LMFDB - Embedded newform 275.4.b.f.199.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [275,4,Mod(199,275)] 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("275.199"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 275 = 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	275.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,-80] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$16.2255252516$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 80x^{8} + 2296x^{6} + 27417x^{4} + 110472x^{2} + 21904 $ x^10 + 80x^8 + 2296x^6 + 27417x^4 + 110472x^2 + 21904
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.5
Root			$-0.457079i$ of defining polynomial
Character	$\chi$	$=$	275.199
Dual form			275.4.b.f.199.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.457079i q^{2} -2.45289i q^{3} +7.79108 q^{4} -1.12116 q^{6} +23.1894i q^{7} -7.21777i q^{8} +20.9833 q^{9} -11.0000 q^{11} -19.1106i q^{12} +75.6462i q^{13} +10.5994 q^{14} +59.0295 q^{16} +40.8029i q^{17} -9.59104i q^{18} -61.5212 q^{19} +56.8809 q^{21} +5.02787i q^{22} +86.3856i q^{23} -17.7044 q^{24} +34.5763 q^{26} -117.698i q^{27} +180.670i q^{28} +236.491 q^{29} -237.800 q^{31} -84.7233i q^{32} +26.9818i q^{33} +18.6501 q^{34} +163.483 q^{36} -251.864i q^{37} +28.1200i q^{38} +185.552 q^{39} +446.841 q^{41} -25.9990i q^{42} +263.646i q^{43} -85.7019 q^{44} +39.4850 q^{46} -438.728i q^{47} -144.793i q^{48} -194.746 q^{49} +100.085 q^{51} +589.365i q^{52} +286.114i q^{53} -53.7971 q^{54} +167.375 q^{56} +150.905i q^{57} -108.095i q^{58} +529.452 q^{59} +75.7253 q^{61} +108.693i q^{62} +486.590i q^{63} +433.511 q^{64} +12.3328 q^{66} +384.462i q^{67} +317.898i q^{68} +211.894 q^{69} +7.15821 q^{71} -151.453i q^{72} -590.216i q^{73} -115.122 q^{74} -479.316 q^{76} -255.083i q^{77} -84.8117i q^{78} +139.112 q^{79} +277.851 q^{81} -204.242i q^{82} -719.121i q^{83} +443.164 q^{84} +120.507 q^{86} -580.086i q^{87} +79.3954i q^{88} -1647.00 q^{89} -1754.19 q^{91} +673.037i q^{92} +583.297i q^{93} -200.533 q^{94} -207.817 q^{96} -939.129i q^{97} +89.0144i q^{98} -230.817 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 80 q^{4} - 84 q^{6} - 62 q^{9} - 110 q^{11} + 266 q^{14} + 416 q^{16} - 46 q^{19} + 564 q^{21} + 982 q^{24} + 644 q^{26} + 366 q^{29} - 2 q^{31} + 1300 q^{34} + 2676 q^{36} + 540 q^{39} + 188 q^{41}+ \cdots + 682 q^{99}+O(q^{100}) $ 10 * q - 80 * q^4 - 84 * q^6 - 62 * q^9 - 110 * q^11 + 266 * q^14 + 416 * q^16 - 46 * q^19 + 564 * q^21 + 982 * q^24 + 644 * q^26 + 366 * q^29 - 2 * q^31 + 1300 * q^34 + 2676 * q^36 + 540 * q^39 + 188 * q^41 + 880 * q^44 + 2012 * q^46 - 454 * q^49 - 2256 * q^51 + 2410 * q^54 - 1670 * q^56 - 2348 * q^59 + 1280 * q^61 + 1898 * q^64 + 924 * q^66 + 3292 * q^69 - 1346 * q^71 + 1122 * q^74 + 2496 * q^76 - 208 * q^79 - 2438 * q^81 - 8706 * q^84 + 4994 * q^86 - 5642 * q^89 + 960 * q^91 + 3574 * q^94 - 3638 * q^96 + 682 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$.

$n$	$101$	$177$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 0.457079i	− 0.161602i	−0.996730	−	0.0808009i	$-0.974252\pi$
$2$	− 0.457079i	− 0.161602i	0.996730	−	0.0808009i	$-0.0257478\pi$
$3$	− 2.45289i	− 0.472059i	−0.971746	−	0.236029i	$-0.924154\pi$
$3$	− 2.45289i	− 0.472059i	0.971746	−	0.236029i	$-0.0758461\pi$
$4$	7.79108	0.973885
$5$	0	0
$5$	0	0
$6$	−1.12116	−0.0762855
$7$	23.1894i	1.25211i	0.779780	+	0.626054i	$0.215331\pi$
$7$	23.1894i	1.25211i	−0.779780	+	0.626054i	$0.784669\pi$
$8$	− 7.21777i	− 0.318983i
$9$	20.9833	0.777161
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	− 19.1106i	− 0.459731i
$13$	75.6462i	1.61388i	0.590631	+	0.806941i	$0.298879\pi$
$13$	75.6462i	1.61388i	−0.590631	+	0.806941i	$0.701121\pi$
$14$	10.5994	0.202343
$15$	0	0
$16$	59.0295	0.922337
$17$	40.8029i	0.582126i	0.956704	+	0.291063i	$0.0940090\pi$
$17$	40.8029i	0.582126i	−0.956704	+	0.291063i	$0.905991\pi$
$18$	− 9.59104i	− 0.125591i
$19$	−61.5212	−0.742838	−0.371419	−	0.928465i	$-0.621129\pi$
$19$	−61.5212	−0.742838	−0.371419	+	0.928465i	$0.621129\pi$
$20$	0	0
$21$	56.8809	0.591068
$22$	5.02787i	0.0487247i
$23$	86.3856i	0.783159i	0.920144	+	0.391579i	$0.128071\pi$
$23$	86.3856i	0.783159i	−0.920144	+	0.391579i	$0.871929\pi$
$24$	−17.7044	−0.150579
$25$	0	0
$26$	34.5763	0.260806
$27$	− 117.698i	− 0.838924i
$28$	180.670i	1.21941i
$29$	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	$0.741895\pi$
$32$	− 84.7233i	− 0.468034i
$33$	26.9818i	0.142331i
$34$	18.6501	0.0940726
$35$	0	0
$36$	163.483	0.756865
$37$	− 251.864i	− 1.11909i	−0.828801	−	0.559543i	$-0.810976\pi$
$37$	− 251.864i	− 1.11909i	0.828801	−	0.559543i	$-0.189024\pi$
$38$	28.1200i	0.120044i
$39$	185.552	0.761847
$40$	0	0
$41$	446.841	1.70207	0.851035	−	0.525108i	$-0.175975\pi$
$41$	446.841	1.70207	0.851035	+	0.525108i	$0.175975\pi$
$42$	− 25.9990i	− 0.0955176i
$43$	263.646i	0.935015i	0.883989	+	0.467507i	$0.154848\pi$
$43$	263.646i	0.935015i	−0.883989	+	0.467507i	$0.845152\pi$
$44$	−85.7019	−0.293637
$45$	0	0
$46$	39.4850	0.126560
$47$	− 438.728i	− 1.36160i	−0.732471	−	0.680799i	$-0.761633\pi$
$47$	− 438.728i	− 1.36160i	0.732471	−	0.680799i	$-0.238367\pi$
$48$	− 144.793i	− 0.435397i
$49$	−194.746	−0.567773
$50$	0	0
$51$	100.085	0.274798
$52$	589.365i	1.57174i
$53$	286.114i	0.741525i	0.928728	+	0.370763i	$0.120904\pi$
$53$	286.114i	0.741525i	−0.928728	+	0.370763i	$0.879096\pi$
$54$	−53.7971	−0.135572
$55$	0	0
$56$	167.375	0.399401
$57$	150.905i	0.350663i
$58$	− 108.095i	− 0.244717i
$59$	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	$0.474676\pi$
$62$	108.693i	0.222646i
$63$	486.590i	0.973089i
$64$	433.511	0.846702
$65$	0	0
$66$	12.3328	0.0230009
$67$	384.462i	0.701037i	0.936556	+	0.350518i	$0.113995\pi$
$67$	384.462i	0.701037i	−0.936556	+	0.350518i	$0.886005\pi$
$68$	317.898i	0.566924i
$69$	211.894	0.369697
$70$	0	0
$71$	7.15821	0.0119651	0.00598256	−	0.999982i	$-0.498096\pi$
$71$	7.15821	0.0119651	0.00598256	+	0.999982i	$0.498096\pi$
$72$	− 151.453i	− 0.247901i
$73$	− 590.216i	− 0.946295i	−0.880983	−	0.473148i	$-0.843118\pi$
$73$	− 590.216i	− 0.946295i	0.880983	−	0.473148i	$-0.156882\pi$
$74$	−115.122	−0.180846
$75$	0	0
$76$	−479.316	−0.723439
$77$	− 255.083i	− 0.377525i
$78$	− 84.8117i	− 0.123116i
$79$	139.112	0.198118	0.0990589	−	0.995082i	$-0.468417\pi$
$79$	139.112	0.198118	0.0990589	+	0.995082i	$0.468417\pi$
$80$	0	0
$81$	277.851	0.381140
$82$	− 204.242i	− 0.275058i
$83$	− 719.121i	− 0.951009i	−0.879713	−	0.475505i	$-0.842265\pi$
$83$	− 719.121i	− 0.951009i	0.879713	−	0.475505i	$-0.157735\pi$
$84$	443.164	0.575632
$85$	0	0
$86$	120.507	0.151100
$87$	− 580.086i	− 0.714848i
$88$	79.3954i	0.0961770i
$89$	−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
$92$	673.037i	0.762706i
$93$	583.297i	0.650377i
$94$	−200.533	−0.220036
$95$	0	0
$96$	−207.817	−0.220940
$97$	− 939.129i	− 0.983032i	−0.870868	−	0.491516i	$-0.836443\pi$
$97$	− 939.129i	− 0.983032i	0.870868	−	0.491516i	$-0.163557\pi$
$98$	89.0144i	0.0917532i
$99$	−230.817	−0.234323
$100$	0	0
$101$	−611.529	−0.602470	−0.301235	−	0.953550i	$-0.597399\pi$
$101$	−611.529	−0.602470	−0.301235	+	0.953550i	$0.597399\pi$
$102$	− 45.7467i	− 0.0444078i
$103$	− 1039.22i	− 0.994148i	−0.867708	−	0.497074i	$-0.834408\pi$
$103$	− 1039.22i	− 0.994148i	0.867708	−	0.497074i	$-0.165592\pi$
$104$	545.996	0.514801
$105$	0	0
$106$	130.777	0.119832
$107$	272.253i	0.245979i	0.992408	+	0.122989i	$0.0392481\pi$
$107$	272.253i	0.245979i	−0.992408	+	0.122989i	$0.960752\pi$
$108$	− 916.993i	− 0.817015i
$109$	−1320.39	−1.16028	−0.580139	−	0.814517i	$-0.697002\pi$
$109$	−1320.39	−1.16028	−0.580139	+	0.814517i	$0.697002\pi$
$110$	0	0
$111$	−617.795	−0.528275
$112$	1368.86i	1.15486i
$113$	904.125i	0.752681i	0.926481	+	0.376341i	$0.122818\pi$
$113$	904.125i	0.752681i	−0.926481	+	0.376341i	$0.877182\pi$
$114$	68.9752	0.0566677
$115$	0	0
$116$	1842.52	1.47477
$117$	1587.31i	1.25425i
$118$	− 242.001i	− 0.188797i
$119$	−946.192	−0.728885
$120$	0	0
$121$	121.000	0.0909091
$122$	− 34.6124i	− 0.0256858i
$123$	− 1096.05i	− 0.803477i
$124$	−1852.72	−1.34177
$125$	0	0
$126$	222.410	0.157253
$127$	− 2059.56i	− 1.43903i	−0.694478	−	0.719514i	$-0.744365\pi$
$127$	− 2059.56i	− 1.43903i	0.694478	−	0.719514i	$-0.255635\pi$
$128$	− 875.935i	− 0.604863i
$129$	646.694	0.441382
$130$	0	0
$131$	−1019.17	−0.679737	−0.339868	−	0.940473i	$-0.610383\pi$
$131$	−1019.17	−0.679737	−0.339868	+	0.940473i	$0.610383\pi$
$132$	210.217i	0.138614i
$133$	− 1426.64i	− 0.930113i
$134$	175.729	0.113289
$135$	0	0
$136$	294.506	0.185689
$137$	1924.56i	1.20019i	0.799928	+	0.600096i	$0.204871\pi$
$137$	1924.56i	1.20019i	−0.799928	+	0.600096i	$0.795129\pi$
$138$	− 96.8523i	− 0.0597436i
$139$	−814.169	−0.496812	−0.248406	−	0.968656i	$-0.579907\pi$
$139$	−814.169	−0.496812	−0.248406	+	0.968656i	$0.579907\pi$
$140$	0	0
$141$	−1076.15	−0.642754
$142$	− 3.27186i	− 0.00193358i
$143$	− 832.108i	− 0.486604i
$144$	1238.64	0.716804
$145$	0	0
$146$	−269.775	−0.152923
$147$	477.691i	0.268022i
$148$	− 1962.29i	− 1.08986i
$149$	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	$0.457240\pi$
$152$	444.045i	0.236953i
$153$	856.180i	0.452406i
$154$	−116.593	−0.0610086
$155$	0	0
$156$	1445.65	0.741951
$157$	− 1652.41i	− 0.839977i	−0.907529	−	0.419989i	$-0.862034\pi$
$157$	− 1652.41i	− 0.839977i	0.907529	−	0.419989i	$-0.137966\pi$
$158$	− 63.5851i	− 0.0320162i
$159$	701.807	0.350043
$160$	0	0
$161$	−2003.23	−0.980599
$162$	− 127.000i	− 0.0615928i
$163$	1586.38i	0.762299i	0.924513	+	0.381150i	$0.124472\pi$
$163$	1586.38i	0.762299i	−0.924513	+	0.381150i	$0.875528\pi$
$164$	3481.38	1.65762
$165$	0	0
$166$	−328.695	−0.153685
$167$	259.897i	0.120428i	0.998185	+	0.0602139i	$0.0191783\pi$
$167$	259.897i	0.120428i	−0.998185	+	0.0602139i	$0.980822\pi$
$168$	− 410.553i	− 0.188541i
$169$	−3525.35	−1.60462
$170$	0	0
$171$	−1290.92	−0.577305
$172$	2054.09i	0.910597i
$173$	− 896.130i	− 0.393824i	−0.980421	−	0.196912i	$-0.936909\pi$
$173$	− 896.130i	− 0.393824i	0.980421	−	0.196912i	$-0.0630912\pi$
$174$	−265.145	−0.115521
$175$	0	0
$176$	−649.325	−0.278095
$177$	− 1298.69i	− 0.551499i
$178$	752.807i	0.316996i
$179$	−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	$0.433438\pi$
$182$	801.801i	0.326557i
$183$	− 185.746i	− 0.0750312i
$184$	623.511	0.249814
$185$	0	0
$186$	266.613	0.105102
$187$	− 448.832i	− 0.175518i
$188$	− 3418.17i	− 1.32604i
$189$	2729.34	1.05042
$190$	0	0
$191$	4659.09	1.76503	0.882514	−	0.470287i	$-0.155850\pi$
$191$	4659.09	1.76503	0.882514	+	0.470287i	$0.155850\pi$
$192$	− 1063.35i	− 0.399693i
$193$	4808.84i	1.79351i	0.442524	+	0.896757i	$0.354083\pi$
$193$	4808.84i	1.79351i	−0.442524	+	0.896757i	$0.645917\pi$
$194$	−429.256	−0.158860
$195$	0	0
$196$	−1517.28	−0.552946
$197$	− 2661.39i	− 0.962518i	−0.876578	−	0.481259i	$-0.840180\pi$
$197$	− 2661.39i	− 0.962518i	0.876578	−	0.481259i	$-0.159820\pi$
$198$	105.501i	0.0378670i
$199$	−745.497	−0.265562	−0.132781	−	0.991145i	$-0.542391\pi$
$199$	−745.497	−0.265562	−0.132781	+	0.991145i	$0.542391\pi$
$200$	0	0
$201$	943.041	0.330930
$202$	279.517i	0.0973601i
$203$	5484.08i	1.89609i
$204$	779.769	0.267621
$205$	0	0
$206$	−475.005	−0.160656
$207$	1812.66i	0.608640i
$208$	4465.36i	1.48854i
$209$	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	$0.671783\pi$
$212$	2229.14i	0.722160i
$213$	− 17.5583i	− 0.00564823i
$214$	124.441	0.0397506
$215$	0	0
$216$	−849.515	−0.267603
$217$	− 5514.43i	− 1.72509i
$218$	603.522i	0.187503i
$219$	−1447.73	−0.446707
$220$	0	0
$221$	−3086.58	−0.939484
$222$	282.381i	0.0853701i
$223$	− 2003.22i	− 0.601549i	−0.953695	−	0.300774i	$-0.902755\pi$
$223$	− 2003.22i	− 0.601549i	0.953695	−	0.300774i	$-0.0972451\pi$
$224$	1964.68	0.586029
$225$	0	0
$226$	413.256	0.121635
$227$	− 957.035i	− 0.279827i	−0.990164	−	0.139913i	$-0.955318\pi$
$227$	− 957.035i	− 0.279827i	0.990164	−	0.139913i	$-0.0446824\pi$
$228$	1175.71i	0.341505i
$229$	3125.34	0.901869	0.450934	−	0.892557i	$-0.351091\pi$
$229$	3125.34	0.901869	0.450934	+	0.892557i	$0.351091\pi$
$230$	0	0
$231$	−625.690	−0.178214
$232$	− 1706.94i	− 0.483043i
$233$	834.501i	0.234635i	0.993094	+	0.117318i	$0.0374295\pi$
$233$	834.501i	0.234635i	−0.993094	+	0.117318i	$0.962570\pi$
$234$	725.525	0.202688
$235$	0	0
$236$	4125.00	1.13777
$237$	− 341.226i	− 0.0935232i
$238$	432.484i	0.117789i
$239$	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	$0.271458\pi$
$242$	− 55.3065i	− 0.0146911i
$243$	− 3859.38i	− 1.01884i
$244$	589.982	0.154794
$245$	0	0
$246$	−500.982	−0.129843
$247$	− 4653.84i	− 1.19885i
$248$	1716.39i	0.439478i
$249$	−1763.92	−0.448932
$250$	0	0
$251$	4824.17	1.21314	0.606572	−	0.795029i	$-0.292544\pi$
$251$	4824.17	1.21314	0.606572	+	0.795029i	$0.292544\pi$
$252$	3791.06i	0.947677i
$253$	− 950.242i	− 0.236131i
$254$	−941.382	−0.232549
$255$	0	0
$256$	3067.72	0.748955
$257$	− 1571.39i	− 0.381404i	−0.981648	−	0.190702i	$-0.938924\pi$
$257$	− 1571.39i	− 0.381404i	0.981648	−	0.190702i	$-0.0610765\pi$
$258$	− 295.590i	− 0.0713280i
$259$	5840.57	1.40122
$260$	0	0
$261$	4962.37	1.17687
$262$	465.842i	0.109847i
$263$	− 749.271i	− 0.175673i	−0.996135	−	0.0878366i	$-0.972005\pi$
$263$	− 749.271i	− 0.175673i	0.996135	−	0.0878366i	$-0.0279953\pi$
$264$	194.748	0.0454012
$265$	0	0
$266$	−652.085	−0.150308
$267$	4039.90i	0.925984i
$268$	2995.37i	0.682729i
$269$	−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	$0.654903\pi$
$272$	2408.57i	0.536917i
$273$	4302.82i	0.953915i
$274$	879.676	0.193953
$275$	0	0
$276$	1650.88	0.360042
$277$	− 3788.49i	− 0.821763i	−0.911689	−	0.410881i	$-0.865221\pi$
$277$	− 3788.49i	− 0.821763i	0.911689	−	0.410881i	$-0.134779\pi$
$278$	372.139i	0.0802857i
$279$	−4989.84	−1.07073
$280$	0	0
$281$	8894.59	1.88828	0.944140	−	0.329543i	$-0.106895\pi$
$281$	8894.59	1.88828	0.944140	+	0.329543i	$0.106895\pi$
$282$	491.886i	0.103870i
$283$	8365.30i	1.75712i	0.477631	+	0.878560i	$0.341496\pi$
$283$	8365.30i	1.75712i	−0.477631	+	0.878560i	$0.658504\pi$
$284$	55.7702	0.0116526
$285$	0	0
$286$	−380.339	−0.0786360
$287$	10362.0i	2.13118i
$288$	− 1777.78i	− 0.363738i
$289$	3248.13	0.661129
$290$	0	0
$291$	−2303.58	−0.464049
$292$	− 4598.42i	− 0.921583i
$293$	− 7094.63i	− 1.41458i	−0.706923	−	0.707291i	$-0.749917\pi$
$293$	− 7094.63i	− 1.41458i	0.706923	−	0.707291i	$-0.250083\pi$
$294$	218.342	0.0433129
$295$	0	0
$296$	−1817.90	−0.356970
$297$	1294.68i	0.252945i
$298$	− 287.583i	− 0.0559035i
$299$	−6534.74	−1.26393
$300$	0	0
$301$	−6113.78	−1.17074
$302$	− 227.180i	− 0.0432872i
$303$	1500.01i	0.284401i
$304$	−3631.57	−0.685147
$305$	0	0
$306$	391.342	0.0731096
$307$	167.179i	0.0310795i	0.999879	+	0.0155397i	$0.00494665\pi$
$307$	167.179i	0.0310795i	−0.999879	+	0.0155397i	$0.995053\pi$
$308$	− 1987.37i	− 0.367666i
$309$	−2549.09	−0.469296
$310$	0	0
$311$	−5182.66	−0.944956	−0.472478	−	0.881342i	$-0.656640\pi$
$311$	−5182.66	−0.944956	−0.472478	+	0.881342i	$0.656640\pi$
$312$	− 1339.27i	− 0.243016i
$313$	7854.87i	1.41848i	0.704968	+	0.709239i	$0.250961\pi$
$313$	7854.87i	1.41848i	−0.704968	+	0.709239i	$0.749039\pi$
$314$	−755.280	−0.135742
$315$	0	0
$316$	1083.83	0.192944
$317$	− 6898.30i	− 1.22223i	−0.791542	−	0.611115i	$-0.790721\pi$
$317$	− 6898.30i	− 1.22223i	0.791542	−	0.611115i	$-0.209279\pi$
$318$	− 320.781i	− 0.0565676i
$319$	−2601.40	−0.456585
$320$	0	0
$321$	667.807	0.116116
$322$	915.632i	0.158466i
$323$	− 2510.24i	− 0.432426i
$324$	2164.76	0.371186
$325$	0	0
$326$	725.100	0.123189
$327$	3238.77i	0.547719i
$328$	− 3225.20i	− 0.542932i
$329$	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	$0.890261\pi$
$332$	− 5602.73i	− 0.926174i
$333$	− 5284.95i	− 0.869710i
$334$	118.794	0.0194614
$335$	0	0
$336$	3357.65	0.545164
$337$	− 5783.14i	− 0.934801i	−0.884046	−	0.467400i	$-0.845191\pi$
$337$	− 5783.14i	− 0.934801i	0.884046	−	0.467400i	$-0.154809\pi$
$338$	1611.36i	0.259309i
$339$	2217.72	0.355310
$340$	0	0
$341$	2615.80	0.415407
$342$	590.052i	0.0932934i
$343$	3437.91i	0.541194i
$344$	1902.94	0.298254
$345$	0	0
$346$	−409.602	−0.0636426
$347$	− 1737.90i	− 0.268863i	−0.990923	−	0.134432i	$-0.957079\pi$
$347$	− 1737.90i	− 0.268863i	0.990923	−	0.134432i	$-0.0429208\pi$
$348$	− 4519.50i	− 0.696180i
$349$	−6935.84	−1.06380	−0.531901	−	0.846806i	$-0.678522\pi$
$349$	−6935.84	−1.06380	−0.531901	+	0.846806i	$0.678522\pi$
$350$	0	0
$351$	8903.39	1.35392
$352$	931.956i	0.141118i
$353$	− 11417.5i	− 1.72151i	−0.509023	−	0.860753i	$-0.669993\pi$
$353$	− 11417.5i	− 1.72151i	0.509023	−	0.860753i	$-0.330007\pi$
$354$	−593.602	−0.0891231
$355$	0	0
$356$	−12831.9	−1.91036
$357$	2320.90i	0.344076i
$358$	1401.97i	0.206973i
$359$	−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
$362$	− 462.108i	− 0.0670936i
$363$	− 296.799i	− 0.0429144i
$364$	−13667.0	−1.96798
$365$	0	0
$366$	−84.9004	−0.0121252
$367$	− 2235.51i	− 0.317963i	−0.987282	−	0.158982i	$-0.949179\pi$
$367$	− 2235.51i	− 0.317963i	0.987282	−	0.158982i	$-0.0508210\pi$
$368$	5099.30i	0.722336i
$369$	9376.23	1.32278
$370$	0	0
$371$	−6634.81	−0.928470
$372$	4544.51i	0.633393i
$373$	5243.63i	0.727895i	0.931420	+	0.363947i	$0.118571\pi$
$373$	5243.63i	0.727895i	−0.931420	+	0.363947i	$0.881429\pi$
$374$	−205.151	−0.0283640
$375$	0	0
$376$	−3166.64	−0.434327
$377$	17889.7i	2.44394i
$378$	− 1247.52i	− 0.169750i
$379$	5262.50	0.713237	0.356618	−	0.934250i	$-0.383930\pi$
$379$	5262.50	0.713237	0.356618	+	0.934250i	$0.383930\pi$
$380$	0	0
$381$	−5051.87	−0.679305
$382$	− 2129.57i	− 0.285231i
$383$	− 12845.0i	− 1.71370i	−0.515564	−	0.856851i	$-0.672418\pi$
$383$	− 12845.0i	− 1.71370i	0.515564	−	0.856851i	$-0.327582\pi$
$384$	−2148.57	−0.285531
$385$	0	0
$386$	2198.02	0.289835
$387$	5532.17i	0.726657i
$388$	− 7316.83i	− 0.957360i
$389$	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
$392$	1405.63i	0.181110i
$393$	2499.92i	0.320876i
$394$	−1216.46	−0.155545
$395$	0	0
$396$	−1798.31	−0.228203
$397$	4136.07i	0.522880i	0.965220	+	0.261440i	$0.0841974\pi$
$397$	4136.07i	0.522880i	−0.965220	+	0.261440i	$0.915803\pi$
$398$	340.751i	0.0429153i
$399$	−3499.38	−0.439068
$400$	0	0
$401$	3334.81	0.415292	0.207646	−	0.978204i	$-0.433420\pi$
$401$	3334.81	0.415292	0.207646	+	0.978204i	$0.433420\pi$
$402$	− 431.044i	− 0.0534789i
$403$	− 17988.7i	− 2.22352i
$404$	−4764.47	−0.586736
$405$	0	0
$406$	2506.65	0.306412
$407$	2770.51i	0.337417i
$408$	− 722.389i	− 0.0876559i
$409$	−11146.6	−1.34758	−0.673792	−	0.738921i	$-0.735336\pi$
$409$	−11146.6	−1.34758	−0.673792	+	0.738921i	$0.735336\pi$
$410$	0	0
$411$	4720.73	0.566561
$412$	− 8096.63i	− 0.968186i
$413$	12277.7i	1.46282i
$414$	828.528	0.0983573
$415$	0	0
$416$	6408.99	0.755353
$417$	1997.06i	0.234524i
$418$	− 309.320i	− 0.0361946i
$419$	−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	$0.488586\pi$
$422$	1439.75i	0.166080i
$423$	− 9205.98i	− 1.05818i
$424$	2065.11	0.236534
$425$	0	0
$426$	−8.02551	−0.000912764 0
$427$	1756.02i	0.199016i
$428$	2121.15i	0.239555i
$429$	−2041.07	−0.229706
$430$	0	0
$431$	9544.05	1.06664	0.533319	−	0.845914i	$-0.320945\pi$
$431$	9544.05	1.06664	0.533319	+	0.845914i	$0.320945\pi$
$432$	− 6947.65i	− 0.773770i
$433$	− 2043.50i	− 0.226800i	−0.993549	−	0.113400i	$-0.963826\pi$
$433$	− 2043.50i	− 0.226800i	0.993549	−	0.113400i	$-0.0361741\pi$
$434$	−2520.53	−0.278777
$435$	0	0
$436$	−10287.3	−1.12998
$437$	− 5314.54i	− 0.581760i
$438$	661.728i	0.0721886i
$439$	3972.17	0.431848	0.215924	−	0.976410i	$-0.430724\pi$
$439$	3972.17	0.431848	0.215924	+	0.976410i	$0.430724\pi$
$440$	0	0
$441$	−4086.43	−0.441251
$442$	1410.81i	0.151822i
$443$	− 11124.2i	− 1.19306i	−0.802589	−	0.596532i	$-0.796545\pi$
$443$	− 11124.2i	− 1.19306i	0.802589	−	0.596532i	$-0.203455\pi$
$444$	−4813.29	−0.514479
$445$	0	0
$446$	−915.628	−0.0972113
$447$	− 1543.30i	− 0.163301i
$448$	10052.8i	1.06016i
$449$	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
$452$	7044.11i	0.733025i
$453$	− 1219.15i	− 0.126447i
$454$	−437.440	−0.0452204
$455$	0	0
$456$	1089.19	0.111856
$457$	1723.31i	0.176396i	0.996103	+	0.0881979i	$0.0281108\pi$
$457$	1723.31i	0.176396i	−0.996103	+	0.0881979i	$0.971889\pi$
$458$	− 1428.52i	− 0.145744i
$459$	4802.41	0.488360
$460$	0	0
$461$	7282.40	0.735738	0.367869	−	0.929878i	$-0.380088\pi$
$461$	7282.40	0.735738	0.367869	+	0.929878i	$0.380088\pi$
$462$	285.989i	0.0287996i
$463$	− 15632.3i	− 1.56910i	−0.620065	−	0.784550i	$-0.712894\pi$
$463$	− 15632.3i	− 1.56910i	0.620065	−	0.784550i	$-0.287106\pi$
$464$	13960.0	1.39671
$465$	0	0
$466$	381.433	0.0379174
$467$	7075.86i	0.701139i	0.936537	+	0.350569i	$0.114012\pi$
$467$	7075.86i	0.701139i	−0.936537	+	0.350569i	$0.885988\pi$
$468$	12366.9i	1.22149i
$469$	−8915.42	−0.877773
$470$	0	0
$471$	−4053.17	−0.396518
$472$	− 3821.46i	− 0.372663i
$473$	− 2900.11i	− 0.281918i
$474$	−155.967	−0.0151135
$475$	0	0
$476$	−7371.86	−0.709850
$477$	6003.64i	0.576284i
$478$	− 2098.37i	− 0.200789i
$479$	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
$482$	− 2250.02i	− 0.212626i
$483$	4913.69i	0.462900i
$484$	942.721	0.0885350
$485$	0	0
$486$	−1764.04	−0.164647
$487$	− 19796.8i	− 1.84205i	−0.389502	−	0.921026i	$-0.627353\pi$
$487$	− 19796.8i	− 1.84205i	0.389502	−	0.921026i	$-0.372647\pi$
$488$	− 546.568i	− 0.0507007i
$489$	3891.21	0.359850
$490$	0	0
$491$	−7113.02	−0.653780	−0.326890	−	0.945062i	$-0.606001\pi$
$491$	−7113.02	−0.653780	−0.326890	+	0.945062i	$0.606001\pi$
$492$	− 8539.43i	− 0.782494i
$493$	9649.52i	0.881526i
$494$	−2127.17	−0.193737
$495$	0	0
$496$	−14037.2	−1.27075
$497$	165.994i	0.0149816i
$498$	806.251i	0.0725482i
$499$	−7116.10	−0.638398	−0.319199	−	0.947688i	$-0.603414\pi$
$499$	−7116.10	−0.638398	−0.319199	+	0.947688i	$0.603414\pi$
$500$	0	0
$501$	637.499	0.0568490
$502$	− 2205.03i	− 0.196046i
$503$	759.231i	0.0673010i	0.999434	+	0.0336505i	$0.0107133\pi$
$503$	759.231i	0.0673010i	−0.999434	+	0.0336505i	$0.989287\pi$
$504$	3512.09	0.310399
$505$	0	0
$506$	−434.335	−0.0381592
$507$	8647.28i	0.757473i
$508$	− 16046.2i	− 1.40145i
$509$	−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
$512$	− 8409.67i	− 0.725895i
$513$	7240.90i	0.623185i
$514$	−718.251	−0.0616356
$515$	0	0
$516$	5038.45	0.429855
$517$	4826.01i	0.410537i
$518$	− 2669.60i	− 0.226439i
$519$	−2198.11	−0.185908
$520$	0	0
$521$	−6037.10	−0.507659	−0.253829	−	0.967249i	$-0.581690\pi$
$521$	−6037.10	−0.507659	−0.253829	+	0.967249i	$0.581690\pi$
$522$	− 2268.20i	− 0.190184i
$523$	20359.6i	1.70222i	0.524987	+	0.851110i	$0.324070\pi$
$523$	20359.6i	1.70222i	−0.524987	+	0.851110i	$0.675930\pi$
$524$	−7940.45	−0.661985
$525$	0	0
$526$	−342.476	−0.0283891
$527$	− 9702.93i	− 0.802023i
$528$	1592.72i	0.131277i
$529$	4704.52	0.386663
$530$	0	0
$531$	11109.7	0.907945
$532$	− 11115.0i	− 0.905823i
$533$	33801.8i	2.74694i
$534$	1846.55	0.149641
$535$	0	0
$536$	2774.95	0.223619
$537$	7523.60i	0.604595i
$538$	139.324i	0.0111649i
$539$	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	$0.376500\pi$
$542$	1907.24i	0.151150i
$543$	− 2479.88i	− 0.195989i
$544$	3456.95	0.272455
$545$	0	0
$546$	1966.73	0.154154
$547$	− 408.097i	− 0.0318994i	−0.999873	−	0.0159497i	$-0.994923\pi$
$547$	− 408.097i	− 0.0318994i	0.999873	−	0.0159497i	$-0.00507716\pi$
$548$	14994.4i	1.16885i
$549$	1588.97	0.123526
$550$	0	0
$551$	−14549.2	−1.12490
$552$	− 1529.40i	− 0.117927i
$553$	3225.92i	0.248065i
$554$	−1731.64	−0.132798
$555$	0	0
$556$	−6343.25	−0.483838
$557$	21981.3i	1.67213i	0.548627	+	0.836067i	$0.315151\pi$
$557$	21981.3i	1.67213i	−0.548627	+	0.836067i	$0.684849\pi$
$558$	2280.75i	0.173032i
$559$	−19943.8	−1.50900
$560$	0	0
$561$	−1100.93	−0.0828546
$562$	− 4065.53i	− 0.305149i
$563$	6384.70i	0.477945i	0.971026	+	0.238973i	$0.0768106\pi$
$563$	6384.70i	0.477945i	−0.971026	+	0.238973i	$0.923189\pi$
$564$	−8384.38	−0.625968
$565$	0	0
$566$	3823.60	0.283954
$567$	6443.18i	0.477228i
$568$	− 51.6663i	− 0.00381667i
$569$	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	$0.633707\pi$
$572$	− 6483.02i	− 0.473896i
$573$	− 11428.2i	− 0.833196i
$574$	4736.23	0.344402
$575$	0	0
$576$	9096.51	0.658023
$577$	− 1957.55i	− 0.141237i	−0.997503	−	0.0706185i	$-0.977503\pi$
$577$	− 1957.55i	− 0.141237i	0.997503	−	0.0706185i	$-0.0224973\pi$
$578$	− 1484.65i	− 0.106840i
$579$	11795.6	0.846643
$580$	0	0
$581$	16675.9	1.19077
$582$	1052.92i	0.0749911i
$583$	− 3147.26i	− 0.223578i
$584$	−4260.04	−0.301852
$585$	0	0
$586$	−3242.80	−0.228599
$587$	26875.7i	1.88974i	0.327447	+	0.944870i	$0.393812\pi$
$587$	26875.7i	1.88974i	−0.327447	+	0.944870i	$0.606188\pi$
$588$	3721.73i	0.261023i
$589$	14629.7	1.02344
$590$	0	0
$591$	−6528.09	−0.454365
$592$	− 14867.4i	− 1.03217i
$593$	18231.3i	1.26251i	0.775575	+	0.631255i	$0.217460\pi$
$593$	18231.3i	1.26251i	−0.775575	+	0.631255i	$0.782540\pi$
$594$	591.768	0.0408764
$595$	0	0
$596$	4901.96	0.336900
$597$	1828.62i	0.125361i
$598$	2986.89i	0.204253i
$599$	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	$0.501000\pi$
$602$	2794.48i	0.189193i
$603$	8067.29i	0.544818i
$604$	3872.36	0.260868
$605$	0	0
$606$	685.624	0.0459597
$607$	− 6711.75i	− 0.448800i	−0.974497	−	0.224400i	$-0.927958\pi$
$607$	− 6711.75i	− 0.448800i	0.974497	−	0.224400i	$-0.0720422\pi$
$608$	5212.27i	0.347674i
$609$	13451.8	0.895067
$610$	0	0
$611$	33188.1	2.19746
$612$	6670.57i	0.440591i
$613$	− 1130.01i	− 0.0744549i	−0.999307	−	0.0372274i	$-0.988147\pi$
$613$	− 1130.01i	− 0.0744549i	0.999307	−	0.0372274i	$-0.0118526\pi$
$614$	76.4139	0.00502250
$615$	0	0
$616$	−1841.13	−0.120424
$617$	− 19798.5i	− 1.29183i	−0.763411	−	0.645913i	$-0.776477\pi$
$617$	− 19798.5i	− 1.29183i	0.763411	−	0.645913i	$-0.223523\pi$
$618$	1165.13i	0.0758390i
$619$	−7620.34	−0.494810	−0.247405	−	0.968912i	$-0.579578\pi$
$619$	−7620.34	−0.494810	−0.247405	+	0.968912i	$0.579578\pi$
$620$	0	0
$621$	10167.4	0.657010
$622$	2368.88i	0.152707i
$623$	− 38192.8i	− 2.45612i
$624$	10953.0	0.702680
$625$	0	0
$626$	3590.30	0.229229
$627$	− 1659.95i	− 0.105729i
$628$	− 12874.0i	− 0.818041i
$629$	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	$0.423535\pi$
$632$	− 1004.08i	− 0.0631963i
$633$	7726.32i	0.485140i
$634$	−3153.06	−0.197515
$635$	0	0
$636$	5467.83	0.340902
$637$	− 14731.8i	− 0.916320i
$638$	1189.05i	0.0737849i
$639$	150.203	0.00929881
$640$	0	0
$641$	−27329.5	−1.68401	−0.842005	−	0.539470i	$-0.818625\pi$
$641$	−27329.5	−1.68401	−0.842005	+	0.539470i	$0.818625\pi$
$642$	− 305.240i	− 0.0187646i
$643$	− 8792.26i	− 0.539243i	−0.962966	−	0.269621i	$-0.913101\pi$
$643$	− 8792.26i	− 0.539243i	0.962966	−	0.269621i	$-0.0868985\pi$
$644$	−15607.3	−0.954990
$645$	0	0
$646$	−1147.38	−0.0698807
$647$	− 8420.01i	− 0.511630i	−0.966726	−	0.255815i	$-0.917656\pi$
$647$	− 8420.01i	− 0.511630i	0.966726	−	0.255815i	$-0.0823438\pi$
$648$	− 2005.46i	− 0.121577i
$649$	−5823.97	−0.352251
$650$	0	0
$651$	−13526.3	−0.814343
$652$	12359.6i	0.742392i
$653$	23596.6i	1.41410i	0.707164	+	0.707050i	$0.249974\pi$
$653$	23596.6i	1.41410i	−0.707164	+	0.707050i	$0.750026\pi$
$654$	1480.37	0.0885124
$655$	0	0
$656$	26376.8	1.56988
$657$	− 12384.7i	− 0.735424i
$658$	− 4650.24i	− 0.275509i
$659$	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	$0.323310\pi$
$662$	5181.14i	0.304186i
$663$	7571.04i	0.443491i
$664$	−5190.45	−0.303356
$665$	0	0
$666$	−2415.64	−0.140547
$667$	20429.4i	1.18595i
$668$	2024.88i	0.117283i
$669$	−4913.67	−0.283966
$670$	0	0
$671$	−832.978	−0.0479237
$672$	− 4819.14i	− 0.276640i
$673$	942.349i	0.0539746i	0.999636	+	0.0269873i	$0.00859137\pi$
$673$	942.349i	0.0539746i	−0.999636	+	0.0269873i	$0.991409\pi$
$674$	−2643.35	−0.151065
$675$	0	0
$676$	−27466.2	−1.56271
$677$	− 13681.1i	− 0.776673i	−0.921518	−	0.388336i	$-0.873050\pi$
$677$	− 13681.1i	− 0.776673i	0.921518	−	0.388336i	$-0.126950\pi$
$678$	− 1013.67i	− 0.0574186i
$679$	21777.8	1.23086
$680$	0	0
$681$	−2347.50	−0.132094
$682$	− 1195.63i	− 0.0671304i
$683$	− 7909.32i	− 0.443107i	−0.975148	−	0.221553i	$-0.928887\pi$
$683$	− 7909.32i	− 0.443107i	0.975148	−	0.221553i	$-0.0711127\pi$
$684$	−10057.7	−0.562228
$685$	0	0
$686$	1571.39	0.0874579
$687$	− 7666.10i	− 0.425735i
$688$	15562.9i	0.862399i
$689$	−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	$0.600815\pi$
$692$	− 6981.82i	− 0.383539i
$693$	− 5352.49i	− 0.293397i
$694$	−794.358	−0.0434487
$695$	0	0
$696$	−4186.93	−0.228024
$697$	18232.4i	0.990820i
$698$	3170.23i	0.171912i
$699$	2046.94	0.110762
$700$	0	0
$701$	−5833.23	−0.314291	−0.157146	−	0.987575i	$-0.550229\pi$
$701$	−5833.23	−0.314291	−0.157146	+	0.987575i	$0.550229\pi$
$702$	− 4069.55i	− 0.218797i
$703$	15495.0i	0.831300i
$704$	−4768.62	−0.255290
$705$	0	0
$706$	−5218.69	−0.278198
$707$	− 14181.0i	− 0.754357i
$708$	− 10118.2i	− 0.537096i
$709$	−31717.5	−1.68008	−0.840038	−	0.542527i	$-0.817468\pi$
$709$	−31717.5	−1.68008	−0.840038	+	0.542527i	$0.817468\pi$
$710$	0	0
$711$	2919.03	0.153969
$712$	11887.6i	0.625714i
$713$	− 20542.5i	− 1.07899i
$714$	1060.84	0.0556033
$715$	0	0
$716$	−23897.1	−1.24731
$717$	− 11260.8i	− 0.586530i
$718$	3758.80i	0.195372i
$719$	−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
$722$	1405.13i	0.0724285i
$723$	− 12074.6i	− 0.621105i
$724$	7876.81	0.404336
$725$	0	0
$726$	−135.661	−0.00693504
$727$	23555.4i	1.20168i	0.799370	+	0.600839i	$0.205167\pi$
$727$	23555.4i	1.20168i	−0.799370	+	0.600839i	$0.794833\pi$
$728$	12661.3i	0.644587i
$729$	−1964.65	−0.0998144
$730$	0	0
$731$	−10757.5	−0.544297
$732$	− 1447.16i	− 0.0730718i
$733$	32919.2i	1.65880i	0.558656	+	0.829399i	$0.311317\pi$
$733$	32919.2i	1.65880i	−0.558656	+	0.829399i	$0.688683\pi$
$734$	−1021.80	−0.0513834
$735$	0	0
$736$	7318.87	0.366545
$737$	− 4229.08i	− 0.211370i
$738$	− 4285.67i	− 0.213764i
$739$	−7979.80	−0.397215	−0.198607	−	0.980079i	$-0.563642\pi$
$739$	−7979.80	−0.397215	−0.198607	+	0.980079i	$0.563642\pi$
$740$	0	0
$741$	−11415.4	−0.565929
$742$	3032.63i	0.150042i
$743$	− 36491.6i	− 1.80181i	−0.434013	−	0.900907i	$-0.642903\pi$
$743$	− 36491.6i	− 1.80181i	0.434013	−	0.900907i	$-0.357097\pi$
$744$	4210.10	0.207459
$745$	0	0
$746$	2396.75	0.117629
$747$	− 15089.6i	− 0.739087i
$748$	− 3496.88i	− 0.170934i
$749$	−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	$0.562767\pi$
$752$	− 25897.9i	− 1.25585i
$753$	− 11833.2i	− 0.572675i
$754$	8176.98	0.394944
$755$	0	0
$756$	21264.5	1.02299
$757$	164.260i	0.00788659i	0.999992	+	0.00394329i	$0.00125519\pi$
$757$	164.260i	0.00788659i	−0.999992	+	0.00394329i	$0.998745\pi$
$758$	− 2405.38i	− 0.115260i
$759$	−2330.84	−0.111468
$760$	0	0
$761$	6387.72	0.304277	0.152138	−	0.988359i	$-0.451384\pi$
$761$	6387.72	0.304277	0.152138	+	0.988359i	$0.451384\pi$
$762$	2309.10i	0.109777i
$763$	− 30619.0i	− 1.45279i
$764$	36299.4	1.71893
$765$	0	0
$766$	−5871.17	−0.276937
$767$	40051.0i	1.88547i
$768$	− 7524.77i	− 0.353550i
$769$	4920.10	0.230719	0.115360	−	0.993324i	$-0.463198\pi$
$769$	4920.10	0.230719	0.115360	+	0.993324i	$0.463198\pi$
$770$	0	0
$771$	−3854.46	−0.180045
$772$	37466.1i	1.74668i
$773$	− 25930.2i	− 1.20653i	−0.797542	−	0.603264i	$-0.793867\pi$
$773$	− 25930.2i	− 1.20653i	0.797542	−	0.603264i	$-0.206133\pi$
$774$	2528.64	0.117429
$775$	0	0
$776$	−6778.41	−0.313571
$777$	− 14326.3i	− 0.661457i
$778$	− 5700.22i	− 0.262677i
$779$	−27490.2	−1.26436
$780$	0	0
$781$	−78.7403	−0.00360762
$782$	1611.10i	0.0736738i
$783$	− 27834.5i	− 1.27040i
$784$	−11495.8	−0.523678
$785$	0	0
$786$	1142.66	0.0518540
$787$	− 876.332i	− 0.0396923i	−0.999803	−	0.0198462i	$-0.993682\pi$
$787$	− 876.332i	− 0.0396923i	0.999803	−	0.0198462i	$-0.00631764\pi$
$788$	− 20735.1i	− 0.937382i
$789$	−1837.88	−0.0829280
$790$	0	0
$791$	−20966.1	−0.942438
$792$	1665.98i	0.0747450i
$793$	5728.33i	0.256518i
$794$	1890.51	0.0844983
$795$	0	0
$796$	−5808.22	−0.258627
$797$	− 20900.9i	− 0.928917i	−0.885595	−	0.464458i	$-0.846249\pi$
$797$	− 20900.9i	− 0.928917i	0.885595	−	0.464458i	$-0.153751\pi$
$798$	1599.49i	0.0709541i
$799$	17901.4	0.792622
$800$	0	0
$801$	−34559.5	−1.52447
$802$	− 1524.27i	− 0.0671120i
$803$	6492.38i	0.285319i
$804$	7347.31	0.322288
$805$	0	0
$806$	−8222.24	−0.359325
$807$	747.676i	0.0326139i
$808$	4413.87i	0.192178i
$809$	−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	$0.529185\pi$
$812$	42726.9i	1.84658i
$813$	10235.1i	0.441527i
$814$	1266.34	0.0545272
$815$	0	0
$816$	5907.96	0.253456
$817$	− 16219.8i	− 0.694565i
$818$	5094.85i	0.217772i
$819$	−36808.7	−1.57045
$820$	0	0
$821$	−31556.0	−1.34143	−0.670713	−	0.741717i	$-0.734012\pi$
$821$	−31556.0	−1.34143	−0.670713	+	0.741717i	$0.734012\pi$
$822$	− 2157.75i	− 0.0915572i
$823$	36827.7i	1.55982i	0.625891	+	0.779910i	$0.284735\pi$
$823$	36827.7i	1.55982i	−0.625891	+	0.779910i	$0.715265\pi$
$824$	−7500.83	−0.317116
$825$	0	0
$826$	5611.85	0.236394
$827$	1188.45i	0.0499714i	0.999688	+	0.0249857i	$0.00795402\pi$
$827$	1188.45i	0.0499714i	−0.999688	+	0.0249857i	$0.992046\pi$
$828$	14122.6i	0.592745i
$829$	37008.8	1.55051	0.775253	−	0.631651i	$-0.217622\pi$
$829$	37008.8	1.55051	0.775253	+	0.631651i	$0.217622\pi$
$830$	0	0
$831$	−9292.74	−0.387920
$832$	32793.5i	1.36648i
$833$	− 7946.21i	− 0.330516i
$834$	912.815	0.0378995
$835$	0	0
$836$	5272.48	0.218125
$837$	27988.5i	1.15583i
$838$	4074.72i	0.167970i
$839$	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
$842$	− 283.098i	− 0.0115869i
$843$	− 21817.4i	− 0.891379i
$844$	−24541.0	−1.00087
$845$	0	0
$846$	−4207.86	−0.171004
$847$	2805.91i	0.113828i
$848$	16889.2i	0.683936i
$849$	20519.1	0.829464
$850$	0	0
$851$	21757.4	0.876423
$852$	− 136.798i	− 0.00550073i
$853$	5084.37i	0.204086i	0.994780	+	0.102043i	$0.0325379\pi$
$853$	5084.37i	0.204086i	−0.994780	+	0.102043i	$0.967462\pi$
$854$	802.640	0.0321613
$855$	0	0
$856$	1965.06	0.0784631
$857$	36268.5i	1.44563i	0.691040	+	0.722816i	$0.257153\pi$
$857$	36268.5i	1.44563i	−0.691040	+	0.722816i	$0.742847\pi$
$858$	932.929i	0.0371208i
$859$	−36158.6	−1.43622	−0.718111	−	0.695928i	$-0.754993\pi$
$859$	−36158.6	−1.43622	−0.718111	+	0.695928i	$0.754993\pi$
$860$	0	0
$861$	25416.7	1.00604
$862$	− 4362.38i	− 0.172370i
$863$	− 5105.14i	− 0.201368i	−0.994918	−	0.100684i	$-0.967897\pi$
$863$	− 5105.14i	− 0.201368i	0.994918	−	0.100684i	$-0.0321032\pi$
$864$	−9971.74	−0.392645
$865$	0	0
$866$	−934.039	−0.0366512
$867$	− 7967.29i	− 0.312091i
$868$	− 42963.4i	− 1.68004i
$869$	−1530.23	−0.0597348
$870$	0	0
$871$	−29083.0	−1.13139
$872$	9530.26i	0.370109i
$873$	− 19706.1i	− 0.763974i
$874$	−2429.16	−0.0940134
$875$	0	0
$876$	−11279.4	−0.435041
$877$	− 33922.6i	− 1.30614i	−0.757297	−	0.653071i	$-0.773480\pi$
$877$	− 33922.6i	− 1.30614i	0.757297	−	0.653071i	$-0.226520\pi$
$878$	− 1815.59i	− 0.0697874i
$879$	−17402.3	−0.667765
$880$	0	0
$881$	12610.2	0.482235	0.241117	−	0.970496i	$-0.422486\pi$
$881$	12610.2	0.482235	0.241117	+	0.970496i	$0.422486\pi$
$882$	1867.82i	0.0713070i
$883$	− 40762.8i	− 1.55354i	−0.629783	−	0.776771i	$-0.716856\pi$
$883$	− 40762.8i	− 1.55354i	0.629783	−	0.776771i	$-0.283144\pi$
$884$	−24047.8	−0.914949
$885$	0	0
$886$	−5084.64	−0.192801
$887$	− 29954.3i	− 1.13390i	−0.823753	−	0.566949i	$-0.808123\pi$
$887$	− 29954.3i	− 1.13390i	0.823753	−	0.566949i	$-0.191877\pi$
$888$	4459.10i	0.168511i
$889$	47759.9	1.80182
$890$	0	0
$891$	−3056.36	−0.114918
$892$	− 15607.2i	− 0.585839i
$893$	26991.1i	1.01145i
$894$	−705.409	−0.0263897
$895$	0	0
$896$	20312.4	0.757353
$897$	16029.0i	0.596647i
$898$	− 3210.18i	− 0.119293i
$899$	−56237.6	−2.08635
$900$	0	0
$901$	−11674.3	−0.431662
$902$	2246.66i	0.0829330i
$903$	14996.4i	0.552657i
$904$	6525.76	0.240093
$905$	0	0
$906$	−557.246	−0.0204341
$907$	29053.0i	1.06360i	0.846869	+	0.531802i	$0.178485\pi$
$907$	29053.0i	1.06360i	−0.846869	+	0.531802i	$0.821515\pi$
$908$	− 7456.33i	− 0.272519i
$909$	−12831.9	−0.468216
$910$	0	0
$911$	−3562.14	−0.129549	−0.0647744	−	0.997900i	$-0.520633\pi$
$911$	−3562.14	−0.129549	−0.0647744	+	0.997900i	$0.520633\pi$
$912$	8907.83i	0.323429i
$913$	7910.33i	0.286740i
$914$	787.687	0.0285059
$915$	0	0
$916$	24349.7	0.878317
$917$	− 23634.0i	− 0.851104i
$918$	− 2195.08i	− 0.0789198i
$919$	−2936.50	−0.105404	−0.0527020	−	0.998610i	$-0.516783\pi$
$919$	−2936.50	−0.105404	−0.0527020	+	0.998610i	$0.516783\pi$
$920$	0	0
$921$	410.071	0.0146713
$922$	− 3328.63i	− 0.118896i
$923$	541.491i	0.0193103i
$924$	−4874.80	−0.173560
$925$	0	0
$926$	−7145.18	−0.253569
$927$	− 21806.3i	− 0.772613i
$928$	− 20036.3i	− 0.708754i
$929$	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
$932$	6501.66i	0.228508i
$933$	12712.5i	0.446075i
$934$	3234.22	0.113305
$935$	0	0
$936$	11456.8	0.400084
$937$	− 19863.3i	− 0.692537i	−0.938135	−	0.346269i	$-0.887449\pi$
$937$	− 19863.3i	− 0.692537i	0.938135	−	0.346269i	$-0.112551\pi$
$938$	4075.05i	0.141850i
$939$	19267.1	0.669605
$940$	0	0
$941$	−39408.8	−1.36524	−0.682620	−	0.730773i	$-0.739160\pi$
$941$	−39408.8	−1.36524	−0.682620	+	0.730773i	$0.739160\pi$
$942$	1852.62i	0.0640781i
$943$	38600.7i	1.33299i
$944$	31253.3	1.07755
$945$	0	0
$946$	−1325.58	−0.0455584
$947$	− 40190.4i	− 1.37910i	−0.724236	−	0.689552i	$-0.757807\pi$
$947$	− 40190.4i	− 1.37910i	0.724236	−	0.689552i	$-0.242193\pi$
$948$	− 2658.52i	− 0.0910809i
$949$	44647.6	1.52721
$950$	0	0
$951$	−16920.8	−0.576964
$952$	6829.39i	0.232502i
$953$	43114.9i	1.46551i	0.680493	+	0.732754i	$0.261765\pi$
$953$	43114.9i	1.46551i	−0.680493	+	0.732754i	$0.738235\pi$
$954$	2744.13	0.0931286
$955$	0	0
$956$	35767.5	1.21005
$957$	6380.95i	0.215535i
$958$	− 1938.20i	− 0.0653657i
$959$	−44629.3	−1.50277
$960$	0	0
$961$	26757.9	0.898188
$962$	− 8708.52i	− 0.291865i
$963$	5712.78i	0.191165i
$964$	38352.4	1.28138
$965$	0	0
$966$	2245.94	0.0748054
$967$	15536.2i	0.516659i	0.966057	+	0.258330i	$0.0831721\pi$
$967$	15536.2i	0.516659i	−0.966057	+	0.258330i	$0.916828\pi$
$968$	− 873.350i	− 0.0289985i
$969$	−6157.34	−0.204130
$970$	0	0
$971$	−18466.6	−0.610320	−0.305160	−	0.952301i	$-0.598710\pi$
$971$	−18466.6	−0.610320	−0.305160	+	0.952301i	$0.598710\pi$
$972$	− 30068.7i	− 0.992237i
$973$	− 18880.0i	− 0.622062i
$974$	−9048.70	−0.297679
$975$	0	0
$976$	4470.03	0.146601
$977$	− 13352.4i	− 0.437236i	−0.975810	−	0.218618i	$-0.929845\pi$
$977$	− 13352.4i	− 0.437236i	0.975810	−	0.218618i	$-0.0701549\pi$
$978$	− 1778.59i	− 0.0581524i
$979$	18117.0	0.591441
$980$	0	0
$981$	−27706.2	−0.901723
$982$	3251.21i	0.105652i
$983$	9970.61i	0.323513i	0.986831	+	0.161756i	$0.0517159\pi$
$983$	9970.61i	0.323513i	−0.986831	+	0.161756i	$0.948284\pi$
$984$	−7911.05	−0.256296
$985$	0	0
$986$	4410.59	0.142456
$987$	− 24955.3i	− 0.804797i
$988$	− 36258.4i	− 1.16755i
$989$	−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	$0.303640\pi$
$992$	20147.2i	0.644833i
$993$	27804.3i	0.888563i
$994$	75.8724	0.00242105
$995$	0	0
$996$	−13742.9	−0.437208
$997$	− 51210.1i	− 1.62672i	−0.581760	−	0.813360i	$-0.697636\pi$
$997$	− 51210.1i	− 1.62672i	0.581760	−	0.813360i	$-0.302364\pi$
$998$	3252.62i	0.103166i
$999$	−29643.9	−0.938829

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.4.a.g.1.3	✓	5	5.3	odd	4
275.4.a.h.1.3	yes	5	5.2	odd	4
275.4.b.f.199.5		10	1.1	even	1	trivial
275.4.b.f.199.6		10	5.4	even	2	inner
2475.4.a.bh.1.3		5	15.2	even	4
2475.4.a.bl.1.3		5	15.8	even	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:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	275.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.457079i q^{2} -2.45289i q^{3} +7.79108 q^{4} -1.12116 q^{6} +23.1894i q^{7} -7.21777i q^{8} +20.9833 q^{9} -11.0000 q^{11} -19.1106i q^{12} +75.6462i q^{13} +10.5994 q^{14} +59.0295 q^{16} +40.8029i q^{17} -9.59104i q^{18} -61.5212 q^{19} +56.8809 q^{21} +5.02787i q^{22} +86.3856i q^{23} -17.7044 q^{24} +34.5763 q^{26} -117.698i q^{27} +180.670i q^{28} +236.491 q^{29} -237.800 q^{31} -84.7233i q^{32} +26.9818i q^{33} +18.6501 q^{34} +163.483 q^{36} -251.864i q^{37} +28.1200i q^{38} +185.552 q^{39} +446.841 q^{41} -25.9990i q^{42} +263.646i q^{43} -85.7019 q^{44} +39.4850 q^{46} -438.728i q^{47} -144.793i q^{48} -194.746 q^{49} +100.085 q^{51} +589.365i q^{52} +286.114i q^{53} -53.7971 q^{54} +167.375 q^{56} +150.905i q^{57} -108.095i q^{58} +529.452 q^{59} +75.7253 q^{61} +108.693i q^{62} +486.590i q^{63} +433.511 q^{64} +12.3328 q^{66} +384.462i q^{67} +317.898i q^{68} +211.894 q^{69} +7.15821 q^{71} -151.453i q^{72} -590.216i q^{73} -115.122 q^{74} -479.316 q^{76} -255.083i q^{77} -84.8117i q^{78} +139.112 q^{79} +277.851 q^{81} -204.242i q^{82} -719.121i q^{83} +443.164 q^{84} +120.507 q^{86} -580.086i q^{87} +79.3954i q^{88} -1647.00 q^{89} -1754.19 q^{91} +673.037i q^{92} +583.297i q^{93} -200.533 q^{94} -207.817 q^{96} -939.129i q^{97} +89.0144i q^{98} -230.817 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 80 q^{4} - 84 q^{6} - 62 q^{9} - 110 q^{11} + 266 q^{14} + 416 q^{16} - 46 q^{19} + 564 q^{21} + 982 q^{24} + 644 q^{26} + 366 q^{29} - 2 q^{31} + 1300 q^{34} + 2676 q^{36} + 540 q^{39} + 188 q^{41}+ \cdots + 682 q^{99}+O(q^{100}) \) 10 * q - 80 * q^4 - 84 * q^6 - 62 * q^9 - 110 * q^11 + 266 * q^14 + 416 * q^16 - 46 * q^19 + 564 * q^21 + 982 * q^24 + 644 * q^26 + 366 * q^29 - 2 * q^31 + 1300 * q^34 + 2676 * q^36 + 540 * q^39 + 188 * q^41 + 880 * q^44 + 2012 * q^46 - 454 * q^49 - 2256 * q^51 + 2410 * q^54 - 1670 * q^56 - 2348 * q^59 + 1280 * q^61 + 1898 * q^64 + 924 * q^66 + 3292 * q^69 - 1346 * q^71 + 1122 * q^74 + 2496 * q^76 - 208 * q^79 - 2438 * q^81 - 8706 * q^84 + 4994 * q^86 - 5642 * q^89 + 960 * q^91 + 3574 * q^94 - 3638 * q^96 + 682 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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