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Properties

Label 425.4.b.a.324.2
Level $425$
Weight $4$
Character 425.324
Analytic conductor $25.076$
Analytic rank $0$
Dimension $2$
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] = [425,4,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,-42] 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: \(2\)
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.a.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)
 
\(f(q)\) \(=\) \(q+3.00000i q^{2} +7.00000i q^{3} -1.00000 q^{4} -21.0000 q^{6} -22.0000i q^{7} +21.0000i q^{8} -22.0000 q^{9} -64.0000 q^{11} -7.00000i q^{12} -73.0000i q^{13} +66.0000 q^{14} -71.0000 q^{16} -17.0000i q^{17} -66.0000i q^{18} +49.0000 q^{19} +154.000 q^{21} -192.000i q^{22} -110.000i q^{23} -147.000 q^{24} +219.000 q^{26} +35.0000i q^{27} +22.0000i q^{28} -155.000 q^{29} -197.000 q^{31} -45.0000i q^{32} -448.000i q^{33} +51.0000 q^{34} +22.0000 q^{36} -372.000i q^{37} +147.000i q^{38} +511.000 q^{39} -262.000 q^{41} +462.000i q^{42} -258.000i q^{43} +64.0000 q^{44} +330.000 q^{46} -13.0000i q^{47} -497.000i q^{48} -141.000 q^{49} +119.000 q^{51} +73.0000i q^{52} +653.000i q^{53} -105.000 q^{54} +462.000 q^{56} +343.000i q^{57} -465.000i q^{58} +333.000 q^{59} -355.000 q^{61} -591.000i q^{62} +484.000i q^{63} -433.000 q^{64} +1344.00 q^{66} +814.000i q^{67} +17.0000i q^{68} +770.000 q^{69} +47.0000 q^{71} -462.000i q^{72} +437.000i q^{73} +1116.00 q^{74} -49.0000 q^{76} +1408.00i q^{77} +1533.00i q^{78} +384.000 q^{79} -839.000 q^{81} -786.000i q^{82} +736.000i q^{83} -154.000 q^{84} +774.000 q^{86} -1085.00i q^{87} -1344.00i q^{88} -511.000 q^{89} -1606.00 q^{91} +110.000i q^{92} -1379.00i q^{93} +39.0000 q^{94} +315.000 q^{96} +537.000i q^{97} -423.000i q^{98} +1408.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} - 42 q^{6} - 44 q^{9} - 128 q^{11} + 132 q^{14} - 142 q^{16} + 98 q^{19} + 308 q^{21} - 294 q^{24} + 438 q^{26} - 310 q^{29} - 394 q^{31} + 102 q^{34} + 44 q^{36} + 1022 q^{39} - 524 q^{41}+ \cdots + 2816 q^{99}+O(q^{100}) \) 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 \(0.822068\pi\)
\(3\) 7.00000i 1.34715i 0.739119 + 0.673575i \(0.235242\pi\)
−0.739119 + 0.673575i \(0.764758\pi\)
\(4\) −1.00000 −0.125000
\(5\) 0 0
\(6\) −21.0000 −1.42887
\(7\) − 22.0000i − 1.18789i −0.804506 − 0.593944i \(-0.797570\pi\)
0.804506 − 0.593944i \(-0.202430\pi\)
\(8\) 21.0000i 0.928078i
\(9\) −22.0000 −0.814815
\(10\) 0 0
\(11\) −64.0000 −1.75425 −0.877124 − 0.480264i \(-0.840541\pi\)
−0.877124 + 0.480264i \(0.840541\pi\)
\(12\) − 7.00000i − 0.168394i
\(13\) − 73.0000i − 1.55743i −0.627379 − 0.778714i \(-0.715872\pi\)
0.627379 − 0.778714i \(-0.284128\pi\)
\(14\) 66.0000 1.25995
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) − 17.0000i − 0.242536i
\(18\) − 66.0000i − 0.864242i
\(19\) 49.0000 0.591651 0.295826 − 0.955242i \(-0.404405\pi\)
0.295826 + 0.955242i \(0.404405\pi\)
\(20\) 0 0
\(21\) 154.000 1.60026
\(22\) − 192.000i − 1.86066i
\(23\) − 110.000i − 0.997243i −0.866820 − 0.498621i \(-0.833840\pi\)
0.866820 − 0.498621i \(-0.166160\pi\)
\(24\) −147.000 −1.25026
\(25\) 0 0
\(26\) 219.000 1.65190
\(27\) 35.0000i 0.249472i
\(28\) 22.0000i 0.148486i
\(29\) −155.000 −0.992510 −0.496255 − 0.868177i \(-0.665292\pi\)
−0.496255 + 0.868177i \(0.665292\pi\)
\(30\) 0 0
\(31\) −197.000 −1.14136 −0.570681 − 0.821172i \(-0.693321\pi\)
−0.570681 + 0.821172i \(0.693321\pi\)
\(32\) − 45.0000i − 0.248592i
\(33\) − 448.000i − 2.36324i
\(34\) 51.0000 0.257248
\(35\) 0 0
\(36\) 22.0000 0.101852
\(37\) − 372.000i − 1.65288i −0.563028 − 0.826438i \(-0.690364\pi\)
0.563028 − 0.826438i \(-0.309636\pi\)
\(38\) 147.000i 0.627541i
\(39\) 511.000 2.09809
\(40\) 0 0
\(41\) −262.000 −0.997988 −0.498994 − 0.866605i \(-0.666297\pi\)
−0.498994 + 0.866605i \(0.666297\pi\)
\(42\) 462.000i 1.69734i
\(43\) − 258.000i − 0.914991i −0.889212 − 0.457496i \(-0.848747\pi\)
0.889212 − 0.457496i \(-0.151253\pi\)
\(44\) 64.0000 0.219281
\(45\) 0 0
\(46\) 330.000 1.05774
\(47\) − 13.0000i − 0.0403456i −0.999797 − 0.0201728i \(-0.993578\pi\)
0.999797 − 0.0201728i \(-0.00642164\pi\)
\(48\) − 497.000i − 1.49450i
\(49\) −141.000 −0.411079
\(50\) 0 0
\(51\) 119.000 0.326732
\(52\) 73.0000i 0.194678i
\(53\) 653.000i 1.69239i 0.532877 + 0.846193i \(0.321111\pi\)
−0.532877 + 0.846193i \(0.678889\pi\)
\(54\) −105.000 −0.264605
\(55\) 0 0
\(56\) 462.000 1.10245
\(57\) 343.000i 0.797043i
\(58\) − 465.000i − 1.05272i
\(59\) 333.000 0.734795 0.367398 − 0.930064i \(-0.380249\pi\)
0.367398 + 0.930064i \(0.380249\pi\)
\(60\) 0 0
\(61\) −355.000 −0.745133 −0.372566 − 0.928006i \(-0.621522\pi\)
−0.372566 + 0.928006i \(0.621522\pi\)
\(62\) − 591.000i − 1.21060i
\(63\) 484.000i 0.967909i
\(64\) −433.000 −0.845703
\(65\) 0 0
\(66\) 1344.00 2.50659
\(67\) 814.000i 1.48427i 0.670252 + 0.742134i \(0.266186\pi\)
−0.670252 + 0.742134i \(0.733814\pi\)
\(68\) 17.0000i 0.0303170i
\(69\) 770.000 1.34344
\(70\) 0 0
\(71\) 47.0000 0.0785616 0.0392808 − 0.999228i \(-0.487493\pi\)
0.0392808 + 0.999228i \(0.487493\pi\)
\(72\) − 462.000i − 0.756211i
\(73\) 437.000i 0.700644i 0.936629 + 0.350322i \(0.113928\pi\)
−0.936629 + 0.350322i \(0.886072\pi\)
\(74\) 1116.00 1.75314
\(75\) 0 0
\(76\) −49.0000 −0.0739564
\(77\) 1408.00i 2.08385i
\(78\) 1533.00i 2.22536i
\(79\) 384.000 0.546878 0.273439 − 0.961889i \(-0.411839\pi\)
0.273439 + 0.961889i \(0.411839\pi\)
\(80\) 0 0
\(81\) −839.000 −1.15089
\(82\) − 786.000i − 1.05853i
\(83\) 736.000i 0.973331i 0.873588 + 0.486666i \(0.161787\pi\)
−0.873588 + 0.486666i \(0.838213\pi\)
\(84\) −154.000 −0.200033
\(85\) 0 0
\(86\) 774.000 0.970495
\(87\) − 1085.00i − 1.33706i
\(88\) − 1344.00i − 1.62808i
\(89\) −511.000 −0.608606 −0.304303 − 0.952575i \(-0.598423\pi\)
−0.304303 + 0.952575i \(0.598423\pi\)
\(90\) 0 0
\(91\) −1606.00 −1.85005
\(92\) 110.000i 0.124655i
\(93\) − 1379.00i − 1.53759i
\(94\) 39.0000 0.0427930
\(95\) 0 0
\(96\) 315.000 0.334891
\(97\) 537.000i 0.562104i 0.959693 + 0.281052i \(0.0906833\pi\)
−0.959693 + 0.281052i \(0.909317\pi\)
\(98\) − 423.000i − 0.436015i
\(99\) 1408.00 1.42939
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.a.324.2 2
5.2 odd 4 425.4.a.c.1.1 1
5.3 odd 4 85.4.a.a.1.1 ✓ 1
5.4 even 2 inner 425.4.b.a.324.1 2
15.8 even 4 765.4.a.b.1.1 1
20.3 even 4 1360.4.a.i.1.1 1
85.33 odd 4 1445.4.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.a.1.1 ✓ 1 5.3 odd 4
425.4.a.c.1.1 1 5.2 odd 4
425.4.b.a.324.1 2 5.4 even 2 inner
425.4.b.a.324.2 2 1.1 even 1 trivial
765.4.a.b.1.1 1 15.8 even 4
1360.4.a.i.1.1 1 20.3 even 4
1445.4.a.h.1.1 1 85.33 odd 4