LMFDB - Embedded newform 1449.4.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,4,Mod(1,1449)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, 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("1449.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1449.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:	$85.4937675983$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 51x^{7} + 34x^{6} + 861x^{5} - 401x^{4} - 5403x^{3} + 1772x^{2} + 8716x - 192 $ x^9 - x^8 - 51x^7 + 34x^6 + 861x^5 - 401x^4 - 5403x^3 + 1772x^2 + 8716*x - 192
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.47249$ of defining polynomial
Character	$\chi$	$=$	1449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.47249 q^{2} +21.9481 q^{4} -8.93744 q^{5} +7.00000 q^{7} -76.3311 q^{8} +48.9100 q^{10} +64.9462 q^{11} +34.6747 q^{13} -38.3074 q^{14} +242.136 q^{16} +30.9018 q^{17} -138.470 q^{19} -196.160 q^{20} -355.417 q^{22} +23.0000 q^{23} -45.1222 q^{25} -189.757 q^{26} +153.637 q^{28} -16.0755 q^{29} -142.943 q^{31} -714.437 q^{32} -169.110 q^{34} -62.5621 q^{35} +368.798 q^{37} +757.774 q^{38} +682.204 q^{40} +79.8823 q^{41} -441.379 q^{43} +1425.45 q^{44} -125.867 q^{46} -197.161 q^{47} +49.0000 q^{49} +246.931 q^{50} +761.046 q^{52} -38.9761 q^{53} -580.453 q^{55} -534.317 q^{56} +87.9728 q^{58} -854.084 q^{59} +435.416 q^{61} +782.257 q^{62} +1972.66 q^{64} -309.903 q^{65} -796.063 q^{67} +678.238 q^{68} +342.370 q^{70} +236.179 q^{71} +646.398 q^{73} -2018.25 q^{74} -3039.15 q^{76} +454.623 q^{77} -859.016 q^{79} -2164.07 q^{80} -437.155 q^{82} -821.976 q^{83} -276.183 q^{85} +2415.44 q^{86} -4957.41 q^{88} -576.608 q^{89} +242.723 q^{91} +504.807 q^{92} +1078.96 q^{94} +1237.56 q^{95} +1651.08 q^{97} -268.152 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 8 q^{2} + 38 q^{4} - 39 q^{5} + 63 q^{7} - 135 q^{8} + 81 q^{10} - 38 q^{11} + 107 q^{13} - 56 q^{14} + 178 q^{16} - 170 q^{17} + 20 q^{19} - 300 q^{20} - 129 q^{22} + 207 q^{23} + 354 q^{25} + 114 q^{26}+ \cdots - 392 q^{98}+O(q^{100}) $ 9 * q - 8 * q^2 + 38 * q^4 - 39 * q^5 + 63 * q^7 - 135 * q^8 + 81 * q^10 - 38 * q^11 + 107 * q^13 - 56 * q^14 + 178 * q^16 - 170 * q^17 + 20 * q^19 - 300 * q^20 - 129 * q^22 + 207 * q^23 + 354 * q^25 + 114 * q^26 + 266 * q^28 - 405 * q^29 - 182 * q^31 - 648 * q^32 - 277 * q^34 - 273 * q^35 + 347 * q^37 + 291 * q^38 + 1066 * q^40 - 717 * q^41 - 77 * q^43 + 705 * q^44 - 184 * q^46 - 799 * q^47 + 441 * q^49 - 406 * q^50 + 519 * q^52 - 780 * q^53 - 754 * q^55 - 945 * q^56 - 508 * q^58 - 766 * q^59 + 1836 * q^61 - 173 * q^62 + 805 * q^64 - 655 * q^65 - 1290 * q^67 - 317 * q^68 + 567 * q^70 - 1802 * q^71 + 598 * q^73 - 578 * q^74 - 1323 * q^76 - 266 * q^77 - 1150 * q^79 - 795 * q^80 - 1040 * q^82 - 1758 * q^83 + 1078 * q^85 + 1920 * q^86 - 3056 * q^88 - 1400 * q^89 + 749 * q^91 + 874 * q^92 + 799 * q^94 - 2186 * q^95 + 1123 * q^97 - 392 * q^98

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 $

$2$	−5.47249	−1.93482	−0.967409	−	0.253220i	$-0.918510\pi$
$2$	−5.47249	−1.93482	−0.967409	+	0.253220i	$0.918510\pi$
$3$	0	0
$3$	0	0
$4$	21.9481	2.74352
$5$	−8.93744	−0.799389	−0.399694	−	0.916648i	$-0.630884\pi$
$5$	−8.93744	−0.799389	−0.399694	+	0.916648i	$0.630884\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−76.3311	−3.37339
$9$	0	0
$10$	48.9100	1.54667
$11$	64.9462	1.78018	0.890092	−	0.455782i	$-0.150640\pi$
$11$	64.9462	1.78018	0.890092	+	0.455782i	$0.150640\pi$
$12$	0	0
$13$	34.6747	0.739772	0.369886	−	0.929077i	$-0.379397\pi$
$13$	34.6747	0.739772	0.369886	+	0.929077i	$0.379397\pi$
$14$	−38.3074	−0.731292
$15$	0	0
$16$	242.136	3.78337
$17$	30.9018	0.440870	0.220435	−	0.975402i	$-0.429252\pi$
$17$	30.9018	0.440870	0.220435	+	0.975402i	$0.429252\pi$
$18$	0	0
$19$	−138.470	−1.67195	−0.835977	−	0.548765i	$-0.815098\pi$
$19$	−138.470	−1.67195	−0.835977	+	0.548765i	$0.815098\pi$
$20$	−196.160	−2.19314
$21$	0	0
$22$	−355.417	−3.44433
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	0	0
$25$	−45.1222	−0.360978
$26$	−189.757	−1.43132
$27$	0	0
$28$	153.637	1.03695
$29$	−16.0755	−0.102936	−0.0514679	−	0.998675i	$-0.516390\pi$
$29$	−16.0755	−0.102936	−0.0514679	+	0.998675i	$0.516390\pi$
$30$	0	0
$31$	−142.943	−0.828174	−0.414087	−	0.910237i	$-0.635899\pi$
$31$	−142.943	−0.828174	−0.414087	+	0.910237i	$0.635899\pi$
$32$	−714.437	−3.94675
$33$	0	0
$34$	−169.110	−0.853003
$35$	−62.5621	−0.302141
$36$	0	0
$37$	368.798	1.63865	0.819325	−	0.573329i	$-0.194348\pi$
$37$	368.798	1.63865	0.819325	+	0.573329i	$0.194348\pi$
$38$	757.774	3.23493
$39$	0	0
$40$	682.204	2.69665
$41$	79.8823	0.304281	0.152140	−	0.988359i	$-0.451383\pi$
$41$	79.8823	0.304281	0.152140	+	0.988359i	$0.451383\pi$
$42$	0	0
$43$	−441.379	−1.56534	−0.782670	−	0.622437i	$-0.786143\pi$
$43$	−441.379	−1.56534	−0.782670	+	0.622437i	$0.786143\pi$
$44$	1425.45	4.88396
$45$	0	0
$46$	−125.867	−0.403437
$47$	−197.161	−0.611893	−0.305946	−	0.952049i	$-0.598973\pi$
$47$	−197.161	−0.611893	−0.305946	+	0.952049i	$0.598973\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	246.931	0.698426
$51$	0	0
$52$	761.046	2.02958
$53$	−38.9761	−0.101015	−0.0505073	−	0.998724i	$-0.516084\pi$
$53$	−38.9761	−0.101015	−0.0505073	+	0.998724i	$0.516084\pi$
$54$	0	0
$55$	−580.453	−1.42306
$56$	−534.317	−1.27502
$57$	0	0
$58$	87.9728	0.199162
$59$	−854.084	−1.88461	−0.942307	−	0.334750i	$-0.891348\pi$
$59$	−854.084	−1.88461	−0.942307	+	0.334750i	$0.891348\pi$
$60$	0	0
$61$	435.416	0.913923	0.456962	−	0.889486i	$-0.348938\pi$
$61$	435.416	0.913923	0.456962	+	0.889486i	$0.348938\pi$
$62$	782.257	1.60237
$63$	0	0
$64$	1972.66	3.85286
$65$	−309.903	−0.591366
$66$	0	0
$67$	−796.063	−1.45156	−0.725781	−	0.687926i	$-0.758521\pi$
$67$	−796.063	−1.45156	−0.725781	+	0.687926i	$0.758521\pi$
$68$	678.238	1.20954
$69$	0	0
$70$	342.370	0.584587
$71$	236.179	0.394779	0.197389	−	0.980325i	$-0.436754\pi$
$71$	236.179	0.394779	0.197389	+	0.980325i	$0.436754\pi$
$72$	0	0
$73$	646.398	1.03637	0.518186	−	0.855268i	$-0.326608\pi$
$73$	646.398	1.03637	0.518186	+	0.855268i	$0.326608\pi$
$74$	−2018.25	−3.17049
$75$	0	0
$76$	−3039.15	−4.58704
$77$	454.623	0.672846
$78$	0	0
$79$	−859.016	−1.22338	−0.611689	−	0.791098i	$-0.709510\pi$
$79$	−859.016	−1.22338	−0.611689	+	0.791098i	$0.709510\pi$
$80$	−2164.07	−3.02439
$81$	0	0
$82$	−437.155	−0.588728
$83$	−821.976	−1.08703	−0.543516	−	0.839399i	$-0.682907\pi$
$83$	−821.976	−1.08703	−0.543516	+	0.839399i	$0.682907\pi$
$84$	0	0
$85$	−276.183	−0.352427
$86$	2415.44	3.02865
$87$	0	0
$88$	−4957.41	−6.00525
$89$	−576.608	−0.686745	−0.343373	−	0.939199i	$-0.611569\pi$
$89$	−576.608	−0.686745	−0.343373	+	0.939199i	$0.611569\pi$
$90$	0	0
$91$	242.723	0.279608
$92$	504.807	0.572063
$93$	0	0
$94$	1078.96	1.18390
$95$	1237.56	1.33654
$96$	0	0
$97$	1651.08	1.72826	0.864131	−	0.503268i	$-0.167869\pi$
$97$	1651.08	1.72826	0.864131	+	0.503268i	$0.167869\pi$
$98$	−268.152	−0.276402
$99$	0	0
$100$	−990.349	−0.990349
$101$	1177.39	1.15995	0.579974	−	0.814635i	$-0.303063\pi$
$101$	1177.39	1.15995	0.579974	+	0.814635i	$0.303063\pi$
$102$	0	0
$103$	1358.79	1.29986	0.649930	−	0.759994i	$-0.274798\pi$
$103$	1358.79	1.29986	0.649930	+	0.759994i	$0.274798\pi$
$104$	−2646.76	−2.49554
$105$	0	0
$106$	213.296	0.195445
$107$	550.175	0.497079	0.248539	−	0.968622i	$-0.420049\pi$
$107$	550.175	0.497079	0.248539	+	0.968622i	$0.420049\pi$
$108$	0	0
$109$	−996.931	−0.876042	−0.438021	−	0.898965i	$-0.644321\pi$
$109$	−996.931	−0.876042	−0.438021	+	0.898965i	$0.644321\pi$
$110$	3176.52	2.75336
$111$	0	0
$112$	1694.95	1.42998
$113$	−1890.29	−1.57366	−0.786828	−	0.617172i	$-0.788278\pi$
$113$	−1890.29	−1.57366	−0.786828	+	0.617172i	$0.788278\pi$
$114$	0	0
$115$	−205.561	−0.166684
$116$	−352.827	−0.282406
$117$	0	0
$118$	4673.96	3.64638
$119$	216.313	0.166633
$120$	0	0
$121$	2887.01	2.16905
$122$	−2382.81	−1.76827
$123$	0	0
$124$	−3137.34	−2.27211
$125$	1520.46	1.08795
$126$	0	0
$127$	−2062.71	−1.44123	−0.720613	−	0.693337i	$-0.756140\pi$
$127$	−2062.71	−1.44123	−0.720613	+	0.693337i	$0.756140\pi$
$128$	−5079.89	−3.50783
$129$	0	0
$130$	1695.94	1.14418
$131$	1112.85	0.742218	0.371109	−	0.928589i	$-0.378978\pi$
$131$	1112.85	0.742218	0.371109	+	0.928589i	$0.378978\pi$
$132$	0	0
$133$	−969.288	−0.631939
$134$	4356.45	2.80851
$135$	0	0
$136$	−2358.77	−1.48723
$137$	456.121	0.284446	0.142223	−	0.989835i	$-0.454575\pi$
$137$	456.121	0.284446	0.142223	+	0.989835i	$0.454575\pi$
$138$	0	0
$139$	618.754	0.377569	0.188784	−	0.982019i	$-0.439545\pi$
$139$	618.754	0.377569	0.188784	+	0.982019i	$0.439545\pi$
$140$	−1373.12	−0.828928
$141$	0	0
$142$	−1292.49	−0.763824
$143$	2251.99	1.31693
$144$	0	0
$145$	143.674	0.0822858
$146$	−3537.40	−2.00519
$147$	0	0
$148$	8094.44	4.49567
$149$	−191.696	−0.105398	−0.0526992	−	0.998610i	$-0.516782\pi$
$149$	−191.696	−0.105398	−0.0526992	+	0.998610i	$0.516782\pi$
$150$	0	0
$151$	−1503.50	−0.810288	−0.405144	−	0.914253i	$-0.632779\pi$
$151$	−1503.50	−0.810288	−0.405144	+	0.914253i	$0.632779\pi$
$152$	10569.5	5.64015
$153$	0	0
$154$	−2487.92	−1.30183
$155$	1277.55	0.662033
$156$	0	0
$157$	−1360.53	−0.691605	−0.345803	−	0.938307i	$-0.612393\pi$
$157$	−1360.53	−0.691605	−0.345803	+	0.938307i	$0.612393\pi$
$158$	4700.96	2.36701
$159$	0	0
$160$	6385.24	3.15498
$161$	161.000	0.0788110
$162$	0	0
$163$	2283.61	1.09734	0.548670	−	0.836039i	$-0.315134\pi$
$163$	2283.61	1.09734	0.548670	+	0.836039i	$0.315134\pi$
$164$	1753.27	0.834800
$165$	0	0
$166$	4498.26	2.10321
$167$	970.287	0.449599	0.224800	−	0.974405i	$-0.427827\pi$
$167$	970.287	0.449599	0.224800	+	0.974405i	$0.427827\pi$
$168$	0	0
$169$	−994.664	−0.452737
$170$	1511.41	0.681881
$171$	0	0
$172$	−9687.45	−4.29454
$173$	2175.48	0.956062	0.478031	−	0.878343i	$-0.341351\pi$
$173$	2175.48	0.956062	0.478031	+	0.878343i	$0.341351\pi$
$174$	0	0
$175$	−315.855	−0.136437
$176$	15725.8	6.73510
$177$	0	0
$178$	3155.48	1.32873
$179$	−3254.30	−1.35887	−0.679435	−	0.733736i	$-0.737775\pi$
$179$	−3254.30	−1.35887	−0.679435	+	0.733736i	$0.737775\pi$
$180$	0	0
$181$	1147.73	0.471325	0.235663	−	0.971835i	$-0.424274\pi$
$181$	1147.73	0.471325	0.235663	+	0.971835i	$0.424274\pi$
$182$	−1328.30	−0.540990
$183$	0	0
$184$	−1755.61	−0.703400
$185$	−3296.11	−1.30992
$186$	0	0
$187$	2006.96	0.784830
$188$	−4327.33	−1.67874
$189$	0	0
$190$	−6772.56	−2.58596
$191$	865.071	0.327719	0.163859	−	0.986484i	$-0.447606\pi$
$191$	865.071	0.327719	0.163859	+	0.986484i	$0.447606\pi$
$192$	0	0
$193$	−2005.48	−0.747969	−0.373984	−	0.927435i	$-0.622009\pi$
$193$	−2005.48	−0.747969	−0.373984	+	0.927435i	$0.622009\pi$
$194$	−9035.49	−3.34387
$195$	0	0
$196$	1075.46	0.391931
$197$	218.707	0.0790975	0.0395488	−	0.999218i	$-0.487408\pi$
$197$	218.707	0.0790975	0.0395488	+	0.999218i	$0.487408\pi$
$198$	0	0
$199$	−273.538	−0.0974401	−0.0487201	−	0.998812i	$-0.515514\pi$
$199$	−273.538	−0.0974401	−0.0487201	+	0.998812i	$0.515514\pi$
$200$	3444.23	1.21772
$201$	0	0
$202$	−6443.26	−2.24429
$203$	−112.528	−0.0389061
$204$	0	0
$205$	−713.943	−0.243239
$206$	−7435.96	−2.51499
$207$	0	0
$208$	8395.99	2.79883
$209$	−8993.08	−2.97638
$210$	0	0
$211$	899.743	0.293559	0.146779	−	0.989169i	$-0.453109\pi$
$211$	899.743	0.293559	0.146779	+	0.989169i	$0.453109\pi$
$212$	−855.453	−0.277136
$213$	0	0
$214$	−3010.83	−0.961756
$215$	3944.80	1.25132
$216$	0	0
$217$	−1000.60	−0.313020
$218$	5455.69	1.69498
$219$	0	0
$220$	−12739.9	−3.90419
$221$	1071.51	0.326144
$222$	0	0
$223$	−827.023	−0.248348	−0.124174	−	0.992260i	$-0.539628\pi$
$223$	−827.023	−0.248348	−0.124174	+	0.992260i	$0.539628\pi$
$224$	−5001.06	−1.49173
$225$	0	0
$226$	10344.6	3.04474
$227$	3323.67	0.971806	0.485903	−	0.874013i	$-0.338491\pi$
$227$	3323.67	0.971806	0.485903	+	0.874013i	$0.338491\pi$
$228$	0	0
$229$	4620.63	1.33336	0.666681	−	0.745343i	$-0.267714\pi$
$229$	4620.63	1.33336	0.666681	+	0.745343i	$0.267714\pi$
$230$	1124.93	0.322503
$231$	0	0
$232$	1227.06	0.347243
$233$	2523.32	0.709478	0.354739	−	0.934965i	$-0.384570\pi$
$233$	2523.32	0.709478	0.354739	+	0.934965i	$0.384570\pi$
$234$	0	0
$235$	1762.12	0.489140
$236$	−18745.5	−5.17047
$237$	0	0
$238$	−1183.77	−0.322405
$239$	−4250.48	−1.15038	−0.575190	−	0.818020i	$-0.695072\pi$
$239$	−4250.48	−1.15038	−0.575190	+	0.818020i	$0.695072\pi$
$240$	0	0
$241$	−7054.93	−1.88568	−0.942838	−	0.333250i	$-0.891855\pi$
$241$	−7054.93	−1.88568	−0.942838	+	0.333250i	$0.891855\pi$
$242$	−15799.1	−4.19672
$243$	0	0
$244$	9556.58	2.50736
$245$	−437.934	−0.114198
$246$	0	0
$247$	−4801.40	−1.23686
$248$	10911.0	2.79375
$249$	0	0
$250$	−8320.68	−2.10498
$251$	−5829.72	−1.46601	−0.733005	−	0.680223i	$-0.761883\pi$
$251$	−5829.72	−1.46601	−0.733005	+	0.680223i	$0.761883\pi$
$252$	0	0
$253$	1493.76	0.371194
$254$	11288.2	2.78851
$255$	0	0
$256$	12018.3	2.93416
$257$	1639.22	0.397866	0.198933	−	0.980013i	$-0.436252\pi$
$257$	1639.22	0.397866	0.198933	+	0.980013i	$0.436252\pi$
$258$	0	0
$259$	2581.59	0.619352
$260$	−6801.80	−1.62242
$261$	0	0
$262$	−6090.09	−1.43606
$263$	273.365	0.0640927	0.0320464	−	0.999486i	$-0.489798\pi$
$263$	273.365	0.0640927	0.0320464	+	0.999486i	$0.489798\pi$
$264$	0	0
$265$	348.346	0.0807500
$266$	5304.42	1.22269
$267$	0	0
$268$	−17472.1	−3.98238
$269$	−7443.39	−1.68711	−0.843553	−	0.537046i	$-0.819540\pi$
$269$	−7443.39	−1.68711	−0.843553	+	0.537046i	$0.819540\pi$
$270$	0	0
$271$	3102.28	0.695388	0.347694	−	0.937608i	$-0.386965\pi$
$271$	3102.28	0.695388	0.347694	+	0.937608i	$0.386965\pi$
$272$	7482.44	1.66798
$273$	0	0
$274$	−2496.12	−0.550350
$275$	−2930.52	−0.642606
$276$	0	0
$277$	2151.15	0.466608	0.233304	−	0.972404i	$-0.425046\pi$
$277$	2151.15	0.466608	0.233304	+	0.972404i	$0.425046\pi$
$278$	−3386.13	−0.730527
$279$	0	0
$280$	4775.43	1.01924
$281$	−664.784	−0.141131	−0.0705653	−	0.997507i	$-0.522480\pi$
$281$	−664.784	−0.141131	−0.0705653	+	0.997507i	$0.522480\pi$
$282$	0	0
$283$	2923.45	0.614067	0.307033	−	0.951699i	$-0.400664\pi$
$283$	2923.45	0.614067	0.307033	+	0.951699i	$0.400664\pi$
$284$	5183.69	1.08308
$285$	0	0
$286$	−12324.0	−2.54802
$287$	559.176	0.115007
$288$	0	0
$289$	−3958.08	−0.805633
$290$	−786.252	−0.159208
$291$	0	0
$292$	14187.2	2.84330
$293$	1765.02	0.351924	0.175962	−	0.984397i	$-0.443696\pi$
$293$	1765.02	0.351924	0.175962	+	0.984397i	$0.443696\pi$
$294$	0	0
$295$	7633.32	1.50654
$296$	−28150.8	−5.52781
$297$	0	0
$298$	1049.06	0.203927
$299$	797.519	0.154253
$300$	0	0
$301$	−3089.65	−0.591643
$302$	8227.91	1.56776
$303$	0	0
$304$	−33528.5	−6.32562
$305$	−3891.50	−0.730580
$306$	0	0
$307$	−4091.63	−0.760657	−0.380328	−	0.924852i	$-0.624189\pi$
$307$	−4091.63	−0.760657	−0.380328	+	0.924852i	$0.624189\pi$
$308$	9978.14	1.84596
$309$	0	0
$310$	−6991.37	−1.28091
$311$	−2508.33	−0.457346	−0.228673	−	0.973503i	$-0.573439\pi$
$311$	−2508.33	−0.457346	−0.228673	+	0.973503i	$0.573439\pi$
$312$	0	0
$313$	−2276.71	−0.411142	−0.205571	−	0.978642i	$-0.565905\pi$
$313$	−2276.71	−0.411142	−0.205571	+	0.978642i	$0.565905\pi$
$314$	7445.48	1.33813
$315$	0	0
$316$	−18853.8	−3.35636
$317$	−2282.31	−0.404377	−0.202188	−	0.979347i	$-0.564805\pi$
$317$	−2282.31	−0.404377	−0.202188	+	0.979347i	$0.564805\pi$
$318$	0	0
$319$	−1044.04	−0.183245
$320$	−17630.6	−3.07993
$321$	0	0
$322$	−881.071	−0.152485
$323$	−4278.97	−0.737115
$324$	0	0
$325$	−1564.60	−0.267041
$326$	−12497.0	−2.12315
$327$	0	0
$328$	−6097.50	−1.02646
$329$	−1380.13	−0.231274
$330$	0	0
$331$	−6834.36	−1.13490	−0.567448	−	0.823409i	$-0.692069\pi$
$331$	−6834.36	−1.13490	−0.567448	+	0.823409i	$0.692069\pi$
$332$	−18040.9	−2.98229
$333$	0	0
$334$	−5309.89	−0.869892
$335$	7114.76	1.16036
$336$	0	0
$337$	−2946.37	−0.476258	−0.238129	−	0.971234i	$-0.576534\pi$
$337$	−2946.37	−0.476258	−0.238129	+	0.971234i	$0.576534\pi$
$338$	5443.29	0.875964
$339$	0	0
$340$	−6061.71	−0.966889
$341$	−9283.63	−1.47430
$342$	0	0
$343$	343.000	0.0539949
$344$	33690.9	5.28050
$345$	0	0
$346$	−11905.3	−1.84981
$347$	−10980.7	−1.69878	−0.849391	−	0.527764i	$-0.823030\pi$
$347$	−10980.7	−1.69878	−0.849391	+	0.527764i	$0.823030\pi$
$348$	0	0
$349$	−1299.92	−0.199379	−0.0996894	−	0.995019i	$-0.531785\pi$
$349$	−1299.92	−0.199379	−0.0996894	+	0.995019i	$0.531785\pi$
$350$	1728.52	0.263980
$351$	0	0
$352$	−46400.0	−7.02593
$353$	5177.82	0.780702	0.390351	−	0.920666i	$-0.372354\pi$
$353$	5177.82	0.780702	0.390351	+	0.920666i	$0.372354\pi$
$354$	0	0
$355$	−2110.83	−0.315582
$356$	−12655.5	−1.88410
$357$	0	0
$358$	17809.1	2.62916
$359$	−11913.7	−1.75148	−0.875740	−	0.482784i	$-0.839626\pi$
$359$	−11913.7	−1.75148	−0.875740	+	0.482784i	$0.839626\pi$
$360$	0	0
$361$	12314.9	1.79543
$362$	−6280.93	−0.911929
$363$	0	0
$364$	5327.32	0.767108
$365$	−5777.14	−0.828464
$366$	0	0
$367$	2003.44	0.284956	0.142478	−	0.989798i	$-0.454493\pi$
$367$	2003.44	0.284956	0.142478	+	0.989798i	$0.454493\pi$
$368$	5569.12	0.788888
$369$	0	0
$370$	18037.9	2.53445
$371$	−272.833	−0.0381800
$372$	0	0
$373$	5882.77	0.816618	0.408309	−	0.912844i	$-0.366119\pi$
$373$	5882.77	0.816618	0.408309	+	0.912844i	$0.366119\pi$
$374$	−10983.0	−1.51850
$375$	0	0
$376$	15049.5	2.06415
$377$	−557.412	−0.0761491
$378$	0	0
$379$	−11601.1	−1.57232	−0.786162	−	0.618020i	$-0.787935\pi$
$379$	−11601.1	−1.57232	−0.786162	+	0.618020i	$0.787935\pi$
$380$	27162.2	3.66682
$381$	0	0
$382$	−4734.09	−0.634076
$383$	1653.63	0.220618	0.110309	−	0.993897i	$-0.464816\pi$
$383$	1653.63	0.220618	0.110309	+	0.993897i	$0.464816\pi$
$384$	0	0
$385$	−4063.17	−0.537865
$386$	10975.0	1.44718
$387$	0	0
$388$	36238.0	4.74152
$389$	11585.3	1.51002	0.755008	−	0.655716i	$-0.227633\pi$
$389$	11585.3	1.51002	0.755008	+	0.655716i	$0.227633\pi$
$390$	0	0
$391$	710.742	0.0919278
$392$	−3740.22	−0.481913
$393$	0	0
$394$	−1196.87	−0.153039
$395$	7677.41	0.977955
$396$	0	0
$397$	−2639.48	−0.333682	−0.166841	−	0.985984i	$-0.553357\pi$
$397$	−2639.48	−0.333682	−0.166841	+	0.985984i	$0.553357\pi$
$398$	1496.93	0.188529
$399$	0	0
$400$	−10925.7	−1.36571
$401$	−2043.37	−0.254466	−0.127233	−	0.991873i	$-0.540610\pi$
$401$	−2043.37	−0.254466	−0.127233	+	0.991873i	$0.540610\pi$
$402$	0	0
$403$	−4956.52	−0.612660
$404$	25841.5	3.18234
$405$	0	0
$406$	615.810	0.0752762
$407$	23952.0	2.91710
$408$	0	0
$409$	2054.04	0.248327	0.124163	−	0.992262i	$-0.460375\pi$
$409$	2054.04	0.248327	0.124163	+	0.992262i	$0.460375\pi$
$410$	3907.04	0.470622
$411$	0	0
$412$	29822.9	3.56619
$413$	−5978.59	−0.712317
$414$	0	0
$415$	7346.36	0.868961
$416$	−24772.9	−2.91969
$417$	0	0
$418$	49214.5	5.75876
$419$	3741.09	0.436191	0.218095	−	0.975927i	$-0.430016\pi$
$419$	3741.09	0.436191	0.218095	+	0.975927i	$0.430016\pi$
$420$	0	0
$421$	−7830.25	−0.906468	−0.453234	−	0.891392i	$-0.649730\pi$
$421$	−7830.25	−0.906468	−0.453234	+	0.891392i	$0.649730\pi$
$422$	−4923.83	−0.567982
$423$	0	0
$424$	2975.09	0.340762
$425$	−1394.36	−0.159144
$426$	0	0
$427$	3047.91	0.345431
$428$	12075.3	1.36374
$429$	0	0
$430$	−21587.9	−2.42107
$431$	−14614.5	−1.63331	−0.816654	−	0.577128i	$-0.804173\pi$
$431$	−14614.5	−1.63331	−0.816654	+	0.577128i	$0.804173\pi$
$432$	0	0
$433$	−1920.46	−0.213144	−0.106572	−	0.994305i	$-0.533987\pi$
$433$	−1920.46	−0.213144	−0.106572	+	0.994305i	$0.533987\pi$
$434$	5475.80	0.605637
$435$	0	0
$436$	−21880.8	−2.40344
$437$	−3184.80	−0.348626
$438$	0	0
$439$	4703.07	0.511310	0.255655	−	0.966768i	$-0.417709\pi$
$439$	4703.07	0.511310	0.255655	+	0.966768i	$0.417709\pi$
$440$	44306.6	4.80053
$441$	0	0
$442$	−5863.84	−0.631028
$443$	−5855.09	−0.627954	−0.313977	−	0.949431i	$-0.601662\pi$
$443$	−5855.09	−0.627954	−0.313977	+	0.949431i	$0.601662\pi$
$444$	0	0
$445$	5153.40	0.548976
$446$	4525.87	0.480507
$447$	0	0
$448$	13808.6	1.45624
$449$	−14581.6	−1.53262	−0.766312	−	0.642469i	$-0.777910\pi$
$449$	−14581.6	−1.53262	−0.766312	+	0.642469i	$0.777910\pi$
$450$	0	0
$451$	5188.05	0.541676
$452$	−41488.3	−4.31735
$453$	0	0
$454$	−18188.8	−1.88027
$455$	−2169.32	−0.223515
$456$	0	0
$457$	−1552.87	−0.158950	−0.0794752	−	0.996837i	$-0.525324\pi$
$457$	−1552.87	−0.158950	−0.0794752	+	0.996837i	$0.525324\pi$
$458$	−25286.4	−2.57981
$459$	0	0
$460$	−4511.68	−0.457301
$461$	−11609.2	−1.17287	−0.586436	−	0.809996i	$-0.699469\pi$
$461$	−11609.2	−1.17287	−0.586436	+	0.809996i	$0.699469\pi$
$462$	0	0
$463$	−15818.5	−1.58779	−0.793897	−	0.608053i	$-0.791951\pi$
$463$	−15818.5	−1.58779	−0.793897	+	0.608053i	$0.791951\pi$
$464$	−3892.45	−0.389445
$465$	0	0
$466$	−13808.9	−1.37271
$467$	−11797.8	−1.16903	−0.584516	−	0.811382i	$-0.698715\pi$
$467$	−11797.8	−1.16903	−0.584516	+	0.811382i	$0.698715\pi$
$468$	0	0
$469$	−5572.44	−0.548639
$470$	−9643.17	−0.946397
$471$	0	0
$472$	65193.1	6.35753
$473$	−28665.9	−2.78659
$474$	0	0
$475$	6248.06	0.603538
$476$	4747.66	0.457161
$477$	0	0
$478$	23260.7	2.22578
$479$	9158.14	0.873582	0.436791	−	0.899563i	$-0.356115\pi$
$479$	9158.14	0.873582	0.436791	+	0.899563i	$0.356115\pi$
$480$	0	0
$481$	12788.0	1.21223
$482$	38608.0	3.64844
$483$	0	0
$484$	63364.5	5.95083
$485$	−14756.4	−1.38155
$486$	0	0
$487$	16332.9	1.51974	0.759871	−	0.650074i	$-0.225262\pi$
$487$	16332.9	1.51974	0.759871	+	0.650074i	$0.225262\pi$
$488$	−33235.8	−3.08302
$489$	0	0
$490$	2396.59	0.220953
$491$	−17333.9	−1.59321	−0.796607	−	0.604498i	$-0.793374\pi$
$491$	−17333.9	−1.59321	−0.796607	+	0.604498i	$0.793374\pi$
$492$	0	0
$493$	−496.761	−0.0453814
$494$	26275.6	2.39311
$495$	0	0
$496$	−34611.7	−3.13329
$497$	1653.25	0.149212
$498$	0	0
$499$	9399.30	0.843227	0.421614	−	0.906776i	$-0.361464\pi$
$499$	9399.30	0.843227	0.421614	+	0.906776i	$0.361464\pi$
$500$	33371.2	2.98481
$501$	0	0
$502$	31903.1	2.83646
$503$	−16313.9	−1.44612	−0.723062	−	0.690783i	$-0.757266\pi$
$503$	−16313.9	−1.44612	−0.723062	+	0.690783i	$0.757266\pi$
$504$	0	0
$505$	−10522.9	−0.927249
$506$	−8174.60	−0.718192
$507$	0	0
$508$	−45272.6	−3.95403
$509$	−14553.0	−1.26729	−0.633643	−	0.773626i	$-0.718441\pi$
$509$	−14553.0	−1.26729	−0.633643	+	0.773626i	$0.718441\pi$
$510$	0	0
$511$	4524.78	0.391712
$512$	−25131.0	−2.16922
$513$	0	0
$514$	−8970.59	−0.769797
$515$	−12144.1	−1.03909
$516$	0	0
$517$	−12804.9	−1.08928
$518$	−14127.7	−1.19833
$519$	0	0
$520$	23655.2	1.99491
$521$	−3658.55	−0.307647	−0.153823	−	0.988098i	$-0.549159\pi$
$521$	−3658.55	−0.307647	−0.153823	+	0.988098i	$0.549159\pi$
$522$	0	0
$523$	9660.17	0.807667	0.403834	−	0.914832i	$-0.367677\pi$
$523$	9660.17	0.807667	0.403834	+	0.914832i	$0.367677\pi$
$524$	24425.1	2.03629
$525$	0	0
$526$	−1495.99	−0.124008
$527$	−4417.21	−0.365117
$528$	0	0
$529$	529.000	0.0434783
$530$	−1906.32	−0.156236
$531$	0	0
$532$	−21274.1	−1.73374
$533$	2769.90	0.225098
$534$	0	0
$535$	−4917.15	−0.397359
$536$	60764.4	4.89668
$537$	0	0
$538$	40733.9	3.26424
$539$	3182.36	0.254312
$540$	0	0
$541$	−11525.7	−0.915950	−0.457975	−	0.888965i	$-0.651425\pi$
$541$	−11525.7	−0.915950	−0.457975	+	0.888965i	$0.651425\pi$
$542$	−16977.2	−1.34545
$543$	0	0
$544$	−22077.4	−1.74000
$545$	8910.00	0.700298
$546$	0	0
$547$	−18887.0	−1.47633	−0.738163	−	0.674623i	$-0.764306\pi$
$547$	−18887.0	−1.47633	−0.738163	+	0.674623i	$0.764306\pi$
$548$	10011.0	0.780381
$549$	0	0
$550$	16037.2	1.24333
$551$	2225.97	0.172104
$552$	0	0
$553$	−6013.11	−0.462394
$554$	−11772.2	−0.902801
$555$	0	0
$556$	13580.5	1.03587
$557$	13486.0	1.02589	0.512943	−	0.858423i	$-0.328555\pi$
$557$	13486.0	1.02589	0.512943	+	0.858423i	$0.328555\pi$
$558$	0	0
$559$	−15304.7	−1.15800
$560$	−15148.5	−1.14311
$561$	0	0
$562$	3638.02	0.273062
$563$	16192.7	1.21215	0.606075	−	0.795408i	$-0.292743\pi$
$563$	16192.7	1.21215	0.606075	+	0.795408i	$0.292743\pi$
$564$	0	0
$565$	16894.3	1.25796
$566$	−15998.5	−1.18811
$567$	0	0
$568$	−18027.8	−1.33174
$569$	−9293.58	−0.684722	−0.342361	−	0.939568i	$-0.611227\pi$
$569$	−9293.58	−0.684722	−0.342361	+	0.939568i	$0.611227\pi$
$570$	0	0
$571$	15286.5	1.12035	0.560175	−	0.828374i	$-0.310734\pi$
$571$	15286.5	1.12035	0.560175	+	0.828374i	$0.310734\pi$
$572$	49427.0	3.61302
$573$	0	0
$574$	−3060.08	−0.222518
$575$	−1037.81	−0.0752690
$576$	0	0
$577$	−10993.2	−0.793161	−0.396581	−	0.918000i	$-0.629803\pi$
$577$	−10993.2	−0.793161	−0.396581	+	0.918000i	$0.629803\pi$
$578$	21660.5	1.55875
$579$	0	0
$580$	3153.37	0.225752
$581$	−5753.83	−0.410859
$582$	0	0
$583$	−2531.35	−0.179825
$584$	−49340.2	−3.49608
$585$	0	0
$586$	−9659.06	−0.680908
$587$	5939.95	0.417663	0.208831	−	0.977952i	$-0.433034\pi$
$587$	5939.95	0.417663	0.208831	+	0.977952i	$0.433034\pi$
$588$	0	0
$589$	19793.3	1.38467
$590$	−41773.3	−2.91488
$591$	0	0
$592$	89299.3	6.19963
$593$	11704.3	0.810518	0.405259	−	0.914202i	$-0.367181\pi$
$593$	11704.3	0.810518	0.405259	+	0.914202i	$0.367181\pi$
$594$	0	0
$595$	−1933.28	−0.133205
$596$	−4207.37	−0.289162
$597$	0	0
$598$	−4364.41	−0.298452
$599$	3212.92	0.219159	0.109580	−	0.993978i	$-0.465050\pi$
$599$	3212.92	0.219159	0.109580	+	0.993978i	$0.465050\pi$
$600$	0	0
$601$	24141.0	1.63849	0.819246	−	0.573442i	$-0.194392\pi$
$601$	24141.0	1.63849	0.819246	+	0.573442i	$0.194392\pi$
$602$	16908.1	1.14472
$603$	0	0
$604$	−32999.1	−2.22304
$605$	−25802.5	−1.73392
$606$	0	0
$607$	−8240.03	−0.550993	−0.275496	−	0.961302i	$-0.588842\pi$
$607$	−8240.03	−0.550993	−0.275496	+	0.961302i	$0.588842\pi$
$608$	98927.9	6.59878
$609$	0	0
$610$	21296.2	1.41354
$611$	−6836.52	−0.452661
$612$	0	0
$613$	−24760.8	−1.63145	−0.815726	−	0.578438i	$-0.803662\pi$
$613$	−24760.8	−1.63145	−0.815726	+	0.578438i	$0.803662\pi$
$614$	22391.4	1.47173
$615$	0	0
$616$	−34701.9	−2.26977
$617$	4624.30	0.301730	0.150865	−	0.988554i	$-0.451794\pi$
$617$	4624.30	0.301730	0.150865	+	0.988554i	$0.451794\pi$
$618$	0	0
$619$	−1330.77	−0.0864103	−0.0432051	−	0.999066i	$-0.513757\pi$
$619$	−1330.77	−0.0864103	−0.0432051	+	0.999066i	$0.513757\pi$
$620$	28039.8	1.81630
$621$	0	0
$622$	13726.8	0.884880
$623$	−4036.25	−0.259565
$624$	0	0
$625$	−7948.71	−0.508717
$626$	12459.3	0.795484
$627$	0	0
$628$	−29861.1	−1.89743
$629$	11396.5	0.722432
$630$	0	0
$631$	−29390.6	−1.85423	−0.927117	−	0.374773i	$-0.877721\pi$
$631$	−29390.6	−1.85423	−0.927117	+	0.374773i	$0.877721\pi$
$632$	65569.6	4.12693
$633$	0	0
$634$	12489.9	0.782396
$635$	18435.3	1.15210
$636$	0	0
$637$	1699.06	0.105682
$638$	5713.50	0.354545
$639$	0	0
$640$	45401.2	2.80412
$641$	8899.76	0.548392	0.274196	−	0.961674i	$-0.411588\pi$
$641$	8899.76	0.548392	0.274196	+	0.961674i	$0.411588\pi$
$642$	0	0
$643$	24093.0	1.47766	0.738830	−	0.673892i	$-0.235379\pi$
$643$	24093.0	1.47766	0.738830	+	0.673892i	$0.235379\pi$
$644$	3533.65	0.216219
$645$	0	0
$646$	23416.6	1.42618
$647$	15696.8	0.953796	0.476898	−	0.878959i	$-0.341761\pi$
$647$	15696.8	0.953796	0.476898	+	0.878959i	$0.341761\pi$
$648$	0	0
$649$	−55469.5	−3.35496
$650$	8562.26	0.516676
$651$	0	0
$652$	50121.1	3.01057
$653$	−8664.84	−0.519267	−0.259634	−	0.965707i	$-0.583602\pi$
$653$	−8664.84	−0.519267	−0.259634	+	0.965707i	$0.583602\pi$
$654$	0	0
$655$	−9946.07	−0.593321
$656$	19342.4	1.15121
$657$	0	0
$658$	7552.75	0.447472
$659$	18667.0	1.10343	0.551716	−	0.834032i	$-0.313973\pi$
$659$	18667.0	1.10343	0.551716	+	0.834032i	$0.313973\pi$
$660$	0	0
$661$	14121.6	0.830964	0.415482	−	0.909601i	$-0.363613\pi$
$661$	14121.6	0.830964	0.415482	+	0.909601i	$0.363613\pi$
$662$	37401.0	2.19582
$663$	0	0
$664$	62742.3	3.66698
$665$	8662.95	0.505165
$666$	0	0
$667$	−369.736	−0.0214636
$668$	21296.0	1.23348
$669$	0	0
$670$	−38935.5	−2.24509
$671$	28278.6	1.62695
$672$	0	0
$673$	31238.5	1.78923	0.894617	−	0.446833i	$-0.147448\pi$
$673$	31238.5	1.78923	0.894617	+	0.446833i	$0.147448\pi$
$674$	16124.0	0.921473
$675$	0	0
$676$	−21831.0	−1.24209
$677$	−23202.3	−1.31719	−0.658596	−	0.752497i	$-0.728849\pi$
$677$	−23202.3	−1.31719	−0.658596	+	0.752497i	$0.728849\pi$
$678$	0	0
$679$	11557.5	0.653221
$680$	21081.4	1.18887
$681$	0	0
$682$	50804.6	2.85250
$683$	11108.7	0.622348	0.311174	−	0.950353i	$-0.399278\pi$
$683$	11108.7	0.622348	0.311174	+	0.950353i	$0.399278\pi$
$684$	0	0
$685$	−4076.55	−0.227383
$686$	−1877.06	−0.104470
$687$	0	0
$688$	−106874.	−5.92227
$689$	−1351.49	−0.0747278
$690$	0	0
$691$	−32784.6	−1.80490	−0.902449	−	0.430797i	$-0.858233\pi$
$691$	−32784.6	−1.80490	−0.902449	+	0.430797i	$0.858233\pi$
$692$	47747.8	2.62297
$693$	0	0
$694$	60092.0	3.28683
$695$	−5530.08	−0.301824
$696$	0	0
$697$	2468.51	0.134148
$698$	7113.80	0.385761
$699$	0	0
$700$	−6932.44	−0.374317
$701$	2027.49	0.109240	0.0546200	−	0.998507i	$-0.482605\pi$
$701$	2027.49	0.109240	0.0546200	+	0.998507i	$0.482605\pi$
$702$	0	0
$703$	−51067.4	−2.73975
$704$	128117.	6.85880
$705$	0	0
$706$	−28335.6	−1.51052
$707$	8241.73	0.438419
$708$	0	0
$709$	−31315.4	−1.65878	−0.829389	−	0.558671i	$-0.811311\pi$
$709$	−31315.4	−1.65878	−0.829389	+	0.558671i	$0.811311\pi$
$710$	11551.5	0.610593
$711$	0	0
$712$	44013.1	2.31666
$713$	−3287.70	−0.172686
$714$	0	0
$715$	−20127.0	−1.05274
$716$	−71425.8	−3.72808
$717$	0	0
$718$	65197.6	3.38879
$719$	−13785.9	−0.715060	−0.357530	−	0.933902i	$-0.616381\pi$
$719$	−13785.9	−0.715060	−0.357530	+	0.933902i	$0.616381\pi$
$720$	0	0
$721$	9511.53	0.491301
$722$	−67392.9	−3.47383
$723$	0	0
$724$	25190.5	1.29309
$725$	725.361	0.0371575
$726$	0	0
$727$	11853.4	0.604700	0.302350	−	0.953197i	$-0.402229\pi$
$727$	11853.4	0.604700	0.302350	+	0.953197i	$0.402229\pi$
$728$	−18527.3	−0.943225
$729$	0	0
$730$	31615.3	1.60293
$731$	−13639.4	−0.690112
$732$	0	0
$733$	−6857.02	−0.345525	−0.172762	−	0.984964i	$-0.555269\pi$
$733$	−6857.02	−0.345525	−0.172762	+	0.984964i	$0.555269\pi$
$734$	−10963.8	−0.551338
$735$	0	0
$736$	−16432.1	−0.822953
$737$	−51701.3	−2.58404
$738$	0	0
$739$	7095.95	0.353219	0.176609	−	0.984281i	$-0.443487\pi$
$739$	7095.95	0.353219	0.176609	+	0.984281i	$0.443487\pi$
$740$	−72343.5	−3.59379
$741$	0	0
$742$	1493.07	0.0738712
$743$	26319.1	1.29953	0.649767	−	0.760133i	$-0.274866\pi$
$743$	26319.1	1.29953	0.649767	+	0.760133i	$0.274866\pi$
$744$	0	0
$745$	1713.27	0.0842543
$746$	−32193.4	−1.58001
$747$	0	0
$748$	44049.0	2.15319
$749$	3851.22	0.187878
$750$	0	0
$751$	27498.1	1.33611	0.668055	−	0.744112i	$-0.267127\pi$
$751$	27498.1	1.33611	0.668055	+	0.744112i	$0.267127\pi$
$752$	−47739.9	−2.31502
$753$	0	0
$754$	3050.43	0.147335
$755$	13437.5	0.647735
$756$	0	0
$757$	28018.2	1.34523	0.672614	−	0.739994i	$-0.265172\pi$
$757$	28018.2	1.34523	0.672614	+	0.739994i	$0.265172\pi$
$758$	63487.2	3.04216
$759$	0	0
$760$	−94464.6	−4.50867
$761$	30614.8	1.45833	0.729163	−	0.684340i	$-0.239909\pi$
$761$	30614.8	1.45833	0.729163	+	0.684340i	$0.239909\pi$
$762$	0	0
$763$	−6978.51	−0.331113
$764$	18986.7	0.899103
$765$	0	0
$766$	−9049.48	−0.426855
$767$	−29615.1	−1.39418
$768$	0	0
$769$	−36535.2	−1.71326	−0.856628	−	0.515935i	$-0.827445\pi$
$769$	−36535.2	−1.71326	−0.856628	+	0.515935i	$0.827445\pi$
$770$	22235.6	1.04067
$771$	0	0
$772$	−44016.7	−2.05207
$773$	2926.76	0.136181	0.0680907	−	0.997679i	$-0.478309\pi$
$773$	2926.76	0.136181	0.0680907	+	0.997679i	$0.478309\pi$
$774$	0	0
$775$	6449.92	0.298952
$776$	−126028.	−5.83010
$777$	0	0
$778$	−63400.2	−2.92160
$779$	−11061.3	−0.508743
$780$	0	0
$781$	15338.9	0.702778
$782$	−3889.53	−0.177864
$783$	0	0
$784$	11864.7	0.540482
$785$	12159.6	0.552861
$786$	0	0
$787$	−15882.5	−0.719376	−0.359688	−	0.933073i	$-0.617117\pi$
$787$	−15882.5	−0.719376	−0.359688	+	0.933073i	$0.617117\pi$
$788$	4800.21	0.217005
$789$	0	0
$790$	−42014.5	−1.89216
$791$	−13232.0	−0.594786
$792$	0	0
$793$	15097.9	0.676095
$794$	14444.5	0.645614
$795$	0	0
$796$	−6003.65	−0.267329
$797$	16842.7	0.748556	0.374278	−	0.927316i	$-0.377891\pi$
$797$	16842.7	0.748556	0.374278	+	0.927316i	$0.377891\pi$
$798$	0	0
$799$	−6092.65	−0.269765
$800$	32237.0	1.42469
$801$	0	0
$802$	11182.3	0.492345
$803$	41981.1	1.84493
$804$	0	0
$805$	−1438.93	−0.0630007
$806$	27124.5	1.18539
$807$	0	0
$808$	−89871.4	−3.91295
$809$	−18173.9	−0.789815	−0.394907	−	0.918721i	$-0.629223\pi$
$809$	−18173.9	−0.789815	−0.394907	+	0.918721i	$0.629223\pi$
$810$	0	0
$811$	−13426.0	−0.581321	−0.290661	−	0.956826i	$-0.593875\pi$
$811$	−13426.0	−0.581321	−0.290661	+	0.956826i	$0.593875\pi$
$812$	−2469.79	−0.106740
$813$	0	0
$814$	−131077.	−5.64405
$815$	−20409.7	−0.877201
$816$	0	0
$817$	61117.6	2.61718
$818$	−11240.7	−0.480467
$819$	0	0
$820$	−15669.7	−0.667330
$821$	21961.4	0.933569	0.466784	−	0.884371i	$-0.345412\pi$
$821$	21961.4	0.933569	0.466784	+	0.884371i	$0.345412\pi$
$822$	0	0
$823$	−5614.14	−0.237784	−0.118892	−	0.992907i	$-0.537934\pi$
$823$	−5614.14	−0.237784	−0.118892	+	0.992907i	$0.537934\pi$
$824$	−103718.	−4.38493
$825$	0	0
$826$	32717.7	1.37820
$827$	−10174.5	−0.427815	−0.213907	−	0.976854i	$-0.568619\pi$
$827$	−10174.5	−0.427815	−0.213907	+	0.976854i	$0.568619\pi$
$828$	0	0
$829$	−27711.0	−1.16097	−0.580485	−	0.814271i	$-0.697137\pi$
$829$	−27711.0	−1.16097	−0.580485	+	0.814271i	$0.697137\pi$
$830$	−40202.9	−1.68128
$831$	0	0
$832$	68401.6	2.85024
$833$	1514.19	0.0629815
$834$	0	0
$835$	−8671.88	−0.359404
$836$	−197381.	−8.16576
$837$	0	0
$838$	−20473.1	−0.843950
$839$	1474.25	0.0606637	0.0303319	−	0.999540i	$-0.490344\pi$
$839$	1474.25	0.0606637	0.0303319	+	0.999540i	$0.490344\pi$
$840$	0	0
$841$	−24130.6	−0.989404
$842$	42850.9	1.75385
$843$	0	0
$844$	19747.7	0.805383
$845$	8889.74	0.361913
$846$	0	0
$847$	20209.1	0.819825
$848$	−9437.51	−0.382176
$849$	0	0
$850$	7630.61	0.307915
$851$	8482.36	0.341682
$852$	0	0
$853$	−31045.8	−1.24617	−0.623087	−	0.782152i	$-0.714122\pi$
$853$	−31045.8	−1.24617	−0.623087	+	0.782152i	$0.714122\pi$
$854$	−16679.7	−0.668345
$855$	0	0
$856$	−41995.4	−1.67684
$857$	41409.9	1.65057	0.825283	−	0.564720i	$-0.191016\pi$
$857$	41409.9	1.65057	0.825283	+	0.564720i	$0.191016\pi$
$858$	0	0
$859$	4188.80	0.166380	0.0831898	−	0.996534i	$-0.473489\pi$
$859$	4188.80	0.166380	0.0831898	+	0.996534i	$0.473489\pi$
$860$	86580.9	3.43301
$861$	0	0
$862$	79977.7	3.16015
$863$	14097.9	0.556082	0.278041	−	0.960569i	$-0.410315\pi$
$863$	14097.9	0.556082	0.278041	+	0.960569i	$0.410315\pi$
$864$	0	0
$865$	−19443.2	−0.764265
$866$	10509.7	0.412395
$867$	0	0
$868$	−21961.4	−0.858777
$869$	−55789.8	−2.17784
$870$	0	0
$871$	−27603.3	−1.07382
$872$	76096.8	2.95523
$873$	0	0
$874$	17428.8	0.674528
$875$	10643.2	0.411207
$876$	0	0
$877$	−26701.7	−1.02811	−0.514055	−	0.857757i	$-0.671857\pi$
$877$	−26701.7	−1.02811	−0.514055	+	0.857757i	$0.671857\pi$
$878$	−25737.5	−0.989291
$879$	0	0
$880$	−140548.	−5.38396
$881$	12186.4	0.466029	0.233014	−	0.972473i	$-0.425141\pi$
$881$	12186.4	0.466029	0.233014	+	0.972473i	$0.425141\pi$
$882$	0	0
$883$	−43820.5	−1.67008	−0.835039	−	0.550191i	$-0.814555\pi$
$883$	−43820.5	−1.67008	−0.835039	+	0.550191i	$0.814555\pi$
$884$	23517.7	0.894781
$885$	0	0
$886$	32041.9	1.21498
$887$	12779.0	0.483741	0.241871	−	0.970309i	$-0.422239\pi$
$887$	12779.0	0.483741	0.241871	+	0.970309i	$0.422239\pi$
$888$	0	0
$889$	−14439.0	−0.544732
$890$	−28201.9	−1.06217
$891$	0	0
$892$	−18151.6	−0.681346
$893$	27300.9	1.02306
$894$	0	0
$895$	29085.1	1.08626
$896$	−35559.2	−1.32584
$897$	0	0
$898$	79797.6	2.96535
$899$	2297.88	0.0852488
$900$	0	0
$901$	−1204.43	−0.0445344
$902$	−28391.5	−1.04804
$903$	0	0
$904$	144288.	5.30855
$905$	−10257.7	−0.376772
$906$	0	0
$907$	−7244.34	−0.265209	−0.132604	−	0.991169i	$-0.542334\pi$
$907$	−7244.34	−0.265209	−0.132604	+	0.991169i	$0.542334\pi$
$908$	72948.5	2.66617
$909$	0	0
$910$	11871.6	0.432461
$911$	19922.1	0.724531	0.362266	−	0.932075i	$-0.382003\pi$
$911$	19922.1	0.724531	0.362266	+	0.932075i	$0.382003\pi$
$912$	0	0
$913$	−53384.2	−1.93512
$914$	8498.09	0.307540
$915$	0	0
$916$	101414.	3.65810
$917$	7789.98	0.280532
$918$	0	0
$919$	−12768.2	−0.458306	−0.229153	−	0.973390i	$-0.573596\pi$
$919$	−12768.2	−0.458306	−0.229153	+	0.973390i	$0.573596\pi$
$920$	15690.7	0.562290
$921$	0	0
$922$	63531.2	2.26929
$923$	8189.44	0.292046
$924$	0	0
$925$	−16641.0	−0.591516
$926$	86566.6	3.07209
$927$	0	0
$928$	11484.9	0.406262
$929$	−27889.7	−0.984964	−0.492482	−	0.870323i	$-0.663910\pi$
$929$	−27889.7	−0.984964	−0.492482	+	0.870323i	$0.663910\pi$
$930$	0	0
$931$	−6785.01	−0.238851
$932$	55382.3	1.94647
$933$	0	0
$934$	64563.4	2.26186
$935$	−17937.0	−0.627384
$936$	0	0
$937$	−33958.2	−1.18395	−0.591977	−	0.805955i	$-0.701652\pi$
$937$	−33958.2	−1.18395	−0.591977	+	0.805955i	$0.701652\pi$
$938$	30495.1	1.06152
$939$	0	0
$940$	38675.2	1.34196
$941$	−39002.5	−1.35116	−0.675582	−	0.737285i	$-0.736108\pi$
$941$	−39002.5	−1.35116	−0.675582	+	0.737285i	$0.736108\pi$
$942$	0	0
$943$	1837.29	0.0634469
$944$	−206804.	−7.13020
$945$	0	0
$946$	156874.	5.39155
$947$	6375.90	0.218784	0.109392	−	0.993999i	$-0.465110\pi$
$947$	6375.90	0.218784	0.109392	+	0.993999i	$0.465110\pi$
$948$	0	0
$949$	22413.7	0.766679
$950$	−34192.4	−1.16774
$951$	0	0
$952$	−16511.4	−0.562119
$953$	−11469.9	−0.389871	−0.194935	−	0.980816i	$-0.562450\pi$
$953$	−11469.9	−0.389871	−0.194935	+	0.980816i	$0.562450\pi$
$954$	0	0
$955$	−7731.52	−0.261975
$956$	−93290.2	−3.15609
$957$	0	0
$958$	−50117.8	−1.69022
$959$	3192.85	0.107510
$960$	0	0
$961$	−9358.17	−0.314127
$962$	−69982.1	−2.34544
$963$	0	0
$964$	−154843.	−5.17339
$965$	17923.9	0.597918
$966$	0	0
$967$	−11097.4	−0.369045	−0.184523	−	0.982828i	$-0.559074\pi$
$967$	−11097.4	−0.369045	−0.184523	+	0.982828i	$0.559074\pi$
$968$	−220368.	−7.31705
$969$	0	0
$970$	80754.2	2.67305
$971$	−19350.8	−0.639545	−0.319773	−	0.947494i	$-0.603606\pi$
$971$	−19350.8	−0.639545	−0.319773	+	0.947494i	$0.603606\pi$
$972$	0	0
$973$	4331.28	0.142708
$974$	−89381.7	−2.94042
$975$	0	0
$976$	105430.	3.45771
$977$	−19020.2	−0.622834	−0.311417	−	0.950273i	$-0.600804\pi$
$977$	−19020.2	−0.622834	−0.311417	+	0.950273i	$0.600804\pi$
$978$	0	0
$979$	−37448.5	−1.22253
$980$	−9611.85	−0.313305
$981$	0	0
$982$	94859.6	3.08258
$983$	−48405.2	−1.57059	−0.785293	−	0.619125i	$-0.787488\pi$
$983$	−48405.2	−1.57059	−0.785293	+	0.619125i	$0.787488\pi$
$984$	0	0
$985$	−1954.68	−0.0632297
$986$	2718.52	0.0878046
$987$	0	0
$988$	−105382.	−3.39336
$989$	−10151.7	−0.326396
$990$	0	0
$991$	31822.7	1.02006	0.510031	−	0.860156i	$-0.329634\pi$
$991$	31822.7	1.02006	0.510031	+	0.860156i	$0.329634\pi$
$992$	102124.	3.26859
$993$	0	0
$994$	−9047.41	−0.288699
$995$	2444.73	0.0778926
$996$	0	0
$997$	−7520.08	−0.238880	−0.119440	−	0.992841i	$-0.538110\pi$
$997$	−7520.08	−0.238880	−0.119440	+	0.992841i	$0.538110\pi$
$998$	−51437.6	−1.63149
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.4.a.j.1.1		9
3.2	odd	2		483.4.a.i.1.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.4.a.i.1.9	✓	9	3.2	odd	2
1449.4.a.j.1.1		9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1449.a (trivial)

\(f(q)\)	\(=\)	\(q-5.47249 q^{2} +21.9481 q^{4} -8.93744 q^{5} +7.00000 q^{7} -76.3311 q^{8} +48.9100 q^{10} +64.9462 q^{11} +34.6747 q^{13} -38.3074 q^{14} +242.136 q^{16} +30.9018 q^{17} -138.470 q^{19} -196.160 q^{20} -355.417 q^{22} +23.0000 q^{23} -45.1222 q^{25} -189.757 q^{26} +153.637 q^{28} -16.0755 q^{29} -142.943 q^{31} -714.437 q^{32} -169.110 q^{34} -62.5621 q^{35} +368.798 q^{37} +757.774 q^{38} +682.204 q^{40} +79.8823 q^{41} -441.379 q^{43} +1425.45 q^{44} -125.867 q^{46} -197.161 q^{47} +49.0000 q^{49} +246.931 q^{50} +761.046 q^{52} -38.9761 q^{53} -580.453 q^{55} -534.317 q^{56} +87.9728 q^{58} -854.084 q^{59} +435.416 q^{61} +782.257 q^{62} +1972.66 q^{64} -309.903 q^{65} -796.063 q^{67} +678.238 q^{68} +342.370 q^{70} +236.179 q^{71} +646.398 q^{73} -2018.25 q^{74} -3039.15 q^{76} +454.623 q^{77} -859.016 q^{79} -2164.07 q^{80} -437.155 q^{82} -821.976 q^{83} -276.183 q^{85} +2415.44 q^{86} -4957.41 q^{88} -576.608 q^{89} +242.723 q^{91} +504.807 q^{92} +1078.96 q^{94} +1237.56 q^{95} +1651.08 q^{97} -268.152 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 8 q^{2} + 38 q^{4} - 39 q^{5} + 63 q^{7} - 135 q^{8} + 81 q^{10} - 38 q^{11} + 107 q^{13} - 56 q^{14} + 178 q^{16} - 170 q^{17} + 20 q^{19} - 300 q^{20} - 129 q^{22} + 207 q^{23} + 354 q^{25} + 114 q^{26}+ \cdots - 392 q^{98}+O(q^{100}) \) 9 * q - 8 * q^2 + 38 * q^4 - 39 * q^5 + 63 * q^7 - 135 * q^8 + 81 * q^10 - 38 * q^11 + 107 * q^13 - 56 * q^14 + 178 * q^16 - 170 * q^17 + 20 * q^19 - 300 * q^20 - 129 * q^22 + 207 * q^23 + 354 * q^25 + 114 * q^26 + 266 * q^28 - 405 * q^29 - 182 * q^31 - 648 * q^32 - 277 * q^34 - 273 * q^35 + 347 * q^37 + 291 * q^38 + 1066 * q^40 - 717 * q^41 - 77 * q^43 + 705 * q^44 - 184 * q^46 - 799 * q^47 + 441 * q^49 - 406 * q^50 + 519 * q^52 - 780 * q^53 - 754 * q^55 - 945 * q^56 - 508 * q^58 - 766 * q^59 + 1836 * q^61 - 173 * q^62 + 805 * q^64 - 655 * q^65 - 1290 * q^67 - 317 * q^68 + 567 * q^70 - 1802 * q^71 + 598 * q^73 - 578 * q^74 - 1323 * q^76 - 266 * q^77 - 1150 * q^79 - 795 * q^80 - 1040 * q^82 - 1758 * q^83 + 1078 * q^85 + 1920 * q^86 - 3056 * q^88 - 1400 * q^89 + 749 * q^91 + 874 * q^92 + 799 * q^94 - 2186 * q^95 + 1123 * q^97 - 392 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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