LMFDB - Embedded newform 495.4.a.p.1.3

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			1.3
Root			$2.29255$ 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)

$f(q)$	$=$	$q-1.29255 q^{2} -6.32931 q^{4} +5.00000 q^{5} -23.9463 q^{7} +18.5214 q^{8} +O(q^{10})$	\(q-1.29255 q^{2} -6.32931 q^{4} +5.00000 q^{5} -23.9463 q^{7} +18.5214 q^{8} -6.46276 q^{10} +11.0000 q^{11} +11.9206 q^{13} +30.9519 q^{14} +26.6946 q^{16} -94.7243 q^{17} -139.174 q^{19} -31.6465 q^{20} -14.2181 q^{22} +85.7733 q^{23} +25.0000 q^{25} -15.4080 q^{26} +151.564 q^{28} -194.111 q^{29} +220.612 q^{31} -182.675 q^{32} +122.436 q^{34} -119.732 q^{35} +323.796 q^{37} +179.890 q^{38} +92.6069 q^{40} +146.253 q^{41} -48.9314 q^{43} -69.6224 q^{44} -110.866 q^{46} +365.420 q^{47} +230.427 q^{49} -32.3138 q^{50} -75.4492 q^{52} -298.207 q^{53} +55.0000 q^{55} -443.519 q^{56} +250.898 q^{58} -216.540 q^{59} +504.865 q^{61} -285.152 q^{62} +22.5601 q^{64} +59.6030 q^{65} -267.177 q^{67} +599.539 q^{68} +154.759 q^{70} +568.648 q^{71} +647.751 q^{73} -418.524 q^{74} +880.875 q^{76} -263.410 q^{77} +895.113 q^{79} +133.473 q^{80} -189.040 q^{82} +1338.79 q^{83} -473.621 q^{85} +63.2464 q^{86} +203.735 q^{88} +894.190 q^{89} -285.455 q^{91} -542.886 q^{92} -472.324 q^{94} -695.870 q^{95} +1656.84 q^{97} -297.838 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.29255	−0.456986	−0.228493	−	0.973546i	$-0.573380\pi$
$2$	−1.29255	−0.456986	−0.228493	+	0.973546i	$0.573380\pi$
$3$	0	0
$3$	0	0
$4$	−6.32931	−0.791164
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−23.9463	−1.29298	−0.646490	−	0.762922i	$-0.723764\pi$
$7$	−23.9463	−1.29298	−0.646490	+	0.762922i	$0.723764\pi$
$8$	18.5214	0.818537
$9$	0	0
$10$	−6.46276	−0.204370
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	11.9206	0.254322	0.127161	−	0.991882i	$-0.459414\pi$
$13$	11.9206	0.254322	0.127161	+	0.991882i	$0.459414\pi$
$14$	30.9519	0.590874
$15$	0	0
$16$	26.6946	0.417104
$17$	−94.7243	−1.35141	−0.675706	−	0.737171i	$-0.736161\pi$
$17$	−94.7243	−1.35141	−0.675706	+	0.737171i	$0.736161\pi$
$18$	0	0
$19$	−139.174	−1.68046	−0.840229	−	0.542232i	$-0.817579\pi$
$19$	−139.174	−1.68046	−0.840229	+	0.542232i	$0.817579\pi$
$20$	−31.6465	−0.353819
$21$	0	0
$22$	−14.2181	−0.137786
$23$	85.7733	0.777607	0.388804	−	0.921321i	$-0.372888\pi$
$23$	85.7733	0.777607	0.388804	+	0.921321i	$0.372888\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−15.4080	−0.116221
$27$	0	0
$28$	151.564	1.02296
$29$	−194.111	−1.24295	−0.621473	−	0.783436i	$-0.713465\pi$
$29$	−194.111	−1.24295	−0.621473	+	0.783436i	$0.713465\pi$
$30$	0	0
$31$	220.612	1.27816	0.639082	−	0.769139i	$-0.279315\pi$
$31$	220.612	1.27816	0.639082	+	0.769139i	$0.279315\pi$
$32$	−182.675	−1.00915
$33$	0	0
$34$	122.436	0.617577
$35$	−119.732	−0.578238
$36$	0	0
$37$	323.796	1.43870	0.719349	−	0.694649i	$-0.244440\pi$
$37$	323.796	1.43870	0.719349	+	0.694649i	$0.244440\pi$
$38$	179.890	0.767946
$39$	0	0
$40$	92.6069	0.366061
$41$	146.253	0.557095	0.278548	−	0.960422i	$-0.410147\pi$
$41$	146.253	0.557095	0.278548	+	0.960422i	$0.410147\pi$
$42$	0	0
$43$	−48.9314	−0.173534	−0.0867670	−	0.996229i	$-0.527654\pi$
$43$	−48.9314	−0.173534	−0.0867670	+	0.996229i	$0.527654\pi$
$44$	−69.6224	−0.238545
$45$	0	0
$46$	−110.866	−0.355356
$47$	365.420	1.13408	0.567042	−	0.823689i	$-0.308088\pi$
$47$	365.420	1.13408	0.567042	+	0.823689i	$0.308088\pi$
$48$	0	0
$49$	230.427	0.671798
$50$	−32.3138	−0.0913972
$51$	0	0
$52$	−75.4492	−0.201210
$53$	−298.207	−0.772865	−0.386432	−	0.922318i	$-0.626293\pi$
$53$	−298.207	−0.772865	−0.386432	+	0.922318i	$0.626293\pi$
$54$	0	0
$55$	55.0000	0.134840
$56$	−443.519	−1.05835
$57$	0	0
$58$	250.898	0.568009
$59$	−216.540	−0.477816	−0.238908	−	0.971042i	$-0.576789\pi$
$59$	−216.540	−0.477816	−0.238908	+	0.971042i	$0.576789\pi$
$60$	0	0
$61$	504.865	1.05969	0.529847	−	0.848093i	$-0.322249\pi$
$61$	504.865	1.05969	0.529847	+	0.848093i	$0.322249\pi$
$62$	−285.152	−0.584103
$63$	0	0
$64$	22.5601	0.0440627
$65$	59.6030	0.113736
$66$	0	0
$67$	−267.177	−0.487177	−0.243589	−	0.969879i	$-0.578325\pi$
$67$	−267.177	−0.487177	−0.243589	+	0.969879i	$0.578325\pi$
$68$	599.539	1.06919
$69$	0	0
$70$	154.759	0.264247
$71$	568.648	0.950508	0.475254	−	0.879849i	$-0.342356\pi$
$71$	568.648	0.950508	0.475254	+	0.879849i	$0.342356\pi$
$72$	0	0
$73$	647.751	1.03854	0.519271	−	0.854610i	$-0.326204\pi$
$73$	647.751	1.03854	0.519271	+	0.854610i	$0.326204\pi$
$74$	−418.524	−0.657465
$75$	0	0
$76$	880.875	1.32952
$77$	−263.410	−0.389848
$78$	0	0
$79$	895.113	1.27479	0.637393	−	0.770539i	$-0.280013\pi$
$79$	895.113	1.27479	0.637393	+	0.770539i	$0.280013\pi$
$80$	133.473	0.186534
$81$	0	0
$82$	−189.040	−0.254585
$83$	1338.79	1.77049	0.885247	−	0.465122i	$-0.153989\pi$
$83$	1338.79	1.77049	0.885247	+	0.465122i	$0.153989\pi$
$84$	0	0
$85$	−473.621	−0.604370
$86$	63.2464	0.0793027
$87$	0	0
$88$	203.735	0.246798
$89$	894.190	1.06499	0.532494	−	0.846434i	$-0.321255\pi$
$89$	894.190	1.06499	0.532494	+	0.846434i	$0.321255\pi$
$90$	0	0
$91$	−285.455	−0.328833
$92$	−542.886	−0.615215
$93$	0	0
$94$	−472.324	−0.518260
$95$	−695.870	−0.751524
$96$	0	0
$97$	1656.84	1.73430	0.867150	−	0.498047i	$-0.165949\pi$
$97$	1656.84	1.73430	0.867150	+	0.498047i	$0.165949\pi$
$98$	−297.838	−0.307002
$99$	0	0
$100$	−158.233	−0.158233
$101$	−599.514	−0.590632	−0.295316	−	0.955400i	$-0.595425\pi$
$101$	−599.514	−0.590632	−0.295316	+	0.955400i	$0.595425\pi$
$102$	0	0
$103$	−695.644	−0.665475	−0.332737	−	0.943020i	$-0.607972\pi$
$103$	−695.644	−0.665475	−0.332737	+	0.943020i	$0.607972\pi$
$104$	220.786	0.208172
$105$	0	0
$106$	385.448	0.353189
$107$	−810.718	−0.732477	−0.366239	−	0.930521i	$-0.619355\pi$
$107$	−810.718	−0.732477	−0.366239	+	0.930521i	$0.619355\pi$
$108$	0	0
$109$	687.886	0.604472	0.302236	−	0.953233i	$-0.402267\pi$
$109$	687.886	0.604472	0.302236	+	0.953233i	$0.402267\pi$
$110$	−71.0904	−0.0616200
$111$	0	0
$112$	−639.239	−0.539307
$113$	−41.1356	−0.0342453	−0.0171226	−	0.999853i	$-0.505451\pi$
$113$	−41.1356	−0.0342453	−0.0171226	+	0.999853i	$0.505451\pi$
$114$	0	0
$115$	428.866	0.347757
$116$	1228.59	0.983374
$117$	0	0
$118$	279.890	0.218355
$119$	2268.30	1.74735
$120$	0	0
$121$	121.000	0.0909091
$122$	−652.564	−0.484265
$123$	0	0
$124$	−1396.32	−1.01124
$125$	125.000	0.0894427
$126$	0	0
$127$	1287.60	0.899657	0.449828	−	0.893115i	$-0.351485\pi$
$127$	1287.60	0.899657	0.449828	+	0.893115i	$0.351485\pi$
$128$	1432.24	0.989011
$129$	0	0
$130$	−77.0400	−0.0519758
$131$	−2478.44	−1.65300	−0.826499	−	0.562939i	$-0.809671\pi$
$131$	−2478.44	−1.65300	−0.826499	+	0.562939i	$0.809671\pi$
$132$	0	0
$133$	3332.71	2.17280
$134$	345.340	0.222633
$135$	0	0
$136$	−1754.42	−1.10618
$137$	1095.06	0.682899	0.341450	−	0.939900i	$-0.389082\pi$
$137$	1095.06	0.682899	0.341450	+	0.939900i	$0.389082\pi$
$138$	0	0
$139$	−2907.99	−1.77448	−0.887240	−	0.461308i	$-0.847380\pi$
$139$	−2907.99	−1.77448	−0.887240	+	0.461308i	$0.847380\pi$
$140$	757.819	0.457481
$141$	0	0
$142$	−735.007	−0.434369
$143$	131.127	0.0766809
$144$	0	0
$145$	−970.553	−0.555862
$146$	−837.252	−0.474599
$147$	0	0
$148$	−2049.41	−1.13825
$149$	2666.40	1.46604	0.733019	−	0.680208i	$-0.238111\pi$
$149$	2666.40	1.46604	0.733019	+	0.680208i	$0.238111\pi$
$150$	0	0
$151$	856.810	0.461763	0.230881	−	0.972982i	$-0.425839\pi$
$151$	856.810	0.461763	0.230881	+	0.972982i	$0.425839\pi$
$152$	−2577.69	−1.37552
$153$	0	0
$154$	340.471	0.178155
$155$	1103.06	0.571612
$156$	0	0
$157$	1612.91	0.819900	0.409950	−	0.912108i	$-0.365546\pi$
$157$	1612.91	0.819900	0.409950	+	0.912108i	$0.365546\pi$
$158$	−1156.98	−0.582559
$159$	0	0
$160$	−913.376	−0.451304
$161$	−2053.96	−1.00543
$162$	0	0
$163$	989.643	0.475551	0.237776	−	0.971320i	$-0.423582\pi$
$163$	989.643	0.475551	0.237776	+	0.971320i	$0.423582\pi$
$164$	−925.681	−0.440753
$165$	0	0
$166$	−1730.45	−0.809091
$167$	−982.176	−0.455108	−0.227554	−	0.973765i	$-0.573073\pi$
$167$	−982.176	−0.455108	−0.227554	+	0.973765i	$0.573073\pi$
$168$	0	0
$169$	−2054.90	−0.935320
$170$	612.180	0.276189
$171$	0	0
$172$	309.702	0.137294
$173$	1455.43	0.639620	0.319810	−	0.947482i	$-0.396381\pi$
$173$	1455.43	0.639620	0.319810	+	0.947482i	$0.396381\pi$
$174$	0	0
$175$	−598.658	−0.258596
$176$	293.641	0.125761
$177$	0	0
$178$	−1155.79	−0.486685
$179$	−3440.47	−1.43661	−0.718305	−	0.695729i	$-0.755082\pi$
$179$	−3440.47	−1.43661	−0.718305	+	0.695729i	$0.755082\pi$
$180$	0	0
$181$	872.495	0.358299	0.179149	−	0.983822i	$-0.442665\pi$
$181$	872.495	0.358299	0.179149	+	0.983822i	$0.442665\pi$
$182$	368.965	0.150272
$183$	0	0
$184$	1588.64	0.636500
$185$	1618.98	0.643405
$186$	0	0
$187$	−1041.97	−0.407466
$188$	−2312.85	−0.897246
$189$	0	0
$190$	899.448	0.343436
$191$	−2371.20	−0.898293	−0.449146	−	0.893458i	$-0.648272\pi$
$191$	−2371.20	−0.898293	−0.449146	+	0.893458i	$0.648272\pi$
$192$	0	0
$193$	1173.91	0.437825	0.218913	−	0.975744i	$-0.429749\pi$
$193$	1173.91	0.437825	0.218913	+	0.975744i	$0.429749\pi$
$194$	−2141.56	−0.792551
$195$	0	0
$196$	−1458.44	−0.531502
$197$	2835.33	1.02543	0.512714	−	0.858560i	$-0.328640\pi$
$197$	2835.33	1.02543	0.512714	+	0.858560i	$0.328640\pi$
$198$	0	0
$199$	2852.59	1.01615	0.508077	−	0.861312i	$-0.330357\pi$
$199$	2852.59	1.01615	0.508077	+	0.861312i	$0.330357\pi$
$200$	463.034	0.163707
$201$	0	0
$202$	774.903	0.269911
$203$	4648.24	1.60710
$204$	0	0
$205$	731.266	0.249141
$206$	899.157	0.304113
$207$	0	0
$208$	318.216	0.106079
$209$	−1530.91	−0.506677
$210$	0	0
$211$	−1412.00	−0.460691	−0.230346	−	0.973109i	$-0.573986\pi$
$211$	−1412.00	−0.460691	−0.230346	+	0.973109i	$0.573986\pi$
$212$	1887.44	0.611463
$213$	0	0
$214$	1047.90	0.334732
$215$	−244.657	−0.0776068
$216$	0	0
$217$	−5282.84	−1.65264
$218$	−889.128	−0.276235
$219$	0	0
$220$	−348.112	−0.106680
$221$	−1129.17	−0.343694
$222$	0	0
$223$	−3318.57	−0.996537	−0.498269	−	0.867023i	$-0.666031\pi$
$223$	−3318.57	−0.996537	−0.498269	+	0.867023i	$0.666031\pi$
$224$	4374.40	1.30481
$225$	0	0
$226$	53.1700	0.0156496
$227$	−4668.50	−1.36502	−0.682510	−	0.730877i	$-0.739111\pi$
$227$	−4668.50	−1.36502	−0.682510	+	0.730877i	$0.739111\pi$
$228$	0	0
$229$	−4958.20	−1.43077	−0.715387	−	0.698729i	$-0.753749\pi$
$229$	−4958.20	−1.43077	−0.715387	+	0.698729i	$0.753749\pi$
$230$	−554.332	−0.158920
$231$	0	0
$232$	−3595.19	−1.01740
$233$	−2578.36	−0.724954	−0.362477	−	0.931993i	$-0.618069\pi$
$233$	−2578.36	−0.724954	−0.362477	+	0.931993i	$0.618069\pi$
$234$	0	0
$235$	1827.10	0.507178
$236$	1370.55	0.378031
$237$	0	0
$238$	−2931.89	−0.798515
$239$	733.970	0.198647	0.0993234	−	0.995055i	$-0.468332\pi$
$239$	733.970	0.198647	0.0993234	+	0.995055i	$0.468332\pi$
$240$	0	0
$241$	2920.84	0.780695	0.390348	−	0.920668i	$-0.372355\pi$
$241$	2920.84	0.780695	0.390348	+	0.920668i	$0.372355\pi$
$242$	−156.399	−0.0415442
$243$	0	0
$244$	−3195.45	−0.838391
$245$	1152.13	0.300437
$246$	0	0
$247$	−1659.04	−0.427377
$248$	4086.03	1.04622
$249$	0	0
$250$	−161.569	−0.0408741
$251$	−6217.42	−1.56351	−0.781753	−	0.623589i	$-0.785674\pi$
$251$	−6217.42	−1.56351	−0.781753	+	0.623589i	$0.785674\pi$
$252$	0	0
$253$	943.506	0.234457
$254$	−1664.30	−0.411131
$255$	0	0
$256$	−2031.73	−0.496027
$257$	4834.97	1.17353	0.586765	−	0.809757i	$-0.300401\pi$
$257$	4834.97	1.17353	0.586765	+	0.809757i	$0.300401\pi$
$258$	0	0
$259$	−7753.74	−1.86021
$260$	−377.246	−0.0899839
$261$	0	0
$262$	3203.52	0.755397
$263$	5833.19	1.36764	0.683821	−	0.729649i	$-0.260317\pi$
$263$	5833.19	1.36764	0.683821	+	0.729649i	$0.260317\pi$
$264$	0	0
$265$	−1491.03	−0.345636
$266$	−4307.69	−0.992939
$267$	0	0
$268$	1691.05	0.385437
$269$	−6046.60	−1.37051	−0.685256	−	0.728302i	$-0.740310\pi$
$269$	−6046.60	−1.37051	−0.685256	+	0.728302i	$0.740310\pi$
$270$	0	0
$271$	1878.10	0.420983	0.210492	−	0.977596i	$-0.432494\pi$
$271$	1878.10	0.420983	0.210492	+	0.977596i	$0.432494\pi$
$272$	−2528.63	−0.563679
$273$	0	0
$274$	−1415.42	−0.312075
$275$	275.000	0.0603023
$276$	0	0
$277$	−5070.95	−1.09994	−0.549970	−	0.835184i	$-0.685361\pi$
$277$	−5070.95	−1.09994	−0.549970	+	0.835184i	$0.685361\pi$
$278$	3758.73	0.810913
$279$	0	0
$280$	−2217.59	−0.473309
$281$	−4874.35	−1.03480	−0.517401	−	0.855743i	$-0.673101\pi$
$281$	−4874.35	−1.03480	−0.517401	+	0.855743i	$0.673101\pi$
$282$	0	0
$283$	−2316.73	−0.486626	−0.243313	−	0.969948i	$-0.578234\pi$
$283$	−2316.73	−0.486626	−0.243313	+	0.969948i	$0.578234\pi$
$284$	−3599.15	−0.752008
$285$	0	0
$286$	−169.488	−0.0350421
$287$	−3502.23	−0.720313
$288$	0	0
$289$	4059.69	0.826316
$290$	1254.49	0.254021
$291$	0	0
$292$	−4099.82	−0.821657
$293$	3466.19	0.691116	0.345558	−	0.938397i	$-0.387690\pi$
$293$	3466.19	0.691116	0.345558	+	0.938397i	$0.387690\pi$
$294$	0	0
$295$	−1082.70	−0.213686
$296$	5997.16	1.17763
$297$	0	0
$298$	−3446.45	−0.669959
$299$	1022.47	0.197762
$300$	0	0
$301$	1171.73	0.224376
$302$	−1107.47	−0.211019
$303$	0	0
$304$	−3715.20	−0.700925
$305$	2524.32	0.473910
$306$	0	0
$307$	3350.79	0.622930	0.311465	−	0.950258i	$-0.399180\pi$
$307$	3350.79	0.622930	0.311465	+	0.950258i	$0.399180\pi$
$308$	1667.20	0.308434
$309$	0	0
$310$	−1425.76	−0.261219
$311$	−6675.62	−1.21717	−0.608585	−	0.793489i	$-0.708262\pi$
$311$	−6675.62	−1.21717	−0.608585	+	0.793489i	$0.708262\pi$
$312$	0	0
$313$	8110.22	1.46459	0.732295	−	0.680988i	$-0.238449\pi$
$313$	8110.22	1.46459	0.732295	+	0.680988i	$0.238449\pi$
$314$	−2084.77	−0.374683
$315$	0	0
$316$	−5665.45	−1.00856
$317$	5126.97	0.908389	0.454195	−	0.890903i	$-0.349927\pi$
$317$	5126.97	0.908389	0.454195	+	0.890903i	$0.349927\pi$
$318$	0	0
$319$	−2135.22	−0.374762
$320$	112.801	0.0197054
$321$	0	0
$322$	2654.84	0.459468
$323$	13183.2	2.27099
$324$	0	0
$325$	298.015	0.0508643
$326$	−1279.17	−0.217320
$327$	0	0
$328$	2708.81	0.456003
$329$	−8750.46	−1.46635
$330$	0	0
$331$	7190.21	1.19399	0.596994	−	0.802246i	$-0.296362\pi$
$331$	7190.21	1.19399	0.596994	+	0.802246i	$0.296362\pi$
$332$	−8473.60	−1.40075
$333$	0	0
$334$	1269.51	0.207978
$335$	−1335.89	−0.217872
$336$	0	0
$337$	6701.75	1.08329	0.541644	−	0.840608i	$-0.317802\pi$
$337$	6701.75	1.08329	0.541644	+	0.840608i	$0.317802\pi$
$338$	2656.06	0.427428
$339$	0	0
$340$	2997.70	0.478156
$341$	2426.73	0.385381
$342$	0	0
$343$	2695.72	0.424359
$344$	−906.277	−0.142044
$345$	0	0
$346$	−1881.22	−0.292297
$347$	9748.82	1.50820	0.754098	−	0.656762i	$-0.228074\pi$
$347$	9748.82	1.50820	0.754098	+	0.656762i	$0.228074\pi$
$348$	0	0
$349$	9682.14	1.48502	0.742512	−	0.669833i	$-0.233634\pi$
$349$	9682.14	1.48502	0.742512	+	0.669833i	$0.233634\pi$
$350$	773.797	0.118175
$351$	0	0
$352$	−2009.43	−0.304269
$353$	5516.77	0.831807	0.415904	−	0.909409i	$-0.363465\pi$
$353$	5516.77	0.831807	0.415904	+	0.909409i	$0.363465\pi$
$354$	0	0
$355$	2843.24	0.425080
$356$	−5659.60	−0.842580
$357$	0	0
$358$	4446.99	0.656511
$359$	−9005.89	−1.32399	−0.661996	−	0.749508i	$-0.730290\pi$
$359$	−9005.89	−1.32399	−0.661996	+	0.749508i	$0.730290\pi$
$360$	0	0
$361$	12510.4	1.82394
$362$	−1127.75	−0.163737
$363$	0	0
$364$	1806.73	0.260161
$365$	3238.76	0.464450
$366$	0	0
$367$	2003.75	0.285000	0.142500	−	0.989795i	$-0.454486\pi$
$367$	2003.75	0.285000	0.142500	+	0.989795i	$0.454486\pi$
$368$	2289.69	0.324343
$369$	0	0
$370$	−2092.62	−0.294027
$371$	7140.96	0.999299
$372$	0	0
$373$	−1645.62	−0.228437	−0.114218	−	0.993456i	$-0.536436\pi$
$373$	−1645.62	−0.228437	−0.114218	+	0.993456i	$0.536436\pi$
$374$	1346.80	0.186206
$375$	0	0
$376$	6768.07	0.928289
$377$	−2313.92	−0.316108
$378$	0	0
$379$	10870.6	1.47331	0.736653	−	0.676271i	$-0.236405\pi$
$379$	10870.6	1.47331	0.736653	+	0.676271i	$0.236405\pi$
$380$	4404.38	0.594578
$381$	0	0
$382$	3064.90	0.410507
$383$	1397.77	0.186483	0.0932414	−	0.995644i	$-0.470277\pi$
$383$	1397.77	0.186483	0.0932414	+	0.995644i	$0.470277\pi$
$384$	0	0
$385$	−1317.05	−0.174345
$386$	−1517.35	−0.200080
$387$	0	0
$388$	−10486.7	−1.37212
$389$	11786.8	1.53629	0.768145	−	0.640276i	$-0.221180\pi$
$389$	11786.8	1.53629	0.768145	+	0.640276i	$0.221180\pi$
$390$	0	0
$391$	−8124.81	−1.05087
$392$	4267.82	0.549891
$393$	0	0
$394$	−3664.82	−0.468606
$395$	4475.56	0.570102
$396$	0	0
$397$	11543.3	1.45930	0.729649	−	0.683822i	$-0.239683\pi$
$397$	11543.3	1.45930	0.729649	+	0.683822i	$0.239683\pi$
$398$	−3687.12	−0.464368
$399$	0	0
$400$	667.366	0.0834207
$401$	−10332.6	−1.28675	−0.643376	−	0.765551i	$-0.722467\pi$
$401$	−10332.6	−1.28675	−0.643376	+	0.765551i	$0.722467\pi$
$402$	0	0
$403$	2629.83	0.325065
$404$	3794.51	0.467287
$405$	0	0
$406$	−6008.09	−0.734424
$407$	3561.76	0.433784
$408$	0	0
$409$	506.035	0.0611781	0.0305890	−	0.999532i	$-0.490262\pi$
$409$	506.035	0.0611781	0.0305890	+	0.999532i	$0.490262\pi$
$410$	−945.199	−0.113854
$411$	0	0
$412$	4402.95	0.526499
$413$	5185.35	0.617807
$414$	0	0
$415$	6693.93	0.791789
$416$	−2177.60	−0.256648
$417$	0	0
$418$	1978.79	0.231544
$419$	−2687.19	−0.313312	−0.156656	−	0.987653i	$-0.550071\pi$
$419$	−2687.19	−0.313312	−0.156656	+	0.987653i	$0.550071\pi$
$420$	0	0
$421$	−7001.75	−0.810557	−0.405279	−	0.914193i	$-0.632826\pi$
$421$	−7001.75	−0.810557	−0.405279	+	0.914193i	$0.632826\pi$
$422$	1825.08	0.210530
$423$	0	0
$424$	−5523.20	−0.632619
$425$	−2368.11	−0.270283
$426$	0	0
$427$	−12089.7	−1.37016
$428$	5131.29	0.579509
$429$	0	0
$430$	316.232	0.0354652
$431$	−312.157	−0.0348865	−0.0174433	−	0.999848i	$-0.505553\pi$
$431$	−312.157	−0.0348865	−0.0174433	+	0.999848i	$0.505553\pi$
$432$	0	0
$433$	−7815.74	−0.867438	−0.433719	−	0.901048i	$-0.642799\pi$
$433$	−7815.74	−0.867438	−0.433719	+	0.901048i	$0.642799\pi$
$434$	6828.35	0.755233
$435$	0	0
$436$	−4353.84	−0.478237
$437$	−11937.4	−1.30674
$438$	0	0
$439$	8798.68	0.956579	0.478290	−	0.878202i	$-0.341257\pi$
$439$	8798.68	0.956579	0.478290	+	0.878202i	$0.341257\pi$
$440$	1018.68	0.110371
$441$	0	0
$442$	1459.51	0.157063
$443$	−11005.1	−1.18029	−0.590144	−	0.807298i	$-0.700929\pi$
$443$	−11005.1	−1.18029	−0.590144	+	0.807298i	$0.700929\pi$
$444$	0	0
$445$	4470.95	0.476277
$446$	4289.42	0.455404
$447$	0	0
$448$	−540.232	−0.0569722
$449$	−3499.00	−0.367768	−0.183884	−	0.982948i	$-0.558867\pi$
$449$	−3499.00	−0.367768	−0.183884	+	0.982948i	$0.558867\pi$
$450$	0	0
$451$	1608.78	0.167971
$452$	260.360	0.0270936
$453$	0	0
$454$	6034.28	0.623795
$455$	−1427.27	−0.147059
$456$	0	0
$457$	13407.3	1.37236	0.686179	−	0.727433i	$-0.259287\pi$
$457$	13407.3	1.37236	0.686179	+	0.727433i	$0.259287\pi$
$458$	6408.73	0.653844
$459$	0	0
$460$	−2714.43	−0.275132
$461$	−11744.2	−1.18651	−0.593254	−	0.805016i	$-0.702157\pi$
$461$	−11744.2	−1.18651	−0.593254	+	0.805016i	$0.702157\pi$
$462$	0	0
$463$	1661.76	0.166800	0.0834000	−	0.996516i	$-0.473422\pi$
$463$	1661.76	0.166800	0.0834000	+	0.996516i	$0.473422\pi$
$464$	−5181.71	−0.518437
$465$	0	0
$466$	3332.67	0.331294
$467$	−13439.7	−1.33173	−0.665863	−	0.746074i	$-0.731936\pi$
$467$	−13439.7	−1.33173	−0.665863	+	0.746074i	$0.731936\pi$
$468$	0	0
$469$	6397.91	0.629911
$470$	−2361.62	−0.231773
$471$	0	0
$472$	−4010.63	−0.391110
$473$	−538.245	−0.0523225
$474$	0	0
$475$	−3479.35	−0.336092
$476$	−14356.8	−1.38244
$477$	0	0
$478$	−948.694	−0.0907788
$479$	3573.86	0.340906	0.170453	−	0.985366i	$-0.445477\pi$
$479$	3573.86	0.340906	0.170453	+	0.985366i	$0.445477\pi$
$480$	0	0
$481$	3859.85	0.365892
$482$	−3775.33	−0.356767
$483$	0	0
$484$	−765.846	−0.0719240
$485$	8284.22	0.775603
$486$	0	0
$487$	−9057.06	−0.842740	−0.421370	−	0.906889i	$-0.638451\pi$
$487$	−9057.06	−0.842740	−0.421370	+	0.906889i	$0.638451\pi$
$488$	9350.79	0.867399
$489$	0	0
$490$	−1489.19	−0.137296
$491$	−21528.2	−1.97872	−0.989360	−	0.145485i	$-0.953526\pi$
$491$	−21528.2	−1.97872	−0.989360	+	0.145485i	$0.953526\pi$
$492$	0	0
$493$	18387.0	1.67973
$494$	2144.39	0.195305
$495$	0	0
$496$	5889.15	0.533127
$497$	−13617.0	−1.22899
$498$	0	0
$499$	18590.7	1.66780	0.833902	−	0.551912i	$-0.186101\pi$
$499$	18590.7	1.66780	0.833902	+	0.551912i	$0.186101\pi$
$500$	−791.164	−0.0707638
$501$	0	0
$502$	8036.33	0.714500
$503$	4207.37	0.372957	0.186479	−	0.982459i	$-0.440292\pi$
$503$	4207.37	0.372957	0.186479	+	0.982459i	$0.440292\pi$
$504$	0	0
$505$	−2997.57	−0.264139
$506$	−1219.53	−0.107144
$507$	0	0
$508$	−8149.65	−0.711776
$509$	21708.8	1.89043	0.945213	−	0.326455i	$-0.105854\pi$
$509$	21708.8	1.89043	0.945213	+	0.326455i	$0.105854\pi$
$510$	0	0
$511$	−15511.3	−1.34281
$512$	−8831.82	−0.762334
$513$	0	0
$514$	−6249.46	−0.536287
$515$	−3478.22	−0.297609
$516$	0	0
$517$	4019.61	0.341939
$518$	10022.1	0.850089
$519$	0	0
$520$	1103.93	0.0930972
$521$	16149.6	1.35802	0.679008	−	0.734131i	$-0.262410\pi$
$521$	16149.6	1.35802	0.679008	+	0.734131i	$0.262410\pi$
$522$	0	0
$523$	14762.0	1.23422	0.617110	−	0.786877i	$-0.288303\pi$
$523$	14762.0	1.23422	0.617110	+	0.786877i	$0.288303\pi$
$524$	15686.8	1.30779
$525$	0	0
$526$	−7539.70	−0.624994
$527$	−20897.3	−1.72733
$528$	0	0
$529$	−4809.94	−0.395327
$530$	1927.24	0.157951
$531$	0	0
$532$	−21093.7	−1.71904
$533$	1743.43	0.141681
$534$	0	0
$535$	−4053.59	−0.327574
$536$	−4948.49	−0.398773
$537$	0	0
$538$	7815.55	0.626305
$539$	2534.69	0.202555
$540$	0	0
$541$	−5965.01	−0.474040	−0.237020	−	0.971505i	$-0.576171\pi$
$541$	−5965.01	−0.474040	−0.237020	+	0.971505i	$0.576171\pi$
$542$	−2427.54	−0.192383
$543$	0	0
$544$	17303.8	1.36377
$545$	3439.43	0.270328
$546$	0	0
$547$	19294.4	1.50817	0.754084	−	0.656778i	$-0.228081\pi$
$547$	19294.4	1.50817	0.754084	+	0.656778i	$0.228081\pi$
$548$	−6930.97	−0.540285
$549$	0	0
$550$	−355.452	−0.0275573
$551$	27015.1	2.08872
$552$	0	0
$553$	−21434.7	−1.64827
$554$	6554.46	0.502658
$555$	0	0
$556$	18405.6	1.40390
$557$	−108.806	−0.00827698	−0.00413849	−	0.999991i	$-0.501317\pi$
$557$	−108.806	−0.00827698	−0.00413849	+	0.999991i	$0.501317\pi$
$558$	0	0
$559$	−583.292	−0.0441335
$560$	−3196.19	−0.241185
$561$	0	0
$562$	6300.35	0.472890
$563$	5471.91	0.409616	0.204808	−	0.978802i	$-0.434343\pi$
$563$	5471.91	0.409616	0.204808	+	0.978802i	$0.434343\pi$
$564$	0	0
$565$	−205.678	−0.0153150
$566$	2994.49	0.222381
$567$	0	0
$568$	10532.1	0.778026
$569$	−16531.0	−1.21796	−0.608978	−	0.793187i	$-0.708420\pi$
$569$	−16531.0	−1.21796	−0.608978	+	0.793187i	$0.708420\pi$
$570$	0	0
$571$	11982.3	0.878182	0.439091	−	0.898443i	$-0.355301\pi$
$571$	11982.3	0.878182	0.439091	+	0.898443i	$0.355301\pi$
$572$	−829.942	−0.0606671
$573$	0	0
$574$	4526.81	0.329173
$575$	2144.33	0.155521
$576$	0	0
$577$	−22727.3	−1.63977	−0.819887	−	0.572525i	$-0.805964\pi$
$577$	−22727.3	−1.63977	−0.819887	+	0.572525i	$0.805964\pi$
$578$	−5247.36	−0.377615
$579$	0	0
$580$	6142.93	0.439778
$581$	−32059.0	−2.28921
$582$	0	0
$583$	−3280.27	−0.233028
$584$	11997.2	0.850085
$585$	0	0
$586$	−4480.23	−0.315830
$587$	21427.2	1.50664	0.753318	−	0.657657i	$-0.228452\pi$
$587$	21427.2	1.50664	0.753318	+	0.657657i	$0.228452\pi$
$588$	0	0
$589$	−30703.4	−2.14790
$590$	1399.45	0.0976515
$591$	0	0
$592$	8643.63	0.600086
$593$	−22434.5	−1.55358	−0.776791	−	0.629758i	$-0.783154\pi$
$593$	−22434.5	−1.55358	−0.776791	+	0.629758i	$0.783154\pi$
$594$	0	0
$595$	11341.5	0.781439
$596$	−16876.4	−1.15988
$597$	0	0
$598$	−1321.60	−0.0903747
$599$	11183.4	0.762839	0.381419	−	0.924402i	$-0.375435\pi$
$599$	11183.4	0.762839	0.381419	+	0.924402i	$0.375435\pi$
$600$	0	0
$601$	14021.3	0.951648	0.475824	−	0.879541i	$-0.342150\pi$
$601$	14021.3	0.951648	0.475824	+	0.879541i	$0.342150\pi$
$602$	−1514.52	−0.102537
$603$	0	0
$604$	−5423.01	−0.365330
$605$	605.000	0.0406558
$606$	0	0
$607$	−15679.6	−1.04846	−0.524230	−	0.851577i	$-0.675647\pi$
$607$	−15679.6	−1.04846	−0.524230	+	0.851577i	$0.675647\pi$
$608$	25423.6	1.69583
$609$	0	0
$610$	−3262.82	−0.216570
$611$	4356.02	0.288422
$612$	0	0
$613$	−22771.0	−1.50034	−0.750172	−	0.661243i	$-0.770029\pi$
$613$	−22771.0	−1.50034	−0.750172	+	0.661243i	$0.770029\pi$
$614$	−4331.06	−0.284670
$615$	0	0
$616$	−4878.71	−0.319105
$617$	−3452.49	−0.225270	−0.112635	−	0.993636i	$-0.535929\pi$
$617$	−3452.49	−0.225270	−0.112635	+	0.993636i	$0.535929\pi$
$618$	0	0
$619$	7022.36	0.455981	0.227991	−	0.973663i	$-0.426784\pi$
$619$	7022.36	0.455981	0.227991	+	0.973663i	$0.426784\pi$
$620$	−6981.60	−0.452239
$621$	0	0
$622$	8628.58	0.556229
$623$	−21412.6	−1.37701
$624$	0	0
$625$	625.000	0.0400000
$626$	−10482.9	−0.669297
$627$	0	0
$628$	−10208.6	−0.648675
$629$	−30671.4	−1.94427
$630$	0	0
$631$	−2306.63	−0.145524	−0.0727619	−	0.997349i	$-0.523181\pi$
$631$	−2306.63	−0.145524	−0.0727619	+	0.997349i	$0.523181\pi$
$632$	16578.7	1.04346
$633$	0	0
$634$	−6626.88	−0.415121
$635$	6438.02	0.402339
$636$	0	0
$637$	2746.83	0.170853
$638$	2759.88	0.171261
$639$	0	0
$640$	7161.21	0.442299
$641$	16132.1	0.994038	0.497019	−	0.867740i	$-0.334428\pi$
$641$	16132.1	0.994038	0.497019	+	0.867740i	$0.334428\pi$
$642$	0	0
$643$	−24654.0	−1.51207	−0.756033	−	0.654534i	$-0.772865\pi$
$643$	−24654.0	−1.51207	−0.756033	+	0.654534i	$0.772865\pi$
$644$	13000.1	0.795460
$645$	0	0
$646$	−17039.9	−1.03781
$647$	7179.94	0.436279	0.218140	−	0.975918i	$-0.430001\pi$
$647$	7179.94	0.436279	0.218140	+	0.975918i	$0.430001\pi$
$648$	0	0
$649$	−2381.94	−0.144067
$650$	−385.200	−0.0232443
$651$	0	0
$652$	−6263.76	−0.376239
$653$	22995.6	1.37808	0.689040	−	0.724723i	$-0.258032\pi$
$653$	22995.6	1.37808	0.689040	+	0.724723i	$0.258032\pi$
$654$	0	0
$655$	−12392.2	−0.739243
$656$	3904.17	0.232366
$657$	0	0
$658$	11310.4	0.670100
$659$	−15857.1	−0.937336	−0.468668	−	0.883375i	$-0.655266\pi$
$659$	−15857.1	−0.937336	−0.468668	+	0.883375i	$0.655266\pi$
$660$	0	0
$661$	−12595.8	−0.741178	−0.370589	−	0.928797i	$-0.620844\pi$
$661$	−12595.8	−0.741178	−0.370589	+	0.928797i	$0.620844\pi$
$662$	−9293.72	−0.545636
$663$	0	0
$664$	24796.2	1.44921
$665$	16663.5	0.971705
$666$	0	0
$667$	−16649.5	−0.966524
$668$	6216.50	0.360065
$669$	0	0
$670$	1726.70	0.0995646
$671$	5553.51	0.319510
$672$	0	0
$673$	5198.03	0.297726	0.148863	−	0.988858i	$-0.452439\pi$
$673$	5198.03	0.297726	0.148863	+	0.988858i	$0.452439\pi$
$674$	−8662.37	−0.495047
$675$	0	0
$676$	13006.1	0.739992
$677$	10878.4	0.617566	0.308783	−	0.951132i	$-0.400078\pi$
$677$	10878.4	0.617566	0.308783	+	0.951132i	$0.400078\pi$
$678$	0	0
$679$	−39675.4	−2.24242
$680$	−8772.12	−0.494699
$681$	0	0
$682$	−3136.67	−0.176114
$683$	17991.0	1.00792	0.503958	−	0.863728i	$-0.331877\pi$
$683$	17991.0	1.00792	0.503958	+	0.863728i	$0.331877\pi$
$684$	0	0
$685$	5475.29	0.305402
$686$	−3484.36	−0.193926
$687$	0	0
$688$	−1306.21	−0.0723817
$689$	−3554.81	−0.196556
$690$	0	0
$691$	8114.92	0.446753	0.223376	−	0.974732i	$-0.428292\pi$
$691$	8114.92	0.446753	0.223376	+	0.974732i	$0.428292\pi$
$692$	−9211.86	−0.506044
$693$	0	0
$694$	−12600.9	−0.689225
$695$	−14540.0	−0.793572
$696$	0	0
$697$	−13853.7	−0.752865
$698$	−12514.7	−0.678635
$699$	0	0
$700$	3789.09	0.204592
$701$	−373.037	−0.0200990	−0.0100495	−	0.999950i	$-0.503199\pi$
$701$	−373.037	−0.0200990	−0.0100495	+	0.999950i	$0.503199\pi$
$702$	0	0
$703$	−45064.0	−2.41767
$704$	248.161	0.0132854
$705$	0	0
$706$	−7130.71	−0.380124
$707$	14356.2	0.763676
$708$	0	0
$709$	25154.1	1.33241	0.666206	−	0.745768i	$-0.267917\pi$
$709$	25154.1	1.33241	0.666206	+	0.745768i	$0.267917\pi$
$710$	−3675.03	−0.194256
$711$	0	0
$712$	16561.6	0.871732
$713$	18922.6	0.993909
$714$	0	0
$715$	655.634	0.0342927
$716$	21775.8	1.13659
$717$	0	0
$718$	11640.6	0.605046
$719$	−23564.5	−1.22227	−0.611133	−	0.791528i	$-0.709286\pi$
$719$	−23564.5	−1.22227	−0.611133	+	0.791528i	$0.709286\pi$
$720$	0	0
$721$	16658.1	0.860445
$722$	−16170.3	−0.833514
$723$	0	0
$724$	−5522.29	−0.283473
$725$	−4852.76	−0.248589
$726$	0	0
$727$	−534.682	−0.0272769	−0.0136384	−	0.999907i	$-0.504341\pi$
$727$	−534.682	−0.0272769	−0.0136384	+	0.999907i	$0.504341\pi$
$728$	−5287.02	−0.269162
$729$	0	0
$730$	−4186.26	−0.212247
$731$	4634.99	0.234516
$732$	0	0
$733$	11735.8	0.591365	0.295683	−	0.955286i	$-0.404453\pi$
$733$	11735.8	0.591365	0.295683	+	0.955286i	$0.404453\pi$
$734$	−2589.96	−0.130241
$735$	0	0
$736$	−15668.7	−0.784720
$737$	−2938.95	−0.146889
$738$	0	0
$739$	10127.1	0.504103	0.252052	−	0.967714i	$-0.418895\pi$
$739$	10127.1	0.504103	0.252052	+	0.967714i	$0.418895\pi$
$740$	−10247.0	−0.509039
$741$	0	0
$742$	−9230.06	−0.456666
$743$	20613.3	1.01780	0.508901	−	0.860825i	$-0.330052\pi$
$743$	20613.3	1.01780	0.508901	+	0.860825i	$0.330052\pi$
$744$	0	0
$745$	13332.0	0.655632
$746$	2127.05	0.104392
$747$	0	0
$748$	6594.93	0.322372
$749$	19413.7	0.947079
$750$	0	0
$751$	−28558.8	−1.38765	−0.693826	−	0.720142i	$-0.744076\pi$
$751$	−28558.8	−1.38765	−0.693826	+	0.720142i	$0.744076\pi$
$752$	9754.74	0.473030
$753$	0	0
$754$	2990.86	0.144457
$755$	4284.05	0.206507
$756$	0	0
$757$	−16283.2	−0.781802	−0.390901	−	0.920433i	$-0.627837\pi$
$757$	−16283.2	−0.781802	−0.390901	+	0.920433i	$0.627837\pi$
$758$	−14050.8	−0.673280
$759$	0	0
$760$	−12888.5	−0.615150
$761$	14869.8	0.708319	0.354160	−	0.935185i	$-0.384767\pi$
$761$	14869.8	0.708319	0.354160	+	0.935185i	$0.384767\pi$
$762$	0	0
$763$	−16472.3	−0.781571
$764$	15008.0	0.710697
$765$	0	0
$766$	−1806.69	−0.0852200
$767$	−2581.29	−0.121519
$768$	0	0
$769$	6908.25	0.323951	0.161975	−	0.986795i	$-0.448214\pi$
$769$	6908.25	0.323951	0.161975	+	0.986795i	$0.448214\pi$
$770$	1702.35	0.0796734
$771$	0	0
$772$	−7430.07	−0.346391
$773$	−13969.8	−0.650012	−0.325006	−	0.945712i	$-0.605366\pi$
$773$	−13969.8	−0.650012	−0.325006	+	0.945712i	$0.605366\pi$
$774$	0	0
$775$	5515.30	0.255633
$776$	30687.0	1.41959
$777$	0	0
$778$	−15235.1	−0.702063
$779$	−20354.6	−0.936175
$780$	0	0
$781$	6255.13	0.286589
$782$	10501.7	0.480232
$783$	0	0
$784$	6151.16	0.280209
$785$	8064.56	0.366671
$786$	0	0
$787$	−17219.3	−0.779926	−0.389963	−	0.920831i	$-0.627512\pi$
$787$	−17219.3	−0.779926	−0.389963	+	0.920831i	$0.627512\pi$
$788$	−17945.7	−0.811281
$789$	0	0
$790$	−5784.90	−0.260528
$791$	985.048	0.0442785
$792$	0	0
$793$	6018.30	0.269503
$794$	−14920.3	−0.666879
$795$	0	0
$796$	−18054.9	−0.803944
$797$	11574.4	0.514410	0.257205	−	0.966357i	$-0.417198\pi$
$797$	11574.4	0.514410	0.257205	+	0.966357i	$0.417198\pi$
$798$	0	0
$799$	−34614.1	−1.53261
$800$	−4566.88	−0.201830
$801$	0	0
$802$	13355.5	0.588027
$803$	7125.26	0.313132
$804$	0	0
$805$	−10269.8	−0.449642
$806$	−3399.19	−0.148550
$807$	0	0
$808$	−11103.8	−0.483454
$809$	33007.8	1.43448	0.717238	−	0.696828i	$-0.245406\pi$
$809$	33007.8	1.43448	0.717238	+	0.696828i	$0.245406\pi$
$810$	0	0
$811$	−15818.5	−0.684909	−0.342455	−	0.939534i	$-0.611258\pi$
$811$	−15818.5	−0.684909	−0.342455	+	0.939534i	$0.611258\pi$
$812$	−29420.1	−1.27148
$813$	0	0
$814$	−4603.76	−0.198233
$815$	4948.22	0.212673
$816$	0	0
$817$	6809.97	0.291617
$818$	−654.077	−0.0279575
$819$	0	0
$820$	−4628.41	−0.197111
$821$	−13434.7	−0.571102	−0.285551	−	0.958364i	$-0.592177\pi$
$821$	−13434.7	−0.571102	−0.285551	+	0.958364i	$0.592177\pi$
$822$	0	0
$823$	19453.4	0.823940	0.411970	−	0.911197i	$-0.364841\pi$
$823$	19453.4	0.823940	0.411970	+	0.911197i	$0.364841\pi$
$824$	−12884.3	−0.544716
$825$	0	0
$826$	−6702.33	−0.282329
$827$	16814.0	0.706991	0.353495	−	0.935436i	$-0.384993\pi$
$827$	16814.0	0.706991	0.353495	+	0.935436i	$0.384993\pi$
$828$	0	0
$829$	11008.1	0.461192	0.230596	−	0.973050i	$-0.425932\pi$
$829$	11008.1	0.461192	0.230596	+	0.973050i	$0.425932\pi$
$830$	−8652.26	−0.361836
$831$	0	0
$832$	268.930	0.0112061
$833$	−21827.0	−0.907876
$834$	0	0
$835$	−4910.88	−0.203531
$836$	9689.63	0.400865
$837$	0	0
$838$	3473.33	0.143179
$839$	−30822.2	−1.26829	−0.634147	−	0.773212i	$-0.718649\pi$
$839$	−30822.2	−1.26829	−0.634147	+	0.773212i	$0.718649\pi$
$840$	0	0
$841$	13289.9	0.544914
$842$	9050.13	0.370413
$843$	0	0
$844$	8936.96	0.364482
$845$	−10274.5	−0.418288
$846$	0	0
$847$	−2897.51	−0.117544
$848$	−7960.52	−0.322365
$849$	0	0
$850$	3060.90	0.123515
$851$	27773.1	1.11874
$852$	0	0
$853$	−18592.3	−0.746294	−0.373147	−	0.927772i	$-0.621721\pi$
$853$	−18592.3	−0.746294	−0.373147	+	0.927772i	$0.621721\pi$
$854$	15626.5	0.626146
$855$	0	0
$856$	−15015.6	−0.599560
$857$	−10618.3	−0.423236	−0.211618	−	0.977352i	$-0.567873\pi$
$857$	−10618.3	−0.423236	−0.211618	+	0.977352i	$0.567873\pi$
$858$	0	0
$859$	−5247.82	−0.208444	−0.104222	−	0.994554i	$-0.533235\pi$
$859$	−5247.82	−0.208444	−0.104222	+	0.994554i	$0.533235\pi$
$860$	1548.51	0.0613997
$861$	0	0
$862$	403.479	0.0159427
$863$	−28392.0	−1.11990	−0.559950	−	0.828526i	$-0.689180\pi$
$863$	−28392.0	−1.11990	−0.559950	+	0.828526i	$0.689180\pi$
$864$	0	0
$865$	7277.14	0.286047
$866$	10102.3	0.396407
$867$	0	0
$868$	33436.8	1.30751
$869$	9846.24	0.384362
$870$	0	0
$871$	−3184.91	−0.123900
$872$	12740.6	0.494783
$873$	0	0
$874$	15429.7	0.597160
$875$	−2993.29	−0.115648
$876$	0	0
$877$	37113.8	1.42901	0.714506	−	0.699630i	$-0.246652\pi$
$877$	37113.8	1.42901	0.714506	+	0.699630i	$0.246652\pi$
$878$	−11372.8	−0.437143
$879$	0	0
$880$	1468.21	0.0562423
$881$	−3637.60	−0.139107	−0.0695537	−	0.997578i	$-0.522158\pi$
$881$	−3637.60	−0.139107	−0.0695537	+	0.997578i	$0.522158\pi$
$882$	0	0
$883$	8700.02	0.331573	0.165787	−	0.986162i	$-0.446984\pi$
$883$	8700.02	0.331573	0.165787	+	0.986162i	$0.446984\pi$
$884$	7146.87	0.271918
$885$	0	0
$886$	14224.6	0.539375
$887$	30756.8	1.16428	0.582139	−	0.813090i	$-0.302216\pi$
$887$	30756.8	1.16428	0.582139	+	0.813090i	$0.302216\pi$
$888$	0	0
$889$	−30833.4	−1.16324
$890$	−5778.93	−0.217652
$891$	0	0
$892$	21004.3	0.788424
$893$	−50856.9	−1.90578
$894$	0	0
$895$	−17202.4	−0.642471
$896$	−34296.9	−1.27877
$897$	0	0
$898$	4522.64	0.168065
$899$	−42823.1	−1.58869
$900$	0	0
$901$	28247.4	1.04446
$902$	−2079.44	−0.0767602
$903$	0	0
$904$	−761.889	−0.0280310
$905$	4362.48	0.160236
$906$	0	0
$907$	−34784.7	−1.27344	−0.636718	−	0.771097i	$-0.719708\pi$
$907$	−34784.7	−1.27344	−0.636718	+	0.771097i	$0.719708\pi$
$908$	29548.4	1.07995
$909$	0	0
$910$	1844.83	0.0672037
$911$	28762.5	1.04604	0.523022	−	0.852319i	$-0.324805\pi$
$911$	28762.5	1.04604	0.523022	+	0.852319i	$0.324805\pi$
$912$	0	0
$913$	14726.7	0.533824
$914$	−17329.6	−0.627149
$915$	0	0
$916$	31382.0	1.13198
$917$	59349.6	2.13729
$918$	0	0
$919$	−32891.5	−1.18062	−0.590310	−	0.807177i	$-0.700994\pi$
$919$	−32891.5	−1.18062	−0.590310	+	0.807177i	$0.700994\pi$
$920$	7943.20	0.284652
$921$	0	0
$922$	15179.9	0.542217
$923$	6778.63	0.241735
$924$	0	0
$925$	8094.91	0.287739
$926$	−2147.91	−0.0762253
$927$	0	0
$928$	35459.2	1.25432
$929$	28294.3	0.999254	0.499627	−	0.866241i	$-0.333470\pi$
$929$	28294.3	0.999254	0.499627	+	0.866241i	$0.333470\pi$
$930$	0	0
$931$	−32069.4	−1.12893
$932$	16319.3	0.573557
$933$	0	0
$934$	17371.5	0.608580
$935$	−5209.84	−0.182224
$936$	0	0
$937$	18847.9	0.657135	0.328567	−	0.944481i	$-0.393434\pi$
$937$	18847.9	0.657135	0.328567	+	0.944481i	$0.393434\pi$
$938$	−8269.63	−0.287860
$939$	0	0
$940$	−11564.3	−0.401260
$941$	−20646.6	−0.715260	−0.357630	−	0.933863i	$-0.616415\pi$
$941$	−20646.6	−0.715260	−0.357630	+	0.933863i	$0.616415\pi$
$942$	0	0
$943$	12544.6	0.433201
$944$	−5780.47	−0.199299
$945$	0	0
$946$	695.710	0.0239107
$947$	41421.3	1.42134	0.710671	−	0.703525i	$-0.248392\pi$
$947$	41421.3	1.42134	0.710671	+	0.703525i	$0.248392\pi$
$948$	0	0
$949$	7721.59	0.264124
$950$	4497.24	0.153589
$951$	0	0
$952$	42012.0	1.43027
$953$	−37462.3	−1.27337	−0.636685	−	0.771124i	$-0.719695\pi$
$953$	−37462.3	−1.27337	−0.636685	+	0.771124i	$0.719695\pi$
$954$	0	0
$955$	−11856.0	−0.401729
$956$	−4645.52	−0.157162
$957$	0	0
$958$	−4619.41	−0.155789
$959$	−26222.6	−0.882975
$960$	0	0
$961$	18878.6	0.633701
$962$	−4989.06	−0.167208
$963$	0	0
$964$	−18486.9	−0.617658
$965$	5869.57	0.195801
$966$	0	0
$967$	−15660.8	−0.520804	−0.260402	−	0.965500i	$-0.583855\pi$
$967$	−15660.8	−0.520804	−0.260402	+	0.965500i	$0.583855\pi$
$968$	2241.09	0.0744124
$969$	0	0
$970$	−10707.8	−0.354440
$971$	−14937.4	−0.493681	−0.246841	−	0.969056i	$-0.579392\pi$
$971$	−14937.4	−0.493681	−0.246841	+	0.969056i	$0.579392\pi$
$972$	0	0
$973$	69635.8	2.29437
$974$	11706.7	0.385121
$975$	0	0
$976$	13477.2	0.442002
$977$	−25321.2	−0.829169	−0.414584	−	0.910011i	$-0.636073\pi$
$977$	−25321.2	−0.829169	−0.414584	+	0.910011i	$0.636073\pi$
$978$	0	0
$979$	9836.09	0.321106
$980$	−7292.21	−0.237695
$981$	0	0
$982$	27826.3	0.904248
$983$	23799.7	0.772219	0.386110	−	0.922453i	$-0.373819\pi$
$983$	23799.7	0.772219	0.386110	+	0.922453i	$0.373819\pi$
$984$	0	0
$985$	14176.7	0.458585
$986$	−23766.1	−0.767614
$987$	0	0
$988$	10500.6	0.338125
$989$	−4197.01	−0.134941
$990$	0	0
$991$	30988.7	0.993328	0.496664	−	0.867943i	$-0.334558\pi$
$991$	30988.7	0.993328	0.496664	+	0.867943i	$0.334558\pi$
$992$	−40300.3	−1.28986
$993$	0	0
$994$	17600.7	0.561631
$995$	14262.9	0.454438
$996$	0	0
$997$	808.632	0.0256867	0.0128433	−	0.999918i	$-0.495912\pi$
$997$	808.632	0.0256867	0.0128433	+	0.999918i	$0.495912\pi$
$998$	−24029.5	−0.762164
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.29255 q^{2} -6.32931 q^{4} +5.00000 q^{5} -23.9463 q^{7} +18.5214 q^{8} +O(q^{10})\)	\(q-1.29255 q^{2} -6.32931 q^{4} +5.00000 q^{5} -23.9463 q^{7} +18.5214 q^{8} -6.46276 q^{10} +11.0000 q^{11} +11.9206 q^{13} +30.9519 q^{14} +26.6946 q^{16} -94.7243 q^{17} -139.174 q^{19} -31.6465 q^{20} -14.2181 q^{22} +85.7733 q^{23} +25.0000 q^{25} -15.4080 q^{26} +151.564 q^{28} -194.111 q^{29} +220.612 q^{31} -182.675 q^{32} +122.436 q^{34} -119.732 q^{35} +323.796 q^{37} +179.890 q^{38} +92.6069 q^{40} +146.253 q^{41} -48.9314 q^{43} -69.6224 q^{44} -110.866 q^{46} +365.420 q^{47} +230.427 q^{49} -32.3138 q^{50} -75.4492 q^{52} -298.207 q^{53} +55.0000 q^{55} -443.519 q^{56} +250.898 q^{58} -216.540 q^{59} +504.865 q^{61} -285.152 q^{62} +22.5601 q^{64} +59.6030 q^{65} -267.177 q^{67} +599.539 q^{68} +154.759 q^{70} +568.648 q^{71} +647.751 q^{73} -418.524 q^{74} +880.875 q^{76} -263.410 q^{77} +895.113 q^{79} +133.473 q^{80} -189.040 q^{82} +1338.79 q^{83} -473.621 q^{85} +63.2464 q^{86} +203.735 q^{88} +894.190 q^{89} -285.455 q^{91} -542.886 q^{92} -472.324 q^{94} -695.870 q^{95} +1656.84 q^{97} -297.838 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 5 q^{2} + 33 q^{4} + 35 q^{5} + 30 q^{7} + 45 q^{8}+O(q^{10}) \) 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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