LMFDB - Embedded newform 1089.3.c.j.604.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,3,Mod(604,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1089.604");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1089.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:	$29.6731007888$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.41108373504.15
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 4x^{6} + 20x^{5} + 20x^{4} - 28x^{3} + 4x^{2} + 12x + 33 $ x^8 - 4x^7 - 4x^6 + 20x^5 + 20x^4 - 28x^3 + 4x^2 + 12*x + 33
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 363)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.8
Root			$-1.64430 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.j.604.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.73960i q^{2} -9.98460 q^{4} +0.556540 q^{5} +12.0391i q^{7} -22.3800i q^{8} +O(q^{10})$	$q+3.73960i q^{2} -9.98460 q^{4} +0.556540 q^{5} +12.0391i q^{7} -22.3800i q^{8} +2.08124i q^{10} +7.96452i q^{13} -45.0215 q^{14} +43.7538 q^{16} -10.8970i q^{17} +9.70455i q^{19} -5.55683 q^{20} -27.3098 q^{23} -24.6903 q^{25} -29.7841 q^{26} -120.206i q^{28} +11.5608i q^{29} -36.3959 q^{31} +74.1017i q^{32} +40.7506 q^{34} +6.70025i q^{35} +32.8223 q^{37} -36.2911 q^{38} -12.4554i q^{40} -57.0728i q^{41} +18.9306i q^{43} -102.128i q^{46} +24.7138 q^{47} -95.9403 q^{49} -92.3317i q^{50} -79.5225i q^{52} -52.5362 q^{53} +269.435 q^{56} -43.2327 q^{58} -5.04423 q^{59} +8.25704i q^{61} -136.106i q^{62} -102.095 q^{64} +4.43258i q^{65} +56.3510 q^{67} +108.803i q^{68} -25.0563 q^{70} +107.479 q^{71} +37.8722i q^{73} +122.742i q^{74} -96.8960i q^{76} -118.635i q^{79} +24.3508 q^{80} +213.429 q^{82} +126.178i q^{83} -6.06464i q^{85} -70.7930 q^{86} -38.4568 q^{89} -95.8858 q^{91} +272.677 q^{92} +92.4199i q^{94} +5.40097i q^{95} +79.9978 q^{97} -358.778i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 24 q^{4} - 16 q^{5}+O(q^{10}) $ 8 * q - 24 * q^4 - 16 * q^5	$ 8 q - 24 q^{4} - 16 q^{5} - 136 q^{14} + 72 q^{16} + 16 q^{20} - 88 q^{25} + 120 q^{26} - 128 q^{31} + 96 q^{34} + 80 q^{37} + 216 q^{38} + 32 q^{47} - 152 q^{49} - 80 q^{53} + 696 q^{56} - 176 q^{58} + 64 q^{59} + 8 q^{64} + 464 q^{67} + 304 q^{70} + 128 q^{71} - 80 q^{80} + 528 q^{82} - 24 q^{86} + 720 q^{89} + 80 q^{91} + 1248 q^{92} + 416 q^{97}+O(q^{100}) $ 8 * q - 24 * q^4 - 16 * q^5 - 136 * q^14 + 72 * q^16 + 16 * q^20 - 88 * q^25 + 120 * q^26 - 128 * q^31 + 96 * q^34 + 80 * q^37 + 216 * q^38 + 32 * q^47 - 152 * q^49 - 80 * q^53 + 696 * q^56 - 176 * q^58 + 64 * q^59 + 8 * q^64 + 464 * q^67 + 304 * q^70 + 128 * q^71 - 80 * q^80 + 528 * q^82 - 24 * q^86 + 720 * q^89 + 80 * q^91 + 1248 * q^92 + 416 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1089\mathbb{Z}\right)^\times$.

$n$	$244$	$848$
$\chi(n)$	$-1$	$1$

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 $

$2$	3.73960i	1.86980i	0.354912	+	0.934900i	$0.384511\pi$
$2$	3.73960i	1.86980i	−0.354912	+	0.934900i	$0.615489\pi$
$3$	0	0
$3$	0	0
$4$	−9.98460	−2.49615
$5$	0.556540	0.111308	0.0556540	−	0.998450i	$-0.482276\pi$
$5$	0.556540	0.111308	0.0556540	+	0.998450i	$0.482276\pi$
$6$	0	0
$7$	12.0391i	1.71987i	0.510400	+	0.859937i	$0.329497\pi$
$7$	12.0391i	1.71987i	−0.510400	+	0.859937i	$0.670503\pi$
$8$	− 22.3800i	− 2.79750i
$9$	0	0
$10$	2.08124i	0.208124i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	7.96452i	0.612655i	0.951926	+	0.306328i	$0.0991003\pi$
$13$	7.96452i	0.612655i	−0.951926	+	0.306328i	$0.900900\pi$
$14$	−45.0215	−3.21582
$15$	0	0
$16$	43.7538	2.73461
$17$	− 10.8970i	− 0.641003i	−0.947248	−	0.320501i	$-0.896149\pi$
$17$	− 10.8970i	− 0.641003i	0.947248	−	0.320501i	$-0.103851\pi$
$18$	0	0
$19$	9.70455i	0.510766i	0.966840	+	0.255383i	$0.0822015\pi$
$19$	9.70455i	0.510766i	−0.966840	+	0.255383i	$0.917799\pi$
$20$	−5.55683	−0.277842
$21$	0	0
$22$	0	0
$23$	−27.3098	−1.18738	−0.593691	−	0.804693i	$-0.702330\pi$
$23$	−27.3098	−1.18738	−0.593691	+	0.804693i	$0.702330\pi$
$24$	0	0
$25$	−24.6903	−0.987611
$26$	−29.7841	−1.14554
$27$	0	0
$28$	− 120.206i	− 4.29306i
$29$	11.5608i	0.398648i	0.979934	+	0.199324i	$0.0638746\pi$
$29$	11.5608i	0.398648i	−0.979934	+	0.199324i	$0.936125\pi$
$30$	0	0
$31$	−36.3959	−1.17406	−0.587031	−	0.809564i	$-0.699703\pi$
$31$	−36.3959	−1.17406	−0.587031	+	0.809564i	$0.699703\pi$
$32$	74.1017i	2.31568i
$33$	0	0
$34$	40.7506	1.19855
$35$	6.70025i	0.191436i
$36$	0	0
$37$	32.8223	0.887088	0.443544	−	0.896253i	$-0.353721\pi$
$37$	32.8223	0.887088	0.443544	+	0.896253i	$0.353721\pi$
$38$	−36.2911	−0.955029
$39$	0	0
$40$	− 12.4554i	− 0.311384i
$41$	− 57.0728i	− 1.39202i	−0.718033	−	0.696010i	$-0.754957\pi$
$41$	− 57.0728i	− 1.39202i	0.718033	−	0.696010i	$-0.245043\pi$
$42$	0	0
$43$	18.9306i	0.440247i	0.975472	+	0.220124i	$0.0706461\pi$
$43$	18.9306i	0.440247i	−0.975472	+	0.220124i	$0.929354\pi$
$44$	0	0
$45$	0	0
$46$	− 102.128i	− 2.22016i
$47$	24.7138	0.525827	0.262913	−	0.964819i	$-0.415317\pi$
$47$	24.7138	0.525827	0.262913	+	0.964819i	$0.415317\pi$
$48$	0	0
$49$	−95.9403	−1.95797
$50$	− 92.3317i	− 1.84663i
$51$	0	0
$52$	− 79.5225i	− 1.52928i
$53$	−52.5362	−0.991249	−0.495625	−	0.868537i	$-0.665061\pi$
$53$	−52.5362	−0.991249	−0.495625	+	0.868537i	$0.665061\pi$
$54$	0	0
$55$	0	0
$56$	269.435	4.81135
$57$	0	0
$58$	−43.2327	−0.745392
$59$	−5.04423	−0.0854954	−0.0427477	−	0.999086i	$-0.513611\pi$
$59$	−5.04423	−0.0854954	−0.0427477	+	0.999086i	$0.513611\pi$
$60$	0	0
$61$	8.25704i	0.135361i	0.997707	+	0.0676807i	$0.0215599\pi$
$61$	8.25704i	0.135361i	−0.997707	+	0.0676807i	$0.978440\pi$
$62$	− 136.106i	− 2.19526i
$63$	0	0
$64$	−102.095	−1.59524
$65$	4.43258i	0.0681935i
$66$	0	0
$67$	56.3510	0.841060	0.420530	−	0.907279i	$-0.361844\pi$
$67$	56.3510	0.841060	0.420530	+	0.907279i	$0.361844\pi$
$68$	108.803i	1.60004i
$69$	0	0
$70$	−25.0563	−0.357947
$71$	107.479	1.51378	0.756892	−	0.653540i	$-0.226717\pi$
$71$	107.479	1.51378	0.756892	+	0.653540i	$0.226717\pi$
$72$	0	0
$73$	37.8722i	0.518797i	0.965770	+	0.259399i	$0.0835243\pi$
$73$	37.8722i	0.518797i	−0.965770	+	0.259399i	$0.916476\pi$
$74$	122.742i	1.65868i
$75$	0	0
$76$	− 96.8960i	− 1.27495i
$77$	0	0
$78$	0	0
$79$	− 118.635i	− 1.50171i	−0.660467	−	0.750855i	$-0.729642\pi$
$79$	− 118.635i	− 1.50171i	0.660467	−	0.750855i	$-0.270358\pi$
$80$	24.3508	0.304384
$81$	0	0
$82$	213.429	2.60280
$83$	126.178i	1.52021i	0.649800	+	0.760106i	$0.274853\pi$
$83$	126.178i	1.52021i	−0.649800	+	0.760106i	$0.725147\pi$
$84$	0	0
$85$	− 6.06464i	− 0.0713488i
$86$	−70.7930	−0.823174
$87$	0	0
$88$	0	0
$89$	−38.4568	−0.432099	−0.216049	−	0.976382i	$-0.569317\pi$
$89$	−38.4568	−0.432099	−0.216049	+	0.976382i	$0.569317\pi$
$90$	0	0
$91$	−95.8858	−1.05369
$92$	272.677	2.96388
$93$	0	0
$94$	92.4199i	0.983190i
$95$	5.40097i	0.0568523i
$96$	0	0
$97$	79.9978	0.824720	0.412360	−	0.911021i	$-0.364705\pi$
$97$	79.9978	0.824720	0.412360	+	0.911021i	$0.364705\pi$
$98$	− 358.778i	− 3.66100i
$99$	0	0
$100$	246.522	2.46522
$101$	− 196.531i	− 1.94586i	−0.231108	−	0.972928i	$-0.574235\pi$
$101$	− 196.531i	− 1.94586i	0.231108	−	0.972928i	$-0.425765\pi$
$102$	0	0
$103$	82.6136	0.802074	0.401037	−	0.916062i	$-0.368650\pi$
$103$	82.6136	0.802074	0.401037	+	0.916062i	$0.368650\pi$
$104$	178.246	1.71390
$105$	0	0
$106$	− 196.464i	− 1.85344i
$107$	− 182.713i	− 1.70759i	−0.520606	−	0.853797i	$-0.674294\pi$
$107$	− 182.713i	− 1.70759i	0.520606	−	0.853797i	$-0.325706\pi$
$108$	0	0
$109$	− 17.1701i	− 0.157523i	−0.996893	−	0.0787617i	$-0.974903\pi$
$109$	− 17.1701i	− 0.157523i	0.996893	−	0.0787617i	$-0.0250966\pi$
$110$	0	0
$111$	0	0
$112$	526.757i	4.70319i
$113$	25.6172	0.226701	0.113350	−	0.993555i	$-0.463842\pi$
$113$	25.6172	0.226701	0.113350	+	0.993555i	$0.463842\pi$
$114$	0	0
$115$	−15.1990	−0.132165
$116$	− 115.430i	− 0.995085i
$117$	0	0
$118$	− 18.8634i	− 0.159859i
$119$	131.191	1.10244
$120$	0	0
$121$	0	0
$122$	−30.8780	−0.253099
$123$	0	0
$124$	363.399	2.93063
$125$	−27.6546	−0.221237
$126$	0	0
$127$	132.307i	1.04179i	0.853622	+	0.520893i	$0.174401\pi$
$127$	132.307i	1.04179i	−0.853622	+	0.520893i	$0.825599\pi$
$128$	− 85.3889i	− 0.667101i
$129$	0	0
$130$	−16.5761	−0.127508
$131$	62.0270i	0.473489i	0.971572	+	0.236744i	$0.0760804\pi$
$131$	62.0270i	0.473489i	−0.971572	+	0.236744i	$0.923920\pi$
$132$	0	0
$133$	−116.834	−0.878452
$134$	210.730i	1.57261i
$135$	0	0
$136$	−243.876	−1.79320
$137$	−171.552	−1.25220	−0.626101	−	0.779742i	$-0.715350\pi$
$137$	−171.552	−1.25220	−0.626101	+	0.779742i	$0.715350\pi$
$138$	0	0
$139$	25.6741i	0.184706i	0.995726	+	0.0923528i	$0.0294388\pi$
$139$	25.6741i	0.184706i	−0.995726	+	0.0923528i	$0.970561\pi$
$140$	− 66.8993i	− 0.477852i
$141$	0	0
$142$	401.927i	2.83047i
$143$	0	0
$144$	0	0
$145$	6.43405i	0.0443727i
$146$	−141.627	−0.970047
$147$	0	0
$148$	−327.717	−2.21431
$149$	61.7507i	0.414434i	0.978295	+	0.207217i	$0.0664407\pi$
$149$	61.7507i	0.414434i	−0.978295	+	0.207217i	$0.933559\pi$
$150$	0	0
$151$	− 135.741i	− 0.898950i	−0.893293	−	0.449475i	$-0.851611\pi$
$151$	− 135.741i	− 0.898950i	0.893293	−	0.449475i	$-0.148389\pi$
$152$	217.188	1.42887
$153$	0	0
$154$	0	0
$155$	−20.2558	−0.130683
$156$	0	0
$157$	−94.4481	−0.601580	−0.300790	−	0.953690i	$-0.597250\pi$
$157$	−94.4481	−0.601580	−0.300790	+	0.953690i	$0.597250\pi$
$158$	443.648	2.80790
$159$	0	0
$160$	41.2406i	0.257754i
$161$	− 328.785i	− 2.04215i
$162$	0	0
$163$	−42.6523	−0.261671	−0.130835	−	0.991404i	$-0.541766\pi$
$163$	−42.6523	−0.261671	−0.130835	+	0.991404i	$0.541766\pi$
$164$	569.849i	3.47469i
$165$	0	0
$166$	−471.853	−2.84249
$167$	10.6464i	0.0637511i	0.999492	+	0.0318756i	$0.0101480\pi$
$167$	10.6464i	0.0637511i	−0.999492	+	0.0318756i	$0.989852\pi$
$168$	0	0
$169$	105.566	0.624653
$170$	22.6793	0.133408
$171$	0	0
$172$	− 189.015i	− 1.09892i
$173$	161.521i	0.933648i	0.884350	+	0.466824i	$0.154602\pi$
$173$	161.521i	0.933648i	−0.884350	+	0.466824i	$0.845398\pi$
$174$	0	0
$175$	− 297.249i	− 1.69857i
$176$	0	0
$177$	0	0
$178$	− 143.813i	− 0.807938i
$179$	−328.354	−1.83438	−0.917190	−	0.398449i	$-0.869548\pi$
$179$	−328.354	−1.83438	−0.917190	+	0.398449i	$0.869548\pi$
$180$	0	0
$181$	−298.805	−1.65086	−0.825429	−	0.564506i	$-0.809067\pi$
$181$	−298.805	−1.65086	−0.825429	+	0.564506i	$0.809067\pi$
$182$	− 358.574i	− 1.97019i
$183$	0	0
$184$	611.192i	3.32170i
$185$	18.2669	0.0987401
$186$	0	0
$187$	0	0
$188$	−246.758	−1.31254
$189$	0	0
$190$	−20.1975	−0.106302
$191$	−246.018	−1.28805	−0.644027	−	0.765003i	$-0.722738\pi$
$191$	−246.018	−1.28805	−0.644027	+	0.765003i	$0.722738\pi$
$192$	0	0
$193$	− 91.0180i	− 0.471596i	−0.971802	−	0.235798i	$-0.924230\pi$
$193$	− 91.0180i	− 0.471596i	0.971802	−	0.235798i	$-0.0757703\pi$
$194$	299.160i	1.54206i
$195$	0	0
$196$	957.926	4.88738
$197$	91.6248i	0.465101i	0.972584	+	0.232550i	$0.0747070\pi$
$197$	91.6248i	0.465101i	−0.972584	+	0.232550i	$0.925293\pi$
$198$	0	0
$199$	−236.382	−1.18785	−0.593925	−	0.804521i	$-0.702422\pi$
$199$	−236.382	−1.18785	−0.593925	+	0.804521i	$0.702422\pi$
$200$	552.568i	2.76284i
$201$	0	0
$202$	734.949	3.63836
$203$	−139.182	−0.685624
$204$	0	0
$205$	− 31.7633i	− 0.154943i
$206$	308.942i	1.49972i
$207$	0	0
$208$	348.478i	1.67538i
$209$	0	0
$210$	0	0
$211$	297.231i	1.40868i	0.709865	+	0.704338i	$0.248756\pi$
$211$	297.231i	1.40868i	−0.709865	+	0.704338i	$0.751244\pi$
$212$	524.553	2.47431
$213$	0	0
$214$	683.272	3.19286
$215$	10.5357i	0.0490031i
$216$	0	0
$217$	− 438.175i	− 2.01924i
$218$	64.2091	0.294537
$219$	0	0
$220$	0	0
$221$	86.7897	0.392714
$222$	0	0
$223$	−378.854	−1.69890	−0.849449	−	0.527670i	$-0.823066\pi$
$223$	−378.854	−1.69890	−0.849449	+	0.527670i	$0.823066\pi$
$224$	−892.119	−3.98267
$225$	0	0
$226$	95.7979i	0.423885i
$227$	438.387i	1.93122i	0.259995	+	0.965610i	$0.416279\pi$
$227$	438.387i	1.93122i	−0.259995	+	0.965610i	$0.583721\pi$
$228$	0	0
$229$	−175.836	−0.767844	−0.383922	−	0.923366i	$-0.625427\pi$
$229$	−175.836	−0.767844	−0.383922	+	0.923366i	$0.625427\pi$
$230$	− 56.8381i	− 0.247122i
$231$	0	0
$232$	258.730	1.11522
$233$	75.2903i	0.323134i	0.986862	+	0.161567i	$0.0516549\pi$
$233$	75.2903i	0.323134i	−0.986862	+	0.161567i	$0.948345\pi$
$234$	0	0
$235$	13.7543	0.0585287
$236$	50.3646	0.213409
$237$	0	0
$238$	490.601i	2.06135i
$239$	− 146.421i	− 0.612641i	−0.951928	−	0.306321i	$-0.900902\pi$
$239$	− 146.421i	− 0.612641i	0.951928	−	0.306321i	$-0.0990980\pi$
$240$	0	0
$241$	− 230.674i	− 0.957155i	−0.878045	−	0.478577i	$-0.841153\pi$
$241$	− 230.674i	− 0.957155i	0.878045	−	0.478577i	$-0.158847\pi$
$242$	0	0
$243$	0	0
$244$	− 82.4433i	− 0.337882i
$245$	−53.3947	−0.217937
$246$	0	0
$247$	−77.2920	−0.312923
$248$	814.541i	3.28444i
$249$	0	0
$250$	− 103.417i	− 0.413669i
$251$	337.886	1.34616	0.673079	−	0.739571i	$-0.264971\pi$
$251$	337.886	1.34616	0.673079	+	0.739571i	$0.264971\pi$
$252$	0	0
$253$	0	0
$254$	−494.775	−1.94793
$255$	0	0
$256$	−89.0611	−0.347895
$257$	213.303	0.829972	0.414986	−	0.909828i	$-0.363786\pi$
$257$	213.303	0.829972	0.414986	+	0.909828i	$0.363786\pi$
$258$	0	0
$259$	395.151i	1.52568i
$260$	− 44.2575i	− 0.170221i
$261$	0	0
$262$	−231.956	−0.885329
$263$	− 69.7400i	− 0.265171i	−0.991172	−	0.132585i	$-0.957672\pi$
$263$	− 69.7400i	− 0.265171i	0.991172	−	0.132585i	$-0.0423279\pi$
$264$	0	0
$265$	−29.2385	−0.110334
$266$	− 436.913i	− 1.64253i
$267$	0	0
$268$	−562.642	−2.09941
$269$	−362.767	−1.34858	−0.674289	−	0.738468i	$-0.735550\pi$
$269$	−362.767	−1.34858	−0.674289	+	0.738468i	$0.735550\pi$
$270$	0	0
$271$	− 307.805i	− 1.13581i	−0.823094	−	0.567905i	$-0.807754\pi$
$271$	− 307.805i	− 1.13581i	0.823094	−	0.567905i	$-0.192246\pi$
$272$	− 476.787i	− 1.75289i
$273$	0	0
$274$	− 641.535i	− 2.34137i
$275$	0	0
$276$	0	0
$277$	− 324.946i	− 1.17309i	−0.809916	−	0.586546i	$-0.800487\pi$
$277$	− 324.946i	− 1.17309i	0.809916	−	0.586546i	$-0.199513\pi$
$278$	−96.0107	−0.345362
$279$	0	0
$280$	149.952	0.535542
$281$	335.001i	1.19217i	0.802920	+	0.596087i	$0.203279\pi$
$281$	335.001i	1.19217i	−0.802920	+	0.596087i	$0.796721\pi$
$282$	0	0
$283$	125.961i	0.445093i	0.974922	+	0.222547i	$0.0714370\pi$
$283$	125.961i	0.445093i	−0.974922	+	0.222547i	$0.928563\pi$
$284$	−1073.13	−3.77863
$285$	0	0
$286$	0	0
$287$	687.106	2.39410
$288$	0	0
$289$	170.254	0.589116
$290$	−24.0608	−0.0829681
$291$	0	0
$292$	− 378.139i	− 1.29500i
$293$	− 323.127i	− 1.10282i	−0.834234	−	0.551411i	$-0.814090\pi$
$293$	− 323.127i	− 1.10282i	0.834234	−	0.551411i	$-0.185910\pi$
$294$	0	0
$295$	−2.80732	−0.00951632
$296$	− 734.562i	− 2.48163i
$297$	0	0
$298$	−230.923	−0.774909
$299$	− 217.509i	− 0.727455i
$300$	0	0
$301$	−227.908	−0.757170
$302$	507.619	1.68086
$303$	0	0
$304$	424.611i	1.39675i
$305$	4.59538i	0.0150668i
$306$	0	0
$307$	137.877i	0.449109i	0.974461	+	0.224555i	$0.0720927\pi$
$307$	137.877i	0.449109i	−0.974461	+	0.224555i	$0.927907\pi$
$308$	0	0
$309$	0	0
$310$	− 75.7486i	− 0.244350i
$311$	194.185	0.624389	0.312194	−	0.950018i	$-0.398936\pi$
$311$	194.185	0.624389	0.312194	+	0.950018i	$0.398936\pi$
$312$	0	0
$313$	72.5476	0.231782	0.115891	−	0.993262i	$-0.463028\pi$
$313$	72.5476	0.231782	0.115891	+	0.993262i	$0.463028\pi$
$314$	− 353.198i	− 1.12483i
$315$	0	0
$316$	1184.52i	3.74849i
$317$	−533.314	−1.68238	−0.841190	−	0.540740i	$-0.818144\pi$
$317$	−533.314	−1.68238	−0.841190	+	0.540740i	$0.818144\pi$
$318$	0	0
$319$	0	0
$320$	−56.8202	−0.177563
$321$	0	0
$322$	1229.53	3.81840
$323$	105.751	0.327402
$324$	0	0
$325$	− 196.646i	− 0.605065i
$326$	− 159.503i	− 0.489272i
$327$	0	0
$328$	−1277.29	−3.89417
$329$	297.533i	0.904355i
$330$	0	0
$331$	122.679	0.370632	0.185316	−	0.982679i	$-0.440669\pi$
$331$	122.679	0.370632	0.185316	+	0.982679i	$0.440669\pi$
$332$	− 1259.83i	− 3.79467i
$333$	0	0
$334$	−39.8134	−0.119202
$335$	31.3616	0.0936168
$336$	0	0
$337$	64.3041i	0.190813i	0.995438	+	0.0954067i	$0.0304151\pi$
$337$	64.3041i	0.190813i	−0.995438	+	0.0954067i	$0.969585\pi$
$338$	394.776i	1.16798i
$339$	0	0
$340$	60.5530i	0.178097i
$341$	0	0
$342$	0	0
$343$	− 565.120i	− 1.64758i
$344$	423.667	1.23159
$345$	0	0
$346$	−604.024	−1.74573
$347$	99.0680i	0.285499i	0.989759	+	0.142749i	$0.0455943\pi$
$347$	99.0680i	0.285499i	−0.989759	+	0.142749i	$0.954406\pi$
$348$	0	0
$349$	− 450.615i	− 1.29116i	−0.763693	−	0.645580i	$-0.776616\pi$
$349$	− 450.615i	− 1.29116i	0.763693	−	0.645580i	$-0.223384\pi$
$350$	1111.59	3.17598
$351$	0	0
$352$	0	0
$353$	−450.850	−1.27720	−0.638598	−	0.769541i	$-0.720485\pi$
$353$	−450.850	−1.27720	−0.638598	+	0.769541i	$0.720485\pi$
$354$	0	0
$355$	59.8162	0.168496
$356$	383.975	1.07858
$357$	0	0
$358$	− 1227.91i	− 3.42992i
$359$	− 142.450i	− 0.396796i	−0.980122	−	0.198398i	$-0.936426\pi$
$359$	− 142.450i	− 0.396796i	0.980122	−	0.198398i	$-0.0635738\pi$
$360$	0	0
$361$	266.822	0.739119
$362$	− 1117.41i	− 3.08677i
$363$	0	0
$364$	957.381	2.63017
$365$	21.0774i	0.0577463i
$366$	0	0
$367$	30.6981	0.0836461	0.0418230	−	0.999125i	$-0.486683\pi$
$367$	30.6981	0.0836461	0.0418230	+	0.999125i	$0.486683\pi$
$368$	−1194.91	−3.24703
$369$	0	0
$370$	68.3109i	0.184624i
$371$	− 632.490i	− 1.70482i
$372$	0	0
$373$	409.938i	1.09903i	0.835484	+	0.549515i	$0.185188\pi$
$373$	409.938i	1.09903i	−0.835484	+	0.549515i	$0.814812\pi$
$374$	0	0
$375$	0	0
$376$	− 553.096i	− 1.47100i
$377$	−92.0762	−0.244234
$378$	0	0
$379$	544.141	1.43573	0.717864	−	0.696183i	$-0.245120\pi$
$379$	544.141	1.43573	0.717864	+	0.696183i	$0.245120\pi$
$380$	− 53.9265i	− 0.141912i
$381$	0	0
$382$	− 920.010i	− 2.40840i
$383$	−122.612	−0.320137	−0.160068	−	0.987106i	$-0.551171\pi$
$383$	−122.612	−0.320137	−0.160068	+	0.987106i	$0.551171\pi$
$384$	0	0
$385$	0	0
$386$	340.371	0.881789
$387$	0	0
$388$	−798.746	−2.05862
$389$	−226.982	−0.583502	−0.291751	−	0.956494i	$-0.594238\pi$
$389$	−226.982	−0.583502	−0.291751	+	0.956494i	$0.594238\pi$
$390$	0	0
$391$	297.596i	0.761114i
$392$	2147.14i	5.47741i
$393$	0	0
$394$	−342.640	−0.869645
$395$	− 66.0252i	− 0.167152i
$396$	0	0
$397$	277.969	0.700174	0.350087	−	0.936717i	$-0.386152\pi$
$397$	277.969	0.700174	0.350087	+	0.936717i	$0.386152\pi$
$398$	− 883.974i	− 2.22104i
$399$	0	0
$400$	−1080.29	−2.70073
$401$	−277.583	−0.692226	−0.346113	−	0.938193i	$-0.612499\pi$
$401$	−277.583	−0.692226	−0.346113	+	0.938193i	$0.612499\pi$
$402$	0	0
$403$	− 289.876i	− 0.719295i
$404$	1962.29i	4.85715i
$405$	0	0
$406$	− 520.484i	− 1.28198i
$407$	0	0
$408$	0	0
$409$	177.805i	0.434732i	0.976090	+	0.217366i	$0.0697464\pi$
$409$	177.805i	0.434732i	−0.976090	+	0.217366i	$0.930254\pi$
$410$	118.782	0.289712
$411$	0	0
$412$	−824.864	−2.00210
$413$	− 60.7280i	− 0.147041i
$414$	0	0
$415$	70.2229i	0.169212i
$416$	−590.184	−1.41871
$417$	0	0
$418$	0	0
$419$	−132.913	−0.317215	−0.158608	−	0.987342i	$-0.550700\pi$
$419$	−132.913	−0.317215	−0.158608	+	0.987342i	$0.550700\pi$
$420$	0	0
$421$	−65.2568	−0.155004	−0.0775022	−	0.996992i	$-0.524694\pi$
$421$	−65.2568	−0.155004	−0.0775022	+	0.996992i	$0.524694\pi$
$422$	−1111.52	−2.63394
$423$	0	0
$424$	1175.76i	2.77302i
$425$	269.051i	0.633061i
$426$	0	0
$427$	−99.4075	−0.232804
$428$	1824.31i	4.26241i
$429$	0	0
$430$	−39.3991	−0.0916259
$431$	544.138i	1.26250i	0.775579	+	0.631251i	$0.217458\pi$
$431$	544.138i	1.26250i	−0.775579	+	0.631251i	$0.782542\pi$
$432$	0	0
$433$	257.530	0.594758	0.297379	−	0.954760i	$-0.403888\pi$
$433$	257.530	0.594758	0.297379	+	0.954760i	$0.403888\pi$
$434$	1638.60	3.77557
$435$	0	0
$436$	171.436i	0.393202i
$437$	− 265.029i	− 0.606473i
$438$	0	0
$439$	764.391i	1.74121i	0.491983	+	0.870605i	$0.336272\pi$
$439$	764.391i	1.74121i	−0.491983	+	0.870605i	$0.663728\pi$
$440$	0	0
$441$	0	0
$442$	324.559i	0.734296i
$443$	131.890	0.297721	0.148860	−	0.988858i	$-0.452439\pi$
$443$	131.890	0.297721	0.148860	+	0.988858i	$0.452439\pi$
$444$	0	0
$445$	−21.4027	−0.0480961
$446$	− 1416.76i	− 3.17660i
$447$	0	0
$448$	− 1229.14i	− 2.74361i
$449$	−481.543	−1.07248	−0.536239	−	0.844066i	$-0.680155\pi$
$449$	−481.543	−1.07248	−0.536239	+	0.844066i	$0.680155\pi$
$450$	0	0
$451$	0	0
$452$	−255.777	−0.565879
$453$	0	0
$454$	−1639.39	−3.61099
$455$	−53.3643	−0.117284
$456$	0	0
$457$	317.701i	0.695189i	0.937645	+	0.347594i	$0.113001\pi$
$457$	317.701i	0.695189i	−0.937645	+	0.347594i	$0.886999\pi$
$458$	− 657.557i	− 1.43571i
$459$	0	0
$460$	151.756	0.329904
$461$	− 284.463i	− 0.617056i	−0.951215	−	0.308528i	$-0.900164\pi$
$461$	− 284.463i	− 0.617056i	0.951215	−	0.308528i	$-0.0998364\pi$
$462$	0	0
$463$	254.965	0.550681	0.275341	−	0.961347i	$-0.411209\pi$
$463$	254.965	0.550681	0.275341	+	0.961347i	$0.411209\pi$
$464$	505.829i	1.09015i
$465$	0	0
$466$	−281.556	−0.604197
$467$	38.0838	0.0815499	0.0407749	−	0.999168i	$-0.487017\pi$
$467$	38.0838	0.0815499	0.0407749	+	0.999168i	$0.487017\pi$
$468$	0	0
$469$	678.417i	1.44652i
$470$	51.4354i	0.109437i
$471$	0	0
$472$	112.890i	0.239173i
$473$	0	0
$474$	0	0
$475$	− 239.608i	− 0.504437i
$476$	−1309.89	−2.75186
$477$	0	0
$478$	547.557	1.14552
$479$	303.397i	0.633396i	0.948526	+	0.316698i	$0.102574\pi$
$479$	303.397i	0.633396i	−0.948526	+	0.316698i	$0.897426\pi$
$480$	0	0
$481$	261.414i	0.543479i
$482$	862.629	1.78969
$483$	0	0
$484$	0	0
$485$	44.5220	0.0917980
$486$	0	0
$487$	−226.899	−0.465912	−0.232956	−	0.972487i	$-0.574840\pi$
$487$	−226.899	−0.465912	−0.232956	+	0.972487i	$0.574840\pi$
$488$	184.793	0.378673
$489$	0	0
$490$	− 199.675i	− 0.407499i
$491$	111.952i	0.228008i	0.993480	+	0.114004i	$0.0363676\pi$
$491$	111.952i	0.228008i	−0.993480	+	0.114004i	$0.963632\pi$
$492$	0	0
$493$	125.978	0.255534
$494$	− 289.041i	− 0.585104i
$495$	0	0
$496$	−1592.46	−3.21061
$497$	1293.95i	2.60352i
$498$	0	0
$499$	−502.425	−1.00686	−0.503432	−	0.864035i	$-0.667930\pi$
$499$	−502.425	−1.00686	−0.503432	+	0.864035i	$0.667930\pi$
$500$	276.120	0.552241
$501$	0	0
$502$	1263.56i	2.51704i
$503$	− 213.204i	− 0.423865i	−0.977284	−	0.211932i	$-0.932024\pi$
$503$	− 213.204i	− 0.423865i	0.977284	−	0.211932i	$-0.0679757\pi$
$504$	0	0
$505$	− 109.378i	− 0.216589i
$506$	0	0
$507$	0	0
$508$	− 1321.03i	− 2.60045i
$509$	−474.102	−0.931437	−0.465719	−	0.884933i	$-0.654204\pi$
$509$	−474.102	−0.931437	−0.465719	+	0.884933i	$0.654204\pi$
$510$	0	0
$511$	−455.948	−0.892266
$512$	− 674.609i	− 1.31760i
$513$	0	0
$514$	797.667i	1.55188i
$515$	45.9778	0.0892773
$516$	0	0
$517$	0	0
$518$	−1477.71	−2.85272
$519$	0	0
$520$	99.2010	0.190771
$521$	−509.660	−0.978233	−0.489117	−	0.872218i	$-0.662681\pi$
$521$	−509.660	−0.978233	−0.489117	+	0.872218i	$0.662681\pi$
$522$	0	0
$523$	− 483.196i	− 0.923892i	−0.886908	−	0.461946i	$-0.847151\pi$
$523$	− 483.196i	− 0.923892i	0.886908	−	0.461946i	$-0.152849\pi$
$524$	− 619.315i	− 1.18190i
$525$	0	0
$526$	260.799	0.495816
$527$	396.608i	0.752577i
$528$	0	0
$529$	216.823	0.409874
$530$	− 109.340i	− 0.206303i
$531$	0	0
$532$	1166.54	2.19275
$533$	454.557	0.852828
$534$	0	0
$535$	− 101.687i	− 0.190069i
$536$	− 1261.14i	− 2.35286i
$537$	0	0
$538$	− 1356.60i	− 2.52157i
$539$	0	0
$540$	0	0
$541$	929.346i	1.71783i	0.512118	+	0.858915i	$0.328861\pi$
$541$	929.346i	1.71783i	−0.512118	+	0.858915i	$0.671139\pi$
$542$	1151.07	2.12374
$543$	0	0
$544$	807.489	1.48436
$545$	− 9.55583i	− 0.0175336i
$546$	0	0
$547$	330.141i	0.603549i	0.953379	+	0.301774i	$0.0975790\pi$
$547$	330.141i	0.603549i	−0.953379	+	0.301774i	$0.902421\pi$
$548$	1712.88	3.12569
$549$	0	0
$550$	0	0
$551$	−112.192	−0.203616
$552$	0	0
$553$	1428.26	2.58275
$554$	1215.17	2.19345
$555$	0	0
$556$	− 256.345i	− 0.461053i
$557$	309.905i	0.556382i	0.960526	+	0.278191i	$0.0897349\pi$
$557$	309.905i	0.556382i	−0.960526	+	0.278191i	$0.910265\pi$
$558$	0	0
$559$	−150.773	−0.269720
$560$	293.162i	0.523503i
$561$	0	0
$562$	−1252.77	−2.22912
$563$	− 84.0902i	− 0.149361i	−0.997208	−	0.0746805i	$-0.976206\pi$
$563$	− 84.0902i	− 0.149361i	0.997208	−	0.0746805i	$-0.0237937\pi$
$564$	0	0
$565$	14.2570	0.0252336
$566$	−471.045	−0.832235
$567$	0	0
$568$	− 2405.37i	− 4.23481i
$569$	536.624i	0.943100i	0.881839	+	0.471550i	$0.156305\pi$
$569$	536.624i	0.943100i	−0.881839	+	0.471550i	$0.843695\pi$
$570$	0	0
$571$	831.674i	1.45652i	0.685300	+	0.728261i	$0.259671\pi$
$571$	831.674i	1.45652i	−0.685300	+	0.728261i	$0.740329\pi$
$572$	0	0
$573$	0	0
$574$	2569.50i	4.47648i
$575$	674.285	1.17267
$576$	0	0
$577$	−893.097	−1.54783	−0.773914	−	0.633291i	$-0.781704\pi$
$577$	−893.097	−1.54783	−0.773914	+	0.633291i	$0.781704\pi$
$578$	636.683i	1.10153i
$579$	0	0
$580$	− 64.2414i	− 0.110761i
$581$	−1519.07	−2.61457
$582$	0	0
$583$	0	0
$584$	847.580	1.45133
$585$	0	0
$586$	1208.37	2.06206
$587$	−254.456	−0.433485	−0.216743	−	0.976229i	$-0.569543\pi$
$587$	−254.456	−0.433485	−0.216743	+	0.976229i	$0.569543\pi$
$588$	0	0
$589$	− 353.206i	− 0.599671i
$590$	− 10.4982i	− 0.0177936i
$591$	0	0
$592$	1436.10	2.42584
$593$	− 575.572i	− 0.970610i	−0.874345	−	0.485305i	$-0.838709\pi$
$593$	− 575.572i	− 0.970610i	0.874345	−	0.485305i	$-0.161291\pi$
$594$	0	0
$595$	73.0130	0.122711
$596$	− 616.556i	− 1.03449i
$597$	0	0
$598$	813.397	1.36020
$599$	−572.742	−0.956163	−0.478081	−	0.878316i	$-0.658668\pi$
$599$	−572.742	−0.956163	−0.478081	+	0.878316i	$0.658668\pi$
$600$	0	0
$601$	− 84.4365i	− 0.140493i	−0.997530	−	0.0702467i	$-0.977621\pi$
$601$	− 84.4365i	− 0.140493i	0.997530	−	0.0702467i	$-0.0223786\pi$
$602$	− 852.285i	− 1.41576i
$603$	0	0
$604$	1355.32i	2.24391i
$605$	0	0
$606$	0	0
$607$	75.7837i	0.124850i	0.998050	+	0.0624248i	$0.0198834\pi$
$607$	75.7837i	0.124850i	−0.998050	+	0.0624248i	$0.980117\pi$
$608$	−719.123	−1.18277
$609$	0	0
$610$	−17.1849	−0.0281719
$611$	196.834i	0.322150i
$612$	0	0
$613$	314.002i	0.512238i	0.966645	+	0.256119i	$0.0824439\pi$
$613$	314.002i	0.512238i	−0.966645	+	0.256119i	$0.917556\pi$
$614$	−515.603	−0.839744
$615$	0	0
$616$	0	0
$617$	−474.235	−0.768614	−0.384307	−	0.923205i	$-0.625560\pi$
$617$	−474.235	−0.768614	−0.384307	+	0.923205i	$0.625560\pi$
$618$	0	0
$619$	−142.570	−0.230323	−0.115161	−	0.993347i	$-0.536738\pi$
$619$	−142.570	−0.230323	−0.115161	+	0.993347i	$0.536738\pi$
$620$	202.246	0.326203
$621$	0	0
$622$	726.174i	1.16748i
$623$	− 462.986i	− 0.743155i
$624$	0	0
$625$	601.866	0.962985
$626$	271.299i	0.433385i
$627$	0	0
$628$	943.026	1.50163
$629$	− 357.666i	− 0.568626i
$630$	0	0
$631$	−930.792	−1.47511	−0.737553	−	0.675289i	$-0.764019\pi$
$631$	−930.792	−1.47511	−0.737553	+	0.675289i	$0.764019\pi$
$632$	−2655.05	−4.20103
$633$	0	0
$634$	− 1994.38i	− 3.14571i
$635$	73.6341i	0.115959i
$636$	0	0
$637$	− 764.119i	− 1.19956i
$638$	0	0
$639$	0	0
$640$	− 47.5224i	− 0.0742537i
$641$	−901.410	−1.40626	−0.703128	−	0.711063i	$-0.748214\pi$
$641$	−901.410	−1.40626	−0.703128	+	0.711063i	$0.748214\pi$
$642$	0	0
$643$	855.033	1.32976	0.664878	−	0.746952i	$-0.268484\pi$
$643$	855.033	1.32976	0.664878	+	0.746952i	$0.268484\pi$
$644$	3282.79i	5.09750i
$645$	0	0
$646$	395.466i	0.612176i
$647$	753.022	1.16387	0.581934	−	0.813236i	$-0.302296\pi$
$647$	753.022	1.16387	0.581934	+	0.813236i	$0.302296\pi$
$648$	0	0
$649$	0	0
$650$	735.377	1.13135
$651$	0	0
$652$	425.866	0.653169
$653$	70.0016	0.107200	0.0536000	−	0.998562i	$-0.482930\pi$
$653$	70.0016	0.107200	0.0536000	+	0.998562i	$0.482930\pi$
$654$	0	0
$655$	34.5205i	0.0527031i
$656$	− 2497.15i	− 3.80663i
$657$	0	0
$658$	−1112.65	−1.69096
$659$	102.122i	0.154964i	0.996994	+	0.0774822i	$0.0246881\pi$
$659$	102.122i	0.154964i	−0.996994	+	0.0774822i	$0.975312\pi$
$660$	0	0
$661$	633.087	0.957772	0.478886	−	0.877877i	$-0.341041\pi$
$661$	633.087	0.957772	0.478886	+	0.877877i	$0.341041\pi$
$662$	458.772i	0.693008i
$663$	0	0
$664$	2823.85	4.25279
$665$	−65.0229	−0.0977788
$666$	0	0
$667$	− 315.723i	− 0.473347i
$668$	− 106.300i	− 0.159132i
$669$	0	0
$670$	117.280i	0.175045i
$671$	0	0
$672$	0	0
$673$	516.957i	0.768139i	0.923304	+	0.384069i	$0.125478\pi$
$673$	516.957i	0.768139i	−0.923304	+	0.384069i	$0.874522\pi$
$674$	−240.471	−0.356783
$675$	0	0
$676$	−1054.04	−1.55923
$677$	11.0364i	0.0163019i	0.999967	+	0.00815095i	$0.00259456\pi$
$677$	11.0364i	0.0163019i	−0.999967	+	0.00815095i	$0.997405\pi$
$678$	0	0
$679$	963.103i	1.41841i
$680$	−135.727	−0.199598
$681$	0	0
$682$	0	0
$683$	−104.720	−0.153323	−0.0766617	−	0.997057i	$-0.524426\pi$
$683$	−104.720	−0.153323	−0.0766617	+	0.997057i	$0.524426\pi$
$684$	0	0
$685$	−95.4755	−0.139380
$686$	2113.32	3.08064
$687$	0	0
$688$	828.287i	1.20391i
$689$	− 418.426i	− 0.607294i
$690$	0	0
$691$	−360.681	−0.521969	−0.260985	−	0.965343i	$-0.584047\pi$
$691$	−360.681	−0.521969	−0.260985	+	0.965343i	$0.584047\pi$
$692$	− 1612.72i	− 2.33052i
$693$	0	0
$694$	−370.475	−0.533825
$695$	14.2887i	0.0205592i
$696$	0	0
$697$	−621.925	−0.892288
$698$	1685.12	2.41421
$699$	0	0
$700$	2967.91i	4.23987i
$701$	736.170i	1.05017i	0.851050	+	0.525085i	$0.175967\pi$
$701$	736.170i	1.05017i	−0.851050	+	0.525085i	$0.824033\pi$
$702$	0	0
$703$	318.525i	0.453094i
$704$	0	0
$705$	0	0
$706$	− 1686.00i	− 2.38810i
$707$	2366.07	3.34663
$708$	0	0
$709$	645.278	0.910124	0.455062	−	0.890460i	$-0.349617\pi$
$709$	645.278	0.910124	0.455062	+	0.890460i	$0.349617\pi$
$710$	223.689i	0.315055i
$711$	0	0
$712$	860.662i	1.20880i
$713$	993.964	1.39406
$714$	0	0
$715$	0	0
$716$	3278.48	4.57889
$717$	0	0
$718$	532.705	0.741928
$719$	−565.301	−0.786233	−0.393116	−	0.919489i	$-0.628603\pi$
$719$	−565.301	−0.786233	−0.393116	+	0.919489i	$0.628603\pi$
$720$	0	0
$721$	994.595i	1.37947i
$722$	997.806i	1.38200i
$723$	0	0
$724$	2983.45	4.12079
$725$	− 285.439i	− 0.393709i
$726$	0	0
$727$	−559.139	−0.769105	−0.384552	−	0.923103i	$-0.625644\pi$
$727$	−559.139	−0.769105	−0.384552	+	0.923103i	$0.625644\pi$
$728$	2145.92i	2.94770i
$729$	0	0
$730$	−78.8210	−0.107974
$731$	206.288	0.282200
$732$	0	0
$733$	132.503i	0.180768i	0.995907	+	0.0903841i	$0.0288095\pi$
$733$	132.503i	0.180768i	−0.995907	+	0.0903841i	$0.971191\pi$
$734$	114.799i	0.156401i
$735$	0	0
$736$	− 2023.70i	− 2.74959i
$737$	0	0
$738$	0	0
$739$	422.782i	0.572100i	0.958215	+	0.286050i	$0.0923424\pi$
$739$	422.782i	0.572100i	−0.958215	+	0.286050i	$0.907658\pi$
$740$	−182.388	−0.246470
$741$	0	0
$742$	2365.26	3.18768
$743$	− 1096.23i	− 1.47541i	−0.675124	−	0.737704i	$-0.735910\pi$
$743$	− 1096.23i	− 1.47541i	0.675124	−	0.737704i	$-0.264090\pi$
$744$	0	0
$745$	34.3668i	0.0461299i
$746$	−1533.00	−2.05497
$747$	0	0
$748$	0	0
$749$	2199.70	2.93685
$750$	0	0
$751$	1232.07	1.64058	0.820288	−	0.571950i	$-0.193813\pi$
$751$	1232.07	1.64058	0.820288	+	0.571950i	$0.193813\pi$
$752$	1081.32	1.43793
$753$	0	0
$754$	− 344.328i	− 0.456668i
$755$	− 75.5456i	− 0.100060i
$756$	0	0
$757$	365.767	0.483180	0.241590	−	0.970378i	$-0.422331\pi$
$757$	365.767	0.483180	0.241590	+	0.970378i	$0.422331\pi$
$758$	2034.87i	2.68452i
$759$	0	0
$760$	120.874	0.159044
$761$	410.374i	0.539257i	0.962964	+	0.269628i	$0.0869008\pi$
$761$	410.374i	0.539257i	−0.962964	+	0.269628i	$0.913099\pi$
$762$	0	0
$763$	206.712	0.270920
$764$	2456.39	3.21518
$765$	0	0
$766$	− 458.521i	− 0.598592i
$767$	− 40.1748i	− 0.0523792i
$768$	0	0
$769$	− 506.197i	− 0.658254i	−0.944286	−	0.329127i	$-0.893246\pi$
$769$	− 506.197i	− 0.658254i	0.944286	−	0.329127i	$-0.106754\pi$
$770$	0	0
$771$	0	0
$772$	908.778i	1.17717i
$773$	−488.947	−0.632532	−0.316266	−	0.948671i	$-0.602429\pi$
$773$	−488.947	−0.632532	−0.316266	+	0.948671i	$0.602429\pi$
$774$	0	0
$775$	898.625	1.15952
$776$	− 1790.35i	− 2.30715i
$777$	0	0
$778$	− 848.823i	− 1.09103i
$779$	553.865	0.710995
$780$	0	0
$781$	0	0
$782$	−1112.89	−1.42313
$783$	0	0
$784$	−4197.75	−5.35428
$785$	−52.5642	−0.0669607
$786$	0	0
$787$	137.770i	0.175057i	0.996162	+	0.0875286i	$0.0278969\pi$
$787$	137.770i	0.175057i	−0.996162	+	0.0875286i	$0.972103\pi$
$788$	− 914.837i	− 1.16096i
$789$	0	0
$790$	246.908	0.312541
$791$	308.408i	0.389896i
$792$	0	0
$793$	−65.7634	−0.0829299
$794$	1039.49i	1.30918i
$795$	0	0
$796$	2360.18	2.96505
$797$	−539.765	−0.677246	−0.338623	−	0.940922i	$-0.609961\pi$
$797$	−539.765	−0.677246	−0.338623	+	0.940922i	$0.609961\pi$
$798$	0	0
$799$	− 269.308i	− 0.337056i
$800$	− 1829.59i	− 2.28699i
$801$	0	0
$802$	− 1038.05i	− 1.29432i
$803$	0	0
$804$	0	0
$805$	− 182.982i	− 0.227307i
$806$	1084.02	1.34494
$807$	0	0
$808$	−4398.37	−5.44353
$809$	1231.26i	1.52196i	0.648777	+	0.760979i	$0.275281\pi$
$809$	1231.26i	1.52196i	−0.648777	+	0.760979i	$0.724719\pi$
$810$	0	0
$811$	− 1085.55i	− 1.33853i	−0.743025	−	0.669264i	$-0.766609\pi$
$811$	− 1085.55i	− 1.33853i	0.743025	−	0.669264i	$-0.233391\pi$
$812$	1389.67	1.71142
$813$	0	0
$814$	0	0
$815$	−23.7377	−0.0291260
$816$	0	0
$817$	−183.713	−0.224863
$818$	−664.920	−0.812861
$819$	0	0
$820$	317.144i	0.386761i
$821$	457.770i	0.557576i	0.960353	+	0.278788i	$0.0899328\pi$
$821$	457.770i	0.557576i	−0.960353	+	0.278788i	$0.910067\pi$
$822$	0	0
$823$	−355.076	−0.431441	−0.215721	−	0.976455i	$-0.569210\pi$
$823$	−355.076	−0.431441	−0.215721	+	0.976455i	$0.569210\pi$
$824$	− 1848.89i	− 2.24380i
$825$	0	0
$826$	227.098	0.274938
$827$	− 1039.23i	− 1.25662i	−0.777963	−	0.628311i	$-0.783747\pi$
$827$	− 1039.23i	− 1.25662i	0.777963	−	0.628311i	$-0.216253\pi$
$828$	0	0
$829$	−995.808	−1.20122	−0.600608	−	0.799544i	$-0.705075\pi$
$829$	−995.808	−1.20122	−0.600608	+	0.799544i	$0.705075\pi$
$830$	−262.605	−0.316392
$831$	0	0
$832$	− 813.141i	− 0.977332i
$833$	1045.47i	1.25506i
$834$	0	0
$835$	5.92517i	0.00709601i
$836$	0	0
$837$	0	0
$838$	− 497.042i	− 0.593129i
$839$	−1018.40	−1.21382	−0.606912	−	0.794769i	$-0.707592\pi$
$839$	−1018.40	−1.21382	−0.606912	+	0.794769i	$0.707592\pi$
$840$	0	0
$841$	707.348	0.841080
$842$	− 244.034i	− 0.289827i
$843$	0	0
$844$	− 2967.73i	− 3.51626i
$845$	58.7520	0.0695290
$846$	0	0
$847$	0	0
$848$	−2298.66	−2.71068
$849$	0	0
$850$	−1006.14	−1.18370
$851$	−896.368	−1.05331
$852$	0	0
$853$	− 1416.32i	− 1.66040i	−0.557463	−	0.830202i	$-0.688225\pi$
$853$	− 1416.32i	− 1.66040i	0.557463	−	0.830202i	$-0.311775\pi$
$854$	− 371.744i	− 0.435298i
$855$	0	0
$856$	−4089.11	−4.77699
$857$	1262.20i	1.47281i	0.676542	+	0.736404i	$0.263478\pi$
$857$	1262.20i	1.47281i	−0.676542	+	0.736404i	$0.736522\pi$
$858$	0	0
$859$	−371.365	−0.432322	−0.216161	−	0.976358i	$-0.569354\pi$
$859$	−371.365	−0.432322	−0.216161	+	0.976358i	$0.569354\pi$
$860$	− 105.194i	− 0.122319i
$861$	0	0
$862$	−2034.86	−2.36062
$863$	96.9188	0.112304	0.0561522	−	0.998422i	$-0.482117\pi$
$863$	96.9188	0.112304	0.0561522	+	0.998422i	$0.482117\pi$
$864$	0	0
$865$	89.8930i	0.103923i
$866$	963.060i	1.11208i
$867$	0	0
$868$	4375.00i	5.04032i
$869$	0	0
$870$	0	0
$871$	448.809i	0.515280i
$872$	−384.266	−0.440672
$873$	0	0
$874$	991.102	1.13398
$875$	− 332.937i	− 0.380500i
$876$	0	0
$877$	− 840.092i	− 0.957915i	−0.877838	−	0.478958i	$-0.841015\pi$
$877$	− 840.092i	− 0.957915i	0.877838	−	0.478958i	$-0.158985\pi$
$878$	−2858.52	−3.25571
$879$	0	0
$880$	0	0
$881$	−1737.76	−1.97249	−0.986245	−	0.165291i	$-0.947144\pi$
$881$	−1737.76	−1.97249	−0.986245	+	0.165291i	$0.947144\pi$
$882$	0	0
$883$	450.198	0.509850	0.254925	−	0.966961i	$-0.417949\pi$
$883$	450.198	0.509850	0.254925	+	0.966961i	$0.417949\pi$
$884$	−866.560	−0.980272
$885$	0	0
$886$	493.217i	0.556678i
$887$	− 726.352i	− 0.818886i	−0.912336	−	0.409443i	$-0.865723\pi$
$887$	− 726.352i	− 0.818886i	0.912336	−	0.409443i	$-0.134277\pi$
$888$	0	0
$889$	−1592.86	−1.79174
$890$	− 80.0377i	− 0.0899300i
$891$	0	0
$892$	3782.71	4.24070
$893$	239.837i	0.268574i
$894$	0	0
$895$	−182.742	−0.204181
$896$	1028.01	1.14733
$897$	0	0
$898$	− 1800.78i	− 2.00532i
$899$	− 420.766i	− 0.468038i
$900$	0	0
$901$	572.489i	0.635393i
$902$	0	0
$903$	0	0
$904$	− 573.312i	− 0.634195i
$905$	−166.297	−0.183754
$906$	0	0
$907$	−1368.26	−1.50855	−0.754276	−	0.656558i	$-0.772012\pi$
$907$	−1368.26	−1.50855	−0.754276	+	0.656558i	$0.772012\pi$
$908$	− 4377.12i	− 4.82061i
$909$	0	0
$910$	− 199.561i	− 0.219298i
$911$	613.094	0.672990	0.336495	−	0.941685i	$-0.390758\pi$
$911$	613.094	0.672990	0.336495	+	0.941685i	$0.390758\pi$
$912$	0	0
$913$	0	0
$914$	−1188.07	−1.29986
$915$	0	0
$916$	1755.65	1.91665
$917$	−746.750	−0.814341
$918$	0	0
$919$	− 1137.66i	− 1.23794i	−0.785415	−	0.618969i	$-0.787551\pi$
$919$	− 1137.66i	− 1.23794i	0.785415	−	0.618969i	$-0.212449\pi$
$920$	340.153i	0.369732i
$921$	0	0
$922$	1063.78	1.15377
$923$	856.016i	0.927428i
$924$	0	0
$925$	−810.390	−0.876098
$926$	953.468i	1.02966i
$927$	0	0
$928$	−856.674	−0.923140
$929$	569.803	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$929$	569.803	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$930$	0	0
$931$	− 931.057i	− 1.00006i
$932$	− 751.744i	− 0.806592i
$933$	0	0
$934$	142.418i	0.152482i
$935$	0	0
$936$	0	0
$937$	323.287i	0.345023i	0.985007	+	0.172512i	$0.0551882\pi$
$937$	323.287i	0.345023i	−0.985007	+	0.172512i	$0.944812\pi$
$938$	−2537.01	−2.70470
$939$	0	0
$940$	−137.331	−0.146096
$941$	− 636.630i	− 0.676547i	−0.941048	−	0.338273i	$-0.890157\pi$
$941$	− 636.630i	− 0.676547i	0.941048	−	0.338273i	$-0.109843\pi$
$942$	0	0
$943$	1558.64i	1.65286i
$944$	−220.704	−0.233797
$945$	0	0
$946$	0	0
$947$	1740.33	1.83773	0.918866	−	0.394570i	$-0.129106\pi$
$947$	1740.33	1.83773	0.918866	+	0.394570i	$0.129106\pi$
$948$	0	0
$949$	−301.634	−0.317844
$950$	896.037	0.943197
$951$	0	0
$952$	− 2936.05i	− 3.08409i
$953$	1577.33i	1.65512i	0.561379	+	0.827559i	$0.310271\pi$
$953$	1577.33i	1.65512i	−0.561379	+	0.827559i	$0.689729\pi$
$954$	0	0
$955$	−136.919	−0.143371
$956$	1461.96i	1.52924i
$957$	0	0
$958$	−1134.58	−1.18432
$959$	− 2065.33i	− 2.15363i
$960$	0	0
$961$	363.664	0.378422
$962$	−977.582	−1.01620
$963$	0	0
$964$	2303.19i	2.38920i
$965$	− 50.6552i	− 0.0524924i
$966$	0	0
$967$	− 745.733i	− 0.771182i	−0.922670	−	0.385591i	$-0.873998\pi$
$967$	− 745.733i	− 0.771182i	0.922670	−	0.385591i	$-0.126002\pi$
$968$	0	0
$969$	0	0
$970$	166.494i	0.171644i
$971$	891.046	0.917658	0.458829	−	0.888525i	$-0.348269\pi$
$971$	891.046	0.917658	0.458829	+	0.888525i	$0.348269\pi$
$972$	0	0
$973$	−309.093	−0.317670
$974$	− 848.511i	− 0.871161i
$975$	0	0
$976$	361.277i	0.370161i
$977$	771.533	0.789696	0.394848	−	0.918746i	$-0.370797\pi$
$977$	771.533	0.789696	0.394848	+	0.918746i	$0.370797\pi$
$978$	0	0
$979$	0	0
$980$	533.124	0.544004
$981$	0	0
$982$	−418.655	−0.426329
$983$	1662.66	1.69142	0.845709	−	0.533645i	$-0.179178\pi$
$983$	1662.66	1.69142	0.845709	+	0.533645i	$0.179178\pi$
$984$	0	0
$985$	50.9929i	0.0517694i
$986$	471.109i	0.477798i
$987$	0	0
$988$	771.730	0.781103
$989$	− 516.991i	− 0.522741i
$990$	0	0
$991$	−1215.61	−1.22665	−0.613327	−	0.789829i	$-0.710169\pi$
$991$	−1215.61	−1.22665	−0.613327	+	0.789829i	$0.710169\pi$
$992$	− 2697.00i	− 2.71875i
$993$	0	0
$994$	−4838.85	−4.86806
$995$	−131.556	−0.132217
$996$	0	0
$997$	1275.53i	1.27936i	0.768640	+	0.639682i	$0.220934\pi$
$997$	1275.53i	1.27936i	−0.768640	+	0.639682i	$0.779066\pi$
$998$	− 1878.87i	− 1.88263i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.j.604.8		8
3.2	odd	2		363.3.c.d.241.1	✓	8
11.10	odd	2	inner	1089.3.c.j.604.1		8
33.2	even	10		363.3.g.h.40.8		32
33.5	odd	10		363.3.g.h.118.8		32
33.8	even	10		363.3.g.h.112.8		32
33.14	odd	10		363.3.g.h.112.1		32
33.17	even	10		363.3.g.h.118.1		32
33.20	odd	10		363.3.g.h.40.1		32
33.26	odd	10		363.3.g.h.94.8		32
33.29	even	10		363.3.g.h.94.1		32
33.32	even	2		363.3.c.d.241.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.3.c.d.241.1	✓	8	3.2	odd	2
363.3.c.d.241.8	yes	8	33.32	even	2
363.3.g.h.40.1		32	33.20	odd	10
363.3.g.h.40.8		32	33.2	even	10
363.3.g.h.94.1		32	33.29	even	10
363.3.g.h.94.8		32	33.26	odd	10
363.3.g.h.112.1		32	33.14	odd	10
363.3.g.h.112.8		32	33.8	even	10
363.3.g.h.118.1		32	33.17	even	10
363.3.g.h.118.8		32	33.5	odd	10
1089.3.c.j.604.1		8	11.10	odd	2	inner
1089.3.c.j.604.8		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.73960i q^{2} -9.98460 q^{4} +0.556540 q^{5} +12.0391i q^{7} -22.3800i q^{8} +O(q^{10})\)	\(q+3.73960i q^{2} -9.98460 q^{4} +0.556540 q^{5} +12.0391i q^{7} -22.3800i q^{8} +2.08124i q^{10} +7.96452i q^{13} -45.0215 q^{14} +43.7538 q^{16} -10.8970i q^{17} +9.70455i q^{19} -5.55683 q^{20} -27.3098 q^{23} -24.6903 q^{25} -29.7841 q^{26} -120.206i q^{28} +11.5608i q^{29} -36.3959 q^{31} +74.1017i q^{32} +40.7506 q^{34} +6.70025i q^{35} +32.8223 q^{37} -36.2911 q^{38} -12.4554i q^{40} -57.0728i q^{41} +18.9306i q^{43} -102.128i q^{46} +24.7138 q^{47} -95.9403 q^{49} -92.3317i q^{50} -79.5225i q^{52} -52.5362 q^{53} +269.435 q^{56} -43.2327 q^{58} -5.04423 q^{59} +8.25704i q^{61} -136.106i q^{62} -102.095 q^{64} +4.43258i q^{65} +56.3510 q^{67} +108.803i q^{68} -25.0563 q^{70} +107.479 q^{71} +37.8722i q^{73} +122.742i q^{74} -96.8960i q^{76} -118.635i q^{79} +24.3508 q^{80} +213.429 q^{82} +126.178i q^{83} -6.06464i q^{85} -70.7930 q^{86} -38.4568 q^{89} -95.8858 q^{91} +272.677 q^{92} +92.4199i q^{94} +5.40097i q^{95} +79.9978 q^{97} -358.778i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{4} - 16 q^{5}+O(q^{10}) \) 8 * q - 24 * q^4 - 16 * q^5	\( 8 q - 24 q^{4} - 16 q^{5} - 136 q^{14} + 72 q^{16} + 16 q^{20} - 88 q^{25} + 120 q^{26} - 128 q^{31} + 96 q^{34} + 80 q^{37} + 216 q^{38} + 32 q^{47} - 152 q^{49} - 80 q^{53} + 696 q^{56} - 176 q^{58} + 64 q^{59} + 8 q^{64} + 464 q^{67} + 304 q^{70} + 128 q^{71} - 80 q^{80} + 528 q^{82} - 24 q^{86} + 720 q^{89} + 80 q^{91} + 1248 q^{92} + 416 q^{97}+O(q^{100}) \) 8 * q - 24 * q^4 - 16 * q^5 - 136 * q^14 + 72 * q^16 + 16 * q^20 - 88 * q^25 + 120 * q^26 - 128 * q^31 + 96 * q^34 + 80 * q^37 + 216 * q^38 + 32 * q^47 - 152 * q^49 - 80 * q^53 + 696 * q^56 - 176 * q^58 + 64 * q^59 + 8 * q^64 + 464 * q^67 + 304 * q^70 + 128 * q^71 - 80 * q^80 + 528 * q^82 - 24 * q^86 + 720 * q^89 + 80 * q^91 + 1248 * q^92 + 416 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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