LMFDB - Embedded newform 3312.2.a.ba.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3312,2,Mod(1,3312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3312, 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("3312.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3312 = 2^{4} \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3312.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:	$26.4464531494$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 23)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	3312.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23607 q^{5} -3.23607 q^{7} +O(q^{10})$	$q-1.23607 q^{5} -3.23607 q^{7} -5.23607 q^{11} +3.00000 q^{13} -0.763932 q^{17} +2.00000 q^{19} +1.00000 q^{23} -3.47214 q^{25} +3.00000 q^{29} -6.70820 q^{31} +4.00000 q^{35} -1.23607 q^{37} +3.47214 q^{41} -2.23607 q^{47} +3.47214 q^{49} -0.472136 q^{53} +6.47214 q^{55} +6.47214 q^{59} -6.94427 q^{61} -3.70820 q^{65} +2.76393 q^{67} +12.2361 q^{71} +6.52786 q^{73} +16.9443 q^{77} +10.9443 q^{79} -8.76393 q^{83} +0.944272 q^{85} +10.4721 q^{89} -9.70820 q^{91} -2.47214 q^{95} +17.7082 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - 2 * q^7	$ 2 q + 2 q^{5} - 2 q^{7} - 6 q^{11} + 6 q^{13} - 6 q^{17} + 4 q^{19} + 2 q^{23} + 2 q^{25} + 6 q^{29} + 8 q^{35} + 2 q^{37} - 2 q^{41} - 2 q^{49} + 8 q^{53} + 4 q^{55} + 4 q^{59} + 4 q^{61} + 6 q^{65} + 10 q^{67} + 20 q^{71} + 22 q^{73} + 16 q^{77} + 4 q^{79} - 22 q^{83} - 16 q^{85} + 12 q^{89} - 6 q^{91} + 4 q^{95} + 22 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 2 * q^7 - 6 * q^11 + 6 * q^13 - 6 * q^17 + 4 * q^19 + 2 * q^23 + 2 * q^25 + 6 * q^29 + 8 * q^35 + 2 * q^37 - 2 * q^41 - 2 * q^49 + 8 * q^53 + 4 * q^55 + 4 * q^59 + 4 * q^61 + 6 * q^65 + 10 * q^67 + 20 * q^71 + 22 * q^73 + 16 * q^77 + 4 * q^79 - 22 * q^83 - 16 * q^85 + 12 * q^89 - 6 * q^91 + 4 * q^95 + 22 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	0	0
$7$	−3.23607	−1.22312	−0.611559	−	0.791199i	$-0.709457\pi$
$7$	−3.23607	−1.22312	−0.611559	+	0.791199i	$0.709457\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.23607	−1.57873	−0.789367	−	0.613922i	$-0.789591\pi$
$11$	−5.23607	−1.57873	−0.789367	+	0.613922i	$0.789591\pi$
$12$	0	0
$13$	3.00000	0.832050	0.416025	−	0.909353i	$-0.363423\pi$
$13$	3.00000	0.832050	0.416025	+	0.909353i	$0.363423\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.763932	−0.185281	−0.0926404	−	0.995700i	$-0.529531\pi$
$17$	−0.763932	−0.185281	−0.0926404	+	0.995700i	$0.529531\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	−6.70820	−1.20483	−0.602414	−	0.798183i	$-0.705795\pi$
$31$	−6.70820	−1.20483	−0.602414	+	0.798183i	$0.705795\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.00000	0.676123
$36$	0	0
$37$	−1.23607	−0.203208	−0.101604	−	0.994825i	$-0.532398\pi$
$37$	−1.23607	−0.203208	−0.101604	+	0.994825i	$0.532398\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.47214	0.542257	0.271128	−	0.962543i	$-0.412603\pi$
$41$	3.47214	0.542257	0.271128	+	0.962543i	$0.412603\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.23607	−0.326164	−0.163082	−	0.986613i	$-0.552144\pi$
$47$	−2.23607	−0.326164	−0.163082	+	0.986613i	$0.552144\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	0	0
$55$	6.47214	0.872703
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.47214	0.842600	0.421300	−	0.906921i	$-0.361574\pi$
$59$	6.47214	0.842600	0.421300	+	0.906921i	$0.361574\pi$
$60$	0	0
$61$	−6.94427	−0.889123	−0.444561	−	0.895748i	$-0.646640\pi$
$61$	−6.94427	−0.889123	−0.444561	+	0.895748i	$0.646640\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.70820	−0.459946
$66$	0	0
$67$	2.76393	0.337668	0.168834	−	0.985644i	$-0.446000\pi$
$67$	2.76393	0.337668	0.168834	+	0.985644i	$0.446000\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.2361	1.45215	0.726077	−	0.687613i	$-0.241342\pi$
$71$	12.2361	1.45215	0.726077	+	0.687613i	$0.241342\pi$
$72$	0	0
$73$	6.52786	0.764029	0.382014	−	0.924156i	$-0.375230\pi$
$73$	6.52786	0.764029	0.382014	+	0.924156i	$0.375230\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	16.9443	1.93098
$78$	0	0
$79$	10.9443	1.23133	0.615663	−	0.788009i	$-0.288888\pi$
$79$	10.9443	1.23133	0.615663	+	0.788009i	$0.288888\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.76393	−0.961967	−0.480983	−	0.876730i	$-0.659720\pi$
$83$	−8.76393	−0.961967	−0.480983	+	0.876730i	$0.659720\pi$
$84$	0	0
$85$	0.944272	0.102421
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.4721	1.11004	0.555022	−	0.831836i	$-0.312710\pi$
$89$	10.4721	1.11004	0.555022	+	0.831836i	$0.312710\pi$
$90$	0	0
$91$	−9.70820	−1.01770
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.47214	−0.253636
$96$	0	0
$97$	17.7082	1.79800	0.898998	−	0.437953i	$-0.144296\pi$
$97$	17.7082	1.79800	0.898998	+	0.437953i	$0.144296\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−4.47214	−0.444994	−0.222497	−	0.974933i	$-0.571421\pi$
$101$	−4.47214	−0.444994	−0.222497	+	0.974933i	$0.571421\pi$
$102$	0	0
$103$	4.18034	0.411901	0.205951	−	0.978562i	$-0.433971\pi$
$103$	4.18034	0.411901	0.205951	+	0.978562i	$0.433971\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	13.4164	1.29701	0.648507	−	0.761209i	$-0.275394\pi$
$107$	13.4164	1.29701	0.648507	+	0.761209i	$0.275394\pi$
$108$	0	0
$109$	0	0		−	1.00000i	$-0.5\pi$
$109$	0	0			1.00000i	$0.5\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.76393	−0.824441	−0.412221	−	0.911084i	$-0.635247\pi$
$113$	−8.76393	−0.824441	−0.412221	+	0.911084i	$0.635247\pi$
$114$	0	0
$115$	−1.23607	−0.115264
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.47214	0.226620
$120$	0	0
$121$	16.4164	1.49240
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10.4721	0.936656
$126$	0	0
$127$	7.29180	0.647042	0.323521	−	0.946221i	$-0.395133\pi$
$127$	7.29180	0.647042	0.323521	+	0.946221i	$0.395133\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.7082	1.63454	0.817272	−	0.576253i	$-0.195486\pi$
$131$	18.7082	1.63454	0.817272	+	0.576253i	$0.195486\pi$
$132$	0	0
$133$	−6.47214	−0.561205
$134$	0	0
$135$	0	0
$136$	0	0
$137$	21.8885	1.87006	0.935032	−	0.354563i	$-0.115370\pi$
$137$	21.8885	1.87006	0.935032	+	0.354563i	$0.115370\pi$
$138$	0	0
$139$	10.7082	0.908258	0.454129	−	0.890936i	$-0.349951\pi$
$139$	10.7082	0.908258	0.454129	+	0.890936i	$0.349951\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−15.7082	−1.31359
$144$	0	0
$145$	−3.70820	−0.307950
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−23.8885	−1.95703	−0.978513	−	0.206186i	$-0.933895\pi$
$149$	−23.8885	−1.95703	−0.978513	+	0.206186i	$0.933895\pi$
$150$	0	0
$151$	−4.23607	−0.344726	−0.172363	−	0.985033i	$-0.555140\pi$
$151$	−4.23607	−0.344726	−0.172363	+	0.985033i	$0.555140\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.29180	0.666013
$156$	0	0
$157$	−11.4164	−0.911129	−0.455564	−	0.890203i	$-0.650562\pi$
$157$	−11.4164	−0.911129	−0.455564	+	0.890203i	$0.650562\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.23607	−0.255038
$162$	0	0
$163$	5.76393	0.451466	0.225733	−	0.974189i	$-0.427522\pi$
$163$	5.76393	0.451466	0.225733	+	0.974189i	$0.427522\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.52786	0.118230	0.0591148	−	0.998251i	$-0.481172\pi$
$167$	1.52786	0.118230	0.0591148	+	0.998251i	$0.481172\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−22.9443	−1.74442	−0.872210	−	0.489131i	$-0.837314\pi$
$173$	−22.9443	−1.74442	−0.872210	+	0.489131i	$0.837314\pi$
$174$	0	0
$175$	11.2361	0.849367
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0.708204	0.0529336	0.0264668	−	0.999650i	$-0.491574\pi$
$179$	0.708204	0.0529336	0.0264668	+	0.999650i	$0.491574\pi$
$180$	0	0
$181$	16.6525	1.23777	0.618884	−	0.785482i	$-0.287585\pi$
$181$	16.6525	1.23777	0.618884	+	0.785482i	$0.287585\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.52786	0.112331
$186$	0	0
$187$	4.00000	0.292509
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−26.1803	−1.89434	−0.947171	−	0.320728i	$-0.896073\pi$
$191$	−26.1803	−1.89434	−0.947171	+	0.320728i	$0.896073\pi$
$192$	0	0
$193$	9.94427	0.715804	0.357902	−	0.933759i	$-0.383492\pi$
$193$	9.94427	0.715804	0.357902	+	0.933759i	$0.383492\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.47214	0.104885	0.0524427	−	0.998624i	$-0.483299\pi$
$197$	1.47214	0.104885	0.0524427	+	0.998624i	$0.483299\pi$
$198$	0	0
$199$	12.2918	0.871342	0.435671	−	0.900106i	$-0.356511\pi$
$199$	12.2918	0.871342	0.435671	+	0.900106i	$0.356511\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−9.70820	−0.681382
$204$	0	0
$205$	−4.29180	−0.299752
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−10.4721	−0.724373
$210$	0	0
$211$	23.4164	1.61205	0.806026	−	0.591880i	$-0.201614\pi$
$211$	23.4164	1.61205	0.806026	+	0.591880i	$0.201614\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	21.7082	1.47365
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−2.29180	−0.154163
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.1803	−0.808438	−0.404219	−	0.914662i	$-0.632457\pi$
$227$	−12.1803	−0.808438	−0.404219	+	0.914662i	$0.632457\pi$
$228$	0	0
$229$	−12.0000	−0.792982	−0.396491	−	0.918039i	$-0.629772\pi$
$229$	−12.0000	−0.792982	−0.396491	+	0.918039i	$0.629772\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.52786	0.427655	0.213827	−	0.976871i	$-0.431407\pi$
$233$	6.52786	0.427655	0.213827	+	0.976871i	$0.431407\pi$
$234$	0	0
$235$	2.76393	0.180299
$236$	0	0
$237$	0	0
$238$	0	0
$239$	13.7639	0.890315	0.445157	−	0.895452i	$-0.353148\pi$
$239$	13.7639	0.890315	0.445157	+	0.895452i	$0.353148\pi$
$240$	0	0
$241$	−23.1246	−1.48959	−0.744794	−	0.667295i	$-0.767452\pi$
$241$	−23.1246	−1.48959	−0.744794	+	0.667295i	$0.767452\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.29180	−0.274193
$246$	0	0
$247$	6.00000	0.381771
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.29180	0.144657	0.0723284	−	0.997381i	$-0.476957\pi$
$251$	2.29180	0.144657	0.0723284	+	0.997381i	$0.476957\pi$
$252$	0	0
$253$	−5.23607	−0.329189
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.47214	0.466099	0.233050	−	0.972465i	$-0.425130\pi$
$257$	7.47214	0.466099	0.233050	+	0.972465i	$0.425130\pi$
$258$	0	0
$259$	4.00000	0.248548
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.94427	0.181552	0.0907758	−	0.995871i	$-0.471065\pi$
$263$	2.94427	0.181552	0.0907758	+	0.995871i	$0.471065\pi$
$264$	0	0
$265$	0.583592	0.0358498
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7.94427	0.484371	0.242185	−	0.970230i	$-0.422136\pi$
$269$	7.94427	0.484371	0.242185	+	0.970230i	$0.422136\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	18.1803	1.09632
$276$	0	0
$277$	15.4721	0.929631	0.464815	−	0.885408i	$-0.346121\pi$
$277$	15.4721	0.929631	0.464815	+	0.885408i	$0.346121\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	8.76393	0.522812	0.261406	−	0.965229i	$-0.415814\pi$
$281$	8.76393	0.522812	0.261406	+	0.965229i	$0.415814\pi$
$282$	0	0
$283$	−27.7082	−1.64708	−0.823541	−	0.567257i	$-0.808005\pi$
$283$	−27.7082	−1.64708	−0.823541	+	0.567257i	$0.808005\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−11.2361	−0.663244
$288$	0	0
$289$	−16.4164	−0.965671
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.52786	0.0892588	0.0446294	−	0.999004i	$-0.485789\pi$
$293$	1.52786	0.0892588	0.0446294	+	0.999004i	$0.485789\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.00000	0.173494
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.58359	0.491495
$306$	0	0
$307$	−9.52786	−0.543784	−0.271892	−	0.962328i	$-0.587649\pi$
$307$	−9.52786	−0.543784	−0.271892	+	0.962328i	$0.587649\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	13.1803	0.747389	0.373694	−	0.927552i	$-0.378091\pi$
$311$	13.1803	0.747389	0.373694	+	0.927552i	$0.378091\pi$
$312$	0	0
$313$	24.3607	1.37695	0.688474	−	0.725261i	$-0.258281\pi$
$313$	24.3607	1.37695	0.688474	+	0.725261i	$0.258281\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−25.4164	−1.42753	−0.713764	−	0.700386i	$-0.753011\pi$
$317$	−25.4164	−1.42753	−0.713764	+	0.700386i	$0.753011\pi$
$318$	0	0
$319$	−15.7082	−0.879491
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.52786	−0.0850126
$324$	0	0
$325$	−10.4164	−0.577798
$326$	0	0
$327$	0	0
$328$	0	0
$329$	7.23607	0.398937
$330$	0	0
$331$	19.6525	1.08020	0.540099	−	0.841602i	$-0.318387\pi$
$331$	19.6525	1.08020	0.540099	+	0.841602i	$0.318387\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.41641	−0.186658
$336$	0	0
$337$	23.4164	1.27557	0.637787	−	0.770213i	$-0.279850\pi$
$337$	23.4164	1.27557	0.637787	+	0.770213i	$0.279850\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	35.1246	1.90210
$342$	0	0
$343$	11.4164	0.616428
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−9.88854	−0.530845	−0.265422	−	0.964132i	$-0.585511\pi$
$347$	−9.88854	−0.530845	−0.265422	+	0.964132i	$0.585511\pi$
$348$	0	0
$349$	24.4164	1.30698	0.653490	−	0.756935i	$-0.273304\pi$
$349$	24.4164	1.30698	0.653490	+	0.756935i	$0.273304\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9.36068	−0.498219	−0.249109	−	0.968475i	$-0.580138\pi$
$353$	−9.36068	−0.498219	−0.249109	+	0.968475i	$0.580138\pi$
$354$	0	0
$355$	−15.1246	−0.802731
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−19.8885	−1.04968	−0.524839	−	0.851202i	$-0.675874\pi$
$359$	−19.8885	−1.04968	−0.524839	+	0.851202i	$0.675874\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.06888	−0.422345
$366$	0	0
$367$	4.18034	0.218212	0.109106	−	0.994030i	$-0.465201\pi$
$367$	4.18034	0.218212	0.109106	+	0.994030i	$0.465201\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.52786	0.0793227
$372$	0	0
$373$	7.70820	0.399116	0.199558	−	0.979886i	$-0.436049\pi$
$373$	7.70820	0.399116	0.199558	+	0.979886i	$0.436049\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.00000	0.463524
$378$	0	0
$379$	−24.3607	−1.25132	−0.625662	−	0.780094i	$-0.715171\pi$
$379$	−24.3607	−1.25132	−0.625662	+	0.780094i	$0.715171\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.05573	0.360531	0.180265	−	0.983618i	$-0.442304\pi$
$383$	7.05573	0.360531	0.180265	+	0.983618i	$0.442304\pi$
$384$	0	0
$385$	−20.9443	−1.06742
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−25.5279	−1.29431	−0.647157	−	0.762357i	$-0.724042\pi$
$389$	−25.5279	−1.29431	−0.647157	+	0.762357i	$0.724042\pi$
$390$	0	0
$391$	−0.763932	−0.0386337
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−13.5279	−0.680661
$396$	0	0
$397$	−24.4164	−1.22542	−0.612712	−	0.790306i	$-0.709922\pi$
$397$	−24.4164	−1.22542	−0.612712	+	0.790306i	$0.709922\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	14.1803	0.708132	0.354066	−	0.935220i	$-0.384799\pi$
$401$	14.1803	0.708132	0.354066	+	0.935220i	$0.384799\pi$
$402$	0	0
$403$	−20.1246	−1.00248
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.47214	0.320812
$408$	0	0
$409$	21.3607	1.05622	0.528109	−	0.849177i	$-0.322901\pi$
$409$	21.3607	1.05622	0.528109	+	0.849177i	$0.322901\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−20.9443	−1.03060
$414$	0	0
$415$	10.8328	0.531762
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.58359	−0.223923	−0.111962	−	0.993713i	$-0.535713\pi$
$419$	−4.58359	−0.223923	−0.111962	+	0.993713i	$0.535713\pi$
$420$	0	0
$421$	−10.2918	−0.501591	−0.250796	−	0.968040i	$-0.580692\pi$
$421$	−10.2918	−0.501591	−0.250796	+	0.968040i	$0.580692\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2.65248	0.128664
$426$	0	0
$427$	22.4721	1.08750
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.5279	−0.844288	−0.422144	−	0.906529i	$-0.638722\pi$
$431$	−17.5279	−0.844288	−0.422144	+	0.906529i	$0.638722\pi$
$432$	0	0
$433$	17.8197	0.856358	0.428179	−	0.903694i	$-0.359155\pi$
$433$	17.8197	0.856358	0.428179	+	0.903694i	$0.359155\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.00000	0.0956730
$438$	0	0
$439$	18.7082	0.892894	0.446447	−	0.894810i	$-0.352689\pi$
$439$	18.7082	0.892894	0.446447	+	0.894810i	$0.352689\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	38.1246	1.81135	0.905677	−	0.423967i	$-0.139363\pi$
$443$	38.1246	1.81135	0.905677	+	0.423967i	$0.139363\pi$
$444$	0	0
$445$	−12.9443	−0.613617
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14.9443	0.705264	0.352632	−	0.935762i	$-0.385287\pi$
$449$	14.9443	0.705264	0.352632	+	0.935762i	$0.385287\pi$
$450$	0	0
$451$	−18.1803	−0.856079
$452$	0	0
$453$	0	0
$454$	0	0
$455$	12.0000	0.562569
$456$	0	0
$457$	−5.12461	−0.239719	−0.119860	−	0.992791i	$-0.538244\pi$
$457$	−5.12461	−0.239719	−0.119860	+	0.992791i	$0.538244\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.47214	0.0685642	0.0342821	−	0.999412i	$-0.489086\pi$
$461$	1.47214	0.0685642	0.0342821	+	0.999412i	$0.489086\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−13.0557	−0.604147	−0.302074	−	0.953285i	$-0.597679\pi$
$467$	−13.0557	−0.604147	−0.302074	+	0.953285i	$0.597679\pi$
$468$	0	0
$469$	−8.94427	−0.413008
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−6.94427	−0.318625
$476$	0	0
$477$	0	0
$478$	0	0
$479$	31.5967	1.44369	0.721846	−	0.692054i	$-0.243294\pi$
$479$	31.5967	1.44369	0.721846	+	0.692054i	$0.243294\pi$
$480$	0	0
$481$	−3.70820	−0.169080
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−21.8885	−0.993908
$486$	0	0
$487$	14.7082	0.666492	0.333246	−	0.942840i	$-0.391856\pi$
$487$	14.7082	0.666492	0.333246	+	0.942840i	$0.391856\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8.34752	0.376718	0.188359	−	0.982100i	$-0.439683\pi$
$491$	8.34752	0.376718	0.188359	+	0.982100i	$0.439683\pi$
$492$	0	0
$493$	−2.29180	−0.103217
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−39.5967	−1.77616
$498$	0	0
$499$	−19.2918	−0.863619	−0.431810	−	0.901965i	$-0.642125\pi$
$499$	−19.2918	−0.863619	−0.431810	+	0.901965i	$0.642125\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−26.9443	−1.20139	−0.600693	−	0.799480i	$-0.705109\pi$
$503$	−26.9443	−1.20139	−0.600693	+	0.799480i	$0.705109\pi$
$504$	0	0
$505$	5.52786	0.245987
$506$	0	0
$507$	0	0
$508$	0	0
$509$	28.3050	1.25459	0.627297	−	0.778780i	$-0.284161\pi$
$509$	28.3050	1.25459	0.627297	+	0.778780i	$0.284161\pi$
$510$	0	0
$511$	−21.1246	−0.934498
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−5.16718	−0.227693
$516$	0	0
$517$	11.7082	0.514926
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−31.4164	−1.37638	−0.688189	−	0.725532i	$-0.741594\pi$
$521$	−31.4164	−1.37638	−0.688189	+	0.725532i	$0.741594\pi$
$522$	0	0
$523$	−41.1246	−1.79825	−0.899127	−	0.437688i	$-0.855797\pi$
$523$	−41.1246	−1.79825	−0.899127	+	0.437688i	$0.855797\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.12461	0.223232
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.4164	0.451185
$534$	0	0
$535$	−16.5836	−0.716971
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−18.1803	−0.783083
$540$	0	0
$541$	−34.4164	−1.47968	−0.739838	−	0.672785i	$-0.765098\pi$
$541$	−34.4164	−1.47968	−0.739838	+	0.672785i	$0.765098\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	29.5410	1.26308	0.631541	−	0.775342i	$-0.282423\pi$
$547$	29.5410	1.26308	0.631541	+	0.775342i	$0.282423\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	−35.4164	−1.50606
$554$	0	0
$555$	0	0
$556$	0	0
$557$	7.41641	0.314243	0.157122	−	0.987579i	$-0.449779\pi$
$557$	7.41641	0.314243	0.157122	+	0.987579i	$0.449779\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−32.9443	−1.38844	−0.694218	−	0.719765i	$-0.744250\pi$
$563$	−32.9443	−1.38844	−0.694218	+	0.719765i	$0.744250\pi$
$564$	0	0
$565$	10.8328	0.455740
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22.1803	0.929848	0.464924	−	0.885351i	$-0.346082\pi$
$569$	22.1803	0.929848	0.464924	+	0.885351i	$0.346082\pi$
$570$	0	0
$571$	14.2918	0.598093	0.299047	−	0.954239i	$-0.403331\pi$
$571$	14.2918	0.598093	0.299047	+	0.954239i	$0.403331\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.47214	−0.144798
$576$	0	0
$577$	22.8885	0.952863	0.476431	−	0.879212i	$-0.341930\pi$
$577$	22.8885	0.952863	0.476431	+	0.879212i	$0.341930\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	28.3607	1.17660
$582$	0	0
$583$	2.47214	0.102385
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.7082	−1.01982	−0.509908	−	0.860229i	$-0.670321\pi$
$587$	−24.7082	−1.01982	−0.509908	+	0.860229i	$0.670321\pi$
$588$	0	0
$589$	−13.4164	−0.552813
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.94427	0.120907	0.0604534	−	0.998171i	$-0.480745\pi$
$593$	2.94427	0.120907	0.0604534	+	0.998171i	$0.480745\pi$
$594$	0	0
$595$	−3.05573	−0.125273
$596$	0	0
$597$	0	0
$598$	0	0
$599$	33.8885	1.38465	0.692324	−	0.721587i	$-0.256587\pi$
$599$	33.8885	1.38465	0.692324	+	0.721587i	$0.256587\pi$
$600$	0	0
$601$	46.8885	1.91262	0.956312	−	0.292349i	$-0.0944368\pi$
$601$	46.8885	1.91262	0.956312	+	0.292349i	$0.0944368\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−20.2918	−0.824979
$606$	0	0
$607$	−26.4721	−1.07447	−0.537235	−	0.843432i	$-0.680531\pi$
$607$	−26.4721	−1.07447	−0.537235	+	0.843432i	$0.680531\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.70820	−0.271385
$612$	0	0
$613$	5.70820	0.230552	0.115276	−	0.993333i	$-0.463225\pi$
$613$	5.70820	0.230552	0.115276	+	0.993333i	$0.463225\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.52786	0.303060	0.151530	−	0.988453i	$-0.451580\pi$
$617$	7.52786	0.303060	0.151530	+	0.988453i	$0.451580\pi$
$618$	0	0
$619$	−19.4164	−0.780411	−0.390206	−	0.920728i	$-0.627596\pi$
$619$	−19.4164	−0.780411	−0.390206	+	0.920728i	$0.627596\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−33.8885	−1.35772
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.944272	0.0376506
$630$	0	0
$631$	−12.3607	−0.492071	−0.246035	−	0.969261i	$-0.579128\pi$
$631$	−12.3607	−0.492071	−0.246035	+	0.969261i	$0.579128\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−9.01316	−0.357676
$636$	0	0
$637$	10.4164	0.412713
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17.3050	0.683504	0.341752	−	0.939790i	$-0.388980\pi$
$641$	17.3050	0.683504	0.341752	+	0.939790i	$0.388980\pi$
$642$	0	0
$643$	29.5967	1.16718	0.583591	−	0.812048i	$-0.301647\pi$
$643$	29.5967	1.16718	0.583591	+	0.812048i	$0.301647\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.70820	0.263727	0.131863	−	0.991268i	$-0.457904\pi$
$647$	6.70820	0.263727	0.131863	+	0.991268i	$0.457904\pi$
$648$	0	0
$649$	−33.8885	−1.33024
$650$	0	0
$651$	0	0
$652$	0	0
$653$	38.3050	1.49899	0.749494	−	0.662011i	$-0.230297\pi$
$653$	38.3050	1.49899	0.749494	+	0.662011i	$0.230297\pi$
$654$	0	0
$655$	−23.1246	−0.903553
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−10.6525	−0.414962	−0.207481	−	0.978239i	$-0.566526\pi$
$659$	−10.6525	−0.414962	−0.207481	+	0.978239i	$0.566526\pi$
$660$	0	0
$661$	−22.9443	−0.892429	−0.446214	−	0.894926i	$-0.647228\pi$
$661$	−22.9443	−0.892429	−0.446214	+	0.894926i	$0.647228\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	3.00000	0.116160
$668$	0	0
$669$	0	0
$670$	0	0
$671$	36.3607	1.40369
$672$	0	0
$673$	3.00000	0.115642	0.0578208	−	0.998327i	$-0.481585\pi$
$673$	3.00000	0.115642	0.0578208	+	0.998327i	$0.481585\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−57.3050	−2.19916
$680$	0	0
$681$	0	0
$682$	0	0
$683$	26.5967	1.01770	0.508848	−	0.860856i	$-0.330071\pi$
$683$	26.5967	1.01770	0.508848	+	0.860856i	$0.330071\pi$
$684$	0	0
$685$	−27.0557	−1.03375
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.41641	−0.0539608
$690$	0	0
$691$	−7.05573	−0.268413	−0.134206	−	0.990953i	$-0.542848\pi$
$691$	−7.05573	−0.268413	−0.134206	+	0.990953i	$0.542848\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−13.2361	−0.502073
$696$	0	0
$697$	−2.65248	−0.100470
$698$	0	0
$699$	0	0
$700$	0	0
$701$	3.81966	0.144267	0.0721333	−	0.997395i	$-0.477019\pi$
$701$	3.81966	0.144267	0.0721333	+	0.997395i	$0.477019\pi$
$702$	0	0
$703$	−2.47214	−0.0932384
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.4721	0.544281
$708$	0	0
$709$	−42.0689	−1.57993	−0.789965	−	0.613152i	$-0.789901\pi$
$709$	−42.0689	−1.57993	−0.789965	+	0.613152i	$0.789901\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−6.70820	−0.251224
$714$	0	0
$715$	19.4164	0.726132
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−3.05573	−0.113959	−0.0569797	−	0.998375i	$-0.518147\pi$
$719$	−3.05573	−0.113959	−0.0569797	+	0.998375i	$0.518147\pi$
$720$	0	0
$721$	−13.5279	−0.503804
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−10.4164	−0.386856
$726$	0	0
$727$	27.7082	1.02764	0.513820	−	0.857898i	$-0.328230\pi$
$727$	27.7082	1.02764	0.513820	+	0.857898i	$0.328230\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−31.2361	−1.15373	−0.576865	−	0.816839i	$-0.695724\pi$
$733$	−31.2361	−1.15373	−0.576865	+	0.816839i	$0.695724\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−14.4721	−0.533088
$738$	0	0
$739$	−26.8197	−0.986577	−0.493289	−	0.869866i	$-0.664205\pi$
$739$	−26.8197	−0.986577	−0.493289	+	0.869866i	$0.664205\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	41.1246	1.50872	0.754358	−	0.656463i	$-0.227948\pi$
$743$	41.1246	1.50872	0.754358	+	0.656463i	$0.227948\pi$
$744$	0	0
$745$	29.5279	1.08182
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−43.4164	−1.58640
$750$	0	0
$751$	−0.360680	−0.0131614	−0.00658070	−	0.999978i	$-0.502095\pi$
$751$	−0.360680	−0.0131614	−0.00658070	+	0.999978i	$0.502095\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.23607	0.190560
$756$	0	0
$757$	1.59675	0.0580348	0.0290174	−	0.999579i	$-0.490762\pi$
$757$	1.59675	0.0580348	0.0290174	+	0.999579i	$0.490762\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−46.3050	−1.67855	−0.839277	−	0.543705i	$-0.817021\pi$
$761$	−46.3050	−1.67855	−0.839277	+	0.543705i	$0.817021\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	19.4164	0.701086
$768$	0	0
$769$	−23.1246	−0.833895	−0.416947	−	0.908931i	$-0.636900\pi$
$769$	−23.1246	−0.833895	−0.416947	+	0.908931i	$0.636900\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	5.52786	0.198823	0.0994117	−	0.995046i	$-0.468304\pi$
$773$	5.52786	0.198823	0.0994117	+	0.995046i	$0.468304\pi$
$774$	0	0
$775$	23.2918	0.836666
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.94427	0.248804
$780$	0	0
$781$	−64.0689	−2.29256
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.1115	0.503659
$786$	0	0
$787$	−24.5836	−0.876310	−0.438155	−	0.898899i	$-0.644368\pi$
$787$	−24.5836	−0.876310	−0.438155	+	0.898899i	$0.644368\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	28.3607	1.00839
$792$	0	0
$793$	−20.8328	−0.739795
$794$	0	0
$795$	0	0
$796$	0	0
$797$	34.3607	1.21712	0.608559	−	0.793509i	$-0.291748\pi$
$797$	34.3607	1.21712	0.608559	+	0.793509i	$0.291748\pi$
$798$	0	0
$799$	1.70820	0.0604319
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−34.1803	−1.20620
$804$	0	0
$805$	4.00000	0.140981
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−12.1115	−0.425816	−0.212908	−	0.977072i	$-0.568293\pi$
$809$	−12.1115	−0.425816	−0.212908	+	0.977072i	$0.568293\pi$
$810$	0	0
$811$	24.3475	0.854957	0.427479	−	0.904025i	$-0.359402\pi$
$811$	24.3475	0.854957	0.427479	+	0.904025i	$0.359402\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−7.12461	−0.249564
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	38.9443	1.35916	0.679582	−	0.733599i	$-0.262161\pi$
$821$	38.9443	1.35916	0.679582	+	0.733599i	$0.262161\pi$
$822$	0	0
$823$	39.5410	1.37831	0.689157	−	0.724612i	$-0.257981\pi$
$823$	39.5410	1.37831	0.689157	+	0.724612i	$0.257981\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.52786	0.0531290	0.0265645	−	0.999647i	$-0.491543\pi$
$827$	1.52786	0.0531290	0.0265645	+	0.999647i	$0.491543\pi$
$828$	0	0
$829$	40.2492	1.39791	0.698957	−	0.715164i	$-0.253648\pi$
$829$	40.2492	1.39791	0.698957	+	0.715164i	$0.253648\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−2.65248	−0.0919028
$834$	0	0
$835$	−1.88854	−0.0653558
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−41.1246	−1.41978	−0.709890	−	0.704313i	$-0.751255\pi$
$839$	−41.1246	−1.41978	−0.709890	+	0.704313i	$0.751255\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.94427	0.170088
$846$	0	0
$847$	−53.1246	−1.82538
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.23607	−0.0423719
$852$	0	0
$853$	−10.5836	−0.362375	−0.181188	−	0.983449i	$-0.557994\pi$
$853$	−10.5836	−0.362375	−0.181188	+	0.983449i	$0.557994\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.47214	−0.0502872	−0.0251436	−	0.999684i	$-0.508004\pi$
$857$	−1.47214	−0.0502872	−0.0251436	+	0.999684i	$0.508004\pi$
$858$	0	0
$859$	16.7082	0.570077	0.285038	−	0.958516i	$-0.407994\pi$
$859$	16.7082	0.570077	0.285038	+	0.958516i	$0.407994\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−21.5410	−0.733265	−0.366632	−	0.930366i	$-0.619489\pi$
$863$	−21.5410	−0.733265	−0.366632	+	0.930366i	$0.619489\pi$
$864$	0	0
$865$	28.3607	0.964292
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−57.3050	−1.94394
$870$	0	0
$871$	8.29180	0.280957
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−33.8885	−1.14564
$876$	0	0
$877$	−36.4721	−1.23158	−0.615788	−	0.787912i	$-0.711162\pi$
$877$	−36.4721	−1.23158	−0.615788	+	0.787912i	$0.711162\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−44.1803	−1.48847	−0.744237	−	0.667916i	$-0.767187\pi$
$881$	−44.1803	−1.48847	−0.744237	+	0.667916i	$0.767187\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.0689	0.774577	0.387289	−	0.921959i	$-0.373412\pi$
$887$	23.0689	0.774577	0.387289	+	0.921959i	$0.373412\pi$
$888$	0	0
$889$	−23.5967	−0.791410
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.47214	−0.149654
$894$	0	0
$895$	−0.875388	−0.0292610
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−20.1246	−0.671193
$900$	0	0
$901$	0.360680	0.0120160
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−20.5836	−0.684222
$906$	0	0
$907$	40.2492	1.33645	0.668227	−	0.743958i	$-0.267054\pi$
$907$	40.2492	1.33645	0.668227	+	0.743958i	$0.267054\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	31.3050	1.03718	0.518590	−	0.855023i	$-0.326457\pi$
$911$	31.3050	1.03718	0.518590	+	0.855023i	$0.326457\pi$
$912$	0	0
$913$	45.8885	1.51869
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−60.5410	−1.99924
$918$	0	0
$919$	−41.1246	−1.35658	−0.678288	−	0.734796i	$-0.737278\pi$
$919$	−41.1246	−1.35658	−0.678288	+	0.734796i	$0.737278\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	36.7082	1.20827
$924$	0	0
$925$	4.29180	0.141113
$926$	0	0
$927$	0	0
$928$	0	0
$929$	24.0557	0.789243	0.394621	−	0.918844i	$-0.370876\pi$
$929$	24.0557	0.789243	0.394621	+	0.918844i	$0.370876\pi$
$930$	0	0
$931$	6.94427	0.227589
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.94427	−0.161695
$936$	0	0
$937$	34.1803	1.11662	0.558312	−	0.829631i	$-0.311449\pi$
$937$	34.1803	1.11662	0.558312	+	0.829631i	$0.311449\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−6.65248	−0.216865	−0.108432	−	0.994104i	$-0.534583\pi$
$941$	−6.65248	−0.216865	−0.108432	+	0.994104i	$0.534583\pi$
$942$	0	0
$943$	3.47214	0.113068
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−10.8197	−0.351592	−0.175796	−	0.984427i	$-0.556250\pi$
$947$	−10.8197	−0.351592	−0.175796	+	0.984427i	$0.556250\pi$
$948$	0	0
$949$	19.5836	0.635710
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−20.4721	−0.663158	−0.331579	−	0.943428i	$-0.607581\pi$
$953$	−20.4721	−0.663158	−0.331579	+	0.943428i	$0.607581\pi$
$954$	0	0
$955$	32.3607	1.04717
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−70.8328	−2.28731
$960$	0	0
$961$	14.0000	0.451613
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−12.2918	−0.395687
$966$	0	0
$967$	−27.5410	−0.885659	−0.442830	−	0.896606i	$-0.646025\pi$
$967$	−27.5410	−0.885659	−0.442830	+	0.896606i	$0.646025\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	16.4721	0.528616	0.264308	−	0.964438i	$-0.414856\pi$
$971$	16.4721	0.528616	0.264308	+	0.964438i	$0.414856\pi$
$972$	0	0
$973$	−34.6525	−1.11091
$974$	0	0
$975$	0	0
$976$	0	0
$977$	23.3475	0.746953	0.373477	−	0.927640i	$-0.378166\pi$
$977$	23.3475	0.746953	0.373477	+	0.927640i	$0.378166\pi$
$978$	0	0
$979$	−54.8328	−1.75246
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−40.4721	−1.29086	−0.645430	−	0.763819i	$-0.723322\pi$
$983$	−40.4721	−1.29086	−0.645430	+	0.763819i	$0.723322\pi$
$984$	0	0
$985$	−1.81966	−0.0579792
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−15.1935	−0.481666
$996$	0	0
$997$	16.8328	0.533101	0.266550	−	0.963821i	$-0.414116\pi$
$997$	16.8328	0.533101	0.266550	+	0.963821i	$0.414116\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3312.2.a.ba.1.1		2
3.2	odd	2		368.2.a.h.1.2		2
4.3	odd	2		207.2.a.d.1.1		2
12.11	even	2		23.2.a.a.1.2	✓	2
15.14	odd	2		9200.2.a.bt.1.1		2
20.19	odd	2		5175.2.a.be.1.2		2
24.5	odd	2		1472.2.a.s.1.1		2
24.11	even	2		1472.2.a.t.1.2		2
60.23	odd	4		575.2.b.d.24.2		4
60.47	odd	4		575.2.b.d.24.3		4
60.59	even	2		575.2.a.f.1.1		2
69.68	even	2		8464.2.a.bb.1.2		2
84.83	odd	2		1127.2.a.c.1.2		2
92.91	even	2		4761.2.a.w.1.1		2
132.131	odd	2		2783.2.a.c.1.1		2
156.155	even	2		3887.2.a.i.1.1		2
204.203	even	2		6647.2.a.b.1.2		2
228.227	odd	2		8303.2.a.e.1.1		2
276.11	odd	22		529.2.c.n.466.2		20
276.35	even	22		529.2.c.o.466.2		20
276.59	even	22		529.2.c.o.399.1		20
276.71	even	22		529.2.c.o.487.2		20
276.83	odd	22		529.2.c.n.334.2		20
276.95	even	22		529.2.c.o.170.1		20
276.107	odd	22		529.2.c.n.501.1		20
276.119	even	22		529.2.c.o.177.1		20
276.131	even	22		529.2.c.o.118.1		20
276.143	odd	22		529.2.c.n.255.2		20
276.155	odd	22		529.2.c.n.266.1		20
276.167	even	22		529.2.c.o.266.1		20
276.179	even	22		529.2.c.o.255.2		20
276.191	odd	22		529.2.c.n.118.1		20
276.203	odd	22		529.2.c.n.177.1		20
276.215	even	22		529.2.c.o.501.1		20
276.227	odd	22		529.2.c.n.170.1		20
276.239	even	22		529.2.c.o.334.2		20
276.251	odd	22		529.2.c.n.487.2		20
276.263	odd	22		529.2.c.n.399.1		20
276.275	odd	2		529.2.a.a.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.2.a.a.1.2	✓	2	12.11	even	2
207.2.a.d.1.1		2	4.3	odd	2
368.2.a.h.1.2		2	3.2	odd	2
529.2.a.a.1.2		2	276.275	odd	2
529.2.c.n.118.1		20	276.191	odd	22
529.2.c.n.170.1		20	276.227	odd	22
529.2.c.n.177.1		20	276.203	odd	22
529.2.c.n.255.2		20	276.143	odd	22
529.2.c.n.266.1		20	276.155	odd	22
529.2.c.n.334.2		20	276.83	odd	22
529.2.c.n.399.1		20	276.263	odd	22
529.2.c.n.466.2		20	276.11	odd	22
529.2.c.n.487.2		20	276.251	odd	22
529.2.c.n.501.1		20	276.107	odd	22
529.2.c.o.118.1		20	276.131	even	22
529.2.c.o.170.1		20	276.95	even	22
529.2.c.o.177.1		20	276.119	even	22
529.2.c.o.255.2		20	276.179	even	22
529.2.c.o.266.1		20	276.167	even	22
529.2.c.o.334.2		20	276.239	even	22
529.2.c.o.399.1		20	276.59	even	22
529.2.c.o.466.2		20	276.35	even	22
529.2.c.o.487.2		20	276.71	even	22
529.2.c.o.501.1		20	276.215	even	22
575.2.a.f.1.1		2	60.59	even	2
575.2.b.d.24.2		4	60.23	odd	4
575.2.b.d.24.3		4	60.47	odd	4
1127.2.a.c.1.2		2	84.83	odd	2
1472.2.a.s.1.1		2	24.5	odd	2
1472.2.a.t.1.2		2	24.11	even	2
2783.2.a.c.1.1		2	132.131	odd	2
3312.2.a.ba.1.1		2	1.1	even	1	trivial
3887.2.a.i.1.1		2	156.155	even	2
4761.2.a.w.1.1		2	92.91	even	2
5175.2.a.be.1.2		2	20.19	odd	2
6647.2.a.b.1.2		2	204.203	even	2
8303.2.a.e.1.1		2	228.227	odd	2
8464.2.a.bb.1.2		2	69.68	even	2
9200.2.a.bt.1.1		2	15.14	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:	\( N \)	\(=\)	\( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3312.a (trivial)

\(f(q)\)	\(=\)	\(q-1.23607 q^{5} -3.23607 q^{7} +O(q^{10})\)	\(q-1.23607 q^{5} -3.23607 q^{7} -5.23607 q^{11} +3.00000 q^{13} -0.763932 q^{17} +2.00000 q^{19} +1.00000 q^{23} -3.47214 q^{25} +3.00000 q^{29} -6.70820 q^{31} +4.00000 q^{35} -1.23607 q^{37} +3.47214 q^{41} -2.23607 q^{47} +3.47214 q^{49} -0.472136 q^{53} +6.47214 q^{55} +6.47214 q^{59} -6.94427 q^{61} -3.70820 q^{65} +2.76393 q^{67} +12.2361 q^{71} +6.52786 q^{73} +16.9443 q^{77} +10.9443 q^{79} -8.76393 q^{83} +0.944272 q^{85} +10.4721 q^{89} -9.70820 q^{91} -2.47214 q^{95} +17.7082 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - 2 * q^7	\( 2 q + 2 q^{5} - 2 q^{7} - 6 q^{11} + 6 q^{13} - 6 q^{17} + 4 q^{19} + 2 q^{23} + 2 q^{25} + 6 q^{29} + 8 q^{35} + 2 q^{37} - 2 q^{41} - 2 q^{49} + 8 q^{53} + 4 q^{55} + 4 q^{59} + 4 q^{61} + 6 q^{65} + 10 q^{67} + 20 q^{71} + 22 q^{73} + 16 q^{77} + 4 q^{79} - 22 q^{83} - 16 q^{85} + 12 q^{89} - 6 q^{91} + 4 q^{95} + 22 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 2 * q^7 - 6 * q^11 + 6 * q^13 - 6 * q^17 + 4 * q^19 + 2 * q^23 + 2 * q^25 + 6 * q^29 + 8 * q^35 + 2 * q^37 - 2 * q^41 - 2 * q^49 + 8 * q^53 + 4 * q^55 + 4 * q^59 + 4 * q^61 + 6 * q^65 + 10 * q^67 + 20 * q^71 + 22 * q^73 + 16 * q^77 + 4 * q^79 - 22 * q^83 - 16 * q^85 + 12 * q^89 - 6 * q^91 + 4 * q^95 + 22 * q^97

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