LMFDB - Embedded newform 1176.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1176,2,Mod(1,1176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1176, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1176.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1176.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:	$9.39040727770$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
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			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1176.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -0.585786 q^{5} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.585786 q^{5} +1.00000 q^{9} -0.828427 q^{11} +1.41421 q^{13} +0.585786 q^{15} -2.24264 q^{17} -6.82843 q^{19} +4.82843 q^{23} -4.65685 q^{25} -1.00000 q^{27} +8.48528 q^{29} -5.17157 q^{31} +0.828427 q^{33} +1.65685 q^{37} -1.41421 q^{39} +0.585786 q^{41} -8.00000 q^{43} -0.585786 q^{45} -6.82843 q^{47} +2.24264 q^{51} -13.3137 q^{53} +0.485281 q^{55} +6.82843 q^{57} -5.17157 q^{59} -13.8995 q^{61} -0.828427 q^{65} -8.00000 q^{67} -4.82843 q^{69} -0.828427 q^{71} +11.0711 q^{73} +4.65685 q^{75} -2.34315 q^{79} +1.00000 q^{81} -15.3137 q^{83} +1.31371 q^{85} -8.48528 q^{87} +10.7279 q^{89} +5.17157 q^{93} +4.00000 q^{95} +7.75736 q^{97} -0.828427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} + 4 q^{11} + 4 q^{15} + 4 q^{17} - 8 q^{19} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 16 q^{31} - 4 q^{33} - 8 q^{37} + 4 q^{41} - 16 q^{43} - 4 q^{45} - 8 q^{47} - 4 q^{51} - 4 q^{53} - 16 q^{55} + 8 q^{57} - 16 q^{59} - 8 q^{61} + 4 q^{65} - 16 q^{67} - 4 q^{69} + 4 q^{71} + 8 q^{73} - 2 q^{75} - 16 q^{79} + 2 q^{81} - 8 q^{83} - 20 q^{85} - 4 q^{89} + 16 q^{93} + 8 q^{95} + 24 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 + 4 * q^11 + 4 * q^15 + 4 * q^17 - 8 * q^19 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 16 * q^31 - 4 * q^33 - 8 * q^37 + 4 * q^41 - 16 * q^43 - 4 * q^45 - 8 * q^47 - 4 * q^51 - 4 * q^53 - 16 * q^55 + 8 * q^57 - 16 * q^59 - 8 * q^61 + 4 * q^65 - 16 * q^67 - 4 * q^69 + 4 * q^71 + 8 * q^73 - 2 * q^75 - 16 * q^79 + 2 * q^81 - 8 * q^83 - 20 * q^85 - 4 * q^89 + 16 * q^93 + 8 * q^95 + 24 * q^97 + 4 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−0.585786	−0.261972	−0.130986	−	0.991384i	$-0.541814\pi$
$5$	−0.585786	−0.261972	−0.130986	+	0.991384i	$0.541814\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	0	0
$13$	1.41421	0.392232	0.196116	−	0.980581i	$-0.437167\pi$
$13$	1.41421	0.392232	0.196116	+	0.980581i	$0.437167\pi$
$14$	0	0
$15$	0.585786	0.151249
$16$	0	0
$17$	−2.24264	−0.543920	−0.271960	−	0.962309i	$-0.587672\pi$
$17$	−2.24264	−0.543920	−0.271960	+	0.962309i	$0.587672\pi$
$18$	0	0
$19$	−6.82843	−1.56655	−0.783274	−	0.621676i	$-0.786452\pi$
$19$	−6.82843	−1.56655	−0.783274	+	0.621676i	$0.786452\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.82843	1.00680	0.503398	−	0.864054i	$-0.332083\pi$
$23$	4.82843	1.00680	0.503398	+	0.864054i	$0.332083\pi$
$24$	0	0
$25$	−4.65685	−0.931371
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.48528	1.57568	0.787839	−	0.615882i	$-0.211200\pi$
$29$	8.48528	1.57568	0.787839	+	0.615882i	$0.211200\pi$
$30$	0	0
$31$	−5.17157	−0.928842	−0.464421	−	0.885615i	$-0.653738\pi$
$31$	−5.17157	−0.928842	−0.464421	+	0.885615i	$0.653738\pi$
$32$	0	0
$33$	0.828427	0.144211
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.65685	0.272385	0.136193	−	0.990682i	$-0.456513\pi$
$37$	1.65685	0.272385	0.136193	+	0.990682i	$0.456513\pi$
$38$	0	0
$39$	−1.41421	−0.226455
$40$	0	0
$41$	0.585786	0.0914845	0.0457422	−	0.998953i	$-0.485435\pi$
$41$	0.585786	0.0914845	0.0457422	+	0.998953i	$0.485435\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	−0.585786	−0.0873239
$46$	0	0
$47$	−6.82843	−0.996028	−0.498014	−	0.867169i	$-0.665937\pi$
$47$	−6.82843	−0.996028	−0.498014	+	0.867169i	$0.665937\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	2.24264	0.314033
$52$	0	0
$53$	−13.3137	−1.82878	−0.914389	−	0.404836i	$-0.867329\pi$
$53$	−13.3137	−1.82878	−0.914389	+	0.404836i	$0.867329\pi$
$54$	0	0
$55$	0.485281	0.0654353
$56$	0	0
$57$	6.82843	0.904447
$58$	0	0
$59$	−5.17157	−0.673281	−0.336641	−	0.941633i	$-0.609291\pi$
$59$	−5.17157	−0.673281	−0.336641	+	0.941633i	$0.609291\pi$
$60$	0	0
$61$	−13.8995	−1.77965	−0.889824	−	0.456304i	$-0.849173\pi$
$61$	−13.8995	−1.77965	−0.889824	+	0.456304i	$0.849173\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.828427	−0.102754
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	−4.82843	−0.581274
$70$	0	0
$71$	−0.828427	−0.0983162	−0.0491581	−	0.998791i	$-0.515654\pi$
$71$	−0.828427	−0.0983162	−0.0491581	+	0.998791i	$0.515654\pi$
$72$	0	0
$73$	11.0711	1.29577	0.647885	−	0.761738i	$-0.275654\pi$
$73$	11.0711	1.29577	0.647885	+	0.761738i	$0.275654\pi$
$74$	0	0
$75$	4.65685	0.537727
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.34315	−0.263624	−0.131812	−	0.991275i	$-0.542080\pi$
$79$	−2.34315	−0.263624	−0.131812	+	0.991275i	$0.542080\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−15.3137	−1.68090	−0.840449	−	0.541891i	$-0.817709\pi$
$83$	−15.3137	−1.68090	−0.840449	+	0.541891i	$0.817709\pi$
$84$	0	0
$85$	1.31371	0.142492
$86$	0	0
$87$	−8.48528	−0.909718
$88$	0	0
$89$	10.7279	1.13716	0.568579	−	0.822629i	$-0.307493\pi$
$89$	10.7279	1.13716	0.568579	+	0.822629i	$0.307493\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	5.17157	0.536267
$94$	0	0
$95$	4.00000	0.410391
$96$	0	0
$97$	7.75736	0.787641	0.393820	−	0.919187i	$-0.371153\pi$
$97$	7.75736	0.787641	0.393820	+	0.919187i	$0.371153\pi$
$98$	0	0
$99$	−0.828427	−0.0832601
$100$	0	0
$101$	−3.41421	−0.339727	−0.169863	−	0.985468i	$-0.554333\pi$
$101$	−3.41421	−0.339727	−0.169863	+	0.985468i	$0.554333\pi$
$102$	0	0
$103$	−10.8284	−1.06696	−0.533478	−	0.845814i	$-0.679115\pi$
$103$	−10.8284	−1.06696	−0.533478	+	0.845814i	$0.679115\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	14.4853	1.40035	0.700173	−	0.713974i	$-0.253106\pi$
$107$	14.4853	1.40035	0.700173	+	0.713974i	$0.253106\pi$
$108$	0	0
$109$	−11.3137	−1.08366	−0.541828	−	0.840489i	$-0.682268\pi$
$109$	−11.3137	−1.08366	−0.541828	+	0.840489i	$0.682268\pi$
$110$	0	0
$111$	−1.65685	−0.157262
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	−2.82843	−0.263752
$116$	0	0
$117$	1.41421	0.130744
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	−0.585786	−0.0528186
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	−15.3137	−1.35887	−0.679436	−	0.733735i	$-0.737775\pi$
$127$	−15.3137	−1.35887	−0.679436	+	0.733735i	$0.737775\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	7.31371	0.639002	0.319501	−	0.947586i	$-0.396485\pi$
$131$	7.31371	0.639002	0.319501	+	0.947586i	$0.396485\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0.585786	0.0504165
$136$	0	0
$137$	4.48528	0.383203	0.191602	−	0.981473i	$-0.438632\pi$
$137$	4.48528	0.383203	0.191602	+	0.981473i	$0.438632\pi$
$138$	0	0
$139$	1.65685	0.140533	0.0702663	−	0.997528i	$-0.477615\pi$
$139$	1.65685	0.140533	0.0702663	+	0.997528i	$0.477615\pi$
$140$	0	0
$141$	6.82843	0.575057
$142$	0	0
$143$	−1.17157	−0.0979718
$144$	0	0
$145$	−4.97056	−0.412783
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	0	0
$151$	9.65685	0.785864	0.392932	−	0.919568i	$-0.371461\pi$
$151$	9.65685	0.785864	0.392932	+	0.919568i	$0.371461\pi$
$152$	0	0
$153$	−2.24264	−0.181307
$154$	0	0
$155$	3.02944	0.243330
$156$	0	0
$157$	13.8995	1.10930	0.554650	−	0.832084i	$-0.312852\pi$
$157$	13.8995	1.10930	0.554650	+	0.832084i	$0.312852\pi$
$158$	0	0
$159$	13.3137	1.05585
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−13.6569	−1.06969	−0.534844	−	0.844951i	$-0.679630\pi$
$163$	−13.6569	−1.06969	−0.534844	+	0.844951i	$0.679630\pi$
$164$	0	0
$165$	−0.485281	−0.0377791
$166$	0	0
$167$	−1.17157	−0.0906590	−0.0453295	−	0.998972i	$-0.514434\pi$
$167$	−1.17157	−0.0906590	−0.0453295	+	0.998972i	$0.514434\pi$
$168$	0	0
$169$	−11.0000	−0.846154
$170$	0	0
$171$	−6.82843	−0.522183
$172$	0	0
$173$	3.41421	0.259578	0.129789	−	0.991542i	$-0.458570\pi$
$173$	3.41421	0.259578	0.129789	+	0.991542i	$0.458570\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.17157	0.388719
$178$	0	0
$179$	17.7990	1.33036	0.665179	−	0.746684i	$-0.268355\pi$
$179$	17.7990	1.33036	0.665179	+	0.746684i	$0.268355\pi$
$180$	0	0
$181$	9.89949	0.735824	0.367912	−	0.929861i	$-0.380073\pi$
$181$	9.89949	0.735824	0.367912	+	0.929861i	$0.380073\pi$
$182$	0	0
$183$	13.8995	1.02748
$184$	0	0
$185$	−0.970563	−0.0713572
$186$	0	0
$187$	1.85786	0.135860
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.1716	1.09778	0.548888	−	0.835896i	$-0.315051\pi$
$191$	15.1716	1.09778	0.548888	+	0.835896i	$0.315051\pi$
$192$	0	0
$193$	24.6274	1.77272	0.886360	−	0.462996i	$-0.153226\pi$
$193$	24.6274	1.77272	0.886360	+	0.462996i	$0.153226\pi$
$194$	0	0
$195$	0.828427	0.0593249
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	−5.65685	−0.401004	−0.200502	−	0.979693i	$-0.564257\pi$
$199$	−5.65685	−0.401004	−0.200502	+	0.979693i	$0.564257\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−0.343146	−0.0239663
$206$	0	0
$207$	4.82843	0.335599
$208$	0	0
$209$	5.65685	0.391293
$210$	0	0
$211$	−14.3431	−0.987423	−0.493711	−	0.869626i	$-0.664360\pi$
$211$	−14.3431	−0.987423	−0.493711	+	0.869626i	$0.664360\pi$
$212$	0	0
$213$	0.828427	0.0567629
$214$	0	0
$215$	4.68629	0.319602
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−11.0711	−0.748113
$220$	0	0
$221$	−3.17157	−0.213343
$222$	0	0
$223$	13.6569	0.914531	0.457265	−	0.889330i	$-0.348829\pi$
$223$	13.6569	0.914531	0.457265	+	0.889330i	$0.348829\pi$
$224$	0	0
$225$	−4.65685	−0.310457
$226$	0	0
$227$	19.7990	1.31411	0.657053	−	0.753845i	$-0.271803\pi$
$227$	19.7990	1.31411	0.657053	+	0.753845i	$0.271803\pi$
$228$	0	0
$229$	−15.0711	−0.995924	−0.497962	−	0.867199i	$-0.665918\pi$
$229$	−15.0711	−0.995924	−0.497962	+	0.867199i	$0.665918\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−28.4853	−1.86613	−0.933066	−	0.359704i	$-0.882878\pi$
$233$	−28.4853	−1.86613	−0.933066	+	0.359704i	$0.882878\pi$
$234$	0	0
$235$	4.00000	0.260931
$236$	0	0
$237$	2.34315	0.152204
$238$	0	0
$239$	−3.17157	−0.205152	−0.102576	−	0.994725i	$-0.532708\pi$
$239$	−3.17157	−0.205152	−0.102576	+	0.994725i	$0.532708\pi$
$240$	0	0
$241$	21.8995	1.41067	0.705335	−	0.708874i	$-0.250796\pi$
$241$	21.8995	1.41067	0.705335	+	0.708874i	$0.250796\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−9.65685	−0.614451
$248$	0	0
$249$	15.3137	0.970467
$250$	0	0
$251$	−8.48528	−0.535586	−0.267793	−	0.963476i	$-0.586294\pi$
$251$	−8.48528	−0.535586	−0.267793	+	0.963476i	$0.586294\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	0	0
$255$	−1.31371	−0.0822676
$256$	0	0
$257$	30.2426	1.88648	0.943242	−	0.332106i	$-0.107759\pi$
$257$	30.2426	1.88648	0.943242	+	0.332106i	$0.107759\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	8.48528	0.525226
$262$	0	0
$263$	24.8284	1.53099	0.765493	−	0.643444i	$-0.222495\pi$
$263$	24.8284	1.53099	0.765493	+	0.643444i	$0.222495\pi$
$264$	0	0
$265$	7.79899	0.479088
$266$	0	0
$267$	−10.7279	−0.656538
$268$	0	0
$269$	−30.0416	−1.83167	−0.915835	−	0.401554i	$-0.868470\pi$
$269$	−30.0416	−1.83167	−0.915835	+	0.401554i	$0.868470\pi$
$270$	0	0
$271$	13.1716	0.800116	0.400058	−	0.916490i	$-0.368990\pi$
$271$	13.1716	0.800116	0.400058	+	0.916490i	$0.368990\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.85786	0.232638
$276$	0	0
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	0	0
$279$	−5.17157	−0.309614
$280$	0	0
$281$	−12.4853	−0.744809	−0.372405	−	0.928070i	$-0.621467\pi$
$281$	−12.4853	−0.744809	−0.372405	+	0.928070i	$0.621467\pi$
$282$	0	0
$283$	14.8284	0.881458	0.440729	−	0.897640i	$-0.354720\pi$
$283$	14.8284	0.881458	0.440729	+	0.897640i	$0.354720\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−11.9706	−0.704151
$290$	0	0
$291$	−7.75736	−0.454744
$292$	0	0
$293$	−1.07107	−0.0625724	−0.0312862	−	0.999510i	$-0.509960\pi$
$293$	−1.07107	−0.0625724	−0.0312862	+	0.999510i	$0.509960\pi$
$294$	0	0
$295$	3.02944	0.176381
$296$	0	0
$297$	0.828427	0.0480702
$298$	0	0
$299$	6.82843	0.394898
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3.41421	0.196141
$304$	0	0
$305$	8.14214	0.466217
$306$	0	0
$307$	11.5147	0.657180	0.328590	−	0.944473i	$-0.393427\pi$
$307$	11.5147	0.657180	0.328590	+	0.944473i	$0.393427\pi$
$308$	0	0
$309$	10.8284	0.616008
$310$	0	0
$311$	26.1421	1.48238	0.741192	−	0.671293i	$-0.234261\pi$
$311$	26.1421	1.48238	0.741192	+	0.671293i	$0.234261\pi$
$312$	0	0
$313$	17.4142	0.984310	0.492155	−	0.870508i	$-0.336209\pi$
$313$	17.4142	0.984310	0.492155	+	0.870508i	$0.336209\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.3137	1.19710	0.598549	−	0.801087i	$-0.295744\pi$
$317$	21.3137	1.19710	0.598549	+	0.801087i	$0.295744\pi$
$318$	0	0
$319$	−7.02944	−0.393573
$320$	0	0
$321$	−14.4853	−0.808490
$322$	0	0
$323$	15.3137	0.852078
$324$	0	0
$325$	−6.58579	−0.365314
$326$	0	0
$327$	11.3137	0.625650
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−8.68629	−0.477442	−0.238721	−	0.971088i	$-0.576728\pi$
$331$	−8.68629	−0.477442	−0.238721	+	0.971088i	$0.576728\pi$
$332$	0	0
$333$	1.65685	0.0907951
$334$	0	0
$335$	4.68629	0.256039
$336$	0	0
$337$	−16.9706	−0.924445	−0.462223	−	0.886764i	$-0.652948\pi$
$337$	−16.9706	−0.924445	−0.462223	+	0.886764i	$0.652948\pi$
$338$	0	0
$339$	−6.00000	−0.325875
$340$	0	0
$341$	4.28427	0.232006
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2.82843	0.152277
$346$	0	0
$347$	−4.14214	−0.222361	−0.111181	−	0.993800i	$-0.535463\pi$
$347$	−4.14214	−0.222361	−0.111181	+	0.993800i	$0.535463\pi$
$348$	0	0
$349$	−30.3848	−1.62646	−0.813230	−	0.581943i	$-0.802293\pi$
$349$	−30.3848	−1.62646	−0.813230	+	0.581943i	$0.802293\pi$
$350$	0	0
$351$	−1.41421	−0.0754851
$352$	0	0
$353$	15.8995	0.846245	0.423122	−	0.906073i	$-0.360934\pi$
$353$	15.8995	0.846245	0.423122	+	0.906073i	$0.360934\pi$
$354$	0	0
$355$	0.485281	0.0257561
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−35.4558	−1.87129	−0.935644	−	0.352945i	$-0.885180\pi$
$359$	−35.4558	−1.87129	−0.935644	+	0.352945i	$0.885180\pi$
$360$	0	0
$361$	27.6274	1.45407
$362$	0	0
$363$	10.3137	0.541329
$364$	0	0
$365$	−6.48528	−0.339455
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	0.585786	0.0304948
$370$	0	0
$371$	0	0
$372$	0	0
$373$	14.6863	0.760427	0.380214	−	0.924899i	$-0.375850\pi$
$373$	14.6863	0.760427	0.380214	+	0.924899i	$0.375850\pi$
$374$	0	0
$375$	−5.65685	−0.292119
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	−0.686292	−0.0352524	−0.0176262	−	0.999845i	$-0.505611\pi$
$379$	−0.686292	−0.0352524	−0.0176262	+	0.999845i	$0.505611\pi$
$380$	0	0
$381$	15.3137	0.784545
$382$	0	0
$383$	−24.9706	−1.27594	−0.637968	−	0.770063i	$-0.720225\pi$
$383$	−24.9706	−1.27594	−0.637968	+	0.770063i	$0.720225\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−8.00000	−0.406663
$388$	0	0
$389$	−14.1421	−0.717035	−0.358517	−	0.933523i	$-0.616718\pi$
$389$	−14.1421	−0.717035	−0.358517	+	0.933523i	$0.616718\pi$
$390$	0	0
$391$	−10.8284	−0.547617
$392$	0	0
$393$	−7.31371	−0.368928
$394$	0	0
$395$	1.37258	0.0690621
$396$	0	0
$397$	−7.27208	−0.364975	−0.182488	−	0.983208i	$-0.558415\pi$
$397$	−7.27208	−0.364975	−0.182488	+	0.983208i	$0.558415\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−12.4853	−0.623485	−0.311743	−	0.950167i	$-0.600913\pi$
$401$	−12.4853	−0.623485	−0.311743	+	0.950167i	$0.600913\pi$
$402$	0	0
$403$	−7.31371	−0.364322
$404$	0	0
$405$	−0.585786	−0.0291080
$406$	0	0
$407$	−1.37258	−0.0680364
$408$	0	0
$409$	−16.2426	−0.803147	−0.401573	−	0.915827i	$-0.631537\pi$
$409$	−16.2426	−0.803147	−0.401573	+	0.915827i	$0.631537\pi$
$410$	0	0
$411$	−4.48528	−0.221243
$412$	0	0
$413$	0	0
$414$	0	0
$415$	8.97056	0.440348
$416$	0	0
$417$	−1.65685	−0.0811365
$418$	0	0
$419$	−10.8284	−0.529003	−0.264502	−	0.964385i	$-0.585207\pi$
$419$	−10.8284	−0.529003	−0.264502	+	0.964385i	$0.585207\pi$
$420$	0	0
$421$	6.68629	0.325870	0.162935	−	0.986637i	$-0.447904\pi$
$421$	6.68629	0.325870	0.162935	+	0.986637i	$0.447904\pi$
$422$	0	0
$423$	−6.82843	−0.332009
$424$	0	0
$425$	10.4437	0.506591
$426$	0	0
$427$	0	0
$428$	0	0
$429$	1.17157	0.0565641
$430$	0	0
$431$	−30.4853	−1.46842	−0.734212	−	0.678920i	$-0.762448\pi$
$431$	−30.4853	−1.46842	−0.734212	+	0.678920i	$0.762448\pi$
$432$	0	0
$433$	−26.3848	−1.26797	−0.633986	−	0.773345i	$-0.718582\pi$
$433$	−26.3848	−1.26797	−0.633986	+	0.773345i	$0.718582\pi$
$434$	0	0
$435$	4.97056	0.238320
$436$	0	0
$437$	−32.9706	−1.57720
$438$	0	0
$439$	3.31371	0.158155	0.0790773	−	0.996868i	$-0.474803\pi$
$439$	3.31371	0.158155	0.0790773	+	0.996868i	$0.474803\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	38.4853	1.82849	0.914245	−	0.405161i	$-0.132784\pi$
$443$	38.4853	1.82849	0.914245	+	0.405161i	$0.132784\pi$
$444$	0	0
$445$	−6.28427	−0.297903
$446$	0	0
$447$	10.0000	0.472984
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	−0.485281	−0.0228510
$452$	0	0
$453$	−9.65685	−0.453719
$454$	0	0
$455$	0	0
$456$	0	0
$457$	20.6274	0.964910	0.482455	−	0.875921i	$-0.339745\pi$
$457$	20.6274	0.964910	0.482455	+	0.875921i	$0.339745\pi$
$458$	0	0
$459$	2.24264	0.104678
$460$	0	0
$461$	−18.2426	−0.849644	−0.424822	−	0.905277i	$-0.639663\pi$
$461$	−18.2426	−0.849644	−0.424822	+	0.905277i	$0.639663\pi$
$462$	0	0
$463$	20.9706	0.974585	0.487292	−	0.873239i	$-0.337985\pi$
$463$	20.9706	0.974585	0.487292	+	0.873239i	$0.337985\pi$
$464$	0	0
$465$	−3.02944	−0.140487
$466$	0	0
$467$	10.8284	0.501080	0.250540	−	0.968106i	$-0.419392\pi$
$467$	10.8284	0.501080	0.250540	+	0.968106i	$0.419392\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−13.8995	−0.640455
$472$	0	0
$473$	6.62742	0.304729
$474$	0	0
$475$	31.7990	1.45904
$476$	0	0
$477$	−13.3137	−0.609593
$478$	0	0
$479$	43.1127	1.96987	0.984935	−	0.172926i	$-0.0553223\pi$
$479$	43.1127	1.96987	0.984935	+	0.172926i	$0.0553223\pi$
$480$	0	0
$481$	2.34315	0.106838
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4.54416	−0.206339
$486$	0	0
$487$	−20.9706	−0.950267	−0.475133	−	0.879914i	$-0.657600\pi$
$487$	−20.9706	−0.950267	−0.475133	+	0.879914i	$0.657600\pi$
$488$	0	0
$489$	13.6569	0.617584
$490$	0	0
$491$	24.1421	1.08952	0.544760	−	0.838592i	$-0.316621\pi$
$491$	24.1421	1.08952	0.544760	+	0.838592i	$0.316621\pi$
$492$	0	0
$493$	−19.0294	−0.857043
$494$	0	0
$495$	0.485281	0.0218118
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−28.2843	−1.26618	−0.633089	−	0.774079i	$-0.718213\pi$
$499$	−28.2843	−1.26618	−0.633089	+	0.774079i	$0.718213\pi$
$500$	0	0
$501$	1.17157	0.0523420
$502$	0	0
$503$	14.6274	0.652204	0.326102	−	0.945335i	$-0.394265\pi$
$503$	14.6274	0.652204	0.326102	+	0.945335i	$0.394265\pi$
$504$	0	0
$505$	2.00000	0.0889988
$506$	0	0
$507$	11.0000	0.488527
$508$	0	0
$509$	−37.0711	−1.64315	−0.821573	−	0.570103i	$-0.806903\pi$
$509$	−37.0711	−1.64315	−0.821573	+	0.570103i	$0.806903\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	6.82843	0.301482
$514$	0	0
$515$	6.34315	0.279512
$516$	0	0
$517$	5.65685	0.248788
$518$	0	0
$519$	−3.41421	−0.149867
$520$	0	0
$521$	−30.7279	−1.34621	−0.673107	−	0.739545i	$-0.735041\pi$
$521$	−30.7279	−1.34621	−0.673107	+	0.739545i	$0.735041\pi$
$522$	0	0
$523$	−9.65685	−0.422265	−0.211132	−	0.977457i	$-0.567715\pi$
$523$	−9.65685	−0.422265	−0.211132	+	0.977457i	$0.567715\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.5980	0.505216
$528$	0	0
$529$	0.313708	0.0136395
$530$	0	0
$531$	−5.17157	−0.224427
$532$	0	0
$533$	0.828427	0.0358832
$534$	0	0
$535$	−8.48528	−0.366851
$536$	0	0
$537$	−17.7990	−0.768083
$538$	0	0
$539$	0	0
$540$	0	0
$541$	36.6274	1.57474	0.787368	−	0.616483i	$-0.211443\pi$
$541$	36.6274	1.57474	0.787368	+	0.616483i	$0.211443\pi$
$542$	0	0
$543$	−9.89949	−0.424828
$544$	0	0
$545$	6.62742	0.283887
$546$	0	0
$547$	28.9706	1.23869	0.619346	−	0.785118i	$-0.287398\pi$
$547$	28.9706	1.23869	0.619346	+	0.785118i	$0.287398\pi$
$548$	0	0
$549$	−13.8995	−0.593216
$550$	0	0
$551$	−57.9411	−2.46837
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.970563	0.0411981
$556$	0	0
$557$	−23.9411	−1.01442	−0.507209	−	0.861823i	$-0.669323\pi$
$557$	−23.9411	−1.01442	−0.507209	+	0.861823i	$0.669323\pi$
$558$	0	0
$559$	−11.3137	−0.478519
$560$	0	0
$561$	−1.85786	−0.0784391
$562$	0	0
$563$	−21.1716	−0.892275	−0.446138	−	0.894964i	$-0.647201\pi$
$563$	−21.1716	−0.892275	−0.446138	+	0.894964i	$0.647201\pi$
$564$	0	0
$565$	−3.51472	−0.147865
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.17157	0.0491149	0.0245574	−	0.999698i	$-0.492182\pi$
$569$	1.17157	0.0491149	0.0245574	+	0.999698i	$0.492182\pi$
$570$	0	0
$571$	−16.2843	−0.681476	−0.340738	−	0.940158i	$-0.610677\pi$
$571$	−16.2843	−0.681476	−0.340738	+	0.940158i	$0.610677\pi$
$572$	0	0
$573$	−15.1716	−0.633802
$574$	0	0
$575$	−22.4853	−0.937701
$576$	0	0
$577$	6.58579	0.274170	0.137085	−	0.990559i	$-0.456227\pi$
$577$	6.58579	0.274170	0.137085	+	0.990559i	$0.456227\pi$
$578$	0	0
$579$	−24.6274	−1.02348
$580$	0	0
$581$	0	0
$582$	0	0
$583$	11.0294	0.456793
$584$	0	0
$585$	−0.828427	−0.0342512
$586$	0	0
$587$	21.1716	0.873844	0.436922	−	0.899499i	$-0.356069\pi$
$587$	21.1716	0.873844	0.436922	+	0.899499i	$0.356069\pi$
$588$	0	0
$589$	35.3137	1.45508
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	0	0
$593$	−14.9289	−0.613058	−0.306529	−	0.951861i	$-0.599168\pi$
$593$	−14.9289	−0.613058	−0.306529	+	0.951861i	$0.599168\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.65685	0.231520
$598$	0	0
$599$	−12.1421	−0.496114	−0.248057	−	0.968745i	$-0.579792\pi$
$599$	−12.1421	−0.496114	−0.248057	+	0.968745i	$0.579792\pi$
$600$	0	0
$601$	3.75736	0.153266	0.0766329	−	0.997059i	$-0.475583\pi$
$601$	3.75736	0.153266	0.0766329	+	0.997059i	$0.475583\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	0	0
$605$	6.04163	0.245627
$606$	0	0
$607$	47.5980	1.93194	0.965971	−	0.258650i	$-0.0832775\pi$
$607$	47.5980	1.93194	0.965971	+	0.258650i	$0.0832775\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.65685	−0.390675
$612$	0	0
$613$	−13.6569	−0.551595	−0.275798	−	0.961216i	$-0.588942\pi$
$613$	−13.6569	−0.551595	−0.275798	+	0.961216i	$0.588942\pi$
$614$	0	0
$615$	0.343146	0.0138370
$616$	0	0
$617$	−29.4558	−1.18585	−0.592924	−	0.805259i	$-0.702026\pi$
$617$	−29.4558	−1.18585	−0.592924	+	0.805259i	$0.702026\pi$
$618$	0	0
$619$	−16.2843	−0.654520	−0.327260	−	0.944934i	$-0.606125\pi$
$619$	−16.2843	−0.654520	−0.327260	+	0.944934i	$0.606125\pi$
$620$	0	0
$621$	−4.82843	−0.193758
$622$	0	0
$623$	0	0
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	−5.65685	−0.225913
$628$	0	0
$629$	−3.71573	−0.148156
$630$	0	0
$631$	48.2843	1.92217	0.961083	−	0.276259i	$-0.0890947\pi$
$631$	48.2843	1.92217	0.961083	+	0.276259i	$0.0890947\pi$
$632$	0	0
$633$	14.3431	0.570089
$634$	0	0
$635$	8.97056	0.355986
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−0.828427	−0.0327721
$640$	0	0
$641$	30.8284	1.21765	0.608825	−	0.793305i	$-0.291641\pi$
$641$	30.8284	1.21765	0.608825	+	0.793305i	$0.291641\pi$
$642$	0	0
$643$	−36.4853	−1.43884	−0.719420	−	0.694576i	$-0.755592\pi$
$643$	−36.4853	−1.43884	−0.719420	+	0.694576i	$0.755592\pi$
$644$	0	0
$645$	−4.68629	−0.184523
$646$	0	0
$647$	−9.17157	−0.360572	−0.180286	−	0.983614i	$-0.557702\pi$
$647$	−9.17157	−0.360572	−0.180286	+	0.983614i	$0.557702\pi$
$648$	0	0
$649$	4.28427	0.168172
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.82843	−0.110685	−0.0553425	−	0.998467i	$-0.517625\pi$
$653$	−2.82843	−0.110685	−0.0553425	+	0.998467i	$0.517625\pi$
$654$	0	0
$655$	−4.28427	−0.167400
$656$	0	0
$657$	11.0711	0.431923
$658$	0	0
$659$	42.4853	1.65499	0.827496	−	0.561472i	$-0.189765\pi$
$659$	42.4853	1.65499	0.827496	+	0.561472i	$0.189765\pi$
$660$	0	0
$661$	−47.3553	−1.84191	−0.920955	−	0.389670i	$-0.872589\pi$
$661$	−47.3553	−1.84191	−0.920955	+	0.389670i	$0.872589\pi$
$662$	0	0
$663$	3.17157	0.123174
$664$	0	0
$665$	0	0
$666$	0	0
$667$	40.9706	1.58639
$668$	0	0
$669$	−13.6569	−0.528004
$670$	0	0
$671$	11.5147	0.444521
$672$	0	0
$673$	−7.31371	−0.281923	−0.140961	−	0.990015i	$-0.545019\pi$
$673$	−7.31371	−0.281923	−0.140961	+	0.990015i	$0.545019\pi$
$674$	0	0
$675$	4.65685	0.179242
$676$	0	0
$677$	−29.5563	−1.13594	−0.567971	−	0.823048i	$-0.692272\pi$
$677$	−29.5563	−1.13594	−0.567971	+	0.823048i	$0.692272\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−19.7990	−0.758699
$682$	0	0
$683$	−16.1421	−0.617662	−0.308831	−	0.951117i	$-0.599938\pi$
$683$	−16.1421	−0.617662	−0.308831	+	0.951117i	$0.599938\pi$
$684$	0	0
$685$	−2.62742	−0.100388
$686$	0	0
$687$	15.0711	0.574997
$688$	0	0
$689$	−18.8284	−0.717306
$690$	0	0
$691$	−28.0000	−1.06517	−0.532585	−	0.846376i	$-0.678779\pi$
$691$	−28.0000	−1.06517	−0.532585	+	0.846376i	$0.678779\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.970563	−0.0368155
$696$	0	0
$697$	−1.31371	−0.0497603
$698$	0	0
$699$	28.4853	1.07741
$700$	0	0
$701$	−5.17157	−0.195328	−0.0976638	−	0.995219i	$-0.531137\pi$
$701$	−5.17157	−0.195328	−0.0976638	+	0.995219i	$0.531137\pi$
$702$	0	0
$703$	−11.3137	−0.426705
$704$	0	0
$705$	−4.00000	−0.150649
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−8.00000	−0.300446	−0.150223	−	0.988652i	$-0.547999\pi$
$709$	−8.00000	−0.300446	−0.150223	+	0.988652i	$0.547999\pi$
$710$	0	0
$711$	−2.34315	−0.0878748
$712$	0	0
$713$	−24.9706	−0.935155
$714$	0	0
$715$	0.686292	0.0256658
$716$	0	0
$717$	3.17157	0.118445
$718$	0	0
$719$	4.68629	0.174769	0.0873846	−	0.996175i	$-0.472149\pi$
$719$	4.68629	0.174769	0.0873846	+	0.996175i	$0.472149\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−21.8995	−0.814451
$724$	0	0
$725$	−39.5147	−1.46754
$726$	0	0
$727$	−25.4558	−0.944105	−0.472052	−	0.881570i	$-0.656487\pi$
$727$	−25.4558	−0.944105	−0.472052	+	0.881570i	$0.656487\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	17.9411	0.663576
$732$	0	0
$733$	−11.7574	−0.434268	−0.217134	−	0.976142i	$-0.569671\pi$
$733$	−11.7574	−0.434268	−0.217134	+	0.976142i	$0.569671\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.62742	0.244124
$738$	0	0
$739$	20.2843	0.746169	0.373084	−	0.927797i	$-0.378300\pi$
$739$	20.2843	0.746169	0.373084	+	0.927797i	$0.378300\pi$
$740$	0	0
$741$	9.65685	0.354753
$742$	0	0
$743$	−11.1716	−0.409845	−0.204923	−	0.978778i	$-0.565694\pi$
$743$	−11.1716	−0.409845	−0.204923	+	0.978778i	$0.565694\pi$
$744$	0	0
$745$	5.85786	0.214616
$746$	0	0
$747$	−15.3137	−0.560299
$748$	0	0
$749$	0	0
$750$	0	0
$751$	29.6569	1.08219	0.541097	−	0.840960i	$-0.318009\pi$
$751$	29.6569	1.08219	0.541097	+	0.840960i	$0.318009\pi$
$752$	0	0
$753$	8.48528	0.309221
$754$	0	0
$755$	−5.65685	−0.205874
$756$	0	0
$757$	−11.3137	−0.411204	−0.205602	−	0.978636i	$-0.565915\pi$
$757$	−11.3137	−0.411204	−0.205602	+	0.978636i	$0.565915\pi$
$758$	0	0
$759$	4.00000	0.145191
$760$	0	0
$761$	9.75736	0.353704	0.176852	−	0.984237i	$-0.443409\pi$
$761$	9.75736	0.353704	0.176852	+	0.984237i	$0.443409\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	1.31371	0.0474972
$766$	0	0
$767$	−7.31371	−0.264083
$768$	0	0
$769$	−15.0711	−0.543477	−0.271738	−	0.962371i	$-0.587599\pi$
$769$	−15.0711	−0.543477	−0.271738	+	0.962371i	$0.587599\pi$
$770$	0	0
$771$	−30.2426	−1.08916
$772$	0	0
$773$	3.41421	0.122801	0.0614004	−	0.998113i	$-0.480443\pi$
$773$	3.41421	0.122801	0.0614004	+	0.998113i	$0.480443\pi$
$774$	0	0
$775$	24.0833	0.865096
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−4.00000	−0.143315
$780$	0	0
$781$	0.686292	0.0245574
$782$	0	0
$783$	−8.48528	−0.303239
$784$	0	0
$785$	−8.14214	−0.290605
$786$	0	0
$787$	10.6274	0.378827	0.189413	−	0.981897i	$-0.439341\pi$
$787$	10.6274	0.378827	0.189413	+	0.981897i	$0.439341\pi$
$788$	0	0
$789$	−24.8284	−0.883915
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−19.6569	−0.698035
$794$	0	0
$795$	−7.79899	−0.276602
$796$	0	0
$797$	43.2132	1.53069	0.765345	−	0.643620i	$-0.222568\pi$
$797$	43.2132	1.53069	0.765345	+	0.643620i	$0.222568\pi$
$798$	0	0
$799$	15.3137	0.541760
$800$	0	0
$801$	10.7279	0.379052
$802$	0	0
$803$	−9.17157	−0.323658
$804$	0	0
$805$	0	0
$806$	0	0
$807$	30.0416	1.05752
$808$	0	0
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	20.9706	0.736376	0.368188	−	0.929751i	$-0.379978\pi$
$811$	20.9706	0.736376	0.368188	+	0.929751i	$0.379978\pi$
$812$	0	0
$813$	−13.1716	−0.461947
$814$	0	0
$815$	8.00000	0.280228
$816$	0	0
$817$	54.6274	1.91117
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10.0000	0.349002	0.174501	−	0.984657i	$-0.444169\pi$
$821$	10.0000	0.349002	0.174501	+	0.984657i	$0.444169\pi$
$822$	0	0
$823$	−2.34315	−0.0816769	−0.0408385	−	0.999166i	$-0.513003\pi$
$823$	−2.34315	−0.0816769	−0.0408385	+	0.999166i	$0.513003\pi$
$824$	0	0
$825$	−3.85786	−0.134314
$826$	0	0
$827$	−19.4558	−0.676546	−0.338273	−	0.941048i	$-0.609843\pi$
$827$	−19.4558	−0.676546	−0.338273	+	0.941048i	$0.609843\pi$
$828$	0	0
$829$	−3.27208	−0.113644	−0.0568220	−	0.998384i	$-0.518097\pi$
$829$	−3.27208	−0.113644	−0.0568220	+	0.998384i	$0.518097\pi$
$830$	0	0
$831$	6.00000	0.208138
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0.686292	0.0237501
$836$	0	0
$837$	5.17157	0.178756
$838$	0	0
$839$	−49.1716	−1.69759	−0.848796	−	0.528721i	$-0.822672\pi$
$839$	−49.1716	−1.69759	−0.848796	+	0.528721i	$0.822672\pi$
$840$	0	0
$841$	43.0000	1.48276
$842$	0	0
$843$	12.4853	0.430016
$844$	0	0
$845$	6.44365	0.221668
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−14.8284	−0.508910
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	−36.0416	−1.23404	−0.617021	−	0.786947i	$-0.711661\pi$
$853$	−36.0416	−1.23404	−0.617021	+	0.786947i	$0.711661\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	43.6985	1.49271	0.746356	−	0.665547i	$-0.231802\pi$
$857$	43.6985	1.49271	0.746356	+	0.665547i	$0.231802\pi$
$858$	0	0
$859$	38.8284	1.32481	0.662404	−	0.749146i	$-0.269536\pi$
$859$	38.8284	1.32481	0.662404	+	0.749146i	$0.269536\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−29.5147	−1.00469	−0.502346	−	0.864666i	$-0.667530\pi$
$863$	−29.5147	−1.00469	−0.502346	+	0.864666i	$0.667530\pi$
$864$	0	0
$865$	−2.00000	−0.0680020
$866$	0	0
$867$	11.9706	0.406542
$868$	0	0
$869$	1.94113	0.0658482
$870$	0	0
$871$	−11.3137	−0.383350
$872$	0	0
$873$	7.75736	0.262547
$874$	0	0
$875$	0	0
$876$	0	0
$877$	20.2843	0.684951	0.342476	−	0.939527i	$-0.388735\pi$
$877$	20.2843	0.684951	0.342476	+	0.939527i	$0.388735\pi$
$878$	0	0
$879$	1.07107	0.0361262
$880$	0	0
$881$	25.0711	0.844666	0.422333	−	0.906441i	$-0.361211\pi$
$881$	25.0711	0.844666	0.422333	+	0.906441i	$0.361211\pi$
$882$	0	0
$883$	18.3431	0.617296	0.308648	−	0.951176i	$-0.400123\pi$
$883$	18.3431	0.617296	0.308648	+	0.951176i	$0.400123\pi$
$884$	0	0
$885$	−3.02944	−0.101833
$886$	0	0
$887$	−21.4558	−0.720417	−0.360208	−	0.932872i	$-0.617294\pi$
$887$	−21.4558	−0.720417	−0.360208	+	0.932872i	$0.617294\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−0.828427	−0.0277534
$892$	0	0
$893$	46.6274	1.56033
$894$	0	0
$895$	−10.4264	−0.348516
$896$	0	0
$897$	−6.82843	−0.227995
$898$	0	0
$899$	−43.8823	−1.46356
$900$	0	0
$901$	29.8579	0.994710
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.79899	−0.192765
$906$	0	0
$907$	−7.02944	−0.233409	−0.116704	−	0.993167i	$-0.537233\pi$
$907$	−7.02944	−0.233409	−0.116704	+	0.993167i	$0.537233\pi$
$908$	0	0
$909$	−3.41421	−0.113242
$910$	0	0
$911$	−54.4853	−1.80518	−0.902589	−	0.430503i	$-0.858336\pi$
$911$	−54.4853	−1.80518	−0.902589	+	0.430503i	$0.858336\pi$
$912$	0	0
$913$	12.6863	0.419855
$914$	0	0
$915$	−8.14214	−0.269171
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−49.2548	−1.62477	−0.812384	−	0.583123i	$-0.801830\pi$
$919$	−49.2548	−1.62477	−0.812384	+	0.583123i	$0.801830\pi$
$920$	0	0
$921$	−11.5147	−0.379423
$922$	0	0
$923$	−1.17157	−0.0385628
$924$	0	0
$925$	−7.71573	−0.253692
$926$	0	0
$927$	−10.8284	−0.355652
$928$	0	0
$929$	−60.8701	−1.99708	−0.998541	−	0.0540006i	$-0.982803\pi$
$929$	−60.8701	−1.99708	−0.998541	+	0.0540006i	$0.982803\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−26.1421	−0.855855
$934$	0	0
$935$	−1.08831	−0.0355916
$936$	0	0
$937$	6.10051	0.199295	0.0996474	−	0.995023i	$-0.468229\pi$
$937$	6.10051	0.199295	0.0996474	+	0.995023i	$0.468229\pi$
$938$	0	0
$939$	−17.4142	−0.568291
$940$	0	0
$941$	43.8995	1.43108	0.715541	−	0.698570i	$-0.246180\pi$
$941$	43.8995	1.43108	0.715541	+	0.698570i	$0.246180\pi$
$942$	0	0
$943$	2.82843	0.0921063
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20.1421	0.654531	0.327266	−	0.944932i	$-0.393873\pi$
$947$	20.1421	0.654531	0.327266	+	0.944932i	$0.393873\pi$
$948$	0	0
$949$	15.6569	0.508243
$950$	0	0
$951$	−21.3137	−0.691144
$952$	0	0
$953$	−7.37258	−0.238821	−0.119411	−	0.992845i	$-0.538101\pi$
$953$	−7.37258	−0.238821	−0.119411	+	0.992845i	$0.538101\pi$
$954$	0	0
$955$	−8.88730	−0.287586
$956$	0	0
$957$	7.02944	0.227229
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−4.25483	−0.137253
$962$	0	0
$963$	14.4853	0.466782
$964$	0	0
$965$	−14.4264	−0.464402
$966$	0	0
$967$	−34.6274	−1.11354	−0.556771	−	0.830666i	$-0.687960\pi$
$967$	−34.6274	−1.11354	−0.556771	+	0.830666i	$0.687960\pi$
$968$	0	0
$969$	−15.3137	−0.491947
$970$	0	0
$971$	57.2548	1.83740	0.918698	−	0.394962i	$-0.129242\pi$
$971$	57.2548	1.83740	0.918698	+	0.394962i	$0.129242\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	6.58579	0.210914
$976$	0	0
$977$	21.8579	0.699295	0.349648	−	0.936881i	$-0.386301\pi$
$977$	21.8579	0.699295	0.349648	+	0.936881i	$0.386301\pi$
$978$	0	0
$979$	−8.88730	−0.284039
$980$	0	0
$981$	−11.3137	−0.361219
$982$	0	0
$983$	14.6274	0.466542	0.233271	−	0.972412i	$-0.425057\pi$
$983$	14.6274	0.466542	0.233271	+	0.972412i	$0.425057\pi$
$984$	0	0
$985$	−1.17157	−0.0373294
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−38.6274	−1.22828
$990$	0	0
$991$	23.3137	0.740584	0.370292	−	0.928915i	$-0.379258\pi$
$991$	23.3137	0.740584	0.370292	+	0.928915i	$0.379258\pi$
$992$	0	0
$993$	8.68629	0.275651
$994$	0	0
$995$	3.31371	0.105052
$996$	0	0
$997$	34.5858	1.09534	0.547671	−	0.836693i	$-0.315514\pi$
$997$	34.5858	1.09534	0.547671	+	0.836693i	$0.315514\pi$
$998$	0	0
$999$	−1.65685	−0.0524205

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.2.a.j.1.2	✓	2
3.2	odd	2		3528.2.a.bl.1.1		2
4.3	odd	2		2352.2.a.bd.1.2		2
7.2	even	3		1176.2.q.o.361.1		4
7.3	odd	6		1176.2.q.k.961.2		4
7.4	even	3		1176.2.q.o.961.1		4
7.5	odd	6		1176.2.q.k.361.2		4
7.6	odd	2		1176.2.a.o.1.1	yes	2
8.3	odd	2		9408.2.a.ds.1.1		2
8.5	even	2		9408.2.a.ee.1.1		2
12.11	even	2		7056.2.a.cx.1.1		2
21.2	odd	6		3528.2.s.bd.361.2		4
21.5	even	6		3528.2.s.bm.361.1		4
21.11	odd	6		3528.2.s.bd.3313.2		4
21.17	even	6		3528.2.s.bm.3313.1		4
21.20	even	2		3528.2.a.bb.1.2		2
28.3	even	6		2352.2.q.be.961.2		4
28.11	odd	6		2352.2.q.bc.961.1		4
28.19	even	6		2352.2.q.be.1537.2		4
28.23	odd	6		2352.2.q.bc.1537.1		4
28.27	even	2		2352.2.a.bb.1.1		2
56.13	odd	2		9408.2.a.dg.1.2		2
56.27	even	2		9408.2.a.du.1.2		2
84.83	odd	2		7056.2.a.cg.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1176.2.a.j.1.2	✓	2	1.1	even	1	trivial
1176.2.a.o.1.1	yes	2	7.6	odd	2
1176.2.q.k.361.2		4	7.5	odd	6
1176.2.q.k.961.2		4	7.3	odd	6
1176.2.q.o.361.1		4	7.2	even	3
1176.2.q.o.961.1		4	7.4	even	3
2352.2.a.bb.1.1		2	28.27	even	2
2352.2.a.bd.1.2		2	4.3	odd	2
2352.2.q.bc.961.1		4	28.11	odd	6
2352.2.q.bc.1537.1		4	28.23	odd	6
2352.2.q.be.961.2		4	28.3	even	6
2352.2.q.be.1537.2		4	28.19	even	6
3528.2.a.bb.1.2		2	21.20	even	2
3528.2.a.bl.1.1		2	3.2	odd	2
3528.2.s.bd.361.2		4	21.2	odd	6
3528.2.s.bd.3313.2		4	21.11	odd	6
3528.2.s.bm.361.1		4	21.5	even	6
3528.2.s.bm.3313.1		4	21.17	even	6
7056.2.a.cg.1.2		2	84.83	odd	2
7056.2.a.cx.1.1		2	12.11	even	2
9408.2.a.dg.1.2		2	56.13	odd	2
9408.2.a.ds.1.1		2	8.3	odd	2
9408.2.a.du.1.2		2	56.27	even	2
9408.2.a.ee.1.1		2	8.5	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:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1176.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.585786 q^{5} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.585786 q^{5} +1.00000 q^{9} -0.828427 q^{11} +1.41421 q^{13} +0.585786 q^{15} -2.24264 q^{17} -6.82843 q^{19} +4.82843 q^{23} -4.65685 q^{25} -1.00000 q^{27} +8.48528 q^{29} -5.17157 q^{31} +0.828427 q^{33} +1.65685 q^{37} -1.41421 q^{39} +0.585786 q^{41} -8.00000 q^{43} -0.585786 q^{45} -6.82843 q^{47} +2.24264 q^{51} -13.3137 q^{53} +0.485281 q^{55} +6.82843 q^{57} -5.17157 q^{59} -13.8995 q^{61} -0.828427 q^{65} -8.00000 q^{67} -4.82843 q^{69} -0.828427 q^{71} +11.0711 q^{73} +4.65685 q^{75} -2.34315 q^{79} +1.00000 q^{81} -15.3137 q^{83} +1.31371 q^{85} -8.48528 q^{87} +10.7279 q^{89} +5.17157 q^{93} +4.00000 q^{95} +7.75736 q^{97} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} + 4 q^{11} + 4 q^{15} + 4 q^{17} - 8 q^{19} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 16 q^{31} - 4 q^{33} - 8 q^{37} + 4 q^{41} - 16 q^{43} - 4 q^{45} - 8 q^{47} - 4 q^{51} - 4 q^{53} - 16 q^{55} + 8 q^{57} - 16 q^{59} - 8 q^{61} + 4 q^{65} - 16 q^{67} - 4 q^{69} + 4 q^{71} + 8 q^{73} - 2 q^{75} - 16 q^{79} + 2 q^{81} - 8 q^{83} - 20 q^{85} - 4 q^{89} + 16 q^{93} + 8 q^{95} + 24 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 + 4 * q^11 + 4 * q^15 + 4 * q^17 - 8 * q^19 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 16 * q^31 - 4 * q^33 - 8 * q^37 + 4 * q^41 - 16 * q^43 - 4 * q^45 - 8 * q^47 - 4 * q^51 - 4 * q^53 - 16 * q^55 + 8 * q^57 - 16 * q^59 - 8 * q^61 + 4 * q^65 - 16 * q^67 - 4 * q^69 + 4 * q^71 + 8 * q^73 - 2 * q^75 - 16 * q^79 + 2 * q^81 - 8 * q^83 - 20 * q^85 - 4 * q^89 + 16 * q^93 + 8 * q^95 + 24 * q^97 + 4 * q^99

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