LMFDB - Embedded newform 495.4.a.p.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [495,4,Mod(1,495)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(495, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("495.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	495.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:	$29.2059454528$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 41x^{5} + 40x^{4} + 424x^{3} - 168x^{2} - 1042x - 388 $ x^7 - 2x^6 - 41x^5 + 40x^4 + 424x^3 - 168x^2 - 1042x - 388
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.30247$ of defining polynomial
Character	$\chi$	$=$	495.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30247 q^{2} -2.69863 q^{4} +5.00000 q^{5} +33.7164 q^{7} -24.6333 q^{8} +O(q^{10})$	\(q+2.30247 q^{2} -2.69863 q^{4} +5.00000 q^{5} +33.7164 q^{7} -24.6333 q^{8} +11.5123 q^{10} +11.0000 q^{11} +37.9156 q^{13} +77.6309 q^{14} -35.1283 q^{16} -44.2817 q^{17} -42.3986 q^{19} -13.4932 q^{20} +25.3272 q^{22} +30.6793 q^{23} +25.0000 q^{25} +87.2995 q^{26} -90.9881 q^{28} +194.548 q^{29} -82.9601 q^{31} +116.184 q^{32} -101.957 q^{34} +168.582 q^{35} +445.236 q^{37} -97.6216 q^{38} -123.166 q^{40} +206.184 q^{41} +197.228 q^{43} -29.6850 q^{44} +70.6382 q^{46} -275.301 q^{47} +793.792 q^{49} +57.5617 q^{50} -102.320 q^{52} -382.718 q^{53} +55.0000 q^{55} -830.544 q^{56} +447.941 q^{58} +771.622 q^{59} -210.576 q^{61} -191.013 q^{62} +548.537 q^{64} +189.578 q^{65} +452.672 q^{67} +119.500 q^{68} +388.154 q^{70} -310.139 q^{71} -174.350 q^{73} +1025.14 q^{74} +114.418 q^{76} +370.880 q^{77} +1191.11 q^{79} -175.641 q^{80} +474.732 q^{82} -183.669 q^{83} -221.409 q^{85} +454.111 q^{86} -270.966 q^{88} -613.710 q^{89} +1278.38 q^{91} -82.7923 q^{92} -633.873 q^{94} -211.993 q^{95} -1204.77 q^{97} +1827.68 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 5 q^{2} + 33 q^{4} + 35 q^{5} + 30 q^{7} + 45 q^{8}+O(q^{10}) $ 7 * q + 5 * q^2 + 33 * q^4 + 35 * q^5 + 30 * q^7 + 45 * q^8	$ 7 q + 5 q^{2} + 33 q^{4} + 35 q^{5} + 30 q^{7} + 45 q^{8} + 25 q^{10} + 77 q^{11} + 38 q^{13} + 20 q^{14} + 309 q^{16} + 12 q^{17} + 226 q^{19} + 165 q^{20} + 55 q^{22} + 334 q^{23} + 175 q^{25} - 372 q^{26} + 812 q^{28} - 258 q^{29} + 336 q^{31} + 485 q^{32} + 78 q^{34} + 150 q^{35} + 466 q^{37} - 494 q^{38} + 225 q^{40} - 258 q^{41} + 308 q^{43} + 363 q^{44} + 98 q^{46} + 546 q^{47} + 735 q^{49} + 125 q^{50} + 512 q^{52} + 110 q^{53} + 385 q^{55} + 20 q^{56} + 1362 q^{58} - 68 q^{59} + 1096 q^{61} + 356 q^{62} + 2761 q^{64} + 190 q^{65} + 2268 q^{67} - 1186 q^{68} + 100 q^{70} - 166 q^{71} + 200 q^{73} - 1710 q^{74} + 3310 q^{76} + 330 q^{77} + 2152 q^{79} + 1545 q^{80} - 1006 q^{82} + 370 q^{83} + 60 q^{85} + 106 q^{86} + 495 q^{88} - 252 q^{89} + 2768 q^{91} + 3774 q^{92} + 2218 q^{94} + 1130 q^{95} + 3698 q^{97} + 697 q^{98}+O(q^{100}) $ 7 * q + 5 * q^2 + 33 * q^4 + 35 * q^5 + 30 * q^7 + 45 * q^8 + 25 * q^10 + 77 * q^11 + 38 * q^13 + 20 * q^14 + 309 * q^16 + 12 * q^17 + 226 * q^19 + 165 * q^20 + 55 * q^22 + 334 * q^23 + 175 * q^25 - 372 * q^26 + 812 * q^28 - 258 * q^29 + 336 * q^31 + 485 * q^32 + 78 * q^34 + 150 * q^35 + 466 * q^37 - 494 * q^38 + 225 * q^40 - 258 * q^41 + 308 * q^43 + 363 * q^44 + 98 * q^46 + 546 * q^47 + 735 * q^49 + 125 * q^50 + 512 * q^52 + 110 * q^53 + 385 * q^55 + 20 * q^56 + 1362 * q^58 - 68 * q^59 + 1096 * q^61 + 356 * q^62 + 2761 * q^64 + 190 * q^65 + 2268 * q^67 - 1186 * q^68 + 100 * q^70 - 166 * q^71 + 200 * q^73 - 1710 * q^74 + 3310 * q^76 + 330 * q^77 + 2152 * q^79 + 1545 * q^80 - 1006 * q^82 + 370 * q^83 + 60 * q^85 + 106 * q^86 + 495 * q^88 - 252 * q^89 + 2768 * q^91 + 3774 * q^92 + 2218 * q^94 + 1130 * q^95 + 3698 * q^97 + 697 * 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 $

$2$	2.30247	0.814046	0.407023	−	0.913418i	$-0.366567\pi$
$2$	2.30247	0.814046	0.407023	+	0.913418i	$0.366567\pi$
$3$	0	0
$3$	0	0
$4$	−2.69863	−0.337329
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	33.7164	1.82051	0.910256	−	0.414046i	$-0.135885\pi$
$7$	33.7164	1.82051	0.910256	+	0.414046i	$0.135885\pi$
$8$	−24.6333	−1.08865
$9$	0	0
$10$	11.5123	0.364052
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	37.9156	0.808915	0.404457	−	0.914557i	$-0.367460\pi$
$13$	37.9156	0.808915	0.404457	+	0.914557i	$0.367460\pi$
$14$	77.6309	1.48198
$15$	0	0
$16$	−35.1283	−0.548880
$17$	−44.2817	−0.631758	−0.315879	−	0.948799i	$-0.602299\pi$
$17$	−44.2817	−0.631758	−0.315879	+	0.948799i	$0.602299\pi$
$18$	0	0
$19$	−42.3986	−0.511943	−0.255971	−	0.966684i	$-0.582395\pi$
$19$	−42.3986	−0.511943	−0.255971	+	0.966684i	$0.582395\pi$
$20$	−13.4932	−0.150858
$21$	0	0
$22$	25.3272	0.245444
$23$	30.6793	0.278134	0.139067	−	0.990283i	$-0.455590\pi$
$23$	30.6793	0.278134	0.139067	+	0.990283i	$0.455590\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	87.2995	0.658494
$27$	0	0
$28$	−90.9881	−0.614112
$29$	194.548	1.24575	0.622873	−	0.782323i	$-0.285965\pi$
$29$	194.548	1.24575	0.622873	+	0.782323i	$0.285965\pi$
$30$	0	0
$31$	−82.9601	−0.480648	−0.240324	−	0.970693i	$-0.577254\pi$
$31$	−82.9601	−0.480648	−0.240324	+	0.970693i	$0.577254\pi$
$32$	116.184	0.641834
$33$	0	0
$34$	−101.957	−0.514280
$35$	168.582	0.814158
$36$	0	0
$37$	445.236	1.97828	0.989140	−	0.146980i	$-0.0469552\pi$
$37$	445.236	1.97828	0.989140	+	0.146980i	$0.0469552\pi$
$38$	−97.6216	−0.416745
$39$	0	0
$40$	−123.166	−0.486858
$41$	206.184	0.785378	0.392689	−	0.919671i	$-0.371545\pi$
$41$	206.184	0.785378	0.392689	+	0.919671i	$0.371545\pi$
$42$	0	0
$43$	197.228	0.699464	0.349732	−	0.936850i	$-0.386273\pi$
$43$	197.228	0.699464	0.349732	+	0.936850i	$0.386273\pi$
$44$	−29.6850	−0.101709
$45$	0	0
$46$	70.6382	0.226414
$47$	−275.301	−0.854401	−0.427201	−	0.904157i	$-0.640500\pi$
$47$	−275.301	−0.854401	−0.427201	+	0.904157i	$0.640500\pi$
$48$	0	0
$49$	793.792	2.31426
$50$	57.5617	0.162809
$51$	0	0
$52$	−102.320	−0.272871
$53$	−382.718	−0.991894	−0.495947	−	0.868353i	$-0.665179\pi$
$53$	−382.718	−0.991894	−0.495947	+	0.868353i	$0.665179\pi$
$54$	0	0
$55$	55.0000	0.134840
$56$	−830.544	−1.98190
$57$	0	0
$58$	447.941	1.01409
$59$	771.622	1.70265	0.851327	−	0.524635i	$-0.175798\pi$
$59$	771.622	1.70265	0.851327	+	0.524635i	$0.175798\pi$
$60$	0	0
$61$	−210.576	−0.441992	−0.220996	−	0.975275i	$-0.570931\pi$
$61$	−210.576	−0.441992	−0.220996	+	0.975275i	$0.570931\pi$
$62$	−191.013	−0.391269
$63$	0	0
$64$	548.537	1.07136
$65$	189.578	0.361758
$66$	0	0
$67$	452.672	0.825413	0.412707	−	0.910864i	$-0.364584\pi$
$67$	452.672	0.825413	0.412707	+	0.910864i	$0.364584\pi$
$68$	119.500	0.213111
$69$	0	0
$70$	388.154	0.662762
$71$	−310.139	−0.518405	−0.259203	−	0.965823i	$-0.583460\pi$
$71$	−310.139	−0.518405	−0.259203	+	0.965823i	$0.583460\pi$
$72$	0	0
$73$	−174.350	−0.279536	−0.139768	−	0.990184i	$-0.544636\pi$
$73$	−174.350	−0.279536	−0.139768	+	0.990184i	$0.544636\pi$
$74$	1025.14	1.61041
$75$	0	0
$76$	114.418	0.172693
$77$	370.880	0.548905
$78$	0	0
$79$	1191.11	1.69634	0.848168	−	0.529727i	$-0.177706\pi$
$79$	1191.11	1.69634	0.848168	+	0.529727i	$0.177706\pi$
$80$	−175.641	−0.245466
$81$	0	0
$82$	474.732	0.639334
$83$	−183.669	−0.242896	−0.121448	−	0.992598i	$-0.538754\pi$
$83$	−183.669	−0.242896	−0.121448	+	0.992598i	$0.538754\pi$
$84$	0	0
$85$	−221.409	−0.282531
$86$	454.111	0.569396
$87$	0	0
$88$	−270.966	−0.328240
$89$	−613.710	−0.730934	−0.365467	−	0.930824i	$-0.619091\pi$
$89$	−613.710	−0.730934	−0.365467	+	0.930824i	$0.619091\pi$
$90$	0	0
$91$	1278.38	1.47264
$92$	−82.7923	−0.0938227
$93$	0	0
$94$	−633.873	−0.695522
$95$	−211.993	−0.228948
$96$	0	0
$97$	−1204.77	−1.26109	−0.630545	−	0.776152i	$-0.717169\pi$
$97$	−1204.77	−1.26109	−0.630545	+	0.776152i	$0.717169\pi$
$98$	1827.68	1.88392
$99$	0	0
$100$	−67.4659	−0.0674659
$101$	−1389.76	−1.36917	−0.684586	−	0.728932i	$-0.740017\pi$
$101$	−1389.76	−1.36917	−0.684586	+	0.728932i	$0.740017\pi$
$102$	0	0
$103$	−981.947	−0.939361	−0.469680	−	0.882837i	$-0.655631\pi$
$103$	−981.947	−0.939361	−0.469680	+	0.882837i	$0.655631\pi$
$104$	−933.985	−0.880623
$105$	0	0
$106$	−881.197	−0.807448
$107$	−738.649	−0.667364	−0.333682	−	0.942686i	$-0.608291\pi$
$107$	−738.649	−0.667364	−0.333682	+	0.942686i	$0.608291\pi$
$108$	0	0
$109$	599.887	0.527144	0.263572	−	0.964640i	$-0.415099\pi$
$109$	599.887	0.527144	0.263572	+	0.964640i	$0.415099\pi$
$110$	126.636	0.109766
$111$	0	0
$112$	−1184.40	−0.999242
$113$	−817.177	−0.680297	−0.340149	−	0.940372i	$-0.610477\pi$
$113$	−817.177	−0.680297	−0.340149	+	0.940372i	$0.610477\pi$
$114$	0	0
$115$	153.397	0.124385
$116$	−525.014	−0.420227
$117$	0	0
$118$	1776.64	1.38604
$119$	−1493.02	−1.15012
$120$	0	0
$121$	121.000	0.0909091
$122$	−484.845	−0.359802
$123$	0	0
$124$	223.879	0.162137
$125$	125.000	0.0894427
$126$	0	0
$127$	916.072	0.640065	0.320033	−	0.947407i	$-0.396306\pi$
$127$	916.072	0.640065	0.320033	+	0.947407i	$0.396306\pi$
$128$	333.515	0.230304
$129$	0	0
$130$	436.497	0.294487
$131$	903.159	0.602361	0.301181	−	0.953567i	$-0.402619\pi$
$131$	903.159	0.602361	0.301181	+	0.953567i	$0.402619\pi$
$132$	0	0
$133$	−1429.53	−0.931998
$134$	1042.26	0.671924
$135$	0	0
$136$	1090.80	0.687762
$137$	−2373.28	−1.48002	−0.740009	−	0.672597i	$-0.765179\pi$
$137$	−2373.28	−1.48002	−0.740009	+	0.672597i	$0.765179\pi$
$138$	0	0
$139$	126.777	0.0773601	0.0386801	−	0.999252i	$-0.487685\pi$
$139$	126.777	0.0773601	0.0386801	+	0.999252i	$0.487685\pi$
$140$	−454.941	−0.274639
$141$	0	0
$142$	−714.086	−0.422006
$143$	417.071	0.243897
$144$	0	0
$145$	972.739	0.557115
$146$	−401.436	−0.227555
$147$	0	0
$148$	−1201.53	−0.667332
$149$	−1448.83	−0.796596	−0.398298	−	0.917256i	$-0.630399\pi$
$149$	−1448.83	−0.796596	−0.398298	+	0.917256i	$0.630399\pi$
$150$	0	0
$151$	1536.43	0.828033	0.414016	−	0.910269i	$-0.364126\pi$
$151$	1536.43	0.828033	0.414016	+	0.910269i	$0.364126\pi$
$152$	1044.42	0.557325
$153$	0	0
$154$	853.940	0.446834
$155$	−414.800	−0.214952
$156$	0	0
$157$	2928.34	1.48858	0.744289	−	0.667857i	$-0.232788\pi$
$157$	2928.34	1.48858	0.744289	+	0.667857i	$0.232788\pi$
$158$	2742.50	1.38090
$159$	0	0
$160$	580.922	0.287037
$161$	1034.39	0.506346
$162$	0	0
$163$	−2064.31	−0.991957	−0.495979	−	0.868335i	$-0.665191\pi$
$163$	−2064.31	−0.991957	−0.495979	+	0.868335i	$0.665191\pi$
$164$	−556.415	−0.264931
$165$	0	0
$166$	−422.893	−0.197728
$167$	−2896.40	−1.34210	−0.671049	−	0.741413i	$-0.734156\pi$
$167$	−2896.40	−1.34210	−0.671049	+	0.741413i	$0.734156\pi$
$168$	0	0
$169$	−759.408	−0.345657
$170$	−509.786	−0.229993
$171$	0	0
$172$	−532.245	−0.235950
$173$	2938.08	1.29120	0.645601	−	0.763675i	$-0.276607\pi$
$173$	2938.08	1.29120	0.645601	+	0.763675i	$0.276607\pi$
$174$	0	0
$175$	842.909	0.364102
$176$	−386.411	−0.165493
$177$	0	0
$178$	−1413.05	−0.595013
$179$	−3352.50	−1.39987	−0.699937	−	0.714204i	$-0.746789\pi$
$179$	−3352.50	−1.39987	−0.699937	+	0.714204i	$0.746789\pi$
$180$	0	0
$181$	−4167.19	−1.71130	−0.855649	−	0.517556i	$-0.826842\pi$
$181$	−4167.19	−1.71130	−0.855649	+	0.517556i	$0.826842\pi$
$182$	2943.42	1.19880
$183$	0	0
$184$	−755.732	−0.302790
$185$	2226.18	0.884713
$186$	0	0
$187$	−487.099	−0.190482
$188$	742.938	0.288215
$189$	0	0
$190$	−488.108	−0.186374
$191$	1663.88	0.630337	0.315169	−	0.949036i	$-0.397939\pi$
$191$	1663.88	0.630337	0.315169	+	0.949036i	$0.397939\pi$
$192$	0	0
$193$	−3148.25	−1.17418	−0.587088	−	0.809523i	$-0.699726\pi$
$193$	−3148.25	−1.17418	−0.587088	+	0.809523i	$0.699726\pi$
$194$	−2773.94	−1.02659
$195$	0	0
$196$	−2142.16	−0.780669
$197$	5034.38	1.82074	0.910368	−	0.413800i	$-0.135799\pi$
$197$	5034.38	1.82074	0.910368	+	0.413800i	$0.135799\pi$
$198$	0	0
$199$	1918.95	0.683571	0.341785	−	0.939778i	$-0.388968\pi$
$199$	1918.95	0.683571	0.341785	+	0.939778i	$0.388968\pi$
$200$	−615.832	−0.217729
$201$	0	0
$202$	−3199.88	−1.11457
$203$	6559.44	2.26790
$204$	0	0
$205$	1030.92	0.351232
$206$	−2260.90	−0.764683
$207$	0	0
$208$	−1331.91	−0.443997
$209$	−466.385	−0.154357
$210$	0	0
$211$	−1200.62	−0.391725	−0.195862	−	0.980631i	$-0.562751\pi$
$211$	−1200.62	−0.391725	−0.195862	+	0.980631i	$0.562751\pi$
$212$	1032.82	0.334595
$213$	0	0
$214$	−1700.72	−0.543265
$215$	986.138	0.312810
$216$	0	0
$217$	−2797.11	−0.875025
$218$	1381.22	0.429120
$219$	0	0
$220$	−148.425	−0.0454855
$221$	−1678.97	−0.511039
$222$	0	0
$223$	3956.12	1.18799	0.593994	−	0.804469i	$-0.297550\pi$
$223$	3956.12	1.18799	0.593994	+	0.804469i	$0.297550\pi$
$224$	3917.31	1.16847
$225$	0	0
$226$	−1881.53	−0.553793
$227$	−5849.85	−1.71043	−0.855216	−	0.518272i	$-0.826576\pi$
$227$	−5849.85	−1.71043	−0.855216	+	0.518272i	$0.826576\pi$
$228$	0	0
$229$	483.283	0.139459	0.0697297	−	0.997566i	$-0.477786\pi$
$229$	483.283	0.139459	0.0697297	+	0.997566i	$0.477786\pi$
$230$	353.191	0.101255
$231$	0	0
$232$	−4792.35	−1.35618
$233$	656.670	0.184635	0.0923173	−	0.995730i	$-0.470573\pi$
$233$	656.670	0.184635	0.0923173	+	0.995730i	$0.470573\pi$
$234$	0	0
$235$	−1376.51	−0.382100
$236$	−2082.33	−0.574355
$237$	0	0
$238$	−3437.63	−0.936253
$239$	2394.73	0.648125	0.324063	−	0.946036i	$-0.394951\pi$
$239$	2394.73	0.648125	0.324063	+	0.946036i	$0.394951\pi$
$240$	0	0
$241$	−1308.22	−0.349668	−0.174834	−	0.984598i	$-0.555939\pi$
$241$	−1308.22	−0.349668	−0.174834	+	0.984598i	$0.555939\pi$
$242$	278.599	0.0740042
$243$	0	0
$244$	568.268	0.149097
$245$	3968.96	1.03497
$246$	0	0
$247$	−1607.57	−0.414118
$248$	2043.58	0.523256
$249$	0	0
$250$	287.809	0.0728105
$251$	−5173.00	−1.30086	−0.650432	−	0.759565i	$-0.725412\pi$
$251$	−5173.00	−1.30086	−0.650432	+	0.759565i	$0.725412\pi$
$252$	0	0
$253$	337.472	0.0838605
$254$	2109.23	0.521042
$255$	0	0
$256$	−3620.39	−0.883884
$257$	−7589.68	−1.84215	−0.921073	−	0.389390i	$-0.872686\pi$
$257$	−7589.68	−1.84215	−0.921073	+	0.389390i	$0.872686\pi$
$258$	0	0
$259$	15011.7	3.60148
$260$	−511.602	−0.122031
$261$	0	0
$262$	2079.50	0.490350
$263$	−6237.14	−1.46235	−0.731176	−	0.682189i	$-0.761028\pi$
$263$	−6237.14	−1.46235	−0.731176	+	0.682189i	$0.761028\pi$
$264$	0	0
$265$	−1913.59	−0.443589
$266$	−3291.44	−0.758689
$267$	0	0
$268$	−1221.60	−0.278436
$269$	2250.12	0.510009	0.255004	−	0.966940i	$-0.417923\pi$
$269$	2250.12	0.510009	0.255004	+	0.966940i	$0.417923\pi$
$270$	0	0
$271$	−2157.04	−0.483508	−0.241754	−	0.970338i	$-0.577723\pi$
$271$	−2157.04	−0.483508	−0.241754	+	0.970338i	$0.577723\pi$
$272$	1555.54	0.346759
$273$	0	0
$274$	−5464.39	−1.20480
$275$	275.000	0.0603023
$276$	0	0
$277$	6886.28	1.49371	0.746853	−	0.664990i	$-0.231564\pi$
$277$	6886.28	1.49371	0.746853	+	0.664990i	$0.231564\pi$
$278$	291.899	0.0629747
$279$	0	0
$280$	−4152.72	−0.886331
$281$	2279.62	0.483953	0.241977	−	0.970282i	$-0.422204\pi$
$281$	2279.62	0.483953	0.241977	+	0.970282i	$0.422204\pi$
$282$	0	0
$283$	−5289.42	−1.11104	−0.555519	−	0.831504i	$-0.687480\pi$
$283$	−5289.42	−1.11104	−0.555519	+	0.831504i	$0.687480\pi$
$284$	836.953	0.174873
$285$	0	0
$286$	960.294	0.198543
$287$	6951.77	1.42979
$288$	0	0
$289$	−2952.13	−0.600881
$290$	2239.70	0.453517
$291$	0	0
$292$	470.508	0.0942958
$293$	6130.70	1.22239	0.611193	−	0.791482i	$-0.290690\pi$
$293$	6130.70	1.22239	0.611193	+	0.791482i	$0.290690\pi$
$294$	0	0
$295$	3858.11	0.761450
$296$	−10967.6	−2.15365
$297$	0	0
$298$	−3335.89	−0.648465
$299$	1163.22	0.224987
$300$	0	0
$301$	6649.80	1.27338
$302$	3537.58	0.674057
$303$	0	0
$304$	1489.39	0.280995
$305$	−1052.88	−0.197665
$306$	0	0
$307$	−3627.26	−0.674328	−0.337164	−	0.941446i	$-0.609468\pi$
$307$	−3627.26	−0.674328	−0.337164	+	0.941446i	$0.609468\pi$
$308$	−1000.87	−0.185162
$309$	0	0
$310$	−955.065	−0.174981
$311$	−701.327	−0.127873	−0.0639367	−	0.997954i	$-0.520366\pi$
$311$	−701.327	−0.127873	−0.0639367	+	0.997954i	$0.520366\pi$
$312$	0	0
$313$	7338.97	1.32531	0.662657	−	0.748923i	$-0.269429\pi$
$313$	7338.97	1.32531	0.662657	+	0.748923i	$0.269429\pi$
$314$	6742.41	1.21177
$315$	0	0
$316$	−3214.38	−0.572224
$317$	−7376.07	−1.30688	−0.653440	−	0.756978i	$-0.726675\pi$
$317$	−7376.07	−1.30688	−0.653440	+	0.756978i	$0.726675\pi$
$318$	0	0
$319$	2140.03	0.375607
$320$	2742.69	0.479128
$321$	0	0
$322$	2381.66	0.412189
$323$	1877.48	0.323424
$324$	0	0
$325$	947.890	0.161783
$326$	−4753.01	−0.807499
$327$	0	0
$328$	−5078.98	−0.855000
$329$	−9282.16	−1.55545
$330$	0	0
$331$	−7801.26	−1.29546	−0.647729	−	0.761871i	$-0.724281\pi$
$331$	−7801.26	−1.29546	−0.647729	+	0.761871i	$0.724281\pi$
$332$	495.657	0.0819358
$333$	0	0
$334$	−6668.88	−1.09253
$335$	2263.36	0.369136
$336$	0	0
$337$	2829.95	0.457439	0.228720	−	0.973492i	$-0.426546\pi$
$337$	2829.95	0.457439	0.228720	+	0.973492i	$0.426546\pi$
$338$	−1748.51	−0.281381
$339$	0	0
$340$	597.501	0.0953060
$341$	−912.561	−0.144921
$342$	0	0
$343$	15199.1	2.39263
$344$	−4858.36	−0.761469
$345$	0	0
$346$	6764.83	1.05110
$347$	−1629.68	−0.252120	−0.126060	−	0.992023i	$-0.540233\pi$
$347$	−1629.68	−0.252120	−0.126060	+	0.992023i	$0.540233\pi$
$348$	0	0
$349$	−2409.67	−0.369590	−0.184795	−	0.982777i	$-0.559162\pi$
$349$	−2409.67	−0.369590	−0.184795	+	0.982777i	$0.559162\pi$
$350$	1940.77	0.296396
$351$	0	0
$352$	1278.03	0.193520
$353$	720.287	0.108603	0.0543017	−	0.998525i	$-0.482707\pi$
$353$	720.287	0.108603	0.0543017	+	0.998525i	$0.482707\pi$
$354$	0	0
$355$	−1550.70	−0.231838
$356$	1656.18	0.246565
$357$	0	0
$358$	−7719.03	−1.13956
$359$	3839.51	0.564461	0.282230	−	0.959347i	$-0.408926\pi$
$359$	3839.51	0.564461	0.282230	+	0.959347i	$0.408926\pi$
$360$	0	0
$361$	−5061.36	−0.737914
$362$	−9594.84	−1.39308
$363$	0	0
$364$	−3449.87	−0.496764
$365$	−871.751	−0.125012
$366$	0	0
$367$	3599.11	0.511913	0.255957	−	0.966688i	$-0.417610\pi$
$367$	3599.11	0.511913	0.255957	+	0.966688i	$0.417610\pi$
$368$	−1077.71	−0.152662
$369$	0	0
$370$	5125.71	0.720197
$371$	−12903.9	−1.80576
$372$	0	0
$373$	−2214.60	−0.307419	−0.153710	−	0.988116i	$-0.549122\pi$
$373$	−2214.60	−0.307419	−0.153710	+	0.988116i	$0.549122\pi$
$374$	−1121.53	−0.155061
$375$	0	0
$376$	6781.58	0.930142
$377$	7376.40	1.00770
$378$	0	0
$379$	−13631.2	−1.84747	−0.923733	−	0.383037i	$-0.874878\pi$
$379$	−13631.2	−1.84747	−0.923733	+	0.383037i	$0.874878\pi$
$380$	572.092	0.0772308
$381$	0	0
$382$	3831.04	0.513124
$383$	−7154.42	−0.954501	−0.477250	−	0.878767i	$-0.658366\pi$
$383$	−7154.42	−0.954501	−0.477250	+	0.878767i	$0.658366\pi$
$384$	0	0
$385$	1854.40	0.245478
$386$	−7248.75	−0.955833
$387$	0	0
$388$	3251.23	0.425403
$389$	−3300.42	−0.430175	−0.215088	−	0.976595i	$-0.569004\pi$
$389$	−3300.42	−0.430175	−0.215088	+	0.976595i	$0.569004\pi$
$390$	0	0
$391$	−1358.53	−0.175713
$392$	−19553.7	−2.51942
$393$	0	0
$394$	11591.5	1.48216
$395$	5955.56	0.758625
$396$	0	0
$397$	3855.87	0.487457	0.243729	−	0.969843i	$-0.421629\pi$
$397$	3855.87	0.487457	0.243729	+	0.969843i	$0.421629\pi$
$398$	4418.32	0.556458
$399$	0	0
$400$	−878.207	−0.109776
$401$	9412.25	1.17213	0.586066	−	0.810263i	$-0.300676\pi$
$401$	9412.25	1.17213	0.586066	+	0.810263i	$0.300676\pi$
$402$	0	0
$403$	−3145.48	−0.388803
$404$	3750.46	0.461862
$405$	0	0
$406$	15102.9	1.84617
$407$	4897.59	0.596474
$408$	0	0
$409$	10135.1	1.22530	0.612649	−	0.790355i	$-0.290104\pi$
$409$	10135.1	1.22530	0.612649	+	0.790355i	$0.290104\pi$
$410$	2373.66	0.285919
$411$	0	0
$412$	2649.92	0.316874
$413$	26016.3	3.09970
$414$	0	0
$415$	−918.347	−0.108626
$416$	4405.20	0.519189
$417$	0	0
$418$	−1073.84	−0.125653
$419$	6764.61	0.788718	0.394359	−	0.918956i	$-0.370967\pi$
$419$	6764.61	0.788718	0.394359	+	0.918956i	$0.370967\pi$
$420$	0	0
$421$	12417.9	1.43756	0.718778	−	0.695240i	$-0.244702\pi$
$421$	12417.9	1.43756	0.718778	+	0.695240i	$0.244702\pi$
$422$	−2764.38	−0.318882
$423$	0	0
$424$	9427.61	1.07982
$425$	−1107.04	−0.126352
$426$	0	0
$427$	−7099.86	−0.804652
$428$	1993.34	0.225121
$429$	0	0
$430$	2270.55	0.254641
$431$	−15819.9	−1.76802	−0.884010	−	0.467467i	$-0.845167\pi$
$431$	−15819.9	−1.76802	−0.884010	+	0.467467i	$0.845167\pi$
$432$	0	0
$433$	15554.9	1.72637	0.863185	−	0.504887i	$-0.168466\pi$
$433$	15554.9	1.72637	0.863185	+	0.504887i	$0.168466\pi$
$434$	−6440.26	−0.712310
$435$	0	0
$436$	−1618.87	−0.177821
$437$	−1300.76	−0.142389
$438$	0	0
$439$	−3128.47	−0.340123	−0.170061	−	0.985433i	$-0.554397\pi$
$439$	−3128.47	−0.340123	−0.170061	+	0.985433i	$0.554397\pi$
$440$	−1354.83	−0.146793
$441$	0	0
$442$	−3865.77	−0.416009
$443$	10581.2	1.13482	0.567411	−	0.823435i	$-0.307945\pi$
$443$	10581.2	1.13482	0.567411	+	0.823435i	$0.307945\pi$
$444$	0	0
$445$	−3068.55	−0.326883
$446$	9108.84	0.967077
$447$	0	0
$448$	18494.7	1.95043
$449$	−8549.76	−0.898637	−0.449318	−	0.893372i	$-0.648333\pi$
$449$	−8549.76	−0.898637	−0.449318	+	0.893372i	$0.648333\pi$
$450$	0	0
$451$	2268.02	0.236800
$452$	2205.26	0.229484
$453$	0	0
$454$	−13469.1	−1.39237
$455$	6391.88	0.658584
$456$	0	0
$457$	10920.2	1.11778	0.558888	−	0.829243i	$-0.311228\pi$
$457$	10920.2	1.11778	0.558888	+	0.829243i	$0.311228\pi$
$458$	1112.74	0.113526
$459$	0	0
$460$	−413.961	−0.0419588
$461$	−10700.4	−1.08106	−0.540528	−	0.841326i	$-0.681775\pi$
$461$	−10700.4	−1.08106	−0.540528	+	0.841326i	$0.681775\pi$
$462$	0	0
$463$	713.866	0.0716548	0.0358274	−	0.999358i	$-0.488593\pi$
$463$	713.866	0.0716548	0.0358274	+	0.999358i	$0.488593\pi$
$464$	−6834.14	−0.683765
$465$	0	0
$466$	1511.96	0.150301
$467$	11686.7	1.15802	0.579010	−	0.815320i	$-0.303439\pi$
$467$	11686.7	1.15802	0.579010	+	0.815320i	$0.303439\pi$
$468$	0	0
$469$	15262.4	1.50267
$470$	−3169.37	−0.311047
$471$	0	0
$472$	−19007.6	−1.85359
$473$	2169.50	0.210896
$474$	0	0
$475$	−1059.97	−0.102389
$476$	4029.11	0.387970
$477$	0	0
$478$	5513.78	0.527604
$479$	1443.85	0.137727	0.0688633	−	0.997626i	$-0.478063\pi$
$479$	1443.85	0.137727	0.0688633	+	0.997626i	$0.478063\pi$
$480$	0	0
$481$	16881.4	1.60026
$482$	−3012.15	−0.284646
$483$	0	0
$484$	−326.535	−0.0306663
$485$	−6023.85	−0.563977
$486$	0	0
$487$	7658.37	0.712595	0.356297	−	0.934373i	$-0.384039\pi$
$487$	7658.37	0.712595	0.356297	+	0.934373i	$0.384039\pi$
$488$	5187.18	0.481174
$489$	0	0
$490$	9138.41	0.842513
$491$	3394.27	0.311978	0.155989	−	0.987759i	$-0.450144\pi$
$491$	3394.27	0.311978	0.155989	+	0.987759i	$0.450144\pi$
$492$	0	0
$493$	−8614.91	−0.787010
$494$	−3701.38	−0.337111
$495$	0	0
$496$	2914.25	0.263818
$497$	−10456.8	−0.943763
$498$	0	0
$499$	−11997.3	−1.07630	−0.538149	−	0.842850i	$-0.680876\pi$
$499$	−11997.3	−1.07630	−0.538149	+	0.842850i	$0.680876\pi$
$500$	−337.329	−0.0301717
$501$	0	0
$502$	−11910.7	−1.05896
$503$	11969.4	1.06101	0.530506	−	0.847681i	$-0.322002\pi$
$503$	11969.4	1.06101	0.530506	+	0.847681i	$0.322002\pi$
$504$	0	0
$505$	−6948.81	−0.612313
$506$	777.020	0.0682663
$507$	0	0
$508$	−2472.14	−0.215913
$509$	−18724.8	−1.63057	−0.815286	−	0.579059i	$-0.803420\pi$
$509$	−18724.8	−1.63057	−0.815286	+	0.579059i	$0.803420\pi$
$510$	0	0
$511$	−5878.45	−0.508899
$512$	−11004.0	−0.949826
$513$	0	0
$514$	−17475.0	−1.49959
$515$	−4909.74	−0.420095
$516$	0	0
$517$	−3028.32	−0.257612
$518$	34564.0	2.93177
$519$	0	0
$520$	−4669.93	−0.393827
$521$	18285.5	1.53762	0.768812	−	0.639475i	$-0.220848\pi$
$521$	18285.5	1.53762	0.768812	+	0.639475i	$0.220848\pi$
$522$	0	0
$523$	−19025.4	−1.59068	−0.795338	−	0.606167i	$-0.792706\pi$
$523$	−19025.4	−1.59068	−0.795338	+	0.606167i	$0.792706\pi$
$524$	−2437.30	−0.203194
$525$	0	0
$526$	−14360.8	−1.19042
$527$	3673.61	0.303653
$528$	0	0
$529$	−11225.8	−0.922642
$530$	−4405.99	−0.361102
$531$	0	0
$532$	3857.77	0.314390
$533$	7817.58	0.635304
$534$	0	0
$535$	−3693.25	−0.298454
$536$	−11150.8	−0.898584
$537$	0	0
$538$	5180.84	0.415171
$539$	8731.72	0.697777
$540$	0	0
$541$	4886.50	0.388331	0.194166	−	0.980969i	$-0.437800\pi$
$541$	4886.50	0.388331	0.194166	+	0.980969i	$0.437800\pi$
$542$	−4966.51	−0.393598
$543$	0	0
$544$	−5144.84	−0.405484
$545$	2999.43	0.235746
$546$	0	0
$547$	12672.6	0.990570	0.495285	−	0.868731i	$-0.335064\pi$
$547$	12672.6	0.990570	0.495285	+	0.868731i	$0.335064\pi$
$548$	6404.60	0.499254
$549$	0	0
$550$	633.179	0.0490888
$551$	−8248.57	−0.637751
$552$	0	0
$553$	40159.9	3.08820
$554$	15855.4	1.21594
$555$	0	0
$556$	−342.124	−0.0260958
$557$	8274.65	0.629458	0.314729	−	0.949182i	$-0.398086\pi$
$557$	8274.65	0.629458	0.314729	+	0.949182i	$0.398086\pi$
$558$	0	0
$559$	7478.00	0.565807
$560$	−5921.99	−0.446875
$561$	0	0
$562$	5248.76	0.393960
$563$	−9174.35	−0.686772	−0.343386	−	0.939194i	$-0.611574\pi$
$563$	−9174.35	−0.686772	−0.343386	+	0.939194i	$0.611574\pi$
$564$	0	0
$565$	−4085.89	−0.304238
$566$	−12178.7	−0.904435
$567$	0	0
$568$	7639.75	0.564360
$569$	4686.05	0.345254	0.172627	−	0.984987i	$-0.444774\pi$
$569$	4686.05	0.345254	0.172627	+	0.984987i	$0.444774\pi$
$570$	0	0
$571$	−19892.5	−1.45792	−0.728962	−	0.684554i	$-0.759997\pi$
$571$	−19892.5	−1.45792	−0.728962	+	0.684554i	$0.759997\pi$
$572$	−1125.52	−0.0822736
$573$	0	0
$574$	16006.2	1.16391
$575$	766.983	0.0556268
$576$	0	0
$577$	−2566.17	−0.185149	−0.0925746	−	0.995706i	$-0.529510\pi$
$577$	−2566.17	−0.185149	−0.0925746	+	0.995706i	$0.529510\pi$
$578$	−6797.19	−0.489145
$579$	0	0
$580$	−2625.07	−0.187931
$581$	−6192.66	−0.442194
$582$	0	0
$583$	−4209.90	−0.299067
$584$	4294.82	0.304316
$585$	0	0
$586$	14115.7	0.995078
$587$	19097.4	1.34282	0.671409	−	0.741087i	$-0.265690\pi$
$587$	19097.4	1.34282	0.671409	+	0.741087i	$0.265690\pi$
$588$	0	0
$589$	3517.40	0.246064
$590$	8883.18	0.619855
$591$	0	0
$592$	−15640.4	−1.08584
$593$	−598.368	−0.0414368	−0.0207184	−	0.999785i	$-0.506595\pi$
$593$	−598.368	−0.0414368	−0.0207184	+	0.999785i	$0.506595\pi$
$594$	0	0
$595$	−7465.09	−0.514351
$596$	3909.86	0.268715
$597$	0	0
$598$	2678.29	0.183149
$599$	1927.27	0.131463	0.0657315	−	0.997837i	$-0.479062\pi$
$599$	1927.27	0.131463	0.0657315	+	0.997837i	$0.479062\pi$
$600$	0	0
$601$	11578.8	0.785869	0.392935	−	0.919566i	$-0.371460\pi$
$601$	11578.8	0.785869	0.392935	+	0.919566i	$0.371460\pi$
$602$	15311.0	1.03659
$603$	0	0
$604$	−4146.27	−0.279320
$605$	605.000	0.0406558
$606$	0	0
$607$	1284.90	0.0859182	0.0429591	−	0.999077i	$-0.486321\pi$
$607$	1284.90	0.0859182	0.0429591	+	0.999077i	$0.486321\pi$
$608$	−4926.06	−0.328582
$609$	0	0
$610$	−2424.23	−0.160908
$611$	−10438.2	−0.691138
$612$	0	0
$613$	24325.7	1.60278	0.801391	−	0.598141i	$-0.204094\pi$
$613$	24325.7	1.60278	0.801391	+	0.598141i	$0.204094\pi$
$614$	−8351.65	−0.548934
$615$	0	0
$616$	−9135.99	−0.597564
$617$	18016.9	1.17558	0.587789	−	0.809014i	$-0.299998\pi$
$617$	18016.9	1.17558	0.587789	+	0.809014i	$0.299998\pi$
$618$	0	0
$619$	−20542.5	−1.33388	−0.666941	−	0.745110i	$-0.732397\pi$
$619$	−20542.5	−1.33388	−0.666941	+	0.745110i	$0.732397\pi$
$620$	1119.39	0.0725097
$621$	0	0
$622$	−1614.79	−0.104095
$623$	−20692.0	−1.33067
$624$	0	0
$625$	625.000	0.0400000
$626$	16897.8	1.07887
$627$	0	0
$628$	−7902.52	−0.502141
$629$	−19715.8	−1.24979
$630$	0	0
$631$	5441.98	0.343331	0.171665	−	0.985155i	$-0.445085\pi$
$631$	5441.98	0.343331	0.171665	+	0.985155i	$0.445085\pi$
$632$	−29341.0	−1.84671
$633$	0	0
$634$	−16983.2	−1.06386
$635$	4580.36	0.286246
$636$	0	0
$637$	30097.1	1.87204
$638$	4927.35	0.305761
$639$	0	0
$640$	1667.58	0.102995
$641$	4499.67	0.277264	0.138632	−	0.990344i	$-0.455729\pi$
$641$	4499.67	0.277264	0.138632	+	0.990344i	$0.455729\pi$
$642$	0	0
$643$	2929.30	0.179659	0.0898293	−	0.995957i	$-0.471368\pi$
$643$	2929.30	0.179659	0.0898293	+	0.995957i	$0.471368\pi$
$644$	−2791.45	−0.170805
$645$	0	0
$646$	4322.85	0.263282
$647$	−20128.9	−1.22311	−0.611554	−	0.791203i	$-0.709455\pi$
$647$	−20128.9	−1.22311	−0.611554	+	0.791203i	$0.709455\pi$
$648$	0	0
$649$	8487.84	0.513370
$650$	2182.49	0.131699
$651$	0	0
$652$	5570.81	0.334616
$653$	13802.2	0.827141	0.413571	−	0.910472i	$-0.364281\pi$
$653$	13802.2	0.827141	0.413571	+	0.910472i	$0.364281\pi$
$654$	0	0
$655$	4515.79	0.269384
$656$	−7242.88	−0.431078
$657$	0	0
$658$	−21371.9	−1.26621
$659$	29377.3	1.73653	0.868267	−	0.496097i	$-0.165234\pi$
$659$	29377.3	1.73653	0.868267	+	0.496097i	$0.165234\pi$
$660$	0	0
$661$	−22438.5	−1.32036	−0.660179	−	0.751108i	$-0.729520\pi$
$661$	−22438.5	−1.32036	−0.660179	+	0.751108i	$0.729520\pi$
$662$	−17962.2	−1.05456
$663$	0	0
$664$	4524.38	0.264428
$665$	−7147.64	−0.416802
$666$	0	0
$667$	5968.60	0.346484
$668$	7816.33	0.452729
$669$	0	0
$670$	5211.32	0.300494
$671$	−2316.34	−0.133266
$672$	0	0
$673$	−31739.0	−1.81790	−0.908951	−	0.416903i	$-0.863116\pi$
$673$	−31739.0	−1.81790	−0.908951	+	0.416903i	$0.863116\pi$
$674$	6515.87	0.372377
$675$	0	0
$676$	2049.36	0.116600
$677$	28959.9	1.64404	0.822022	−	0.569456i	$-0.192846\pi$
$677$	28959.9	1.64404	0.822022	+	0.569456i	$0.192846\pi$
$678$	0	0
$679$	−40620.4	−2.29583
$680$	5454.02	0.307577
$681$	0	0
$682$	−2101.14	−0.117972
$683$	−20978.4	−1.17528	−0.587640	−	0.809122i	$-0.699943\pi$
$683$	−20978.4	−1.17528	−0.587640	+	0.809122i	$0.699943\pi$
$684$	0	0
$685$	−11866.4	−0.661885
$686$	34995.4	1.94771
$687$	0	0
$688$	−6928.27	−0.383921
$689$	−14511.0	−0.802358
$690$	0	0
$691$	−14713.7	−0.810037	−0.405018	−	0.914309i	$-0.632735\pi$
$691$	−14713.7	−0.810037	−0.405018	+	0.914309i	$0.632735\pi$
$692$	−7928.80	−0.435560
$693$	0	0
$694$	−3752.29	−0.205238
$695$	633.883	0.0345965
$696$	0	0
$697$	−9130.17	−0.496169
$698$	−5548.20	−0.300863
$699$	0	0
$700$	−2274.70	−0.122822
$701$	−30033.8	−1.61821	−0.809103	−	0.587667i	$-0.800046\pi$
$701$	−30033.8	−1.61821	−0.809103	+	0.587667i	$0.800046\pi$
$702$	0	0
$703$	−18877.4	−1.01277
$704$	6033.91	0.323028
$705$	0	0
$706$	1658.44	0.0884082
$707$	−46857.7	−2.49260
$708$	0	0
$709$	6202.27	0.328535	0.164267	−	0.986416i	$-0.447474\pi$
$709$	6202.27	0.328535	0.164267	+	0.986416i	$0.447474\pi$
$710$	−3570.43	−0.188727
$711$	0	0
$712$	15117.7	0.795729
$713$	−2545.16	−0.133684
$714$	0	0
$715$	2085.36	0.109074
$716$	9047.17	0.472219
$717$	0	0
$718$	8840.35	0.459497
$719$	28650.6	1.48608	0.743038	−	0.669250i	$-0.233384\pi$
$719$	28650.6	1.48608	0.743038	+	0.669250i	$0.233384\pi$
$720$	0	0
$721$	−33107.7	−1.71012
$722$	−11653.6	−0.600696
$723$	0	0
$724$	11245.7	0.577271
$725$	4863.70	0.249149
$726$	0	0
$727$	−11144.9	−0.568558	−0.284279	−	0.958742i	$-0.591754\pi$
$727$	−11144.9	−0.568558	−0.284279	+	0.958742i	$0.591754\pi$
$728$	−31490.6	−1.60318
$729$	0	0
$730$	−2007.18	−0.101766
$731$	−8733.58	−0.441892
$732$	0	0
$733$	−28546.2	−1.43844	−0.719221	−	0.694781i	$-0.755501\pi$
$733$	−28546.2	−1.43844	−0.719221	+	0.694781i	$0.755501\pi$
$734$	8286.84	0.416721
$735$	0	0
$736$	3564.46	0.178516
$737$	4979.39	0.248871
$738$	0	0
$739$	−19992.4	−0.995175	−0.497587	−	0.867414i	$-0.665781\pi$
$739$	−19992.4	−0.995175	−0.497587	+	0.867414i	$0.665781\pi$
$740$	−6007.64	−0.298440
$741$	0	0
$742$	−29710.8	−1.46997
$743$	−10763.2	−0.531446	−0.265723	−	0.964049i	$-0.585611\pi$
$743$	−10763.2	−0.531446	−0.265723	+	0.964049i	$0.585611\pi$
$744$	0	0
$745$	−7244.15	−0.356248
$746$	−5099.04	−0.250253
$747$	0	0
$748$	1314.50	0.0642553
$749$	−24904.6	−1.21494
$750$	0	0
$751$	3243.24	0.157586	0.0787932	−	0.996891i	$-0.474893\pi$
$751$	3243.24	0.157586	0.0787932	+	0.996891i	$0.474893\pi$
$752$	9670.87	0.468963
$753$	0	0
$754$	16983.9	0.820316
$755$	7682.15	0.370308
$756$	0	0
$757$	11503.0	0.552288	0.276144	−	0.961116i	$-0.410943\pi$
$757$	11503.0	0.552288	0.276144	+	0.961116i	$0.410943\pi$
$758$	−31385.5	−1.50392
$759$	0	0
$760$	5222.09	0.249243
$761$	−36993.2	−1.76216	−0.881080	−	0.472968i	$-0.843183\pi$
$761$	−36993.2	−1.76216	−0.881080	+	0.472968i	$0.843183\pi$
$762$	0	0
$763$	20226.0	0.959672
$764$	−4490.22	−0.212631
$765$	0	0
$766$	−16472.8	−0.777007
$767$	29256.5	1.37730
$768$	0	0
$769$	−21702.1	−1.01768	−0.508841	−	0.860861i	$-0.669926\pi$
$769$	−21702.1	−1.01768	−0.508841	+	0.860861i	$0.669926\pi$
$770$	4269.70	0.199830
$771$	0	0
$772$	8495.98	0.396084
$773$	−14968.0	−0.696459	−0.348230	−	0.937409i	$-0.613217\pi$
$773$	−14968.0	−0.696459	−0.348230	+	0.937409i	$0.613217\pi$
$774$	0	0
$775$	−2074.00	−0.0961295
$776$	29677.4	1.37288
$777$	0	0
$778$	−7599.13	−0.350182
$779$	−8741.91	−0.402069
$780$	0	0
$781$	−3411.53	−0.156305
$782$	−3127.98	−0.143039
$783$	0	0
$784$	−27884.6	−1.27025
$785$	14641.7	0.665713
$786$	0	0
$787$	−43780.7	−1.98299	−0.991494	−	0.130149i	$-0.958455\pi$
$787$	−43780.7	−1.98299	−0.991494	+	0.130149i	$0.958455\pi$
$788$	−13586.0	−0.614187
$789$	0	0
$790$	13712.5	0.617555
$791$	−27552.2	−1.23849
$792$	0	0
$793$	−7984.12	−0.357534
$794$	8878.02	0.396813
$795$	0	0
$796$	−5178.54	−0.230588
$797$	4398.73	0.195497	0.0977484	−	0.995211i	$-0.468836\pi$
$797$	4398.73	0.195497	0.0977484	+	0.995211i	$0.468836\pi$
$798$	0	0
$799$	12190.8	0.539775
$800$	2904.61	0.128367
$801$	0	0
$802$	21671.4	0.954170
$803$	−1917.85	−0.0842834
$804$	0	0
$805$	5171.97	0.226445
$806$	−7242.37	−0.316503
$807$	0	0
$808$	34234.4	1.49055
$809$	164.162	0.00713427	0.00356714	−	0.999994i	$-0.498865\pi$
$809$	164.162	0.00713427	0.00356714	+	0.999994i	$0.498865\pi$
$810$	0	0
$811$	−35517.8	−1.53785	−0.768926	−	0.639338i	$-0.779209\pi$
$811$	−35517.8	−1.53785	−0.768926	+	0.639338i	$0.779209\pi$
$812$	−17701.5	−0.765028
$813$	0	0
$814$	11276.6	0.485557
$815$	−10321.5	−0.443617
$816$	0	0
$817$	−8362.18	−0.358085
$818$	23335.7	0.997449
$819$	0	0
$820$	−2782.07	−0.118481
$821$	8770.33	0.372822	0.186411	−	0.982472i	$-0.440314\pi$
$821$	8770.33	0.372822	0.186411	+	0.982472i	$0.440314\pi$
$822$	0	0
$823$	9344.99	0.395803	0.197902	−	0.980222i	$-0.436587\pi$
$823$	9344.99	0.395803	0.197902	+	0.980222i	$0.436587\pi$
$824$	24188.6	1.02263
$825$	0	0
$826$	59901.7	2.52330
$827$	37820.4	1.59026	0.795130	−	0.606439i	$-0.207403\pi$
$827$	37820.4	1.59026	0.795130	+	0.606439i	$0.207403\pi$
$828$	0	0
$829$	8605.31	0.360525	0.180262	−	0.983619i	$-0.442305\pi$
$829$	8605.31	0.360525	0.180262	+	0.983619i	$0.442305\pi$
$830$	−2114.47	−0.0884267
$831$	0	0
$832$	20798.1	0.866641
$833$	−35150.5	−1.46206
$834$	0	0
$835$	−14482.0	−0.600205
$836$	1258.60	0.0520690
$837$	0	0
$838$	15575.3	0.642053
$839$	46477.5	1.91249	0.956247	−	0.292560i	$-0.0945071\pi$
$839$	46477.5	1.91249	0.956247	+	0.292560i	$0.0945071\pi$
$840$	0	0
$841$	13459.9	0.551883
$842$	28591.8	1.17024
$843$	0	0
$844$	3240.03	0.132140
$845$	−3797.04	−0.154582
$846$	0	0
$847$	4079.68	0.165501
$848$	13444.2	0.544431
$849$	0	0
$850$	−2548.93	−0.102856
$851$	13659.5	0.550226
$852$	0	0
$853$	3867.32	0.155234	0.0776169	−	0.996983i	$-0.475269\pi$
$853$	3867.32	0.155234	0.0776169	+	0.996983i	$0.475269\pi$
$854$	−16347.2	−0.655024
$855$	0	0
$856$	18195.4	0.726524
$857$	−37944.2	−1.51243	−0.756213	−	0.654325i	$-0.772953\pi$
$857$	−37944.2	−1.51243	−0.756213	+	0.654325i	$0.772953\pi$
$858$	0	0
$859$	9438.58	0.374901	0.187451	−	0.982274i	$-0.439978\pi$
$859$	9438.58	0.374901	0.187451	+	0.982274i	$0.439978\pi$
$860$	−2661.23	−0.105520
$861$	0	0
$862$	−36424.8	−1.43925
$863$	2626.80	0.103612	0.0518061	−	0.998657i	$-0.483502\pi$
$863$	2626.80	0.103612	0.0518061	+	0.998657i	$0.483502\pi$
$864$	0	0
$865$	14690.4	0.577443
$866$	35814.6	1.40534
$867$	0	0
$868$	7548.38	0.295171
$869$	13102.2	0.511465
$870$	0	0
$871$	17163.3	0.667689
$872$	−14777.2	−0.573874
$873$	0	0
$874$	−2994.96	−0.115911
$875$	4214.54	0.162832
$876$	0	0
$877$	−39926.8	−1.53732	−0.768662	−	0.639655i	$-0.779077\pi$
$877$	−39926.8	−1.53732	−0.768662	+	0.639655i	$0.779077\pi$
$878$	−7203.22	−0.276876
$879$	0	0
$880$	−1932.06	−0.0740109
$881$	48387.5	1.85041	0.925207	−	0.379462i	$-0.123891\pi$
$881$	48387.5	1.85041	0.925207	+	0.379462i	$0.123891\pi$
$882$	0	0
$883$	−24690.2	−0.940986	−0.470493	−	0.882404i	$-0.655924\pi$
$883$	−24690.2	−0.940986	−0.470493	+	0.882404i	$0.655924\pi$
$884$	4530.92	0.172388
$885$	0	0
$886$	24362.8	0.923797
$887$	25504.5	0.965453	0.482726	−	0.875771i	$-0.339647\pi$
$887$	25504.5	0.965453	0.482726	+	0.875771i	$0.339647\pi$
$888$	0	0
$889$	30886.6	1.16525
$890$	−7065.24	−0.266098
$891$	0	0
$892$	−10676.1	−0.400743
$893$	11672.4	0.437405
$894$	0	0
$895$	−16762.5	−0.626043
$896$	11244.9	0.419271
$897$	0	0
$898$	−19685.6	−0.731532
$899$	−16139.7	−0.598765
$900$	0	0
$901$	16947.4	0.626638
$902$	5222.05	0.192766
$903$	0	0
$904$	20129.8	0.740604
$905$	−20836.0	−0.765316
$906$	0	0
$907$	4050.02	0.148268	0.0741338	−	0.997248i	$-0.476381\pi$
$907$	4050.02	0.148268	0.0741338	+	0.997248i	$0.476381\pi$
$908$	15786.6	0.576979
$909$	0	0
$910$	14717.1	0.536118
$911$	10405.9	0.378445	0.189223	−	0.981934i	$-0.439403\pi$
$911$	10405.9	0.378445	0.189223	+	0.981934i	$0.439403\pi$
$912$	0	0
$913$	−2020.36	−0.0732358
$914$	25143.4	0.909922
$915$	0	0
$916$	−1304.20	−0.0470438
$917$	30451.2	1.09661
$918$	0	0
$919$	31719.1	1.13854	0.569268	−	0.822152i	$-0.307227\pi$
$919$	31719.1	1.13854	0.569268	+	0.822152i	$0.307227\pi$
$920$	−3778.66	−0.135412
$921$	0	0
$922$	−24637.3	−0.880029
$923$	−11759.1	−0.419346
$924$	0	0
$925$	11130.9	0.395656
$926$	1643.65	0.0583303
$927$	0	0
$928$	22603.4	0.799562
$929$	−36958.3	−1.30523	−0.652617	−	0.757688i	$-0.726329\pi$
$929$	−36958.3	−1.30523	−0.652617	+	0.757688i	$0.726329\pi$
$930$	0	0
$931$	−33655.7	−1.18477
$932$	−1772.11	−0.0622827
$933$	0	0
$934$	26908.2	0.942681
$935$	−2435.49	−0.0851863
$936$	0	0
$937$	−10365.2	−0.361383	−0.180691	−	0.983540i	$-0.557833\pi$
$937$	−10365.2	−0.361383	−0.180691	+	0.983540i	$0.557833\pi$
$938$	35141.3	1.22325
$939$	0	0
$940$	3714.69	0.128893
$941$	10350.4	0.358570	0.179285	−	0.983797i	$-0.442622\pi$
$941$	10350.4	0.358570	0.179285	+	0.983797i	$0.442622\pi$
$942$	0	0
$943$	6325.58	0.218440
$944$	−27105.8	−0.934552
$945$	0	0
$946$	4995.22	0.171679
$947$	24145.1	0.828521	0.414260	−	0.910158i	$-0.364040\pi$
$947$	24145.1	0.828521	0.414260	+	0.910158i	$0.364040\pi$
$948$	0	0
$949$	−6610.59	−0.226121
$950$	−2440.54	−0.0833490
$951$	0	0
$952$	36777.9	1.25208
$953$	−53268.6	−1.81064	−0.905319	−	0.424732i	$-0.860368\pi$
$953$	−53268.6	−1.81064	−0.905319	+	0.424732i	$0.860368\pi$
$954$	0	0
$955$	8319.42	0.281895
$956$	−6462.49	−0.218632
$957$	0	0
$958$	3324.41	0.112116
$959$	−80018.2	−2.69439
$960$	0	0
$961$	−22908.6	−0.768978
$962$	38868.9	1.30268
$963$	0	0
$964$	3530.42	0.117953
$965$	−15741.2	−0.525107
$966$	0	0
$967$	−10029.7	−0.333541	−0.166771	−	0.985996i	$-0.553334\pi$
$967$	−10029.7	−0.333541	−0.166771	+	0.985996i	$0.553334\pi$
$968$	−2980.63	−0.0989679
$969$	0	0
$970$	−13869.7	−0.459103
$971$	42152.5	1.39314	0.696569	−	0.717489i	$-0.254709\pi$
$971$	42152.5	1.39314	0.696569	+	0.717489i	$0.254709\pi$
$972$	0	0
$973$	4274.45	0.140835
$974$	17633.2	0.580085
$975$	0	0
$976$	7397.18	0.242601
$977$	40437.3	1.32416	0.662079	−	0.749434i	$-0.269674\pi$
$977$	40437.3	1.32416	0.662079	+	0.749434i	$0.269674\pi$
$978$	0	0
$979$	−6750.81	−0.220385
$980$	−10710.8	−0.349126
$981$	0	0
$982$	7815.21	0.253965
$983$	−40951.5	−1.32874	−0.664369	−	0.747405i	$-0.731299\pi$
$983$	−40951.5	−1.32874	−0.664369	+	0.747405i	$0.731299\pi$
$984$	0	0
$985$	25171.9	0.814258
$986$	−19835.6	−0.640663
$987$	0	0
$988$	4338.24	0.139694
$989$	6050.81	0.194545
$990$	0	0
$991$	255.910	0.00820306	0.00410153	−	0.999992i	$-0.498694\pi$
$991$	255.910	0.00820306	0.00410153	+	0.999992i	$0.498694\pi$
$992$	−9638.67	−0.308496
$993$	0	0
$994$	−24076.4	−0.768266
$995$	9594.73	0.305702
$996$	0	0
$997$	14729.2	0.467883	0.233941	−	0.972251i	$-0.424838\pi$
$997$	14729.2	0.467883	0.233941	+	0.972251i	$0.424838\pi$
$998$	−27623.4	−0.876156
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.4.a.p.1.5	yes	7
3.2	odd	2		495.4.a.o.1.3	✓	7
5.4	even	2		2475.4.a.bp.1.3		7
15.14	odd	2		2475.4.a.bt.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.4.a.o.1.3	✓	7	3.2	odd	2
495.4.a.p.1.5	yes	7	1.1	even	1	trivial
2475.4.a.bp.1.3		7	5.4	even	2
2475.4.a.bt.1.5		7	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	495.a (trivial)

\(f(q)\)	\(=\)	\(q+2.30247 q^{2} -2.69863 q^{4} +5.00000 q^{5} +33.7164 q^{7} -24.6333 q^{8} +O(q^{10})\)	\(q+2.30247 q^{2} -2.69863 q^{4} +5.00000 q^{5} +33.7164 q^{7} -24.6333 q^{8} +11.5123 q^{10} +11.0000 q^{11} +37.9156 q^{13} +77.6309 q^{14} -35.1283 q^{16} -44.2817 q^{17} -42.3986 q^{19} -13.4932 q^{20} +25.3272 q^{22} +30.6793 q^{23} +25.0000 q^{25} +87.2995 q^{26} -90.9881 q^{28} +194.548 q^{29} -82.9601 q^{31} +116.184 q^{32} -101.957 q^{34} +168.582 q^{35} +445.236 q^{37} -97.6216 q^{38} -123.166 q^{40} +206.184 q^{41} +197.228 q^{43} -29.6850 q^{44} +70.6382 q^{46} -275.301 q^{47} +793.792 q^{49} +57.5617 q^{50} -102.320 q^{52} -382.718 q^{53} +55.0000 q^{55} -830.544 q^{56} +447.941 q^{58} +771.622 q^{59} -210.576 q^{61} -191.013 q^{62} +548.537 q^{64} +189.578 q^{65} +452.672 q^{67} +119.500 q^{68} +388.154 q^{70} -310.139 q^{71} -174.350 q^{73} +1025.14 q^{74} +114.418 q^{76} +370.880 q^{77} +1191.11 q^{79} -175.641 q^{80} +474.732 q^{82} -183.669 q^{83} -221.409 q^{85} +454.111 q^{86} -270.966 q^{88} -613.710 q^{89} +1278.38 q^{91} -82.7923 q^{92} -633.873 q^{94} -211.993 q^{95} -1204.77 q^{97} +1827.68 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 5 q^{2} + 33 q^{4} + 35 q^{5} + 30 q^{7} + 45 q^{8}+O(q^{10}) \) 7 * q + 5 * q^2 + 33 * q^4 + 35 * q^5 + 30 * q^7 + 45 * q^8	\( 7 q + 5 q^{2} + 33 q^{4} + 35 q^{5} + 30 q^{7} + 45 q^{8} + 25 q^{10} + 77 q^{11} + 38 q^{13} + 20 q^{14} + 309 q^{16} + 12 q^{17} + 226 q^{19} + 165 q^{20} + 55 q^{22} + 334 q^{23} + 175 q^{25} - 372 q^{26} + 812 q^{28} - 258 q^{29} + 336 q^{31} + 485 q^{32} + 78 q^{34} + 150 q^{35} + 466 q^{37} - 494 q^{38} + 225 q^{40} - 258 q^{41} + 308 q^{43} + 363 q^{44} + 98 q^{46} + 546 q^{47} + 735 q^{49} + 125 q^{50} + 512 q^{52} + 110 q^{53} + 385 q^{55} + 20 q^{56} + 1362 q^{58} - 68 q^{59} + 1096 q^{61} + 356 q^{62} + 2761 q^{64} + 190 q^{65} + 2268 q^{67} - 1186 q^{68} + 100 q^{70} - 166 q^{71} + 200 q^{73} - 1710 q^{74} + 3310 q^{76} + 330 q^{77} + 2152 q^{79} + 1545 q^{80} - 1006 q^{82} + 370 q^{83} + 60 q^{85} + 106 q^{86} + 495 q^{88} - 252 q^{89} + 2768 q^{91} + 3774 q^{92} + 2218 q^{94} + 1130 q^{95} + 3698 q^{97} + 697 q^{98}+O(q^{100}) \) 7 * q + 5 * q^2 + 33 * q^4 + 35 * q^5 + 30 * q^7 + 45 * q^8 + 25 * q^10 + 77 * q^11 + 38 * q^13 + 20 * q^14 + 309 * q^16 + 12 * q^17 + 226 * q^19 + 165 * q^20 + 55 * q^22 + 334 * q^23 + 175 * q^25 - 372 * q^26 + 812 * q^28 - 258 * q^29 + 336 * q^31 + 485 * q^32 + 78 * q^34 + 150 * q^35 + 466 * q^37 - 494 * q^38 + 225 * q^40 - 258 * q^41 + 308 * q^43 + 363 * q^44 + 98 * q^46 + 546 * q^47 + 735 * q^49 + 125 * q^50 + 512 * q^52 + 110 * q^53 + 385 * q^55 + 20 * q^56 + 1362 * q^58 - 68 * q^59 + 1096 * q^61 + 356 * q^62 + 2761 * q^64 + 190 * q^65 + 2268 * q^67 - 1186 * q^68 + 100 * q^70 - 166 * q^71 + 200 * q^73 - 1710 * q^74 + 3310 * q^76 + 330 * q^77 + 2152 * q^79 + 1545 * q^80 - 1006 * q^82 + 370 * q^83 + 60 * q^85 + 106 * q^86 + 495 * q^88 - 252 * q^89 + 2768 * q^91 + 3774 * q^92 + 2218 * q^94 + 1130 * q^95 + 3698 * q^97 + 697 * q^98

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