
\(15\) 0 0
\(16\) 30.4076 0.475119
\(17\) − 17.0000i − 0.242536i
\(18\) − 31.3484i − 0.410494i
\(19\) −85.9592 −1.03792 −0.518958 − 0.854800i \(-0.673680\pi\)
−0.518958 + 0.854800i \(0.673680\pi\)
\(20\) 0 0
\(21\) −16.9860 −0.176507
\(22\) 87.1964i 0.845015i
\(23\) − 7.17872i − 0.0650811i −0.999470 − 0.0325406i \(-0.989640\pi\)
0.999470 − 0.0325406i \(-0.0103598\pi\)
\(24\) −20.5154 −0.174487
\(25\) 0 0
\(26\) −34.7000 −0.261739
\(27\) − 60.9556i − 0.434478i
\(28\) − 95.4934i − 0.644520i
\(29\) 27.1610 0.173919 0.0869597 − 0.996212i \(-0.472285\pi\)
0.0869597 + 0.996212i \(0.472285\pi\)
\(30\) 0 0
\(31\) −15.6170 −0.0904803 −0.0452401 − 0.998976i \(-0.514405\pi\)
−0.0452401 + 0.998976i \(0.514405\pi\)
\(32\) − 178.936i − 0.988493i
\(33\) 82.6178i 0.435815i
\(34\) −20.7685 −0.104758
\(35\) 0 0
\(36\) 166.983 0.773071
\(37\) 67.0036i 0.297712i 0.988859 + 0.148856i \(0.0475590\pi\)
−0.988859 + 0.148856i \(0.952441\pi\)
\(38\) 105.014i 0.448305i
\(39\) −32.8779 −0.134992
\(40\) 0 0
\(41\) −38.5701 −0.146918 −0.0734590 − 0.997298i \(-0.523404\pi\)
−0.0734590 + 0.997298i \(0.523404\pi\)
\(42\) 20.7514i 0.0762383i
\(43\) − 251.495i − 0.891920i −0.895053 − 0.445960i \(-0.852862\pi\)
0.895053 − 0.445960i \(-0.147138\pi\)
\(44\) −464.469 −1.59139
\(45\) 0 0
\(46\) −8.77008 −0.0281104
\(47\) − 57.9121i − 0.179731i −0.995954 − 0.0898654i \(-0.971356\pi\)
0.995954 − 0.0898654i \(-0.0286437\pi\)
\(48\) − 35.1977i − 0.105841i
\(49\) 127.663 0.372197
\(50\) 0 0
\(51\) −19.6780 −0.0540288
\(52\) − 184.836i − 0.492926i
\(53\) − 677.807i − 1.75668i −0.478039 − 0.878339i \(-0.658652\pi\)
0.478039 − 0.878339i \(-0.341348\pi\)
\(54\) −74.4680 −0.187663
\(55\) 0 0
\(56\) −260.081 −0.620620
\(57\) 99.5002i 0.231213i
\(58\) − 33.1819i − 0.0751207i
\(59\) −598.645 −1.32097 −0.660483 − 0.750841i \(-0.729648\pi\)
−0.660483 + 0.750841i \(0.729648\pi\)
\(60\) 0 0
\(61\) 346.063 0.726375 0.363187 − 0.931716i \(-0.381689\pi\)
0.363187 + 0.931716i \(0.381689\pi\)
\(62\) 19.0789i 0.0390810i
\(63\) − 376.546i − 0.753021i
\(64\) 24.6588 0.0481617
\(65\) 0 0
\(66\) 100.932 0.188241
\(67\) 849.923i 1.54977i 0.632102 + 0.774885i \(0.282192\pi\)
−0.632102 + 0.774885i \(0.717808\pi\)
\(68\) − 110.628i − 0.197288i
\(69\) −8.30957 −0.0144979
\(70\) 0 0
\(71\) −911.836 −1.52415 −0.762077 − 0.647486i \(-0.775820\pi\)
−0.762077 + 0.647486i \(0.775820\pi\)
\(72\) − 454.787i − 0.744405i
\(73\) 704.352i 1.12929i 0.825334 + 0.564645i \(0.190987\pi\)
−0.825334 + 0.564645i \(0.809013\pi\)
\(74\) 81.8568 0.128590
\(75\) 0 0
\(76\) −559.380 −0.844280
\(77\) 1047.37i 1.55012i
\(78\) 40.1662i 0.0583067i
\(79\) −55.8414 −0.0795272 −0.0397636 − 0.999209i \(-0.512660\pi\)
−0.0397636 + 0.999209i \(0.512660\pi\)
\(80\) 0 0
\(81\) 622.266 0.853588
\(82\) 47.1202i 0.0634580i
\(83\) − 1235.19i − 1.63350i −0.576995 − 0.816748i \(-0.695775\pi\)
0.576995 − 0.816748i \(-0.304225\pi\)
\(84\) −110.536 −0.143577
\(85\) 0 0
\(86\) −307.245 −0.385246
\(87\) − 31.4396i − 0.0387434i
\(88\) 1265.00i 1.53238i
\(89\) 72.8069 0.0867136 0.0433568 − 0.999060i \(-0.486195\pi\)
0.0433568 + 0.999060i \(0.486195\pi\)
\(90\) 0 0
\(91\) −416.803 −0.480141
\(92\) − 46.7155i − 0.0529395i
\(93\) 18.0771i 0.0201560i
\(94\) −70.7499 −0.0776308
\(95\) 0 0
\(96\) −207.124 −0.220203
\(97\) − 972.939i − 1.01842i −0.860642 − 0.509211i \(-0.829937\pi\)
0.860642 − 0.509211i \(-0.170063\pi\)
\(98\) − 155.964i − 0.160762i
\(99\) −1831.47 −1.85929
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 425.4.b.g.324.3 6
5.2 odd 4 85.4.a.e.1.3 ✓ 3
5.3 odd 4 425.4.a.h.1.1 3
5.4 even 2 inner 425.4.b.g.324.4 6
15.2 even 4 765.4.a.l.1.1 3
20.7 even 4 1360.4.a.s.1.2 3
85.67 odd 4 1445.4.a.j.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.e.1.3 ✓ 3 5.2 odd 4
425.4.a.h.1.1 3 5.3 odd 4
425.4.b.g.324.3 6 1.1 even 1 trivial
425.4.b.g.324.4 6 5.4 even 2 inner
765.4.a.l.1.1 3 15.2 even 4
1360.4.a.s.1.2 3 20.7 even 4
1445.4.a.j.1.3 3 85.67 odd 4