LMFDB - Embedded newform 189.3.b.a.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:	4.0.1166592.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 20x^{2} + 93 $ x^4 + 20*x^2 + 93
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			134.3
Root			$2.71187i$ of defining polynomial
Character	$\chi$	$=$	189.134
Dual form			189.3.b.a.134.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.71187i q^{2} -3.35425 q^{4} +7.17494i q^{5} +2.64575 q^{7} +1.75119i q^{8} +O(q^{10})$	\(q+2.71187i q^{2} -3.35425 q^{4} +7.17494i q^{5} +2.64575 q^{7} +1.75119i q^{8} -19.4575 q^{10} -2.71187i q^{11} +6.35425 q^{13} +7.17494i q^{14} -18.1660 q^{16} -6.38442i q^{17} -30.5203 q^{19} -24.0665i q^{20} +7.35425 q^{22} +27.9092i q^{23} -26.4797 q^{25} +17.2319i q^{26} -8.87451 q^{28} -37.0055i q^{29} +44.3948 q^{31} -42.2591i q^{32} +17.3137 q^{34} +18.9831i q^{35} +71.2065 q^{37} -82.7670i q^{38} -12.5647 q^{40} +28.8699i q^{41} -4.68627 q^{43} +9.09629i q^{44} -75.6863 q^{46} +65.7053i q^{47} +7.00000 q^{49} -71.8097i q^{50} -21.3137 q^{52} -36.0449i q^{53} +19.4575 q^{55} +4.63323i q^{56} +100.354 q^{58} +101.750i q^{59} +45.5425 q^{61} +120.393i q^{62} +41.9373 q^{64} +45.5913i q^{65} +93.8523 q^{67} +21.4149i q^{68} -51.4797 q^{70} -80.8457i q^{71} -60.2288 q^{73} +193.103i q^{74} +102.373 q^{76} -7.17494i q^{77} +20.6863 q^{79} -130.340i q^{80} -78.2915 q^{82} +94.7454i q^{83} +45.8078 q^{85} -12.7086i q^{86} +4.74902 q^{88} -52.1459i q^{89} +16.8118 q^{91} -93.6145i q^{92} -178.184 q^{94} -218.981i q^{95} +98.2065 q^{97} +18.9831i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 24 q^{4}+O(q^{10}) $ 4 * q - 24 * q^4	$ 4 q - 24 q^{4} + 28 q^{10} + 36 q^{13} + 12 q^{16} - 48 q^{19} + 40 q^{22} - 180 q^{25} + 28 q^{28} + 40 q^{31} + 228 q^{34} + 52 q^{37} - 336 q^{40} + 140 q^{43} - 144 q^{46} + 28 q^{49} - 244 q^{52} - 28 q^{55} + 412 q^{58} + 288 q^{61} + 136 q^{64} + 132 q^{67} - 280 q^{70} - 188 q^{73} + 92 q^{76} - 76 q^{79} - 292 q^{82} - 420 q^{85} - 108 q^{88} - 28 q^{91} - 300 q^{94} + 160 q^{97}+O(q^{100}) $ 4 * q - 24 * q^4 + 28 * q^10 + 36 * q^13 + 12 * q^16 - 48 * q^19 + 40 * q^22 - 180 * q^25 + 28 * q^28 + 40 * q^31 + 228 * q^34 + 52 * q^37 - 336 * q^40 + 140 * q^43 - 144 * q^46 + 28 * q^49 - 244 * q^52 - 28 * q^55 + 412 * q^58 + 288 * q^61 + 136 * q^64 + 132 * q^67 - 280 * q^70 - 188 * q^73 + 92 * q^76 - 76 * q^79 - 292 * q^82 - 420 * q^85 - 108 * q^88 - 28 * q^91 - 300 * q^94 + 160 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	2.71187i	1.35594i	0.735091	+	0.677968i	$0.237139\pi$
$2$	2.71187i	1.35594i	−0.735091	+	0.677968i	$0.762861\pi$
$3$	0	0
$3$	0	0
$4$	−3.35425	−0.838562
$5$	7.17494i	1.43499i	0.696565	+	0.717494i	$0.254711\pi$
$5$	7.17494i	1.43499i	−0.696565	+	0.717494i	$0.745289\pi$
$6$	0	0
$7$	2.64575	0.377964
$7$	2.64575	0.377964
$8$	1.75119i	0.218899i
$9$	0	0
$10$	−19.4575	−1.94575
$11$	− 2.71187i	− 0.246534i	−0.992374	−	0.123267i	$-0.960663\pi$
$11$	− 2.71187i	− 0.246534i	0.992374	−	0.123267i	$-0.0393371\pi$
$12$	0	0
$13$	6.35425	0.488788	0.244394	−	0.969676i	$-0.421411\pi$
$13$	6.35425	0.488788	0.244394	+	0.969676i	$0.421411\pi$
$14$	7.17494i	0.512496i
$15$	0	0
$16$	−18.1660	−1.13538
$17$	− 6.38442i	− 0.375554i	−0.982212	−	0.187777i	$-0.939872\pi$
$17$	− 6.38442i	− 0.375554i	0.982212	−	0.187777i	$-0.0601283\pi$
$18$	0	0
$19$	−30.5203	−1.60633	−0.803165	−	0.595757i	$-0.796852\pi$
$19$	−30.5203	−1.60633	−0.803165	+	0.595757i	$0.796852\pi$
$20$	− 24.0665i	− 1.20333i
$21$	0	0
$22$	7.35425	0.334284
$23$	27.9092i	1.21345i	0.794914	+	0.606723i	$0.207516\pi$
$23$	27.9092i	1.21345i	−0.794914	+	0.606723i	$0.792484\pi$
$24$	0	0
$25$	−26.4797	−1.05919
$26$	17.2319i	0.662766i
$27$	0	0
$28$	−8.87451	−0.316947
$29$	− 37.0055i	− 1.27605i	−0.770015	−	0.638026i	$-0.779751\pi$
$29$	− 37.0055i	− 1.27605i	0.770015	−	0.638026i	$-0.220249\pi$
$30$	0	0
$31$	44.3948	1.43209	0.716045	−	0.698055i	$-0.245951\pi$
$31$	44.3948	1.43209	0.716045	+	0.698055i	$0.245951\pi$
$32$	− 42.2591i	− 1.32060i
$33$	0	0
$34$	17.3137	0.509227
$35$	18.9831i	0.542374i
$36$	0	0
$37$	71.2065	1.92450	0.962250	−	0.272166i	$-0.0877399\pi$
$37$	71.2065	1.92450	0.962250	+	0.272166i	$0.0877399\pi$
$38$	− 82.7670i	− 2.17808i
$39$	0	0
$40$	−12.5647	−0.314118
$41$	28.8699i	0.704144i	0.935973	+	0.352072i	$0.114523\pi$
$41$	28.8699i	0.704144i	−0.935973	+	0.352072i	$0.885477\pi$
$42$	0	0
$43$	−4.68627	−0.108983	−0.0544915	−	0.998514i	$-0.517354\pi$
$43$	−4.68627	−0.108983	−0.0544915	+	0.998514i	$0.517354\pi$
$44$	9.09629i	0.206734i
$45$	0	0
$46$	−75.6863	−1.64535
$47$	65.7053i	1.39798i	0.715129	+	0.698992i	$0.246368\pi$
$47$	65.7053i	1.39798i	−0.715129	+	0.698992i	$0.753632\pi$
$48$	0	0
$49$	7.00000	0.142857
$50$	− 71.8097i	− 1.43619i
$51$	0	0
$52$	−21.3137	−0.409879
$53$	− 36.0449i	− 0.680092i	−0.940409	−	0.340046i	$-0.889557\pi$
$53$	− 36.0449i	− 0.680092i	0.940409	−	0.340046i	$-0.110443\pi$
$54$	0	0
$55$	19.4575	0.353773
$56$	4.63323i	0.0827362i
$57$	0	0
$58$	100.354	1.73025
$59$	101.750i	1.72458i	0.506416	+	0.862289i	$0.330970\pi$
$59$	101.750i	1.72458i	−0.506416	+	0.862289i	$0.669030\pi$
$60$	0	0
$61$	45.5425	0.746598	0.373299	−	0.927711i	$-0.378227\pi$
$61$	45.5425	0.746598	0.373299	+	0.927711i	$0.378227\pi$
$62$	120.393i	1.94182i
$63$	0	0
$64$	41.9373	0.655270
$65$	45.5913i	0.701405i
$66$	0	0
$67$	93.8523	1.40078	0.700390	−	0.713760i	$-0.253009\pi$
$67$	93.8523	1.40078	0.700390	+	0.713760i	$0.253009\pi$
$68$	21.4149i	0.314926i
$69$	0	0
$70$	−51.4797	−0.735425
$71$	− 80.8457i	− 1.13867i	−0.822105	−	0.569336i	$-0.807200\pi$
$71$	− 80.8457i	− 1.13867i	0.822105	−	0.569336i	$-0.192800\pi$
$72$	0	0
$73$	−60.2288	−0.825051	−0.412526	−	0.910946i	$-0.635353\pi$
$73$	−60.2288	−0.825051	−0.412526	+	0.910946i	$0.635353\pi$
$74$	193.103i	2.60950i
$75$	0	0
$76$	102.373	1.34701
$77$	− 7.17494i	− 0.0931810i
$78$	0	0
$79$	20.6863	0.261852	0.130926	−	0.991392i	$-0.458205\pi$
$79$	20.6863	0.261852	0.130926	+	0.991392i	$0.458205\pi$
$80$	− 130.340i	− 1.62925i
$81$	0	0
$82$	−78.2915	−0.954774
$83$	94.7454i	1.14151i	0.821120	+	0.570755i	$0.193349\pi$
$83$	94.7454i	1.14151i	−0.821120	+	0.570755i	$0.806651\pi$
$84$	0	0
$85$	45.8078	0.538916
$86$	− 12.7086i	− 0.147774i
$87$	0	0
$88$	4.74902	0.0539661
$89$	− 52.1459i	− 0.585909i	−0.956126	−	0.292955i	$-0.905362\pi$
$89$	− 52.1459i	− 0.585909i	0.956126	−	0.292955i	$-0.0946385\pi$
$90$	0	0
$91$	16.8118	0.184745
$92$	− 93.6145i	− 1.01755i
$93$	0	0
$94$	−178.184	−1.89558
$95$	− 218.981i	− 2.30506i
$96$	0	0
$97$	98.2065	1.01244	0.506219	−	0.862405i	$-0.331043\pi$
$97$	98.2065	1.01244	0.506219	+	0.862405i	$0.331043\pi$
$98$	18.9831i	0.193705i
$99$	0	0
$100$	88.8196	0.888196
$101$	− 21.0746i	− 0.208660i	−0.994543	−	0.104330i	$-0.966730\pi$
$101$	− 21.0746i	− 0.208660i	0.994543	−	0.104330i	$-0.0332697\pi$
$102$	0	0
$103$	−117.435	−1.14015	−0.570074	−	0.821593i	$-0.693086\pi$
$103$	−117.435	−1.14015	−0.570074	+	0.821593i	$0.693086\pi$
$104$	11.1275i	0.106995i
$105$	0	0
$106$	97.7490	0.922161
$107$	− 146.381i	− 1.36804i	−0.729461	−	0.684022i	$-0.760229\pi$
$107$	− 146.381i	− 1.36804i	0.729461	−	0.684022i	$-0.239771\pi$
$108$	0	0
$109$	52.8967	0.485291	0.242646	−	0.970115i	$-0.421985\pi$
$109$	52.8967	0.485291	0.242646	+	0.970115i	$0.421985\pi$
$110$	52.7663i	0.479693i
$111$	0	0
$112$	−48.0627	−0.429132
$113$	− 169.036i	− 1.49590i	−0.663756	−	0.747949i	$-0.731039\pi$
$113$	− 169.036i	− 1.49590i	0.663756	−	0.747949i	$-0.268961\pi$
$114$	0	0
$115$	−200.247	−1.74128
$116$	124.126i	1.07005i
$117$	0	0
$118$	−275.933	−2.33842
$119$	− 16.8916i	− 0.141946i
$120$	0	0
$121$	113.646	0.939221
$122$	123.505i	1.01234i
$123$	0	0
$124$	−148.911	−1.20090
$125$	− 10.6170i	− 0.0849364i
$126$	0	0
$127$	−105.727	−0.832494	−0.416247	−	0.909251i	$-0.636655\pi$
$127$	−105.727	−0.832494	−0.416247	+	0.909251i	$0.636655\pi$
$128$	− 55.3080i	− 0.432094i
$129$	0	0
$130$	−123.638	−0.951061
$131$	− 6.72474i	− 0.0513339i	−0.999671	−	0.0256669i	$-0.991829\pi$
$131$	− 6.72474i	− 0.0513339i	0.999671	−	0.0256669i	$-0.00817094\pi$
$132$	0	0
$133$	−80.7490	−0.607135
$134$	254.515i	1.89937i
$135$	0	0
$136$	11.1804	0.0822086
$137$	− 71.4693i	− 0.521674i	−0.965383	−	0.260837i	$-0.916001\pi$
$137$	− 71.4693i	− 0.521674i	0.965383	−	0.260837i	$-0.0839985\pi$
$138$	0	0
$139$	−22.7895	−0.163953	−0.0819767	−	0.996634i	$-0.526123\pi$
$139$	−22.7895	−0.163953	−0.0819767	+	0.996634i	$0.526123\pi$
$140$	− 63.6740i	− 0.454815i
$141$	0	0
$142$	219.243	1.54397
$143$	− 17.2319i	− 0.120503i
$144$	0	0
$145$	265.512	1.83112
$146$	− 163.333i	− 1.11872i
$147$	0	0
$148$	−238.844	−1.61381
$149$	197.736i	1.32709i	0.748137	+	0.663544i	$0.230949\pi$
$149$	197.736i	1.32709i	−0.748137	+	0.663544i	$0.769051\pi$
$150$	0	0
$151$	−204.369	−1.35343	−0.676717	−	0.736243i	$-0.736598\pi$
$151$	−204.369	−1.35343	−0.676717	+	0.736243i	$0.736598\pi$
$152$	− 53.4469i	− 0.351624i
$153$	0	0
$154$	19.4575	0.126347
$155$	318.530i	2.05503i
$156$	0	0
$157$	89.6013	0.570709	0.285354	−	0.958422i	$-0.407889\pi$
$157$	89.6013	0.570709	0.285354	+	0.958422i	$0.407889\pi$
$158$	56.0985i	0.355054i
$159$	0	0
$160$	303.207	1.89504
$161$	73.8409i	0.458639i
$162$	0	0
$163$	120.354	0.738370	0.369185	−	0.929356i	$-0.379637\pi$
$163$	120.354	0.738370	0.369185	+	0.929356i	$0.379637\pi$
$164$	− 96.8369i	− 0.590469i
$165$	0	0
$166$	−256.937	−1.54781
$167$	− 196.836i	− 1.17866i	−0.807893	−	0.589329i	$-0.799392\pi$
$167$	− 196.836i	− 1.17866i	0.807893	−	0.589329i	$-0.200608\pi$
$168$	0	0
$169$	−128.624	−0.761086
$170$	124.225i	0.730735i
$171$	0	0
$172$	15.7189	0.0913890
$173$	− 250.843i	− 1.44996i	−0.688771	−	0.724979i	$-0.741849\pi$
$173$	− 250.843i	− 1.44996i	0.688771	−	0.724979i	$-0.258151\pi$
$174$	0	0
$175$	−70.0588	−0.400336
$176$	49.2639i	0.279908i
$177$	0	0
$178$	141.413	0.794455
$179$	17.2319i	0.0962676i	0.998841	+	0.0481338i	$0.0153274\pi$
$179$	17.2319i	0.0962676i	−0.998841	+	0.0481338i	$0.984673\pi$
$180$	0	0
$181$	58.5385	0.323417	0.161709	−	0.986839i	$-0.448300\pi$
$181$	58.5385	0.323417	0.161709	+	0.986839i	$0.448300\pi$
$182$	45.5913i	0.250502i
$183$	0	0
$184$	−48.8745	−0.265622
$185$	510.902i	2.76163i
$186$	0	0
$187$	−17.3137	−0.0925868
$188$	− 220.392i	− 1.17230i
$189$	0	0
$190$	593.848	3.12552
$191$	− 125.877i	− 0.659042i	−0.944148	−	0.329521i	$-0.893113\pi$
$191$	− 125.877i	− 0.659042i	0.944148	−	0.329521i	$-0.106887\pi$
$192$	0	0
$193$	−180.125	−0.933293	−0.466646	−	0.884444i	$-0.654538\pi$
$193$	−180.125	−0.933293	−0.466646	+	0.884444i	$0.654538\pi$
$194$	266.324i	1.37280i
$195$	0	0
$196$	−23.4797	−0.119795
$197$	344.858i	1.75055i	0.483628	+	0.875274i	$0.339319\pi$
$197$	344.858i	1.75055i	−0.483628	+	0.875274i	$0.660681\pi$
$198$	0	0
$199$	−148.911	−0.748297	−0.374148	−	0.927369i	$-0.622065\pi$
$199$	−148.911	−0.748297	−0.374148	+	0.927369i	$0.622065\pi$
$200$	− 46.3712i	− 0.231856i
$201$	0	0
$202$	57.1517	0.282929
$203$	− 97.9074i	− 0.482303i
$204$	0	0
$205$	−207.140	−1.01044
$206$	− 318.469i	− 1.54597i
$207$	0	0
$208$	−115.431	−0.554958
$209$	82.7670i	0.396014i
$210$	0	0
$211$	83.9111	0.397683	0.198841	−	0.980032i	$-0.436282\pi$
$211$	83.9111	0.397683	0.198841	+	0.980032i	$0.436282\pi$
$212$	120.903i	0.570299i
$213$	0	0
$214$	396.966	1.85498
$215$	− 33.6237i	− 0.156389i
$216$	0	0
$217$	117.458	0.541279
$218$	143.449i	0.658024i
$219$	0	0
$220$	−65.2653	−0.296661
$221$	− 40.5682i	− 0.183566i
$222$	0	0
$223$	167.170	0.749641	0.374821	−	0.927097i	$-0.377704\pi$
$223$	167.170	0.749641	0.374821	+	0.927097i	$0.377704\pi$
$224$	− 111.807i	− 0.499139i
$225$	0	0
$226$	458.405	2.02834
$227$	− 94.3448i	− 0.415616i	−0.978170	−	0.207808i	$-0.933367\pi$
$227$	− 94.3448i	− 0.415616i	0.978170	−	0.207808i	$-0.0666329\pi$
$228$	0	0
$229$	114.251	0.498913	0.249456	−	0.968386i	$-0.419748\pi$
$229$	114.251	0.498913	0.249456	+	0.968386i	$0.419748\pi$
$230$	− 543.044i	− 2.36106i
$231$	0	0
$232$	64.8039	0.279327
$233$	− 341.916i	− 1.46745i	−0.679447	−	0.733724i	$-0.737780\pi$
$233$	− 341.916i	− 1.46745i	0.679447	−	0.733724i	$-0.262220\pi$
$234$	0	0
$235$	−471.431	−2.00609
$236$	− 341.295i	− 1.44617i
$237$	0	0
$238$	45.8078	0.192470
$239$	206.602i	0.864444i	0.901767	+	0.432222i	$0.142270\pi$
$239$	206.602i	0.864444i	−0.901767	+	0.432222i	$0.857730\pi$
$240$	0	0
$241$	93.8157	0.389277	0.194638	−	0.980875i	$-0.437647\pi$
$241$	93.8157	0.389277	0.194638	+	0.980875i	$0.437647\pi$
$242$	308.193i	1.27352i
$243$	0	0
$244$	−152.761	−0.626069
$245$	50.2246i	0.204998i
$246$	0	0
$247$	−193.933	−0.785155
$248$	77.7439i	0.313483i
$249$	0	0
$250$	28.7921	0.115168
$251$	127.568i	0.508238i	0.967173	+	0.254119i	$0.0817856\pi$
$251$	127.568i	0.508238i	−0.967173	+	0.254119i	$0.918214\pi$
$252$	0	0
$253$	75.6863	0.299155
$254$	− 286.717i	− 1.12881i
$255$	0	0
$256$	317.737	1.24116
$257$	− 391.520i	− 1.52342i	−0.647916	−	0.761712i	$-0.724359\pi$
$257$	− 391.520i	− 1.52342i	0.647916	−	0.761712i	$-0.275641\pi$
$258$	0	0
$259$	188.395	0.727393
$260$	− 152.925i	− 0.588172i
$261$	0	0
$262$	18.2366	0.0696055
$263$	− 430.507i	− 1.63691i	−0.574572	−	0.818454i	$-0.694831\pi$
$263$	− 430.507i	− 1.63691i	0.574572	−	0.818454i	$-0.305169\pi$
$264$	0	0
$265$	258.620	0.975923
$266$	− 218.981i	− 0.823237i
$267$	0	0
$268$	−314.804	−1.17464
$269$	128.018i	0.475904i	0.971277	+	0.237952i	$0.0764760\pi$
$269$	128.018i	0.475904i	−0.971277	+	0.237952i	$0.923524\pi$
$270$	0	0
$271$	−32.7006	−0.120667	−0.0603333	−	0.998178i	$-0.519216\pi$
$271$	−32.7006	−0.120667	−0.0603333	+	0.998178i	$0.519216\pi$
$272$	115.979i	0.426395i
$273$	0	0
$274$	193.816	0.707357
$275$	71.8097i	0.261126i
$276$	0	0
$277$	361.162	1.30383	0.651917	−	0.758290i	$-0.273965\pi$
$277$	361.162	1.30383	0.651917	+	0.758290i	$0.273965\pi$
$278$	− 61.8023i	− 0.222310i
$279$	0	0
$280$	−33.2431	−0.118725
$281$	220.732i	0.785524i	0.919640	+	0.392762i	$0.128480\pi$
$281$	220.732i	0.785524i	−0.919640	+	0.392762i	$0.871520\pi$
$282$	0	0
$283$	−168.332	−0.594813	−0.297406	−	0.954751i	$-0.596122\pi$
$283$	−168.332	−0.594813	−0.297406	+	0.954751i	$0.596122\pi$
$284$	271.177i	0.954847i
$285$	0	0
$286$	46.7307	0.163394
$287$	76.3826i	0.266141i
$288$	0	0
$289$	248.239	0.858959
$290$	720.036i	2.48288i
$291$	0	0
$292$	202.022	0.691857
$293$	242.767i	0.828558i	0.910150	+	0.414279i	$0.135966\pi$
$293$	242.767i	0.828558i	−0.910150	+	0.414279i	$0.864034\pi$
$294$	0	0
$295$	−730.051	−2.47475
$296$	124.697i	0.421272i
$297$	0	0
$298$	−536.235	−1.79945
$299$	177.342i	0.593118i
$300$	0	0
$301$	−12.3987	−0.0411917
$302$	− 554.221i	− 1.83517i
$303$	0	0
$304$	554.431	1.82379
$305$	326.765i	1.07136i
$306$	0	0
$307$	−363.508	−1.18407	−0.592033	−	0.805913i	$-0.701675\pi$
$307$	−363.508	−1.18407	−0.592033	+	0.805913i	$0.701675\pi$
$308$	24.0665i	0.0781381i
$309$	0	0
$310$	−863.812	−2.78649
$311$	− 140.276i	− 0.451050i	−0.974237	−	0.225525i	$-0.927590\pi$
$311$	− 140.276i	− 0.451050i	0.974237	−	0.225525i	$-0.0724097\pi$
$312$	0	0
$313$	−488.804	−1.56167	−0.780837	−	0.624735i	$-0.785207\pi$
$313$	−488.804	−1.56167	−0.780837	+	0.624735i	$0.785207\pi$
$314$	242.987i	0.773845i
$315$	0	0
$316$	−69.3869	−0.219579
$317$	− 49.2143i	− 0.155250i	−0.996983	−	0.0776251i	$-0.975266\pi$
$317$	− 49.2143i	− 0.155250i	0.996983	−	0.0776251i	$-0.0247337\pi$
$318$	0	0
$319$	−100.354	−0.314590
$320$	300.897i	0.940304i
$321$	0	0
$322$	−200.247	−0.621885
$323$	194.854i	0.603264i
$324$	0	0
$325$	−168.259	−0.517720
$326$	326.385i	1.00118i
$327$	0	0
$328$	−50.5568	−0.154137
$329$	173.840i	0.528389i
$330$	0	0
$331$	−38.4941	−0.116296	−0.0581482	−	0.998308i	$-0.518520\pi$
$331$	−38.4941	−0.116296	−0.0581482	+	0.998308i	$0.518520\pi$
$332$	− 317.799i	− 0.957227i
$333$	0	0
$334$	533.793	1.59818
$335$	673.384i	2.01010i
$336$	0	0
$337$	−525.199	−1.55845	−0.779226	−	0.626742i	$-0.784388\pi$
$337$	−525.199	−1.55845	−0.779226	+	0.626742i	$0.784388\pi$
$338$	− 348.811i	− 1.03198i
$339$	0	0
$340$	−153.651	−0.451914
$341$	− 120.393i	− 0.353058i
$342$	0	0
$343$	18.5203	0.0539949
$344$	− 8.20657i	− 0.0238563i
$345$	0	0
$346$	680.254	1.96605
$347$	58.9309i	0.169830i	0.996388	+	0.0849149i	$0.0270618\pi$
$347$	58.9309i	0.169830i	−0.996388	+	0.0849149i	$0.972938\pi$
$348$	0	0
$349$	−385.705	−1.10517	−0.552585	−	0.833456i	$-0.686359\pi$
$349$	−385.705	−1.10517	−0.552585	+	0.833456i	$0.686359\pi$
$350$	− 189.991i	− 0.542830i
$351$	0	0
$352$	−114.601	−0.325572
$353$	− 269.656i	− 0.763897i	−0.924184	−	0.381949i	$-0.875253\pi$
$353$	− 269.656i	− 0.763897i	0.924184	−	0.381949i	$-0.124747\pi$
$354$	0	0
$355$	580.063	1.63398
$356$	174.910i	0.491321i
$357$	0	0
$358$	−46.7307	−0.130533
$359$	− 228.297i	− 0.635925i	−0.948103	−	0.317962i	$-0.897001\pi$
$359$	− 228.297i	− 0.635925i	0.948103	−	0.317962i	$-0.102999\pi$
$360$	0	0
$361$	570.486	1.58029
$362$	158.749i	0.438533i
$363$	0	0
$364$	−56.3908	−0.154920
$365$	− 432.138i	− 1.18394i
$366$	0	0
$367$	−67.9229	−0.185076	−0.0925380	−	0.995709i	$-0.529498\pi$
$367$	−67.9229	−0.185076	−0.0925380	+	0.995709i	$0.529498\pi$
$368$	− 506.999i	− 1.37772i
$369$	0	0
$370$	−1385.50	−3.74460
$371$	− 95.3657i	− 0.257050i
$372$	0	0
$373$	506.055	1.35672	0.678358	−	0.734732i	$-0.262692\pi$
$373$	506.055	1.35672	0.678358	+	0.734732i	$0.262692\pi$
$374$	− 46.9526i	− 0.125542i
$375$	0	0
$376$	−115.063	−0.306018
$377$	− 235.142i	− 0.623720i
$378$	0	0
$379$	50.5163	0.133288	0.0666442	−	0.997777i	$-0.478771\pi$
$379$	50.5163	0.133288	0.0666442	+	0.997777i	$0.478771\pi$
$380$	734.517i	1.93294i
$381$	0	0
$382$	341.362	0.893618
$383$	14.0096i	0.0365785i	0.999833	+	0.0182892i	$0.00582197\pi$
$383$	14.0096i	0.0365785i	−0.999833	+	0.0182892i	$0.994178\pi$
$384$	0	0
$385$	51.4797	0.133714
$386$	− 488.477i	− 1.26549i
$387$	0	0
$388$	−329.409	−0.848993
$389$	− 198.757i	− 0.510944i	−0.966816	−	0.255472i	$-0.917769\pi$
$389$	− 198.757i	− 0.510944i	0.966816	−	0.255472i	$-0.0822308\pi$
$390$	0	0
$391$	178.184	0.455714
$392$	12.2584i	0.0312713i
$393$	0	0
$394$	−935.210	−2.37363
$395$	148.423i	0.375754i
$396$	0	0
$397$	313.587	0.789892	0.394946	−	0.918704i	$-0.370763\pi$
$397$	313.587	0.789892	0.394946	+	0.918704i	$0.370763\pi$
$398$	− 403.828i	− 1.01464i
$399$	0	0
$400$	481.031	1.20258
$401$	81.9765i	0.204430i	0.994762	+	0.102215i	$0.0325930\pi$
$401$	81.9765i	0.204430i	−0.994762	+	0.102215i	$0.967407\pi$
$402$	0	0
$403$	282.095	0.699989
$404$	70.6895i	0.174974i
$405$	0	0
$406$	265.512	0.653971
$407$	− 193.103i	− 0.474454i
$408$	0	0
$409$	76.8118	0.187804	0.0939019	−	0.995581i	$-0.470066\pi$
$409$	76.8118	0.187804	0.0939019	+	0.995581i	$0.470066\pi$
$410$	− 561.737i	− 1.37009i
$411$	0	0
$412$	393.907	0.956085
$413$	269.206i	0.651829i
$414$	0	0
$415$	−679.792	−1.63805
$416$	− 268.525i	− 0.645493i
$417$	0	0
$418$	−224.454	−0.536970
$419$	88.1412i	0.210361i	0.994453	+	0.105180i	$0.0335420\pi$
$419$	88.1412i	0.210361i	−0.994453	+	0.105180i	$0.966458\pi$
$420$	0	0
$421$	−26.8824	−0.0638536	−0.0319268	−	0.999490i	$-0.510164\pi$
$421$	−26.8824	−0.0638536	−0.0319268	+	0.999490i	$0.510164\pi$
$422$	227.556i	0.539233i
$423$	0	0
$424$	63.1216	0.148872
$425$	169.058i	0.397783i
$426$	0	0
$427$	120.494	0.282188
$428$	490.998i	1.14719i
$429$	0	0
$430$	91.1832	0.212054
$431$	− 124.516i	− 0.288899i	−0.989512	−	0.144450i	$-0.953859\pi$
$431$	− 124.516i	− 0.288899i	0.989512	−	0.144450i	$-0.0461412\pi$
$432$	0	0
$433$	−2.01435	−0.00465209	−0.00232604	−	0.999997i	$-0.500740\pi$
$433$	−2.01435	−0.00465209	−0.00232604	+	0.999997i	$0.500740\pi$
$434$	318.530i	0.733939i
$435$	0	0
$436$	−177.429	−0.406947
$437$	− 851.797i	− 1.94919i
$438$	0	0
$439$	97.2105	0.221436	0.110718	−	0.993852i	$-0.464685\pi$
$439$	97.2105	0.221436	0.110718	+	0.993852i	$0.464685\pi$
$440$	34.0739i	0.0774407i
$441$	0	0
$442$	110.016	0.248904
$443$	− 237.174i	− 0.535381i	−0.963505	−	0.267690i	$-0.913740\pi$
$443$	− 237.174i	− 0.535381i	0.963505	−	0.267690i	$-0.0862604\pi$
$444$	0	0
$445$	374.144	0.840773
$446$	453.343i	1.01647i
$447$	0	0
$448$	110.956	0.247669
$449$	− 801.012i	− 1.78399i	−0.452043	−	0.891996i	$-0.649305\pi$
$449$	− 801.012i	− 1.78399i	0.452043	−	0.891996i	$-0.350695\pi$
$450$	0	0
$451$	78.2915	0.173595
$452$	566.990i	1.25440i
$453$	0	0
$454$	255.851	0.563548
$455$	120.623i	0.265106i
$456$	0	0
$457$	320.535	0.701389	0.350694	−	0.936490i	$-0.385946\pi$
$457$	320.535	0.701389	0.350694	+	0.936490i	$0.385946\pi$
$458$	309.834i	0.676493i
$459$	0	0
$460$	671.678	1.46017
$461$	− 589.256i	− 1.27821i	−0.769118	−	0.639106i	$-0.779304\pi$
$461$	− 589.256i	− 1.27821i	0.769118	−	0.639106i	$-0.220696\pi$
$462$	0	0
$463$	−713.069	−1.54011	−0.770053	−	0.637980i	$-0.779770\pi$
$463$	−713.069	−1.54011	−0.770053	+	0.637980i	$0.779770\pi$
$464$	672.243i	1.44880i
$465$	0	0
$466$	927.231	1.98977
$467$	636.879i	1.36377i	0.731461	+	0.681883i	$0.238839\pi$
$467$	636.879i	1.36377i	−0.731461	+	0.681883i	$0.761161\pi$
$468$	0	0
$469$	248.310	0.529445
$470$	− 1278.46i	− 2.72013i
$471$	0	0
$472$	−178.184	−0.377509
$473$	12.7086i	0.0268680i
$474$	0	0
$475$	808.169	1.70141
$476$	56.6586i	0.119031i
$477$	0	0
$478$	−560.278	−1.17213
$479$	− 686.993i	− 1.43422i	−0.696958	−	0.717112i	$-0.745464\pi$
$479$	− 686.993i	− 1.43422i	0.696958	−	0.717112i	$-0.254536\pi$
$480$	0	0
$481$	452.464	0.940674
$482$	254.416i	0.527834i
$483$	0	0
$484$	−381.196	−0.787595
$485$	704.626i	1.45284i
$486$	0	0
$487$	−316.516	−0.649931	−0.324965	−	0.945726i	$-0.605353\pi$
$487$	−316.516	−0.649931	−0.324965	+	0.945726i	$0.605353\pi$
$488$	79.7538i	0.163430i
$489$	0	0
$490$	−136.203	−0.277964
$491$	350.211i	0.713260i	0.934246	+	0.356630i	$0.116074\pi$
$491$	350.211i	0.713260i	−0.934246	+	0.356630i	$0.883926\pi$
$492$	0	0
$493$	−236.259	−0.479227
$494$	− 525.922i	− 1.06462i
$495$	0	0
$496$	−806.476	−1.62596
$497$	− 213.898i	− 0.430377i
$498$	0	0
$499$	−37.2850	−0.0747195	−0.0373597	−	0.999302i	$-0.511895\pi$
$499$	−37.2850	−0.0747195	−0.0373597	+	0.999302i	$0.511895\pi$
$500$	35.6122i	0.0712244i
$501$	0	0
$502$	−345.948	−0.689139
$503$	484.234i	0.962692i	0.876531	+	0.481346i	$0.159852\pi$
$503$	484.234i	0.962692i	−0.876531	+	0.481346i	$0.840148\pi$
$504$	0	0
$505$	151.209	0.299424
$506$	205.251i	0.405635i
$507$	0	0
$508$	354.634	0.698098
$509$	− 561.528i	− 1.10320i	−0.834110	−	0.551599i	$-0.814018\pi$
$509$	− 561.528i	− 1.10320i	0.834110	−	0.551599i	$-0.185982\pi$
$510$	0	0
$511$	−159.350	−0.311840
$512$	640.431i	1.25084i
$513$	0	0
$514$	1061.75	2.06566
$515$	− 842.591i	− 1.63610i
$516$	0	0
$517$	178.184	0.344650
$518$	510.902i	0.986298i
$519$	0	0
$520$	−79.8393	−0.153537
$521$	516.486i	0.991335i	0.868512	+	0.495668i	$0.165077\pi$
$521$	516.486i	0.991335i	−0.868512	+	0.495668i	$0.834923\pi$
$522$	0	0
$523$	−31.0968	−0.0594585	−0.0297292	−	0.999558i	$-0.509465\pi$
$523$	−31.0968	−0.0594585	−0.0297292	+	0.999558i	$0.509465\pi$
$524$	22.5564i	0.0430467i
$525$	0	0
$526$	1167.48	2.21954
$527$	− 283.435i	− 0.537827i
$528$	0	0
$529$	−249.925	−0.472449
$530$	701.343i	1.32329i
$531$	0	0
$532$	270.852	0.509121
$533$	183.447i	0.344177i
$534$	0	0
$535$	1050.27	1.96313
$536$	164.354i	0.306630i
$537$	0	0
$538$	−347.169	−0.645295
$539$	− 18.9831i	− 0.0352191i
$540$	0	0
$541$	115.369	0.213251	0.106625	−	0.994299i	$-0.465995\pi$
$541$	115.369	0.213251	0.106625	+	0.994299i	$0.465995\pi$
$542$	− 88.6799i	− 0.163616i
$543$	0	0
$544$	−269.800	−0.495956
$545$	379.531i	0.696387i
$546$	0	0
$547$	505.150	0.923492	0.461746	−	0.887012i	$-0.347223\pi$
$547$	505.150	0.923492	0.461746	+	0.887012i	$0.347223\pi$
$548$	239.726i	0.437456i
$549$	0	0
$550$	−194.739	−0.354070
$551$	1129.42i	2.04976i
$552$	0	0
$553$	54.7307	0.0989706
$554$	979.425i	1.76792i
$555$	0	0
$556$	76.4418	0.137485
$557$	191.192i	0.343254i	0.985162	+	0.171627i	$0.0549023\pi$
$557$	191.192i	0.343254i	−0.985162	+	0.171627i	$0.945098\pi$
$558$	0	0
$559$	−29.7777	−0.0532696
$560$	− 344.847i	− 0.615799i
$561$	0	0
$562$	−598.597	−1.06512
$563$	22.5351i	0.0400268i	0.999800	+	0.0200134i	$0.00637089\pi$
$563$	22.5351i	0.0400268i	−0.999800	+	0.0200134i	$0.993629\pi$
$564$	0	0
$565$	1212.83	2.14659
$566$	− 456.495i	− 0.806528i
$567$	0	0
$568$	141.577	0.249254
$569$	257.518i	0.452580i	0.974060	+	0.226290i	$0.0726597\pi$
$569$	257.518i	0.452580i	−0.974060	+	0.226290i	$0.927340\pi$
$570$	0	0
$571$	−939.247	−1.64492	−0.822458	−	0.568826i	$-0.807398\pi$
$571$	−939.247	−1.64492	−0.822458	+	0.568826i	$0.807398\pi$
$572$	57.8001i	0.101049i
$573$	0	0
$574$	−207.140	−0.360871
$575$	− 739.029i	− 1.28527i
$576$	0	0
$577$	130.029	0.225353	0.112677	−	0.993632i	$-0.464058\pi$
$577$	130.029	0.225353	0.112677	+	0.993632i	$0.464058\pi$
$578$	673.193i	1.16469i
$579$	0	0
$580$	−890.595	−1.53551
$581$	250.673i	0.431450i
$582$	0	0
$583$	−97.7490	−0.167666
$584$	− 105.472i	− 0.180603i
$585$	0	0
$586$	−658.354	−1.12347
$587$	− 500.225i	− 0.852172i	−0.904683	−	0.426086i	$-0.859892\pi$
$587$	− 500.225i	− 0.852172i	0.904683	−	0.426086i	$-0.140108\pi$
$588$	0	0
$589$	−1354.94	−2.30041
$590$	− 1979.80i	− 3.35560i
$591$	0	0
$592$	−1293.54	−2.18503
$593$	− 944.281i	− 1.59238i	−0.605047	−	0.796190i	$-0.706846\pi$
$593$	− 944.281i	− 1.59238i	0.605047	−	0.796190i	$-0.293154\pi$
$594$	0	0
$595$	121.196	0.203691
$596$	− 663.256i	− 1.11285i
$597$	0	0
$598$	−480.929	−0.804230
$599$	655.482i	1.09429i	0.837036	+	0.547147i	$0.184286\pi$
$599$	655.482i	1.09429i	−0.837036	+	0.547147i	$0.815714\pi$
$600$	0	0
$601$	265.654	0.442019	0.221010	−	0.975272i	$-0.429065\pi$
$601$	265.654	0.442019	0.221010	+	0.975272i	$0.429065\pi$
$602$	− 33.6237i	− 0.0558533i
$603$	0	0
$604$	685.503	1.13494
$605$	815.401i	1.34777i
$606$	0	0
$607$	−337.012	−0.555209	−0.277604	−	0.960695i	$-0.589540\pi$
$607$	−337.012	−0.555209	−0.277604	+	0.960695i	$0.589540\pi$
$608$	1289.76i	2.12131i
$609$	0	0
$610$	−886.144	−1.45269
$611$	417.508i	0.683319i
$612$	0	0
$613$	−354.387	−0.578119	−0.289059	−	0.957311i	$-0.593343\pi$
$613$	−354.387	−0.578119	−0.289059	+	0.957311i	$0.593343\pi$
$614$	− 985.788i	− 1.60552i
$615$	0	0
$616$	12.5647	0.0203973
$617$	− 480.150i	− 0.778201i	−0.921195	−	0.389101i	$-0.872786\pi$
$617$	− 480.150i	− 0.778201i	0.921195	−	0.389101i	$-0.127214\pi$
$618$	0	0
$619$	685.162	1.10689	0.553443	−	0.832887i	$-0.313314\pi$
$619$	685.162	1.10689	0.553443	+	0.832887i	$0.313314\pi$
$620$	− 1068.43i	− 1.72327i
$621$	0	0
$622$	380.412	0.611594
$623$	− 137.965i	− 0.221453i
$624$	0	0
$625$	−585.817	−0.937307
$626$	− 1325.57i	− 2.11753i
$627$	0	0
$628$	−300.545	−0.478575
$629$	− 454.612i	− 0.722754i
$630$	0	0
$631$	325.314	0.515553	0.257776	−	0.966205i	$-0.417010\pi$
$631$	325.314	0.515553	0.257776	+	0.966205i	$0.417010\pi$
$632$	36.2257i	0.0573191i
$633$	0	0
$634$	133.463	0.210509
$635$	− 758.583i	− 1.19462i
$636$	0	0
$637$	44.4797	0.0698269
$638$	− 272.148i	− 0.426564i
$639$	0	0
$640$	396.831	0.620049
$641$	908.346i	1.41708i	0.705673	+	0.708538i	$0.250645\pi$
$641$	908.346i	1.41708i	−0.705673	+	0.708538i	$0.749355\pi$
$642$	0	0
$643$	109.302	0.169987	0.0849937	−	0.996381i	$-0.472913\pi$
$643$	109.302	0.169987	0.0849937	+	0.996381i	$0.472913\pi$
$644$	− 247.681i	− 0.384597i
$645$	0	0
$646$	−528.420	−0.817987
$647$	487.056i	0.752791i	0.926459	+	0.376395i	$0.122837\pi$
$647$	487.056i	0.752791i	−0.926459	+	0.376395i	$0.877163\pi$
$648$	0	0
$649$	275.933	0.425167
$650$	− 456.296i	− 0.701995i
$651$	0	0
$652$	−403.698	−0.619169
$653$	542.973i	0.831506i	0.909478	+	0.415753i	$0.136482\pi$
$653$	542.973i	0.831506i	−0.909478	+	0.415753i	$0.863518\pi$
$654$	0	0
$655$	48.2496	0.0736635
$656$	− 524.451i	− 0.799468i
$657$	0	0
$658$	−471.431	−0.716461
$659$	− 181.355i	− 0.275197i	−0.990488	−	0.137599i	$-0.956062\pi$
$659$	− 181.355i	− 0.275197i	0.990488	−	0.137599i	$-0.0439384\pi$
$660$	0	0
$661$	−1073.62	−1.62424	−0.812120	−	0.583491i	$-0.801686\pi$
$661$	−1073.62	−1.62424	−0.812120	+	0.583491i	$0.801686\pi$
$662$	− 104.391i	− 0.157690i
$663$	0	0
$664$	−165.918	−0.249876
$665$	− 579.369i	− 0.871232i
$666$	0	0
$667$	1032.80	1.54842
$668$	660.236i	0.988378i
$669$	0	0
$670$	−1826.13	−2.72557
$671$	− 123.505i	− 0.184062i
$672$	0	0
$673$	495.652	0.736482	0.368241	−	0.929730i	$-0.379960\pi$
$673$	495.652	0.736482	0.368241	+	0.929730i	$0.379960\pi$
$674$	− 1424.27i	− 2.11316i
$675$	0	0
$676$	431.435	0.638218
$677$	1124.69i	1.66128i	0.556811	+	0.830640i	$0.312025\pi$
$677$	1124.69i	1.66128i	−0.556811	+	0.830640i	$0.687975\pi$
$678$	0	0
$679$	259.830	0.382666
$680$	80.2184i	0.117968i
$681$	0	0
$682$	326.490	0.478725
$683$	− 399.085i	− 0.584311i	−0.956371	−	0.292156i	$-0.905627\pi$
$683$	− 399.085i	− 0.584311i	0.956371	−	0.292156i	$-0.0943725\pi$
$684$	0	0
$685$	512.788	0.748596
$686$	50.2246i	0.0732137i
$687$	0	0
$688$	85.1308	0.123737
$689$	− 229.038i	− 0.332421i
$690$	0	0
$691$	309.106	0.447331	0.223666	−	0.974666i	$-0.428198\pi$
$691$	309.106	0.447331	0.223666	+	0.974666i	$0.428198\pi$
$692$	841.389i	1.21588i
$693$	0	0
$694$	−159.813	−0.230278
$695$	− 163.514i	− 0.235271i
$696$	0	0
$697$	184.318	0.264444
$698$	− 1045.98i	− 1.49854i
$699$	0	0
$700$	234.995	0.335707
$701$	837.096i	1.19415i	0.802187	+	0.597073i	$0.203670\pi$
$701$	837.096i	1.19415i	−0.802187	+	0.597073i	$0.796330\pi$
$702$	0	0
$703$	−2173.24	−3.09138
$704$	− 113.728i	− 0.161546i
$705$	0	0
$706$	731.272	1.03580
$707$	− 55.7582i	− 0.0788659i
$708$	0	0
$709$	−1320.08	−1.86190	−0.930948	−	0.365153i	$-0.881017\pi$
$709$	−1320.08	−1.86190	−0.930948	+	0.365153i	$0.881017\pi$
$710$	1573.06i	2.21557i
$711$	0	0
$712$	91.3177	0.128255
$713$	1239.02i	1.73776i
$714$	0	0
$715$	123.638	0.172920
$716$	− 57.8001i	− 0.0807264i
$717$	0	0
$718$	619.112	0.862273
$719$	299.267i	0.416226i	0.978105	+	0.208113i	$0.0667322\pi$
$719$	299.267i	0.416226i	−0.978105	+	0.208113i	$0.933268\pi$
$720$	0	0
$721$	−310.705	−0.430936
$722$	1547.09i	2.14278i
$723$	0	0
$724$	−196.353	−0.271206
$725$	979.897i	1.35158i
$726$	0	0
$727$	724.988	0.997233	0.498616	−	0.866823i	$-0.333842\pi$
$727$	724.988	0.997233	0.498616	+	0.866823i	$0.333842\pi$
$728$	29.4407i	0.0404405i
$729$	0	0
$730$	1171.90	1.60534
$731$	29.9191i	0.0409290i
$732$	0	0
$733$	82.3987	0.112413	0.0562065	−	0.998419i	$-0.482099\pi$
$733$	82.3987	0.112413	0.0562065	+	0.998419i	$0.482099\pi$
$734$	− 184.198i	− 0.250951i
$735$	0	0
$736$	1179.42	1.60247
$737$	− 254.515i	− 0.345340i
$738$	0	0
$739$	154.465	0.209019	0.104510	−	0.994524i	$-0.466673\pi$
$739$	154.465	0.209019	0.104510	+	0.994524i	$0.466673\pi$
$740$	− 1713.69i	− 2.31580i
$741$	0	0
$742$	258.620	0.348544
$743$	1393.71i	1.87579i	0.346922	+	0.937894i	$0.387227\pi$
$743$	1393.71i	1.87579i	−0.346922	+	0.937894i	$0.612773\pi$
$744$	0	0
$745$	−1418.75	−1.90436
$746$	1372.36i	1.83962i
$747$	0	0
$748$	58.0746	0.0776398
$749$	− 387.287i	− 0.517072i
$750$	0	0
$751$	857.735	1.14212	0.571062	−	0.820907i	$-0.306532\pi$
$751$	857.735	1.14212	0.571062	+	0.820907i	$0.306532\pi$
$752$	− 1193.60i	− 1.58724i
$753$	0	0
$754$	637.676	0.845724
$755$	− 1466.33i	− 1.94216i
$756$	0	0
$757$	−3.00394	−0.00396821	−0.00198411	−	0.999998i	$-0.500632\pi$
$757$	−3.00394	−0.00396821	−0.00198411	+	0.999998i	$0.500632\pi$
$758$	136.994i	0.180731i
$759$	0	0
$760$	383.478	0.504577
$761$	705.686i	0.927314i	0.886015	+	0.463657i	$0.153463\pi$
$761$	705.686i	0.927314i	−0.886015	+	0.463657i	$0.846537\pi$
$762$	0	0
$763$	139.952	0.183423
$764$	422.223i	0.552647i
$765$	0	0
$766$	−37.9921	−0.0495981
$767$	646.546i	0.842954i
$768$	0	0
$769$	−1227.42	−1.59613	−0.798065	−	0.602572i	$-0.794143\pi$
$769$	−1227.42	−1.59613	−0.798065	+	0.602572i	$0.794143\pi$
$770$	139.606i	0.181307i
$771$	0	0
$772$	604.186	0.782624
$773$	502.997i	0.650708i	0.945592	+	0.325354i	$0.105483\pi$
$773$	502.997i	0.650708i	−0.945592	+	0.325354i	$0.894517\pi$
$774$	0	0
$775$	−1175.56	−1.51685
$776$	171.979i	0.221622i
$777$	0	0
$778$	539.004	0.692807
$779$	− 881.117i	− 1.13109i
$780$	0	0
$781$	−219.243	−0.280721
$782$	483.213i	0.617919i
$783$	0	0
$784$	−127.162	−0.162197
$785$	642.884i	0.818960i
$786$	0	0
$787$	931.527	1.18364	0.591821	−	0.806069i	$-0.298409\pi$
$787$	931.527	1.18364	0.591821	+	0.806069i	$0.298409\pi$
$788$	− 1156.74i	− 1.46794i
$789$	0	0
$790$	−402.503	−0.509498
$791$	− 447.228i	− 0.565396i
$792$	0	0
$793$	289.388	0.364928
$794$	850.408i	1.07104i
$795$	0	0
$796$	499.485	0.627493
$797$	− 1535.97i	− 1.92719i	−0.267370	−	0.963594i	$-0.586155\pi$
$797$	− 1535.97i	− 1.92719i	0.267370	−	0.963594i	$-0.413845\pi$
$798$	0	0
$799$	419.490	0.525019
$800$	1119.01i	1.39876i
$801$	0	0
$802$	−222.310	−0.277194
$803$	163.333i	0.203403i
$804$	0	0
$805$	−529.804	−0.658141
$806$	765.007i	0.949140i
$807$	0	0
$808$	36.9058	0.0456754
$809$	− 274.168i	− 0.338898i	−0.985539	−	0.169449i	$-0.945801\pi$
$809$	− 274.168i	− 0.338898i	0.985539	−	0.169449i	$-0.0541988\pi$
$810$	0	0
$811$	178.757	0.220415	0.110208	−	0.993909i	$-0.464848\pi$
$811$	178.757	0.220415	0.110208	+	0.993909i	$0.464848\pi$
$812$	328.406i	0.404441i
$813$	0	0
$814$	523.671	0.643330
$815$	863.534i	1.05955i
$816$	0	0
$817$	143.026	0.175063
$818$	208.304i	0.254650i
$819$	0	0
$820$	694.799	0.847315
$821$	− 476.449i	− 0.580328i	−0.956977	−	0.290164i	$-0.906290\pi$
$821$	− 476.449i	− 0.580328i	0.956977	−	0.290164i	$-0.0937099\pi$
$822$	0	0
$823$	−990.515	−1.20354	−0.601771	−	0.798669i	$-0.705538\pi$
$823$	−990.515	−1.20354	−0.601771	+	0.798669i	$0.705538\pi$
$824$	− 205.652i	− 0.249578i
$825$	0	0
$826$	−730.051	−0.883839
$827$	− 1384.88i	− 1.67459i	−0.546753	−	0.837294i	$-0.684137\pi$
$827$	− 1384.88i	− 1.67459i	0.546753	−	0.837294i	$-0.315863\pi$
$828$	0	0
$829$	357.949	0.431784	0.215892	−	0.976417i	$-0.430734\pi$
$829$	357.949	0.431784	0.215892	+	0.976417i	$0.430734\pi$
$830$	− 1843.51i	− 2.22110i
$831$	0	0
$832$	266.480	0.320288
$833$	− 44.6909i	− 0.0536506i
$834$	0	0
$835$	1412.28	1.69136
$836$	− 277.621i	− 0.332083i
$837$	0	0
$838$	−239.028	−0.285236
$839$	− 1128.86i	− 1.34548i	−0.739879	−	0.672740i	$-0.765117\pi$
$839$	− 1128.86i	− 1.34548i	0.739879	−	0.672740i	$-0.234883\pi$
$840$	0	0
$841$	−528.409	−0.628310
$842$	− 72.9016i	− 0.0865814i
$843$	0	0
$844$	−281.459	−0.333482
$845$	− 922.866i	− 1.09215i
$846$	0	0
$847$	300.678	0.354992
$848$	654.791i	0.772159i
$849$	0	0
$850$	−458.463	−0.539368
$851$	1987.32i	2.33528i
$852$	0	0
$853$	−155.106	−0.181836	−0.0909178	−	0.995858i	$-0.528980\pi$
$853$	−155.106	−0.181836	−0.0909178	+	0.995858i	$0.528980\pi$
$854$	326.765i	0.382628i
$855$	0	0
$856$	256.341	0.299464
$857$	231.690i	0.270350i	0.990822	+	0.135175i	$0.0431596\pi$
$857$	231.690i	0.270350i	−0.990822	+	0.135175i	$0.956840\pi$
$858$	0	0
$859$	545.577	0.635130	0.317565	−	0.948237i	$-0.397135\pi$
$859$	545.577	0.635130	0.317565	+	0.948237i	$0.397135\pi$
$860$	112.782i	0.131142i
$861$	0	0
$862$	337.671	0.391729
$863$	1369.08i	1.58642i	0.608946	+	0.793212i	$0.291593\pi$
$863$	1369.08i	1.58642i	−0.608946	+	0.793212i	$0.708407\pi$
$864$	0	0
$865$	1799.78	2.08067
$866$	− 5.46267i	− 0.00630793i
$867$	0	0
$868$	−393.982	−0.453896
$869$	− 56.0985i	− 0.0645552i
$870$	0	0
$871$	596.361	0.684685
$872$	92.6325i	0.106230i
$873$	0	0
$874$	2309.96	2.64298
$875$	− 28.0901i	− 0.0321029i
$876$	0	0
$877$	1080.88	1.23248	0.616239	−	0.787559i	$-0.288655\pi$
$877$	1080.88	1.23248	0.616239	+	0.787559i	$0.288655\pi$
$878$	263.622i	0.300253i
$879$	0	0
$880$	−353.465	−0.401665
$881$	− 468.654i	− 0.531957i	−0.963979	−	0.265978i	$-0.914305\pi$
$881$	− 468.654i	− 0.531957i	0.963979	−	0.265978i	$-0.0856950\pi$
$882$	0	0
$883$	−1255.34	−1.42168	−0.710839	−	0.703355i	$-0.751684\pi$
$883$	−1255.34	−1.42168	−0.710839	+	0.703355i	$0.751684\pi$
$884$	136.076i	0.153932i
$885$	0	0
$886$	643.184	0.725942
$887$	− 1409.97i	− 1.58960i	−0.606874	−	0.794798i	$-0.707577\pi$
$887$	− 1409.97i	− 1.58960i	0.606874	−	0.794798i	$-0.292423\pi$
$888$	0	0
$889$	−279.727	−0.314653
$890$	1014.63i	1.14003i
$891$	0	0
$892$	−560.730	−0.628621
$893$	− 2005.34i	− 2.24562i
$894$	0	0
$895$	−123.638	−0.138143
$896$	− 146.331i	− 0.163316i
$897$	0	0
$898$	2172.24	2.41898
$899$	− 1642.85i	− 1.82742i
$900$	0	0
$901$	−230.125	−0.255411
$902$	212.317i	0.235384i
$903$	0	0
$904$	296.016	0.327451
$905$	420.010i	0.464100i
$906$	0	0
$907$	658.102	0.725581	0.362790	−	0.931871i	$-0.381824\pi$
$907$	658.102	0.725581	0.362790	+	0.931871i	$0.381824\pi$
$908$	316.456i	0.348520i
$909$	0	0
$910$	−327.115	−0.359467
$911$	− 173.039i	− 0.189944i	−0.995480	−	0.0949718i	$-0.969724\pi$
$911$	− 173.039i	− 0.189944i	0.995480	−	0.0949718i	$-0.0302761\pi$
$912$	0	0
$913$	256.937	0.281421
$914$	869.249i	0.951038i
$915$	0	0
$916$	−383.226	−0.418369
$917$	− 17.7920i	− 0.0194024i
$918$	0	0
$919$	−1775.99	−1.93253	−0.966263	−	0.257556i	$-0.917083\pi$
$919$	−1775.99	−1.93253	−0.966263	+	0.257556i	$0.917083\pi$
$920$	− 350.672i	− 0.381165i
$921$	0	0
$922$	1597.99	1.73317
$923$	− 513.714i	− 0.556569i
$924$	0	0
$925$	−1885.53	−2.03841
$926$	− 1933.75i	− 2.08829i
$927$	0	0
$928$	−1563.82	−1.68515
$929$	94.0647i	0.101254i	0.998718	+	0.0506269i	$0.0161219\pi$
$929$	94.0647i	0.101254i	−0.998718	+	0.0506269i	$0.983878\pi$
$930$	0	0
$931$	−213.642	−0.229476
$932$	1146.87i	1.23055i
$933$	0	0
$934$	−1727.13	−1.84918
$935$	− 124.225i	− 0.132861i
$936$	0	0
$937$	−806.797	−0.861043	−0.430522	−	0.902580i	$-0.641670\pi$
$937$	−806.797	−0.861043	−0.430522	+	0.902580i	$0.641670\pi$
$938$	673.384i	0.717894i
$939$	0	0
$940$	1581.30	1.68223
$941$	− 914.280i	− 0.971605i	−0.874069	−	0.485802i	$-0.838527\pi$
$941$	− 914.280i	− 0.971605i	0.874069	−	0.485802i	$-0.161473\pi$
$942$	0	0
$943$	−805.737	−0.854440
$944$	− 1848.39i	− 1.95804i
$945$	0	0
$946$	−34.4640	−0.0364313
$947$	68.9986i	0.0728602i	0.999336	+	0.0364301i	$0.0115986\pi$
$947$	68.9986i	0.0728602i	−0.999336	+	0.0364301i	$0.988401\pi$
$948$	0	0
$949$	−382.708	−0.403276
$950$	2191.65i	2.30700i
$951$	0	0
$952$	29.5805	0.0310719
$953$	267.504i	0.280697i	0.990102	+	0.140348i	$0.0448222\pi$
$953$	267.504i	0.280697i	−0.990102	+	0.140348i	$0.955178\pi$
$954$	0	0
$955$	903.159	0.945717
$956$	− 692.995i	− 0.724890i
$957$	0	0
$958$	1863.04	1.94472
$959$	− 189.090i	− 0.197174i
$960$	0	0
$961$	1009.90	1.05088
$962$	1227.02i	1.27549i
$963$	0	0
$964$	−314.681	−0.326433
$965$	− 1292.39i	− 1.33926i
$966$	0	0
$967$	−95.3596	−0.0986138	−0.0493069	−	0.998784i	$-0.515701\pi$
$967$	−95.3596	−0.0986138	−0.0493069	+	0.998784i	$0.515701\pi$
$968$	199.016i	0.205595i
$969$	0	0
$970$	−1910.85	−1.96995
$971$	662.998i	0.682799i	0.939918	+	0.341399i	$0.110901\pi$
$971$	662.998i	0.682799i	−0.939918	+	0.341399i	$0.889099\pi$
$972$	0	0
$973$	−60.2954	−0.0619686
$974$	− 858.352i	− 0.881265i
$975$	0	0
$976$	−827.325	−0.847669
$977$	85.6597i	0.0876763i	0.999039	+	0.0438381i	$0.0139586\pi$
$977$	85.6597i	0.0876763i	−0.999039	+	0.0438381i	$0.986041\pi$
$978$	0	0
$979$	−141.413	−0.144446
$980$	− 168.466i	− 0.171904i
$981$	0	0
$982$	−949.727	−0.967135
$983$	800.392i	0.814234i	0.913376	+	0.407117i	$0.133466\pi$
$983$	800.392i	0.814234i	−0.913376	+	0.407117i	$0.866534\pi$
$984$	0	0
$985$	−2474.33	−2.51201
$986$	− 640.704i	− 0.649801i
$987$	0	0
$988$	650.501	0.658401
$989$	− 130.790i	− 0.132245i
$990$	0	0
$991$	−691.565	−0.697845	−0.348923	−	0.937152i	$-0.613452\pi$
$991$	−691.565	−0.697845	−0.348923	+	0.937152i	$0.613452\pi$
$992$	− 1876.08i	− 1.89121i
$993$	0	0
$994$	580.063	0.583564
$995$	− 1068.43i	− 1.07380i
$996$	0	0
$997$	−1398.79	−1.40300	−0.701502	−	0.712668i	$-0.747487\pi$
$997$	−1398.79	−1.40300	−0.701502	+	0.712668i	$0.747487\pi$
$998$	− 101.112i	− 0.101315i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	189.3.b.a.134.3	yes	4
3.2	odd	2	inner	189.3.b.a.134.2	✓	4
4.3	odd	2		3024.3.d.f.1457.3		4
9.2	odd	6		567.3.r.d.134.2		8
9.4	even	3		567.3.r.d.512.2		8
9.5	odd	6		567.3.r.d.512.3		8
9.7	even	3		567.3.r.d.134.3		8
12.11	even	2		3024.3.d.f.1457.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.3.b.a.134.2	✓	4	3.2	odd	2	inner
189.3.b.a.134.3	yes	4	1.1	even	1	trivial
567.3.r.d.134.2		8	9.2	odd	6
567.3.r.d.134.3		8	9.7	even	3
567.3.r.d.512.2		8	9.4	even	3
567.3.r.d.512.3		8	9.5	odd	6
3024.3.d.f.1457.2		4	12.11	even	2
3024.3.d.f.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)

\(f(q)\)	\(=\)	\(q+2.71187i q^{2} -3.35425 q^{4} +7.17494i q^{5} +2.64575 q^{7} +1.75119i q^{8} +O(q^{10})\)	\(q+2.71187i q^{2} -3.35425 q^{4} +7.17494i q^{5} +2.64575 q^{7} +1.75119i q^{8} -19.4575 q^{10} -2.71187i q^{11} +6.35425 q^{13} +7.17494i q^{14} -18.1660 q^{16} -6.38442i q^{17} -30.5203 q^{19} -24.0665i q^{20} +7.35425 q^{22} +27.9092i q^{23} -26.4797 q^{25} +17.2319i q^{26} -8.87451 q^{28} -37.0055i q^{29} +44.3948 q^{31} -42.2591i q^{32} +17.3137 q^{34} +18.9831i q^{35} +71.2065 q^{37} -82.7670i q^{38} -12.5647 q^{40} +28.8699i q^{41} -4.68627 q^{43} +9.09629i q^{44} -75.6863 q^{46} +65.7053i q^{47} +7.00000 q^{49} -71.8097i q^{50} -21.3137 q^{52} -36.0449i q^{53} +19.4575 q^{55} +4.63323i q^{56} +100.354 q^{58} +101.750i q^{59} +45.5425 q^{61} +120.393i q^{62} +41.9373 q^{64} +45.5913i q^{65} +93.8523 q^{67} +21.4149i q^{68} -51.4797 q^{70} -80.8457i q^{71} -60.2288 q^{73} +193.103i q^{74} +102.373 q^{76} -7.17494i q^{77} +20.6863 q^{79} -130.340i q^{80} -78.2915 q^{82} +94.7454i q^{83} +45.8078 q^{85} -12.7086i q^{86} +4.74902 q^{88} -52.1459i q^{89} +16.8118 q^{91} -93.6145i q^{92} -178.184 q^{94} -218.981i q^{95} +98.2065 q^{97} +18.9831i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 24 q^{4}+O(q^{10}) \) 4 * q - 24 * q^4	\( 4 q - 24 q^{4} + 28 q^{10} + 36 q^{13} + 12 q^{16} - 48 q^{19} + 40 q^{22} - 180 q^{25} + 28 q^{28} + 40 q^{31} + 228 q^{34} + 52 q^{37} - 336 q^{40} + 140 q^{43} - 144 q^{46} + 28 q^{49} - 244 q^{52} - 28 q^{55} + 412 q^{58} + 288 q^{61} + 136 q^{64} + 132 q^{67} - 280 q^{70} - 188 q^{73} + 92 q^{76} - 76 q^{79} - 292 q^{82} - 420 q^{85} - 108 q^{88} - 28 q^{91} - 300 q^{94} + 160 q^{97}+O(q^{100}) \) 4 * q - 24 * q^4 + 28 * q^10 + 36 * q^13 + 12 * q^16 - 48 * q^19 + 40 * q^22 - 180 * q^25 + 28 * q^28 + 40 * q^31 + 228 * q^34 + 52 * q^37 - 336 * q^40 + 140 * q^43 - 144 * q^46 + 28 * q^49 - 244 * q^52 - 28 * q^55 + 412 * q^58 + 288 * q^61 + 136 * q^64 + 132 * q^67 - 280 * q^70 - 188 * q^73 + 92 * q^76 - 76 * q^79 - 292 * q^82 - 420 * q^85 - 108 * q^88 - 28 * q^91 - 300 * q^94 + 160 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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