LMFDB - Embedded newform 2368.4.a.r.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,4,Mod(1,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, 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("2368.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2368.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:	$139.716522894$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 27x^{3} + 3x^{2} + 176x + 144 $ x^5 - x^4 - 27x^3 + 3x^2 + 176*x + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 37)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$3.56768$ of defining polynomial
Character	$\chi$	$=$	2368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+9.45274 q^{3} -1.50802 q^{5} -12.6891 q^{7} +62.3543 q^{9} +O(q^{10})$	$q+9.45274 q^{3} -1.50802 q^{5} -12.6891 q^{7} +62.3543 q^{9} -3.77312 q^{11} +71.0065 q^{13} -14.2549 q^{15} +38.8146 q^{17} -64.4749 q^{19} -119.947 q^{21} +197.638 q^{23} -122.726 q^{25} +334.195 q^{27} -255.479 q^{29} +97.3830 q^{31} -35.6664 q^{33} +19.1354 q^{35} +37.0000 q^{37} +671.206 q^{39} +350.356 q^{41} +138.813 q^{43} -94.0313 q^{45} -305.653 q^{47} -181.986 q^{49} +366.904 q^{51} +310.167 q^{53} +5.68993 q^{55} -609.465 q^{57} +112.918 q^{59} +595.518 q^{61} -791.221 q^{63} -107.079 q^{65} +899.470 q^{67} +1868.22 q^{69} +450.353 q^{71} +469.956 q^{73} -1160.10 q^{75} +47.8776 q^{77} -263.304 q^{79} +1475.49 q^{81} -314.892 q^{83} -58.5331 q^{85} -2414.98 q^{87} -772.521 q^{89} -901.010 q^{91} +920.536 q^{93} +97.2293 q^{95} +190.846 q^{97} -235.270 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 13 q^{3} - 11 q^{5} - 24 q^{7} + 46 q^{9}+O(q^{10}) $ 5 * q + 13 * q^3 - 11 * q^5 - 24 * q^7 + 46 * q^9	$ 5 q + 13 q^{3} - 11 q^{5} - 24 q^{7} + 46 q^{9} + 61 q^{11} + 37 q^{13} + 116 q^{15} + 130 q^{17} - 22 q^{19} + 44 q^{21} - 73 q^{23} + 26 q^{25} + 472 q^{27} - 271 q^{29} - 363 q^{31} + 198 q^{33} + 604 q^{35} + 185 q^{37} + 65 q^{39} + 381 q^{41} - 408 q^{43} + 704 q^{45} - 276 q^{47} - 949 q^{49} - 38 q^{51} - 156 q^{53} + 843 q^{55} - 1618 q^{57} + 100 q^{59} + 1711 q^{61} - 94 q^{63} - 890 q^{65} + 787 q^{67} + 2335 q^{69} - 1578 q^{71} - 313 q^{73} + 684 q^{75} + 342 q^{77} - 569 q^{79} + 385 q^{81} + 2422 q^{83} + 2210 q^{85} - 2371 q^{87} - 2466 q^{89} - 1678 q^{91} + 1142 q^{93} - 794 q^{95} - 2406 q^{97} + 1746 q^{99}+O(q^{100}) $ 5 * q + 13 * q^3 - 11 * q^5 - 24 * q^7 + 46 * q^9 + 61 * q^11 + 37 * q^13 + 116 * q^15 + 130 * q^17 - 22 * q^19 + 44 * q^21 - 73 * q^23 + 26 * q^25 + 472 * q^27 - 271 * q^29 - 363 * q^31 + 198 * q^33 + 604 * q^35 + 185 * q^37 + 65 * q^39 + 381 * q^41 - 408 * q^43 + 704 * q^45 - 276 * q^47 - 949 * q^49 - 38 * q^51 - 156 * q^53 + 843 * q^55 - 1618 * q^57 + 100 * q^59 + 1711 * q^61 - 94 * q^63 - 890 * q^65 + 787 * q^67 + 2335 * q^69 - 1578 * q^71 - 313 * q^73 + 684 * q^75 + 342 * q^77 - 569 * q^79 + 385 * q^81 + 2422 * q^83 + 2210 * q^85 - 2371 * q^87 - 2466 * q^89 - 1678 * q^91 + 1142 * q^93 - 794 * q^95 - 2406 * q^97 + 1746 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	9.45274	1.81918	0.909590	−	0.415506i	$-0.136396\pi$
$3$	9.45274	1.81918	0.909590	+	0.415506i	$0.136396\pi$
$4$	0	0
$5$	−1.50802	−0.134881	−0.0674405	−	0.997723i	$-0.521483\pi$
$5$	−1.50802	−0.134881	−0.0674405	+	0.997723i	$0.521483\pi$
$6$	0	0
$7$	−12.6891	−0.685148	−0.342574	−	0.939491i	$-0.611299\pi$
$7$	−12.6891	−0.685148	−0.342574	+	0.939491i	$0.611299\pi$
$8$	0	0
$9$	62.3543	2.30942
$10$	0	0
$11$	−3.77312	−0.103422	−0.0517109	−	0.998662i	$-0.516467\pi$
$11$	−3.77312	−0.103422	−0.0517109	+	0.998662i	$0.516467\pi$
$12$	0	0
$13$	71.0065	1.51490	0.757449	−	0.652895i	$-0.226446\pi$
$13$	71.0065	1.51490	0.757449	+	0.652895i	$0.226446\pi$
$14$	0	0
$15$	−14.2549	−0.245373
$16$	0	0
$17$	38.8146	0.553760	0.276880	−	0.960904i	$-0.410700\pi$
$17$	38.8146	0.553760	0.276880	+	0.960904i	$0.410700\pi$
$18$	0	0
$19$	−64.4749	−0.778503	−0.389252	−	0.921131i	$-0.627266\pi$
$19$	−64.4749	−0.778503	−0.389252	+	0.921131i	$0.627266\pi$
$20$	0	0
$21$	−119.947	−1.24641
$22$	0	0
$23$	197.638	1.79175	0.895877	−	0.444302i	$-0.146548\pi$
$23$	197.638	1.79175	0.895877	+	0.444302i	$0.146548\pi$
$24$	0	0
$25$	−122.726	−0.981807
$26$	0	0
$27$	334.195	2.38207
$28$	0	0
$29$	−255.479	−1.63591	−0.817954	−	0.575283i	$-0.804892\pi$
$29$	−255.479	−1.63591	−0.817954	+	0.575283i	$0.804892\pi$
$30$	0	0
$31$	97.3830	0.564210	0.282105	−	0.959384i	$-0.408967\pi$
$31$	97.3830	0.564210	0.282105	+	0.959384i	$0.408967\pi$
$32$	0	0
$33$	−35.6664	−0.188143
$34$	0	0
$35$	19.1354	0.0924135
$36$	0	0
$37$	37.0000	0.164399
$37$	37.0000	0.164399
$38$	0	0
$39$	671.206	2.75587
$40$	0	0
$41$	350.356	1.33455	0.667273	−	0.744813i	$-0.267461\pi$
$41$	350.356	1.33455	0.667273	+	0.744813i	$0.267461\pi$
$42$	0	0
$43$	138.813	0.492296	0.246148	−	0.969232i	$-0.420835\pi$
$43$	138.813	0.492296	0.246148	+	0.969232i	$0.420835\pi$
$44$	0	0
$45$	−94.0313	−0.311497
$46$	0	0
$47$	−305.653	−0.948597	−0.474299	−	0.880364i	$-0.657298\pi$
$47$	−305.653	−0.948597	−0.474299	+	0.880364i	$0.657298\pi$
$48$	0	0
$49$	−181.986	−0.530572
$50$	0	0
$51$	366.904	1.00739
$52$	0	0
$53$	310.167	0.803864	0.401932	−	0.915670i	$-0.368339\pi$
$53$	310.167	0.803864	0.401932	+	0.915670i	$0.368339\pi$
$54$	0	0
$55$	5.68993	0.0139496
$56$	0	0
$57$	−609.465	−1.41624
$58$	0	0
$59$	112.918	0.249163	0.124582	−	0.992209i	$-0.460241\pi$
$59$	112.918	0.249163	0.124582	+	0.992209i	$0.460241\pi$
$60$	0	0
$61$	595.518	1.24997	0.624986	−	0.780636i	$-0.285105\pi$
$61$	595.518	1.24997	0.624986	+	0.780636i	$0.285105\pi$
$62$	0	0
$63$	−791.221	−1.58229
$64$	0	0
$65$	−107.079	−0.204331
$66$	0	0
$67$	899.470	1.64012	0.820058	−	0.572280i	$-0.193941\pi$
$67$	899.470	1.64012	0.820058	+	0.572280i	$0.193941\pi$
$68$	0	0
$69$	1868.22	3.25952
$70$	0	0
$71$	450.353	0.752775	0.376388	−	0.926462i	$-0.377166\pi$
$71$	450.353	0.752775	0.376388	+	0.926462i	$0.377166\pi$
$72$	0	0
$73$	469.956	0.753482	0.376741	−	0.926319i	$-0.377045\pi$
$73$	469.956	0.753482	0.376741	+	0.926319i	$0.377045\pi$
$74$	0	0
$75$	−1160.10	−1.78608
$76$	0	0
$77$	47.8776	0.0708593
$78$	0	0
$79$	−263.304	−0.374987	−0.187493	−	0.982266i	$-0.560036\pi$
$79$	−263.304	−0.374987	−0.187493	+	0.982266i	$0.560036\pi$
$80$	0	0
$81$	1475.49	2.02399
$82$	0	0
$83$	−314.892	−0.416432	−0.208216	−	0.978083i	$-0.566766\pi$
$83$	−314.892	−0.416432	−0.208216	+	0.978083i	$0.566766\pi$
$84$	0	0
$85$	−58.5331	−0.0746918
$86$	0	0
$87$	−2414.98	−2.97601
$88$	0	0
$89$	−772.521	−0.920079	−0.460039	−	0.887898i	$-0.652165\pi$
$89$	−772.521	−0.920079	−0.460039	+	0.887898i	$0.652165\pi$
$90$	0	0
$91$	−901.010	−1.03793
$92$	0	0
$93$	920.536	1.02640
$94$	0	0
$95$	97.2293	0.105005
$96$	0	0
$97$	190.846	0.199768	0.0998838	−	0.994999i	$-0.468153\pi$
$97$	190.846	0.199768	0.0998838	+	0.994999i	$0.468153\pi$
$98$	0	0
$99$	−235.270	−0.238844
$100$	0	0
$101$	−219.256	−0.216008	−0.108004	−	0.994150i	$-0.534446\pi$
$101$	−219.256	−0.216008	−0.108004	+	0.994150i	$0.534446\pi$
$102$	0	0
$103$	−1064.84	−1.01866	−0.509330	−	0.860571i	$-0.670107\pi$
$103$	−1064.84	−1.01866	−0.509330	+	0.860571i	$0.670107\pi$
$104$	0	0
$105$	180.882	0.168117
$106$	0	0
$107$	948.801	0.857234	0.428617	−	0.903486i	$-0.359001\pi$
$107$	948.801	0.857234	0.428617	+	0.903486i	$0.359001\pi$
$108$	0	0
$109$	−836.883	−0.735402	−0.367701	−	0.929944i	$-0.619855\pi$
$109$	−836.883	−0.735402	−0.367701	+	0.929944i	$0.619855\pi$
$110$	0	0
$111$	349.751	0.299071
$112$	0	0
$113$	−791.819	−0.659187	−0.329593	−	0.944123i	$-0.606912\pi$
$113$	−791.819	−0.659187	−0.329593	+	0.944123i	$0.606912\pi$
$114$	0	0
$115$	−298.041	−0.241674
$116$	0	0
$117$	4427.56	3.49853
$118$	0	0
$119$	−492.523	−0.379408
$120$	0	0
$121$	−1316.76	−0.989304
$122$	0	0
$123$	3311.82	2.42778
$124$	0	0
$125$	373.575	0.267308
$126$	0	0
$127$	1562.59	1.09179	0.545896	−	0.837853i	$-0.316189\pi$
$127$	1562.59	1.09179	0.545896	+	0.837853i	$0.316189\pi$
$128$	0	0
$129$	1312.16	0.895575
$130$	0	0
$131$	784.197	0.523020	0.261510	−	0.965201i	$-0.415780\pi$
$131$	784.197	0.523020	0.261510	+	0.965201i	$0.415780\pi$
$132$	0	0
$133$	818.130	0.533390
$134$	0	0
$135$	−503.971	−0.321296
$136$	0	0
$137$	2107.26	1.31413	0.657064	−	0.753835i	$-0.271798\pi$
$137$	2107.26	1.31413	0.657064	+	0.753835i	$0.271798\pi$
$138$	0	0
$139$	2025.93	1.23624	0.618119	−	0.786085i	$-0.287895\pi$
$139$	2025.93	1.23624	0.618119	+	0.786085i	$0.287895\pi$
$140$	0	0
$141$	−2889.26	−1.72567
$142$	0	0
$143$	−267.916	−0.156673
$144$	0	0
$145$	385.267	0.220653
$146$	0	0
$147$	−1720.27	−0.965206
$148$	0	0
$149$	−745.838	−0.410077	−0.205038	−	0.978754i	$-0.565732\pi$
$149$	−745.838	−0.410077	−0.205038	+	0.978754i	$0.565732\pi$
$150$	0	0
$151$	−658.206	−0.354729	−0.177365	−	0.984145i	$-0.556757\pi$
$151$	−658.206	−0.354729	−0.177365	+	0.984145i	$0.556757\pi$
$152$	0	0
$153$	2420.26	1.27886
$154$	0	0
$155$	−146.855	−0.0761012
$156$	0	0
$157$	2059.43	1.04688	0.523441	−	0.852062i	$-0.324648\pi$
$157$	2059.43	1.04688	0.523441	+	0.852062i	$0.324648\pi$
$158$	0	0
$159$	2931.93	1.46237
$160$	0	0
$161$	−2507.85	−1.22762
$162$	0	0
$163$	1229.58	0.590848	0.295424	−	0.955366i	$-0.404539\pi$
$163$	1229.58	0.590848	0.295424	+	0.955366i	$0.404539\pi$
$164$	0	0
$165$	53.7855	0.0253769
$166$	0	0
$167$	152.311	0.0705761	0.0352881	−	0.999377i	$-0.488765\pi$
$167$	152.311	0.0705761	0.0352881	+	0.999377i	$0.488765\pi$
$168$	0	0
$169$	2844.93	1.29491
$170$	0	0
$171$	−4020.29	−1.79789
$172$	0	0
$173$	−1156.14	−0.508089	−0.254044	−	0.967193i	$-0.581761\pi$
$173$	−1156.14	−0.508089	−0.254044	+	0.967193i	$0.581761\pi$
$174$	0	0
$175$	1557.28	0.672683
$176$	0	0
$177$	1067.38	0.453273
$178$	0	0
$179$	332.020	0.138639	0.0693194	−	0.997595i	$-0.477917\pi$
$179$	332.020	0.138639	0.0693194	+	0.997595i	$0.477917\pi$
$180$	0	0
$181$	3340.60	1.37185	0.685926	−	0.727671i	$-0.259397\pi$
$181$	3340.60	1.37185	0.685926	+	0.727671i	$0.259397\pi$
$182$	0	0
$183$	5629.28	2.27393
$184$	0	0
$185$	−55.7966	−0.0221743
$186$	0	0
$187$	−146.452	−0.0572709
$188$	0	0
$189$	−4240.64	−1.63207
$190$	0	0
$191$	−172.721	−0.0654328	−0.0327164	−	0.999465i	$-0.510416\pi$
$191$	−172.721	−0.0654328	−0.0327164	+	0.999465i	$0.510416\pi$
$192$	0	0
$193$	3533.01	1.31768	0.658838	−	0.752285i	$-0.271048\pi$
$193$	3533.01	1.31768	0.658838	+	0.752285i	$0.271048\pi$
$194$	0	0
$195$	−1012.19	−0.371715
$196$	0	0
$197$	−2038.45	−0.737227	−0.368613	−	0.929583i	$-0.620167\pi$
$197$	−2038.45	−0.737227	−0.368613	+	0.929583i	$0.620167\pi$
$198$	0	0
$199$	−3418.61	−1.21778	−0.608891	−	0.793254i	$-0.708385\pi$
$199$	−3418.61	−1.21778	−0.608891	+	0.793254i	$0.708385\pi$
$200$	0	0
$201$	8502.46	2.98367
$202$	0	0
$203$	3241.81	1.12084
$204$	0	0
$205$	−528.342	−0.180005
$206$	0	0
$207$	12323.6	4.13791
$208$	0	0
$209$	243.272	0.0805142
$210$	0	0
$211$	−4651.74	−1.51772	−0.758861	−	0.651253i	$-0.774244\pi$
$211$	−4651.74	−1.51772	−0.758861	+	0.651253i	$0.774244\pi$
$212$	0	0
$213$	4257.07	1.36943
$214$	0	0
$215$	−209.332	−0.0664014
$216$	0	0
$217$	−1235.70	−0.386567
$218$	0	0
$219$	4442.37	1.37072
$220$	0	0
$221$	2756.09	0.838890
$222$	0	0
$223$	−280.807	−0.0843240	−0.0421620	−	0.999111i	$-0.513425\pi$
$223$	−280.807	−0.0843240	−0.0421620	+	0.999111i	$0.513425\pi$
$224$	0	0
$225$	−7652.48	−2.26740
$226$	0	0
$227$	3749.27	1.09624	0.548122	−	0.836398i	$-0.315343\pi$
$227$	3749.27	1.09624	0.548122	+	0.836398i	$0.315343\pi$
$228$	0	0
$229$	5265.97	1.51959	0.759793	−	0.650165i	$-0.225300\pi$
$229$	5265.97	1.51959	0.759793	+	0.650165i	$0.225300\pi$
$230$	0	0
$231$	452.575	0.128906
$232$	0	0
$233$	−4555.42	−1.28084	−0.640420	−	0.768025i	$-0.721240\pi$
$233$	−4555.42	−1.28084	−0.640420	+	0.768025i	$0.721240\pi$
$234$	0	0
$235$	460.930	0.127948
$236$	0	0
$237$	−2488.94	−0.682169
$238$	0	0
$239$	2746.85	0.743426	0.371713	−	0.928348i	$-0.378770\pi$
$239$	2746.85	0.743426	0.371713	+	0.928348i	$0.378770\pi$
$240$	0	0
$241$	2224.32	0.594528	0.297264	−	0.954795i	$-0.403926\pi$
$241$	2224.32	0.594528	0.297264	+	0.954795i	$0.403926\pi$
$242$	0	0
$243$	4924.17	1.29994
$244$	0	0
$245$	274.438	0.0715641
$246$	0	0
$247$	−4578.14	−1.17935
$248$	0	0
$249$	−2976.59	−0.757565
$250$	0	0
$251$	3928.85	0.987996	0.493998	−	0.869463i	$-0.335535\pi$
$251$	3928.85	0.987996	0.493998	+	0.869463i	$0.335535\pi$
$252$	0	0
$253$	−745.712	−0.185306
$254$	0	0
$255$	−553.298	−0.135878
$256$	0	0
$257$	−3962.37	−0.961734	−0.480867	−	0.876794i	$-0.659678\pi$
$257$	−3962.37	−0.961734	−0.480867	+	0.876794i	$0.659678\pi$
$258$	0	0
$259$	−469.498	−0.112638
$260$	0	0
$261$	−15930.2	−3.77800
$262$	0	0
$263$	−37.4419	−0.00877857	−0.00438929	−	0.999990i	$-0.501397\pi$
$263$	−37.4419	−0.00877857	−0.00438929	+	0.999990i	$0.501397\pi$
$264$	0	0
$265$	−467.738	−0.108426
$266$	0	0
$267$	−7302.44	−1.67379
$268$	0	0
$269$	2913.59	0.660390	0.330195	−	0.943913i	$-0.392886\pi$
$269$	2913.59	0.660390	0.330195	+	0.943913i	$0.392886\pi$
$270$	0	0
$271$	6604.08	1.48033	0.740165	−	0.672425i	$-0.234747\pi$
$271$	6604.08	1.48033	0.740165	+	0.672425i	$0.234747\pi$
$272$	0	0
$273$	−8517.02	−1.88818
$274$	0	0
$275$	463.060	0.101540
$276$	0	0
$277$	−1587.03	−0.344242	−0.172121	−	0.985076i	$-0.555062\pi$
$277$	−1587.03	−0.344242	−0.172121	+	0.985076i	$0.555062\pi$
$278$	0	0
$279$	6072.25	1.30300
$280$	0	0
$281$	8590.39	1.82370	0.911850	−	0.410524i	$-0.134654\pi$
$281$	8590.39	1.82370	0.911850	+	0.410524i	$0.134654\pi$
$282$	0	0
$283$	613.606	0.128887	0.0644437	−	0.997921i	$-0.479473\pi$
$283$	613.606	0.128887	0.0644437	+	0.997921i	$0.479473\pi$
$284$	0	0
$285$	919.083	0.191024
$286$	0	0
$287$	−4445.71	−0.914362
$288$	0	0
$289$	−3406.43	−0.693350
$290$	0	0
$291$	1804.02	0.363413
$292$	0	0
$293$	−4502.56	−0.897755	−0.448878	−	0.893593i	$-0.648176\pi$
$293$	−4502.56	−0.897755	−0.448878	+	0.893593i	$0.648176\pi$
$294$	0	0
$295$	−170.282	−0.0336074
$296$	0	0
$297$	−1260.96	−0.246358
$298$	0	0
$299$	14033.6	2.71432
$300$	0	0
$301$	−1761.41	−0.337296
$302$	0	0
$303$	−2072.57	−0.392958
$304$	0	0
$305$	−898.051	−0.168598
$306$	0	0
$307$	1490.79	0.277146	0.138573	−	0.990352i	$-0.455748\pi$
$307$	1490.79	0.277146	0.138573	+	0.990352i	$0.455748\pi$
$308$	0	0
$309$	−10065.7	−1.85313
$310$	0	0
$311$	−213.697	−0.0389636	−0.0194818	−	0.999810i	$-0.506202\pi$
$311$	−213.697	−0.0389636	−0.0194818	+	0.999810i	$0.506202\pi$
$312$	0	0
$313$	−8886.22	−1.60472	−0.802362	−	0.596838i	$-0.796424\pi$
$313$	−8886.22	−1.60472	−0.802362	+	0.596838i	$0.796424\pi$
$314$	0	0
$315$	1193.17	0.213421
$316$	0	0
$317$	−2408.56	−0.426745	−0.213373	−	0.976971i	$-0.568445\pi$
$317$	−2408.56	−0.426745	−0.213373	+	0.976971i	$0.568445\pi$
$318$	0	0
$319$	963.956	0.169189
$320$	0	0
$321$	8968.76	1.55946
$322$	0	0
$323$	−2502.57	−0.431104
$324$	0	0
$325$	−8714.34	−1.48734
$326$	0	0
$327$	−7910.84	−1.33783
$328$	0	0
$329$	3878.47	0.649930
$330$	0	0
$331$	−1592.84	−0.264503	−0.132252	−	0.991216i	$-0.542221\pi$
$331$	−1592.84	−0.264503	−0.132252	+	0.991216i	$0.542221\pi$
$332$	0	0
$333$	2307.11	0.379666
$334$	0	0
$335$	−1356.42	−0.221221
$336$	0	0
$337$	−4151.95	−0.671131	−0.335565	−	0.942017i	$-0.608927\pi$
$337$	−4151.95	−0.671131	−0.335565	+	0.942017i	$0.608927\pi$
$338$	0	0
$339$	−7484.86	−1.19918
$340$	0	0
$341$	−367.438	−0.0583516
$342$	0	0
$343$	6661.61	1.04867
$344$	0	0
$345$	−2817.31	−0.439648
$346$	0	0
$347$	6076.73	0.940103	0.470051	−	0.882639i	$-0.344235\pi$
$347$	6076.73	0.940103	0.470051	+	0.882639i	$0.344235\pi$
$348$	0	0
$349$	1581.40	0.242551	0.121276	−	0.992619i	$-0.461302\pi$
$349$	1581.40	0.242551	0.121276	+	0.992619i	$0.461302\pi$
$350$	0	0
$351$	23730.0	3.60859
$352$	0	0
$353$	−2015.99	−0.303967	−0.151984	−	0.988383i	$-0.548566\pi$
$353$	−2015.99	−0.303967	−0.151984	+	0.988383i	$0.548566\pi$
$354$	0	0
$355$	−679.140	−0.101535
$356$	0	0
$357$	−4655.69	−0.690211
$358$	0	0
$359$	−3176.91	−0.467050	−0.233525	−	0.972351i	$-0.575026\pi$
$359$	−3176.91	−0.467050	−0.233525	+	0.972351i	$0.575026\pi$
$360$	0	0
$361$	−2701.98	−0.393932
$362$	0	0
$363$	−12447.0	−1.79972
$364$	0	0
$365$	−708.701	−0.101630
$366$	0	0
$367$	−3963.25	−0.563706	−0.281853	−	0.959458i	$-0.590949\pi$
$367$	−3963.25	−0.563706	−0.281853	+	0.959458i	$0.590949\pi$
$368$	0	0
$369$	21846.2	3.08202
$370$	0	0
$371$	−3935.75	−0.550766
$372$	0	0
$373$	11018.6	1.52955	0.764776	−	0.644296i	$-0.222850\pi$
$373$	11018.6	1.52955	0.764776	+	0.644296i	$0.222850\pi$
$374$	0	0
$375$	3531.30	0.486282
$376$	0	0
$377$	−18140.7	−2.47823
$378$	0	0
$379$	−1215.73	−0.164769	−0.0823847	−	0.996601i	$-0.526254\pi$
$379$	−1215.73	−0.164769	−0.0823847	+	0.996601i	$0.526254\pi$
$380$	0	0
$381$	14770.8	1.98617
$382$	0	0
$383$	−8873.92	−1.18391	−0.591953	−	0.805972i	$-0.701643\pi$
$383$	−8873.92	−1.18391	−0.591953	+	0.805972i	$0.701643\pi$
$384$	0	0
$385$	−72.2003	−0.00955757
$386$	0	0
$387$	8655.56	1.13692
$388$	0	0
$389$	5901.38	0.769182	0.384591	−	0.923087i	$-0.374343\pi$
$389$	5901.38	0.769182	0.384591	+	0.923087i	$0.374343\pi$
$390$	0	0
$391$	7671.24	0.992202
$392$	0	0
$393$	7412.81	0.951468
$394$	0	0
$395$	397.066	0.0505786
$396$	0	0
$397$	−13938.7	−1.76212	−0.881060	−	0.473004i	$-0.843170\pi$
$397$	−13938.7	−1.76212	−0.881060	+	0.473004i	$0.843170\pi$
$398$	0	0
$399$	7733.57	0.970333
$400$	0	0
$401$	−11862.8	−1.47730	−0.738652	−	0.674087i	$-0.764537\pi$
$401$	−11862.8	−1.47730	−0.738652	+	0.674087i	$0.764537\pi$
$402$	0	0
$403$	6914.83	0.854720
$404$	0	0
$405$	−2225.06	−0.272998
$406$	0	0
$407$	−139.606	−0.0170024
$408$	0	0
$409$	−15104.9	−1.82614	−0.913068	−	0.407808i	$-0.866293\pi$
$409$	−15104.9	−1.82614	−0.913068	+	0.407808i	$0.866293\pi$
$410$	0	0
$411$	19919.4	2.39064
$412$	0	0
$413$	−1432.83	−0.170714
$414$	0	0
$415$	474.862	0.0561688
$416$	0	0
$417$	19150.6	2.24894
$418$	0	0
$419$	−16878.5	−1.96794	−0.983970	−	0.178334i	$-0.942929\pi$
$419$	−16878.5	−1.96794	−0.983970	+	0.178334i	$0.942929\pi$
$420$	0	0
$421$	−9479.38	−1.09738	−0.548690	−	0.836026i	$-0.684873\pi$
$421$	−9479.38	−1.09738	−0.548690	+	0.836026i	$0.684873\pi$
$422$	0	0
$423$	−19058.8	−2.19071
$424$	0	0
$425$	−4763.56	−0.543686
$426$	0	0
$427$	−7556.60	−0.856416
$428$	0	0
$429$	−2532.54	−0.285017
$430$	0	0
$431$	11023.1	1.23193	0.615965	−	0.787774i	$-0.288766\pi$
$431$	11023.1	1.23193	0.615965	+	0.787774i	$0.288766\pi$
$432$	0	0
$433$	2955.66	0.328037	0.164018	−	0.986457i	$-0.447554\pi$
$433$	2955.66	0.328037	0.164018	+	0.986457i	$0.447554\pi$
$434$	0	0
$435$	3641.83	0.401408
$436$	0	0
$437$	−12742.7	−1.39489
$438$	0	0
$439$	−4573.78	−0.497255	−0.248627	−	0.968599i	$-0.579979\pi$
$439$	−4573.78	−0.497255	−0.248627	+	0.968599i	$0.579979\pi$
$440$	0	0
$441$	−11347.6	−1.22531
$442$	0	0
$443$	−1029.07	−0.110367	−0.0551836	−	0.998476i	$-0.517574\pi$
$443$	−1029.07	−0.110367	−0.0551836	+	0.998476i	$0.517574\pi$
$444$	0	0
$445$	1164.97	0.124101
$446$	0	0
$447$	−7050.21	−0.746003
$448$	0	0
$449$	−1172.94	−0.123284	−0.0616420	−	0.998098i	$-0.519634\pi$
$449$	−1172.94	−0.123284	−0.0616420	+	0.998098i	$0.519634\pi$
$450$	0	0
$451$	−1321.94	−0.138021
$452$	0	0
$453$	−6221.85	−0.645316
$454$	0	0
$455$	1358.74	0.139997
$456$	0	0
$457$	−11020.8	−1.12808	−0.564040	−	0.825747i	$-0.690754\pi$
$457$	−11020.8	−1.12808	−0.564040	+	0.825747i	$0.690754\pi$
$458$	0	0
$459$	12971.6	1.31909
$460$	0	0
$461$	2568.56	0.259501	0.129750	−	0.991547i	$-0.458582\pi$
$461$	2568.56	0.259501	0.129750	+	0.991547i	$0.458582\pi$
$462$	0	0
$463$	−2479.09	−0.248841	−0.124420	−	0.992230i	$-0.539707\pi$
$463$	−2479.09	−0.248841	−0.124420	+	0.992230i	$0.539707\pi$
$464$	0	0
$465$	−1388.18	−0.138442
$466$	0	0
$467$	−9158.15	−0.907470	−0.453735	−	0.891137i	$-0.649909\pi$
$467$	−9158.15	−0.907470	−0.453735	+	0.891137i	$0.649909\pi$
$468$	0	0
$469$	−11413.5	−1.12372
$470$	0	0
$471$	19467.3	1.90447
$472$	0	0
$473$	−523.757	−0.0509141
$474$	0	0
$475$	7912.74	0.764340
$476$	0	0
$477$	19340.3	1.85646
$478$	0	0
$479$	9855.94	0.940144	0.470072	−	0.882628i	$-0.344228\pi$
$479$	9855.94	0.940144	0.470072	+	0.882628i	$0.344228\pi$
$480$	0	0
$481$	2627.24	0.249048
$482$	0	0
$483$	−23706.1	−2.23326
$484$	0	0
$485$	−287.799	−0.0269449
$486$	0	0
$487$	−9612.71	−0.894443	−0.447221	−	0.894423i	$-0.647586\pi$
$487$	−9612.71	−0.894443	−0.447221	+	0.894423i	$0.647586\pi$
$488$	0	0
$489$	11622.9	1.07486
$490$	0	0
$491$	19063.0	1.75214	0.876071	−	0.482183i	$-0.160156\pi$
$491$	19063.0	1.75214	0.876071	+	0.482183i	$0.160156\pi$
$492$	0	0
$493$	−9916.33	−0.905901
$494$	0	0
$495$	354.792	0.0322156
$496$	0	0
$497$	−5714.58	−0.515763
$498$	0	0
$499$	−6598.57	−0.591969	−0.295985	−	0.955193i	$-0.595648\pi$
$499$	−6598.57	−0.591969	−0.295985	+	0.955193i	$0.595648\pi$
$500$	0	0
$501$	1439.76	0.128391
$502$	0	0
$503$	4191.61	0.371560	0.185780	−	0.982591i	$-0.440519\pi$
$503$	4191.61	0.371560	0.185780	+	0.982591i	$0.440519\pi$
$504$	0	0
$505$	330.642	0.0291354
$506$	0	0
$507$	26892.3	2.35568
$508$	0	0
$509$	9553.58	0.831935	0.415967	−	0.909379i	$-0.363443\pi$
$509$	9553.58	0.831935	0.415967	+	0.909379i	$0.363443\pi$
$510$	0	0
$511$	−5963.33	−0.516247
$512$	0	0
$513$	−21547.2	−1.85445
$514$	0	0
$515$	1605.80	0.137398
$516$	0	0
$517$	1153.27	0.0981057
$518$	0	0
$519$	−10928.6	−0.924305
$520$	0	0
$521$	1074.39	0.0903456	0.0451728	−	0.998979i	$-0.485616\pi$
$521$	1074.39	0.0903456	0.0451728	+	0.998979i	$0.485616\pi$
$522$	0	0
$523$	5493.14	0.459270	0.229635	−	0.973277i	$-0.426247\pi$
$523$	5493.14	0.459270	0.229635	+	0.973277i	$0.426247\pi$
$524$	0	0
$525$	14720.6	1.22373
$526$	0	0
$527$	3779.88	0.312437
$528$	0	0
$529$	26893.7	2.21038
$530$	0	0
$531$	7040.90	0.575422
$532$	0	0
$533$	24877.5	2.02170
$534$	0	0
$535$	−1430.81	−0.115625
$536$	0	0
$537$	3138.50	0.252209
$538$	0	0
$539$	686.656	0.0548727
$540$	0	0
$541$	3776.15	0.300091	0.150046	−	0.988679i	$-0.452058\pi$
$541$	3776.15	0.300091	0.150046	+	0.988679i	$0.452058\pi$
$542$	0	0
$543$	31577.9	2.49565
$544$	0	0
$545$	1262.03	0.0991919
$546$	0	0
$547$	502.622	0.0392880	0.0196440	−	0.999807i	$-0.493747\pi$
$547$	502.622	0.0392880	0.0196440	+	0.999807i	$0.493747\pi$
$548$	0	0
$549$	37133.1	2.88671
$550$	0	0
$551$	16472.0	1.27356
$552$	0	0
$553$	3341.09	0.256922
$554$	0	0
$555$	−527.431	−0.0403391
$556$	0	0
$557$	−2600.54	−0.197825	−0.0989125	−	0.995096i	$-0.531536\pi$
$557$	−2600.54	−0.197825	−0.0989125	+	0.995096i	$0.531536\pi$
$558$	0	0
$559$	9856.60	0.745777
$560$	0	0
$561$	−1384.38	−0.104186
$562$	0	0
$563$	10974.5	0.821530	0.410765	−	0.911741i	$-0.365262\pi$
$563$	10974.5	0.821530	0.410765	+	0.911741i	$0.365262\pi$
$564$	0	0
$565$	1194.08	0.0889118
$566$	0	0
$567$	−18722.7	−1.38673
$568$	0	0
$569$	−6044.57	−0.445346	−0.222673	−	0.974893i	$-0.571478\pi$
$569$	−6044.57	−0.445346	−0.222673	+	0.974893i	$0.571478\pi$
$570$	0	0
$571$	−1398.76	−0.102515	−0.0512577	−	0.998685i	$-0.516323\pi$
$571$	−1398.76	−0.102515	−0.0512577	+	0.998685i	$0.516323\pi$
$572$	0	0
$573$	−1632.69	−0.119034
$574$	0	0
$575$	−24255.3	−1.75916
$576$	0	0
$577$	10841.3	0.782199	0.391100	−	0.920348i	$-0.372095\pi$
$577$	10841.3	0.782199	0.391100	+	0.920348i	$0.372095\pi$
$578$	0	0
$579$	33396.6	2.39709
$580$	0	0
$581$	3995.70	0.285318
$582$	0	0
$583$	−1170.30	−0.0831370
$584$	0	0
$585$	−6676.83	−0.471886
$586$	0	0
$587$	19183.2	1.34885	0.674424	−	0.738344i	$-0.264392\pi$
$587$	19183.2	1.34885	0.674424	+	0.738344i	$0.264392\pi$
$588$	0	0
$589$	−6278.76	−0.439239
$590$	0	0
$591$	−19269.0	−1.34115
$592$	0	0
$593$	−2984.87	−0.206701	−0.103351	−	0.994645i	$-0.532956\pi$
$593$	−2984.87	−0.206701	−0.103351	+	0.994645i	$0.532956\pi$
$594$	0	0
$595$	742.733	0.0511749
$596$	0	0
$597$	−32315.2	−2.21536
$598$	0	0
$599$	−26488.6	−1.80684	−0.903418	−	0.428760i	$-0.858950\pi$
$599$	−26488.6	−1.80684	−0.903418	+	0.428760i	$0.858950\pi$
$600$	0	0
$601$	−11038.5	−0.749202	−0.374601	−	0.927186i	$-0.622220\pi$
$601$	−11038.5	−0.749202	−0.374601	+	0.927186i	$0.622220\pi$
$602$	0	0
$603$	56085.8	3.78771
$604$	0	0
$605$	1985.70	0.133438
$606$	0	0
$607$	−1147.97	−0.0767621	−0.0383811	−	0.999263i	$-0.512220\pi$
$607$	−1147.97	−0.0767621	−0.0383811	+	0.999263i	$0.512220\pi$
$608$	0	0
$609$	30644.0	2.03901
$610$	0	0
$611$	−21703.4	−1.43703
$612$	0	0
$613$	−15341.5	−1.01083	−0.505413	−	0.862878i	$-0.668660\pi$
$613$	−15341.5	−1.01083	−0.505413	+	0.862878i	$0.668660\pi$
$614$	0	0
$615$	−4994.28	−0.327462
$616$	0	0
$617$	−4831.63	−0.315258	−0.157629	−	0.987498i	$-0.550385\pi$
$617$	−4831.63	−0.315258	−0.157629	+	0.987498i	$0.550385\pi$
$618$	0	0
$619$	−13437.2	−0.872514	−0.436257	−	0.899822i	$-0.643696\pi$
$619$	−13437.2	−0.872514	−0.436257	+	0.899822i	$0.643696\pi$
$620$	0	0
$621$	66049.5	4.26808
$622$	0	0
$623$	9802.61	0.630390
$624$	0	0
$625$	14777.4	0.945752
$626$	0	0
$627$	2299.59	0.146470
$628$	0	0
$629$	1436.14	0.0910376
$630$	0	0
$631$	−2826.33	−0.178311	−0.0891556	−	0.996018i	$-0.528417\pi$
$631$	−2826.33	−0.178311	−0.0891556	+	0.996018i	$0.528417\pi$
$632$	0	0
$633$	−43971.7	−2.76101
$634$	0	0
$635$	−2356.41	−0.147262
$636$	0	0
$637$	−12922.2	−0.803762
$638$	0	0
$639$	28081.4	1.73847
$640$	0	0
$641$	−26104.6	−1.60853	−0.804266	−	0.594270i	$-0.797441\pi$
$641$	−26104.6	−1.60853	−0.804266	+	0.594270i	$0.797441\pi$
$642$	0	0
$643$	8811.39	0.540416	0.270208	−	0.962802i	$-0.412908\pi$
$643$	8811.39	0.540416	0.270208	+	0.962802i	$0.412908\pi$
$644$	0	0
$645$	−1978.76	−0.120796
$646$	0	0
$647$	−20932.0	−1.27191	−0.635953	−	0.771728i	$-0.719393\pi$
$647$	−20932.0	−1.27191	−0.635953	+	0.771728i	$0.719393\pi$
$648$	0	0
$649$	−426.052	−0.0257689
$650$	0	0
$651$	−11680.8	−0.703236
$652$	0	0
$653$	15214.9	0.911802	0.455901	−	0.890031i	$-0.349317\pi$
$653$	15214.9	0.911802	0.455901	+	0.890031i	$0.349317\pi$
$654$	0	0
$655$	−1182.58	−0.0705455
$656$	0	0
$657$	29303.8	1.74010
$658$	0	0
$659$	1499.89	0.0886609	0.0443305	−	0.999017i	$-0.485885\pi$
$659$	1499.89	0.0886609	0.0443305	+	0.999017i	$0.485885\pi$
$660$	0	0
$661$	5665.19	0.333359	0.166680	−	0.986011i	$-0.446695\pi$
$661$	5665.19	0.333359	0.166680	+	0.986011i	$0.446695\pi$
$662$	0	0
$663$	26052.6	1.52609
$664$	0	0
$665$	−1233.75	−0.0719443
$666$	0	0
$667$	−50492.4	−2.93115
$668$	0	0
$669$	−2654.40	−0.153401
$670$	0	0
$671$	−2246.96	−0.129274
$672$	0	0
$673$	−10083.0	−0.577522	−0.288761	−	0.957401i	$-0.593243\pi$
$673$	−10083.0	−0.577522	−0.288761	+	0.957401i	$0.593243\pi$
$674$	0	0
$675$	−41014.3	−2.33873
$676$	0	0
$677$	−15720.0	−0.892421	−0.446210	−	0.894928i	$-0.647227\pi$
$677$	−15720.0	−0.892421	−0.446210	+	0.894928i	$0.647227\pi$
$678$	0	0
$679$	−2421.67	−0.136870
$680$	0	0
$681$	35440.8	1.99427
$682$	0	0
$683$	12308.4	0.689557	0.344778	−	0.938684i	$-0.387954\pi$
$683$	12308.4	0.689557	0.344778	+	0.938684i	$0.387954\pi$
$684$	0	0
$685$	−3177.79	−0.177251
$686$	0	0
$687$	49777.9	2.76440
$688$	0	0
$689$	22023.9	1.21777
$690$	0	0
$691$	23345.1	1.28522	0.642611	−	0.766193i	$-0.277851\pi$
$691$	23345.1	1.28522	0.642611	+	0.766193i	$0.277851\pi$
$692$	0	0
$693$	2985.38	0.163644
$694$	0	0
$695$	−3055.13	−0.166745
$696$	0	0
$697$	13598.9	0.739018
$698$	0	0
$699$	−43061.2	−2.33008
$700$	0	0
$701$	−23813.7	−1.28307	−0.641534	−	0.767094i	$-0.721702\pi$
$701$	−23813.7	−1.28307	−0.641534	+	0.767094i	$0.721702\pi$
$702$	0	0
$703$	−2385.57	−0.127985
$704$	0	0
$705$	4357.05	0.232760
$706$	0	0
$707$	2782.17	0.147998
$708$	0	0
$709$	17351.3	0.919101	0.459550	−	0.888152i	$-0.348011\pi$
$709$	17351.3	0.919101	0.459550	+	0.888152i	$0.348011\pi$
$710$	0	0
$711$	−16418.1	−0.866001
$712$	0	0
$713$	19246.6	1.01093
$714$	0	0
$715$	404.022	0.0211323
$716$	0	0
$717$	25965.3	1.35243
$718$	0	0
$719$	−8786.46	−0.455743	−0.227872	−	0.973691i	$-0.573177\pi$
$719$	−8786.46	−0.455743	−0.227872	+	0.973691i	$0.573177\pi$
$720$	0	0
$721$	13511.9	0.697933
$722$	0	0
$723$	21026.0	1.08155
$724$	0	0
$725$	31353.9	1.60615
$726$	0	0
$727$	−1973.85	−0.100696	−0.0503480	−	0.998732i	$-0.516033\pi$
$727$	−1973.85	−0.100696	−0.0503480	+	0.998732i	$0.516033\pi$
$728$	0	0
$729$	6708.62	0.340833
$730$	0	0
$731$	5387.95	0.272614
$732$	0	0
$733$	16371.2	0.824943	0.412471	−	0.910971i	$-0.364666\pi$
$733$	16371.2	0.824943	0.412471	+	0.910971i	$0.364666\pi$
$734$	0	0
$735$	2594.19	0.130188
$736$	0	0
$737$	−3393.81	−0.169624
$738$	0	0
$739$	−28981.5	−1.44263	−0.721314	−	0.692608i	$-0.756462\pi$
$739$	−28981.5	−1.44263	−0.721314	+	0.692608i	$0.756462\pi$
$740$	0	0
$741$	−43276.0	−2.14546
$742$	0	0
$743$	25088.6	1.23878	0.619389	−	0.785084i	$-0.287380\pi$
$743$	25088.6	1.23878	0.619389	+	0.785084i	$0.287380\pi$
$744$	0	0
$745$	1124.74	0.0553116
$746$	0	0
$747$	−19634.8	−0.961715
$748$	0	0
$749$	−12039.4	−0.587332
$750$	0	0
$751$	16568.9	0.805069	0.402534	−	0.915405i	$-0.368129\pi$
$751$	16568.9	0.805069	0.402534	+	0.915405i	$0.368129\pi$
$752$	0	0
$753$	37138.4	1.79734
$754$	0	0
$755$	992.586	0.0478462
$756$	0	0
$757$	−5430.63	−0.260739	−0.130370	−	0.991465i	$-0.541616\pi$
$757$	−5430.63	−0.260739	−0.130370	+	0.991465i	$0.541616\pi$
$758$	0	0
$759$	−7049.02	−0.337106
$760$	0	0
$761$	−7111.25	−0.338742	−0.169371	−	0.985552i	$-0.554174\pi$
$761$	−7111.25	−0.338742	−0.169371	+	0.985552i	$0.554174\pi$
$762$	0	0
$763$	10619.3	0.503860
$764$	0	0
$765$	−3649.79	−0.172495
$766$	0	0
$767$	8017.89	0.377457
$768$	0	0
$769$	−31140.3	−1.46027	−0.730136	−	0.683302i	$-0.760543\pi$
$769$	−31140.3	−1.46027	−0.730136	+	0.683302i	$0.760543\pi$
$770$	0	0
$771$	−37455.2	−1.74957
$772$	0	0
$773$	−34520.2	−1.60622	−0.803109	−	0.595832i	$-0.796822\pi$
$773$	−34520.2	−1.60622	−0.803109	+	0.595832i	$0.796822\pi$
$774$	0	0
$775$	−11951.4	−0.553945
$776$	0	0
$777$	−4438.04	−0.204908
$778$	0	0
$779$	−22589.2	−1.03895
$780$	0	0
$781$	−1699.24	−0.0778534
$782$	0	0
$783$	−85379.9	−3.89684
$784$	0	0
$785$	−3105.66	−0.141205
$786$	0	0
$787$	−31271.2	−1.41639	−0.708195	−	0.706017i	$-0.750490\pi$
$787$	−31271.2	−1.41639	−0.708195	+	0.706017i	$0.750490\pi$
$788$	0	0
$789$	−353.928	−0.0159698
$790$	0	0
$791$	10047.5	0.451641
$792$	0	0
$793$	42285.7	1.89358
$794$	0	0
$795$	−4421.40	−0.197246
$796$	0	0
$797$	−37179.6	−1.65241	−0.826205	−	0.563370i	$-0.809505\pi$
$797$	−37179.6	−1.65241	−0.826205	+	0.563370i	$0.809505\pi$
$798$	0	0
$799$	−11863.8	−0.525296
$800$	0	0
$801$	−48170.0	−2.12485
$802$	0	0
$803$	−1773.20	−0.0779264
$804$	0	0
$805$	3781.88	0.165582
$806$	0	0
$807$	27541.4	1.20137
$808$	0	0
$809$	2118.84	0.0920820	0.0460410	−	0.998940i	$-0.485340\pi$
$809$	2118.84	0.0920820	0.0460410	+	0.998940i	$0.485340\pi$
$810$	0	0
$811$	18596.6	0.805198	0.402599	−	0.915377i	$-0.368107\pi$
$811$	18596.6	0.805198	0.402599	+	0.915377i	$0.368107\pi$
$812$	0	0
$813$	62426.7	2.69299
$814$	0	0
$815$	−1854.23	−0.0796942
$816$	0	0
$817$	−8949.93	−0.383254
$818$	0	0
$819$	−56181.9	−2.39701
$820$	0	0
$821$	−2969.97	−0.126252	−0.0631258	−	0.998006i	$-0.520107\pi$
$821$	−2969.97	−0.126252	−0.0631258	+	0.998006i	$0.520107\pi$
$822$	0	0
$823$	−21859.2	−0.925835	−0.462918	−	0.886401i	$-0.653197\pi$
$823$	−21859.2	−0.925835	−0.462918	+	0.886401i	$0.653197\pi$
$824$	0	0
$825$	4377.19	0.184720
$826$	0	0
$827$	−40220.4	−1.69117	−0.845587	−	0.533837i	$-0.820750\pi$
$827$	−40220.4	−1.69117	−0.845587	+	0.533837i	$0.820750\pi$
$828$	0	0
$829$	−16095.9	−0.674346	−0.337173	−	0.941443i	$-0.609471\pi$
$829$	−16095.9	−0.674346	−0.337173	+	0.941443i	$0.609471\pi$
$830$	0	0
$831$	−15001.7	−0.626239
$832$	0	0
$833$	−7063.72	−0.293810
$834$	0	0
$835$	−229.688	−0.00951938
$836$	0	0
$837$	32544.9	1.34399
$838$	0	0
$839$	7811.35	0.321428	0.160714	−	0.987001i	$-0.448620\pi$
$839$	7811.35	0.321428	0.160714	+	0.987001i	$0.448620\pi$
$840$	0	0
$841$	40880.7	1.67620
$842$	0	0
$843$	81202.7	3.31764
$844$	0	0
$845$	−4290.19	−0.174659
$846$	0	0
$847$	16708.6	0.677820
$848$	0	0
$849$	5800.26	0.234469
$850$	0	0
$851$	7312.60	0.294563
$852$	0	0
$853$	24196.0	0.971224	0.485612	−	0.874175i	$-0.338597\pi$
$853$	24196.0	0.971224	0.485612	+	0.874175i	$0.338597\pi$
$854$	0	0
$855$	6062.66	0.242501
$856$	0	0
$857$	−18625.1	−0.742380	−0.371190	−	0.928557i	$-0.621050\pi$
$857$	−18625.1	−0.742380	−0.371190	+	0.928557i	$0.621050\pi$
$858$	0	0
$859$	31480.8	1.25042	0.625210	−	0.780457i	$-0.285013\pi$
$859$	31480.8	1.25042	0.625210	+	0.780457i	$0.285013\pi$
$860$	0	0
$861$	−42024.1	−1.66339
$862$	0	0
$863$	8968.69	0.353763	0.176882	−	0.984232i	$-0.443399\pi$
$863$	8968.69	0.353763	0.176882	+	0.984232i	$0.443399\pi$
$864$	0	0
$865$	1743.47	0.0685316
$866$	0	0
$867$	−32200.1	−1.26133
$868$	0	0
$869$	993.477	0.0387818
$870$	0	0
$871$	63868.2	2.48461
$872$	0	0
$873$	11900.1	0.461347
$874$	0	0
$875$	−4740.33	−0.183146
$876$	0	0
$877$	−44884.3	−1.72820	−0.864101	−	0.503318i	$-0.832113\pi$
$877$	−44884.3	−1.72820	−0.864101	+	0.503318i	$0.832113\pi$
$878$	0	0
$879$	−42561.5	−1.63318
$880$	0	0
$881$	−29887.5	−1.14295	−0.571473	−	0.820621i	$-0.693628\pi$
$881$	−29887.5	−1.14295	−0.571473	+	0.820621i	$0.693628\pi$
$882$	0	0
$883$	34321.4	1.30805	0.654024	−	0.756474i	$-0.273080\pi$
$883$	34321.4	1.30805	0.654024	+	0.756474i	$0.273080\pi$
$884$	0	0
$885$	−1609.63	−0.0611379
$886$	0	0
$887$	−29971.4	−1.13455	−0.567273	−	0.823530i	$-0.692001\pi$
$887$	−29971.4	−1.13455	−0.567273	+	0.823530i	$0.692001\pi$
$888$	0	0
$889$	−19827.9	−0.748039
$890$	0	0
$891$	−5567.21	−0.209325
$892$	0	0
$893$	19707.0	0.738486
$894$	0	0
$895$	−500.691	−0.0186997
$896$	0	0
$897$	132656.	4.93784
$898$	0	0
$899$	−24879.4	−0.922995
$900$	0	0
$901$	12039.0	0.445148
$902$	0	0
$903$	−16650.1	−0.613601
$904$	0	0
$905$	−5037.69	−0.185037
$906$	0	0
$907$	−53353.2	−1.95321	−0.976607	−	0.215032i	$-0.931014\pi$
$907$	−53353.2	−1.95321	−0.976607	+	0.215032i	$0.931014\pi$
$908$	0	0
$909$	−13671.6	−0.498853
$910$	0	0
$911$	36185.6	1.31601	0.658004	−	0.753014i	$-0.271401\pi$
$911$	36185.6	1.31601	0.658004	+	0.753014i	$0.271401\pi$
$912$	0	0
$913$	1188.13	0.0430681
$914$	0	0
$915$	−8489.04	−0.306709
$916$	0	0
$917$	−9950.78	−0.358346
$918$	0	0
$919$	−38631.3	−1.38665	−0.693324	−	0.720626i	$-0.743854\pi$
$919$	−38631.3	−1.38665	−0.693324	+	0.720626i	$0.743854\pi$
$920$	0	0
$921$	14092.1	0.504179
$922$	0	0
$923$	31978.0	1.14038
$924$	0	0
$925$	−4540.86	−0.161408
$926$	0	0
$927$	−66397.5	−2.35251
$928$	0	0
$929$	25067.5	0.885294	0.442647	−	0.896696i	$-0.354040\pi$
$929$	25067.5	0.885294	0.442647	+	0.896696i	$0.354040\pi$
$930$	0	0
$931$	11733.5	0.413052
$932$	0	0
$933$	−2020.03	−0.0708818
$934$	0	0
$935$	220.853	0.00772476
$936$	0	0
$937$	−5069.83	−0.176760	−0.0883801	−	0.996087i	$-0.528169\pi$
$937$	−5069.83	−0.176760	−0.0883801	+	0.996087i	$0.528169\pi$
$938$	0	0
$939$	−83999.1	−2.91928
$940$	0	0
$941$	6417.99	0.222338	0.111169	−	0.993801i	$-0.464540\pi$
$941$	6417.99	0.222338	0.111169	+	0.993801i	$0.464540\pi$
$942$	0	0
$943$	69243.6	2.39118
$944$	0	0
$945$	6394.95	0.220135
$946$	0	0
$947$	11504.2	0.394760	0.197380	−	0.980327i	$-0.436757\pi$
$947$	11504.2	0.394760	0.197380	+	0.980327i	$0.436757\pi$
$948$	0	0
$949$	33369.9	1.14145
$950$	0	0
$951$	−22767.5	−0.776327
$952$	0	0
$953$	−39729.1	−1.35042	−0.675212	−	0.737624i	$-0.735948\pi$
$953$	−39729.1	−1.35042	−0.675212	+	0.737624i	$0.735948\pi$
$954$	0	0
$955$	260.466	0.00882564
$956$	0	0
$957$	9112.02	0.307785
$958$	0	0
$959$	−26739.3	−0.900372
$960$	0	0
$961$	−20307.6	−0.681667
$962$	0	0
$963$	59161.8	1.97971
$964$	0	0
$965$	−5327.83	−0.177730
$966$	0	0
$967$	54102.8	1.79920	0.899602	−	0.436711i	$-0.143857\pi$
$967$	54102.8	1.79920	0.899602	+	0.436711i	$0.143857\pi$
$968$	0	0
$969$	−23656.1	−0.784256
$970$	0	0
$971$	−11991.7	−0.396325	−0.198162	−	0.980169i	$-0.563497\pi$
$971$	−11991.7	−0.396325	−0.198162	+	0.980169i	$0.563497\pi$
$972$	0	0
$973$	−25707.3	−0.847006
$974$	0	0
$975$	−82374.4	−2.70573
$976$	0	0
$977$	−30943.4	−1.01327	−0.506636	−	0.862160i	$-0.669111\pi$
$977$	−30943.4	−1.01327	−0.506636	+	0.862160i	$0.669111\pi$
$978$	0	0
$979$	2914.82	0.0951562
$980$	0	0
$981$	−52183.2	−1.69835
$982$	0	0
$983$	−25930.5	−0.841359	−0.420679	−	0.907209i	$-0.638208\pi$
$983$	−25930.5	−0.841359	−0.420679	+	0.907209i	$0.638208\pi$
$984$	0	0
$985$	3074.02	0.0994379
$986$	0	0
$987$	36662.2	1.18234
$988$	0	0
$989$	27434.6	0.882073
$990$	0	0
$991$	15448.4	0.495192	0.247596	−	0.968863i	$-0.420359\pi$
$991$	15448.4	0.495192	0.247596	+	0.968863i	$0.420359\pi$
$992$	0	0
$993$	−15056.7	−0.481179
$994$	0	0
$995$	5155.31	0.164256
$996$	0	0
$997$	−4372.65	−0.138900	−0.0694499	−	0.997585i	$-0.522124\pi$
$997$	−4372.65	−0.138900	−0.0694499	+	0.997585i	$0.522124\pi$
$998$	0	0
$999$	12365.2	0.391609

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.4.a.r.1.5		5
4.3	odd	2		2368.4.a.m.1.1		5
8.3	odd	2		37.4.a.b.1.2	✓	5
8.5	even	2		592.4.a.g.1.1		5
24.11	even	2		333.4.a.f.1.4		5
40.19	odd	2		925.4.a.b.1.4		5
56.27	even	2		1813.4.a.c.1.2		5
296.147	odd	2		1369.4.a.d.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
37.4.a.b.1.2	✓	5	8.3	odd	2
333.4.a.f.1.4		5	24.11	even	2
592.4.a.g.1.1		5	8.5	even	2
925.4.a.b.1.4		5	40.19	odd	2
1369.4.a.d.1.4		5	296.147	odd	2
1813.4.a.c.1.2		5	56.27	even	2
2368.4.a.m.1.1		5	4.3	odd	2
2368.4.a.r.1.5		5	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:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2368.a (trivial)

\(f(q)\)	\(=\)	\(q+9.45274 q^{3} -1.50802 q^{5} -12.6891 q^{7} +62.3543 q^{9} +O(q^{10})\)	\(q+9.45274 q^{3} -1.50802 q^{5} -12.6891 q^{7} +62.3543 q^{9} -3.77312 q^{11} +71.0065 q^{13} -14.2549 q^{15} +38.8146 q^{17} -64.4749 q^{19} -119.947 q^{21} +197.638 q^{23} -122.726 q^{25} +334.195 q^{27} -255.479 q^{29} +97.3830 q^{31} -35.6664 q^{33} +19.1354 q^{35} +37.0000 q^{37} +671.206 q^{39} +350.356 q^{41} +138.813 q^{43} -94.0313 q^{45} -305.653 q^{47} -181.986 q^{49} +366.904 q^{51} +310.167 q^{53} +5.68993 q^{55} -609.465 q^{57} +112.918 q^{59} +595.518 q^{61} -791.221 q^{63} -107.079 q^{65} +899.470 q^{67} +1868.22 q^{69} +450.353 q^{71} +469.956 q^{73} -1160.10 q^{75} +47.8776 q^{77} -263.304 q^{79} +1475.49 q^{81} -314.892 q^{83} -58.5331 q^{85} -2414.98 q^{87} -772.521 q^{89} -901.010 q^{91} +920.536 q^{93} +97.2293 q^{95} +190.846 q^{97} -235.270 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 13 q^{3} - 11 q^{5} - 24 q^{7} + 46 q^{9}+O(q^{10}) \) 5 * q + 13 * q^3 - 11 * q^5 - 24 * q^7 + 46 * q^9	\( 5 q + 13 q^{3} - 11 q^{5} - 24 q^{7} + 46 q^{9} + 61 q^{11} + 37 q^{13} + 116 q^{15} + 130 q^{17} - 22 q^{19} + 44 q^{21} - 73 q^{23} + 26 q^{25} + 472 q^{27} - 271 q^{29} - 363 q^{31} + 198 q^{33} + 604 q^{35} + 185 q^{37} + 65 q^{39} + 381 q^{41} - 408 q^{43} + 704 q^{45} - 276 q^{47} - 949 q^{49} - 38 q^{51} - 156 q^{53} + 843 q^{55} - 1618 q^{57} + 100 q^{59} + 1711 q^{61} - 94 q^{63} - 890 q^{65} + 787 q^{67} + 2335 q^{69} - 1578 q^{71} - 313 q^{73} + 684 q^{75} + 342 q^{77} - 569 q^{79} + 385 q^{81} + 2422 q^{83} + 2210 q^{85} - 2371 q^{87} - 2466 q^{89} - 1678 q^{91} + 1142 q^{93} - 794 q^{95} - 2406 q^{97} + 1746 q^{99}+O(q^{100}) \) 5 * q + 13 * q^3 - 11 * q^5 - 24 * q^7 + 46 * q^9 + 61 * q^11 + 37 * q^13 + 116 * q^15 + 130 * q^17 - 22 * q^19 + 44 * q^21 - 73 * q^23 + 26 * q^25 + 472 * q^27 - 271 * q^29 - 363 * q^31 + 198 * q^33 + 604 * q^35 + 185 * q^37 + 65 * q^39 + 381 * q^41 - 408 * q^43 + 704 * q^45 - 276 * q^47 - 949 * q^49 - 38 * q^51 - 156 * q^53 + 843 * q^55 - 1618 * q^57 + 100 * q^59 + 1711 * q^61 - 94 * q^63 - 890 * q^65 + 787 * q^67 + 2335 * q^69 - 1578 * q^71 - 313 * q^73 + 684 * q^75 + 342 * q^77 - 569 * q^79 + 385 * q^81 + 2422 * q^83 + 2210 * q^85 - 2371 * q^87 - 2466 * q^89 - 1678 * q^91 + 1142 * q^93 - 794 * q^95 - 2406 * q^97 + 1746 * q^99

Introduction

L-functions

Modular forms

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