LMFDB - Embedded newform 2169.4.a.c.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2169,4,Mod(1,2169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2169, 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("2169.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2169 = 3^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2169.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:	$127.975142802$
Analytic rank:	$1$
Dimension:	$29$
Twist minimal:	no (minimal twist has level 723)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	2169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.93958 q^{2} +0.641136 q^{4} +15.8431 q^{5} -32.0159 q^{7} +21.6320 q^{8} +O(q^{10})$	\(q-2.93958 q^{2} +0.641136 q^{4} +15.8431 q^{5} -32.0159 q^{7} +21.6320 q^{8} -46.5721 q^{10} +50.5188 q^{11} +44.4539 q^{13} +94.1133 q^{14} -68.7180 q^{16} +94.4066 q^{17} -55.5515 q^{19} +10.1576 q^{20} -148.504 q^{22} -117.305 q^{23} +126.004 q^{25} -130.676 q^{26} -20.5265 q^{28} -121.214 q^{29} -101.115 q^{31} +28.9464 q^{32} -277.516 q^{34} -507.231 q^{35} -143.753 q^{37} +163.298 q^{38} +342.718 q^{40} +248.700 q^{41} -460.581 q^{43} +32.3894 q^{44} +344.829 q^{46} -122.365 q^{47} +682.017 q^{49} -370.399 q^{50} +28.5010 q^{52} +247.212 q^{53} +800.374 q^{55} -692.567 q^{56} +356.318 q^{58} -750.776 q^{59} -750.847 q^{61} +297.235 q^{62} +464.654 q^{64} +704.288 q^{65} -18.3196 q^{67} +60.5275 q^{68} +1491.05 q^{70} +795.363 q^{71} +607.301 q^{73} +422.573 q^{74} -35.6161 q^{76} -1617.40 q^{77} -617.074 q^{79} -1088.71 q^{80} -731.073 q^{82} -120.040 q^{83} +1495.69 q^{85} +1353.91 q^{86} +1092.82 q^{88} +410.157 q^{89} -1423.23 q^{91} -75.2087 q^{92} +359.702 q^{94} -880.108 q^{95} +727.521 q^{97} -2004.84 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 29 q - 9 q^{2} + 97 q^{4} - 62 q^{5} - 30 q^{7} - 108 q^{8}+O(q^{10}) $ 29 * q - 9 * q^2 + 97 * q^4 - 62 * q^5 - 30 * q^7 - 108 * q^8	$ 29 q - 9 q^{2} + 97 q^{4} - 62 q^{5} - 30 q^{7} - 108 q^{8} + 51 q^{10} - 46 q^{11} + 250 q^{13} - 84 q^{14} + 333 q^{16} - 128 q^{17} + 58 q^{19} - 405 q^{20} - 48 q^{22} - 232 q^{23} + 707 q^{25} - 238 q^{26} - 89 q^{28} - 590 q^{29} - 468 q^{31} - 1068 q^{32} + 287 q^{34} - 474 q^{35} + 842 q^{37} - 160 q^{38} + 434 q^{40} - 814 q^{41} + 20 q^{43} - 150 q^{44} - 37 q^{46} - 1004 q^{47} + 1239 q^{49} - 839 q^{50} + 1928 q^{52} - 2192 q^{53} + 432 q^{55} - 437 q^{56} - 28 q^{58} - 1288 q^{59} + 1502 q^{61} - 3059 q^{62} + 3372 q^{64} - 2312 q^{65} + 358 q^{67} - 4990 q^{68} + 5366 q^{70} - 1938 q^{71} + 3266 q^{73} - 2510 q^{74} + 3591 q^{76} - 5098 q^{77} - 292 q^{79} - 8235 q^{80} + 4511 q^{82} - 4256 q^{83} + 1998 q^{85} - 6860 q^{86} + 5935 q^{88} - 6428 q^{89} - 1650 q^{91} - 6823 q^{92} + 3025 q^{94} - 1802 q^{95} + 5040 q^{97} - 9410 q^{98}+O(q^{100}) $ 29 * q - 9 * q^2 + 97 * q^4 - 62 * q^5 - 30 * q^7 - 108 * q^8 + 51 * q^10 - 46 * q^11 + 250 * q^13 - 84 * q^14 + 333 * q^16 - 128 * q^17 + 58 * q^19 - 405 * q^20 - 48 * q^22 - 232 * q^23 + 707 * q^25 - 238 * q^26 - 89 * q^28 - 590 * q^29 - 468 * q^31 - 1068 * q^32 + 287 * q^34 - 474 * q^35 + 842 * q^37 - 160 * q^38 + 434 * q^40 - 814 * q^41 + 20 * q^43 - 150 * q^44 - 37 * q^46 - 1004 * q^47 + 1239 * q^49 - 839 * q^50 + 1928 * q^52 - 2192 * q^53 + 432 * q^55 - 437 * q^56 - 28 * q^58 - 1288 * q^59 + 1502 * q^61 - 3059 * q^62 + 3372 * q^64 - 2312 * q^65 + 358 * q^67 - 4990 * q^68 + 5366 * q^70 - 1938 * q^71 + 3266 * q^73 - 2510 * q^74 + 3591 * q^76 - 5098 * q^77 - 292 * q^79 - 8235 * q^80 + 4511 * q^82 - 4256 * q^83 + 1998 * q^85 - 6860 * q^86 + 5935 * q^88 - 6428 * q^89 - 1650 * q^91 - 6823 * q^92 + 3025 * q^94 - 1802 * q^95 + 5040 * q^97 - 9410 * 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 $

$2$	−2.93958	−1.03930	−0.519649	−	0.854380i	$-0.673937\pi$
$2$	−2.93958	−1.03930	−0.519649	+	0.854380i	$0.673937\pi$
$3$	0	0
$3$	0	0
$4$	0.641136	0.0801420
$5$	15.8431	1.41705	0.708525	−	0.705685i	$-0.249361\pi$
$5$	15.8431	1.41705	0.708525	+	0.705685i	$0.249361\pi$
$6$	0	0
$7$	−32.0159	−1.72870	−0.864348	−	0.502895i	$-0.832268\pi$
$7$	−32.0159	−1.72870	−0.864348	+	0.502895i	$0.832268\pi$
$8$	21.6320	0.956007
$9$	0	0
$10$	−46.5721	−1.47274
$11$	50.5188	1.38473	0.692363	−	0.721550i	$-0.256570\pi$
$11$	50.5188	1.38473	0.692363	+	0.721550i	$0.256570\pi$
$12$	0	0
$13$	44.4539	0.948408	0.474204	−	0.880415i	$-0.342736\pi$
$13$	44.4539	0.948408	0.474204	+	0.880415i	$0.342736\pi$
$14$	94.1133	1.79663
$15$	0	0
$16$	−68.7180	−1.07372
$17$	94.4066	1.34688	0.673440	−	0.739242i	$-0.264816\pi$
$17$	94.4066	1.34688	0.673440	+	0.739242i	$0.264816\pi$
$18$	0	0
$19$	−55.5515	−0.670757	−0.335379	−	0.942083i	$-0.608864\pi$
$19$	−55.5515	−0.670757	−0.335379	+	0.942083i	$0.608864\pi$
$20$	10.1576	0.113565
$21$	0	0
$22$	−148.504	−1.43914
$23$	−117.305	−1.06347	−0.531737	−	0.846910i	$-0.678460\pi$
$23$	−117.305	−1.06347	−0.531737	+	0.846910i	$0.678460\pi$
$24$	0	0
$25$	126.004	1.00803
$26$	−130.676	−0.985679
$27$	0	0
$28$	−20.5265	−0.138541
$29$	−121.214	−0.776167	−0.388084	−	0.921624i	$-0.626863\pi$
$29$	−121.214	−0.776167	−0.388084	+	0.921624i	$0.626863\pi$
$30$	0	0
$31$	−101.115	−0.585830	−0.292915	−	0.956138i	$-0.594625\pi$
$31$	−101.115	−0.585830	−0.292915	+	0.956138i	$0.594625\pi$
$32$	28.9464	0.159908
$33$	0	0
$34$	−277.516	−1.39981
$35$	−507.231	−2.44965
$36$	0	0
$37$	−143.753	−0.638725	−0.319362	−	0.947633i	$-0.603469\pi$
$37$	−143.753	−0.638725	−0.319362	+	0.947633i	$0.603469\pi$
$38$	163.298	0.697117
$39$	0	0
$40$	342.718	1.35471
$41$	248.700	0.947326	0.473663	−	0.880706i	$-0.342931\pi$
$41$	248.700	0.947326	0.473663	+	0.880706i	$0.342931\pi$
$42$	0	0
$43$	−460.581	−1.63344	−0.816720	−	0.577034i	$-0.804210\pi$
$43$	−460.581	−1.63344	−0.816720	+	0.577034i	$0.804210\pi$
$44$	32.3894	0.110975
$45$	0	0
$46$	344.829	1.10527
$47$	−122.365	−0.379761	−0.189881	−	0.981807i	$-0.560810\pi$
$47$	−122.365	−0.379761	−0.189881	+	0.981807i	$0.560810\pi$
$48$	0	0
$49$	682.017	1.98839
$50$	−370.399	−1.04765
$51$	0	0
$52$	28.5010	0.0760073
$53$	247.212	0.640703	0.320351	−	0.947299i	$-0.396199\pi$
$53$	247.212	0.640703	0.320351	+	0.947299i	$0.396199\pi$
$54$	0	0
$55$	800.374	1.96223
$56$	−692.567	−1.65265
$57$	0	0
$58$	356.318	0.806669
$59$	−750.776	−1.65666	−0.828328	−	0.560243i	$-0.810708\pi$
$59$	−750.776	−1.65666	−0.828328	+	0.560243i	$0.810708\pi$
$60$	0	0
$61$	−750.847	−1.57600	−0.788001	−	0.615674i	$-0.788884\pi$
$61$	−750.847	−1.57600	−0.788001	+	0.615674i	$0.788884\pi$
$62$	297.235	0.608853
$63$	0	0
$64$	464.654	0.907527
$65$	704.288	1.34394
$66$	0	0
$67$	−18.3196	−0.0334045	−0.0167022	−	0.999861i	$-0.505317\pi$
$67$	−18.3196	−0.0334045	−0.0167022	+	0.999861i	$0.505317\pi$
$68$	60.5275	0.107942
$69$	0	0
$70$	1491.05	2.54592
$71$	795.363	1.32947	0.664734	−	0.747080i	$-0.268545\pi$
$71$	795.363	1.32947	0.664734	+	0.747080i	$0.268545\pi$
$72$	0	0
$73$	607.301	0.973688	0.486844	−	0.873489i	$-0.338148\pi$
$73$	607.301	0.973688	0.486844	+	0.873489i	$0.338148\pi$
$74$	422.573	0.663826
$75$	0	0
$76$	−35.6161	−0.0537558
$77$	−1617.40	−2.39377
$78$	0	0
$79$	−617.074	−0.878814	−0.439407	−	0.898288i	$-0.644811\pi$
$79$	−617.074	−0.878814	−0.439407	+	0.898288i	$0.644811\pi$
$80$	−1088.71	−1.52151
$81$	0	0
$82$	−731.073	−0.984555
$83$	−120.040	−0.158749	−0.0793743	−	0.996845i	$-0.525292\pi$
$83$	−120.040	−0.158749	−0.0793743	+	0.996845i	$0.525292\pi$
$84$	0	0
$85$	1495.69	1.90860
$86$	1353.91	1.69763
$87$	0	0
$88$	1092.82	1.32381
$89$	410.157	0.488501	0.244250	−	0.969712i	$-0.421458\pi$
$89$	410.157	0.488501	0.244250	+	0.969712i	$0.421458\pi$
$90$	0	0
$91$	−1423.23	−1.63951
$92$	−75.2087	−0.0852288
$93$	0	0
$94$	359.702	0.394685
$95$	−880.108	−0.950497
$96$	0	0
$97$	727.521	0.761531	0.380766	−	0.924672i	$-0.375660\pi$
$97$	727.521	0.761531	0.380766	+	0.924672i	$0.375660\pi$
$98$	−2004.84	−2.06653
$99$	0	0
$100$	80.7857	0.0807857
$101$	699.405	0.689044	0.344522	−	0.938778i	$-0.388041\pi$
$101$	699.405	0.689044	0.344522	+	0.938778i	$0.388041\pi$
$102$	0	0
$103$	−1878.94	−1.79745	−0.898724	−	0.438514i	$-0.855505\pi$
$103$	−1878.94	−1.79745	−0.898724	+	0.438514i	$0.855505\pi$
$104$	961.626	0.906685
$105$	0	0
$106$	−726.701	−0.665881
$107$	1410.02	1.27394	0.636969	−	0.770890i	$-0.280188\pi$
$107$	1410.02	1.27394	0.636969	+	0.770890i	$0.280188\pi$
$108$	0	0
$109$	−462.533	−0.406447	−0.203223	−	0.979132i	$-0.565142\pi$
$109$	−462.533	−0.406447	−0.203223	+	0.979132i	$0.565142\pi$
$110$	−2352.76	−2.03934
$111$	0	0
$112$	2200.07	1.85613
$113$	−1727.77	−1.43836	−0.719181	−	0.694823i	$-0.755483\pi$
$113$	−1727.77	−1.43836	−0.719181	+	0.694823i	$0.755483\pi$
$114$	0	0
$115$	−1858.48	−1.50700
$116$	−77.7145	−0.0622036
$117$	0	0
$118$	2206.97	1.72176
$119$	−3022.51	−2.32835
$120$	0	0
$121$	1221.15	0.917464
$122$	2207.18	1.63794
$123$	0	0
$124$	−64.8283	−0.0469496
$125$	15.9076	0.0113825
$126$	0	0
$127$	−2479.42	−1.73239	−0.866194	−	0.499709i	$-0.833440\pi$
$127$	−2479.42	−1.73239	−0.866194	+	0.499709i	$0.833440\pi$
$128$	−1597.46	−1.10310
$129$	0	0
$130$	−2070.31	−1.39676
$131$	479.694	0.319932	0.159966	−	0.987123i	$-0.448862\pi$
$131$	479.694	0.319932	0.159966	+	0.987123i	$0.448862\pi$
$132$	0	0
$133$	1778.53	1.15953
$134$	53.8521	0.0347172
$135$	0	0
$136$	2042.20	1.28763
$137$	1846.95	1.15179	0.575897	−	0.817522i	$-0.304653\pi$
$137$	1846.95	1.15179	0.575897	+	0.817522i	$0.304653\pi$
$138$	0	0
$139$	513.670	0.313446	0.156723	−	0.987643i	$-0.449907\pi$
$139$	513.670	0.313446	0.156723	+	0.987643i	$0.449907\pi$
$140$	−325.204	−0.196320
$141$	0	0
$142$	−2338.03	−1.38171
$143$	2245.76	1.31328
$144$	0	0
$145$	−1920.40	−1.09987
$146$	−1785.21	−1.01195
$147$	0	0
$148$	−92.1651	−0.0511887
$149$	−3015.29	−1.65787	−0.828934	−	0.559346i	$-0.811052\pi$
$149$	−3015.29	−1.65787	−0.828934	+	0.559346i	$0.811052\pi$
$150$	0	0
$151$	−409.048	−0.220449	−0.110225	−	0.993907i	$-0.535157\pi$
$151$	−409.048	−0.220449	−0.110225	+	0.993907i	$0.535157\pi$
$152$	−1201.69	−0.641249
$153$	0	0
$154$	4754.49	2.48784
$155$	−1601.97	−0.830151
$156$	0	0
$157$	−1116.90	−0.567762	−0.283881	−	0.958860i	$-0.591622\pi$
$157$	−1116.90	−0.567762	−0.283881	+	0.958860i	$0.591622\pi$
$158$	1813.94	0.913350
$159$	0	0
$160$	458.601	0.226597
$161$	3755.64	1.83842
$162$	0	0
$163$	3035.74	1.45876	0.729378	−	0.684111i	$-0.239810\pi$
$163$	3035.74	1.45876	0.729378	+	0.684111i	$0.239810\pi$
$164$	159.450	0.0759206
$165$	0	0
$166$	352.868	0.164987
$167$	1285.18	0.595511	0.297756	−	0.954642i	$-0.403762\pi$
$167$	1285.18	0.595511	0.297756	+	0.954642i	$0.403762\pi$
$168$	0	0
$169$	−220.848	−0.100522
$170$	−4396.71	−1.98360
$171$	0	0
$172$	−295.295	−0.130907
$173$	−1257.83	−0.552780	−0.276390	−	0.961046i	$-0.589138\pi$
$173$	−1257.83	−0.552780	−0.276390	+	0.961046i	$0.589138\pi$
$174$	0	0
$175$	−4034.13	−1.74258
$176$	−3471.55	−1.48681
$177$	0	0
$178$	−1205.69	−0.507698
$179$	−385.265	−0.160872	−0.0804359	−	0.996760i	$-0.525631\pi$
$179$	−385.265	−0.160872	−0.0804359	+	0.996760i	$0.525631\pi$
$180$	0	0
$181$	3328.45	1.36686	0.683431	−	0.730015i	$-0.260487\pi$
$181$	3328.45	1.36686	0.683431	+	0.730015i	$0.260487\pi$
$182$	4183.71	1.70394
$183$	0	0
$184$	−2537.55	−1.01669
$185$	−2277.49	−0.905105
$186$	0	0
$187$	4769.31	1.86506
$188$	−78.4526	−0.0304348
$189$	0	0
$190$	2587.15	0.987850
$191$	−3433.83	−1.30086	−0.650428	−	0.759568i	$-0.725410\pi$
$191$	−3433.83	−1.30086	−0.650428	+	0.759568i	$0.725410\pi$
$192$	0	0
$193$	−781.837	−0.291595	−0.145798	−	0.989314i	$-0.546575\pi$
$193$	−781.837	−0.291595	−0.145798	+	0.989314i	$0.546575\pi$
$194$	−2138.61	−0.791459
$195$	0	0
$196$	437.265	0.159353
$197$	3565.81	1.28961	0.644805	−	0.764347i	$-0.276938\pi$
$197$	3565.81	1.28961	0.644805	+	0.764347i	$0.276938\pi$
$198$	0	0
$199$	−1881.95	−0.670391	−0.335195	−	0.942149i	$-0.608802\pi$
$199$	−1881.95	−0.670391	−0.335195	+	0.942149i	$0.608802\pi$
$200$	2725.72	0.963687
$201$	0	0
$202$	−2055.96	−0.716122
$203$	3880.77	1.34176
$204$	0	0
$205$	3940.18	1.34241
$206$	5523.29	1.86809
$207$	0	0
$208$	−3054.79	−1.01832
$209$	−2806.39	−0.928815
$210$	0	0
$211$	−1705.06	−0.556310	−0.278155	−	0.960536i	$-0.589723\pi$
$211$	−1705.06	−0.556310	−0.278155	+	0.960536i	$0.589723\pi$
$212$	158.497	0.0513472
$213$	0	0
$214$	−4144.85	−1.32400
$215$	−7297.03	−2.31467
$216$	0	0
$217$	3237.28	1.01272
$218$	1359.65	0.422419
$219$	0	0
$220$	513.149	0.157257
$221$	4196.75	1.27739
$222$	0	0
$223$	3717.54	1.11634	0.558172	−	0.829725i	$-0.311503\pi$
$223$	3717.54	1.11634	0.558172	+	0.829725i	$0.311503\pi$
$224$	−926.745	−0.276432
$225$	0	0
$226$	5078.92	1.49489
$227$	3881.15	1.13481	0.567403	−	0.823440i	$-0.307948\pi$
$227$	3881.15	1.13481	0.567403	+	0.823440i	$0.307948\pi$
$228$	0	0
$229$	4899.77	1.41391	0.706956	−	0.707258i	$-0.250068\pi$
$229$	4899.77	1.41391	0.706956	+	0.707258i	$0.250068\pi$
$230$	5463.16	1.56622
$231$	0	0
$232$	−2622.09	−0.742021
$233$	1929.09	0.542400	0.271200	−	0.962523i	$-0.412580\pi$
$233$	1929.09	0.542400	0.271200	+	0.962523i	$0.412580\pi$
$234$	0	0
$235$	−1938.64	−0.538141
$236$	−481.350	−0.132768
$237$	0	0
$238$	8884.92	2.41985
$239$	4618.54	1.24999	0.624997	−	0.780628i	$-0.285100\pi$
$239$	4618.54	1.24999	0.624997	+	0.780628i	$0.285100\pi$
$240$	0	0
$241$	−241.000	−0.0644157
$241$	−241.000	−0.0644157
$242$	−3589.65	−0.953520
$243$	0	0
$244$	−481.395	−0.126304
$245$	10805.3	2.81764
$246$	0	0
$247$	−2469.48	−0.636152
$248$	−2187.31	−0.560058
$249$	0	0
$250$	−46.7616	−0.0118299
$251$	−458.737	−0.115359	−0.0576797	−	0.998335i	$-0.518370\pi$
$251$	−458.737	−0.115359	−0.0576797	+	0.998335i	$0.518370\pi$
$252$	0	0
$253$	−5926.13	−1.47262
$254$	7288.46	1.80047
$255$	0	0
$256$	978.629	0.238923
$257$	−3096.58	−0.751593	−0.375796	−	0.926702i	$-0.622631\pi$
$257$	−3096.58	−0.751593	−0.375796	+	0.926702i	$0.622631\pi$
$258$	0	0
$259$	4602.37	1.10416
$260$	451.545	0.107706
$261$	0	0
$262$	−1410.10	−0.332505
$263$	−6137.10	−1.43890	−0.719449	−	0.694545i	$-0.755605\pi$
$263$	−6137.10	−1.43890	−0.719449	+	0.694545i	$0.755605\pi$
$264$	0	0
$265$	3916.61	0.907908
$266$	−5228.13	−1.20510
$267$	0	0
$268$	−11.7454	−0.00267710
$269$	3133.97	0.710339	0.355170	−	0.934802i	$-0.384423\pi$
$269$	3133.97	0.710339	0.355170	+	0.934802i	$0.384423\pi$
$270$	0	0
$271$	−633.329	−0.141963	−0.0709816	−	0.997478i	$-0.522613\pi$
$271$	−633.329	−0.141963	−0.0709816	+	0.997478i	$0.522613\pi$
$272$	−6487.44	−1.44617
$273$	0	0
$274$	−5429.27	−1.19706
$275$	6365.57	1.39585
$276$	0	0
$277$	−6365.11	−1.38066	−0.690329	−	0.723496i	$-0.742534\pi$
$277$	−6365.11	−1.38066	−0.690329	+	0.723496i	$0.742534\pi$
$278$	−1509.98	−0.325764
$279$	0	0
$280$	−10972.4	−2.34188
$281$	3025.76	0.642355	0.321178	−	0.947019i	$-0.395921\pi$
$281$	3025.76	0.642355	0.321178	+	0.947019i	$0.395921\pi$
$282$	0	0
$283$	−8571.59	−1.80045	−0.900226	−	0.435423i	$-0.856599\pi$
$283$	−8571.59	−1.80045	−0.900226	+	0.435423i	$0.856599\pi$
$284$	509.936	0.106546
$285$	0	0
$286$	−6601.59	−1.36489
$287$	−7962.34	−1.63764
$288$	0	0
$289$	3999.61	0.814088
$290$	5645.18	1.14309
$291$	0	0
$292$	389.362	0.0780333
$293$	−3370.86	−0.672108	−0.336054	−	0.941843i	$-0.609092\pi$
$293$	−3370.86	−0.672108	−0.336054	+	0.941843i	$0.609092\pi$
$294$	0	0
$295$	−11894.6	−2.34757
$296$	−3109.66	−0.610625
$297$	0	0
$298$	8863.69	1.72302
$299$	−5214.69	−1.00861
$300$	0	0
$301$	14745.9	2.82372
$302$	1202.43	0.229113
$303$	0	0
$304$	3817.39	0.720205
$305$	−11895.7	−2.23327
$306$	0	0
$307$	3562.21	0.662235	0.331117	−	0.943590i	$-0.392574\pi$
$307$	3562.21	0.662235	0.331117	+	0.943590i	$0.392574\pi$
$308$	−1036.97	−0.191841
$309$	0	0
$310$	4709.12	0.862775
$311$	1555.58	0.283630	0.141815	−	0.989893i	$-0.454706\pi$
$311$	1555.58	0.283630	0.141815	+	0.989893i	$0.454706\pi$
$312$	0	0
$313$	−9941.44	−1.79528	−0.897641	−	0.440728i	$-0.854720\pi$
$313$	−9941.44	−1.79528	−0.897641	+	0.440728i	$0.854720\pi$
$314$	3283.23	0.590074
$315$	0	0
$316$	−395.629	−0.0704299
$317$	−6046.39	−1.07129	−0.535645	−	0.844443i	$-0.679932\pi$
$317$	−6046.39	−1.07129	−0.535645	+	0.844443i	$0.679932\pi$
$318$	0	0
$319$	−6123.57	−1.07478
$320$	7361.56	1.28601
$321$	0	0
$322$	−11040.0	−1.91067
$323$	−5244.43	−0.903430
$324$	0	0
$325$	5601.38	0.956026
$326$	−8923.79	−1.51608
$327$	0	0
$328$	5379.87	0.905651
$329$	3917.62	0.656491
$330$	0	0
$331$	−5429.80	−0.901658	−0.450829	−	0.892610i	$-0.648872\pi$
$331$	−5429.80	−0.901658	−0.450829	+	0.892610i	$0.648872\pi$
$332$	−76.9621	−0.0127224
$333$	0	0
$334$	−3777.90	−0.618914
$335$	−290.240	−0.0473358
$336$	0	0
$337$	7048.47	1.13933	0.569666	−	0.821877i	$-0.307073\pi$
$337$	7048.47	1.13933	0.569666	+	0.821877i	$0.307073\pi$
$338$	649.200	0.104473
$339$	0	0
$340$	958.943	0.152959
$341$	−5108.19	−0.811214
$342$	0	0
$343$	−10853.9	−1.70862
$344$	−9963.28	−1.56158
$345$	0	0
$346$	3697.49	0.574504
$347$	−6201.31	−0.959377	−0.479689	−	0.877439i	$-0.659250\pi$
$347$	−6201.31	−0.959377	−0.479689	+	0.877439i	$0.659250\pi$
$348$	0	0
$349$	−3684.93	−0.565186	−0.282593	−	0.959240i	$-0.591195\pi$
$349$	−3684.93	−0.565186	−0.282593	+	0.959240i	$0.591195\pi$
$350$	11858.7	1.81106
$351$	0	0
$352$	1462.34	0.221428
$353$	−10018.5	−1.51056	−0.755281	−	0.655401i	$-0.772500\pi$
$353$	−10018.5	−1.51056	−0.755281	+	0.655401i	$0.772500\pi$
$354$	0	0
$355$	12601.0	1.88392
$356$	262.966	0.0391494
$357$	0	0
$358$	1132.52	0.167194
$359$	−4672.15	−0.686871	−0.343435	−	0.939176i	$-0.611591\pi$
$359$	−4672.15	−0.686871	−0.343435	+	0.939176i	$0.611591\pi$
$360$	0	0
$361$	−3773.03	−0.550085
$362$	−9784.25	−1.42058
$363$	0	0
$364$	−912.485	−0.131393
$365$	9621.54	1.37976
$366$	0	0
$367$	−886.106	−0.126034	−0.0630168	−	0.998012i	$-0.520072\pi$
$367$	−886.106	−0.126034	−0.0630168	+	0.998012i	$0.520072\pi$
$368$	8061.00	1.14187
$369$	0	0
$370$	6694.87	0.940675
$371$	−7914.72	−1.10758
$372$	0	0
$373$	−11654.3	−1.61780	−0.808898	−	0.587949i	$-0.799936\pi$
$373$	−11654.3	−1.61780	−0.808898	+	0.587949i	$0.799936\pi$
$374$	−14019.8	−1.93835
$375$	0	0
$376$	−2647.00	−0.363054
$377$	−5388.43	−0.736123
$378$	0	0
$379$	4034.28	0.546773	0.273386	−	0.961904i	$-0.411856\pi$
$379$	4034.28	0.546773	0.273386	+	0.961904i	$0.411856\pi$
$380$	−564.269	−0.0761747
$381$	0	0
$382$	10094.0	1.35198
$383$	1920.05	0.256162	0.128081	−	0.991764i	$-0.459118\pi$
$383$	1920.05	0.256162	0.128081	+	0.991764i	$0.459118\pi$
$384$	0	0
$385$	−25624.7	−3.39209
$386$	2298.27	0.303054
$387$	0	0
$388$	466.440	0.0610306
$389$	−1982.43	−0.258388	−0.129194	−	0.991619i	$-0.541239\pi$
$389$	−1982.43	−0.258388	−0.129194	+	0.991619i	$0.541239\pi$
$390$	0	0
$391$	−11074.4	−1.43237
$392$	14753.4	1.90091
$393$	0	0
$394$	−10482.0	−1.34029
$395$	−9776.38	−1.24532
$396$	0	0
$397$	974.756	0.123228	0.0616141	−	0.998100i	$-0.480375\pi$
$397$	974.756	0.123228	0.0616141	+	0.998100i	$0.480375\pi$
$398$	5532.14	0.696736
$399$	0	0
$400$	−8658.75	−1.08234
$401$	−12750.8	−1.58789	−0.793946	−	0.607988i	$-0.791977\pi$
$401$	−12750.8	−1.58789	−0.793946	+	0.607988i	$0.791977\pi$
$402$	0	0
$403$	−4494.95	−0.555606
$404$	448.414	0.0552213
$405$	0	0
$406$	−11407.8	−1.39449
$407$	−7262.21	−0.884458
$408$	0	0
$409$	−13813.5	−1.67001	−0.835006	−	0.550241i	$-0.814536\pi$
$409$	−13813.5	−1.67001	−0.835006	+	0.550241i	$0.814536\pi$
$410$	−11582.5	−1.39516
$411$	0	0
$412$	−1204.65	−0.144051
$413$	24036.8	2.86385
$414$	0	0
$415$	−1901.81	−0.224955
$416$	1286.78	0.151658
$417$	0	0
$418$	8249.62	0.965316
$419$	7104.49	0.828346	0.414173	−	0.910198i	$-0.364071\pi$
$419$	7104.49	0.828346	0.414173	+	0.910198i	$0.364071\pi$
$420$	0	0
$421$	−4869.83	−0.563756	−0.281878	−	0.959450i	$-0.590957\pi$
$421$	−4869.83	−0.563756	−0.281878	+	0.959450i	$0.590957\pi$
$422$	5012.17	0.578173
$423$	0	0
$424$	5347.69	0.612516
$425$	11895.6	1.35770
$426$	0	0
$427$	24039.0	2.72443
$428$	904.011	0.102096
$429$	0	0
$430$	21450.2	2.40563
$431$	7056.78	0.788661	0.394331	−	0.918969i	$-0.370976\pi$
$431$	7056.78	0.788661	0.394331	+	0.918969i	$0.370976\pi$
$432$	0	0
$433$	−48.6144	−0.00539552	−0.00269776	−	0.999996i	$-0.500859\pi$
$433$	−48.6144	−0.00539552	−0.00269776	+	0.999996i	$0.500859\pi$
$434$	−9516.24	−1.05252
$435$	0	0
$436$	−296.547	−0.0325734
$437$	6516.50	0.713332
$438$	0	0
$439$	−9394.24	−1.02133	−0.510663	−	0.859781i	$-0.670600\pi$
$439$	−9394.24	−1.02133	−0.510663	+	0.859781i	$0.670600\pi$
$440$	17313.7	1.87590
$441$	0	0
$442$	−12336.7	−1.32759
$443$	12659.2	1.35769	0.678844	−	0.734283i	$-0.262481\pi$
$443$	12659.2	1.35769	0.678844	+	0.734283i	$0.262481\pi$
$444$	0	0
$445$	6498.16	0.692230
$446$	−10928.0	−1.16021
$447$	0	0
$448$	−14876.3	−1.56884
$449$	−2727.17	−0.286644	−0.143322	−	0.989676i	$-0.545778\pi$
$449$	−2727.17	−0.286644	−0.143322	+	0.989676i	$0.545778\pi$
$450$	0	0
$451$	12564.0	1.31179
$452$	−1107.73	−0.115273
$453$	0	0
$454$	−11408.9	−1.17940
$455$	−22548.4	−2.32327
$456$	0	0
$457$	13163.9	1.34744	0.673721	−	0.738985i	$-0.264695\pi$
$457$	13163.9	1.34744	0.673721	+	0.738985i	$0.264695\pi$
$458$	−14403.3	−1.46948
$459$	0	0
$460$	−1191.54	−0.120774
$461$	−18454.7	−1.86448	−0.932238	−	0.361847i	$-0.882146\pi$
$461$	−18454.7	−1.86448	−0.932238	+	0.361847i	$0.882146\pi$
$462$	0	0
$463$	13773.5	1.38253	0.691263	−	0.722604i	$-0.257055\pi$
$463$	13773.5	1.38253	0.691263	+	0.722604i	$0.257055\pi$
$464$	8329.58	0.833386
$465$	0	0
$466$	−5670.73	−0.563715
$467$	−15544.4	−1.54028	−0.770138	−	0.637877i	$-0.779813\pi$
$467$	−15544.4	−1.54028	−0.770138	+	0.637877i	$0.779813\pi$
$468$	0	0
$469$	586.519	0.0577462
$470$	5698.79	0.559289
$471$	0	0
$472$	−16240.8	−1.58378
$473$	−23268.0	−2.26187
$474$	0	0
$475$	−6999.72	−0.676145
$476$	−1937.84	−0.186598
$477$	0	0
$478$	−13576.6	−1.29912
$479$	100.784	0.00961363	0.00480681	−	0.999988i	$-0.498470\pi$
$479$	100.784	0.00961363	0.00480681	+	0.999988i	$0.498470\pi$
$480$	0	0
$481$	−6390.38	−0.605771
$482$	708.439	0.0669471
$483$	0	0
$484$	782.920	0.0735274
$485$	11526.2	1.07913
$486$	0	0
$487$	−2974.57	−0.276777	−0.138389	−	0.990378i	$-0.544192\pi$
$487$	−2974.57	−0.276777	−0.138389	+	0.990378i	$0.544192\pi$
$488$	−16242.3	−1.50667
$489$	0	0
$490$	−31762.9	−2.92837
$491$	12934.4	1.18884	0.594422	−	0.804153i	$-0.297381\pi$
$491$	12934.4	1.18884	0.594422	+	0.804153i	$0.297381\pi$
$492$	0	0
$493$	−11443.4	−1.04540
$494$	7259.24	0.661152
$495$	0	0
$496$	6948.41	0.629017
$497$	−25464.2	−2.29824
$498$	0	0
$499$	−13183.8	−1.18274	−0.591370	−	0.806401i	$-0.701413\pi$
$499$	−13183.8	−1.18274	−0.591370	+	0.806401i	$0.701413\pi$
$500$	10.1989	0.000912219 0
$501$	0	0
$502$	1348.49	0.119893
$503$	7036.87	0.623774	0.311887	−	0.950119i	$-0.399039\pi$
$503$	7036.87	0.623774	0.311887	+	0.950119i	$0.399039\pi$
$504$	0	0
$505$	11080.7	0.976410
$506$	17420.3	1.53049
$507$	0	0
$508$	−1589.65	−0.138837
$509$	−6501.80	−0.566183	−0.283092	−	0.959093i	$-0.591360\pi$
$509$	−6501.80	−0.566183	−0.283092	+	0.959093i	$0.591360\pi$
$510$	0	0
$511$	−19443.3	−1.68321
$512$	9902.92	0.854787
$513$	0	0
$514$	9102.65	0.781130
$515$	−29768.2	−2.54708
$516$	0	0
$517$	−6181.73	−0.525865
$518$	−13529.0	−1.14755
$519$	0	0
$520$	15235.2	1.28482
$521$	10509.8	0.883770	0.441885	−	0.897072i	$-0.354310\pi$
$521$	10509.8	0.883770	0.441885	+	0.897072i	$0.354310\pi$
$522$	0	0
$523$	7760.96	0.648878	0.324439	−	0.945907i	$-0.394824\pi$
$523$	7760.96	0.648878	0.324439	+	0.945907i	$0.394824\pi$
$524$	307.549	0.0256400
$525$	0	0
$526$	18040.5	1.49544
$527$	−9545.90	−0.789044
$528$	0	0
$529$	1593.58	0.130975
$530$	−11513.2	−0.943588
$531$	0	0
$532$	1140.28	0.0929274
$533$	11055.7	0.898452
$534$	0	0
$535$	22339.0	1.80523
$536$	−396.290	−0.0319349
$537$	0	0
$538$	−9212.55	−0.738255
$539$	34454.6	2.75337
$540$	0	0
$541$	4999.95	0.397347	0.198674	−	0.980066i	$-0.436337\pi$
$541$	4999.95	0.397347	0.198674	+	0.980066i	$0.436337\pi$
$542$	1861.72	0.147542
$543$	0	0
$544$	2732.73	0.215377
$545$	−7327.97	−0.575955
$546$	0	0
$547$	21313.4	1.66599	0.832994	−	0.553282i	$-0.186625\pi$
$547$	21313.4	1.66599	0.832994	+	0.553282i	$0.186625\pi$
$548$	1184.15	0.0923070
$549$	0	0
$550$	−18712.1	−1.45070
$551$	6733.61	0.520620
$552$	0	0
$553$	19756.2	1.51920
$554$	18710.7	1.43492
$555$	0	0
$556$	329.332	0.0251202
$557$	883.006	0.0671708	0.0335854	−	0.999436i	$-0.489307\pi$
$557$	883.006	0.0671708	0.0335854	+	0.999436i	$0.489307\pi$
$558$	0	0
$559$	−20474.6	−1.54917
$560$	34855.9	2.63023
$561$	0	0
$562$	−8894.47	−0.667599
$563$	8321.39	0.622921	0.311461	−	0.950259i	$-0.399182\pi$
$563$	8321.39	0.622921	0.311461	+	0.950259i	$0.399182\pi$
$564$	0	0
$565$	−27373.2	−2.03823
$566$	25196.9	1.87121
$567$	0	0
$568$	17205.3	1.27098
$569$	−22827.9	−1.68189	−0.840946	−	0.541118i	$-0.818001\pi$
$569$	−22827.9	−1.68189	−0.840946	+	0.541118i	$0.818001\pi$
$570$	0	0
$571$	−14356.5	−1.05219	−0.526095	−	0.850426i	$-0.676344\pi$
$571$	−14356.5	−1.05219	−0.526095	+	0.850426i	$0.676344\pi$
$572$	1439.84	0.105249
$573$	0	0
$574$	23405.9	1.70200
$575$	−14781.0	−1.07202
$576$	0	0
$577$	−3183.15	−0.229665	−0.114832	−	0.993385i	$-0.536633\pi$
$577$	−3183.15	−0.229665	−0.114832	+	0.993385i	$0.536633\pi$
$578$	−11757.2	−0.846080
$579$	0	0
$580$	−1231.24	−0.0881456
$581$	3843.19	0.274428
$582$	0	0
$583$	12488.9	0.887197
$584$	13137.1	0.930853
$585$	0	0
$586$	9908.91	0.698521
$587$	13687.6	0.962432	0.481216	−	0.876602i	$-0.340195\pi$
$587$	13687.6	0.962432	0.481216	+	0.876602i	$0.340195\pi$
$588$	0	0
$589$	5617.08	0.392950
$590$	34965.2	2.43982
$591$	0	0
$592$	9878.41	0.685811
$593$	−19299.7	−1.33650	−0.668248	−	0.743938i	$-0.732956\pi$
$593$	−19299.7	−1.33650	−0.668248	+	0.743938i	$0.732956\pi$
$594$	0	0
$595$	−47886.0	−3.29938
$596$	−1933.21	−0.132865
$597$	0	0
$598$	15329.0	1.04824
$599$	−14474.0	−0.987301	−0.493651	−	0.869660i	$-0.664338\pi$
$599$	−14474.0	−0.987301	−0.493651	+	0.869660i	$0.664338\pi$
$600$	0	0
$601$	9664.73	0.655961	0.327981	−	0.944684i	$-0.393632\pi$
$601$	9664.73	0.655961	0.327981	+	0.944684i	$0.393632\pi$
$602$	−43346.8	−2.93469
$603$	0	0
$604$	−262.255	−0.0176672
$605$	19346.7	1.30009
$606$	0	0
$607$	−404.394	−0.0270409	−0.0135205	−	0.999909i	$-0.504304\pi$
$607$	−404.394	−0.0270409	−0.0135205	+	0.999909i	$0.504304\pi$
$608$	−1608.02	−0.107259
$609$	0	0
$610$	34968.5	2.32104
$611$	−5439.61	−0.360168
$612$	0	0
$613$	−582.269	−0.0383648	−0.0191824	−	0.999816i	$-0.506106\pi$
$613$	−582.269	−0.0383648	−0.0191824	+	0.999816i	$0.506106\pi$
$614$	−10471.4	−0.688260
$615$	0	0
$616$	−34987.6	−2.28846
$617$	23691.9	1.54587	0.772933	−	0.634488i	$-0.218789\pi$
$617$	23691.9	1.54587	0.772933	+	0.634488i	$0.218789\pi$
$618$	0	0
$619$	−18571.4	−1.20589	−0.602946	−	0.797782i	$-0.706006\pi$
$619$	−18571.4	−1.20589	−0.602946	+	0.797782i	$0.706006\pi$
$620$	−1027.08	−0.0665300
$621$	0	0
$622$	−4572.75	−0.294776
$623$	−13131.5	−0.844469
$624$	0	0
$625$	−15498.5	−0.991903
$626$	29223.7	1.86583
$627$	0	0
$628$	−716.087	−0.0455015
$629$	−13571.2	−0.860286
$630$	0	0
$631$	−6866.44	−0.433199	−0.216600	−	0.976260i	$-0.569497\pi$
$631$	−6866.44	−0.433199	−0.216600	+	0.976260i	$0.569497\pi$
$632$	−13348.5	−0.840153
$633$	0	0
$634$	17773.9	1.11339
$635$	−39281.8	−2.45488
$636$	0	0
$637$	30318.3	1.88580
$638$	18000.7	1.11702
$639$	0	0
$640$	−25308.7	−1.56315
$641$	9671.90	0.595970	0.297985	−	0.954570i	$-0.403685\pi$
$641$	9671.90	0.595970	0.297985	+	0.954570i	$0.403685\pi$
$642$	0	0
$643$	8325.27	0.510601	0.255301	−	0.966862i	$-0.417826\pi$
$643$	8325.27	0.510601	0.255301	+	0.966862i	$0.417826\pi$
$644$	2407.87	0.147335
$645$	0	0
$646$	15416.4	0.938934
$647$	−13818.9	−0.839686	−0.419843	−	0.907597i	$-0.637915\pi$
$647$	−13818.9	−0.839686	−0.419843	+	0.907597i	$0.637915\pi$
$648$	0	0
$649$	−37928.3	−2.29401
$650$	−16465.7	−0.993597
$651$	0	0
$652$	1946.32	0.116908
$653$	5658.77	0.339119	0.169560	−	0.985520i	$-0.445766\pi$
$653$	5658.77	0.339119	0.169560	+	0.985520i	$0.445766\pi$
$654$	0	0
$655$	7599.85	0.453360
$656$	−17090.2	−1.01716
$657$	0	0
$658$	−11516.2	−0.682290
$659$	−24952.4	−1.47498	−0.737488	−	0.675361i	$-0.763988\pi$
$659$	−24952.4	−1.47498	−0.737488	+	0.675361i	$0.763988\pi$
$660$	0	0
$661$	−13398.0	−0.788385	−0.394193	−	0.919028i	$-0.628976\pi$
$661$	−13398.0	−0.788385	−0.394193	+	0.919028i	$0.628976\pi$
$662$	15961.3	0.937092
$663$	0	0
$664$	−2596.71	−0.151765
$665$	28177.4	1.64312
$666$	0	0
$667$	14219.0	0.825433
$668$	823.976	0.0477255
$669$	0	0
$670$	853.184	0.0491961
$671$	−37931.9	−2.18233
$672$	0	0
$673$	10385.4	0.594841	0.297421	−	0.954747i	$-0.403874\pi$
$673$	10385.4	0.594841	0.297421	+	0.954747i	$0.403874\pi$
$674$	−20719.5	−1.18411
$675$	0	0
$676$	−141.593	−0.00805607
$677$	−24567.5	−1.39469	−0.697345	−	0.716736i	$-0.745635\pi$
$677$	−24567.5	−1.39469	−0.697345	+	0.716736i	$0.745635\pi$
$678$	0	0
$679$	−23292.2	−1.31646
$680$	32354.8	1.82463
$681$	0	0
$682$	15015.9	0.843094
$683$	−4123.85	−0.231032	−0.115516	−	0.993306i	$-0.536852\pi$
$683$	−4123.85	−0.231032	−0.115516	+	0.993306i	$0.536852\pi$
$684$	0	0
$685$	29261.5	1.63215
$686$	31906.0	1.77577
$687$	0	0
$688$	31650.2	1.75386
$689$	10989.6	0.607647
$690$	0	0
$691$	29489.0	1.62346	0.811732	−	0.584029i	$-0.198525\pi$
$691$	29489.0	1.62346	0.811732	+	0.584029i	$0.198525\pi$
$692$	−806.439	−0.0443009
$693$	0	0
$694$	18229.3	0.997080
$695$	8138.13	0.444168
$696$	0	0
$697$	23478.9	1.27594
$698$	10832.1	0.587397
$699$	0	0
$700$	−2586.43	−0.139654
$701$	13861.8	0.746865	0.373432	−	0.927657i	$-0.378181\pi$
$701$	13861.8	0.746865	0.373432	+	0.927657i	$0.378181\pi$
$702$	0	0
$703$	7985.68	0.428429
$704$	23473.7	1.25668
$705$	0	0
$706$	29450.1	1.56993
$707$	−22392.1	−1.19115
$708$	0	0
$709$	−6025.25	−0.319158	−0.159579	−	0.987185i	$-0.551014\pi$
$709$	−6025.25	−0.319158	−0.159579	+	0.987185i	$0.551014\pi$
$710$	−37041.7	−1.95796
$711$	0	0
$712$	8872.51	0.467010
$713$	11861.3	0.623015
$714$	0	0
$715$	35579.8	1.86099
$716$	−247.007	−0.0128926
$717$	0	0
$718$	13734.2	0.713864
$719$	−37185.7	−1.92878	−0.964390	−	0.264484i	$-0.914798\pi$
$719$	−37185.7	−1.92878	−0.964390	+	0.264484i	$0.914798\pi$
$720$	0	0
$721$	60155.8	3.10724
$722$	11091.1	0.571702
$723$	0	0
$724$	2133.99	0.109543
$725$	−15273.4	−0.782402
$726$	0	0
$727$	29163.4	1.48777	0.743887	−	0.668306i	$-0.232980\pi$
$727$	29163.4	1.48777	0.743887	+	0.668306i	$0.232980\pi$
$728$	−30787.3	−1.56738
$729$	0	0
$730$	−28283.3	−1.43399
$731$	−43481.9	−2.20005
$732$	0	0
$733$	33123.0	1.66907	0.834533	−	0.550958i	$-0.185738\pi$
$733$	33123.0	1.66907	0.834533	+	0.550958i	$0.185738\pi$
$734$	2604.78	0.130987
$735$	0	0
$736$	−3395.57	−0.170058
$737$	−925.486	−0.0462560
$738$	0	0
$739$	−20048.3	−0.997956	−0.498978	−	0.866615i	$-0.666291\pi$
$739$	−20048.3	−0.997956	−0.498978	+	0.866615i	$0.666291\pi$
$740$	−1460.18	−0.0725369
$741$	0	0
$742$	23266.0	1.15111
$743$	3525.08	0.174055	0.0870274	−	0.996206i	$-0.472263\pi$
$743$	3525.08	0.174055	0.0870274	+	0.996206i	$0.472263\pi$
$744$	0	0
$745$	−47771.6	−2.34928
$746$	34258.8	1.68137
$747$	0	0
$748$	3057.77	0.149470
$749$	−45142.9	−2.20225
$750$	0	0
$751$	15466.1	0.751485	0.375742	−	0.926724i	$-0.377388\pi$
$751$	15466.1	0.751485	0.375742	+	0.926724i	$0.377388\pi$
$752$	8408.68	0.407757
$753$	0	0
$754$	15839.7	0.765052
$755$	−6480.58	−0.312388
$756$	0	0
$757$	−28743.4	−1.38005	−0.690024	−	0.723787i	$-0.742400\pi$
$757$	−28743.4	−1.38005	−0.690024	+	0.723787i	$0.742400\pi$
$758$	−11859.1	−0.568260
$759$	0	0
$760$	−19038.5	−0.908682
$761$	16512.5	0.786565	0.393283	−	0.919418i	$-0.371339\pi$
$761$	16512.5	0.786565	0.393283	+	0.919418i	$0.371339\pi$
$762$	0	0
$763$	14808.4	0.702622
$764$	−2201.55	−0.104253
$765$	0	0
$766$	−5644.14	−0.266229
$767$	−33375.0	−1.57119
$768$	0	0
$769$	−7716.02	−0.361829	−0.180915	−	0.983499i	$-0.557906\pi$
$769$	−7716.02	−0.361829	−0.180915	+	0.983499i	$0.557906\pi$
$770$	75325.8	3.52540
$771$	0	0
$772$	−501.264	−0.0233690
$773$	−40944.4	−1.90513	−0.952566	−	0.304333i	$-0.901567\pi$
$773$	−40944.4	−1.90513	−0.952566	+	0.304333i	$0.901567\pi$
$774$	0	0
$775$	−12740.9	−0.590536
$776$	15737.7	0.728030
$777$	0	0
$778$	5827.50	0.268542
$779$	−13815.6	−0.635426
$780$	0	0
$781$	40180.8	1.84095
$782$	32554.1	1.48866
$783$	0	0
$784$	−46866.8	−2.13497
$785$	−17695.2	−0.804547
$786$	0	0
$787$	−19083.7	−0.864373	−0.432186	−	0.901784i	$-0.642258\pi$
$787$	−19083.7	−0.864373	−0.432186	+	0.901784i	$0.642258\pi$
$788$	2286.17	0.103352
$789$	0	0
$790$	28738.5	1.29426
$791$	55316.1	2.48649
$792$	0	0
$793$	−33378.1	−1.49469
$794$	−2865.38	−0.128071
$795$	0	0
$796$	−1206.58	−0.0537264
$797$	−25384.4	−1.12818	−0.564092	−	0.825712i	$-0.690774\pi$
$797$	−25384.4	−1.12818	−0.564092	+	0.825712i	$0.690774\pi$
$798$	0	0
$799$	−11552.1	−0.511493
$800$	3647.36	0.161192
$801$	0	0
$802$	37482.0	1.65029
$803$	30680.1	1.34829
$804$	0	0
$805$	59501.0	2.60514
$806$	13213.3	0.577441
$807$	0	0
$808$	15129.5	0.658731
$809$	38774.0	1.68507	0.842536	−	0.538641i	$-0.181062\pi$
$809$	38774.0	1.68507	0.842536	+	0.538641i	$0.181062\pi$
$810$	0	0
$811$	15243.3	0.660004	0.330002	−	0.943980i	$-0.392951\pi$
$811$	15243.3	0.660004	0.330002	+	0.943980i	$0.392951\pi$
$812$	2488.10	0.107531
$813$	0	0
$814$	21347.9	0.919216
$815$	48095.5	2.06713
$816$	0	0
$817$	25586.0	1.09564
$818$	40606.0	1.73564
$819$	0	0
$820$	2526.19	0.107583
$821$	44991.4	1.91256	0.956279	−	0.292454i	$-0.0944719\pi$
$821$	44991.4	1.91256	0.956279	+	0.292454i	$0.0944719\pi$
$822$	0	0
$823$	−36088.4	−1.52851	−0.764254	−	0.644915i	$-0.776893\pi$
$823$	−36088.4	−1.52851	−0.764254	+	0.644915i	$0.776893\pi$
$824$	−40645.1	−1.71837
$825$	0	0
$826$	−70658.0	−2.97640
$827$	17633.1	0.741431	0.370716	−	0.928746i	$-0.379112\pi$
$827$	17633.1	0.741431	0.370716	+	0.928746i	$0.379112\pi$
$828$	0	0
$829$	37725.8	1.58054	0.790272	−	0.612756i	$-0.209939\pi$
$829$	37725.8	1.58054	0.790272	+	0.612756i	$0.209939\pi$
$830$	5590.53	0.233795
$831$	0	0
$832$	20655.7	0.860706
$833$	64386.9	2.67812
$834$	0	0
$835$	20361.3	0.843870
$836$	−1799.28	−0.0744370
$837$	0	0
$838$	−20884.2	−0.860899
$839$	14423.7	0.593518	0.296759	−	0.954952i	$-0.404094\pi$
$839$	14423.7	0.593518	0.296759	+	0.954952i	$0.404094\pi$
$840$	0	0
$841$	−9696.21	−0.397565
$842$	14315.3	0.585910
$843$	0	0
$844$	−1093.18	−0.0445838
$845$	−3498.92	−0.142445
$846$	0	0
$847$	−39096.0	−1.58602
$848$	−16988.0	−0.687935
$849$	0	0
$850$	−34968.1	−1.41106
$851$	16863.0	0.679267
$852$	0	0
$853$	−12637.9	−0.507285	−0.253642	−	0.967298i	$-0.581629\pi$
$853$	−12637.9	−0.507285	−0.253642	+	0.967298i	$0.581629\pi$
$854$	−70664.7	−2.83149
$855$	0	0
$856$	30501.4	1.21789
$857$	11718.1	0.467074	0.233537	−	0.972348i	$-0.424970\pi$
$857$	11718.1	0.467074	0.233537	+	0.972348i	$0.424970\pi$
$858$	0	0
$859$	−37732.7	−1.49875	−0.749373	−	0.662148i	$-0.769645\pi$
$859$	−37732.7	−1.49875	−0.749373	+	0.662148i	$0.769645\pi$
$860$	−4678.39	−0.185502
$861$	0	0
$862$	−20744.0	−0.819655
$863$	−7865.60	−0.310253	−0.155126	−	0.987895i	$-0.549578\pi$
$863$	−7865.60	−0.310253	−0.155126	+	0.987895i	$0.549578\pi$
$864$	0	0
$865$	−19927.9	−0.783317
$866$	142.906	0.00560755
$867$	0	0
$868$	2075.53	0.0811615
$869$	−31173.8	−1.21692
$870$	0	0
$871$	−814.380	−0.0316811
$872$	−10005.5	−0.388566
$873$	0	0
$874$	−19155.8	−0.741366
$875$	−509.295	−0.0196769
$876$	0	0
$877$	−47385.9	−1.82452	−0.912262	−	0.409608i	$-0.865665\pi$
$877$	−47385.9	−1.82452	−0.912262	+	0.409608i	$0.865665\pi$
$878$	27615.1	1.06146
$879$	0	0
$880$	−55000.1	−2.10688
$881$	−41917.4	−1.60299	−0.801495	−	0.598001i	$-0.795962\pi$
$881$	−41917.4	−1.60299	−0.801495	+	0.598001i	$0.795962\pi$
$882$	0	0
$883$	23413.2	0.892320	0.446160	−	0.894953i	$-0.352791\pi$
$883$	23413.2	0.892320	0.446160	+	0.894953i	$0.352791\pi$
$884$	2690.68	0.102373
$885$	0	0
$886$	−37212.7	−1.41104
$887$	26129.6	0.989118	0.494559	−	0.869144i	$-0.335330\pi$
$887$	26129.6	0.989118	0.494559	+	0.869144i	$0.335330\pi$
$888$	0	0
$889$	79380.9	2.99477
$890$	−19101.9	−0.719434
$891$	0	0
$892$	2383.45	0.0894660
$893$	6797.56	0.254728
$894$	0	0
$895$	−6103.79	−0.227963
$896$	51144.1	1.90692
$897$	0	0
$898$	8016.74	0.297909
$899$	12256.5	0.454702
$900$	0	0
$901$	23338.5	0.862950
$902$	−36932.9	−1.36334
$903$	0	0
$904$	−37375.1	−1.37508
$905$	52733.0	1.93691
$906$	0	0
$907$	1826.86	0.0668795	0.0334398	−	0.999441i	$-0.489354\pi$
$907$	1826.86	0.0668795	0.0334398	+	0.999441i	$0.489354\pi$
$908$	2488.34	0.0909456
$909$	0	0
$910$	66282.9	2.41457
$911$	18206.9	0.662153	0.331077	−	0.943604i	$-0.392588\pi$
$911$	18206.9	0.662153	0.331077	+	0.943604i	$0.392588\pi$
$912$	0	0
$913$	−6064.28	−0.219823
$914$	−38696.4	−1.40040
$915$	0	0
$916$	3141.42	0.113314
$917$	−15357.8	−0.553065
$918$	0	0
$919$	−14810.6	−0.531616	−0.265808	−	0.964026i	$-0.585639\pi$
$919$	−14810.6	−0.531616	−0.265808	+	0.964026i	$0.585639\pi$
$920$	−40202.7	−1.44070
$921$	0	0
$922$	54249.2	1.93775
$923$	35357.0	1.26088
$924$	0	0
$925$	−18113.4	−0.643855
$926$	−40488.3	−1.43686
$927$	0	0
$928$	−3508.70	−0.124115
$929$	−6753.04	−0.238493	−0.119247	−	0.992865i	$-0.538048\pi$
$929$	−6753.04	−0.238493	−0.119247	+	0.992865i	$0.538048\pi$
$930$	0	0
$931$	−37887.1	−1.33372
$932$	1236.81	0.0434690
$933$	0	0
$934$	45694.0	1.60081
$935$	75560.6	2.64288
$936$	0	0
$937$	3881.72	0.135336	0.0676682	−	0.997708i	$-0.478444\pi$
$937$	3881.72	0.135336	0.0676682	+	0.997708i	$0.478444\pi$
$938$	−1724.12	−0.0600155
$939$	0	0
$940$	−1242.93	−0.0431277
$941$	−40413.2	−1.40003	−0.700017	−	0.714127i	$-0.746824\pi$
$941$	−40413.2	−1.40003	−0.700017	+	0.714127i	$0.746824\pi$
$942$	0	0
$943$	−29173.8	−1.00746
$944$	51591.9	1.77878
$945$	0	0
$946$	68398.1	2.35075
$947$	25695.5	0.881724	0.440862	−	0.897575i	$-0.354673\pi$
$947$	25695.5	0.881724	0.440862	+	0.897575i	$0.354673\pi$
$948$	0	0
$949$	26996.9	0.923453
$950$	20576.2	0.702717
$951$	0	0
$952$	−65382.9	−2.22592
$953$	15226.2	0.517549	0.258775	−	0.965938i	$-0.416681\pi$
$953$	15226.2	0.517549	0.258775	+	0.965938i	$0.416681\pi$
$954$	0	0
$955$	−54402.6	−1.84338
$956$	2961.11	0.100177
$957$	0	0
$958$	−296.262	−0.00999143
$959$	−59131.8	−1.99110
$960$	0	0
$961$	−19566.8	−0.656803
$962$	18785.0	0.629578
$963$	0	0
$964$	−154.514	−0.00516240
$965$	−12386.7	−0.413205
$966$	0	0
$967$	4905.52	0.163134	0.0815671	−	0.996668i	$-0.474007\pi$
$967$	4905.52	0.163134	0.0815671	+	0.996668i	$0.474007\pi$
$968$	26415.8	0.877103
$969$	0	0
$970$	−33882.2	−1.12154
$971$	−10540.0	−0.348346	−0.174173	−	0.984715i	$-0.555725\pi$
$971$	−10540.0	−0.348346	−0.174173	+	0.984715i	$0.555725\pi$
$972$	0	0
$973$	−16445.6	−0.541852
$974$	8743.98	0.287654
$975$	0	0
$976$	51596.7	1.69218
$977$	−4535.47	−0.148519	−0.0742593	−	0.997239i	$-0.523659\pi$
$977$	−4535.47	−0.148519	−0.0742593	+	0.997239i	$0.523659\pi$
$978$	0	0
$979$	20720.6	0.676439
$980$	6927.64	0.225812
$981$	0	0
$982$	−38021.8	−1.23556
$983$	53423.0	1.73340	0.866699	−	0.498832i	$-0.166238\pi$
$983$	53423.0	1.73340	0.866699	+	0.498832i	$0.166238\pi$
$984$	0	0
$985$	56493.5	1.82744
$986$	33638.8	1.08649
$987$	0	0
$988$	−1583.27	−0.0509824
$989$	54028.7	1.73712
$990$	0	0
$991$	−52827.4	−1.69336	−0.846678	−	0.532105i	$-0.821401\pi$
$991$	−52827.4	−1.69336	−0.846678	+	0.532105i	$0.821401\pi$
$992$	−2926.91	−0.0936788
$993$	0	0
$994$	74854.2	2.38856
$995$	−29815.9	−0.949977
$996$	0	0
$997$	41095.6	1.30543	0.652714	−	0.757605i	$-0.273630\pi$
$997$	41095.6	1.30543	0.652714	+	0.757605i	$0.273630\pi$
$998$	38754.8	1.22922
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2169.4.a.c.1.9		29
3.2	odd	2		723.4.a.b.1.21	✓	29

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
723.4.a.b.1.21	✓	29	3.2	odd	2
2169.4.a.c.1.9		29	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:	\( N \)	\(=\)	\( 2169 = 3^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2169.a (trivial)

\(f(q)\)	\(=\)	\(q-2.93958 q^{2} +0.641136 q^{4} +15.8431 q^{5} -32.0159 q^{7} +21.6320 q^{8} +O(q^{10})\)	\(q-2.93958 q^{2} +0.641136 q^{4} +15.8431 q^{5} -32.0159 q^{7} +21.6320 q^{8} -46.5721 q^{10} +50.5188 q^{11} +44.4539 q^{13} +94.1133 q^{14} -68.7180 q^{16} +94.4066 q^{17} -55.5515 q^{19} +10.1576 q^{20} -148.504 q^{22} -117.305 q^{23} +126.004 q^{25} -130.676 q^{26} -20.5265 q^{28} -121.214 q^{29} -101.115 q^{31} +28.9464 q^{32} -277.516 q^{34} -507.231 q^{35} -143.753 q^{37} +163.298 q^{38} +342.718 q^{40} +248.700 q^{41} -460.581 q^{43} +32.3894 q^{44} +344.829 q^{46} -122.365 q^{47} +682.017 q^{49} -370.399 q^{50} +28.5010 q^{52} +247.212 q^{53} +800.374 q^{55} -692.567 q^{56} +356.318 q^{58} -750.776 q^{59} -750.847 q^{61} +297.235 q^{62} +464.654 q^{64} +704.288 q^{65} -18.3196 q^{67} +60.5275 q^{68} +1491.05 q^{70} +795.363 q^{71} +607.301 q^{73} +422.573 q^{74} -35.6161 q^{76} -1617.40 q^{77} -617.074 q^{79} -1088.71 q^{80} -731.073 q^{82} -120.040 q^{83} +1495.69 q^{85} +1353.91 q^{86} +1092.82 q^{88} +410.157 q^{89} -1423.23 q^{91} -75.2087 q^{92} +359.702 q^{94} -880.108 q^{95} +727.521 q^{97} -2004.84 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 29 q - 9 q^{2} + 97 q^{4} - 62 q^{5} - 30 q^{7} - 108 q^{8}+O(q^{10}) \) 29 * q - 9 * q^2 + 97 * q^4 - 62 * q^5 - 30 * q^7 - 108 * q^8	\( 29 q - 9 q^{2} + 97 q^{4} - 62 q^{5} - 30 q^{7} - 108 q^{8} + 51 q^{10} - 46 q^{11} + 250 q^{13} - 84 q^{14} + 333 q^{16} - 128 q^{17} + 58 q^{19} - 405 q^{20} - 48 q^{22} - 232 q^{23} + 707 q^{25} - 238 q^{26} - 89 q^{28} - 590 q^{29} - 468 q^{31} - 1068 q^{32} + 287 q^{34} - 474 q^{35} + 842 q^{37} - 160 q^{38} + 434 q^{40} - 814 q^{41} + 20 q^{43} - 150 q^{44} - 37 q^{46} - 1004 q^{47} + 1239 q^{49} - 839 q^{50} + 1928 q^{52} - 2192 q^{53} + 432 q^{55} - 437 q^{56} - 28 q^{58} - 1288 q^{59} + 1502 q^{61} - 3059 q^{62} + 3372 q^{64} - 2312 q^{65} + 358 q^{67} - 4990 q^{68} + 5366 q^{70} - 1938 q^{71} + 3266 q^{73} - 2510 q^{74} + 3591 q^{76} - 5098 q^{77} - 292 q^{79} - 8235 q^{80} + 4511 q^{82} - 4256 q^{83} + 1998 q^{85} - 6860 q^{86} + 5935 q^{88} - 6428 q^{89} - 1650 q^{91} - 6823 q^{92} + 3025 q^{94} - 1802 q^{95} + 5040 q^{97} - 9410 q^{98}+O(q^{100}) \) 29 * q - 9 * q^2 + 97 * q^4 - 62 * q^5 - 30 * q^7 - 108 * q^8 + 51 * q^10 - 46 * q^11 + 250 * q^13 - 84 * q^14 + 333 * q^16 - 128 * q^17 + 58 * q^19 - 405 * q^20 - 48 * q^22 - 232 * q^23 + 707 * q^25 - 238 * q^26 - 89 * q^28 - 590 * q^29 - 468 * q^31 - 1068 * q^32 + 287 * q^34 - 474 * q^35 + 842 * q^37 - 160 * q^38 + 434 * q^40 - 814 * q^41 + 20 * q^43 - 150 * q^44 - 37 * q^46 - 1004 * q^47 + 1239 * q^49 - 839 * q^50 + 1928 * q^52 - 2192 * q^53 + 432 * q^55 - 437 * q^56 - 28 * q^58 - 1288 * q^59 + 1502 * q^61 - 3059 * q^62 + 3372 * q^64 - 2312 * q^65 + 358 * q^67 - 4990 * q^68 + 5366 * q^70 - 1938 * q^71 + 3266 * q^73 - 2510 * q^74 + 3591 * q^76 - 5098 * q^77 - 292 * q^79 - 8235 * q^80 + 4511 * q^82 - 4256 * q^83 + 1998 * q^85 - 6860 * q^86 + 5935 * q^88 - 6428 * q^89 - 1650 * q^91 - 6823 * q^92 + 3025 * q^94 - 1802 * q^95 + 5040 * q^97 - 9410 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more