LMFDB - Embedded newform 189.3.b.b.134.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,3,Mod(134,189)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("189.134");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 189 = 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	189.b (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:	$5.14987699641$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			134.3
Root			$1.16372i$ of defining polynomial
Character	$\chi$	$=$	189.134
Dual form			189.3.b.b.134.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.16372i q^{2} +2.64575 q^{4} -5.40636i q^{5} +2.64575 q^{7} +7.73381i q^{8} +6.29150 q^{10} -18.1343i q^{11} +10.2288 q^{13} +3.07892i q^{14} +1.58301 q^{16} -4.24264i q^{17} +3.35425 q^{19} -14.3039i q^{20} +21.1033 q^{22} +28.9470i q^{23} -4.22876 q^{25} +11.9034i q^{26} +7.00000 q^{28} +36.0024i q^{29} +34.5203 q^{31} +32.7774i q^{32} +4.93725 q^{34} -14.3039i q^{35} -60.0405 q^{37} +3.90341i q^{38} +41.8118 q^{40} -71.9317i q^{41} -46.6863 q^{43} -47.9788i q^{44} -33.6863 q^{46} -18.2073i q^{47} +7.00000 q^{49} -4.92110i q^{50} +27.0627 q^{52} +43.3969i q^{53} -98.0405 q^{55} +20.4617i q^{56} -41.8967 q^{58} +53.1190i q^{59} -65.9555 q^{61} +40.1720i q^{62} -31.8118 q^{64} -55.3004i q^{65} +36.3542 q^{67} -11.2250i q^{68} +16.6458 q^{70} +0.219057i q^{71} -11.8523 q^{73} -69.8705i q^{74} +8.87451 q^{76} -47.9788i q^{77} -144.812 q^{79} -8.55830i q^{80} +83.7085 q^{82} +99.0837i q^{83} -22.9373 q^{85} -54.3298i q^{86} +140.247 q^{88} -92.7327i q^{89} +27.0627 q^{91} +76.5866i q^{92} +21.1882 q^{94} -18.1343i q^{95} -96.5385 q^{97} +8.14605i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{10} - 12 q^{13} - 36 q^{16} + 24 q^{19} - 32 q^{22} + 36 q^{25} + 28 q^{28} + 64 q^{31} - 12 q^{34} - 92 q^{37} + 72 q^{40} - 28 q^{43} + 24 q^{46} + 28 q^{49} + 140 q^{52} - 244 q^{55} - 284 q^{58}+ \cdots + 16 q^{97}+O(q^{100}) $ 4 * q + 4 * q^10 - 12 * q^13 - 36 * q^16 + 24 * q^19 - 32 * q^22 + 36 * q^25 + 28 * q^28 + 64 * q^31 - 12 * q^34 - 92 * q^37 + 72 * q^40 - 28 * q^43 + 24 * q^46 + 28 * q^49 + 140 * q^52 - 244 * q^55 - 284 * q^58 + 96 * q^61 - 32 * q^64 + 156 * q^67 + 56 * q^70 + 196 * q^73 - 28 * q^76 - 484 * q^79 + 356 * q^82 - 60 * q^85 + 180 * q^88 + 140 * q^91 + 180 * q^94 + 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$.

$n$	$29$	$136$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.16372i	0.581861i	0.956744	+	0.290930i	$0.0939648\pi$
$2$	1.16372i	0.581861i	−0.956744	+	0.290930i	$0.906035\pi$
$3$	0	0
$3$	0	0
$4$	2.64575	0.661438
$5$	− 5.40636i	− 1.08127i	−0.841256	−	0.540636i	$-0.818184\pi$
$5$	− 5.40636i	− 1.08127i	0.841256	−	0.540636i	$-0.181816\pi$
$6$	0	0
$7$	2.64575	0.377964
$7$	2.64575	0.377964
$8$	7.73381i	0.966726i
$9$	0	0
$10$	6.29150	0.629150
$11$	− 18.1343i	− 1.64857i	−0.566174	−	0.824286i	$-0.691577\pi$
$11$	− 18.1343i	− 1.64857i	0.566174	−	0.824286i	$-0.308423\pi$
$12$	0	0
$13$	10.2288	0.786827	0.393414	−	0.919362i	$-0.371294\pi$
$13$	10.2288	0.786827	0.393414	+	0.919362i	$0.371294\pi$
$14$	3.07892i	0.219923i
$15$	0	0
$16$	1.58301	0.0989378
$17$	− 4.24264i	− 0.249567i	−0.992184	−	0.124784i	$-0.960176\pi$
$17$	− 4.24264i	− 0.249567i	0.992184	−	0.124784i	$-0.0398236\pi$
$18$	0	0
$19$	3.35425	0.176539	0.0882697	−	0.996097i	$-0.471866\pi$
$19$	3.35425	0.176539	0.0882697	+	0.996097i	$0.471866\pi$
$20$	− 14.3039i	− 0.715195i
$21$	0	0
$22$	21.1033	0.959239
$23$	28.9470i	1.25857i	0.777176	+	0.629283i	$0.216651\pi$
$23$	28.9470i	1.25857i	−0.777176	+	0.629283i	$0.783349\pi$
$24$	0	0
$25$	−4.22876	−0.169150
$26$	11.9034i	0.457824i
$27$	0	0
$28$	7.00000	0.250000
$29$	36.0024i	1.24146i	0.784024	+	0.620730i	$0.213164\pi$
$29$	36.0024i	1.24146i	−0.784024	+	0.620730i	$0.786836\pi$
$30$	0	0
$31$	34.5203	1.11356	0.556778	−	0.830661i	$-0.312037\pi$
$31$	34.5203	1.11356	0.556778	+	0.830661i	$0.312037\pi$
$32$	32.7774i	1.02429i
$33$	0	0
$34$	4.93725	0.145213
$35$	− 14.3039i	− 0.408683i
$36$	0	0
$37$	−60.0405	−1.62272	−0.811358	−	0.584549i	$-0.801271\pi$
$37$	−60.0405	−1.62272	−0.811358	+	0.584549i	$0.801271\pi$
$38$	3.90341i	0.102721i
$39$	0	0
$40$	41.8118	1.04529
$41$	− 71.9317i	− 1.75443i	−0.480096	−	0.877216i	$-0.659398\pi$
$41$	− 71.9317i	− 1.75443i	0.480096	−	0.877216i	$-0.340602\pi$
$42$	0	0
$43$	−46.6863	−1.08573	−0.542864	−	0.839821i	$-0.682660\pi$
$43$	−46.6863	−1.08573	−0.542864	+	0.839821i	$0.682660\pi$
$44$	− 47.9788i	− 1.09043i
$45$	0	0
$46$	−33.6863	−0.732310
$47$	− 18.2073i	− 0.387389i	−0.981062	−	0.193695i	$-0.937953\pi$
$47$	− 18.2073i	− 0.387389i	0.981062	−	0.193695i	$-0.0620471\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	− 4.92110i	− 0.0984219i
$51$	0	0
$52$	27.0627	0.520437
$53$	43.3969i	0.818810i	0.912353	+	0.409405i	$0.134264\pi$
$53$	43.3969i	0.818810i	−0.912353	+	0.409405i	$0.865736\pi$
$54$	0	0
$55$	−98.0405	−1.78255
$56$	20.4617i	0.365388i
$57$	0	0
$58$	−41.8967	−0.722358
$59$	53.1190i	0.900321i	0.892948	+	0.450161i	$0.148633\pi$
$59$	53.1190i	0.900321i	−0.892948	+	0.450161i	$0.851367\pi$
$60$	0	0
$61$	−65.9555	−1.08124	−0.540619	−	0.841267i	$-0.681810\pi$
$61$	−65.9555	−1.08124	−0.540619	+	0.841267i	$0.681810\pi$
$62$	40.1720i	0.647935i
$63$	0	0
$64$	−31.8118	−0.497059
$65$	− 55.3004i	− 0.850775i
$66$	0	0
$67$	36.3542	0.542601	0.271300	−	0.962495i	$-0.412546\pi$
$67$	36.3542	0.542601	0.271300	+	0.962495i	$0.412546\pi$
$68$	− 11.2250i	− 0.165073i
$69$	0	0
$70$	16.6458	0.237796
$71$	0.219057i	0.00308531i	0.999999	+	0.00154265i	$0.000491042\pi$
$71$	0.219057i	0.00308531i	−0.999999	+	0.00154265i	$0.999509\pi$
$72$	0	0
$73$	−11.8523	−0.162360	−0.0811800	−	0.996699i	$-0.525869\pi$
$73$	−11.8523	−0.162360	−0.0811800	+	0.996699i	$0.525869\pi$
$74$	− 69.8705i	− 0.944195i
$75$	0	0
$76$	8.87451	0.116770
$77$	− 47.9788i	− 0.623101i
$78$	0	0
$79$	−144.812	−1.83306	−0.916530	−	0.399966i	$-0.869022\pi$
$79$	−144.812	−1.83306	−0.916530	+	0.399966i	$0.869022\pi$
$80$	− 8.55830i	− 0.106979i
$81$	0	0
$82$	83.7085	1.02084
$83$	99.0837i	1.19378i	0.802323	+	0.596890i	$0.203597\pi$
$83$	99.0837i	1.19378i	−0.802323	+	0.596890i	$0.796403\pi$
$84$	0	0
$85$	−22.9373	−0.269850
$86$	− 54.3298i	− 0.631742i
$87$	0	0
$88$	140.247	1.59372
$89$	− 92.7327i	− 1.04194i	−0.853575	−	0.520970i	$-0.825570\pi$
$89$	− 92.7327i	− 1.04194i	0.853575	−	0.520970i	$-0.174430\pi$
$90$	0	0
$91$	27.0627	0.297393
$92$	76.5866i	0.832463i
$93$	0	0
$94$	21.1882	0.225407
$95$	− 18.1343i	− 0.190887i
$96$	0	0
$97$	−96.5385	−0.995243	−0.497621	−	0.867394i	$-0.665793\pi$
$97$	−96.5385	−0.995243	−0.497621	+	0.867394i	$0.665793\pi$
$98$	8.14605i	0.0831230i
$99$	0	0
$100$	−11.1882	−0.111882
$101$	108.247i	1.07176i	0.844295	+	0.535878i	$0.180019\pi$
$101$	108.247i	1.07176i	−0.844295	+	0.535878i	$0.819981\pi$
$102$	0	0
$103$	154.933	1.50421	0.752103	−	0.659045i	$-0.229039\pi$
$103$	154.933	1.50421	0.752103	+	0.659045i	$0.229039\pi$
$104$	79.1072i	0.760646i
$105$	0	0
$106$	−50.5020	−0.476434
$107$	65.9671i	0.616515i	0.951303	+	0.308257i	$0.0997458\pi$
$107$	65.9671i	0.616515i	−0.951303	+	0.308257i	$0.900254\pi$
$108$	0	0
$109$	58.1438	0.533429	0.266715	−	0.963776i	$-0.414062\pi$
$109$	58.1438	0.533429	0.266715	+	0.963776i	$0.414062\pi$
$110$	− 114.092i	− 1.03720i
$111$	0	0
$112$	4.18824	0.0373950
$113$	− 57.5076i	− 0.508917i	−0.967084	−	0.254459i	$-0.918103\pi$
$113$	− 57.5076i	− 0.508917i	0.967084	−	0.254459i	$-0.0818973\pi$
$114$	0	0
$115$	156.498	1.36085
$116$	95.2533i	0.821149i
$117$	0	0
$118$	−61.8157	−0.523862
$119$	− 11.2250i	− 0.0943275i
$120$	0	0
$121$	−207.852	−1.71779
$122$	− 76.7539i	− 0.629130i
$123$	0	0
$124$	91.3320	0.736549
$125$	− 112.297i	− 0.898375i
$126$	0	0
$127$	147.269	1.15960	0.579800	−	0.814759i	$-0.303131\pi$
$127$	147.269	1.15960	0.579800	+	0.814759i	$0.303131\pi$
$128$	94.0896i	0.735075i
$129$	0	0
$130$	64.3542	0.495033
$131$	232.134i	1.77202i	0.463668	+	0.886009i	$0.346533\pi$
$131$	232.134i	1.77202i	−0.463668	+	0.886009i	$0.653467\pi$
$132$	0	0
$133$	8.87451	0.0667256
$134$	42.3062i	0.315718i
$135$	0	0
$136$	32.8118	0.241263
$137$	130.989i	0.956127i	0.878325	+	0.478064i	$0.158661\pi$
$137$	130.989i	0.956127i	−0.878325	+	0.478064i	$0.841339\pi$
$138$	0	0
$139$	−23.5425	−0.169370	−0.0846852	−	0.996408i	$-0.526988\pi$
$139$	−23.5425	−0.169370	−0.0846852	+	0.996408i	$0.526988\pi$
$140$	− 37.8445i	− 0.270318i
$141$	0	0
$142$	−0.254921	−0.00179522
$143$	− 185.491i	− 1.29714i
$144$	0	0
$145$	194.642	1.34236
$146$	− 13.7928i	− 0.0944709i
$147$	0	0
$148$	−158.852	−1.07333
$149$	97.7526i	0.656058i	0.944668	+	0.328029i	$0.106384\pi$
$149$	97.7526i	0.656058i	−0.944668	+	0.328029i	$0.893616\pi$
$150$	0	0
$151$	149.875	0.992546	0.496273	−	0.868166i	$-0.334701\pi$
$151$	149.875	0.992546	0.496273	+	0.868166i	$0.334701\pi$
$152$	25.9411i	0.170665i
$153$	0	0
$154$	55.8340	0.362558
$155$	− 186.629i	− 1.20406i
$156$	0	0
$157$	114.731	0.730769	0.365384	−	0.930857i	$-0.380938\pi$
$157$	114.731	0.730769	0.365384	+	0.930857i	$0.380938\pi$
$158$	− 168.521i	− 1.06659i
$159$	0	0
$160$	177.207	1.10754
$161$	76.5866i	0.475693i
$162$	0	0
$163$	129.099	0.792020	0.396010	−	0.918246i	$-0.370394\pi$
$163$	129.099	0.792020	0.396010	+	0.918246i	$0.370394\pi$
$164$	− 190.313i	− 1.16045i
$165$	0	0
$166$	−115.306	−0.694614
$167$	− 152.349i	− 0.912268i	−0.889911	−	0.456134i	$-0.849234\pi$
$167$	− 152.349i	− 0.912268i	0.889911	−	0.456134i	$-0.150766\pi$
$168$	0	0
$169$	−64.3725	−0.380903
$170$	− 26.6926i	− 0.157015i
$171$	0	0
$172$	−123.520	−0.718141
$173$	112.881i	0.652491i	0.945285	+	0.326246i	$0.105784\pi$
$173$	112.881i	0.652491i	−0.945285	+	0.326246i	$0.894216\pi$
$174$	0	0
$175$	−11.1882	−0.0639328
$176$	− 28.7067i	− 0.163106i
$177$	0	0
$178$	107.915	0.606264
$179$	− 130.405i	− 0.728521i	−0.931297	−	0.364261i	$-0.881322\pi$
$179$	− 130.405i	− 0.728521i	0.931297	−	0.364261i	$-0.118678\pi$
$180$	0	0
$181$	−328.207	−1.81330	−0.906648	−	0.421888i	$-0.861367\pi$
$181$	−328.207	−1.81330	−0.906648	+	0.421888i	$0.861367\pi$
$182$	31.4935i	0.173041i
$183$	0	0
$184$	−223.871	−1.21669
$185$	324.601i	1.75460i
$186$	0	0
$187$	−76.9373	−0.411429
$188$	− 48.1720i	− 0.256234i
$189$	0	0
$190$	21.1033	0.111070
$191$	− 215.769i	− 1.12968i	−0.825200	−	0.564841i	$-0.808938\pi$
$191$	− 215.769i	− 1.12968i	0.825200	−	0.564841i	$-0.191062\pi$
$192$	0	0
$193$	−59.3725	−0.307630	−0.153815	−	0.988100i	$-0.549156\pi$
$193$	−59.3725	−0.307630	−0.153815	+	0.988100i	$0.549156\pi$
$194$	− 112.344i	− 0.579093i
$195$	0	0
$196$	18.5203	0.0944911
$197$	147.110i	0.746749i	0.927681	+	0.373375i	$0.121799\pi$
$197$	147.110i	0.746749i	−0.927681	+	0.373375i	$0.878201\pi$
$198$	0	0
$199$	142.077	0.713955	0.356978	−	0.934113i	$-0.383807\pi$
$199$	142.077	0.713955	0.356978	+	0.934113i	$0.383807\pi$
$200$	− 32.7044i	− 0.163522i
$201$	0	0
$202$	−125.970	−0.623613
$203$	95.2533i	0.469228i
$204$	0	0
$205$	−388.889	−1.89702
$206$	180.299i	0.875239i
$207$	0	0
$208$	16.1922	0.0778470
$209$	− 60.8269i	− 0.291038i
$210$	0	0
$211$	16.9150	0.0801660	0.0400830	−	0.999196i	$-0.487238\pi$
$211$	16.9150	0.0801660	0.0400830	+	0.999196i	$0.487238\pi$
$212$	114.818i	0.541592i
$213$	0	0
$214$	−76.7673	−0.358726
$215$	252.403i	1.17397i
$216$	0	0
$217$	91.3320	0.420885
$218$	67.6632i	0.310382i
$219$	0	0
$220$	−259.391	−1.17905
$221$	− 43.3969i	− 0.196366i
$222$	0	0
$223$	350.158	1.57022	0.785108	−	0.619359i	$-0.212607\pi$
$223$	350.158	1.57022	0.785108	+	0.619359i	$0.212607\pi$
$224$	86.7209i	0.387147i
$225$	0	0
$226$	66.9229	0.296119
$227$	− 331.802i	− 1.46168i	−0.682547	−	0.730842i	$-0.739128\pi$
$227$	− 331.802i	− 1.46168i	0.682547	−	0.730842i	$-0.260872\pi$
$228$	0	0
$229$	−308.494	−1.34714	−0.673568	−	0.739125i	$-0.735239\pi$
$229$	−308.494	−1.34714	−0.673568	+	0.739125i	$0.735239\pi$
$230$	182.120i	0.791827i
$231$	0	0
$232$	−278.435	−1.20015
$233$	109.776i	0.471143i	0.971857	+	0.235571i	$0.0756961\pi$
$233$	109.776i	0.471143i	−0.971857	+	0.235571i	$0.924304\pi$
$234$	0	0
$235$	−98.4353	−0.418874
$236$	140.540i	0.595507i
$237$	0	0
$238$	13.0627	0.0548855
$239$	− 329.501i	− 1.37866i	−0.724446	−	0.689332i	$-0.757904\pi$
$239$	− 329.501i	− 1.37866i	0.724446	−	0.689332i	$-0.242096\pi$
$240$	0	0
$241$	−137.306	−0.569734	−0.284867	−	0.958567i	$-0.591949\pi$
$241$	−137.306	−0.569734	−0.284867	+	0.958567i	$0.591949\pi$
$242$	− 241.882i	− 0.999513i
$243$	0	0
$244$	−174.502	−0.715172
$245$	− 37.8445i	− 0.154468i
$246$	0	0
$247$	34.3098	0.138906
$248$	266.973i	1.07650i
$249$	0	0
$250$	130.682	0.522729
$251$	356.141i	1.41889i	0.704761	+	0.709445i	$0.251054\pi$
$251$	356.141i	1.41889i	−0.704761	+	0.709445i	$0.748946\pi$
$252$	0	0
$253$	524.933	2.07484
$254$	171.380i	0.674726i
$255$	0	0
$256$	−236.741	−0.924770
$257$	− 384.650i	− 1.49669i	−0.663307	−	0.748347i	$-0.730848\pi$
$257$	− 384.650i	− 1.49669i	0.663307	−	0.748347i	$-0.269152\pi$
$258$	0	0
$259$	−158.852	−0.613329
$260$	− 146.311i	− 0.562735i
$261$	0	0
$262$	−270.140	−1.03107
$263$	290.247i	1.10360i	0.833976	+	0.551801i	$0.186059\pi$
$263$	290.247i	1.10360i	−0.833976	+	0.551801i	$0.813941\pi$
$264$	0	0
$265$	234.620	0.885357
$266$	10.3275i	0.0388250i
$267$	0	0
$268$	96.1843	0.358897
$269$	− 532.928i	− 1.98115i	−0.136985	−	0.990573i	$-0.543741\pi$
$269$	− 532.928i	− 1.98115i	0.136985	−	0.990573i	$-0.456259\pi$
$270$	0	0
$271$	73.0405	0.269522	0.134761	−	0.990878i	$-0.456973\pi$
$271$	73.0405	0.269522	0.134761	+	0.990878i	$0.456973\pi$
$272$	− 6.71612i	− 0.0246916i
$273$	0	0
$274$	−152.435	−0.556333
$275$	76.6855i	0.278856i
$276$	0	0
$277$	−68.5791	−0.247578	−0.123789	−	0.992309i	$-0.539505\pi$
$277$	−68.5791	−0.247578	−0.123789	+	0.992309i	$0.539505\pi$
$278$	− 27.3969i	− 0.0985500i
$279$	0	0
$280$	110.624	0.395084
$281$	− 89.9199i	− 0.320000i	−0.987117	−	0.160000i	$-0.948851\pi$
$281$	− 89.9199i	− 0.320000i	0.987117	−	0.160000i	$-0.0511494\pi$
$282$	0	0
$283$	372.907	1.31769	0.658847	−	0.752277i	$-0.271045\pi$
$283$	372.907	1.31769	0.658847	+	0.752277i	$0.271045\pi$
$284$	0.579570i	0.00204074i
$285$	0	0
$286$	215.860	0.754756
$287$	− 190.313i	− 0.663113i
$288$	0	0
$289$	271.000	0.937716
$290$	226.509i	0.781065i
$291$	0	0
$292$	−31.3582	−0.107391
$293$	86.7680i	0.296137i	0.988977	+	0.148068i	$0.0473055\pi$
$293$	86.7680i	0.296137i	−0.988977	+	0.148068i	$0.952694\pi$
$294$	0	0
$295$	287.180	0.973493
$296$	− 464.342i	− 1.56872i
$297$	0	0
$298$	−113.757	−0.381735
$299$	296.092i	0.990274i
$300$	0	0
$301$	−123.520	−0.410366
$302$	174.412i	0.577524i
$303$	0	0
$304$	5.30979	0.0174664
$305$	356.580i	1.16911i
$306$	0	0
$307$	−448.395	−1.46057	−0.730285	−	0.683143i	$-0.760613\pi$
$307$	−448.395	−1.46057	−0.730285	+	0.683143i	$0.760613\pi$
$308$	− 126.940i	− 0.412143i
$309$	0	0
$310$	217.184	0.700595
$311$	− 259.333i	− 0.833870i	−0.908936	−	0.416935i	$-0.863104\pi$
$311$	− 259.333i	− 0.833870i	0.908936	−	0.416935i	$-0.136896\pi$
$312$	0	0
$313$	−311.439	−0.995013	−0.497507	−	0.867460i	$-0.665751\pi$
$313$	−311.439	−0.995013	−0.497507	+	0.867460i	$0.665751\pi$
$314$	133.515i	0.425206i
$315$	0	0
$316$	−383.136	−1.21246
$317$	236.351i	0.745587i	0.927914	+	0.372794i	$0.121600\pi$
$317$	236.351i	0.745587i	−0.927914	+	0.372794i	$0.878400\pi$
$318$	0	0
$319$	652.877	2.04664
$320$	171.986i	0.537456i
$321$	0	0
$322$	−89.1255	−0.276787
$323$	− 14.2309i	− 0.0440584i
$324$	0	0
$325$	−43.2549	−0.133092
$326$	150.236i	0.460846i
$327$	0	0
$328$	556.306	1.69605
$329$	− 48.1720i	− 0.146419i
$330$	0	0
$331$	163.247	0.493193	0.246597	−	0.969118i	$-0.420688\pi$
$331$	163.247	0.493193	0.246597	+	0.969118i	$0.420688\pi$
$332$	262.151i	0.789611i
$333$	0	0
$334$	177.292	0.530813
$335$	− 196.544i	− 0.586699i
$336$	0	0
$337$	41.5464	0.123283	0.0616416	−	0.998098i	$-0.480366\pi$
$337$	41.5464	0.123283	0.0616416	+	0.998098i	$0.480366\pi$
$338$	− 74.9117i	− 0.221632i
$339$	0	0
$340$	−60.6863	−0.178489
$341$	− 626.000i	− 1.83578i
$342$	0	0
$343$	18.5203	0.0539949
$344$	− 361.063i	− 1.04960i
$345$	0	0
$346$	−131.362	−0.379659
$347$	− 148.394i	− 0.427647i	−0.976872	−	0.213824i	$-0.931408\pi$
$347$	− 148.394i	− 0.427647i	0.976872	−	0.213824i	$-0.0685918\pi$
$348$	0	0
$349$	254.546	0.729359	0.364680	−	0.931133i	$-0.381178\pi$
$349$	254.546	0.729359	0.364680	+	0.931133i	$0.381178\pi$
$350$	− 13.0200i	− 0.0372000i
$351$	0	0
$352$	594.395	1.68862
$353$	− 154.071i	− 0.436461i	−0.975897	−	0.218230i	$-0.929972\pi$
$353$	− 154.071i	− 0.436461i	0.975897	−	0.218230i	$-0.0700284\pi$
$354$	0	0
$355$	1.18430	0.00333606
$356$	− 245.348i	− 0.689179i
$357$	0	0
$358$	151.755	0.423898
$359$	34.8858i	0.0971749i	0.998819	+	0.0485875i	$0.0154720\pi$
$359$	34.8858i	0.0971749i	−0.998819	+	0.0485875i	$0.984528\pi$
$360$	0	0
$361$	−349.749	−0.968834
$362$	− 381.941i	− 1.05509i
$363$	0	0
$364$	71.6013	0.196707
$365$	64.0777i	0.175555i
$366$	0	0
$367$	−187.413	−0.510662	−0.255331	−	0.966854i	$-0.582184\pi$
$367$	−187.413	−0.510662	−0.255331	+	0.966854i	$0.582184\pi$
$368$	45.8233i	0.124520i
$369$	0	0
$370$	−377.745	−1.02093
$371$	114.818i	0.309481i
$372$	0	0
$373$	706.047	1.89289	0.946444	−	0.322869i	$-0.104647\pi$
$373$	706.047	1.89289	0.946444	+	0.322869i	$0.104647\pi$
$374$	− 89.5336i	− 0.239395i
$375$	0	0
$376$	140.812	0.374499
$377$	368.259i	0.976815i
$378$	0	0
$379$	462.524	1.22038	0.610190	−	0.792255i	$-0.291093\pi$
$379$	462.524	1.22038	0.610190	+	0.792255i	$0.291093\pi$
$380$	− 47.9788i	− 0.126260i
$381$	0	0
$382$	251.095	0.657318
$383$	184.735i	0.482337i	0.970483	+	0.241169i	$0.0775307\pi$
$383$	184.735i	0.482337i	−0.970483	+	0.241169i	$0.922469\pi$
$384$	0	0
$385$	−259.391	−0.673742
$386$	− 69.0931i	− 0.178998i
$387$	0	0
$388$	−255.417	−0.658291
$389$	− 134.262i	− 0.345145i	−0.984997	−	0.172573i	$-0.944792\pi$
$389$	− 134.262i	− 0.345145i	0.984997	−	0.172573i	$-0.0552080\pi$
$390$	0	0
$391$	122.812	0.314097
$392$	54.1366i	0.138104i
$393$	0	0
$394$	−171.195	−0.434504
$395$	782.905i	1.98204i
$396$	0	0
$397$	−649.158	−1.63516	−0.817580	−	0.575816i	$-0.804685\pi$
$397$	−649.158	−1.63516	−0.817580	+	0.575816i	$0.804685\pi$
$398$	165.338i	0.415423i
$399$	0	0
$400$	−6.69414	−0.0167354
$401$	− 353.986i	− 0.882758i	−0.897321	−	0.441379i	$-0.854489\pi$
$401$	− 353.986i	− 0.882758i	0.897321	−	0.441379i	$-0.145511\pi$
$402$	0	0
$403$	353.099	0.876177
$404$	286.396i	0.708900i
$405$	0	0
$406$	−110.848	−0.273025
$407$	1088.79i	2.67516i
$408$	0	0
$409$	−4.55290	−0.0111318	−0.00556590	−	0.999985i	$-0.501772\pi$
$409$	−4.55290	−0.0111318	−0.00556590	+	0.999985i	$0.501772\pi$
$410$	− 452.558i	− 1.10380i
$411$	0	0
$412$	409.915	0.994939
$413$	140.540i	0.340289i
$414$	0	0
$415$	535.682	1.29080
$416$	335.272i	0.805943i
$417$	0	0
$418$	70.7856	0.169344
$419$	111.906i	0.267079i	0.991044	+	0.133539i	$0.0426342\pi$
$419$	111.906i	0.267079i	−0.991044	+	0.133539i	$0.957366\pi$
$420$	0	0
$421$	−376.608	−0.894555	−0.447278	−	0.894395i	$-0.647606\pi$
$421$	−376.608	−0.894555	−0.447278	+	0.894395i	$0.647606\pi$
$422$	19.6844i	0.0466455i
$423$	0	0
$424$	−335.624	−0.791565
$425$	17.9411i	0.0422143i
$426$	0	0
$427$	−174.502	−0.408670
$428$	174.532i	0.407786i
$429$	0	0
$430$	−293.727	−0.683086
$431$	13.5737i	0.0314935i	0.999876	+	0.0157468i	$0.00501255\pi$
$431$	13.5737i	0.0314935i	−0.999876	+	0.0157468i	$0.994987\pi$
$432$	0	0
$433$	−354.642	−0.819034	−0.409517	−	0.912302i	$-0.634303\pi$
$433$	−354.642	−0.819034	−0.409517	+	0.912302i	$0.634303\pi$
$434$	106.285i	0.244896i
$435$	0	0
$436$	153.834	0.352830
$437$	97.0955i	0.222186i
$438$	0	0
$439$	−313.048	−0.713094	−0.356547	−	0.934277i	$-0.616046\pi$
$439$	−313.048	−0.713094	−0.356547	+	0.934277i	$0.616046\pi$
$440$	− 758.226i	− 1.72324i
$441$	0	0
$442$	50.5020	0.114258
$443$	− 537.875i	− 1.21417i	−0.794639	−	0.607083i	$-0.792340\pi$
$443$	− 537.875i	− 1.21417i	0.794639	−	0.607083i	$-0.207660\pi$
$444$	0	0
$445$	−501.346	−1.12662
$446$	407.487i	0.913647i
$447$	0	0
$448$	−84.1660	−0.187871
$449$	60.3675i	0.134449i	0.997738	+	0.0672244i	$0.0214143\pi$
$449$	60.3675i	0.134449i	−0.997738	+	0.0672244i	$0.978586\pi$
$450$	0	0
$451$	−1304.43	−2.89231
$452$	− 152.151i	− 0.336617i
$453$	0	0
$454$	386.125	0.850497
$455$	− 146.311i	− 0.321563i
$456$	0	0
$457$	−463.450	−1.01411	−0.507056	−	0.861913i	$-0.669266\pi$
$457$	−463.450	−1.01411	−0.507056	+	0.861913i	$0.669266\pi$
$458$	− 359.001i	− 0.783846i
$459$	0	0
$460$	414.055	0.900119
$461$	− 646.097i	− 1.40151i	−0.713401	−	0.700756i	$-0.752846\pi$
$461$	− 646.097i	− 1.40151i	0.713401	−	0.700756i	$-0.247154\pi$
$462$	0	0
$463$	333.409	0.720106	0.360053	−	0.932932i	$-0.382759\pi$
$463$	333.409	0.720106	0.360053	+	0.932932i	$0.382759\pi$
$464$	56.9919i	0.122827i
$465$	0	0
$466$	−127.749	−0.274140
$467$	− 291.051i	− 0.623235i	−0.950208	−	0.311617i	$-0.899129\pi$
$467$	− 291.051i	− 0.623235i	0.950208	−	0.311617i	$-0.100871\pi$
$468$	0	0
$469$	96.1843	0.205084
$470$	− 114.551i	− 0.243726i
$471$	0	0
$472$	−410.812	−0.870364
$473$	846.622i	1.78990i
$474$	0	0
$475$	−14.1843	−0.0298617
$476$	− 29.6985i	− 0.0623918i
$477$	0	0
$478$	383.447	0.802191
$479$	8.53702i	0.0178226i	0.999960	+	0.00891129i	$0.00283659\pi$
$479$	8.53702i	0.0178226i	−0.999960	+	0.00891129i	$0.997163\pi$
$480$	0	0
$481$	−614.140	−1.27680
$482$	− 159.786i	− 0.331506i
$483$	0	0
$484$	−549.925	−1.13621
$485$	521.922i	1.07613i
$486$	0	0
$487$	−423.254	−0.869104	−0.434552	−	0.900647i	$-0.643093\pi$
$487$	−423.254	−0.869104	−0.434552	+	0.900647i	$0.643093\pi$
$488$	− 510.087i	− 1.04526i
$489$	0	0
$490$	44.0405	0.0898786
$491$	800.705i	1.63076i	0.578924	+	0.815381i	$0.303473\pi$
$491$	800.705i	1.63076i	−0.578924	+	0.815381i	$0.696527\pi$
$492$	0	0
$493$	152.745	0.309828
$494$	39.9271i	0.0808240i
$495$	0	0
$496$	54.6458	0.110173
$497$	0.579570i	0.00116614i
$498$	0	0
$499$	−347.003	−0.695396	−0.347698	−	0.937607i	$-0.613037\pi$
$499$	−347.003	−0.695396	−0.347698	+	0.937607i	$0.613037\pi$
$500$	− 297.110i	− 0.594219i
$501$	0	0
$502$	−414.450	−0.825597
$503$	− 979.329i	− 1.94698i	−0.228738	−	0.973488i	$-0.573460\pi$
$503$	− 979.329i	− 1.94698i	0.228738	−	0.973488i	$-0.426540\pi$
$504$	0	0
$505$	585.225	1.15886
$506$	610.876i	1.20727i
$507$	0	0
$508$	389.638	0.767004
$509$	− 29.8020i	− 0.0585500i	−0.999571	−	0.0292750i	$-0.990680\pi$
$509$	− 29.8020i	− 0.0585500i	0.999571	−	0.0292750i	$-0.00931985\pi$
$510$	0	0
$511$	−31.3582	−0.0613663
$512$	100.857i	0.196987i
$513$	0	0
$514$	447.626	0.870868
$515$	− 837.626i	− 1.62646i
$516$	0	0
$517$	−330.176	−0.638639
$518$	− 184.860i	− 0.356872i
$519$	0	0
$520$	427.682	0.822466
$521$	− 215.812i	− 0.414226i	−0.978317	−	0.207113i	$-0.933593\pi$
$521$	− 215.812i	− 0.414226i	0.978317	−	0.207113i	$-0.0664068\pi$
$522$	0	0
$523$	−278.069	−0.531681	−0.265841	−	0.964017i	$-0.585649\pi$
$523$	−278.069	−0.531681	−0.265841	+	0.964017i	$0.585649\pi$
$524$	614.170i	1.17208i
$525$	0	0
$526$	−337.767	−0.642143
$527$	− 146.457i	− 0.277907i
$528$	0	0
$529$	−308.929	−0.583987
$530$	273.032i	0.515155i
$531$	0	0
$532$	23.4797	0.0441349
$533$	− 735.772i	− 1.38044i
$534$	0	0
$535$	356.642	0.666620
$536$	281.157i	0.524546i
$537$	0	0
$538$	620.180	1.15275
$539$	− 126.940i	− 0.235510i
$540$	0	0
$541$	375.118	0.693378	0.346689	−	0.937980i	$-0.387306\pi$
$541$	375.118	0.693378	0.346689	+	0.937980i	$0.387306\pi$
$542$	84.9989i	0.156824i
$543$	0	0
$544$	139.063	0.255630
$545$	− 314.346i	− 0.576782i
$546$	0	0
$547$	276.664	0.505784	0.252892	−	0.967494i	$-0.418618\pi$
$547$	276.664	0.505784	0.252892	+	0.967494i	$0.418618\pi$
$548$	346.565i	0.632419i
$549$	0	0
$550$	−89.2406	−0.162256
$551$	120.761i	0.219167i
$552$	0	0
$553$	−383.136	−0.692832
$554$	− 79.8070i	− 0.144056i
$555$	0	0
$556$	−62.2876	−0.112028
$557$	83.2981i	0.149548i	0.997201	+	0.0747739i	$0.0238235\pi$
$557$	83.2981i	0.149548i	−0.997201	+	0.0747739i	$0.976176\pi$
$558$	0	0
$559$	−477.542	−0.854280
$560$	− 22.6431i	− 0.0404342i
$561$	0	0
$562$	104.642	0.186195
$563$	− 723.319i	− 1.28476i	−0.766387	−	0.642379i	$-0.777947\pi$
$563$	− 723.319i	− 1.28476i	0.766387	−	0.642379i	$-0.222053\pi$
$564$	0	0
$565$	−310.907	−0.550278
$566$	433.960i	0.766714i
$567$	0	0
$568$	−1.69414	−0.00298265
$569$	− 57.0695i	− 0.100298i	−0.998742	−	0.0501490i	$-0.984030\pi$
$569$	− 57.0695i	− 0.100298i	0.998742	−	0.0501490i	$-0.0159696\pi$
$570$	0	0
$571$	−619.231	−1.08447	−0.542234	−	0.840227i	$-0.682421\pi$
$571$	−619.231	−1.08447	−0.542234	+	0.840227i	$0.682421\pi$
$572$	− 490.764i	− 0.857978i
$573$	0	0
$574$	221.472	0.385839
$575$	− 122.410i	− 0.212887i
$576$	0	0
$577$	382.029	0.662095	0.331047	−	0.943614i	$-0.392598\pi$
$577$	382.029	0.662095	0.331047	+	0.943614i	$0.392598\pi$
$578$	315.369i	0.545620i
$579$	0	0
$580$	514.974	0.887886
$581$	262.151i	0.451206i
$582$	0	0
$583$	786.972	1.34987
$584$	− 91.6632i	− 0.156958i
$585$	0	0
$586$	−100.974	−0.172310
$587$	633.494i	1.07921i	0.841920	+	0.539603i	$0.181426\pi$
$587$	633.494i	1.07921i	−0.841920	+	0.539603i	$0.818574\pi$
$588$	0	0
$589$	115.790	0.196587
$590$	334.198i	0.566437i
$591$	0	0
$592$	−95.0445	−0.160548
$593$	841.310i	1.41874i	0.704839	+	0.709368i	$0.251019\pi$
$593$	841.310i	1.41874i	−0.704839	+	0.709368i	$0.748981\pi$
$594$	0	0
$595$	−60.6863	−0.101994
$596$	258.629i	0.433942i
$597$	0	0
$598$	−344.569	−0.576202
$599$	854.398i	1.42637i	0.700973	+	0.713187i	$0.252749\pi$
$599$	854.398i	1.42637i	−0.700973	+	0.713187i	$0.747251\pi$
$600$	0	0
$601$	585.512	0.974230	0.487115	−	0.873338i	$-0.338049\pi$
$601$	585.512	0.974230	0.487115	+	0.873338i	$0.338049\pi$
$602$	− 143.743i	− 0.238776i
$603$	0	0
$604$	396.531	0.656508
$605$	1123.72i	1.85740i
$606$	0	0
$607$	−200.494	−0.330303	−0.165152	−	0.986268i	$-0.552811\pi$
$607$	−200.494	−0.330303	−0.165152	+	0.986268i	$0.552811\pi$
$608$	109.944i	0.180828i
$609$	0	0
$610$	−414.959	−0.680261
$611$	− 186.238i	− 0.304809i
$612$	0	0
$613$	139.848	0.228138	0.114069	−	0.993473i	$-0.463612\pi$
$613$	139.848	0.228138	0.114069	+	0.993473i	$0.463612\pi$
$614$	− 521.807i	− 0.849848i
$615$	0	0
$616$	371.059	0.602368
$617$	− 606.165i	− 0.982440i	−0.871036	−	0.491220i	$-0.836551\pi$
$617$	− 606.165i	− 0.982440i	0.871036	−	0.491220i	$-0.163449\pi$
$618$	0	0
$619$	130.660	0.211083	0.105541	−	0.994415i	$-0.466342\pi$
$619$	130.660	0.211083	0.105541	+	0.994415i	$0.466342\pi$
$620$	− 493.774i	− 0.796410i
$621$	0	0
$622$	301.792	0.485196
$623$	− 245.348i	− 0.393816i
$624$	0	0
$625$	−712.837	−1.14054
$626$	− 362.429i	− 0.578960i
$627$	0	0
$628$	303.549	0.483358
$629$	254.730i	0.404977i
$630$	0	0
$631$	277.557	0.439868	0.219934	−	0.975515i	$-0.429416\pi$
$631$	277.557	0.439868	0.219934	+	0.975515i	$0.429416\pi$
$632$	− 1119.95i	− 1.77207i
$633$	0	0
$634$	−275.047	−0.433828
$635$	− 796.191i	− 1.25384i
$636$	0	0
$637$	71.6013	0.112404
$638$	759.767i	1.19086i
$639$	0	0
$640$	508.682	0.794816
$641$	− 1005.30i	− 1.56833i	−0.620551	−	0.784166i	$-0.713091\pi$
$641$	− 1005.30i	− 1.56833i	0.620551	−	0.784166i	$-0.286909\pi$
$642$	0	0
$643$	−219.569	−0.341475	−0.170738	−	0.985317i	$-0.554615\pi$
$643$	−219.569	−0.341475	−0.170738	+	0.985317i	$0.554615\pi$
$644$	202.629i	0.314641i
$645$	0	0
$646$	16.5608	0.0256359
$647$	657.567i	1.01633i	0.861259	+	0.508166i	$0.169676\pi$
$647$	657.567i	1.01633i	−0.861259	+	0.508166i	$0.830324\pi$
$648$	0	0
$649$	963.274	1.48424
$650$	− 50.3367i	− 0.0774411i
$651$	0	0
$652$	341.565	0.523872
$653$	− 79.6396i	− 0.121960i	−0.998139	−	0.0609798i	$-0.980577\pi$
$653$	− 79.6396i	− 0.121960i	0.998139	−	0.0609798i	$-0.0194225\pi$
$654$	0	0
$655$	1255.00	1.91603
$656$	− 113.868i	− 0.173580i
$657$	0	0
$658$	56.0588	0.0851958
$659$	368.547i	0.559252i	0.960109	+	0.279626i	$0.0902104\pi$
$659$	368.547i	0.559252i	−0.960109	+	0.279626i	$0.909790\pi$
$660$	0	0
$661$	345.233	0.522288	0.261144	−	0.965300i	$-0.415900\pi$
$661$	345.233	0.522288	0.261144	+	0.965300i	$0.415900\pi$
$662$	189.974i	0.286970i
$663$	0	0
$664$	−766.294	−1.15406
$665$	− 47.9788i	− 0.0721486i
$666$	0	0
$667$	−1042.16	−1.56246
$668$	− 403.077i	− 0.603408i
$669$	0	0
$670$	228.723	0.341377
$671$	1196.06i	1.78250i
$672$	0	0
$673$	1130.90	1.68039	0.840193	−	0.542288i	$-0.182442\pi$
$673$	1130.90	1.68039	0.840193	+	0.542288i	$0.182442\pi$
$674$	48.3485i	0.0717337i
$675$	0	0
$676$	−170.314	−0.251943
$677$	563.065i	0.831706i	0.909432	+	0.415853i	$0.136517\pi$
$677$	563.065i	0.831706i	−0.909432	+	0.415853i	$0.863483\pi$
$678$	0	0
$679$	−255.417	−0.376166
$680$	− 177.392i	− 0.260871i
$681$	0	0
$682$	728.490	1.06817
$683$	− 765.225i	− 1.12039i	−0.828361	−	0.560194i	$-0.810726\pi$
$683$	− 765.225i	− 1.12039i	0.828361	−	0.560194i	$-0.189274\pi$
$684$	0	0
$685$	708.176	1.03383
$686$	21.5524i	0.0314175i
$687$	0	0
$688$	−73.9046	−0.107419
$689$	443.897i	0.644262i
$690$	0	0
$691$	−63.8666	−0.0924264	−0.0462132	−	0.998932i	$-0.514715\pi$
$691$	−63.8666	−0.0924264	−0.0462132	+	0.998932i	$0.514715\pi$
$692$	298.655i	0.431583i
$693$	0	0
$694$	172.689	0.248831
$695$	127.279i	0.183136i
$696$	0	0
$697$	−305.180	−0.437848
$698$	296.221i	0.424386i
$699$	0	0
$700$	−29.6013	−0.0422876
$701$	− 1125.08i	− 1.60497i	−0.596674	−	0.802483i	$-0.703512\pi$
$701$	− 1125.08i	− 1.60497i	0.596674	−	0.802483i	$-0.296488\pi$
$702$	0	0
$703$	−201.391	−0.286473
$704$	576.884i	0.819437i
$705$	0	0
$706$	179.295	0.253960
$707$	286.396i	0.405086i
$708$	0	0
$709$	−383.354	−0.540697	−0.270349	−	0.962763i	$-0.587139\pi$
$709$	−383.354	−0.540697	−0.270349	+	0.962763i	$0.587139\pi$
$710$	1.37820i	0.00194112i
$711$	0	0
$712$	717.176	1.00727
$713$	999.258i	1.40148i
$714$	0	0
$715$	−1002.83	−1.40256
$716$	− 345.020i	− 0.481871i
$717$	0	0
$718$	−40.5974	−0.0565423
$719$	− 55.6867i	− 0.0774503i	−0.999250	−	0.0387251i	$-0.987670\pi$
$719$	− 55.6867i	− 0.0774503i	0.999250	−	0.0387251i	$-0.0123297\pi$
$720$	0	0
$721$	409.915	0.568537
$722$	− 407.011i	− 0.563727i
$723$	0	0
$724$	−868.353	−1.19938
$725$	− 152.245i	− 0.209993i
$726$	0	0
$727$	54.2274	0.0745906	0.0372953	−	0.999304i	$-0.488126\pi$
$727$	54.2274	0.0745906	0.0372953	+	0.999304i	$0.488126\pi$
$728$	209.298i	0.287497i
$729$	0	0
$730$	−74.5687	−0.102149
$731$	198.073i	0.270962i
$732$	0	0
$733$	117.269	0.159985	0.0799927	−	0.996795i	$-0.474510\pi$
$733$	117.269	0.159985	0.0799927	+	0.996795i	$0.474510\pi$
$734$	− 218.097i	− 0.297134i
$735$	0	0
$736$	−948.808	−1.28914
$737$	− 659.258i	− 0.894516i
$738$	0	0
$739$	803.705	1.08756	0.543778	−	0.839229i	$-0.316993\pi$
$739$	803.705	1.08756	0.543778	+	0.839229i	$0.316993\pi$
$740$	858.813i	1.16056i
$741$	0	0
$742$	−133.616	−0.180075
$743$	− 1085.61i	− 1.46112i	−0.682847	−	0.730561i	$-0.739259\pi$
$743$	− 1085.61i	− 1.46112i	0.682847	−	0.730561i	$-0.260741\pi$
$744$	0	0
$745$	528.486	0.709377
$746$	821.642i	1.10140i
$747$	0	0
$748$	−203.557	−0.272135
$749$	174.532i	0.233021i
$750$	0	0
$751$	−13.0183	−0.0173346	−0.00866730	−	0.999962i	$-0.502759\pi$
$751$	−13.0183	−0.0173346	−0.00866730	+	0.999962i	$0.502759\pi$
$752$	− 28.8223i	− 0.0383275i
$753$	0	0
$754$	−428.552	−0.568371
$755$	− 810.276i	− 1.07321i
$756$	0	0
$757$	252.761	0.333898	0.166949	−	0.985966i	$-0.446608\pi$
$757$	252.761	0.333898	0.166949	+	0.985966i	$0.446608\pi$
$758$	538.250i	0.710092i
$759$	0	0
$760$	140.247	0.184536
$761$	− 620.912i	− 0.815916i	−0.913001	−	0.407958i	$-0.866241\pi$
$761$	− 620.912i	− 0.815916i	0.913001	−	0.407958i	$-0.133759\pi$
$762$	0	0
$763$	153.834	0.201617
$764$	− 570.872i	− 0.747214i
$765$	0	0
$766$	−214.980	−0.280653
$767$	543.341i	0.708398i
$768$	0	0
$769$	−1075.30	−1.39831	−0.699153	−	0.714972i	$-0.746440\pi$
$769$	−1075.30	−1.39831	−0.699153	+	0.714972i	$0.746440\pi$
$770$	− 301.859i	− 0.392024i
$771$	0	0
$772$	−157.085	−0.203478
$773$	− 116.144i	− 0.150251i	−0.997174	−	0.0751255i	$-0.976064\pi$
$773$	− 116.144i	− 0.150251i	0.997174	−	0.0751255i	$-0.0239357\pi$
$774$	0	0
$775$	−145.978	−0.188358
$776$	− 746.610i	− 0.962127i
$777$	0	0
$778$	156.243	0.200827
$779$	− 241.277i	− 0.309726i
$780$	0	0
$781$	3.97244	0.00508635
$782$	142.919i	0.182761i
$783$	0	0
$784$	11.0810	0.0141340
$785$	− 620.276i	− 0.790160i
$786$	0	0
$787$	−70.9359	−0.0901345	−0.0450673	−	0.998984i	$-0.514350\pi$
$787$	−70.9359	−0.0901345	−0.0450673	+	0.998984i	$0.514350\pi$
$788$	389.216i	0.493928i
$789$	0	0
$790$	−911.084	−1.15327
$791$	− 152.151i	− 0.192353i
$792$	0	0
$793$	−674.643	−0.850748
$794$	− 755.440i	− 0.951435i
$795$	0	0
$796$	375.901	0.472237
$797$	− 667.791i	− 0.837880i	−0.908014	−	0.418940i	$-0.862402\pi$
$797$	− 667.791i	− 0.837880i	0.908014	−	0.418940i	$-0.137598\pi$
$798$	0	0
$799$	−77.2470	−0.0966797
$800$	− 138.608i	− 0.173260i
$801$	0	0
$802$	411.941	0.513642
$803$	214.933i	0.267662i
$804$	0	0
$805$	414.055	0.514354
$806$	410.909i	0.509813i
$807$	0	0
$808$	−837.165	−1.03609
$809$	− 625.232i	− 0.772846i	−0.922322	−	0.386423i	$-0.873711\pi$
$809$	− 625.232i	− 0.772846i	0.922322	−	0.386423i	$-0.126289\pi$
$810$	0	0
$811$	−876.467	−1.08072	−0.540362	−	0.841433i	$-0.681713\pi$
$811$	−876.467	−1.08072	−0.540362	+	0.841433i	$0.681713\pi$
$812$	252.017i	0.310365i
$813$	0	0
$814$	−1267.05	−1.55657
$815$	− 697.958i	− 0.856390i
$816$	0	0
$817$	−156.597	−0.191674
$818$	− 5.29831i	− 0.00647716i
$819$	0	0
$820$	−1028.90	−1.25476
$821$	1252.79i	1.52593i	0.646439	+	0.762966i	$0.276258\pi$
$821$	1252.79i	1.52593i	−0.646439	+	0.762966i	$0.723742\pi$
$822$	0	0
$823$	177.956	0.216228	0.108114	−	0.994139i	$-0.465519\pi$
$823$	177.956	0.216228	0.108114	+	0.994139i	$0.465519\pi$
$824$	1198.22i	1.45416i
$825$	0	0
$826$	−163.549	−0.198001
$827$	− 10.9892i	− 0.0132880i	−0.999978	−	0.00664402i	$-0.997885\pi$
$827$	− 10.9892i	− 0.0132880i	0.999978	−	0.00664402i	$-0.00211487\pi$
$828$	0	0
$829$	930.294	1.12219	0.561094	−	0.827752i	$-0.310381\pi$
$829$	930.294	1.12219	0.561094	+	0.827752i	$0.310381\pi$
$830$	623.385i	0.751067i
$831$	0	0
$832$	−325.395	−0.391099
$833$	− 29.6985i	− 0.0356524i
$834$	0	0
$835$	−823.652	−0.986410
$836$	− 160.933i	− 0.192503i
$837$	0	0
$838$	−130.227	−0.155403
$839$	− 908.054i	− 1.08231i	−0.840924	−	0.541153i	$-0.817988\pi$
$839$	− 908.054i	− 1.08231i	0.840924	−	0.541153i	$-0.182012\pi$
$840$	0	0
$841$	−455.170	−0.541225
$842$	− 438.267i	− 0.520507i
$843$	0	0
$844$	44.7530	0.0530248
$845$	348.021i	0.411860i
$846$	0	0
$847$	−549.925	−0.649263
$848$	68.6976i	0.0810113i
$849$	0	0
$850$	−20.8784	−0.0245629
$851$	− 1737.99i	− 2.04230i
$852$	0	0
$853$	−172.376	−0.202083	−0.101041	−	0.994882i	$-0.532217\pi$
$853$	−172.376	−0.202083	−0.101041	+	0.994882i	$0.532217\pi$
$854$	− 203.072i	− 0.237789i
$855$	0	0
$856$	−510.176	−0.596000
$857$	− 1250.42i	− 1.45907i	−0.683943	−	0.729536i	$-0.739736\pi$
$857$	− 1250.42i	− 1.45907i	0.683943	−	0.729536i	$-0.260264\pi$
$858$	0	0
$859$	692.674	0.806373	0.403187	−	0.915118i	$-0.367903\pi$
$859$	692.674	0.806373	0.403187	+	0.915118i	$0.367903\pi$
$860$	667.795i	0.776506i
$861$	0	0
$862$	−15.7960	−0.0183248
$863$	1347.01i	1.56085i	0.625250	+	0.780425i	$0.284997\pi$
$863$	1347.01i	1.56085i	−0.625250	+	0.780425i	$0.715003\pi$
$864$	0	0
$865$	610.276	0.705521
$866$	− 412.704i	− 0.476564i
$867$	0	0
$868$	241.642	0.278389
$869$	2626.06i	3.02193i
$870$	0	0
$871$	371.859	0.426933
$872$	449.673i	0.515680i
$873$	0	0
$874$	−112.992	−0.129282
$875$	− 297.110i	− 0.339554i
$876$	0	0
$877$	234.154	0.266995	0.133497	−	0.991049i	$-0.457379\pi$
$877$	234.154	0.266995	0.133497	+	0.991049i	$0.457379\pi$
$878$	− 364.301i	− 0.414922i
$879$	0	0
$880$	−155.199	−0.176362
$881$	1673.19i	1.89920i	0.313465	+	0.949600i	$0.398510\pi$
$881$	1673.19i	1.89920i	−0.313465	+	0.949600i	$0.601490\pi$
$882$	0	0
$883$	−1415.59	−1.60316	−0.801581	−	0.597886i	$-0.796007\pi$
$883$	−1415.59	−1.60316	−0.801581	+	0.597886i	$0.796007\pi$
$884$	− 114.818i	− 0.129884i
$885$	0	0
$886$	625.937	0.706475
$887$	753.223i	0.849181i	0.905386	+	0.424590i	$0.139582\pi$
$887$	753.223i	0.849181i	−0.905386	+	0.424590i	$0.860418\pi$
$888$	0	0
$889$	389.638	0.438288
$890$	− 583.428i	− 0.655537i
$891$	0	0
$892$	926.431	1.03860
$893$	− 61.0718i	− 0.0683895i
$894$	0	0
$895$	−705.018	−0.787730
$896$	248.938i	0.277832i
$897$	0	0
$898$	−70.2510	−0.0782305
$899$	1242.81i	1.38244i
$900$	0	0
$901$	184.118	0.204348
$902$	− 1517.99i	− 1.68292i
$903$	0	0
$904$	444.753	0.491983
$905$	1774.40i	1.96067i
$906$	0	0
$907$	575.388	0.634386	0.317193	−	0.948361i	$-0.397260\pi$
$907$	575.388	0.634386	0.317193	+	0.948361i	$0.397260\pi$
$908$	− 877.866i	− 0.966813i
$909$	0	0
$910$	170.265	0.187105
$911$	− 110.231i	− 0.121000i	−0.998168	−	0.0605000i	$-0.980730\pi$
$911$	− 110.231i	− 0.121000i	0.998168	−	0.0605000i	$-0.0192695\pi$
$912$	0	0
$913$	1796.81	1.96803
$914$	− 539.327i	− 0.590073i
$915$	0	0
$916$	−816.199	−0.891047
$917$	614.170i	0.669760i
$918$	0	0
$919$	−1304.49	−1.41947	−0.709736	−	0.704468i	$-0.751186\pi$
$919$	−1304.49	−1.41947	−0.709736	+	0.704468i	$0.751186\pi$
$920$	1210.33i	1.31557i
$921$	0	0
$922$	751.877	0.815485
$923$	2.24068i	0.00242761i
$924$	0	0
$925$	253.897	0.274483
$926$	387.995i	0.419002i
$927$	0	0
$928$	−1180.06	−1.27162
$929$	− 451.154i	− 0.485634i	−0.970072	−	0.242817i	$-0.921928\pi$
$929$	− 451.154i	− 0.485634i	0.970072	−	0.242817i	$-0.0780715\pi$
$930$	0	0
$931$	23.4797	0.0252199
$932$	290.441i	0.311632i
$933$	0	0
$934$	338.702	0.362636
$935$	415.951i	0.444867i
$936$	0	0
$937$	1173.69	1.25260	0.626301	−	0.779581i	$-0.284568\pi$
$937$	1173.69	1.25260	0.626301	+	0.779581i	$0.284568\pi$
$938$	111.932i	0.119330i
$939$	0	0
$940$	−260.435	−0.277059
$941$	− 484.988i	− 0.515396i	−0.966226	−	0.257698i	$-0.917036\pi$
$941$	− 484.988i	− 0.515396i	0.966226	−	0.257698i	$-0.0829639\pi$
$942$	0	0
$943$	2082.21	2.20807
$944$	84.0876i	0.0890758i
$945$	0	0
$946$	−985.233	−1.04147
$947$	470.938i	0.497294i	0.968594	+	0.248647i	$0.0799860\pi$
$947$	470.938i	0.497294i	−0.968594	+	0.248647i	$0.920014\pi$
$948$	0	0
$949$	−121.234	−0.127749
$950$	− 16.5066i	− 0.0173753i
$951$	0	0
$952$	86.8118	0.0911888
$953$	− 100.784i	− 0.105755i	−0.998601	−	0.0528774i	$-0.983161\pi$
$953$	− 100.784i	− 0.105755i	0.998601	−	0.0528774i	$-0.0168393\pi$
$954$	0	0
$955$	−1166.53	−1.22149
$956$	− 871.777i	− 0.911900i
$957$	0	0
$958$	−9.93471	−0.0103703
$959$	346.565i	0.361382i
$960$	0	0
$961$	230.648	0.240009
$962$	− 714.688i	− 0.742919i
$963$	0	0
$964$	−363.277	−0.376844
$965$	320.989i	0.332632i
$966$	0	0
$967$	472.891	0.489029	0.244515	−	0.969646i	$-0.421371\pi$
$967$	472.891	0.489029	0.244515	+	0.969646i	$0.421371\pi$
$968$	− 1607.49i	− 1.66063i
$969$	0	0
$970$	−607.373	−0.626157
$971$	− 1162.81i	− 1.19754i	−0.800921	−	0.598770i	$-0.795656\pi$
$971$	− 1162.81i	− 1.19754i	0.800921	−	0.598770i	$-0.204344\pi$
$972$	0	0
$973$	−62.2876	−0.0640160
$974$	− 492.549i	− 0.505698i
$975$	0	0
$976$	−104.408	−0.106975
$977$	− 1162.45i	− 1.18982i	−0.803794	−	0.594908i	$-0.797188\pi$
$977$	− 1162.45i	− 1.18982i	0.803794	−	0.594908i	$-0.202812\pi$
$978$	0	0
$979$	−1681.64	−1.71771
$980$	− 100.127i	− 0.102171i
$981$	0	0
$982$	−931.797	−0.948877
$983$	− 491.159i	− 0.499653i	−0.968291	−	0.249827i	$-0.919626\pi$
$983$	− 491.159i	− 0.499653i	0.968291	−	0.249827i	$-0.0803736\pi$
$984$	0	0
$985$	795.328	0.807440
$986$	177.753i	0.180277i
$987$	0	0
$988$	90.7752	0.0918777
$989$	− 1351.43i	− 1.36646i
$990$	0	0
$991$	1496.93	1.51052	0.755262	−	0.655423i	$-0.227510\pi$
$991$	1496.93	1.51052	0.755262	+	0.655423i	$0.227510\pi$
$992$	1131.48i	1.14061i
$993$	0	0
$994$	−0.674458	−0.000678530 0
$995$	− 768.120i	− 0.771980i
$996$	0	0
$997$	−283.814	−0.284668	−0.142334	−	0.989819i	$-0.545461\pi$
$997$	−283.814	−0.284668	−0.142334	+	0.989819i	$0.545461\pi$
$998$	− 403.814i	− 0.404624i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.3.b.b.134.3	yes	4
3.2	odd	2	inner	189.3.b.b.134.2	✓	4
4.3	odd	2		3024.3.d.c.1457.1		4
9.2	odd	6		567.3.r.b.134.2		8
9.4	even	3		567.3.r.b.512.2		8
9.5	odd	6		567.3.r.b.512.3		8
9.7	even	3		567.3.r.b.134.3		8
12.11	even	2		3024.3.d.c.1457.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.3.b.b.134.2	✓	4	3.2	odd	2	inner
189.3.b.b.134.3	yes	4	1.1	even	1	trivial
567.3.r.b.134.2		8	9.2	odd	6
567.3.r.b.134.3		8	9.7	even	3
567.3.r.b.512.2		8	9.4	even	3
567.3.r.b.512.3		8	9.5	odd	6
3024.3.d.c.1457.1		4	4.3	odd	2
3024.3.d.c.1457.4		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 189 = 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	189.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.16372i q^{2} +2.64575 q^{4} -5.40636i q^{5} +2.64575 q^{7} +7.73381i q^{8} +6.29150 q^{10} -18.1343i q^{11} +10.2288 q^{13} +3.07892i q^{14} +1.58301 q^{16} -4.24264i q^{17} +3.35425 q^{19} -14.3039i q^{20} +21.1033 q^{22} +28.9470i q^{23} -4.22876 q^{25} +11.9034i q^{26} +7.00000 q^{28} +36.0024i q^{29} +34.5203 q^{31} +32.7774i q^{32} +4.93725 q^{34} -14.3039i q^{35} -60.0405 q^{37} +3.90341i q^{38} +41.8118 q^{40} -71.9317i q^{41} -46.6863 q^{43} -47.9788i q^{44} -33.6863 q^{46} -18.2073i q^{47} +7.00000 q^{49} -4.92110i q^{50} +27.0627 q^{52} +43.3969i q^{53} -98.0405 q^{55} +20.4617i q^{56} -41.8967 q^{58} +53.1190i q^{59} -65.9555 q^{61} +40.1720i q^{62} -31.8118 q^{64} -55.3004i q^{65} +36.3542 q^{67} -11.2250i q^{68} +16.6458 q^{70} +0.219057i q^{71} -11.8523 q^{73} -69.8705i q^{74} +8.87451 q^{76} -47.9788i q^{77} -144.812 q^{79} -8.55830i q^{80} +83.7085 q^{82} +99.0837i q^{83} -22.9373 q^{85} -54.3298i q^{86} +140.247 q^{88} -92.7327i q^{89} +27.0627 q^{91} +76.5866i q^{92} +21.1882 q^{94} -18.1343i q^{95} -96.5385 q^{97} +8.14605i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{10} - 12 q^{13} - 36 q^{16} + 24 q^{19} - 32 q^{22} + 36 q^{25} + 28 q^{28} + 64 q^{31} - 12 q^{34} - 92 q^{37} + 72 q^{40} - 28 q^{43} + 24 q^{46} + 28 q^{49} + 140 q^{52} - 244 q^{55} - 284 q^{58}+ \cdots + 16 q^{97}+O(q^{100}) \) 4 * q + 4 * q^10 - 12 * q^13 - 36 * q^16 + 24 * q^19 - 32 * q^22 + 36 * q^25 + 28 * q^28 + 64 * q^31 - 12 * q^34 - 92 * q^37 + 72 * q^40 - 28 * q^43 + 24 * q^46 + 28 * q^49 + 140 * q^52 - 244 * q^55 - 284 * q^58 + 96 * q^61 - 32 * q^64 + 156 * q^67 + 56 * q^70 + 196 * q^73 - 28 * q^76 - 484 * q^79 + 356 * q^82 - 60 * q^85 + 180 * q^88 + 140 * q^91 + 180 * q^94 + 16 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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