↑

Properties

Label 1445.4.a.i.1.2
Level $1445$
Weight $4$
Character 1445.1
Self dual yes
Analytic conductor $85.258$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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] = [1445,4,Mod(1,1445)] 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("1445.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1445 = 5 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1445.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-4,2,-2,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(85.2577599583\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 85)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1445.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.267949 q^{2} -0.732051 q^{3} -7.92820 q^{4} -5.00000 q^{5} +0.196152 q^{6} +14.5885 q^{7} +4.26795 q^{8} -26.4641 q^{9} +1.33975 q^{10} -3.51666 q^{11} +5.80385 q^{12} -88.4974 q^{13} -3.90897 q^{14} +3.66025 q^{15} +62.2820 q^{16} +7.09103 q^{18} -79.6077 q^{19} +39.6410 q^{20} -10.6795 q^{21} +0.942286 q^{22} -153.937 q^{23} -3.12436 q^{24} +25.0000 q^{25} +23.7128 q^{26} +39.1384 q^{27} -115.660 q^{28} +28.3257 q^{29} -0.980762 q^{30} +209.219 q^{31} -50.8320 q^{32} +2.57437 q^{33} -72.9423 q^{35} +209.813 q^{36} -359.597 q^{37} +21.3308 q^{38} +64.7846 q^{39} -21.3397 q^{40} -417.951 q^{41} +2.86156 q^{42} +243.520 q^{43} +27.8808 q^{44} +132.321 q^{45} +41.2473 q^{46} -160.144 q^{47} -45.5936 q^{48} -130.177 q^{49} -6.69873 q^{50} +701.626 q^{52} +28.8513 q^{53} -10.4871 q^{54} +17.5833 q^{55} +62.2628 q^{56} +58.2769 q^{57} -7.58984 q^{58} -832.315 q^{59} -29.0192 q^{60} +502.190 q^{61} -56.0601 q^{62} -386.070 q^{63} -484.636 q^{64} +442.487 q^{65} -0.689801 q^{66} -555.472 q^{67} +112.690 q^{69} +19.5448 q^{70} +961.114 q^{71} -112.947 q^{72} -512.823 q^{73} +96.3538 q^{74} -18.3013 q^{75} +631.146 q^{76} -51.3027 q^{77} -17.3590 q^{78} -277.906 q^{79} -311.410 q^{80} +685.879 q^{81} +111.990 q^{82} -288.228 q^{83} +84.6692 q^{84} -65.2511 q^{86} -20.7358 q^{87} -15.0089 q^{88} -1387.18 q^{89} -35.4552 q^{90} -1291.04 q^{91} +1220.44 q^{92} -153.159 q^{93} +42.9103 q^{94} +398.038 q^{95} +37.2116 q^{96} +249.174 q^{97} +34.8808 q^{98} +93.0653 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 2 q^{3} - 2 q^{4} - 10 q^{5} - 10 q^{6} - 2 q^{7} + 12 q^{8} - 46 q^{9} + 20 q^{10} + 38 q^{11} + 22 q^{12} - 80 q^{13} + 58 q^{14} - 10 q^{15} - 14 q^{16} + 80 q^{18} - 180 q^{19} + 10 q^{20}+ \cdots - 718 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.267949 −0.0947343 −0.0473672 − 0.998878i \(-0.515083\pi\)
−0.0473672 + 0.998878i \(0.515083\pi\)
\(3\) −0.732051 −0.140883 −0.0704416 − 0.997516i \(-0.522441\pi\)
−0.0704416 + 0.997516i \(0.522441\pi\)
\(4\) −7.92820 −0.991025
\(5\) −5.00000 −0.447214
\(6\) 0.196152 0.0133465
\(7\) 14.5885 0.787703 0.393851 − 0.919174i \(-0.371142\pi\)
0.393851 + 0.919174i \(0.371142\pi\)
\(8\) 4.26795 0.188618
\(9\) −26.4641 −0.980152
\(10\) 1.33975 0.0423665
\(11\) −3.51666 −0.0963921 −0.0481960 − 0.998838i \(-0.515347\pi\)
−0.0481960 + 0.998838i \(0.515347\pi\)
\(12\) 5.80385 0.139619
\(13\) −88.4974 −1.88806 −0.944030 − 0.329861i \(-0.892998\pi\)
−0.944030 + 0.329861i \(0.892998\pi\)
\(14\) −3.90897 −0.0746225
\(15\) 3.66025 0.0630049
\(16\) 62.2820 0.973157
\(17\) 0 0
\(18\) 7.09103 0.0928540
\(19\) −79.6077 −0.961224 −0.480612 − 0.876933i \(-0.659585\pi\)
−0.480612 + 0.876933i \(0.659585\pi\)
\(20\) 39.6410 0.443200
\(21\) −10.6795 −0.110974
\(22\) 0.942286 0.00913164
\(23\) −153.937 −1.39557 −0.697785 − 0.716307i \(-0.745831\pi\)
−0.697785 + 0.716307i \(0.745831\pi\)
\(24\) −3.12436 −0.0265732
\(25\) 25.0000 0.200000
\(26\) 23.7128 0.178864
\(27\) 39.1384 0.278970
\(28\) −115.660 −0.780633
\(29\) 28.3257 0.181377 0.0906887 − 0.995879i \(-0.471093\pi\)
0.0906887 + 0.995879i \(0.471093\pi\)
\(30\) −0.980762 −0.00596873
\(31\) 209.219 1.21216 0.606079 − 0.795405i \(-0.292742\pi\)
0.606079 + 0.795405i \(0.292742\pi\)
\(32\) −50.8320 −0.280810
\(33\) 2.57437 0.0135800
\(34\) 0 0
\(35\) −72.9423 −0.352271
\(36\) 209.813 0.971355
\(37\) −359.597 −1.59777 −0.798884 − 0.601485i \(-0.794576\pi\)
−0.798884 + 0.601485i \(0.794576\pi\)
\(38\) 21.3308 0.0910609
\(39\) 64.7846 0.265996
\(40\) −21.3397 −0.0843528
\(41\) −417.951 −1.59202 −0.796012 − 0.605280i \(-0.793061\pi\)
−0.796012 + 0.605280i \(0.793061\pi\)
\(42\) 2.86156 0.0105131
\(43\) 243.520 0.863640 0.431820 − 0.901960i \(-0.357872\pi\)
0.431820 + 0.901960i \(0.357872\pi\)
\(44\) 27.8808 0.0955270
\(45\) 132.321 0.438337
\(46\) 41.2473 0.132208
\(47\) −160.144 −0.497007 −0.248504 − 0.968631i \(-0.579939\pi\)
−0.248504 + 0.968631i \(0.579939\pi\)
\(48\) −45.5936 −0.137101
\(49\) −130.177 −0.379525
\(50\) −6.69873 −0.0189469
\(51\) 0 0
\(52\) 701.626 1.87111
\(53\) 28.8513 0.0747740 0.0373870 − 0.999301i \(-0.488097\pi\)
0.0373870 + 0.999301i \(0.488097\pi\)
\(54\) −10.4871 −0.0264281
\(55\) 17.5833 0.0431079
\(56\) 62.2628 0.148575
\(57\) 58.2769 0.135420
\(58\) −7.58984 −0.0171827
\(59\) −832.315 −1.83658 −0.918290 − 0.395908i \(-0.870430\pi\)
−0.918290 + 0.395908i \(0.870430\pi\)
\(60\) −29.0192 −0.0624395
\(61\) 502.190 1.05408 0.527039 − 0.849841i \(-0.323302\pi\)
0.527039 + 0.849841i \(0.323302\pi\)
\(62\) −56.0601 −0.114833
\(63\) −386.070 −0.772068
\(64\) −484.636 −0.946554
\(65\) 442.487 0.844366
\(66\) −0.689801 −0.00128650
\(67\) −555.472 −1.01286 −0.506430 − 0.862281i \(-0.669035\pi\)
−0.506430 + 0.862281i \(0.669035\pi\)
\(68\) 0 0
\(69\) 112.690 0.196612
\(70\) 19.5448 0.0333722
\(71\) 961.114 1.60652 0.803262 − 0.595625i \(-0.203096\pi\)
0.803262 + 0.595625i \(0.203096\pi\)
\(72\) −112.947 −0.184875
\(73\) −512.823 −0.822211 −0.411105 − 0.911588i \(-0.634857\pi\)
−0.411105 + 0.911588i \(0.634857\pi\)
\(74\) 96.3538 0.151364
\(75\) −18.3013 −0.0281766
\(76\) 631.146 0.952597
\(77\) −51.3027 −0.0759283
\(78\) −17.3590 −0.0251989
\(79\) −277.906 −0.395783 −0.197892 − 0.980224i \(-0.563409\pi\)
−0.197892 + 0.980224i \(0.563409\pi\)
\(80\) −311.410 −0.435209
\(81\) 685.879 0.940850
\(82\) 111.990 0.150819
\(83\) −288.228 −0.381170 −0.190585 − 0.981671i \(-0.561039\pi\)
−0.190585 + 0.981671i \(0.561039\pi\)
\(84\) 84.6692 0.109978
\(85\) 0 0
\(86\) −65.2511 −0.0818164
\(87\) −20.7358 −0.0255530
\(88\) −15.0089 −0.0181813
\(89\) −1387.18 −1.65215 −0.826073 − 0.563563i \(-0.809430\pi\)
−0.826073 + 0.563563i \(0.809430\pi\)
\(90\) −35.4552 −0.0415256
\(91\) −1291.04 −1.48723
\(92\) 1220.44 1.38305
\(93\) −153.159 −0.170773
\(94\) 42.9103 0.0470837
\(95\) 398.038 0.429872
\(96\) 37.2116 0.0395614
\(97\) 249.174 0.260823 0.130411 − 0.991460i \(-0.458370\pi\)
0.130411 + 0.991460i \(0.458370\pi\)
\(98\) 34.8808 0.0359540
\(99\) 93.0653 0.0944789
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 1445.4.a.i.1.2 2
17.16 even 2 85.4.a.d.1.2 ✓ 2
51.50 odd 2 765.4.a.i.1.1 2
68.67 odd 2 1360.4.a.m.1.1 2
85.33 odd 4 425.4.b.e.324.3 4
85.67 odd 4 425.4.b.e.324.2 4
85.84 even 2 425.4.a.e.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
85.4.a.d.1.2 ✓ 2 17.16 even 2
425.4.a.e.1.1 2 85.84 even 2
425.4.b.e.324.2 4 85.67 odd 4
425.4.b.e.324.3 4 85.33 odd 4
765.4.a.i.1.1 2 51.50 odd 2
1360.4.a.m.1.1 2 68.67 odd 2
1445.4.a.i.1.2 2 1.1 even 1 trivial