LMFDB - Embedded newform 975.4.a.bb.1.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [975,4,Mod(1,975)] 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("975.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	975.a (trivial)

Newform invariants

comment:select newform

sage:traces = [11,-3,33,55,0,-9,20] 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:	$57.5268622556$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 3 x^{10} - 67 x^{9} + 177 x^{8} + 1527 x^{7} - 3289 x^{6} - 14195 x^{5} + 20777 x^{4} + \cdots - 5840 $ x^11 - 3x^10 - 67x^9 + 177x^8 + 1527x^7 - 3289x^6 - 14195x^5 + 20777x^4 + 56526x^3 - 34078x^2 - 76096x - 5840
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 5^{3} $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-2.70833$ of defining polynomial
Character	$\chi$	$=$	975.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.70833 q^{2} +3.00000 q^{3} -0.664940 q^{4} +8.12499 q^{6} +5.41772 q^{7} -23.4675 q^{8} +9.00000 q^{9} +3.07004 q^{11} -1.99482 q^{12} -13.0000 q^{13} +14.6730 q^{14} -58.2383 q^{16} +107.736 q^{17} +24.3750 q^{18} +91.3112 q^{19} +16.2532 q^{21} +8.31469 q^{22} -140.680 q^{23} -70.4026 q^{24} -35.2083 q^{26} +27.0000 q^{27} -3.60246 q^{28} +205.961 q^{29} +36.4603 q^{31} +30.0115 q^{32} +9.21012 q^{33} +291.785 q^{34} -5.98446 q^{36} -29.7849 q^{37} +247.301 q^{38} -39.0000 q^{39} -230.912 q^{41} +44.0190 q^{42} +431.281 q^{43} -2.04139 q^{44} -381.008 q^{46} +471.533 q^{47} -174.715 q^{48} -313.648 q^{49} +323.209 q^{51} +8.64422 q^{52} +371.850 q^{53} +73.1250 q^{54} -127.141 q^{56} +273.934 q^{57} +557.810 q^{58} +461.053 q^{59} +729.260 q^{61} +98.7466 q^{62} +48.7595 q^{63} +547.188 q^{64} +24.9441 q^{66} +231.079 q^{67} -71.6381 q^{68} -422.040 q^{69} -169.865 q^{71} -211.208 q^{72} +147.889 q^{73} -80.6674 q^{74} -60.7165 q^{76} +16.6326 q^{77} -105.625 q^{78} -426.831 q^{79} +81.0000 q^{81} -625.386 q^{82} -500.300 q^{83} -10.8074 q^{84} +1168.05 q^{86} +617.883 q^{87} -72.0463 q^{88} -1359.33 q^{89} -70.4304 q^{91} +93.5438 q^{92} +109.381 q^{93} +1277.07 q^{94} +90.0346 q^{96} +1033.12 q^{97} -849.463 q^{98} +27.6304 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 3 q^{2} + 33 q^{3} + 55 q^{4} - 9 q^{6} + 20 q^{7} - 51 q^{8} + 99 q^{9} + 100 q^{11} + 165 q^{12} - 143 q^{13} + 150 q^{14} + 511 q^{16} - 56 q^{17} - 27 q^{18} + 44 q^{19} + 60 q^{21} + 262 q^{22}+ \cdots + 900 q^{99}+O(q^{100}) $ 11 * q - 3 * q^2 + 33 * q^3 + 55 * q^4 - 9 * q^6 + 20 * q^7 - 51 * q^8 + 99 * q^9 + 100 * q^11 + 165 * q^12 - 143 * q^13 + 150 * q^14 + 511 * q^16 - 56 * q^17 - 27 * q^18 + 44 * q^19 + 60 * q^21 + 262 * q^22 + 98 * q^23 - 153 * q^24 + 39 * q^26 + 297 * q^27 + 590 * q^28 + 280 * q^29 + 256 * q^31 - 311 * q^32 + 300 * q^33 + 78 * q^34 + 495 * q^36 + 196 * q^37 + 180 * q^38 - 429 * q^39 + 700 * q^41 + 450 * q^42 + 1032 * q^43 + 1674 * q^44 + 620 * q^46 - 772 * q^47 + 1533 * q^48 + 1993 * q^49 - 168 * q^51 - 715 * q^52 + 164 * q^53 - 81 * q^54 + 1314 * q^56 + 132 * q^57 + 288 * q^58 + 1882 * q^59 + 1312 * q^61 - 100 * q^62 + 180 * q^63 + 4595 * q^64 + 786 * q^66 + 1720 * q^67 - 746 * q^68 + 294 * q^69 + 790 * q^71 - 459 * q^72 + 272 * q^73 - 342 * q^74 - 608 * q^76 + 1570 * q^77 + 117 * q^78 + 1766 * q^79 + 891 * q^81 + 1502 * q^82 + 864 * q^83 + 1770 * q^84 - 1332 * q^86 + 840 * q^87 + 3282 * q^88 + 3644 * q^89 - 260 * q^91 - 6472 * q^92 + 768 * q^93 + 6716 * q^94 - 933 * q^96 - 558 * q^97 - 8027 * q^98 + 900 * 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 $

$2$	2.70833	0.957540	0.478770	−	0.877940i	$-0.341083\pi$
$2$	2.70833	0.957540	0.478770	+	0.877940i	$0.341083\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−0.664940	−0.0831175
$5$	0	0
$5$	0	0
$6$	8.12499	0.552836
$7$	5.41772	0.292530	0.146265	−	0.989245i	$-0.453275\pi$
$7$	5.41772	0.292530	0.146265	+	0.989245i	$0.453275\pi$
$8$	−23.4675	−1.03713
$9$	9.00000	0.333333
$10$	0	0
$11$	3.07004	0.0841502	0.0420751	−	0.999114i	$-0.486603\pi$
$11$	3.07004	0.0841502	0.0420751	+	0.999114i	$0.486603\pi$
$12$	−1.99482	−0.0479879
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	14.6730	0.280109
$15$	0	0
$16$	−58.2383	−0.909974
$17$	107.736	1.53705	0.768526	−	0.639819i	$-0.220991\pi$
$17$	107.736	1.53705	0.768526	+	0.639819i	$0.220991\pi$
$18$	24.3750	0.319180
$19$	91.3112	1.10254	0.551269	−	0.834327i	$-0.314144\pi$
$19$	91.3112	1.10254	0.551269	+	0.834327i	$0.314144\pi$
$20$	0	0
$21$	16.2532	0.168892
$22$	8.31469	0.0805771
$23$	−140.680	−1.27538	−0.637692	−	0.770291i	$-0.720111\pi$
$23$	−140.680	−1.27538	−0.637692	+	0.770291i	$0.720111\pi$
$24$	−70.4026	−0.598786
$25$	0	0
$26$	−35.2083	−0.265574
$27$	27.0000	0.192450
$28$	−3.60246	−0.0243143
$29$	205.961	1.31883	0.659413	−	0.751780i	$-0.270805\pi$
$29$	205.961	1.31883	0.659413	+	0.751780i	$0.270805\pi$
$30$	0	0
$31$	36.4603	0.211241	0.105620	−	0.994407i	$-0.466317\pi$
$31$	36.4603	0.211241	0.105620	+	0.994407i	$0.466317\pi$
$32$	30.0115	0.165792
$33$	9.21012	0.0485841
$34$	291.785	1.47179
$35$	0	0
$36$	−5.98446	−0.0277058
$37$	−29.7849	−0.132341	−0.0661704	−	0.997808i	$-0.521078\pi$
$37$	−29.7849	−0.132341	−0.0661704	+	0.997808i	$0.521078\pi$
$38$	247.301	1.05572
$39$	−39.0000	−0.160128
$40$	0	0
$41$	−230.912	−0.879570	−0.439785	−	0.898103i	$-0.644945\pi$
$41$	−230.912	−0.879570	−0.439785	+	0.898103i	$0.644945\pi$
$42$	44.0190	0.161721
$43$	431.281	1.52953	0.764765	−	0.644310i	$-0.222855\pi$
$43$	431.281	1.52953	0.764765	+	0.644310i	$0.222855\pi$
$44$	−2.04139	−0.00699435
$45$	0	0
$46$	−381.008	−1.22123
$47$	471.533	1.46341	0.731704	−	0.681623i	$-0.238725\pi$
$47$	471.533	1.46341	0.731704	+	0.681623i	$0.238725\pi$
$48$	−174.715	−0.525374
$49$	−313.648	−0.914426
$50$	0	0
$51$	323.209	0.887417
$52$	8.64422	0.0230526
$53$	371.850	0.963726	0.481863	−	0.876247i	$-0.339960\pi$
$53$	371.850	0.963726	0.481863	+	0.876247i	$0.339960\pi$
$54$	73.1250	0.184279
$55$	0	0
$56$	−127.141	−0.303391
$57$	273.934	0.636551
$58$	557.810	1.26283
$59$	461.053	1.01736	0.508678	−	0.860957i	$-0.330134\pi$
$59$	461.053	1.01736	0.508678	+	0.860957i	$0.330134\pi$
$60$	0	0
$61$	729.260	1.53069	0.765346	−	0.643619i	$-0.222568\pi$
$61$	729.260	1.53069	0.765346	+	0.643619i	$0.222568\pi$
$62$	98.7466	0.202271
$63$	48.7595	0.0975099
$64$	547.188	1.06873
$65$	0	0
$66$	24.9441	0.0465212
$67$	231.079	0.421356	0.210678	−	0.977556i	$-0.432433\pi$
$67$	231.079	0.421356	0.210678	+	0.977556i	$0.432433\pi$
$68$	−71.6381	−0.127756
$69$	−422.040	−0.736343
$70$	0	0
$71$	−169.865	−0.283933	−0.141966	−	0.989871i	$-0.545342\pi$
$71$	−169.865	−0.283933	−0.141966	+	0.989871i	$0.545342\pi$
$72$	−211.208	−0.345709
$73$	147.889	0.237111	0.118556	−	0.992947i	$-0.462174\pi$
$73$	147.889	0.237111	0.118556	+	0.992947i	$0.462174\pi$
$74$	−80.6674	−0.126722
$75$	0	0
$76$	−60.7165	−0.0916403
$77$	16.6326	0.0246164
$78$	−105.625	−0.153329
$79$	−426.831	−0.607876	−0.303938	−	0.952692i	$-0.598302\pi$
$79$	−426.831	−0.607876	−0.303938	+	0.952692i	$0.598302\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−625.386	−0.842223
$83$	−500.300	−0.661627	−0.330813	−	0.943696i	$-0.607323\pi$
$83$	−500.300	−0.661627	−0.330813	+	0.943696i	$0.607323\pi$
$84$	−10.8074	−0.0140379
$85$	0	0
$86$	1168.05	1.46459
$87$	617.883	0.761425
$88$	−72.0463	−0.0872745
$89$	−1359.33	−1.61898	−0.809490	−	0.587134i	$-0.800256\pi$
$89$	−1359.33	−1.61898	−0.809490	+	0.587134i	$0.800256\pi$
$90$	0	0
$91$	−70.4304	−0.0811331
$92$	93.5438	0.106007
$93$	109.381	0.121960
$94$	1277.07	1.40127
$95$	0	0
$96$	90.0346	0.0957200
$97$	1033.12	1.08142	0.540711	−	0.841209i	$-0.318155\pi$
$97$	1033.12	1.08142	0.540711	+	0.841209i	$0.318155\pi$
$98$	−849.463	−0.875600
$99$	27.6304	0.0280501
$100$	0	0
$101$	1452.89	1.43137	0.715684	−	0.698424i	$-0.246115\pi$
$101$	1452.89	1.43137	0.715684	+	0.698424i	$0.246115\pi$
$102$	875.356	0.849737
$103$	−181.209	−0.173350	−0.0866749	−	0.996237i	$-0.527624\pi$
$103$	−181.209	−0.173350	−0.0866749	+	0.996237i	$0.527624\pi$
$104$	305.078	0.287648
$105$	0	0
$106$	1007.09	0.922806
$107$	−1026.35	−0.927300	−0.463650	−	0.886019i	$-0.653460\pi$
$107$	−1026.35	−0.927300	−0.463650	+	0.886019i	$0.653460\pi$
$108$	−17.9534	−0.0159960
$109$	1487.64	1.30725	0.653623	−	0.756821i	$-0.273248\pi$
$109$	1487.64	1.30725	0.653623	+	0.756821i	$0.273248\pi$
$110$	0	0
$111$	−89.3547	−0.0764070
$112$	−315.519	−0.266194
$113$	−63.9600	−0.0532464	−0.0266232	−	0.999646i	$-0.508475\pi$
$113$	−63.9600	−0.0532464	−0.0266232	+	0.999646i	$0.508475\pi$
$114$	741.903	0.609523
$115$	0	0
$116$	−136.952	−0.109618
$117$	−117.000	−0.0924500
$118$	1248.68	0.974158
$119$	583.685	0.449633
$120$	0	0
$121$	−1321.57	−0.992919
$122$	1975.08	1.46570
$123$	−692.736	−0.507820
$124$	−24.2439	−0.0175578
$125$	0	0
$126$	132.057	0.0933696
$127$	−1090.42	−0.761883	−0.380941	−	0.924599i	$-0.624400\pi$
$127$	−1090.42	−0.761883	−0.380941	+	0.924599i	$0.624400\pi$
$128$	1241.87	0.857556
$129$	1293.84	0.883074
$130$	0	0
$131$	1577.19	1.05191	0.525954	−	0.850513i	$-0.323708\pi$
$131$	1577.19	1.05191	0.525954	+	0.850513i	$0.323708\pi$
$132$	−6.12418	−0.00403819
$133$	494.699	0.322525
$134$	625.840	0.403465
$135$	0	0
$136$	−2528.30	−1.59412
$137$	−200.368	−0.124953	−0.0624766	−	0.998046i	$-0.519900\pi$
$137$	−200.368	−0.124953	−0.0624766	+	0.998046i	$0.519900\pi$
$138$	−1143.03	−0.705078
$139$	−503.261	−0.307094	−0.153547	−	0.988141i	$-0.549070\pi$
$139$	−503.261	−0.307094	−0.153547	+	0.988141i	$0.549070\pi$
$140$	0	0
$141$	1414.60	0.844899
$142$	−460.050	−0.271877
$143$	−39.9105	−0.0233391
$144$	−524.145	−0.303325
$145$	0	0
$146$	400.533	0.227044
$147$	−940.945	−0.527944
$148$	19.8052	0.0109998
$149$	3103.97	1.70662	0.853312	−	0.521400i	$-0.174590\pi$
$149$	3103.97	1.70662	0.853312	+	0.521400i	$0.174590\pi$
$150$	0	0
$151$	−1638.87	−0.883241	−0.441621	−	0.897202i	$-0.645596\pi$
$151$	−1638.87	−0.883241	−0.441621	+	0.897202i	$0.645596\pi$
$152$	−2142.85	−1.14347
$153$	969.626	0.512351
$154$	45.0467	0.0235712
$155$	0	0
$156$	25.9327	0.0133095
$157$	986.717	0.501583	0.250792	−	0.968041i	$-0.419309\pi$
$157$	986.717	0.501583	0.250792	+	0.968041i	$0.419309\pi$
$158$	−1156.00	−0.582065
$159$	1115.55	0.556408
$160$	0	0
$161$	−762.166	−0.373088
$162$	219.375	0.106393
$163$	164.386	0.0789922	0.0394961	−	0.999220i	$-0.487425\pi$
$163$	164.386	0.0789922	0.0394961	+	0.999220i	$0.487425\pi$
$164$	153.543	0.0731077
$165$	0	0
$166$	−1354.98	−0.633534
$167$	−3722.38	−1.72483	−0.862414	−	0.506203i	$-0.831049\pi$
$167$	−3722.38	−1.72483	−0.862414	+	0.506203i	$0.831049\pi$
$168$	−381.422	−0.175163
$169$	169.000	0.0769231
$170$	0	0
$171$	821.801	0.367513
$172$	−286.776	−0.127131
$173$	−3048.06	−1.33954	−0.669769	−	0.742570i	$-0.733607\pi$
$173$	−3048.06	−1.33954	−0.669769	+	0.742570i	$0.733607\pi$
$174$	1673.43	0.729095
$175$	0	0
$176$	−178.794	−0.0765745
$177$	1383.16	0.587370
$178$	−3681.53	−1.55024
$179$	3867.45	1.61490	0.807449	−	0.589938i	$-0.200848\pi$
$179$	3867.45	1.61490	0.807449	+	0.589938i	$0.200848\pi$
$180$	0	0
$181$	−1241.98	−0.510032	−0.255016	−	0.966937i	$-0.582081\pi$
$181$	−1241.98	−0.510032	−0.255016	+	0.966937i	$0.582081\pi$
$182$	−190.749	−0.0776882
$183$	2187.78	0.883745
$184$	3301.41	1.32274
$185$	0	0
$186$	296.240	0.116781
$187$	330.755	0.129343
$188$	−313.541	−0.121635
$189$	146.279	0.0562974
$190$	0	0
$191$	4548.65	1.72319	0.861593	−	0.507600i	$-0.169467\pi$
$191$	4548.65	1.72319	0.861593	+	0.507600i	$0.169467\pi$
$192$	1641.56	0.617029
$193$	−1560.79	−0.582115	−0.291057	−	0.956706i	$-0.594007\pi$
$193$	−1560.79	−0.582115	−0.291057	+	0.956706i	$0.594007\pi$
$194$	2798.04	1.03550
$195$	0	0
$196$	208.557	0.0760048
$197$	−34.2839	−0.0123991	−0.00619956	−	0.999981i	$-0.501973\pi$
$197$	−34.2839	−0.0123991	−0.00619956	+	0.999981i	$0.501973\pi$
$198$	74.8322	0.0268590
$199$	−4550.33	−1.62093	−0.810463	−	0.585790i	$-0.800785\pi$
$199$	−4550.33	−1.62093	−0.810463	+	0.585790i	$0.800785\pi$
$200$	0	0
$201$	693.238	0.243270
$202$	3934.91	1.37059
$203$	1115.84	0.385796
$204$	−214.914	−0.0737599
$205$	0	0
$206$	−490.773	−0.165989
$207$	−1266.12	−0.425128
$208$	757.098	0.252381
$209$	280.329	0.0927788
$210$	0	0
$211$	−7.91124	−0.00258119	−0.00129060	−	0.999999i	$-0.500411\pi$
$211$	−7.91124	−0.00258119	−0.00129060	+	0.999999i	$0.500411\pi$
$212$	−247.258	−0.0801025
$213$	−509.594	−0.163929
$214$	−2779.70	−0.887926
$215$	0	0
$216$	−633.623	−0.199595
$217$	197.532	0.0617942
$218$	4029.01	1.25174
$219$	443.668	0.136896
$220$	0	0
$221$	−1400.57	−0.426301
$222$	−242.002	−0.0731627
$223$	−4124.30	−1.23849	−0.619245	−	0.785197i	$-0.712561\pi$
$223$	−4124.30	−1.23849	−0.619245	+	0.785197i	$0.712561\pi$
$224$	162.594	0.0484990
$225$	0	0
$226$	−173.225	−0.0509856
$227$	1241.38	0.362967	0.181483	−	0.983394i	$-0.441910\pi$
$227$	1241.38	0.362967	0.181483	+	0.983394i	$0.441910\pi$
$228$	−182.150	−0.0529085
$229$	94.8108	0.0273593	0.0136796	−	0.999906i	$-0.495645\pi$
$229$	94.8108	0.0273593	0.0136796	+	0.999906i	$0.495645\pi$
$230$	0	0
$231$	49.8979	0.0142123
$232$	−4833.39	−1.36779
$233$	−1419.38	−0.399083	−0.199542	−	0.979889i	$-0.563945\pi$
$233$	−1419.38	−0.399083	−0.199542	+	0.979889i	$0.563945\pi$
$234$	−316.875	−0.0885246
$235$	0	0
$236$	−306.573	−0.0845600
$237$	−1280.49	−0.350957
$238$	1580.81	0.430542
$239$	−3166.49	−0.857001	−0.428501	−	0.903542i	$-0.640958\pi$
$239$	−3166.49	−0.857001	−0.428501	+	0.903542i	$0.640958\pi$
$240$	0	0
$241$	3386.63	0.905194	0.452597	−	0.891715i	$-0.350498\pi$
$241$	3386.63	0.905194	0.452597	+	0.891715i	$0.350498\pi$
$242$	−3579.26	−0.950759
$243$	243.000	0.0641500
$244$	−484.914	−0.127227
$245$	0	0
$246$	−1876.16	−0.486258
$247$	−1187.05	−0.305789
$248$	−855.633	−0.219084
$249$	−1500.90	−0.381991
$250$	0	0
$251$	426.852	0.107341	0.0536706	−	0.998559i	$-0.482908\pi$
$251$	426.852	0.107341	0.0536706	+	0.998559i	$0.482908\pi$
$252$	−32.4222	−0.00810478
$253$	−431.894	−0.107324
$254$	−2953.22	−0.729533
$255$	0	0
$256$	−1014.10	−0.247582
$257$	−5944.49	−1.44283	−0.721415	−	0.692503i	$-0.756508\pi$
$257$	−5944.49	−1.44283	−0.721415	+	0.692503i	$0.756508\pi$
$258$	3504.16	0.845579
$259$	−161.366	−0.0387136
$260$	0	0
$261$	1853.65	0.439609
$262$	4271.56	1.00724
$263$	140.731	0.0329957	0.0164978	−	0.999864i	$-0.494748\pi$
$263$	140.731	0.0329957	0.0164978	+	0.999864i	$0.494748\pi$
$264$	−216.139	−0.0503880
$265$	0	0
$266$	1339.81	0.308831
$267$	−4078.00	−0.934718
$268$	−153.654	−0.0350221
$269$	478.062	0.108357	0.0541783	−	0.998531i	$-0.482746\pi$
$269$	478.062	0.108357	0.0541783	+	0.998531i	$0.482746\pi$
$270$	0	0
$271$	3095.39	0.693843	0.346921	−	0.937894i	$-0.387227\pi$
$271$	3095.39	0.693843	0.346921	+	0.937894i	$0.387227\pi$
$272$	−6274.38	−1.39868
$273$	−211.291	−0.0468422
$274$	−542.663	−0.119648
$275$	0	0
$276$	280.631	0.0612030
$277$	1439.02	0.312138	0.156069	−	0.987746i	$-0.450118\pi$
$277$	1439.02	0.312138	0.156069	+	0.987746i	$0.450118\pi$
$278$	−1363.00	−0.294054
$279$	328.143	0.0704136
$280$	0	0
$281$	−7641.31	−1.62221	−0.811107	−	0.584897i	$-0.801135\pi$
$281$	−7641.31	−1.62221	−0.811107	+	0.584897i	$0.801135\pi$
$282$	3831.20	0.809024
$283$	2139.12	0.449319	0.224660	−	0.974437i	$-0.427873\pi$
$283$	2139.12	0.449319	0.224660	+	0.974437i	$0.427873\pi$
$284$	112.950	0.0235998
$285$	0	0
$286$	−108.091	−0.0223481
$287$	−1251.02	−0.257300
$288$	270.104	0.0552639
$289$	6694.10	1.36253
$290$	0	0
$291$	3099.37	0.624359
$292$	−98.3375	−0.0197081
$293$	−8705.33	−1.73574	−0.867869	−	0.496794i	$-0.834511\pi$
$293$	−8705.33	−1.73574	−0.867869	+	0.496794i	$0.834511\pi$
$294$	−2548.39	−0.505528
$295$	0	0
$296$	698.978	0.137254
$297$	82.8911	0.0161947
$298$	8406.58	1.63416
$299$	1828.84	0.353728
$300$	0	0
$301$	2336.56	0.447433
$302$	−4438.61	−0.845738
$303$	4358.68	0.826401
$304$	−5317.82	−1.00328
$305$	0	0
$306$	2626.07	0.490596
$307$	7898.09	1.46830	0.734150	−	0.678988i	$-0.237581\pi$
$307$	7898.09	1.46830	0.734150	+	0.678988i	$0.237581\pi$
$308$	−11.0597	−0.00204606
$309$	−543.626	−0.100084
$310$	0	0
$311$	4176.48	0.761501	0.380750	−	0.924678i	$-0.375666\pi$
$311$	4176.48	0.761501	0.380750	+	0.924678i	$0.375666\pi$
$312$	915.234	0.166073
$313$	−5630.55	−1.01680	−0.508399	−	0.861122i	$-0.669762\pi$
$313$	−5630.55	−1.01680	−0.508399	+	0.861122i	$0.669762\pi$
$314$	2672.36	0.480286
$315$	0	0
$316$	283.817	0.0505251
$317$	−5820.53	−1.03127	−0.515637	−	0.856807i	$-0.672445\pi$
$317$	−5820.53	−1.03127	−0.515637	+	0.856807i	$0.672445\pi$
$318$	3021.28	0.532782
$319$	632.308	0.110980
$320$	0	0
$321$	−3079.05	−0.535377
$322$	−2064.20	−0.357246
$323$	9837.53	1.69466
$324$	−53.8601	−0.00923528
$325$	0	0
$326$	445.213	0.0756382
$327$	4462.91	0.754738
$328$	5418.93	0.912227
$329$	2554.64	0.428090
$330$	0	0
$331$	958.521	0.159169	0.0795847	−	0.996828i	$-0.474641\pi$
$331$	958.521	0.159169	0.0795847	+	0.996828i	$0.474641\pi$
$332$	332.669	0.0549928
$333$	−268.064	−0.0441136
$334$	−10081.4	−1.65159
$335$	0	0
$336$	−946.558	−0.153687
$337$	12029.2	1.94442	0.972211	−	0.234105i	$-0.0752161\pi$
$337$	12029.2	1.94442	0.972211	+	0.234105i	$0.0752161\pi$
$338$	457.708	0.0736569
$339$	−191.880	−0.0307418
$340$	0	0
$341$	111.935	0.0177759
$342$	2225.71	0.351908
$343$	−3557.54	−0.560026
$344$	−10121.1	−1.58632
$345$	0	0
$346$	−8255.17	−1.28266
$347$	9167.14	1.41821	0.709103	−	0.705105i	$-0.249100\pi$
$347$	9167.14	1.41821	0.709103	+	0.705105i	$0.249100\pi$
$348$	−410.855	−0.0632878
$349$	−1439.35	−0.220765	−0.110382	−	0.993889i	$-0.535208\pi$
$349$	−1439.35	−0.220765	−0.110382	+	0.993889i	$0.535208\pi$
$350$	0	0
$351$	−351.000	−0.0533761
$352$	92.1366	0.0139514
$353$	−3422.82	−0.516086	−0.258043	−	0.966133i	$-0.583078\pi$
$353$	−3422.82	−0.516086	−0.258043	+	0.966133i	$0.583078\pi$
$354$	3746.05	0.562431
$355$	0	0
$356$	903.876	0.134566
$357$	1751.06	0.259596
$358$	10474.3	1.54633
$359$	−8285.14	−1.21803	−0.609015	−	0.793159i	$-0.708435\pi$
$359$	−8285.14	−1.21803	−0.609015	+	0.793159i	$0.708435\pi$
$360$	0	0
$361$	1478.74	0.215592
$362$	−3363.70	−0.488376
$363$	−3964.72	−0.573262
$364$	46.8320	0.00674358
$365$	0	0
$366$	5925.24	0.846221
$367$	−5545.58	−0.788765	−0.394383	−	0.918946i	$-0.629042\pi$
$367$	−5545.58	−0.788765	−0.394383	+	0.918946i	$0.629042\pi$
$368$	8192.97	1.16057
$369$	−2078.21	−0.293190
$370$	0	0
$371$	2014.58	0.281918
$372$	−72.7317	−0.0101370
$373$	−12633.8	−1.75376	−0.876878	−	0.480712i	$-0.840378\pi$
$373$	−12633.8	−1.75376	−0.876878	+	0.480712i	$0.840378\pi$
$374$	895.793	0.123851
$375$	0	0
$376$	−11065.7	−1.51774
$377$	−2677.49	−0.365777
$378$	396.171	0.0539070
$379$	828.732	0.112319	0.0561597	−	0.998422i	$-0.482114\pi$
$379$	828.732	0.112319	0.0561597	+	0.998422i	$0.482114\pi$
$380$	0	0
$381$	−3271.26	−0.439873
$382$	12319.2	1.65002
$383$	−4358.99	−0.581551	−0.290775	−	0.956791i	$-0.593913\pi$
$383$	−4358.99	−0.581551	−0.290775	+	0.956791i	$0.593913\pi$
$384$	3725.62	0.495110
$385$	0	0
$386$	−4227.14	−0.557398
$387$	3881.53	0.509843
$388$	−686.966	−0.0898850
$389$	7099.72	0.925372	0.462686	−	0.886522i	$-0.346886\pi$
$389$	7099.72	0.925372	0.462686	+	0.886522i	$0.346886\pi$
$390$	0	0
$391$	−15156.3	−1.96033
$392$	7360.55	0.948377
$393$	4731.58	0.607320
$394$	−92.8522	−0.0118726
$395$	0	0
$396$	−18.3725	−0.00233145
$397$	−3687.56	−0.466180	−0.233090	−	0.972455i	$-0.574884\pi$
$397$	−3687.56	−0.466180	−0.233090	+	0.972455i	$0.574884\pi$
$398$	−12323.8	−1.55210
$399$	1484.10	0.186210
$400$	0	0
$401$	−4891.98	−0.609211	−0.304606	−	0.952479i	$-0.598525\pi$
$401$	−4891.98	−0.609211	−0.304606	+	0.952479i	$0.598525\pi$
$402$	1877.52	0.232941
$403$	−473.984	−0.0585876
$404$	−966.086	−0.118972
$405$	0	0
$406$	3022.06	0.369415
$407$	−91.4408	−0.0111365
$408$	−7584.91	−0.920365
$409$	12871.0	1.55607	0.778033	−	0.628223i	$-0.216218\pi$
$409$	12871.0	1.55607	0.778033	+	0.628223i	$0.216218\pi$
$410$	0	0
$411$	−601.104	−0.0721418
$412$	120.493	0.0144084
$413$	2497.86	0.297607
$414$	−3429.08	−0.407077
$415$	0	0
$416$	−390.150	−0.0459824
$417$	−1509.78	−0.177301
$418$	759.224	0.0888394
$419$	−13348.5	−1.55637	−0.778183	−	0.628038i	$-0.783858\pi$
$419$	−13348.5	−1.55637	−0.778183	+	0.628038i	$0.783858\pi$
$420$	0	0
$421$	12034.2	1.39314	0.696571	−	0.717488i	$-0.254708\pi$
$421$	12034.2	1.39314	0.696571	+	0.717488i	$0.254708\pi$
$422$	−21.4263	−0.00247160
$423$	4243.80	0.487802
$424$	−8726.39	−0.999508
$425$	0	0
$426$	−1380.15	−0.156968
$427$	3950.93	0.447773
$428$	682.462	0.0770748
$429$	−119.732	−0.0134748
$430$	0	0
$431$	7224.57	0.807413	0.403707	−	0.914888i	$-0.367722\pi$
$431$	7224.57	0.807413	0.403707	+	0.914888i	$0.367722\pi$
$432$	−1572.44	−0.175125
$433$	−4808.44	−0.533669	−0.266835	−	0.963742i	$-0.585978\pi$
$433$	−4808.44	−0.533669	−0.266835	+	0.963742i	$0.585978\pi$
$434$	534.982	0.0591704
$435$	0	0
$436$	−989.189	−0.108655
$437$	−12845.7	−1.40616
$438$	1201.60	0.131084
$439$	−813.739	−0.0884684	−0.0442342	−	0.999021i	$-0.514085\pi$
$439$	−813.739	−0.0884684	−0.0442342	+	0.999021i	$0.514085\pi$
$440$	0	0
$441$	−2822.83	−0.304809
$442$	−3793.21	−0.408201
$443$	3538.13	0.379462	0.189731	−	0.981836i	$-0.439238\pi$
$443$	3538.13	0.379462	0.189731	+	0.981836i	$0.439238\pi$
$444$	59.4155	0.00635076
$445$	0	0
$446$	−11170.0	−1.18590
$447$	9311.91	0.985320
$448$	2964.51	0.312634
$449$	−2525.20	−0.265415	−0.132708	−	0.991155i	$-0.542367\pi$
$449$	−2525.20	−0.265415	−0.132708	+	0.991155i	$0.542367\pi$
$450$	0	0
$451$	−708.909	−0.0740160
$452$	42.5295	0.00442571
$453$	−4916.61	−0.509939
$454$	3362.08	0.347555
$455$	0	0
$456$	−6428.55	−0.660185
$457$	6979.12	0.714375	0.357188	−	0.934033i	$-0.383736\pi$
$457$	6979.12	0.714375	0.357188	+	0.934033i	$0.383736\pi$
$458$	256.779	0.0261976
$459$	2908.88	0.295806
$460$	0	0
$461$	13807.6	1.39498	0.697489	−	0.716595i	$-0.254301\pi$
$461$	13807.6	1.39498	0.697489	+	0.716595i	$0.254301\pi$
$462$	135.140	0.0136088
$463$	5081.40	0.510049	0.255024	−	0.966935i	$-0.417917\pi$
$463$	5081.40	0.510049	0.255024	+	0.966935i	$0.417917\pi$
$464$	−11994.8	−1.20010
$465$	0	0
$466$	−3844.14	−0.382138
$467$	−6741.20	−0.667978	−0.333989	−	0.942577i	$-0.608395\pi$
$467$	−6741.20	−0.667978	−0.333989	+	0.942577i	$0.608395\pi$
$468$	77.7980	0.00768422
$469$	1251.92	0.123259
$470$	0	0
$471$	2960.15	0.289589
$472$	−10819.8	−1.05513
$473$	1324.05	0.128710
$474$	−3468.00	−0.336056
$475$	0	0
$476$	−388.116	−0.0373724
$477$	3346.65	0.321242
$478$	−8575.91	−0.820613
$479$	3261.59	0.311119	0.155559	−	0.987827i	$-0.450282\pi$
$479$	3261.59	0.311119	0.155559	+	0.987827i	$0.450282\pi$
$480$	0	0
$481$	387.204	0.0367047
$482$	9172.11	0.866760
$483$	−2286.50	−0.215402
$484$	878.768	0.0825289
$485$	0	0
$486$	658.125	0.0614262
$487$	6121.95	0.569634	0.284817	−	0.958582i	$-0.408067\pi$
$487$	6121.95	0.569634	0.284817	+	0.958582i	$0.408067\pi$
$488$	−17113.9	−1.58752
$489$	493.159	0.0456062
$490$	0	0
$491$	−13314.3	−1.22376	−0.611880	−	0.790951i	$-0.709586\pi$
$491$	−13314.3	−1.22376	−0.611880	+	0.790951i	$0.709586\pi$
$492$	460.628	0.0422087
$493$	22189.5	2.02711
$494$	−3214.91	−0.292805
$495$	0	0
$496$	−2123.39	−0.192224
$497$	−920.280	−0.0830587
$498$	−4064.93	−0.365771
$499$	−15727.4	−1.41093	−0.705467	−	0.708743i	$-0.749263\pi$
$499$	−15727.4	−1.41093	−0.705467	+	0.708743i	$0.749263\pi$
$500$	0	0
$501$	−11167.1	−0.995830
$502$	1156.06	0.102784
$503$	−13039.0	−1.15583	−0.577913	−	0.816098i	$-0.696133\pi$
$503$	−13039.0	−1.15583	−0.577913	+	0.816098i	$0.696133\pi$
$504$	−1144.27	−0.101130
$505$	0	0
$506$	−1169.71	−0.102767
$507$	507.000	0.0444116
$508$	725.064	0.0633258
$509$	−6419.60	−0.559025	−0.279513	−	0.960142i	$-0.590173\pi$
$509$	−6419.60	−0.559025	−0.279513	+	0.960142i	$0.590173\pi$
$510$	0	0
$511$	801.223	0.0693621
$512$	−12681.5	−1.09463
$513$	2465.40	0.212184
$514$	−16099.7	−1.38157
$515$	0	0
$516$	−860.329	−0.0733989
$517$	1447.63	0.123146
$518$	−437.034	−0.0370698
$519$	−9144.19	−0.773382
$520$	0	0
$521$	−10256.4	−0.862460	−0.431230	−	0.902242i	$-0.641920\pi$
$521$	−10256.4	−0.862460	−0.431230	+	0.902242i	$0.641920\pi$
$522$	5020.29	0.420943
$523$	16477.9	1.37768	0.688839	−	0.724914i	$-0.258120\pi$
$523$	16477.9	1.37768	0.688839	+	0.724914i	$0.258120\pi$
$524$	−1048.74	−0.0874320
$525$	0	0
$526$	381.147	0.0315947
$527$	3928.10	0.324688
$528$	−536.382	−0.0442103
$529$	7623.89	0.626604
$530$	0	0
$531$	4149.48	0.339118
$532$	−328.945	−0.0268075
$533$	3001.85	0.243949
$534$	−11044.6	−0.895030
$535$	0	0
$536$	−5422.86	−0.437000
$537$	11602.3	0.932362
$538$	1294.75	0.103756
$539$	−962.913	−0.0769491
$540$	0	0
$541$	4990.81	0.396621	0.198310	−	0.980139i	$-0.436455\pi$
$541$	4990.81	0.396621	0.198310	+	0.980139i	$0.436455\pi$
$542$	8383.33	0.664382
$543$	−3725.95	−0.294467
$544$	3233.33	0.254831
$545$	0	0
$546$	−572.247	−0.0448533
$547$	1666.16	0.130237	0.0651187	−	0.997878i	$-0.479257\pi$
$547$	1666.16	0.130237	0.0651187	+	0.997878i	$0.479257\pi$
$548$	133.233	0.0103858
$549$	6563.34	0.510231
$550$	0	0
$551$	18806.5	1.45406
$552$	9904.24	0.763682
$553$	−2312.45	−0.177822
$554$	3897.34	0.298885
$555$	0	0
$556$	334.638	0.0255249
$557$	−19693.7	−1.49811	−0.749057	−	0.662506i	$-0.769493\pi$
$557$	−19693.7	−1.49811	−0.749057	+	0.662506i	$0.769493\pi$
$558$	888.719	0.0674238
$559$	−5606.66	−0.424215
$560$	0	0
$561$	992.264	0.0746763
$562$	−20695.2	−1.55334
$563$	−17759.0	−1.32940	−0.664700	−	0.747110i	$-0.731441\pi$
$563$	−17759.0	−1.32940	−0.664700	+	0.747110i	$0.731441\pi$
$564$	−940.623	−0.0702259
$565$	0	0
$566$	5793.44	0.430241
$567$	438.836	0.0325033
$568$	3986.30	0.294475
$569$	3979.03	0.293163	0.146581	−	0.989199i	$-0.453173\pi$
$569$	3979.03	0.293163	0.146581	+	0.989199i	$0.453173\pi$
$570$	0	0
$571$	25536.4	1.87156	0.935782	−	0.352578i	$-0.114695\pi$
$571$	25536.4	1.87156	0.935782	+	0.352578i	$0.114695\pi$
$572$	26.5381	0.00193988
$573$	13645.9	0.994882
$574$	−3388.17	−0.246375
$575$	0	0
$576$	4924.69	0.356242
$577$	2840.80	0.204964	0.102482	−	0.994735i	$-0.467322\pi$
$577$	2840.80	0.204964	0.102482	+	0.994735i	$0.467322\pi$
$578$	18129.8	1.30467
$579$	−4682.37	−0.336084
$580$	0	0
$581$	−2710.49	−0.193546
$582$	8394.13	0.597848
$583$	1141.59	0.0810977
$584$	−3470.60	−0.245915
$585$	0	0
$586$	−23576.9	−1.66204
$587$	−25276.9	−1.77732	−0.888661	−	0.458565i	$-0.848364\pi$
$587$	−25276.9	−1.77732	−0.888661	+	0.458565i	$0.848364\pi$
$588$	625.672	0.0438814
$589$	3329.23	0.232901
$590$	0	0
$591$	−102.852	−0.00715863
$592$	1734.62	0.120427
$593$	7804.98	0.540492	0.270246	−	0.962791i	$-0.412895\pi$
$593$	7804.98	0.540492	0.270246	+	0.962791i	$0.412895\pi$
$594$	224.497	0.0155071
$595$	0	0
$596$	−2063.95	−0.141850
$597$	−13651.0	−0.935842
$598$	4953.11	0.338708
$599$	−25470.2	−1.73737	−0.868684	−	0.495367i	$-0.835034\pi$
$599$	−25470.2	−1.73737	−0.868684	+	0.495367i	$0.835034\pi$
$600$	0	0
$601$	24252.1	1.64603	0.823014	−	0.568021i	$-0.192291\pi$
$601$	24252.1	1.64603	0.823014	+	0.568021i	$0.192291\pi$
$602$	6328.19	0.428435
$603$	2079.71	0.140452
$604$	1089.75	0.0734128
$605$	0	0
$606$	11804.7	0.791312
$607$	2614.50	0.174826	0.0874128	−	0.996172i	$-0.472140\pi$
$607$	2614.50	0.174826	0.0874128	+	0.996172i	$0.472140\pi$
$608$	2740.39	0.182792
$609$	3347.52	0.222739
$610$	0	0
$611$	−6129.93	−0.405876
$612$	−644.743	−0.0425853
$613$	−17271.2	−1.13797	−0.568987	−	0.822347i	$-0.692664\pi$
$613$	−17271.2	−1.13797	−0.568987	+	0.822347i	$0.692664\pi$
$614$	21390.6	1.40595
$615$	0	0
$616$	−390.327	−0.0255304
$617$	5488.99	0.358150	0.179075	−	0.983835i	$-0.442690\pi$
$617$	5488.99	0.358150	0.179075	+	0.983835i	$0.442690\pi$
$618$	−1472.32	−0.0958340
$619$	−1624.78	−0.105502	−0.0527509	−	0.998608i	$-0.516799\pi$
$619$	−1624.78	−0.105502	−0.0527509	+	0.998608i	$0.516799\pi$
$620$	0	0
$621$	−3798.36	−0.245448
$622$	11311.3	0.729167
$623$	−7364.50	−0.473600
$624$	2271.30	0.145712
$625$	0	0
$626$	−15249.4	−0.973624
$627$	840.988	0.0535659
$628$	−656.108	−0.0416904
$629$	−3208.91	−0.203415
$630$	0	0
$631$	6502.32	0.410227	0.205114	−	0.978738i	$-0.434244\pi$
$631$	6502.32	0.410227	0.205114	+	0.978738i	$0.434244\pi$
$632$	10016.7	0.630445
$633$	−23.7337	−0.00149025
$634$	−15763.9	−0.987485
$635$	0	0
$636$	−741.773	−0.0462472
$637$	4077.43	0.253616
$638$	1712.50	0.106267
$639$	−1528.78	−0.0946442
$640$	0	0
$641$	3787.91	0.233406	0.116703	−	0.993167i	$-0.462767\pi$
$641$	3787.91	0.233406	0.116703	+	0.993167i	$0.462767\pi$
$642$	−8339.09	−0.512644
$643$	13088.1	0.802710	0.401355	−	0.915922i	$-0.368539\pi$
$643$	13088.1	0.802710	0.401355	+	0.915922i	$0.368539\pi$
$644$	506.795	0.0310101
$645$	0	0
$646$	26643.3	1.62270
$647$	−25582.4	−1.55448	−0.777239	−	0.629205i	$-0.783381\pi$
$647$	−25582.4	−1.55448	−0.777239	+	0.629205i	$0.783381\pi$
$648$	−1900.87	−0.115236
$649$	1415.45	0.0856106
$650$	0	0
$651$	592.595	0.0356769
$652$	−109.307	−0.00656564
$653$	−26811.4	−1.60675	−0.803376	−	0.595472i	$-0.796965\pi$
$653$	−26811.4	−1.60675	−0.803376	+	0.595472i	$0.796965\pi$
$654$	12087.0	0.722692
$655$	0	0
$656$	13447.9	0.800386
$657$	1331.00	0.0790371
$658$	6918.80	0.409913
$659$	30688.3	1.81403	0.907014	−	0.421100i	$-0.138356\pi$
$659$	30688.3	1.81403	0.907014	+	0.421100i	$0.138356\pi$
$660$	0	0
$661$	−7558.15	−0.444747	−0.222374	−	0.974962i	$-0.571380\pi$
$661$	−7558.15	−0.444747	−0.222374	+	0.974962i	$0.571380\pi$
$662$	2595.99	0.152411
$663$	−4201.71	−0.246125
$664$	11740.8	0.686192
$665$	0	0
$666$	−726.006	−0.0422405
$667$	−28974.6	−1.68201
$668$	2475.16	0.143363
$669$	−12372.9	−0.715043
$670$	0	0
$671$	2238.86	0.128808
$672$	487.783	0.0280009
$673$	26209.2	1.50118	0.750589	−	0.660770i	$-0.229770\pi$
$673$	26209.2	1.50118	0.750589	+	0.660770i	$0.229770\pi$
$674$	32579.0	1.86186
$675$	0	0
$676$	−112.375	−0.00639365
$677$	8585.42	0.487392	0.243696	−	0.969852i	$-0.421640\pi$
$677$	8585.42	0.487392	0.243696	+	0.969852i	$0.421640\pi$
$678$	−519.674	−0.0294365
$679$	5597.18	0.316348
$680$	0	0
$681$	3724.15	0.209559
$682$	303.156	0.0170212
$683$	10219.8	0.572548	0.286274	−	0.958148i	$-0.407583\pi$
$683$	10219.8	0.572548	0.286274	+	0.958148i	$0.407583\pi$
$684$	−546.449	−0.0305468
$685$	0	0
$686$	−9635.00	−0.536248
$687$	284.433	0.0157959
$688$	−25117.1	−1.39183
$689$	−4834.05	−0.267290
$690$	0	0
$691$	−12144.3	−0.668583	−0.334292	−	0.942470i	$-0.608497\pi$
$691$	−12144.3	−0.668583	−0.334292	+	0.942470i	$0.608497\pi$
$692$	2026.78	0.111339
$693$	149.694	0.00820547
$694$	24827.6	1.35799
$695$	0	0
$696$	−14500.2	−0.789695
$697$	−24877.6	−1.35194
$698$	−3898.25	−0.211391
$699$	−4258.13	−0.230411
$700$	0	0
$701$	−18328.1	−0.987505	−0.493753	−	0.869602i	$-0.664375\pi$
$701$	−18328.1	−0.987505	−0.493753	+	0.869602i	$0.664375\pi$
$702$	−950.624	−0.0511097
$703$	−2719.70	−0.145911
$704$	1679.89	0.0899335
$705$	0	0
$706$	−9270.14	−0.494173
$707$	7871.37	0.418718
$708$	−919.718	−0.0488208
$709$	−8350.50	−0.442327	−0.221163	−	0.975237i	$-0.570985\pi$
$709$	−8350.50	−0.442327	−0.221163	+	0.975237i	$0.570985\pi$
$710$	0	0
$711$	−3841.48	−0.202625
$712$	31900.2	1.67909
$713$	−5129.24	−0.269413
$714$	4742.44	0.248573
$715$	0	0
$716$	−2571.62	−0.134226
$717$	−9499.47	−0.494790
$718$	−22438.9	−1.16631
$719$	−4163.60	−0.215961	−0.107980	−	0.994153i	$-0.534438\pi$
$719$	−4163.60	−0.215961	−0.107980	+	0.994153i	$0.534438\pi$
$720$	0	0
$721$	−981.739	−0.0507100
$722$	4004.93	0.206438
$723$	10159.9	0.522614
$724$	825.844	0.0423926
$725$	0	0
$726$	−10737.8	−0.548921
$727$	3086.79	0.157473	0.0787365	−	0.996895i	$-0.474911\pi$
$727$	3086.79	0.157473	0.0787365	+	0.996895i	$0.474911\pi$
$728$	1652.83	0.0841454
$729$	729.000	0.0370370
$730$	0	0
$731$	46464.6	2.35097
$732$	−1454.74	−0.0734547
$733$	−19694.1	−0.992387	−0.496194	−	0.868212i	$-0.665269\pi$
$733$	−19694.1	−0.992387	−0.496194	+	0.868212i	$0.665269\pi$
$734$	−15019.3	−0.755274
$735$	0	0
$736$	−4222.02	−0.211448
$737$	709.423	0.0354572
$738$	−5628.47	−0.280741
$739$	30767.8	1.53155	0.765773	−	0.643111i	$-0.222357\pi$
$739$	30767.8	1.53155	0.765773	+	0.643111i	$0.222357\pi$
$740$	0	0
$741$	−3561.14	−0.176548
$742$	5456.15	0.269948
$743$	18338.7	0.905494	0.452747	−	0.891639i	$-0.350444\pi$
$743$	18338.7	0.905494	0.452747	+	0.891639i	$0.350444\pi$
$744$	−2566.90	−0.126488
$745$	0	0
$746$	−34216.4	−1.67929
$747$	−4502.70	−0.220542
$748$	−219.932	−0.0107507
$749$	−5560.49	−0.271263
$750$	0	0
$751$	28944.7	1.40640	0.703199	−	0.710993i	$-0.251754\pi$
$751$	28944.7	1.40640	0.703199	+	0.710993i	$0.251754\pi$
$752$	−27461.3	−1.33166
$753$	1280.56	0.0619735
$754$	−7251.54	−0.350246
$755$	0	0
$756$	−97.2665	−0.00467930
$757$	−19155.4	−0.919704	−0.459852	−	0.887996i	$-0.652097\pi$
$757$	−19155.4	−0.919704	−0.459852	+	0.887996i	$0.652097\pi$
$758$	2244.48	0.107550
$759$	−1295.68	−0.0619634
$760$	0	0
$761$	−7064.25	−0.336503	−0.168252	−	0.985744i	$-0.553812\pi$
$761$	−7064.25	−0.336503	−0.168252	+	0.985744i	$0.553812\pi$
$762$	−8859.66	−0.421196
$763$	8059.61	0.382408
$764$	−3024.58	−0.143227
$765$	0	0
$766$	−11805.6	−0.556858
$767$	−5993.69	−0.282164
$768$	−3042.29	−0.142942
$769$	31907.6	1.49625	0.748126	−	0.663556i	$-0.230954\pi$
$769$	31907.6	1.49625	0.748126	+	0.663556i	$0.230954\pi$
$770$	0	0
$771$	−17833.5	−0.833018
$772$	1037.83	0.0483839
$773$	4714.82	0.219379	0.109690	−	0.993966i	$-0.465014\pi$
$773$	4714.82	0.219379	0.109690	+	0.993966i	$0.465014\pi$
$774$	10512.5	0.488195
$775$	0	0
$776$	−24244.9	−1.12157
$777$	−484.099	−0.0223513
$778$	19228.4	0.886081
$779$	−21084.9	−0.969760
$780$	0	0
$781$	−521.491	−0.0238930
$782$	−41048.4	−1.87709
$783$	5560.94	0.253808
$784$	18266.4	0.832104
$785$	0	0
$786$	12814.7	0.581533
$787$	13924.7	0.630701	0.315351	−	0.948975i	$-0.397878\pi$
$787$	13924.7	0.630701	0.315351	+	0.948975i	$0.397878\pi$
$788$	22.7967	0.00103058
$789$	422.194	0.0190501
$790$	0	0
$791$	−346.517	−0.0155762
$792$	−648.416	−0.0290915
$793$	−9480.38	−0.424538
$794$	−9987.14	−0.446386
$795$	0	0
$796$	3025.70	0.134727
$797$	3504.66	0.155761	0.0778805	−	0.996963i	$-0.475185\pi$
$797$	3504.66	0.155761	0.0778805	+	0.996963i	$0.475185\pi$
$798$	4019.43	0.178304
$799$	50801.2	2.24933
$800$	0	0
$801$	−12234.0	−0.539660
$802$	−13249.1	−0.583344
$803$	454.026	0.0199530
$804$	−460.962	−0.0202200
$805$	0	0
$806$	−1283.71	−0.0561000
$807$	1434.19	0.0625597
$808$	−34095.8	−1.48451
$809$	34408.4	1.49534	0.747672	−	0.664068i	$-0.231172\pi$
$809$	34408.4	1.49534	0.747672	+	0.664068i	$0.231172\pi$
$810$	0	0
$811$	43517.1	1.88421	0.942104	−	0.335320i	$-0.108844\pi$
$811$	43517.1	1.88421	0.942104	+	0.335320i	$0.108844\pi$
$812$	−741.966	−0.0320664
$813$	9286.16	0.400590
$814$	−247.652	−0.0106636
$815$	0	0
$816$	−18823.1	−0.807527
$817$	39380.8	1.68637
$818$	34859.0	1.49000
$819$	−633.874	−0.0270444
$820$	0	0
$821$	22906.2	0.973728	0.486864	−	0.873478i	$-0.338141\pi$
$821$	22906.2	0.973728	0.486864	+	0.873478i	$0.338141\pi$
$822$	−1627.99	−0.0690787
$823$	19227.0	0.814351	0.407175	−	0.913350i	$-0.366514\pi$
$823$	19227.0	0.814351	0.407175	+	0.913350i	$0.366514\pi$
$824$	4252.52	0.179786
$825$	0	0
$826$	6765.03	0.284970
$827$	−11679.2	−0.491082	−0.245541	−	0.969386i	$-0.578966\pi$
$827$	−11679.2	−0.491082	−0.245541	+	0.969386i	$0.578966\pi$
$828$	841.894	0.0353356
$829$	6818.05	0.285646	0.142823	−	0.989748i	$-0.454382\pi$
$829$	6818.05	0.285646	0.142823	+	0.989748i	$0.454382\pi$
$830$	0	0
$831$	4317.06	0.180213
$832$	−7113.44	−0.296411
$833$	−33791.3	−1.40552
$834$	−4088.99	−0.169772
$835$	0	0
$836$	−186.402	−0.00771155
$837$	984.428	0.0406533
$838$	−36152.2	−1.49028
$839$	−9262.57	−0.381143	−0.190572	−	0.981673i	$-0.561034\pi$
$839$	−9262.57	−0.381143	−0.190572	+	0.981673i	$0.561034\pi$
$840$	0	0
$841$	18030.9	0.739304
$842$	32592.7	1.33399
$843$	−22923.9	−0.936586
$844$	5.26050	0.000214542 0
$845$	0	0
$846$	11493.6	0.467090
$847$	−7159.93	−0.290458
$848$	−21655.9	−0.876966
$849$	6417.35	0.259415
$850$	0	0
$851$	4190.14	0.168785
$852$	338.849	0.0136253
$853$	−33963.4	−1.36329	−0.681644	−	0.731684i	$-0.738735\pi$
$853$	−33963.4	−1.36329	−0.681644	+	0.731684i	$0.738735\pi$
$854$	10700.4	0.428760
$855$	0	0
$856$	24085.9	0.961729
$857$	28317.8	1.12872	0.564362	−	0.825527i	$-0.309122\pi$
$857$	28317.8	1.12872	0.564362	+	0.825527i	$0.309122\pi$
$858$	−324.273	−0.0129027
$859$	5332.22	0.211796	0.105898	−	0.994377i	$-0.466228\pi$
$859$	5332.22	0.211796	0.105898	+	0.994377i	$0.466228\pi$
$860$	0	0
$861$	−3753.05	−0.148552
$862$	19566.5	0.773130
$863$	14812.3	0.584261	0.292131	−	0.956378i	$-0.405636\pi$
$863$	14812.3	0.584261	0.292131	+	0.956378i	$0.405636\pi$
$864$	810.311	0.0319067
$865$	0	0
$866$	−13022.8	−0.511009
$867$	20082.3	0.786656
$868$	−131.347	−0.00513618
$869$	−1310.39	−0.0511529
$870$	0	0
$871$	−3004.03	−0.116863
$872$	−34911.2	−1.35578
$873$	9298.12	0.360474
$874$	−34790.3	−1.34645
$875$	0	0
$876$	−295.013	−0.0113785
$877$	33819.1	1.30216	0.651078	−	0.759011i	$-0.274317\pi$
$877$	33819.1	1.30216	0.651078	+	0.759011i	$0.274317\pi$
$878$	−2203.87	−0.0847120
$879$	−26116.0	−1.00213
$880$	0	0
$881$	−7929.69	−0.303244	−0.151622	−	0.988439i	$-0.548450\pi$
$881$	−7929.69	−0.303244	−0.151622	+	0.988439i	$0.548450\pi$
$882$	−7645.17	−0.291867
$883$	5420.80	0.206596	0.103298	−	0.994650i	$-0.467060\pi$
$883$	5420.80	0.206596	0.103298	+	0.994650i	$0.467060\pi$
$884$	931.296	0.0354331
$885$	0	0
$886$	9582.43	0.363350
$887$	32323.1	1.22357	0.611784	−	0.791025i	$-0.290452\pi$
$887$	32323.1	1.22357	0.611784	+	0.791025i	$0.290452\pi$
$888$	2096.93	0.0792438
$889$	−5907.60	−0.222873
$890$	0	0
$891$	248.673	0.00935002
$892$	2742.41	0.102940
$893$	43056.3	1.61346
$894$	25219.7	0.943483
$895$	0	0
$896$	6728.13	0.250861
$897$	5486.52	0.204225
$898$	−6839.08	−0.254146
$899$	7509.39	0.278590
$900$	0	0
$901$	40061.7	1.48130
$902$	−1919.96	−0.0708733
$903$	7009.69	0.258325
$904$	1500.98	0.0552234
$905$	0	0
$906$	−13315.8	−0.488287
$907$	7275.07	0.266334	0.133167	−	0.991094i	$-0.457485\pi$
$907$	7275.07	0.266334	0.133167	+	0.991094i	$0.457485\pi$
$908$	−825.445	−0.0301689
$909$	13076.0	0.477123
$910$	0	0
$911$	−9811.78	−0.356837	−0.178419	−	0.983955i	$-0.557098\pi$
$911$	−9811.78	−0.356837	−0.178419	+	0.983955i	$0.557098\pi$
$912$	−15953.4	−0.579245
$913$	−1535.94	−0.0556760
$914$	18901.8	0.684043
$915$	0	0
$916$	−63.0435	−0.00227404
$917$	8544.80	0.307714
$918$	7878.21	0.283246
$919$	−34026.6	−1.22137	−0.610683	−	0.791875i	$-0.709105\pi$
$919$	−34026.6	−1.22137	−0.610683	+	0.791875i	$0.709105\pi$
$920$	0	0
$921$	23694.3	0.847723
$922$	37395.6	1.33575
$923$	2208.24	0.0787488
$924$	−33.1791	−0.00118129
$925$	0	0
$926$	13762.1	0.488392
$927$	−1630.88	−0.0577833
$928$	6181.20	0.218651
$929$	52275.9	1.84620	0.923098	−	0.384565i	$-0.125648\pi$
$929$	52275.9	1.84620	0.923098	+	0.384565i	$0.125648\pi$
$930$	0	0
$931$	−28639.6	−1.00819
$932$	943.800	0.0331708
$933$	12529.5	0.439653
$934$	−18257.4	−0.639616
$935$	0	0
$936$	2745.70	0.0958825
$937$	−2069.73	−0.0721613	−0.0360807	−	0.999349i	$-0.511487\pi$
$937$	−2069.73	−0.0721613	−0.0360807	+	0.999349i	$0.511487\pi$
$938$	3390.63	0.118025
$939$	−16891.7	−0.587048
$940$	0	0
$941$	37728.9	1.30704	0.653522	−	0.756908i	$-0.273291\pi$
$941$	37728.9	1.30704	0.653522	+	0.756908i	$0.273291\pi$
$942$	8017.07	0.277293
$943$	32484.7	1.12179
$944$	−26851.0	−0.925767
$945$	0	0
$946$	3585.97	0.123245
$947$	−33141.5	−1.13723	−0.568614	−	0.822604i	$-0.692520\pi$
$947$	−33141.5	−1.13723	−0.568614	+	0.822604i	$0.692520\pi$
$948$	851.450	0.0291707
$949$	−1922.56	−0.0657629
$950$	0	0
$951$	−17461.6	−0.595406
$952$	−13697.7	−0.466327
$953$	17146.0	0.582807	0.291403	−	0.956600i	$-0.405878\pi$
$953$	17146.0	0.582807	0.291403	+	0.956600i	$0.405878\pi$
$954$	9063.83	0.307602
$955$	0	0
$956$	2105.53	0.0712318
$957$	1896.92	0.0640741
$958$	8833.47	0.297909
$959$	−1085.54	−0.0365525
$960$	0	0
$961$	−28461.6	−0.955377
$962$	1048.68	0.0351462
$963$	−9237.16	−0.309100
$964$	−2251.90	−0.0752375
$965$	0	0
$966$	−6192.60	−0.206256
$967$	−26129.9	−0.868957	−0.434478	−	0.900682i	$-0.643067\pi$
$967$	−26129.9	−0.868957	−0.434478	+	0.900682i	$0.643067\pi$
$968$	31014.1	1.02978
$969$	29512.6	0.978412
$970$	0	0
$971$	−4062.76	−0.134274	−0.0671371	−	0.997744i	$-0.521386\pi$
$971$	−4062.76	−0.134274	−0.0671371	+	0.997744i	$0.521386\pi$
$972$	−161.580	−0.00533199
$973$	−2726.53	−0.0898340
$974$	16580.3	0.545447
$975$	0	0
$976$	−42470.9	−1.39289
$977$	−11364.6	−0.372146	−0.186073	−	0.982536i	$-0.559576\pi$
$977$	−11364.6	−0.372146	−0.186073	+	0.982536i	$0.559576\pi$
$978$	1335.64	0.0436697
$979$	−4173.21	−0.136237
$980$	0	0
$981$	13388.7	0.435748
$982$	−36059.5	−1.17180
$983$	−23502.7	−0.762585	−0.381293	−	0.924454i	$-0.624521\pi$
$983$	−23502.7	−0.762585	−0.381293	+	0.924454i	$0.624521\pi$
$984$	16256.8	0.526675
$985$	0	0
$986$	60096.4	1.94103
$987$	7663.91	0.247158
$988$	789.315	0.0254164
$989$	−60672.7	−1.95074
$990$	0	0
$991$	15653.1	0.501752	0.250876	−	0.968019i	$-0.419281\pi$
$991$	15653.1	0.501752	0.250876	+	0.968019i	$0.419281\pi$
$992$	1094.23	0.0350220
$993$	2875.56	0.0918965
$994$	−2492.42	−0.0795320
$995$	0	0
$996$	998.008	0.0317501
$997$	−12714.3	−0.403876	−0.201938	−	0.979398i	$-0.564724\pi$
$997$	−12714.3	−0.403876	−0.201938	+	0.979398i	$0.564724\pi$
$998$	−42595.1	−1.35103
$999$	−804.192	−0.0254690

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.4.a.bb.1.9		11
5.2	odd	4		195.4.c.c.79.16	yes	22
5.3	odd	4		195.4.c.c.79.7	✓	22
5.4	even	2		975.4.a.bc.1.3		11
15.2	even	4		585.4.c.e.469.7		22
15.8	even	4		585.4.c.e.469.16		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.4.c.c.79.7	✓	22	5.3	odd	4
195.4.c.c.79.16	yes	22	5.2	odd	4
585.4.c.e.469.7		22	15.2	even	4
585.4.c.e.469.16		22	15.8	even	4
975.4.a.bb.1.9		11	1.1	even	1	trivial
975.4.a.bc.1.3		11	5.4	even	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:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	975.a (trivial)

\(f(q)\)	\(=\)	\(q+2.70833 q^{2} +3.00000 q^{3} -0.664940 q^{4} +8.12499 q^{6} +5.41772 q^{7} -23.4675 q^{8} +9.00000 q^{9} +3.07004 q^{11} -1.99482 q^{12} -13.0000 q^{13} +14.6730 q^{14} -58.2383 q^{16} +107.736 q^{17} +24.3750 q^{18} +91.3112 q^{19} +16.2532 q^{21} +8.31469 q^{22} -140.680 q^{23} -70.4026 q^{24} -35.2083 q^{26} +27.0000 q^{27} -3.60246 q^{28} +205.961 q^{29} +36.4603 q^{31} +30.0115 q^{32} +9.21012 q^{33} +291.785 q^{34} -5.98446 q^{36} -29.7849 q^{37} +247.301 q^{38} -39.0000 q^{39} -230.912 q^{41} +44.0190 q^{42} +431.281 q^{43} -2.04139 q^{44} -381.008 q^{46} +471.533 q^{47} -174.715 q^{48} -313.648 q^{49} +323.209 q^{51} +8.64422 q^{52} +371.850 q^{53} +73.1250 q^{54} -127.141 q^{56} +273.934 q^{57} +557.810 q^{58} +461.053 q^{59} +729.260 q^{61} +98.7466 q^{62} +48.7595 q^{63} +547.188 q^{64} +24.9441 q^{66} +231.079 q^{67} -71.6381 q^{68} -422.040 q^{69} -169.865 q^{71} -211.208 q^{72} +147.889 q^{73} -80.6674 q^{74} -60.7165 q^{76} +16.6326 q^{77} -105.625 q^{78} -426.831 q^{79} +81.0000 q^{81} -625.386 q^{82} -500.300 q^{83} -10.8074 q^{84} +1168.05 q^{86} +617.883 q^{87} -72.0463 q^{88} -1359.33 q^{89} -70.4304 q^{91} +93.5438 q^{92} +109.381 q^{93} +1277.07 q^{94} +90.0346 q^{96} +1033.12 q^{97} -849.463 q^{98} +27.6304 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 3 q^{2} + 33 q^{3} + 55 q^{4} - 9 q^{6} + 20 q^{7} - 51 q^{8} + 99 q^{9} + 100 q^{11} + 165 q^{12} - 143 q^{13} + 150 q^{14} + 511 q^{16} - 56 q^{17} - 27 q^{18} + 44 q^{19} + 60 q^{21} + 262 q^{22}+ \cdots + 900 q^{99}+O(q^{100}) \) 11 * q - 3 * q^2 + 33 * q^3 + 55 * q^4 - 9 * q^6 + 20 * q^7 - 51 * q^8 + 99 * q^9 + 100 * q^11 + 165 * q^12 - 143 * q^13 + 150 * q^14 + 511 * q^16 - 56 * q^17 - 27 * q^18 + 44 * q^19 + 60 * q^21 + 262 * q^22 + 98 * q^23 - 153 * q^24 + 39 * q^26 + 297 * q^27 + 590 * q^28 + 280 * q^29 + 256 * q^31 - 311 * q^32 + 300 * q^33 + 78 * q^34 + 495 * q^36 + 196 * q^37 + 180 * q^38 - 429 * q^39 + 700 * q^41 + 450 * q^42 + 1032 * q^43 + 1674 * q^44 + 620 * q^46 - 772 * q^47 + 1533 * q^48 + 1993 * q^49 - 168 * q^51 - 715 * q^52 + 164 * q^53 - 81 * q^54 + 1314 * q^56 + 132 * q^57 + 288 * q^58 + 1882 * q^59 + 1312 * q^61 - 100 * q^62 + 180 * q^63 + 4595 * q^64 + 786 * q^66 + 1720 * q^67 - 746 * q^68 + 294 * q^69 + 790 * q^71 - 459 * q^72 + 272 * q^73 - 342 * q^74 - 608 * q^76 + 1570 * q^77 + 117 * q^78 + 1766 * q^79 + 891 * q^81 + 1502 * q^82 + 864 * q^83 + 1770 * q^84 - 1332 * q^86 + 840 * q^87 + 3282 * q^88 + 3644 * q^89 - 260 * q^91 - 6472 * q^92 + 768 * q^93 + 6716 * q^94 - 933 * q^96 - 558 * q^97 - 8027 * q^98 + 900 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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