LMFDB - Embedded newform 5265.2.a.bj.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5265,2,Mod(1,5265)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5265, 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("5265.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5265 = 3^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5265.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:	$42.0412366642$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 2 x^{13} - 20 x^{12} + 36 x^{11} + 156 x^{10} - 242 x^{9} - 601 x^{8} + 750 x^{7} + 1188 x^{6} + \cdots + 6 $ x^14 - 2x^13 - 20x^12 + 36x^11 + 156x^10 - 242x^9 - 601x^8 + 750x^7 + 1188x^6 - 1038x^5 - 1133x^4 + 438x^3 + 423x^2 + 90*x + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$2.74898$ of defining polynomial
Character	$\chi$	$=$	5265.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.74898 q^{2} +5.55687 q^{4} +1.00000 q^{5} +0.361779 q^{7} +9.77774 q^{8} +O(q^{10})$	\(q+2.74898 q^{2} +5.55687 q^{4} +1.00000 q^{5} +0.361779 q^{7} +9.77774 q^{8} +2.74898 q^{10} -4.19949 q^{11} +1.00000 q^{13} +0.994523 q^{14} +15.7650 q^{16} -5.94605 q^{17} +2.86554 q^{19} +5.55687 q^{20} -11.5443 q^{22} +7.69849 q^{23} +1.00000 q^{25} +2.74898 q^{26} +2.01036 q^{28} +5.35430 q^{29} +5.91693 q^{31} +23.7822 q^{32} -16.3455 q^{34} +0.361779 q^{35} -1.10532 q^{37} +7.87731 q^{38} +9.77774 q^{40} +5.02976 q^{41} -11.5766 q^{43} -23.3360 q^{44} +21.1630 q^{46} +10.1871 q^{47} -6.86912 q^{49} +2.74898 q^{50} +5.55687 q^{52} +9.60923 q^{53} -4.19949 q^{55} +3.53739 q^{56} +14.7188 q^{58} +6.22705 q^{59} -1.38514 q^{61} +16.2655 q^{62} +33.8466 q^{64} +1.00000 q^{65} +4.84843 q^{67} -33.0414 q^{68} +0.994523 q^{70} +0.306717 q^{71} +5.22456 q^{73} -3.03849 q^{74} +15.9234 q^{76} -1.51929 q^{77} -10.1599 q^{79} +15.7650 q^{80} +13.8267 q^{82} -2.65806 q^{83} -5.94605 q^{85} -31.8239 q^{86} -41.0615 q^{88} -10.7990 q^{89} +0.361779 q^{91} +42.7795 q^{92} +28.0040 q^{94} +2.86554 q^{95} +6.20571 q^{97} -18.8830 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 2 q^{2} + 16 q^{4} + 14 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) $ 14 * q + 2 * q^2 + 16 * q^4 + 14 * q^5 + 4 * q^7 + 12 * q^8	$ 14 q + 2 q^{2} + 16 q^{4} + 14 q^{5} + 4 q^{7} + 12 q^{8} + 2 q^{10} + 8 q^{11} + 14 q^{13} + 12 q^{14} + 16 q^{16} + 10 q^{17} + 10 q^{19} + 16 q^{20} - 10 q^{22} + 34 q^{23} + 14 q^{25} + 2 q^{26} + 2 q^{28} + 16 q^{29} + 6 q^{31} + 26 q^{32} + 8 q^{34} + 4 q^{35} + 4 q^{37} + 12 q^{38} + 12 q^{40} + 10 q^{41} - 8 q^{43} + 8 q^{44} - 16 q^{46} + 46 q^{47} + 2 q^{49} + 2 q^{50} + 16 q^{52} + 14 q^{53} + 8 q^{55} + 20 q^{56} - 28 q^{58} + 16 q^{59} + 30 q^{62} + 14 q^{64} + 14 q^{65} + 6 q^{67} + 4 q^{68} + 12 q^{70} + 34 q^{71} - 4 q^{73} + 8 q^{74} + 50 q^{76} + 24 q^{77} - 14 q^{79} + 16 q^{80} - 16 q^{82} + 4 q^{83} + 10 q^{85} + 6 q^{86} - 68 q^{88} + 18 q^{89} + 4 q^{91} + 90 q^{92} + 10 q^{95} + 12 q^{97} - 16 q^{98}+O(q^{100}) $ 14 * q + 2 * q^2 + 16 * q^4 + 14 * q^5 + 4 * q^7 + 12 * q^8 + 2 * q^10 + 8 * q^11 + 14 * q^13 + 12 * q^14 + 16 * q^16 + 10 * q^17 + 10 * q^19 + 16 * q^20 - 10 * q^22 + 34 * q^23 + 14 * q^25 + 2 * q^26 + 2 * q^28 + 16 * q^29 + 6 * q^31 + 26 * q^32 + 8 * q^34 + 4 * q^35 + 4 * q^37 + 12 * q^38 + 12 * q^40 + 10 * q^41 - 8 * q^43 + 8 * q^44 - 16 * q^46 + 46 * q^47 + 2 * q^49 + 2 * q^50 + 16 * q^52 + 14 * q^53 + 8 * q^55 + 20 * q^56 - 28 * q^58 + 16 * q^59 + 30 * q^62 + 14 * q^64 + 14 * q^65 + 6 * q^67 + 4 * q^68 + 12 * q^70 + 34 * q^71 - 4 * q^73 + 8 * q^74 + 50 * q^76 + 24 * q^77 - 14 * q^79 + 16 * q^80 - 16 * q^82 + 4 * q^83 + 10 * q^85 + 6 * q^86 - 68 * q^88 + 18 * q^89 + 4 * q^91 + 90 * q^92 + 10 * q^95 + 12 * q^97 - 16 * 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 $

$2$	2.74898	1.94382	0.971910	−	0.235354i	$-0.0756251\pi$
$2$	2.74898	1.94382	0.971910	+	0.235354i	$0.0756251\pi$
$3$	0	0
$3$	0	0
$4$	5.55687	2.77843
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.361779	0.136740	0.0683699	−	0.997660i	$-0.478220\pi$
$7$	0.361779	0.136740	0.0683699	+	0.997660i	$0.478220\pi$
$8$	9.77774	3.45695
$9$	0	0
$10$	2.74898	0.869302
$11$	−4.19949	−1.26619	−0.633097	−	0.774073i	$-0.718217\pi$
$11$	−4.19949	−1.26619	−0.633097	+	0.774073i	$0.718217\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0.994523	0.265797
$15$	0	0
$16$	15.7650	3.94126
$17$	−5.94605	−1.44213	−0.721064	−	0.692869i	$-0.756347\pi$
$17$	−5.94605	−1.44213	−0.721064	+	0.692869i	$0.756347\pi$
$18$	0	0
$19$	2.86554	0.657401	0.328700	−	0.944434i	$-0.393389\pi$
$19$	2.86554	0.657401	0.328700	+	0.944434i	$0.393389\pi$
$20$	5.55687	1.24255
$21$	0	0
$22$	−11.5443	−2.46125
$23$	7.69849	1.60525	0.802623	−	0.596487i	$-0.203437\pi$
$23$	7.69849	1.60525	0.802623	+	0.596487i	$0.203437\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.74898	0.539118
$27$	0	0
$28$	2.01036	0.379922
$29$	5.35430	0.994268	0.497134	−	0.867674i	$-0.334386\pi$
$29$	5.35430	0.994268	0.497134	+	0.867674i	$0.334386\pi$
$30$	0	0
$31$	5.91693	1.06271	0.531356	−	0.847149i	$-0.321683\pi$
$31$	5.91693	1.06271	0.531356	+	0.847149i	$0.321683\pi$
$32$	23.7822	4.20414
$33$	0	0
$34$	−16.3455	−2.80324
$35$	0.361779	0.0611519
$36$	0	0
$37$	−1.10532	−0.181713	−0.0908565	−	0.995864i	$-0.528960\pi$
$37$	−1.10532	−0.181713	−0.0908565	+	0.995864i	$0.528960\pi$
$38$	7.87731	1.27787
$39$	0	0
$40$	9.77774	1.54600
$41$	5.02976	0.785516	0.392758	−	0.919642i	$-0.371521\pi$
$41$	5.02976	0.785516	0.392758	+	0.919642i	$0.371521\pi$
$42$	0	0
$43$	−11.5766	−1.76542	−0.882710	−	0.469918i	$-0.844284\pi$
$43$	−11.5766	−1.76542	−0.882710	+	0.469918i	$0.844284\pi$
$44$	−23.3360	−3.51803
$45$	0	0
$46$	21.1630	3.12031
$47$	10.1871	1.48593	0.742967	−	0.669328i	$-0.233418\pi$
$47$	10.1871	1.48593	0.742967	+	0.669328i	$0.233418\pi$
$48$	0	0
$49$	−6.86912	−0.981302
$50$	2.74898	0.388764
$51$	0	0
$52$	5.55687	0.770599
$53$	9.60923	1.31993	0.659964	−	0.751297i	$-0.270571\pi$
$53$	9.60923	1.31993	0.659964	+	0.751297i	$0.270571\pi$
$54$	0	0
$55$	−4.19949	−0.566259
$56$	3.53739	0.472703
$57$	0	0
$58$	14.7188	1.93268
$59$	6.22705	0.810692	0.405346	−	0.914163i	$-0.367151\pi$
$59$	6.22705	0.810692	0.405346	+	0.914163i	$0.367151\pi$
$60$	0	0
$61$	−1.38514	−0.177349	−0.0886746	−	0.996061i	$-0.528263\pi$
$61$	−1.38514	−0.177349	−0.0886746	+	0.996061i	$0.528263\pi$
$62$	16.2655	2.06572
$63$	0	0
$64$	33.8466	4.23083
$65$	1.00000	0.124035
$66$	0	0
$67$	4.84843	0.592330	0.296165	−	0.955137i	$-0.404292\pi$
$67$	4.84843	0.592330	0.296165	+	0.955137i	$0.404292\pi$
$68$	−33.0414	−4.00686
$69$	0	0
$70$	0.994523	0.118868
$71$	0.306717	0.0364006	0.0182003	−	0.999834i	$-0.494206\pi$
$71$	0.306717	0.0364006	0.0182003	+	0.999834i	$0.494206\pi$
$72$	0	0
$73$	5.22456	0.611488	0.305744	−	0.952114i	$-0.401095\pi$
$73$	5.22456	0.611488	0.305744	+	0.952114i	$0.401095\pi$
$74$	−3.03849	−0.353217
$75$	0	0
$76$	15.9234	1.82654
$77$	−1.51929	−0.173139
$78$	0	0
$79$	−10.1599	−1.14308	−0.571541	−	0.820574i	$-0.693654\pi$
$79$	−10.1599	−1.14308	−0.571541	+	0.820574i	$0.693654\pi$
$80$	15.7650	1.76258
$81$	0	0
$82$	13.8267	1.52690
$83$	−2.65806	−0.291760	−0.145880	−	0.989302i	$-0.546601\pi$
$83$	−2.65806	−0.291760	−0.145880	+	0.989302i	$0.546601\pi$
$84$	0	0
$85$	−5.94605	−0.644939
$86$	−31.8239	−3.43166
$87$	0	0
$88$	−41.0615	−4.37717
$89$	−10.7990	−1.14469	−0.572343	−	0.820014i	$-0.693966\pi$
$89$	−10.7990	−1.14469	−0.572343	+	0.820014i	$0.693966\pi$
$90$	0	0
$91$	0.361779	0.0379248
$92$	42.7795	4.46007
$93$	0	0
$94$	28.0040	2.88839
$95$	2.86554	0.293999
$96$	0	0
$97$	6.20571	0.630094	0.315047	−	0.949076i	$-0.397980\pi$
$97$	6.20571	0.630094	0.315047	+	0.949076i	$0.397980\pi$
$98$	−18.8830	−1.90747
$99$	0	0
$100$	5.55687	0.555687
$101$	−5.12191	−0.509649	−0.254825	−	0.966987i	$-0.582018\pi$
$101$	−5.12191	−0.509649	−0.254825	+	0.966987i	$0.582018\pi$
$102$	0	0
$103$	−12.7183	−1.25317	−0.626585	−	0.779353i	$-0.715548\pi$
$103$	−12.7183	−1.25317	−0.626585	+	0.779353i	$0.715548\pi$
$104$	9.77774	0.958786
$105$	0	0
$106$	26.4155	2.56570
$107$	−14.9256	−1.44291	−0.721456	−	0.692460i	$-0.756527\pi$
$107$	−14.9256	−1.44291	−0.721456	+	0.692460i	$0.756527\pi$
$108$	0	0
$109$	−6.60395	−0.632543	−0.316272	−	0.948669i	$-0.602431\pi$
$109$	−6.60395	−0.632543	−0.316272	+	0.948669i	$0.602431\pi$
$110$	−11.5443	−1.10071
$111$	0	0
$112$	5.70346	0.538927
$113$	−15.4355	−1.45205	−0.726026	−	0.687667i	$-0.758635\pi$
$113$	−15.4355	−1.45205	−0.726026	+	0.687667i	$0.758635\pi$
$114$	0	0
$115$	7.69849	0.717888
$116$	29.7531	2.76251
$117$	0	0
$118$	17.1180	1.57584
$119$	−2.15116	−0.197196
$120$	0	0
$121$	6.63571	0.603247
$122$	−3.80772	−0.344735
$123$	0	0
$124$	32.8796	2.95267
$125$	1.00000	0.0894427
$126$	0	0
$127$	−18.9201	−1.67889	−0.839446	−	0.543443i	$-0.817120\pi$
$127$	−18.9201	−1.67889	−0.839446	+	0.543443i	$0.817120\pi$
$128$	45.4792	4.01983
$129$	0	0
$130$	2.74898	0.241101
$131$	11.1433	0.973594	0.486797	−	0.873515i	$-0.338165\pi$
$131$	11.1433	0.973594	0.486797	+	0.873515i	$0.338165\pi$
$132$	0	0
$133$	1.03670	0.0898929
$134$	13.3282	1.15138
$135$	0	0
$136$	−58.1389	−4.98537
$137$	15.4443	1.31950	0.659749	−	0.751486i	$-0.270663\pi$
$137$	15.4443	1.31950	0.659749	+	0.751486i	$0.270663\pi$
$138$	0	0
$139$	−13.5812	−1.15194	−0.575969	−	0.817471i	$-0.695375\pi$
$139$	−13.5812	−1.15194	−0.575969	+	0.817471i	$0.695375\pi$
$140$	2.01036	0.169906
$141$	0	0
$142$	0.843156	0.0707561
$143$	−4.19949	−0.351179
$144$	0	0
$145$	5.35430	0.444650
$146$	14.3622	1.18862
$147$	0	0
$148$	−6.14210	−0.504878
$149$	4.17557	0.342076	0.171038	−	0.985264i	$-0.445288\pi$
$149$	4.17557	0.342076	0.171038	+	0.985264i	$0.445288\pi$
$150$	0	0
$151$	−6.81775	−0.554821	−0.277410	−	0.960752i	$-0.589476\pi$
$151$	−6.81775	−0.554821	−0.277410	+	0.960752i	$0.589476\pi$
$152$	28.0185	2.27260
$153$	0	0
$154$	−4.17649	−0.336551
$155$	5.91693	0.475259
$156$	0	0
$157$	−24.8185	−1.98073	−0.990366	−	0.138476i	$-0.955780\pi$
$157$	−24.8185	−1.98073	−0.990366	+	0.138476i	$0.955780\pi$
$158$	−27.9294	−2.22194
$159$	0	0
$160$	23.7822	1.88015
$161$	2.78516	0.219501
$162$	0	0
$163$	2.36922	0.185572	0.0927860	−	0.995686i	$-0.470423\pi$
$163$	2.36922	0.185572	0.0927860	+	0.995686i	$0.470423\pi$
$164$	27.9497	2.18250
$165$	0	0
$166$	−7.30695	−0.567129
$167$	−22.4675	−1.73858	−0.869292	−	0.494298i	$-0.835425\pi$
$167$	−22.4675	−1.73858	−0.869292	+	0.494298i	$0.835425\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−16.3455	−1.25365
$171$	0	0
$172$	−64.3298	−4.90510
$173$	8.61113	0.654692	0.327346	−	0.944905i	$-0.393846\pi$
$173$	8.61113	0.654692	0.327346	+	0.944905i	$0.393846\pi$
$174$	0	0
$175$	0.361779	0.0273480
$176$	−66.2051	−4.99040
$177$	0	0
$178$	−29.6861	−2.22506
$179$	−21.4105	−1.60030	−0.800150	−	0.599801i	$-0.795247\pi$
$179$	−21.4105	−1.60030	−0.800150	+	0.599801i	$0.795247\pi$
$180$	0	0
$181$	1.63616	0.121615	0.0608076	−	0.998150i	$-0.480632\pi$
$181$	1.63616	0.121615	0.0608076	+	0.998150i	$0.480632\pi$
$182$	0.994523	0.0737189
$183$	0	0
$184$	75.2738	5.54926
$185$	−1.10532	−0.0812646
$186$	0	0
$187$	24.9704	1.82601
$188$	56.6081	4.12857
$189$	0	0
$190$	7.87731	0.571480
$191$	3.43505	0.248551	0.124276	−	0.992248i	$-0.460339\pi$
$191$	3.43505	0.248551	0.124276	+	0.992248i	$0.460339\pi$
$192$	0	0
$193$	−25.6109	−1.84351	−0.921755	−	0.387773i	$-0.873244\pi$
$193$	−25.6109	−1.84351	−0.921755	+	0.387773i	$0.873244\pi$
$194$	17.0593	1.22479
$195$	0	0
$196$	−38.1708	−2.72648
$197$	0.494416	0.0352257	0.0176129	−	0.999845i	$-0.494393\pi$
$197$	0.494416	0.0352257	0.0176129	+	0.999845i	$0.494393\pi$
$198$	0	0
$199$	6.52918	0.462842	0.231421	−	0.972854i	$-0.425663\pi$
$199$	6.52918	0.462842	0.231421	+	0.972854i	$0.425663\pi$
$200$	9.77774	0.691390
$201$	0	0
$202$	−14.0800	−0.990666
$203$	1.93707	0.135956
$204$	0	0
$205$	5.02976	0.351293
$206$	−34.9623	−2.43594
$207$	0	0
$208$	15.7650	1.09311
$209$	−12.0338	−0.832397
$210$	0	0
$211$	5.55265	0.382260	0.191130	−	0.981565i	$-0.438785\pi$
$211$	5.55265	0.382260	0.191130	+	0.981565i	$0.438785\pi$
$212$	53.3972	3.66733
$213$	0	0
$214$	−41.0301	−2.80476
$215$	−11.5766	−0.789520
$216$	0	0
$217$	2.14062	0.145315
$218$	−18.1541	−1.22955
$219$	0	0
$220$	−23.3360	−1.57331
$221$	−5.94605	−0.399974
$222$	0	0
$223$	−5.32491	−0.356582	−0.178291	−	0.983978i	$-0.557057\pi$
$223$	−5.32491	−0.356582	−0.178291	+	0.983978i	$0.557057\pi$
$224$	8.60391	0.574873
$225$	0	0
$226$	−42.4319	−2.82253
$227$	−6.04250	−0.401055	−0.200528	−	0.979688i	$-0.564266\pi$
$227$	−6.04250	−0.401055	−0.200528	+	0.979688i	$0.564266\pi$
$228$	0	0
$229$	23.1205	1.52785	0.763924	−	0.645306i	$-0.223270\pi$
$229$	23.1205	1.52785	0.763924	+	0.645306i	$0.223270\pi$
$230$	21.1630	1.39544
$231$	0	0
$232$	52.3529	3.43714
$233$	6.42953	0.421213	0.210606	−	0.977571i	$-0.432456\pi$
$233$	6.42953	0.421213	0.210606	+	0.977571i	$0.432456\pi$
$234$	0	0
$235$	10.1871	0.664530
$236$	34.6029	2.25245
$237$	0	0
$238$	−5.91348	−0.383314
$239$	−9.02413	−0.583722	−0.291861	−	0.956461i	$-0.594275\pi$
$239$	−9.02413	−0.583722	−0.291861	+	0.956461i	$0.594275\pi$
$240$	0	0
$241$	−5.23003	−0.336896	−0.168448	−	0.985711i	$-0.553875\pi$
$241$	−5.23003	−0.336896	−0.168448	+	0.985711i	$0.553875\pi$
$242$	18.2414	1.17260
$243$	0	0
$244$	−7.69705	−0.492753
$245$	−6.86912	−0.438852
$246$	0	0
$247$	2.86554	0.182330
$248$	57.8542	3.67374
$249$	0	0
$250$	2.74898	0.173860
$251$	−18.3627	−1.15904	−0.579521	−	0.814958i	$-0.696760\pi$
$251$	−18.3627	−1.15904	−0.579521	+	0.814958i	$0.696760\pi$
$252$	0	0
$253$	−32.3297	−2.03255
$254$	−52.0110	−3.26346
$255$	0	0
$256$	57.3278	3.58299
$257$	−17.7814	−1.10917	−0.554586	−	0.832126i	$-0.687123\pi$
$257$	−17.7814	−1.10917	−0.554586	+	0.832126i	$0.687123\pi$
$258$	0	0
$259$	−0.399881	−0.0248474
$260$	5.55687	0.344622
$261$	0	0
$262$	30.6326	1.89249
$263$	2.76621	0.170572	0.0852859	−	0.996357i	$-0.472820\pi$
$263$	2.76621	0.170572	0.0852859	+	0.996357i	$0.472820\pi$
$264$	0	0
$265$	9.60923	0.590290
$266$	2.84985	0.174735
$267$	0	0
$268$	26.9421	1.64575
$269$	−1.84680	−0.112601	−0.0563007	−	0.998414i	$-0.517931\pi$
$269$	−1.84680	−0.112601	−0.0563007	+	0.998414i	$0.517931\pi$
$270$	0	0
$271$	−0.777893	−0.0472536	−0.0236268	−	0.999721i	$-0.507521\pi$
$271$	−0.777893	−0.0472536	−0.0236268	+	0.999721i	$0.507521\pi$
$272$	−93.7396	−5.68380
$273$	0	0
$274$	42.4561	2.56487
$275$	−4.19949	−0.253239
$276$	0	0
$277$	−23.2237	−1.39538	−0.697689	−	0.716401i	$-0.745788\pi$
$277$	−23.2237	−1.39538	−0.697689	+	0.716401i	$0.745788\pi$
$278$	−37.3343	−2.23916
$279$	0	0
$280$	3.53739	0.211399
$281$	10.3340	0.616476	0.308238	−	0.951309i	$-0.400261\pi$
$281$	10.3340	0.616476	0.308238	+	0.951309i	$0.400261\pi$
$282$	0	0
$283$	−1.81167	−0.107693	−0.0538464	−	0.998549i	$-0.517148\pi$
$283$	−1.81167	−0.107693	−0.0538464	+	0.998549i	$0.517148\pi$
$284$	1.70438	0.101137
$285$	0	0
$286$	−11.5443	−0.682628
$287$	1.81966	0.107411
$288$	0	0
$289$	18.3555	1.07973
$290$	14.7188	0.864320
$291$	0	0
$292$	29.0322	1.69898
$293$	−3.09487	−0.180804	−0.0904021	−	0.995905i	$-0.528815\pi$
$293$	−3.09487	−0.180804	−0.0904021	+	0.995905i	$0.528815\pi$
$294$	0	0
$295$	6.22705	0.362553
$296$	−10.8075	−0.628174
$297$	0	0
$298$	11.4785	0.664934
$299$	7.69849	0.445215
$300$	0	0
$301$	−4.18819	−0.241403
$302$	−18.7418	−1.07847
$303$	0	0
$304$	45.1754	2.59099
$305$	−1.38514	−0.0793130
$306$	0	0
$307$	20.6159	1.17661	0.588307	−	0.808638i	$-0.299795\pi$
$307$	20.6159	1.17661	0.588307	+	0.808638i	$0.299795\pi$
$308$	−8.44249	−0.481055
$309$	0	0
$310$	16.2655	0.923818
$311$	24.4797	1.38812	0.694059	−	0.719918i	$-0.255821\pi$
$311$	24.4797	1.38812	0.694059	+	0.719918i	$0.255821\pi$
$312$	0	0
$313$	24.5316	1.38661	0.693305	−	0.720644i	$-0.256154\pi$
$313$	24.5316	1.38661	0.693305	+	0.720644i	$0.256154\pi$
$314$	−68.2254	−3.85018
$315$	0	0
$316$	−56.4574	−3.17598
$317$	−2.04187	−0.114683	−0.0573414	−	0.998355i	$-0.518262\pi$
$317$	−2.04187	−0.114683	−0.0573414	+	0.998355i	$0.518262\pi$
$318$	0	0
$319$	−22.4853	−1.25894
$320$	33.8466	1.89208
$321$	0	0
$322$	7.65632	0.426670
$323$	−17.0387	−0.948056
$324$	0	0
$325$	1.00000	0.0554700
$326$	6.51294	0.360718
$327$	0	0
$328$	49.1796	2.71549
$329$	3.68547	0.203186
$330$	0	0
$331$	−0.991268	−0.0544850	−0.0272425	−	0.999629i	$-0.508673\pi$
$331$	−0.991268	−0.0544850	−0.0272425	+	0.999629i	$0.508673\pi$
$332$	−14.7705	−0.810637
$333$	0	0
$334$	−61.7625	−3.37949
$335$	4.84843	0.264898
$336$	0	0
$337$	−6.65469	−0.362504	−0.181252	−	0.983437i	$-0.558015\pi$
$337$	−6.65469	−0.362504	−0.181252	+	0.983437i	$0.558015\pi$
$338$	2.74898	0.149525
$339$	0	0
$340$	−33.0414	−1.79192
$341$	−24.8481	−1.34560
$342$	0	0
$343$	−5.01756	−0.270923
$344$	−113.193	−6.10297
$345$	0	0
$346$	23.6718	1.27260
$347$	33.9811	1.82420	0.912101	−	0.409966i	$-0.134460\pi$
$347$	33.9811	1.82420	0.912101	+	0.409966i	$0.134460\pi$
$348$	0	0
$349$	−11.4437	−0.612567	−0.306283	−	0.951940i	$-0.599086\pi$
$349$	−11.4437	−0.612567	−0.306283	+	0.951940i	$0.599086\pi$
$350$	0.994523	0.0531595
$351$	0	0
$352$	−99.8731	−5.32326
$353$	33.7077	1.79408	0.897039	−	0.441951i	$-0.145713\pi$
$353$	33.7077	1.79408	0.897039	+	0.441951i	$0.145713\pi$
$354$	0	0
$355$	0.306717	0.0162788
$356$	−60.0083	−3.18044
$357$	0	0
$358$	−58.8571	−3.11069
$359$	−4.35260	−0.229721	−0.114861	−	0.993382i	$-0.536642\pi$
$359$	−4.35260	−0.229721	−0.114861	+	0.993382i	$0.536642\pi$
$360$	0	0
$361$	−10.7887	−0.567824
$362$	4.49778	0.236398
$363$	0	0
$364$	2.01036	0.105372
$365$	5.22456	0.273466
$366$	0	0
$367$	21.1645	1.10478	0.552388	−	0.833587i	$-0.313717\pi$
$367$	21.1645	1.10478	0.552388	+	0.833587i	$0.313717\pi$
$368$	121.367	6.32669
$369$	0	0
$370$	−3.03849	−0.157964
$371$	3.47642	0.180487
$372$	0	0
$373$	25.8303	1.33744	0.668722	−	0.743513i	$-0.266842\pi$
$373$	25.8303	1.33744	0.668722	+	0.743513i	$0.266842\pi$
$374$	68.6429	3.54944
$375$	0	0
$376$	99.6064	5.13681
$377$	5.35430	0.275760
$378$	0	0
$379$	−12.6346	−0.648998	−0.324499	−	0.945886i	$-0.605196\pi$
$379$	−12.6346	−0.648998	−0.324499	+	0.945886i	$0.605196\pi$
$380$	15.9234	0.816856
$381$	0	0
$382$	9.44287	0.483139
$383$	35.3817	1.80792	0.903961	−	0.427614i	$-0.140646\pi$
$383$	35.3817	1.80792	0.903961	+	0.427614i	$0.140646\pi$
$384$	0	0
$385$	−1.51929	−0.0774302
$386$	−70.4036	−3.58345
$387$	0	0
$388$	34.4843	1.75068
$389$	−13.5143	−0.685200	−0.342600	−	0.939481i	$-0.611308\pi$
$389$	−13.5143	−0.685200	−0.342600	+	0.939481i	$0.611308\pi$
$390$	0	0
$391$	−45.7756	−2.31497
$392$	−67.1644	−3.39232
$393$	0	0
$394$	1.35914	0.0684724
$395$	−10.1599	−0.511202
$396$	0	0
$397$	8.64789	0.434025	0.217013	−	0.976169i	$-0.430369\pi$
$397$	8.64789	0.434025	0.217013	+	0.976169i	$0.430369\pi$
$398$	17.9486	0.899680
$399$	0	0
$400$	15.7650	0.788251
$401$	−32.2333	−1.60966	−0.804828	−	0.593508i	$-0.797742\pi$
$401$	−32.2333	−1.60966	−0.804828	+	0.593508i	$0.797742\pi$
$402$	0	0
$403$	5.91693	0.294743
$404$	−28.4618	−1.41603
$405$	0	0
$406$	5.32497	0.264274
$407$	4.64177	0.230084
$408$	0	0
$409$	16.0629	0.794257	0.397129	−	0.917763i	$-0.370007\pi$
$409$	16.0629	0.794257	0.397129	+	0.917763i	$0.370007\pi$
$410$	13.8267	0.682851
$411$	0	0
$412$	−70.6739	−3.48185
$413$	2.25282	0.110854
$414$	0	0
$415$	−2.65806	−0.130479
$416$	23.7822	1.16602
$417$	0	0
$418$	−33.0807	−1.61803
$419$	39.0180	1.90615	0.953077	−	0.302728i	$-0.0978974\pi$
$419$	39.0180	1.90615	0.953077	+	0.302728i	$0.0978974\pi$
$420$	0	0
$421$	−8.43085	−0.410895	−0.205447	−	0.978668i	$-0.565865\pi$
$421$	−8.43085	−0.410895	−0.205447	+	0.978668i	$0.565865\pi$
$422$	15.2641	0.743044
$423$	0	0
$424$	93.9565	4.56293
$425$	−5.94605	−0.288426
$426$	0	0
$427$	−0.501116	−0.0242507
$428$	−82.9395	−4.00903
$429$	0	0
$430$	−31.8239	−1.53468
$431$	19.6990	0.948870	0.474435	−	0.880291i	$-0.342652\pi$
$431$	19.6990	0.948870	0.474435	+	0.880291i	$0.342652\pi$
$432$	0	0
$433$	−19.1177	−0.918738	−0.459369	−	0.888246i	$-0.651924\pi$
$433$	−19.1177	−0.918738	−0.459369	+	0.888246i	$0.651924\pi$
$434$	5.88452	0.282466
$435$	0	0
$436$	−36.6972	−1.75748
$437$	22.0604	1.05529
$438$	0	0
$439$	33.6002	1.60365	0.801826	−	0.597557i	$-0.203862\pi$
$439$	33.6002	1.60365	0.801826	+	0.597557i	$0.203862\pi$
$440$	−41.0615	−1.95753
$441$	0	0
$442$	−16.3455	−0.777478
$443$	31.4950	1.49637	0.748187	−	0.663487i	$-0.230924\pi$
$443$	31.4950	1.49637	0.748187	+	0.663487i	$0.230924\pi$
$444$	0	0
$445$	−10.7990	−0.511919
$446$	−14.6380	−0.693131
$447$	0	0
$448$	12.2450	0.578523
$449$	35.8250	1.69068	0.845342	−	0.534225i	$-0.179397\pi$
$449$	35.8250	1.69068	0.845342	+	0.534225i	$0.179397\pi$
$450$	0	0
$451$	−21.1224	−0.994616
$452$	−85.7732	−4.03443
$453$	0	0
$454$	−16.6107	−0.779579
$455$	0.361779	0.0169605
$456$	0	0
$457$	28.1045	1.31467	0.657337	−	0.753597i	$-0.271683\pi$
$457$	28.1045	1.31467	0.657337	+	0.753597i	$0.271683\pi$
$458$	63.5578	2.96986
$459$	0	0
$460$	42.7795	1.99460
$461$	−14.0133	−0.652665	−0.326333	−	0.945255i	$-0.605813\pi$
$461$	−14.0133	−0.652665	−0.326333	+	0.945255i	$0.605813\pi$
$462$	0	0
$463$	22.6140	1.05096	0.525480	−	0.850806i	$-0.323886\pi$
$463$	22.6140	1.05096	0.525480	+	0.850806i	$0.323886\pi$
$464$	84.4106	3.91867
$465$	0	0
$466$	17.6746	0.818761
$467$	−7.37482	−0.341266	−0.170633	−	0.985335i	$-0.554581\pi$
$467$	−7.37482	−0.341266	−0.170633	+	0.985335i	$0.554581\pi$
$468$	0	0
$469$	1.75406	0.0809951
$470$	28.0040	1.29173
$471$	0	0
$472$	60.8864	2.80252
$473$	48.6160	2.23536
$474$	0	0
$475$	2.86554	0.131480
$476$	−11.9537	−0.547897
$477$	0	0
$478$	−24.8071	−1.13465
$479$	−1.80534	−0.0824881	−0.0412441	−	0.999149i	$-0.513132\pi$
$479$	−1.80534	−0.0824881	−0.0412441	+	0.999149i	$0.513132\pi$
$480$	0	0
$481$	−1.10532	−0.0503981
$482$	−14.3772	−0.654864
$483$	0	0
$484$	36.8738	1.67608
$485$	6.20571	0.281787
$486$	0	0
$487$	−19.8330	−0.898720	−0.449360	−	0.893351i	$-0.648348\pi$
$487$	−19.8330	−0.898720	−0.449360	+	0.893351i	$0.648348\pi$
$488$	−13.5436	−0.613088
$489$	0	0
$490$	−18.8830	−0.853048
$491$	28.8360	1.30135	0.650676	−	0.759356i	$-0.274486\pi$
$491$	28.8360	1.30135	0.650676	+	0.759356i	$0.274486\pi$
$492$	0	0
$493$	−31.8369	−1.43386
$494$	7.87731	0.354417
$495$	0	0
$496$	93.2806	4.18842
$497$	0.110964	0.00497740
$498$	0	0
$499$	−10.4478	−0.467706	−0.233853	−	0.972272i	$-0.575133\pi$
$499$	−10.4478	−0.467706	−0.233853	+	0.972272i	$0.575133\pi$
$500$	5.55687	0.248511
$501$	0	0
$502$	−50.4785	−2.25297
$503$	10.8873	0.485440	0.242720	−	0.970096i	$-0.421960\pi$
$503$	10.8873	0.485440	0.242720	+	0.970096i	$0.421960\pi$
$504$	0	0
$505$	−5.12191	−0.227922
$506$	−88.8736	−3.95091
$507$	0	0
$508$	−105.137	−4.66469
$509$	−5.74094	−0.254463	−0.127231	−	0.991873i	$-0.540609\pi$
$509$	−5.74094	−0.254463	−0.127231	+	0.991873i	$0.540609\pi$
$510$	0	0
$511$	1.89014	0.0836148
$512$	66.6345	2.94486
$513$	0	0
$514$	−48.8806	−2.15603
$515$	−12.7183	−0.560435
$516$	0	0
$517$	−42.7804	−1.88148
$518$	−1.09926	−0.0482989
$519$	0	0
$520$	9.77774	0.428782
$521$	27.5745	1.20806	0.604029	−	0.796962i	$-0.293561\pi$
$521$	27.5745	1.20806	0.604029	+	0.796962i	$0.293561\pi$
$522$	0	0
$523$	30.3171	1.32567	0.662837	−	0.748764i	$-0.269352\pi$
$523$	30.3171	1.32567	0.662837	+	0.748764i	$0.269352\pi$
$524$	61.9218	2.70507
$525$	0	0
$526$	7.60424	0.331561
$527$	−35.1823	−1.53257
$528$	0	0
$529$	36.2667	1.57681
$530$	26.4155	1.14742
$531$	0	0
$532$	5.76078	0.249761
$533$	5.02976	0.217863
$534$	0	0
$535$	−14.9256	−0.645290
$536$	47.4067	2.04766
$537$	0	0
$538$	−5.07681	−0.218877
$539$	28.8468	1.24252
$540$	0	0
$541$	−42.3179	−1.81939	−0.909695	−	0.415276i	$-0.863685\pi$
$541$	−42.3179	−1.81939	−0.909695	+	0.415276i	$0.863685\pi$
$542$	−2.13841	−0.0918525
$543$	0	0
$544$	−141.410	−6.06291
$545$	−6.60395	−0.282882
$546$	0	0
$547$	−1.51426	−0.0647451	−0.0323726	−	0.999476i	$-0.510306\pi$
$547$	−1.51426	−0.0647451	−0.0323726	+	0.999476i	$0.510306\pi$
$548$	85.8221	3.66614
$549$	0	0
$550$	−11.5443	−0.492250
$551$	15.3430	0.653633
$552$	0	0
$553$	−3.67565	−0.156305
$554$	−63.8414	−2.71236
$555$	0	0
$556$	−75.4686	−3.20058
$557$	−38.4393	−1.62873	−0.814363	−	0.580355i	$-0.802914\pi$
$557$	−38.4393	−1.62873	−0.814363	+	0.580355i	$0.802914\pi$
$558$	0	0
$559$	−11.5766	−0.489640
$560$	5.70346	0.241015
$561$	0	0
$562$	28.4080	1.19832
$563$	−31.6827	−1.33527	−0.667634	−	0.744489i	$-0.732693\pi$
$563$	−31.6827	−1.33527	−0.667634	+	0.744489i	$0.732693\pi$
$564$	0	0
$565$	−15.4355	−0.649378
$566$	−4.98024	−0.209335
$567$	0	0
$568$	2.99899	0.125835
$569$	19.0442	0.798375	0.399188	−	0.916869i	$-0.369292\pi$
$569$	19.0442	0.798375	0.399188	+	0.916869i	$0.369292\pi$
$570$	0	0
$571$	−20.3034	−0.849670	−0.424835	−	0.905271i	$-0.639668\pi$
$571$	−20.3034	−0.849670	−0.424835	+	0.905271i	$0.639668\pi$
$572$	−23.3360	−0.975727
$573$	0	0
$574$	5.00221	0.208788
$575$	7.69849	0.321049
$576$	0	0
$577$	17.5844	0.732050	0.366025	−	0.930605i	$-0.380718\pi$
$577$	17.5844	0.732050	0.366025	+	0.930605i	$0.380718\pi$
$578$	50.4587	2.09880
$579$	0	0
$580$	29.7531	1.23543
$581$	−0.961633	−0.0398953
$582$	0	0
$583$	−40.3539	−1.67129
$584$	51.0844	2.11389
$585$	0	0
$586$	−8.50772	−0.351451
$587$	3.07247	0.126814	0.0634072	−	0.997988i	$-0.479803\pi$
$587$	3.07247	0.126814	0.0634072	+	0.997988i	$0.479803\pi$
$588$	0	0
$589$	16.9552	0.698628
$590$	17.1180	0.704737
$591$	0	0
$592$	−17.4254	−0.716178
$593$	−15.3464	−0.630200	−0.315100	−	0.949058i	$-0.602038\pi$
$593$	−15.3464	−0.630200	−0.315100	+	0.949058i	$0.602038\pi$
$594$	0	0
$595$	−2.15116	−0.0881888
$596$	23.2031	0.950435
$597$	0	0
$598$	21.1630	0.865418
$599$	2.66009	0.108689	0.0543443	−	0.998522i	$-0.482693\pi$
$599$	2.66009	0.108689	0.0543443	+	0.998522i	$0.482693\pi$
$600$	0	0
$601$	−10.2965	−0.420005	−0.210002	−	0.977701i	$-0.567347\pi$
$601$	−10.2965	−0.420005	−0.210002	+	0.977701i	$0.567347\pi$
$602$	−11.5132	−0.469244
$603$	0	0
$604$	−37.8853	−1.54153
$605$	6.63571	0.269780
$606$	0	0
$607$	2.14532	0.0870757	0.0435379	−	0.999052i	$-0.486137\pi$
$607$	2.14532	0.0870757	0.0435379	+	0.999052i	$0.486137\pi$
$608$	68.1490	2.76381
$609$	0	0
$610$	−3.80772	−0.154170
$611$	10.1871	0.412124
$612$	0	0
$613$	−41.9012	−1.69237	−0.846186	−	0.532888i	$-0.821107\pi$
$613$	−41.9012	−1.69237	−0.846186	+	0.532888i	$0.821107\pi$
$614$	56.6727	2.28712
$615$	0	0
$616$	−14.8552	−0.598534
$617$	−4.44053	−0.178769	−0.0893845	−	0.995997i	$-0.528490\pi$
$617$	−4.44053	−0.178769	−0.0893845	+	0.995997i	$0.528490\pi$
$618$	0	0
$619$	−0.410218	−0.0164881	−0.00824403	−	0.999966i	$-0.502624\pi$
$619$	−0.410218	−0.0164881	−0.00824403	+	0.999966i	$0.502624\pi$
$620$	32.8796	1.32048
$621$	0	0
$622$	67.2942	2.69825
$623$	−3.90684	−0.156524
$624$	0	0
$625$	1.00000	0.0400000
$626$	67.4368	2.69532
$627$	0	0
$628$	−137.913	−5.50333
$629$	6.57227	0.262054
$630$	0	0
$631$	−22.6747	−0.902664	−0.451332	−	0.892356i	$-0.649051\pi$
$631$	−22.6747	−0.902664	−0.451332	+	0.892356i	$0.649051\pi$
$632$	−99.3411	−3.95158
$633$	0	0
$634$	−5.61305	−0.222923
$635$	−18.9201	−0.750823
$636$	0	0
$637$	−6.86912	−0.272164
$638$	−61.8116	−2.44714
$639$	0	0
$640$	45.4792	1.79772
$641$	−30.2124	−1.19332	−0.596660	−	0.802494i	$-0.703506\pi$
$641$	−30.2124	−1.19332	−0.596660	+	0.802494i	$0.703506\pi$
$642$	0	0
$643$	−26.6397	−1.05057	−0.525283	−	0.850927i	$-0.676041\pi$
$643$	−26.6397	−1.05057	−0.525283	+	0.850927i	$0.676041\pi$
$644$	15.4767	0.609869
$645$	0	0
$646$	−46.8389	−1.84285
$647$	−41.4434	−1.62931	−0.814654	−	0.579947i	$-0.803073\pi$
$647$	−41.4434	−1.62931	−0.814654	+	0.579947i	$0.803073\pi$
$648$	0	0
$649$	−26.1504	−1.02649
$650$	2.74898	0.107824
$651$	0	0
$652$	13.1655	0.515599
$653$	22.7582	0.890597	0.445298	−	0.895382i	$-0.353098\pi$
$653$	22.7582	0.890597	0.445298	+	0.895382i	$0.353098\pi$
$654$	0	0
$655$	11.1433	0.435404
$656$	79.2943	3.09592
$657$	0	0
$658$	10.1313	0.394958
$659$	31.0803	1.21072	0.605358	−	0.795953i	$-0.293030\pi$
$659$	31.0803	1.21072	0.605358	+	0.795953i	$0.293030\pi$
$660$	0	0
$661$	−19.4859	−0.757915	−0.378957	−	0.925414i	$-0.623717\pi$
$661$	−19.4859	−0.757915	−0.378957	+	0.925414i	$0.623717\pi$
$662$	−2.72497	−0.105909
$663$	0	0
$664$	−25.9899	−1.00860
$665$	1.03670	0.0402013
$666$	0	0
$667$	41.2200	1.59604
$668$	−124.849	−4.83054
$669$	0	0
$670$	13.3282	0.514914
$671$	5.81689	0.224558
$672$	0	0
$673$	12.4972	0.481731	0.240866	−	0.970558i	$-0.422569\pi$
$673$	12.4972	0.481731	0.240866	+	0.970558i	$0.422569\pi$
$674$	−18.2936	−0.704642
$675$	0	0
$676$	5.55687	0.213726
$677$	−19.3589	−0.744024	−0.372012	−	0.928228i	$-0.621332\pi$
$677$	−19.3589	−0.744024	−0.372012	+	0.928228i	$0.621332\pi$
$678$	0	0
$679$	2.24510	0.0861590
$680$	−58.1389	−2.22952
$681$	0	0
$682$	−68.3068	−2.61560
$683$	−25.1513	−0.962387	−0.481194	−	0.876614i	$-0.659797\pi$
$683$	−25.1513	−0.962387	−0.481194	+	0.876614i	$0.659797\pi$
$684$	0	0
$685$	15.4443	0.590098
$686$	−13.7932	−0.526625
$687$	0	0
$688$	−182.506	−6.95798
$689$	9.60923	0.366082
$690$	0	0
$691$	2.03770	0.0775177	0.0387588	−	0.999249i	$-0.487660\pi$
$691$	2.03770	0.0775177	0.0387588	+	0.999249i	$0.487660\pi$
$692$	47.8509	1.81902
$693$	0	0
$694$	93.4132	3.54592
$695$	−13.5812	−0.515162
$696$	0	0
$697$	−29.9072	−1.13281
$698$	−31.4584	−1.19072
$699$	0	0
$700$	2.01036	0.0759845
$701$	−45.8395	−1.73133	−0.865667	−	0.500621i	$-0.833105\pi$
$701$	−45.8395	−1.73133	−0.865667	+	0.500621i	$0.833105\pi$
$702$	0	0
$703$	−3.16734	−0.119458
$704$	−142.139	−5.35705
$705$	0	0
$706$	92.6616	3.48736
$707$	−1.85300	−0.0696894
$708$	0	0
$709$	10.4273	0.391604	0.195802	−	0.980643i	$-0.437269\pi$
$709$	10.4273	0.391604	0.195802	+	0.980643i	$0.437269\pi$
$710$	0.843156	0.0316431
$711$	0	0
$712$	−105.589	−3.95713
$713$	45.5514	1.70591
$714$	0	0
$715$	−4.19949	−0.157052
$716$	−118.976	−4.44632
$717$	0	0
$718$	−11.9652	−0.446537
$719$	24.5010	0.913733	0.456867	−	0.889535i	$-0.348972\pi$
$719$	24.5010	0.913733	0.456867	+	0.889535i	$0.348972\pi$
$720$	0	0
$721$	−4.60122	−0.171358
$722$	−29.6577	−1.10375
$723$	0	0
$724$	9.09195	0.337900
$725$	5.35430	0.198854
$726$	0	0
$727$	1.28715	0.0477377	0.0238688	−	0.999715i	$-0.492402\pi$
$727$	1.28715	0.0477377	0.0238688	+	0.999715i	$0.492402\pi$
$728$	3.53739	0.131104
$729$	0	0
$730$	14.3622	0.531568
$731$	68.8352	2.54596
$732$	0	0
$733$	1.17550	0.0434181	0.0217091	−	0.999764i	$-0.493089\pi$
$733$	1.17550	0.0434181	0.0217091	+	0.999764i	$0.493089\pi$
$734$	58.1806	2.14748
$735$	0	0
$736$	183.087	6.74868
$737$	−20.3609	−0.750005
$738$	0	0
$739$	41.0193	1.50892	0.754460	−	0.656346i	$-0.227899\pi$
$739$	41.0193	1.50892	0.754460	+	0.656346i	$0.227899\pi$
$740$	−6.14210	−0.225788
$741$	0	0
$742$	9.55660	0.350834
$743$	−27.1237	−0.995071	−0.497535	−	0.867444i	$-0.665762\pi$
$743$	−27.1237	−0.995071	−0.497535	+	0.867444i	$0.665762\pi$
$744$	0	0
$745$	4.17557	0.152981
$746$	71.0069	2.59975
$747$	0	0
$748$	138.757	5.07346
$749$	−5.39977	−0.197303
$750$	0	0
$751$	23.5828	0.860548	0.430274	−	0.902698i	$-0.358417\pi$
$751$	23.5828	0.860548	0.430274	+	0.902698i	$0.358417\pi$
$752$	160.599	5.85645
$753$	0	0
$754$	14.7188	0.536028
$755$	−6.81775	−0.248123
$756$	0	0
$757$	−46.4510	−1.68829	−0.844146	−	0.536113i	$-0.819892\pi$
$757$	−46.4510	−1.68829	−0.844146	+	0.536113i	$0.819892\pi$
$758$	−34.7323	−1.26153
$759$	0	0
$760$	28.0185	1.01634
$761$	−54.0307	−1.95861	−0.979306	−	0.202384i	$-0.935131\pi$
$761$	−54.0307	−1.95861	−0.979306	+	0.202384i	$0.935131\pi$
$762$	0	0
$763$	−2.38917	−0.0864938
$764$	19.0881	0.690584
$765$	0	0
$766$	97.2635	3.51427
$767$	6.22705	0.224846
$768$	0	0
$769$	−31.1513	−1.12334	−0.561672	−	0.827360i	$-0.689842\pi$
$769$	−31.1513	−1.12334	−0.561672	+	0.827360i	$0.689842\pi$
$770$	−4.17649	−0.150510
$771$	0	0
$772$	−142.316	−5.12207
$773$	8.19382	0.294711	0.147356	−	0.989084i	$-0.452924\pi$
$773$	8.19382	0.294711	0.147356	+	0.989084i	$0.452924\pi$
$774$	0	0
$775$	5.91693	0.212542
$776$	60.6778	2.17821
$777$	0	0
$778$	−37.1504	−1.33191
$779$	14.4130	0.516399
$780$	0	0
$781$	−1.28805	−0.0460902
$782$	−125.836	−4.49988
$783$	0	0
$784$	−108.292	−3.86756
$785$	−24.8185	−0.885810
$786$	0	0
$787$	44.0351	1.56968	0.784841	−	0.619697i	$-0.212745\pi$
$787$	44.0351	1.56968	0.784841	+	0.619697i	$0.212745\pi$
$788$	2.74741	0.0978723
$789$	0	0
$790$	−27.9294	−0.993683
$791$	−5.58426	−0.198553
$792$	0	0
$793$	−1.38514	−0.0491878
$794$	23.7728	0.843666
$795$	0	0
$796$	36.2818	1.28597
$797$	21.6435	0.766652	0.383326	−	0.923613i	$-0.374779\pi$
$797$	21.6435	0.766652	0.383326	+	0.923613i	$0.374779\pi$
$798$	0	0
$799$	−60.5727	−2.14291
$800$	23.7822	0.840828
$801$	0	0
$802$	−88.6086	−3.12888
$803$	−21.9405	−0.774263
$804$	0	0
$805$	2.78516	0.0981638
$806$	16.2655	0.572928
$807$	0	0
$808$	−50.0807	−1.76183
$809$	−2.46809	−0.0867732	−0.0433866	−	0.999058i	$-0.513815\pi$
$809$	−2.46809	−0.0867732	−0.0433866	+	0.999058i	$0.513815\pi$
$810$	0	0
$811$	25.5080	0.895708	0.447854	−	0.894107i	$-0.352188\pi$
$811$	25.5080	0.895708	0.447854	+	0.894107i	$0.352188\pi$
$812$	10.7641	0.377745
$813$	0	0
$814$	12.7601	0.447242
$815$	2.36922	0.0829903
$816$	0	0
$817$	−33.1734	−1.16059
$818$	44.1564	1.54389
$819$	0	0
$820$	27.9497	0.976045
$821$	−16.0559	−0.560354	−0.280177	−	0.959948i	$-0.590393\pi$
$821$	−16.0559	−0.560354	−0.280177	+	0.959948i	$0.590393\pi$
$822$	0	0
$823$	4.69098	0.163517	0.0817586	−	0.996652i	$-0.473946\pi$
$823$	4.69098	0.163517	0.0817586	+	0.996652i	$0.473946\pi$
$824$	−124.356	−4.33215
$825$	0	0
$826$	6.19294	0.215480
$827$	−1.72852	−0.0601066	−0.0300533	−	0.999548i	$-0.509568\pi$
$827$	−1.72852	−0.0601066	−0.0300533	+	0.999548i	$0.509568\pi$
$828$	0	0
$829$	−31.7676	−1.10334	−0.551668	−	0.834064i	$-0.686008\pi$
$829$	−31.7676	−1.10334	−0.551668	+	0.834064i	$0.686008\pi$
$830$	−7.30695	−0.253628
$831$	0	0
$832$	33.8466	1.17342
$833$	40.8441	1.41516
$834$	0	0
$835$	−22.4675	−0.777519
$836$	−66.8704	−2.31276
$837$	0	0
$838$	107.260	3.70522
$839$	8.74269	0.301831	0.150916	−	0.988547i	$-0.451778\pi$
$839$	8.74269	0.301831	0.150916	+	0.988547i	$0.451778\pi$
$840$	0	0
$841$	−0.331505	−0.0114312
$842$	−23.1762	−0.798705
$843$	0	0
$844$	30.8553	1.06208
$845$	1.00000	0.0344010
$846$	0	0
$847$	2.40067	0.0824878
$848$	151.490	5.20218
$849$	0	0
$850$	−16.3455	−0.560647
$851$	−8.50927	−0.291694
$852$	0	0
$853$	33.0301	1.13093	0.565464	−	0.824773i	$-0.308697\pi$
$853$	33.0301	1.13093	0.565464	+	0.824773i	$0.308697\pi$
$854$	−1.37756	−0.0471390
$855$	0	0
$856$	−145.939	−4.98808
$857$	−18.2006	−0.621721	−0.310860	−	0.950456i	$-0.600617\pi$
$857$	−18.2006	−0.621721	−0.310860	+	0.950456i	$0.600617\pi$
$858$	0	0
$859$	12.5794	0.429204	0.214602	−	0.976702i	$-0.431155\pi$
$859$	12.5794	0.429204	0.214602	+	0.976702i	$0.431155\pi$
$860$	−64.3298	−2.19363
$861$	0	0
$862$	54.1522	1.84443
$863$	38.9076	1.32443	0.662215	−	0.749314i	$-0.269617\pi$
$863$	38.9076	1.32443	0.662215	+	0.749314i	$0.269617\pi$
$864$	0	0
$865$	8.61113	0.292787
$866$	−52.5541	−1.78586
$867$	0	0
$868$	11.8952	0.403748
$869$	42.6665	1.44736
$870$	0	0
$871$	4.84843	0.164283
$872$	−64.5717	−2.18667
$873$	0	0
$874$	60.6434	2.05129
$875$	0.361779	0.0122304
$876$	0	0
$877$	−26.1320	−0.882415	−0.441207	−	0.897405i	$-0.645450\pi$
$877$	−26.1320	−0.882415	−0.441207	+	0.897405i	$0.645450\pi$
$878$	92.3663	3.11721
$879$	0	0
$880$	−66.2051	−2.23177
$881$	43.8952	1.47887	0.739433	−	0.673230i	$-0.235094\pi$
$881$	43.8952	1.47887	0.739433	+	0.673230i	$0.235094\pi$
$882$	0	0
$883$	−10.9543	−0.368641	−0.184320	−	0.982866i	$-0.559008\pi$
$883$	−10.9543	−0.368641	−0.184320	+	0.982866i	$0.559008\pi$
$884$	−33.0414	−1.11130
$885$	0	0
$886$	86.5791	2.90868
$887$	−13.1222	−0.440600	−0.220300	−	0.975432i	$-0.570704\pi$
$887$	−13.1222	−0.440600	−0.220300	+	0.975432i	$0.570704\pi$
$888$	0	0
$889$	−6.84492	−0.229571
$890$	−29.6861	−0.995079
$891$	0	0
$892$	−29.5898	−0.990739
$893$	29.1915	0.976855
$894$	0	0
$895$	−21.4105	−0.715675
$896$	16.4534	0.549671
$897$	0	0
$898$	98.4819	3.28638
$899$	31.6810	1.05662
$900$	0	0
$901$	−57.1369	−1.90351
$902$	−58.0650	−1.93335
$903$	0	0
$904$	−150.925	−5.01968
$905$	1.63616	0.0543879
$906$	0	0
$907$	−14.2071	−0.471740	−0.235870	−	0.971785i	$-0.575794\pi$
$907$	−14.2071	−0.471740	−0.235870	+	0.971785i	$0.575794\pi$
$908$	−33.5774	−1.11430
$909$	0	0
$910$	0.994523	0.0329681
$911$	28.3629	0.939705	0.469853	−	0.882745i	$-0.344307\pi$
$911$	28.3629	0.939705	0.469853	+	0.882745i	$0.344307\pi$
$912$	0	0
$913$	11.1625	0.369425
$914$	77.2586	2.55549
$915$	0	0
$916$	128.478	4.24502
$917$	4.03141	0.133129
$918$	0	0
$919$	−7.13020	−0.235204	−0.117602	−	0.993061i	$-0.537521\pi$
$919$	−7.13020	−0.235204	−0.117602	+	0.993061i	$0.537521\pi$
$920$	75.2738	2.48170
$921$	0	0
$922$	−38.5223	−1.26866
$923$	0.306717	0.0100957
$924$	0	0
$925$	−1.10532	−0.0363426
$926$	62.1653	2.04288
$927$	0	0
$928$	127.337	4.18004
$929$	52.3628	1.71797	0.858983	−	0.512004i	$-0.171097\pi$
$929$	52.3628	1.71797	0.858983	+	0.512004i	$0.171097\pi$
$930$	0	0
$931$	−19.6838	−0.645109
$932$	35.7281	1.17031
$933$	0	0
$934$	−20.2732	−0.663359
$935$	24.9704	0.816618
$936$	0	0
$937$	6.94716	0.226954	0.113477	−	0.993541i	$-0.463801\pi$
$937$	6.94716	0.226954	0.113477	+	0.993541i	$0.463801\pi$
$938$	4.82188	0.157440
$939$	0	0
$940$	56.6081	1.84635
$941$	5.12999	0.167233	0.0836164	−	0.996498i	$-0.473353\pi$
$941$	5.12999	0.167233	0.0836164	+	0.996498i	$0.473353\pi$
$942$	0	0
$943$	38.7215	1.26095
$944$	98.1696	3.19515
$945$	0	0
$946$	133.644	4.34514
$947$	41.1737	1.33797	0.668983	−	0.743277i	$-0.266730\pi$
$947$	41.1737	1.33797	0.668983	+	0.743277i	$0.266730\pi$
$948$	0	0
$949$	5.22456	0.169596
$950$	7.87731	0.255574
$951$	0	0
$952$	−21.0335	−0.681698
$953$	−44.1169	−1.42909	−0.714543	−	0.699592i	$-0.753365\pi$
$953$	−44.1169	−1.42909	−0.714543	+	0.699592i	$0.753365\pi$
$954$	0	0
$955$	3.43505	0.111156
$956$	−50.1459	−1.62183
$957$	0	0
$958$	−4.96284	−0.160342
$959$	5.58744	0.180428
$960$	0	0
$961$	4.01005	0.129356
$962$	−3.03849	−0.0979649
$963$	0	0
$964$	−29.0626	−0.936042
$965$	−25.6109	−0.824443
$966$	0	0
$967$	−36.8736	−1.18577	−0.592887	−	0.805285i	$-0.702012\pi$
$967$	−36.8736	−1.18577	−0.592887	+	0.805285i	$0.702012\pi$
$968$	64.8823	2.08540
$969$	0	0
$970$	17.0593	0.547743
$971$	28.7526	0.922714	0.461357	−	0.887214i	$-0.347363\pi$
$971$	28.7526	0.922714	0.461357	+	0.887214i	$0.347363\pi$
$972$	0	0
$973$	−4.91338	−0.157516
$974$	−54.5205	−1.74695
$975$	0	0
$976$	−21.8368	−0.698979
$977$	59.5974	1.90669	0.953345	−	0.301883i	$-0.0976151\pi$
$977$	59.5974	1.90669	0.953345	+	0.301883i	$0.0976151\pi$
$978$	0	0
$979$	45.3501	1.44940
$980$	−38.1708	−1.21932
$981$	0	0
$982$	79.2695	2.52959
$983$	17.4044	0.555116	0.277558	−	0.960709i	$-0.410475\pi$
$983$	17.4044	0.555116	0.277558	+	0.960709i	$0.410475\pi$
$984$	0	0
$985$	0.494416	0.0157534
$986$	−87.5188	−2.78717
$987$	0	0
$988$	15.9234	0.506592
$989$	−89.1226	−2.83393
$990$	0	0
$991$	24.3641	0.773952	0.386976	−	0.922090i	$-0.373520\pi$
$991$	24.3641	0.773952	0.386976	+	0.922090i	$0.373520\pi$
$992$	140.718	4.46779
$993$	0	0
$994$	0.305037	0.00967517
$995$	6.52918	0.206989
$996$	0	0
$997$	13.8719	0.439329	0.219665	−	0.975575i	$-0.429504\pi$
$997$	13.8719	0.439329	0.219665	+	0.975575i	$0.429504\pi$
$998$	−28.7206	−0.909135
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.bj.1.14	yes	14
3.2	odd	2		5265.2.a.bi.1.1	✓	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5265.2.a.bi.1.1	✓	14	3.2	odd	2
5265.2.a.bj.1.14	yes	14	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:	\( N \)	\(=\)	\( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5265.a (trivial)

\(f(q)\)	\(=\)	\(q+2.74898 q^{2} +5.55687 q^{4} +1.00000 q^{5} +0.361779 q^{7} +9.77774 q^{8} +O(q^{10})\)	\(q+2.74898 q^{2} +5.55687 q^{4} +1.00000 q^{5} +0.361779 q^{7} +9.77774 q^{8} +2.74898 q^{10} -4.19949 q^{11} +1.00000 q^{13} +0.994523 q^{14} +15.7650 q^{16} -5.94605 q^{17} +2.86554 q^{19} +5.55687 q^{20} -11.5443 q^{22} +7.69849 q^{23} +1.00000 q^{25} +2.74898 q^{26} +2.01036 q^{28} +5.35430 q^{29} +5.91693 q^{31} +23.7822 q^{32} -16.3455 q^{34} +0.361779 q^{35} -1.10532 q^{37} +7.87731 q^{38} +9.77774 q^{40} +5.02976 q^{41} -11.5766 q^{43} -23.3360 q^{44} +21.1630 q^{46} +10.1871 q^{47} -6.86912 q^{49} +2.74898 q^{50} +5.55687 q^{52} +9.60923 q^{53} -4.19949 q^{55} +3.53739 q^{56} +14.7188 q^{58} +6.22705 q^{59} -1.38514 q^{61} +16.2655 q^{62} +33.8466 q^{64} +1.00000 q^{65} +4.84843 q^{67} -33.0414 q^{68} +0.994523 q^{70} +0.306717 q^{71} +5.22456 q^{73} -3.03849 q^{74} +15.9234 q^{76} -1.51929 q^{77} -10.1599 q^{79} +15.7650 q^{80} +13.8267 q^{82} -2.65806 q^{83} -5.94605 q^{85} -31.8239 q^{86} -41.0615 q^{88} -10.7990 q^{89} +0.361779 q^{91} +42.7795 q^{92} +28.0040 q^{94} +2.86554 q^{95} +6.20571 q^{97} -18.8830 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 2 q^{2} + 16 q^{4} + 14 q^{5} + 4 q^{7} + 12 q^{8}+O(q^{10}) \) 14 * q + 2 * q^2 + 16 * q^4 + 14 * q^5 + 4 * q^7 + 12 * q^8	\( 14 q + 2 q^{2} + 16 q^{4} + 14 q^{5} + 4 q^{7} + 12 q^{8} + 2 q^{10} + 8 q^{11} + 14 q^{13} + 12 q^{14} + 16 q^{16} + 10 q^{17} + 10 q^{19} + 16 q^{20} - 10 q^{22} + 34 q^{23} + 14 q^{25} + 2 q^{26} + 2 q^{28} + 16 q^{29} + 6 q^{31} + 26 q^{32} + 8 q^{34} + 4 q^{35} + 4 q^{37} + 12 q^{38} + 12 q^{40} + 10 q^{41} - 8 q^{43} + 8 q^{44} - 16 q^{46} + 46 q^{47} + 2 q^{49} + 2 q^{50} + 16 q^{52} + 14 q^{53} + 8 q^{55} + 20 q^{56} - 28 q^{58} + 16 q^{59} + 30 q^{62} + 14 q^{64} + 14 q^{65} + 6 q^{67} + 4 q^{68} + 12 q^{70} + 34 q^{71} - 4 q^{73} + 8 q^{74} + 50 q^{76} + 24 q^{77} - 14 q^{79} + 16 q^{80} - 16 q^{82} + 4 q^{83} + 10 q^{85} + 6 q^{86} - 68 q^{88} + 18 q^{89} + 4 q^{91} + 90 q^{92} + 10 q^{95} + 12 q^{97} - 16 q^{98}+O(q^{100}) \) 14 * q + 2 * q^2 + 16 * q^4 + 14 * q^5 + 4 * q^7 + 12 * q^8 + 2 * q^10 + 8 * q^11 + 14 * q^13 + 12 * q^14 + 16 * q^16 + 10 * q^17 + 10 * q^19 + 16 * q^20 - 10 * q^22 + 34 * q^23 + 14 * q^25 + 2 * q^26 + 2 * q^28 + 16 * q^29 + 6 * q^31 + 26 * q^32 + 8 * q^34 + 4 * q^35 + 4 * q^37 + 12 * q^38 + 12 * q^40 + 10 * q^41 - 8 * q^43 + 8 * q^44 - 16 * q^46 + 46 * q^47 + 2 * q^49 + 2 * q^50 + 16 * q^52 + 14 * q^53 + 8 * q^55 + 20 * q^56 - 28 * q^58 + 16 * q^59 + 30 * q^62 + 14 * q^64 + 14 * q^65 + 6 * q^67 + 4 * q^68 + 12 * q^70 + 34 * q^71 - 4 * q^73 + 8 * q^74 + 50 * q^76 + 24 * q^77 - 14 * q^79 + 16 * q^80 - 16 * q^82 + 4 * q^83 + 10 * q^85 + 6 * q^86 - 68 * q^88 + 18 * q^89 + 4 * q^91 + 90 * q^92 + 10 * q^95 + 12 * q^97 - 16 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more