LMFDB - Embedded newform 2151.4.a.e.1.20

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:	$32$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q+1.64226 q^{2} -5.30297 q^{4} +14.9594 q^{5} +8.99120 q^{7} -21.8470 q^{8} +O(q^{10})$	\(q+1.64226 q^{2} -5.30297 q^{4} +14.9594 q^{5} +8.99120 q^{7} -21.8470 q^{8} +24.5672 q^{10} +38.0853 q^{11} +48.4818 q^{13} +14.7659 q^{14} +6.54525 q^{16} +67.1804 q^{17} -150.323 q^{19} -79.3290 q^{20} +62.5461 q^{22} +178.955 q^{23} +98.7825 q^{25} +79.6200 q^{26} -47.6800 q^{28} +207.458 q^{29} +3.02502 q^{31} +185.525 q^{32} +110.328 q^{34} +134.503 q^{35} -202.734 q^{37} -246.871 q^{38} -326.817 q^{40} -137.802 q^{41} -3.56013 q^{43} -201.965 q^{44} +293.891 q^{46} -29.9753 q^{47} -262.158 q^{49} +162.227 q^{50} -257.098 q^{52} +199.697 q^{53} +569.732 q^{55} -196.431 q^{56} +340.701 q^{58} -6.04685 q^{59} -75.2305 q^{61} +4.96789 q^{62} +252.319 q^{64} +725.257 q^{65} +1014.53 q^{67} -356.255 q^{68} +220.889 q^{70} -326.257 q^{71} -533.435 q^{73} -332.942 q^{74} +797.161 q^{76} +342.433 q^{77} +416.413 q^{79} +97.9127 q^{80} -226.307 q^{82} +22.6681 q^{83} +1004.98 q^{85} -5.84667 q^{86} -832.049 q^{88} +950.440 q^{89} +435.910 q^{91} -948.993 q^{92} -49.2273 q^{94} -2248.74 q^{95} -1232.55 q^{97} -430.533 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8}+O(q^{10}) $ 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8	$ 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8} + 32 q^{10} - 154 q^{11} + 100 q^{13} - 42 q^{14} + 719 q^{16} - 32 q^{17} + 202 q^{19} + 132 q^{20} + 265 q^{22} - 552 q^{23} + 1086 q^{25} + 280 q^{26} + 390 q^{28} + 154 q^{29} + 560 q^{31} - 444 q^{32} + 156 q^{34} - 394 q^{35} + 914 q^{37} - 111 q^{38} + 257 q^{40} + 914 q^{41} + 1722 q^{43} - 1243 q^{44} + 584 q^{46} - 380 q^{47} + 2446 q^{49} + 454 q^{50} + 1552 q^{52} - 370 q^{53} + 918 q^{55} + 499 q^{56} + 2446 q^{58} - 492 q^{59} + 668 q^{61} - 578 q^{62} + 6475 q^{64} - 736 q^{65} + 4548 q^{67} - 5253 q^{68} + 7793 q^{70} - 258 q^{71} + 3096 q^{73} - 449 q^{74} + 6814 q^{76} - 3804 q^{77} + 2864 q^{79} + 1052 q^{80} + 14145 q^{82} - 2364 q^{83} + 3088 q^{85} - 2811 q^{86} + 8329 q^{88} + 4172 q^{89} + 7350 q^{91} - 13644 q^{92} + 6122 q^{94} - 3336 q^{95} + 6370 q^{97} - 1572 q^{98}+O(q^{100}) $ 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8 + 32 * q^10 - 154 * q^11 + 100 * q^13 - 42 * q^14 + 719 * q^16 - 32 * q^17 + 202 * q^19 + 132 * q^20 + 265 * q^22 - 552 * q^23 + 1086 * q^25 + 280 * q^26 + 390 * q^28 + 154 * q^29 + 560 * q^31 - 444 * q^32 + 156 * q^34 - 394 * q^35 + 914 * q^37 - 111 * q^38 + 257 * q^40 + 914 * q^41 + 1722 * q^43 - 1243 * q^44 + 584 * q^46 - 380 * q^47 + 2446 * q^49 + 454 * q^50 + 1552 * q^52 - 370 * q^53 + 918 * q^55 + 499 * q^56 + 2446 * q^58 - 492 * q^59 + 668 * q^61 - 578 * q^62 + 6475 * q^64 - 736 * q^65 + 4548 * q^67 - 5253 * q^68 + 7793 * q^70 - 258 * q^71 + 3096 * q^73 - 449 * q^74 + 6814 * q^76 - 3804 * q^77 + 2864 * q^79 + 1052 * q^80 + 14145 * q^82 - 2364 * q^83 + 3088 * q^85 - 2811 * q^86 + 8329 * q^88 + 4172 * q^89 + 7350 * q^91 - 13644 * q^92 + 6122 * q^94 - 3336 * q^95 + 6370 * q^97 - 1572 * 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 $

$2$	1.64226	0.580628	0.290314	−	0.956931i	$-0.406240\pi$
$2$	1.64226	0.580628	0.290314	+	0.956931i	$0.406240\pi$
$3$	0	0
$3$	0	0
$4$	−5.30297	−0.662871
$5$	14.9594	1.33801	0.669003	−	0.743260i	$-0.266721\pi$
$5$	14.9594	1.33801	0.669003	+	0.743260i	$0.266721\pi$
$6$	0	0
$7$	8.99120	0.485479	0.242739	−	0.970092i	$-0.421954\pi$
$7$	8.99120	0.485479	0.242739	+	0.970092i	$0.421954\pi$
$8$	−21.8470	−0.965509
$9$	0	0
$10$	24.5672	0.776884
$11$	38.0853	1.04392	0.521962	−	0.852969i	$-0.325200\pi$
$11$	38.0853	1.04392	0.521962	+	0.852969i	$0.325200\pi$
$12$	0	0
$13$	48.4818	1.03434	0.517171	−	0.855882i	$-0.326985\pi$
$13$	48.4818	1.03434	0.517171	+	0.855882i	$0.326985\pi$
$14$	14.7659	0.281883
$15$	0	0
$16$	6.54525	0.102269
$17$	67.1804	0.958449	0.479224	−	0.877692i	$-0.340918\pi$
$17$	67.1804	0.958449	0.479224	+	0.877692i	$0.340918\pi$
$18$	0	0
$19$	−150.323	−1.81508	−0.907541	−	0.419963i	$-0.862043\pi$
$19$	−150.323	−1.81508	−0.907541	+	0.419963i	$0.862043\pi$
$20$	−79.3290	−0.886926
$21$	0	0
$22$	62.5461	0.606131
$23$	178.955	1.62238	0.811189	−	0.584784i	$-0.198821\pi$
$23$	178.955	1.62238	0.811189	+	0.584784i	$0.198821\pi$
$24$	0	0
$25$	98.7825	0.790260
$26$	79.6200	0.600568
$27$	0	0
$28$	−47.6800	−0.321810
$29$	207.458	1.32841	0.664207	−	0.747548i	$-0.268769\pi$
$29$	207.458	1.32841	0.664207	+	0.747548i	$0.268769\pi$
$30$	0	0
$31$	3.02502	0.0175261	0.00876307	−	0.999962i	$-0.497211\pi$
$31$	3.02502	0.0175261	0.00876307	+	0.999962i	$0.497211\pi$
$32$	185.525	1.02489
$33$	0	0
$34$	110.328	0.556502
$35$	134.503	0.649574
$36$	0	0
$37$	−202.734	−0.900790	−0.450395	−	0.892829i	$-0.648717\pi$
$37$	−202.734	−0.900790	−0.450395	+	0.892829i	$0.648717\pi$
$38$	−246.871	−1.05389
$39$	0	0
$40$	−326.817	−1.29186
$41$	−137.802	−0.524904	−0.262452	−	0.964945i	$-0.584531\pi$
$41$	−137.802	−0.524904	−0.262452	+	0.964945i	$0.584531\pi$
$42$	0	0
$43$	−3.56013	−0.0126259	−0.00631295	−	0.999980i	$-0.502009\pi$
$43$	−3.56013	−0.0126259	−0.00631295	+	0.999980i	$0.502009\pi$
$44$	−201.965	−0.691987
$45$	0	0
$46$	293.891	0.941998
$47$	−29.9753	−0.0930286	−0.0465143	−	0.998918i	$-0.514811\pi$
$47$	−29.9753	−0.0930286	−0.0465143	+	0.998918i	$0.514811\pi$
$48$	0	0
$49$	−262.158	−0.764310
$50$	162.227	0.458847
$51$	0	0
$52$	−257.098	−0.685635
$53$	199.697	0.517556	0.258778	−	0.965937i	$-0.416680\pi$
$53$	199.697	0.517556	0.258778	+	0.965937i	$0.416680\pi$
$54$	0	0
$55$	569.732	1.39678
$56$	−196.431	−0.468734
$57$	0	0
$58$	340.701	0.771315
$59$	−6.04685	−0.0133429	−0.00667146	−	0.999978i	$-0.502124\pi$
$59$	−6.04685	−0.0133429	−0.00667146	+	0.999978i	$0.502124\pi$
$60$	0	0
$61$	−75.2305	−0.157906	−0.0789531	−	0.996878i	$-0.525158\pi$
$61$	−75.2305	−0.157906	−0.0789531	+	0.996878i	$0.525158\pi$
$62$	4.96789	0.0101762
$63$	0	0
$64$	252.319	0.492810
$65$	725.257	1.38396
$66$	0	0
$67$	1014.53	1.84991	0.924956	−	0.380074i	$-0.124101\pi$
$67$	1014.53	1.84991	0.924956	+	0.380074i	$0.124101\pi$
$68$	−356.255	−0.635328
$69$	0	0
$70$	220.889	0.377161
$71$	−326.257	−0.545347	−0.272673	−	0.962107i	$-0.587908\pi$
$71$	−326.257	−0.545347	−0.272673	+	0.962107i	$0.587908\pi$
$72$	0	0
$73$	−533.435	−0.855257	−0.427629	−	0.903954i	$-0.640651\pi$
$73$	−533.435	−0.855257	−0.427629	+	0.903954i	$0.640651\pi$
$74$	−332.942	−0.523024
$75$	0	0
$76$	797.161	1.20317
$77$	342.433	0.506803
$78$	0	0
$79$	416.413	0.593040	0.296520	−	0.955027i	$-0.404174\pi$
$79$	416.413	0.593040	0.296520	+	0.955027i	$0.404174\pi$
$80$	97.9127	0.136837
$81$	0	0
$82$	−226.307	−0.304774
$83$	22.6681	0.0299776	0.0149888	−	0.999888i	$-0.495229\pi$
$83$	22.6681	0.0299776	0.0149888	+	0.999888i	$0.495229\pi$
$84$	0	0
$85$	1004.98	1.28241
$86$	−5.84667	−0.00733096
$87$	0	0
$88$	−832.049	−1.00792
$89$	950.440	1.13198	0.565991	−	0.824411i	$-0.308494\pi$
$89$	950.440	1.13198	0.565991	+	0.824411i	$0.308494\pi$
$90$	0	0
$91$	435.910	0.502151
$92$	−948.993	−1.07543
$93$	0	0
$94$	−49.2273	−0.0540150
$95$	−2248.74	−2.42859
$96$	0	0
$97$	−1232.55	−1.29017	−0.645087	−	0.764109i	$-0.723179\pi$
$97$	−1232.55	−1.29017	−0.645087	+	0.764109i	$0.723179\pi$
$98$	−430.533	−0.443780
$99$	0	0
$100$	−523.841	−0.523841
$101$	−1177.20	−1.15976	−0.579878	−	0.814703i	$-0.696900\pi$
$101$	−1177.20	−1.15976	−0.579878	+	0.814703i	$0.696900\pi$
$102$	0	0
$103$	−689.326	−0.659430	−0.329715	−	0.944080i	$-0.606953\pi$
$103$	−689.326	−0.659430	−0.329715	+	0.944080i	$0.606953\pi$
$104$	−1059.18	−0.998667
$105$	0	0
$106$	327.955	0.300508
$107$	1289.98	1.16548	0.582741	−	0.812658i	$-0.301980\pi$
$107$	1289.98	1.16548	0.582741	+	0.812658i	$0.301980\pi$
$108$	0	0
$109$	953.605	0.837971	0.418985	−	0.907993i	$-0.362386\pi$
$109$	953.605	0.837971	0.418985	+	0.907993i	$0.362386\pi$
$110$	935.650	0.811007
$111$	0	0
$112$	58.8496	0.0496497
$113$	918.318	0.764496	0.382248	−	0.924060i	$-0.375150\pi$
$113$	918.318	0.764496	0.382248	+	0.924060i	$0.375150\pi$
$114$	0	0
$115$	2677.05	2.17075
$116$	−1100.14	−0.880568
$117$	0	0
$118$	−9.93052	−0.00774727
$119$	604.032	0.465307
$120$	0	0
$121$	119.492	0.0897760
$122$	−123.548	−0.0916847
$123$	0	0
$124$	−16.0416	−0.0116176
$125$	−392.197	−0.280633
$126$	0	0
$127$	−1899.40	−1.32712	−0.663562	−	0.748121i	$-0.730956\pi$
$127$	−1899.40	−1.32712	−0.663562	+	0.748121i	$0.730956\pi$
$128$	−1069.83	−0.738751
$129$	0	0
$130$	1191.06	0.803563
$131$	974.544	0.649972	0.324986	−	0.945719i	$-0.394640\pi$
$131$	974.544	0.649972	0.324986	+	0.945719i	$0.394640\pi$
$132$	0	0
$133$	−1351.59	−0.881184
$134$	1666.12	1.07411
$135$	0	0
$136$	−1467.69	−0.925391
$137$	−806.685	−0.503064	−0.251532	−	0.967849i	$-0.580934\pi$
$137$	−806.685	−0.503064	−0.251532	+	0.967849i	$0.580934\pi$
$138$	0	0
$139$	774.780	0.472777	0.236388	−	0.971659i	$-0.424036\pi$
$139$	774.780	0.472777	0.236388	+	0.971659i	$0.424036\pi$
$140$	−713.263	−0.430584
$141$	0	0
$142$	−535.801	−0.316644
$143$	1846.45	1.07977
$144$	0	0
$145$	3103.44	1.77743
$146$	−876.040	−0.496586
$147$	0	0
$148$	1075.09	0.597108
$149$	2899.05	1.59395	0.796977	−	0.604010i	$-0.206431\pi$
$149$	2899.05	1.59395	0.796977	+	0.604010i	$0.206431\pi$
$150$	0	0
$151$	−1733.07	−0.934010	−0.467005	−	0.884255i	$-0.654667\pi$
$151$	−1733.07	−0.934010	−0.467005	+	0.884255i	$0.654667\pi$
$152$	3284.11	1.75248
$153$	0	0
$154$	562.365	0.294264
$155$	45.2524	0.0234501
$156$	0	0
$157$	3278.07	1.66636	0.833180	−	0.553002i	$-0.186518\pi$
$157$	3278.07	1.66636	0.833180	+	0.553002i	$0.186518\pi$
$158$	683.860	0.344335
$159$	0	0
$160$	2775.33	1.37131
$161$	1609.02	0.787630
$162$	0	0
$163$	1798.26	0.864114	0.432057	−	0.901846i	$-0.357788\pi$
$163$	1798.26	0.864114	0.432057	+	0.901846i	$0.357788\pi$
$164$	730.759	0.347943
$165$	0	0
$166$	37.2269	0.0174058
$167$	1631.92	0.756176	0.378088	−	0.925770i	$-0.376582\pi$
$167$	1631.92	0.756176	0.378088	+	0.925770i	$0.376582\pi$
$168$	0	0
$169$	153.489	0.0698628
$170$	1650.43	0.744603
$171$	0	0
$172$	18.8792	0.00836935
$173$	−1016.02	−0.446510	−0.223255	−	0.974760i	$-0.571668\pi$
$173$	−1016.02	−0.446510	−0.223255	+	0.974760i	$0.571668\pi$
$174$	0	0
$175$	888.173	0.383655
$176$	249.278	0.106762
$177$	0	0
$178$	1560.87	0.657260
$179$	−3509.96	−1.46562	−0.732812	−	0.680431i	$-0.761792\pi$
$179$	−3509.96	−1.46562	−0.732812	+	0.680431i	$0.761792\pi$
$180$	0	0
$181$	4599.76	1.88894	0.944469	−	0.328601i	$-0.106577\pi$
$181$	4599.76	1.88894	0.944469	+	0.328601i	$0.106577\pi$
$182$	715.879	0.291563
$183$	0	0
$184$	−3909.63	−1.56642
$185$	−3032.77	−1.20526
$186$	0	0
$187$	2558.59	1.00055
$188$	158.958	0.0616660
$189$	0	0
$190$	−3693.03	−1.41011
$191$	−2044.74	−0.774617	−0.387309	−	0.921950i	$-0.626595\pi$
$191$	−2044.74	−0.774617	−0.387309	+	0.921950i	$0.626595\pi$
$192$	0	0
$193$	3380.44	1.26077	0.630387	−	0.776281i	$-0.282896\pi$
$193$	3380.44	1.26077	0.630387	+	0.776281i	$0.282896\pi$
$194$	−2024.18	−0.749111
$195$	0	0
$196$	1390.22	0.506639
$197$	−3648.78	−1.31962	−0.659809	−	0.751434i	$-0.729363\pi$
$197$	−3648.78	−1.31962	−0.659809	+	0.751434i	$0.729363\pi$
$198$	0	0
$199$	2062.88	0.734841	0.367421	−	0.930055i	$-0.380241\pi$
$199$	2062.88	0.734841	0.367421	+	0.930055i	$0.380241\pi$
$200$	−2158.10	−0.763004
$201$	0	0
$202$	−1933.27	−0.673387
$203$	1865.30	0.644917
$204$	0	0
$205$	−2061.43	−0.702324
$206$	−1132.06	−0.382884
$207$	0	0
$208$	317.326	0.105782
$209$	−5725.12	−1.89481
$210$	0	0
$211$	2331.38	0.760657	0.380329	−	0.924851i	$-0.375811\pi$
$211$	2331.38	0.760657	0.380329	+	0.924851i	$0.375811\pi$
$212$	−1058.99	−0.343073
$213$	0	0
$214$	2118.48	0.676712
$215$	−53.2572	−0.0168935
$216$	0	0
$217$	27.1986	0.00850857
$218$	1566.07	0.486549
$219$	0	0
$220$	−3021.27	−0.925882
$221$	3257.03	0.991364
$222$	0	0
$223$	−2341.99	−0.703280	−0.351640	−	0.936135i	$-0.614376\pi$
$223$	−2341.99	−0.703280	−0.351640	+	0.936135i	$0.614376\pi$
$224$	1668.09	0.497562
$225$	0	0
$226$	1508.12	0.443888
$227$	2210.03	0.646190	0.323095	−	0.946367i	$-0.395277\pi$
$227$	2210.03	0.646190	0.323095	+	0.946367i	$0.395277\pi$
$228$	0	0
$229$	5629.97	1.62462	0.812312	−	0.583223i	$-0.198209\pi$
$229$	5629.97	1.62462	0.812312	+	0.583223i	$0.198209\pi$
$230$	4396.43	1.26040
$231$	0	0
$232$	−4532.34	−1.28260
$233$	4407.81	1.23933	0.619667	−	0.784865i	$-0.287268\pi$
$233$	4407.81	1.23933	0.619667	+	0.784865i	$0.287268\pi$
$234$	0	0
$235$	−448.411	−0.124473
$236$	32.0662	0.00884464
$237$	0	0
$238$	991.979	0.270170
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−1737.76	−0.464476	−0.232238	−	0.972659i	$-0.574605\pi$
$241$	−1737.76	−0.464476	−0.232238	+	0.972659i	$0.574605\pi$
$242$	196.237	0.0521264
$243$	0	0
$244$	398.945	0.104671
$245$	−3921.72	−1.02265
$246$	0	0
$247$	−7287.96	−1.87742
$248$	−66.0877	−0.0169217
$249$	0	0
$250$	−644.091	−0.162943
$251$	−7688.15	−1.93335	−0.966677	−	0.256000i	$-0.917595\pi$
$251$	−7688.15	−1.93335	−0.966677	+	0.256000i	$0.917595\pi$
$252$	0	0
$253$	6815.56	1.69364
$254$	−3119.32	−0.770566
$255$	0	0
$256$	−3775.49	−0.921749
$257$	4397.59	1.06737	0.533685	−	0.845683i	$-0.320807\pi$
$257$	4397.59	1.06737	0.533685	+	0.845683i	$0.320807\pi$
$258$	0	0
$259$	−1822.82	−0.437315
$260$	−3846.02	−0.917384
$261$	0	0
$262$	1600.46	0.377392
$263$	6359.16	1.49096	0.745480	−	0.666528i	$-0.232220\pi$
$263$	6359.16	1.49096	0.745480	+	0.666528i	$0.232220\pi$
$264$	0	0
$265$	2987.34	0.692494
$266$	−2219.66	−0.511640
$267$	0	0
$268$	−5380.00	−1.22625
$269$	2753.97	0.624209	0.312105	−	0.950048i	$-0.398966\pi$
$269$	2753.97	0.624209	0.312105	+	0.950048i	$0.398966\pi$
$270$	0	0
$271$	−5798.34	−1.29972	−0.649860	−	0.760054i	$-0.725173\pi$
$271$	−5798.34	−1.29972	−0.649860	+	0.760054i	$0.725173\pi$
$272$	439.712	0.0980201
$273$	0	0
$274$	−1324.79	−0.292093
$275$	3762.16	0.824971
$276$	0	0
$277$	2154.92	0.467424	0.233712	−	0.972306i	$-0.424913\pi$
$277$	2154.92	0.467424	0.233712	+	0.972306i	$0.424913\pi$
$278$	1272.39	0.274507
$279$	0	0
$280$	−2938.48	−0.627170
$281$	3133.55	0.665239	0.332619	−	0.943061i	$-0.392068\pi$
$281$	3133.55	0.665239	0.332619	+	0.943061i	$0.392068\pi$
$282$	0	0
$283$	3007.19	0.631656	0.315828	−	0.948816i	$-0.397718\pi$
$283$	3007.19	0.631656	0.315828	+	0.948816i	$0.397718\pi$
$284$	1730.13	0.361495
$285$	0	0
$286$	3032.35	0.626947
$287$	−1239.00	−0.254830
$288$	0	0
$289$	−399.799	−0.0813758
$290$	5096.67	1.03202
$291$	0	0
$292$	2828.79	0.566925
$293$	7154.40	1.42650	0.713250	−	0.700909i	$-0.247222\pi$
$293$	7154.40	1.42650	0.713250	+	0.700909i	$0.247222\pi$
$294$	0	0
$295$	−90.4570	−0.0178529
$296$	4429.12	0.869721
$297$	0	0
$298$	4761.00	0.925494
$299$	8676.07	1.67809
$300$	0	0
$301$	−32.0098	−0.00612961
$302$	−2846.16	−0.542312
$303$	0	0
$304$	−983.904	−0.185628
$305$	−1125.40	−0.211279
$306$	0	0
$307$	5822.52	1.08244	0.541219	−	0.840881i	$-0.317963\pi$
$307$	5822.52	1.08244	0.541219	+	0.840881i	$0.317963\pi$
$308$	−1815.91	−0.335945
$309$	0	0
$310$	74.3164	0.0136158
$311$	−3594.31	−0.655353	−0.327677	−	0.944790i	$-0.606266\pi$
$311$	−3594.31	−0.655353	−0.327677	+	0.944790i	$0.606266\pi$
$312$	0	0
$313$	3075.96	0.555475	0.277737	−	0.960657i	$-0.410416\pi$
$313$	3075.96	0.555475	0.277737	+	0.960657i	$0.410416\pi$
$314$	5383.46	0.967535
$315$	0	0
$316$	−2208.23	−0.393109
$317$	−3341.97	−0.592124	−0.296062	−	0.955169i	$-0.595674\pi$
$317$	−3341.97	−0.592124	−0.296062	+	0.955169i	$0.595674\pi$
$318$	0	0
$319$	7901.12	1.38676
$320$	3774.53	0.659383
$321$	0	0
$322$	2642.43	0.457320
$323$	−10098.8	−1.73966
$324$	0	0
$325$	4789.16	0.817399
$326$	2953.22	0.501728
$327$	0	0
$328$	3010.56	0.506799
$329$	−269.514	−0.0451634
$330$	0	0
$331$	−9215.66	−1.53033	−0.765164	−	0.643835i	$-0.777342\pi$
$331$	−9215.66	−1.53033	−0.765164	+	0.643835i	$0.777342\pi$
$332$	−120.208	−0.0198713
$333$	0	0
$334$	2680.04	0.439057
$335$	15176.7	2.47519
$336$	0	0
$337$	−1866.06	−0.301635	−0.150817	−	0.988562i	$-0.548191\pi$
$337$	−1866.06	−0.301635	−0.150817	+	0.988562i	$0.548191\pi$
$338$	252.069	0.0405643
$339$	0	0
$340$	−5329.35	−0.850073
$341$	115.209	0.0182959
$342$	0	0
$343$	−5441.10	−0.856535
$344$	77.7780	0.0121904
$345$	0	0
$346$	−1668.57	−0.259256
$347$	2268.15	0.350896	0.175448	−	0.984489i	$-0.443863\pi$
$347$	2268.15	0.350896	0.175448	+	0.984489i	$0.443863\pi$
$348$	0	0
$349$	−7339.41	−1.12570	−0.562851	−	0.826559i	$-0.690295\pi$
$349$	−7339.41	−1.12570	−0.562851	+	0.826559i	$0.690295\pi$
$350$	1458.61	0.222761
$351$	0	0
$352$	7065.78	1.06991
$353$	11086.8	1.67165	0.835825	−	0.548997i	$-0.184990\pi$
$353$	11086.8	1.67165	0.835825	+	0.548997i	$0.184990\pi$
$354$	0	0
$355$	−4880.60	−0.729677
$356$	−5040.15	−0.750358
$357$	0	0
$358$	−5764.28	−0.850982
$359$	−5236.20	−0.769794	−0.384897	−	0.922960i	$-0.625763\pi$
$359$	−5236.20	−0.769794	−0.384897	+	0.922960i	$0.625763\pi$
$360$	0	0
$361$	15738.1	2.29452
$362$	7554.03	1.09677
$363$	0	0
$364$	−2311.62	−0.332862
$365$	−7979.84	−1.14434
$366$	0	0
$367$	10418.7	1.48189	0.740944	−	0.671567i	$-0.234378\pi$
$367$	10418.7	1.48189	0.740944	+	0.671567i	$0.234378\pi$
$368$	1171.30	0.165920
$369$	0	0
$370$	−4980.61	−0.699809
$371$	1795.51	0.251263
$372$	0	0
$373$	988.257	0.137185	0.0685925	−	0.997645i	$-0.478149\pi$
$373$	988.257	0.137185	0.0685925	+	0.997645i	$0.478149\pi$
$374$	4201.87	0.580946
$375$	0	0
$376$	654.870	0.0898200
$377$	10058.0	1.37403
$378$	0	0
$379$	−3852.85	−0.522183	−0.261091	−	0.965314i	$-0.584082\pi$
$379$	−3852.85	−0.522183	−0.261091	+	0.965314i	$0.584082\pi$
$380$	11925.0	1.60984
$381$	0	0
$382$	−3357.99	−0.449764
$383$	−8404.65	−1.12130	−0.560650	−	0.828053i	$-0.689449\pi$
$383$	−8404.65	−1.12130	−0.560650	+	0.828053i	$0.689449\pi$
$384$	0	0
$385$	5122.57	0.678105
$386$	5551.57	0.732040
$387$	0	0
$388$	6536.19	0.855219
$389$	5118.26	0.667110	0.333555	−	0.942731i	$-0.391752\pi$
$389$	5118.26	0.667110	0.333555	+	0.942731i	$0.391752\pi$
$390$	0	0
$391$	12022.3	1.55497
$392$	5727.37	0.737949
$393$	0	0
$394$	−5992.25	−0.766207
$395$	6229.27	0.793490
$396$	0	0
$397$	7068.63	0.893613	0.446806	−	0.894631i	$-0.352561\pi$
$397$	7068.63	0.893613	0.446806	+	0.894631i	$0.352561\pi$
$398$	3387.79	0.426669
$399$	0	0
$400$	646.556	0.0808195
$401$	−3923.50	−0.488605	−0.244302	−	0.969699i	$-0.578559\pi$
$401$	−3923.50	−0.488605	−0.244302	+	0.969699i	$0.578559\pi$
$402$	0	0
$403$	146.659	0.0181280
$404$	6242.64	0.768769
$405$	0	0
$406$	3063.31	0.374457
$407$	−7721.18	−0.940356
$408$	0	0
$409$	−2838.79	−0.343201	−0.171601	−	0.985167i	$-0.554894\pi$
$409$	−2838.79	−0.343201	−0.171601	+	0.985167i	$0.554894\pi$
$410$	−3385.41	−0.407789
$411$	0	0
$412$	3655.48	0.437117
$413$	−54.3684	−0.00647771
$414$	0	0
$415$	339.100	0.0401102
$416$	8994.59	1.06009
$417$	0	0
$418$	−9402.15	−1.10018
$419$	−3407.48	−0.397294	−0.198647	−	0.980071i	$-0.563655\pi$
$419$	−3407.48	−0.397294	−0.198647	+	0.980071i	$0.563655\pi$
$420$	0	0
$421$	3512.14	0.406582	0.203291	−	0.979118i	$-0.434836\pi$
$421$	3512.14	0.406582	0.203291	+	0.979118i	$0.434836\pi$
$422$	3828.74	0.441659
$423$	0	0
$424$	−4362.78	−0.499706
$425$	6636.25	0.757424
$426$	0	0
$427$	−676.412	−0.0766601
$428$	−6840.70	−0.772565
$429$	0	0
$430$	−87.4624	−0.00980886
$431$	−13307.9	−1.48729	−0.743643	−	0.668577i	$-0.766904\pi$
$431$	−13307.9	−1.48729	−0.743643	+	0.668577i	$0.766904\pi$
$432$	0	0
$433$	−3485.08	−0.386795	−0.193398	−	0.981120i	$-0.561951\pi$
$433$	−3485.08	−0.386795	−0.193398	+	0.981120i	$0.561951\pi$
$434$	44.6672	0.00494031
$435$	0	0
$436$	−5056.94	−0.555467
$437$	−26901.1	−2.94475
$438$	0	0
$439$	−574.466	−0.0624551	−0.0312275	−	0.999512i	$-0.509942\pi$
$439$	−574.466	−0.0624551	−0.0312275	+	0.999512i	$0.509942\pi$
$440$	−12446.9	−1.34860
$441$	0	0
$442$	5348.90	0.575613
$443$	−3009.96	−0.322817	−0.161408	−	0.986888i	$-0.551604\pi$
$443$	−3009.96	−0.322817	−0.161408	+	0.986888i	$0.551604\pi$
$444$	0	0
$445$	14218.0	1.51460
$446$	−3846.17	−0.408344
$447$	0	0
$448$	2268.65	0.239249
$449$	−7945.65	−0.835141	−0.417571	−	0.908645i	$-0.637118\pi$
$449$	−7945.65	−0.835141	−0.417571	+	0.908645i	$0.637118\pi$
$450$	0	0
$451$	−5248.23	−0.547959
$452$	−4869.81	−0.506763
$453$	0	0
$454$	3629.46	0.375196
$455$	6520.93	0.671881
$456$	0	0
$457$	−18716.7	−1.91582	−0.957912	−	0.287061i	$-0.907322\pi$
$457$	−18716.7	−1.91582	−0.957912	+	0.287061i	$0.907322\pi$
$458$	9245.90	0.943302
$459$	0	0
$460$	−14196.3	−1.43893
$461$	16100.6	1.62663	0.813317	−	0.581820i	$-0.197659\pi$
$461$	16100.6	1.62663	0.813317	+	0.581820i	$0.197659\pi$
$462$	0	0
$463$	877.711	0.0881008	0.0440504	−	0.999029i	$-0.485974\pi$
$463$	877.711	0.0881008	0.0440504	+	0.999029i	$0.485974\pi$
$464$	1357.87	0.135856
$465$	0	0
$466$	7238.78	0.719592
$467$	665.934	0.0659866	0.0329933	−	0.999456i	$-0.489496\pi$
$467$	665.934	0.0659866	0.0329933	+	0.999456i	$0.489496\pi$
$468$	0	0
$469$	9121.80	0.898093
$470$	−736.409	−0.0722724
$471$	0	0
$472$	132.105	0.0128827
$473$	−135.589	−0.0131805
$474$	0	0
$475$	−14849.3	−1.43439
$476$	−3203.16	−0.308438
$477$	0	0
$478$	392.501	0.0375577
$479$	−3377.20	−0.322147	−0.161073	−	0.986942i	$-0.551496\pi$
$479$	−3377.20	−0.322147	−0.161073	+	0.986942i	$0.551496\pi$
$480$	0	0
$481$	−9828.91	−0.931725
$482$	−2853.85	−0.269688
$483$	0	0
$484$	−633.662	−0.0595099
$485$	−18438.2	−1.72626
$486$	0	0
$487$	12467.8	1.16011	0.580053	−	0.814579i	$-0.303032\pi$
$487$	12467.8	1.16011	0.580053	+	0.814579i	$0.303032\pi$
$488$	1643.56	0.152460
$489$	0	0
$490$	−6440.50	−0.593780
$491$	766.987	0.0704962	0.0352481	−	0.999379i	$-0.488778\pi$
$491$	766.987	0.0704962	0.0352481	+	0.999379i	$0.488778\pi$
$492$	0	0
$493$	13937.1	1.27322
$494$	−11968.7	−1.09008
$495$	0	0
$496$	19.7995	0.00179239
$497$	−2933.44	−0.264754
$498$	0	0
$499$	−420.150	−0.0376924	−0.0188462	−	0.999822i	$-0.505999\pi$
$499$	−420.150	−0.0376924	−0.0188462	+	0.999822i	$0.505999\pi$
$500$	2079.81	0.186024
$501$	0	0
$502$	−12626.0	−1.12256
$503$	5011.25	0.444216	0.222108	−	0.975022i	$-0.428706\pi$
$503$	5011.25	0.444216	0.222108	+	0.975022i	$0.428706\pi$
$504$	0	0
$505$	−17610.1	−1.55176
$506$	11192.9	0.983374
$507$	0	0
$508$	10072.5	0.879713
$509$	17388.1	1.51417	0.757086	−	0.653315i	$-0.226622\pi$
$509$	17388.1	1.51417	0.757086	+	0.653315i	$0.226622\pi$
$510$	0	0
$511$	−4796.21	−0.415209
$512$	2358.26	0.203557
$513$	0	0
$514$	7222.01	0.619745
$515$	−10311.9	−0.882322
$516$	0	0
$517$	−1141.62	−0.0971148
$518$	−2993.55	−0.253917
$519$	0	0
$520$	−15844.7	−1.33622
$521$	−875.228	−0.0735977	−0.0367989	−	0.999323i	$-0.511716\pi$
$521$	−875.228	−0.0735977	−0.0367989	+	0.999323i	$0.511716\pi$
$522$	0	0
$523$	3548.98	0.296723	0.148361	−	0.988933i	$-0.452600\pi$
$523$	3548.98	0.296723	0.148361	+	0.988933i	$0.452600\pi$
$524$	−5167.98	−0.430848
$525$	0	0
$526$	10443.4	0.865693
$527$	203.222	0.0167979
$528$	0	0
$529$	19857.9	1.63211
$530$	4906.00	0.402081
$531$	0	0
$532$	7167.43	0.584112
$533$	−6680.89	−0.542930
$534$	0	0
$535$	19297.2	1.55942
$536$	−22164.3	−1.78611
$537$	0	0
$538$	4522.74	0.362433
$539$	−9984.39	−0.797881
$540$	0	0
$541$	−12240.3	−0.972740	−0.486370	−	0.873753i	$-0.661679\pi$
$541$	−12240.3	−0.972740	−0.486370	+	0.873753i	$0.661679\pi$
$542$	−9522.41	−0.754654
$543$	0	0
$544$	12463.6	0.982305
$545$	14265.3	1.12121
$546$	0	0
$547$	−21323.8	−1.66680	−0.833399	−	0.552672i	$-0.813608\pi$
$547$	−21323.8	−1.66680	−0.833399	+	0.552672i	$0.813608\pi$
$548$	4277.82	0.333466
$549$	0	0
$550$	6178.47	0.479001
$551$	−31185.8	−2.41118
$552$	0	0
$553$	3744.05	0.287908
$554$	3538.94	0.271400
$555$	0	0
$556$	−4108.63	−0.313390
$557$	11995.9	0.912536	0.456268	−	0.889843i	$-0.349186\pi$
$557$	11995.9	0.912536	0.456268	+	0.889843i	$0.349186\pi$
$558$	0	0
$559$	−172.601	−0.0130595
$560$	880.353	0.0664316
$561$	0	0
$562$	5146.12	0.386256
$563$	−10754.9	−0.805090	−0.402545	−	0.915400i	$-0.631874\pi$
$563$	−10754.9	−0.805090	−0.402545	+	0.915400i	$0.631874\pi$
$564$	0	0
$565$	13737.5	1.02290
$566$	4938.59	0.366757
$567$	0	0
$568$	7127.74	0.526538
$569$	−10126.6	−0.746098	−0.373049	−	0.927812i	$-0.621688\pi$
$569$	−10126.6	−0.746098	−0.373049	+	0.927812i	$0.621688\pi$
$570$	0	0
$571$	−6144.82	−0.450355	−0.225178	−	0.974318i	$-0.572296\pi$
$571$	−6144.82	−0.450355	−0.225178	+	0.974318i	$0.572296\pi$
$572$	−9791.65	−0.715751
$573$	0	0
$574$	−2034.77	−0.147961
$575$	17677.6	1.28210
$576$	0	0
$577$	−8486.65	−0.612312	−0.306156	−	0.951981i	$-0.599043\pi$
$577$	−8486.65	−0.612312	−0.306156	+	0.951981i	$0.599043\pi$
$578$	−656.576	−0.0472491
$579$	0	0
$580$	−16457.5	−1.17821
$581$	203.813	0.0145535
$582$	0	0
$583$	7605.52	0.540289
$584$	11653.9	0.825759
$585$	0	0
$586$	11749.4	0.828266
$587$	4366.08	0.306997	0.153499	−	0.988149i	$-0.450946\pi$
$587$	4366.08	0.306997	0.153499	+	0.988149i	$0.450946\pi$
$588$	0	0
$589$	−454.732	−0.0318114
$590$	−148.554	−0.0103659
$591$	0	0
$592$	−1326.94	−0.0921233
$593$	−2469.49	−0.171012	−0.0855058	−	0.996338i	$-0.527251\pi$
$593$	−2469.49	−0.171012	−0.0855058	+	0.996338i	$0.527251\pi$
$594$	0	0
$595$	9035.93	0.622583
$596$	−15373.5	−1.05659
$597$	0	0
$598$	14248.4	0.974348
$599$	15117.8	1.03121	0.515607	−	0.856825i	$-0.327567\pi$
$599$	15117.8	1.03121	0.515607	+	0.856825i	$0.327567\pi$
$600$	0	0
$601$	−29028.4	−1.97020	−0.985102	−	0.171972i	$-0.944986\pi$
$601$	−29028.4	−1.97020	−0.985102	+	0.171972i	$0.944986\pi$
$602$	−52.5685	−0.00355902
$603$	0	0
$604$	9190.43	0.619128
$605$	1787.52	0.120121
$606$	0	0
$607$	13185.9	0.881709	0.440854	−	0.897579i	$-0.354676\pi$
$607$	13185.9	0.881709	0.440854	+	0.897579i	$0.354676\pi$
$608$	−27888.7	−1.86026
$609$	0	0
$610$	−1848.20	−0.122675
$611$	−1453.26	−0.0962234
$612$	0	0
$613$	9485.46	0.624983	0.312491	−	0.949921i	$-0.398837\pi$
$613$	9485.46	0.624983	0.312491	+	0.949921i	$0.398837\pi$
$614$	9562.11	0.628494
$615$	0	0
$616$	−7481.12	−0.489323
$617$	−17702.0	−1.15503	−0.577517	−	0.816379i	$-0.695978\pi$
$617$	−17702.0	−1.15503	−0.577517	+	0.816379i	$0.695978\pi$
$618$	0	0
$619$	−18669.7	−1.21228	−0.606139	−	0.795359i	$-0.707282\pi$
$619$	−18669.7	−1.21228	−0.606139	+	0.795359i	$0.707282\pi$
$620$	−239.972	−0.0155444
$621$	0	0
$622$	−5902.81	−0.380516
$623$	8545.59	0.549553
$624$	0	0
$625$	−18214.8	−1.16575
$626$	5051.54	0.322524
$627$	0	0
$628$	−17383.5	−1.10458
$629$	−13619.7	−0.863361
$630$	0	0
$631$	22169.6	1.39866	0.699332	−	0.714797i	$-0.253481\pi$
$631$	22169.6	1.39866	0.699332	+	0.714797i	$0.253481\pi$
$632$	−9097.37	−0.572585
$633$	0	0
$634$	−5488.39	−0.343804
$635$	−28413.9	−1.77570
$636$	0	0
$637$	−12709.9	−0.790558
$638$	12975.7	0.805194
$639$	0	0
$640$	−16003.9	−0.988453
$641$	−27439.7	−1.69080	−0.845400	−	0.534133i	$-0.820638\pi$
$641$	−27439.7	−1.69080	−0.845400	+	0.534133i	$0.820638\pi$
$642$	0	0
$643$	21997.7	1.34915	0.674576	−	0.738205i	$-0.264326\pi$
$643$	21997.7	1.34915	0.674576	+	0.738205i	$0.264326\pi$
$644$	−8532.58	−0.522097
$645$	0	0
$646$	−16584.9	−1.01010
$647$	5991.40	0.364059	0.182030	−	0.983293i	$-0.441733\pi$
$647$	5991.40	0.364059	0.182030	+	0.983293i	$0.441733\pi$
$648$	0	0
$649$	−230.296	−0.0139290
$650$	7865.06	0.474605
$651$	0	0
$652$	−9536.11	−0.572796
$653$	−17418.9	−1.04388	−0.521940	−	0.852982i	$-0.674791\pi$
$653$	−17418.9	−1.04388	−0.521940	+	0.852982i	$0.674791\pi$
$654$	0	0
$655$	14578.6	0.869666
$656$	−901.948	−0.0536816
$657$	0	0
$658$	−442.613	−0.0262232
$659$	19621.0	1.15983	0.579914	−	0.814678i	$-0.303086\pi$
$659$	19621.0	1.15983	0.579914	+	0.814678i	$0.303086\pi$
$660$	0	0
$661$	−20513.7	−1.20710	−0.603548	−	0.797327i	$-0.706247\pi$
$661$	−20513.7	−1.20710	−0.603548	+	0.797327i	$0.706247\pi$
$662$	−15134.5	−0.888551
$663$	0	0
$664$	−495.229	−0.0289437
$665$	−20218.9	−1.17903
$666$	0	0
$667$	37125.7	2.15519
$668$	−8654.00	−0.501247
$669$	0	0
$670$	24924.1	1.43717
$671$	−2865.18	−0.164842
$672$	0	0
$673$	20349.5	1.16555	0.582777	−	0.812632i	$-0.301966\pi$
$673$	20349.5	1.16555	0.582777	+	0.812632i	$0.301966\pi$
$674$	−3064.57	−0.175138
$675$	0	0
$676$	−813.946	−0.0463101
$677$	−8294.64	−0.470885	−0.235442	−	0.971888i	$-0.575654\pi$
$677$	−8294.64	−0.470885	−0.235442	+	0.971888i	$0.575654\pi$
$678$	0	0
$679$	−11082.1	−0.626352
$680$	−21955.7	−1.23818
$681$	0	0
$682$	189.204	0.0106231
$683$	−9553.32	−0.535209	−0.267604	−	0.963529i	$-0.586232\pi$
$683$	−9553.32	−0.535209	−0.267604	+	0.963529i	$0.586232\pi$
$684$	0	0
$685$	−12067.5	−0.673102
$686$	−8935.72	−0.497328
$687$	0	0
$688$	−23.3019	−0.00129125
$689$	9681.68	0.535330
$690$	0	0
$691$	−23725.8	−1.30618	−0.653090	−	0.757280i	$-0.726528\pi$
$691$	−23725.8	−1.30618	−0.653090	+	0.757280i	$0.726528\pi$
$692$	5387.90	0.295979
$693$	0	0
$694$	3724.91	0.203740
$695$	11590.2	0.632578
$696$	0	0
$697$	−9257.58	−0.503093
$698$	−12053.3	−0.653614
$699$	0	0
$700$	−4709.95	−0.254314
$701$	−16086.7	−0.866743	−0.433371	−	0.901215i	$-0.642676\pi$
$701$	−16086.7	−0.866743	−0.433371	+	0.901215i	$0.642676\pi$
$702$	0	0
$703$	30475.6	1.63501
$704$	9609.64	0.514456
$705$	0	0
$706$	18207.5	0.970606
$707$	−10584.4	−0.563037
$708$	0	0
$709$	−17961.1	−0.951399	−0.475699	−	0.879608i	$-0.657805\pi$
$709$	−17961.1	−0.951399	−0.475699	+	0.879608i	$0.657805\pi$
$710$	−8015.24	−0.423671
$711$	0	0
$712$	−20764.2	−1.09294
$713$	541.343	0.0284340
$714$	0	0
$715$	27621.7	1.44474
$716$	18613.2	0.971520
$717$	0	0
$718$	−8599.22	−0.446964
$719$	−33773.7	−1.75180	−0.875901	−	0.482490i	$-0.839732\pi$
$719$	−33773.7	−1.75180	−0.875901	+	0.482490i	$0.839732\pi$
$720$	0	0
$721$	−6197.87	−0.320139
$722$	25846.2	1.33226
$723$	0	0
$724$	−24392.4	−1.25212
$725$	20493.3	1.04979
$726$	0	0
$727$	11828.2	0.603418	0.301709	−	0.953400i	$-0.402443\pi$
$727$	11828.2	0.603418	0.301709	+	0.953400i	$0.402443\pi$
$728$	−9523.31	−0.484832
$729$	0	0
$730$	−13105.0	−0.664435
$731$	−239.171	−0.0121013
$732$	0	0
$733$	29880.4	1.50567	0.752837	−	0.658207i	$-0.228685\pi$
$733$	29880.4	1.50567	0.752837	+	0.658207i	$0.228685\pi$
$734$	17110.3	0.860425
$735$	0	0
$736$	33200.6	1.66276
$737$	38638.6	1.93117
$738$	0	0
$739$	−36271.3	−1.80550	−0.902748	−	0.430169i	$-0.858454\pi$
$739$	−36271.3	−1.80550	−0.902748	+	0.430169i	$0.858454\pi$
$740$	16082.7	0.798934
$741$	0	0
$742$	2948.71	0.145890
$743$	−10602.0	−0.523486	−0.261743	−	0.965138i	$-0.584297\pi$
$743$	−10602.0	−0.523486	−0.261743	+	0.965138i	$0.584297\pi$
$744$	0	0
$745$	43367.9	2.13272
$746$	1622.98	0.0796535
$747$	0	0
$748$	−13568.1	−0.663234
$749$	11598.4	0.565817
$750$	0	0
$751$	−33482.9	−1.62691	−0.813453	−	0.581630i	$-0.802415\pi$
$751$	−33482.9	−1.62691	−0.813453	+	0.581630i	$0.802415\pi$
$752$	−196.196	−0.00951399
$753$	0	0
$754$	16517.8	0.797803
$755$	−25925.7	−1.24971
$756$	0	0
$757$	−11711.1	−0.562281	−0.281140	−	0.959667i	$-0.590713\pi$
$757$	−11711.1	−0.562281	−0.281140	+	0.959667i	$0.590713\pi$
$758$	−6327.39	−0.303194
$759$	0	0
$760$	49128.2	2.34483
$761$	−13075.4	−0.622843	−0.311422	−	0.950272i	$-0.600805\pi$
$761$	−13075.4	−0.622843	−0.311422	+	0.950272i	$0.600805\pi$
$762$	0	0
$763$	8574.05	0.406817
$764$	10843.2	0.513471
$765$	0	0
$766$	−13802.7	−0.651058
$767$	−293.162	−0.0138011
$768$	0	0
$769$	7140.61	0.334847	0.167423	−	0.985885i	$-0.446455\pi$
$769$	7140.61	0.334847	0.167423	+	0.985885i	$0.446455\pi$
$770$	8412.62	0.393727
$771$	0	0
$772$	−17926.4	−0.835730
$773$	−7854.25	−0.365456	−0.182728	−	0.983163i	$-0.558493\pi$
$773$	−7854.25	−0.365456	−0.182728	+	0.983163i	$0.558493\pi$
$774$	0	0
$775$	298.819	0.0138502
$776$	26927.6	1.24567
$777$	0	0
$778$	8405.52	0.387343
$779$	20714.9	0.952743
$780$	0	0
$781$	−12425.6	−0.569300
$782$	19743.7	0.902857
$783$	0	0
$784$	−1715.89	−0.0781656
$785$	49037.9	2.22960
$786$	0	0
$787$	−5451.59	−0.246923	−0.123461	−	0.992349i	$-0.539400\pi$
$787$	−5451.59	−0.246923	−0.123461	+	0.992349i	$0.539400\pi$
$788$	19349.4	0.874736
$789$	0	0
$790$	10230.1	0.460723
$791$	8256.78	0.371147
$792$	0	0
$793$	−3647.31	−0.163329
$794$	11608.5	0.518856
$795$	0	0
$796$	−10939.4	−0.487105
$797$	38627.5	1.71676	0.858378	−	0.513018i	$-0.171472\pi$
$797$	38627.5	1.71676	0.858378	+	0.513018i	$0.171472\pi$
$798$	0	0
$799$	−2013.75	−0.0891632
$800$	18326.6	0.809930
$801$	0	0
$802$	−6443.43	−0.283698
$803$	−20316.0	−0.892823
$804$	0	0
$805$	24069.9	1.05385
$806$	240.852	0.0105256
$807$	0	0
$808$	25718.2	1.11976
$809$	5448.03	0.236764	0.118382	−	0.992968i	$-0.462229\pi$
$809$	5448.03	0.236764	0.118382	+	0.992968i	$0.462229\pi$
$810$	0	0
$811$	711.355	0.0308003	0.0154002	−	0.999881i	$-0.495098\pi$
$811$	711.355	0.0308003	0.0154002	+	0.999881i	$0.495098\pi$
$812$	−9891.62	−0.427497
$813$	0	0
$814$	−12680.2	−0.545997
$815$	26900.8	1.15619
$816$	0	0
$817$	535.170	0.0229171
$818$	−4662.05	−0.199272
$819$	0	0
$820$	10931.7	0.465550
$821$	−8748.41	−0.371890	−0.185945	−	0.982560i	$-0.559535\pi$
$821$	−8748.41	−0.371890	−0.185945	+	0.982560i	$0.559535\pi$
$822$	0	0
$823$	19292.1	0.817110	0.408555	−	0.912734i	$-0.366033\pi$
$823$	19292.1	0.817110	0.408555	+	0.912734i	$0.366033\pi$
$824$	15059.7	0.636686
$825$	0	0
$826$	−89.2872	−0.00376114
$827$	−23666.9	−0.995136	−0.497568	−	0.867425i	$-0.665774\pi$
$827$	−23666.9	−0.995136	−0.497568	+	0.867425i	$0.665774\pi$
$828$	0	0
$829$	−16671.7	−0.698469	−0.349235	−	0.937035i	$-0.613558\pi$
$829$	−16671.7	−0.698469	−0.349235	+	0.937035i	$0.613558\pi$
$830$	556.891	0.0232891
$831$	0	0
$832$	12232.9	0.509734
$833$	−17611.9	−0.732552
$834$	0	0
$835$	24412.4	1.01177
$836$	30360.1	1.25601
$837$	0	0
$838$	−5595.98	−0.230680
$839$	36306.3	1.49396	0.746981	−	0.664845i	$-0.231503\pi$
$839$	36306.3	1.49396	0.746981	+	0.664845i	$0.231503\pi$
$840$	0	0
$841$	18649.9	0.764686
$842$	5767.86	0.236073
$843$	0	0
$844$	−12363.2	−0.504218
$845$	2296.09	0.0934769
$846$	0	0
$847$	1074.37	0.0435844
$848$	1307.07	0.0529302
$849$	0	0
$850$	10898.5	0.439781
$851$	−36280.2	−1.46142
$852$	0	0
$853$	−29371.3	−1.17896	−0.589480	−	0.807783i	$-0.700667\pi$
$853$	−29371.3	−1.17896	−0.589480	+	0.807783i	$0.700667\pi$
$854$	−1110.85	−0.0445110
$855$	0	0
$856$	−28182.1	−1.12528
$857$	28023.6	1.11700	0.558500	−	0.829505i	$-0.311377\pi$
$857$	28023.6	1.11700	0.558500	+	0.829505i	$0.311377\pi$
$858$	0	0
$859$	−16280.3	−0.646657	−0.323328	−	0.946287i	$-0.604802\pi$
$859$	−16280.3	−0.646657	−0.323328	+	0.946287i	$0.604802\pi$
$860$	282.421	0.0111982
$861$	0	0
$862$	−21855.1	−0.863560
$863$	12738.9	0.502476	0.251238	−	0.967925i	$-0.419162\pi$
$863$	12738.9	0.502476	0.251238	+	0.967925i	$0.419162\pi$
$864$	0	0
$865$	−15199.0	−0.597433
$866$	−5723.42	−0.224584
$867$	0	0
$868$	−144.233	−0.00564009
$869$	15859.2	0.619088
$870$	0	0
$871$	49186.1	1.91344
$872$	−20833.4	−0.809069
$873$	0	0
$874$	−44178.7	−1.70980
$875$	−3526.32	−0.136241
$876$	0	0
$877$	3975.43	0.153068	0.0765340	−	0.997067i	$-0.475615\pi$
$877$	3975.43	0.153068	0.0765340	+	0.997067i	$0.475615\pi$
$878$	−943.425	−0.0362632
$879$	0	0
$880$	3729.04	0.142848
$881$	−22853.1	−0.873940	−0.436970	−	0.899476i	$-0.643948\pi$
$881$	−22853.1	−0.873940	−0.436970	+	0.899476i	$0.643948\pi$
$882$	0	0
$883$	−31388.3	−1.19626	−0.598131	−	0.801398i	$-0.704090\pi$
$883$	−31388.3	−1.19626	−0.598131	+	0.801398i	$0.704090\pi$
$884$	−17271.9	−0.657146
$885$	0	0
$886$	−4943.16	−0.187436
$887$	−12606.9	−0.477224	−0.238612	−	0.971115i	$-0.576692\pi$
$887$	−12606.9	−0.477224	−0.238612	+	0.971115i	$0.576692\pi$
$888$	0	0
$889$	−17077.9	−0.644291
$890$	23349.7	0.879418
$891$	0	0
$892$	12419.5	0.466184
$893$	4505.99	0.168855
$894$	0	0
$895$	−52506.7	−1.96101
$896$	−9619.01	−0.358648
$897$	0	0
$898$	−13048.9	−0.484906
$899$	627.566	0.0232820
$900$	0	0
$901$	13415.7	0.496051
$902$	−8618.98	−0.318160
$903$	0	0
$904$	−20062.5	−0.738128
$905$	68809.5	2.52741
$906$	0	0
$907$	−8371.33	−0.306467	−0.153233	−	0.988190i	$-0.548969\pi$
$907$	−8371.33	−0.306467	−0.153233	+	0.988190i	$0.548969\pi$
$908$	−11719.7	−0.428340
$909$	0	0
$910$	10709.1	0.390113
$911$	−45405.6	−1.65132	−0.825661	−	0.564166i	$-0.809198\pi$
$911$	−45405.6	−1.65132	−0.825661	+	0.564166i	$0.809198\pi$
$912$	0	0
$913$	863.320	0.0312943
$914$	−30737.8	−1.11238
$915$	0	0
$916$	−29855.6	−1.07692
$917$	8762.32	0.315548
$918$	0	0
$919$	3823.47	0.137241	0.0686206	−	0.997643i	$-0.478140\pi$
$919$	3823.47	0.137241	0.0686206	+	0.997643i	$0.478140\pi$
$920$	−58485.5	−2.09588
$921$	0	0
$922$	26441.4	0.944469
$923$	−15817.6	−0.564075
$924$	0	0
$925$	−20026.6	−0.711858
$926$	1441.43	0.0511538
$927$	0	0
$928$	38488.7	1.36148
$929$	49363.0	1.74332	0.871662	−	0.490107i	$-0.163042\pi$
$929$	49363.0	1.74332	0.871662	+	0.490107i	$0.163042\pi$
$930$	0	0
$931$	39408.5	1.38729
$932$	−23374.5	−0.821519
$933$	0	0
$934$	1093.64	0.0383137
$935$	38274.8	1.33874
$936$	0	0
$937$	−38810.6	−1.35314	−0.676568	−	0.736380i	$-0.736533\pi$
$937$	−38810.6	−1.35314	−0.676568	+	0.736380i	$0.736533\pi$
$938$	14980.4	0.521458
$939$	0	0
$940$	2377.91	0.0825095
$941$	−6095.71	−0.211174	−0.105587	−	0.994410i	$-0.533672\pi$
$941$	−6095.71	−0.211174	−0.105587	+	0.994410i	$0.533672\pi$
$942$	0	0
$943$	−24660.3	−0.851592
$944$	−39.5781	−0.00136457
$945$	0	0
$946$	−222.672	−0.00765296
$947$	−31292.3	−1.07377	−0.536886	−	0.843655i	$-0.680399\pi$
$947$	−31292.3	−1.07377	−0.536886	+	0.843655i	$0.680399\pi$
$948$	0	0
$949$	−25861.9	−0.884628
$950$	−24386.5	−0.832845
$951$	0	0
$952$	−13196.3	−0.449258
$953$	26473.9	0.899867	0.449934	−	0.893062i	$-0.351448\pi$
$953$	26473.9	0.899867	0.449934	+	0.893062i	$0.351448\pi$
$954$	0	0
$955$	−30587.9	−1.03644
$956$	−1267.41	−0.0428776
$957$	0	0
$958$	−5546.26	−0.187047
$959$	−7253.06	−0.244227
$960$	0	0
$961$	−29781.8	−0.999693
$962$	−16141.7	−0.540985
$963$	0	0
$964$	9215.27	0.307888
$965$	50569.2	1.68692
$966$	0	0
$967$	39952.3	1.32862	0.664312	−	0.747456i	$-0.268725\pi$
$967$	39952.3	1.32862	0.664312	+	0.747456i	$0.268725\pi$
$968$	−2610.54	−0.0866796
$969$	0	0
$970$	−30280.4	−1.00231
$971$	26581.2	0.878508	0.439254	−	0.898363i	$-0.355243\pi$
$971$	26581.2	0.878508	0.439254	+	0.898363i	$0.355243\pi$
$972$	0	0
$973$	6966.20	0.229523
$974$	20475.5	0.673589
$975$	0	0
$976$	−492.402	−0.0161490
$977$	−43965.8	−1.43970	−0.719852	−	0.694127i	$-0.755791\pi$
$977$	−43965.8	−1.43970	−0.719852	+	0.694127i	$0.755791\pi$
$978$	0	0
$979$	36197.8	1.18170
$980$	20796.8	0.677886
$981$	0	0
$982$	1259.59	0.0409320
$983$	−5641.21	−0.183038	−0.0915191	−	0.995803i	$-0.529172\pi$
$983$	−5641.21	−0.183038	−0.0915191	+	0.995803i	$0.529172\pi$
$984$	0	0
$985$	−54583.4	−1.76566
$986$	22888.4	0.739266
$987$	0	0
$988$	38647.8	1.24448
$989$	−637.102	−0.0204840
$990$	0	0
$991$	−51711.9	−1.65760	−0.828800	−	0.559545i	$-0.810976\pi$
$991$	−51711.9	−1.65760	−0.828800	+	0.559545i	$0.810976\pi$
$992$	561.217	0.0179624
$993$	0	0
$994$	−4817.49	−0.153724
$995$	30859.3	0.983222
$996$	0	0
$997$	−30628.5	−0.972934	−0.486467	−	0.873699i	$-0.661715\pi$
$997$	−30628.5	−0.972934	−0.486467	+	0.873699i	$0.661715\pi$
$998$	−689.997	−0.0218852
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.e.1.20		32
3.2	odd	2		717.4.a.c.1.13	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.c.1.13	✓	32	3.2	odd	2
2151.4.a.e.1.20		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)

\(f(q)\)	\(=\)	\(q+1.64226 q^{2} -5.30297 q^{4} +14.9594 q^{5} +8.99120 q^{7} -21.8470 q^{8} +O(q^{10})\)	\(q+1.64226 q^{2} -5.30297 q^{4} +14.9594 q^{5} +8.99120 q^{7} -21.8470 q^{8} +24.5672 q^{10} +38.0853 q^{11} +48.4818 q^{13} +14.7659 q^{14} +6.54525 q^{16} +67.1804 q^{17} -150.323 q^{19} -79.3290 q^{20} +62.5461 q^{22} +178.955 q^{23} +98.7825 q^{25} +79.6200 q^{26} -47.6800 q^{28} +207.458 q^{29} +3.02502 q^{31} +185.525 q^{32} +110.328 q^{34} +134.503 q^{35} -202.734 q^{37} -246.871 q^{38} -326.817 q^{40} -137.802 q^{41} -3.56013 q^{43} -201.965 q^{44} +293.891 q^{46} -29.9753 q^{47} -262.158 q^{49} +162.227 q^{50} -257.098 q^{52} +199.697 q^{53} +569.732 q^{55} -196.431 q^{56} +340.701 q^{58} -6.04685 q^{59} -75.2305 q^{61} +4.96789 q^{62} +252.319 q^{64} +725.257 q^{65} +1014.53 q^{67} -356.255 q^{68} +220.889 q^{70} -326.257 q^{71} -533.435 q^{73} -332.942 q^{74} +797.161 q^{76} +342.433 q^{77} +416.413 q^{79} +97.9127 q^{80} -226.307 q^{82} +22.6681 q^{83} +1004.98 q^{85} -5.84667 q^{86} -832.049 q^{88} +950.440 q^{89} +435.910 q^{91} -948.993 q^{92} -49.2273 q^{94} -2248.74 q^{95} -1232.55 q^{97} -430.533 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8}+O(q^{10}) \) 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8	\( 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8} + 32 q^{10} - 154 q^{11} + 100 q^{13} - 42 q^{14} + 719 q^{16} - 32 q^{17} + 202 q^{19} + 132 q^{20} + 265 q^{22} - 552 q^{23} + 1086 q^{25} + 280 q^{26} + 390 q^{28} + 154 q^{29} + 560 q^{31} - 444 q^{32} + 156 q^{34} - 394 q^{35} + 914 q^{37} - 111 q^{38} + 257 q^{40} + 914 q^{41} + 1722 q^{43} - 1243 q^{44} + 584 q^{46} - 380 q^{47} + 2446 q^{49} + 454 q^{50} + 1552 q^{52} - 370 q^{53} + 918 q^{55} + 499 q^{56} + 2446 q^{58} - 492 q^{59} + 668 q^{61} - 578 q^{62} + 6475 q^{64} - 736 q^{65} + 4548 q^{67} - 5253 q^{68} + 7793 q^{70} - 258 q^{71} + 3096 q^{73} - 449 q^{74} + 6814 q^{76} - 3804 q^{77} + 2864 q^{79} + 1052 q^{80} + 14145 q^{82} - 2364 q^{83} + 3088 q^{85} - 2811 q^{86} + 8329 q^{88} + 4172 q^{89} + 7350 q^{91} - 13644 q^{92} + 6122 q^{94} - 3336 q^{95} + 6370 q^{97} - 1572 q^{98}+O(q^{100}) \) 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8 + 32 * q^10 - 154 * q^11 + 100 * q^13 - 42 * q^14 + 719 * q^16 - 32 * q^17 + 202 * q^19 + 132 * q^20 + 265 * q^22 - 552 * q^23 + 1086 * q^25 + 280 * q^26 + 390 * q^28 + 154 * q^29 + 560 * q^31 - 444 * q^32 + 156 * q^34 - 394 * q^35 + 914 * q^37 - 111 * q^38 + 257 * q^40 + 914 * q^41 + 1722 * q^43 - 1243 * q^44 + 584 * q^46 - 380 * q^47 + 2446 * q^49 + 454 * q^50 + 1552 * q^52 - 370 * q^53 + 918 * q^55 + 499 * q^56 + 2446 * q^58 - 492 * q^59 + 668 * q^61 - 578 * q^62 + 6475 * q^64 - 736 * q^65 + 4548 * q^67 - 5253 * q^68 + 7793 * q^70 - 258 * q^71 + 3096 * q^73 - 449 * q^74 + 6814 * q^76 - 3804 * q^77 + 2864 * q^79 + 1052 * q^80 + 14145 * q^82 - 2364 * q^83 + 3088 * q^85 - 2811 * q^86 + 8329 * q^88 + 4172 * q^89 + 7350 * q^91 - 13644 * q^92 + 6122 * q^94 - 3336 * q^95 + 6370 * q^97 - 1572 * q^98

Introduction

L-functions

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