LMFDB - Embedded newform 850.4.a.q.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [850,4,Mod(1,850)] 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("850.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(850, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 850 = 2 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	850.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-10,1,20,0,-2,-53] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$50.1516235049$
Analytic rank:	$1$
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} - 126x^{3} + 88x^{2} + 2675x + 213 $ x^5 - x^4 - 126x^3 + 88x^2 + 2675*x + 213
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.0798599$ of defining polynomial
Character	$\chi$	$=$	850.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} -0.0798599 q^{3} +4.00000 q^{4} +0.159720 q^{6} -2.42339 q^{7} -8.00000 q^{8} -26.9936 q^{9} +26.6650 q^{11} -0.319440 q^{12} +16.1070 q^{13} +4.84677 q^{14} +16.0000 q^{16} +17.0000 q^{17} +53.9872 q^{18} +3.43476 q^{19} +0.193531 q^{21} -53.3301 q^{22} -164.232 q^{23} +0.638880 q^{24} -32.2140 q^{26} +4.31193 q^{27} -9.69354 q^{28} +220.009 q^{29} +251.143 q^{31} -32.0000 q^{32} -2.12947 q^{33} -34.0000 q^{34} -107.974 q^{36} +58.4324 q^{37} -6.86952 q^{38} -1.28631 q^{39} -250.581 q^{41} -0.387063 q^{42} -394.934 q^{43} +106.660 q^{44} +328.464 q^{46} -59.4596 q^{47} -1.27776 q^{48} -337.127 q^{49} -1.35762 q^{51} +64.4281 q^{52} -238.378 q^{53} -8.62386 q^{54} +19.3871 q^{56} -0.274300 q^{57} -440.017 q^{58} -397.099 q^{59} +418.044 q^{61} -502.285 q^{62} +65.4160 q^{63} +64.0000 q^{64} +4.25894 q^{66} -556.567 q^{67} +68.0000 q^{68} +13.1155 q^{69} +959.142 q^{71} +215.949 q^{72} +22.1159 q^{73} -116.865 q^{74} +13.7390 q^{76} -64.6197 q^{77} +2.57261 q^{78} +711.293 q^{79} +728.483 q^{81} +501.163 q^{82} +962.217 q^{83} +0.774126 q^{84} +789.867 q^{86} -17.5699 q^{87} -213.320 q^{88} -1326.07 q^{89} -39.0335 q^{91} -656.927 q^{92} -20.0562 q^{93} +118.919 q^{94} +2.55552 q^{96} -202.787 q^{97} +674.254 q^{98} -719.786 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{2} + q^{3} + 20 q^{4} - 2 q^{6} - 53 q^{7} - 40 q^{8} + 118 q^{9} - 30 q^{11} + 4 q^{12} - 123 q^{13} + 106 q^{14} + 80 q^{16} + 85 q^{17} - 236 q^{18} + 18 q^{19} + 119 q^{21} + 60 q^{22}+ \cdots - 4118 q^{99}+O(q^{100}) $ 5 * q - 10 * q^2 + q^3 + 20 * q^4 - 2 * q^6 - 53 * q^7 - 40 * q^8 + 118 * q^9 - 30 * q^11 + 4 * q^12 - 123 * q^13 + 106 * q^14 + 80 * q^16 + 85 * q^17 - 236 * q^18 + 18 * q^19 + 119 * q^21 + 60 * q^22 - 226 * q^23 - 8 * q^24 + 246 * q^26 + 61 * q^27 - 212 * q^28 + 86 * q^29 - 7 * q^31 - 160 * q^32 - 132 * q^33 - 170 * q^34 + 472 * q^36 - 186 * q^37 - 36 * q^38 - 443 * q^39 + 636 * q^41 - 238 * q^42 - 432 * q^43 - 120 * q^44 + 452 * q^46 - 18 * q^47 + 16 * q^48 - 378 * q^49 + 17 * q^51 - 492 * q^52 - 299 * q^53 - 122 * q^54 + 424 * q^56 - 692 * q^57 - 172 * q^58 + 1602 * q^61 + 14 * q^62 - 2382 * q^63 + 320 * q^64 + 264 * q^66 + 448 * q^67 + 340 * q^68 - 1200 * q^69 + 447 * q^71 - 944 * q^72 - 602 * q^73 + 372 * q^74 + 72 * q^76 - 1396 * q^77 + 886 * q^78 - 1697 * q^79 + 4357 * q^81 - 1272 * q^82 - 2988 * q^83 + 476 * q^84 + 864 * q^86 - 1636 * q^87 + 240 * q^88 - 86 * q^89 + 3237 * q^91 - 904 * q^92 - 2331 * q^93 + 36 * q^94 - 32 * q^96 + 664 * q^97 + 756 * q^98 - 4118 * 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
$3$	−0.0798599	−0.0153691	−0.00768453	−	0.999970i	$-0.502446\pi$
$3$	−0.0798599	−0.0153691	−0.00768453	+	0.999970i	$0.502446\pi$
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0.159720	0.0108676
$7$	−2.42339	−0.130851	−0.0654253	−	0.997857i	$-0.520840\pi$
$7$	−2.42339	−0.130851	−0.0654253	+	0.997857i	$0.520840\pi$
$8$	−8.00000	−0.353553
$9$	−26.9936	−0.999764
$10$	0	0
$11$	26.6650	0.730892	0.365446	−	0.930833i	$-0.380916\pi$
$11$	26.6650	0.730892	0.365446	+	0.930833i	$0.380916\pi$
$12$	−0.319440	−0.00768453
$13$	16.1070	0.343637	0.171819	−	0.985129i	$-0.445036\pi$
$13$	16.1070	0.343637	0.171819	+	0.985129i	$0.445036\pi$
$14$	4.84677	0.0925253
$15$	0	0
$16$	16.0000	0.250000
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	53.9872	0.706940
$19$	3.43476	0.0414730	0.0207365	−	0.999785i	$-0.493399\pi$
$19$	3.43476	0.0414730	0.0207365	+	0.999785i	$0.493399\pi$
$20$	0	0
$21$	0.193531	0.00201105
$22$	−53.3301	−0.516819
$23$	−164.232	−1.48890	−0.744450	−	0.667678i	$-0.767288\pi$
$23$	−164.232	−1.48890	−0.744450	+	0.667678i	$0.767288\pi$
$24$	0.638880	0.00543378
$25$	0	0
$26$	−32.2140	−0.242988
$27$	4.31193	0.0307345
$28$	−9.69354	−0.0654253
$29$	220.009	1.40878	0.704389	−	0.709814i	$-0.251221\pi$
$29$	220.009	1.40878	0.704389	+	0.709814i	$0.251221\pi$
$30$	0	0
$31$	251.143	1.45505	0.727525	−	0.686081i	$-0.240671\pi$
$31$	251.143	1.45505	0.727525	+	0.686081i	$0.240671\pi$
$32$	−32.0000	−0.176777
$33$	−2.12947	−0.0112331
$34$	−34.0000	−0.171499
$35$	0	0
$36$	−107.974	−0.499882
$37$	58.4324	0.259628	0.129814	−	0.991538i	$-0.458562\pi$
$37$	58.4324	0.259628	0.129814	+	0.991538i	$0.458562\pi$
$38$	−6.86952	−0.0293259
$39$	−1.28631	−0.00528138
$40$	0	0
$41$	−250.581	−0.954494	−0.477247	−	0.878769i	$-0.658365\pi$
$41$	−250.581	−0.954494	−0.477247	+	0.878769i	$0.658365\pi$
$42$	−0.387063	−0.00142203
$43$	−394.934	−1.40062	−0.700312	−	0.713837i	$-0.746956\pi$
$43$	−394.934	−1.40062	−0.700312	+	0.713837i	$0.746956\pi$
$44$	106.660	0.365446
$45$	0	0
$46$	328.464	1.05281
$47$	−59.4596	−0.184533	−0.0922667	−	0.995734i	$-0.529411\pi$
$47$	−59.4596	−0.184533	−0.0922667	+	0.995734i	$0.529411\pi$
$48$	−1.27776	−0.00384226
$49$	−337.127	−0.982878
$50$	0	0
$51$	−1.35762	−0.00372754
$52$	64.4281	0.171819
$53$	−238.378	−0.617806	−0.308903	−	0.951094i	$-0.599962\pi$
$53$	−238.378	−0.617806	−0.308903	+	0.951094i	$0.599962\pi$
$54$	−8.62386	−0.0217326
$55$	0	0
$56$	19.3871	0.0462627
$57$	−0.274300	−0.000637401 0
$58$	−440.017	−0.996157
$59$	−397.099	−0.876235	−0.438118	−	0.898918i	$-0.644355\pi$
$59$	−397.099	−0.876235	−0.438118	+	0.898918i	$0.644355\pi$
$60$	0	0
$61$	418.044	0.877460	0.438730	−	0.898619i	$-0.355428\pi$
$61$	418.044	0.877460	0.438730	+	0.898619i	$0.355428\pi$
$62$	−502.285	−1.02888
$63$	65.4160	0.130820
$64$	64.0000	0.125000
$65$	0	0
$66$	4.25894	0.00794302
$67$	−556.567	−1.01486	−0.507429	−	0.861694i	$-0.669404\pi$
$67$	−556.567	−1.01486	−0.507429	+	0.861694i	$0.669404\pi$
$68$	68.0000	0.121268
$69$	13.1155	0.0228830
$70$	0	0
$71$	959.142	1.60323	0.801615	−	0.597841i	$-0.203975\pi$
$71$	959.142	1.60323	0.801615	+	0.597841i	$0.203975\pi$
$72$	215.949	0.353470
$73$	22.1159	0.0354586	0.0177293	−	0.999843i	$-0.494356\pi$
$73$	22.1159	0.0354586	0.0177293	+	0.999843i	$0.494356\pi$
$74$	−116.865	−0.183585
$75$	0	0
$76$	13.7390	0.0207365
$77$	−64.6197	−0.0956376
$78$	2.57261	0.00373450
$79$	711.293	1.01300	0.506498	−	0.862241i	$-0.330940\pi$
$79$	711.293	1.01300	0.506498	+	0.862241i	$0.330940\pi$
$80$	0	0
$81$	728.483	0.999291
$82$	501.163	0.674929
$83$	962.217	1.27249	0.636247	−	0.771485i	$-0.280486\pi$
$83$	962.217	1.27249	0.636247	+	0.771485i	$0.280486\pi$
$84$	0.774126	0.00100552
$85$	0	0
$86$	789.867	0.990391
$87$	−17.5699	−0.0216516
$88$	−213.320	−0.258409
$89$	−1326.07	−1.57936	−0.789678	−	0.613522i	$-0.789752\pi$
$89$	−1326.07	−1.57936	−0.789678	+	0.613522i	$0.789752\pi$
$90$	0	0
$91$	−39.0335	−0.0449651
$92$	−656.927	−0.744450
$93$	−20.0562	−0.0223627
$94$	118.919	0.130485
$95$	0	0
$96$	2.55552	0.00271689
$97$	−202.787	−0.212268	−0.106134	−	0.994352i	$-0.533847\pi$
$97$	−202.787	−0.212268	−0.106134	+	0.994352i	$0.533847\pi$
$98$	674.254	0.695000
$99$	−719.786	−0.730720
$100$	0	0
$101$	189.230	0.186427	0.0932133	−	0.995646i	$-0.470286\pi$
$101$	189.230	0.186427	0.0932133	+	0.995646i	$0.470286\pi$
$102$	2.71524	0.00263577
$103$	−1349.77	−1.29123	−0.645615	−	0.763663i	$-0.723399\pi$
$103$	−1349.77	−1.29123	−0.645615	+	0.763663i	$0.723399\pi$
$104$	−128.856	−0.121494
$105$	0	0
$106$	476.755	0.436855
$107$	−907.578	−0.819989	−0.409995	−	0.912088i	$-0.634469\pi$
$107$	−907.578	−0.819989	−0.409995	+	0.912088i	$0.634469\pi$
$108$	17.2477	0.0153672
$109$	−1211.59	−1.06468	−0.532338	−	0.846532i	$-0.678686\pi$
$109$	−1211.59	−1.06468	−0.532338	+	0.846532i	$0.678686\pi$
$110$	0	0
$111$	−4.66641	−0.00399024
$112$	−38.7742	−0.0327126
$113$	−1843.42	−1.53464	−0.767319	−	0.641266i	$-0.778409\pi$
$113$	−1843.42	−1.53464	−0.767319	+	0.641266i	$0.778409\pi$
$114$	0.548599	0.000450711 0
$115$	0	0
$116$	880.034	0.704389
$117$	−434.787	−0.343556
$118$	794.198	0.619592
$119$	−41.1976	−0.0317359
$120$	0	0
$121$	−619.975	−0.465797
$122$	−836.088	−0.620458
$123$	20.0114	0.0146697
$124$	1004.57	0.727525
$125$	0	0
$126$	−130.832	−0.0925034
$127$	−556.175	−0.388603	−0.194301	−	0.980942i	$-0.562244\pi$
$127$	−556.175	−0.388603	−0.194301	+	0.980942i	$0.562244\pi$
$128$	−128.000	−0.0883883
$129$	31.5394	0.0215263
$130$	0	0
$131$	−842.465	−0.561882	−0.280941	−	0.959725i	$-0.590646\pi$
$131$	−842.465	−0.561882	−0.280941	+	0.959725i	$0.590646\pi$
$132$	−8.51788	−0.00561656
$133$	−8.32375	−0.00542677
$134$	1113.13	0.717612
$135$	0	0
$136$	−136.000	−0.0857493
$137$	477.926	0.298043	0.149022	−	0.988834i	$-0.452388\pi$
$137$	477.926	0.298043	0.149022	+	0.988834i	$0.452388\pi$
$138$	−26.2311	−0.0161807
$139$	−521.247	−0.318069	−0.159035	−	0.987273i	$-0.550838\pi$
$139$	−521.247	−0.318069	−0.159035	+	0.987273i	$0.550838\pi$
$140$	0	0
$141$	4.74844	0.00283610
$142$	−1918.28	−1.13365
$143$	429.495	0.251162
$144$	−431.898	−0.249941
$145$	0	0
$146$	−44.2319	−0.0250730
$147$	26.9230	0.0151059
$148$	233.730	0.129814
$149$	−2629.16	−1.44557	−0.722783	−	0.691075i	$-0.757138\pi$
$149$	−2629.16	−1.44557	−0.722783	+	0.691075i	$0.757138\pi$
$150$	0	0
$151$	2775.41	1.49576	0.747880	−	0.663834i	$-0.231072\pi$
$151$	2775.41	1.49576	0.747880	+	0.663834i	$0.231072\pi$
$152$	−27.4781	−0.0146629
$153$	−458.892	−0.242478
$154$	129.239	0.0676260
$155$	0	0
$156$	−5.14522	−0.00264069
$157$	−1753.90	−0.891572	−0.445786	−	0.895140i	$-0.647076\pi$
$157$	−1753.90	−0.891572	−0.445786	+	0.895140i	$0.647076\pi$
$158$	−1422.59	−0.716297
$159$	19.0368	0.00949509
$160$	0	0
$161$	397.997	0.194823
$162$	−1456.97	−0.706606
$163$	−1640.51	−0.788309	−0.394155	−	0.919044i	$-0.628963\pi$
$163$	−1640.51	−0.788309	−0.394155	+	0.919044i	$0.628963\pi$
$164$	−1002.33	−0.477247
$165$	0	0
$166$	−1924.43	−0.899789
$167$	−1834.34	−0.849972	−0.424986	−	0.905200i	$-0.639721\pi$
$167$	−1834.34	−0.849972	−0.424986	+	0.905200i	$0.639721\pi$
$168$	−1.54825	−0.000711013 0
$169$	−1937.56	−0.881913
$170$	0	0
$171$	−92.7166	−0.0414632
$172$	−1579.73	−0.700312
$173$	−2509.39	−1.10280	−0.551402	−	0.834240i	$-0.685907\pi$
$173$	−2509.39	−1.10280	−0.551402	+	0.834240i	$0.685907\pi$
$174$	35.1397	0.0153100
$175$	0	0
$176$	426.641	0.182723
$177$	31.7123	0.0134669
$178$	2652.13	1.11677
$179$	−675.459	−0.282046	−0.141023	−	0.990006i	$-0.545039\pi$
$179$	−675.459	−0.282046	−0.141023	+	0.990006i	$0.545039\pi$
$180$	0	0
$181$	−1645.36	−0.675683	−0.337842	−	0.941203i	$-0.609697\pi$
$181$	−1645.36	−0.675683	−0.337842	+	0.941203i	$0.609697\pi$
$182$	78.0671	0.0317951
$183$	−33.3850	−0.0134857
$184$	1313.85	0.526406
$185$	0	0
$186$	40.1125	0.0158128
$187$	453.306	0.177267
$188$	−237.838	−0.0922667
$189$	−10.4495	−0.00402162
$190$	0	0
$191$	−2413.96	−0.914493	−0.457246	−	0.889340i	$-0.651164\pi$
$191$	−2413.96	−0.914493	−0.457246	+	0.889340i	$0.651164\pi$
$192$	−5.11104	−0.00192113
$193$	2718.50	1.01390	0.506948	−	0.861976i	$-0.330773\pi$
$193$	2718.50	1.01390	0.506948	+	0.861976i	$0.330773\pi$
$194$	405.575	0.150096
$195$	0	0
$196$	−1348.51	−0.491439
$197$	−1830.72	−0.662097	−0.331049	−	0.943614i	$-0.607402\pi$
$197$	−1830.72	−0.662097	−0.331049	+	0.943614i	$0.607402\pi$
$198$	1439.57	0.516697
$199$	−4012.92	−1.42949	−0.714745	−	0.699385i	$-0.753457\pi$
$199$	−4012.92	−1.42949	−0.714745	+	0.699385i	$0.753457\pi$
$200$	0	0
$201$	44.4474	0.0155974
$202$	−378.460	−0.131823
$203$	−533.166	−0.184339
$204$	−5.43048	−0.00186377
$205$	0	0
$206$	2699.54	0.913037
$207$	4433.21	1.48855
$208$	257.712	0.0859093
$209$	91.5880	0.0303123
$210$	0	0
$211$	−1889.94	−0.616628	−0.308314	−	0.951285i	$-0.599765\pi$
$211$	−1889.94	−0.616628	−0.308314	+	0.951285i	$0.599765\pi$
$212$	−953.511	−0.308903
$213$	−76.5971	−0.0246401
$214$	1815.16	0.579820
$215$	0	0
$216$	−34.4954	−0.0108663
$217$	−608.615	−0.190394
$218$	2423.19	0.752839
$219$	−1.76618	−0.000544965 0
$220$	0	0
$221$	273.819	0.0833443
$222$	9.33282	0.00282152
$223$	330.737	0.0993173	0.0496587	−	0.998766i	$-0.484187\pi$
$223$	330.737	0.0993173	0.0496587	+	0.998766i	$0.484187\pi$
$224$	77.5483	0.0231313
$225$	0	0
$226$	3686.83	1.08515
$227$	−3997.78	−1.16891	−0.584453	−	0.811427i	$-0.698691\pi$
$227$	−3997.78	−1.16891	−0.584453	+	0.811427i	$0.698691\pi$
$228$	−1.09720	−0.000318701 0
$229$	3900.22	1.12547	0.562737	−	0.826636i	$-0.309748\pi$
$229$	3900.22	1.12547	0.562737	+	0.826636i	$0.309748\pi$
$230$	0	0
$231$	5.16053	0.00146986
$232$	−1760.07	−0.498078
$233$	2859.91	0.804117	0.402058	−	0.915614i	$-0.368295\pi$
$233$	2859.91	0.804117	0.402058	+	0.915614i	$0.368295\pi$
$234$	869.574	0.242931
$235$	0	0
$236$	−1588.40	−0.438118
$237$	−56.8038	−0.0155688
$238$	82.3951	0.0224407
$239$	−1072.36	−0.290230	−0.145115	−	0.989415i	$-0.546355\pi$
$239$	−1072.36	−0.290230	−0.145115	+	0.989415i	$0.546355\pi$
$240$	0	0
$241$	4242.27	1.13389	0.566947	−	0.823754i	$-0.308124\pi$
$241$	4242.27	1.13389	0.566947	+	0.823754i	$0.308124\pi$
$242$	1239.95	0.329368
$243$	−174.599	−0.0460926
$244$	1672.18	0.438730
$245$	0	0
$246$	−40.0228	−0.0103730
$247$	55.3237	0.0142517
$248$	−2009.14	−0.514438
$249$	−76.8426	−0.0195570
$250$	0	0
$251$	−7042.64	−1.77103	−0.885513	−	0.464614i	$-0.846193\pi$
$251$	−7042.64	−1.77103	−0.885513	+	0.464614i	$0.846193\pi$
$252$	261.664	0.0654098
$253$	−4379.25	−1.08823
$254$	1112.35	0.274784
$255$	0	0
$256$	256.000	0.0625000
$257$	2914.12	0.707307	0.353653	−	0.935377i	$-0.384939\pi$
$257$	2914.12	0.707307	0.353653	+	0.935377i	$0.384939\pi$
$258$	−63.0788	−0.0152214
$259$	−141.604	−0.0339725
$260$	0	0
$261$	−5938.83	−1.40845
$262$	1684.93	0.397310
$263$	−2030.12	−0.475979	−0.237990	−	0.971268i	$-0.576488\pi$
$263$	−2030.12	−0.475979	−0.237990	+	0.971268i	$0.576488\pi$
$264$	17.0358	0.00397151
$265$	0	0
$266$	16.6475	0.00383731
$267$	105.900	0.0242732
$268$	−2226.27	−0.507429
$269$	−485.865	−0.110125	−0.0550627	−	0.998483i	$-0.517536\pi$
$269$	−485.865	−0.110125	−0.0550627	+	0.998483i	$0.517536\pi$
$270$	0	0
$271$	−401.360	−0.0899663	−0.0449831	−	0.998988i	$-0.514323\pi$
$271$	−401.360	−0.0899663	−0.0449831	+	0.998988i	$0.514323\pi$
$272$	272.000	0.0606339
$273$	3.11722	0.000691071 0
$274$	−955.851	−0.210749
$275$	0	0
$276$	52.4622	0.0114415
$277$	7397.62	1.60462	0.802311	−	0.596907i	$-0.203604\pi$
$277$	7397.62	1.60462	0.802311	+	0.596907i	$0.203604\pi$
$278$	1042.49	0.224909
$279$	−6779.25	−1.45471
$280$	0	0
$281$	3319.96	0.704812	0.352406	−	0.935847i	$-0.385364\pi$
$281$	3319.96	0.704812	0.352406	+	0.935847i	$0.385364\pi$
$282$	−9.49688	−0.00200543
$283$	6075.80	1.27621	0.638107	−	0.769947i	$-0.279718\pi$
$283$	6075.80	1.27621	0.638107	+	0.769947i	$0.279718\pi$
$284$	3836.57	0.801615
$285$	0	0
$286$	−858.989	−0.177598
$287$	607.255	0.124896
$288$	863.796	0.176735
$289$	289.000	0.0588235
$290$	0	0
$291$	16.1946	0.00326235
$292$	88.4638	0.0177293
$293$	−6006.45	−1.19761	−0.598806	−	0.800894i	$-0.704358\pi$
$293$	−6006.45	−1.19761	−0.598806	+	0.800894i	$0.704358\pi$
$294$	−53.8459	−0.0106815
$295$	0	0
$296$	−467.459	−0.0917923
$297$	114.978	0.0224636
$298$	5258.33	1.02217
$299$	−2645.29	−0.511642
$300$	0	0
$301$	957.077	0.183272
$302$	−5550.82	−1.05766
$303$	−15.1119	−0.00286520
$304$	54.9562	0.0103683
$305$	0	0
$306$	917.783	0.171458
$307$	1572.83	0.292398	0.146199	−	0.989255i	$-0.453296\pi$
$307$	1572.83	0.292398	0.146199	+	0.989255i	$0.453296\pi$
$308$	−258.479	−0.0478188
$309$	107.792	0.0198450
$310$	0	0
$311$	−2348.12	−0.428134	−0.214067	−	0.976819i	$-0.568671\pi$
$311$	−2348.12	−0.428134	−0.214067	+	0.976819i	$0.568671\pi$
$312$	10.2904	0.00186725
$313$	1688.80	0.304974	0.152487	−	0.988305i	$-0.451272\pi$
$313$	1688.80	0.304974	0.152487	+	0.988305i	$0.451272\pi$
$314$	3507.81	0.630437
$315$	0	0
$316$	2845.17	0.506498
$317$	6944.29	1.23038	0.615189	−	0.788379i	$-0.289080\pi$
$317$	6944.29	1.23038	0.615189	+	0.788379i	$0.289080\pi$
$318$	−38.0737	−0.00671404
$319$	5866.54	1.02966
$320$	0	0
$321$	72.4791	0.0126025
$322$	−795.994	−0.137761
$323$	58.3909	0.0100587
$324$	2913.93	0.499646
$325$	0	0
$326$	3281.01	0.557419
$327$	96.7578	0.0163631
$328$	2004.65	0.337464
$329$	144.094	0.0241463
$330$	0	0
$331$	−1944.41	−0.322884	−0.161442	−	0.986882i	$-0.551615\pi$
$331$	−1944.41	−0.322884	−0.161442	+	0.986882i	$0.551615\pi$
$332$	3848.87	0.636247
$333$	−1577.30	−0.259567
$334$	3668.68	0.601021
$335$	0	0
$336$	3.09650	0.000502762 0
$337$	6607.88	1.06811	0.534057	−	0.845449i	$-0.320667\pi$
$337$	6607.88	1.06811	0.534057	+	0.845449i	$0.320667\pi$
$338$	3875.13	0.623607
$339$	147.215	0.0235859
$340$	0	0
$341$	6696.73	1.06348
$342$	185.433	0.0293189
$343$	1648.21	0.259461
$344$	3159.47	0.495195
$345$	0	0
$346$	5018.77	0.779801
$347$	20.6665	0.00319723	0.00159861	−	0.999999i	$-0.499491\pi$
$347$	20.6665	0.00319723	0.00159861	+	0.999999i	$0.499491\pi$
$348$	−70.2795	−0.0108258
$349$	−5935.84	−0.910425	−0.455213	−	0.890383i	$-0.650437\pi$
$349$	−5935.84	−0.910425	−0.455213	+	0.890383i	$0.650437\pi$
$350$	0	0
$351$	69.4523	0.0105615
$352$	−853.282	−0.129205
$353$	235.646	0.0355303	0.0177651	−	0.999842i	$-0.494345\pi$
$353$	235.646	0.0355303	0.0177651	+	0.999842i	$0.494345\pi$
$354$	−63.4246	−0.00952254
$355$	0	0
$356$	−5304.26	−0.789678
$357$	3.29003	0.000487751 0
$358$	1350.92	0.199436
$359$	−3683.68	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$359$	−3683.68	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$360$	0	0
$361$	−6847.20	−0.998280
$362$	3290.72	0.477780
$363$	49.5112	0.00715885
$364$	−156.134	−0.0224826
$365$	0	0
$366$	66.7699	0.00953585
$367$	9215.27	1.31072	0.655358	−	0.755318i	$-0.272518\pi$
$367$	9215.27	1.31072	0.655358	+	0.755318i	$0.272518\pi$
$368$	−2627.71	−0.372225
$369$	6764.10	0.954268
$370$	0	0
$371$	577.681	0.0808402
$372$	−80.2249	−0.0111814
$373$	5587.98	0.775696	0.387848	−	0.921723i	$-0.373219\pi$
$373$	5587.98	0.775696	0.387848	+	0.921723i	$0.373219\pi$
$374$	−906.612	−0.125347
$375$	0	0
$376$	475.677	0.0652424
$377$	3543.68	0.484109
$378$	20.8989	0.00284372
$379$	6978.28	0.945779	0.472890	−	0.881122i	$-0.343211\pi$
$379$	6978.28	0.945779	0.472890	+	0.881122i	$0.343211\pi$
$380$	0	0
$381$	44.4161	0.00597246
$382$	4827.92	0.646644
$383$	−9035.04	−1.20540	−0.602701	−	0.797967i	$-0.705909\pi$
$383$	−9035.04	−1.20540	−0.602701	+	0.797967i	$0.705909\pi$
$384$	10.2221	0.00135845
$385$	0	0
$386$	−5437.00	−0.716933
$387$	10660.7	1.40029
$388$	−811.150	−0.106134
$389$	6080.85	0.792573	0.396287	−	0.918127i	$-0.370299\pi$
$389$	6080.85	0.792573	0.396287	+	0.918127i	$0.370299\pi$
$390$	0	0
$391$	−2791.94	−0.361111
$392$	2697.02	0.347500
$393$	67.2792	0.00863559
$394$	3661.43	0.468174
$395$	0	0
$396$	−2879.14	−0.365360
$397$	7221.63	0.912955	0.456478	−	0.889735i	$-0.349111\pi$
$397$	7221.63	0.912955	0.456478	+	0.889735i	$0.349111\pi$
$398$	8025.85	1.01080
$399$	0.664734	8.34043e−5 0
$400$	0	0
$401$	1537.31	0.191445	0.0957226	−	0.995408i	$-0.469484\pi$
$401$	1537.31	0.191445	0.0957226	+	0.995408i	$0.469484\pi$
$402$	−88.8948	−0.0110290
$403$	4045.16	0.500009
$404$	756.920	0.0932133
$405$	0	0
$406$	1066.33	0.130348
$407$	1558.10	0.189760
$408$	10.8610	0.00131789
$409$	5999.41	0.725310	0.362655	−	0.931924i	$-0.381870\pi$
$409$	5999.41	0.725310	0.362655	+	0.931924i	$0.381870\pi$
$410$	0	0
$411$	−38.1671	−0.00458065
$412$	−5399.07	−0.645615
$413$	962.324	0.114656
$414$	−8866.42	−1.05256
$415$	0	0
$416$	−515.425	−0.0607471
$417$	41.6268	0.00488842
$418$	−183.176	−0.0214341
$419$	−4075.16	−0.475142	−0.237571	−	0.971370i	$-0.576351\pi$
$419$	−4075.16	−0.475142	−0.237571	+	0.971370i	$0.576351\pi$
$420$	0	0
$421$	3993.72	0.462332	0.231166	−	0.972914i	$-0.425746\pi$
$421$	3993.72	0.462332	0.231166	+	0.972914i	$0.425746\pi$
$422$	3779.87	0.436022
$423$	1605.03	0.184490
$424$	1907.02	0.218427
$425$	0	0
$426$	153.194	0.0174232
$427$	−1013.08	−0.114816
$428$	−3630.31	−0.409995
$429$	−34.2994	−0.00386012
$430$	0	0
$431$	−3004.07	−0.335733	−0.167867	−	0.985810i	$-0.553688\pi$
$431$	−3004.07	−0.335733	−0.167867	+	0.985810i	$0.553688\pi$
$432$	68.9908	0.00768362
$433$	−5153.01	−0.571912	−0.285956	−	0.958243i	$-0.592311\pi$
$433$	−5153.01	−0.571912	−0.285956	+	0.958243i	$0.592311\pi$
$434$	1217.23	0.134629
$435$	0	0
$436$	−4846.38	−0.532338
$437$	−564.097	−0.0617492
$438$	3.53236	0.000385348 0
$439$	−795.820	−0.0865203	−0.0432602	−	0.999064i	$-0.513774\pi$
$439$	−795.820	−0.0865203	−0.0432602	+	0.999064i	$0.513774\pi$
$440$	0	0
$441$	9100.28	0.982646
$442$	−547.639	−0.0589333
$443$	−13950.6	−1.49619	−0.748097	−	0.663590i	$-0.769032\pi$
$443$	−13950.6	−1.49619	−0.748097	+	0.663590i	$0.769032\pi$
$444$	−18.6656	−0.00199512
$445$	0	0
$446$	−661.473	−0.0702280
$447$	209.965	0.0222170
$448$	−155.097	−0.0163563
$449$	13542.7	1.42343	0.711717	−	0.702467i	$-0.247918\pi$
$449$	13542.7	1.42343	0.711717	+	0.702467i	$0.247918\pi$
$450$	0	0
$451$	−6681.76	−0.697632
$452$	−7373.66	−0.767319
$453$	−221.644	−0.0229884
$454$	7995.55	0.826542
$455$	0	0
$456$	2.19440	0.000225355 0
$457$	−14617.8	−1.49626	−0.748131	−	0.663551i	$-0.769049\pi$
$457$	−14617.8	−1.49626	−0.748131	+	0.663551i	$0.769049\pi$
$458$	−7800.44	−0.795831
$459$	73.3028	0.00745421
$460$	0	0
$461$	2094.62	0.211619	0.105809	−	0.994386i	$-0.466257\pi$
$461$	2094.62	0.211619	0.105809	+	0.994386i	$0.466257\pi$
$462$	−10.3211	−0.00103935
$463$	12885.2	1.29336	0.646680	−	0.762761i	$-0.276157\pi$
$463$	12885.2	1.29336	0.646680	+	0.762761i	$0.276157\pi$
$464$	3520.14	0.352195
$465$	0	0
$466$	−5719.83	−0.568596
$467$	11038.8	1.09382	0.546912	−	0.837190i	$-0.315803\pi$
$467$	11038.8	1.09382	0.546912	+	0.837190i	$0.315803\pi$
$468$	−1739.15	−0.171778
$469$	1348.78	0.132795
$470$	0	0
$471$	140.067	0.0137026
$472$	3176.79	0.309796
$473$	−10530.9	−1.02371
$474$	113.608	0.0110088
$475$	0	0
$476$	−164.790	−0.0158680
$477$	6434.68	0.617660
$478$	2144.71	0.205223
$479$	13071.1	1.24684	0.623418	−	0.781888i	$-0.285743\pi$
$479$	13071.1	1.24684	0.623418	+	0.781888i	$0.285743\pi$
$480$	0	0
$481$	941.173	0.0892178
$482$	−8484.54	−0.801784
$483$	−31.7840	−0.00299425
$484$	−2479.90	−0.232898
$485$	0	0
$486$	349.197	0.0325924
$487$	−5104.65	−0.474977	−0.237488	−	0.971390i	$-0.576324\pi$
$487$	−5104.65	−0.474977	−0.237488	+	0.971390i	$0.576324\pi$
$488$	−3344.35	−0.310229
$489$	131.011	0.0121156
$490$	0	0
$491$	−19047.4	−1.75071	−0.875353	−	0.483484i	$-0.839371\pi$
$491$	−19047.4	−1.75071	−0.875353	+	0.483484i	$0.839371\pi$
$492$	80.0457	0.00733483
$493$	3740.15	0.341679
$494$	−110.647	−0.0100775
$495$	0	0
$496$	4018.28	0.363762
$497$	−2324.37	−0.209783
$498$	153.685	0.0138289
$499$	−559.986	−0.0502373	−0.0251186	−	0.999684i	$-0.507996\pi$
$499$	−559.986	−0.0502373	−0.0251186	+	0.999684i	$0.507996\pi$
$500$	0	0
$501$	146.490	0.0130633
$502$	14085.3	1.25230
$503$	4974.15	0.440927	0.220464	−	0.975395i	$-0.429243\pi$
$503$	4974.15	0.440927	0.220464	+	0.975395i	$0.429243\pi$
$504$	−523.328	−0.0462517
$505$	0	0
$506$	8758.50	0.769492
$507$	154.734	0.0135542
$508$	−2224.70	−0.194301
$509$	−19902.6	−1.73314	−0.866571	−	0.499054i	$-0.833681\pi$
$509$	−19902.6	−1.73314	−0.866571	+	0.499054i	$0.833681\pi$
$510$	0	0
$511$	−53.5955	−0.00463977
$512$	−512.000	−0.0441942
$513$	14.8104	0.00127465
$514$	−5828.24	−0.500141
$515$	0	0
$516$	126.158	0.0107631
$517$	−1585.49	−0.134874
$518$	283.209	0.0240222
$519$	200.400	0.0169491
$520$	0	0
$521$	−17650.7	−1.48424	−0.742122	−	0.670265i	$-0.766181\pi$
$521$	−17650.7	−1.48424	−0.742122	+	0.670265i	$0.766181\pi$
$522$	11877.7	0.995921
$523$	18495.1	1.54634	0.773168	−	0.634201i	$-0.218671\pi$
$523$	18495.1	1.54634	0.773168	+	0.634201i	$0.218671\pi$
$524$	−3369.86	−0.280941
$525$	0	0
$526$	4060.24	0.336568
$527$	4269.42	0.352901
$528$	−34.0715	−0.00280828
$529$	14805.1	1.21682
$530$	0	0
$531$	10719.1	0.876028
$532$	−33.2950	−0.00271339
$533$	−4036.12	−0.328000
$534$	−211.799	−0.0171637
$535$	0	0
$536$	4452.53	0.358806
$537$	53.9421	0.00433478
$538$	971.730	0.0778704
$539$	−8989.51	−0.718378
$540$	0	0
$541$	−6646.18	−0.528173	−0.264087	−	0.964499i	$-0.585070\pi$
$541$	−6646.18	−0.528173	−0.264087	+	0.964499i	$0.585070\pi$
$542$	802.719	0.0636158
$543$	131.398	0.0103846
$544$	−544.000	−0.0428746
$545$	0	0
$546$	−6.23443	−0.000488661 0
$547$	−20607.6	−1.61082	−0.805408	−	0.592721i	$-0.798054\pi$
$547$	−20607.6	−1.61082	−0.805408	+	0.592721i	$0.798054\pi$
$548$	1911.70	0.149022
$549$	−11284.5	−0.877253
$550$	0	0
$551$	755.676	0.0584263
$552$	−104.924	−0.00809036
$553$	−1723.74	−0.132551
$554$	−14795.2	−1.13464
$555$	0	0
$556$	−2084.99	−0.159035
$557$	11619.5	0.883901	0.441951	−	0.897039i	$-0.354287\pi$
$557$	11619.5	0.883901	0.441951	+	0.897039i	$0.354287\pi$
$558$	13558.5	1.02863
$559$	−6361.21	−0.481307
$560$	0	0
$561$	−36.2010	−0.00272443
$562$	−6639.91	−0.498377
$563$	−5942.49	−0.444842	−0.222421	−	0.974951i	$-0.571396\pi$
$563$	−5942.49	−0.444842	−0.222421	+	0.974951i	$0.571396\pi$
$564$	18.9938	0.00141805
$565$	0	0
$566$	−12151.6	−0.902420
$567$	−1765.40	−0.130758
$568$	−7673.14	−0.566827
$569$	7106.52	0.523587	0.261793	−	0.965124i	$-0.415686\pi$
$569$	7106.52	0.523587	0.261793	+	0.965124i	$0.415686\pi$
$570$	0	0
$571$	−1730.50	−0.126829	−0.0634144	−	0.997987i	$-0.520199\pi$
$571$	−1730.50	−0.126829	−0.0634144	+	0.997987i	$0.520199\pi$
$572$	1717.98	0.125581
$573$	192.779	0.0140549
$574$	−1214.51	−0.0883148
$575$	0	0
$576$	−1727.59	−0.124970
$577$	−6281.16	−0.453186	−0.226593	−	0.973990i	$-0.572759\pi$
$577$	−6281.16	−0.453186	−0.226593	+	0.973990i	$0.572759\pi$
$578$	−578.000	−0.0415945
$579$	−217.099	−0.0155826
$580$	0	0
$581$	−2331.82	−0.166507
$582$	−32.3892	−0.00230683
$583$	−6356.35	−0.451549
$584$	−176.928	−0.0125365
$585$	0	0
$586$	12012.9	0.846840
$587$	−24877.1	−1.74921	−0.874607	−	0.484833i	$-0.838880\pi$
$587$	−24877.1	−1.74921	−0.874607	+	0.484833i	$0.838880\pi$
$588$	107.692	0.00755295
$589$	862.614	0.0603453
$590$	0	0
$591$	146.201	0.0101758
$592$	934.919	0.0649070
$593$	21543.7	1.49189	0.745947	−	0.666005i	$-0.231997\pi$
$593$	21543.7	1.49189	0.745947	+	0.666005i	$0.231997\pi$
$594$	−229.956	−0.0158842
$595$	0	0
$596$	−10516.7	−0.722783
$597$	320.472	0.0219699
$598$	5290.57	0.361785
$599$	10994.9	0.749984	0.374992	−	0.927028i	$-0.377645\pi$
$599$	10994.9	0.749984	0.374992	+	0.927028i	$0.377645\pi$
$600$	0	0
$601$	13218.0	0.897124	0.448562	−	0.893752i	$-0.351936\pi$
$601$	13218.0	0.897124	0.448562	+	0.893752i	$0.351936\pi$
$602$	−1914.15	−0.129593
$603$	15023.8	1.01462
$604$	11101.6	0.747880
$605$	0	0
$606$	30.2238	0.00202600
$607$	−1986.24	−0.132816	−0.0664078	−	0.997793i	$-0.521154\pi$
$607$	−1986.24	−0.132816	−0.0664078	+	0.997793i	$0.521154\pi$
$608$	−109.912	−0.00733147
$609$	42.5786	0.00283312
$610$	0	0
$611$	−957.717	−0.0634126
$612$	−1835.57	−0.121239
$613$	19585.4	1.29045	0.645227	−	0.763991i	$-0.276763\pi$
$613$	19585.4	1.29045	0.645227	+	0.763991i	$0.276763\pi$
$614$	−3145.66	−0.206757
$615$	0	0
$616$	516.958	0.0338130
$617$	27941.0	1.82311	0.911557	−	0.411175i	$-0.134881\pi$
$617$	27941.0	1.82311	0.911557	+	0.411175i	$0.134881\pi$
$618$	−215.585	−0.0140325
$619$	−10134.4	−0.658052	−0.329026	−	0.944321i	$-0.606720\pi$
$619$	−10134.4	−0.658052	−0.329026	+	0.944321i	$0.606720\pi$
$620$	0	0
$621$	−708.156	−0.0457606
$622$	4696.24	0.302737
$623$	3213.57	0.206660
$624$	−20.5809	−0.00132034
$625$	0	0
$626$	−3377.61	−0.215649
$627$	−7.31421	−0.000465872 0
$628$	−7015.62	−0.445786
$629$	993.351	0.0629690
$630$	0	0
$631$	10019.3	0.632113	0.316057	−	0.948740i	$-0.397641\pi$
$631$	10019.3	0.632113	0.316057	+	0.948740i	$0.397641\pi$
$632$	−5690.35	−0.358148
$633$	150.930	0.00947699
$634$	−13888.6	−0.870009
$635$	0	0
$636$	76.1473	0.00474754
$637$	−5430.12	−0.337754
$638$	−11733.1	−0.728083
$639$	−25890.7	−1.60285
$640$	0	0
$641$	−5966.58	−0.367653	−0.183827	−	0.982959i	$-0.558849\pi$
$641$	−5966.58	−0.367653	−0.183827	+	0.982959i	$0.558849\pi$
$642$	−144.958	−0.00891128
$643$	11345.8	0.695856	0.347928	−	0.937521i	$-0.386885\pi$
$643$	11345.8	0.695856	0.347928	+	0.937521i	$0.386885\pi$
$644$	1591.99	0.0974117
$645$	0	0
$646$	−116.782	−0.00711257
$647$	−28300.9	−1.71967	−0.859833	−	0.510575i	$-0.829433\pi$
$647$	−28300.9	−1.71967	−0.859833	+	0.510575i	$0.829433\pi$
$648$	−5827.87	−0.353303
$649$	−10588.7	−0.640433
$650$	0	0
$651$	48.6040	0.00292618
$652$	−6562.03	−0.394155
$653$	−16197.0	−0.970657	−0.485329	−	0.874332i	$-0.661300\pi$
$653$	−16197.0	−0.970657	−0.485329	+	0.874332i	$0.661300\pi$
$654$	−193.516	−0.0115704
$655$	0	0
$656$	−4009.30	−0.238623
$657$	−596.989	−0.0354502
$658$	−288.187	−0.0170740
$659$	8088.41	0.478118	0.239059	−	0.971005i	$-0.423161\pi$
$659$	8088.41	0.478118	0.239059	+	0.971005i	$0.423161\pi$
$660$	0	0
$661$	27454.6	1.61552	0.807760	−	0.589511i	$-0.200680\pi$
$661$	27454.6	1.61552	0.807760	+	0.589511i	$0.200680\pi$
$662$	3888.83	0.228314
$663$	−21.8672	−0.00128092
$664$	−7697.73	−0.449895
$665$	0	0
$666$	3154.61	0.183541
$667$	−36132.4	−2.09753
$668$	−7337.35	−0.424986
$669$	−26.4126	−0.00152641
$670$	0	0
$671$	11147.2	0.641329
$672$	−6.19301	−0.000355507 0
$673$	−23867.0	−1.36702	−0.683512	−	0.729939i	$-0.739548\pi$
$673$	−23867.0	−1.36702	−0.683512	+	0.729939i	$0.739548\pi$
$674$	−13215.8	−0.755270
$675$	0	0
$676$	−7750.26	−0.440957
$677$	−26115.3	−1.48256	−0.741278	−	0.671198i	$-0.765780\pi$
$677$	−26115.3	−1.48256	−0.741278	+	0.671198i	$0.765780\pi$
$678$	−294.430	−0.0166778
$679$	491.432	0.0277753
$680$	0	0
$681$	319.262	0.0179650
$682$	−13393.5	−0.751997
$683$	18950.9	1.06169	0.530845	−	0.847469i	$-0.321875\pi$
$683$	18950.9	1.06169	0.530845	+	0.847469i	$0.321875\pi$
$684$	−370.866	−0.0207316
$685$	0	0
$686$	−3296.42	−0.183466
$687$	−311.471	−0.0172975
$688$	−6318.94	−0.350156
$689$	−3839.56	−0.212301
$690$	0	0
$691$	−11976.4	−0.659340	−0.329670	−	0.944096i	$-0.606938\pi$
$691$	−11976.4	−0.659340	−0.329670	+	0.944096i	$0.606938\pi$
$692$	−10037.5	−0.551402
$693$	1744.32	0.0956150
$694$	−41.3331	−0.00226078
$695$	0	0
$696$	140.559	0.00765499
$697$	−4259.88	−0.231499
$698$	11871.7	0.643768
$699$	−228.393	−0.0123585
$700$	0	0
$701$	30759.6	1.65731	0.828656	−	0.559758i	$-0.189106\pi$
$701$	30759.6	1.65731	0.828656	+	0.559758i	$0.189106\pi$
$702$	−138.905	−0.00746812
$703$	200.701	0.0107676
$704$	1706.56	0.0913615
$705$	0	0
$706$	−471.293	−0.0251237
$707$	−458.577	−0.0243940
$708$	126.849	0.00673345
$709$	24780.5	1.31263	0.656313	−	0.754489i	$-0.272115\pi$
$709$	24780.5	1.31263	0.656313	+	0.754489i	$0.272115\pi$
$710$	0	0
$711$	−19200.4	−1.01276
$712$	10608.5	0.558387
$713$	−41245.6	−2.16642
$714$	−6.58007	−0.000344892 0
$715$	0	0
$716$	−2701.84	−0.141023
$717$	85.6382	0.00446056
$718$	7367.36	0.382935
$719$	25967.9	1.34693	0.673463	−	0.739221i	$-0.264806\pi$
$719$	25967.9	1.34693	0.673463	+	0.739221i	$0.264806\pi$
$720$	0	0
$721$	3271.01	0.168958
$722$	13694.4	0.705891
$723$	−338.787	−0.0174269
$724$	−6581.44	−0.337842
$725$	0	0
$726$	−99.0224	−0.00506207
$727$	3589.05	0.183095	0.0915477	−	0.995801i	$-0.470819\pi$
$727$	3589.05	0.183095	0.0915477	+	0.995801i	$0.470819\pi$
$728$	312.268	0.0158976
$729$	−19655.1	−0.998583
$730$	0	0
$731$	−6713.87	−0.339701
$732$	−133.540	−0.00674286
$733$	4672.05	0.235425	0.117712	−	0.993048i	$-0.462444\pi$
$733$	4672.05	0.235425	0.117712	+	0.993048i	$0.462444\pi$
$734$	−18430.5	−0.926817
$735$	0	0
$736$	5255.42	0.263203
$737$	−14840.9	−0.741751
$738$	−13528.2	−0.674770
$739$	22452.2	1.11762	0.558808	−	0.829297i	$-0.311259\pi$
$739$	22452.2	1.11762	0.558808	+	0.829297i	$0.311259\pi$
$740$	0	0
$741$	−4.41815	−0.000219035 0
$742$	−1155.36	−0.0571627
$743$	−83.9369	−0.00414448	−0.00207224	−	0.999998i	$-0.500660\pi$
$743$	−83.9369	−0.00414448	−0.00207224	+	0.999998i	$0.500660\pi$
$744$	160.450	0.00790642
$745$	0	0
$746$	−11176.0	−0.548500
$747$	−25973.7	−1.27219
$748$	1813.22	0.0886337
$749$	2199.41	0.107296
$750$	0	0
$751$	−3202.70	−0.155617	−0.0778084	−	0.996968i	$-0.524792\pi$
$751$	−3202.70	−0.155617	−0.0778084	+	0.996968i	$0.524792\pi$
$752$	−951.353	−0.0461334
$753$	562.425	0.0272190
$754$	−7087.37	−0.342317
$755$	0	0
$756$	−41.7979	−0.00201081
$757$	14268.7	0.685081	0.342541	−	0.939503i	$-0.388713\pi$
$757$	14268.7	0.685081	0.342541	+	0.939503i	$0.388713\pi$
$758$	−13956.6	−0.668767
$759$	349.727	0.0167250
$760$	0	0
$761$	11451.7	0.545498	0.272749	−	0.962085i	$-0.412067\pi$
$761$	11451.7	0.545498	0.272749	+	0.962085i	$0.412067\pi$
$762$	−88.8322	−0.00422316
$763$	2936.16	0.139313
$764$	−9655.85	−0.457246
$765$	0	0
$766$	18070.1	0.852348
$767$	−6396.08	−0.301107
$768$	−20.4441	−0.000960566 0
$769$	−2550.07	−0.119581	−0.0597907	−	0.998211i	$-0.519043\pi$
$769$	−2550.07	−0.119581	−0.0597907	+	0.998211i	$0.519043\pi$
$770$	0	0
$771$	−232.721	−0.0108706
$772$	10874.0	0.506948
$773$	−19549.6	−0.909637	−0.454818	−	0.890584i	$-0.650296\pi$
$773$	−19549.6	−0.909637	−0.454818	+	0.890584i	$0.650296\pi$
$774$	−21321.4	−0.990157
$775$	0	0
$776$	1622.30	0.0750479
$777$	11.3085	0.000522124 0
$778$	−12161.7	−0.560434
$779$	−860.687	−0.0395858
$780$	0	0
$781$	25575.6	1.17179
$782$	5583.88	0.255344
$783$	948.661	0.0432981
$784$	−5394.04	−0.245720
$785$	0	0
$786$	−134.558	−0.00610628
$787$	10714.3	0.485292	0.242646	−	0.970115i	$-0.421985\pi$
$787$	10714.3	0.485292	0.242646	+	0.970115i	$0.421985\pi$
$788$	−7322.87	−0.331049
$789$	162.125	0.00731535
$790$	0	0
$791$	4467.31	0.200808
$792$	5758.29	0.258348
$793$	6733.44	0.301528
$794$	−14443.3	−0.645557
$795$	0	0
$796$	−16051.7	−0.714745
$797$	−25134.8	−1.11709	−0.558544	−	0.829475i	$-0.688640\pi$
$797$	−25134.8	−1.11709	−0.558544	+	0.829475i	$0.688640\pi$
$798$	−1.32947	−5.89758e−5 0
$799$	−1010.81	−0.0447559
$800$	0	0
$801$	35795.3	1.57898
$802$	−3074.62	−0.135372
$803$	589.723	0.0259164
$804$	177.790	0.00779870
$805$	0	0
$806$	−8090.32	−0.353560
$807$	38.8012	0.00169252
$808$	−1513.84	−0.0659117
$809$	−15870.0	−0.689689	−0.344844	−	0.938660i	$-0.612068\pi$
$809$	−15870.0	−0.689689	−0.344844	+	0.938660i	$0.612068\pi$
$810$	0	0
$811$	8629.24	0.373630	0.186815	−	0.982395i	$-0.440184\pi$
$811$	8629.24	0.373630	0.186815	+	0.982395i	$0.440184\pi$
$812$	−2132.66	−0.0921697
$813$	32.0526	0.00138270
$814$	−3116.21	−0.134181
$815$	0	0
$816$	−21.7219	−0.000931886 0
$817$	−1356.50	−0.0580881
$818$	−11998.8	−0.512871
$819$	1053.66	0.0449545
$820$	0	0
$821$	−8572.87	−0.364428	−0.182214	−	0.983259i	$-0.558326\pi$
$821$	−8572.87	−0.364428	−0.182214	+	0.983259i	$0.558326\pi$
$822$	76.3342	0.00323901
$823$	19418.4	0.822457	0.411228	−	0.911532i	$-0.365100\pi$
$823$	19418.4	0.822457	0.411228	+	0.911532i	$0.365100\pi$
$824$	10798.1	0.456518
$825$	0	0
$826$	−1924.65	−0.0810739
$827$	−6320.73	−0.265772	−0.132886	−	0.991131i	$-0.542424\pi$
$827$	−6320.73	−0.265772	−0.132886	+	0.991131i	$0.542424\pi$
$828$	17732.8	0.744274
$829$	27394.8	1.14772	0.573860	−	0.818953i	$-0.305445\pi$
$829$	27394.8	1.14772	0.573860	+	0.818953i	$0.305445\pi$
$830$	0	0
$831$	−590.774	−0.0246615
$832$	1030.85	0.0429547
$833$	−5731.16	−0.238383
$834$	−83.2535	−0.00345664
$835$	0	0
$836$	366.352	0.0151562
$837$	1082.91	0.0447202
$838$	8150.32	0.335976
$839$	−8708.72	−0.358353	−0.179177	−	0.983817i	$-0.557343\pi$
$839$	−8708.72	−0.358353	−0.179177	+	0.983817i	$0.557343\pi$
$840$	0	0
$841$	24014.8	0.984656
$842$	−7987.43	−0.326918
$843$	−265.132	−0.0108323
$844$	−7559.74	−0.308314
$845$	0	0
$846$	−3210.06	−0.130454
$847$	1502.44	0.0609497
$848$	−3814.04	−0.154451
$849$	−485.213	−0.0196142
$850$	0	0
$851$	−9596.46	−0.386560
$852$	−306.388	−0.0123201
$853$	−6876.02	−0.276003	−0.138002	−	0.990432i	$-0.544068\pi$
$853$	−6876.02	−0.276003	−0.138002	+	0.990432i	$0.544068\pi$
$854$	2026.16	0.0811872
$855$	0	0
$856$	7260.62	0.289910
$857$	29987.0	1.19526	0.597630	−	0.801772i	$-0.296109\pi$
$857$	29987.0	1.19526	0.597630	+	0.801772i	$0.296109\pi$
$858$	68.5988	0.00272952
$859$	−36647.4	−1.45564	−0.727818	−	0.685770i	$-0.759466\pi$
$859$	−36647.4	−1.45564	−0.727818	+	0.685770i	$0.759466\pi$
$860$	0	0
$861$	−48.4954	−0.00191953
$862$	6008.14	0.237399
$863$	−28823.6	−1.13693	−0.568463	−	0.822709i	$-0.692462\pi$
$863$	−28823.6	−1.13693	−0.568463	+	0.822709i	$0.692462\pi$
$864$	−137.982	−0.00543314
$865$	0	0
$866$	10306.0	0.404403
$867$	−23.0795	−0.000904062 0
$868$	−2434.46	−0.0951970
$869$	18966.7	0.740391
$870$	0	0
$871$	−8964.63	−0.348743
$872$	9692.75	0.376420
$873$	5473.97	0.212217
$874$	1128.19	0.0436633
$875$	0	0
$876$	−7.06471	−0.000272482 0
$877$	−30956.1	−1.19192	−0.595961	−	0.803014i	$-0.703229\pi$
$877$	−30956.1	−1.19192	−0.595961	+	0.803014i	$0.703229\pi$
$878$	1591.64	0.0611791
$879$	479.675	0.0184062
$880$	0	0
$881$	5460.90	0.208833	0.104417	−	0.994534i	$-0.466702\pi$
$881$	5460.90	0.208833	0.104417	+	0.994534i	$0.466702\pi$
$882$	−18200.6	−0.694836
$883$	−13503.3	−0.514636	−0.257318	−	0.966327i	$-0.582839\pi$
$883$	−13503.3	−0.514636	−0.257318	+	0.966327i	$0.582839\pi$
$884$	1095.28	0.0416721
$885$	0	0
$886$	27901.2	1.05797
$887$	−26835.4	−1.01583	−0.507917	−	0.861406i	$-0.669584\pi$
$887$	−26835.4	−1.01583	−0.507917	+	0.861406i	$0.669584\pi$
$888$	37.3313	0.00141076
$889$	1347.83	0.0508489
$890$	0	0
$891$	19425.0	0.730374
$892$	1322.95	0.0496587
$893$	−204.229	−0.00765316
$894$	−419.930	−0.0157098
$895$	0	0
$896$	310.193	0.0115657
$897$	211.252	0.00786345
$898$	−27085.5	−1.00652
$899$	55253.5	2.04984
$900$	0	0
$901$	−4052.42	−0.149840
$902$	13363.5	0.493300
$903$	−76.4321	−0.00281672
$904$	14747.3	0.542576
$905$	0	0
$906$	443.288	0.0162553
$907$	−8310.85	−0.304253	−0.152126	−	0.988361i	$-0.548612\pi$
$907$	−8310.85	−0.304253	−0.152126	+	0.988361i	$0.548612\pi$
$908$	−15991.1	−0.584453
$909$	−5108.00	−0.186383
$910$	0	0
$911$	−10942.6	−0.397964	−0.198982	−	0.980003i	$-0.563764\pi$
$911$	−10942.6	−0.397964	−0.198982	+	0.980003i	$0.563764\pi$
$912$	−4.38880	−0.000159350 0
$913$	25657.6	0.930056
$914$	29235.6	1.05802
$915$	0	0
$916$	15600.9	0.562737
$917$	2041.62	0.0735225
$918$	−146.606	−0.00527092
$919$	−19691.4	−0.706810	−0.353405	−	0.935470i	$-0.614976\pi$
$919$	−19691.4	−0.706810	−0.353405	+	0.935470i	$0.614976\pi$
$920$	0	0
$921$	−125.606	−0.00449388
$922$	−4189.24	−0.149637
$923$	15448.9	0.550929
$924$	20.6421	0.000734930 0
$925$	0	0
$926$	−25770.4	−0.914544
$927$	36435.1	1.29092
$928$	−7040.27	−0.249039
$929$	−2746.09	−0.0969821	−0.0484910	−	0.998824i	$-0.515441\pi$
$929$	−2746.09	−0.0969821	−0.0484910	+	0.998824i	$0.515441\pi$
$930$	0	0
$931$	−1157.95	−0.0407629
$932$	11439.7	0.402058
$933$	187.521	0.00658002
$934$	−22077.7	−0.773451
$935$	0	0
$936$	3478.30	0.121465
$937$	−27740.8	−0.967185	−0.483592	−	0.875293i	$-0.660668\pi$
$937$	−27740.8	−0.967185	−0.483592	+	0.875293i	$0.660668\pi$
$938$	−2697.55	−0.0939000
$939$	−134.868	−0.00468716
$940$	0	0
$941$	29396.7	1.01839	0.509195	−	0.860651i	$-0.329943\pi$
$941$	29396.7	1.01839	0.509195	+	0.860651i	$0.329943\pi$
$942$	−280.133	−0.00968921
$943$	41153.4	1.42115
$944$	−6353.58	−0.219059
$945$	0	0
$946$	21061.9	0.723869
$947$	−3229.30	−0.110811	−0.0554056	−	0.998464i	$-0.517645\pi$
$947$	−3229.30	−0.110811	−0.0554056	+	0.998464i	$0.517645\pi$
$948$	−227.215	−0.00778440
$949$	356.222	0.0121849
$950$	0	0
$951$	−554.570	−0.0189098
$952$	329.580	0.0112203
$953$	−34669.0	−1.17842	−0.589212	−	0.807978i	$-0.700562\pi$
$953$	−34669.0	−1.17842	−0.589212	+	0.807978i	$0.700562\pi$
$954$	−12869.4	−0.436751
$955$	0	0
$956$	−4289.42	−0.145115
$957$	−468.501	−0.0158250
$958$	−26142.2	−0.881647
$959$	−1158.20	−0.0389991
$960$	0	0
$961$	33281.6	1.11717
$962$	−1882.35	−0.0630865
$963$	24498.8	0.819796
$964$	16969.1	0.566947
$965$	0	0
$966$	63.5680	0.00211725
$967$	−39902.4	−1.32697	−0.663483	−	0.748192i	$-0.730922\pi$
$967$	−39902.4	−1.32697	−0.663483	+	0.748192i	$0.730922\pi$
$968$	4959.80	0.164684
$969$	−4.66309	−0.000154593 0
$970$	0	0
$971$	−44625.9	−1.47489	−0.737443	−	0.675409i	$-0.763967\pi$
$971$	−44625.9	−1.47489	−0.737443	+	0.675409i	$0.763967\pi$
$972$	−698.395	−0.0230463
$973$	1263.18	0.0416195
$974$	10209.3	0.335859
$975$	0	0
$976$	6688.70	0.219365
$977$	12483.6	0.408788	0.204394	−	0.978889i	$-0.434478\pi$
$977$	12483.6	0.408788	0.204394	+	0.978889i	$0.434478\pi$
$978$	−262.021	−0.00856700
$979$	−35359.6	−1.15434
$980$	0	0
$981$	32705.3	1.06442
$982$	38094.8	1.23794
$983$	−31684.5	−1.02805	−0.514027	−	0.857774i	$-0.671847\pi$
$983$	−31684.5	−1.02805	−0.514027	+	0.857774i	$0.671847\pi$
$984$	−160.091	−0.00518651
$985$	0	0
$986$	−7480.29	−0.241603
$987$	−11.5073	−0.000371106 0
$988$	221.295	0.00712584
$989$	64860.7	2.08539
$990$	0	0
$991$	−51910.5	−1.66397	−0.831984	−	0.554800i	$-0.812795\pi$
$991$	−51910.5	−1.66397	−0.831984	+	0.554800i	$0.812795\pi$
$992$	−8036.56	−0.257219
$993$	155.281	0.00496243
$994$	4648.74	0.148339
$995$	0	0
$996$	−307.370	−0.00977851
$997$	54606.9	1.73462	0.867311	−	0.497767i	$-0.165847\pi$
$997$	54606.9	1.73462	0.867311	+	0.497767i	$0.165847\pi$
$998$	1119.97	0.0355231
$999$	251.956	0.00797953

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	850.4.a.q.1.3	✓	5
5.2	odd	4		850.4.c.n.749.3		10
5.3	odd	4		850.4.c.n.749.8		10
5.4	even	2		850.4.a.t.1.3	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
850.4.a.q.1.3	✓	5	1.1	even	1	trivial
850.4.a.t.1.3	yes	5	5.4	even	2
850.4.c.n.749.3		10	5.2	odd	4
850.4.c.n.749.8		10	5.3	odd	4

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:	\( N \)	\(=\)	\( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	850.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -0.0798599 q^{3} +4.00000 q^{4} +0.159720 q^{6} -2.42339 q^{7} -8.00000 q^{8} -26.9936 q^{9} +26.6650 q^{11} -0.319440 q^{12} +16.1070 q^{13} +4.84677 q^{14} +16.0000 q^{16} +17.0000 q^{17} +53.9872 q^{18} +3.43476 q^{19} +0.193531 q^{21} -53.3301 q^{22} -164.232 q^{23} +0.638880 q^{24} -32.2140 q^{26} +4.31193 q^{27} -9.69354 q^{28} +220.009 q^{29} +251.143 q^{31} -32.0000 q^{32} -2.12947 q^{33} -34.0000 q^{34} -107.974 q^{36} +58.4324 q^{37} -6.86952 q^{38} -1.28631 q^{39} -250.581 q^{41} -0.387063 q^{42} -394.934 q^{43} +106.660 q^{44} +328.464 q^{46} -59.4596 q^{47} -1.27776 q^{48} -337.127 q^{49} -1.35762 q^{51} +64.4281 q^{52} -238.378 q^{53} -8.62386 q^{54} +19.3871 q^{56} -0.274300 q^{57} -440.017 q^{58} -397.099 q^{59} +418.044 q^{61} -502.285 q^{62} +65.4160 q^{63} +64.0000 q^{64} +4.25894 q^{66} -556.567 q^{67} +68.0000 q^{68} +13.1155 q^{69} +959.142 q^{71} +215.949 q^{72} +22.1159 q^{73} -116.865 q^{74} +13.7390 q^{76} -64.6197 q^{77} +2.57261 q^{78} +711.293 q^{79} +728.483 q^{81} +501.163 q^{82} +962.217 q^{83} +0.774126 q^{84} +789.867 q^{86} -17.5699 q^{87} -213.320 q^{88} -1326.07 q^{89} -39.0335 q^{91} -656.927 q^{92} -20.0562 q^{93} +118.919 q^{94} +2.55552 q^{96} -202.787 q^{97} +674.254 q^{98} -719.786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{2} + q^{3} + 20 q^{4} - 2 q^{6} - 53 q^{7} - 40 q^{8} + 118 q^{9} - 30 q^{11} + 4 q^{12} - 123 q^{13} + 106 q^{14} + 80 q^{16} + 85 q^{17} - 236 q^{18} + 18 q^{19} + 119 q^{21} + 60 q^{22}+ \cdots - 4118 q^{99}+O(q^{100}) \) 5 * q - 10 * q^2 + q^3 + 20 * q^4 - 2 * q^6 - 53 * q^7 - 40 * q^8 + 118 * q^9 - 30 * q^11 + 4 * q^12 - 123 * q^13 + 106 * q^14 + 80 * q^16 + 85 * q^17 - 236 * q^18 + 18 * q^19 + 119 * q^21 + 60 * q^22 - 226 * q^23 - 8 * q^24 + 246 * q^26 + 61 * q^27 - 212 * q^28 + 86 * q^29 - 7 * q^31 - 160 * q^32 - 132 * q^33 - 170 * q^34 + 472 * q^36 - 186 * q^37 - 36 * q^38 - 443 * q^39 + 636 * q^41 - 238 * q^42 - 432 * q^43 - 120 * q^44 + 452 * q^46 - 18 * q^47 + 16 * q^48 - 378 * q^49 + 17 * q^51 - 492 * q^52 - 299 * q^53 - 122 * q^54 + 424 * q^56 - 692 * q^57 - 172 * q^58 + 1602 * q^61 + 14 * q^62 - 2382 * q^63 + 320 * q^64 + 264 * q^66 + 448 * q^67 + 340 * q^68 - 1200 * q^69 + 447 * q^71 - 944 * q^72 - 602 * q^73 + 372 * q^74 + 72 * q^76 - 1396 * q^77 + 886 * q^78 - 1697 * q^79 + 4357 * q^81 - 1272 * q^82 - 2988 * q^83 + 476 * q^84 + 864 * q^86 - 1636 * q^87 + 240 * q^88 - 86 * q^89 + 3237 * q^91 - 904 * q^92 - 2331 * q^93 + 36 * q^94 - 32 * q^96 + 664 * q^97 + 756 * q^98 - 4118 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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