LMFDB - Embedded newform 3344.2.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, 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("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.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:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 22x^{7} + 22x^{6} + 152x^{5} - 136x^{4} - 341x^{3} + 169x^{2} + 196x + 32 $ x^9 - x^8 - 22x^7 + 22x^6 + 152x^5 - 136x^4 - 341x^3 + 169x^2 + 196*x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1672)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.34973$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.34973 q^{3} +3.03417 q^{5} +3.94780 q^{7} +2.52125 q^{9} +O(q^{10})$	$q-2.34973 q^{3} +3.03417 q^{5} +3.94780 q^{7} +2.52125 q^{9} -1.00000 q^{11} +2.12092 q^{13} -7.12948 q^{15} +5.59559 q^{17} +1.00000 q^{19} -9.27627 q^{21} +2.55094 q^{23} +4.20616 q^{25} +1.12494 q^{27} +7.80160 q^{29} -10.6007 q^{31} +2.34973 q^{33} +11.9783 q^{35} +7.46905 q^{37} -4.98361 q^{39} -9.34486 q^{41} +8.94365 q^{43} +7.64989 q^{45} +2.06833 q^{47} +8.58511 q^{49} -13.1482 q^{51} +11.2322 q^{53} -3.03417 q^{55} -2.34973 q^{57} -4.01392 q^{59} -10.4044 q^{61} +9.95338 q^{63} +6.43524 q^{65} -15.3402 q^{67} -5.99403 q^{69} +5.75463 q^{71} +1.09907 q^{73} -9.88337 q^{75} -3.94780 q^{77} -6.32301 q^{79} -10.2070 q^{81} +6.77829 q^{83} +16.9780 q^{85} -18.3317 q^{87} -16.4997 q^{89} +8.37298 q^{91} +24.9088 q^{93} +3.03417 q^{95} -14.5196 q^{97} -2.52125 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9}+O(q^{10}) $ 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9	$ 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9} - 9 q^{11} + q^{13} - 6 q^{15} + 7 q^{17} + 9 q^{19} + 3 q^{21} - 13 q^{23} + 31 q^{25} + 5 q^{27} + 9 q^{29} + 4 q^{31} + q^{33} + 4 q^{35} + 24 q^{37} - 13 q^{39} - 6 q^{41} + 14 q^{43} + 26 q^{45} - 24 q^{47} + 20 q^{49} + 33 q^{51} + 19 q^{53} - 6 q^{55} - q^{57} + 19 q^{59} + 28 q^{61} - 16 q^{63} + 16 q^{65} - 5 q^{67} + 35 q^{69} - 16 q^{71} + 15 q^{73} - 3 q^{75} + 3 q^{77} - 2 q^{79} + 37 q^{81} - 8 q^{83} + 20 q^{85} - 23 q^{87} + 12 q^{89} + 29 q^{91} + 44 q^{93} + 6 q^{95} - 4 q^{97} - 18 q^{99}+O(q^{100}) $ 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9 - 9 * q^11 + q^13 - 6 * q^15 + 7 * q^17 + 9 * q^19 + 3 * q^21 - 13 * q^23 + 31 * q^25 + 5 * q^27 + 9 * q^29 + 4 * q^31 + q^33 + 4 * q^35 + 24 * q^37 - 13 * q^39 - 6 * q^41 + 14 * q^43 + 26 * q^45 - 24 * q^47 + 20 * q^49 + 33 * q^51 + 19 * q^53 - 6 * q^55 - q^57 + 19 * q^59 + 28 * q^61 - 16 * q^63 + 16 * q^65 - 5 * q^67 + 35 * q^69 - 16 * q^71 + 15 * q^73 - 3 * q^75 + 3 * q^77 - 2 * q^79 + 37 * q^81 - 8 * q^83 + 20 * q^85 - 23 * q^87 + 12 * q^89 + 29 * q^91 + 44 * q^93 + 6 * q^95 - 4 * q^97 - 18 * 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
$3$	−2.34973	−1.35662	−0.678310	−	0.734776i	$-0.737287\pi$
$3$	−2.34973	−1.35662	−0.678310	+	0.734776i	$0.737287\pi$
$4$	0	0
$5$	3.03417	1.35692	0.678460	−	0.734637i	$-0.262648\pi$
$5$	3.03417	1.35692	0.678460	+	0.734637i	$0.262648\pi$
$6$	0	0
$7$	3.94780	1.49213	0.746064	−	0.665875i	$-0.231941\pi$
$7$	3.94780	1.49213	0.746064	+	0.665875i	$0.231941\pi$
$8$	0	0
$9$	2.52125	0.840416
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	2.12092	0.588239	0.294119	−	0.955769i	$-0.404974\pi$
$13$	2.12092	0.588239	0.294119	+	0.955769i	$0.404974\pi$
$14$	0	0
$15$	−7.12948	−1.84082
$16$	0	0
$17$	5.59559	1.35713	0.678566	−	0.734540i	$-0.262602\pi$
$17$	5.59559	1.35713	0.678566	+	0.734540i	$0.262602\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−9.27627	−2.02425
$22$	0	0
$23$	2.55094	0.531908	0.265954	−	0.963986i	$-0.414313\pi$
$23$	2.55094	0.531908	0.265954	+	0.963986i	$0.414313\pi$
$24$	0	0
$25$	4.20616	0.841233
$26$	0	0
$27$	1.12494	0.216494
$28$	0	0
$29$	7.80160	1.44872	0.724361	−	0.689421i	$-0.242135\pi$
$29$	7.80160	1.44872	0.724361	+	0.689421i	$0.242135\pi$
$30$	0	0
$31$	−10.6007	−1.90394	−0.951970	−	0.306192i	$-0.900945\pi$
$31$	−10.6007	−1.90394	−0.951970	+	0.306192i	$0.900945\pi$
$32$	0	0
$33$	2.34973	0.409036
$34$	0	0
$35$	11.9783	2.02470
$36$	0	0
$37$	7.46905	1.22790	0.613952	−	0.789343i	$-0.289579\pi$
$37$	7.46905	1.22790	0.613952	+	0.789343i	$0.289579\pi$
$38$	0	0
$39$	−4.98361	−0.798016
$40$	0	0
$41$	−9.34486	−1.45942	−0.729711	−	0.683756i	$-0.760345\pi$
$41$	−9.34486	−1.45942	−0.729711	+	0.683756i	$0.760345\pi$
$42$	0	0
$43$	8.94365	1.36389	0.681947	−	0.731401i	$-0.261133\pi$
$43$	8.94365	1.36389	0.681947	+	0.731401i	$0.261133\pi$
$44$	0	0
$45$	7.64989	1.14038
$46$	0	0
$47$	2.06833	0.301697	0.150849	−	0.988557i	$-0.451799\pi$
$47$	2.06833	0.301697	0.150849	+	0.988557i	$0.451799\pi$
$48$	0	0
$49$	8.58511	1.22644
$50$	0	0
$51$	−13.1482	−1.84111
$52$	0	0
$53$	11.2322	1.54286	0.771428	−	0.636316i	$-0.219543\pi$
$53$	11.2322	1.54286	0.771428	+	0.636316i	$0.219543\pi$
$54$	0	0
$55$	−3.03417	−0.409127
$56$	0	0
$57$	−2.34973	−0.311230
$58$	0	0
$59$	−4.01392	−0.522568	−0.261284	−	0.965262i	$-0.584146\pi$
$59$	−4.01392	−0.522568	−0.261284	+	0.965262i	$0.584146\pi$
$60$	0	0
$61$	−10.4044	−1.33214	−0.666071	−	0.745888i	$-0.732025\pi$
$61$	−10.4044	−1.33214	−0.666071	+	0.745888i	$0.732025\pi$
$62$	0	0
$63$	9.95338	1.25401
$64$	0	0
$65$	6.43524	0.798193
$66$	0	0
$67$	−15.3402	−1.87410	−0.937050	−	0.349196i	$-0.886455\pi$
$67$	−15.3402	−1.87410	−0.937050	+	0.349196i	$0.886455\pi$
$68$	0	0
$69$	−5.99403	−0.721596
$70$	0	0
$71$	5.75463	0.682949	0.341475	−	0.939891i	$-0.389074\pi$
$71$	5.75463	0.682949	0.341475	+	0.939891i	$0.389074\pi$
$72$	0	0
$73$	1.09907	0.128636	0.0643182	−	0.997929i	$-0.479513\pi$
$73$	1.09907	0.128636	0.0643182	+	0.997929i	$0.479513\pi$
$74$	0	0
$75$	−9.88337	−1.14123
$76$	0	0
$77$	−3.94780	−0.449893
$78$	0	0
$79$	−6.32301	−0.711394	−0.355697	−	0.934601i	$-0.615756\pi$
$79$	−6.32301	−0.711394	−0.355697	+	0.934601i	$0.615756\pi$
$80$	0	0
$81$	−10.2070	−1.13412
$82$	0	0
$83$	6.77829	0.744014	0.372007	−	0.928230i	$-0.378670\pi$
$83$	6.77829	0.744014	0.372007	+	0.928230i	$0.378670\pi$
$84$	0	0
$85$	16.9780	1.84152
$86$	0	0
$87$	−18.3317	−1.96536
$88$	0	0
$89$	−16.4997	−1.74897	−0.874485	−	0.485053i	$-0.838800\pi$
$89$	−16.4997	−1.74897	−0.874485	+	0.485053i	$0.838800\pi$
$90$	0	0
$91$	8.37298	0.877727
$92$	0	0
$93$	24.9088	2.58292
$94$	0	0
$95$	3.03417	0.311299
$96$	0	0
$97$	−14.5196	−1.47425	−0.737123	−	0.675759i	$-0.763816\pi$
$97$	−14.5196	−1.47425	−0.737123	+	0.675759i	$0.763816\pi$
$98$	0	0
$99$	−2.52125	−0.253395
$100$	0	0
$101$	−6.92092	−0.688657	−0.344328	−	0.938849i	$-0.611893\pi$
$101$	−6.92092	−0.688657	−0.344328	+	0.938849i	$0.611893\pi$
$102$	0	0
$103$	9.67227	0.953038	0.476519	−	0.879164i	$-0.341898\pi$
$103$	9.67227	0.953038	0.476519	+	0.879164i	$0.341898\pi$
$104$	0	0
$105$	−28.1458	−2.74674
$106$	0	0
$107$	3.34092	0.322979	0.161489	−	0.986874i	$-0.448370\pi$
$107$	3.34092	0.322979	0.161489	+	0.986874i	$0.448370\pi$
$108$	0	0
$109$	6.92633	0.663422	0.331711	−	0.943381i	$-0.392374\pi$
$109$	6.92633	0.663422	0.331711	+	0.943381i	$0.392374\pi$
$110$	0	0
$111$	−17.5503	−1.66580
$112$	0	0
$113$	−17.3274	−1.63003	−0.815014	−	0.579441i	$-0.803271\pi$
$113$	−17.3274	−1.63003	−0.815014	+	0.579441i	$0.803271\pi$
$114$	0	0
$115$	7.73997	0.721756
$116$	0	0
$117$	5.34738	0.494365
$118$	0	0
$119$	22.0903	2.02501
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	21.9579	1.97988
$124$	0	0
$125$	−2.40863	−0.215435
$126$	0	0
$127$	5.99987	0.532402	0.266201	−	0.963918i	$-0.414232\pi$
$127$	5.99987	0.532402	0.266201	+	0.963918i	$0.414232\pi$
$128$	0	0
$129$	−21.0152	−1.85029
$130$	0	0
$131$	−6.39855	−0.559044	−0.279522	−	0.960139i	$-0.590176\pi$
$131$	−6.39855	−0.559044	−0.279522	+	0.960139i	$0.590176\pi$
$132$	0	0
$133$	3.94780	0.342317
$134$	0	0
$135$	3.41324	0.293765
$136$	0	0
$137$	−5.90309	−0.504335	−0.252167	−	0.967684i	$-0.581143\pi$
$137$	−5.90309	−0.504335	−0.252167	+	0.967684i	$0.581143\pi$
$138$	0	0
$139$	18.5162	1.57053	0.785264	−	0.619162i	$-0.212527\pi$
$139$	18.5162	1.57053	0.785264	+	0.619162i	$0.212527\pi$
$140$	0	0
$141$	−4.86003	−0.409288
$142$	0	0
$143$	−2.12092	−0.177361
$144$	0	0
$145$	23.6714	1.96580
$146$	0	0
$147$	−20.1727	−1.66382
$148$	0	0
$149$	3.10970	0.254757	0.127378	−	0.991854i	$-0.459344\pi$
$149$	3.10970	0.254757	0.127378	+	0.991854i	$0.459344\pi$
$150$	0	0
$151$	6.89839	0.561383	0.280691	−	0.959798i	$-0.409436\pi$
$151$	6.89839	0.561383	0.280691	+	0.959798i	$0.409436\pi$
$152$	0	0
$153$	14.1079	1.14056
$154$	0	0
$155$	−32.1642	−2.58349
$156$	0	0
$157$	16.4585	1.31353	0.656766	−	0.754095i	$-0.271924\pi$
$157$	16.4585	1.31353	0.656766	+	0.754095i	$0.271924\pi$
$158$	0	0
$159$	−26.3926	−2.09307
$160$	0	0
$161$	10.0706	0.793674
$162$	0	0
$163$	−16.4555	−1.28890	−0.644448	−	0.764648i	$-0.722913\pi$
$163$	−16.4555	−1.28890	−0.644448	+	0.764648i	$0.722913\pi$
$164$	0	0
$165$	7.12948	0.555029
$166$	0	0
$167$	20.7199	1.60336	0.801678	−	0.597756i	$-0.203941\pi$
$167$	20.7199	1.60336	0.801678	+	0.597756i	$0.203941\pi$
$168$	0	0
$169$	−8.50168	−0.653975
$170$	0	0
$171$	2.52125	0.192805
$172$	0	0
$173$	−8.73689	−0.664253	−0.332127	−	0.943235i	$-0.607766\pi$
$173$	−8.73689	−0.664253	−0.332127	+	0.943235i	$0.607766\pi$
$174$	0	0
$175$	16.6051	1.25523
$176$	0	0
$177$	9.43164	0.708925
$178$	0	0
$179$	−4.79152	−0.358135	−0.179067	−	0.983837i	$-0.557308\pi$
$179$	−4.79152	−0.358135	−0.179067	+	0.983837i	$0.557308\pi$
$180$	0	0
$181$	23.2251	1.72631	0.863154	−	0.504941i	$-0.168486\pi$
$181$	23.2251	1.72631	0.863154	+	0.504941i	$0.168486\pi$
$182$	0	0
$183$	24.4475	1.80721
$184$	0	0
$185$	22.6623	1.66617
$186$	0	0
$187$	−5.59559	−0.409190
$188$	0	0
$189$	4.44102	0.323037
$190$	0	0
$191$	2.98316	0.215854	0.107927	−	0.994159i	$-0.465579\pi$
$191$	2.98316	0.215854	0.107927	+	0.994159i	$0.465579\pi$
$192$	0	0
$193$	9.11979	0.656457	0.328228	−	0.944598i	$-0.393548\pi$
$193$	9.11979	0.656457	0.328228	+	0.944598i	$0.393548\pi$
$194$	0	0
$195$	−15.1211	−1.08284
$196$	0	0
$197$	2.90203	0.206761	0.103381	−	0.994642i	$-0.467034\pi$
$197$	2.90203	0.206761	0.103381	+	0.994642i	$0.467034\pi$
$198$	0	0
$199$	−1.73021	−0.122651	−0.0613255	−	0.998118i	$-0.519533\pi$
$199$	−1.73021	−0.122651	−0.0613255	+	0.998118i	$0.519533\pi$
$200$	0	0
$201$	36.0453	2.54244
$202$	0	0
$203$	30.7992	2.16168
$204$	0	0
$205$	−28.3539	−1.98032
$206$	0	0
$207$	6.43155	0.447024
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	11.8044	0.812651	0.406325	−	0.913729i	$-0.366810\pi$
$211$	11.8044	0.812651	0.406325	+	0.913729i	$0.366810\pi$
$212$	0	0
$213$	−13.5219	−0.926503
$214$	0	0
$215$	27.1365	1.85070
$216$	0	0
$217$	−41.8494	−2.84092
$218$	0	0
$219$	−2.58252	−0.174511
$220$	0	0
$221$	11.8678	0.798317
$222$	0	0
$223$	4.88320	0.327003	0.163502	−	0.986543i	$-0.447721\pi$
$223$	4.88320	0.327003	0.163502	+	0.986543i	$0.447721\pi$
$224$	0	0
$225$	10.6048	0.706986
$226$	0	0
$227$	−26.3527	−1.74909	−0.874544	−	0.484946i	$-0.838839\pi$
$227$	−26.3527	−1.74909	−0.874544	+	0.484946i	$0.838839\pi$
$228$	0	0
$229$	−9.96059	−0.658214	−0.329107	−	0.944293i	$-0.606748\pi$
$229$	−9.96059	−0.658214	−0.329107	+	0.944293i	$0.606748\pi$
$230$	0	0
$231$	9.27627	0.610334
$232$	0	0
$233$	27.4440	1.79792	0.898959	−	0.438033i	$-0.144325\pi$
$233$	27.4440	1.79792	0.898959	+	0.438033i	$0.144325\pi$
$234$	0	0
$235$	6.27566	0.409379
$236$	0	0
$237$	14.8574	0.965091
$238$	0	0
$239$	2.62370	0.169713	0.0848565	−	0.996393i	$-0.472957\pi$
$239$	2.62370	0.169713	0.0848565	+	0.996393i	$0.472957\pi$
$240$	0	0
$241$	−4.61737	−0.297431	−0.148716	−	0.988880i	$-0.547514\pi$
$241$	−4.61737	−0.297431	−0.148716	+	0.988880i	$0.547514\pi$
$242$	0	0
$243$	20.6090	1.32207
$244$	0	0
$245$	26.0486	1.66419
$246$	0	0
$247$	2.12092	0.134951
$248$	0	0
$249$	−15.9272	−1.00934
$250$	0	0
$251$	−12.1513	−0.766986	−0.383493	−	0.923544i	$-0.625279\pi$
$251$	−12.1513	−0.766986	−0.383493	+	0.923544i	$0.625279\pi$
$252$	0	0
$253$	−2.55094	−0.160376
$254$	0	0
$255$	−39.8937	−2.49824
$256$	0	0
$257$	8.85419	0.552309	0.276154	−	0.961113i	$-0.410940\pi$
$257$	8.85419	0.552309	0.276154	+	0.961113i	$0.410940\pi$
$258$	0	0
$259$	29.4863	1.83219
$260$	0	0
$261$	19.6698	1.21753
$262$	0	0
$263$	−6.60378	−0.407206	−0.203603	−	0.979053i	$-0.565265\pi$
$263$	−6.60378	−0.407206	−0.203603	+	0.979053i	$0.565265\pi$
$264$	0	0
$265$	34.0802	2.09353
$266$	0	0
$267$	38.7700	2.37269
$268$	0	0
$269$	−13.5020	−0.823229	−0.411614	−	0.911358i	$-0.635035\pi$
$269$	−13.5020	−0.823229	−0.411614	+	0.911358i	$0.635035\pi$
$270$	0	0
$271$	21.9522	1.33350	0.666751	−	0.745281i	$-0.267684\pi$
$271$	21.9522	1.33350	0.666751	+	0.745281i	$0.267684\pi$
$272$	0	0
$273$	−19.6743	−1.19074
$274$	0	0
$275$	−4.20616	−0.253641
$276$	0	0
$277$	2.11463	0.127056	0.0635278	−	0.997980i	$-0.479765\pi$
$277$	2.11463	0.127056	0.0635278	+	0.997980i	$0.479765\pi$
$278$	0	0
$279$	−26.7270	−1.60010
$280$	0	0
$281$	−7.06167	−0.421264	−0.210632	−	0.977565i	$-0.567552\pi$
$281$	−7.06167	−0.421264	−0.210632	+	0.977565i	$0.567552\pi$
$282$	0	0
$283$	25.4377	1.51212	0.756058	−	0.654505i	$-0.227123\pi$
$283$	25.4377	1.51212	0.756058	+	0.654505i	$0.227123\pi$
$284$	0	0
$285$	−7.12948	−0.422314
$286$	0	0
$287$	−36.8916	−2.17764
$288$	0	0
$289$	14.3107	0.841805
$290$	0	0
$291$	34.1173	1.99999
$292$	0	0
$293$	23.6572	1.38207	0.691033	−	0.722823i	$-0.257156\pi$
$293$	23.6572	1.38207	0.691033	+	0.722823i	$0.257156\pi$
$294$	0	0
$295$	−12.1789	−0.709083
$296$	0	0
$297$	−1.12494	−0.0652754
$298$	0	0
$299$	5.41035	0.312889
$300$	0	0
$301$	35.3077	2.03510
$302$	0	0
$303$	16.2623	0.934245
$304$	0	0
$305$	−31.5686	−1.80761
$306$	0	0
$307$	−9.72085	−0.554798	−0.277399	−	0.960755i	$-0.589472\pi$
$307$	−9.72085	−0.554798	−0.277399	+	0.960755i	$0.589472\pi$
$308$	0	0
$309$	−22.7273	−1.29291
$310$	0	0
$311$	5.52113	0.313075	0.156537	−	0.987672i	$-0.449967\pi$
$311$	5.52113	0.313075	0.156537	+	0.987672i	$0.449967\pi$
$312$	0	0
$313$	−20.0907	−1.13560	−0.567798	−	0.823168i	$-0.692205\pi$
$313$	−20.0907	−1.13560	−0.567798	+	0.823168i	$0.692205\pi$
$314$	0	0
$315$	30.2002	1.70159
$316$	0	0
$317$	10.2757	0.577142	0.288571	−	0.957458i	$-0.406820\pi$
$317$	10.2757	0.577142	0.288571	+	0.957458i	$0.406820\pi$
$318$	0	0
$319$	−7.80160	−0.436806
$320$	0	0
$321$	−7.85027	−0.438159
$322$	0	0
$323$	5.59559	0.311347
$324$	0	0
$325$	8.92096	0.494846
$326$	0	0
$327$	−16.2750	−0.900012
$328$	0	0
$329$	8.16536	0.450171
$330$	0	0
$331$	−13.2163	−0.726434	−0.363217	−	0.931705i	$-0.618322\pi$
$331$	−13.2163	−0.726434	−0.363217	+	0.931705i	$0.618322\pi$
$332$	0	0
$333$	18.8313	1.03195
$334$	0	0
$335$	−46.5446	−2.54300
$336$	0	0
$337$	25.6604	1.39781	0.698905	−	0.715215i	$-0.253671\pi$
$337$	25.6604	1.39781	0.698905	+	0.715215i	$0.253671\pi$
$338$	0	0
$339$	40.7149	2.21133
$340$	0	0
$341$	10.6007	0.574059
$342$	0	0
$343$	6.25768	0.337883
$344$	0	0
$345$	−18.1869	−0.979149
$346$	0	0
$347$	−8.01803	−0.430430	−0.215215	−	0.976567i	$-0.569045\pi$
$347$	−8.01803	−0.430430	−0.215215	+	0.976567i	$0.569045\pi$
$348$	0	0
$349$	6.85825	0.367113	0.183557	−	0.983009i	$-0.441239\pi$
$349$	6.85825	0.367113	0.183557	+	0.983009i	$0.441239\pi$
$350$	0	0
$351$	2.38591	0.127350
$352$	0	0
$353$	−22.8714	−1.21732	−0.608660	−	0.793431i	$-0.708293\pi$
$353$	−22.8714	−1.21732	−0.608660	+	0.793431i	$0.708293\pi$
$354$	0	0
$355$	17.4605	0.926708
$356$	0	0
$357$	−51.9063	−2.74717
$358$	0	0
$359$	−0.390305	−0.0205995	−0.0102997	−	0.999947i	$-0.503279\pi$
$359$	−0.390305	−0.0205995	−0.0102997	+	0.999947i	$0.503279\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−2.34973	−0.123329
$364$	0	0
$365$	3.33476	0.174549
$366$	0	0
$367$	17.1342	0.894400	0.447200	−	0.894434i	$-0.352421\pi$
$367$	17.1342	0.894400	0.447200	+	0.894434i	$0.352421\pi$
$368$	0	0
$369$	−23.5607	−1.22652
$370$	0	0
$371$	44.3423	2.30214
$372$	0	0
$373$	−25.9426	−1.34325	−0.671627	−	0.740889i	$-0.734404\pi$
$373$	−25.9426	−1.34325	−0.671627	+	0.740889i	$0.734404\pi$
$374$	0	0
$375$	5.65964	0.292263
$376$	0	0
$377$	16.5466	0.852194
$378$	0	0
$379$	19.3100	0.991886	0.495943	−	0.868355i	$-0.334822\pi$
$379$	19.3100	0.991886	0.495943	+	0.868355i	$0.334822\pi$
$380$	0	0
$381$	−14.0981	−0.722267
$382$	0	0
$383$	18.6657	0.953771	0.476886	−	0.878965i	$-0.341766\pi$
$383$	18.6657	0.953771	0.476886	+	0.878965i	$0.341766\pi$
$384$	0	0
$385$	−11.9783	−0.610469
$386$	0	0
$387$	22.5492	1.14624
$388$	0	0
$389$	−14.2819	−0.724121	−0.362061	−	0.932155i	$-0.617927\pi$
$389$	−14.2819	−0.724121	−0.362061	+	0.932155i	$0.617927\pi$
$390$	0	0
$391$	14.2740	0.721868
$392$	0	0
$393$	15.0349	0.758410
$394$	0	0
$395$	−19.1851	−0.965305
$396$	0	0
$397$	12.3867	0.621670	0.310835	−	0.950464i	$-0.399391\pi$
$397$	12.3867	0.621670	0.310835	+	0.950464i	$0.399391\pi$
$398$	0	0
$399$	−9.27627	−0.464395
$400$	0	0
$401$	−7.11792	−0.355452	−0.177726	−	0.984080i	$-0.556874\pi$
$401$	−7.11792	−0.355452	−0.177726	+	0.984080i	$0.556874\pi$
$402$	0	0
$403$	−22.4833	−1.11997
$404$	0	0
$405$	−30.9699	−1.53891
$406$	0	0
$407$	−7.46905	−0.370227
$408$	0	0
$409$	−13.1427	−0.649865	−0.324933	−	0.945737i	$-0.605342\pi$
$409$	−13.1427	−0.649865	−0.324933	+	0.945737i	$0.605342\pi$
$410$	0	0
$411$	13.8707	0.684191
$412$	0	0
$413$	−15.8461	−0.779737
$414$	0	0
$415$	20.5665	1.00957
$416$	0	0
$417$	−43.5082	−2.13061
$418$	0	0
$419$	23.1170	1.12934	0.564669	−	0.825317i	$-0.309004\pi$
$419$	23.1170	1.12934	0.564669	+	0.825317i	$0.309004\pi$
$420$	0	0
$421$	28.4447	1.38631	0.693156	−	0.720788i	$-0.256220\pi$
$421$	28.4447	1.38631	0.693156	+	0.720788i	$0.256220\pi$
$422$	0	0
$423$	5.21478	0.253551
$424$	0	0
$425$	23.5360	1.14166
$426$	0	0
$427$	−41.0743	−1.98773
$428$	0	0
$429$	4.98361	0.240611
$430$	0	0
$431$	−16.2860	−0.784468	−0.392234	−	0.919865i	$-0.628298\pi$
$431$	−16.2860	−0.784468	−0.392234	+	0.919865i	$0.628298\pi$
$432$	0	0
$433$	−28.2273	−1.35652	−0.678259	−	0.734823i	$-0.737265\pi$
$433$	−28.2273	−1.35652	−0.678259	+	0.734823i	$0.737265\pi$
$434$	0	0
$435$	−55.6214	−2.66684
$436$	0	0
$437$	2.55094	0.122028
$438$	0	0
$439$	−21.9516	−1.04769	−0.523847	−	0.851813i	$-0.675504\pi$
$439$	−21.9516	−1.04769	−0.523847	+	0.851813i	$0.675504\pi$
$440$	0	0
$441$	21.6452	1.03072
$442$	0	0
$443$	−15.2592	−0.724989	−0.362494	−	0.931986i	$-0.618075\pi$
$443$	−15.2592	−0.724989	−0.362494	+	0.931986i	$0.618075\pi$
$444$	0	0
$445$	−50.0630	−2.37321
$446$	0	0
$447$	−7.30697	−0.345608
$448$	0	0
$449$	−21.3528	−1.00770	−0.503852	−	0.863790i	$-0.668084\pi$
$449$	−21.3528	−1.00770	−0.503852	+	0.863790i	$0.668084\pi$
$450$	0	0
$451$	9.34486	0.440032
$452$	0	0
$453$	−16.2094	−0.761583
$454$	0	0
$455$	25.4050	1.19101
$456$	0	0
$457$	1.04050	0.0486728	0.0243364	−	0.999704i	$-0.492253\pi$
$457$	1.04050	0.0486728	0.0243364	+	0.999704i	$0.492253\pi$
$458$	0	0
$459$	6.29469	0.293811
$460$	0	0
$461$	−17.9731	−0.837091	−0.418546	−	0.908196i	$-0.637460\pi$
$461$	−17.9731	−0.837091	−0.418546	+	0.908196i	$0.637460\pi$
$462$	0	0
$463$	−12.6975	−0.590102	−0.295051	−	0.955482i	$-0.595337\pi$
$463$	−12.6975	−0.590102	−0.295051	+	0.955482i	$0.595337\pi$
$464$	0	0
$465$	75.5774	3.50482
$466$	0	0
$467$	5.33115	0.246696	0.123348	−	0.992363i	$-0.460637\pi$
$467$	5.33115	0.246696	0.123348	+	0.992363i	$0.460637\pi$
$468$	0	0
$469$	−60.5599	−2.79639
$470$	0	0
$471$	−38.6731	−1.78196
$472$	0	0
$473$	−8.94365	−0.411230
$474$	0	0
$475$	4.20616	0.192992
$476$	0	0
$477$	28.3191	1.29664
$478$	0	0
$479$	−3.54726	−0.162079	−0.0810393	−	0.996711i	$-0.525824\pi$
$479$	−3.54726	−0.162079	−0.0810393	+	0.996711i	$0.525824\pi$
$480$	0	0
$481$	15.8413	0.722300
$482$	0	0
$483$	−23.6632	−1.07671
$484$	0	0
$485$	−44.0550	−2.00043
$486$	0	0
$487$	−10.8424	−0.491317	−0.245659	−	0.969356i	$-0.579004\pi$
$487$	−10.8424	−0.491317	−0.245659	+	0.969356i	$0.579004\pi$
$488$	0	0
$489$	38.6661	1.74854
$490$	0	0
$491$	−33.5878	−1.51579	−0.757897	−	0.652374i	$-0.773773\pi$
$491$	−33.5878	−1.51579	−0.757897	+	0.652374i	$0.773773\pi$
$492$	0	0
$493$	43.6546	1.96610
$494$	0	0
$495$	−7.64989	−0.343837
$496$	0	0
$497$	22.7181	1.01905
$498$	0	0
$499$	30.8511	1.38108	0.690542	−	0.723292i	$-0.257372\pi$
$499$	30.8511	1.38108	0.690542	+	0.723292i	$0.257372\pi$
$500$	0	0
$501$	−48.6863	−2.17514
$502$	0	0
$503$	−10.3520	−0.461575	−0.230787	−	0.973004i	$-0.574130\pi$
$503$	−10.3520	−0.461575	−0.230787	+	0.973004i	$0.574130\pi$
$504$	0	0
$505$	−20.9992	−0.934453
$506$	0	0
$507$	19.9767	0.887196
$508$	0	0
$509$	21.4961	0.952797	0.476399	−	0.879229i	$-0.341942\pi$
$509$	21.4961	0.952797	0.476399	+	0.879229i	$0.341942\pi$
$510$	0	0
$511$	4.33891	0.191942
$512$	0	0
$513$	1.12494	0.0496672
$514$	0	0
$515$	29.3473	1.29320
$516$	0	0
$517$	−2.06833	−0.0909651
$518$	0	0
$519$	20.5294	0.901139
$520$	0	0
$521$	18.5951	0.814665	0.407332	−	0.913280i	$-0.366459\pi$
$521$	18.5951	0.814665	0.407332	+	0.913280i	$0.366459\pi$
$522$	0	0
$523$	14.7620	0.645499	0.322749	−	0.946484i	$-0.395393\pi$
$523$	14.7620	0.645499	0.322749	+	0.946484i	$0.395393\pi$
$524$	0	0
$525$	−39.0175	−1.70286
$526$	0	0
$527$	−59.3171	−2.58390
$528$	0	0
$529$	−16.4927	−0.717074
$530$	0	0
$531$	−10.1201	−0.439174
$532$	0	0
$533$	−19.8197	−0.858489
$534$	0	0
$535$	10.1369	0.438257
$536$	0	0
$537$	11.2588	0.485852
$538$	0	0
$539$	−8.58511	−0.369787
$540$	0	0
$541$	20.1167	0.864883	0.432442	−	0.901662i	$-0.357652\pi$
$541$	20.1167	0.864883	0.432442	+	0.901662i	$0.357652\pi$
$542$	0	0
$543$	−54.5728	−2.34194
$544$	0	0
$545$	21.0156	0.900211
$546$	0	0
$547$	46.4756	1.98715	0.993577	−	0.113159i	$-0.0360970\pi$
$547$	46.4756	1.98715	0.993577	+	0.113159i	$0.0360970\pi$
$548$	0	0
$549$	−26.2320	−1.11955
$550$	0	0
$551$	7.80160	0.332359
$552$	0	0
$553$	−24.9620	−1.06149
$554$	0	0
$555$	−53.2504	−2.26036
$556$	0	0
$557$	20.3891	0.863914	0.431957	−	0.901894i	$-0.357823\pi$
$557$	20.3891	0.863914	0.431957	+	0.901894i	$0.357823\pi$
$558$	0	0
$559$	18.9688	0.802295
$560$	0	0
$561$	13.1482	0.555116
$562$	0	0
$563$	−31.3556	−1.32148	−0.660741	−	0.750614i	$-0.729758\pi$
$563$	−31.3556	−1.32148	−0.660741	+	0.750614i	$0.729758\pi$
$564$	0	0
$565$	−52.5743	−2.21182
$566$	0	0
$567$	−40.2954	−1.69225
$568$	0	0
$569$	5.99002	0.251115	0.125557	−	0.992086i	$-0.459928\pi$
$569$	5.99002	0.251115	0.125557	+	0.992086i	$0.459928\pi$
$570$	0	0
$571$	−47.4250	−1.98467	−0.992336	−	0.123566i	$-0.960567\pi$
$571$	−47.4250	−1.98467	−0.992336	+	0.123566i	$0.960567\pi$
$572$	0	0
$573$	−7.00962	−0.292831
$574$	0	0
$575$	10.7297	0.447458
$576$	0	0
$577$	−9.15079	−0.380952	−0.190476	−	0.981692i	$-0.561003\pi$
$577$	−9.15079	−0.380952	−0.190476	+	0.981692i	$0.561003\pi$
$578$	0	0
$579$	−21.4291	−0.890562
$580$	0	0
$581$	26.7593	1.11016
$582$	0	0
$583$	−11.2322	−0.465189
$584$	0	0
$585$	16.2248	0.670814
$586$	0	0
$587$	−6.53946	−0.269912	−0.134956	−	0.990852i	$-0.543089\pi$
$587$	−6.53946	−0.269912	−0.134956	+	0.990852i	$0.543089\pi$
$588$	0	0
$589$	−10.6007	−0.436794
$590$	0	0
$591$	−6.81901	−0.280496
$592$	0	0
$593$	−18.9136	−0.776689	−0.388345	−	0.921514i	$-0.626953\pi$
$593$	−18.9136	−0.776689	−0.388345	+	0.921514i	$0.626953\pi$
$594$	0	0
$595$	67.0256	2.74778
$596$	0	0
$597$	4.06552	0.166391
$598$	0	0
$599$	11.2920	0.461378	0.230689	−	0.973028i	$-0.425902\pi$
$599$	11.2920	0.461378	0.230689	+	0.973028i	$0.425902\pi$
$600$	0	0
$601$	−19.7078	−0.803899	−0.401949	−	0.915662i	$-0.631667\pi$
$601$	−19.7078	−0.803899	−0.401949	+	0.915662i	$0.631667\pi$
$602$	0	0
$603$	−38.6764	−1.57502
$604$	0	0
$605$	3.03417	0.123356
$606$	0	0
$607$	3.69504	0.149977	0.0749884	−	0.997184i	$-0.476108\pi$
$607$	3.69504	0.149977	0.0749884	+	0.997184i	$0.476108\pi$
$608$	0	0
$609$	−72.3698	−2.93257
$610$	0	0
$611$	4.38678	0.177470
$612$	0	0
$613$	15.2905	0.617579	0.308790	−	0.951130i	$-0.400076\pi$
$613$	15.2905	0.617579	0.308790	+	0.951130i	$0.400076\pi$
$614$	0	0
$615$	66.6240	2.68654
$616$	0	0
$617$	43.8991	1.76731	0.883656	−	0.468137i	$-0.155075\pi$
$617$	43.8991	1.76731	0.883656	+	0.468137i	$0.155075\pi$
$618$	0	0
$619$	−3.66902	−0.147470	−0.0737351	−	0.997278i	$-0.523492\pi$
$619$	−3.66902	−0.147470	−0.0737351	+	0.997278i	$0.523492\pi$
$620$	0	0
$621$	2.86965	0.115155
$622$	0	0
$623$	−65.1377	−2.60969
$624$	0	0
$625$	−28.3390	−1.13356
$626$	0	0
$627$	2.34973	0.0938393
$628$	0	0
$629$	41.7938	1.66643
$630$	0	0
$631$	6.88194	0.273966	0.136983	−	0.990573i	$-0.456259\pi$
$631$	6.88194	0.273966	0.136983	+	0.990573i	$0.456259\pi$
$632$	0	0
$633$	−27.7373	−1.10246
$634$	0	0
$635$	18.2046	0.722427
$636$	0	0
$637$	18.2084	0.721442
$638$	0	0
$639$	14.5089	0.573962
$640$	0	0
$641$	−5.29991	−0.209334	−0.104667	−	0.994507i	$-0.533378\pi$
$641$	−5.29991	−0.209334	−0.104667	+	0.994507i	$0.533378\pi$
$642$	0	0
$643$	10.9741	0.432776	0.216388	−	0.976307i	$-0.430572\pi$
$643$	10.9741	0.432776	0.216388	+	0.976307i	$0.430572\pi$
$644$	0	0
$645$	−63.7636	−2.51069
$646$	0	0
$647$	−41.2101	−1.62013	−0.810067	−	0.586337i	$-0.800569\pi$
$647$	−41.2101	−1.62013	−0.810067	+	0.586337i	$0.800569\pi$
$648$	0	0
$649$	4.01392	0.157560
$650$	0	0
$651$	98.3349	3.85405
$652$	0	0
$653$	−14.6595	−0.573672	−0.286836	−	0.957980i	$-0.592604\pi$
$653$	−14.6595	−0.573672	−0.286836	+	0.957980i	$0.592604\pi$
$654$	0	0
$655$	−19.4143	−0.758579
$656$	0	0
$657$	2.77103	0.108108
$658$	0	0
$659$	35.6405	1.38836	0.694179	−	0.719803i	$-0.255768\pi$
$659$	35.6405	1.38836	0.694179	+	0.719803i	$0.255768\pi$
$660$	0	0
$661$	−18.4306	−0.716867	−0.358434	−	0.933555i	$-0.616689\pi$
$661$	−18.4306	−0.716867	−0.358434	+	0.933555i	$0.616689\pi$
$662$	0	0
$663$	−27.8863	−1.08301
$664$	0	0
$665$	11.9783	0.464498
$666$	0	0
$667$	19.9014	0.770586
$668$	0	0
$669$	−11.4742	−0.443619
$670$	0	0
$671$	10.4044	0.401656
$672$	0	0
$673$	−33.3166	−1.28426	−0.642130	−	0.766596i	$-0.721949\pi$
$673$	−33.3166	−1.28426	−0.642130	+	0.766596i	$0.721949\pi$
$674$	0	0
$675$	4.73167	0.182122
$676$	0	0
$677$	−10.9352	−0.420273	−0.210136	−	0.977672i	$-0.567391\pi$
$677$	−10.9352	−0.420273	−0.210136	+	0.977672i	$0.567391\pi$
$678$	0	0
$679$	−57.3206	−2.19976
$680$	0	0
$681$	61.9218	2.37285
$682$	0	0
$683$	−32.0803	−1.22752	−0.613759	−	0.789494i	$-0.710343\pi$
$683$	−32.0803	−1.22752	−0.613759	+	0.789494i	$0.710343\pi$
$684$	0	0
$685$	−17.9110	−0.684342
$686$	0	0
$687$	23.4047	0.892946
$688$	0	0
$689$	23.8226	0.907568
$690$	0	0
$691$	−31.7939	−1.20950	−0.604748	−	0.796417i	$-0.706726\pi$
$691$	−31.7939	−1.20950	−0.604748	+	0.796417i	$0.706726\pi$
$692$	0	0
$693$	−9.95338	−0.378098
$694$	0	0
$695$	56.1814	2.13108
$696$	0	0
$697$	−52.2901	−1.98063
$698$	0	0
$699$	−64.4861	−2.43909
$700$	0	0
$701$	27.4906	1.03830	0.519152	−	0.854682i	$-0.326248\pi$
$701$	27.4906	1.03830	0.519152	+	0.854682i	$0.326248\pi$
$702$	0	0
$703$	7.46905	0.281700
$704$	0	0
$705$	−14.7461	−0.555372
$706$	0	0
$707$	−27.3224	−1.02756
$708$	0	0
$709$	−49.4778	−1.85818	−0.929089	−	0.369857i	$-0.879407\pi$
$709$	−49.4778	−1.85818	−0.929089	+	0.369857i	$0.879407\pi$
$710$	0	0
$711$	−15.9419	−0.597867
$712$	0	0
$713$	−27.0417	−1.01272
$714$	0	0
$715$	−6.43524	−0.240664
$716$	0	0
$717$	−6.16500	−0.230236
$718$	0	0
$719$	−2.85931	−0.106634	−0.0533171	−	0.998578i	$-0.516979\pi$
$719$	−2.85931	−0.106634	−0.0533171	+	0.998578i	$0.516979\pi$
$720$	0	0
$721$	38.1842	1.42205
$722$	0	0
$723$	10.8496	0.403501
$724$	0	0
$725$	32.8148	1.21871
$726$	0	0
$727$	5.60475	0.207869	0.103934	−	0.994584i	$-0.466857\pi$
$727$	5.60475	0.207869	0.103934	+	0.994584i	$0.466857\pi$
$728$	0	0
$729$	−17.8046	−0.659430
$730$	0	0
$731$	50.0451	1.85098
$732$	0	0
$733$	−9.16290	−0.338440	−0.169220	−	0.985578i	$-0.554125\pi$
$733$	−9.16290	−0.338440	−0.169220	+	0.985578i	$0.554125\pi$
$734$	0	0
$735$	−61.2074	−2.25767
$736$	0	0
$737$	15.3402	0.565062
$738$	0	0
$739$	−18.4943	−0.680325	−0.340162	−	0.940367i	$-0.610482\pi$
$739$	−18.4943	−0.680325	−0.340162	+	0.940367i	$0.610482\pi$
$740$	0	0
$741$	−4.98361	−0.183077
$742$	0	0
$743$	27.2595	1.00005	0.500027	−	0.866010i	$-0.333323\pi$
$743$	27.2595	1.00005	0.500027	+	0.866010i	$0.333323\pi$
$744$	0	0
$745$	9.43536	0.345685
$746$	0	0
$747$	17.0898	0.625282
$748$	0	0
$749$	13.1893	0.481926
$750$	0	0
$751$	2.89811	0.105754	0.0528768	−	0.998601i	$-0.483161\pi$
$751$	2.89811	0.105754	0.0528768	+	0.998601i	$0.483161\pi$
$752$	0	0
$753$	28.5524	1.04051
$754$	0	0
$755$	20.9309	0.761752
$756$	0	0
$757$	43.0637	1.56518	0.782588	−	0.622539i	$-0.213899\pi$
$757$	43.0637	1.56518	0.782588	+	0.622539i	$0.213899\pi$
$758$	0	0
$759$	5.99403	0.217569
$760$	0	0
$761$	−25.2568	−0.915557	−0.457778	−	0.889066i	$-0.651355\pi$
$761$	−25.2568	−0.915557	−0.457778	+	0.889066i	$0.651355\pi$
$762$	0	0
$763$	27.3438	0.989911
$764$	0	0
$765$	42.8057	1.54764
$766$	0	0
$767$	−8.51322	−0.307394
$768$	0	0
$769$	26.4769	0.954782	0.477391	−	0.878691i	$-0.341583\pi$
$769$	26.4769	0.954782	0.477391	+	0.878691i	$0.341583\pi$
$770$	0	0
$771$	−20.8050	−0.749273
$772$	0	0
$773$	19.3848	0.697225	0.348612	−	0.937267i	$-0.386653\pi$
$773$	19.3848	0.697225	0.348612	+	0.937267i	$0.386653\pi$
$774$	0	0
$775$	−44.5882	−1.60166
$776$	0	0
$777$	−69.2849	−2.48558
$778$	0	0
$779$	−9.34486	−0.334814
$780$	0	0
$781$	−5.75463	−0.205917
$782$	0	0
$783$	8.77631	0.313640
$784$	0	0
$785$	49.9378	1.78236
$786$	0	0
$787$	−50.7457	−1.80889	−0.904444	−	0.426592i	$-0.859714\pi$
$787$	−50.7457	−1.80889	−0.904444	+	0.426592i	$0.859714\pi$
$788$	0	0
$789$	15.5171	0.552424
$790$	0	0
$791$	−68.4052	−2.43221
$792$	0	0
$793$	−22.0669	−0.783618
$794$	0	0
$795$	−80.0795	−2.84013
$796$	0	0
$797$	−29.5246	−1.04581	−0.522907	−	0.852390i	$-0.675153\pi$
$797$	−29.5246	−1.04581	−0.522907	+	0.852390i	$0.675153\pi$
$798$	0	0
$799$	11.5735	0.409443
$800$	0	0
$801$	−41.6000	−1.46986
$802$	0	0
$803$	−1.09907	−0.0387853
$804$	0	0
$805$	30.5559	1.07695
$806$	0	0
$807$	31.7260	1.11681
$808$	0	0
$809$	−19.2549	−0.676966	−0.338483	−	0.940973i	$-0.609914\pi$
$809$	−19.2549	−0.676966	−0.338483	+	0.940973i	$0.609914\pi$
$810$	0	0
$811$	−27.3572	−0.960640	−0.480320	−	0.877093i	$-0.659479\pi$
$811$	−27.3572	−0.960640	−0.480320	+	0.877093i	$0.659479\pi$
$812$	0	0
$813$	−51.5819	−1.80905
$814$	0	0
$815$	−49.9288	−1.74893
$816$	0	0
$817$	8.94365	0.312899
$818$	0	0
$819$	21.1104	0.737656
$820$	0	0
$821$	−38.5540	−1.34554	−0.672771	−	0.739851i	$-0.734896\pi$
$821$	−38.5540	−1.34554	−0.672771	+	0.739851i	$0.734896\pi$
$822$	0	0
$823$	32.3633	1.12811	0.564057	−	0.825736i	$-0.309240\pi$
$823$	32.3633	1.12811	0.564057	+	0.825736i	$0.309240\pi$
$824$	0	0
$825$	9.88337	0.344095
$826$	0	0
$827$	−27.0957	−0.942209	−0.471104	−	0.882077i	$-0.656144\pi$
$827$	−27.0957	−0.942209	−0.471104	+	0.882077i	$0.656144\pi$
$828$	0	0
$829$	48.3400	1.67892	0.839460	−	0.543422i	$-0.182872\pi$
$829$	48.3400	1.67892	0.839460	+	0.543422i	$0.182872\pi$
$830$	0	0
$831$	−4.96881	−0.172366
$832$	0	0
$833$	48.0388	1.66444
$834$	0	0
$835$	62.8677	2.17563
$836$	0	0
$837$	−11.9251	−0.412192
$838$	0	0
$839$	−42.4763	−1.46645	−0.733223	−	0.679989i	$-0.761985\pi$
$839$	−42.4763	−1.46645	−0.733223	+	0.679989i	$0.761985\pi$
$840$	0	0
$841$	31.8650	1.09879
$842$	0	0
$843$	16.5931	0.571495
$844$	0	0
$845$	−25.7955	−0.887392
$846$	0	0
$847$	3.94780	0.135648
$848$	0	0
$849$	−59.7719	−2.05137
$850$	0	0
$851$	19.0531	0.653131
$852$	0	0
$853$	22.4008	0.766990	0.383495	−	0.923543i	$-0.374720\pi$
$853$	22.4008	0.766990	0.383495	+	0.923543i	$0.374720\pi$
$854$	0	0
$855$	7.64989	0.261621
$856$	0	0
$857$	−9.03285	−0.308556	−0.154278	−	0.988027i	$-0.549305\pi$
$857$	−9.03285	−0.308556	−0.154278	+	0.988027i	$0.549305\pi$
$858$	0	0
$859$	4.43059	0.151170	0.0755849	−	0.997139i	$-0.475918\pi$
$859$	4.43059	0.151170	0.0755849	+	0.997139i	$0.475918\pi$
$860$	0	0
$861$	86.6855	2.95423
$862$	0	0
$863$	−2.58093	−0.0878559	−0.0439280	−	0.999035i	$-0.513987\pi$
$863$	−2.58093	−0.0878559	−0.0439280	+	0.999035i	$0.513987\pi$
$864$	0	0
$865$	−26.5092	−0.901339
$866$	0	0
$867$	−33.6263	−1.14201
$868$	0	0
$869$	6.32301	0.214493
$870$	0	0
$871$	−32.5353	−1.10242
$872$	0	0
$873$	−36.6076	−1.23898
$874$	0	0
$875$	−9.50879	−0.321456
$876$	0	0
$877$	−4.06424	−0.137240	−0.0686198	−	0.997643i	$-0.521860\pi$
$877$	−4.06424	−0.137240	−0.0686198	+	0.997643i	$0.521860\pi$
$878$	0	0
$879$	−55.5880	−1.87494
$880$	0	0
$881$	−5.45199	−0.183682	−0.0918411	−	0.995774i	$-0.529275\pi$
$881$	−5.45199	−0.183682	−0.0918411	+	0.995774i	$0.529275\pi$
$882$	0	0
$883$	33.4756	1.12654	0.563272	−	0.826272i	$-0.309542\pi$
$883$	33.4756	1.12654	0.563272	+	0.826272i	$0.309542\pi$
$884$	0	0
$885$	28.6172	0.961955
$886$	0	0
$887$	32.2836	1.08398	0.541989	−	0.840386i	$-0.317672\pi$
$887$	32.2836	1.08398	0.541989	+	0.840386i	$0.317672\pi$
$888$	0	0
$889$	23.6863	0.794411
$890$	0	0
$891$	10.2070	0.341949
$892$	0	0
$893$	2.06833	0.0692141
$894$	0	0
$895$	−14.5383	−0.485960
$896$	0	0
$897$	−12.7129	−0.424471
$898$	0	0
$899$	−82.7024	−2.75828
$900$	0	0
$901$	62.8506	2.09386
$902$	0	0
$903$	−82.9638	−2.76086
$904$	0	0
$905$	70.4688	2.34246
$906$	0	0
$907$	49.6085	1.64722	0.823611	−	0.567155i	$-0.191956\pi$
$907$	49.6085	1.64722	0.823611	+	0.567155i	$0.191956\pi$
$908$	0	0
$909$	−17.4494	−0.578759
$910$	0	0
$911$	20.6641	0.684632	0.342316	−	0.939585i	$-0.388789\pi$
$911$	20.6641	0.684632	0.342316	+	0.939585i	$0.388789\pi$
$912$	0	0
$913$	−6.77829	−0.224329
$914$	0	0
$915$	74.1778	2.45224
$916$	0	0
$917$	−25.2602	−0.834165
$918$	0	0
$919$	3.87795	0.127922	0.0639608	−	0.997952i	$-0.479627\pi$
$919$	3.87795	0.127922	0.0639608	+	0.997952i	$0.479627\pi$
$920$	0	0
$921$	22.8414	0.752650
$922$	0	0
$923$	12.2051	0.401737
$924$	0	0
$925$	31.4160	1.03295
$926$	0	0
$927$	24.3862	0.800948
$928$	0	0
$929$	27.5500	0.903885	0.451943	−	0.892047i	$-0.350731\pi$
$929$	27.5500	0.903885	0.451943	+	0.892047i	$0.350731\pi$
$930$	0	0
$931$	8.58511	0.281365
$932$	0	0
$933$	−12.9732	−0.424723
$934$	0	0
$935$	−16.9780	−0.555239
$936$	0	0
$937$	24.6035	0.803760	0.401880	−	0.915692i	$-0.368357\pi$
$937$	24.6035	0.803760	0.401880	+	0.915692i	$0.368357\pi$
$938$	0	0
$939$	47.2079	1.54057
$940$	0	0
$941$	41.5187	1.35347	0.676736	−	0.736226i	$-0.263394\pi$
$941$	41.5187	1.35347	0.676736	+	0.736226i	$0.263394\pi$
$942$	0	0
$943$	−23.8382	−0.776278
$944$	0	0
$945$	13.4748	0.438335
$946$	0	0
$947$	5.22088	0.169656	0.0848279	−	0.996396i	$-0.472966\pi$
$947$	5.22088	0.169656	0.0848279	+	0.996396i	$0.472966\pi$
$948$	0	0
$949$	2.33104	0.0756689
$950$	0	0
$951$	−24.1452	−0.782963
$952$	0	0
$953$	42.8532	1.38815	0.694076	−	0.719902i	$-0.255813\pi$
$953$	42.8532	1.38815	0.694076	+	0.719902i	$0.255813\pi$
$954$	0	0
$955$	9.05139	0.292896
$956$	0	0
$957$	18.3317	0.592579
$958$	0	0
$959$	−23.3042	−0.752532
$960$	0	0
$961$	81.3746	2.62499
$962$	0	0
$963$	8.42329	0.271437
$964$	0	0
$965$	27.6710	0.890760
$966$	0	0
$967$	−52.3312	−1.68286	−0.841429	−	0.540367i	$-0.818285\pi$
$967$	−52.3312	−1.68286	−0.841429	+	0.540367i	$0.818285\pi$
$968$	0	0
$969$	−13.1482	−0.422380
$970$	0	0
$971$	18.5866	0.596472	0.298236	−	0.954492i	$-0.403602\pi$
$971$	18.5866	0.596472	0.298236	+	0.954492i	$0.403602\pi$
$972$	0	0
$973$	73.0984	2.34343
$974$	0	0
$975$	−20.9619	−0.671317
$976$	0	0
$977$	−29.8788	−0.955906	−0.477953	−	0.878385i	$-0.658621\pi$
$977$	−29.8788	−0.955906	−0.477953	+	0.878385i	$0.658621\pi$
$978$	0	0
$979$	16.4997	0.527334
$980$	0	0
$981$	17.4630	0.557551
$982$	0	0
$983$	−40.7083	−1.29839	−0.649196	−	0.760621i	$-0.724895\pi$
$983$	−40.7083	−1.29839	−0.649196	+	0.760621i	$0.724895\pi$
$984$	0	0
$985$	8.80525	0.280559
$986$	0	0
$987$	−19.1864	−0.610710
$988$	0	0
$989$	22.8147	0.725466
$990$	0	0
$991$	36.3914	1.15601	0.578005	−	0.816033i	$-0.303832\pi$
$991$	36.3914	1.15601	0.578005	+	0.816033i	$0.303832\pi$
$992$	0	0
$993$	31.0548	0.985494
$994$	0	0
$995$	−5.24973	−0.166428
$996$	0	0
$997$	15.3132	0.484974	0.242487	−	0.970155i	$-0.422037\pi$
$997$	15.3132	0.484974	0.242487	+	0.970155i	$0.422037\pi$
$998$	0	0
$999$	8.40220	0.265834

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.bb.1.3		9
4.3	odd	2		1672.2.a.k.1.7	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1672.2.a.k.1.7	✓	9	4.3	odd	2
3344.2.a.bb.1.3		9	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q-2.34973 q^{3} +3.03417 q^{5} +3.94780 q^{7} +2.52125 q^{9} +O(q^{10})\)	\(q-2.34973 q^{3} +3.03417 q^{5} +3.94780 q^{7} +2.52125 q^{9} -1.00000 q^{11} +2.12092 q^{13} -7.12948 q^{15} +5.59559 q^{17} +1.00000 q^{19} -9.27627 q^{21} +2.55094 q^{23} +4.20616 q^{25} +1.12494 q^{27} +7.80160 q^{29} -10.6007 q^{31} +2.34973 q^{33} +11.9783 q^{35} +7.46905 q^{37} -4.98361 q^{39} -9.34486 q^{41} +8.94365 q^{43} +7.64989 q^{45} +2.06833 q^{47} +8.58511 q^{49} -13.1482 q^{51} +11.2322 q^{53} -3.03417 q^{55} -2.34973 q^{57} -4.01392 q^{59} -10.4044 q^{61} +9.95338 q^{63} +6.43524 q^{65} -15.3402 q^{67} -5.99403 q^{69} +5.75463 q^{71} +1.09907 q^{73} -9.88337 q^{75} -3.94780 q^{77} -6.32301 q^{79} -10.2070 q^{81} +6.77829 q^{83} +16.9780 q^{85} -18.3317 q^{87} -16.4997 q^{89} +8.37298 q^{91} +24.9088 q^{93} +3.03417 q^{95} -14.5196 q^{97} -2.52125 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9}+O(q^{10}) \) 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9	\( 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9} - 9 q^{11} + q^{13} - 6 q^{15} + 7 q^{17} + 9 q^{19} + 3 q^{21} - 13 q^{23} + 31 q^{25} + 5 q^{27} + 9 q^{29} + 4 q^{31} + q^{33} + 4 q^{35} + 24 q^{37} - 13 q^{39} - 6 q^{41} + 14 q^{43} + 26 q^{45} - 24 q^{47} + 20 q^{49} + 33 q^{51} + 19 q^{53} - 6 q^{55} - q^{57} + 19 q^{59} + 28 q^{61} - 16 q^{63} + 16 q^{65} - 5 q^{67} + 35 q^{69} - 16 q^{71} + 15 q^{73} - 3 q^{75} + 3 q^{77} - 2 q^{79} + 37 q^{81} - 8 q^{83} + 20 q^{85} - 23 q^{87} + 12 q^{89} + 29 q^{91} + 44 q^{93} + 6 q^{95} - 4 q^{97} - 18 q^{99}+O(q^{100}) \) 9 * q - q^3 + 6 * q^5 - 3 * q^7 + 18 * q^9 - 9 * q^11 + q^13 - 6 * q^15 + 7 * q^17 + 9 * q^19 + 3 * q^21 - 13 * q^23 + 31 * q^25 + 5 * q^27 + 9 * q^29 + 4 * q^31 + q^33 + 4 * q^35 + 24 * q^37 - 13 * q^39 - 6 * q^41 + 14 * q^43 + 26 * q^45 - 24 * q^47 + 20 * q^49 + 33 * q^51 + 19 * q^53 - 6 * q^55 - q^57 + 19 * q^59 + 28 * q^61 - 16 * q^63 + 16 * q^65 - 5 * q^67 + 35 * q^69 - 16 * q^71 + 15 * q^73 - 3 * q^75 + 3 * q^77 - 2 * q^79 + 37 * q^81 - 8 * q^83 + 20 * q^85 - 23 * q^87 + 12 * q^89 + 29 * q^91 + 44 * q^93 + 6 * q^95 - 4 * q^97 - 18 * q^99

Introduction

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