LMFDB - Embedded newform 152.4.a.c.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [152,4,Mod(1,152)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("152.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 152 = 2^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	152.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$8.96829032087$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.7057.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 22x + 32 $ x^3 - x^2 - 22*x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.86867$ of defining polynomial
Character	$\chi$	$=$	152.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+8.28632 q^{3} +2.38427 q^{5} +5.83529 q^{7} +41.6631 q^{9} -7.33364 q^{11} +55.6319 q^{13} +19.7568 q^{15} -10.0356 q^{17} -19.0000 q^{19} +48.3531 q^{21} -9.26357 q^{23} -119.315 q^{25} +121.503 q^{27} -83.9613 q^{29} +202.424 q^{31} -60.7689 q^{33} +13.9129 q^{35} +95.2017 q^{37} +460.983 q^{39} -25.9916 q^{41} -119.579 q^{43} +99.3358 q^{45} +467.824 q^{47} -308.949 q^{49} -83.1579 q^{51} -764.970 q^{53} -17.4853 q^{55} -157.440 q^{57} +69.1344 q^{59} -398.549 q^{61} +243.116 q^{63} +132.641 q^{65} -243.650 q^{67} -76.7609 q^{69} -781.354 q^{71} -711.911 q^{73} -988.684 q^{75} -42.7939 q^{77} +723.944 q^{79} -118.092 q^{81} -1227.94 q^{83} -23.9275 q^{85} -695.730 q^{87} -653.687 q^{89} +324.628 q^{91} +1677.35 q^{93} -45.3010 q^{95} +1692.40 q^{97} -305.542 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 21 q^{9} + 103 q^{11} + 32 q^{13} + 122 q^{15} + 11 q^{17} - 57 q^{19} + 114 q^{21} + 316 q^{23} + 162 q^{25} + 178 q^{27} - 138 q^{29} + 420 q^{31} - 330 q^{33} + 333 q^{35}+ \cdots - 1305 q^{99}+O(q^{100}) $ 3 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 21 * q^9 + 103 * q^11 + 32 * q^13 + 122 * q^15 + 11 * q^17 - 57 * q^19 + 114 * q^21 + 316 * q^23 + 162 * q^25 + 178 * q^27 - 138 * q^29 + 420 * q^31 - 330 * q^33 + 333 * q^35 + 102 * q^37 + 164 * q^39 - 370 * q^41 + 431 * q^43 - 429 * q^45 - 199 * q^47 - 802 * q^49 + 272 * q^51 - 308 * q^53 + 85 * q^55 - 76 * q^57 - 188 * q^59 - 609 * q^61 - 61 * q^63 - 1536 * q^65 - 246 * q^67 - 892 * q^69 - 954 * q^71 - 629 * q^73 - 1074 * q^75 - 71 * q^77 + 452 * q^79 - 361 * q^81 - 780 * q^83 + 1883 * q^85 - 96 * q^87 - 1356 * q^89 - 670 * q^91 + 1092 * q^93 - 133 * q^95 - 548 * q^97 - 1305 * q^99

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
$3$	8.28632	1.59470	0.797351	−	0.603516i	$-0.206234\pi$
$3$	8.28632	1.59470	0.797351	+	0.603516i	$0.206234\pi$
$4$	0	0
$5$	2.38427	0.213255	0.106628	−	0.994299i	$-0.465995\pi$
$5$	2.38427	0.213255	0.106628	+	0.994299i	$0.465995\pi$
$6$	0	0
$7$	5.83529	0.315076	0.157538	−	0.987513i	$-0.449644\pi$
$7$	5.83529	0.315076	0.157538	+	0.987513i	$0.449644\pi$
$8$	0	0
$9$	41.6631	1.54308
$10$	0	0
$11$	−7.33364	−0.201016	−0.100508	−	0.994936i	$-0.532047\pi$
$11$	−7.33364	−0.201016	−0.100508	+	0.994936i	$0.532047\pi$
$12$	0	0
$13$	55.6319	1.18688	0.593442	−	0.804876i	$-0.297768\pi$
$13$	55.6319	1.18688	0.593442	+	0.804876i	$0.297768\pi$
$14$	0	0
$15$	19.7568	0.340079
$16$	0	0
$17$	−10.0356	−0.143176	−0.0715878	−	0.997434i	$-0.522807\pi$
$17$	−10.0356	−0.143176	−0.0715878	+	0.997434i	$0.522807\pi$
$18$	0	0
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	0	0
$21$	48.3531	0.502453
$22$	0	0
$23$	−9.26357	−0.0839821	−0.0419911	−	0.999118i	$-0.513370\pi$
$23$	−9.26357	−0.0839821	−0.0419911	+	0.999118i	$0.513370\pi$
$24$	0	0
$25$	−119.315	−0.954522
$26$	0	0
$27$	121.503	0.866045
$28$	0	0
$29$	−83.9613	−0.537628	−0.268814	−	0.963192i	$-0.586632\pi$
$29$	−83.9613	−0.537628	−0.268814	+	0.963192i	$0.586632\pi$
$30$	0	0
$31$	202.424	1.17279	0.586394	−	0.810026i	$-0.300547\pi$
$31$	202.424	1.17279	0.586394	+	0.810026i	$0.300547\pi$
$32$	0	0
$33$	−60.7689	−0.320561
$34$	0	0
$35$	13.9129	0.0671916
$36$	0	0
$37$	95.2017	0.423002	0.211501	−	0.977378i	$-0.432165\pi$
$37$	95.2017	0.423002	0.211501	+	0.977378i	$0.432165\pi$
$38$	0	0
$39$	460.983	1.89273
$40$	0	0
$41$	−25.9916	−0.0990049	−0.0495024	−	0.998774i	$-0.515764\pi$
$41$	−25.9916	−0.0990049	−0.0495024	+	0.998774i	$0.515764\pi$
$42$	0	0
$43$	−119.579	−0.424084	−0.212042	−	0.977261i	$-0.568011\pi$
$43$	−119.579	−0.424084	−0.212042	+	0.977261i	$0.568011\pi$
$44$	0	0
$45$	99.3358	0.329069
$46$	0	0
$47$	467.824	1.45190	0.725948	−	0.687749i	$-0.241401\pi$
$47$	467.824	1.45190	0.725948	+	0.687749i	$0.241401\pi$
$48$	0	0
$49$	−308.949	−0.900727
$50$	0	0
$51$	−83.1579	−0.228322
$52$	0	0
$53$	−764.970	−1.98258	−0.991290	−	0.131695i	$-0.957958\pi$
$53$	−764.970	−1.98258	−0.991290	+	0.131695i	$0.957958\pi$
$54$	0	0
$55$	−17.4853	−0.0428677
$56$	0	0
$57$	−157.440	−0.365850
$58$	0	0
$59$	69.1344	0.152551	0.0762757	−	0.997087i	$-0.475697\pi$
$59$	69.1344	0.152551	0.0762757	+	0.997087i	$0.475697\pi$
$60$	0	0
$61$	−398.549	−0.836541	−0.418271	−	0.908322i	$-0.637364\pi$
$61$	−398.549	−0.836541	−0.418271	+	0.908322i	$0.637364\pi$
$62$	0	0
$63$	243.116	0.486186
$64$	0	0
$65$	132.641	0.253109
$66$	0	0
$67$	−243.650	−0.444278	−0.222139	−	0.975015i	$-0.571304\pi$
$67$	−243.650	−0.444278	−0.222139	+	0.975015i	$0.571304\pi$
$68$	0	0
$69$	−76.7609	−0.133926
$70$	0	0
$71$	−781.354	−1.30605	−0.653026	−	0.757336i	$-0.726501\pi$
$71$	−781.354	−1.30605	−0.653026	+	0.757336i	$0.726501\pi$
$72$	0	0
$73$	−711.911	−1.14141	−0.570705	−	0.821155i	$-0.693330\pi$
$73$	−711.911	−1.14141	−0.570705	+	0.821155i	$0.693330\pi$
$74$	0	0
$75$	−988.684	−1.52218
$76$	0	0
$77$	−42.7939	−0.0633353
$78$	0	0
$79$	723.944	1.03101	0.515507	−	0.856885i	$-0.327604\pi$
$79$	723.944	1.03101	0.515507	+	0.856885i	$0.327604\pi$
$80$	0	0
$81$	−118.092	−0.161992
$82$	0	0
$83$	−1227.94	−1.62391	−0.811953	−	0.583723i	$-0.801595\pi$
$83$	−1227.94	−1.62391	−0.811953	+	0.583723i	$0.801595\pi$
$84$	0	0
$85$	−23.9275	−0.0305329
$86$	0	0
$87$	−695.730	−0.857357
$88$	0	0
$89$	−653.687	−0.778548	−0.389274	−	0.921122i	$-0.627274\pi$
$89$	−653.687	−0.778548	−0.389274	+	0.921122i	$0.627274\pi$
$90$	0	0
$91$	324.628	0.373959
$92$	0	0
$93$	1677.35	1.87025
$94$	0	0
$95$	−45.3010	−0.0489241
$96$	0	0
$97$	1692.40	1.77152	0.885758	−	0.464148i	$-0.153639\pi$
$97$	1692.40	1.77152	0.885758	+	0.464148i	$0.153639\pi$
$98$	0	0
$99$	−305.542	−0.310183
$100$	0	0
$101$	839.619	0.827180	0.413590	−	0.910463i	$-0.364275\pi$
$101$	839.619	0.827180	0.413590	+	0.910463i	$0.364275\pi$
$102$	0	0
$103$	352.945	0.337638	0.168819	−	0.985647i	$-0.446005\pi$
$103$	352.945	0.337638	0.168819	+	0.985647i	$0.446005\pi$
$104$	0	0
$105$	115.287	0.107151
$106$	0	0
$107$	375.247	0.339033	0.169516	−	0.985527i	$-0.445779\pi$
$107$	375.247	0.339033	0.169516	+	0.985527i	$0.445779\pi$
$108$	0	0
$109$	1297.69	1.14033	0.570164	−	0.821531i	$-0.306880\pi$
$109$	1297.69	1.14033	0.570164	+	0.821531i	$0.306880\pi$
$110$	0	0
$111$	788.871	0.674562
$112$	0	0
$113$	−83.3803	−0.0694138	−0.0347069	−	0.999398i	$-0.511050\pi$
$113$	−83.3803	−0.0694138	−0.0347069	+	0.999398i	$0.511050\pi$
$114$	0	0
$115$	−22.0868	−0.0179096
$116$	0	0
$117$	2317.79	1.83145
$118$	0	0
$119$	−58.5605	−0.0451112
$120$	0	0
$121$	−1277.22	−0.959593
$122$	0	0
$123$	−215.374	−0.157883
$124$	0	0
$125$	−582.513	−0.416812
$126$	0	0
$127$	−1083.10	−0.756772	−0.378386	−	0.925648i	$-0.623521\pi$
$127$	−1083.10	−0.756772	−0.378386	+	0.925648i	$0.623521\pi$
$128$	0	0
$129$	−990.870	−0.676288
$130$	0	0
$131$	2400.48	1.60100	0.800500	−	0.599333i	$-0.204567\pi$
$131$	2400.48	1.60100	0.800500	+	0.599333i	$0.204567\pi$
$132$	0	0
$133$	−110.871	−0.0722834
$134$	0	0
$135$	289.695	0.184689
$136$	0	0
$137$	−2349.46	−1.46517	−0.732584	−	0.680677i	$-0.761686\pi$
$137$	−2349.46	−1.46517	−0.732584	+	0.680677i	$0.761686\pi$
$138$	0	0
$139$	2828.01	1.72568	0.862838	−	0.505481i	$-0.168685\pi$
$139$	2828.01	1.72568	0.862838	+	0.505481i	$0.168685\pi$
$140$	0	0
$141$	3876.54	2.31534
$142$	0	0
$143$	−407.984	−0.238583
$144$	0	0
$145$	−200.186	−0.114652
$146$	0	0
$147$	−2560.05	−1.43639
$148$	0	0
$149$	−1609.68	−0.885033	−0.442517	−	0.896760i	$-0.645914\pi$
$149$	−1609.68	−0.885033	−0.442517	+	0.896760i	$0.645914\pi$
$150$	0	0
$151$	1625.10	0.875819	0.437910	−	0.899019i	$-0.355719\pi$
$151$	1625.10	0.875819	0.437910	+	0.899019i	$0.355719\pi$
$152$	0	0
$153$	−418.113	−0.220931
$154$	0	0
$155$	482.633	0.250103
$156$	0	0
$157$	1747.90	0.888521	0.444261	−	0.895898i	$-0.353466\pi$
$157$	1747.90	0.888521	0.444261	+	0.895898i	$0.353466\pi$
$158$	0	0
$159$	−6338.79	−3.16163
$160$	0	0
$161$	−54.0556	−0.0264608
$162$	0	0
$163$	805.324	0.386981	0.193490	−	0.981102i	$-0.438019\pi$
$163$	805.324	0.386981	0.193490	+	0.981102i	$0.438019\pi$
$164$	0	0
$165$	−144.889	−0.0683612
$166$	0	0
$167$	1297.79	0.601354	0.300677	−	0.953726i	$-0.402787\pi$
$167$	1297.79	0.601354	0.300677	+	0.953726i	$0.402787\pi$
$168$	0	0
$169$	897.905	0.408696
$170$	0	0
$171$	−791.598	−0.354006
$172$	0	0
$173$	−1244.83	−0.547068	−0.273534	−	0.961862i	$-0.588193\pi$
$173$	−1244.83	−0.547068	−0.273534	+	0.961862i	$0.588193\pi$
$174$	0	0
$175$	−696.239	−0.300747
$176$	0	0
$177$	572.870	0.243274
$178$	0	0
$179$	1715.51	0.716331	0.358166	−	0.933658i	$-0.383402\pi$
$179$	1715.51	0.716331	0.358166	+	0.933658i	$0.383402\pi$
$180$	0	0
$181$	2382.52	0.978406	0.489203	−	0.872170i	$-0.337288\pi$
$181$	2382.52	0.978406	0.489203	+	0.872170i	$0.337288\pi$
$182$	0	0
$183$	−3302.51	−1.33403
$184$	0	0
$185$	226.986	0.0902073
$186$	0	0
$187$	73.5973	0.0287806
$188$	0	0
$189$	709.004	0.272870
$190$	0	0
$191$	1209.12	0.458056	0.229028	−	0.973420i	$-0.426445\pi$
$191$	1209.12	0.458056	0.229028	+	0.973420i	$0.426445\pi$
$192$	0	0
$193$	3304.21	1.23234	0.616172	−	0.787612i	$-0.288683\pi$
$193$	3304.21	1.23234	0.616172	+	0.787612i	$0.288683\pi$
$194$	0	0
$195$	1099.11	0.403634
$196$	0	0
$197$	−263.227	−0.0951989	−0.0475994	−	0.998867i	$-0.515157\pi$
$197$	−263.227	−0.0951989	−0.0475994	+	0.998867i	$0.515157\pi$
$198$	0	0
$199$	1172.29	0.417593	0.208797	−	0.977959i	$-0.433045\pi$
$199$	1172.29	0.417593	0.208797	+	0.977959i	$0.433045\pi$
$200$	0	0
$201$	−2018.96	−0.708491
$202$	0	0
$203$	−489.939	−0.169394
$204$	0	0
$205$	−61.9708	−0.0211133
$206$	0	0
$207$	−385.949	−0.129591
$208$	0	0
$209$	139.339	0.0461162
$210$	0	0
$211$	4970.45	1.62170	0.810852	−	0.585251i	$-0.199004\pi$
$211$	4970.45	1.62170	0.810852	+	0.585251i	$0.199004\pi$
$212$	0	0
$213$	−6474.55	−2.08276
$214$	0	0
$215$	−285.108	−0.0904382
$216$	0	0
$217$	1181.20	0.369518
$218$	0	0
$219$	−5899.12	−1.82021
$220$	0	0
$221$	−558.298	−0.169933
$222$	0	0
$223$	−4348.86	−1.30592	−0.652962	−	0.757391i	$-0.726474\pi$
$223$	−4348.86	−1.30592	−0.652962	+	0.757391i	$0.726474\pi$
$224$	0	0
$225$	−4971.04	−1.47290
$226$	0	0
$227$	−735.340	−0.215006	−0.107503	−	0.994205i	$-0.534285\pi$
$227$	−735.340	−0.215006	−0.107503	+	0.994205i	$0.534285\pi$
$228$	0	0
$229$	392.347	0.113218	0.0566092	−	0.998396i	$-0.481971\pi$
$229$	392.347	0.113218	0.0566092	+	0.998396i	$0.481971\pi$
$230$	0	0
$231$	−354.604	−0.101001
$232$	0	0
$233$	−4007.01	−1.12664	−0.563322	−	0.826237i	$-0.690477\pi$
$233$	−4007.01	−1.12664	−0.563322	+	0.826237i	$0.690477\pi$
$234$	0	0
$235$	1115.42	0.309625
$236$	0	0
$237$	5998.83	1.64416
$238$	0	0
$239$	−5551.22	−1.50242	−0.751211	−	0.660062i	$-0.770530\pi$
$239$	−5551.22	−1.50242	−0.751211	+	0.660062i	$0.770530\pi$
$240$	0	0
$241$	3185.68	0.851486	0.425743	−	0.904844i	$-0.360013\pi$
$241$	3185.68	0.851486	0.425743	+	0.904844i	$0.360013\pi$
$242$	0	0
$243$	−4259.12	−1.12437
$244$	0	0
$245$	−736.617	−0.192085
$246$	0	0
$247$	−1057.01	−0.272290
$248$	0	0
$249$	−10175.1	−2.58965
$250$	0	0
$251$	5916.38	1.48780	0.743902	−	0.668289i	$-0.232973\pi$
$251$	5916.38	1.48780	0.743902	+	0.668289i	$0.232973\pi$
$252$	0	0
$253$	67.9357	0.0168817
$254$	0	0
$255$	−198.271	−0.0486909
$256$	0	0
$257$	−1094.22	−0.265586	−0.132793	−	0.991144i	$-0.542395\pi$
$257$	−1094.22	−0.265586	−0.132793	+	0.991144i	$0.542395\pi$
$258$	0	0
$259$	555.530	0.133278
$260$	0	0
$261$	−3498.08	−0.829601
$262$	0	0
$263$	5474.53	1.28355	0.641775	−	0.766893i	$-0.278198\pi$
$263$	5474.53	1.28355	0.641775	+	0.766893i	$0.278198\pi$
$264$	0	0
$265$	−1823.89	−0.422796
$266$	0	0
$267$	−5416.66	−1.24155
$268$	0	0
$269$	5031.11	1.14034	0.570172	−	0.821525i	$-0.306877\pi$
$269$	5031.11	1.14034	0.570172	+	0.821525i	$0.306877\pi$
$270$	0	0
$271$	7531.38	1.68819	0.844093	−	0.536196i	$-0.180139\pi$
$271$	7531.38	1.68819	0.844093	+	0.536196i	$0.180139\pi$
$272$	0	0
$273$	2689.97	0.596354
$274$	0	0
$275$	875.015	0.191874
$276$	0	0
$277$	4082.11	0.885452	0.442726	−	0.896657i	$-0.354011\pi$
$277$	4082.11	0.885452	0.442726	+	0.896657i	$0.354011\pi$
$278$	0	0
$279$	8433.60	1.80970
$280$	0	0
$281$	−8485.56	−1.80145	−0.900723	−	0.434395i	$-0.856962\pi$
$281$	−8485.56	−1.80145	−0.900723	+	0.434395i	$0.856962\pi$
$282$	0	0
$283$	−966.510	−0.203014	−0.101507	−	0.994835i	$-0.532366\pi$
$283$	−966.510	−0.203014	−0.101507	+	0.994835i	$0.532366\pi$
$284$	0	0
$285$	−375.379	−0.0780194
$286$	0	0
$287$	−151.668	−0.0311941
$288$	0	0
$289$	−4812.29	−0.979501
$290$	0	0
$291$	14023.7	2.82504
$292$	0	0
$293$	−1361.87	−0.271540	−0.135770	−	0.990740i	$-0.543351\pi$
$293$	−1361.87	−0.271540	−0.135770	+	0.990740i	$0.543351\pi$
$294$	0	0
$295$	164.835	0.0325324
$296$	0	0
$297$	−891.057	−0.174089
$298$	0	0
$299$	−515.350	−0.0996771
$300$	0	0
$301$	−697.778	−0.133619
$302$	0	0
$303$	6957.35	1.31911
$304$	0	0
$305$	−950.248	−0.178397
$306$	0	0
$307$	1526.15	0.283720	0.141860	−	0.989887i	$-0.454692\pi$
$307$	1526.15	0.283720	0.141860	+	0.989887i	$0.454692\pi$
$308$	0	0
$309$	2924.61	0.538432
$310$	0	0
$311$	1893.60	0.345261	0.172630	−	0.984987i	$-0.444773\pi$
$311$	1893.60	0.345261	0.172630	+	0.984987i	$0.444773\pi$
$312$	0	0
$313$	9049.39	1.63419	0.817096	−	0.576502i	$-0.195583\pi$
$313$	9049.39	1.63419	0.817096	+	0.576502i	$0.195583\pi$
$314$	0	0
$315$	579.653	0.103682
$316$	0	0
$317$	1317.33	0.233403	0.116702	−	0.993167i	$-0.462768\pi$
$317$	1317.33	0.233403	0.116702	+	0.993167i	$0.462768\pi$
$318$	0	0
$319$	615.742	0.108072
$320$	0	0
$321$	3109.42	0.540656
$322$	0	0
$323$	190.676	0.0328467
$324$	0	0
$325$	−6637.73	−1.13291
$326$	0	0
$327$	10753.0	1.81848
$328$	0	0
$329$	2729.89	0.457458
$330$	0	0
$331$	−9896.94	−1.64346	−0.821730	−	0.569877i	$-0.806991\pi$
$331$	−9896.94	−1.64346	−0.821730	+	0.569877i	$0.806991\pi$
$332$	0	0
$333$	3966.39	0.652724
$334$	0	0
$335$	−580.927	−0.0947446
$336$	0	0
$337$	1263.75	0.204276	0.102138	−	0.994770i	$-0.467432\pi$
$337$	1263.75	0.204276	0.102138	+	0.994770i	$0.467432\pi$
$338$	0	0
$339$	−690.915	−0.110694
$340$	0	0
$341$	−1484.50	−0.235749
$342$	0	0
$343$	−3804.31	−0.598874
$344$	0	0
$345$	−183.018	−0.0285605
$346$	0	0
$347$	7582.42	1.17304	0.586521	−	0.809934i	$-0.300497\pi$
$347$	7582.42	1.17304	0.586521	+	0.809934i	$0.300497\pi$
$348$	0	0
$349$	10227.9	1.56872	0.784362	−	0.620303i	$-0.212991\pi$
$349$	10227.9	1.56872	0.784362	+	0.620303i	$0.212991\pi$
$350$	0	0
$351$	6759.42	1.02790
$352$	0	0
$353$	7056.98	1.06404	0.532019	−	0.846732i	$-0.321434\pi$
$353$	7056.98	1.06404	0.532019	+	0.846732i	$0.321434\pi$
$354$	0	0
$355$	−1862.96	−0.278522
$356$	0	0
$357$	−485.251	−0.0719389
$358$	0	0
$359$	−8459.22	−1.24362	−0.621811	−	0.783167i	$-0.713603\pi$
$359$	−8459.22	−1.24362	−0.621811	+	0.783167i	$0.713603\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	−10583.4	−1.53026
$364$	0	0
$365$	−1697.39	−0.243412
$366$	0	0
$367$	−1344.85	−0.191282	−0.0956410	−	0.995416i	$-0.530490\pi$
$367$	−1344.85	−0.191282	−0.0956410	+	0.995416i	$0.530490\pi$
$368$	0	0
$369$	−1082.89	−0.152772
$370$	0	0
$371$	−4463.83	−0.624664
$372$	0	0
$373$	−3928.85	−0.545384	−0.272692	−	0.962101i	$-0.587914\pi$
$373$	−3928.85	−0.545384	−0.272692	+	0.962101i	$0.587914\pi$
$374$	0	0
$375$	−4826.88	−0.664691
$376$	0	0
$377$	−4670.92	−0.638103
$378$	0	0
$379$	−10874.9	−1.47390	−0.736951	−	0.675947i	$-0.763735\pi$
$379$	−10874.9	−1.47390	−0.736951	+	0.675947i	$0.763735\pi$
$380$	0	0
$381$	−8974.95	−1.20683
$382$	0	0
$383$	−8209.99	−1.09533	−0.547664	−	0.836698i	$-0.684483\pi$
$383$	−8209.99	−1.09533	−0.547664	+	0.836698i	$0.684483\pi$
$384$	0	0
$385$	−102.032	−0.0135066
$386$	0	0
$387$	−4982.03	−0.654394
$388$	0	0
$389$	−6353.98	−0.828174	−0.414087	−	0.910237i	$-0.635899\pi$
$389$	−6353.98	−0.828174	−0.414087	+	0.910237i	$0.635899\pi$
$390$	0	0
$391$	92.9652	0.0120242
$392$	0	0
$393$	19891.2	2.55312
$394$	0	0
$395$	1726.08	0.219869
$396$	0	0
$397$	4184.09	0.528951	0.264476	−	0.964392i	$-0.414801\pi$
$397$	4184.09	0.528951	0.264476	+	0.964392i	$0.414801\pi$
$398$	0	0
$399$	−918.708	−0.115271
$400$	0	0
$401$	−12492.3	−1.55569	−0.777847	−	0.628453i	$-0.783688\pi$
$401$	−12492.3	−1.55569	−0.777847	+	0.628453i	$0.783688\pi$
$402$	0	0
$403$	11261.2	1.39196
$404$	0	0
$405$	−281.563	−0.0345457
$406$	0	0
$407$	−698.175	−0.0850301
$408$	0	0
$409$	7559.01	0.913861	0.456931	−	0.889502i	$-0.348949\pi$
$409$	7559.01	0.913861	0.456931	+	0.889502i	$0.348949\pi$
$410$	0	0
$411$	−19468.4	−2.33651
$412$	0	0
$413$	403.420	0.0480653
$414$	0	0
$415$	−2927.74	−0.346306
$416$	0	0
$417$	23433.8	2.75194
$418$	0	0
$419$	−12805.0	−1.49300	−0.746498	−	0.665388i	$-0.768266\pi$
$419$	−12805.0	−1.49300	−0.746498	+	0.665388i	$0.768266\pi$
$420$	0	0
$421$	−6126.65	−0.709251	−0.354626	−	0.935008i	$-0.615392\pi$
$421$	−6126.65	−0.709251	−0.354626	+	0.935008i	$0.615392\pi$
$422$	0	0
$423$	19491.0	2.24039
$424$	0	0
$425$	1197.40	0.136664
$426$	0	0
$427$	−2325.65	−0.263574
$428$	0	0
$429$	−3380.68	−0.380469
$430$	0	0
$431$	9784.87	1.09355	0.546776	−	0.837279i	$-0.315855\pi$
$431$	9784.87	1.09355	0.546776	+	0.837279i	$0.315855\pi$
$432$	0	0
$433$	9483.80	1.05257	0.526285	−	0.850308i	$-0.323585\pi$
$433$	9483.80	1.05257	0.526285	+	0.850308i	$0.323585\pi$
$434$	0	0
$435$	−1658.80	−0.182836
$436$	0	0
$437$	176.008	0.0192668
$438$	0	0
$439$	−16627.0	−1.80766	−0.903828	−	0.427895i	$-0.859255\pi$
$439$	−16627.0	−1.80766	−0.903828	+	0.427895i	$0.859255\pi$
$440$	0	0
$441$	−12871.8	−1.38989
$442$	0	0
$443$	12294.8	1.31861	0.659304	−	0.751877i	$-0.270851\pi$
$443$	12294.8	1.31861	0.659304	+	0.751877i	$0.270851\pi$
$444$	0	0
$445$	−1558.56	−0.166029
$446$	0	0
$447$	−13338.3	−1.41136
$448$	0	0
$449$	−4523.75	−0.475476	−0.237738	−	0.971329i	$-0.576406\pi$
$449$	−4523.75	−0.475476	−0.237738	+	0.971329i	$0.576406\pi$
$450$	0	0
$451$	190.613	0.0199016
$452$	0	0
$453$	13466.1	1.39667
$454$	0	0
$455$	774.000	0.0797487
$456$	0	0
$457$	−14738.0	−1.50856	−0.754281	−	0.656551i	$-0.772015\pi$
$457$	−14738.0	−1.50856	−0.754281	+	0.656551i	$0.772015\pi$
$458$	0	0
$459$	−1219.35	−0.123996
$460$	0	0
$461$	10875.2	1.09872	0.549360	−	0.835586i	$-0.314872\pi$
$461$	10875.2	1.09872	0.549360	+	0.835586i	$0.314872\pi$
$462$	0	0
$463$	−12721.9	−1.27697	−0.638484	−	0.769635i	$-0.720438\pi$
$463$	−12721.9	−1.27697	−0.638484	+	0.769635i	$0.720438\pi$
$464$	0	0
$465$	3999.25	0.398840
$466$	0	0
$467$	5255.10	0.520721	0.260361	−	0.965511i	$-0.416159\pi$
$467$	5255.10	0.520721	0.260361	+	0.965511i	$0.416159\pi$
$468$	0	0
$469$	−1421.77	−0.139981
$470$	0	0
$471$	14483.7	1.41693
$472$	0	0
$473$	876.949	0.0852477
$474$	0	0
$475$	2266.99	0.218982
$476$	0	0
$477$	−31871.0	−3.05927
$478$	0	0
$479$	−6261.62	−0.597288	−0.298644	−	0.954365i	$-0.596534\pi$
$479$	−6261.62	−0.597288	−0.298644	+	0.954365i	$0.596534\pi$
$480$	0	0
$481$	5296.25	0.502054
$482$	0	0
$483$	−447.922	−0.0421970
$484$	0	0
$485$	4035.13	0.377785
$486$	0	0
$487$	−9957.10	−0.926487	−0.463243	−	0.886231i	$-0.653314\pi$
$487$	−9957.10	−0.926487	−0.463243	+	0.886231i	$0.653314\pi$
$488$	0	0
$489$	6673.17	0.617119
$490$	0	0
$491$	3112.32	0.286064	0.143032	−	0.989718i	$-0.454315\pi$
$491$	3112.32	0.286064	0.143032	+	0.989718i	$0.454315\pi$
$492$	0	0
$493$	842.600	0.0769752
$494$	0	0
$495$	−728.493	−0.0661481
$496$	0	0
$497$	−4559.43	−0.411506
$498$	0	0
$499$	1278.95	0.114737	0.0573685	−	0.998353i	$-0.481729\pi$
$499$	1278.95	0.114737	0.0573685	+	0.998353i	$0.481729\pi$
$500$	0	0
$501$	10753.9	0.958981
$502$	0	0
$503$	10223.8	0.906272	0.453136	−	0.891441i	$-0.350305\pi$
$503$	10223.8	0.906272	0.453136	+	0.891441i	$0.350305\pi$
$504$	0	0
$505$	2001.88	0.176401
$506$	0	0
$507$	7440.32	0.651748
$508$	0	0
$509$	−8872.97	−0.772667	−0.386333	−	0.922359i	$-0.626259\pi$
$509$	−8872.97	−0.772667	−0.386333	+	0.922359i	$0.626259\pi$
$510$	0	0
$511$	−4154.21	−0.359631
$512$	0	0
$513$	−2308.55	−0.198684
$514$	0	0
$515$	841.515	0.0720031
$516$	0	0
$517$	−3430.85	−0.291854
$518$	0	0
$519$	−10315.1	−0.872411
$520$	0	0
$521$	−9651.60	−0.811601	−0.405801	−	0.913962i	$-0.633007\pi$
$521$	−9651.60	−0.811601	−0.405801	+	0.913962i	$0.633007\pi$
$522$	0	0
$523$	2707.90	0.226402	0.113201	−	0.993572i	$-0.463890\pi$
$523$	2707.90	0.226402	0.113201	+	0.993572i	$0.463890\pi$
$524$	0	0
$525$	−5769.26	−0.479602
$526$	0	0
$527$	−2031.44	−0.167915
$528$	0	0
$529$	−12081.2	−0.992947
$530$	0	0
$531$	2880.35	0.235399
$532$	0	0
$533$	−1445.96	−0.117507
$534$	0	0
$535$	894.689	0.0723005
$536$	0	0
$537$	14215.3	1.14234
$538$	0	0
$539$	2265.72	0.181060
$540$	0	0
$541$	−14478.5	−1.15061	−0.575304	−	0.817940i	$-0.695116\pi$
$541$	−14478.5	−1.15061	−0.575304	+	0.817940i	$0.695116\pi$
$542$	0	0
$543$	19742.3	1.56027
$544$	0	0
$545$	3094.03	0.243181
$546$	0	0
$547$	−10220.5	−0.798895	−0.399448	−	0.916756i	$-0.630798\pi$
$547$	−10220.5	−0.798895	−0.399448	+	0.916756i	$0.630798\pi$
$548$	0	0
$549$	−16604.8	−1.29085
$550$	0	0
$551$	1595.26	0.123340
$552$	0	0
$553$	4224.43	0.324848
$554$	0	0
$555$	1880.88	0.143854
$556$	0	0
$557$	11008.2	0.837400	0.418700	−	0.908125i	$-0.362486\pi$
$557$	11008.2	0.837400	0.418700	+	0.908125i	$0.362486\pi$
$558$	0	0
$559$	−6652.40	−0.503339
$560$	0	0
$561$	609.850	0.0458964
$562$	0	0
$563$	5553.05	0.415689	0.207845	−	0.978162i	$-0.433355\pi$
$563$	5553.05	0.415689	0.207845	+	0.978162i	$0.433355\pi$
$564$	0	0
$565$	−198.801	−0.0148028
$566$	0	0
$567$	−689.103	−0.0510399
$568$	0	0
$569$	14737.2	1.08580	0.542898	−	0.839799i	$-0.317327\pi$
$569$	14737.2	1.08580	0.542898	+	0.839799i	$0.317327\pi$
$570$	0	0
$571$	3141.77	0.230261	0.115131	−	0.993350i	$-0.463271\pi$
$571$	3141.77	0.230261	0.115131	+	0.993350i	$0.463271\pi$
$572$	0	0
$573$	10019.1	0.730464
$574$	0	0
$575$	1105.29	0.0801628
$576$	0	0
$577$	3639.45	0.262586	0.131293	−	0.991344i	$-0.458087\pi$
$577$	3639.45	0.262586	0.131293	+	0.991344i	$0.458087\pi$
$578$	0	0
$579$	27379.8	1.96522
$580$	0	0
$581$	−7165.40	−0.511654
$582$	0	0
$583$	5610.02	0.398530
$584$	0	0
$585$	5526.24	0.390567
$586$	0	0
$587$	14028.9	0.986433	0.493216	−	0.869907i	$-0.335821\pi$
$587$	14028.9	0.986433	0.493216	+	0.869907i	$0.335821\pi$
$588$	0	0
$589$	−3846.06	−0.269056
$590$	0	0
$591$	−2181.19	−0.151814
$592$	0	0
$593$	20191.0	1.39822	0.699110	−	0.715014i	$-0.253580\pi$
$593$	20191.0	1.39822	0.699110	+	0.715014i	$0.253580\pi$
$594$	0	0
$595$	−139.624	−0.00962020
$596$	0	0
$597$	9713.93	0.665937
$598$	0	0
$599$	3131.35	0.213595	0.106798	−	0.994281i	$-0.465940\pi$
$599$	3131.35	0.213595	0.106798	+	0.994281i	$0.465940\pi$
$600$	0	0
$601$	18210.1	1.23595	0.617976	−	0.786197i	$-0.287953\pi$
$601$	18210.1	1.23595	0.617976	+	0.786197i	$0.287953\pi$
$602$	0	0
$603$	−10151.2	−0.685555
$604$	0	0
$605$	−3045.23	−0.204638
$606$	0	0
$607$	−12605.1	−0.842875	−0.421437	−	0.906858i	$-0.638474\pi$
$607$	−12605.1	−0.842875	−0.421437	+	0.906858i	$0.638474\pi$
$608$	0	0
$609$	−4059.79	−0.270133
$610$	0	0
$611$	26025.9	1.72323
$612$	0	0
$613$	10096.7	0.665256	0.332628	−	0.943058i	$-0.392065\pi$
$613$	10096.7	0.665256	0.332628	+	0.943058i	$0.392065\pi$
$614$	0	0
$615$	−513.510	−0.0336694
$616$	0	0
$617$	5008.97	0.326829	0.163415	−	0.986557i	$-0.447749\pi$
$617$	5008.97	0.326829	0.163415	+	0.986557i	$0.447749\pi$
$618$	0	0
$619$	6083.17	0.394997	0.197499	−	0.980303i	$-0.436718\pi$
$619$	6083.17	0.394997	0.197499	+	0.980303i	$0.436718\pi$
$620$	0	0
$621$	−1125.55	−0.0727323
$622$	0	0
$623$	−3814.46	−0.245302
$624$	0	0
$625$	13525.5	0.865635
$626$	0	0
$627$	1154.61	0.0735416
$628$	0	0
$629$	−955.403	−0.0605635
$630$	0	0
$631$	3719.15	0.234639	0.117319	−	0.993094i	$-0.462570\pi$
$631$	3719.15	0.234639	0.117319	+	0.993094i	$0.462570\pi$
$632$	0	0
$633$	41186.7	2.58614
$634$	0	0
$635$	−2582.41	−0.161386
$636$	0	0
$637$	−17187.4	−1.06906
$638$	0	0
$639$	−32553.6	−2.01534
$640$	0	0
$641$	−10626.9	−0.654816	−0.327408	−	0.944883i	$-0.606175\pi$
$641$	−10626.9	−0.654816	−0.327408	+	0.944883i	$0.606175\pi$
$642$	0	0
$643$	1914.95	0.117447	0.0587234	−	0.998274i	$-0.481297\pi$
$643$	1914.95	0.117447	0.0587234	+	0.998274i	$0.481297\pi$
$644$	0	0
$645$	−2362.50	−0.144222
$646$	0	0
$647$	3804.40	0.231169	0.115584	−	0.993298i	$-0.463126\pi$
$647$	3804.40	0.231169	0.115584	+	0.993298i	$0.463126\pi$
$648$	0	0
$649$	−507.007	−0.0306653
$650$	0	0
$651$	9787.83	0.589271
$652$	0	0
$653$	15110.4	0.905534	0.452767	−	0.891629i	$-0.350437\pi$
$653$	15110.4	0.905534	0.452767	+	0.891629i	$0.350437\pi$
$654$	0	0
$655$	5723.39	0.341422
$656$	0	0
$657$	−29660.4	−1.76128
$658$	0	0
$659$	25627.7	1.51489	0.757445	−	0.652899i	$-0.226447\pi$
$659$	25627.7	1.51489	0.757445	+	0.652899i	$0.226447\pi$
$660$	0	0
$661$	−22255.1	−1.30956	−0.654782	−	0.755818i	$-0.727239\pi$
$661$	−22255.1	−1.30956	−0.654782	+	0.755818i	$0.727239\pi$
$662$	0	0
$663$	−4626.23	−0.270992
$664$	0	0
$665$	−264.345	−0.0154148
$666$	0	0
$667$	777.781	0.0451511
$668$	0	0
$669$	−36036.0	−2.08256
$670$	0	0
$671$	2922.82	0.168158
$672$	0	0
$673$	−20867.0	−1.19519	−0.597595	−	0.801798i	$-0.703877\pi$
$673$	−20867.0	−1.19519	−0.597595	+	0.801798i	$0.703877\pi$
$674$	0	0
$675$	−14497.1	−0.826659
$676$	0	0
$677$	13538.0	0.768551	0.384275	−	0.923219i	$-0.374451\pi$
$677$	13538.0	0.768551	0.384275	+	0.923219i	$0.374451\pi$
$678$	0	0
$679$	9875.64	0.558162
$680$	0	0
$681$	−6093.26	−0.342870
$682$	0	0
$683$	−19790.9	−1.10875	−0.554377	−	0.832266i	$-0.687043\pi$
$683$	−19790.9	−1.10875	−0.554377	+	0.832266i	$0.687043\pi$
$684$	0	0
$685$	−5601.74	−0.312455
$686$	0	0
$687$	3251.11	0.180550
$688$	0	0
$689$	−42556.7	−2.35309
$690$	0	0
$691$	−4170.34	−0.229591	−0.114795	−	0.993389i	$-0.536621\pi$
$691$	−4170.34	−0.229591	−0.114795	+	0.993389i	$0.536621\pi$
$692$	0	0
$693$	−1782.93	−0.0977312
$694$	0	0
$695$	6742.73	0.368009
$696$	0	0
$697$	260.840	0.0141751
$698$	0	0
$699$	−33203.4	−1.79666
$700$	0	0
$701$	1836.49	0.0989491	0.0494746	−	0.998775i	$-0.484245\pi$
$701$	1836.49	0.0989491	0.0494746	+	0.998775i	$0.484245\pi$
$702$	0	0
$703$	−1808.83	−0.0970432
$704$	0	0
$705$	9242.70	0.493759
$706$	0	0
$707$	4899.42	0.260625
$708$	0	0
$709$	9458.42	0.501013	0.250507	−	0.968115i	$-0.419403\pi$
$709$	9458.42	0.501013	0.250507	+	0.968115i	$0.419403\pi$
$710$	0	0
$711$	30161.7	1.59093
$712$	0	0
$713$	−1875.17	−0.0984932
$714$	0	0
$715$	−972.742	−0.0508790
$716$	0	0
$717$	−45999.2	−2.39592
$718$	0	0
$719$	−28024.8	−1.45362	−0.726808	−	0.686840i	$-0.758997\pi$
$719$	−28024.8	−1.45362	−0.726808	+	0.686840i	$0.758997\pi$
$720$	0	0
$721$	2059.54	0.106382
$722$	0	0
$723$	26397.6	1.35787
$724$	0	0
$725$	10017.9	0.513178
$726$	0	0
$727$	−729.786	−0.0372301	−0.0186150	−	0.999827i	$-0.505926\pi$
$727$	−729.786	−0.0372301	−0.0186150	+	0.999827i	$0.505926\pi$
$728$	0	0
$729$	−32104.0	−1.63105
$730$	0	0
$731$	1200.04	0.0607185
$732$	0	0
$733$	−28958.9	−1.45924	−0.729618	−	0.683855i	$-0.760302\pi$
$733$	−28958.9	−1.45924	−0.729618	+	0.683855i	$0.760302\pi$
$734$	0	0
$735$	−6103.85	−0.306318
$736$	0	0
$737$	1786.84	0.0893070
$738$	0	0
$739$	4906.85	0.244251	0.122125	−	0.992515i	$-0.461029\pi$
$739$	4906.85	0.244251	0.122125	+	0.992515i	$0.461029\pi$
$740$	0	0
$741$	−8758.68	−0.434222
$742$	0	0
$743$	29014.2	1.43261	0.716304	−	0.697788i	$-0.245832\pi$
$743$	29014.2	1.43261	0.716304	+	0.697788i	$0.245832\pi$
$744$	0	0
$745$	−3837.90	−0.188738
$746$	0	0
$747$	−51159.8	−2.50581
$748$	0	0
$749$	2189.68	0.106821
$750$	0	0
$751$	−14611.4	−0.709958	−0.354979	−	0.934874i	$-0.615512\pi$
$751$	−14611.4	−0.709958	−0.354979	+	0.934874i	$0.615512\pi$
$752$	0	0
$753$	49025.0	2.37261
$754$	0	0
$755$	3874.67	0.186773
$756$	0	0
$757$	34632.6	1.66281	0.831403	−	0.555670i	$-0.187538\pi$
$757$	34632.6	1.66281	0.831403	+	0.555670i	$0.187538\pi$
$758$	0	0
$759$	562.937	0.0269214
$760$	0	0
$761$	−1887.07	−0.0898899	−0.0449449	−	0.998989i	$-0.514311\pi$
$761$	−1887.07	−0.0898899	−0.0449449	+	0.998989i	$0.514311\pi$
$762$	0	0
$763$	7572.37	0.359290
$764$	0	0
$765$	−996.892	−0.0471146
$766$	0	0
$767$	3846.08	0.181061
$768$	0	0
$769$	−16854.3	−0.790351	−0.395176	−	0.918606i	$-0.629316\pi$
$769$	−16854.3	−0.790351	−0.395176	+	0.918606i	$0.629316\pi$
$770$	0	0
$771$	−9067.05	−0.423530
$772$	0	0
$773$	−37853.8	−1.76133	−0.880665	−	0.473740i	$-0.842904\pi$
$773$	−37853.8	−1.76133	−0.880665	+	0.473740i	$0.842904\pi$
$774$	0	0
$775$	−24152.3	−1.11945
$776$	0	0
$777$	4603.29	0.212538
$778$	0	0
$779$	493.840	0.0227133
$780$	0	0
$781$	5730.17	0.262537
$782$	0	0
$783$	−10201.5	−0.465610
$784$	0	0
$785$	4167.47	0.189482
$786$	0	0
$787$	40230.0	1.82216	0.911082	−	0.412225i	$-0.135248\pi$
$787$	40230.0	1.82216	0.911082	+	0.412225i	$0.135248\pi$
$788$	0	0
$789$	45363.7	2.04688
$790$	0	0
$791$	−486.548	−0.0218706
$792$	0	0
$793$	−22172.0	−0.992878
$794$	0	0
$795$	−15113.4	−0.674233
$796$	0	0
$797$	−14861.6	−0.660507	−0.330253	−	0.943892i	$-0.607134\pi$
$797$	−14861.6	−0.660507	−0.330253	+	0.943892i	$0.607134\pi$
$798$	0	0
$799$	−4694.88	−0.207876
$800$	0	0
$801$	−27234.6	−1.20136
$802$	0	0
$803$	5220.90	0.229442
$804$	0	0
$805$	−128.883	−0.00564289
$806$	0	0
$807$	41689.4	1.81851
$808$	0	0
$809$	17346.5	0.753857	0.376928	−	0.926242i	$-0.376980\pi$
$809$	17346.5	0.753857	0.376928	+	0.926242i	$0.376980\pi$
$810$	0	0
$811$	−19405.3	−0.840210	−0.420105	−	0.907475i	$-0.638007\pi$
$811$	−19405.3	−0.840210	−0.420105	+	0.907475i	$0.638007\pi$
$812$	0	0
$813$	62407.4	2.69216
$814$	0	0
$815$	1920.11	0.0825256
$816$	0	0
$817$	2272.00	0.0972916
$818$	0	0
$819$	13525.0	0.577047
$820$	0	0
$821$	9008.08	0.382928	0.191464	−	0.981500i	$-0.438676\pi$
$821$	9008.08	0.382928	0.191464	+	0.981500i	$0.438676\pi$
$822$	0	0
$823$	22387.4	0.948207	0.474103	−	0.880469i	$-0.342772\pi$
$823$	22387.4	0.948207	0.474103	+	0.880469i	$0.342772\pi$
$824$	0	0
$825$	7250.65	0.305982
$826$	0	0
$827$	−8244.50	−0.346662	−0.173331	−	0.984864i	$-0.555453\pi$
$827$	−8244.50	−0.346662	−0.173331	+	0.984864i	$0.555453\pi$
$828$	0	0
$829$	−19708.0	−0.825680	−0.412840	−	0.910804i	$-0.635463\pi$
$829$	−19708.0	−0.825680	−0.412840	+	0.910804i	$0.635463\pi$
$830$	0	0
$831$	33825.6	1.41203
$832$	0	0
$833$	3100.48	0.128962
$834$	0	0
$835$	3094.28	0.128242
$836$	0	0
$837$	24595.1	1.01569
$838$	0	0
$839$	24561.7	1.01069	0.505343	−	0.862919i	$-0.331366\pi$
$839$	24561.7	1.01069	0.505343	+	0.862919i	$0.331366\pi$
$840$	0	0
$841$	−17339.5	−0.710956
$842$	0	0
$843$	−70314.1	−2.87277
$844$	0	0
$845$	2140.84	0.0871565
$846$	0	0
$847$	−7452.94	−0.302345
$848$	0	0
$849$	−8008.81	−0.323747
$850$	0	0
$851$	−881.908	−0.0355246
$852$	0	0
$853$	13832.7	0.555243	0.277622	−	0.960690i	$-0.410454\pi$
$853$	13832.7	0.555243	0.277622	+	0.960690i	$0.410454\pi$
$854$	0	0
$855$	−1887.38	−0.0754936
$856$	0	0
$857$	22597.5	0.900720	0.450360	−	0.892847i	$-0.351296\pi$
$857$	22597.5	0.900720	0.450360	+	0.892847i	$0.351296\pi$
$858$	0	0
$859$	−33731.1	−1.33980	−0.669902	−	0.742449i	$-0.733664\pi$
$859$	−33731.1	−1.33980	−0.669902	+	0.742449i	$0.733664\pi$
$860$	0	0
$861$	−1256.77	−0.0497453
$862$	0	0
$863$	15230.6	0.600759	0.300379	−	0.953820i	$-0.402887\pi$
$863$	15230.6	0.600759	0.300379	+	0.953820i	$0.402887\pi$
$864$	0	0
$865$	−2968.01	−0.116665
$866$	0	0
$867$	−39876.1	−1.56201
$868$	0	0
$869$	−5309.15	−0.207250
$870$	0	0
$871$	−13554.7	−0.527307
$872$	0	0
$873$	70510.5	2.73358
$874$	0	0
$875$	−3399.13	−0.131328
$876$	0	0
$877$	44693.6	1.72086	0.860432	−	0.509566i	$-0.170194\pi$
$877$	44693.6	1.72086	0.860432	+	0.509566i	$0.170194\pi$
$878$	0	0
$879$	−11284.9	−0.433025
$880$	0	0
$881$	−45125.9	−1.72569	−0.862844	−	0.505471i	$-0.831319\pi$
$881$	−45125.9	−1.72569	−0.862844	+	0.505471i	$0.831319\pi$
$882$	0	0
$883$	−35570.8	−1.35567	−0.677833	−	0.735216i	$-0.737081\pi$
$883$	−35570.8	−1.35567	−0.677833	+	0.735216i	$0.737081\pi$
$884$	0	0
$885$	1365.87	0.0518795
$886$	0	0
$887$	−34224.2	−1.29553	−0.647766	−	0.761840i	$-0.724296\pi$
$887$	−34224.2	−1.29553	−0.647766	+	0.761840i	$0.724296\pi$
$888$	0	0
$889$	−6320.23	−0.238441
$890$	0	0
$891$	866.046	0.0325630
$892$	0	0
$893$	−8888.66	−0.333088
$894$	0	0
$895$	4090.23	0.152761
$896$	0	0
$897$	−4270.35	−0.158955
$898$	0	0
$899$	−16995.8	−0.630524
$900$	0	0
$901$	7676.92	0.283857
$902$	0	0
$903$	−5782.01	−0.213082
$904$	0	0
$905$	5680.57	0.208650
$906$	0	0
$907$	35101.3	1.28503	0.642514	−	0.766274i	$-0.277891\pi$
$907$	35101.3	1.28503	0.642514	+	0.766274i	$0.277891\pi$
$908$	0	0
$909$	34981.1	1.27640
$910$	0	0
$911$	−35322.9	−1.28463	−0.642317	−	0.766439i	$-0.722027\pi$
$911$	−35322.9	−1.28463	−0.642317	+	0.766439i	$0.722027\pi$
$912$	0	0
$913$	9005.28	0.326431
$914$	0	0
$915$	−7874.05	−0.284490
$916$	0	0
$917$	14007.5	0.504437
$918$	0	0
$919$	21222.7	0.761778	0.380889	−	0.924621i	$-0.375618\pi$
$919$	21222.7	0.761778	0.380889	+	0.924621i	$0.375618\pi$
$920$	0	0
$921$	12646.2	0.452450
$922$	0	0
$923$	−43468.2	−1.55013
$924$	0	0
$925$	−11359.0	−0.403764
$926$	0	0
$927$	14704.8	0.521001
$928$	0	0
$929$	49566.4	1.75051	0.875253	−	0.483666i	$-0.160695\pi$
$929$	49566.4	1.75051	0.875253	+	0.483666i	$0.160695\pi$
$930$	0	0
$931$	5870.04	0.206641
$932$	0	0
$933$	15691.0	0.550589
$934$	0	0
$935$	175.475	0.00613760
$936$	0	0
$937$	23500.7	0.819354	0.409677	−	0.912231i	$-0.365641\pi$
$937$	23500.7	0.819354	0.409677	+	0.912231i	$0.365641\pi$
$938$	0	0
$939$	74986.1	2.60605
$940$	0	0
$941$	31868.2	1.10401	0.552005	−	0.833841i	$-0.313863\pi$
$941$	31868.2	1.10401	0.552005	+	0.833841i	$0.313863\pi$
$942$	0	0
$943$	240.775	0.00831464
$944$	0	0
$945$	1690.45	0.0581910
$946$	0	0
$947$	−19262.6	−0.660983	−0.330491	−	0.943809i	$-0.607214\pi$
$947$	−19262.6	−0.660983	−0.330491	+	0.943809i	$0.607214\pi$
$948$	0	0
$949$	−39605.0	−1.35472
$950$	0	0
$951$	10915.8	0.372209
$952$	0	0
$953$	−32426.9	−1.10221	−0.551106	−	0.834435i	$-0.685794\pi$
$953$	−32426.9	−1.10221	−0.551106	+	0.834435i	$0.685794\pi$
$954$	0	0
$955$	2882.86	0.0976829
$956$	0	0
$957$	5102.23	0.172342
$958$	0	0
$959$	−13709.8	−0.461639
$960$	0	0
$961$	11184.5	0.375432
$962$	0	0
$963$	15633.9	0.523153
$964$	0	0
$965$	7878.12	0.262804
$966$	0	0
$967$	25418.9	0.845313	0.422656	−	0.906290i	$-0.361098\pi$
$967$	25418.9	0.845313	0.422656	+	0.906290i	$0.361098\pi$
$968$	0	0
$969$	1580.00	0.0523807
$970$	0	0
$971$	11477.7	0.379339	0.189670	−	0.981848i	$-0.439258\pi$
$971$	11477.7	0.379339	0.189670	+	0.981848i	$0.439258\pi$
$972$	0	0
$973$	16502.3	0.543719
$974$	0	0
$975$	−55002.4	−1.80665
$976$	0	0
$977$	−31621.3	−1.03547	−0.517735	−	0.855541i	$-0.673225\pi$
$977$	−31621.3	−1.03547	−0.517735	+	0.855541i	$0.673225\pi$
$978$	0	0
$979$	4793.91	0.156500
$980$	0	0
$981$	54065.6	1.75961
$982$	0	0
$983$	22000.0	0.713827	0.356913	−	0.934137i	$-0.383829\pi$
$983$	22000.0	0.713827	0.356913	+	0.934137i	$0.383829\pi$
$984$	0	0
$985$	−627.604	−0.0203017
$986$	0	0
$987$	22620.7	0.729509
$988$	0	0
$989$	1107.73	0.0356155
$990$	0	0
$991$	31554.0	1.01145	0.505724	−	0.862695i	$-0.331225\pi$
$991$	31554.0	1.01145	0.505724	+	0.862695i	$0.331225\pi$
$992$	0	0
$993$	−82009.2	−2.62083
$994$	0	0
$995$	2795.04	0.0890540
$996$	0	0
$997$	−57792.3	−1.83581	−0.917904	−	0.396802i	$-0.870120\pi$
$997$	−57792.3	−1.83581	−0.917904	+	0.396802i	$0.870120\pi$
$998$	0	0
$999$	11567.3	0.366338

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	152.4.a.c.1.3	✓	3
3.2	odd	2		1368.4.a.d.1.2		3
4.3	odd	2		304.4.a.g.1.1		3
8.3	odd	2		1216.4.a.w.1.3		3
8.5	even	2		1216.4.a.r.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.4.a.c.1.3	✓	3	1.1	even	1	trivial
304.4.a.g.1.1		3	4.3	odd	2
1216.4.a.r.1.1		3	8.5	even	2
1216.4.a.w.1.3		3	8.3	odd	2
1368.4.a.d.1.2		3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 152 = 2^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	152.a (trivial)

\(f(q)\)	\(=\)	\(q+8.28632 q^{3} +2.38427 q^{5} +5.83529 q^{7} +41.6631 q^{9} -7.33364 q^{11} +55.6319 q^{13} +19.7568 q^{15} -10.0356 q^{17} -19.0000 q^{19} +48.3531 q^{21} -9.26357 q^{23} -119.315 q^{25} +121.503 q^{27} -83.9613 q^{29} +202.424 q^{31} -60.7689 q^{33} +13.9129 q^{35} +95.2017 q^{37} +460.983 q^{39} -25.9916 q^{41} -119.579 q^{43} +99.3358 q^{45} +467.824 q^{47} -308.949 q^{49} -83.1579 q^{51} -764.970 q^{53} -17.4853 q^{55} -157.440 q^{57} +69.1344 q^{59} -398.549 q^{61} +243.116 q^{63} +132.641 q^{65} -243.650 q^{67} -76.7609 q^{69} -781.354 q^{71} -711.911 q^{73} -988.684 q^{75} -42.7939 q^{77} +723.944 q^{79} -118.092 q^{81} -1227.94 q^{83} -23.9275 q^{85} -695.730 q^{87} -653.687 q^{89} +324.628 q^{91} +1677.35 q^{93} -45.3010 q^{95} +1692.40 q^{97} -305.542 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{3} + 7 q^{5} + 7 q^{7} + 21 q^{9} + 103 q^{11} + 32 q^{13} + 122 q^{15} + 11 q^{17} - 57 q^{19} + 114 q^{21} + 316 q^{23} + 162 q^{25} + 178 q^{27} - 138 q^{29} + 420 q^{31} - 330 q^{33} + 333 q^{35}+ \cdots - 1305 q^{99}+O(q^{100}) \) 3 * q + 4 * q^3 + 7 * q^5 + 7 * q^7 + 21 * q^9 + 103 * q^11 + 32 * q^13 + 122 * q^15 + 11 * q^17 - 57 * q^19 + 114 * q^21 + 316 * q^23 + 162 * q^25 + 178 * q^27 - 138 * q^29 + 420 * q^31 - 330 * q^33 + 333 * q^35 + 102 * q^37 + 164 * q^39 - 370 * q^41 + 431 * q^43 - 429 * q^45 - 199 * q^47 - 802 * q^49 + 272 * q^51 - 308 * q^53 + 85 * q^55 - 76 * q^57 - 188 * q^59 - 609 * q^61 - 61 * q^63 - 1536 * q^65 - 246 * q^67 - 892 * q^69 - 954 * q^71 - 629 * q^73 - 1074 * q^75 - 71 * q^77 + 452 * q^79 - 361 * q^81 - 780 * q^83 + 1883 * q^85 - 96 * q^87 - 1356 * q^89 - 670 * q^91 + 1092 * q^93 - 133 * q^95 - 548 * q^97 - 1305 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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