LMFDB - Embedded newform 8037.2.a.r.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.28005 q^{2} -0.361479 q^{4} +4.05982 q^{5} -2.17032 q^{7} +3.02280 q^{8} +O(q^{10})$	\(q-1.28005 q^{2} -0.361479 q^{4} +4.05982 q^{5} -2.17032 q^{7} +3.02280 q^{8} -5.19677 q^{10} -3.17495 q^{11} +2.18283 q^{13} +2.77812 q^{14} -3.14637 q^{16} -2.58799 q^{17} +1.00000 q^{19} -1.46754 q^{20} +4.06408 q^{22} +7.11637 q^{23} +11.4822 q^{25} -2.79413 q^{26} +0.784527 q^{28} +7.08282 q^{29} +4.78570 q^{31} -2.01810 q^{32} +3.31275 q^{34} -8.81113 q^{35} -5.35811 q^{37} -1.28005 q^{38} +12.2721 q^{40} -2.87373 q^{41} +5.09752 q^{43} +1.14768 q^{44} -9.10928 q^{46} +1.00000 q^{47} -2.28969 q^{49} -14.6977 q^{50} -0.789049 q^{52} -11.4217 q^{53} -12.8897 q^{55} -6.56047 q^{56} -9.06634 q^{58} +12.0536 q^{59} +0.245349 q^{61} -6.12592 q^{62} +8.87601 q^{64} +8.86192 q^{65} +7.94060 q^{67} +0.935505 q^{68} +11.2787 q^{70} +5.97488 q^{71} -1.67259 q^{73} +6.85863 q^{74} -0.361479 q^{76} +6.89067 q^{77} -10.8832 q^{79} -12.7737 q^{80} +3.67851 q^{82} -9.35097 q^{83} -10.5068 q^{85} -6.52507 q^{86} -9.59725 q^{88} -17.0723 q^{89} -4.73745 q^{91} -2.57242 q^{92} -1.28005 q^{94} +4.05982 q^{95} +9.71823 q^{97} +2.93092 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - q^{2} + 31 q^{4} - q^{5} + 15 q^{7}+O(q^{10}) $ 23 * q - q^2 + 31 * q^4 - q^5 + 15 * q^7	$ 23 q - q^{2} + 31 q^{4} - q^{5} + 15 q^{7} + 5 q^{10} - 2 q^{11} + 13 q^{13} - 6 q^{14} + 35 q^{16} - 12 q^{17} + 23 q^{19} - 3 q^{20} - 4 q^{22} - 13 q^{23} + 46 q^{25} + 7 q^{26} + 11 q^{28} + 18 q^{29} + 22 q^{31} + 4 q^{32} - 20 q^{34} - 25 q^{35} + 8 q^{37} - q^{38} - 16 q^{40} + 16 q^{41} + 68 q^{43} + 18 q^{44} + 13 q^{46} + 23 q^{47} + 52 q^{49} + 21 q^{50} + 54 q^{52} + 7 q^{53} + 32 q^{55} - 33 q^{56} + 6 q^{58} - 10 q^{59} + 28 q^{61} + 24 q^{62} + 40 q^{64} + 18 q^{65} + 55 q^{67} - 41 q^{68} - 40 q^{70} + 3 q^{71} + 48 q^{73} + 55 q^{74} + 31 q^{76} + 14 q^{77} - 31 q^{79} + 3 q^{80} + 48 q^{82} - 47 q^{83} - 5 q^{85} + 71 q^{86} + 9 q^{88} + 6 q^{89} + 40 q^{91} - 35 q^{92} - q^{94} - q^{95} - 18 q^{97} + 68 q^{98}+O(q^{100}) $ 23 * q - q^2 + 31 * q^4 - q^5 + 15 * q^7 + 5 * q^10 - 2 * q^11 + 13 * q^13 - 6 * q^14 + 35 * q^16 - 12 * q^17 + 23 * q^19 - 3 * q^20 - 4 * q^22 - 13 * q^23 + 46 * q^25 + 7 * q^26 + 11 * q^28 + 18 * q^29 + 22 * q^31 + 4 * q^32 - 20 * q^34 - 25 * q^35 + 8 * q^37 - q^38 - 16 * q^40 + 16 * q^41 + 68 * q^43 + 18 * q^44 + 13 * q^46 + 23 * q^47 + 52 * q^49 + 21 * q^50 + 54 * q^52 + 7 * q^53 + 32 * q^55 - 33 * q^56 + 6 * q^58 - 10 * q^59 + 28 * q^61 + 24 * q^62 + 40 * q^64 + 18 * q^65 + 55 * q^67 - 41 * q^68 - 40 * q^70 + 3 * q^71 + 48 * q^73 + 55 * q^74 + 31 * q^76 + 14 * q^77 - 31 * q^79 + 3 * q^80 + 48 * q^82 - 47 * q^83 - 5 * q^85 + 71 * q^86 + 9 * q^88 + 6 * q^89 + 40 * q^91 - 35 * q^92 - q^94 - q^95 - 18 * q^97 + 68 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.28005	−0.905130	−0.452565	−	0.891731i	$-0.649491\pi$
$2$	−1.28005	−0.905130	−0.452565	+	0.891731i	$0.649491\pi$
$3$	0	0
$3$	0	0
$4$	−0.361479	−0.180740
$5$	4.05982	1.81561	0.907804	−	0.419394i	$-0.137758\pi$
$5$	4.05982	1.81561	0.907804	+	0.419394i	$0.137758\pi$
$6$	0	0
$7$	−2.17032	−0.820305	−0.410153	−	0.912017i	$-0.634525\pi$
$7$	−2.17032	−0.820305	−0.410153	+	0.912017i	$0.634525\pi$
$8$	3.02280	1.06872
$9$	0	0
$10$	−5.19677	−1.64336
$11$	−3.17495	−0.957283	−0.478642	−	0.878010i	$-0.658871\pi$
$11$	−3.17495	−0.957283	−0.478642	+	0.878010i	$0.658871\pi$
$12$	0	0
$13$	2.18283	0.605409	0.302704	−	0.953084i	$-0.402111\pi$
$13$	2.18283	0.605409	0.302704	+	0.953084i	$0.402111\pi$
$14$	2.77812	0.742483
$15$	0	0
$16$	−3.14637	−0.786594
$17$	−2.58799	−0.627680	−0.313840	−	0.949476i	$-0.601616\pi$
$17$	−2.58799	−0.627680	−0.313840	+	0.949476i	$0.601616\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.46754	−0.328152
$21$	0	0
$22$	4.06408	0.866466
$23$	7.11637	1.48386	0.741932	−	0.670475i	$-0.233909\pi$
$23$	7.11637	1.48386	0.741932	+	0.670475i	$0.233909\pi$
$24$	0	0
$25$	11.4822	2.29643
$26$	−2.79413	−0.547974
$27$	0	0
$28$	0.784527	0.148262
$29$	7.08282	1.31525	0.657623	−	0.753347i	$-0.271562\pi$
$29$	7.08282	1.31525	0.657623	+	0.753347i	$0.271562\pi$
$30$	0	0
$31$	4.78570	0.859536	0.429768	−	0.902939i	$-0.358595\pi$
$31$	4.78570	0.859536	0.429768	+	0.902939i	$0.358595\pi$
$32$	−2.01810	−0.356753
$33$	0	0
$34$	3.31275	0.568132
$35$	−8.81113	−1.48935
$36$	0	0
$37$	−5.35811	−0.880868	−0.440434	−	0.897785i	$-0.645175\pi$
$37$	−5.35811	−0.880868	−0.440434	+	0.897785i	$0.645175\pi$
$38$	−1.28005	−0.207651
$39$	0	0
$40$	12.2721	1.94038
$41$	−2.87373	−0.448801	−0.224401	−	0.974497i	$-0.572042\pi$
$41$	−2.87373	−0.448801	−0.224401	+	0.974497i	$0.572042\pi$
$42$	0	0
$43$	5.09752	0.777365	0.388682	−	0.921372i	$-0.372930\pi$
$43$	5.09752	0.777365	0.388682	+	0.921372i	$0.372930\pi$
$44$	1.14768	0.173019
$45$	0	0
$46$	−9.10928	−1.34309
$47$	1.00000	0.145865
$47$	1.00000	0.145865
$48$	0	0
$49$	−2.28969	−0.327099
$50$	−14.6977	−2.07857
$51$	0	0
$52$	−0.789049	−0.109421
$53$	−11.4217	−1.56889	−0.784447	−	0.620196i	$-0.787053\pi$
$53$	−11.4217	−1.56889	−0.784447	+	0.620196i	$0.787053\pi$
$54$	0	0
$55$	−12.8897	−1.73805
$56$	−6.56047	−0.876679
$57$	0	0
$58$	−9.06634	−1.19047
$59$	12.0536	1.56924	0.784619	−	0.619978i	$-0.212858\pi$
$59$	12.0536	1.56924	0.784619	+	0.619978i	$0.212858\pi$
$60$	0	0
$61$	0.245349	0.0314137	0.0157069	−	0.999877i	$-0.495000\pi$
$61$	0.245349	0.0314137	0.0157069	+	0.999877i	$0.495000\pi$
$62$	−6.12592	−0.777992
$63$	0	0
$64$	8.87601	1.10950
$65$	8.86192	1.09919
$66$	0	0
$67$	7.94060	0.970099	0.485050	−	0.874487i	$-0.338802\pi$
$67$	7.94060	0.970099	0.485050	+	0.874487i	$0.338802\pi$
$68$	0.935505	0.113447
$69$	0	0
$70$	11.2787	1.34806
$71$	5.97488	0.709087	0.354544	−	0.935039i	$-0.384636\pi$
$71$	5.97488	0.709087	0.354544	+	0.935039i	$0.384636\pi$
$72$	0	0
$73$	−1.67259	−0.195762	−0.0978810	−	0.995198i	$-0.531206\pi$
$73$	−1.67259	−0.195762	−0.0978810	+	0.995198i	$0.531206\pi$
$74$	6.85863	0.797300
$75$	0	0
$76$	−0.361479	−0.0414645
$77$	6.89067	0.785264
$78$	0	0
$79$	−10.8832	−1.22446	−0.612230	−	0.790680i	$-0.709727\pi$
$79$	−10.8832	−1.22446	−0.612230	+	0.790680i	$0.709727\pi$
$80$	−12.7737	−1.42815
$81$	0	0
$82$	3.67851	0.406224
$83$	−9.35097	−1.02640	−0.513201	−	0.858268i	$-0.671541\pi$
$83$	−9.35097	−1.02640	−0.513201	+	0.858268i	$0.671541\pi$
$84$	0	0
$85$	−10.5068	−1.13962
$86$	−6.52507	−0.703616
$87$	0	0
$88$	−9.59725	−1.02307
$89$	−17.0723	−1.80966	−0.904832	−	0.425770i	$-0.860003\pi$
$89$	−17.0723	−1.80966	−0.904832	+	0.425770i	$0.860003\pi$
$90$	0	0
$91$	−4.73745	−0.496620
$92$	−2.57242	−0.268193
$93$	0	0
$94$	−1.28005	−0.132027
$95$	4.05982	0.416529
$96$	0	0
$97$	9.71823	0.986737	0.493368	−	0.869820i	$-0.335765\pi$
$97$	9.71823	0.986737	0.493368	+	0.869820i	$0.335765\pi$
$98$	2.93092	0.296067
$99$	0	0
$100$	−4.15056	−0.415056
$101$	−1.29156	−0.128515	−0.0642576	−	0.997933i	$-0.520468\pi$
$101$	−1.29156	−0.128515	−0.0642576	+	0.997933i	$0.520468\pi$
$102$	0	0
$103$	12.2420	1.20624	0.603121	−	0.797650i	$-0.293924\pi$
$103$	12.2420	1.20624	0.603121	+	0.797650i	$0.293924\pi$
$104$	6.59828	0.647014
$105$	0	0
$106$	14.6203	1.42005
$107$	−12.6551	−1.22341	−0.611705	−	0.791086i	$-0.709516\pi$
$107$	−12.6551	−1.22341	−0.611705	+	0.791086i	$0.709516\pi$
$108$	0	0
$109$	−1.25626	−0.120328	−0.0601640	−	0.998189i	$-0.519162\pi$
$109$	−1.25626	−0.120328	−0.0601640	+	0.998189i	$0.519162\pi$
$110$	16.4995	1.57316
$111$	0	0
$112$	6.82865	0.645247
$113$	9.43251	0.887336	0.443668	−	0.896191i	$-0.353677\pi$
$113$	9.43251	0.887336	0.443668	+	0.896191i	$0.353677\pi$
$114$	0	0
$115$	28.8912	2.69412
$116$	−2.56029	−0.237717
$117$	0	0
$118$	−15.4291	−1.42037
$119$	5.61678	0.514889
$120$	0	0
$121$	−0.919699	−0.0836090
$122$	−0.314058	−0.0284335
$123$	0	0
$124$	−1.72993	−0.155352
$125$	26.3165	2.35382
$126$	0	0
$127$	18.8859	1.67586	0.837928	−	0.545780i	$-0.183767\pi$
$127$	18.8859	1.67586	0.837928	+	0.545780i	$0.183767\pi$
$128$	−7.32551	−0.647490
$129$	0	0
$130$	−11.3437	−0.994906
$131$	−12.6119	−1.10191	−0.550953	−	0.834536i	$-0.685736\pi$
$131$	−12.6119	−1.10191	−0.550953	+	0.834536i	$0.685736\pi$
$132$	0	0
$133$	−2.17032	−0.188191
$134$	−10.1643	−0.878066
$135$	0	0
$136$	−7.82299	−0.670816
$137$	−2.11080	−0.180338	−0.0901691	−	0.995926i	$-0.528741\pi$
$137$	−2.11080	−0.180338	−0.0901691	+	0.995926i	$0.528741\pi$
$138$	0	0
$139$	−2.07343	−0.175866	−0.0879331	−	0.996126i	$-0.528026\pi$
$139$	−2.07343	−0.175866	−0.0879331	+	0.996126i	$0.528026\pi$
$140$	3.18504	0.269185
$141$	0	0
$142$	−7.64812	−0.641816
$143$	−6.93038	−0.579548
$144$	0	0
$145$	28.7550	2.38797
$146$	2.14100	0.177190
$147$	0	0
$148$	1.93685	0.159208
$149$	13.4014	1.09789	0.548943	−	0.835860i	$-0.315030\pi$
$149$	13.4014	1.09789	0.548943	+	0.835860i	$0.315030\pi$
$150$	0	0
$151$	18.9236	1.53998	0.769989	−	0.638057i	$-0.220261\pi$
$151$	18.9236	1.53998	0.769989	+	0.638057i	$0.220261\pi$
$152$	3.02280	0.245182
$153$	0	0
$154$	−8.82038	−0.710766
$155$	19.4291	1.56058
$156$	0	0
$157$	−7.79375	−0.622009	−0.311004	−	0.950409i	$-0.600665\pi$
$157$	−7.79375	−0.622009	−0.311004	+	0.950409i	$0.600665\pi$
$158$	13.9311	1.10830
$159$	0	0
$160$	−8.19314	−0.647724
$161$	−15.4448	−1.21722
$162$	0	0
$163$	24.7943	1.94204	0.971020	−	0.238996i	$-0.0768184\pi$
$163$	24.7943	1.94204	0.971020	+	0.238996i	$0.0768184\pi$
$164$	1.03879	0.0811162
$165$	0	0
$166$	11.9697	0.929028
$167$	14.7193	1.13901	0.569505	−	0.821988i	$-0.307135\pi$
$167$	14.7193	1.13901	0.569505	+	0.821988i	$0.307135\pi$
$168$	0	0
$169$	−8.23524	−0.633480
$170$	13.4492	1.03151
$171$	0	0
$172$	−1.84265	−0.140501
$173$	−13.0554	−0.992581	−0.496290	−	0.868157i	$-0.665305\pi$
$173$	−13.0554	−0.992581	−0.496290	+	0.868157i	$0.665305\pi$
$174$	0	0
$175$	−24.9200	−1.88378
$176$	9.98958	0.752993
$177$	0	0
$178$	21.8534	1.63798
$179$	1.07386	0.0802639	0.0401320	−	0.999194i	$-0.487222\pi$
$179$	1.07386	0.0802639	0.0401320	+	0.999194i	$0.487222\pi$
$180$	0	0
$181$	21.8894	1.62703	0.813514	−	0.581545i	$-0.197552\pi$
$181$	21.8894	1.62703	0.813514	+	0.581545i	$0.197552\pi$
$182$	6.06416	0.449506
$183$	0	0
$184$	21.5114	1.58584
$185$	−21.7530	−1.59931
$186$	0	0
$187$	8.21674	0.600868
$188$	−0.361479	−0.0263636
$189$	0	0
$190$	−5.19677	−0.377013
$191$	22.1171	1.60034	0.800170	−	0.599773i	$-0.204743\pi$
$191$	22.1171	1.60034	0.800170	+	0.599773i	$0.204743\pi$
$192$	0	0
$193$	−16.2889	−1.17250	−0.586252	−	0.810129i	$-0.699397\pi$
$193$	−16.2889	−1.17250	−0.586252	+	0.810129i	$0.699397\pi$
$194$	−12.4398	−0.893125
$195$	0	0
$196$	0.827677	0.0591198
$197$	6.43379	0.458388	0.229194	−	0.973381i	$-0.426391\pi$
$197$	6.43379	0.458388	0.229194	+	0.973381i	$0.426391\pi$
$198$	0	0
$199$	3.47820	0.246563	0.123282	−	0.992372i	$-0.460658\pi$
$199$	3.47820	0.246563	0.123282	+	0.992372i	$0.460658\pi$
$200$	34.7083	2.45425
$201$	0	0
$202$	1.65326	0.116323
$203$	−15.3720	−1.07890
$204$	0	0
$205$	−11.6668	−0.814848
$206$	−15.6704	−1.09181
$207$	0	0
$208$	−6.86801	−0.476211
$209$	−3.17495	−0.219616
$210$	0	0
$211$	−7.09757	−0.488617	−0.244308	−	0.969698i	$-0.578561\pi$
$211$	−7.09757	−0.488617	−0.244308	+	0.969698i	$0.578561\pi$
$212$	4.12871	0.283561
$213$	0	0
$214$	16.1991	1.10735
$215$	20.6950	1.41139
$216$	0	0
$217$	−10.3865	−0.705082
$218$	1.60807	0.108912
$219$	0	0
$220$	4.65937	0.314135
$221$	−5.64915	−0.380003
$222$	0	0
$223$	−10.6130	−0.710701	−0.355351	−	0.934733i	$-0.615638\pi$
$223$	−10.6130	−0.710701	−0.355351	+	0.934733i	$0.615638\pi$
$224$	4.37994	0.292647
$225$	0	0
$226$	−12.0741	−0.803154
$227$	7.08790	0.470440	0.235220	−	0.971942i	$-0.424419\pi$
$227$	7.08790	0.470440	0.235220	+	0.971942i	$0.424419\pi$
$228$	0	0
$229$	10.2588	0.677919	0.338960	−	0.940801i	$-0.389925\pi$
$229$	10.2588	0.677919	0.338960	+	0.940801i	$0.389925\pi$
$230$	−36.9821	−2.43853
$231$	0	0
$232$	21.4100	1.40563
$233$	7.79158	0.510443	0.255222	−	0.966883i	$-0.417852\pi$
$233$	7.79158	0.510443	0.255222	+	0.966883i	$0.417852\pi$
$234$	0	0
$235$	4.05982	0.264834
$236$	−4.35711	−0.283624
$237$	0	0
$238$	−7.18974	−0.466042
$239$	3.75354	0.242796	0.121398	−	0.992604i	$-0.461262\pi$
$239$	3.75354	0.242796	0.121398	+	0.992604i	$0.461262\pi$
$240$	0	0
$241$	4.18544	0.269608	0.134804	−	0.990872i	$-0.456960\pi$
$241$	4.18544	0.269608	0.134804	+	0.990872i	$0.456960\pi$
$242$	1.17726	0.0756770
$243$	0	0
$244$	−0.0886886	−0.00567771
$245$	−9.29575	−0.593884
$246$	0	0
$247$	2.18283	0.138890
$248$	14.4662	0.918606
$249$	0	0
$250$	−33.6863	−2.13051
$251$	−27.7580	−1.75207	−0.876035	−	0.482248i	$-0.839821\pi$
$251$	−27.7580	−1.75207	−0.876035	+	0.482248i	$0.839821\pi$
$252$	0	0
$253$	−22.5941	−1.42048
$254$	−24.1749	−1.51687
$255$	0	0
$256$	−8.37503	−0.523439
$257$	4.62252	0.288345	0.144172	−	0.989553i	$-0.453948\pi$
$257$	4.62252	0.288345	0.144172	+	0.989553i	$0.453948\pi$
$258$	0	0
$259$	11.6288	0.722581
$260$	−3.20340	−0.198666
$261$	0	0
$262$	16.1438	0.997369
$263$	−30.8673	−1.90336	−0.951678	−	0.307097i	$-0.900642\pi$
$263$	−30.8673	−1.90336	−0.951678	+	0.307097i	$0.900642\pi$
$264$	0	0
$265$	−46.3702	−2.84850
$266$	2.77812	0.170337
$267$	0	0
$268$	−2.87036	−0.175335
$269$	13.8358	0.843581	0.421790	−	0.906693i	$-0.361402\pi$
$269$	13.8358	0.843581	0.421790	+	0.906693i	$0.361402\pi$
$270$	0	0
$271$	−27.9938	−1.70050	−0.850252	−	0.526376i	$-0.823551\pi$
$271$	−27.9938	−1.70050	−0.850252	+	0.526376i	$0.823551\pi$
$272$	8.14279	0.493729
$273$	0	0
$274$	2.70193	0.163230
$275$	−36.4553	−2.19834
$276$	0	0
$277$	9.99039	0.600264	0.300132	−	0.953898i	$-0.402969\pi$
$277$	9.99039	0.600264	0.300132	+	0.953898i	$0.402969\pi$
$278$	2.65409	0.159182
$279$	0	0
$280$	−26.6343	−1.59171
$281$	−3.55679	−0.212181	−0.106090	−	0.994357i	$-0.533833\pi$
$281$	−3.55679	−0.212181	−0.106090	+	0.994357i	$0.533833\pi$
$282$	0	0
$283$	20.0791	1.19358	0.596789	−	0.802398i	$-0.296443\pi$
$283$	20.0791	1.19358	0.596789	+	0.802398i	$0.296443\pi$
$284$	−2.15979	−0.128160
$285$	0	0
$286$	8.87122	0.524566
$287$	6.23693	0.368154
$288$	0	0
$289$	−10.3023	−0.606018
$290$	−36.8077	−2.16143
$291$	0	0
$292$	0.604607	0.0353820
$293$	−3.20064	−0.186984	−0.0934918	−	0.995620i	$-0.529803\pi$
$293$	−3.20064	−0.186984	−0.0934918	+	0.995620i	$0.529803\pi$
$294$	0	0
$295$	48.9353	2.84912
$296$	−16.1965	−0.941404
$297$	0	0
$298$	−17.1544	−0.993729
$299$	15.5338	0.898345
$300$	0	0
$301$	−11.0633	−0.637676
$302$	−24.2231	−1.39388
$303$	0	0
$304$	−3.14637	−0.180457
$305$	0.996074	0.0570350
$306$	0	0
$307$	16.4446	0.938542	0.469271	−	0.883054i	$-0.344517\pi$
$307$	16.4446	0.938542	0.469271	+	0.883054i	$0.344517\pi$
$308$	−2.49083	−0.141928
$309$	0	0
$310$	−24.8701	−1.41253
$311$	−24.4684	−1.38748	−0.693738	−	0.720227i	$-0.744038\pi$
$311$	−24.4684	−1.38748	−0.693738	+	0.720227i	$0.744038\pi$
$312$	0	0
$313$	0.384563	0.0217368	0.0108684	−	0.999941i	$-0.496540\pi$
$313$	0.384563	0.0217368	0.0108684	+	0.999941i	$0.496540\pi$
$314$	9.97636	0.562999
$315$	0	0
$316$	3.93407	0.221309
$317$	−10.9884	−0.617171	−0.308586	−	0.951197i	$-0.599856\pi$
$317$	−10.9884	−0.617171	−0.308586	+	0.951197i	$0.599856\pi$
$318$	0	0
$319$	−22.4876	−1.25906
$320$	36.0351	2.01442
$321$	0	0
$322$	19.7701	1.10174
$323$	−2.58799	−0.144000
$324$	0	0
$325$	25.0636	1.39028
$326$	−31.7379	−1.75780
$327$	0	0
$328$	−8.68673	−0.479644
$329$	−2.17032	−0.119654
$330$	0	0
$331$	2.44087	0.134162	0.0670811	−	0.997748i	$-0.478631\pi$
$331$	2.44087	0.134162	0.0670811	+	0.997748i	$0.478631\pi$
$332$	3.38018	0.185512
$333$	0	0
$334$	−18.8413	−1.03095
$335$	32.2375	1.76132
$336$	0	0
$337$	2.11250	0.115075	0.0575375	−	0.998343i	$-0.481675\pi$
$337$	2.11250	0.115075	0.0575375	+	0.998343i	$0.481675\pi$
$338$	10.5415	0.573382
$339$	0	0
$340$	3.79799	0.205975
$341$	−15.1943	−0.822820
$342$	0	0
$343$	20.1616	1.08863
$344$	15.4088	0.830787
$345$	0	0
$346$	16.7115	0.898414
$347$	0.872665	0.0468471	0.0234236	−	0.999726i	$-0.492543\pi$
$347$	0.872665	0.0468471	0.0234236	+	0.999726i	$0.492543\pi$
$348$	0	0
$349$	23.4602	1.25580	0.627898	−	0.778295i	$-0.283915\pi$
$349$	23.4602	1.25580	0.627898	+	0.778295i	$0.283915\pi$
$350$	31.8988	1.70506
$351$	0	0
$352$	6.40737	0.341514
$353$	23.4185	1.24644	0.623221	−	0.782045i	$-0.285824\pi$
$353$	23.4185	1.24644	0.623221	+	0.782045i	$0.285824\pi$
$354$	0	0
$355$	24.2569	1.28742
$356$	6.17129	0.327078
$357$	0	0
$358$	−1.37459	−0.0726493
$359$	21.3228	1.12538	0.562688	−	0.826669i	$-0.309767\pi$
$359$	21.3228	1.12538	0.562688	+	0.826669i	$0.309767\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−28.0195	−1.47267
$363$	0	0
$364$	1.71249	0.0897589
$365$	−6.79043	−0.355427
$366$	0	0
$367$	28.2181	1.47297	0.736486	−	0.676453i	$-0.236484\pi$
$367$	28.2181	1.47297	0.736486	+	0.676453i	$0.236484\pi$
$368$	−22.3908	−1.16720
$369$	0	0
$370$	27.8448	1.44758
$371$	24.7888	1.28697
$372$	0	0
$373$	−32.0909	−1.66160	−0.830802	−	0.556569i	$-0.812118\pi$
$373$	−32.0909	−1.66160	−0.830802	+	0.556569i	$0.812118\pi$
$374$	−10.5178	−0.543863
$375$	0	0
$376$	3.02280	0.155889
$377$	15.4606	0.796262
$378$	0	0
$379$	9.74080	0.500352	0.250176	−	0.968200i	$-0.419512\pi$
$379$	9.74080	0.500352	0.250176	+	0.968200i	$0.419512\pi$
$380$	−1.46754	−0.0752833
$381$	0	0
$382$	−28.3110	−1.44852
$383$	−17.8003	−0.909555	−0.454777	−	0.890605i	$-0.650281\pi$
$383$	−17.8003	−0.909555	−0.454777	+	0.890605i	$0.650281\pi$
$384$	0	0
$385$	27.9749	1.42573
$386$	20.8506	1.06127
$387$	0	0
$388$	−3.51294	−0.178342
$389$	24.0700	1.22040	0.610198	−	0.792249i	$-0.291090\pi$
$389$	24.0700	1.22040	0.610198	+	0.792249i	$0.291090\pi$
$390$	0	0
$391$	−18.4171	−0.931392
$392$	−6.92130	−0.349578
$393$	0	0
$394$	−8.23555	−0.414901
$395$	−44.1840	−2.22314
$396$	0	0
$397$	22.6135	1.13494	0.567469	−	0.823395i	$-0.307923\pi$
$397$	22.6135	1.13494	0.567469	+	0.823395i	$0.307923\pi$
$398$	−4.45226	−0.223172
$399$	0	0
$400$	−36.1272	−1.80636
$401$	12.4096	0.619708	0.309854	−	0.950784i	$-0.399720\pi$
$401$	12.4096	0.619708	0.309854	+	0.950784i	$0.399720\pi$
$402$	0	0
$403$	10.4464	0.520371
$404$	0.466873	0.0232278
$405$	0	0
$406$	19.6769	0.976548
$407$	17.0117	0.843240
$408$	0	0
$409$	31.2296	1.54421	0.772103	−	0.635498i	$-0.219205\pi$
$409$	31.2296	1.54421	0.772103	+	0.635498i	$0.219205\pi$
$410$	14.9341	0.737543
$411$	0	0
$412$	−4.42524	−0.218016
$413$	−26.1601	−1.28726
$414$	0	0
$415$	−37.9633	−1.86355
$416$	−4.40518	−0.215982
$417$	0	0
$418$	4.06408	0.198781
$419$	11.3031	0.552191	0.276096	−	0.961130i	$-0.410959\pi$
$419$	11.3031	0.552191	0.276096	+	0.961130i	$0.410959\pi$
$420$	0	0
$421$	−1.63420	−0.0796458	−0.0398229	−	0.999207i	$-0.512679\pi$
$421$	−1.63420	−0.0796458	−0.0398229	+	0.999207i	$0.512679\pi$
$422$	9.08522	0.442262
$423$	0	0
$424$	−34.5256	−1.67671
$425$	−29.7157	−1.44143
$426$	0	0
$427$	−0.532487	−0.0257689
$428$	4.57454	0.221119
$429$	0	0
$430$	−26.4906	−1.27749
$431$	4.48073	0.215829	0.107914	−	0.994160i	$-0.465583\pi$
$431$	4.48073	0.215829	0.107914	+	0.994160i	$0.465583\pi$
$432$	0	0
$433$	−35.1763	−1.69046	−0.845232	−	0.534399i	$-0.820538\pi$
$433$	−35.1763	−1.69046	−0.845232	+	0.534399i	$0.820538\pi$
$434$	13.2952	0.638191
$435$	0	0
$436$	0.454112	0.0217480
$437$	7.11637	0.340422
$438$	0	0
$439$	22.7912	1.08776	0.543882	−	0.839162i	$-0.316954\pi$
$439$	22.7912	1.08776	0.543882	+	0.839162i	$0.316954\pi$
$440$	−38.9631	−1.85750
$441$	0	0
$442$	7.23118	0.343952
$443$	−9.94593	−0.472545	−0.236273	−	0.971687i	$-0.575926\pi$
$443$	−9.94593	−0.472545	−0.236273	+	0.971687i	$0.575926\pi$
$444$	0	0
$445$	−69.3106	−3.28564
$446$	13.5852	0.643277
$447$	0	0
$448$	−19.2638	−0.910130
$449$	27.9070	1.31701	0.658507	−	0.752575i	$-0.271188\pi$
$449$	27.9070	1.31701	0.658507	+	0.752575i	$0.271188\pi$
$450$	0	0
$451$	9.12395	0.429630
$452$	−3.40966	−0.160377
$453$	0	0
$454$	−9.07284	−0.425810
$455$	−19.2332	−0.901668
$456$	0	0
$457$	−14.1126	−0.660157	−0.330079	−	0.943953i	$-0.607075\pi$
$457$	−14.1126	−0.660157	−0.330079	+	0.943953i	$0.607075\pi$
$458$	−13.1317	−0.613605
$459$	0	0
$460$	−10.4436	−0.486934
$461$	0.901698	0.0419963	0.0209981	−	0.999780i	$-0.493316\pi$
$461$	0.901698	0.0419963	0.0209981	+	0.999780i	$0.493316\pi$
$462$	0	0
$463$	−12.8687	−0.598059	−0.299030	−	0.954244i	$-0.596663\pi$
$463$	−12.8687	−0.598059	−0.299030	+	0.954244i	$0.596663\pi$
$464$	−22.2852	−1.03456
$465$	0	0
$466$	−9.97359	−0.462018
$467$	−15.4568	−0.715256	−0.357628	−	0.933864i	$-0.616414\pi$
$467$	−15.4568	−0.715256	−0.357628	+	0.933864i	$0.616414\pi$
$468$	0	0
$469$	−17.2337	−0.795778
$470$	−5.19677	−0.239709
$471$	0	0
$472$	36.4355	1.67708
$473$	−16.1844	−0.744158
$474$	0	0
$475$	11.4822	0.526838
$476$	−2.03035	−0.0930609
$477$	0	0
$478$	−4.80471	−0.219762
$479$	−14.2271	−0.650053	−0.325026	−	0.945705i	$-0.605373\pi$
$479$	−14.2271	−0.650053	−0.325026	+	0.945705i	$0.605373\pi$
$480$	0	0
$481$	−11.6959	−0.533285
$482$	−5.35756	−0.244030
$483$	0	0
$484$	0.332452	0.0151115
$485$	39.4543	1.79153
$486$	0	0
$487$	42.5931	1.93008	0.965039	−	0.262105i	$-0.0844167\pi$
$487$	42.5931	1.93008	0.965039	+	0.262105i	$0.0844167\pi$
$488$	0.741642	0.0335726
$489$	0	0
$490$	11.8990	0.537542
$491$	23.8393	1.07585	0.537927	−	0.842991i	$-0.319208\pi$
$491$	23.8393	1.07585	0.537927	+	0.842991i	$0.319208\pi$
$492$	0	0
$493$	−18.3303	−0.825554
$494$	−2.79413	−0.125714
$495$	0	0
$496$	−15.0576	−0.676106
$497$	−12.9674	−0.581668
$498$	0	0
$499$	17.5317	0.784826	0.392413	−	0.919789i	$-0.371640\pi$
$499$	17.5317	0.784826	0.392413	+	0.919789i	$0.371640\pi$
$500$	−9.51285	−0.425428
$501$	0	0
$502$	35.5316	1.58585
$503$	28.8250	1.28524	0.642622	−	0.766183i	$-0.277847\pi$
$503$	28.8250	1.28524	0.642622	+	0.766183i	$0.277847\pi$
$504$	0	0
$505$	−5.24351	−0.233333
$506$	28.9215	1.28572
$507$	0	0
$508$	−6.82688	−0.302894
$509$	−10.3568	−0.459059	−0.229530	−	0.973302i	$-0.573719\pi$
$509$	−10.3568	−0.459059	−0.229530	+	0.973302i	$0.573719\pi$
$510$	0	0
$511$	3.63007	0.160585
$512$	25.3715	1.12127
$513$	0	0
$514$	−5.91704	−0.260989
$515$	49.7004	2.19006
$516$	0	0
$517$	−3.17495	−0.139634
$518$	−14.8855	−0.654029
$519$	0	0
$520$	26.7878	1.17472
$521$	−15.9829	−0.700223	−0.350111	−	0.936708i	$-0.613856\pi$
$521$	−15.9829	−0.700223	−0.350111	+	0.936708i	$0.613856\pi$
$522$	0	0
$523$	−3.13384	−0.137033	−0.0685167	−	0.997650i	$-0.521827\pi$
$523$	−3.13384	−0.137033	−0.0685167	+	0.997650i	$0.521827\pi$
$524$	4.55894	0.199158
$525$	0	0
$526$	39.5115	1.72279
$527$	−12.3853	−0.539514
$528$	0	0
$529$	27.6427	1.20185
$530$	59.3560	2.57826
$531$	0	0
$532$	0.784527	0.0340136
$533$	−6.27287	−0.271708
$534$	0	0
$535$	−51.3773	−2.22123
$536$	24.0029	1.03677
$537$	0	0
$538$	−17.7104	−0.763550
$539$	7.26966	0.313126
$540$	0	0
$541$	8.35329	0.359136	0.179568	−	0.983746i	$-0.442530\pi$
$541$	8.35329	0.359136	0.179568	+	0.983746i	$0.442530\pi$
$542$	35.8334	1.53918
$543$	0	0
$544$	5.22283	0.223927
$545$	−5.10020	−0.218468
$546$	0	0
$547$	11.4555	0.489801	0.244900	−	0.969548i	$-0.421245\pi$
$547$	11.4555	0.489801	0.244900	+	0.969548i	$0.421245\pi$
$548$	0.763012	0.0325943
$549$	0	0
$550$	46.6645	1.98978
$551$	7.08282	0.301738
$552$	0	0
$553$	23.6202	1.00443
$554$	−12.7882	−0.543317
$555$	0	0
$556$	0.749503	0.0317860
$557$	−26.9605	−1.14235	−0.571176	−	0.820828i	$-0.693512\pi$
$557$	−26.9605	−1.14235	−0.571176	+	0.820828i	$0.693512\pi$
$558$	0	0
$559$	11.1270	0.470623
$560$	27.7231	1.17152
$561$	0	0
$562$	4.55286	0.192051
$563$	−12.7925	−0.539139	−0.269569	−	0.962981i	$-0.586881\pi$
$563$	−12.7925	−0.539139	−0.269569	+	0.962981i	$0.586881\pi$
$564$	0	0
$565$	38.2943	1.61105
$566$	−25.7022	−1.08034
$567$	0	0
$568$	18.0609	0.757818
$569$	−19.2823	−0.808356	−0.404178	−	0.914680i	$-0.632442\pi$
$569$	−19.2823	−0.808356	−0.404178	+	0.914680i	$0.632442\pi$
$570$	0	0
$571$	26.6814	1.11658	0.558291	−	0.829645i	$-0.311457\pi$
$571$	26.6814	1.11658	0.558291	+	0.829645i	$0.311457\pi$
$572$	2.50519	0.104747
$573$	0	0
$574$	−7.98356	−0.333227
$575$	81.7113	3.40760
$576$	0	0
$577$	12.7848	0.532236	0.266118	−	0.963940i	$-0.414259\pi$
$577$	12.7848	0.532236	0.266118	+	0.963940i	$0.414259\pi$
$578$	13.1874	0.548525
$579$	0	0
$580$	−10.3943	−0.431601
$581$	20.2946	0.841964
$582$	0	0
$583$	36.2634	1.50188
$584$	−5.05592	−0.209215
$585$	0	0
$586$	4.09698	0.169245
$587$	0.661563	0.0273056	0.0136528	−	0.999907i	$-0.495654\pi$
$587$	0.661563	0.0273056	0.0136528	+	0.999907i	$0.495654\pi$
$588$	0	0
$589$	4.78570	0.197191
$590$	−62.6395	−2.57883
$591$	0	0
$592$	16.8586	0.692885
$593$	15.5087	0.636866	0.318433	−	0.947945i	$-0.396843\pi$
$593$	15.5087	0.636866	0.318433	+	0.947945i	$0.396843\pi$
$594$	0	0
$595$	22.8031	0.934837
$596$	−4.84433	−0.198431
$597$	0	0
$598$	−19.8840	−0.813119
$599$	−7.61579	−0.311173	−0.155586	−	0.987822i	$-0.549727\pi$
$599$	−7.61579	−0.311173	−0.155586	+	0.987822i	$0.549727\pi$
$600$	0	0
$601$	−19.9601	−0.814188	−0.407094	−	0.913386i	$-0.633458\pi$
$601$	−19.9601	−0.814188	−0.407094	+	0.913386i	$0.633458\pi$
$602$	14.1615	0.577180
$603$	0	0
$604$	−6.84048	−0.278335
$605$	−3.73381	−0.151801
$606$	0	0
$607$	−14.7946	−0.600493	−0.300246	−	0.953862i	$-0.597069\pi$
$607$	−14.7946	−0.600493	−0.300246	+	0.953862i	$0.597069\pi$
$608$	−2.01810	−0.0818448
$609$	0	0
$610$	−1.27502	−0.0516241
$611$	2.18283	0.0883080
$612$	0	0
$613$	−25.4465	−1.02778	−0.513888	−	0.857857i	$-0.671795\pi$
$613$	−25.4465	−1.02778	−0.513888	+	0.857857i	$0.671795\pi$
$614$	−21.0498	−0.849503
$615$	0	0
$616$	20.8291	0.839230
$617$	−3.20384	−0.128982	−0.0644908	−	0.997918i	$-0.520542\pi$
$617$	−3.20384	−0.128982	−0.0644908	+	0.997918i	$0.520542\pi$
$618$	0	0
$619$	22.7081	0.912717	0.456359	−	0.889796i	$-0.349153\pi$
$619$	22.7081	0.912717	0.456359	+	0.889796i	$0.349153\pi$
$620$	−7.02321	−0.282059
$621$	0	0
$622$	31.3207	1.25585
$623$	37.0525	1.48448
$624$	0	0
$625$	49.4293	1.97717
$626$	−0.492258	−0.0196746
$627$	0	0
$628$	2.81728	0.112422
$629$	13.8667	0.552903
$630$	0	0
$631$	22.1237	0.880731	0.440366	−	0.897818i	$-0.354849\pi$
$631$	22.1237	0.880731	0.440366	+	0.897818i	$0.354849\pi$
$632$	−32.8979	−1.30861
$633$	0	0
$634$	14.0657	0.558620
$635$	76.6736	3.04270
$636$	0	0
$637$	−4.99802	−0.198029
$638$	28.7852	1.13962
$639$	0	0
$640$	−29.7403	−1.17559
$641$	4.80428	0.189758	0.0948788	−	0.995489i	$-0.469754\pi$
$641$	4.80428	0.189758	0.0948788	+	0.995489i	$0.469754\pi$
$642$	0	0
$643$	27.0688	1.06749	0.533744	−	0.845646i	$-0.320785\pi$
$643$	27.0688	1.06749	0.533744	+	0.845646i	$0.320785\pi$
$644$	5.58298	0.220000
$645$	0	0
$646$	3.31275	0.130338
$647$	−20.7842	−0.817111	−0.408555	−	0.912733i	$-0.633967\pi$
$647$	−20.7842	−0.817111	−0.408555	+	0.912733i	$0.633967\pi$
$648$	0	0
$649$	−38.2694	−1.50221
$650$	−32.0827	−1.25839
$651$	0	0
$652$	−8.96263	−0.351004
$653$	−36.3128	−1.42103	−0.710515	−	0.703682i	$-0.751538\pi$
$653$	−36.3128	−1.42103	−0.710515	+	0.703682i	$0.751538\pi$
$654$	0	0
$655$	−51.2021	−2.00063
$656$	9.04183	0.353024
$657$	0	0
$658$	2.77812	0.108302
$659$	−33.9632	−1.32302	−0.661510	−	0.749937i	$-0.730084\pi$
$659$	−33.9632	−1.32302	−0.661510	+	0.749937i	$0.730084\pi$
$660$	0	0
$661$	−1.66353	−0.0647039	−0.0323520	−	0.999477i	$-0.510300\pi$
$661$	−1.66353	−0.0647039	−0.0323520	+	0.999477i	$0.510300\pi$
$662$	−3.12442	−0.121434
$663$	0	0
$664$	−28.2662	−1.09694
$665$	−8.81113	−0.341681
$666$	0	0
$667$	50.4039	1.95165
$668$	−5.32071	−0.205864
$669$	0	0
$670$	−41.2655	−1.59422
$671$	−0.778971	−0.0300718
$672$	0	0
$673$	−8.42733	−0.324850	−0.162425	−	0.986721i	$-0.551932\pi$
$673$	−8.42733	−0.324850	−0.162425	+	0.986721i	$0.551932\pi$
$674$	−2.70409	−0.104158
$675$	0	0
$676$	2.97687	0.114495
$677$	4.59108	0.176450	0.0882248	−	0.996101i	$-0.471881\pi$
$677$	4.59108	0.176450	0.0882248	+	0.996101i	$0.471881\pi$
$678$	0	0
$679$	−21.0917	−0.809426
$680$	−31.7600	−1.21794
$681$	0	0
$682$	19.4495	0.744759
$683$	24.2974	0.929714	0.464857	−	0.885386i	$-0.346106\pi$
$683$	24.2974	0.929714	0.464857	+	0.885386i	$0.346106\pi$
$684$	0	0
$685$	−8.56949	−0.327424
$686$	−25.8079	−0.985349
$687$	0	0
$688$	−16.0387	−0.611470
$689$	−24.9317	−0.949822
$690$	0	0
$691$	−13.5531	−0.515584	−0.257792	−	0.966200i	$-0.582995\pi$
$691$	−13.5531	−0.515584	−0.257792	+	0.966200i	$0.582995\pi$
$692$	4.71924	0.179399
$693$	0	0
$694$	−1.11705	−0.0424027
$695$	−8.41777	−0.319304
$696$	0	0
$697$	7.43719	0.281704
$698$	−30.0302	−1.13666
$699$	0	0
$700$	9.00807	0.340473
$701$	22.7152	0.857939	0.428970	−	0.903319i	$-0.358877\pi$
$701$	22.7152	0.857939	0.428970	+	0.903319i	$0.358877\pi$
$702$	0	0
$703$	−5.35811	−0.202085
$704$	−28.1809	−1.06211
$705$	0	0
$706$	−29.9768	−1.12819
$707$	2.80311	0.105422
$708$	0	0
$709$	39.0000	1.46468	0.732339	−	0.680941i	$-0.238429\pi$
$709$	39.0000	1.46468	0.732339	+	0.680941i	$0.238429\pi$
$710$	−31.0500	−1.16529
$711$	0	0
$712$	−51.6063	−1.93403
$713$	34.0568	1.27544
$714$	0	0
$715$	−28.1361	−1.05223
$716$	−0.388177	−0.0145069
$717$	0	0
$718$	−27.2942	−1.01861
$719$	−20.7863	−0.775196	−0.387598	−	0.921828i	$-0.626695\pi$
$719$	−20.7863	−0.775196	−0.387598	+	0.921828i	$0.626695\pi$
$720$	0	0
$721$	−26.5691	−0.989487
$722$	−1.28005	−0.0476384
$723$	0	0
$724$	−7.91257	−0.294068
$725$	81.3261	3.02038
$726$	0	0
$727$	5.46993	0.202868	0.101434	−	0.994842i	$-0.467657\pi$
$727$	5.46993	0.202868	0.101434	+	0.994842i	$0.467657\pi$
$728$	−14.3204	−0.530749
$729$	0	0
$730$	8.69207	0.321708
$731$	−13.1923	−0.487936
$732$	0	0
$733$	21.6545	0.799828	0.399914	−	0.916553i	$-0.369040\pi$
$733$	21.6545	0.799828	0.399914	+	0.916553i	$0.369040\pi$
$734$	−36.1205	−1.33323
$735$	0	0
$736$	−14.3616	−0.529374
$737$	−25.2110	−0.928660
$738$	0	0
$739$	40.3484	1.48424	0.742121	−	0.670266i	$-0.233820\pi$
$739$	40.3484	1.48424	0.742121	+	0.670266i	$0.233820\pi$
$740$	7.86325	0.289059
$741$	0	0
$742$	−31.7309	−1.16488
$743$	8.55801	0.313963	0.156981	−	0.987602i	$-0.449824\pi$
$743$	8.55801	0.313963	0.156981	+	0.987602i	$0.449824\pi$
$744$	0	0
$745$	54.4073	1.99333
$746$	41.0779	1.50397
$747$	0	0
$748$	−2.97018	−0.108601
$749$	27.4656	1.00357
$750$	0	0
$751$	43.0619	1.57135	0.785675	−	0.618640i	$-0.212316\pi$
$751$	43.0619	1.57135	0.785675	+	0.618640i	$0.212316\pi$
$752$	−3.14637	−0.114736
$753$	0	0
$754$	−19.7903	−0.720720
$755$	76.8264	2.79600
$756$	0	0
$757$	22.6553	0.823422	0.411711	−	0.911314i	$-0.364931\pi$
$757$	22.6553	0.823422	0.411711	+	0.911314i	$0.364931\pi$
$758$	−12.4687	−0.452883
$759$	0	0
$760$	12.2721	0.445154
$761$	4.35713	0.157946	0.0789730	−	0.996877i	$-0.474836\pi$
$761$	4.35713	0.157946	0.0789730	+	0.996877i	$0.474836\pi$
$762$	0	0
$763$	2.72649	0.0987057
$764$	−7.99488	−0.289245
$765$	0	0
$766$	22.7853	0.823265
$767$	26.3109	0.950031
$768$	0	0
$769$	−13.5337	−0.488038	−0.244019	−	0.969771i	$-0.578466\pi$
$769$	−13.5337	−0.488038	−0.244019	+	0.969771i	$0.578466\pi$
$770$	−35.8092	−1.29047
$771$	0	0
$772$	5.88812	0.211918
$773$	−11.8027	−0.424513	−0.212257	−	0.977214i	$-0.568081\pi$
$773$	−11.8027	−0.424513	−0.212257	+	0.977214i	$0.568081\pi$
$774$	0	0
$775$	54.9502	1.97387
$776$	29.3763	1.05455
$777$	0	0
$778$	−30.8107	−1.10462
$779$	−2.87373	−0.102962
$780$	0	0
$781$	−18.9699	−0.678797
$782$	23.5747	0.843031
$783$	0	0
$784$	7.20423	0.257294
$785$	−31.6412	−1.12932
$786$	0	0
$787$	−48.7498	−1.73774	−0.868872	−	0.495037i	$-0.835154\pi$
$787$	−48.7498	−1.73774	−0.868872	+	0.495037i	$0.835154\pi$
$788$	−2.32568	−0.0828489
$789$	0	0
$790$	56.5577	2.01223
$791$	−20.4716	−0.727886
$792$	0	0
$793$	0.535556	0.0190182
$794$	−28.9463	−1.02727
$795$	0	0
$796$	−1.25730	−0.0445638
$797$	48.2420	1.70882	0.854409	−	0.519601i	$-0.173919\pi$
$797$	48.2420	1.70882	0.854409	+	0.519601i	$0.173919\pi$
$798$	0	0
$799$	−2.58799	−0.0915565
$800$	−23.1722	−0.819260
$801$	0	0
$802$	−15.8849	−0.560916
$803$	5.31039	0.187400
$804$	0	0
$805$	−62.7032	−2.21000
$806$	−13.3718	−0.471003
$807$	0	0
$808$	−3.90414	−0.137347
$809$	−22.6647	−0.796849	−0.398425	−	0.917201i	$-0.630443\pi$
$809$	−22.6647	−0.796849	−0.398425	+	0.917201i	$0.630443\pi$
$810$	0	0
$811$	6.94422	0.243844	0.121922	−	0.992540i	$-0.461094\pi$
$811$	6.94422	0.243844	0.121922	+	0.992540i	$0.461094\pi$
$812$	5.55666	0.195001
$813$	0	0
$814$	−21.7758	−0.763242
$815$	100.661	3.52599
$816$	0	0
$817$	5.09752	0.178340
$818$	−39.9754	−1.39771
$819$	0	0
$820$	4.21732	0.147275
$821$	17.7375	0.619042	0.309521	−	0.950893i	$-0.399831\pi$
$821$	17.7375	0.619042	0.309521	+	0.950893i	$0.399831\pi$
$822$	0	0
$823$	−42.6440	−1.48648	−0.743238	−	0.669027i	$-0.766711\pi$
$823$	−42.6440	−1.48648	−0.743238	+	0.669027i	$0.766711\pi$
$824$	37.0052	1.28914
$825$	0	0
$826$	33.4862	1.16513
$827$	47.2681	1.64367	0.821837	−	0.569723i	$-0.192950\pi$
$827$	47.2681	1.64367	0.821837	+	0.569723i	$0.192950\pi$
$828$	0	0
$829$	−23.9647	−0.832327	−0.416163	−	0.909290i	$-0.636626\pi$
$829$	−23.9647	−0.832327	−0.416163	+	0.909290i	$0.636626\pi$
$830$	48.5948	1.68675
$831$	0	0
$832$	19.3749	0.671702
$833$	5.92571	0.205314
$834$	0	0
$835$	59.7576	2.06800
$836$	1.14768	0.0396933
$837$	0	0
$838$	−14.4685	−0.499805
$839$	26.7090	0.922099	0.461049	−	0.887375i	$-0.347473\pi$
$839$	26.7090	0.922099	0.461049	+	0.887375i	$0.347473\pi$
$840$	0	0
$841$	21.1663	0.729873
$842$	2.09185	0.0720898
$843$	0	0
$844$	2.56562	0.0883124
$845$	−33.4336	−1.15015
$846$	0	0
$847$	1.99604	0.0685849
$848$	35.9370	1.23408
$849$	0	0
$850$	38.0376	1.30468
$851$	−38.1303	−1.30709
$852$	0	0
$853$	−24.5641	−0.841058	−0.420529	−	0.907279i	$-0.638156\pi$
$853$	−24.5641	−0.841058	−0.420529	+	0.907279i	$0.638156\pi$
$854$	0.681609	0.0233242
$855$	0	0
$856$	−38.2538	−1.30749
$857$	55.4756	1.89501	0.947506	−	0.319739i	$-0.103595\pi$
$857$	55.4756	1.89501	0.947506	+	0.319739i	$0.103595\pi$
$858$	0	0
$859$	15.4264	0.526340	0.263170	−	0.964749i	$-0.415232\pi$
$859$	15.4264	0.526340	0.263170	+	0.964749i	$0.415232\pi$
$860$	−7.48082	−0.255094
$861$	0	0
$862$	−5.73554	−0.195353
$863$	−6.45671	−0.219789	−0.109894	−	0.993943i	$-0.535051\pi$
$863$	−6.45671	−0.219789	−0.109894	+	0.993943i	$0.535051\pi$
$864$	0	0
$865$	−53.0025	−1.80214
$866$	45.0273	1.53009
$867$	0	0
$868$	3.75451	0.127436
$869$	34.5537	1.17216
$870$	0	0
$871$	17.3330	0.587307
$872$	−3.79743	−0.128597
$873$	0	0
$874$	−9.10928	−0.308126
$875$	−57.1152	−1.93085
$876$	0	0
$877$	−25.6882	−0.867429	−0.433715	−	0.901050i	$-0.642797\pi$
$877$	−25.6882	−0.867429	−0.433715	+	0.901050i	$0.642797\pi$
$878$	−29.1738	−0.984567
$879$	0	0
$880$	40.5559	1.36714
$881$	27.4064	0.923343	0.461672	−	0.887051i	$-0.347250\pi$
$881$	27.4064	0.923343	0.461672	+	0.887051i	$0.347250\pi$
$882$	0	0
$883$	9.45444	0.318167	0.159084	−	0.987265i	$-0.449146\pi$
$883$	9.45444	0.318167	0.159084	+	0.987265i	$0.449146\pi$
$884$	2.04205	0.0686816
$885$	0	0
$886$	12.7313	0.427715
$887$	41.2723	1.38579	0.692894	−	0.721039i	$-0.256335\pi$
$887$	41.2723	1.38579	0.692894	+	0.721039i	$0.256335\pi$
$888$	0	0
$889$	−40.9886	−1.37471
$890$	88.7209	2.97393
$891$	0	0
$892$	3.83639	0.128452
$893$	1.00000	0.0334637
$894$	0	0
$895$	4.35967	0.145728
$896$	15.8987	0.531140
$897$	0	0
$898$	−35.7223	−1.19207
$899$	33.8962	1.13050
$900$	0	0
$901$	29.5593	0.984763
$902$	−11.6791	−0.388871
$903$	0	0
$904$	28.5126	0.948316
$905$	88.8672	2.95405
$906$	0	0
$907$	−51.5866	−1.71290	−0.856452	−	0.516226i	$-0.827336\pi$
$907$	−51.5866	−1.71290	−0.856452	+	0.516226i	$0.827336\pi$
$908$	−2.56213	−0.0850272
$909$	0	0
$910$	24.6194	0.816126
$911$	30.3580	1.00580	0.502902	−	0.864343i	$-0.332266\pi$
$911$	30.3580	1.00580	0.502902	+	0.864343i	$0.332266\pi$
$912$	0	0
$913$	29.6889	0.982558
$914$	18.0647	0.597528
$915$	0	0
$916$	−3.70834	−0.122527
$917$	27.3719	0.903900
$918$	0	0
$919$	7.64481	0.252179	0.126090	−	0.992019i	$-0.459757\pi$
$919$	7.64481	0.252179	0.126090	+	0.992019i	$0.459757\pi$
$920$	87.3324	2.87926
$921$	0	0
$922$	−1.15422	−0.0380121
$923$	13.0422	0.429288
$924$	0	0
$925$	−61.5227	−2.02285
$926$	16.4725	0.541322
$927$	0	0
$928$	−14.2938	−0.469219
$929$	7.83453	0.257043	0.128521	−	0.991707i	$-0.458977\pi$
$929$	7.83453	0.257043	0.128521	+	0.991707i	$0.458977\pi$
$930$	0	0
$931$	−2.28969	−0.0750417
$932$	−2.81649	−0.0922573
$933$	0	0
$934$	19.7854	0.647400
$935$	33.3585	1.09094
$936$	0	0
$937$	9.03707	0.295228	0.147614	−	0.989045i	$-0.452841\pi$
$937$	9.03707	0.295228	0.147614	+	0.989045i	$0.452841\pi$
$938$	22.0599	0.720282
$939$	0	0
$940$	−1.46754	−0.0478659
$941$	53.2348	1.73540	0.867702	−	0.497085i	$-0.165596\pi$
$941$	53.2348	1.73540	0.867702	+	0.497085i	$0.165596\pi$
$942$	0	0
$943$	−20.4505	−0.665961
$944$	−37.9250	−1.23435
$945$	0	0
$946$	20.7168	0.673560
$947$	−53.6947	−1.74484	−0.872421	−	0.488755i	$-0.837451\pi$
$947$	−53.6947	−1.74484	−0.872421	+	0.488755i	$0.837451\pi$
$948$	0	0
$949$	−3.65099	−0.118516
$950$	−14.6977	−0.476857
$951$	0	0
$952$	16.9784	0.550274
$953$	−45.4578	−1.47252	−0.736261	−	0.676698i	$-0.763410\pi$
$953$	−45.4578	−1.47252	−0.736261	+	0.676698i	$0.763410\pi$
$954$	0	0
$955$	89.7917	2.90559
$956$	−1.35683	−0.0438829
$957$	0	0
$958$	18.2114	0.588382
$959$	4.58113	0.147932
$960$	0	0
$961$	−8.09712	−0.261197
$962$	14.9712	0.482692
$963$	0	0
$964$	−1.51295	−0.0487288
$965$	−66.1302	−2.12881
$966$	0	0
$967$	50.6775	1.62968	0.814840	−	0.579686i	$-0.196825\pi$
$967$	50.6775	1.62968	0.814840	+	0.579686i	$0.196825\pi$
$968$	−2.78007	−0.0893548
$969$	0	0
$970$	−50.5034	−1.62157
$971$	−8.83039	−0.283381	−0.141690	−	0.989911i	$-0.545254\pi$
$971$	−8.83039	−0.283381	−0.141690	+	0.989911i	$0.545254\pi$
$972$	0	0
$973$	4.50002	0.144264
$974$	−54.5212	−1.74697
$975$	0	0
$976$	−0.771960	−0.0247098
$977$	−32.6529	−1.04466	−0.522329	−	0.852744i	$-0.674937\pi$
$977$	−32.6529	−1.04466	−0.522329	+	0.852744i	$0.674937\pi$
$978$	0	0
$979$	54.2038	1.73236
$980$	3.36022	0.107338
$981$	0	0
$982$	−30.5155	−0.973788
$983$	22.6204	0.721479	0.360739	−	0.932667i	$-0.382524\pi$
$983$	22.6204	0.721479	0.360739	+	0.932667i	$0.382524\pi$
$984$	0	0
$985$	26.1200	0.832254
$986$	23.4636	0.747234
$987$	0	0
$988$	−0.789049	−0.0251030
$989$	36.2758	1.15350
$990$	0	0
$991$	−15.6864	−0.498296	−0.249148	−	0.968465i	$-0.580151\pi$
$991$	−15.6864	−0.498296	−0.249148	+	0.968465i	$0.580151\pi$
$992$	−9.65802	−0.306642
$993$	0	0
$994$	16.5989	0.526485
$995$	14.1209	0.447662
$996$	0	0
$997$	42.8762	1.35790	0.678951	−	0.734183i	$-0.262435\pi$
$997$	42.8762	1.35790	0.678951	+	0.734183i	$0.262435\pi$
$998$	−22.4414	−0.710370
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.r.1.8		23
3.2	odd	2		893.2.a.d.1.16	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.d.1.16	✓	23	3.2	odd	2
8037.2.a.r.1.8		23	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:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q-1.28005 q^{2} -0.361479 q^{4} +4.05982 q^{5} -2.17032 q^{7} +3.02280 q^{8} +O(q^{10})\)	\(q-1.28005 q^{2} -0.361479 q^{4} +4.05982 q^{5} -2.17032 q^{7} +3.02280 q^{8} -5.19677 q^{10} -3.17495 q^{11} +2.18283 q^{13} +2.77812 q^{14} -3.14637 q^{16} -2.58799 q^{17} +1.00000 q^{19} -1.46754 q^{20} +4.06408 q^{22} +7.11637 q^{23} +11.4822 q^{25} -2.79413 q^{26} +0.784527 q^{28} +7.08282 q^{29} +4.78570 q^{31} -2.01810 q^{32} +3.31275 q^{34} -8.81113 q^{35} -5.35811 q^{37} -1.28005 q^{38} +12.2721 q^{40} -2.87373 q^{41} +5.09752 q^{43} +1.14768 q^{44} -9.10928 q^{46} +1.00000 q^{47} -2.28969 q^{49} -14.6977 q^{50} -0.789049 q^{52} -11.4217 q^{53} -12.8897 q^{55} -6.56047 q^{56} -9.06634 q^{58} +12.0536 q^{59} +0.245349 q^{61} -6.12592 q^{62} +8.87601 q^{64} +8.86192 q^{65} +7.94060 q^{67} +0.935505 q^{68} +11.2787 q^{70} +5.97488 q^{71} -1.67259 q^{73} +6.85863 q^{74} -0.361479 q^{76} +6.89067 q^{77} -10.8832 q^{79} -12.7737 q^{80} +3.67851 q^{82} -9.35097 q^{83} -10.5068 q^{85} -6.52507 q^{86} -9.59725 q^{88} -17.0723 q^{89} -4.73745 q^{91} -2.57242 q^{92} -1.28005 q^{94} +4.05982 q^{95} +9.71823 q^{97} +2.93092 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - q^{2} + 31 q^{4} - q^{5} + 15 q^{7}+O(q^{10}) \) 23 * q - q^2 + 31 * q^4 - q^5 + 15 * q^7	\( 23 q - q^{2} + 31 q^{4} - q^{5} + 15 q^{7} + 5 q^{10} - 2 q^{11} + 13 q^{13} - 6 q^{14} + 35 q^{16} - 12 q^{17} + 23 q^{19} - 3 q^{20} - 4 q^{22} - 13 q^{23} + 46 q^{25} + 7 q^{26} + 11 q^{28} + 18 q^{29} + 22 q^{31} + 4 q^{32} - 20 q^{34} - 25 q^{35} + 8 q^{37} - q^{38} - 16 q^{40} + 16 q^{41} + 68 q^{43} + 18 q^{44} + 13 q^{46} + 23 q^{47} + 52 q^{49} + 21 q^{50} + 54 q^{52} + 7 q^{53} + 32 q^{55} - 33 q^{56} + 6 q^{58} - 10 q^{59} + 28 q^{61} + 24 q^{62} + 40 q^{64} + 18 q^{65} + 55 q^{67} - 41 q^{68} - 40 q^{70} + 3 q^{71} + 48 q^{73} + 55 q^{74} + 31 q^{76} + 14 q^{77} - 31 q^{79} + 3 q^{80} + 48 q^{82} - 47 q^{83} - 5 q^{85} + 71 q^{86} + 9 q^{88} + 6 q^{89} + 40 q^{91} - 35 q^{92} - q^{94} - q^{95} - 18 q^{97} + 68 q^{98}+O(q^{100}) \) 23 * q - q^2 + 31 * q^4 - q^5 + 15 * q^7 + 5 * q^10 - 2 * q^11 + 13 * q^13 - 6 * q^14 + 35 * q^16 - 12 * q^17 + 23 * q^19 - 3 * q^20 - 4 * q^22 - 13 * q^23 + 46 * q^25 + 7 * q^26 + 11 * q^28 + 18 * q^29 + 22 * q^31 + 4 * q^32 - 20 * q^34 - 25 * q^35 + 8 * q^37 - q^38 - 16 * q^40 + 16 * q^41 + 68 * q^43 + 18 * q^44 + 13 * q^46 + 23 * q^47 + 52 * q^49 + 21 * q^50 + 54 * q^52 + 7 * q^53 + 32 * q^55 - 33 * q^56 + 6 * q^58 - 10 * q^59 + 28 * q^61 + 24 * q^62 + 40 * q^64 + 18 * q^65 + 55 * q^67 - 41 * q^68 - 40 * q^70 + 3 * q^71 + 48 * q^73 + 55 * q^74 + 31 * q^76 + 14 * q^77 - 31 * q^79 + 3 * q^80 + 48 * q^82 - 47 * q^83 - 5 * q^85 + 71 * q^86 + 9 * q^88 + 6 * q^89 + 40 * q^91 - 35 * q^92 - q^94 - q^95 - 18 * q^97 + 68 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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