LMFDB - Embedded newform 1521.4.a.bj.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 48x^{7} + 29x^{6} + 772x^{5} - 150x^{4} - 4745x^{3} - 966x^{2} + 9428x + 5144 $ x^9 - x^8 - 48x^7 + 29x^6 + 772x^5 - 150x^4 - 4745x^3 - 966x^2 + 9428*x + 5144
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	no (minimal twist has level 507)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.73419$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73419 q^{2} -0.524213 q^{4} +21.1246 q^{5} -25.8618 q^{7} -23.3068 q^{8} +O(q^{10})$	$q+2.73419 q^{2} -0.524213 q^{4} +21.1246 q^{5} -25.8618 q^{7} -23.3068 q^{8} +57.7586 q^{10} +6.96892 q^{11} -70.7111 q^{14} -59.5315 q^{16} -122.879 q^{17} +43.1340 q^{19} -11.0738 q^{20} +19.0543 q^{22} +75.5492 q^{23} +321.248 q^{25} +13.5571 q^{28} +163.764 q^{29} +139.421 q^{31} +23.6841 q^{32} -335.975 q^{34} -546.320 q^{35} +2.80616 q^{37} +117.936 q^{38} -492.347 q^{40} +300.555 q^{41} +363.145 q^{43} -3.65320 q^{44} +206.566 q^{46} +41.2660 q^{47} +325.834 q^{49} +878.354 q^{50} +125.763 q^{53} +147.216 q^{55} +602.756 q^{56} +447.762 q^{58} +407.311 q^{59} +536.710 q^{61} +381.204 q^{62} +541.009 q^{64} +340.155 q^{67} +64.4150 q^{68} -1493.74 q^{70} -514.831 q^{71} -491.231 q^{73} +7.67257 q^{74} -22.6114 q^{76} -180.229 q^{77} +762.869 q^{79} -1257.58 q^{80} +821.774 q^{82} -345.948 q^{83} -2595.78 q^{85} +992.907 q^{86} -162.423 q^{88} -362.482 q^{89} -39.6039 q^{92} +112.829 q^{94} +911.188 q^{95} -276.297 q^{97} +890.890 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 8 q^{2} + 32 q^{4} + 41 q^{5} - q^{7} + 111 q^{8}+O(q^{10}) $ 9 * q + 8 * q^2 + 32 * q^4 + 41 * q^5 - q^7 + 111 * q^8	$ 9 q + 8 q^{2} + 32 q^{4} + 41 q^{5} - q^{7} + 111 q^{8} + 198 q^{10} + 37 q^{11} - 98 q^{14} + 32 q^{16} + 134 q^{17} + 72 q^{19} + 356 q^{20} + 274 q^{22} - 226 q^{23} + 612 q^{25} - 132 q^{28} + 547 q^{29} + 521 q^{31} + 721 q^{32} + 100 q^{34} - 138 q^{35} - 584 q^{37} + 416 q^{38} + 1342 q^{40} + 482 q^{41} + 158 q^{43} + 1453 q^{44} - 1537 q^{46} + 1500 q^{47} + 642 q^{49} + 2777 q^{50} - 1399 q^{53} - 1408 q^{55} + 616 q^{56} - 1455 q^{58} + 1541 q^{59} + 2092 q^{61} + 293 q^{62} + 2481 q^{64} - 252 q^{67} + 1579 q^{68} - 2492 q^{70} + 2352 q^{71} - 903 q^{73} - 1037 q^{74} + 485 q^{76} + 1686 q^{77} - 115 q^{79} + 5701 q^{80} - 5147 q^{82} + 1207 q^{83} - 4308 q^{85} + 5691 q^{86} - 484 q^{88} + 2336 q^{89} - 2087 q^{92} - 468 q^{94} + 222 q^{95} - 2155 q^{97} + 5593 q^{98}+O(q^{100}) $ 9 * q + 8 * q^2 + 32 * q^4 + 41 * q^5 - q^7 + 111 * q^8 + 198 * q^10 + 37 * q^11 - 98 * q^14 + 32 * q^16 + 134 * q^17 + 72 * q^19 + 356 * q^20 + 274 * q^22 - 226 * q^23 + 612 * q^25 - 132 * q^28 + 547 * q^29 + 521 * q^31 + 721 * q^32 + 100 * q^34 - 138 * q^35 - 584 * q^37 + 416 * q^38 + 1342 * q^40 + 482 * q^41 + 158 * q^43 + 1453 * q^44 - 1537 * q^46 + 1500 * q^47 + 642 * q^49 + 2777 * q^50 - 1399 * q^53 - 1408 * q^55 + 616 * q^56 - 1455 * q^58 + 1541 * q^59 + 2092 * q^61 + 293 * q^62 + 2481 * q^64 - 252 * q^67 + 1579 * q^68 - 2492 * q^70 + 2352 * q^71 - 903 * q^73 - 1037 * q^74 + 485 * q^76 + 1686 * q^77 - 115 * q^79 + 5701 * q^80 - 5147 * q^82 + 1207 * q^83 - 4308 * q^85 + 5691 * q^86 - 484 * q^88 + 2336 * q^89 - 2087 * q^92 - 468 * q^94 + 222 * q^95 - 2155 * q^97 + 5593 * 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 $

$2$	2.73419	0.966682	0.483341	−	0.875432i	$-0.339423\pi$
$2$	2.73419	0.966682	0.483341	+	0.875432i	$0.339423\pi$
$3$	0	0
$3$	0	0
$4$	−0.524213	−0.0655266
$5$	21.1246	1.88944	0.944721	−	0.327877i	$-0.106333\pi$
$5$	21.1246	1.88944	0.944721	+	0.327877i	$0.106333\pi$
$6$	0	0
$7$	−25.8618	−1.39641	−0.698203	−	0.715899i	$-0.746017\pi$
$7$	−25.8618	−1.39641	−0.698203	+	0.715899i	$0.746017\pi$
$8$	−23.3068	−1.03003
$9$	0	0
$10$	57.7586	1.82649
$11$	6.96892	0.191019	0.0955095	−	0.995429i	$-0.469552\pi$
$11$	6.96892	0.191019	0.0955095	+	0.995429i	$0.469552\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−70.7111	−1.34988
$15$	0	0
$16$	−59.5315	−0.930180
$17$	−122.879	−1.75310	−0.876548	−	0.481315i	$-0.840159\pi$
$17$	−122.879	−1.75310	−0.876548	+	0.481315i	$0.840159\pi$
$18$	0	0
$19$	43.1340	0.520822	0.260411	−	0.965498i	$-0.416142\pi$
$19$	43.1340	0.520822	0.260411	+	0.965498i	$0.416142\pi$
$20$	−11.0738	−0.123809
$21$	0	0
$22$	19.0543	0.184655
$23$	75.5492	0.684918	0.342459	−	0.939533i	$-0.388740\pi$
$23$	75.5492	0.684918	0.342459	+	0.939533i	$0.388740\pi$
$24$	0	0
$25$	321.248	2.56999
$26$	0	0
$27$	0	0
$28$	13.5571	0.0915018
$29$	163.764	1.04863	0.524314	−	0.851525i	$-0.324322\pi$
$29$	163.764	1.04863	0.524314	+	0.851525i	$0.324322\pi$
$30$	0	0
$31$	139.421	0.807767	0.403884	−	0.914810i	$-0.367660\pi$
$31$	139.421	0.807767	0.403884	+	0.914810i	$0.367660\pi$
$32$	23.6841	0.130837
$33$	0	0
$34$	−335.975	−1.69469
$35$	−546.320	−2.63843
$36$	0	0
$37$	2.80616	0.0124684	0.00623419	−	0.999981i	$-0.498016\pi$
$37$	2.80616	0.0124684	0.00623419	+	0.999981i	$0.498016\pi$
$38$	117.936	0.503469
$39$	0	0
$40$	−492.347	−1.94617
$41$	300.555	1.14485	0.572425	−	0.819957i	$-0.306003\pi$
$41$	300.555	1.14485	0.572425	+	0.819957i	$0.306003\pi$
$42$	0	0
$43$	363.145	1.28789	0.643943	−	0.765073i	$-0.277297\pi$
$43$	363.145	1.28789	0.643943	+	0.765073i	$0.277297\pi$
$44$	−3.65320	−0.0125168
$45$	0	0
$46$	206.566	0.662097
$47$	41.2660	0.128070	0.0640348	−	0.997948i	$-0.479603\pi$
$47$	41.2660	0.128070	0.0640348	+	0.997948i	$0.479603\pi$
$48$	0	0
$49$	325.834	0.949952
$50$	878.354	2.48436
$51$	0	0
$52$	0	0
$53$	125.763	0.325941	0.162971	−	0.986631i	$-0.447892\pi$
$53$	125.763	0.325941	0.162971	+	0.986631i	$0.447892\pi$
$54$	0	0
$55$	147.216	0.360919
$56$	602.756	1.43833
$57$	0	0
$58$	447.762	1.01369
$59$	407.311	0.898768	0.449384	−	0.893339i	$-0.351643\pi$
$59$	407.311	0.898768	0.449384	+	0.893339i	$0.351643\pi$
$60$	0	0
$61$	536.710	1.12654	0.563268	−	0.826274i	$-0.309544\pi$
$61$	536.710	1.12654	0.563268	+	0.826274i	$0.309544\pi$
$62$	381.204	0.780854
$63$	0	0
$64$	541.009	1.05666
$65$	0	0
$66$	0	0
$67$	340.155	0.620247	0.310124	−	0.950696i	$-0.399630\pi$
$67$	340.155	0.620247	0.310124	+	0.950696i	$0.399630\pi$
$68$	64.4150	0.114874
$69$	0	0
$70$	−1493.74	−2.55052
$71$	−514.831	−0.860552	−0.430276	−	0.902697i	$-0.641584\pi$
$71$	−514.831	−0.860552	−0.430276	+	0.902697i	$0.641584\pi$
$72$	0	0
$73$	−491.231	−0.787592	−0.393796	−	0.919198i	$-0.628838\pi$
$73$	−491.231	−0.787592	−0.393796	+	0.919198i	$0.628838\pi$
$74$	7.67257	0.0120529
$75$	0	0
$76$	−22.6114	−0.0341277
$77$	−180.229	−0.266740
$78$	0	0
$79$	762.869	1.08645	0.543225	−	0.839587i	$-0.317203\pi$
$79$	762.869	1.08645	0.543225	+	0.839587i	$0.317203\pi$
$80$	−1257.58	−1.75752
$81$	0	0
$82$	821.774	1.10670
$83$	−345.948	−0.457503	−0.228752	−	0.973485i	$-0.573464\pi$
$83$	−345.948	−0.457503	−0.228752	+	0.973485i	$0.573464\pi$
$84$	0	0
$85$	−2595.78	−3.31237
$86$	992.907	1.24498
$87$	0	0
$88$	−162.423	−0.196754
$89$	−362.482	−0.431720	−0.215860	−	0.976424i	$-0.569255\pi$
$89$	−362.482	−0.431720	−0.215860	+	0.976424i	$0.569255\pi$
$90$	0	0
$91$	0	0
$92$	−39.6039	−0.0448803
$93$	0	0
$94$	112.829	0.123803
$95$	911.188	0.984062
$96$	0	0
$97$	−276.297	−0.289213	−0.144607	−	0.989489i	$-0.546192\pi$
$97$	−276.297	−0.289213	−0.144607	+	0.989489i	$0.546192\pi$
$98$	890.890	0.918301
$99$	0	0
$100$	−168.403	−0.168403
$101$	−313.657	−0.309010	−0.154505	−	0.987992i	$-0.549378\pi$
$101$	−313.657	−0.309010	−0.154505	+	0.987992i	$0.549378\pi$
$102$	0	0
$103$	507.209	0.485212	0.242606	−	0.970125i	$-0.421998\pi$
$103$	507.209	0.485212	0.242606	+	0.970125i	$0.421998\pi$
$104$	0	0
$105$	0	0
$106$	343.860	0.315082
$107$	1262.50	1.14066	0.570330	−	0.821416i	$-0.306815\pi$
$107$	1262.50	1.14066	0.570330	+	0.821416i	$0.306815\pi$
$108$	0	0
$109$	−469.356	−0.412442	−0.206221	−	0.978505i	$-0.566117\pi$
$109$	−469.356	−0.412442	−0.206221	+	0.978505i	$0.566117\pi$
$110$	402.515	0.348894
$111$	0	0
$112$	1539.59	1.29891
$113$	1110.69	0.924649	0.462325	−	0.886711i	$-0.347015\pi$
$113$	1110.69	0.924649	0.462325	+	0.886711i	$0.347015\pi$
$114$	0	0
$115$	1595.95	1.29411
$116$	−85.8472	−0.0687131
$117$	0	0
$118$	1113.66	0.868823
$119$	3177.88	2.44803
$120$	0	0
$121$	−1282.43	−0.963512
$122$	1467.47	1.08900
$123$	0	0
$124$	−73.0864	−0.0529303
$125$	4145.67	2.96640
$126$	0	0
$127$	−2547.15	−1.77971	−0.889853	−	0.456248i	$-0.849193\pi$
$127$	−2547.15	−1.77971	−0.889853	+	0.456248i	$0.849193\pi$
$128$	1289.75	0.890614
$129$	0	0
$130$	0	0
$131$	701.027	0.467550	0.233775	−	0.972291i	$-0.424892\pi$
$131$	701.027	0.467550	0.233775	+	0.972291i	$0.424892\pi$
$132$	0	0
$133$	−1115.52	−0.727279
$134$	930.048	0.599581
$135$	0	0
$136$	2863.93	1.80573
$137$	−2576.33	−1.60665	−0.803325	−	0.595541i	$-0.796938\pi$
$137$	−2576.33	−1.60665	−0.803325	+	0.595541i	$0.796938\pi$
$138$	0	0
$139$	233.539	0.142508	0.0712538	−	0.997458i	$-0.477300\pi$
$139$	233.539	0.142508	0.0712538	+	0.997458i	$0.477300\pi$
$140$	286.388	0.172887
$141$	0	0
$142$	−1407.64	−0.831880
$143$	0	0
$144$	0	0
$145$	3459.45	1.98132
$146$	−1343.12	−0.761350
$147$	0	0
$148$	−1.47102	−0.000817010 0
$149$	−1595.97	−0.877495	−0.438748	−	0.898610i	$-0.644578\pi$
$149$	−1595.97	−0.877495	−0.438748	+	0.898610i	$0.644578\pi$
$150$	0	0
$151$	3494.13	1.88310	0.941552	−	0.336869i	$-0.109368\pi$
$151$	3494.13	1.88310	0.941552	+	0.336869i	$0.109368\pi$
$152$	−1005.32	−0.536459
$153$	0	0
$154$	−492.780	−0.257853
$155$	2945.22	1.52623
$156$	0	0
$157$	−2203.71	−1.12022	−0.560112	−	0.828417i	$-0.689242\pi$
$157$	−2203.71	−1.12022	−0.560112	+	0.828417i	$0.689242\pi$
$158$	2085.83	1.05025
$159$	0	0
$160$	500.317	0.247210
$161$	−1953.84	−0.956424
$162$	0	0
$163$	758.697	0.364575	0.182288	−	0.983245i	$-0.441650\pi$
$163$	758.697	0.364575	0.182288	+	0.983245i	$0.441650\pi$
$164$	−157.555	−0.0750181
$165$	0	0
$166$	−945.888	−0.442260
$167$	855.272	0.396305	0.198152	−	0.980171i	$-0.436506\pi$
$167$	855.272	0.396305	0.198152	+	0.980171i	$0.436506\pi$
$168$	0	0
$169$	0	0
$170$	−7097.34	−3.20201
$171$	0	0
$172$	−190.365	−0.0843908
$173$	76.6923	0.0337041	0.0168521	−	0.999858i	$-0.494636\pi$
$173$	76.6923	0.0337041	0.0168521	+	0.999858i	$0.494636\pi$
$174$	0	0
$175$	−8308.07	−3.58875
$176$	−414.870	−0.177682
$177$	0	0
$178$	−991.095	−0.417335
$179$	−3533.09	−1.47528	−0.737642	−	0.675193i	$-0.764061\pi$
$179$	−3533.09	−1.47528	−0.737642	+	0.675193i	$0.764061\pi$
$180$	0	0
$181$	2352.77	0.966189	0.483095	−	0.875568i	$-0.339513\pi$
$181$	2352.77	0.966189	0.483095	+	0.875568i	$0.339513\pi$
$182$	0	0
$183$	0	0
$184$	−1760.81	−0.705482
$185$	59.2790	0.0235583
$186$	0	0
$187$	−856.337	−0.334875
$188$	−21.6322	−0.00839197
$189$	0	0
$190$	2491.36	0.951275
$191$	1532.91	0.580719	0.290360	−	0.956918i	$-0.406225\pi$
$191$	1532.91	0.580719	0.290360	+	0.956918i	$0.406225\pi$
$192$	0	0
$193$	4611.82	1.72003	0.860015	−	0.510269i	$-0.170454\pi$
$193$	4611.82	1.72003	0.860015	+	0.510269i	$0.170454\pi$
$194$	−755.447	−0.279577
$195$	0	0
$196$	−170.806	−0.0622471
$197$	1914.37	0.692353	0.346176	−	0.938169i	$-0.387480\pi$
$197$	1914.37	0.692353	0.346176	+	0.938169i	$0.387480\pi$
$198$	0	0
$199$	2304.94	0.821070	0.410535	−	0.911845i	$-0.365342\pi$
$199$	2304.94	0.821070	0.410535	+	0.911845i	$0.365342\pi$
$200$	−7487.27	−2.64715
$201$	0	0
$202$	−857.596	−0.298714
$203$	−4235.24	−1.46431
$204$	0	0
$205$	6349.10	2.16312
$206$	1386.81	0.469045
$207$	0	0
$208$	0	0
$209$	300.597	0.0994868
$210$	0	0
$211$	−3500.56	−1.14213	−0.571064	−	0.820906i	$-0.693469\pi$
$211$	−3500.56	−1.14213	−0.571064	+	0.820906i	$0.693469\pi$
$212$	−65.9266	−0.0213578
$213$	0	0
$214$	3451.92	1.10266
$215$	7671.29	2.43339
$216$	0	0
$217$	−3605.69	−1.12797
$218$	−1283.31	−0.398700
$219$	0	0
$220$	−77.1723	−0.0236498
$221$	0	0
$222$	0	0
$223$	1255.99	0.377163	0.188582	−	0.982058i	$-0.439611\pi$
$223$	1255.99	0.377163	0.188582	+	0.982058i	$0.439611\pi$
$224$	−612.514	−0.182702
$225$	0	0
$226$	3036.85	0.893842
$227$	3408.52	0.996615	0.498307	−	0.867000i	$-0.333955\pi$
$227$	3408.52	0.996615	0.498307	+	0.867000i	$0.333955\pi$
$228$	0	0
$229$	−4135.13	−1.19326	−0.596631	−	0.802515i	$-0.703495\pi$
$229$	−4135.13	−1.19326	−0.596631	+	0.802515i	$0.703495\pi$
$230$	4363.62	1.25099
$231$	0	0
$232$	−3816.82	−1.08011
$233$	−3788.88	−1.06531	−0.532656	−	0.846332i	$-0.678806\pi$
$233$	−3788.88	−1.06531	−0.532656	+	0.846332i	$0.678806\pi$
$234$	0	0
$235$	871.728	0.241980
$236$	−213.518	−0.0588933
$237$	0	0
$238$	8688.93	2.36647
$239$	−1543.75	−0.417810	−0.208905	−	0.977936i	$-0.566990\pi$
$239$	−1543.75	−0.417810	−0.208905	+	0.977936i	$0.566990\pi$
$240$	0	0
$241$	6295.40	1.68266	0.841332	−	0.540518i	$-0.181772\pi$
$241$	6295.40	1.68266	0.841332	+	0.540518i	$0.181772\pi$
$242$	−3506.42	−0.931409
$243$	0	0
$244$	−281.350	−0.0738181
$245$	6883.10	1.79488
$246$	0	0
$247$	0	0
$248$	−3249.46	−0.832021
$249$	0	0
$250$	11335.0	2.86756
$251$	−2241.93	−0.563781	−0.281891	−	0.959447i	$-0.590962\pi$
$251$	−2241.93	−0.563781	−0.281891	+	0.959447i	$0.590962\pi$
$252$	0	0
$253$	526.497	0.130832
$254$	−6964.38	−1.72041
$255$	0	0
$256$	−801.658	−0.195717
$257$	−3603.80	−0.874703	−0.437351	−	0.899291i	$-0.644083\pi$
$257$	−3603.80	−0.874703	−0.437351	+	0.899291i	$0.644083\pi$
$258$	0	0
$259$	−72.5724	−0.0174109
$260$	0	0
$261$	0	0
$262$	1916.74	0.451972
$263$	2131.06	0.499646	0.249823	−	0.968292i	$-0.419628\pi$
$263$	2131.06	0.499646	0.249823	+	0.968292i	$0.419628\pi$
$264$	0	0
$265$	2656.69	0.615847
$266$	−3050.05	−0.703047
$267$	0	0
$268$	−178.314	−0.0406427
$269$	2505.79	0.567958	0.283979	−	0.958830i	$-0.408345\pi$
$269$	2505.79	0.567958	0.283979	+	0.958830i	$0.408345\pi$
$270$	0	0
$271$	−230.220	−0.0516046	−0.0258023	−	0.999667i	$-0.508214\pi$
$271$	−230.220	−0.0516046	−0.0258023	+	0.999667i	$0.508214\pi$
$272$	7315.19	1.63069
$273$	0	0
$274$	−7044.18	−1.55312
$275$	2238.75	0.490916
$276$	0	0
$277$	−7750.38	−1.68114	−0.840569	−	0.541705i	$-0.817779\pi$
$277$	−7750.38	−1.68114	−0.840569	+	0.541705i	$0.817779\pi$
$278$	638.541	0.137760
$279$	0	0
$280$	12733.0	2.71765
$281$	5179.21	1.09952	0.549761	−	0.835322i	$-0.314719\pi$
$281$	5179.21	1.09952	0.549761	+	0.835322i	$0.314719\pi$
$282$	0	0
$283$	3799.93	0.798171	0.399085	−	0.916914i	$-0.369328\pi$
$283$	3799.93	0.798171	0.399085	+	0.916914i	$0.369328\pi$
$284$	269.881	0.0563891
$285$	0	0
$286$	0	0
$287$	−7772.90	−1.59868
$288$	0	0
$289$	10186.3	2.07334
$290$	9458.78	1.91531
$291$	0	0
$292$	257.509	0.0516082
$293$	5840.85	1.16459	0.582297	−	0.812976i	$-0.302154\pi$
$293$	5840.85	1.16459	0.582297	+	0.812976i	$0.302154\pi$
$294$	0	0
$295$	8604.27	1.69817
$296$	−65.4026	−0.0128427
$297$	0	0
$298$	−4363.68	−0.848258
$299$	0	0
$300$	0	0
$301$	−9391.59	−1.79841
$302$	9553.62	1.82036
$303$	0	0
$304$	−2567.83	−0.484458
$305$	11337.8	2.12852
$306$	0	0
$307$	1686.29	0.313490	0.156745	−	0.987639i	$-0.449900\pi$
$307$	1686.29	0.313490	0.156745	+	0.987639i	$0.449900\pi$
$308$	94.4783	0.0174786
$309$	0	0
$310$	8052.78	1.47538
$311$	10601.1	1.93290	0.966449	−	0.256860i	$-0.0826881\pi$
$311$	10601.1	1.93290	0.966449	+	0.256860i	$0.0826881\pi$
$312$	0	0
$313$	−5889.99	−1.06365	−0.531824	−	0.846855i	$-0.678493\pi$
$313$	−5889.99	−1.06365	−0.531824	+	0.846855i	$0.678493\pi$
$314$	−6025.35	−1.08290
$315$	0	0
$316$	−399.906	−0.0711913
$317$	2432.98	0.431071	0.215536	−	0.976496i	$-0.430850\pi$
$317$	2432.98	0.431071	0.215536	+	0.976496i	$0.430850\pi$
$318$	0	0
$319$	1141.26	0.200308
$320$	11428.6	1.99649
$321$	0	0
$322$	−5342.17	−0.924557
$323$	−5300.28	−0.913050
$324$	0	0
$325$	0	0
$326$	2074.42	0.352428
$327$	0	0
$328$	−7004.98	−1.17922
$329$	−1067.21	−0.178837
$330$	0	0
$331$	−757.534	−0.125794	−0.0628971	−	0.998020i	$-0.520034\pi$
$331$	−757.534	−0.125794	−0.0628971	+	0.998020i	$0.520034\pi$
$332$	181.351	0.0299786
$333$	0	0
$334$	2338.47	0.383101
$335$	7185.64	1.17192
$336$	0	0
$337$	−889.307	−0.143750	−0.0718748	−	0.997414i	$-0.522898\pi$
$337$	−889.307	−0.143750	−0.0718748	+	0.997414i	$0.522898\pi$
$338$	0	0
$339$	0	0
$340$	1360.74	0.217048
$341$	971.615	0.154299
$342$	0	0
$343$	443.956	0.0698874
$344$	−8463.75	−1.32655
$345$	0	0
$346$	209.691	0.0325811
$347$	9644.48	1.49205	0.746027	−	0.665916i	$-0.231959\pi$
$347$	9644.48	1.49205	0.746027	+	0.665916i	$0.231959\pi$
$348$	0	0
$349$	1807.89	0.277289	0.138645	−	0.990342i	$-0.455725\pi$
$349$	1807.89	0.277289	0.138645	+	0.990342i	$0.455725\pi$
$350$	−22715.8	−3.46918
$351$	0	0
$352$	165.053	0.0249924
$353$	10681.8	1.61058	0.805289	−	0.592883i	$-0.202010\pi$
$353$	10681.8	1.61058	0.805289	+	0.592883i	$0.202010\pi$
$354$	0	0
$355$	−10875.6	−1.62596
$356$	190.018	0.0282891
$357$	0	0
$358$	−9660.14	−1.42613
$359$	−8195.95	−1.20492	−0.602459	−	0.798149i	$-0.705812\pi$
$359$	−8195.95	−1.20492	−0.602459	+	0.798149i	$0.705812\pi$
$360$	0	0
$361$	−4998.46	−0.728745
$362$	6432.92	0.933997
$363$	0	0
$364$	0	0
$365$	−10377.0	−1.48811
$366$	0	0
$367$	−2555.29	−0.363447	−0.181724	−	0.983350i	$-0.558168\pi$
$367$	−2555.29	−0.363447	−0.181724	+	0.983350i	$0.558168\pi$
$368$	−4497.56	−0.637096
$369$	0	0
$370$	162.080	0.0227733
$371$	−3252.46	−0.455147
$372$	0	0
$373$	−600.757	−0.0833941	−0.0416971	−	0.999130i	$-0.513276\pi$
$373$	−600.757	−0.0833941	−0.0416971	+	0.999130i	$0.513276\pi$
$374$	−2341.39	−0.323717
$375$	0	0
$376$	−961.779	−0.131915
$377$	0	0
$378$	0	0
$379$	863.612	0.117047	0.0585234	−	0.998286i	$-0.481361\pi$
$379$	863.612	0.117047	0.0585234	+	0.998286i	$0.481361\pi$
$380$	−477.656	−0.0644822
$381$	0	0
$382$	4191.26	0.561371
$383$	−9089.38	−1.21265	−0.606326	−	0.795216i	$-0.707357\pi$
$383$	−9089.38	−1.21265	−0.606326	+	0.795216i	$0.707357\pi$
$384$	0	0
$385$	−3807.26	−0.503990
$386$	12609.6	1.66272
$387$	0	0
$388$	144.838	0.0189511
$389$	1513.16	0.197224	0.0986120	−	0.995126i	$-0.468560\pi$
$389$	1513.16	0.197224	0.0986120	+	0.995126i	$0.468560\pi$
$390$	0	0
$391$	−9283.44	−1.20073
$392$	−7594.14	−0.978474
$393$	0	0
$394$	5234.26	0.669285
$395$	16115.3	2.05278
$396$	0	0
$397$	−7572.77	−0.957346	−0.478673	−	0.877993i	$-0.658882\pi$
$397$	−7572.77	−0.957346	−0.478673	+	0.877993i	$0.658882\pi$
$398$	6302.15	0.793714
$399$	0	0
$400$	−19124.4	−2.39055
$401$	−11075.5	−1.37926	−0.689630	−	0.724162i	$-0.742227\pi$
$401$	−11075.5	−1.37926	−0.689630	+	0.724162i	$0.742227\pi$
$402$	0	0
$403$	0	0
$404$	164.423	0.0202484
$405$	0	0
$406$	−11579.9	−1.41552
$407$	19.5559	0.00238170
$408$	0	0
$409$	11082.8	1.33988	0.669940	−	0.742415i	$-0.266320\pi$
$409$	11082.8	1.33988	0.669940	+	0.742415i	$0.266320\pi$
$410$	17359.6	2.09105
$411$	0	0
$412$	−265.886	−0.0317943
$413$	−10533.8	−1.25505
$414$	0	0
$415$	−7308.02	−0.864425
$416$	0	0
$417$	0	0
$418$	821.890	0.0961721
$419$	−13035.6	−1.51988	−0.759940	−	0.649993i	$-0.774772\pi$
$419$	−13035.6	−1.51988	−0.759940	+	0.649993i	$0.774772\pi$
$420$	0	0
$421$	−14664.0	−1.69757	−0.848787	−	0.528735i	$-0.822666\pi$
$421$	−14664.0	−1.69757	−0.848787	+	0.528735i	$0.822666\pi$
$422$	−9571.20	−1.10407
$423$	0	0
$424$	−2931.14	−0.335728
$425$	−39474.8	−4.50543
$426$	0	0
$427$	−13880.3	−1.57310
$428$	−661.820	−0.0747436
$429$	0	0
$430$	20974.8	2.35231
$431$	−1454.87	−0.162596	−0.0812979	−	0.996690i	$-0.525907\pi$
$431$	−1454.87	−0.162596	−0.0812979	+	0.996690i	$0.525907\pi$
$432$	0	0
$433$	−8982.19	−0.996897	−0.498449	−	0.866919i	$-0.666097\pi$
$433$	−8982.19	−0.996897	−0.498449	+	0.866919i	$0.666097\pi$
$434$	−9858.62	−1.09039
$435$	0	0
$436$	246.042	0.0270259
$437$	3258.74	0.356720
$438$	0	0
$439$	3348.05	0.363995	0.181997	−	0.983299i	$-0.441744\pi$
$439$	3348.05	0.363995	0.181997	+	0.983299i	$0.441744\pi$
$440$	−3431.13	−0.371756
$441$	0	0
$442$	0	0
$443$	15671.4	1.68075	0.840375	−	0.542006i	$-0.182335\pi$
$443$	15671.4	1.68075	0.840375	+	0.542006i	$0.182335\pi$
$444$	0	0
$445$	−7657.29	−0.815709
$446$	3434.12	0.364597
$447$	0	0
$448$	−13991.5	−1.47552
$449$	12265.8	1.28922	0.644611	−	0.764511i	$-0.277019\pi$
$449$	12265.8	1.28922	0.644611	+	0.764511i	$0.277019\pi$
$450$	0	0
$451$	2094.54	0.218688
$452$	−582.240	−0.0605891
$453$	0	0
$454$	9319.54	0.963409
$455$	0	0
$456$	0	0
$457$	8382.54	0.858027	0.429014	−	0.903298i	$-0.358861\pi$
$457$	8382.54	0.858027	0.429014	+	0.903298i	$0.358861\pi$
$458$	−11306.2	−1.15351
$459$	0	0
$460$	−836.616	−0.0847987
$461$	2308.36	0.233212	0.116606	−	0.993178i	$-0.462798\pi$
$461$	2308.36	0.233212	0.116606	+	0.993178i	$0.462798\pi$
$462$	0	0
$463$	2330.53	0.233928	0.116964	−	0.993136i	$-0.462684\pi$
$463$	2330.53	0.233928	0.116964	+	0.993136i	$0.462684\pi$
$464$	−9749.12	−0.975413
$465$	0	0
$466$	−10359.5	−1.02982
$467$	−3437.89	−0.340657	−0.170328	−	0.985387i	$-0.554483\pi$
$467$	−3437.89	−0.340657	−0.170328	+	0.985387i	$0.554483\pi$
$468$	0	0
$469$	−8797.03	−0.866117
$470$	2383.47	0.233918
$471$	0	0
$472$	−9493.11	−0.925754
$473$	2530.73	0.246011
$474$	0	0
$475$	13856.7	1.33851
$476$	−1665.89	−0.160411
$477$	0	0
$478$	−4220.89	−0.403889
$479$	15820.0	1.50905	0.754525	−	0.656272i	$-0.227867\pi$
$479$	15820.0	1.50905	0.754525	+	0.656272i	$0.227867\pi$
$480$	0	0
$481$	0	0
$482$	17212.8	1.62660
$483$	0	0
$484$	672.268	0.0631357
$485$	−5836.65	−0.546451
$486$	0	0
$487$	−14504.1	−1.34957	−0.674787	−	0.738012i	$-0.735765\pi$
$487$	−14504.1	−1.34957	−0.674787	+	0.738012i	$0.735765\pi$
$488$	−12509.0	−1.16036
$489$	0	0
$490$	18819.7	1.73508
$491$	17513.3	1.60970	0.804850	−	0.593479i	$-0.202246\pi$
$491$	17513.3	1.60970	0.804850	+	0.593479i	$0.202246\pi$
$492$	0	0
$493$	−20123.2	−1.83835
$494$	0	0
$495$	0	0
$496$	−8299.95	−0.751369
$497$	13314.5	1.20168
$498$	0	0
$499$	3856.26	0.345952	0.172976	−	0.984926i	$-0.444662\pi$
$499$	3856.26	0.345952	0.172976	+	0.984926i	$0.444662\pi$
$500$	−2173.21	−0.194378
$501$	0	0
$502$	−6129.85	−0.544997
$503$	1822.27	0.161533	0.0807666	−	0.996733i	$-0.474263\pi$
$503$	1822.27	0.161533	0.0807666	+	0.996733i	$0.474263\pi$
$504$	0	0
$505$	−6625.87	−0.583856
$506$	1439.54	0.126473
$507$	0	0
$508$	1335.25	0.116618
$509$	3738.01	0.325510	0.162755	−	0.986667i	$-0.447962\pi$
$509$	3738.01	0.325510	0.162755	+	0.986667i	$0.447962\pi$
$510$	0	0
$511$	12704.1	1.09980
$512$	−12509.9	−1.07981
$513$	0	0
$514$	−9853.46	−0.845559
$515$	10714.6	0.916779
$516$	0	0
$517$	287.580	0.0244637
$518$	−198.427	−0.0168308
$519$	0	0
$520$	0	0
$521$	−17735.5	−1.49138	−0.745690	−	0.666294i	$-0.767880\pi$
$521$	−17735.5	−1.49138	−0.745690	+	0.666294i	$0.767880\pi$
$522$	0	0
$523$	−6099.99	−0.510007	−0.255004	−	0.966940i	$-0.582077\pi$
$523$	−6099.99	−0.510007	−0.255004	+	0.966940i	$0.582077\pi$
$524$	−367.487	−0.0306370
$525$	0	0
$526$	5826.72	0.482998
$527$	−17132.0	−1.41609
$528$	0	0
$529$	−6459.31	−0.530888
$530$	7263.90	0.595328
$531$	0	0
$532$	584.772	0.0476561
$533$	0	0
$534$	0	0
$535$	26669.8	2.15521
$536$	−7927.93	−0.638870
$537$	0	0
$538$	6851.30	0.549035
$539$	2270.71	0.181459
$540$	0	0
$541$	4453.47	0.353918	0.176959	−	0.984218i	$-0.443374\pi$
$541$	4453.47	0.353918	0.176959	+	0.984218i	$0.443374\pi$
$542$	−629.464	−0.0498852
$543$	0	0
$544$	−2910.29	−0.229371
$545$	−9914.95	−0.779284
$546$	0	0
$547$	−17915.1	−1.40035	−0.700176	−	0.713970i	$-0.746895\pi$
$547$	−17915.1	−1.40035	−0.700176	+	0.713970i	$0.746895\pi$
$548$	1350.55	0.105278
$549$	0	0
$550$	6121.18	0.474560
$551$	7063.79	0.546148
$552$	0	0
$553$	−19729.2	−1.51712
$554$	−21191.0	−1.62513
$555$	0	0
$556$	−122.424	−0.00933804
$557$	3127.12	0.237882	0.118941	−	0.992901i	$-0.462050\pi$
$557$	3127.12	0.237882	0.118941	+	0.992901i	$0.462050\pi$
$558$	0	0
$559$	0	0
$560$	32523.3	2.45421
$561$	0	0
$562$	14160.9	1.06289
$563$	−12978.0	−0.971507	−0.485753	−	0.874096i	$-0.661455\pi$
$563$	−12978.0	−0.971507	−0.485753	+	0.874096i	$0.661455\pi$
$564$	0	0
$565$	23463.0	1.74707
$566$	10389.7	0.771577
$567$	0	0
$568$	11999.1	0.886390
$569$	−1275.65	−0.0939857	−0.0469928	−	0.998895i	$-0.514964\pi$
$569$	−1275.65	−0.0939857	−0.0469928	+	0.998895i	$0.514964\pi$
$570$	0	0
$571$	26406.4	1.93533	0.967663	−	0.252245i	$-0.0811689\pi$
$571$	26406.4	1.93533	0.967663	+	0.252245i	$0.0811689\pi$
$572$	0	0
$573$	0	0
$574$	−21252.6	−1.54541
$575$	24270.1	1.76023
$576$	0	0
$577$	−14705.2	−1.06098	−0.530489	−	0.847692i	$-0.677992\pi$
$577$	−14705.2	−1.06098	−0.530489	+	0.847692i	$0.677992\pi$
$578$	27851.4	2.00426
$579$	0	0
$580$	−1813.49	−0.129829
$581$	8946.85	0.638860
$582$	0	0
$583$	876.433	0.0622610
$584$	11449.0	0.811239
$585$	0	0
$586$	15970.0	1.12579
$587$	−18627.3	−1.30976	−0.654881	−	0.755732i	$-0.727281\pi$
$587$	−18627.3	−1.30976	−0.654881	+	0.755732i	$0.727281\pi$
$588$	0	0
$589$	6013.79	0.420703
$590$	23525.7	1.64159
$591$	0	0
$592$	−167.055	−0.0115978
$593$	−4495.16	−0.311288	−0.155644	−	0.987813i	$-0.549745\pi$
$593$	−4495.16	−0.311288	−0.155644	+	0.987813i	$0.549745\pi$
$594$	0	0
$595$	67131.5	4.62542
$596$	836.627	0.0574993
$597$	0	0
$598$	0	0
$599$	−3262.10	−0.222514	−0.111257	−	0.993792i	$-0.535488\pi$
$599$	−3262.10	−0.222514	−0.111257	+	0.993792i	$0.535488\pi$
$600$	0	0
$601$	−5618.35	−0.381327	−0.190663	−	0.981655i	$-0.561064\pi$
$601$	−5618.35	−0.381327	−0.190663	+	0.981655i	$0.561064\pi$
$602$	−25678.4	−1.73849
$603$	0	0
$604$	−1831.67	−0.123393
$605$	−27090.9	−1.82050
$606$	0	0
$607$	−19129.2	−1.27913	−0.639563	−	0.768738i	$-0.720885\pi$
$607$	−19129.2	−1.27913	−0.639563	+	0.768738i	$0.720885\pi$
$608$	1021.59	0.0681430
$609$	0	0
$610$	30999.6	2.05760
$611$	0	0
$612$	0	0
$613$	−16054.6	−1.05781	−0.528907	−	0.848679i	$-0.677398\pi$
$613$	−16054.6	−1.05781	−0.528907	+	0.848679i	$0.677398\pi$
$614$	4610.62	0.303045
$615$	0	0
$616$	4200.56	0.274749
$617$	18669.4	1.21816	0.609079	−	0.793110i	$-0.291539\pi$
$617$	18669.4	1.21816	0.609079	+	0.793110i	$0.291539\pi$
$618$	0	0
$619$	2488.87	0.161609	0.0808045	−	0.996730i	$-0.474251\pi$
$619$	2488.87	0.161609	0.0808045	+	0.996730i	$0.474251\pi$
$620$	−1543.92	−0.100009
$621$	0	0
$622$	28985.3	1.86850
$623$	9374.45	0.602856
$624$	0	0
$625$	47419.5	3.03485
$626$	−16104.4	−1.02821
$627$	0	0
$628$	1155.21	0.0734044
$629$	−344.819	−0.0218582
$630$	0	0
$631$	−3595.22	−0.226820	−0.113410	−	0.993548i	$-0.536177\pi$
$631$	−3595.22	−0.226820	−0.113410	+	0.993548i	$0.536177\pi$
$632$	−17780.0	−1.11907
$633$	0	0
$634$	6652.22	0.416709
$635$	−53807.4	−3.36265
$636$	0	0
$637$	0	0
$638$	3120.42	0.193634
$639$	0	0
$640$	27245.4	1.68276
$641$	4048.14	0.249441	0.124721	−	0.992192i	$-0.460197\pi$
$641$	4048.14	0.249441	0.124721	+	0.992192i	$0.460197\pi$
$642$	0	0
$643$	20768.9	1.27379	0.636894	−	0.770951i	$-0.280219\pi$
$643$	20768.9	1.27379	0.636894	+	0.770951i	$0.280219\pi$
$644$	1024.23	0.0626712
$645$	0	0
$646$	−14492.0	−0.882629
$647$	−14788.6	−0.898610	−0.449305	−	0.893378i	$-0.648328\pi$
$647$	−14788.6	−0.898610	−0.449305	+	0.893378i	$0.648328\pi$
$648$	0	0
$649$	2838.52	0.171682
$650$	0	0
$651$	0	0
$652$	−397.719	−0.0238894
$653$	−13026.1	−0.780628	−0.390314	−	0.920682i	$-0.627633\pi$
$653$	−13026.1	−0.780628	−0.390314	+	0.920682i	$0.627633\pi$
$654$	0	0
$655$	14808.9	0.883408
$656$	−17892.5	−1.06492
$657$	0	0
$658$	−2917.97	−0.172879
$659$	−1633.07	−0.0965330	−0.0482665	−	0.998834i	$-0.515370\pi$
$659$	−1633.07	−0.0965330	−0.0482665	+	0.998834i	$0.515370\pi$
$660$	0	0
$661$	−14205.6	−0.835908	−0.417954	−	0.908468i	$-0.637253\pi$
$661$	−14205.6	−0.835908	−0.417954	+	0.908468i	$0.637253\pi$
$662$	−2071.24	−0.121603
$663$	0	0
$664$	8062.95	0.471240
$665$	−23565.0	−1.37415
$666$	0	0
$667$	12372.2	0.718224
$668$	−448.345	−0.0259685
$669$	0	0
$670$	19646.9	1.13287
$671$	3740.29	0.215190
$672$	0	0
$673$	−27400.1	−1.56939	−0.784693	−	0.619885i	$-0.787179\pi$
$673$	−27400.1	−1.56939	−0.784693	+	0.619885i	$0.787179\pi$
$674$	−2431.53	−0.138960
$675$	0	0
$676$	0	0
$677$	2463.68	0.139862	0.0699311	−	0.997552i	$-0.477722\pi$
$677$	2463.68	0.139862	0.0699311	+	0.997552i	$0.477722\pi$
$678$	0	0
$679$	7145.53	0.403859
$680$	60499.3	3.41182
$681$	0	0
$682$	2656.58	0.149158
$683$	−21663.1	−1.21364	−0.606820	−	0.794839i	$-0.707555\pi$
$683$	−21663.1	−1.21364	−0.606820	+	0.794839i	$0.707555\pi$
$684$	0	0
$685$	−54424.0	−3.03567
$686$	1213.86	0.0675589
$687$	0	0
$688$	−21618.6	−1.19797
$689$	0	0
$690$	0	0
$691$	3787.41	0.208509	0.104254	−	0.994551i	$-0.466754\pi$
$691$	3787.41	0.208509	0.104254	+	0.994551i	$0.466754\pi$
$692$	−40.2031	−0.00220852
$693$	0	0
$694$	26369.8	1.44234
$695$	4933.43	0.269260
$696$	0	0
$697$	−36932.0	−2.00703
$698$	4943.10	0.268051
$699$	0	0
$700$	4355.20	0.235158
$701$	12593.4	0.678527	0.339263	−	0.940691i	$-0.389822\pi$
$701$	12593.4	0.678527	0.339263	+	0.940691i	$0.389822\pi$
$702$	0	0
$703$	121.041	0.00649380
$704$	3770.25	0.201842
$705$	0	0
$706$	29206.0	1.55692
$707$	8111.73	0.431504
$708$	0	0
$709$	−22849.8	−1.21036	−0.605178	−	0.796090i	$-0.706898\pi$
$709$	−22849.8	−1.21036	−0.605178	+	0.796090i	$0.706898\pi$
$710$	−29735.9	−1.57179
$711$	0	0
$712$	8448.30	0.444682
$713$	10533.2	0.553254
$714$	0	0
$715$	0	0
$716$	1852.09	0.0966703
$717$	0	0
$718$	−22409.3	−1.16477
$719$	−7269.04	−0.377037	−0.188518	−	0.982070i	$-0.560368\pi$
$719$	−7269.04	−0.377037	−0.188518	+	0.982070i	$0.560368\pi$
$720$	0	0
$721$	−13117.4	−0.677553
$722$	−13666.7	−0.704464
$723$	0	0
$724$	−1233.35	−0.0633111
$725$	52608.9	2.69496
$726$	0	0
$727$	21380.1	1.09071	0.545355	−	0.838205i	$-0.316395\pi$
$727$	21380.1	1.09071	0.545355	+	0.838205i	$0.316395\pi$
$728$	0	0
$729$	0	0
$730$	−28372.8	−1.43853
$731$	−44623.0	−2.25779
$732$	0	0
$733$	−5430.11	−0.273623	−0.136811	−	0.990597i	$-0.543685\pi$
$733$	−5430.11	−0.273623	−0.136811	+	0.990597i	$0.543685\pi$
$734$	−6986.65	−0.351338
$735$	0	0
$736$	1789.32	0.0896128
$737$	2370.51	0.118479
$738$	0	0
$739$	30838.6	1.53507	0.767534	−	0.641008i	$-0.221483\pi$
$739$	30838.6	1.53507	0.767534	+	0.641008i	$0.221483\pi$
$740$	−31.0748	−0.00154369
$741$	0	0
$742$	−8892.85	−0.439982
$743$	31665.8	1.56353	0.781767	−	0.623571i	$-0.214319\pi$
$743$	31665.8	1.56353	0.781767	+	0.623571i	$0.214319\pi$
$744$	0	0
$745$	−33714.2	−1.65798
$746$	−1642.58	−0.0806156
$747$	0	0
$748$	448.903	0.0219432
$749$	−32650.6	−1.59283
$750$	0	0
$751$	−23714.1	−1.15225	−0.576124	−	0.817362i	$-0.695436\pi$
$751$	−23714.1	−1.15225	−0.576124	+	0.817362i	$0.695436\pi$
$752$	−2456.63	−0.119128
$753$	0	0
$754$	0	0
$755$	73812.2	3.55801
$756$	0	0
$757$	−19213.1	−0.922474	−0.461237	−	0.887277i	$-0.652594\pi$
$757$	−19213.1	−0.922474	−0.461237	+	0.887277i	$0.652594\pi$
$758$	2361.28	0.113147
$759$	0	0
$760$	−21236.9	−1.01361
$761$	−11406.5	−0.543345	−0.271673	−	0.962390i	$-0.587577\pi$
$761$	−11406.5	−0.543345	−0.271673	+	0.962390i	$0.587577\pi$
$762$	0	0
$763$	12138.4	0.575936
$764$	−803.570	−0.0380526
$765$	0	0
$766$	−24852.1	−1.17225
$767$	0	0
$768$	0	0
$769$	−6384.27	−0.299379	−0.149690	−	0.988733i	$-0.547827\pi$
$769$	−6384.27	−0.299379	−0.149690	+	0.988733i	$0.547827\pi$
$770$	−10409.8	−0.487198
$771$	0	0
$772$	−2417.57	−0.112708
$773$	1561.96	0.0726776	0.0363388	−	0.999340i	$-0.488430\pi$
$773$	1561.96	0.0726776	0.0363388	+	0.999340i	$0.488430\pi$
$774$	0	0
$775$	44788.8	2.07595
$776$	6439.59	0.297897
$777$	0	0
$778$	4137.26	0.190653
$779$	12964.1	0.596262
$780$	0	0
$781$	−3587.82	−0.164382
$782$	−25382.7	−1.16072
$783$	0	0
$784$	−19397.4	−0.883626
$785$	−46552.4	−2.11660
$786$	0	0
$787$	11134.6	0.504326	0.252163	−	0.967685i	$-0.418858\pi$
$787$	11134.6	0.504326	0.252163	+	0.967685i	$0.418858\pi$
$788$	−1003.54	−0.0453675
$789$	0	0
$790$	44062.3	1.98439
$791$	−28724.6	−1.29119
$792$	0	0
$793$	0	0
$794$	−20705.4	−0.925448
$795$	0	0
$796$	−1208.28	−0.0538020
$797$	−13798.3	−0.613250	−0.306625	−	0.951830i	$-0.599200\pi$
$797$	−13798.3	−0.613250	−0.306625	+	0.951830i	$0.599200\pi$
$798$	0	0
$799$	−5070.74	−0.224518
$800$	7608.48	0.336251
$801$	0	0
$802$	−30282.5	−1.33331
$803$	−3423.35	−0.150445
$804$	0	0
$805$	−41274.1	−1.80711
$806$	0	0
$807$	0	0
$808$	7310.33	0.318288
$809$	−8271.72	−0.359479	−0.179739	−	0.983714i	$-0.557525\pi$
$809$	−8271.72	−0.359479	−0.179739	+	0.983714i	$0.557525\pi$
$810$	0	0
$811$	31496.6	1.36374	0.681872	−	0.731472i	$-0.261166\pi$
$811$	31496.6	1.36374	0.681872	+	0.731472i	$0.261166\pi$
$812$	2220.16	0.0959514
$813$	0	0
$814$	53.4695	0.00230234
$815$	16027.2	0.688843
$816$	0	0
$817$	15663.9	0.670759
$818$	30302.6	1.29524
$819$	0	0
$820$	−3328.28	−0.141742
$821$	12125.3	0.515441	0.257721	−	0.966219i	$-0.417029\pi$
$821$	12125.3	0.515441	0.257721	+	0.966219i	$0.417029\pi$
$822$	0	0
$823$	6237.45	0.264184	0.132092	−	0.991237i	$-0.457831\pi$
$823$	6237.45	0.264184	0.132092	+	0.991237i	$0.457831\pi$
$824$	−11821.4	−0.499780
$825$	0	0
$826$	−28801.4	−1.21323
$827$	31790.0	1.33669	0.668347	−	0.743850i	$-0.267002\pi$
$827$	31790.0	1.33669	0.668347	+	0.743850i	$0.267002\pi$
$828$	0	0
$829$	−23138.6	−0.969404	−0.484702	−	0.874679i	$-0.661072\pi$
$829$	−23138.6	−0.969404	−0.484702	+	0.874679i	$0.661072\pi$
$830$	−19981.5	−0.835624
$831$	0	0
$832$	0	0
$833$	−40038.2	−1.66536
$834$	0	0
$835$	18067.3	0.748795
$836$	−157.577	−0.00651904
$837$	0	0
$838$	−35641.7	−1.46924
$839$	3357.69	0.138165	0.0690825	−	0.997611i	$-0.477993\pi$
$839$	3357.69	0.138165	0.0690825	+	0.997611i	$0.477993\pi$
$840$	0	0
$841$	2429.66	0.0996211
$842$	−40094.1	−1.64101
$843$	0	0
$844$	1835.04	0.0748397
$845$	0	0
$846$	0	0
$847$	33166.1	1.34545
$848$	−7486.87	−0.303184
$849$	0	0
$850$	−107932.	−4.35532
$851$	212.003	0.00853981
$852$	0	0
$853$	16262.6	0.652778	0.326389	−	0.945236i	$-0.394168\pi$
$853$	16262.6	0.652778	0.326389	+	0.945236i	$0.394168\pi$
$854$	−37951.4	−1.52069
$855$	0	0
$856$	−29424.9	−1.17491
$857$	−33293.4	−1.32705	−0.663524	−	0.748155i	$-0.730940\pi$
$857$	−33293.4	−1.32705	−0.663524	+	0.748155i	$0.730940\pi$
$858$	0	0
$859$	26170.5	1.03950	0.519748	−	0.854320i	$-0.326026\pi$
$859$	26170.5	1.03950	0.519748	+	0.854320i	$0.326026\pi$
$860$	−4021.39	−0.159451
$861$	0	0
$862$	−3977.90	−0.157178
$863$	37186.2	1.46678	0.733391	−	0.679807i	$-0.237936\pi$
$863$	37186.2	1.46678	0.733391	+	0.679807i	$0.237936\pi$
$864$	0	0
$865$	1620.09	0.0636819
$866$	−24559.0	−0.963682
$867$	0	0
$868$	1890.15	0.0739122
$869$	5316.37	0.207532
$870$	0	0
$871$	0	0
$872$	10939.2	0.424825
$873$	0	0
$874$	8910.01	0.344835
$875$	−107215.	−4.14230
$876$	0	0
$877$	19008.1	0.731880	0.365940	−	0.930638i	$-0.380748\pi$
$877$	19008.1	0.731880	0.365940	+	0.930638i	$0.380748\pi$
$878$	9154.20	0.351867
$879$	0	0
$880$	−8763.97	−0.335720
$881$	11259.4	0.430578	0.215289	−	0.976550i	$-0.430931\pi$
$881$	11259.4	0.430578	0.215289	+	0.976550i	$0.430931\pi$
$882$	0	0
$883$	34604.5	1.31884	0.659419	−	0.751776i	$-0.270802\pi$
$883$	34604.5	1.31884	0.659419	+	0.751776i	$0.270802\pi$
$884$	0	0
$885$	0	0
$886$	42848.6	1.62475
$887$	−36352.6	−1.37610	−0.688050	−	0.725663i	$-0.741533\pi$
$887$	−36352.6	−1.37610	−0.688050	+	0.725663i	$0.741533\pi$
$888$	0	0
$889$	65873.8	2.48519
$890$	−20936.5	−0.788531
$891$	0	0
$892$	−658.407	−0.0247142
$893$	1779.97	0.0667014
$894$	0	0
$895$	−74635.1	−2.78746
$896$	−33355.2	−1.24366
$897$	0	0
$898$	33537.1	1.24627
$899$	22832.2	0.847048
$900$	0	0
$901$	−15453.7	−0.571406
$902$	5726.88	0.211402
$903$	0	0
$904$	−25886.7	−0.952412
$905$	49701.4	1.82556
$906$	0	0
$907$	−36358.6	−1.33106	−0.665528	−	0.746373i	$-0.731794\pi$
$907$	−36358.6	−1.33106	−0.665528	+	0.746373i	$0.731794\pi$
$908$	−1786.79	−0.0653048
$909$	0	0
$910$	0	0
$911$	−16789.4	−0.610603	−0.305301	−	0.952256i	$-0.598757\pi$
$911$	−16789.4	−0.610603	−0.305301	+	0.952256i	$0.598757\pi$
$912$	0	0
$913$	−2410.89	−0.0873918
$914$	22919.4	0.829439
$915$	0	0
$916$	2167.69	0.0781905
$917$	−18129.8	−0.652890
$918$	0	0
$919$	42322.8	1.51915	0.759575	−	0.650419i	$-0.225407\pi$
$919$	42322.8	1.51915	0.759575	+	0.650419i	$0.225407\pi$
$920$	−37196.4	−1.33297
$921$	0	0
$922$	6311.48	0.225442
$923$	0	0
$924$	0	0
$925$	901.474	0.0320436
$926$	6372.11	0.226134
$927$	0	0
$928$	3878.60	0.137200
$929$	15413.1	0.544337	0.272168	−	0.962250i	$-0.412259\pi$
$929$	15413.1	0.544337	0.272168	+	0.962250i	$0.412259\pi$
$930$	0	0
$931$	14054.5	0.494756
$932$	1986.18	0.0698063
$933$	0	0
$934$	−9399.85	−0.329307
$935$	−18089.8	−0.632726
$936$	0	0
$937$	45360.4	1.58149	0.790747	−	0.612143i	$-0.209692\pi$
$937$	45360.4	1.58149	0.790747	+	0.612143i	$0.209692\pi$
$938$	−24052.7	−0.837260
$939$	0	0
$940$	−456.971	−0.0158561
$941$	37604.6	1.30274	0.651368	−	0.758762i	$-0.274195\pi$
$941$	37604.6	1.30274	0.651368	+	0.758762i	$0.274195\pi$
$942$	0	0
$943$	22706.7	0.784127
$944$	−24247.8	−0.836016
$945$	0	0
$946$	6919.49	0.237814
$947$	28270.9	0.970095	0.485048	−	0.874488i	$-0.338802\pi$
$947$	28270.9	0.970095	0.485048	+	0.874488i	$0.338802\pi$
$948$	0	0
$949$	0	0
$950$	37886.9	1.29391
$951$	0	0
$952$	−74066.3	−2.52154
$953$	−21396.1	−0.727268	−0.363634	−	0.931542i	$-0.618464\pi$
$953$	−21396.1	−0.727268	−0.363634	+	0.931542i	$0.618464\pi$
$954$	0	0
$955$	32382.1	1.09723
$956$	809.252	0.0273777
$957$	0	0
$958$	43254.9	1.45877
$959$	66628.7	2.24354
$960$	0	0
$961$	−10352.7	−0.347512
$962$	0	0
$963$	0	0
$964$	−3300.13	−0.110259
$965$	97422.7	3.24990
$966$	0	0
$967$	898.300	0.0298732	0.0149366	−	0.999888i	$-0.495245\pi$
$967$	898.300	0.0298732	0.0149366	+	0.999888i	$0.495245\pi$
$968$	29889.4	0.992441
$969$	0	0
$970$	−15958.5	−0.528244
$971$	−3749.51	−0.123921	−0.0619607	−	0.998079i	$-0.519735\pi$
$971$	−3749.51	−0.123921	−0.0619607	+	0.998079i	$0.519735\pi$
$972$	0	0
$973$	−6039.76	−0.198999
$974$	−39656.9	−1.30461
$975$	0	0
$976$	−31951.2	−1.04788
$977$	−2717.19	−0.0889771	−0.0444885	−	0.999010i	$-0.514166\pi$
$977$	−2717.19	−0.0889771	−0.0444885	+	0.999010i	$0.514166\pi$
$978$	0	0
$979$	−2526.11	−0.0824666
$980$	−3608.21	−0.117612
$981$	0	0
$982$	47884.6	1.55607
$983$	−4179.98	−0.135626	−0.0678132	−	0.997698i	$-0.521602\pi$
$983$	−4179.98	−0.135626	−0.0678132	+	0.997698i	$0.521602\pi$
$984$	0	0
$985$	40440.4	1.30816
$986$	−55020.7	−1.77709
$987$	0	0
$988$	0	0
$989$	27435.3	0.882096
$990$	0	0
$991$	−15297.9	−0.490367	−0.245184	−	0.969477i	$-0.578848\pi$
$991$	−15297.9	−0.490367	−0.245184	+	0.969477i	$0.578848\pi$
$992$	3302.07	0.105686
$993$	0	0
$994$	36404.3	1.16164
$995$	48691.0	1.55136
$996$	0	0
$997$	9164.26	0.291108	0.145554	−	0.989350i	$-0.453504\pi$
$997$	9164.26	0.291108	0.145554	+	0.989350i	$0.453504\pi$
$998$	10543.7	0.334425
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.bj.1.6		9
3.2	odd	2		507.4.a.n.1.4	✓	9
13.12	even	2		1521.4.a.be.1.4		9
39.5	even	4		507.4.b.j.337.13		18
39.8	even	4		507.4.b.j.337.6		18
39.38	odd	2		507.4.a.q.1.6	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.n.1.4	✓	9	3.2	odd	2
507.4.a.q.1.6	yes	9	39.38	odd	2
507.4.b.j.337.6		18	39.8	even	4
507.4.b.j.337.13		18	39.5	even	4
1521.4.a.be.1.4		9	13.12	even	2
1521.4.a.bj.1.6		9	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+2.73419 q^{2} -0.524213 q^{4} +21.1246 q^{5} -25.8618 q^{7} -23.3068 q^{8} +O(q^{10})\)	\(q+2.73419 q^{2} -0.524213 q^{4} +21.1246 q^{5} -25.8618 q^{7} -23.3068 q^{8} +57.7586 q^{10} +6.96892 q^{11} -70.7111 q^{14} -59.5315 q^{16} -122.879 q^{17} +43.1340 q^{19} -11.0738 q^{20} +19.0543 q^{22} +75.5492 q^{23} +321.248 q^{25} +13.5571 q^{28} +163.764 q^{29} +139.421 q^{31} +23.6841 q^{32} -335.975 q^{34} -546.320 q^{35} +2.80616 q^{37} +117.936 q^{38} -492.347 q^{40} +300.555 q^{41} +363.145 q^{43} -3.65320 q^{44} +206.566 q^{46} +41.2660 q^{47} +325.834 q^{49} +878.354 q^{50} +125.763 q^{53} +147.216 q^{55} +602.756 q^{56} +447.762 q^{58} +407.311 q^{59} +536.710 q^{61} +381.204 q^{62} +541.009 q^{64} +340.155 q^{67} +64.4150 q^{68} -1493.74 q^{70} -514.831 q^{71} -491.231 q^{73} +7.67257 q^{74} -22.6114 q^{76} -180.229 q^{77} +762.869 q^{79} -1257.58 q^{80} +821.774 q^{82} -345.948 q^{83} -2595.78 q^{85} +992.907 q^{86} -162.423 q^{88} -362.482 q^{89} -39.6039 q^{92} +112.829 q^{94} +911.188 q^{95} -276.297 q^{97} +890.890 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 8 q^{2} + 32 q^{4} + 41 q^{5} - q^{7} + 111 q^{8}+O(q^{10}) \) 9 * q + 8 * q^2 + 32 * q^4 + 41 * q^5 - q^7 + 111 * q^8	\( 9 q + 8 q^{2} + 32 q^{4} + 41 q^{5} - q^{7} + 111 q^{8} + 198 q^{10} + 37 q^{11} - 98 q^{14} + 32 q^{16} + 134 q^{17} + 72 q^{19} + 356 q^{20} + 274 q^{22} - 226 q^{23} + 612 q^{25} - 132 q^{28} + 547 q^{29} + 521 q^{31} + 721 q^{32} + 100 q^{34} - 138 q^{35} - 584 q^{37} + 416 q^{38} + 1342 q^{40} + 482 q^{41} + 158 q^{43} + 1453 q^{44} - 1537 q^{46} + 1500 q^{47} + 642 q^{49} + 2777 q^{50} - 1399 q^{53} - 1408 q^{55} + 616 q^{56} - 1455 q^{58} + 1541 q^{59} + 2092 q^{61} + 293 q^{62} + 2481 q^{64} - 252 q^{67} + 1579 q^{68} - 2492 q^{70} + 2352 q^{71} - 903 q^{73} - 1037 q^{74} + 485 q^{76} + 1686 q^{77} - 115 q^{79} + 5701 q^{80} - 5147 q^{82} + 1207 q^{83} - 4308 q^{85} + 5691 q^{86} - 484 q^{88} + 2336 q^{89} - 2087 q^{92} - 468 q^{94} + 222 q^{95} - 2155 q^{97} + 5593 q^{98}+O(q^{100}) \) 9 * q + 8 * q^2 + 32 * q^4 + 41 * q^5 - q^7 + 111 * q^8 + 198 * q^10 + 37 * q^11 - 98 * q^14 + 32 * q^16 + 134 * q^17 + 72 * q^19 + 356 * q^20 + 274 * q^22 - 226 * q^23 + 612 * q^25 - 132 * q^28 + 547 * q^29 + 521 * q^31 + 721 * q^32 + 100 * q^34 - 138 * q^35 - 584 * q^37 + 416 * q^38 + 1342 * q^40 + 482 * q^41 + 158 * q^43 + 1453 * q^44 - 1537 * q^46 + 1500 * q^47 + 642 * q^49 + 2777 * q^50 - 1399 * q^53 - 1408 * q^55 + 616 * q^56 - 1455 * q^58 + 1541 * q^59 + 2092 * q^61 + 293 * q^62 + 2481 * q^64 - 252 * q^67 + 1579 * q^68 - 2492 * q^70 + 2352 * q^71 - 903 * q^73 - 1037 * q^74 + 485 * q^76 + 1686 * q^77 - 115 * q^79 + 5701 * q^80 - 5147 * q^82 + 1207 * q^83 - 4308 * q^85 + 5691 * q^86 - 484 * q^88 + 2336 * q^89 - 2087 * q^92 - 468 * q^94 + 222 * q^95 - 2155 * q^97 + 5593 * q^98

Introduction

L-functions

Modular forms

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Database

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