LMFDB - Embedded newform 5733.2.a.bu.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5733,2,Mod(1,5733)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5733, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5733.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5733 = 3^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5733.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:	$45.7782354788$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.4507648.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 1 $ x^6 - 2x^5 - 5x^4 + 8x^3 + 7x^2 - 6*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 637)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.35100$ of defining polynomial
Character	$\chi$	$=$	5733.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17619 q^{2} -0.616586 q^{4} +3.14862 q^{5} -3.07759 q^{8} +O(q^{10})$	$q+1.17619 q^{2} -0.616586 q^{4} +3.14862 q^{5} -3.07759 q^{8} +3.70337 q^{10} +0.773390 q^{11} -1.00000 q^{13} -2.38665 q^{16} -5.75340 q^{17} +1.22298 q^{19} -1.94140 q^{20} +0.909650 q^{22} +2.99182 q^{23} +4.91383 q^{25} -1.17619 q^{26} +2.46882 q^{29} +6.13487 q^{31} +3.34804 q^{32} -6.76707 q^{34} +4.99933 q^{37} +1.43845 q^{38} -9.69018 q^{40} -2.55981 q^{41} -2.73150 q^{43} -0.476861 q^{44} +3.51894 q^{46} +5.37169 q^{47} +5.77958 q^{50} +0.616586 q^{52} +9.79015 q^{53} +2.43511 q^{55} +2.90379 q^{58} +2.50456 q^{59} +10.9167 q^{61} +7.21575 q^{62} +8.71122 q^{64} -3.14862 q^{65} +4.32518 q^{67} +3.54746 q^{68} -10.6649 q^{71} +5.17450 q^{73} +5.88014 q^{74} -0.754072 q^{76} -0.542038 q^{79} -7.51467 q^{80} -3.01081 q^{82} +15.2259 q^{83} -18.1153 q^{85} -3.21275 q^{86} -2.38018 q^{88} +9.23208 q^{89} -1.84471 q^{92} +6.31811 q^{94} +3.85070 q^{95} +1.26291 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{4} + 6 q^{5}+O(q^{10}) $ 6 * q + 4 * q^4 + 6 * q^5	$ 6 q + 4 q^{4} + 6 q^{5} - 4 q^{10} - 4 q^{11} - 6 q^{13} + 16 q^{17} - 2 q^{19} + 16 q^{20} - 12 q^{22} + 6 q^{23} - 4 q^{25} + 6 q^{29} - 6 q^{31} + 20 q^{32} + 8 q^{38} - 4 q^{40} - 8 q^{41} + 2 q^{43} + 4 q^{44} + 8 q^{46} + 30 q^{47} - 8 q^{50} - 4 q^{52} + 14 q^{53} + 8 q^{55} - 8 q^{58} + 24 q^{59} + 28 q^{62} - 20 q^{64} - 6 q^{65} + 16 q^{67} + 28 q^{68} - 8 q^{71} + 6 q^{73} + 12 q^{74} + 16 q^{76} - 22 q^{79} - 28 q^{80} + 40 q^{82} + 50 q^{83} - 8 q^{85} + 16 q^{86} - 44 q^{88} + 26 q^{89} - 20 q^{92} + 32 q^{94} + 6 q^{95} + 14 q^{97}+O(q^{100}) $ 6 * q + 4 * q^4 + 6 * q^5 - 4 * q^10 - 4 * q^11 - 6 * q^13 + 16 * q^17 - 2 * q^19 + 16 * q^20 - 12 * q^22 + 6 * q^23 - 4 * q^25 + 6 * q^29 - 6 * q^31 + 20 * q^32 + 8 * q^38 - 4 * q^40 - 8 * q^41 + 2 * q^43 + 4 * q^44 + 8 * q^46 + 30 * q^47 - 8 * q^50 - 4 * q^52 + 14 * q^53 + 8 * q^55 - 8 * q^58 + 24 * q^59 + 28 * q^62 - 20 * q^64 - 6 * q^65 + 16 * q^67 + 28 * q^68 - 8 * q^71 + 6 * q^73 + 12 * q^74 + 16 * q^76 - 22 * q^79 - 28 * q^80 + 40 * q^82 + 50 * q^83 - 8 * q^85 + 16 * q^86 - 44 * q^88 + 26 * q^89 - 20 * q^92 + 32 * q^94 + 6 * q^95 + 14 * 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 $

$2$	1.17619	0.831689	0.415845	−	0.909436i	$-0.363486\pi$
$2$	1.17619	0.831689	0.415845	+	0.909436i	$0.363486\pi$
$3$	0	0
$3$	0	0
$4$	−0.616586	−0.308293
$5$	3.14862	1.40811	0.704054	−	0.710147i	$-0.251371\pi$
$5$	3.14862	1.40811	0.704054	+	0.710147i	$0.251371\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−3.07759	−1.08809
$9$	0	0
$10$	3.70337	1.17111
$11$	0.773390	0.233186	0.116593	−	0.993180i	$-0.462803\pi$
$11$	0.773390	0.233186	0.116593	+	0.993180i	$0.462803\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	−2.38665	−0.596663
$17$	−5.75340	−1.39540	−0.697702	−	0.716388i	$-0.745794\pi$
$17$	−5.75340	−1.39540	−0.697702	+	0.716388i	$0.745794\pi$
$18$	0	0
$19$	1.22298	0.280571	0.140285	−	0.990111i	$-0.455198\pi$
$19$	1.22298	0.280571	0.140285	+	0.990111i	$0.455198\pi$
$20$	−1.94140	−0.434109
$21$	0	0
$22$	0.909650	0.193938
$23$	2.99182	0.623838	0.311919	−	0.950109i	$-0.399028\pi$
$23$	2.99182	0.623838	0.311919	+	0.950109i	$0.399028\pi$
$24$	0	0
$25$	4.91383	0.982767
$26$	−1.17619	−0.230669
$27$	0	0
$28$	0	0
$29$	2.46882	0.458449	0.229224	−	0.973374i	$-0.426381\pi$
$29$	2.46882	0.458449	0.229224	+	0.973374i	$0.426381\pi$
$30$	0	0
$31$	6.13487	1.10185	0.550927	−	0.834553i	$-0.314274\pi$
$31$	6.13487	1.10185	0.550927	+	0.834553i	$0.314274\pi$
$32$	3.34804	0.591855
$33$	0	0
$34$	−6.76707	−1.16054
$35$	0	0
$36$	0	0
$37$	4.99933	0.821884	0.410942	−	0.911661i	$-0.365200\pi$
$37$	4.99933	0.821884	0.410942	+	0.911661i	$0.365200\pi$
$38$	1.43845	0.233348
$39$	0	0
$40$	−9.69018	−1.53215
$41$	−2.55981	−0.399774	−0.199887	−	0.979819i	$-0.564058\pi$
$41$	−2.55981	−0.399774	−0.199887	+	0.979819i	$0.564058\pi$
$42$	0	0
$43$	−2.73150	−0.416550	−0.208275	−	0.978070i	$-0.566785\pi$
$43$	−2.73150	−0.416550	−0.208275	+	0.978070i	$0.566785\pi$
$44$	−0.476861	−0.0718895
$45$	0	0
$46$	3.51894	0.518839
$47$	5.37169	0.783542	0.391771	−	0.920063i	$-0.371863\pi$
$47$	5.37169	0.783542	0.391771	+	0.920063i	$0.371863\pi$
$48$	0	0
$49$	0	0
$50$	5.77958	0.817357
$51$	0	0
$52$	0.616586	0.0855050
$53$	9.79015	1.34478	0.672390	−	0.740197i	$-0.265268\pi$
$53$	9.79015	1.34478	0.672390	+	0.740197i	$0.265268\pi$
$54$	0	0
$55$	2.43511	0.328351
$56$	0	0
$57$	0	0
$58$	2.90379	0.381287
$59$	2.50456	0.326066	0.163033	−	0.986621i	$-0.447872\pi$
$59$	2.50456	0.326066	0.163033	+	0.986621i	$0.447872\pi$
$60$	0	0
$61$	10.9167	1.39774	0.698870	−	0.715248i	$-0.253686\pi$
$61$	10.9167	1.39774	0.698870	+	0.715248i	$0.253686\pi$
$62$	7.21575	0.916401
$63$	0	0
$64$	8.71122	1.08890
$65$	−3.14862	−0.390539
$66$	0	0
$67$	4.32518	0.528404	0.264202	−	0.964467i	$-0.414891\pi$
$67$	4.32518	0.528404	0.264202	+	0.964467i	$0.414891\pi$
$68$	3.54746	0.430193
$69$	0	0
$70$	0	0
$71$	−10.6649	−1.26570	−0.632848	−	0.774276i	$-0.718114\pi$
$71$	−10.6649	−1.26570	−0.632848	+	0.774276i	$0.718114\pi$
$72$	0	0
$73$	5.17450	0.605630	0.302815	−	0.953049i	$-0.402074\pi$
$73$	5.17450	0.605630	0.302815	+	0.953049i	$0.402074\pi$
$74$	5.88014	0.683552
$75$	0	0
$76$	−0.754072	−0.0864979
$77$	0	0
$78$	0	0
$79$	−0.542038	−0.0609841	−0.0304920	−	0.999535i	$-0.509707\pi$
$79$	−0.542038	−0.0609841	−0.0304920	+	0.999535i	$0.509707\pi$
$80$	−7.51467	−0.840165
$81$	0	0
$82$	−3.01081	−0.332488
$83$	15.2259	1.67125	0.835627	−	0.549297i	$-0.185104\pi$
$83$	15.2259	1.67125	0.835627	+	0.549297i	$0.185104\pi$
$84$	0	0
$85$	−18.1153	−1.96488
$86$	−3.21275	−0.346440
$87$	0	0
$88$	−2.38018	−0.253728
$89$	9.23208	0.978599	0.489299	−	0.872116i	$-0.337253\pi$
$89$	9.23208	0.978599	0.489299	+	0.872116i	$0.337253\pi$
$90$	0	0
$91$	0	0
$92$	−1.84471	−0.192325
$93$	0	0
$94$	6.31811	0.651663
$95$	3.85070	0.395074
$96$	0	0
$97$	1.26291	0.128229	0.0641145	−	0.997943i	$-0.479578\pi$
$97$	1.26291	0.128229	0.0641145	+	0.997943i	$0.479578\pi$
$98$	0	0
$99$	0	0
$100$	−3.02980	−0.302980
$101$	0.605447	0.0602443	0.0301221	−	0.999546i	$-0.490410\pi$
$101$	0.605447	0.0602443	0.0301221	+	0.999546i	$0.490410\pi$
$102$	0	0
$103$	−7.64804	−0.753584	−0.376792	−	0.926298i	$-0.622973\pi$
$103$	−7.64804	−0.753584	−0.376792	+	0.926298i	$0.622973\pi$
$104$	3.07759	0.301783
$105$	0	0
$106$	11.5150	1.11844
$107$	−4.82965	−0.466900	−0.233450	−	0.972369i	$-0.575001\pi$
$107$	−4.82965	−0.466900	−0.233450	+	0.972369i	$0.575001\pi$
$108$	0	0
$109$	7.23092	0.692596	0.346298	−	0.938125i	$-0.387439\pi$
$109$	7.23092	0.692596	0.346298	+	0.938125i	$0.387439\pi$
$110$	2.86415	0.273086
$111$	0	0
$112$	0	0
$113$	9.19375	0.864875	0.432438	−	0.901664i	$-0.357654\pi$
$113$	9.19375	0.864875	0.432438	+	0.901664i	$0.357654\pi$
$114$	0	0
$115$	9.42012	0.878430
$116$	−1.52224	−0.141336
$117$	0	0
$118$	2.94583	0.271186
$119$	0	0
$120$	0	0
$121$	−10.4019	−0.945624
$122$	12.8401	1.16249
$123$	0	0
$124$	−3.78267	−0.339694
$125$	−0.271305	−0.0242663
$126$	0	0
$127$	−11.2118	−0.994888	−0.497444	−	0.867496i	$-0.665728\pi$
$127$	−11.2118	−0.994888	−0.497444	+	0.867496i	$0.665728\pi$
$128$	3.54994	0.313773
$129$	0	0
$130$	−3.70337	−0.324807
$131$	15.7380	1.37503	0.687517	−	0.726169i	$-0.258701\pi$
$131$	15.7380	1.37503	0.687517	+	0.726169i	$0.258701\pi$
$132$	0	0
$133$	0	0
$134$	5.08721	0.439468
$135$	0	0
$136$	17.7066	1.51833
$137$	18.6210	1.59090	0.795449	−	0.606020i	$-0.207235\pi$
$137$	18.6210	1.59090	0.795449	+	0.606020i	$0.207235\pi$
$138$	0	0
$139$	11.9137	1.01050	0.505252	−	0.862972i	$-0.331399\pi$
$139$	11.9137	1.01050	0.505252	+	0.862972i	$0.331399\pi$
$140$	0	0
$141$	0	0
$142$	−12.5440	−1.05267
$143$	−0.773390	−0.0646741
$144$	0	0
$145$	7.77339	0.645545
$146$	6.08618	0.503696
$147$	0	0
$148$	−3.08251	−0.253381
$149$	−11.3753	−0.931904	−0.465952	−	0.884810i	$-0.654288\pi$
$149$	−11.3753	−0.931904	−0.465952	+	0.884810i	$0.654288\pi$
$150$	0	0
$151$	−3.69250	−0.300492	−0.150246	−	0.988649i	$-0.548007\pi$
$151$	−3.69250	−0.300492	−0.150246	+	0.988649i	$0.548007\pi$
$152$	−3.76383	−0.305287
$153$	0	0
$154$	0	0
$155$	19.3164	1.55153
$156$	0	0
$157$	−1.12065	−0.0894374	−0.0447187	−	0.999000i	$-0.514239\pi$
$157$	−1.12065	−0.0894374	−0.0447187	+	0.999000i	$0.514239\pi$
$158$	−0.637538	−0.0507198
$159$	0	0
$160$	10.5417	0.833396
$161$	0	0
$162$	0	0
$163$	−20.3435	−1.59342	−0.796712	−	0.604359i	$-0.793429\pi$
$163$	−20.3435	−1.59342	−0.796712	+	0.604359i	$0.793429\pi$
$164$	1.57834	0.123248
$165$	0	0
$166$	17.9084	1.38996
$167$	13.0063	1.00646	0.503228	−	0.864154i	$-0.332146\pi$
$167$	13.0063	1.00646	0.503228	+	0.864154i	$0.332146\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−21.3070	−1.63417
$171$	0	0
$172$	1.68420	0.128419
$173$	−25.9110	−1.96998	−0.984989	−	0.172616i	$-0.944778\pi$
$173$	−25.9110	−1.96998	−0.984989	+	0.172616i	$0.944778\pi$
$174$	0	0
$175$	0	0
$176$	−1.84581	−0.139133
$177$	0	0
$178$	10.8587	0.813890
$179$	−9.64163	−0.720649	−0.360325	−	0.932827i	$-0.617334\pi$
$179$	−9.64163	−0.720649	−0.360325	+	0.932827i	$0.617334\pi$
$180$	0	0
$181$	2.92683	0.217550	0.108775	−	0.994066i	$-0.465307\pi$
$181$	2.92683	0.217550	0.108775	+	0.994066i	$0.465307\pi$
$182$	0	0
$183$	0	0
$184$	−9.20760	−0.678793
$185$	15.7410	1.15730
$186$	0	0
$187$	−4.44962	−0.325388
$188$	−3.31211	−0.241560
$189$	0	0
$190$	4.52914	0.328579
$191$	16.1364	1.16759	0.583796	−	0.811901i	$-0.301567\pi$
$191$	16.1364	1.16759	0.583796	+	0.811901i	$0.301567\pi$
$192$	0	0
$193$	−0.0863817	−0.00621789	−0.00310894	−	0.999995i	$-0.500990\pi$
$193$	−0.0863817	−0.00621789	−0.00310894	+	0.999995i	$0.500990\pi$
$194$	1.48542	0.106647
$195$	0	0
$196$	0	0
$197$	−15.0589	−1.07290	−0.536451	−	0.843932i	$-0.680235\pi$
$197$	−15.0589	−1.07290	−0.536451	+	0.843932i	$0.680235\pi$
$198$	0	0
$199$	−15.4466	−1.09498	−0.547492	−	0.836811i	$-0.684417\pi$
$199$	−15.4466	−1.09498	−0.547492	+	0.836811i	$0.684417\pi$
$200$	−15.1228	−1.06934
$201$	0	0
$202$	0.712119	0.0501045
$203$	0	0
$204$	0	0
$205$	−8.05987	−0.562925
$206$	−8.99552	−0.626748
$207$	0	0
$208$	2.38665	0.165484
$209$	0.945840	0.0654251
$210$	0	0
$211$	−25.0561	−1.72493	−0.862466	−	0.506115i	$-0.831081\pi$
$211$	−25.0561	−1.72493	−0.862466	+	0.506115i	$0.831081\pi$
$212$	−6.03647	−0.414586
$213$	0	0
$214$	−5.68057	−0.388316
$215$	−8.60047	−0.586547
$216$	0	0
$217$	0	0
$218$	8.50491	0.576025
$219$	0	0
$220$	−1.50146	−0.101228
$221$	5.75340	0.387015
$222$	0	0
$223$	20.6640	1.38376	0.691881	−	0.722012i	$-0.256782\pi$
$223$	20.6640	1.38376	0.691881	+	0.722012i	$0.256782\pi$
$224$	0	0
$225$	0	0
$226$	10.8136	0.719307
$227$	0.982378	0.0652027	0.0326013	−	0.999468i	$-0.489621\pi$
$227$	0.982378	0.0652027	0.0326013	+	0.999468i	$0.489621\pi$
$228$	0	0
$229$	14.2742	0.943269	0.471634	−	0.881794i	$-0.343664\pi$
$229$	14.2742	0.943269	0.471634	+	0.881794i	$0.343664\pi$
$230$	11.0798	0.730581
$231$	0	0
$232$	−7.59802	−0.498835
$233$	−1.39534	−0.0914119	−0.0457060	−	0.998955i	$-0.514554\pi$
$233$	−1.39534	−0.0914119	−0.0457060	+	0.998955i	$0.514554\pi$
$234$	0	0
$235$	16.9134	1.10331
$236$	−1.54428	−0.100524
$237$	0	0
$238$	0	0
$239$	−2.54875	−0.164865	−0.0824326	−	0.996597i	$-0.526269\pi$
$239$	−2.54875	−0.164865	−0.0824326	+	0.996597i	$0.526269\pi$
$240$	0	0
$241$	−29.6746	−1.91151	−0.955755	−	0.294164i	$-0.904959\pi$
$241$	−29.6746	−1.91151	−0.955755	+	0.294164i	$0.904959\pi$
$242$	−12.2345	−0.786466
$243$	0	0
$244$	−6.73108	−0.430913
$245$	0	0
$246$	0	0
$247$	−1.22298	−0.0778163
$248$	−18.8806	−1.19892
$249$	0	0
$250$	−0.319106	−0.0201820
$251$	18.0858	1.14157	0.570784	−	0.821100i	$-0.306639\pi$
$251$	18.0858	1.14157	0.570784	+	0.821100i	$0.306639\pi$
$252$	0	0
$253$	2.31384	0.145470
$254$	−13.1872	−0.827437
$255$	0	0
$256$	−13.2470	−0.827940
$257$	14.7403	0.919473	0.459737	−	0.888055i	$-0.347944\pi$
$257$	14.7403	0.919473	0.459737	+	0.888055i	$0.347944\pi$
$258$	0	0
$259$	0	0
$260$	1.94140	0.120400
$261$	0	0
$262$	18.5108	1.14360
$263$	8.15922	0.503119	0.251560	−	0.967842i	$-0.419057\pi$
$263$	8.15922	0.503119	0.251560	+	0.967842i	$0.419057\pi$
$264$	0	0
$265$	30.8255	1.89360
$266$	0	0
$267$	0	0
$268$	−2.66684	−0.162903
$269$	0.442582	0.0269847	0.0134924	−	0.999909i	$-0.495705\pi$
$269$	0.442582	0.0269847	0.0134924	+	0.999909i	$0.495705\pi$
$270$	0	0
$271$	21.0831	1.28071	0.640354	−	0.768080i	$-0.278788\pi$
$271$	21.0831	1.28071	0.640354	+	0.768080i	$0.278788\pi$
$272$	13.7314	0.832586
$273$	0	0
$274$	21.9018	1.32313
$275$	3.80031	0.229167
$276$	0	0
$277$	3.59421	0.215955	0.107977	−	0.994153i	$-0.465563\pi$
$277$	3.59421	0.215955	0.107977	+	0.994153i	$0.465563\pi$
$278$	14.0127	0.840426
$279$	0	0
$280$	0	0
$281$	9.01252	0.537642	0.268821	−	0.963190i	$-0.413366\pi$
$281$	9.01252	0.537642	0.268821	+	0.963190i	$0.413366\pi$
$282$	0	0
$283$	−27.7411	−1.64904	−0.824520	−	0.565833i	$-0.808555\pi$
$283$	−27.7411	−1.64904	−0.824520	+	0.565833i	$0.808555\pi$
$284$	6.57585	0.390205
$285$	0	0
$286$	−0.909650	−0.0537887
$287$	0	0
$288$	0	0
$289$	16.1016	0.947153
$290$	9.14296	0.536893
$291$	0	0
$292$	−3.19052	−0.186711
$293$	12.7409	0.744333	0.372167	−	0.928166i	$-0.378615\pi$
$293$	12.7409	0.744333	0.372167	+	0.928166i	$0.378615\pi$
$294$	0	0
$295$	7.88593	0.459137
$296$	−15.3859	−0.894287
$297$	0	0
$298$	−13.3795	−0.775055
$299$	−2.99182	−0.173021
$300$	0	0
$301$	0	0
$302$	−4.34307	−0.249916
$303$	0	0
$304$	−2.91883	−0.167406
$305$	34.3726	1.96817
$306$	0	0
$307$	−10.1384	−0.578626	−0.289313	−	0.957235i	$-0.593427\pi$
$307$	−10.1384	−0.578626	−0.289313	+	0.957235i	$0.593427\pi$
$308$	0	0
$309$	0	0
$310$	22.7197	1.29039
$311$	20.6013	1.16819	0.584096	−	0.811685i	$-0.301449\pi$
$311$	20.6013	1.16819	0.584096	+	0.811685i	$0.301449\pi$
$312$	0	0
$313$	−16.5456	−0.935213	−0.467606	−	0.883937i	$-0.654883\pi$
$313$	−16.5456	−0.935213	−0.467606	+	0.883937i	$0.654883\pi$
$314$	−1.31809	−0.0743841
$315$	0	0
$316$	0.334213	0.0188010
$317$	21.2340	1.19262	0.596310	−	0.802754i	$-0.296633\pi$
$317$	21.2340	1.19262	0.596310	+	0.802754i	$0.296633\pi$
$318$	0	0
$319$	1.90936	0.106904
$320$	27.4284	1.53329
$321$	0	0
$322$	0	0
$323$	−7.03629	−0.391510
$324$	0	0
$325$	−4.91383	−0.272570
$326$	−23.9277	−1.32523
$327$	0	0
$328$	7.87804	0.434992
$329$	0	0
$330$	0	0
$331$	16.2415	0.892712	0.446356	−	0.894855i	$-0.352721\pi$
$331$	16.2415	0.892712	0.446356	+	0.894855i	$0.352721\pi$
$332$	−9.38804	−0.515236
$333$	0	0
$334$	15.2978	0.837058
$335$	13.6184	0.744050
$336$	0	0
$337$	6.42141	0.349797	0.174898	−	0.984586i	$-0.444040\pi$
$337$	6.42141	0.349797	0.174898	+	0.984586i	$0.444040\pi$
$338$	1.17619	0.0639761
$339$	0	0
$340$	11.1696	0.605758
$341$	4.74464	0.256937
$342$	0	0
$343$	0	0
$344$	8.40645	0.453245
$345$	0	0
$346$	−30.4762	−1.63841
$347$	−14.4809	−0.777376	−0.388688	−	0.921369i	$-0.627072\pi$
$347$	−14.4809	−0.777376	−0.388688	+	0.921369i	$0.627072\pi$
$348$	0	0
$349$	−1.74500	−0.0934079	−0.0467040	−	0.998909i	$-0.514872\pi$
$349$	−1.74500	−0.0934079	−0.0467040	+	0.998909i	$0.514872\pi$
$350$	0	0
$351$	0	0
$352$	2.58934	0.138012
$353$	15.9580	0.849358	0.424679	−	0.905344i	$-0.360387\pi$
$353$	15.9580	0.849358	0.424679	+	0.905344i	$0.360387\pi$
$354$	0	0
$355$	−33.5799	−1.78224
$356$	−5.69237	−0.301695
$357$	0	0
$358$	−11.3404	−0.599356
$359$	−8.81400	−0.465185	−0.232593	−	0.972574i	$-0.574721\pi$
$359$	−8.81400	−0.465185	−0.232593	+	0.972574i	$0.574721\pi$
$360$	0	0
$361$	−17.5043	−0.921280
$362$	3.44250	0.180934
$363$	0	0
$364$	0	0
$365$	16.2926	0.852792
$366$	0	0
$367$	19.7080	1.02875	0.514374	−	0.857566i	$-0.328024\pi$
$367$	19.7080	1.02875	0.514374	+	0.857566i	$0.328024\pi$
$368$	−7.14043	−0.372221
$369$	0	0
$370$	18.5144	0.962515
$371$	0	0
$372$	0	0
$373$	−0.365792	−0.0189400	−0.00947000	−	0.999955i	$-0.503014\pi$
$373$	−0.365792	−0.0189400	−0.00947000	+	0.999955i	$0.503014\pi$
$374$	−5.23358	−0.270622
$375$	0	0
$376$	−16.5319	−0.852566
$377$	−2.46882	−0.127151
$378$	0	0
$379$	7.39215	0.379709	0.189855	−	0.981812i	$-0.439198\pi$
$379$	7.39215	0.379709	0.189855	+	0.981812i	$0.439198\pi$
$380$	−2.37429	−0.121798
$381$	0	0
$382$	18.9795	0.971073
$383$	−3.35364	−0.171363	−0.0856814	−	0.996323i	$-0.527307\pi$
$383$	−3.35364	−0.171363	−0.0856814	+	0.996323i	$0.527307\pi$
$384$	0	0
$385$	0	0
$386$	−0.101601	−0.00517135
$387$	0	0
$388$	−0.778692	−0.0395321
$389$	2.59344	0.131492	0.0657462	−	0.997836i	$-0.479057\pi$
$389$	2.59344	0.131492	0.0657462	+	0.997836i	$0.479057\pi$
$390$	0	0
$391$	−17.2131	−0.870506
$392$	0	0
$393$	0	0
$394$	−17.7121	−0.892321
$395$	−1.70668	−0.0858721
$396$	0	0
$397$	33.4914	1.68088	0.840441	−	0.541902i	$-0.182296\pi$
$397$	33.4914	1.68088	0.840441	+	0.541902i	$0.182296\pi$
$398$	−18.1681	−0.910686
$399$	0	0
$400$	−11.7276	−0.586380
$401$	−39.5156	−1.97331	−0.986657	−	0.162811i	$-0.947944\pi$
$401$	−39.5156	−1.97331	−0.986657	+	0.162811i	$0.947944\pi$
$402$	0	0
$403$	−6.13487	−0.305599
$404$	−0.373310	−0.0185729
$405$	0	0
$406$	0	0
$407$	3.86643	0.191652
$408$	0	0
$409$	10.9799	0.542922	0.271461	−	0.962449i	$-0.412493\pi$
$409$	10.9799	0.542922	0.271461	+	0.962449i	$0.412493\pi$
$410$	−9.47990	−0.468179
$411$	0	0
$412$	4.71567	0.232324
$413$	0	0
$414$	0	0
$415$	47.9405	2.35331
$416$	−3.34804	−0.164151
$417$	0	0
$418$	1.11248	0.0544134
$419$	31.5621	1.54191	0.770954	−	0.636891i	$-0.219780\pi$
$419$	31.5621	1.54191	0.770954	+	0.636891i	$0.219780\pi$
$420$	0	0
$421$	−17.7055	−0.862914	−0.431457	−	0.902134i	$-0.642000\pi$
$421$	−17.7055	−0.862914	−0.431457	+	0.902134i	$0.642000\pi$
$422$	−29.4706	−1.43461
$423$	0	0
$424$	−30.1301	−1.46325
$425$	−28.2712	−1.37136
$426$	0	0
$427$	0	0
$428$	2.97789	0.143942
$429$	0	0
$430$	−10.1158	−0.487825
$431$	21.4816	1.03473	0.517367	−	0.855764i	$-0.326912\pi$
$431$	21.4816	1.03473	0.517367	+	0.855764i	$0.326912\pi$
$432$	0	0
$433$	−34.7200	−1.66854	−0.834269	−	0.551358i	$-0.814110\pi$
$433$	−34.7200	−1.66854	−0.834269	+	0.551358i	$0.814110\pi$
$434$	0	0
$435$	0	0
$436$	−4.45848	−0.213522
$437$	3.65894	0.175031
$438$	0	0
$439$	−0.728505	−0.0347697	−0.0173848	−	0.999849i	$-0.505534\pi$
$439$	−0.728505	−0.0347697	−0.0173848	+	0.999849i	$0.505534\pi$
$440$	−7.49428	−0.357276
$441$	0	0
$442$	6.76707	0.321877
$443$	−0.837291	−0.0397809	−0.0198904	−	0.999802i	$-0.506332\pi$
$443$	−0.837291	−0.0397809	−0.0198904	+	0.999802i	$0.506332\pi$
$444$	0	0
$445$	29.0684	1.37797
$446$	24.3047	1.15086
$447$	0	0
$448$	0	0
$449$	−26.4312	−1.24737	−0.623683	−	0.781677i	$-0.714365\pi$
$449$	−26.4312	−1.24737	−0.623683	+	0.781677i	$0.714365\pi$
$450$	0	0
$451$	−1.97973	−0.0932217
$452$	−5.66873	−0.266635
$453$	0	0
$454$	1.15546	0.0542284
$455$	0	0
$456$	0	0
$457$	−2.40475	−0.112489	−0.0562447	−	0.998417i	$-0.517913\pi$
$457$	−2.40475	−0.112489	−0.0562447	+	0.998417i	$0.517913\pi$
$458$	16.7892	0.784507
$459$	0	0
$460$	−5.80831	−0.270814
$461$	−22.2702	−1.03722	−0.518612	−	0.855010i	$-0.673551\pi$
$461$	−22.2702	−1.03722	−0.518612	+	0.855010i	$0.673551\pi$
$462$	0	0
$463$	32.3085	1.50151	0.750753	−	0.660583i	$-0.229691\pi$
$463$	32.3085	1.50151	0.750753	+	0.660583i	$0.229691\pi$
$464$	−5.89221	−0.273539
$465$	0	0
$466$	−1.64118	−0.0760263
$467$	−12.8744	−0.595756	−0.297878	−	0.954604i	$-0.596279\pi$
$467$	−12.8744	−0.595756	−0.297878	+	0.954604i	$0.596279\pi$
$468$	0	0
$469$	0	0
$470$	19.8934	0.917612
$471$	0	0
$472$	−7.70802	−0.354791
$473$	−2.11251	−0.0971335
$474$	0	0
$475$	6.00952	0.275736
$476$	0	0
$477$	0	0
$478$	−2.99781	−0.137117
$479$	10.5419	0.481670	0.240835	−	0.970566i	$-0.422579\pi$
$479$	10.5419	0.481670	0.240835	+	0.970566i	$0.422579\pi$
$480$	0	0
$481$	−4.99933	−0.227950
$482$	−34.9029	−1.58978
$483$	0	0
$484$	6.41364	0.291529
$485$	3.97643	0.180560
$486$	0	0
$487$	38.2416	1.73289	0.866446	−	0.499271i	$-0.166399\pi$
$487$	38.2416	1.73289	0.866446	+	0.499271i	$0.166399\pi$
$488$	−33.5972	−1.52087
$489$	0	0
$490$	0	0
$491$	15.4291	0.696306	0.348153	−	0.937438i	$-0.386809\pi$
$491$	15.4291	0.696306	0.348153	+	0.937438i	$0.386809\pi$
$492$	0	0
$493$	−14.2041	−0.639721
$494$	−1.43845	−0.0647190
$495$	0	0
$496$	−14.6418	−0.657436
$497$	0	0
$498$	0	0
$499$	−38.8212	−1.73788	−0.868938	−	0.494921i	$-0.835197\pi$
$499$	−38.8212	−1.73788	−0.868938	+	0.494921i	$0.835197\pi$
$500$	0.167283	0.00748112
$501$	0	0
$502$	21.2723	0.949430
$503$	−27.0935	−1.20804	−0.604020	−	0.796969i	$-0.706435\pi$
$503$	−27.0935	−1.20804	−0.604020	+	0.796969i	$0.706435\pi$
$504$	0	0
$505$	1.90633	0.0848304
$506$	2.72151	0.120986
$507$	0	0
$508$	6.91304	0.306717
$509$	−15.7693	−0.698961	−0.349480	−	0.936944i	$-0.613642\pi$
$509$	−15.7693	−0.698961	−0.349480	+	0.936944i	$0.613642\pi$
$510$	0	0
$511$	0	0
$512$	−22.6809	−1.00236
$513$	0	0
$514$	17.3373	0.764716
$515$	−24.0808	−1.06113
$516$	0	0
$517$	4.15441	0.182711
$518$	0	0
$519$	0	0
$520$	9.69018	0.424943
$521$	7.30737	0.320142	0.160071	−	0.987106i	$-0.448828\pi$
$521$	7.30737	0.320142	0.160071	+	0.987106i	$0.448828\pi$
$522$	0	0
$523$	−14.0826	−0.615788	−0.307894	−	0.951421i	$-0.599624\pi$
$523$	−14.0826	−0.615788	−0.307894	+	0.951421i	$0.599624\pi$
$524$	−9.70381	−0.423913
$525$	0	0
$526$	9.59677	0.418439
$527$	−35.2963	−1.53753
$528$	0	0
$529$	−14.0490	−0.610827
$530$	36.2565	1.57488
$531$	0	0
$532$	0	0
$533$	2.55981	0.110877
$534$	0	0
$535$	−15.2067	−0.657445
$536$	−13.3111	−0.574953
$537$	0	0
$538$	0.520559	0.0224429
$539$	0	0
$540$	0	0
$541$	−39.1750	−1.68426	−0.842132	−	0.539271i	$-0.818700\pi$
$541$	−39.1750	−1.68426	−0.842132	+	0.539271i	$0.818700\pi$
$542$	24.7977	1.06515
$543$	0	0
$544$	−19.2626	−0.825877
$545$	22.7674	0.975250
$546$	0	0
$547$	38.7917	1.65862	0.829308	−	0.558792i	$-0.188735\pi$
$547$	38.7917	1.65862	0.829308	+	0.558792i	$0.188735\pi$
$548$	−11.4814	−0.490463
$549$	0	0
$550$	4.46987	0.190596
$551$	3.01932	0.128627
$552$	0	0
$553$	0	0
$554$	4.22746	0.179607
$555$	0	0
$556$	−7.34580	−0.311531
$557$	27.4693	1.16391	0.581956	−	0.813220i	$-0.302287\pi$
$557$	27.4693	1.16391	0.581956	+	0.813220i	$0.302287\pi$
$558$	0	0
$559$	2.73150	0.115530
$560$	0	0
$561$	0	0
$562$	10.6004	0.447151
$563$	14.1210	0.595129	0.297565	−	0.954702i	$-0.403826\pi$
$563$	14.1210	0.595129	0.297565	+	0.954702i	$0.403826\pi$
$564$	0	0
$565$	28.9477	1.21784
$566$	−32.6288	−1.37149
$567$	0	0
$568$	32.8223	1.37720
$569$	8.28649	0.347388	0.173694	−	0.984800i	$-0.444430\pi$
$569$	8.28649	0.347388	0.173694	+	0.984800i	$0.444430\pi$
$570$	0	0
$571$	−23.5274	−0.984591	−0.492295	−	0.870428i	$-0.663842\pi$
$571$	−23.5274	−0.984591	−0.492295	+	0.870428i	$0.663842\pi$
$572$	0.476861	0.0199386
$573$	0	0
$574$	0	0
$575$	14.7013	0.613087
$576$	0	0
$577$	17.7732	0.739906	0.369953	−	0.929050i	$-0.379374\pi$
$577$	17.7732	0.739906	0.369953	+	0.929050i	$0.379374\pi$
$578$	18.9385	0.787737
$579$	0	0
$580$	−4.79296	−0.199017
$581$	0	0
$582$	0	0
$583$	7.57160	0.313584
$584$	−15.9250	−0.658982
$585$	0	0
$586$	14.9857	0.619054
$587$	−6.64096	−0.274102	−0.137051	−	0.990564i	$-0.543762\pi$
$587$	−6.64096	−0.274102	−0.137051	+	0.990564i	$0.543762\pi$
$588$	0	0
$589$	7.50282	0.309148
$590$	9.27532	0.381859
$591$	0	0
$592$	−11.9317	−0.490388
$593$	34.0504	1.39828	0.699141	−	0.714984i	$-0.253566\pi$
$593$	34.0504	1.39828	0.699141	+	0.714984i	$0.253566\pi$
$594$	0	0
$595$	0	0
$596$	7.01387	0.287299
$597$	0	0
$598$	−3.51894	−0.143900
$599$	−9.38441	−0.383437	−0.191718	−	0.981450i	$-0.561406\pi$
$599$	−9.38441	−0.383437	−0.191718	+	0.981450i	$0.561406\pi$
$600$	0	0
$601$	−8.80294	−0.359079	−0.179540	−	0.983751i	$-0.557461\pi$
$601$	−8.80294	−0.359079	−0.179540	+	0.983751i	$0.557461\pi$
$602$	0	0
$603$	0	0
$604$	2.27675	0.0926394
$605$	−32.7516	−1.33154
$606$	0	0
$607$	−17.7720	−0.721342	−0.360671	−	0.932693i	$-0.617452\pi$
$607$	−17.7720	−0.721342	−0.360671	+	0.932693i	$0.617452\pi$
$608$	4.09458	0.166057
$609$	0	0
$610$	40.4286	1.63691
$611$	−5.37169	−0.217315
$612$	0	0
$613$	10.6081	0.428457	0.214228	−	0.976784i	$-0.431276\pi$
$613$	10.6081	0.428457	0.214228	+	0.976784i	$0.431276\pi$
$614$	−11.9246	−0.481237
$615$	0	0
$616$	0	0
$617$	−49.3483	−1.98669	−0.993344	−	0.115188i	$-0.963253\pi$
$617$	−49.3483	−1.98669	−0.993344	+	0.115188i	$0.963253\pi$
$618$	0	0
$619$	−7.42203	−0.298317	−0.149158	−	0.988813i	$-0.547656\pi$
$619$	−7.42203	−0.298317	−0.149158	+	0.988813i	$0.547656\pi$
$620$	−11.9102	−0.478325
$621$	0	0
$622$	24.2309	0.971573
$623$	0	0
$624$	0	0
$625$	−25.4234	−1.01694
$626$	−19.4607	−0.777806
$627$	0	0
$628$	0.690975	0.0275729
$629$	−28.7631	−1.14686
$630$	0	0
$631$	35.5184	1.41396	0.706982	−	0.707231i	$-0.250056\pi$
$631$	35.5184	1.41396	0.706982	+	0.707231i	$0.250056\pi$
$632$	1.66817	0.0663564
$633$	0	0
$634$	24.9752	0.991890
$635$	−35.3018	−1.40091
$636$	0	0
$637$	0	0
$638$	2.24576	0.0889106
$639$	0	0
$640$	11.1774	0.441827
$641$	−37.1554	−1.46755	−0.733775	−	0.679393i	$-0.762243\pi$
$641$	−37.1554	−1.46755	−0.733775	+	0.679393i	$0.762243\pi$
$642$	0	0
$643$	−39.7694	−1.56835	−0.784176	−	0.620538i	$-0.786914\pi$
$643$	−39.7694	−1.56835	−0.784176	+	0.620538i	$0.786914\pi$
$644$	0	0
$645$	0	0
$646$	−8.27599	−0.325614
$647$	22.9754	0.903256	0.451628	−	0.892206i	$-0.350843\pi$
$647$	22.9754	0.903256	0.451628	+	0.892206i	$0.350843\pi$
$648$	0	0
$649$	1.93700	0.0760340
$650$	−5.77958	−0.226694
$651$	0	0
$652$	12.5435	0.491241
$653$	−18.1757	−0.711268	−0.355634	−	0.934625i	$-0.615735\pi$
$653$	−18.1757	−0.711268	−0.355634	+	0.934625i	$0.615735\pi$
$654$	0	0
$655$	49.5530	1.93619
$656$	6.10936	0.238531
$657$	0	0
$658$	0	0
$659$	−14.1044	−0.549431	−0.274716	−	0.961526i	$-0.588584\pi$
$659$	−14.1044	−0.549431	−0.274716	+	0.961526i	$0.588584\pi$
$660$	0	0
$661$	12.8557	0.500027	0.250013	−	0.968242i	$-0.419565\pi$
$661$	12.8557	0.500027	0.250013	+	0.968242i	$0.419565\pi$
$662$	19.1030	0.742459
$663$	0	0
$664$	−46.8590	−1.81848
$665$	0	0
$666$	0	0
$667$	7.38627	0.285997
$668$	−8.01948	−0.310283
$669$	0	0
$670$	16.0177	0.618819
$671$	8.44287	0.325933
$672$	0	0
$673$	−45.6138	−1.75828	−0.879141	−	0.476561i	$-0.841883\pi$
$673$	−45.6138	−1.75828	−0.879141	+	0.476561i	$0.841883\pi$
$674$	7.55278	0.290922
$675$	0	0
$676$	−0.616586	−0.0237148
$677$	10.2469	0.393821	0.196910	−	0.980421i	$-0.436909\pi$
$677$	10.2469	0.393821	0.196910	+	0.980421i	$0.436909\pi$
$678$	0	0
$679$	0	0
$680$	55.7515	2.13797
$681$	0	0
$682$	5.58058	0.213692
$683$	−2.90040	−0.110981	−0.0554904	−	0.998459i	$-0.517672\pi$
$683$	−2.90040	−0.110981	−0.0554904	+	0.998459i	$0.517672\pi$
$684$	0	0
$685$	58.6305	2.24016
$686$	0	0
$687$	0	0
$688$	6.51914	0.248540
$689$	−9.79015	−0.372975
$690$	0	0
$691$	−6.91470	−0.263048	−0.131524	−	0.991313i	$-0.541987\pi$
$691$	−6.91470	−0.263048	−0.131524	+	0.991313i	$0.541987\pi$
$692$	15.9764	0.607330
$693$	0	0
$694$	−17.0323	−0.646536
$695$	37.5117	1.42290
$696$	0	0
$697$	14.7276	0.557847
$698$	−2.05245	−0.0776864
$699$	0	0
$700$	0	0
$701$	26.0973	0.985682	0.492841	−	0.870119i	$-0.335958\pi$
$701$	26.0973	0.985682	0.492841	+	0.870119i	$0.335958\pi$
$702$	0	0
$703$	6.11408	0.230597
$704$	6.73717	0.253916
$705$	0	0
$706$	18.7696	0.706402
$707$	0	0
$708$	0	0
$709$	−3.38472	−0.127116	−0.0635580	−	0.997978i	$-0.520245\pi$
$709$	−3.38472	−0.127116	−0.0635580	+	0.997978i	$0.520245\pi$
$710$	−39.4962	−1.48227
$711$	0	0
$712$	−28.4126	−1.06481
$713$	18.3544	0.687378
$714$	0	0
$715$	−2.43511	−0.0910681
$716$	5.94489	0.222171
$717$	0	0
$718$	−10.3669	−0.386890
$719$	−2.41574	−0.0900918	−0.0450459	−	0.998985i	$-0.514343\pi$
$719$	−2.41574	−0.0900918	−0.0450459	+	0.998985i	$0.514343\pi$
$720$	0	0
$721$	0	0
$722$	−20.5883	−0.766219
$723$	0	0
$724$	−1.80464	−0.0670691
$725$	12.1314	0.450548
$726$	0	0
$727$	−17.0150	−0.631050	−0.315525	−	0.948917i	$-0.602181\pi$
$727$	−17.0150	−0.631050	−0.315525	+	0.948917i	$0.602181\pi$
$728$	0	0
$729$	0	0
$730$	19.1631	0.709258
$731$	15.7154	0.581256
$732$	0	0
$733$	−4.18453	−0.154559	−0.0772795	−	0.997009i	$-0.524623\pi$
$733$	−4.18453	−0.154559	−0.0772795	+	0.997009i	$0.524623\pi$
$734$	23.1803	0.855599
$735$	0	0
$736$	10.0167	0.369221
$737$	3.34505	0.123216
$738$	0	0
$739$	−28.0794	−1.03292	−0.516458	−	0.856312i	$-0.672750\pi$
$739$	−28.0794	−1.03292	−0.516458	+	0.856312i	$0.672750\pi$
$740$	−9.70567	−0.356788
$741$	0	0
$742$	0	0
$743$	26.5210	0.972961	0.486481	−	0.873691i	$-0.338280\pi$
$743$	26.5210	0.972961	0.486481	+	0.873691i	$0.338280\pi$
$744$	0	0
$745$	−35.8167	−1.31222
$746$	−0.430240	−0.0157522
$747$	0	0
$748$	2.74357	0.100315
$749$	0	0
$750$	0	0
$751$	−23.7162	−0.865418	−0.432709	−	0.901534i	$-0.642442\pi$
$751$	−23.7162	−0.865418	−0.432709	+	0.901534i	$0.642442\pi$
$752$	−12.8204	−0.467510
$753$	0	0
$754$	−2.90379	−0.105750
$755$	−11.6263	−0.423125
$756$	0	0
$757$	−52.3661	−1.90328	−0.951639	−	0.307219i	$-0.900602\pi$
$757$	−52.3661	−1.90328	−0.951639	+	0.307219i	$0.900602\pi$
$758$	8.69454	0.315800
$759$	0	0
$760$	−11.8509	−0.429877
$761$	23.8695	0.865268	0.432634	−	0.901570i	$-0.357584\pi$
$761$	23.8695	0.865268	0.432634	+	0.901570i	$0.357584\pi$
$762$	0	0
$763$	0	0
$764$	−9.94949	−0.359960
$765$	0	0
$766$	−3.94450	−0.142521
$767$	−2.50456	−0.0904345
$768$	0	0
$769$	−41.7599	−1.50590	−0.752950	−	0.658077i	$-0.771370\pi$
$769$	−41.7599	−1.50590	−0.752950	+	0.658077i	$0.771370\pi$
$770$	0	0
$771$	0	0
$772$	0.0532617	0.00191693
$773$	2.97178	0.106888	0.0534438	−	0.998571i	$-0.482980\pi$
$773$	2.97178	0.106888	0.0534438	+	0.998571i	$0.482980\pi$
$774$	0	0
$775$	30.1457	1.08287
$776$	−3.88672	−0.139525
$777$	0	0
$778$	3.05037	0.109361
$779$	−3.13059	−0.112165
$780$	0	0
$781$	−8.24815	−0.295142
$782$	−20.2459	−0.723990
$783$	0	0
$784$	0	0
$785$	−3.52850	−0.125937
$786$	0	0
$787$	48.6142	1.73291	0.866454	−	0.499256i	$-0.166393\pi$
$787$	48.6142	1.73291	0.866454	+	0.499256i	$0.166393\pi$
$788$	9.28509	0.330768
$789$	0	0
$790$	−2.00737	−0.0714189
$791$	0	0
$792$	0	0
$793$	−10.9167	−0.387664
$794$	39.3921	1.39797
$795$	0	0
$796$	9.52418	0.337576
$797$	25.6470	0.908463	0.454232	−	0.890884i	$-0.349914\pi$
$797$	25.6470	0.908463	0.454232	+	0.890884i	$0.349914\pi$
$798$	0	0
$799$	−30.9055	−1.09336
$800$	16.4517	0.581655
$801$	0	0
$802$	−46.4777	−1.64118
$803$	4.00191	0.141224
$804$	0	0
$805$	0	0
$806$	−7.21575	−0.254164
$807$	0	0
$808$	−1.86332	−0.0655514
$809$	11.9205	0.419101	0.209550	−	0.977798i	$-0.432800\pi$
$809$	11.9205	0.419101	0.209550	+	0.977798i	$0.432800\pi$
$810$	0	0
$811$	10.5564	0.370685	0.185343	−	0.982674i	$-0.440661\pi$
$811$	10.5564	0.370685	0.185343	+	0.982674i	$0.440661\pi$
$812$	0	0
$813$	0	0
$814$	4.54764	0.159395
$815$	−64.0540	−2.24371
$816$	0	0
$817$	−3.34057	−0.116872
$818$	12.9144	0.451543
$819$	0	0
$820$	4.96960	0.173546
$821$	46.2139	1.61288	0.806438	−	0.591319i	$-0.201393\pi$
$821$	46.2139	1.61288	0.806438	+	0.591319i	$0.201393\pi$
$822$	0	0
$823$	41.2171	1.43674	0.718368	−	0.695663i	$-0.244889\pi$
$823$	41.2171	1.43674	0.718368	+	0.695663i	$0.244889\pi$
$824$	23.5375	0.819969
$825$	0	0
$826$	0	0
$827$	−5.94430	−0.206704	−0.103352	−	0.994645i	$-0.532957\pi$
$827$	−5.94430	−0.206704	−0.103352	+	0.994645i	$0.532957\pi$
$828$	0	0
$829$	−34.8106	−1.20902	−0.604511	−	0.796596i	$-0.706632\pi$
$829$	−34.8106	−1.20902	−0.604511	+	0.796596i	$0.706632\pi$
$830$	56.3869	1.95722
$831$	0	0
$832$	−8.71122	−0.302007
$833$	0	0
$834$	0	0
$835$	40.9519	1.41720
$836$	−0.583191	−0.0201701
$837$	0	0
$838$	37.1229	1.28239
$839$	10.7896	0.372500	0.186250	−	0.982502i	$-0.440367\pi$
$839$	10.7896	0.372500	0.186250	+	0.982502i	$0.440367\pi$
$840$	0	0
$841$	−22.9049	−0.789825
$842$	−20.8250	−0.717676
$843$	0	0
$844$	15.4492	0.531784
$845$	3.14862	0.108316
$846$	0	0
$847$	0	0
$848$	−23.3657	−0.802381
$849$	0	0
$850$	−33.2523	−1.14054
$851$	14.9571	0.512722
$852$	0	0
$853$	−39.1053	−1.33894	−0.669470	−	0.742839i	$-0.733479\pi$
$853$	−39.1053	−1.33894	−0.669470	+	0.742839i	$0.733479\pi$
$854$	0	0
$855$	0	0
$856$	14.8637	0.508030
$857$	−33.3654	−1.13974	−0.569870	−	0.821735i	$-0.693006\pi$
$857$	−33.3654	−1.13974	−0.569870	+	0.821735i	$0.693006\pi$
$858$	0	0
$859$	29.4914	1.00623	0.503116	−	0.864219i	$-0.332187\pi$
$859$	29.4914	1.00623	0.503116	+	0.864219i	$0.332187\pi$
$860$	5.30293	0.180828
$861$	0	0
$862$	25.2664	0.860578
$863$	24.6224	0.838157	0.419079	−	0.907950i	$-0.362353\pi$
$863$	24.6224	0.838157	0.419079	+	0.907950i	$0.362353\pi$
$864$	0	0
$865$	−81.5841	−2.77394
$866$	−40.8372	−1.38771
$867$	0	0
$868$	0	0
$869$	−0.419207	−0.0142206
$870$	0	0
$871$	−4.32518	−0.146553
$872$	−22.2538	−0.753609
$873$	0	0
$874$	4.30359	0.145571
$875$	0	0
$876$	0	0
$877$	−4.62556	−0.156194	−0.0780971	−	0.996946i	$-0.524884\pi$
$877$	−4.62556	−0.156194	−0.0780971	+	0.996946i	$0.524884\pi$
$878$	−0.856858	−0.0289176
$879$	0	0
$880$	−5.81177	−0.195915
$881$	−42.1720	−1.42081	−0.710405	−	0.703793i	$-0.751488\pi$
$881$	−42.1720	−1.42081	−0.710405	+	0.703793i	$0.751488\pi$
$882$	0	0
$883$	20.7992	0.699950	0.349975	−	0.936759i	$-0.386190\pi$
$883$	20.7992	0.699950	0.349975	+	0.936759i	$0.386190\pi$
$884$	−3.54746	−0.119314
$885$	0	0
$886$	−0.984810	−0.0330853
$887$	34.4889	1.15802	0.579012	−	0.815319i	$-0.303438\pi$
$887$	34.4889	1.15802	0.579012	+	0.815319i	$0.303438\pi$
$888$	0	0
$889$	0	0
$890$	34.1898	1.14605
$891$	0	0
$892$	−12.7411	−0.426604
$893$	6.56947	0.219839
$894$	0	0
$895$	−30.3579	−1.01475
$896$	0	0
$897$	0	0
$898$	−31.0880	−1.03742
$899$	15.1459	0.505144
$900$	0	0
$901$	−56.3267	−1.87651
$902$	−2.32853	−0.0775315
$903$	0	0
$904$	−28.2946	−0.941065
$905$	9.21550	0.306334
$906$	0	0
$907$	13.3619	0.443675	0.221838	−	0.975084i	$-0.428795\pi$
$907$	13.3619	0.443675	0.221838	+	0.975084i	$0.428795\pi$
$908$	−0.605720	−0.0201015
$909$	0	0
$910$	0	0
$911$	−5.29058	−0.175285	−0.0876424	−	0.996152i	$-0.527933\pi$
$911$	−5.29058	−0.175285	−0.0876424	+	0.996152i	$0.527933\pi$
$912$	0	0
$913$	11.7755	0.389713
$914$	−2.82843	−0.0935562
$915$	0	0
$916$	−8.80129	−0.290803
$917$	0	0
$918$	0	0
$919$	−18.8306	−0.621164	−0.310582	−	0.950547i	$-0.600524\pi$
$919$	−18.8306	−0.621164	−0.310582	+	0.950547i	$0.600524\pi$
$920$	−28.9913	−0.955814
$921$	0	0
$922$	−26.1939	−0.862649
$923$	10.6649	0.351041
$924$	0	0
$925$	24.5659	0.807721
$926$	38.0009	1.24879
$927$	0	0
$928$	8.26571	0.271335
$929$	−44.9537	−1.47488	−0.737442	−	0.675411i	$-0.763966\pi$
$929$	−44.9537	−1.47488	−0.737442	+	0.675411i	$0.763966\pi$
$930$	0	0
$931$	0	0
$932$	0.860348	0.0281816
$933$	0	0
$934$	−15.1427	−0.495484
$935$	−14.0102	−0.458182
$936$	0	0
$937$	−37.9272	−1.23903	−0.619514	−	0.784986i	$-0.712670\pi$
$937$	−37.9272	−1.23903	−0.619514	+	0.784986i	$0.712670\pi$
$938$	0	0
$939$	0	0
$940$	−10.4286	−0.340143
$941$	35.7869	1.16662	0.583310	−	0.812250i	$-0.301757\pi$
$941$	35.7869	1.16662	0.583310	+	0.812250i	$0.301757\pi$
$942$	0	0
$943$	−7.65848	−0.249394
$944$	−5.97752	−0.194552
$945$	0	0
$946$	−2.48471	−0.0807849
$947$	14.7443	0.479125	0.239563	−	0.970881i	$-0.422996\pi$
$947$	14.7443	0.479125	0.239563	+	0.970881i	$0.422996\pi$
$948$	0	0
$949$	−5.17450	−0.167971
$950$	7.06831	0.229326
$951$	0	0
$952$	0	0
$953$	−0.649669	−0.0210448	−0.0105224	−	0.999945i	$-0.503349\pi$
$953$	−0.649669	−0.0210448	−0.0105224	+	0.999945i	$0.503349\pi$
$954$	0	0
$955$	50.8076	1.64409
$956$	1.57153	0.0508268
$957$	0	0
$958$	12.3992	0.400600
$959$	0	0
$960$	0	0
$961$	6.63659	0.214083
$962$	−5.88014	−0.189583
$963$	0	0
$964$	18.2969	0.589305
$965$	−0.271983	−0.00875545
$966$	0	0
$967$	−13.0802	−0.420632	−0.210316	−	0.977633i	$-0.567449\pi$
$967$	−13.0802	−0.420632	−0.210316	+	0.977633i	$0.567449\pi$
$968$	32.0127	1.02893
$969$	0	0
$970$	4.67702	0.150170
$971$	−29.1203	−0.934515	−0.467258	−	0.884121i	$-0.654758\pi$
$971$	−29.1203	−0.934515	−0.467258	+	0.884121i	$0.654758\pi$
$972$	0	0
$973$	0	0
$974$	44.9792	1.44123
$975$	0	0
$976$	−26.0544	−0.833980
$977$	18.5000	0.591868	0.295934	−	0.955208i	$-0.404369\pi$
$977$	18.5000	0.591868	0.295934	+	0.955208i	$0.404369\pi$
$978$	0	0
$979$	7.14000	0.228195
$980$	0	0
$981$	0	0
$982$	18.1475	0.579111
$983$	−11.0462	−0.352318	−0.176159	−	0.984362i	$-0.556367\pi$
$983$	−11.0462	−0.352318	−0.176159	+	0.984362i	$0.556367\pi$
$984$	0	0
$985$	−47.4148	−1.51076
$986$	−16.7067	−0.532049
$987$	0	0
$988$	0.754072	0.0239902
$989$	−8.17216	−0.259860
$990$	0	0
$991$	−40.4757	−1.28575	−0.642877	−	0.765969i	$-0.722259\pi$
$991$	−40.4757	−1.28575	−0.642877	+	0.765969i	$0.722259\pi$
$992$	20.5398	0.652138
$993$	0	0
$994$	0	0
$995$	−48.6357	−1.54185
$996$	0	0
$997$	12.0623	0.382018	0.191009	−	0.981588i	$-0.438824\pi$
$997$	12.0623	0.382018	0.191009	+	0.981588i	$0.438824\pi$
$998$	−45.6610	−1.44537
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.bu.1.5		6
3.2	odd	2		637.2.a.m.1.2	✓	6
7.6	odd	2		5733.2.a.br.1.5		6
21.2	odd	6		637.2.e.o.508.5		12
21.5	even	6		637.2.e.n.508.5		12
21.11	odd	6		637.2.e.o.79.5		12
21.17	even	6		637.2.e.n.79.5		12
21.20	even	2		637.2.a.n.1.2	yes	6
39.38	odd	2		8281.2.a.cc.1.5		6
273.272	even	2		8281.2.a.cd.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.a.m.1.2	✓	6	3.2	odd	2
637.2.a.n.1.2	yes	6	21.20	even	2
637.2.e.n.79.5		12	21.17	even	6
637.2.e.n.508.5		12	21.5	even	6
637.2.e.o.79.5		12	21.11	odd	6
637.2.e.o.508.5		12	21.2	odd	6
5733.2.a.br.1.5		6	7.6	odd	2
5733.2.a.bu.1.5		6	1.1	even	1	trivial
8281.2.a.cc.1.5		6	39.38	odd	2
8281.2.a.cd.1.5		6	273.272	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5733.a (trivial)

\(f(q)\)	\(=\)	\(q+1.17619 q^{2} -0.616586 q^{4} +3.14862 q^{5} -3.07759 q^{8} +O(q^{10})\)	\(q+1.17619 q^{2} -0.616586 q^{4} +3.14862 q^{5} -3.07759 q^{8} +3.70337 q^{10} +0.773390 q^{11} -1.00000 q^{13} -2.38665 q^{16} -5.75340 q^{17} +1.22298 q^{19} -1.94140 q^{20} +0.909650 q^{22} +2.99182 q^{23} +4.91383 q^{25} -1.17619 q^{26} +2.46882 q^{29} +6.13487 q^{31} +3.34804 q^{32} -6.76707 q^{34} +4.99933 q^{37} +1.43845 q^{38} -9.69018 q^{40} -2.55981 q^{41} -2.73150 q^{43} -0.476861 q^{44} +3.51894 q^{46} +5.37169 q^{47} +5.77958 q^{50} +0.616586 q^{52} +9.79015 q^{53} +2.43511 q^{55} +2.90379 q^{58} +2.50456 q^{59} +10.9167 q^{61} +7.21575 q^{62} +8.71122 q^{64} -3.14862 q^{65} +4.32518 q^{67} +3.54746 q^{68} -10.6649 q^{71} +5.17450 q^{73} +5.88014 q^{74} -0.754072 q^{76} -0.542038 q^{79} -7.51467 q^{80} -3.01081 q^{82} +15.2259 q^{83} -18.1153 q^{85} -3.21275 q^{86} -2.38018 q^{88} +9.23208 q^{89} -1.84471 q^{92} +6.31811 q^{94} +3.85070 q^{95} +1.26291 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{4} + 6 q^{5}+O(q^{10}) \) 6 * q + 4 * q^4 + 6 * q^5	\( 6 q + 4 q^{4} + 6 q^{5} - 4 q^{10} - 4 q^{11} - 6 q^{13} + 16 q^{17} - 2 q^{19} + 16 q^{20} - 12 q^{22} + 6 q^{23} - 4 q^{25} + 6 q^{29} - 6 q^{31} + 20 q^{32} + 8 q^{38} - 4 q^{40} - 8 q^{41} + 2 q^{43} + 4 q^{44} + 8 q^{46} + 30 q^{47} - 8 q^{50} - 4 q^{52} + 14 q^{53} + 8 q^{55} - 8 q^{58} + 24 q^{59} + 28 q^{62} - 20 q^{64} - 6 q^{65} + 16 q^{67} + 28 q^{68} - 8 q^{71} + 6 q^{73} + 12 q^{74} + 16 q^{76} - 22 q^{79} - 28 q^{80} + 40 q^{82} + 50 q^{83} - 8 q^{85} + 16 q^{86} - 44 q^{88} + 26 q^{89} - 20 q^{92} + 32 q^{94} + 6 q^{95} + 14 q^{97}+O(q^{100}) \) 6 * q + 4 * q^4 + 6 * q^5 - 4 * q^10 - 4 * q^11 - 6 * q^13 + 16 * q^17 - 2 * q^19 + 16 * q^20 - 12 * q^22 + 6 * q^23 - 4 * q^25 + 6 * q^29 - 6 * q^31 + 20 * q^32 + 8 * q^38 - 4 * q^40 - 8 * q^41 + 2 * q^43 + 4 * q^44 + 8 * q^46 + 30 * q^47 - 8 * q^50 - 4 * q^52 + 14 * q^53 + 8 * q^55 - 8 * q^58 + 24 * q^59 + 28 * q^62 - 20 * q^64 - 6 * q^65 + 16 * q^67 + 28 * q^68 - 8 * q^71 + 6 * q^73 + 12 * q^74 + 16 * q^76 - 22 * q^79 - 28 * q^80 + 40 * q^82 + 50 * q^83 - 8 * q^85 + 16 * q^86 - 44 * q^88 + 26 * q^89 - 20 * q^92 + 32 * q^94 + 6 * q^95 + 14 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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