LMFDB - Embedded newform 1089.3.c.l.604.7

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} - 21x^{14} + 227x^{12} - 1488x^{10} + 24225x^{8} - 62832x^{6} + 64372x^{4} + 7986x^{2} + 14641 $ x^16 - 21x^14 + 227x^12 - 1488x^10 + 24225x^8 - 62832x^6 + 64372x^4 + 7986*x^2 + 14641
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 11^{4} $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			604.7
Root			$-0.386583 + 0.532086i$ of defining polynomial
Character	$\chi$	$=$	1089.604
Dual form			1089.3.c.l.604.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.657695i q^{2} +3.56744 q^{4} -8.10135 q^{5} +4.08611i q^{7} -4.97707i q^{8} +O(q^{10})$	$q-0.657695i q^{2} +3.56744 q^{4} -8.10135 q^{5} +4.08611i q^{7} -4.97707i q^{8} +5.32822i q^{10} -14.7880i q^{13} +2.68741 q^{14} +10.9964 q^{16} +29.4413i q^{17} -12.7965i q^{19} -28.9011 q^{20} -12.6293 q^{23} +40.6319 q^{25} -9.72602 q^{26} +14.5769i q^{28} +13.3705i q^{29} +35.5591 q^{31} -27.1405i q^{32} +19.3634 q^{34} -33.1030i q^{35} -17.9636 q^{37} -8.41621 q^{38} +40.3210i q^{40} +16.1818i q^{41} +44.2415i q^{43} +8.30624i q^{46} -9.59750 q^{47} +32.3037 q^{49} -26.7234i q^{50} -52.7554i q^{52} +57.7439 q^{53} +20.3368 q^{56} +8.79374 q^{58} +100.258 q^{59} +83.1124i q^{61} -23.3871i q^{62} +26.1353 q^{64} +119.803i q^{65} +87.5688 q^{67} +105.030i q^{68} -21.7717 q^{70} +68.4732 q^{71} +25.9942i q^{73} +11.8146i q^{74} -45.6508i q^{76} +65.6023i q^{79} -89.0854 q^{80} +10.6427 q^{82} -87.2922i q^{83} -238.514i q^{85} +29.0974 q^{86} +29.3444 q^{89} +60.4255 q^{91} -45.0543 q^{92} +6.31223i q^{94} +103.669i q^{95} -36.7844 q^{97} -21.2460i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 44 q^{4}+O(q^{10}) $ 16 * q - 44 * q^4	$ 16 q - 44 q^{4} + 244 q^{16} + 16 q^{25} - 80 q^{31} + 328 q^{34} - 280 q^{37} + 436 q^{49} + 140 q^{58} - 656 q^{64} + 300 q^{67} + 308 q^{70} + 580 q^{82} + 768 q^{91} - 720 q^{97}+O(q^{100}) $ 16 * q - 44 * q^4 + 244 * q^16 + 16 * q^25 - 80 * q^31 + 328 * q^34 - 280 * q^37 + 436 * q^49 + 140 * q^58 - 656 * q^64 + 300 * q^67 + 308 * q^70 + 580 * q^82 + 768 * q^91 - 720 * 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 $

$2$	− 0.657695i	− 0.328847i	−0.986390	−	0.164424i	$-0.947424\pi$
$2$	− 0.657695i	− 0.328847i	0.986390	−	0.164424i	$-0.0525764\pi$
$3$	0	0
$3$	0	0
$4$	3.56744	0.891859
$5$	−8.10135	−1.62027	−0.810135	−	0.586243i	$-0.800606\pi$
$5$	−8.10135	−1.62027	−0.810135	+	0.586243i	$0.800606\pi$
$6$	0	0
$7$	4.08611i	0.583729i	0.956460	+	0.291865i	$0.0942757\pi$
$7$	4.08611i	0.583729i	−0.956460	+	0.291865i	$0.905724\pi$
$8$	− 4.97707i	− 0.622133i
$9$	0	0
$10$	5.32822i	0.532822i
$11$	0	0
$11$	0	0
$12$	0	0
$13$	− 14.7880i	− 1.13754i	−0.822496	−	0.568771i	$-0.807419\pi$
$13$	− 14.7880i	− 1.13754i	0.822496	−	0.568771i	$-0.192581\pi$
$14$	2.68741	0.191958
$15$	0	0
$16$	10.9964	0.687272
$17$	29.4413i	1.73184i	0.500181	+	0.865921i	$0.333267\pi$
$17$	29.4413i	1.73184i	−0.500181	+	0.865921i	$0.666733\pi$
$18$	0	0
$19$	− 12.7965i	− 0.673501i	−0.941594	−	0.336751i	$-0.890672\pi$
$19$	− 12.7965i	− 0.673501i	0.941594	−	0.336751i	$-0.109328\pi$
$20$	−28.9011	−1.44505
$21$	0	0
$22$	0	0
$23$	−12.6293	−0.549101	−0.274551	−	0.961573i	$-0.588529\pi$
$23$	−12.6293	−0.549101	−0.274551	+	0.961573i	$0.588529\pi$
$24$	0	0
$25$	40.6319	1.62528
$26$	−9.72602	−0.374078
$27$	0	0
$28$	14.5769i	0.520605i
$29$	13.3705i	0.461053i	0.973066	+	0.230527i	$0.0740449\pi$
$29$	13.3705i	0.461053i	−0.973066	+	0.230527i	$0.925955\pi$
$30$	0	0
$31$	35.5591	1.14707	0.573534	−	0.819182i	$-0.305572\pi$
$31$	35.5591	1.14707	0.573534	+	0.819182i	$0.305572\pi$
$32$	− 27.1405i	− 0.848141i
$33$	0	0
$34$	19.3634	0.569512
$35$	− 33.1030i	− 0.945800i
$36$	0	0
$37$	−17.9636	−0.485503	−0.242751	−	0.970089i	$-0.578050\pi$
$37$	−17.9636	−0.485503	−0.242751	+	0.970089i	$0.578050\pi$
$38$	−8.41621	−0.221479
$39$	0	0
$40$	40.3210i	1.00802i
$41$	16.1818i	0.394679i	0.980335	+	0.197339i	$0.0632301\pi$
$41$	16.1818i	0.394679i	−0.980335	+	0.197339i	$0.936770\pi$
$42$	0	0
$43$	44.2415i	1.02887i	0.857529	+	0.514436i	$0.171999\pi$
$43$	44.2415i	1.02887i	−0.857529	+	0.514436i	$0.828001\pi$
$44$	0	0
$45$	0	0
$46$	8.30624i	0.180570i
$47$	−9.59750	−0.204202	−0.102101	−	0.994774i	$-0.532557\pi$
$47$	−9.59750	−0.204202	−0.102101	+	0.994774i	$0.532557\pi$
$48$	0	0
$49$	32.3037	0.659260
$50$	− 26.7234i	− 0.534468i
$51$	0	0
$52$	− 52.7554i	− 1.01453i
$53$	57.7439	1.08951	0.544754	−	0.838596i	$-0.316623\pi$
$53$	57.7439	1.08951	0.544754	+	0.838596i	$0.316623\pi$
$54$	0	0
$55$	0	0
$56$	20.3368	0.363157
$57$	0	0
$58$	8.79374	0.151616
$59$	100.258	1.69929	0.849644	−	0.527356i	$-0.176817\pi$
$59$	100.258	1.69929	0.849644	+	0.527356i	$0.176817\pi$
$60$	0	0
$61$	83.1124i	1.36250i	0.732052	+	0.681249i	$0.238563\pi$
$61$	83.1124i	1.36250i	−0.732052	+	0.681249i	$0.761437\pi$
$62$	− 23.3871i	− 0.377210i
$63$	0	0
$64$	26.1353	0.408363
$65$	119.803i	1.84312i
$66$	0	0
$67$	87.5688	1.30700	0.653499	−	0.756928i	$-0.273301\pi$
$67$	87.5688	1.30700	0.653499	+	0.756928i	$0.273301\pi$
$68$	105.030i	1.54456i
$69$	0	0
$70$	−21.7717	−0.311024
$71$	68.4732	0.964411	0.482206	−	0.876058i	$-0.339836\pi$
$71$	68.4732	0.964411	0.482206	+	0.876058i	$0.339836\pi$
$72$	0	0
$73$	25.9942i	0.356085i	0.984023	+	0.178043i	$0.0569765\pi$
$73$	25.9942i	0.356085i	−0.984023	+	0.178043i	$0.943023\pi$
$74$	11.8146i	0.159656i
$75$	0	0
$76$	− 45.6508i	− 0.600668i
$77$	0	0
$78$	0	0
$79$	65.6023i	0.830409i	0.909728	+	0.415205i	$0.136290\pi$
$79$	65.6023i	0.830409i	−0.909728	+	0.415205i	$0.863710\pi$
$80$	−89.0854	−1.11357
$81$	0	0
$82$	10.6427	0.129789
$83$	− 87.2922i	− 1.05171i	−0.850573	−	0.525857i	$-0.823745\pi$
$83$	− 87.2922i	− 1.05171i	0.850573	−	0.525857i	$-0.176255\pi$
$84$	0	0
$85$	− 238.514i	− 2.80605i
$86$	29.0974	0.338342
$87$	0	0
$88$	0	0
$89$	29.3444	0.329713	0.164856	−	0.986318i	$-0.447284\pi$
$89$	29.3444	0.329713	0.164856	+	0.986318i	$0.447284\pi$
$90$	0	0
$91$	60.4255	0.664016
$92$	−45.0543	−0.489721
$93$	0	0
$94$	6.31223i	0.0671514i
$95$	103.669i	1.09125i
$96$	0	0
$97$	−36.7844	−0.379221	−0.189610	−	0.981859i	$-0.560723\pi$
$97$	−36.7844	−0.379221	−0.189610	+	0.981859i	$0.560723\pi$
$98$	− 21.2460i	− 0.216796i
$99$	0	0
$100$	144.952	1.44952
$101$	− 90.6215i	− 0.897243i	−0.893722	−	0.448621i	$-0.851915\pi$
$101$	− 90.6215i	− 0.897243i	0.893722	−	0.448621i	$-0.148085\pi$
$102$	0	0
$103$	−153.500	−1.49029	−0.745144	−	0.666904i	$-0.767619\pi$
$103$	−153.500	−1.49029	−0.745144	+	0.666904i	$0.767619\pi$
$104$	−73.6010	−0.707702
$105$	0	0
$106$	− 37.9779i	− 0.358282i
$107$	67.4966i	0.630809i	0.948957	+	0.315405i	$0.102140\pi$
$107$	67.4966i	0.630809i	−0.948957	+	0.315405i	$0.897860\pi$
$108$	0	0
$109$	150.536i	1.38106i	0.723304	+	0.690530i	$0.242623\pi$
$109$	150.536i	1.38106i	−0.723304	+	0.690530i	$0.757377\pi$
$110$	0	0
$111$	0	0
$112$	44.9323i	0.401181i
$113$	146.800	1.29911	0.649556	−	0.760313i	$-0.274955\pi$
$113$	146.800	1.29911	0.649556	+	0.760313i	$0.274955\pi$
$114$	0	0
$115$	102.315	0.889692
$116$	47.6986i	0.411195i
$117$	0	0
$118$	− 65.9392i	− 0.558807i
$119$	−120.300	−1.01093
$120$	0	0
$121$	0	0
$122$	54.6626	0.448054
$123$	0	0
$124$	126.855	1.02302
$125$	−126.640	−1.01312
$126$	0	0
$127$	64.5405i	0.508193i	0.967179	+	0.254096i	$0.0817780\pi$
$127$	64.5405i	0.508193i	−0.967179	+	0.254096i	$0.918222\pi$
$128$	− 125.751i	− 0.982430i
$129$	0	0
$130$	78.7939	0.606107
$131$	− 222.809i	− 1.70083i	−0.526110	−	0.850416i	$-0.676350\pi$
$131$	− 222.809i	− 1.70083i	0.526110	−	0.850416i	$-0.323650\pi$
$132$	0	0
$133$	52.2879	0.393142
$134$	− 57.5936i	− 0.429803i
$135$	0	0
$136$	146.531	1.07744
$137$	50.9196	0.371676	0.185838	−	0.982580i	$-0.440500\pi$
$137$	50.9196	0.371676	0.185838	+	0.982580i	$0.440500\pi$
$138$	0	0
$139$	132.747i	0.955015i	0.878628	+	0.477507i	$0.158460\pi$
$139$	132.747i	0.955015i	−0.878628	+	0.477507i	$0.841540\pi$
$140$	− 118.093i	− 0.843520i
$141$	0	0
$142$	− 45.0345i	− 0.317144i
$143$	0	0
$144$	0	0
$145$	− 108.320i	− 0.747031i
$146$	17.0963	0.117098
$147$	0	0
$148$	−64.0840	−0.433000
$149$	263.354i	1.76748i	0.467980	+	0.883739i	$0.344982\pi$
$149$	263.354i	1.76748i	−0.467980	+	0.883739i	$0.655018\pi$
$150$	0	0
$151$	− 50.4791i	− 0.334299i	−0.985932	−	0.167149i	$-0.946544\pi$
$151$	− 50.4791i	− 0.334299i	0.985932	−	0.167149i	$-0.0534562\pi$
$152$	−63.6891	−0.419007
$153$	0	0
$154$	0	0
$155$	−288.077	−1.85856
$156$	0	0
$157$	30.1238	0.191871	0.0959355	−	0.995388i	$-0.469416\pi$
$157$	30.1238	0.191871	0.0959355	+	0.995388i	$0.469416\pi$
$158$	43.1463	0.273078
$159$	0	0
$160$	219.875i	1.37422i
$161$	− 51.6048i	− 0.320526i
$162$	0	0
$163$	102.782	0.630562	0.315281	−	0.948998i	$-0.397901\pi$
$163$	102.782	0.630562	0.315281	+	0.948998i	$0.397901\pi$
$164$	57.7276i	0.351998i
$165$	0	0
$166$	−57.4117	−0.345853
$167$	133.469i	0.799214i	0.916687	+	0.399607i	$0.130853\pi$
$167$	133.469i	0.799214i	−0.916687	+	0.399607i	$0.869147\pi$
$168$	0	0
$169$	−49.6860	−0.294000
$170$	−156.870	−0.922763
$171$	0	0
$172$	157.829i	0.917610i
$173$	− 246.633i	− 1.42563i	−0.701354	−	0.712814i	$-0.747421\pi$
$173$	− 246.633i	− 1.42563i	0.701354	−	0.712814i	$-0.252579\pi$
$174$	0	0
$175$	166.026i	0.948722i
$176$	0	0
$177$	0	0
$178$	− 19.2997i	− 0.108425i
$179$	−66.2080	−0.369877	−0.184938	−	0.982750i	$-0.559209\pi$
$179$	−66.2080	−0.369877	−0.184938	+	0.982750i	$0.559209\pi$
$180$	0	0
$181$	158.399	0.875133	0.437567	−	0.899186i	$-0.355840\pi$
$181$	158.399	0.875133	0.437567	+	0.899186i	$0.355840\pi$
$182$	− 39.7415i	− 0.218360i
$183$	0	0
$184$	62.8570i	0.341614i
$185$	145.529	0.786646
$186$	0	0
$187$	0	0
$188$	−34.2385	−0.182120
$189$	0	0
$190$	68.1827	0.358856
$191$	−148.825	−0.779190	−0.389595	−	0.920986i	$-0.627385\pi$
$191$	−148.825	−0.779190	−0.389595	+	0.920986i	$0.627385\pi$
$192$	0	0
$193$	− 214.567i	− 1.11174i	−0.831268	−	0.555872i	$-0.812384\pi$
$193$	− 214.567i	− 1.11174i	0.831268	−	0.555872i	$-0.187616\pi$
$194$	24.1929i	0.124706i
$195$	0	0
$196$	115.242	0.587967
$197$	54.8802i	0.278580i	0.990252	+	0.139290i	$0.0444820\pi$
$197$	54.8802i	0.278580i	−0.990252	+	0.139290i	$0.955518\pi$
$198$	0	0
$199$	−58.1547	−0.292235	−0.146117	−	0.989267i	$-0.546678\pi$
$199$	−58.1547	−0.292235	−0.146117	+	0.989267i	$0.546678\pi$
$200$	− 202.228i	− 1.01114i
$201$	0	0
$202$	−59.6013	−0.295056
$203$	−54.6335	−0.269130
$204$	0	0
$205$	− 131.095i	− 0.639486i
$206$	100.956i	0.490077i
$207$	0	0
$208$	− 162.615i	− 0.781801i
$209$	0	0
$210$	0	0
$211$	− 226.348i	− 1.07274i	−0.843983	−	0.536370i	$-0.819795\pi$
$211$	− 226.348i	− 1.07274i	0.843983	−	0.536370i	$-0.180205\pi$
$212$	205.998	0.971688
$213$	0	0
$214$	44.3922	0.207440
$215$	− 358.416i	− 1.66705i
$216$	0	0
$217$	145.298i	0.669577i
$218$	99.0065	0.454158
$219$	0	0
$220$	0	0
$221$	435.379	1.97004
$222$	0	0
$223$	−82.4568	−0.369761	−0.184881	−	0.982761i	$-0.559190\pi$
$223$	−82.4568	−0.369761	−0.184881	+	0.982761i	$0.559190\pi$
$224$	110.899	0.495085
$225$	0	0
$226$	− 96.5494i	− 0.427210i
$227$	− 176.871i	− 0.779169i	−0.920991	−	0.389585i	$-0.872619\pi$
$227$	− 176.871i	− 0.779169i	0.920991	−	0.389585i	$-0.127381\pi$
$228$	0	0
$229$	−2.92139	−0.0127571	−0.00637857	−	0.999980i	$-0.502030\pi$
$229$	−2.92139	−0.0127571	−0.00637857	+	0.999980i	$0.502030\pi$
$230$	− 67.2918i	− 0.292573i
$231$	0	0
$232$	66.5461	0.286837
$233$	294.504i	1.26396i	0.774983	+	0.631982i	$0.217758\pi$
$233$	294.504i	1.26396i	−0.774983	+	0.631982i	$0.782242\pi$
$234$	0	0
$235$	77.7528	0.330863
$236$	357.664	1.51553
$237$	0	0
$238$	79.1209i	0.332441i
$239$	342.885i	1.43467i	0.696731	+	0.717333i	$0.254637\pi$
$239$	342.885i	1.43467i	−0.696731	+	0.717333i	$0.745363\pi$
$240$	0	0
$241$	− 43.2741i	− 0.179561i	−0.995962	−	0.0897804i	$-0.971383\pi$
$241$	− 43.2741i	− 0.179561i	0.995962	−	0.0897804i	$-0.0286165\pi$
$242$	0	0
$243$	0	0
$244$	296.498i	1.21516i
$245$	−261.704	−1.06818
$246$	0	0
$247$	−189.235	−0.766135
$248$	− 176.980i	− 0.713629i
$249$	0	0
$250$	83.2903i	0.333161i
$251$	−117.004	−0.466152	−0.233076	−	0.972458i	$-0.574879\pi$
$251$	−117.004	−0.466152	−0.233076	+	0.972458i	$0.574879\pi$
$252$	0	0
$253$	0	0
$254$	42.4479	0.167118
$255$	0	0
$256$	21.8352	0.0852937
$257$	28.2538	0.109937	0.0549685	−	0.998488i	$-0.482494\pi$
$257$	28.2538	0.109937	0.0549685	+	0.998488i	$0.482494\pi$
$258$	0	0
$259$	− 73.4012i	− 0.283402i
$260$	427.390i	1.64381i
$261$	0	0
$262$	−146.540	−0.559314
$263$	247.683i	0.941759i	0.882198	+	0.470879i	$0.156063\pi$
$263$	247.683i	0.941759i	−0.882198	+	0.470879i	$0.843937\pi$
$264$	0	0
$265$	−467.804	−1.76530
$266$	− 34.3895i	− 0.129284i
$267$	0	0
$268$	312.396	1.16566
$269$	277.531	1.03171	0.515856	−	0.856675i	$-0.327474\pi$
$269$	277.531	1.03171	0.515856	+	0.856675i	$0.327474\pi$
$270$	0	0
$271$	− 395.232i	− 1.45842i	−0.684290	−	0.729210i	$-0.739888\pi$
$271$	− 395.232i	− 1.45842i	0.684290	−	0.729210i	$-0.260112\pi$
$272$	323.747i	1.19025i
$273$	0	0
$274$	− 33.4896i	− 0.122225i
$275$	0	0
$276$	0	0
$277$	− 210.196i	− 0.758831i	−0.925226	−	0.379415i	$-0.876125\pi$
$277$	− 210.196i	− 0.758831i	0.925226	−	0.379415i	$-0.123875\pi$
$278$	87.3071	0.314054
$279$	0	0
$280$	−164.756	−0.588413
$281$	− 26.1508i	− 0.0930635i	−0.998917	−	0.0465318i	$-0.985183\pi$
$281$	− 26.1508i	− 0.0930635i	0.998917	−	0.0465318i	$-0.0148169\pi$
$282$	0	0
$283$	295.112i	1.04280i	0.853313	+	0.521399i	$0.174590\pi$
$283$	295.112i	1.04280i	−0.853313	+	0.521399i	$0.825410\pi$
$284$	244.274	0.860119
$285$	0	0
$286$	0	0
$287$	−66.1206	−0.230385
$288$	0	0
$289$	−577.791	−1.99928
$290$	−71.2412	−0.245659
$291$	0	0
$292$	92.7328i	0.317578i
$293$	− 62.1824i	− 0.212227i	−0.994354	−	0.106113i	$-0.966159\pi$
$293$	− 62.1824i	− 0.212227i	0.994354	−	0.106113i	$-0.0338406\pi$
$294$	0	0
$295$	−812.226	−2.75331
$296$	89.4060i	0.302047i
$297$	0	0
$298$	173.207	0.581231
$299$	186.763i	0.624625i
$300$	0	0
$301$	−180.776	−0.600583
$302$	−33.1999	−0.109933
$303$	0	0
$304$	− 140.715i	− 0.462879i
$305$	− 673.323i	− 2.20761i
$306$	0	0
$307$	119.712i	0.389941i	0.980809	+	0.194970i	$0.0624610\pi$
$307$	119.712i	0.389941i	−0.980809	+	0.194970i	$0.937539\pi$
$308$	0	0
$309$	0	0
$310$	189.467i	0.611183i
$311$	−313.161	−1.00695	−0.503474	−	0.864010i	$-0.667945\pi$
$311$	−313.161	−1.00695	−0.503474	+	0.864010i	$0.667945\pi$
$312$	0	0
$313$	−392.583	−1.25426	−0.627130	−	0.778915i	$-0.715770\pi$
$313$	−392.583	−1.25426	−0.627130	+	0.778915i	$0.715770\pi$
$314$	− 19.8122i	− 0.0630963i
$315$	0	0
$316$	234.032i	0.740608i
$317$	−35.4623	−0.111869	−0.0559343	−	0.998434i	$-0.517814\pi$
$317$	−35.4623	−0.111869	−0.0559343	+	0.998434i	$0.517814\pi$
$318$	0	0
$319$	0	0
$320$	−211.731	−0.661659
$321$	0	0
$322$	−33.9402	−0.105404
$323$	376.746	1.16640
$324$	0	0
$325$	− 600.866i	− 1.84882i
$326$	− 67.5990i	− 0.207359i
$327$	0	0
$328$	80.5380	0.245543
$329$	− 39.2164i	− 0.119199i
$330$	0	0
$331$	−437.250	−1.32100	−0.660498	−	0.750828i	$-0.729655\pi$
$331$	−437.250	−1.32100	−0.660498	+	0.750828i	$0.729655\pi$
$332$	− 311.410i	− 0.937981i
$333$	0	0
$334$	87.7817	0.262820
$335$	−709.426	−2.11769
$336$	0	0
$337$	− 163.851i	− 0.486205i	−0.970001	−	0.243102i	$-0.921835\pi$
$337$	− 163.851i	− 0.486205i	0.970001	−	0.243102i	$-0.0781651\pi$
$338$	32.6782i	0.0966811i
$339$	0	0
$340$	− 850.885i	− 2.50260i
$341$	0	0
$342$	0	0
$343$	332.216i	0.968559i
$344$	220.193	0.640096
$345$	0	0
$346$	−162.210	−0.468814
$347$	499.597i	1.43976i	0.694099	+	0.719880i	$0.255803\pi$
$347$	499.597i	1.43976i	−0.694099	+	0.719880i	$0.744197\pi$
$348$	0	0
$349$	294.050i	0.842551i	0.906933	+	0.421276i	$0.138417\pi$
$349$	294.050i	0.842551i	−0.906933	+	0.421276i	$0.861583\pi$
$350$	109.195	0.311985
$351$	0	0
$352$	0	0
$353$	−36.8161	−0.104295	−0.0521474	−	0.998639i	$-0.516607\pi$
$353$	−36.8161	−0.104295	−0.0521474	+	0.998639i	$0.516607\pi$
$354$	0	0
$355$	−554.726	−1.56261
$356$	104.684	0.294057
$357$	0	0
$358$	43.5446i	0.121633i
$359$	− 372.705i	− 1.03817i	−0.854721	−	0.519087i	$-0.826272\pi$
$359$	− 372.705i	− 1.03817i	0.854721	−	0.519087i	$-0.173728\pi$
$360$	0	0
$361$	197.249	0.546396
$362$	− 104.178i	− 0.287785i
$363$	0	0
$364$	215.564	0.592209
$365$	− 210.588i	− 0.576955i
$366$	0	0
$367$	−213.277	−0.581137	−0.290569	−	0.956854i	$-0.593844\pi$
$367$	−213.277	−0.581137	−0.290569	+	0.956854i	$0.593844\pi$
$368$	−138.877	−0.377382
$369$	0	0
$370$	− 95.7140i	− 0.258686i
$371$	235.948i	0.635978i
$372$	0	0
$373$	− 32.7478i	− 0.0877957i	−0.999036	−	0.0438978i	$-0.986022\pi$
$373$	− 32.7478i	− 0.0877957i	0.999036	−	0.0438978i	$-0.0139776\pi$
$374$	0	0
$375$	0	0
$376$	47.7674i	0.127041i
$377$	197.724	0.524467
$378$	0	0
$379$	542.952	1.43259	0.716296	−	0.697796i	$-0.245836\pi$
$379$	542.952	1.43259	0.716296	+	0.697796i	$0.245836\pi$
$380$	369.833i	0.973245i
$381$	0	0
$382$	97.8817i	0.256235i
$383$	−598.643	−1.56304	−0.781518	−	0.623883i	$-0.785554\pi$
$383$	−598.643	−1.56304	−0.781518	+	0.623883i	$0.785554\pi$
$384$	0	0
$385$	0	0
$386$	−141.119	−0.365594
$387$	0	0
$388$	−131.226	−0.338212
$389$	−220.631	−0.567175	−0.283587	−	0.958946i	$-0.591525\pi$
$389$	−220.631	−0.567175	−0.283587	+	0.958946i	$0.591525\pi$
$390$	0	0
$391$	− 371.824i	− 0.950956i
$392$	− 160.778i	− 0.410148i
$393$	0	0
$394$	36.0944	0.0916102
$395$	− 531.468i	− 1.34549i
$396$	0	0
$397$	−734.263	−1.84953	−0.924764	−	0.380541i	$-0.875738\pi$
$397$	−734.263	−1.84953	−0.924764	+	0.380541i	$0.875738\pi$
$398$	38.2480i	0.0961006i
$399$	0	0
$400$	446.803	1.11701
$401$	90.9686	0.226854	0.113427	−	0.993546i	$-0.463817\pi$
$401$	90.9686	0.226854	0.113427	+	0.993546i	$0.463817\pi$
$402$	0	0
$403$	− 525.849i	− 1.30484i
$404$	− 323.287i	− 0.800214i
$405$	0	0
$406$	35.9322i	0.0885029i
$407$	0	0
$408$	0	0
$409$	− 173.866i	− 0.425099i	−0.977150	−	0.212550i	$-0.931823\pi$
$409$	− 173.866i	− 0.425099i	0.977150	−	0.212550i	$-0.0681767\pi$
$410$	−86.2203	−0.210293
$411$	0	0
$412$	−547.600	−1.32913
$413$	409.665i	0.991925i
$414$	0	0
$415$	707.185i	1.70406i
$416$	−401.355	−0.964795
$417$	0	0
$418$	0	0
$419$	−374.418	−0.893599	−0.446800	−	0.894634i	$-0.647436\pi$
$419$	−374.418	−0.893599	−0.446800	+	0.894634i	$0.647436\pi$
$420$	0	0
$421$	−0.114071	−0.000270952 0	−0.000135476	−	1.00000i	$-0.500043\pi$
$421$	−0.114071	−0.000270952 0	−0.000135476		1.00000i	$0.500043\pi$
$422$	−148.868	−0.352768
$423$	0	0
$424$	− 287.395i	− 0.677819i
$425$	1196.26i	2.81472i
$426$	0	0
$427$	−339.606	−0.795330
$428$	240.790i	0.562593i
$429$	0	0
$430$	−235.729	−0.548206
$431$	− 265.064i	− 0.614997i	−0.951549	−	0.307499i	$-0.900508\pi$
$431$	− 265.064i	− 0.614997i	0.951549	−	0.307499i	$-0.0994919\pi$
$432$	0	0
$433$	−180.197	−0.416158	−0.208079	−	0.978112i	$-0.566721\pi$
$433$	−180.197	−0.416158	−0.208079	+	0.978112i	$0.566721\pi$
$434$	95.5620	0.220189
$435$	0	0
$436$	537.026i	1.23171i
$437$	161.611i	0.369820i
$438$	0	0
$439$	410.931i	0.936062i	0.883712	+	0.468031i	$0.155037\pi$
$439$	410.931i	0.936062i	−0.883712	+	0.468031i	$0.844963\pi$
$440$	0	0
$441$	0	0
$442$	− 286.347i	− 0.647843i
$443$	−21.9986	−0.0496583	−0.0248291	−	0.999692i	$-0.507904\pi$
$443$	−21.9986	−0.0496583	−0.0248291	+	0.999692i	$0.507904\pi$
$444$	0	0
$445$	−237.730	−0.534224
$446$	54.2314i	0.121595i
$447$	0	0
$448$	106.791i	0.238374i
$449$	−144.214	−0.321189	−0.160594	−	0.987020i	$-0.551341\pi$
$449$	−144.214	−0.321189	−0.160594	+	0.987020i	$0.551341\pi$
$450$	0	0
$451$	0	0
$452$	523.699	1.15863
$453$	0	0
$454$	−116.327	−0.256228
$455$	−489.528	−1.07589
$456$	0	0
$457$	734.981i	1.60827i	0.594445	+	0.804137i	$0.297372\pi$
$457$	734.981i	1.60827i	−0.594445	+	0.804137i	$0.702628\pi$
$458$	1.92138i	0.00419515i
$459$	0	0
$460$	365.001	0.793480
$461$	− 128.454i	− 0.278643i	−0.990247	−	0.139321i	$-0.955508\pi$
$461$	− 128.454i	− 0.278643i	0.990247	−	0.139321i	$-0.0444921\pi$
$462$	0	0
$463$	7.10283	0.0153409	0.00767044	−	0.999971i	$-0.497558\pi$
$463$	7.10283	0.0153409	0.00767044	+	0.999971i	$0.497558\pi$
$464$	147.027i	0.316869i
$465$	0	0
$466$	193.694	0.415651
$467$	285.646	0.611662	0.305831	−	0.952086i	$-0.401066\pi$
$467$	285.646	0.611662	0.305831	+	0.952086i	$0.401066\pi$
$468$	0	0
$469$	357.815i	0.762933i
$470$	− 51.1376i	− 0.108803i
$471$	0	0
$472$	− 498.991i	− 1.05718i
$473$	0	0
$474$	0	0
$475$	− 519.947i	− 1.09463i
$476$	−429.164	−0.901605
$477$	0	0
$478$	225.514	0.471786
$479$	− 392.628i	− 0.819682i	−0.912157	−	0.409841i	$-0.865584\pi$
$479$	− 392.628i	− 0.819682i	0.912157	−	0.409841i	$-0.134416\pi$
$480$	0	0
$481$	265.646i	0.552279i
$482$	−28.4612	−0.0590481
$483$	0	0
$484$	0	0
$485$	298.004	0.614440
$486$	0	0
$487$	515.788	1.05911	0.529557	−	0.848275i	$-0.322358\pi$
$487$	515.788	1.05911	0.529557	+	0.848275i	$0.322358\pi$
$488$	413.656	0.847655
$489$	0	0
$490$	172.121i	0.351268i
$491$	− 308.778i	− 0.628875i	−0.949278	−	0.314438i	$-0.898184\pi$
$491$	− 308.778i	− 0.628875i	0.949278	−	0.314438i	$-0.101816\pi$
$492$	0	0
$493$	−393.646	−0.798472
$494$	124.459i	0.251942i
$495$	0	0
$496$	391.021	0.788348
$497$	279.789i	0.562955i
$498$	0	0
$499$	579.959	1.16224	0.581122	−	0.813817i	$-0.302614\pi$
$499$	579.959	1.16224	0.581122	+	0.813817i	$0.302614\pi$
$500$	−451.779	−0.903558
$501$	0	0
$502$	76.9531i	0.153293i
$503$	780.859i	1.55240i	0.630484	+	0.776202i	$0.282856\pi$
$503$	780.859i	1.55240i	−0.630484	+	0.776202i	$0.717144\pi$
$504$	0	0
$505$	734.157i	1.45378i
$506$	0	0
$507$	0	0
$508$	230.244i	0.453236i
$509$	677.104	1.33026	0.665131	−	0.746726i	$-0.268376\pi$
$509$	677.104	1.33026	0.665131	+	0.746726i	$0.268376\pi$
$510$	0	0
$511$	−106.215	−0.207857
$512$	− 517.365i	− 1.01048i
$513$	0	0
$514$	− 18.5824i	− 0.0361525i
$515$	1243.55	2.41467
$516$	0	0
$517$	0	0
$518$	−48.2756	−0.0931961
$519$	0	0
$520$	596.268	1.14667
$521$	665.534	1.27742	0.638708	−	0.769449i	$-0.279469\pi$
$521$	665.534	1.27742	0.638708	+	0.769449i	$0.279469\pi$
$522$	0	0
$523$	− 900.914i	− 1.72259i	−0.508106	−	0.861295i	$-0.669654\pi$
$523$	− 900.914i	− 1.72259i	0.508106	−	0.861295i	$-0.330346\pi$
$524$	− 794.857i	− 1.51690i
$525$	0	0
$526$	162.900	0.309695
$527$	1046.91i	1.98654i
$528$	0	0
$529$	−369.500	−0.698488
$530$	307.672i	0.580514i
$531$	0	0
$532$	186.534	0.350628
$533$	239.297	0.448963
$534$	0	0
$535$	− 546.814i	− 1.02208i
$536$	− 435.836i	− 0.813126i
$537$	0	0
$538$	− 182.531i	− 0.339276i
$539$	0	0
$540$	0	0
$541$	− 389.421i	− 0.719818i	−0.932988	−	0.359909i	$-0.882808\pi$
$541$	− 389.421i	− 0.719818i	0.932988	−	0.359909i	$-0.117192\pi$
$542$	−259.942	−0.479598
$543$	0	0
$544$	799.052	1.46885
$545$	− 1219.54i	− 2.23769i
$546$	0	0
$547$	− 705.860i	− 1.29042i	−0.764005	−	0.645210i	$-0.776770\pi$
$547$	− 705.860i	− 1.29042i	0.764005	−	0.645210i	$-0.223230\pi$
$548$	181.652	0.331483
$549$	0	0
$550$	0	0
$551$	171.097	0.310520
$552$	0	0
$553$	−268.058	−0.484734
$554$	−138.245	−0.249540
$555$	0	0
$556$	473.567i	0.851739i
$557$	− 165.549i	− 0.297215i	−0.988896	−	0.148608i	$-0.952521\pi$
$557$	− 165.549i	− 0.297215i	0.988896	−	0.148608i	$-0.0474792\pi$
$558$	0	0
$559$	654.245	1.17039
$560$	− 364.012i	− 0.650022i
$561$	0	0
$562$	−17.1993	−0.0306037
$563$	− 363.473i	− 0.645600i	−0.946467	−	0.322800i	$-0.895376\pi$
$563$	− 363.473i	− 0.645600i	0.946467	−	0.322800i	$-0.104624\pi$
$564$	0	0
$565$	−1189.28	−2.10491
$566$	194.093	0.342921
$567$	0	0
$568$	− 340.796i	− 0.599992i
$569$	− 231.938i	− 0.407624i	−0.979010	−	0.203812i	$-0.934667\pi$
$569$	− 231.938i	− 0.407624i	0.979010	−	0.203812i	$-0.0653332\pi$
$570$	0	0
$571$	306.665i	0.537067i	0.963270	+	0.268533i	$0.0865390\pi$
$571$	306.665i	0.537067i	−0.963270	+	0.268533i	$0.913461\pi$
$572$	0	0
$573$	0	0
$574$	43.4872i	0.0757617i
$575$	−513.154	−0.892441
$576$	0	0
$577$	85.1608	0.147592	0.0737961	−	0.997273i	$-0.476489\pi$
$577$	85.1608	0.147592	0.0737961	+	0.997273i	$0.476489\pi$
$578$	380.010i	0.657457i
$579$	0	0
$580$	− 386.423i	− 0.666247i
$581$	356.685	0.613916
$582$	0	0
$583$	0	0
$584$	129.375	0.221532
$585$	0	0
$586$	−40.8970	−0.0697902
$587$	615.176	1.04800	0.524000	−	0.851718i	$-0.324439\pi$
$587$	615.176	1.04800	0.524000	+	0.851718i	$0.324439\pi$
$588$	0	0
$589$	− 455.033i	− 0.772552i
$590$	534.197i	0.905418i
$591$	0	0
$592$	−197.534	−0.333673
$593$	74.4809i	0.125600i	0.998026	+	0.0628001i	$0.0200030\pi$
$593$	74.4809i	0.125600i	−0.998026	+	0.0628001i	$0.979997\pi$
$594$	0	0
$595$	974.595	1.63798
$596$	939.500i	1.57634i
$597$	0	0
$598$	122.833	0.205406
$599$	741.739	1.23830	0.619148	−	0.785275i	$-0.287478\pi$
$599$	741.739	1.23830	0.619148	+	0.785275i	$0.287478\pi$
$600$	0	0
$601$	1121.84i	1.86663i	0.359059	+	0.933315i	$0.383098\pi$
$601$	1121.84i	1.86663i	−0.359059	+	0.933315i	$0.616902\pi$
$602$	118.895i	0.197500i
$603$	0	0
$604$	− 180.081i	− 0.298147i
$605$	0	0
$606$	0	0
$607$	− 11.0765i	− 0.0182479i	−0.999958	−	0.00912394i	$-0.997096\pi$
$607$	− 11.0765i	− 0.0182479i	0.999958	−	0.00912394i	$-0.00290428\pi$
$608$	−347.304	−0.571224
$609$	0	0
$610$	−442.841	−0.725969
$611$	141.928i	0.232288i
$612$	0	0
$613$	− 747.900i	− 1.22006i	−0.792376	−	0.610032i	$-0.791156\pi$
$613$	− 747.900i	− 1.22006i	0.792376	−	0.610032i	$-0.208844\pi$
$614$	78.7338	0.128231
$615$	0	0
$616$	0	0
$617$	−771.423	−1.25028	−0.625140	−	0.780513i	$-0.714958\pi$
$617$	−771.423	−1.25028	−0.625140	+	0.780513i	$0.714958\pi$
$618$	0	0
$619$	175.333	0.283252	0.141626	−	0.989920i	$-0.454767\pi$
$619$	175.333	0.283252	0.141626	+	0.989920i	$0.454767\pi$
$620$	−1027.70	−1.65757
$621$	0	0
$622$	205.964i	0.331132i
$623$	119.904i	0.192463i
$624$	0	0
$625$	10.1549	0.0162478
$626$	258.200i	0.412460i
$627$	0	0
$628$	107.465	0.171122
$629$	− 528.872i	− 0.840814i
$630$	0	0
$631$	265.472	0.420717	0.210359	−	0.977624i	$-0.432537\pi$
$631$	265.472	0.420717	0.210359	+	0.977624i	$0.432537\pi$
$632$	326.507	0.516625
$633$	0	0
$634$	23.3234i	0.0367877i
$635$	− 522.865i	− 0.823410i
$636$	0	0
$637$	− 477.709i	− 0.749935i
$638$	0	0
$639$	0	0
$640$	1018.75i	1.59180i
$641$	1143.64	1.78415	0.892077	−	0.451884i	$-0.149248\pi$
$641$	1143.64	1.78415	0.892077	+	0.451884i	$0.149248\pi$
$642$	0	0
$643$	563.315	0.876073	0.438036	−	0.898957i	$-0.355674\pi$
$643$	563.315	0.876073	0.438036	+	0.898957i	$0.355674\pi$
$644$	− 184.097i	− 0.285864i
$645$	0	0
$646$	− 247.784i	− 0.383567i
$647$	−56.7108	−0.0876520	−0.0438260	−	0.999039i	$-0.513955\pi$
$647$	−56.7108	−0.0876520	−0.0438260	+	0.999039i	$0.513955\pi$
$648$	0	0
$649$	0	0
$650$	−395.187	−0.607980
$651$	0	0
$652$	366.667	0.562373
$653$	−762.352	−1.16746	−0.583731	−	0.811947i	$-0.698408\pi$
$653$	−762.352	−1.16746	−0.583731	+	0.811947i	$0.698408\pi$
$654$	0	0
$655$	1805.05i	2.75581i
$656$	177.941i	0.271252i
$657$	0	0
$658$	−25.7924	−0.0391982
$659$	− 1137.55i	− 1.72617i	−0.505056	−	0.863086i	$-0.668528\pi$
$659$	− 1137.55i	− 1.72617i	0.505056	−	0.863086i	$-0.331472\pi$
$660$	0	0
$661$	814.127	1.23166	0.615829	−	0.787879i	$-0.288821\pi$
$661$	814.127	1.23166	0.615829	+	0.787879i	$0.288821\pi$
$662$	287.577i	0.434406i
$663$	0	0
$664$	−434.459	−0.654306
$665$	−423.603	−0.636997
$666$	0	0
$667$	− 168.861i	− 0.253165i
$668$	476.141i	0.712787i
$669$	0	0
$670$	466.586i	0.696397i
$671$	0	0
$672$	0	0
$673$	317.298i	0.471468i	0.971818	+	0.235734i	$0.0757494\pi$
$673$	317.298i	0.471468i	−0.971818	+	0.235734i	$0.924251\pi$
$674$	−107.764	−0.159887
$675$	0	0
$676$	−177.252	−0.262206
$677$	491.843i	0.726504i	0.931691	+	0.363252i	$0.118334\pi$
$677$	491.843i	0.726504i	−0.931691	+	0.363252i	$0.881666\pi$
$678$	0	0
$679$	− 150.305i	− 0.221362i
$680$	−1187.10	−1.74574
$681$	0	0
$682$	0	0
$683$	1267.89	1.85635	0.928176	−	0.372142i	$-0.121376\pi$
$683$	1267.89	1.85635	0.928176	+	0.372142i	$0.121376\pi$
$684$	0	0
$685$	−412.518	−0.602215
$686$	218.497	0.318508
$687$	0	0
$688$	486.496i	0.707116i
$689$	− 853.919i	− 1.23936i
$690$	0	0
$691$	−232.404	−0.336329	−0.168165	−	0.985759i	$-0.553784\pi$
$691$	−232.404	−0.336329	−0.168165	+	0.985759i	$0.553784\pi$
$692$	− 879.850i	− 1.27146i
$693$	0	0
$694$	328.582	0.473461
$695$	− 1075.43i	− 1.54738i
$696$	0	0
$697$	−476.414	−0.683521
$698$	193.395	0.277071
$699$	0	0
$700$	592.288i	0.846126i
$701$	443.675i	0.632918i	0.948606	+	0.316459i	$0.102494\pi$
$701$	443.675i	0.632918i	−0.948606	+	0.316459i	$0.897506\pi$
$702$	0	0
$703$	229.872i	0.326987i
$704$	0	0
$705$	0	0
$706$	24.2137i	0.0342971i
$707$	370.289	0.523747
$708$	0	0
$709$	−410.502	−0.578988	−0.289494	−	0.957180i	$-0.593487\pi$
$709$	−410.502	−0.578988	−0.289494	+	0.957180i	$0.593487\pi$
$710$	364.840i	0.513859i
$711$	0	0
$712$	− 146.049i	− 0.205125i
$713$	−449.088	−0.629856
$714$	0	0
$715$	0	0
$716$	−236.193	−0.329878
$717$	0	0
$718$	−245.126	−0.341401
$719$	−232.189	−0.322933	−0.161467	−	0.986878i	$-0.551622\pi$
$719$	−232.189	−0.322933	−0.161467	+	0.986878i	$0.551622\pi$
$720$	0	0
$721$	− 627.216i	− 0.869925i
$722$	− 129.730i	− 0.179681i
$723$	0	0
$724$	565.079	0.780496
$725$	543.271i	0.749339i
$726$	0	0
$727$	−523.474	−0.720047	−0.360024	−	0.932943i	$-0.617231\pi$
$727$	−523.474	−0.720047	−0.360024	+	0.932943i	$0.617231\pi$
$728$	− 300.742i	− 0.413106i
$729$	0	0
$730$	−138.503	−0.189730
$731$	−1302.53	−1.78185
$732$	0	0
$733$	− 316.538i	− 0.431838i	−0.976411	−	0.215919i	$-0.930725\pi$
$733$	− 316.538i	− 0.431838i	0.976411	−	0.215919i	$-0.0692748\pi$
$734$	140.271i	0.191105i
$735$	0	0
$736$	342.766i	0.465715i
$737$	0	0
$738$	0	0
$739$	706.414i	0.955905i	0.878386	+	0.477953i	$0.158621\pi$
$739$	706.414i	0.955905i	−0.878386	+	0.477953i	$0.841379\pi$
$740$	519.167	0.701577
$741$	0	0
$742$	155.182	0.209140
$743$	322.350i	0.433849i	0.976188	+	0.216924i	$0.0696025\pi$
$743$	322.350i	0.433849i	−0.976188	+	0.216924i	$0.930398\pi$
$744$	0	0
$745$	− 2133.53i	− 2.86379i
$746$	−21.5381	−0.0288714
$747$	0	0
$748$	0	0
$749$	−275.798	−0.368222
$750$	0	0
$751$	1319.20	1.75660	0.878298	−	0.478114i	$-0.158679\pi$
$751$	1319.20	1.75660	0.878298	+	0.478114i	$0.158679\pi$
$752$	−105.538	−0.140343
$753$	0	0
$754$	− 130.042i	− 0.172470i
$755$	408.949i	0.541654i
$756$	0	0
$757$	911.592	1.20422	0.602108	−	0.798414i	$-0.294328\pi$
$757$	911.592	1.20422	0.602108	+	0.798414i	$0.294328\pi$
$758$	− 357.097i	− 0.471104i
$759$	0	0
$760$	515.968	0.678905
$761$	175.557i	0.230692i	0.993325	+	0.115346i	$0.0367977\pi$
$761$	175.557i	0.230692i	−0.993325	+	0.115346i	$0.963202\pi$
$762$	0	0
$763$	−615.104	−0.806165
$764$	−530.925	−0.694928
$765$	0	0
$766$	393.724i	0.514000i
$767$	− 1482.62i	− 1.93301i
$768$	0	0
$769$	692.657i	0.900725i	0.892846	+	0.450362i	$0.148705\pi$
$769$	692.657i	0.900725i	−0.892846	+	0.450362i	$0.851295\pi$
$770$	0	0
$771$	0	0
$772$	− 765.453i	− 0.991520i
$773$	−419.149	−0.542237	−0.271119	−	0.962546i	$-0.587394\pi$
$773$	−419.149	−0.542237	−0.271119	+	0.962546i	$0.587394\pi$
$774$	0	0
$775$	1444.84	1.86430
$776$	183.078i	0.235926i
$777$	0	0
$778$	145.108i	0.186514i
$779$	207.071	0.265816
$780$	0	0
$781$	0	0
$782$	−244.547	−0.312720
$783$	0	0
$784$	355.224	0.453091
$785$	−244.043	−0.310883
$786$	0	0
$787$	985.114i	1.25173i	0.779930	+	0.625866i	$0.215254\pi$
$787$	985.114i	1.25173i	−0.779930	+	0.625866i	$0.784746\pi$
$788$	195.782i	0.248454i
$789$	0	0
$790$	−349.544	−0.442460
$791$	599.839i	0.758330i
$792$	0	0
$793$	1229.07	1.54990
$794$	482.921i	0.608213i
$795$	0	0
$796$	−207.463	−0.260632
$797$	−750.217	−0.941301	−0.470650	−	0.882320i	$-0.655981\pi$
$797$	−750.217	−0.941301	−0.470650	+	0.882320i	$0.655981\pi$
$798$	0	0
$799$	− 282.563i	− 0.353646i
$800$	− 1102.77i	− 1.37846i
$801$	0	0
$802$	− 59.8296i	− 0.0746005i
$803$	0	0
$804$	0	0
$805$	418.068i	0.519340i
$806$	−345.849	−0.429092
$807$	0	0
$808$	−451.029	−0.558204
$809$	729.962i	0.902302i	0.892448	+	0.451151i	$0.148986\pi$
$809$	729.962i	0.902302i	−0.892448	+	0.451151i	$0.851014\pi$
$810$	0	0
$811$	− 23.9687i	− 0.0295545i	−0.999891	−	0.0147773i	$-0.995296\pi$
$811$	− 23.9687i	− 0.0295545i	0.999891	−	0.0147773i	$-0.00470392\pi$
$812$	−194.902	−0.240026
$813$	0	0
$814$	0	0
$815$	−832.671	−1.02168
$816$	0	0
$817$	566.138	0.692947
$818$	−114.351	−0.139793
$819$	0	0
$820$	− 467.672i	− 0.570332i
$821$	− 2.62512i	− 0.00319747i	−0.999999	−	0.00159874i	$-0.999491\pi$
$821$	− 2.62512i	− 0.00319747i	0.999999	−	0.00159874i	$-0.000508893\pi$
$822$	0	0
$823$	33.7690	0.0410316	0.0205158	−	0.999790i	$-0.493469\pi$
$823$	33.7690	0.0410316	0.0205158	+	0.999790i	$0.493469\pi$
$824$	763.978i	0.927157i
$825$	0	0
$826$	269.435	0.326192
$827$	1191.15i	1.44033i	0.693803	+	0.720165i	$0.255934\pi$
$827$	1191.15i	1.44033i	−0.693803	+	0.720165i	$0.744066\pi$
$828$	0	0
$829$	470.485	0.567533	0.283767	−	0.958893i	$-0.408416\pi$
$829$	470.485	0.567533	0.283767	+	0.958893i	$0.408416\pi$
$830$	465.112	0.560376
$831$	0	0
$832$	− 386.489i	− 0.464530i
$833$	951.064i	1.14173i
$834$	0	0
$835$	− 1081.28i	− 1.29494i
$836$	0	0
$837$	0	0
$838$	246.253i	0.293858i
$839$	−542.993	−0.647190	−0.323595	−	0.946196i	$-0.604892\pi$
$839$	−542.993	−0.647190	−0.323595	+	0.946196i	$0.604892\pi$
$840$	0	0
$841$	662.228	0.787430
$842$	0.0750236i	0 8.91017e-5i
$843$	0	0
$844$	− 807.483i	− 0.956733i
$845$	402.523	0.476359
$846$	0	0
$847$	0	0
$848$	634.973	0.748789
$849$	0	0
$850$	786.772	0.925614
$851$	226.868	0.266590
$852$	0	0
$853$	432.271i	0.506765i	0.967366	+	0.253383i	$0.0815432\pi$
$853$	432.271i	0.506765i	−0.967366	+	0.253383i	$0.918457\pi$
$854$	223.357i	0.261542i
$855$	0	0
$856$	335.935	0.392447
$857$	− 267.129i	− 0.311703i	−0.987781	−	0.155851i	$-0.950188\pi$
$857$	− 267.129i	− 0.311703i	0.987781	−	0.155851i	$-0.0498121\pi$
$858$	0	0
$859$	537.909	0.626203	0.313102	−	0.949720i	$-0.398632\pi$
$859$	537.909	0.626203	0.313102	+	0.949720i	$0.398632\pi$
$860$	− 1278.63i	− 1.48678i
$861$	0	0
$862$	−174.331	−0.202240
$863$	1137.76	1.31838	0.659191	−	0.751975i	$-0.270899\pi$
$863$	1137.76	1.31838	0.659191	+	0.751975i	$0.270899\pi$
$864$	0	0
$865$	1998.06i	2.30990i
$866$	118.514i	0.136853i
$867$	0	0
$868$	518.343i	0.597169i
$869$	0	0
$870$	0	0
$871$	− 1294.97i	− 1.48676i
$872$	749.225	0.859203
$873$	0	0
$874$	106.291	0.121614
$875$	− 517.463i	− 0.591386i
$876$	0	0
$877$	− 679.083i	− 0.774324i	−0.922012	−	0.387162i	$-0.873455\pi$
$877$	− 679.083i	− 0.774324i	0.922012	−	0.387162i	$-0.126545\pi$
$878$	270.267	0.307822
$879$	0	0
$880$	0	0
$881$	−419.212	−0.475837	−0.237918	−	0.971285i	$-0.576465\pi$
$881$	−419.212	−0.475837	−0.237918	+	0.971285i	$0.576465\pi$
$882$	0	0
$883$	−361.858	−0.409806	−0.204903	−	0.978782i	$-0.565688\pi$
$883$	−361.858	−0.409806	−0.204903	+	0.978782i	$0.565688\pi$
$884$	1553.19	1.75700
$885$	0	0
$886$	14.4684i	0.0163300i
$887$	42.4684i	0.0478786i	0.999713	+	0.0239393i	$0.00762085\pi$
$887$	42.4684i	0.0478786i	−0.999713	+	0.0239393i	$0.992379\pi$
$888$	0	0
$889$	−263.719	−0.296647
$890$	156.354i	0.175678i
$891$	0	0
$892$	−294.159	−0.329775
$893$	122.815i	0.137530i
$894$	0	0
$895$	536.374	0.599301
$896$	513.832	0.573473
$897$	0	0
$898$	94.8487i	0.105622i
$899$	475.445i	0.528860i
$900$	0	0
$901$	1700.06i	1.88685i
$902$	0	0
$903$	0	0
$904$	− 730.632i	− 0.808221i
$905$	−1283.25	−1.41795
$906$	0	0
$907$	−61.6587	−0.0679810	−0.0339905	−	0.999422i	$-0.510822\pi$
$907$	−61.6587	−0.0679810	−0.0339905	+	0.999422i	$0.510822\pi$
$908$	− 630.978i	− 0.694909i
$909$	0	0
$910$	321.960i	0.353802i
$911$	−271.238	−0.297737	−0.148868	−	0.988857i	$-0.547563\pi$
$911$	−271.238	−0.297737	−0.148868	+	0.988857i	$0.547563\pi$
$912$	0	0
$913$	0	0
$914$	483.393	0.528877
$915$	0	0
$916$	−10.4219	−0.0113776
$917$	910.421	0.992826
$918$	0	0
$919$	126.069i	0.137180i	0.997645	+	0.0685902i	$0.0218501\pi$
$919$	126.069i	0.137180i	−0.997645	+	0.0685902i	$0.978150\pi$
$920$	− 509.227i	− 0.553507i
$921$	0	0
$922$	−84.4838	−0.0916310
$923$	− 1012.58i	− 1.09706i
$924$	0	0
$925$	−729.895	−0.789076
$926$	− 4.67150i	− 0.00504481i
$927$	0	0
$928$	362.884	0.391038
$929$	275.277	0.296315	0.148158	−	0.988964i	$-0.452666\pi$
$929$	275.277	0.296315	0.148158	+	0.988964i	$0.452666\pi$
$930$	0	0
$931$	− 413.376i	− 0.444012i
$932$	1050.62i	1.12728i
$933$	0	0
$934$	− 187.868i	− 0.201144i
$935$	0	0
$936$	0	0
$937$	− 1544.67i	− 1.64853i	−0.566205	−	0.824265i	$-0.691589\pi$
$937$	− 1544.67i	− 1.64853i	0.566205	−	0.824265i	$-0.308411\pi$
$938$	235.333	0.250888
$939$	0	0
$940$	277.378	0.295083
$941$	− 1070.76i	− 1.13789i	−0.822374	−	0.568947i	$-0.807351\pi$
$941$	− 1070.76i	− 1.13789i	0.822374	−	0.568947i	$-0.192649\pi$
$942$	0	0
$943$	− 204.365i	− 0.216718i
$944$	1102.47	1.16787
$945$	0	0
$946$	0	0
$947$	1480.48	1.56333	0.781667	−	0.623696i	$-0.214370\pi$
$947$	1480.48	1.56333	0.781667	+	0.623696i	$0.214370\pi$
$948$	0	0
$949$	384.404	0.405062
$950$	−341.967	−0.359965
$951$	0	0
$952$	598.742i	0.628931i
$953$	− 432.963i	− 0.454316i	−0.973858	−	0.227158i	$-0.927057\pi$
$953$	− 432.963i	− 0.454316i	0.973858	−	0.227158i	$-0.0729434\pi$
$954$	0	0
$955$	1205.69	1.26250
$956$	1223.22i	1.27952i
$957$	0	0
$958$	−258.229	−0.269550
$959$	208.063i	0.216958i
$960$	0	0
$961$	303.451	0.315766
$962$	174.714	0.181616
$963$	0	0
$964$	− 154.378i	− 0.160143i
$965$	1738.28i	1.80133i
$966$	0	0
$967$	− 211.542i	− 0.218761i	−0.994000	−	0.109381i	$-0.965113\pi$
$967$	− 211.542i	− 0.218761i	0.994000	−	0.109381i	$-0.0348867\pi$
$968$	0	0
$969$	0	0
$970$	− 195.995i	− 0.202057i
$971$	1391.67	1.43323	0.716617	−	0.697466i	$-0.245689\pi$
$971$	1391.67	1.43323	0.716617	+	0.697466i	$0.245689\pi$
$972$	0	0
$973$	−542.419	−0.557470
$974$	− 339.231i	− 0.348287i
$975$	0	0
$976$	913.933i	0.936407i
$977$	32.5334	0.0332993	0.0166496	−	0.999861i	$-0.494700\pi$
$977$	32.5334	0.0332993	0.0166496	+	0.999861i	$0.494700\pi$
$978$	0	0
$979$	0	0
$980$	−933.613	−0.952666
$981$	0	0
$982$	−203.082	−0.206804
$983$	−1212.45	−1.23342	−0.616709	−	0.787191i	$-0.711534\pi$
$983$	−1212.45	−1.23342	−0.616709	+	0.787191i	$0.711534\pi$
$984$	0	0
$985$	− 444.604i	− 0.451374i
$986$	258.899i	0.262575i
$987$	0	0
$988$	−675.085	−0.683285
$989$	− 558.741i	− 0.564955i
$990$	0	0
$991$	−106.677	−0.107645	−0.0538227	−	0.998551i	$-0.517141\pi$
$991$	−106.677	−0.107645	−0.0538227	+	0.998551i	$0.517141\pi$
$992$	− 965.093i	− 0.972876i
$993$	0	0
$994$	184.016	0.185126
$995$	471.132	0.473499
$996$	0	0
$997$	1291.81i	1.29570i	0.761768	+	0.647850i	$0.224332\pi$
$997$	1291.81i	1.29570i	−0.761768	+	0.647850i	$0.775668\pi$
$998$	− 381.436i	− 0.382201i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.3.c.l.604.7		16
3.2	odd	2	inner	1089.3.c.l.604.10		16
11.4	even	5		99.3.k.b.28.3	yes	16
11.8	odd	10		99.3.k.b.46.3	yes	16
11.10	odd	2	inner	1089.3.c.l.604.9		16
33.8	even	10		99.3.k.b.46.2	yes	16
33.26	odd	10		99.3.k.b.28.2	✓	16
33.32	even	2	inner	1089.3.c.l.604.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.k.b.28.2	✓	16	33.26	odd	10
99.3.k.b.28.3	yes	16	11.4	even	5
99.3.k.b.46.2	yes	16	33.8	even	10
99.3.k.b.46.3	yes	16	11.8	odd	10
1089.3.c.l.604.7		16	1.1	even	1	trivial
1089.3.c.l.604.8		16	33.32	even	2	inner
1089.3.c.l.604.9		16	11.10	odd	2	inner
1089.3.c.l.604.10		16	3.2	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)

\(f(q)\)	\(=\)	\(q-0.657695i q^{2} +3.56744 q^{4} -8.10135 q^{5} +4.08611i q^{7} -4.97707i q^{8} +O(q^{10})\)	\(q-0.657695i q^{2} +3.56744 q^{4} -8.10135 q^{5} +4.08611i q^{7} -4.97707i q^{8} +5.32822i q^{10} -14.7880i q^{13} +2.68741 q^{14} +10.9964 q^{16} +29.4413i q^{17} -12.7965i q^{19} -28.9011 q^{20} -12.6293 q^{23} +40.6319 q^{25} -9.72602 q^{26} +14.5769i q^{28} +13.3705i q^{29} +35.5591 q^{31} -27.1405i q^{32} +19.3634 q^{34} -33.1030i q^{35} -17.9636 q^{37} -8.41621 q^{38} +40.3210i q^{40} +16.1818i q^{41} +44.2415i q^{43} +8.30624i q^{46} -9.59750 q^{47} +32.3037 q^{49} -26.7234i q^{50} -52.7554i q^{52} +57.7439 q^{53} +20.3368 q^{56} +8.79374 q^{58} +100.258 q^{59} +83.1124i q^{61} -23.3871i q^{62} +26.1353 q^{64} +119.803i q^{65} +87.5688 q^{67} +105.030i q^{68} -21.7717 q^{70} +68.4732 q^{71} +25.9942i q^{73} +11.8146i q^{74} -45.6508i q^{76} +65.6023i q^{79} -89.0854 q^{80} +10.6427 q^{82} -87.2922i q^{83} -238.514i q^{85} +29.0974 q^{86} +29.3444 q^{89} +60.4255 q^{91} -45.0543 q^{92} +6.31223i q^{94} +103.669i q^{95} -36.7844 q^{97} -21.2460i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 44 q^{4}+O(q^{10}) \) 16 * q - 44 * q^4	\( 16 q - 44 q^{4} + 244 q^{16} + 16 q^{25} - 80 q^{31} + 328 q^{34} - 280 q^{37} + 436 q^{49} + 140 q^{58} - 656 q^{64} + 300 q^{67} + 308 q^{70} + 580 q^{82} + 768 q^{91} - 720 q^{97}+O(q^{100}) \) 16 * q - 44 * q^4 + 244 * q^16 + 16 * q^25 - 80 * q^31 + 328 * q^34 - 280 * q^37 + 436 * q^49 + 140 * q^58 - 656 * q^64 + 300 * q^67 + 308 * q^70 + 580 * q^82 + 768 * q^91 - 720 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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