LMFDB - Embedded newform 825.4.a.w.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [825,4,Mod(1,825)] 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("825.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(825, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	825.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-4,-15,46] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$48.6765757547$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 41x^{3} + 3x^{2} + 294x - 200 $ x^5 - x^4 - 41x^3 + 3x^2 + 294*x - 200
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.729174$ of defining polynomial
Character	$\chi$	$=$	825.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.270826 q^{2} -3.00000 q^{3} -7.92665 q^{4} +0.812479 q^{6} +33.4133 q^{7} +4.31336 q^{8} +9.00000 q^{9} +11.0000 q^{11} +23.7800 q^{12} +33.5174 q^{13} -9.04920 q^{14} +62.2451 q^{16} +71.1540 q^{17} -2.43744 q^{18} -48.9667 q^{19} -100.240 q^{21} -2.97909 q^{22} -66.7490 q^{23} -12.9401 q^{24} -9.07738 q^{26} -27.0000 q^{27} -264.856 q^{28} -66.8864 q^{29} +145.759 q^{31} -51.3644 q^{32} -33.0000 q^{33} -19.2704 q^{34} -71.3399 q^{36} +37.8491 q^{37} +13.2615 q^{38} -100.552 q^{39} -344.036 q^{41} +27.1476 q^{42} -34.4830 q^{43} -87.1932 q^{44} +18.0774 q^{46} +270.887 q^{47} -186.735 q^{48} +773.448 q^{49} -213.462 q^{51} -265.681 q^{52} -666.088 q^{53} +7.31231 q^{54} +144.123 q^{56} +146.900 q^{57} +18.1146 q^{58} +876.971 q^{59} +783.523 q^{61} -39.4753 q^{62} +300.720 q^{63} -484.050 q^{64} +8.93727 q^{66} -876.724 q^{67} -564.013 q^{68} +200.247 q^{69} -523.428 q^{71} +38.8202 q^{72} -91.0973 q^{73} -10.2505 q^{74} +388.142 q^{76} +367.546 q^{77} +27.2322 q^{78} -96.9287 q^{79} +81.0000 q^{81} +93.1741 q^{82} +1395.68 q^{83} +794.567 q^{84} +9.33889 q^{86} +200.659 q^{87} +47.4469 q^{88} +508.117 q^{89} +1119.93 q^{91} +529.096 q^{92} -437.276 q^{93} -73.3634 q^{94} +154.093 q^{96} +644.082 q^{97} -209.470 q^{98} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{2} - 15 q^{3} + 46 q^{4} + 12 q^{6} + 38 q^{7} - 72 q^{8} + 45 q^{9} + 55 q^{11} - 138 q^{12} - 43 q^{13} + 55 q^{14} + 634 q^{16} - 150 q^{17} - 36 q^{18} - 337 q^{19} - 114 q^{21} - 44 q^{22}+ \cdots + 495 q^{99}+O(q^{100}) $ 5 * q - 4 * q^2 - 15 * q^3 + 46 * q^4 + 12 * q^6 + 38 * q^7 - 72 * q^8 + 45 * q^9 + 55 * q^11 - 138 * q^12 - 43 * q^13 + 55 * q^14 + 634 * q^16 - 150 * q^17 - 36 * q^18 - 337 * q^19 - 114 * q^21 - 44 * q^22 + 201 * q^23 + 216 * q^24 + 359 * q^26 - 135 * q^27 + 441 * q^28 - 153 * q^29 + 263 * q^31 - 280 * q^32 - 165 * q^33 + 517 * q^34 + 414 * q^36 + 350 * q^37 - 786 * q^38 + 129 * q^39 - 88 * q^41 - 165 * q^42 - 161 * q^43 + 506 * q^44 + 436 * q^46 - 490 * q^47 - 1902 * q^48 + 2381 * q^49 + 450 * q^51 - 289 * q^52 - 558 * q^53 + 108 * q^54 + 211 * q^56 + 1011 * q^57 + 969 * q^58 + 410 * q^59 - 1130 * q^61 - 463 * q^62 + 342 * q^63 + 3170 * q^64 + 132 * q^66 - 1130 * q^67 - 3333 * q^68 - 603 * q^69 + 1249 * q^71 - 648 * q^72 + 44 * q^73 - 2985 * q^74 - 2556 * q^76 + 418 * q^77 - 1077 * q^78 + 2258 * q^79 + 405 * q^81 + 3737 * q^82 + 957 * q^83 - 1323 * q^84 - 1435 * q^86 + 459 * q^87 - 792 * q^88 + 347 * q^89 - 612 * q^91 + 6006 * q^92 - 789 * q^93 + 907 * q^94 + 840 * q^96 + 2355 * q^97 - 9417 * q^98 + 495 * 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 $

$2$	−0.270826	−0.0957515	−0.0478758	−	0.998853i	$-0.515245\pi$
$2$	−0.270826	−0.0957515	−0.0478758	+	0.998853i	$0.515245\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−7.92665	−0.990832
$5$	0	0
$5$	0	0
$6$	0.812479	0.0552822
$7$	33.4133	1.80415	0.902074	−	0.431581i	$-0.142044\pi$
$7$	33.4133	1.80415	0.902074	+	0.431581i	$0.142044\pi$
$8$	4.31336	0.190625
$9$	9.00000	0.333333
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	23.7800	0.572057
$13$	33.5174	0.715081	0.357540	−	0.933898i	$-0.383615\pi$
$13$	33.5174	0.715081	0.357540	+	0.933898i	$0.383615\pi$
$14$	−9.04920	−0.172750
$15$	0	0
$16$	62.2451	0.972579
$17$	71.1540	1.01514	0.507570	−	0.861610i	$-0.330544\pi$
$17$	71.1540	1.01514	0.507570	+	0.861610i	$0.330544\pi$
$18$	−2.43744	−0.0319172
$19$	−48.9667	−0.591249	−0.295625	−	0.955304i	$-0.595528\pi$
$19$	−48.9667	−0.591249	−0.295625	+	0.955304i	$0.595528\pi$
$20$	0	0
$21$	−100.240	−1.04163
$22$	−2.97909	−0.0288702
$23$	−66.7490	−0.605136	−0.302568	−	0.953128i	$-0.597844\pi$
$23$	−66.7490	−0.605136	−0.302568	+	0.953128i	$0.597844\pi$
$24$	−12.9401	−0.110058
$25$	0	0
$26$	−9.07738	−0.0684701
$27$	−27.0000	−0.192450
$28$	−264.856	−1.78761
$29$	−66.8864	−0.428293	−0.214146	−	0.976802i	$-0.568697\pi$
$29$	−66.8864	−0.428293	−0.214146	+	0.976802i	$0.568697\pi$
$30$	0	0
$31$	145.759	0.844486	0.422243	−	0.906483i	$-0.361243\pi$
$31$	145.759	0.844486	0.422243	+	0.906483i	$0.361243\pi$
$32$	−51.3644	−0.283751
$33$	−33.0000	−0.174078
$34$	−19.2704	−0.0972013
$35$	0	0
$36$	−71.3399	−0.330277
$37$	37.8491	0.168172	0.0840859	−	0.996459i	$-0.473203\pi$
$37$	37.8491	0.168172	0.0840859	+	0.996459i	$0.473203\pi$
$38$	13.2615	0.0566130
$39$	−100.552	−0.412852
$40$	0	0
$41$	−344.036	−1.31047	−0.655237	−	0.755423i	$-0.727431\pi$
$41$	−344.036	−1.31047	−0.655237	+	0.755423i	$0.727431\pi$
$42$	27.1476	0.0997373
$43$	−34.4830	−0.122293	−0.0611465	−	0.998129i	$-0.519476\pi$
$43$	−34.4830	−0.122293	−0.0611465	+	0.998129i	$0.519476\pi$
$44$	−87.1932	−0.298747
$45$	0	0
$46$	18.0774	0.0579427
$47$	270.887	0.840701	0.420351	−	0.907362i	$-0.361907\pi$
$47$	270.887	0.840701	0.420351	+	0.907362i	$0.361907\pi$
$48$	−186.735	−0.561519
$49$	773.448	2.25495
$50$	0	0
$51$	−213.462	−0.586092
$52$	−265.681	−0.708524
$53$	−666.088	−1.72631	−0.863154	−	0.504941i	$-0.831514\pi$
$53$	−666.088	−1.72631	−0.863154	+	0.504941i	$0.831514\pi$
$54$	7.31231	0.0184274
$55$	0	0
$56$	144.123	0.343916
$57$	146.900	0.341358
$58$	18.1146	0.0410097
$59$	876.971	1.93512	0.967559	−	0.252645i	$-0.0813006\pi$
$59$	876.971	1.93512	0.967559	+	0.252645i	$0.0813006\pi$
$60$	0	0
$61$	783.523	1.64459	0.822294	−	0.569063i	$-0.192694\pi$
$61$	783.523	1.64459	0.822294	+	0.569063i	$0.192694\pi$
$62$	−39.4753	−0.0808608
$63$	300.720	0.601383
$64$	−484.050	−0.945409
$65$	0	0
$66$	8.93727	0.0166682
$67$	−876.724	−1.59864	−0.799320	−	0.600906i	$-0.794807\pi$
$67$	−876.724	−1.59864	−0.799320	+	0.600906i	$0.794807\pi$
$68$	−564.013	−1.00583
$69$	200.247	0.349375
$70$	0	0
$71$	−523.428	−0.874922	−0.437461	−	0.899237i	$-0.644122\pi$
$71$	−523.428	−0.874922	−0.437461	+	0.899237i	$0.644122\pi$
$72$	38.8202	0.0635417
$73$	−91.0973	−0.146057	−0.0730283	−	0.997330i	$-0.523266\pi$
$73$	−91.0973	−0.146057	−0.0730283	+	0.997330i	$0.523266\pi$
$74$	−10.2505	−0.0161027
$75$	0	0
$76$	388.142	0.585829
$77$	367.546	0.543971
$78$	27.2322	0.0395312
$79$	−96.9287	−0.138042	−0.0690211	−	0.997615i	$-0.521988\pi$
$79$	−96.9287	−0.138042	−0.0690211	+	0.997615i	$0.521988\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	93.1741	0.125480
$83$	1395.68	1.84574	0.922868	−	0.385116i	$-0.125839\pi$
$83$	1395.68	1.84574	0.922868	+	0.385116i	$0.125839\pi$
$84$	794.567	1.03208
$85$	0	0
$86$	9.33889	0.0117097
$87$	200.659	0.247275
$88$	47.4469	0.0574757
$89$	508.117	0.605172	0.302586	−	0.953122i	$-0.402150\pi$
$89$	508.117	0.605172	0.302586	+	0.953122i	$0.402150\pi$
$90$	0	0
$91$	1119.93	1.29011
$92$	529.096	0.599588
$93$	−437.276	−0.487564
$94$	−73.3634	−0.0804984
$95$	0	0
$96$	154.093	0.163824
$97$	644.082	0.674192	0.337096	−	0.941470i	$-0.390555\pi$
$97$	644.082	0.674192	0.337096	+	0.941470i	$0.390555\pi$
$98$	−209.470	−0.215915
$99$	99.0000	0.100504
$100$	0	0
$101$	1485.24	1.46324	0.731618	−	0.681715i	$-0.238766\pi$
$101$	1485.24	1.46324	0.731618	+	0.681715i	$0.238766\pi$
$102$	57.8111	0.0561192
$103$	−605.952	−0.579672	−0.289836	−	0.957076i	$-0.593601\pi$
$103$	−605.952	−0.579672	−0.289836	+	0.957076i	$0.593601\pi$
$104$	144.572	0.136312
$105$	0	0
$106$	180.394	0.165297
$107$	352.908	0.318850	0.159425	−	0.987210i	$-0.449036\pi$
$107$	352.908	0.318850	0.159425	+	0.987210i	$0.449036\pi$
$108$	214.020	0.190686
$109$	1413.29	1.24191	0.620957	−	0.783844i	$-0.286744\pi$
$109$	1413.29	1.24191	0.620957	+	0.783844i	$0.286744\pi$
$110$	0	0
$111$	−113.547	−0.0970940
$112$	2079.81	1.75468
$113$	−103.753	−0.0863741	−0.0431870	−	0.999067i	$-0.513751\pi$
$113$	−103.753	−0.0863741	−0.0431870	+	0.999067i	$0.513751\pi$
$114$	−39.7844	−0.0326856
$115$	0	0
$116$	530.185	0.424366
$117$	301.656	0.238360
$118$	−237.507	−0.185291
$119$	2377.49	1.83146
$120$	0	0
$121$	121.000	0.0909091
$122$	−212.199	−0.157472
$123$	1032.11	0.756603
$124$	−1155.38	−0.836743
$125$	0	0
$126$	−81.4428	−0.0575833
$127$	1576.66	1.10162	0.550809	−	0.834631i	$-0.314319\pi$
$127$	1576.66	1.10162	0.550809	+	0.834631i	$0.314319\pi$
$128$	542.009	0.374276
$129$	103.449	0.0706059
$130$	0	0
$131$	−1085.16	−0.723747	−0.361873	−	0.932227i	$-0.617863\pi$
$131$	−1085.16	−0.723747	−0.361873	+	0.932227i	$0.617863\pi$
$132$	261.580	0.172482
$133$	−1636.14	−1.06670
$134$	237.440	0.153072
$135$	0	0
$136$	306.913	0.193511
$137$	−1389.46	−0.866496	−0.433248	−	0.901275i	$-0.642633\pi$
$137$	−1389.46	−0.866496	−0.433248	+	0.901275i	$0.642633\pi$
$138$	−54.2321	−0.0334532
$139$	500.066	0.305144	0.152572	−	0.988292i	$-0.451244\pi$
$139$	500.066	0.305144	0.152572	+	0.988292i	$0.451244\pi$
$140$	0	0
$141$	−812.662	−0.485379
$142$	141.758	0.0837751
$143$	368.691	0.215605
$144$	560.205	0.324193
$145$	0	0
$146$	24.6715	0.0139851
$147$	−2320.35	−1.30190
$148$	−300.017	−0.166630
$149$	685.689	0.377005	0.188503	−	0.982073i	$-0.439637\pi$
$149$	685.689	0.377005	0.188503	+	0.982073i	$0.439637\pi$
$150$	0	0
$151$	−1208.96	−0.651550	−0.325775	−	0.945447i	$-0.605625\pi$
$151$	−1208.96	−0.651550	−0.325775	+	0.945447i	$0.605625\pi$
$152$	−211.211	−0.112707
$153$	640.386	0.338380
$154$	−99.5412	−0.0520861
$155$	0	0
$156$	797.042	0.409067
$157$	3322.47	1.68893	0.844465	−	0.535610i	$-0.179918\pi$
$157$	3322.47	1.68893	0.844465	+	0.535610i	$0.179918\pi$
$158$	26.2509	0.0132178
$159$	1998.27	0.996684
$160$	0	0
$161$	−2230.30	−1.09176
$162$	−21.9369	−0.0106391
$163$	−2448.31	−1.17648	−0.588240	−	0.808686i	$-0.700179\pi$
$163$	−2448.31	−1.17648	−0.588240	+	0.808686i	$0.700179\pi$
$164$	2727.06	1.29846
$165$	0	0
$166$	−377.988	−0.176732
$167$	1518.71	0.703720	0.351860	−	0.936053i	$-0.385549\pi$
$167$	1518.71	0.703720	0.351860	+	0.936053i	$0.385549\pi$
$168$	−432.370	−0.198560
$169$	−1073.59	−0.488660
$170$	0	0
$171$	−440.701	−0.197083
$172$	273.334	0.121172
$173$	−2636.80	−1.15880	−0.579399	−	0.815044i	$-0.696713\pi$
$173$	−2636.80	−1.15880	−0.579399	+	0.815044i	$0.696713\pi$
$174$	−54.3438	−0.0236770
$175$	0	0
$176$	684.696	0.293244
$177$	−2630.91	−1.11724
$178$	−137.611	−0.0579461
$179$	−1773.31	−0.740466	−0.370233	−	0.928939i	$-0.620722\pi$
$179$	−1773.31	−0.740466	−0.370233	+	0.928939i	$0.620722\pi$
$180$	0	0
$181$	529.751	0.217547	0.108774	−	0.994067i	$-0.465308\pi$
$181$	529.751	0.217547	0.108774	+	0.994067i	$0.465308\pi$
$182$	−303.305	−0.123530
$183$	−2350.57	−0.949503
$184$	−287.912	−0.115354
$185$	0	0
$186$	118.426	0.0466850
$187$	782.694	0.306076
$188$	−2147.23	−0.832993
$189$	−902.159	−0.347209
$190$	0	0
$191$	2061.93	0.781130	0.390565	−	0.920575i	$-0.372280\pi$
$191$	2061.93	0.781130	0.390565	+	0.920575i	$0.372280\pi$
$192$	1452.15	0.545832
$193$	1637.24	0.610627	0.305314	−	0.952252i	$-0.401239\pi$
$193$	1637.24	0.610627	0.305314	+	0.952252i	$0.401239\pi$
$194$	−174.434	−0.0645550
$195$	0	0
$196$	−6130.86	−2.23428
$197$	−3414.86	−1.23502	−0.617510	−	0.786563i	$-0.711858\pi$
$197$	−3414.86	−1.23502	−0.617510	+	0.786563i	$0.711858\pi$
$198$	−26.8118	−0.00962339
$199$	3938.93	1.40313	0.701566	−	0.712605i	$-0.252485\pi$
$199$	3938.93	1.40313	0.701566	+	0.712605i	$0.252485\pi$
$200$	0	0
$201$	2630.17	0.922975
$202$	−402.242	−0.140107
$203$	−2234.90	−0.772704
$204$	1692.04	0.580718
$205$	0	0
$206$	164.108	0.0555045
$207$	−600.741	−0.201712
$208$	2086.29	0.695472
$209$	−538.634	−0.178268
$210$	0	0
$211$	3543.14	1.15602	0.578009	−	0.816031i	$-0.303830\pi$
$211$	3543.14	1.15602	0.578009	+	0.816031i	$0.303830\pi$
$212$	5279.85	1.71048
$213$	1570.28	0.505136
$214$	−95.5768	−0.0305304
$215$	0	0
$216$	−116.461	−0.0366858
$217$	4870.28	1.52358
$218$	−382.756	−0.118915
$219$	273.292	0.0843258
$220$	0	0
$221$	2384.90	0.725907
$222$	30.7516	0.00929690
$223$	850.930	0.255527	0.127763	−	0.991805i	$-0.459220\pi$
$223$	850.930	0.255527	0.127763	+	0.991805i	$0.459220\pi$
$224$	−1716.26	−0.511929
$225$	0	0
$226$	28.0991	0.00827045
$227$	−5040.56	−1.47381	−0.736903	−	0.675999i	$-0.763712\pi$
$227$	−5040.56	−1.47381	−0.736903	+	0.675999i	$0.763712\pi$
$228$	−1164.43	−0.338228
$229$	−3624.34	−1.04587	−0.522933	−	0.852374i	$-0.675162\pi$
$229$	−3624.34	−1.04587	−0.522933	+	0.852374i	$0.675162\pi$
$230$	0	0
$231$	−1102.64	−0.314062
$232$	−288.505	−0.0816434
$233$	2984.56	0.839164	0.419582	−	0.907718i	$-0.362177\pi$
$233$	2984.56	0.839164	0.419582	+	0.907718i	$0.362177\pi$
$234$	−81.6965	−0.0228234
$235$	0	0
$236$	−6951.45	−1.91738
$237$	290.786	0.0796987
$238$	−643.887	−0.175366
$239$	1298.11	0.351331	0.175665	−	0.984450i	$-0.443792\pi$
$239$	1298.11	0.351331	0.175665	+	0.984450i	$0.443792\pi$
$240$	0	0
$241$	3978.71	1.06345	0.531725	−	0.846917i	$-0.321544\pi$
$241$	3978.71	1.06345	0.531725	+	0.846917i	$0.321544\pi$
$242$	−32.7700	−0.00870469
$243$	−243.000	−0.0641500
$244$	−6210.71	−1.62951
$245$	0	0
$246$	−279.522	−0.0724459
$247$	−1641.24	−0.422791
$248$	628.710	0.160980
$249$	−4187.05	−1.06564
$250$	0	0
$251$	−1884.05	−0.473784	−0.236892	−	0.971536i	$-0.576129\pi$
$251$	−1884.05	−0.473784	−0.236892	+	0.971536i	$0.576129\pi$
$252$	−2383.70	−0.595869
$253$	−734.239	−0.182455
$254$	−427.000	−0.105482
$255$	0	0
$256$	3725.61	0.909572
$257$	4012.23	0.973836	0.486918	−	0.873448i	$-0.338121\pi$
$257$	4012.23	0.973836	0.486918	+	0.873448i	$0.338121\pi$
$258$	−28.0167	−0.00676063
$259$	1264.66	0.303407
$260$	0	0
$261$	−601.978	−0.142764
$262$	293.890	0.0692999
$263$	−290.429	−0.0680935	−0.0340468	−	0.999420i	$-0.510840\pi$
$263$	−290.429	−0.0680935	−0.0340468	+	0.999420i	$0.510840\pi$
$264$	−142.341	−0.0331836
$265$	0	0
$266$	443.110	0.102138
$267$	−1524.35	−0.349396
$268$	6949.48	1.58398
$269$	−6743.15	−1.52839	−0.764196	−	0.644985i	$-0.776864\pi$
$269$	−6743.15	−1.52839	−0.764196	+	0.644985i	$0.776864\pi$
$270$	0	0
$271$	7182.11	1.60990	0.804949	−	0.593344i	$-0.202193\pi$
$271$	7182.11	1.60990	0.804949	+	0.593344i	$0.202193\pi$
$272$	4428.99	0.987304
$273$	−3359.78	−0.744846
$274$	376.304	0.0829683
$275$	0	0
$276$	−1587.29	−0.346172
$277$	5708.35	1.23820	0.619100	−	0.785312i	$-0.287498\pi$
$277$	5708.35	1.23820	0.619100	+	0.785312i	$0.287498\pi$
$278$	−135.431	−0.0292180
$279$	1311.83	0.281495
$280$	0	0
$281$	8740.94	1.85566	0.927831	−	0.373002i	$-0.121672\pi$
$281$	8740.94	1.85566	0.927831	+	0.373002i	$0.121672\pi$
$282$	220.090	0.0464758
$283$	7883.57	1.65593	0.827967	−	0.560777i	$-0.189497\pi$
$283$	7883.57	1.65593	0.827967	+	0.560777i	$0.189497\pi$
$284$	4149.03	0.866900
$285$	0	0
$286$	−99.8512	−0.0206445
$287$	−11495.4	−2.36429
$288$	−462.280	−0.0945837
$289$	149.897	0.0305102
$290$	0	0
$291$	−1932.25	−0.389245
$292$	722.097	0.144718
$293$	−2328.54	−0.464282	−0.232141	−	0.972682i	$-0.574573\pi$
$293$	−2328.54	−0.464282	−0.232141	+	0.972682i	$0.574573\pi$
$294$	628.410	0.124659
$295$	0	0
$296$	163.257	0.0320578
$297$	−297.000	−0.0580259
$298$	−185.703	−0.0360989
$299$	−2237.25	−0.432721
$300$	0	0
$301$	−1152.19	−0.220635
$302$	327.419	0.0623869
$303$	−4455.72	−0.844799
$304$	−3047.94	−0.575037
$305$	0	0
$306$	−173.433	−0.0324004
$307$	−2830.31	−0.526170	−0.263085	−	0.964773i	$-0.584740\pi$
$307$	−2830.31	−0.526170	−0.263085	+	0.964773i	$0.584740\pi$
$308$	−2913.41	−0.538984
$309$	1817.86	0.334674
$310$	0	0
$311$	−6580.67	−1.19986	−0.599929	−	0.800053i	$-0.704804\pi$
$311$	−6580.67	−1.19986	−0.599929	+	0.800053i	$0.704804\pi$
$312$	−433.717	−0.0787000
$313$	7837.96	1.41542	0.707712	−	0.706501i	$-0.249728\pi$
$313$	7837.96	1.41542	0.707712	+	0.706501i	$0.249728\pi$
$314$	−899.813	−0.161718
$315$	0	0
$316$	768.321	0.136777
$317$	−5853.61	−1.03713	−0.518567	−	0.855037i	$-0.673534\pi$
$317$	−5853.61	−1.03713	−0.518567	+	0.855037i	$0.673534\pi$
$318$	−541.183	−0.0954340
$319$	−735.751	−0.129135
$320$	0	0
$321$	−1058.72	−0.184088
$322$	604.025	0.104537
$323$	−3484.18	−0.600201
$324$	−642.059	−0.110092
$325$	0	0
$326$	663.066	0.112650
$327$	−4239.87	−0.717020
$328$	−1483.95	−0.249809
$329$	9051.23	1.51675
$330$	0	0
$331$	5347.61	0.888010	0.444005	−	0.896024i	$-0.353557\pi$
$331$	5347.61	0.888010	0.444005	+	0.896024i	$0.353557\pi$
$332$	−11063.1	−1.82881
$333$	340.642	0.0560573
$334$	−411.306	−0.0673823
$335$	0	0
$336$	−6239.44	−1.01306
$337$	−2763.99	−0.446779	−0.223389	−	0.974729i	$-0.571712\pi$
$337$	−2763.99	−0.446779	−0.223389	+	0.974729i	$0.571712\pi$
$338$	290.755	0.0467899
$339$	311.259	0.0498681
$340$	0	0
$341$	1603.35	0.254622
$342$	119.353	0.0188710
$343$	14382.7	2.26412
$344$	−148.737	−0.0233121
$345$	0	0
$346$	714.114	0.110957
$347$	−5038.66	−0.779509	−0.389754	−	0.920919i	$-0.627440\pi$
$347$	−5038.66	−0.779509	−0.389754	+	0.920919i	$0.627440\pi$
$348$	−1590.56	−0.245008
$349$	−7429.49	−1.13952	−0.569759	−	0.821812i	$-0.692963\pi$
$349$	−7429.49	−1.13952	−0.569759	+	0.821812i	$0.692963\pi$
$350$	0	0
$351$	−904.969	−0.137617
$352$	−565.009	−0.0855542
$353$	3151.84	0.475228	0.237614	−	0.971360i	$-0.423635\pi$
$353$	3151.84	0.475228	0.237614	+	0.971360i	$0.423635\pi$
$354$	712.521	0.106978
$355$	0	0
$356$	−4027.67	−0.599623
$357$	−7132.47	−1.05740
$358$	480.259	0.0709008
$359$	903.954	0.132894	0.0664469	−	0.997790i	$-0.478834\pi$
$359$	903.954	0.132894	0.0664469	+	0.997790i	$0.478834\pi$
$360$	0	0
$361$	−4461.26	−0.650424
$362$	−143.470	−0.0208305
$363$	−363.000	−0.0524864
$364$	−8877.26	−1.27828
$365$	0	0
$366$	636.596	0.0909164
$367$	−3978.31	−0.565848	−0.282924	−	0.959142i	$-0.591304\pi$
$367$	−3978.31	−0.565848	−0.282924	+	0.959142i	$0.591304\pi$
$368$	−4154.79	−0.588543
$369$	−3096.33	−0.436825
$370$	0	0
$371$	−22256.2	−3.11451
$372$	3466.14	0.483094
$373$	5493.67	0.762605	0.381302	−	0.924450i	$-0.375476\pi$
$373$	5493.67	0.762605	0.381302	+	0.924450i	$0.375476\pi$
$374$	−211.974	−0.0293073
$375$	0	0
$376$	1168.43	0.160259
$377$	−2241.86	−0.306264
$378$	244.328	0.0332458
$379$	3035.93	0.411465	0.205733	−	0.978608i	$-0.434042\pi$
$379$	3035.93	0.411465	0.205733	+	0.978608i	$0.434042\pi$
$380$	0	0
$381$	−4729.97	−0.636020
$382$	−558.424	−0.0747944
$383$	3556.77	0.474524	0.237262	−	0.971446i	$-0.423750\pi$
$383$	3556.77	0.474524	0.237262	+	0.971446i	$0.423750\pi$
$384$	−1626.03	−0.216088
$385$	0	0
$386$	−443.408	−0.0584685
$387$	−310.347	−0.0407644
$388$	−5105.42	−0.668011
$389$	−5211.56	−0.679272	−0.339636	−	0.940557i	$-0.610304\pi$
$389$	−5211.56	−0.679272	−0.339636	+	0.940557i	$0.610304\pi$
$390$	0	0
$391$	−4749.46	−0.614298
$392$	3336.16	0.429851
$393$	3255.48	0.417855
$394$	924.835	0.118255
$395$	0	0
$396$	−784.739	−0.0995823
$397$	6547.28	0.827704	0.413852	−	0.910344i	$-0.364183\pi$
$397$	6547.28	0.827704	0.413852	+	0.910344i	$0.364183\pi$
$398$	−1066.76	−0.134352
$399$	4908.42	0.615860
$400$	0	0
$401$	−1661.59	−0.206922	−0.103461	−	0.994634i	$-0.532992\pi$
$401$	−1661.59	−0.206922	−0.103461	+	0.994634i	$0.532992\pi$
$402$	−712.319	−0.0883763
$403$	4885.45	0.603875
$404$	−11773.0	−1.44982
$405$	0	0
$406$	605.268	0.0739876
$407$	416.340	0.0507057
$408$	−920.738	−0.111724
$409$	7278.96	0.880004	0.440002	−	0.897997i	$-0.354978\pi$
$409$	7278.96	0.880004	0.440002	+	0.897997i	$0.354978\pi$
$410$	0	0
$411$	4168.39	0.500272
$412$	4803.17	0.574357
$413$	29302.5	3.49124
$414$	162.696	0.0193142
$415$	0	0
$416$	−1721.60	−0.202905
$417$	−1500.20	−0.176175
$418$	145.876	0.0170695
$419$	−5892.15	−0.686994	−0.343497	−	0.939154i	$-0.611611\pi$
$419$	−5892.15	−0.686994	−0.343497	+	0.939154i	$0.611611\pi$
$420$	0	0
$421$	11086.3	1.28340	0.641700	−	0.766955i	$-0.278229\pi$
$421$	11086.3	1.28340	0.641700	+	0.766955i	$0.278229\pi$
$422$	−959.575	−0.110690
$423$	2437.98	0.280234
$424$	−2873.08	−0.329078
$425$	0	0
$426$	−425.274	−0.0483676
$427$	26180.1	2.96708
$428$	−2797.38	−0.315926
$429$	−1106.07	−0.124480
$430$	0	0
$431$	5658.06	0.632341	0.316170	−	0.948702i	$-0.397603\pi$
$431$	5658.06	0.632341	0.316170	+	0.948702i	$0.397603\pi$
$432$	−1680.62	−0.187173
$433$	−3065.03	−0.340176	−0.170088	−	0.985429i	$-0.554405\pi$
$433$	−3065.03	−0.340176	−0.170088	+	0.985429i	$0.554405\pi$
$434$	−1319.00	−0.145885
$435$	0	0
$436$	−11202.7	−1.23053
$437$	3268.48	0.357786
$438$	−74.0146	−0.00807433
$439$	−3017.50	−0.328058	−0.164029	−	0.986456i	$-0.552449\pi$
$439$	−3017.50	−0.328058	−0.164029	+	0.986456i	$0.552449\pi$
$440$	0	0
$441$	6961.04	0.751650
$442$	−645.893	−0.0695067
$443$	−17098.3	−1.83378	−0.916888	−	0.399144i	$-0.869307\pi$
$443$	−17098.3	−1.83378	−0.916888	+	0.399144i	$0.869307\pi$
$444$	900.050	0.0962038
$445$	0	0
$446$	−230.454	−0.0244671
$447$	−2057.07	−0.217664
$448$	−16173.7	−1.70566
$449$	−10823.4	−1.13762	−0.568808	−	0.822470i	$-0.692595\pi$
$449$	−10823.4	−1.13762	−0.568808	+	0.822470i	$0.692595\pi$
$450$	0	0
$451$	−3784.40	−0.395123
$452$	822.415	0.0855821
$453$	3626.89	0.376172
$454$	1365.12	0.141119
$455$	0	0
$456$	633.633	0.0650714
$457$	6859.01	0.702081	0.351040	−	0.936360i	$-0.385828\pi$
$457$	6859.01	0.702081	0.351040	+	0.936360i	$0.385828\pi$
$458$	981.567	0.100143
$459$	−1921.16	−0.195364
$460$	0	0
$461$	9651.72	0.975109	0.487555	−	0.873093i	$-0.337889\pi$
$461$	9651.72	0.975109	0.487555	+	0.873093i	$0.337889\pi$
$462$	298.624	0.0300719
$463$	4744.49	0.476232	0.238116	−	0.971237i	$-0.423470\pi$
$463$	4744.49	0.476232	0.238116	+	0.971237i	$0.423470\pi$
$464$	−4163.35	−0.416549
$465$	0	0
$466$	−808.298	−0.0803512
$467$	11038.1	1.09376	0.546878	−	0.837213i	$-0.315816\pi$
$467$	11038.1	1.09376	0.546878	+	0.837213i	$0.315816\pi$
$468$	−2391.13	−0.236175
$469$	−29294.2	−2.88418
$470$	0	0
$471$	−9967.42	−0.975105
$472$	3782.69	0.368882
$473$	−379.313	−0.0368727
$474$	−78.7526	−0.00763128
$475$	0	0
$476$	−18845.5	−1.81467
$477$	−5994.80	−0.575436
$478$	−351.563	−0.0336404
$479$	17476.9	1.66710	0.833550	−	0.552444i	$-0.186305\pi$
$479$	17476.9	1.66710	0.833550	+	0.552444i	$0.186305\pi$
$480$	0	0
$481$	1268.60	0.120256
$482$	−1077.54	−0.101827
$483$	6690.91	0.630325
$484$	−959.125	−0.0900756
$485$	0	0
$486$	65.8108	0.00614246
$487$	−17014.6	−1.58317	−0.791586	−	0.611058i	$-0.790744\pi$
$487$	−17014.6	−1.58317	−0.791586	+	0.611058i	$0.790744\pi$
$488$	3379.61	0.313500
$489$	7344.92	0.679241
$490$	0	0
$491$	19673.8	1.80828	0.904142	−	0.427233i	$-0.140512\pi$
$491$	19673.8	1.80828	0.904142	+	0.427233i	$0.140512\pi$
$492$	−8181.17	−0.749666
$493$	−4759.24	−0.434778
$494$	444.490	0.0404829
$495$	0	0
$496$	9072.76	0.821329
$497$	−17489.4	−1.57849
$498$	1133.96	0.102036
$499$	−15754.6	−1.41337	−0.706687	−	0.707526i	$-0.749811\pi$
$499$	−15754.6	−1.41337	−0.706687	+	0.707526i	$0.749811\pi$
$500$	0	0
$501$	−4556.13	−0.406293
$502$	510.249	0.0453656
$503$	16457.2	1.45883	0.729414	−	0.684073i	$-0.239793\pi$
$503$	16457.2	1.45883	0.729414	+	0.684073i	$0.239793\pi$
$504$	1297.11	0.114639
$505$	0	0
$506$	198.851	0.0174704
$507$	3220.76	0.282128
$508$	−12497.6	−1.09152
$509$	13852.0	1.20625	0.603124	−	0.797647i	$-0.293922\pi$
$509$	13852.0	1.20625	0.603124	+	0.797647i	$0.293922\pi$
$510$	0	0
$511$	−3043.86	−0.263508
$512$	−5345.06	−0.461368
$513$	1322.10	0.113786
$514$	−1086.62	−0.0932463
$515$	0	0
$516$	−820.003	−0.0699586
$517$	2979.76	0.253481
$518$	−342.504	−0.0290517
$519$	7910.39	0.669032
$520$	0	0
$521$	−12295.8	−1.03395	−0.516976	−	0.856000i	$-0.672942\pi$
$521$	−12295.8	−1.03395	−0.516976	+	0.856000i	$0.672942\pi$
$522$	163.031	0.0136699
$523$	−1971.83	−0.164861	−0.0824305	−	0.996597i	$-0.526268\pi$
$523$	−1971.83	−0.164861	−0.0824305	+	0.996597i	$0.526268\pi$
$524$	8601.68	0.717111
$525$	0	0
$526$	78.6557	0.00652006
$527$	10371.3	0.857271
$528$	−2054.09	−0.169304
$529$	−7711.57	−0.633810
$530$	0	0
$531$	7892.74	0.645039
$532$	12969.1	1.05692
$533$	−11531.2	−0.937095
$534$	412.834	0.0334552
$535$	0	0
$536$	−3781.62	−0.304741
$537$	5319.93	0.427508
$538$	1826.22	0.146346
$539$	8507.93	0.679893
$540$	0	0
$541$	−8849.45	−0.703267	−0.351634	−	0.936138i	$-0.614374\pi$
$541$	−8849.45	−0.703267	−0.351634	+	0.936138i	$0.614374\pi$
$542$	−1945.10	−0.154150
$543$	−1589.25	−0.125601
$544$	−3654.79	−0.288047
$545$	0	0
$546$	909.916	0.0713202
$547$	−10687.9	−0.835435	−0.417717	−	0.908577i	$-0.637170\pi$
$547$	−10687.9	−0.835435	−0.417717	+	0.908577i	$0.637170\pi$
$548$	11013.8	0.858552
$549$	7051.71	0.548196
$550$	0	0
$551$	3275.21	0.253228
$552$	863.737	0.0665998
$553$	−3238.71	−0.249049
$554$	−1545.97	−0.118560
$555$	0	0
$556$	−3963.85	−0.302346
$557$	−23864.2	−1.81537	−0.907684	−	0.419655i	$-0.862151\pi$
$557$	−23864.2	−1.81537	−0.907684	+	0.419655i	$0.862151\pi$
$558$	−355.278	−0.0269536
$559$	−1155.78	−0.0874494
$560$	0	0
$561$	−2348.08	−0.176713
$562$	−2367.28	−0.177682
$563$	−13610.5	−1.01886	−0.509428	−	0.860513i	$-0.670143\pi$
$563$	−13610.5	−1.01886	−0.509428	+	0.860513i	$0.670143\pi$
$564$	6441.69	0.480929
$565$	0	0
$566$	−2135.08	−0.158558
$567$	2706.48	0.200461
$568$	−2257.73	−0.166782
$569$	8564.56	0.631011	0.315505	−	0.948924i	$-0.397826\pi$
$569$	8564.56	0.631011	0.315505	+	0.948924i	$0.397826\pi$
$570$	0	0
$571$	15329.9	1.12353	0.561764	−	0.827298i	$-0.310123\pi$
$571$	15329.9	1.12353	0.561764	+	0.827298i	$0.310123\pi$
$572$	−2922.49	−0.213628
$573$	−6185.78	−0.450985
$574$	3113.25	0.226384
$575$	0	0
$576$	−4356.45	−0.315136
$577$	9594.47	0.692241	0.346120	−	0.938190i	$-0.387499\pi$
$577$	9594.47	0.692241	0.346120	+	0.938190i	$0.387499\pi$
$578$	−40.5959	−0.00292140
$579$	−4911.72	−0.352546
$580$	0	0
$581$	46634.4	3.32998
$582$	523.303	0.0372708
$583$	−7326.97	−0.520501
$584$	−392.935	−0.0278421
$585$	0	0
$586$	630.629	0.0444557
$587$	−16659.5	−1.17140	−0.585698	−	0.810530i	$-0.699179\pi$
$587$	−16659.5	−1.17140	−0.585698	+	0.810530i	$0.699179\pi$
$588$	18392.6	1.28996
$589$	−7137.33	−0.499302
$590$	0	0
$591$	10244.6	0.713039
$592$	2355.92	0.163560
$593$	−459.030	−0.0317877	−0.0158939	−	0.999874i	$-0.505059\pi$
$593$	−459.030	−0.0317877	−0.0158939	+	0.999874i	$0.505059\pi$
$594$	80.4354	0.00555607
$595$	0	0
$596$	−5435.22	−0.373549
$597$	−11816.8	−0.810098
$598$	605.906	0.0414337
$599$	869.305	0.0592969	0.0296485	−	0.999560i	$-0.490561\pi$
$599$	869.305	0.0592969	0.0296485	+	0.999560i	$0.490561\pi$
$600$	0	0
$601$	−5455.29	−0.370259	−0.185130	−	0.982714i	$-0.559271\pi$
$601$	−5455.29	−0.370259	−0.185130	+	0.982714i	$0.559271\pi$
$602$	312.043	0.0211261
$603$	−7890.51	−0.532880
$604$	9583.03	0.645576
$605$	0	0
$606$	1206.72	0.0808908
$607$	−27152.9	−1.81565	−0.907826	−	0.419347i	$-0.862259\pi$
$607$	−27152.9	−1.81565	−0.907826	+	0.419347i	$0.862259\pi$
$608$	2515.15	0.167768
$609$	6704.69	0.446121
$610$	0	0
$611$	9079.43	0.601169
$612$	−5076.12	−0.335278
$613$	10313.6	0.679546	0.339773	−	0.940507i	$-0.389650\pi$
$613$	10313.6	0.679546	0.339773	+	0.940507i	$0.389650\pi$
$614$	766.521	0.0503816
$615$	0	0
$616$	1585.36	0.103695
$617$	27148.4	1.77140	0.885699	−	0.464259i	$-0.153679\pi$
$617$	27148.4	1.77140	0.885699	+	0.464259i	$0.153679\pi$
$618$	−492.323	−0.0320455
$619$	13110.4	0.851294	0.425647	−	0.904889i	$-0.360047\pi$
$619$	13110.4	0.851294	0.425647	+	0.904889i	$0.360047\pi$
$620$	0	0
$621$	1802.22	0.116458
$622$	1782.22	0.114888
$623$	16977.9	1.09182
$624$	−6258.87	−0.401531
$625$	0	0
$626$	−2122.72	−0.135529
$627$	1615.90	0.102923
$628$	−26336.1	−1.67345
$629$	2693.12	0.170718
$630$	0	0
$631$	30199.9	1.90529	0.952647	−	0.304078i	$-0.0983482\pi$
$631$	30199.9	1.90529	0.952647	+	0.304078i	$0.0983482\pi$
$632$	−418.088	−0.0263143
$633$	−10629.4	−0.667427
$634$	1585.31	0.0993072
$635$	0	0
$636$	−15839.6	−0.987546
$637$	25924.0	1.61247
$638$	199.261	0.0123649
$639$	−4710.85	−0.291641
$640$	0	0
$641$	−8889.43	−0.547756	−0.273878	−	0.961764i	$-0.588306\pi$
$641$	−8889.43	−0.547756	−0.273878	+	0.961764i	$0.588306\pi$
$642$	286.730	0.0176267
$643$	16570.2	1.01627	0.508137	−	0.861276i	$-0.330334\pi$
$643$	16570.2	1.01627	0.508137	+	0.861276i	$0.330334\pi$
$644$	17678.8	1.08175
$645$	0	0
$646$	943.608	0.0574702
$647$	13356.3	0.811576	0.405788	−	0.913967i	$-0.366997\pi$
$647$	13356.3	0.811576	0.405788	+	0.913967i	$0.366997\pi$
$648$	349.382	0.0211806
$649$	9646.69	0.583460
$650$	0	0
$651$	−14610.8	−0.879638
$652$	19406.9	1.16569
$653$	−32689.5	−1.95902	−0.979510	−	0.201394i	$-0.935453\pi$
$653$	−32689.5	−1.95902	−0.979510	+	0.201394i	$0.935453\pi$
$654$	1148.27	0.0686558
$655$	0	0
$656$	−21414.6	−1.27454
$657$	−819.876	−0.0486855
$658$	−2451.31	−0.145231
$659$	16123.3	0.953071	0.476536	−	0.879155i	$-0.341892\pi$
$659$	16123.3	0.953071	0.476536	+	0.879155i	$0.341892\pi$
$660$	0	0
$661$	22340.9	1.31461	0.657306	−	0.753624i	$-0.271696\pi$
$661$	22340.9	1.31461	0.657306	+	0.753624i	$0.271696\pi$
$662$	−1448.27	−0.0850283
$663$	−7154.69	−0.419103
$664$	6020.08	0.351844
$665$	0	0
$666$	−92.2548	−0.00536757
$667$	4464.60	0.259175
$668$	−12038.3	−0.697268
$669$	−2552.79	−0.147528
$670$	0	0
$671$	8618.75	0.495862
$672$	5148.77	0.295562
$673$	3415.65	0.195637	0.0978184	−	0.995204i	$-0.468814\pi$
$673$	3415.65	0.195637	0.0978184	+	0.995204i	$0.468814\pi$
$674$	748.562	0.0427797
$675$	0	0
$676$	8509.94	0.484180
$677$	−32841.9	−1.86442	−0.932212	−	0.361912i	$-0.882124\pi$
$677$	−32841.9	−1.86442	−0.932212	+	0.361912i	$0.882124\pi$
$678$	−84.2972	−0.00477495
$679$	21520.9	1.21634
$680$	0	0
$681$	15121.7	0.850902
$682$	−434.228	−0.0243804
$683$	7227.43	0.404905	0.202452	−	0.979292i	$-0.435109\pi$
$683$	7227.43	0.404905	0.202452	+	0.979292i	$0.435109\pi$
$684$	3493.28	0.195276
$685$	0	0
$686$	−3895.21	−0.216793
$687$	10873.0	0.603831
$688$	−2146.39	−0.118940
$689$	−22325.5	−1.23445
$690$	0	0
$691$	15397.5	0.847682	0.423841	−	0.905737i	$-0.360681\pi$
$691$	15397.5	0.847682	0.423841	+	0.905737i	$0.360681\pi$
$692$	20901.0	1.14817
$693$	3307.92	0.181324
$694$	1364.60	0.0746392
$695$	0	0
$696$	865.515	0.0471369
$697$	−24479.6	−1.33032
$698$	2012.10	0.109111
$699$	−8953.68	−0.484491
$700$	0	0
$701$	−22130.1	−1.19236	−0.596180	−	0.802851i	$-0.703315\pi$
$701$	−22130.1	−1.19236	−0.596180	+	0.802851i	$0.703315\pi$
$702$	245.089	0.0131771
$703$	−1853.35	−0.0994315
$704$	−5324.55	−0.285052
$705$	0	0
$706$	−853.601	−0.0455038
$707$	49626.7	2.63989
$708$	20854.3	1.10700
$709$	10062.6	0.533018	0.266509	−	0.963832i	$-0.414130\pi$
$709$	10062.6	0.533018	0.266509	+	0.963832i	$0.414130\pi$
$710$	0	0
$711$	−872.359	−0.0460141
$712$	2191.69	0.115361
$713$	−9729.25	−0.511029
$714$	1931.66	0.101247
$715$	0	0
$716$	14056.4	0.733677
$717$	−3894.34	−0.202841
$718$	−244.814	−0.0127248
$719$	−18629.7	−0.966303	−0.483151	−	0.875537i	$-0.660508\pi$
$719$	−18629.7	−0.966303	−0.483151	+	0.875537i	$0.660508\pi$
$720$	0	0
$721$	−20246.9	−1.04581
$722$	1208.23	0.0622791
$723$	−11936.1	−0.613983
$724$	−4199.15	−0.215553
$725$	0	0
$726$	98.3099	0.00502565
$727$	21605.8	1.10222	0.551112	−	0.834431i	$-0.314204\pi$
$727$	21605.8	1.10222	0.551112	+	0.834431i	$0.314204\pi$
$728$	4830.64	0.245928
$729$	729.000	0.0370370
$730$	0	0
$731$	−2453.60	−0.124145
$732$	18632.1	0.940798
$733$	−23668.1	−1.19264	−0.596318	−	0.802749i	$-0.703370\pi$
$733$	−23668.1	−1.19264	−0.596318	+	0.802749i	$0.703370\pi$
$734$	1077.43	0.0541808
$735$	0	0
$736$	3428.52	0.171708
$737$	−9643.96	−0.482008
$738$	838.567	0.0418267
$739$	−8475.82	−0.421906	−0.210953	−	0.977496i	$-0.567657\pi$
$739$	−8475.82	−0.421906	−0.210953	+	0.977496i	$0.567657\pi$
$740$	0	0
$741$	4923.71	0.244098
$742$	6027.57	0.298220
$743$	−14491.2	−0.715520	−0.357760	−	0.933813i	$-0.616459\pi$
$743$	−14491.2	−0.715520	−0.357760	+	0.933813i	$0.616459\pi$
$744$	−1886.13	−0.0929420
$745$	0	0
$746$	−1487.83	−0.0730206
$747$	12561.1	0.615245
$748$	−6204.15	−0.303270
$749$	11791.8	0.575252
$750$	0	0
$751$	11780.5	0.572408	0.286204	−	0.958169i	$-0.407607\pi$
$751$	11780.5	0.572408	0.286204	+	0.958169i	$0.407607\pi$
$752$	16861.4	0.817648
$753$	5652.14	0.273540
$754$	607.154	0.0293252
$755$	0	0
$756$	7151.10	0.344025
$757$	−22616.7	−1.08589	−0.542945	−	0.839768i	$-0.682691\pi$
$757$	−22616.7	−1.08589	−0.542945	+	0.839768i	$0.682691\pi$
$758$	−822.210	−0.0393984
$759$	2202.72	0.105341
$760$	0	0
$761$	2037.48	0.0970547	0.0485273	−	0.998822i	$-0.484547\pi$
$761$	2037.48	0.0970547	0.0485273	+	0.998822i	$0.484547\pi$
$762$	1281.00	0.0608999
$763$	47222.7	2.24060
$764$	−16344.2	−0.773968
$765$	0	0
$766$	−963.268	−0.0454364
$767$	29393.8	1.38377
$768$	−11176.8	−0.525142
$769$	16192.1	0.759298	0.379649	−	0.925131i	$-0.376045\pi$
$769$	16192.1	0.759298	0.379649	+	0.925131i	$0.376045\pi$
$770$	0	0
$771$	−12036.7	−0.562245
$772$	−12977.8	−0.605029
$773$	−39726.3	−1.84845	−0.924227	−	0.381844i	$-0.875289\pi$
$773$	−39726.3	−1.84845	−0.924227	+	0.381844i	$0.875289\pi$
$774$	84.0500	0.00390325
$775$	0	0
$776$	2778.16	0.128518
$777$	−3793.99	−0.175172
$778$	1411.43	0.0650413
$779$	16846.3	0.774817
$780$	0	0
$781$	−5757.71	−0.263799
$782$	1286.28	0.0588200
$783$	1805.93	0.0824250
$784$	48143.3	2.19312
$785$	0	0
$786$	−881.669	−0.0400103
$787$	5160.53	0.233739	0.116870	−	0.993147i	$-0.462714\pi$
$787$	5160.53	0.233739	0.116870	+	0.993147i	$0.462714\pi$
$788$	27068.4	1.22370
$789$	871.286	0.0393138
$790$	0	0
$791$	−3466.73	−0.155832
$792$	427.022	0.0191586
$793$	26261.6	1.17601
$794$	−1773.17	−0.0792539
$795$	0	0
$796$	−31222.5	−1.39027
$797$	−9247.76	−0.411007	−0.205504	−	0.978656i	$-0.565883\pi$
$797$	−9247.76	−0.411007	−0.205504	+	0.978656i	$0.565883\pi$
$798$	−1329.33	−0.0589696
$799$	19274.7	0.853430
$800$	0	0
$801$	4573.05	0.201724
$802$	450.002	0.0198131
$803$	−1002.07	−0.0440377
$804$	−20848.5	−0.914513
$805$	0	0
$806$	−1323.11	−0.0578220
$807$	20229.5	0.882417
$808$	6406.36	0.278930
$809$	29359.7	1.27594	0.637968	−	0.770063i	$-0.279775\pi$
$809$	29359.7	1.27594	0.637968	+	0.770063i	$0.279775\pi$
$810$	0	0
$811$	13935.0	0.603358	0.301679	−	0.953410i	$-0.402453\pi$
$811$	13935.0	0.603358	0.301679	+	0.953410i	$0.402453\pi$
$812$	17715.2	0.765620
$813$	−21546.3	−0.929475
$814$	−112.756	−0.00485515
$815$	0	0
$816$	−13287.0	−0.570020
$817$	1688.52	0.0723057
$818$	−1971.33	−0.0842617
$819$	10079.3	0.430037
$820$	0	0
$821$	9965.28	0.423618	0.211809	−	0.977311i	$-0.432064\pi$
$821$	9965.28	0.423618	0.211809	+	0.977311i	$0.432064\pi$
$822$	−1128.91	−0.0479018
$823$	2339.58	0.0990920	0.0495460	−	0.998772i	$-0.484223\pi$
$823$	2339.58	0.0990920	0.0495460	+	0.998772i	$0.484223\pi$
$824$	−2613.69	−0.110500
$825$	0	0
$826$	−7935.89	−0.334292
$827$	15883.4	0.667861	0.333931	−	0.942598i	$-0.391625\pi$
$827$	15883.4	0.667861	0.333931	+	0.942598i	$0.391625\pi$
$828$	4761.87	0.199863
$829$	4388.04	0.183839	0.0919196	−	0.995766i	$-0.470700\pi$
$829$	4388.04	0.183839	0.0919196	+	0.995766i	$0.470700\pi$
$830$	0	0
$831$	−17125.0	−0.714875
$832$	−16224.1	−0.676044
$833$	55034.0	2.28909
$834$	406.293	0.0168690
$835$	0	0
$836$	4269.57	0.176634
$837$	−3935.49	−0.162521
$838$	1595.75	0.0657807
$839$	33830.2	1.39207	0.696036	−	0.718007i	$-0.254945\pi$
$839$	33830.2	1.39207	0.696036	+	0.718007i	$0.254945\pi$
$840$	0	0
$841$	−19915.2	−0.816565
$842$	−3002.45	−0.122888
$843$	−26222.8	−1.07137
$844$	−28085.2	−1.14542
$845$	0	0
$846$	−660.270	−0.0268328
$847$	4043.01	0.164013
$848$	−41460.7	−1.67897
$849$	−23650.7	−0.956054
$850$	0	0
$851$	−2526.39	−0.101767
$852$	−12447.1	−0.500505
$853$	−26047.8	−1.04556	−0.522778	−	0.852469i	$-0.675104\pi$
$853$	−26047.8	−1.04556	−0.522778	+	0.852469i	$0.675104\pi$
$854$	−7090.25	−0.284102
$855$	0	0
$856$	1522.22	0.0607808
$857$	−17210.7	−0.686006	−0.343003	−	0.939334i	$-0.611444\pi$
$857$	−17210.7	−0.686006	−0.343003	+	0.939334i	$0.611444\pi$
$858$	299.554	0.0119191
$859$	27367.9	1.08706	0.543528	−	0.839391i	$-0.317088\pi$
$859$	27367.9	1.08706	0.543528	+	0.839391i	$0.317088\pi$
$860$	0	0
$861$	34486.2	1.36502
$862$	−1532.35	−0.0605476
$863$	−34684.4	−1.36810	−0.684049	−	0.729436i	$-0.739783\pi$
$863$	−34684.4	−1.36810	−0.684049	+	0.729436i	$0.739783\pi$
$864$	1386.84	0.0546079
$865$	0	0
$866$	830.092	0.0325724
$867$	−449.690	−0.0176151
$868$	−38605.0	−1.50961
$869$	−1066.22	−0.0416213
$870$	0	0
$871$	−29385.5	−1.14316
$872$	6096.03	0.236740
$873$	5796.74	0.224731
$874$	−885.190	−0.0342586
$875$	0	0
$876$	−2166.29	−0.0835527
$877$	−45816.6	−1.76410	−0.882051	−	0.471153i	$-0.843838\pi$
$877$	−45816.6	−1.76410	−0.882051	+	0.471153i	$0.843838\pi$
$878$	817.217	0.0314120
$879$	6985.61	0.268053
$880$	0	0
$881$	−26404.8	−1.00976	−0.504881	−	0.863189i	$-0.668464\pi$
$881$	−26404.8	−1.00976	−0.504881	+	0.863189i	$0.668464\pi$
$882$	−1885.23	−0.0719717
$883$	−33977.1	−1.29493	−0.647463	−	0.762097i	$-0.724170\pi$
$883$	−33977.1	−1.29493	−0.647463	+	0.762097i	$0.724170\pi$
$884$	−18904.2	−0.719252
$885$	0	0
$886$	4630.66	0.175587
$887$	3023.60	0.114456	0.0572280	−	0.998361i	$-0.481774\pi$
$887$	3023.60	0.114456	0.0572280	+	0.998361i	$0.481774\pi$
$888$	−489.770	−0.0185086
$889$	52681.2	1.98748
$890$	0	0
$891$	891.000	0.0335013
$892$	−6745.03	−0.253184
$893$	−13264.5	−0.497064
$894$	557.108	0.0208417
$895$	0	0
$896$	18110.3	0.675249
$897$	6711.75	0.249832
$898$	2931.27	0.108929
$899$	−9749.28	−0.361687
$900$	0	0
$901$	−47394.9	−1.75244
$902$	1024.92	0.0378336
$903$	3456.57	0.127384
$904$	−447.524	−0.0164651
$905$	0	0
$906$	−982.257	−0.0360191
$907$	−23951.0	−0.876825	−0.438412	−	0.898774i	$-0.644459\pi$
$907$	−23951.0	−0.876825	−0.438412	+	0.898774i	$0.644459\pi$
$908$	39954.8	1.46029
$909$	13367.1	0.487745
$910$	0	0
$911$	24198.9	0.880071	0.440035	−	0.897980i	$-0.354966\pi$
$911$	24198.9	0.880071	0.440035	+	0.897980i	$0.354966\pi$
$912$	9143.81	0.331998
$913$	15352.5	0.556510
$914$	−1857.60	−0.0672253
$915$	0	0
$916$	28728.9	1.03628
$917$	−36258.8	−1.30575
$918$	520.300	0.0187064
$919$	11056.9	0.396880	0.198440	−	0.980113i	$-0.436413\pi$
$919$	11056.9	0.396880	0.198440	+	0.980113i	$0.436413\pi$
$920$	0	0
$921$	8490.92	0.303784
$922$	−2613.94	−0.0933682
$923$	−17543.9	−0.625639
$924$	8740.24	0.311182
$925$	0	0
$926$	−1284.93	−0.0455999
$927$	−5453.57	−0.193224
$928$	3435.58	0.121529
$929$	−48242.5	−1.70375	−0.851876	−	0.523744i	$-0.824535\pi$
$929$	−48242.5	−1.70375	−0.851876	+	0.523744i	$0.824535\pi$
$930$	0	0
$931$	−37873.2	−1.33324
$932$	−23657.6	−0.831470
$933$	19742.0	0.692738
$934$	−2989.42	−0.104729
$935$	0	0
$936$	1301.15	0.0454375
$937$	48721.9	1.69869	0.849346	−	0.527836i	$-0.176996\pi$
$937$	48721.9	1.69869	0.849346	+	0.527836i	$0.176996\pi$
$938$	7933.65	0.276165
$939$	−23513.9	−0.817195
$940$	0	0
$941$	−4978.66	−0.172476	−0.0862379	−	0.996275i	$-0.527485\pi$
$941$	−4978.66	−0.172476	−0.0862379	+	0.996275i	$0.527485\pi$
$942$	2699.44	0.0933678
$943$	22964.1	0.793015
$944$	54587.1	1.88205
$945$	0	0
$946$	102.728	0.00353062
$947$	−27711.4	−0.950896	−0.475448	−	0.879744i	$-0.657714\pi$
$947$	−27711.4	−0.950896	−0.475448	+	0.879744i	$0.657714\pi$
$948$	−2304.96	−0.0789680
$949$	−3053.34	−0.104442
$950$	0	0
$951$	17560.8	0.598790
$952$	10255.0	0.349123
$953$	49175.6	1.67152	0.835758	−	0.549098i	$-0.185029\pi$
$953$	49175.6	1.67152	0.835758	+	0.549098i	$0.185029\pi$
$954$	1623.55	0.0550989
$955$	0	0
$956$	−10289.7	−0.348109
$957$	2207.25	0.0745562
$958$	−4733.21	−0.159627
$959$	−46426.6	−1.56329
$960$	0	0
$961$	−8545.37	−0.286844
$962$	−343.571	−0.0115147
$963$	3176.17	0.106283
$964$	−31537.9	−1.05370
$965$	0	0
$966$	−1812.07	−0.0603546
$967$	52678.5	1.75184	0.875918	−	0.482461i	$-0.160257\pi$
$967$	52678.5	1.75184	0.875918	+	0.482461i	$0.160257\pi$
$968$	521.916	0.0173296
$969$	10452.5	0.346526
$970$	0	0
$971$	46877.5	1.54930	0.774651	−	0.632390i	$-0.217926\pi$
$971$	46877.5	1.54930	0.774651	+	0.632390i	$0.217926\pi$
$972$	1926.18	0.0635619
$973$	16708.8	0.550525
$974$	4608.00	0.151591
$975$	0	0
$976$	48770.4	1.59949
$977$	−1335.33	−0.0437267	−0.0218633	−	0.999761i	$-0.506960\pi$
$977$	−1335.33	−0.0437267	−0.0218633	+	0.999761i	$0.506960\pi$
$978$	−1989.20	−0.0650384
$979$	5589.29	0.182466
$980$	0	0
$981$	12719.6	0.413972
$982$	−5328.19	−0.173146
$983$	−15102.0	−0.490011	−0.245005	−	0.969522i	$-0.578790\pi$
$983$	−15102.0	−0.490011	−0.245005	+	0.969522i	$0.578790\pi$
$984$	4451.85	0.144228
$985$	0	0
$986$	1288.93	0.0416306
$987$	−27153.7	−0.875696
$988$	13009.5	0.418915
$989$	2301.70	0.0740039
$990$	0	0
$991$	−45849.8	−1.46969	−0.734847	−	0.678233i	$-0.762746\pi$
$991$	−45849.8	−1.46969	−0.734847	+	0.678233i	$0.762746\pi$
$992$	−7486.82	−0.239624
$993$	−16042.8	−0.512693
$994$	4736.60	0.151143
$995$	0	0
$996$	33189.3	1.05587
$997$	−49662.4	−1.57756	−0.788779	−	0.614678i	$-0.789286\pi$
$997$	−49662.4	−1.57756	−0.788779	+	0.614678i	$0.789286\pi$
$998$	4266.77	0.135333
$999$	−1021.93	−0.0323647

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.4.a.w.1.3	✓	5
3.2	odd	2		2475.4.a.bm.1.3		5
5.2	odd	4		825.4.c.r.199.5		10
5.3	odd	4		825.4.c.r.199.6		10
5.4	even	2		825.4.a.z.1.3	yes	5
15.14	odd	2		2475.4.a.bf.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.4.a.w.1.3	✓	5	1.1	even	1	trivial
825.4.a.z.1.3	yes	5	5.4	even	2
825.4.c.r.199.5		10	5.2	odd	4
825.4.c.r.199.6		10	5.3	odd	4
2475.4.a.bf.1.3		5	15.14	odd	2
2475.4.a.bm.1.3		5	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:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)

\(f(q)\)	\(=\)	\(q-0.270826 q^{2} -3.00000 q^{3} -7.92665 q^{4} +0.812479 q^{6} +33.4133 q^{7} +4.31336 q^{8} +9.00000 q^{9} +11.0000 q^{11} +23.7800 q^{12} +33.5174 q^{13} -9.04920 q^{14} +62.2451 q^{16} +71.1540 q^{17} -2.43744 q^{18} -48.9667 q^{19} -100.240 q^{21} -2.97909 q^{22} -66.7490 q^{23} -12.9401 q^{24} -9.07738 q^{26} -27.0000 q^{27} -264.856 q^{28} -66.8864 q^{29} +145.759 q^{31} -51.3644 q^{32} -33.0000 q^{33} -19.2704 q^{34} -71.3399 q^{36} +37.8491 q^{37} +13.2615 q^{38} -100.552 q^{39} -344.036 q^{41} +27.1476 q^{42} -34.4830 q^{43} -87.1932 q^{44} +18.0774 q^{46} +270.887 q^{47} -186.735 q^{48} +773.448 q^{49} -213.462 q^{51} -265.681 q^{52} -666.088 q^{53} +7.31231 q^{54} +144.123 q^{56} +146.900 q^{57} +18.1146 q^{58} +876.971 q^{59} +783.523 q^{61} -39.4753 q^{62} +300.720 q^{63} -484.050 q^{64} +8.93727 q^{66} -876.724 q^{67} -564.013 q^{68} +200.247 q^{69} -523.428 q^{71} +38.8202 q^{72} -91.0973 q^{73} -10.2505 q^{74} +388.142 q^{76} +367.546 q^{77} +27.2322 q^{78} -96.9287 q^{79} +81.0000 q^{81} +93.1741 q^{82} +1395.68 q^{83} +794.567 q^{84} +9.33889 q^{86} +200.659 q^{87} +47.4469 q^{88} +508.117 q^{89} +1119.93 q^{91} +529.096 q^{92} -437.276 q^{93} -73.3634 q^{94} +154.093 q^{96} +644.082 q^{97} -209.470 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{2} - 15 q^{3} + 46 q^{4} + 12 q^{6} + 38 q^{7} - 72 q^{8} + 45 q^{9} + 55 q^{11} - 138 q^{12} - 43 q^{13} + 55 q^{14} + 634 q^{16} - 150 q^{17} - 36 q^{18} - 337 q^{19} - 114 q^{21} - 44 q^{22}+ \cdots + 495 q^{99}+O(q^{100}) \) 5 * q - 4 * q^2 - 15 * q^3 + 46 * q^4 + 12 * q^6 + 38 * q^7 - 72 * q^8 + 45 * q^9 + 55 * q^11 - 138 * q^12 - 43 * q^13 + 55 * q^14 + 634 * q^16 - 150 * q^17 - 36 * q^18 - 337 * q^19 - 114 * q^21 - 44 * q^22 + 201 * q^23 + 216 * q^24 + 359 * q^26 - 135 * q^27 + 441 * q^28 - 153 * q^29 + 263 * q^31 - 280 * q^32 - 165 * q^33 + 517 * q^34 + 414 * q^36 + 350 * q^37 - 786 * q^38 + 129 * q^39 - 88 * q^41 - 165 * q^42 - 161 * q^43 + 506 * q^44 + 436 * q^46 - 490 * q^47 - 1902 * q^48 + 2381 * q^49 + 450 * q^51 - 289 * q^52 - 558 * q^53 + 108 * q^54 + 211 * q^56 + 1011 * q^57 + 969 * q^58 + 410 * q^59 - 1130 * q^61 - 463 * q^62 + 342 * q^63 + 3170 * q^64 + 132 * q^66 - 1130 * q^67 - 3333 * q^68 - 603 * q^69 + 1249 * q^71 - 648 * q^72 + 44 * q^73 - 2985 * q^74 - 2556 * q^76 + 418 * q^77 - 1077 * q^78 + 2258 * q^79 + 405 * q^81 + 3737 * q^82 + 957 * q^83 - 1323 * q^84 - 1435 * q^86 + 459 * q^87 - 792 * q^88 + 347 * q^89 - 612 * q^91 + 6006 * q^92 - 789 * q^93 + 907 * q^94 + 840 * q^96 + 2355 * q^97 - 9417 * q^98 + 495 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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