LMFDB - Embedded newform 2128.2.a.l.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2128,2,Mod(1,2128)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2128.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2128, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2128 = 2^{4} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2128.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,3,0,-6,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$16.9921655501$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 133)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	2128.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.302776 q^{3} -3.00000 q^{5} -1.00000 q^{7} -2.90833 q^{9} +0.697224 q^{11} -5.60555 q^{13} +0.908327 q^{15} -5.30278 q^{17} -1.00000 q^{19} +0.302776 q^{21} +3.00000 q^{23} +4.00000 q^{25} +1.78890 q^{27} +9.90833 q^{29} -1.30278 q^{31} -0.211103 q^{33} +3.00000 q^{35} +3.60555 q^{37} +1.69722 q^{39} +0.697224 q^{41} +10.0000 q^{43} +8.72498 q^{45} -6.21110 q^{47} +1.00000 q^{49} +1.60555 q^{51} -6.90833 q^{53} -2.09167 q^{55} +0.302776 q^{57} +6.21110 q^{59} -4.21110 q^{61} +2.90833 q^{63} +16.8167 q^{65} +1.90833 q^{67} -0.908327 q^{69} -12.2111 q^{71} +1.51388 q^{73} -1.21110 q^{75} -0.697224 q^{77} -11.2111 q^{79} +8.18335 q^{81} +12.9083 q^{83} +15.9083 q^{85} -3.00000 q^{87} -10.6056 q^{89} +5.60555 q^{91} +0.394449 q^{93} +3.00000 q^{95} +9.60555 q^{97} -2.02776 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} - 6 q^{5} - 2 q^{7} + 5 q^{9} + 5 q^{11} - 4 q^{13} - 9 q^{15} - 7 q^{17} - 2 q^{19} - 3 q^{21} + 6 q^{23} + 8 q^{25} + 18 q^{27} + 9 q^{29} + q^{31} + 14 q^{33} + 6 q^{35} + 7 q^{39} + 5 q^{41}+ \cdots + 32 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 - 6 * q^5 - 2 * q^7 + 5 * q^9 + 5 * q^11 - 4 * q^13 - 9 * q^15 - 7 * q^17 - 2 * q^19 - 3 * q^21 + 6 * q^23 + 8 * q^25 + 18 * q^27 + 9 * q^29 + q^31 + 14 * q^33 + 6 * q^35 + 7 * q^39 + 5 * q^41 + 20 * q^43 - 15 * q^45 + 2 * q^47 + 2 * q^49 - 4 * q^51 - 3 * q^53 - 15 * q^55 - 3 * q^57 - 2 * q^59 + 6 * q^61 - 5 * q^63 + 12 * q^65 - 7 * q^67 + 9 * q^69 - 10 * q^71 - 15 * q^73 + 12 * q^75 - 5 * q^77 - 8 * q^79 + 38 * q^81 + 15 * q^83 + 21 * q^85 - 6 * q^87 - 14 * q^89 + 4 * q^91 + 8 * q^93 + 6 * q^95 + 12 * q^97 + 32 * q^99

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
$3$	−0.302776	−0.174808	−0.0874038	−	0.996173i	$-0.527857\pi$
$3$	−0.302776	−0.174808	−0.0874038	+	0.996173i	$0.527857\pi$
$4$	0	0
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.90833	−0.969442
$10$	0	0
$11$	0.697224	0.210221	0.105111	−	0.994461i	$-0.466480\pi$
$11$	0.697224	0.210221	0.105111	+	0.994461i	$0.466480\pi$
$12$	0	0
$13$	−5.60555	−1.55470	−0.777350	−	0.629068i	$-0.783437\pi$
$13$	−5.60555	−1.55470	−0.777350	+	0.629068i	$0.783437\pi$
$14$	0	0
$15$	0.908327	0.234529
$16$	0	0
$17$	−5.30278	−1.28611	−0.643056	−	0.765819i	$-0.722334\pi$
$17$	−5.30278	−1.28611	−0.643056	+	0.765819i	$0.722334\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0.302776	0.0660711
$22$	0	0
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	1.78890	0.344273
$28$	0	0
$29$	9.90833	1.83993	0.919965	−	0.392000i	$-0.128217\pi$
$29$	9.90833	1.83993	0.919965	+	0.392000i	$0.128217\pi$
$30$	0	0
$31$	−1.30278	−0.233985	−0.116993	−	0.993133i	$-0.537325\pi$
$31$	−1.30278	−0.233985	−0.116993	+	0.993133i	$0.537325\pi$
$32$	0	0
$33$	−0.211103	−0.0367482
$34$	0	0
$35$	3.00000	0.507093
$36$	0	0
$37$	3.60555	0.592749	0.296374	−	0.955072i	$-0.404222\pi$
$37$	3.60555	0.592749	0.296374	+	0.955072i	$0.404222\pi$
$38$	0	0
$39$	1.69722	0.271773
$40$	0	0
$41$	0.697224	0.108888	0.0544441	−	0.998517i	$-0.482661\pi$
$41$	0.697224	0.108888	0.0544441	+	0.998517i	$0.482661\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	0	0
$45$	8.72498	1.30064
$46$	0	0
$47$	−6.21110	−0.905982	−0.452991	−	0.891515i	$-0.649643\pi$
$47$	−6.21110	−0.905982	−0.452991	+	0.891515i	$0.649643\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	1.60555	0.224822
$52$	0	0
$53$	−6.90833	−0.948932	−0.474466	−	0.880274i	$-0.657359\pi$
$53$	−6.90833	−0.948932	−0.474466	+	0.880274i	$0.657359\pi$
$54$	0	0
$55$	−2.09167	−0.282041
$56$	0	0
$57$	0.302776	0.0401036
$58$	0	0
$59$	6.21110	0.808617	0.404308	−	0.914623i	$-0.367512\pi$
$59$	6.21110	0.808617	0.404308	+	0.914623i	$0.367512\pi$
$60$	0	0
$61$	−4.21110	−0.539176	−0.269588	−	0.962976i	$-0.586888\pi$
$61$	−4.21110	−0.539176	−0.269588	+	0.962976i	$0.586888\pi$
$62$	0	0
$63$	2.90833	0.366415
$64$	0	0
$65$	16.8167	2.08585
$66$	0	0
$67$	1.90833	0.233139	0.116570	−	0.993183i	$-0.462810\pi$
$67$	1.90833	0.233139	0.116570	+	0.993183i	$0.462810\pi$
$68$	0	0
$69$	−0.908327	−0.109350
$70$	0	0
$71$	−12.2111	−1.44919	−0.724596	−	0.689174i	$-0.757973\pi$
$71$	−12.2111	−1.44919	−0.724596	+	0.689174i	$0.757973\pi$
$72$	0	0
$73$	1.51388	0.177186	0.0885930	−	0.996068i	$-0.471763\pi$
$73$	1.51388	0.177186	0.0885930	+	0.996068i	$0.471763\pi$
$74$	0	0
$75$	−1.21110	−0.139846
$76$	0	0
$77$	−0.697224	−0.0794561
$78$	0	0
$79$	−11.2111	−1.26135	−0.630674	−	0.776048i	$-0.717221\pi$
$79$	−11.2111	−1.26135	−0.630674	+	0.776048i	$0.717221\pi$
$80$	0	0
$81$	8.18335	0.909261
$82$	0	0
$83$	12.9083	1.41687	0.708436	−	0.705775i	$-0.249401\pi$
$83$	12.9083	1.41687	0.708436	+	0.705775i	$0.249401\pi$
$84$	0	0
$85$	15.9083	1.72550
$86$	0	0
$87$	−3.00000	−0.321634
$88$	0	0
$89$	−10.6056	−1.12419	−0.562093	−	0.827074i	$-0.690004\pi$
$89$	−10.6056	−1.12419	−0.562093	+	0.827074i	$0.690004\pi$
$90$	0	0
$91$	5.60555	0.587621
$92$	0	0
$93$	0.394449	0.0409024
$94$	0	0
$95$	3.00000	0.307794
$96$	0	0
$97$	9.60555	0.975296	0.487648	−	0.873040i	$-0.337855\pi$
$97$	9.60555	0.975296	0.487648	+	0.873040i	$0.337855\pi$
$98$	0	0
$99$	−2.02776	−0.203797
$100$	0	0
$101$	−4.39445	−0.437264	−0.218632	−	0.975807i	$-0.570159\pi$
$101$	−4.39445	−0.437264	−0.218632	+	0.975807i	$0.570159\pi$
$102$	0	0
$103$	16.2111	1.59733	0.798664	−	0.601778i	$-0.205541\pi$
$103$	16.2111	1.59733	0.798664	+	0.601778i	$0.205541\pi$
$104$	0	0
$105$	−0.908327	−0.0886436
$106$	0	0
$107$	5.78890	0.559634	0.279817	−	0.960053i	$-0.409726\pi$
$107$	5.78890	0.559634	0.279817	+	0.960053i	$0.409726\pi$
$108$	0	0
$109$	−10.2111	−0.978046	−0.489023	−	0.872271i	$-0.662647\pi$
$109$	−10.2111	−0.978046	−0.489023	+	0.872271i	$0.662647\pi$
$110$	0	0
$111$	−1.09167	−0.103617
$112$	0	0
$113$	9.69722	0.912238	0.456119	−	0.889919i	$-0.349239\pi$
$113$	9.69722	0.912238	0.456119	+	0.889919i	$0.349239\pi$
$114$	0	0
$115$	−9.00000	−0.839254
$116$	0	0
$117$	16.3028	1.50719
$118$	0	0
$119$	5.30278	0.486105
$120$	0	0
$121$	−10.5139	−0.955807
$122$	0	0
$123$	−0.211103	−0.0190345
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	11.8167	1.04856	0.524279	−	0.851546i	$-0.324335\pi$
$127$	11.8167	1.04856	0.524279	+	0.851546i	$0.324335\pi$
$128$	0	0
$129$	−3.02776	−0.266579
$130$	0	0
$131$	20.5139	1.79231	0.896153	−	0.443745i	$-0.146351\pi$
$131$	20.5139	1.79231	0.896153	+	0.443745i	$0.146351\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	−5.36669	−0.461891
$136$	0	0
$137$	21.6333	1.84826	0.924129	−	0.382080i	$-0.124792\pi$
$137$	21.6333	1.84826	0.924129	+	0.382080i	$0.124792\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	1.88057	0.158373
$142$	0	0
$143$	−3.90833	−0.326831
$144$	0	0
$145$	−29.7250	−2.46853
$146$	0	0
$147$	−0.302776	−0.0249725
$148$	0	0
$149$	−6.21110	−0.508833	−0.254417	−	0.967095i	$-0.581883\pi$
$149$	−6.21110	−0.508833	−0.254417	+	0.967095i	$0.581883\pi$
$150$	0	0
$151$	−5.90833	−0.480813	−0.240406	−	0.970672i	$-0.577281\pi$
$151$	−5.90833	−0.480813	−0.240406	+	0.970672i	$0.577281\pi$
$152$	0	0
$153$	15.4222	1.24681
$154$	0	0
$155$	3.90833	0.313924
$156$	0	0
$157$	4.51388	0.360247	0.180123	−	0.983644i	$-0.442350\pi$
$157$	4.51388	0.360247	0.180123	+	0.983644i	$0.442350\pi$
$158$	0	0
$159$	2.09167	0.165880
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	−5.69722	−0.446241	−0.223121	−	0.974791i	$-0.571624\pi$
$163$	−5.69722	−0.446241	−0.223121	+	0.974791i	$0.571624\pi$
$164$	0	0
$165$	0.633308	0.0493029
$166$	0	0
$167$	−4.39445	−0.340053	−0.170026	−	0.985440i	$-0.554385\pi$
$167$	−4.39445	−0.340053	−0.170026	+	0.985440i	$0.554385\pi$
$168$	0	0
$169$	18.4222	1.41709
$170$	0	0
$171$	2.90833	0.222405
$172$	0	0
$173$	7.81665	0.594289	0.297145	−	0.954832i	$-0.403966\pi$
$173$	7.81665	0.594289	0.297145	+	0.954832i	$0.403966\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	−1.88057	−0.141352
$178$	0	0
$179$	11.7250	0.876366	0.438183	−	0.898886i	$-0.355622\pi$
$179$	11.7250	0.876366	0.438183	+	0.898886i	$0.355622\pi$
$180$	0	0
$181$	−10.6972	−0.795118	−0.397559	−	0.917577i	$-0.630143\pi$
$181$	−10.6972	−0.795118	−0.397559	+	0.917577i	$0.630143\pi$
$182$	0	0
$183$	1.27502	0.0942521
$184$	0	0
$185$	−10.8167	−0.795256
$186$	0	0
$187$	−3.69722	−0.270368
$188$	0	0
$189$	−1.78890	−0.130123
$190$	0	0
$191$	−25.3305	−1.83285	−0.916426	−	0.400203i	$-0.868940\pi$
$191$	−25.3305	−1.83285	−0.916426	+	0.400203i	$0.868940\pi$
$192$	0	0
$193$	−23.1194	−1.66417	−0.832086	−	0.554646i	$-0.812854\pi$
$193$	−23.1194	−1.66417	−0.832086	+	0.554646i	$0.812854\pi$
$194$	0	0
$195$	−5.09167	−0.364622
$196$	0	0
$197$	−6.90833	−0.492198	−0.246099	−	0.969245i	$-0.579149\pi$
$197$	−6.90833	−0.492198	−0.246099	+	0.969245i	$0.579149\pi$
$198$	0	0
$199$	13.4222	0.951475	0.475737	−	0.879587i	$-0.342181\pi$
$199$	13.4222	0.951475	0.475737	+	0.879587i	$0.342181\pi$
$200$	0	0
$201$	−0.577795	−0.0407545
$202$	0	0
$203$	−9.90833	−0.695428
$204$	0	0
$205$	−2.09167	−0.146089
$206$	0	0
$207$	−8.72498	−0.606428
$208$	0	0
$209$	−0.697224	−0.0482280
$210$	0	0
$211$	7.69722	0.529899	0.264949	−	0.964262i	$-0.414645\pi$
$211$	7.69722	0.529899	0.264949	+	0.964262i	$0.414645\pi$
$212$	0	0
$213$	3.69722	0.253330
$214$	0	0
$215$	−30.0000	−2.04598
$216$	0	0
$217$	1.30278	0.0884382
$218$	0	0
$219$	−0.458365	−0.0309735
$220$	0	0
$221$	29.7250	1.99952
$222$	0	0
$223$	−12.8167	−0.858267	−0.429133	−	0.903241i	$-0.641181\pi$
$223$	−12.8167	−0.858267	−0.429133	+	0.903241i	$0.641181\pi$
$224$	0	0
$225$	−11.6333	−0.775554
$226$	0	0
$227$	25.1194	1.66724	0.833618	−	0.552342i	$-0.186266\pi$
$227$	25.1194	1.66724	0.833618	+	0.552342i	$0.186266\pi$
$228$	0	0
$229$	−0.788897	−0.0521318	−0.0260659	−	0.999660i	$-0.508298\pi$
$229$	−0.788897	−0.0521318	−0.0260659	+	0.999660i	$0.508298\pi$
$230$	0	0
$231$	0.211103	0.0138895
$232$	0	0
$233$	−2.09167	−0.137030	−0.0685150	−	0.997650i	$-0.521826\pi$
$233$	−2.09167	−0.137030	−0.0685150	+	0.997650i	$0.521826\pi$
$234$	0	0
$235$	18.6333	1.21550
$236$	0	0
$237$	3.39445	0.220493
$238$	0	0
$239$	27.4222	1.77379	0.886897	−	0.461966i	$-0.152856\pi$
$239$	27.4222	1.77379	0.886897	+	0.461966i	$0.152856\pi$
$240$	0	0
$241$	−20.6056	−1.32732	−0.663660	−	0.748034i	$-0.730998\pi$
$241$	−20.6056	−1.32732	−0.663660	+	0.748034i	$0.730998\pi$
$242$	0	0
$243$	−7.84441	−0.503219
$244$	0	0
$245$	−3.00000	−0.191663
$246$	0	0
$247$	5.60555	0.356673
$248$	0	0
$249$	−3.90833	−0.247680
$250$	0	0
$251$	15.9083	1.00412	0.502062	−	0.864831i	$-0.332575\pi$
$251$	15.9083	1.00412	0.502062	+	0.864831i	$0.332575\pi$
$252$	0	0
$253$	2.09167	0.131502
$254$	0	0
$255$	−4.81665	−0.301631
$256$	0	0
$257$	−0.908327	−0.0566599	−0.0283299	−	0.999599i	$-0.509019\pi$
$257$	−0.908327	−0.0566599	−0.0283299	+	0.999599i	$0.509019\pi$
$258$	0	0
$259$	−3.60555	−0.224038
$260$	0	0
$261$	−28.8167	−1.78371
$262$	0	0
$263$	9.69722	0.597956	0.298978	−	0.954260i	$-0.403354\pi$
$263$	9.69722	0.597956	0.298978	+	0.954260i	$0.403354\pi$
$264$	0	0
$265$	20.7250	1.27313
$266$	0	0
$267$	3.21110	0.196516
$268$	0	0
$269$	20.7250	1.26362	0.631812	−	0.775122i	$-0.282311\pi$
$269$	20.7250	1.26362	0.631812	+	0.775122i	$0.282311\pi$
$270$	0	0
$271$	−17.9083	−1.08785	−0.543927	−	0.839133i	$-0.683063\pi$
$271$	−17.9083	−1.08785	−0.543927	+	0.839133i	$0.683063\pi$
$272$	0	0
$273$	−1.69722	−0.102721
$274$	0	0
$275$	2.78890	0.168177
$276$	0	0
$277$	0.605551	0.0363840	0.0181920	−	0.999835i	$-0.494209\pi$
$277$	0.605551	0.0363840	0.0181920	+	0.999835i	$0.494209\pi$
$278$	0	0
$279$	3.78890	0.226835
$280$	0	0
$281$	3.00000	0.178965	0.0894825	−	0.995988i	$-0.471479\pi$
$281$	3.00000	0.178965	0.0894825	+	0.995988i	$0.471479\pi$
$282$	0	0
$283$	−27.3305	−1.62463	−0.812316	−	0.583218i	$-0.801793\pi$
$283$	−27.3305	−1.62463	−0.812316	+	0.583218i	$0.801793\pi$
$284$	0	0
$285$	−0.908327	−0.0538046
$286$	0	0
$287$	−0.697224	−0.0411559
$288$	0	0
$289$	11.1194	0.654084
$290$	0	0
$291$	−2.90833	−0.170489
$292$	0	0
$293$	−12.6333	−0.738046	−0.369023	−	0.929420i	$-0.620308\pi$
$293$	−12.6333	−0.738046	−0.369023	+	0.929420i	$0.620308\pi$
$294$	0	0
$295$	−18.6333	−1.08487
$296$	0	0
$297$	1.24726	0.0723735
$298$	0	0
$299$	−16.8167	−0.972532
$300$	0	0
$301$	−10.0000	−0.576390
$302$	0	0
$303$	1.33053	0.0764371
$304$	0	0
$305$	12.6333	0.723381
$306$	0	0
$307$	−14.6972	−0.838815	−0.419407	−	0.907798i	$-0.637762\pi$
$307$	−14.6972	−0.838815	−0.419407	+	0.907798i	$0.637762\pi$
$308$	0	0
$309$	−4.90833	−0.279225
$310$	0	0
$311$	2.51388	0.142549	0.0712745	−	0.997457i	$-0.477293\pi$
$311$	2.51388	0.142549	0.0712745	+	0.997457i	$0.477293\pi$
$312$	0	0
$313$	−27.0278	−1.52770	−0.763850	−	0.645394i	$-0.776693\pi$
$313$	−27.0278	−1.52770	−0.763850	+	0.645394i	$0.776693\pi$
$314$	0	0
$315$	−8.72498	−0.491597
$316$	0	0
$317$	8.78890	0.493634	0.246817	−	0.969062i	$-0.420615\pi$
$317$	8.78890	0.493634	0.246817	+	0.969062i	$0.420615\pi$
$318$	0	0
$319$	6.90833	0.386792
$320$	0	0
$321$	−1.75274	−0.0978282
$322$	0	0
$323$	5.30278	0.295054
$324$	0	0
$325$	−22.4222	−1.24376
$326$	0	0
$327$	3.09167	0.170970
$328$	0	0
$329$	6.21110	0.342429
$330$	0	0
$331$	−11.6972	−0.642938	−0.321469	−	0.946920i	$-0.604177\pi$
$331$	−11.6972	−0.642938	−0.321469	+	0.946920i	$0.604177\pi$
$332$	0	0
$333$	−10.4861	−0.574636
$334$	0	0
$335$	−5.72498	−0.312789
$336$	0	0
$337$	7.72498	0.420807	0.210403	−	0.977615i	$-0.432522\pi$
$337$	7.72498	0.420807	0.210403	+	0.977615i	$0.432522\pi$
$338$	0	0
$339$	−2.93608	−0.159466
$340$	0	0
$341$	−0.908327	−0.0491887
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	2.72498	0.146708
$346$	0	0
$347$	28.5416	1.53220	0.766098	−	0.642724i	$-0.222196\pi$
$347$	28.5416	1.53220	0.766098	+	0.642724i	$0.222196\pi$
$348$	0	0
$349$	7.51388	0.402209	0.201104	−	0.979570i	$-0.435547\pi$
$349$	7.51388	0.402209	0.201104	+	0.979570i	$0.435547\pi$
$350$	0	0
$351$	−10.0278	−0.535242
$352$	0	0
$353$	9.90833	0.527367	0.263684	−	0.964609i	$-0.415063\pi$
$353$	9.90833	0.527367	0.263684	+	0.964609i	$0.415063\pi$
$354$	0	0
$355$	36.6333	1.94429
$356$	0	0
$357$	−1.60555	−0.0849748
$358$	0	0
$359$	25.5416	1.34804	0.674018	−	0.738715i	$-0.264567\pi$
$359$	25.5416	1.34804	0.674018	+	0.738715i	$0.264567\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	3.18335	0.167082
$364$	0	0
$365$	−4.54163	−0.237720
$366$	0	0
$367$	0.0277564	0.00144887	0.000724436	−	1.00000i	$-0.499769\pi$
$367$	0.0277564	0.00144887	0.000724436		1.00000i	$0.499769\pi$
$368$	0	0
$369$	−2.02776	−0.105561
$370$	0	0
$371$	6.90833	0.358662
$372$	0	0
$373$	24.1194	1.24886	0.624428	−	0.781082i	$-0.285332\pi$
$373$	24.1194	1.24886	0.624428	+	0.781082i	$0.285332\pi$
$374$	0	0
$375$	−0.908327	−0.0469058
$376$	0	0
$377$	−55.5416	−2.86054
$378$	0	0
$379$	−6.81665	−0.350148	−0.175074	−	0.984555i	$-0.556016\pi$
$379$	−6.81665	−0.350148	−0.175074	+	0.984555i	$0.556016\pi$
$380$	0	0
$381$	−3.57779	−0.183296
$382$	0	0
$383$	−6.63331	−0.338946	−0.169473	−	0.985535i	$-0.554207\pi$
$383$	−6.63331	−0.338946	−0.169473	+	0.985535i	$0.554207\pi$
$384$	0	0
$385$	2.09167	0.106602
$386$	0	0
$387$	−29.0833	−1.47839
$388$	0	0
$389$	−24.1472	−1.22431	−0.612155	−	0.790737i	$-0.709697\pi$
$389$	−24.1472	−1.22431	−0.612155	+	0.790737i	$0.709697\pi$
$390$	0	0
$391$	−15.9083	−0.804519
$392$	0	0
$393$	−6.21110	−0.313309
$394$	0	0
$395$	33.6333	1.69228
$396$	0	0
$397$	37.0278	1.85837	0.929185	−	0.369615i	$-0.120510\pi$
$397$	37.0278	1.85837	0.929185	+	0.369615i	$0.120510\pi$
$398$	0	0
$399$	−0.302776	−0.0151577
$400$	0	0
$401$	−3.48612	−0.174089	−0.0870443	−	0.996204i	$-0.527742\pi$
$401$	−3.48612	−0.174089	−0.0870443	+	0.996204i	$0.527742\pi$
$402$	0	0
$403$	7.30278	0.363777
$404$	0	0
$405$	−24.5500	−1.21990
$406$	0	0
$407$	2.51388	0.124608
$408$	0	0
$409$	20.9083	1.03385	0.516925	−	0.856031i	$-0.327077\pi$
$409$	20.9083	1.03385	0.516925	+	0.856031i	$0.327077\pi$
$410$	0	0
$411$	−6.55004	−0.323090
$412$	0	0
$413$	−6.21110	−0.305628
$414$	0	0
$415$	−38.7250	−1.90093
$416$	0	0
$417$	1.51388	0.0741349
$418$	0	0
$419$	−15.0000	−0.732798	−0.366399	−	0.930458i	$-0.619409\pi$
$419$	−15.0000	−0.732798	−0.366399	+	0.930458i	$0.619409\pi$
$420$	0	0
$421$	−29.6056	−1.44289	−0.721443	−	0.692474i	$-0.756521\pi$
$421$	−29.6056	−1.44289	−0.721443	+	0.692474i	$0.756521\pi$
$422$	0	0
$423$	18.0639	0.878298
$424$	0	0
$425$	−21.2111	−1.02889
$426$	0	0
$427$	4.21110	0.203790
$428$	0	0
$429$	1.18335	0.0571325
$430$	0	0
$431$	3.21110	0.154673	0.0773367	−	0.997005i	$-0.475358\pi$
$431$	3.21110	0.154673	0.0773367	+	0.997005i	$0.475358\pi$
$432$	0	0
$433$	−0.577795	−0.0277671	−0.0138835	−	0.999904i	$-0.504419\pi$
$433$	−0.577795	−0.0277671	−0.0138835	+	0.999904i	$0.504419\pi$
$434$	0	0
$435$	9.00000	0.431517
$436$	0	0
$437$	−3.00000	−0.143509
$438$	0	0
$439$	−22.7889	−1.08765	−0.543827	−	0.839197i	$-0.683025\pi$
$439$	−22.7889	−1.08765	−0.543827	+	0.839197i	$0.683025\pi$
$440$	0	0
$441$	−2.90833	−0.138492
$442$	0	0
$443$	16.1194	0.765857	0.382929	−	0.923778i	$-0.374916\pi$
$443$	16.1194	0.765857	0.382929	+	0.923778i	$0.374916\pi$
$444$	0	0
$445$	31.8167	1.50825
$446$	0	0
$447$	1.88057	0.0889479
$448$	0	0
$449$	−0.486122	−0.0229415	−0.0114708	−	0.999934i	$-0.503651\pi$
$449$	−0.486122	−0.0229415	−0.0114708	+	0.999934i	$0.503651\pi$
$450$	0	0
$451$	0.486122	0.0228906
$452$	0	0
$453$	1.78890	0.0840497
$454$	0	0
$455$	−16.8167	−0.788377
$456$	0	0
$457$	22.3028	1.04328	0.521640	−	0.853166i	$-0.325320\pi$
$457$	22.3028	1.04328	0.521640	+	0.853166i	$0.325320\pi$
$458$	0	0
$459$	−9.48612	−0.442774
$460$	0	0
$461$	0.908327	0.0423050	0.0211525	−	0.999776i	$-0.493266\pi$
$461$	0.908327	0.0423050	0.0211525	+	0.999776i	$0.493266\pi$
$462$	0	0
$463$	36.4500	1.69397	0.846987	−	0.531614i	$-0.178414\pi$
$463$	36.4500	1.69397	0.846987	+	0.531614i	$0.178414\pi$
$464$	0	0
$465$	−1.18335	−0.0548764
$466$	0	0
$467$	−16.5416	−0.765456	−0.382728	−	0.923861i	$-0.625015\pi$
$467$	−16.5416	−0.765456	−0.382728	+	0.923861i	$0.625015\pi$
$468$	0	0
$469$	−1.90833	−0.0881183
$470$	0	0
$471$	−1.36669	−0.0629739
$472$	0	0
$473$	6.97224	0.320584
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	20.0917	0.919935
$478$	0	0
$479$	0.697224	0.0318570	0.0159285	−	0.999873i	$-0.494930\pi$
$479$	0.697224	0.0318570	0.0159285	+	0.999873i	$0.494930\pi$
$480$	0	0
$481$	−20.2111	−0.921547
$482$	0	0
$483$	0.908327	0.0413303
$484$	0	0
$485$	−28.8167	−1.30850
$486$	0	0
$487$	−4.57779	−0.207440	−0.103720	−	0.994607i	$-0.533075\pi$
$487$	−4.57779	−0.207440	−0.103720	+	0.994607i	$0.533075\pi$
$488$	0	0
$489$	1.72498	0.0780063
$490$	0	0
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	−52.5416	−2.36636
$494$	0	0
$495$	6.08327	0.273423
$496$	0	0
$497$	12.2111	0.547743
$498$	0	0
$499$	−19.9361	−0.892462	−0.446231	−	0.894918i	$-0.647234\pi$
$499$	−19.9361	−0.892462	−0.446231	+	0.894918i	$0.647234\pi$
$500$	0	0
$501$	1.33053	0.0594438
$502$	0	0
$503$	−6.42221	−0.286352	−0.143176	−	0.989697i	$-0.545731\pi$
$503$	−6.42221	−0.286352	−0.143176	+	0.989697i	$0.545731\pi$
$504$	0	0
$505$	13.1833	0.586651
$506$	0	0
$507$	−5.57779	−0.247719
$508$	0	0
$509$	33.6333	1.49077	0.745385	−	0.666634i	$-0.232266\pi$
$509$	33.6333	1.49077	0.745385	+	0.666634i	$0.232266\pi$
$510$	0	0
$511$	−1.51388	−0.0669700
$512$	0	0
$513$	−1.78890	−0.0789818
$514$	0	0
$515$	−48.6333	−2.14304
$516$	0	0
$517$	−4.33053	−0.190457
$518$	0	0
$519$	−2.36669	−0.103886
$520$	0	0
$521$	36.6333	1.60493	0.802467	−	0.596696i	$-0.203520\pi$
$521$	36.6333	1.60493	0.802467	+	0.596696i	$0.203520\pi$
$522$	0	0
$523$	−37.6611	−1.64680	−0.823402	−	0.567459i	$-0.807927\pi$
$523$	−37.6611	−1.64680	−0.823402	+	0.567459i	$0.807927\pi$
$524$	0	0
$525$	1.21110	0.0528568
$526$	0	0
$527$	6.90833	0.300931
$528$	0	0
$529$	−14.0000	−0.608696
$530$	0	0
$531$	−18.0639	−0.783907
$532$	0	0
$533$	−3.90833	−0.169288
$534$	0	0
$535$	−17.3667	−0.750828
$536$	0	0
$537$	−3.55004	−0.153195
$538$	0	0
$539$	0.697224	0.0300316
$540$	0	0
$541$	16.7889	0.721811	0.360906	−	0.932602i	$-0.382468\pi$
$541$	16.7889	0.721811	0.360906	+	0.932602i	$0.382468\pi$
$542$	0	0
$543$	3.23886	0.138993
$544$	0	0
$545$	30.6333	1.31219
$546$	0	0
$547$	10.4861	0.448354	0.224177	−	0.974548i	$-0.428031\pi$
$547$	10.4861	0.448354	0.224177	+	0.974548i	$0.428031\pi$
$548$	0	0
$549$	12.2473	0.522700
$550$	0	0
$551$	−9.90833	−0.422109
$552$	0	0
$553$	11.2111	0.476745
$554$	0	0
$555$	3.27502	0.139017
$556$	0	0
$557$	−1.11943	−0.0474317	−0.0237159	−	0.999719i	$-0.507550\pi$
$557$	−1.11943	−0.0474317	−0.0237159	+	0.999719i	$0.507550\pi$
$558$	0	0
$559$	−56.0555	−2.37090
$560$	0	0
$561$	1.11943	0.0472623
$562$	0	0
$563$	24.2111	1.02038	0.510188	−	0.860063i	$-0.329576\pi$
$563$	24.2111	1.02038	0.510188	+	0.860063i	$0.329576\pi$
$564$	0	0
$565$	−29.0917	−1.22390
$566$	0	0
$567$	−8.18335	−0.343668
$568$	0	0
$569$	7.18335	0.301142	0.150571	−	0.988599i	$-0.451889\pi$
$569$	7.18335	0.301142	0.150571	+	0.988599i	$0.451889\pi$
$570$	0	0
$571$	2.39445	0.100205	0.0501023	−	0.998744i	$-0.484045\pi$
$571$	2.39445	0.100205	0.0501023	+	0.998744i	$0.484045\pi$
$572$	0	0
$573$	7.66947	0.320397
$574$	0	0
$575$	12.0000	0.500435
$576$	0	0
$577$	19.5139	0.812373	0.406187	−	0.913790i	$-0.366858\pi$
$577$	19.5139	0.812373	0.406187	+	0.913790i	$0.366858\pi$
$578$	0	0
$579$	7.00000	0.290910
$580$	0	0
$581$	−12.9083	−0.535528
$582$	0	0
$583$	−4.81665	−0.199485
$584$	0	0
$585$	−48.9083	−2.02211
$586$	0	0
$587$	−0.422205	−0.0174263	−0.00871313	−	0.999962i	$-0.502774\pi$
$587$	−0.422205	−0.0174263	−0.00871313	+	0.999962i	$0.502774\pi$
$588$	0	0
$589$	1.30278	0.0536799
$590$	0	0
$591$	2.09167	0.0860399
$592$	0	0
$593$	−4.81665	−0.197796	−0.0988981	−	0.995098i	$-0.531532\pi$
$593$	−4.81665	−0.197796	−0.0988981	+	0.995098i	$0.531532\pi$
$594$	0	0
$595$	−15.9083	−0.652178
$596$	0	0
$597$	−4.06392	−0.166325
$598$	0	0
$599$	0.908327	0.0371132	0.0185566	−	0.999828i	$-0.494093\pi$
$599$	0.908327	0.0371132	0.0185566	+	0.999828i	$0.494093\pi$
$600$	0	0
$601$	−27.5139	−1.12231	−0.561157	−	0.827709i	$-0.689644\pi$
$601$	−27.5139	−1.12231	−0.561157	+	0.827709i	$0.689644\pi$
$602$	0	0
$603$	−5.55004	−0.226015
$604$	0	0
$605$	31.5416	1.28235
$606$	0	0
$607$	13.0000	0.527654	0.263827	−	0.964570i	$-0.415015\pi$
$607$	13.0000	0.527654	0.263827	+	0.964570i	$0.415015\pi$
$608$	0	0
$609$	3.00000	0.121566
$610$	0	0
$611$	34.8167	1.40853
$612$	0	0
$613$	−0.724981	−0.0292817	−0.0146408	−	0.999893i	$-0.504660\pi$
$613$	−0.724981	−0.0292817	−0.0146408	+	0.999893i	$0.504660\pi$
$614$	0	0
$615$	0.633308	0.0255374
$616$	0	0
$617$	−10.8806	−0.438035	−0.219018	−	0.975721i	$-0.570285\pi$
$617$	−10.8806	−0.438035	−0.219018	+	0.975721i	$0.570285\pi$
$618$	0	0
$619$	41.3305	1.66121	0.830607	−	0.556859i	$-0.187994\pi$
$619$	41.3305	1.66121	0.830607	+	0.556859i	$0.187994\pi$
$620$	0	0
$621$	5.36669	0.215358
$622$	0	0
$623$	10.6056	0.424902
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0.211103	0.00843062
$628$	0	0
$629$	−19.1194	−0.762342
$630$	0	0
$631$	−13.2389	−0.527031	−0.263515	−	0.964655i	$-0.584882\pi$
$631$	−13.2389	−0.527031	−0.263515	+	0.964655i	$0.584882\pi$
$632$	0	0
$633$	−2.33053	−0.0926303
$634$	0	0
$635$	−35.4500	−1.40679
$636$	0	0
$637$	−5.60555	−0.222100
$638$	0	0
$639$	35.5139	1.40491
$640$	0	0
$641$	9.90833	0.391355	0.195678	−	0.980668i	$-0.437309\pi$
$641$	9.90833	0.391355	0.195678	+	0.980668i	$0.437309\pi$
$642$	0	0
$643$	−30.3944	−1.19864	−0.599320	−	0.800510i	$-0.704562\pi$
$643$	−30.3944	−1.19864	−0.599320	+	0.800510i	$0.704562\pi$
$644$	0	0
$645$	9.08327	0.357653
$646$	0	0
$647$	−15.4222	−0.606309	−0.303155	−	0.952941i	$-0.598040\pi$
$647$	−15.4222	−0.606309	−0.303155	+	0.952941i	$0.598040\pi$
$648$	0	0
$649$	4.33053	0.169988
$650$	0	0
$651$	−0.394449	−0.0154597
$652$	0	0
$653$	−16.8167	−0.658087	−0.329043	−	0.944315i	$-0.606726\pi$
$653$	−16.8167	−0.658087	−0.329043	+	0.944315i	$0.606726\pi$
$654$	0	0
$655$	−61.5416	−2.40463
$656$	0	0
$657$	−4.40285	−0.171772
$658$	0	0
$659$	−16.3305	−0.636147	−0.318074	−	0.948066i	$-0.603036\pi$
$659$	−16.3305	−0.636147	−0.318074	+	0.948066i	$0.603036\pi$
$660$	0	0
$661$	−0.788897	−0.0306846	−0.0153423	−	0.999882i	$-0.504884\pi$
$661$	−0.788897	−0.0306846	−0.0153423	+	0.999882i	$0.504884\pi$
$662$	0	0
$663$	−9.00000	−0.349531
$664$	0	0
$665$	−3.00000	−0.116335
$666$	0	0
$667$	29.7250	1.15096
$668$	0	0
$669$	3.88057	0.150032
$670$	0	0
$671$	−2.93608	−0.113346
$672$	0	0
$673$	14.9083	0.574674	0.287337	−	0.957830i	$-0.407230\pi$
$673$	14.9083	0.574674	0.287337	+	0.957830i	$0.407230\pi$
$674$	0	0
$675$	7.15559	0.275419
$676$	0	0
$677$	17.9361	0.689340	0.344670	−	0.938724i	$-0.387991\pi$
$677$	17.9361	0.689340	0.344670	+	0.938724i	$0.387991\pi$
$678$	0	0
$679$	−9.60555	−0.368627
$680$	0	0
$681$	−7.60555	−0.291445
$682$	0	0
$683$	−12.2111	−0.467245	−0.233622	−	0.972327i	$-0.575058\pi$
$683$	−12.2111	−0.467245	−0.233622	+	0.972327i	$0.575058\pi$
$684$	0	0
$685$	−64.8999	−2.47970
$686$	0	0
$687$	0.238859	0.00911304
$688$	0	0
$689$	38.7250	1.47530
$690$	0	0
$691$	41.8167	1.59078	0.795390	−	0.606098i	$-0.207266\pi$
$691$	41.8167	1.59078	0.795390	+	0.606098i	$0.207266\pi$
$692$	0	0
$693$	2.02776	0.0770281
$694$	0	0
$695$	15.0000	0.568982
$696$	0	0
$697$	−3.69722	−0.140042
$698$	0	0
$699$	0.633308	0.0239539
$700$	0	0
$701$	21.2111	0.801132	0.400566	−	0.916268i	$-0.368813\pi$
$701$	21.2111	0.801132	0.400566	+	0.916268i	$0.368813\pi$
$702$	0	0
$703$	−3.60555	−0.135986
$704$	0	0
$705$	−5.64171	−0.212479
$706$	0	0
$707$	4.39445	0.165270
$708$	0	0
$709$	−11.6056	−0.435856	−0.217928	−	0.975965i	$-0.569930\pi$
$709$	−11.6056	−0.435856	−0.217928	+	0.975965i	$0.569930\pi$
$710$	0	0
$711$	32.6056	1.22280
$712$	0	0
$713$	−3.90833	−0.146368
$714$	0	0
$715$	11.7250	0.438489
$716$	0	0
$717$	−8.30278	−0.310073
$718$	0	0
$719$	27.6333	1.03055	0.515274	−	0.857025i	$-0.327690\pi$
$719$	27.6333	1.03055	0.515274	+	0.857025i	$0.327690\pi$
$720$	0	0
$721$	−16.2111	−0.603733
$722$	0	0
$723$	6.23886	0.232026
$724$	0	0
$725$	39.6333	1.47194
$726$	0	0
$727$	27.6611	1.02589	0.512946	−	0.858421i	$-0.328554\pi$
$727$	27.6611	1.02589	0.512946	+	0.858421i	$0.328554\pi$
$728$	0	0
$729$	−22.1749	−0.821294
$730$	0	0
$731$	−53.0278	−1.96130
$732$	0	0
$733$	32.0000	1.18195	0.590973	−	0.806691i	$-0.298744\pi$
$733$	32.0000	1.18195	0.590973	+	0.806691i	$0.298744\pi$
$734$	0	0
$735$	0.908327	0.0335041
$736$	0	0
$737$	1.33053	0.0490108
$738$	0	0
$739$	18.5778	0.683395	0.341698	−	0.939810i	$-0.388998\pi$
$739$	18.5778	0.683395	0.341698	+	0.939810i	$0.388998\pi$
$740$	0	0
$741$	−1.69722	−0.0623491
$742$	0	0
$743$	12.6333	0.463471	0.231736	−	0.972779i	$-0.425560\pi$
$743$	12.6333	0.463471	0.231736	+	0.972779i	$0.425560\pi$
$744$	0	0
$745$	18.6333	0.682672
$746$	0	0
$747$	−37.5416	−1.37358
$748$	0	0
$749$	−5.78890	−0.211522
$750$	0	0
$751$	−26.3583	−0.961828	−0.480914	−	0.876768i	$-0.659695\pi$
$751$	−26.3583	−0.961828	−0.480914	+	0.876768i	$0.659695\pi$
$752$	0	0
$753$	−4.81665	−0.175529
$754$	0	0
$755$	17.7250	0.645078
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	−0.633308	−0.0229876
$760$	0	0
$761$	−24.2111	−0.877652	−0.438826	−	0.898572i	$-0.644606\pi$
$761$	−24.2111	−0.877652	−0.438826	+	0.898572i	$0.644606\pi$
$762$	0	0
$763$	10.2111	0.369667
$764$	0	0
$765$	−46.2666	−1.67277
$766$	0	0
$767$	−34.8167	−1.25716
$768$	0	0
$769$	−25.2111	−0.909136	−0.454568	−	0.890712i	$-0.650206\pi$
$769$	−25.2111	−0.909136	−0.454568	+	0.890712i	$0.650206\pi$
$770$	0	0
$771$	0.275019	0.00990458
$772$	0	0
$773$	−22.1833	−0.797880	−0.398940	−	0.916977i	$-0.630622\pi$
$773$	−22.1833	−0.797880	−0.398940	+	0.916977i	$0.630622\pi$
$774$	0	0
$775$	−5.21110	−0.187188
$776$	0	0
$777$	1.09167	0.0391636
$778$	0	0
$779$	−0.697224	−0.0249807
$780$	0	0
$781$	−8.51388	−0.304651
$782$	0	0
$783$	17.7250	0.633439
$784$	0	0
$785$	−13.5416	−0.483322
$786$	0	0
$787$	31.6333	1.12761	0.563803	−	0.825909i	$-0.309338\pi$
$787$	31.6333	1.12761	0.563803	+	0.825909i	$0.309338\pi$
$788$	0	0
$789$	−2.93608	−0.104527
$790$	0	0
$791$	−9.69722	−0.344794
$792$	0	0
$793$	23.6056	0.838258
$794$	0	0
$795$	−6.27502	−0.222552
$796$	0	0
$797$	26.0917	0.924214	0.462107	−	0.886824i	$-0.347094\pi$
$797$	26.0917	0.924214	0.462107	+	0.886824i	$0.347094\pi$
$798$	0	0
$799$	32.9361	1.16519
$800$	0	0
$801$	30.8444	1.08983
$802$	0	0
$803$	1.05551	0.0372482
$804$	0	0
$805$	9.00000	0.317208
$806$	0	0
$807$	−6.27502	−0.220891
$808$	0	0
$809$	−26.0278	−0.915087	−0.457544	−	0.889187i	$-0.651271\pi$
$809$	−26.0278	−0.915087	−0.457544	+	0.889187i	$0.651271\pi$
$810$	0	0
$811$	32.0555	1.12562	0.562811	−	0.826586i	$-0.309720\pi$
$811$	32.0555	1.12562	0.562811	+	0.826586i	$0.309720\pi$
$812$	0	0
$813$	5.42221	0.190165
$814$	0	0
$815$	17.0917	0.598695
$816$	0	0
$817$	−10.0000	−0.349856
$818$	0	0
$819$	−16.3028	−0.569665
$820$	0	0
$821$	−51.6333	−1.80201	−0.901007	−	0.433804i	$-0.857171\pi$
$821$	−51.6333	−1.80201	−0.901007	+	0.433804i	$0.857171\pi$
$822$	0	0
$823$	−16.2389	−0.566051	−0.283026	−	0.959112i	$-0.591338\pi$
$823$	−16.2389	−0.566051	−0.283026	+	0.959112i	$0.591338\pi$
$824$	0	0
$825$	−0.844410	−0.0293986
$826$	0	0
$827$	45.0000	1.56480	0.782402	−	0.622774i	$-0.213994\pi$
$827$	45.0000	1.56480	0.782402	+	0.622774i	$0.213994\pi$
$828$	0	0
$829$	54.0555	1.87743	0.938713	−	0.344700i	$-0.112019\pi$
$829$	54.0555	1.87743	0.938713	+	0.344700i	$0.112019\pi$
$830$	0	0
$831$	−0.183346	−0.00636021
$832$	0	0
$833$	−5.30278	−0.183730
$834$	0	0
$835$	13.1833	0.456229
$836$	0	0
$837$	−2.33053	−0.0805550
$838$	0	0
$839$	2.78890	0.0962834	0.0481417	−	0.998841i	$-0.484670\pi$
$839$	2.78890	0.0962834	0.0481417	+	0.998841i	$0.484670\pi$
$840$	0	0
$841$	69.1749	2.38534
$842$	0	0
$843$	−0.908327	−0.0312844
$844$	0	0
$845$	−55.2666	−1.90123
$846$	0	0
$847$	10.5139	0.361261
$848$	0	0
$849$	8.27502	0.283998
$850$	0	0
$851$	10.8167	0.370790
$852$	0	0
$853$	3.33053	0.114035	0.0570176	−	0.998373i	$-0.481841\pi$
$853$	3.33053	0.114035	0.0570176	+	0.998373i	$0.481841\pi$
$854$	0	0
$855$	−8.72498	−0.298388
$856$	0	0
$857$	32.0917	1.09623	0.548115	−	0.836403i	$-0.315345\pi$
$857$	32.0917	1.09623	0.548115	+	0.836403i	$0.315345\pi$
$858$	0	0
$859$	27.3028	0.931559	0.465779	−	0.884901i	$-0.345774\pi$
$859$	27.3028	0.931559	0.465779	+	0.884901i	$0.345774\pi$
$860$	0	0
$861$	0.211103	0.00719436
$862$	0	0
$863$	16.5416	0.563084	0.281542	−	0.959549i	$-0.409154\pi$
$863$	16.5416	0.563084	0.281542	+	0.959549i	$0.409154\pi$
$864$	0	0
$865$	−23.4500	−0.797323
$866$	0	0
$867$	−3.36669	−0.114339
$868$	0	0
$869$	−7.81665	−0.265162
$870$	0	0
$871$	−10.6972	−0.362462
$872$	0	0
$873$	−27.9361	−0.945493
$874$	0	0
$875$	−3.00000	−0.101419
$876$	0	0
$877$	−8.18335	−0.276332	−0.138166	−	0.990409i	$-0.544121\pi$
$877$	−8.18335	−0.276332	−0.138166	+	0.990409i	$0.544121\pi$
$878$	0	0
$879$	3.82506	0.129016
$880$	0	0
$881$	−19.7527	−0.665487	−0.332743	−	0.943017i	$-0.607974\pi$
$881$	−19.7527	−0.665487	−0.332743	+	0.943017i	$0.607974\pi$
$882$	0	0
$883$	46.2111	1.55513	0.777564	−	0.628804i	$-0.216455\pi$
$883$	46.2111	1.55513	0.777564	+	0.628804i	$0.216455\pi$
$884$	0	0
$885$	5.64171	0.189644
$886$	0	0
$887$	−19.1833	−0.644114	−0.322057	−	0.946720i	$-0.604374\pi$
$887$	−19.1833	−0.644114	−0.322057	+	0.946720i	$0.604374\pi$
$888$	0	0
$889$	−11.8167	−0.396318
$890$	0	0
$891$	5.70563	0.191146
$892$	0	0
$893$	6.21110	0.207847
$894$	0	0
$895$	−35.1749	−1.17577
$896$	0	0
$897$	5.09167	0.170006
$898$	0	0
$899$	−12.9083	−0.430517
$900$	0	0
$901$	36.6333	1.22043
$902$	0	0
$903$	3.02776	0.100757
$904$	0	0
$905$	32.0917	1.06676
$906$	0	0
$907$	5.18335	0.172110	0.0860551	−	0.996290i	$-0.472574\pi$
$907$	5.18335	0.172110	0.0860551	+	0.996290i	$0.472574\pi$
$908$	0	0
$909$	12.7805	0.423902
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	0	0
$913$	9.00000	0.297857
$914$	0	0
$915$	−3.82506	−0.126453
$916$	0	0
$917$	−20.5139	−0.677428
$918$	0	0
$919$	7.97224	0.262980	0.131490	−	0.991317i	$-0.458024\pi$
$919$	7.97224	0.262980	0.131490	+	0.991317i	$0.458024\pi$
$920$	0	0
$921$	4.44996	0.146631
$922$	0	0
$923$	68.4500	2.25306
$924$	0	0
$925$	14.4222	0.474199
$926$	0	0
$927$	−47.1472	−1.54852
$928$	0	0
$929$	11.5139	0.377758	0.188879	−	0.982000i	$-0.439515\pi$
$929$	11.5139	0.377758	0.188879	+	0.982000i	$0.439515\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	−0.761141	−0.0249186
$934$	0	0
$935$	11.0917	0.362736
$936$	0	0
$937$	21.1194	0.689942	0.344971	−	0.938613i	$-0.387889\pi$
$937$	21.1194	0.689942	0.344971	+	0.938613i	$0.387889\pi$
$938$	0	0
$939$	8.18335	0.267053
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	2.09167	0.0681142
$944$	0	0
$945$	5.36669	0.174579
$946$	0	0
$947$	−36.9083	−1.19936	−0.599680	−	0.800240i	$-0.704705\pi$
$947$	−36.9083	−1.19936	−0.599680	+	0.800240i	$0.704705\pi$
$948$	0	0
$949$	−8.48612	−0.275471
$950$	0	0
$951$	−2.66106	−0.0862909
$952$	0	0
$953$	4.33053	0.140280	0.0701398	−	0.997537i	$-0.477655\pi$
$953$	4.33053	0.140280	0.0701398	+	0.997537i	$0.477655\pi$
$954$	0	0
$955$	75.9916	2.45903
$956$	0	0
$957$	−2.09167	−0.0676142
$958$	0	0
$959$	−21.6333	−0.698576
$960$	0	0
$961$	−29.3028	−0.945251
$962$	0	0
$963$	−16.8360	−0.542533
$964$	0	0
$965$	69.3583	2.23272
$966$	0	0
$967$	−26.6972	−0.858525	−0.429262	−	0.903180i	$-0.641226\pi$
$967$	−26.6972	−0.858525	−0.429262	+	0.903180i	$0.641226\pi$
$968$	0	0
$969$	−1.60555	−0.0515777
$970$	0	0
$971$	18.6333	0.597971	0.298986	−	0.954258i	$-0.403352\pi$
$971$	18.6333	0.597971	0.298986	+	0.954258i	$0.403352\pi$
$972$	0	0
$973$	5.00000	0.160293
$974$	0	0
$975$	6.78890	0.217419
$976$	0	0
$977$	−17.4500	−0.558274	−0.279137	−	0.960251i	$-0.590048\pi$
$977$	−17.4500	−0.558274	−0.279137	+	0.960251i	$0.590048\pi$
$978$	0	0
$979$	−7.39445	−0.236328
$980$	0	0
$981$	29.6972	0.948159
$982$	0	0
$983$	49.0555	1.56463	0.782314	−	0.622884i	$-0.214039\pi$
$983$	49.0555	1.56463	0.782314	+	0.622884i	$0.214039\pi$
$984$	0	0
$985$	20.7250	0.660353
$986$	0	0
$987$	−1.88057	−0.0598592
$988$	0	0
$989$	30.0000	0.953945
$990$	0	0
$991$	16.9722	0.539141	0.269571	−	0.962981i	$-0.413118\pi$
$991$	16.9722	0.539141	0.269571	+	0.962981i	$0.413118\pi$
$992$	0	0
$993$	3.54163	0.112390
$994$	0	0
$995$	−40.2666	−1.27654
$996$	0	0
$997$	5.27502	0.167062	0.0835308	−	0.996505i	$-0.473380\pi$
$997$	5.27502	0.167062	0.0835308	+	0.996505i	$0.473380\pi$
$998$	0	0
$999$	6.44996	0.204068

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2128.2.a.l.1.1		2
4.3	odd	2		133.2.a.b.1.1	✓	2
8.3	odd	2		8512.2.a.bh.1.1		2
8.5	even	2		8512.2.a.l.1.2		2
12.11	even	2		1197.2.a.h.1.2		2
20.19	odd	2		3325.2.a.n.1.2		2
28.3	even	6		931.2.f.g.324.2		4
28.11	odd	6		931.2.f.h.324.2		4
28.19	even	6		931.2.f.g.704.2		4
28.23	odd	6		931.2.f.h.704.2		4
28.27	even	2		931.2.a.g.1.1		2
76.75	even	2		2527.2.a.d.1.2		2
84.83	odd	2		8379.2.a.bf.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
133.2.a.b.1.1	✓	2	4.3	odd	2
931.2.a.g.1.1		2	28.27	even	2
931.2.f.g.324.2		4	28.3	even	6
931.2.f.g.704.2		4	28.19	even	6
931.2.f.h.324.2		4	28.11	odd	6
931.2.f.h.704.2		4	28.23	odd	6
1197.2.a.h.1.2		2	12.11	even	2
2128.2.a.l.1.1		2	1.1	even	1	trivial
2527.2.a.d.1.2		2	76.75	even	2
3325.2.a.n.1.2		2	20.19	odd	2
8379.2.a.bf.1.2		2	84.83	odd	2
8512.2.a.l.1.2		2	8.5	even	2
8512.2.a.bh.1.1		2	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2128 = 2^{4} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2128.a (trivial)

\(f(q)\)	\(=\)	\(q-0.302776 q^{3} -3.00000 q^{5} -1.00000 q^{7} -2.90833 q^{9} +0.697224 q^{11} -5.60555 q^{13} +0.908327 q^{15} -5.30278 q^{17} -1.00000 q^{19} +0.302776 q^{21} +3.00000 q^{23} +4.00000 q^{25} +1.78890 q^{27} +9.90833 q^{29} -1.30278 q^{31} -0.211103 q^{33} +3.00000 q^{35} +3.60555 q^{37} +1.69722 q^{39} +0.697224 q^{41} +10.0000 q^{43} +8.72498 q^{45} -6.21110 q^{47} +1.00000 q^{49} +1.60555 q^{51} -6.90833 q^{53} -2.09167 q^{55} +0.302776 q^{57} +6.21110 q^{59} -4.21110 q^{61} +2.90833 q^{63} +16.8167 q^{65} +1.90833 q^{67} -0.908327 q^{69} -12.2111 q^{71} +1.51388 q^{73} -1.21110 q^{75} -0.697224 q^{77} -11.2111 q^{79} +8.18335 q^{81} +12.9083 q^{83} +15.9083 q^{85} -3.00000 q^{87} -10.6056 q^{89} +5.60555 q^{91} +0.394449 q^{93} +3.00000 q^{95} +9.60555 q^{97} -2.02776 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} - 6 q^{5} - 2 q^{7} + 5 q^{9} + 5 q^{11} - 4 q^{13} - 9 q^{15} - 7 q^{17} - 2 q^{19} - 3 q^{21} + 6 q^{23} + 8 q^{25} + 18 q^{27} + 9 q^{29} + q^{31} + 14 q^{33} + 6 q^{35} + 7 q^{39} + 5 q^{41}+ \cdots + 32 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 - 6 * q^5 - 2 * q^7 + 5 * q^9 + 5 * q^11 - 4 * q^13 - 9 * q^15 - 7 * q^17 - 2 * q^19 - 3 * q^21 + 6 * q^23 + 8 * q^25 + 18 * q^27 + 9 * q^29 + q^31 + 14 * q^33 + 6 * q^35 + 7 * q^39 + 5 * q^41 + 20 * q^43 - 15 * q^45 + 2 * q^47 + 2 * q^49 - 4 * q^51 - 3 * q^53 - 15 * q^55 - 3 * q^57 - 2 * q^59 + 6 * q^61 - 5 * q^63 + 12 * q^65 - 7 * q^67 + 9 * q^69 - 10 * q^71 - 15 * q^73 + 12 * q^75 - 5 * q^77 - 8 * q^79 + 38 * q^81 + 15 * q^83 + 21 * q^85 - 6 * q^87 - 14 * q^89 + 4 * q^91 + 8 * q^93 + 6 * q^95 + 12 * q^97 + 32 * q^99

Introduction

L-functions

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