LMFDB - Embedded newform 495.4.a.p.1.4

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			1.4
Root			$-0.434062$ 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)

$f(q)$	$=$	$q+1.43406 q^{2} -5.94347 q^{4} +5.00000 q^{5} -23.1010 q^{7} -19.9958 q^{8} +O(q^{10})$	\(q+1.43406 q^{2} -5.94347 q^{4} +5.00000 q^{5} -23.1010 q^{7} -19.9958 q^{8} +7.17031 q^{10} +11.0000 q^{11} -85.3360 q^{13} -33.1283 q^{14} +18.8725 q^{16} +109.083 q^{17} +121.498 q^{19} -29.7173 q^{20} +15.7747 q^{22} +171.169 q^{23} +25.0000 q^{25} -122.377 q^{26} +137.300 q^{28} +80.1854 q^{29} -255.029 q^{31} +187.031 q^{32} +156.432 q^{34} -115.505 q^{35} +103.684 q^{37} +174.236 q^{38} -99.9790 q^{40} -194.671 q^{41} +434.049 q^{43} -65.3781 q^{44} +245.467 q^{46} -459.463 q^{47} +190.656 q^{49} +35.8516 q^{50} +507.191 q^{52} +656.774 q^{53} +55.0000 q^{55} +461.923 q^{56} +114.991 q^{58} -158.957 q^{59} +107.192 q^{61} -365.727 q^{62} +117.234 q^{64} -426.680 q^{65} +685.810 q^{67} -648.332 q^{68} -165.641 q^{70} +671.377 q^{71} -260.221 q^{73} +148.689 q^{74} -722.121 q^{76} -254.111 q^{77} +361.102 q^{79} +94.3625 q^{80} -279.170 q^{82} +706.521 q^{83} +545.416 q^{85} +622.453 q^{86} -219.954 q^{88} -557.720 q^{89} +1971.35 q^{91} -1017.34 q^{92} -658.898 q^{94} +607.491 q^{95} +482.897 q^{97} +273.413 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.43406	0.507018	0.253509	−	0.967333i	$-0.418415\pi$
$2$	1.43406	0.507018	0.253509	+	0.967333i	$0.418415\pi$
$3$	0	0
$3$	0	0
$4$	−5.94347	−0.742933
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−23.1010	−1.24734	−0.623668	−	0.781689i	$-0.714358\pi$
$7$	−23.1010	−1.24734	−0.623668	+	0.781689i	$0.714358\pi$
$8$	−19.9958	−0.883698
$9$	0	0
$10$	7.17031	0.226745
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−85.3360	−1.82061	−0.910305	−	0.413938i	$-0.864153\pi$
$13$	−85.3360	−1.82061	−0.910305	+	0.413938i	$0.864153\pi$
$14$	−33.1283	−0.632422
$15$	0	0
$16$	18.8725	0.294883
$17$	109.083	1.55627	0.778134	−	0.628098i	$-0.216167\pi$
$17$	109.083	1.55627	0.778134	+	0.628098i	$0.216167\pi$
$18$	0	0
$19$	121.498	1.46703	0.733516	−	0.679672i	$-0.237878\pi$
$19$	121.498	1.46703	0.733516	+	0.679672i	$0.237878\pi$
$20$	−29.7173	−0.332250
$21$	0	0
$22$	15.7747	0.152872
$23$	171.169	1.55179	0.775896	−	0.630860i	$-0.217298\pi$
$23$	171.169	1.55179	0.775896	+	0.630860i	$0.217298\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−122.377	−0.923082
$27$	0	0
$28$	137.300	0.926688
$29$	80.1854	0.513450	0.256725	−	0.966485i	$-0.417357\pi$
$29$	80.1854	0.513450	0.256725	+	0.966485i	$0.417357\pi$
$30$	0	0
$31$	−255.029	−1.47756	−0.738782	−	0.673944i	$-0.764599\pi$
$31$	−255.029	−1.47756	−0.738782	+	0.673944i	$0.764599\pi$
$32$	187.031	1.03321
$33$	0	0
$34$	156.432	0.789055
$35$	−115.505	−0.557826
$36$	0	0
$37$	103.684	0.460689	0.230345	−	0.973109i	$-0.426015\pi$
$37$	103.684	0.460689	0.230345	+	0.973109i	$0.426015\pi$
$38$	174.236	0.743811
$39$	0	0
$40$	−99.9790	−0.395202
$41$	−194.671	−0.741523	−0.370761	−	0.928728i	$-0.620903\pi$
$41$	−194.671	−0.741523	−0.370761	+	0.928728i	$0.620903\pi$
$42$	0	0
$43$	434.049	1.53934	0.769672	−	0.638439i	$-0.220420\pi$
$43$	434.049	1.53934	0.769672	+	0.638439i	$0.220420\pi$
$44$	−65.3781	−0.224003
$45$	0	0
$46$	245.467	0.786786
$47$	−459.463	−1.42595	−0.712973	−	0.701191i	$-0.752652\pi$
$47$	−459.463	−1.42595	−0.712973	+	0.701191i	$0.752652\pi$
$48$	0	0
$49$	190.656	0.555849
$50$	35.8516	0.101404
$51$	0	0
$52$	507.191	1.35259
$53$	656.774	1.70217	0.851084	−	0.525029i	$-0.175946\pi$
$53$	656.774	1.70217	0.851084	+	0.525029i	$0.175946\pi$
$54$	0	0
$55$	55.0000	0.134840
$56$	461.923	1.10227
$57$	0	0
$58$	114.991	0.260328
$59$	−158.957	−0.350753	−0.175377	−	0.984501i	$-0.556114\pi$
$59$	−158.957	−0.350753	−0.175377	+	0.984501i	$0.556114\pi$
$60$	0	0
$61$	107.192	0.224992	0.112496	−	0.993652i	$-0.464115\pi$
$61$	107.192	0.224992	0.112496	+	0.993652i	$0.464115\pi$
$62$	−365.727	−0.749151
$63$	0	0
$64$	117.234	0.228972
$65$	−426.680	−0.814202
$66$	0	0
$67$	685.810	1.25052	0.625261	−	0.780416i	$-0.284992\pi$
$67$	685.810	1.25052	0.625261	+	0.780416i	$0.284992\pi$
$68$	−648.332	−1.15620
$69$	0	0
$70$	−165.641	−0.282828
$71$	671.377	1.12222	0.561111	−	0.827741i	$-0.310374\pi$
$71$	671.377	1.12222	0.561111	+	0.827741i	$0.310374\pi$
$72$	0	0
$73$	−260.221	−0.417213	−0.208606	−	0.978000i	$-0.566893\pi$
$73$	−260.221	−0.417213	−0.208606	+	0.978000i	$0.566893\pi$
$74$	148.689	0.233578
$75$	0	0
$76$	−722.121	−1.08991
$77$	−254.111	−0.376086
$78$	0	0
$79$	361.102	0.514267	0.257134	−	0.966376i	$-0.417222\pi$
$79$	361.102	0.514267	0.257134	+	0.966376i	$0.417222\pi$
$80$	94.3625	0.131876
$81$	0	0
$82$	−279.170	−0.375965
$83$	706.521	0.934346	0.467173	−	0.884166i	$-0.345273\pi$
$83$	706.521	0.934346	0.467173	+	0.884166i	$0.345273\pi$
$84$	0	0
$85$	545.416	0.695984
$86$	622.453	0.780475
$87$	0	0
$88$	−219.954	−0.266445
$89$	−557.720	−0.664249	−0.332124	−	0.943236i	$-0.607765\pi$
$89$	−557.720	−0.664249	−0.332124	+	0.943236i	$0.607765\pi$
$90$	0	0
$91$	1971.35	2.27091
$92$	−1017.34	−1.15288
$93$	0	0
$94$	−658.898	−0.722980
$95$	607.491	0.656077
$96$	0	0
$97$	482.897	0.505472	0.252736	−	0.967535i	$-0.418670\pi$
$97$	482.897	0.505472	0.252736	+	0.967535i	$0.418670\pi$
$98$	273.413	0.281825
$99$	0	0
$100$	−148.587	−0.148587
$101$	−900.150	−0.886815	−0.443408	−	0.896320i	$-0.646231\pi$
$101$	−900.150	−0.886815	−0.443408	+	0.896320i	$0.646231\pi$
$102$	0	0
$103$	757.943	0.725072	0.362536	−	0.931970i	$-0.381911\pi$
$103$	757.943	0.725072	0.362536	+	0.931970i	$0.381911\pi$
$104$	1706.36	1.60887
$105$	0	0
$106$	941.856	0.863029
$107$	−793.166	−0.716619	−0.358310	−	0.933603i	$-0.616647\pi$
$107$	−793.166	−0.716619	−0.358310	+	0.933603i	$0.616647\pi$
$108$	0	0
$109$	1232.47	1.08302	0.541508	−	0.840696i	$-0.317854\pi$
$109$	1232.47	1.08302	0.541508	+	0.840696i	$0.317854\pi$
$110$	78.8734	0.0683662
$111$	0	0
$112$	−435.974	−0.367818
$113$	1255.66	1.04533	0.522664	−	0.852539i	$-0.324938\pi$
$113$	1255.66	1.04533	0.522664	+	0.852539i	$0.324938\pi$
$114$	0	0
$115$	855.846	0.693983
$116$	−476.579	−0.381459
$117$	0	0
$118$	−227.954	−0.177838
$119$	−2519.93	−1.94119
$120$	0	0
$121$	121.000	0.0909091
$122$	153.720	0.114075
$123$	0	0
$124$	1515.75	1.09773
$125$	125.000	0.0894427
$126$	0	0
$127$	−2021.58	−1.41249	−0.706245	−	0.707968i	$-0.749612\pi$
$127$	−2021.58	−1.41249	−0.706245	+	0.707968i	$0.749612\pi$
$128$	−1328.13	−0.917116
$129$	0	0
$130$	−611.885	−0.412815
$131$	−2782.03	−1.85548	−0.927738	−	0.373232i	$-0.878250\pi$
$131$	−2782.03	−1.85548	−0.927738	+	0.373232i	$0.878250\pi$
$132$	0	0
$133$	−2806.73	−1.82988
$134$	983.494	0.634037
$135$	0	0
$136$	−2181.20	−1.37527
$137$	481.615	0.300344	0.150172	−	0.988660i	$-0.452017\pi$
$137$	481.615	0.300344	0.150172	+	0.988660i	$0.452017\pi$
$138$	0	0
$139$	−252.051	−0.153803	−0.0769017	−	0.997039i	$-0.524503\pi$
$139$	−252.051	−0.153803	−0.0769017	+	0.997039i	$0.524503\pi$
$140$	686.500	0.414427
$141$	0	0
$142$	962.796	0.568986
$143$	−938.696	−0.548935
$144$	0	0
$145$	400.927	0.229622
$146$	−373.173	−0.211534
$147$	0	0
$148$	−616.241	−0.342261
$149$	2393.86	1.31619	0.658096	−	0.752934i	$-0.271362\pi$
$149$	2393.86	1.31619	0.658096	+	0.752934i	$0.271362\pi$
$150$	0	0
$151$	−2799.55	−1.50877	−0.754384	−	0.656434i	$-0.772064\pi$
$151$	−2799.55	−1.50877	−0.754384	+	0.656434i	$0.772064\pi$
$152$	−2429.45	−1.29641
$153$	0	0
$154$	−364.411	−0.190682
$155$	−1275.14	−0.660787
$156$	0	0
$157$	−811.381	−0.412454	−0.206227	−	0.978504i	$-0.566118\pi$
$157$	−811.381	−0.412454	−0.206227	+	0.978504i	$0.566118\pi$
$158$	517.842	0.260743
$159$	0	0
$160$	935.154	0.462065
$161$	−3954.18	−1.93561
$162$	0	0
$163$	401.857	0.193103	0.0965517	−	0.995328i	$-0.469219\pi$
$163$	401.857	0.193103	0.0965517	+	0.995328i	$0.469219\pi$
$164$	1157.02	0.550902
$165$	0	0
$166$	1013.19	0.473730
$167$	2483.79	1.15091	0.575454	−	0.817834i	$-0.304825\pi$
$167$	2483.79	1.15091	0.575454	+	0.817834i	$0.304825\pi$
$168$	0	0
$169$	5085.23	2.31462
$170$	782.160	0.352876
$171$	0	0
$172$	−2579.75	−1.14363
$173$	2068.32	0.908967	0.454483	−	0.890755i	$-0.349824\pi$
$173$	2068.32	0.908967	0.454483	+	0.890755i	$0.349824\pi$
$174$	0	0
$175$	−577.525	−0.249467
$176$	207.598	0.0889105
$177$	0	0
$178$	−799.805	−0.336786
$179$	2584.76	1.07930	0.539648	−	0.841891i	$-0.318557\pi$
$179$	2584.76	1.07930	0.539648	+	0.841891i	$0.318557\pi$
$180$	0	0
$181$	2522.42	1.03586	0.517929	−	0.855424i	$-0.326703\pi$
$181$	2522.42	1.03586	0.517929	+	0.855424i	$0.326703\pi$
$182$	2827.03	1.15139
$183$	0	0
$184$	−3422.66	−1.37132
$185$	518.419	0.206027
$186$	0	0
$187$	1199.91	0.469232
$188$	2730.80	1.05938
$189$	0	0
$190$	871.180	0.332642
$191$	2240.14	0.848644	0.424322	−	0.905511i	$-0.360513\pi$
$191$	2240.14	0.848644	0.424322	+	0.905511i	$0.360513\pi$
$192$	0	0
$193$	2692.31	1.00413	0.502064	−	0.864831i	$-0.332574\pi$
$193$	2692.31	1.00413	0.502064	+	0.864831i	$0.332574\pi$
$194$	692.504	0.256283
$195$	0	0
$196$	−1133.16	−0.412959
$197$	−1843.20	−0.666612	−0.333306	−	0.942819i	$-0.608164\pi$
$197$	−1843.20	−0.666612	−0.333306	+	0.942819i	$0.608164\pi$
$198$	0	0
$199$	−3094.01	−1.10215	−0.551076	−	0.834455i	$-0.685783\pi$
$199$	−3094.01	−1.10215	−0.551076	+	0.834455i	$0.685783\pi$
$200$	−499.895	−0.176740
$201$	0	0
$202$	−1290.87	−0.449631
$203$	−1852.36	−0.640445
$204$	0	0
$205$	−973.353	−0.331619
$206$	1086.94	0.367624
$207$	0	0
$208$	−1610.50	−0.536867
$209$	1336.48	0.442327
$210$	0	0
$211$	−380.009	−0.123985	−0.0619927	−	0.998077i	$-0.519746\pi$
$211$	−380.009	−0.123985	−0.0619927	+	0.998077i	$0.519746\pi$
$212$	−3903.52	−1.26460
$213$	0	0
$214$	−1137.45	−0.363338
$215$	2170.24	0.688416
$216$	0	0
$217$	5891.42	1.84302
$218$	1767.43	0.549108
$219$	0	0
$220$	−326.891	−0.100177
$221$	−9308.72	−2.83336
$222$	0	0
$223$	−2016.16	−0.605436	−0.302718	−	0.953080i	$-0.597894\pi$
$223$	−2016.16	−0.605436	−0.302718	+	0.953080i	$0.597894\pi$
$224$	−4320.60	−1.28876
$225$	0	0
$226$	1800.69	0.530000
$227$	−1813.60	−0.530278	−0.265139	−	0.964210i	$-0.585418\pi$
$227$	−1813.60	−0.530278	−0.265139	+	0.964210i	$0.585418\pi$
$228$	0	0
$229$	−895.092	−0.258294	−0.129147	−	0.991625i	$-0.541224\pi$
$229$	−895.092	−0.258294	−0.129147	+	0.991625i	$0.541224\pi$
$230$	1227.34	0.351862
$231$	0	0
$232$	−1603.37	−0.453735
$233$	4981.99	1.40078	0.700389	−	0.713761i	$-0.253010\pi$
$233$	4981.99	1.40078	0.700389	+	0.713761i	$0.253010\pi$
$234$	0	0
$235$	−2297.31	−0.637703
$236$	944.756	0.260586
$237$	0	0
$238$	−3613.74	−0.984217
$239$	−2692.91	−0.728829	−0.364415	−	0.931237i	$-0.618731\pi$
$239$	−2692.91	−0.728829	−0.364415	+	0.931237i	$0.618731\pi$
$240$	0	0
$241$	−6472.03	−1.72988	−0.864938	−	0.501878i	$-0.832643\pi$
$241$	−6472.03	−1.72988	−0.864938	+	0.501878i	$0.832643\pi$
$242$	173.522	0.0460925
$243$	0	0
$244$	−637.090	−0.167154
$245$	953.281	0.248583
$246$	0	0
$247$	−10368.2	−2.67089
$248$	5099.50	1.30572
$249$	0	0
$250$	179.258	0.0453490
$251$	5949.61	1.49616	0.748080	−	0.663608i	$-0.230976\pi$
$251$	5949.61	1.49616	0.748080	+	0.663608i	$0.230976\pi$
$252$	0	0
$253$	1882.86	0.467883
$254$	−2899.07	−0.716157
$255$	0	0
$256$	−2842.48	−0.693966
$257$	4983.32	1.20954	0.604768	−	0.796401i	$-0.293266\pi$
$257$	4983.32	1.20954	0.604768	+	0.796401i	$0.293266\pi$
$258$	0	0
$259$	−2395.20	−0.574635
$260$	2535.96	0.604897
$261$	0	0
$262$	−3989.61	−0.940759
$263$	415.329	0.0973775	0.0486888	−	0.998814i	$-0.484496\pi$
$263$	415.329	0.0973775	0.0486888	+	0.998814i	$0.484496\pi$
$264$	0	0
$265$	3283.87	0.761233
$266$	−4025.03	−0.927783
$267$	0	0
$268$	−4076.09	−0.929054
$269$	−959.649	−0.217512	−0.108756	−	0.994068i	$-0.534687\pi$
$269$	−959.649	−0.217512	−0.108756	+	0.994068i	$0.534687\pi$
$270$	0	0
$271$	−997.792	−0.223659	−0.111829	−	0.993727i	$-0.535671\pi$
$271$	−997.792	−0.223659	−0.111829	+	0.993727i	$0.535671\pi$
$272$	2058.67	0.458917
$273$	0	0
$274$	690.666	0.152280
$275$	275.000	0.0603023
$276$	0	0
$277$	7652.53	1.65991	0.829957	−	0.557828i	$-0.188365\pi$
$277$	7652.53	1.65991	0.829957	+	0.557828i	$0.188365\pi$
$278$	−361.457	−0.0779811
$279$	0	0
$280$	2309.61	0.492950
$281$	807.627	0.171455	0.0857277	−	0.996319i	$-0.472678\pi$
$281$	807.627	0.171455	0.0857277	+	0.996319i	$0.472678\pi$
$282$	0	0
$283$	−5795.36	−1.21731	−0.608654	−	0.793435i	$-0.708290\pi$
$283$	−5795.36	−1.21731	−0.608654	+	0.793435i	$0.708290\pi$
$284$	−3990.30	−0.833736
$285$	0	0
$286$	−1346.15	−0.278320
$287$	4497.08	0.924929
$288$	0	0
$289$	6986.13	1.42197
$290$	574.954	0.116422
$291$	0	0
$292$	1546.61	0.309961
$293$	−4286.00	−0.854576	−0.427288	−	0.904115i	$-0.640531\pi$
$293$	−4286.00	−0.854576	−0.427288	+	0.904115i	$0.640531\pi$
$294$	0	0
$295$	−794.786	−0.156862
$296$	−2073.24	−0.407110
$297$	0	0
$298$	3432.95	0.667333
$299$	−14606.9	−2.82521
$300$	0	0
$301$	−10027.0	−1.92008
$302$	−4014.72	−0.764972
$303$	0	0
$304$	2292.98	0.432603
$305$	535.959	0.100619
$306$	0	0
$307$	8830.25	1.64159	0.820796	−	0.571221i	$-0.193530\pi$
$307$	8830.25	1.64159	0.820796	+	0.571221i	$0.193530\pi$
$308$	1510.30	0.279407
$309$	0	0
$310$	−1828.64	−0.335031
$311$	−3446.59	−0.628419	−0.314210	−	0.949354i	$-0.601739\pi$
$311$	−3446.59	−0.628419	−0.314210	+	0.949354i	$0.601739\pi$
$312$	0	0
$313$	−4359.69	−0.787298	−0.393649	−	0.919261i	$-0.628787\pi$
$313$	−4359.69	−0.787298	−0.393649	+	0.919261i	$0.628787\pi$
$314$	−1163.57	−0.209121
$315$	0	0
$316$	−2146.20	−0.382066
$317$	−3954.65	−0.700679	−0.350340	−	0.936623i	$-0.613934\pi$
$317$	−3954.65	−0.700679	−0.350340	+	0.936623i	$0.613934\pi$
$318$	0	0
$319$	882.039	0.154811
$320$	586.169	0.102399
$321$	0	0
$322$	−5670.54	−0.981387
$323$	13253.4	2.28309
$324$	0	0
$325$	−2133.40	−0.364122
$326$	576.287	0.0979068
$327$	0	0
$328$	3892.59	0.655282
$329$	10614.0	1.77864
$330$	0	0
$331$	2845.54	0.472523	0.236262	−	0.971689i	$-0.424078\pi$
$331$	2845.54	0.472523	0.236262	+	0.971689i	$0.424078\pi$
$332$	−4199.18	−0.694157
$333$	0	0
$334$	3561.92	0.583531
$335$	3429.05	0.559251
$336$	0	0
$337$	5477.00	0.885314	0.442657	−	0.896691i	$-0.354036\pi$
$337$	5477.00	0.885314	0.442657	+	0.896691i	$0.354036\pi$
$338$	7292.53	1.17355
$339$	0	0
$340$	−3241.66	−0.517070
$341$	−2805.32	−0.445503
$342$	0	0
$343$	3519.29	0.554006
$344$	−8679.15	−1.36032
$345$	0	0
$346$	2966.10	0.460862
$347$	−4121.77	−0.637660	−0.318830	−	0.947812i	$-0.603290\pi$
$347$	−4121.77	−0.637660	−0.318830	+	0.947812i	$0.603290\pi$
$348$	0	0
$349$	10299.6	1.57973	0.789867	−	0.613279i	$-0.210150\pi$
$349$	10299.6	1.57973	0.789867	+	0.613279i	$0.210150\pi$
$350$	−828.207	−0.126484
$351$	0	0
$352$	2057.34	0.311524
$353$	6574.93	0.991355	0.495677	−	0.868507i	$-0.334920\pi$
$353$	6574.93	0.991355	0.495677	+	0.868507i	$0.334920\pi$
$354$	0	0
$355$	3356.88	0.501873
$356$	3314.79	0.493493
$357$	0	0
$358$	3706.71	0.547222
$359$	−8334.97	−1.22536	−0.612678	−	0.790332i	$-0.709908\pi$
$359$	−8334.97	−1.22536	−0.612678	+	0.790332i	$0.709908\pi$
$360$	0	0
$361$	7902.83	1.15218
$362$	3617.31	0.525198
$363$	0	0
$364$	−11716.6	−1.68714
$365$	−1301.10	−0.186583
$366$	0	0
$367$	1298.89	0.184746	0.0923729	−	0.995724i	$-0.470555\pi$
$367$	1298.89	0.184746	0.0923729	+	0.995724i	$0.470555\pi$
$368$	3230.39	0.457597
$369$	0	0
$370$	743.445	0.104459
$371$	−15172.1	−2.12318
$372$	0	0
$373$	−5715.86	−0.793448	−0.396724	−	0.917938i	$-0.629853\pi$
$373$	−5715.86	−0.793448	−0.396724	+	0.917938i	$0.629853\pi$
$374$	1720.75	0.237909
$375$	0	0
$376$	9187.32	1.26011
$377$	−6842.69	−0.934792
$378$	0	0
$379$	3446.62	0.467126	0.233563	−	0.972342i	$-0.424961\pi$
$379$	3446.62	0.467126	0.233563	+	0.972342i	$0.424961\pi$
$380$	−3610.60	−0.487421
$381$	0	0
$382$	3212.50	0.430277
$383$	5307.81	0.708137	0.354069	−	0.935219i	$-0.384798\pi$
$383$	5307.81	0.708137	0.354069	+	0.935219i	$0.384798\pi$
$384$	0	0
$385$	−1270.55	−0.168191
$386$	3860.94	0.509110
$387$	0	0
$388$	−2870.08	−0.375532
$389$	8148.22	1.06203	0.531017	−	0.847361i	$-0.321810\pi$
$389$	8148.22	1.06203	0.531017	+	0.847361i	$0.321810\pi$
$390$	0	0
$391$	18671.7	2.41501
$392$	−3812.32	−0.491202
$393$	0	0
$394$	−2643.26	−0.337984
$395$	1805.51	0.229987
$396$	0	0
$397$	−9697.26	−1.22592	−0.612962	−	0.790113i	$-0.710022\pi$
$397$	−9697.26	−1.22592	−0.612962	+	0.790113i	$0.710022\pi$
$398$	−4437.00	−0.558811
$399$	0	0
$400$	471.813	0.0589766
$401$	−15273.3	−1.90202	−0.951012	−	0.309153i	$-0.899954\pi$
$401$	−15273.3	−1.90202	−0.951012	+	0.309153i	$0.899954\pi$
$402$	0	0
$403$	21763.1	2.69007
$404$	5350.01	0.658844
$405$	0	0
$406$	−2656.40	−0.324717
$407$	1140.52	0.138903
$408$	0	0
$409$	4890.84	0.591287	0.295644	−	0.955298i	$-0.404466\pi$
$409$	4890.84	0.591287	0.295644	+	0.955298i	$0.404466\pi$
$410$	−1395.85	−0.168137
$411$	0	0
$412$	−4504.81	−0.538680
$413$	3672.07	0.437508
$414$	0	0
$415$	3532.60	0.417852
$416$	−15960.4	−1.88107
$417$	0	0
$418$	1916.60	0.224267
$419$	9513.26	1.10920	0.554598	−	0.832118i	$-0.312872\pi$
$419$	9513.26	1.10920	0.554598	+	0.832118i	$0.312872\pi$
$420$	0	0
$421$	9213.00	1.06654	0.533271	−	0.845945i	$-0.320963\pi$
$421$	9213.00	1.06654	0.533271	+	0.845945i	$0.320963\pi$
$422$	−544.957	−0.0628627
$423$	0	0
$424$	−13132.7	−1.50420
$425$	2727.08	0.311254
$426$	0	0
$427$	−2476.24	−0.280640
$428$	4714.15	0.532400
$429$	0	0
$430$	3112.26	0.349039
$431$	−9388.13	−1.04921	−0.524606	−	0.851345i	$-0.675787\pi$
$431$	−9388.13	−1.04921	−0.524606	+	0.851345i	$0.675787\pi$
$432$	0	0
$433$	12962.4	1.43864	0.719321	−	0.694678i	$-0.244453\pi$
$433$	12962.4	1.43864	0.719321	+	0.694678i	$0.244453\pi$
$434$	8448.66	0.934444
$435$	0	0
$436$	−7325.11	−0.804609
$437$	20796.8	2.27653
$438$	0	0
$439$	−1076.83	−0.117071	−0.0585357	−	0.998285i	$-0.518643\pi$
$439$	−1076.83	−0.117071	−0.0585357	+	0.998285i	$0.518643\pi$
$440$	−1099.77	−0.119158
$441$	0	0
$442$	−13349.3	−1.43656
$443$	2508.84	0.269072	0.134536	−	0.990909i	$-0.457046\pi$
$443$	2508.84	0.269072	0.134536	+	0.990909i	$0.457046\pi$
$444$	0	0
$445$	−2788.60	−0.297061
$446$	−2891.30	−0.306967
$447$	0	0
$448$	−2708.22	−0.285605
$449$	−976.211	−0.102606	−0.0513032	−	0.998683i	$-0.516337\pi$
$449$	−976.211	−0.102606	−0.0513032	+	0.998683i	$0.516337\pi$
$450$	0	0
$451$	−2141.38	−0.223578
$452$	−7462.94	−0.776609
$453$	0	0
$454$	−2600.82	−0.268860
$455$	9856.73	1.01558
$456$	0	0
$457$	9967.11	1.02022	0.510111	−	0.860108i	$-0.329604\pi$
$457$	9967.11	1.02022	0.510111	+	0.860108i	$0.329604\pi$
$458$	−1283.62	−0.130960
$459$	0	0
$460$	−5086.69	−0.515583
$461$	−2123.41	−0.214527	−0.107264	−	0.994231i	$-0.534209\pi$
$461$	−2123.41	−0.214527	−0.107264	+	0.994231i	$0.534209\pi$
$462$	0	0
$463$	−7343.50	−0.737109	−0.368555	−	0.929606i	$-0.620147\pi$
$463$	−7343.50	−0.737109	−0.368555	+	0.929606i	$0.620147\pi$
$464$	1513.30	0.151408
$465$	0	0
$466$	7144.49	0.710219
$467$	7351.06	0.728408	0.364204	−	0.931319i	$-0.381341\pi$
$467$	7351.06	0.728408	0.364204	+	0.931319i	$0.381341\pi$
$468$	0	0
$469$	−15842.9	−1.55982
$470$	−3294.49	−0.323327
$471$	0	0
$472$	3178.47	0.309960
$473$	4774.54	0.464130
$474$	0	0
$475$	3037.46	0.293406
$476$	14977.1	1.44217
$477$	0	0
$478$	−3861.81	−0.369529
$479$	327.620	0.0312512	0.0156256	−	0.999878i	$-0.495026\pi$
$479$	327.620	0.0312512	0.0156256	+	0.999878i	$0.495026\pi$
$480$	0	0
$481$	−8847.95	−0.838736
$482$	−9281.30	−0.877078
$483$	0	0
$484$	−719.159	−0.0675394
$485$	2414.48	0.226054
$486$	0	0
$487$	12591.3	1.17159	0.585797	−	0.810458i	$-0.300782\pi$
$487$	12591.3	1.17159	0.585797	+	0.810458i	$0.300782\pi$
$488$	−2143.38	−0.198825
$489$	0	0
$490$	1367.06	0.126036
$491$	2124.55	0.195274	0.0976369	−	0.995222i	$-0.468872\pi$
$491$	2124.55	0.195274	0.0976369	+	0.995222i	$0.468872\pi$
$492$	0	0
$493$	8746.87	0.799065
$494$	−14868.6	−1.35419
$495$	0	0
$496$	−4813.03	−0.435708
$497$	−15509.5	−1.39979
$498$	0	0
$499$	8119.15	0.728383	0.364192	−	0.931324i	$-0.381345\pi$
$499$	8119.15	0.728383	0.364192	+	0.931324i	$0.381345\pi$
$500$	−742.933	−0.0664500
$501$	0	0
$502$	8532.12	0.758580
$503$	−8466.22	−0.750477	−0.375238	−	0.926928i	$-0.622439\pi$
$503$	−8466.22	−0.750477	−0.375238	+	0.926928i	$0.622439\pi$
$504$	0	0
$505$	−4500.75	−0.396596
$506$	2700.14	0.237225
$507$	0	0
$508$	12015.2	1.04939
$509$	−3047.74	−0.265400	−0.132700	−	0.991156i	$-0.542365\pi$
$509$	−3047.74	−0.265400	−0.132700	+	0.991156i	$0.542365\pi$
$510$	0	0
$511$	6011.36	0.520405
$512$	6548.70	0.565263
$513$	0	0
$514$	7146.39	0.613256
$515$	3789.72	0.324262
$516$	0	0
$517$	−5054.09	−0.429939
$518$	−3434.86	−0.291350
$519$	0	0
$520$	8531.80	0.719508
$521$	18982.8	1.59626	0.798130	−	0.602486i	$-0.205823\pi$
$521$	18982.8	1.59626	0.798130	+	0.602486i	$0.205823\pi$
$522$	0	0
$523$	4340.00	0.362859	0.181429	−	0.983404i	$-0.441928\pi$
$523$	4340.00	0.362859	0.181429	+	0.983404i	$0.441928\pi$
$524$	16534.9	1.37849
$525$	0	0
$526$	595.608	0.0493721
$527$	−27819.3	−2.29949
$528$	0	0
$529$	17131.9	1.40806
$530$	4709.28	0.385958
$531$	0	0
$532$	16681.7	1.35948
$533$	16612.4	1.35002
$534$	0	0
$535$	−3965.83	−0.320482
$536$	−13713.3	−1.10508
$537$	0	0
$538$	−1376.20	−0.110283
$539$	2097.22	0.167595
$540$	0	0
$541$	10372.3	0.824288	0.412144	−	0.911119i	$-0.364780\pi$
$541$	10372.3	0.824288	0.412144	+	0.911119i	$0.364780\pi$
$542$	−1430.90	−0.113399
$543$	0	0
$544$	20401.9	1.60795
$545$	6162.33	0.484340
$546$	0	0
$547$	−5859.58	−0.458021	−0.229010	−	0.973424i	$-0.573549\pi$
$547$	−5859.58	−0.458021	−0.229010	+	0.973424i	$0.573549\pi$
$548$	−2862.46	−0.223136
$549$	0	0
$550$	394.367	0.0305743
$551$	9742.38	0.753248
$552$	0	0
$553$	−8341.81	−0.641464
$554$	10974.2	0.841606
$555$	0	0
$556$	1498.06	0.114266
$557$	−16395.2	−1.24720	−0.623598	−	0.781745i	$-0.714330\pi$
$557$	−16395.2	−1.24720	−0.623598	+	0.781745i	$0.714330\pi$
$558$	0	0
$559$	−37040.0	−2.80255
$560$	−2179.87	−0.164493
$561$	0	0
$562$	1158.19	0.0869309
$563$	7660.01	0.573412	0.286706	−	0.958019i	$-0.407440\pi$
$563$	7660.01	0.573412	0.286706	+	0.958019i	$0.407440\pi$
$564$	0	0
$565$	6278.28	0.467485
$566$	−8310.91	−0.617197
$567$	0	0
$568$	−13424.7	−0.991705
$569$	12844.6	0.946353	0.473177	−	0.880968i	$-0.343107\pi$
$569$	12844.6	0.946353	0.473177	+	0.880968i	$0.343107\pi$
$570$	0	0
$571$	−19440.0	−1.42476	−0.712382	−	0.701792i	$-0.752384\pi$
$571$	−19440.0	−1.42476	−0.712382	+	0.701792i	$0.752384\pi$
$572$	5579.10	0.407822
$573$	0	0
$574$	6449.10	0.468955
$575$	4279.23	0.310359
$576$	0	0
$577$	977.160	0.0705021	0.0352510	−	0.999378i	$-0.488777\pi$
$577$	977.160	0.0705021	0.0352510	+	0.999378i	$0.488777\pi$
$578$	10018.6	0.720963
$579$	0	0
$580$	−2382.89	−0.170594
$581$	−16321.3	−1.16544
$582$	0	0
$583$	7224.52	0.513223
$584$	5203.32	0.368690
$585$	0	0
$586$	−6146.39	−0.433285
$587$	−7212.61	−0.507149	−0.253574	−	0.967316i	$-0.581606\pi$
$587$	−7212.61	−0.507149	−0.253574	+	0.967316i	$0.581606\pi$
$588$	0	0
$589$	−30985.5	−2.16763
$590$	−1139.77	−0.0795317
$591$	0	0
$592$	1956.77	0.135849
$593$	−21076.0	−1.45950	−0.729752	−	0.683712i	$-0.760365\pi$
$593$	−21076.0	−1.45950	−0.729752	+	0.683712i	$0.760365\pi$
$594$	0	0
$595$	−12599.6	−0.868127
$596$	−14227.8	−0.977843
$597$	0	0
$598$	−20947.2	−1.43243
$599$	−25691.7	−1.75248	−0.876238	−	0.481878i	$-0.839955\pi$
$599$	−25691.7	−1.75248	−0.876238	+	0.481878i	$0.839955\pi$
$600$	0	0
$601$	8638.39	0.586302	0.293151	−	0.956066i	$-0.405296\pi$
$601$	8638.39	0.586302	0.293151	+	0.956066i	$0.405296\pi$
$602$	−14379.3	−0.973515
$603$	0	0
$604$	16639.0	1.12091
$605$	605.000	0.0406558
$606$	0	0
$607$	−7794.30	−0.521188	−0.260594	−	0.965449i	$-0.583918\pi$
$607$	−7794.30	−0.521188	−0.260594	+	0.965449i	$0.583918\pi$
$608$	22723.9	1.51575
$609$	0	0
$610$	768.598	0.0510158
$611$	39208.7	2.59609
$612$	0	0
$613$	6321.35	0.416504	0.208252	−	0.978075i	$-0.433223\pi$
$613$	6321.35	0.416504	0.208252	+	0.978075i	$0.433223\pi$
$614$	12663.1	0.832316
$615$	0	0
$616$	5081.15	0.332347
$617$	24939.9	1.62730	0.813650	−	0.581356i	$-0.197477\pi$
$617$	24939.9	1.62730	0.813650	+	0.581356i	$0.197477\pi$
$618$	0	0
$619$	5992.55	0.389113	0.194556	−	0.980891i	$-0.437673\pi$
$619$	5992.55	0.389113	0.194556	+	0.980891i	$0.437673\pi$
$620$	7578.77	0.490921
$621$	0	0
$622$	−4942.63	−0.318620
$623$	12883.9	0.828542
$624$	0	0
$625$	625.000	0.0400000
$626$	−6252.07	−0.399174
$627$	0	0
$628$	4822.41	0.306426
$629$	11310.2	0.716956
$630$	0	0
$631$	−7767.52	−0.490048	−0.245024	−	0.969517i	$-0.578796\pi$
$631$	−7767.52	−0.490048	−0.245024	+	0.969517i	$0.578796\pi$
$632$	−7220.52	−0.454457
$633$	0	0
$634$	−5671.22	−0.355257
$635$	−10107.9	−0.631685
$636$	0	0
$637$	−16269.8	−1.01198
$638$	1264.90	0.0784919
$639$	0	0
$640$	−6640.63	−0.410147
$641$	−16348.1	−1.00735	−0.503674	−	0.863894i	$-0.668019\pi$
$641$	−16348.1	−1.00735	−0.503674	+	0.863894i	$0.668019\pi$
$642$	0	0
$643$	1423.25	0.0872902	0.0436451	−	0.999047i	$-0.486103\pi$
$643$	1423.25	0.0872902	0.0436451	+	0.999047i	$0.486103\pi$
$644$	23501.5	1.43803
$645$	0	0
$646$	19006.2	1.15757
$647$	−2225.13	−0.135207	−0.0676035	−	0.997712i	$-0.521535\pi$
$647$	−2225.13	−0.135207	−0.0676035	+	0.997712i	$0.521535\pi$
$648$	0	0
$649$	−1748.53	−0.105756
$650$	−3059.43	−0.184616
$651$	0	0
$652$	−2388.42	−0.143463
$653$	8257.95	0.494883	0.247441	−	0.968903i	$-0.420410\pi$
$653$	8257.95	0.494883	0.247441	+	0.968903i	$0.420410\pi$
$654$	0	0
$655$	−13910.2	−0.829794
$656$	−3673.92	−0.218662
$657$	0	0
$658$	15221.2	0.901800
$659$	−27126.3	−1.60348	−0.801739	−	0.597674i	$-0.796092\pi$
$659$	−27126.3	−1.60348	−0.801739	+	0.597674i	$0.796092\pi$
$660$	0	0
$661$	2629.79	0.154746	0.0773728	−	0.997002i	$-0.475347\pi$
$661$	2629.79	0.154746	0.0773728	+	0.997002i	$0.475347\pi$
$662$	4080.68	0.239578
$663$	0	0
$664$	−14127.4	−0.825680
$665$	−14033.7	−0.818349
$666$	0	0
$667$	13725.3	0.796768
$668$	−14762.3	−0.855048
$669$	0	0
$670$	4917.47	0.283550
$671$	1179.11	0.0678376
$672$	0	0
$673$	−3289.98	−0.188439	−0.0942196	−	0.995551i	$-0.530036\pi$
$673$	−3289.98	−0.188439	−0.0942196	+	0.995551i	$0.530036\pi$
$674$	7854.35	0.448870
$675$	0	0
$676$	−30223.9	−1.71961
$677$	−4469.65	−0.253741	−0.126871	−	0.991919i	$-0.540493\pi$
$677$	−4469.65	−0.253741	−0.126871	+	0.991919i	$0.540493\pi$
$678$	0	0
$679$	−11155.4	−0.630493
$680$	−10906.0	−0.615040
$681$	0	0
$682$	−4023.00	−0.225878
$683$	27927.6	1.56460	0.782298	−	0.622905i	$-0.214048\pi$
$683$	27927.6	1.56460	0.782298	+	0.622905i	$0.214048\pi$
$684$	0	0
$685$	2408.08	0.134318
$686$	5046.89	0.280891
$687$	0	0
$688$	8191.58	0.453926
$689$	−56046.5	−3.09899
$690$	0	0
$691$	−30950.9	−1.70395	−0.851974	−	0.523584i	$-0.824595\pi$
$691$	−30950.9	−1.70395	−0.851974	+	0.523584i	$0.824595\pi$
$692$	−12293.0	−0.675301
$693$	0	0
$694$	−5910.87	−0.323305
$695$	−1260.25	−0.0687830
$696$	0	0
$697$	−21235.3	−1.15401
$698$	14770.3	0.800953
$699$	0	0
$700$	3432.50	0.185338
$701$	−9614.69	−0.518034	−0.259017	−	0.965873i	$-0.583399\pi$
$701$	−9614.69	−0.518034	−0.259017	+	0.965873i	$0.583399\pi$
$702$	0	0
$703$	12597.4	0.675846
$704$	1289.57	0.0690377
$705$	0	0
$706$	9428.86	0.502634
$707$	20794.4	1.10616
$708$	0	0
$709$	9294.67	0.492340	0.246170	−	0.969227i	$-0.420828\pi$
$709$	9294.67	0.492340	0.246170	+	0.969227i	$0.420828\pi$
$710$	4813.98	0.254458
$711$	0	0
$712$	11152.0	0.586995
$713$	−43653.0	−2.29287
$714$	0	0
$715$	−4693.48	−0.245491
$716$	−15362.4	−0.801845
$717$	0	0
$718$	−11952.9	−0.621277
$719$	12434.2	0.644948	0.322474	−	0.946578i	$-0.395486\pi$
$719$	12434.2	0.644948	0.322474	+	0.946578i	$0.395486\pi$
$720$	0	0
$721$	−17509.2	−0.904408
$722$	11333.1	0.584177
$723$	0	0
$724$	−14991.9	−0.769573
$725$	2004.63	0.102690
$726$	0	0
$727$	−2146.65	−0.109511	−0.0547556	−	0.998500i	$-0.517438\pi$
$727$	−2146.65	−0.109511	−0.0547556	+	0.998500i	$0.517438\pi$
$728$	−39418.6	−2.00680
$729$	0	0
$730$	−1865.86	−0.0946009
$731$	47347.4	2.39563
$732$	0	0
$733$	−11617.8	−0.585423	−0.292711	−	0.956201i	$-0.594558\pi$
$733$	−11617.8	−0.585423	−0.292711	+	0.956201i	$0.594558\pi$
$734$	1862.69	0.0936693
$735$	0	0
$736$	32013.9	1.60333
$737$	7543.91	0.377047
$738$	0	0
$739$	23974.7	1.19340	0.596702	−	0.802463i	$-0.296477\pi$
$739$	23974.7	1.19340	0.596702	+	0.802463i	$0.296477\pi$
$740$	−3081.20	−0.153064
$741$	0	0
$742$	−21757.8	−1.07649
$743$	13294.7	0.656443	0.328221	−	0.944601i	$-0.393551\pi$
$743$	13294.7	0.656443	0.328221	+	0.944601i	$0.393551\pi$
$744$	0	0
$745$	11969.3	0.588619
$746$	−8196.91	−0.402292
$747$	0	0
$748$	−7131.65	−0.348608
$749$	18322.9	0.893865
$750$	0	0
$751$	−25319.4	−1.23025	−0.615124	−	0.788430i	$-0.710894\pi$
$751$	−25319.4	−1.23025	−0.615124	+	0.788430i	$0.710894\pi$
$752$	−8671.21	−0.420487
$753$	0	0
$754$	−9812.85	−0.473956
$755$	−13997.7	−0.674741
$756$	0	0
$757$	16978.3	0.815173	0.407586	−	0.913167i	$-0.366371\pi$
$757$	16978.3	0.815173	0.407586	+	0.913167i	$0.366371\pi$
$758$	4942.66	0.236841
$759$	0	0
$760$	−12147.3	−0.579774
$761$	−16484.5	−0.785232	−0.392616	−	0.919702i	$-0.628430\pi$
$761$	−16484.5	−0.785232	−0.392616	+	0.919702i	$0.628430\pi$
$762$	0	0
$763$	−28471.2	−1.35089
$764$	−13314.2	−0.630485
$765$	0	0
$766$	7611.74	0.359038
$767$	13564.8	0.638585
$768$	0	0
$769$	−34803.2	−1.63204	−0.816019	−	0.578025i	$-0.803824\pi$
$769$	−34803.2	−1.63204	−0.816019	+	0.578025i	$0.803824\pi$
$770$	−1822.06	−0.0852757
$771$	0	0
$772$	−16001.6	−0.745999
$773$	−33850.8	−1.57507	−0.787535	−	0.616270i	$-0.788643\pi$
$773$	−33850.8	−1.57507	−0.787535	+	0.616270i	$0.788643\pi$
$774$	0	0
$775$	−6375.72	−0.295513
$776$	−9655.91	−0.446684
$777$	0	0
$778$	11685.1	0.538470
$779$	−23652.1	−1.08784
$780$	0	0
$781$	7385.14	0.338363
$782$	26776.3	1.22445
$783$	0	0
$784$	3598.16	0.163910
$785$	−4056.90	−0.184455
$786$	0	0
$787$	17983.2	0.814526	0.407263	−	0.913311i	$-0.366483\pi$
$787$	17983.2	0.814526	0.407263	+	0.913311i	$0.366483\pi$
$788$	10955.0	0.495248
$789$	0	0
$790$	2589.21	0.116608
$791$	−29006.9	−1.30388
$792$	0	0
$793$	−9147.31	−0.409622
$794$	−13906.5	−0.621565
$795$	0	0
$796$	18389.1	0.818826
$797$	20935.7	0.930464	0.465232	−	0.885189i	$-0.345971\pi$
$797$	20935.7	0.930464	0.465232	+	0.885189i	$0.345971\pi$
$798$	0	0
$799$	−50119.6	−2.21915
$800$	4675.77	0.206642
$801$	0	0
$802$	−21902.9	−0.964360
$803$	−2862.43	−0.125794
$804$	0	0
$805$	−19770.9	−0.865630
$806$	31209.7	1.36391
$807$	0	0
$808$	17999.2	0.783676
$809$	−37360.3	−1.62363	−0.811815	−	0.583914i	$-0.801520\pi$
$809$	−37360.3	−1.62363	−0.811815	+	0.583914i	$0.801520\pi$
$810$	0	0
$811$	−32067.8	−1.38847	−0.694237	−	0.719747i	$-0.744258\pi$
$811$	−32067.8	−1.38847	−0.694237	+	0.719747i	$0.744258\pi$
$812$	11009.4	0.475808
$813$	0	0
$814$	1635.58	0.0704263
$815$	2009.28	0.0863584
$816$	0	0
$817$	52736.2	2.25827
$818$	7013.77	0.299793
$819$	0	0
$820$	5785.09	0.246371
$821$	3988.65	0.169555	0.0847777	−	0.996400i	$-0.472982\pi$
$821$	3988.65	0.169555	0.0847777	+	0.996400i	$0.472982\pi$
$822$	0	0
$823$	−7867.82	−0.333238	−0.166619	−	0.986021i	$-0.553285\pi$
$823$	−7867.82	−0.333238	−0.166619	+	0.986021i	$0.553285\pi$
$824$	−15155.7	−0.640744
$825$	0	0
$826$	5265.97	0.221824
$827$	−9219.74	−0.387669	−0.193834	−	0.981034i	$-0.562092\pi$
$827$	−9219.74	−0.387669	−0.193834	+	0.981034i	$0.562092\pi$
$828$	0	0
$829$	35423.8	1.48410	0.742051	−	0.670344i	$-0.233853\pi$
$829$	35423.8	1.48410	0.742051	+	0.670344i	$0.233853\pi$
$830$	5065.97	0.211858
$831$	0	0
$832$	−10004.3	−0.416869
$833$	20797.4	0.865050
$834$	0	0
$835$	12419.0	0.514702
$836$	−7943.33	−0.328619
$837$	0	0
$838$	13642.6	0.562382
$839$	159.682	0.00657071	0.00328536	−	0.999995i	$-0.498954\pi$
$839$	159.682	0.00657071	0.00328536	+	0.999995i	$0.498954\pi$
$840$	0	0
$841$	−17959.3	−0.736369
$842$	13212.0	0.540755
$843$	0	0
$844$	2258.57	0.0921128
$845$	25426.1	1.03513
$846$	0	0
$847$	−2795.22	−0.113394
$848$	12395.0	0.501940
$849$	0	0
$850$	3910.80	0.157811
$851$	17747.5	0.714895
$852$	0	0
$853$	26602.4	1.06782	0.533910	−	0.845541i	$-0.320722\pi$
$853$	26602.4	1.06782	0.533910	+	0.845541i	$0.320722\pi$
$854$	−3551.08	−0.142290
$855$	0	0
$856$	15860.0	0.633275
$857$	36913.3	1.47133	0.735667	−	0.677343i	$-0.236869\pi$
$857$	36913.3	1.47133	0.735667	+	0.677343i	$0.236869\pi$
$858$	0	0
$859$	4372.37	0.173671	0.0868354	−	0.996223i	$-0.472325\pi$
$859$	4372.37	0.173671	0.0868354	+	0.996223i	$0.472325\pi$
$860$	−12898.8	−0.511447
$861$	0	0
$862$	−13463.2	−0.531969
$863$	4398.69	0.173503	0.0867515	−	0.996230i	$-0.472351\pi$
$863$	4398.69	0.173503	0.0867515	+	0.996230i	$0.472351\pi$
$864$	0	0
$865$	10341.6	0.406502
$866$	18588.8	0.729417
$867$	0	0
$868$	−35015.4	−1.36924
$869$	3972.12	0.155057
$870$	0	0
$871$	−58524.2	−2.27671
$872$	−24644.1	−0.957059
$873$	0	0
$874$	29823.8	1.15424
$875$	−2887.62	−0.111565
$876$	0	0
$877$	−34751.4	−1.33805	−0.669025	−	0.743240i	$-0.733288\pi$
$877$	−34751.4	−1.33805	−0.669025	+	0.743240i	$0.733288\pi$
$878$	−1544.24	−0.0593573
$879$	0	0
$880$	1037.99	0.0397620
$881$	31026.1	1.18649	0.593244	−	0.805023i	$-0.297847\pi$
$881$	31026.1	1.18649	0.593244	+	0.805023i	$0.297847\pi$
$882$	0	0
$883$	−46496.8	−1.77207	−0.886037	−	0.463615i	$-0.846552\pi$
$883$	−46496.8	−1.77207	−0.886037	+	0.463615i	$0.846552\pi$
$884$	55326.0	2.10500
$885$	0	0
$886$	3597.84	0.136424
$887$	−27215.4	−1.03022	−0.515110	−	0.857124i	$-0.672249\pi$
$887$	−27215.4	−1.03022	−0.515110	+	0.857124i	$0.672249\pi$
$888$	0	0
$889$	46700.5	1.76185
$890$	−3999.02	−0.150615
$891$	0	0
$892$	11983.0	0.449799
$893$	−55823.9	−2.09191
$894$	0	0
$895$	12923.8	0.482676
$896$	30681.0	1.14395
$897$	0	0
$898$	−1399.95	−0.0520232
$899$	−20449.6	−0.758655
$900$	0	0
$901$	71643.0	2.64903
$902$	−3070.87	−0.113358
$903$	0	0
$904$	−25107.8	−0.923754
$905$	12612.1	0.463250
$906$	0	0
$907$	−19319.3	−0.707263	−0.353632	−	0.935385i	$-0.615053\pi$
$907$	−19319.3	−0.707263	−0.353632	+	0.935385i	$0.615053\pi$
$908$	10779.1	0.393961
$909$	0	0
$910$	14135.2	0.514919
$911$	44063.1	1.60250	0.801249	−	0.598332i	$-0.204169\pi$
$911$	44063.1	1.60250	0.801249	+	0.598332i	$0.204169\pi$
$912$	0	0
$913$	7771.73	0.281716
$914$	14293.5	0.517271
$915$	0	0
$916$	5319.95	0.191895
$917$	64267.7	2.31440
$918$	0	0
$919$	11766.0	0.422332	0.211166	−	0.977450i	$-0.432274\pi$
$919$	11766.0	0.422332	0.211166	+	0.977450i	$0.432274\pi$
$920$	−17113.3	−0.613271
$921$	0	0
$922$	−3045.10	−0.108769
$923$	−57292.6	−2.04313
$924$	0	0
$925$	2592.09	0.0921379
$926$	−10531.0	−0.373727
$927$	0	0
$928$	14997.1	0.530501
$929$	3733.33	0.131848	0.0659239	−	0.997825i	$-0.479001\pi$
$929$	3733.33	0.131848	0.0659239	+	0.997825i	$0.479001\pi$
$930$	0	0
$931$	23164.4	0.815448
$932$	−29610.3	−1.04068
$933$	0	0
$934$	10541.9	0.369315
$935$	5999.57	0.209847
$936$	0	0
$937$	4865.84	0.169648	0.0848239	−	0.996396i	$-0.472967\pi$
$937$	4865.84	0.169648	0.0848239	+	0.996396i	$0.472967\pi$
$938$	−22719.7	−0.790857
$939$	0	0
$940$	13654.0	0.473771
$941$	16216.8	0.561797	0.280898	−	0.959737i	$-0.409368\pi$
$941$	16216.8	0.561797	0.280898	+	0.959737i	$0.409368\pi$
$942$	0	0
$943$	−33321.6	−1.15069
$944$	−2999.92	−0.103431
$945$	0	0
$946$	6846.98	0.235322
$947$	−1297.47	−0.0445216	−0.0222608	−	0.999752i	$-0.507086\pi$
$947$	−1297.47	−0.0445216	−0.0222608	+	0.999752i	$0.507086\pi$
$948$	0	0
$949$	22206.2	0.759582
$950$	4355.90	0.148762
$951$	0	0
$952$	50388.0	1.71543
$953$	−23938.9	−0.813701	−0.406851	−	0.913495i	$-0.633373\pi$
$953$	−23938.9	−0.813701	−0.406851	+	0.913495i	$0.633373\pi$
$954$	0	0
$955$	11200.7	0.379525
$956$	16005.2	0.541471
$957$	0	0
$958$	469.827	0.0158449
$959$	−11125.8	−0.374630
$960$	0	0
$961$	35248.6	1.18320
$962$	−12688.5	−0.425254
$963$	0	0
$964$	38466.3	1.28518
$965$	13461.5	0.449059
$966$	0	0
$967$	−27087.0	−0.900786	−0.450393	−	0.892830i	$-0.648716\pi$
$967$	−27087.0	−0.900786	−0.450393	+	0.892830i	$0.648716\pi$
$968$	−2419.49	−0.0803362
$969$	0	0
$970$	3462.52	0.114613
$971$	−36157.4	−1.19500	−0.597500	−	0.801869i	$-0.703839\pi$
$971$	−36157.4	−1.19500	−0.597500	+	0.801869i	$0.703839\pi$
$972$	0	0
$973$	5822.63	0.191845
$974$	18056.7	0.594019
$975$	0	0
$976$	2022.98	0.0663462
$977$	−34426.9	−1.12734	−0.563671	−	0.825999i	$-0.690611\pi$
$977$	−34426.9	−1.12734	−0.563671	+	0.825999i	$0.690611\pi$
$978$	0	0
$979$	−6134.92	−0.200279
$980$	−5665.79	−0.184681
$981$	0	0
$982$	3046.73	0.0990072
$983$	−22704.7	−0.736692	−0.368346	−	0.929689i	$-0.620076\pi$
$983$	−22704.7	−0.736692	−0.368346	+	0.929689i	$0.620076\pi$
$984$	0	0
$985$	−9216.00	−0.298118
$986$	12543.6	0.405140
$987$	0	0
$988$	61622.9	1.98430
$989$	74295.7	2.38874
$990$	0	0
$991$	8792.88	0.281852	0.140926	−	0.990020i	$-0.454992\pi$
$991$	8792.88	0.281852	0.140926	+	0.990020i	$0.454992\pi$
$992$	−47698.2	−1.52663
$993$	0	0
$994$	−22241.6	−0.709717
$995$	−15470.0	−0.492898
$996$	0	0
$997$	14403.0	0.457519	0.228760	−	0.973483i	$-0.426533\pi$
$997$	14403.0	0.457519	0.228760	+	0.973483i	$0.426533\pi$
$998$	11643.4	0.369303
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.4.a.p.1.4	yes	7
3.2	odd	2		495.4.a.o.1.4	✓	7
5.4	even	2		2475.4.a.bp.1.4		7
15.14	odd	2		2475.4.a.bt.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.4.a.o.1.4	✓	7	3.2	odd	2
495.4.a.p.1.4	yes	7	1.1	even	1	trivial
2475.4.a.bp.1.4		7	5.4	even	2
2475.4.a.bt.1.4		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)

\(f(q)\)	\(=\)	\(q+1.43406 q^{2} -5.94347 q^{4} +5.00000 q^{5} -23.1010 q^{7} -19.9958 q^{8} +O(q^{10})\)	\(q+1.43406 q^{2} -5.94347 q^{4} +5.00000 q^{5} -23.1010 q^{7} -19.9958 q^{8} +7.17031 q^{10} +11.0000 q^{11} -85.3360 q^{13} -33.1283 q^{14} +18.8725 q^{16} +109.083 q^{17} +121.498 q^{19} -29.7173 q^{20} +15.7747 q^{22} +171.169 q^{23} +25.0000 q^{25} -122.377 q^{26} +137.300 q^{28} +80.1854 q^{29} -255.029 q^{31} +187.031 q^{32} +156.432 q^{34} -115.505 q^{35} +103.684 q^{37} +174.236 q^{38} -99.9790 q^{40} -194.671 q^{41} +434.049 q^{43} -65.3781 q^{44} +245.467 q^{46} -459.463 q^{47} +190.656 q^{49} +35.8516 q^{50} +507.191 q^{52} +656.774 q^{53} +55.0000 q^{55} +461.923 q^{56} +114.991 q^{58} -158.957 q^{59} +107.192 q^{61} -365.727 q^{62} +117.234 q^{64} -426.680 q^{65} +685.810 q^{67} -648.332 q^{68} -165.641 q^{70} +671.377 q^{71} -260.221 q^{73} +148.689 q^{74} -722.121 q^{76} -254.111 q^{77} +361.102 q^{79} +94.3625 q^{80} -279.170 q^{82} +706.521 q^{83} +545.416 q^{85} +622.453 q^{86} -219.954 q^{88} -557.720 q^{89} +1971.35 q^{91} -1017.34 q^{92} -658.898 q^{94} +607.491 q^{95} +482.897 q^{97} +273.413 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 5 q^{2} + 33 q^{4} + 35 q^{5} + 30 q^{7} + 45 q^{8}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more