LMFDB - Embedded newform 2394.4.a.bh.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2394,4,Mod(1,2394)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2394, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2394.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2394.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:	$141.250572554$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 429x^{5} + 1799x^{4} + 59687x^{3} - 308117x^{2} - 2682459x + 15997617 $ x^7 - 3x^6 - 429x^5 + 1799x^4 + 59687x^3 - 308117x^2 - 2682459x + 15997617
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-10.1491$ of defining polynomial
Character	$\chi$	$=$	2394.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +4.00000 q^{4} -13.1491 q^{5} -7.00000 q^{7} -8.00000 q^{8} +O(q^{10})$	\(q-2.00000 q^{2} +4.00000 q^{4} -13.1491 q^{5} -7.00000 q^{7} -8.00000 q^{8} +26.2982 q^{10} -65.3890 q^{11} +43.4823 q^{13} +14.0000 q^{14} +16.0000 q^{16} +13.8938 q^{17} +19.0000 q^{19} -52.5964 q^{20} +130.778 q^{22} +39.4886 q^{23} +47.8991 q^{25} -86.9647 q^{26} -28.0000 q^{28} -109.096 q^{29} +124.397 q^{31} -32.0000 q^{32} -27.7876 q^{34} +92.0438 q^{35} -231.001 q^{37} -38.0000 q^{38} +105.193 q^{40} -225.325 q^{41} +553.936 q^{43} -261.556 q^{44} -78.9773 q^{46} -65.2308 q^{47} +49.0000 q^{49} -95.7982 q^{50} +173.929 q^{52} +641.849 q^{53} +859.808 q^{55} +56.0000 q^{56} +218.193 q^{58} +598.264 q^{59} +109.297 q^{61} -248.794 q^{62} +64.0000 q^{64} -571.754 q^{65} +306.189 q^{67} +55.5752 q^{68} -184.088 q^{70} +429.432 q^{71} -879.248 q^{73} +462.001 q^{74} +76.0000 q^{76} +457.723 q^{77} +220.909 q^{79} -210.386 q^{80} +450.650 q^{82} -532.470 q^{83} -182.691 q^{85} -1107.87 q^{86} +523.112 q^{88} -106.803 q^{89} -304.376 q^{91} +157.955 q^{92} +130.462 q^{94} -249.833 q^{95} +538.521 q^{97} -98.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 14 q^{2} + 28 q^{4} - 18 q^{5} - 49 q^{7} - 56 q^{8}+O(q^{10}) $ 7 * q - 14 * q^2 + 28 * q^4 - 18 * q^5 - 49 * q^7 - 56 * q^8	$ 7 q - 14 q^{2} + 28 q^{4} - 18 q^{5} - 49 q^{7} - 56 q^{8} + 36 q^{10} - 28 q^{11} + 44 q^{13} + 98 q^{14} + 112 q^{16} - 26 q^{17} + 133 q^{19} - 72 q^{20} + 56 q^{22} - 56 q^{23} + 37 q^{25} - 88 q^{26} - 196 q^{28} - 270 q^{29} + 64 q^{31} - 224 q^{32} + 52 q^{34} + 126 q^{35} + 458 q^{37} - 266 q^{38} + 144 q^{40} - 110 q^{41} + 296 q^{43} - 112 q^{44} + 112 q^{46} + 142 q^{47} + 343 q^{49} - 74 q^{50} + 176 q^{52} - 330 q^{53} + 596 q^{55} + 392 q^{56} + 540 q^{58} - 236 q^{59} + 882 q^{61} - 128 q^{62} + 448 q^{64} + 180 q^{65} + 1622 q^{67} - 104 q^{68} - 252 q^{70} - 820 q^{71} + 1130 q^{73} - 916 q^{74} + 532 q^{76} + 196 q^{77} + 1694 q^{79} - 288 q^{80} + 220 q^{82} - 890 q^{83} + 1540 q^{85} - 592 q^{86} + 224 q^{88} - 686 q^{89} - 308 q^{91} - 224 q^{92} - 284 q^{94} - 342 q^{95} + 1772 q^{97} - 686 q^{98}+O(q^{100}) $ 7 * q - 14 * q^2 + 28 * q^4 - 18 * q^5 - 49 * q^7 - 56 * q^8 + 36 * q^10 - 28 * q^11 + 44 * q^13 + 98 * q^14 + 112 * q^16 - 26 * q^17 + 133 * q^19 - 72 * q^20 + 56 * q^22 - 56 * q^23 + 37 * q^25 - 88 * q^26 - 196 * q^28 - 270 * q^29 + 64 * q^31 - 224 * q^32 + 52 * q^34 + 126 * q^35 + 458 * q^37 - 266 * q^38 + 144 * q^40 - 110 * q^41 + 296 * q^43 - 112 * q^44 + 112 * q^46 + 142 * q^47 + 343 * q^49 - 74 * q^50 + 176 * q^52 - 330 * q^53 + 596 * q^55 + 392 * q^56 + 540 * q^58 - 236 * q^59 + 882 * q^61 - 128 * q^62 + 448 * q^64 + 180 * q^65 + 1622 * q^67 - 104 * q^68 - 252 * q^70 - 820 * q^71 + 1130 * q^73 - 916 * q^74 + 532 * q^76 + 196 * q^77 + 1694 * q^79 - 288 * q^80 + 220 * q^82 - 890 * q^83 + 1540 * q^85 - 592 * q^86 + 224 * q^88 - 686 * q^89 - 308 * q^91 - 224 * q^92 - 284 * q^94 - 342 * q^95 + 1772 * q^97 - 686 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−13.1491	−1.17609	−0.588046	−	0.808827i	$-0.700103\pi$
$5$	−13.1491	−1.17609	−0.588046	+	0.808827i	$0.700103\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	0	0
$10$	26.2982	0.831623
$11$	−65.3890	−1.79232	−0.896161	−	0.443730i	$-0.853655\pi$
$11$	−65.3890	−1.79232	−0.896161	+	0.443730i	$0.853655\pi$
$12$	0	0
$13$	43.4823	0.927680	0.463840	−	0.885919i	$-0.346471\pi$
$13$	43.4823	0.927680	0.463840	+	0.885919i	$0.346471\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	13.8938	0.198220	0.0991101	−	0.995076i	$-0.468400\pi$
$17$	13.8938	0.198220	0.0991101	+	0.995076i	$0.468400\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−52.5964	−0.588046
$21$	0	0
$22$	130.778	1.26736
$23$	39.4886	0.357998	0.178999	−	0.983849i	$-0.442714\pi$
$23$	39.4886	0.357998	0.178999	+	0.983849i	$0.442714\pi$
$24$	0	0
$25$	47.8991	0.383193
$26$	−86.9647	−0.655968
$27$	0	0
$28$	−28.0000	−0.188982
$29$	−109.096	−0.698576	−0.349288	−	0.937016i	$-0.613576\pi$
$29$	−109.096	−0.698576	−0.349288	+	0.937016i	$0.613576\pi$
$30$	0	0
$31$	124.397	0.720723	0.360361	−	0.932813i	$-0.382653\pi$
$31$	124.397	0.720723	0.360361	+	0.932813i	$0.382653\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−27.7876	−0.140163
$35$	92.0438	0.444521
$36$	0	0
$37$	−231.001	−1.02639	−0.513193	−	0.858273i	$-0.671537\pi$
$37$	−231.001	−1.02639	−0.513193	+	0.858273i	$0.671537\pi$
$38$	−38.0000	−0.162221
$39$	0	0
$40$	105.193	0.415811
$41$	−225.325	−0.858289	−0.429144	−	0.903236i	$-0.641185\pi$
$41$	−225.325	−0.858289	−0.429144	+	0.903236i	$0.641185\pi$
$42$	0	0
$43$	553.936	1.96452	0.982261	−	0.187518i	$-0.0600444\pi$
$43$	553.936	1.96452	0.982261	+	0.187518i	$0.0600444\pi$
$44$	−261.556	−0.896161
$45$	0	0
$46$	−78.9773	−0.253143
$47$	−65.2308	−0.202445	−0.101222	−	0.994864i	$-0.532275\pi$
$47$	−65.2308	−0.202445	−0.101222	+	0.994864i	$0.532275\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−95.7982	−0.270958
$51$	0	0
$52$	173.929	0.463840
$53$	641.849	1.66349	0.831743	−	0.555160i	$-0.187343\pi$
$53$	641.849	1.66349	0.831743	+	0.555160i	$0.187343\pi$
$54$	0	0
$55$	859.808	2.10793
$56$	56.0000	0.133631
$57$	0	0
$58$	218.193	0.493967
$59$	598.264	1.32013	0.660063	−	0.751211i	$-0.270530\pi$
$59$	598.264	1.32013	0.660063	+	0.751211i	$0.270530\pi$
$60$	0	0
$61$	109.297	0.229410	0.114705	−	0.993400i	$-0.463408\pi$
$61$	109.297	0.229410	0.114705	+	0.993400i	$0.463408\pi$
$62$	−248.794	−0.509628
$63$	0	0
$64$	64.0000	0.125000
$65$	−571.754	−1.09104
$66$	0	0
$67$	306.189	0.558313	0.279157	−	0.960246i	$-0.409945\pi$
$67$	306.189	0.558313	0.279157	+	0.960246i	$0.409945\pi$
$68$	55.5752	0.0991101
$69$	0	0
$70$	−184.088	−0.314324
$71$	429.432	0.717806	0.358903	−	0.933375i	$-0.383151\pi$
$71$	429.432	0.717806	0.358903	+	0.933375i	$0.383151\pi$
$72$	0	0
$73$	−879.248	−1.40970	−0.704851	−	0.709356i	$-0.748986\pi$
$73$	−879.248	−1.40970	−0.704851	+	0.709356i	$0.748986\pi$
$74$	462.001	0.725764
$75$	0	0
$76$	76.0000	0.114708
$77$	457.723	0.677434
$78$	0	0
$79$	220.909	0.314609	0.157305	−	0.987550i	$-0.449720\pi$
$79$	220.909	0.314609	0.157305	+	0.987550i	$0.449720\pi$
$80$	−210.386	−0.294023
$81$	0	0
$82$	450.650	0.606902
$83$	−532.470	−0.704171	−0.352086	−	0.935968i	$-0.614527\pi$
$83$	−532.470	−0.704171	−0.352086	+	0.935968i	$0.614527\pi$
$84$	0	0
$85$	−182.691	−0.233125
$86$	−1107.87	−1.38913
$87$	0	0
$88$	523.112	0.633681
$89$	−106.803	−0.127203	−0.0636015	−	0.997975i	$-0.520259\pi$
$89$	−106.803	−0.127203	−0.0636015	+	0.997975i	$0.520259\pi$
$90$	0	0
$91$	−304.376	−0.350630
$92$	157.955	0.178999
$93$	0	0
$94$	130.462	0.143150
$95$	−249.833	−0.269814
$96$	0	0
$97$	538.521	0.563697	0.281848	−	0.959459i	$-0.409053\pi$
$97$	538.521	0.563697	0.281848	+	0.959459i	$0.409053\pi$
$98$	−98.0000	−0.101015
$99$	0	0
$100$	191.596	0.191596
$101$	−386.951	−0.381218	−0.190609	−	0.981666i	$-0.561046\pi$
$101$	−386.951	−0.381218	−0.190609	+	0.981666i	$0.561046\pi$
$102$	0	0
$103$	19.5460	0.0186983	0.00934915	−	0.999956i	$-0.497024\pi$
$103$	19.5460	0.0186983	0.00934915	+	0.999956i	$0.497024\pi$
$104$	−347.859	−0.327984
$105$	0	0
$106$	−1283.70	−1.17626
$107$	−38.5487	−0.0348284	−0.0174142	−	0.999848i	$-0.505543\pi$
$107$	−38.5487	−0.0348284	−0.0174142	+	0.999848i	$0.505543\pi$
$108$	0	0
$109$	267.695	0.235234	0.117617	−	0.993059i	$-0.462474\pi$
$109$	267.695	0.235234	0.117617	+	0.993059i	$0.462474\pi$
$110$	−1719.62	−1.49054
$111$	0	0
$112$	−112.000	−0.0944911
$113$	541.771	0.451022	0.225511	−	0.974241i	$-0.427595\pi$
$113$	541.771	0.451022	0.225511	+	0.974241i	$0.427595\pi$
$114$	0	0
$115$	−519.240	−0.421038
$116$	−436.386	−0.349288
$117$	0	0
$118$	−1196.53	−0.933469
$119$	−97.2567	−0.0749202
$120$	0	0
$121$	2944.73	2.21242
$122$	−218.593	−0.162217
$123$	0	0
$124$	497.589	0.360361
$125$	1013.81	0.725422
$126$	0	0
$127$	−1054.06	−0.736479	−0.368239	−	0.929731i	$-0.620039\pi$
$127$	−1054.06	−0.736479	−0.368239	+	0.929731i	$0.620039\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	1143.51	0.771479
$131$	2836.14	1.89156	0.945781	−	0.324806i	$-0.105299\pi$
$131$	2836.14	1.89156	0.945781	+	0.324806i	$0.105299\pi$
$132$	0	0
$133$	−133.000	−0.0867110
$134$	−612.379	−0.394787
$135$	0	0
$136$	−111.150	−0.0700814
$137$	−838.111	−0.522662	−0.261331	−	0.965249i	$-0.584161\pi$
$137$	−838.111	−0.522662	−0.261331	+	0.965249i	$0.584161\pi$
$138$	0	0
$139$	−707.442	−0.431687	−0.215843	−	0.976428i	$-0.569250\pi$
$139$	−707.442	−0.431687	−0.215843	+	0.976428i	$0.569250\pi$
$140$	368.175	0.222261
$141$	0	0
$142$	−858.864	−0.507565
$143$	−2843.27	−1.66270
$144$	0	0
$145$	1434.52	0.821589
$146$	1758.50	0.996809
$147$	0	0
$148$	−924.003	−0.513193
$149$	−294.733	−0.162050	−0.0810251	−	0.996712i	$-0.525819\pi$
$149$	−294.733	−0.162050	−0.0810251	+	0.996712i	$0.525819\pi$
$150$	0	0
$151$	2235.62	1.20485	0.602424	−	0.798176i	$-0.294202\pi$
$151$	2235.62	1.20485	0.602424	+	0.798176i	$0.294202\pi$
$152$	−152.000	−0.0811107
$153$	0	0
$154$	−915.446	−0.479018
$155$	−1635.71	−0.847636
$156$	0	0
$157$	−890.904	−0.452878	−0.226439	−	0.974025i	$-0.572708\pi$
$157$	−890.904	−0.452878	−0.226439	+	0.974025i	$0.572708\pi$
$158$	−441.817	−0.222462
$159$	0	0
$160$	420.772	0.207906
$161$	−276.420	−0.135310
$162$	0	0
$163$	−3148.27	−1.51283	−0.756417	−	0.654090i	$-0.773052\pi$
$163$	−3148.27	−1.51283	−0.756417	+	0.654090i	$0.773052\pi$
$164$	−901.300	−0.429144
$165$	0	0
$166$	1064.94	0.497924
$167$	−2462.58	−1.14108	−0.570540	−	0.821270i	$-0.693266\pi$
$167$	−2462.58	−1.14108	−0.570540	+	0.821270i	$0.693266\pi$
$168$	0	0
$169$	−306.285	−0.139411
$170$	365.383	0.164844
$171$	0	0
$172$	2215.74	0.982261
$173$	−3727.29	−1.63804	−0.819018	−	0.573768i	$-0.805481\pi$
$173$	−3727.29	−1.63804	−0.819018	+	0.573768i	$0.805481\pi$
$174$	0	0
$175$	−335.294	−0.144833
$176$	−1046.22	−0.448080
$177$	0	0
$178$	213.606	0.0899461
$179$	−308.870	−0.128972	−0.0644861	−	0.997919i	$-0.520541\pi$
$179$	−308.870	−0.128972	−0.0644861	+	0.997919i	$0.520541\pi$
$180$	0	0
$181$	3150.48	1.29377	0.646887	−	0.762586i	$-0.276070\pi$
$181$	3150.48	1.29377	0.646887	+	0.762586i	$0.276070\pi$
$182$	608.753	0.247933
$183$	0	0
$184$	−315.909	−0.126571
$185$	3037.45	1.20712
$186$	0	0
$187$	−908.503	−0.355274
$188$	−260.923	−0.101222
$189$	0	0
$190$	499.666	0.190787
$191$	−1278.57	−0.484366	−0.242183	−	0.970231i	$-0.577863\pi$
$191$	−1278.57	−0.484366	−0.242183	+	0.970231i	$0.577863\pi$
$192$	0	0
$193$	2389.31	0.891122	0.445561	−	0.895251i	$-0.353004\pi$
$193$	2389.31	0.891122	0.445561	+	0.895251i	$0.353004\pi$
$194$	−1077.04	−0.398594
$195$	0	0
$196$	196.000	0.0714286
$197$	−998.054	−0.360956	−0.180478	−	0.983579i	$-0.557765\pi$
$197$	−998.054	−0.360956	−0.180478	+	0.983579i	$0.557765\pi$
$198$	0	0
$199$	−2790.34	−0.993979	−0.496989	−	0.867757i	$-0.665561\pi$
$199$	−2790.34	−0.993979	−0.496989	+	0.867757i	$0.665561\pi$
$200$	−383.193	−0.135479
$201$	0	0
$202$	773.902	0.269562
$203$	763.675	0.264037
$204$	0	0
$205$	2962.82	1.00943
$206$	−39.0920	−0.0132217
$207$	0	0
$208$	695.718	0.231920
$209$	−1242.39	−0.411187
$210$	0	0
$211$	3176.06	1.03625	0.518125	−	0.855305i	$-0.326630\pi$
$211$	3176.06	1.03625	0.518125	+	0.855305i	$0.326630\pi$
$212$	2567.40	0.831743
$213$	0	0
$214$	77.0974	0.0246274
$215$	−7283.77	−2.31046
$216$	0	0
$217$	−870.781	−0.272408
$218$	−535.390	−0.166336
$219$	0	0
$220$	3439.23	1.05397
$221$	604.136	0.183885
$222$	0	0
$223$	−3279.21	−0.984719	−0.492360	−	0.870392i	$-0.663865\pi$
$223$	−3279.21	−0.984719	−0.492360	+	0.870392i	$0.663865\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	−1083.54	−0.318921
$227$	−1782.69	−0.521240	−0.260620	−	0.965441i	$-0.583927\pi$
$227$	−1782.69	−0.521240	−0.260620	+	0.965441i	$0.583927\pi$
$228$	0	0
$229$	1709.92	0.493427	0.246714	−	0.969088i	$-0.420649\pi$
$229$	1709.92	0.493427	0.246714	+	0.969088i	$0.420649\pi$
$230$	1038.48	0.297719
$231$	0	0
$232$	872.771	0.246984
$233$	−317.917	−0.0893881	−0.0446940	−	0.999001i	$-0.514231\pi$
$233$	−317.917	−0.0893881	−0.0446940	+	0.999001i	$0.514231\pi$
$234$	0	0
$235$	857.727	0.238093
$236$	2393.06	0.660063
$237$	0	0
$238$	194.513	0.0529766
$239$	−1458.97	−0.394865	−0.197432	−	0.980317i	$-0.563260\pi$
$239$	−1458.97	−0.394865	−0.197432	+	0.980317i	$0.563260\pi$
$240$	0	0
$241$	1459.07	0.389988	0.194994	−	0.980804i	$-0.437531\pi$
$241$	1459.07	0.389988	0.194994	+	0.980804i	$0.437531\pi$
$242$	−5889.45	−1.56441
$243$	0	0
$244$	437.187	0.114705
$245$	−644.306	−0.168013
$246$	0	0
$247$	826.165	0.212824
$248$	−995.178	−0.254814
$249$	0	0
$250$	−2027.62	−0.512951
$251$	−570.597	−0.143489	−0.0717445	−	0.997423i	$-0.522857\pi$
$251$	−570.597	−0.143489	−0.0717445	+	0.997423i	$0.522857\pi$
$252$	0	0
$253$	−2582.12	−0.641647
$254$	2108.12	0.520769
$255$	0	0
$256$	256.000	0.0625000
$257$	−1073.69	−0.260603	−0.130301	−	0.991474i	$-0.541594\pi$
$257$	−1073.69	−0.260603	−0.130301	+	0.991474i	$0.541594\pi$
$258$	0	0
$259$	1617.00	0.387937
$260$	−2287.02	−0.545518
$261$	0	0
$262$	−5672.28	−1.33754
$263$	6721.78	1.57598	0.787990	−	0.615689i	$-0.211122\pi$
$263$	6721.78	1.57598	0.787990	+	0.615689i	$0.211122\pi$
$264$	0	0
$265$	−8439.75	−1.95641
$266$	266.000	0.0613139
$267$	0	0
$268$	1224.76	0.279157
$269$	197.406	0.0447437	0.0223718	−	0.999750i	$-0.492878\pi$
$269$	197.406	0.0447437	0.0223718	+	0.999750i	$0.492878\pi$
$270$	0	0
$271$	−3323.52	−0.744979	−0.372489	−	0.928036i	$-0.621496\pi$
$271$	−3323.52	−0.744979	−0.372489	+	0.928036i	$0.621496\pi$
$272$	222.301	0.0495551
$273$	0	0
$274$	1676.22	0.369578
$275$	−3132.07	−0.686804
$276$	0	0
$277$	2777.18	0.602399	0.301199	−	0.953561i	$-0.402613\pi$
$277$	2777.18	0.602399	0.301199	+	0.953561i	$0.402613\pi$
$278$	1414.88	0.305249
$279$	0	0
$280$	−736.350	−0.157162
$281$	3781.11	0.802713	0.401357	−	0.915922i	$-0.368539\pi$
$281$	3781.11	0.802713	0.401357	+	0.915922i	$0.368539\pi$
$282$	0	0
$283$	−1916.01	−0.402456	−0.201228	−	0.979544i	$-0.564493\pi$
$283$	−1916.01	−0.402456	−0.201228	+	0.979544i	$0.564493\pi$
$284$	1717.73	0.358903
$285$	0	0
$286$	5686.54	1.17571
$287$	1577.27	0.324403
$288$	0	0
$289$	−4719.96	−0.960709
$290$	−2869.04	−0.580951
$291$	0	0
$292$	−3516.99	−0.704851
$293$	2611.85	0.520771	0.260386	−	0.965505i	$-0.416150\pi$
$293$	2611.85	0.520771	0.260386	+	0.965505i	$0.416150\pi$
$294$	0	0
$295$	−7866.64	−1.55259
$296$	1848.01	0.362882
$297$	0	0
$298$	589.466	0.114587
$299$	1717.06	0.332107
$300$	0	0
$301$	−3877.55	−0.742520
$302$	−4471.24	−0.851956
$303$	0	0
$304$	304.000	0.0573539
$305$	−1437.15	−0.269807
$306$	0	0
$307$	2674.16	0.497140	0.248570	−	0.968614i	$-0.420039\pi$
$307$	2674.16	0.497140	0.248570	+	0.968614i	$0.420039\pi$
$308$	1830.89	0.338717
$309$	0	0
$310$	3271.43	0.599369
$311$	−4599.94	−0.838710	−0.419355	−	0.907822i	$-0.637744\pi$
$311$	−4599.94	−0.838710	−0.419355	+	0.907822i	$0.637744\pi$
$312$	0	0
$313$	207.634	0.0374957	0.0187479	−	0.999824i	$-0.494032\pi$
$313$	207.634	0.0374957	0.0187479	+	0.999824i	$0.494032\pi$
$314$	1781.81	0.320233
$315$	0	0
$316$	883.634	0.157305
$317$	−1905.55	−0.337623	−0.168812	−	0.985648i	$-0.553993\pi$
$317$	−1905.55	−0.337623	−0.168812	+	0.985648i	$0.553993\pi$
$318$	0	0
$319$	7133.71	1.25207
$320$	−841.543	−0.147012
$321$	0	0
$322$	552.841	0.0956790
$323$	263.982	0.0454748
$324$	0	0
$325$	2082.76	0.355480
$326$	6296.55	1.06973
$327$	0	0
$328$	1802.60	0.303451
$329$	456.616	0.0765169
$330$	0	0
$331$	−6301.75	−1.04645	−0.523226	−	0.852194i	$-0.675272\pi$
$331$	−6301.75	−1.04645	−0.523226	+	0.852194i	$0.675272\pi$
$332$	−2129.88	−0.352086
$333$	0	0
$334$	4925.16	0.806865
$335$	−4026.12	−0.656628
$336$	0	0
$337$	6633.68	1.07228	0.536142	−	0.844128i	$-0.319881\pi$
$337$	6633.68	1.07228	0.536142	+	0.844128i	$0.319881\pi$
$338$	612.571	0.0985783
$339$	0	0
$340$	−730.765	−0.116563
$341$	−8134.22	−1.29177
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−4431.49	−0.694563
$345$	0	0
$346$	7454.57	1.15827
$347$	−225.448	−0.0348781	−0.0174391	−	0.999848i	$-0.505551\pi$
$347$	−225.448	−0.0348781	−0.0174391	+	0.999848i	$0.505551\pi$
$348$	0	0
$349$	8061.50	1.23645	0.618227	−	0.786000i	$-0.287851\pi$
$349$	8061.50	1.23645	0.618227	+	0.786000i	$0.287851\pi$
$350$	670.587	0.102413
$351$	0	0
$352$	2092.45	0.316841
$353$	3007.65	0.453488	0.226744	−	0.973954i	$-0.427192\pi$
$353$	3007.65	0.453488	0.226744	+	0.973954i	$0.427192\pi$
$354$	0	0
$355$	−5646.65	−0.844205
$356$	−427.211	−0.0636015
$357$	0	0
$358$	617.740	0.0911971
$359$	−8069.53	−1.18633	−0.593166	−	0.805080i	$-0.702122\pi$
$359$	−8069.53	−1.18633	−0.593166	+	0.805080i	$0.702122\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	−6300.95	−0.914837
$363$	0	0
$364$	−1217.51	−0.175315
$365$	11561.3	1.65794
$366$	0	0
$367$	−9277.94	−1.31963	−0.659815	−	0.751428i	$-0.729365\pi$
$367$	−9277.94	−1.31963	−0.659815	+	0.751428i	$0.729365\pi$
$368$	631.818	0.0894995
$369$	0	0
$370$	−6074.91	−0.853566
$371$	−4492.95	−0.628739
$372$	0	0
$373$	−8904.71	−1.23611	−0.618054	−	0.786136i	$-0.712079\pi$
$373$	−8904.71	−1.23611	−0.618054	+	0.786136i	$0.712079\pi$
$374$	1817.01	0.251217
$375$	0	0
$376$	521.847	0.0715750
$377$	−4743.77	−0.648054
$378$	0	0
$379$	−10580.9	−1.43404	−0.717021	−	0.697051i	$-0.754495\pi$
$379$	−10580.9	−1.43404	−0.717021	+	0.697051i	$0.754495\pi$
$380$	−999.332	−0.134907
$381$	0	0
$382$	2557.13	0.342498
$383$	−189.111	−0.0252300	−0.0126150	−	0.999920i	$-0.504016\pi$
$383$	−189.111	−0.0252300	−0.0126150	+	0.999920i	$0.504016\pi$
$384$	0	0
$385$	−6018.65	−0.796725
$386$	−4778.63	−0.630119
$387$	0	0
$388$	2154.09	0.281848
$389$	−11102.1	−1.44704	−0.723522	−	0.690301i	$-0.757478\pi$
$389$	−11102.1	−1.44704	−0.723522	+	0.690301i	$0.757478\pi$
$390$	0	0
$391$	548.648	0.0709624
$392$	−392.000	−0.0505076
$393$	0	0
$394$	1996.11	0.255235
$395$	−2904.75	−0.370010
$396$	0	0
$397$	−9358.10	−1.18305	−0.591523	−	0.806288i	$-0.701473\pi$
$397$	−9358.10	−1.18305	−0.591523	+	0.806288i	$0.701473\pi$
$398$	5580.68	0.702849
$399$	0	0
$400$	766.385	0.0957982
$401$	−5709.93	−0.711073	−0.355536	−	0.934662i	$-0.615702\pi$
$401$	−5709.93	−0.711073	−0.355536	+	0.934662i	$0.615702\pi$
$402$	0	0
$403$	5409.08	0.668600
$404$	−1547.80	−0.190609
$405$	0	0
$406$	−1527.35	−0.186702
$407$	15104.9	1.83961
$408$	0	0
$409$	−1999.94	−0.241786	−0.120893	−	0.992666i	$-0.538576\pi$
$409$	−1999.94	−0.241786	−0.120893	+	0.992666i	$0.538576\pi$
$410$	−5925.64	−0.713772
$411$	0	0
$412$	78.1840	0.00934915
$413$	−4187.85	−0.498960
$414$	0	0
$415$	7001.51	0.828170
$416$	−1391.44	−0.163992
$417$	0	0
$418$	2484.78	0.290753
$419$	−7065.43	−0.823792	−0.411896	−	0.911231i	$-0.635133\pi$
$419$	−7065.43	−0.823792	−0.411896	+	0.911231i	$0.635133\pi$
$420$	0	0
$421$	5462.41	0.632355	0.316177	−	0.948700i	$-0.397601\pi$
$421$	5462.41	0.632355	0.316177	+	0.948700i	$0.397601\pi$
$422$	−6352.11	−0.732739
$423$	0	0
$424$	−5134.80	−0.588131
$425$	665.501	0.0759565
$426$	0	0
$427$	−765.077	−0.0867088
$428$	−154.195	−0.0174142
$429$	0	0
$430$	14567.5	1.63374
$431$	3617.53	0.404293	0.202146	−	0.979355i	$-0.435208\pi$
$431$	3617.53	0.404293	0.202146	+	0.979355i	$0.435208\pi$
$432$	0	0
$433$	2090.81	0.232050	0.116025	−	0.993246i	$-0.462985\pi$
$433$	2090.81	0.232050	0.116025	+	0.993246i	$0.462985\pi$
$434$	1741.56	0.192621
$435$	0	0
$436$	1070.78	0.117617
$437$	750.284	0.0821303
$438$	0	0
$439$	10140.6	1.10247	0.551234	−	0.834351i	$-0.314157\pi$
$439$	10140.6	1.10247	0.551234	+	0.834351i	$0.314157\pi$
$440$	−6878.46	−0.745268
$441$	0	0
$442$	−1208.27	−0.130026
$443$	−10839.6	−1.16254	−0.581272	−	0.813709i	$-0.697445\pi$
$443$	−10839.6	−1.16254	−0.581272	+	0.813709i	$0.697445\pi$
$444$	0	0
$445$	1404.36	0.149602
$446$	6558.43	0.696302
$447$	0	0
$448$	−448.000	−0.0472456
$449$	9494.25	0.997910	0.498955	−	0.866628i	$-0.333717\pi$
$449$	9494.25	0.997910	0.498955	+	0.866628i	$0.333717\pi$
$450$	0	0
$451$	14733.8	1.53833
$452$	2167.08	0.225511
$453$	0	0
$454$	3565.38	0.368572
$455$	4002.28	0.412373
$456$	0	0
$457$	12729.9	1.30302	0.651510	−	0.758640i	$-0.274136\pi$
$457$	12729.9	1.30302	0.651510	+	0.758640i	$0.274136\pi$
$458$	−3419.85	−0.348906
$459$	0	0
$460$	−2076.96	−0.210519
$461$	−5547.34	−0.560445	−0.280223	−	0.959935i	$-0.590408\pi$
$461$	−5547.34	−0.560445	−0.280223	+	0.959935i	$0.590408\pi$
$462$	0	0
$463$	−9422.73	−0.945813	−0.472907	−	0.881113i	$-0.656795\pi$
$463$	−9422.73	−0.945813	−0.472907	+	0.881113i	$0.656795\pi$
$464$	−1745.54	−0.174644
$465$	0	0
$466$	635.833	0.0632069
$467$	−4431.05	−0.439067	−0.219534	−	0.975605i	$-0.570454\pi$
$467$	−4431.05	−0.439067	−0.219534	+	0.975605i	$0.570454\pi$
$468$	0	0
$469$	−2143.32	−0.211022
$470$	−1715.45	−0.168358
$471$	0	0
$472$	−4786.11	−0.466735
$473$	−36221.3	−3.52106
$474$	0	0
$475$	910.082	0.0879104
$476$	−389.027	−0.0374601
$477$	0	0
$478$	2917.93	0.279211
$479$	−13131.8	−1.25263	−0.626314	−	0.779571i	$-0.715437\pi$
$479$	−13131.8	−1.25263	−0.626314	+	0.779571i	$0.715437\pi$
$480$	0	0
$481$	−10044.5	−0.952157
$482$	−2918.14	−0.275763
$483$	0	0
$484$	11778.9	1.10621
$485$	−7081.08	−0.662959
$486$	0	0
$487$	15368.7	1.43002	0.715012	−	0.699112i	$-0.246421\pi$
$487$	15368.7	1.43002	0.715012	+	0.699112i	$0.246421\pi$
$488$	−874.373	−0.0811086
$489$	0	0
$490$	1288.61	0.118803
$491$	−17352.8	−1.59495	−0.797475	−	0.603352i	$-0.793832\pi$
$491$	−17352.8	−1.59495	−0.797475	+	0.603352i	$0.793832\pi$
$492$	0	0
$493$	−1515.76	−0.138472
$494$	−1652.33	−0.150489
$495$	0	0
$496$	1990.36	0.180181
$497$	−3006.02	−0.271305
$498$	0	0
$499$	−15615.3	−1.40087	−0.700436	−	0.713715i	$-0.747011\pi$
$499$	−15615.3	−1.40087	−0.700436	+	0.713715i	$0.747011\pi$
$500$	4055.23	0.362711
$501$	0	0
$502$	1141.19	0.101462
$503$	16220.3	1.43783	0.718915	−	0.695098i	$-0.244639\pi$
$503$	16220.3	1.43783	0.718915	+	0.695098i	$0.244639\pi$
$504$	0	0
$505$	5088.06	0.448348
$506$	5164.25	0.453713
$507$	0	0
$508$	−4216.25	−0.368239
$509$	102.658	0.00893956	0.00446978	−	0.999990i	$-0.498577\pi$
$509$	102.658	0.00893956	0.00446978	+	0.999990i	$0.498577\pi$
$510$	0	0
$511$	6154.74	0.532817
$512$	−512.000	−0.0441942
$513$	0	0
$514$	2147.38	0.184274
$515$	−257.013	−0.0219909
$516$	0	0
$517$	4265.38	0.362846
$518$	−3234.01	−0.274313
$519$	0	0
$520$	4574.03	0.385740
$521$	−887.985	−0.0746705	−0.0373352	−	0.999303i	$-0.511887\pi$
$521$	−887.985	−0.0746705	−0.0373352	+	0.999303i	$0.511887\pi$
$522$	0	0
$523$	−18319.5	−1.53165	−0.765826	−	0.643047i	$-0.777670\pi$
$523$	−18319.5	−1.53165	−0.765826	+	0.643047i	$0.777670\pi$
$524$	11344.6	0.945781
$525$	0	0
$526$	−13443.6	−1.11439
$527$	1728.35	0.142862
$528$	0	0
$529$	−10607.6	−0.871838
$530$	16879.5	1.38339
$531$	0	0
$532$	−532.000	−0.0433555
$533$	−9797.66	−0.796217
$534$	0	0
$535$	506.881	0.0409614
$536$	−2449.51	−0.197393
$537$	0	0
$538$	−394.812	−0.0316385
$539$	−3204.06	−0.256046
$540$	0	0
$541$	−10446.2	−0.830163	−0.415081	−	0.909784i	$-0.636247\pi$
$541$	−10446.2	−0.830163	−0.415081	+	0.909784i	$0.636247\pi$
$542$	6647.03	0.526780
$543$	0	0
$544$	−444.602	−0.0350407
$545$	−3519.95	−0.276657
$546$	0	0
$547$	2394.80	0.187192	0.0935962	−	0.995610i	$-0.470164\pi$
$547$	2394.80	0.187192	0.0935962	+	0.995610i	$0.470164\pi$
$548$	−3352.44	−0.261331
$549$	0	0
$550$	6264.15	0.485644
$551$	−2072.83	−0.160264
$552$	0	0
$553$	−1546.36	−0.118911
$554$	−5554.36	−0.425960
$555$	0	0
$556$	−2829.77	−0.215843
$557$	−21022.1	−1.59916	−0.799581	−	0.600558i	$-0.794945\pi$
$557$	−21022.1	−1.59916	−0.799581	+	0.600558i	$0.794945\pi$
$558$	0	0
$559$	24086.4	1.82245
$560$	1472.70	0.111130
$561$	0	0
$562$	−7562.23	−0.567604
$563$	23997.6	1.79641	0.898204	−	0.439579i	$-0.144872\pi$
$563$	23997.6	1.79641	0.898204	+	0.439579i	$0.144872\pi$
$564$	0	0
$565$	−7123.81	−0.530444
$566$	3832.02	0.284579
$567$	0	0
$568$	−3435.46	−0.253783
$569$	−9163.79	−0.675160	−0.337580	−	0.941297i	$-0.609608\pi$
$569$	−9163.79	−0.675160	−0.337580	+	0.941297i	$0.609608\pi$
$570$	0	0
$571$	−4022.46	−0.294807	−0.147403	−	0.989076i	$-0.547092\pi$
$571$	−4022.46	−0.294807	−0.147403	+	0.989076i	$0.547092\pi$
$572$	−11373.1	−0.831350
$573$	0	0
$574$	−3154.55	−0.229387
$575$	1891.47	0.137182
$576$	0	0
$577$	−4767.23	−0.343955	−0.171978	−	0.985101i	$-0.555016\pi$
$577$	−4767.23	−0.343955	−0.171978	+	0.985101i	$0.555016\pi$
$578$	9439.92	0.679324
$579$	0	0
$580$	5738.08	0.410795
$581$	3727.29	0.266152
$582$	0	0
$583$	−41969.9	−2.98150
$584$	7033.98	0.498405
$585$	0	0
$586$	−5223.70	−0.368241
$587$	3889.64	0.273496	0.136748	−	0.990606i	$-0.456335\pi$
$587$	3889.64	0.273496	0.136748	+	0.990606i	$0.456335\pi$
$588$	0	0
$589$	2363.55	0.165345
$590$	15733.3	1.09785
$591$	0	0
$592$	−3696.01	−0.256596
$593$	−1824.09	−0.126317	−0.0631587	−	0.998003i	$-0.520117\pi$
$593$	−1824.09	−0.126317	−0.0631587	+	0.998003i	$0.520117\pi$
$594$	0	0
$595$	1278.84	0.0881131
$596$	−1178.93	−0.0810251
$597$	0	0
$598$	−3434.12	−0.234835
$599$	13082.7	0.892396	0.446198	−	0.894934i	$-0.352778\pi$
$599$	13082.7	0.892396	0.446198	+	0.894934i	$0.352778\pi$
$600$	0	0
$601$	23214.2	1.57559	0.787794	−	0.615938i	$-0.211223\pi$
$601$	23214.2	1.57559	0.787794	+	0.615938i	$0.211223\pi$
$602$	7755.10	0.525041
$603$	0	0
$604$	8942.47	0.602424
$605$	−38720.5	−2.60200
$606$	0	0
$607$	−14885.5	−0.995360	−0.497680	−	0.867361i	$-0.665815\pi$
$607$	−14885.5	−0.995360	−0.497680	+	0.867361i	$0.665815\pi$
$608$	−608.000	−0.0405554
$609$	0	0
$610$	2874.31	0.190782
$611$	−2836.39	−0.187804
$612$	0	0
$613$	−17221.2	−1.13468	−0.567340	−	0.823484i	$-0.692027\pi$
$613$	−17221.2	−1.13468	−0.567340	+	0.823484i	$0.692027\pi$
$614$	−5348.31	−0.351531
$615$	0	0
$616$	−3661.79	−0.239509
$617$	17920.3	1.16928	0.584638	−	0.811294i	$-0.301236\pi$
$617$	17920.3	1.16928	0.584638	+	0.811294i	$0.301236\pi$
$618$	0	0
$619$	−17116.8	−1.11144	−0.555721	−	0.831369i	$-0.687558\pi$
$619$	−17116.8	−1.11144	−0.555721	+	0.831369i	$0.687558\pi$
$620$	−6542.85	−0.423818
$621$	0	0
$622$	9199.88	0.593057
$623$	747.619	0.0480782
$624$	0	0
$625$	−19318.1	−1.23636
$626$	−415.268	−0.0265135
$627$	0	0
$628$	−3563.62	−0.226439
$629$	−3209.48	−0.203450
$630$	0	0
$631$	−8051.01	−0.507933	−0.253966	−	0.967213i	$-0.581735\pi$
$631$	−8051.01	−0.507933	−0.253966	+	0.967213i	$0.581735\pi$
$632$	−1767.27	−0.111231
$633$	0	0
$634$	3811.11	0.238736
$635$	13860.0	0.866167
$636$	0	0
$637$	2130.64	0.132526
$638$	−14267.4	−0.885348
$639$	0	0
$640$	1683.09	0.103953
$641$	5522.25	0.340274	0.170137	−	0.985420i	$-0.445579\pi$
$641$	5522.25	0.340274	0.170137	+	0.985420i	$0.445579\pi$
$642$	0	0
$643$	−22117.4	−1.35649	−0.678247	−	0.734834i	$-0.737260\pi$
$643$	−22117.4	−1.35649	−0.678247	+	0.734834i	$0.737260\pi$
$644$	−1105.68	−0.0676552
$645$	0	0
$646$	−527.965	−0.0321556
$647$	27786.9	1.68843	0.844215	−	0.536005i	$-0.180067\pi$
$647$	27786.9	1.68843	0.844215	+	0.536005i	$0.180067\pi$
$648$	0	0
$649$	−39119.9	−2.36609
$650$	−4165.53	−0.251362
$651$	0	0
$652$	−12593.1	−0.756417
$653$	−23910.1	−1.43289	−0.716443	−	0.697645i	$-0.754231\pi$
$653$	−23910.1	−1.43289	−0.716443	+	0.697645i	$0.754231\pi$
$654$	0	0
$655$	−37292.7	−2.22465
$656$	−3605.20	−0.214572
$657$	0	0
$658$	−913.232	−0.0541056
$659$	6542.30	0.386725	0.193363	−	0.981127i	$-0.438061\pi$
$659$	6542.30	0.386725	0.193363	+	0.981127i	$0.438061\pi$
$660$	0	0
$661$	21097.8	1.24146	0.620732	−	0.784023i	$-0.286836\pi$
$661$	21097.8	1.24146	0.620732	+	0.784023i	$0.286836\pi$
$662$	12603.5	0.739953
$663$	0	0
$664$	4259.76	0.248962
$665$	1748.83	0.101980
$666$	0	0
$667$	−4308.07	−0.250089
$668$	−9850.33	−0.570540
$669$	0	0
$670$	8052.23	0.464306
$671$	−7146.80	−0.411176
$672$	0	0
$673$	−18336.7	−1.05026	−0.525131	−	0.851021i	$-0.675984\pi$
$673$	−18336.7	−1.05026	−0.525131	+	0.851021i	$0.675984\pi$
$674$	−13267.4	−0.758219
$675$	0	0
$676$	−1225.14	−0.0697054
$677$	−20526.1	−1.16526	−0.582630	−	0.812737i	$-0.697976\pi$
$677$	−20526.1	−1.16526	−0.582630	+	0.812737i	$0.697976\pi$
$678$	0	0
$679$	−3769.65	−0.213057
$680$	1461.53	0.0824222
$681$	0	0
$682$	16268.4	0.913417
$683$	−24875.3	−1.39359	−0.696797	−	0.717268i	$-0.745392\pi$
$683$	−24875.3	−1.39359	−0.696797	+	0.717268i	$0.745392\pi$
$684$	0	0
$685$	11020.4	0.614698
$686$	686.000	0.0381802
$687$	0	0
$688$	8862.98	0.491131
$689$	27909.1	1.54318
$690$	0	0
$691$	4839.64	0.266438	0.133219	−	0.991087i	$-0.457469\pi$
$691$	4839.64	0.266438	0.133219	+	0.991087i	$0.457469\pi$
$692$	−14909.1	−0.819018
$693$	0	0
$694$	450.897	0.0246625
$695$	9302.23	0.507703
$696$	0	0
$697$	−3130.62	−0.170130
$698$	−16123.0	−0.874305
$699$	0	0
$700$	−1341.17	−0.0724166
$701$	−21064.0	−1.13492	−0.567458	−	0.823402i	$-0.692073\pi$
$701$	−21064.0	−1.13492	−0.567458	+	0.823402i	$0.692073\pi$
$702$	0	0
$703$	−4389.01	−0.235469
$704$	−4184.90	−0.224040
$705$	0	0
$706$	−6015.31	−0.320665
$707$	2708.66	0.144087
$708$	0	0
$709$	−865.527	−0.0458470	−0.0229235	−	0.999737i	$-0.507297\pi$
$709$	−865.527	−0.0458470	−0.0229235	+	0.999737i	$0.507297\pi$
$710$	11293.3	0.596943
$711$	0	0
$712$	854.422	0.0449731
$713$	4912.28	0.258017
$714$	0	0
$715$	37386.5	1.95549
$716$	−1235.48	−0.0644861
$717$	0	0
$718$	16139.1	0.838864
$719$	36094.4	1.87217	0.936086	−	0.351770i	$-0.114420\pi$
$719$	36094.4	1.87217	0.936086	+	0.351770i	$0.114420\pi$
$720$	0	0
$721$	−136.822	−0.00706729
$722$	−722.000	−0.0372161
$723$	0	0
$724$	12601.9	0.646887
$725$	−5225.62	−0.267689
$726$	0	0
$727$	18184.3	0.927672	0.463836	−	0.885921i	$-0.346473\pi$
$727$	18184.3	0.927672	0.463836	+	0.885921i	$0.346473\pi$
$728$	2435.01	0.123966
$729$	0	0
$730$	−23122.7	−1.17234
$731$	7696.28	0.389408
$732$	0	0
$733$	−26254.6	−1.32297	−0.661483	−	0.749960i	$-0.730073\pi$
$733$	−26254.6	−1.32297	−0.661483	+	0.749960i	$0.730073\pi$
$734$	18555.9	0.933120
$735$	0	0
$736$	−1263.64	−0.0632857
$737$	−20021.4	−1.00068
$738$	0	0
$739$	−24039.3	−1.19662	−0.598309	−	0.801265i	$-0.704161\pi$
$739$	−24039.3	−1.19662	−0.598309	+	0.801265i	$0.704161\pi$
$740$	12149.8	0.603562
$741$	0	0
$742$	8985.89	0.444586
$743$	2755.02	0.136032	0.0680162	−	0.997684i	$-0.478333\pi$
$743$	2755.02	0.136032	0.0680162	+	0.997684i	$0.478333\pi$
$744$	0	0
$745$	3875.48	0.190586
$746$	17809.4	0.874061
$747$	0	0
$748$	−3634.01	−0.177637
$749$	269.841	0.0131639
$750$	0	0
$751$	−18746.2	−0.910864	−0.455432	−	0.890271i	$-0.650515\pi$
$751$	−18746.2	−0.910864	−0.455432	+	0.890271i	$0.650515\pi$
$752$	−1043.69	−0.0506111
$753$	0	0
$754$	9487.53	0.458244
$755$	−29396.4	−1.41701
$756$	0	0
$757$	39333.3	1.88850	0.944250	−	0.329230i	$-0.106789\pi$
$757$	39333.3	1.88850	0.944250	+	0.329230i	$0.106789\pi$
$758$	21161.7	1.01402
$759$	0	0
$760$	1998.66	0.0953937
$761$	41265.6	1.96567	0.982837	−	0.184475i	$-0.0590584\pi$
$761$	41265.6	1.96567	0.982837	+	0.184475i	$0.0590584\pi$
$762$	0	0
$763$	−1873.87	−0.0889102
$764$	−5114.27	−0.242183
$765$	0	0
$766$	378.221	0.0178403
$767$	26013.9	1.22465
$768$	0	0
$769$	−14262.6	−0.668820	−0.334410	−	0.942428i	$-0.608537\pi$
$769$	−14262.6	−0.668820	−0.334410	+	0.942428i	$0.608537\pi$
$770$	12037.3	0.563369
$771$	0	0
$772$	9557.26	0.445561
$773$	25749.9	1.19814	0.599068	−	0.800698i	$-0.295538\pi$
$773$	25749.9	1.19814	0.599068	+	0.800698i	$0.295538\pi$
$774$	0	0
$775$	5958.51	0.276176
$776$	−4308.17	−0.199297
$777$	0	0
$778$	22204.3	1.02322
$779$	−4281.17	−0.196905
$780$	0	0
$781$	−28080.1	−1.28654
$782$	−1097.30	−0.0501780
$783$	0	0
$784$	784.000	0.0357143
$785$	11714.6	0.532627
$786$	0	0
$787$	27191.2	1.23159	0.615795	−	0.787906i	$-0.288835\pi$
$787$	27191.2	1.23159	0.615795	+	0.787906i	$0.288835\pi$
$788$	−3992.22	−0.180478
$789$	0	0
$790$	5809.50	0.261636
$791$	−3792.40	−0.170470
$792$	0	0
$793$	4752.48	0.212819
$794$	18716.2	0.836540
$795$	0	0
$796$	−11161.4	−0.496989
$797$	40696.4	1.80871	0.904354	−	0.426783i	$-0.140353\pi$
$797$	40696.4	1.80871	0.904354	+	0.426783i	$0.140353\pi$
$798$	0	0
$799$	−906.305	−0.0401286
$800$	−1532.77	−0.0677395
$801$	0	0
$802$	11419.9	0.502804
$803$	57493.2	2.52664
$804$	0	0
$805$	3634.68	0.159138
$806$	−10818.2	−0.472771
$807$	0	0
$808$	3095.61	0.134781
$809$	−2219.46	−0.0964549	−0.0482275	−	0.998836i	$-0.515357\pi$
$809$	−2219.46	−0.0964549	−0.0482275	+	0.998836i	$0.515357\pi$
$810$	0	0
$811$	−18624.2	−0.806393	−0.403197	−	0.915113i	$-0.632101\pi$
$811$	−18624.2	−0.806393	−0.403197	+	0.915113i	$0.632101\pi$
$812$	3054.70	0.132018
$813$	0	0
$814$	−30209.8	−1.30080
$815$	41397.0	1.77923
$816$	0	0
$817$	10524.8	0.450692
$818$	3999.87	0.170969
$819$	0	0
$820$	11851.3	0.504713
$821$	−9903.68	−0.421000	−0.210500	−	0.977594i	$-0.567509\pi$
$821$	−9903.68	−0.421000	−0.210500	+	0.977594i	$0.567509\pi$
$822$	0	0
$823$	−25420.4	−1.07667	−0.538334	−	0.842731i	$-0.680946\pi$
$823$	−25420.4	−1.07667	−0.538334	+	0.842731i	$0.680946\pi$
$824$	−156.368	−0.00661085
$825$	0	0
$826$	8375.70	0.352818
$827$	−537.233	−0.0225894	−0.0112947	−	0.999936i	$-0.503595\pi$
$827$	−537.233	−0.0225894	−0.0112947	+	0.999936i	$0.503595\pi$
$828$	0	0
$829$	9001.21	0.377111	0.188555	−	0.982063i	$-0.439619\pi$
$829$	9001.21	0.377111	0.188555	+	0.982063i	$0.439619\pi$
$830$	−14003.0	−0.585605
$831$	0	0
$832$	2782.87	0.115960
$833$	680.797	0.0283172
$834$	0	0
$835$	32380.8	1.34202
$836$	−4969.57	−0.205593
$837$	0	0
$838$	14130.9	0.582509
$839$	−23860.0	−0.981808	−0.490904	−	0.871214i	$-0.663333\pi$
$839$	−23860.0	−0.981808	−0.490904	+	0.871214i	$0.663333\pi$
$840$	0	0
$841$	−12487.0	−0.511992
$842$	−10924.8	−0.447142
$843$	0	0
$844$	12704.2	0.518125
$845$	4027.38	0.163960
$846$	0	0
$847$	−20613.1	−0.836215
$848$	10269.6	0.415872
$849$	0	0
$850$	−1331.00	−0.0537094
$851$	−9121.90	−0.367444
$852$	0	0
$853$	−1578.55	−0.0633630	−0.0316815	−	0.999498i	$-0.510086\pi$
$853$	−1578.55	−0.0633630	−0.0316815	+	0.999498i	$0.510086\pi$
$854$	1530.15	0.0613124
$855$	0	0
$856$	308.390	0.0123137
$857$	35432.8	1.41232	0.706162	−	0.708050i	$-0.250425\pi$
$857$	35432.8	1.41232	0.706162	+	0.708050i	$0.250425\pi$
$858$	0	0
$859$	4235.52	0.168235	0.0841176	−	0.996456i	$-0.473193\pi$
$859$	4235.52	0.168235	0.0841176	+	0.996456i	$0.473193\pi$
$860$	−29135.1	−1.15523
$861$	0	0
$862$	−7235.06	−0.285878
$863$	14231.5	0.561352	0.280676	−	0.959803i	$-0.409441\pi$
$863$	14231.5	0.561352	0.280676	+	0.959803i	$0.409441\pi$
$864$	0	0
$865$	49010.5	1.92648
$866$	−4181.61	−0.164084
$867$	0	0
$868$	−3483.12	−0.136204
$869$	−14445.0	−0.563881
$870$	0	0
$871$	13313.8	0.517936
$872$	−2141.56	−0.0831679
$873$	0	0
$874$	−1500.57	−0.0580749
$875$	−7096.66	−0.274184
$876$	0	0
$877$	−13085.5	−0.503836	−0.251918	−	0.967749i	$-0.581061\pi$
$877$	−13085.5	−0.503836	−0.251918	+	0.967749i	$0.581061\pi$
$878$	−20281.2	−0.779563
$879$	0	0
$880$	13756.9	0.526984
$881$	13579.6	0.519305	0.259652	−	0.965702i	$-0.416392\pi$
$881$	13579.6	0.519305	0.259652	+	0.965702i	$0.416392\pi$
$882$	0	0
$883$	26249.7	1.00042	0.500211	−	0.865904i	$-0.333256\pi$
$883$	26249.7	1.00042	0.500211	+	0.865904i	$0.333256\pi$
$884$	2416.54	0.0919424
$885$	0	0
$886$	21679.3	0.822043
$887$	−6737.30	−0.255036	−0.127518	−	0.991836i	$-0.540701\pi$
$887$	−6737.30	−0.255036	−0.127518	+	0.991836i	$0.540701\pi$
$888$	0	0
$889$	7378.43	0.278363
$890$	−2808.72	−0.105785
$891$	0	0
$892$	−13116.9	−0.492360
$893$	−1239.39	−0.0464440
$894$	0	0
$895$	4061.37	0.151683
$896$	896.000	0.0334077
$897$	0	0
$898$	−18988.5	−0.705629
$899$	−13571.3	−0.503479
$900$	0	0
$901$	8917.74	0.329737
$902$	−29467.6	−1.08776
$903$	0	0
$904$	−4334.17	−0.159461
$905$	−41426.0	−1.52160
$906$	0	0
$907$	−2592.28	−0.0949010	−0.0474505	−	0.998874i	$-0.515110\pi$
$907$	−2592.28	−0.0949010	−0.0474505	+	0.998874i	$0.515110\pi$
$908$	−7130.77	−0.260620
$909$	0	0
$910$	−8004.56	−0.291592
$911$	27332.4	0.994033	0.497016	−	0.867741i	$-0.334429\pi$
$911$	27332.4	0.994033	0.497016	+	0.867741i	$0.334429\pi$
$912$	0	0
$913$	34817.7	1.26210
$914$	−25459.8	−0.921374
$915$	0	0
$916$	6839.69	0.246714
$917$	−19853.0	−0.714943
$918$	0	0
$919$	−41245.2	−1.48047	−0.740236	−	0.672347i	$-0.765286\pi$
$919$	−41245.2	−1.48047	−0.740236	+	0.672347i	$0.765286\pi$
$920$	4153.92	0.148860
$921$	0	0
$922$	11094.7	0.396295
$923$	18672.7	0.665894
$924$	0	0
$925$	−11064.7	−0.393303
$926$	18845.5	0.668791
$927$	0	0
$928$	3491.08	0.123492
$929$	24739.7	0.873719	0.436859	−	0.899530i	$-0.356091\pi$
$929$	24739.7	0.873719	0.436859	+	0.899530i	$0.356091\pi$
$930$	0	0
$931$	931.000	0.0327737
$932$	−1271.67	−0.0446940
$933$	0	0
$934$	8862.09	0.310467
$935$	11946.0	0.417835
$936$	0	0
$937$	−25446.0	−0.887175	−0.443588	−	0.896231i	$-0.646295\pi$
$937$	−25446.0	−0.887175	−0.443588	+	0.896231i	$0.646295\pi$
$938$	4286.65	0.149215
$939$	0	0
$940$	3430.91	0.119047
$941$	8692.84	0.301146	0.150573	−	0.988599i	$-0.451888\pi$
$941$	8692.84	0.301146	0.150573	+	0.988599i	$0.451888\pi$
$942$	0	0
$943$	−8897.77	−0.307266
$944$	9572.23	0.330031
$945$	0	0
$946$	72442.7	2.48976
$947$	21884.7	0.750957	0.375478	−	0.926831i	$-0.377478\pi$
$947$	21884.7	0.750957	0.375478	+	0.926831i	$0.377478\pi$
$948$	0	0
$949$	−38231.8	−1.30775
$950$	−1820.16	−0.0621621
$951$	0	0
$952$	778.053	0.0264883
$953$	14282.7	0.485481	0.242740	−	0.970091i	$-0.421954\pi$
$953$	14282.7	0.485481	0.242740	+	0.970091i	$0.421954\pi$
$954$	0	0
$955$	16812.0	0.569659
$956$	−5835.86	−0.197432
$957$	0	0
$958$	26263.7	0.885742
$959$	5866.78	0.197548
$960$	0	0
$961$	−14316.3	−0.480559
$962$	20088.9	0.673277
$963$	0	0
$964$	5836.28	0.194994
$965$	−31417.4	−1.04804
$966$	0	0
$967$	−24360.4	−0.810111	−0.405055	−	0.914292i	$-0.632748\pi$
$967$	−24360.4	−0.810111	−0.405055	+	0.914292i	$0.632748\pi$
$968$	−23557.8	−0.782207
$969$	0	0
$970$	14162.2	0.468783
$971$	33869.5	1.11939	0.559693	−	0.828700i	$-0.310919\pi$
$971$	33869.5	1.11939	0.559693	+	0.828700i	$0.310919\pi$
$972$	0	0
$973$	4952.09	0.163162
$974$	−30737.4	−1.01118
$975$	0	0
$976$	1748.75	0.0573525
$977$	56444.9	1.84835	0.924173	−	0.381975i	$-0.124756\pi$
$977$	56444.9	1.84835	0.924173	+	0.381975i	$0.124756\pi$
$978$	0	0
$979$	6983.73	0.227989
$980$	−2577.23	−0.0840066
$981$	0	0
$982$	34705.6	1.12780
$983$	54268.9	1.76084	0.880422	−	0.474190i	$-0.157259\pi$
$983$	54268.9	1.76084	0.880422	+	0.474190i	$0.157259\pi$
$984$	0	0
$985$	13123.5	0.424518
$986$	3031.53	0.0979144
$987$	0	0
$988$	3304.66	0.106412
$989$	21874.2	0.703295
$990$	0	0
$991$	30602.7	0.980957	0.490478	−	0.871453i	$-0.336822\pi$
$991$	30602.7	0.980957	0.490478	+	0.871453i	$0.336822\pi$
$992$	−3980.71	−0.127407
$993$	0	0
$994$	6012.05	0.191842
$995$	36690.5	1.16901
$996$	0	0
$997$	17116.8	0.543726	0.271863	−	0.962336i	$-0.412360\pi$
$997$	17116.8	0.543726	0.271863	+	0.962336i	$0.412360\pi$
$998$	31230.5	0.990566
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2394.4.a.bh.1.3	✓	7
3.2	odd	2		2394.4.a.bi.1.5	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2394.4.a.bh.1.3	✓	7	1.1	even	1	trivial
2394.4.a.bi.1.5	yes	7	3.2	odd	2

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:	\( N \)	\(=\)	\( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2394.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} -13.1491 q^{5} -7.00000 q^{7} -8.00000 q^{8} +O(q^{10})\)	\(q-2.00000 q^{2} +4.00000 q^{4} -13.1491 q^{5} -7.00000 q^{7} -8.00000 q^{8} +26.2982 q^{10} -65.3890 q^{11} +43.4823 q^{13} +14.0000 q^{14} +16.0000 q^{16} +13.8938 q^{17} +19.0000 q^{19} -52.5964 q^{20} +130.778 q^{22} +39.4886 q^{23} +47.8991 q^{25} -86.9647 q^{26} -28.0000 q^{28} -109.096 q^{29} +124.397 q^{31} -32.0000 q^{32} -27.7876 q^{34} +92.0438 q^{35} -231.001 q^{37} -38.0000 q^{38} +105.193 q^{40} -225.325 q^{41} +553.936 q^{43} -261.556 q^{44} -78.9773 q^{46} -65.2308 q^{47} +49.0000 q^{49} -95.7982 q^{50} +173.929 q^{52} +641.849 q^{53} +859.808 q^{55} +56.0000 q^{56} +218.193 q^{58} +598.264 q^{59} +109.297 q^{61} -248.794 q^{62} +64.0000 q^{64} -571.754 q^{65} +306.189 q^{67} +55.5752 q^{68} -184.088 q^{70} +429.432 q^{71} -879.248 q^{73} +462.001 q^{74} +76.0000 q^{76} +457.723 q^{77} +220.909 q^{79} -210.386 q^{80} +450.650 q^{82} -532.470 q^{83} -182.691 q^{85} -1107.87 q^{86} +523.112 q^{88} -106.803 q^{89} -304.376 q^{91} +157.955 q^{92} +130.462 q^{94} -249.833 q^{95} +538.521 q^{97} -98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 14 q^{2} + 28 q^{4} - 18 q^{5} - 49 q^{7} - 56 q^{8}+O(q^{10}) \) 7 * q - 14 * q^2 + 28 * q^4 - 18 * q^5 - 49 * q^7 - 56 * q^8	\( 7 q - 14 q^{2} + 28 q^{4} - 18 q^{5} - 49 q^{7} - 56 q^{8} + 36 q^{10} - 28 q^{11} + 44 q^{13} + 98 q^{14} + 112 q^{16} - 26 q^{17} + 133 q^{19} - 72 q^{20} + 56 q^{22} - 56 q^{23} + 37 q^{25} - 88 q^{26} - 196 q^{28} - 270 q^{29} + 64 q^{31} - 224 q^{32} + 52 q^{34} + 126 q^{35} + 458 q^{37} - 266 q^{38} + 144 q^{40} - 110 q^{41} + 296 q^{43} - 112 q^{44} + 112 q^{46} + 142 q^{47} + 343 q^{49} - 74 q^{50} + 176 q^{52} - 330 q^{53} + 596 q^{55} + 392 q^{56} + 540 q^{58} - 236 q^{59} + 882 q^{61} - 128 q^{62} + 448 q^{64} + 180 q^{65} + 1622 q^{67} - 104 q^{68} - 252 q^{70} - 820 q^{71} + 1130 q^{73} - 916 q^{74} + 532 q^{76} + 196 q^{77} + 1694 q^{79} - 288 q^{80} + 220 q^{82} - 890 q^{83} + 1540 q^{85} - 592 q^{86} + 224 q^{88} - 686 q^{89} - 308 q^{91} - 224 q^{92} - 284 q^{94} - 342 q^{95} + 1772 q^{97} - 686 q^{98}+O(q^{100}) \) 7 * q - 14 * q^2 + 28 * q^4 - 18 * q^5 - 49 * q^7 - 56 * q^8 + 36 * q^10 - 28 * q^11 + 44 * q^13 + 98 * q^14 + 112 * q^16 - 26 * q^17 + 133 * q^19 - 72 * q^20 + 56 * q^22 - 56 * q^23 + 37 * q^25 - 88 * q^26 - 196 * q^28 - 270 * q^29 + 64 * q^31 - 224 * q^32 + 52 * q^34 + 126 * q^35 + 458 * q^37 - 266 * q^38 + 144 * q^40 - 110 * q^41 + 296 * q^43 - 112 * q^44 + 112 * q^46 + 142 * q^47 + 343 * q^49 - 74 * q^50 + 176 * q^52 - 330 * q^53 + 596 * q^55 + 392 * q^56 + 540 * q^58 - 236 * q^59 + 882 * q^61 - 128 * q^62 + 448 * q^64 + 180 * q^65 + 1622 * q^67 - 104 * q^68 - 252 * q^70 - 820 * q^71 + 1130 * q^73 - 916 * q^74 + 532 * q^76 + 196 * q^77 + 1694 * q^79 - 288 * q^80 + 220 * q^82 - 890 * q^83 + 1540 * q^85 - 592 * q^86 + 224 * q^88 - 686 * q^89 - 308 * q^91 - 224 * q^92 - 284 * q^94 - 342 * q^95 + 1772 * q^97 - 686 * q^98

Introduction

L-functions

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