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Properties

Label 425.4.b.a.324.1
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 324.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 425.324
Dual form 425.4.b.a.324.2

$q$-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 \(-0.822068\pi\)
0.847791 − 0.530330i \(-0.177932\pi\)
\(3\) − 7.00000i − 1.34715i −0.739119 − 0.673575i \(-0.764758\pi\)
0.739119 − 0.673575i \(-0.235242\pi\)
\(4\) −1.00000 −0.125000
\(5\) 0 0
\(6\) −21.0000 −1.42887
\(7\) 22.0000i 1.18789i 0.804506 + 0.593944i \(0.202430\pi\)
−0.804506 + 0.593944i \(0.797570\pi\)
\(8\) − 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 \(0.284128\pi\)
−0.627379 + 0.778714i \(0.715872\pi\)
\(14\) 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 \(0.166160\pi\)
−0.866820 + 0.498621i \(0.833840\pi\)
\(24\) −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 \(0.309636\pi\)
−0.563028 + 0.826438i \(0.690364\pi\)
\(38\) − 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 \(0.151253\pi\)
−0.889212 + 0.457496i \(0.848747\pi\)
\(44\) 64.0000 0.219281
\(45\) 0 0
\(46\) 330.000 1.05774
\(47\) 13.0000i 0.0403456i 0.999797 + 0.0201728i \(0.00642164\pi\)
−0.999797 + 0.0201728i \(0.993578\pi\)
\(48\) 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 \(-0.678889\pi\)
0.532877 − 0.846193i \(-0.321111\pi\)
\(54\) −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 \(-0.733814\pi\)
0.670252 − 0.742134i \(-0.266186\pi\)
\(68\) − 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 \(-0.886072\pi\)
0.936629 − 0.350322i \(-0.113928\pi\)
\(74\) 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 \(-0.838213\pi\)
0.873588 − 0.486666i \(-0.161787\pi\)
\(84\) −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 \(-0.909317\pi\)
0.959693 − 0.281052i \(-0.0906833\pi\)
\(98\) 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.324.1 2
5.2 odd 4 85.4.a.a.1.1 ✓ 1
5.3 odd 4 425.4.a.c.1.1 1
5.4 even 2 inner 425.4.b.a.324.2 2
15.2 even 4 765.4.a.b.1.1 1
20.7 even 4 1360.4.a.i.1.1 1
85.67 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 5.2 odd 4
425.4.a.c.1.1 1 5.3 odd 4
425.4.b.a.324.1 2 1.1 even 1 trivial
425.4.b.a.324.2 2 5.4 even 2 inner
765.4.a.b.1.1 1 15.2 even 4
1360.4.a.i.1.1 1 20.7 even 4
1445.4.a.h.1.1 1 85.67 odd 4