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Properties

Label 425.2.m.b.26.5
Level $425$
Weight $2$
Character 425.26
Analytic conductor $3.394$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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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] = [425,2,Mod(26,425)] 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("425.26"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.m (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,0,0,0,-8,0,0,-24,0,-8,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(12)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.39364208590\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 26.5
Character \(\chi\) \(=\) 425.26
Dual form 425.2.m.b.376.5

$q$-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+(1.27691 - 1.27691i) q^{2} +(-0.635552 - 0.263254i) q^{3} -1.26102i q^{4} +(-1.14770 + 0.475393i) q^{6} +(-1.66158 - 4.01142i) q^{7} +(0.943613 + 0.943613i) q^{8} +(-1.78670 - 1.78670i) q^{9} +(0.0485041 - 0.0200910i) q^{11} +(-0.331970 + 0.801445i) q^{12} -3.02508i q^{13} +(-7.24394 - 3.00054i) q^{14} +4.93187 q^{16} +(-3.12202 + 2.69314i) q^{17} -4.56292 q^{18} +(5.52988 - 5.52988i) q^{19} +2.98689i q^{21} +(0.0362810 - 0.0875901i) q^{22} +(-0.962654 + 0.398744i) q^{23} +(-0.351305 - 0.848125i) q^{24} +(-3.86277 - 3.86277i) q^{26} +(1.45495 + 3.51255i) q^{27} +(-5.05848 + 2.09529i) q^{28} +(-0.161016 + 0.388726i) q^{29} +(-1.27892 - 0.529745i) q^{31} +(4.41035 - 4.41035i) q^{32} -0.0361159 q^{33} +(-0.547638 + 7.42546i) q^{34} +(-2.25306 + 2.25306i) q^{36} +(-0.311301 - 0.128945i) q^{37} -14.1224i q^{38} +(-0.796365 + 1.92260i) q^{39} +(2.52291 + 6.09084i) q^{41} +(3.81400 + 3.81400i) q^{42} +(7.06729 + 7.06729i) q^{43} +(-0.0253352 - 0.0611647i) q^{44} +(-0.720064 + 1.73839i) q^{46} +6.13168i q^{47} +(-3.13446 - 1.29834i) q^{48} +(-8.38087 + 8.38087i) q^{49} +(2.69319 - 0.889748i) q^{51} -3.81469 q^{52} +(8.52974 - 8.52974i) q^{53} +(6.34307 + 2.62739i) q^{54} +(2.21733 - 5.35312i) q^{56} +(-4.97030 + 2.05876i) q^{57} +(0.290767 + 0.701974i) q^{58} +(3.60468 + 3.60468i) q^{59} +(-2.28486 - 5.51614i) q^{61} +(-2.30951 + 0.956630i) q^{62} +(-4.19844 + 10.1359i) q^{63} -1.39954i q^{64} +(-0.0461170 + 0.0461170i) q^{66} -0.916040 q^{67} +(3.39611 + 3.93693i) q^{68} +0.716788 q^{69} +(3.86169 + 1.59956i) q^{71} -3.37190i q^{72} +(-2.06289 + 4.98025i) q^{73} +(-0.562156 + 0.232853i) q^{74} +(-6.97330 - 6.97330i) q^{76} +(-0.161187 - 0.161187i) q^{77} +(1.43810 + 3.47188i) q^{78} +(9.22305 - 3.82031i) q^{79} +4.96488i q^{81} +(10.9990 + 4.55595i) q^{82} +(4.61746 - 4.61746i) q^{83} +3.76653 q^{84} +18.0487 q^{86} +(0.204668 - 0.204668i) q^{87} +(0.0647272 + 0.0268109i) q^{88} -10.2159i q^{89} +(-12.1349 + 5.02642i) q^{91} +(0.502825 + 1.21393i) q^{92} +(0.673362 + 0.673362i) q^{93} +(7.82963 + 7.82963i) q^{94} +(-3.96405 + 1.64196i) q^{96} +(7.35663 - 17.7605i) q^{97} +21.4033i q^{98} +(-0.122559 - 0.0507655i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 8 q^{6} - 24 q^{9} - 8 q^{11} - 24 q^{12} - 24 q^{16} + 8 q^{17} - 8 q^{18} - 8 q^{19} + 32 q^{22} + 16 q^{23} - 8 q^{24} + 16 q^{26} - 24 q^{27} - 48 q^{28} - 8 q^{29} + 16 q^{34} - 24 q^{36} - 24 q^{37}+ \cdots - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(52\) \(326\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{8}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.27691 − 1.27691i 0.902915 − 0.902915i −0.0927724 − 0.995687i \(-0.529573\pi\)
0.995687 + 0.0927724i \(0.0295729\pi\)
\(3\) −0.635552 − 0.263254i −0.366936 − 0.151990i 0.191594 − 0.981474i \(-0.438634\pi\)
−0.558530 + 0.829484i \(0.688634\pi\)
\(4\) − 1.26102i − 0.630511i
\(5\) 0 0
\(6\) −1.14770 + 0.475393i −0.468546 + 0.194078i
\(7\) −1.66158 − 4.01142i −0.628020 − 1.51617i −0.842080 − 0.539353i \(-0.818669\pi\)
0.214060 − 0.976821i \(-0.431331\pi\)
\(8\) 0.943613 + 0.943613i 0.333617 + 0.333617i
\(9\) −1.78670 − 1.78670i −0.595565 − 0.595565i
\(10\) 0 0
\(11\) 0.0485041 − 0.0200910i 0.0146245 − 0.00605768i −0.375359 − 0.926879i \(-0.622481\pi\)
0.389984 + 0.920822i \(0.372481\pi\)
\(12\) −0.331970 + 0.801445i −0.0958313 + 0.231357i
\(13\) − 3.02508i − 0.839006i −0.907754 − 0.419503i \(-0.862204\pi\)
0.907754 − 0.419503i \(-0.137796\pi\)
\(14\) −7.24394 − 3.00054i −1.93602 − 0.801928i
\(15\) 0 0
\(16\) 4.93187 1.23297
\(17\) −3.12202 + 2.69314i −0.757200 + 0.653183i
\(18\) −4.56292 −1.07549
\(19\) 5.52988 − 5.52988i 1.26864 − 1.26864i 0.321851 − 0.946790i \(-0.395695\pi\)
0.946790 − 0.321851i \(-0.104305\pi\)
\(20\) 0 0
\(21\) 2.98689i 0.651792i
\(22\) 0.0362810 − 0.0875901i 0.00773514 − 0.0186743i
\(23\) −0.962654 + 0.398744i −0.200727 + 0.0831439i −0.480782 − 0.876840i \(-0.659647\pi\)
0.280055 + 0.959984i \(0.409647\pi\)
\(24\) −0.351305 − 0.848125i −0.0717098 − 0.173123i
\(25\) 0 0
\(26\) −3.86277 − 3.86277i −0.757551 − 0.757551i
\(27\) 1.45495 + 3.51255i 0.280005 + 0.675991i
\(28\) −5.05848 + 2.09529i −0.955964 + 0.395973i
\(29\) −0.161016 + 0.388726i −0.0298999 + 0.0721847i −0.938124 − 0.346299i \(-0.887438\pi\)
0.908224 + 0.418484i \(0.137438\pi\)
\(30\) 0 0
\(31\) −1.27892 − 0.529745i −0.229701 − 0.0951451i 0.264865 − 0.964286i \(-0.414673\pi\)
−0.494565 + 0.869141i \(0.664673\pi\)
\(32\) 4.41035 − 4.41035i 0.779647 − 0.779647i
\(33\) −0.0361159 −0.00628698
\(34\) −0.547638 + 7.42546i −0.0939192 + 1.27346i
\(35\) 0 0
\(36\) −2.25306 + 2.25306i −0.375510 + 0.375510i
\(37\) −0.311301 − 0.128945i −0.0511775 − 0.0211984i 0.356948 − 0.934124i \(-0.383817\pi\)
−0.408125 + 0.912926i \(0.633817\pi\)
\(38\) − 14.1224i − 2.29095i
\(39\) −0.796365 + 1.92260i −0.127520 + 0.307862i
\(40\) 0 0
\(41\) 2.52291 + 6.09084i 0.394012 + 0.951230i 0.989057 + 0.147537i \(0.0471345\pi\)
−0.595044 + 0.803693i \(0.702865\pi\)
\(42\) 3.81400 + 3.81400i 0.588513 + 0.588513i
\(43\) 7.06729 + 7.06729i 1.07775 + 1.07775i 0.996711 + 0.0810414i \(0.0258246\pi\)
0.0810414 + 0.996711i \(0.474175\pi\)
\(44\) −0.0253352 − 0.0611647i −0.00381943 − 0.00922092i
\(45\) 0 0
\(46\) −0.720064 + 1.73839i −0.106168 + 0.256311i
\(47\) 6.13168i 0.894398i 0.894435 + 0.447199i \(0.147578\pi\)
−0.894435 + 0.447199i \(0.852422\pi\)
\(48\) −3.13446 − 1.29834i −0.452420 − 0.187399i
\(49\) −8.38087 + 8.38087i −1.19727 + 1.19727i
\(50\) 0 0
\(51\) 2.69319 − 0.889748i 0.377122 − 0.124590i
\(52\) −3.81469 −0.529002
\(53\) 8.52974 − 8.52974i 1.17165 − 1.17165i 0.189834 − 0.981816i \(-0.439205\pi\)
0.981816 − 0.189834i \(-0.0607949\pi\)
\(54\) 6.34307 + 2.62739i 0.863183 + 0.357542i
\(55\) 0 0
\(56\) 2.21733 − 5.35312i 0.296304 − 0.715340i
\(57\) −4.97030 + 2.05876i −0.658332 + 0.272690i
\(58\) 0.290767 + 0.701974i 0.0381796 + 0.0921737i
\(59\) 3.60468 + 3.60468i 0.469290 + 0.469290i 0.901684 − 0.432395i \(-0.142331\pi\)
−0.432395 + 0.901684i \(0.642331\pi\)
\(60\) 0 0
\(61\) −2.28486 − 5.51614i −0.292547 − 0.706270i 0.707453 − 0.706760i \(-0.249844\pi\)
−1.00000 0.000490243i \(0.999844\pi\)
\(62\) −2.30951 + 0.956630i −0.293308 + 0.121492i
\(63\) −4.19844 + 10.1359i −0.528954 + 1.27701i
\(64\) − 1.39954i − 0.174943i
\(65\) 0 0
\(66\) −0.0461170 + 0.0461170i −0.00567661 + 0.00567661i
\(67\) −0.916040 −0.111912 −0.0559561 − 0.998433i \(-0.517821\pi\)
−0.0559561 + 0.998433i \(0.517821\pi\)
\(68\) 3.39611 + 3.93693i 0.411839 + 0.477423i
\(69\) 0.716788 0.0862912
\(70\) 0 0
\(71\) 3.86169 + 1.59956i 0.458298 + 0.189833i 0.599875 − 0.800094i \(-0.295217\pi\)
−0.141577 + 0.989927i \(0.545217\pi\)
\(72\) − 3.37190i − 0.397382i
\(73\) −2.06289 + 4.98025i −0.241443 + 0.582895i −0.997427 − 0.0716959i \(-0.977159\pi\)
0.755984 + 0.654590i \(0.227159\pi\)
\(74\) −0.562156 + 0.232853i −0.0653493 + 0.0270686i
\(75\) 0 0
\(76\) −6.97330 − 6.97330i −0.799892 − 0.799892i
\(77\) −0.161187 − 0.161187i −0.0183690 − 0.0183690i
\(78\) 1.43810 + 3.47188i 0.162833 + 0.393113i
\(79\) 9.22305 − 3.82031i 1.03767 − 0.429819i 0.202198 − 0.979345i \(-0.435192\pi\)
0.835477 + 0.549526i \(0.185192\pi\)
\(80\) 0 0
\(81\) 4.96488i 0.551653i
\(82\) 10.9990 + 4.55595i 1.21464 + 0.503120i
\(83\) 4.61746 − 4.61746i 0.506833 − 0.506833i −0.406720 − 0.913553i \(-0.633328\pi\)
0.913553 + 0.406720i \(0.133328\pi\)
\(84\) 3.76653 0.410962
\(85\) 0 0
\(86\) 18.0487 1.94624
\(87\) 0.204668 − 0.204668i 0.0219427 − 0.0219427i
\(88\) 0.0647272 + 0.0268109i 0.00689994 + 0.00285805i
\(89\) − 10.2159i − 1.08289i −0.840738 − 0.541443i \(-0.817878\pi\)
0.840738 − 0.541443i \(-0.182122\pi\)
\(90\) 0 0
\(91\) −12.1349 + 5.02642i −1.27208 + 0.526912i
\(92\) 0.502825 + 1.21393i 0.0524231 + 0.126561i
\(93\) 0.673362 + 0.673362i 0.0698244 + 0.0698244i
\(94\) 7.82963 + 7.82963i 0.807565 + 0.807565i
\(95\) 0 0
\(96\) −3.96405 + 1.64196i −0.404579 + 0.167582i
\(97\) 7.35663 − 17.7605i 0.746952 − 1.80330i 0.172036 − 0.985091i \(-0.444965\pi\)
0.574916 − 0.818212i \(-0.305035\pi\)
\(98\) 21.4033i 2.16206i
\(99\) −0.122559 − 0.0507655i −0.0123176 − 0.00510212i
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 425.2.m.b.26.5 24
5.2 odd 4 425.2.n.c.349.5 24
5.3 odd 4 425.2.n.f.349.2 24
5.4 even 2 85.2.l.a.26.2 ✓ 24
15.14 odd 2 765.2.be.b.451.5 24
17.2 even 8 inner 425.2.m.b.376.5 24
17.6 odd 16 7225.2.a.bs.1.10 12
17.11 odd 16 7225.2.a.bq.1.10 12
85.2 odd 8 425.2.n.f.274.2 24
85.19 even 8 85.2.l.a.36.2 yes 24
85.24 odd 16 1445.2.d.j.866.20 24
85.44 odd 16 1445.2.d.j.866.19 24
85.53 odd 8 425.2.n.c.274.5 24
85.74 odd 16 1445.2.a.p.1.3 12
85.79 odd 16 1445.2.a.q.1.3 12
255.104 odd 8 765.2.be.b.631.5 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.l.a.26.2 ✓ 24 5.4 even 2
85.2.l.a.36.2 yes 24 85.19 even 8
425.2.m.b.26.5 24 1.1 even 1 trivial
425.2.m.b.376.5 24 17.2 even 8 inner
425.2.n.c.274.5 24 85.53 odd 8
425.2.n.c.349.5 24 5.2 odd 4
425.2.n.f.274.2 24 85.2 odd 8
425.2.n.f.349.2 24 5.3 odd 4
765.2.be.b.451.5 24 15.14 odd 2
765.2.be.b.631.5 24 255.104 odd 8
1445.2.a.p.1.3 12 85.74 odd 16
1445.2.a.q.1.3 12 85.79 odd 16
1445.2.d.j.866.19 24 85.44 odd 16
1445.2.d.j.866.20 24 85.24 odd 16
7225.2.a.bq.1.10 12 17.11 odd 16
7225.2.a.bs.1.10 12 17.6 odd 16