LMFDB - Embedded newform 6044.2.a.b.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6044,2,Mod(1,6044)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6044, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6044.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6044 = 2^{2} \cdot 1511 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6044.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:	$48.2615829817$
Analytic rank:	$0$
Dimension:	$63$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	6044.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69290 q^{3} -2.32986 q^{5} -1.27542 q^{7} +4.25169 q^{9} +O(q^{10})$	$q-2.69290 q^{3} -2.32986 q^{5} -1.27542 q^{7} +4.25169 q^{9} +4.79770 q^{11} -2.77694 q^{13} +6.27407 q^{15} +0.827835 q^{17} -2.18324 q^{19} +3.43456 q^{21} +0.943429 q^{23} +0.428249 q^{25} -3.37066 q^{27} +5.95542 q^{29} -1.77507 q^{31} -12.9197 q^{33} +2.97154 q^{35} +1.36507 q^{37} +7.47801 q^{39} +5.18688 q^{41} -7.44663 q^{43} -9.90583 q^{45} -5.53106 q^{47} -5.37331 q^{49} -2.22927 q^{51} +10.0806 q^{53} -11.1780 q^{55} +5.87924 q^{57} +9.00966 q^{59} -9.70088 q^{61} -5.42267 q^{63} +6.46989 q^{65} -6.55332 q^{67} -2.54056 q^{69} +11.6341 q^{71} -8.41140 q^{73} -1.15323 q^{75} -6.11906 q^{77} -2.65091 q^{79} -3.67823 q^{81} -6.26750 q^{83} -1.92874 q^{85} -16.0373 q^{87} -16.5255 q^{89} +3.54176 q^{91} +4.78006 q^{93} +5.08665 q^{95} -1.58095 q^{97} +20.3983 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) $ 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9	$ 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9} + 21 q^{11} + 17 q^{13} + 26 q^{15} - 5 q^{17} + 57 q^{19} + 18 q^{21} + 16 q^{23} + 60 q^{25} + 14 q^{27} + 22 q^{29} + 36 q^{31} - q^{33} + 28 q^{35} + 21 q^{37} + 69 q^{39} - 3 q^{41} + 86 q^{43} + 39 q^{45} + 23 q^{47} + 63 q^{49} + 67 q^{51} + 18 q^{53} + 67 q^{55} + 27 q^{59} + 62 q^{61} + 58 q^{63} - 13 q^{65} + 62 q^{67} + 45 q^{69} + 29 q^{71} - q^{73} + 39 q^{75} + 19 q^{77} + 132 q^{79} + 51 q^{81} + 35 q^{83} + 60 q^{85} + 55 q^{87} - 2 q^{89} + 84 q^{91} + 29 q^{93} + 53 q^{95} - 10 q^{97} + 95 q^{99}+O(q^{100}) $ 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9 + 21 * q^11 + 17 * q^13 + 26 * q^15 - 5 * q^17 + 57 * q^19 + 18 * q^21 + 16 * q^23 + 60 * q^25 + 14 * q^27 + 22 * q^29 + 36 * q^31 - q^33 + 28 * q^35 + 21 * q^37 + 69 * q^39 - 3 * q^41 + 86 * q^43 + 39 * q^45 + 23 * q^47 + 63 * q^49 + 67 * q^51 + 18 * q^53 + 67 * q^55 + 27 * q^59 + 62 * q^61 + 58 * q^63 - 13 * q^65 + 62 * q^67 + 45 * q^69 + 29 * q^71 - q^73 + 39 * q^75 + 19 * q^77 + 132 * q^79 + 51 * q^81 + 35 * q^83 + 60 * q^85 + 55 * q^87 - 2 * q^89 + 84 * q^91 + 29 * q^93 + 53 * q^95 - 10 * q^97 + 95 * 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
$3$	−2.69290	−1.55474	−0.777372	−	0.629041i	$-0.783448\pi$
$3$	−2.69290	−1.55474	−0.777372	+	0.629041i	$0.783448\pi$
$4$	0	0
$5$	−2.32986	−1.04195	−0.520973	−	0.853573i	$-0.674431\pi$
$5$	−2.32986	−1.04195	−0.520973	+	0.853573i	$0.674431\pi$
$6$	0	0
$7$	−1.27542	−0.482062	−0.241031	−	0.970517i	$-0.577486\pi$
$7$	−1.27542	−0.482062	−0.241031	+	0.970517i	$0.577486\pi$
$8$	0	0
$9$	4.25169	1.41723
$10$	0	0
$11$	4.79770	1.44656	0.723280	−	0.690555i	$-0.242634\pi$
$11$	4.79770	1.44656	0.723280	+	0.690555i	$0.242634\pi$
$12$	0	0
$13$	−2.77694	−0.770185	−0.385093	−	0.922878i	$-0.625830\pi$
$13$	−2.77694	−0.770185	−0.385093	+	0.922878i	$0.625830\pi$
$14$	0	0
$15$	6.27407	1.61996
$16$	0	0
$17$	0.827835	0.200780	0.100390	−	0.994948i	$-0.467991\pi$
$17$	0.827835	0.200780	0.100390	+	0.994948i	$0.467991\pi$
$18$	0	0
$19$	−2.18324	−0.500870	−0.250435	−	0.968133i	$-0.580574\pi$
$19$	−2.18324	−0.500870	−0.250435	+	0.968133i	$0.580574\pi$
$20$	0	0
$21$	3.43456	0.749483
$22$	0	0
$23$	0.943429	0.196719	0.0983593	−	0.995151i	$-0.468641\pi$
$23$	0.943429	0.196719	0.0983593	+	0.995151i	$0.468641\pi$
$24$	0	0
$25$	0.428249	0.0856497
$26$	0	0
$27$	−3.37066	−0.648683
$28$	0	0
$29$	5.95542	1.10589	0.552947	−	0.833216i	$-0.313503\pi$
$29$	5.95542	1.10589	0.552947	+	0.833216i	$0.313503\pi$
$30$	0	0
$31$	−1.77507	−0.318811	−0.159406	−	0.987213i	$-0.550958\pi$
$31$	−1.77507	−0.318811	−0.159406	+	0.987213i	$0.550958\pi$
$32$	0	0
$33$	−12.9197	−2.24903
$34$	0	0
$35$	2.97154	0.502282
$36$	0	0
$37$	1.36507	0.224417	0.112208	−	0.993685i	$-0.464208\pi$
$37$	1.36507	0.224417	0.112208	+	0.993685i	$0.464208\pi$
$38$	0	0
$39$	7.47801	1.19744
$40$	0	0
$41$	5.18688	0.810054	0.405027	−	0.914305i	$-0.367262\pi$
$41$	5.18688	0.810054	0.405027	+	0.914305i	$0.367262\pi$
$42$	0	0
$43$	−7.44663	−1.13560	−0.567800	−	0.823166i	$-0.692205\pi$
$43$	−7.44663	−1.13560	−0.567800	+	0.823166i	$0.692205\pi$
$44$	0	0
$45$	−9.90583	−1.47667
$46$	0	0
$47$	−5.53106	−0.806788	−0.403394	−	0.915026i	$-0.632170\pi$
$47$	−5.53106	−0.806788	−0.403394	+	0.915026i	$0.632170\pi$
$48$	0	0
$49$	−5.37331	−0.767616
$50$	0	0
$51$	−2.22927	−0.312161
$52$	0	0
$53$	10.0806	1.38468	0.692341	−	0.721570i	$-0.256579\pi$
$53$	10.0806	1.38468	0.692341	+	0.721570i	$0.256579\pi$
$54$	0	0
$55$	−11.1780	−1.50724
$56$	0	0
$57$	5.87924	0.778725
$58$	0	0
$59$	9.00966	1.17296	0.586479	−	0.809964i	$-0.300514\pi$
$59$	9.00966	1.17296	0.586479	+	0.809964i	$0.300514\pi$
$60$	0	0
$61$	−9.70088	−1.24207	−0.621035	−	0.783782i	$-0.713288\pi$
$61$	−9.70088	−1.24207	−0.621035	+	0.783782i	$0.713288\pi$
$62$	0	0
$63$	−5.42267	−0.683192
$64$	0	0
$65$	6.46989	0.802491
$66$	0	0
$67$	−6.55332	−0.800616	−0.400308	−	0.916381i	$-0.631097\pi$
$67$	−6.55332	−0.800616	−0.400308	+	0.916381i	$0.631097\pi$
$68$	0	0
$69$	−2.54056	−0.305847
$70$	0	0
$71$	11.6341	1.38071	0.690354	−	0.723472i	$-0.257455\pi$
$71$	11.6341	1.38071	0.690354	+	0.723472i	$0.257455\pi$
$72$	0	0
$73$	−8.41140	−0.984480	−0.492240	−	0.870460i	$-0.663822\pi$
$73$	−8.41140	−0.984480	−0.492240	+	0.870460i	$0.663822\pi$
$74$	0	0
$75$	−1.15323	−0.133163
$76$	0	0
$77$	−6.11906	−0.697331
$78$	0	0
$79$	−2.65091	−0.298251	−0.149125	−	0.988818i	$-0.547646\pi$
$79$	−2.65091	−0.298251	−0.149125	+	0.988818i	$0.547646\pi$
$80$	0	0
$81$	−3.67823	−0.408692
$82$	0	0
$83$	−6.26750	−0.687948	−0.343974	−	0.938979i	$-0.611773\pi$
$83$	−6.26750	−0.687948	−0.343974	+	0.938979i	$0.611773\pi$
$84$	0	0
$85$	−1.92874	−0.209201
$86$	0	0
$87$	−16.0373	−1.71938
$88$	0	0
$89$	−16.5255	−1.75170	−0.875852	−	0.482580i	$-0.839700\pi$
$89$	−16.5255	−1.75170	−0.875852	+	0.482580i	$0.839700\pi$
$90$	0	0
$91$	3.54176	0.371277
$92$	0	0
$93$	4.78006	0.495670
$94$	0	0
$95$	5.08665	0.521879
$96$	0	0
$97$	−1.58095	−0.160521	−0.0802606	−	0.996774i	$-0.525575\pi$
$97$	−1.58095	−0.160521	−0.0802606	+	0.996774i	$0.525575\pi$
$98$	0	0
$99$	20.3983	2.05011
$100$	0	0
$101$	−7.18540	−0.714974	−0.357487	−	0.933918i	$-0.616366\pi$
$101$	−7.18540	−0.714974	−0.357487	+	0.933918i	$0.616366\pi$
$102$	0	0
$103$	−1.70351	−0.167852	−0.0839259	−	0.996472i	$-0.526746\pi$
$103$	−1.70351	−0.167852	−0.0839259	+	0.996472i	$0.526746\pi$
$104$	0	0
$105$	−8.00205	−0.780920
$106$	0	0
$107$	−3.79232	−0.366617	−0.183309	−	0.983055i	$-0.558681\pi$
$107$	−3.79232	−0.366617	−0.183309	+	0.983055i	$0.558681\pi$
$108$	0	0
$109$	−5.10392	−0.488867	−0.244433	−	0.969666i	$-0.578602\pi$
$109$	−5.10392	−0.488867	−0.244433	+	0.969666i	$0.578602\pi$
$110$	0	0
$111$	−3.67600	−0.348911
$112$	0	0
$113$	−4.46958	−0.420463	−0.210231	−	0.977652i	$-0.567422\pi$
$113$	−4.46958	−0.420463	−0.210231	+	0.977652i	$0.567422\pi$
$114$	0	0
$115$	−2.19806	−0.204970
$116$	0	0
$117$	−11.8067	−1.09153
$118$	0	0
$119$	−1.05583	−0.0967882
$120$	0	0
$121$	12.0179	1.09254
$122$	0	0
$123$	−13.9677	−1.25943
$124$	0	0
$125$	10.6515	0.952703
$126$	0	0
$127$	−7.56365	−0.671166	−0.335583	−	0.942011i	$-0.608933\pi$
$127$	−7.56365	−0.671166	−0.335583	+	0.942011i	$0.608933\pi$
$128$	0	0
$129$	20.0530	1.76557
$130$	0	0
$131$	−2.40705	−0.210305	−0.105153	−	0.994456i	$-0.533533\pi$
$131$	−2.40705	−0.210305	−0.105153	+	0.994456i	$0.533533\pi$
$132$	0	0
$133$	2.78454	0.241450
$134$	0	0
$135$	7.85316	0.675892
$136$	0	0
$137$	2.92776	0.250135	0.125068	−	0.992148i	$-0.460085\pi$
$137$	2.92776	0.250135	0.125068	+	0.992148i	$0.460085\pi$
$138$	0	0
$139$	0.672992	0.0570824	0.0285412	−	0.999593i	$-0.490914\pi$
$139$	0.672992	0.0570824	0.0285412	+	0.999593i	$0.490914\pi$
$140$	0	0
$141$	14.8946	1.25435
$142$	0	0
$143$	−13.3229	−1.11412
$144$	0	0
$145$	−13.8753	−1.15228
$146$	0	0
$147$	14.4698	1.19345
$148$	0	0
$149$	−0.256676	−0.0210277	−0.0105139	−	0.999945i	$-0.503347\pi$
$149$	−0.256676	−0.0210277	−0.0105139	+	0.999945i	$0.503347\pi$
$150$	0	0
$151$	18.6726	1.51955	0.759776	−	0.650185i	$-0.225309\pi$
$151$	18.6726	1.51955	0.759776	+	0.650185i	$0.225309\pi$
$152$	0	0
$153$	3.51970	0.284551
$154$	0	0
$155$	4.13565	0.332184
$156$	0	0
$157$	−19.5705	−1.56189	−0.780947	−	0.624598i	$-0.785263\pi$
$157$	−19.5705	−1.56189	−0.780947	+	0.624598i	$0.785263\pi$
$158$	0	0
$159$	−27.1461	−2.15283
$160$	0	0
$161$	−1.20326	−0.0948306
$162$	0	0
$163$	17.2669	1.35245	0.676223	−	0.736697i	$-0.263615\pi$
$163$	17.2669	1.35245	0.676223	+	0.736697i	$0.263615\pi$
$164$	0	0
$165$	30.1011	2.34337
$166$	0	0
$167$	−10.0915	−0.780905	−0.390453	−	0.920623i	$-0.627681\pi$
$167$	−10.0915	−0.780905	−0.390453	+	0.920623i	$0.627681\pi$
$168$	0	0
$169$	−5.28859	−0.406815
$170$	0	0
$171$	−9.28246	−0.709848
$172$	0	0
$173$	6.49667	0.493932	0.246966	−	0.969024i	$-0.420566\pi$
$173$	6.49667	0.493932	0.246966	+	0.969024i	$0.420566\pi$
$174$	0	0
$175$	−0.546195	−0.0412885
$176$	0	0
$177$	−24.2621	−1.82365
$178$	0	0
$179$	13.9764	1.04464	0.522322	−	0.852748i	$-0.325066\pi$
$179$	13.9764	1.04464	0.522322	+	0.852748i	$0.325066\pi$
$180$	0	0
$181$	20.6685	1.53628	0.768140	−	0.640282i	$-0.221182\pi$
$181$	20.6685	1.53628	0.768140	+	0.640282i	$0.221182\pi$
$182$	0	0
$183$	26.1235	1.93110
$184$	0	0
$185$	−3.18043	−0.233830
$186$	0	0
$187$	3.97170	0.290440
$188$	0	0
$189$	4.29899	0.312705
$190$	0	0
$191$	−0.739804	−0.0535303	−0.0267652	−	0.999642i	$-0.508521\pi$
$191$	−0.739804	−0.0535303	−0.0267652	+	0.999642i	$0.508521\pi$
$192$	0	0
$193$	11.7333	0.844579	0.422290	−	0.906461i	$-0.361226\pi$
$193$	11.7333	0.844579	0.422290	+	0.906461i	$0.361226\pi$
$194$	0	0
$195$	−17.4227	−1.24767
$196$	0	0
$197$	−5.29387	−0.377173	−0.188586	−	0.982057i	$-0.560391\pi$
$197$	−5.29387	−0.377173	−0.188586	+	0.982057i	$0.560391\pi$
$198$	0	0
$199$	12.5144	0.887124	0.443562	−	0.896244i	$-0.353715\pi$
$199$	12.5144	0.887124	0.443562	+	0.896244i	$0.353715\pi$
$200$	0	0
$201$	17.6474	1.24475
$202$	0	0
$203$	−7.59564	−0.533110
$204$	0	0
$205$	−12.0847	−0.844032
$206$	0	0
$207$	4.01117	0.278795
$208$	0	0
$209$	−10.4745	−0.724539
$210$	0	0
$211$	21.4781	1.47861	0.739306	−	0.673370i	$-0.235154\pi$
$211$	21.4781	1.47861	0.739306	+	0.673370i	$0.235154\pi$
$212$	0	0
$213$	−31.3293	−2.14665
$214$	0	0
$215$	17.3496	1.18323
$216$	0	0
$217$	2.26395	0.153687
$218$	0	0
$219$	22.6510	1.53061
$220$	0	0
$221$	−2.29885	−0.154637
$222$	0	0
$223$	10.4454	0.699479	0.349739	−	0.936847i	$-0.386270\pi$
$223$	10.4454	0.699479	0.349739	+	0.936847i	$0.386270\pi$
$224$	0	0
$225$	1.82078	0.121385
$226$	0	0
$227$	−13.3394	−0.885367	−0.442684	−	0.896678i	$-0.645973\pi$
$227$	−13.3394	−0.885367	−0.442684	+	0.896678i	$0.645973\pi$
$228$	0	0
$229$	12.1433	0.802454	0.401227	−	0.915979i	$-0.368584\pi$
$229$	12.1433	0.802454	0.401227	+	0.915979i	$0.368584\pi$
$230$	0	0
$231$	16.4780	1.08417
$232$	0	0
$233$	−11.7759	−0.771462	−0.385731	−	0.922611i	$-0.626051\pi$
$233$	−11.7759	−0.771462	−0.385731	+	0.922611i	$0.626051\pi$
$234$	0	0
$235$	12.8866	0.840629
$236$	0	0
$237$	7.13862	0.463703
$238$	0	0
$239$	−20.2618	−1.31063	−0.655313	−	0.755358i	$-0.727463\pi$
$239$	−20.2618	−1.31063	−0.655313	+	0.755358i	$0.727463\pi$
$240$	0	0
$241$	−8.54687	−0.550552	−0.275276	−	0.961365i	$-0.588769\pi$
$241$	−8.54687	−0.550552	−0.275276	+	0.961365i	$0.588769\pi$
$242$	0	0
$243$	20.0171	1.28409
$244$	0	0
$245$	12.5191	0.799814
$246$	0	0
$247$	6.06274	0.385763
$248$	0	0
$249$	16.8777	1.06958
$250$	0	0
$251$	6.85822	0.432887	0.216444	−	0.976295i	$-0.430554\pi$
$251$	6.85822	0.432887	0.216444	+	0.976295i	$0.430554\pi$
$252$	0	0
$253$	4.52629	0.284565
$254$	0	0
$255$	5.19390	0.325254
$256$	0	0
$257$	9.60245	0.598984	0.299492	−	0.954099i	$-0.403183\pi$
$257$	9.60245	0.598984	0.299492	+	0.954099i	$0.403183\pi$
$258$	0	0
$259$	−1.74104	−0.108183
$260$	0	0
$261$	25.3206	1.56731
$262$	0	0
$263$	13.7178	0.845874	0.422937	−	0.906159i	$-0.360999\pi$
$263$	13.7178	0.845874	0.422937	+	0.906159i	$0.360999\pi$
$264$	0	0
$265$	−23.4865	−1.44276
$266$	0	0
$267$	44.5016	2.72345
$268$	0	0
$269$	−7.37958	−0.449941	−0.224971	−	0.974366i	$-0.572229\pi$
$269$	−7.37958	−0.449941	−0.224971	+	0.974366i	$0.572229\pi$
$270$	0	0
$271$	−2.16812	−0.131704	−0.0658520	−	0.997829i	$-0.520977\pi$
$271$	−2.16812	−0.131704	−0.0658520	+	0.997829i	$0.520977\pi$
$272$	0	0
$273$	−9.53758	−0.577240
$274$	0	0
$275$	2.05461	0.123897
$276$	0	0
$277$	4.73220	0.284330	0.142165	−	0.989843i	$-0.454594\pi$
$277$	4.73220	0.284330	0.142165	+	0.989843i	$0.454594\pi$
$278$	0	0
$279$	−7.54702	−0.451828
$280$	0	0
$281$	26.2665	1.56693	0.783463	−	0.621438i	$-0.213451\pi$
$281$	26.2665	1.56693	0.783463	+	0.621438i	$0.213451\pi$
$282$	0	0
$283$	32.8431	1.95232	0.976158	−	0.217059i	$-0.0696464\pi$
$283$	32.8431	1.95232	0.976158	+	0.217059i	$0.0696464\pi$
$284$	0	0
$285$	−13.6978	−0.811389
$286$	0	0
$287$	−6.61543	−0.390496
$288$	0	0
$289$	−16.3147	−0.959688
$290$	0	0
$291$	4.25734	0.249569
$292$	0	0
$293$	6.83631	0.399381	0.199691	−	0.979859i	$-0.436006\pi$
$293$	6.83631	0.399381	0.199691	+	0.979859i	$0.436006\pi$
$294$	0	0
$295$	−20.9912	−1.22216
$296$	0	0
$297$	−16.1714	−0.938359
$298$	0	0
$299$	−2.61985	−0.151510
$300$	0	0
$301$	9.49755	0.547430
$302$	0	0
$303$	19.3495	1.11160
$304$	0	0
$305$	22.6017	1.29417
$306$	0	0
$307$	21.4675	1.22521	0.612606	−	0.790388i	$-0.290121\pi$
$307$	21.4675	1.22521	0.612606	+	0.790388i	$0.290121\pi$
$308$	0	0
$309$	4.58737	0.260966
$310$	0	0
$311$	5.21018	0.295442	0.147721	−	0.989029i	$-0.452806\pi$
$311$	5.21018	0.295442	0.147721	+	0.989029i	$0.452806\pi$
$312$	0	0
$313$	9.10553	0.514675	0.257337	−	0.966322i	$-0.417155\pi$
$313$	9.10553	0.514675	0.257337	+	0.966322i	$0.417155\pi$
$314$	0	0
$315$	12.6341	0.711848
$316$	0	0
$317$	−24.5434	−1.37849	−0.689247	−	0.724527i	$-0.742058\pi$
$317$	−24.5434	−1.37849	−0.689247	+	0.724527i	$0.742058\pi$
$318$	0	0
$319$	28.5723	1.59974
$320$	0	0
$321$	10.2123	0.569996
$322$	0	0
$323$	−1.80737	−0.100565
$324$	0	0
$325$	−1.18922	−0.0659661
$326$	0	0
$327$	13.7443	0.760062
$328$	0	0
$329$	7.05440	0.388922
$330$	0	0
$331$	6.83276	0.375562	0.187781	−	0.982211i	$-0.439870\pi$
$331$	6.83276	0.375562	0.187781	+	0.982211i	$0.439870\pi$
$332$	0	0
$333$	5.80387	0.318050
$334$	0	0
$335$	15.2683	0.834198
$336$	0	0
$337$	16.8719	0.919074	0.459537	−	0.888159i	$-0.348015\pi$
$337$	16.8719	0.919074	0.459537	+	0.888159i	$0.348015\pi$
$338$	0	0
$339$	12.0361	0.653711
$340$	0	0
$341$	−8.51623	−0.461179
$342$	0	0
$343$	15.7811	0.852100
$344$	0	0
$345$	5.91914	0.318676
$346$	0	0
$347$	20.7466	1.11373	0.556867	−	0.830602i	$-0.312003\pi$
$347$	20.7466	1.11373	0.556867	+	0.830602i	$0.312003\pi$
$348$	0	0
$349$	10.8248	0.579438	0.289719	−	0.957112i	$-0.406438\pi$
$349$	10.8248	0.579438	0.289719	+	0.957112i	$0.406438\pi$
$350$	0	0
$351$	9.36012	0.499606
$352$	0	0
$353$	−33.3155	−1.77321	−0.886603	−	0.462530i	$-0.846942\pi$
$353$	−33.3155	−1.77321	−0.886603	+	0.462530i	$0.846942\pi$
$354$	0	0
$355$	−27.1057	−1.43862
$356$	0	0
$357$	2.84325	0.150481
$358$	0	0
$359$	−10.4814	−0.553188	−0.276594	−	0.960987i	$-0.589206\pi$
$359$	−10.4814	−0.553188	−0.276594	+	0.960987i	$0.589206\pi$
$360$	0	0
$361$	−14.2334	−0.749129
$362$	0	0
$363$	−32.3630	−1.69861
$364$	0	0
$365$	19.5974	1.02577
$366$	0	0
$367$	1.75054	0.0913773	0.0456887	−	0.998956i	$-0.485452\pi$
$367$	1.75054	0.0913773	0.0456887	+	0.998956i	$0.485452\pi$
$368$	0	0
$369$	22.0530	1.14803
$370$	0	0
$371$	−12.8570	−0.667503
$372$	0	0
$373$	−16.1038	−0.833821	−0.416911	−	0.908947i	$-0.636887\pi$
$373$	−16.1038	−0.833821	−0.416911	+	0.908947i	$0.636887\pi$
$374$	0	0
$375$	−28.6835	−1.48121
$376$	0	0
$377$	−16.5379	−0.851744
$378$	0	0
$379$	2.27266	0.116739	0.0583694	−	0.998295i	$-0.481410\pi$
$379$	2.27266	0.116739	0.0583694	+	0.998295i	$0.481410\pi$
$380$	0	0
$381$	20.3681	1.04349
$382$	0	0
$383$	13.7328	0.701713	0.350856	−	0.936429i	$-0.385891\pi$
$383$	13.7328	0.701713	0.350856	+	0.936429i	$0.385891\pi$
$384$	0	0
$385$	14.2566	0.726581
$386$	0	0
$387$	−31.6607	−1.60941
$388$	0	0
$389$	−4.48838	−0.227570	−0.113785	−	0.993505i	$-0.536297\pi$
$389$	−4.48838	−0.227570	−0.113785	+	0.993505i	$0.536297\pi$
$390$	0	0
$391$	0.781004	0.0394971
$392$	0	0
$393$	6.48195	0.326971
$394$	0	0
$395$	6.17625	0.310761
$396$	0	0
$397$	10.8100	0.542539	0.271270	−	0.962503i	$-0.412557\pi$
$397$	10.8100	0.542539	0.271270	+	0.962503i	$0.412557\pi$
$398$	0	0
$399$	−7.49848	−0.375394
$400$	0	0
$401$	11.5163	0.575097	0.287549	−	0.957766i	$-0.407160\pi$
$401$	11.5163	0.575097	0.287549	+	0.957766i	$0.407160\pi$
$402$	0	0
$403$	4.92925	0.245544
$404$	0	0
$405$	8.56976	0.425835
$406$	0	0
$407$	6.54922	0.324633
$408$	0	0
$409$	2.22889	0.110211	0.0551057	−	0.998481i	$-0.482450\pi$
$409$	2.22889	0.110211	0.0551057	+	0.998481i	$0.482450\pi$
$410$	0	0
$411$	−7.88415	−0.388896
$412$	0	0
$413$	−11.4911	−0.565438
$414$	0	0
$415$	14.6024	0.716804
$416$	0	0
$417$	−1.81230	−0.0887485
$418$	0	0
$419$	16.8438	0.822872	0.411436	−	0.911439i	$-0.365027\pi$
$419$	16.8438	0.822872	0.411436	+	0.911439i	$0.365027\pi$
$420$	0	0
$421$	10.1105	0.492757	0.246379	−	0.969174i	$-0.420759\pi$
$421$	10.1105	0.492757	0.246379	+	0.969174i	$0.420759\pi$
$422$	0	0
$423$	−23.5163	−1.14340
$424$	0	0
$425$	0.354519	0.0171967
$426$	0	0
$427$	12.3727	0.598755
$428$	0	0
$429$	35.8773	1.73217
$430$	0	0
$431$	−3.31994	−0.159916	−0.0799580	−	0.996798i	$-0.525479\pi$
$431$	−3.31994	−0.159916	−0.0799580	+	0.996798i	$0.525479\pi$
$432$	0	0
$433$	−33.2961	−1.60011	−0.800055	−	0.599926i	$-0.795197\pi$
$433$	−33.2961	−1.60011	−0.800055	+	0.599926i	$0.795197\pi$
$434$	0	0
$435$	37.3647	1.79150
$436$	0	0
$437$	−2.05974	−0.0985305
$438$	0	0
$439$	10.5733	0.504638	0.252319	−	0.967644i	$-0.418807\pi$
$439$	10.5733	0.504638	0.252319	+	0.967644i	$0.418807\pi$
$440$	0	0
$441$	−22.8456	−1.08789
$442$	0	0
$443$	29.3935	1.39653	0.698263	−	0.715841i	$-0.253957\pi$
$443$	29.3935	1.39653	0.698263	+	0.715841i	$0.253957\pi$
$444$	0	0
$445$	38.5022	1.82518
$446$	0	0
$447$	0.691201	0.0326927
$448$	0	0
$449$	7.02174	0.331376	0.165688	−	0.986178i	$-0.447015\pi$
$449$	7.02174	0.331376	0.165688	+	0.986178i	$0.447015\pi$
$450$	0	0
$451$	24.8851	1.17179
$452$	0	0
$453$	−50.2832	−2.36251
$454$	0	0
$455$	−8.25180	−0.386850
$456$	0	0
$457$	−37.3694	−1.74807	−0.874035	−	0.485864i	$-0.838505\pi$
$457$	−37.3694	−1.74807	−0.874035	+	0.485864i	$0.838505\pi$
$458$	0	0
$459$	−2.79035	−0.130242
$460$	0	0
$461$	26.9809	1.25663	0.628313	−	0.777960i	$-0.283746\pi$
$461$	26.9809	1.25663	0.628313	+	0.777960i	$0.283746\pi$
$462$	0	0
$463$	14.4477	0.671443	0.335721	−	0.941961i	$-0.391020\pi$
$463$	14.4477	0.671443	0.335721	+	0.941961i	$0.391020\pi$
$464$	0	0
$465$	−11.1369	−0.516461
$466$	0	0
$467$	22.0434	1.02005	0.510023	−	0.860161i	$-0.329637\pi$
$467$	22.0434	1.02005	0.510023	+	0.860161i	$0.329637\pi$
$468$	0	0
$469$	8.35821	0.385946
$470$	0	0
$471$	52.7012	2.42834
$472$	0	0
$473$	−35.7267	−1.64272
$474$	0	0
$475$	−0.934971	−0.0428994
$476$	0	0
$477$	42.8597	1.96241
$478$	0	0
$479$	33.9102	1.54940	0.774699	−	0.632331i	$-0.217902\pi$
$479$	33.9102	1.54940	0.774699	+	0.632331i	$0.217902\pi$
$480$	0	0
$481$	−3.79073	−0.172843
$482$	0	0
$483$	3.24027	0.147437
$484$	0	0
$485$	3.68340	0.167254
$486$	0	0
$487$	−1.63780	−0.0742160	−0.0371080	−	0.999311i	$-0.511815\pi$
$487$	−1.63780	−0.0742160	−0.0371080	+	0.999311i	$0.511815\pi$
$488$	0	0
$489$	−46.4979	−2.10271
$490$	0	0
$491$	10.6999	0.482878	0.241439	−	0.970416i	$-0.422381\pi$
$491$	10.6999	0.482878	0.241439	+	0.970416i	$0.422381\pi$
$492$	0	0
$493$	4.93011	0.222041
$494$	0	0
$495$	−47.5252	−2.13610
$496$	0	0
$497$	−14.8383	−0.665587
$498$	0	0
$499$	−28.1693	−1.26103	−0.630515	−	0.776177i	$-0.717156\pi$
$499$	−28.1693	−1.26103	−0.630515	+	0.776177i	$0.717156\pi$
$500$	0	0
$501$	27.1754	1.21411
$502$	0	0
$503$	20.9137	0.932496	0.466248	−	0.884654i	$-0.345605\pi$
$503$	20.9137	0.932496	0.466248	+	0.884654i	$0.345605\pi$
$504$	0	0
$505$	16.7410	0.744964
$506$	0	0
$507$	14.2416	0.632493
$508$	0	0
$509$	12.3230	0.546207	0.273104	−	0.961985i	$-0.411950\pi$
$509$	12.3230	0.546207	0.273104	+	0.961985i	$0.411950\pi$
$510$	0	0
$511$	10.7280	0.474580
$512$	0	0
$513$	7.35896	0.324906
$514$	0	0
$515$	3.96894	0.174892
$516$	0	0
$517$	−26.5364	−1.16707
$518$	0	0
$519$	−17.4948	−0.767938
$520$	0	0
$521$	−6.94115	−0.304097	−0.152049	−	0.988373i	$-0.548587\pi$
$521$	−6.94115	−0.304097	−0.152049	+	0.988373i	$0.548587\pi$
$522$	0	0
$523$	−8.21401	−0.359174	−0.179587	−	0.983742i	$-0.557476\pi$
$523$	−8.21401	−0.359174	−0.179587	+	0.983742i	$0.557476\pi$
$524$	0	0
$525$	1.47085	0.0641930
$526$	0	0
$527$	−1.46946	−0.0640108
$528$	0	0
$529$	−22.1099	−0.961302
$530$	0	0
$531$	38.3062	1.66235
$532$	0	0
$533$	−14.4037	−0.623892
$534$	0	0
$535$	8.83557	0.381995
$536$	0	0
$537$	−37.6370	−1.62415
$538$	0	0
$539$	−25.7795	−1.11040
$540$	0	0
$541$	−0.916041	−0.0393837	−0.0196918	−	0.999806i	$-0.506269\pi$
$541$	−0.916041	−0.0393837	−0.0196918	+	0.999806i	$0.506269\pi$
$542$	0	0
$543$	−55.6582	−2.38852
$544$	0	0
$545$	11.8914	0.509372
$546$	0	0
$547$	3.64956	0.156044	0.0780220	−	0.996952i	$-0.475140\pi$
$547$	3.64956	0.156044	0.0780220	+	0.996952i	$0.475140\pi$
$548$	0	0
$549$	−41.2451	−1.76030
$550$	0	0
$551$	−13.0021	−0.553910
$552$	0	0
$553$	3.38101	0.143775
$554$	0	0
$555$	8.56457	0.363546
$556$	0	0
$557$	−13.3594	−0.566055	−0.283028	−	0.959112i	$-0.591339\pi$
$557$	−13.3594	−0.566055	−0.283028	+	0.959112i	$0.591339\pi$
$558$	0	0
$559$	20.6789	0.874623
$560$	0	0
$561$	−10.6954	−0.451559
$562$	0	0
$563$	−21.4172	−0.902627	−0.451314	−	0.892365i	$-0.649044\pi$
$563$	−21.4172	−0.902627	−0.451314	+	0.892365i	$0.649044\pi$
$564$	0	0
$565$	10.4135	0.438099
$566$	0	0
$567$	4.69127	0.197015
$568$	0	0
$569$	15.2833	0.640711	0.320356	−	0.947297i	$-0.396198\pi$
$569$	15.2833	0.640711	0.320356	+	0.947297i	$0.396198\pi$
$570$	0	0
$571$	−37.4915	−1.56897	−0.784486	−	0.620147i	$-0.787073\pi$
$571$	−37.4915	−1.56897	−0.784486	+	0.620147i	$0.787073\pi$
$572$	0	0
$573$	1.99221	0.0832260
$574$	0	0
$575$	0.404022	0.0168489
$576$	0	0
$577$	−43.9257	−1.82865	−0.914326	−	0.404979i	$-0.867279\pi$
$577$	−43.9257	−1.82865	−0.914326	+	0.404979i	$0.867279\pi$
$578$	0	0
$579$	−31.5965	−1.31310
$580$	0	0
$581$	7.99367	0.331633
$582$	0	0
$583$	48.3639	2.00303
$584$	0	0
$585$	27.5079	1.13731
$586$	0	0
$587$	−7.92252	−0.326997	−0.163499	−	0.986544i	$-0.552278\pi$
$587$	−7.92252	−0.326997	−0.163499	+	0.986544i	$0.552278\pi$
$588$	0	0
$589$	3.87540	0.159683
$590$	0	0
$591$	14.2558	0.586407
$592$	0	0
$593$	15.4515	0.634515	0.317258	−	0.948339i	$-0.397238\pi$
$593$	15.4515	0.634515	0.317258	+	0.948339i	$0.397238\pi$
$594$	0	0
$595$	2.45995	0.100848
$596$	0	0
$597$	−33.7001	−1.37925
$598$	0	0
$599$	28.5048	1.16467	0.582336	−	0.812948i	$-0.302139\pi$
$599$	28.5048	1.16467	0.582336	+	0.812948i	$0.302139\pi$
$600$	0	0
$601$	−17.0069	−0.693728	−0.346864	−	0.937916i	$-0.612753\pi$
$601$	−17.0069	−0.693728	−0.346864	+	0.937916i	$0.612753\pi$
$602$	0	0
$603$	−27.8627	−1.13466
$604$	0	0
$605$	−28.0000	−1.13836
$606$	0	0
$607$	−22.4578	−0.911536	−0.455768	−	0.890099i	$-0.650635\pi$
$607$	−22.4578	−0.911536	−0.455768	+	0.890099i	$0.650635\pi$
$608$	0	0
$609$	20.4543	0.828849
$610$	0	0
$611$	15.3594	0.621376
$612$	0	0
$613$	−18.6592	−0.753639	−0.376820	−	0.926287i	$-0.622982\pi$
$613$	−18.6592	−0.753639	−0.376820	+	0.926287i	$0.622982\pi$
$614$	0	0
$615$	32.5428	1.31225
$616$	0	0
$617$	−11.9558	−0.481324	−0.240662	−	0.970609i	$-0.577364\pi$
$617$	−11.9558	−0.481324	−0.240662	+	0.970609i	$0.577364\pi$
$618$	0	0
$619$	21.5950	0.867976	0.433988	−	0.900919i	$-0.357106\pi$
$619$	21.5950	0.867976	0.433988	+	0.900919i	$0.357106\pi$
$620$	0	0
$621$	−3.17998	−0.127608
$622$	0	0
$623$	21.0769	0.844430
$624$	0	0
$625$	−26.9578	−1.07831
$626$	0	0
$627$	28.2068	1.12647
$628$	0	0
$629$	1.13006	0.0450583
$630$	0	0
$631$	−21.6380	−0.861393	−0.430697	−	0.902497i	$-0.641732\pi$
$631$	−21.6380	−0.861393	−0.430697	+	0.902497i	$0.641732\pi$
$632$	0	0
$633$	−57.8382	−2.29886
$634$	0	0
$635$	17.6223	0.699318
$636$	0	0
$637$	14.9214	0.591207
$638$	0	0
$639$	49.4643	1.95678
$640$	0	0
$641$	−32.0498	−1.26589	−0.632946	−	0.774196i	$-0.718154\pi$
$641$	−32.0498	−1.26589	−0.632946	+	0.774196i	$0.718154\pi$
$642$	0	0
$643$	−21.7262	−0.856798	−0.428399	−	0.903590i	$-0.640922\pi$
$643$	−21.7262	−0.856798	−0.428399	+	0.903590i	$0.640922\pi$
$644$	0	0
$645$	−46.7207	−1.83963
$646$	0	0
$647$	18.0406	0.709247	0.354624	−	0.935009i	$-0.384609\pi$
$647$	18.0406	0.709247	0.354624	+	0.935009i	$0.384609\pi$
$648$	0	0
$649$	43.2256	1.69675
$650$	0	0
$651$	−6.09657	−0.238943
$652$	0	0
$653$	41.9291	1.64081	0.820406	−	0.571782i	$-0.193748\pi$
$653$	41.9291	1.64081	0.820406	+	0.571782i	$0.193748\pi$
$654$	0	0
$655$	5.60810	0.219127
$656$	0	0
$657$	−35.7626	−1.39523
$658$	0	0
$659$	−35.4317	−1.38023	−0.690113	−	0.723702i	$-0.742439\pi$
$659$	−35.4317	−1.38023	−0.690113	+	0.723702i	$0.742439\pi$
$660$	0	0
$661$	32.9626	1.28209	0.641047	−	0.767501i	$-0.278500\pi$
$661$	32.9626	1.28209	0.641047	+	0.767501i	$0.278500\pi$
$662$	0	0
$663$	6.19057	0.240422
$664$	0	0
$665$	−6.48759	−0.251578
$666$	0	0
$667$	5.61852	0.217550
$668$	0	0
$669$	−28.1285	−1.08751
$670$	0	0
$671$	−46.5419	−1.79673
$672$	0	0
$673$	−5.90789	−0.227732	−0.113866	−	0.993496i	$-0.536324\pi$
$673$	−5.90789	−0.227732	−0.113866	+	0.993496i	$0.536324\pi$
$674$	0	0
$675$	−1.44348	−0.0555595
$676$	0	0
$677$	−4.69632	−0.180494	−0.0902472	−	0.995919i	$-0.528766\pi$
$677$	−4.69632	−0.180494	−0.0902472	+	0.995919i	$0.528766\pi$
$678$	0	0
$679$	2.01637	0.0773812
$680$	0	0
$681$	35.9216	1.37652
$682$	0	0
$683$	−3.98827	−0.152607	−0.0763034	−	0.997085i	$-0.524312\pi$
$683$	−3.98827	−0.152607	−0.0763034	+	0.997085i	$0.524312\pi$
$684$	0	0
$685$	−6.82127	−0.260627
$686$	0	0
$687$	−32.7007	−1.24761
$688$	0	0
$689$	−27.9934	−1.06646
$690$	0	0
$691$	−4.51591	−0.171793	−0.0858966	−	0.996304i	$-0.527375\pi$
$691$	−4.51591	−0.171793	−0.0858966	+	0.996304i	$0.527375\pi$
$692$	0	0
$693$	−26.0163	−0.988278
$694$	0	0
$695$	−1.56798	−0.0594767
$696$	0	0
$697$	4.29388	0.162642
$698$	0	0
$699$	31.7111	1.19943
$700$	0	0
$701$	23.9095	0.903049	0.451524	−	0.892259i	$-0.350880\pi$
$701$	23.9095	0.903049	0.451524	+	0.892259i	$0.350880\pi$
$702$	0	0
$703$	−2.98029	−0.112404
$704$	0	0
$705$	−34.7023	−1.30696
$706$	0	0
$707$	9.16437	0.344662
$708$	0	0
$709$	−2.11298	−0.0793548	−0.0396774	−	0.999213i	$-0.512633\pi$
$709$	−2.11298	−0.0793548	−0.0396774	+	0.999213i	$0.512633\pi$
$710$	0	0
$711$	−11.2708	−0.422689
$712$	0	0
$713$	−1.67465	−0.0627161
$714$	0	0
$715$	31.0406	1.16085
$716$	0	0
$717$	54.5629	2.03769
$718$	0	0
$719$	−24.5609	−0.915967	−0.457984	−	0.888961i	$-0.651428\pi$
$719$	−24.5609	−0.915967	−0.457984	+	0.888961i	$0.651428\pi$
$720$	0	0
$721$	2.17268	0.0809149
$722$	0	0
$723$	23.0158	0.855968
$724$	0	0
$725$	2.55040	0.0947195
$726$	0	0
$727$	1.55032	0.0574981	0.0287490	−	0.999587i	$-0.490848\pi$
$727$	1.55032	0.0574981	0.0287490	+	0.999587i	$0.490848\pi$
$728$	0	0
$729$	−42.8692	−1.58775
$730$	0	0
$731$	−6.16459	−0.228005
$732$	0	0
$733$	50.2058	1.85439	0.927196	−	0.374575i	$-0.122211\pi$
$733$	50.2058	1.85439	0.927196	+	0.374575i	$0.122211\pi$
$734$	0	0
$735$	−33.7126	−1.24351
$736$	0	0
$737$	−31.4409	−1.15814
$738$	0	0
$739$	25.3783	0.933557	0.466778	−	0.884374i	$-0.345415\pi$
$739$	25.3783	0.933557	0.466778	+	0.884374i	$0.345415\pi$
$740$	0	0
$741$	−16.3263	−0.599762
$742$	0	0
$743$	−14.2868	−0.524130	−0.262065	−	0.965050i	$-0.584404\pi$
$743$	−14.2868	−0.524130	−0.262065	+	0.965050i	$0.584404\pi$
$744$	0	0
$745$	0.598019	0.0219097
$746$	0	0
$747$	−26.6475	−0.974979
$748$	0	0
$749$	4.83678	0.176732
$750$	0	0
$751$	22.8361	0.833302	0.416651	−	0.909066i	$-0.363204\pi$
$751$	22.8361	0.833302	0.416651	+	0.909066i	$0.363204\pi$
$752$	0	0
$753$	−18.4685	−0.673029
$754$	0	0
$755$	−43.5044	−1.58329
$756$	0	0
$757$	25.0600	0.910823	0.455411	−	0.890281i	$-0.349492\pi$
$757$	25.0600	0.910823	0.455411	+	0.890281i	$0.349492\pi$
$758$	0	0
$759$	−12.1888	−0.442426
$760$	0	0
$761$	−11.8936	−0.431143	−0.215572	−	0.976488i	$-0.569161\pi$
$761$	−11.8936	−0.431143	−0.215572	+	0.976488i	$0.569161\pi$
$762$	0	0
$763$	6.50962	0.235664
$764$	0	0
$765$	−8.20040	−0.296486
$766$	0	0
$767$	−25.0193	−0.903395
$768$	0	0
$769$	40.0073	1.44270	0.721350	−	0.692571i	$-0.243522\pi$
$769$	40.0073	1.44270	0.721350	+	0.692571i	$0.243522\pi$
$770$	0	0
$771$	−25.8584	−0.931267
$772$	0	0
$773$	33.8244	1.21658	0.608289	−	0.793715i	$-0.291856\pi$
$773$	33.8244	1.21658	0.608289	+	0.793715i	$0.291856\pi$
$774$	0	0
$775$	−0.760169	−0.0273061
$776$	0	0
$777$	4.68843	0.168197
$778$	0	0
$779$	−11.3242	−0.405732
$780$	0	0
$781$	55.8167	1.99728
$782$	0	0
$783$	−20.0737	−0.717375
$784$	0	0
$785$	45.5965	1.62741
$786$	0	0
$787$	22.1686	0.790224	0.395112	−	0.918633i	$-0.370706\pi$
$787$	22.1686	0.790224	0.395112	+	0.918633i	$0.370706\pi$
$788$	0	0
$789$	−36.9405	−1.31512
$790$	0	0
$791$	5.70057	0.202689
$792$	0	0
$793$	26.9388	0.956625
$794$	0	0
$795$	63.2467	2.24313
$796$	0	0
$797$	−42.6840	−1.51194	−0.755972	−	0.654604i	$-0.772835\pi$
$797$	−42.6840	−1.51194	−0.755972	+	0.654604i	$0.772835\pi$
$798$	0	0
$799$	−4.57881	−0.161987
$800$	0	0
$801$	−70.2614	−2.48256
$802$	0	0
$803$	−40.3553	−1.42411
$804$	0	0
$805$	2.80344	0.0988082
$806$	0	0
$807$	19.8724	0.699543
$808$	0	0
$809$	39.7301	1.39684	0.698418	−	0.715691i	$-0.253888\pi$
$809$	39.7301	1.39684	0.698418	+	0.715691i	$0.253888\pi$
$810$	0	0
$811$	25.4585	0.893968	0.446984	−	0.894542i	$-0.352498\pi$
$811$	25.4585	0.893968	0.446984	+	0.894542i	$0.352498\pi$
$812$	0	0
$813$	5.83852	0.204766
$814$	0	0
$815$	−40.2294	−1.40918
$816$	0	0
$817$	16.2578	0.568789
$818$	0	0
$819$	15.0584	0.526184
$820$	0	0
$821$	31.0407	1.08333	0.541663	−	0.840596i	$-0.317795\pi$
$821$	31.0407	1.08333	0.541663	+	0.840596i	$0.317795\pi$
$822$	0	0
$823$	3.09070	0.107735	0.0538675	−	0.998548i	$-0.482845\pi$
$823$	3.09070	0.107735	0.0538675	+	0.998548i	$0.482845\pi$
$824$	0	0
$825$	−5.53284	−0.192629
$826$	0	0
$827$	3.75642	0.130624	0.0653118	−	0.997865i	$-0.479196\pi$
$827$	3.75642	0.130624	0.0653118	+	0.997865i	$0.479196\pi$
$828$	0	0
$829$	1.05808	0.0367488	0.0183744	−	0.999831i	$-0.494151\pi$
$829$	1.05808	0.0367488	0.0183744	+	0.999831i	$0.494151\pi$
$830$	0	0
$831$	−12.7433	−0.442061
$832$	0	0
$833$	−4.44822	−0.154122
$834$	0	0
$835$	23.5118	0.813660
$836$	0	0
$837$	5.98314	0.206807
$838$	0	0
$839$	14.4879	0.500179	0.250089	−	0.968223i	$-0.419540\pi$
$839$	14.4879	0.500179	0.250089	+	0.968223i	$0.419540\pi$
$840$	0	0
$841$	6.46708	0.223003
$842$	0	0
$843$	−70.7329	−2.43617
$844$	0	0
$845$	12.3217	0.423879
$846$	0	0
$847$	−15.3278	−0.526670
$848$	0	0
$849$	−88.4429	−3.03535
$850$	0	0
$851$	1.28785	0.0441470
$852$	0	0
$853$	−0.561346	−0.0192201	−0.00961006	−	0.999954i	$-0.503059\pi$
$853$	−0.561346	−0.0192201	−0.00961006	+	0.999954i	$0.503059\pi$
$854$	0	0
$855$	21.6268	0.739622
$856$	0	0
$857$	−7.48017	−0.255518	−0.127759	−	0.991805i	$-0.540778\pi$
$857$	−7.48017	−0.255518	−0.127759	+	0.991805i	$0.540778\pi$
$858$	0	0
$859$	46.7693	1.59575	0.797873	−	0.602825i	$-0.205958\pi$
$859$	46.7693	1.59575	0.797873	+	0.602825i	$0.205958\pi$
$860$	0	0
$861$	17.8146	0.607122
$862$	0	0
$863$	−29.2694	−0.996341	−0.498170	−	0.867079i	$-0.665995\pi$
$863$	−29.2694	−0.996341	−0.498170	+	0.867079i	$0.665995\pi$
$864$	0	0
$865$	−15.1363	−0.514650
$866$	0	0
$867$	43.9337	1.49207
$868$	0	0
$869$	−12.7183	−0.431438
$870$	0	0
$871$	18.1982	0.616622
$872$	0	0
$873$	−6.72171	−0.227495
$874$	0	0
$875$	−13.5851	−0.459262
$876$	0	0
$877$	35.4190	1.19601	0.598007	−	0.801491i	$-0.295959\pi$
$877$	35.4190	1.19601	0.598007	+	0.801491i	$0.295959\pi$
$878$	0	0
$879$	−18.4095	−0.620935
$880$	0	0
$881$	49.9708	1.68356	0.841779	−	0.539823i	$-0.181509\pi$
$881$	49.9708	1.68356	0.841779	+	0.539823i	$0.181509\pi$
$882$	0	0
$883$	4.72749	0.159093	0.0795463	−	0.996831i	$-0.474653\pi$
$883$	4.72749	0.159093	0.0795463	+	0.996831i	$0.474653\pi$
$884$	0	0
$885$	56.5272	1.90014
$886$	0	0
$887$	13.5331	0.454396	0.227198	−	0.973849i	$-0.427044\pi$
$887$	13.5331	0.454396	0.227198	+	0.973849i	$0.427044\pi$
$888$	0	0
$889$	9.64680	0.323543
$890$	0	0
$891$	−17.6470	−0.591198
$892$	0	0
$893$	12.0757	0.404096
$894$	0	0
$895$	−32.5630	−1.08846
$896$	0	0
$897$	7.05498	0.235559
$898$	0	0
$899$	−10.5713	−0.352571
$900$	0	0
$901$	8.34511	0.278016
$902$	0	0
$903$	−25.5759	−0.851113
$904$	0	0
$905$	−48.1548	−1.60072
$906$	0	0
$907$	49.2806	1.63633	0.818167	−	0.574980i	$-0.194990\pi$
$907$	49.2806	1.63633	0.818167	+	0.574980i	$0.194990\pi$
$908$	0	0
$909$	−30.5501	−1.01328
$910$	0	0
$911$	40.8842	1.35455	0.677277	−	0.735728i	$-0.263160\pi$
$911$	40.8842	1.35455	0.677277	+	0.735728i	$0.263160\pi$
$912$	0	0
$913$	−30.0696	−0.995158
$914$	0	0
$915$	−60.8640	−2.01210
$916$	0	0
$917$	3.07000	0.101380
$918$	0	0
$919$	−23.5633	−0.777282	−0.388641	−	0.921389i	$-0.627055\pi$
$919$	−23.5633	−0.777282	−0.388641	+	0.921389i	$0.627055\pi$
$920$	0	0
$921$	−57.8096	−1.90489
$922$	0	0
$923$	−32.3071	−1.06340
$924$	0	0
$925$	0.584591	0.0192212
$926$	0	0
$927$	−7.24278	−0.237884
$928$	0	0
$929$	−11.7882	−0.386758	−0.193379	−	0.981124i	$-0.561945\pi$
$929$	−11.7882	−0.386758	−0.193379	+	0.981124i	$0.561945\pi$
$930$	0	0
$931$	11.7313	0.384476
$932$	0	0
$933$	−14.0305	−0.459337
$934$	0	0
$935$	−9.25352	−0.302622
$936$	0	0
$937$	−5.57797	−0.182224	−0.0911122	−	0.995841i	$-0.529042\pi$
$937$	−5.57797	−0.182224	−0.0911122	+	0.995841i	$0.529042\pi$
$938$	0	0
$939$	−24.5202	−0.800188
$940$	0	0
$941$	44.3663	1.44630	0.723151	−	0.690690i	$-0.242693\pi$
$941$	44.3663	1.44630	0.723151	+	0.690690i	$0.242693\pi$
$942$	0	0
$943$	4.89345	0.159353
$944$	0	0
$945$	−10.0160	−0.325822
$946$	0	0
$947$	−49.0061	−1.59248	−0.796242	−	0.604979i	$-0.793182\pi$
$947$	−49.0061	−1.59248	−0.796242	+	0.604979i	$0.793182\pi$
$948$	0	0
$949$	23.3580	0.758232
$950$	0	0
$951$	66.0927	2.14320
$952$	0	0
$953$	−9.16043	−0.296735	−0.148368	−	0.988932i	$-0.547402\pi$
$953$	−9.16043	−0.296735	−0.148368	+	0.988932i	$0.547402\pi$
$954$	0	0
$955$	1.72364	0.0557757
$956$	0	0
$957$	−76.9423	−2.48719
$958$	0	0
$959$	−3.73411	−0.120581
$960$	0	0
$961$	−27.8491	−0.898359
$962$	0	0
$963$	−16.1237	−0.519580
$964$	0	0
$965$	−27.3369	−0.880005
$966$	0	0
$967$	35.6235	1.14557	0.572787	−	0.819704i	$-0.305862\pi$
$967$	35.6235	1.14557	0.572787	+	0.819704i	$0.305862\pi$
$968$	0	0
$969$	4.86705	0.156352
$970$	0	0
$971$	−46.5885	−1.49510	−0.747548	−	0.664207i	$-0.768769\pi$
$971$	−46.5885	−1.49510	−0.747548	+	0.664207i	$0.768769\pi$
$972$	0	0
$973$	−0.858344	−0.0275172
$974$	0	0
$975$	3.20245	0.102560
$976$	0	0
$977$	18.4366	0.589839	0.294920	−	0.955522i	$-0.404707\pi$
$977$	18.4366	0.589839	0.294920	+	0.955522i	$0.404707\pi$
$978$	0	0
$979$	−79.2845	−2.53395
$980$	0	0
$981$	−21.7002	−0.692836
$982$	0	0
$983$	−1.50522	−0.0480090	−0.0240045	−	0.999712i	$-0.507642\pi$
$983$	−1.50522	−0.0480090	−0.0240045	+	0.999712i	$0.507642\pi$
$984$	0	0
$985$	12.3340	0.392993
$986$	0	0
$987$	−18.9968	−0.604674
$988$	0	0
$989$	−7.02537	−0.223394
$990$	0	0
$991$	−26.8361	−0.852478	−0.426239	−	0.904611i	$-0.640162\pi$
$991$	−26.8361	−0.852478	−0.426239	+	0.904611i	$0.640162\pi$
$992$	0	0
$993$	−18.3999	−0.583903
$994$	0	0
$995$	−29.1569	−0.924335
$996$	0	0
$997$	−20.2951	−0.642753	−0.321377	−	0.946951i	$-0.604146\pi$
$997$	−20.2951	−0.642753	−0.321377	+	0.946951i	$0.604146\pi$
$998$	0	0
$999$	−4.60120	−0.145575

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6044.2.a.b.1.5	✓	63

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6044.2.a.b.1.5	✓	63	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:	\( N \)	\(=\)	\( 6044 = 2^{2} \cdot 1511 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6044.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69290 q^{3} -2.32986 q^{5} -1.27542 q^{7} +4.25169 q^{9} +O(q^{10})\)	\(q-2.69290 q^{3} -2.32986 q^{5} -1.27542 q^{7} +4.25169 q^{9} +4.79770 q^{11} -2.77694 q^{13} +6.27407 q^{15} +0.827835 q^{17} -2.18324 q^{19} +3.43456 q^{21} +0.943429 q^{23} +0.428249 q^{25} -3.37066 q^{27} +5.95542 q^{29} -1.77507 q^{31} -12.9197 q^{33} +2.97154 q^{35} +1.36507 q^{37} +7.47801 q^{39} +5.18688 q^{41} -7.44663 q^{43} -9.90583 q^{45} -5.53106 q^{47} -5.37331 q^{49} -2.22927 q^{51} +10.0806 q^{53} -11.1780 q^{55} +5.87924 q^{57} +9.00966 q^{59} -9.70088 q^{61} -5.42267 q^{63} +6.46989 q^{65} -6.55332 q^{67} -2.54056 q^{69} +11.6341 q^{71} -8.41140 q^{73} -1.15323 q^{75} -6.11906 q^{77} -2.65091 q^{79} -3.67823 q^{81} -6.26750 q^{83} -1.92874 q^{85} -16.0373 q^{87} -16.5255 q^{89} +3.54176 q^{91} +4.78006 q^{93} +5.08665 q^{95} -1.58095 q^{97} +20.3983 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9}+O(q^{10}) \) 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9	\( 63 q + 5 q^{3} + 5 q^{5} + 22 q^{7} + 62 q^{9} + 21 q^{11} + 17 q^{13} + 26 q^{15} - 5 q^{17} + 57 q^{19} + 18 q^{21} + 16 q^{23} + 60 q^{25} + 14 q^{27} + 22 q^{29} + 36 q^{31} - q^{33} + 28 q^{35} + 21 q^{37} + 69 q^{39} - 3 q^{41} + 86 q^{43} + 39 q^{45} + 23 q^{47} + 63 q^{49} + 67 q^{51} + 18 q^{53} + 67 q^{55} + 27 q^{59} + 62 q^{61} + 58 q^{63} - 13 q^{65} + 62 q^{67} + 45 q^{69} + 29 q^{71} - q^{73} + 39 q^{75} + 19 q^{77} + 132 q^{79} + 51 q^{81} + 35 q^{83} + 60 q^{85} + 55 q^{87} - 2 q^{89} + 84 q^{91} + 29 q^{93} + 53 q^{95} - 10 q^{97} + 95 q^{99}+O(q^{100}) \) 63 * q + 5 * q^3 + 5 * q^5 + 22 * q^7 + 62 * q^9 + 21 * q^11 + 17 * q^13 + 26 * q^15 - 5 * q^17 + 57 * q^19 + 18 * q^21 + 16 * q^23 + 60 * q^25 + 14 * q^27 + 22 * q^29 + 36 * q^31 - q^33 + 28 * q^35 + 21 * q^37 + 69 * q^39 - 3 * q^41 + 86 * q^43 + 39 * q^45 + 23 * q^47 + 63 * q^49 + 67 * q^51 + 18 * q^53 + 67 * q^55 + 27 * q^59 + 62 * q^61 + 58 * q^63 - 13 * q^65 + 62 * q^67 + 45 * q^69 + 29 * q^71 - q^73 + 39 * q^75 + 19 * q^77 + 132 * q^79 + 51 * q^81 + 35 * q^83 + 60 * q^85 + 55 * q^87 - 2 * q^89 + 84 * q^91 + 29 * q^93 + 53 * q^95 - 10 * q^97 + 95 * q^99

Introduction

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