LMFDB - Embedded newform 184.4.h.b.91.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [184,4,Mod(91,184)] 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("184.91"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(184, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 184 = 2^{3} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	184.h (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$10.8563514411$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-23})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} - 6x + 36 $ x^4 - x^3 - 5x^2 - 6x + 36
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			91.2
Root			$2.32666 - 0.765945i$ of defining polynomial
Character	$\chi$	$=$	184.91
Dual form			184.4.h.b.91.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.82666 + 0.100080i) q^{2} -6.30662 q^{3} +(7.97997 - 0.565785i) q^{4} +(17.8267 - 0.631168i) q^{6} +(-22.5000 + 2.39792i) q^{8} +12.7735 q^{9} +(-50.3267 + 3.56819i) q^{12} +92.8609i q^{13} +(63.3598 - 9.02989i) q^{16} +(-36.1063 + 1.27838i) q^{18} -110.304i q^{23} +(141.899 - 15.1228i) q^{24} -125.000 q^{25} +(-9.29354 - 262.486i) q^{26} +89.7212 q^{27} -177.078i q^{29} -283.525i q^{31} +(-178.193 + 31.8654i) q^{32} +(101.932 - 7.22705i) q^{36} -585.639i q^{39} +478.812 q^{41} +(11.0393 + 311.792i) q^{46} -362.890i q^{47} +(-399.586 + 56.9481i) q^{48} -343.000 q^{49} +(353.332 - 12.5100i) q^{50} +(52.5393 + 741.027i) q^{52} +(-253.611 + 8.97931i) q^{54} +(17.7220 + 500.537i) q^{58} -396.000 q^{59} +(28.3753 + 801.428i) q^{62} +(500.500 - 107.906i) q^{64} +695.647i q^{69} -1128.75i q^{71} +(-287.404 + 30.6298i) q^{72} +413.641 q^{73} +788.328 q^{75} +(58.6108 + 1655.40i) q^{78} -910.722 q^{81} +(-1353.44 + 47.9196i) q^{82} +1116.76i q^{87} +(-62.4084 - 880.223i) q^{92} +1788.09i q^{93} +(36.3181 + 1025.76i) q^{94} +(1123.79 - 200.963i) q^{96} +(969.543 - 34.3275i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{2} + 8 q^{3} + 7 q^{4} + 63 q^{6} - 90 q^{8} + 184 q^{9} - 193 q^{12} + 79 q^{16} + 138 q^{18} - 180 q^{24} - 500 q^{25} - 195 q^{26} + 1256 q^{27} - 123 q^{32} - 506 q^{36} + 852 q^{41} - 529 q^{46}+ \cdots + 1029 q^{98}+O(q^{100}) $ 4 * q - 3 * q^2 + 8 * q^3 + 7 * q^4 + 63 * q^6 - 90 * q^8 + 184 * q^9 - 193 * q^12 + 79 * q^16 + 138 * q^18 - 180 * q^24 - 500 * q^25 - 195 * q^26 + 1256 * q^27 - 123 * q^32 - 506 * q^36 + 852 * q^41 - 529 * q^46 - 1291 * q^48 - 1372 * q^49 + 375 * q^50 - 363 * q^52 + 921 * q^54 + 1591 * q^58 - 1584 * q^59 + 1617 * q^62 + 2002 * q^64 - 4140 * q^72 + 2452 * q^73 - 1000 * q^75 - 1701 * q^78 + 4996 * q^81 - 2847 * q^82 - 1587 * q^92 - 1325 * q^94 + 4653 * q^96 + 1029 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/184\mathbb{Z}\right)^\times$.

$n$	$47$	$93$	$97$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$	−2.82666	+	0.100080i	−0.999374	+	0.0353837i
$2$	−2.82666	+	0.100080i	−0.999374	+	0.0353837i
$3$	−6.30662			−1.21371			−0.606855	−	0.794812i	$-0.707569\pi$
$3$	−6.30662			−1.21371			−0.606855	+	0.794812i	$0.707569\pi$
$4$	7.97997	−	0.565785i	0.997496	−	0.0707231i
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	17.8267	−	0.631168i	1.21295	−	0.0429456i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	−22.5000	+	2.39792i	−0.994369	+	0.105974i
$9$	12.7735			0.473093
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$	−50.3267	+	3.56819i	−1.21067	+	0.0858373i
$13$			92.8609i			1.98115i	0.136966	+	0.990576i	$0.456265\pi$
$13$			92.8609i			1.98115i	−0.136966	+	0.990576i	$0.543735\pi$
$14$		0			0
$15$		0			0
$16$	63.3598	−	9.02989i	0.989996	−	0.141092i
$17$		0			0		1.00000			$0$
$17$		0			0		−1.00000			$\pi$
$18$	−36.1063	+	1.27838i	−0.472797	+	0.0167398i
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	110.304i		−	1.00000i
$23$		−	110.304i		−	1.00000i
$24$	141.899	−	15.1228i	1.20688	−	0.128622i
$25$	−125.000			−1.00000
$26$	−9.29354	−	262.486i	−0.0701005	−	1.97991i
$27$	89.7212			0.639513
$28$		0			0
$29$		−	177.078i		−	1.13388i	−0.823760	−	0.566939i	$-0.808127\pi$
$29$		−	177.078i		−	1.13388i	0.823760	−	0.566939i	$-0.191873\pi$
$30$		0			0
$31$		−	283.525i		−	1.64267i	−0.570449	−	0.821333i	$-0.693231\pi$
$31$		−	283.525i		−	1.64267i	0.570449	−	0.821333i	$-0.306769\pi$
$32$	−178.193	+	31.8654i	−0.984384	+	0.176033i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	101.932	−	7.22705i	0.471908	−	0.0334586i
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		−	585.639i		−	2.40454i
$40$		0			0
$41$	478.812			1.82385			0.911925	−	0.410356i	$-0.134596\pi$
$41$	478.812			1.82385			0.911925	+	0.410356i	$0.134596\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$		0			0
$46$	11.0393	+	311.792i	0.0353837	+	0.999374i
$47$		−	362.890i		−	1.12623i	−0.826378	−	0.563116i	$-0.809602\pi$
$47$		−	362.890i		−	1.12623i	0.826378	−	0.563116i	$-0.190398\pi$
$48$	−399.586	+	56.9481i	−1.20157	+	0.171245i
$49$	−343.000			−1.00000
$50$	353.332	−	12.5100i	0.999374	−	0.0353837i
$51$		0			0
$52$	52.5393	+	741.027i	0.140113	+	1.97619i
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	−253.611	+	8.97931i	−0.639112	+	0.0226283i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	17.7220	+	500.537i	0.0401208	+	1.13317i
$59$	−396.000			−0.873810			−0.436905	−	0.899508i	$-0.643925\pi$
$59$	−396.000			−0.873810			−0.436905	+	0.899508i	$0.643925\pi$
$60$		0			0
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$	28.3753	+	801.428i	0.0581236	+	1.64164i
$63$		0			0
$64$	500.500	−	107.906i	0.977539	−	0.210754i
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$			695.647i			1.21371i
$70$		0			0
$71$		−	1128.75i		−	1.88673i	−0.331754	−	0.943366i	$-0.607640\pi$
$71$		−	1128.75i		−	1.88673i	0.331754	−	0.943366i	$-0.392360\pi$
$72$	−287.404	+	30.6298i	−0.470429	+	0.0501355i
$73$	413.641			0.663192			0.331596	−	0.943421i	$-0.392413\pi$
$73$	413.641			0.663192			0.331596	+	0.943421i	$0.392413\pi$
$74$		0			0
$75$	788.328			1.21371
$76$		0			0
$77$		0			0
$78$	58.6108	+	1655.40i	0.0850817	+	2.40304i
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−910.722			−1.24928
$82$	−1353.44	+	47.9196i	−1.82271	+	0.0645346i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$			1116.76i			1.37620i
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$	−62.4084	−	880.223i	−0.0707231	−	0.997496i
$93$			1788.09i			1.99372i
$94$	36.3181	+	1025.76i	0.0398503	+	1.12553i
$95$		0			0
$96$	1123.79	−	200.963i	1.19476	−	0.213654i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$	969.543	−	34.3275i	0.999374	−	0.0353837i
$99$		0			0
$100$	−997.496	+	70.7231i	−0.997496	+	0.0707231i
$101$		−	1246.92i		−	1.22844i	−0.789133	−	0.614222i	$-0.789470\pi$
$101$		−	1246.92i		−	1.22844i	0.789133	−	0.614222i	$-0.210530\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$	−222.673	−	2089.37i	−0.209950	−	1.97000i
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$	715.972	−	50.7629i	0.637911	−	0.0452283i
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$	−100.188	−	1413.07i	−0.0801914	−	1.13104i
$117$			1186.16i			0.937268i
$118$	1119.36	−	39.6318i	0.873263	−	0.0309186i
$119$		0			0
$120$		0			0
$121$	1331.00			1.00000
$122$		0			0
$123$	−3019.69			−2.21363
$124$	−160.414	−	2262.52i	−0.116174	−	1.63855i
$125$		0			0
$126$		0			0
$127$			1668.71i			1.16594i	0.812495	+	0.582968i	$0.198109\pi$
$127$			1668.71i			1.16594i	−0.812495	+	0.582968i	$0.801891\pi$
$128$	−1403.94	+	355.104i	−0.969470	+	0.245211i
$129$		0			0
$130$		0			0
$131$	−459.352			−0.306364			−0.153182	−	0.988198i	$-0.548952\pi$
$131$	−459.352			−0.306364			−0.153182	+	0.988198i	$0.548952\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$	−69.6205	−	1966.35i	−0.0429456	−	1.21295i
$139$	3021.95			1.84401			0.922007	−	0.387172i	$-0.126548\pi$
$139$	3021.95			1.84401			0.922007	+	0.387172i	$0.126548\pi$
$140$		0			0
$141$			2288.61i			1.36692i
$142$	112.966	+	3190.59i	0.0667596	+	1.88555i
$143$		0			0
$144$	809.326	−	115.343i	0.468360	−	0.0667496i
$145$		0			0
$146$	−1169.22	+	41.3973i	−0.662777	+	0.0234662i
$147$	2163.17			1.21371
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	−2228.33	+	78.8960i	−1.21295	+	0.0429456i
$151$		−	2714.31i		−	1.46283i	−0.681932	−	0.731415i	$-0.738860\pi$
$151$		−	2714.31i		−	1.46283i	0.681932	−	0.731415i	$-0.261140\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−331.345	−	4673.38i	−0.170057	−	2.39852i
$157$		0			0			−	1.00000i	$-0.5\pi$
$157$		0			0				1.00000i	$0.5\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	2574.30	−	91.1453i	1.24849	−	0.0442040i
$163$	−2496.34			−1.19956			−0.599781	−	0.800164i	$-0.704746\pi$
$163$	−2496.34			−1.19956			−0.599781	+	0.800164i	$0.704746\pi$
$164$	3820.90	−	270.904i	1.81928	−	0.128988i
$165$		0			0
$166$		0			0
$167$		−	3922.99i		−	1.81778i	−0.417030	−	0.908892i	$-0.636929\pi$
$167$		−	3922.99i		−	1.81778i	0.417030	−	0.908892i	$-0.363071\pi$
$168$		0			0
$169$	−6426.14			−2.92496
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	2896.68i		−	1.27301i	−0.771273	−	0.636505i	$-0.780380\pi$
$173$		−	2896.68i		−	1.27301i	0.771273	−	0.636505i	$-0.219620\pi$
$174$	−111.766	−	3156.70i	−0.0486950	−	1.37534i
$175$		0			0
$176$		0			0
$177$	2497.42			1.06055
$178$		0			0
$179$	2023.41			0.844898			0.422449	−	0.906387i	$-0.361171\pi$
$179$	2023.41			0.844898			0.422449	+	0.906387i	$0.361171\pi$
$180$		0			0
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$	264.500	+	2481.84i	0.105974	+	0.994369i
$185$		0			0
$186$	−178.952	−	5054.31i	−0.0705452	−	1.99247i
$187$		0			0
$188$	−205.318	−	2895.85i	−0.0796507	−	1.12341i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	−3156.47	+	680.524i	−1.18645	+	0.255795i
$193$	4934.03			1.84020			0.920101	−	0.391681i	$-0.128106\pi$
$193$	4934.03			1.84020			0.920101	+	0.391681i	$0.128106\pi$
$194$		0			0
$195$		0			0
$196$	−2737.13	+	194.064i	−0.997496	+	0.0707231i
$197$			5043.03i			1.82386i	0.410342	+	0.911931i	$0.365409\pi$
$197$			5043.03i			1.82386i	−0.410342	+	0.911931i	$0.634591\pi$
$198$		0			0
$199$		0			0			−	1.00000i	$-0.5\pi$
$199$		0			0				1.00000i	$0.5\pi$
$200$	2812.50	−	299.739i	0.994369	−	0.105974i
$201$		0			0
$202$	124.792	+	3524.60i	0.0434669	+	1.22767i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	1408.97i		−	0.473093i
$208$	838.523	+	5883.64i	0.279525	+	1.96133i
$209$		0			0
$210$		0			0
$211$	−2468.00			−0.805233			−0.402616	−	0.915369i	$-0.631899\pi$
$211$	−2468.00			−0.805233			−0.402616	+	0.915369i	$0.631899\pi$
$212$		0			0
$213$			7118.60i			2.28995i
$214$		0			0
$215$		0			0
$216$	−2018.73	+	215.144i	−0.635912	+	0.0677717i
$217$		0			0
$218$		0			0
$219$	−2608.68			−0.804923
$220$		0			0
$221$		0			0
$222$		0			0
$223$			4575.22i			1.37390i	0.726705	+	0.686950i	$0.241051\pi$
$223$			4575.22i			1.37390i	−0.726705	+	0.686950i	$0.758949\pi$
$224$		0			0
$225$	−1596.69			−0.473093
$226$		0			0
$227$		0			0		1.00000			$0$
$227$		0			0		−1.00000			$\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$	424.617	+	3984.24i	0.120162	+	1.12749i
$233$	744.688			0.209383			0.104691	−	0.994505i	$-0.466615\pi$
$233$	744.688			0.209383			0.104691	+	0.994505i	$0.466615\pi$
$234$	−118.711	−	3352.86i	−0.0331640	−	0.936682i
$235$		0			0
$236$	−3160.07	+	224.051i	−0.871622	+	0.0617986i
$237$		0			0
$238$		0			0
$239$			6663.83i			1.80355i	0.432211	+	0.901773i	$0.357734\pi$
$239$			6663.83i			1.80355i	−0.432211	+	0.901773i	$0.642266\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$	−3762.28	+	133.207i	−0.999374	+	0.0353837i
$243$	3321.11			0.876746
$244$		0			0
$245$		0			0
$246$	8535.62	−	302.211i	2.21224	−	0.0783263i
$247$		0			0
$248$	679.870	+	6379.32i	0.174080	+	1.63342i
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$	−167.005	−	4716.86i	−0.0412551	−	1.16521i
$255$		0			0
$256$	3932.92	−	1144.26i	0.960186	−	0.279361i
$257$	−4904.62			−1.19043			−0.595217	−	0.803565i	$-0.702934\pi$
$257$	−4904.62			−1.19043			−0.595217	+	0.803565i	$0.702934\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		−	2261.90i		−	0.536430i
$262$	1298.43	−	45.9720i	0.306172	−	0.0108403i
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	3515.63i		−	0.796847i	−0.917202	−	0.398424i	$-0.869557\pi$
$269$		−	3515.63i		−	0.796847i	0.917202	−	0.398424i	$-0.130443\pi$
$270$		0			0
$271$		−	431.625i		−	0.0967503i	−0.998829	−	0.0483752i	$-0.984596\pi$
$271$		−	431.625i		−	0.0967503i	0.998829	−	0.0483752i	$-0.0154043\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	393.586	+	5551.24i	0.0858373	+	1.21067i
$277$		−	6109.43i		−	1.32520i	−0.748974	−	0.662599i	$-0.769453\pi$
$277$		−	6109.43i		−	1.32520i	0.748974	−	0.662599i	$-0.230547\pi$
$278$	−8542.00	+	302.437i	−1.84286	+	0.0652481i
$279$		−	3621.61i		−	0.777133i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$	−229.045	−	6469.11i	−0.0483667	−	1.36606i
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$	−638.630	−	9007.39i	−0.133436	−	1.88201i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−2276.14	+	407.033i	−0.465705	+	0.0832801i
$289$	4913.00			1.00000
$290$		0			0
$291$		0			0
$292$	3300.84	−	234.032i	0.661531	−	0.0469030i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	−6114.54	+	216.491i	−1.21295	+	0.0429456i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$	10242.9			1.98115
$300$	6290.83	−	446.024i	1.21067	−	0.0858373i
$301$		0			0
$302$	271.649	+	7672.42i	0.0517604	+	1.46191i
$303$			7863.83i			1.49097i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−10420.0			−1.93714			−0.968568	−	0.248749i	$-0.919981\pi$
$307$	−10420.0			−1.93714			−0.968568	+	0.248749i	$0.919981\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	433.876i		−	0.0791088i	−0.999217	−	0.0395544i	$-0.987406\pi$
$311$		−	433.876i		−	0.0791088i	0.999217	−	0.0395544i	$-0.0125938\pi$
$312$	1404.31	+	13176.9i	0.254819	+	2.39100i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	11183.9i		−	1.98154i	−0.135542	−	0.990772i	$-0.543277\pi$
$317$		−	11183.9i		−	1.98154i	0.135542	−	0.990772i	$-0.456723\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−7267.53	+	515.273i	−1.24615	+	0.0883527i
$325$		−	11607.6i		−	1.98115i
$326$	7056.30	−	249.835i	1.19881	−	0.0424450i
$327$		0			0
$328$	−10773.3	+	1148.15i	−1.81358	+	0.193281i
$329$		0			0
$330$		0			0
$331$	6046.92			1.00414			0.502068	−	0.864828i	$-0.332573\pi$
$331$	6046.92			1.00414			0.502068	+	0.864828i	$0.332573\pi$
$332$		0			0
$333$		0			0
$334$	392.614	+	11088.9i	0.0643200	+	1.81665i
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$	18164.5	−	643.130i	2.92313	−	0.103496i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	289.901	+	8187.92i	0.0450438	+	1.27221i
$347$	−9180.00			−1.42020			−0.710098	−	0.704103i	$-0.751350\pi$
$347$	−9180.00			−1.42020			−0.710098	+	0.704103i	$0.751350\pi$
$348$	631.847	+	8911.72i	0.0973291	+	1.37275i
$349$		−	12764.5i		−	1.95779i	−0.204360	−	0.978896i	$-0.565511\pi$
$349$		−	12764.5i		−	1.95779i	0.204360	−	0.978896i	$-0.434489\pi$
$350$		0			0
$351$			8331.58i			1.26697i
$352$		0			0
$353$	−12810.6			−1.93156			−0.965782	−	0.259355i	$-0.916490\pi$
$353$	−12810.6			−1.93156			−0.965782	+	0.259355i	$0.916490\pi$
$354$	−7059.36	+	249.943i	−1.05989	+	0.0375263i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−5719.48	+	202.503i	−0.844369	+	0.0298956i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	6859.00			1.00000
$362$		0			0
$363$	−8394.12			−1.21371
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	−996.034	−	6988.84i	−0.141092	−	0.989996i
$369$	6116.11			0.862850
$370$		0			0
$371$		0			0
$372$	1011.67	+	14268.9i	0.141002	+	1.98873i
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$	870.179	+	8165.02i	0.119351	+	1.11989i
$377$	16443.6			2.24638
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$		−	10523.9i		−	1.41511i
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	8854.13	−	2239.51i	1.17666	−	0.297615i
$385$		0			0
$386$	−13946.8	+	493.798i	−1.83905	+	0.0651132i
$387$		0			0
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$	7717.50	−	822.485i	0.994369	−	0.105974i
$393$	2896.96			0.371838
$394$	−504.708	−	14254.9i	−0.0645350	−	1.82272i
$395$		0			0
$396$		0			0
$397$		−	15449.7i		−	1.95315i	−0.215185	−	0.976573i	$-0.569035\pi$
$397$		−	15449.7i		−	1.95315i	0.215185	−	0.976573i	$-0.430965\pi$
$398$		0			0
$399$		0			0
$400$	−7919.97	+	1128.74i	−0.989996	+	0.141092i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	26328.4			3.25437
$404$	−705.486	−	9950.35i	−0.0868793	−	1.22537i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−16157.1			−1.95334			−0.976671	−	0.214742i	$-0.931109\pi$
$409$	−16157.1			−1.95334			−0.976671	+	0.214742i	$0.931109\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	141.010	+	3982.67i	0.0167398	+	0.472797i
$415$		0			0
$416$	−2959.05	−	16547.1i	−0.348749	−	1.95021i
$417$	−19058.3			−2.23810
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$		0			0			−	1.00000i	$-0.5\pi$
$421$		0			0				1.00000i	$0.5\pi$
$422$	6976.19	−	246.998i	0.804729	−	0.0284921i
$423$		−	4635.38i		−	0.532813i
$424$		0			0
$425$		0			0
$426$	−712.431	−	20121.8i	−0.0810268	−	2.28851i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	5684.71	−	810.172i	0.633115	−	0.0902301i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	7373.84	−	261.077i	0.804419	−	0.0284812i
$439$			2133.87i			0.231991i	0.993250	+	0.115995i	$0.0370058\pi$
$439$			2133.87i			0.231991i	−0.993250	+	0.115995i	$0.962994\pi$
$440$		0			0
$441$	−4381.31			−0.473093
$442$		0			0
$443$	−13724.1			−1.47191			−0.735953	−	0.677033i	$-0.763265\pi$
$443$	−13724.1			−1.47191			−0.735953	+	0.677033i	$0.763265\pi$
$444$		0			0
$445$		0			0
$446$	−457.889	−	12932.6i	−0.0486137	−	1.37304i
$447$		0			0
$448$		0			0
$449$	18414.0			1.93544			0.967718	−	0.252037i	$-0.0811005\pi$
$449$	18414.0			1.93544			0.967718	+	0.252037i	$0.0811005\pi$
$450$	4513.29	−	159.797i	0.472797	−	0.0167398i
$451$		0			0
$452$		0			0
$453$			17118.1i			1.77545i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			12481.1i			1.26096i	0.776205	+	0.630480i	$0.217142\pi$
$461$			12481.1i			1.26096i	−0.776205	+	0.630480i	$0.782858\pi$
$462$		0			0
$463$			16142.8i			1.62034i	0.586194	+	0.810171i	$0.300626\pi$
$463$			16142.8i			1.62034i	−0.586194	+	0.810171i	$0.699374\pi$
$464$	−1598.99	−	11219.6i	−0.159981	−	1.12254i
$465$		0			0
$466$	−2104.98	+	74.5285i	−0.209251	+	0.00740873i
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	671.110	+	9465.51i	0.0662865	+	0.934922i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	8910.00	−	949.575i	0.868890	−	0.0926011i
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−666.918	−	18836.4i	−0.0638161	−	1.80242i
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	10621.3	−	753.059i	0.997496	−	0.0707231i
$485$		0			0
$486$	−9387.64	+	332.378i	−0.876197	+	0.0310225i
$487$		−	21367.1i		−	1.98816i	−0.108639	−	0.994081i	$-0.534649\pi$
$487$		−	21367.1i		−	1.98816i	0.108639	−	0.994081i	$-0.465351\pi$
$488$		0			0
$489$	15743.5			1.45592
$490$		0			0
$491$	−16210.8			−1.48998			−0.744992	−	0.667073i	$-0.767547\pi$
$491$	−16210.8			−1.48998			−0.744992	+	0.667073i	$0.767547\pi$
$492$	−24097.0	+	1708.49i	−2.20808	+	0.156554i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−2560.20	−	17964.1i	−0.231767	−	1.62623i
$497$		0			0
$498$		0			0
$499$	−21525.2			−1.93106			−0.965530	−	0.260292i	$-0.916181\pi$
$499$	−21525.2			−1.93106			−0.965530	+	0.260292i	$0.916181\pi$
$500$		0			0
$501$			24740.8i			2.20626i
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	40527.3			3.55006
$508$	944.129	+	13316.2i	0.0824586	+	1.16302i
$509$			17666.7i			1.53844i	0.638986	+	0.769218i	$0.279354\pi$
$509$			17666.7i			1.53844i	−0.638986	+	0.769218i	$0.720646\pi$
$510$		0			0
$511$		0			0
$512$	−11002.5	+	3628.05i	−0.949700	+	0.313161i
$513$		0			0
$514$	13863.7	−	490.855i	1.18969	−	0.0421220i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$			18268.3i			1.54506i
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$	226.372	+	6393.61i	0.0189809	+	0.536094i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$	−3665.61	+	259.894i	−0.305597	+	0.0216670i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−12167.0			−1.00000
$530$		0			0
$531$	−5058.31			−0.413393
$532$		0			0
$533$			44462.9i			3.61332i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−12760.9			−1.02546
$538$	351.845	+	9937.48i	0.0281954	+	0.796348i
$539$		0			0
$540$		0			0
$541$			13383.5i			1.06359i	0.846873	+	0.531796i	$0.178483\pi$
$541$			13383.5i			1.06359i	−0.846873	+	0.531796i	$0.821517\pi$
$542$	43.1971	+	1220.05i	0.00342339	+	0.0966898i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1569.66			−0.122694			−0.0613471	−	0.998116i	$-0.519540\pi$
$547$	−1569.66			−0.122694			−0.0613471	+	0.998116i	$0.519540\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−1668.10	−	15652.0i	−0.128622	−	1.20688i
$553$		0			0
$554$	611.433	+	17269.3i	0.0468904	+	1.32437i
$555$		0			0
$556$	24115.0	−	1709.77i	1.83940	−	0.130414i
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$	362.452	+	10237.0i	0.0274979	+	0.776647i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		1.00000			$0$
$563$		0			0		−1.00000			$\pi$
$564$	1294.86	+	18263.0i	0.0966728	+	1.36350i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	2706.65	+	25396.9i	0.199944	+	1.87611i
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$		0			0		1.00000			$0$
$571$		0			0		−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			13788.0i			1.00000i
$576$	6393.14	−	1378.34i	0.462467	−	0.0997063i
$577$	25741.2			1.85723			0.928615	−	0.371044i	$-0.121000\pi$
$577$	25741.2			1.85723			0.928615	+	0.371044i	$0.121000\pi$
$578$	−13887.4	+	491.694i	−0.999374	+	0.0353837i
$579$	−31117.0			−2.23347
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−9306.92	+	991.876i	−0.659458	+	0.0702811i
$585$		0			0
$586$		0			0
$587$	23488.7			1.65159			0.825794	−	0.563972i	$-0.190728\pi$
$587$	23488.7			1.65159			0.825794	+	0.563972i	$0.190728\pi$
$588$	17262.0	−	1223.89i	1.21067	−	0.0858373i
$589$		0			0
$590$		0			0
$591$		−	31804.5i		−	2.21364i
$592$		0			0
$593$	−26370.0			−1.82611			−0.913057	−	0.407831i	$-0.866285\pi$
$593$	−26370.0			−1.82611			−0.913057	+	0.407831i	$0.866285\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−28953.3	+	1025.12i	−1.97991	+	0.0701005i
$599$			16353.8i			1.11552i	0.830002	+	0.557761i	$0.188339\pi$
$599$			16353.8i			1.11552i	−0.830002	+	0.557761i	$0.811661\pi$
$600$	−17737.4	+	1890.34i	−1.20688	+	0.128622i
$601$	−25680.9			−1.74301			−0.871504	−	0.490388i	$-0.836855\pi$
$601$	−25680.9			−1.74301			−0.871504	+	0.490388i	$0.836855\pi$
$602$		0			0
$603$		0			0
$604$	−1535.72	−	21660.1i	−0.103456	−	1.45917i
$605$		0			0
$606$	−787.014	−	22228.3i	−0.0527562	−	1.49004i
$607$		−	28573.6i		−	1.91065i	−0.295556	−	0.955326i	$-0.595505\pi$
$607$		−	28573.6i		−	1.91065i	0.295556	−	0.955326i	$-0.404495\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	33698.3			2.23124
$612$		0			0
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$	29453.8	−	1042.84i	1.93592	−	0.0685431i
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$		−	9896.61i		−	0.639513i
$622$	43.4224	+	1226.42i	0.00279916	+	0.0790592i
$623$		0			0
$624$	−5288.25	−	37105.9i	−0.339262	−	2.38049i
$625$	15625.0			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0			−	1.00000i	$-0.5\pi$
$631$		0			0				1.00000i	$0.5\pi$
$632$		0			0
$633$	15564.7			0.977319
$634$	1119.29	+	31613.0i	0.0701143	+	1.98030i
$635$		0			0
$636$		0			0
$637$		−	31851.3i		−	1.98115i
$638$		0			0
$639$		−	14418.1i		−	0.892599i
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$		0			0		1.00000			$0$
$643$		0			0		−1.00000			$\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		−	32913.8i		−	1.99996i	−0.00630906	−	0.999980i	$-0.502008\pi$
$647$		−	32913.8i		−	1.99996i	0.00630906	−	0.999980i	$-0.497992\pi$
$648$	20491.2	−	2183.84i	1.24224	−	0.132391i
$649$		0			0
$650$	1161.69	+	32810.7i	0.0701005	+	1.97991i
$651$		0			0
$652$	−19920.7	+	1412.39i	−1.19656	+	0.0848368i
$653$			31611.4i			1.89441i	0.320631	+	0.947204i	$0.396105\pi$
$653$			31611.4i			1.89441i	−0.320631	+	0.947204i	$0.603895\pi$
$654$		0			0
$655$		0			0
$656$	30337.4	−	4323.62i	1.80561	−	0.257331i
$657$	5283.65			0.313751
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$	−17092.6	+	605.177i	−1.00351	+	0.0355300i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−19532.4			−1.13388
$668$	−2219.57	−	31305.3i	−0.128559	−	1.81323i
$669$		−	28854.2i		−	1.66752i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	25066.4			1.43572			0.717859	−	0.696188i	$-0.245122\pi$
$673$	25066.4			1.43572			0.717859	+	0.696188i	$0.245122\pi$
$674$		0			0
$675$	−11215.1			−0.639513
$676$	−51280.4	+	3635.81i	−2.91764	+	0.206862i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−21910.1			−1.22748			−0.613738	−	0.789510i	$-0.710335\pi$
$683$	−21910.1			−1.22748			−0.613738	+	0.789510i	$0.710335\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	6028.00			0.331861			0.165930	−	0.986137i	$-0.446937\pi$
$691$	6028.00			0.331861			0.165930	+	0.986137i	$0.446937\pi$
$692$	−1638.90	−	23115.4i	−0.0900312	−	1.26982i
$693$		0			0
$694$	25948.7	−	918.736i	1.41931	−	0.0502518i
$695$		0			0
$696$	−2677.90	−	25127.1i	−0.145841	−	1.36845i
$697$		0			0
$698$	1277.48	+	36080.9i	0.0692739	+	1.95657i
$699$	−4696.47			−0.254130
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	−833.827	−	23550.5i	−0.0448302	−	1.26618i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	36211.3	−	1282.09i	1.93035	−	0.0683459i
$707$		0			0
$708$	19929.4	−	1413.00i	1.05790	−	0.0750056i
$709$		0			0			−	1.00000i	$-0.5\pi$
$709$		0			0				1.00000i	$0.5\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−31274.0			−1.64267
$714$		0			0
$715$		0			0
$716$	16146.7	−	1144.81i	0.842782	−	0.0597538i
$717$		−	42026.3i		−	2.18898i
$718$		0			0
$719$		−	6858.04i		−	0.355719i	−0.984056	−	0.177859i	$-0.943083\pi$
$719$		−	6858.04i		−	0.355719i	0.984056	−	0.177859i	$-0.0569172\pi$
$720$		0			0
$721$		0			0
$722$	−19388.0	+	686.450i	−0.999374	+	0.0353837i
$723$		0			0
$724$		0			0
$725$			22134.7i			1.13388i
$726$	23727.3	−	840.085i	1.21295	−	0.0429456i
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$	3644.50			0.185160
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$	3514.89	+	19655.4i	0.176033	+	0.984384i
$737$		0			0
$738$	−17288.1	+	612.101i	−0.862310	+	0.0305308i
$739$	37876.3			1.88539			0.942695	−	0.333656i	$-0.108282\pi$
$739$	37876.3			1.88539			0.942695	+	0.333656i	$0.108282\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$	−4287.68	−	40232.0i	−0.211282	−	1.98249i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0			−	1.00000i	$-0.5\pi$
$751$		0			0				1.00000i	$0.5\pi$
$752$	−3276.86	−	22992.6i	−0.158902	−	1.11497i
$753$		0			0
$754$	−46480.3	+	1645.68i	−2.24498	+	0.0794854i
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	6372.08			0.303532			0.151766	−	0.988416i	$-0.451504\pi$
$761$	6372.08			0.303532			0.151766	+	0.988416i	$0.451504\pi$
$762$	1053.24	+	29747.5i	0.0500718	+	1.41422i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	36772.9i		−	1.73115i
$768$	−24803.5	+	7216.44i	−1.16539	+	0.339064i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	30931.6			1.44484
$772$	39373.4	−	2791.60i	1.83559	−	0.130145i
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$			35440.7i			1.64267i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		−	15887.6i		−	0.725130i
$784$	−21732.4	+	3097.25i	−0.989996	+	0.141092i
$785$		0			0
$786$	−8188.70	+	289.928i	−0.371605	+	0.0131570i
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$	2853.27	+	40243.2i	0.128989	+	1.81930i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$	1546.21	+	43671.0i	0.0691096	+	1.95192i
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$	22274.1	−	3983.18i	0.984384	−	0.176033i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$	−74421.3	+	2634.95i	−3.25233	+	0.115152i
$807$			22171.8i			0.967142i
$808$	2990.00	+	28055.6i	0.130183	+	1.22153i
$809$	−27846.0			−1.21015			−0.605076	−	0.796168i	$-0.706857\pi$
$809$	−27846.0			−1.21015			−0.605076	+	0.796168i	$0.706857\pi$
$810$		0			0
$811$	−41578.7			−1.80028			−0.900139	−	0.435602i	$-0.856536\pi$
$811$	−41578.7			−1.80028			−0.900139	+	0.435602i	$0.856536\pi$
$812$		0			0
$813$			2722.10i			0.117427i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	45670.5	−	1617.01i	1.95212	−	0.0691165i
$819$		0			0
$820$		0			0
$821$			40189.1i			1.70841i	0.519933	+	0.854207i	$0.325957\pi$
$821$			40189.1i			1.70841i	−0.519933	+	0.854207i	$0.674043\pi$
$822$		0			0
$823$		−	6683.96i		−	0.283096i	−0.989931	−	0.141548i	$-0.954792\pi$
$823$		−	6683.96i		−	0.283096i	0.989931	−	0.141548i	$-0.0452080\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$	−797.174	−	11243.5i	−0.0334586	−	0.471908i
$829$		−	47478.7i		−	1.98915i	−0.104027	−	0.994575i	$-0.533173\pi$
$829$		−	47478.7i		−	1.98915i	0.104027	−	0.994575i	$-0.466827\pi$
$830$		0			0
$831$			38529.9i			1.60841i
$832$	10020.3	+	46476.9i	0.417536	+	1.93665i
$833$		0			0
$834$	53871.2	−	1907.36i	2.23670	−	0.0791923i
$835$		0			0
$836$		0			0
$837$		−	25438.2i		−	1.05051i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−6967.45			−0.285680
$842$		0			0
$843$		0			0
$844$	−19694.6	+	1396.36i	−0.803217	+	0.0569486i
$845$		0			0
$846$	463.909	+	13102.6i	0.0188529	+	0.532479i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	4027.60	+	56806.2i	0.161952	+	2.28421i
$853$			43335.1i			1.73947i	0.493520	+	0.869734i	$0.335710\pi$
$853$			43335.1i			1.73947i	−0.493520	+	0.869734i	$0.664290\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	50108.5			1.99728			0.998642	−	0.0520955i	$-0.0165900\pi$
$857$	50108.5			1.99728			0.998642	+	0.0520955i	$0.0165900\pi$
$858$		0			0
$859$	24600.8			0.977148			0.488574	−	0.872523i	$-0.337517\pi$
$859$	24600.8			0.977148			0.488574	+	0.872523i	$0.337517\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	50434.3i		−	1.98935i	−0.103084	−	0.994673i	$-0.532871\pi$
$863$		−	50434.3i		−	1.98935i	0.103084	−	0.994673i	$-0.467129\pi$
$864$	−15987.6	+	2859.00i	−0.629526	+	0.112576i
$865$		0			0
$866$		0			0
$867$	−30984.4			−1.21371
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−20817.2	+	1475.95i	−0.802908	+	0.0569266i
$877$		−	39191.5i		−	1.50901i	−0.656293	−	0.754506i	$-0.727876\pi$
$877$		−	39191.5i		−	1.50901i	0.656293	−	0.754506i	$-0.272124\pi$
$878$	−213.558	−	6031.71i	−0.00820869	−	0.231845i
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$	12384.5	−	438.483i	0.472797	−	0.0167398i
$883$	2860.00			0.109000			0.0544998	−	0.998514i	$-0.482644\pi$
$883$	2860.00			0.109000			0.0544998	+	0.998514i	$0.482644\pi$
$884$		0			0
$885$		0			0
$886$	38793.4	−	1373.52i	1.47098	−	0.0520815i
$887$		−	24375.4i		−	0.922711i	−0.887215	−	0.461355i	$-0.847363\pi$
$887$		−	24375.4i		−	0.922711i	0.887215	−	0.461355i	$-0.152637\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	2588.59	+	36510.1i	0.0971664	+	1.37046i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−64598.4			−2.40454
$898$	−52050.0	+	1842.88i	−1.93422	+	0.0684829i
$899$	−50205.9			−1.86258
$900$	−12741.5	+	903.382i	−0.471908	+	0.0334586i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	−1713.19	−	48387.1i	−0.0628221	−	1.77434i
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$		−	15927.5i		−	0.581168i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	65715.0			2.35112
$922$	−1249.11	−	35279.8i	−0.0446175	−	1.26017i
$923$	104817.			3.73790
$924$		0			0
$925$		0			0
$926$	−1615.57	−	45630.1i	−0.0573337	−	1.61933i
$927$		0			0
$928$	5642.65	+	31553.9i	0.199600	+	1.11617i
$929$	48618.9			1.71704			0.858521	−	0.512778i	$-0.171384\pi$
$929$	48618.9			1.71704			0.858521	+	0.512778i	$0.171384\pi$
$930$		0			0
$931$		0			0
$932$	5942.59	−	421.333i	0.208858	−	0.0148082i
$933$			2736.29i			0.0960152i
$934$		0			0
$935$		0			0
$936$	−2844.31	−	26688.6i	−0.0993260	−	0.931991i
$937$		0			0		1.00000			$0$
$937$		0			0		−1.00000			$\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		−	52814.9i		−	1.82385i
$944$	−25090.5	+	3575.84i	−0.865069	+	0.123288i
$945$		0			0
$946$		0			0
$947$	−57261.1			−1.96487			−0.982437	−	0.186594i	$-0.940255\pi$
$947$	−57261.1			−1.96487			−0.982437	+	0.186594i	$0.940255\pi$
$948$		0			0
$949$			38411.1i			1.31388i
$950$		0			0
$951$			70532.5i			2.40502i
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	3770.29	+	53177.1i	0.127552	+	1.79903i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−50595.6			−1.69835
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$			22396.1i			0.744789i	0.928075	+	0.372394i	$0.121463\pi$
$967$			22396.1i			0.744789i	−0.928075	+	0.372394i	$0.878537\pi$
$968$	−29947.5	+	3191.63i	−0.994369	+	0.105974i
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$	26502.4	−	1879.03i	0.874551	−	0.0620062i
$973$		0			0
$974$	2138.42	+	60397.4i	0.0703486	+	1.98692i
$975$			73204.8i			2.40454i
$976$		0			0
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$	−44501.4	+	1575.61i	−1.45501	+	0.0515159i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	45822.3	−	1622.38i	1.48905	−	0.0527212i
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	67943.0	−	7240.95i	2.20116	−	0.234587i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	61722.4i		−	1.97848i	−0.146297	−	0.989241i	$-0.546735\pi$
$991$		−	61722.4i		−	1.97848i	0.146297	−	0.989241i	$-0.453265\pi$
$992$	9034.66	+	50522.1i	0.289164	+	1.61701i
$993$	−38135.6			−1.21873
$994$		0			0
$995$		0			0
$996$		0			0
$997$			18473.5i			0.586823i	0.955986	+	0.293412i	$0.0947907\pi$
$997$			18473.5i			0.586823i	−0.955986	+	0.293412i	$0.905209\pi$
$998$	60844.2	−	2154.24i	1.92985	−	0.0683281i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	184.4.h.b.91.2	yes	4
4.3	odd	2		736.4.h.b.367.4		4
8.3	odd	2	inner	184.4.h.b.91.1	✓	4
8.5	even	2		736.4.h.b.367.3		4
23.22	odd	2	CM	184.4.h.b.91.2	yes	4
92.91	even	2		736.4.h.b.367.4		4
184.45	odd	2		736.4.h.b.367.3		4
184.91	even	2	inner	184.4.h.b.91.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.4.h.b.91.1	✓	4	8.3	odd	2	inner
184.4.h.b.91.1	✓	4	184.91	even	2	inner
184.4.h.b.91.2	yes	4	1.1	even	1	trivial
184.4.h.b.91.2	yes	4	23.22	odd	2	CM
736.4.h.b.367.3		4	8.5	even	2
736.4.h.b.367.3		4	184.45	odd	2
736.4.h.b.367.4		4	4.3	odd	2
736.4.h.b.367.4		4	92.91	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 184 = 2^{3} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	184.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.82666 + 0.100080i) q^{2} -6.30662 q^{3} +(7.97997 - 0.565785i) q^{4} +(17.8267 - 0.631168i) q^{6} +(-22.5000 + 2.39792i) q^{8} +12.7735 q^{9} +(-50.3267 + 3.56819i) q^{12} +92.8609i q^{13} +(63.3598 - 9.02989i) q^{16} +(-36.1063 + 1.27838i) q^{18} -110.304i q^{23} +(141.899 - 15.1228i) q^{24} -125.000 q^{25} +(-9.29354 - 262.486i) q^{26} +89.7212 q^{27} -177.078i q^{29} -283.525i q^{31} +(-178.193 + 31.8654i) q^{32} +(101.932 - 7.22705i) q^{36} -585.639i q^{39} +478.812 q^{41} +(11.0393 + 311.792i) q^{46} -362.890i q^{47} +(-399.586 + 56.9481i) q^{48} -343.000 q^{49} +(353.332 - 12.5100i) q^{50} +(52.5393 + 741.027i) q^{52} +(-253.611 + 8.97931i) q^{54} +(17.7220 + 500.537i) q^{58} -396.000 q^{59} +(28.3753 + 801.428i) q^{62} +(500.500 - 107.906i) q^{64} +695.647i q^{69} -1128.75i q^{71} +(-287.404 + 30.6298i) q^{72} +413.641 q^{73} +788.328 q^{75} +(58.6108 + 1655.40i) q^{78} -910.722 q^{81} +(-1353.44 + 47.9196i) q^{82} +1116.76i q^{87} +(-62.4084 - 880.223i) q^{92} +1788.09i q^{93} +(36.3181 + 1025.76i) q^{94} +(1123.79 - 200.963i) q^{96} +(969.543 - 34.3275i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} + 8 q^{3} + 7 q^{4} + 63 q^{6} - 90 q^{8} + 184 q^{9} - 193 q^{12} + 79 q^{16} + 138 q^{18} - 180 q^{24} - 500 q^{25} - 195 q^{26} + 1256 q^{27} - 123 q^{32} - 506 q^{36} + 852 q^{41} - 529 q^{46}+ \cdots + 1029 q^{98}+O(q^{100}) \) 4 * q - 3 * q^2 + 8 * q^3 + 7 * q^4 + 63 * q^6 - 90 * q^8 + 184 * q^9 - 193 * q^12 + 79 * q^16 + 138 * q^18 - 180 * q^24 - 500 * q^25 - 195 * q^26 + 1256 * q^27 - 123 * q^32 - 506 * q^36 + 852 * q^41 - 529 * q^46 - 1291 * q^48 - 1372 * q^49 + 375 * q^50 - 363 * q^52 + 921 * q^54 + 1591 * q^58 - 1584 * q^59 + 1617 * q^62 + 2002 * q^64 - 4140 * q^72 + 2452 * q^73 - 1000 * q^75 - 1701 * q^78 + 4996 * q^81 - 2847 * q^82 - 1587 * q^92 - 1325 * q^94 + 4653 * q^96 + 1029 * q^98

Introduction

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