LMFDB - Embedded newform 69.4.c.b.68.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,4,Mod(68,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("69.68");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	69.c (of order $2$, degree $1$, 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:	$4.07113179040$
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:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			68.2
Root			$2.32666 + 0.765945i$ of defining polynomial
Character	$\chi$	$=$	69.68
Dual form			69.4.c.b.68.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.200160i q^{2} +(3.15331 + 4.12997i) q^{3} +7.95994 q^{4} +(0.826656 - 0.631168i) q^{6} -3.19455i q^{8} +(-7.11325 + 26.0461i) q^{9} +O(q^{10})$	$q-0.200160i q^{2} +(3.15331 + 4.12997i) q^{3} +7.95994 q^{4} +(0.826656 - 0.631168i) q^{6} -3.19455i q^{8} +(-7.11325 + 26.0461i) q^{9} +(25.1002 + 32.8743i) q^{12} +12.8397 q^{13} +63.0401 q^{16} +(5.21341 + 1.42379i) q^{18} -110.304i q^{23} +(13.1934 - 10.0734i) q^{24} -125.000 q^{25} -2.57001i q^{26} +(-130.000 + 52.7541i) q^{27} -177.078i q^{29} -196.920 q^{31} -38.1745i q^{32} +(-56.6210 + 207.326i) q^{36} +(40.4877 + 53.0277i) q^{39} -215.460i q^{41} -22.0785 q^{46} -362.890i q^{47} +(198.785 + 260.353i) q^{48} +343.000 q^{49} +25.0201i q^{50} +102.204 q^{52} +(10.5593 + 26.0209i) q^{54} -35.4439 q^{58} +815.291i q^{59} +39.4156i q^{62} +496.679 q^{64} +(455.552 - 347.823i) q^{69} +1128.75i q^{71} +(83.2057 + 22.7236i) q^{72} -413.641 q^{73} +(-394.164 - 516.246i) q^{75} +(10.6141 - 8.10404i) q^{78} +(-627.803 - 370.545i) q^{81} -43.1266 q^{82} +(731.324 - 558.381i) q^{87} -878.014i q^{92} +(-620.950 - 813.272i) q^{93} -72.6362 q^{94} +(157.659 - 120.376i) q^{96} -68.6550i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 18 q^{4} - 5 q^{6} + 38 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 18 * q^4 - 5 * q^6 + 38 * q^9	$ 4 q - 4 q^{3} - 18 q^{4} - 5 q^{6} + 38 q^{9} + 225 q^{12} - 148 q^{13} + 302 q^{16} + 79 q^{18} + 86 q^{24} - 500 q^{25} - 520 q^{27} - 688 q^{31} - 999 q^{36} + 976 q^{39} + 1058 q^{46} - 509 q^{48} + 1372 q^{49} + 3150 q^{52} - 506 q^{54} - 3182 q^{58} + 1588 q^{64} + 1058 q^{69} - 448 q^{72} - 2452 q^{73} + 500 q^{75} + 599 q^{78} + 14 q^{81} - 5306 q^{82} + 1048 q^{87} + 274 q^{93} + 2650 q^{94} + 207 q^{96}+O(q^{100}) $ 4 * q - 4 * q^3 - 18 * q^4 - 5 * q^6 + 38 * q^9 + 225 * q^12 - 148 * q^13 + 302 * q^16 + 79 * q^18 + 86 * q^24 - 500 * q^25 - 520 * q^27 - 688 * q^31 - 999 * q^36 + 976 * q^39 + 1058 * q^46 - 509 * q^48 + 1372 * q^49 + 3150 * q^52 - 506 * q^54 - 3182 * q^58 + 1588 * q^64 + 1058 * q^69 - 448 * q^72 - 2452 * q^73 + 500 * q^75 + 599 * q^78 + 14 * q^81 - 5306 * q^82 + 1048 * q^87 + 274 * q^93 + 2650 * q^94 + 207 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$-1$

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 $

$2$		−	0.200160i		−	0.0707674i	−0.999374	−	0.0353837i	$-0.988735\pi$
$2$		−	0.200160i		−	0.0707674i	0.999374	−	0.0353837i	$-0.0112653\pi$
$3$	3.15331	+	4.12997i	0.606855	+	0.794812i
$3$	3.15331	+	4.12997i	0.606855	+	0.794812i
$4$	7.95994			0.994992
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	0.826656	−	0.631168i	0.0562468	−	0.0429456i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		−	3.19455i		−	0.141180i
$9$	−7.11325	+	26.0461i	−0.263454	+	0.964672i
$10$		0			0
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	25.1002	+	32.8743i	0.603816	+	0.790832i
$13$	12.8397			0.273931			0.136966	−	0.990576i	$-0.456265\pi$
$13$	12.8397			0.273931			0.136966	+	0.990576i	$0.456265\pi$
$14$		0			0
$15$		0			0
$16$	63.0401			0.985001
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$	5.21341	+	1.42379i	0.0682673	+	0.0186439i
$19$		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
$24$	13.1934	−	10.0734i	0.112212	−	0.0856761i
$25$	−125.000			−1.00000
$26$		−	2.57001i		−	0.0193854i
$27$	−130.000	+	52.7541i	−0.926612	+	0.376020i
$28$		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
$31$	−196.920			−1.14090			−0.570449	−	0.821333i	$-0.693231\pi$
$31$	−196.920			−1.14090			−0.570449	+	0.821333i	$0.693231\pi$
$32$		−	38.1745i		−	0.210886i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−56.6210	+	207.326i	−0.262134	+	0.959841i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$	40.4877	+	53.0277i	0.166236	+	0.217724i
$40$		0			0
$41$		−	215.460i		−	0.820713i	−0.911925	−	0.410356i	$-0.865404\pi$
$41$		−	215.460i		−	0.820713i	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
$46$	−22.0785			−0.0707674
$47$		−	362.890i		−	1.12623i	−0.826378	−	0.563116i	$-0.809602\pi$
$47$		−	362.890i		−	1.12623i	0.826378	−	0.563116i	$-0.190398\pi$
$48$	198.785	+	260.353i	0.597753	+	0.782891i
$49$	343.000			1.00000
$50$			25.0201i			0.0707674i
$51$		0			0
$52$	102.204			0.272559
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	10.5593	+	26.0209i	0.0266100	+	0.0655739i
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−35.4439			−0.0802416
$59$			815.291i			1.79902i	0.436905	+	0.899508i	$0.356075\pi$
$59$			815.291i			1.79902i	−0.436905	+	0.899508i	$0.643925\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$			39.4156i			0.0807384i
$63$		0			0
$64$	496.679			0.970077
$65$		0			0
$66$		0			0
$67$		0			0		1.00000			$0$
$67$		0			0		−1.00000			$\pi$
$68$		0			0
$69$	455.552	−	347.823i	0.794812	−	0.606855i
$70$		0			0
$71$			1128.75i			1.88673i	0.331754	+	0.943366i	$0.392360\pi$
$71$			1128.75i			1.88673i	−0.331754	+	0.943366i	$0.607640\pi$
$72$	83.2057	+	22.7236i	0.136193	+	0.0371945i
$73$	−413.641			−0.663192			−0.331596	−	0.943421i	$-0.607587\pi$
$73$	−413.641			−0.663192			−0.331596	+	0.943421i	$0.607587\pi$
$74$		0			0
$75$	−394.164	−	516.246i	−0.606855	−	0.794812i
$76$		0			0
$77$		0			0
$78$	10.6141	−	8.10404i	0.0154078	−	0.0117641i
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−627.803	−	370.545i	−0.861184	−	0.508293i
$82$	−43.1266			−0.0580797
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	731.324	−	558.381i	0.901221	−	0.688100i
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$		−	878.014i		−	0.994992i
$93$	−620.950	−	813.272i	−0.692360	−	0.906800i
$94$	−72.6362			−0.0797006
$95$		0			0
$96$	157.659	−	120.376i	0.167615	−	0.127977i
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		−	68.6550i		−	0.0707674i
$99$		0			0
$100$	−994.992			−0.994992
$101$			1246.92i			1.22844i	0.789133	+	0.614222i	$0.210530\pi$
$101$			1246.92i			1.22844i	−0.789133	+	0.614222i	$0.789470\pi$
$102$		0			0
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		−	41.0172i		−	0.0386737i
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$	−1034.79	+	419.920i	−0.921971	+	0.374137i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$		−	1409.53i		−	1.12820i
$117$	−91.3323	+	334.426i	−0.0721681	+	0.264254i
$118$	163.189			0.127312
$119$		0			0
$120$		0			0
$121$	−1331.00			−1.00000
$122$		0			0
$123$	889.843	−	679.413i	0.652313	−	0.498054i
$124$	−1567.47			−1.13518
$125$		0			0
$126$		0			0
$127$	2325.71			1.62499			0.812495	−	0.582968i	$-0.198109\pi$
$127$	2325.71			1.62499			0.812495	+	0.582968i	$0.198109\pi$
$128$		−	404.812i		−	0.279536i
$129$		0			0
$130$		0			0
$131$			2963.34i			1.97640i	0.153182	+	0.988198i	$0.451048\pi$
$131$			2963.34i			1.97640i	−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			−	1.00000i	$-0.5\pi$
$137$		0			0				1.00000i	$0.5\pi$
$138$	−69.6205	−	91.1836i	−0.0429456	−	0.0562468i
$139$	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
$141$	1498.72	−	1144.31i	0.895144	−	0.683460i
$142$	225.931			0.133519
$143$		0			0
$144$	−448.420	+	1641.95i	−0.259502	+	0.950203i
$145$		0			0
$146$			82.7946i			0.0469324i
$147$	1081.59	+	1416.58i	0.606855	+	0.794812i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	−103.332	+	78.8960i	−0.0562468	+	0.0429456i
$151$	2530.68			1.36386			0.681932	−	0.731415i	$-0.261140\pi$
$151$	2530.68			1.36386			0.681932	+	0.731415i	$0.261140\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	322.280	+	422.097i	0.165404	+	0.216633i
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−74.1685	+	125.661i	−0.0359706	+	0.0609438i
$163$	−2496.34			−1.19956			−0.599781	−	0.800164i	$-0.704746\pi$
$163$	−2496.34			−1.19956			−0.599781	+	0.800164i	$0.704746\pi$
$164$		−	1715.05i		−	0.816603i
$165$		0			0
$166$		0			0
$167$			3922.99i			1.81778i	0.417030	+	0.908892i	$0.363071\pi$
$167$			3922.99i			1.81778i	−0.417030	+	0.908892i	$0.636929\pi$
$168$		0			0
$169$	−2032.14			−0.924962
$170$		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	−	146.382i	−0.0486950	−	0.0637770i
$175$		0			0
$176$		0			0
$177$	−3367.13	+	2570.87i	−1.42988	+	1.09174i
$178$		0			0
$179$		−	4341.33i		−	1.81277i	−0.422449	−	0.906387i	$-0.638829\pi$
$179$		−	4341.33i		−	1.81277i	0.422449	−	0.906387i	$-0.361171\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$	−352.372			−0.141180
$185$		0			0
$186$	−162.785	+	124.290i	−0.0641719	+	0.0489965i
$187$		0			0
$188$		−	2888.58i		−	1.12059i
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	1566.19	+	2051.27i	0.588696	+	0.771029i
$193$	−4934.03			−1.84020			−0.920101	−	0.391681i	$-0.871894\pi$
$193$	−4934.03			−1.84020			−0.920101	+	0.391681i	$0.871894\pi$
$194$		0			0
$195$		0			0
$196$	2730.26			0.994992
$197$		−	5043.03i		−	1.82386i	−0.410342	−	0.911931i	$-0.634591\pi$
$197$		−	5043.03i		−	1.82386i	0.410342	−	0.911931i	$-0.365409\pi$
$198$		0			0
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$			399.318i			0.141180i
$201$		0			0
$202$	249.583			0.0869338
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	2873.00	+	784.621i	0.964672	+	0.263454i
$208$	809.418			0.269822
$209$		0			0
$210$		0			0
$211$	2468.00			0.805233			0.402616	−	0.915369i	$-0.368101\pi$
$211$	2468.00			0.805233			0.402616	+	0.915369i	$0.368101\pi$
$212$		0			0
$213$	−4661.70	+	3559.30i	−1.49960	+	1.14497i
$214$		0			0
$215$		0			0
$216$	168.526	+	415.291i	0.0530867	+	0.130819i
$217$		0			0
$218$		0			0
$219$	−1304.34	−	1708.32i	−0.402462	−	0.527113i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−4840.00			−1.45341			−0.726705	−	0.686950i	$-0.758949\pi$
$223$	−4840.00			−1.45341			−0.726705	+	0.686950i	$0.758949\pi$
$224$		0			0
$225$	889.156	−	3255.77i	0.263454	−	0.964672i
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		0			0
$231$		0			0
$232$	−565.683			−0.160081
$233$		−	7074.09i		−	1.98901i	−0.104691	−	0.994505i	$-0.533385\pi$
$233$		−	7074.09i		−	1.98901i	0.104691	−	0.994505i	$-0.466615\pi$
$234$	66.9388	+	18.2811i	0.0187005	+	0.00510715i
$235$		0			0
$236$			6489.67i			1.79001i
$237$		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			$\pi$
$242$			266.414i			0.0707674i
$243$	−449.320	−	3761.25i	−0.118617	−	0.992940i
$244$		0			0
$245$		0			0
$246$	−135.992	−	178.111i	−0.0352460	−	0.0461625i
$247$		0			0
$248$			629.070i			0.161073i
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$		0			0
$254$		−	465.516i		−	0.114996i
$255$		0			0
$256$	3892.41			0.950295
$257$			6621.41i			1.60713i	0.595217	+	0.803565i	$0.297066\pi$
$257$			6621.41i			1.60713i	−0.595217	+	0.803565i	$0.702934\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	4612.19	+	1259.60i	1.09382	+	0.298724i
$262$	593.143			0.139864
$263$		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	$0.130443\pi$
$269$			3515.63i			0.796847i	−0.917202	+	0.398424i	$0.869557\pi$
$270$		0			0
$271$	8912.00			1.99766			0.998829	−	0.0483752i	$-0.0154043\pi$
$271$	8912.00			1.99766			0.998829	+	0.0483752i	$0.0154043\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$	3626.17	−	2768.65i	0.790832	−	0.603816i
$277$	6905.84			1.49795			0.748974	−	0.662599i	$-0.230547\pi$
$277$	6905.84			1.49795			0.748974	+	0.662599i	$0.230547\pi$
$278$		−	604.874i		−	0.130496i
$279$	1400.74	−	5129.00i	0.300574	−	1.10059i
$280$		0			0
$281$		0			0			−	1.00000i	$-0.5\pi$
$281$		0			0				1.00000i	$0.5\pi$
$282$	−229.045	−	299.985i	−0.0483667	−	0.0633470i
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$			8984.78i			1.87728i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	994.299	+	271.545i	0.203436	+	0.0555588i
$289$	−4913.00			−1.00000
$290$		0			0
$291$		0			0
$292$	−3292.56			−0.659871
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	283.543	−	216.491i	0.0562468	−	0.0429456i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	1416.28i		−	0.273931i
$300$	−3137.52	−	4109.28i	−0.603816	−	0.790832i
$301$		0			0
$302$		−	506.541i		−	0.0965171i
$303$	−5149.72	+	3931.92i	−0.976382	+	0.745487i
$304$		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	$0.0125938\pi$
$311$			433.876i			0.0791088i	−0.999217	+	0.0395544i	$0.987406\pi$
$312$	169.400	−	129.340i	0.0307383	−	0.0234693i
$313$		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
$324$	−4997.27	−	2949.52i	−0.856872	−	0.505747i
$325$	−1604.97			−0.273931
$326$			499.669i			0.0848899i
$327$		0			0
$328$	−688.298			−0.115869
$329$		0			0
$330$		0			0
$331$	−6046.92			−1.00414			−0.502068	−	0.864828i	$-0.667427\pi$
$331$	−6046.92			−1.00414			−0.502068	+	0.864828i	$0.667427\pi$
$332$		0			0
$333$		0			0
$334$	785.227			0.128640
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$			406.754i			0.0654571i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−579.801			−0.0900876
$347$			9102.49i			1.40821i	0.710098	+	0.704103i	$0.248650\pi$
$347$			9102.49i			1.40821i	−0.710098	+	0.704103i	$0.751350\pi$
$348$	5821.29	−	4444.67i	0.896707	−	0.684654i
$349$	2664.80			0.408721			0.204360	−	0.978896i	$-0.434489\pi$
$349$	2664.80			0.408721			0.204360	+	0.978896i	$0.434489\pi$
$350$		0			0
$351$	−1669.17	+	677.350i	−0.253828	+	0.103004i
$352$		0			0
$353$		−	3440.23i		−	0.518711i	−0.965782	−	0.259355i	$-0.916490\pi$
$353$		−	3440.23i		−	0.518711i	0.965782	−	0.259355i	$-0.0835101\pi$
$354$	514.586	+	673.965i	0.0772597	+	0.101189i
$355$		0			0
$356$		0			0
$357$		0			0
$358$	−868.963			−0.128285
$359$		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
$363$	−4197.06	−	5496.99i	−0.606855	−	0.794812i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		−	6953.58i		−	0.985001i
$369$	5611.91	+	1532.62i	0.791719	+	0.216220i
$370$		0			0
$371$		0			0
$372$	−4942.72	−	6473.60i	−0.688893	−	0.902259i
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$	−1159.27			−0.159002
$377$		−	2273.63i		−	0.310605i
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$		0			0
$381$	7333.70	+	9605.12i	0.986134	+	1.29156i
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	1671.86	−	1276.50i	0.222179	−	0.169638i
$385$		0			0
$386$			987.597i			0.130226i
$387$		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
$392$		−	1095.73i		−	0.141180i
$393$	−12238.5	+	9344.32i	−1.57086	+	1.19939i
$394$	−1009.42			−0.129070
$395$		0			0
$396$		0			0
$397$	3404.30			0.430370			0.215185	−	0.976573i	$-0.430965\pi$
$397$	3404.30			0.430370			0.215185	+	0.976573i	$0.430965\pi$
$398$		0			0
$399$		0			0
$400$	−7880.01			−0.985001
$401$		0			0			−	1.00000i	$-0.5\pi$
$401$		0			0				1.00000i	$0.5\pi$
$402$		0			0
$403$	−2528.40			−0.312528
$404$			9925.37i			1.22229i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	16157.1			1.95334			0.976671	−	0.214742i	$-0.0688912\pi$
$409$	16157.1			1.95334			0.976671	+	0.214742i	$0.0688912\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	157.050	−	575.060i	0.0186439	−	0.0682673i
$415$		0			0
$416$		−	490.151i		−	0.0577683i
$417$	9529.14	+	12480.5i	1.11905	+	1.46565i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		−	493.996i		−	0.0569842i
$423$	9451.88	+	2581.33i	1.08645	+	0.296710i
$424$		0			0
$425$		0			0
$426$	712.431	+	933.088i	0.0810268	+	0.106123i
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	−8195.21	+	3325.62i	−0.912713	+	0.370380i
$433$		0			0		1.00000			$0$
$433$		0			0		−1.00000			$\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−341.939	+	261.077i	−0.0373024	+	0.0284812i
$439$	18272.0			1.98650			0.993250	−	0.115995i	$-0.0370058\pi$
$439$	18272.0			1.98650			0.993250	+	0.115995i	$0.0370058\pi$
$440$		0			0
$441$	−2439.84	+	8933.83i	−0.263454	+	0.964672i
$442$		0			0
$443$		−	12625.4i		−	1.35407i	−0.735953	−	0.677033i	$-0.763265\pi$
$443$		−	12625.4i		−	1.35407i	0.735953	−	0.677033i	$-0.236735\pi$
$444$		0			0
$445$		0			0
$446$			968.777i			0.102854i
$447$		0			0
$448$		0			0
$449$		−	4795.83i		−	0.504074i	−0.967718	−	0.252037i	$-0.918900\pi$
$449$		−	4795.83i		−	0.504074i	0.967718	−	0.252037i	$-0.0811005\pi$
$450$	−651.676	−	177.974i	−0.0682673	−	0.0186439i
$451$		0			0
$452$		0			0
$453$	7980.01	+	10451.6i	0.827668	+	1.08402i
$454$		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	$-0.782858\pi$
$461$		−	12481.1i		−	1.26096i	0.776205	−	0.630480i	$-0.217142\pi$
$462$		0			0
$463$	−11680.0			−1.17239			−0.586194	−	0.810171i	$-0.699374\pi$
$463$	−11680.0			−1.17239			−0.586194	+	0.810171i	$0.699374\pi$
$464$		−	11163.0i		−	1.11687i
$465$		0			0
$466$	−1415.95			−0.140757
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$	−726.999	+	2662.01i	−0.0718067	+	0.262930i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	2604.49			0.253986
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	1333.84			0.127632
$479$		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
$484$	−10594.7			−0.994992
$485$		0			0
$486$	−752.854	+	89.9361i	−0.0702678	+	0.00839421i
$487$	2335.12			0.217278			0.108639	−	0.994081i	$-0.465351\pi$
$487$	2335.12			0.217278			0.108639	+	0.994081i	$0.465351\pi$
$488$		0			0
$489$	−7871.75	−	10309.8i	−0.727961	−	0.953427i
$490$		0			0
$491$		−	14515.3i		−	1.33415i	−0.744992	−	0.667073i	$-0.767547\pi$
$491$		−	14515.3i		−	1.33415i	0.744992	−	0.667073i	$-0.232453\pi$
$492$	7083.10	−	5408.09i	0.649046	−	0.495560i
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−12413.8			−1.12379
$497$		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
$501$	−16201.8	+	12370.4i	−1.44480	+	1.10313i
$502$		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
$507$	−6407.97	−	8392.67i	−0.561318	−	0.735171i
$508$	18512.5			1.61685
$509$			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
$512$		−	4017.60i		−	0.346786i
$513$		0			0
$514$	1325.34			0.113732
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	11963.2	−	9134.14i	1.01180	−	0.772532i
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$	252.121	−	923.177i	0.0211399	−	0.0774069i
$523$		0			0		1.00000			$0$
$523$		0			0		−1.00000			$\pi$
$524$			23588.0i			1.96650i
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−12167.0			−1.00000
$530$		0			0
$531$	−21235.2	−	5799.37i	−1.73546	−	0.473957i
$532$		0			0
$533$		−	2766.45i		−	0.224819i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	17929.6	−	13689.6i	1.44081	−	1.10009i
$538$	703.691			0.0563908
$539$		0			0
$540$		0			0
$541$	−21313.0			−1.69375			−0.846873	−	0.531796i	$-0.821517\pi$
$541$	−21313.0			−1.69375			−0.846873	+	0.531796i	$0.821517\pi$
$542$		−	1783.83i		−	0.141369i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1569.66			0.122694			0.0613471	−	0.998116i	$-0.480460\pi$
$547$	1569.66			0.122694			0.0613471	+	0.998116i	$0.480460\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$	−1111.14	−	1455.28i	−0.0856761	−	0.112212i
$553$		0			0
$554$		−	1382.28i		−	0.106006i
$555$		0			0
$556$	24054.5			1.83478
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$	−1026.62	−	280.373i	−0.0778861	−	0.0212708i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$	11929.7	−	9108.60i	0.890661	−	0.680037i
$565$		0			0
$566$		0			0
$567$		0			0
$568$	3605.85			0.266370
$569$		0			0			−	1.00000i	$-0.5\pi$
$569$		0			0				1.00000i	$0.5\pi$
$570$		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
$576$	−3533.00	+	12936.6i	−0.255570	+	0.935806i
$577$	25741.2			1.85723			0.928615	−	0.371044i	$-0.121000\pi$
$577$	25741.2			1.85723			0.928615	+	0.371044i	$0.121000\pi$
$578$			983.388i			0.0707674i
$579$	−15558.5	−	20377.4i	−1.11674	−	1.46262i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$			1321.40i			0.0936297i
$585$		0			0
$586$		0			0
$587$			16041.5i			1.12794i	0.825794	+	0.563972i	$0.190728\pi$
$587$			16041.5i			1.12794i	−0.825794	+	0.563972i	$0.809272\pi$
$588$	8609.36	+	11275.9i	0.603816	+	0.790832i
$589$		0			0
$590$		0			0
$591$	20827.5	−	15902.2i	1.44963	−	1.10682i
$592$		0			0
$593$			11778.6i			0.815662i	0.913057	+	0.407831i	$0.133715\pi$
$593$			11778.6i			0.815662i	−0.913057	+	0.407831i	$0.866285\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$	−283.483			−0.0193854
$599$			16353.8i			1.11552i	0.830002	+	0.557761i	$0.188339\pi$
$599$			16353.8i			1.11552i	−0.830002	+	0.557761i	$0.811661\pi$
$600$	−1649.17	+	1259.18i	−0.112212	+	0.0856761i
$601$	−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
$604$	20144.0			1.35703
$605$		0			0
$606$	787.014	+	1030.77i	0.0527562	+	0.0690960i
$607$	8840.00			0.591111			0.295556	−	0.955326i	$-0.404495\pi$
$607$	8840.00			0.591111			0.295556	+	0.955326i	$0.404495\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	4659.41i		−	0.308510i
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$			2085.67i			0.137086i
$615$		0			0
$616$		0			0
$617$		0			0			−	1.00000i	$-0.5\pi$
$617$		0			0				1.00000i	$0.5\pi$
$618$		0			0
$619$		0			0		1.00000			$0$
$619$		0			0		−1.00000			$\pi$
$620$		0			0
$621$	5819.00	+	14339.5i	0.376020	+	0.926612i
$622$	86.8448			0.00559832
$623$		0			0
$624$	2552.35	+	3342.87i	0.163743	+	0.214458i
$625$	15625.0			1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	7782.37	+	10192.8i	0.488660	+	0.640009i
$634$	−2238.57			−0.140229
$635$		0			0
$636$		0			0
$637$	4404.03			0.273931
$638$		0			0
$639$	−29399.6	−	8029.08i	−1.82008	−	0.497066i
$640$		0			0
$641$		0			0			−	1.00000i	$-0.5\pi$
$641$		0			0				1.00000i	$0.5\pi$
$642$		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	$0.497992\pi$
$647$			32913.8i			1.99996i	−0.00630906	+	0.999980i	$0.502008\pi$
$648$	−1183.72	+	2005.55i	−0.0717610	+	0.121582i
$649$		0			0
$650$			321.251i			0.0193854i
$651$		0			0
$652$	−19870.7			−1.19355
$653$		−	31611.4i		−	1.89441i	−0.320631	−	0.947204i	$-0.603895\pi$
$653$		−	31611.4i		−	1.89441i	0.320631	−	0.947204i	$-0.396105\pi$
$654$		0			0
$655$		0			0
$656$		−	13582.6i		−	0.808403i
$657$	2942.33	−	10773.8i	0.174720	−	0.639763i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$			1210.35i			0.0710600i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−19532.4			−1.13388
$668$			31226.8i			1.80868i
$669$	−15262.0	−	19989.0i	−0.882009	−	1.15519i
$670$		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
$675$	16250.0	−	6594.27i	0.926612	−	0.376020i
$676$	−16175.7			−0.920330
$677$		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
$683$		−	28185.0i		−	1.57902i	−0.613738	−	0.789510i	$-0.710335\pi$
$683$		−	28185.0i		−	1.57902i	0.613738	−	0.789510i	$-0.289665\pi$
$684$		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	$-0.553063\pi$
$691$	−6028.00			−0.331861			−0.165930	+	0.986137i	$0.553063\pi$
$692$		−	23057.4i		−	1.26663i
$693$		0			0
$694$	1821.96			0.0996550
$695$		0			0
$696$	−1783.77	−	2336.25i	−0.0971462	−	0.127235i
$697$		0			0
$698$		−	533.388i		−	0.0289241i
$699$	29215.8	−	22306.8i	1.58089	−	1.20704i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$	135.579	+	334.101i	0.00728930	+	0.0179627i
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−688.598			−0.0367078
$707$		0			0
$708$	−26802.1	+	20463.9i	−1.42272	+	1.08627i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			21721.1i			1.14090i
$714$		0			0
$715$		0			0
$716$		−	34556.7i		−	1.80370i
$717$	−27521.4	+	21013.1i	−1.43348	+	1.09449i
$718$		0			0
$719$			6858.04i			0.355719i	0.984056	+	0.177859i	$0.0569172\pi$
$719$			6858.04i			0.355719i	−0.984056	+	0.177859i	$0.943083\pi$
$720$		0			0
$721$		0			0
$722$		−	1372.90i		−	0.0707674i
$723$		0			0
$724$		0			0
$725$			22134.7i			1.13388i
$726$	−1100.28	+	840.085i	−0.0562468	+	0.0429456i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	14117.0	−	13716.1i	0.717218	−	0.696849i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$	−4210.81			−0.210886
$737$		0			0
$738$	306.770	−	1123.28i	0.0153013	−	0.0560279i
$739$	−37876.3			−1.88539			−0.942695	−	0.333656i	$-0.891718\pi$
$739$	−37876.3			−1.88539			−0.942695	+	0.333656i	$0.891718\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$	−2598.04	+	1983.65i	−0.128022	+	0.0977477i
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		−	22876.6i		−	1.10934i
$753$		0			0
$754$	−455.091			−0.0219807
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	41499.9i		−	1.97683i	−0.151766	−	0.988416i	$-0.548496\pi$
$761$		−	41499.9i		−	1.97683i	0.151766	−	0.988416i	$-0.451504\pi$
$762$	1922.57	−	1467.92i	0.0914005	−	0.0697861i
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$			10468.1i			0.492806i
$768$	12274.0	+	16075.5i	0.576691	+	0.755306i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	−27346.2	+	20879.4i	−1.27737	+	0.975295i
$772$	−39274.5			−1.83099
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$	24615.0			1.14090
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$	9341.57	+	23020.1i	0.426361	+	1.05066i
$784$	21622.7			0.985001
$785$		0			0
$786$	1870.36	+	2449.66i	0.0848774	+	0.111166i
$787$		0			0		1.00000			$0$
$787$		0			0		−1.00000			$\pi$
$788$		−	40142.2i		−	1.81473i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		−	681.406i		−	0.0304562i
$795$		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
$800$			4771.81i			0.210886i
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$			506.086i			0.0221168i
$807$	−14519.4	+	11085.9i	−0.633344	+	0.483571i
$808$	3983.33			0.173432
$809$			36640.2i			1.59234i	0.605076	+	0.796168i	$0.293143\pi$
$809$			36640.2i			1.59234i	−0.605076	+	0.796168i	$0.706857\pi$
$810$		0			0
$811$	41578.7			1.80028			0.900139	−	0.435602i	$-0.143464\pi$
$811$	41578.7			1.80028			0.900139	+	0.435602i	$0.143464\pi$
$812$		0			0
$813$	28102.3	+	36806.3i	1.21229	+	1.58776i
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$		−	3234.01i		−	0.138233i
$819$		0			0
$820$		0			0
$821$		−	40189.1i		−	1.70841i	−0.519933	−	0.854207i	$-0.674043\pi$
$821$		−	40189.1i		−	1.70841i	0.519933	−	0.854207i	$-0.325957\pi$
$822$		0			0
$823$	46745.0			1.97986			0.989931	−	0.141548i	$-0.0452080\pi$
$823$	46745.0			1.97986			0.989931	+	0.141548i	$0.0452080\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$	22868.9	+	6245.53i	0.959841	+	0.262134i
$829$	−4966.00			−0.208053			−0.104027	−	0.994575i	$-0.533173\pi$
$829$	−4966.00			−0.208053			−0.104027	+	0.994575i	$0.533173\pi$
$830$		0			0
$831$	21776.3	+	28520.9i	0.909038	+	1.19059i
$832$	6377.24			0.265734
$833$		0			0
$834$	2498.11	−	1907.36i	0.103720	−	0.0791923i
$835$		0			0
$836$		0			0
$837$	25599.6	−	10388.3i	1.05717	−	0.429001i
$838$		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
$844$	19645.1			0.801200
$845$		0			0
$846$	516.679	−	1891.89i	0.0209974	−	0.0768849i
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$	−37106.8	+	28331.8i	−1.49209	+	1.13924i
$853$	24590.0			0.987041			0.493520	−	0.869734i	$-0.335710\pi$
$853$	24590.0			0.987041			0.493520	+	0.869734i	$0.335710\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	2613.97i		−	0.104191i	−0.998642	−	0.0520955i	$-0.983410\pi$
$857$		−	2613.97i		−	0.104191i	0.998642	−	0.0520955i	$-0.0165900\pi$
$858$		0			0
$859$	−24600.8			−0.977148			−0.488574	−	0.872523i	$-0.662483\pi$
$859$	−24600.8			−0.977148			−0.488574	+	0.872523i	$0.662483\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			50434.3i			1.98935i	0.103084	+	0.994673i	$0.467129\pi$
$863$			50434.3i			1.98935i	−0.103084	+	0.994673i	$0.532871\pi$
$864$	2013.86	+	4962.69i	0.0792975	+	0.195410i
$865$		0			0
$866$		0			0
$867$	−15492.2	−	20290.5i	−0.606855	−	0.794812i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$	−10382.5	−	13598.1i	−0.400446	−	0.524474i
$877$	−34090.0			−1.31259			−0.656293	−	0.754506i	$-0.727876\pi$
$877$	−34090.0			−1.31259			−0.656293	+	0.754506i	$0.727876\pi$
$878$		−	3657.32i		−	0.140579i
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$	1788.20	+	488.360i	0.0682673	+	0.0186439i
$883$	−2860.00			−0.109000			−0.0544998	−	0.998514i	$-0.517356\pi$
$883$	−2860.00			−0.109000			−0.0544998	+	0.998514i	$0.517356\pi$
$884$		0			0
$885$		0			0
$886$	−2527.11			−0.0958238
$887$		−	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
$892$	−38526.1			−1.44613
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$	5849.18	−	4465.96i	0.217724	−	0.166236i
$898$	−959.936			−0.0356720
$899$			34870.1i			1.29364i
$900$	7077.62	−	25915.7i	0.262134	−	0.959841i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$	2092.00	−	1597.28i	0.0767130	−	0.0585719i
$907$		0			0		1.00000			$0$
$907$		0			0		−1.00000			$\pi$
$908$		0			0
$909$	−32477.4	−	8869.62i	−1.18505	−	0.323638i
$910$		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		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	−32857.5	−	43034.3i	−1.17556	−	1.53966i
$922$	−2498.22			−0.0892349
$923$			14492.9i			0.516835i
$924$		0			0
$925$		0			0
$926$			2337.87i			0.0829669i
$927$		0			0
$928$	−6759.85			−0.239119
$929$		−	29039.1i		−	1.02556i	−0.858521	−	0.512778i	$-0.828616\pi$
$929$		−	29039.1i		−	1.02556i	0.858521	−	0.512778i	$-0.171384\pi$
$930$		0			0
$931$		0			0
$932$		−	56309.3i		−	1.97905i
$933$	−1791.89	+	1368.15i	−0.0628766	+	0.0480076i
$934$		0			0
$935$		0			0
$936$	1068.34	+	291.765i	0.0373074	+	0.0101887i
$937$		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
$943$	−23766.2			−0.820713
$944$			51396.0i			1.77203i
$945$		0			0
$946$		0			0
$947$		−	10875.6i		−	0.373189i	−0.982437	−	0.186594i	$-0.940255\pi$
$947$		−	10875.6i		−	0.373189i	0.982437	−	0.186594i	$-0.0597450\pi$
$948$		0			0
$949$	−5311.04			−0.181669
$950$		0			0
$951$	46189.0	−	35266.3i	1.57496	−	1.20251i
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$		0			0
$956$			53043.7i			1.79451i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	8986.44			0.301649
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	55815.2			1.85615			0.928075	−	0.372394i	$-0.121463\pi$
$967$	55815.2			1.85615			0.928075	+	0.372394i	$0.121463\pi$
$968$			4251.94i			0.141180i
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$	−3576.56	−	29939.3i	−0.118023	−	0.987967i
$973$		0			0
$974$		−	467.398i		−	0.0153762i
$975$	−5060.96	−	6628.46i	−0.166236	−	0.217724i
$976$		0			0
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$	−2063.62	+	1575.61i	−0.0674716	+	0.0515159i
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−2905.39			−0.0944141
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	−2170.42	−	2842.65i	−0.0703155	−	0.0920938i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	9128.00			0.292594			0.146297	−	0.989241i	$-0.453265\pi$
$991$	9128.00			0.292594			0.146297	+	0.989241i	$0.453265\pi$
$992$			7517.32i			0.240600i
$993$	−19067.8	−	24973.6i	−0.609365	−	0.798099i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−60190.0			−1.91197			−0.955986	−	0.293412i	$-0.905209\pi$
$997$	−60190.0			−1.91197			−0.955986	+	0.293412i	$0.905209\pi$
$998$			4308.49i			0.136656i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.4.c.b.68.2	✓	4
3.2	odd	2	inner	69.4.c.b.68.3	yes	4
23.22	odd	2	CM	69.4.c.b.68.2	✓	4
69.68	even	2	inner	69.4.c.b.68.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.4.c.b.68.2	✓	4	1.1	even	1	trivial
69.4.c.b.68.2	✓	4	23.22	odd	2	CM
69.4.c.b.68.3	yes	4	3.2	odd	2	inner
69.4.c.b.68.3	yes	4	69.68	even	2	inner

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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	69.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.200160i q^{2} +(3.15331 + 4.12997i) q^{3} +7.95994 q^{4} +(0.826656 - 0.631168i) q^{6} -3.19455i q^{8} +(-7.11325 + 26.0461i) q^{9} +O(q^{10})\)	\(q-0.200160i q^{2} +(3.15331 + 4.12997i) q^{3} +7.95994 q^{4} +(0.826656 - 0.631168i) q^{6} -3.19455i q^{8} +(-7.11325 + 26.0461i) q^{9} +(25.1002 + 32.8743i) q^{12} +12.8397 q^{13} +63.0401 q^{16} +(5.21341 + 1.42379i) q^{18} -110.304i q^{23} +(13.1934 - 10.0734i) q^{24} -125.000 q^{25} -2.57001i q^{26} +(-130.000 + 52.7541i) q^{27} -177.078i q^{29} -196.920 q^{31} -38.1745i q^{32} +(-56.6210 + 207.326i) q^{36} +(40.4877 + 53.0277i) q^{39} -215.460i q^{41} -22.0785 q^{46} -362.890i q^{47} +(198.785 + 260.353i) q^{48} +343.000 q^{49} +25.0201i q^{50} +102.204 q^{52} +(10.5593 + 26.0209i) q^{54} -35.4439 q^{58} +815.291i q^{59} +39.4156i q^{62} +496.679 q^{64} +(455.552 - 347.823i) q^{69} +1128.75i q^{71} +(83.2057 + 22.7236i) q^{72} -413.641 q^{73} +(-394.164 - 516.246i) q^{75} +(10.6141 - 8.10404i) q^{78} +(-627.803 - 370.545i) q^{81} -43.1266 q^{82} +(731.324 - 558.381i) q^{87} -878.014i q^{92} +(-620.950 - 813.272i) q^{93} -72.6362 q^{94} +(157.659 - 120.376i) q^{96} -68.6550i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 18 q^{4} - 5 q^{6} + 38 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 18 * q^4 - 5 * q^6 + 38 * q^9	\( 4 q - 4 q^{3} - 18 q^{4} - 5 q^{6} + 38 q^{9} + 225 q^{12} - 148 q^{13} + 302 q^{16} + 79 q^{18} + 86 q^{24} - 500 q^{25} - 520 q^{27} - 688 q^{31} - 999 q^{36} + 976 q^{39} + 1058 q^{46} - 509 q^{48} + 1372 q^{49} + 3150 q^{52} - 506 q^{54} - 3182 q^{58} + 1588 q^{64} + 1058 q^{69} - 448 q^{72} - 2452 q^{73} + 500 q^{75} + 599 q^{78} + 14 q^{81} - 5306 q^{82} + 1048 q^{87} + 274 q^{93} + 2650 q^{94} + 207 q^{96}+O(q^{100}) \) 4 * q - 4 * q^3 - 18 * q^4 - 5 * q^6 + 38 * q^9 + 225 * q^12 - 148 * q^13 + 302 * q^16 + 79 * q^18 + 86 * q^24 - 500 * q^25 - 520 * q^27 - 688 * q^31 - 999 * q^36 + 976 * q^39 + 1058 * q^46 - 509 * q^48 + 1372 * q^49 + 3150 * q^52 - 506 * q^54 - 3182 * q^58 + 1588 * q^64 + 1058 * q^69 - 448 * q^72 - 2452 * q^73 + 500 * q^75 + 599 * q^78 + 14 * q^81 - 5306 * q^82 + 1048 * q^87 + 274 * q^93 + 2650 * q^94 + 207 * q^96

Introduction

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