LMFDB - Embedded newform 189.3.b.b.134.4

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			134.4
Root			$2.57794i$ of defining polynomial
Character	$\chi$	$=$	189.134
Dual form			189.3.b.b.134.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.57794i q^{2} -2.64575 q^{4} +1.66471i q^{5} -2.64575 q^{7} +3.49117i q^{8} -4.29150 q^{10} +14.3926i q^{11} -16.2288 q^{13} -6.82058i q^{14} -19.5830 q^{16} +4.24264i q^{17} +8.64575 q^{19} -4.40440i q^{20} -37.1033 q^{22} -17.7220i q^{23} +22.2288 q^{25} -41.8367i q^{26} +7.00000 q^{28} +38.8308i q^{29} -2.52026 q^{31} -36.5191i q^{32} -10.9373 q^{34} -4.40440i q^{35} +14.0405 q^{37} +22.2882i q^{38} -5.81176 q^{40} -36.5764i q^{41} +32.6863 q^{43} -38.0793i q^{44} +45.6863 q^{46} -26.6926i q^{47} +7.00000 q^{49} +57.3043i q^{50} +42.9373 q^{52} +68.8528i q^{53} -23.9595 q^{55} -9.23676i q^{56} -100.103 q^{58} +104.031i q^{59} +113.956 q^{61} -6.49707i q^{62} +15.8118 q^{64} -27.0161i q^{65} +41.6458 q^{67} -11.2250i q^{68} +11.3542 q^{70} +123.256i q^{71} +109.852 q^{73} +36.1956i q^{74} -22.8745 q^{76} -38.0793i q^{77} -97.1882 q^{79} -32.5999i q^{80} +94.2915 q^{82} -121.534i q^{83} -7.06275 q^{85} +84.2631i q^{86} -50.2470 q^{88} -0.808777i q^{89} +42.9373 q^{91} +46.8881i q^{92} +68.8118 q^{94} +14.3926i q^{95} +104.539 q^{97} +18.0455i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.57794i	1.28897i	0.764618	+	0.644484i	$0.222928\pi$
$2$	2.57794i	1.28897i	−0.764618	+	0.644484i	$0.777072\pi$
$3$	0	0
$3$	0	0
$4$	−2.64575	−0.661438
$5$	1.66471i	0.332941i	0.986046	+	0.166471i	$0.0532371\pi$
$5$	1.66471i	0.332941i	−0.986046	+	0.166471i	$0.946763\pi$
$6$	0	0
$7$	−2.64575	−0.377964
$7$	−2.64575	−0.377964
$8$	3.49117i	0.436396i
$9$	0	0
$10$	−4.29150	−0.429150
$11$	14.3926i	1.30842i	0.756313	+	0.654210i	$0.226999\pi$
$11$	14.3926i	1.30842i	−0.756313	+	0.654210i	$0.773001\pi$
$12$	0	0
$13$	−16.2288	−1.24837	−0.624183	−	0.781278i	$-0.714568\pi$
$13$	−16.2288	−1.24837	−0.624183	+	0.781278i	$0.714568\pi$
$14$	− 6.82058i	− 0.487184i
$15$	0	0
$16$	−19.5830	−1.22394
$17$	4.24264i	0.249567i	0.992184	+	0.124784i	$0.0398236\pi$
$17$	4.24264i	0.249567i	−0.992184	+	0.124784i	$0.960176\pi$
$18$	0	0
$19$	8.64575	0.455040	0.227520	−	0.973773i	$-0.426938\pi$
$19$	8.64575	0.455040	0.227520	+	0.973773i	$0.426938\pi$
$20$	− 4.40440i	− 0.220220i
$21$	0	0
$22$	−37.1033	−1.68651
$23$	− 17.7220i	− 0.770523i	−0.922807	−	0.385262i	$-0.874111\pi$
$23$	− 17.7220i	− 0.770523i	0.922807	−	0.385262i	$-0.125889\pi$
$24$	0	0
$25$	22.2288	0.889150
$26$	− 41.8367i	− 1.60910i
$27$	0	0
$28$	7.00000	0.250000
$29$	38.8308i	1.33899i	0.742815	+	0.669496i	$0.233490\pi$
$29$	38.8308i	1.33899i	−0.742815	+	0.669496i	$0.766510\pi$
$30$	0	0
$31$	−2.52026	−0.0812987	−0.0406493	−	0.999173i	$-0.512943\pi$
$31$	−2.52026	−0.0812987	−0.0406493	+	0.999173i	$0.512943\pi$
$32$	− 36.5191i	− 1.14122i
$33$	0	0
$34$	−10.9373	−0.321684
$35$	− 4.40440i	− 0.125840i
$36$	0	0
$37$	14.0405	0.379473	0.189737	−	0.981835i	$-0.439237\pi$
$37$	14.0405	0.379473	0.189737	+	0.981835i	$0.439237\pi$
$38$	22.2882i	0.586531i
$39$	0	0
$40$	−5.81176	−0.145294
$41$	− 36.5764i	− 0.892106i	−0.895006	−	0.446053i	$-0.852829\pi$
$41$	− 36.5764i	− 0.892106i	0.895006	−	0.446053i	$-0.147171\pi$
$42$	0	0
$43$	32.6863	0.760146	0.380073	−	0.924957i	$-0.375899\pi$
$43$	32.6863	0.760146	0.380073	+	0.924957i	$0.375899\pi$
$44$	− 38.0793i	− 0.865439i
$45$	0	0
$46$	45.6863	0.993180
$47$	− 26.6926i	− 0.567927i	−0.958835	−	0.283964i	$-0.908350\pi$
$47$	− 26.6926i	− 0.567927i	0.958835	−	0.283964i	$-0.0916495\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	57.3043i	1.14609i
$51$	0	0
$52$	42.9373	0.825716
$53$	68.8528i	1.29911i	0.760315	+	0.649555i	$0.225045\pi$
$53$	68.8528i	1.29911i	−0.760315	+	0.649555i	$0.774955\pi$
$54$	0	0
$55$	−23.9595	−0.435627
$56$	− 9.23676i	− 0.164942i
$57$	0	0
$58$	−100.103	−1.72592
$59$	104.031i	1.76323i	0.471968	+	0.881616i	$0.343544\pi$
$59$	104.031i	1.76323i	−0.471968	+	0.881616i	$0.656456\pi$
$60$	0	0
$61$	113.956	1.86812	0.934062	−	0.357111i	$-0.116238\pi$
$61$	113.956	1.86812	0.934062	+	0.357111i	$0.116238\pi$
$62$	− 6.49707i	− 0.104791i
$63$	0	0
$64$	15.8118	0.247059
$65$	− 27.0161i	− 0.415632i
$66$	0	0
$67$	41.6458	0.621578	0.310789	−	0.950479i	$-0.399407\pi$
$67$	41.6458	0.621578	0.310789	+	0.950479i	$0.399407\pi$
$68$	− 11.2250i	− 0.165073i
$69$	0	0
$70$	11.3542	0.162204
$71$	123.256i	1.73599i	0.496569	+	0.867997i	$0.334593\pi$
$71$	123.256i	1.73599i	−0.496569	+	0.867997i	$0.665407\pi$
$72$	0	0
$73$	109.852	1.50483	0.752413	−	0.658692i	$-0.228890\pi$
$73$	109.852	1.50483	0.752413	+	0.658692i	$0.228890\pi$
$74$	36.1956i	0.489129i
$75$	0	0
$76$	−22.8745	−0.300980
$77$	− 38.0793i	− 0.494537i
$78$	0	0
$79$	−97.1882	−1.23023	−0.615115	−	0.788437i	$-0.710891\pi$
$79$	−97.1882	−1.23023	−0.615115	+	0.788437i	$0.710891\pi$
$80$	− 32.5999i	− 0.407499i
$81$	0	0
$82$	94.2915	1.14990
$83$	− 121.534i	− 1.46426i	−0.681165	−	0.732130i	$-0.738526\pi$
$83$	− 121.534i	− 1.46426i	0.681165	−	0.732130i	$-0.261474\pi$
$84$	0	0
$85$	−7.06275	−0.0830911
$86$	84.2631i	0.979803i
$87$	0	0
$88$	−50.2470	−0.570989
$89$	− 0.808777i	− 0.00908738i	−0.999990	−	0.00454369i	$-0.998554\pi$
$89$	− 0.808777i	− 0.00908738i	0.999990	−	0.00454369i	$-0.00144631\pi$
$90$	0	0
$91$	42.9373	0.471838
$92$	46.8881i	0.509653i
$93$	0	0
$94$	68.8118	0.732040
$95$	14.3926i	0.151501i
$96$	0	0
$97$	104.539	1.07772	0.538859	−	0.842396i	$-0.318856\pi$
$97$	104.539	1.07772	0.538859	+	0.842396i	$0.318856\pi$
$98$	18.0455i	0.184138i
$99$	0	0
$100$	−58.8118	−0.588118
$101$	− 183.081i	− 1.81268i	−0.422550	−	0.906339i	$-0.638865\pi$
$101$	− 183.081i	− 1.81268i	0.422550	−	0.906339i	$-0.361135\pi$
$102$	0	0
$103$	−114.933	−1.11586	−0.557929	−	0.829889i	$-0.688404\pi$
$103$	−114.933	−1.11586	−0.557929	+	0.829889i	$0.688404\pi$
$104$	− 56.6573i	− 0.544782i
$105$	0	0
$106$	−177.498	−1.67451
$107$	− 58.4837i	− 0.546577i	−0.961932	−	0.273289i	$-0.911889\pi$
$107$	− 58.4837i	− 0.546577i	0.961932	−	0.273289i	$-0.0881114\pi$
$108$	0	0
$109$	−74.1438	−0.680218	−0.340109	−	0.940386i	$-0.610464\pi$
$109$	−74.1438	−0.680218	−0.340109	+	0.940386i	$0.610464\pi$
$110$	− 61.7660i	− 0.561509i
$111$	0	0
$112$	51.8118	0.462605
$113$	− 181.958i	− 1.61025i	−0.593104	−	0.805126i	$-0.702098\pi$
$113$	− 181.958i	− 1.61025i	0.593104	−	0.805126i	$-0.297902\pi$
$114$	0	0
$115$	29.5020	0.256539
$116$	− 102.737i	− 0.885660i
$117$	0	0
$118$	−268.184	−2.27275
$119$	− 11.2250i	− 0.0943275i
$120$	0	0
$121$	−86.1477	−0.711965
$122$	293.770i	2.40795i
$123$	0	0
$124$	6.66798	0.0537740
$125$	78.6220i	0.628976i
$126$	0	0
$127$	46.7307	0.367958	0.183979	−	0.982930i	$-0.441102\pi$
$127$	46.7307	0.367958	0.183979	+	0.982930i	$0.441102\pi$
$128$	− 105.315i	− 0.822770i
$129$	0	0
$130$	69.6458	0.535737
$131$	− 45.0515i	− 0.343905i	−0.985105	−	0.171952i	$-0.944993\pi$
$131$	− 45.0515i	− 0.343905i	0.985105	−	0.171952i	$-0.0550075\pi$
$132$	0	0
$133$	−22.8745	−0.171989
$134$	107.360i	0.801194i
$135$	0	0
$136$	−14.8118	−0.108910
$137$	3.71022i	0.0270819i	0.999908	+	0.0135410i	$0.00431035\pi$
$137$	3.71022i	0.0270819i	−0.999908	+	0.0135410i	$0.995690\pi$
$138$	0	0
$139$	−76.4575	−0.550054	−0.275027	−	0.961436i	$-0.588687\pi$
$139$	−76.4575	−0.550054	−0.275027	+	0.961436i	$0.588687\pi$
$140$	11.6529i	0.0832353i
$141$	0	0
$142$	−317.745	−2.23764
$143$	− 233.574i	− 1.63339i
$144$	0	0
$145$	−64.6418	−0.445806
$146$	283.192i	1.93967i
$147$	0	0
$148$	−37.1477	−0.250998
$149$	216.547i	1.45333i	0.686991	+	0.726666i	$0.258931\pi$
$149$	216.547i	1.45333i	−0.686991	+	0.726666i	$0.741069\pi$
$150$	0	0
$151$	118.125	0.782288	0.391144	−	0.920329i	$-0.372079\pi$
$151$	118.125	0.782288	0.391144	+	0.920329i	$0.372079\pi$
$152$	30.1838i	0.198577i
$153$	0	0
$154$	98.1660	0.637442
$155$	− 4.19549i	− 0.0270677i
$156$	0	0
$157$	215.269	1.37114	0.685571	−	0.728006i	$-0.259553\pi$
$157$	215.269	1.37114	0.685571	+	0.728006i	$0.259553\pi$
$158$	− 250.545i	− 1.58573i
$159$	0	0
$160$	60.7935	0.379959
$161$	46.8881i	0.291230i
$162$	0	0
$163$	−183.099	−1.12331	−0.561654	−	0.827372i	$-0.689835\pi$
$163$	−183.099	−1.12331	−0.561654	+	0.827372i	$0.689835\pi$
$164$	96.7720i	0.590073i
$165$	0	0
$166$	313.306	1.88738
$167$	− 64.6674i	− 0.387230i	−0.981078	−	0.193615i	$-0.937979\pi$
$167$	− 64.6674i	− 0.387230i	0.981078	−	0.193615i	$-0.0620213\pi$
$168$	0	0
$169$	94.3725	0.558417
$170$	− 18.2073i	− 0.107102i
$171$	0	0
$172$	−86.4797	−0.502789
$173$	250.060i	1.44543i	0.691145	+	0.722716i	$0.257107\pi$
$173$	250.060i	1.44543i	−0.691145	+	0.722716i	$0.742893\pi$
$174$	0	0
$175$	−58.8118	−0.336067
$176$	− 281.851i	− 1.60143i
$177$	0	0
$178$	2.08497	0.0117133
$179$	324.971i	1.81548i	0.419530	+	0.907742i	$0.362195\pi$
$179$	324.971i	1.81548i	−0.419530	+	0.907742i	$0.637805\pi$
$180$	0	0
$181$	−211.793	−1.17013	−0.585065	−	0.810987i	$-0.698931\pi$
$181$	−211.793	−1.17013	−0.585065	+	0.810987i	$0.698931\pi$
$182$	110.689i	0.608184i
$183$	0	0
$184$	61.8706	0.336253
$185$	23.3733i	0.126342i
$186$	0	0
$187$	−61.0627	−0.326539
$188$	70.6219i	0.375649i
$189$	0	0
$190$	−37.1033	−0.195280
$191$	122.228i	0.639936i	0.947428	+	0.319968i	$0.103672\pi$
$191$	122.228i	0.639936i	−0.947428	+	0.319968i	$0.896328\pi$
$192$	0	0
$193$	99.3725	0.514884	0.257442	−	0.966294i	$-0.417120\pi$
$193$	99.3725	0.514884	0.257442	+	0.966294i	$0.417120\pi$
$194$	269.494i	1.38914i
$195$	0	0
$196$	−18.5203	−0.0944911
$197$	− 274.326i	− 1.39252i	−0.717791	−	0.696259i	$-0.754847\pi$
$197$	− 274.326i	− 1.39252i	0.717791	−	0.696259i	$-0.245153\pi$
$198$	0	0
$199$	−260.077	−1.30692	−0.653460	−	0.756961i	$-0.726683\pi$
$199$	−260.077	−1.30692	−0.653460	+	0.756961i	$0.726683\pi$
$200$	77.6043i	0.388021i
$201$	0	0
$202$	471.970	2.33648
$203$	− 102.737i	− 0.506092i
$204$	0	0
$205$	60.8889	0.297019
$206$	− 296.291i	− 1.43830i
$207$	0	0
$208$	317.808	1.52792
$209$	124.435i	0.595383i
$210$	0	0
$211$	−88.9150	−0.421398	−0.210699	−	0.977551i	$-0.567574\pi$
$211$	−88.9150	−0.421398	−0.210699	+	0.977551i	$0.567574\pi$
$212$	− 182.167i	− 0.859280i
$213$	0	0
$214$	150.767	0.704520
$215$	54.4130i	0.253084i
$216$	0	0
$217$	6.66798	0.0307280
$218$	− 191.138i	− 0.876779i
$219$	0	0
$220$	63.3908	0.288140
$221$	− 68.8528i	− 0.311551i
$222$	0	0
$223$	−200.158	−0.897570	−0.448785	−	0.893640i	$-0.648143\pi$
$223$	−200.158	−0.897570	−0.448785	+	0.893640i	$0.648143\pi$
$224$	96.6204i	0.431341i
$225$	0	0
$226$	469.077	2.07556
$227$	− 162.097i	− 0.714082i	−0.934089	−	0.357041i	$-0.883786\pi$
$227$	− 162.097i	− 0.714082i	0.934089	−	0.357041i	$-0.116214\pi$
$228$	0	0
$229$	72.4941	0.316568	0.158284	−	0.987394i	$-0.449404\pi$
$229$	72.4941	0.316568	0.158284	+	0.987394i	$0.449404\pi$
$230$	76.0542i	0.330670i
$231$	0	0
$232$	−135.565	−0.584331
$233$	24.9234i	0.106967i	0.998569	+	0.0534837i	$0.0170325\pi$
$233$	24.9234i	0.106967i	−0.998569	+	0.0534837i	$0.982967\pi$
$234$	0	0
$235$	44.4353	0.189086
$236$	− 275.239i	− 1.16627i
$237$	0	0
$238$	28.9373	0.121585
$239$	− 388.898i	− 1.62719i	−0.581434	−	0.813593i	$-0.697508\pi$
$239$	− 388.898i	− 1.62719i	0.581434	−	0.813593i	$-0.302492\pi$
$240$	0	0
$241$	291.306	1.20874	0.604369	−	0.796705i	$-0.293425\pi$
$241$	291.306	1.20874	0.604369	+	0.796705i	$0.293425\pi$
$242$	− 222.083i	− 0.917699i
$243$	0	0
$244$	−301.498	−1.23565
$245$	11.6529i	0.0475630i
$246$	0	0
$247$	−140.310	−0.568056
$248$	− 8.79864i	− 0.0354784i
$249$	0	0
$250$	−202.682	−0.810729
$251$	− 56.8089i	− 0.226330i	−0.993576	−	0.113165i	$-0.963901\pi$
$251$	− 56.8089i	− 0.226330i	0.993576	−	0.113165i	$-0.0360989\pi$
$252$	0	0
$253$	255.067	1.00817
$254$	120.469i	0.474287i
$255$	0	0
$256$	334.741	1.30758
$257$	321.042i	1.24919i	0.780948	+	0.624596i	$0.214736\pi$
$257$	321.042i	1.24919i	−0.780948	+	0.624596i	$0.785264\pi$
$258$	0	0
$259$	−37.1477	−0.143427
$260$	71.4779i	0.274915i
$261$	0	0
$262$	116.140	0.443282
$263$	42.7601i	0.162586i	0.996690	+	0.0812929i	$0.0259049\pi$
$263$	42.7601i	0.162586i	−0.996690	+	0.0812929i	$0.974095\pi$
$264$	0	0
$265$	−114.620	−0.432527
$266$	− 58.9690i	− 0.221688i
$267$	0	0
$268$	−110.184	−0.411135
$269$	− 61.9952i	− 0.230465i	−0.993339	−	0.115233i	$-0.963239\pi$
$269$	− 61.9952i	− 0.230465i	0.993339	−	0.115233i	$-0.0367614\pi$
$270$	0	0
$271$	−1.04052	−0.00383955	−0.00191978	−	0.999998i	$-0.500611\pi$
$271$	−1.04052	−0.00383955	−0.00191978	+	0.999998i	$0.500611\pi$
$272$	− 83.0837i	− 0.305455i
$273$	0	0
$274$	−9.56471	−0.0349077
$275$	319.930i	1.16338i
$276$	0	0
$277$	206.579	0.745773	0.372886	−	0.927877i	$-0.378368\pi$
$277$	206.579	0.745773	0.372886	+	0.927877i	$0.378368\pi$
$278$	− 197.103i	− 0.709002i
$279$	0	0
$280$	15.3765	0.0549160
$281$	59.9867i	0.213476i	0.994287	+	0.106738i	$0.0340406\pi$
$281$	59.9867i	0.213476i	−0.994287	+	0.106738i	$0.965959\pi$
$282$	0	0
$283$	−240.907	−0.851262	−0.425631	−	0.904897i	$-0.639948\pi$
$283$	−240.907	−0.851262	−0.425631	+	0.904897i	$0.639948\pi$
$284$	− 326.104i	− 1.14825i
$285$	0	0
$286$	602.140	2.10538
$287$	96.7720i	0.337185i
$288$	0	0
$289$	271.000	0.937716
$290$	− 166.642i	− 0.574629i
$291$	0	0
$292$	−290.642	−0.995349
$293$	− 94.2513i	− 0.321677i	−0.986981	−	0.160838i	$-0.948580\pi$
$293$	− 94.2513i	− 0.321677i	0.986981	−	0.160838i	$-0.0514198\pi$
$294$	0	0
$295$	−173.180	−0.587052
$296$	49.0178i	0.165601i
$297$	0	0
$298$	−558.243	−1.87330
$299$	287.607i	0.961895i
$300$	0	0
$301$	−86.4797	−0.287308
$302$	304.520i	1.00834i
$303$	0	0
$304$	−169.310	−0.556940
$305$	189.702i	0.621975i
$306$	0	0
$307$	−379.605	−1.23650	−0.618250	−	0.785982i	$-0.712158\pi$
$307$	−379.605	−1.23650	−0.618250	+	0.785982i	$0.712158\pi$
$308$	100.748i	0.327105i
$309$	0	0
$310$	10.8157	0.0348894
$311$	394.033i	1.26699i	0.773748	+	0.633494i	$0.218380\pi$
$311$	394.033i	1.26699i	−0.773748	+	0.633494i	$0.781620\pi$
$312$	0	0
$313$	−422.561	−1.35003	−0.675017	−	0.737802i	$-0.735864\pi$
$313$	−422.561	−1.35003	−0.675017	+	0.737802i	$0.735864\pi$
$314$	554.950i	1.76736i
$315$	0	0
$316$	257.136	0.813721
$317$	− 281.251i	− 0.887227i	−0.896218	−	0.443614i	$-0.853696\pi$
$317$	− 281.251i	− 0.887227i	0.896218	−	0.443614i	$-0.146304\pi$
$318$	0	0
$319$	−558.877	−1.75197
$320$	26.3219i	0.0822560i
$321$	0	0
$322$	−120.875	−0.375387
$323$	36.6808i	0.113563i
$324$	0	0
$325$	−360.745	−1.10998
$326$	− 472.018i	− 1.44791i
$327$	0	0
$328$	127.694	0.389311
$329$	70.6219i	0.214656i
$330$	0	0
$331$	−27.2470	−0.0823174	−0.0411587	−	0.999153i	$-0.513105\pi$
$331$	−27.2470	−0.0823174	−0.0411587	+	0.999153i	$0.513105\pi$
$332$	321.548i	0.968517i
$333$	0	0
$334$	166.708	0.499127
$335$	69.3279i	0.206949i
$336$	0	0
$337$	348.454	1.03399	0.516993	−	0.855989i	$-0.327051\pi$
$337$	348.454	1.03399	0.516993	+	0.855989i	$0.327051\pi$
$338$	243.286i	0.719782i
$339$	0	0
$340$	18.6863	0.0549596
$341$	− 36.2732i	− 0.106373i
$342$	0	0
$343$	−18.5203	−0.0539949
$344$	114.113i	0.331724i
$345$	0	0
$346$	−644.638	−1.86312
$347$	421.535i	1.21480i	0.794397	+	0.607398i	$0.207787\pi$
$347$	421.535i	1.21480i	−0.794397	+	0.607398i	$0.792213\pi$
$348$	0	0
$349$	561.454	1.60875	0.804375	−	0.594122i	$-0.202500\pi$
$349$	561.454	1.60875	0.804375	+	0.594122i	$0.202500\pi$
$350$	− 151.613i	− 0.433180i
$351$	0	0
$352$	525.605	1.49320
$353$	− 163.970i	− 0.464505i	−0.972656	−	0.232252i	$-0.925390\pi$
$353$	− 163.970i	− 0.464505i	0.972656	−	0.232252i	$-0.0746095\pi$
$354$	0	0
$355$	−205.184	−0.577984
$356$	2.13982i	0.00601073i
$357$	0	0
$358$	−837.755	−2.34010
$359$	− 154.619i	− 0.430693i	−0.976538	−	0.215347i	$-0.930912\pi$
$359$	− 154.619i	− 0.430693i	0.976538	−	0.215347i	$-0.0690881\pi$
$360$	0	0
$361$	−286.251	−0.792939
$362$	− 545.990i	− 1.50826i
$363$	0	0
$364$	−113.601	−0.312091
$365$	182.872i	0.501018i
$366$	0	0
$367$	45.4131	0.123741	0.0618707	−	0.998084i	$-0.480293\pi$
$367$	45.4131	0.123741	0.0618707	+	0.998084i	$0.480293\pi$
$368$	347.051i	0.943073i
$369$	0	0
$370$	−60.2549	−0.162851
$371$	− 182.167i	− 0.491017i
$372$	0	0
$373$	−294.047	−0.788330	−0.394165	−	0.919040i	$-0.628966\pi$
$373$	−294.047	−0.788330	−0.394165	+	0.919040i	$0.628966\pi$
$374$	− 157.416i	− 0.420898i
$375$	0	0
$376$	93.1882	0.247841
$377$	− 630.175i	− 1.67155i
$378$	0	0
$379$	679.476	1.79281	0.896406	−	0.443234i	$-0.146169\pi$
$379$	679.476	1.79281	0.896406	+	0.443234i	$0.146169\pi$
$380$	− 38.0793i	− 0.100209i
$381$	0	0
$382$	−315.095	−0.824857
$383$	− 409.235i	− 1.06850i	−0.845327	−	0.534249i	$-0.820595\pi$
$383$	− 409.235i	− 1.06850i	0.845327	−	0.534249i	$-0.179405\pi$
$384$	0	0
$385$	63.3908	0.164652
$386$	256.176i	0.663668i
$387$	0	0
$388$	−276.583	−0.712843
$389$	111.812i	0.287433i	0.989619	+	0.143717i	$0.0459054\pi$
$389$	111.812i	0.287433i	−0.989619	+	0.143717i	$0.954095\pi$
$390$	0	0
$391$	75.1882	0.192297
$392$	24.4382i	0.0623422i
$393$	0	0
$394$	707.195	1.79491
$395$	− 161.790i	− 0.409594i
$396$	0	0
$397$	−98.8419	−0.248972	−0.124486	−	0.992221i	$-0.539728\pi$
$397$	−98.8419	−0.248972	−0.124486	+	0.992221i	$0.539728\pi$
$398$	− 670.462i	− 1.68458i
$399$	0	0
$400$	−435.306	−1.08826
$401$	− 252.163i	− 0.628834i	−0.949285	−	0.314417i	$-0.898191\pi$
$401$	− 252.163i	− 0.628834i	0.949285	−	0.314417i	$-0.101809\pi$
$402$	0	0
$403$	40.9007	0.101491
$404$	484.386i	1.19897i
$405$	0	0
$406$	264.848	0.652336
$407$	202.080i	0.496511i
$408$	0	0
$409$	614.553	1.50257	0.751287	−	0.659975i	$-0.229433\pi$
$409$	614.553	1.50257	0.751287	+	0.659975i	$0.229433\pi$
$410$	156.968i	0.382848i
$411$	0	0
$412$	304.085	0.738070
$413$	− 275.239i	− 0.666439i
$414$	0	0
$415$	202.318	0.487512
$416$	592.659i	1.42466i
$417$	0	0
$418$	−320.786	−0.767430
$419$	− 516.005i	− 1.23152i	−0.787936	−	0.615758i	$-0.788850\pi$
$419$	− 516.005i	− 1.23152i	0.787936	−	0.615758i	$-0.211150\pi$
$420$	0	0
$421$	734.608	1.74491	0.872456	−	0.488693i	$-0.162526\pi$
$421$	734.608	1.74491	0.872456	+	0.488693i	$0.162526\pi$
$422$	− 229.217i	− 0.543169i
$423$	0	0
$424$	−240.376	−0.566926
$425$	94.3086i	0.221903i
$426$	0	0
$427$	−301.498	−0.706084
$428$	154.733i	0.361527i
$429$	0	0
$430$	−140.273	−0.326217
$431$	− 406.448i	− 0.943034i	−0.881857	−	0.471517i	$-0.843707\pi$
$431$	− 406.448i	− 0.943034i	0.881857	−	0.471517i	$-0.156293\pi$
$432$	0	0
$433$	−95.3582	−0.220227	−0.110113	−	0.993919i	$-0.535121\pi$
$433$	−95.3582	−0.220227	−0.110113	+	0.993919i	$0.535121\pi$
$434$	17.1896i	0.0396074i
$435$	0	0
$436$	196.166	0.449922
$437$	− 153.220i	− 0.350619i
$438$	0	0
$439$	−746.952	−1.70148	−0.850742	−	0.525583i	$-0.823847\pi$
$439$	−746.952	−1.70148	−0.850742	+	0.525583i	$0.823847\pi$
$440$	− 83.6465i	− 0.190106i
$441$	0	0
$442$	177.498	0.401579
$443$	− 236.648i	− 0.534194i	−0.963670	−	0.267097i	$-0.913936\pi$
$443$	− 236.648i	− 0.534194i	0.963670	−	0.267097i	$-0.0860644\pi$
$444$	0	0
$445$	1.34637	0.00302556
$446$	− 515.995i	− 1.15694i
$447$	0	0
$448$	−41.8340	−0.0933794
$449$	51.8822i	0.115551i	0.998330	+	0.0577753i	$0.0184007\pi$
$449$	51.8822i	0.115551i	−0.998330	+	0.0577753i	$0.981599\pi$
$450$	0	0
$451$	526.430	1.16725
$452$	481.417i	1.06508i
$453$	0	0
$454$	417.875	0.920428
$455$	71.4779i	0.157094i
$456$	0	0
$457$	97.4496	0.213238	0.106619	−	0.994300i	$-0.465998\pi$
$457$	97.4496	0.213238	0.106619	+	0.994300i	$0.465998\pi$
$458$	186.885i	0.408046i
$459$	0	0
$460$	−78.0549	−0.169685
$461$	178.390i	0.386962i	0.981104	+	0.193481i	$0.0619779\pi$
$461$	178.390i	0.386962i	−0.981104	+	0.193481i	$0.938022\pi$
$462$	0	0
$463$	−153.409	−0.331337	−0.165669	−	0.986181i	$-0.552978\pi$
$463$	−153.409	−0.331337	−0.165669	+	0.986181i	$0.552978\pi$
$464$	− 760.424i	− 1.63884i
$465$	0	0
$466$	−64.2510	−0.137878
$467$	− 494.697i	− 1.05931i	−0.848213	−	0.529655i	$-0.822322\pi$
$467$	− 494.697i	− 1.05931i	0.848213	−	0.529655i	$-0.177678\pi$
$468$	0	0
$469$	−110.184	−0.234935
$470$	114.551i	0.243726i
$471$	0	0
$472$	−363.188	−0.769467
$473$	470.441i	0.994590i
$474$	0	0
$475$	192.184	0.404599
$476$	29.6985i	0.0623918i
$477$	0	0
$478$	1002.55	2.09739
$479$	455.429i	0.950790i	0.879772	+	0.475395i	$0.157695\pi$
$479$	455.429i	0.950790i	−0.879772	+	0.475395i	$0.842305\pi$
$480$	0	0
$481$	−227.860	−0.473722
$482$	750.968i	1.55802i
$483$	0	0
$484$	227.925	0.470920
$485$	174.026i	0.358816i
$486$	0	0
$487$	693.254	1.42352	0.711759	−	0.702423i	$-0.247899\pi$
$487$	693.254	1.42352	0.711759	+	0.702423i	$0.247899\pi$
$488$	397.838i	0.815241i
$489$	0	0
$490$	−30.0405	−0.0613072
$491$	505.134i	1.02879i	0.857555	+	0.514393i	$0.171983\pi$
$491$	505.134i	1.02879i	−0.857555	+	0.514393i	$0.828017\pi$
$492$	0	0
$493$	−164.745	−0.334169
$494$	− 361.710i	− 0.732206i
$495$	0	0
$496$	49.3542	0.0995045
$497$	− 326.104i	− 0.656144i
$498$	0	0
$499$	833.003	1.66934	0.834672	−	0.550748i	$-0.185657\pi$
$499$	833.003	1.66934	0.834672	+	0.550748i	$0.185657\pi$
$500$	− 208.014i	− 0.416028i
$501$	0	0
$502$	146.450	0.291732
$503$	81.3312i	0.161692i	0.996727	+	0.0808461i	$0.0257622\pi$
$503$	81.3312i	0.161692i	−0.996727	+	0.0808461i	$0.974238\pi$
$504$	0	0
$505$	304.775	0.603515
$506$	657.545i	1.29950i
$507$	0	0
$508$	−123.638	−0.243382
$509$	− 898.129i	− 1.76450i	−0.470784	−	0.882249i	$-0.656029\pi$
$509$	− 898.129i	− 1.76450i	0.470784	−	0.882249i	$-0.343971\pi$
$510$	0	0
$511$	−290.642	−0.568771
$512$	441.683i	0.862662i
$513$	0	0
$514$	−827.626	−1.61017
$515$	− 191.330i	− 0.371515i
$516$	0	0
$517$	384.176	0.743088
$518$	− 95.7644i	− 0.184873i
$519$	0	0
$520$	94.3177	0.181380
$521$	967.885i	1.85774i	0.370400	+	0.928872i	$0.379221\pi$
$521$	967.885i	1.85774i	−0.370400	+	0.928872i	$0.620779\pi$
$522$	0	0
$523$	632.069	1.20855	0.604273	−	0.796778i	$-0.293464\pi$
$523$	632.069	1.20855	0.604273	+	0.796778i	$0.293464\pi$
$524$	119.195i	0.227471i
$525$	0	0
$526$	−110.233	−0.209568
$527$	− 10.6926i	− 0.0202895i
$528$	0	0
$529$	214.929	0.406294
$530$	− 295.482i	− 0.557513i
$531$	0	0
$532$	60.5203	0.113760
$533$	593.589i	1.11368i
$534$	0	0
$535$	97.3582	0.181978
$536$	145.392i	0.271254i
$537$	0	0
$538$	159.820	0.297063
$539$	100.748i	0.186917i
$540$	0	0
$541$	−101.118	−0.186909	−0.0934544	−	0.995624i	$-0.529791\pi$
$541$	−101.118	−0.186909	−0.0934544	+	0.995624i	$0.529791\pi$
$542$	− 2.68239i	− 0.00494906i
$543$	0	0
$544$	154.937	0.284811
$545$	− 123.428i	− 0.226473i
$546$	0	0
$547$	107.336	0.196227	0.0981133	−	0.995175i	$-0.468719\pi$
$547$	107.336	0.196227	0.0981133	+	0.995175i	$0.468719\pi$
$548$	− 9.81633i	− 0.0179130i
$549$	0	0
$550$	−824.759	−1.49956
$551$	335.721i	0.609295i
$552$	0	0
$553$	257.136	0.464984
$554$	532.548i	0.961277i
$555$	0	0
$556$	202.288	0.363827
$557$	− 524.814i	− 0.942215i	−0.882076	−	0.471107i	$-0.843854\pi$
$557$	− 524.814i	− 0.942215i	0.882076	−	0.471107i	$-0.156146\pi$
$558$	0	0
$559$	−530.458	−0.948940
$560$	86.2513i	0.154020i
$561$	0	0
$562$	−154.642	−0.275163
$563$	− 661.094i	− 1.17423i	−0.809502	−	0.587117i	$-0.800263\pi$
$563$	− 661.094i	− 1.17423i	0.809502	−	0.587117i	$-0.199737\pi$
$564$	0	0
$565$	302.907	0.536119
$566$	− 621.043i	− 1.09725i
$567$	0	0
$568$	−430.306	−0.757581
$569$	64.5528i	0.113450i	0.998390	+	0.0567248i	$0.0180658\pi$
$569$	64.5528i	0.113450i	−0.998390	+	0.0567248i	$0.981934\pi$
$570$	0	0
$571$	587.231	1.02843	0.514213	−	0.857662i	$-0.328084\pi$
$571$	587.231	1.02843	0.514213	+	0.857662i	$0.328084\pi$
$572$	617.980i	1.08038i
$573$	0	0
$574$	−249.472	−0.434620
$575$	− 393.939i	− 0.685111i
$576$	0	0
$577$	−454.029	−0.786878	−0.393439	−	0.919351i	$-0.628715\pi$
$577$	−454.029	−0.786878	−0.393439	+	0.919351i	$0.628715\pi$
$578$	698.621i	1.20869i
$579$	0	0
$580$	171.026	0.294873
$581$	321.548i	0.553438i
$582$	0	0
$583$	−990.972	−1.69978
$584$	383.513i	0.656700i
$585$	0	0
$586$	242.974	0.414631
$587$	339.337i	0.578087i	0.957316	+	0.289044i	$0.0933373\pi$
$587$	339.337i	0.578087i	−0.957316	+	0.289044i	$0.906663\pi$
$588$	0	0
$589$	−21.7895	−0.0369941
$590$	− 446.448i	− 0.756691i
$591$	0	0
$592$	−274.956	−0.464452
$593$	90.3626i	0.152382i	0.997093	+	0.0761911i	$0.0242759\pi$
$593$	90.3626i	0.152382i	−0.997093	+	0.0761911i	$0.975724\pi$
$594$	0	0
$595$	18.6863	0.0314055
$596$	− 572.928i	− 0.961289i
$597$	0	0
$598$	−741.431	−1.23985
$599$	− 371.725i	− 0.620575i	−0.950643	−	0.310288i	$-0.899575\pi$
$599$	− 371.725i	− 0.620575i	0.950643	−	0.310288i	$-0.100425\pi$
$600$	0	0
$601$	40.4876	0.0673671	0.0336835	−	0.999433i	$-0.489276\pi$
$601$	40.4876	0.0673671	0.0336835	+	0.999433i	$0.489276\pi$
$602$	− 222.939i	− 0.370331i
$603$	0	0
$604$	−312.531	−0.517435
$605$	− 143.411i	− 0.237042i
$606$	0	0
$607$	180.494	0.297354	0.148677	−	0.988886i	$-0.452498\pi$
$607$	180.494	0.297354	0.148677	+	0.988886i	$0.452498\pi$
$608$	− 315.735i	− 0.519301i
$609$	0	0
$610$	−489.041	−0.801706
$611$	433.187i	0.708981i
$612$	0	0
$613$	−235.848	−0.384744	−0.192372	−	0.981322i	$-0.561618\pi$
$613$	−235.848	−0.384744	−0.192372	+	0.981322i	$0.561618\pi$
$614$	− 978.598i	− 1.59381i
$615$	0	0
$616$	132.941	0.215814
$617$	471.466i	0.764126i	0.924136	+	0.382063i	$0.124786\pi$
$617$	471.466i	0.764126i	−0.924136	+	0.382063i	$0.875214\pi$
$618$	0	0
$619$	−292.660	−0.472795	−0.236397	−	0.971656i	$-0.575967\pi$
$619$	−292.660	−0.472795	−0.236397	+	0.971656i	$0.575967\pi$
$620$	11.1002i	0.0179036i
$621$	0	0
$622$	−1015.79	−1.63311
$623$	2.13982i	0.00343471i
$624$	0	0
$625$	424.837	0.679738
$626$	− 1089.33i	− 1.74015i
$627$	0	0
$628$	−569.549	−0.906925
$629$	59.5689i	0.0947041i
$630$	0	0
$631$	−87.5568	−0.138759	−0.0693794	−	0.997590i	$-0.522102\pi$
$631$	−87.5568	−0.138759	−0.0693794	+	0.997590i	$0.522102\pi$
$632$	− 339.300i	− 0.536867i
$633$	0	0
$634$	725.047	1.14361
$635$	77.7929i	0.122508i
$636$	0	0
$637$	−113.601	−0.178338
$638$	− 1440.75i	− 2.25823i
$639$	0	0
$640$	175.318	0.273934
$641$	− 835.595i	− 1.30358i	−0.758399	−	0.651790i	$-0.774018\pi$
$641$	− 835.595i	− 1.30358i	0.758399	−	0.651790i	$-0.225982\pi$
$642$	0	0
$643$	−616.431	−0.958680	−0.479340	−	0.877629i	$-0.659124\pi$
$643$	−616.431	−0.958680	−0.479340	+	0.877629i	$0.659124\pi$
$644$	− 124.054i	− 0.192631i
$645$	0	0
$646$	−94.5608	−0.146379
$647$	188.048i	0.290646i	0.989384	+	0.145323i	$0.0464221\pi$
$647$	188.048i	0.290646i	−0.989384	+	0.145323i	$0.953578\pi$
$648$	0	0
$649$	−1497.27	−2.30705
$650$	− 929.978i	− 1.43073i
$651$	0	0
$652$	484.435	0.742999
$653$	191.889i	0.293858i	0.989147	+	0.146929i	$0.0469389\pi$
$653$	191.889i	0.293858i	−0.989147	+	0.146929i	$0.953061\pi$
$654$	0	0
$655$	74.9975	0.114500
$656$	716.275i	1.09188i
$657$	0	0
$658$	−182.059	−0.276685
$659$	− 1120.62i	− 1.70049i	−0.526391	−	0.850243i	$-0.676455\pi$
$659$	− 1120.62i	− 1.70049i	0.526391	−	0.850243i	$-0.323545\pi$
$660$	0	0
$661$	572.767	0.866516	0.433258	−	0.901270i	$-0.357364\pi$
$661$	572.767	0.866516	0.433258	+	0.901270i	$0.357364\pi$
$662$	− 70.2411i	− 0.106104i
$663$	0	0
$664$	424.294	0.638997
$665$	− 38.0793i	− 0.0572621i
$666$	0	0
$667$	688.161	1.03173
$668$	171.094i	0.256129i
$669$	0	0
$670$	−178.723	−0.266751
$671$	1640.12i	2.44429i
$672$	0	0
$673$	9.10072	0.0135226	0.00676131	−	0.999977i	$-0.497848\pi$
$673$	9.10072	0.0135226	0.00676131	+	0.999977i	$0.497848\pi$
$674$	898.291i	1.33278i
$675$	0	0
$676$	−249.686	−0.369358
$677$	278.808i	0.411829i	0.978570	+	0.205914i	$0.0660168\pi$
$677$	278.808i	0.411829i	−0.978570	+	0.205914i	$0.933983\pi$
$678$	0	0
$679$	−276.583	−0.407339
$680$	− 24.6572i	− 0.0362606i
$681$	0	0
$682$	93.5098	0.137111
$683$	978.500i	1.43265i	0.697767	+	0.716325i	$0.254177\pi$
$683$	978.500i	1.43265i	−0.697767	+	0.716325i	$0.745823\pi$
$684$	0	0
$685$	−6.17643	−0.00901668
$686$	− 47.7440i	− 0.0695977i
$687$	0	0
$688$	−640.095	−0.930371
$689$	− 1117.40i	− 1.62176i
$690$	0	0
$691$	475.867	0.688664	0.344332	−	0.938848i	$-0.388105\pi$
$691$	475.867	0.688664	0.344332	+	0.938848i	$0.388105\pi$
$692$	− 661.596i	− 0.956063i
$693$	0	0
$694$	−1086.69	−1.56583
$695$	− 127.279i	− 0.183136i
$696$	0	0
$697$	155.180	0.222640
$698$	1447.39i	2.07363i
$699$	0	0
$700$	155.601	0.222288
$701$	249.534i	0.355968i	0.984033	+	0.177984i	$0.0569576\pi$
$701$	249.534i	0.355968i	−0.984033	+	0.177984i	$0.943042\pi$
$702$	0	0
$703$	121.391	0.172675
$704$	227.573i	0.323257i
$705$	0	0
$706$	422.705	0.598732
$707$	484.386i	0.685128i
$708$	0	0
$709$	−388.646	−0.548160	−0.274080	−	0.961707i	$-0.588373\pi$
$709$	−388.646	−0.548160	−0.274080	+	0.961707i	$0.588373\pi$
$710$	− 528.952i	− 0.745003i
$711$	0	0
$712$	2.82357	0.00396569
$713$	44.6641i	0.0626425i
$714$	0	0
$715$	388.833	0.543822
$716$	− 859.794i	− 1.20083i
$717$	0	0
$718$	398.597	0.555150
$719$	190.386i	0.264793i	0.991197	+	0.132397i	$0.0422673\pi$
$719$	190.386i	0.264793i	−0.991197	+	0.132397i	$0.957733\pi$
$720$	0	0
$721$	304.085	0.421754
$722$	− 737.937i	− 1.02207i
$723$	0	0
$724$	560.353	0.773968
$725$	863.160i	1.19057i
$726$	0	0
$727$	−1406.23	−1.93429	−0.967144	−	0.254229i	$-0.918178\pi$
$727$	−1406.23	−1.93429	−0.967144	+	0.254229i	$0.918178\pi$
$728$	149.901i	0.205908i
$729$	0	0
$730$	−471.431	−0.645796
$731$	138.676i	0.189707i
$732$	0	0
$733$	16.7307	0.0228250	0.0114125	−	0.999935i	$-0.496367\pi$
$733$	16.7307	0.0228250	0.0114125	+	0.999935i	$0.496367\pi$
$734$	117.072i	0.159499i
$735$	0	0
$736$	−647.192	−0.879337
$737$	599.392i	0.813286i
$738$	0	0
$739$	560.295	0.758181	0.379090	−	0.925360i	$-0.376237\pi$
$739$	560.295	0.758181	0.379090	+	0.925360i	$0.376237\pi$
$740$	− 61.8400i	− 0.0835676i
$741$	0	0
$742$	469.616	0.632905
$743$	64.1416i	0.0863278i	0.999068	+	0.0431639i	$0.0137438\pi$
$743$	64.1416i	0.0863278i	−0.999068	+	0.0431639i	$0.986256\pi$
$744$	0	0
$745$	−360.486	−0.483874
$746$	− 758.034i	− 1.01613i
$747$	0	0
$748$	161.557	0.215985
$749$	154.733i	0.206587i
$750$	0	0
$751$	151.018	0.201090	0.100545	−	0.994933i	$-0.467941\pi$
$751$	151.018	0.201090	0.100545	+	0.994933i	$0.467941\pi$
$752$	522.721i	0.695108i
$753$	0	0
$754$	1624.55	2.15458
$755$	196.644i	0.260456i
$756$	0	0
$757$	951.239	1.25659	0.628295	−	0.777975i	$-0.283753\pi$
$757$	951.239	1.25659	0.628295	+	0.777975i	$0.283753\pi$
$758$	1751.64i	2.31088i
$759$	0	0
$760$	−50.2470	−0.0661145
$761$	− 434.236i	− 0.570612i	−0.958437	−	0.285306i	$-0.907905\pi$
$761$	− 434.236i	− 0.570612i	0.958437	−	0.285306i	$-0.0920952\pi$
$762$	0	0
$763$	196.166	0.257098
$764$	− 323.384i	− 0.423278i
$765$	0	0
$766$	1054.98	1.37726
$767$	− 1688.29i	− 2.20116i
$768$	0	0
$769$	−138.702	−0.180367	−0.0901834	−	0.995925i	$-0.528745\pi$
$769$	−138.702	−0.180367	−0.0901834	+	0.995925i	$0.528745\pi$
$770$	163.417i	0.212230i
$771$	0	0
$772$	−262.915	−0.340564
$773$	1175.03i	1.52009i	0.649868	+	0.760047i	$0.274824\pi$
$773$	1175.03i	1.52009i	−0.649868	+	0.760047i	$0.725176\pi$
$774$	0	0
$775$	−56.0222	−0.0722867
$776$	364.961i	0.470311i
$777$	0	0
$778$	−288.243	−0.370492
$779$	− 316.230i	− 0.405944i
$780$	0	0
$781$	−1773.97	−2.27141
$782$	193.830i	0.247865i
$783$	0	0
$784$	−137.081	−0.174848
$785$	358.360i	0.456509i
$786$	0	0
$787$	1378.94	1.75214	0.876071	−	0.482182i	$-0.160156\pi$
$787$	1378.94	1.75214	0.876071	+	0.482182i	$0.160156\pi$
$788$	725.798i	0.921064i
$789$	0	0
$790$	417.084	0.527954
$791$	481.417i	0.608618i
$792$	0	0
$793$	−1849.36	−2.33210
$794$	− 254.808i	− 0.320917i
$795$	0	0
$796$	688.099	0.864446
$797$	798.749i	1.00219i	0.865391	+	0.501097i	$0.167070\pi$
$797$	798.749i	1.00219i	−0.865391	+	0.501097i	$0.832930\pi$
$798$	0	0
$799$	113.247	0.141736
$800$	− 811.773i	− 1.01472i
$801$	0	0
$802$	650.059	0.810547
$803$	1581.06i	1.96895i
$804$	0	0
$805$	−78.0549	−0.0969626
$806$	105.439i	0.130818i
$807$	0	0
$808$	639.165	0.791045
$809$	− 1125.86i	− 1.39167i	−0.718200	−	0.695837i	$-0.755034\pi$
$809$	− 1125.86i	− 1.39167i	0.718200	−	0.695837i	$-0.244966\pi$
$810$	0	0
$811$	1282.47	1.58134	0.790670	−	0.612243i	$-0.209732\pi$
$811$	1282.47	1.58134	0.790670	+	0.612243i	$0.209732\pi$
$812$	271.816i	0.334748i
$813$	0	0
$814$	−520.949	−0.639987
$815$	− 304.806i	− 0.373996i
$816$	0	0
$817$	282.597	0.345896
$818$	1584.28i	1.93677i
$819$	0	0
$820$	−161.097	−0.196459
$821$	− 1439.87i	− 1.75380i	−0.480670	−	0.876902i	$-0.659606\pi$
$821$	− 1439.87i	− 1.75380i	0.480670	−	0.876902i	$-0.340394\pi$
$822$	0	0
$823$	−1.95554	−0.00237612	−0.00118806	−	0.999999i	$-0.500378\pi$
$823$	−1.95554	−0.00237612	−0.00118806	+	0.999999i	$0.500378\pi$
$824$	− 401.251i	− 0.486955i
$825$	0	0
$826$	709.549	0.859018
$827$	− 965.583i	− 1.16757i	−0.811907	−	0.583787i	$-0.801570\pi$
$827$	− 965.583i	− 1.16757i	0.811907	−	0.583787i	$-0.198430\pi$
$828$	0	0
$829$	−260.294	−0.313986	−0.156993	−	0.987600i	$-0.550180\pi$
$829$	−260.294	−0.313986	−0.156993	+	0.987600i	$0.550180\pi$
$830$	521.562i	0.628388i
$831$	0	0
$832$	−256.605	−0.308420
$833$	29.6985i	0.0356524i
$834$	0	0
$835$	107.652	0.128925
$836$	− 329.224i	− 0.393809i
$837$	0	0
$838$	1330.23	1.58738
$839$	− 251.859i	− 0.300190i	−0.988672	−	0.150095i	$-0.952042\pi$
$839$	− 251.859i	− 0.300190i	0.988672	−	0.150095i	$-0.0479579\pi$
$840$	0	0
$841$	−666.830	−0.792901
$842$	1893.77i	2.24913i
$843$	0	0
$844$	235.247	0.278729
$845$	157.102i	0.185920i
$846$	0	0
$847$	227.925	0.269097
$848$	− 1348.34i	− 1.59003i
$849$	0	0
$850$	−243.122	−0.286025
$851$	− 248.827i	− 0.292393i
$852$	0	0
$853$	−267.624	−0.313744	−0.156872	−	0.987619i	$-0.550141\pi$
$853$	−267.624	−0.313744	−0.156872	+	0.987619i	$0.550141\pi$
$854$	− 777.242i	− 0.910120i
$855$	0	0
$856$	204.176	0.238524
$857$	− 55.4140i	− 0.0646604i	−0.999477	−	0.0323302i	$-0.989707\pi$
$857$	− 55.4140i	− 0.0646604i	0.999477	−	0.0323302i	$-0.0102928\pi$
$858$	0	0
$859$	−148.674	−0.173079	−0.0865393	−	0.996248i	$-0.527581\pi$
$859$	−148.674	−0.173079	−0.0865393	+	0.996248i	$0.527581\pi$
$860$	− 143.963i	− 0.167399i
$861$	0	0
$862$	1047.80	1.21554
$863$	269.383i	0.312147i	0.987745	+	0.156073i	$0.0498836\pi$
$863$	269.383i	0.312147i	−0.987745	+	0.156073i	$0.950116\pi$
$864$	0	0
$865$	−416.276	−0.481244
$866$	− 245.827i	− 0.283865i
$867$	0	0
$868$	−17.6418	−0.0203247
$869$	− 1398.79i	− 1.60966i
$870$	0	0
$871$	−675.859	−0.775957
$872$	− 258.848i	− 0.296844i
$873$	0	0
$874$	394.992	0.451936
$875$	− 208.014i	− 0.237730i
$876$	0	0
$877$	−570.154	−0.650119	−0.325059	−	0.945694i	$-0.605384\pi$
$877$	−570.154	−0.650119	−0.325059	+	0.945694i	$0.605384\pi$
$878$	− 1925.59i	− 2.19316i
$879$	0	0
$880$	469.199	0.533180
$881$	964.674i	1.09498i	0.836814	+	0.547488i	$0.184416\pi$
$881$	964.674i	1.09498i	−0.836814	+	0.547488i	$0.815584\pi$
$882$	0	0
$883$	711.592	0.805880	0.402940	−	0.915226i	$-0.367988\pi$
$883$	711.592	0.805880	0.402940	+	0.915226i	$0.367988\pi$
$884$	182.167i	0.206072i
$885$	0	0
$886$	610.063	0.688558
$887$	− 1209.71i	− 1.36382i	−0.731438	−	0.681908i	$-0.761150\pi$
$887$	− 1209.71i	− 1.36382i	0.731438	−	0.681908i	$-0.238850\pi$
$888$	0	0
$889$	−123.638	−0.139075
$890$	3.47087i	0.00389985i
$891$	0	0
$892$	529.569	0.593687
$893$	− 230.777i	− 0.258429i
$894$	0	0
$895$	−540.982	−0.604449
$896$	278.636i	0.310978i
$897$	0	0
$898$	−133.749	−0.148941
$899$	− 97.8636i	− 0.108858i
$900$	0	0
$901$	−292.118	−0.324215
$902$	1357.10i	1.50455i
$903$	0	0
$904$	635.247	0.702707
$905$	− 352.574i	− 0.389584i
$906$	0	0
$907$	1432.61	1.57951	0.789753	−	0.613425i	$-0.210209\pi$
$907$	1432.61	1.57951	0.789753	+	0.613425i	$0.210209\pi$
$908$	428.867i	0.472321i
$909$	0	0
$910$	−184.265	−0.202489
$911$	806.179i	0.884939i	0.896784	+	0.442469i	$0.145897\pi$
$911$	806.179i	0.884939i	−0.896784	+	0.442469i	$0.854103\pi$
$912$	0	0
$913$	1749.19	1.91587
$914$	251.219i	0.274857i
$915$	0	0
$916$	−191.801	−0.209390
$917$	119.195i	0.129984i
$918$	0	0
$919$	−923.506	−1.00490	−0.502452	−	0.864605i	$-0.667568\pi$
$919$	−923.506	−1.00490	−0.502452	+	0.864605i	$0.667568\pi$
$920$	102.996i	0.111952i
$921$	0	0
$922$	−459.877	−0.498782
$923$	− 2000.29i	− 2.16716i
$924$	0	0
$925$	312.103	0.337409
$926$	− 395.479i	− 0.427083i
$927$	0	0
$928$	1418.06	1.52809
$929$	− 140.027i	− 0.150729i	−0.997156	−	0.0753646i	$-0.975988\pi$
$929$	− 140.027i	− 0.150729i	0.997156	−	0.0753646i	$-0.0240121\pi$
$930$	0	0
$931$	60.5203	0.0650056
$932$	− 65.9412i	− 0.0707523i
$933$	0	0
$934$	1275.30	1.36542
$935$	− 101.651i	− 0.108718i
$936$	0	0
$937$	−85.6888	−0.0914502	−0.0457251	−	0.998954i	$-0.514560\pi$
$937$	−85.6888	−0.0914502	−0.0457251	+	0.998954i	$0.514560\pi$
$938$	− 284.048i	− 0.302823i
$939$	0	0
$940$	−117.565	−0.125069
$941$	1476.53i	1.56910i	0.620063	+	0.784552i	$0.287107\pi$
$941$	1476.53i	1.56910i	−0.620063	+	0.784552i	$0.712893\pi$
$942$	0	0
$943$	−648.208	−0.687389
$944$	− 2037.23i	− 2.15809i
$945$	0	0
$946$	−1212.77	−1.28200
$947$	183.852i	0.194142i	0.995277	+	0.0970709i	$0.0309474\pi$
$947$	183.852i	0.194142i	−0.995277	+	0.0970709i	$0.969053\pi$
$948$	0	0
$949$	−1782.77	−1.87857
$950$	495.439i	0.521514i
$951$	0	0
$952$	39.1882	0.0411641
$953$	− 400.598i	− 0.420354i	−0.977663	−	0.210177i	$-0.932596\pi$
$953$	− 400.598i	− 0.420354i	0.977663	−	0.210177i	$-0.0674041\pi$
$954$	0	0
$955$	−203.473	−0.213061
$956$	1028.93i	1.07628i
$957$	0	0
$958$	−1174.07	−1.22554
$959$	− 9.81633i	− 0.0102360i
$960$	0	0
$961$	−954.648	−0.993391
$962$	− 587.409i	− 0.610612i
$963$	0	0
$964$	−770.723	−0.799505
$965$	165.426i	0.171426i
$966$	0	0
$967$	−1156.89	−1.19637	−0.598186	−	0.801357i	$-0.704112\pi$
$967$	−1156.89	−1.19637	−0.598186	+	0.801357i	$0.704112\pi$
$968$	− 300.756i	− 0.310698i
$969$	0	0
$970$	−448.627	−0.462503
$971$	− 543.385i	− 0.559614i	−0.960056	−	0.279807i	$-0.909730\pi$
$971$	− 543.385i	− 0.559614i	0.960056	−	0.279807i	$-0.0902705\pi$
$972$	0	0
$973$	202.288	0.207901
$974$	1787.16i	1.83487i
$975$	0	0
$976$	−2231.59	−2.28647
$977$	− 992.745i	− 1.01612i	−0.861323	−	0.508058i	$-0.830364\pi$
$977$	− 992.745i	− 1.01612i	0.861323	−	0.508058i	$-0.169636\pi$
$978$	0	0
$979$	11.6404	0.0118901
$980$	− 30.8308i	− 0.0314600i
$981$	0	0
$982$	−1302.20	−1.32607
$983$	− 474.189i	− 0.482389i	−0.970477	−	0.241195i	$-0.922461\pi$
$983$	− 474.189i	− 0.482389i	0.970477	−	0.241195i	$-0.0775392\pi$
$984$	0	0
$985$	456.672	0.463626
$986$	− 424.702i	− 0.430732i
$987$	0	0
$988$	371.225	0.375734
$989$	− 579.267i	− 0.585710i
$990$	0	0
$991$	973.071	0.981908	0.490954	−	0.871186i	$-0.336648\pi$
$991$	973.071	0.981908	0.490954	+	0.871186i	$0.336648\pi$
$992$	92.0375i	0.0927797i
$993$	0	0
$994$	840.674	0.845749
$995$	− 432.952i	− 0.435127i
$996$	0	0
$997$	943.814	0.946654	0.473327	−	0.880887i	$-0.343053\pi$
$997$	943.814	0.946654	0.473327	+	0.880887i	$0.343053\pi$
$998$	2147.43i	2.15173i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.3.b.b.134.4	yes	4
3.2	odd	2	inner	189.3.b.b.134.1	✓	4
4.3	odd	2		3024.3.d.c.1457.3		4
9.2	odd	6		567.3.r.b.134.1		8
9.4	even	3		567.3.r.b.512.1		8
9.5	odd	6		567.3.r.b.512.4		8
9.7	even	3		567.3.r.b.134.4		8
12.11	even	2		3024.3.d.c.1457.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.3.b.b.134.1	✓	4	3.2	odd	2	inner
189.3.b.b.134.4	yes	4	1.1	even	1	trivial
567.3.r.b.134.1		8	9.2	odd	6
567.3.r.b.134.4		8	9.7	even	3
567.3.r.b.512.1		8	9.4	even	3
567.3.r.b.512.4		8	9.5	odd	6
3024.3.d.c.1457.2		4	12.11	even	2
3024.3.d.c.1457.3		4	4.3	odd	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)

\(f(q)\)	\(=\)	\(q+2.57794i q^{2} -2.64575 q^{4} +1.66471i q^{5} -2.64575 q^{7} +3.49117i q^{8} -4.29150 q^{10} +14.3926i q^{11} -16.2288 q^{13} -6.82058i q^{14} -19.5830 q^{16} +4.24264i q^{17} +8.64575 q^{19} -4.40440i q^{20} -37.1033 q^{22} -17.7220i q^{23} +22.2288 q^{25} -41.8367i q^{26} +7.00000 q^{28} +38.8308i q^{29} -2.52026 q^{31} -36.5191i q^{32} -10.9373 q^{34} -4.40440i q^{35} +14.0405 q^{37} +22.2882i q^{38} -5.81176 q^{40} -36.5764i q^{41} +32.6863 q^{43} -38.0793i q^{44} +45.6863 q^{46} -26.6926i q^{47} +7.00000 q^{49} +57.3043i q^{50} +42.9373 q^{52} +68.8528i q^{53} -23.9595 q^{55} -9.23676i q^{56} -100.103 q^{58} +104.031i q^{59} +113.956 q^{61} -6.49707i q^{62} +15.8118 q^{64} -27.0161i q^{65} +41.6458 q^{67} -11.2250i q^{68} +11.3542 q^{70} +123.256i q^{71} +109.852 q^{73} +36.1956i q^{74} -22.8745 q^{76} -38.0793i q^{77} -97.1882 q^{79} -32.5999i q^{80} +94.2915 q^{82} -121.534i q^{83} -7.06275 q^{85} +84.2631i q^{86} -50.2470 q^{88} -0.808777i q^{89} +42.9373 q^{91} +46.8881i q^{92} +68.8118 q^{94} +14.3926i q^{95} +104.539 q^{97} +18.0455i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{10} - 12 q^{13} - 36 q^{16} + 24 q^{19} - 32 q^{22} + 36 q^{25} + 28 q^{28} + 64 q^{31} - 12 q^{34} - 92 q^{37} + 72 q^{40} - 28 q^{43} + 24 q^{46} + 28 q^{49} + 140 q^{52} - 244 q^{55} - 284 q^{58}+ \cdots + 16 q^{97}+O(q^{100}) \) 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

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Database

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