↑

Properties

Label 425.2.n.c.274.5
Level $425$
Weight $2$
Character 425.274
Analytic conductor $3.394$
Analytic rank $0$
Dimension $24$
Inner twists $2$

Related objects

Downloads

Learn more

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(49,425)] 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("425.49"); 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([4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 425.n (of order \(8\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == 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 274.5
Character \(\chi\) \(=\) 425.274
Dual form 425.2.n.c.349.5

$q$-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} +(-0.263254 - 0.635552i) q^{3} -1.26102i q^{4} +(-1.14770 - 0.475393i) q^{6} +(4.01142 + 1.66158i) q^{7} +(0.943613 + 0.943613i) q^{8} +(1.78670 - 1.78670i) q^{9} +(0.0485041 + 0.0200910i) q^{11} +(-0.801445 + 0.331970i) q^{12} -3.02508 q^{13} +(7.24394 - 3.00054i) q^{14} +4.93187 q^{16} +(-2.69314 + 3.12202i) q^{17} -4.56292i q^{18} +(-5.52988 - 5.52988i) q^{19} -2.98689i q^{21} +(0.0875901 - 0.0362810i) q^{22} +(0.398744 - 0.962654i) q^{23} +(0.351305 - 0.848125i) q^{24} +(-3.86277 + 3.86277i) q^{26} +(-3.51255 - 1.45495i) q^{27} +(2.09529 - 5.05848i) q^{28} +(0.161016 + 0.388726i) q^{29} +(-1.27892 + 0.529745i) q^{31} +(4.41035 - 4.41035i) q^{32} -0.0361159i q^{33} +(0.547638 + 7.42546i) q^{34} +(-2.25306 - 2.25306i) q^{36} +(0.128945 + 0.311301i) q^{37} -14.1224 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} -6.13168 q^{47} +(-1.29834 - 3.13446i) q^{48} +(8.38087 + 8.38087i) q^{49} +(2.69319 + 0.889748i) q^{51} +3.81469i q^{52} +(-8.52974 + 8.52974i) q^{53} +(-6.34307 + 2.62739i) q^{54} +(2.21733 + 5.35312i) q^{56} +(-2.05876 + 4.97030i) q^{57} +(0.701974 + 0.290767i) q^{58} +(-3.60468 + 3.60468i) q^{59} +(-2.28486 + 5.51614i) q^{61} +(-0.956630 + 2.30951i) q^{62} +(10.1359 - 4.19844i) q^{63} -1.39954i q^{64} +(-0.0461170 - 0.0461170i) q^{66} +0.916040i q^{67} +(3.93693 + 3.39611i) q^{68} -0.716788 q^{69} +(3.86169 - 1.59956i) q^{71} +3.37190 q^{72} +(4.98025 - 2.06289i) q^{73} +(0.562156 + 0.232853i) q^{74} +(-6.97330 + 6.97330i) q^{76} +(0.161187 + 0.161187i) q^{77} +(3.47188 + 1.43810i) q^{78} +(-9.22305 - 3.82031i) q^{79} -4.96488i q^{81} +(-4.55595 - 10.9990i) q^{82} +(-4.61746 + 4.61746i) q^{83} -3.76653 q^{84} +18.0487 q^{86} +(0.204668 - 0.204668i) q^{87} +(0.0268109 + 0.0647272i) q^{88} -10.2159i q^{89} +(-12.1349 - 5.02642i) q^{91} +(-1.21393 - 0.502825i) q^{92} +(0.673362 + 0.673362i) q^{93} +(-7.82963 + 7.82963i) q^{94} +(-3.96405 - 1.64196i) q^{96} +(17.7605 - 7.35663i) q^{97} +21.4033 q^{98} +(0.122559 - 0.0507655i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 8 q^{3} - 8 q^{6} + 24 q^{9} - 8 q^{11} + 40 q^{12} + 16 q^{13} - 24 q^{16} + 8 q^{19} - 24 q^{22} + 8 q^{23} + 8 q^{24} + 16 q^{26} + 16 q^{27} - 40 q^{28} + 8 q^{29} - 16 q^{34} - 24 q^{36} - 16 q^{37}+ \cdots + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(e\left(\frac{7}{8}\right)\)

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 \(0.0295729\pi\)
\(3\) −0.263254 − 0.635552i −0.151990 − 0.366936i 0.829484 − 0.558530i \(-0.188634\pi\)
−0.981474 + 0.191594i \(0.938634\pi\)
\(4\) − 1.26102i − 0.630511i
\(5\) 0 0
\(6\) −1.14770 − 0.475393i −0.468546 − 0.194078i
\(7\) 4.01142 + 1.66158i 1.51617 + 0.628020i 0.976821 − 0.214060i \(-0.0686688\pi\)
0.539353 + 0.842080i \(0.318669\pi\)
\(8\) 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 0.389984 − 0.920822i \(-0.372481\pi\)
−0.375359 + 0.926879i \(0.622481\pi\)
\(12\) −0.801445 + 0.331970i −0.231357 + 0.0958313i
\(13\) −3.02508 −0.839006 −0.419503 − 0.907754i \(-0.637796\pi\)
−0.419503 + 0.907754i \(0.637796\pi\)
\(14\) 7.24394 − 3.00054i 1.93602 − 0.801928i
\(15\) 0 0
\(16\) 4.93187 1.23297
\(17\) −2.69314 + 3.12202i −0.653183 + 0.757200i
\(18\) − 4.56292i − 1.07549i
\(19\) −5.52988 − 5.52988i −1.26864 − 1.26864i −0.946790 − 0.321851i \(-0.895695\pi\)
−0.321851 − 0.946790i \(-0.604305\pi\)
\(20\) 0 0
\(21\) − 2.98689i − 0.651792i
\(22\) 0.0875901 − 0.0362810i 0.0186743 − 0.00773514i
\(23\) 0.398744 − 0.962654i 0.0831439 − 0.200727i −0.876840 − 0.480782i \(-0.840353\pi\)
0.959984 + 0.280055i \(0.0903527\pi\)
\(24\) 0.351305 − 0.848125i 0.0717098 − 0.173123i
\(25\) 0 0
\(26\) −3.86277 + 3.86277i −0.757551 + 0.757551i
\(27\) −3.51255 − 1.45495i −0.675991 − 0.280005i
\(28\) 2.09529 − 5.05848i 0.395973 − 0.955964i
\(29\) 0.161016 + 0.388726i 0.0298999 + 0.0721847i 0.938124 − 0.346299i \(-0.112562\pi\)
−0.908224 + 0.418484i \(0.862562\pi\)
\(30\) 0 0
\(31\) −1.27892 + 0.529745i −0.229701 + 0.0951451i −0.494565 − 0.869141i \(-0.664673\pi\)
0.264865 + 0.964286i \(0.414673\pi\)
\(32\) 4.41035 − 4.41035i 0.779647 − 0.779647i
\(33\) − 0.0361159i − 0.00628698i
\(34\) 0.547638 + 7.42546i 0.0939192 + 1.27346i
\(35\) 0 0
\(36\) −2.25306 − 2.25306i −0.375510 − 0.375510i
\(37\) 0.128945 + 0.311301i 0.0211984 + 0.0511775i 0.934124 − 0.356948i \(-0.116183\pi\)
−0.912926 + 0.408125i \(0.866183\pi\)
\(38\) −14.1224 −2.29095
\(39\) 0.796365 + 1.92260i 0.127520 + 0.307862i
\(40\) 0 0
\(41\) 2.52291 − 6.09084i 0.394012 − 0.951230i −0.595044 − 0.803693i \(-0.702865\pi\)
0.989057 − 0.147537i \(-0.0471345\pi\)
\(42\) −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
\(47\) −6.13168 −0.894398 −0.447199 − 0.894435i \(-0.647578\pi\)
−0.447199 + 0.894435i \(0.647578\pi\)
\(48\) −1.29834 − 3.13446i −0.187399 − 0.452420i
\(49\) 8.38087 + 8.38087i 1.19727 + 1.19727i
\(50\) 0 0
\(51\) 2.69319 + 0.889748i 0.377122 + 0.124590i
\(52\) 3.81469i 0.529002i
\(53\) −8.52974 + 8.52974i −1.17165 + 1.17165i −0.189834 + 0.981816i \(0.560795\pi\)
−0.981816 + 0.189834i \(0.939205\pi\)
\(54\) −6.34307 + 2.62739i −0.863183 + 0.357542i
\(55\) 0 0
\(56\) 2.21733 + 5.35312i 0.296304 + 0.715340i
\(57\) −2.05876 + 4.97030i −0.272690 + 0.658332i
\(58\) 0.701974 + 0.290767i 0.0921737 + 0.0381796i
\(59\) −3.60468 + 3.60468i −0.469290 + 0.469290i −0.901684 − 0.432395i \(-0.857669\pi\)
0.432395 + 0.901684i \(0.357669\pi\)
\(60\) 0 0
\(61\) −2.28486 + 5.51614i −0.292547 + 0.706270i −1.00000 0.000490243i \(-0.999844\pi\)
0.707453 + 0.706760i \(0.249844\pi\)
\(62\) −0.956630 + 2.30951i −0.121492 + 0.293308i
\(63\) 10.1359 − 4.19844i 1.27701 − 0.528954i
\(64\) − 1.39954i − 0.174943i
\(65\) 0 0
\(66\) −0.0461170 − 0.0461170i −0.00567661 − 0.00567661i
\(67\) 0.916040i 0.111912i 0.998433 + 0.0559561i \(0.0178207\pi\)
−0.998433 + 0.0559561i \(0.982179\pi\)
\(68\) 3.93693 + 3.39611i 0.477423 + 0.411839i
\(69\) −0.716788 −0.0862912
\(70\) 0 0
\(71\) 3.86169 − 1.59956i 0.458298 − 0.189833i −0.141577 − 0.989927i \(-0.545217\pi\)
0.599875 + 0.800094i \(0.295217\pi\)
\(72\) 3.37190 0.397382
\(73\) 4.98025 − 2.06289i 0.582895 − 0.241443i −0.0716959 − 0.997427i \(-0.522841\pi\)
0.654590 + 0.755984i \(0.272841\pi\)
\(74\) 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
\(78\) 3.47188 + 1.43810i 0.393113 + 0.162833i
\(79\) −9.22305 − 3.82031i −1.03767 − 0.429819i −0.202198 − 0.979345i \(-0.564808\pi\)
−0.835477 + 0.549526i \(0.814808\pi\)
\(80\) 0 0
\(81\) − 4.96488i − 0.551653i
\(82\) −4.55595 − 10.9990i −0.503120 − 1.21464i
\(83\) −4.61746 + 4.61746i −0.506833 + 0.506833i −0.913553 − 0.406720i \(-0.866672\pi\)
0.406720 + 0.913553i \(0.366672\pi\)
\(84\) −3.76653 −0.410962
\(85\) 0 0
\(86\) 18.0487 1.94624
\(87\) 0.204668 − 0.204668i 0.0219427 − 0.0219427i
\(88\) 0.0268109 + 0.0647272i 0.00285805 + 0.00689994i
\(89\) − 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
\(92\) −1.21393 − 0.502825i −0.126561 − 0.0524231i
\(93\) 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
\(97\) 17.7605 − 7.35663i 1.80330 − 0.746952i 0.818212 − 0.574916i \(-0.194965\pi\)
0.985091 − 0.172036i \(-0.0550346\pi\)
\(98\) 21.4033 2.16206
\(99\) 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.n.c.274.5 24
5.2 odd 4 425.2.m.b.376.5 24
5.3 odd 4 85.2.l.a.36.2 yes 24
5.4 even 2 425.2.n.f.274.2 24
15.8 even 4 765.2.be.b.631.5 24
17.9 even 8 425.2.n.f.349.2 24
85.3 even 16 1445.2.a.p.1.3 12
85.9 even 8 inner 425.2.n.c.349.5 24
85.37 even 16 7225.2.a.bs.1.10 12
85.43 odd 8 85.2.l.a.26.2 ✓ 24
85.48 even 16 1445.2.a.q.1.3 12
85.63 even 16 1445.2.d.j.866.20 24
85.73 even 16 1445.2.d.j.866.19 24
85.77 odd 8 425.2.m.b.26.5 24
85.82 even 16 7225.2.a.bq.1.10 12
255.128 even 8 765.2.be.b.451.5 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.2.l.a.26.2 ✓ 24 85.43 odd 8
85.2.l.a.36.2 yes 24 5.3 odd 4
425.2.m.b.26.5 24 85.77 odd 8
425.2.m.b.376.5 24 5.2 odd 4
425.2.n.c.274.5 24 1.1 even 1 trivial
425.2.n.c.349.5 24 85.9 even 8 inner
425.2.n.f.274.2 24 5.4 even 2
425.2.n.f.349.2 24 17.9 even 8
765.2.be.b.451.5 24 255.128 even 8
765.2.be.b.631.5 24 15.8 even 4
1445.2.a.p.1.3 12 85.3 even 16
1445.2.a.q.1.3 12 85.48 even 16
1445.2.d.j.866.19 24 85.73 even 16
1445.2.d.j.866.20 24 85.63 even 16
7225.2.a.bq.1.10 12 85.82 even 16
7225.2.a.bs.1.10 12 85.37 even 16