LMFDB - Embedded newform 1089.3.c.m.604.2

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 77 x^{12} + 88 x^{11} - 577 x^{10} + 578 x^{9} + 1520 x^{8} + \cdots + 83521 $ x^16 - 2x^15 + 3x^14 - 4x^13 + 77x^12 + 88x^11 - 577x^10 + 578x^9 + 1520x^8 + 1868x^7 - 1619x^6 - 16804x^5 + 32427x^4 + 43316x^3 - 71672x^2 + 83521
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 11^{4} $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.2
Root			$2.24350 - 2.23726i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.m.604.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.88468i q^{2} -4.32139 q^{4} +0.441126 q^{5} +10.5820i q^{7} +0.927105i q^{8} +O(q^{10})$	$q-2.88468i q^{2} -4.32139 q^{4} +0.441126 q^{5} +10.5820i q^{7} +0.927105i q^{8} -1.27251i q^{10} +6.44736i q^{13} +30.5257 q^{14} -14.6112 q^{16} -18.8981i q^{17} -27.5450i q^{19} -1.90628 q^{20} -6.29263 q^{23} -24.8054 q^{25} +18.5986 q^{26} -45.7290i q^{28} +44.1676i q^{29} -27.1033 q^{31} +45.8569i q^{32} -54.5151 q^{34} +4.66800i q^{35} -0.853144 q^{37} -79.4585 q^{38} +0.408970i q^{40} +13.2319i q^{41} +68.8186i q^{43} +18.1522i q^{46} +16.1381 q^{47} -62.9789 q^{49} +71.5557i q^{50} -27.8616i q^{52} -38.6822 q^{53} -9.81064 q^{56} +127.409 q^{58} -76.9113 q^{59} +28.6652i q^{61} +78.1844i q^{62} +73.8381 q^{64} +2.84410i q^{65} -78.0944 q^{67} +81.6662i q^{68} +13.4657 q^{70} +27.1193 q^{71} +12.9807i q^{73} +2.46105i q^{74} +119.033i q^{76} +94.2544i q^{79} -6.44536 q^{80} +38.1699 q^{82} -115.975i q^{83} -8.33646i q^{85} +198.520 q^{86} -65.2779 q^{89} -68.2261 q^{91} +27.1929 q^{92} -46.5532i q^{94} -12.1508i q^{95} -70.2994 q^{97} +181.674i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 20 q^{4} + 4 q^{5}+O(q^{10}) $ 16 * q - 20 * q^4 + 4 * q^5	$ 16 q - 20 q^{4} + 4 q^{5} + 52 q^{14} - 44 q^{16} + 108 q^{20} - 132 q^{23} + 88 q^{25} + 4 q^{26} + 40 q^{31} - 368 q^{34} - 16 q^{37} - 280 q^{38} - 80 q^{47} - 140 q^{49} + 128 q^{53} - 524 q^{56} + 140 q^{58} + 220 q^{59} - 8 q^{64} + 36 q^{67} - 100 q^{70} - 644 q^{71} - 264 q^{80} - 476 q^{82} - 76 q^{86} - 76 q^{89} - 624 q^{91} - 120 q^{92} + 216 q^{97}+O(q^{100}) $ 16 * q - 20 * q^4 + 4 * q^5 + 52 * q^14 - 44 * q^16 + 108 * q^20 - 132 * q^23 + 88 * q^25 + 4 * q^26 + 40 * q^31 - 368 * q^34 - 16 * q^37 - 280 * q^38 - 80 * q^47 - 140 * q^49 + 128 * q^53 - 524 * q^56 + 140 * q^58 + 220 * q^59 - 8 * q^64 + 36 * q^67 - 100 * q^70 - 644 * q^71 - 264 * q^80 - 476 * q^82 - 76 * q^86 - 76 * q^89 - 624 * q^91 - 120 * q^92 + 216 * 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 $

$2$	− 2.88468i	− 1.44234i	−0.692758	−	0.721170i	$-0.743604\pi$
$2$	− 2.88468i	− 1.44234i	0.692758	−	0.721170i	$-0.256396\pi$
$3$	0	0
$3$	0	0
$4$	−4.32139	−1.08035
$5$	0.441126	0.0882252	0.0441126	−	0.999027i	$-0.485954\pi$
$5$	0.441126	0.0882252	0.0441126	+	0.999027i	$0.485954\pi$
$6$	0	0
$7$	10.5820i	1.51172i	0.654736	+	0.755858i	$0.272780\pi$
$7$	10.5820i	1.51172i	−0.654736	+	0.755858i	$0.727220\pi$
$8$	0.927105i	0.115888i
$9$	0	0
$10$	− 1.27251i	− 0.127251i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	6.44736i	0.495951i	0.968766	+	0.247976i	$0.0797653\pi$
$13$	6.44736i	0.495951i	−0.968766	+	0.247976i	$0.920235\pi$
$14$	30.5257	2.18041
$15$	0	0
$16$	−14.6112	−0.913197
$17$	− 18.8981i	− 1.11165i	−0.831298	−	0.555827i	$-0.812402\pi$
$17$	− 18.8981i	− 1.11165i	0.831298	−	0.555827i	$-0.187598\pi$
$18$	0	0
$19$	− 27.5450i	− 1.44974i	−0.688888	−	0.724868i	$-0.741901\pi$
$19$	− 27.5450i	− 1.44974i	0.688888	−	0.724868i	$-0.258099\pi$
$20$	−1.90628	−0.0953139
$21$	0	0
$22$	0	0
$23$	−6.29263	−0.273593	−0.136796	−	0.990599i	$-0.543681\pi$
$23$	−6.29263	−0.273593	−0.136796	+	0.990599i	$0.543681\pi$
$24$	0	0
$25$	−24.8054	−0.992216
$26$	18.5986	0.715331
$27$	0	0
$28$	− 45.7290i	− 1.63318i
$29$	44.1676i	1.52302i	0.648153	+	0.761510i	$0.275542\pi$
$29$	44.1676i	1.52302i	−0.648153	+	0.761510i	$0.724458\pi$
$30$	0	0
$31$	−27.1033	−0.874300	−0.437150	−	0.899389i	$-0.644012\pi$
$31$	−27.1033	−0.874300	−0.437150	+	0.899389i	$0.644012\pi$
$32$	45.8569i	1.43303i
$33$	0	0
$34$	−54.5151	−1.60338
$35$	4.66800i	0.133371i
$36$	0	0
$37$	−0.853144	−0.0230579	−0.0115290	−	0.999934i	$-0.503670\pi$
$37$	−0.853144	−0.0230579	−0.0115290	+	0.999934i	$0.503670\pi$
$38$	−79.4585	−2.09101
$39$	0	0
$40$	0.408970i	0.0102243i
$41$	13.2319i	0.322730i	0.986895	+	0.161365i	$0.0515897\pi$
$41$	13.2319i	0.322730i	−0.986895	+	0.161365i	$0.948410\pi$
$42$	0	0
$43$	68.8186i	1.60043i	0.599712	+	0.800216i	$0.295282\pi$
$43$	68.8186i	1.60043i	−0.599712	+	0.800216i	$0.704718\pi$
$44$	0	0
$45$	0	0
$46$	18.1522i	0.394614i
$47$	16.1381	0.343363	0.171682	−	0.985152i	$-0.445080\pi$
$47$	16.1381	0.343363	0.171682	+	0.985152i	$0.445080\pi$
$48$	0	0
$49$	−62.9789	−1.28528
$50$	71.5557i	1.43111i
$51$	0	0
$52$	− 27.8616i	− 0.535799i
$53$	−38.6822	−0.729853	−0.364926	−	0.931036i	$-0.618906\pi$
$53$	−38.6822	−0.729853	−0.364926	+	0.931036i	$0.618906\pi$
$54$	0	0
$55$	0	0
$56$	−9.81064	−0.175190
$57$	0	0
$58$	127.409	2.19671
$59$	−76.9113	−1.30358	−0.651791	−	0.758399i	$-0.725982\pi$
$59$	−76.9113	−1.30358	−0.651791	+	0.758399i	$0.725982\pi$
$60$	0	0
$61$	28.6652i	0.469921i	0.972005	+	0.234961i	$0.0754961\pi$
$61$	28.6652i	0.469921i	−0.972005	+	0.234961i	$0.924504\pi$
$62$	78.1844i	1.26104i
$63$	0	0
$64$	73.8381	1.15372
$65$	2.84410i	0.0437554i
$66$	0	0
$67$	−78.0944	−1.16559	−0.582794	−	0.812620i	$-0.698041\pi$
$67$	−78.0944	−1.16559	−0.582794	+	0.812620i	$0.698041\pi$
$68$	81.6662i	1.20097i
$69$	0	0
$70$	13.4657	0.192367
$71$	27.1193	0.381963	0.190981	−	0.981594i	$-0.438833\pi$
$71$	27.1193	0.381963	0.190981	+	0.981594i	$0.438833\pi$
$72$	0	0
$73$	12.9807i	0.177818i	0.996040	+	0.0889092i	$0.0283381\pi$
$73$	12.9807i	0.177818i	−0.996040	+	0.0889092i	$0.971662\pi$
$74$	2.46105i	0.0332574i
$75$	0	0
$76$	119.033i	1.56622i
$77$	0	0
$78$	0	0
$79$	94.2544i	1.19309i	0.802578	+	0.596547i	$0.203461\pi$
$79$	94.2544i	1.19309i	−0.802578	+	0.596547i	$0.796539\pi$
$80$	−6.44536	−0.0805670
$81$	0	0
$82$	38.1699	0.465487
$83$	− 115.975i	− 1.39729i	−0.715468	−	0.698645i	$-0.753787\pi$
$83$	− 115.975i	− 1.39729i	0.715468	−	0.698645i	$-0.246213\pi$
$84$	0	0
$85$	− 8.33646i	− 0.0980760i
$86$	198.520	2.30837
$87$	0	0
$88$	0	0
$89$	−65.2779	−0.733460	−0.366730	−	0.930327i	$-0.619523\pi$
$89$	−65.2779	−0.733460	−0.366730	+	0.930327i	$0.619523\pi$
$90$	0	0
$91$	−68.2261	−0.749737
$92$	27.1929	0.295575
$93$	0	0
$94$	− 46.5532i	− 0.495247i
$95$	− 12.1508i	− 0.127903i
$96$	0	0
$97$	−70.2994	−0.724736	−0.362368	−	0.932035i	$-0.618032\pi$
$97$	−70.2994	−0.724736	−0.362368	+	0.932035i	$0.618032\pi$
$98$	181.674i	1.85382i
$99$	0	0
$100$	107.194	1.07194
$101$	47.4114i	0.469420i	0.972065	+	0.234710i	$0.0754141\pi$
$101$	47.4114i	0.469420i	−0.972065	+	0.234710i	$0.924586\pi$
$102$	0	0
$103$	−94.9775	−0.922112	−0.461056	−	0.887371i	$-0.652529\pi$
$103$	−94.9775	−0.922112	−0.461056	+	0.887371i	$0.652529\pi$
$104$	−5.97739	−0.0574749
$105$	0	0
$106$	111.586i	1.05270i
$107$	30.7567i	0.287446i	0.989618	+	0.143723i	$0.0459074\pi$
$107$	30.7567i	0.287446i	−0.989618	+	0.143723i	$0.954093\pi$
$108$	0	0
$109$	− 79.2234i	− 0.726820i	−0.931629	−	0.363410i	$-0.881612\pi$
$109$	− 79.2234i	− 0.726820i	0.931629	−	0.363410i	$-0.118388\pi$
$110$	0	0
$111$	0	0
$112$	− 154.615i	− 1.38049i
$113$	26.4704	0.234251	0.117126	−	0.993117i	$-0.462632\pi$
$113$	26.4704	0.234251	0.117126	+	0.993117i	$0.462632\pi$
$114$	0	0
$115$	−2.77584	−0.0241378
$116$	− 190.865i	− 1.64539i
$117$	0	0
$118$	221.865i	1.88021i
$119$	199.980	1.68051
$120$	0	0
$121$	0	0
$122$	82.6900	0.677787
$123$	0	0
$124$	117.124	0.944547
$125$	−21.9705	−0.175764
$126$	0	0
$127$	141.744i	1.11609i	0.829810	+	0.558046i	$0.188449\pi$
$127$	141.744i	1.11609i	−0.829810	+	0.558046i	$0.811551\pi$
$128$	− 29.5716i	− 0.231028i
$129$	0	0
$130$	8.20433	0.0631102
$131$	− 92.6286i	− 0.707088i	−0.935418	−	0.353544i	$-0.884976\pi$
$131$	− 92.6286i	− 0.707088i	0.935418	−	0.353544i	$-0.115024\pi$
$132$	0	0
$133$	291.481	2.19159
$134$	225.277i	1.68118i
$135$	0	0
$136$	17.5206	0.128828
$137$	21.9256	0.160041	0.0800204	−	0.996793i	$-0.474501\pi$
$137$	21.9256	0.160041	0.0800204	+	0.996793i	$0.474501\pi$
$138$	0	0
$139$	189.655i	1.36442i	0.731154	+	0.682212i	$0.238982\pi$
$139$	189.655i	1.36442i	−0.731154	+	0.682212i	$0.761018\pi$
$140$	− 20.1722i	− 0.144087i
$141$	0	0
$142$	− 78.2307i	− 0.550920i
$143$	0	0
$144$	0	0
$145$	19.4835i	0.134369i
$146$	37.4453	0.256475
$147$	0	0
$148$	3.68677	0.0249106
$149$	− 13.3105i	− 0.0893323i	−0.999002	−	0.0446661i	$-0.985778\pi$
$149$	− 13.3105i	− 0.0893323i	0.999002	−	0.0446661i	$-0.0142224\pi$
$150$	0	0
$151$	82.3554i	0.545400i	0.962099	+	0.272700i	$0.0879167\pi$
$151$	82.3554i	0.545400i	−0.962099	+	0.272700i	$0.912083\pi$
$152$	25.5371	0.168007
$153$	0	0
$154$	0	0
$155$	−11.9560	−0.0771353
$156$	0	0
$157$	−92.4364	−0.588767	−0.294383	−	0.955687i	$-0.595114\pi$
$157$	−92.4364	−0.588767	−0.294383	+	0.955687i	$0.595114\pi$
$158$	271.894	1.72085
$159$	0	0
$160$	20.2287i	0.126429i
$161$	− 66.5887i	− 0.413594i
$162$	0	0
$163$	233.616	1.43322	0.716612	−	0.697472i	$-0.245692\pi$
$163$	233.616	1.43322	0.716612	+	0.697472i	$0.245692\pi$
$164$	− 57.1804i	− 0.348661i
$165$	0	0
$166$	−334.551	−2.01537
$167$	− 192.987i	− 1.15561i	−0.816175	−	0.577805i	$-0.803910\pi$
$167$	− 192.987i	− 1.15561i	0.816175	−	0.577805i	$-0.196090\pi$
$168$	0	0
$169$	127.431	0.754033
$170$	−24.0480	−0.141459
$171$	0	0
$172$	− 297.392i	− 1.72902i
$173$	− 88.1075i	− 0.509292i	−0.967034	−	0.254646i	$-0.918041\pi$
$173$	− 88.1075i	− 0.509292i	0.967034	−	0.254646i	$-0.0819590\pi$
$174$	0	0
$175$	− 262.491i	− 1.49995i
$176$	0	0
$177$	0	0
$178$	188.306i	1.05790i
$179$	−317.289	−1.77256	−0.886282	−	0.463146i	$-0.846720\pi$
$179$	−317.289	−1.77256	−0.886282	+	0.463146i	$0.846720\pi$
$180$	0	0
$181$	−15.2796	−0.0844179	−0.0422089	−	0.999109i	$-0.513440\pi$
$181$	−15.2796	−0.0844179	−0.0422089	+	0.999109i	$0.513440\pi$
$182$	196.811i	1.08138i
$183$	0	0
$184$	− 5.83393i	− 0.0317062i
$185$	−0.376344	−0.00203429
$186$	0	0
$187$	0	0
$188$	−69.7389	−0.370952
$189$	0	0
$190$	−35.0512	−0.184480
$191$	−20.4709	−0.107178	−0.0535888	−	0.998563i	$-0.517066\pi$
$191$	−20.4709	−0.107178	−0.0535888	+	0.998563i	$0.517066\pi$
$192$	0	0
$193$	− 189.238i	− 0.980508i	−0.871580	−	0.490254i	$-0.836904\pi$
$193$	− 189.238i	− 0.980508i	0.871580	−	0.490254i	$-0.163096\pi$
$194$	202.791i	1.04532i
$195$	0	0
$196$	272.156	1.38855
$197$	− 380.855i	− 1.93327i	−0.256150	−	0.966637i	$-0.582454\pi$
$197$	− 380.855i	− 1.93327i	0.256150	−	0.966637i	$-0.417546\pi$
$198$	0	0
$199$	291.989	1.46728	0.733642	−	0.679537i	$-0.237819\pi$
$199$	291.989	1.46728	0.733642	+	0.679537i	$0.237819\pi$
$200$	− 22.9972i	− 0.114986i
$201$	0	0
$202$	136.767	0.677064
$203$	−467.382	−2.30237
$204$	0	0
$205$	5.83696i	0.0284730i
$206$	273.980i	1.33000i
$207$	0	0
$208$	− 94.2034i	− 0.452901i
$209$	0	0
$210$	0	0
$211$	− 33.3954i	− 0.158272i	−0.996864	−	0.0791359i	$-0.974784\pi$
$211$	− 33.3954i	− 0.158272i	0.996864	−	0.0791359i	$-0.0252161\pi$
$212$	167.161	0.788495
$213$	0	0
$214$	88.7232	0.414595
$215$	30.3577i	0.141199i
$216$	0	0
$217$	− 286.807i	− 1.32169i
$218$	−228.534	−1.04832
$219$	0	0
$220$	0	0
$221$	121.843	0.551326
$222$	0	0
$223$	−370.973	−1.66356	−0.831779	−	0.555107i	$-0.812677\pi$
$223$	−370.973	−1.66356	−0.831779	+	0.555107i	$0.812677\pi$
$224$	−485.259	−2.16633
$225$	0	0
$226$	− 76.3587i	− 0.337871i
$227$	114.155i	0.502886i	0.967872	+	0.251443i	$0.0809050\pi$
$227$	114.155i	0.502886i	−0.967872	+	0.251443i	$0.919095\pi$
$228$	0	0
$229$	−53.6484	−0.234272	−0.117136	−	0.993116i	$-0.537371\pi$
$229$	−53.6484	−0.234272	−0.117136	+	0.993116i	$0.537371\pi$
$230$	8.00743i	0.0348149i
$231$	0	0
$232$	−40.9480	−0.176500
$233$	193.960i	0.832446i	0.909263	+	0.416223i	$0.136646\pi$
$233$	193.960i	0.832446i	−0.909263	+	0.416223i	$0.863354\pi$
$234$	0	0
$235$	7.11893	0.0302933
$236$	332.364	1.40832
$237$	0	0
$238$	− 576.879i	− 2.42386i
$239$	− 150.146i	− 0.628225i	−0.949386	−	0.314113i	$-0.898293\pi$
$239$	− 150.146i	− 0.628225i	0.949386	−	0.314113i	$-0.101707\pi$
$240$	0	0
$241$	135.128i	0.560696i	0.959898	+	0.280348i	$0.0904499\pi$
$241$	135.128i	0.560696i	−0.959898	+	0.280348i	$0.909550\pi$
$242$	0	0
$243$	0	0
$244$	− 123.873i	− 0.507678i
$245$	−27.7817	−0.113394
$246$	0	0
$247$	177.592	0.718998
$248$	− 25.1276i	− 0.101321i
$249$	0	0
$250$	63.3778i	0.253511i
$251$	−235.203	−0.937062	−0.468531	−	0.883447i	$-0.655217\pi$
$251$	−235.203	−0.937062	−0.468531	+	0.883447i	$0.655217\pi$
$252$	0	0
$253$	0	0
$254$	408.885	1.60978
$255$	0	0
$256$	210.048	0.820499
$257$	399.203	1.55332	0.776660	−	0.629921i	$-0.216913\pi$
$257$	399.203	1.55332	0.776660	+	0.629921i	$0.216913\pi$
$258$	0	0
$259$	− 9.02797i	− 0.0348570i
$260$	− 12.2905i	− 0.0472710i
$261$	0	0
$262$	−267.204	−1.01986
$263$	281.116i	1.06888i	0.845206	+	0.534441i	$0.179478\pi$
$263$	281.116i	1.06888i	−0.845206	+	0.534441i	$0.820522\pi$
$264$	0	0
$265$	−17.0637	−0.0643914
$266$	− 840.830i	− 3.16102i
$267$	0	0
$268$	337.476	1.25924
$269$	444.987	1.65423	0.827114	−	0.562034i	$-0.189981\pi$
$269$	444.987	1.65423	0.827114	+	0.562034i	$0.189981\pi$
$270$	0	0
$271$	214.084i	0.789979i	0.918686	+	0.394990i	$0.129252\pi$
$271$	214.084i	0.789979i	−0.918686	+	0.394990i	$0.870748\pi$
$272$	276.123i	1.01516i
$273$	0	0
$274$	− 63.2483i	− 0.230833i
$275$	0	0
$276$	0	0
$277$	18.0389i	0.0651226i	0.999470	+	0.0325613i	$0.0103664\pi$
$277$	18.0389i	0.0651226i	−0.999470	+	0.0325613i	$0.989634\pi$
$278$	547.094	1.96796
$279$	0	0
$280$	−4.32773	−0.0154562
$281$	200.710i	0.714272i	0.934052	+	0.357136i	$0.116247\pi$
$281$	200.710i	0.714272i	−0.934052	+	0.357136i	$0.883753\pi$
$282$	0	0
$283$	188.893i	0.667467i	0.942668	+	0.333733i	$0.108309\pi$
$283$	188.893i	0.667467i	−0.942668	+	0.333733i	$0.891691\pi$
$284$	−117.193	−0.412652
$285$	0	0
$286$	0	0
$287$	−140.021	−0.487876
$288$	0	0
$289$	−68.1392	−0.235776
$290$	56.2036	0.193806
$291$	0	0
$292$	− 56.0949i	− 0.192106i
$293$	− 195.784i	− 0.668206i	−0.942537	−	0.334103i	$-0.891567\pi$
$293$	− 195.784i	− 0.668206i	0.942537	−	0.334103i	$-0.108433\pi$
$294$	0	0
$295$	−33.9276	−0.115009
$296$	− 0.790954i	− 0.00267214i
$297$	0	0
$298$	−38.3966	−0.128848
$299$	− 40.5709i	− 0.135689i
$300$	0	0
$301$	−728.239	−2.41940
$302$	237.569	0.786653
$303$	0	0
$304$	402.464i	1.32389i
$305$	12.6450i	0.0414589i
$306$	0	0
$307$	115.995i	0.377832i	0.981993	+	0.188916i	$0.0604974\pi$
$307$	115.995i	0.377832i	−0.981993	+	0.188916i	$0.939503\pi$
$308$	0	0
$309$	0	0
$310$	34.4892i	0.111255i
$311$	−273.924	−0.880785	−0.440392	−	0.897805i	$-0.645161\pi$
$311$	−273.924	−0.880785	−0.440392	+	0.897805i	$0.645161\pi$
$312$	0	0
$313$	9.03378	0.0288619	0.0144310	−	0.999896i	$-0.495406\pi$
$313$	9.03378	0.0288619	0.0144310	+	0.999896i	$0.495406\pi$
$314$	266.649i	0.849202i
$315$	0	0
$316$	− 407.310i	− 1.28896i
$317$	162.560	0.512808	0.256404	−	0.966570i	$-0.417462\pi$
$317$	162.560	0.512808	0.256404	+	0.966570i	$0.417462\pi$
$318$	0	0
$319$	0	0
$320$	32.5719	0.101787
$321$	0	0
$322$	−192.087	−0.596544
$323$	−520.548	−1.61161
$324$	0	0
$325$	− 159.930i	− 0.492091i
$326$	− 673.907i	− 2.06720i
$327$	0	0
$328$	−12.2674	−0.0374006
$329$	170.773i	0.519068i
$330$	0	0
$331$	−448.559	−1.35516	−0.677581	−	0.735448i	$-0.736972\pi$
$331$	−448.559	−1.35516	−0.677581	+	0.735448i	$0.736972\pi$
$332$	501.174i	1.50956i
$333$	0	0
$334$	−556.705	−1.66678
$335$	−34.4495	−0.102834
$336$	0	0
$337$	− 42.5410i	− 0.126234i	−0.998006	−	0.0631172i	$-0.979896\pi$
$337$	− 42.5410i	− 0.126234i	0.998006	−	0.0631172i	$-0.0201042\pi$
$338$	− 367.599i	− 1.08757i
$339$	0	0
$340$	36.0251i	0.105956i
$341$	0	0
$342$	0	0
$343$	− 147.925i	− 0.431269i
$344$	−63.8021	−0.185471
$345$	0	0
$346$	−254.162	−0.734573
$347$	169.303i	0.487904i	0.969787	+	0.243952i	$0.0784439\pi$
$347$	169.303i	0.487904i	−0.969787	+	0.243952i	$0.921556\pi$
$348$	0	0
$349$	− 297.042i	− 0.851123i	−0.904929	−	0.425562i	$-0.860077\pi$
$349$	− 297.042i	− 0.851123i	0.904929	−	0.425562i	$-0.139923\pi$
$350$	−757.203	−2.16344
$351$	0	0
$352$	0	0
$353$	−639.289	−1.81102	−0.905509	−	0.424327i	$-0.860511\pi$
$353$	−639.289	−1.81102	−0.905509	+	0.424327i	$0.860511\pi$
$354$	0	0
$355$	11.9630	0.0336987
$356$	282.091	0.792392
$357$	0	0
$358$	915.277i	2.55664i
$359$	− 383.235i	− 1.06751i	−0.845641	−	0.533753i	$-0.820781\pi$
$359$	− 383.235i	− 1.06751i	0.845641	−	0.533753i	$-0.179219\pi$
$360$	0	0
$361$	−397.726	−1.10173
$362$	44.0769i	0.121759i
$363$	0	0
$364$	294.831	0.809976
$365$	5.72615i	0.0156881i
$366$	0	0
$367$	354.233	0.965211	0.482606	−	0.875838i	$-0.339690\pi$
$367$	354.233	0.965211	0.482606	+	0.875838i	$0.339690\pi$
$368$	91.9426	0.249844
$369$	0	0
$370$	1.08563i	0.00293414i
$371$	− 409.335i	− 1.10333i
$372$	0	0
$373$	− 185.060i	− 0.496140i	−0.968742	−	0.248070i	$-0.920204\pi$
$373$	− 185.060i	− 0.496140i	0.968742	−	0.248070i	$-0.0797962\pi$
$374$	0	0
$375$	0	0
$376$	14.9617i	0.0397918i
$377$	−284.764	−0.755343
$378$	0	0
$379$	−252.585	−0.666452	−0.333226	−	0.942847i	$-0.608137\pi$
$379$	−252.585	−0.666452	−0.333226	+	0.942847i	$0.608137\pi$
$380$	52.5084i	0.138180i
$381$	0	0
$382$	59.0521i	0.154587i
$383$	251.069	0.655532	0.327766	−	0.944759i	$-0.393704\pi$
$383$	251.069	0.655532	0.327766	+	0.944759i	$0.393704\pi$
$384$	0	0
$385$	0	0
$386$	−545.891	−1.41423
$387$	0	0
$388$	303.791	0.782966
$389$	150.224	0.386180	0.193090	−	0.981181i	$-0.438149\pi$
$389$	150.224	0.386180	0.193090	+	0.981181i	$0.438149\pi$
$390$	0	0
$391$	118.919i	0.304141i
$392$	− 58.3881i	− 0.148949i
$393$	0	0
$394$	−1098.65	−2.78844
$395$	41.5781i	0.105261i
$396$	0	0
$397$	−516.245	−1.30036	−0.650182	−	0.759778i	$-0.725307\pi$
$397$	−516.245	−1.30036	−0.650182	+	0.759778i	$0.725307\pi$
$398$	− 842.296i	− 2.11632i
$399$	0	0
$400$	362.436	0.906089
$401$	16.6182	0.0414418	0.0207209	−	0.999785i	$-0.493404\pi$
$401$	16.6182	0.0414418	0.0207209	+	0.999785i	$0.493404\pi$
$402$	0	0
$403$	− 174.745i	− 0.433610i
$404$	− 204.883i	− 0.507137i
$405$	0	0
$406$	1348.25i	3.32081i
$407$	0	0
$408$	0	0
$409$	− 405.773i	− 0.992109i	−0.868291	−	0.496054i	$-0.834782\pi$
$409$	− 405.773i	− 0.992109i	0.868291	−	0.496054i	$-0.165218\pi$
$410$	16.8378	0.0410677
$411$	0	0
$412$	410.435	0.996201
$413$	− 813.876i	− 1.97065i
$414$	0	0
$415$	− 51.1597i	− 0.123276i
$416$	−295.656	−0.710713
$417$	0	0
$418$	0	0
$419$	628.759	1.50062	0.750309	−	0.661087i	$-0.229905\pi$
$419$	628.759	1.50062	0.750309	+	0.661087i	$0.229905\pi$
$420$	0	0
$421$	605.034	1.43713	0.718567	−	0.695458i	$-0.244798\pi$
$421$	605.034	1.43713	0.718567	+	0.695458i	$0.244798\pi$
$422$	−96.3350	−0.228282
$423$	0	0
$424$	− 35.8625i	− 0.0845813i
$425$	468.776i	1.10300i
$426$	0	0
$427$	−303.335	−0.710387
$428$	− 132.912i	− 0.310541i
$429$	0	0
$430$	87.5723	0.203656
$431$	726.164i	1.68484i	0.538825	+	0.842418i	$0.318868\pi$
$431$	726.164i	1.68484i	−0.538825	+	0.842418i	$0.681132\pi$
$432$	0	0
$433$	−234.892	−0.542475	−0.271238	−	0.962512i	$-0.587433\pi$
$433$	−234.892	−0.542475	−0.271238	+	0.962512i	$0.587433\pi$
$434$	−827.348	−1.90633
$435$	0	0
$436$	342.355i	0.785218i
$437$	173.330i	0.396637i
$438$	0	0
$439$	− 142.495i	− 0.324589i	−0.986742	−	0.162295i	$-0.948111\pi$
$439$	− 142.495i	− 0.324589i	0.986742	−	0.162295i	$-0.0518895\pi$
$440$	0	0
$441$	0	0
$442$	− 351.479i	− 0.795201i
$443$	−390.409	−0.881285	−0.440643	−	0.897683i	$-0.645249\pi$
$443$	−390.409	−0.881285	−0.440643	+	0.897683i	$0.645249\pi$
$444$	0	0
$445$	−28.7958	−0.0647097
$446$	1070.14i	2.39942i
$447$	0	0
$448$	781.355i	1.74410i
$449$	−716.748	−1.59632	−0.798160	−	0.602445i	$-0.794193\pi$
$449$	−716.748	−1.59632	−0.798160	+	0.602445i	$0.794193\pi$
$450$	0	0
$451$	0	0
$452$	−114.389	−0.253073
$453$	0	0
$454$	329.301	0.725332
$455$	−30.0963	−0.0661457
$456$	0	0
$457$	− 190.455i	− 0.416752i	−0.978049	−	0.208376i	$-0.933182\pi$
$457$	− 190.455i	− 0.416752i	0.978049	−	0.208376i	$-0.0668177\pi$
$458$	154.758i	0.337901i
$459$	0	0
$460$	11.9955	0.0260772
$461$	− 857.726i	− 1.86058i	−0.366828	−	0.930289i	$-0.619556\pi$
$461$	− 857.726i	− 1.86058i	0.366828	−	0.930289i	$-0.380444\pi$
$462$	0	0
$463$	−33.4807	−0.0723126	−0.0361563	−	0.999346i	$-0.511511\pi$
$463$	−33.4807	−0.0723126	−0.0361563	+	0.999346i	$0.511511\pi$
$464$	− 645.339i	− 1.39082i
$465$	0	0
$466$	559.513	1.20067
$467$	263.349	0.563916	0.281958	−	0.959427i	$-0.409016\pi$
$467$	263.349	0.563916	0.281958	+	0.959427i	$0.409016\pi$
$468$	0	0
$469$	− 826.396i	− 1.76204i
$470$	− 20.5358i	− 0.0436933i
$471$	0	0
$472$	− 71.3049i	− 0.151070i
$473$	0	0
$474$	0	0
$475$	683.264i	1.43845i
$476$	−864.192	−1.81553
$477$	0	0
$478$	−433.123	−0.906115
$479$	− 615.130i	− 1.28420i	−0.766622	−	0.642098i	$-0.778064\pi$
$479$	− 615.130i	− 1.28420i	0.766622	−	0.642098i	$-0.221936\pi$
$480$	0	0
$481$	− 5.50053i	− 0.0114356i
$482$	389.801	0.808715
$483$	0	0
$484$	0	0
$485$	−31.0109	−0.0639400
$486$	0	0
$487$	298.390	0.612710	0.306355	−	0.951917i	$-0.400891\pi$
$487$	298.390	0.612710	0.306355	+	0.951917i	$0.400891\pi$
$488$	−26.5757	−0.0544583
$489$	0	0
$490$	80.1412i	0.163554i
$491$	− 465.746i	− 0.948567i	−0.880372	−	0.474284i	$-0.842707\pi$
$491$	− 465.746i	− 0.948567i	0.880372	−	0.474284i	$-0.157293\pi$
$492$	0	0
$493$	834.684	1.69307
$494$	− 512.298i	− 1.03704i
$495$	0	0
$496$	396.010	0.798408
$497$	286.977i	0.577419i
$498$	0	0
$499$	545.872	1.09393	0.546965	−	0.837155i	$-0.315783\pi$
$499$	545.872	1.09393	0.546965	+	0.837155i	$0.315783\pi$
$500$	94.9429	0.189886
$501$	0	0
$502$	678.484i	1.35156i
$503$	− 625.729i	− 1.24399i	−0.783020	−	0.621997i	$-0.786322\pi$
$503$	− 625.729i	− 1.24399i	0.783020	−	0.621997i	$-0.213678\pi$
$504$	0	0
$505$	20.9144i	0.0414147i
$506$	0	0
$507$	0	0
$508$	− 612.529i	− 1.20577i
$509$	208.095	0.408830	0.204415	−	0.978884i	$-0.434471\pi$
$509$	208.095	0.408830	0.204415	+	0.978884i	$0.434471\pi$
$510$	0	0
$511$	−137.362	−0.268811
$512$	− 724.207i	− 1.41447i
$513$	0	0
$514$	− 1151.57i	− 2.24042i
$515$	−41.8971	−0.0813535
$516$	0	0
$517$	0	0
$518$	−26.0428	−0.0502757
$519$	0	0
$520$	−2.63678	−0.00507073
$521$	−315.168	−0.604929	−0.302464	−	0.953161i	$-0.597809\pi$
$521$	−315.168	−0.604929	−0.302464	+	0.953161i	$0.597809\pi$
$522$	0	0
$523$	− 472.827i	− 0.904067i	−0.892001	−	0.452034i	$-0.850699\pi$
$523$	− 472.827i	− 0.904067i	0.892001	−	0.452034i	$-0.149301\pi$
$524$	400.284i	0.763901i
$525$	0	0
$526$	810.930	1.54169
$527$	512.201i	0.971919i
$528$	0	0
$529$	−489.403	−0.925147
$530$	49.2234i	0.0928744i
$531$	0	0
$532$	−1259.60	−2.36768
$533$	−85.3112	−0.160058
$534$	0	0
$535$	13.5676i	0.0253600i
$536$	− 72.4017i	− 0.135078i
$537$	0	0
$538$	− 1283.65i	− 2.38596i
$539$	0	0
$540$	0	0
$541$	14.1746i	0.0262008i	0.999914	+	0.0131004i	$0.00417010\pi$
$541$	14.1746i	0.0262008i	−0.999914	+	0.0131004i	$0.995830\pi$
$542$	617.565	1.13942
$543$	0	0
$544$	866.610	1.59303
$545$	− 34.9475i	− 0.0641239i
$546$	0	0
$547$	640.079i	1.17016i	0.810975	+	0.585081i	$0.198937\pi$
$547$	640.079i	1.17016i	−0.810975	+	0.585081i	$0.801063\pi$
$548$	−94.7490	−0.172900
$549$	0	0
$550$	0	0
$551$	1216.59	2.20798
$552$	0	0
$553$	−997.401	−1.80362
$554$	52.0366	0.0939289
$555$	0	0
$556$	− 819.573i	− 1.47405i
$557$	570.143i	1.02360i	0.859106	+	0.511798i	$0.171020\pi$
$557$	570.143i	1.02360i	−0.859106	+	0.511798i	$0.828980\pi$
$558$	0	0
$559$	−443.699	−0.793736
$560$	− 68.2049i	− 0.121794i
$561$	0	0
$562$	578.986	1.03022
$563$	− 235.923i	− 0.419046i	−0.977804	−	0.209523i	$-0.932809\pi$
$563$	− 235.923i	− 0.419046i	0.977804	−	0.209523i	$-0.0671910\pi$
$564$	0	0
$565$	11.6768	0.0206669
$566$	544.896	0.962715
$567$	0	0
$568$	25.1425i	0.0442649i
$569$	− 396.044i	− 0.696035i	−0.937488	−	0.348018i	$-0.886855\pi$
$569$	− 396.044i	− 0.696035i	0.937488	−	0.348018i	$-0.113145\pi$
$570$	0	0
$571$	− 67.0903i	− 0.117496i	−0.998273	−	0.0587481i	$-0.981289\pi$
$571$	− 67.0903i	− 0.117496i	0.998273	−	0.0587481i	$-0.0187109\pi$
$572$	0	0
$573$	0	0
$574$	403.915i	0.703684i
$575$	156.091	0.271463
$576$	0	0
$577$	−160.995	−0.279020	−0.139510	−	0.990221i	$-0.544553\pi$
$577$	−160.995	−0.279020	−0.139510	+	0.990221i	$0.544553\pi$
$578$	196.560i	0.340069i
$579$	0	0
$580$	− 84.1957i	− 0.145165i
$581$	1227.25	2.11231
$582$	0	0
$583$	0	0
$584$	−12.0345	−0.0206071
$585$	0	0
$586$	−564.776	−0.963781
$587$	607.600	1.03509	0.517547	−	0.855655i	$-0.326845\pi$
$587$	607.600	1.03509	0.517547	+	0.855655i	$0.326845\pi$
$588$	0	0
$589$	746.559i	1.26750i
$590$	97.8703i	0.165882i
$591$	0	0
$592$	12.4654	0.0210564
$593$	691.234i	1.16566i	0.812596	+	0.582828i	$0.198054\pi$
$593$	691.234i	1.16566i	−0.812596	+	0.582828i	$0.801946\pi$
$594$	0	0
$595$	88.2165	0.148263
$596$	57.5199i	0.0965099i
$597$	0	0
$598$	−117.034	−0.195709
$599$	−241.924	−0.403880	−0.201940	−	0.979398i	$-0.564725\pi$
$599$	−241.924	−0.403880	−0.201940	+	0.979398i	$0.564725\pi$
$600$	0	0
$601$	− 63.6092i	− 0.105839i	−0.998599	−	0.0529194i	$-0.983147\pi$
$601$	− 63.6092i	− 0.105839i	0.998599	−	0.0529194i	$-0.0168527\pi$
$602$	2100.74i	3.48960i
$603$	0	0
$604$	− 355.890i	− 0.589222i
$605$	0	0
$606$	0	0
$607$	515.315i	0.848953i	0.905439	+	0.424477i	$0.139542\pi$
$607$	515.315i	0.848953i	−0.905439	+	0.424477i	$0.860458\pi$
$608$	1263.13	2.07751
$609$	0	0
$610$	36.4767	0.0597979
$611$	104.048i	0.170291i
$612$	0	0
$613$	38.8301i	0.0633443i	0.999498	+	0.0316722i	$0.0100833\pi$
$613$	38.8301i	0.0633443i	−0.999498	+	0.0316722i	$0.989917\pi$
$614$	334.607	0.544963
$615$	0	0
$616$	0	0
$617$	−636.272	−1.03124	−0.515618	−	0.856819i	$-0.672438\pi$
$617$	−636.272	−1.03124	−0.515618	+	0.856819i	$0.672438\pi$
$618$	0	0
$619$	−42.7230	−0.0690193	−0.0345097	−	0.999404i	$-0.510987\pi$
$619$	−42.7230	−0.0690193	−0.0345097	+	0.999404i	$0.510987\pi$
$620$	51.6664	0.0833329
$621$	0	0
$622$	790.184i	1.27039i
$623$	− 690.772i	− 1.10878i
$624$	0	0
$625$	610.443	0.976710
$626$	− 26.0596i	− 0.0416287i
$627$	0	0
$628$	399.454	0.636072
$629$	16.1228i	0.0256325i
$630$	0	0
$631$	584.880	0.926910	0.463455	−	0.886120i	$-0.346610\pi$
$631$	584.880	0.926910	0.463455	+	0.886120i	$0.346610\pi$
$632$	−87.3838	−0.138265
$633$	0	0
$634$	− 468.934i	− 0.739643i
$635$	62.5268i	0.0984674i
$636$	0	0
$637$	− 406.048i	− 0.637438i
$638$	0	0
$639$	0	0
$640$	− 13.0448i	− 0.0203825i
$641$	205.562	0.320689	0.160345	−	0.987061i	$-0.448739\pi$
$641$	205.562	0.320689	0.160345	+	0.987061i	$0.448739\pi$
$642$	0	0
$643$	120.317	0.187118	0.0935592	−	0.995614i	$-0.470176\pi$
$643$	120.317	0.187118	0.0935592	+	0.995614i	$0.470176\pi$
$644$	287.756i	0.446826i
$645$	0	0
$646$	1501.62i	2.32448i
$647$	583.513	0.901874	0.450937	−	0.892556i	$-0.351090\pi$
$647$	583.513	0.901874	0.450937	+	0.892556i	$0.351090\pi$
$648$	0	0
$649$	0	0
$650$	−461.346	−0.709763
$651$	0	0
$652$	−1009.54	−1.54838
$653$	438.744	0.671890	0.335945	−	0.941882i	$-0.390944\pi$
$653$	438.744	0.671890	0.335945	+	0.941882i	$0.390944\pi$
$654$	0	0
$655$	− 40.8609i	− 0.0623830i
$656$	− 193.334i	− 0.294716i
$657$	0	0
$658$	492.627	0.748673
$659$	598.225i	0.907777i	0.891058	+	0.453888i	$0.149964\pi$
$659$	598.225i	0.907777i	−0.891058	+	0.453888i	$0.850036\pi$
$660$	0	0
$661$	811.159	1.22717	0.613585	−	0.789629i	$-0.289727\pi$
$661$	811.159	1.22717	0.613585	+	0.789629i	$0.289727\pi$
$662$	1293.95i	1.95461i
$663$	0	0
$664$	107.521	0.161929
$665$	128.580	0.193353
$666$	0	0
$667$	− 277.930i	− 0.416687i
$668$	833.971i	1.24846i
$669$	0	0
$670$	99.3758i	0.148322i
$671$	0	0
$672$	0	0
$673$	− 933.510i	− 1.38709i	−0.720414	−	0.693544i	$-0.756048\pi$
$673$	− 933.510i	− 1.38709i	0.720414	−	0.693544i	$-0.243952\pi$
$674$	−122.717	−0.182073
$675$	0	0
$676$	−550.681	−0.814617
$677$	795.030i	1.17434i	0.809463	+	0.587171i	$0.199758\pi$
$677$	795.030i	1.17434i	−0.809463	+	0.587171i	$0.800242\pi$
$678$	0	0
$679$	− 743.908i	− 1.09559i
$680$	7.72878	0.0113658
$681$	0	0
$682$	0	0
$683$	−364.827	−0.534154	−0.267077	−	0.963675i	$-0.586058\pi$
$683$	−364.827	−0.534154	−0.267077	+	0.963675i	$0.586058\pi$
$684$	0	0
$685$	9.67195	0.0141196
$686$	−426.717	−0.622036
$687$	0	0
$688$	− 1005.52i	− 1.46151i
$689$	− 249.398i	− 0.361971i
$690$	0	0
$691$	882.970	1.27781	0.638907	−	0.769284i	$-0.279387\pi$
$691$	882.970	1.27781	0.638907	+	0.769284i	$0.279387\pi$
$692$	380.747i	0.550212i
$693$	0	0
$694$	488.384	0.703724
$695$	83.6618i	0.120377i
$696$	0	0
$697$	250.059	0.358765
$698$	−856.872	−1.22761
$699$	0	0
$700$	1134.33i	1.62047i
$701$	1100.51i	1.56991i	0.619553	+	0.784955i	$0.287314\pi$
$701$	1100.51i	1.56991i	−0.619553	+	0.784955i	$0.712686\pi$
$702$	0	0
$703$	23.4998i	0.0334279i
$704$	0	0
$705$	0	0
$706$	1844.15i	2.61211i
$707$	−501.708	−0.709630
$708$	0	0
$709$	1390.19	1.96077	0.980384	−	0.197094i	$-0.0631506\pi$
$709$	1390.19	1.96077	0.980384	+	0.197094i	$0.0631506\pi$
$710$	− 34.5096i	− 0.0486051i
$711$	0	0
$712$	− 60.5195i	− 0.0849993i
$713$	170.551	0.239202
$714$	0	0
$715$	0	0
$716$	1371.13	1.91498
$717$	0	0
$718$	−1105.51	−1.53971
$719$	59.7926	0.0831607	0.0415804	−	0.999135i	$-0.486761\pi$
$719$	59.7926	0.0831607	0.0415804	+	0.999135i	$0.486761\pi$
$720$	0	0
$721$	− 1005.05i	− 1.39397i
$722$	1147.31i	1.58907i
$723$	0	0
$724$	66.0293	0.0912006
$725$	− 1095.59i	− 1.51116i
$726$	0	0
$727$	−1173.95	−1.61479	−0.807396	−	0.590010i	$-0.799124\pi$
$727$	−1173.95	−1.61479	−0.807396	+	0.590010i	$0.799124\pi$
$728$	− 63.2528i	− 0.0868857i
$729$	0	0
$730$	16.5181	0.0226276
$731$	1300.54	1.77913
$732$	0	0
$733$	− 576.624i	− 0.786663i	−0.919397	−	0.393331i	$-0.871323\pi$
$733$	− 576.624i	− 0.786663i	0.919397	−	0.393331i	$-0.128677\pi$
$734$	− 1021.85i	− 1.39216i
$735$	0	0
$736$	− 288.561i	− 0.392066i
$737$	0	0
$738$	0	0
$739$	171.619i	0.232231i	0.993236	+	0.116116i	$0.0370443\pi$
$739$	171.619i	0.232231i	−0.993236	+	0.116116i	$0.962956\pi$
$740$	1.62633	0.00219774
$741$	0	0
$742$	−1180.80	−1.59138
$743$	251.057i	0.337896i	0.985625	+	0.168948i	$0.0540370\pi$
$743$	251.057i	0.337896i	−0.985625	+	0.168948i	$0.945963\pi$
$744$	0	0
$745$	− 5.87162i	− 0.00788136i
$746$	−533.839	−0.715602
$747$	0	0
$748$	0	0
$749$	−325.468	−0.434536
$750$	0	0
$751$	403.716	0.537571	0.268786	−	0.963200i	$-0.413378\pi$
$751$	403.716	0.537571	0.268786	+	0.963200i	$0.413378\pi$
$752$	−235.796	−0.313558
$753$	0	0
$754$	821.455i	1.08946i
$755$	36.3291i	0.0481181i
$756$	0	0
$757$	1080.23	1.42699	0.713494	−	0.700662i	$-0.247112\pi$
$757$	1080.23	1.42699	0.713494	+	0.700662i	$0.247112\pi$
$758$	728.628i	0.961250i
$759$	0	0
$760$	11.2651	0.0148225
$761$	− 1359.77i	− 1.78682i	−0.449238	−	0.893412i	$-0.648305\pi$
$761$	− 1359.77i	− 1.78682i	0.449238	−	0.893412i	$-0.351695\pi$
$762$	0	0
$763$	838.342	1.09875
$764$	88.4628	0.115789
$765$	0	0
$766$	− 724.254i	− 0.945501i
$767$	− 495.875i	− 0.646513i
$768$	0	0
$769$	925.077i	1.20296i	0.798888	+	0.601480i	$0.205422\pi$
$769$	925.077i	1.20296i	−0.798888	+	0.601480i	$0.794578\pi$
$770$	0	0
$771$	0	0
$772$	817.771i	1.05929i
$773$	−363.454	−0.470186	−0.235093	−	0.971973i	$-0.575540\pi$
$773$	−363.454	−0.470186	−0.235093	+	0.971973i	$0.575540\pi$
$774$	0	0
$775$	672.308	0.867494
$776$	− 65.1749i	− 0.0839883i
$777$	0	0
$778$	− 433.348i	− 0.557003i
$779$	364.474	0.467874
$780$	0	0
$781$	0	0
$782$	343.043	0.438674
$783$	0	0
$784$	920.195	1.17372
$785$	−40.7761	−0.0519441
$786$	0	0
$787$	− 112.181i	− 0.142542i	−0.997457	−	0.0712712i	$-0.977294\pi$
$787$	− 112.181i	− 0.142542i	0.997457	−	0.0712712i	$-0.0227056\pi$
$788$	1645.82i	2.08861i
$789$	0	0
$790$	119.940	0.151822
$791$	280.110i	0.354122i
$792$	0	0
$793$	−184.815	−0.233058
$794$	1489.20i	1.87557i
$795$	0	0
$796$	−1261.80	−1.58518
$797$	−1444.35	−1.81224	−0.906120	−	0.423021i	$-0.860969\pi$
$797$	−1444.35	−1.81224	−0.906120	+	0.423021i	$0.860969\pi$
$798$	0	0
$799$	− 304.979i	− 0.381701i
$800$	− 1137.50i	− 1.42188i
$801$	0	0
$802$	− 47.9381i	− 0.0597732i
$803$	0	0
$804$	0	0
$805$	− 29.3740i	− 0.0364895i
$806$	−504.083	−0.625413
$807$	0	0
$808$	−43.9554	−0.0544002
$809$	− 593.021i	− 0.733029i	−0.930412	−	0.366515i	$-0.880551\pi$
$809$	− 593.021i	− 0.733029i	0.930412	−	0.366515i	$-0.119449\pi$
$810$	0	0
$811$	432.443i	0.533222i	0.963804	+	0.266611i	$0.0859038\pi$
$811$	432.443i	0.533222i	−0.963804	+	0.266611i	$0.914096\pi$
$812$	2019.74	2.48736
$813$	0	0
$814$	0	0
$815$	103.054	0.126447
$816$	0	0
$817$	1895.61	2.32020
$818$	−1170.52	−1.43096
$819$	0	0
$820$	− 25.2238i	− 0.0307607i
$821$	− 583.606i	− 0.710848i	−0.934705	−	0.355424i	$-0.884337\pi$
$821$	− 583.606i	− 0.710848i	0.934705	−	0.355424i	$-0.115663\pi$
$822$	0	0
$823$	0.274319	0.000333316 0	0.000166658	−	1.00000i	$-0.499947\pi$
$823$	0.274319	0.000333316 0	0.000166658		1.00000i	$0.499947\pi$
$824$	− 88.0541i	− 0.106862i
$825$	0	0
$826$	−2347.77	−2.84234
$827$	1481.38i	1.79127i	0.444789	+	0.895635i	$0.353279\pi$
$827$	1481.38i	1.79127i	−0.444789	+	0.895635i	$0.646721\pi$
$828$	0	0
$829$	−707.744	−0.853732	−0.426866	−	0.904315i	$-0.640382\pi$
$829$	−707.744	−0.853732	−0.426866	+	0.904315i	$0.640382\pi$
$830$	−147.579	−0.177806
$831$	0	0
$832$	476.061i	0.572189i
$833$	1190.18i	1.42879i
$834$	0	0
$835$	− 85.1315i	− 0.101954i
$836$	0	0
$837$	0	0
$838$	− 1813.77i	− 2.16440i
$839$	1350.18	1.60927	0.804636	−	0.593768i	$-0.202360\pi$
$839$	1350.18	1.60927	0.804636	+	0.593768i	$0.202360\pi$
$840$	0	0
$841$	−1109.77	−1.31959
$842$	− 1745.33i	− 2.07284i
$843$	0	0
$844$	144.314i	0.170989i
$845$	56.2134	0.0665247
$846$	0	0
$847$	0	0
$848$	565.192	0.666499
$849$	0	0
$850$	1352.27	1.59090
$851$	5.36852	0.00630848
$852$	0	0
$853$	− 427.682i	− 0.501386i	−0.968067	−	0.250693i	$-0.919341\pi$
$853$	− 427.682i	− 0.501386i	0.968067	−	0.250693i	$-0.0806585\pi$
$854$	875.026i	1.02462i
$855$	0	0
$856$	−28.5147	−0.0333116
$857$	− 281.918i	− 0.328959i	−0.986380	−	0.164480i	$-0.947405\pi$
$857$	− 281.918i	− 0.328959i	0.986380	−	0.164480i	$-0.0525945\pi$
$858$	0	0
$859$	1367.66	1.59215	0.796075	−	0.605198i	$-0.206906\pi$
$859$	1367.66	1.59215	0.796075	+	0.605198i	$0.206906\pi$
$860$	− 131.187i	− 0.152543i
$861$	0	0
$862$	2094.75	2.43011
$863$	−1144.02	−1.32564	−0.662818	−	0.748780i	$-0.730640\pi$
$863$	−1144.02	−1.32564	−0.662818	+	0.748780i	$0.730640\pi$
$864$	0	0
$865$	− 38.8665i	− 0.0449324i
$866$	677.588i	0.782434i
$867$	0	0
$868$	1239.41i	1.42789i
$869$	0	0
$870$	0	0
$871$	− 503.503i	− 0.578075i
$872$	73.4484	0.0842298
$873$	0	0
$874$	500.003	0.572086
$875$	− 232.492i	− 0.265705i
$876$	0	0
$877$	445.784i	0.508306i	0.967164	+	0.254153i	$0.0817966\pi$
$877$	445.784i	0.508306i	−0.967164	+	0.254153i	$0.918203\pi$
$878$	−411.052	−0.468168
$879$	0	0
$880$	0	0
$881$	−1122.27	−1.27386	−0.636931	−	0.770921i	$-0.719796\pi$
$881$	−1122.27	−1.27386	−0.636931	+	0.770921i	$0.719796\pi$
$882$	0	0
$883$	−8.40671	−0.00952063	−0.00476031	−	0.999989i	$-0.501515\pi$
$883$	−8.40671	−0.00952063	−0.00476031	+	0.999989i	$0.501515\pi$
$884$	−526.532	−0.595624
$885$	0	0
$886$	1126.21i	1.27111i
$887$	1330.63i	1.50014i	0.661356	+	0.750072i	$0.269981\pi$
$887$	1330.63i	1.50014i	−0.661356	+	0.750072i	$0.730019\pi$
$888$	0	0
$889$	−1499.93	−1.68721
$890$	83.0667i	0.0933334i
$891$	0	0
$892$	1603.12	1.79722
$893$	− 444.523i	− 0.497786i
$894$	0	0
$895$	−139.964	−0.156385
$896$	312.927	0.349249
$897$	0	0
$898$	2067.59i	2.30244i
$899$	− 1197.09i	− 1.33158i
$900$	0	0
$901$	731.021i	0.811344i
$902$	0	0
$903$	0	0
$904$	24.5409i	0.0271470i
$905$	−6.74025	−0.00744779
$906$	0	0
$907$	36.0706	0.0397691	0.0198846	−	0.999802i	$-0.493670\pi$
$907$	36.0706	0.0397691	0.0198846	+	0.999802i	$0.493670\pi$
$908$	− 493.308i	− 0.543291i
$909$	0	0
$910$	86.8183i	0.0954047i
$911$	−1506.82	−1.65403	−0.827014	−	0.562182i	$-0.809962\pi$
$911$	−1506.82	−1.65403	−0.827014	+	0.562182i	$0.809962\pi$
$912$	0	0
$913$	0	0
$914$	−549.403	−0.601098
$915$	0	0
$916$	231.835	0.253095
$917$	980.197	1.06892
$918$	0	0
$919$	− 449.899i	− 0.489553i	−0.969580	−	0.244776i	$-0.921285\pi$
$919$	− 449.899i	− 0.489553i	0.969580	−	0.244776i	$-0.0787145\pi$
$920$	− 2.57350i	− 0.00279728i
$921$	0	0
$922$	−2474.27	−2.68359
$923$	174.848i	0.189435i
$924$	0	0
$925$	21.1626	0.0228785
$926$	96.5813i	0.104299i
$927$	0	0
$928$	−2025.39	−2.18253
$929$	184.104	0.198175	0.0990874	−	0.995079i	$-0.468408\pi$
$929$	184.104	0.198175	0.0990874	+	0.995079i	$0.468408\pi$
$930$	0	0
$931$	1734.75i	1.86332i
$932$	− 838.176i	− 0.899331i
$933$	0	0
$934$	− 759.677i	− 0.813359i
$935$	0	0
$936$	0	0
$937$	− 397.162i	− 0.423866i	−0.977284	−	0.211933i	$-0.932024\pi$
$937$	− 397.162i	− 0.423866i	0.977284	−	0.211933i	$-0.0679758\pi$
$938$	−2383.89	−2.54146
$939$	0	0
$940$	−30.7637	−0.0327273
$941$	− 964.377i	− 1.02484i	−0.858734	−	0.512421i	$-0.828749\pi$
$941$	− 964.377i	− 1.02484i	0.858734	−	0.512421i	$-0.171251\pi$
$942$	0	0
$943$	− 83.2637i	− 0.0882967i
$944$	1123.76	1.19043
$945$	0	0
$946$	0	0
$947$	−154.326	−0.162963	−0.0814815	−	0.996675i	$-0.525965\pi$
$947$	−154.326	−0.162963	−0.0814815	+	0.996675i	$0.525965\pi$
$948$	0	0
$949$	−83.6916	−0.0881893
$950$	1971.00	2.07474
$951$	0	0
$952$	185.403i	0.194751i
$953$	− 697.639i	− 0.732045i	−0.930606	−	0.366022i	$-0.880719\pi$
$953$	− 697.639i	− 0.732045i	0.930606	−	0.366022i	$-0.119281\pi$
$954$	0	0
$955$	−9.03026	−0.00945577
$956$	648.838i	0.678701i
$957$	0	0
$958$	−1774.45	−1.85225
$959$	232.017i	0.241936i
$960$	0	0
$961$	−226.412	−0.235600
$962$	−15.8673	−0.0164940
$963$	0	0
$964$	− 583.940i	− 0.605747i
$965$	− 83.4778i	− 0.0865055i
$966$	0	0
$967$	1569.06i	1.62260i	0.584627	+	0.811302i	$0.301241\pi$
$967$	1569.06i	1.62260i	−0.584627	+	0.811302i	$0.698759\pi$
$968$	0	0
$969$	0	0
$970$	89.4565i	0.0922232i
$971$	1239.12	1.27613	0.638066	−	0.769982i	$-0.279735\pi$
$971$	1239.12	1.27613	0.638066	+	0.769982i	$0.279735\pi$
$972$	0	0
$973$	−2006.93	−2.06262
$974$	− 860.760i	− 0.883737i
$975$	0	0
$976$	− 418.832i	− 0.429131i
$977$	−71.3624	−0.0730423	−0.0365212	−	0.999333i	$-0.511628\pi$
$977$	−71.3624	−0.0730423	−0.0365212	+	0.999333i	$0.511628\pi$
$978$	0	0
$979$	0	0
$980$	120.055	0.122505
$981$	0	0
$982$	−1343.53	−1.36816
$983$	294.338	0.299428	0.149714	−	0.988729i	$-0.452165\pi$
$983$	294.338	0.299428	0.149714	+	0.988729i	$0.452165\pi$
$984$	0	0
$985$	− 168.005i	− 0.170564i
$986$	− 2407.80i	− 2.44199i
$987$	0	0
$988$	−767.446	−0.776767
$989$	− 433.050i	− 0.437867i
$990$	0	0
$991$	1230.78	1.24195	0.620977	−	0.783829i	$-0.286736\pi$
$991$	1230.78	1.24195	0.620977	+	0.783829i	$0.286736\pi$
$992$	− 1242.87i	− 1.25290i
$993$	0	0
$994$	827.838	0.832835
$995$	128.804	0.129451
$996$	0	0
$997$	− 161.251i	− 0.161736i	−0.996725	−	0.0808679i	$-0.974231\pi$
$997$	− 161.251i	− 0.161736i	0.996725	−	0.0808679i	$-0.0257692\pi$
$998$	− 1574.67i	− 1.57782i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.m.604.2		16
3.2	odd	2		363.3.c.e.241.15		16
11.6	odd	10		99.3.k.c.19.1		16
11.9	even	5		99.3.k.c.73.1		16
11.10	odd	2	inner	1089.3.c.m.604.15		16
33.2	even	10		363.3.g.f.40.1		16
33.5	odd	10		363.3.g.f.118.1		16
33.8	even	10		363.3.g.a.112.1		16
33.14	odd	10		363.3.g.g.112.4		16
33.17	even	10		33.3.g.a.19.4	yes	16
33.20	odd	10		33.3.g.a.7.4	✓	16
33.26	odd	10		363.3.g.a.94.1		16
33.29	even	10		363.3.g.g.94.4		16
33.32	even	2		363.3.c.e.241.2		16
132.83	odd	10		528.3.bf.b.481.4		16
132.119	even	10		528.3.bf.b.337.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.g.a.7.4	✓	16	33.20	odd	10
33.3.g.a.19.4	yes	16	33.17	even	10
99.3.k.c.19.1		16	11.6	odd	10
99.3.k.c.73.1		16	11.9	even	5
363.3.c.e.241.2		16	33.32	even	2
363.3.c.e.241.15		16	3.2	odd	2
363.3.g.a.94.1		16	33.26	odd	10
363.3.g.a.112.1		16	33.8	even	10
363.3.g.f.40.1		16	33.2	even	10
363.3.g.f.118.1		16	33.5	odd	10
363.3.g.g.94.4		16	33.29	even	10
363.3.g.g.112.4		16	33.14	odd	10
528.3.bf.b.337.4		16	132.119	even	10
528.3.bf.b.481.4		16	132.83	odd	10
1089.3.c.m.604.2		16	1.1	even	1	trivial
1089.3.c.m.604.15		16	11.10	odd	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:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1089.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.88468i q^{2} -4.32139 q^{4} +0.441126 q^{5} +10.5820i q^{7} +0.927105i q^{8} +O(q^{10})\)	\(q-2.88468i q^{2} -4.32139 q^{4} +0.441126 q^{5} +10.5820i q^{7} +0.927105i q^{8} -1.27251i q^{10} +6.44736i q^{13} +30.5257 q^{14} -14.6112 q^{16} -18.8981i q^{17} -27.5450i q^{19} -1.90628 q^{20} -6.29263 q^{23} -24.8054 q^{25} +18.5986 q^{26} -45.7290i q^{28} +44.1676i q^{29} -27.1033 q^{31} +45.8569i q^{32} -54.5151 q^{34} +4.66800i q^{35} -0.853144 q^{37} -79.4585 q^{38} +0.408970i q^{40} +13.2319i q^{41} +68.8186i q^{43} +18.1522i q^{46} +16.1381 q^{47} -62.9789 q^{49} +71.5557i q^{50} -27.8616i q^{52} -38.6822 q^{53} -9.81064 q^{56} +127.409 q^{58} -76.9113 q^{59} +28.6652i q^{61} +78.1844i q^{62} +73.8381 q^{64} +2.84410i q^{65} -78.0944 q^{67} +81.6662i q^{68} +13.4657 q^{70} +27.1193 q^{71} +12.9807i q^{73} +2.46105i q^{74} +119.033i q^{76} +94.2544i q^{79} -6.44536 q^{80} +38.1699 q^{82} -115.975i q^{83} -8.33646i q^{85} +198.520 q^{86} -65.2779 q^{89} -68.2261 q^{91} +27.1929 q^{92} -46.5532i q^{94} -12.1508i q^{95} -70.2994 q^{97} +181.674i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 20 q^{4} + 4 q^{5}+O(q^{10}) \) 16 * q - 20 * q^4 + 4 * q^5	\( 16 q - 20 q^{4} + 4 q^{5} + 52 q^{14} - 44 q^{16} + 108 q^{20} - 132 q^{23} + 88 q^{25} + 4 q^{26} + 40 q^{31} - 368 q^{34} - 16 q^{37} - 280 q^{38} - 80 q^{47} - 140 q^{49} + 128 q^{53} - 524 q^{56} + 140 q^{58} + 220 q^{59} - 8 q^{64} + 36 q^{67} - 100 q^{70} - 644 q^{71} - 264 q^{80} - 476 q^{82} - 76 q^{86} - 76 q^{89} - 624 q^{91} - 120 q^{92} + 216 q^{97}+O(q^{100}) \) 16 * q - 20 * q^4 + 4 * q^5 + 52 * q^14 - 44 * q^16 + 108 * q^20 - 132 * q^23 + 88 * q^25 + 4 * q^26 + 40 * q^31 - 368 * q^34 - 16 * q^37 - 280 * q^38 - 80 * q^47 - 140 * q^49 + 128 * q^53 - 524 * q^56 + 140 * q^58 + 220 * q^59 - 8 * q^64 + 36 * q^67 - 100 * q^70 - 644 * q^71 - 264 * q^80 - 476 * q^82 - 76 * q^86 - 76 * q^89 - 624 * q^91 - 120 * q^92 + 216 * q^97

Introduction

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