LMFDB - Embedded newform 875.2.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [875,2,Mod(1,875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(875, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("875.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 875 = 5^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	875.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:	$6.98691017686$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 10x^{6} + 30x^{5} + 29x^{4} - 79x^{3} - 43x^{2} + 62x + 29 $ x^8 - 3x^7 - 10x^6 + 30x^5 + 29x^4 - 79x^3 - 43x^2 + 62*x + 29
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.26874$ of defining polynomial
Character	$\chi$	$=$	875.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.64338 q^{2} -0.833911 q^{3} +4.98745 q^{4} +2.20434 q^{6} -1.00000 q^{7} -7.89697 q^{8} -2.30459 q^{9} -4.54519 q^{11} -4.15909 q^{12} -3.85791 q^{13} +2.64338 q^{14} +10.8998 q^{16} +4.62525 q^{17} +6.09191 q^{18} +6.66244 q^{19} +0.833911 q^{21} +12.0147 q^{22} -4.15079 q^{23} +6.58537 q^{24} +10.1979 q^{26} +4.42356 q^{27} -4.98745 q^{28} -8.32707 q^{29} -0.269795 q^{31} -13.0183 q^{32} +3.79028 q^{33} -12.2263 q^{34} -11.4940 q^{36} +3.02342 q^{37} -17.6114 q^{38} +3.21716 q^{39} -2.70751 q^{41} -2.20434 q^{42} +0.570645 q^{43} -22.6689 q^{44} +10.9721 q^{46} +6.18001 q^{47} -9.08944 q^{48} +1.00000 q^{49} -3.85704 q^{51} -19.2412 q^{52} +12.2587 q^{53} -11.6931 q^{54} +7.89697 q^{56} -5.55588 q^{57} +22.0116 q^{58} -0.419019 q^{59} +11.5307 q^{61} +0.713171 q^{62} +2.30459 q^{63} +12.6127 q^{64} -10.0192 q^{66} +0.289042 q^{67} +23.0682 q^{68} +3.46139 q^{69} -7.27856 q^{71} +18.1993 q^{72} -9.34497 q^{73} -7.99204 q^{74} +33.2286 q^{76} +4.54519 q^{77} -8.50417 q^{78} +4.37430 q^{79} +3.22492 q^{81} +7.15697 q^{82} -11.3312 q^{83} +4.15909 q^{84} -1.50843 q^{86} +6.94404 q^{87} +35.8932 q^{88} +14.6783 q^{89} +3.85791 q^{91} -20.7018 q^{92} +0.224985 q^{93} -16.3361 q^{94} +10.8561 q^{96} +9.86723 q^{97} -2.64338 q^{98} +10.4748 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} - 8 q^{3} + 13 q^{4} + 2 q^{6} - 8 q^{7} + 12 q^{8} + 18 q^{9} - 5 q^{11} - 20 q^{12} - 6 q^{13} - q^{14} + 35 q^{16} + 13 q^{17} + 3 q^{18} + 13 q^{19} + 8 q^{21} + 22 q^{22} - 5 q^{23} - 3 q^{24}+ \cdots - 31 q^{99}+O(q^{100}) $ 8 * q + q^2 - 8 * q^3 + 13 * q^4 + 2 * q^6 - 8 * q^7 + 12 * q^8 + 18 * q^9 - 5 * q^11 - 20 * q^12 - 6 * q^13 - q^14 + 35 * q^16 + 13 * q^17 + 3 * q^18 + 13 * q^19 + 8 * q^21 + 22 * q^22 - 5 * q^23 - 3 * q^24 - 5 * q^26 - 29 * q^27 - 13 * q^28 + 22 * q^29 + 3 * q^31 + 35 * q^32 - 10 * q^33 + 19 * q^34 + 27 * q^36 + 7 * q^37 - 5 * q^38 + 7 * q^39 + 12 * q^41 - 2 * q^42 - q^43 - 41 * q^44 + 11 * q^46 - 2 * q^47 + 12 * q^48 + 8 * q^49 + 10 * q^51 - 49 * q^52 + 10 * q^53 - 31 * q^54 - 12 * q^56 + 17 * q^57 + 58 * q^58 - 11 * q^59 + 45 * q^61 + 26 * q^62 - 18 * q^63 + 82 * q^64 - 42 * q^66 + 6 * q^67 + 58 * q^68 + 25 * q^69 - 15 * q^71 + 85 * q^72 - 32 * q^73 + 24 * q^74 + 26 * q^76 + 5 * q^77 - 32 * q^78 + 12 * q^79 + 36 * q^81 - 15 * q^83 + 20 * q^84 - 48 * q^86 - 35 * q^87 - 2 * q^88 + 12 * q^89 + 6 * q^91 - 78 * q^92 + 33 * q^93 - 38 * q^94 - 30 * q^96 + 20 * q^97 + q^98 - 31 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.64338	−1.86915	−0.934576	−	0.355765i	$-0.884220\pi$
$2$	−2.64338	−1.86915	−0.934576	+	0.355765i	$0.884220\pi$
$3$	−0.833911	−0.481459	−0.240729	−	0.970592i	$-0.577387\pi$
$3$	−0.833911	−0.481459	−0.240729	+	0.970592i	$0.577387\pi$
$4$	4.98745	2.49373
$5$	0	0
$5$	0	0
$6$	2.20434	0.899919
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−7.89697	−2.79200
$9$	−2.30459	−0.768197
$10$	0	0
$11$	−4.54519	−1.37043	−0.685213	−	0.728343i	$-0.740291\pi$
$11$	−4.54519	−1.37043	−0.685213	+	0.728343i	$0.740291\pi$
$12$	−4.15909	−1.20063
$13$	−3.85791	−1.06999	−0.534996	−	0.844854i	$-0.679687\pi$
$13$	−3.85791	−1.06999	−0.534996	+	0.844854i	$0.679687\pi$
$14$	2.64338	0.706473
$15$	0	0
$16$	10.8998	2.72494
$17$	4.62525	1.12179	0.560893	−	0.827888i	$-0.310458\pi$
$17$	4.62525	1.12179	0.560893	+	0.827888i	$0.310458\pi$
$18$	6.09191	1.43588
$19$	6.66244	1.52847	0.764235	−	0.644938i	$-0.223117\pi$
$19$	6.66244	1.52847	0.764235	+	0.644938i	$0.223117\pi$
$20$	0	0
$21$	0.833911	0.181974
$22$	12.0147	2.56153
$23$	−4.15079	−0.865499	−0.432749	−	0.901514i	$-0.642456\pi$
$23$	−4.15079	−0.865499	−0.432749	+	0.901514i	$0.642456\pi$
$24$	6.58537	1.34423
$25$	0	0
$26$	10.1979	1.99998
$27$	4.42356	0.851314
$28$	−4.98745	−0.942540
$29$	−8.32707	−1.54630	−0.773149	−	0.634224i	$-0.781320\pi$
$29$	−8.32707	−1.54630	−0.773149	+	0.634224i	$0.781320\pi$
$30$	0	0
$31$	−0.269795	−0.0484567	−0.0242283	−	0.999706i	$-0.507713\pi$
$31$	−0.269795	−0.0484567	−0.0242283	+	0.999706i	$0.507713\pi$
$32$	−13.0183	−2.30133
$33$	3.79028	0.659803
$34$	−12.2263	−2.09679
$35$	0	0
$36$	−11.4940	−1.91567
$37$	3.02342	0.497047	0.248524	−	0.968626i	$-0.420055\pi$
$37$	3.02342	0.497047	0.248524	+	0.968626i	$0.420055\pi$
$38$	−17.6114	−2.85694
$39$	3.21716	0.515158
$40$	0	0
$41$	−2.70751	−0.422842	−0.211421	−	0.977395i	$-0.567809\pi$
$41$	−2.70751	−0.422842	−0.211421	+	0.977395i	$0.567809\pi$
$42$	−2.20434	−0.340138
$43$	0.570645	0.0870226	0.0435113	−	0.999053i	$-0.486146\pi$
$43$	0.570645	0.0870226	0.0435113	+	0.999053i	$0.486146\pi$
$44$	−22.6689	−3.41747
$45$	0	0
$46$	10.9721	1.61775
$47$	6.18001	0.901448	0.450724	−	0.892663i	$-0.351166\pi$
$47$	6.18001	0.901448	0.450724	+	0.892663i	$0.351166\pi$
$48$	−9.08944	−1.31195
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.85704	−0.540094
$52$	−19.2412	−2.66827
$53$	12.2587	1.68387	0.841934	−	0.539581i	$-0.181417\pi$
$53$	12.2587	1.68387	0.841934	+	0.539581i	$0.181417\pi$
$54$	−11.6931	−1.59123
$55$	0	0
$56$	7.89697	1.05528
$57$	−5.55588	−0.735895
$58$	22.0116	2.89026
$59$	−0.419019	−0.0545516	−0.0272758	−	0.999628i	$-0.508683\pi$
$59$	−0.419019	−0.0545516	−0.0272758	+	0.999628i	$0.508683\pi$
$60$	0	0
$61$	11.5307	1.47636	0.738180	−	0.674604i	$-0.235686\pi$
$61$	11.5307	1.47636	0.738180	+	0.674604i	$0.235686\pi$
$62$	0.713171	0.0905728
$63$	2.30459	0.290351
$64$	12.6127	1.57659
$65$	0	0
$66$	−10.0192	−1.23327
$67$	0.289042	0.0353121	0.0176560	−	0.999844i	$-0.494380\pi$
$67$	0.289042	0.0353121	0.0176560	+	0.999844i	$0.494380\pi$
$68$	23.0682	2.79743
$69$	3.46139	0.416702
$70$	0	0
$71$	−7.27856	−0.863806	−0.431903	−	0.901920i	$-0.642158\pi$
$71$	−7.27856	−0.863806	−0.431903	+	0.901920i	$0.642158\pi$
$72$	18.1993	2.14481
$73$	−9.34497	−1.09375	−0.546873	−	0.837215i	$-0.684182\pi$
$73$	−9.34497	−1.09375	−0.546873	+	0.837215i	$0.684182\pi$
$74$	−7.99204	−0.929056
$75$	0	0
$76$	33.2286	3.81158
$77$	4.54519	0.517972
$78$	−8.50417	−0.962907
$79$	4.37430	0.492148	0.246074	−	0.969251i	$-0.420859\pi$
$79$	4.37430	0.492148	0.246074	+	0.969251i	$0.420859\pi$
$80$	0	0
$81$	3.22492	0.358325
$82$	7.15697	0.790355
$83$	−11.3312	−1.24376	−0.621879	−	0.783114i	$-0.713630\pi$
$83$	−11.3312	−1.24376	−0.621879	+	0.783114i	$0.713630\pi$
$84$	4.15909	0.453794
$85$	0	0
$86$	−1.50843	−0.162658
$87$	6.94404	0.744479
$88$	35.8932	3.82623
$89$	14.6783	1.55589	0.777947	−	0.628330i	$-0.216261\pi$
$89$	14.6783	1.55589	0.777947	+	0.628330i	$0.216261\pi$
$90$	0	0
$91$	3.85791	0.404419
$92$	−20.7018	−2.15832
$93$	0.224985	0.0233299
$94$	−16.3361	−1.68494
$95$	0	0
$96$	10.8561	1.10800
$97$	9.86723	1.00187	0.500933	−	0.865486i	$-0.332990\pi$
$97$	9.86723	1.00187	0.500933	+	0.865486i	$0.332990\pi$
$98$	−2.64338	−0.267022
$99$	10.4748	1.05276
$100$	0	0
$101$	6.27767	0.624651	0.312326	−	0.949975i	$-0.398892\pi$
$101$	6.27767	0.624651	0.312326	+	0.949975i	$0.398892\pi$
$102$	10.1956	1.00952
$103$	−2.61593	−0.257756	−0.128878	−	0.991660i	$-0.541138\pi$
$103$	−2.61593	−0.257756	−0.128878	+	0.991660i	$0.541138\pi$
$104$	30.4658	2.98742
$105$	0	0
$106$	−32.4045	−3.14740
$107$	0.176723	0.0170845	0.00854225	−	0.999964i	$-0.497281\pi$
$107$	0.176723	0.0170845	0.00854225	+	0.999964i	$0.497281\pi$
$108$	22.0623	2.12294
$109$	10.7849	1.03301	0.516503	−	0.856285i	$-0.327233\pi$
$109$	10.7849	1.03301	0.516503	+	0.856285i	$0.327233\pi$
$110$	0	0
$111$	−2.52126	−0.239308
$112$	−10.8998	−1.02993
$113$	8.12932	0.764742	0.382371	−	0.924009i	$-0.375108\pi$
$113$	8.12932	0.764742	0.382371	+	0.924009i	$0.375108\pi$
$114$	14.6863	1.37550
$115$	0	0
$116$	−41.5309	−3.85604
$117$	8.89092	0.821966
$118$	1.10762	0.101965
$119$	−4.62525	−0.423996
$120$	0	0
$121$	9.65873	0.878066
$122$	−30.4801	−2.75954
$123$	2.25782	0.203581
$124$	−1.34559	−0.120838
$125$	0	0
$126$	−6.09191	−0.542710
$127$	19.1636	1.70050	0.850249	−	0.526380i	$-0.176451\pi$
$127$	19.1636	1.70050	0.850249	+	0.526380i	$0.176451\pi$
$128$	−7.30364	−0.645557
$129$	−0.475867	−0.0418978
$130$	0	0
$131$	−18.0366	−1.57587	−0.787934	−	0.615760i	$-0.788849\pi$
$131$	−18.0366	−1.57587	−0.787934	+	0.615760i	$0.788849\pi$
$132$	18.9038	1.64537
$133$	−6.66244	−0.577707
$134$	−0.764047	−0.0660036
$135$	0	0
$136$	−36.5254	−3.13203
$137$	−5.79298	−0.494927	−0.247464	−	0.968897i	$-0.579597\pi$
$137$	−5.79298	−0.494927	−0.247464	+	0.968897i	$0.579597\pi$
$138$	−9.14976	−0.778879
$139$	4.68119	0.397053	0.198527	−	0.980095i	$-0.436384\pi$
$139$	4.68119	0.397053	0.198527	+	0.980095i	$0.436384\pi$
$140$	0	0
$141$	−5.15358	−0.434010
$142$	19.2400	1.61458
$143$	17.5349	1.46635
$144$	−25.1195	−2.09329
$145$	0	0
$146$	24.7023	2.04438
$147$	−0.833911	−0.0687798
$148$	15.0792	1.23950
$149$	19.6161	1.60701	0.803506	−	0.595297i	$-0.202965\pi$
$149$	19.6161	1.60701	0.803506	+	0.595297i	$0.202965\pi$
$150$	0	0
$151$	16.5712	1.34855	0.674274	−	0.738481i	$-0.264457\pi$
$151$	16.5712	1.34855	0.674274	+	0.738481i	$0.264457\pi$
$152$	−52.6131	−4.26748
$153$	−10.6593	−0.861754
$154$	−12.0147	−0.968168
$155$	0	0
$156$	16.0454	1.28466
$157$	10.4246	0.831972	0.415986	−	0.909371i	$-0.363436\pi$
$157$	10.4246	0.831972	0.415986	+	0.909371i	$0.363436\pi$
$158$	−11.5629	−0.919899
$159$	−10.2227	−0.810713
$160$	0	0
$161$	4.15079	0.327128
$162$	−8.52469	−0.669763
$163$	4.02969	0.315630	0.157815	−	0.987469i	$-0.449555\pi$
$163$	4.02969	0.315630	0.157815	+	0.987469i	$0.449555\pi$
$164$	−13.5036	−1.05445
$165$	0	0
$166$	29.9526	2.32477
$167$	−22.2680	−1.72315	−0.861573	−	0.507633i	$-0.830521\pi$
$167$	−22.2680	−1.72315	−0.861573	+	0.507633i	$0.830521\pi$
$168$	−6.58537	−0.508072
$169$	1.88350	0.144885
$170$	0	0
$171$	−15.3542	−1.17417
$172$	2.84606	0.217010
$173$	17.1363	1.30285	0.651426	−	0.758712i	$-0.274171\pi$
$173$	17.1363	1.30285	0.651426	+	0.758712i	$0.274171\pi$
$174$	−18.3557	−1.39154
$175$	0	0
$176$	−49.5415	−3.73433
$177$	0.349424	0.0262643
$178$	−38.8003	−2.90820
$179$	15.3534	1.14757	0.573783	−	0.819007i	$-0.305475\pi$
$179$	15.3534	1.14757	0.573783	+	0.819007i	$0.305475\pi$
$180$	0	0
$181$	−6.20953	−0.461551	−0.230775	−	0.973007i	$-0.574126\pi$
$181$	−6.20953	−0.461551	−0.230775	+	0.973007i	$0.574126\pi$
$182$	−10.1979	−0.755921
$183$	−9.61561	−0.710806
$184$	32.7786	2.41647
$185$	0	0
$186$	−0.594721	−0.0436071
$187$	−21.0226	−1.53733
$188$	30.8225	2.24796
$189$	−4.42356	−0.321767
$190$	0	0
$191$	7.32070	0.529707	0.264854	−	0.964289i	$-0.414676\pi$
$191$	7.32070	0.529707	0.264854	+	0.964289i	$0.414676\pi$
$192$	−10.5179	−0.759064
$193$	−23.6227	−1.70040	−0.850200	−	0.526459i	$-0.823519\pi$
$193$	−23.6227	−1.70040	−0.850200	+	0.526459i	$0.823519\pi$
$194$	−26.0828	−1.87264
$195$	0	0
$196$	4.98745	0.356247
$197$	8.09174	0.576513	0.288256	−	0.957553i	$-0.406924\pi$
$197$	8.09174	0.576513	0.288256	+	0.957553i	$0.406924\pi$
$198$	−27.6889	−1.96776
$199$	−24.6744	−1.74912	−0.874561	−	0.484915i	$-0.838851\pi$
$199$	−24.6744	−1.74912	−0.874561	+	0.484915i	$0.838851\pi$
$200$	0	0
$201$	−0.241035	−0.0170013
$202$	−16.5943	−1.16757
$203$	8.32707	0.584446
$204$	−19.2368	−1.34685
$205$	0	0
$206$	6.91491	0.481784
$207$	9.56587	0.664874
$208$	−42.0504	−2.91567
$209$	−30.2820	−2.09465
$210$	0	0
$211$	−5.08112	−0.349798	−0.174899	−	0.984586i	$-0.555960\pi$
$211$	−5.08112	−0.349798	−0.174899	+	0.984586i	$0.555960\pi$
$212$	61.1399	4.19910
$213$	6.06967	0.415887
$214$	−0.467147	−0.0319335
$215$	0	0
$216$	−34.9327	−2.37687
$217$	0.269795	0.0183149
$218$	−28.5086	−1.93085
$219$	7.79288	0.526594
$220$	0	0
$221$	−17.8438	−1.20030
$222$	6.66465	0.447302
$223$	1.25572	0.0840896	0.0420448	−	0.999116i	$-0.486613\pi$
$223$	1.25572	0.0840896	0.0420448	+	0.999116i	$0.486613\pi$
$224$	13.0183	0.869821
$225$	0	0
$226$	−21.4889	−1.42942
$227$	−6.42534	−0.426465	−0.213232	−	0.977002i	$-0.568399\pi$
$227$	−6.42534	−0.426465	−0.213232	+	0.977002i	$0.568399\pi$
$228$	−27.7097	−1.83512
$229$	−0.848045	−0.0560404	−0.0280202	−	0.999607i	$-0.508920\pi$
$229$	−0.848045	−0.0560404	−0.0280202	+	0.999607i	$0.508920\pi$
$230$	0	0
$231$	−3.79028	−0.249382
$232$	65.7586	4.31726
$233$	−1.13822	−0.0745672	−0.0372836	−	0.999305i	$-0.511870\pi$
$233$	−1.13822	−0.0745672	−0.0372836	+	0.999305i	$0.511870\pi$
$234$	−23.5021	−1.53638
$235$	0	0
$236$	−2.08983	−0.136037
$237$	−3.64778	−0.236949
$238$	12.2263	0.792512
$239$	6.74663	0.436403	0.218202	−	0.975904i	$-0.429981\pi$
$239$	6.74663	0.436403	0.218202	+	0.975904i	$0.429981\pi$
$240$	0	0
$241$	6.87736	0.443010	0.221505	−	0.975159i	$-0.428903\pi$
$241$	6.87736	0.443010	0.221505	+	0.975159i	$0.428903\pi$
$242$	−25.5317	−1.64124
$243$	−15.9600	−1.02383
$244$	57.5090	3.68164
$245$	0	0
$246$	−5.96827	−0.380523
$247$	−25.7031	−1.63545
$248$	2.13056	0.135291
$249$	9.44919	0.598818
$250$	0	0
$251$	27.9049	1.76134	0.880671	−	0.473728i	$-0.157092\pi$
$251$	27.9049	1.76134	0.880671	+	0.473728i	$0.157092\pi$
$252$	11.4940	0.724057
$253$	18.8661	1.18610
$254$	−50.6568	−3.17849
$255$	0	0
$256$	−5.91916	−0.369948
$257$	1.48316	0.0925169	0.0462585	−	0.998930i	$-0.485270\pi$
$257$	1.48316	0.0925169	0.0462585	+	0.998930i	$0.485270\pi$
$258$	1.25790	0.0783133
$259$	−3.02342	−0.187866
$260$	0	0
$261$	19.1905	1.18786
$262$	47.6776	2.94553
$263$	−0.390033	−0.0240504	−0.0120252	−	0.999928i	$-0.503828\pi$
$263$	−0.390033	−0.0240504	−0.0120252	+	0.999928i	$0.503828\pi$
$264$	−29.9317	−1.84217
$265$	0	0
$266$	17.6114	1.07982
$267$	−12.2404	−0.749099
$268$	1.44158	0.0880587
$269$	29.5173	1.79970	0.899851	−	0.436197i	$-0.143675\pi$
$269$	29.5173	1.79970	0.899851	+	0.436197i	$0.143675\pi$
$270$	0	0
$271$	1.82944	0.111131	0.0555654	−	0.998455i	$-0.482304\pi$
$271$	1.82944	0.111131	0.0555654	+	0.998455i	$0.482304\pi$
$272$	50.4141	3.05680
$273$	−3.21716	−0.194711
$274$	15.3130	0.925094
$275$	0	0
$276$	17.2635	1.03914
$277$	1.49518	0.0898364	0.0449182	−	0.998991i	$-0.485697\pi$
$277$	1.49518	0.0898364	0.0449182	+	0.998991i	$0.485697\pi$
$278$	−12.3742	−0.742153
$279$	0.621768	0.0372243
$280$	0	0
$281$	16.0659	0.958412	0.479206	−	0.877703i	$-0.340925\pi$
$281$	16.0659	0.958412	0.479206	+	0.877703i	$0.340925\pi$
$282$	13.6229	0.811230
$283$	17.1301	1.01828	0.509139	−	0.860684i	$-0.329964\pi$
$283$	17.1301	1.01828	0.509139	+	0.860684i	$0.329964\pi$
$284$	−36.3015	−2.15410
$285$	0	0
$286$	−46.3515	−2.74082
$287$	2.70751	0.159819
$288$	30.0018	1.76788
$289$	4.39290	0.258406
$290$	0	0
$291$	−8.22840	−0.482357
$292$	−46.6076	−2.72750
$293$	−5.47224	−0.319691	−0.159846	−	0.987142i	$-0.551100\pi$
$293$	−5.47224	−0.319691	−0.159846	+	0.987142i	$0.551100\pi$
$294$	2.20434	0.128560
$295$	0	0
$296$	−23.8758	−1.38776
$297$	−20.1059	−1.16666
$298$	−51.8527	−3.00375
$299$	16.0134	0.926077
$300$	0	0
$301$	−0.570645	−0.0328914
$302$	−43.8041	−2.52064
$303$	−5.23502	−0.300744
$304$	72.6191	4.16499
$305$	0	0
$306$	28.1766	1.61075
$307$	0.329124	0.0187841	0.00939205	−	0.999956i	$-0.497010\pi$
$307$	0.329124	0.0187841	0.00939205	+	0.999956i	$0.497010\pi$
$308$	22.6689	1.29168
$309$	2.18146	0.124099
$310$	0	0
$311$	−2.26993	−0.128716	−0.0643579	−	0.997927i	$-0.520500\pi$
$311$	−2.26993	−0.128716	−0.0643579	+	0.997927i	$0.520500\pi$
$312$	−25.4058	−1.43832
$313$	−13.2388	−0.748301	−0.374150	−	0.927368i	$-0.622066\pi$
$313$	−13.2388	−0.748301	−0.374150	+	0.927368i	$0.622066\pi$
$314$	−27.5561	−1.55508
$315$	0	0
$316$	21.8166	1.22728
$317$	−1.18652	−0.0666417	−0.0333209	−	0.999445i	$-0.510608\pi$
$317$	−1.18652	−0.0666417	−0.0333209	+	0.999445i	$0.510608\pi$
$318$	27.0225	1.51534
$319$	37.8481	2.11909
$320$	0	0
$321$	−0.147372	−0.00822548
$322$	−10.9721	−0.611451
$323$	30.8154	1.71462
$324$	16.0841	0.893563
$325$	0	0
$326$	−10.6520	−0.589960
$327$	−8.99366	−0.497350
$328$	21.3811	1.18057
$329$	−6.18001	−0.340715
$330$	0	0
$331$	−24.5431	−1.34901	−0.674504	−	0.738271i	$-0.735643\pi$
$331$	−24.5431	−1.34901	−0.674504	+	0.738271i	$0.735643\pi$
$332$	−56.5136	−3.10159
$333$	−6.96775	−0.381830
$334$	58.8627	3.22082
$335$	0	0
$336$	9.08944	0.495870
$337$	−21.3287	−1.16185	−0.580925	−	0.813957i	$-0.697309\pi$
$337$	−21.3287	−1.16185	−0.580925	+	0.813957i	$0.697309\pi$
$338$	−4.97881	−0.270812
$339$	−6.77913	−0.368192
$340$	0	0
$341$	1.22627	0.0664063
$342$	40.5870	2.19469
$343$	−1.00000	−0.0539949
$344$	−4.50637	−0.242967
$345$	0	0
$346$	−45.2979	−2.43523
$347$	−9.25146	−0.496644	−0.248322	−	0.968678i	$-0.579879\pi$
$347$	−9.25146	−0.496644	−0.248322	+	0.968678i	$0.579879\pi$
$348$	34.6330	1.85653
$349$	−28.6781	−1.53510	−0.767552	−	0.640987i	$-0.778525\pi$
$349$	−28.6781	−1.53510	−0.767552	+	0.640987i	$0.778525\pi$
$350$	0	0
$351$	−17.0657	−0.910900
$352$	59.1706	3.15380
$353$	8.63065	0.459363	0.229682	−	0.973266i	$-0.426231\pi$
$353$	8.63065	0.459363	0.229682	+	0.973266i	$0.426231\pi$
$354$	−0.923661	−0.0490920
$355$	0	0
$356$	73.2072	3.87997
$357$	3.85704	0.204136
$358$	−40.5848	−2.14497
$359$	−20.5429	−1.08421	−0.542106	−	0.840310i	$-0.682373\pi$
$359$	−20.5429	−1.08421	−0.542106	+	0.840310i	$0.682373\pi$
$360$	0	0
$361$	25.3881	1.33622
$362$	16.4141	0.862708
$363$	−8.05452	−0.422753
$364$	19.2412	1.00851
$365$	0	0
$366$	25.4177	1.32860
$367$	−2.41855	−0.126247	−0.0631236	−	0.998006i	$-0.520106\pi$
$367$	−2.41855	−0.126247	−0.0631236	+	0.998006i	$0.520106\pi$
$368$	−45.2426	−2.35843
$369$	6.23970	0.324826
$370$	0	0
$371$	−12.2587	−0.636442
$372$	1.12210	0.0581783
$373$	−5.97422	−0.309333	−0.154667	−	0.987967i	$-0.549430\pi$
$373$	−5.97422	−0.309333	−0.154667	+	0.987967i	$0.549430\pi$
$374$	55.5707	2.87349
$375$	0	0
$376$	−48.8034	−2.51684
$377$	32.1251	1.65453
$378$	11.6931	0.601430
$379$	−31.8336	−1.63518	−0.817591	−	0.575799i	$-0.804691\pi$
$379$	−31.8336	−1.63518	−0.817591	+	0.575799i	$0.804691\pi$
$380$	0	0
$381$	−15.9808	−0.818720
$382$	−19.3514	−0.990103
$383$	−21.3560	−1.09124	−0.545619	−	0.838033i	$-0.683705\pi$
$383$	−21.3560	−1.09124	−0.545619	+	0.838033i	$0.683705\pi$
$384$	6.09059	0.310809
$385$	0	0
$386$	62.4438	3.17831
$387$	−1.31510	−0.0668505
$388$	49.2124	2.49838
$389$	30.6084	1.55191	0.775954	−	0.630789i	$-0.217269\pi$
$389$	30.6084	1.55191	0.775954	+	0.630789i	$0.217269\pi$
$390$	0	0
$391$	−19.1984	−0.970905
$392$	−7.89697	−0.398857
$393$	15.0409	0.758715
$394$	−21.3895	−1.07759
$395$	0	0
$396$	52.2426	2.62529
$397$	15.0292	0.754294	0.377147	−	0.926153i	$-0.376905\pi$
$397$	15.0292	0.754294	0.377147	+	0.926153i	$0.376905\pi$
$398$	65.2238	3.26937
$399$	5.55588	0.278142
$400$	0	0
$401$	12.0828	0.603387	0.301693	−	0.953405i	$-0.402448\pi$
$401$	12.0828	0.603387	0.301693	+	0.953405i	$0.402448\pi$
$402$	0.637147	0.0317780
$403$	1.04085	0.0518483
$404$	31.3096	1.55771
$405$	0	0
$406$	−22.0116	−1.09242
$407$	−13.7420	−0.681166
$408$	30.4589	1.50794
$409$	−10.8652	−0.537248	−0.268624	−	0.963245i	$-0.586569\pi$
$409$	−10.8652	−0.537248	−0.268624	+	0.963245i	$0.586569\pi$
$410$	0	0
$411$	4.83083	0.238287
$412$	−13.0468	−0.642772
$413$	0.419019	0.0206186
$414$	−25.2862	−1.24275
$415$	0	0
$416$	50.2234	2.46241
$417$	−3.90370	−0.191165
$418$	80.0469	3.91522
$419$	−9.71688	−0.474700	−0.237350	−	0.971424i	$-0.576279\pi$
$419$	−9.71688	−0.474700	−0.237350	+	0.971424i	$0.576279\pi$
$420$	0	0
$421$	−6.31066	−0.307563	−0.153781	−	0.988105i	$-0.549145\pi$
$421$	−6.31066	−0.307563	−0.153781	+	0.988105i	$0.549145\pi$
$422$	13.4313	0.653826
$423$	−14.2424	−0.692490
$424$	−96.8069	−4.70136
$425$	0	0
$426$	−16.0444	−0.777356
$427$	−11.5307	−0.558011
$428$	0.881399	0.0426040
$429$	−14.6226	−0.705985
$430$	0	0
$431$	−4.92653	−0.237303	−0.118651	−	0.992936i	$-0.537857\pi$
$431$	−4.92653	−0.237303	−0.118651	+	0.992936i	$0.537857\pi$
$432$	48.2158	2.31978
$433$	37.1049	1.78315	0.891575	−	0.452874i	$-0.149601\pi$
$433$	37.1049	1.78315	0.891575	+	0.452874i	$0.149601\pi$
$434$	−0.713171	−0.0342333
$435$	0	0
$436$	53.7892	2.57604
$437$	−27.6544	−1.32289
$438$	−20.5995	−0.984284
$439$	−9.50092	−0.453454	−0.226727	−	0.973958i	$-0.572803\pi$
$439$	−9.50092	−0.453454	−0.226727	+	0.973958i	$0.572803\pi$
$440$	0	0
$441$	−2.30459	−0.109742
$442$	47.1679	2.24355
$443$	13.2415	0.629123	0.314561	−	0.949237i	$-0.398143\pi$
$443$	13.2415	0.629123	0.314561	+	0.949237i	$0.398143\pi$
$444$	−12.5747	−0.596768
$445$	0	0
$446$	−3.31936	−0.157176
$447$	−16.3581	−0.773710
$448$	−12.6127	−0.595895
$449$	−20.2167	−0.954085	−0.477043	−	0.878880i	$-0.658291\pi$
$449$	−20.2167	−0.954085	−0.477043	+	0.878880i	$0.658291\pi$
$450$	0	0
$451$	12.3061	0.579473
$452$	40.5446	1.90706
$453$	−13.8189	−0.649271
$454$	16.9846	0.797127
$455$	0	0
$456$	43.8746	2.05462
$457$	−23.1239	−1.08169	−0.540845	−	0.841122i	$-0.681895\pi$
$457$	−23.1239	−1.08169	−0.540845	+	0.841122i	$0.681895\pi$
$458$	2.24170	0.104748
$459$	20.4600	0.954993
$460$	0	0
$461$	7.94071	0.369836	0.184918	−	0.982754i	$-0.440798\pi$
$461$	7.94071	0.369836	0.184918	+	0.982754i	$0.440798\pi$
$462$	10.0192	0.466133
$463$	15.2021	0.706503	0.353252	−	0.935528i	$-0.385076\pi$
$463$	15.2021	0.706503	0.353252	+	0.935528i	$0.385076\pi$
$464$	−90.7631	−4.21357
$465$	0	0
$466$	3.00874	0.139377
$467$	3.86560	0.178879	0.0894393	−	0.995992i	$-0.471493\pi$
$467$	3.86560	0.178879	0.0894393	+	0.995992i	$0.471493\pi$
$468$	44.3430	2.04976
$469$	−0.289042	−0.0133467
$470$	0	0
$471$	−8.69318	−0.400560
$472$	3.30898	0.152308
$473$	−2.59369	−0.119258
$474$	9.64247	0.442893
$475$	0	0
$476$	−23.0682	−1.05733
$477$	−28.2514	−1.29354
$478$	−17.8339	−0.815703
$479$	−36.4902	−1.66728	−0.833641	−	0.552307i	$-0.813748\pi$
$479$	−36.4902	−1.66728	−0.833641	+	0.552307i	$0.813748\pi$
$480$	0	0
$481$	−11.6641	−0.531837
$482$	−18.1795	−0.828052
$483$	−3.46139	−0.157499
$484$	48.1724	2.18966
$485$	0	0
$486$	42.1883	1.91370
$487$	28.0516	1.27114	0.635570	−	0.772044i	$-0.280765\pi$
$487$	28.0516	1.27114	0.635570	+	0.772044i	$0.280765\pi$
$488$	−91.0578	−4.12199
$489$	−3.36041	−0.151963
$490$	0	0
$491$	1.94688	0.0878616	0.0439308	−	0.999035i	$-0.486012\pi$
$491$	1.94688	0.0878616	0.0439308	+	0.999035i	$0.486012\pi$
$492$	11.2608	0.507675
$493$	−38.5147	−1.73462
$494$	67.9431	3.05691
$495$	0	0
$496$	−2.94071	−0.132042
$497$	7.27856	0.326488
$498$	−24.9778	−1.11928
$499$	9.04201	0.404776	0.202388	−	0.979305i	$-0.435130\pi$
$499$	9.04201	0.404776	0.202388	+	0.979305i	$0.435130\pi$
$500$	0	0
$501$	18.5695	0.829624
$502$	−73.7632	−3.29221
$503$	−8.34548	−0.372107	−0.186053	−	0.982540i	$-0.559570\pi$
$503$	−8.34548	−0.372107	−0.186053	+	0.982540i	$0.559570\pi$
$504$	−18.1993	−0.810661
$505$	0	0
$506$	−49.8702	−2.21700
$507$	−1.57067	−0.0697561
$508$	95.5778	4.24058
$509$	29.1093	1.29025	0.645123	−	0.764079i	$-0.276806\pi$
$509$	29.1093	1.29025	0.645123	+	0.764079i	$0.276806\pi$
$510$	0	0
$511$	9.34497	0.413397
$512$	30.2539	1.33705
$513$	29.4717	1.30121
$514$	−3.92055	−0.172928
$515$	0	0
$516$	−2.37337	−0.104482
$517$	−28.0893	−1.23537
$518$	7.99204	0.351150
$519$	−14.2902	−0.627270
$520$	0	0
$521$	42.4571	1.86008	0.930039	−	0.367461i	$-0.119773\pi$
$521$	42.4571	1.86008	0.930039	+	0.367461i	$0.119773\pi$
$522$	−50.7278	−2.22029
$523$	−2.39751	−0.104836	−0.0524179	−	0.998625i	$-0.516693\pi$
$523$	−2.39751	−0.104836	−0.0524179	+	0.998625i	$0.516693\pi$
$524$	−89.9568	−3.92978
$525$	0	0
$526$	1.03100	0.0449539
$527$	−1.24787	−0.0543581
$528$	41.3132	1.79793
$529$	−5.77098	−0.250912
$530$	0	0
$531$	0.965667	0.0419064
$532$	−33.2286	−1.44064
$533$	10.4453	0.452438
$534$	32.3560	1.40018
$535$	0	0
$536$	−2.28255	−0.0985913
$537$	−12.8034	−0.552506
$538$	−78.0254	−3.36392
$539$	−4.54519	−0.195775
$540$	0	0
$541$	−10.2880	−0.442314	−0.221157	−	0.975238i	$-0.570983\pi$
$541$	−10.2880	−0.442314	−0.221157	+	0.975238i	$0.570983\pi$
$542$	−4.83591	−0.207720
$543$	5.17819	0.222218
$544$	−60.2128	−2.58160
$545$	0	0
$546$	8.50417	0.363945
$547$	36.1810	1.54699	0.773494	−	0.633804i	$-0.218507\pi$
$547$	36.1810	1.54699	0.773494	+	0.633804i	$0.218507\pi$
$548$	−28.8922	−1.23421
$549$	−26.5736	−1.13414
$550$	0	0
$551$	−55.4786	−2.36347
$552$	−27.3345	−1.16343
$553$	−4.37430	−0.186014
$554$	−3.95232	−0.167918
$555$	0	0
$556$	23.3472	0.990142
$557$	−37.5288	−1.59015	−0.795073	−	0.606514i	$-0.792568\pi$
$557$	−37.5288	−1.59015	−0.795073	+	0.606514i	$0.792568\pi$
$558$	−1.64357	−0.0695778
$559$	−2.20150	−0.0931135
$560$	0	0
$561$	17.5310	0.740159
$562$	−42.4683	−1.79142
$563$	5.50239	0.231898	0.115949	−	0.993255i	$-0.463009\pi$
$563$	5.50239	0.231898	0.115949	+	0.993255i	$0.463009\pi$
$564$	−25.7032	−1.08230
$565$	0	0
$566$	−45.2814	−1.90332
$567$	−3.22492	−0.135434
$568$	57.4785	2.41175
$569$	12.1804	0.510629	0.255314	−	0.966858i	$-0.417821\pi$
$569$	12.1804	0.510629	0.255314	+	0.966858i	$0.417821\pi$
$570$	0	0
$571$	−10.3473	−0.433019	−0.216510	−	0.976281i	$-0.569467\pi$
$571$	−10.3473	−0.433019	−0.216510	+	0.976281i	$0.569467\pi$
$572$	87.4547	3.65666
$573$	−6.10481	−0.255032
$574$	−7.15697	−0.298726
$575$	0	0
$576$	−29.0672	−1.21113
$577$	29.3627	1.22238	0.611192	−	0.791482i	$-0.290690\pi$
$577$	29.3627	1.22238	0.611192	+	0.791482i	$0.290690\pi$
$578$	−11.6121	−0.482999
$579$	19.6992	0.818673
$580$	0	0
$581$	11.3312	0.470096
$582$	21.7508	0.901598
$583$	−55.7183	−2.30762
$584$	73.7969	3.05374
$585$	0	0
$586$	14.4652	0.597552
$587$	−18.1347	−0.748498	−0.374249	−	0.927328i	$-0.622099\pi$
$587$	−18.1347	−0.748498	−0.374249	+	0.927328i	$0.622099\pi$
$588$	−4.15909	−0.171518
$589$	−1.79750	−0.0740645
$590$	0	0
$591$	−6.74779	−0.277567
$592$	32.9546	1.35442
$593$	−23.7551	−0.975505	−0.487752	−	0.872982i	$-0.662183\pi$
$593$	−23.7551	−0.975505	−0.487752	+	0.872982i	$0.662183\pi$
$594$	53.1475	2.18067
$595$	0	0
$596$	97.8343	4.00745
$597$	20.5763	0.842131
$598$	−42.3294	−1.73098
$599$	9.08142	0.371057	0.185528	−	0.982639i	$-0.440600\pi$
$599$	9.08142	0.371057	0.185528	+	0.982639i	$0.440600\pi$
$600$	0	0
$601$	27.1808	1.10873	0.554364	−	0.832274i	$-0.312962\pi$
$601$	27.1808	1.10873	0.554364	+	0.832274i	$0.312962\pi$
$602$	1.50843	0.0614791
$603$	−0.666124	−0.0271267
$604$	82.6483	3.36291
$605$	0	0
$606$	13.8381	0.562136
$607$	8.67473	0.352096	0.176048	−	0.984382i	$-0.443669\pi$
$607$	8.67473	0.352096	0.176048	+	0.984382i	$0.443669\pi$
$608$	−86.7336	−3.51751
$609$	−6.94404	−0.281387
$610$	0	0
$611$	−23.8420	−0.964543
$612$	−53.1628	−2.14898
$613$	10.2136	0.412522	0.206261	−	0.978497i	$-0.433870\pi$
$613$	10.2136	0.412522	0.206261	+	0.978497i	$0.433870\pi$
$614$	−0.869999	−0.0351103
$615$	0	0
$616$	−35.8932	−1.44618
$617$	8.10309	0.326218	0.163109	−	0.986608i	$-0.447848\pi$
$617$	8.10309	0.326218	0.163109	+	0.986608i	$0.447848\pi$
$618$	−5.76642	−0.231959
$619$	9.49309	0.381559	0.190780	−	0.981633i	$-0.438898\pi$
$619$	9.49309	0.381559	0.190780	+	0.981633i	$0.438898\pi$
$620$	0	0
$621$	−18.3612	−0.736811
$622$	6.00028	0.240589
$623$	−14.6783	−0.588073
$624$	35.0663	1.40377
$625$	0	0
$626$	34.9951	1.39869
$627$	25.2525	1.00849
$628$	51.9921	2.07471
$629$	13.9841	0.557581
$630$	0	0
$631$	−19.0891	−0.759925	−0.379962	−	0.925002i	$-0.624063\pi$
$631$	−19.0891	−0.759925	−0.379962	+	0.925002i	$0.624063\pi$
$632$	−34.5437	−1.37408
$633$	4.23720	0.168414
$634$	3.13643	0.124563
$635$	0	0
$636$	−50.9852	−2.02170
$637$	−3.85791	−0.152856
$638$	−100.047	−3.96089
$639$	16.7741	0.663574
$640$	0	0
$641$	−0.0611719	−0.00241614	−0.00120807	−	0.999999i	$-0.500385\pi$
$641$	−0.0611719	−0.00241614	−0.00120807	+	0.999999i	$0.500385\pi$
$642$	0.389559	0.0153747
$643$	−8.28259	−0.326634	−0.163317	−	0.986574i	$-0.552219\pi$
$643$	−8.28259	−0.326634	−0.163317	+	0.986574i	$0.552219\pi$
$644$	20.7018	0.815767
$645$	0	0
$646$	−81.4569	−3.20488
$647$	43.9925	1.72952	0.864762	−	0.502182i	$-0.167469\pi$
$647$	43.9925	1.72952	0.864762	+	0.502182i	$0.167469\pi$
$648$	−25.4671	−1.00044
$649$	1.90452	0.0747589
$650$	0	0
$651$	−0.224985	−0.00881787
$652$	20.0979	0.787095
$653$	18.6070	0.728147	0.364073	−	0.931370i	$-0.381386\pi$
$653$	18.6070	0.728147	0.364073	+	0.931370i	$0.381386\pi$
$654$	23.7736	0.929623
$655$	0	0
$656$	−29.5112	−1.15222
$657$	21.5364	0.840213
$658$	16.3361	0.636848
$659$	−42.8021	−1.66733	−0.833667	−	0.552268i	$-0.813763\pi$
$659$	−42.8021	−1.66733	−0.833667	+	0.552268i	$0.813763\pi$
$660$	0	0
$661$	17.0003	0.661234	0.330617	−	0.943765i	$-0.392743\pi$
$661$	17.0003	0.661234	0.330617	+	0.943765i	$0.392743\pi$
$662$	64.8766	2.52150
$663$	14.8801	0.577897
$664$	89.4818	3.47257
$665$	0	0
$666$	18.4184	0.713698
$667$	34.5639	1.33832
$668$	−111.060	−4.29706
$669$	−1.04716	−0.0404857
$670$	0	0
$671$	−52.4093	−2.02324
$672$	−10.8561	−0.418783
$673$	−5.43715	−0.209587	−0.104793	−	0.994494i	$-0.533418\pi$
$673$	−5.43715	−0.209587	−0.104793	+	0.994494i	$0.533418\pi$
$674$	56.3799	2.17167
$675$	0	0
$676$	9.39388	0.361303
$677$	5.00554	0.192378	0.0961892	−	0.995363i	$-0.469335\pi$
$677$	5.00554	0.192378	0.0961892	+	0.995363i	$0.469335\pi$
$678$	17.9198	0.688206
$679$	−9.86723	−0.378670
$680$	0	0
$681$	5.35816	0.205325
$682$	−3.24150	−0.124123
$683$	39.6432	1.51691	0.758453	−	0.651728i	$-0.225956\pi$
$683$	39.6432	1.51691	0.758453	+	0.651728i	$0.225956\pi$
$684$	−76.5784	−2.92805
$685$	0	0
$686$	2.64338	0.100925
$687$	0.707194	0.0269811
$688$	6.21990	0.237131
$689$	−47.2932	−1.80173
$690$	0	0
$691$	19.5149	0.742383	0.371191	−	0.928556i	$-0.378949\pi$
$691$	19.5149	0.742383	0.371191	+	0.928556i	$0.378949\pi$
$692$	85.4667	3.24896
$693$	−10.4748	−0.397905
$694$	24.4551	0.928303
$695$	0	0
$696$	−54.8368	−2.07858
$697$	−12.5229	−0.474338
$698$	75.8071	2.86934
$699$	0.949173	0.0359010
$700$	0	0
$701$	21.8539	0.825412	0.412706	−	0.910864i	$-0.364584\pi$
$701$	21.8539	0.825412	0.412706	+	0.910864i	$0.364584\pi$
$702$	45.1111	1.70261
$703$	20.1434	0.759721
$704$	−57.3272	−2.16060
$705$	0	0
$706$	−22.8141	−0.858620
$707$	−6.27767	−0.236096
$708$	1.74274	0.0654961
$709$	40.7664	1.53101	0.765507	−	0.643427i	$-0.222488\pi$
$709$	40.7664	1.53101	0.765507	+	0.643427i	$0.222488\pi$
$710$	0	0
$711$	−10.0810	−0.378067
$712$	−115.914	−4.34406
$713$	1.11986	0.0419392
$714$	−10.1956	−0.381562
$715$	0	0
$716$	76.5743	2.86172
$717$	−5.62609	−0.210110
$718$	54.3026	2.02656
$719$	20.4371	0.762176	0.381088	−	0.924539i	$-0.375549\pi$
$719$	20.4371	0.762176	0.381088	+	0.924539i	$0.375549\pi$
$720$	0	0
$721$	2.61593	0.0974225
$722$	−67.1105	−2.49759
$723$	−5.73511	−0.213291
$724$	−30.9697	−1.15098
$725$	0	0
$726$	21.2911	0.790189
$727$	−32.0766	−1.18966	−0.594828	−	0.803853i	$-0.702780\pi$
$727$	−32.0766	−1.18966	−0.594828	+	0.803853i	$0.702780\pi$
$728$	−30.4658	−1.12914
$729$	3.63443	0.134609
$730$	0	0
$731$	2.63937	0.0976208
$732$	−47.9574	−1.77256
$733$	−44.5082	−1.64395	−0.821974	−	0.569525i	$-0.807127\pi$
$733$	−44.5082	−1.64395	−0.821974	+	0.569525i	$0.807127\pi$
$734$	6.39314	0.235975
$735$	0	0
$736$	54.0361	1.99180
$737$	−1.31375	−0.0483926
$738$	−16.4939	−0.607149
$739$	0.0772344	0.00284111	0.00142056	−	0.999999i	$-0.499548\pi$
$739$	0.0772344	0.00284111	0.00142056	+	0.999999i	$0.499548\pi$
$740$	0	0
$741$	21.4341	0.787402
$742$	32.4045	1.18961
$743$	14.8925	0.546353	0.273177	−	0.961964i	$-0.411926\pi$
$743$	14.8925	0.546353	0.273177	+	0.961964i	$0.411926\pi$
$744$	−1.77670	−0.0651370
$745$	0	0
$746$	15.7921	0.578191
$747$	26.1137	0.955451
$748$	−104.849	−3.83367
$749$	−0.176723	−0.00645733
$750$	0	0
$751$	16.8102	0.613414	0.306707	−	0.951804i	$-0.400773\pi$
$751$	16.8102	0.613414	0.306707	+	0.951804i	$0.400773\pi$
$752$	67.3607	2.45639
$753$	−23.2702	−0.848014
$754$	−84.9189	−3.09256
$755$	0	0
$756$	−22.0623	−0.802398
$757$	12.0534	0.438090	0.219045	−	0.975715i	$-0.429706\pi$
$757$	12.0534	0.438090	0.219045	+	0.975715i	$0.429706\pi$
$758$	84.1483	3.05640
$759$	−15.7327	−0.571059
$760$	0	0
$761$	13.6602	0.495182	0.247591	−	0.968865i	$-0.420361\pi$
$761$	13.6602	0.495182	0.247591	+	0.968865i	$0.420361\pi$
$762$	42.2433	1.53031
$763$	−10.7849	−0.390440
$764$	36.5116	1.32094
$765$	0	0
$766$	56.4519	2.03969
$767$	1.61654	0.0583698
$768$	4.93606	0.178115
$769$	−37.1200	−1.33858	−0.669290	−	0.743001i	$-0.733402\pi$
$769$	−37.1200	−1.33858	−0.669290	+	0.743001i	$0.733402\pi$
$770$	0	0
$771$	−1.23682	−0.0445431
$772$	−117.817	−4.24033
$773$	29.4332	1.05864	0.529318	−	0.848423i	$-0.322448\pi$
$773$	29.4332	1.05864	0.529318	+	0.848423i	$0.322448\pi$
$774$	3.47632	0.124954
$775$	0	0
$776$	−77.9212	−2.79721
$777$	2.52126	0.0904498
$778$	−80.9096	−2.90075
$779$	−18.0386	−0.646300
$780$	0	0
$781$	33.0824	1.18378
$782$	50.7487	1.81477
$783$	−36.8353	−1.31639
$784$	10.8998	0.389278
$785$	0	0
$786$	−39.7589	−1.41815
$787$	−37.0603	−1.32106	−0.660529	−	0.750801i	$-0.729668\pi$
$787$	−37.0603	−1.32106	−0.660529	+	0.750801i	$0.729668\pi$
$788$	40.3572	1.43766
$789$	0.325253	0.0115793
$790$	0	0
$791$	−8.12932	−0.289045
$792$	−82.7192	−2.93930
$793$	−44.4846	−1.57969
$794$	−39.7279	−1.40989
$795$	0	0
$796$	−123.062	−4.36183
$797$	−4.72508	−0.167371	−0.0836855	−	0.996492i	$-0.526669\pi$
$797$	−4.72508	−0.167371	−0.0836855	+	0.996492i	$0.526669\pi$
$798$	−14.6863	−0.519890
$799$	28.5841	1.01123
$800$	0	0
$801$	−33.8274	−1.19523
$802$	−31.9394	−1.12782
$803$	42.4747	1.49890
$804$	−1.20215	−0.0423966
$805$	0	0
$806$	−2.75135	−0.0969123
$807$	−24.6148	−0.866483
$808$	−49.5745	−1.74403
$809$	−27.4587	−0.965397	−0.482699	−	0.875786i	$-0.660343\pi$
$809$	−27.4587	−0.965397	−0.482699	+	0.875786i	$0.660343\pi$
$810$	0	0
$811$	−22.2780	−0.782286	−0.391143	−	0.920330i	$-0.627920\pi$
$811$	−22.2780	−0.782286	−0.391143	+	0.920330i	$0.627920\pi$
$812$	41.5309	1.45745
$813$	−1.52559	−0.0535049
$814$	36.3253	1.27320
$815$	0	0
$816$	−42.0409	−1.47173
$817$	3.80189	0.133011
$818$	28.7208	1.00420
$819$	−8.89092	−0.310674
$820$	0	0
$821$	20.7446	0.723991	0.361996	−	0.932180i	$-0.382096\pi$
$821$	20.7446	0.723991	0.361996	+	0.932180i	$0.382096\pi$
$822$	−12.7697	−0.445395
$823$	35.7195	1.24510	0.622552	−	0.782578i	$-0.286096\pi$
$823$	35.7195	1.24510	0.622552	+	0.782578i	$0.286096\pi$
$824$	20.6579	0.719654
$825$	0	0
$826$	−1.10762	−0.0385392
$827$	−25.4541	−0.885126	−0.442563	−	0.896737i	$-0.645931\pi$
$827$	−25.4541	−0.885126	−0.442563	+	0.896737i	$0.645931\pi$
$828$	47.7093	1.65801
$829$	−13.4046	−0.465561	−0.232780	−	0.972529i	$-0.574782\pi$
$829$	−13.4046	−0.465561	−0.232780	+	0.972529i	$0.574782\pi$
$830$	0	0
$831$	−1.24684	−0.0432525
$832$	−48.6588	−1.68694
$833$	4.62525	0.160255
$834$	10.3189	0.357316
$835$	0	0
$836$	−151.030	−5.22349
$837$	−1.19346	−0.0412519
$838$	25.6854	0.887287
$839$	−10.6811	−0.368754	−0.184377	−	0.982856i	$-0.559027\pi$
$839$	−10.6811	−0.368754	−0.184377	+	0.982856i	$0.559027\pi$
$840$	0	0
$841$	40.3401	1.39104
$842$	16.6815	0.574881
$843$	−13.3975	−0.461436
$844$	−25.3418	−0.872301
$845$	0	0
$846$	37.6481	1.29437
$847$	−9.65873	−0.331878
$848$	133.617	4.58844
$849$	−14.2850	−0.490259
$850$	0	0
$851$	−12.5496	−0.430194
$852$	30.2722	1.03711
$853$	−44.8085	−1.53422	−0.767108	−	0.641518i	$-0.778305\pi$
$853$	−44.8085	−1.53422	−0.767108	+	0.641518i	$0.778305\pi$
$854$	30.4801	1.04301
$855$	0	0
$856$	−1.39558	−0.0476999
$857$	36.9801	1.26322	0.631608	−	0.775288i	$-0.282395\pi$
$857$	36.9801	1.26322	0.631608	+	0.775288i	$0.282395\pi$
$858$	38.6530	1.31959
$859$	−46.8718	−1.59924	−0.799622	−	0.600504i	$-0.794967\pi$
$859$	−46.8718	−1.59924	−0.799622	+	0.600504i	$0.794967\pi$
$860$	0	0
$861$	−2.25782	−0.0769463
$862$	13.0227	0.443554
$863$	15.3188	0.521459	0.260730	−	0.965412i	$-0.416037\pi$
$863$	15.3188	0.521459	0.260730	+	0.965412i	$0.416037\pi$
$864$	−57.5872	−1.95916
$865$	0	0
$866$	−98.0824	−3.33298
$867$	−3.66329	−0.124412
$868$	1.34559	0.0456723
$869$	−19.8820	−0.674452
$870$	0	0
$871$	−1.11510	−0.0377837
$872$	−85.1681	−2.88415
$873$	−22.7400	−0.769631
$874$	73.1010	2.47268
$875$	0	0
$876$	38.8666	1.31318
$877$	43.4733	1.46799	0.733994	−	0.679156i	$-0.237654\pi$
$877$	43.4733	1.46799	0.733994	+	0.679156i	$0.237654\pi$
$878$	25.1145	0.847575
$879$	4.56336	0.153918
$880$	0	0
$881$	−44.7337	−1.50712	−0.753558	−	0.657381i	$-0.771664\pi$
$881$	−44.7337	−1.50712	−0.753558	+	0.657381i	$0.771664\pi$
$882$	6.09191	0.205125
$883$	−32.1748	−1.08277	−0.541384	−	0.840776i	$-0.682099\pi$
$883$	−32.1748	−1.08277	−0.541384	+	0.840776i	$0.682099\pi$
$884$	−88.9951	−2.99323
$885$	0	0
$886$	−35.0023	−1.17593
$887$	32.3009	1.08456	0.542279	−	0.840199i	$-0.317562\pi$
$887$	32.3009	1.08456	0.542279	+	0.840199i	$0.317562\pi$
$888$	19.9103	0.668147
$889$	−19.1636	−0.642728
$890$	0	0
$891$	−14.6579	−0.491057
$892$	6.26287	0.209696
$893$	41.1740	1.37783
$894$	43.2406	1.44618
$895$	0	0
$896$	7.30364	0.243998
$897$	−13.3537	−0.445868
$898$	53.4404	1.78333
$899$	2.24660	0.0749284
$900$	0	0
$901$	56.6997	1.88894
$902$	−32.5298	−1.08312
$903$	0.475867	0.0158359
$904$	−64.1969	−2.13516
$905$	0	0
$906$	36.5287	1.21358
$907$	32.6364	1.08367	0.541836	−	0.840484i	$-0.317729\pi$
$907$	32.6364	1.08367	0.541836	+	0.840484i	$0.317729\pi$
$908$	−32.0461	−1.06349
$909$	−14.4675	−0.479855
$910$	0	0
$911$	14.5262	0.481276	0.240638	−	0.970615i	$-0.422643\pi$
$911$	14.5262	0.481276	0.240638	+	0.970615i	$0.422643\pi$
$912$	−60.5579	−2.00527
$913$	51.5023	1.70448
$914$	61.1252	2.02184
$915$	0	0
$916$	−4.22958	−0.139749
$917$	18.0366	0.595622
$918$	−54.0836	−1.78503
$919$	−28.2136	−0.930681	−0.465341	−	0.885132i	$-0.654068\pi$
$919$	−28.2136	−0.930681	−0.465341	+	0.885132i	$0.654068\pi$
$920$	0	0
$921$	−0.274460	−0.00904377
$922$	−20.9903	−0.691279
$923$	28.0801	0.924266
$924$	−18.9038	−0.621891
$925$	0	0
$926$	−40.1850	−1.32056
$927$	6.02866	0.198007
$928$	108.404	3.55854
$929$	−2.45383	−0.0805075	−0.0402537	−	0.999189i	$-0.512817\pi$
$929$	−2.45383	−0.0805075	−0.0402537	+	0.999189i	$0.512817\pi$
$930$	0	0
$931$	6.66244	0.218353
$932$	−5.67681	−0.185950
$933$	1.89292	0.0619713
$934$	−10.2182	−0.334351
$935$	0	0
$936$	−70.2113	−2.29493
$937$	36.6194	1.19630	0.598151	−	0.801383i	$-0.295902\pi$
$937$	36.6194	1.19630	0.598151	+	0.801383i	$0.295902\pi$
$938$	0.764047	0.0249470
$939$	11.0400	0.360276
$940$	0	0
$941$	14.7576	0.481085	0.240543	−	0.970639i	$-0.422675\pi$
$941$	14.7576	0.481085	0.240543	+	0.970639i	$0.422675\pi$
$942$	22.9794	0.748708
$943$	11.2383	0.365969
$944$	−4.56721	−0.148650
$945$	0	0
$946$	6.85610	0.222911
$947$	54.3926	1.76752	0.883761	−	0.467939i	$-0.155003\pi$
$947$	54.3926	1.76752	0.883761	+	0.467939i	$0.155003\pi$
$948$	−18.1931	−0.590886
$949$	36.0521	1.17030
$950$	0	0
$951$	0.989454	0.0320852
$952$	36.5254	1.18380
$953$	42.6983	1.38313	0.691566	−	0.722313i	$-0.256921\pi$
$953$	42.6983	1.38313	0.691566	+	0.722313i	$0.256921\pi$
$954$	74.6792	2.41783
$955$	0	0
$956$	33.6485	1.08827
$957$	−31.5619	−1.02025
$958$	96.4575	3.11640
$959$	5.79298	0.187065
$960$	0	0
$961$	−30.9272	−0.997652
$962$	30.8326	0.994083
$963$	−0.407275	−0.0131243
$964$	34.3005	1.10475
$965$	0	0
$966$	9.14976	0.294389
$967$	44.4649	1.42989	0.714947	−	0.699179i	$-0.246451\pi$
$967$	44.4649	1.42989	0.714947	+	0.699179i	$0.246451\pi$
$968$	−76.2746	−2.45156
$969$	−25.6973	−0.825517
$970$	0	0
$971$	26.5710	0.852704	0.426352	−	0.904557i	$-0.359799\pi$
$971$	26.5710	0.852704	0.426352	+	0.904557i	$0.359799\pi$
$972$	−79.5996	−2.55316
$973$	−4.68119	−0.150072
$974$	−74.1510	−2.37595
$975$	0	0
$976$	125.682	4.02299
$977$	33.1445	1.06038	0.530192	−	0.847877i	$-0.322120\pi$
$977$	33.1445	1.06038	0.530192	+	0.847877i	$0.322120\pi$
$978$	8.88283	0.284042
$979$	−66.7155	−2.13224
$980$	0	0
$981$	−24.8548	−0.793553
$982$	−5.14635	−0.164227
$983$	−15.1403	−0.482900	−0.241450	−	0.970413i	$-0.577623\pi$
$983$	−15.1403	−0.482900	−0.241450	+	0.970413i	$0.577623\pi$
$984$	−17.8299	−0.568397
$985$	0	0
$986$	101.809	3.24226
$987$	5.15358	0.164040
$988$	−128.193	−4.07837
$989$	−2.36863	−0.0753179
$990$	0	0
$991$	3.95460	0.125622	0.0628109	−	0.998025i	$-0.479993\pi$
$991$	3.95460	0.125622	0.0628109	+	0.998025i	$0.479993\pi$
$992$	3.51227	0.111515
$993$	20.4667	0.649492
$994$	−19.2400	−0.610255
$995$	0	0
$996$	47.1274	1.49329
$997$	−1.11672	−0.0353668	−0.0176834	−	0.999844i	$-0.505629\pi$
$997$	−1.11672	−0.0353668	−0.0176834	+	0.999844i	$0.505629\pi$
$998$	−23.9015	−0.756587
$999$	13.3743	0.423143

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	875.2.a.j.1.1	yes	8
3.2	odd	2		7875.2.a.w.1.8		8
5.2	odd	4		875.2.b.e.624.3		16
5.3	odd	4		875.2.b.e.624.14		16
5.4	even	2		875.2.a.i.1.8	✓	8
7.6	odd	2		6125.2.a.w.1.1		8
15.14	odd	2		7875.2.a.bb.1.1		8
35.34	odd	2		6125.2.a.v.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
875.2.a.i.1.8	✓	8	5.4	even	2
875.2.a.j.1.1	yes	8	1.1	even	1	trivial
875.2.b.e.624.3		16	5.2	odd	4
875.2.b.e.624.14		16	5.3	odd	4
6125.2.a.v.1.8		8	35.34	odd	2
6125.2.a.w.1.1		8	7.6	odd	2
7875.2.a.w.1.8		8	3.2	odd	2
7875.2.a.bb.1.1		8	15.14	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 875 = 5^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	875.a (trivial)

\(f(q)\)	\(=\)	\(q-2.64338 q^{2} -0.833911 q^{3} +4.98745 q^{4} +2.20434 q^{6} -1.00000 q^{7} -7.89697 q^{8} -2.30459 q^{9} -4.54519 q^{11} -4.15909 q^{12} -3.85791 q^{13} +2.64338 q^{14} +10.8998 q^{16} +4.62525 q^{17} +6.09191 q^{18} +6.66244 q^{19} +0.833911 q^{21} +12.0147 q^{22} -4.15079 q^{23} +6.58537 q^{24} +10.1979 q^{26} +4.42356 q^{27} -4.98745 q^{28} -8.32707 q^{29} -0.269795 q^{31} -13.0183 q^{32} +3.79028 q^{33} -12.2263 q^{34} -11.4940 q^{36} +3.02342 q^{37} -17.6114 q^{38} +3.21716 q^{39} -2.70751 q^{41} -2.20434 q^{42} +0.570645 q^{43} -22.6689 q^{44} +10.9721 q^{46} +6.18001 q^{47} -9.08944 q^{48} +1.00000 q^{49} -3.85704 q^{51} -19.2412 q^{52} +12.2587 q^{53} -11.6931 q^{54} +7.89697 q^{56} -5.55588 q^{57} +22.0116 q^{58} -0.419019 q^{59} +11.5307 q^{61} +0.713171 q^{62} +2.30459 q^{63} +12.6127 q^{64} -10.0192 q^{66} +0.289042 q^{67} +23.0682 q^{68} +3.46139 q^{69} -7.27856 q^{71} +18.1993 q^{72} -9.34497 q^{73} -7.99204 q^{74} +33.2286 q^{76} +4.54519 q^{77} -8.50417 q^{78} +4.37430 q^{79} +3.22492 q^{81} +7.15697 q^{82} -11.3312 q^{83} +4.15909 q^{84} -1.50843 q^{86} +6.94404 q^{87} +35.8932 q^{88} +14.6783 q^{89} +3.85791 q^{91} -20.7018 q^{92} +0.224985 q^{93} -16.3361 q^{94} +10.8561 q^{96} +9.86723 q^{97} -2.64338 q^{98} +10.4748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} - 8 q^{3} + 13 q^{4} + 2 q^{6} - 8 q^{7} + 12 q^{8} + 18 q^{9} - 5 q^{11} - 20 q^{12} - 6 q^{13} - q^{14} + 35 q^{16} + 13 q^{17} + 3 q^{18} + 13 q^{19} + 8 q^{21} + 22 q^{22} - 5 q^{23} - 3 q^{24}+ \cdots - 31 q^{99}+O(q^{100}) \) 8 * q + q^2 - 8 * q^3 + 13 * q^4 + 2 * q^6 - 8 * q^7 + 12 * q^8 + 18 * q^9 - 5 * q^11 - 20 * q^12 - 6 * q^13 - q^14 + 35 * q^16 + 13 * q^17 + 3 * q^18 + 13 * q^19 + 8 * q^21 + 22 * q^22 - 5 * q^23 - 3 * q^24 - 5 * q^26 - 29 * q^27 - 13 * q^28 + 22 * q^29 + 3 * q^31 + 35 * q^32 - 10 * q^33 + 19 * q^34 + 27 * q^36 + 7 * q^37 - 5 * q^38 + 7 * q^39 + 12 * q^41 - 2 * q^42 - q^43 - 41 * q^44 + 11 * q^46 - 2 * q^47 + 12 * q^48 + 8 * q^49 + 10 * q^51 - 49 * q^52 + 10 * q^53 - 31 * q^54 - 12 * q^56 + 17 * q^57 + 58 * q^58 - 11 * q^59 + 45 * q^61 + 26 * q^62 - 18 * q^63 + 82 * q^64 - 42 * q^66 + 6 * q^67 + 58 * q^68 + 25 * q^69 - 15 * q^71 + 85 * q^72 - 32 * q^73 + 24 * q^74 + 26 * q^76 + 5 * q^77 - 32 * q^78 + 12 * q^79 + 36 * q^81 - 15 * q^83 + 20 * q^84 - 48 * q^86 - 35 * q^87 - 2 * q^88 + 12 * q^89 + 6 * q^91 - 78 * q^92 + 33 * q^93 - 38 * q^94 - 30 * q^96 + 20 * q^97 + q^98 - 31 * q^99

Introduction

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