LMFDB - Embedded newform 2475.4.a.bl.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2475,4,Mod(1,2475)] 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("2475.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2475, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2475 = 3^{2} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2475.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,2,0,40,0,0,-40,21,0,0,55] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$146.029727264$
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} - 2x^{4} - 38x^{3} + 61x^{2} + 304x - 148 $ x^5 - 2x^4 - 38x^3 + 61x^2 + 304x - 148
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 275)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.457079$ of defining polynomial
Character	$\chi$	$=$	2475.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.457079 q^{2} -7.79108 q^{4} +23.1894 q^{7} -7.21777 q^{8} +11.0000 q^{11} -75.6462 q^{13} +10.5994 q^{14} +59.0295 q^{16} -40.8029 q^{17} +61.5212 q^{19} +5.02787 q^{22} +86.3856 q^{23} -34.5763 q^{26} -180.670 q^{28} +236.491 q^{29} -237.800 q^{31} +84.7233 q^{32} -18.6501 q^{34} -251.864 q^{37} +28.1200 q^{38} -446.841 q^{41} -263.646 q^{43} -85.7019 q^{44} +39.4850 q^{46} +438.728 q^{47} +194.746 q^{49} +589.365 q^{52} +286.114 q^{53} -167.375 q^{56} +108.095 q^{58} +529.452 q^{59} +75.7253 q^{61} -108.693 q^{62} -433.511 q^{64} +384.462 q^{67} +317.898 q^{68} -7.15821 q^{71} +590.216 q^{73} -115.122 q^{74} -479.316 q^{76} +255.083 q^{77} -139.112 q^{79} -204.242 q^{82} -719.121 q^{83} -120.507 q^{86} -79.3954 q^{88} -1647.00 q^{89} -1754.19 q^{91} -673.037 q^{92} +200.533 q^{94} -939.129 q^{97} +89.0144 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 2 q^{2} + 40 q^{4} - 40 q^{7} + 21 q^{8} + 55 q^{11} - 211 q^{13} + 133 q^{14} + 208 q^{16} + 72 q^{17} + 23 q^{19} + 22 q^{22} + 57 q^{23} - 322 q^{26} - 1050 q^{28} + 183 q^{29} - q^{31} - 430 q^{32}+ \cdots - 5586 q^{98}+O(q^{100}) $ 5 * q + 2 * q^2 + 40 * q^4 - 40 * q^7 + 21 * q^8 + 55 * q^11 - 211 * q^13 + 133 * q^14 + 208 * q^16 + 72 * q^17 + 23 * q^19 + 22 * q^22 + 57 * q^23 - 322 * q^26 - 1050 * q^28 + 183 * q^29 - q^31 - 430 * q^32 - 650 * q^34 - 856 * q^37 + 348 * q^38 - 94 * q^41 - 497 * q^43 + 440 * q^44 + 1006 * q^46 + 26 * q^47 + 227 * q^49 - 2086 * q^52 + 264 * q^53 + 835 * q^56 - 843 * q^58 - 1174 * q^59 + 640 * q^61 + 1454 * q^62 - 949 * q^64 - 98 * q^67 + 1767 * q^68 + 673 * q^71 - 1462 * q^73 + 561 * q^74 + 1248 * q^76 - 440 * q^77 + 104 * q^79 + 221 * q^82 - 747 * q^83 - 2497 * q^86 + 231 * q^88 - 2821 * q^89 + 480 * q^91 - 1503 * q^92 - 1787 * q^94 - 615 * q^97 - 5586 * q^98

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 $

$2$	0.457079	0.161602	0.0808009	−	0.996730i	$-0.474252\pi$
$2$	0.457079	0.161602	0.0808009	+	0.996730i	$0.474252\pi$
$3$	0	0
$3$	0	0
$4$	−7.79108	−0.973885
$5$	0	0
$5$	0	0
$6$	0	0
$7$	23.1894	1.25211	0.626054	−	0.779780i	$-0.284669\pi$
$7$	23.1894	1.25211	0.626054	+	0.779780i	$0.284669\pi$
$8$	−7.21777	−0.318983
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−75.6462	−1.61388	−0.806941	−	0.590631i	$-0.798879\pi$
$13$	−75.6462	−1.61388	−0.806941	+	0.590631i	$0.798879\pi$
$14$	10.5994	0.202343
$15$	0	0
$16$	59.0295	0.922337
$17$	−40.8029	−0.582126	−0.291063	−	0.956704i	$-0.594009\pi$
$17$	−40.8029	−0.582126	−0.291063	+	0.956704i	$0.594009\pi$
$18$	0	0
$19$	61.5212	0.742838	0.371419	−	0.928465i	$-0.378871\pi$
$19$	61.5212	0.742838	0.371419	+	0.928465i	$0.378871\pi$
$20$	0	0
$21$	0	0
$22$	5.02787	0.0487247
$23$	86.3856	0.783159	0.391579	−	0.920144i	$-0.371929\pi$
$23$	86.3856	0.783159	0.391579	+	0.920144i	$0.371929\pi$
$24$	0	0
$25$	0	0
$26$	−34.5763	−0.260806
$27$	0	0
$28$	−180.670	−1.21941
$29$	236.491	1.51432	0.757160	−	0.653229i	$-0.226586\pi$
$29$	236.491	1.51432	0.757160	+	0.653229i	$0.226586\pi$
$30$	0	0
$31$	−237.800	−1.37775	−0.688874	−	0.724881i	$-0.741895\pi$
$31$	−237.800	−1.37775	−0.688874	+	0.724881i	$0.741895\pi$
$32$	84.7233	0.468034
$33$	0	0
$34$	−18.6501	−0.0940726
$35$	0	0
$36$	0	0
$37$	−251.864	−1.11909	−0.559543	−	0.828801i	$-0.689024\pi$
$37$	−251.864	−1.11909	−0.559543	+	0.828801i	$0.689024\pi$
$38$	28.1200	0.120044
$39$	0	0
$40$	0	0
$41$	−446.841	−1.70207	−0.851035	−	0.525108i	$-0.824025\pi$
$41$	−446.841	−1.70207	−0.851035	+	0.525108i	$0.824025\pi$
$42$	0	0
$43$	−263.646	−0.935015	−0.467507	−	0.883989i	$-0.654848\pi$
$43$	−263.646	−0.935015	−0.467507	+	0.883989i	$0.654848\pi$
$44$	−85.7019	−0.293637
$45$	0	0
$46$	39.4850	0.126560
$47$	438.728	1.36160	0.680799	−	0.732471i	$-0.261633\pi$
$47$	438.728	1.36160	0.680799	+	0.732471i	$0.261633\pi$
$48$	0	0
$49$	194.746	0.567773
$50$	0	0
$51$	0	0
$52$	589.365	1.57174
$53$	286.114	0.741525	0.370763	−	0.928728i	$-0.379096\pi$
$53$	286.114	0.741525	0.370763	+	0.928728i	$0.379096\pi$
$54$	0	0
$55$	0	0
$56$	−167.375	−0.399401
$57$	0	0
$58$	108.095	0.244717
$59$	529.452	1.16828	0.584142	−	0.811651i	$-0.301431\pi$
$59$	529.452	1.16828	0.584142	+	0.811651i	$0.301431\pi$
$60$	0	0
$61$	75.7253	0.158945	0.0794724	−	0.996837i	$-0.474676\pi$
$61$	75.7253	0.158945	0.0794724	+	0.996837i	$0.474676\pi$
$62$	−108.693	−0.222646
$63$	0	0
$64$	−433.511	−0.846702
$65$	0	0
$66$	0	0
$67$	384.462	0.701037	0.350518	−	0.936556i	$-0.386005\pi$
$67$	384.462	0.701037	0.350518	+	0.936556i	$0.386005\pi$
$68$	317.898	0.566924
$69$	0	0
$70$	0	0
$71$	−7.15821	−0.0119651	−0.00598256	−	0.999982i	$-0.501904\pi$
$71$	−7.15821	−0.0119651	−0.00598256	+	0.999982i	$0.501904\pi$
$72$	0	0
$73$	590.216	0.946295	0.473148	−	0.880983i	$-0.343118\pi$
$73$	590.216	0.946295	0.473148	+	0.880983i	$0.343118\pi$
$74$	−115.122	−0.180846
$75$	0	0
$76$	−479.316	−0.723439
$77$	255.083	0.377525
$78$	0	0
$79$	−139.112	−0.198118	−0.0990589	−	0.995082i	$-0.531583\pi$
$79$	−139.112	−0.198118	−0.0990589	+	0.995082i	$0.531583\pi$
$80$	0	0
$81$	0	0
$82$	−204.242	−0.275058
$83$	−719.121	−0.951009	−0.475505	−	0.879713i	$-0.657735\pi$
$83$	−719.121	−0.951009	−0.475505	+	0.879713i	$0.657735\pi$
$84$	0	0
$85$	0	0
$86$	−120.507	−0.151100
$87$	0	0
$88$	−79.3954	−0.0961770
$89$	−1647.00	−1.96159	−0.980794	−	0.195046i	$-0.937515\pi$
$89$	−1647.00	−1.96159	−0.980794	+	0.195046i	$0.937515\pi$
$90$	0	0
$91$	−1754.19	−2.02075
$92$	−673.037	−0.762706
$93$	0	0
$94$	200.533	0.220036
$95$	0	0
$96$	0	0
$97$	−939.129	−0.983032	−0.491516	−	0.870868i	$-0.663557\pi$
$97$	−939.129	−0.983032	−0.491516	+	0.870868i	$0.663557\pi$
$98$	89.0144	0.0917532
$99$	0	0
$100$	0	0
$101$	611.529	0.602470	0.301235	−	0.953550i	$-0.402601\pi$
$101$	611.529	0.602470	0.301235	+	0.953550i	$0.402601\pi$
$102$	0	0
$103$	1039.22	0.994148	0.497074	−	0.867708i	$-0.334408\pi$
$103$	1039.22	0.994148	0.497074	+	0.867708i	$0.334408\pi$
$104$	545.996	0.514801
$105$	0	0
$106$	130.777	0.119832
$107$	−272.253	−0.245979	−0.122989	−	0.992408i	$-0.539248\pi$
$107$	−272.253	−0.245979	−0.122989	+	0.992408i	$0.539248\pi$
$108$	0	0
$109$	1320.39	1.16028	0.580139	−	0.814517i	$-0.302998\pi$
$109$	1320.39	1.16028	0.580139	+	0.814517i	$0.302998\pi$
$110$	0	0
$111$	0	0
$112$	1368.86	1.15486
$113$	904.125	0.752681	0.376341	−	0.926481i	$-0.377182\pi$
$113$	904.125	0.752681	0.376341	+	0.926481i	$0.377182\pi$
$114$	0	0
$115$	0	0
$116$	−1842.52	−1.47477
$117$	0	0
$118$	242.001	0.188797
$119$	−946.192	−0.728885
$120$	0	0
$121$	121.000	0.0909091
$122$	34.6124	0.0256858
$123$	0	0
$124$	1852.72	1.34177
$125$	0	0
$126$	0	0
$127$	−2059.56	−1.43903	−0.719514	−	0.694478i	$-0.755635\pi$
$127$	−2059.56	−1.43903	−0.719514	+	0.694478i	$0.755635\pi$
$128$	−875.935	−0.604863
$129$	0	0
$130$	0	0
$131$	1019.17	0.679737	0.339868	−	0.940473i	$-0.389617\pi$
$131$	1019.17	0.679737	0.339868	+	0.940473i	$0.389617\pi$
$132$	0	0
$133$	1426.64	0.930113
$134$	175.729	0.113289
$135$	0	0
$136$	294.506	0.185689
$137$	−1924.56	−1.20019	−0.600096	−	0.799928i	$-0.704871\pi$
$137$	−1924.56	−1.20019	−0.600096	+	0.799928i	$0.704871\pi$
$138$	0	0
$139$	814.169	0.496812	0.248406	−	0.968656i	$-0.420093\pi$
$139$	814.169	0.496812	0.248406	+	0.968656i	$0.420093\pi$
$140$	0	0
$141$	0	0
$142$	−3.27186	−0.00193358
$143$	−832.108	−0.486604
$144$	0	0
$145$	0	0
$146$	269.775	0.152923
$147$	0	0
$148$	1962.29	1.08986
$149$	629.177	0.345934	0.172967	−	0.984928i	$-0.444665\pi$
$149$	629.177	0.345934	0.172967	+	0.984928i	$0.444665\pi$
$150$	0	0
$151$	497.025	0.267863	0.133932	−	0.990991i	$-0.457240\pi$
$151$	497.025	0.267863	0.133932	+	0.990991i	$0.457240\pi$
$152$	−444.045	−0.236953
$153$	0	0
$154$	116.593	0.0610086
$155$	0	0
$156$	0	0
$157$	−1652.41	−0.839977	−0.419989	−	0.907529i	$-0.637966\pi$
$157$	−1652.41	−0.839977	−0.419989	+	0.907529i	$0.637966\pi$
$158$	−63.5851	−0.0320162
$159$	0	0
$160$	0	0
$161$	2003.23	0.980599
$162$	0	0
$163$	−1586.38	−0.762299	−0.381150	−	0.924513i	$-0.624472\pi$
$163$	−1586.38	−0.762299	−0.381150	+	0.924513i	$0.624472\pi$
$164$	3481.38	1.65762
$165$	0	0
$166$	−328.695	−0.153685
$167$	−259.897	−0.120428	−0.0602139	−	0.998185i	$-0.519178\pi$
$167$	−259.897	−0.120428	−0.0602139	+	0.998185i	$0.519178\pi$
$168$	0	0
$169$	3525.35	1.60462
$170$	0	0
$171$	0	0
$172$	2054.09	0.910597
$173$	−896.130	−0.393824	−0.196912	−	0.980421i	$-0.563091\pi$
$173$	−896.130	−0.393824	−0.196912	+	0.980421i	$0.563091\pi$
$174$	0	0
$175$	0	0
$176$	649.325	0.278095
$177$	0	0
$178$	−752.807	−0.316996
$179$	−3067.24	−1.28076	−0.640381	−	0.768057i	$-0.721224\pi$
$179$	−3067.24	−1.28076	−0.640381	+	0.768057i	$0.721224\pi$
$180$	0	0
$181$	1011.00	0.415179	0.207589	−	0.978216i	$-0.433438\pi$
$181$	1011.00	0.415179	0.207589	+	0.978216i	$0.433438\pi$
$182$	−801.801	−0.326557
$183$	0	0
$184$	−623.511	−0.249814
$185$	0	0
$186$	0	0
$187$	−448.832	−0.175518
$188$	−3418.17	−1.32604
$189$	0	0
$190$	0	0
$191$	−4659.09	−1.76503	−0.882514	−	0.470287i	$-0.844150\pi$
$191$	−4659.09	−1.76503	−0.882514	+	0.470287i	$0.844150\pi$
$192$	0	0
$193$	−4808.84	−1.79351	−0.896757	−	0.442524i	$-0.854083\pi$
$193$	−4808.84	−1.79351	−0.896757	+	0.442524i	$0.854083\pi$
$194$	−429.256	−0.158860
$195$	0	0
$196$	−1517.28	−0.552946
$197$	2661.39	0.962518	0.481259	−	0.876578i	$-0.340180\pi$
$197$	2661.39	0.962518	0.481259	+	0.876578i	$0.340180\pi$
$198$	0	0
$199$	745.497	0.265562	0.132781	−	0.991145i	$-0.457609\pi$
$199$	745.497	0.265562	0.132781	+	0.991145i	$0.457609\pi$
$200$	0	0
$201$	0	0
$202$	279.517	0.0973601
$203$	5484.08	1.89609
$204$	0	0
$205$	0	0
$206$	475.005	0.160656
$207$	0	0
$208$	−4465.36	−1.48854
$209$	676.733	0.223974
$210$	0	0
$211$	−3149.89	−1.02771	−0.513856	−	0.857876i	$-0.671783\pi$
$211$	−3149.89	−1.02771	−0.513856	+	0.857876i	$0.671783\pi$
$212$	−2229.14	−0.722160
$213$	0	0
$214$	−124.441	−0.0397506
$215$	0	0
$216$	0	0
$217$	−5514.43	−1.72509
$218$	603.522	0.187503
$219$	0	0
$220$	0	0
$221$	3086.58	0.939484
$222$	0	0
$223$	2003.22	0.601549	0.300774	−	0.953695i	$-0.402755\pi$
$223$	2003.22	0.601549	0.300774	+	0.953695i	$0.402755\pi$
$224$	1964.68	0.586029
$225$	0	0
$226$	413.256	0.121635
$227$	957.035	0.279827	0.139913	−	0.990164i	$-0.455318\pi$
$227$	957.035	0.279827	0.139913	+	0.990164i	$0.455318\pi$
$228$	0	0
$229$	−3125.34	−0.901869	−0.450934	−	0.892557i	$-0.648909\pi$
$229$	−3125.34	−0.901869	−0.450934	+	0.892557i	$0.648909\pi$
$230$	0	0
$231$	0	0
$232$	−1706.94	−0.483043
$233$	834.501	0.234635	0.117318	−	0.993094i	$-0.462570\pi$
$233$	834.501	0.234635	0.117318	+	0.993094i	$0.462570\pi$
$234$	0	0
$235$	0	0
$236$	−4125.00	−1.13777
$237$	0	0
$238$	−432.484	−0.117789
$239$	4590.83	1.24249	0.621247	−	0.783615i	$-0.286626\pi$
$239$	4590.83	1.24249	0.621247	+	0.783615i	$0.286626\pi$
$240$	0	0
$241$	4922.60	1.31574	0.657869	−	0.753132i	$-0.271458\pi$
$241$	4922.60	1.31574	0.657869	+	0.753132i	$0.271458\pi$
$242$	55.3065	0.0146911
$243$	0	0
$244$	−589.982	−0.154794
$245$	0	0
$246$	0	0
$247$	−4653.84	−1.19885
$248$	1716.39	0.439478
$249$	0	0
$250$	0	0
$251$	−4824.17	−1.21314	−0.606572	−	0.795029i	$-0.707456\pi$
$251$	−4824.17	−1.21314	−0.606572	+	0.795029i	$0.707456\pi$
$252$	0	0
$253$	950.242	0.236131
$254$	−941.382	−0.232549
$255$	0	0
$256$	3067.72	0.748955
$257$	1571.39	0.381404	0.190702	−	0.981648i	$-0.438924\pi$
$257$	1571.39	0.381404	0.190702	+	0.981648i	$0.438924\pi$
$258$	0	0
$259$	−5840.57	−1.40122
$260$	0	0
$261$	0	0
$262$	465.842	0.109847
$263$	−749.271	−0.175673	−0.0878366	−	0.996135i	$-0.527995\pi$
$263$	−749.271	−0.175673	−0.0878366	+	0.996135i	$0.527995\pi$
$264$	0	0
$265$	0	0
$266$	652.085	0.150308
$267$	0	0
$268$	−2995.37	−0.682729
$269$	−304.815	−0.0690888	−0.0345444	−	0.999403i	$-0.510998\pi$
$269$	−304.815	−0.0690888	−0.0345444	+	0.999403i	$0.510998\pi$
$270$	0	0
$271$	−4172.68	−0.935322	−0.467661	−	0.883908i	$-0.654903\pi$
$271$	−4172.68	−0.935322	−0.467661	+	0.883908i	$0.654903\pi$
$272$	−2408.57	−0.536917
$273$	0	0
$274$	−879.676	−0.193953
$275$	0	0
$276$	0	0
$277$	−3788.49	−0.821763	−0.410881	−	0.911689i	$-0.634779\pi$
$277$	−3788.49	−0.821763	−0.410881	+	0.911689i	$0.634779\pi$
$278$	372.139	0.0802857
$279$	0	0
$280$	0	0
$281$	−8894.59	−1.88828	−0.944140	−	0.329543i	$-0.893105\pi$
$281$	−8894.59	−1.88828	−0.944140	+	0.329543i	$0.893105\pi$
$282$	0	0
$283$	−8365.30	−1.75712	−0.878560	−	0.477631i	$-0.841496\pi$
$283$	−8365.30	−1.75712	−0.878560	+	0.477631i	$0.841496\pi$
$284$	55.7702	0.0116526
$285$	0	0
$286$	−380.339	−0.0786360
$287$	−10362.0	−2.13118
$288$	0	0
$289$	−3248.13	−0.661129
$290$	0	0
$291$	0	0
$292$	−4598.42	−0.921583
$293$	−7094.63	−1.41458	−0.707291	−	0.706923i	$-0.750083\pi$
$293$	−7094.63	−1.41458	−0.707291	+	0.706923i	$0.750083\pi$
$294$	0	0
$295$	0	0
$296$	1817.90	0.356970
$297$	0	0
$298$	287.583	0.0559035
$299$	−6534.74	−1.26393
$300$	0	0
$301$	−6113.78	−1.17074
$302$	227.180	0.0432872
$303$	0	0
$304$	3631.57	0.685147
$305$	0	0
$306$	0	0
$307$	167.179	0.0310795	0.0155397	−	0.999879i	$-0.495053\pi$
$307$	167.179	0.0310795	0.0155397	+	0.999879i	$0.495053\pi$
$308$	−1987.37	−0.367666
$309$	0	0
$310$	0	0
$311$	5182.66	0.944956	0.472478	−	0.881342i	$-0.343360\pi$
$311$	5182.66	0.944956	0.472478	+	0.881342i	$0.343360\pi$
$312$	0	0
$313$	−7854.87	−1.41848	−0.709239	−	0.704968i	$-0.750961\pi$
$313$	−7854.87	−1.41848	−0.709239	+	0.704968i	$0.750961\pi$
$314$	−755.280	−0.135742
$315$	0	0
$316$	1083.83	0.192944
$317$	6898.30	1.22223	0.611115	−	0.791542i	$-0.290721\pi$
$317$	6898.30	1.22223	0.611115	+	0.791542i	$0.290721\pi$
$318$	0	0
$319$	2601.40	0.456585
$320$	0	0
$321$	0	0
$322$	915.632	0.158466
$323$	−2510.24	−0.432426
$324$	0	0
$325$	0	0
$326$	−725.100	−0.123189
$327$	0	0
$328$	3225.20	0.542932
$329$	10173.8	1.70487
$330$	0	0
$331$	−11335.3	−1.88232	−0.941158	−	0.337967i	$-0.890261\pi$
$331$	−11335.3	−1.88232	−0.941158	+	0.337967i	$0.890261\pi$
$332$	5602.73	0.926174
$333$	0	0
$334$	−118.794	−0.0194614
$335$	0	0
$336$	0	0
$337$	−5783.14	−0.934801	−0.467400	−	0.884046i	$-0.654809\pi$
$337$	−5783.14	−0.934801	−0.467400	+	0.884046i	$0.654809\pi$
$338$	1611.36	0.259309
$339$	0	0
$340$	0	0
$341$	−2615.80	−0.415407
$342$	0	0
$343$	−3437.91	−0.541194
$344$	1902.94	0.298254
$345$	0	0
$346$	−409.602	−0.0636426
$347$	1737.90	0.268863	0.134432	−	0.990923i	$-0.457079\pi$
$347$	1737.90	0.268863	0.134432	+	0.990923i	$0.457079\pi$
$348$	0	0
$349$	6935.84	1.06380	0.531901	−	0.846806i	$-0.321478\pi$
$349$	6935.84	1.06380	0.531901	+	0.846806i	$0.321478\pi$
$350$	0	0
$351$	0	0
$352$	931.956	0.141118
$353$	−11417.5	−1.72151	−0.860753	−	0.509023i	$-0.830007\pi$
$353$	−11417.5	−1.72151	−0.860753	+	0.509023i	$0.830007\pi$
$354$	0	0
$355$	0	0
$356$	12831.9	1.91036
$357$	0	0
$358$	−1401.97	−0.206973
$359$	−8223.54	−1.20897	−0.604487	−	0.796615i	$-0.706622\pi$
$359$	−8223.54	−1.20897	−0.604487	+	0.796615i	$0.706622\pi$
$360$	0	0
$361$	−3074.15	−0.448192
$362$	462.108	0.0670936
$363$	0	0
$364$	13667.0	1.96798
$365$	0	0
$366$	0	0
$367$	−2235.51	−0.317963	−0.158982	−	0.987282i	$-0.550821\pi$
$367$	−2235.51	−0.317963	−0.158982	+	0.987282i	$0.550821\pi$
$368$	5099.30	0.722336
$369$	0	0
$370$	0	0
$371$	6634.81	0.928470
$372$	0	0
$373$	−5243.63	−0.727895	−0.363947	−	0.931420i	$-0.618571\pi$
$373$	−5243.63	−0.727895	−0.363947	+	0.931420i	$0.618571\pi$
$374$	−205.151	−0.0283640
$375$	0	0
$376$	−3166.64	−0.434327
$377$	−17889.7	−2.44394
$378$	0	0
$379$	−5262.50	−0.713237	−0.356618	−	0.934250i	$-0.616070\pi$
$379$	−5262.50	−0.713237	−0.356618	+	0.934250i	$0.616070\pi$
$380$	0	0
$381$	0	0
$382$	−2129.57	−0.285231
$383$	−12845.0	−1.71370	−0.856851	−	0.515564i	$-0.827582\pi$
$383$	−12845.0	−1.71370	−0.856851	+	0.515564i	$0.827582\pi$
$384$	0	0
$385$	0	0
$386$	−2198.02	−0.289835
$387$	0	0
$388$	7316.83	0.957360
$389$	12471.0	1.62546	0.812731	−	0.582640i	$-0.197980\pi$
$389$	12471.0	1.62546	0.812731	+	0.582640i	$0.197980\pi$
$390$	0	0
$391$	−3524.78	−0.455897
$392$	−1405.63	−0.181110
$393$	0	0
$394$	1216.46	0.155545
$395$	0	0
$396$	0	0
$397$	4136.07	0.522880	0.261440	−	0.965220i	$-0.415803\pi$
$397$	4136.07	0.522880	0.261440	+	0.965220i	$0.415803\pi$
$398$	340.751	0.0429153
$399$	0	0
$400$	0	0
$401$	−3334.81	−0.415292	−0.207646	−	0.978204i	$-0.566580\pi$
$401$	−3334.81	−0.415292	−0.207646	+	0.978204i	$0.566580\pi$
$402$	0	0
$403$	17988.7	2.22352
$404$	−4764.47	−0.586736
$405$	0	0
$406$	2506.65	0.306412
$407$	−2770.51	−0.337417
$408$	0	0
$409$	11146.6	1.34758	0.673792	−	0.738921i	$-0.264664\pi$
$409$	11146.6	1.34758	0.673792	+	0.738921i	$0.264664\pi$
$410$	0	0
$411$	0	0
$412$	−8096.63	−0.968186
$413$	12277.7	1.46282
$414$	0	0
$415$	0	0
$416$	−6408.99	−0.755353
$417$	0	0
$418$	309.320	0.0361946
$419$	−8914.70	−1.03941	−0.519703	−	0.854347i	$-0.673958\pi$
$419$	−8914.70	−1.03941	−0.519703	+	0.854347i	$0.673958\pi$
$420$	0	0
$421$	619.364	0.0717006	0.0358503	−	0.999357i	$-0.488586\pi$
$421$	619.364	0.0717006	0.0358503	+	0.999357i	$0.488586\pi$
$422$	−1439.75	−0.166080
$423$	0	0
$424$	−2065.11	−0.236534
$425$	0	0
$426$	0	0
$427$	1756.02	0.199016
$428$	2121.15	0.239555
$429$	0	0
$430$	0	0
$431$	−9544.05	−1.06664	−0.533319	−	0.845914i	$-0.679055\pi$
$431$	−9544.05	−1.06664	−0.533319	+	0.845914i	$0.679055\pi$
$432$	0	0
$433$	2043.50	0.226800	0.113400	−	0.993549i	$-0.463826\pi$
$433$	2043.50	0.226800	0.113400	+	0.993549i	$0.463826\pi$
$434$	−2520.53	−0.278777
$435$	0	0
$436$	−10287.3	−1.12998
$437$	5314.54	0.581760
$438$	0	0
$439$	−3972.17	−0.431848	−0.215924	−	0.976410i	$-0.569276\pi$
$439$	−3972.17	−0.431848	−0.215924	+	0.976410i	$0.569276\pi$
$440$	0	0
$441$	0	0
$442$	1410.81	0.151822
$443$	−11124.2	−1.19306	−0.596532	−	0.802589i	$-0.703455\pi$
$443$	−11124.2	−1.19306	−0.596532	+	0.802589i	$0.703455\pi$
$444$	0	0
$445$	0	0
$446$	915.628	0.0972113
$447$	0	0
$448$	−10052.8	−1.06016
$449$	7023.26	0.738192	0.369096	−	0.929391i	$-0.379667\pi$
$449$	7023.26	0.738192	0.369096	+	0.929391i	$0.379667\pi$
$450$	0	0
$451$	−4915.26	−0.513194
$452$	−7044.11	−0.733025
$453$	0	0
$454$	437.440	0.0452204
$455$	0	0
$456$	0	0
$457$	1723.31	0.176396	0.0881979	−	0.996103i	$-0.471889\pi$
$457$	1723.31	0.176396	0.0881979	+	0.996103i	$0.471889\pi$
$458$	−1428.52	−0.145744
$459$	0	0
$460$	0	0
$461$	−7282.40	−0.735738	−0.367869	−	0.929878i	$-0.619912\pi$
$461$	−7282.40	−0.735738	−0.367869	+	0.929878i	$0.619912\pi$
$462$	0	0
$463$	15632.3	1.56910	0.784550	−	0.620065i	$-0.212894\pi$
$463$	15632.3	1.56910	0.784550	+	0.620065i	$0.212894\pi$
$464$	13960.0	1.39671
$465$	0	0
$466$	381.433	0.0379174
$467$	−7075.86	−0.701139	−0.350569	−	0.936537i	$-0.614012\pi$
$467$	−7075.86	−0.701139	−0.350569	+	0.936537i	$0.614012\pi$
$468$	0	0
$469$	8915.42	0.877773
$470$	0	0
$471$	0	0
$472$	−3821.46	−0.372663
$473$	−2900.11	−0.281918
$474$	0	0
$475$	0	0
$476$	7371.86	0.709850
$477$	0	0
$478$	2098.37	0.200789
$479$	4240.40	0.404486	0.202243	−	0.979335i	$-0.435177\pi$
$479$	4240.40	0.404486	0.202243	+	0.979335i	$0.435177\pi$
$480$	0	0
$481$	19052.6	1.80608
$482$	2250.02	0.212626
$483$	0	0
$484$	−942.721	−0.0885350
$485$	0	0
$486$	0	0
$487$	−19796.8	−1.84205	−0.921026	−	0.389502i	$-0.872647\pi$
$487$	−19796.8	−1.84205	−0.921026	+	0.389502i	$0.872647\pi$
$488$	−546.568	−0.0507007
$489$	0	0
$490$	0	0
$491$	7113.02	0.653780	0.326890	−	0.945062i	$-0.393999\pi$
$491$	7113.02	0.653780	0.326890	+	0.945062i	$0.393999\pi$
$492$	0	0
$493$	−9649.52	−0.881526
$494$	−2127.17	−0.193737
$495$	0	0
$496$	−14037.2	−1.27075
$497$	−165.994	−0.0149816
$498$	0	0
$499$	7116.10	0.638398	0.319199	−	0.947688i	$-0.396586\pi$
$499$	7116.10	0.638398	0.319199	+	0.947688i	$0.396586\pi$
$500$	0	0
$501$	0	0
$502$	−2205.03	−0.196046
$503$	759.231	0.0673010	0.0336505	−	0.999434i	$-0.489287\pi$
$503$	759.231	0.0673010	0.0336505	+	0.999434i	$0.489287\pi$
$504$	0	0
$505$	0	0
$506$	434.335	0.0381592
$507$	0	0
$508$	16046.2	1.40145
$509$	−4259.50	−0.370922	−0.185461	−	0.982652i	$-0.559378\pi$
$509$	−4259.50	−0.370922	−0.185461	+	0.982652i	$0.559378\pi$
$510$	0	0
$511$	13686.7	1.18486
$512$	8409.67	0.725895
$513$	0	0
$514$	718.251	0.0616356
$515$	0	0
$516$	0	0
$517$	4826.01	0.410537
$518$	−2669.60	−0.226439
$519$	0	0
$520$	0	0
$521$	6037.10	0.507659	0.253829	−	0.967249i	$-0.418310\pi$
$521$	6037.10	0.507659	0.253829	+	0.967249i	$0.418310\pi$
$522$	0	0
$523$	−20359.6	−1.70222	−0.851110	−	0.524987i	$-0.824070\pi$
$523$	−20359.6	−1.70222	−0.851110	+	0.524987i	$0.824070\pi$
$524$	−7940.45	−0.661985
$525$	0	0
$526$	−342.476	−0.0283891
$527$	9702.93	0.802023
$528$	0	0
$529$	−4704.52	−0.386663
$530$	0	0
$531$	0	0
$532$	−11115.0	−0.905823
$533$	33801.8	2.74694
$534$	0	0
$535$	0	0
$536$	−2774.95	−0.223619
$537$	0	0
$538$	−139.324	−0.0111649
$539$	2142.21	0.171190
$540$	0	0
$541$	9521.19	0.756650	0.378325	−	0.925673i	$-0.376500\pi$
$541$	9521.19	0.756650	0.378325	+	0.925673i	$0.376500\pi$
$542$	−1907.24	−0.151150
$543$	0	0
$544$	−3456.95	−0.272455
$545$	0	0
$546$	0	0
$547$	−408.097	−0.0318994	−0.0159497	−	0.999873i	$-0.505077\pi$
$547$	−408.097	−0.0318994	−0.0159497	+	0.999873i	$0.505077\pi$
$548$	14994.4	1.16885
$549$	0	0
$550$	0	0
$551$	14549.2	1.12490
$552$	0	0
$553$	−3225.92	−0.248065
$554$	−1731.64	−0.132798
$555$	0	0
$556$	−6343.25	−0.483838
$557$	−21981.3	−1.67213	−0.836067	−	0.548627i	$-0.815151\pi$
$557$	−21981.3	−1.67213	−0.836067	+	0.548627i	$0.815151\pi$
$558$	0	0
$559$	19943.8	1.50900
$560$	0	0
$561$	0	0
$562$	−4065.53	−0.305149
$563$	6384.70	0.477945	0.238973	−	0.971026i	$-0.423189\pi$
$563$	6384.70	0.477945	0.238973	+	0.971026i	$0.423189\pi$
$564$	0	0
$565$	0	0
$566$	−3823.60	−0.283954
$567$	0	0
$568$	51.6663	0.00381667
$569$	25115.1	1.85040	0.925202	−	0.379475i	$-0.123896\pi$
$569$	25115.1	1.85040	0.925202	+	0.379475i	$0.123896\pi$
$570$	0	0
$571$	−11128.6	−0.815616	−0.407808	−	0.913068i	$-0.633707\pi$
$571$	−11128.6	−0.815616	−0.407808	+	0.913068i	$0.633707\pi$
$572$	6483.02	0.473896
$573$	0	0
$574$	−4736.23	−0.344402
$575$	0	0
$576$	0	0
$577$	−1957.55	−0.141237	−0.0706185	−	0.997503i	$-0.522497\pi$
$577$	−1957.55	−0.141237	−0.0706185	+	0.997503i	$0.522497\pi$
$578$	−1484.65	−0.106840
$579$	0	0
$580$	0	0
$581$	−16675.9	−1.19077
$582$	0	0
$583$	3147.26	0.223578
$584$	−4260.04	−0.301852
$585$	0	0
$586$	−3242.80	−0.228599
$587$	−26875.7	−1.88974	−0.944870	−	0.327447i	$-0.893812\pi$
$587$	−26875.7	−1.88974	−0.944870	+	0.327447i	$0.893812\pi$
$588$	0	0
$589$	−14629.7	−1.02344
$590$	0	0
$591$	0	0
$592$	−14867.4	−1.03217
$593$	18231.3	1.26251	0.631255	−	0.775575i	$-0.282540\pi$
$593$	18231.3	1.26251	0.631255	+	0.775575i	$0.282540\pi$
$594$	0	0
$595$	0	0
$596$	−4901.96	−0.336900
$597$	0	0
$598$	−2986.89	−0.204253
$599$	24200.0	1.65073	0.825364	−	0.564601i	$-0.190970\pi$
$599$	24200.0	1.65073	0.825364	+	0.564601i	$0.190970\pi$
$600$	0	0
$601$	−92.5601	−0.00628221	−0.00314110	−	0.999995i	$-0.501000\pi$
$601$	−92.5601	−0.00628221	−0.00314110	+	0.999995i	$0.501000\pi$
$602$	−2794.48	−0.189193
$603$	0	0
$604$	−3872.36	−0.260868
$605$	0	0
$606$	0	0
$607$	−6711.75	−0.448800	−0.224400	−	0.974497i	$-0.572042\pi$
$607$	−6711.75	−0.448800	−0.224400	+	0.974497i	$0.572042\pi$
$608$	5212.27	0.347674
$609$	0	0
$610$	0	0
$611$	−33188.1	−2.19746
$612$	0	0
$613$	1130.01	0.0744549	0.0372274	−	0.999307i	$-0.488147\pi$
$613$	1130.01	0.0744549	0.0372274	+	0.999307i	$0.488147\pi$
$614$	76.4139	0.00502250
$615$	0	0
$616$	−1841.13	−0.120424
$617$	19798.5	1.29183	0.645913	−	0.763411i	$-0.276477\pi$
$617$	19798.5	1.29183	0.645913	+	0.763411i	$0.276477\pi$
$618$	0	0
$619$	7620.34	0.494810	0.247405	−	0.968912i	$-0.420422\pi$
$619$	7620.34	0.494810	0.247405	+	0.968912i	$0.420422\pi$
$620$	0	0
$621$	0	0
$622$	2368.88	0.152707
$623$	−38192.8	−2.45612
$624$	0	0
$625$	0	0
$626$	−3590.30	−0.229229
$627$	0	0
$628$	12874.0	0.818041
$629$	10276.8	0.651450
$630$	0	0
$631$	7542.23	0.475834	0.237917	−	0.971285i	$-0.423535\pi$
$631$	7542.23	0.475834	0.237917	+	0.971285i	$0.423535\pi$
$632$	1004.08	0.0631963
$633$	0	0
$634$	3153.06	0.197515
$635$	0	0
$636$	0	0
$637$	−14731.8	−0.916320
$638$	1189.05	0.0737849
$639$	0	0
$640$	0	0
$641$	27329.5	1.68401	0.842005	−	0.539470i	$-0.181375\pi$
$641$	27329.5	1.68401	0.842005	+	0.539470i	$0.181375\pi$
$642$	0	0
$643$	8792.26	0.539243	0.269621	−	0.962966i	$-0.413101\pi$
$643$	8792.26	0.539243	0.269621	+	0.962966i	$0.413101\pi$
$644$	−15607.3	−0.954990
$645$	0	0
$646$	−1147.38	−0.0698807
$647$	8420.01	0.511630	0.255815	−	0.966726i	$-0.417656\pi$
$647$	8420.01	0.511630	0.255815	+	0.966726i	$0.417656\pi$
$648$	0	0
$649$	5823.97	0.352251
$650$	0	0
$651$	0	0
$652$	12359.6	0.742392
$653$	23596.6	1.41410	0.707050	−	0.707164i	$-0.250026\pi$
$653$	23596.6	1.41410	0.707050	+	0.707164i	$0.250026\pi$
$654$	0	0
$655$	0	0
$656$	−26376.8	−1.56988
$657$	0	0
$658$	4650.24	0.275509
$659$	804.250	0.0475404	0.0237702	−	0.999717i	$-0.492433\pi$
$659$	804.250	0.0475404	0.0237702	+	0.999717i	$0.492433\pi$
$660$	0	0
$661$	17912.6	1.05404	0.527019	−	0.849854i	$-0.323310\pi$
$661$	17912.6	1.05404	0.527019	+	0.849854i	$0.323310\pi$
$662$	−5181.14	−0.304186
$663$	0	0
$664$	5190.45	0.303356
$665$	0	0
$666$	0	0
$667$	20429.4	1.18595
$668$	2024.88	0.117283
$669$	0	0
$670$	0	0
$671$	832.978	0.0479237
$672$	0	0
$673$	−942.349	−0.0539746	−0.0269873	−	0.999636i	$-0.508591\pi$
$673$	−942.349	−0.0539746	−0.0269873	+	0.999636i	$0.508591\pi$
$674$	−2643.35	−0.151065
$675$	0	0
$676$	−27466.2	−1.56271
$677$	13681.1	0.776673	0.388336	−	0.921518i	$-0.373050\pi$
$677$	13681.1	0.776673	0.388336	+	0.921518i	$0.373050\pi$
$678$	0	0
$679$	−21777.8	−1.23086
$680$	0	0
$681$	0	0
$682$	−1195.63	−0.0671304
$683$	−7909.32	−0.443107	−0.221553	−	0.975148i	$-0.571113\pi$
$683$	−7909.32	−0.443107	−0.221553	+	0.975148i	$0.571113\pi$
$684$	0	0
$685$	0	0
$686$	−1571.39	−0.0874579
$687$	0	0
$688$	−15562.9	−0.862399
$689$	−21643.5	−1.19674
$690$	0	0
$691$	−11314.6	−0.622903	−0.311452	−	0.950262i	$-0.600815\pi$
$691$	−11314.6	−0.622903	−0.311452	+	0.950262i	$0.600815\pi$
$692$	6981.82	0.383539
$693$	0	0
$694$	794.358	0.0434487
$695$	0	0
$696$	0	0
$697$	18232.4	0.990820
$698$	3170.23	0.171912
$699$	0	0
$700$	0	0
$701$	5833.23	0.314291	0.157146	−	0.987575i	$-0.449771\pi$
$701$	5833.23	0.314291	0.157146	+	0.987575i	$0.449771\pi$
$702$	0	0
$703$	−15495.0	−0.831300
$704$	−4768.62	−0.255290
$705$	0	0
$706$	−5218.69	−0.278198
$707$	14181.0	0.754357
$708$	0	0
$709$	31717.5	1.68008	0.840038	−	0.542527i	$-0.182532\pi$
$709$	31717.5	1.68008	0.840038	+	0.542527i	$0.182532\pi$
$710$	0	0
$711$	0	0
$712$	11887.6	0.625714
$713$	−20542.5	−1.07899
$714$	0	0
$715$	0	0
$716$	23897.1	1.24731
$717$	0	0
$718$	−3758.80	−0.195372
$719$	−20176.6	−1.04654	−0.523269	−	0.852167i	$-0.675288\pi$
$719$	−20176.6	−1.04654	−0.523269	+	0.852167i	$0.675288\pi$
$720$	0	0
$721$	24098.8	1.24478
$722$	−1405.13	−0.0724285
$723$	0	0
$724$	−7876.81	−0.404336
$725$	0	0
$726$	0	0
$727$	23555.4	1.20168	0.600839	−	0.799370i	$-0.294833\pi$
$727$	23555.4	1.20168	0.600839	+	0.799370i	$0.294833\pi$
$728$	12661.3	0.644587
$729$	0	0
$730$	0	0
$731$	10757.5	0.544297
$732$	0	0
$733$	−32919.2	−1.65880	−0.829399	−	0.558656i	$-0.811317\pi$
$733$	−32919.2	−1.65880	−0.829399	+	0.558656i	$0.811317\pi$
$734$	−1021.80	−0.0513834
$735$	0	0
$736$	7318.87	0.366545
$737$	4229.08	0.211370
$738$	0	0
$739$	7979.80	0.397215	0.198607	−	0.980079i	$-0.436358\pi$
$739$	7979.80	0.397215	0.198607	+	0.980079i	$0.436358\pi$
$740$	0	0
$741$	0	0
$742$	3032.63	0.150042
$743$	−36491.6	−1.80181	−0.900907	−	0.434013i	$-0.857097\pi$
$743$	−36491.6	−1.80181	−0.900907	+	0.434013i	$0.857097\pi$
$744$	0	0
$745$	0	0
$746$	−2396.75	−0.117629
$747$	0	0
$748$	3496.88	0.170934
$749$	−6313.38	−0.307992
$750$	0	0
$751$	−8064.10	−0.391828	−0.195914	−	0.980621i	$-0.562767\pi$
$751$	−8064.10	−0.391828	−0.195914	+	0.980621i	$0.562767\pi$
$752$	25897.9	1.25585
$753$	0	0
$754$	−8176.98	−0.394944
$755$	0	0
$756$	0	0
$757$	164.260	0.00788659	0.00394329	−	0.999992i	$-0.498745\pi$
$757$	164.260	0.00788659	0.00394329	+	0.999992i	$0.498745\pi$
$758$	−2405.38	−0.115260
$759$	0	0
$760$	0	0
$761$	−6387.72	−0.304277	−0.152138	−	0.988359i	$-0.548616\pi$
$761$	−6387.72	−0.304277	−0.152138	+	0.988359i	$0.548616\pi$
$762$	0	0
$763$	30619.0	1.45279
$764$	36299.4	1.71893
$765$	0	0
$766$	−5871.17	−0.276937
$767$	−40051.0	−1.88547
$768$	0	0
$769$	−4920.10	−0.230719	−0.115360	−	0.993324i	$-0.536802\pi$
$769$	−4920.10	−0.230719	−0.115360	+	0.993324i	$0.536802\pi$
$770$	0	0
$771$	0	0
$772$	37466.1	1.74668
$773$	−25930.2	−1.20653	−0.603264	−	0.797542i	$-0.706133\pi$
$773$	−25930.2	−1.20653	−0.603264	+	0.797542i	$0.706133\pi$
$774$	0	0
$775$	0	0
$776$	6778.41	0.313571
$777$	0	0
$778$	5700.22	0.262677
$779$	−27490.2	−1.26436
$780$	0	0
$781$	−78.7403	−0.00360762
$782$	−1611.10	−0.0736738
$783$	0	0
$784$	11495.8	0.523678
$785$	0	0
$786$	0	0
$787$	−876.332	−0.0396923	−0.0198462	−	0.999803i	$-0.506318\pi$
$787$	−876.332	−0.0396923	−0.0198462	+	0.999803i	$0.506318\pi$
$788$	−20735.1	−0.937382
$789$	0	0
$790$	0	0
$791$	20966.1	0.942438
$792$	0	0
$793$	−5728.33	−0.256518
$794$	1890.51	0.0844983
$795$	0	0
$796$	−5808.22	−0.258627
$797$	20900.9	0.928917	0.464458	−	0.885595i	$-0.346249\pi$
$797$	20900.9	0.928917	0.464458	+	0.885595i	$0.346249\pi$
$798$	0	0
$799$	−17901.4	−0.792622
$800$	0	0
$801$	0	0
$802$	−1524.27	−0.0671120
$803$	6492.38	0.285319
$804$	0	0
$805$	0	0
$806$	8222.24	0.359325
$807$	0	0
$808$	−4413.87	−0.192178
$809$	−7164.89	−0.311377	−0.155689	−	0.987806i	$-0.549760\pi$
$809$	−7164.89	−0.311377	−0.155689	+	0.987806i	$0.549760\pi$
$810$	0	0
$811$	−4229.19	−0.183116	−0.0915580	−	0.995800i	$-0.529185\pi$
$811$	−4229.19	−0.183116	−0.0915580	+	0.995800i	$0.529185\pi$
$812$	−42726.9	−1.84658
$813$	0	0
$814$	−1266.34	−0.0545272
$815$	0	0
$816$	0	0
$817$	−16219.8	−0.694565
$818$	5094.85	0.217772
$819$	0	0
$820$	0	0
$821$	31556.0	1.34143	0.670713	−	0.741717i	$-0.265988\pi$
$821$	31556.0	1.34143	0.670713	+	0.741717i	$0.265988\pi$
$822$	0	0
$823$	−36827.7	−1.55982	−0.779910	−	0.625891i	$-0.784735\pi$
$823$	−36827.7	−1.55982	−0.779910	+	0.625891i	$0.784735\pi$
$824$	−7500.83	−0.317116
$825$	0	0
$826$	5611.85	0.236394
$827$	−1188.45	−0.0499714	−0.0249857	−	0.999688i	$-0.507954\pi$
$827$	−1188.45	−0.0499714	−0.0249857	+	0.999688i	$0.507954\pi$
$828$	0	0
$829$	−37008.8	−1.55051	−0.775253	−	0.631651i	$-0.782378\pi$
$829$	−37008.8	−1.55051	−0.775253	+	0.631651i	$0.782378\pi$
$830$	0	0
$831$	0	0
$832$	32793.5	1.36648
$833$	−7946.21	−0.330516
$834$	0	0
$835$	0	0
$836$	−5272.48	−0.218125
$837$	0	0
$838$	−4074.72	−0.167970
$839$	11427.0	0.470206	0.235103	−	0.971970i	$-0.424457\pi$
$839$	11427.0	0.470206	0.235103	+	0.971970i	$0.424457\pi$
$840$	0	0
$841$	31539.1	1.29317
$842$	283.098	0.0115869
$843$	0	0
$844$	24541.0	1.00087
$845$	0	0
$846$	0	0
$847$	2805.91	0.113828
$848$	16889.2	0.683936
$849$	0	0
$850$	0	0
$851$	−21757.4	−0.876423
$852$	0	0
$853$	−5084.37	−0.204086	−0.102043	−	0.994780i	$-0.532538\pi$
$853$	−5084.37	−0.204086	−0.102043	+	0.994780i	$0.532538\pi$
$854$	802.640	0.0321613
$855$	0	0
$856$	1965.06	0.0784631
$857$	−36268.5	−1.44563	−0.722816	−	0.691040i	$-0.757153\pi$
$857$	−36268.5	−1.44563	−0.722816	+	0.691040i	$0.757153\pi$
$858$	0	0
$859$	36158.6	1.43622	0.718111	−	0.695928i	$-0.245007\pi$
$859$	36158.6	1.43622	0.718111	+	0.695928i	$0.245007\pi$
$860$	0	0
$861$	0	0
$862$	−4362.38	−0.172370
$863$	−5105.14	−0.201368	−0.100684	−	0.994918i	$-0.532103\pi$
$863$	−5105.14	−0.201368	−0.100684	+	0.994918i	$0.532103\pi$
$864$	0	0
$865$	0	0
$866$	934.039	0.0366512
$867$	0	0
$868$	42963.4	1.68004
$869$	−1530.23	−0.0597348
$870$	0	0
$871$	−29083.0	−1.13139
$872$	−9530.26	−0.370109
$873$	0	0
$874$	2429.16	0.0940134
$875$	0	0
$876$	0	0
$877$	−33922.6	−1.30614	−0.653071	−	0.757297i	$-0.726520\pi$
$877$	−33922.6	−1.30614	−0.653071	+	0.757297i	$0.726520\pi$
$878$	−1815.59	−0.0697874
$879$	0	0
$880$	0	0
$881$	−12610.2	−0.482235	−0.241117	−	0.970496i	$-0.577514\pi$
$881$	−12610.2	−0.482235	−0.241117	+	0.970496i	$0.577514\pi$
$882$	0	0
$883$	40762.8	1.55354	0.776771	−	0.629783i	$-0.216856\pi$
$883$	40762.8	1.55354	0.776771	+	0.629783i	$0.216856\pi$
$884$	−24047.8	−0.914949
$885$	0	0
$886$	−5084.64	−0.192801
$887$	29954.3	1.13390	0.566949	−	0.823753i	$-0.308123\pi$
$887$	29954.3	1.13390	0.566949	+	0.823753i	$0.308123\pi$
$888$	0	0
$889$	−47759.9	−1.80182
$890$	0	0
$891$	0	0
$892$	−15607.2	−0.585839
$893$	26991.1	1.01145
$894$	0	0
$895$	0	0
$896$	−20312.4	−0.757353
$897$	0	0
$898$	3210.18	0.119293
$899$	−56237.6	−2.08635
$900$	0	0
$901$	−11674.3	−0.431662
$902$	−2246.66	−0.0829330
$903$	0	0
$904$	−6525.76	−0.240093
$905$	0	0
$906$	0	0
$907$	29053.0	1.06360	0.531802	−	0.846869i	$-0.321515\pi$
$907$	29053.0	1.06360	0.531802	+	0.846869i	$0.321515\pi$
$908$	−7456.33	−0.272519
$909$	0	0
$910$	0	0
$911$	3562.14	0.129549	0.0647744	−	0.997900i	$-0.479367\pi$
$911$	3562.14	0.129549	0.0647744	+	0.997900i	$0.479367\pi$
$912$	0	0
$913$	−7910.33	−0.286740
$914$	787.687	0.0285059
$915$	0	0
$916$	24349.7	0.878317
$917$	23634.0	0.851104
$918$	0	0
$919$	2936.50	0.105404	0.0527020	−	0.998610i	$-0.483217\pi$
$919$	2936.50	0.105404	0.0527020	+	0.998610i	$0.483217\pi$
$920$	0	0
$921$	0	0
$922$	−3328.63	−0.118896
$923$	541.491	0.0193103
$924$	0	0
$925$	0	0
$926$	7145.18	0.253569
$927$	0	0
$928$	20036.3	0.708754
$929$	16941.3	0.598306	0.299153	−	0.954205i	$-0.403296\pi$
$929$	16941.3	0.598306	0.299153	+	0.954205i	$0.403296\pi$
$930$	0	0
$931$	11981.0	0.421764
$932$	−6501.66	−0.228508
$933$	0	0
$934$	−3234.22	−0.113305
$935$	0	0
$936$	0	0
$937$	−19863.3	−0.692537	−0.346269	−	0.938135i	$-0.612551\pi$
$937$	−19863.3	−0.692537	−0.346269	+	0.938135i	$0.612551\pi$
$938$	4075.05	0.141850
$939$	0	0
$940$	0	0
$941$	39408.8	1.36524	0.682620	−	0.730773i	$-0.260840\pi$
$941$	39408.8	1.36524	0.682620	+	0.730773i	$0.260840\pi$
$942$	0	0
$943$	−38600.7	−1.33299
$944$	31253.3	1.07755
$945$	0	0
$946$	−1325.58	−0.0455584
$947$	40190.4	1.37910	0.689552	−	0.724236i	$-0.257807\pi$
$947$	40190.4	1.37910	0.689552	+	0.724236i	$0.257807\pi$
$948$	0	0
$949$	−44647.6	−1.52721
$950$	0	0
$951$	0	0
$952$	6829.39	0.232502
$953$	43114.9	1.46551	0.732754	−	0.680493i	$-0.238235\pi$
$953$	43114.9	1.46551	0.732754	+	0.680493i	$0.238235\pi$
$954$	0	0
$955$	0	0
$956$	−35767.5	−1.21005
$957$	0	0
$958$	1938.20	0.0653657
$959$	−44629.3	−1.50277
$960$	0	0
$961$	26757.9	0.898188
$962$	8708.52	0.291865
$963$	0	0
$964$	−38352.4	−1.28138
$965$	0	0
$966$	0	0
$967$	15536.2	0.516659	0.258330	−	0.966057i	$-0.416828\pi$
$967$	15536.2	0.516659	0.258330	+	0.966057i	$0.416828\pi$
$968$	−873.350	−0.0289985
$969$	0	0
$970$	0	0
$971$	18466.6	0.610320	0.305160	−	0.952301i	$-0.401290\pi$
$971$	18466.6	0.610320	0.305160	+	0.952301i	$0.401290\pi$
$972$	0	0
$973$	18880.0	0.622062
$974$	−9048.70	−0.297679
$975$	0	0
$976$	4470.03	0.146601
$977$	13352.4	0.437236	0.218618	−	0.975810i	$-0.429845\pi$
$977$	13352.4	0.437236	0.218618	+	0.975810i	$0.429845\pi$
$978$	0	0
$979$	−18117.0	−0.591441
$980$	0	0
$981$	0	0
$982$	3251.21	0.105652
$983$	9970.61	0.323513	0.161756	−	0.986831i	$-0.448284\pi$
$983$	9970.61	0.323513	0.161756	+	0.986831i	$0.448284\pi$
$984$	0	0
$985$	0	0
$986$	−4410.59	−0.142456
$987$	0	0
$988$	36258.4	1.16755
$989$	−22775.2	−0.732265
$990$	0	0
$991$	36094.5	1.15699	0.578496	−	0.815685i	$-0.303640\pi$
$991$	36094.5	1.15699	0.578496	+	0.815685i	$0.303640\pi$
$992$	−20147.2	−0.644833
$993$	0	0
$994$	−75.8724	−0.00242105
$995$	0	0
$996$	0	0
$997$	−51210.1	−1.62672	−0.813360	−	0.581760i	$-0.802364\pi$
$997$	−51210.1	−1.62672	−0.813360	+	0.581760i	$0.802364\pi$
$998$	3252.62	0.103166
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2475.4.a.bl.1.3		5
3.2	odd	2		275.4.a.g.1.3	✓	5
5.4	even	2		2475.4.a.bh.1.3		5
15.2	even	4		275.4.b.f.199.5		10
15.8	even	4		275.4.b.f.199.6		10
15.14	odd	2		275.4.a.h.1.3	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.4.a.g.1.3	✓	5	3.2	odd	2
275.4.a.h.1.3	yes	5	15.14	odd	2
275.4.b.f.199.5		10	15.2	even	4
275.4.b.f.199.6		10	15.8	even	4
2475.4.a.bh.1.3		5	5.4	even	2
2475.4.a.bl.1.3		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2475.a (trivial)

\(f(q)\)	\(=\)	\(q+0.457079 q^{2} -7.79108 q^{4} +23.1894 q^{7} -7.21777 q^{8} +11.0000 q^{11} -75.6462 q^{13} +10.5994 q^{14} +59.0295 q^{16} -40.8029 q^{17} +61.5212 q^{19} +5.02787 q^{22} +86.3856 q^{23} -34.5763 q^{26} -180.670 q^{28} +236.491 q^{29} -237.800 q^{31} +84.7233 q^{32} -18.6501 q^{34} -251.864 q^{37} +28.1200 q^{38} -446.841 q^{41} -263.646 q^{43} -85.7019 q^{44} +39.4850 q^{46} +438.728 q^{47} +194.746 q^{49} +589.365 q^{52} +286.114 q^{53} -167.375 q^{56} +108.095 q^{58} +529.452 q^{59} +75.7253 q^{61} -108.693 q^{62} -433.511 q^{64} +384.462 q^{67} +317.898 q^{68} -7.15821 q^{71} +590.216 q^{73} -115.122 q^{74} -479.316 q^{76} +255.083 q^{77} -139.112 q^{79} -204.242 q^{82} -719.121 q^{83} -120.507 q^{86} -79.3954 q^{88} -1647.00 q^{89} -1754.19 q^{91} -673.037 q^{92} +200.533 q^{94} -939.129 q^{97} +89.0144 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 2 q^{2} + 40 q^{4} - 40 q^{7} + 21 q^{8} + 55 q^{11} - 211 q^{13} + 133 q^{14} + 208 q^{16} + 72 q^{17} + 23 q^{19} + 22 q^{22} + 57 q^{23} - 322 q^{26} - 1050 q^{28} + 183 q^{29} - q^{31} - 430 q^{32}+ \cdots - 5586 q^{98}+O(q^{100}) \) 5 * q + 2 * q^2 + 40 * q^4 - 40 * q^7 + 21 * q^8 + 55 * q^11 - 211 * q^13 + 133 * q^14 + 208 * q^16 + 72 * q^17 + 23 * q^19 + 22 * q^22 + 57 * q^23 - 322 * q^26 - 1050 * q^28 + 183 * q^29 - q^31 - 430 * q^32 - 650 * q^34 - 856 * q^37 + 348 * q^38 - 94 * q^41 - 497 * q^43 + 440 * q^44 + 1006 * q^46 + 26 * q^47 + 227 * q^49 - 2086 * q^52 + 264 * q^53 + 835 * q^56 - 843 * q^58 - 1174 * q^59 + 640 * q^61 + 1454 * q^62 - 949 * q^64 - 98 * q^67 + 1767 * q^68 + 673 * q^71 - 1462 * q^73 + 561 * q^74 + 1248 * q^76 - 440 * q^77 + 104 * q^79 + 221 * q^82 - 747 * q^83 - 2497 * q^86 + 231 * q^88 - 2821 * q^89 + 480 * q^91 - 1503 * q^92 - 1787 * q^94 - 615 * q^97 - 5586 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more