LMFDB - Embedded newform 2151.4.a.d.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$1$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.74193 q^{2} -0.481825 q^{4} +19.5291 q^{5} -9.73710 q^{7} +23.2566 q^{8} +O(q^{10})$	\(q-2.74193 q^{2} -0.481825 q^{4} +19.5291 q^{5} -9.73710 q^{7} +23.2566 q^{8} -53.5474 q^{10} -60.6297 q^{11} -74.0676 q^{13} +26.6984 q^{14} -59.9132 q^{16} +14.0475 q^{17} +116.505 q^{19} -9.40962 q^{20} +166.242 q^{22} -190.650 q^{23} +256.386 q^{25} +203.088 q^{26} +4.69158 q^{28} +180.429 q^{29} +245.201 q^{31} -21.7747 q^{32} -38.5173 q^{34} -190.157 q^{35} +145.419 q^{37} -319.448 q^{38} +454.180 q^{40} +54.5565 q^{41} +132.222 q^{43} +29.2129 q^{44} +522.748 q^{46} +552.349 q^{47} -248.189 q^{49} -702.992 q^{50} +35.6876 q^{52} -274.951 q^{53} -1184.04 q^{55} -226.452 q^{56} -494.724 q^{58} -319.285 q^{59} +379.722 q^{61} -672.322 q^{62} +539.010 q^{64} -1446.47 q^{65} +262.184 q^{67} -6.76846 q^{68} +521.397 q^{70} -67.3812 q^{71} +447.523 q^{73} -398.729 q^{74} -56.1350 q^{76} +590.357 q^{77} -637.949 q^{79} -1170.05 q^{80} -149.590 q^{82} +222.848 q^{83} +274.336 q^{85} -362.544 q^{86} -1410.04 q^{88} -1561.61 q^{89} +721.203 q^{91} +91.8600 q^{92} -1514.50 q^{94} +2275.24 q^{95} -932.258 q^{97} +680.516 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	$ 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) $ 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.74193	−0.969418	−0.484709	−	0.874675i	$-0.661075\pi$
$2$	−2.74193	−0.969418	−0.484709	+	0.874675i	$0.661075\pi$
$3$	0	0
$3$	0	0
$4$	−0.481825	−0.0602282
$5$	19.5291	1.74674	0.873368	−	0.487060i	$-0.161931\pi$
$5$	19.5291	1.74674	0.873368	+	0.487060i	$0.161931\pi$
$6$	0	0
$7$	−9.73710	−0.525754	−0.262877	−	0.964829i	$-0.584671\pi$
$7$	−9.73710	−0.525754	−0.262877	+	0.964829i	$0.584671\pi$
$8$	23.2566	1.02780
$9$	0	0
$10$	−53.5474	−1.69332
$11$	−60.6297	−1.66187	−0.830934	−	0.556371i	$-0.812193\pi$
$11$	−60.6297	−1.66187	−0.830934	+	0.556371i	$0.812193\pi$
$12$	0	0
$13$	−74.0676	−1.58020	−0.790102	−	0.612976i	$-0.789972\pi$
$13$	−74.0676	−1.58020	−0.790102	+	0.612976i	$0.789972\pi$
$14$	26.6984	0.509676
$15$	0	0
$16$	−59.9132	−0.936144
$17$	14.0475	0.200413	0.100207	−	0.994967i	$-0.468050\pi$
$17$	14.0475	0.200413	0.100207	+	0.994967i	$0.468050\pi$
$18$	0	0
$19$	116.505	1.40674	0.703370	−	0.710824i	$-0.251678\pi$
$19$	116.505	1.40674	0.703370	+	0.710824i	$0.251678\pi$
$20$	−9.40962	−0.105203
$21$	0	0
$22$	166.242	1.61104
$23$	−190.650	−1.72840	−0.864201	−	0.503147i	$-0.832176\pi$
$23$	−190.650	−1.72840	−0.864201	+	0.503147i	$0.832176\pi$
$24$	0	0
$25$	256.386	2.05109
$26$	203.088	1.53188
$27$	0	0
$28$	4.69158	0.0316652
$29$	180.429	1.15534	0.577670	−	0.816271i	$-0.303962\pi$
$29$	180.429	1.15534	0.577670	+	0.816271i	$0.303962\pi$
$30$	0	0
$31$	245.201	1.42062	0.710312	−	0.703887i	$-0.248554\pi$
$31$	245.201	1.42062	0.710312	+	0.703887i	$0.248554\pi$
$32$	−21.7747	−0.120289
$33$	0	0
$34$	−38.5173	−0.194284
$35$	−190.157	−0.918354
$36$	0	0
$37$	145.419	0.646129	0.323064	−	0.946377i	$-0.395287\pi$
$37$	145.419	0.646129	0.323064	+	0.946377i	$0.395287\pi$
$38$	−319.448	−1.36372
$39$	0	0
$40$	454.180	1.79530
$41$	54.5565	0.207812	0.103906	−	0.994587i	$-0.466866\pi$
$41$	54.5565	0.207812	0.103906	+	0.994587i	$0.466866\pi$
$42$	0	0
$43$	132.222	0.468924	0.234462	−	0.972125i	$-0.424667\pi$
$43$	132.222	0.468924	0.234462	+	0.972125i	$0.424667\pi$
$44$	29.2129	0.100091
$45$	0	0
$46$	522.748	1.67554
$47$	552.349	1.71422	0.857110	−	0.515133i	$-0.172258\pi$
$47$	552.349	1.71422	0.857110	+	0.515133i	$0.172258\pi$
$48$	0	0
$49$	−248.189	−0.723583
$50$	−702.992	−1.98836
$51$	0	0
$52$	35.6876	0.0951728
$53$	−274.951	−0.712594	−0.356297	−	0.934373i	$-0.615961\pi$
$53$	−274.951	−0.712594	−0.356297	+	0.934373i	$0.615961\pi$
$54$	0	0
$55$	−1184.04	−2.90284
$56$	−226.452	−0.540372
$57$	0	0
$58$	−494.724	−1.12001
$59$	−319.285	−0.704531	−0.352265	−	0.935900i	$-0.614589\pi$
$59$	−319.285	−0.704531	−0.352265	+	0.935900i	$0.614589\pi$
$60$	0	0
$61$	379.722	0.797023	0.398511	−	0.917163i	$-0.369527\pi$
$61$	379.722	0.797023	0.398511	+	0.917163i	$0.369527\pi$
$62$	−672.322	−1.37718
$63$	0	0
$64$	539.010	1.05275
$65$	−1446.47	−2.76020
$66$	0	0
$67$	262.184	0.478072	0.239036	−	0.971011i	$-0.423169\pi$
$67$	262.184	0.478072	0.239036	+	0.971011i	$0.423169\pi$
$68$	−6.76846	−0.0120705
$69$	0	0
$70$	521.397	0.890269
$71$	−67.3812	−0.112629	−0.0563146	−	0.998413i	$-0.517935\pi$
$71$	−67.3812	−0.112629	−0.0563146	+	0.998413i	$0.517935\pi$
$72$	0	0
$73$	447.523	0.717515	0.358758	−	0.933431i	$-0.383200\pi$
$73$	447.523	0.717515	0.358758	+	0.933431i	$0.383200\pi$
$74$	−398.729	−0.626369
$75$	0	0
$76$	−56.1350	−0.0847254
$77$	590.357	0.873733
$78$	0	0
$79$	−637.949	−0.908543	−0.454272	−	0.890863i	$-0.650100\pi$
$79$	−637.949	−0.908543	−0.454272	+	0.890863i	$0.650100\pi$
$80$	−1170.05	−1.63520
$81$	0	0
$82$	−149.590	−0.201457
$83$	222.848	0.294708	0.147354	−	0.989084i	$-0.452924\pi$
$83$	222.848	0.294708	0.147354	+	0.989084i	$0.452924\pi$
$84$	0	0
$85$	274.336	0.350069
$86$	−362.544	−0.454583
$87$	0	0
$88$	−1410.04	−1.70807
$89$	−1561.61	−1.85989	−0.929947	−	0.367693i	$-0.880148\pi$
$89$	−1561.61	−1.85989	−0.929947	+	0.367693i	$0.880148\pi$
$90$	0	0
$91$	721.203	0.830799
$92$	91.8600	0.104099
$93$	0	0
$94$	−1514.50	−1.66180
$95$	2275.24	2.45720
$96$	0	0
$97$	−932.258	−0.975840	−0.487920	−	0.872888i	$-0.662244\pi$
$97$	−932.258	−0.975840	−0.487920	+	0.872888i	$0.662244\pi$
$98$	680.516	0.701454
$99$	0	0
$100$	−123.533	−0.123533
$101$	−26.0994	−0.0257128	−0.0128564	−	0.999917i	$-0.504092\pi$
$101$	−26.0994	−0.0257128	−0.0128564	+	0.999917i	$0.504092\pi$
$102$	0	0
$103$	−489.572	−0.468340	−0.234170	−	0.972196i	$-0.575237\pi$
$103$	−489.572	−0.468340	−0.234170	+	0.972196i	$0.575237\pi$
$104$	−1722.56	−1.62414
$105$	0	0
$106$	753.897	0.690802
$107$	−67.2513	−0.0607610	−0.0303805	−	0.999538i	$-0.509672\pi$
$107$	−67.2513	−0.0607610	−0.0303805	+	0.999538i	$0.509672\pi$
$108$	0	0
$109$	−1680.92	−1.47709	−0.738546	−	0.674203i	$-0.764487\pi$
$109$	−1680.92	−1.47709	−0.738546	+	0.674203i	$0.764487\pi$
$110$	3246.56	2.81407
$111$	0	0
$112$	583.381	0.492182
$113$	−1821.17	−1.51612	−0.758059	−	0.652186i	$-0.773852\pi$
$113$	−1821.17	−1.51612	−0.758059	+	0.652186i	$0.773852\pi$
$114$	0	0
$115$	−3723.22	−3.01906
$116$	−86.9353	−0.0695840
$117$	0	0
$118$	875.456	0.682985
$119$	−136.782	−0.105368
$120$	0	0
$121$	2344.96	1.76180
$122$	−1041.17	−0.772648
$123$	0	0
$124$	−118.144	−0.0855616
$125$	2565.85	1.83598
$126$	0	0
$127$	2624.24	1.83357	0.916786	−	0.399379i	$-0.130774\pi$
$127$	2624.24	1.83357	0.916786	+	0.399379i	$0.130774\pi$
$128$	−1303.73	−0.900271
$129$	0	0
$130$	3966.13	2.67579
$131$	516.941	0.344774	0.172387	−	0.985029i	$-0.444852\pi$
$131$	516.941	0.344774	0.172387	+	0.985029i	$0.444852\pi$
$132$	0	0
$133$	−1134.42	−0.739599
$134$	−718.889	−0.463452
$135$	0	0
$136$	326.697	0.205986
$137$	1326.48	0.827221	0.413610	−	0.910454i	$-0.364268\pi$
$137$	1326.48	0.827221	0.413610	+	0.910454i	$0.364268\pi$
$138$	0	0
$139$	412.107	0.251471	0.125735	−	0.992064i	$-0.459871\pi$
$139$	412.107	0.251471	0.125735	+	0.992064i	$0.459871\pi$
$140$	91.6224	0.0553108
$141$	0	0
$142$	184.754	0.109185
$143$	4490.69	2.62609
$144$	0	0
$145$	3523.62	2.01807
$146$	−1227.08	−0.695573
$147$	0	0
$148$	−70.0667	−0.0389152
$149$	−1185.38	−0.651746	−0.325873	−	0.945414i	$-0.605658\pi$
$149$	−1185.38	−0.651746	−0.325873	+	0.945414i	$0.605658\pi$
$150$	0	0
$151$	−806.040	−0.434401	−0.217201	−	0.976127i	$-0.569693\pi$
$151$	−806.040	−0.434401	−0.217201	+	0.976127i	$0.569693\pi$
$152$	2709.50	1.44585
$153$	0	0
$154$	−1618.72	−0.847013
$155$	4788.55	2.48145
$156$	0	0
$157$	−1723.88	−0.876312	−0.438156	−	0.898899i	$-0.644368\pi$
$157$	−1723.88	−0.876312	−0.438156	+	0.898899i	$0.644368\pi$
$158$	1749.21	0.880758
$159$	0	0
$160$	−425.240	−0.210113
$161$	1856.38	0.908714
$162$	0	0
$163$	3673.15	1.76505	0.882525	−	0.470265i	$-0.155842\pi$
$163$	3673.15	1.76505	0.882525	+	0.470265i	$0.155842\pi$
$164$	−26.2867	−0.0125161
$165$	0	0
$166$	−611.035	−0.285696
$167$	−3934.27	−1.82301	−0.911505	−	0.411288i	$-0.865079\pi$
$167$	−3934.27	−1.82301	−0.911505	+	0.411288i	$0.865079\pi$
$168$	0	0
$169$	3289.01	1.49704
$170$	−752.209	−0.339364
$171$	0	0
$172$	−63.7081	−0.0282424
$173$	−3905.24	−1.71624	−0.858121	−	0.513448i	$-0.828368\pi$
$173$	−3905.24	−1.71624	−0.858121	+	0.513448i	$0.828368\pi$
$174$	0	0
$175$	−2496.46	−1.07837
$176$	3632.52	1.55575
$177$	0	0
$178$	4281.83	1.80302
$179$	−120.733	−0.0504135	−0.0252068	−	0.999682i	$-0.508024\pi$
$179$	−120.733	−0.0504135	−0.0252068	+	0.999682i	$0.508024\pi$
$180$	0	0
$181$	−3327.76	−1.36658	−0.683288	−	0.730149i	$-0.739451\pi$
$181$	−3327.76	−1.36658	−0.683288	+	0.730149i	$0.739451\pi$
$182$	−1977.49	−0.805391
$183$	0	0
$184$	−4433.86	−1.77646
$185$	2839.91	1.12862
$186$	0	0
$187$	−851.698	−0.333060
$188$	−266.136	−0.103244
$189$	0	0
$190$	−6238.54	−2.38206
$191$	585.904	0.221961	0.110980	−	0.993823i	$-0.464601\pi$
$191$	585.904	0.221961	0.110980	+	0.993823i	$0.464601\pi$
$192$	0	0
$193$	−2273.54	−0.847942	−0.423971	−	0.905676i	$-0.639364\pi$
$193$	−2273.54	−0.847942	−0.423971	+	0.905676i	$0.639364\pi$
$194$	2556.19	0.945997
$195$	0	0
$196$	119.584	0.0435801
$197$	−2316.23	−0.837687	−0.418844	−	0.908058i	$-0.637564\pi$
$197$	−2316.23	−0.837687	−0.418844	+	0.908058i	$0.637564\pi$
$198$	0	0
$199$	1639.03	0.583858	0.291929	−	0.956440i	$-0.405703\pi$
$199$	1639.03	0.583858	0.291929	+	0.956440i	$0.405703\pi$
$200$	5962.66	2.10812
$201$	0	0
$202$	71.5628	0.0249264
$203$	−1756.86	−0.607424
$204$	0	0
$205$	1065.44	0.362993
$206$	1342.37	0.454017
$207$	0	0
$208$	4437.63	1.47930
$209$	−7063.66	−2.33781
$210$	0	0
$211$	−4300.19	−1.40302	−0.701510	−	0.712660i	$-0.747490\pi$
$211$	−4300.19	−1.40302	−0.701510	+	0.712660i	$0.747490\pi$
$212$	132.479	0.0429183
$213$	0	0
$214$	184.398	0.0589028
$215$	2582.19	0.819086
$216$	0	0
$217$	−2387.54	−0.746898
$218$	4608.97	1.43192
$219$	0	0
$220$	570.503	0.174833
$221$	−1040.47	−0.316694
$222$	0	0
$223$	2715.22	0.815358	0.407679	−	0.913125i	$-0.366338\pi$
$223$	2715.22	0.815358	0.407679	+	0.913125i	$0.366338\pi$
$224$	212.022	0.0632425
$225$	0	0
$226$	4993.52	1.46975
$227$	4107.46	1.20098	0.600488	−	0.799634i	$-0.294973\pi$
$227$	4107.46	1.20098	0.600488	+	0.799634i	$0.294973\pi$
$228$	0	0
$229$	−517.144	−0.149231	−0.0746154	−	0.997212i	$-0.523773\pi$
$229$	−517.144	−0.149231	−0.0746154	+	0.997212i	$0.523773\pi$
$230$	10208.8	2.92674
$231$	0	0
$232$	4196.16	1.18746
$233$	−6592.48	−1.85359	−0.926797	−	0.375561i	$-0.877450\pi$
$233$	−6592.48	−1.85359	−0.926797	+	0.375561i	$0.877450\pi$
$234$	0	0
$235$	10786.9	2.99429
$236$	153.839	0.0424326
$237$	0	0
$238$	375.047	0.102146
$239$	−239.000	−0.0646846
$239$	−239.000	−0.0646846
$240$	0	0
$241$	−4057.72	−1.08457	−0.542283	−	0.840196i	$-0.682440\pi$
$241$	−4057.72	−1.08457	−0.542283	+	0.840196i	$0.682440\pi$
$242$	−6429.71	−1.70792
$243$	0	0
$244$	−182.960	−0.0480032
$245$	−4846.91	−1.26391
$246$	0	0
$247$	−8629.23	−2.22294
$248$	5702.52	1.46012
$249$	0	0
$250$	−7035.39	−1.77983
$251$	−3777.07	−0.949828	−0.474914	−	0.880032i	$-0.657521\pi$
$251$	−3777.07	−0.949828	−0.474914	+	0.880032i	$0.657521\pi$
$252$	0	0
$253$	11559.0	2.87238
$254$	−7195.48	−1.77750
$255$	0	0
$256$	−737.345	−0.180016
$257$	1992.34	0.483575	0.241788	−	0.970329i	$-0.422266\pi$
$257$	1992.34	0.483575	0.241788	+	0.970329i	$0.422266\pi$
$258$	0	0
$259$	−1415.96	−0.339705
$260$	696.948	0.166242
$261$	0	0
$262$	−1417.42	−0.334230
$263$	−272.936	−0.0639922	−0.0319961	−	0.999488i	$-0.510186\pi$
$263$	−272.936	−0.0639922	−0.0319961	+	0.999488i	$0.510186\pi$
$264$	0	0
$265$	−5369.56	−1.24471
$266$	3110.50	0.716981
$267$	0	0
$268$	−126.327	−0.0287934
$269$	−1961.33	−0.444552	−0.222276	−	0.974984i	$-0.571349\pi$
$269$	−1961.33	−0.444552	−0.222276	+	0.974984i	$0.571349\pi$
$270$	0	0
$271$	−5139.13	−1.15196	−0.575978	−	0.817465i	$-0.695379\pi$
$271$	−5139.13	−1.15196	−0.575978	+	0.817465i	$0.695379\pi$
$272$	−841.633	−0.187616
$273$	0	0
$274$	−3637.13	−0.801923
$275$	−15544.6	−3.40864
$276$	0	0
$277$	8952.28	1.94184	0.970921	−	0.239398i	$-0.0769502\pi$
$277$	8952.28	1.94184	0.970921	+	0.239398i	$0.0769502\pi$
$278$	−1129.97	−0.243780
$279$	0	0
$280$	−4422.40	−0.943888
$281$	−2841.07	−0.603146	−0.301573	−	0.953443i	$-0.597512\pi$
$281$	−2841.07	−0.603146	−0.301573	+	0.953443i	$0.597512\pi$
$282$	0	0
$283$	−1402.90	−0.294677	−0.147339	−	0.989086i	$-0.547071\pi$
$283$	−1402.90	−0.294677	−0.147339	+	0.989086i	$0.547071\pi$
$284$	32.4660	0.00678345
$285$	0	0
$286$	−12313.2	−2.54578
$287$	−531.222	−0.109258
$288$	0	0
$289$	−4715.67	−0.959834
$290$	−9661.51	−1.95636
$291$	0	0
$292$	−215.628	−0.0432147
$293$	−9536.05	−1.90137	−0.950686	−	0.310155i	$-0.899619\pi$
$293$	−9536.05	−1.90137	−0.950686	+	0.310155i	$0.899619\pi$
$294$	0	0
$295$	−6235.34	−1.23063
$296$	3381.95	0.664094
$297$	0	0
$298$	3250.23	0.631815
$299$	14121.0	2.73123
$300$	0	0
$301$	−1287.46	−0.246539
$302$	2210.10	0.421117
$303$	0	0
$304$	−6980.18	−1.31691
$305$	7415.63	1.39219
$306$	0	0
$307$	4596.63	0.854539	0.427270	−	0.904124i	$-0.359476\pi$
$307$	4596.63	0.854539	0.427270	+	0.904124i	$0.359476\pi$
$308$	−284.449	−0.0526234
$309$	0	0
$310$	−13129.9	−2.40557
$311$	4604.84	0.839603	0.419802	−	0.907616i	$-0.362100\pi$
$311$	4604.84	0.839603	0.419802	+	0.907616i	$0.362100\pi$
$312$	0	0
$313$	7137.95	1.28901	0.644506	−	0.764599i	$-0.277063\pi$
$313$	7137.95	1.28901	0.644506	+	0.764599i	$0.277063\pi$
$314$	4726.77	0.849513
$315$	0	0
$316$	307.380	0.0547199
$317$	−1644.03	−0.291288	−0.145644	−	0.989337i	$-0.546525\pi$
$317$	−1644.03	−0.291288	−0.145644	+	0.989337i	$0.546525\pi$
$318$	0	0
$319$	−10939.4	−1.92002
$320$	10526.4	1.83889
$321$	0	0
$322$	−5090.05	−0.880924
$323$	1636.61	0.281929
$324$	0	0
$325$	−18989.9	−3.24114
$326$	−10071.5	−1.71107
$327$	0	0
$328$	1268.80	0.213590
$329$	−5378.28	−0.901258
$330$	0	0
$331$	5435.59	0.902620	0.451310	−	0.892367i	$-0.350957\pi$
$331$	5435.59	0.902620	0.451310	+	0.892367i	$0.350957\pi$
$332$	−107.374	−0.0177497
$333$	0	0
$334$	10787.5	1.76726
$335$	5120.21	0.835066
$336$	0	0
$337$	−2097.15	−0.338988	−0.169494	−	0.985531i	$-0.554213\pi$
$337$	−2097.15	−0.338988	−0.169494	+	0.985531i	$0.554213\pi$
$338$	−9018.22	−1.45126
$339$	0	0
$340$	−132.182	−0.0210840
$341$	−14866.4	−2.36089
$342$	0	0
$343$	5756.47	0.906181
$344$	3075.04	0.481962
$345$	0	0
$346$	10707.9	1.66376
$347$	−7869.15	−1.21740	−0.608700	−	0.793400i	$-0.708309\pi$
$347$	−7869.15	−1.21740	−0.608700	+	0.793400i	$0.708309\pi$
$348$	0	0
$349$	−8546.77	−1.31088	−0.655442	−	0.755246i	$-0.727518\pi$
$349$	−8546.77	−1.31088	−0.655442	+	0.755246i	$0.727518\pi$
$350$	6845.11	1.04539
$351$	0	0
$352$	1320.19	0.199905
$353$	5711.09	0.861107	0.430553	−	0.902565i	$-0.358318\pi$
$353$	5711.09	0.861107	0.430553	+	0.902565i	$0.358318\pi$
$354$	0	0
$355$	−1315.89	−0.196734
$356$	752.425	0.112018
$357$	0	0
$358$	331.042	0.0488718
$359$	−7906.05	−1.16230	−0.581150	−	0.813797i	$-0.697397\pi$
$359$	−7906.05	−1.16230	−0.581150	+	0.813797i	$0.697397\pi$
$360$	0	0
$361$	6714.39	0.978916
$362$	9124.48	1.32478
$363$	0	0
$364$	−347.494	−0.0500375
$365$	8739.73	1.25331
$366$	0	0
$367$	−6606.21	−0.939622	−0.469811	−	0.882767i	$-0.655678\pi$
$367$	−6606.21	−0.939622	−0.469811	+	0.882767i	$0.655678\pi$
$368$	11422.5	1.61803
$369$	0	0
$370$	−7786.82	−1.09410
$371$	2677.23	0.374649
$372$	0	0
$373$	5328.50	0.739676	0.369838	−	0.929096i	$-0.379413\pi$
$373$	5328.50	0.739676	0.369838	+	0.929096i	$0.379413\pi$
$374$	2335.29	0.322875
$375$	0	0
$376$	12845.7	1.76188
$377$	−13363.9	−1.82567
$378$	0	0
$379$	−13220.4	−1.79178	−0.895892	−	0.444272i	$-0.853463\pi$
$379$	−13220.4	−1.79178	−0.895892	+	0.444272i	$0.853463\pi$
$380$	−1096.27	−0.147993
$381$	0	0
$382$	−1606.51	−0.215173
$383$	7089.62	0.945855	0.472928	−	0.881101i	$-0.343197\pi$
$383$	7089.62	0.945855	0.472928	+	0.881101i	$0.343197\pi$
$384$	0	0
$385$	11529.2	1.52618
$386$	6233.88	0.822010
$387$	0	0
$388$	449.186	0.0587731
$389$	−1268.59	−0.165348	−0.0826738	−	0.996577i	$-0.526346\pi$
$389$	−1268.59	−0.165348	−0.0826738	+	0.996577i	$0.526346\pi$
$390$	0	0
$391$	−2678.16	−0.346395
$392$	−5772.02	−0.743702
$393$	0	0
$394$	6350.93	0.812069
$395$	−12458.6	−1.58699
$396$	0	0
$397$	−6612.84	−0.835992	−0.417996	−	0.908449i	$-0.637267\pi$
$397$	−6612.84	−0.835992	−0.417996	+	0.908449i	$0.637267\pi$
$398$	−4494.10	−0.566003
$399$	0	0
$400$	−15360.9	−1.92012
$401$	4669.41	0.581494	0.290747	−	0.956800i	$-0.406096\pi$
$401$	4669.41	0.581494	0.290747	+	0.956800i	$0.406096\pi$
$402$	0	0
$403$	−18161.4	−2.24487
$404$	12.5754	0.00154863
$405$	0	0
$406$	4817.17	0.588848
$407$	−8816.72	−1.07378
$408$	0	0
$409$	−9960.99	−1.20425	−0.602126	−	0.798401i	$-0.705680\pi$
$409$	−9960.99	−1.20425	−0.602126	+	0.798401i	$0.705680\pi$
$410$	−2921.36	−0.351892
$411$	0	0
$412$	235.888	0.0282072
$413$	3108.91	0.370410
$414$	0	0
$415$	4352.03	0.514778
$416$	1612.80	0.190081
$417$	0	0
$418$	19368.0	2.26632
$419$	967.817	0.112842	0.0564212	−	0.998407i	$-0.482031\pi$
$419$	967.817	0.112842	0.0564212	+	0.998407i	$0.482031\pi$
$420$	0	0
$421$	2070.08	0.239643	0.119821	−	0.992795i	$-0.461768\pi$
$421$	2070.08	0.239643	0.119821	+	0.992795i	$0.461768\pi$
$422$	11790.8	1.36011
$423$	0	0
$424$	−6394.43	−0.732408
$425$	3601.59	0.411066
$426$	0	0
$427$	−3697.39	−0.419038
$428$	32.4034	0.00365952
$429$	0	0
$430$	−7080.17	−0.794037
$431$	−6945.65	−0.776242	−0.388121	−	0.921608i	$-0.626876\pi$
$431$	−6945.65	−0.776242	−0.388121	+	0.921608i	$0.626876\pi$
$432$	0	0
$433$	10937.9	1.21395	0.606975	−	0.794721i	$-0.292383\pi$
$433$	10937.9	1.21395	0.606975	+	0.794721i	$0.292383\pi$
$434$	6546.47	0.724057
$435$	0	0
$436$	809.911	0.0889626
$437$	−22211.6	−2.43141
$438$	0	0
$439$	7967.26	0.866188	0.433094	−	0.901349i	$-0.357422\pi$
$439$	7967.26	0.866188	0.433094	+	0.901349i	$0.357422\pi$
$440$	−27536.8	−2.98356
$441$	0	0
$442$	2852.89	0.307009
$443$	−15343.6	−1.64559	−0.822795	−	0.568339i	$-0.807586\pi$
$443$	−15343.6	−1.64559	−0.822795	+	0.568339i	$0.807586\pi$
$444$	0	0
$445$	−30496.9	−3.24875
$446$	−7444.95	−0.790423
$447$	0	0
$448$	−5248.40	−0.553490
$449$	10968.2	1.15283	0.576417	−	0.817156i	$-0.304451\pi$
$449$	10968.2	1.15283	0.576417	+	0.817156i	$0.304451\pi$
$450$	0	0
$451$	−3307.75	−0.345356
$452$	877.487	0.0913131
$453$	0	0
$454$	−11262.4	−1.16425
$455$	14084.5	1.45119
$456$	0	0
$457$	988.407	0.101172	0.0505862	−	0.998720i	$-0.483891\pi$
$457$	988.407	0.101172	0.0505862	+	0.998720i	$0.483891\pi$
$458$	1417.97	0.144667
$459$	0	0
$460$	1793.94	0.181833
$461$	13611.9	1.37521	0.687604	−	0.726086i	$-0.258662\pi$
$461$	13611.9	1.37521	0.687604	+	0.726086i	$0.258662\pi$
$462$	0	0
$463$	−5065.31	−0.508434	−0.254217	−	0.967147i	$-0.581818\pi$
$463$	−5065.31	−0.508434	−0.254217	+	0.967147i	$0.581818\pi$
$464$	−10810.1	−1.08156
$465$	0	0
$466$	18076.1	1.79691
$467$	−3877.55	−0.384222	−0.192111	−	0.981373i	$-0.561533\pi$
$467$	−3877.55	−0.384222	−0.192111	+	0.981373i	$0.561533\pi$
$468$	0	0
$469$	−2552.91	−0.251348
$470$	−29576.9	−2.90272
$471$	0	0
$472$	−7425.46	−0.724120
$473$	−8016.60	−0.779289
$474$	0	0
$475$	29870.2	2.88535
$476$	65.9052	0.00634613
$477$	0	0
$478$	655.321	0.0627065
$479$	11124.3	1.06113	0.530566	−	0.847643i	$-0.321979\pi$
$479$	11124.3	1.06113	0.530566	+	0.847643i	$0.321979\pi$
$480$	0	0
$481$	−10770.8	−1.02102
$482$	11126.0	1.05140
$483$	0	0
$484$	−1129.86	−0.106110
$485$	−18206.2	−1.70454
$486$	0	0
$487$	14456.6	1.34516	0.672580	−	0.740025i	$-0.265186\pi$
$487$	14456.6	1.34516	0.672580	+	0.740025i	$0.265186\pi$
$488$	8831.02	0.819184
$489$	0	0
$490$	13289.9	1.22526
$491$	1070.87	0.0984273	0.0492136	−	0.998788i	$-0.484328\pi$
$491$	1070.87	0.0984273	0.0492136	+	0.998788i	$0.484328\pi$
$492$	0	0
$493$	2534.58	0.231545
$494$	23660.7	2.15495
$495$	0	0
$496$	−14690.8	−1.32991
$497$	656.097	0.0592152
$498$	0	0
$499$	21.3100	0.00191175	0.000955876	−	1.00000i	$-0.499696\pi$
$499$	21.3100	0.00191175	0.000955876		1.00000i	$0.499696\pi$
$500$	−1236.29	−0.110577
$501$	0	0
$502$	10356.5	0.920781
$503$	15202.3	1.34759	0.673794	−	0.738919i	$-0.264663\pi$
$503$	15202.3	1.34759	0.673794	+	0.738919i	$0.264663\pi$
$504$	0	0
$505$	−509.699	−0.0449134
$506$	−31694.1	−2.78453
$507$	0	0
$508$	−1264.43	−0.110433
$509$	−7617.23	−0.663316	−0.331658	−	0.943400i	$-0.607608\pi$
$509$	−7617.23	−0.663316	−0.331658	+	0.943400i	$0.607608\pi$
$510$	0	0
$511$	−4357.58	−0.377237
$512$	12451.6	1.07478
$513$	0	0
$514$	−5462.86	−0.468787
$515$	−9560.91	−0.818066
$516$	0	0
$517$	−33488.7	−2.84881
$518$	3882.47	0.329316
$519$	0	0
$520$	−33640.0	−2.83695
$521$	−7649.97	−0.643285	−0.321642	−	0.946861i	$-0.604235\pi$
$521$	−7649.97	−0.643285	−0.321642	+	0.946861i	$0.604235\pi$
$522$	0	0
$523$	−13069.8	−1.09274	−0.546371	−	0.837543i	$-0.683991\pi$
$523$	−13069.8	−1.09274	−0.546371	+	0.837543i	$0.683991\pi$
$524$	−249.075	−0.0207651
$525$	0	0
$526$	748.371	0.0620352
$527$	3444.46	0.284712
$528$	0	0
$529$	24180.4	1.98737
$530$	14722.9	1.20665
$531$	0	0
$532$	546.592	0.0445447
$533$	−4040.87	−0.328385
$534$	0	0
$535$	−1313.36	−0.106133
$536$	6097.49	0.491365
$537$	0	0
$538$	5377.83	0.430956
$539$	15047.6	1.20250
$540$	0	0
$541$	6418.05	0.510043	0.255022	−	0.966935i	$-0.417917\pi$
$541$	6418.05	0.510043	0.255022	+	0.966935i	$0.417917\pi$
$542$	14091.1	1.11673
$543$	0	0
$544$	−305.880	−0.0241075
$545$	−32826.9	−2.58009
$546$	0	0
$547$	−7886.45	−0.616454	−0.308227	−	0.951313i	$-0.599736\pi$
$547$	−7886.45	−0.616454	−0.308227	+	0.951313i	$0.599736\pi$
$548$	−639.134	−0.0498220
$549$	0	0
$550$	42622.2	3.30440
$551$	21020.9	1.62526
$552$	0	0
$553$	6211.78	0.477670
$554$	−24546.5	−1.88246
$555$	0	0
$556$	−198.563	−0.0151456
$557$	11871.7	0.903085	0.451542	−	0.892250i	$-0.350874\pi$
$557$	11871.7	0.903085	0.451542	+	0.892250i	$0.350874\pi$
$558$	0	0
$559$	−9793.39	−0.740995
$560$	11392.9	0.859712
$561$	0	0
$562$	7790.01	0.584701
$563$	−11690.2	−0.875105	−0.437553	−	0.899193i	$-0.644155\pi$
$563$	−11690.2	−0.875105	−0.437553	+	0.899193i	$0.644155\pi$
$564$	0	0
$565$	−35565.8	−2.64826
$566$	3846.65	0.285666
$567$	0	0
$568$	−1567.05	−0.115761
$569$	−14108.3	−1.03946	−0.519729	−	0.854331i	$-0.673967\pi$
$569$	−14108.3	−1.03946	−0.519729	+	0.854331i	$0.673967\pi$
$570$	0	0
$571$	−15365.2	−1.12612	−0.563060	−	0.826416i	$-0.690376\pi$
$571$	−15365.2	−1.12612	−0.563060	+	0.826416i	$0.690376\pi$
$572$	−2163.73	−0.158165
$573$	0	0
$574$	1456.57	0.105917
$575$	−48880.0	−3.54511
$576$	0	0
$577$	7978.65	0.575660	0.287830	−	0.957682i	$-0.407066\pi$
$577$	7978.65	0.575660	0.287830	+	0.957682i	$0.407066\pi$
$578$	12930.0	0.930481
$579$	0	0
$580$	−1697.77	−0.121545
$581$	−2169.90	−0.154944
$582$	0	0
$583$	16670.2	1.18424
$584$	10407.9	0.737466
$585$	0	0
$586$	26147.2	1.84322
$587$	−311.493	−0.0219024	−0.0109512	−	0.999940i	$-0.503486\pi$
$587$	−311.493	−0.0219024	−0.0109512	+	0.999940i	$0.503486\pi$
$588$	0	0
$589$	28567.1	1.99845
$590$	17096.9	1.19300
$591$	0	0
$592$	−8712.53	−0.604870
$593$	−15524.1	−1.07504	−0.537518	−	0.843252i	$-0.680638\pi$
$593$	−15524.1	−1.07504	−0.537518	+	0.843252i	$0.680638\pi$
$594$	0	0
$595$	−2671.24	−0.184050
$596$	571.147	0.0392535
$597$	0	0
$598$	−38718.7	−2.64770
$599$	−92.8924	−0.00633636	−0.00316818	−	0.999995i	$-0.501008\pi$
$599$	−92.8924	−0.00633636	−0.00316818	+	0.999995i	$0.501008\pi$
$600$	0	0
$601$	−13738.9	−0.932480	−0.466240	−	0.884658i	$-0.654392\pi$
$601$	−13738.9	−0.932480	−0.466240	+	0.884658i	$0.654392\pi$
$602$	3530.13	0.238999
$603$	0	0
$604$	388.371	0.0261632
$605$	45795.0	3.07741
$606$	0	0
$607$	−26296.5	−1.75839	−0.879195	−	0.476461i	$-0.841919\pi$
$607$	−26296.5	−1.75839	−0.879195	+	0.476461i	$0.841919\pi$
$608$	−2536.85	−0.169215
$609$	0	0
$610$	−20333.1	−1.34961
$611$	−40911.1	−2.70882
$612$	0	0
$613$	10996.8	0.724563	0.362281	−	0.932069i	$-0.381998\pi$
$613$	10996.8	0.724563	0.362281	+	0.932069i	$0.381998\pi$
$614$	−12603.6	−0.828406
$615$	0	0
$616$	13729.7	0.898027
$617$	−10951.9	−0.714599	−0.357300	−	0.933990i	$-0.616303\pi$
$617$	−10951.9	−0.714599	−0.357300	+	0.933990i	$0.616303\pi$
$618$	0	0
$619$	19494.8	1.26585	0.632926	−	0.774213i	$-0.281854\pi$
$619$	19494.8	1.26585	0.632926	+	0.774213i	$0.281854\pi$
$620$	−2307.24	−0.149453
$621$	0	0
$622$	−12626.1	−0.813927
$623$	15205.6	0.977847
$624$	0	0
$625$	18060.6	1.15588
$626$	−19571.8	−1.24959
$627$	0	0
$628$	830.611	0.0527787
$629$	2042.78	0.129493
$630$	0	0
$631$	2857.88	0.180302	0.0901509	−	0.995928i	$-0.471265\pi$
$631$	2857.88	0.180302	0.0901509	+	0.995928i	$0.471265\pi$
$632$	−14836.5	−0.933805
$633$	0	0
$634$	4507.82	0.282379
$635$	51249.1	3.20277
$636$	0	0
$637$	18382.7	1.14341
$638$	29994.9	1.86130
$639$	0	0
$640$	−25460.7	−1.57254
$641$	−2757.41	−0.169908	−0.0849541	−	0.996385i	$-0.527074\pi$
$641$	−2757.41	−0.169908	−0.0849541	+	0.996385i	$0.527074\pi$
$642$	0	0
$643$	25284.7	1.55075	0.775374	−	0.631502i	$-0.217561\pi$
$643$	25284.7	1.55075	0.775374	+	0.631502i	$0.217561\pi$
$644$	−894.450	−0.0547302
$645$	0	0
$646$	−4487.46	−0.273307
$647$	−21870.4	−1.32893	−0.664463	−	0.747322i	$-0.731340\pi$
$647$	−21870.4	−1.32893	−0.664463	+	0.747322i	$0.731340\pi$
$648$	0	0
$649$	19358.1	1.17084
$650$	52068.9	3.14202
$651$	0	0
$652$	−1769.82	−0.106306
$653$	−557.209	−0.0333924	−0.0166962	−	0.999861i	$-0.505315\pi$
$653$	−557.209	−0.0333924	−0.0166962	+	0.999861i	$0.505315\pi$
$654$	0	0
$655$	10095.4	0.602229
$656$	−3268.66	−0.194542
$657$	0	0
$658$	14746.8	0.873696
$659$	4019.92	0.237623	0.118812	−	0.992917i	$-0.462092\pi$
$659$	4019.92	0.237623	0.118812	+	0.992917i	$0.462092\pi$
$660$	0	0
$661$	4108.49	0.241758	0.120879	−	0.992667i	$-0.461429\pi$
$661$	4108.49	0.241758	0.120879	+	0.992667i	$0.461429\pi$
$662$	−14904.0	−0.875016
$663$	0	0
$664$	5182.69	0.302903
$665$	−22154.2	−1.29188
$666$	0	0
$667$	−34398.8	−1.99689
$668$	1895.63	0.109797
$669$	0	0
$670$	−14039.3	−0.809528
$671$	−23022.4	−1.32455
$672$	0	0
$673$	−16280.1	−0.932470	−0.466235	−	0.884661i	$-0.654390\pi$
$673$	−16280.1	−0.932470	−0.466235	+	0.884661i	$0.654390\pi$
$674$	5750.24	0.328621
$675$	0	0
$676$	−1584.73	−0.0901642
$677$	−19531.1	−1.10877	−0.554387	−	0.832259i	$-0.687047\pi$
$677$	−19531.1	−1.10877	−0.554387	+	0.832259i	$0.687047\pi$
$678$	0	0
$679$	9077.49	0.513052
$680$	6380.11	0.359803
$681$	0	0
$682$	40762.7	2.28869
$683$	−20173.7	−1.13020	−0.565099	−	0.825023i	$-0.691162\pi$
$683$	−20173.7	−1.13020	−0.565099	+	0.825023i	$0.691162\pi$
$684$	0	0
$685$	25905.1	1.44494
$686$	−15783.8	−0.878468
$687$	0	0
$688$	−7921.87	−0.438980
$689$	20365.0	1.12604
$690$	0	0
$691$	−3520.11	−0.193793	−0.0968967	−	0.995294i	$-0.530892\pi$
$691$	−3520.11	−0.193793	−0.0968967	+	0.995294i	$0.530892\pi$
$692$	1881.64	0.103366
$693$	0	0
$694$	21576.6	1.18017
$695$	8048.07	0.439253
$696$	0	0
$697$	766.384	0.0416483
$698$	23434.6	1.27079
$699$	0	0
$700$	1202.86	0.0649482
$701$	26094.5	1.40596	0.702979	−	0.711211i	$-0.251853\pi$
$701$	26094.5	1.40596	0.702979	+	0.711211i	$0.251853\pi$
$702$	0	0
$703$	16942.0	0.908935
$704$	−32680.0	−1.74954
$705$	0	0
$706$	−15659.4	−0.834773
$707$	254.133	0.0135186
$708$	0	0
$709$	−11558.3	−0.612247	−0.306123	−	0.951992i	$-0.599032\pi$
$709$	−11558.3	−0.612247	−0.306123	+	0.951992i	$0.599032\pi$
$710$	3608.09	0.190717
$711$	0	0
$712$	−36317.7	−1.91161
$713$	−46747.5	−2.45541
$714$	0	0
$715$	87699.3	4.58709
$716$	58.1723	0.00303632
$717$	0	0
$718$	21677.8	1.12675
$719$	897.836	0.0465697	0.0232849	−	0.999729i	$-0.492588\pi$
$719$	897.836	0.0465697	0.0232849	+	0.999729i	$0.492588\pi$
$720$	0	0
$721$	4767.01	0.246231
$722$	−18410.4	−0.948979
$723$	0	0
$724$	1603.40	0.0823064
$725$	46259.5	2.36970
$726$	0	0
$727$	−36780.3	−1.87635	−0.938175	−	0.346161i	$-0.887485\pi$
$727$	−36780.3	−1.87635	−0.938175	+	0.346161i	$0.887485\pi$
$728$	16772.7	0.853899
$729$	0	0
$730$	−23963.7	−1.21498
$731$	1857.40	0.0939786
$732$	0	0
$733$	9559.23	0.481689	0.240845	−	0.970564i	$-0.422576\pi$
$733$	9559.23	0.481689	0.240845	+	0.970564i	$0.422576\pi$
$734$	18113.8	0.910887
$735$	0	0
$736$	4151.34	0.207908
$737$	−15896.1	−0.794492
$738$	0	0
$739$	17170.1	0.854684	0.427342	−	0.904090i	$-0.359450\pi$
$739$	17170.1	0.854684	0.427342	+	0.904090i	$0.359450\pi$
$740$	−1368.34	−0.0679745
$741$	0	0
$742$	−7340.78	−0.363192
$743$	10115.0	0.499438	0.249719	−	0.968318i	$-0.419662\pi$
$743$	10115.0	0.499438	0.249719	+	0.968318i	$0.419662\pi$
$744$	0	0
$745$	−23149.4	−1.13843
$746$	−14610.4	−0.717056
$747$	0	0
$748$	410.370	0.0200596
$749$	654.833	0.0319453
$750$	0	0
$751$	416.195	0.0202226	0.0101113	−	0.999949i	$-0.496781\pi$
$751$	416.195	0.0202226	0.0101113	+	0.999949i	$0.496781\pi$
$752$	−33093.0	−1.60476
$753$	0	0
$754$	36643.0	1.76984
$755$	−15741.2	−0.758785
$756$	0	0
$757$	31208.1	1.49838	0.749191	−	0.662353i	$-0.230442\pi$
$757$	31208.1	1.49838	0.749191	+	0.662353i	$0.230442\pi$
$758$	36249.4	1.73699
$759$	0	0
$760$	52914.2	2.52553
$761$	−13575.8	−0.646681	−0.323340	−	0.946283i	$-0.604806\pi$
$761$	−13575.8	−0.646681	−0.323340	+	0.946283i	$0.604806\pi$
$762$	0	0
$763$	16367.3	0.776587
$764$	−282.303	−0.0133683
$765$	0	0
$766$	−19439.2	−0.916929
$767$	23648.6	1.11330
$768$	0	0
$769$	−10596.7	−0.496912	−0.248456	−	0.968643i	$-0.579923\pi$
$769$	−10596.7	−0.496912	−0.248456	+	0.968643i	$0.579923\pi$
$770$	−31612.1	−1.47951
$771$	0	0
$772$	1095.45	0.0510700
$773$	−12803.2	−0.595729	−0.297865	−	0.954608i	$-0.596274\pi$
$773$	−12803.2	−0.595729	−0.297865	+	0.954608i	$0.596274\pi$
$774$	0	0
$775$	62866.0	2.91382
$776$	−21681.1	−1.00297
$777$	0	0
$778$	3478.39	0.160291
$779$	6356.10	0.292338
$780$	0	0
$781$	4085.30	0.187175
$782$	7343.33	0.335802
$783$	0	0
$784$	14869.8	0.677378
$785$	−33665.9	−1.53069
$786$	0	0
$787$	35380.1	1.60250	0.801248	−	0.598333i	$-0.204170\pi$
$787$	35380.1	1.60250	0.801248	+	0.598333i	$0.204170\pi$
$788$	1116.02	0.0504524
$789$	0	0
$790$	34160.6	1.53845
$791$	17732.9	0.797105
$792$	0	0
$793$	−28125.1	−1.25946
$794$	18131.9	0.810426
$795$	0	0
$796$	−789.727	−0.0351647
$797$	−11785.9	−0.523811	−0.261905	−	0.965094i	$-0.584351\pi$
$797$	−11785.9	−0.523811	−0.261905	+	0.965094i	$0.584351\pi$
$798$	0	0
$799$	7759.14	0.343553
$800$	−5582.72	−0.246724
$801$	0	0
$802$	−12803.2	−0.563711
$803$	−27133.2	−1.19242
$804$	0	0
$805$	36253.4	1.58728
$806$	49797.3	2.17622
$807$	0	0
$808$	−606.983	−0.0264277
$809$	6641.65	0.288638	0.144319	−	0.989531i	$-0.453901\pi$
$809$	6641.65	0.288638	0.144319	+	0.989531i	$0.453901\pi$
$810$	0	0
$811$	10471.4	0.453390	0.226695	−	0.973966i	$-0.427208\pi$
$811$	10471.4	0.453390	0.226695	+	0.973966i	$0.427208\pi$
$812$	846.498	0.0365841
$813$	0	0
$814$	24174.8	1.04094
$815$	71733.3	3.08308
$816$	0	0
$817$	15404.6	0.659654
$818$	27312.3	1.16742
$819$	0	0
$820$	−513.356	−0.0218624
$821$	13524.8	0.574932	0.287466	−	0.957791i	$-0.407187\pi$
$821$	13524.8	0.574932	0.287466	+	0.957791i	$0.407187\pi$
$822$	0	0
$823$	−22843.6	−0.967532	−0.483766	−	0.875197i	$-0.660731\pi$
$823$	−22843.6	−0.967532	−0.483766	+	0.875197i	$0.660731\pi$
$824$	−11385.8	−0.481362
$825$	0	0
$826$	−8524.40	−0.359082
$827$	43072.5	1.81110	0.905548	−	0.424243i	$-0.139460\pi$
$827$	43072.5	1.81110	0.905548	+	0.424243i	$0.139460\pi$
$828$	0	0
$829$	−43785.0	−1.83440	−0.917200	−	0.398428i	$-0.869556\pi$
$829$	−43785.0	−1.83440	−0.917200	+	0.398428i	$0.869556\pi$
$830$	−11933.0	−0.499035
$831$	0	0
$832$	−39923.2	−1.66357
$833$	−3486.44	−0.145016
$834$	0	0
$835$	−76832.8	−3.18432
$836$	3403.45	0.140802
$837$	0	0
$838$	−2653.69	−0.109391
$839$	22354.7	0.919869	0.459934	−	0.887953i	$-0.347873\pi$
$839$	22354.7	0.919869	0.459934	+	0.887953i	$0.347873\pi$
$840$	0	0
$841$	8165.64	0.334809
$842$	−5676.01	−0.232314
$843$	0	0
$844$	2071.94	0.0845013
$845$	64231.3	2.61494
$846$	0	0
$847$	−22833.1	−0.926275
$848$	16473.2	0.667091
$849$	0	0
$850$	−9875.31	−0.398494
$851$	−27724.1	−1.11677
$852$	0	0
$853$	−8126.38	−0.326192	−0.163096	−	0.986610i	$-0.552148\pi$
$853$	−8126.38	−0.326192	−0.163096	+	0.986610i	$0.552148\pi$
$854$	10138.0	0.406223
$855$	0	0
$856$	−1564.03	−0.0624504
$857$	21577.6	0.860064	0.430032	−	0.902814i	$-0.358502\pi$
$857$	21577.6	0.860064	0.430032	+	0.902814i	$0.358502\pi$
$858$	0	0
$859$	−34511.1	−1.37079	−0.685393	−	0.728174i	$-0.740369\pi$
$859$	−34511.1	−1.37079	−0.685393	+	0.728174i	$0.740369\pi$
$860$	−1244.16	−0.0493321
$861$	0	0
$862$	19044.5	0.752503
$863$	−12851.5	−0.506919	−0.253460	−	0.967346i	$-0.581568\pi$
$863$	−12851.5	−0.506919	−0.253460	+	0.967346i	$0.581568\pi$
$864$	0	0
$865$	−76265.8	−2.99782
$866$	−29990.8	−1.17683
$867$	0	0
$868$	1150.38	0.0449843
$869$	38678.7	1.50988
$870$	0	0
$871$	−19419.3	−0.755451
$872$	−39092.5	−1.51816
$873$	0	0
$874$	60902.7	2.35706
$875$	−24984.0	−0.965271
$876$	0	0
$877$	25258.4	0.972538	0.486269	−	0.873809i	$-0.338358\pi$
$877$	25258.4	0.972538	0.486269	+	0.873809i	$0.338358\pi$
$878$	−21845.6	−0.839698
$879$	0	0
$880$	70939.9	2.71748
$881$	37545.8	1.43581	0.717905	−	0.696141i	$-0.245101\pi$
$881$	37545.8	1.43581	0.717905	+	0.696141i	$0.245101\pi$
$882$	0	0
$883$	−12829.4	−0.488952	−0.244476	−	0.969655i	$-0.578616\pi$
$883$	−12829.4	−0.488952	−0.244476	+	0.969655i	$0.578616\pi$
$884$	501.323	0.0190739
$885$	0	0
$886$	42071.0	1.59526
$887$	12124.6	0.458968	0.229484	−	0.973312i	$-0.426296\pi$
$887$	12124.6	0.458968	0.229484	+	0.973312i	$0.426296\pi$
$888$	0	0
$889$	−25552.5	−0.964008
$890$	83620.4	3.14939
$891$	0	0
$892$	−1308.26	−0.0491075
$893$	64351.3	2.41146
$894$	0	0
$895$	−2357.81	−0.0880591
$896$	12694.6	0.473321
$897$	0	0
$898$	−30074.1	−1.11758
$899$	44241.3	1.64130
$900$	0	0
$901$	−3862.39	−0.142813
$902$	9069.60	0.334795
$903$	0	0
$904$	−42354.2	−1.55827
$905$	−64988.2	−2.38705
$906$	0	0
$907$	20246.6	0.741208	0.370604	−	0.928791i	$-0.379151\pi$
$907$	20246.6	0.741208	0.370604	+	0.928791i	$0.379151\pi$
$908$	−1979.08	−0.0723326
$909$	0	0
$910$	−38618.6	−1.40681
$911$	−24698.5	−0.898241	−0.449120	−	0.893471i	$-0.648263\pi$
$911$	−24698.5	−0.898241	−0.449120	+	0.893471i	$0.648263\pi$
$912$	0	0
$913$	−13511.2	−0.489766
$914$	−2710.14	−0.0980783
$915$	0	0
$916$	249.173	0.00898790
$917$	−5033.51	−0.181266
$918$	0	0
$919$	38637.6	1.38688	0.693438	−	0.720517i	$-0.256095\pi$
$919$	38637.6	1.38688	0.693438	+	0.720517i	$0.256095\pi$
$920$	−86589.4	−3.10301
$921$	0	0
$922$	−37323.0	−1.33315
$923$	4990.76	0.177977
$924$	0	0
$925$	37283.5	1.32527
$926$	13888.7	0.492885
$927$	0	0
$928$	−3928.78	−0.138975
$929$	−27521.0	−0.971943	−0.485972	−	0.873975i	$-0.661534\pi$
$929$	−27521.0	−0.971943	−0.485972	+	0.873975i	$0.661534\pi$
$930$	0	0
$931$	−28915.2	−1.01789
$932$	3176.42	0.111639
$933$	0	0
$934$	10632.0	0.372472
$935$	−16632.9	−0.581769
$936$	0	0
$937$	22140.5	0.771931	0.385965	−	0.922513i	$-0.373868\pi$
$937$	22140.5	0.771931	0.385965	+	0.922513i	$0.373868\pi$
$938$	6999.89	0.243662
$939$	0	0
$940$	−5197.39	−0.180341
$941$	−30946.5	−1.07208	−0.536039	−	0.844193i	$-0.680080\pi$
$941$	−30946.5	−1.07208	−0.536039	+	0.844193i	$0.680080\pi$
$942$	0	0
$943$	−10401.2	−0.359183
$944$	19129.4	0.659543
$945$	0	0
$946$	21981.0	0.755457
$947$	−19451.6	−0.667467	−0.333733	−	0.942667i	$-0.608309\pi$
$947$	−19451.6	−0.667467	−0.333733	+	0.942667i	$0.608309\pi$
$948$	0	0
$949$	−33147.0	−1.13382
$950$	−81902.1	−2.79711
$951$	0	0
$952$	−3181.08	−0.108298
$953$	−883.526	−0.0300317	−0.0150159	−	0.999887i	$-0.504780\pi$
$953$	−883.526	−0.0300317	−0.0150159	+	0.999887i	$0.504780\pi$
$954$	0	0
$955$	11442.2	0.387707
$956$	115.156	0.00389584
$957$	0	0
$958$	−30502.1	−1.02868
$959$	−12916.1	−0.434915
$960$	0	0
$961$	30332.3	1.01817
$962$	29532.9	0.989791
$963$	0	0
$964$	1955.11	0.0653215
$965$	−44400.1	−1.48113
$966$	0	0
$967$	−20976.0	−0.697563	−0.348781	−	0.937204i	$-0.613404\pi$
$967$	−20976.0	−0.697563	−0.348781	+	0.937204i	$0.613404\pi$
$968$	54535.7	1.81079
$969$	0	0
$970$	49920.0	1.65241
$971$	−18857.7	−0.623245	−0.311623	−	0.950206i	$-0.600872\pi$
$971$	−18857.7	−0.623245	−0.311623	+	0.950206i	$0.600872\pi$
$972$	0	0
$973$	−4012.72	−0.132212
$974$	−39639.1	−1.30402
$975$	0	0
$976$	−22750.4	−0.746128
$977$	−16356.4	−0.535606	−0.267803	−	0.963474i	$-0.586298\pi$
$977$	−16356.4	−0.535606	−0.267803	+	0.963474i	$0.586298\pi$
$978$	0	0
$979$	94680.1	3.09090
$980$	2335.36	0.0761229
$981$	0	0
$982$	−2936.26	−0.0954172
$983$	342.193	0.0111030	0.00555151	−	0.999985i	$-0.498233\pi$
$983$	342.193	0.0111030	0.00555151	+	0.999985i	$0.498233\pi$
$984$	0	0
$985$	−45233.9	−1.46322
$986$	−6949.65	−0.224464
$987$	0	0
$988$	4157.78	0.133883
$989$	−25208.2	−0.810489
$990$	0	0
$991$	30023.8	0.962400	0.481200	−	0.876611i	$-0.340201\pi$
$991$	30023.8	0.962400	0.481200	+	0.876611i	$0.340201\pi$
$992$	−5339.16	−0.170885
$993$	0	0
$994$	−1798.97	−0.0574043
$995$	32008.8	1.01985
$996$	0	0
$997$	25541.5	0.811340	0.405670	−	0.914020i	$-0.367038\pi$
$997$	25541.5	0.811340	0.405670	+	0.914020i	$0.367038\pi$
$998$	−58.4304	−0.00185329
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.d.1.10		32
3.2	odd	2		717.4.a.d.1.23	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.d.1.23	✓	32	3.2	odd	2
2151.4.a.d.1.10		32	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-2.74193 q^{2} -0.481825 q^{4} +19.5291 q^{5} -9.73710 q^{7} +23.2566 q^{8} +O(q^{10})\)	\(q-2.74193 q^{2} -0.481825 q^{4} +19.5291 q^{5} -9.73710 q^{7} +23.2566 q^{8} -53.5474 q^{10} -60.6297 q^{11} -74.0676 q^{13} +26.6984 q^{14} -59.9132 q^{16} +14.0475 q^{17} +116.505 q^{19} -9.40962 q^{20} +166.242 q^{22} -190.650 q^{23} +256.386 q^{25} +203.088 q^{26} +4.69158 q^{28} +180.429 q^{29} +245.201 q^{31} -21.7747 q^{32} -38.5173 q^{34} -190.157 q^{35} +145.419 q^{37} -319.448 q^{38} +454.180 q^{40} +54.5565 q^{41} +132.222 q^{43} +29.2129 q^{44} +522.748 q^{46} +552.349 q^{47} -248.189 q^{49} -702.992 q^{50} +35.6876 q^{52} -274.951 q^{53} -1184.04 q^{55} -226.452 q^{56} -494.724 q^{58} -319.285 q^{59} +379.722 q^{61} -672.322 q^{62} +539.010 q^{64} -1446.47 q^{65} +262.184 q^{67} -6.76846 q^{68} +521.397 q^{70} -67.3812 q^{71} +447.523 q^{73} -398.729 q^{74} -56.1350 q^{76} +590.357 q^{77} -637.949 q^{79} -1170.05 q^{80} -149.590 q^{82} +222.848 q^{83} +274.336 q^{85} -362.544 q^{86} -1410.04 q^{88} -1561.61 q^{89} +721.203 q^{91} +91.8600 q^{92} -1514.50 q^{94} +2275.24 q^{95} -932.258 q^{97} +680.516 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8}+O(q^{10}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8	\( 32 q - 11 q^{2} + 147 q^{4} - 66 q^{5} + 58 q^{7} - 153 q^{8} + 52 q^{10} - 270 q^{11} + 48 q^{13} - 184 q^{14} + 775 q^{16} - 384 q^{17} + 216 q^{19} - 534 q^{20} + 437 q^{22} - 712 q^{23} + 1190 q^{25} - 436 q^{26} + 598 q^{28} - 562 q^{29} + 384 q^{31} - 1770 q^{32} + 452 q^{34} - 1026 q^{35} + 770 q^{37} - 733 q^{38} + 877 q^{40} - 1648 q^{41} + 1592 q^{43} - 1595 q^{44} + 532 q^{46} - 1540 q^{47} + 2134 q^{49} - 1646 q^{50} - 144 q^{52} - 1708 q^{53} + 1282 q^{55} - 2155 q^{56} + 1086 q^{58} - 2396 q^{59} + 364 q^{61} - 2180 q^{62} + 1663 q^{64} - 1520 q^{65} + 2728 q^{67} - 1545 q^{68} - 4609 q^{70} - 3322 q^{71} - 188 q^{73} - 1111 q^{74} - 3134 q^{76} - 556 q^{77} - 462 q^{79} - 6076 q^{80} - 7965 q^{82} - 4604 q^{83} - 852 q^{85} - 549 q^{86} - 1127 q^{88} - 6742 q^{89} + 1390 q^{91} - 1802 q^{92} - 2796 q^{94} - 448 q^{95} - 1322 q^{97} - 1000 q^{98}+O(q^{100}) \) 32 * q - 11 * q^2 + 147 * q^4 - 66 * q^5 + 58 * q^7 - 153 * q^8 + 52 * q^10 - 270 * q^11 + 48 * q^13 - 184 * q^14 + 775 * q^16 - 384 * q^17 + 216 * q^19 - 534 * q^20 + 437 * q^22 - 712 * q^23 + 1190 * q^25 - 436 * q^26 + 598 * q^28 - 562 * q^29 + 384 * q^31 - 1770 * q^32 + 452 * q^34 - 1026 * q^35 + 770 * q^37 - 733 * q^38 + 877 * q^40 - 1648 * q^41 + 1592 * q^43 - 1595 * q^44 + 532 * q^46 - 1540 * q^47 + 2134 * q^49 - 1646 * q^50 - 144 * q^52 - 1708 * q^53 + 1282 * q^55 - 2155 * q^56 + 1086 * q^58 - 2396 * q^59 + 364 * q^61 - 2180 * q^62 + 1663 * q^64 - 1520 * q^65 + 2728 * q^67 - 1545 * q^68 - 4609 * q^70 - 3322 * q^71 - 188 * q^73 - 1111 * q^74 - 3134 * q^76 - 556 * q^77 - 462 * q^79 - 6076 * q^80 - 7965 * q^82 - 4604 * q^83 - 852 * q^85 - 549 * q^86 - 1127 * q^88 - 6742 * q^89 + 1390 * q^91 - 1802 * q^92 - 2796 * q^94 - 448 * q^95 - 1322 * q^97 - 1000 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more