LMFDB - Embedded newform 546.4.a.p.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [546,4,Mod(1,546)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("546.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(546, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 546 = 2 \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	546.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,6,-9,12,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.2150428631$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.118088.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 50x - 32 $ x^3 - 50*x - 32
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-6.72626$ of defining polynomial
Character	$\chi$	$=$	546.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -17.0738 q^{5} -6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -34.1475 q^{10} -61.9788 q^{11} -12.0000 q^{12} +13.0000 q^{13} -14.0000 q^{14} +51.2213 q^{15} +16.0000 q^{16} +36.7987 q^{17} +18.0000 q^{18} +102.254 q^{19} -68.2951 q^{20} +21.0000 q^{21} -123.958 q^{22} -197.116 q^{23} -24.0000 q^{24} +166.514 q^{25} +26.0000 q^{26} -27.0000 q^{27} -28.0000 q^{28} +249.306 q^{29} +102.443 q^{30} +119.035 q^{31} +32.0000 q^{32} +185.936 q^{33} +73.5974 q^{34} +119.516 q^{35} +36.0000 q^{36} +135.869 q^{37} +204.508 q^{38} -39.0000 q^{39} -136.590 q^{40} +241.310 q^{41} +42.0000 q^{42} -132.816 q^{43} -247.915 q^{44} -153.664 q^{45} -394.233 q^{46} -22.5528 q^{47} -48.0000 q^{48} +49.0000 q^{49} +333.028 q^{50} -110.396 q^{51} +52.0000 q^{52} -342.041 q^{53} -54.0000 q^{54} +1058.21 q^{55} -56.0000 q^{56} -306.762 q^{57} +498.613 q^{58} -136.318 q^{59} +204.885 q^{60} +800.634 q^{61} +238.070 q^{62} -63.0000 q^{63} +64.0000 q^{64} -221.959 q^{65} +371.873 q^{66} +994.938 q^{67} +147.195 q^{68} +591.349 q^{69} +239.033 q^{70} +526.503 q^{71} +72.0000 q^{72} -1020.45 q^{73} +271.737 q^{74} -499.541 q^{75} +409.015 q^{76} +433.852 q^{77} -78.0000 q^{78} -90.4321 q^{79} -273.180 q^{80} +81.0000 q^{81} +482.620 q^{82} +1070.66 q^{83} +84.0000 q^{84} -628.293 q^{85} -265.632 q^{86} -747.919 q^{87} -495.830 q^{88} +205.188 q^{89} -307.328 q^{90} -91.0000 q^{91} -788.466 q^{92} -357.106 q^{93} -45.1055 q^{94} -1745.86 q^{95} -96.0000 q^{96} +1013.11 q^{97} +98.0000 q^{98} -557.809 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 7 q^{5} - 18 q^{6} - 21 q^{7} + 24 q^{8} + 27 q^{9} + 14 q^{10} - 47 q^{11} - 36 q^{12} + 39 q^{13} - 42 q^{14} - 21 q^{15} + 48 q^{16} + 119 q^{17} + 54 q^{18} + 101 q^{19}+ \cdots - 423 q^{99}+O(q^{100}) $ 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 + 7 * q^5 - 18 * q^6 - 21 * q^7 + 24 * q^8 + 27 * q^9 + 14 * q^10 - 47 * q^11 - 36 * q^12 + 39 * q^13 - 42 * q^14 - 21 * q^15 + 48 * q^16 + 119 * q^17 + 54 * q^18 + 101 * q^19 + 28 * q^20 + 63 * q^21 - 94 * q^22 - 27 * q^23 - 72 * q^24 + 266 * q^25 + 78 * q^26 - 81 * q^27 - 84 * q^28 + 345 * q^29 - 42 * q^30 + 152 * q^31 + 96 * q^32 + 141 * q^33 + 238 * q^34 - 49 * q^35 + 108 * q^36 + 227 * q^37 + 202 * q^38 - 117 * q^39 + 56 * q^40 + 452 * q^41 + 126 * q^42 - 415 * q^43 - 188 * q^44 + 63 * q^45 - 54 * q^46 + 240 * q^47 - 144 * q^48 + 147 * q^49 + 532 * q^50 - 357 * q^51 + 156 * q^52 + 874 * q^53 - 162 * q^54 + 1123 * q^55 - 168 * q^56 - 303 * q^57 + 690 * q^58 + 948 * q^59 - 84 * q^60 + 951 * q^61 + 304 * q^62 - 189 * q^63 + 192 * q^64 + 91 * q^65 + 282 * q^66 + 742 * q^67 + 476 * q^68 + 81 * q^69 - 98 * q^70 + 732 * q^71 + 216 * q^72 - 585 * q^73 + 454 * q^74 - 798 * q^75 + 404 * q^76 + 329 * q^77 - 234 * q^78 - 618 * q^79 + 112 * q^80 + 243 * q^81 + 904 * q^82 + 774 * q^83 + 252 * q^84 + 1273 * q^85 - 830 * q^86 - 1035 * q^87 - 376 * q^88 + 730 * q^89 + 126 * q^90 - 273 * q^91 - 108 * q^92 - 456 * q^93 + 480 * q^94 - 2617 * q^95 - 288 * q^96 + 2518 * q^97 + 294 * q^98 - 423 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	−17.0738	−1.52712	−0.763562	−	0.645734i	$-0.776551\pi$
$5$	−17.0738	−1.52712	−0.763562	+	0.645734i	$0.776551\pi$
$6$	−6.00000	−0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	−34.1475	−1.07984
$11$	−61.9788	−1.69885	−0.849423	−	0.527712i	$-0.823050\pi$
$11$	−61.9788	−1.69885	−0.849423	+	0.527712i	$0.823050\pi$
$12$	−12.0000	−0.288675
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	−14.0000	−0.267261
$15$	51.2213	0.881686
$16$	16.0000	0.250000
$17$	36.7987	0.525000	0.262500	−	0.964932i	$-0.415453\pi$
$17$	36.7987	0.525000	0.262500	+	0.964932i	$0.415453\pi$
$18$	18.0000	0.235702
$19$	102.254	1.23467	0.617333	−	0.786702i	$-0.288213\pi$
$19$	102.254	1.23467	0.617333	+	0.786702i	$0.288213\pi$
$20$	−68.2951	−0.763562
$21$	21.0000	0.218218
$22$	−123.958	−1.20127
$23$	−197.116	−1.78703	−0.893514	−	0.449036i	$-0.851767\pi$
$23$	−197.116	−1.78703	−0.893514	+	0.449036i	$0.851767\pi$
$24$	−24.0000	−0.204124
$25$	166.514	1.33211
$26$	26.0000	0.196116
$27$	−27.0000	−0.192450
$28$	−28.0000	−0.188982
$29$	249.306	1.59638	0.798190	−	0.602405i	$-0.205791\pi$
$29$	249.306	1.59638	0.798190	+	0.602405i	$0.205791\pi$
$30$	102.443	0.623446
$31$	119.035	0.689656	0.344828	−	0.938666i	$-0.387937\pi$
$31$	119.035	0.689656	0.344828	+	0.938666i	$0.387937\pi$
$32$	32.0000	0.176777
$33$	185.936	0.980829
$34$	73.5974	0.371231
$35$	119.516	0.577199
$36$	36.0000	0.166667
$37$	135.869	0.603694	0.301847	−	0.953356i	$-0.402397\pi$
$37$	135.869	0.603694	0.301847	+	0.953356i	$0.402397\pi$
$38$	204.508	0.873040
$39$	−39.0000	−0.160128
$40$	−136.590	−0.539920
$41$	241.310	0.919179	0.459589	−	0.888132i	$-0.347997\pi$
$41$	241.310	0.919179	0.459589	+	0.888132i	$0.347997\pi$
$42$	42.0000	0.154303
$43$	−132.816	−0.471029	−0.235515	−	0.971871i	$-0.575678\pi$
$43$	−132.816	−0.471029	−0.235515	+	0.971871i	$0.575678\pi$
$44$	−247.915	−0.849423
$45$	−153.664	−0.509042
$46$	−394.233	−1.26362
$47$	−22.5528	−0.0699927	−0.0349964	−	0.999387i	$-0.511142\pi$
$47$	−22.5528	−0.0699927	−0.0349964	+	0.999387i	$0.511142\pi$
$48$	−48.0000	−0.144338
$49$	49.0000	0.142857
$50$	333.028	0.941944
$51$	−110.396	−0.303109
$52$	52.0000	0.138675
$53$	−342.041	−0.886470	−0.443235	−	0.896405i	$-0.646169\pi$
$53$	−342.041	−0.886470	−0.443235	+	0.896405i	$0.646169\pi$
$54$	−54.0000	−0.136083
$55$	1058.21	2.59435
$56$	−56.0000	−0.133631
$57$	−306.762	−0.712834
$58$	498.613	1.12881
$59$	−136.318	−0.300798	−0.150399	−	0.988625i	$-0.548056\pi$
$59$	−136.318	−0.300798	−0.150399	+	0.988625i	$0.548056\pi$
$60$	204.885	0.440843
$61$	800.634	1.68050	0.840252	−	0.542196i	$-0.182407\pi$
$61$	800.634	1.68050	0.840252	+	0.542196i	$0.182407\pi$
$62$	238.070	0.487661
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	−221.959	−0.423548
$66$	371.873	0.693551
$67$	994.938	1.81419	0.907097	−	0.420921i	$-0.138293\pi$
$67$	994.938	1.81419	0.907097	+	0.420921i	$0.138293\pi$
$68$	147.195	0.262500
$69$	591.349	1.03174
$70$	239.033	0.408141
$71$	526.503	0.880062	0.440031	−	0.897983i	$-0.354967\pi$
$71$	526.503	0.880062	0.440031	+	0.897983i	$0.354967\pi$
$72$	72.0000	0.117851
$73$	−1020.45	−1.63609	−0.818047	−	0.575151i	$-0.804943\pi$
$73$	−1020.45	−1.63609	−0.818047	+	0.575151i	$0.804943\pi$
$74$	271.737	0.426876
$75$	−499.541	−0.769094
$76$	409.015	0.617333
$77$	433.852	0.642104
$78$	−78.0000	−0.113228
$79$	−90.4321	−0.128790	−0.0643950	−	0.997924i	$-0.520512\pi$
$79$	−90.4321	−0.128790	−0.0643950	+	0.997924i	$0.520512\pi$
$80$	−273.180	−0.381781
$81$	81.0000	0.111111
$82$	482.620	0.649957
$83$	1070.66	1.41590	0.707950	−	0.706263i	$-0.249620\pi$
$83$	1070.66	1.41590	0.707950	+	0.706263i	$0.249620\pi$
$84$	84.0000	0.109109
$85$	−628.293	−0.801740
$86$	−265.632	−0.333068
$87$	−747.919	−0.921671
$88$	−495.830	−0.600633
$89$	205.188	0.244380	0.122190	−	0.992507i	$-0.461008\pi$
$89$	205.188	0.244380	0.122190	+	0.992507i	$0.461008\pi$
$90$	−307.328	−0.359947
$91$	−91.0000	−0.104828
$92$	−788.466	−0.893514
$93$	−357.106	−0.398173
$94$	−45.1055	−0.0494923
$95$	−1745.86	−1.88549
$96$	−96.0000	−0.102062
$97$	1013.11	1.06047	0.530236	−	0.847850i	$-0.322103\pi$
$97$	1013.11	1.06047	0.530236	+	0.847850i	$0.322103\pi$
$98$	98.0000	0.101015
$99$	−557.809	−0.566282
$100$	666.055	0.666055
$101$	535.507	0.527573	0.263787	−	0.964581i	$-0.415028\pi$
$101$	535.507	0.527573	0.263787	+	0.964581i	$0.415028\pi$
$102$	−220.792	−0.214330
$103$	−315.647	−0.301957	−0.150979	−	0.988537i	$-0.548242\pi$
$103$	−315.647	−0.301957	−0.150979	+	0.988537i	$0.548242\pi$
$104$	104.000	0.0980581
$105$	−358.549	−0.333246
$106$	−684.081	−0.626829
$107$	−1370.10	−1.23787	−0.618936	−	0.785441i	$-0.712436\pi$
$107$	−1370.10	−1.23787	−0.618936	+	0.785441i	$0.712436\pi$
$108$	−108.000	−0.0962250
$109$	−1401.36	−1.23143	−0.615717	−	0.787967i	$-0.711134\pi$
$109$	−1401.36	−1.23143	−0.615717	+	0.787967i	$0.711134\pi$
$110$	2116.42	1.83448
$111$	−407.606	−0.348543
$112$	−112.000	−0.0944911
$113$	−1277.86	−1.06381	−0.531907	−	0.846803i	$-0.678525\pi$
$113$	−1277.86	−1.06381	−0.531907	+	0.846803i	$0.678525\pi$
$114$	−613.523	−0.504050
$115$	3365.52	2.72901
$116$	997.226	0.798190
$117$	117.000	0.0924500
$118$	−272.636	−0.212697
$119$	−257.591	−0.198431
$120$	409.771	0.311723
$121$	2510.37	1.88608
$122$	1601.27	1.18830
$123$	−723.931	−0.530688
$124$	476.141	0.344828
$125$	−708.797	−0.507174
$126$	−126.000	−0.0890871
$127$	−2510.76	−1.75428	−0.877140	−	0.480235i	$-0.840551\pi$
$127$	−2510.76	−1.75428	−0.877140	+	0.480235i	$0.840551\pi$
$128$	128.000	0.0883883
$129$	398.448	0.271949
$130$	−443.918	−0.299494
$131$	1259.62	0.840102	0.420051	−	0.907500i	$-0.362012\pi$
$131$	1259.62	0.840102	0.420051	+	0.907500i	$0.362012\pi$
$132$	743.746	0.490415
$133$	−715.777	−0.466660
$134$	1989.88	1.28283
$135$	460.992	0.293895
$136$	294.390	0.185616
$137$	2030.16	1.26605	0.633023	−	0.774133i	$-0.281814\pi$
$137$	2030.16	1.26605	0.633023	+	0.774133i	$0.281814\pi$
$138$	1182.70	0.729551
$139$	−280.659	−0.171260	−0.0856302	−	0.996327i	$-0.527290\pi$
$139$	−280.659	−0.171260	−0.0856302	+	0.996327i	$0.527290\pi$
$140$	478.066	0.288599
$141$	67.6583	0.0404103
$142$	1053.01	0.622298
$143$	−805.724	−0.471175
$144$	144.000	0.0833333
$145$	−4256.60	−2.43787
$146$	−2040.90	−1.15689
$147$	−147.000	−0.0824786
$148$	543.475	0.301847
$149$	−1058.80	−0.582149	−0.291074	−	0.956700i	$-0.594013\pi$
$149$	−1058.80	−0.582149	−0.291074	+	0.956700i	$0.594013\pi$
$150$	−999.083	−0.543832
$151$	−484.835	−0.261294	−0.130647	−	0.991429i	$-0.541705\pi$
$151$	−484.835	−0.261294	−0.130647	+	0.991429i	$0.541705\pi$
$152$	818.031	0.436520
$153$	331.188	0.175000
$154$	867.703	0.454036
$155$	−2032.38	−1.05319
$156$	−156.000	−0.0800641
$157$	1954.94	0.993766	0.496883	−	0.867818i	$-0.334478\pi$
$157$	1954.94	0.993766	0.496883	+	0.867818i	$0.334478\pi$
$158$	−180.864	−0.0910683
$159$	1026.12	0.511804
$160$	−546.361	−0.269960
$161$	1379.82	0.675433
$162$	162.000	0.0785674
$163$	1653.01	0.794315	0.397158	−	0.917750i	$-0.369997\pi$
$163$	1653.01	0.794315	0.397158	+	0.917750i	$0.369997\pi$
$164$	965.241	0.459589
$165$	−3174.64	−1.49785
$166$	2141.31	1.00119
$167$	2942.41	1.36341	0.681707	−	0.731625i	$-0.261238\pi$
$167$	2942.41	1.36341	0.681707	+	0.731625i	$0.261238\pi$
$168$	168.000	0.0771517
$169$	169.000	0.0769231
$170$	−1256.59	−0.566916
$171$	920.285	0.411555
$172$	−531.264	−0.235515
$173$	−576.533	−0.253370	−0.126685	−	0.991943i	$-0.540434\pi$
$173$	−576.533	−0.253370	−0.126685	+	0.991943i	$0.540434\pi$
$174$	−1495.84	−0.651720
$175$	−1165.60	−0.503490
$176$	−991.661	−0.424712
$177$	408.954	0.173666
$178$	410.375	0.172803
$179$	−1198.16	−0.500307	−0.250154	−	0.968206i	$-0.580481\pi$
$179$	−1198.16	−0.500307	−0.250154	+	0.968206i	$0.580481\pi$
$180$	−614.656	−0.254521
$181$	−1433.92	−0.588852	−0.294426	−	0.955674i	$-0.595128\pi$
$181$	−1433.92	−0.588852	−0.294426	+	0.955674i	$0.595128\pi$
$182$	−182.000	−0.0741249
$183$	−2401.90	−0.970239
$184$	−1576.93	−0.631810
$185$	−2319.79	−0.921916
$186$	−714.211	−0.281551
$187$	−2280.74	−0.891894
$188$	−90.2110	−0.0349964
$189$	189.000	0.0727393
$190$	−3491.72	−1.33324
$191$	1693.94	0.641724	0.320862	−	0.947126i	$-0.396027\pi$
$191$	1693.94	0.641724	0.320862	+	0.947126i	$0.396027\pi$
$192$	−192.000	−0.0721688
$193$	3169.79	1.18221	0.591106	−	0.806594i	$-0.298692\pi$
$193$	3169.79	1.18221	0.591106	+	0.806594i	$0.298692\pi$
$194$	2026.22	0.749867
$195$	665.877	0.244536
$196$	196.000	0.0714286
$197$	3579.40	1.29453	0.647264	−	0.762266i	$-0.275913\pi$
$197$	3579.40	1.29453	0.647264	+	0.762266i	$0.275913\pi$
$198$	−1115.62	−0.400422
$199$	−1268.84	−0.451989	−0.225995	−	0.974129i	$-0.572563\pi$
$199$	−1268.84	−0.451989	−0.225995	+	0.974129i	$0.572563\pi$
$200$	1332.11	0.470972
$201$	−2984.81	−1.04743
$202$	1071.01	0.373051
$203$	−1745.15	−0.603375
$204$	−441.585	−0.151554
$205$	−4120.08	−1.40370
$206$	−631.294	−0.213516
$207$	−1774.05	−0.595676
$208$	208.000	0.0693375
$209$	−6337.57	−2.09751
$210$	−717.099	−0.235640
$211$	−1463.14	−0.477379	−0.238690	−	0.971096i	$-0.576718\pi$
$211$	−1463.14	−0.477379	−0.238690	+	0.971096i	$0.576718\pi$
$212$	−1368.16	−0.443235
$213$	−1579.51	−0.508104
$214$	−2740.20	−0.875308
$215$	2267.67	0.719321
$216$	−216.000	−0.0680414
$217$	−833.246	−0.260666
$218$	−2802.73	−0.870756
$219$	3061.36	0.944599
$220$	4232.85	1.29718
$221$	478.383	0.145609
$222$	−815.212	−0.246457
$223$	2073.00	0.622504	0.311252	−	0.950327i	$-0.399252\pi$
$223$	2073.00	0.622504	0.311252	+	0.950327i	$0.399252\pi$
$224$	−224.000	−0.0668153
$225$	1498.62	0.444037
$226$	−2555.72	−0.752231
$227$	−1475.91	−0.431540	−0.215770	−	0.976444i	$-0.569226\pi$
$227$	−1475.91	−0.431540	−0.215770	+	0.976444i	$0.569226\pi$
$228$	−1227.05	−0.356417
$229$	3075.67	0.887536	0.443768	−	0.896142i	$-0.353641\pi$
$229$	3075.67	0.887536	0.443768	+	0.896142i	$0.353641\pi$
$230$	6731.05	1.92970
$231$	−1301.55	−0.370719
$232$	1994.45	0.564406
$233$	−3653.73	−1.02731	−0.513656	−	0.857996i	$-0.671709\pi$
$233$	−3653.73	−1.02731	−0.513656	+	0.857996i	$0.671709\pi$
$234$	234.000	0.0653720
$235$	385.061	0.106888
$236$	−545.272	−0.150399
$237$	271.296	0.0743569
$238$	−515.182	−0.140312
$239$	−2664.71	−0.721194	−0.360597	−	0.932722i	$-0.617427\pi$
$239$	−2664.71	−0.721194	−0.360597	+	0.932722i	$0.617427\pi$
$240$	819.541	0.220421
$241$	−4458.00	−1.19156	−0.595778	−	0.803149i	$-0.703156\pi$
$241$	−4458.00	−1.19156	−0.595778	+	0.803149i	$0.703156\pi$
$242$	5020.74	1.33366
$243$	−243.000	−0.0641500
$244$	3202.54	0.840252
$245$	−836.615	−0.218161
$246$	−1447.86	−0.375253
$247$	1329.30	0.342435
$248$	952.281	0.243830
$249$	−3211.97	−0.817470
$250$	−1417.59	−0.358626
$251$	6877.21	1.72943	0.864713	−	0.502267i	$-0.167500\pi$
$251$	6877.21	1.72943	0.864713	+	0.502267i	$0.167500\pi$
$252$	−252.000	−0.0629941
$253$	12217.0	3.03589
$254$	−5021.51	−1.24046
$255$	1884.88	0.462885
$256$	256.000	0.0625000
$257$	−3617.75	−0.878090	−0.439045	−	0.898465i	$-0.644683\pi$
$257$	−3617.75	−0.878090	−0.439045	+	0.898465i	$0.644683\pi$
$258$	796.897	0.192297
$259$	−951.081	−0.228175
$260$	−887.836	−0.211774
$261$	2243.76	0.532127
$262$	2519.24	0.594042
$263$	−2511.67	−0.588884	−0.294442	−	0.955669i	$-0.595134\pi$
$263$	−2511.67	−0.588884	−0.294442	+	0.955669i	$0.595134\pi$
$264$	1487.49	0.346776
$265$	5839.93	1.35375
$266$	−1431.55	−0.329978
$267$	−615.563	−0.141093
$268$	3979.75	0.907097
$269$	−8125.12	−1.84163	−0.920813	−	0.390004i	$-0.872474\pi$
$269$	−8125.12	−1.84163	−0.920813	+	0.390004i	$0.872474\pi$
$270$	921.984	0.207815
$271$	583.537	0.130802	0.0654010	−	0.997859i	$-0.479167\pi$
$271$	583.537	0.130802	0.0654010	+	0.997859i	$0.479167\pi$
$272$	588.779	0.131250
$273$	273.000	0.0605228
$274$	4060.32	0.895230
$275$	−10320.3	−2.26305
$276$	2365.40	0.515870
$277$	6336.54	1.37446	0.687231	−	0.726439i	$-0.258826\pi$
$277$	6336.54	1.37446	0.687231	+	0.726439i	$0.258826\pi$
$278$	−561.318	−0.121099
$279$	1071.32	0.229885
$280$	956.131	0.204071
$281$	3421.04	0.726271	0.363135	−	0.931736i	$-0.381706\pi$
$281$	3421.04	0.726271	0.363135	+	0.931736i	$0.381706\pi$
$282$	135.317	0.0285744
$283$	−639.384	−0.134302	−0.0671510	−	0.997743i	$-0.521391\pi$
$283$	−639.384	−0.134302	−0.0671510	+	0.997743i	$0.521391\pi$
$284$	2106.01	0.440031
$285$	5237.58	1.08859
$286$	−1611.45	−0.333171
$287$	−1689.17	−0.347417
$288$	288.000	0.0589256
$289$	−3558.85	−0.724375
$290$	−8513.20	−1.72384
$291$	−3039.33	−0.612264
$292$	−4081.81	−0.818047
$293$	1124.30	0.224172	0.112086	−	0.993699i	$-0.464247\pi$
$293$	1124.30	0.224172	0.112086	+	0.993699i	$0.464247\pi$
$294$	−294.000	−0.0583212
$295$	2327.46	0.459357
$296$	1086.95	0.213438
$297$	1673.43	0.326943
$298$	−2117.60	−0.411641
$299$	−2562.51	−0.495632
$300$	−1998.17	−0.384547
$301$	929.713	0.178032
$302$	−969.671	−0.184762
$303$	−1606.52	−0.304595
$304$	1636.06	0.308666
$305$	−13669.9	−2.56634
$306$	662.377	0.123744
$307$	2099.44	0.390297	0.195149	−	0.980774i	$-0.437481\pi$
$307$	2099.44	0.390297	0.195149	+	0.980774i	$0.437481\pi$
$308$	1735.41	0.321052
$309$	946.941	0.174335
$310$	−4064.76	−0.744719
$311$	−6543.26	−1.19304	−0.596518	−	0.802600i	$-0.703449\pi$
$311$	−6543.26	−1.19304	−0.596518	+	0.802600i	$0.703449\pi$
$312$	−312.000	−0.0566139
$313$	4668.63	0.843088	0.421544	−	0.906808i	$-0.361488\pi$
$313$	4668.63	0.843088	0.421544	+	0.906808i	$0.361488\pi$
$314$	3909.88	0.702699
$315$	1075.65	0.192400
$316$	−361.728	−0.0643950
$317$	−2594.43	−0.459677	−0.229838	−	0.973229i	$-0.573820\pi$
$317$	−2594.43	−0.459677	−0.229838	+	0.973229i	$0.573820\pi$
$318$	2052.24	0.361900
$319$	−15451.7	−2.71201
$320$	−1092.72	−0.190891
$321$	4110.29	0.714686
$322$	2759.63	0.477603
$323$	3762.81	0.648199
$324$	324.000	0.0555556
$325$	2164.68	0.369461
$326$	3306.01	0.561666
$327$	4204.09	0.710969
$328$	1930.48	0.324979
$329$	157.869	0.0264548
$330$	−6349.27	−1.05914
$331$	10963.3	1.82054	0.910269	−	0.414016i	$-0.135874\pi$
$331$	10963.3	1.82054	0.910269	+	0.414016i	$0.135874\pi$
$332$	4282.62	0.707950
$333$	1222.82	0.201231
$334$	5884.81	0.964079
$335$	−16987.3	−2.77050
$336$	336.000	0.0545545
$337$	12159.6	1.96550	0.982750	−	0.184941i	$-0.0592093\pi$
$337$	12159.6	1.96550	0.982750	+	0.184941i	$0.0592093\pi$
$338$	338.000	0.0543928
$339$	3833.58	0.614194
$340$	−2513.17	−0.400870
$341$	−7377.66	−1.17162
$342$	1840.57	0.291013
$343$	−343.000	−0.0539949
$344$	−1062.53	−0.166534
$345$	−10096.6	−1.57560
$346$	−1153.07	−0.179159
$347$	10622.5	1.64336	0.821680	−	0.569949i	$-0.193037\pi$
$347$	10622.5	1.64336	0.821680	+	0.569949i	$0.193037\pi$
$348$	−2991.68	−0.460835
$349$	11033.0	1.69221	0.846106	−	0.533015i	$-0.178941\pi$
$349$	11033.0	1.69221	0.846106	+	0.533015i	$0.178941\pi$
$350$	−2331.19	−0.356021
$351$	−351.000	−0.0533761
$352$	−1983.32	−0.300316
$353$	2980.26	0.449358	0.224679	−	0.974433i	$-0.427867\pi$
$353$	2980.26	0.449358	0.224679	+	0.974433i	$0.427867\pi$
$354$	817.909	0.122800
$355$	−8989.39	−1.34396
$356$	820.751	0.122190
$357$	772.773	0.114564
$358$	−2396.33	−0.353771
$359$	−1227.97	−0.180529	−0.0902644	−	0.995918i	$-0.528771\pi$
$359$	−1227.97	−0.180529	−0.0902644	+	0.995918i	$0.528771\pi$
$360$	−1229.31	−0.179973
$361$	3596.85	0.524399
$362$	−2867.83	−0.416381
$363$	−7531.12	−1.08893
$364$	−364.000	−0.0524142
$365$	17423.0	2.49852
$366$	−4803.81	−0.686063
$367$	−7686.76	−1.09331	−0.546656	−	0.837357i	$-0.684100\pi$
$367$	−7686.76	−1.09331	−0.546656	+	0.837357i	$0.684100\pi$
$368$	−3153.86	−0.446757
$369$	2171.79	0.306393
$370$	−4639.58	−0.651893
$371$	2394.28	0.335054
$372$	−1428.42	−0.199087
$373$	5627.67	0.781205	0.390603	−	0.920559i	$-0.372267\pi$
$373$	5627.67	0.781205	0.390603	+	0.920559i	$0.372267\pi$
$374$	−4561.48	−0.630665
$375$	2126.39	0.292817
$376$	−180.422	−0.0247462
$377$	3240.98	0.442756
$378$	378.000	0.0514344
$379$	−8251.93	−1.11840	−0.559199	−	0.829033i	$-0.688891\pi$
$379$	−8251.93	−1.11840	−0.559199	+	0.829033i	$0.688891\pi$
$380$	−6983.44	−0.942744
$381$	7532.27	1.01283
$382$	3387.88	0.453767
$383$	−14076.1	−1.87795	−0.938973	−	0.343990i	$-0.888221\pi$
$383$	−14076.1	−1.87795	−0.938973	+	0.343990i	$0.888221\pi$
$384$	−384.000	−0.0510310
$385$	−7407.48	−0.980572
$386$	6339.59	0.835950
$387$	−1195.35	−0.157010
$388$	4052.44	0.530236
$389$	9155.59	1.19333	0.596667	−	0.802489i	$-0.296491\pi$
$389$	9155.59	1.19333	0.596667	+	0.802489i	$0.296491\pi$
$390$	1331.75	0.172913
$391$	−7253.63	−0.938189
$392$	392.000	0.0505076
$393$	−3778.86	−0.485033
$394$	7158.80	0.915369
$395$	1544.02	0.196678
$396$	−2231.24	−0.283141
$397$	3276.88	0.414262	0.207131	−	0.978313i	$-0.433587\pi$
$397$	3276.88	0.414262	0.207131	+	0.978313i	$0.433587\pi$
$398$	−2537.69	−0.319605
$399$	2147.33	0.269426
$400$	2664.22	0.333028
$401$	2440.20	0.303884	0.151942	−	0.988389i	$-0.451447\pi$
$401$	2440.20	0.303884	0.151942	+	0.988389i	$0.451447\pi$
$402$	−5969.63	−0.740642
$403$	1547.46	0.191276
$404$	2142.03	0.263787
$405$	−1382.98	−0.169681
$406$	−3490.29	−0.426651
$407$	−8420.98	−1.02558
$408$	−883.169	−0.107165
$409$	12614.9	1.52510	0.762549	−	0.646930i	$-0.223947\pi$
$409$	12614.9	1.52510	0.762549	+	0.646930i	$0.223947\pi$
$410$	−8240.15	−0.992566
$411$	−6090.48	−0.730952
$412$	−1262.59	−0.150979
$413$	954.227	0.113691
$414$	−3548.10	−0.421206
$415$	−18280.1	−2.16226
$416$	416.000	0.0490290
$417$	841.978	0.0988773
$418$	−12675.1	−1.48316
$419$	5686.49	0.663015	0.331507	−	0.943453i	$-0.392443\pi$
$419$	5686.49	0.663015	0.331507	+	0.943453i	$0.392443\pi$
$420$	−1434.20	−0.166623
$421$	15505.0	1.79493	0.897466	−	0.441084i	$-0.145406\pi$
$421$	15505.0	1.79493	0.897466	+	0.441084i	$0.145406\pi$
$422$	−2926.29	−0.337558
$423$	−202.975	−0.0233309
$424$	−2736.33	−0.313414
$425$	6127.49	0.699358
$426$	−3159.02	−0.359284
$427$	−5604.44	−0.635171
$428$	−5480.39	−0.618936
$429$	2417.17	0.272033
$430$	4535.35	0.508637
$431$	8759.64	0.978973	0.489486	−	0.872011i	$-0.337184\pi$
$431$	8759.64	0.978973	0.489486	+	0.872011i	$0.337184\pi$
$432$	−432.000	−0.0481125
$433$	5734.40	0.636438	0.318219	−	0.948017i	$-0.396915\pi$
$433$	5734.40	0.636438	0.318219	+	0.948017i	$0.396915\pi$
$434$	−1666.49	−0.184318
$435$	12769.8	1.40751
$436$	−5605.46	−0.615717
$437$	−20155.9	−2.20638
$438$	6122.71	0.667933
$439$	18116.3	1.96958	0.984790	−	0.173746i	$-0.0555873\pi$
$439$	18116.3	1.96958	0.984790	+	0.173746i	$0.0555873\pi$
$440$	8465.70	0.917241
$441$	441.000	0.0476190
$442$	956.767	0.102961
$443$	−5949.20	−0.638048	−0.319024	−	0.947747i	$-0.603355\pi$
$443$	−5949.20	−0.638048	−0.319024	+	0.947747i	$0.603355\pi$
$444$	−1630.42	−0.174271
$445$	−3503.33	−0.373199
$446$	4146.00	0.440177
$447$	3176.40	0.336104
$448$	−448.000	−0.0472456
$449$	14495.8	1.52361	0.761804	−	0.647807i	$-0.224314\pi$
$449$	14495.8	1.52361	0.761804	+	0.647807i	$0.224314\pi$
$450$	2997.25	0.313981
$451$	−14956.1	−1.56154
$452$	−5111.44	−0.531907
$453$	1454.51	0.150858
$454$	−2951.82	−0.305145
$455$	1553.71	0.160086
$456$	−2454.09	−0.252025
$457$	−9883.54	−1.01167	−0.505834	−	0.862631i	$-0.668815\pi$
$457$	−9883.54	−1.01167	−0.505834	+	0.862631i	$0.668815\pi$
$458$	6151.33	0.627583
$459$	−993.565	−0.101036
$460$	13462.1	1.36451
$461$	2106.11	0.212779	0.106389	−	0.994325i	$-0.466071\pi$
$461$	2106.11	0.212779	0.106389	+	0.994325i	$0.466071\pi$
$462$	−2603.11	−0.262138
$463$	−14047.3	−1.41001	−0.705006	−	0.709202i	$-0.749056\pi$
$463$	−14047.3	−1.41001	−0.705006	+	0.709202i	$0.749056\pi$
$464$	3988.90	0.399095
$465$	6097.14	0.608060
$466$	−7307.46	−0.726419
$467$	10584.3	1.04878	0.524390	−	0.851478i	$-0.324293\pi$
$467$	10584.3	1.04878	0.524390	+	0.851478i	$0.324293\pi$
$468$	468.000	0.0462250
$469$	−6964.57	−0.685701
$470$	770.121	0.0755810
$471$	−5864.82	−0.573751
$472$	−1090.54	−0.106348
$473$	8231.78	0.800207
$474$	542.593	0.0525783
$475$	17026.7	1.64471
$476$	−1030.36	−0.0992157
$477$	−3078.37	−0.295490
$478$	−5329.41	−0.509961
$479$	3703.05	0.353229	0.176614	−	0.984280i	$-0.443485\pi$
$479$	3703.05	0.353229	0.176614	+	0.984280i	$0.443485\pi$
$480$	1639.08	0.155862
$481$	1766.29	0.167435
$482$	−8916.00	−0.842558
$483$	−4139.45	−0.389961
$484$	10041.5	0.943040
$485$	−17297.6	−1.61947
$486$	−486.000	−0.0453609
$487$	7026.98	0.653845	0.326923	−	0.945051i	$-0.393988\pi$
$487$	7026.98	0.653845	0.326923	+	0.945051i	$0.393988\pi$
$488$	6405.08	0.594148
$489$	−4959.02	−0.458598
$490$	−1673.23	−0.154263
$491$	14083.2	1.29444	0.647218	−	0.762305i	$-0.275932\pi$
$491$	14083.2	1.29444	0.647218	+	0.762305i	$0.275932\pi$
$492$	−2895.72	−0.265344
$493$	9174.16	0.838100
$494$	2658.60	0.242138
$495$	9523.91	0.864784
$496$	1904.56	0.172414
$497$	−3685.52	−0.332632
$498$	−6423.93	−0.578039
$499$	−1412.86	−0.126751	−0.0633753	−	0.997990i	$-0.520187\pi$
$499$	−1412.86	−0.126751	−0.0633753	+	0.997990i	$0.520187\pi$
$500$	−2835.19	−0.253587
$501$	−8827.22	−0.787168
$502$	13754.4	1.22289
$503$	−3233.54	−0.286633	−0.143317	−	0.989677i	$-0.545777\pi$
$503$	−3233.54	−0.286633	−0.143317	+	0.989677i	$0.545777\pi$
$504$	−504.000	−0.0445435
$505$	−9143.12	−0.805670
$506$	24434.1	2.14669
$507$	−507.000	−0.0444116
$508$	−10043.0	−0.877140
$509$	18731.4	1.63114	0.815572	−	0.578655i	$-0.196422\pi$
$509$	18731.4	1.63114	0.815572	+	0.578655i	$0.196422\pi$
$510$	3769.76	0.327309
$511$	7143.16	0.618385
$512$	512.000	0.0441942
$513$	−2760.85	−0.237611
$514$	−7235.50	−0.620903
$515$	5389.28	0.461127
$516$	1593.79	0.135975
$517$	1397.79	0.118907
$518$	−1902.16	−0.161344
$519$	1729.60	0.146283
$520$	−1775.67	−0.149747
$521$	−4629.78	−0.389317	−0.194659	−	0.980871i	$-0.562360\pi$
$521$	−4629.78	−0.389317	−0.194659	+	0.980871i	$0.562360\pi$
$522$	4487.52	0.376271
$523$	18556.6	1.55148	0.775738	−	0.631055i	$-0.217378\pi$
$523$	18556.6	1.55148	0.775738	+	0.631055i	$0.217378\pi$
$524$	5038.48	0.420051
$525$	3496.79	0.290690
$526$	−5023.35	−0.416404
$527$	4380.34	0.362070
$528$	2974.98	0.245207
$529$	26687.9	2.19347
$530$	11679.9	0.957246
$531$	−1226.86	−0.100266
$532$	−2863.11	−0.233330
$533$	3137.03	0.254934
$534$	−1231.13	−0.0997679
$535$	23392.7	1.89039
$536$	7959.51	0.641415
$537$	3594.49	0.288853
$538$	−16250.2	−1.30223
$539$	−3036.96	−0.242692
$540$	1843.97	0.146948
$541$	899.838	0.0715103	0.0357552	−	0.999361i	$-0.488616\pi$
$541$	899.838	0.0715103	0.0357552	+	0.999361i	$0.488616\pi$
$542$	1167.07	0.0924910
$543$	4301.75	0.339974
$544$	1177.56	0.0928078
$545$	23926.6	1.88055
$546$	546.000	0.0427960
$547$	9046.01	0.707092	0.353546	−	0.935417i	$-0.384976\pi$
$547$	9046.01	0.707092	0.353546	+	0.935417i	$0.384976\pi$
$548$	8120.65	0.633023
$549$	7205.71	0.560168
$550$	−20640.6	−1.60022
$551$	25492.5	1.97100
$552$	4730.80	0.364775
$553$	633.025	0.0486780
$554$	12673.1	0.971892
$555$	6959.37	0.532268
$556$	−1122.64	−0.0856302
$557$	−3109.07	−0.236509	−0.118254	−	0.992983i	$-0.537730\pi$
$557$	−3109.07	−0.236509	−0.118254	+	0.992983i	$0.537730\pi$
$558$	2142.63	0.162554
$559$	−1726.61	−0.130640
$560$	1912.26	0.144300
$561$	6842.22	0.514935
$562$	6842.08	0.513551
$563$	−13282.2	−0.994273	−0.497137	−	0.867672i	$-0.665615\pi$
$563$	−13282.2	−0.994273	−0.497137	+	0.867672i	$0.665615\pi$
$564$	270.633	0.0202052
$565$	21817.9	1.62458
$566$	−1278.77	−0.0949658
$567$	−567.000	−0.0419961
$568$	4212.02	0.311149
$569$	11184.4	0.824029	0.412015	−	0.911177i	$-0.364825\pi$
$569$	11184.4	0.824029	0.412015	+	0.911177i	$0.364825\pi$
$570$	10475.2	0.769747
$571$	−14237.9	−1.04350	−0.521749	−	0.853099i	$-0.674720\pi$
$571$	−14237.9	−1.04350	−0.521749	+	0.853099i	$0.674720\pi$
$572$	−3222.90	−0.235588
$573$	−5081.82	−0.370499
$574$	−3378.34	−0.245661
$575$	−32822.6	−2.38052
$576$	576.000	0.0416667
$577$	4281.58	0.308916	0.154458	−	0.987999i	$-0.450637\pi$
$577$	4281.58	0.308916	0.154458	+	0.987999i	$0.450637\pi$
$578$	−7117.71	−0.512211
$579$	−9509.38	−0.682550
$580$	−17026.4	−1.21894
$581$	−7494.59	−0.535160
$582$	−6078.66	−0.432936
$583$	21199.3	1.50598
$584$	−8163.62	−0.578447
$585$	−1997.63	−0.141183
$586$	2248.60	0.158513
$587$	21316.5	1.49886	0.749428	−	0.662086i	$-0.230329\pi$
$587$	21316.5	1.49886	0.749428	+	0.662086i	$0.230329\pi$
$588$	−588.000	−0.0412393
$589$	12171.8	0.851495
$590$	4654.93	0.324814
$591$	−10738.2	−0.747396
$592$	2173.90	0.150923
$593$	−18966.5	−1.31342	−0.656711	−	0.754142i	$-0.728053\pi$
$593$	−18966.5	−1.31342	−0.656711	+	0.754142i	$0.728053\pi$
$594$	3346.86	0.231184
$595$	4398.05	0.303029
$596$	−4235.20	−0.291074
$597$	3806.53	0.260956
$598$	−5125.03	−0.350465
$599$	1209.51	0.0825031	0.0412516	−	0.999149i	$-0.486865\pi$
$599$	1209.51	0.0825031	0.0412516	+	0.999149i	$0.486865\pi$
$600$	−3996.33	−0.271916
$601$	4321.42	0.293302	0.146651	−	0.989188i	$-0.453151\pi$
$601$	4321.42	0.293302	0.146651	+	0.989188i	$0.453151\pi$
$602$	1859.43	0.125888
$603$	8954.44	0.604732
$604$	−1939.34	−0.130647
$605$	−42861.5	−2.88028
$606$	−3213.04	−0.215381
$607$	−16960.5	−1.13411	−0.567056	−	0.823679i	$-0.691918\pi$
$607$	−16960.5	−1.13411	−0.567056	+	0.823679i	$0.691918\pi$
$608$	3272.12	0.218260
$609$	5235.44	0.348359
$610$	−27339.7	−1.81468
$611$	−293.186	−0.0194125
$612$	1324.75	0.0875000
$613$	−6287.42	−0.414268	−0.207134	−	0.978313i	$-0.566414\pi$
$613$	−6287.42	−0.414268	−0.207134	+	0.978313i	$0.566414\pi$
$614$	4198.88	0.275982
$615$	12360.2	0.810427
$616$	3470.81	0.227018
$617$	−17692.3	−1.15440	−0.577199	−	0.816604i	$-0.695854\pi$
$617$	−17692.3	−1.15440	−0.577199	+	0.816604i	$0.695854\pi$
$618$	1893.88	0.123274
$619$	−18983.7	−1.23267	−0.616333	−	0.787486i	$-0.711382\pi$
$619$	−18983.7	−1.23267	−0.616333	+	0.787486i	$0.711382\pi$
$620$	−8129.52	−0.526596
$621$	5322.15	0.343914
$622$	−13086.5	−0.843603
$623$	−1436.31	−0.0923671
$624$	−624.000	−0.0400320
$625$	−8712.38	−0.557592
$626$	9337.25	0.596153
$627$	19012.7	1.21100
$628$	7819.76	0.496883
$629$	4999.79	0.316939
$630$	2151.30	0.136047
$631$	−10042.7	−0.633586	−0.316793	−	0.948495i	$-0.602606\pi$
$631$	−10042.7	−0.633586	−0.316793	+	0.948495i	$0.602606\pi$
$632$	−723.457	−0.0455341
$633$	4389.43	0.275615
$634$	−5188.86	−0.325041
$635$	42868.1	2.67900
$636$	4104.49	0.255902
$637$	637.000	0.0396214
$638$	−30903.4	−1.91768
$639$	4738.52	0.293354
$640$	−2185.44	−0.134980
$641$	−7468.57	−0.460204	−0.230102	−	0.973167i	$-0.573906\pi$
$641$	−7468.57	−0.460204	−0.230102	+	0.973167i	$0.573906\pi$
$642$	8220.59	0.505359
$643$	−14175.1	−0.869382	−0.434691	−	0.900580i	$-0.643142\pi$
$643$	−14175.1	−0.869382	−0.434691	+	0.900580i	$0.643142\pi$
$644$	5519.26	0.337716
$645$	−6803.02	−0.415300
$646$	7525.62	0.458346
$647$	−13871.4	−0.842875	−0.421438	−	0.906857i	$-0.638474\pi$
$647$	−13871.4	−0.842875	−0.421438	+	0.906857i	$0.638474\pi$
$648$	648.000	0.0392837
$649$	8448.83	0.511010
$650$	4329.36	0.261248
$651$	2499.74	0.150495
$652$	6612.02	0.397158
$653$	13542.5	0.811574	0.405787	−	0.913968i	$-0.366997\pi$
$653$	13542.5	0.811574	0.405787	+	0.913968i	$0.366997\pi$
$654$	8408.19	0.502731
$655$	−21506.4	−1.28294
$656$	3860.96	0.229795
$657$	−9184.07	−0.545365
$658$	315.739	0.0187063
$659$	11753.3	0.694756	0.347378	−	0.937725i	$-0.387072\pi$
$659$	11753.3	0.694756	0.347378	+	0.937725i	$0.387072\pi$
$660$	−12698.5	−0.748925
$661$	−26836.1	−1.57913	−0.789565	−	0.613667i	$-0.789694\pi$
$661$	−26836.1	−1.57913	−0.789565	+	0.613667i	$0.789694\pi$
$662$	21926.6	1.28732
$663$	−1435.15	−0.0840673
$664$	8565.24	0.500596
$665$	12221.0	0.712648
$666$	2445.64	0.142292
$667$	−49142.4	−2.85278
$668$	11769.6	0.681707
$669$	−6219.01	−0.359403
$670$	−33974.7	−1.95904
$671$	−49622.4	−2.85492
$672$	672.000	0.0385758
$673$	19156.4	1.09722	0.548608	−	0.836079i	$-0.315158\pi$
$673$	19156.4	1.09722	0.548608	+	0.836079i	$0.315158\pi$
$674$	24319.1	1.38982
$675$	−4495.87	−0.256365
$676$	676.000	0.0384615
$677$	14046.7	0.797427	0.398714	−	0.917075i	$-0.369457\pi$
$677$	14046.7	0.797427	0.398714	+	0.917075i	$0.369457\pi$
$678$	7667.17	0.434301
$679$	−7091.77	−0.400821
$680$	−5026.34	−0.283458
$681$	4427.73	0.249150
$682$	−14755.3	−0.828461
$683$	−34041.0	−1.90709	−0.953544	−	0.301254i	$-0.902595\pi$
$683$	−34041.0	−1.90709	−0.953544	+	0.301254i	$0.902595\pi$
$684$	3681.14	0.205778
$685$	−34662.5	−1.93341
$686$	−686.000	−0.0381802
$687$	−9227.00	−0.512419
$688$	−2125.06	−0.117757
$689$	−4446.53	−0.245863
$690$	−20193.1	−1.11412
$691$	−18838.8	−1.03714	−0.518568	−	0.855037i	$-0.673535\pi$
$691$	−18838.8	−1.03714	−0.518568	+	0.855037i	$0.673535\pi$
$692$	−2306.13	−0.126685
$693$	3904.66	0.214035
$694$	21245.0	1.16203
$695$	4791.91	0.261536
$696$	−5983.35	−0.325860
$697$	8879.91	0.482569
$698$	22066.0	1.19657
$699$	10961.2	0.593119
$700$	−4662.39	−0.251745
$701$	7313.66	0.394056	0.197028	−	0.980398i	$-0.436871\pi$
$701$	7313.66	0.394056	0.197028	+	0.980398i	$0.436871\pi$
$702$	−702.000	−0.0377426
$703$	13893.1	0.745360
$704$	−3966.64	−0.212356
$705$	−1155.18	−0.0617116
$706$	5960.52	0.317744
$707$	−3748.55	−0.199404
$708$	1635.82	0.0868330
$709$	−22090.9	−1.17016	−0.585079	−	0.810976i	$-0.698937\pi$
$709$	−22090.9	−1.17016	−0.585079	+	0.810976i	$0.698937\pi$
$710$	−17978.8	−0.950326
$711$	−813.889	−0.0429300
$712$	1641.50	0.0864015
$713$	−23463.8	−1.23243
$714$	1545.55	0.0810093
$715$	13756.8	0.719543
$716$	−4792.66	−0.250154
$717$	7994.12	0.416382
$718$	−2455.94	−0.127653
$719$	4726.04	0.245134	0.122567	−	0.992460i	$-0.460887\pi$
$719$	4726.04	0.245134	0.122567	+	0.992460i	$0.460887\pi$
$720$	−2458.62	−0.127260
$721$	2209.53	0.114129
$722$	7193.70	0.370806
$723$	13374.0	0.687946
$724$	−5735.67	−0.294426
$725$	41513.0	2.12656
$726$	−15062.2	−0.769989
$727$	4096.77	0.208997	0.104499	−	0.994525i	$-0.466676\pi$
$727$	4096.77	0.208997	0.104499	+	0.994525i	$0.466676\pi$
$728$	−728.000	−0.0370625
$729$	729.000	0.0370370
$730$	34845.9	1.76672
$731$	−4887.46	−0.247290
$732$	−9607.61	−0.485120
$733$	−37484.5	−1.88884	−0.944421	−	0.328740i	$-0.893376\pi$
$733$	−37484.5	−1.88884	−0.944421	+	0.328740i	$0.893376\pi$
$734$	−15373.5	−0.773089
$735$	2509.84	0.125955
$736$	−6307.73	−0.315905
$737$	−61665.1	−3.08204
$738$	4343.58	0.216652
$739$	14414.1	0.717498	0.358749	−	0.933434i	$-0.383203\pi$
$739$	14414.1	0.717498	0.358749	+	0.933434i	$0.383203\pi$
$740$	−9279.17	−0.460958
$741$	−3987.90	−0.197705
$742$	4788.57	0.236919
$743$	−20016.9	−0.988356	−0.494178	−	0.869361i	$-0.664531\pi$
$743$	−20016.9	−0.988356	−0.494178	+	0.869361i	$0.664531\pi$
$744$	−2856.84	−0.140776
$745$	18077.7	0.889014
$746$	11255.3	0.552396
$747$	9635.90	0.471967
$748$	−9122.96	−0.445947
$749$	9590.68	0.467872
$750$	4252.78	0.207053
$751$	−33004.0	−1.60364	−0.801820	−	0.597566i	$-0.796135\pi$
$751$	−33004.0	−1.60364	−0.801820	+	0.597566i	$0.796135\pi$
$752$	−360.844	−0.0174982
$753$	−20631.6	−0.998484
$754$	6481.97	0.313076
$755$	8277.97	0.399028
$756$	756.000	0.0363696
$757$	−6242.19	−0.299705	−0.149852	−	0.988708i	$-0.547880\pi$
$757$	−6242.19	−0.299705	−0.149852	+	0.988708i	$0.547880\pi$
$758$	−16503.9	−0.790827
$759$	−36651.1	−1.75277
$760$	−13966.9	−0.666621
$761$	−10648.7	−0.507246	−0.253623	−	0.967303i	$-0.581622\pi$
$761$	−10648.7	−0.507246	−0.253623	+	0.967303i	$0.581622\pi$
$762$	15064.5	0.716182
$763$	9809.55	0.465439
$764$	6775.76	0.320862
$765$	−5654.64	−0.267247
$766$	−28152.1	−1.32791
$767$	−1772.14	−0.0834265
$768$	−768.000	−0.0360844
$769$	36927.0	1.73163	0.865813	−	0.500368i	$-0.166802\pi$
$769$	36927.0	1.73163	0.865813	+	0.500368i	$0.166802\pi$
$770$	−14815.0	−0.693369
$771$	10853.3	0.506965
$772$	12679.2	0.591106
$773$	26341.3	1.22565	0.612827	−	0.790217i	$-0.290032\pi$
$773$	26341.3	1.22565	0.612827	+	0.790217i	$0.290032\pi$
$774$	−2390.69	−0.111023
$775$	19821.0	0.918698
$776$	8104.88	0.374933
$777$	2853.24	0.131737
$778$	18311.2	0.843814
$779$	24674.9	1.13488
$780$	2663.51	0.122268
$781$	−32632.0	−1.49509
$782$	−14507.3	−0.663400
$783$	−6731.27	−0.307224
$784$	784.000	0.0357143
$785$	−33378.2	−1.51760
$786$	−7557.71	−0.342970
$787$	4276.36	0.193692	0.0968462	−	0.995299i	$-0.469125\pi$
$787$	4276.36	0.193692	0.0968462	+	0.995299i	$0.469125\pi$
$788$	14317.6	0.647264
$789$	7535.02	0.339992
$790$	3088.04	0.139073
$791$	8945.03	0.402084
$792$	−4462.47	−0.200211
$793$	10408.2	0.466088
$794$	6553.76	0.292927
$795$	−17519.8	−0.781588
$796$	−5075.37	−0.225995
$797$	5959.46	0.264862	0.132431	−	0.991192i	$-0.457722\pi$
$797$	5959.46	0.264862	0.132431	+	0.991192i	$0.457722\pi$
$798$	4294.66	0.190513
$799$	−829.912	−0.0367462
$800$	5328.44	0.235486
$801$	1846.69	0.0814601
$802$	4880.39	0.214879
$803$	63246.4	2.77947
$804$	−11939.3	−0.523713
$805$	−23558.7	−1.03147
$806$	3094.91	0.135253
$807$	24375.4	1.06326
$808$	4284.05	0.186525
$809$	33285.9	1.44656	0.723282	−	0.690553i	$-0.242633\pi$
$809$	33285.9	1.44656	0.723282	+	0.690553i	$0.242633\pi$
$810$	−2765.95	−0.119982
$811$	32994.5	1.42860	0.714299	−	0.699841i	$-0.246746\pi$
$811$	32994.5	1.42860	0.714299	+	0.699841i	$0.246746\pi$
$812$	−6980.58	−0.301688
$813$	−1750.61	−0.0755186
$814$	−16842.0	−0.725197
$815$	−28223.0	−1.21302
$816$	−1766.34	−0.0757772
$817$	−13581.0	−0.581564
$818$	25229.8	1.07841
$819$	−819.000	−0.0349428
$820$	−16480.3	−0.701850
$821$	−22164.4	−0.942197	−0.471098	−	0.882081i	$-0.656142\pi$
$821$	−22164.4	−0.942197	−0.471098	+	0.882081i	$0.656142\pi$
$822$	−12181.0	−0.516861
$823$	−19811.9	−0.839125	−0.419563	−	0.907726i	$-0.637817\pi$
$823$	−19811.9	−0.839125	−0.419563	+	0.907726i	$0.637817\pi$
$824$	−2525.18	−0.106758
$825$	30961.0	1.30657
$826$	1908.45	0.0803917
$827$	−1069.76	−0.0449809	−0.0224905	−	0.999747i	$-0.507160\pi$
$827$	−1069.76	−0.0449809	−0.0224905	+	0.999747i	$0.507160\pi$
$828$	−7096.19	−0.297838
$829$	−16444.5	−0.688953	−0.344477	−	0.938795i	$-0.611944\pi$
$829$	−16444.5	−0.688953	−0.344477	+	0.938795i	$0.611944\pi$
$830$	−36560.3	−1.52895
$831$	−19009.6	−0.793546
$832$	832.000	0.0346688
$833$	1803.14	0.0750000
$834$	1683.96	0.0699168
$835$	−50238.0	−2.08210
$836$	−25350.3	−1.04875
$837$	−3213.95	−0.132724
$838$	11373.0	0.468822
$839$	−39286.8	−1.61660	−0.808302	−	0.588769i	$-0.799613\pi$
$839$	−39286.8	−1.61660	−0.808302	+	0.588769i	$0.799613\pi$
$840$	−2868.39	−0.117820
$841$	37764.7	1.54843
$842$	31010.0	1.26921
$843$	−10263.1	−0.419313
$844$	−5852.58	−0.238690
$845$	−2885.47	−0.117471
$846$	−405.950	−0.0164974
$847$	−17572.6	−0.712871
$848$	−5472.65	−0.221617
$849$	1918.15	0.0775393
$850$	12255.0	0.494521
$851$	−26782.0	−1.07882
$852$	−6318.03	−0.254052
$853$	−12690.3	−0.509387	−0.254693	−	0.967022i	$-0.581975\pi$
$853$	−12690.3	−0.509387	−0.254693	+	0.967022i	$0.581975\pi$
$854$	−11208.9	−0.449134
$855$	−15712.7	−0.628496
$856$	−10960.8	−0.437654
$857$	34493.8	1.37490	0.687448	−	0.726233i	$-0.258731\pi$
$857$	34493.8	1.37490	0.687448	+	0.726233i	$0.258731\pi$
$858$	4834.35	0.192356
$859$	−266.560	−0.0105878	−0.00529389	−	0.999986i	$-0.501685\pi$
$859$	−266.560	−0.0105878	−0.00529389	+	0.999986i	$0.501685\pi$
$860$	9070.69	0.359660
$861$	5067.51	0.200581
$862$	17519.3	0.692238
$863$	21061.8	0.830767	0.415384	−	0.909646i	$-0.363647\pi$
$863$	21061.8	0.830767	0.415384	+	0.909646i	$0.363647\pi$
$864$	−864.000	−0.0340207
$865$	9843.59	0.386927
$866$	11468.8	0.450030
$867$	10676.6	0.418218
$868$	−3332.98	−0.130333
$869$	5604.87	0.218794
$870$	25539.6	0.995257
$871$	12934.2	0.503167
$872$	−11210.9	−0.435378
$873$	9117.99	0.353491
$874$	−40311.8	−1.56015
$875$	4961.58	0.191694
$876$	12245.4	0.472300
$877$	5534.39	0.213094	0.106547	−	0.994308i	$-0.466021\pi$
$877$	5534.39	0.213094	0.106547	+	0.994308i	$0.466021\pi$
$878$	36232.7	1.39270
$879$	−3372.90	−0.129426
$880$	16931.4	0.648588
$881$	−6944.05	−0.265552	−0.132776	−	0.991146i	$-0.542389\pi$
$881$	−6944.05	−0.265552	−0.132776	+	0.991146i	$0.542389\pi$
$882$	882.000	0.0336718
$883$	−10208.1	−0.389047	−0.194524	−	0.980898i	$-0.562316\pi$
$883$	−10208.1	−0.389047	−0.194524	+	0.980898i	$0.562316\pi$
$884$	1913.53	0.0728044
$885$	−6982.39	−0.265210
$886$	−11898.4	−0.451168
$887$	7228.00	0.273610	0.136805	−	0.990598i	$-0.456317\pi$
$887$	7228.00	0.273610	0.136805	+	0.990598i	$0.456317\pi$
$888$	−3260.85	−0.123229
$889$	17575.3	0.663055
$890$	−7006.66	−0.263892
$891$	−5020.28	−0.188761
$892$	8292.01	0.311252
$893$	−2306.11	−0.0864176
$894$	6352.79	0.237661
$895$	20457.2	0.764032
$896$	−896.000	−0.0334077
$897$	7687.54	0.286153
$898$	28991.7	1.07735
$899$	29676.2	1.10095
$900$	5994.50	0.222018
$901$	−12586.7	−0.465397
$902$	−29912.2	−1.10418
$903$	−2789.14	−0.102787
$904$	−10222.9	−0.376115
$905$	24482.4	0.899250
$906$	2909.01	0.106673
$907$	−30541.1	−1.11808	−0.559042	−	0.829139i	$-0.688831\pi$
$907$	−30541.1	−1.11808	−0.559042	+	0.829139i	$0.688831\pi$
$908$	−5903.64	−0.215770
$909$	4819.56	0.175858
$910$	3107.43	0.113198
$911$	24457.7	0.889483	0.444742	−	0.895659i	$-0.353296\pi$
$911$	24457.7	0.889483	0.444742	+	0.895659i	$0.353296\pi$
$912$	−4908.19	−0.178209
$913$	−66357.9	−2.40540
$914$	−19767.1	−0.715358
$915$	41009.6	1.48168
$916$	12302.7	0.443768
$917$	−8817.33	−0.317529
$918$	−1987.13	−0.0714434
$919$	−23601.2	−0.847150	−0.423575	−	0.905861i	$-0.639225\pi$
$919$	−23601.2	−0.847150	−0.423575	+	0.905861i	$0.639225\pi$
$920$	26924.2	0.964852
$921$	−6298.32	−0.225338
$922$	4212.21	0.150457
$923$	6844.54	0.244085
$924$	−5206.22	−0.185359
$925$	22624.0	0.804187
$926$	−28094.7	−0.997029
$927$	−2840.82	−0.100652
$928$	7977.81	0.282203
$929$	42564.3	1.50322	0.751609	−	0.659609i	$-0.229278\pi$
$929$	42564.3	1.50322	0.751609	+	0.659609i	$0.229278\pi$
$930$	12194.3	0.429964
$931$	5010.44	0.176381
$932$	−14614.9	−0.513656
$933$	19629.8	0.688799
$934$	21168.5	0.741600
$935$	38940.8	1.36203
$936$	936.000	0.0326860
$937$	−39336.9	−1.37149	−0.685743	−	0.727844i	$-0.740523\pi$
$937$	−39336.9	−1.37149	−0.685743	+	0.727844i	$0.740523\pi$
$938$	−13929.1	−0.484864
$939$	−14005.9	−0.486757
$940$	1540.24	0.0534438
$941$	36044.2	1.24868	0.624340	−	0.781153i	$-0.285368\pi$
$941$	36044.2	1.24868	0.624340	+	0.781153i	$0.285368\pi$
$942$	−11729.6	−0.405703
$943$	−47566.2	−1.64260
$944$	−2181.09	−0.0751996
$945$	−3226.94	−0.111082
$946$	16463.6	0.565832
$947$	−3835.87	−0.131625	−0.0658126	−	0.997832i	$-0.520964\pi$
$947$	−3835.87	−0.131625	−0.0658126	+	0.997832i	$0.520964\pi$
$948$	1085.19	0.0371785
$949$	−13265.9	−0.453771
$950$	34053.4	1.16299
$951$	7783.28	0.265395
$952$	−2060.73	−0.0701561
$953$	−56016.0	−1.90402	−0.952012	−	0.306060i	$-0.900989\pi$
$953$	−56016.0	−1.90402	−0.952012	+	0.306060i	$0.900989\pi$
$954$	−6156.73	−0.208943
$955$	−28922.0	−0.979992
$956$	−10658.8	−0.360597
$957$	46355.1	1.56578
$958$	7406.09	0.249770
$959$	−14211.1	−0.478521
$960$	3278.16	0.110211
$961$	−15621.6	−0.524374
$962$	3532.59	0.118394
$963$	−12330.9	−0.412624
$964$	−17832.0	−0.595778
$965$	−54120.4	−1.80538
$966$	−8278.89	−0.275744
$967$	21356.6	0.710218	0.355109	−	0.934825i	$-0.384444\pi$
$967$	21356.6	0.710218	0.355109	+	0.934825i	$0.384444\pi$
$968$	20083.0	0.666830
$969$	−11288.4	−0.374238
$970$	−34595.2	−1.14514
$971$	−21327.3	−0.704867	−0.352433	−	0.935837i	$-0.614646\pi$
$971$	−21327.3	−0.704867	−0.352433	+	0.935837i	$0.614646\pi$
$972$	−972.000	−0.0320750
$973$	1964.61	0.0647304
$974$	14054.0	0.462339
$975$	−6494.04	−0.213308
$976$	12810.2	0.420126
$977$	−47165.8	−1.54449	−0.772246	−	0.635323i	$-0.780867\pi$
$977$	−47165.8	−1.54449	−0.772246	+	0.635323i	$0.780867\pi$
$978$	−9918.03	−0.324278
$979$	−12717.3	−0.415165
$980$	−3346.46	−0.109080
$981$	−12612.3	−0.410478
$982$	28166.5	0.915304
$983$	17237.7	0.559306	0.279653	−	0.960101i	$-0.409781\pi$
$983$	17237.7	0.559306	0.279653	+	0.960101i	$0.409781\pi$
$984$	−5791.45	−0.187627
$985$	−61113.9	−1.97690
$986$	18348.3	0.592626
$987$	−473.608	−0.0152737
$988$	5317.20	0.171217
$989$	26180.2	0.841743
$990$	19047.8	0.611494
$991$	28068.8	0.899731	0.449865	−	0.893096i	$-0.351472\pi$
$991$	28068.8	0.899731	0.449865	+	0.893096i	$0.351472\pi$
$992$	3809.13	0.121915
$993$	−32890.0	−1.05109
$994$	−7371.04	−0.235206
$995$	21663.9	0.690244
$996$	−12847.9	−0.408735
$997$	−19386.0	−0.615809	−0.307905	−	0.951417i	$-0.599628\pi$
$997$	−19386.0	−0.615809	−0.307905	+	0.951417i	$0.599628\pi$
$998$	−2825.73	−0.0896262
$999$	−3668.45	−0.116181

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	546.4.a.p.1.1	✓	3
3.2	odd	2		1638.4.a.v.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.4.a.p.1.1	✓	3	1.1	even	1	trivial
1638.4.a.v.1.3		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	546.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} -17.0738 q^{5} -6.00000 q^{6} -7.00000 q^{7} +8.00000 q^{8} +9.00000 q^{9} -34.1475 q^{10} -61.9788 q^{11} -12.0000 q^{12} +13.0000 q^{13} -14.0000 q^{14} +51.2213 q^{15} +16.0000 q^{16} +36.7987 q^{17} +18.0000 q^{18} +102.254 q^{19} -68.2951 q^{20} +21.0000 q^{21} -123.958 q^{22} -197.116 q^{23} -24.0000 q^{24} +166.514 q^{25} +26.0000 q^{26} -27.0000 q^{27} -28.0000 q^{28} +249.306 q^{29} +102.443 q^{30} +119.035 q^{31} +32.0000 q^{32} +185.936 q^{33} +73.5974 q^{34} +119.516 q^{35} +36.0000 q^{36} +135.869 q^{37} +204.508 q^{38} -39.0000 q^{39} -136.590 q^{40} +241.310 q^{41} +42.0000 q^{42} -132.816 q^{43} -247.915 q^{44} -153.664 q^{45} -394.233 q^{46} -22.5528 q^{47} -48.0000 q^{48} +49.0000 q^{49} +333.028 q^{50} -110.396 q^{51} +52.0000 q^{52} -342.041 q^{53} -54.0000 q^{54} +1058.21 q^{55} -56.0000 q^{56} -306.762 q^{57} +498.613 q^{58} -136.318 q^{59} +204.885 q^{60} +800.634 q^{61} +238.070 q^{62} -63.0000 q^{63} +64.0000 q^{64} -221.959 q^{65} +371.873 q^{66} +994.938 q^{67} +147.195 q^{68} +591.349 q^{69} +239.033 q^{70} +526.503 q^{71} +72.0000 q^{72} -1020.45 q^{73} +271.737 q^{74} -499.541 q^{75} +409.015 q^{76} +433.852 q^{77} -78.0000 q^{78} -90.4321 q^{79} -273.180 q^{80} +81.0000 q^{81} +482.620 q^{82} +1070.66 q^{83} +84.0000 q^{84} -628.293 q^{85} -265.632 q^{86} -747.919 q^{87} -495.830 q^{88} +205.188 q^{89} -307.328 q^{90} -91.0000 q^{91} -788.466 q^{92} -357.106 q^{93} -45.1055 q^{94} -1745.86 q^{95} -96.0000 q^{96} +1013.11 q^{97} +98.0000 q^{98} -557.809 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 7 q^{5} - 18 q^{6} - 21 q^{7} + 24 q^{8} + 27 q^{9} + 14 q^{10} - 47 q^{11} - 36 q^{12} + 39 q^{13} - 42 q^{14} - 21 q^{15} + 48 q^{16} + 119 q^{17} + 54 q^{18} + 101 q^{19}+ \cdots - 423 q^{99}+O(q^{100}) \) 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 + 7 * q^5 - 18 * q^6 - 21 * q^7 + 24 * q^8 + 27 * q^9 + 14 * q^10 - 47 * q^11 - 36 * q^12 + 39 * q^13 - 42 * q^14 - 21 * q^15 + 48 * q^16 + 119 * q^17 + 54 * q^18 + 101 * q^19 + 28 * q^20 + 63 * q^21 - 94 * q^22 - 27 * q^23 - 72 * q^24 + 266 * q^25 + 78 * q^26 - 81 * q^27 - 84 * q^28 + 345 * q^29 - 42 * q^30 + 152 * q^31 + 96 * q^32 + 141 * q^33 + 238 * q^34 - 49 * q^35 + 108 * q^36 + 227 * q^37 + 202 * q^38 - 117 * q^39 + 56 * q^40 + 452 * q^41 + 126 * q^42 - 415 * q^43 - 188 * q^44 + 63 * q^45 - 54 * q^46 + 240 * q^47 - 144 * q^48 + 147 * q^49 + 532 * q^50 - 357 * q^51 + 156 * q^52 + 874 * q^53 - 162 * q^54 + 1123 * q^55 - 168 * q^56 - 303 * q^57 + 690 * q^58 + 948 * q^59 - 84 * q^60 + 951 * q^61 + 304 * q^62 - 189 * q^63 + 192 * q^64 + 91 * q^65 + 282 * q^66 + 742 * q^67 + 476 * q^68 + 81 * q^69 - 98 * q^70 + 732 * q^71 + 216 * q^72 - 585 * q^73 + 454 * q^74 - 798 * q^75 + 404 * q^76 + 329 * q^77 - 234 * q^78 - 618 * q^79 + 112 * q^80 + 243 * q^81 + 904 * q^82 + 774 * q^83 + 252 * q^84 + 1273 * q^85 - 830 * q^86 - 1035 * q^87 - 376 * q^88 + 730 * q^89 + 126 * q^90 - 273 * q^91 - 108 * q^92 - 456 * q^93 + 480 * q^94 - 2617 * q^95 - 288 * q^96 + 2518 * q^97 + 294 * q^98 - 423 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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