LMFDB - Embedded newform 5265.2.a.w.1.2

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:	$1$
Dimension:	$6$
Coefficient field:	6.6.585163476.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8x^{4} + 13x^{3} + 14x^{2} - 11x - 1 $ x^6 - 2x^5 - 8x^4 + 13x^3 + 14x^2 - 11*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.36425$ 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)

$f(q)$	$=$	$q-2.36425 q^{2} +3.58967 q^{4} -1.00000 q^{5} +3.65786 q^{7} -3.75837 q^{8} +O(q^{10})$	\(q-2.36425 q^{2} +3.58967 q^{4} -1.00000 q^{5} +3.65786 q^{7} -3.75837 q^{8} +2.36425 q^{10} +0.970122 q^{11} -1.00000 q^{13} -8.64808 q^{14} +1.70639 q^{16} -3.19554 q^{17} -2.79468 q^{19} -3.58967 q^{20} -2.29361 q^{22} +0.530951 q^{23} +1.00000 q^{25} +2.36425 q^{26} +13.1305 q^{28} -5.58967 q^{29} +9.60977 q^{31} +3.48242 q^{32} +7.55506 q^{34} -3.65786 q^{35} +7.24079 q^{37} +6.60732 q^{38} +3.75837 q^{40} +0.942246 q^{41} -9.10093 q^{43} +3.48242 q^{44} -1.25530 q^{46} -0.571568 q^{47} +6.37992 q^{49} -2.36425 q^{50} -3.58967 q^{52} -6.82352 q^{53} -0.970122 q^{55} -13.7476 q^{56} +13.2154 q^{58} -2.75392 q^{59} +1.25555 q^{61} -22.7199 q^{62} -11.6461 q^{64} +1.00000 q^{65} -1.73154 q^{67} -11.4709 q^{68} +8.64808 q^{70} -7.29133 q^{71} -16.0855 q^{73} -17.1190 q^{74} -10.0320 q^{76} +3.54857 q^{77} +0.310262 q^{79} -1.70639 q^{80} -2.22770 q^{82} -2.20501 q^{83} +3.19554 q^{85} +21.5169 q^{86} -3.64608 q^{88} -11.5883 q^{89} -3.65786 q^{91} +1.90594 q^{92} +1.35133 q^{94} +2.79468 q^{95} -13.3950 q^{97} -15.0837 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} + 8 q^{4} - 6 q^{5} + 4 q^{7} - 9 q^{8}+O(q^{10}) $ 6 * q - 2 * q^2 + 8 * q^4 - 6 * q^5 + 4 * q^7 - 9 * q^8	$ 6 q - 2 q^{2} + 8 q^{4} - 6 q^{5} + 4 q^{7} - 9 q^{8} + 2 q^{10} - 5 q^{11} - 6 q^{13} + 7 q^{14} + 16 q^{16} - 7 q^{17} + 12 q^{19} - 8 q^{20} - 8 q^{22} - 9 q^{23} + 6 q^{25} + 2 q^{26} - 16 q^{28} - 20 q^{29} + 14 q^{31} - 15 q^{32} + 3 q^{34} - 4 q^{35} - 6 q^{37} - 2 q^{38} + 9 q^{40} - 16 q^{41} + 7 q^{43} - 15 q^{44} - 17 q^{46} - 20 q^{47} + 22 q^{49} - 2 q^{50} - 8 q^{52} + 5 q^{55} + 13 q^{56} + 17 q^{58} - 5 q^{59} + 16 q^{61} - 8 q^{62} - 11 q^{64} + 6 q^{65} - 9 q^{67} - 36 q^{68} - 7 q^{70} - 18 q^{71} + q^{73} - 11 q^{74} + 39 q^{76} - q^{77} + 9 q^{79} - 16 q^{80} - 32 q^{82} + 2 q^{83} + 7 q^{85} - 8 q^{86} + 37 q^{88} + 11 q^{89} - 4 q^{91} + 5 q^{92} + 42 q^{94} - 12 q^{95} - 11 q^{97} - 55 q^{98}+O(q^{100}) $ 6 * q - 2 * q^2 + 8 * q^4 - 6 * q^5 + 4 * q^7 - 9 * q^8 + 2 * q^10 - 5 * q^11 - 6 * q^13 + 7 * q^14 + 16 * q^16 - 7 * q^17 + 12 * q^19 - 8 * q^20 - 8 * q^22 - 9 * q^23 + 6 * q^25 + 2 * q^26 - 16 * q^28 - 20 * q^29 + 14 * q^31 - 15 * q^32 + 3 * q^34 - 4 * q^35 - 6 * q^37 - 2 * q^38 + 9 * q^40 - 16 * q^41 + 7 * q^43 - 15 * q^44 - 17 * q^46 - 20 * q^47 + 22 * q^49 - 2 * q^50 - 8 * q^52 + 5 * q^55 + 13 * q^56 + 17 * q^58 - 5 * q^59 + 16 * q^61 - 8 * q^62 - 11 * q^64 + 6 * q^65 - 9 * q^67 - 36 * q^68 - 7 * q^70 - 18 * q^71 + q^73 - 11 * q^74 + 39 * q^76 - q^77 + 9 * q^79 - 16 * q^80 - 32 * q^82 + 2 * q^83 + 7 * q^85 - 8 * q^86 + 37 * q^88 + 11 * q^89 - 4 * q^91 + 5 * q^92 + 42 * q^94 - 12 * q^95 - 11 * q^97 - 55 * 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 $

$2$	−2.36425	−1.67178	−0.835888	−	0.548900i	$-0.815047\pi$
$2$	−2.36425	−1.67178	−0.835888	+	0.548900i	$0.815047\pi$
$3$	0	0
$3$	0	0
$4$	3.58967	1.79483
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.65786	1.38254	0.691270	−	0.722596i	$-0.257051\pi$
$7$	3.65786	1.38254	0.691270	+	0.722596i	$0.257051\pi$
$8$	−3.75837	−1.32879
$9$	0	0
$10$	2.36425	0.747641
$11$	0.970122	0.292503	0.146251	−	0.989247i	$-0.453279\pi$
$11$	0.970122	0.292503	0.146251	+	0.989247i	$0.453279\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−8.64808	−2.31130
$15$	0	0
$16$	1.70639	0.426598
$17$	−3.19554	−0.775033	−0.387517	−	0.921863i	$-0.626667\pi$
$17$	−3.19554	−0.775033	−0.387517	+	0.921863i	$0.626667\pi$
$18$	0	0
$19$	−2.79468	−0.641144	−0.320572	−	0.947224i	$-0.603875\pi$
$19$	−2.79468	−0.641144	−0.320572	+	0.947224i	$0.603875\pi$
$20$	−3.58967	−0.802675
$21$	0	0
$22$	−2.29361	−0.488999
$23$	0.530951	0.110711	0.0553554	−	0.998467i	$-0.482371\pi$
$23$	0.530951	0.110711	0.0553554	+	0.998467i	$0.482371\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.36425	0.463667
$27$	0	0
$28$	13.1305	2.48143
$29$	−5.58967	−1.03798	−0.518988	−	0.854782i	$-0.673691\pi$
$29$	−5.58967	−1.03798	−0.518988	+	0.854782i	$0.673691\pi$
$30$	0	0
$31$	9.60977	1.72597	0.862983	−	0.505233i	$-0.168593\pi$
$31$	9.60977	1.72597	0.862983	+	0.505233i	$0.168593\pi$
$32$	3.48242	0.615610
$33$	0	0
$34$	7.55506	1.29568
$35$	−3.65786	−0.618291
$36$	0	0
$37$	7.24079	1.19038	0.595189	−	0.803585i	$-0.297077\pi$
$37$	7.24079	1.19038	0.595189	+	0.803585i	$0.297077\pi$
$38$	6.60732	1.07185
$39$	0	0
$40$	3.75837	0.594251
$41$	0.942246	0.147154	0.0735771	−	0.997290i	$-0.476559\pi$
$41$	0.942246	0.147154	0.0735771	+	0.997290i	$0.476559\pi$
$42$	0	0
$43$	−9.10093	−1.38788	−0.693939	−	0.720033i	$-0.744126\pi$
$43$	−9.10093	−1.38788	−0.693939	+	0.720033i	$0.744126\pi$
$44$	3.48242	0.524994
$45$	0	0
$46$	−1.25530	−0.185084
$47$	−0.571568	−0.0833717	−0.0416859	−	0.999131i	$-0.513273\pi$
$47$	−0.571568	−0.0833717	−0.0416859	+	0.999131i	$0.513273\pi$
$48$	0	0
$49$	6.37992	0.911418
$50$	−2.36425	−0.334355
$51$	0	0
$52$	−3.58967	−0.497798
$53$	−6.82352	−0.937283	−0.468642	−	0.883388i	$-0.655256\pi$
$53$	−6.82352	−0.937283	−0.468642	+	0.883388i	$0.655256\pi$
$54$	0	0
$55$	−0.970122	−0.130811
$56$	−13.7476	−1.83710
$57$	0	0
$58$	13.2154	1.73526
$59$	−2.75392	−0.358530	−0.179265	−	0.983801i	$-0.557372\pi$
$59$	−2.75392	−0.358530	−0.179265	+	0.983801i	$0.557372\pi$
$60$	0	0
$61$	1.25555	0.160756	0.0803782	−	0.996764i	$-0.474387\pi$
$61$	1.25555	0.160756	0.0803782	+	0.996764i	$0.474387\pi$
$62$	−22.7199	−2.88543
$63$	0	0
$64$	−11.6461	−1.45576
$65$	1.00000	0.124035
$66$	0	0
$67$	−1.73154	−0.211541	−0.105770	−	0.994391i	$-0.533731\pi$
$67$	−1.73154	−0.211541	−0.105770	+	0.994391i	$0.533731\pi$
$68$	−11.4709	−1.39106
$69$	0	0
$70$	8.64808	1.03364
$71$	−7.29133	−0.865321	−0.432661	−	0.901557i	$-0.642425\pi$
$71$	−7.29133	−0.865321	−0.432661	+	0.901557i	$0.642425\pi$
$72$	0	0
$73$	−16.0855	−1.88266	−0.941330	−	0.337487i	$-0.890423\pi$
$73$	−16.0855	−1.88266	−0.941330	+	0.337487i	$0.890423\pi$
$74$	−17.1190	−1.99005
$75$	0	0
$76$	−10.0320	−1.15075
$77$	3.54857	0.404397
$78$	0	0
$79$	0.310262	0.0349072	0.0174536	−	0.999848i	$-0.494444\pi$
$79$	0.310262	0.0349072	0.0174536	+	0.999848i	$0.494444\pi$
$80$	−1.70639	−0.190780
$81$	0	0
$82$	−2.22770	−0.246009
$83$	−2.20501	−0.242032	−0.121016	−	0.992651i	$-0.538615\pi$
$83$	−2.20501	−0.242032	−0.121016	+	0.992651i	$0.538615\pi$
$84$	0	0
$85$	3.19554	0.346605
$86$	21.5169	2.32022
$87$	0	0
$88$	−3.64608	−0.388674
$89$	−11.5883	−1.22836	−0.614180	−	0.789166i	$-0.710513\pi$
$89$	−11.5883	−1.22836	−0.614180	+	0.789166i	$0.710513\pi$
$90$	0	0
$91$	−3.65786	−0.383448
$92$	1.90594	0.198708
$93$	0	0
$94$	1.35133	0.139379
$95$	2.79468	0.286728
$96$	0	0
$97$	−13.3950	−1.36005	−0.680027	−	0.733187i	$-0.738032\pi$
$97$	−13.3950	−1.36005	−0.680027	+	0.733187i	$0.738032\pi$
$98$	−15.0837	−1.52369
$99$	0	0
$100$	3.58967	0.358967
$101$	9.27955	0.923350	0.461675	−	0.887049i	$-0.347249\pi$
$101$	9.27955	0.923350	0.461675	+	0.887049i	$0.347249\pi$
$102$	0	0
$103$	9.37962	0.924201	0.462101	−	0.886827i	$-0.347096\pi$
$103$	9.37962	0.924201	0.462101	+	0.886827i	$0.347096\pi$
$104$	3.75837	0.368539
$105$	0	0
$106$	16.1325	1.56693
$107$	−8.35447	−0.807658	−0.403829	−	0.914835i	$-0.632321\pi$
$107$	−8.35447	−0.807658	−0.403829	+	0.914835i	$0.632321\pi$
$108$	0	0
$109$	−3.02514	−0.289756	−0.144878	−	0.989450i	$-0.546279\pi$
$109$	−3.02514	−0.289756	−0.144878	+	0.989450i	$0.546279\pi$
$110$	2.29361	0.218687
$111$	0	0
$112$	6.24173	0.589788
$113$	3.45499	0.325018	0.162509	−	0.986707i	$-0.448041\pi$
$113$	3.45499	0.325018	0.162509	+	0.986707i	$0.448041\pi$
$114$	0	0
$115$	−0.530951	−0.0495114
$116$	−20.0651	−1.86299
$117$	0	0
$118$	6.51095	0.599382
$119$	−11.6888	−1.07151
$120$	0	0
$121$	−10.0589	−0.914442
$122$	−2.96843	−0.268749
$123$	0	0
$124$	34.4959	3.09782
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	8.33827	0.739902	0.369951	−	0.929051i	$-0.379374\pi$
$127$	8.33827	0.739902	0.369951	+	0.929051i	$0.379374\pi$
$128$	20.5694	1.81809
$129$	0	0
$130$	−2.36425	−0.207358
$131$	19.1648	1.67444	0.837220	−	0.546867i	$-0.184180\pi$
$131$	19.1648	1.67444	0.837220	+	0.546867i	$0.184180\pi$
$132$	0	0
$133$	−10.2226	−0.886408
$134$	4.09378	0.353649
$135$	0	0
$136$	12.0101	1.02985
$137$	−14.3137	−1.22290	−0.611452	−	0.791282i	$-0.709414\pi$
$137$	−14.3137	−1.22290	−0.611452	+	0.791282i	$0.709414\pi$
$138$	0	0
$139$	16.4011	1.39113	0.695563	−	0.718465i	$-0.255155\pi$
$139$	16.4011	1.39113	0.695563	+	0.718465i	$0.255155\pi$
$140$	−13.1305	−1.10973
$141$	0	0
$142$	17.2385	1.44662
$143$	−0.970122	−0.0811257
$144$	0	0
$145$	5.58967	0.464197
$146$	38.0300	3.14739
$147$	0	0
$148$	25.9921	2.13653
$149$	20.6398	1.69088	0.845439	−	0.534073i	$-0.179339\pi$
$149$	20.6398	1.69088	0.845439	+	0.534073i	$0.179339\pi$
$150$	0	0
$151$	−14.6884	−1.19532	−0.597661	−	0.801749i	$-0.703903\pi$
$151$	−14.6884	−1.19532	−0.597661	+	0.801749i	$0.703903\pi$
$152$	10.5035	0.851943
$153$	0	0
$154$	−8.38970	−0.676061
$155$	−9.60977	−0.771876
$156$	0	0
$157$	−9.25745	−0.738825	−0.369412	−	0.929266i	$-0.620441\pi$
$157$	−9.25745	−0.738825	−0.369412	+	0.929266i	$0.620441\pi$
$158$	−0.733537	−0.0583571
$159$	0	0
$160$	−3.48242	−0.275309
$161$	1.94214	0.153062
$162$	0	0
$163$	−7.92683	−0.620877	−0.310439	−	0.950593i	$-0.600476\pi$
$163$	−7.92683	−0.620877	−0.310439	+	0.950593i	$0.600476\pi$
$164$	3.38235	0.264117
$165$	0	0
$166$	5.21320	0.404623
$167$	−3.30353	−0.255635	−0.127817	−	0.991798i	$-0.540797\pi$
$167$	−3.30353	−0.255635	−0.127817	+	0.991798i	$0.540797\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−7.55506	−0.579447
$171$	0	0
$172$	−32.6693	−2.49101
$173$	15.2464	1.15917	0.579583	−	0.814913i	$-0.303215\pi$
$173$	15.2464	1.15917	0.579583	+	0.814913i	$0.303215\pi$
$174$	0	0
$175$	3.65786	0.276508
$176$	1.65541	0.124781
$177$	0	0
$178$	27.3977	2.05354
$179$	3.84148	0.287126	0.143563	−	0.989641i	$-0.454144\pi$
$179$	3.84148	0.287126	0.143563	+	0.989641i	$0.454144\pi$
$180$	0	0
$181$	−14.7516	−1.09648	−0.548238	−	0.836323i	$-0.684701\pi$
$181$	−14.7516	−1.09648	−0.548238	+	0.836323i	$0.684701\pi$
$182$	8.64808	0.641039
$183$	0	0
$184$	−1.99551	−0.147111
$185$	−7.24079	−0.532354
$186$	0	0
$187$	−3.10007	−0.226699
$188$	−2.05174	−0.149638
$189$	0	0
$190$	−6.60732	−0.479346
$191$	7.79112	0.563745	0.281873	−	0.959452i	$-0.409044\pi$
$191$	7.79112	0.563745	0.281873	+	0.959452i	$0.409044\pi$
$192$	0	0
$193$	−20.6931	−1.48952	−0.744760	−	0.667332i	$-0.767436\pi$
$193$	−20.6931	−1.48952	−0.744760	+	0.667332i	$0.767436\pi$
$194$	31.6691	2.27371
$195$	0	0
$196$	22.9018	1.63584
$197$	14.7501	1.05090	0.525449	−	0.850825i	$-0.323897\pi$
$197$	14.7501	1.05090	0.525449	+	0.850825i	$0.323897\pi$
$198$	0	0
$199$	13.9033	0.985577	0.492789	−	0.870149i	$-0.335978\pi$
$199$	13.9033	0.985577	0.492789	+	0.870149i	$0.335978\pi$
$200$	−3.75837	−0.265757
$201$	0	0
$202$	−21.9392	−1.54363
$203$	−20.4462	−1.43504
$204$	0	0
$205$	−0.942246	−0.0658093
$206$	−22.1757	−1.54506
$207$	0	0
$208$	−1.70639	−0.118317
$209$	−2.71118	−0.187536
$210$	0	0
$211$	−6.49702	−0.447273	−0.223637	−	0.974673i	$-0.571793\pi$
$211$	−6.49702	−0.447273	−0.223637	+	0.974673i	$0.571793\pi$
$212$	−24.4942	−1.68227
$213$	0	0
$214$	19.7521	1.35022
$215$	9.10093	0.620678
$216$	0	0
$217$	35.1512	2.38622
$218$	7.15219	0.484408
$219$	0	0
$220$	−3.48242	−0.234785
$221$	3.19554	0.214956
$222$	0	0
$223$	−13.5266	−0.905810	−0.452905	−	0.891559i	$-0.649612\pi$
$223$	−13.5266	−0.905810	−0.452905	+	0.891559i	$0.649612\pi$
$224$	12.7382	0.851106
$225$	0	0
$226$	−8.16846	−0.543358
$227$	−8.81503	−0.585074	−0.292537	−	0.956254i	$-0.594500\pi$
$227$	−8.81503	−0.585074	−0.292537	+	0.956254i	$0.594500\pi$
$228$	0	0
$229$	−15.1654	−1.00216	−0.501078	−	0.865402i	$-0.667063\pi$
$229$	−15.1654	−1.00216	−0.501078	+	0.865402i	$0.667063\pi$
$230$	1.25530	0.0827720
$231$	0	0
$232$	21.0081	1.37925
$233$	−29.0933	−1.90596	−0.952982	−	0.303026i	$-0.902003\pi$
$233$	−29.0933	−1.90596	−0.952982	+	0.303026i	$0.902003\pi$
$234$	0	0
$235$	0.571568	0.0372850
$236$	−9.88567	−0.643502
$237$	0	0
$238$	27.6353	1.79133
$239$	10.3516	0.669592	0.334796	−	0.942291i	$-0.391333\pi$
$239$	10.3516	0.669592	0.334796	+	0.942291i	$0.391333\pi$
$240$	0	0
$241$	20.4353	1.31635	0.658177	−	0.752863i	$-0.271328\pi$
$241$	20.4353	1.31635	0.658177	+	0.752863i	$0.271328\pi$
$242$	23.7816	1.52874
$243$	0	0
$244$	4.50700	0.288531
$245$	−6.37992	−0.407598
$246$	0	0
$247$	2.79468	0.177821
$248$	−36.1171	−2.29344
$249$	0	0
$250$	2.36425	0.149528
$251$	−7.13541	−0.450383	−0.225192	−	0.974315i	$-0.572301\pi$
$251$	−7.13541	−0.450383	−0.225192	+	0.974315i	$0.572301\pi$
$252$	0	0
$253$	0.515087	0.0323833
$254$	−19.7137	−1.23695
$255$	0	0
$256$	−25.3390	−1.58369
$257$	−10.3089	−0.643053	−0.321526	−	0.946901i	$-0.604196\pi$
$257$	−10.3089	−0.643053	−0.321526	+	0.946901i	$0.604196\pi$
$258$	0	0
$259$	26.4858	1.64575
$260$	3.58967	0.222622
$261$	0	0
$262$	−45.3104	−2.79929
$263$	−10.9067	−0.672534	−0.336267	−	0.941767i	$-0.609164\pi$
$263$	−10.9067	−0.672534	−0.336267	+	0.941767i	$0.609164\pi$
$264$	0	0
$265$	6.82352	0.419166
$266$	24.1686	1.48187
$267$	0	0
$268$	−6.21564	−0.379681
$269$	15.6537	0.954423	0.477212	−	0.878788i	$-0.341648\pi$
$269$	15.6537	0.954423	0.477212	+	0.878788i	$0.341648\pi$
$270$	0	0
$271$	−12.5421	−0.761877	−0.380938	−	0.924600i	$-0.624399\pi$
$271$	−12.5421	−0.761877	−0.380938	+	0.924600i	$0.624399\pi$
$272$	−5.45285	−0.330627
$273$	0	0
$274$	33.8412	2.04442
$275$	0.970122	0.0585006
$276$	0	0
$277$	21.2857	1.27894	0.639468	−	0.768818i	$-0.279155\pi$
$277$	21.2857	1.27894	0.639468	+	0.768818i	$0.279155\pi$
$278$	−38.7764	−2.32565
$279$	0	0
$280$	13.7476	0.821576
$281$	−14.2074	−0.847540	−0.423770	−	0.905770i	$-0.639294\pi$
$281$	−14.2074	−0.847540	−0.423770	+	0.905770i	$0.639294\pi$
$282$	0	0
$283$	−5.29551	−0.314785	−0.157393	−	0.987536i	$-0.550309\pi$
$283$	−5.29551	−0.314785	−0.157393	+	0.987536i	$0.550309\pi$
$284$	−26.1735	−1.55311
$285$	0	0
$286$	2.29361	0.135624
$287$	3.44660	0.203447
$288$	0	0
$289$	−6.78850	−0.399324
$290$	−13.2154	−0.776033
$291$	0	0
$292$	−57.7415	−3.37906
$293$	−4.33188	−0.253071	−0.126536	−	0.991962i	$-0.540386\pi$
$293$	−4.33188	−0.253071	−0.126536	+	0.991962i	$0.540386\pi$
$294$	0	0
$295$	2.75392	0.160340
$296$	−27.2136	−1.58176
$297$	0	0
$298$	−48.7976	−2.82677
$299$	−0.530951	−0.0307057
$300$	0	0
$301$	−33.2899	−1.91880
$302$	34.7269	1.99831
$303$	0	0
$304$	−4.76882	−0.273511
$305$	−1.25555	−0.0718924
$306$	0	0
$307$	18.2209	1.03992	0.519960	−	0.854191i	$-0.325947\pi$
$307$	18.2209	1.03992	0.519960	+	0.854191i	$0.325947\pi$
$308$	12.7382	0.725826
$309$	0	0
$310$	22.7199	1.29040
$311$	26.7716	1.51808	0.759040	−	0.651045i	$-0.225669\pi$
$311$	26.7716	1.51808	0.759040	+	0.651045i	$0.225669\pi$
$312$	0	0
$313$	11.0007	0.621794	0.310897	−	0.950444i	$-0.399371\pi$
$313$	11.0007	0.621794	0.310897	+	0.950444i	$0.399371\pi$
$314$	21.8869	1.23515
$315$	0	0
$316$	1.11374	0.0626527
$317$	−25.6353	−1.43982	−0.719912	−	0.694066i	$-0.755818\pi$
$317$	−25.6353	−1.43982	−0.719912	+	0.694066i	$0.755818\pi$
$318$	0	0
$319$	−5.42266	−0.303611
$320$	11.6461	0.651036
$321$	0	0
$322$	−4.59171	−0.255886
$323$	8.93053	0.496908
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	18.7410	1.03797
$327$	0	0
$328$	−3.54131	−0.195536
$329$	−2.09071	−0.115265
$330$	0	0
$331$	21.5015	1.18183	0.590914	−	0.806734i	$-0.298767\pi$
$331$	21.5015	1.18183	0.590914	+	0.806734i	$0.298767\pi$
$332$	−7.91527	−0.434407
$333$	0	0
$334$	7.81036	0.427364
$335$	1.73154	0.0946039
$336$	0	0
$337$	−22.5893	−1.23052	−0.615259	−	0.788325i	$-0.710948\pi$
$337$	−22.5893	−1.23052	−0.615259	+	0.788325i	$0.710948\pi$
$338$	−2.36425	−0.128598
$339$	0	0
$340$	11.4709	0.622099
$341$	9.32265	0.504850
$342$	0	0
$343$	−2.26815	−0.122469
$344$	34.2047	1.84419
$345$	0	0
$346$	−36.0464	−1.93787
$347$	19.6052	1.05246	0.526232	−	0.850341i	$-0.323604\pi$
$347$	19.6052	1.05246	0.526232	+	0.850341i	$0.323604\pi$
$348$	0	0
$349$	12.0398	0.644475	0.322237	−	0.946659i	$-0.395565\pi$
$349$	12.0398	0.644475	0.322237	+	0.946659i	$0.395565\pi$
$350$	−8.64808	−0.462260
$351$	0	0
$352$	3.37837	0.180068
$353$	−35.9544	−1.91366	−0.956831	−	0.290645i	$-0.906130\pi$
$353$	−35.9544	−1.91366	−0.956831	+	0.290645i	$0.906130\pi$
$354$	0	0
$355$	7.29133	0.386983
$356$	−41.5983	−2.20470
$357$	0	0
$358$	−9.08222	−0.480010
$359$	24.1485	1.27451	0.637255	−	0.770653i	$-0.280070\pi$
$359$	24.1485	1.27451	0.637255	+	0.770653i	$0.280070\pi$
$360$	0	0
$361$	−11.1897	−0.588934
$362$	34.8764	1.83306
$363$	0	0
$364$	−13.1305	−0.688225
$365$	16.0855	0.841951
$366$	0	0
$367$	−28.9994	−1.51376	−0.756879	−	0.653555i	$-0.773277\pi$
$367$	−28.9994	−1.51376	−0.756879	+	0.653555i	$0.773277\pi$
$368$	0.906009	0.0472290
$369$	0	0
$370$	17.1190	0.889976
$371$	−24.9595	−1.29583
$372$	0	0
$373$	8.14632	0.421801	0.210900	−	0.977508i	$-0.432360\pi$
$373$	8.14632	0.421801	0.210900	+	0.977508i	$0.432360\pi$
$374$	7.32933	0.378991
$375$	0	0
$376$	2.14817	0.110783
$377$	5.58967	0.287883
$378$	0	0
$379$	31.2289	1.60412	0.802062	−	0.597241i	$-0.203737\pi$
$379$	31.2289	1.60412	0.802062	+	0.597241i	$0.203737\pi$
$380$	10.0320	0.514630
$381$	0	0
$382$	−18.4201	−0.942456
$383$	14.2820	0.729776	0.364888	−	0.931051i	$-0.381107\pi$
$383$	14.2820	0.729776	0.364888	+	0.931051i	$0.381107\pi$
$384$	0	0
$385$	−3.54857	−0.180852
$386$	48.9236	2.49015
$387$	0	0
$388$	−48.0836	−2.44107
$389$	−11.0870	−0.562135	−0.281067	−	0.959688i	$-0.590688\pi$
$389$	−11.0870	−0.562135	−0.281067	+	0.959688i	$0.590688\pi$
$390$	0	0
$391$	−1.69668	−0.0858046
$392$	−23.9781	−1.21108
$393$	0	0
$394$	−34.8728	−1.75687
$395$	−0.310262	−0.0156110
$396$	0	0
$397$	−16.4731	−0.826762	−0.413381	−	0.910558i	$-0.635652\pi$
$397$	−16.4731	−0.826762	−0.413381	+	0.910558i	$0.635652\pi$
$398$	−32.8708	−1.64766
$399$	0	0
$400$	1.70639	0.0853195
$401$	−20.0926	−1.00338	−0.501689	−	0.865048i	$-0.667288\pi$
$401$	−20.0926	−1.00338	−0.501689	+	0.865048i	$0.667288\pi$
$402$	0	0
$403$	−9.60977	−0.478697
$404$	33.3105	1.65726
$405$	0	0
$406$	48.3399	2.39907
$407$	7.02445	0.348189
$408$	0	0
$409$	−16.8678	−0.834061	−0.417030	−	0.908892i	$-0.636929\pi$
$409$	−16.8678	−0.834061	−0.417030	+	0.908892i	$0.636929\pi$
$410$	2.22770	0.110018
$411$	0	0
$412$	33.6697	1.65879
$413$	−10.0735	−0.495682
$414$	0	0
$415$	2.20501	0.108240
$416$	−3.48242	−0.170740
$417$	0	0
$418$	6.40991	0.313519
$419$	−17.5575	−0.857740	−0.428870	−	0.903366i	$-0.641088\pi$
$419$	−17.5575	−0.857740	−0.428870	+	0.903366i	$0.641088\pi$
$420$	0	0
$421$	15.1723	0.739454	0.369727	−	0.929140i	$-0.379451\pi$
$421$	15.1723	0.739454	0.369727	+	0.929140i	$0.379451\pi$
$422$	15.3606	0.747741
$423$	0	0
$424$	25.6454	1.24545
$425$	−3.19554	−0.155007
$426$	0	0
$427$	4.59261	0.222252
$428$	−29.9898	−1.44961
$429$	0	0
$430$	−21.5169	−1.03763
$431$	13.5409	0.652242	0.326121	−	0.945328i	$-0.394258\pi$
$431$	13.5409	0.652242	0.326121	+	0.945328i	$0.394258\pi$
$432$	0	0
$433$	12.7689	0.613633	0.306816	−	0.951769i	$-0.400736\pi$
$433$	12.7689	0.613633	0.306816	+	0.951769i	$0.400736\pi$
$434$	−83.1061	−3.98922
$435$	0	0
$436$	−10.8593	−0.520065
$437$	−1.48384	−0.0709816
$438$	0	0
$439$	−5.28597	−0.252286	−0.126143	−	0.992012i	$-0.540260\pi$
$439$	−5.28597	−0.252286	−0.126143	+	0.992012i	$0.540260\pi$
$440$	3.64608	0.173820
$441$	0	0
$442$	−7.55506	−0.359358
$443$	22.0446	1.04737	0.523684	−	0.851912i	$-0.324557\pi$
$443$	22.0446	1.04737	0.523684	+	0.851912i	$0.324557\pi$
$444$	0	0
$445$	11.5883	0.549339
$446$	31.9803	1.51431
$447$	0	0
$448$	−42.5997	−2.01265
$449$	−8.12628	−0.383503	−0.191751	−	0.981444i	$-0.561417\pi$
$449$	−8.12628	−0.383503	−0.191751	+	0.981444i	$0.561417\pi$
$450$	0	0
$451$	0.914094	0.0430430
$452$	12.4023	0.583354
$453$	0	0
$454$	20.8409	0.978113
$455$	3.65786	0.171483
$456$	0	0
$457$	−22.4701	−1.05111	−0.525553	−	0.850761i	$-0.676142\pi$
$457$	−22.4701	−1.05111	−0.525553	+	0.850761i	$0.676142\pi$
$458$	35.8547	1.67538
$459$	0	0
$460$	−1.90594	−0.0888648
$461$	4.39803	0.204837	0.102418	−	0.994741i	$-0.467342\pi$
$461$	4.39803	0.204837	0.102418	+	0.994741i	$0.467342\pi$
$462$	0	0
$463$	−24.5627	−1.14152	−0.570762	−	0.821116i	$-0.693352\pi$
$463$	−24.5627	−1.14152	−0.570762	+	0.821116i	$0.693352\pi$
$464$	−9.53816	−0.442798
$465$	0	0
$466$	68.7838	3.18635
$467$	−39.1880	−1.81340	−0.906702	−	0.421771i	$-0.861409\pi$
$467$	−39.1880	−1.81340	−0.906702	+	0.421771i	$0.861409\pi$
$468$	0	0
$469$	−6.33371	−0.292464
$470$	−1.35133	−0.0623321
$471$	0	0
$472$	10.3503	0.476410
$473$	−8.82901	−0.405958
$474$	0	0
$475$	−2.79468	−0.128229
$476$	−41.9591	−1.92319
$477$	0	0
$478$	−24.4738	−1.11941
$479$	−10.2858	−0.469968	−0.234984	−	0.971999i	$-0.575504\pi$
$479$	−10.2858	−0.469968	−0.234984	+	0.971999i	$0.575504\pi$
$480$	0	0
$481$	−7.24079	−0.330152
$482$	−48.3141	−2.20065
$483$	0	0
$484$	−36.1080	−1.64127
$485$	13.3950	0.608235
$486$	0	0
$487$	39.5500	1.79218	0.896090	−	0.443872i	$-0.146395\pi$
$487$	39.5500	1.79218	0.896090	+	0.443872i	$0.146395\pi$
$488$	−4.71882	−0.213611
$489$	0	0
$490$	15.0837	0.681413
$491$	29.8222	1.34586	0.672929	−	0.739707i	$-0.265036\pi$
$491$	29.8222	1.34586	0.672929	+	0.739707i	$0.265036\pi$
$492$	0	0
$493$	17.8620	0.804466
$494$	−6.60732	−0.297278
$495$	0	0
$496$	16.3980	0.736293
$497$	−26.6706	−1.19634
$498$	0	0
$499$	−20.4921	−0.917350	−0.458675	−	0.888604i	$-0.651676\pi$
$499$	−20.4921	−0.917350	−0.458675	+	0.888604i	$0.651676\pi$
$500$	−3.58967	−0.160535
$501$	0	0
$502$	16.8699	0.752940
$503$	−4.09721	−0.182686	−0.0913428	−	0.995820i	$-0.529116\pi$
$503$	−4.09721	−0.182686	−0.0913428	+	0.995820i	$0.529116\pi$
$504$	0	0
$505$	−9.27955	−0.412935
$506$	−1.21779	−0.0541375
$507$	0	0
$508$	29.9316	1.32800
$509$	−21.6398	−0.959167	−0.479584	−	0.877496i	$-0.659212\pi$
$509$	−21.6398	−0.959167	−0.479584	+	0.877496i	$0.659212\pi$
$510$	0	0
$511$	−58.8383	−2.60285
$512$	18.7689	0.829475
$513$	0	0
$514$	24.3728	1.07504
$515$	−9.37962	−0.413315
$516$	0	0
$517$	−0.554491	−0.0243865
$518$	−62.6190	−2.75132
$519$	0	0
$520$	−3.75837	−0.164816
$521$	1.34401	0.0588820	0.0294410	−	0.999567i	$-0.490627\pi$
$521$	1.34401	0.0588820	0.0294410	+	0.999567i	$0.490627\pi$
$522$	0	0
$523$	−16.4164	−0.717841	−0.358920	−	0.933368i	$-0.616855\pi$
$523$	−16.4164	−0.717841	−0.358920	+	0.933368i	$0.616855\pi$
$524$	68.7954	3.00534
$525$	0	0
$526$	25.7861	1.12433
$527$	−30.7085	−1.33768
$528$	0	0
$529$	−22.7181	−0.987743
$530$	−16.1325	−0.700751
$531$	0	0
$532$	−36.6956	−1.59096
$533$	−0.942246	−0.0408132
$534$	0	0
$535$	8.35447	0.361195
$536$	6.50776	0.281092
$537$	0	0
$538$	−37.0092	−1.59558
$539$	6.18931	0.266592
$540$	0	0
$541$	−34.1917	−1.47002	−0.735009	−	0.678058i	$-0.762822\pi$
$541$	−34.1917	−1.47002	−0.735009	+	0.678058i	$0.762822\pi$
$542$	29.6526	1.27369
$543$	0	0
$544$	−11.1282	−0.477118
$545$	3.02514	0.129583
$546$	0	0
$547$	4.62449	0.197729	0.0988645	−	0.995101i	$-0.468479\pi$
$547$	4.62449	0.197729	0.0988645	+	0.995101i	$0.468479\pi$
$548$	−51.3815	−2.19491
$549$	0	0
$550$	−2.29361	−0.0977998
$551$	15.6214	0.665492
$552$	0	0
$553$	1.13490	0.0482607
$554$	−50.3248	−2.13809
$555$	0	0
$556$	58.8747	2.49684
$557$	35.1728	1.49032	0.745159	−	0.666886i	$-0.232373\pi$
$557$	35.1728	1.49032	0.745159	+	0.666886i	$0.232373\pi$
$558$	0	0
$559$	9.10093	0.384928
$560$	−6.24173	−0.263761
$561$	0	0
$562$	33.5897	1.41690
$563$	−0.694887	−0.0292860	−0.0146430	−	0.999893i	$-0.504661\pi$
$563$	−0.694887	−0.0292860	−0.0146430	+	0.999893i	$0.504661\pi$
$564$	0	0
$565$	−3.45499	−0.145353
$566$	12.5199	0.526250
$567$	0	0
$568$	27.4035	1.14983
$569$	1.83728	0.0770227	0.0385113	−	0.999258i	$-0.487738\pi$
$569$	1.83728	0.0770227	0.0385113	+	0.999258i	$0.487738\pi$
$570$	0	0
$571$	−32.4711	−1.35887	−0.679436	−	0.733734i	$-0.737776\pi$
$571$	−32.4711	−1.35887	−0.679436	+	0.733734i	$0.737776\pi$
$572$	−3.48242	−0.145607
$573$	0	0
$574$	−8.14862	−0.340117
$575$	0.530951	0.0221422
$576$	0	0
$577$	−35.3592	−1.47202	−0.736011	−	0.676970i	$-0.763293\pi$
$577$	−35.3592	−1.47202	−0.736011	+	0.676970i	$0.763293\pi$
$578$	16.0497	0.667579
$579$	0	0
$580$	20.0651	0.833157
$581$	−8.06562	−0.334618
$582$	0	0
$583$	−6.61965	−0.274158
$584$	60.4552	2.50165
$585$	0	0
$586$	10.2417	0.423079
$587$	23.2474	0.959521	0.479761	−	0.877399i	$-0.340724\pi$
$587$	23.2474	0.959521	0.479761	+	0.877399i	$0.340724\pi$
$588$	0	0
$589$	−26.8563	−1.10659
$590$	−6.51095	−0.268052
$591$	0	0
$592$	12.3556	0.507813
$593$	−31.0702	−1.27590	−0.637949	−	0.770078i	$-0.720217\pi$
$593$	−31.0702	−1.27590	−0.637949	+	0.770078i	$0.720217\pi$
$594$	0	0
$595$	11.6888	0.479196
$596$	74.0900	3.03485
$597$	0	0
$598$	1.25530	0.0513330
$599$	−41.2097	−1.68378	−0.841892	−	0.539646i	$-0.818558\pi$
$599$	−41.2097	−1.68378	−0.841892	+	0.539646i	$0.818558\pi$
$600$	0	0
$601$	5.70892	0.232872	0.116436	−	0.993198i	$-0.462853\pi$
$601$	5.70892	0.232872	0.116436	+	0.993198i	$0.462853\pi$
$602$	78.7056	3.20780
$603$	0	0
$604$	−52.7263	−2.14540
$605$	10.0589	0.408951
$606$	0	0
$607$	7.09539	0.287993	0.143996	−	0.989578i	$-0.454005\pi$
$607$	7.09539	0.287993	0.143996	+	0.989578i	$0.454005\pi$
$608$	−9.73225	−0.394695
$609$	0	0
$610$	2.96843	0.120188
$611$	0.571568	0.0231232
$612$	0	0
$613$	−8.80161	−0.355494	−0.177747	−	0.984076i	$-0.556881\pi$
$613$	−8.80161	−0.355494	−0.177747	+	0.984076i	$0.556881\pi$
$614$	−43.0786	−1.73851
$615$	0	0
$616$	−13.3369	−0.537357
$617$	42.2026	1.69901	0.849507	−	0.527578i	$-0.176900\pi$
$617$	42.2026	1.69901	0.849507	+	0.527578i	$0.176900\pi$
$618$	0	0
$619$	−41.3733	−1.66293	−0.831467	−	0.555574i	$-0.812499\pi$
$619$	−41.3733	−1.66293	−0.831467	+	0.555574i	$0.812499\pi$
$620$	−34.4959	−1.38539
$621$	0	0
$622$	−63.2947	−2.53789
$623$	−42.3885	−1.69826
$624$	0	0
$625$	1.00000	0.0400000
$626$	−26.0083	−1.03950
$627$	0	0
$628$	−33.2312	−1.32607
$629$	−23.1383	−0.922583
$630$	0	0
$631$	−12.4199	−0.494429	−0.247215	−	0.968961i	$-0.579515\pi$
$631$	−12.4199	−0.494429	−0.247215	+	0.968961i	$0.579515\pi$
$632$	−1.16608	−0.0463842
$633$	0	0
$634$	60.6083	2.40706
$635$	−8.33827	−0.330894
$636$	0	0
$637$	−6.37992	−0.252782
$638$	12.8205	0.507569
$639$	0	0
$640$	−20.5694	−0.813077
$641$	−9.37834	−0.370422	−0.185211	−	0.982699i	$-0.559297\pi$
$641$	−9.37834	−0.370422	−0.185211	+	0.982699i	$0.559297\pi$
$642$	0	0
$643$	2.12183	0.0836769	0.0418384	−	0.999124i	$-0.486679\pi$
$643$	2.12183	0.0836769	0.0418384	+	0.999124i	$0.486679\pi$
$644$	6.97165	0.274722
$645$	0	0
$646$	−21.1140	−0.830719
$647$	−5.07011	−0.199327	−0.0996633	−	0.995021i	$-0.531777\pi$
$647$	−5.07011	−0.199327	−0.0996633	+	0.995021i	$0.531777\pi$
$648$	0	0
$649$	−2.67164	−0.104871
$650$	2.36425	0.0927334
$651$	0	0
$652$	−28.4547	−1.11437
$653$	8.94581	0.350077	0.175038	−	0.984562i	$-0.443995\pi$
$653$	8.94581	0.350077	0.175038	+	0.984562i	$0.443995\pi$
$654$	0	0
$655$	−19.1648	−0.748832
$656$	1.60784	0.0627756
$657$	0	0
$658$	4.94297	0.192697
$659$	34.2143	1.33280	0.666400	−	0.745595i	$-0.267834\pi$
$659$	34.2143	1.33280	0.666400	+	0.745595i	$0.267834\pi$
$660$	0	0
$661$	49.6529	1.93127	0.965637	−	0.259895i	$-0.0836880\pi$
$661$	49.6529	1.93127	0.965637	+	0.259895i	$0.0836880\pi$
$662$	−50.8349	−1.97575
$663$	0	0
$664$	8.28726	0.321608
$665$	10.2226	0.396414
$666$	0	0
$667$	−2.96784	−0.114915
$668$	−11.8586	−0.458822
$669$	0	0
$670$	−4.09378	−0.158156
$671$	1.21803	0.0470217
$672$	0	0
$673$	−40.8340	−1.57403	−0.787017	−	0.616932i	$-0.788375\pi$
$673$	−40.8340	−1.57403	−0.787017	+	0.616932i	$0.788375\pi$
$674$	53.4067	2.05715
$675$	0	0
$676$	3.58967	0.138064
$677$	29.0974	1.11831	0.559153	−	0.829065i	$-0.311127\pi$
$677$	29.0974	1.11831	0.559153	+	0.829065i	$0.311127\pi$
$678$	0	0
$679$	−48.9970	−1.88033
$680$	−12.0101	−0.460564
$681$	0	0
$682$	−22.0411	−0.843996
$683$	−23.1161	−0.884512	−0.442256	−	0.896889i	$-0.645822\pi$
$683$	−23.1161	−0.884512	−0.442256	+	0.896889i	$0.645822\pi$
$684$	0	0
$685$	14.3137	0.546899
$686$	5.36247	0.204740
$687$	0	0
$688$	−15.5297	−0.592066
$689$	6.82352	0.259956
$690$	0	0
$691$	15.4605	0.588144	0.294072	−	0.955783i	$-0.404989\pi$
$691$	15.4605	0.588144	0.294072	+	0.955783i	$0.404989\pi$
$692$	54.7297	2.08051
$693$	0	0
$694$	−46.3516	−1.75948
$695$	−16.4011	−0.622131
$696$	0	0
$697$	−3.01099	−0.114049
$698$	−28.4650	−1.07742
$699$	0	0
$700$	13.1305	0.496286
$701$	−30.5208	−1.15276	−0.576378	−	0.817183i	$-0.695534\pi$
$701$	−30.5208	−1.15276	−0.576378	+	0.817183i	$0.695534\pi$
$702$	0	0
$703$	−20.2357	−0.763205
$704$	−11.2981	−0.425814
$705$	0	0
$706$	85.0052	3.19921
$707$	33.9433	1.27657
$708$	0	0
$709$	32.7428	1.22968	0.614840	−	0.788652i	$-0.289220\pi$
$709$	32.7428	1.22968	0.614840	+	0.788652i	$0.289220\pi$
$710$	−17.2385	−0.646950
$711$	0	0
$712$	43.5533	1.63223
$713$	5.10232	0.191083
$714$	0	0
$715$	0.970122	0.0362805
$716$	13.7897	0.515344
$717$	0	0
$718$	−57.0931	−2.13070
$719$	−31.6026	−1.17858	−0.589290	−	0.807922i	$-0.700592\pi$
$719$	−31.6026	−1.17858	−0.589290	+	0.807922i	$0.700592\pi$
$720$	0	0
$721$	34.3093	1.27775
$722$	26.4553	0.984566
$723$	0	0
$724$	−52.9533	−1.96799
$725$	−5.58967	−0.207595
$726$	0	0
$727$	13.2880	0.492823	0.246412	−	0.969165i	$-0.420748\pi$
$727$	13.2880	0.492823	0.246412	+	0.969165i	$0.420748\pi$
$728$	13.7476	0.509520
$729$	0	0
$730$	−38.0300	−1.40755
$731$	29.0824	1.07565
$732$	0	0
$733$	−25.4281	−0.939208	−0.469604	−	0.882877i	$-0.655603\pi$
$733$	−25.4281	−0.939208	−0.469604	+	0.882877i	$0.655603\pi$
$734$	68.5619	2.53067
$735$	0	0
$736$	1.84899	0.0681548
$737$	−1.67980	−0.0618762
$738$	0	0
$739$	12.2903	0.452107	0.226053	−	0.974115i	$-0.427418\pi$
$739$	12.2903	0.452107	0.226053	+	0.974115i	$0.427418\pi$
$740$	−25.9921	−0.955487
$741$	0	0
$742$	59.0104	2.16634
$743$	−6.78861	−0.249050	−0.124525	−	0.992216i	$-0.539741\pi$
$743$	−6.78861	−0.249050	−0.124525	+	0.992216i	$0.539741\pi$
$744$	0	0
$745$	−20.6398	−0.756183
$746$	−19.2599	−0.705156
$747$	0	0
$748$	−11.1282	−0.406888
$749$	−30.5595	−1.11662
$750$	0	0
$751$	−28.5593	−1.04214	−0.521071	−	0.853513i	$-0.674467\pi$
$751$	−28.5593	−1.04214	−0.521071	+	0.853513i	$0.674467\pi$
$752$	−0.975318	−0.0355662
$753$	0	0
$754$	−13.2154	−0.481275
$755$	14.6884	0.534564
$756$	0	0
$757$	−8.94453	−0.325095	−0.162547	−	0.986701i	$-0.551971\pi$
$757$	−8.94453	−0.325095	−0.162547	+	0.986701i	$0.551971\pi$
$758$	−73.8330	−2.68173
$759$	0	0
$760$	−10.5035	−0.381001
$761$	−23.8040	−0.862894	−0.431447	−	0.902138i	$-0.641997\pi$
$761$	−23.8040	−0.862894	−0.431447	+	0.902138i	$0.641997\pi$
$762$	0	0
$763$	−11.0655	−0.400600
$764$	27.9675	1.01183
$765$	0	0
$766$	−33.7662	−1.22002
$767$	2.75392	0.0994383
$768$	0	0
$769$	−18.6750	−0.673438	−0.336719	−	0.941605i	$-0.609317\pi$
$769$	−18.6750	−0.673438	−0.336719	+	0.941605i	$0.609317\pi$
$770$	8.38970	0.302344
$771$	0	0
$772$	−74.2813	−2.67344
$773$	11.1058	0.399448	0.199724	−	0.979852i	$-0.435995\pi$
$773$	11.1058	0.399448	0.199724	+	0.979852i	$0.435995\pi$
$774$	0	0
$775$	9.60977	0.345193
$776$	50.3434	1.80722
$777$	0	0
$778$	26.2125	0.939764
$779$	−2.63328	−0.0943470
$780$	0	0
$781$	−7.07348	−0.253109
$782$	4.01136	0.143446
$783$	0	0
$784$	10.8866	0.388809
$785$	9.25745	0.330412
$786$	0	0
$787$	−15.0046	−0.534857	−0.267429	−	0.963578i	$-0.586174\pi$
$787$	−15.0046	−0.534857	−0.267429	+	0.963578i	$0.586174\pi$
$788$	52.9478	1.88619
$789$	0	0
$790$	0.733537	0.0260981
$791$	12.6379	0.449351
$792$	0	0
$793$	−1.25555	−0.0445858
$794$	38.9465	1.38216
$795$	0	0
$796$	49.9082	1.76895
$797$	16.2153	0.574376	0.287188	−	0.957874i	$-0.407280\pi$
$797$	16.2153	0.574376	0.287188	+	0.957874i	$0.407280\pi$
$798$	0	0
$799$	1.82647	0.0646159
$800$	3.48242	0.123122
$801$	0	0
$802$	47.5040	1.67742
$803$	−15.6049	−0.550683
$804$	0	0
$805$	−1.94214	−0.0684515
$806$	22.7199	0.800274
$807$	0	0
$808$	−34.8760	−1.22693
$809$	−29.1661	−1.02542	−0.512712	−	0.858561i	$-0.671359\pi$
$809$	−29.1661	−1.02542	−0.512712	+	0.858561i	$0.671359\pi$
$810$	0	0
$811$	48.7400	1.71149	0.855747	−	0.517395i	$-0.173098\pi$
$811$	48.7400	1.71149	0.855747	+	0.517395i	$0.173098\pi$
$812$	−73.3952	−2.57567
$813$	0	0
$814$	−16.6076	−0.582094
$815$	7.92683	0.277665
$816$	0	0
$817$	25.4342	0.889830
$818$	39.8798	1.39436
$819$	0	0
$820$	−3.38235	−0.118117
$821$	32.7799	1.14403	0.572013	−	0.820245i	$-0.306163\pi$
$821$	32.7799	1.14403	0.572013	+	0.820245i	$0.306163\pi$
$822$	0	0
$823$	27.5718	0.961094	0.480547	−	0.876969i	$-0.340438\pi$
$823$	27.5718	0.961094	0.480547	+	0.876969i	$0.340438\pi$
$824$	−35.2521	−1.22807
$825$	0	0
$826$	23.8161	0.828670
$827$	28.9299	1.00599	0.502995	−	0.864289i	$-0.332231\pi$
$827$	28.9299	1.00599	0.502995	+	0.864289i	$0.332231\pi$
$828$	0	0
$829$	−43.4313	−1.50843	−0.754216	−	0.656627i	$-0.771983\pi$
$829$	−43.4313	−1.50843	−0.754216	+	0.656627i	$0.771983\pi$
$830$	−5.21320	−0.180953
$831$	0	0
$832$	11.6461	0.403755
$833$	−20.3873	−0.706379
$834$	0	0
$835$	3.30353	0.114323
$836$	−9.73225	−0.336597
$837$	0	0
$838$	41.5103	1.43395
$839$	−50.4633	−1.74218	−0.871092	−	0.491119i	$-0.836588\pi$
$839$	−50.4633	−1.74218	−0.871092	+	0.491119i	$0.836588\pi$
$840$	0	0
$841$	2.24441	0.0773935
$842$	−35.8712	−1.23620
$843$	0	0
$844$	−23.3222	−0.802782
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−36.7939	−1.26425
$848$	−11.6436	−0.399843
$849$	0	0
$850$	7.55506	0.259136
$851$	3.84450	0.131788
$852$	0	0
$853$	−50.3863	−1.72519	−0.862597	−	0.505892i	$-0.831164\pi$
$853$	−50.3863	−1.72519	−0.862597	+	0.505892i	$0.831164\pi$
$854$	−10.8581	−0.371556
$855$	0	0
$856$	31.3992	1.07320
$857$	−35.4435	−1.21073	−0.605363	−	0.795949i	$-0.706972\pi$
$857$	−35.4435	−1.21073	−0.605363	+	0.795949i	$0.706972\pi$
$858$	0	0
$859$	21.8839	0.746668	0.373334	−	0.927697i	$-0.378215\pi$
$859$	21.8839	0.746668	0.373334	+	0.927697i	$0.378215\pi$
$860$	32.6693	1.11402
$861$	0	0
$862$	−32.0140	−1.09040
$863$	−46.3182	−1.57669	−0.788345	−	0.615233i	$-0.789062\pi$
$863$	−46.3182	−1.57669	−0.788345	+	0.615233i	$0.789062\pi$
$864$	0	0
$865$	−15.2464	−0.518395
$866$	−30.1888	−1.02586
$867$	0	0
$868$	126.181	4.28287
$869$	0.300992	0.0102105
$870$	0	0
$871$	1.73154	0.0586708
$872$	11.3696	0.385024
$873$	0	0
$874$	3.50816	0.118665
$875$	−3.65786	−0.123658
$876$	0	0
$877$	−43.8257	−1.47989	−0.739945	−	0.672668i	$-0.765148\pi$
$877$	−43.8257	−1.47989	−0.739945	+	0.672668i	$0.765148\pi$
$878$	12.4974	0.421765
$879$	0	0
$880$	−1.65541	−0.0558038
$881$	22.9529	0.773304	0.386652	−	0.922226i	$-0.373631\pi$
$881$	22.9529	0.773304	0.386652	+	0.922226i	$0.373631\pi$
$882$	0	0
$883$	−21.0194	−0.707359	−0.353680	−	0.935367i	$-0.615070\pi$
$883$	−21.0194	−0.707359	−0.353680	+	0.935367i	$0.615070\pi$
$884$	11.4709	0.385810
$885$	0	0
$886$	−52.1188	−1.75097
$887$	52.7321	1.77057	0.885285	−	0.465049i	$-0.153963\pi$
$887$	52.7321	1.77057	0.885285	+	0.465049i	$0.153963\pi$
$888$	0	0
$889$	30.5002	1.02294
$890$	−27.3977	−0.918373
$891$	0	0
$892$	−48.5561	−1.62578
$893$	1.59735	0.0534533
$894$	0	0
$895$	−3.84148	−0.128407
$896$	75.2399	2.51359
$897$	0	0
$898$	19.2125	0.641131
$899$	−53.7155	−1.79151
$900$	0	0
$901$	21.8049	0.726426
$902$	−2.16115	−0.0719583
$903$	0	0
$904$	−12.9852	−0.431880
$905$	14.7516	0.490359
$906$	0	0
$907$	−57.4118	−1.90633	−0.953164	−	0.302453i	$-0.902194\pi$
$907$	−57.4118	−1.90633	−0.953164	+	0.302453i	$0.902194\pi$
$908$	−31.6431	−1.05011
$909$	0	0
$910$	−8.64808	−0.286681
$911$	−24.0357	−0.796337	−0.398168	−	0.917312i	$-0.630354\pi$
$911$	−24.0357	−0.796337	−0.398168	+	0.917312i	$0.630354\pi$
$912$	0	0
$913$	−2.13913	−0.0707949
$914$	53.1249	1.75721
$915$	0	0
$916$	−54.4387	−1.79871
$917$	70.1022	2.31498
$918$	0	0
$919$	3.74118	0.123410	0.0617050	−	0.998094i	$-0.480346\pi$
$919$	3.74118	0.123410	0.0617050	+	0.998094i	$0.480346\pi$
$920$	1.99551	0.0657901
$921$	0	0
$922$	−10.3980	−0.342441
$923$	7.29133	0.239997
$924$	0	0
$925$	7.24079	0.238076
$926$	58.0722	1.90837
$927$	0	0
$928$	−19.4656	−0.638989
$929$	36.7692	1.20636	0.603179	−	0.797606i	$-0.293900\pi$
$929$	36.7692	1.20636	0.603179	+	0.797606i	$0.293900\pi$
$930$	0	0
$931$	−17.8299	−0.584350
$932$	−104.435	−3.42089
$933$	0	0
$934$	92.6502	3.03161
$935$	3.10007	0.101383
$936$	0	0
$937$	19.8684	0.649073	0.324537	−	0.945873i	$-0.394792\pi$
$937$	19.8684	0.649073	0.324537	+	0.945873i	$0.394792\pi$
$938$	14.9745	0.488934
$939$	0	0
$940$	2.05174	0.0669204
$941$	47.1179	1.53600	0.768000	−	0.640450i	$-0.221252\pi$
$941$	47.1179	1.53600	0.768000	+	0.640450i	$0.221252\pi$
$942$	0	0
$943$	0.500286	0.0162916
$944$	−4.69927	−0.152948
$945$	0	0
$946$	20.8740	0.678672
$947$	0.760752	0.0247211	0.0123606	−	0.999924i	$-0.496065\pi$
$947$	0.760752	0.0247211	0.0123606	+	0.999924i	$0.496065\pi$
$948$	0	0
$949$	16.0855	0.522156
$950$	6.60732	0.214370
$951$	0	0
$952$	43.9311	1.42381
$953$	−25.3889	−0.822428	−0.411214	−	0.911539i	$-0.634895\pi$
$953$	−25.3889	−0.822428	−0.411214	+	0.911539i	$0.634895\pi$
$954$	0	0
$955$	−7.79112	−0.252115
$956$	37.1590	1.20181
$957$	0	0
$958$	24.3181	0.785682
$959$	−52.3575	−1.69071
$960$	0	0
$961$	61.3478	1.97896
$962$	17.1190	0.551940
$963$	0	0
$964$	73.3560	2.36264
$965$	20.6931	0.666134
$966$	0	0
$967$	−13.9709	−0.449275	−0.224637	−	0.974442i	$-0.572120\pi$
$967$	−13.9709	−0.449275	−0.224637	+	0.974442i	$0.572120\pi$
$968$	37.8050	1.21510
$969$	0	0
$970$	−31.6691	−1.01683
$971$	−57.9564	−1.85991	−0.929954	−	0.367675i	$-0.880154\pi$
$971$	−57.9564	−1.85991	−0.929954	+	0.367675i	$0.880154\pi$
$972$	0	0
$973$	59.9930	1.92329
$974$	−93.5060	−2.99612
$975$	0	0
$976$	2.14245	0.0685783
$977$	−40.1238	−1.28367	−0.641837	−	0.766841i	$-0.721828\pi$
$977$	−40.1238	−1.28367	−0.641837	+	0.766841i	$0.721828\pi$
$978$	0	0
$979$	−11.2421	−0.359299
$980$	−22.9018	−0.731572
$981$	0	0
$982$	−70.5072	−2.24997
$983$	−21.6907	−0.691826	−0.345913	−	0.938267i	$-0.612431\pi$
$983$	−21.6907	−0.691826	−0.345913	+	0.938267i	$0.612431\pi$
$984$	0	0
$985$	−14.7501	−0.469976
$986$	−42.2303	−1.34489
$987$	0	0
$988$	10.0320	0.319160
$989$	−4.83215	−0.153653
$990$	0	0
$991$	41.9890	1.33382	0.666912	−	0.745137i	$-0.267616\pi$
$991$	41.9890	1.33382	0.666912	+	0.745137i	$0.267616\pi$
$992$	33.4653	1.06252
$993$	0	0
$994$	63.0560	2.00002
$995$	−13.9033	−0.440764
$996$	0	0
$997$	−16.5928	−0.525500	−0.262750	−	0.964864i	$-0.584629\pi$
$997$	−16.5928	−0.525500	−0.262750	+	0.964864i	$0.584629\pi$
$998$	48.4483	1.53360
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5265.2.a.w.1.2	✓	6
3.2	odd	2		5265.2.a.x.1.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5265.2.a.w.1.2	✓	6	1.1	even	1	trivial
5265.2.a.x.1.5	yes	6	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5265 = 3^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5265.a (trivial)

\(f(q)\)	\(=\)	\(q-2.36425 q^{2} +3.58967 q^{4} -1.00000 q^{5} +3.65786 q^{7} -3.75837 q^{8} +O(q^{10})\)	\(q-2.36425 q^{2} +3.58967 q^{4} -1.00000 q^{5} +3.65786 q^{7} -3.75837 q^{8} +2.36425 q^{10} +0.970122 q^{11} -1.00000 q^{13} -8.64808 q^{14} +1.70639 q^{16} -3.19554 q^{17} -2.79468 q^{19} -3.58967 q^{20} -2.29361 q^{22} +0.530951 q^{23} +1.00000 q^{25} +2.36425 q^{26} +13.1305 q^{28} -5.58967 q^{29} +9.60977 q^{31} +3.48242 q^{32} +7.55506 q^{34} -3.65786 q^{35} +7.24079 q^{37} +6.60732 q^{38} +3.75837 q^{40} +0.942246 q^{41} -9.10093 q^{43} +3.48242 q^{44} -1.25530 q^{46} -0.571568 q^{47} +6.37992 q^{49} -2.36425 q^{50} -3.58967 q^{52} -6.82352 q^{53} -0.970122 q^{55} -13.7476 q^{56} +13.2154 q^{58} -2.75392 q^{59} +1.25555 q^{61} -22.7199 q^{62} -11.6461 q^{64} +1.00000 q^{65} -1.73154 q^{67} -11.4709 q^{68} +8.64808 q^{70} -7.29133 q^{71} -16.0855 q^{73} -17.1190 q^{74} -10.0320 q^{76} +3.54857 q^{77} +0.310262 q^{79} -1.70639 q^{80} -2.22770 q^{82} -2.20501 q^{83} +3.19554 q^{85} +21.5169 q^{86} -3.64608 q^{88} -11.5883 q^{89} -3.65786 q^{91} +1.90594 q^{92} +1.35133 q^{94} +2.79468 q^{95} -13.3950 q^{97} -15.0837 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} + 8 q^{4} - 6 q^{5} + 4 q^{7} - 9 q^{8}+O(q^{10}) \) 6 * q - 2 * q^2 + 8 * q^4 - 6 * q^5 + 4 * q^7 - 9 * q^8	\( 6 q - 2 q^{2} + 8 q^{4} - 6 q^{5} + 4 q^{7} - 9 q^{8} + 2 q^{10} - 5 q^{11} - 6 q^{13} + 7 q^{14} + 16 q^{16} - 7 q^{17} + 12 q^{19} - 8 q^{20} - 8 q^{22} - 9 q^{23} + 6 q^{25} + 2 q^{26} - 16 q^{28} - 20 q^{29} + 14 q^{31} - 15 q^{32} + 3 q^{34} - 4 q^{35} - 6 q^{37} - 2 q^{38} + 9 q^{40} - 16 q^{41} + 7 q^{43} - 15 q^{44} - 17 q^{46} - 20 q^{47} + 22 q^{49} - 2 q^{50} - 8 q^{52} + 5 q^{55} + 13 q^{56} + 17 q^{58} - 5 q^{59} + 16 q^{61} - 8 q^{62} - 11 q^{64} + 6 q^{65} - 9 q^{67} - 36 q^{68} - 7 q^{70} - 18 q^{71} + q^{73} - 11 q^{74} + 39 q^{76} - q^{77} + 9 q^{79} - 16 q^{80} - 32 q^{82} + 2 q^{83} + 7 q^{85} - 8 q^{86} + 37 q^{88} + 11 q^{89} - 4 q^{91} + 5 q^{92} + 42 q^{94} - 12 q^{95} - 11 q^{97} - 55 q^{98}+O(q^{100}) \) 6 * q - 2 * q^2 + 8 * q^4 - 6 * q^5 + 4 * q^7 - 9 * q^8 + 2 * q^10 - 5 * q^11 - 6 * q^13 + 7 * q^14 + 16 * q^16 - 7 * q^17 + 12 * q^19 - 8 * q^20 - 8 * q^22 - 9 * q^23 + 6 * q^25 + 2 * q^26 - 16 * q^28 - 20 * q^29 + 14 * q^31 - 15 * q^32 + 3 * q^34 - 4 * q^35 - 6 * q^37 - 2 * q^38 + 9 * q^40 - 16 * q^41 + 7 * q^43 - 15 * q^44 - 17 * q^46 - 20 * q^47 + 22 * q^49 - 2 * q^50 - 8 * q^52 + 5 * q^55 + 13 * q^56 + 17 * q^58 - 5 * q^59 + 16 * q^61 - 8 * q^62 - 11 * q^64 + 6 * q^65 - 9 * q^67 - 36 * q^68 - 7 * q^70 - 18 * q^71 + q^73 - 11 * q^74 + 39 * q^76 - q^77 + 9 * q^79 - 16 * q^80 - 32 * q^82 + 2 * q^83 + 7 * q^85 - 8 * q^86 + 37 * q^88 + 11 * q^89 - 4 * q^91 + 5 * q^92 + 42 * q^94 - 12 * q^95 - 11 * q^97 - 55 * q^98

Introduction

L-functions

Modular forms

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Database

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