↑

Properties

Label 425.4.b.d.324.2
Level $425$
Weight $4$
Character 425.324
Analytic conductor $25.076$
Analytic rank $1$
Dimension $2$
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] = [425,4,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 = [2,0,0,-2,0,60] 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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 85)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 324.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 425.324
Dual form 425.4.b.d.324.1

$q$-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)
 
\(f(q)\) \(=\) \(q+3.00000i q^{2} -10.0000i q^{3} -1.00000 q^{4} +30.0000 q^{6} -22.0000i q^{7} +21.0000i q^{8} -73.0000 q^{9} -30.0000 q^{11} +10.0000i q^{12} +46.0000i q^{13} +66.0000 q^{14} -71.0000 q^{16} +17.0000i q^{17} -219.000i q^{18} -104.000 q^{19} -220.000 q^{21} -90.0000i q^{22} -42.0000i q^{23} +210.000 q^{24} -138.000 q^{26} +460.000i q^{27} +22.0000i q^{28} +66.0000 q^{29} +194.000 q^{31} -45.0000i q^{32} +300.000i q^{33} -51.0000 q^{34} +73.0000 q^{36} +206.000i q^{37} -312.000i q^{38} +460.000 q^{39} -126.000 q^{41} -660.000i q^{42} +388.000i q^{43} +30.0000 q^{44} +126.000 q^{46} -540.000i q^{47} +710.000i q^{48} -141.000 q^{49} +170.000 q^{51} -46.0000i q^{52} -78.0000i q^{53} -1380.00 q^{54} +462.000 q^{56} +1040.00i q^{57} +198.000i q^{58} -432.000 q^{59} -610.000 q^{61} +582.000i q^{62} +1606.00i q^{63} -433.000 q^{64} -900.000 q^{66} +848.000i q^{67} -17.0000i q^{68} -420.000 q^{69} -174.000 q^{71} -1533.00i q^{72} -362.000i q^{73} -618.000 q^{74} +104.000 q^{76} +660.000i q^{77} +1380.00i q^{78} -398.000 q^{79} +2629.00 q^{81} -378.000i q^{82} -828.000i q^{83} +220.000 q^{84} -1164.00 q^{86} -660.000i q^{87} -630.000i q^{88} -630.000 q^{89} +1012.00 q^{91} +42.0000i q^{92} -1940.00i q^{93} +1620.00 q^{94} -450.000 q^{96} -1486.00i q^{97} -423.000i q^{98} +2190.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 60 q^{6} - 146 q^{9} - 60 q^{11} + 132 q^{14} - 142 q^{16} - 208 q^{19} - 440 q^{21} + 420 q^{24} - 276 q^{26} + 132 q^{29} + 388 q^{31} - 102 q^{34} + 146 q^{36} + 920 q^{39} - 252 q^{41}+ \cdots + 4380 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.00000i 1.06066i 0.847791 + 0.530330i \(0.177932\pi\)
−0.847791 + 0.530330i \(0.822068\pi\)
\(3\) − 10.0000i − 1.92450i −0.272166 − 0.962250i \(-0.587740\pi\)
0.272166 − 0.962250i \(-0.412260\pi\)
\(4\) −1.00000 −0.125000
\(5\) 0 0
\(6\) 30.0000 2.04124
\(7\) − 22.0000i − 1.18789i −0.804506 − 0.593944i \(-0.797570\pi\)
0.804506 − 0.593944i \(-0.202430\pi\)
\(8\) 21.0000i 0.928078i
\(9\) −73.0000 −2.70370
\(10\) 0 0
\(11\) −30.0000 −0.822304 −0.411152 − 0.911567i \(-0.634873\pi\)
−0.411152 + 0.911567i \(0.634873\pi\)
\(12\) 10.0000i 0.240563i
\(13\) 46.0000i 0.981393i 0.871331 + 0.490696i \(0.163258\pi\)
−0.871331 + 0.490696i \(0.836742\pi\)
\(14\) 66.0000 1.25995
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) 17.0000i 0.242536i
\(18\) − 219.000i − 2.86771i
\(19\) −104.000 −1.25575 −0.627875 − 0.778314i \(-0.716075\pi\)
−0.627875 + 0.778314i \(0.716075\pi\)
\(20\) 0 0
\(21\) −220.000 −2.28609
\(22\) − 90.0000i − 0.872185i
\(23\) − 42.0000i − 0.380765i −0.981710 − 0.190383i \(-0.939027\pi\)
0.981710 − 0.190383i \(-0.0609729\pi\)
\(24\) 210.000 1.78609
\(25\) 0 0
\(26\) −138.000 −1.04092
\(27\) 460.000i 3.27878i
\(28\) 22.0000i 0.148486i
\(29\) 66.0000 0.422617 0.211308 − 0.977419i \(-0.432228\pi\)
0.211308 + 0.977419i \(0.432228\pi\)
\(30\) 0 0
\(31\) 194.000 1.12398 0.561991 − 0.827143i \(-0.310036\pi\)
0.561991 + 0.827143i \(0.310036\pi\)
\(32\) − 45.0000i − 0.248592i
\(33\) 300.000i 1.58252i
\(34\) −51.0000 −0.257248
\(35\) 0 0
\(36\) 73.0000 0.337963
\(37\) 206.000i 0.915302i 0.889132 + 0.457651i \(0.151309\pi\)
−0.889132 + 0.457651i \(0.848691\pi\)
\(38\) − 312.000i − 1.33192i
\(39\) 460.000 1.88869
\(40\) 0 0
\(41\) −126.000 −0.479949 −0.239974 − 0.970779i \(-0.577139\pi\)
−0.239974 + 0.970779i \(0.577139\pi\)
\(42\) − 660.000i − 2.42477i
\(43\) 388.000i 1.37603i 0.725695 + 0.688017i \(0.241518\pi\)
−0.725695 + 0.688017i \(0.758482\pi\)
\(44\) 30.0000 0.102788
\(45\) 0 0
\(46\) 126.000 0.403863
\(47\) − 540.000i − 1.67590i −0.545750 − 0.837948i \(-0.683755\pi\)
0.545750 − 0.837948i \(-0.316245\pi\)
\(48\) 710.000i 2.13499i
\(49\) −141.000 −0.411079
\(50\) 0 0
\(51\) 170.000 0.466760
\(52\) − 46.0000i − 0.122674i
\(53\) − 78.0000i − 0.202153i −0.994879 − 0.101077i \(-0.967771\pi\)
0.994879 − 0.101077i \(-0.0322287\pi\)
\(54\) −1380.00 −3.47767
\(55\) 0 0
\(56\) 462.000 1.10245
\(57\) 1040.00i 2.41669i
\(58\) 198.000i 0.448253i
\(59\) −432.000 −0.953248 −0.476624 − 0.879107i \(-0.658140\pi\)
−0.476624 + 0.879107i \(0.658140\pi\)
\(60\) 0 0
\(61\) −610.000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 582.000i 1.19216i
\(63\) 1606.00i 3.21170i
\(64\) −433.000 −0.845703
\(65\) 0 0
\(66\) −900.000 −1.67852
\(67\) 848.000i 1.54626i 0.634245 + 0.773132i \(0.281311\pi\)
−0.634245 + 0.773132i \(0.718689\pi\)
\(68\) − 17.0000i − 0.0303170i
\(69\) −420.000 −0.732783
\(70\) 0 0
\(71\) −174.000 −0.290845 −0.145423 − 0.989370i \(-0.546454\pi\)
−0.145423 + 0.989370i \(0.546454\pi\)
\(72\) − 1533.00i − 2.50925i
\(73\) − 362.000i − 0.580396i −0.956967 − 0.290198i \(-0.906279\pi\)
0.956967 − 0.290198i \(-0.0937211\pi\)
\(74\) −618.000 −0.970825
\(75\) 0 0
\(76\) 104.000 0.156969
\(77\) 660.000i 0.976805i
\(78\) 1380.00i 2.00326i
\(79\) −398.000 −0.566816 −0.283408 − 0.958999i \(-0.591465\pi\)
−0.283408 + 0.958999i \(0.591465\pi\)
\(80\) 0 0
\(81\) 2629.00 3.60631
\(82\) − 378.000i − 0.509062i
\(83\) − 828.000i − 1.09500i −0.836806 − 0.547499i \(-0.815580\pi\)
0.836806 − 0.547499i \(-0.184420\pi\)
\(84\) 220.000 0.285762
\(85\) 0 0
\(86\) −1164.00 −1.45950
\(87\) − 660.000i − 0.813327i
\(88\) − 630.000i − 0.763162i
\(89\) −630.000 −0.750336 −0.375168 − 0.926957i \(-0.622415\pi\)
−0.375168 + 0.926957i \(0.622415\pi\)
\(90\) 0 0
\(91\) 1012.00 1.16578
\(92\) 42.0000i 0.0475957i
\(93\) − 1940.00i − 2.16310i
\(94\) 1620.00 1.77756
\(95\) 0 0
\(96\) −450.000 −0.478416
\(97\) − 1486.00i − 1.55547i −0.628593 − 0.777734i \(-0.716369\pi\)
0.628593 − 0.777734i \(-0.283631\pi\)
\(98\) − 423.000i − 0.436015i
\(99\) 2190.00 2.22327
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.d.324.2 2
5.2 odd 4 425.4.a.a.1.1 1
5.3 odd 4 85.4.a.c.1.1 ✓ 1
5.4 even 2 inner 425.4.b.d.324.1 2
15.8 even 4 765.4.a.a.1.1 1
20.3 even 4 1360.4.a.a.1.1 1
85.33 odd 4 1445.4.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.c.1.1 ✓ 1 5.3 odd 4
425.4.a.a.1.1 1 5.2 odd 4
425.4.b.d.324.1 2 5.4 even 2 inner
425.4.b.d.324.2 2 1.1 even 1 trivial
765.4.a.a.1.1 1 15.8 even 4
1360.4.a.a.1.1 1 20.3 even 4
1445.4.a.f.1.1 1 85.33 odd 4