LMFDB - Embedded newform 5265.2.a.x.1.5

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:	$6$
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			1.5
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	$-0.184953\pi$
$2$	2.36425	1.67178	0.835888	+	0.548900i	$0.184953\pi$
$3$	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	$-0.546721\pi$
$11$	−0.970122	−0.292503	−0.146251	+	0.989247i	$0.546721\pi$
$12$	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	$-0.373333\pi$
$17$	3.19554	0.775033	0.387517	+	0.921863i	$0.373333\pi$
$18$	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	$-0.517629\pi$
$23$	−0.530951	−0.110711	−0.0553554	+	0.998467i	$0.517629\pi$
$24$	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	$-0.326309\pi$
$29$	5.58967	1.03798	0.518988	+	0.854782i	$0.326309\pi$
$30$	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	$-0.523441\pi$
$41$	−0.942246	−0.147154	−0.0735771	+	0.997290i	$0.523441\pi$
$42$	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	$-0.486727\pi$
$47$	0.571568	0.0833717	0.0416859	+	0.999131i	$0.486727\pi$
$48$	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	$-0.344744\pi$
$53$	6.82352	0.937283	0.468642	+	0.883388i	$0.344744\pi$
$54$	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	$-0.442628\pi$
$59$	2.75392	0.358530	0.179265	+	0.983801i	$0.442628\pi$
$60$	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	$-0.357575\pi$
$71$	7.29133	0.865321	0.432661	+	0.901557i	$0.357575\pi$
$72$	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	$-0.461385\pi$
$83$	2.20501	0.242032	0.121016	+	0.992651i	$0.461385\pi$
$84$	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	$-0.289487\pi$
$89$	11.5883	1.22836	0.614180	+	0.789166i	$0.289487\pi$
$90$	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	$-0.652751\pi$
$101$	−9.27955	−0.923350	−0.461675	+	0.887049i	$0.652751\pi$
$102$	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	$-0.367679\pi$
$107$	8.35447	0.807658	0.403829	+	0.914835i	$0.367679\pi$
$108$	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	$-0.551959\pi$
$113$	−3.45499	−0.325018	−0.162509	+	0.986707i	$0.551959\pi$
$114$	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	$-0.815820\pi$
$131$	−19.1648	−1.67444	−0.837220	+	0.546867i	$0.815820\pi$
$132$	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	$-0.290586\pi$
$137$	14.3137	1.22290	0.611452	+	0.791282i	$0.290586\pi$
$138$	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	$-0.820661\pi$
$149$	−20.6398	−1.69088	−0.845439	+	0.534073i	$0.820661\pi$
$150$	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	$-0.459203\pi$
$167$	3.30353	0.255635	0.127817	+	0.991798i	$0.459203\pi$
$168$	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	$-0.696785\pi$
$173$	−15.2464	−1.15917	−0.579583	+	0.814913i	$0.696785\pi$
$174$	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	$-0.545856\pi$
$179$	−3.84148	−0.287126	−0.143563	+	0.989641i	$0.545856\pi$
$180$	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	$-0.590956\pi$
$191$	−7.79112	−0.563745	−0.281873	+	0.959452i	$0.590956\pi$
$192$	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	$-0.676103\pi$
$197$	−14.7501	−1.05090	−0.525449	+	0.850825i	$0.676103\pi$
$198$	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	$-0.405500\pi$
$227$	8.81503	0.585074	0.292537	+	0.956254i	$0.405500\pi$
$228$	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	$-0.0979968\pi$
$233$	29.0933	1.90596	0.952982	+	0.303026i	$0.0979968\pi$
$234$	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	$-0.608667\pi$
$239$	−10.3516	−0.669592	−0.334796	+	0.942291i	$0.608667\pi$
$240$	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	$-0.427699\pi$
$251$	7.13541	0.450383	0.225192	+	0.974315i	$0.427699\pi$
$252$	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	$-0.395804\pi$
$257$	10.3089	0.643053	0.321526	+	0.946901i	$0.395804\pi$
$258$	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	$-0.390836\pi$
$263$	10.9067	0.672534	0.336267	+	0.941767i	$0.390836\pi$
$264$	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	$-0.658352\pi$
$269$	−15.6537	−0.954423	−0.477212	+	0.878788i	$0.658352\pi$
$270$	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	$-0.360706\pi$
$281$	14.2074	0.847540	0.423770	+	0.905770i	$0.360706\pi$
$282$	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	$-0.459614\pi$
$293$	4.33188	0.253071	0.126536	+	0.991962i	$0.459614\pi$
$294$	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	$-0.774331\pi$
$311$	−26.7716	−1.51808	−0.759040	+	0.651045i	$0.774331\pi$
$312$	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	$-0.244182\pi$
$317$	25.6353	1.43982	0.719912	+	0.694066i	$0.244182\pi$
$318$	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	$-0.676396\pi$
$347$	−19.6052	−1.05246	−0.526232	+	0.850341i	$0.676396\pi$
$348$	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	$-0.0938699\pi$
$353$	35.9544	1.91366	0.956831	+	0.290645i	$0.0938699\pi$
$354$	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	$-0.719930\pi$
$359$	−24.1485	−1.27451	−0.637255	+	0.770653i	$0.719930\pi$
$360$	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	$-0.618893\pi$
$383$	−14.2820	−0.729776	−0.364888	+	0.931051i	$0.618893\pi$
$384$	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	$-0.409312\pi$
$389$	11.0870	0.562135	0.281067	+	0.959688i	$0.409312\pi$
$390$	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	$-0.332712\pi$
$401$	20.0926	1.00338	0.501689	+	0.865048i	$0.332712\pi$
$402$	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	$-0.358912\pi$
$419$	17.5575	0.857740	0.428870	+	0.903366i	$0.358912\pi$
$420$	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	$-0.605742\pi$
$431$	−13.5409	−0.652242	−0.326121	+	0.945328i	$0.605742\pi$
$432$	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	$-0.675443\pi$
$443$	−22.0446	−1.04737	−0.523684	+	0.851912i	$0.675443\pi$
$444$	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	$-0.438583\pi$
$449$	8.12628	0.383503	0.191751	+	0.981444i	$0.438583\pi$
$450$	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	$-0.532658\pi$
$461$	−4.39803	−0.204837	−0.102418	+	0.994741i	$0.532658\pi$
$462$	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	$-0.138591\pi$
$467$	39.1880	1.81340	0.906702	+	0.421771i	$0.138591\pi$
$468$	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	$-0.424496\pi$
$479$	10.2858	0.469968	0.234984	+	0.971999i	$0.424496\pi$
$480$	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	$-0.734964\pi$
$491$	−29.8222	−1.34586	−0.672929	+	0.739707i	$0.734964\pi$
$492$	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	$-0.470884\pi$
$503$	4.09721	0.182686	0.0913428	+	0.995820i	$0.470884\pi$
$504$	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	$-0.340788\pi$
$509$	21.6398	0.959167	0.479584	+	0.877496i	$0.340788\pi$
$510$	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	$-0.509373\pi$
$521$	−1.34401	−0.0588820	−0.0294410	+	0.999567i	$0.509373\pi$
$522$	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	$-0.767627\pi$
$557$	−35.1728	−1.49032	−0.745159	+	0.666886i	$0.767627\pi$
$558$	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	$-0.495339\pi$
$563$	0.694887	0.0292860	0.0146430	+	0.999893i	$0.495339\pi$
$564$	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	$-0.512262\pi$
$569$	−1.83728	−0.0770227	−0.0385113	+	0.999258i	$0.512262\pi$
$570$	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	$-0.659276\pi$
$587$	−23.2474	−0.959521	−0.479761	+	0.877399i	$0.659276\pi$
$588$	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	$-0.279783\pi$
$593$	31.0702	1.27590	0.637949	+	0.770078i	$0.279783\pi$
$594$	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	$-0.181442\pi$
$599$	41.2097	1.68378	0.841892	+	0.539646i	$0.181442\pi$
$600$	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	$-0.823100\pi$
$617$	−42.2026	−1.69901	−0.849507	+	0.527578i	$0.823100\pi$
$618$	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	$-0.440703\pi$
$641$	9.37834	0.370422	0.185211	+	0.982699i	$0.440703\pi$
$642$	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	$-0.468223\pi$
$647$	5.07011	0.199327	0.0996633	+	0.995021i	$0.468223\pi$
$648$	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	$-0.556005\pi$
$653$	−8.94581	−0.350077	−0.175038	+	0.984562i	$0.556005\pi$
$654$	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	$-0.732166\pi$
$659$	−34.2143	−1.33280	−0.666400	+	0.745595i	$0.732166\pi$
$660$	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	$-0.688873\pi$
$677$	−29.0974	−1.11831	−0.559153	+	0.829065i	$0.688873\pi$
$678$	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	$-0.354178\pi$
$683$	23.1161	0.884512	0.442256	+	0.896889i	$0.354178\pi$
$684$	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	$-0.304466\pi$
$701$	30.5208	1.15276	0.576378	+	0.817183i	$0.304466\pi$
$702$	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	$-0.299408\pi$
$719$	31.6026	1.17858	0.589290	+	0.807922i	$0.299408\pi$
$720$	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	$-0.460259\pi$
$743$	6.78861	0.249050	0.124525	+	0.992216i	$0.460259\pi$
$744$	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	$-0.358003\pi$
$761$	23.8040	0.862894	0.431447	+	0.902138i	$0.358003\pi$
$762$	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	$-0.564005\pi$
$773$	−11.1058	−0.399448	−0.199724	+	0.979852i	$0.564005\pi$
$774$	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	$-0.592720\pi$
$797$	−16.2153	−0.574376	−0.287188	+	0.957874i	$0.592720\pi$
$798$	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	$-0.328641\pi$
$809$	29.1661	1.02542	0.512712	+	0.858561i	$0.328641\pi$
$810$	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	$-0.693837\pi$
$821$	−32.7799	−1.14403	−0.572013	+	0.820245i	$0.693837\pi$
$822$	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	$-0.667769\pi$
$827$	−28.9299	−1.00599	−0.502995	+	0.864289i	$0.667769\pi$
$828$	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	$-0.163412\pi$
$839$	50.4633	1.74218	0.871092	+	0.491119i	$0.163412\pi$
$840$	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	$-0.293028\pi$
$857$	35.4435	1.21073	0.605363	+	0.795949i	$0.293028\pi$
$858$	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	$-0.210938\pi$
$863$	46.3182	1.57669	0.788345	+	0.615233i	$0.210938\pi$
$864$	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	$-0.626369\pi$
$881$	−22.9529	−0.773304	−0.386652	+	0.922226i	$0.626369\pi$
$882$	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	$-0.846037\pi$
$887$	−52.7321	−1.77057	−0.885285	+	0.465049i	$0.846037\pi$
$888$	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	$-0.369646\pi$
$911$	24.0357	0.796337	0.398168	+	0.917312i	$0.369646\pi$
$912$	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	$-0.706100\pi$
$929$	−36.7692	−1.20636	−0.603179	+	0.797606i	$0.706100\pi$
$930$	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	$-0.778748\pi$
$941$	−47.1179	−1.53600	−0.768000	+	0.640450i	$0.778748\pi$
$942$	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	$-0.503935\pi$
$947$	−0.760752	−0.0247211	−0.0123606	+	0.999924i	$0.503935\pi$
$948$	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	$-0.365105\pi$
$953$	25.3889	0.822428	0.411214	+	0.911539i	$0.365105\pi$
$954$	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	$-0.119846\pi$
$971$	57.9564	1.85991	0.929954	+	0.367675i	$0.119846\pi$
$972$	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	$-0.278172\pi$
$977$	40.1238	1.28367	0.641837	+	0.766841i	$0.278172\pi$
$978$	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	$-0.387569\pi$
$983$	21.6907	0.691826	0.345913	+	0.938267i	$0.387569\pi$
$984$	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.x.1.5	yes	6
3.2	odd	2		5265.2.a.w.1.2	✓	6

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

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

Varieties

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Representations

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Database

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