LMFDB - Embedded newform 2151.4.a.f.1.16

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:	$0$
Dimension:	$37$
Twist minimal:	no (minimal twist has level 239)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
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)

$f(q)$	$=$	$q-1.09413 q^{2} -6.80288 q^{4} +10.8819 q^{5} +4.64141 q^{7} +16.1963 q^{8} +O(q^{10})$	\(q-1.09413 q^{2} -6.80288 q^{4} +10.8819 q^{5} +4.64141 q^{7} +16.1963 q^{8} -11.9062 q^{10} -67.3054 q^{11} +34.3652 q^{13} -5.07830 q^{14} +36.7022 q^{16} -68.2171 q^{17} -0.737622 q^{19} -74.0284 q^{20} +73.6409 q^{22} -98.7512 q^{23} -6.58380 q^{25} -37.6000 q^{26} -31.5750 q^{28} -1.29726 q^{29} -15.8224 q^{31} -169.727 q^{32} +74.6383 q^{34} +50.5074 q^{35} +60.8933 q^{37} +0.807054 q^{38} +176.247 q^{40} +235.425 q^{41} +386.475 q^{43} +457.871 q^{44} +108.047 q^{46} -392.361 q^{47} -321.457 q^{49} +7.20353 q^{50} -233.782 q^{52} +646.053 q^{53} -732.412 q^{55} +75.1735 q^{56} +1.41937 q^{58} -489.966 q^{59} +840.303 q^{61} +17.3117 q^{62} -107.914 q^{64} +373.959 q^{65} +187.166 q^{67} +464.072 q^{68} -55.2617 q^{70} -412.368 q^{71} -544.229 q^{73} -66.6252 q^{74} +5.01795 q^{76} -312.392 q^{77} +1065.32 q^{79} +399.391 q^{80} -257.585 q^{82} -1342.25 q^{83} -742.333 q^{85} -422.854 q^{86} -1090.10 q^{88} +964.071 q^{89} +159.503 q^{91} +671.793 q^{92} +429.294 q^{94} -8.02674 q^{95} +798.587 q^{97} +351.716 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8}+O(q^{10}) $ 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8	$ 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8} + 147 q^{10} - 55 q^{11} + 250 q^{13} - 169 q^{14} + 918 q^{16} - 189 q^{17} + 550 q^{19} - 486 q^{20} + 226 q^{22} - 74 q^{23} + 1604 q^{25} - 560 q^{26} + 829 q^{28} - 389 q^{29} + 1107 q^{31} - 125 q^{32} + 1423 q^{34} - 270 q^{35} + 1002 q^{37} - 1037 q^{38} + 1536 q^{40} - 1518 q^{41} + 1098 q^{43} - 1037 q^{44} + 1030 q^{46} - 1214 q^{47} + 4663 q^{49} - 929 q^{50} + 2895 q^{52} - 904 q^{53} + 1350 q^{55} - 2556 q^{56} + 1396 q^{58} - 1658 q^{59} + 2313 q^{61} + 4519 q^{62} + 3807 q^{64} + 56 q^{65} + 1535 q^{67} + 6526 q^{68} - 4099 q^{70} + 3255 q^{71} + 3154 q^{73} + 2629 q^{74} + 1981 q^{76} + 3734 q^{77} + 2260 q^{79} + 8242 q^{80} - 9898 q^{82} + 939 q^{83} + 1272 q^{85} + 3457 q^{86} - 1808 q^{88} - 1486 q^{89} + 174 q^{91} + 14076 q^{92} - 984 q^{94} + 1828 q^{95} + 6148 q^{97} + 6243 q^{98}+O(q^{100}) $ 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8 + 147 * q^10 - 55 * q^11 + 250 * q^13 - 169 * q^14 + 918 * q^16 - 189 * q^17 + 550 * q^19 - 486 * q^20 + 226 * q^22 - 74 * q^23 + 1604 * q^25 - 560 * q^26 + 829 * q^28 - 389 * q^29 + 1107 * q^31 - 125 * q^32 + 1423 * q^34 - 270 * q^35 + 1002 * q^37 - 1037 * q^38 + 1536 * q^40 - 1518 * q^41 + 1098 * q^43 - 1037 * q^44 + 1030 * q^46 - 1214 * q^47 + 4663 * q^49 - 929 * q^50 + 2895 * q^52 - 904 * q^53 + 1350 * q^55 - 2556 * q^56 + 1396 * q^58 - 1658 * q^59 + 2313 * q^61 + 4519 * q^62 + 3807 * q^64 + 56 * q^65 + 1535 * q^67 + 6526 * q^68 - 4099 * q^70 + 3255 * q^71 + 3154 * q^73 + 2629 * q^74 + 1981 * q^76 + 3734 * q^77 + 2260 * q^79 + 8242 * q^80 - 9898 * q^82 + 939 * q^83 + 1272 * q^85 + 3457 * q^86 - 1808 * q^88 - 1486 * q^89 + 174 * q^91 + 14076 * q^92 - 984 * q^94 + 1828 * q^95 + 6148 * q^97 + 6243 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.09413	−0.386833	−0.193417	−	0.981117i	$-0.561957\pi$
$2$	−1.09413	−0.386833	−0.193417	+	0.981117i	$0.561957\pi$
$3$	0	0
$3$	0	0
$4$	−6.80288	−0.850360
$5$	10.8819	0.973309	0.486654	−	0.873595i	$-0.338217\pi$
$5$	10.8819	0.973309	0.486654	+	0.873595i	$0.338217\pi$
$6$	0	0
$7$	4.64141	0.250613	0.125306	−	0.992118i	$-0.460009\pi$
$7$	4.64141	0.250613	0.125306	+	0.992118i	$0.460009\pi$
$8$	16.1963	0.715781
$9$	0	0
$10$	−11.9062	−0.376508
$11$	−67.3054	−1.84485	−0.922425	−	0.386176i	$-0.873796\pi$
$11$	−67.3054	−1.84485	−0.922425	+	0.386176i	$0.873796\pi$
$12$	0	0
$13$	34.3652	0.733169	0.366584	−	0.930385i	$-0.380527\pi$
$13$	34.3652	0.733169	0.366584	+	0.930385i	$0.380527\pi$
$14$	−5.07830	−0.0969452
$15$	0	0
$16$	36.7022	0.573472
$17$	−68.2171	−0.973239	−0.486620	−	0.873614i	$-0.661770\pi$
$17$	−68.2171	−0.973239	−0.486620	+	0.873614i	$0.661770\pi$
$18$	0	0
$19$	−0.737622	−0.00890642	−0.00445321	−	0.999990i	$-0.501418\pi$
$19$	−0.737622	−0.00890642	−0.00445321	+	0.999990i	$0.501418\pi$
$20$	−74.0284	−0.827663
$21$	0	0
$22$	73.6409	0.713649
$23$	−98.7512	−0.895263	−0.447632	−	0.894218i	$-0.647732\pi$
$23$	−98.7512	−0.895263	−0.447632	+	0.894218i	$0.647732\pi$
$24$	0	0
$25$	−6.58380	−0.0526704
$26$	−37.6000	−0.283614
$27$	0	0
$28$	−31.5750	−0.213111
$29$	−1.29726	−0.00830672	−0.00415336	−	0.999991i	$-0.501322\pi$
$29$	−1.29726	−0.00830672	−0.00415336	+	0.999991i	$0.501322\pi$
$30$	0	0
$31$	−15.8224	−0.0916704	−0.0458352	−	0.998949i	$-0.514595\pi$
$31$	−15.8224	−0.0916704	−0.0458352	+	0.998949i	$0.514595\pi$
$32$	−169.727	−0.937619
$33$	0	0
$34$	74.6383	0.376481
$35$	50.5074	0.243923
$36$	0	0
$37$	60.8933	0.270562	0.135281	−	0.990807i	$-0.456806\pi$
$37$	60.8933	0.270562	0.135281	+	0.990807i	$0.456806\pi$
$38$	0.807054	0.00344530
$39$	0	0
$40$	176.247	0.696675
$41$	235.425	0.896759	0.448380	−	0.893843i	$-0.352001\pi$
$41$	235.425	0.896759	0.448380	+	0.893843i	$0.352001\pi$
$42$	0	0
$43$	386.475	1.37062	0.685312	−	0.728249i	$-0.259666\pi$
$43$	386.475	1.37062	0.685312	+	0.728249i	$0.259666\pi$
$44$	457.871	1.56879
$45$	0	0
$46$	108.047	0.346317
$47$	−392.361	−1.21770	−0.608849	−	0.793286i	$-0.708368\pi$
$47$	−392.361	−1.21770	−0.608849	+	0.793286i	$0.708368\pi$
$48$	0	0
$49$	−321.457	−0.937193
$50$	7.20353	0.0203747
$51$	0	0
$52$	−233.782	−0.623457
$53$	646.053	1.67438	0.837191	−	0.546910i	$-0.184196\pi$
$53$	646.053	1.67438	0.837191	+	0.546910i	$0.184196\pi$
$54$	0	0
$55$	−732.412	−1.79561
$56$	75.1735	0.179384
$57$	0	0
$58$	1.41937	0.00321331
$59$	−489.966	−1.08116	−0.540578	−	0.841294i	$-0.681794\pi$
$59$	−489.966	−1.08116	−0.540578	+	0.841294i	$0.681794\pi$
$60$	0	0
$61$	840.303	1.76377	0.881884	−	0.471467i	$-0.156275\pi$
$61$	840.303	1.76377	0.881884	+	0.471467i	$0.156275\pi$
$62$	17.3117	0.0354611
$63$	0	0
$64$	−107.914	−0.210770
$65$	373.959	0.713599
$66$	0	0
$67$	187.166	0.341283	0.170642	−	0.985333i	$-0.445416\pi$
$67$	187.166	0.341283	0.170642	+	0.985333i	$0.445416\pi$
$68$	464.072	0.827604
$69$	0	0
$70$	−55.2617	−0.0943576
$71$	−412.368	−0.689282	−0.344641	−	0.938735i	$-0.611999\pi$
$71$	−412.368	−0.689282	−0.344641	+	0.938735i	$0.611999\pi$
$72$	0	0
$73$	−544.229	−0.872565	−0.436282	−	0.899810i	$-0.643705\pi$
$73$	−544.229	−0.872565	−0.436282	+	0.899810i	$0.643705\pi$
$74$	−66.6252	−0.104662
$75$	0	0
$76$	5.01795	0.00757367
$77$	−312.392	−0.462343
$78$	0	0
$79$	1065.32	1.51719	0.758596	−	0.651562i	$-0.225886\pi$
$79$	1065.32	1.51719	0.758596	+	0.651562i	$0.225886\pi$
$80$	399.391	0.558166
$81$	0	0
$82$	−257.585	−0.346896
$83$	−1342.25	−1.77508	−0.887539	−	0.460733i	$-0.847587\pi$
$83$	−1342.25	−1.77508	−0.887539	+	0.460733i	$0.847587\pi$
$84$	0	0
$85$	−742.333	−0.947262
$86$	−422.854	−0.530203
$87$	0	0
$88$	−1090.10	−1.32051
$89$	964.071	1.14822	0.574108	−	0.818779i	$-0.305349\pi$
$89$	964.071	1.14822	0.574108	+	0.818779i	$0.305349\pi$
$90$	0	0
$91$	159.503	0.183741
$92$	671.793	0.761296
$93$	0	0
$94$	429.294	0.471046
$95$	−8.02674	−0.00866870
$96$	0	0
$97$	798.587	0.835920	0.417960	−	0.908466i	$-0.362745\pi$
$97$	798.587	0.835920	0.417960	+	0.908466i	$0.362745\pi$
$98$	351.716	0.362537
$99$	0	0
$100$	44.7888	0.0447888
$101$	790.770	0.779055	0.389527	−	0.921015i	$-0.372638\pi$
$101$	790.770	0.779055	0.389527	+	0.921015i	$0.372638\pi$
$102$	0	0
$103$	−443.164	−0.423944	−0.211972	−	0.977276i	$-0.567989\pi$
$103$	−443.164	−0.423944	−0.211972	+	0.977276i	$0.567989\pi$
$104$	556.588	0.524788
$105$	0	0
$106$	−706.866	−0.647707
$107$	−803.437	−0.725899	−0.362950	−	0.931809i	$-0.618230\pi$
$107$	−803.437	−0.725899	−0.362950	+	0.931809i	$0.618230\pi$
$108$	0	0
$109$	−1546.26	−1.35876	−0.679379	−	0.733787i	$-0.737751\pi$
$109$	−1546.26	−1.35876	−0.679379	+	0.733787i	$0.737751\pi$
$110$	801.354	0.694601
$111$	0	0
$112$	170.350	0.143719
$113$	1951.87	1.62492	0.812462	−	0.583014i	$-0.198127\pi$
$113$	1951.87	1.62492	0.812462	+	0.583014i	$0.198127\pi$
$114$	0	0
$115$	−1074.60	−0.871367
$116$	8.82509	0.00706370
$117$	0	0
$118$	536.087	0.418227
$119$	−316.623	−0.243906
$120$	0	0
$121$	3199.02	2.40347
$122$	−919.400	−0.682284
$123$	0	0
$124$	107.638	0.0779528
$125$	−1431.88	−1.02457
$126$	0	0
$127$	911.440	0.636828	0.318414	−	0.947952i	$-0.396850\pi$
$127$	911.440	0.636828	0.318414	+	0.947952i	$0.396850\pi$
$128$	1475.89	1.01915
$129$	0	0
$130$	−409.160	−0.276044
$131$	1525.04	1.01712	0.508562	−	0.861026i	$-0.330177\pi$
$131$	1525.04	1.01712	0.508562	+	0.861026i	$0.330177\pi$
$132$	0	0
$133$	−3.42360	−0.00223206
$134$	−204.784	−0.132020
$135$	0	0
$136$	−1104.86	−0.696626
$137$	−2788.78	−1.73914	−0.869569	−	0.493811i	$-0.835603\pi$
$137$	−2788.78	−1.73914	−0.869569	+	0.493811i	$0.835603\pi$
$138$	0	0
$139$	136.743	0.0834414	0.0417207	−	0.999129i	$-0.486716\pi$
$139$	136.743	0.0834414	0.0417207	+	0.999129i	$0.486716\pi$
$140$	−343.596	−0.207423
$141$	0	0
$142$	451.184	0.266637
$143$	−2312.97	−1.35259
$144$	0	0
$145$	−14.1167	−0.00808500
$146$	595.457	0.337537
$147$	0	0
$148$	−414.250	−0.230075
$149$	508.330	0.279490	0.139745	−	0.990188i	$-0.455372\pi$
$149$	508.330	0.279490	0.139745	+	0.990188i	$0.455372\pi$
$150$	0	0
$151$	−2628.90	−1.41680	−0.708400	−	0.705811i	$-0.750583\pi$
$151$	−2628.90	−1.41680	−0.708400	+	0.705811i	$0.750583\pi$
$152$	−11.9467	−0.00637505
$153$	0	0
$154$	341.797	0.178849
$155$	−172.178	−0.0892236
$156$	0	0
$157$	−851.906	−0.433054	−0.216527	−	0.976277i	$-0.569473\pi$
$157$	−851.906	−0.433054	−0.216527	+	0.976277i	$0.569473\pi$
$158$	−1165.60	−0.586900
$159$	0	0
$160$	−1846.96	−0.912592
$161$	−458.345	−0.224364
$162$	0	0
$163$	1387.75	0.666854	0.333427	−	0.942776i	$-0.391795\pi$
$163$	1387.75	0.666854	0.333427	+	0.942776i	$0.391795\pi$
$164$	−1601.56	−0.762568
$165$	0	0
$166$	1468.60	0.686659
$167$	2872.74	1.33113	0.665566	−	0.746339i	$-0.268190\pi$
$167$	2872.74	1.33113	0.665566	+	0.746339i	$0.268190\pi$
$168$	0	0
$169$	−1016.03	−0.462464
$170$	812.208	0.366432
$171$	0	0
$172$	−2629.14	−1.16552
$173$	−3344.60	−1.46986	−0.734929	−	0.678144i	$-0.762785\pi$
$173$	−3344.60	−1.46986	−0.734929	+	0.678144i	$0.762785\pi$
$174$	0	0
$175$	−30.5581	−0.0131999
$176$	−2470.26	−1.05797
$177$	0	0
$178$	−1054.82	−0.444168
$179$	−1530.75	−0.639183	−0.319592	−	0.947555i	$-0.603546\pi$
$179$	−1530.75	−0.639183	−0.319592	+	0.947555i	$0.603546\pi$
$180$	0	0
$181$	4070.15	1.67145	0.835724	−	0.549150i	$-0.185049\pi$
$181$	4070.15	1.67145	0.835724	+	0.549150i	$0.185049\pi$
$182$	−174.517	−0.0710772
$183$	0	0
$184$	−1599.40	−0.640812
$185$	662.637	0.263341
$186$	0	0
$187$	4591.38	1.79548
$188$	2669.19	1.03548
$189$	0	0
$190$	8.78229	0.00335334
$191$	−60.4630	−0.0229055	−0.0114527	−	0.999934i	$-0.503646\pi$
$191$	−60.4630	−0.0229055	−0.0114527	+	0.999934i	$0.503646\pi$
$192$	0	0
$193$	4066.18	1.51653	0.758264	−	0.651948i	$-0.226048\pi$
$193$	4066.18	1.51653	0.758264	+	0.651948i	$0.226048\pi$
$194$	−873.757	−0.323361
$195$	0	0
$196$	2186.84	0.796952
$197$	1011.06	0.365661	0.182830	−	0.983144i	$-0.441474\pi$
$197$	1011.06	0.365661	0.182830	+	0.983144i	$0.441474\pi$
$198$	0	0
$199$	−330.381	−0.117689	−0.0588444	−	0.998267i	$-0.518742\pi$
$199$	−330.381	−0.117689	−0.0588444	+	0.998267i	$0.518742\pi$
$200$	−106.633	−0.0377005
$201$	0	0
$202$	−865.204	−0.301364
$203$	−6.02111	−0.00208177
$204$	0	0
$205$	2561.87	0.872824
$206$	484.879	0.163996
$207$	0	0
$208$	1261.28	0.420452
$209$	49.6460	0.0164310
$210$	0	0
$211$	1079.95	0.352355	0.176177	−	0.984358i	$-0.443627\pi$
$211$	1079.95	0.352355	0.176177	+	0.984358i	$0.443627\pi$
$212$	−4395.02	−1.42383
$213$	0	0
$214$	879.064	0.280802
$215$	4205.59	1.33404
$216$	0	0
$217$	−73.4381	−0.0229737
$218$	1691.81	0.525613
$219$	0	0
$220$	4982.51	1.52691
$221$	−2344.29	−0.713549
$222$	0	0
$223$	1118.02	0.335730	0.167865	−	0.985810i	$-0.446313\pi$
$223$	1118.02	0.335730	0.167865	+	0.985810i	$0.446313\pi$
$224$	−787.773	−0.234979
$225$	0	0
$226$	−2135.60	−0.628575
$227$	−889.341	−0.260034	−0.130017	−	0.991512i	$-0.541503\pi$
$227$	−889.341	−0.260034	−0.130017	+	0.991512i	$0.541503\pi$
$228$	0	0
$229$	1760.24	0.507948	0.253974	−	0.967211i	$-0.418262\pi$
$229$	1760.24	0.507948	0.253974	+	0.967211i	$0.418262\pi$
$230$	1175.75	0.337074
$231$	0	0
$232$	−21.0107	−0.00594579
$233$	−5426.36	−1.52572	−0.762860	−	0.646564i	$-0.776205\pi$
$233$	−5426.36	−1.52572	−0.762860	+	0.646564i	$0.776205\pi$
$234$	0	0
$235$	−4269.64	−1.18520
$236$	3333.18	0.919372
$237$	0	0
$238$	346.427	0.0943509
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	982.482	0.262602	0.131301	−	0.991343i	$-0.458084\pi$
$241$	982.482	0.262602	0.131301	+	0.991343i	$0.458084\pi$
$242$	−3500.14	−0.929743
$243$	0	0
$244$	−5716.48	−1.49984
$245$	−3498.07	−0.912178
$246$	0	0
$247$	−25.3485	−0.00652991
$248$	−256.263	−0.0656159
$249$	0	0
$250$	1566.67	0.396339
$251$	5103.90	1.28349	0.641744	−	0.766919i	$-0.278211\pi$
$251$	5103.90	1.28349	0.641744	+	0.766919i	$0.278211\pi$
$252$	0	0
$253$	6646.49	1.65163
$254$	−997.233	−0.246346
$255$	0	0
$256$	−751.499	−0.183471
$257$	709.669	0.172249	0.0861243	−	0.996284i	$-0.472552\pi$
$257$	709.669	0.172249	0.0861243	+	0.996284i	$0.472552\pi$
$258$	0	0
$259$	282.631	0.0678063
$260$	−2544.00	−0.606816
$261$	0	0
$262$	−1668.59	−0.393457
$263$	3227.33	0.756675	0.378338	−	0.925668i	$-0.376496\pi$
$263$	3227.33	0.756675	0.378338	+	0.925668i	$0.376496\pi$
$264$	0	0
$265$	7030.30	1.62969
$266$	3.74587	0.000863435 0
$267$	0	0
$268$	−1273.27	−0.290214
$269$	7387.87	1.67452	0.837261	−	0.546803i	$-0.184155\pi$
$269$	7387.87	1.67452	0.837261	+	0.546803i	$0.184155\pi$
$270$	0	0
$271$	1795.75	0.402525	0.201263	−	0.979537i	$-0.435496\pi$
$271$	1795.75	0.402525	0.201263	+	0.979537i	$0.435496\pi$
$272$	−2503.72	−0.558126
$273$	0	0
$274$	3051.29	0.672756
$275$	443.126	0.0971691
$276$	0	0
$277$	880.503	0.190990	0.0954951	−	0.995430i	$-0.469557\pi$
$277$	880.503	0.190990	0.0954951	+	0.995430i	$0.469557\pi$
$278$	−149.614	−0.0322779
$279$	0	0
$280$	818.032	0.174596
$281$	4446.44	0.943959	0.471979	−	0.881610i	$-0.343540\pi$
$281$	4446.44	0.943959	0.471979	+	0.881610i	$0.343540\pi$
$282$	0	0
$283$	6792.54	1.42677	0.713383	−	0.700775i	$-0.247162\pi$
$283$	6792.54	1.42677	0.713383	+	0.700775i	$0.247162\pi$
$284$	2805.29	0.586138
$285$	0	0
$286$	2530.68	0.523225
$287$	1092.70	0.224739
$288$	0	0
$289$	−259.434	−0.0528056
$290$	15.4455	0.00312755
$291$	0	0
$292$	3702.33	0.741994
$293$	4815.54	0.960159	0.480080	−	0.877225i	$-0.340608\pi$
$293$	4815.54	0.960159	0.480080	+	0.877225i	$0.340608\pi$
$294$	0	0
$295$	−5331.78	−1.05230
$296$	986.245	0.193663
$297$	0	0
$298$	−556.178	−0.108116
$299$	−3393.61	−0.656379
$300$	0	0
$301$	1793.79	0.343496
$302$	2876.36	0.548066
$303$	0	0
$304$	−27.0724	−0.00510759
$305$	9144.11	1.71669
$306$	0	0
$307$	6036.97	1.12231	0.561153	−	0.827712i	$-0.310358\pi$
$307$	6036.97	1.12231	0.561153	+	0.827712i	$0.310358\pi$
$308$	2125.17	0.393158
$309$	0	0
$310$	188.385	0.0345146
$311$	1104.93	0.201463	0.100731	−	0.994914i	$-0.467882\pi$
$311$	1104.93	0.201463	0.100731	+	0.994914i	$0.467882\pi$
$312$	0	0
$313$	−845.716	−0.152724	−0.0763622	−	0.997080i	$-0.524331\pi$
$313$	−845.716	−0.152724	−0.0763622	+	0.997080i	$0.524331\pi$
$314$	932.096	0.167520
$315$	0	0
$316$	−7247.26	−1.29016
$317$	3736.62	0.662048	0.331024	−	0.943622i	$-0.392606\pi$
$317$	3736.62	0.662048	0.331024	+	0.943622i	$0.392606\pi$
$318$	0	0
$319$	87.3125	0.0153247
$320$	−1174.32	−0.205145
$321$	0	0
$322$	501.489	0.0867915
$323$	50.3184	0.00866808
$324$	0	0
$325$	−226.254	−0.0386163
$326$	−1518.38	−0.257961
$327$	0	0
$328$	3813.00	0.641883
$329$	−1821.11	−0.305170
$330$	0	0
$331$	4637.38	0.770070	0.385035	−	0.922902i	$-0.374189\pi$
$331$	4637.38	0.770070	0.385035	+	0.922902i	$0.374189\pi$
$332$	9131.19	1.50946
$333$	0	0
$334$	−3143.15	−0.514926
$335$	2036.73	0.332174
$336$	0	0
$337$	48.4702	0.00783484	0.00391742	−	0.999992i	$-0.498753\pi$
$337$	48.4702	0.00783484	0.00391742	+	0.999992i	$0.498753\pi$
$338$	1111.67	0.178896
$339$	0	0
$340$	5050.00	0.805514
$341$	1064.93	0.169118
$342$	0	0
$343$	−3084.02	−0.485485
$344$	6259.45	0.981067
$345$	0	0
$346$	3659.43	0.568590
$347$	10453.2	1.61717	0.808587	−	0.588376i	$-0.200233\pi$
$347$	10453.2	1.61717	0.808587	+	0.588376i	$0.200233\pi$
$348$	0	0
$349$	2656.25	0.407409	0.203704	−	0.979032i	$-0.434702\pi$
$349$	2656.25	0.407409	0.203704	+	0.979032i	$0.434702\pi$
$350$	33.4345	0.00510615
$351$	0	0
$352$	11423.6	1.72977
$353$	3053.04	0.460332	0.230166	−	0.973151i	$-0.426073\pi$
$353$	3053.04	0.460332	0.230166	+	0.973151i	$0.426073\pi$
$354$	0	0
$355$	−4487.35	−0.670884
$356$	−6558.46	−0.976398
$357$	0	0
$358$	1674.84	0.247257
$359$	2448.35	0.359941	0.179971	−	0.983672i	$-0.442400\pi$
$359$	2448.35	0.359941	0.179971	+	0.983672i	$0.442400\pi$
$360$	0	0
$361$	−6858.46	−0.999921
$362$	−4453.27	−0.646571
$363$	0	0
$364$	−1085.08	−0.156246
$365$	−5922.26	−0.849275
$366$	0	0
$367$	−957.834	−0.136236	−0.0681179	−	0.997677i	$-0.521699\pi$
$367$	−957.834	−0.136236	−0.0681179	+	0.997677i	$0.521699\pi$
$368$	−3624.39	−0.513409
$369$	0	0
$370$	−725.010	−0.101869
$371$	2998.60	0.419621
$372$	0	0
$373$	−5576.90	−0.774158	−0.387079	−	0.922047i	$-0.626516\pi$
$373$	−5576.90	−0.774158	−0.387079	+	0.922047i	$0.626516\pi$
$374$	−5023.56	−0.694551
$375$	0	0
$376$	−6354.79	−0.871604
$377$	−44.5806	−0.00609023
$378$	0	0
$379$	−9642.35	−1.30685	−0.653423	−	0.756993i	$-0.726668\pi$
$379$	−9642.35	−1.30685	−0.653423	+	0.756993i	$0.726668\pi$
$380$	54.6050	0.00737151
$381$	0	0
$382$	66.1543	0.00886060
$383$	3525.73	0.470382	0.235191	−	0.971949i	$-0.424428\pi$
$383$	3525.73	0.470382	0.235191	+	0.971949i	$0.424428\pi$
$384$	0	0
$385$	−3399.43	−0.450002
$386$	−4448.93	−0.586643
$387$	0	0
$388$	−5432.69	−0.710833
$389$	9585.95	1.24943	0.624713	−	0.780854i	$-0.285216\pi$
$389$	9585.95	1.24943	0.624713	+	0.780854i	$0.285216\pi$
$390$	0	0
$391$	6736.52	0.871305
$392$	−5206.41	−0.670825
$393$	0	0
$394$	−1106.23	−0.141450
$395$	11592.8	1.47670
$396$	0	0
$397$	−8546.13	−1.08040	−0.540199	−	0.841537i	$-0.681651\pi$
$397$	−8546.13	−1.08040	−0.540199	+	0.841537i	$0.681651\pi$
$398$	361.479	0.0455259
$399$	0	0
$400$	−241.640	−0.0302050
$401$	11403.1	1.42006	0.710030	−	0.704171i	$-0.248681\pi$
$401$	11403.1	1.42006	0.710030	+	0.704171i	$0.248681\pi$
$402$	0	0
$403$	−543.739	−0.0672099
$404$	−5379.51	−0.662477
$405$	0	0
$406$	6.58787	0.000805297 0
$407$	−4098.45	−0.499147
$408$	0	0
$409$	10736.4	1.29800	0.648999	−	0.760790i	$-0.275188\pi$
$409$	10736.4	1.29800	0.648999	+	0.760790i	$0.275188\pi$
$410$	−2803.02	−0.337637
$411$	0	0
$412$	3014.79	0.360505
$413$	−2274.13	−0.270951
$414$	0	0
$415$	−14606.3	−1.72770
$416$	−5832.71	−0.687433
$417$	0	0
$418$	−54.3191	−0.00635606
$419$	−16639.6	−1.94008	−0.970042	−	0.242937i	$-0.921889\pi$
$419$	−16639.6	−1.94008	−0.970042	+	0.242937i	$0.921889\pi$
$420$	0	0
$421$	16025.3	1.85517	0.927586	−	0.373610i	$-0.121880\pi$
$421$	16025.3	1.85517	0.927586	+	0.373610i	$0.121880\pi$
$422$	−1181.61	−0.136303
$423$	0	0
$424$	10463.7	1.19849
$425$	449.128	0.0512609
$426$	0	0
$427$	3900.19	0.442022
$428$	5465.69	0.617276
$429$	0	0
$430$	−4601.46	−0.516051
$431$	−8182.10	−0.914427	−0.457213	−	0.889357i	$-0.651152\pi$
$431$	−8182.10	−0.914427	−0.457213	+	0.889357i	$0.651152\pi$
$432$	0	0
$433$	−5944.36	−0.659740	−0.329870	−	0.944026i	$-0.607005\pi$
$433$	−5944.36	−0.659740	−0.329870	+	0.944026i	$0.607005\pi$
$434$	80.3508	0.00888701
$435$	0	0
$436$	10519.0	1.15543
$437$	72.8410	0.00797359
$438$	0	0
$439$	9040.26	0.982843	0.491422	−	0.870922i	$-0.336477\pi$
$439$	9040.26	0.982843	0.491422	+	0.870922i	$0.336477\pi$
$440$	−11862.3	−1.28526
$441$	0	0
$442$	2564.96	0.276024
$443$	15600.6	1.67316	0.836579	−	0.547847i	$-0.184552\pi$
$443$	15600.6	1.67316	0.836579	+	0.547847i	$0.184552\pi$
$444$	0	0
$445$	10490.9	1.11757
$446$	−1223.25	−0.129872
$447$	0	0
$448$	−500.875	−0.0528217
$449$	−8601.86	−0.904113	−0.452057	−	0.891989i	$-0.649309\pi$
$449$	−8601.86	−0.904113	−0.452057	+	0.891989i	$0.649309\pi$
$450$	0	0
$451$	−15845.4	−1.65439
$452$	−13278.3	−1.38177
$453$	0	0
$454$	973.055	0.100590
$455$	1735.70	0.178837
$456$	0	0
$457$	7875.42	0.806119	0.403060	−	0.915174i	$-0.367947\pi$
$457$	7875.42	0.806119	0.403060	+	0.915174i	$0.367947\pi$
$458$	−1925.93	−0.196491
$459$	0	0
$460$	7310.39	0.740976
$461$	13956.3	1.41000	0.705000	−	0.709207i	$-0.250947\pi$
$461$	13956.3	1.41000	0.705000	+	0.709207i	$0.250947\pi$
$462$	0	0
$463$	−8909.22	−0.894269	−0.447134	−	0.894467i	$-0.647555\pi$
$463$	−8909.22	−0.894269	−0.447134	+	0.894467i	$0.647555\pi$
$464$	−47.6123	−0.00476367
$465$	0	0
$466$	5937.14	0.590199
$467$	8777.52	0.869755	0.434877	−	0.900490i	$-0.356792\pi$
$467$	8777.52	0.869755	0.434877	+	0.900490i	$0.356792\pi$
$468$	0	0
$469$	868.715	0.0855299
$470$	4671.54	0.458473
$471$	0	0
$472$	−7935.63	−0.773870
$473$	−26011.9	−2.52860
$474$	0	0
$475$	4.85636	0.000469105 0
$476$	2153.95	0.207408
$477$	0	0
$478$	−261.497	−0.0250222
$479$	9250.75	0.882417	0.441208	−	0.897405i	$-0.354550\pi$
$479$	9250.75	0.882417	0.441208	+	0.897405i	$0.354550\pi$
$480$	0	0
$481$	2092.61	0.198368
$482$	−1074.96	−0.101583
$483$	0	0
$484$	−21762.6	−2.04382
$485$	8690.16	0.813608
$486$	0	0
$487$	1638.97	0.152502	0.0762512	−	0.997089i	$-0.475705\pi$
$487$	1638.97	0.152502	0.0762512	+	0.997089i	$0.475705\pi$
$488$	13609.8	1.26247
$489$	0	0
$490$	3827.34	0.352861
$491$	−13591.3	−1.24922	−0.624609	−	0.780938i	$-0.714742\pi$
$491$	−13591.3	−1.24922	−0.624609	+	0.780938i	$0.714742\pi$
$492$	0	0
$493$	88.4951	0.00808442
$494$	27.7346	0.00252599
$495$	0	0
$496$	−580.716	−0.0525704
$497$	−1913.97	−0.172743
$498$	0	0
$499$	−8364.58	−0.750401	−0.375200	−	0.926944i	$-0.622426\pi$
$499$	−8364.58	−0.750401	−0.375200	+	0.926944i	$0.622426\pi$
$500$	9740.94	0.871256
$501$	0	0
$502$	−5584.33	−0.496496
$503$	−15091.4	−1.33776	−0.668881	−	0.743369i	$-0.733226\pi$
$503$	−15091.4	−1.33776	−0.668881	+	0.743369i	$0.733226\pi$
$504$	0	0
$505$	8605.09	0.758261
$506$	−7272.12	−0.638904
$507$	0	0
$508$	−6200.42	−0.541533
$509$	2866.71	0.249636	0.124818	−	0.992180i	$-0.460165\pi$
$509$	2866.71	0.249636	0.124818	+	0.992180i	$0.460165\pi$
$510$	0	0
$511$	−2525.99	−0.218676
$512$	−10984.9	−0.948179
$513$	0	0
$514$	−776.469	−0.0666315
$515$	−4822.47	−0.412628
$516$	0	0
$517$	26408.0	2.24647
$518$	−309.235	−0.0262297
$519$	0	0
$520$	6056.75	0.510781
$521$	205.931	0.0173167	0.00865836	−	0.999963i	$-0.497244\pi$
$521$	205.931	0.0173167	0.00865836	+	0.999963i	$0.497244\pi$
$522$	0	0
$523$	5482.74	0.458401	0.229200	−	0.973379i	$-0.426389\pi$
$523$	5482.74	0.458401	0.229200	+	0.973379i	$0.426389\pi$
$524$	−10374.6	−0.864921
$525$	0	0
$526$	−3531.12	−0.292707
$527$	1079.36	0.0892172
$528$	0	0
$529$	−2415.20	−0.198504
$530$	−7692.06	−0.630418
$531$	0	0
$532$	23.2904	0.00189806
$533$	8090.41	0.657476
$534$	0	0
$535$	−8742.94	−0.706524
$536$	3031.39	0.244284
$537$	0	0
$538$	−8083.29	−0.647761
$539$	21635.8	1.72898
$540$	0	0
$541$	11822.2	0.939510	0.469755	−	0.882797i	$-0.344342\pi$
$541$	11822.2	0.939510	0.469755	+	0.882797i	$0.344342\pi$
$542$	−1964.79	−0.155710
$543$	0	0
$544$	11578.3	0.912527
$545$	−16826.3	−1.32249
$546$	0	0
$547$	24338.0	1.90241	0.951205	−	0.308559i	$-0.0998468\pi$
$547$	24338.0	1.90241	0.951205	+	0.308559i	$0.0998468\pi$
$548$	18971.8	1.47889
$549$	0	0
$550$	−484.837	−0.0375882
$551$	0.956886	7.39831e−5 0
$552$	0	0
$553$	4944.60	0.380227
$554$	−963.384	−0.0738814
$555$	0	0
$556$	−930.244	−0.0709553
$557$	24272.4	1.84642	0.923208	−	0.384301i	$-0.125558\pi$
$557$	24272.4	1.84642	0.923208	+	0.384301i	$0.125558\pi$
$558$	0	0
$559$	13281.3	1.00490
$560$	1853.74	0.139883
$561$	0	0
$562$	−4864.98	−0.365155
$563$	1455.15	0.108929	0.0544646	−	0.998516i	$-0.482655\pi$
$563$	1455.15	0.108929	0.0544646	+	0.998516i	$0.482655\pi$
$564$	0	0
$565$	21240.1	1.58155
$566$	−7431.92	−0.551920
$567$	0	0
$568$	−6678.82	−0.493375
$569$	−7426.69	−0.547176	−0.273588	−	0.961847i	$-0.588210\pi$
$569$	−7426.69	−0.547176	−0.273588	+	0.961847i	$0.588210\pi$
$570$	0	0
$571$	−11161.1	−0.817999	−0.408999	−	0.912535i	$-0.634122\pi$
$571$	−11161.1	−0.817999	−0.408999	+	0.912535i	$0.634122\pi$
$572$	15734.8	1.15019
$573$	0	0
$574$	−1195.56	−0.0869366
$575$	650.159	0.0471539
$576$	0	0
$577$	−7069.73	−0.510081	−0.255040	−	0.966930i	$-0.582089\pi$
$577$	−7069.73	−0.510081	−0.255040	+	0.966930i	$0.582089\pi$
$578$	283.854	0.0204270
$579$	0	0
$580$	96.0340	0.00687516
$581$	−6229.95	−0.444857
$582$	0	0
$583$	−43482.9	−3.08898
$584$	−8814.48	−0.624565
$585$	0	0
$586$	−5268.82	−0.371421
$587$	11023.7	0.775119	0.387560	−	0.921845i	$-0.373318\pi$
$587$	11023.7	0.775119	0.387560	+	0.921845i	$0.373318\pi$
$588$	0	0
$589$	11.6709	0.000816455 0
$590$	5833.65	0.407064
$591$	0	0
$592$	2234.92	0.155160
$593$	−2572.68	−0.178158	−0.0890788	−	0.996025i	$-0.528392\pi$
$593$	−2572.68	−0.178158	−0.0890788	+	0.996025i	$0.528392\pi$
$594$	0	0
$595$	−3445.47	−0.237396
$596$	−3458.11	−0.237667
$597$	0	0
$598$	3713.04	0.253909
$599$	2147.47	0.146483	0.0732415	−	0.997314i	$-0.476666\pi$
$599$	2147.47	0.146483	0.0732415	+	0.997314i	$0.476666\pi$
$600$	0	0
$601$	13262.0	0.900112	0.450056	−	0.893000i	$-0.351404\pi$
$601$	13262.0	0.900112	0.450056	+	0.893000i	$0.351404\pi$
$602$	−1962.64	−0.132876
$603$	0	0
$604$	17884.1	1.20479
$605$	34811.5	2.33932
$606$	0	0
$607$	7164.95	0.479104	0.239552	−	0.970884i	$-0.422999\pi$
$607$	7164.95	0.479104	0.239552	+	0.970884i	$0.422999\pi$
$608$	125.194	0.00835083
$609$	0	0
$610$	−10004.8	−0.664072
$611$	−13483.6	−0.892778
$612$	0	0
$613$	7149.45	0.471066	0.235533	−	0.971866i	$-0.424316\pi$
$613$	7149.45	0.471066	0.235533	+	0.971866i	$0.424316\pi$
$614$	−6605.23	−0.434145
$615$	0	0
$616$	−5059.59	−0.330936
$617$	5172.56	0.337503	0.168752	−	0.985659i	$-0.446026\pi$
$617$	5172.56	0.337503	0.168752	+	0.985659i	$0.446026\pi$
$618$	0	0
$619$	−14428.9	−0.936908	−0.468454	−	0.883488i	$-0.655189\pi$
$619$	−14428.9	−0.936908	−0.468454	+	0.883488i	$0.655189\pi$
$620$	1171.30	0.0758722
$621$	0	0
$622$	−1208.94	−0.0779325
$623$	4474.65	0.287757
$624$	0	0
$625$	−14758.7	−0.944555
$626$	925.323	0.0590788
$627$	0	0
$628$	5795.42	0.368252
$629$	−4153.96	−0.263322
$630$	0	0
$631$	3374.41	0.212889	0.106444	−	0.994319i	$-0.466053\pi$
$631$	3374.41	0.212889	0.106444	+	0.994319i	$0.466053\pi$
$632$	17254.2	1.08598
$633$	0	0
$634$	−4088.34	−0.256102
$635$	9918.22	0.619831
$636$	0	0
$637$	−11046.9	−0.687121
$638$	−95.5312	−0.00592808
$639$	0	0
$640$	16060.5	0.991949
$641$	−26213.7	−1.61525	−0.807627	−	0.589693i	$-0.799249\pi$
$641$	−26213.7	−1.61525	−0.807627	+	0.589693i	$0.799249\pi$
$642$	0	0
$643$	9113.03	0.558916	0.279458	−	0.960158i	$-0.409845\pi$
$643$	9113.03	0.558916	0.279458	+	0.960158i	$0.409845\pi$
$644$	3118.06	0.190790
$645$	0	0
$646$	−55.0548	−0.00335310
$647$	3190.50	0.193866	0.0969331	−	0.995291i	$-0.469097\pi$
$647$	3190.50	0.193866	0.0969331	+	0.995291i	$0.469097\pi$
$648$	0	0
$649$	32977.4	1.99457
$650$	247.551	0.0149381
$651$	0	0
$652$	−9440.72	−0.567066
$653$	8453.13	0.506580	0.253290	−	0.967390i	$-0.418487\pi$
$653$	8453.13	0.506580	0.253290	+	0.967390i	$0.418487\pi$
$654$	0	0
$655$	16595.3	0.989975
$656$	8640.61	0.514267
$657$	0	0
$658$	1992.53	0.118050
$659$	−2899.65	−0.171403	−0.0857013	−	0.996321i	$-0.527313\pi$
$659$	−2899.65	−0.171403	−0.0857013	+	0.996321i	$0.527313\pi$
$660$	0	0
$661$	−1693.07	−0.0996261	−0.0498131	−	0.998759i	$-0.515863\pi$
$661$	−1693.07	−0.0996261	−0.0498131	+	0.998759i	$0.515863\pi$
$662$	−5073.89	−0.297889
$663$	0	0
$664$	−21739.5	−1.27057
$665$	−37.2554	−0.00217248
$666$	0	0
$667$	128.106	0.00743670
$668$	−19542.9	−1.13194
$669$	0	0
$670$	−2228.44	−0.128496
$671$	−56557.0	−3.25389
$672$	0	0
$673$	9692.83	0.555173	0.277586	−	0.960701i	$-0.410466\pi$
$673$	9692.83	0.555173	0.277586	+	0.960701i	$0.410466\pi$
$674$	−53.0327	−0.00303078
$675$	0	0
$676$	6911.95	0.393261
$677$	30104.1	1.70900	0.854501	−	0.519450i	$-0.173863\pi$
$677$	30104.1	1.70900	0.854501	+	0.519450i	$0.173863\pi$
$678$	0	0
$679$	3706.57	0.209492
$680$	−12023.0	−0.678032
$681$	0	0
$682$	−1165.17	−0.0654205
$683$	4228.00	0.236867	0.118433	−	0.992962i	$-0.462213\pi$
$683$	4228.00	0.236867	0.118433	+	0.992962i	$0.462213\pi$
$684$	0	0
$685$	−30347.3	−1.69272
$686$	3374.32	0.187802
$687$	0	0
$688$	14184.5	0.786015
$689$	22201.8	1.22760
$690$	0	0
$691$	−8614.62	−0.474263	−0.237132	−	0.971478i	$-0.576207\pi$
$691$	−8614.62	−0.474263	−0.237132	+	0.971478i	$0.576207\pi$
$692$	22752.9	1.24991
$693$	0	0
$694$	−11437.2	−0.625577
$695$	1488.02	0.0812143
$696$	0	0
$697$	−16060.0	−0.872761
$698$	−2906.28	−0.157599
$699$	0	0
$700$	207.883	0.0112246
$701$	−7937.47	−0.427666	−0.213833	−	0.976870i	$-0.568595\pi$
$701$	−7937.47	−0.427666	−0.213833	+	0.976870i	$0.568595\pi$
$702$	0	0
$703$	−44.9163	−0.00240974
$704$	7263.23	0.388840
$705$	0	0
$706$	−3340.43	−0.178072
$707$	3670.29	0.195241
$708$	0	0
$709$	−25213.5	−1.33556	−0.667781	−	0.744358i	$-0.732756\pi$
$709$	−25213.5	−1.33556	−0.667781	+	0.744358i	$0.732756\pi$
$710$	4909.74	0.259520
$711$	0	0
$712$	15614.3	0.821871
$713$	1562.48	0.0820691
$714$	0	0
$715$	−25169.5	−1.31648
$716$	10413.5	0.543536
$717$	0	0
$718$	−2678.81	−0.139237
$719$	−34905.6	−1.81051	−0.905256	−	0.424866i	$-0.860321\pi$
$719$	−34905.6	−1.81051	−0.905256	+	0.424866i	$0.860321\pi$
$720$	0	0
$721$	−2056.90	−0.106246
$722$	7504.04	0.386802
$723$	0	0
$724$	−27688.7	−1.42133
$725$	8.54089	0.000437518 0
$726$	0	0
$727$	−34706.6	−1.77056	−0.885279	−	0.465061i	$-0.846033\pi$
$727$	−34706.6	−1.77056	−0.885279	+	0.465061i	$0.846033\pi$
$728$	2583.35	0.131518
$729$	0	0
$730$	6479.72	0.328528
$731$	−26364.2	−1.33395
$732$	0	0
$733$	6254.39	0.315159	0.157579	−	0.987506i	$-0.449631\pi$
$733$	6254.39	0.315159	0.157579	+	0.987506i	$0.449631\pi$
$734$	1047.99	0.0527005
$735$	0	0
$736$	16760.8	0.839415
$737$	−12597.3	−0.629617
$738$	0	0
$739$	−18709.4	−0.931310	−0.465655	−	0.884966i	$-0.654181\pi$
$739$	−18709.4	−0.931310	−0.465655	+	0.884966i	$0.654181\pi$
$740$	−4507.84	−0.223934
$741$	0	0
$742$	−3280.85	−0.162323
$743$	−3875.44	−0.191354	−0.0956771	−	0.995412i	$-0.530502\pi$
$743$	−3875.44	−0.191354	−0.0956771	+	0.995412i	$0.530502\pi$
$744$	0	0
$745$	5531.60	0.272030
$746$	6101.85	0.299470
$747$	0	0
$748$	−31234.6	−1.52680
$749$	−3729.08	−0.181919
$750$	0	0
$751$	16851.7	0.818811	0.409405	−	0.912353i	$-0.365736\pi$
$751$	16851.7	0.818811	0.409405	+	0.912353i	$0.365736\pi$
$752$	−14400.5	−0.698316
$753$	0	0
$754$	48.7769	0.00235590
$755$	−28607.5	−1.37898
$756$	0	0
$757$	4696.07	0.225471	0.112735	−	0.993625i	$-0.464039\pi$
$757$	4696.07	0.225471	0.112735	+	0.993625i	$0.464039\pi$
$758$	10550.0	0.505531
$759$	0	0
$760$	−130.003	−0.00620489
$761$	−3182.11	−0.151579	−0.0757895	−	0.997124i	$-0.524148\pi$
$761$	−3182.11	−0.151579	−0.0757895	+	0.997124i	$0.524148\pi$
$762$	0	0
$763$	−7176.82	−0.340522
$764$	411.322	0.0194779
$765$	0	0
$766$	−3857.61	−0.181959
$767$	−16837.8	−0.792670
$768$	0	0
$769$	−9035.71	−0.423714	−0.211857	−	0.977301i	$-0.567951\pi$
$769$	−9035.71	−0.423714	−0.211857	+	0.977301i	$0.567951\pi$
$770$	3719.41	0.174076
$771$	0	0
$772$	−27661.7	−1.28960
$773$	−808.070	−0.0375993	−0.0187997	−	0.999823i	$-0.505984\pi$
$773$	−808.070	−0.0375993	−0.0187997	+	0.999823i	$0.505984\pi$
$774$	0	0
$775$	104.171	0.00482832
$776$	12934.1	0.598335
$777$	0	0
$778$	−10488.3	−0.483320
$779$	−173.654	−0.00798692
$780$	0	0
$781$	27754.6	1.27162
$782$	−7370.62	−0.337050
$783$	0	0
$784$	−11798.2	−0.537454
$785$	−9270.38	−0.421495
$786$	0	0
$787$	13171.4	0.596580	0.298290	−	0.954475i	$-0.403584\pi$
$787$	13171.4	0.596580	0.298290	+	0.954475i	$0.403584\pi$
$788$	−6878.13	−0.310943
$789$	0	0
$790$	−12684.0	−0.571235
$791$	9059.42	0.407226
$792$	0	0
$793$	28877.2	1.29314
$794$	9350.58	0.417934
$795$	0	0
$796$	2247.54	0.100078
$797$	−21572.8	−0.958780	−0.479390	−	0.877602i	$-0.659142\pi$
$797$	−21572.8	−0.958780	−0.479390	+	0.877602i	$0.659142\pi$
$798$	0	0
$799$	26765.7	1.18511
$800$	1117.45	0.0493848
$801$	0	0
$802$	−12476.5	−0.549327
$803$	36629.6	1.60975
$804$	0	0
$805$	−4987.67	−0.218376
$806$	594.921	0.0259990
$807$	0	0
$808$	12807.5	0.557632
$809$	−30769.0	−1.33718	−0.668592	−	0.743630i	$-0.733103\pi$
$809$	−30769.0	−1.33718	−0.668592	+	0.743630i	$0.733103\pi$
$810$	0	0
$811$	39420.6	1.70684	0.853419	−	0.521225i	$-0.174525\pi$
$811$	39420.6	1.70684	0.853419	+	0.521225i	$0.174525\pi$
$812$	40.9609	0.00177025
$813$	0	0
$814$	4484.24	0.193087
$815$	15101.4	0.649055
$816$	0	0
$817$	−285.072	−0.0122074
$818$	−11747.0	−0.502108
$819$	0	0
$820$	−17428.1	−0.742214
$821$	34902.0	1.48366	0.741832	−	0.670586i	$-0.233957\pi$
$821$	34902.0	1.48366	0.741832	+	0.670586i	$0.233957\pi$
$822$	0	0
$823$	−6907.65	−0.292570	−0.146285	−	0.989242i	$-0.546732\pi$
$823$	−6907.65	−0.292570	−0.146285	+	0.989242i	$0.546732\pi$
$824$	−7177.60	−0.303451
$825$	0	0
$826$	2488.20	0.104813
$827$	24598.3	1.03430	0.517149	−	0.855895i	$-0.326993\pi$
$827$	24598.3	1.03430	0.517149	+	0.855895i	$0.326993\pi$
$828$	0	0
$829$	−22877.7	−0.958473	−0.479237	−	0.877686i	$-0.659086\pi$
$829$	−22877.7	−0.958473	−0.479237	+	0.877686i	$0.659086\pi$
$830$	15981.2	0.668331
$831$	0	0
$832$	−3708.50	−0.154530
$833$	21928.9	0.912113
$834$	0	0
$835$	31260.9	1.29560
$836$	−337.736	−0.0139723
$837$	0	0
$838$	18205.8	0.750489
$839$	−27005.5	−1.11124	−0.555621	−	0.831436i	$-0.687519\pi$
$839$	−27005.5	−1.11124	−0.555621	+	0.831436i	$0.687519\pi$
$840$	0	0
$841$	−24387.3	−0.999931
$842$	−17533.8	−0.717642
$843$	0	0
$844$	−7346.78	−0.299629
$845$	−11056.4	−0.450120
$846$	0	0
$847$	14848.0	0.602340
$848$	23711.6	0.960212
$849$	0	0
$850$	−491.404	−0.0198294
$851$	−6013.29	−0.242224
$852$	0	0
$853$	12202.4	0.489803	0.244901	−	0.969548i	$-0.421244\pi$
$853$	12202.4	0.489803	0.244901	+	0.969548i	$0.421244\pi$
$854$	−4267.31	−0.170989
$855$	0	0
$856$	−13012.7	−0.519585
$857$	−36132.2	−1.44020	−0.720101	−	0.693869i	$-0.755905\pi$
$857$	−36132.2	−1.44020	−0.720101	+	0.693869i	$0.755905\pi$
$858$	0	0
$859$	45284.7	1.79871	0.899357	−	0.437215i	$-0.144035\pi$
$859$	45284.7	1.79871	0.899357	+	0.437215i	$0.144035\pi$
$860$	−28610.1	−1.13442
$861$	0	0
$862$	8952.28	0.353731
$863$	22750.1	0.897362	0.448681	−	0.893692i	$-0.351894\pi$
$863$	22750.1	0.897362	0.448681	+	0.893692i	$0.351894\pi$
$864$	0	0
$865$	−36395.7	−1.43063
$866$	6503.90	0.255209
$867$	0	0
$868$	499.591	0.0195360
$869$	−71702.0	−2.79899
$870$	0	0
$871$	6432.00	0.250218
$872$	−25043.6	−0.972573
$873$	0	0
$874$	−79.6975	−0.00308445
$875$	−6645.96	−0.256771
$876$	0	0
$877$	−12208.3	−0.470063	−0.235032	−	0.971988i	$-0.575519\pi$
$877$	−12208.3	−0.470063	−0.235032	+	0.971988i	$0.575519\pi$
$878$	−9891.22	−0.380196
$879$	0	0
$880$	−26881.2	−1.02973
$881$	−12772.9	−0.488456	−0.244228	−	0.969718i	$-0.578535\pi$
$881$	−12772.9	−0.488456	−0.244228	+	0.969718i	$0.578535\pi$
$882$	0	0
$883$	−9712.93	−0.370177	−0.185088	−	0.982722i	$-0.559257\pi$
$883$	−9712.93	−0.370177	−0.185088	+	0.982722i	$0.559257\pi$
$884$	15947.9	0.606773
$885$	0	0
$886$	−17069.1	−0.647233
$887$	−46140.7	−1.74662	−0.873310	−	0.487165i	$-0.838031\pi$
$887$	−46140.7	−1.74662	−0.873310	+	0.487165i	$0.838031\pi$
$888$	0	0
$889$	4230.36	0.159597
$890$	−11478.4	−0.432313
$891$	0	0
$892$	−7605.72	−0.285492
$893$	289.414	0.0108453
$894$	0	0
$895$	−16657.5	−0.622122
$896$	6850.21	0.255412
$897$	0	0
$898$	9411.55	0.349741
$899$	20.5257	0.000761480 0
$900$	0	0
$901$	−44071.9	−1.62957
$902$	17336.9	0.639972
$903$	0	0
$904$	31613.0	1.16309
$905$	44291.1	1.62683
$906$	0	0
$907$	−13802.7	−0.505303	−0.252651	−	0.967557i	$-0.581303\pi$
$907$	−13802.7	−0.505303	−0.252651	+	0.967557i	$0.581303\pi$
$908$	6050.08	0.221122
$909$	0	0
$910$	−1899.08	−0.0691801
$911$	33604.9	1.22215	0.611076	−	0.791572i	$-0.290737\pi$
$911$	33604.9	1.22215	0.611076	+	0.791572i	$0.290737\pi$
$912$	0	0
$913$	90341.0	3.27475
$914$	−8616.73	−0.311834
$915$	0	0
$916$	−11974.7	−0.431939
$917$	7078.32	0.254904
$918$	0	0
$919$	−27312.3	−0.980359	−0.490179	−	0.871622i	$-0.663069\pi$
$919$	−27312.3	−0.980359	−0.490179	+	0.871622i	$0.663069\pi$
$920$	−17404.6	−0.623708
$921$	0	0
$922$	−15270.0	−0.545435
$923$	−14171.1	−0.505360
$924$	0	0
$925$	−400.910	−0.0142506
$926$	9747.84	0.345933
$927$	0	0
$928$	220.180	0.00778854
$929$	−48509.2	−1.71317	−0.856585	−	0.516006i	$-0.827418\pi$
$929$	−48509.2	−1.71317	−0.856585	+	0.516006i	$0.827418\pi$
$930$	0	0
$931$	237.114	0.00834704
$932$	36914.9	1.29741
$933$	0	0
$934$	−9603.75	−0.336450
$935$	49963.0	1.74756
$936$	0	0
$937$	19954.7	0.695722	0.347861	−	0.937546i	$-0.386908\pi$
$937$	19954.7	0.695722	0.347861	+	0.937546i	$0.386908\pi$
$938$	−950.486	−0.0330858
$939$	0	0
$940$	29045.9	1.00784
$941$	3883.02	0.134520	0.0672598	−	0.997735i	$-0.478574\pi$
$941$	3883.02	0.134520	0.0672598	+	0.997735i	$0.478574\pi$
$942$	0	0
$943$	−23248.5	−0.802836
$944$	−17982.9	−0.620013
$945$	0	0
$946$	28460.3	0.978145
$947$	47665.6	1.63561	0.817806	−	0.575495i	$-0.195190\pi$
$947$	47665.6	1.63561	0.817806	+	0.575495i	$0.195190\pi$
$948$	0	0
$949$	−18702.6	−0.639737
$950$	−5.31348	−0.000181465 0
$951$	0	0
$952$	−5128.11	−0.174583
$953$	6552.28	0.222717	0.111358	−	0.993780i	$-0.464480\pi$
$953$	6552.28	0.222717	0.111358	+	0.993780i	$0.464480\pi$
$954$	0	0
$955$	−657.953	−0.0222941
$956$	−1625.89	−0.0550052
$957$	0	0
$958$	−10121.5	−0.341348
$959$	−12943.9	−0.435850
$960$	0	0
$961$	−29540.7	−0.991597
$962$	−2289.59	−0.0767353
$963$	0	0
$964$	−6683.70	−0.223307
$965$	44247.8	1.47605
$966$	0	0
$967$	6674.94	0.221977	0.110988	−	0.993822i	$-0.464598\pi$
$967$	6674.94	0.221977	0.110988	+	0.993822i	$0.464598\pi$
$968$	51812.2	1.72036
$969$	0	0
$970$	−9508.16	−0.314730
$971$	−38364.0	−1.26793	−0.633965	−	0.773362i	$-0.718574\pi$
$971$	−38364.0	−1.26793	−0.633965	+	0.773362i	$0.718574\pi$
$972$	0	0
$973$	634.679	0.0209115
$974$	−1793.24	−0.0589930
$975$	0	0
$976$	30841.0	1.01147
$977$	2079.68	0.0681012	0.0340506	−	0.999420i	$-0.489159\pi$
$977$	2079.68	0.0681012	0.0340506	+	0.999420i	$0.489159\pi$
$978$	0	0
$979$	−64887.2	−2.11829
$980$	23797.0	0.775680
$981$	0	0
$982$	14870.6	0.483239
$983$	38376.0	1.24517	0.622587	−	0.782551i	$-0.286082\pi$
$983$	38376.0	1.24517	0.622587	+	0.782551i	$0.286082\pi$
$984$	0	0
$985$	11002.3	0.355901
$986$	−96.8251	−0.00312732
$987$	0	0
$988$	172.443	0.00555278
$989$	−38164.9	−1.22707
$990$	0	0
$991$	30557.5	0.979508	0.489754	−	0.871861i	$-0.337087\pi$
$991$	30557.5	0.979508	0.489754	+	0.871861i	$0.337087\pi$
$992$	2685.49	0.0859519
$993$	0	0
$994$	2094.13	0.0668226
$995$	−3595.18	−0.114547
$996$	0	0
$997$	11331.8	0.359961	0.179980	−	0.983670i	$-0.442397\pi$
$997$	11331.8	0.359961	0.179980	+	0.983670i	$0.442397\pi$
$998$	9151.93	0.290280
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.f.1.16		37
3.2	odd	2		239.4.a.b.1.22	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
239.4.a.b.1.22	✓	37	3.2	odd	2
2151.4.a.f.1.16		37	1.1	even	1	trivial

Overview	Random
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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)

\(f(q)\)	\(=\)	\(q-1.09413 q^{2} -6.80288 q^{4} +10.8819 q^{5} +4.64141 q^{7} +16.1963 q^{8} +O(q^{10})\)	\(q-1.09413 q^{2} -6.80288 q^{4} +10.8819 q^{5} +4.64141 q^{7} +16.1963 q^{8} -11.9062 q^{10} -67.3054 q^{11} +34.3652 q^{13} -5.07830 q^{14} +36.7022 q^{16} -68.2171 q^{17} -0.737622 q^{19} -74.0284 q^{20} +73.6409 q^{22} -98.7512 q^{23} -6.58380 q^{25} -37.6000 q^{26} -31.5750 q^{28} -1.29726 q^{29} -15.8224 q^{31} -169.727 q^{32} +74.6383 q^{34} +50.5074 q^{35} +60.8933 q^{37} +0.807054 q^{38} +176.247 q^{40} +235.425 q^{41} +386.475 q^{43} +457.871 q^{44} +108.047 q^{46} -392.361 q^{47} -321.457 q^{49} +7.20353 q^{50} -233.782 q^{52} +646.053 q^{53} -732.412 q^{55} +75.1735 q^{56} +1.41937 q^{58} -489.966 q^{59} +840.303 q^{61} +17.3117 q^{62} -107.914 q^{64} +373.959 q^{65} +187.166 q^{67} +464.072 q^{68} -55.2617 q^{70} -412.368 q^{71} -544.229 q^{73} -66.6252 q^{74} +5.01795 q^{76} -312.392 q^{77} +1065.32 q^{79} +399.391 q^{80} -257.585 q^{82} -1342.25 q^{83} -742.333 q^{85} -422.854 q^{86} -1090.10 q^{88} +964.071 q^{89} +159.503 q^{91} +671.793 q^{92} +429.294 q^{94} -8.02674 q^{95} +798.587 q^{97} +351.716 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8}+O(q^{10}) \) 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8	\( 37 q - 4 q^{2} + 170 q^{4} - 43 q^{5} + 60 q^{7} - 27 q^{8} + 147 q^{10} - 55 q^{11} + 250 q^{13} - 169 q^{14} + 918 q^{16} - 189 q^{17} + 550 q^{19} - 486 q^{20} + 226 q^{22} - 74 q^{23} + 1604 q^{25} - 560 q^{26} + 829 q^{28} - 389 q^{29} + 1107 q^{31} - 125 q^{32} + 1423 q^{34} - 270 q^{35} + 1002 q^{37} - 1037 q^{38} + 1536 q^{40} - 1518 q^{41} + 1098 q^{43} - 1037 q^{44} + 1030 q^{46} - 1214 q^{47} + 4663 q^{49} - 929 q^{50} + 2895 q^{52} - 904 q^{53} + 1350 q^{55} - 2556 q^{56} + 1396 q^{58} - 1658 q^{59} + 2313 q^{61} + 4519 q^{62} + 3807 q^{64} + 56 q^{65} + 1535 q^{67} + 6526 q^{68} - 4099 q^{70} + 3255 q^{71} + 3154 q^{73} + 2629 q^{74} + 1981 q^{76} + 3734 q^{77} + 2260 q^{79} + 8242 q^{80} - 9898 q^{82} + 939 q^{83} + 1272 q^{85} + 3457 q^{86} - 1808 q^{88} - 1486 q^{89} + 174 q^{91} + 14076 q^{92} - 984 q^{94} + 1828 q^{95} + 6148 q^{97} + 6243 q^{98}+O(q^{100}) \) 37 * q - 4 * q^2 + 170 * q^4 - 43 * q^5 + 60 * q^7 - 27 * q^8 + 147 * q^10 - 55 * q^11 + 250 * q^13 - 169 * q^14 + 918 * q^16 - 189 * q^17 + 550 * q^19 - 486 * q^20 + 226 * q^22 - 74 * q^23 + 1604 * q^25 - 560 * q^26 + 829 * q^28 - 389 * q^29 + 1107 * q^31 - 125 * q^32 + 1423 * q^34 - 270 * q^35 + 1002 * q^37 - 1037 * q^38 + 1536 * q^40 - 1518 * q^41 + 1098 * q^43 - 1037 * q^44 + 1030 * q^46 - 1214 * q^47 + 4663 * q^49 - 929 * q^50 + 2895 * q^52 - 904 * q^53 + 1350 * q^55 - 2556 * q^56 + 1396 * q^58 - 1658 * q^59 + 2313 * q^61 + 4519 * q^62 + 3807 * q^64 + 56 * q^65 + 1535 * q^67 + 6526 * q^68 - 4099 * q^70 + 3255 * q^71 + 3154 * q^73 + 2629 * q^74 + 1981 * q^76 + 3734 * q^77 + 2260 * q^79 + 8242 * q^80 - 9898 * q^82 + 939 * q^83 + 1272 * q^85 + 3457 * q^86 - 1808 * q^88 - 1486 * q^89 + 174 * q^91 + 14076 * q^92 - 984 * q^94 + 1828 * q^95 + 6148 * q^97 + 6243 * q^98

Introduction

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