LMFDB - Embedded newform 1089.3.c.j.604.5

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			604.5
Root			$0.912245 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.j.604.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.124105i q^{2} +3.98460 q^{4} -4.55654 q^{5} -5.06958i q^{7} +0.990926i q^{8} +O(q^{10})$	$q+0.124105i q^{2} +3.98460 q^{4} -4.55654 q^{5} -5.06958i q^{7} +0.990926i q^{8} -0.565488i q^{10} -7.20664i q^{13} +0.629159 q^{14} +15.8154 q^{16} +23.3204i q^{17} +26.1041i q^{19} -18.1560 q^{20} +41.1662 q^{23} -4.23794 q^{25} +0.894378 q^{26} -20.2003i q^{28} -49.6430i q^{29} -9.46048 q^{31} +5.92647i q^{32} -2.89417 q^{34} +23.0998i q^{35} +49.5316 q^{37} -3.23964 q^{38} -4.51519i q^{40} -69.5972i q^{41} -47.5666i q^{43} +5.10891i q^{46} -2.85744 q^{47} +23.2993 q^{49} -0.525948i q^{50} -28.7156i q^{52} +18.6798 q^{53} +5.02358 q^{56} +6.16093 q^{58} +7.18782 q^{59} -42.5499i q^{61} -1.17409i q^{62} +62.5262 q^{64} +32.8374i q^{65} +66.5772 q^{67} +92.9223i q^{68} -2.86679 q^{70} +49.2289 q^{71} -35.8561i q^{73} +6.14710i q^{74} +104.014i q^{76} +105.196i q^{79} -72.0636 q^{80} +8.63734 q^{82} -127.287i q^{83} -106.260i q^{85} +5.90324 q^{86} +93.7491 q^{89} -36.5347 q^{91} +164.031 q^{92} -0.354622i q^{94} -118.944i q^{95} -52.2081 q^{97} +2.89155i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.124105i	0.0620523i	0.999519	+	0.0310262i	$0.00987752\pi$
$2$	0.124105i	0.0620523i	−0.999519	+	0.0310262i	$0.990122\pi$
$3$	0	0
$3$	0	0
$4$	3.98460	0.996150
$5$	−4.55654	−0.911308	−0.455654	−	0.890157i	$-0.650595\pi$
$5$	−4.55654	−0.911308	−0.455654	+	0.890157i	$0.650595\pi$
$6$	0	0
$7$	− 5.06958i	− 0.724226i	−0.932134	−	0.362113i	$-0.882055\pi$
$7$	− 5.06958i	− 0.724226i	0.932134	−	0.362113i	$-0.117945\pi$
$8$	0.990926i	0.123866i
$9$	0	0
$10$	− 0.565488i	− 0.0565488i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 7.20664i	− 0.554357i	−0.960818	−	0.277179i	$-0.910601\pi$
$13$	− 7.20664i	− 0.554357i	0.960818	−	0.277179i	$-0.0893994\pi$
$14$	0.629159	0.0449399
$15$	0	0
$16$	15.8154	0.988463
$17$	23.3204i	1.37179i	0.727703	+	0.685893i	$0.240588\pi$
$17$	23.3204i	1.37179i	−0.727703	+	0.685893i	$0.759412\pi$
$18$	0	0
$19$	26.1041i	1.37390i	0.726706	+	0.686949i	$0.241050\pi$
$19$	26.1041i	1.37390i	−0.726706	+	0.686949i	$0.758950\pi$
$20$	−18.1560	−0.907799
$21$	0	0
$22$	0	0
$23$	41.1662	1.78983	0.894917	−	0.446233i	$-0.147235\pi$
$23$	41.1662	1.78983	0.894917	+	0.446233i	$0.147235\pi$
$24$	0	0
$25$	−4.23794	−0.169518
$26$	0.894378	0.0343992
$27$	0	0
$28$	− 20.2003i	− 0.721438i
$29$	− 49.6430i	− 1.71183i	−0.517118	−	0.855914i	$-0.672995\pi$
$29$	− 49.6430i	− 1.71183i	0.517118	−	0.855914i	$-0.327005\pi$
$30$	0	0
$31$	−9.46048	−0.305177	−0.152588	−	0.988290i	$-0.548761\pi$
$31$	−9.46048	−0.305177	−0.152588	+	0.988290i	$0.548761\pi$
$32$	5.92647i	0.185202i
$33$	0	0
$34$	−2.89417	−0.0851225
$35$	23.0998i	0.659993i
$36$	0	0
$37$	49.5316	1.33869	0.669345	−	0.742951i	$-0.266575\pi$
$37$	49.5316	1.33869	0.669345	+	0.742951i	$0.266575\pi$
$38$	−3.23964	−0.0852536
$39$	0	0
$40$	− 4.51519i	− 0.112880i
$41$	− 69.5972i	− 1.69749i	−0.528800	−	0.848747i	$-0.677358\pi$
$41$	− 69.5972i	− 1.69749i	0.528800	−	0.848747i	$-0.322642\pi$
$42$	0	0
$43$	− 47.5666i	− 1.10620i	−0.833115	−	0.553100i	$-0.813445\pi$
$43$	− 47.5666i	− 1.10620i	0.833115	−	0.553100i	$-0.186555\pi$
$44$	0	0
$45$	0	0
$46$	5.10891i	0.111063i
$47$	−2.85744	−0.0607966	−0.0303983	−	0.999538i	$-0.509678\pi$
$47$	−2.85744	−0.0607966	−0.0303983	+	0.999538i	$0.509678\pi$
$48$	0	0
$49$	23.2993	0.475496
$50$	− 0.525948i	− 0.0105190i
$51$	0	0
$52$	− 28.7156i	− 0.552223i
$53$	18.6798	0.352449	0.176225	−	0.984350i	$-0.443611\pi$
$53$	18.6798	0.352449	0.176225	+	0.984350i	$0.443611\pi$
$54$	0	0
$55$	0	0
$56$	5.02358	0.0897068
$57$	0	0
$58$	6.16093	0.106223
$59$	7.18782	0.121827	0.0609137	−	0.998143i	$-0.480599\pi$
$59$	7.18782	0.121827	0.0609137	+	0.998143i	$0.480599\pi$
$60$	0	0
$61$	− 42.5499i	− 0.697539i	−0.937208	−	0.348770i	$-0.886600\pi$
$61$	− 42.5499i	− 0.697539i	0.937208	−	0.348770i	$-0.113400\pi$
$62$	− 1.17409i	− 0.0189369i
$63$	0	0
$64$	62.5262	0.976971
$65$	32.8374i	0.505190i
$66$	0	0
$67$	66.5772	0.993689	0.496845	−	0.867840i	$-0.334492\pi$
$67$	66.5772	0.993689	0.496845	+	0.867840i	$0.334492\pi$
$68$	92.9223i	1.36650i
$69$	0	0
$70$	−2.86679	−0.0409541
$71$	49.2289	0.693365	0.346683	−	0.937982i	$-0.387308\pi$
$71$	49.2289	0.693365	0.346683	+	0.937982i	$0.387308\pi$
$72$	0	0
$73$	− 35.8561i	− 0.491179i	−0.969374	−	0.245589i	$-0.921018\pi$
$73$	− 35.8561i	− 0.491179i	0.969374	−	0.245589i	$-0.0789815\pi$
$74$	6.14710i	0.0830689i
$75$	0	0
$76$	104.014i	1.36861i
$77$	0	0
$78$	0	0
$79$	105.196i	1.33160i	0.746130	+	0.665800i	$0.231910\pi$
$79$	105.196i	1.33160i	−0.746130	+	0.665800i	$0.768090\pi$
$80$	−72.0636	−0.900795
$81$	0	0
$82$	8.63734	0.105333
$83$	− 127.287i	− 1.53358i	−0.641898	−	0.766790i	$-0.721853\pi$
$83$	− 127.287i	− 1.53358i	0.641898	−	0.766790i	$-0.278147\pi$
$84$	0	0
$85$	− 106.260i	− 1.25012i
$86$	5.90324	0.0686423
$87$	0	0
$88$	0	0
$89$	93.7491	1.05336	0.526680	−	0.850063i	$-0.323436\pi$
$89$	93.7491	1.05336	0.526680	+	0.850063i	$0.323436\pi$
$90$	0	0
$91$	−36.5347	−0.401480
$92$	164.031	1.78294
$93$	0	0
$94$	− 0.354622i	− 0.00377257i
$95$	− 118.944i	− 1.25204i
$96$	0	0
$97$	−52.2081	−0.538227	−0.269114	−	0.963108i	$-0.586731\pi$
$97$	−52.2081	−0.538227	−0.269114	+	0.963108i	$0.586731\pi$
$98$	2.89155i	0.0295056i
$99$	0	0
$100$	−16.8865	−0.168865
$101$	45.8271i	0.453734i	0.973926	+	0.226867i	$0.0728482\pi$
$101$	45.8271i	0.453734i	−0.973926	+	0.226867i	$0.927152\pi$
$102$	0	0
$103$	−50.3265	−0.488606	−0.244303	−	0.969699i	$-0.578559\pi$
$103$	−50.3265	−0.488606	−0.244303	+	0.969699i	$0.578559\pi$
$104$	7.14125	0.0686659
$105$	0	0
$106$	2.31825i	0.0218703i
$107$	− 24.6692i	− 0.230553i	−0.993333	−	0.115276i	$-0.963225\pi$
$107$	− 24.6692i	− 0.230553i	0.993333	−	0.115276i	$-0.0367754\pi$
$108$	0	0
$109$	25.5067i	0.234006i	0.993132	+	0.117003i	$0.0373288\pi$
$109$	25.5067i	0.234006i	−0.993132	+	0.117003i	$0.962671\pi$
$110$	0	0
$111$	0	0
$112$	− 80.1776i	− 0.715871i
$113$	−73.9044	−0.654021	−0.327010	−	0.945021i	$-0.606041\pi$
$113$	−73.9044	−0.654021	−0.327010	+	0.945021i	$0.606041\pi$
$114$	0	0
$115$	−187.575	−1.63109
$116$	− 197.807i	− 1.70524i
$117$	0	0
$118$	0.892042i	0.00755968i
$119$	118.225	0.993483
$120$	0	0
$121$	0	0
$122$	5.28064	0.0432839
$123$	0	0
$124$	−37.6962	−0.304002
$125$	133.224	1.06579
$126$	0	0
$127$	175.883i	1.38490i	0.721464	+	0.692452i	$0.243470\pi$
$127$	175.883i	1.38490i	−0.721464	+	0.692452i	$0.756530\pi$
$128$	31.4657i	0.245826i
$129$	0	0
$130$	−4.07527	−0.0313482
$131$	124.649i	0.951521i	0.879575	+	0.475760i	$0.157827\pi$
$131$	124.649i	0.951521i	−0.879575	+	0.475760i	$0.842173\pi$
$132$	0	0
$133$	132.337	0.995013
$134$	8.26254i	0.0616607i
$135$	0	0
$136$	−23.1087	−0.169917
$137$	212.834	1.55353	0.776766	−	0.629790i	$-0.216859\pi$
$137$	212.834	1.55353	0.776766	+	0.629790i	$0.216859\pi$
$138$	0	0
$139$	− 135.725i	− 0.976441i	−0.872720	−	0.488220i	$-0.837646\pi$
$139$	− 135.725i	− 0.976441i	0.872720	−	0.488220i	$-0.162354\pi$
$140$	92.0433i	0.657452i
$141$	0	0
$142$	6.10954i	0.0430249i
$143$	0	0
$144$	0	0
$145$	226.200i	1.56000i
$146$	4.44990	0.0304788
$147$	0	0
$148$	197.363	1.33354
$149$	192.808i	1.29401i	0.762485	+	0.647006i	$0.223979\pi$
$149$	192.808i	1.29401i	−0.762485	+	0.647006i	$0.776021\pi$
$150$	0	0
$151$	7.64995i	0.0506619i	0.999679	+	0.0253310i	$0.00806396\pi$
$151$	7.64995i	0.0506619i	−0.999679	+	0.0253310i	$0.991936\pi$
$152$	−25.8672	−0.170179
$153$	0	0
$154$	0	0
$155$	43.1071	0.278110
$156$	0	0
$157$	−45.0545	−0.286971	−0.143486	−	0.989652i	$-0.545831\pi$
$157$	−45.0545	−0.286971	−0.143486	+	0.989652i	$0.545831\pi$
$158$	−13.0554	−0.0826289
$159$	0	0
$160$	− 27.0042i	− 0.168776i
$161$	− 208.695i	− 1.29624i
$162$	0	0
$163$	−186.553	−1.14450	−0.572248	−	0.820081i	$-0.693928\pi$
$163$	−186.553	−1.14450	−0.572248	+	0.820081i	$0.693928\pi$
$164$	− 277.317i	− 1.69096i
$165$	0	0
$166$	15.7969	0.0951622
$167$	− 48.6198i	− 0.291137i	−0.989348	−	0.145568i	$-0.953499\pi$
$167$	− 48.6198i	− 0.291137i	0.989348	−	0.145568i	$-0.0465011\pi$
$168$	0	0
$169$	117.064	0.692688
$170$	13.1874	0.0775728
$171$	0	0
$172$	− 189.534i	− 1.10194i
$173$	− 17.1770i	− 0.0992891i	−0.998767	−	0.0496445i	$-0.984191\pi$
$173$	− 17.1770i	− 0.0992891i	0.998767	−	0.0496445i	$-0.0158088\pi$
$174$	0	0
$175$	21.4846i	0.122769i
$176$	0	0
$177$	0	0
$178$	11.6347i	0.0653635i
$179$	148.067	0.827190	0.413595	−	0.910461i	$-0.364273\pi$
$179$	148.067	0.827190	0.413595	+	0.910461i	$0.364273\pi$
$180$	0	0
$181$	−28.1792	−0.155686	−0.0778430	−	0.996966i	$-0.524803\pi$
$181$	−28.1792	−0.155686	−0.0778430	+	0.996966i	$0.524803\pi$
$182$	− 4.53413i	− 0.0249128i
$183$	0	0
$184$	40.7926i	0.221699i
$185$	−225.693	−1.21996
$186$	0	0
$187$	0	0
$188$	−11.3858	−0.0605625
$189$	0	0
$190$	14.7615	0.0776923
$191$	−51.5509	−0.269900	−0.134950	−	0.990852i	$-0.543087\pi$
$191$	−51.5509	−0.269900	−0.134950	+	0.990852i	$0.543087\pi$
$192$	0	0
$193$	− 60.2956i	− 0.312412i	−0.987724	−	0.156206i	$-0.950074\pi$
$193$	− 60.2956i	− 0.312412i	0.987724	−	0.156206i	$-0.0499264\pi$
$194$	− 6.47926i	− 0.0333983i
$195$	0	0
$196$	92.8384	0.473665
$197$	9.38629i	0.0476461i	0.999716	+	0.0238231i	$0.00758384\pi$
$197$	9.38629i	0.0476461i	−0.999716	+	0.0238231i	$0.992416\pi$
$198$	0	0
$199$	−174.756	−0.878173	−0.439086	−	0.898445i	$-0.644698\pi$
$199$	−174.756	−0.878173	−0.439086	+	0.898445i	$0.644698\pi$
$200$	− 4.19948i	− 0.0209974i
$201$	0	0
$202$	−5.68736	−0.0281552
$203$	−251.670	−1.23975
$204$	0	0
$205$	317.123i	1.54694i
$206$	− 6.24575i	− 0.0303192i
$207$	0	0
$208$	− 113.976i	− 0.547962i
$209$	0	0
$210$	0	0
$211$	− 360.511i	− 1.70858i	−0.519794	−	0.854292i	$-0.673991\pi$
$211$	− 360.511i	− 1.70858i	0.519794	−	0.854292i	$-0.326009\pi$
$212$	74.4315	0.351092
$213$	0	0
$214$	3.06156	0.0143063
$215$	216.739i	1.00809i
$216$	0	0
$217$	47.9607i	0.221017i
$218$	−3.16550	−0.0145206
$219$	0	0
$220$	0	0
$221$	168.062	0.760459
$222$	0	0
$223$	155.449	0.697082	0.348541	−	0.937293i	$-0.386677\pi$
$223$	155.449	0.697082	0.348541	+	0.937293i	$0.386677\pi$
$224$	30.0447	0.134128
$225$	0	0
$226$	− 9.17187i	− 0.0405835i
$227$	151.224i	0.666185i	0.942894	+	0.333093i	$0.108092\pi$
$227$	151.224i	0.666185i	−0.942894	+	0.333093i	$0.891908\pi$
$228$	0	0
$229$	307.605	1.34326	0.671628	−	0.740889i	$-0.265596\pi$
$229$	307.605	1.34326	0.671628	+	0.740889i	$0.265596\pi$
$230$	− 23.2790i	− 0.101213i
$231$	0	0
$232$	49.1926	0.212037
$233$	100.478i	0.431237i	0.976478	+	0.215619i	$0.0691768\pi$
$233$	100.478i	0.431237i	−0.976478	+	0.215619i	$0.930823\pi$
$234$	0	0
$235$	13.0200	0.0554045
$236$	28.6406	0.121358
$237$	0	0
$238$	14.6722i	0.0616480i
$239$	− 460.051i	− 1.92490i	−0.271457	−	0.962451i	$-0.587505\pi$
$239$	− 460.051i	− 1.92490i	0.271457	−	0.962451i	$-0.412495\pi$
$240$	0	0
$241$	− 251.848i	− 1.04501i	−0.852635	−	0.522506i	$-0.824997\pi$
$241$	− 251.848i	− 1.04501i	0.852635	−	0.522506i	$-0.175003\pi$
$242$	0	0
$243$	0	0
$244$	− 169.544i	− 0.694853i
$245$	−106.164	−0.433323
$246$	0	0
$247$	188.123	0.761630
$248$	− 9.37463i	− 0.0378009i
$249$	0	0
$250$	16.5337i	0.0661348i
$251$	84.2374	0.335607	0.167804	−	0.985820i	$-0.446333\pi$
$251$	84.2374	0.335607	0.167804	+	0.985820i	$0.446333\pi$
$252$	0	0
$253$	0	0
$254$	−21.8279	−0.0859365
$255$	0	0
$256$	246.200	0.961717
$257$	−495.580	−1.92833	−0.964163	−	0.265311i	$-0.914526\pi$
$257$	−495.580	−1.92833	−0.964163	+	0.265311i	$0.914526\pi$
$258$	0	0
$259$	− 251.104i	− 0.969515i
$260$	130.844i	0.503245i
$261$	0	0
$262$	−15.4695	−0.0590441
$263$	303.296i	1.15322i	0.817021	+	0.576608i	$0.195624\pi$
$263$	303.296i	1.15322i	−0.817021	+	0.576608i	$0.804376\pi$
$264$	0	0
$265$	−85.1153	−0.321190
$266$	16.4236i	0.0617429i
$267$	0	0
$268$	265.283	0.989863
$269$	−393.079	−1.46126	−0.730630	−	0.682774i	$-0.760773\pi$
$269$	−393.079	−1.46126	−0.730630	+	0.682774i	$0.760773\pi$
$270$	0	0
$271$	121.331i	0.447717i	0.974622	+	0.223859i	$0.0718654\pi$
$271$	121.331i	0.447717i	−0.974622	+	0.223859i	$0.928135\pi$
$272$	368.821i	1.35596i
$273$	0	0
$274$	26.4137i	0.0964003i
$275$	0	0
$276$	0	0
$277$	349.144i	1.26045i	0.776414	+	0.630224i	$0.217037\pi$
$277$	349.144i	1.26045i	−0.776414	+	0.630224i	$0.782963\pi$
$278$	16.8441	0.0605904
$279$	0	0
$280$	−22.8902	−0.0817506
$281$	214.011i	0.761607i	0.924656	+	0.380803i	$0.124353\pi$
$281$	214.011i	0.761607i	−0.924656	+	0.380803i	$0.875647\pi$
$282$	0	0
$283$	130.356i	0.460621i	0.973117	+	0.230310i	$0.0739741\pi$
$283$	130.356i	0.460621i	−0.973117	+	0.230310i	$0.926026\pi$
$284$	196.158	0.690696
$285$	0	0
$286$	0	0
$287$	−352.829	−1.22937
$288$	0	0
$289$	−254.839	−0.881796
$290$	−28.0725	−0.0968018
$291$	0	0
$292$	− 142.872i	− 0.489288i
$293$	343.644i	1.17285i	0.810005	+	0.586423i	$0.199464\pi$
$293$	343.644i	1.17285i	−0.810005	+	0.586423i	$0.800536\pi$
$294$	0	0
$295$	−32.7516	−0.111022
$296$	49.0821i	0.165818i
$297$	0	0
$298$	−23.9283	−0.0802964
$299$	− 296.670i	− 0.992207i
$300$	0	0
$301$	−241.143	−0.801140
$302$	−0.949395	−0.00314369
$303$	0	0
$304$	412.847i	1.35805i
$305$	193.880i	0.635673i
$306$	0	0
$307$	368.749i	1.20114i	0.799573	+	0.600568i	$0.205059\pi$
$307$	368.749i	1.20114i	−0.799573	+	0.600568i	$0.794941\pi$
$308$	0	0
$309$	0	0
$310$	5.34979i	0.0172574i
$311$	143.959	0.462890	0.231445	−	0.972848i	$-0.425655\pi$
$311$	143.959	0.462890	0.231445	+	0.972848i	$0.425655\pi$
$312$	0	0
$313$	−300.681	−0.960642	−0.480321	−	0.877093i	$-0.659480\pi$
$313$	−300.681	−0.960642	−0.480321	+	0.877093i	$0.659480\pi$
$314$	− 5.59147i	− 0.0178072i
$315$	0	0
$316$	419.165i	1.32647i
$317$	−198.224	−0.625312	−0.312656	−	0.949866i	$-0.601219\pi$
$317$	−198.224	−0.625312	−0.312656	+	0.949866i	$0.601219\pi$
$318$	0	0
$319$	0	0
$320$	−284.903	−0.890322
$321$	0	0
$322$	25.9001	0.0804350
$323$	−608.756	−1.88469
$324$	0	0
$325$	30.5413i	0.0939733i
$326$	− 23.1521i	− 0.0710186i
$327$	0	0
$328$	68.9657	0.210261
$329$	14.4860i	0.0440305i
$330$	0	0
$331$	−332.018	−1.00307	−0.501537	−	0.865136i	$-0.667232\pi$
$331$	−332.018	−1.00307	−0.501537	+	0.865136i	$0.667232\pi$
$332$	− 507.188i	− 1.52767i
$333$	0	0
$334$	6.03395	0.0180657
$335$	−303.362	−0.905557
$336$	0	0
$337$	− 234.104i	− 0.694671i	−0.937741	−	0.347335i	$-0.887087\pi$
$337$	− 234.104i	− 0.694671i	0.937741	−	0.347335i	$-0.112913\pi$
$338$	14.5282i	0.0429829i
$339$	0	0
$340$	− 423.404i	− 1.24531i
$341$	0	0
$342$	0	0
$343$	− 366.527i	− 1.06859i
$344$	47.1350	0.137020
$345$	0	0
$346$	2.13175	0.00616112
$347$	57.6196i	0.166051i	0.996547	+	0.0830254i	$0.0264583\pi$
$347$	57.6196i	0.166051i	−0.996547	+	0.0830254i	$0.973542\pi$
$348$	0	0
$349$	− 611.166i	− 1.75119i	−0.483045	−	0.875596i	$-0.660469\pi$
$349$	− 611.166i	− 1.75119i	0.483045	−	0.875596i	$-0.339531\pi$
$350$	−2.66634	−0.00761811
$351$	0	0
$352$	0	0
$353$	−462.545	−1.31032	−0.655162	−	0.755488i	$-0.727400\pi$
$353$	−462.545	−1.31032	−0.655162	+	0.755488i	$0.727400\pi$
$354$	0	0
$355$	−224.314	−0.631869
$356$	373.553	1.04930
$357$	0	0
$358$	18.3758i	0.0513290i
$359$	598.189i	1.66626i	0.553074	+	0.833132i	$0.313455\pi$
$359$	598.189i	1.66626i	−0.553074	+	0.833132i	$0.686545\pi$
$360$	0	0
$361$	−320.422	−0.887595
$362$	− 3.49717i	− 0.00966068i
$363$	0	0
$364$	−145.576	−0.399934
$365$	163.380i	0.447615i
$366$	0	0
$367$	−549.990	−1.49861	−0.749306	−	0.662224i	$-0.769613\pi$
$367$	−549.990	−1.49861	−0.749306	+	0.662224i	$0.769613\pi$
$368$	651.060	1.76918
$369$	0	0
$370$	− 28.0095i	− 0.0757013i
$371$	− 94.6989i	− 0.255253i
$372$	0	0
$373$	394.767i	1.05836i	0.848511	+	0.529178i	$0.177500\pi$
$373$	394.767i	1.05836i	−0.848511	+	0.529178i	$0.822500\pi$
$374$	0	0
$375$	0	0
$376$	− 2.83151i	− 0.00753062i
$377$	−357.760	−0.948965
$378$	0	0
$379$	201.197	0.530864	0.265432	−	0.964130i	$-0.414485\pi$
$379$	201.197	0.530864	0.265432	+	0.964130i	$0.414485\pi$
$380$	− 473.945i	− 1.24722i
$381$	0	0
$382$	− 6.39771i	− 0.0167479i
$383$	80.4688	0.210101	0.105051	−	0.994467i	$-0.466500\pi$
$383$	80.4688	0.210101	0.105051	+	0.994467i	$0.466500\pi$
$384$	0	0
$385$	0	0
$386$	7.48296	0.0193859
$387$	0	0
$388$	−208.028	−0.536155
$389$	−99.8895	−0.256785	−0.128393	−	0.991723i	$-0.540982\pi$
$389$	−99.8895	−0.256785	−0.128393	+	0.991723i	$0.540982\pi$
$390$	0	0
$391$	960.010i	2.45527i
$392$	23.0879i	0.0588977i
$393$	0	0
$394$	−1.16488	−0.00295655
$395$	− 479.332i	− 1.21350i
$396$	0	0
$397$	195.426	0.492256	0.246128	−	0.969237i	$-0.420842\pi$
$397$	195.426	0.492256	0.246128	+	0.969237i	$0.420842\pi$
$398$	− 21.6881i	− 0.0544927i
$399$	0	0
$400$	−67.0248	−0.167562
$401$	21.7160	0.0541547	0.0270773	−	0.999633i	$-0.491380\pi$
$401$	21.7160	0.0541547	0.0270773	+	0.999633i	$0.491380\pi$
$402$	0	0
$403$	68.1783i	0.169177i
$404$	182.603i	0.451987i
$405$	0	0
$406$	− 31.2334i	− 0.0769295i
$407$	0	0
$408$	0	0
$409$	− 140.738i	− 0.344103i	−0.985088	−	0.172052i	$-0.944960\pi$
$409$	− 140.738i	− 0.344103i	0.985088	−	0.172052i	$-0.0550396\pi$
$410$	−39.3564	−0.0959912
$411$	0	0
$412$	−200.531	−0.486725
$413$	− 36.4393i	− 0.0882307i
$414$	0	0
$415$	579.989i	1.39756i
$416$	42.7100	0.102668
$417$	0	0
$418$	0	0
$419$	−205.230	−0.489810	−0.244905	−	0.969547i	$-0.578757\pi$
$419$	−205.230	−0.489810	−0.244905	+	0.969547i	$0.578757\pi$
$420$	0	0
$421$	775.170	1.84126	0.920629	−	0.390439i	$-0.127677\pi$
$421$	775.170	1.84126	0.920629	+	0.390439i	$0.127677\pi$
$422$	44.7411	0.106022
$423$	0	0
$424$	18.5103i	0.0436564i
$425$	− 98.8303i	− 0.232542i
$426$	0	0
$427$	−215.710	−0.505176
$428$	− 98.2966i	− 0.229665i
$429$	0	0
$430$	−26.8984	−0.0625543
$431$	54.6468i	0.126791i	0.997988	+	0.0633954i	$0.0201929\pi$
$431$	54.6468i	0.126791i	−0.997988	+	0.0633954i	$0.979807\pi$
$432$	0	0
$433$	−498.812	−1.15199	−0.575996	−	0.817453i	$-0.695385\pi$
$433$	−498.812	−1.15199	−0.575996	+	0.817453i	$0.695385\pi$
$434$	−5.95215	−0.0137146
$435$	0	0
$436$	101.634i	0.233105i
$437$	1074.60i	2.45905i
$438$	0	0
$439$	125.316i	0.285457i	0.989762	+	0.142729i	$0.0455876\pi$
$439$	125.316i	0.285457i	−0.989762	+	0.142729i	$0.954412\pi$
$440$	0	0
$441$	0	0
$442$	20.8572i	0.0471883i
$443$	613.792	1.38553	0.692767	−	0.721161i	$-0.256391\pi$
$443$	613.792	1.38553	0.692767	+	0.721161i	$0.256391\pi$
$444$	0	0
$445$	−427.172	−0.959936
$446$	19.2920i	0.0432556i
$447$	0	0
$448$	− 316.982i	− 0.707548i
$449$	−385.155	−0.857806	−0.428903	−	0.903351i	$-0.641100\pi$
$449$	−385.155	−0.857806	−0.428903	+	0.903351i	$0.641100\pi$
$450$	0	0
$451$	0	0
$452$	−294.479	−0.651503
$453$	0	0
$454$	−18.7676	−0.0413383
$455$	166.472	0.365872
$456$	0	0
$457$	44.6202i	0.0976373i	0.998808	+	0.0488186i	$0.0155456\pi$
$457$	44.6202i	0.0976373i	−0.998808	+	0.0488186i	$0.984454\pi$
$458$	38.1753i	0.0833521i
$459$	0	0
$460$	−747.412	−1.62481
$461$	− 392.409i	− 0.851212i	−0.904909	−	0.425606i	$-0.860061\pi$
$461$	− 392.409i	− 0.851212i	0.904909	−	0.425606i	$-0.139939\pi$
$462$	0	0
$463$	−98.5450	−0.212840	−0.106420	−	0.994321i	$-0.533939\pi$
$463$	−98.5450	−0.212840	−0.106420	+	0.994321i	$0.533939\pi$
$464$	− 785.125i	− 1.69208i
$465$	0	0
$466$	−12.4698	−0.0267593
$467$	−269.776	−0.577679	−0.288839	−	0.957378i	$-0.593269\pi$
$467$	−269.776	−0.577679	−0.288839	+	0.957378i	$0.593269\pi$
$468$	0	0
$469$	− 337.519i	− 0.719656i
$470$	1.61585i	0.00343798i
$471$	0	0
$472$	7.12260i	0.0150902i
$473$	0	0
$474$	0	0
$475$	− 110.627i	− 0.232900i
$476$	471.077	0.989658
$477$	0	0
$478$	57.0945	0.119445
$479$	408.366i	0.852539i	0.904596	+	0.426270i	$0.140173\pi$
$479$	408.366i	0.852539i	−0.904596	+	0.426270i	$0.859827\pi$
$480$	0	0
$481$	− 356.956i	− 0.742113i
$482$	31.2555	0.0648455
$483$	0	0
$484$	0	0
$485$	237.888	0.490491
$486$	0	0
$487$	−494.516	−1.01543	−0.507717	−	0.861524i	$-0.669510\pi$
$487$	−494.516	−1.01543	−0.507717	+	0.861524i	$0.669510\pi$
$488$	42.1638	0.0864012
$489$	0	0
$490$	− 13.1755i	− 0.0268887i
$491$	− 37.3031i	− 0.0759738i	−0.999278	−	0.0379869i	$-0.987905\pi$
$491$	− 37.3031i	− 0.0759738i	0.999278	−	0.0379869i	$-0.0120945\pi$
$492$	0	0
$493$	1157.69	2.34826
$494$	23.3469i	0.0472609i
$495$	0	0
$496$	−149.621	−0.301656
$497$	− 249.570i	− 0.502154i
$498$	0	0
$499$	202.446	0.405703	0.202851	−	0.979210i	$-0.434979\pi$
$499$	202.446	0.405703	0.202851	+	0.979210i	$0.434979\pi$
$500$	530.844	1.06169
$501$	0	0
$502$	10.4543i	0.0208252i
$503$	− 369.829i	− 0.735247i	−0.929975	−	0.367623i	$-0.880172\pi$
$503$	− 369.829i	− 0.735247i	0.929975	−	0.367623i	$-0.119828\pi$
$504$	0	0
$505$	− 208.813i	− 0.413491i
$506$	0	0
$507$	0	0
$508$	700.822i	1.37957i
$509$	−76.0419	−0.149395	−0.0746974	−	0.997206i	$-0.523799\pi$
$509$	−76.0419	−0.149395	−0.0746974	+	0.997206i	$0.523799\pi$
$510$	0	0
$511$	−181.775	−0.355725
$512$	156.417i	0.305502i
$513$	0	0
$514$	− 61.5038i	− 0.119657i
$515$	229.315	0.445271
$516$	0	0
$517$	0	0
$518$	31.1632	0.0601607
$519$	0	0
$520$	−32.5394	−0.0625758
$521$	339.649	0.651918	0.325959	−	0.945384i	$-0.394313\pi$
$521$	339.649	0.651918	0.325959	+	0.945384i	$0.394313\pi$
$522$	0	0
$523$	102.616i	0.196207i	0.995176	+	0.0981035i	$0.0312776\pi$
$523$	102.616i	0.196207i	−0.995176	+	0.0981035i	$0.968722\pi$
$524$	496.677i	0.947857i
$525$	0	0
$526$	−37.6404	−0.0715597
$527$	− 220.622i	− 0.418637i
$528$	0	0
$529$	1165.65	2.20350
$530$	− 10.5632i	− 0.0199306i
$531$	0	0
$532$	527.309	0.991182
$533$	−501.562	−0.941018
$534$	0	0
$535$	112.406i	0.210105i
$536$	65.9730i	0.123084i
$537$	0	0
$538$	− 48.7829i	− 0.0906745i
$539$	0	0
$540$	0	0
$541$	− 431.271i	− 0.797173i	−0.917131	−	0.398587i	$-0.869501\pi$
$541$	− 431.271i	− 0.797173i	0.917131	−	0.398587i	$-0.130499\pi$
$542$	−15.0578	−0.0277819
$543$	0	0
$544$	−138.207	−0.254058
$545$	− 116.222i	− 0.213252i
$546$	0	0
$547$	− 518.823i	− 0.948488i	−0.880393	−	0.474244i	$-0.842721\pi$
$547$	− 518.823i	− 0.948488i	0.880393	−	0.474244i	$-0.157279\pi$
$548$	848.057	1.54755
$549$	0	0
$550$	0	0
$551$	1295.88	2.35188
$552$	0	0
$553$	533.302	0.964380
$554$	−43.3304	−0.0782137
$555$	0	0
$556$	− 540.811i	− 0.972681i
$557$	− 487.784i	− 0.875734i	−0.899040	−	0.437867i	$-0.855734\pi$
$557$	− 487.784i	− 0.875734i	0.899040	−	0.437867i	$-0.144266\pi$
$558$	0	0
$559$	−342.796	−0.613230
$560$	365.332i	0.652379i
$561$	0	0
$562$	−26.5598	−0.0472595
$563$	203.073i	0.360697i	0.983603	+	0.180349i	$0.0577226\pi$
$563$	203.073i	0.360697i	−0.983603	+	0.180349i	$0.942277\pi$
$564$	0	0
$565$	336.748	0.596014
$566$	−16.1777	−0.0285826
$567$	0	0
$568$	48.7822i	0.0858842i
$569$	− 728.660i	− 1.28060i	−0.768126	−	0.640299i	$-0.778811\pi$
$569$	− 728.660i	− 1.28060i	0.768126	−	0.640299i	$-0.221189\pi$
$570$	0	0
$571$	− 384.133i	− 0.672738i	−0.941730	−	0.336369i	$-0.890801\pi$
$571$	− 384.133i	− 0.672738i	0.941730	−	0.336369i	$-0.109199\pi$
$572$	0	0
$573$	0	0
$574$	− 43.7877i	− 0.0762852i
$575$	−174.460	−0.303408
$576$	0	0
$577$	523.738	0.907691	0.453846	−	0.891080i	$-0.350052\pi$
$577$	523.738	0.907691	0.453846	+	0.891080i	$0.350052\pi$
$578$	− 31.6267i	− 0.0547175i
$579$	0	0
$580$	901.318i	1.55400i
$581$	−645.293	−1.11066
$582$	0	0
$583$	0	0
$584$	35.5307	0.0608402
$585$	0	0
$586$	−42.6478	−0.0727779
$587$	−645.790	−1.10015	−0.550077	−	0.835114i	$-0.685402\pi$
$587$	−645.790	−1.10015	−0.550077	+	0.835114i	$0.685402\pi$
$588$	0	0
$589$	− 246.957i	− 0.419282i
$590$	− 4.06463i	− 0.00688920i
$591$	0	0
$592$	783.362	1.32325
$593$	640.096i	1.07942i	0.841851	+	0.539710i	$0.181466\pi$
$593$	640.096i	1.07942i	−0.841851	+	0.539710i	$0.818534\pi$
$594$	0	0
$595$	−538.695	−0.905369
$596$	768.261i	1.28903i
$597$	0	0
$598$	36.8181	0.0615688
$599$	126.085	0.210493	0.105247	−	0.994446i	$-0.466437\pi$
$599$	126.085	0.210493	0.105247	+	0.994446i	$0.466437\pi$
$600$	0	0
$601$	1070.82i	1.78174i	0.454260	+	0.890869i	$0.349904\pi$
$601$	1070.82i	1.78174i	−0.454260	+	0.890869i	$0.650096\pi$
$602$	− 29.9270i	− 0.0497126i
$603$	0	0
$604$	30.4820i	0.0504669i
$605$	0	0
$606$	0	0
$607$	175.511i	0.289145i	0.989494	+	0.144572i	$0.0461807\pi$
$607$	175.511i	0.289145i	−0.989494	+	0.144572i	$0.953819\pi$
$608$	−154.705	−0.254449
$609$	0	0
$610$	−24.0615	−0.0394450
$611$	20.5926i	0.0337031i
$612$	0	0
$613$	175.385i	0.286109i	0.989715	+	0.143055i	$0.0456924\pi$
$613$	175.385i	0.286109i	−0.989715	+	0.143055i	$0.954308\pi$
$614$	−45.7635	−0.0745333
$615$	0	0
$616$	0	0
$617$	−377.991	−0.612627	−0.306313	−	0.951931i	$-0.599096\pi$
$617$	−377.991	−0.612627	−0.306313	+	0.951931i	$0.599096\pi$
$618$	0	0
$619$	866.149	1.39927	0.699636	−	0.714500i	$-0.253346\pi$
$619$	866.149	1.39927	0.699636	+	0.714500i	$0.253346\pi$
$620$	171.764	0.277039
$621$	0	0
$622$	17.8659i	0.0287234i
$623$	− 475.269i	− 0.762872i
$624$	0	0
$625$	−501.091	−0.801746
$626$	− 37.3159i	− 0.0596101i
$627$	0	0
$628$	−179.524	−0.285866
$629$	1155.09i	1.83640i
$630$	0	0
$631$	−437.321	−0.693060	−0.346530	−	0.938039i	$-0.612640\pi$
$631$	−437.321	−0.693060	−0.346530	+	0.938039i	$0.612640\pi$
$632$	−104.242	−0.164940
$633$	0	0
$634$	− 24.6005i	− 0.0388021i
$635$	− 801.417i	− 1.26207i
$636$	0	0
$637$	− 167.910i	− 0.263595i
$638$	0	0
$639$	0	0
$640$	− 143.375i	− 0.224023i
$641$	−928.077	−1.44786	−0.723929	−	0.689875i	$-0.757666\pi$
$641$	−928.077	−1.44786	−0.723929	+	0.689875i	$0.757666\pi$
$642$	0	0
$643$	616.947	0.959482	0.479741	−	0.877410i	$-0.340731\pi$
$643$	616.947	0.959482	0.479741	+	0.877410i	$0.340731\pi$
$644$	− 831.567i	− 1.29125i
$645$	0	0
$646$	− 75.5495i	− 0.116950i
$647$	−473.750	−0.732226	−0.366113	−	0.930570i	$-0.619312\pi$
$647$	−473.750	−0.732226	−0.366113	+	0.930570i	$0.619312\pi$
$648$	0	0
$649$	0	0
$650$	−3.79032	−0.00583126
$651$	0	0
$652$	−743.338	−1.14009
$653$	489.445	0.749532	0.374766	−	0.927119i	$-0.377723\pi$
$653$	489.445	0.749532	0.374766	+	0.927119i	$0.377723\pi$
$654$	0	0
$655$	− 567.969i	− 0.867128i
$656$	− 1100.71i	− 1.67791i
$657$	0	0
$658$	−1.79779	−0.00273220
$659$	597.704i	0.906986i	0.891260	+	0.453493i	$0.149822\pi$
$659$	597.704i	0.906986i	−0.891260	+	0.453493i	$0.850178\pi$
$660$	0	0
$661$	−510.846	−0.772838	−0.386419	−	0.922323i	$-0.626288\pi$
$661$	−510.846	−0.772838	−0.386419	+	0.922323i	$0.626288\pi$
$662$	− 41.2049i	− 0.0622431i
$663$	0	0
$664$	126.132	0.189958
$665$	−602.998	−0.906763
$666$	0	0
$667$	− 2043.61i	− 3.06389i
$668$	− 193.731i	− 0.290016i
$669$	0	0
$670$	− 37.6486i	− 0.0561919i
$671$	0	0
$672$	0	0
$673$	236.608i	0.351572i	0.984428	+	0.175786i	$0.0562467\pi$
$673$	236.608i	0.351572i	−0.984428	+	0.175786i	$0.943753\pi$
$674$	29.0534	0.0431059
$675$	0	0
$676$	466.454	0.690021
$677$	1252.79i	1.85050i	0.379353	+	0.925252i	$0.376147\pi$
$677$	1252.79i	1.85050i	−0.379353	+	0.925252i	$0.623853\pi$
$678$	0	0
$679$	264.673i	0.389799i
$680$	105.296	0.154847
$681$	0	0
$682$	0	0
$683$	144.720	0.211888	0.105944	−	0.994372i	$-0.466213\pi$
$683$	144.720	0.211888	0.105944	+	0.994372i	$0.466213\pi$
$684$	0	0
$685$	−969.786	−1.41575
$686$	45.4878	0.0663087
$687$	0	0
$688$	− 752.286i	− 1.09344i
$689$	− 134.619i	− 0.195383i
$690$	0	0
$691$	1115.11	1.61376	0.806882	−	0.590712i	$-0.201153\pi$
$691$	1115.11	1.61376	0.806882	+	0.590712i	$0.201153\pi$
$692$	− 68.4435i	− 0.0989068i
$693$	0	0
$694$	−7.15087	−0.0103038
$695$	618.438i	0.889838i
$696$	0	0
$697$	1623.03	2.32860
$698$	75.8485	0.108665
$699$	0	0
$700$	85.6075i	0.122296i
$701$	− 422.309i	− 0.602437i	−0.953555	−	0.301219i	$-0.902607\pi$
$701$	− 422.309i	− 0.602437i	0.953555	−	0.301219i	$-0.0973934\pi$
$702$	0	0
$703$	1292.97i	1.83922i
$704$	0	0
$705$	0	0
$706$	− 57.4039i	− 0.0813087i
$707$	232.324	0.328606
$708$	0	0
$709$	406.871	0.573865	0.286933	−	0.957951i	$-0.407364\pi$
$709$	406.871	0.573865	0.286933	+	0.957951i	$0.407364\pi$
$710$	− 27.8384i	− 0.0392090i
$711$	0	0
$712$	92.8984i	0.130475i
$713$	−389.452	−0.546216
$714$	0	0
$715$	0	0
$716$	589.987	0.824005
$717$	0	0
$718$	−74.2380	−0.103396
$719$	−192.781	−0.268123	−0.134062	−	0.990973i	$-0.542802\pi$
$719$	−192.781	−0.268123	−0.134062	+	0.990973i	$0.542802\pi$
$720$	0	0
$721$	255.134i	0.353862i
$722$	− 39.7659i	− 0.0550774i
$723$	0	0
$724$	−112.283	−0.155087
$725$	210.384i	0.290185i
$726$	0	0
$727$	778.698	1.07111	0.535556	−	0.844500i	$-0.320102\pi$
$727$	778.698	1.07111	0.535556	+	0.844500i	$0.320102\pi$
$728$	− 36.2032i	− 0.0497296i
$729$	0	0
$730$	−20.2762	−0.0277756
$731$	1109.27	1.51747
$732$	0	0
$733$	− 66.1405i	− 0.0902325i	−0.998982	−	0.0451163i	$-0.985634\pi$
$733$	− 66.1405i	− 0.0902325i	0.998982	−	0.0451163i	$-0.0143658\pi$
$734$	− 68.2564i	− 0.0929923i
$735$	0	0
$736$	243.970i	0.331481i
$737$	0	0
$738$	0	0
$739$	515.469i	0.697522i	0.937212	+	0.348761i	$0.113397\pi$
$739$	515.469i	0.697522i	−0.937212	+	0.348761i	$0.886603\pi$
$740$	−899.294	−1.21526
$741$	0	0
$742$	11.7526	0.0158390
$743$	− 586.425i	− 0.789266i	−0.918839	−	0.394633i	$-0.870872\pi$
$743$	− 586.425i	− 0.789266i	0.918839	−	0.394633i	$-0.129128\pi$
$744$	0	0
$745$	− 878.536i	− 1.17924i
$746$	−48.9924	−0.0656735
$747$	0	0
$748$	0	0
$749$	−125.062	−0.166972
$750$	0	0
$751$	−975.919	−1.29949	−0.649746	−	0.760151i	$-0.725125\pi$
$751$	−975.919	−1.29949	−0.649746	+	0.760151i	$0.725125\pi$
$752$	−45.1916	−0.0600952
$753$	0	0
$754$	− 44.3996i	− 0.0588855i
$755$	− 34.8573i	− 0.0461686i
$756$	0	0
$757$	−1018.99	−1.34609	−0.673047	−	0.739600i	$-0.735015\pi$
$757$	−1018.99	−1.34609	−0.673047	+	0.739600i	$0.735015\pi$
$758$	24.9695i	0.0329413i
$759$	0	0
$760$	117.865	0.155085
$761$	636.194i	0.835998i	0.908448	+	0.417999i	$0.137268\pi$
$761$	636.194i	0.835998i	−0.908448	+	0.417999i	$0.862732\pi$
$762$	0	0
$763$	129.308	0.169474
$764$	−205.410	−0.268861
$765$	0	0
$766$	9.98655i	0.0130373i
$767$	− 51.8001i	− 0.0675359i
$768$	0	0
$769$	− 1120.79i	− 1.45747i	−0.684796	−	0.728735i	$-0.740109\pi$
$769$	− 1120.79i	− 1.45747i	0.684796	−	0.728735i	$-0.259891\pi$
$770$	0	0
$771$	0	0
$772$	− 240.254i	− 0.311209i
$773$	−411.320	−0.532108	−0.266054	−	0.963958i	$-0.585720\pi$
$773$	−411.320	−0.532108	−0.266054	+	0.963958i	$0.585720\pi$
$774$	0	0
$775$	40.0929	0.0517328
$776$	− 51.7343i	− 0.0666679i
$777$	0	0
$778$	− 12.3968i	− 0.0159341i
$779$	1816.77	2.33218
$780$	0	0
$781$	0	0
$782$	−119.142	−0.152355
$783$	0	0
$784$	368.488	0.470011
$785$	205.292	0.261519
$786$	0	0
$787$	− 98.1176i	− 0.124673i	−0.998055	−	0.0623365i	$-0.980145\pi$
$787$	− 98.1176i	− 0.124673i	0.998055	−	0.0623365i	$-0.0198552\pi$
$788$	37.4006i	0.0474627i
$789$	0	0
$790$	59.4873	0.0753004
$791$	374.664i	0.473659i
$792$	0	0
$793$	−306.642	−0.386686
$794$	24.2532i	0.0305456i
$795$	0	0
$796$	−696.334	−0.874791
$797$	27.5903	0.0346178	0.0173089	−	0.999850i	$-0.494490\pi$
$797$	27.5903	0.0346178	0.0173089	+	0.999850i	$0.494490\pi$
$798$	0	0
$799$	− 66.6366i	− 0.0833999i
$800$	− 25.1160i	− 0.0313950i
$801$	0	0
$802$	2.69506i	0.00336042i
$803$	0	0
$804$	0	0
$805$	950.929i	1.18128i
$806$	−8.46125	−0.0104978
$807$	0	0
$808$	−45.4113	−0.0562021
$809$	− 1113.72i	− 1.37666i	−0.725398	−	0.688330i	$-0.758344\pi$
$809$	− 1113.72i	− 1.37666i	0.725398	−	0.688330i	$-0.241656\pi$
$810$	0	0
$811$	1459.55i	1.79969i	0.436211	+	0.899844i	$0.356320\pi$
$811$	1459.55i	1.79969i	−0.436211	+	0.899844i	$0.643680\pi$
$812$	−1002.80	−1.23498
$813$	0	0
$814$	0	0
$815$	850.035	1.04299
$816$	0	0
$817$	1241.68	1.51981
$818$	17.4663	0.0213524
$819$	0	0
$820$	1263.61i	1.54098i
$821$	− 1130.31i	− 1.37675i	−0.725355	−	0.688375i	$-0.758324\pi$
$821$	− 1130.31i	− 1.37675i	0.725355	−	0.688375i	$-0.241676\pi$
$822$	0	0
$823$	−980.288	−1.19112	−0.595558	−	0.803313i	$-0.703069\pi$
$823$	−980.288	−1.19112	−0.595558	+	0.803313i	$0.703069\pi$
$824$	− 49.8698i	− 0.0605216i
$825$	0	0
$826$	4.52228	0.00547492
$827$	− 1041.92i	− 1.25988i	−0.776643	−	0.629942i	$-0.783079\pi$
$827$	− 1041.92i	− 1.25988i	0.776643	−	0.629942i	$-0.216921\pi$
$828$	0	0
$829$	45.3982	0.0547626	0.0273813	−	0.999625i	$-0.491283\pi$
$829$	45.3982	0.0547626	0.0273813	+	0.999625i	$0.491283\pi$
$830$	−71.9793	−0.0867221
$831$	0	0
$832$	− 450.604i	− 0.541591i
$833$	543.348i	0.652279i
$834$	0	0
$835$	221.538i	0.265315i
$836$	0	0
$837$	0	0
$838$	− 25.4701i	− 0.0303939i
$839$	896.214	1.06819	0.534096	−	0.845424i	$-0.320652\pi$
$839$	896.214	1.06819	0.534096	+	0.845424i	$0.320652\pi$
$840$	0	0
$841$	−1623.43	−1.93036
$842$	96.2022i	0.114254i
$843$	0	0
$844$	− 1436.49i	− 1.70200i
$845$	−533.408	−0.631252
$846$	0	0
$847$	0	0
$848$	295.429	0.348383
$849$	0	0
$850$	12.2653	0.0144298
$851$	2039.02	2.39603
$852$	0	0
$853$	− 522.189i	− 0.612180i	−0.952003	−	0.306090i	$-0.900979\pi$
$853$	− 522.189i	− 0.612180i	0.952003	−	0.306090i	$-0.0990208\pi$
$854$	− 26.7707i	− 0.0313474i
$855$	0	0
$856$	24.4453	0.0285576
$857$	116.799i	0.136288i	0.997675	+	0.0681442i	$0.0217078\pi$
$857$	116.799i	0.136288i	−0.997675	+	0.0681442i	$0.978292\pi$
$858$	0	0
$859$	1359.69	1.58287	0.791436	−	0.611251i	$-0.209334\pi$
$859$	1359.69	1.58287	0.791436	+	0.611251i	$0.209334\pi$
$860$	863.619i	1.00421i
$861$	0	0
$862$	−6.78193	−0.00786766
$863$	1194.91	1.38460	0.692298	−	0.721612i	$-0.256598\pi$
$863$	1194.91	1.38460	0.692298	+	0.721612i	$0.256598\pi$
$864$	0	0
$865$	78.2677i	0.0904829i
$866$	− 61.9049i	− 0.0714837i
$867$	0	0
$868$	191.104i	0.220166i
$869$	0	0
$870$	0	0
$871$	− 479.798i	− 0.550859i
$872$	−25.2752	−0.0289854
$873$	0	0
$874$	−133.363	−0.152590
$875$	− 675.390i	− 0.771874i
$876$	0	0
$877$	564.773i	0.643983i	0.946743	+	0.321992i	$0.104352\pi$
$877$	564.773i	0.643983i	−0.946743	+	0.321992i	$0.895648\pi$
$878$	−15.5523	−0.0177133
$879$	0	0
$880$	0	0
$881$	1276.50	1.44892	0.724462	−	0.689315i	$-0.242088\pi$
$881$	1276.50	1.44892	0.724462	+	0.689315i	$0.242088\pi$
$882$	0	0
$883$	−1357.43	−1.53730	−0.768649	−	0.639671i	$-0.779071\pi$
$883$	−1357.43	−1.53730	−0.768649	+	0.639671i	$0.779071\pi$
$884$	669.658	0.757531
$885$	0	0
$886$	76.1744i	0.0859756i
$887$	− 710.054i	− 0.800512i	−0.916403	−	0.400256i	$-0.868921\pi$
$887$	− 710.054i	− 0.800512i	0.916403	−	0.400256i	$-0.131079\pi$
$888$	0	0
$889$	891.653	1.00298
$890$	− 53.0140i	− 0.0595663i
$891$	0	0
$892$	619.403	0.694398
$893$	− 74.5908i	− 0.0835284i
$894$	0	0
$895$	−674.673	−0.753825
$896$	159.518	0.178033
$897$	0	0
$898$	− 47.7995i	− 0.0532288i
$899$	469.647i	0.522410i
$900$	0	0
$901$	435.620i	0.483485i
$902$	0	0
$903$	0	0
$904$	− 73.2337i	− 0.0810108i
$905$	128.400	0.141878
$906$	0	0
$907$	462.174	0.509564	0.254782	−	0.966999i	$-0.417996\pi$
$907$	462.174	0.509564	0.254782	+	0.966999i	$0.417996\pi$
$908$	602.567i	0.663620i
$909$	0	0
$910$	20.6599i	0.0227032i
$911$	1419.86	1.55857	0.779286	−	0.626668i	$-0.215582\pi$
$911$	1419.86	1.55857	0.779286	+	0.626668i	$0.215582\pi$
$912$	0	0
$913$	0	0
$914$	−5.53758	−0.00605862
$915$	0	0
$916$	1225.68	1.33808
$917$	631.920	0.689116
$918$	0	0
$919$	1235.25i	1.34413i	0.740493	+	0.672064i	$0.234592\pi$
$919$	1235.25i	1.34413i	−0.740493	+	0.672064i	$0.765408\pi$
$920$	− 185.873i	− 0.202036i
$921$	0	0
$922$	48.6997	0.0528197
$923$	− 354.775i	− 0.384372i
$924$	0	0
$925$	−209.912	−0.226932
$926$	− 12.2299i	− 0.0132072i
$927$	0	0
$928$	294.208	0.317034
$929$	97.7349	0.105204	0.0526022	−	0.998616i	$-0.483248\pi$
$929$	97.7349	0.105204	0.0526022	+	0.998616i	$0.483248\pi$
$930$	0	0
$931$	608.207i	0.653283i
$932$	400.366i	0.429577i
$933$	0	0
$934$	− 33.4805i	− 0.0358463i
$935$	0	0
$936$	0	0
$937$	523.196i	0.558374i	0.960237	+	0.279187i	$0.0900649\pi$
$937$	523.196i	0.558374i	−0.960237	+	0.279187i	$0.909935\pi$
$938$	41.8876	0.0446563
$939$	0	0
$940$	51.8797	0.0551911
$941$	− 86.4170i	− 0.0918353i	−0.998945	−	0.0459177i	$-0.985379\pi$
$941$	− 86.4170i	− 0.0918353i	0.998945	−	0.0459177i	$-0.0146212\pi$
$942$	0	0
$943$	− 2865.05i	− 3.03823i
$944$	113.678	0.120422
$945$	0	0
$946$	0	0
$947$	−5.94234	−0.00627491	−0.00313746	−	0.999995i	$-0.500999\pi$
$947$	−5.94234	−0.00627491	−0.00313746	+	0.999995i	$0.500999\pi$
$948$	0	0
$949$	−258.402	−0.272289
$950$	13.7294	0.0144520
$951$	0	0
$952$	117.152i	0.123059i
$953$	1251.07i	1.31277i	0.754426	+	0.656386i	$0.227916\pi$
$953$	1251.07i	1.31277i	−0.754426	+	0.656386i	$0.772084\pi$
$954$	0	0
$955$	234.894	0.245962
$956$	− 1833.12i	− 1.91749i
$957$	0	0
$958$	−50.6802	−0.0529020
$959$	− 1078.98i	− 1.12511i
$960$	0	0
$961$	−871.499	−0.906867
$962$	44.2999	0.0460498
$963$	0	0
$964$	− 1003.51i	− 1.04099i
$965$	274.739i	0.284704i
$966$	0	0
$967$	4.40274i	0.00455299i	0.999997	+	0.00227650i	$0.000724632\pi$
$967$	4.40274i	0.00455299i	−0.999997	+	0.00227650i	$0.999275\pi$
$968$	0	0
$969$	0	0
$970$	29.5230i	0.0304361i
$971$	332.226	0.342148	0.171074	−	0.985258i	$-0.445276\pi$
$971$	332.226	0.342148	0.171074	+	0.985258i	$0.445276\pi$
$972$	0	0
$973$	−688.071	−0.707164
$974$	− 61.3718i	− 0.0630100i
$975$	0	0
$976$	− 672.944i	− 0.689492i
$977$	−865.872	−0.886256	−0.443128	−	0.896458i	$-0.646131\pi$
$977$	−865.872	−0.886256	−0.443128	+	0.896458i	$0.646131\pi$
$978$	0	0
$979$	0	0
$980$	−423.022	−0.431655
$981$	0	0
$982$	4.62949	0.00471435
$983$	−53.4433	−0.0543676	−0.0271838	−	0.999630i	$-0.508654\pi$
$983$	−53.4433	−0.0543676	−0.0271838	+	0.999630i	$0.508654\pi$
$984$	0	0
$985$	− 42.7690i	− 0.0434203i
$986$	143.675i	0.145715i
$987$	0	0
$988$	749.593	0.758698
$989$	− 1958.14i	− 1.97992i
$990$	0	0
$991$	−624.037	−0.629704	−0.314852	−	0.949141i	$-0.601955\pi$
$991$	−624.037	−0.629704	−0.314852	+	0.949141i	$0.601955\pi$
$992$	− 56.0672i	− 0.0565194i
$993$	0	0
$994$	30.9728	0.0311598
$995$	796.284	0.800286
$996$	0	0
$997$	1255.44i	1.25922i	0.776912	+	0.629609i	$0.216785\pi$
$997$	1255.44i	1.25922i	−0.776912	+	0.629609i	$0.783215\pi$
$998$	25.1245i	0.0251748i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.j.604.5		8
3.2	odd	2		363.3.c.d.241.4	✓	8
11.10	odd	2	inner	1089.3.c.j.604.4		8
33.2	even	10		363.3.g.h.40.5		32
33.5	odd	10		363.3.g.h.118.5		32
33.8	even	10		363.3.g.h.112.5		32
33.14	odd	10		363.3.g.h.112.4		32
33.17	even	10		363.3.g.h.118.4		32
33.20	odd	10		363.3.g.h.40.4		32
33.26	odd	10		363.3.g.h.94.5		32
33.29	even	10		363.3.g.h.94.4		32
33.32	even	2		363.3.c.d.241.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.3.c.d.241.4	✓	8	3.2	odd	2
363.3.c.d.241.5	yes	8	33.32	even	2
363.3.g.h.40.4		32	33.20	odd	10
363.3.g.h.40.5		32	33.2	even	10
363.3.g.h.94.4		32	33.29	even	10
363.3.g.h.94.5		32	33.26	odd	10
363.3.g.h.112.4		32	33.14	odd	10
363.3.g.h.112.5		32	33.8	even	10
363.3.g.h.118.4		32	33.17	even	10
363.3.g.h.118.5		32	33.5	odd	10
1089.3.c.j.604.4		8	11.10	odd	2	inner
1089.3.c.j.604.5		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)

\(f(q)\)	\(=\)	\(q+0.124105i q^{2} +3.98460 q^{4} -4.55654 q^{5} -5.06958i q^{7} +0.990926i q^{8} +O(q^{10})\)	\(q+0.124105i q^{2} +3.98460 q^{4} -4.55654 q^{5} -5.06958i q^{7} +0.990926i q^{8} -0.565488i q^{10} -7.20664i q^{13} +0.629159 q^{14} +15.8154 q^{16} +23.3204i q^{17} +26.1041i q^{19} -18.1560 q^{20} +41.1662 q^{23} -4.23794 q^{25} +0.894378 q^{26} -20.2003i q^{28} -49.6430i q^{29} -9.46048 q^{31} +5.92647i q^{32} -2.89417 q^{34} +23.0998i q^{35} +49.5316 q^{37} -3.23964 q^{38} -4.51519i q^{40} -69.5972i q^{41} -47.5666i q^{43} +5.10891i q^{46} -2.85744 q^{47} +23.2993 q^{49} -0.525948i q^{50} -28.7156i q^{52} +18.6798 q^{53} +5.02358 q^{56} +6.16093 q^{58} +7.18782 q^{59} -42.5499i q^{61} -1.17409i q^{62} +62.5262 q^{64} +32.8374i q^{65} +66.5772 q^{67} +92.9223i q^{68} -2.86679 q^{70} +49.2289 q^{71} -35.8561i q^{73} +6.14710i q^{74} +104.014i q^{76} +105.196i q^{79} -72.0636 q^{80} +8.63734 q^{82} -127.287i q^{83} -106.260i q^{85} +5.90324 q^{86} +93.7491 q^{89} -36.5347 q^{91} +164.031 q^{92} -0.354622i q^{94} -118.944i q^{95} -52.2081 q^{97} +2.89155i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 24 q^{4} - 16 q^{5}+O(q^{10}) \) 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

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Database

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