LMFDB - Embedded newform 1134.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1134,2,Mod(1,1134)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1134, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1134.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1134 = 2 \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1134.a (trivial)

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:	yes
Analytic conductor:	$9.05503558921$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	1134.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.73205 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.73205 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.73205 q^{10} +4.73205 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.46410 q^{17} -6.19615 q^{19} -1.73205 q^{20} +4.73205 q^{22} +1.26795 q^{23} -2.00000 q^{25} -1.00000 q^{26} +1.00000 q^{28} +6.46410 q^{29} +4.19615 q^{31} +1.00000 q^{32} +6.46410 q^{34} -1.73205 q^{35} -3.19615 q^{37} -6.19615 q^{38} -1.73205 q^{40} +2.53590 q^{41} +12.3923 q^{43} +4.73205 q^{44} +1.26795 q^{46} +2.19615 q^{47} +1.00000 q^{49} -2.00000 q^{50} -1.00000 q^{52} +9.46410 q^{53} -8.19615 q^{55} +1.00000 q^{56} +6.46410 q^{58} -8.19615 q^{59} +9.39230 q^{61} +4.19615 q^{62} +1.00000 q^{64} +1.73205 q^{65} +4.19615 q^{67} +6.46410 q^{68} -1.73205 q^{70} -4.39230 q^{71} -9.19615 q^{73} -3.19615 q^{74} -6.19615 q^{76} +4.73205 q^{77} -6.19615 q^{79} -1.73205 q^{80} +2.53590 q^{82} +1.26795 q^{83} -11.1962 q^{85} +12.3923 q^{86} +4.73205 q^{88} -12.4641 q^{89} -1.00000 q^{91} +1.26795 q^{92} +2.19615 q^{94} +10.7321 q^{95} -16.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 6 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} + 6 q^{17} - 2 q^{19} + 6 q^{22} + 6 q^{23} - 4 q^{25} - 2 q^{26} + 2 q^{28} + 6 q^{29} - 2 q^{31} + 2 q^{32} + 6 q^{34} + 4 q^{37} - 2 q^{38} + 12 q^{41} + 4 q^{43} + 6 q^{44} + 6 q^{46} - 6 q^{47} + 2 q^{49} - 4 q^{50} - 2 q^{52} + 12 q^{53} - 6 q^{55} + 2 q^{56} + 6 q^{58} - 6 q^{59} - 2 q^{61} - 2 q^{62} + 2 q^{64} - 2 q^{67} + 6 q^{68} + 12 q^{71} - 8 q^{73} + 4 q^{74} - 2 q^{76} + 6 q^{77} - 2 q^{79} + 12 q^{82} + 6 q^{83} - 12 q^{85} + 4 q^{86} + 6 q^{88} - 18 q^{89} - 2 q^{91} + 6 q^{92} - 6 q^{94} + 18 q^{95} - 32 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8 + 6 * q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 + 6 * q^17 - 2 * q^19 + 6 * q^22 + 6 * q^23 - 4 * q^25 - 2 * q^26 + 2 * q^28 + 6 * q^29 - 2 * q^31 + 2 * q^32 + 6 * q^34 + 4 * q^37 - 2 * q^38 + 12 * q^41 + 4 * q^43 + 6 * q^44 + 6 * q^46 - 6 * q^47 + 2 * q^49 - 4 * q^50 - 2 * q^52 + 12 * q^53 - 6 * q^55 + 2 * q^56 + 6 * q^58 - 6 * q^59 - 2 * q^61 - 2 * q^62 + 2 * q^64 - 2 * q^67 + 6 * q^68 + 12 * q^71 - 8 * q^73 + 4 * q^74 - 2 * q^76 + 6 * q^77 - 2 * q^79 + 12 * q^82 + 6 * q^83 - 12 * q^85 + 4 * q^86 + 6 * q^88 - 18 * q^89 - 2 * q^91 + 6 * q^92 - 6 * q^94 + 18 * q^95 - 32 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.73205	−0.774597	−0.387298	−	0.921954i	$-0.626592\pi$
$5$	−1.73205	−0.774597	−0.387298	+	0.921954i	$0.626592\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.73205	−0.547723
$11$	4.73205	1.42677	0.713384	−	0.700774i	$-0.247162\pi$
$11$	4.73205	1.42677	0.713384	+	0.700774i	$0.247162\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	6.46410	1.56777	0.783887	−	0.620903i	$-0.213234\pi$
$17$	6.46410	1.56777	0.783887	+	0.620903i	$0.213234\pi$
$18$	0	0
$19$	−6.19615	−1.42149	−0.710747	−	0.703447i	$-0.751643\pi$
$19$	−6.19615	−1.42149	−0.710747	+	0.703447i	$0.751643\pi$
$20$	−1.73205	−0.387298
$21$	0	0
$22$	4.73205	1.00888
$23$	1.26795	0.264386	0.132193	−	0.991224i	$-0.457798\pi$
$23$	1.26795	0.264386	0.132193	+	0.991224i	$0.457798\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	−1.00000	−0.196116
$27$	0	0
$28$	1.00000	0.188982
$29$	6.46410	1.20035	0.600177	−	0.799867i	$-0.295097\pi$
$29$	6.46410	1.20035	0.600177	+	0.799867i	$0.295097\pi$
$30$	0	0
$31$	4.19615	0.753651	0.376826	−	0.926284i	$-0.377016\pi$
$31$	4.19615	0.753651	0.376826	+	0.926284i	$0.377016\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.46410	1.10858
$35$	−1.73205	−0.292770
$36$	0	0
$37$	−3.19615	−0.525444	−0.262722	−	0.964872i	$-0.584620\pi$
$37$	−3.19615	−0.525444	−0.262722	+	0.964872i	$0.584620\pi$
$38$	−6.19615	−1.00515
$39$	0	0
$40$	−1.73205	−0.273861
$41$	2.53590	0.396041	0.198020	−	0.980198i	$-0.436549\pi$
$41$	2.53590	0.396041	0.198020	+	0.980198i	$0.436549\pi$
$42$	0	0
$43$	12.3923	1.88981	0.944904	−	0.327346i	$-0.106154\pi$
$43$	12.3923	1.88981	0.944904	+	0.327346i	$0.106154\pi$
$44$	4.73205	0.713384
$45$	0	0
$46$	1.26795	0.186949
$47$	2.19615	0.320342	0.160171	−	0.987089i	$-0.448795\pi$
$47$	2.19615	0.320342	0.160171	+	0.987089i	$0.448795\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.00000	−0.282843
$51$	0	0
$52$	−1.00000	−0.138675
$53$	9.46410	1.29999	0.649997	−	0.759937i	$-0.274770\pi$
$53$	9.46410	1.29999	0.649997	+	0.759937i	$0.274770\pi$
$54$	0	0
$55$	−8.19615	−1.10517
$56$	1.00000	0.133631
$57$	0	0
$58$	6.46410	0.848778
$59$	−8.19615	−1.06705	−0.533524	−	0.845785i	$-0.679133\pi$
$59$	−8.19615	−1.06705	−0.533524	+	0.845785i	$0.679133\pi$
$60$	0	0
$61$	9.39230	1.20256	0.601281	−	0.799038i	$-0.294657\pi$
$61$	9.39230	1.20256	0.601281	+	0.799038i	$0.294657\pi$
$62$	4.19615	0.532912
$63$	0	0
$64$	1.00000	0.125000
$65$	1.73205	0.214834
$66$	0	0
$67$	4.19615	0.512642	0.256321	−	0.966592i	$-0.417490\pi$
$67$	4.19615	0.512642	0.256321	+	0.966592i	$0.417490\pi$
$68$	6.46410	0.783887
$69$	0	0
$70$	−1.73205	−0.207020
$71$	−4.39230	−0.521271	−0.260635	−	0.965437i	$-0.583932\pi$
$71$	−4.39230	−0.521271	−0.260635	+	0.965437i	$0.583932\pi$
$72$	0	0
$73$	−9.19615	−1.07633	−0.538164	−	0.842840i	$-0.680882\pi$
$73$	−9.19615	−1.07633	−0.538164	+	0.842840i	$0.680882\pi$
$74$	−3.19615	−0.371545
$75$	0	0
$76$	−6.19615	−0.710747
$77$	4.73205	0.539267
$78$	0	0
$79$	−6.19615	−0.697122	−0.348561	−	0.937286i	$-0.613330\pi$
$79$	−6.19615	−0.697122	−0.348561	+	0.937286i	$0.613330\pi$
$80$	−1.73205	−0.193649
$81$	0	0
$82$	2.53590	0.280043
$83$	1.26795	0.139176	0.0695878	−	0.997576i	$-0.477832\pi$
$83$	1.26795	0.139176	0.0695878	+	0.997576i	$0.477832\pi$
$84$	0	0
$85$	−11.1962	−1.21439
$86$	12.3923	1.33630
$87$	0	0
$88$	4.73205	0.504438
$89$	−12.4641	−1.32119	−0.660596	−	0.750742i	$-0.729696\pi$
$89$	−12.4641	−1.32119	−0.660596	+	0.750742i	$0.729696\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	1.26795	0.132193
$93$	0	0
$94$	2.19615	0.226516
$95$	10.7321	1.10109
$96$	0	0
$97$	−16.0000	−1.62455	−0.812277	−	0.583272i	$-0.801772\pi$
$97$	−16.0000	−1.62455	−0.812277	+	0.583272i	$0.801772\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	−2.00000	−0.200000
$101$	12.9282	1.28640	0.643202	−	0.765696i	$-0.277605\pi$
$101$	12.9282	1.28640	0.643202	+	0.765696i	$0.277605\pi$
$102$	0	0
$103$	−8.39230	−0.826918	−0.413459	−	0.910523i	$-0.635680\pi$
$103$	−8.39230	−0.826918	−0.413459	+	0.910523i	$0.635680\pi$
$104$	−1.00000	−0.0980581
$105$	0	0
$106$	9.46410	0.919235
$107$	−13.8564	−1.33955	−0.669775	−	0.742564i	$-0.733609\pi$
$107$	−13.8564	−1.33955	−0.669775	+	0.742564i	$0.733609\pi$
$108$	0	0
$109$	−4.80385	−0.460125	−0.230063	−	0.973176i	$-0.573893\pi$
$109$	−4.80385	−0.460125	−0.230063	+	0.973176i	$0.573893\pi$
$110$	−8.19615	−0.781472
$111$	0	0
$112$	1.00000	0.0944911
$113$	−10.2679	−0.965927	−0.482964	−	0.875640i	$-0.660440\pi$
$113$	−10.2679	−0.965927	−0.482964	+	0.875640i	$0.660440\pi$
$114$	0	0
$115$	−2.19615	−0.204792
$116$	6.46410	0.600177
$117$	0	0
$118$	−8.19615	−0.754517
$119$	6.46410	0.592563
$120$	0	0
$121$	11.3923	1.03566
$122$	9.39230	0.850339
$123$	0	0
$124$	4.19615	0.376826
$125$	12.1244	1.08444
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	1.73205	0.151911
$131$	−2.53590	−0.221562	−0.110781	−	0.993845i	$-0.535335\pi$
$131$	−2.53590	−0.221562	−0.110781	+	0.993845i	$0.535335\pi$
$132$	0	0
$133$	−6.19615	−0.537275
$134$	4.19615	0.362492
$135$	0	0
$136$	6.46410	0.554292
$137$	−2.66025	−0.227281	−0.113640	−	0.993522i	$-0.536251\pi$
$137$	−2.66025	−0.227281	−0.113640	+	0.993522i	$0.536251\pi$
$138$	0	0
$139$	−18.1962	−1.54338	−0.771689	−	0.636000i	$-0.780588\pi$
$139$	−18.1962	−1.54338	−0.771689	+	0.636000i	$0.780588\pi$
$140$	−1.73205	−0.146385
$141$	0	0
$142$	−4.39230	−0.368594
$143$	−4.73205	−0.395714
$144$	0	0
$145$	−11.1962	−0.929790
$146$	−9.19615	−0.761079
$147$	0	0
$148$	−3.19615	−0.262722
$149$	−3.92820	−0.321811	−0.160905	−	0.986970i	$-0.551441\pi$
$149$	−3.92820	−0.321811	−0.160905	+	0.986970i	$0.551441\pi$
$150$	0	0
$151$	−22.5885	−1.83822	−0.919111	−	0.393998i	$-0.871092\pi$
$151$	−22.5885	−1.83822	−0.919111	+	0.393998i	$0.871092\pi$
$152$	−6.19615	−0.502574
$153$	0	0
$154$	4.73205	0.381320
$155$	−7.26795	−0.583776
$156$	0	0
$157$	15.3923	1.22844	0.614220	−	0.789135i	$-0.289471\pi$
$157$	15.3923	1.22844	0.614220	+	0.789135i	$0.289471\pi$
$158$	−6.19615	−0.492939
$159$	0	0
$160$	−1.73205	−0.136931
$161$	1.26795	0.0999284
$162$	0	0
$163$	24.3923	1.91055	0.955276	−	0.295714i	$-0.0955577\pi$
$163$	24.3923	1.91055	0.955276	+	0.295714i	$0.0955577\pi$
$164$	2.53590	0.198020
$165$	0	0
$166$	1.26795	0.0984119
$167$	−7.26795	−0.562411	−0.281205	−	0.959648i	$-0.590734\pi$
$167$	−7.26795	−0.562411	−0.281205	+	0.959648i	$0.590734\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	−11.1962	−0.858706
$171$	0	0
$172$	12.3923	0.944904
$173$	−12.8038	−0.973459	−0.486729	−	0.873553i	$-0.661810\pi$
$173$	−12.8038	−0.973459	−0.486729	+	0.873553i	$0.661810\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	4.73205	0.356692
$177$	0	0
$178$	−12.4641	−0.934224
$179$	7.26795	0.543232	0.271616	−	0.962406i	$-0.412442\pi$
$179$	7.26795	0.543232	0.271616	+	0.962406i	$0.412442\pi$
$180$	0	0
$181$	0.392305	0.0291598	0.0145799	−	0.999894i	$-0.495359\pi$
$181$	0.392305	0.0291598	0.0145799	+	0.999894i	$0.495359\pi$
$182$	−1.00000	−0.0741249
$183$	0	0
$184$	1.26795	0.0934745
$185$	5.53590	0.407007
$186$	0	0
$187$	30.5885	2.23685
$188$	2.19615	0.160171
$189$	0	0
$190$	10.7321	0.778585
$191$	24.5885	1.77916	0.889579	−	0.456781i	$-0.150998\pi$
$191$	24.5885	1.77916	0.889579	+	0.456781i	$0.150998\pi$
$192$	0	0
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	−16.0000	−1.14873
$195$	0	0
$196$	1.00000	0.0714286
$197$	−15.0000	−1.06871	−0.534353	−	0.845262i	$-0.679445\pi$
$197$	−15.0000	−1.06871	−0.534353	+	0.845262i	$0.679445\pi$
$198$	0	0
$199$	20.5885	1.45948	0.729739	−	0.683726i	$-0.239642\pi$
$199$	20.5885	1.45948	0.729739	+	0.683726i	$0.239642\pi$
$200$	−2.00000	−0.141421
$201$	0	0
$202$	12.9282	0.909625
$203$	6.46410	0.453691
$204$	0	0
$205$	−4.39230	−0.306772
$206$	−8.39230	−0.584720
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	−29.3205	−2.02814
$210$	0	0
$211$	11.8038	0.812610	0.406305	−	0.913737i	$-0.366817\pi$
$211$	11.8038	0.812610	0.406305	+	0.913737i	$0.366817\pi$
$212$	9.46410	0.649997
$213$	0	0
$214$	−13.8564	−0.947204
$215$	−21.4641	−1.46384
$216$	0	0
$217$	4.19615	0.284853
$218$	−4.80385	−0.325358
$219$	0	0
$220$	−8.19615	−0.552584
$221$	−6.46410	−0.434823
$222$	0	0
$223$	12.3923	0.829850	0.414925	−	0.909856i	$-0.363808\pi$
$223$	12.3923	0.829850	0.414925	+	0.909856i	$0.363808\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−10.2679	−0.683014
$227$	−5.07180	−0.336627	−0.168313	−	0.985734i	$-0.553832\pi$
$227$	−5.07180	−0.336627	−0.168313	+	0.985734i	$0.553832\pi$
$228$	0	0
$229$	−21.7846	−1.43957	−0.719784	−	0.694198i	$-0.755759\pi$
$229$	−21.7846	−1.43957	−0.719784	+	0.694198i	$0.755759\pi$
$230$	−2.19615	−0.144810
$231$	0	0
$232$	6.46410	0.424389
$233$	13.7321	0.899617	0.449808	−	0.893125i	$-0.351492\pi$
$233$	13.7321	0.899617	0.449808	+	0.893125i	$0.351492\pi$
$234$	0	0
$235$	−3.80385	−0.248136
$236$	−8.19615	−0.533524
$237$	0	0
$238$	6.46410	0.419005
$239$	15.1244	0.978313	0.489157	−	0.872196i	$-0.337305\pi$
$239$	15.1244	0.978313	0.489157	+	0.872196i	$0.337305\pi$
$240$	0	0
$241$	−13.5885	−0.875309	−0.437655	−	0.899143i	$-0.644191\pi$
$241$	−13.5885	−0.875309	−0.437655	+	0.899143i	$0.644191\pi$
$242$	11.3923	0.732325
$243$	0	0
$244$	9.39230	0.601281
$245$	−1.73205	−0.110657
$246$	0	0
$247$	6.19615	0.394252
$248$	4.19615	0.266456
$249$	0	0
$250$	12.1244	0.766812
$251$	−3.80385	−0.240097	−0.120048	−	0.992768i	$-0.538305\pi$
$251$	−3.80385	−0.240097	−0.120048	+	0.992768i	$0.538305\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	−4.00000	−0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	−27.9282	−1.74211	−0.871057	−	0.491182i	$-0.836565\pi$
$257$	−27.9282	−1.74211	−0.871057	+	0.491182i	$0.836565\pi$
$258$	0	0
$259$	−3.19615	−0.198599
$260$	1.73205	0.107417
$261$	0	0
$262$	−2.53590	−0.156668
$263$	15.1244	0.932608	0.466304	−	0.884625i	$-0.345585\pi$
$263$	15.1244	0.932608	0.466304	+	0.884625i	$0.345585\pi$
$264$	0	0
$265$	−16.3923	−1.00697
$266$	−6.19615	−0.379910
$267$	0	0
$268$	4.19615	0.256321
$269$	29.4449	1.79529	0.897643	−	0.440724i	$-0.145278\pi$
$269$	29.4449	1.79529	0.897643	+	0.440724i	$0.145278\pi$
$270$	0	0
$271$	28.1962	1.71279	0.856397	−	0.516318i	$-0.172698\pi$
$271$	28.1962	1.71279	0.856397	+	0.516318i	$0.172698\pi$
$272$	6.46410	0.391944
$273$	0	0
$274$	−2.66025	−0.160712
$275$	−9.46410	−0.570707
$276$	0	0
$277$	−18.7846	−1.12866	−0.564329	−	0.825550i	$-0.690865\pi$
$277$	−18.7846	−1.12866	−0.564329	+	0.825550i	$0.690865\pi$
$278$	−18.1962	−1.09133
$279$	0	0
$280$	−1.73205	−0.103510
$281$	−13.7321	−0.819185	−0.409593	−	0.912268i	$-0.634329\pi$
$281$	−13.7321	−0.819185	−0.409593	+	0.912268i	$0.634329\pi$
$282$	0	0
$283$	3.60770	0.214455	0.107228	−	0.994234i	$-0.465803\pi$
$283$	3.60770	0.214455	0.107228	+	0.994234i	$0.465803\pi$
$284$	−4.39230	−0.260635
$285$	0	0
$286$	−4.73205	−0.279812
$287$	2.53590	0.149689
$288$	0	0
$289$	24.7846	1.45792
$290$	−11.1962	−0.657461
$291$	0	0
$292$	−9.19615	−0.538164
$293$	−26.6603	−1.55751	−0.778754	−	0.627329i	$-0.784148\pi$
$293$	−26.6603	−1.55751	−0.778754	+	0.627329i	$0.784148\pi$
$294$	0	0
$295$	14.1962	0.826532
$296$	−3.19615	−0.185773
$297$	0	0
$298$	−3.92820	−0.227555
$299$	−1.26795	−0.0733274
$300$	0	0
$301$	12.3923	0.714281
$302$	−22.5885	−1.29982
$303$	0	0
$304$	−6.19615	−0.355374
$305$	−16.2679	−0.931500
$306$	0	0
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	4.73205	0.269634
$309$	0	0
$310$	−7.26795	−0.412792
$311$	−22.9808	−1.30312	−0.651560	−	0.758597i	$-0.725885\pi$
$311$	−22.9808	−1.30312	−0.651560	+	0.758597i	$0.725885\pi$
$312$	0	0
$313$	21.9808	1.24243	0.621213	−	0.783642i	$-0.286640\pi$
$313$	21.9808	1.24243	0.621213	+	0.783642i	$0.286640\pi$
$314$	15.3923	0.868638
$315$	0	0
$316$	−6.19615	−0.348561
$317$	−1.39230	−0.0781996	−0.0390998	−	0.999235i	$-0.512449\pi$
$317$	−1.39230	−0.0781996	−0.0390998	+	0.999235i	$0.512449\pi$
$318$	0	0
$319$	30.5885	1.71262
$320$	−1.73205	−0.0968246
$321$	0	0
$322$	1.26795	0.0706600
$323$	−40.0526	−2.22858
$324$	0	0
$325$	2.00000	0.110940
$326$	24.3923	1.35096
$327$	0	0
$328$	2.53590	0.140022
$329$	2.19615	0.121078
$330$	0	0
$331$	−29.1769	−1.60371	−0.801854	−	0.597520i	$-0.796153\pi$
$331$	−29.1769	−1.60371	−0.801854	+	0.597520i	$0.796153\pi$
$332$	1.26795	0.0695878
$333$	0	0
$334$	−7.26795	−0.397684
$335$	−7.26795	−0.397090
$336$	0	0
$337$	−32.3923	−1.76452	−0.882261	−	0.470761i	$-0.843979\pi$
$337$	−32.3923	−1.76452	−0.882261	+	0.470761i	$0.843979\pi$
$338$	−12.0000	−0.652714
$339$	0	0
$340$	−11.1962	−0.607197
$341$	19.8564	1.07528
$342$	0	0
$343$	1.00000	0.0539949
$344$	12.3923	0.668148
$345$	0	0
$346$	−12.8038	−0.688339
$347$	16.3923	0.879985	0.439993	−	0.898001i	$-0.354981\pi$
$347$	16.3923	0.879985	0.439993	+	0.898001i	$0.354981\pi$
$348$	0	0
$349$	12.3923	0.663345	0.331672	−	0.943395i	$-0.392387\pi$
$349$	12.3923	0.663345	0.331672	+	0.943395i	$0.392387\pi$
$350$	−2.00000	−0.106904
$351$	0	0
$352$	4.73205	0.252219
$353$	4.14359	0.220541	0.110271	−	0.993902i	$-0.464828\pi$
$353$	4.14359	0.220541	0.110271	+	0.993902i	$0.464828\pi$
$354$	0	0
$355$	7.60770	0.403775
$356$	−12.4641	−0.660596
$357$	0	0
$358$	7.26795	0.384123
$359$	18.9282	0.998992	0.499496	−	0.866316i	$-0.333518\pi$
$359$	18.9282	0.998992	0.499496	+	0.866316i	$0.333518\pi$
$360$	0	0
$361$	19.3923	1.02065
$362$	0.392305	0.0206191
$363$	0	0
$364$	−1.00000	−0.0524142
$365$	15.9282	0.833720
$366$	0	0
$367$	−4.58846	−0.239516	−0.119758	−	0.992803i	$-0.538212\pi$
$367$	−4.58846	−0.239516	−0.119758	+	0.992803i	$0.538212\pi$
$368$	1.26795	0.0660964
$369$	0	0
$370$	5.53590	0.287798
$371$	9.46410	0.491352
$372$	0	0
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	30.5885	1.58169
$375$	0	0
$376$	2.19615	0.113258
$377$	−6.46410	−0.332918
$378$	0	0
$379$	−16.5885	−0.852092	−0.426046	−	0.904702i	$-0.640094\pi$
$379$	−16.5885	−0.852092	−0.426046	+	0.904702i	$0.640094\pi$
$380$	10.7321	0.550543
$381$	0	0
$382$	24.5885	1.25805
$383$	−10.1436	−0.518313	−0.259157	−	0.965835i	$-0.583445\pi$
$383$	−10.1436	−0.518313	−0.259157	+	0.965835i	$0.583445\pi$
$384$	0	0
$385$	−8.19615	−0.417715
$386$	−19.0000	−0.967075
$387$	0	0
$388$	−16.0000	−0.812277
$389$	−30.2487	−1.53367	−0.766835	−	0.641844i	$-0.778170\pi$
$389$	−30.2487	−1.53367	−0.766835	+	0.641844i	$0.778170\pi$
$390$	0	0
$391$	8.19615	0.414497
$392$	1.00000	0.0505076
$393$	0	0
$394$	−15.0000	−0.755689
$395$	10.7321	0.539988
$396$	0	0
$397$	−29.3923	−1.47516	−0.737579	−	0.675261i	$-0.764031\pi$
$397$	−29.3923	−1.47516	−0.737579	+	0.675261i	$0.764031\pi$
$398$	20.5885	1.03201
$399$	0	0
$400$	−2.00000	−0.100000
$401$	22.2679	1.11201	0.556004	−	0.831180i	$-0.312334\pi$
$401$	22.2679	1.11201	0.556004	+	0.831180i	$0.312334\pi$
$402$	0	0
$403$	−4.19615	−0.209025
$404$	12.9282	0.643202
$405$	0	0
$406$	6.46410	0.320808
$407$	−15.1244	−0.749686
$408$	0	0
$409$	−21.1962	−1.04808	−0.524041	−	0.851693i	$-0.675576\pi$
$409$	−21.1962	−1.04808	−0.524041	+	0.851693i	$0.675576\pi$
$410$	−4.39230	−0.216920
$411$	0	0
$412$	−8.39230	−0.413459
$413$	−8.19615	−0.403306
$414$	0	0
$415$	−2.19615	−0.107805
$416$	−1.00000	−0.0490290
$417$	0	0
$418$	−29.3205	−1.43411
$419$	23.3205	1.13928	0.569641	−	0.821894i	$-0.307082\pi$
$419$	23.3205	1.13928	0.569641	+	0.821894i	$0.307082\pi$
$420$	0	0
$421$	13.1962	0.643141	0.321571	−	0.946886i	$-0.395789\pi$
$421$	13.1962	0.643141	0.321571	+	0.946886i	$0.395789\pi$
$422$	11.8038	0.574602
$423$	0	0
$424$	9.46410	0.459617
$425$	−12.9282	−0.627110
$426$	0	0
$427$	9.39230	0.454525
$428$	−13.8564	−0.669775
$429$	0	0
$430$	−21.4641	−1.03509
$431$	0.679492	0.0327300	0.0163650	−	0.999866i	$-0.494791\pi$
$431$	0.679492	0.0327300	0.0163650	+	0.999866i	$0.494791\pi$
$432$	0	0
$433$	−31.5885	−1.51804	−0.759022	−	0.651065i	$-0.774323\pi$
$433$	−31.5885	−1.51804	−0.759022	+	0.651065i	$0.774323\pi$
$434$	4.19615	0.201422
$435$	0	0
$436$	−4.80385	−0.230063
$437$	−7.85641	−0.375823
$438$	0	0
$439$	21.1769	1.01072	0.505359	−	0.862909i	$-0.331360\pi$
$439$	21.1769	1.01072	0.505359	+	0.862909i	$0.331360\pi$
$440$	−8.19615	−0.390736
$441$	0	0
$442$	−6.46410	−0.307466
$443$	2.19615	0.104342	0.0521712	−	0.998638i	$-0.483386\pi$
$443$	2.19615	0.104342	0.0521712	+	0.998638i	$0.483386\pi$
$444$	0	0
$445$	21.5885	1.02339
$446$	12.3923	0.586793
$447$	0	0
$448$	1.00000	0.0472456
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	12.0000	0.565058
$452$	−10.2679	−0.482964
$453$	0	0
$454$	−5.07180	−0.238031
$455$	1.73205	0.0811998
$456$	0	0
$457$	−1.00000	−0.0467780	−0.0233890	−	0.999726i	$-0.507446\pi$
$457$	−1.00000	−0.0467780	−0.0233890	+	0.999726i	$0.507446\pi$
$458$	−21.7846	−1.01793
$459$	0	0
$460$	−2.19615	−0.102396
$461$	11.0718	0.515665	0.257832	−	0.966190i	$-0.416992\pi$
$461$	11.0718	0.515665	0.257832	+	0.966190i	$0.416992\pi$
$462$	0	0
$463$	5.80385	0.269728	0.134864	−	0.990864i	$-0.456940\pi$
$463$	5.80385	0.269728	0.134864	+	0.990864i	$0.456940\pi$
$464$	6.46410	0.300088
$465$	0	0
$466$	13.7321	0.636125
$467$	−1.26795	−0.0586737	−0.0293368	−	0.999570i	$-0.509340\pi$
$467$	−1.26795	−0.0586737	−0.0293368	+	0.999570i	$0.509340\pi$
$468$	0	0
$469$	4.19615	0.193760
$470$	−3.80385	−0.175458
$471$	0	0
$472$	−8.19615	−0.377258
$473$	58.6410	2.69632
$474$	0	0
$475$	12.3923	0.568598
$476$	6.46410	0.296282
$477$	0	0
$478$	15.1244	0.691772
$479$	16.7321	0.764507	0.382253	−	0.924058i	$-0.375148\pi$
$479$	16.7321	0.764507	0.382253	+	0.924058i	$0.375148\pi$
$480$	0	0
$481$	3.19615	0.145732
$482$	−13.5885	−0.618937
$483$	0	0
$484$	11.3923	0.517832
$485$	27.7128	1.25837
$486$	0	0
$487$	−6.19615	−0.280774	−0.140387	−	0.990097i	$-0.544835\pi$
$487$	−6.19615	−0.280774	−0.140387	+	0.990097i	$0.544835\pi$
$488$	9.39230	0.425170
$489$	0	0
$490$	−1.73205	−0.0782461
$491$	11.3205	0.510887	0.255444	−	0.966824i	$-0.417778\pi$
$491$	11.3205	0.510887	0.255444	+	0.966824i	$0.417778\pi$
$492$	0	0
$493$	41.7846	1.88188
$494$	6.19615	0.278778
$495$	0	0
$496$	4.19615	0.188413
$497$	−4.39230	−0.197022
$498$	0	0
$499$	20.5885	0.921666	0.460833	−	0.887487i	$-0.347551\pi$
$499$	20.5885	0.921666	0.460833	+	0.887487i	$0.347551\pi$
$500$	12.1244	0.542218
$501$	0	0
$502$	−3.80385	−0.169774
$503$	−22.9808	−1.02466	−0.512331	−	0.858788i	$-0.671218\pi$
$503$	−22.9808	−1.02466	−0.512331	+	0.858788i	$0.671218\pi$
$504$	0	0
$505$	−22.3923	−0.996444
$506$	6.00000	0.266733
$507$	0	0
$508$	−4.00000	−0.177471
$509$	−21.7128	−0.962404	−0.481202	−	0.876610i	$-0.659800\pi$
$509$	−21.7128	−0.962404	−0.481202	+	0.876610i	$0.659800\pi$
$510$	0	0
$511$	−9.19615	−0.406814
$512$	1.00000	0.0441942
$513$	0	0
$514$	−27.9282	−1.23186
$515$	14.5359	0.640528
$516$	0	0
$517$	10.3923	0.457053
$518$	−3.19615	−0.140431
$519$	0	0
$520$	1.73205	0.0759555
$521$	43.8564	1.92138	0.960692	−	0.277616i	$-0.0895444\pi$
$521$	43.8564	1.92138	0.960692	+	0.277616i	$0.0895444\pi$
$522$	0	0
$523$	15.6077	0.682477	0.341238	−	0.939977i	$-0.389154\pi$
$523$	15.6077	0.682477	0.341238	+	0.939977i	$0.389154\pi$
$524$	−2.53590	−0.110781
$525$	0	0
$526$	15.1244	0.659453
$527$	27.1244	1.18156
$528$	0	0
$529$	−21.3923	−0.930100
$530$	−16.3923	−0.712036
$531$	0	0
$532$	−6.19615	−0.268637
$533$	−2.53590	−0.109842
$534$	0	0
$535$	24.0000	1.03761
$536$	4.19615	0.181246
$537$	0	0
$538$	29.4449	1.26946
$539$	4.73205	0.203824
$540$	0	0
$541$	−31.5885	−1.35809	−0.679047	−	0.734095i	$-0.737607\pi$
$541$	−31.5885	−1.35809	−0.679047	+	0.734095i	$0.737607\pi$
$542$	28.1962	1.21113
$543$	0	0
$544$	6.46410	0.277146
$545$	8.32051	0.356411
$546$	0	0
$547$	−19.8038	−0.846751	−0.423376	−	0.905954i	$-0.639155\pi$
$547$	−19.8038	−0.846751	−0.423376	+	0.905954i	$0.639155\pi$
$548$	−2.66025	−0.113640
$549$	0	0
$550$	−9.46410	−0.403551
$551$	−40.0526	−1.70630
$552$	0	0
$553$	−6.19615	−0.263487
$554$	−18.7846	−0.798082
$555$	0	0
$556$	−18.1962	−0.771689
$557$	40.8564	1.73114	0.865571	−	0.500787i	$-0.166956\pi$
$557$	40.8564	1.73114	0.865571	+	0.500787i	$0.166956\pi$
$558$	0	0
$559$	−12.3923	−0.524139
$560$	−1.73205	−0.0731925
$561$	0	0
$562$	−13.7321	−0.579252
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	17.7846	0.748204
$566$	3.60770	0.151643
$567$	0	0
$568$	−4.39230	−0.184297
$569$	−7.98076	−0.334571	−0.167285	−	0.985908i	$-0.553500\pi$
$569$	−7.98076	−0.334571	−0.167285	+	0.985908i	$0.553500\pi$
$570$	0	0
$571$	−19.8038	−0.828765	−0.414383	−	0.910103i	$-0.636002\pi$
$571$	−19.8038	−0.828765	−0.414383	+	0.910103i	$0.636002\pi$
$572$	−4.73205	−0.197857
$573$	0	0
$574$	2.53590	0.105846
$575$	−2.53590	−0.105754
$576$	0	0
$577$	31.1962	1.29871	0.649356	−	0.760484i	$-0.275038\pi$
$577$	31.1962	1.29871	0.649356	+	0.760484i	$0.275038\pi$
$578$	24.7846	1.03090
$579$	0	0
$580$	−11.1962	−0.464895
$581$	1.26795	0.0526034
$582$	0	0
$583$	44.7846	1.85479
$584$	−9.19615	−0.380539
$585$	0	0
$586$	−26.6603	−1.10132
$587$	−10.7321	−0.442959	−0.221480	−	0.975165i	$-0.571089\pi$
$587$	−10.7321	−0.442959	−0.221480	+	0.975165i	$0.571089\pi$
$588$	0	0
$589$	−26.0000	−1.07131
$590$	14.1962	0.584446
$591$	0	0
$592$	−3.19615	−0.131361
$593$	−40.8564	−1.67777	−0.838886	−	0.544308i	$-0.816792\pi$
$593$	−40.8564	−1.67777	−0.838886	+	0.544308i	$0.816792\pi$
$594$	0	0
$595$	−11.1962	−0.458997
$596$	−3.92820	−0.160905
$597$	0	0
$598$	−1.26795	−0.0518503
$599$	−3.80385	−0.155421	−0.0777105	−	0.996976i	$-0.524761\pi$
$599$	−3.80385	−0.155421	−0.0777105	+	0.996976i	$0.524761\pi$
$600$	0	0
$601$	−9.19615	−0.375119	−0.187559	−	0.982253i	$-0.560058\pi$
$601$	−9.19615	−0.375119	−0.187559	+	0.982253i	$0.560058\pi$
$602$	12.3923	0.505073
$603$	0	0
$604$	−22.5885	−0.919111
$605$	−19.7321	−0.802222
$606$	0	0
$607$	−28.5885	−1.16037	−0.580185	−	0.814485i	$-0.697020\pi$
$607$	−28.5885	−1.16037	−0.580185	+	0.814485i	$0.697020\pi$
$608$	−6.19615	−0.251287
$609$	0	0
$610$	−16.2679	−0.658670
$611$	−2.19615	−0.0888468
$612$	0	0
$613$	−6.78461	−0.274028	−0.137014	−	0.990569i	$-0.543751\pi$
$613$	−6.78461	−0.274028	−0.137014	+	0.990569i	$0.543751\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	4.73205	0.190660
$617$	16.5167	0.664936	0.332468	−	0.943115i	$-0.392119\pi$
$617$	16.5167	0.664936	0.332468	+	0.943115i	$0.392119\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	−7.26795	−0.291888
$621$	0	0
$622$	−22.9808	−0.921445
$623$	−12.4641	−0.499364
$624$	0	0
$625$	−11.0000	−0.440000
$626$	21.9808	0.878528
$627$	0	0
$628$	15.3923	0.614220
$629$	−20.6603	−0.823778
$630$	0	0
$631$	−22.5885	−0.899232	−0.449616	−	0.893222i	$-0.648439\pi$
$631$	−22.5885	−0.899232	−0.449616	+	0.893222i	$0.648439\pi$
$632$	−6.19615	−0.246470
$633$	0	0
$634$	−1.39230	−0.0552955
$635$	6.92820	0.274937
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	30.5885	1.21101
$639$	0	0
$640$	−1.73205	−0.0684653
$641$	9.33975	0.368898	0.184449	−	0.982842i	$-0.440950\pi$
$641$	9.33975	0.368898	0.184449	+	0.982842i	$0.440950\pi$
$642$	0	0
$643$	17.8038	0.702115	0.351058	−	0.936354i	$-0.385822\pi$
$643$	17.8038	0.702115	0.351058	+	0.936354i	$0.385822\pi$
$644$	1.26795	0.0499642
$645$	0	0
$646$	−40.0526	−1.57585
$647$	−23.3205	−0.916824	−0.458412	−	0.888740i	$-0.651582\pi$
$647$	−23.3205	−0.916824	−0.458412	+	0.888740i	$0.651582\pi$
$648$	0	0
$649$	−38.7846	−1.52243
$650$	2.00000	0.0784465
$651$	0	0
$652$	24.3923	0.955276
$653$	−35.3205	−1.38220	−0.691099	−	0.722760i	$-0.742873\pi$
$653$	−35.3205	−1.38220	−0.691099	+	0.722760i	$0.742873\pi$
$654$	0	0
$655$	4.39230	0.171622
$656$	2.53590	0.0990102
$657$	0	0
$658$	2.19615	0.0856149
$659$	0.679492	0.0264692	0.0132346	−	0.999912i	$-0.495787\pi$
$659$	0.679492	0.0264692	0.0132346	+	0.999912i	$0.495787\pi$
$660$	0	0
$661$	33.3923	1.29881	0.649405	−	0.760443i	$-0.275018\pi$
$661$	33.3923	1.29881	0.649405	+	0.760443i	$0.275018\pi$
$662$	−29.1769	−1.13399
$663$	0	0
$664$	1.26795	0.0492060
$665$	10.7321	0.416171
$666$	0	0
$667$	8.19615	0.317356
$668$	−7.26795	−0.281205
$669$	0	0
$670$	−7.26795	−0.280785
$671$	44.4449	1.71577
$672$	0	0
$673$	18.1769	0.700669	0.350334	−	0.936625i	$-0.386068\pi$
$673$	18.1769	0.700669	0.350334	+	0.936625i	$0.386068\pi$
$674$	−32.3923	−1.24770
$675$	0	0
$676$	−12.0000	−0.461538
$677$	44.7846	1.72121	0.860606	−	0.509271i	$-0.170085\pi$
$677$	44.7846	1.72121	0.860606	+	0.509271i	$0.170085\pi$
$678$	0	0
$679$	−16.0000	−0.614024
$680$	−11.1962	−0.429353
$681$	0	0
$682$	19.8564	0.760341
$683$	−39.7128	−1.51957	−0.759784	−	0.650175i	$-0.774695\pi$
$683$	−39.7128	−1.51957	−0.759784	+	0.650175i	$0.774695\pi$
$684$	0	0
$685$	4.60770	0.176051
$686$	1.00000	0.0381802
$687$	0	0
$688$	12.3923	0.472452
$689$	−9.46410	−0.360554
$690$	0	0
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	−12.8038	−0.486729
$693$	0	0
$694$	16.3923	0.622243
$695$	31.5167	1.19550
$696$	0	0
$697$	16.3923	0.620903
$698$	12.3923	0.469056
$699$	0	0
$700$	−2.00000	−0.0755929
$701$	46.1769	1.74408	0.872039	−	0.489436i	$-0.162797\pi$
$701$	46.1769	1.74408	0.872039	+	0.489436i	$0.162797\pi$
$702$	0	0
$703$	19.8038	0.746916
$704$	4.73205	0.178346
$705$	0	0
$706$	4.14359	0.155946
$707$	12.9282	0.486215
$708$	0	0
$709$	−7.58846	−0.284990	−0.142495	−	0.989795i	$-0.545513\pi$
$709$	−7.58846	−0.284990	−0.142495	+	0.989795i	$0.545513\pi$
$710$	7.60770	0.285512
$711$	0	0
$712$	−12.4641	−0.467112
$713$	5.32051	0.199255
$714$	0	0
$715$	8.19615	0.306519
$716$	7.26795	0.271616
$717$	0	0
$718$	18.9282	0.706394
$719$	23.3205	0.869708	0.434854	−	0.900501i	$-0.356800\pi$
$719$	23.3205	0.869708	0.434854	+	0.900501i	$0.356800\pi$
$720$	0	0
$721$	−8.39230	−0.312546
$722$	19.3923	0.721707
$723$	0	0
$724$	0.392305	0.0145799
$725$	−12.9282	−0.480141
$726$	0	0
$727$	15.6077	0.578857	0.289429	−	0.957200i	$-0.406535\pi$
$727$	15.6077	0.578857	0.289429	+	0.957200i	$0.406535\pi$
$728$	−1.00000	−0.0370625
$729$	0	0
$730$	15.9282	0.589529
$731$	80.1051	2.96279
$732$	0	0
$733$	−29.1769	−1.07767	−0.538837	−	0.842410i	$-0.681136\pi$
$733$	−29.1769	−1.07767	−0.538837	+	0.842410i	$0.681136\pi$
$734$	−4.58846	−0.169363
$735$	0	0
$736$	1.26795	0.0467372
$737$	19.8564	0.731420
$738$	0	0
$739$	−24.1962	−0.890070	−0.445035	−	0.895513i	$-0.646809\pi$
$739$	−24.1962	−0.890070	−0.445035	+	0.895513i	$0.646809\pi$
$740$	5.53590	0.203504
$741$	0	0
$742$	9.46410	0.347438
$743$	−21.4641	−0.787442	−0.393721	−	0.919230i	$-0.628812\pi$
$743$	−21.4641	−0.787442	−0.393721	+	0.919230i	$0.628812\pi$
$744$	0	0
$745$	6.80385	0.249274
$746$	20.0000	0.732252
$747$	0	0
$748$	30.5885	1.11842
$749$	−13.8564	−0.506302
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	2.19615	0.0800854
$753$	0	0
$754$	−6.46410	−0.235409
$755$	39.1244	1.42388
$756$	0	0
$757$	4.78461	0.173900	0.0869498	−	0.996213i	$-0.472288\pi$
$757$	4.78461	0.173900	0.0869498	+	0.996213i	$0.472288\pi$
$758$	−16.5885	−0.602520
$759$	0	0
$760$	10.7321	0.389292
$761$	−8.32051	−0.301618	−0.150809	−	0.988563i	$-0.548188\pi$
$761$	−8.32051	−0.301618	−0.150809	+	0.988563i	$0.548188\pi$
$762$	0	0
$763$	−4.80385	−0.173911
$764$	24.5885	0.889579
$765$	0	0
$766$	−10.1436	−0.366503
$767$	8.19615	0.295946
$768$	0	0
$769$	7.19615	0.259500	0.129750	−	0.991547i	$-0.458583\pi$
$769$	7.19615	0.259500	0.129750	+	0.991547i	$0.458583\pi$
$770$	−8.19615	−0.295369
$771$	0	0
$772$	−19.0000	−0.683825
$773$	−21.3397	−0.767537	−0.383769	−	0.923429i	$-0.625374\pi$
$773$	−21.3397	−0.767537	−0.383769	+	0.923429i	$0.625374\pi$
$774$	0	0
$775$	−8.39230	−0.301460
$776$	−16.0000	−0.574367
$777$	0	0
$778$	−30.2487	−1.08447
$779$	−15.7128	−0.562970
$780$	0	0
$781$	−20.7846	−0.743732
$782$	8.19615	0.293094
$783$	0	0
$784$	1.00000	0.0357143
$785$	−26.6603	−0.951545
$786$	0	0
$787$	45.1769	1.61038	0.805192	−	0.593015i	$-0.202062\pi$
$787$	45.1769	1.61038	0.805192	+	0.593015i	$0.202062\pi$
$788$	−15.0000	−0.534353
$789$	0	0
$790$	10.7321	0.381829
$791$	−10.2679	−0.365086
$792$	0	0
$793$	−9.39230	−0.333531
$794$	−29.3923	−1.04309
$795$	0	0
$796$	20.5885	0.729739
$797$	21.3397	0.755893	0.377946	−	0.925828i	$-0.376630\pi$
$797$	21.3397	0.755893	0.377946	+	0.925828i	$0.376630\pi$
$798$	0	0
$799$	14.1962	0.502224
$800$	−2.00000	−0.0707107
$801$	0	0
$802$	22.2679	0.786309
$803$	−43.5167	−1.53567
$804$	0	0
$805$	−2.19615	−0.0774042
$806$	−4.19615	−0.147803
$807$	0	0
$808$	12.9282	0.454813
$809$	−0.124356	−0.00437211	−0.00218606	−	0.999998i	$-0.500696\pi$
$809$	−0.124356	−0.00437211	−0.00218606	+	0.999998i	$0.500696\pi$
$810$	0	0
$811$	54.9808	1.93064	0.965318	−	0.261078i	$-0.0840778\pi$
$811$	54.9808	1.93064	0.965318	+	0.261078i	$0.0840778\pi$
$812$	6.46410	0.226845
$813$	0	0
$814$	−15.1244	−0.530108
$815$	−42.2487	−1.47991
$816$	0	0
$817$	−76.7846	−2.68635
$818$	−21.1962	−0.741106
$819$	0	0
$820$	−4.39230	−0.153386
$821$	−15.0000	−0.523504	−0.261752	−	0.965135i	$-0.584300\pi$
$821$	−15.0000	−0.523504	−0.261752	+	0.965135i	$0.584300\pi$
$822$	0	0
$823$	−24.7846	−0.863937	−0.431969	−	0.901889i	$-0.642181\pi$
$823$	−24.7846	−0.863937	−0.431969	+	0.901889i	$0.642181\pi$
$824$	−8.39230	−0.292360
$825$	0	0
$826$	−8.19615	−0.285181
$827$	19.6077	0.681826	0.340913	−	0.940095i	$-0.389264\pi$
$827$	19.6077	0.681826	0.340913	+	0.940095i	$0.389264\pi$
$828$	0	0
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	−2.19615	−0.0762296
$831$	0	0
$832$	−1.00000	−0.0346688
$833$	6.46410	0.223968
$834$	0	0
$835$	12.5885	0.435642
$836$	−29.3205	−1.01407
$837$	0	0
$838$	23.3205	0.805594
$839$	42.2487	1.45859	0.729294	−	0.684201i	$-0.239849\pi$
$839$	42.2487	1.45859	0.729294	+	0.684201i	$0.239849\pi$
$840$	0	0
$841$	12.7846	0.440849
$842$	13.1962	0.454769
$843$	0	0
$844$	11.8038	0.406305
$845$	20.7846	0.715012
$846$	0	0
$847$	11.3923	0.391444
$848$	9.46410	0.324999
$849$	0	0
$850$	−12.9282	−0.443434
$851$	−4.05256	−0.138920
$852$	0	0
$853$	2.00000	0.0684787	0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000	0.0684787	0.0342393	+	0.999414i	$0.489099\pi$
$854$	9.39230	0.321398
$855$	0	0
$856$	−13.8564	−0.473602
$857$	−18.4641	−0.630722	−0.315361	−	0.948972i	$-0.602126\pi$
$857$	−18.4641	−0.630722	−0.315361	+	0.948972i	$0.602126\pi$
$858$	0	0
$859$	0.392305	0.0133853	0.00669263	−	0.999978i	$-0.497870\pi$
$859$	0.392305	0.0133853	0.00669263	+	0.999978i	$0.497870\pi$
$860$	−21.4641	−0.731920
$861$	0	0
$862$	0.679492	0.0231436
$863$	23.9090	0.813871	0.406935	−	0.913457i	$-0.366597\pi$
$863$	23.9090	0.813871	0.406935	+	0.913457i	$0.366597\pi$
$864$	0	0
$865$	22.1769	0.754038
$866$	−31.5885	−1.07342
$867$	0	0
$868$	4.19615	0.142427
$869$	−29.3205	−0.994630
$870$	0	0
$871$	−4.19615	−0.142181
$872$	−4.80385	−0.162679
$873$	0	0
$874$	−7.85641	−0.265747
$875$	12.1244	0.409878
$876$	0	0
$877$	31.1962	1.05342	0.526710	−	0.850045i	$-0.323426\pi$
$877$	31.1962	1.05342	0.526710	+	0.850045i	$0.323426\pi$
$878$	21.1769	0.714686
$879$	0	0
$880$	−8.19615	−0.276292
$881$	−25.1769	−0.848232	−0.424116	−	0.905608i	$-0.639415\pi$
$881$	−25.1769	−0.848232	−0.424116	+	0.905608i	$0.639415\pi$
$882$	0	0
$883$	12.9808	0.436837	0.218419	−	0.975855i	$-0.429910\pi$
$883$	12.9808	0.436837	0.218419	+	0.975855i	$0.429910\pi$
$884$	−6.46410	−0.217411
$885$	0	0
$886$	2.19615	0.0737812
$887$	35.6603	1.19735	0.598677	−	0.800990i	$-0.295693\pi$
$887$	35.6603	1.19735	0.598677	+	0.800990i	$0.295693\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	21.5885	0.723647
$891$	0	0
$892$	12.3923	0.414925
$893$	−13.6077	−0.455364
$894$	0	0
$895$	−12.5885	−0.420786
$896$	1.00000	0.0334077
$897$	0	0
$898$	6.00000	0.200223
$899$	27.1244	0.904648
$900$	0	0
$901$	61.1769	2.03810
$902$	12.0000	0.399556
$903$	0	0
$904$	−10.2679	−0.341507
$905$	−0.679492	−0.0225871
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	−5.07180	−0.168313
$909$	0	0
$910$	1.73205	0.0574169
$911$	−32.1051	−1.06369	−0.531845	−	0.846842i	$-0.678501\pi$
$911$	−32.1051	−1.06369	−0.531845	+	0.846842i	$0.678501\pi$
$912$	0	0
$913$	6.00000	0.198571
$914$	−1.00000	−0.0330771
$915$	0	0
$916$	−21.7846	−0.719784
$917$	−2.53590	−0.0837427
$918$	0	0
$919$	−37.8038	−1.24703	−0.623517	−	0.781810i	$-0.714297\pi$
$919$	−37.8038	−1.24703	−0.623517	+	0.781810i	$0.714297\pi$
$920$	−2.19615	−0.0724050
$921$	0	0
$922$	11.0718	0.364630
$923$	4.39230	0.144574
$924$	0	0
$925$	6.39230	0.210178
$926$	5.80385	0.190726
$927$	0	0
$928$	6.46410	0.212195
$929$	7.39230	0.242534	0.121267	−	0.992620i	$-0.461304\pi$
$929$	7.39230	0.242534	0.121267	+	0.992620i	$0.461304\pi$
$930$	0	0
$931$	−6.19615	−0.203071
$932$	13.7321	0.449808
$933$	0	0
$934$	−1.26795	−0.0414886
$935$	−52.9808	−1.73266
$936$	0	0
$937$	−22.8038	−0.744969	−0.372485	−	0.928038i	$-0.621494\pi$
$937$	−22.8038	−0.744969	−0.372485	+	0.928038i	$0.621494\pi$
$938$	4.19615	0.137009
$939$	0	0
$940$	−3.80385	−0.124068
$941$	−46.7654	−1.52451	−0.762254	−	0.647278i	$-0.775907\pi$
$941$	−46.7654	−1.52451	−0.762254	+	0.647278i	$0.775907\pi$
$942$	0	0
$943$	3.21539	0.104708
$944$	−8.19615	−0.266762
$945$	0	0
$946$	58.6410	1.90658
$947$	−7.60770	−0.247217	−0.123608	−	0.992331i	$-0.539447\pi$
$947$	−7.60770	−0.247217	−0.123608	+	0.992331i	$0.539447\pi$
$948$	0	0
$949$	9.19615	0.298520
$950$	12.3923	0.402059
$951$	0	0
$952$	6.46410	0.209503
$953$	−55.9808	−1.81339	−0.906697	−	0.421782i	$-0.861405\pi$
$953$	−55.9808	−1.81339	−0.906697	+	0.421782i	$0.861405\pi$
$954$	0	0
$955$	−42.5885	−1.37813
$956$	15.1244	0.489157
$957$	0	0
$958$	16.7321	0.540588
$959$	−2.66025	−0.0859041
$960$	0	0
$961$	−13.3923	−0.432010
$962$	3.19615	0.103048
$963$	0	0
$964$	−13.5885	−0.437655
$965$	32.9090	1.05938
$966$	0	0
$967$	−22.5885	−0.726396	−0.363198	−	0.931712i	$-0.618315\pi$
$967$	−22.5885	−0.726396	−0.363198	+	0.931712i	$0.618315\pi$
$968$	11.3923	0.366163
$969$	0	0
$970$	27.7128	0.889805
$971$	3.12436	0.100265	0.0501327	−	0.998743i	$-0.484036\pi$
$971$	3.12436	0.100265	0.0501327	+	0.998743i	$0.484036\pi$
$972$	0	0
$973$	−18.1962	−0.583342
$974$	−6.19615	−0.198538
$975$	0	0
$976$	9.39230	0.300640
$977$	35.5692	1.13796	0.568980	−	0.822351i	$-0.307338\pi$
$977$	35.5692	1.13796	0.568980	+	0.822351i	$0.307338\pi$
$978$	0	0
$979$	−58.9808	−1.88503
$980$	−1.73205	−0.0553283
$981$	0	0
$982$	11.3205	0.361252
$983$	−41.0718	−1.30999	−0.654993	−	0.755635i	$-0.727329\pi$
$983$	−41.0718	−1.30999	−0.654993	+	0.755635i	$0.727329\pi$
$984$	0	0
$985$	25.9808	0.827816
$986$	41.7846	1.33069
$987$	0	0
$988$	6.19615	0.197126
$989$	15.7128	0.499638
$990$	0	0
$991$	42.9808	1.36533	0.682664	−	0.730732i	$-0.260821\pi$
$991$	42.9808	1.36533	0.682664	+	0.730732i	$0.260821\pi$
$992$	4.19615	0.133228
$993$	0	0
$994$	−4.39230	−0.139315
$995$	−35.6603	−1.13051
$996$	0	0
$997$	−21.7846	−0.689926	−0.344963	−	0.938616i	$-0.612108\pi$
$997$	−21.7846	−0.689926	−0.344963	+	0.938616i	$0.612108\pi$
$998$	20.5885	0.651716
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1134.2.a.o.1.1	yes	2
3.2	odd	2		1134.2.a.j.1.2	✓	2
4.3	odd	2		9072.2.a.bf.1.1		2
7.6	odd	2		7938.2.a.br.1.2		2
9.2	odd	6		1134.2.f.t.757.1		4
9.4	even	3		1134.2.f.q.379.2		4
9.5	odd	6		1134.2.f.t.379.1		4
9.7	even	3		1134.2.f.q.757.2		4
12.11	even	2		9072.2.a.bi.1.2		2
21.20	even	2		7938.2.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1134.2.a.j.1.2	✓	2	3.2	odd	2
1134.2.a.o.1.1	yes	2	1.1	even	1	trivial
1134.2.f.q.379.2		4	9.4	even	3
1134.2.f.q.757.2		4	9.7	even	3
1134.2.f.t.379.1		4	9.5	odd	6
1134.2.f.t.757.1		4	9.2	odd	6
7938.2.a.bi.1.1		2	21.20	even	2
7938.2.a.br.1.2		2	7.6	odd	2
9072.2.a.bf.1.1		2	4.3	odd	2
9072.2.a.bi.1.2		2	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:	\( N \)	\(=\)	\( 1134 = 2 \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1134.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.73205 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.73205 q^{5} +1.00000 q^{7} +1.00000 q^{8} -1.73205 q^{10} +4.73205 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} +6.46410 q^{17} -6.19615 q^{19} -1.73205 q^{20} +4.73205 q^{22} +1.26795 q^{23} -2.00000 q^{25} -1.00000 q^{26} +1.00000 q^{28} +6.46410 q^{29} +4.19615 q^{31} +1.00000 q^{32} +6.46410 q^{34} -1.73205 q^{35} -3.19615 q^{37} -6.19615 q^{38} -1.73205 q^{40} +2.53590 q^{41} +12.3923 q^{43} +4.73205 q^{44} +1.26795 q^{46} +2.19615 q^{47} +1.00000 q^{49} -2.00000 q^{50} -1.00000 q^{52} +9.46410 q^{53} -8.19615 q^{55} +1.00000 q^{56} +6.46410 q^{58} -8.19615 q^{59} +9.39230 q^{61} +4.19615 q^{62} +1.00000 q^{64} +1.73205 q^{65} +4.19615 q^{67} +6.46410 q^{68} -1.73205 q^{70} -4.39230 q^{71} -9.19615 q^{73} -3.19615 q^{74} -6.19615 q^{76} +4.73205 q^{77} -6.19615 q^{79} -1.73205 q^{80} +2.53590 q^{82} +1.26795 q^{83} -11.1962 q^{85} +12.3923 q^{86} +4.73205 q^{88} -12.4641 q^{89} -1.00000 q^{91} +1.26795 q^{92} +2.19615 q^{94} +10.7321 q^{95} -16.0000 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 6 q^{11} - 2 q^{13} + 2 q^{14} + 2 q^{16} + 6 q^{17} - 2 q^{19} + 6 q^{22} + 6 q^{23} - 4 q^{25} - 2 q^{26} + 2 q^{28} + 6 q^{29} - 2 q^{31} + 2 q^{32} + 6 q^{34} + 4 q^{37} - 2 q^{38} + 12 q^{41} + 4 q^{43} + 6 q^{44} + 6 q^{46} - 6 q^{47} + 2 q^{49} - 4 q^{50} - 2 q^{52} + 12 q^{53} - 6 q^{55} + 2 q^{56} + 6 q^{58} - 6 q^{59} - 2 q^{61} - 2 q^{62} + 2 q^{64} - 2 q^{67} + 6 q^{68} + 12 q^{71} - 8 q^{73} + 4 q^{74} - 2 q^{76} + 6 q^{77} - 2 q^{79} + 12 q^{82} + 6 q^{83} - 12 q^{85} + 4 q^{86} + 6 q^{88} - 18 q^{89} - 2 q^{91} + 6 q^{92} - 6 q^{94} + 18 q^{95} - 32 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8 + 6 * q^11 - 2 * q^13 + 2 * q^14 + 2 * q^16 + 6 * q^17 - 2 * q^19 + 6 * q^22 + 6 * q^23 - 4 * q^25 - 2 * q^26 + 2 * q^28 + 6 * q^29 - 2 * q^31 + 2 * q^32 + 6 * q^34 + 4 * q^37 - 2 * q^38 + 12 * q^41 + 4 * q^43 + 6 * q^44 + 6 * q^46 - 6 * q^47 + 2 * q^49 - 4 * q^50 - 2 * q^52 + 12 * q^53 - 6 * q^55 + 2 * q^56 + 6 * q^58 - 6 * q^59 - 2 * q^61 - 2 * q^62 + 2 * q^64 - 2 * q^67 + 6 * q^68 + 12 * q^71 - 8 * q^73 + 4 * q^74 - 2 * q^76 + 6 * q^77 - 2 * q^79 + 12 * q^82 + 6 * q^83 - 12 * q^85 + 4 * q^86 + 6 * q^88 - 18 * q^89 - 2 * q^91 + 6 * q^92 - 6 * q^94 + 18 * q^95 - 32 * q^97 + 2 * q^98

Introduction

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