LMFDB - Embedded newform 882.3.c.g.685.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,3,Mod(685,882)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(882, 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("882.685");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	882.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:	$24.0327593166$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.339738624.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 $ x^8 - 4x^6 + 14x^4 - 8*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 294)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.6
Root			$0.662827 - 0.382683i$ of defining polynomial
Character	$\chi$	$=$	882.685
Dual form			882.3.c.g.685.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} -1.52049i q^{5} +2.82843 q^{8} +O(q^{10})$	$q+1.41421 q^{2} +2.00000 q^{4} -1.52049i q^{5} +2.82843 q^{8} -2.15029i q^{10} -13.7227 q^{11} +17.9544i q^{13} +4.00000 q^{16} +25.8234i q^{17} +19.0113i q^{19} -3.04097i q^{20} -19.4069 q^{22} -19.3410 q^{23} +22.6881 q^{25} +25.3913i q^{26} +10.5199 q^{29} -15.0866i q^{31} +5.65685 q^{32} +36.5198i q^{34} +58.5711 q^{37} +26.8861i q^{38} -4.30059i q^{40} +54.0746i q^{41} +73.8818 q^{43} -27.4455 q^{44} -27.3523 q^{46} +1.08839i q^{47} +32.0858 q^{50} +35.9087i q^{52} -77.0637 q^{53} +20.8652i q^{55} +14.8774 q^{58} -68.4969i q^{59} +71.1518i q^{61} -21.3357i q^{62} +8.00000 q^{64} +27.2994 q^{65} -118.319 q^{67} +51.6468i q^{68} +11.0021 q^{71} -34.2015i q^{73} +82.8321 q^{74} +38.0227i q^{76} -61.6661 q^{79} -6.08195i q^{80} +76.4731i q^{82} +97.0029i q^{83} +39.2641 q^{85} +104.485 q^{86} -38.8138 q^{88} +0.0239519i q^{89} -38.6820 q^{92} +1.53922i q^{94} +28.9065 q^{95} +20.1381i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4}+O(q^{10}) $ 8 * q + 16 * q^4	$ 8 q + 16 q^{4} + 32 q^{16} + 32 q^{22} - 32 q^{23} - 72 q^{25} + 160 q^{29} + 128 q^{37} + 320 q^{43} - 96 q^{46} - 16 q^{50} - 384 q^{53} + 80 q^{58} + 64 q^{64} + 288 q^{65} - 352 q^{67} + 256 q^{71} - 48 q^{74} + 288 q^{79} + 768 q^{85} - 256 q^{86} + 64 q^{88} - 64 q^{92} + 32 q^{95}+O(q^{100}) $ 8 * q + 16 * q^4 + 32 * q^16 + 32 * q^22 - 32 * q^23 - 72 * q^25 + 160 * q^29 + 128 * q^37 + 320 * q^43 - 96 * q^46 - 16 * q^50 - 384 * q^53 + 80 * q^58 + 64 * q^64 + 288 * q^65 - 352 * q^67 + 256 * q^71 - 48 * q^74 + 288 * q^79 + 768 * q^85 - 256 * q^86 + 64 * q^88 - 64 * q^92 + 32 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/882\mathbb{Z}\right)^\times$.

$n$	$199$	$785$
$\chi(n)$	$-1$	$1$

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 $

$2$	1.41421	0.707107
$2$	1.41421	0.707107
$3$	0	0
$3$	0	0
$4$	2.00000	0.500000
$5$	− 1.52049i	− 0.304097i	−0.988373	−	0.152049i	$-0.951413\pi$
$5$	− 1.52049i	− 0.304097i	0.988373	−	0.152049i	$-0.0485870\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.82843	0.353553
$9$	0	0
$10$	− 2.15029i	− 0.215029i
$11$	−13.7227	−1.24752	−0.623761	−	0.781615i	$-0.714396\pi$
$11$	−13.7227	−1.24752	−0.623761	+	0.781615i	$0.714396\pi$
$12$	0	0
$13$	17.9544i	1.38110i	0.723282	+	0.690552i	$0.242633\pi$
$13$	17.9544i	1.38110i	−0.723282	+	0.690552i	$0.757367\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	0.250000
$17$	25.8234i	1.51902i	0.650494	+	0.759512i	$0.274562\pi$
$17$	25.8234i	1.51902i	−0.650494	+	0.759512i	$0.725438\pi$
$18$	0	0
$19$	19.0113i	1.00060i	0.865853	+	0.500299i	$0.166776\pi$
$19$	19.0113i	1.00060i	−0.865853	+	0.500299i	$0.833224\pi$
$20$	− 3.04097i	− 0.152049i
$21$	0	0
$22$	−19.4069	−0.882131
$23$	−19.3410	−0.840913	−0.420457	−	0.907313i	$-0.638130\pi$
$23$	−19.3410	−0.840913	−0.420457	+	0.907313i	$0.638130\pi$
$24$	0	0
$25$	22.6881	0.907525
$26$	25.3913i	0.976589i
$27$	0	0
$28$	0	0
$29$	10.5199	0.362755	0.181378	−	0.983414i	$-0.441944\pi$
$29$	10.5199	0.362755	0.181378	+	0.983414i	$0.441944\pi$
$30$	0	0
$31$	− 15.0866i	− 0.486666i	−0.969943	−	0.243333i	$-0.921759\pi$
$31$	− 15.0866i	− 0.486666i	0.969943	−	0.243333i	$-0.0782408\pi$
$32$	5.65685	0.176777
$33$	0	0
$34$	36.5198i	1.07411i
$35$	0	0
$36$	0	0
$37$	58.5711	1.58300	0.791502	−	0.611167i	$-0.209300\pi$
$37$	58.5711	1.58300	0.791502	+	0.611167i	$0.209300\pi$
$38$	26.8861i	0.707529i
$39$	0	0
$40$	− 4.30059i	− 0.107515i
$41$	54.0746i	1.31889i	0.751751	+	0.659447i	$0.229209\pi$
$41$	54.0746i	1.31889i	−0.751751	+	0.659447i	$0.770791\pi$
$42$	0	0
$43$	73.8818	1.71818	0.859091	−	0.511823i	$-0.171030\pi$
$43$	73.8818	1.71818	0.859091	+	0.511823i	$0.171030\pi$
$44$	−27.4455	−0.623761
$45$	0	0
$46$	−27.3523	−0.594615
$47$	1.08839i	0.0231573i	0.999933	+	0.0115787i	$0.00368569\pi$
$47$	1.08839i	0.0231573i	−0.999933	+	0.0115787i	$0.996314\pi$
$48$	0	0
$49$	0	0
$50$	32.0858	0.641717
$51$	0	0
$52$	35.9087i	0.690552i
$53$	−77.0637	−1.45403	−0.727016	−	0.686620i	$-0.759094\pi$
$53$	−77.0637	−1.45403	−0.727016	+	0.686620i	$0.759094\pi$
$54$	0	0
$55$	20.8652i	0.379368i
$56$	0	0
$57$	0	0
$58$	14.8774	0.256507
$59$	− 68.4969i	− 1.16096i	−0.814273	−	0.580482i	$-0.802864\pi$
$59$	− 68.4969i	− 1.16096i	0.814273	−	0.580482i	$-0.197136\pi$
$60$	0	0
$61$	71.1518i	1.16642i	0.812320	+	0.583212i	$0.198204\pi$
$61$	71.1518i	1.16642i	−0.812320	+	0.583212i	$0.801796\pi$
$62$	− 21.3357i	− 0.344125i
$63$	0	0
$64$	8.00000	0.125000
$65$	27.2994	0.419990
$66$	0	0
$67$	−118.319	−1.76595	−0.882974	−	0.469421i	$-0.844463\pi$
$67$	−118.319	−1.76595	−0.882974	+	0.469421i	$0.844463\pi$
$68$	51.6468i	0.759512i
$69$	0	0
$70$	0	0
$71$	11.0021	0.154960	0.0774799	−	0.996994i	$-0.475313\pi$
$71$	11.0021	0.154960	0.0774799	+	0.996994i	$0.475313\pi$
$72$	0	0
$73$	− 34.2015i	− 0.468513i	−0.972175	−	0.234257i	$-0.924734\pi$
$73$	− 34.2015i	− 0.468513i	0.972175	−	0.234257i	$-0.0752656\pi$
$74$	82.8321	1.11935
$75$	0	0
$76$	38.0227i	0.500299i
$77$	0	0
$78$	0	0
$79$	−61.6661	−0.780584	−0.390292	−	0.920691i	$-0.627626\pi$
$79$	−61.6661	−0.780584	−0.390292	+	0.920691i	$0.627626\pi$
$80$	− 6.08195i	− 0.0760243i
$81$	0	0
$82$	76.4731i	0.932598i
$83$	97.0029i	1.16871i	0.811498	+	0.584355i	$0.198652\pi$
$83$	97.0029i	1.16871i	−0.811498	+	0.584355i	$0.801348\pi$
$84$	0	0
$85$	39.2641	0.461931
$86$	104.485	1.21494
$87$	0	0
$88$	−38.8138	−0.441066
$89$	0.0239519i	0 0.000269123i	1.00000		0.000134561i	$4.28322e-5\pi$
$89$	0.0239519i	0 0.000269123i	−1.00000		0.000134561i	$0.999957\pi$
$90$	0	0
$91$	0	0
$92$	−38.6820	−0.420457
$93$	0	0
$94$	1.53922i	0.0163747i
$95$	28.9065	0.304279
$96$	0	0
$97$	20.1381i	0.207609i	0.994598	+	0.103805i	$0.0331017\pi$
$97$	20.1381i	0.207609i	−0.994598	+	0.103805i	$0.966898\pi$
$98$	0	0
$99$	0	0
$100$	45.3762	0.453762
$101$	− 44.4035i	− 0.439638i	−0.975541	−	0.219819i	$-0.929453\pi$
$101$	− 44.4035i	− 0.439638i	0.975541	−	0.219819i	$-0.0705467\pi$
$102$	0	0
$103$	122.439i	1.18872i	0.804197	+	0.594362i	$0.202596\pi$
$103$	122.439i	1.18872i	−0.804197	+	0.594362i	$0.797404\pi$
$104$	50.7826i	0.488294i
$105$	0	0
$106$	−108.985	−1.02816
$107$	144.668	1.35204	0.676020	−	0.736883i	$-0.263703\pi$
$107$	144.668	1.35204	0.676020	+	0.736883i	$0.263703\pi$
$108$	0	0
$109$	−151.932	−1.39387	−0.696935	−	0.717134i	$-0.745453\pi$
$109$	−151.932	−1.39387	−0.696935	+	0.717134i	$0.745453\pi$
$110$	29.5079i	0.268254i
$111$	0	0
$112$	0	0
$113$	80.0819	0.708690	0.354345	−	0.935115i	$-0.384704\pi$
$113$	80.0819	0.708690	0.354345	+	0.935115i	$0.384704\pi$
$114$	0	0
$115$	29.4077i	0.255719i
$116$	21.0398	0.181378
$117$	0	0
$118$	− 96.8692i	− 0.820925i
$119$	0	0
$120$	0	0
$121$	67.3136	0.556311
$122$	100.624i	0.824786i
$123$	0	0
$124$	− 30.1733i	− 0.243333i
$125$	− 72.5092i	− 0.580073i
$126$	0	0
$127$	2.74320	0.0216000	0.0108000	−	0.999942i	$-0.496562\pi$
$127$	2.74320	0.0216000	0.0108000	+	0.999942i	$0.496562\pi$
$128$	11.3137	0.0883883
$129$	0	0
$130$	38.6071	0.296978
$131$	− 148.292i	− 1.13200i	−0.824406	−	0.565999i	$-0.808491\pi$
$131$	− 148.292i	− 1.13200i	0.824406	−	0.565999i	$-0.191509\pi$
$132$	0	0
$133$	0	0
$134$	−167.328	−1.24871
$135$	0	0
$136$	73.0396i	0.537056i
$137$	−136.578	−0.996916	−0.498458	−	0.866914i	$-0.666100\pi$
$137$	−136.578	−0.996916	−0.498458	+	0.866914i	$0.666100\pi$
$138$	0	0
$139$	− 42.8524i	− 0.308291i	−0.988048	−	0.154145i	$-0.950738\pi$
$139$	− 42.8524i	− 0.308291i	0.988048	−	0.154145i	$-0.0492625\pi$
$140$	0	0
$141$	0	0
$142$	15.5594	0.109573
$143$	− 246.383i	− 1.72296i
$144$	0	0
$145$	− 15.9954i	− 0.110313i
$146$	− 48.3682i	− 0.331289i
$147$	0	0
$148$	117.142	0.791502
$149$	−140.787	−0.944878	−0.472439	−	0.881363i	$-0.656626\pi$
$149$	−140.787	−0.944878	−0.472439	+	0.881363i	$0.656626\pi$
$150$	0	0
$151$	−2.48894	−0.0164830	−0.00824152	−	0.999966i	$-0.502623\pi$
$151$	−2.48894	−0.0164830	−0.00824152	+	0.999966i	$0.502623\pi$
$152$	53.7722i	0.353764i
$153$	0	0
$154$	0	0
$155$	−22.9390	−0.147994
$156$	0	0
$157$	− 151.040i	− 0.962039i	−0.876710	−	0.481020i	$-0.840267\pi$
$157$	− 151.040i	− 0.962039i	0.876710	−	0.481020i	$-0.159733\pi$
$158$	−87.2091	−0.551956
$159$	0	0
$160$	− 8.60117i	− 0.0537573i
$161$	0	0
$162$	0	0
$163$	95.8728	0.588176	0.294088	−	0.955778i	$-0.404984\pi$
$163$	95.8728	0.588176	0.294088	+	0.955778i	$0.404984\pi$
$164$	108.149i	0.659447i
$165$	0	0
$166$	137.183i	0.826403i
$167$	− 161.441i	− 0.966715i	−0.875423	−	0.483357i	$-0.839417\pi$
$167$	− 161.441i	− 0.966715i	0.875423	−	0.483357i	$-0.160583\pi$
$168$	0	0
$169$	−153.359	−0.907450
$170$	55.5279	0.326634
$171$	0	0
$172$	147.764	0.859091
$173$	267.848i	1.54825i	0.633031	+	0.774126i	$0.281810\pi$
$173$	267.848i	1.54825i	−0.633031	+	0.774126i	$0.718190\pi$
$174$	0	0
$175$	0	0
$176$	−54.8910	−0.311880
$177$	0	0
$178$	0.0338732i	0 0.000190299i
$179$	131.937	0.737076	0.368538	−	0.929613i	$-0.379858\pi$
$179$	131.937	0.737076	0.368538	+	0.929613i	$0.379858\pi$
$180$	0	0
$181$	297.949i	1.64613i	0.567948	+	0.823064i	$0.307737\pi$
$181$	297.949i	1.64613i	−0.567948	+	0.823064i	$0.692263\pi$
$182$	0	0
$183$	0	0
$184$	−54.7046	−0.297308
$185$	− 89.0566i	− 0.481387i
$186$	0	0
$187$	− 354.368i	− 1.89501i
$188$	2.17679i	0.0115787i
$189$	0	0
$190$	40.8800	0.215158
$191$	−56.5872	−0.296268	−0.148134	−	0.988967i	$-0.547327\pi$
$191$	−56.5872	−0.296268	−0.148134	+	0.988967i	$0.547327\pi$
$192$	0	0
$193$	203.869	1.05632	0.528158	−	0.849146i	$-0.322883\pi$
$193$	203.869	1.05632	0.528158	+	0.849146i	$0.322883\pi$
$194$	28.4796i	0.146802i
$195$	0	0
$196$	0	0
$197$	51.6961	0.262417	0.131208	−	0.991355i	$-0.458114\pi$
$197$	51.6961	0.262417	0.131208	+	0.991355i	$0.458114\pi$
$198$	0	0
$199$	78.7228i	0.395592i	0.980243	+	0.197796i	$0.0633784\pi$
$199$	78.7228i	0.395592i	−0.980243	+	0.197796i	$0.936622\pi$
$200$	64.1717	0.320858
$201$	0	0
$202$	− 62.7960i	− 0.310871i
$203$	0	0
$204$	0	0
$205$	82.2197	0.401072
$206$	173.154i	0.840555i
$207$	0	0
$208$	71.8174i	0.345276i
$209$	− 260.888i	− 1.24827i
$210$	0	0
$211$	−138.198	−0.654967	−0.327484	−	0.944857i	$-0.606201\pi$
$211$	−138.198	−0.654967	−0.327484	+	0.944857i	$0.606201\pi$
$212$	−154.127	−0.727016
$213$	0	0
$214$	204.592	0.956037
$215$	− 112.336i	− 0.522494i
$216$	0	0
$217$	0	0
$218$	−214.864	−0.985615
$219$	0	0
$220$	41.7305i	0.189684i
$221$	−463.643	−2.09793
$222$	0	0
$223$	− 395.864i	− 1.77518i	−0.460638	−	0.887588i	$-0.652379\pi$
$223$	− 395.864i	− 1.77518i	0.460638	−	0.887588i	$-0.347621\pi$
$224$	0	0
$225$	0	0
$226$	113.253	0.501119
$227$	288.507i	1.27095i	0.772120	+	0.635477i	$0.219197\pi$
$227$	288.507i	1.27095i	−0.772120	+	0.635477i	$0.780803\pi$
$228$	0	0
$229$	− 326.001i	− 1.42359i	−0.702390	−	0.711793i	$-0.747884\pi$
$229$	− 326.001i	− 1.42359i	0.702390	−	0.711793i	$-0.252116\pi$
$230$	41.5888i	0.180821i
$231$	0	0
$232$	29.7548	0.128253
$233$	−98.0020	−0.420609	−0.210305	−	0.977636i	$-0.567446\pi$
$233$	−98.0020	−0.420609	−0.210305	+	0.977636i	$0.567446\pi$
$234$	0	0
$235$	1.65489	0.00704208
$236$	− 136.994i	− 0.580482i
$237$	0	0
$238$	0	0
$239$	453.258	1.89648	0.948239	−	0.317558i	$-0.102863\pi$
$239$	453.258	1.89648	0.948239	+	0.317558i	$0.102863\pi$
$240$	0	0
$241$	− 315.057i	− 1.30729i	−0.756801	−	0.653646i	$-0.773239\pi$
$241$	− 315.057i	− 1.30729i	0.756801	−	0.653646i	$-0.226761\pi$
$242$	95.1958	0.393371
$243$	0	0
$244$	142.304i	0.583212i
$245$	0	0
$246$	0	0
$247$	−341.337	−1.38193
$248$	− 42.6715i	− 0.172062i
$249$	0	0
$250$	− 102.543i	− 0.410174i
$251$	18.5549i	0.0739238i	0.999317	+	0.0369619i	$0.0117680\pi$
$251$	18.5549i	0.0739238i	−0.999317	+	0.0369619i	$0.988232\pi$
$252$	0	0
$253$	265.411	1.04906
$254$	3.87947	0.0152735
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 294.003i	− 1.14398i	−0.820260	−	0.571991i	$-0.806171\pi$
$257$	− 294.003i	− 1.14398i	0.820260	−	0.571991i	$-0.193829\pi$
$258$	0	0
$259$	0	0
$260$	54.5987	0.209995
$261$	0	0
$262$	− 209.716i	− 0.800443i
$263$	407.574	1.54971	0.774856	−	0.632138i	$-0.217822\pi$
$263$	407.574	1.54971	0.774856	+	0.632138i	$0.217822\pi$
$264$	0	0
$265$	117.174i	0.442168i
$266$	0	0
$267$	0	0
$268$	−236.637	−0.882974
$269$	9.73145i	0.0361764i	0.999836	+	0.0180882i	$0.00575796\pi$
$269$	9.73145i	0.0361764i	−0.999836	+	0.0180882i	$0.994242\pi$
$270$	0	0
$271$	4.44248i	0.0163929i	0.999966	+	0.00819645i	$0.00260904\pi$
$271$	4.44248i	0.0163929i	−0.999966	+	0.00819645i	$0.997391\pi$
$272$	103.294i	0.379756i
$273$	0	0
$274$	−193.150	−0.704926
$275$	−311.343	−1.13216
$276$	0	0
$277$	284.806	1.02818	0.514090	−	0.857736i	$-0.328130\pi$
$277$	284.806	1.02818	0.514090	+	0.857736i	$0.328130\pi$
$278$	− 60.6025i	− 0.217995i
$279$	0	0
$280$	0	0
$281$	80.8445	0.287703	0.143851	−	0.989599i	$-0.454051\pi$
$281$	80.8445	0.287703	0.143851	+	0.989599i	$0.454051\pi$
$282$	0	0
$283$	161.678i	0.571300i	0.958334	+	0.285650i	$0.0922095\pi$
$283$	161.678i	0.571300i	−0.958334	+	0.285650i	$0.907791\pi$
$284$	22.0043	0.0774799
$285$	0	0
$286$	− 348.438i	− 1.21832i
$287$	0	0
$288$	0	0
$289$	−377.848	−1.30743
$290$	− 22.6209i	− 0.0780030i
$291$	0	0
$292$	− 68.4029i	− 0.234257i
$293$	376.872i	1.28625i	0.765760	+	0.643126i	$0.222363\pi$
$293$	376.872i	1.28625i	−0.765760	+	0.643126i	$0.777637\pi$
$294$	0	0
$295$	−104.149	−0.353046
$296$	165.664	0.559676
$297$	0	0
$298$	−199.103	−0.668129
$299$	− 347.255i	− 1.16139i
$300$	0	0
$301$	0	0
$302$	−3.51989	−0.0116553
$303$	0	0
$304$	76.0454i	0.250149i
$305$	108.185	0.354706
$306$	0	0
$307$	− 82.0663i	− 0.267317i	−0.991027	−	0.133658i	$-0.957327\pi$
$307$	− 82.0663i	− 0.267317i	0.991027	−	0.133658i	$-0.0426725\pi$
$308$	0	0
$309$	0	0
$310$	−32.4407	−0.104647
$311$	− 404.897i	− 1.30192i	−0.759113	−	0.650959i	$-0.774367\pi$
$311$	− 404.897i	− 1.30192i	0.759113	−	0.650959i	$-0.225633\pi$
$312$	0	0
$313$	− 320.312i	− 1.02336i	−0.859176	−	0.511680i	$-0.829023\pi$
$313$	− 320.312i	− 1.02336i	0.859176	−	0.511680i	$-0.170977\pi$
$314$	− 213.603i	− 0.680264i
$315$	0	0
$316$	−123.332	−0.390292
$317$	152.214	0.480171	0.240085	−	0.970752i	$-0.422825\pi$
$317$	152.214	0.480171	0.240085	+	0.970752i	$0.422825\pi$
$318$	0	0
$319$	−144.362	−0.452545
$320$	− 12.1639i	− 0.0380122i
$321$	0	0
$322$	0	0
$323$	−490.937	−1.51993
$324$	0	0
$325$	407.351i	1.25339i
$326$	135.585	0.415904
$327$	0	0
$328$	152.946i	0.466299i
$329$	0	0
$330$	0	0
$331$	−583.530	−1.76293	−0.881465	−	0.472250i	$-0.843442\pi$
$331$	−583.530	−1.76293	−0.881465	+	0.472250i	$0.843442\pi$
$332$	194.006i	0.584355i
$333$	0	0
$334$	− 228.313i	− 0.683571i
$335$	179.902i	0.537020i
$336$	0	0
$337$	−21.9285	−0.0650699	−0.0325349	−	0.999471i	$-0.510358\pi$
$337$	−21.9285	−0.0650699	−0.0325349	+	0.999471i	$0.510358\pi$
$338$	−216.883	−0.641664
$339$	0	0
$340$	78.5283	0.230965
$341$	207.030i	0.607126i
$342$	0	0
$343$	0	0
$344$	208.969	0.607469
$345$	0	0
$346$	378.794i	1.09478i
$347$	410.857	1.18402	0.592012	−	0.805929i	$-0.298334\pi$
$347$	410.857	1.18402	0.592012	+	0.805929i	$0.298334\pi$
$348$	0	0
$349$	− 268.097i	− 0.768186i	−0.923294	−	0.384093i	$-0.874514\pi$
$349$	− 268.097i	− 0.768186i	0.923294	−	0.384093i	$-0.125486\pi$
$350$	0	0
$351$	0	0
$352$	−77.6275	−0.220533
$353$	411.611i	1.16604i	0.812459	+	0.583018i	$0.198128\pi$
$353$	411.611i	1.16604i	−0.812459	+	0.583018i	$0.801872\pi$
$354$	0	0
$355$	− 16.7286i	− 0.0471229i
$356$	0.0479039i	0 0.000134561i
$357$	0	0
$358$	186.587	0.521192
$359$	−140.272	−0.390730	−0.195365	−	0.980731i	$-0.562589\pi$
$359$	−140.272	−0.390730	−0.195365	+	0.980731i	$0.562589\pi$
$360$	0	0
$361$	−0.431214	−0.00119450
$362$	421.364i	1.16399i
$363$	0	0
$364$	0	0
$365$	−52.0029	−0.142474
$366$	0	0
$367$	318.965i	0.869114i	0.900644	+	0.434557i	$0.143095\pi$
$367$	318.965i	0.869114i	−0.900644	+	0.434557i	$0.856905\pi$
$368$	−77.3640	−0.210228
$369$	0	0
$370$	− 125.945i	− 0.340392i
$371$	0	0
$372$	0	0
$373$	709.625	1.90248	0.951240	−	0.308452i	$-0.0998109\pi$
$373$	709.625	1.90248	0.951240	+	0.308452i	$0.0998109\pi$
$374$	− 501.152i	− 1.33998i
$375$	0	0
$376$	3.07845i	0.00818735i
$377$	188.878i	0.501003i
$378$	0	0
$379$	269.499	0.711080	0.355540	−	0.934661i	$-0.384297\pi$
$379$	269.499	0.711080	0.355540	+	0.934661i	$0.384297\pi$
$380$	57.8130	0.152139
$381$	0	0
$382$	−80.0265	−0.209493
$383$	− 509.854i	− 1.33121i	−0.746304	−	0.665606i	$-0.768173\pi$
$383$	− 509.854i	− 1.33121i	0.746304	−	0.665606i	$-0.231827\pi$
$384$	0	0
$385$	0	0
$386$	288.314	0.746928
$387$	0	0
$388$	40.2762i	0.103805i
$389$	273.302	0.702575	0.351288	−	0.936268i	$-0.385744\pi$
$389$	273.302	0.702575	0.351288	+	0.936268i	$0.385744\pi$
$390$	0	0
$391$	− 499.450i	− 1.27737i
$392$	0	0
$393$	0	0
$394$	73.1094	0.185557
$395$	93.7625i	0.237374i
$396$	0	0
$397$	713.083i	1.79618i	0.439813	+	0.898090i	$0.355045\pi$
$397$	713.083i	1.79618i	−0.439813	+	0.898090i	$0.644955\pi$
$398$	111.331i	0.279726i
$399$	0	0
$400$	90.7525	0.226881
$401$	546.719	1.36339	0.681694	−	0.731637i	$-0.261244\pi$
$401$	546.719	1.36339	0.681694	+	0.731637i	$0.261244\pi$
$402$	0	0
$403$	270.871	0.672137
$404$	− 88.8069i	− 0.219819i
$405$	0	0
$406$	0	0
$407$	−803.756	−1.97483
$408$	0	0
$409$	− 90.5241i	− 0.221330i	−0.993858	−	0.110665i	$-0.964702\pi$
$409$	− 90.5241i	− 0.221330i	0.993858	−	0.110665i	$-0.0352981\pi$
$410$	116.276	0.283601
$411$	0	0
$412$	244.877i	0.594362i
$413$	0	0
$414$	0	0
$415$	147.492	0.355402
$416$	101.565i	0.244147i
$417$	0	0
$418$	− 368.951i	− 0.882658i
$419$	− 1.39159i	− 0.00332122i	−0.999999	−	0.00166061i	$-0.999471\pi$
$419$	− 1.39159i	− 0.00332122i	0.999999	−	0.00166061i	$-0.000528590\pi$
$420$	0	0
$421$	−49.1331	−0.116706	−0.0583528	−	0.998296i	$-0.518585\pi$
$421$	−49.1331	−0.116706	−0.0583528	+	0.998296i	$0.518585\pi$
$422$	−195.442	−0.463132
$423$	0	0
$424$	−217.969	−0.514078
$425$	585.884i	1.37855i
$426$	0	0
$427$	0	0
$428$	289.337	0.676020
$429$	0	0
$430$	− 158.868i	− 0.369459i
$431$	−231.604	−0.537364	−0.268682	−	0.963229i	$-0.586588\pi$
$431$	−231.604	−0.537364	−0.268682	+	0.963229i	$0.586588\pi$
$432$	0	0
$433$	− 143.495i	− 0.331397i	−0.986176	−	0.165699i	$-0.947012\pi$
$433$	− 143.495i	− 0.331397i	0.986176	−	0.165699i	$-0.0529879\pi$
$434$	0	0
$435$	0	0
$436$	−303.864	−0.696935
$437$	− 367.698i	− 0.841415i
$438$	0	0
$439$	− 357.328i	− 0.813958i	−0.913437	−	0.406979i	$-0.866582\pi$
$439$	− 357.328i	− 0.813958i	0.913437	−	0.406979i	$-0.133418\pi$
$440$	59.0158i	0.134127i
$441$	0	0
$442$	−655.690	−1.48346
$443$	−355.555	−0.802607	−0.401303	−	0.915945i	$-0.631443\pi$
$443$	−355.555	−0.802607	−0.401303	+	0.915945i	$0.631443\pi$
$444$	0	0
$445$	0.0364186	8.18395e−5 0
$446$	− 559.837i	− 1.25524i
$447$	0	0
$448$	0	0
$449$	327.765	0.729989	0.364994	−	0.931010i	$-0.381071\pi$
$449$	327.765	0.729989	0.364994	+	0.931010i	$0.381071\pi$
$450$	0	0
$451$	− 742.052i	− 1.64535i
$452$	160.164	0.354345
$453$	0	0
$454$	408.010i	0.898701i
$455$	0	0
$456$	0	0
$457$	−369.905	−0.809420	−0.404710	−	0.914445i	$-0.632627\pi$
$457$	−369.905	−0.809420	−0.404710	+	0.914445i	$0.632627\pi$
$458$	− 461.035i	− 1.00663i
$459$	0	0
$460$	58.8155i	0.127860i
$461$	− 259.470i	− 0.562842i	−0.959584	−	0.281421i	$-0.909194\pi$
$461$	− 259.470i	− 0.562842i	0.959584	−	0.281421i	$-0.0908058\pi$
$462$	0	0
$463$	−157.954	−0.341154	−0.170577	−	0.985344i	$-0.554563\pi$
$463$	−157.954	−0.341154	−0.170577	+	0.985344i	$0.554563\pi$
$464$	42.0796	0.0906888
$465$	0	0
$466$	−138.596	−0.297416
$467$	62.1703i	0.133127i	0.997782	+	0.0665635i	$0.0212035\pi$
$467$	62.1703i	0.133127i	−0.997782	+	0.0665635i	$0.978797\pi$
$468$	0	0
$469$	0	0
$470$	2.34037	0.00497951
$471$	0	0
$472$	− 193.738i	− 0.410463i
$473$	−1013.86	−2.14347
$474$	0	0
$475$	431.332i	0.908067i
$476$	0	0
$477$	0	0
$478$	641.004	1.34101
$479$	704.864i	1.47153i	0.677235	+	0.735766i	$0.263178\pi$
$479$	704.864i	1.47153i	−0.677235	+	0.735766i	$0.736822\pi$
$480$	0	0
$481$	1051.61i	2.18629i
$482$	− 445.558i	− 0.924395i
$483$	0	0
$484$	134.627	0.278155
$485$	30.6197	0.0631334
$486$	0	0
$487$	384.025	0.788552	0.394276	−	0.918992i	$-0.370995\pi$
$487$	384.025	0.788552	0.394276	+	0.918992i	$0.370995\pi$
$488$	201.248i	0.412393i
$489$	0	0
$490$	0	0
$491$	413.614	0.842391	0.421196	−	0.906970i	$-0.361611\pi$
$491$	413.614	0.842391	0.421196	+	0.906970i	$0.361611\pi$
$492$	0	0
$493$	271.660i	0.551034i
$494$	−482.723	−0.977172
$495$	0	0
$496$	− 60.3466i	− 0.121666i
$497$	0	0
$498$	0	0
$499$	149.628	0.299857	0.149928	−	0.988697i	$-0.452096\pi$
$499$	149.628	0.299857	0.149928	+	0.988697i	$0.452096\pi$
$500$	− 145.018i	− 0.290037i
$501$	0	0
$502$	26.2406i	0.0522720i
$503$	− 867.510i	− 1.72467i	−0.506337	−	0.862336i	$-0.669001\pi$
$503$	− 867.510i	− 1.72467i	0.506337	−	0.862336i	$-0.330999\pi$
$504$	0	0
$505$	−67.5149	−0.133693
$506$	375.349	0.741796
$507$	0	0
$508$	5.48640	0.0108000
$509$	559.540i	1.09929i	0.835398	+	0.549646i	$0.185237\pi$
$509$	559.540i	1.09929i	−0.835398	+	0.549646i	$0.814763\pi$
$510$	0	0
$511$	0	0
$512$	22.6274	0.0441942
$513$	0	0
$514$	− 415.783i	− 0.808917i
$515$	186.166	0.361488
$516$	0	0
$517$	− 14.9358i	− 0.0288893i
$518$	0	0
$519$	0	0
$520$	77.2143	0.148489
$521$	946.154i	1.81603i	0.418932	+	0.908017i	$0.362404\pi$
$521$	946.154i	1.81603i	−0.418932	+	0.908017i	$0.637596\pi$
$522$	0	0
$523$	− 60.6391i	− 0.115945i	−0.998318	−	0.0579724i	$-0.981536\pi$
$523$	− 60.6391i	− 0.115945i	0.998318	−	0.0579724i	$-0.0184635\pi$
$524$	− 296.583i	− 0.565999i
$525$	0	0
$526$	576.397	1.09581
$527$	389.588	0.739257
$528$	0	0
$529$	−154.926	−0.292865
$530$	165.710i	0.312660i
$531$	0	0
$532$	0	0
$533$	−970.875	−1.82153
$534$	0	0
$535$	− 219.966i	− 0.411152i
$536$	−334.655	−0.624357
$537$	0	0
$538$	13.7623i	0.0255806i
$539$	0	0
$540$	0	0
$541$	612.513	1.13219	0.566093	−	0.824341i	$-0.308454\pi$
$541$	612.513	1.13219	0.566093	+	0.824341i	$0.308454\pi$
$542$	6.28261i	0.0115915i
$543$	0	0
$544$	146.079i	0.268528i
$545$	231.010i	0.423872i
$546$	0	0
$547$	−147.297	−0.269282	−0.134641	−	0.990894i	$-0.542988\pi$
$547$	−147.297	−0.269282	−0.134641	+	0.990894i	$0.542988\pi$
$548$	−273.155	−0.498458
$549$	0	0
$550$	−440.306	−0.800556
$551$	199.997i	0.362972i
$552$	0	0
$553$	0	0
$554$	402.777	0.727034
$555$	0	0
$556$	− 85.7049i	− 0.154145i
$557$	314.478	0.564592	0.282296	−	0.959327i	$-0.408904\pi$
$557$	314.478	0.564592	0.282296	+	0.959327i	$0.408904\pi$
$558$	0	0
$559$	1326.50i	2.37299i
$560$	0	0
$561$	0	0
$562$	114.331	0.203437
$563$	− 745.068i	− 1.32339i	−0.749774	−	0.661694i	$-0.769838\pi$
$563$	− 745.068i	− 1.32339i	0.749774	−	0.661694i	$-0.230162\pi$
$564$	0	0
$565$	− 121.764i	− 0.215511i
$566$	228.647i	0.403970i
$567$	0	0
$568$	31.1188	0.0547866
$569$	855.670	1.50381	0.751907	−	0.659269i	$-0.229134\pi$
$569$	855.670	1.50381	0.751907	+	0.659269i	$0.229134\pi$
$570$	0	0
$571$	176.971	0.309931	0.154966	−	0.987920i	$-0.450473\pi$
$571$	176.971	0.309931	0.154966	+	0.987920i	$0.450473\pi$
$572$	− 492.766i	− 0.861479i
$573$	0	0
$574$	0	0
$575$	−438.811	−0.763149
$576$	0	0
$577$	49.7079i	0.0861489i	0.999072	+	0.0430744i	$0.0137153\pi$
$577$	49.7079i	0.0861489i	−0.999072	+	0.0430744i	$0.986285\pi$
$578$	−534.357	−0.924494
$579$	0	0
$580$	− 31.9907i	− 0.0551564i
$581$	0	0
$582$	0	0
$583$	1057.53	1.81394
$584$	− 96.7363i	− 0.165644i
$585$	0	0
$586$	532.978i	0.909518i
$587$	− 355.677i	− 0.605924i	−0.953003	−	0.302962i	$-0.902025\pi$
$587$	− 355.677i	− 0.605924i	0.953003	−	0.302962i	$-0.0979755\pi$
$588$	0	0
$589$	286.817	0.486956
$590$	−147.288	−0.249641
$591$	0	0
$592$	234.285	0.395751
$593$	427.674i	0.721204i	0.932720	+	0.360602i	$0.117429\pi$
$593$	427.674i	0.721204i	−0.932720	+	0.360602i	$0.882571\pi$
$594$	0	0
$595$	0	0
$596$	−281.574	−0.472439
$597$	0	0
$598$	− 491.093i	− 0.821226i
$599$	−838.453	−1.39975	−0.699877	−	0.714263i	$-0.746762\pi$
$599$	−838.453	−1.39975	−0.699877	+	0.714263i	$0.746762\pi$
$600$	0	0
$601$	520.799i	0.866554i	0.901261	+	0.433277i	$0.142643\pi$
$601$	520.799i	0.866554i	−0.901261	+	0.433277i	$0.857357\pi$
$602$	0	0
$603$	0	0
$604$	−4.97788	−0.00824152
$605$	− 102.349i	− 0.169173i
$606$	0	0
$607$	− 51.7830i	− 0.0853098i	−0.999090	−	0.0426549i	$-0.986418\pi$
$607$	− 51.7830i	− 0.0853098i	0.999090	−	0.0426549i	$-0.0135816\pi$
$608$	107.544i	0.176882i
$609$	0	0
$610$	152.997	0.250815
$611$	−19.5414	−0.0319827
$612$	0	0
$613$	−1050.04	−1.71295	−0.856476	−	0.516187i	$-0.827351\pi$
$613$	−1050.04	−1.71295	−0.856476	+	0.516187i	$0.827351\pi$
$614$	− 116.059i	− 0.189022i
$615$	0	0
$616$	0	0
$617$	595.958	0.965897	0.482948	−	0.875649i	$-0.339566\pi$
$617$	595.958	0.965897	0.482948	+	0.875649i	$0.339566\pi$
$618$	0	0
$619$	190.653i	0.308002i	0.988071	+	0.154001i	$0.0492159\pi$
$619$	190.653i	0.308002i	−0.988071	+	0.154001i	$0.950784\pi$
$620$	−45.8781	−0.0739969
$621$	0	0
$622$	− 572.610i	− 0.920596i
$623$	0	0
$624$	0	0
$625$	456.954	0.731126
$626$	− 452.989i	− 0.723625i
$627$	0	0
$628$	− 302.080i	− 0.481020i
$629$	1512.51i	2.40462i
$630$	0	0
$631$	−59.7941	−0.0947609	−0.0473805	−	0.998877i	$-0.515087\pi$
$631$	−59.7941	−0.0947609	−0.0473805	+	0.998877i	$0.515087\pi$
$632$	−174.418	−0.275978
$633$	0	0
$634$	215.263	0.339532
$635$	− 4.17100i	− 0.00656851i
$636$	0	0
$637$	0	0
$638$	−204.159	−0.319998
$639$	0	0
$640$	− 17.2023i	− 0.0268787i
$641$	603.694	0.941801	0.470900	−	0.882186i	$-0.343929\pi$
$641$	603.694	0.941801	0.470900	+	0.882186i	$0.343929\pi$
$642$	0	0
$643$	− 477.650i	− 0.742846i	−0.928464	−	0.371423i	$-0.878870\pi$
$643$	− 477.650i	− 0.742846i	0.928464	−	0.371423i	$-0.121130\pi$
$644$	0	0
$645$	0	0
$646$	−694.290	−1.07475
$647$	753.514i	1.16463i	0.812964	+	0.582314i	$0.197852\pi$
$647$	753.514i	1.16463i	−0.812964	+	0.582314i	$0.802148\pi$
$648$	0	0
$649$	939.965i	1.44833i
$650$	576.081i	0.886278i
$651$	0	0
$652$	191.746	0.294088
$653$	−888.816	−1.36113	−0.680563	−	0.732689i	$-0.738265\pi$
$653$	−888.816	−1.36113	−0.680563	+	0.732689i	$0.738265\pi$
$654$	0	0
$655$	−225.476	−0.344237
$656$	216.298i	0.329723i
$657$	0	0
$658$	0	0
$659$	370.133	0.561659	0.280829	−	0.959758i	$-0.409391\pi$
$659$	370.133	0.561659	0.280829	+	0.959758i	$0.409391\pi$
$660$	0	0
$661$	− 189.411i	− 0.286552i	−0.989683	−	0.143276i	$-0.954236\pi$
$661$	− 189.411i	− 0.286552i	0.989683	−	0.143276i	$-0.0457637\pi$
$662$	−825.235	−1.24658
$663$	0	0
$664$	274.366i	0.413201i
$665$	0	0
$666$	0	0
$667$	−203.465	−0.305046
$668$	− 322.883i	− 0.483357i
$669$	0	0
$670$	254.420i	0.379731i
$671$	− 976.398i	− 1.45514i
$672$	0	0
$673$	−320.118	−0.475659	−0.237829	−	0.971307i	$-0.576436\pi$
$673$	−320.118	−0.475659	−0.237829	+	0.971307i	$0.576436\pi$
$674$	−31.0116	−0.0460113
$675$	0	0
$676$	−306.718	−0.453725
$677$	1322.20i	1.95303i	0.215443	+	0.976516i	$0.430880\pi$
$677$	1322.20i	1.95303i	−0.215443	+	0.976516i	$0.569120\pi$
$678$	0	0
$679$	0	0
$680$	111.056	0.163317
$681$	0	0
$682$	292.785i	0.429303i
$683$	1112.47	1.62880	0.814401	−	0.580303i	$-0.197066\pi$
$683$	1112.47	1.62880	0.814401	+	0.580303i	$0.197066\pi$
$684$	0	0
$685$	207.664i	0.303160i
$686$	0	0
$687$	0	0
$688$	295.527	0.429545
$689$	− 1383.63i	− 2.00817i
$690$	0	0
$691$	830.923i	1.20249i	0.799064	+	0.601247i	$0.205329\pi$
$691$	830.923i	1.20249i	−0.799064	+	0.601247i	$0.794671\pi$
$692$	535.695i	0.774126i
$693$	0	0
$694$	581.039	0.837232
$695$	−65.1566	−0.0937505
$696$	0	0
$697$	−1396.39	−2.00343
$698$	− 379.146i	− 0.543190i
$699$	0	0
$700$	0	0
$701$	349.746	0.498924	0.249462	−	0.968385i	$-0.419746\pi$
$701$	349.746	0.498924	0.249462	+	0.968385i	$0.419746\pi$
$702$	0	0
$703$	1113.52i	1.58395i
$704$	−109.782	−0.155940
$705$	0	0
$706$	582.106i	0.824513i
$707$	0	0
$708$	0	0
$709$	9.81687	0.0138461	0.00692304	−	0.999976i	$-0.497796\pi$
$709$	9.81687	0.0138461	0.00692304	+	0.999976i	$0.497796\pi$
$710$	− 23.6578i	− 0.0333209i
$711$	0	0
$712$	0.0677463i	0 9.51493e-5i
$713$	291.791i	0.409244i
$714$	0	0
$715$	−374.622	−0.523947
$716$	263.873	0.368538
$717$	0	0
$718$	−198.374	−0.276288
$719$	− 245.035i	− 0.340799i	−0.985375	−	0.170400i	$-0.945494\pi$
$719$	− 245.035i	− 0.340799i	0.985375	−	0.170400i	$-0.0545059\pi$
$720$	0	0
$721$	0	0
$722$	−0.609829	−0.000844638 0
$723$	0	0
$724$	595.899i	0.823064i
$725$	238.677	0.329209
$726$	0	0
$727$	− 980.937i	− 1.34929i	−0.738140	−	0.674647i	$-0.764296\pi$
$727$	− 980.937i	− 1.34929i	0.738140	−	0.674647i	$-0.235704\pi$
$728$	0	0
$729$	0	0
$730$	−73.5432	−0.100744
$731$	1907.88i	2.60996i
$732$	0	0
$733$	− 738.088i	− 1.00694i	−0.864012	−	0.503471i	$-0.832056\pi$
$733$	− 738.088i	− 1.00694i	0.864012	−	0.503471i	$-0.167944\pi$
$734$	451.084i	0.614556i
$735$	0	0
$736$	−109.409	−0.148654
$737$	1623.65	2.20306
$738$	0	0
$739$	596.911	0.807728	0.403864	−	0.914819i	$-0.367667\pi$
$739$	596.911	0.807728	0.403864	+	0.914819i	$0.367667\pi$
$740$	− 178.113i	− 0.240694i
$741$	0	0
$742$	0	0
$743$	−167.318	−0.225193	−0.112596	−	0.993641i	$-0.535917\pi$
$743$	−167.318	−0.225193	−0.112596	+	0.993641i	$0.535917\pi$
$744$	0	0
$745$	214.064i	0.287335i
$746$	1003.56	1.34526
$747$	0	0
$748$	− 708.735i	− 0.947507i
$749$	0	0
$750$	0	0
$751$	−81.9734	−0.109152	−0.0545762	−	0.998510i	$-0.517381\pi$
$751$	−81.9734	−0.109152	−0.0545762	+	0.998510i	$0.517381\pi$
$752$	4.35358i	0.00578933i
$753$	0	0
$754$	267.114i	0.354263i
$755$	3.78440i	0.00501245i
$756$	0	0
$757$	187.820	0.248111	0.124055	−	0.992275i	$-0.460410\pi$
$757$	187.820	0.248111	0.124055	+	0.992275i	$0.460410\pi$
$758$	381.130	0.502810
$759$	0	0
$760$	81.7599	0.107579
$761$	− 604.817i	− 0.794766i	−0.917653	−	0.397383i	$-0.869918\pi$
$761$	− 604.817i	− 0.794766i	0.917653	−	0.397383i	$-0.130082\pi$
$762$	0	0
$763$	0	0
$764$	−113.174	−0.148134
$765$	0	0
$766$	− 721.042i	− 0.941309i
$767$	1229.82	1.60341
$768$	0	0
$769$	448.945i	0.583803i	0.956448	+	0.291902i	$0.0942880\pi$
$769$	448.945i	0.583803i	−0.956448	+	0.291902i	$0.905712\pi$
$770$	0	0
$771$	0	0
$772$	407.738	0.528158
$773$	− 855.205i	− 1.10635i	−0.833067	−	0.553173i	$-0.813417\pi$
$773$	− 855.205i	− 1.10635i	0.833067	−	0.553173i	$-0.186583\pi$
$774$	0	0
$775$	− 342.288i	− 0.441661i
$776$	56.9592i	0.0734010i
$777$	0	0
$778$	386.507	0.496796
$779$	−1028.03	−1.31968
$780$	0	0
$781$	−150.980	−0.193316
$782$	− 706.329i	− 0.903234i
$783$	0	0
$784$	0	0
$785$	−229.654	−0.292554
$786$	0	0
$787$	740.656i	0.941113i	0.882370	+	0.470557i	$0.155947\pi$
$787$	740.656i	0.941113i	−0.882370	+	0.470557i	$0.844053\pi$
$788$	103.392	0.131208
$789$	0	0
$790$	132.600i	0.167848i
$791$	0	0
$792$	0	0
$793$	−1277.49	−1.61095
$794$	1008.45i	1.27009i
$795$	0	0
$796$	157.446i	0.197796i
$797$	483.291i	0.606388i	0.952929	+	0.303194i	$0.0980530\pi$
$797$	483.291i	0.606388i	−0.952929	+	0.303194i	$0.901947\pi$
$798$	0	0
$799$	−28.1060	−0.0351765
$800$	128.343	0.160429
$801$	0	0
$802$	773.177	0.964061
$803$	469.338i	0.584480i
$804$	0	0
$805$	0	0
$806$	383.070	0.475272
$807$	0	0
$808$	− 125.592i	− 0.155436i
$809$	−68.3586	−0.0844976	−0.0422488	−	0.999107i	$-0.513452\pi$
$809$	−68.3586	−0.0844976	−0.0422488	+	0.999107i	$0.513452\pi$
$810$	0	0
$811$	1224.65i	1.51004i	0.655699	+	0.755022i	$0.272374\pi$
$811$	1224.65i	1.51004i	−0.655699	+	0.755022i	$0.727626\pi$
$812$	0	0
$813$	0	0
$814$	−1136.68	−1.39642
$815$	− 145.773i	− 0.178863i
$816$	0	0
$817$	1404.59i	1.71921i
$818$	− 128.020i	− 0.156504i
$819$	0	0
$820$	164.439	0.200536
$821$	880.772	1.07280	0.536402	−	0.843963i	$-0.319783\pi$
$821$	880.772	1.07280	0.536402	+	0.843963i	$0.319783\pi$
$822$	0	0
$823$	159.889	0.194276	0.0971378	−	0.995271i	$-0.469031\pi$
$823$	159.889	0.194276	0.0971378	+	0.995271i	$0.469031\pi$
$824$	346.309i	0.420278i
$825$	0	0
$826$	0	0
$827$	658.079	0.795742	0.397871	−	0.917441i	$-0.369749\pi$
$827$	658.079	0.795742	0.397871	+	0.917441i	$0.369749\pi$
$828$	0	0
$829$	1175.00i	1.41738i	0.705522	+	0.708688i	$0.250712\pi$
$829$	1175.00i	1.41738i	−0.705522	+	0.708688i	$0.749288\pi$
$830$	208.585	0.251307
$831$	0	0
$832$	143.635i	0.172638i
$833$	0	0
$834$	0	0
$835$	−245.469	−0.293975
$836$	− 521.775i	− 0.624133i
$837$	0	0
$838$	− 1.96801i	− 0.00234846i
$839$	382.408i	0.455791i	0.973686	+	0.227895i	$0.0731844\pi$
$839$	382.408i	0.455791i	−0.973686	+	0.227895i	$0.926816\pi$
$840$	0	0
$841$	−730.332	−0.868409
$842$	−69.4847	−0.0825234
$843$	0	0
$844$	−276.396	−0.327484
$845$	233.180i	0.275953i
$846$	0	0
$847$	0	0
$848$	−308.255	−0.363508
$849$	0	0
$850$	828.565i	0.974783i
$851$	−1132.82	−1.33117
$852$	0	0
$853$	− 921.219i	− 1.07998i	−0.841673	−	0.539988i	$-0.818429\pi$
$853$	− 921.219i	− 1.07998i	0.841673	−	0.539988i	$-0.181571\pi$
$854$	0	0
$855$	0	0
$856$	409.184	0.478018
$857$	916.408i	1.06932i	0.845067	+	0.534661i	$0.179561\pi$
$857$	916.408i	1.06932i	−0.845067	+	0.534661i	$0.820439\pi$
$858$	0	0
$859$	− 455.657i	− 0.530450i	−0.964186	−	0.265225i	$-0.914554\pi$
$859$	− 455.657i	− 0.530450i	0.964186	−	0.265225i	$-0.0854463\pi$
$860$	− 224.673i	− 0.261247i
$861$	0	0
$862$	−327.537	−0.379974
$863$	−1231.44	−1.42692	−0.713462	−	0.700693i	$-0.752874\pi$
$863$	−1231.44	−1.42692	−0.713462	+	0.700693i	$0.752874\pi$
$864$	0	0
$865$	407.259	0.470819
$866$	− 202.933i	− 0.234333i
$867$	0	0
$868$	0	0
$869$	846.228	0.973796
$870$	0	0
$871$	− 2124.33i	− 2.43896i
$872$	−429.728	−0.492807
$873$	0	0
$874$	− 520.004i	− 0.594970i
$875$	0	0
$876$	0	0
$877$	−357.451	−0.407584	−0.203792	−	0.979014i	$-0.565327\pi$
$877$	−357.451	−0.407584	−0.203792	+	0.979014i	$0.565327\pi$
$878$	− 505.338i	− 0.575555i
$879$	0	0
$880$	83.4610i	0.0948420i
$881$	301.875i	0.342650i	0.985215	+	0.171325i	$0.0548048\pi$
$881$	301.875i	0.342650i	−0.985215	+	0.171325i	$0.945195\pi$
$882$	0	0
$883$	283.240	0.320770	0.160385	−	0.987055i	$-0.448726\pi$
$883$	283.240	0.320770	0.160385	+	0.987055i	$0.448726\pi$
$884$	−927.285	−1.04897
$885$	0	0
$886$	−502.830	−0.567529
$887$	− 1364.80i	− 1.53867i	−0.638845	−	0.769336i	$-0.720587\pi$
$887$	− 1364.80i	− 1.53867i	0.638845	−	0.769336i	$-0.279413\pi$
$888$	0	0
$889$	0	0
$890$	0.0515037	5.78693e−5 0
$891$	0	0
$892$	− 791.729i	− 0.887588i
$893$	−20.6918	−0.0231712
$894$	0	0
$895$	− 200.608i	− 0.224143i
$896$	0	0
$897$	0	0
$898$	463.530	0.516180
$899$	− 158.710i	− 0.176541i
$900$	0	0
$901$	− 1990.05i	− 2.20871i
$902$	− 1049.42i	− 1.16344i
$903$	0	0
$904$	226.506	0.250560
$905$	453.028	0.500583
$906$	0	0
$907$	934.259	1.03005	0.515027	−	0.857174i	$-0.327782\pi$
$907$	934.259	1.03005	0.515027	+	0.857174i	$0.327782\pi$
$908$	577.013i	0.635477i
$909$	0	0
$910$	0	0
$911$	−699.464	−0.767798	−0.383899	−	0.923375i	$-0.625419\pi$
$911$	−699.464	−0.767798	−0.383899	+	0.923375i	$0.625419\pi$
$912$	0	0
$913$	− 1331.15i	− 1.45799i
$914$	−523.124	−0.572346
$915$	0	0
$916$	− 652.002i	− 0.711793i
$917$	0	0
$918$	0	0
$919$	−1067.01	−1.16105	−0.580527	−	0.814241i	$-0.697153\pi$
$919$	−1067.01	−1.16105	−0.580527	+	0.814241i	$0.697153\pi$
$920$	83.1776i	0.0904105i
$921$	0	0
$922$	− 366.946i	− 0.397990i
$923$	197.537i	0.214016i
$924$	0	0
$925$	1328.87	1.43661
$926$	−223.381	−0.241232
$927$	0	0
$928$	59.5095	0.0641267
$929$	− 311.329i	− 0.335123i	−0.985862	−	0.167562i	$-0.946411\pi$
$929$	− 311.329i	− 0.335123i	0.985862	−	0.167562i	$-0.0535893\pi$
$930$	0	0
$931$	0	0
$932$	−196.004	−0.210305
$933$	0	0
$934$	87.9221i	0.0941350i
$935$	−538.811	−0.576269
$936$	0	0
$937$	272.466i	0.290786i	0.989374	+	0.145393i	$0.0464446\pi$
$937$	272.466i	0.290786i	−0.989374	+	0.145393i	$0.953555\pi$
$938$	0	0
$939$	0	0
$940$	3.30978	0.00352104
$941$	− 393.520i	− 0.418193i	−0.977895	−	0.209097i	$-0.932948\pi$
$941$	− 393.520i	− 0.418193i	0.977895	−	0.209097i	$-0.0670523\pi$
$942$	0	0
$943$	− 1045.86i	− 1.10907i
$944$	− 273.987i	− 0.290241i
$945$	0	0
$946$	−1433.82	−1.51566
$947$	−1085.15	−1.14588	−0.572939	−	0.819598i	$-0.694197\pi$
$947$	−1085.15	−1.14588	−0.572939	+	0.819598i	$0.694197\pi$
$948$	0	0
$949$	614.065	0.647066
$950$	609.995i	0.642100i
$951$	0	0
$952$	0	0
$953$	−70.7095	−0.0741968	−0.0370984	−	0.999312i	$-0.511811\pi$
$953$	−70.7095	−0.0741968	−0.0370984	+	0.999312i	$0.511811\pi$
$954$	0	0
$955$	86.0402i	0.0900944i
$956$	906.516	0.948239
$957$	0	0
$958$	996.829i	1.04053i
$959$	0	0
$960$	0	0
$961$	733.393	0.763156
$962$	1487.20i	1.54594i
$963$	0	0
$964$	− 630.115i	− 0.653646i
$965$	− 309.980i	− 0.321223i
$966$	0	0
$967$	−941.750	−0.973888	−0.486944	−	0.873433i	$-0.661888\pi$
$967$	−941.750	−0.973888	−0.486944	+	0.873433i	$0.661888\pi$
$968$	190.392	0.196686
$969$	0	0
$970$	43.3028	0.0446421
$971$	− 269.289i	− 0.277332i	−0.990339	−	0.138666i	$-0.955719\pi$
$971$	− 269.289i	− 0.277332i	0.990339	−	0.138666i	$-0.0442815\pi$
$972$	0	0
$973$	0	0
$974$	543.093	0.557591
$975$	0	0
$976$	284.607i	0.291606i
$977$	84.6451	0.0866377	0.0433189	−	0.999061i	$-0.486207\pi$
$977$	84.6451	0.0866377	0.0433189	+	0.999061i	$0.486207\pi$
$978$	0	0
$979$	− 0.328686i	0 0.000335737i
$980$	0	0
$981$	0	0
$982$	584.939	0.595661
$983$	− 1192.24i	− 1.21286i	−0.795136	−	0.606431i	$-0.792601\pi$
$983$	− 1192.24i	− 1.21286i	0.795136	−	0.606431i	$-0.207399\pi$
$984$	0	0
$985$	− 78.6033i	− 0.0798003i
$986$	384.185i	0.389640i
$987$	0	0
$988$	−682.673	−0.690965
$989$	−1428.95	−1.44484
$990$	0	0
$991$	1059.72	1.06934	0.534670	−	0.845061i	$-0.320436\pi$
$991$	1059.72	1.06934	0.534670	+	0.845061i	$0.320436\pi$
$992$	− 85.3429i	− 0.0860312i
$993$	0	0
$994$	0	0
$995$	119.697	0.120298
$996$	0	0
$997$	1083.64i	1.08690i	0.839443	+	0.543448i	$0.182882\pi$
$997$	1083.64i	1.08690i	−0.839443	+	0.543448i	$0.817118\pi$
$998$	211.607	0.212031
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.3.c.g.685.6		8
3.2	odd	2		294.3.c.b.97.1	✓	8
7.2	even	3		882.3.n.k.325.2		8
7.3	odd	6		882.3.n.k.19.2		8
7.4	even	3		882.3.n.f.19.1		8
7.5	odd	6		882.3.n.f.325.1		8
7.6	odd	2	inner	882.3.c.g.685.7		8
12.11	even	2		2352.3.f.i.97.7		8
21.2	odd	6		294.3.g.e.31.3		8
21.5	even	6		294.3.g.d.31.4		8
21.11	odd	6		294.3.g.d.19.4		8
21.17	even	6		294.3.g.e.19.3		8
21.20	even	2		294.3.c.b.97.4	yes	8
84.83	odd	2		2352.3.f.i.97.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
294.3.c.b.97.1	✓	8	3.2	odd	2
294.3.c.b.97.4	yes	8	21.20	even	2
294.3.g.d.19.4		8	21.11	odd	6
294.3.g.d.31.4		8	21.5	even	6
294.3.g.e.19.3		8	21.17	even	6
294.3.g.e.31.3		8	21.2	odd	6
882.3.c.g.685.6		8	1.1	even	1	trivial
882.3.c.g.685.7		8	7.6	odd	2	inner
882.3.n.f.19.1		8	7.4	even	3
882.3.n.f.325.1		8	7.5	odd	6
882.3.n.k.19.2		8	7.3	odd	6
882.3.n.k.325.2		8	7.2	even	3
2352.3.f.i.97.2		8	84.83	odd	2
2352.3.f.i.97.7		8	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	882.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} -1.52049i q^{5} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} -1.52049i q^{5} +2.82843 q^{8} -2.15029i q^{10} -13.7227 q^{11} +17.9544i q^{13} +4.00000 q^{16} +25.8234i q^{17} +19.0113i q^{19} -3.04097i q^{20} -19.4069 q^{22} -19.3410 q^{23} +22.6881 q^{25} +25.3913i q^{26} +10.5199 q^{29} -15.0866i q^{31} +5.65685 q^{32} +36.5198i q^{34} +58.5711 q^{37} +26.8861i q^{38} -4.30059i q^{40} +54.0746i q^{41} +73.8818 q^{43} -27.4455 q^{44} -27.3523 q^{46} +1.08839i q^{47} +32.0858 q^{50} +35.9087i q^{52} -77.0637 q^{53} +20.8652i q^{55} +14.8774 q^{58} -68.4969i q^{59} +71.1518i q^{61} -21.3357i q^{62} +8.00000 q^{64} +27.2994 q^{65} -118.319 q^{67} +51.6468i q^{68} +11.0021 q^{71} -34.2015i q^{73} +82.8321 q^{74} +38.0227i q^{76} -61.6661 q^{79} -6.08195i q^{80} +76.4731i q^{82} +97.0029i q^{83} +39.2641 q^{85} +104.485 q^{86} -38.8138 q^{88} +0.0239519i q^{89} -38.6820 q^{92} +1.53922i q^{94} +28.9065 q^{95} +20.1381i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4}+O(q^{10}) \) 8 * q + 16 * q^4	\( 8 q + 16 q^{4} + 32 q^{16} + 32 q^{22} - 32 q^{23} - 72 q^{25} + 160 q^{29} + 128 q^{37} + 320 q^{43} - 96 q^{46} - 16 q^{50} - 384 q^{53} + 80 q^{58} + 64 q^{64} + 288 q^{65} - 352 q^{67} + 256 q^{71} - 48 q^{74} + 288 q^{79} + 768 q^{85} - 256 q^{86} + 64 q^{88} - 64 q^{92} + 32 q^{95}+O(q^{100}) \) 8 * q + 16 * q^4 + 32 * q^16 + 32 * q^22 - 32 * q^23 - 72 * q^25 + 160 * q^29 + 128 * q^37 + 320 * q^43 - 96 * q^46 - 16 * q^50 - 384 * q^53 + 80 * q^58 + 64 * q^64 + 288 * q^65 - 352 * q^67 + 256 * q^71 - 48 * q^74 + 288 * q^79 + 768 * q^85 - 256 * q^86 + 64 * q^88 - 64 * q^92 + 32 * q^95

Introduction

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