LMFDB - Embedded newform 2169.2.a.e.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2169,2,Mod(1,2169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2169, 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("2169.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2169 = 3^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2169.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:	$17.3195521984$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	7.7.31056073.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 3x^{5} + 11x^{4} + x^{3} - 9x^{2} + 1 $ x^7 - 3x^6 - 3x^5 + 11x^4 + x^3 - 9x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 241)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-1.60363$ of defining polynomial
Character	$\chi$	$=$	2169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.60363 q^{2} +4.77887 q^{4} +1.69135 q^{5} -1.30586 q^{7} +7.23513 q^{8} +O(q^{10})$	\(q+2.60363 q^{2} +4.77887 q^{4} +1.69135 q^{5} -1.30586 q^{7} +7.23513 q^{8} +4.40364 q^{10} +3.27094 q^{11} +4.30649 q^{13} -3.39998 q^{14} +9.27984 q^{16} +1.02456 q^{17} -7.01250 q^{19} +8.08274 q^{20} +8.51631 q^{22} -0.835873 q^{23} -2.13934 q^{25} +11.2125 q^{26} -6.24054 q^{28} +1.11761 q^{29} -3.97344 q^{31} +9.69098 q^{32} +2.66758 q^{34} -2.20867 q^{35} -11.3098 q^{37} -18.2579 q^{38} +12.2371 q^{40} -1.22869 q^{41} +10.8406 q^{43} +15.6314 q^{44} -2.17630 q^{46} +0.151820 q^{47} -5.29473 q^{49} -5.57003 q^{50} +20.5801 q^{52} -3.02053 q^{53} +5.53231 q^{55} -9.44808 q^{56} +2.90984 q^{58} +4.15373 q^{59} +5.62714 q^{61} -10.3454 q^{62} +6.67199 q^{64} +7.28378 q^{65} +12.9934 q^{67} +4.89626 q^{68} -5.75055 q^{70} +11.2862 q^{71} +11.7148 q^{73} -29.4464 q^{74} -33.5118 q^{76} -4.27140 q^{77} -1.66517 q^{79} +15.6955 q^{80} -3.19906 q^{82} +2.34322 q^{83} +1.73290 q^{85} +28.2250 q^{86} +23.6657 q^{88} -18.1099 q^{89} -5.62368 q^{91} -3.99453 q^{92} +0.395283 q^{94} -11.8606 q^{95} -7.17873 q^{97} -13.7855 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 4 q^{2} + 2 q^{4} + 8 q^{5} - 7 q^{7} + 6 q^{8}+O(q^{10}) $ 7 * q + 4 * q^2 + 2 * q^4 + 8 * q^5 - 7 * q^7 + 6 * q^8	$ 7 q + 4 q^{2} + 2 q^{4} + 8 q^{5} - 7 q^{7} + 6 q^{8} + 3 q^{10} + 18 q^{11} - q^{13} + 6 q^{14} + 4 q^{16} + 2 q^{17} - 6 q^{19} + 8 q^{20} + 10 q^{22} + 22 q^{23} + 5 q^{25} - 8 q^{26} + 9 q^{28} + 16 q^{29} - 18 q^{31} + 6 q^{32} + 11 q^{34} - 7 q^{35} + 8 q^{37} - 16 q^{38} + 14 q^{40} + 15 q^{41} + 14 q^{43} + 4 q^{44} + 11 q^{46} + 10 q^{47} + 6 q^{49} + 4 q^{50} + 27 q^{52} - 15 q^{53} + 29 q^{55} - 13 q^{56} + 17 q^{58} + 18 q^{59} + 4 q^{61} - 13 q^{62} + 2 q^{64} + 7 q^{65} + 18 q^{67} + 15 q^{68} + 8 q^{70} + 50 q^{71} - 10 q^{74} - 20 q^{76} - 17 q^{77} - 15 q^{79} + 11 q^{80} + 45 q^{82} + 24 q^{83} - 2 q^{85} + 23 q^{86} + 8 q^{88} + 13 q^{89} - 12 q^{91} + 10 q^{92} - 32 q^{94} + 41 q^{95} + q^{97} - 9 q^{98}+O(q^{100}) $ 7 * q + 4 * q^2 + 2 * q^4 + 8 * q^5 - 7 * q^7 + 6 * q^8 + 3 * q^10 + 18 * q^11 - q^13 + 6 * q^14 + 4 * q^16 + 2 * q^17 - 6 * q^19 + 8 * q^20 + 10 * q^22 + 22 * q^23 + 5 * q^25 - 8 * q^26 + 9 * q^28 + 16 * q^29 - 18 * q^31 + 6 * q^32 + 11 * q^34 - 7 * q^35 + 8 * q^37 - 16 * q^38 + 14 * q^40 + 15 * q^41 + 14 * q^43 + 4 * q^44 + 11 * q^46 + 10 * q^47 + 6 * q^49 + 4 * q^50 + 27 * q^52 - 15 * q^53 + 29 * q^55 - 13 * q^56 + 17 * q^58 + 18 * q^59 + 4 * q^61 - 13 * q^62 + 2 * q^64 + 7 * q^65 + 18 * q^67 + 15 * q^68 + 8 * q^70 + 50 * q^71 - 10 * q^74 - 20 * q^76 - 17 * q^77 - 15 * q^79 + 11 * q^80 + 45 * q^82 + 24 * q^83 - 2 * q^85 + 23 * q^86 + 8 * q^88 + 13 * q^89 - 12 * q^91 + 10 * q^92 - 32 * q^94 + 41 * q^95 + q^97 - 9 * q^98

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 $

$2$	2.60363	1.84104	0.920521	−	0.390694i	$-0.127765\pi$
$2$	2.60363	1.84104	0.920521	+	0.390694i	$0.127765\pi$
$3$	0	0
$3$	0	0
$4$	4.77887	2.38943
$5$	1.69135	0.756395	0.378197	−	0.925725i	$-0.376544\pi$
$5$	1.69135	0.756395	0.378197	+	0.925725i	$0.376544\pi$
$6$	0	0
$7$	−1.30586	−0.493569	−0.246785	−	0.969070i	$-0.579374\pi$
$7$	−1.30586	−0.493569	−0.246785	+	0.969070i	$0.579374\pi$
$8$	7.23513	2.55801
$9$	0	0
$10$	4.40364	1.39255
$11$	3.27094	0.986226	0.493113	−	0.869965i	$-0.335859\pi$
$11$	3.27094	0.986226	0.493113	+	0.869965i	$0.335859\pi$
$12$	0	0
$13$	4.30649	1.19441	0.597203	−	0.802090i	$-0.296279\pi$
$13$	4.30649	1.19441	0.597203	+	0.802090i	$0.296279\pi$
$14$	−3.39998	−0.908682
$15$	0	0
$16$	9.27984	2.31996
$17$	1.02456	0.248493	0.124247	−	0.992251i	$-0.460349\pi$
$17$	1.02456	0.248493	0.124247	+	0.992251i	$0.460349\pi$
$18$	0	0
$19$	−7.01250	−1.60878	−0.804388	−	0.594104i	$-0.797507\pi$
$19$	−7.01250	−1.60878	−0.804388	+	0.594104i	$0.797507\pi$
$20$	8.08274	1.80736
$21$	0	0
$22$	8.51631	1.81568
$23$	−0.835873	−0.174292	−0.0871458	−	0.996196i	$-0.527775\pi$
$23$	−0.835873	−0.174292	−0.0871458	+	0.996196i	$0.527775\pi$
$24$	0	0
$25$	−2.13934	−0.427867
$26$	11.2125	2.19895
$27$	0	0
$28$	−6.24054	−1.17935
$29$	1.11761	0.207535	0.103767	−	0.994602i	$-0.466910\pi$
$29$	1.11761	0.207535	0.103767	+	0.994602i	$0.466910\pi$
$30$	0	0
$31$	−3.97344	−0.713652	−0.356826	−	0.934171i	$-0.616141\pi$
$31$	−3.97344	−0.713652	−0.356826	+	0.934171i	$0.616141\pi$
$32$	9.69098	1.71314
$33$	0	0
$34$	2.66758	0.457486
$35$	−2.20867	−0.373333
$36$	0	0
$37$	−11.3098	−1.85932	−0.929658	−	0.368423i	$-0.879898\pi$
$37$	−11.3098	−1.85932	−0.929658	+	0.368423i	$0.879898\pi$
$38$	−18.2579	−2.96182
$39$	0	0
$40$	12.2371	1.93486
$41$	−1.22869	−0.191890	−0.0959448	−	0.995387i	$-0.530587\pi$
$41$	−1.22869	−0.191890	−0.0959448	+	0.995387i	$0.530587\pi$
$42$	0	0
$43$	10.8406	1.65318	0.826591	−	0.562804i	$-0.190277\pi$
$43$	10.8406	1.65318	0.826591	+	0.562804i	$0.190277\pi$
$44$	15.6314	2.35652
$45$	0	0
$46$	−2.17630	−0.320878
$47$	0.151820	0.0221453	0.0110726	−	0.999939i	$-0.496475\pi$
$47$	0.151820	0.0221453	0.0110726	+	0.999939i	$0.496475\pi$
$48$	0	0
$49$	−5.29473	−0.756389
$50$	−5.57003	−0.787721
$51$	0	0
$52$	20.5801	2.85395
$53$	−3.02053	−0.414902	−0.207451	−	0.978245i	$-0.566517\pi$
$53$	−3.02053	−0.414902	−0.207451	+	0.978245i	$0.566517\pi$
$54$	0	0
$55$	5.53231	0.745976
$56$	−9.44808	−1.26255
$57$	0	0
$58$	2.90984	0.382080
$59$	4.15373	0.540769	0.270385	−	0.962752i	$-0.412849\pi$
$59$	4.15373	0.540769	0.270385	+	0.962752i	$0.412849\pi$
$60$	0	0
$61$	5.62714	0.720482	0.360241	−	0.932859i	$-0.382694\pi$
$61$	5.62714	0.720482	0.360241	+	0.932859i	$0.382694\pi$
$62$	−10.3454	−1.31386
$63$	0	0
$64$	6.67199	0.833999
$65$	7.28378	0.903442
$66$	0	0
$67$	12.9934	1.58740	0.793700	−	0.608309i	$-0.208152\pi$
$67$	12.9934	1.58740	0.793700	+	0.608309i	$0.208152\pi$
$68$	4.89626	0.593758
$69$	0	0
$70$	−5.75055	−0.687322
$71$	11.2862	1.33942	0.669711	−	0.742622i	$-0.266418\pi$
$71$	11.2862	1.33942	0.669711	+	0.742622i	$0.266418\pi$
$72$	0	0
$73$	11.7148	1.37112	0.685560	−	0.728017i	$-0.259558\pi$
$73$	11.7148	1.37112	0.685560	+	0.728017i	$0.259558\pi$
$74$	−29.4464	−3.42308
$75$	0	0
$76$	−33.5118	−3.84407
$77$	−4.27140	−0.486771
$78$	0	0
$79$	−1.66517	−0.187346	−0.0936732	−	0.995603i	$-0.529861\pi$
$79$	−1.66517	−0.187346	−0.0936732	+	0.995603i	$0.529861\pi$
$80$	15.6955	1.75481
$81$	0	0
$82$	−3.19906	−0.353277
$83$	2.34322	0.257201	0.128601	−	0.991696i	$-0.458951\pi$
$83$	2.34322	0.257201	0.128601	+	0.991696i	$0.458951\pi$
$84$	0	0
$85$	1.73290	0.187959
$86$	28.2250	3.04358
$87$	0	0
$88$	23.6657	2.52277
$89$	−18.1099	−1.91965	−0.959825	−	0.280598i	$-0.909467\pi$
$89$	−18.1099	−1.91965	−0.959825	+	0.280598i	$0.909467\pi$
$90$	0	0
$91$	−5.62368	−0.589522
$92$	−3.99453	−0.416458
$93$	0	0
$94$	0.395283	0.0407703
$95$	−11.8606	−1.21687
$96$	0	0
$97$	−7.17873	−0.728890	−0.364445	−	0.931225i	$-0.618741\pi$
$97$	−7.17873	−0.728890	−0.364445	+	0.931225i	$0.618741\pi$
$98$	−13.7855	−1.39254
$99$	0	0
$100$	−10.2236	−1.02236
$101$	5.48113	0.545393	0.272696	−	0.962100i	$-0.412085\pi$
$101$	5.48113	0.545393	0.272696	+	0.962100i	$0.412085\pi$
$102$	0	0
$103$	−15.7371	−1.55062	−0.775311	−	0.631580i	$-0.782407\pi$
$103$	−15.7371	−1.55062	−0.775311	+	0.631580i	$0.782407\pi$
$104$	31.1580	3.05530
$105$	0	0
$106$	−7.86433	−0.763851
$107$	11.1816	1.08097	0.540483	−	0.841355i	$-0.318242\pi$
$107$	11.1816	1.08097	0.540483	+	0.841355i	$0.318242\pi$
$108$	0	0
$109$	−0.296424	−0.0283922	−0.0141961	−	0.999899i	$-0.504519\pi$
$109$	−0.296424	−0.0283922	−0.0141961	+	0.999899i	$0.504519\pi$
$110$	14.4041	1.37337
$111$	0	0
$112$	−12.1182	−1.14506
$113$	−10.5881	−0.996045	−0.498023	−	0.867164i	$-0.665940\pi$
$113$	−10.5881	−0.996045	−0.498023	+	0.867164i	$0.665940\pi$
$114$	0	0
$115$	−1.41375	−0.131833
$116$	5.34091	0.495891
$117$	0	0
$118$	10.8147	0.995578
$119$	−1.33794	−0.122649
$120$	0	0
$121$	−0.300940	−0.0273582
$122$	14.6510	1.32644
$123$	0	0
$124$	−18.9886	−1.70522
$125$	−12.0751	−1.08003
$126$	0	0
$127$	14.6989	1.30432	0.652160	−	0.758081i	$-0.273863\pi$
$127$	14.6989	1.30432	0.652160	+	0.758081i	$0.273863\pi$
$128$	−2.01059	−0.177713
$129$	0	0
$130$	18.9642	1.66327
$131$	−9.63853	−0.842122	−0.421061	−	0.907032i	$-0.638342\pi$
$131$	−9.63853	−0.842122	−0.421061	+	0.907032i	$0.638342\pi$
$132$	0	0
$133$	9.15735	0.794043
$134$	33.8300	2.92247
$135$	0	0
$136$	7.41286	0.635647
$137$	−23.2376	−1.98532	−0.992661	−	0.120933i	$-0.961411\pi$
$137$	−23.2376	−1.98532	−0.992661	+	0.120933i	$0.961411\pi$
$138$	0	0
$139$	−1.29376	−0.109736	−0.0548678	−	0.998494i	$-0.517474\pi$
$139$	−1.29376	−0.109736	−0.0548678	+	0.998494i	$0.517474\pi$
$140$	−10.5549	−0.892055
$141$	0	0
$142$	29.3850	2.46593
$143$	14.0863	1.17795
$144$	0	0
$145$	1.89027	0.156978
$146$	30.5011	2.52429
$147$	0	0
$148$	−54.0480	−4.44271
$149$	−17.5316	−1.43625	−0.718123	−	0.695916i	$-0.754998\pi$
$149$	−17.5316	−1.43625	−0.718123	+	0.695916i	$0.754998\pi$
$150$	0	0
$151$	5.26990	0.428858	0.214429	−	0.976740i	$-0.431211\pi$
$151$	5.26990	0.428858	0.214429	+	0.976740i	$0.431211\pi$
$152$	−50.7363	−4.11526
$153$	0	0
$154$	−11.1211	−0.896166
$155$	−6.72048	−0.539802
$156$	0	0
$157$	−14.2165	−1.13460	−0.567299	−	0.823512i	$-0.692012\pi$
$157$	−14.2165	−1.13460	−0.567299	+	0.823512i	$0.692012\pi$
$158$	−4.33548	−0.344912
$159$	0	0
$160$	16.3908	1.29581
$161$	1.09153	0.0860250
$162$	0	0
$163$	−15.9688	−1.25077	−0.625386	−	0.780315i	$-0.715059\pi$
$163$	−15.9688	−1.25077	−0.625386	+	0.780315i	$0.715059\pi$
$164$	−5.87176	−0.458508
$165$	0	0
$166$	6.10086	0.473518
$167$	21.5275	1.66585	0.832925	−	0.553386i	$-0.186665\pi$
$167$	21.5275	1.66585	0.832925	+	0.553386i	$0.186665\pi$
$168$	0	0
$169$	5.54585	0.426604
$170$	4.51181	0.346040
$171$	0	0
$172$	51.8060	3.95017
$173$	5.82531	0.442890	0.221445	−	0.975173i	$-0.428923\pi$
$173$	5.82531	0.442890	0.221445	+	0.975173i	$0.428923\pi$
$174$	0	0
$175$	2.79368	0.211182
$176$	30.3538	2.28801
$177$	0	0
$178$	−47.1515	−3.53416
$179$	6.03932	0.451400	0.225700	−	0.974197i	$-0.427533\pi$
$179$	6.03932	0.451400	0.225700	+	0.974197i	$0.427533\pi$
$180$	0	0
$181$	−4.03706	−0.300073	−0.150036	−	0.988680i	$-0.547939\pi$
$181$	−4.03706	−0.300073	−0.150036	+	0.988680i	$0.547939\pi$
$182$	−14.6420	−1.08533
$183$	0	0
$184$	−6.04765	−0.445839
$185$	−19.1288	−1.40638
$186$	0	0
$187$	3.35129	0.245071
$188$	0.725529	0.0529146
$189$	0	0
$190$	−30.8805	−2.24031
$191$	16.4803	1.19247	0.596235	−	0.802810i	$-0.296663\pi$
$191$	16.4803	1.19247	0.596235	+	0.802810i	$0.296663\pi$
$192$	0	0
$193$	−23.0631	−1.66012	−0.830060	−	0.557674i	$-0.811694\pi$
$193$	−23.0631	−1.66012	−0.830060	+	0.557674i	$0.811694\pi$
$194$	−18.6907	−1.34192
$195$	0	0
$196$	−25.3028	−1.80734
$197$	13.5540	0.965680	0.482840	−	0.875709i	$-0.339605\pi$
$197$	13.5540	0.965680	0.482840	+	0.875709i	$0.339605\pi$
$198$	0	0
$199$	−25.6725	−1.81988	−0.909938	−	0.414745i	$-0.863871\pi$
$199$	−25.6725	−1.81988	−0.909938	+	0.414745i	$0.863871\pi$
$200$	−15.4784	−1.09449
$201$	0	0
$202$	14.2708	1.00409
$203$	−1.45944	−0.102433
$204$	0	0
$205$	−2.07815	−0.145144
$206$	−40.9735	−2.85476
$207$	0	0
$208$	39.9635	2.77097
$209$	−22.9375	−1.58662
$210$	0	0
$211$	5.18301	0.356813	0.178407	−	0.983957i	$-0.442906\pi$
$211$	5.18301	0.356813	0.178407	+	0.983957i	$0.442906\pi$
$212$	−14.4347	−0.991380
$213$	0	0
$214$	29.1127	1.99010
$215$	18.3353	1.25046
$216$	0	0
$217$	5.18877	0.352237
$218$	−0.771777	−0.0522713
$219$	0	0
$220$	26.4382	1.78246
$221$	4.41227	0.296802
$222$	0	0
$223$	−6.50914	−0.435884	−0.217942	−	0.975962i	$-0.569934\pi$
$223$	−6.50914	−0.435884	−0.217942	+	0.975962i	$0.569934\pi$
$224$	−12.6551	−0.845553
$225$	0	0
$226$	−27.5675	−1.83376
$227$	−11.0460	−0.733150	−0.366575	−	0.930389i	$-0.619470\pi$
$227$	−11.0460	−0.733150	−0.366575	+	0.930389i	$0.619470\pi$
$228$	0	0
$229$	8.95679	0.591881	0.295941	−	0.955206i	$-0.404367\pi$
$229$	8.95679	0.591881	0.295941	+	0.955206i	$0.404367\pi$
$230$	−3.68089	−0.242710
$231$	0	0
$232$	8.08605	0.530875
$233$	6.46647	0.423632	0.211816	−	0.977310i	$-0.432062\pi$
$233$	6.46647	0.423632	0.211816	+	0.977310i	$0.432062\pi$
$234$	0	0
$235$	0.256781	0.0167506
$236$	19.8501	1.29213
$237$	0	0
$238$	−3.48349	−0.225801
$239$	25.0588	1.62092	0.810460	−	0.585794i	$-0.199217\pi$
$239$	25.0588	1.62092	0.810460	+	0.585794i	$0.199217\pi$
$240$	0	0
$241$	−1.00000	−0.0644157
$241$	−1.00000	−0.0644157
$242$	−0.783536	−0.0503676
$243$	0	0
$244$	26.8914	1.72154
$245$	−8.95523	−0.572129
$246$	0	0
$247$	−30.1992	−1.92153
$248$	−28.7484	−1.82552
$249$	0	0
$250$	−31.4391	−1.98838
$251$	12.5611	0.792847	0.396424	−	0.918068i	$-0.370251\pi$
$251$	12.5611	0.792847	0.396424	+	0.918068i	$0.370251\pi$
$252$	0	0
$253$	−2.73409	−0.171891
$254$	38.2706	2.40131
$255$	0	0
$256$	−18.5788	−1.16117
$257$	−16.0367	−1.00034	−0.500171	−	0.865927i	$-0.666730\pi$
$257$	−16.0367	−1.00034	−0.500171	+	0.865927i	$0.666730\pi$
$258$	0	0
$259$	14.7690	0.917702
$260$	34.8082	2.15871
$261$	0	0
$262$	−25.0951	−1.55038
$263$	−3.33210	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$263$	−3.33210	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$264$	0	0
$265$	−5.10877	−0.313829
$266$	23.8423	1.46187
$267$	0	0
$268$	62.0939	3.79299
$269$	7.44101	0.453686	0.226843	−	0.973931i	$-0.427160\pi$
$269$	7.44101	0.453686	0.226843	+	0.973931i	$0.427160\pi$
$270$	0	0
$271$	−20.6031	−1.25155	−0.625774	−	0.780005i	$-0.715217\pi$
$271$	−20.6031	−1.25155	−0.625774	+	0.780005i	$0.715217\pi$
$272$	9.50779	0.576495
$273$	0	0
$274$	−60.5020	−3.65506
$275$	−6.99764	−0.421974
$276$	0	0
$277$	5.17667	0.311036	0.155518	−	0.987833i	$-0.450295\pi$
$277$	5.17667	0.311036	0.155518	+	0.987833i	$0.450295\pi$
$278$	−3.36848	−0.202028
$279$	0	0
$280$	−15.9800	−0.954988
$281$	14.1821	0.846031	0.423015	−	0.906123i	$-0.360972\pi$
$281$	14.1821	0.846031	0.423015	+	0.906123i	$0.360972\pi$
$282$	0	0
$283$	−7.05196	−0.419196	−0.209598	−	0.977788i	$-0.567215\pi$
$283$	−7.05196	−0.419196	−0.209598	+	0.977788i	$0.567215\pi$
$284$	53.9351	3.20046
$285$	0	0
$286$	36.6754	2.16866
$287$	1.60450	0.0947109
$288$	0	0
$289$	−15.9503	−0.938251
$290$	4.92155	0.289004
$291$	0	0
$292$	55.9837	3.27620
$293$	2.78966	0.162974	0.0814869	−	0.996674i	$-0.474033\pi$
$293$	2.78966	0.162974	0.0814869	+	0.996674i	$0.474033\pi$
$294$	0	0
$295$	7.02540	0.409035
$296$	−81.8278	−4.75614
$297$	0	0
$298$	−45.6458	−2.64419
$299$	−3.59968	−0.208175
$300$	0	0
$301$	−14.1564	−0.815960
$302$	13.7208	0.789546
$303$	0	0
$304$	−65.0749	−3.73230
$305$	9.51747	0.544969
$306$	0	0
$307$	19.8285	1.13167	0.565837	−	0.824517i	$-0.308553\pi$
$307$	19.8285	1.13167	0.565837	+	0.824517i	$0.308553\pi$
$308$	−20.4124	−1.16311
$309$	0	0
$310$	−17.4976	−0.993798
$311$	−33.5109	−1.90023	−0.950114	−	0.311902i	$-0.899034\pi$
$311$	−33.5109	−1.90023	−0.950114	+	0.311902i	$0.899034\pi$
$312$	0	0
$313$	19.4152	1.09741	0.548705	−	0.836016i	$-0.315121\pi$
$313$	19.4152	1.09741	0.548705	+	0.836016i	$0.315121\pi$
$314$	−37.0144	−2.08884
$315$	0	0
$316$	−7.95763	−0.447652
$317$	11.7255	0.658569	0.329284	−	0.944231i	$-0.393193\pi$
$317$	11.7255	0.658569	0.329284	+	0.944231i	$0.393193\pi$
$318$	0	0
$319$	3.65564	0.204676
$320$	11.2847	0.630832
$321$	0	0
$322$	2.84195	0.158376
$323$	−7.18475	−0.399770
$324$	0	0
$325$	−9.21303	−0.511047
$326$	−41.5768	−2.30272
$327$	0	0
$328$	−8.88976	−0.490855
$329$	−0.198256	−0.0109302
$330$	0	0
$331$	4.93072	0.271017	0.135509	−	0.990776i	$-0.456733\pi$
$331$	4.93072	0.271017	0.135509	+	0.990776i	$0.456733\pi$
$332$	11.1979	0.614566
$333$	0	0
$334$	56.0496	3.06690
$335$	21.9764	1.20070
$336$	0	0
$337$	34.4425	1.87620	0.938100	−	0.346364i	$-0.112584\pi$
$337$	34.4425	1.87620	0.938100	+	0.346364i	$0.112584\pi$
$338$	14.4393	0.785395
$339$	0	0
$340$	8.28128	0.449116
$341$	−12.9969	−0.703822
$342$	0	0
$343$	16.0552	0.866900
$344$	78.4334	4.22885
$345$	0	0
$346$	15.1669	0.815378
$347$	23.4553	1.25915	0.629574	−	0.776940i	$-0.283229\pi$
$347$	23.4553	1.25915	0.629574	+	0.776940i	$0.283229\pi$
$348$	0	0
$349$	15.8727	0.849648	0.424824	−	0.905276i	$-0.360336\pi$
$349$	15.8727	0.849648	0.424824	+	0.905276i	$0.360336\pi$
$350$	7.27369	0.388795
$351$	0	0
$352$	31.6986	1.68954
$353$	28.4598	1.51476	0.757382	−	0.652972i	$-0.226478\pi$
$353$	28.4598	1.51476	0.757382	+	0.652972i	$0.226478\pi$
$354$	0	0
$355$	19.0889	1.01313
$356$	−86.5450	−4.58688
$357$	0	0
$358$	15.7241	0.831047
$359$	0.772338	0.0407625	0.0203812	−	0.999792i	$-0.493512\pi$
$359$	0.772338	0.0407625	0.0203812	+	0.999792i	$0.493512\pi$
$360$	0	0
$361$	30.1751	1.58816
$362$	−10.5110	−0.552446
$363$	0	0
$364$	−26.8748	−1.40862
$365$	19.8139	1.03711
$366$	0	0
$367$	−23.3631	−1.21954	−0.609772	−	0.792577i	$-0.708739\pi$
$367$	−23.3631	−1.21954	−0.609772	+	0.792577i	$0.708739\pi$
$368$	−7.75677	−0.404350
$369$	0	0
$370$	−49.8042	−2.58920
$371$	3.94439	0.204783
$372$	0	0
$373$	27.7831	1.43855	0.719277	−	0.694724i	$-0.244473\pi$
$373$	27.7831	1.43855	0.719277	+	0.694724i	$0.244473\pi$
$374$	8.72550	0.451185
$375$	0	0
$376$	1.09844	0.0566477
$377$	4.81297	0.247881
$378$	0	0
$379$	7.33328	0.376685	0.188343	−	0.982103i	$-0.439688\pi$
$379$	7.33328	0.376685	0.188343	+	0.982103i	$0.439688\pi$
$380$	−56.6802	−2.90763
$381$	0	0
$382$	42.9084	2.19539
$383$	33.4931	1.71142	0.855708	−	0.517459i	$-0.173122\pi$
$383$	33.4931	1.71142	0.855708	+	0.517459i	$0.173122\pi$
$384$	0	0
$385$	−7.22443	−0.368191
$386$	−60.0477	−3.05635
$387$	0	0
$388$	−34.3062	−1.74163
$389$	−15.6882	−0.795425	−0.397713	−	0.917510i	$-0.630196\pi$
$389$	−15.6882	−0.795425	−0.397713	+	0.917510i	$0.630196\pi$
$390$	0	0
$391$	−0.856405	−0.0433103
$392$	−38.3080	−1.93485
$393$	0	0
$394$	35.2894	1.77786
$395$	−2.81639	−0.141708
$396$	0	0
$397$	8.34483	0.418815	0.209408	−	0.977828i	$-0.432846\pi$
$397$	8.34483	0.418815	0.209408	+	0.977828i	$0.432846\pi$
$398$	−66.8416	−3.35047
$399$	0	0
$400$	−19.8527	−0.992635
$401$	38.5766	1.92642	0.963211	−	0.268747i	$-0.0866094\pi$
$401$	38.5766	1.92642	0.963211	+	0.268747i	$0.0866094\pi$
$402$	0	0
$403$	−17.1116	−0.852389
$404$	26.1936	1.30318
$405$	0	0
$406$	−3.79985	−0.188583
$407$	−36.9936	−1.83371
$408$	0	0
$409$	−19.5415	−0.966266	−0.483133	−	0.875547i	$-0.660501\pi$
$409$	−19.5415	−0.966266	−0.483133	+	0.875547i	$0.660501\pi$
$410$	−5.41073	−0.267217
$411$	0	0
$412$	−75.2055	−3.70511
$413$	−5.42419	−0.266907
$414$	0	0
$415$	3.96320	0.194546
$416$	41.7341	2.04618
$417$	0	0
$418$	−59.7206	−2.92103
$419$	29.7748	1.45459	0.727297	−	0.686323i	$-0.240776\pi$
$419$	29.7748	1.45459	0.727297	+	0.686323i	$0.240776\pi$
$420$	0	0
$421$	−4.78920	−0.233411	−0.116706	−	0.993167i	$-0.537233\pi$
$421$	−4.78920	−0.233411	−0.116706	+	0.993167i	$0.537233\pi$
$422$	13.4946	0.656908
$423$	0	0
$424$	−21.8539	−1.06132
$425$	−2.19189	−0.106322
$426$	0	0
$427$	−7.34827	−0.355608
$428$	53.4354	2.58290
$429$	0	0
$430$	47.7383	2.30214
$431$	−5.35939	−0.258153	−0.129076	−	0.991635i	$-0.541201\pi$
$431$	−5.35939	−0.258153	−0.129076	+	0.991635i	$0.541201\pi$
$432$	0	0
$433$	21.5028	1.03336	0.516680	−	0.856179i	$-0.327168\pi$
$433$	21.5028	1.03336	0.516680	+	0.856179i	$0.327168\pi$
$434$	13.5096	0.648482
$435$	0	0
$436$	−1.41657	−0.0678414
$437$	5.86156	0.280396
$438$	0	0
$439$	4.22306	0.201556	0.100778	−	0.994909i	$-0.467867\pi$
$439$	4.22306	0.201556	0.100778	+	0.994909i	$0.467867\pi$
$440$	40.0270	1.90821
$441$	0	0
$442$	11.4879	0.546424
$443$	−8.34030	−0.396260	−0.198130	−	0.980176i	$-0.563487\pi$
$443$	−8.34030	−0.396260	−0.198130	+	0.980176i	$0.563487\pi$
$444$	0	0
$445$	−30.6303	−1.45201
$446$	−16.9474	−0.802481
$447$	0	0
$448$	−8.71269	−0.411636
$449$	−15.7796	−0.744684	−0.372342	−	0.928096i	$-0.621445\pi$
$449$	−15.7796	−0.744684	−0.372342	+	0.928096i	$0.621445\pi$
$450$	0	0
$451$	−4.01898	−0.189247
$452$	−50.5992	−2.37998
$453$	0	0
$454$	−28.7597	−1.34976
$455$	−9.51161	−0.445911
$456$	0	0
$457$	−8.71419	−0.407633	−0.203816	−	0.979009i	$-0.565335\pi$
$457$	−8.71419	−0.407633	−0.203816	+	0.979009i	$0.565335\pi$
$458$	23.3201	1.08968
$459$	0	0
$460$	−6.75614	−0.315007
$461$	−24.9140	−1.16036	−0.580181	−	0.814487i	$-0.697018\pi$
$461$	−24.9140	−1.16036	−0.580181	+	0.814487i	$0.697018\pi$
$462$	0	0
$463$	−2.87272	−0.133507	−0.0667534	−	0.997770i	$-0.521264\pi$
$463$	−2.87272	−0.133507	−0.0667534	+	0.997770i	$0.521264\pi$
$464$	10.3712	0.481473
$465$	0	0
$466$	16.8363	0.779925
$467$	−4.87474	−0.225576	−0.112788	−	0.993619i	$-0.535978\pi$
$467$	−4.87474	−0.225576	−0.112788	+	0.993619i	$0.535978\pi$
$468$	0	0
$469$	−16.9676	−0.783492
$470$	0.668562	0.0308385
$471$	0	0
$472$	30.0528	1.38329
$473$	35.4591	1.63041
$474$	0	0
$475$	15.0021	0.688343
$476$	−6.39383	−0.293061
$477$	0	0
$478$	65.2438	2.98418
$479$	−0.194055	−0.00886661	−0.00443331	−	0.999990i	$-0.501411\pi$
$479$	−0.194055	−0.00886661	−0.00443331	+	0.999990i	$0.501411\pi$
$480$	0	0
$481$	−48.7055	−2.22078
$482$	−2.60363	−0.118592
$483$	0	0
$484$	−1.43815	−0.0653707
$485$	−12.1417	−0.551328
$486$	0	0
$487$	−30.7592	−1.39383	−0.696915	−	0.717154i	$-0.745445\pi$
$487$	−30.7592	−1.39383	−0.696915	+	0.717154i	$0.745445\pi$
$488$	40.7131	1.84300
$489$	0	0
$490$	−23.3161	−1.05331
$491$	21.5528	0.972663	0.486332	−	0.873774i	$-0.338335\pi$
$491$	21.5528	0.972663	0.486332	+	0.873774i	$0.338335\pi$
$492$	0	0
$493$	1.14506	0.0515710
$494$	−78.6275	−3.53762
$495$	0	0
$496$	−36.8729	−1.65564
$497$	−14.7382	−0.661097
$498$	0	0
$499$	21.8992	0.980341	0.490170	−	0.871627i	$-0.336935\pi$
$499$	21.8992	0.980341	0.490170	+	0.871627i	$0.336935\pi$
$500$	−57.7054	−2.58066
$501$	0	0
$502$	32.7043	1.45966
$503$	−4.44114	−0.198021	−0.0990104	−	0.995086i	$-0.531568\pi$
$503$	−4.44114	−0.198021	−0.0990104	+	0.995086i	$0.531568\pi$
$504$	0	0
$505$	9.27050	0.412532
$506$	−7.11855	−0.316458
$507$	0	0
$508$	70.2443	3.11659
$509$	−29.6966	−1.31628	−0.658140	−	0.752895i	$-0.728657\pi$
$509$	−29.6966	−1.31628	−0.658140	+	0.752895i	$0.728657\pi$
$510$	0	0
$511$	−15.2980	−0.676742
$512$	−44.3511	−1.96006
$513$	0	0
$514$	−41.7536	−1.84167
$515$	−26.6169	−1.17288
$516$	0	0
$517$	0.496595	0.0218402
$518$	38.4530	1.68953
$519$	0	0
$520$	52.6991	2.31101
$521$	32.5336	1.42532	0.712661	−	0.701509i	$-0.247490\pi$
$521$	32.5336	1.42532	0.712661	+	0.701509i	$0.247490\pi$
$522$	0	0
$523$	−21.6812	−0.948053	−0.474027	−	0.880510i	$-0.657200\pi$
$523$	−21.6812	−0.948053	−0.474027	+	0.880510i	$0.657200\pi$
$524$	−46.0612	−2.01219
$525$	0	0
$526$	−8.67555	−0.378272
$527$	−4.07105	−0.177338
$528$	0	0
$529$	−22.3013	−0.969622
$530$	−13.3013	−0.577773
$531$	0	0
$532$	43.7618	1.89731
$533$	−5.29135	−0.229194
$534$	0	0
$535$	18.9120	0.817637
$536$	94.0092	4.06058
$537$	0	0
$538$	19.3736	0.835255
$539$	−17.3187	−0.745971
$540$	0	0
$541$	6.08311	0.261533	0.130767	−	0.991413i	$-0.458256\pi$
$541$	6.08311	0.261533	0.130767	+	0.991413i	$0.458256\pi$
$542$	−53.6427	−2.30415
$543$	0	0
$544$	9.92903	0.425703
$545$	−0.501356	−0.0214757
$546$	0	0
$547$	−12.1956	−0.521447	−0.260723	−	0.965414i	$-0.583961\pi$
$547$	−12.1956	−0.521447	−0.260723	+	0.965414i	$0.583961\pi$
$548$	−111.049	−4.74379
$549$	0	0
$550$	−18.2192	−0.776871
$551$	−7.83723	−0.333877
$552$	0	0
$553$	2.17448	0.0924684
$554$	13.4781	0.572630
$555$	0	0
$556$	−6.18273	−0.262206
$557$	37.5982	1.59309	0.796544	−	0.604581i	$-0.206659\pi$
$557$	37.5982	1.59309	0.796544	+	0.604581i	$0.206659\pi$
$558$	0	0
$559$	46.6851	1.97457
$560$	−20.4961	−0.866118
$561$	0	0
$562$	36.9248	1.55758
$563$	−20.6037	−0.868342	−0.434171	−	0.900830i	$-0.642959\pi$
$563$	−20.6037	−0.868342	−0.434171	+	0.900830i	$0.642959\pi$
$564$	0	0
$565$	−17.9082	−0.753403
$566$	−18.3607	−0.771756
$567$	0	0
$568$	81.6569	3.42625
$569$	−20.5762	−0.862601	−0.431301	−	0.902208i	$-0.641945\pi$
$569$	−20.5762	−0.862601	−0.431301	+	0.902208i	$0.641945\pi$
$570$	0	0
$571$	37.1320	1.55392	0.776962	−	0.629547i	$-0.216760\pi$
$571$	37.1320	1.55392	0.776962	+	0.629547i	$0.216760\pi$
$572$	67.3165	2.81464
$573$	0	0
$574$	4.17753	0.174367
$575$	1.78821	0.0745736
$576$	0	0
$577$	−34.7064	−1.44484	−0.722422	−	0.691452i	$-0.756971\pi$
$577$	−34.7064	−1.44484	−0.722422	+	0.691452i	$0.756971\pi$
$578$	−41.5285	−1.72736
$579$	0	0
$580$	9.03335	0.375089
$581$	−3.05992	−0.126947
$582$	0	0
$583$	−9.87998	−0.409187
$584$	84.7585	3.50733
$585$	0	0
$586$	7.26324	0.300042
$587$	1.14455	0.0472405	0.0236203	−	0.999721i	$-0.492481\pi$
$587$	1.14455	0.0472405	0.0236203	+	0.999721i	$0.492481\pi$
$588$	0	0
$589$	27.8638	1.14811
$590$	18.2915	0.753050
$591$	0	0
$592$	−104.953	−4.31354
$593$	−3.60735	−0.148136	−0.0740680	−	0.997253i	$-0.523598\pi$
$593$	−3.60735	−0.148136	−0.0740680	+	0.997253i	$0.523598\pi$
$594$	0	0
$595$	−2.26292	−0.0927708
$596$	−83.7813	−3.43181
$597$	0	0
$598$	−9.37222	−0.383258
$599$	31.6723	1.29410	0.647048	−	0.762449i	$-0.276003\pi$
$599$	31.6723	1.29410	0.647048	+	0.762449i	$0.276003\pi$
$600$	0	0
$601$	6.55665	0.267451	0.133726	−	0.991018i	$-0.457306\pi$
$601$	6.55665	0.267451	0.133726	+	0.991018i	$0.457306\pi$
$602$	−36.8579	−1.50222
$603$	0	0
$604$	25.1841	1.02473
$605$	−0.508996	−0.0206936
$606$	0	0
$607$	−9.16648	−0.372056	−0.186028	−	0.982544i	$-0.559562\pi$
$607$	−9.16648	−0.372056	−0.186028	+	0.982544i	$0.559562\pi$
$608$	−67.9579	−2.75606
$609$	0	0
$610$	24.7799	1.00331
$611$	0.653812	0.0264504
$612$	0	0
$613$	7.92345	0.320025	0.160012	−	0.987115i	$-0.448847\pi$
$613$	7.92345	0.320025	0.160012	+	0.987115i	$0.448847\pi$
$614$	51.6260	2.08346
$615$	0	0
$616$	−30.9041	−1.24516
$617$	−16.5959	−0.668126	−0.334063	−	0.942551i	$-0.608420\pi$
$617$	−16.5959	−0.668126	−0.334063	+	0.942551i	$0.608420\pi$
$618$	0	0
$619$	4.49086	0.180503	0.0902514	−	0.995919i	$-0.471233\pi$
$619$	4.49086	0.180503	0.0902514	+	0.995919i	$0.471233\pi$
$620$	−32.1163	−1.28982
$621$	0	0
$622$	−87.2498	−3.49840
$623$	23.6491	0.947481
$624$	0	0
$625$	−9.72656	−0.389063
$626$	50.5499	2.02038
$627$	0	0
$628$	−67.9387	−2.71105
$629$	−11.5876	−0.462028
$630$	0	0
$631$	−5.96553	−0.237484	−0.118742	−	0.992925i	$-0.537886\pi$
$631$	−5.96553	−0.237484	−0.118742	+	0.992925i	$0.537886\pi$
$632$	−12.0477	−0.479233
$633$	0	0
$634$	30.5288	1.21245
$635$	24.8611	0.986581
$636$	0	0
$637$	−22.8017	−0.903435
$638$	9.51791	0.376818
$639$	0	0
$640$	−3.40061	−0.134421
$641$	1.73375	0.0684789	0.0342395	−	0.999414i	$-0.489099\pi$
$641$	1.73375	0.0684789	0.0342395	+	0.999414i	$0.489099\pi$
$642$	0	0
$643$	38.3312	1.51163	0.755817	−	0.654783i	$-0.227240\pi$
$643$	38.3312	1.51163	0.755817	+	0.654783i	$0.227240\pi$
$644$	5.21630	0.205551
$645$	0	0
$646$	−18.7064	−0.735994
$647$	−29.2098	−1.14836	−0.574178	−	0.818730i	$-0.694678\pi$
$647$	−29.2098	−1.14836	−0.574178	+	0.818730i	$0.694678\pi$
$648$	0	0
$649$	13.5866	0.533321
$650$	−23.9873	−0.940858
$651$	0	0
$652$	−76.3127	−2.98864
$653$	42.7952	1.67471	0.837353	−	0.546662i	$-0.184102\pi$
$653$	42.7952	1.67471	0.837353	+	0.546662i	$0.184102\pi$
$654$	0	0
$655$	−16.3021	−0.636976
$656$	−11.4021	−0.445177
$657$	0	0
$658$	−0.516185	−0.0201230
$659$	23.7004	0.923238	0.461619	−	0.887078i	$-0.347269\pi$
$659$	23.7004	0.923238	0.461619	+	0.887078i	$0.347269\pi$
$660$	0	0
$661$	24.1228	0.938268	0.469134	−	0.883127i	$-0.344566\pi$
$661$	24.1228	0.938268	0.469134	+	0.883127i	$0.344566\pi$
$662$	12.8378	0.498954
$663$	0	0
$664$	16.9535	0.657923
$665$	15.4883	0.600610
$666$	0	0
$667$	−0.934180	−0.0361716
$668$	102.877	3.98044
$669$	0	0
$670$	57.2184	2.21054
$671$	18.4061	0.710558
$672$	0	0
$673$	−3.67522	−0.141669	−0.0708346	−	0.997488i	$-0.522566\pi$
$673$	−3.67522	−0.141669	−0.0708346	+	0.997488i	$0.522566\pi$
$674$	89.6753	3.45416
$675$	0	0
$676$	26.5029	1.01934
$677$	−45.7769	−1.75935	−0.879675	−	0.475575i	$-0.842240\pi$
$677$	−45.7769	−1.75935	−0.879675	+	0.475575i	$0.842240\pi$
$678$	0	0
$679$	9.37443	0.359758
$680$	12.5377	0.480800
$681$	0	0
$682$	−33.8391	−1.29577
$683$	1.52419	0.0583215	0.0291608	−	0.999575i	$-0.490717\pi$
$683$	1.52419	0.0583215	0.0291608	+	0.999575i	$0.490717\pi$
$684$	0	0
$685$	−39.3029	−1.50169
$686$	41.8018	1.59600
$687$	0	0
$688$	100.599	3.83532
$689$	−13.0079	−0.495561
$690$	0	0
$691$	20.3036	0.772386	0.386193	−	0.922418i	$-0.373790\pi$
$691$	20.3036	0.772386	0.386193	+	0.922418i	$0.373790\pi$
$692$	27.8384	1.05826
$693$	0	0
$694$	61.0689	2.31814
$695$	−2.18821	−0.0830034
$696$	0	0
$697$	−1.25888	−0.0476833
$698$	41.3267	1.56424
$699$	0	0
$700$	13.3506	0.504606
$701$	10.3863	0.392285	0.196143	−	0.980575i	$-0.437158\pi$
$701$	10.3863	0.392285	0.196143	+	0.980575i	$0.437158\pi$
$702$	0	0
$703$	79.3098	2.99123
$704$	21.8237	0.822511
$705$	0	0
$706$	74.0988	2.78874
$707$	−7.15759	−0.269189
$708$	0	0
$709$	34.9053	1.31090	0.655448	−	0.755240i	$-0.272480\pi$
$709$	34.9053	1.31090	0.655448	+	0.755240i	$0.272480\pi$
$710$	49.7002	1.86522
$711$	0	0
$712$	−131.028	−4.91048
$713$	3.32129	0.124383
$714$	0	0
$715$	23.8248	0.890998
$716$	28.8611	1.07859
$717$	0	0
$718$	2.01088	0.0750454
$719$	−5.17696	−0.193068	−0.0965341	−	0.995330i	$-0.530776\pi$
$719$	−5.17696	−0.193068	−0.0965341	+	0.995330i	$0.530776\pi$
$720$	0	0
$721$	20.5505	0.765339
$722$	78.5646	2.92387
$723$	0	0
$724$	−19.2926	−0.717003
$725$	−2.39094	−0.0887974
$726$	0	0
$727$	−7.00384	−0.259758	−0.129879	−	0.991530i	$-0.541459\pi$
$727$	−7.00384	−0.259758	−0.129879	+	0.991530i	$0.541459\pi$
$728$	−40.6881	−1.50800
$729$	0	0
$730$	51.5880	1.90936
$731$	11.1069	0.410804
$732$	0	0
$733$	−1.22518	−0.0452530	−0.0226265	−	0.999744i	$-0.507203\pi$
$733$	−1.22518	−0.0452530	−0.0226265	+	0.999744i	$0.507203\pi$
$734$	−60.8288	−2.24523
$735$	0	0
$736$	−8.10042	−0.298586
$737$	42.5008	1.56554
$738$	0	0
$739$	−31.4569	−1.15716	−0.578580	−	0.815625i	$-0.696393\pi$
$739$	−31.4569	−1.15716	−0.578580	+	0.815625i	$0.696393\pi$
$740$	−91.4140	−3.36045
$741$	0	0
$742$	10.2697	0.377014
$743$	−17.5176	−0.642659	−0.321330	−	0.946967i	$-0.604130\pi$
$743$	−17.5176	−0.642659	−0.321330	+	0.946967i	$0.604130\pi$
$744$	0	0
$745$	−29.6521	−1.08637
$746$	72.3368	2.64844
$747$	0	0
$748$	16.0154	0.585580
$749$	−14.6016	−0.533532
$750$	0	0
$751$	−35.2893	−1.28773	−0.643863	−	0.765141i	$-0.722669\pi$
$751$	−35.2893	−1.28773	−0.643863	+	0.765141i	$0.722669\pi$
$752$	1.40887	0.0513761
$753$	0	0
$754$	12.5312	0.456359
$755$	8.91324	0.324386
$756$	0	0
$757$	−17.2974	−0.628686	−0.314343	−	0.949309i	$-0.601784\pi$
$757$	−17.2974	−0.628686	−0.314343	+	0.949309i	$0.601784\pi$
$758$	19.0931	0.693493
$759$	0	0
$760$	−85.8129	−3.11276
$761$	−9.53250	−0.345553	−0.172776	−	0.984961i	$-0.555274\pi$
$761$	−9.53250	−0.345553	−0.172776	+	0.984961i	$0.555274\pi$
$762$	0	0
$763$	0.387088	0.0140135
$764$	78.7570	2.84933
$765$	0	0
$766$	87.2034	3.15079
$767$	17.8880	0.645897
$768$	0	0
$769$	−37.0399	−1.33569	−0.667847	−	0.744299i	$-0.732784\pi$
$769$	−37.0399	−1.33569	−0.667847	+	0.744299i	$0.732784\pi$
$770$	−18.8097	−0.677855
$771$	0	0
$772$	−110.216	−3.96675
$773$	6.97450	0.250855	0.125428	−	0.992103i	$-0.459970\pi$
$773$	6.97450	0.250855	0.125428	+	0.992103i	$0.459970\pi$
$774$	0	0
$775$	8.50053	0.305348
$776$	−51.9391	−1.86450
$777$	0	0
$778$	−40.8463	−1.46441
$779$	8.61621	0.308708
$780$	0	0
$781$	36.9164	1.32097
$782$	−2.22976	−0.0797360
$783$	0	0
$784$	−49.1342	−1.75479
$785$	−24.0450	−0.858204
$786$	0	0
$787$	29.9359	1.06710	0.533550	−	0.845768i	$-0.320857\pi$
$787$	29.9359	1.06710	0.533550	+	0.845768i	$0.320857\pi$
$788$	64.7726	2.30743
$789$	0	0
$790$	−7.33281	−0.260890
$791$	13.8266	0.491617
$792$	0	0
$793$	24.2332	0.860547
$794$	21.7268	0.771056
$795$	0	0
$796$	−122.685	−4.34847
$797$	10.7103	0.379380	0.189690	−	0.981844i	$-0.439252\pi$
$797$	10.7103	0.379380	0.189690	+	0.981844i	$0.439252\pi$
$798$	0	0
$799$	0.155550	0.00550295
$800$	−20.7323	−0.732996
$801$	0	0
$802$	100.439	3.54662
$803$	38.3186	1.35223
$804$	0	0
$805$	1.84617	0.0650688
$806$	−44.5522	−1.56928
$807$	0	0
$808$	39.6567	1.39512
$809$	30.4625	1.07100	0.535502	−	0.844534i	$-0.320122\pi$
$809$	30.4625	1.07100	0.535502	+	0.844534i	$0.320122\pi$
$810$	0	0
$811$	−6.38159	−0.224088	−0.112044	−	0.993703i	$-0.535740\pi$
$811$	−6.38159	−0.224088	−0.112044	+	0.993703i	$0.535740\pi$
$812$	−6.97449	−0.244757
$813$	0	0
$814$	−96.3176	−3.37593
$815$	−27.0088	−0.946077
$816$	0	0
$817$	−76.0199	−2.65960
$818$	−50.8788	−1.77894
$819$	0	0
$820$	−9.93121	−0.346813
$821$	31.3814	1.09522	0.547609	−	0.836734i	$-0.315538\pi$
$821$	31.3814	1.09522	0.547609	+	0.836734i	$0.315538\pi$
$822$	0	0
$823$	−8.61720	−0.300377	−0.150188	−	0.988657i	$-0.547988\pi$
$823$	−8.61720	−0.300377	−0.150188	+	0.988657i	$0.547988\pi$
$824$	−113.860	−3.96650
$825$	0	0
$826$	−14.1226	−0.491387
$827$	18.2047	0.633039	0.316520	−	0.948586i	$-0.397486\pi$
$827$	18.2047	0.633039	0.316520	+	0.948586i	$0.397486\pi$
$828$	0	0
$829$	38.0162	1.32036	0.660179	−	0.751108i	$-0.270480\pi$
$829$	38.0162	1.32036	0.660179	+	0.751108i	$0.270480\pi$
$830$	10.3187	0.358167
$831$	0	0
$832$	28.7328	0.996132
$833$	−5.42479	−0.187958
$834$	0	0
$835$	36.4106	1.26004
$836$	−109.615	−3.79112
$837$	0	0
$838$	77.5224	2.67797
$839$	−45.6228	−1.57507	−0.787537	−	0.616267i	$-0.788644\pi$
$839$	−45.6228	−1.57507	−0.787537	+	0.616267i	$0.788644\pi$
$840$	0	0
$841$	−27.7509	−0.956929
$842$	−12.4693	−0.429720
$843$	0	0
$844$	24.7689	0.852582
$845$	9.37997	0.322681
$846$	0	0
$847$	0.392987	0.0135032
$848$	−28.0300	−0.962556
$849$	0	0
$850$	−5.70685	−0.195743
$851$	9.45354	0.324063
$852$	0	0
$853$	17.2406	0.590307	0.295153	−	0.955450i	$-0.404629\pi$
$853$	17.2406	0.590307	0.295153	+	0.955450i	$0.404629\pi$
$854$	−19.1321	−0.654689
$855$	0	0
$856$	80.9003	2.76512
$857$	−9.87638	−0.337371	−0.168685	−	0.985670i	$-0.553952\pi$
$857$	−9.87638	−0.337371	−0.168685	+	0.985670i	$0.553952\pi$
$858$	0	0
$859$	25.8904	0.883369	0.441685	−	0.897170i	$-0.354381\pi$
$859$	25.8904	0.883369	0.441685	+	0.897170i	$0.354381\pi$
$860$	87.6220	2.98789
$861$	0	0
$862$	−13.9538	−0.475270
$863$	14.8445	0.505311	0.252656	−	0.967556i	$-0.418696\pi$
$863$	14.8445	0.505311	0.252656	+	0.967556i	$0.418696\pi$
$864$	0	0
$865$	9.85263	0.334999
$866$	55.9853	1.90246
$867$	0	0
$868$	24.7964	0.841646
$869$	−5.44668	−0.184766
$870$	0	0
$871$	55.9561	1.89600
$872$	−2.14467	−0.0726275
$873$	0	0
$874$	15.2613	0.516221
$875$	15.7684	0.533070
$876$	0	0
$877$	42.0781	1.42088	0.710439	−	0.703759i	$-0.248497\pi$
$877$	42.0781	1.42088	0.710439	+	0.703759i	$0.248497\pi$
$878$	10.9953	0.371072
$879$	0	0
$880$	51.3389	1.73064
$881$	19.0418	0.641534	0.320767	−	0.947158i	$-0.396059\pi$
$881$	19.0418	0.641534	0.320767	+	0.947158i	$0.396059\pi$
$882$	0	0
$883$	6.51087	0.219108	0.109554	−	0.993981i	$-0.465058\pi$
$883$	6.51087	0.219108	0.109554	+	0.993981i	$0.465058\pi$
$884$	21.0857	0.709188
$885$	0	0
$886$	−21.7150	−0.729531
$887$	−5.59140	−0.187741	−0.0938704	−	0.995584i	$-0.529924\pi$
$887$	−5.59140	−0.187741	−0.0938704	+	0.995584i	$0.529924\pi$
$888$	0	0
$889$	−19.1948	−0.643773
$890$	−79.7497	−2.67322
$891$	0	0
$892$	−31.1063	−1.04152
$893$	−1.06464	−0.0356268
$894$	0	0
$895$	10.2146	0.341437
$896$	2.62555	0.0877135
$897$	0	0
$898$	−41.0841	−1.37099
$899$	−4.44076	−0.148108
$900$	0	0
$901$	−3.09473	−0.103100
$902$	−10.4639	−0.348411
$903$	0	0
$904$	−76.6063	−2.54789
$905$	−6.82809	−0.226973
$906$	0	0
$907$	18.3915	0.610681	0.305340	−	0.952243i	$-0.401230\pi$
$907$	18.3915	0.610681	0.305340	+	0.952243i	$0.401230\pi$
$908$	−52.7874	−1.75181
$909$	0	0
$910$	−24.7647	−0.820941
$911$	38.2665	1.26783	0.633913	−	0.773404i	$-0.281448\pi$
$911$	38.2665	1.26783	0.633913	+	0.773404i	$0.281448\pi$
$912$	0	0
$913$	7.66452	0.253659
$914$	−22.6885	−0.750469
$915$	0	0
$916$	42.8033	1.41426
$917$	12.5866	0.415646
$918$	0	0
$919$	−4.89962	−0.161624	−0.0808118	−	0.996729i	$-0.525751\pi$
$919$	−4.89962	−0.161624	−0.0808118	+	0.996729i	$0.525751\pi$
$920$	−10.2287	−0.337230
$921$	0	0
$922$	−64.8669	−2.13628
$923$	48.6038	1.59981
$924$	0	0
$925$	24.1954	0.795541
$926$	−7.47949	−0.245791
$927$	0	0
$928$	10.8307	0.355536
$929$	−17.9381	−0.588528	−0.294264	−	0.955724i	$-0.595075\pi$
$929$	−17.9381	−0.588528	−0.294264	+	0.955724i	$0.595075\pi$
$930$	0	0
$931$	37.1292	1.21686
$932$	30.9024	1.01224
$933$	0	0
$934$	−12.6920	−0.415295
$935$	5.66820	0.185370
$936$	0	0
$937$	29.8369	0.974730	0.487365	−	0.873198i	$-0.337958\pi$
$937$	29.8369	0.974730	0.487365	+	0.873198i	$0.337958\pi$
$938$	−44.1773	−1.44244
$939$	0	0
$940$	1.22712	0.0400243
$941$	−26.8047	−0.873809	−0.436904	−	0.899508i	$-0.643925\pi$
$941$	−26.8047	−0.873809	−0.436904	+	0.899508i	$0.643925\pi$
$942$	0	0
$943$	1.02703	0.0334448
$944$	38.5459	1.25456
$945$	0	0
$946$	92.3222	3.00165
$947$	−2.94622	−0.0957393	−0.0478697	−	0.998854i	$-0.515243\pi$
$947$	−2.94622	−0.0957393	−0.0478697	+	0.998854i	$0.515243\pi$
$948$	0	0
$949$	50.4499	1.63767
$950$	39.0598	1.26727
$951$	0	0
$952$	−9.68017	−0.313736
$953$	4.62861	0.149935	0.0749676	−	0.997186i	$-0.476115\pi$
$953$	4.62861	0.149935	0.0749676	+	0.997186i	$0.476115\pi$
$954$	0	0
$955$	27.8739	0.901978
$956$	119.753	3.87308
$957$	0	0
$958$	−0.505247	−0.0163238
$959$	30.3451	0.979894
$960$	0	0
$961$	−15.2117	−0.490702
$962$	−126.811	−4.08854
$963$	0	0
$964$	−4.77887	−0.153917
$965$	−39.0078	−1.25571
$966$	0	0
$967$	20.1721	0.648689	0.324345	−	0.945939i	$-0.394856\pi$
$967$	20.1721	0.648689	0.324345	+	0.945939i	$0.394856\pi$
$968$	−2.17734	−0.0699825
$969$	0	0
$970$	−31.6126	−1.01502
$971$	−30.0251	−0.963550	−0.481775	−	0.876295i	$-0.660008\pi$
$971$	−30.0251	−0.963550	−0.481775	+	0.876295i	$0.660008\pi$
$972$	0	0
$973$	1.68948	0.0541621
$974$	−80.0853	−2.56610
$975$	0	0
$976$	52.2190	1.67149
$977$	−57.7096	−1.84629	−0.923147	−	0.384448i	$-0.874392\pi$
$977$	−57.7096	−1.84629	−0.923147	+	0.384448i	$0.874392\pi$
$978$	0	0
$979$	−59.2366	−1.89321
$980$	−42.7959	−1.36706
$981$	0	0
$982$	56.1154	1.79071
$983$	3.33817	0.106471	0.0532356	−	0.998582i	$-0.483047\pi$
$983$	3.33817	0.106471	0.0532356	+	0.998582i	$0.483047\pi$
$984$	0	0
$985$	22.9245	0.730435
$986$	2.98132	0.0949444
$987$	0	0
$988$	−144.318	−4.59137
$989$	−9.06139	−0.288136
$990$	0	0
$991$	4.48757	0.142552	0.0712762	−	0.997457i	$-0.477293\pi$
$991$	4.48757	0.142552	0.0712762	+	0.997457i	$0.477293\pi$
$992$	−38.5065	−1.22258
$993$	0	0
$994$	−38.3727	−1.21711
$995$	−43.4212	−1.37654
$996$	0	0
$997$	−47.4116	−1.50154	−0.750770	−	0.660564i	$-0.770317\pi$
$997$	−47.4116	−1.50154	−0.750770	+	0.660564i	$0.770317\pi$
$998$	57.0172	1.80485
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2169.2.a.e.1.7		7
3.2	odd	2		241.2.a.a.1.1	✓	7
12.11	even	2		3856.2.a.j.1.2		7
15.14	odd	2		6025.2.a.f.1.7		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.a.1.1	✓	7	3.2	odd	2
2169.2.a.e.1.7		7	1.1	even	1	trivial
3856.2.a.j.1.2		7	12.11	even	2
6025.2.a.f.1.7		7	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:	\( N \)	\(=\)	\( 2169 = 3^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2169.a (trivial)

\(f(q)\)	\(=\)	\(q+2.60363 q^{2} +4.77887 q^{4} +1.69135 q^{5} -1.30586 q^{7} +7.23513 q^{8} +O(q^{10})\)	\(q+2.60363 q^{2} +4.77887 q^{4} +1.69135 q^{5} -1.30586 q^{7} +7.23513 q^{8} +4.40364 q^{10} +3.27094 q^{11} +4.30649 q^{13} -3.39998 q^{14} +9.27984 q^{16} +1.02456 q^{17} -7.01250 q^{19} +8.08274 q^{20} +8.51631 q^{22} -0.835873 q^{23} -2.13934 q^{25} +11.2125 q^{26} -6.24054 q^{28} +1.11761 q^{29} -3.97344 q^{31} +9.69098 q^{32} +2.66758 q^{34} -2.20867 q^{35} -11.3098 q^{37} -18.2579 q^{38} +12.2371 q^{40} -1.22869 q^{41} +10.8406 q^{43} +15.6314 q^{44} -2.17630 q^{46} +0.151820 q^{47} -5.29473 q^{49} -5.57003 q^{50} +20.5801 q^{52} -3.02053 q^{53} +5.53231 q^{55} -9.44808 q^{56} +2.90984 q^{58} +4.15373 q^{59} +5.62714 q^{61} -10.3454 q^{62} +6.67199 q^{64} +7.28378 q^{65} +12.9934 q^{67} +4.89626 q^{68} -5.75055 q^{70} +11.2862 q^{71} +11.7148 q^{73} -29.4464 q^{74} -33.5118 q^{76} -4.27140 q^{77} -1.66517 q^{79} +15.6955 q^{80} -3.19906 q^{82} +2.34322 q^{83} +1.73290 q^{85} +28.2250 q^{86} +23.6657 q^{88} -18.1099 q^{89} -5.62368 q^{91} -3.99453 q^{92} +0.395283 q^{94} -11.8606 q^{95} -7.17873 q^{97} -13.7855 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 4 q^{2} + 2 q^{4} + 8 q^{5} - 7 q^{7} + 6 q^{8}+O(q^{10}) \) 7 * q + 4 * q^2 + 2 * q^4 + 8 * q^5 - 7 * q^7 + 6 * q^8	\( 7 q + 4 q^{2} + 2 q^{4} + 8 q^{5} - 7 q^{7} + 6 q^{8} + 3 q^{10} + 18 q^{11} - q^{13} + 6 q^{14} + 4 q^{16} + 2 q^{17} - 6 q^{19} + 8 q^{20} + 10 q^{22} + 22 q^{23} + 5 q^{25} - 8 q^{26} + 9 q^{28} + 16 q^{29} - 18 q^{31} + 6 q^{32} + 11 q^{34} - 7 q^{35} + 8 q^{37} - 16 q^{38} + 14 q^{40} + 15 q^{41} + 14 q^{43} + 4 q^{44} + 11 q^{46} + 10 q^{47} + 6 q^{49} + 4 q^{50} + 27 q^{52} - 15 q^{53} + 29 q^{55} - 13 q^{56} + 17 q^{58} + 18 q^{59} + 4 q^{61} - 13 q^{62} + 2 q^{64} + 7 q^{65} + 18 q^{67} + 15 q^{68} + 8 q^{70} + 50 q^{71} - 10 q^{74} - 20 q^{76} - 17 q^{77} - 15 q^{79} + 11 q^{80} + 45 q^{82} + 24 q^{83} - 2 q^{85} + 23 q^{86} + 8 q^{88} + 13 q^{89} - 12 q^{91} + 10 q^{92} - 32 q^{94} + 41 q^{95} + q^{97} - 9 q^{98}+O(q^{100}) \) 7 * q + 4 * q^2 + 2 * q^4 + 8 * q^5 - 7 * q^7 + 6 * q^8 + 3 * q^10 + 18 * q^11 - q^13 + 6 * q^14 + 4 * q^16 + 2 * q^17 - 6 * q^19 + 8 * q^20 + 10 * q^22 + 22 * q^23 + 5 * q^25 - 8 * q^26 + 9 * q^28 + 16 * q^29 - 18 * q^31 + 6 * q^32 + 11 * q^34 - 7 * q^35 + 8 * q^37 - 16 * q^38 + 14 * q^40 + 15 * q^41 + 14 * q^43 + 4 * q^44 + 11 * q^46 + 10 * q^47 + 6 * q^49 + 4 * q^50 + 27 * q^52 - 15 * q^53 + 29 * q^55 - 13 * q^56 + 17 * q^58 + 18 * q^59 + 4 * q^61 - 13 * q^62 + 2 * q^64 + 7 * q^65 + 18 * q^67 + 15 * q^68 + 8 * q^70 + 50 * q^71 - 10 * q^74 - 20 * q^76 - 17 * q^77 - 15 * q^79 + 11 * q^80 + 45 * q^82 + 24 * q^83 - 2 * q^85 + 23 * q^86 + 8 * q^88 + 13 * q^89 - 12 * q^91 + 10 * q^92 - 32 * q^94 + 41 * q^95 + q^97 - 9 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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