LMFDB - Embedded newform 880.4.b.h.529.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [880,4,Mod(529,880)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(880, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("880.529");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 880 = 2^{4} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	880.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$51.9216808051$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 151x^{6} + 7935x^{4} + 171721x^{2} + 1308736 $ x^8 + 151x^6 + 7935x^4 + 171721*x^2 + 1308736
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 110)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			529.1
Root			$-6.63700i$ of defining polynomial
Character	$\chi$	$=$	880.529
Dual form			880.4.b.h.529.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-8.63700i q^{3} +(11.1511 + 0.807757i) q^{5} -0.978517i q^{7} -47.5977 q^{9} +11.0000 q^{11} +89.2090i q^{13} +(6.97659 - 96.3122i) q^{15} +46.3299i q^{17} +44.2390 q^{19} -8.45144 q^{21} +111.380i q^{23} +(123.695 + 18.0148i) q^{25} +177.902i q^{27} -28.6875 q^{29} -83.6519 q^{31} -95.0070i q^{33} +(0.790404 - 10.9116i) q^{35} +222.770i q^{37} +770.498 q^{39} +421.768 q^{41} +273.936i q^{43} +(-530.768 - 38.4474i) q^{45} +469.468i q^{47} +342.043 q^{49} +400.151 q^{51} +220.572i q^{53} +(122.662 + 8.88533i) q^{55} -382.092i q^{57} -180.884 q^{59} -517.166 q^{61} +46.5752i q^{63} +(-72.0592 + 994.780i) q^{65} +193.831i q^{67} +961.988 q^{69} -1016.64 q^{71} -958.858i q^{73} +(155.594 - 1068.35i) q^{75} -10.7637i q^{77} -567.225 q^{79} +251.405 q^{81} -642.773i q^{83} +(-37.4233 + 516.631i) q^{85} +247.774i q^{87} +1406.68 q^{89} +87.2925 q^{91} +722.501i q^{93} +(493.314 + 35.7343i) q^{95} -342.110i q^{97} -523.575 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{5} - 110 q^{9} + 88 q^{11} - 8 q^{15} + 302 q^{19} + 230 q^{21} - 162 q^{25} - 58 q^{29} - 1022 q^{31} + 1058 q^{35} + 320 q^{39} + 452 q^{41} - 622 q^{45} + 222 q^{49} - 834 q^{51} + 176 q^{55}+ \cdots - 1210 q^{99}+O(q^{100}) $ 8 * q + 16 * q^5 - 110 * q^9 + 88 * q^11 - 8 * q^15 + 302 * q^19 + 230 * q^21 - 162 * q^25 - 58 * q^29 - 1022 * q^31 + 1058 * q^35 + 320 * q^39 + 452 * q^41 - 622 * q^45 + 222 * q^49 - 834 * q^51 + 176 * q^55 - 268 * q^59 - 490 * q^61 - 1408 * q^65 + 2820 * q^69 - 2034 * q^71 + 2110 * q^75 - 272 * q^79 - 2488 * q^81 + 330 * q^85 + 5782 * q^89 + 3520 * q^91 - 1268 * q^95 - 1210 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/880\mathbb{Z}\right)^\times$.

$n$	$111$	$177$	$321$	$661$
$\chi(n)$	$1$	$-1$	$1$	$1$

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			0
$2$		0			0
$3$		−	8.63700i		−	1.66219i	−0.556130	−	0.831095i	$-0.687714\pi$
$3$		−	8.63700i		−	1.66219i	0.556130	−	0.831095i	$-0.312286\pi$
$4$		0			0
$5$	11.1511	+	0.807757i	0.997387	+	0.0722480i
$5$	11.1511	+	0.807757i	0.997387	+	0.0722480i
$6$		0			0
$7$		−	0.978517i		−	0.0528349i	−0.999651	−	0.0264175i	$-0.991590\pi$
$7$		−	0.978517i		−	0.0528349i	0.999651	−	0.0264175i	$-0.00840992\pi$
$8$		0			0
$9$	−47.5977			−1.76288
$10$		0			0
$11$	11.0000			0.301511
$11$	11.0000			0.301511
$12$		0			0
$13$			89.2090i			1.90324i	0.307277	+	0.951620i	$0.400582\pi$
$13$			89.2090i			1.90324i	−0.307277	+	0.951620i	$0.599418\pi$
$14$		0			0
$15$	6.97659	−	96.3122i	0.120090	−	1.65785i
$16$		0			0
$17$			46.3299i			0.660980i	0.943810	+	0.330490i	$0.107214\pi$
$17$			46.3299i			0.660980i	−0.943810	+	0.330490i	$0.892786\pi$
$18$		0			0
$19$	44.2390			0.534164			0.267082	−	0.963674i	$-0.413941\pi$
$19$	44.2390			0.534164			0.267082	+	0.963674i	$0.413941\pi$
$20$		0			0
$21$	−8.45144			−0.0878217
$22$		0			0
$23$			111.380i			1.00975i	0.863192	+	0.504876i	$0.168462\pi$
$23$			111.380i			1.00975i	−0.863192	+	0.504876i	$0.831538\pi$
$24$		0			0
$25$	123.695	+	18.0148i	0.989560	+	0.144118i
$26$		0			0
$27$			177.902i			1.26805i
$28$		0			0
$29$	−28.6875			−0.183694			−0.0918472	−	0.995773i	$-0.529277\pi$
$29$	−28.6875			−0.183694			−0.0918472	+	0.995773i	$0.529277\pi$
$30$		0			0
$31$	−83.6519			−0.484655			−0.242328	−	0.970194i	$-0.577911\pi$
$31$	−83.6519			−0.484655			−0.242328	+	0.970194i	$0.577911\pi$
$32$		0			0
$33$		−	95.0070i		−	0.501169i
$34$		0			0
$35$	0.790404	−	10.9116i	0.00381722	−	0.0526969i
$36$		0			0
$37$			222.770i			0.989817i	0.868945	+	0.494909i	$0.164799\pi$
$37$			222.770i			0.989817i	−0.868945	+	0.494909i	$0.835201\pi$
$38$		0			0
$39$	770.498			3.16355
$40$		0			0
$41$	421.768			1.60656			0.803282	−	0.595599i	$-0.203085\pi$
$41$	421.768			1.60656			0.803282	+	0.595599i	$0.203085\pi$
$42$		0			0
$43$			273.936i			0.971509i	0.874095	+	0.485754i	$0.161455\pi$
$43$			273.936i			0.971509i	−0.874095	+	0.485754i	$0.838545\pi$
$44$		0			0
$45$	−530.768	−	38.4474i	−1.75827	−	0.127364i
$46$		0			0
$47$			469.468i			1.45700i	0.685046	+	0.728500i	$0.259782\pi$
$47$			469.468i			1.45700i	−0.685046	+	0.728500i	$0.740218\pi$
$48$		0			0
$49$	342.043			0.997208
$50$		0			0
$51$	400.151			1.09867
$52$		0			0
$53$			220.572i			0.571657i	0.958281	+	0.285829i	$0.0922688\pi$
$53$			220.572i			0.571657i	−0.958281	+	0.285829i	$0.907731\pi$
$54$		0			0
$55$	122.662	+	8.88533i	0.300723	+	0.0217836i
$56$		0			0
$57$		−	382.092i		−	0.887883i
$58$		0			0
$59$	−180.884			−0.399137			−0.199568	−	0.979884i	$-0.563954\pi$
$59$	−180.884			−0.399137			−0.199568	+	0.979884i	$0.563954\pi$
$60$		0			0
$61$	−517.166			−1.08551			−0.542756	−	0.839890i	$-0.682619\pi$
$61$	−517.166			−1.08551			−0.542756	+	0.839890i	$0.682619\pi$
$62$		0			0
$63$			46.5752i			0.0931416i
$64$		0			0
$65$	−72.0592	+	994.780i	−0.137505	+	1.89827i
$66$		0			0
$67$			193.831i			0.353436i	0.984261	+	0.176718i	$0.0565481\pi$
$67$			193.831i			0.353436i	−0.984261	+	0.176718i	$0.943452\pi$
$68$		0			0
$69$	961.988			1.67840
$70$		0			0
$71$	−1016.64			−1.69935			−0.849673	−	0.527311i	$-0.823200\pi$
$71$	−1016.64			−1.69935			−0.849673	+	0.527311i	$0.823200\pi$
$72$		0			0
$73$		−	958.858i		−	1.53734i	−0.639645	−	0.768670i	$-0.720919\pi$
$73$		−	958.858i		−	1.53734i	0.639645	−	0.768670i	$-0.279081\pi$
$74$		0			0
$75$	155.594	−	1068.35i	0.239552	−	1.64484i
$76$		0			0
$77$		−	10.7637i		−	0.0159303i
$78$		0			0
$79$	−567.225			−0.807821			−0.403910	−	0.914799i	$-0.632349\pi$
$79$	−567.225			−0.807821			−0.403910	+	0.914799i	$0.632349\pi$
$80$		0			0
$81$	251.405			0.344862
$82$		0			0
$83$		−	642.773i		−	0.850043i	−0.905183	−	0.425021i	$-0.860267\pi$
$83$		−	642.773i		−	0.850043i	0.905183	−	0.425021i	$-0.139733\pi$
$84$		0			0
$85$	−37.4233	+	516.631i	−0.0477545	+	0.659253i
$86$		0			0
$87$			247.774i			0.305335i
$88$		0			0
$89$	1406.68			1.67537			0.837687	−	0.546151i	$-0.183908\pi$
$89$	1406.68			1.67537			0.837687	+	0.546151i	$0.183908\pi$
$90$		0			0
$91$	87.2925			0.100558
$92$		0			0
$93$			722.501i			0.805590i
$94$		0			0
$95$	493.314	+	35.7343i	0.532768	+	0.0385923i
$96$		0			0
$97$		−	342.110i		−	0.358103i	−0.983840	−	0.179052i	$-0.942697\pi$
$97$		−	342.110i		−	0.358103i	0.983840	−	0.179052i	$-0.0573029\pi$
$98$		0			0
$99$	−523.575			−0.531528
$100$		0			0
$101$	−1765.24			−1.73909			−0.869546	−	0.493851i	$-0.835589\pi$
$101$	−1765.24			−1.73909			−0.869546	+	0.493851i	$0.835589\pi$
$102$		0			0
$103$			1453.44i			1.39041i	0.718813	+	0.695203i	$0.244686\pi$
$103$			1453.44i			1.39041i	−0.718813	+	0.695203i	$0.755314\pi$
$104$		0			0
$105$	−94.2431	−	6.82671i	−0.0875922	−	0.00634494i
$106$		0			0
$107$		−	1094.66i		−	0.989014i	−0.869174	−	0.494507i	$-0.835349\pi$
$107$		−	1094.66i		−	0.989014i	0.869174	−	0.494507i	$-0.164651\pi$
$108$		0			0
$109$	−3.89452			−0.00342227			−0.00171113	−	0.999999i	$-0.500545\pi$
$109$	−3.89452			−0.00342227			−0.00171113	+	0.999999i	$0.500545\pi$
$110$		0			0
$111$	1924.07			1.64527
$112$		0			0
$113$			1941.77i			1.61652i	0.588829	+	0.808258i	$0.299589\pi$
$113$			1941.77i			1.61652i	−0.588829	+	0.808258i	$0.700411\pi$
$114$		0			0
$115$	−89.9679	+	1242.01i	−0.0729526	+	1.00711i
$116$		0			0
$117$		−	4246.14i		−	3.35518i
$118$		0			0
$119$	45.3346			0.0349228
$120$		0			0
$121$	121.000			0.0909091
$122$		0			0
$123$		−	3642.81i		−	2.67042i
$124$		0			0
$125$	1364.79	+	300.801i	0.976562	+	0.215235i
$126$		0			0
$127$		−	1604.89i		−	1.12134i	−0.828038	−	0.560672i	$-0.810543\pi$
$127$		−	1604.89i		−	1.12134i	0.828038	−	0.560672i	$-0.189457\pi$
$128$		0			0
$129$	2365.99			1.61483
$130$		0			0
$131$	1093.94			0.729604			0.364802	−	0.931085i	$-0.381137\pi$
$131$	1093.94			0.729604			0.364802	+	0.931085i	$0.381137\pi$
$132$		0			0
$133$		−	43.2886i		−	0.0282225i
$134$		0			0
$135$	−143.702	+	1983.81i	−0.0916140	+	1.26474i
$136$		0			0
$137$		−	1421.29i		−	0.886344i	−0.896437	−	0.443172i	$-0.853853\pi$
$137$		−	1421.29i		−	0.886344i	0.896437	−	0.443172i	$-0.146147\pi$
$138$		0			0
$139$	1508.41			0.920445			0.460222	−	0.887804i	$-0.347770\pi$
$139$	1508.41			0.920445			0.460222	+	0.887804i	$0.347770\pi$
$140$		0			0
$141$	4054.80			2.42181
$142$		0			0
$143$			981.299i			0.573848i
$144$		0			0
$145$	−319.898	−	23.1725i	−0.183214	−	0.0132715i
$146$		0			0
$147$		−	2954.22i		−	1.65755i
$148$		0			0
$149$	3090.28			1.69910			0.849550	−	0.527509i	$-0.176874\pi$
$149$	3090.28			1.69910			0.849550	+	0.527509i	$0.176874\pi$
$150$		0			0
$151$	1925.90			1.03793			0.518965	−	0.854795i	$-0.326317\pi$
$151$	1925.90			1.03793			0.518965	+	0.854795i	$0.326317\pi$
$152$		0			0
$153$		−	2205.20i		−	1.16523i
$154$		0			0
$155$	−932.812	−	67.5704i	−0.483389	−	0.0350154i
$156$		0			0
$157$		−	2335.62i		−	1.18728i	−0.804730	−	0.593640i	$-0.797690\pi$
$157$		−	2335.62i		−	1.18728i	0.804730	−	0.593640i	$-0.202310\pi$
$158$		0			0
$159$	1905.08			0.950203
$160$		0			0
$161$	108.987			0.0533502
$162$		0			0
$163$		−	1631.87i		−	0.784161i	−0.919931	−	0.392080i	$-0.871756\pi$
$163$		−	1631.87i		−	0.784161i	0.919931	−	0.392080i	$-0.128244\pi$
$164$		0			0
$165$	76.7425	−	1059.43i	0.0362085	−	0.499860i
$166$		0			0
$167$			92.7954i			0.0429983i	0.999769	+	0.0214992i	$0.00684393\pi$
$167$			92.7954i			0.0429983i	−0.999769	+	0.0214992i	$0.993156\pi$
$168$		0			0
$169$	−5761.24			−2.62232
$170$		0			0
$171$	−2105.67			−0.941666
$172$		0			0
$173$			820.888i			0.360757i	0.983597	+	0.180378i	$0.0577323\pi$
$173$			820.888i			0.360757i	−0.983597	+	0.180378i	$0.942268\pi$
$174$		0			0
$175$	17.6278	−	121.038i	0.00761448	−	0.0522834i
$176$		0			0
$177$			1562.29i			0.663442i
$178$		0			0
$179$	3753.53			1.56733			0.783664	−	0.621184i	$-0.213348\pi$
$179$	3753.53			1.56733			0.783664	+	0.621184i	$0.213348\pi$
$180$		0			0
$181$	−2386.11			−0.979877			−0.489939	−	0.871757i	$-0.662981\pi$
$181$	−2386.11			−0.979877			−0.489939	+	0.871757i	$0.662981\pi$
$182$		0			0
$183$			4466.76i			1.80433i
$184$		0			0
$185$	−179.944	+	2484.14i	−0.0715123	+	0.987231i
$186$		0			0
$187$			509.629i			0.199293i
$188$		0			0
$189$	174.081			0.0669973
$190$		0			0
$191$	1157.89			0.438649			0.219324	−	0.975652i	$-0.429615\pi$
$191$	1157.89			0.438649			0.219324	+	0.975652i	$0.429615\pi$
$192$		0			0
$193$		−	1672.44i		−	0.623756i	−0.950122	−	0.311878i	$-0.899042\pi$
$193$		−	1672.44i		−	0.623756i	0.950122	−	0.311878i	$-0.100958\pi$
$194$		0			0
$195$	8591.91	+	622.375i	3.15528	+	0.228560i
$196$		0			0
$197$		−	2072.54i		−	0.749554i	−0.927115	−	0.374777i	$-0.877719\pi$
$197$		−	2072.54i		−	0.749554i	0.927115	−	0.374777i	$-0.122281\pi$
$198$		0			0
$199$	−3351.65			−1.19393			−0.596966	−	0.802267i	$-0.703627\pi$
$199$	−3351.65			−1.19393			−0.596966	+	0.802267i	$0.703627\pi$
$200$		0			0
$201$	1674.12			0.587479
$202$		0			0
$203$			28.0712i			0.00970548i
$204$		0			0
$205$	4703.19	+	340.686i	1.60237	+	0.116071i
$206$		0			0
$207$		−	5301.43i		−	1.78007i
$208$		0			0
$209$	486.629			0.161057
$210$		0			0
$211$	−1466.51			−0.478477			−0.239238	−	0.970961i	$-0.576898\pi$
$211$	−1466.51			−0.478477			−0.239238	+	0.970961i	$0.576898\pi$
$212$		0			0
$213$			8780.76i			2.82464i
$214$		0			0
$215$	−221.274	+	3054.70i	−0.0701896	+	0.968970i
$216$		0			0
$217$			81.8547i			0.0256067i
$218$		0			0
$219$	−8281.65			−2.55535
$220$		0			0
$221$	−4133.05			−1.25800
$222$		0			0
$223$			2199.18i			0.660393i	0.943912	+	0.330197i	$0.107115\pi$
$223$			2199.18i			0.660393i	−0.943912	+	0.330197i	$0.892885\pi$
$224$		0			0
$225$	−5887.60	−	857.463i	−1.74447	−	0.254063i
$226$		0			0
$227$		−	1334.50i		−	0.390193i	−0.980784	−	0.195097i	$-0.937498\pi$
$227$		−	1334.50i		−	0.390193i	0.980784	−	0.195097i	$-0.0625021\pi$
$228$		0			0
$229$	316.001			0.0911876			0.0455938	−	0.998960i	$-0.485482\pi$
$229$	316.001			0.0911876			0.0455938	+	0.998960i	$0.485482\pi$
$230$		0			0
$231$	−92.9659			−0.0264792
$232$		0			0
$233$		−	489.572i		−	0.137652i	−0.997629	−	0.0688261i	$-0.978075\pi$
$233$		−	489.572i		−	0.137652i	0.997629	−	0.0688261i	$-0.0219253\pi$
$234$		0			0
$235$	−379.216	+	5235.10i	−0.105265	+	1.45319i
$236$		0			0
$237$			4899.12i			1.34275i
$238$		0			0
$239$	−3894.02			−1.05390			−0.526952	−	0.849895i	$-0.676665\pi$
$239$	−3894.02			−1.05390			−0.526952	+	0.849895i	$0.676665\pi$
$240$		0			0
$241$	−374.939			−0.100216			−0.0501078	−	0.998744i	$-0.515956\pi$
$241$	−374.939			−0.100216			−0.0501078	+	0.998744i	$0.515956\pi$
$242$		0			0
$243$			2631.99i			0.694823i
$244$		0			0
$245$	3814.16	+	276.287i	0.994602	+	0.0720463i
$246$		0			0
$247$			3946.51i			1.01664i
$248$		0			0
$249$	−5551.63			−1.41293
$250$		0			0
$251$	−578.849			−0.145564			−0.0727821	−	0.997348i	$-0.523188\pi$
$251$	−578.849			−0.145564			−0.0727821	+	0.997348i	$0.523188\pi$
$252$		0			0
$253$			1225.18i			0.304452i
$254$		0			0
$255$	4462.14	+	323.225i	1.09580	+	0.0793770i
$256$		0			0
$257$			1498.89i			0.363806i	0.983316	+	0.181903i	$0.0582257\pi$
$257$			1498.89i			0.363806i	−0.983316	+	0.181903i	$0.941774\pi$
$258$		0			0
$259$	217.985			0.0522969
$260$		0			0
$261$	1365.46			0.323831
$262$		0			0
$263$		−	4807.78i		−	1.12723i	−0.826039	−	0.563613i	$-0.809411\pi$
$263$		−	4807.78i		−	1.12723i	0.826039	−	0.563613i	$-0.190589\pi$
$264$		0			0
$265$	−178.168	+	2459.62i	−0.0413011	+	0.570163i
$266$		0			0
$267$		−	12149.5i		−	2.78479i
$268$		0			0
$269$	2687.00			0.609031			0.304515	−	0.952507i	$-0.401506\pi$
$269$	2687.00			0.609031			0.304515	+	0.952507i	$0.401506\pi$
$270$		0			0
$271$	3564.99			0.799106			0.399553	−	0.916710i	$-0.369165\pi$
$271$	3564.99			0.799106			0.399553	+	0.916710i	$0.369165\pi$
$272$		0			0
$273$		−	753.945i		−	0.167146i
$274$		0			0
$275$	1360.65	+	198.163i	0.298364	+	0.0434533i
$276$		0			0
$277$			4414.81i			0.957618i	0.877919	+	0.478809i	$0.158931\pi$
$277$			4414.81i			0.957618i	−0.877919	+	0.478809i	$0.841069\pi$
$278$		0			0
$279$	3981.64			0.854389
$280$		0			0
$281$	1223.49			0.259740			0.129870	−	0.991531i	$-0.458544\pi$
$281$	1223.49			0.259740			0.129870	+	0.991531i	$0.458544\pi$
$282$		0			0
$283$			1065.06i			0.223714i	0.993724	+	0.111857i	$0.0356799\pi$
$283$			1065.06i			0.223714i	−0.993724	+	0.111857i	$0.964320\pi$
$284$		0			0
$285$	308.637	−	4260.75i	0.0641477	−	0.885562i
$286$		0			0
$287$		−	412.707i		−	0.0848827i
$288$		0			0
$289$	2766.54			0.563106
$290$		0			0
$291$	−2954.80			−0.595236
$292$		0			0
$293$		−	1854.51i		−	0.369766i	−0.982761	−	0.184883i	$-0.940809\pi$
$293$		−	1854.51i		−	0.369766i	0.982761	−	0.184883i	$-0.0591906\pi$
$294$		0			0
$295$	−2017.06	−	146.110i	−0.398094	−	0.0288368i
$296$		0			0
$297$			1956.93i			0.382331i
$298$		0			0
$299$	−9936.08			−1.92180
$300$		0			0
$301$	268.051			0.0513296
$302$		0			0
$303$			15246.4i			2.89070i
$304$		0			0
$305$	−5766.98	−	417.744i	−1.08268	−	0.0784261i
$306$		0			0
$307$			2022.38i			0.375971i	0.982172	+	0.187985i	$0.0601958\pi$
$307$			2022.38i			0.375971i	−0.982172	+	0.187985i	$0.939804\pi$
$308$		0			0
$309$	12553.4			2.31112
$310$		0			0
$311$	−9535.87			−1.73868			−0.869340	−	0.494214i	$-0.835456\pi$
$311$	−9535.87			−1.73868			−0.869340	+	0.494214i	$0.835456\pi$
$312$		0			0
$313$		−	5724.18i		−	1.03370i	−0.856075	−	0.516852i	$-0.827104\pi$
$313$		−	5724.18i		−	1.03370i	0.856075	−	0.516852i	$-0.172896\pi$
$314$		0			0
$315$	−37.6214	+	519.365i	−0.00672929	+	0.0928982i
$316$		0			0
$317$		−	7425.96i		−	1.31572i	−0.753140	−	0.657860i	$-0.771462\pi$
$317$		−	7425.96i		−	1.31572i	0.753140	−	0.657860i	$-0.228538\pi$
$318$		0			0
$319$	−315.563			−0.0553859
$320$		0			0
$321$	−9454.55			−1.64393
$322$		0			0
$323$			2049.59i			0.353072i
$324$		0			0
$325$	−1607.08	+	11034.7i	−0.274292	+	1.88337i
$326$		0			0
$327$			33.6369i			0.00568846i
$328$		0			0
$329$	459.383			0.0769805
$330$		0			0
$331$	2341.24			0.388780			0.194390	−	0.980924i	$-0.437727\pi$
$331$	2341.24			0.388780			0.194390	+	0.980924i	$0.437727\pi$
$332$		0			0
$333$		−	10603.4i		−	1.74493i
$334$		0			0
$335$	−156.568	+	2161.43i	−0.0255351	+	0.352513i
$336$		0			0
$337$			1651.96i			0.267027i	0.991047	+	0.133514i	$0.0426260\pi$
$337$			1651.96i			0.267027i	−0.991047	+	0.133514i	$0.957374\pi$
$338$		0			0
$339$	16771.1			2.68696
$340$		0			0
$341$	−920.171			−0.146129
$342$		0			0
$343$		−	670.325i		−	0.105522i
$344$		0			0
$345$	10727.2	+	777.052i	1.67402	+	0.121261i
$346$		0			0
$347$			7450.35i			1.15261i	0.817235	+	0.576305i	$0.195506\pi$
$347$			7450.35i			1.15261i	−0.817235	+	0.576305i	$0.804494\pi$
$348$		0			0
$349$	1553.54			0.238278			0.119139	−	0.992878i	$-0.461987\pi$
$349$	1553.54			0.238278			0.119139	+	0.992878i	$0.461987\pi$
$350$		0			0
$351$	−15870.5			−2.41340
$352$		0			0
$353$			11352.5i			1.71170i	0.517221	+	0.855852i	$0.326967\pi$
$353$			11352.5i			1.71170i	−0.517221	+	0.855852i	$0.673033\pi$
$354$		0			0
$355$	−11336.7	−	821.202i	−1.69490	−	0.122774i
$356$		0			0
$357$		−	391.555i		−	0.0580484i
$358$		0			0
$359$	4719.79			0.693875			0.346938	−	0.937888i	$-0.387222\pi$
$359$	4719.79			0.693875			0.346938	+	0.937888i	$0.387222\pi$
$360$		0			0
$361$	−4901.91			−0.714669
$362$		0			0
$363$		−	1045.08i		−	0.151108i
$364$		0			0
$365$	774.524	−	10692.3i	0.111070	−	1.53332i
$366$		0			0
$367$			8340.57i			1.18631i	0.805090	+	0.593153i	$0.202117\pi$
$367$			8340.57i			1.18631i	−0.805090	+	0.593153i	$0.797883\pi$
$368$		0			0
$369$	−20075.2			−2.83218
$370$		0			0
$371$	215.833			0.0302035
$372$		0			0
$373$			2980.70i			0.413766i	0.978366	+	0.206883i	$0.0663320\pi$
$373$			2980.70i			0.413766i	−0.978366	+	0.206883i	$0.933668\pi$
$374$		0			0
$375$	2598.01	−	11787.7i	0.357762	−	1.62323i
$376$		0			0
$377$		−	2559.18i		−	0.349614i
$378$		0			0
$379$	3600.70			0.488010			0.244005	−	0.969774i	$-0.421539\pi$
$379$	3600.70			0.488010			0.244005	+	0.969774i	$0.421539\pi$
$380$		0			0
$381$	−13861.4			−1.86389
$382$		0			0
$383$		−	1286.00i		−	0.171571i	−0.996314	−	0.0857853i	$-0.972660\pi$
$383$		−	1286.00i		−	0.171571i	0.996314	−	0.0857853i	$-0.0273399\pi$
$384$		0			0
$385$	8.69444	−	120.027i	0.00115093	−	0.0158887i
$386$		0			0
$387$		−	13038.7i		−	1.71265i
$388$		0			0
$389$	−7592.64			−0.989620			−0.494810	−	0.869001i	$-0.664762\pi$
$389$	−7592.64			−0.989620			−0.494810	+	0.869001i	$0.664762\pi$
$390$		0			0
$391$	−5160.22			−0.667426
$392$		0			0
$393$		−	9448.37i		−	1.21274i
$394$		0			0
$395$	−6325.20	−	458.180i	−0.805710	−	0.0583634i
$396$		0			0
$397$		−	1778.51i		−	0.224838i	−0.993661	−	0.112419i	$-0.964140\pi$
$397$		−	1778.51i		−	0.224838i	0.993661	−	0.112419i	$-0.0358599\pi$
$398$		0			0
$399$	−373.883			−0.0469112
$400$		0			0
$401$	11742.5			1.46233			0.731164	−	0.682202i	$-0.238977\pi$
$401$	11742.5			1.46233			0.731164	+	0.682202i	$0.238977\pi$
$402$		0			0
$403$		−	7462.50i		−	0.922416i
$404$		0			0
$405$	2803.44	+	203.074i	0.343961	+	0.0249156i
$406$		0			0
$407$			2450.48i			0.298441i
$408$		0			0
$409$	−2878.13			−0.347957			−0.173978	−	0.984749i	$-0.555662\pi$
$409$	−2878.13			−0.347957			−0.173978	+	0.984749i	$0.555662\pi$
$410$		0			0
$411$	−12275.7			−1.47327
$412$		0			0
$413$			176.998i			0.0210884i
$414$		0			0
$415$	519.205	−	7167.64i	0.0614139	−	0.847821i
$416$		0			0
$417$		−	13028.2i		−	1.52996i
$418$		0			0
$419$	−7834.84			−0.913501			−0.456750	−	0.889595i	$-0.650987\pi$
$419$	−7834.84			−0.913501			−0.456750	+	0.889595i	$0.650987\pi$
$420$		0			0
$421$	−1895.97			−0.219486			−0.109743	−	0.993960i	$-0.535003\pi$
$421$	−1895.97			−0.219486			−0.109743	+	0.993960i	$0.535003\pi$
$422$		0			0
$423$		−	22345.6i		−	2.56851i
$424$		0			0
$425$	−834.624	+	5730.78i	−0.0952593	+	0.654080i
$426$		0			0
$427$			506.055i			0.0573530i
$428$		0			0
$429$	8475.47			0.953846
$430$		0			0
$431$	8058.24			0.900584			0.450292	−	0.892881i	$-0.351320\pi$
$431$	8058.24			0.900584			0.450292	+	0.892881i	$0.351320\pi$
$432$		0			0
$433$			10461.0i			1.16102i	0.814253	+	0.580510i	$0.197147\pi$
$433$			10461.0i			1.16102i	−0.814253	+	0.580510i	$0.802853\pi$
$434$		0			0
$435$	−200.141	+	2762.96i	−0.0220598	+	0.304537i
$436$		0			0
$437$			4927.33i			0.539374i
$438$		0			0
$439$	16492.0			1.79298			0.896492	−	0.443061i	$-0.146107\pi$
$439$	16492.0			1.79298			0.896492	+	0.443061i	$0.146107\pi$
$440$		0			0
$441$	−16280.4			−1.75796
$442$		0			0
$443$			1914.10i			0.205286i	0.994718	+	0.102643i	$0.0327300\pi$
$443$			1914.10i			0.205286i	−0.994718	+	0.102643i	$0.967270\pi$
$444$		0			0
$445$	15686.1	+	1136.26i	1.67100	+	0.121042i
$446$		0			0
$447$		−	26690.8i		−	2.82423i
$448$		0			0
$449$	6519.69			0.685263			0.342632	−	0.939470i	$-0.388682\pi$
$449$	6519.69			0.685263			0.342632	+	0.939470i	$0.388682\pi$
$450$		0			0
$451$	4639.45			0.484397
$452$		0			0
$453$		−	16634.0i		−	1.72524i
$454$		0			0
$455$	973.409	+	70.5111i	0.100295	+	0.00726508i
$456$		0			0
$457$			5801.01i			0.593785i	0.954911	+	0.296892i	$0.0959502\pi$
$457$			5801.01i			0.593785i	−0.954911	+	0.296892i	$0.904050\pi$
$458$		0			0
$459$	−8242.21			−0.838155
$460$		0			0
$461$	−6575.41			−0.664311			−0.332155	−	0.943225i	$-0.607776\pi$
$461$	−6575.41			−0.664311			−0.332155	+	0.943225i	$0.607776\pi$
$462$		0			0
$463$		−	6527.03i		−	0.655155i	−0.944824	−	0.327577i	$-0.893768\pi$
$463$		−	6527.03i		−	0.655155i	0.944824	−	0.327577i	$-0.106232\pi$
$464$		0			0
$465$	−583.605	+	8056.70i	−0.0582022	+	0.803485i
$466$		0			0
$467$			15800.2i			1.56562i	0.622258	+	0.782812i	$0.286216\pi$
$467$			15800.2i			1.56562i	−0.622258	+	0.782812i	$0.713784\pi$
$468$		0			0
$469$	189.667			0.0186738
$470$		0			0
$471$	−20172.8			−1.97349
$472$		0			0
$473$			3013.30i			0.292921i
$474$		0			0
$475$	5472.14	+	796.956i	0.528588	+	0.0769828i
$476$		0			0
$477$		−	10498.7i		−	1.00776i
$478$		0			0
$479$	−3788.84			−0.361412			−0.180706	−	0.983537i	$-0.557838\pi$
$479$	−3788.84			−0.361412			−0.180706	+	0.983537i	$0.557838\pi$
$480$		0			0
$481$	−19873.1			−1.88386
$482$		0			0
$483$		−	941.321i		−	0.0886782i
$484$		0			0
$485$	276.342	−	3814.91i	0.0258722	−	0.357167i
$486$		0			0
$487$		−	10097.7i		−	0.939570i	−0.882781	−	0.469785i	$-0.844331\pi$
$487$		−	10097.7i		−	0.939570i	0.882781	−	0.469785i	$-0.155669\pi$
$488$		0			0
$489$	−14094.5			−1.30342
$490$		0			0
$491$	4335.49			0.398488			0.199244	−	0.979950i	$-0.436151\pi$
$491$	4335.49			0.398488			0.199244	+	0.979950i	$0.436151\pi$
$492$		0			0
$493$		−	1329.09i		−	0.121418i
$494$		0			0
$495$	−5838.45	−	422.921i	−0.530139	−	0.0384018i
$496$		0			0
$497$			994.804i			0.0897848i
$498$		0			0
$499$	20837.6			1.86938			0.934690	−	0.355463i	$-0.115677\pi$
$499$	20837.6			1.86938			0.934690	+	0.355463i	$0.115677\pi$
$500$		0			0
$501$	801.474			0.0714714
$502$		0			0
$503$			9933.90i			0.880578i	0.897856	+	0.440289i	$0.145124\pi$
$503$			9933.90i			0.880578i	−0.897856	+	0.440289i	$0.854876\pi$
$504$		0			0
$505$	−19684.5	−	1425.89i	−1.73455	−	0.125646i
$506$		0			0
$507$			49759.8i			4.35880i
$508$		0			0
$509$	5123.45			0.446155			0.223078	−	0.974801i	$-0.428390\pi$
$509$	5123.45			0.446155			0.223078	+	0.974801i	$0.428390\pi$
$510$		0			0
$511$	−938.258			−0.0812253
$512$		0			0
$513$			7870.22i			0.677347i
$514$		0			0
$515$	−1174.03	+	16207.5i	−0.100454	+	1.38677i
$516$		0			0
$517$			5164.15i			0.439302i
$518$		0			0
$519$	7090.01			0.599647
$520$		0			0
$521$	−11977.7			−1.00720			−0.503602	−	0.863936i	$-0.667992\pi$
$521$	−11977.7			−1.00720			−0.503602	+	0.863936i	$0.667992\pi$
$522$		0			0
$523$		−	2907.10i		−	0.243056i	−0.992588	−	0.121528i	$-0.961221\pi$
$523$		−	2907.10i		−	0.243056i	0.992588	−	0.121528i	$-0.0387795\pi$
$524$		0			0
$525$	−1045.40	−	152.251i	−0.0869049	−	0.0126567i
$526$		0			0
$527$		−	3875.58i		−	0.320348i
$528$		0			0
$529$	−238.475			−0.0196001
$530$		0			0
$531$	8609.66			0.703630
$532$		0			0
$533$			37625.5i			3.05768i
$534$		0			0
$535$	884.217	−	12206.7i	0.0714543	−	0.986430i
$536$		0			0
$537$		−	32419.2i		−	2.60520i
$538$		0			0
$539$	3762.47			0.300670
$540$		0			0
$541$	19670.0			1.56317			0.781587	−	0.623796i	$-0.214410\pi$
$541$	19670.0			1.56317			0.781587	+	0.623796i	$0.214410\pi$
$542$		0			0
$543$			20608.8i			1.62874i
$544$		0			0
$545$	−43.4282	−	3.14582i	−0.00341332	−	0.000247252i
$546$		0			0
$547$			13942.0i			1.08980i	0.838503	+	0.544898i	$0.183431\pi$
$547$			13942.0i			1.08980i	−0.838503	+	0.544898i	$0.816569\pi$
$548$		0			0
$549$	24615.9			1.91363
$550$		0			0
$551$	−1269.11			−0.0981229
$552$		0			0
$553$			555.039i			0.0426811i
$554$		0			0
$555$	21455.5	+	1554.18i	1.64097	+	0.118867i
$556$		0			0
$557$		−	16663.1i		−	1.26757i	−0.773507	−	0.633787i	$-0.781499\pi$
$557$		−	16663.1i		−	1.26757i	0.773507	−	0.633787i	$-0.218501\pi$
$558$		0			0
$559$	−24437.6			−1.84901
$560$		0			0
$561$	4401.67			0.331263
$562$		0			0
$563$		−	18078.7i		−	1.35333i	−0.736290	−	0.676666i	$-0.763424\pi$
$563$		−	18078.7i		−	1.35333i	0.736290	−	0.676666i	$-0.236576\pi$
$564$		0			0
$565$	−1568.48	+	21652.9i	−0.116790	+	1.61229i
$566$		0			0
$567$		−	246.004i		−	0.0182208i
$568$		0			0
$569$	−13257.7			−0.976789			−0.488394	−	0.872623i	$-0.662417\pi$
$569$	−13257.7			−0.976789			−0.488394	+	0.872623i	$0.662417\pi$
$570$		0			0
$571$	−26187.8			−1.91931			−0.959656	−	0.281178i	$-0.909275\pi$
$571$	−26187.8			−1.91931			−0.959656	+	0.281178i	$0.909275\pi$
$572$		0			0
$573$		−	10000.7i		−	0.729118i
$574$		0			0
$575$	−2006.49	+	13777.1i	−0.145524	+	0.999211i
$576$		0			0
$577$		−	3405.70i		−	0.245721i	−0.992424	−	0.122861i	$-0.960793\pi$
$577$		−	3405.70i		−	0.245721i	0.992424	−	0.122861i	$-0.0392069\pi$
$578$		0			0
$579$	−14444.9			−1.03680
$580$		0			0
$581$	−628.964			−0.0449119
$582$		0			0
$583$			2426.29i			0.172361i
$584$		0			0
$585$	3429.85	−	47349.3i	0.242405	−	3.34641i
$586$		0			0
$587$			17944.4i			1.26175i	0.775885	+	0.630874i	$0.217304\pi$
$587$			17944.4i			1.26175i	−0.775885	+	0.630874i	$0.782696\pi$
$588$		0			0
$589$	−3700.67			−0.258886
$590$		0			0
$591$	−17900.5			−1.24590
$592$		0			0
$593$			8166.27i			0.565512i	0.959192	+	0.282756i	$0.0912487\pi$
$593$			8166.27i			0.565512i	−0.959192	+	0.282756i	$0.908751\pi$
$594$		0			0
$595$	505.532	+	36.6193i	0.0348316	+	0.00252310i
$596$		0			0
$597$			28948.2i			1.98454i
$598$		0			0
$599$	−14949.2			−1.01971			−0.509855	−	0.860260i	$-0.670301\pi$
$599$	−14949.2			−1.01971			−0.509855	+	0.860260i	$0.670301\pi$
$600$		0			0
$601$	21067.5			1.42989			0.714944	−	0.699181i	$-0.246452\pi$
$601$	21067.5			1.42989			0.714944	+	0.699181i	$0.246452\pi$
$602$		0			0
$603$		−	9225.92i		−	0.623065i
$604$		0			0
$605$	1349.29	+	97.7386i	0.0906715	+	0.00656800i
$606$		0			0
$607$			3367.68i			0.225189i	0.993641	+	0.112595i	$0.0359161\pi$
$607$			3367.68i			0.225189i	−0.993641	+	0.112595i	$0.964084\pi$
$608$		0			0
$609$	242.451			0.0161324
$610$		0			0
$611$	−41880.8			−2.77302
$612$		0			0
$613$		−	24536.6i		−	1.61668i	−0.588715	−	0.808341i	$-0.700366\pi$
$613$		−	24536.6i		−	1.61668i	0.588715	−	0.808341i	$-0.299634\pi$
$614$		0			0
$615$	2942.51	−	40621.4i	0.192932	−	2.66344i
$616$		0			0
$617$			14937.9i			0.974682i	0.873212	+	0.487341i	$0.162033\pi$
$617$			14937.9i			0.974682i	−0.873212	+	0.487341i	$0.837967\pi$
$618$		0			0
$619$	12094.4			0.785326			0.392663	−	0.919682i	$-0.371554\pi$
$619$	12094.4			0.785326			0.392663	+	0.919682i	$0.371554\pi$
$620$		0			0
$621$	−19814.8			−1.28042
$622$		0			0
$623$		−	1376.46i		−	0.0885182i
$624$		0			0
$625$	14975.9	+	4456.68i	0.958460	+	0.285228i
$626$		0			0
$627$		−	4203.01i		−	0.267707i
$628$		0			0
$629$	−10320.9			−0.654249
$630$		0			0
$631$	27940.5			1.76275			0.881375	−	0.472418i	$-0.156619\pi$
$631$	27940.5			1.76275			0.881375	+	0.472418i	$0.156619\pi$
$632$		0			0
$633$			12666.2i			0.795320i
$634$		0			0
$635$	1296.36	−	17896.3i	0.0810149	−	1.11841i
$636$		0			0
$637$			30513.3i			1.89793i
$638$		0			0
$639$	48390.0			2.99574
$640$		0			0
$641$	−11648.8			−0.717787			−0.358894	−	0.933378i	$-0.616846\pi$
$641$	−11648.8			−0.717787			−0.358894	+	0.933378i	$0.616846\pi$
$642$		0			0
$643$		−	16950.2i		−	1.03958i	−0.854293	−	0.519791i	$-0.826010\pi$
$643$		−	16950.2i		−	1.03958i	0.854293	−	0.519791i	$-0.173990\pi$
$644$		0			0
$645$	26383.4	+	1911.14i	1.61061	+	0.116668i
$646$		0			0
$647$			2182.00i			0.132586i	0.997800	+	0.0662930i	$0.0211172\pi$
$647$			2182.00i			0.132586i	−0.997800	+	0.0662930i	$0.978883\pi$
$648$		0			0
$649$	−1989.72			−0.120344
$650$		0			0
$651$	706.979			0.0425633
$652$		0			0
$653$		−	218.057i		−	0.0130677i	−0.999979	−	0.00653387i	$-0.997920\pi$
$653$		−	218.057i		−	0.0130677i	0.999979	−	0.00653387i	$-0.00207981\pi$
$654$		0			0
$655$	12198.7	+	883.639i	0.727697	+	0.0527124i
$656$		0			0
$657$			45639.5i			2.71014i
$658$		0			0
$659$	26262.6			1.55242			0.776209	−	0.630475i	$-0.217140\pi$
$659$	26262.6			1.55242			0.776209	+	0.630475i	$0.217140\pi$
$660$		0			0
$661$	25877.8			1.52274			0.761368	−	0.648319i	$-0.224528\pi$
$661$	25877.8			1.52274			0.761368	+	0.648319i	$0.224528\pi$
$662$		0			0
$663$			35697.1i			2.09104i
$664$		0			0
$665$	34.9666	−	482.716i	0.00203902	−	0.0281488i
$666$		0			0
$667$		−	3195.21i		−	0.185486i
$668$		0			0
$669$	18994.3			1.09770
$670$		0			0
$671$	−5688.82			−0.327294
$672$		0			0
$673$		−	5075.20i		−	0.290691i	−0.989381	−	0.145345i	$-0.953571\pi$
$673$		−	5075.20i		−	0.290691i	0.989381	−	0.145345i	$-0.0464293\pi$
$674$		0			0
$675$	−3204.88	+	22005.7i	−0.182749	+	1.25481i
$676$		0			0
$677$			15208.4i			0.863377i	0.902023	+	0.431688i	$0.142082\pi$
$677$			15208.4i			0.863377i	−0.902023	+	0.431688i	$0.857918\pi$
$678$		0			0
$679$	−334.760			−0.0189204
$680$		0			0
$681$	−11526.1			−0.648576
$682$		0			0
$683$		−	12739.6i		−	0.713718i	−0.934158	−	0.356859i	$-0.883848\pi$
$683$		−	12739.6i		−	0.713718i	0.934158	−	0.356859i	$-0.116152\pi$
$684$		0			0
$685$	1148.06	−	15849.0i	0.0640366	−	0.884028i
$686$		0			0
$687$		−	2729.30i		−	0.151571i
$688$		0			0
$689$	−19677.0			−1.08800
$690$		0			0
$691$	24964.7			1.37439			0.687195	−	0.726473i	$-0.258842\pi$
$691$	24964.7			1.37439			0.687195	+	0.726473i	$0.258842\pi$
$692$		0			0
$693$			512.327i			0.0280832i
$694$		0			0
$695$	16820.5	+	1218.43i	0.918040	+	0.0665003i
$696$		0			0
$697$			19540.5i			1.06191i
$698$		0			0
$699$	−4228.43			−0.228804
$700$		0			0
$701$	13561.1			0.730662			0.365331	−	0.930878i	$-0.380956\pi$
$701$	13561.1			0.730662			0.365331	+	0.930878i	$0.380956\pi$
$702$		0			0
$703$			9855.14i			0.528725i
$704$		0			0
$705$	45215.5	+	3275.29i	2.41548	+	0.174971i
$706$		0			0
$707$			1727.32i			0.0918848i
$708$		0			0
$709$	1963.83			0.104024			0.0520121	−	0.998646i	$-0.483437\pi$
$709$	1963.83			0.104024			0.0520121	+	0.998646i	$0.483437\pi$
$710$		0			0
$711$	26998.6			1.42409
$712$		0			0
$713$		−	9317.13i		−	0.489382i
$714$		0			0
$715$	−792.651	+	10942.6i	−0.0414594	+	0.572349i
$716$		0			0
$717$			33632.6i			1.75179i
$718$		0			0
$719$	−17304.4			−0.897558			−0.448779	−	0.893643i	$-0.648141\pi$
$719$	−17304.4			−0.897558			−0.448779	+	0.893643i	$0.648141\pi$
$720$		0			0
$721$	1422.22			0.0734620
$722$		0			0
$723$			3238.35i			0.166577i
$724$		0			0
$725$	−3548.50	−	516.799i	−0.181777	−	0.0264737i
$726$		0			0
$727$			32160.5i			1.64067i	0.571883	+	0.820335i	$0.306213\pi$
$727$			32160.5i			1.64067i	−0.571883	+	0.820335i	$0.693787\pi$
$728$		0			0
$729$	29520.4			1.49979
$730$		0			0
$731$	−12691.4			−0.642148
$732$		0			0
$733$		−	33231.1i		−	1.67451i	−0.546809	−	0.837257i	$-0.684158\pi$
$733$		−	33231.1i		−	1.67451i	0.546809	−	0.837257i	$-0.315842\pi$
$734$		0			0
$735$	2386.29	−	32942.9i	0.119755	−	1.65322i
$736$		0			0
$737$			2132.14i			0.106565i
$738$		0			0
$739$	−12842.4			−0.639262			−0.319631	−	0.947542i	$-0.603559\pi$
$739$	−12842.4			−0.639262			−0.319631	+	0.947542i	$0.603559\pi$
$740$		0			0
$741$	34086.0			1.68985
$742$		0			0
$743$		−	8287.91i		−	0.409225i	−0.978843	−	0.204612i	$-0.934407\pi$
$743$		−	8287.91i		−	0.409225i	0.978843	−	0.204612i	$-0.0655934\pi$
$744$		0			0
$745$	34460.1	+	2496.20i	1.69466	+	0.122756i
$746$		0			0
$747$			30594.5i			1.49852i
$748$		0			0
$749$	−1071.14			−0.0522545
$750$		0			0
$751$	−13051.5			−0.634163			−0.317081	−	0.948398i	$-0.602703\pi$
$751$	−13051.5			−0.634163			−0.317081	+	0.948398i	$0.602703\pi$
$752$		0			0
$753$			4999.52i			0.241956i
$754$		0			0
$755$	21475.9	+	1555.66i	1.03522	+	0.0749884i
$756$		0			0
$757$		−	23237.5i		−	1.11570i	−0.829943	−	0.557848i	$-0.811627\pi$
$757$		−	23237.5i		−	1.11570i	0.829943	−	0.557848i	$-0.188373\pi$
$758$		0			0
$759$	10581.9			0.506057
$760$		0			0
$761$	−10557.8			−0.502919			−0.251460	−	0.967868i	$-0.580911\pi$
$761$	−10557.8			−0.502919			−0.251460	+	0.967868i	$0.580911\pi$
$762$		0			0
$763$			3.81085i			0.000180815i
$764$		0			0
$765$	1781.26	−	24590.4i	0.0841853	−	1.16218i
$766$		0			0
$767$		−	16136.5i		−	0.759653i
$768$		0			0
$769$	23640.8			1.10859			0.554297	−	0.832319i	$-0.312987\pi$
$769$	23640.8			1.10859			0.554297	+	0.832319i	$0.312987\pi$
$770$		0			0
$771$	12945.9			0.604715
$772$		0			0
$773$		−	24827.7i		−	1.15523i	−0.816311	−	0.577613i	$-0.803984\pi$
$773$		−	24827.7i		−	1.15523i	0.816311	−	0.577613i	$-0.196016\pi$
$774$		0			0
$775$	−10347.3	−	1506.97i	−0.479596	−	0.0698477i
$776$		0			0
$777$		−	1882.73i		−	0.0869275i
$778$		0			0
$779$	18658.6			0.858169
$780$		0			0
$781$	−11183.1			−0.512372
$782$		0			0
$783$		−	5103.58i		−	0.232934i
$784$		0			0
$785$	1886.62	−	26044.8i	0.0857786	−	1.18418i
$786$		0			0
$787$		−	16933.3i		−	0.766972i	−0.923547	−	0.383486i	$-0.874723\pi$
$787$		−	16933.3i		−	0.766972i	0.923547	−	0.383486i	$-0.125277\pi$
$788$		0			0
$789$	−41524.8			−1.87366
$790$		0			0
$791$	1900.05			0.0854085
$792$		0			0
$793$		−	46135.8i		−	2.06599i
$794$		0			0
$795$	21243.7	+	1538.84i	0.947720	+	0.0686503i
$796$		0			0
$797$		−	1234.97i		−	0.0548871i	−0.999623	−	0.0274436i	$-0.991263\pi$
$797$		−	1234.97i		−	0.0548871i	0.999623	−	0.0274436i	$-0.00873665\pi$
$798$		0			0
$799$	−21750.4			−0.963048
$800$		0			0
$801$	−66955.0			−2.95348
$802$		0			0
$803$		−	10547.4i		−	0.463526i
$804$		0			0
$805$	1215.33	+	88.0350i	0.0532108	+	0.00385444i
$806$		0			0
$807$		−	23207.6i		−	1.01233i
$808$		0			0
$809$	3622.10			0.157412			0.0787059	−	0.996898i	$-0.474921\pi$
$809$	3622.10			0.157412			0.0787059	+	0.996898i	$0.474921\pi$
$810$		0			0
$811$	5976.02			0.258750			0.129375	−	0.991596i	$-0.458703\pi$
$811$	5976.02			0.258750			0.129375	+	0.991596i	$0.458703\pi$
$812$		0			0
$813$		−	30790.8i		−	1.32827i
$814$		0			0
$815$	1318.16	−	18197.2i	0.0566540	−	0.782111i
$816$		0			0
$817$			12118.7i			0.518945i
$818$		0			0
$819$	−4154.92			−0.177271
$820$		0			0
$821$	−17550.7			−0.746069			−0.373035	−	0.927817i	$-0.621683\pi$
$821$	−17550.7			−0.746069			−0.373035	+	0.927817i	$0.621683\pi$
$822$		0			0
$823$			11061.6i			0.468509i	0.972175	+	0.234254i	$0.0752649\pi$
$823$			11061.6i			0.468509i	−0.972175	+	0.234254i	$0.924735\pi$
$824$		0			0
$825$	1711.53	−	11751.9i	0.0722277	−	0.495937i
$826$		0			0
$827$		−	18647.8i		−	0.784096i	−0.919945	−	0.392048i	$-0.871767\pi$
$827$		−	18647.8i		−	0.784096i	0.919945	−	0.392048i	$-0.128233\pi$
$828$		0			0
$829$	22790.7			0.954828			0.477414	−	0.878679i	$-0.341574\pi$
$829$	22790.7			0.954828			0.477414	+	0.878679i	$0.341574\pi$
$830$		0			0
$831$	38130.7			1.59174
$832$		0			0
$833$			15846.8i			0.659135i
$834$		0			0
$835$	−74.9561	+	1034.77i	−0.00310654	+	0.0428860i
$836$		0			0
$837$		−	14881.9i		−	0.614567i
$838$		0			0
$839$	4391.21			0.180693			0.0903465	−	0.995910i	$-0.471203\pi$
$839$	4391.21			0.180693			0.0903465	+	0.995910i	$0.471203\pi$
$840$		0			0
$841$	−23566.0			−0.966256
$842$		0			0
$843$		−	10567.2i		−	0.431738i
$844$		0			0
$845$	−64244.3	−	4653.68i	−2.61547	−	0.189457i
$846$		0			0
$847$		−	118.401i		−	0.00480318i
$848$		0			0
$849$	9198.91			0.371856
$850$		0			0
$851$	−24812.1			−0.999471
$852$		0			0
$853$		−	24829.8i		−	0.996668i	−0.866985	−	0.498334i	$-0.833945\pi$
$853$		−	24829.8i		−	0.996668i	0.866985	−	0.498334i	$-0.166055\pi$
$854$		0			0
$855$	−23480.6	−	1700.87i	−0.939206	−	0.0680335i
$856$		0			0
$857$		−	31142.7i		−	1.24132i	−0.784079	−	0.620661i	$-0.786864\pi$
$857$		−	31142.7i		−	1.24132i	0.784079	−	0.620661i	$-0.213136\pi$
$858$		0			0
$859$	12063.2			0.479151			0.239575	−	0.970878i	$-0.422992\pi$
$859$	12063.2			0.479151			0.239575	+	0.970878i	$0.422992\pi$
$860$		0			0
$861$	−3564.55			−0.141091
$862$		0			0
$863$			31698.6i			1.25033i	0.780494	+	0.625164i	$0.214968\pi$
$863$			31698.6i			1.25033i	−0.780494	+	0.625164i	$0.785032\pi$
$864$		0			0
$865$	−663.078	+	9153.82i	−0.0260640	+	0.359814i
$866$		0			0
$867$		−	23894.6i		−	0.935989i
$868$		0			0
$869$	−6239.48			−0.243567
$870$		0			0
$871$	−17291.5			−0.672674
$872$		0			0
$873$			16283.7i			0.631292i
$874$		0			0
$875$	294.338	−	1335.47i	0.0113719	−	0.0515966i
$876$		0			0
$877$		−	29044.8i		−	1.11833i	−0.829057	−	0.559164i	$-0.811122\pi$
$877$		−	29044.8i		−	1.11833i	0.829057	−	0.559164i	$-0.188878\pi$
$878$		0			0
$879$	−16017.4			−0.614622
$880$		0			0
$881$	17026.4			0.651116			0.325558	−	0.945522i	$-0.394448\pi$
$881$	17026.4			0.651116			0.325558	+	0.945522i	$0.394448\pi$
$882$		0			0
$883$		−	22720.2i		−	0.865909i	−0.901416	−	0.432954i	$-0.857471\pi$
$883$		−	22720.2i		−	0.865909i	0.901416	−	0.432954i	$-0.142529\pi$
$884$		0			0
$885$	−1261.95	+	17421.3i	−0.0479323	+	0.661708i
$886$		0			0
$887$			32966.9i			1.24794i	0.781450	+	0.623968i	$0.214480\pi$
$887$			32966.9i			1.24794i	−0.781450	+	0.623968i	$0.785520\pi$
$888$		0			0
$889$	−1570.41			−0.0592462
$890$		0			0
$891$	2765.45			0.103980
$892$		0			0
$893$			20768.8i			0.778277i
$894$		0			0
$895$	41856.0	+	3031.94i	1.56323	+	0.113236i
$896$		0			0
$897$			85817.9i			3.19440i
$898$		0			0
$899$	2399.76			0.0890284
$900$		0			0
$901$	−10219.1			−0.377854
$902$		0			0
$903$		−	2315.16i		−	0.0853196i
$904$		0			0
$905$	−26607.7	−	1927.39i	−0.977317	−	0.0707942i
$906$		0			0
$907$		−	8561.99i		−	0.313447i	−0.987643	−	0.156723i	$-0.949907\pi$
$907$		−	8561.99i		−	0.313447i	0.987643	−	0.156723i	$-0.0500931\pi$
$908$		0			0
$909$	84021.6			3.06581
$910$		0			0
$911$	−21637.2			−0.786909			−0.393454	−	0.919344i	$-0.628720\pi$
$911$	−21637.2			−0.786909			−0.393454	+	0.919344i	$0.628720\pi$
$912$		0			0
$913$		−	7070.51i		−	0.256297i
$914$		0			0
$915$	−3608.06	+	49809.4i	−0.130359	+	1.79961i
$916$		0			0
$917$		−	1070.44i		−	0.0385486i
$918$		0			0
$919$	35544.0			1.27583			0.637915	−	0.770107i	$-0.279797\pi$
$919$	35544.0			1.27583			0.637915	+	0.770107i	$0.279797\pi$
$920$		0			0
$921$	17467.3			0.624935
$922$		0			0
$923$		−	90693.8i		−	3.23426i
$924$		0			0
$925$	−4013.16	+	27555.6i	−0.142651	+	0.979484i
$926$		0			0
$927$		−	69180.5i		−	2.45112i
$928$		0			0
$929$	−20189.4			−0.713016			−0.356508	−	0.934292i	$-0.616033\pi$
$929$	−20189.4			−0.713016			−0.356508	+	0.934292i	$0.616033\pi$
$930$		0			0
$931$	15131.6			0.532673
$932$		0			0
$933$			82361.3i			2.89002i
$934$		0			0
$935$	−411.657	+	5682.94i	−0.0143985	+	0.198772i
$936$		0			0
$937$		−	11852.5i		−	0.413237i	−0.978422	−	0.206619i	$-0.933754\pi$
$937$		−	11852.5i		−	0.413237i	0.978422	−	0.206619i	$-0.0662460\pi$
$938$		0			0
$939$	−49439.7			−1.71821
$940$		0			0
$941$	−22307.5			−0.772798			−0.386399	−	0.922332i	$-0.626281\pi$
$941$	−22307.5			−0.772798			−0.386399	+	0.922332i	$0.626281\pi$
$942$		0			0
$943$			46976.5i			1.62223i
$944$		0			0
$945$	1941.19	+	140.615i	0.0668222	+	0.00484042i
$946$		0			0
$947$		−	27787.3i		−	0.953501i	−0.879039	−	0.476750i	$-0.841815\pi$
$947$		−	27787.3i		−	0.953501i	0.879039	−	0.476750i	$-0.158185\pi$
$948$		0			0
$949$	85538.7			2.92593
$950$		0			0
$951$	−64138.0			−2.18698
$952$		0			0
$953$		−	13506.2i		−	0.459086i	−0.973298	−	0.229543i	$-0.926277\pi$
$953$		−	13506.2i		−	0.459086i	0.973298	−	0.229543i	$-0.0737232\pi$
$954$		0			0
$955$	12911.8	+	935.293i	0.437503	+	0.0316915i
$956$		0			0
$957$			2725.51i			0.0920620i
$958$		0			0
$959$	−1390.76			−0.0468299
$960$		0			0
$961$	−22793.4			−0.765109
$962$		0			0
$963$			52103.2i			1.74351i
$964$		0			0
$965$	1350.93	−	18649.6i	0.0450651	−	0.622126i
$966$		0			0
$967$			45849.6i			1.52474i	0.647142	+	0.762370i	$0.275964\pi$
$967$			45849.6i			1.52474i	−0.647142	+	0.762370i	$0.724036\pi$
$968$		0			0
$969$	17702.3			0.586873
$970$		0			0
$971$	28811.8			0.952230			0.476115	−	0.879383i	$-0.342045\pi$
$971$	28811.8			0.952230			0.476115	+	0.879383i	$0.342045\pi$
$972$		0			0
$973$		−	1476.01i		−	0.0486316i
$974$		0			0
$975$	95306.8	+	13880.4i	3.13052	+	0.455925i
$976$		0			0
$977$		−	25775.2i		−	0.844035i	−0.906588	−	0.422017i	$-0.861322\pi$
$977$		−	25775.2i		−	0.844035i	0.906588	−	0.422017i	$-0.138678\pi$
$978$		0			0
$979$	15473.5			0.505144
$980$		0			0
$981$	185.370			0.00603304
$982$		0			0
$983$		−	6675.30i		−	0.216591i	−0.994119	−	0.108295i	$-0.965461\pi$
$983$		−	6675.30i		−	0.216591i	0.994119	−	0.108295i	$-0.0345393\pi$
$984$		0			0
$985$	1674.11	−	23111.1i	0.0541537	−	0.747595i
$986$		0			0
$987$		−	3967.69i		−	0.127956i
$988$		0			0
$989$	−30511.0			−0.980984
$990$		0			0
$991$	20453.3			0.655622			0.327811	−	0.944743i	$-0.393689\pi$
$991$	20453.3			0.655622			0.327811	+	0.944743i	$0.393689\pi$
$992$		0			0
$993$		−	20221.3i		−	0.646227i
$994$		0			0
$995$	−37374.7	−	2707.32i	−1.19081	−	0.0862592i
$996$		0			0
$997$			12458.0i			0.395737i	0.980229	+	0.197868i	$0.0634019\pi$
$997$			12458.0i			0.395737i	−0.980229	+	0.197868i	$0.936598\pi$
$998$		0			0
$999$	−39631.4			−1.25514

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	880.4.b.h.529.1		8
4.3	odd	2		110.4.b.c.89.4	✓	8
5.4	even	2	inner	880.4.b.h.529.8		8
12.11	even	2		990.4.c.i.199.5		8
20.3	even	4		550.4.a.ba.1.1		4
20.7	even	4		550.4.a.bb.1.4		4
20.19	odd	2		110.4.b.c.89.5	yes	8
60.59	even	2		990.4.c.i.199.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.4.b.c.89.4	✓	8	4.3	odd	2
110.4.b.c.89.5	yes	8	20.19	odd	2
550.4.a.ba.1.1		4	20.3	even	4
550.4.a.bb.1.4		4	20.7	even	4
880.4.b.h.529.1		8	1.1	even	1	trivial
880.4.b.h.529.8		8	5.4	even	2	inner
990.4.c.i.199.1		8	60.59	even	2
990.4.c.i.199.5		8	12.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	880.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-8.63700i q^{3} +(11.1511 + 0.807757i) q^{5} -0.978517i q^{7} -47.5977 q^{9} +11.0000 q^{11} +89.2090i q^{13} +(6.97659 - 96.3122i) q^{15} +46.3299i q^{17} +44.2390 q^{19} -8.45144 q^{21} +111.380i q^{23} +(123.695 + 18.0148i) q^{25} +177.902i q^{27} -28.6875 q^{29} -83.6519 q^{31} -95.0070i q^{33} +(0.790404 - 10.9116i) q^{35} +222.770i q^{37} +770.498 q^{39} +421.768 q^{41} +273.936i q^{43} +(-530.768 - 38.4474i) q^{45} +469.468i q^{47} +342.043 q^{49} +400.151 q^{51} +220.572i q^{53} +(122.662 + 8.88533i) q^{55} -382.092i q^{57} -180.884 q^{59} -517.166 q^{61} +46.5752i q^{63} +(-72.0592 + 994.780i) q^{65} +193.831i q^{67} +961.988 q^{69} -1016.64 q^{71} -958.858i q^{73} +(155.594 - 1068.35i) q^{75} -10.7637i q^{77} -567.225 q^{79} +251.405 q^{81} -642.773i q^{83} +(-37.4233 + 516.631i) q^{85} +247.774i q^{87} +1406.68 q^{89} +87.2925 q^{91} +722.501i q^{93} +(493.314 + 35.7343i) q^{95} -342.110i q^{97} -523.575 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{5} - 110 q^{9} + 88 q^{11} - 8 q^{15} + 302 q^{19} + 230 q^{21} - 162 q^{25} - 58 q^{29} - 1022 q^{31} + 1058 q^{35} + 320 q^{39} + 452 q^{41} - 622 q^{45} + 222 q^{49} - 834 q^{51} + 176 q^{55}+ \cdots - 1210 q^{99}+O(q^{100}) \) 8 * q + 16 * q^5 - 110 * q^9 + 88 * q^11 - 8 * q^15 + 302 * q^19 + 230 * q^21 - 162 * q^25 - 58 * q^29 - 1022 * q^31 + 1058 * q^35 + 320 * q^39 + 452 * q^41 - 622 * q^45 + 222 * q^49 - 834 * q^51 + 176 * q^55 - 268 * q^59 - 490 * q^61 - 1408 * q^65 + 2820 * q^69 - 2034 * q^71 + 2110 * q^75 - 272 * q^79 - 2488 * q^81 + 330 * q^85 + 5782 * q^89 + 3520 * q^91 - 1268 * q^95 - 1210 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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