LMFDB - Embedded newform 4851.2.a.ca.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4851,2,Mod(1,4851)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4851, 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("4851.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4851 = 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4851.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:	$38.7354300205$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.3676752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 8x^{3} + 14x^{2} + 11x - 10 $ x^5 - 2x^4 - 8x^3 + 14x^2 + 11x - 10
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.59450$ of defining polynomial
Character	$\chi$	$=$	4851.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.59450 q^{2} +4.73141 q^{4} -2.19202 q^{5} -7.08664 q^{8} +O(q^{10})$	$q-2.59450 q^{2} +4.73141 q^{4} -2.19202 q^{5} -7.08664 q^{8} +5.68719 q^{10} -1.00000 q^{11} -2.95275 q^{13} +8.92343 q^{16} -2.59450 q^{17} -2.35825 q^{19} -10.3713 q^{20} +2.59450 q^{22} -8.92343 q^{23} -0.195048 q^{25} +7.66090 q^{26} -4.48640 q^{29} -1.73141 q^{31} -8.97854 q^{32} +6.73141 q^{34} -4.38101 q^{37} +6.11848 q^{38} +15.5341 q^{40} +9.68719 q^{41} -10.3132 q^{43} -4.73141 q^{44} +23.1518 q^{46} -11.1094 q^{47} +0.506050 q^{50} -13.9707 q^{52} +9.17630 q^{53} +2.19202 q^{55} +11.6399 q^{58} -8.92343 q^{59} +13.4764 q^{61} +4.49214 q^{62} +5.44792 q^{64} +6.47249 q^{65} -3.25287 q^{67} -12.2756 q^{68} -0.994758 q^{71} +0.294879 q^{73} +11.3665 q^{74} -11.1579 q^{76} -15.0734 q^{79} -19.5603 q^{80} -25.1334 q^{82} +6.26778 q^{83} +5.68719 q^{85} +26.7576 q^{86} +7.08664 q^{88} +2.60024 q^{89} -42.2204 q^{92} +28.8233 q^{94} +5.16934 q^{95} +7.51704 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 10 q^{4} + 4 q^{5} - 6 q^{8}+O(q^{10}) $ 5 * q - 2 * q^2 + 10 * q^4 + 4 * q^5 - 6 * q^8	$ 5 q - 2 q^{2} + 10 q^{4} + 4 q^{5} - 6 q^{8} + 2 q^{10} - 5 q^{11} + 5 q^{13} + 16 q^{16} - 2 q^{17} - 3 q^{19} + 8 q^{20} + 2 q^{22} - 16 q^{23} + 7 q^{25} + 10 q^{26} + 5 q^{31} - 4 q^{32} + 20 q^{34} + 15 q^{37} - 6 q^{38} - 6 q^{40} + 22 q^{41} + 3 q^{43} - 10 q^{44} + 16 q^{46} + 2 q^{47} - 34 q^{50} + 40 q^{52} - 6 q^{53} - 4 q^{55} + 12 q^{58} - 16 q^{59} + 12 q^{61} + 4 q^{62} - 4 q^{64} + 28 q^{65} + 7 q^{67} - 10 q^{68} - 24 q^{71} + 17 q^{73} + 36 q^{74} + 30 q^{76} + 7 q^{79} - 16 q^{80} + 8 q^{82} + 12 q^{83} + 2 q^{85} + 18 q^{86} + 6 q^{88} + 6 q^{89} - 68 q^{92} + 82 q^{94} + 18 q^{95} - 14 q^{97}+O(q^{100}) $ 5 * q - 2 * q^2 + 10 * q^4 + 4 * q^5 - 6 * q^8 + 2 * q^10 - 5 * q^11 + 5 * q^13 + 16 * q^16 - 2 * q^17 - 3 * q^19 + 8 * q^20 + 2 * q^22 - 16 * q^23 + 7 * q^25 + 10 * q^26 + 5 * q^31 - 4 * q^32 + 20 * q^34 + 15 * q^37 - 6 * q^38 - 6 * q^40 + 22 * q^41 + 3 * q^43 - 10 * q^44 + 16 * q^46 + 2 * q^47 - 34 * q^50 + 40 * q^52 - 6 * q^53 - 4 * q^55 + 12 * q^58 - 16 * q^59 + 12 * q^61 + 4 * q^62 - 4 * q^64 + 28 * q^65 + 7 * q^67 - 10 * q^68 - 24 * q^71 + 17 * q^73 + 36 * q^74 + 30 * q^76 + 7 * q^79 - 16 * q^80 + 8 * q^82 + 12 * q^83 + 2 * q^85 + 18 * q^86 + 6 * q^88 + 6 * q^89 - 68 * q^92 + 82 * q^94 + 18 * 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 $

$2$	−2.59450	−1.83459	−0.917293	−	0.398213i	$-0.869631\pi$
$2$	−2.59450	−1.83459	−0.917293	+	0.398213i	$0.869631\pi$
$3$	0	0
$3$	0	0
$4$	4.73141	2.36571
$5$	−2.19202	−0.980301	−0.490151	−	0.871638i	$-0.663058\pi$
$5$	−2.19202	−0.980301	−0.490151	+	0.871638i	$0.663058\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−7.08664	−2.50550
$9$	0	0
$10$	5.68719	1.79845
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.95275	−0.818945	−0.409473	−	0.912322i	$-0.634287\pi$
$13$	−2.95275	−0.818945	−0.409473	+	0.912322i	$0.634287\pi$
$14$	0	0
$15$	0	0
$16$	8.92343	2.23086
$17$	−2.59450	−0.629258	−0.314629	−	0.949215i	$-0.601880\pi$
$17$	−2.59450	−0.629258	−0.314629	+	0.949215i	$0.601880\pi$
$18$	0	0
$19$	−2.35825	−0.541020	−0.270510	−	0.962717i	$-0.587192\pi$
$19$	−2.35825	−0.541020	−0.270510	+	0.962717i	$0.587192\pi$
$20$	−10.3713	−2.31910
$21$	0	0
$22$	2.59450	0.553148
$23$	−8.92343	−1.86066	−0.930332	−	0.366718i	$-0.880481\pi$
$23$	−8.92343	−1.86066	−0.930332	+	0.366718i	$0.880481\pi$
$24$	0	0
$25$	−0.195048	−0.0390095
$26$	7.66090	1.50243
$27$	0	0
$28$	0	0
$29$	−4.48640	−0.833103	−0.416551	−	0.909112i	$-0.636761\pi$
$29$	−4.48640	−0.833103	−0.416551	+	0.909112i	$0.636761\pi$
$30$	0	0
$31$	−1.73141	−0.310971	−0.155485	−	0.987838i	$-0.549694\pi$
$31$	−1.73141	−0.310971	−0.155485	+	0.987838i	$0.549694\pi$
$32$	−8.97854	−1.58720
$33$	0	0
$34$	6.73141	1.15443
$35$	0	0
$36$	0	0
$37$	−4.38101	−0.720234	−0.360117	−	0.932907i	$-0.617263\pi$
$37$	−4.38101	−0.720234	−0.360117	+	0.932907i	$0.617263\pi$
$38$	6.11848	0.992548
$39$	0	0
$40$	15.5341	2.45615
$41$	9.68719	1.51288	0.756442	−	0.654060i	$-0.226936\pi$
$41$	9.68719	1.51288	0.756442	+	0.654060i	$0.226936\pi$
$42$	0	0
$43$	−10.3132	−1.57275	−0.786375	−	0.617749i	$-0.788045\pi$
$43$	−10.3132	−1.57275	−0.786375	+	0.617749i	$0.788045\pi$
$44$	−4.73141	−0.713287
$45$	0	0
$46$	23.1518	3.41355
$47$	−11.1094	−1.62047	−0.810236	−	0.586104i	$-0.800661\pi$
$47$	−11.1094	−1.62047	−0.810236	+	0.586104i	$0.800661\pi$
$48$	0	0
$49$	0	0
$50$	0.506050	0.0715663
$51$	0	0
$52$	−13.9707	−1.93738
$53$	9.17630	1.26046	0.630231	−	0.776408i	$-0.282960\pi$
$53$	9.17630	1.26046	0.630231	+	0.776408i	$0.282960\pi$
$54$	0	0
$55$	2.19202	0.295572
$56$	0	0
$57$	0	0
$58$	11.6399	1.52840
$59$	−8.92343	−1.16173	−0.580866	−	0.813999i	$-0.697286\pi$
$59$	−8.92343	−1.16173	−0.580866	+	0.813999i	$0.697286\pi$
$60$	0	0
$61$	13.4764	1.72548	0.862740	−	0.505649i	$-0.168747\pi$
$61$	13.4764	1.72548	0.862740	+	0.505649i	$0.168747\pi$
$62$	4.49214	0.570502
$63$	0	0
$64$	5.44792	0.680990
$65$	6.47249	0.802813
$66$	0	0
$67$	−3.25287	−0.397401	−0.198701	−	0.980060i	$-0.563672\pi$
$67$	−3.25287	−0.397401	−0.198701	+	0.980060i	$0.563672\pi$
$68$	−12.2756	−1.48864
$69$	0	0
$70$	0	0
$71$	−0.994758	−0.118056	−0.0590280	−	0.998256i	$-0.518800\pi$
$71$	−0.994758	−0.118056	−0.0590280	+	0.998256i	$0.518800\pi$
$72$	0	0
$73$	0.294879	0.0345130	0.0172565	−	0.999851i	$-0.494507\pi$
$73$	0.294879	0.0345130	0.0172565	+	0.999851i	$0.494507\pi$
$74$	11.3665	1.32133
$75$	0	0
$76$	−11.1579	−1.27990
$77$	0	0
$78$	0	0
$79$	−15.0734	−1.69589	−0.847947	−	0.530080i	$-0.822162\pi$
$79$	−15.0734	−1.69589	−0.847947	+	0.530080i	$0.822162\pi$
$80$	−19.5603	−2.18691
$81$	0	0
$82$	−25.1334	−2.77552
$83$	6.26778	0.687978	0.343989	−	0.938974i	$-0.388222\pi$
$83$	6.26778	0.687978	0.343989	+	0.938974i	$0.388222\pi$
$84$	0	0
$85$	5.68719	0.616862
$86$	26.7576	2.88535
$87$	0	0
$88$	7.08664	0.755438
$89$	2.60024	0.275625	0.137813	−	0.990458i	$-0.455993\pi$
$89$	2.60024	0.275625	0.137813	+	0.990458i	$0.455993\pi$
$90$	0	0
$91$	0	0
$92$	−42.2204	−4.40178
$93$	0	0
$94$	28.8233	2.97290
$95$	5.16934	0.530363
$96$	0	0
$97$	7.51704	0.763239	0.381620	−	0.924319i	$-0.375366\pi$
$97$	7.51704	0.763239	0.381620	+	0.924319i	$0.375366\pi$
$98$	0	0
$99$	0	0
$100$	−0.922850	−0.0922850
$101$	13.3605	1.32942	0.664708	−	0.747103i	$-0.268556\pi$
$101$	13.3605	1.32942	0.664708	+	0.747103i	$0.268556\pi$
$102$	0	0
$103$	−5.32863	−0.525045	−0.262523	−	0.964926i	$-0.584554\pi$
$103$	−5.32863	−0.525045	−0.262523	+	0.964926i	$0.584554\pi$
$104$	20.9251	2.05187
$105$	0	0
$106$	−23.8079	−2.31243
$107$	−5.89190	−0.569591	−0.284796	−	0.958588i	$-0.591926\pi$
$107$	−5.89190	−0.569591	−0.284796	+	0.958588i	$0.591926\pi$
$108$	0	0
$109$	−16.7996	−1.60911	−0.804556	−	0.593877i	$-0.797596\pi$
$109$	−16.7996	−1.60911	−0.804556	+	0.593877i	$0.797596\pi$
$110$	−5.68719	−0.542252
$111$	0	0
$112$	0	0
$113$	−12.9025	−1.21376	−0.606881	−	0.794793i	$-0.707580\pi$
$113$	−12.9025	−1.21376	−0.606881	+	0.794793i	$0.707580\pi$
$114$	0	0
$115$	19.5603	1.82401
$116$	−21.2270	−1.97088
$117$	0	0
$118$	23.1518	2.13130
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−34.9645	−3.16554
$123$	0	0
$124$	−8.19202	−0.735665
$125$	11.3876	1.01854
$126$	0	0
$127$	14.6951	1.30398	0.651992	−	0.758226i	$-0.273934\pi$
$127$	14.6951	1.30398	0.651992	+	0.758226i	$0.273934\pi$
$128$	3.82247	0.337862
$129$	0	0
$130$	−16.7928	−1.47283
$131$	−10.7660	−0.940627	−0.470314	−	0.882499i	$-0.655859\pi$
$131$	−10.7660	−0.940627	−0.470314	+	0.882499i	$0.655859\pi$
$132$	0	0
$133$	0	0
$134$	8.43956	0.729067
$135$	0	0
$136$	18.3863	1.57661
$137$	−2.61235	−0.223188	−0.111594	−	0.993754i	$-0.535596\pi$
$137$	−2.61235	−0.223188	−0.111594	+	0.993754i	$0.535596\pi$
$138$	0	0
$139$	6.25015	0.530131	0.265066	−	0.964230i	$-0.414606\pi$
$139$	6.25015	0.530131	0.265066	+	0.964230i	$0.414606\pi$
$140$	0	0
$141$	0	0
$142$	2.58090	0.216584
$143$	2.95275	0.246921
$144$	0	0
$145$	9.83427	0.816692
$146$	−0.765062	−0.0633170
$147$	0	0
$148$	−20.7284	−1.70386
$149$	4.12201	0.337688	0.168844	−	0.985643i	$-0.445997\pi$
$149$	4.12201	0.337688	0.168844	+	0.985643i	$0.445997\pi$
$150$	0	0
$151$	12.4353	1.01197	0.505985	−	0.862542i	$-0.331129\pi$
$151$	12.4353	1.01197	0.505985	+	0.862542i	$0.331129\pi$
$152$	16.7121	1.35553
$153$	0	0
$154$	0	0
$155$	3.79529	0.304845
$156$	0	0
$157$	−18.6933	−1.49189	−0.745945	−	0.666007i	$-0.768002\pi$
$157$	−18.6933	−1.49189	−0.745945	+	0.666007i	$0.768002\pi$
$158$	39.1080	3.11126
$159$	0	0
$160$	19.6811	1.55593
$161$	0	0
$162$	0	0
$163$	−1.26475	−0.0990627	−0.0495314	−	0.998773i	$-0.515773\pi$
$163$	−1.26475	−0.0990627	−0.0495314	+	0.998773i	$0.515773\pi$
$164$	45.8341	3.57904
$165$	0	0
$166$	−16.2617	−1.26215
$167$	6.11021	0.472822	0.236411	−	0.971653i	$-0.424029\pi$
$167$	6.11021	0.472822	0.236411	+	0.971653i	$0.424029\pi$
$168$	0	0
$169$	−4.28127	−0.329328
$170$	−14.7554	−1.13169
$171$	0	0
$172$	−48.7961	−3.72067
$173$	17.4649	1.32783	0.663917	−	0.747806i	$-0.268893\pi$
$173$	17.4649	1.32783	0.663917	+	0.747806i	$0.268893\pi$
$174$	0	0
$175$	0	0
$176$	−8.92343	−0.672629
$177$	0	0
$178$	−6.74632	−0.505658
$179$	−0.422545	−0.0315825	−0.0157912	−	0.999875i	$-0.505027\pi$
$179$	−0.422545	−0.0315825	−0.0157912	+	0.999875i	$0.505027\pi$
$180$	0	0
$181$	2.40197	0.178537	0.0892686	−	0.996008i	$-0.471547\pi$
$181$	2.40197	0.178537	0.0892686	+	0.996008i	$0.471547\pi$
$182$	0	0
$183$	0	0
$184$	63.2371	4.66190
$185$	9.60327	0.706046
$186$	0	0
$187$	2.59450	0.189728
$188$	−52.5631	−3.83356
$189$	0	0
$190$	−13.4118	−0.972996
$191$	8.69291	0.628997	0.314498	−	0.949258i	$-0.398164\pi$
$191$	8.69291	0.628997	0.314498	+	0.949258i	$0.398164\pi$
$192$	0	0
$193$	−8.04513	−0.579101	−0.289551	−	0.957163i	$-0.593506\pi$
$193$	−8.04513	−0.579101	−0.289551	+	0.957163i	$0.593506\pi$
$194$	−19.5029	−1.40023
$195$	0	0
$196$	0	0
$197$	4.79921	0.341929	0.170965	−	0.985277i	$-0.445312\pi$
$197$	4.79921	0.341929	0.170965	+	0.985277i	$0.445312\pi$
$198$	0	0
$199$	2.92122	0.207080	0.103540	−	0.994625i	$-0.466983\pi$
$199$	2.92122	0.207080	0.103540	+	0.994625i	$0.466983\pi$
$200$	1.38223	0.0977385
$201$	0	0
$202$	−34.6637	−2.43893
$203$	0	0
$204$	0	0
$205$	−21.2345	−1.48308
$206$	13.8251	0.963240
$207$	0	0
$208$	−26.3487	−1.82695
$209$	2.35825	0.163124
$210$	0	0
$211$	−9.82721	−0.676533	−0.338266	−	0.941050i	$-0.609841\pi$
$211$	−9.82721	−0.676533	−0.338266	+	0.941050i	$0.609841\pi$
$212$	43.4169	2.98188
$213$	0	0
$214$	15.2865	1.04496
$215$	22.6068	1.54177
$216$	0	0
$217$	0	0
$218$	43.5865	2.95205
$219$	0	0
$220$	10.3713	0.699236
$221$	7.66090	0.515328
$222$	0	0
$223$	−16.3127	−1.09238	−0.546190	−	0.837661i	$-0.683922\pi$
$223$	−16.3127	−1.09238	−0.546190	+	0.837661i	$0.683922\pi$
$224$	0	0
$225$	0	0
$226$	33.4754	2.22675
$227$	−22.6089	−1.50060	−0.750302	−	0.661095i	$-0.770092\pi$
$227$	−22.6089	−1.50060	−0.750302	+	0.661095i	$0.770092\pi$
$228$	0	0
$229$	−3.70723	−0.244981	−0.122490	−	0.992470i	$-0.539088\pi$
$229$	−3.70723	−0.244981	−0.122490	+	0.992470i	$0.539088\pi$
$230$	−50.7492	−3.34631
$231$	0	0
$232$	31.7935	2.08734
$233$	7.69899	0.504377	0.252189	−	0.967678i	$-0.418850\pi$
$233$	7.69899	0.504377	0.252189	+	0.967678i	$0.418850\pi$
$234$	0	0
$235$	24.3520	1.58855
$236$	−42.2204	−2.74832
$237$	0	0
$238$	0	0
$239$	4.12986	0.267139	0.133569	−	0.991039i	$-0.457356\pi$
$239$	4.12986	0.267139	0.133569	+	0.991039i	$0.457356\pi$
$240$	0	0
$241$	−14.7560	−0.950516	−0.475258	−	0.879847i	$-0.657645\pi$
$241$	−14.7560	−0.950516	−0.475258	+	0.879847i	$0.657645\pi$
$242$	−2.59450	−0.166781
$243$	0	0
$244$	63.7625	4.08198
$245$	0	0
$246$	0	0
$247$	6.96333	0.443066
$248$	12.2699	0.779138
$249$	0	0
$250$	−29.5452	−1.86860
$251$	13.5521	0.855399	0.427700	−	0.903921i	$-0.359324\pi$
$251$	13.5521	0.855399	0.427700	+	0.903921i	$0.359324\pi$
$252$	0	0
$253$	8.92343	0.561011
$254$	−38.1265	−2.39227
$255$	0	0
$256$	−20.8132	−1.30083
$257$	−8.53112	−0.532157	−0.266078	−	0.963951i	$-0.585728\pi$
$257$	−8.53112	−0.532157	−0.266078	+	0.963951i	$0.585728\pi$
$258$	0	0
$259$	0	0
$260$	30.6240	1.89922
$261$	0	0
$262$	27.9323	1.72566
$263$	−3.55290	−0.219081	−0.109540	−	0.993982i	$-0.534938\pi$
$263$	−3.55290	−0.219081	−0.109540	+	0.993982i	$0.534938\pi$
$264$	0	0
$265$	−20.1146	−1.23563
$266$	0	0
$267$	0	0
$268$	−15.3907	−0.940135
$269$	−11.6234	−0.708692	−0.354346	−	0.935114i	$-0.615296\pi$
$269$	−11.6234	−0.708692	−0.354346	+	0.935114i	$0.615296\pi$
$270$	0	0
$271$	9.87979	0.600154	0.300077	−	0.953915i	$-0.402988\pi$
$271$	9.87979	0.600154	0.300077	+	0.953915i	$0.402988\pi$
$272$	−23.1518	−1.40378
$273$	0	0
$274$	6.77774	0.409458
$275$	0.195048	0.0117618
$276$	0	0
$277$	−1.28210	−0.0770339	−0.0385169	−	0.999258i	$-0.512263\pi$
$277$	−1.28210	−0.0770339	−0.0385169	+	0.999258i	$0.512263\pi$
$278$	−16.2160	−0.972571
$279$	0	0
$280$	0	0
$281$	−25.5914	−1.52665	−0.763327	−	0.646013i	$-0.776435\pi$
$281$	−25.5914	−1.52665	−0.763327	+	0.646013i	$0.776435\pi$
$282$	0	0
$283$	−12.0079	−0.713798	−0.356899	−	0.934143i	$-0.616166\pi$
$283$	−12.0079	−0.713798	−0.356899	+	0.934143i	$0.616166\pi$
$284$	−4.70661	−0.279286
$285$	0	0
$286$	−7.66090	−0.452998
$287$	0	0
$288$	0	0
$289$	−10.2686	−0.604035
$290$	−25.5150	−1.49829
$291$	0	0
$292$	1.39519	0.0816475
$293$	17.6972	1.03388	0.516939	−	0.856022i	$-0.327071\pi$
$293$	17.6972	1.03388	0.516939	+	0.856022i	$0.327071\pi$
$294$	0	0
$295$	19.5603	1.13885
$296$	31.0466	1.80455
$297$	0	0
$298$	−10.6945	−0.619518
$299$	26.3487	1.52378
$300$	0	0
$301$	0	0
$302$	−32.2634	−1.85655
$303$	0	0
$304$	−21.0437	−1.20694
$305$	−29.5406	−1.69149
$306$	0	0
$307$	−8.08311	−0.461327	−0.230664	−	0.973034i	$-0.574090\pi$
$307$	−8.08311	−0.461327	−0.230664	+	0.973034i	$0.574090\pi$
$308$	0	0
$309$	0	0
$310$	−9.84686	−0.559264
$311$	−11.5867	−0.657024	−0.328512	−	0.944500i	$-0.606547\pi$
$311$	−11.5867	−0.657024	−0.328512	+	0.944500i	$0.606547\pi$
$312$	0	0
$313$	1.55592	0.0879460	0.0439730	−	0.999033i	$-0.485998\pi$
$313$	1.55592	0.0879460	0.0439730	+	0.999033i	$0.485998\pi$
$314$	48.4998	2.73700
$315$	0	0
$316$	−71.3187	−4.01199
$317$	−10.4991	−0.589688	−0.294844	−	0.955545i	$-0.595268\pi$
$317$	−10.4991	−0.589688	−0.294844	+	0.955545i	$0.595268\pi$
$318$	0	0
$319$	4.48640	0.251190
$320$	−11.9419	−0.667575
$321$	0	0
$322$	0	0
$323$	6.11848	0.340441
$324$	0	0
$325$	0.575927	0.0319467
$326$	3.28138	0.181739
$327$	0	0
$328$	−68.6496	−3.79054
$329$	0	0
$330$	0	0
$331$	28.9811	1.59294	0.796471	−	0.604676i	$-0.206698\pi$
$331$	28.9811	1.59294	0.796471	+	0.604676i	$0.206698\pi$
$332$	29.6554	1.62755
$333$	0	0
$334$	−15.8529	−0.867433
$335$	7.13036	0.389573
$336$	0	0
$337$	13.9865	0.761893	0.380946	−	0.924597i	$-0.375598\pi$
$337$	13.9865	0.761893	0.380946	+	0.924597i	$0.375598\pi$
$338$	11.1077	0.604181
$339$	0	0
$340$	26.9084	1.45931
$341$	1.73141	0.0937612
$342$	0	0
$343$	0	0
$344$	73.0860	3.94053
$345$	0	0
$346$	−45.3127	−2.43603
$347$	8.94087	0.479971	0.239986	−	0.970776i	$-0.422857\pi$
$347$	8.94087	0.479971	0.239986	+	0.970776i	$0.422857\pi$
$348$	0	0
$349$	0.353903	0.0189440	0.00947199	−	0.999955i	$-0.496985\pi$
$349$	0.353903	0.0189440	0.00947199	+	0.999955i	$0.496985\pi$
$350$	0	0
$351$	0	0
$352$	8.97854	0.478558
$353$	−20.6041	−1.09664	−0.548322	−	0.836267i	$-0.684733\pi$
$353$	−20.6041	−1.09664	−0.548322	+	0.836267i	$0.684733\pi$
$354$	0	0
$355$	2.18053	0.115731
$356$	12.3028	0.652048
$357$	0	0
$358$	1.09629	0.0579408
$359$	−10.5866	−0.558742	−0.279371	−	0.960183i	$-0.590126\pi$
$359$	−10.5866	−0.558742	−0.279371	+	0.960183i	$0.590126\pi$
$360$	0	0
$361$	−13.4386	−0.707297
$362$	−6.23191	−0.327542
$363$	0	0
$364$	0	0
$365$	−0.646381	−0.0338331
$366$	0	0
$367$	10.8793	0.567896	0.283948	−	0.958840i	$-0.408356\pi$
$367$	10.8793	0.567896	0.283948	+	0.958840i	$0.408356\pi$
$368$	−79.6276	−4.15088
$369$	0	0
$370$	−24.9156	−1.29530
$371$	0	0
$372$	0	0
$373$	14.7567	0.764072	0.382036	−	0.924147i	$-0.375223\pi$
$373$	14.7567	0.764072	0.382036	+	0.924147i	$0.375223\pi$
$374$	−6.73141	−0.348073
$375$	0	0
$376$	78.7283	4.06010
$377$	13.2472	0.682266
$378$	0	0
$379$	12.8452	0.659815	0.329908	−	0.944013i	$-0.392982\pi$
$379$	12.8452	0.659815	0.329908	+	0.944013i	$0.392982\pi$
$380$	24.4583	1.25468
$381$	0	0
$382$	−22.5537	−1.15395
$383$	−12.4554	−0.636440	−0.318220	−	0.948017i	$-0.603085\pi$
$383$	−12.4554	−0.636440	−0.318220	+	0.948017i	$0.603085\pi$
$384$	0	0
$385$	0	0
$386$	20.8731	1.06241
$387$	0	0
$388$	35.5662	1.80560
$389$	−13.2108	−0.669812	−0.334906	−	0.942252i	$-0.608705\pi$
$389$	−13.2108	−0.669812	−0.334906	+	0.942252i	$0.608705\pi$
$390$	0	0
$391$	23.1518	1.17084
$392$	0	0
$393$	0	0
$394$	−12.4515	−0.627299
$395$	33.0413	1.66249
$396$	0	0
$397$	18.6942	0.938233	0.469116	−	0.883136i	$-0.344573\pi$
$397$	18.6942	0.938233	0.469116	+	0.883136i	$0.344573\pi$
$398$	−7.57909	−0.379905
$399$	0	0
$400$	−1.74049	−0.0870247
$401$	7.17208	0.358157	0.179078	−	0.983835i	$-0.442688\pi$
$401$	7.17208	0.358157	0.179078	+	0.983835i	$0.442688\pi$
$402$	0	0
$403$	5.11242	0.254668
$404$	63.2139	3.14501
$405$	0	0
$406$	0	0
$407$	4.38101	0.217159
$408$	0	0
$409$	3.04120	0.150377	0.0751887	−	0.997169i	$-0.476044\pi$
$409$	3.04120	0.150377	0.0751887	+	0.997169i	$0.476044\pi$
$410$	55.0929	2.72084
$411$	0	0
$412$	−25.2119	−1.24210
$413$	0	0
$414$	0	0
$415$	−13.7391	−0.674425
$416$	26.5114	1.29983
$417$	0	0
$418$	−6.11848	−0.299265
$419$	−24.9508	−1.21893	−0.609464	−	0.792814i	$-0.708615\pi$
$419$	−24.9508	−1.21893	−0.609464	+	0.792814i	$0.708615\pi$
$420$	0	0
$421$	−11.9958	−0.584638	−0.292319	−	0.956321i	$-0.594427\pi$
$421$	−11.9958	−0.584638	−0.292319	+	0.956321i	$0.594427\pi$
$422$	25.4967	1.24116
$423$	0	0
$424$	−65.0291	−3.15809
$425$	0.506050	0.0245470
$426$	0	0
$427$	0	0
$428$	−27.8770	−1.34749
$429$	0	0
$430$	−58.6532	−2.82851
$431$	17.1701	0.827057	0.413528	−	0.910491i	$-0.364296\pi$
$431$	17.1701	0.827057	0.413528	+	0.910491i	$0.364296\pi$
$432$	0	0
$433$	7.02235	0.337472	0.168736	−	0.985661i	$-0.446031\pi$
$433$	7.02235	0.337472	0.168736	+	0.985661i	$0.446031\pi$
$434$	0	0
$435$	0	0
$436$	−79.4859	−3.80668
$437$	21.0437	1.00666
$438$	0	0
$439$	39.7521	1.89726	0.948631	−	0.316385i	$-0.102469\pi$
$439$	39.7521	1.89726	0.948631	+	0.316385i	$0.102469\pi$
$440$	−15.5341	−0.740557
$441$	0	0
$442$	−19.8762	−0.945413
$443$	−35.0012	−1.66296	−0.831479	−	0.555557i	$-0.812505\pi$
$443$	−35.0012	−1.66296	−0.831479	+	0.555557i	$0.812505\pi$
$444$	0	0
$445$	−5.69978	−0.270196
$446$	42.3233	2.00407
$447$	0	0
$448$	0	0
$449$	10.3780	0.489767	0.244884	−	0.969552i	$-0.421250\pi$
$449$	10.3780	0.489767	0.244884	+	0.969552i	$0.421250\pi$
$450$	0	0
$451$	−9.68719	−0.456152
$452$	−61.0469	−2.87140
$453$	0	0
$454$	58.6587	2.75299
$455$	0	0
$456$	0	0
$457$	−18.6027	−0.870199	−0.435100	−	0.900382i	$-0.643287\pi$
$457$	−18.6027	−0.870199	−0.435100	+	0.900382i	$0.643287\pi$
$458$	9.61840	0.449438
$459$	0	0
$460$	92.5480	4.31507
$461$	17.0730	0.795171	0.397585	−	0.917565i	$-0.369848\pi$
$461$	17.0730	0.795171	0.397585	+	0.917565i	$0.369848\pi$
$462$	0	0
$463$	32.3530	1.50357	0.751786	−	0.659408i	$-0.229193\pi$
$463$	32.3530	1.50357	0.751786	+	0.659408i	$0.229193\pi$
$464$	−40.0340	−1.85853
$465$	0	0
$466$	−19.9750	−0.925324
$467$	0.217409	0.0100605	0.00503024	−	0.999987i	$-0.498399\pi$
$467$	0.217409	0.0100605	0.00503024	+	0.999987i	$0.498399\pi$
$468$	0	0
$469$	0	0
$470$	−63.1812	−2.91433
$471$	0	0
$472$	63.2371	2.91072
$473$	10.3132	0.474202
$474$	0	0
$475$	0.459972	0.0211049
$476$	0	0
$477$	0	0
$478$	−10.7149	−0.490089
$479$	21.4612	0.980587	0.490294	−	0.871557i	$-0.336890\pi$
$479$	21.4612	0.980587	0.490294	+	0.871557i	$0.336890\pi$
$480$	0	0
$481$	12.9360	0.589832
$482$	38.2843	1.74380
$483$	0	0
$484$	4.73141	0.215064
$485$	−16.4775	−0.748205
$486$	0	0
$487$	36.0891	1.63535	0.817676	−	0.575679i	$-0.195262\pi$
$487$	36.0891	1.63535	0.817676	+	0.575679i	$0.195262\pi$
$488$	−95.5025	−4.32320
$489$	0	0
$490$	0	0
$491$	5.10627	0.230443	0.115221	−	0.993340i	$-0.463242\pi$
$491$	5.10627	0.230443	0.115221	+	0.993340i	$0.463242\pi$
$492$	0	0
$493$	11.6399	0.524236
$494$	−18.0663	−0.812843
$495$	0	0
$496$	−15.4501	−0.693731
$497$	0	0
$498$	0	0
$499$	7.73141	0.346106	0.173053	−	0.984913i	$-0.444637\pi$
$499$	7.73141	0.346106	0.173053	+	0.984913i	$0.444637\pi$
$500$	53.8797	2.40957
$501$	0	0
$502$	−35.1608	−1.56930
$503$	−34.5655	−1.54120	−0.770599	−	0.637320i	$-0.780043\pi$
$503$	−34.5655	−1.54120	−0.770599	+	0.637320i	$0.780043\pi$
$504$	0	0
$505$	−29.2864	−1.30323
$506$	−23.1518	−1.02922
$507$	0	0
$508$	69.5288	3.08484
$509$	28.5408	1.26505	0.632524	−	0.774541i	$-0.282019\pi$
$509$	28.5408	1.26505	0.632524	+	0.774541i	$0.282019\pi$
$510$	0	0
$511$	0	0
$512$	46.3549	2.04862
$513$	0	0
$514$	22.1340	0.976287
$515$	11.6805	0.514702
$516$	0	0
$517$	11.1094	0.488591
$518$	0	0
$519$	0	0
$520$	−45.8682	−2.01145
$521$	28.1777	1.23449	0.617244	−	0.786772i	$-0.288249\pi$
$521$	28.1777	1.23449	0.617244	+	0.786772i	$0.288249\pi$
$522$	0	0
$523$	31.4514	1.37527	0.687637	−	0.726054i	$-0.258648\pi$
$523$	31.4514	1.37527	0.687637	+	0.726054i	$0.258648\pi$
$524$	−50.9382	−2.22525
$525$	0	0
$526$	9.21797	0.401923
$527$	4.49214	0.195681
$528$	0	0
$529$	56.6276	2.46207
$530$	52.1874	2.26687
$531$	0	0
$532$	0	0
$533$	−28.6038	−1.23897
$534$	0	0
$535$	12.9152	0.558371
$536$	23.0519	0.995691
$537$	0	0
$538$	30.1569	1.30016
$539$	0	0
$540$	0	0
$541$	37.2790	1.60275	0.801374	−	0.598164i	$-0.204103\pi$
$541$	37.2790	1.60275	0.801374	+	0.598164i	$0.204103\pi$
$542$	−25.6331	−1.10104
$543$	0	0
$544$	23.2948	0.998755
$545$	36.8251	1.57741
$546$	0	0
$547$	−10.7602	−0.460074	−0.230037	−	0.973182i	$-0.573885\pi$
$547$	−10.7602	−0.460074	−0.230037	+	0.973182i	$0.573885\pi$
$548$	−12.3601	−0.527998
$549$	0	0
$550$	−0.506050	−0.0215781
$551$	10.5801	0.450726
$552$	0	0
$553$	0	0
$554$	3.32640	0.141325
$555$	0	0
$556$	29.5720	1.25413
$557$	10.9427	0.463656	0.231828	−	0.972757i	$-0.425529\pi$
$557$	10.9427	0.463656	0.231828	+	0.972757i	$0.425529\pi$
$558$	0	0
$559$	30.4523	1.28800
$560$	0	0
$561$	0	0
$562$	66.3967	2.80078
$563$	−2.44499	−0.103044	−0.0515221	−	0.998672i	$-0.516407\pi$
$563$	−2.44499	−0.103044	−0.0515221	+	0.998672i	$0.516407\pi$
$564$	0	0
$565$	28.2825	1.18985
$566$	31.1545	1.30952
$567$	0	0
$568$	7.04949	0.295790
$569$	−28.5971	−1.19885	−0.599426	−	0.800430i	$-0.704605\pi$
$569$	−28.5971	−1.19885	−0.599426	+	0.800430i	$0.704605\pi$
$570$	0	0
$571$	−1.96272	−0.0821374	−0.0410687	−	0.999156i	$-0.513076\pi$
$571$	−1.96272	−0.0821374	−0.0410687	+	0.999156i	$0.513076\pi$
$572$	13.9707	0.584143
$573$	0	0
$574$	0	0
$575$	1.74049	0.0725836
$576$	0	0
$577$	41.3949	1.72329	0.861646	−	0.507509i	$-0.169434\pi$
$577$	41.3949	1.72329	0.861646	+	0.507509i	$0.169434\pi$
$578$	26.6418	1.10815
$579$	0	0
$580$	46.5300	1.93205
$581$	0	0
$582$	0	0
$583$	−9.17630	−0.380044
$584$	−2.08970	−0.0864724
$585$	0	0
$586$	−45.9152	−1.89674
$587$	−22.5301	−0.929917	−0.464959	−	0.885332i	$-0.653931\pi$
$587$	−22.5301	−0.929917	−0.464959	+	0.885332i	$0.653931\pi$
$588$	0	0
$589$	4.08311	0.168241
$590$	−50.7492	−2.08931
$591$	0	0
$592$	−39.0937	−1.60674
$593$	18.7131	0.768454	0.384227	−	0.923239i	$-0.374468\pi$
$593$	18.7131	0.768454	0.384227	+	0.923239i	$0.374468\pi$
$594$	0	0
$595$	0	0
$596$	19.5029	0.798871
$597$	0	0
$598$	−68.3615	−2.79551
$599$	−45.4471	−1.85692	−0.928459	−	0.371435i	$-0.878866\pi$
$599$	−45.4471	−1.85692	−0.928459	+	0.371435i	$0.878866\pi$
$600$	0	0
$601$	13.6085	0.555103	0.277551	−	0.960711i	$-0.410477\pi$
$601$	13.6085	0.555103	0.277551	+	0.960711i	$0.410477\pi$
$602$	0	0
$603$	0	0
$604$	58.8365	2.39403
$605$	−2.19202	−0.0891183
$606$	0	0
$607$	−29.1530	−1.18329	−0.591643	−	0.806200i	$-0.701520\pi$
$607$	−29.1530	−1.18329	−0.591643	+	0.806200i	$0.701520\pi$
$608$	21.1737	0.858705
$609$	0	0
$610$	76.6430	3.10318
$611$	32.8033	1.32708
$612$	0	0
$613$	20.2424	0.817583	0.408791	−	0.912628i	$-0.365950\pi$
$613$	20.2424	0.817583	0.408791	+	0.912628i	$0.365950\pi$
$614$	20.9716	0.846345
$615$	0	0
$616$	0	0
$617$	6.70281	0.269845	0.134922	−	0.990856i	$-0.456921\pi$
$617$	6.70281	0.269845	0.134922	+	0.990856i	$0.456921\pi$
$618$	0	0
$619$	5.54241	0.222768	0.111384	−	0.993777i	$-0.464472\pi$
$619$	5.54241	0.222768	0.111384	+	0.993777i	$0.464472\pi$
$620$	17.9571	0.721173
$621$	0	0
$622$	30.0617	1.20537
$623$	0	0
$624$	0	0
$625$	−23.9867	−0.959469
$626$	−4.03684	−0.161344
$627$	0	0
$628$	−88.4459	−3.52937
$629$	11.3665	0.453213
$630$	0	0
$631$	−34.6139	−1.37796	−0.688979	−	0.724782i	$-0.741941\pi$
$631$	−34.6139	−1.37796	−0.688979	+	0.724782i	$0.741941\pi$
$632$	106.820	4.24907
$633$	0	0
$634$	27.2399	1.08183
$635$	−32.2121	−1.27830
$636$	0	0
$637$	0	0
$638$	−11.6399	−0.460830
$639$	0	0
$640$	−8.37893	−0.331206
$641$	14.0006	0.552990	0.276495	−	0.961015i	$-0.410827\pi$
$641$	14.0006	0.552990	0.276495	+	0.961015i	$0.410827\pi$
$642$	0	0
$643$	9.03547	0.356324	0.178162	−	0.984001i	$-0.442985\pi$
$643$	9.03547	0.356324	0.178162	+	0.984001i	$0.442985\pi$
$644$	0	0
$645$	0	0
$646$	−15.8744	−0.624569
$647$	39.5406	1.55450	0.777250	−	0.629191i	$-0.216614\pi$
$647$	39.5406	1.55450	0.777250	+	0.629191i	$0.216614\pi$
$648$	0	0
$649$	8.92343	0.350275
$650$	−1.49424	−0.0586089
$651$	0	0
$652$	−5.98404	−0.234353
$653$	−21.6815	−0.848461	−0.424231	−	0.905554i	$-0.639455\pi$
$653$	−21.6815	−0.848461	−0.424231	+	0.905554i	$0.639455\pi$
$654$	0	0
$655$	23.5992	0.922098
$656$	86.4430	3.37503
$657$	0	0
$658$	0	0
$659$	−9.66475	−0.376485	−0.188243	−	0.982123i	$-0.560279\pi$
$659$	−9.66475	−0.376485	−0.188243	+	0.982123i	$0.560279\pi$
$660$	0	0
$661$	−5.78135	−0.224869	−0.112434	−	0.993659i	$-0.535865\pi$
$661$	−5.78135	−0.224869	−0.112434	+	0.993659i	$0.535865\pi$
$662$	−75.1912	−2.92239
$663$	0	0
$664$	−44.4175	−1.72373
$665$	0	0
$666$	0	0
$667$	40.0340	1.55012
$668$	28.9099	1.11856
$669$	0	0
$670$	−18.4997	−0.714705
$671$	−13.4764	−0.520252
$672$	0	0
$673$	−4.33155	−0.166969	−0.0834846	−	0.996509i	$-0.526605\pi$
$673$	−4.33155	−0.166969	−0.0834846	+	0.996509i	$0.526605\pi$
$674$	−36.2879	−1.39776
$675$	0	0
$676$	−20.2565	−0.779094
$677$	−20.6844	−0.794965	−0.397483	−	0.917610i	$-0.630116\pi$
$677$	−20.6844	−0.794965	−0.397483	+	0.917610i	$0.630116\pi$
$678$	0	0
$679$	0	0
$680$	−40.3030	−1.54555
$681$	0	0
$682$	−4.49214	−0.172013
$683$	5.46446	0.209092	0.104546	−	0.994520i	$-0.466661\pi$
$683$	5.46446	0.209092	0.104546	+	0.994520i	$0.466661\pi$
$684$	0	0
$685$	5.72633	0.218792
$686$	0	0
$687$	0	0
$688$	−92.0293	−3.50858
$689$	−27.0953	−1.03225
$690$	0	0
$691$	−43.4221	−1.65185	−0.825927	−	0.563777i	$-0.809348\pi$
$691$	−43.4221	−1.65185	−0.825927	+	0.563777i	$0.809348\pi$
$692$	82.6338	3.14127
$693$	0	0
$694$	−23.1971	−0.880548
$695$	−13.7005	−0.519688
$696$	0	0
$697$	−25.1334	−0.951994
$698$	−0.918199	−0.0347543
$699$	0	0
$700$	0	0
$701$	30.4055	1.14840	0.574200	−	0.818715i	$-0.305313\pi$
$701$	30.4055	1.14840	0.574200	+	0.818715i	$0.305313\pi$
$702$	0	0
$703$	10.3315	0.389661
$704$	−5.44792	−0.205326
$705$	0	0
$706$	53.4572	2.01189
$707$	0	0
$708$	0	0
$709$	18.6885	0.701861	0.350930	−	0.936402i	$-0.385865\pi$
$709$	18.6885	0.701861	0.350930	+	0.936402i	$0.385865\pi$
$710$	−5.65738	−0.212318
$711$	0	0
$712$	−18.4270	−0.690580
$713$	15.4501	0.578612
$714$	0	0
$715$	−6.47249	−0.242057
$716$	−1.99923	−0.0747149
$717$	0	0
$718$	27.4670	1.02506
$719$	32.5232	1.21291	0.606456	−	0.795117i	$-0.292591\pi$
$719$	32.5232	1.21291	0.606456	+	0.795117i	$0.292591\pi$
$720$	0	0
$721$	0	0
$722$	34.8665	1.29760
$723$	0	0
$724$	11.3647	0.422367
$725$	0.875061	0.0324989
$726$	0	0
$727$	−23.7302	−0.880103	−0.440051	−	0.897973i	$-0.645040\pi$
$727$	−23.7302	−0.880103	−0.440051	+	0.897973i	$0.645040\pi$
$728$	0	0
$729$	0	0
$730$	1.67703	0.0620698
$731$	26.7576	0.989666
$732$	0	0
$733$	35.9886	1.32927	0.664635	−	0.747168i	$-0.268587\pi$
$733$	35.9886	1.32927	0.664635	+	0.747168i	$0.268587\pi$
$734$	−28.2263	−1.04185
$735$	0	0
$736$	80.1194	2.95324
$737$	3.25287	0.119821
$738$	0	0
$739$	1.23985	0.0456087	0.0228043	−	0.999740i	$-0.492741\pi$
$739$	1.23985	0.0456087	0.0228043	+	0.999740i	$0.492741\pi$
$740$	45.4370	1.67030
$741$	0	0
$742$	0	0
$743$	−18.6353	−0.683662	−0.341831	−	0.939761i	$-0.611047\pi$
$743$	−18.6353	−0.683662	−0.341831	+	0.939761i	$0.611047\pi$
$744$	0	0
$745$	−9.03553	−0.331036
$746$	−38.2862	−1.40176
$747$	0	0
$748$	12.2756	0.448841
$749$	0	0
$750$	0	0
$751$	−45.3810	−1.65598	−0.827988	−	0.560745i	$-0.810515\pi$
$751$	−45.3810	−1.65598	−0.827988	+	0.560745i	$0.810515\pi$
$752$	−99.1339	−3.61504
$753$	0	0
$754$	−34.3698	−1.25168
$755$	−27.2584	−0.992036
$756$	0	0
$757$	35.8708	1.30375	0.651874	−	0.758327i	$-0.273983\pi$
$757$	35.8708	1.30375	0.651874	+	0.758327i	$0.273983\pi$
$758$	−33.3269	−1.21049
$759$	0	0
$760$	−36.6332	−1.32883
$761$	−18.7213	−0.678645	−0.339323	−	0.940670i	$-0.610198\pi$
$761$	−18.7213	−0.678645	−0.339323	+	0.940670i	$0.610198\pi$
$762$	0	0
$763$	0	0
$764$	41.1297	1.48802
$765$	0	0
$766$	32.3154	1.16760
$767$	26.3487	0.951395
$768$	0	0
$769$	6.00582	0.216576	0.108288	−	0.994120i	$-0.465463\pi$
$769$	6.00582	0.216576	0.108288	+	0.994120i	$0.465463\pi$
$770$	0	0
$771$	0	0
$772$	−38.0648	−1.36998
$773$	14.1425	0.508669	0.254334	−	0.967116i	$-0.418144\pi$
$773$	14.1425	0.508669	0.254334	+	0.967116i	$0.418144\pi$
$774$	0	0
$775$	0.337708	0.0121308
$776$	−53.2705	−1.91230
$777$	0	0
$778$	34.2753	1.22883
$779$	−22.8448	−0.818501
$780$	0	0
$781$	0.994758	0.0355953
$782$	−60.0673	−2.14800
$783$	0	0
$784$	0	0
$785$	40.9762	1.46250
$786$	0	0
$787$	50.0291	1.78334	0.891672	−	0.452681i	$-0.149533\pi$
$787$	50.0291	1.78334	0.891672	+	0.452681i	$0.149533\pi$
$788$	22.7070	0.808904
$789$	0	0
$790$	−85.7255	−3.04998
$791$	0	0
$792$	0	0
$793$	−39.7925	−1.41307
$794$	−48.5019	−1.72127
$795$	0	0
$796$	13.8215	0.489889
$797$	7.24763	0.256724	0.128362	−	0.991727i	$-0.459028\pi$
$797$	7.24763	0.256724	0.128362	+	0.991727i	$0.459028\pi$
$798$	0	0
$799$	28.8233	1.01969
$800$	1.75124	0.0619157
$801$	0	0
$802$	−18.6079	−0.657069
$803$	−0.294879	−0.0104061
$804$	0	0
$805$	0	0
$806$	−13.2642	−0.467210
$807$	0	0
$808$	−94.6808	−3.33086
$809$	−0.512279	−0.0180108	−0.00900538	−	0.999959i	$-0.502867\pi$
$809$	−0.512279	−0.0180108	−0.00900538	+	0.999959i	$0.502867\pi$
$810$	0	0
$811$	40.5892	1.42528	0.712640	−	0.701530i	$-0.247499\pi$
$811$	40.5892	1.42528	0.712640	+	0.701530i	$0.247499\pi$
$812$	0	0
$813$	0	0
$814$	−11.3665	−0.398396
$815$	2.77235	0.0971113
$816$	0	0
$817$	24.3212	0.850890
$818$	−7.89037	−0.275880
$819$	0	0
$820$	−100.469	−3.50854
$821$	22.6189	0.789405	0.394702	−	0.918809i	$-0.370848\pi$
$821$	22.6189	0.789405	0.394702	+	0.918809i	$0.370848\pi$
$822$	0	0
$823$	6.03629	0.210412	0.105206	−	0.994450i	$-0.466450\pi$
$823$	6.03629	0.210412	0.105206	+	0.994450i	$0.466450\pi$
$824$	37.7620	1.31550
$825$	0	0
$826$	0	0
$827$	−4.54161	−0.157927	−0.0789636	−	0.996878i	$-0.525161\pi$
$827$	−4.54161	−0.157927	−0.0789636	+	0.996878i	$0.525161\pi$
$828$	0	0
$829$	25.8717	0.898560	0.449280	−	0.893391i	$-0.351681\pi$
$829$	25.8717	0.898560	0.449280	+	0.893391i	$0.351681\pi$
$830$	35.6460	1.23729
$831$	0	0
$832$	−16.0863	−0.557693
$833$	0	0
$834$	0	0
$835$	−13.3937	−0.463508
$836$	11.1579	0.385903
$837$	0	0
$838$	64.7348	2.23623
$839$	28.9458	0.999319	0.499660	−	0.866222i	$-0.333458\pi$
$839$	28.9458	0.999319	0.499660	+	0.866222i	$0.333458\pi$
$840$	0	0
$841$	−8.87225	−0.305940
$842$	31.1230	1.07257
$843$	0	0
$844$	−46.4966	−1.60048
$845$	9.38463	0.322841
$846$	0	0
$847$	0	0
$848$	81.8841	2.81191
$849$	0	0
$850$	−1.31295	−0.0450337
$851$	39.0937	1.34011
$852$	0	0
$853$	−4.72334	−0.161724	−0.0808620	−	0.996725i	$-0.525767\pi$
$853$	−4.72334	−0.161724	−0.0808620	+	0.996725i	$0.525767\pi$
$854$	0	0
$855$	0	0
$856$	41.7538	1.42711
$857$	−28.2092	−0.963606	−0.481803	−	0.876279i	$-0.660018\pi$
$857$	−28.2092	−0.963606	−0.481803	+	0.876279i	$0.660018\pi$
$858$	0	0
$859$	−7.43580	−0.253706	−0.126853	−	0.991922i	$-0.540488\pi$
$859$	−7.43580	−0.253706	−0.126853	+	0.991922i	$0.540488\pi$
$860$	106.962	3.64737
$861$	0	0
$862$	−44.5479	−1.51731
$863$	−45.6002	−1.55225	−0.776125	−	0.630579i	$-0.782817\pi$
$863$	−45.6002	−1.55225	−0.776125	+	0.630579i	$0.782817\pi$
$864$	0	0
$865$	−38.2835	−1.30168
$866$	−18.2194	−0.619122
$867$	0	0
$868$	0	0
$869$	15.0734	0.511332
$870$	0	0
$871$	9.60491	0.325450
$872$	119.053	4.03164
$873$	0	0
$874$	−54.5978	−1.84680
$875$	0	0
$876$	0	0
$877$	−39.7363	−1.34180	−0.670900	−	0.741548i	$-0.734092\pi$
$877$	−39.7363	−1.34180	−0.670900	+	0.741548i	$0.734092\pi$
$878$	−103.137	−3.48069
$879$	0	0
$880$	19.5603	0.659379
$881$	−28.1813	−0.949452	−0.474726	−	0.880134i	$-0.657453\pi$
$881$	−28.1813	−0.949452	−0.474726	+	0.880134i	$0.657453\pi$
$882$	0	0
$883$	40.6909	1.36936	0.684679	−	0.728845i	$-0.259942\pi$
$883$	40.6909	1.36936	0.684679	+	0.728845i	$0.259942\pi$
$884$	36.2469	1.21911
$885$	0	0
$886$	90.8105	3.05084
$887$	18.4938	0.620960	0.310480	−	0.950580i	$-0.399510\pi$
$887$	18.4938	0.620960	0.310480	+	0.950580i	$0.399510\pi$
$888$	0	0
$889$	0	0
$890$	14.7881	0.495697
$891$	0	0
$892$	−77.1822	−2.58425
$893$	26.1988	0.876708
$894$	0	0
$895$	0.926227	0.0309604
$896$	0	0
$897$	0	0
$898$	−26.9256	−0.898520
$899$	7.76780	0.259071
$900$	0	0
$901$	−23.8079	−0.793156
$902$	25.1334	0.836850
$903$	0	0
$904$	91.4351	3.04109
$905$	−5.26517	−0.175020
$906$	0	0
$907$	−47.9651	−1.59266	−0.796328	−	0.604865i	$-0.793227\pi$
$907$	−47.9651	−1.59266	−0.796328	+	0.604865i	$0.793227\pi$
$908$	−106.972	−3.54999
$909$	0	0
$910$	0	0
$911$	−11.9369	−0.395489	−0.197744	−	0.980254i	$-0.563362\pi$
$911$	−11.9369	−0.395489	−0.197744	+	0.980254i	$0.563362\pi$
$912$	0	0
$913$	−6.26778	−0.207433
$914$	48.2647	1.59646
$915$	0	0
$916$	−17.5404	−0.579552
$917$	0	0
$918$	0	0
$919$	59.0449	1.94771	0.973856	−	0.227167i	$-0.0729462\pi$
$919$	59.0449	1.94771	0.973856	+	0.227167i	$0.0729462\pi$
$920$	−138.617	−4.57007
$921$	0	0
$922$	−44.2959	−1.45881
$923$	2.93727	0.0966815
$924$	0	0
$925$	0.854506	0.0280960
$926$	−83.9397	−2.75843
$927$	0	0
$928$	40.2813	1.32230
$929$	38.0748	1.24919	0.624597	−	0.780948i	$-0.285263\pi$
$929$	38.0748	1.24919	0.624597	+	0.780948i	$0.285263\pi$
$930$	0	0
$931$	0	0
$932$	36.4271	1.19321
$933$	0	0
$934$	−0.564066	−0.0184568
$935$	−5.68719	−0.185991
$936$	0	0
$937$	−19.4407	−0.635099	−0.317550	−	0.948242i	$-0.602860\pi$
$937$	−19.4407	−0.635099	−0.317550	+	0.948242i	$0.602860\pi$
$938$	0	0
$939$	0	0
$940$	115.219	3.75804
$941$	39.7534	1.29592	0.647961	−	0.761673i	$-0.275622\pi$
$941$	39.7534	1.29592	0.647961	+	0.761673i	$0.275622\pi$
$942$	0	0
$943$	−86.4430	−2.81497
$944$	−79.6276	−2.59166
$945$	0	0
$946$	−26.7576	−0.869965
$947$	−32.3542	−1.05137	−0.525686	−	0.850679i	$-0.676191\pi$
$947$	−32.3542	−1.05137	−0.525686	+	0.850679i	$0.676191\pi$
$948$	0	0
$949$	−0.870704	−0.0282642
$950$	−1.19339	−0.0387188
$951$	0	0
$952$	0	0
$953$	−8.56486	−0.277443	−0.138722	−	0.990331i	$-0.544299\pi$
$953$	−8.56486	−0.277443	−0.138722	+	0.990331i	$0.544299\pi$
$954$	0	0
$955$	−19.0550	−0.616606
$956$	19.5401	0.631972
$957$	0	0
$958$	−55.6810	−1.79897
$959$	0	0
$960$	0	0
$961$	−28.0022	−0.903297
$962$	−33.5625	−1.08210
$963$	0	0
$964$	−69.8166	−2.24864
$965$	17.6351	0.567694
$966$	0	0
$967$	11.2215	0.360858	0.180429	−	0.983588i	$-0.442251\pi$
$967$	11.2215	0.360858	0.180429	+	0.983588i	$0.442251\pi$
$968$	−7.08664	−0.227773
$969$	0	0
$970$	42.7508	1.37265
$971$	42.7645	1.37238	0.686188	−	0.727424i	$-0.259283\pi$
$971$	42.7645	1.37238	0.686188	+	0.727424i	$0.259283\pi$
$972$	0	0
$973$	0	0
$974$	−93.6329	−3.00019
$975$	0	0
$976$	120.256	3.84930
$977$	−24.7195	−0.790848	−0.395424	−	0.918499i	$-0.629402\pi$
$977$	−24.7195	−0.790848	−0.395424	+	0.918499i	$0.629402\pi$
$978$	0	0
$979$	−2.60024	−0.0831041
$980$	0	0
$981$	0	0
$982$	−13.2482	−0.422767
$983$	40.7257	1.29895	0.649475	−	0.760383i	$-0.274989\pi$
$983$	40.7257	1.29895	0.649475	+	0.760383i	$0.274989\pi$
$984$	0	0
$985$	−10.5200	−0.335194
$986$	−30.1998	−0.961757
$987$	0	0
$988$	32.9464	1.04816
$989$	92.0293	2.92636
$990$	0	0
$991$	−57.5124	−1.82694	−0.913471	−	0.406905i	$-0.866608\pi$
$991$	−57.5124	−1.82694	−0.913471	+	0.406905i	$0.866608\pi$
$992$	15.5455	0.493571
$993$	0	0
$994$	0	0
$995$	−6.40337	−0.203000
$996$	0	0
$997$	−37.8183	−1.19772	−0.598858	−	0.800855i	$-0.704379\pi$
$997$	−37.8183	−1.19772	−0.598858	+	0.800855i	$0.704379\pi$
$998$	−20.0591	−0.634960
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4851.2.a.ca.1.1		5
3.2	odd	2		1617.2.a.ba.1.5		5
7.2	even	3		693.2.i.j.298.5		10
7.4	even	3		693.2.i.j.100.5		10
7.6	odd	2		4851.2.a.bz.1.1		5
21.2	odd	6		231.2.i.f.67.1	✓	10
21.11	odd	6		231.2.i.f.100.1	yes	10
21.20	even	2		1617.2.a.bb.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.i.f.67.1	✓	10	21.2	odd	6
231.2.i.f.100.1	yes	10	21.11	odd	6
693.2.i.j.100.5		10	7.4	even	3
693.2.i.j.298.5		10	7.2	even	3
1617.2.a.ba.1.5		5	3.2	odd	2
1617.2.a.bb.1.5		5	21.20	even	2
4851.2.a.bz.1.1		5	7.6	odd	2
4851.2.a.ca.1.1		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4851.a (trivial)

\(f(q)\)	\(=\)	\(q-2.59450 q^{2} +4.73141 q^{4} -2.19202 q^{5} -7.08664 q^{8} +O(q^{10})\)	\(q-2.59450 q^{2} +4.73141 q^{4} -2.19202 q^{5} -7.08664 q^{8} +5.68719 q^{10} -1.00000 q^{11} -2.95275 q^{13} +8.92343 q^{16} -2.59450 q^{17} -2.35825 q^{19} -10.3713 q^{20} +2.59450 q^{22} -8.92343 q^{23} -0.195048 q^{25} +7.66090 q^{26} -4.48640 q^{29} -1.73141 q^{31} -8.97854 q^{32} +6.73141 q^{34} -4.38101 q^{37} +6.11848 q^{38} +15.5341 q^{40} +9.68719 q^{41} -10.3132 q^{43} -4.73141 q^{44} +23.1518 q^{46} -11.1094 q^{47} +0.506050 q^{50} -13.9707 q^{52} +9.17630 q^{53} +2.19202 q^{55} +11.6399 q^{58} -8.92343 q^{59} +13.4764 q^{61} +4.49214 q^{62} +5.44792 q^{64} +6.47249 q^{65} -3.25287 q^{67} -12.2756 q^{68} -0.994758 q^{71} +0.294879 q^{73} +11.3665 q^{74} -11.1579 q^{76} -15.0734 q^{79} -19.5603 q^{80} -25.1334 q^{82} +6.26778 q^{83} +5.68719 q^{85} +26.7576 q^{86} +7.08664 q^{88} +2.60024 q^{89} -42.2204 q^{92} +28.8233 q^{94} +5.16934 q^{95} +7.51704 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 10 q^{4} + 4 q^{5} - 6 q^{8}+O(q^{10}) \) 5 * q - 2 * q^2 + 10 * q^4 + 4 * q^5 - 6 * q^8	\( 5 q - 2 q^{2} + 10 q^{4} + 4 q^{5} - 6 q^{8} + 2 q^{10} - 5 q^{11} + 5 q^{13} + 16 q^{16} - 2 q^{17} - 3 q^{19} + 8 q^{20} + 2 q^{22} - 16 q^{23} + 7 q^{25} + 10 q^{26} + 5 q^{31} - 4 q^{32} + 20 q^{34} + 15 q^{37} - 6 q^{38} - 6 q^{40} + 22 q^{41} + 3 q^{43} - 10 q^{44} + 16 q^{46} + 2 q^{47} - 34 q^{50} + 40 q^{52} - 6 q^{53} - 4 q^{55} + 12 q^{58} - 16 q^{59} + 12 q^{61} + 4 q^{62} - 4 q^{64} + 28 q^{65} + 7 q^{67} - 10 q^{68} - 24 q^{71} + 17 q^{73} + 36 q^{74} + 30 q^{76} + 7 q^{79} - 16 q^{80} + 8 q^{82} + 12 q^{83} + 2 q^{85} + 18 q^{86} + 6 q^{88} + 6 q^{89} - 68 q^{92} + 82 q^{94} + 18 q^{95} - 14 q^{97}+O(q^{100}) \) 5 * q - 2 * q^2 + 10 * q^4 + 4 * q^5 - 6 * q^8 + 2 * q^10 - 5 * q^11 + 5 * q^13 + 16 * q^16 - 2 * q^17 - 3 * q^19 + 8 * q^20 + 2 * q^22 - 16 * q^23 + 7 * q^25 + 10 * q^26 + 5 * q^31 - 4 * q^32 + 20 * q^34 + 15 * q^37 - 6 * q^38 - 6 * q^40 + 22 * q^41 + 3 * q^43 - 10 * q^44 + 16 * q^46 + 2 * q^47 - 34 * q^50 + 40 * q^52 - 6 * q^53 - 4 * q^55 + 12 * q^58 - 16 * q^59 + 12 * q^61 + 4 * q^62 - 4 * q^64 + 28 * q^65 + 7 * q^67 - 10 * q^68 - 24 * q^71 + 17 * q^73 + 36 * q^74 + 30 * q^76 + 7 * q^79 - 16 * q^80 + 8 * q^82 + 12 * q^83 + 2 * q^85 + 18 * q^86 + 6 * q^88 + 6 * q^89 - 68 * q^92 + 82 * q^94 + 18 * q^95 - 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more