LMFDB - Embedded newform 5175.2.a.by.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5175,2,Mod(1,5175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5175.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5175 = 3^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5175.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,10,0,0,-6,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.3225830460$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.98838128.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} - x^{3} + 16x^{2} + 5x - 1 $ x^6 - x^5 - 10x^4 - x^3 + 16x^2 + 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1035)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.139251$ of defining polynomial
Character	$\chi$	$=$	5175.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.70812 q^{2} +5.33392 q^{4} -1.21406 q^{7} +9.02867 q^{8} -2.12841 q^{11} +4.32055 q^{13} -3.28783 q^{14} +13.7829 q^{16} -1.98662 q^{17} +5.28783 q^{19} -5.76401 q^{22} +1.00000 q^{23} +11.7006 q^{26} -6.47571 q^{28} +7.10648 q^{29} -2.33392 q^{31} +19.2684 q^{32} -5.38002 q^{34} -5.81865 q^{37} +14.3201 q^{38} +11.5822 q^{41} -1.78112 q^{43} -11.3528 q^{44} +2.70812 q^{46} +7.89242 q^{47} -5.52605 q^{49} +23.0455 q^{52} -4.89046 q^{53} -10.9614 q^{56} +19.2452 q^{58} -0.118364 q^{59} -8.79626 q^{61} -6.32055 q^{62} +24.6154 q^{64} +1.16946 q^{67} -10.5965 q^{68} +14.3135 q^{71} -8.32055 q^{73} -15.7576 q^{74} +28.2049 q^{76} +2.58403 q^{77} +11.3895 q^{79} +31.3660 q^{82} -8.91130 q^{83} -4.82348 q^{86} -19.2167 q^{88} -2.07508 q^{89} -5.24541 q^{91} +5.33392 q^{92} +21.3736 q^{94} +18.9612 q^{97} -14.9652 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 10 q^{4} - 6 q^{7} - 4 q^{11} - 12 q^{13} + 4 q^{14} + 14 q^{16} + 4 q^{17} + 8 q^{19} - 8 q^{22} + 6 q^{23} + 12 q^{26} - 24 q^{28} + 6 q^{29} + 8 q^{31} + 20 q^{32} - 12 q^{34} - 22 q^{37} + 36 q^{38}+ \cdots - 4 q^{98}+O(q^{100}) $ 6 * q + 10 * q^4 - 6 * q^7 - 4 * q^11 - 12 * q^13 + 4 * q^14 + 14 * q^16 + 4 * q^17 + 8 * q^19 - 8 * q^22 + 6 * q^23 + 12 * q^26 - 24 * q^28 + 6 * q^29 + 8 * q^31 + 20 * q^32 - 12 * q^34 - 22 * q^37 + 36 * q^38 + 18 * q^41 - 8 * q^43 - 4 * q^44 + 12 * q^47 + 18 * q^49 + 4 * q^53 - 4 * q^56 + 16 * q^58 + 6 * q^59 + 14 * q^64 - 10 * q^67 + 28 * q^68 + 22 * q^71 - 12 * q^73 + 20 * q^74 + 16 * q^76 - 4 * q^77 + 4 * q^79 + 12 * q^82 + 24 * q^83 - 20 * q^86 + 8 * q^88 + 8 * q^89 + 4 * q^91 + 10 * q^92 + 20 * q^94 - 12 * q^97 - 4 * q^98

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 $

$2$	2.70812	1.91493	0.957466	−	0.288548i	$-0.0931724\pi$
$2$	2.70812	1.91493	0.957466	+	0.288548i	$0.0931724\pi$
$3$	0	0
$3$	0	0
$4$	5.33392	2.66696
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.21406	−0.458872	−0.229436	−	0.973324i	$-0.573688\pi$
$7$	−1.21406	−0.458872	−0.229436	+	0.973324i	$0.573688\pi$
$8$	9.02867	3.19212
$9$	0	0
$10$	0	0
$11$	−2.12841	−0.641741	−0.320871	−	0.947123i	$-0.603975\pi$
$11$	−2.12841	−0.641741	−0.320871	+	0.947123i	$0.603975\pi$
$12$	0	0
$13$	4.32055	1.19830	0.599152	−	0.800635i	$-0.295505\pi$
$13$	4.32055	1.19830	0.599152	+	0.800635i	$0.295505\pi$
$14$	−3.28783	−0.878709
$15$	0	0
$16$	13.7829	3.44572
$17$	−1.98662	−0.481827	−0.240913	−	0.970547i	$-0.577447\pi$
$17$	−1.98662	−0.481827	−0.240913	+	0.970547i	$0.577447\pi$
$18$	0	0
$19$	5.28783	1.21311	0.606556	−	0.795041i	$-0.292551\pi$
$19$	5.28783	1.21311	0.606556	+	0.795041i	$0.292551\pi$
$20$	0	0
$21$	0	0
$22$	−5.76401	−1.22889
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	11.7006	2.29467
$27$	0	0
$28$	−6.47571	−1.22379
$29$	7.10648	1.31964	0.659820	−	0.751423i	$-0.270632\pi$
$29$	7.10648	1.31964	0.659820	+	0.751423i	$0.270632\pi$
$30$	0	0
$31$	−2.33392	−0.419185	−0.209592	−	0.977789i	$-0.567214\pi$
$31$	−2.33392	−0.419185	−0.209592	+	0.977789i	$0.567214\pi$
$32$	19.2684	3.40620
$33$	0	0
$34$	−5.38002	−0.922665
$35$	0	0
$36$	0	0
$37$	−5.81865	−0.956581	−0.478290	−	0.878202i	$-0.658743\pi$
$37$	−5.81865	−0.956581	−0.478290	+	0.878202i	$0.658743\pi$
$38$	14.3201	2.32302
$39$	0	0
$40$	0	0
$41$	11.5822	1.80884	0.904418	−	0.426648i	$-0.140306\pi$
$41$	11.5822	1.80884	0.904418	+	0.426648i	$0.140306\pi$
$42$	0	0
$43$	−1.78112	−0.271618	−0.135809	−	0.990735i	$-0.543363\pi$
$43$	−1.78112	−0.271618	−0.135809	+	0.990735i	$0.543363\pi$
$44$	−11.3528	−1.71150
$45$	0	0
$46$	2.70812	0.399291
$47$	7.89242	1.15123	0.575614	−	0.817722i	$-0.304763\pi$
$47$	7.89242	1.15123	0.575614	+	0.817722i	$0.304763\pi$
$48$	0	0
$49$	−5.52605	−0.789436
$50$	0	0
$51$	0	0
$52$	23.0455	3.19583
$53$	−4.89046	−0.671757	−0.335878	−	0.941905i	$-0.609033\pi$
$53$	−4.89046	−0.671757	−0.335878	+	0.941905i	$0.609033\pi$
$54$	0	0
$55$	0	0
$56$	−10.9614	−1.46477
$57$	0	0
$58$	19.2452	2.52702
$59$	−0.118364	−0.0154097	−0.00770486	−	0.999970i	$-0.502453\pi$
$59$	−0.118364	−0.0154097	−0.00770486	+	0.999970i	$0.502453\pi$
$60$	0	0
$61$	−8.79626	−1.12625	−0.563123	−	0.826373i	$-0.690400\pi$
$61$	−8.79626	−1.12625	−0.563123	+	0.826373i	$0.690400\pi$
$62$	−6.32055	−0.802710
$63$	0	0
$64$	24.6154	3.07692
$65$	0	0
$66$	0	0
$67$	1.16946	0.142873	0.0714364	−	0.997445i	$-0.477242\pi$
$67$	1.16946	0.142873	0.0714364	+	0.997445i	$0.477242\pi$
$68$	−10.5965	−1.28501
$69$	0	0
$70$	0	0
$71$	14.3135	1.69870	0.849349	−	0.527831i	$-0.176995\pi$
$71$	14.3135	1.69870	0.849349	+	0.527831i	$0.176995\pi$
$72$	0	0
$73$	−8.32055	−0.973846	−0.486923	−	0.873445i	$-0.661881\pi$
$73$	−8.32055	−0.973846	−0.486923	+	0.873445i	$0.661881\pi$
$74$	−15.7576	−1.83179
$75$	0	0
$76$	28.2049	3.23532
$77$	2.58403	0.294477
$78$	0	0
$79$	11.3895	1.28142	0.640709	−	0.767784i	$-0.278641\pi$
$79$	11.3895	1.28142	0.640709	+	0.767784i	$0.278641\pi$
$80$	0	0
$81$	0	0
$82$	31.3660	3.46379
$83$	−8.91130	−0.978142	−0.489071	−	0.872244i	$-0.662664\pi$
$83$	−8.91130	−0.978142	−0.489071	+	0.872244i	$0.662664\pi$
$84$	0	0
$85$	0	0
$86$	−4.82348	−0.520129
$87$	0	0
$88$	−19.2167	−2.04851
$89$	−2.07508	−0.219958	−0.109979	−	0.993934i	$-0.535078\pi$
$89$	−2.07508	−0.219958	−0.109979	+	0.993934i	$0.535078\pi$
$90$	0	0
$91$	−5.24541	−0.549868
$92$	5.33392	0.556100
$93$	0	0
$94$	21.3736	2.20452
$95$	0	0
$96$	0	0
$97$	18.9612	1.92522	0.962608	−	0.270899i	$-0.0873211\pi$
$97$	18.9612	1.92522	0.962608	+	0.270899i	$0.0873211\pi$
$98$	−14.9652	−1.51172
$99$	0	0
$100$	0	0
$101$	−7.32537	−0.728901	−0.364451	−	0.931223i	$-0.618743\pi$
$101$	−7.32537	−0.728901	−0.364451	+	0.931223i	$0.618743\pi$
$102$	0	0
$103$	−17.3082	−1.70543	−0.852714	−	0.522378i	$-0.825045\pi$
$103$	−17.3082	−1.70543	−0.852714	+	0.522378i	$0.825045\pi$
$104$	39.0088	3.82512
$105$	0	0
$106$	−13.2440	−1.28637
$107$	6.63363	0.641297	0.320649	−	0.947198i	$-0.396099\pi$
$107$	6.63363	0.641297	0.320649	+	0.947198i	$0.396099\pi$
$108$	0	0
$109$	19.6555	1.88266	0.941328	−	0.337494i	$-0.109579\pi$
$109$	19.6555	1.88266	0.941328	+	0.337494i	$0.109579\pi$
$110$	0	0
$111$	0	0
$112$	−16.7333	−1.58115
$113$	1.17497	0.110532	0.0552660	−	0.998472i	$-0.482399\pi$
$113$	1.17497	0.110532	0.0552660	+	0.998472i	$0.482399\pi$
$114$	0	0
$115$	0	0
$116$	37.9054	3.51943
$117$	0	0
$118$	−0.320545	−0.0295086
$119$	2.41188	0.221097
$120$	0	0
$121$	−6.46985	−0.588168
$122$	−23.8213	−2.15668
$123$	0	0
$124$	−12.4490	−1.11795
$125$	0	0
$126$	0	0
$127$	−1.72778	−0.153316	−0.0766578	−	0.997057i	$-0.524425\pi$
$127$	−1.72778	−0.153316	−0.0766578	+	0.997057i	$0.524425\pi$
$128$	28.1246	2.48589
$129$	0	0
$130$	0	0
$131$	19.3715	1.69249	0.846246	−	0.532793i	$-0.178857\pi$
$131$	19.3715	1.69249	0.846246	+	0.532793i	$0.178857\pi$
$132$	0	0
$133$	−6.41975	−0.556663
$134$	3.16705	0.273592
$135$	0	0
$136$	−17.9366	−1.53805
$137$	−21.3565	−1.82461	−0.912305	−	0.409510i	$-0.865700\pi$
$137$	−21.3565	−1.82461	−0.912305	+	0.409510i	$0.865700\pi$
$138$	0	0
$139$	−23.4240	−1.98680	−0.993398	−	0.114721i	$-0.963402\pi$
$139$	−23.4240	−1.98680	−0.993398	+	0.114721i	$0.963402\pi$
$140$	0	0
$141$	0	0
$142$	38.7627	3.25289
$143$	−9.19591	−0.769001
$144$	0	0
$145$	0	0
$146$	−22.5330	−1.86485
$147$	0	0
$148$	−31.0363	−2.55116
$149$	11.5603	0.947058	0.473529	−	0.880778i	$-0.342980\pi$
$149$	11.5603	0.947058	0.473529	+	0.880778i	$0.342980\pi$
$150$	0	0
$151$	5.42807	0.441730	0.220865	−	0.975304i	$-0.429112\pi$
$151$	5.42807	0.441730	0.220865	+	0.975304i	$0.429112\pi$
$152$	47.7420	3.87239
$153$	0	0
$154$	6.99786	0.563904
$155$	0	0
$156$	0	0
$157$	−8.43891	−0.673498	−0.336749	−	0.941594i	$-0.609327\pi$
$157$	−8.43891	−0.673498	−0.336749	+	0.941594i	$0.609327\pi$
$158$	30.8441	2.45383
$159$	0	0
$160$	0	0
$161$	−1.21406	−0.0956815
$162$	0	0
$163$	−17.3089	−1.35574	−0.677870	−	0.735182i	$-0.737097\pi$
$163$	−17.3089	−1.35574	−0.677870	+	0.735182i	$0.737097\pi$
$164$	61.7785	4.82409
$165$	0	0
$166$	−24.1329	−1.87307
$167$	3.71813	0.287718	0.143859	−	0.989598i	$-0.454049\pi$
$167$	3.71813	0.287718	0.143859	+	0.989598i	$0.454049\pi$
$168$	0	0
$169$	5.66711	0.435932
$170$	0	0
$171$	0	0
$172$	−9.50033	−0.724393
$173$	−4.82348	−0.366722	−0.183361	−	0.983046i	$-0.558698\pi$
$173$	−4.82348	−0.366722	−0.183361	+	0.983046i	$0.558698\pi$
$174$	0	0
$175$	0	0
$176$	−29.3357	−2.21126
$177$	0	0
$178$	−5.61956	−0.421204
$179$	−16.7043	−1.24854	−0.624271	−	0.781208i	$-0.714604\pi$
$179$	−16.7043	−1.24854	−0.624271	+	0.781208i	$0.714604\pi$
$180$	0	0
$181$	12.4764	0.927367	0.463684	−	0.886001i	$-0.346527\pi$
$181$	12.4764	0.927367	0.463684	+	0.886001i	$0.346527\pi$
$182$	−14.2052	−1.05296
$183$	0	0
$184$	9.02867	0.665602
$185$	0	0
$186$	0	0
$187$	4.22836	0.309208
$188$	42.0976	3.07028
$189$	0	0
$190$	0	0
$191$	−21.6287	−1.56500	−0.782501	−	0.622650i	$-0.786056\pi$
$191$	−21.6287	−1.56500	−0.782501	+	0.622650i	$0.786056\pi$
$192$	0	0
$193$	−10.6678	−0.767888	−0.383944	−	0.923356i	$-0.625434\pi$
$193$	−10.6678	−0.767888	−0.383944	+	0.923356i	$0.625434\pi$
$194$	51.3492	3.68665
$195$	0	0
$196$	−29.4755	−2.10540
$197$	−1.49132	−0.106252	−0.0531261	−	0.998588i	$-0.516919\pi$
$197$	−1.49132	−0.106252	−0.0531261	+	0.998588i	$0.516919\pi$
$198$	0	0
$199$	−8.84059	−0.626693	−0.313346	−	0.949639i	$-0.601450\pi$
$199$	−8.84059	−0.626693	−0.313346	+	0.949639i	$0.601450\pi$
$200$	0	0
$201$	0	0
$202$	−19.8380	−1.39580
$203$	−8.62771	−0.605547
$204$	0	0
$205$	0	0
$206$	−46.8727	−3.26578
$207$	0	0
$208$	59.5496	4.12902
$209$	−11.2547	−0.778503
$210$	0	0
$211$	−12.2722	−0.844854	−0.422427	−	0.906397i	$-0.638822\pi$
$211$	−12.2722	−0.844854	−0.422427	+	0.906397i	$0.638822\pi$
$212$	−26.0853	−1.79155
$213$	0	0
$214$	17.9647	1.22804
$215$	0	0
$216$	0	0
$217$	2.83353	0.192352
$218$	53.2295	3.60516
$219$	0	0
$220$	0	0
$221$	−8.58329	−0.577375
$222$	0	0
$223$	3.64396	0.244018	0.122009	−	0.992529i	$-0.461066\pi$
$223$	3.64396	0.244018	0.122009	+	0.992529i	$0.461066\pi$
$224$	−23.3930	−1.56301
$225$	0	0
$226$	3.18197	0.211661
$227$	14.4140	0.956691	0.478346	−	0.878172i	$-0.341237\pi$
$227$	14.4140	0.956691	0.478346	+	0.878172i	$0.341237\pi$
$228$	0	0
$229$	4.86017	0.321169	0.160584	−	0.987022i	$-0.448662\pi$
$229$	4.86017	0.321169	0.160584	+	0.987022i	$0.448662\pi$
$230$	0	0
$231$	0	0
$232$	64.1621	4.21245
$233$	11.0439	0.723510	0.361755	−	0.932273i	$-0.382178\pi$
$233$	11.0439	0.723510	0.361755	+	0.932273i	$0.382178\pi$
$234$	0	0
$235$	0	0
$236$	−0.631346	−0.0410971
$237$	0	0
$238$	6.53167	0.423386
$239$	−23.5741	−1.52488	−0.762441	−	0.647058i	$-0.775999\pi$
$239$	−23.5741	−1.52488	−0.762441	+	0.647058i	$0.775999\pi$
$240$	0	0
$241$	14.9609	0.963716	0.481858	−	0.876249i	$-0.339962\pi$
$241$	14.9609	0.963716	0.481858	+	0.876249i	$0.339962\pi$
$242$	−17.5211	−1.12630
$243$	0	0
$244$	−46.9186	−3.00365
$245$	0	0
$246$	0	0
$247$	22.8463	1.45368
$248$	−21.0722	−1.33809
$249$	0	0
$250$	0	0
$251$	−4.41102	−0.278421	−0.139210	−	0.990263i	$-0.544456\pi$
$251$	−4.41102	−0.278421	−0.139210	+	0.990263i	$0.544456\pi$
$252$	0	0
$253$	−2.12841	−0.133812
$254$	−4.67903	−0.293589
$255$	0	0
$256$	26.9342	1.68339
$257$	1.18789	0.0740983	0.0370491	−	0.999313i	$-0.488204\pi$
$257$	1.18789	0.0740983	0.0370491	+	0.999313i	$0.488204\pi$
$258$	0	0
$259$	7.06421	0.438949
$260$	0	0
$261$	0	0
$262$	52.4602	3.24100
$263$	−12.9484	−0.798434	−0.399217	−	0.916856i	$-0.630718\pi$
$263$	−12.9484	−0.798434	−0.399217	+	0.916856i	$0.630718\pi$
$264$	0	0
$265$	0	0
$266$	−17.3855	−1.06597
$267$	0	0
$268$	6.23783	0.381036
$269$	−3.72600	−0.227178	−0.113589	−	0.993528i	$-0.536235\pi$
$269$	−3.72600	−0.227178	−0.113589	+	0.993528i	$0.536235\pi$
$270$	0	0
$271$	4.11803	0.250152	0.125076	−	0.992147i	$-0.460082\pi$
$271$	4.11803	0.250152	0.125076	+	0.992147i	$0.460082\pi$
$272$	−27.3814	−1.66024
$273$	0	0
$274$	−57.8361	−3.49400
$275$	0	0
$276$	0	0
$277$	−21.1443	−1.27044	−0.635219	−	0.772332i	$-0.719090\pi$
$277$	−21.1443	−1.27044	−0.635219	+	0.772332i	$0.719090\pi$
$278$	−63.4350	−3.80458
$279$	0	0
$280$	0	0
$281$	27.0717	1.61496	0.807480	−	0.589895i	$-0.200831\pi$
$281$	27.0717	1.61496	0.807480	+	0.589895i	$0.200831\pi$
$282$	0	0
$283$	−3.66229	−0.217700	−0.108850	−	0.994058i	$-0.534717\pi$
$283$	−3.66229	−0.217700	−0.108850	+	0.994058i	$0.534717\pi$
$284$	76.3470	4.53036
$285$	0	0
$286$	−24.9036	−1.47258
$287$	−14.0615	−0.830025
$288$	0	0
$289$	−13.0533	−0.767843
$290$	0	0
$291$	0	0
$292$	−44.3811	−2.59721
$293$	7.11899	0.415896	0.207948	−	0.978140i	$-0.433322\pi$
$293$	7.11899	0.415896	0.207948	+	0.978140i	$0.433322\pi$
$294$	0	0
$295$	0	0
$296$	−52.5347	−3.05352
$297$	0	0
$298$	31.3068	1.81355
$299$	4.32055	0.249864
$300$	0	0
$301$	2.16239	0.124638
$302$	14.6999	0.845882
$303$	0	0
$304$	72.8815	4.18004
$305$	0	0
$306$	0	0
$307$	−23.6763	−1.35128	−0.675640	−	0.737232i	$-0.736133\pi$
$307$	−23.6763	−1.35128	−0.675640	+	0.737232i	$0.736133\pi$
$308$	13.7830	0.785360
$309$	0	0
$310$	0	0
$311$	−14.1324	−0.801376	−0.400688	−	0.916215i	$-0.631229\pi$
$311$	−14.1324	−0.801376	−0.400688	+	0.916215i	$0.631229\pi$
$312$	0	0
$313$	−23.6646	−1.33760	−0.668800	−	0.743442i	$-0.733192\pi$
$313$	−23.6646	−1.33760	−0.668800	+	0.743442i	$0.733192\pi$
$314$	−22.8536	−1.28970
$315$	0	0
$316$	60.7506	3.41749
$317$	21.8223	1.22566	0.612832	−	0.790213i	$-0.290030\pi$
$317$	21.8223	1.22566	0.612832	+	0.790213i	$0.290030\pi$
$318$	0	0
$319$	−15.1255	−0.846868
$320$	0	0
$321$	0	0
$322$	−3.28783	−0.183224
$323$	−10.5049	−0.584509
$324$	0	0
$325$	0	0
$326$	−46.8747	−2.59615
$327$	0	0
$328$	104.572	5.77401
$329$	−9.58189	−0.528267
$330$	0	0
$331$	−24.2825	−1.33469	−0.667343	−	0.744751i	$-0.732568\pi$
$331$	−24.2825	−1.33469	−0.667343	+	0.744751i	$0.732568\pi$
$332$	−47.5322	−2.60867
$333$	0	0
$334$	10.0692	0.550960
$335$	0	0
$336$	0	0
$337$	−15.6463	−0.852309	−0.426155	−	0.904650i	$-0.640132\pi$
$337$	−15.6463	−0.852309	−0.426155	+	0.904650i	$0.640132\pi$
$338$	15.3472	0.834779
$339$	0	0
$340$	0	0
$341$	4.96755	0.269008
$342$	0	0
$343$	15.2074	0.821123
$344$	−16.0811	−0.867035
$345$	0	0
$346$	−13.0626	−0.702248
$347$	−28.8979	−1.55132	−0.775661	−	0.631150i	$-0.782583\pi$
$347$	−28.8979	−1.55132	−0.775661	+	0.631150i	$0.782583\pi$
$348$	0	0
$349$	31.8007	1.70225	0.851125	−	0.524962i	$-0.175921\pi$
$349$	31.8007	1.70225	0.851125	+	0.524962i	$0.175921\pi$
$350$	0	0
$351$	0	0
$352$	−41.0111	−2.18590
$353$	16.7406	0.891011	0.445505	−	0.895279i	$-0.353024\pi$
$353$	16.7406	0.891011	0.445505	+	0.895279i	$0.353024\pi$
$354$	0	0
$355$	0	0
$356$	−11.0683	−0.586619
$357$	0	0
$358$	−45.2374	−2.39087
$359$	−16.6840	−0.880546	−0.440273	−	0.897864i	$-0.645118\pi$
$359$	−16.6840	−0.880546	−0.440273	+	0.897864i	$0.645118\pi$
$360$	0	0
$361$	8.96113	0.471638
$362$	33.7877	1.77584
$363$	0	0
$364$	−27.9786	−1.46648
$365$	0	0
$366$	0	0
$367$	14.5917	0.761682	0.380841	−	0.924641i	$-0.375635\pi$
$367$	14.5917	0.761682	0.380841	+	0.924641i	$0.375635\pi$
$368$	13.7829	0.718482
$369$	0	0
$370$	0	0
$371$	5.93733	0.308251
$372$	0	0
$373$	−27.7827	−1.43853	−0.719266	−	0.694735i	$-0.755522\pi$
$373$	−27.7827	−1.43853	−0.719266	+	0.694735i	$0.755522\pi$
$374$	11.4509	0.592112
$375$	0	0
$376$	71.2580	3.67485
$377$	30.7039	1.58133
$378$	0	0
$379$	7.77228	0.399235	0.199618	−	0.979874i	$-0.436030\pi$
$379$	7.77228	0.399235	0.199618	+	0.979874i	$0.436030\pi$
$380$	0	0
$381$	0	0
$382$	−58.5733	−2.99687
$383$	−2.40809	−0.123048	−0.0615239	−	0.998106i	$-0.519596\pi$
$383$	−2.40809	−0.123048	−0.0615239	+	0.998106i	$0.519596\pi$
$384$	0	0
$385$	0	0
$386$	−28.8898	−1.47045
$387$	0	0
$388$	101.137	5.13447
$389$	4.66485	0.236517	0.118259	−	0.992983i	$-0.462269\pi$
$389$	4.66485	0.236517	0.118259	+	0.992983i	$0.462269\pi$
$390$	0	0
$391$	−1.98662	−0.100468
$392$	−49.8929	−2.51997
$393$	0	0
$394$	−4.03868	−0.203466
$395$	0	0
$396$	0	0
$397$	−5.02754	−0.252325	−0.126163	−	0.992010i	$-0.540266\pi$
$397$	−5.02754	−0.252325	−0.126163	+	0.992010i	$0.540266\pi$
$398$	−23.9414	−1.20007
$399$	0	0
$400$	0	0
$401$	−3.70677	−0.185107	−0.0925537	−	0.995708i	$-0.529503\pi$
$401$	−3.70677	−0.185107	−0.0925537	+	0.995708i	$0.529503\pi$
$402$	0	0
$403$	−10.0838	−0.502311
$404$	−39.0729	−1.94395
$405$	0	0
$406$	−23.3649	−1.15958
$407$	12.3845	0.613877
$408$	0	0
$409$	−16.5945	−0.820544	−0.410272	−	0.911963i	$-0.634566\pi$
$409$	−16.5945	−0.820544	−0.410272	+	0.911963i	$0.634566\pi$
$410$	0	0
$411$	0	0
$412$	−92.3206	−4.54831
$413$	0.143702	0.00707110
$414$	0	0
$415$	0	0
$416$	83.2499	4.08166
$417$	0	0
$418$	−30.4791	−1.49078
$419$	−0.355499	−0.0173673	−0.00868363	−	0.999962i	$-0.502764\pi$
$419$	−0.355499	−0.0173673	−0.00868363	+	0.999962i	$0.502764\pi$
$420$	0	0
$421$	−23.6503	−1.15265	−0.576323	−	0.817222i	$-0.695513\pi$
$421$	−23.6503	−1.15265	−0.576323	+	0.817222i	$0.695513\pi$
$422$	−33.2346	−1.61784
$423$	0	0
$424$	−44.1543	−2.14432
$425$	0	0
$426$	0	0
$427$	10.6792	0.516803
$428$	35.3833	1.71032
$429$	0	0
$430$	0	0
$431$	4.97648	0.239709	0.119854	−	0.992791i	$-0.461757\pi$
$431$	4.97648	0.239709	0.119854	+	0.992791i	$0.461757\pi$
$432$	0	0
$433$	21.5379	1.03504	0.517522	−	0.855670i	$-0.326855\pi$
$433$	21.5379	1.03504	0.517522	+	0.855670i	$0.326855\pi$
$434$	7.67354	0.368342
$435$	0	0
$436$	104.841	5.02097
$437$	5.28783	0.252951
$438$	0	0
$439$	−15.7267	−0.750593	−0.375297	−	0.926905i	$-0.622459\pi$
$439$	−15.7267	−0.750593	−0.375297	+	0.926905i	$0.622459\pi$
$440$	0	0
$441$	0	0
$442$	−23.2446	−1.10563
$443$	30.5028	1.44923	0.724617	−	0.689152i	$-0.242017\pi$
$443$	30.5028	1.44923	0.724617	+	0.689152i	$0.242017\pi$
$444$	0	0
$445$	0	0
$446$	9.86829	0.467277
$447$	0	0
$448$	−29.8846	−1.41191
$449$	25.4811	1.20253	0.601264	−	0.799051i	$-0.294664\pi$
$449$	25.4811	1.20253	0.601264	+	0.799051i	$0.294664\pi$
$450$	0	0
$451$	−24.6517	−1.16080
$452$	6.26721	0.294785
$453$	0	0
$454$	39.0349	1.83200
$455$	0	0
$456$	0	0
$457$	28.3815	1.32763	0.663816	−	0.747896i	$-0.268936\pi$
$457$	28.3815	1.32763	0.663816	+	0.747896i	$0.268936\pi$
$458$	13.1619	0.615016
$459$	0	0
$460$	0	0
$461$	−18.8017	−0.875683	−0.437841	−	0.899052i	$-0.644257\pi$
$461$	−18.8017	−0.875683	−0.437841	+	0.899052i	$0.644257\pi$
$462$	0	0
$463$	0.281866	0.0130994	0.00654971	−	0.999979i	$-0.497915\pi$
$463$	0.281866	0.0130994	0.00654971	+	0.999979i	$0.497915\pi$
$464$	97.9478	4.54711
$465$	0	0
$466$	29.9082	1.38547
$467$	−4.12950	−0.191091	−0.0955453	−	0.995425i	$-0.530459\pi$
$467$	−4.12950	−0.191091	−0.0955453	+	0.995425i	$0.530459\pi$
$468$	0	0
$469$	−1.41980	−0.0655604
$470$	0	0
$471$	0	0
$472$	−1.06867	−0.0491896
$473$	3.79095	0.174308
$474$	0	0
$475$	0	0
$476$	12.8648	0.589657
$477$	0	0
$478$	−63.8415	−2.92004
$479$	−32.8864	−1.50262	−0.751308	−	0.659952i	$-0.770577\pi$
$479$	−32.8864	−1.50262	−0.751308	+	0.659952i	$0.770577\pi$
$480$	0	0
$481$	−25.1398	−1.14627
$482$	40.5159	1.84545
$483$	0	0
$484$	−34.5097	−1.56862
$485$	0	0
$486$	0	0
$487$	30.4470	1.37969	0.689844	−	0.723958i	$-0.257679\pi$
$487$	30.4470	1.37969	0.689844	+	0.723958i	$0.257679\pi$
$488$	−79.4185	−3.59511
$489$	0	0
$490$	0	0
$491$	−12.4504	−0.561881	−0.280940	−	0.959725i	$-0.590646\pi$
$491$	−12.4504	−0.561881	−0.280940	+	0.959725i	$0.590646\pi$
$492$	0	0
$493$	−14.1179	−0.635838
$494$	61.8706	2.78369
$495$	0	0
$496$	−32.1682	−1.44439
$497$	−17.3775	−0.779486
$498$	0	0
$499$	14.5298	0.650442	0.325221	−	0.945638i	$-0.394561\pi$
$499$	14.5298	0.650442	0.325221	+	0.945638i	$0.394561\pi$
$500$	0	0
$501$	0	0
$502$	−11.9456	−0.533157
$503$	1.76847	0.0788523	0.0394261	−	0.999222i	$-0.487447\pi$
$503$	1.76847	0.0788523	0.0394261	+	0.999222i	$0.487447\pi$
$504$	0	0
$505$	0	0
$506$	−5.76401	−0.256241
$507$	0	0
$508$	−9.21584	−0.408887
$509$	26.3984	1.17009	0.585045	−	0.811001i	$-0.301077\pi$
$509$	26.3984	1.17009	0.585045	+	0.811001i	$0.301077\pi$
$510$	0	0
$511$	10.1017	0.446871
$512$	16.6918	0.737680
$513$	0	0
$514$	3.21694	0.141893
$515$	0	0
$516$	0	0
$517$	−16.7983	−0.738790
$518$	19.1307	0.840556
$519$	0	0
$520$	0	0
$521$	41.9476	1.83776	0.918878	−	0.394541i	$-0.129096\pi$
$521$	41.9476	1.83776	0.918878	+	0.394541i	$0.129096\pi$
$522$	0	0
$523$	−31.4020	−1.37311	−0.686557	−	0.727076i	$-0.740879\pi$
$523$	−31.4020	−1.37311	−0.686557	+	0.727076i	$0.740879\pi$
$524$	103.326	4.51381
$525$	0	0
$526$	−35.0659	−1.52895
$527$	4.63662	0.201974
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	−34.2425	−1.48460
$533$	50.0414	2.16753
$534$	0	0
$535$	0	0
$536$	10.5587	0.456066
$537$	0	0
$538$	−10.0905	−0.435031
$539$	11.7617	0.506614
$540$	0	0
$541$	−10.1302	−0.435533	−0.217766	−	0.976001i	$-0.569877\pi$
$541$	−10.1302	−0.435533	−0.217766	+	0.976001i	$0.569877\pi$
$542$	11.1521	0.479025
$543$	0	0
$544$	−38.2790	−1.64120
$545$	0	0
$546$	0	0
$547$	7.92182	0.338713	0.169356	−	0.985555i	$-0.445831\pi$
$547$	7.92182	0.338713	0.169356	+	0.985555i	$0.445831\pi$
$548$	−113.914	−4.86617
$549$	0	0
$550$	0	0
$551$	37.5779	1.60087
$552$	0	0
$553$	−13.8275	−0.588007
$554$	−57.2613	−2.43280
$555$	0	0
$556$	−124.942	−5.29871
$557$	10.1854	0.431570	0.215785	−	0.976441i	$-0.430769\pi$
$557$	10.1854	0.431570	0.215785	+	0.976441i	$0.430769\pi$
$558$	0	0
$559$	−7.69539	−0.325480
$560$	0	0
$561$	0	0
$562$	73.3134	3.09254
$563$	35.1935	1.48323	0.741615	−	0.670825i	$-0.234060\pi$
$563$	35.1935	1.48323	0.741615	+	0.670825i	$0.234060\pi$
$564$	0	0
$565$	0	0
$566$	−9.91792	−0.416881
$567$	0	0
$568$	129.232	5.42244
$569$	−36.0976	−1.51329	−0.756644	−	0.653827i	$-0.773162\pi$
$569$	−36.0976	−1.51329	−0.756644	+	0.653827i	$0.773162\pi$
$570$	0	0
$571$	−39.9923	−1.67362	−0.836812	−	0.547490i	$-0.815583\pi$
$571$	−39.9923	−1.67362	−0.836812	+	0.547490i	$0.815583\pi$
$572$	−49.0503	−2.05090
$573$	0	0
$574$	−38.0803	−1.58944
$575$	0	0
$576$	0	0
$577$	30.1288	1.25428	0.627139	−	0.778907i	$-0.284226\pi$
$577$	30.1288	1.25428	0.627139	+	0.778907i	$0.284226\pi$
$578$	−35.3500	−1.47037
$579$	0	0
$580$	0	0
$581$	10.8189	0.448842
$582$	0	0
$583$	10.4089	0.431094
$584$	−75.1234	−3.10863
$585$	0	0
$586$	19.2791	0.796412
$587$	29.8608	1.23249	0.616244	−	0.787555i	$-0.288653\pi$
$587$	29.8608	1.23249	0.616244	+	0.787555i	$0.288653\pi$
$588$	0	0
$589$	−12.3414	−0.508518
$590$	0	0
$591$	0	0
$592$	−80.1978	−3.29611
$593$	1.62566	0.0667577	0.0333788	−	0.999443i	$-0.489373\pi$
$593$	1.62566	0.0667577	0.0333788	+	0.999443i	$0.489373\pi$
$594$	0	0
$595$	0	0
$596$	61.6619	2.52577
$597$	0	0
$598$	11.7006	0.478472
$599$	−23.8860	−0.975957	−0.487979	−	0.872856i	$-0.662266\pi$
$599$	−23.8860	−0.975957	−0.487979	+	0.872856i	$0.662266\pi$
$600$	0	0
$601$	−24.0286	−0.980149	−0.490074	−	0.871681i	$-0.663030\pi$
$601$	−24.0286	−0.980149	−0.490074	+	0.871681i	$0.663030\pi$
$602$	5.85600	0.238673
$603$	0	0
$604$	28.9529	1.17808
$605$	0	0
$606$	0	0
$607$	−13.1188	−0.532476	−0.266238	−	0.963907i	$-0.585781\pi$
$607$	−13.1188	−0.532476	−0.266238	+	0.963907i	$0.585781\pi$
$608$	101.888	4.13210
$609$	0	0
$610$	0	0
$611$	34.0996	1.37952
$612$	0	0
$613$	−45.8575	−1.85217	−0.926084	−	0.377316i	$-0.876847\pi$
$613$	−45.8575	−1.85217	−0.926084	+	0.377316i	$0.876847\pi$
$614$	−64.1184	−2.58761
$615$	0	0
$616$	23.3303	0.940006
$617$	28.9724	1.16638	0.583192	−	0.812334i	$-0.301804\pi$
$617$	28.9724	1.16638	0.583192	+	0.812334i	$0.301804\pi$
$618$	0	0
$619$	−24.7720	−0.995672	−0.497836	−	0.867271i	$-0.665872\pi$
$619$	−24.7720	−0.995672	−0.497836	+	0.867271i	$0.665872\pi$
$620$	0	0
$621$	0	0
$622$	−38.2723	−1.53458
$623$	2.51927	0.100933
$624$	0	0
$625$	0	0
$626$	−64.0865	−2.56141
$627$	0	0
$628$	−45.0125	−1.79619
$629$	11.5595	0.460906
$630$	0	0
$631$	−35.5321	−1.41451	−0.707256	−	0.706958i	$-0.750067\pi$
$631$	−35.5321	−1.41451	−0.707256	+	0.706958i	$0.750067\pi$
$632$	102.832	4.09043
$633$	0	0
$634$	59.0975	2.34706
$635$	0	0
$636$	0	0
$637$	−23.8756	−0.945984
$638$	−40.9618	−1.62169
$639$	0	0
$640$	0	0
$641$	−28.4375	−1.12321	−0.561606	−	0.827405i	$-0.689816\pi$
$641$	−28.4375	−1.12321	−0.561606	+	0.827405i	$0.689816\pi$
$642$	0	0
$643$	41.4662	1.63527	0.817634	−	0.575738i	$-0.195285\pi$
$643$	41.4662	1.63527	0.817634	+	0.575738i	$0.195285\pi$
$644$	−6.47571	−0.255179
$645$	0	0
$646$	−28.4486	−1.11930
$647$	−2.93696	−0.115464	−0.0577320	−	0.998332i	$-0.518387\pi$
$647$	−2.93696	−0.115464	−0.0577320	+	0.998332i	$0.518387\pi$
$648$	0	0
$649$	0.251928	0.00988906
$650$	0	0
$651$	0	0
$652$	−92.3245	−3.61571
$653$	−24.4006	−0.954869	−0.477435	−	0.878667i	$-0.658433\pi$
$653$	−24.4006	−0.954869	−0.477435	+	0.878667i	$0.658433\pi$
$654$	0	0
$655$	0	0
$656$	159.636	6.23274
$657$	0	0
$658$	−25.9489	−1.01159
$659$	−40.5705	−1.58040	−0.790202	−	0.612847i	$-0.790024\pi$
$659$	−40.5705	−1.58040	−0.790202	+	0.612847i	$0.790024\pi$
$660$	0	0
$661$	18.8622	0.733656	0.366828	−	0.930289i	$-0.380444\pi$
$661$	18.8622	0.733656	0.366828	+	0.930289i	$0.380444\pi$
$662$	−65.7599	−2.55583
$663$	0	0
$664$	−80.4571	−3.12234
$665$	0	0
$666$	0	0
$667$	7.10648	0.275164
$668$	19.8322	0.767332
$669$	0	0
$670$	0	0
$671$	18.7221	0.722758
$672$	0	0
$673$	−17.4011	−0.670765	−0.335382	−	0.942082i	$-0.608866\pi$
$673$	−17.4011	−0.670765	−0.335382	+	0.942082i	$0.608866\pi$
$674$	−42.3721	−1.63211
$675$	0	0
$676$	30.2279	1.16261
$677$	31.6827	1.21767	0.608833	−	0.793298i	$-0.291638\pi$
$677$	31.6827	1.21767	0.608833	+	0.793298i	$0.291638\pi$
$678$	0	0
$679$	−23.0200	−0.883428
$680$	0	0
$681$	0	0
$682$	13.4527	0.515132
$683$	−18.8558	−0.721496	−0.360748	−	0.932663i	$-0.617479\pi$
$683$	−18.8558	−0.721496	−0.360748	+	0.932663i	$0.617479\pi$
$684$	0	0
$685$	0	0
$686$	41.1835	1.57239
$687$	0	0
$688$	−24.5489	−0.935918
$689$	−21.1295	−0.804968
$690$	0	0
$691$	47.2210	1.79637	0.898186	−	0.439616i	$-0.144885\pi$
$691$	47.2210	1.79637	0.898186	+	0.439616i	$0.144885\pi$
$692$	−25.7281	−0.978034
$693$	0	0
$694$	−78.2591	−2.97067
$695$	0	0
$696$	0	0
$697$	−23.0095	−0.871545
$698$	86.1201	3.25969
$699$	0	0
$700$	0	0
$701$	14.4222	0.544718	0.272359	−	0.962196i	$-0.412196\pi$
$701$	14.4222	0.544718	0.272359	+	0.962196i	$0.412196\pi$
$702$	0	0
$703$	−30.7680	−1.16044
$704$	−52.3917	−1.97459
$705$	0	0
$706$	45.3355	1.70622
$707$	8.89345	0.334473
$708$	0	0
$709$	26.4996	0.995213	0.497607	−	0.867403i	$-0.334212\pi$
$709$	26.4996	0.995213	0.497607	+	0.867403i	$0.334212\pi$
$710$	0	0
$711$	0	0
$712$	−18.7352	−0.702131
$713$	−2.33392	−0.0874061
$714$	0	0
$715$	0	0
$716$	−89.0997	−3.32981
$717$	0	0
$718$	−45.1822	−1.68619
$719$	−1.39174	−0.0519031	−0.0259516	−	0.999663i	$-0.508262\pi$
$719$	−1.39174	−0.0519031	−0.0259516	+	0.999663i	$0.508262\pi$
$720$	0	0
$721$	21.0132	0.782574
$722$	24.2678	0.903155
$723$	0	0
$724$	66.5484	2.47325
$725$	0	0
$726$	0	0
$727$	33.8951	1.25710	0.628550	−	0.777770i	$-0.283649\pi$
$727$	33.8951	1.25710	0.628550	+	0.777770i	$0.283649\pi$
$728$	−47.3591	−1.75524
$729$	0	0
$730$	0	0
$731$	3.53840	0.130873
$732$	0	0
$733$	31.7370	1.17223	0.586117	−	0.810227i	$-0.300656\pi$
$733$	31.7370	1.17223	0.586117	+	0.810227i	$0.300656\pi$
$734$	39.5162	1.45857
$735$	0	0
$736$	19.2684	0.710242
$737$	−2.48910	−0.0916873
$738$	0	0
$739$	−29.3281	−1.07885	−0.539425	−	0.842034i	$-0.681358\pi$
$739$	−29.3281	−1.07885	−0.539425	+	0.842034i	$0.681358\pi$
$740$	0	0
$741$	0	0
$742$	16.0790	0.590279
$743$	−34.3962	−1.26187	−0.630937	−	0.775834i	$-0.717329\pi$
$743$	−34.3962	−1.26187	−0.630937	+	0.775834i	$0.717329\pi$
$744$	0	0
$745$	0	0
$746$	−75.2388	−2.75469
$747$	0	0
$748$	22.5537	0.824646
$749$	−8.05364	−0.294274
$750$	0	0
$751$	−10.8039	−0.394240	−0.197120	−	0.980379i	$-0.563159\pi$
$751$	−10.8039	−0.394240	−0.197120	+	0.980379i	$0.563159\pi$
$752$	108.780	3.96681
$753$	0	0
$754$	83.1498	3.02814
$755$	0	0
$756$	0	0
$757$	46.6785	1.69656	0.848279	−	0.529549i	$-0.177639\pi$
$757$	46.6785	1.69656	0.848279	+	0.529549i	$0.177639\pi$
$758$	21.0483	0.764508
$759$	0	0
$760$	0	0
$761$	−4.95595	−0.179653	−0.0898264	−	0.995957i	$-0.528631\pi$
$761$	−4.95595	−0.179653	−0.0898264	+	0.995957i	$0.528631\pi$
$762$	0	0
$763$	−23.8630	−0.863899
$764$	−115.366	−4.17380
$765$	0	0
$766$	−6.52141	−0.235628
$767$	−0.511399	−0.0184655
$768$	0	0
$769$	−17.8075	−0.642155	−0.321078	−	0.947053i	$-0.604045\pi$
$769$	−17.8075	−0.642155	−0.321078	+	0.947053i	$0.604045\pi$
$770$	0	0
$771$	0	0
$772$	−56.9015	−2.04793
$773$	−45.9947	−1.65431	−0.827157	−	0.561971i	$-0.810043\pi$
$773$	−45.9947	−1.65431	−0.827157	+	0.561971i	$0.810043\pi$
$774$	0	0
$775$	0	0
$776$	171.194	6.14551
$777$	0	0
$778$	12.6330	0.452915
$779$	61.2447	2.19432
$780$	0	0
$781$	−30.4650	−1.09012
$782$	−5.38002	−0.192389
$783$	0	0
$784$	−76.1649	−2.72018
$785$	0	0
$786$	0	0
$787$	14.1742	0.505255	0.252627	−	0.967564i	$-0.418705\pi$
$787$	14.1742	0.505255	0.252627	+	0.967564i	$0.418705\pi$
$788$	−7.95459	−0.283371
$789$	0	0
$790$	0	0
$791$	−1.42649	−0.0507201
$792$	0	0
$793$	−38.0046	−1.34958
$794$	−13.6152	−0.483185
$795$	0	0
$796$	−47.1550	−1.67136
$797$	35.4453	1.25554	0.627769	−	0.778399i	$-0.283968\pi$
$797$	35.4453	1.25554	0.627769	+	0.778399i	$0.283968\pi$
$798$	0	0
$799$	−15.6793	−0.554692
$800$	0	0
$801$	0	0
$802$	−10.0384	−0.354468
$803$	17.7096	0.624957
$804$	0	0
$805$	0	0
$806$	−27.3082	−0.961890
$807$	0	0
$808$	−66.1383	−2.32674
$809$	38.0117	1.33642	0.668210	−	0.743973i	$-0.267061\pi$
$809$	38.0117	1.33642	0.668210	+	0.743973i	$0.267061\pi$
$810$	0	0
$811$	8.96681	0.314867	0.157434	−	0.987530i	$-0.449678\pi$
$811$	8.96681	0.314867	0.157434	+	0.987530i	$0.449678\pi$
$812$	−46.0196	−1.61497
$813$	0	0
$814$	33.5388	1.17553
$815$	0	0
$816$	0	0
$817$	−9.41823	−0.329502
$818$	−44.9399	−1.57129
$819$	0	0
$820$	0	0
$821$	23.6910	0.826821	0.413411	−	0.910545i	$-0.364337\pi$
$821$	23.6910	0.826821	0.413411	+	0.910545i	$0.364337\pi$
$822$	0	0
$823$	26.9890	0.940776	0.470388	−	0.882460i	$-0.344114\pi$
$823$	26.9890	0.940776	0.470388	+	0.882460i	$0.344114\pi$
$824$	−156.270	−5.44392
$825$	0	0
$826$	0.389162	0.0135407
$827$	17.1055	0.594817	0.297408	−	0.954750i	$-0.403878\pi$
$827$	17.1055	0.594817	0.297408	+	0.954750i	$0.403878\pi$
$828$	0	0
$829$	35.7258	1.24081	0.620404	−	0.784283i	$-0.286969\pi$
$829$	35.7258	1.24081	0.620404	+	0.784283i	$0.286969\pi$
$830$	0	0
$831$	0	0
$832$	106.352	3.68709
$833$	10.9782	0.380371
$834$	0	0
$835$	0	0
$836$	−60.0317	−2.07624
$837$	0	0
$838$	−0.962734	−0.0332571
$839$	35.0696	1.21074	0.605369	−	0.795945i	$-0.293026\pi$
$839$	35.0696	1.21074	0.605369	+	0.795945i	$0.293026\pi$
$840$	0	0
$841$	21.5021	0.741452
$842$	−64.0479	−2.20724
$843$	0	0
$844$	−65.4590	−2.25319
$845$	0	0
$846$	0	0
$847$	7.85480	0.269894
$848$	−67.4047	−2.31469
$849$	0	0
$850$	0	0
$851$	−5.81865	−0.199461
$852$	0	0
$853$	37.2116	1.27410	0.637050	−	0.770823i	$-0.280155\pi$
$853$	37.2116	1.27410	0.637050	+	0.770823i	$0.280155\pi$
$854$	28.9206	0.989642
$855$	0	0
$856$	59.8929	2.04710
$857$	−2.47751	−0.0846302	−0.0423151	−	0.999104i	$-0.513473\pi$
$857$	−2.47751	−0.0846302	−0.0423151	+	0.999104i	$0.513473\pi$
$858$	0	0
$859$	13.2049	0.450545	0.225273	−	0.974296i	$-0.427673\pi$
$859$	13.2049	0.450545	0.225273	+	0.974296i	$0.427673\pi$
$860$	0	0
$861$	0	0
$862$	13.4769	0.459026
$863$	−30.1713	−1.02704	−0.513521	−	0.858077i	$-0.671659\pi$
$863$	−30.1713	−1.02704	−0.513521	+	0.858077i	$0.671659\pi$
$864$	0	0
$865$	0	0
$866$	58.3272	1.98204
$867$	0	0
$868$	15.1138	0.512996
$869$	−24.2416	−0.822338
$870$	0	0
$871$	5.05272	0.171205
$872$	177.463	6.00965
$873$	0	0
$874$	14.3201	0.484384
$875$	0	0
$876$	0	0
$877$	−15.2627	−0.515385	−0.257692	−	0.966227i	$-0.582962\pi$
$877$	−15.2627	−0.515385	−0.257692	+	0.966227i	$0.582962\pi$
$878$	−42.5898	−1.43733
$879$	0	0
$880$	0	0
$881$	9.62761	0.324362	0.162181	−	0.986761i	$-0.448147\pi$
$881$	9.62761	0.324362	0.162181	+	0.986761i	$0.448147\pi$
$882$	0	0
$883$	43.9311	1.47840	0.739199	−	0.673487i	$-0.235204\pi$
$883$	43.9311	1.47840	0.739199	+	0.673487i	$0.235204\pi$
$884$	−45.7826	−1.53984
$885$	0	0
$886$	82.6054	2.77518
$887$	35.4705	1.19098	0.595491	−	0.803362i	$-0.296958\pi$
$887$	35.4705	1.19098	0.595491	+	0.803362i	$0.296958\pi$
$888$	0	0
$889$	2.09763	0.0703523
$890$	0	0
$891$	0	0
$892$	19.4366	0.650786
$893$	41.7338	1.39657
$894$	0	0
$895$	0	0
$896$	−34.1451	−1.14071
$897$	0	0
$898$	69.0059	2.30276
$899$	−16.5860	−0.553173
$900$	0	0
$901$	9.71550	0.323670
$902$	−66.7598	−2.22286
$903$	0	0
$904$	10.6084	0.352831
$905$	0	0
$906$	0	0
$907$	−15.5274	−0.515579	−0.257790	−	0.966201i	$-0.582994\pi$
$907$	−15.5274	−0.515579	−0.257790	+	0.966201i	$0.582994\pi$
$908$	76.8832	2.55146
$909$	0	0
$910$	0	0
$911$	−6.70505	−0.222148	−0.111074	−	0.993812i	$-0.535429\pi$
$911$	−6.70505	−0.222148	−0.111074	+	0.993812i	$0.535429\pi$
$912$	0	0
$913$	18.9669	0.627714
$914$	76.8607	2.54233
$915$	0	0
$916$	25.9238	0.856545
$917$	−23.5182	−0.776638
$918$	0	0
$919$	−0.198341	−0.00654266	−0.00327133	−	0.999995i	$-0.501041\pi$
$919$	−0.198341	−0.00654266	−0.00327133	+	0.999995i	$0.501041\pi$
$920$	0	0
$921$	0	0
$922$	−50.9173	−1.67687
$923$	61.8421	2.03556
$924$	0	0
$925$	0	0
$926$	0.763327	0.0250845
$927$	0	0
$928$	136.930	4.49496
$929$	36.5756	1.20001	0.600003	−	0.799998i	$-0.295166\pi$
$929$	36.5756	1.20001	0.600003	+	0.799998i	$0.295166\pi$
$930$	0	0
$931$	−29.2208	−0.957674
$932$	58.9073	1.92957
$933$	0	0
$934$	−11.1832	−0.365925
$935$	0	0
$936$	0	0
$937$	−34.5234	−1.12783	−0.563915	−	0.825833i	$-0.690706\pi$
$937$	−34.5234	−1.12783	−0.563915	+	0.825833i	$0.690706\pi$
$938$	−3.84500	−0.125544
$939$	0	0
$940$	0	0
$941$	16.3274	0.532257	0.266129	−	0.963938i	$-0.414255\pi$
$941$	16.3274	0.532257	0.266129	+	0.963938i	$0.414255\pi$
$942$	0	0
$943$	11.5822	0.377168
$944$	−1.63140	−0.0530976
$945$	0	0
$946$	10.2664	0.333788
$947$	−23.1333	−0.751731	−0.375865	−	0.926674i	$-0.622654\pi$
$947$	−23.1333	−0.751731	−0.375865	+	0.926674i	$0.622654\pi$
$948$	0	0
$949$	−35.9493	−1.16696
$950$	0	0
$951$	0	0
$952$	21.7761	0.705767
$953$	45.7304	1.48135	0.740677	−	0.671862i	$-0.234505\pi$
$953$	45.7304	1.48135	0.740677	+	0.671862i	$0.234505\pi$
$954$	0	0
$955$	0	0
$956$	−125.742	−4.06680
$957$	0	0
$958$	−89.0603	−2.87741
$959$	25.9282	0.837264
$960$	0	0
$961$	−25.5528	−0.824284
$962$	−68.0815	−2.19504
$963$	0	0
$964$	79.8003	2.57019
$965$	0	0
$966$	0	0
$967$	−0.302509	−0.00972803	−0.00486402	−	0.999988i	$-0.501548\pi$
$967$	−0.302509	−0.00972803	−0.00486402	+	0.999988i	$0.501548\pi$
$968$	−58.4141	−1.87750
$969$	0	0
$970$	0	0
$971$	−9.02535	−0.289637	−0.144819	−	0.989458i	$-0.546260\pi$
$971$	−9.02535	−0.289637	−0.144819	+	0.989458i	$0.546260\pi$
$972$	0	0
$973$	28.4382	0.911686
$974$	82.4543	2.64201
$975$	0	0
$976$	−121.238	−3.88073
$977$	−3.64104	−0.116487	−0.0582436	−	0.998302i	$-0.518550\pi$
$977$	−3.64104	−0.116487	−0.0582436	+	0.998302i	$0.518550\pi$
$978$	0	0
$979$	4.41663	0.141156
$980$	0	0
$981$	0	0
$982$	−33.7173	−1.07596
$983$	−21.0997	−0.672976	−0.336488	−	0.941688i	$-0.609239\pi$
$983$	−21.0997	−0.672976	−0.336488	+	0.941688i	$0.609239\pi$
$984$	0	0
$985$	0	0
$986$	−38.2330	−1.21759
$987$	0	0
$988$	121.860	3.87690
$989$	−1.78112	−0.0566362
$990$	0	0
$991$	−48.9360	−1.55450	−0.777252	−	0.629189i	$-0.783387\pi$
$991$	−48.9360	−1.55450	−0.777252	+	0.629189i	$0.783387\pi$
$992$	−44.9709	−1.42783
$993$	0	0
$994$	−47.0603	−1.49266
$995$	0	0
$996$	0	0
$997$	9.67207	0.306318	0.153159	−	0.988202i	$-0.451055\pi$
$997$	9.67207	0.306318	0.153159	+	0.988202i	$0.451055\pi$
$998$	39.3484	1.24555
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5175.2.a.by.1.6		6
3.2	odd	2		5175.2.a.bz.1.1		6
5.4	even	2		1035.2.a.q.1.1	yes	6
15.14	odd	2		1035.2.a.p.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1035.2.a.p.1.6	✓	6	15.14	odd	2
1035.2.a.q.1.1	yes	6	5.4	even	2
5175.2.a.by.1.6		6	1.1	even	1	trivial
5175.2.a.bz.1.1		6	3.2	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:	\( N \)	\(=\)	\( 5175 = 3^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5175.a (trivial)

\(f(q)\)	\(=\)	\(q+2.70812 q^{2} +5.33392 q^{4} -1.21406 q^{7} +9.02867 q^{8} -2.12841 q^{11} +4.32055 q^{13} -3.28783 q^{14} +13.7829 q^{16} -1.98662 q^{17} +5.28783 q^{19} -5.76401 q^{22} +1.00000 q^{23} +11.7006 q^{26} -6.47571 q^{28} +7.10648 q^{29} -2.33392 q^{31} +19.2684 q^{32} -5.38002 q^{34} -5.81865 q^{37} +14.3201 q^{38} +11.5822 q^{41} -1.78112 q^{43} -11.3528 q^{44} +2.70812 q^{46} +7.89242 q^{47} -5.52605 q^{49} +23.0455 q^{52} -4.89046 q^{53} -10.9614 q^{56} +19.2452 q^{58} -0.118364 q^{59} -8.79626 q^{61} -6.32055 q^{62} +24.6154 q^{64} +1.16946 q^{67} -10.5965 q^{68} +14.3135 q^{71} -8.32055 q^{73} -15.7576 q^{74} +28.2049 q^{76} +2.58403 q^{77} +11.3895 q^{79} +31.3660 q^{82} -8.91130 q^{83} -4.82348 q^{86} -19.2167 q^{88} -2.07508 q^{89} -5.24541 q^{91} +5.33392 q^{92} +21.3736 q^{94} +18.9612 q^{97} -14.9652 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 10 q^{4} - 6 q^{7} - 4 q^{11} - 12 q^{13} + 4 q^{14} + 14 q^{16} + 4 q^{17} + 8 q^{19} - 8 q^{22} + 6 q^{23} + 12 q^{26} - 24 q^{28} + 6 q^{29} + 8 q^{31} + 20 q^{32} - 12 q^{34} - 22 q^{37} + 36 q^{38}+ \cdots - 4 q^{98}+O(q^{100}) \) 6 * q + 10 * q^4 - 6 * q^7 - 4 * q^11 - 12 * q^13 + 4 * q^14 + 14 * q^16 + 4 * q^17 + 8 * q^19 - 8 * q^22 + 6 * q^23 + 12 * q^26 - 24 * q^28 + 6 * q^29 + 8 * q^31 + 20 * q^32 - 12 * q^34 - 22 * q^37 + 36 * q^38 + 18 * q^41 - 8 * q^43 - 4 * q^44 + 12 * q^47 + 18 * q^49 + 4 * q^53 - 4 * q^56 + 16 * q^58 + 6 * q^59 + 14 * q^64 - 10 * q^67 + 28 * q^68 + 22 * q^71 - 12 * q^73 + 20 * q^74 + 16 * q^76 - 4 * q^77 + 4 * q^79 + 12 * q^82 + 24 * q^83 - 20 * q^86 + 8 * q^88 + 8 * q^89 + 4 * q^91 + 10 * q^92 + 20 * q^94 - 12 * q^97 - 4 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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