LMFDB - Embedded newform 387.4.a.h.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,4,Mod(1,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("387.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	387.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:	$22.8337391722$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 32x^{4} - 16x^{3} + 251x^{2} + 276x + 60 $ x^6 - 32x^4 - 16x^3 + 251x^2 + 276x + 60
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.847740$ of defining polynomial
Character	$\chi$	$=$	387.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.84774 q^{2} -4.58586 q^{4} -2.98245 q^{5} -26.0720 q^{7} +23.2554 q^{8} +O(q^{10})$	\(q-1.84774 q^{2} -4.58586 q^{4} -2.98245 q^{5} -26.0720 q^{7} +23.2554 q^{8} +5.51080 q^{10} +36.8506 q^{11} +89.5430 q^{13} +48.1743 q^{14} -6.28309 q^{16} +28.8042 q^{17} -58.8677 q^{19} +13.6771 q^{20} -68.0904 q^{22} -2.63139 q^{23} -116.105 q^{25} -165.452 q^{26} +119.563 q^{28} -173.812 q^{29} +57.9476 q^{31} -174.434 q^{32} -53.2227 q^{34} +77.7586 q^{35} +52.0754 q^{37} +108.772 q^{38} -69.3581 q^{40} -142.704 q^{41} -43.0000 q^{43} -168.992 q^{44} +4.86213 q^{46} +106.853 q^{47} +336.750 q^{49} +214.532 q^{50} -410.631 q^{52} -244.652 q^{53} -109.905 q^{55} -606.315 q^{56} +321.160 q^{58} +127.799 q^{59} -443.613 q^{61} -107.072 q^{62} +372.573 q^{64} -267.058 q^{65} -117.896 q^{67} -132.092 q^{68} -143.678 q^{70} -816.799 q^{71} -620.953 q^{73} -96.2217 q^{74} +269.959 q^{76} -960.770 q^{77} +377.771 q^{79} +18.7390 q^{80} +263.680 q^{82} +1453.23 q^{83} -85.9072 q^{85} +79.4528 q^{86} +856.975 q^{88} -627.993 q^{89} -2334.57 q^{91} +12.0672 q^{92} -197.436 q^{94} +175.570 q^{95} -817.163 q^{97} -622.227 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8}+O(q^{10}) $ 6 * q - 6 * q^2 + 22 * q^4 - 43 * q^5 + 8 * q^7 - 54 * q^8	$ 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8} + 57 q^{10} + 28 q^{11} + 56 q^{13} + 184 q^{14} - 54 q^{16} - 19 q^{17} - 75 q^{19} - 135 q^{20} - 504 q^{22} - 131 q^{23} + 105 q^{25} - 44 q^{26} - 404 q^{28} - 515 q^{29} + 237 q^{31} - 558 q^{32} - 107 q^{34} - 198 q^{35} + 269 q^{37} - 527 q^{38} + 613 q^{40} - 471 q^{41} - 258 q^{43} + 428 q^{44} - 67 q^{46} - 415 q^{47} + 350 q^{49} - 1335 q^{50} - 8 q^{52} - 450 q^{53} - 1732 q^{55} + 780 q^{56} - 1055 q^{58} - 356 q^{59} - 1328 q^{61} - 1603 q^{62} + 466 q^{64} + 62 q^{65} - 632 q^{67} - 571 q^{68} - 1902 q^{70} + 144 q^{71} + 864 q^{73} - 1207 q^{74} + 1005 q^{76} - 2660 q^{77} - 1613 q^{79} - 2399 q^{80} + 1673 q^{82} + 682 q^{83} + 84 q^{85} + 258 q^{86} - 608 q^{88} - 3378 q^{89} - 3900 q^{91} - 3491 q^{92} + 3197 q^{94} + 79 q^{95} - 55 q^{97} - 2398 q^{98}+O(q^{100}) $ 6 * q - 6 * q^2 + 22 * q^4 - 43 * q^5 + 8 * q^7 - 54 * q^8 + 57 * q^10 + 28 * q^11 + 56 * q^13 + 184 * q^14 - 54 * q^16 - 19 * q^17 - 75 * q^19 - 135 * q^20 - 504 * q^22 - 131 * q^23 + 105 * q^25 - 44 * q^26 - 404 * q^28 - 515 * q^29 + 237 * q^31 - 558 * q^32 - 107 * q^34 - 198 * q^35 + 269 * q^37 - 527 * q^38 + 613 * q^40 - 471 * q^41 - 258 * q^43 + 428 * q^44 - 67 * q^46 - 415 * q^47 + 350 * q^49 - 1335 * q^50 - 8 * q^52 - 450 * q^53 - 1732 * q^55 + 780 * q^56 - 1055 * q^58 - 356 * q^59 - 1328 * q^61 - 1603 * q^62 + 466 * q^64 + 62 * q^65 - 632 * q^67 - 571 * q^68 - 1902 * q^70 + 144 * q^71 + 864 * q^73 - 1207 * q^74 + 1005 * q^76 - 2660 * q^77 - 1613 * q^79 - 2399 * q^80 + 1673 * q^82 + 682 * q^83 + 84 * q^85 + 258 * q^86 - 608 * q^88 - 3378 * q^89 - 3900 * q^91 - 3491 * q^92 + 3197 * q^94 + 79 * q^95 - 55 * q^97 - 2398 * 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 $

$2$	−1.84774	−0.653275	−0.326637	−	0.945150i	$-0.605916\pi$
$2$	−1.84774	−0.653275	−0.326637	+	0.945150i	$0.605916\pi$
$3$	0	0
$3$	0	0
$4$	−4.58586	−0.573232
$5$	−2.98245	−0.266759	−0.133379	−	0.991065i	$-0.542583\pi$
$5$	−2.98245	−0.266759	−0.133379	+	0.991065i	$0.542583\pi$
$6$	0	0
$7$	−26.0720	−1.40776	−0.703878	−	0.710321i	$-0.748550\pi$
$7$	−26.0720	−1.40776	−0.703878	+	0.710321i	$0.748550\pi$
$8$	23.2554	1.02775
$9$	0	0
$10$	5.51080	0.174267
$11$	36.8506	1.01008	0.505040	−	0.863096i	$-0.331478\pi$
$11$	36.8506	1.01008	0.505040	+	0.863096i	$0.331478\pi$
$12$	0	0
$13$	89.5430	1.91037	0.955183	−	0.296016i	$-0.0956581\pi$
$13$	89.5430	1.91037	0.955183	+	0.296016i	$0.0956581\pi$
$14$	48.1743	0.919652
$15$	0	0
$16$	−6.28309	−0.0981733
$17$	28.8042	0.410944	0.205472	−	0.978663i	$-0.434127\pi$
$17$	28.8042	0.410944	0.205472	+	0.978663i	$0.434127\pi$
$18$	0	0
$19$	−58.8677	−0.710799	−0.355400	−	0.934714i	$-0.615655\pi$
$19$	−58.8677	−0.710799	−0.355400	+	0.934714i	$0.615655\pi$
$20$	13.6771	0.152915
$21$	0	0
$22$	−68.0904	−0.659860
$23$	−2.63139	−0.0238558	−0.0119279	−	0.999929i	$-0.503797\pi$
$23$	−2.63139	−0.0238558	−0.0119279	+	0.999929i	$0.503797\pi$
$24$	0	0
$25$	−116.105	−0.928840
$26$	−165.452	−1.24799
$27$	0	0
$28$	119.563	0.806971
$29$	−173.812	−1.11297	−0.556486	−	0.830857i	$-0.687851\pi$
$29$	−173.812	−1.11297	−0.556486	+	0.830857i	$0.687851\pi$
$30$	0	0
$31$	57.9476	0.335732	0.167866	−	0.985810i	$-0.446312\pi$
$31$	57.9476	0.335732	0.167866	+	0.985810i	$0.446312\pi$
$32$	−174.434	−0.963619
$33$	0	0
$34$	−53.2227	−0.268459
$35$	77.7586	0.375531
$36$	0	0
$37$	52.0754	0.231382	0.115691	−	0.993285i	$-0.463092\pi$
$37$	52.0754	0.231382	0.115691	+	0.993285i	$0.463092\pi$
$38$	108.772	0.464347
$39$	0	0
$40$	−69.3581	−0.274162
$41$	−142.704	−0.543577	−0.271788	−	0.962357i	$-0.587615\pi$
$41$	−142.704	−0.543577	−0.271788	+	0.962357i	$0.587615\pi$
$42$	0	0
$43$	−43.0000	−0.152499
$43$	−43.0000	−0.152499
$44$	−168.992	−0.579010
$45$	0	0
$46$	4.86213	0.0155844
$47$	106.853	0.331618	0.165809	−	0.986158i	$-0.446976\pi$
$47$	106.853	0.331618	0.165809	+	0.986158i	$0.446976\pi$
$48$	0	0
$49$	336.750	0.981779
$50$	214.532	0.606788
$51$	0	0
$52$	−410.631	−1.09508
$53$	−244.652	−0.634067	−0.317033	−	0.948414i	$-0.602687\pi$
$53$	−244.652	−0.634067	−0.317033	+	0.948414i	$0.602687\pi$
$54$	0	0
$55$	−109.905	−0.269448
$56$	−606.315	−1.44683
$57$	0	0
$58$	321.160	0.727076
$59$	127.799	0.281999	0.141000	−	0.990010i	$-0.454968\pi$
$59$	127.799	0.281999	0.141000	+	0.990010i	$0.454968\pi$
$60$	0	0
$61$	−443.613	−0.931128	−0.465564	−	0.885014i	$-0.654149\pi$
$61$	−443.613	−0.931128	−0.465564	+	0.885014i	$0.654149\pi$
$62$	−107.072	−0.219325
$63$	0	0
$64$	372.573	0.727681
$65$	−267.058	−0.509607
$66$	0	0
$67$	−117.896	−0.214975	−0.107487	−	0.994206i	$-0.534280\pi$
$67$	−117.896	−0.214975	−0.107487	+	0.994206i	$0.534280\pi$
$68$	−132.092	−0.235566
$69$	0	0
$70$	−143.678	−0.245325
$71$	−816.799	−1.36530	−0.682650	−	0.730746i	$-0.739173\pi$
$71$	−816.799	−1.36530	−0.682650	+	0.730746i	$0.739173\pi$
$72$	0	0
$73$	−620.953	−0.995576	−0.497788	−	0.867299i	$-0.665854\pi$
$73$	−620.953	−0.995576	−0.497788	+	0.867299i	$0.665854\pi$
$74$	−96.2217	−0.151156
$75$	0	0
$76$	269.959	0.407453
$77$	−960.770	−1.42195
$78$	0	0
$79$	377.771	0.538007	0.269003	−	0.963139i	$-0.413306\pi$
$79$	377.771	0.538007	0.269003	+	0.963139i	$0.413306\pi$
$80$	18.7390	0.0261886
$81$	0	0
$82$	263.680	0.355105
$83$	1453.23	1.92184	0.960919	−	0.276829i	$-0.0892835\pi$
$83$	1453.23	1.92184	0.960919	+	0.276829i	$0.0892835\pi$
$84$	0	0
$85$	−85.9072	−0.109623
$86$	79.4528	0.0996235
$87$	0	0
$88$	856.975	1.03811
$89$	−627.993	−0.747945	−0.373973	−	0.927440i	$-0.622005\pi$
$89$	−627.993	−0.747945	−0.373973	+	0.927440i	$0.622005\pi$
$90$	0	0
$91$	−2334.57	−2.68933
$92$	12.0672	0.0136749
$93$	0	0
$94$	−197.436	−0.216638
$95$	175.570	0.189612
$96$	0	0
$97$	−817.163	−0.855365	−0.427682	−	0.903929i	$-0.640670\pi$
$97$	−817.163	−0.855365	−0.427682	+	0.903929i	$0.640670\pi$
$98$	−622.227	−0.641372
$99$	0	0
$100$	532.441	0.532441
$101$	−513.438	−0.505832	−0.252916	−	0.967488i	$-0.581390\pi$
$101$	−513.438	−0.505832	−0.252916	+	0.967488i	$0.581390\pi$
$102$	0	0
$103$	689.788	0.659872	0.329936	−	0.944003i	$-0.392973\pi$
$103$	689.788	0.659872	0.329936	+	0.944003i	$0.392973\pi$
$104$	2082.36	1.96338
$105$	0	0
$106$	452.053	0.414220
$107$	−320.710	−0.289759	−0.144879	−	0.989449i	$-0.546279\pi$
$107$	−320.710	−0.289759	−0.144879	+	0.989449i	$0.546279\pi$
$108$	0	0
$109$	−1691.51	−1.48640	−0.743200	−	0.669069i	$-0.766693\pi$
$109$	−1691.51	−1.48640	−0.743200	+	0.669069i	$0.766693\pi$
$110$	203.076	0.176023
$111$	0	0
$112$	163.813	0.138204
$113$	−856.360	−0.712916	−0.356458	−	0.934311i	$-0.616016\pi$
$113$	−856.360	−0.712916	−0.356458	+	0.934311i	$0.616016\pi$
$114$	0	0
$115$	7.84801	0.00636374
$116$	797.079	0.637990
$117$	0	0
$118$	−236.139	−0.184223
$119$	−750.984	−0.578509
$120$	0	0
$121$	26.9674	0.0202610
$122$	819.682	0.608283
$123$	0	0
$124$	−265.739	−0.192452
$125$	719.084	0.514535
$126$	0	0
$127$	−2233.72	−1.56071	−0.780357	−	0.625335i	$-0.784962\pi$
$127$	−2233.72	−1.56071	−0.780357	+	0.625335i	$0.784962\pi$
$128$	707.051	0.488243
$129$	0	0
$130$	493.454	0.332913
$131$	2051.51	1.36825	0.684126	−	0.729364i	$-0.260184\pi$
$131$	2051.51	1.36825	0.684126	+	0.729364i	$0.260184\pi$
$132$	0	0
$133$	1534.80	1.00063
$134$	217.841	0.140437
$135$	0	0
$136$	669.853	0.422349
$137$	−2594.49	−1.61797	−0.808986	−	0.587827i	$-0.799983\pi$
$137$	−2594.49	−1.61797	−0.808986	+	0.587827i	$0.799983\pi$
$138$	0	0
$139$	1140.44	0.695905	0.347953	−	0.937512i	$-0.386877\pi$
$139$	1140.44	0.695905	0.347953	+	0.937512i	$0.386877\pi$
$140$	−356.590	−0.215267
$141$	0	0
$142$	1509.23	0.891916
$143$	3299.71	1.92962
$144$	0	0
$145$	518.388	0.296895
$146$	1147.36	0.650385
$147$	0	0
$148$	−238.810	−0.132636
$149$	2112.03	1.16124	0.580618	−	0.814176i	$-0.302811\pi$
$149$	2112.03	1.16124	0.580618	+	0.814176i	$0.302811\pi$
$150$	0	0
$151$	−1351.31	−0.728265	−0.364132	−	0.931347i	$-0.618634\pi$
$151$	−1351.31	−0.728265	−0.364132	+	0.931347i	$0.618634\pi$
$152$	−1368.99	−0.730526
$153$	0	0
$154$	1775.25	0.928922
$155$	−172.826	−0.0895595
$156$	0	0
$157$	−1506.02	−0.765562	−0.382781	−	0.923839i	$-0.625034\pi$
$157$	−1506.02	−0.765562	−0.382781	+	0.923839i	$0.625034\pi$
$158$	−698.022	−0.351466
$159$	0	0
$160$	520.240	0.257054
$161$	68.6057	0.0335832
$162$	0	0
$163$	−1258.30	−0.604647	−0.302323	−	0.953205i	$-0.597762\pi$
$163$	−1258.30	−0.604647	−0.302323	+	0.953205i	$0.597762\pi$
$164$	654.420	0.311595
$165$	0	0
$166$	−2685.19	−1.25549
$167$	−2764.50	−1.28098	−0.640489	−	0.767967i	$-0.721268\pi$
$167$	−2764.50	−1.28098	−0.640489	+	0.767967i	$0.721268\pi$
$168$	0	0
$169$	5820.95	2.64950
$170$	158.734	0.0716139
$171$	0	0
$172$	197.192	0.0874170
$173$	−1004.21	−0.441322	−0.220661	−	0.975351i	$-0.570821\pi$
$173$	−1004.21	−0.441322	−0.220661	+	0.975351i	$0.570821\pi$
$174$	0	0
$175$	3027.09	1.30758
$176$	−231.536	−0.0991628
$177$	0	0
$178$	1160.37	0.488614
$179$	−2666.39	−1.11338	−0.556692	−	0.830719i	$-0.687930\pi$
$179$	−2666.39	−1.11338	−0.556692	+	0.830719i	$0.687930\pi$
$180$	0	0
$181$	3016.21	1.23863	0.619317	−	0.785141i	$-0.287409\pi$
$181$	3016.21	1.23863	0.619317	+	0.785141i	$0.287409\pi$
$182$	4313.67	1.75687
$183$	0	0
$184$	−61.1941	−0.0245179
$185$	−155.312	−0.0617232
$186$	0	0
$187$	1061.45	0.415086
$188$	−490.010	−0.190094
$189$	0	0
$190$	−324.408	−0.123869
$191$	−1413.15	−0.535352	−0.267676	−	0.963509i	$-0.586256\pi$
$191$	−1413.15	−0.535352	−0.267676	+	0.963509i	$0.586256\pi$
$192$	0	0
$193$	−1246.60	−0.464934	−0.232467	−	0.972604i	$-0.574680\pi$
$193$	−1246.60	−0.464934	−0.232467	+	0.972604i	$0.574680\pi$
$194$	1509.91	0.558788
$195$	0	0
$196$	−1544.29	−0.562787
$197$	4931.17	1.78341	0.891704	−	0.452619i	$-0.149510\pi$
$197$	4931.17	1.78341	0.891704	+	0.452619i	$0.149510\pi$
$198$	0	0
$199$	552.461	0.196799	0.0983993	−	0.995147i	$-0.468628\pi$
$199$	552.461	0.196799	0.0983993	+	0.995147i	$0.468628\pi$
$200$	−2700.07	−0.954618
$201$	0	0
$202$	948.700	0.330447
$203$	4531.64	1.56679
$204$	0	0
$205$	425.609	0.145004
$206$	−1274.55	−0.431078
$207$	0	0
$208$	−562.607	−0.187547
$209$	−2169.31	−0.717964
$210$	0	0
$211$	2302.22	0.751145	0.375572	−	0.926793i	$-0.377446\pi$
$211$	2302.22	0.751145	0.375572	+	0.926793i	$0.377446\pi$
$212$	1121.94	0.363467
$213$	0	0
$214$	592.588	0.189292
$215$	128.246	0.0406803
$216$	0	0
$217$	−1510.81	−0.472629
$218$	3125.48	0.971028
$219$	0	0
$220$	504.010	0.154456
$221$	2579.21	0.785053
$222$	0	0
$223$	2558.41	0.768269	0.384135	−	0.923277i	$-0.374500\pi$
$223$	2558.41	0.768269	0.384135	+	0.923277i	$0.374500\pi$
$224$	4547.84	1.35654
$225$	0	0
$226$	1582.33	0.465730
$227$	−3622.76	−1.05926	−0.529628	−	0.848230i	$-0.677669\pi$
$227$	−3622.76	−1.05926	−0.529628	+	0.848230i	$0.677669\pi$
$228$	0	0
$229$	1155.46	0.333428	0.166714	−	0.986005i	$-0.446684\pi$
$229$	1155.46	0.333428	0.166714	+	0.986005i	$0.446684\pi$
$230$	−14.5011	−0.00415727
$231$	0	0
$232$	−4042.08	−1.14386
$233$	−527.800	−0.148401	−0.0742003	−	0.997243i	$-0.523640\pi$
$233$	−527.800	−0.148401	−0.0742003	+	0.997243i	$0.523640\pi$
$234$	0	0
$235$	−318.683	−0.0884620
$236$	−586.066	−0.161651
$237$	0	0
$238$	1387.62	0.377925
$239$	1341.41	0.363048	0.181524	−	0.983387i	$-0.441897\pi$
$239$	1341.41	0.363048	0.181524	+	0.983387i	$0.441897\pi$
$240$	0	0
$241$	−3738.93	−0.999361	−0.499680	−	0.866210i	$-0.666549\pi$
$241$	−3738.93	−0.999361	−0.499680	+	0.866210i	$0.666549\pi$
$242$	−49.8287	−0.0132360
$243$	0	0
$244$	2034.34	0.533752
$245$	−1004.34	−0.261898
$246$	0	0
$247$	−5271.19	−1.35789
$248$	1347.59	0.345050
$249$	0	0
$250$	−1328.68	−0.336133
$251$	1741.62	0.437969	0.218985	−	0.975728i	$-0.429726\pi$
$251$	1741.62	0.437969	0.218985	+	0.975728i	$0.429726\pi$
$252$	0	0
$253$	−96.9684	−0.0240963
$254$	4127.33	1.01957
$255$	0	0
$256$	−4287.03	−1.04664
$257$	−4121.68	−1.00040	−0.500200	−	0.865910i	$-0.666740\pi$
$257$	−4121.68	−1.00040	−0.500200	+	0.865910i	$0.666740\pi$
$258$	0	0
$259$	−1357.71	−0.325730
$260$	1224.69	0.292123
$261$	0	0
$262$	−3790.65	−0.893844
$263$	−4315.46	−1.01180	−0.505898	−	0.862593i	$-0.668839\pi$
$263$	−4315.46	−1.01180	−0.505898	+	0.862593i	$0.668839\pi$
$264$	0	0
$265$	729.663	0.169143
$266$	−2835.91	−0.653688
$267$	0	0
$268$	540.654	0.123230
$269$	−5195.44	−1.17759	−0.588794	−	0.808283i	$-0.700397\pi$
$269$	−5195.44	−1.17759	−0.588794	+	0.808283i	$0.700397\pi$
$270$	0	0
$271$	−7874.15	−1.76502	−0.882510	−	0.470294i	$-0.844148\pi$
$271$	−7874.15	−1.76502	−0.882510	+	0.470294i	$0.844148\pi$
$272$	−180.979	−0.0403437
$273$	0	0
$274$	4793.94	1.05698
$275$	−4278.54	−0.938202
$276$	0	0
$277$	−175.471	−0.0380615	−0.0190307	−	0.999819i	$-0.506058\pi$
$277$	−175.471	−0.0380615	−0.0190307	+	0.999819i	$0.506058\pi$
$278$	−2107.24	−0.454617
$279$	0	0
$280$	1808.31	0.385954
$281$	−7263.01	−1.54190	−0.770951	−	0.636894i	$-0.780219\pi$
$281$	−7263.01	−1.54190	−0.770951	+	0.636894i	$0.780219\pi$
$282$	0	0
$283$	2314.68	0.486196	0.243098	−	0.970002i	$-0.421836\pi$
$283$	2314.68	0.486196	0.243098	+	0.970002i	$0.421836\pi$
$284$	3745.72	0.782633
$285$	0	0
$286$	−6097.01	−1.26057
$287$	3720.58	0.765224
$288$	0	0
$289$	−4083.32	−0.831125
$290$	−957.846	−0.193954
$291$	0	0
$292$	2847.60	0.570696
$293$	−8723.52	−1.73936	−0.869681	−	0.493614i	$-0.835676\pi$
$293$	−8723.52	−1.73936	−0.869681	+	0.493614i	$0.835676\pi$
$294$	0	0
$295$	−381.153	−0.0752258
$296$	1211.03	0.237804
$297$	0	0
$298$	−3902.48	−0.758607
$299$	−235.623	−0.0455733
$300$	0	0
$301$	1121.10	0.214681
$302$	2496.87	0.475757
$303$	0	0
$304$	369.871	0.0697815
$305$	1323.06	0.248387
$306$	0	0
$307$	1579.68	0.293672	0.146836	−	0.989161i	$-0.453091\pi$
$307$	1579.68	0.293672	0.146836	+	0.989161i	$0.453091\pi$
$308$	4405.95	0.815105
$309$	0	0
$310$	319.338	0.0585070
$311$	5604.24	1.02182	0.510912	−	0.859633i	$-0.329308\pi$
$311$	5604.24	1.02182	0.510912	+	0.859633i	$0.329308\pi$
$312$	0	0
$313$	3429.86	0.619384	0.309692	−	0.950837i	$-0.399774\pi$
$313$	3429.86	0.619384	0.309692	+	0.950837i	$0.399774\pi$
$314$	2782.73	0.500123
$315$	0	0
$316$	−1732.40	−0.308403
$317$	−4493.00	−0.796064	−0.398032	−	0.917372i	$-0.630307\pi$
$317$	−4493.00	−0.796064	−0.398032	+	0.917372i	$0.630307\pi$
$318$	0	0
$319$	−6405.09	−1.12419
$320$	−1111.18	−0.194115
$321$	0	0
$322$	−126.766	−0.0219390
$323$	−1695.64	−0.292098
$324$	0	0
$325$	−10396.4	−1.77442
$326$	2325.01	0.395001
$327$	0	0
$328$	−3318.64	−0.558662
$329$	−2785.86	−0.466837
$330$	0	0
$331$	4433.87	0.736277	0.368139	−	0.929771i	$-0.379995\pi$
$331$	4433.87	0.736277	0.368139	+	0.929771i	$0.379995\pi$
$332$	−6664.30	−1.10166
$333$	0	0
$334$	5108.07	0.836831
$335$	351.620	0.0573463
$336$	0	0
$337$	6498.33	1.05040	0.525202	−	0.850977i	$-0.323990\pi$
$337$	6498.33	1.05040	0.525202	+	0.850977i	$0.323990\pi$
$338$	−10755.6	−1.73085
$339$	0	0
$340$	393.958	0.0628393
$341$	2135.41	0.339116
$342$	0	0
$343$	162.945	0.0256507
$344$	−999.982	−0.156731
$345$	0	0
$346$	1855.52	0.288305
$347$	11973.6	1.85237	0.926187	−	0.377064i	$-0.123066\pi$
$347$	11973.6	1.85237	0.926187	+	0.377064i	$0.123066\pi$
$348$	0	0
$349$	5611.30	0.860648	0.430324	−	0.902674i	$-0.358399\pi$
$349$	5611.30	0.860648	0.430324	+	0.902674i	$0.358399\pi$
$350$	−5593.28	−0.854209
$351$	0	0
$352$	−6427.99	−0.973332
$353$	2022.00	0.304874	0.152437	−	0.988313i	$-0.451288\pi$
$353$	2022.00	0.304874	0.152437	+	0.988313i	$0.451288\pi$
$354$	0	0
$355$	2436.07	0.364206
$356$	2879.88	0.428746
$357$	0	0
$358$	4926.81	0.727346
$359$	−10135.3	−1.49003	−0.745014	−	0.667049i	$-0.767557\pi$
$359$	−10135.3	−1.49003	−0.745014	+	0.667049i	$0.767557\pi$
$360$	0	0
$361$	−3393.59	−0.494765
$362$	−5573.17	−0.809169
$363$	0	0
$364$	10706.0	1.54161
$365$	1851.96	0.265579
$366$	0	0
$367$	−4379.58	−0.622922	−0.311461	−	0.950259i	$-0.600818\pi$
$367$	−4379.58	−0.622922	−0.311461	+	0.950259i	$0.600818\pi$
$368$	16.5333	0.00234200
$369$	0	0
$370$	286.977	0.0403222
$371$	6378.57	0.892612
$372$	0	0
$373$	4392.98	0.609813	0.304906	−	0.952382i	$-0.401375\pi$
$373$	4392.98	0.609813	0.304906	+	0.952382i	$0.401375\pi$
$374$	−1961.29	−0.271165
$375$	0	0
$376$	2484.90	0.340821
$377$	−15563.7	−2.12618
$378$	0	0
$379$	−878.629	−0.119082	−0.0595411	−	0.998226i	$-0.518964\pi$
$379$	−878.629	−0.119082	−0.0595411	+	0.998226i	$0.518964\pi$
$380$	−805.140	−0.108692
$381$	0	0
$382$	2611.14	0.349732
$383$	−5061.60	−0.675289	−0.337644	−	0.941274i	$-0.609630\pi$
$383$	−5061.60	−0.675289	−0.337644	+	0.941274i	$0.609630\pi$
$384$	0	0
$385$	2865.45	0.379317
$386$	2303.39	0.303730
$387$	0	0
$388$	3747.39	0.490322
$389$	3417.17	0.445392	0.222696	−	0.974888i	$-0.428514\pi$
$389$	3417.17	0.445392	0.222696	+	0.974888i	$0.428514\pi$
$390$	0	0
$391$	−75.7952	−0.00980339
$392$	7831.26	1.00903
$393$	0	0
$394$	−9111.52	−1.16506
$395$	−1126.68	−0.143518
$396$	0	0
$397$	7634.34	0.965130	0.482565	−	0.875860i	$-0.339705\pi$
$397$	7634.34	0.965130	0.482565	+	0.875860i	$0.339705\pi$
$398$	−1020.80	−0.128564
$399$	0	0
$400$	729.498	0.0911872
$401$	8402.74	1.04642	0.523208	−	0.852205i	$-0.324735\pi$
$401$	8402.74	1.04642	0.523208	+	0.852205i	$0.324735\pi$
$402$	0	0
$403$	5188.80	0.641372
$404$	2354.55	0.289959
$405$	0	0
$406$	−8373.30	−1.02355
$407$	1919.01	0.233714
$408$	0	0
$409$	−11792.8	−1.42571	−0.712857	−	0.701310i	$-0.752599\pi$
$409$	−11792.8	−1.42571	−0.712857	+	0.701310i	$0.752599\pi$
$410$	−786.414	−0.0947274
$411$	0	0
$412$	−3163.27	−0.378260
$413$	−3331.97	−0.396986
$414$	0	0
$415$	−4334.19	−0.512667
$416$	−15619.3	−1.84086
$417$	0	0
$418$	4008.32	0.469028
$419$	10631.9	1.23962	0.619811	−	0.784751i	$-0.287209\pi$
$419$	10631.9	1.23962	0.619811	+	0.784751i	$0.287209\pi$
$420$	0	0
$421$	−3136.52	−0.363099	−0.181550	−	0.983382i	$-0.558111\pi$
$421$	−3136.52	−0.363099	−0.181550	+	0.983382i	$0.558111\pi$
$422$	−4253.91	−0.490704
$423$	0	0
$424$	−5689.48	−0.651664
$425$	−3344.31	−0.381701
$426$	0	0
$427$	11565.9	1.31080
$428$	1470.73	0.166099
$429$	0	0
$430$	−236.964	−0.0265754
$431$	170.380	0.0190416	0.00952080	−	0.999955i	$-0.496969\pi$
$431$	170.380	0.0190416	0.00952080	+	0.999955i	$0.496969\pi$
$432$	0	0
$433$	2093.65	0.232365	0.116183	−	0.993228i	$-0.462934\pi$
$433$	2093.65	0.232365	0.116183	+	0.993228i	$0.462934\pi$
$434$	2791.59	0.308757
$435$	0	0
$436$	7757.04	0.852052
$437$	154.904	0.0169567
$438$	0	0
$439$	−10860.1	−1.18070	−0.590348	−	0.807149i	$-0.701010\pi$
$439$	−10860.1	−1.18070	−0.590348	+	0.807149i	$0.701010\pi$
$440$	−2555.89	−0.276926
$441$	0	0
$442$	−4765.72	−0.512855
$443$	−8256.28	−0.885480	−0.442740	−	0.896650i	$-0.645994\pi$
$443$	−8256.28	−0.885480	−0.442740	+	0.896650i	$0.645994\pi$
$444$	0	0
$445$	1872.96	0.199521
$446$	−4727.29	−0.501891
$447$	0	0
$448$	−9713.73	−1.02440
$449$	6792.03	0.713888	0.356944	−	0.934126i	$-0.383819\pi$
$449$	6792.03	0.713888	0.356944	+	0.934126i	$0.383819\pi$
$450$	0	0
$451$	−5258.73	−0.549056
$452$	3927.14	0.408666
$453$	0	0
$454$	6693.93	0.691986
$455$	6962.74	0.717403
$456$	0	0
$457$	4004.62	0.409908	0.204954	−	0.978772i	$-0.434295\pi$
$457$	4004.62	0.409908	0.204954	+	0.978772i	$0.434295\pi$
$458$	−2134.99	−0.217820
$459$	0	0
$460$	−35.9898	−0.00364790
$461$	−1164.36	−0.117635	−0.0588175	−	0.998269i	$-0.518733\pi$
$461$	−1164.36	−0.117635	−0.0588175	+	0.998269i	$0.518733\pi$
$462$	0	0
$463$	2566.99	0.257664	0.128832	−	0.991666i	$-0.458877\pi$
$463$	2566.99	0.257664	0.128832	+	0.991666i	$0.458877\pi$
$464$	1092.08	0.109264
$465$	0	0
$466$	975.237	0.0969463
$467$	7654.43	0.758469	0.379234	−	0.925301i	$-0.376187\pi$
$467$	7654.43	0.758469	0.379234	+	0.925301i	$0.376187\pi$
$468$	0	0
$469$	3073.79	0.302632
$470$	588.843	0.0577900
$471$	0	0
$472$	2972.01	0.289826
$473$	−1584.58	−0.154036
$474$	0	0
$475$	6834.84	0.660218
$476$	3443.90	0.331620
$477$	0	0
$478$	−2478.57	−0.237170
$479$	8754.20	0.835051	0.417526	−	0.908665i	$-0.362897\pi$
$479$	8754.20	0.835051	0.417526	+	0.908665i	$0.362897\pi$
$480$	0	0
$481$	4662.98	0.442024
$482$	6908.58	0.652857
$483$	0	0
$484$	−123.668	−0.0116142
$485$	2437.15	0.228176
$486$	0	0
$487$	−9406.39	−0.875245	−0.437623	−	0.899159i	$-0.644179\pi$
$487$	−9406.39	−0.875245	−0.437623	+	0.899159i	$0.644179\pi$
$488$	−10316.4	−0.956970
$489$	0	0
$490$	1855.76	0.171092
$491$	−6362.18	−0.584768	−0.292384	−	0.956301i	$-0.594449\pi$
$491$	−6362.18	−0.584768	−0.292384	+	0.956301i	$0.594449\pi$
$492$	0	0
$493$	−5006.53	−0.457369
$494$	9739.80	0.887073
$495$	0	0
$496$	−364.090	−0.0329599
$497$	21295.6	1.92201
$498$	0	0
$499$	11574.8	1.03840	0.519198	−	0.854654i	$-0.326231\pi$
$499$	11574.8	1.03840	0.519198	+	0.854654i	$0.326231\pi$
$500$	−3297.62	−0.294948
$501$	0	0
$502$	−3218.07	−0.286114
$503$	−11443.5	−1.01439	−0.507196	−	0.861831i	$-0.669318\pi$
$503$	−11443.5	−1.01439	−0.507196	+	0.861831i	$0.669318\pi$
$504$	0	0
$505$	1531.31	0.134935
$506$	179.172	0.0157415
$507$	0	0
$508$	10243.5	0.894651
$509$	17397.9	1.51502	0.757511	−	0.652822i	$-0.226415\pi$
$509$	17397.9	1.51502	0.757511	+	0.652822i	$0.226415\pi$
$510$	0	0
$511$	16189.5	1.40153
$512$	2264.91	0.195499
$513$	0	0
$514$	7615.79	0.653537
$515$	−2057.26	−0.176027
$516$	0	0
$517$	3937.58	0.334961
$518$	2508.70	0.212791
$519$	0	0
$520$	−6210.54	−0.523750
$521$	23236.4	1.95394	0.976972	−	0.213369i	$-0.0684437\pi$
$521$	23236.4	1.95394	0.976972	+	0.213369i	$0.0684437\pi$
$522$	0	0
$523$	−6523.69	−0.545432	−0.272716	−	0.962095i	$-0.587922\pi$
$523$	−6523.69	−0.545432	−0.272716	+	0.962095i	$0.587922\pi$
$524$	−9407.91	−0.784325
$525$	0	0
$526$	7973.85	0.660981
$527$	1669.13	0.137967
$528$	0	0
$529$	−12160.1	−0.999431
$530$	−1348.23	−0.110497
$531$	0	0
$532$	−7038.37	−0.573594
$533$	−12778.2	−1.03843
$534$	0	0
$535$	956.502	0.0772957
$536$	−2741.72	−0.220941
$537$	0	0
$538$	9599.82	0.769289
$539$	12409.5	0.991675
$540$	0	0
$541$	−13311.4	−1.05786	−0.528930	−	0.848666i	$-0.677406\pi$
$541$	−13311.4	−1.05786	−0.528930	+	0.848666i	$0.677406\pi$
$542$	14549.4	1.15304
$543$	0	0
$544$	−5024.42	−0.395993
$545$	5044.86	0.396510
$546$	0	0
$547$	−24529.8	−1.91740	−0.958700	−	0.284420i	$-0.908199\pi$
$547$	−24529.8	−1.91740	−0.958700	+	0.284420i	$0.908199\pi$
$548$	11898.0	0.927474
$549$	0	0
$550$	7905.63	0.612904
$551$	10231.9	0.791099
$552$	0	0
$553$	−9849.25	−0.757383
$554$	324.225	0.0248646
$555$	0	0
$556$	−5229.89	−0.398915
$557$	1732.87	0.131821	0.0659104	−	0.997826i	$-0.479005\pi$
$557$	1732.87	0.131821	0.0659104	+	0.997826i	$0.479005\pi$
$558$	0	0
$559$	−3850.35	−0.291328
$560$	−488.564	−0.0368672
$561$	0	0
$562$	13420.1	1.00729
$563$	−13482.0	−1.00924	−0.504618	−	0.863343i	$-0.668366\pi$
$563$	−13482.0	−1.00924	−0.504618	+	0.863343i	$0.668366\pi$
$564$	0	0
$565$	2554.05	0.190177
$566$	−4276.92	−0.317619
$567$	0	0
$568$	−18995.0	−1.40319
$569$	−23053.6	−1.69852	−0.849261	−	0.527973i	$-0.822952\pi$
$569$	−23053.6	−1.69852	−0.849261	+	0.527973i	$0.822952\pi$
$570$	0	0
$571$	8791.94	0.644363	0.322181	−	0.946678i	$-0.395584\pi$
$571$	8791.94	0.644363	0.322181	+	0.946678i	$0.395584\pi$
$572$	−15132.0	−1.10612
$573$	0	0
$574$	−6874.68	−0.499901
$575$	305.518	0.0221582
$576$	0	0
$577$	−12464.3	−0.899302	−0.449651	−	0.893204i	$-0.648452\pi$
$577$	−12464.3	−0.899302	−0.449651	+	0.893204i	$0.648452\pi$
$578$	7544.91	0.542953
$579$	0	0
$580$	−2377.25	−0.170190
$581$	−37888.6	−2.70548
$582$	0	0
$583$	−9015.58	−0.640458
$584$	−14440.5	−1.02321
$585$	0	0
$586$	16118.8	1.13628
$587$	24395.5	1.71535	0.857674	−	0.514193i	$-0.171909\pi$
$587$	24395.5	1.71535	0.857674	+	0.514193i	$0.171909\pi$
$588$	0	0
$589$	−3411.24	−0.238638
$590$	704.273	0.0491431
$591$	0	0
$592$	−327.194	−0.0227155
$593$	8779.99	0.608012	0.304006	−	0.952670i	$-0.401676\pi$
$593$	8779.99	0.608012	0.304006	+	0.952670i	$0.401676\pi$
$594$	0	0
$595$	2239.77	0.154322
$596$	−9685.46	−0.665658
$597$	0	0
$598$	435.370	0.0297719
$599$	14440.3	0.985001	0.492501	−	0.870312i	$-0.336083\pi$
$599$	14440.3	0.985001	0.492501	+	0.870312i	$0.336083\pi$
$600$	0	0
$601$	7456.08	0.506056	0.253028	−	0.967459i	$-0.418573\pi$
$601$	7456.08	0.506056	0.253028	+	0.967459i	$0.418573\pi$
$602$	−2071.50	−0.140246
$603$	0	0
$604$	6196.91	0.417464
$605$	−80.4290	−0.00540480
$606$	0	0
$607$	−6832.01	−0.456842	−0.228421	−	0.973563i	$-0.573356\pi$
$607$	−6832.01	−0.456842	−0.228421	+	0.973563i	$0.573356\pi$
$608$	10268.5	0.684939
$609$	0	0
$610$	−2444.66	−0.162265
$611$	9567.89	0.633512
$612$	0	0
$613$	12239.8	0.806460	0.403230	−	0.915099i	$-0.367887\pi$
$613$	12239.8	0.806460	0.403230	+	0.915099i	$0.367887\pi$
$614$	−2918.84	−0.191849
$615$	0	0
$616$	−22343.1	−1.46141
$617$	−4307.99	−0.281091	−0.140546	−	0.990074i	$-0.544886\pi$
$617$	−4307.99	−0.281091	−0.140546	+	0.990074i	$0.544886\pi$
$618$	0	0
$619$	21923.8	1.42357	0.711786	−	0.702396i	$-0.247887\pi$
$619$	21923.8	1.42357	0.711786	+	0.702396i	$0.247887\pi$
$620$	792.556	0.0513384
$621$	0	0
$622$	−10355.2	−0.667531
$623$	16373.0	1.05292
$624$	0	0
$625$	12368.5	0.791583
$626$	−6337.49	−0.404628
$627$	0	0
$628$	6906.38	0.438845
$629$	1499.99	0.0950850
$630$	0	0
$631$	13249.3	0.835891	0.417945	−	0.908472i	$-0.362750\pi$
$631$	13249.3	0.835891	0.417945	+	0.908472i	$0.362750\pi$
$632$	8785.21	0.552938
$633$	0	0
$634$	8301.91	0.520049
$635$	6661.97	0.416334
$636$	0	0
$637$	30153.6	1.87556
$638$	11835.0	0.734405
$639$	0	0
$640$	−2108.75	−0.130243
$641$	6897.79	0.425033	0.212517	−	0.977157i	$-0.431834\pi$
$641$	6897.79	0.425033	0.212517	+	0.977157i	$0.431834\pi$
$642$	0	0
$643$	−570.068	−0.0349631	−0.0174816	−	0.999847i	$-0.505565\pi$
$643$	−570.068	−0.0349631	−0.0174816	+	0.999847i	$0.505565\pi$
$644$	−314.616	−0.0192509
$645$	0	0
$646$	3133.10	0.190821
$647$	−15788.1	−0.959341	−0.479670	−	0.877449i	$-0.659244\pi$
$647$	−15788.1	−0.959341	−0.479670	+	0.877449i	$0.659244\pi$
$648$	0	0
$649$	4709.46	0.284842
$650$	19209.8	1.15919
$651$	0	0
$652$	5770.37	0.346603
$653$	−21960.3	−1.31604	−0.658019	−	0.753001i	$-0.728606\pi$
$653$	−21960.3	−1.31604	−0.658019	+	0.753001i	$0.728606\pi$
$654$	0	0
$655$	−6118.52	−0.364993
$656$	896.623	0.0533647
$657$	0	0
$658$	5147.55	0.304973
$659$	12142.6	0.717766	0.358883	−	0.933383i	$-0.383158\pi$
$659$	12142.6	0.717766	0.358883	+	0.933383i	$0.383158\pi$
$660$	0	0
$661$	3554.01	0.209130	0.104565	−	0.994518i	$-0.466655\pi$
$661$	3554.01	0.209130	0.104565	+	0.994518i	$0.466655\pi$
$662$	−8192.65	−0.480992
$663$	0	0
$664$	33795.4	1.97518
$665$	−4577.47	−0.266927
$666$	0	0
$667$	457.369	0.0265508
$668$	12677.6	0.734297
$669$	0	0
$670$	−649.702	−0.0374629
$671$	−16347.4	−0.940514
$672$	0	0
$673$	20271.0	1.16106	0.580528	−	0.814240i	$-0.302846\pi$
$673$	20271.0	1.16106	0.580528	+	0.814240i	$0.302846\pi$
$674$	−12007.2	−0.686203
$675$	0	0
$676$	−26694.0	−1.51878
$677$	−26668.2	−1.51395	−0.756973	−	0.653447i	$-0.773322\pi$
$677$	−26668.2	−1.51395	−0.756973	+	0.653447i	$0.773322\pi$
$678$	0	0
$679$	21305.1	1.20415
$680$	−1997.81	−0.112665
$681$	0	0
$682$	−3945.67	−0.221536
$683$	5584.15	0.312843	0.156421	−	0.987690i	$-0.450004\pi$
$683$	5584.15	0.312843	0.156421	+	0.987690i	$0.450004\pi$
$684$	0	0
$685$	7737.95	0.431608
$686$	−301.079	−0.0167569
$687$	0	0
$688$	270.173	0.0149713
$689$	−21906.9	−1.21130
$690$	0	0
$691$	−2702.39	−0.148776	−0.0743878	−	0.997229i	$-0.523700\pi$
$691$	−2702.39	−0.148776	−0.0743878	+	0.997229i	$0.523700\pi$
$692$	4605.17	0.252980
$693$	0	0
$694$	−22124.0	−1.21011
$695$	−3401.31	−0.185639
$696$	0	0
$697$	−4110.48	−0.223379
$698$	−10368.2	−0.562240
$699$	0	0
$700$	−13881.8	−0.749547
$701$	19885.1	1.07140	0.535700	−	0.844408i	$-0.320048\pi$
$701$	19885.1	1.07140	0.535700	+	0.844408i	$0.320048\pi$
$702$	0	0
$703$	−3065.56	−0.164466
$704$	13729.5	0.735016
$705$	0	0
$706$	−3736.14	−0.199166
$707$	13386.4	0.712088
$708$	0	0
$709$	−7798.56	−0.413090	−0.206545	−	0.978437i	$-0.566222\pi$
$709$	−7798.56	−0.413090	−0.206545	+	0.978437i	$0.566222\pi$
$710$	−4501.22	−0.237926
$711$	0	0
$712$	−14604.2	−0.768703
$713$	−152.483	−0.00800916
$714$	0	0
$715$	−9841.24	−0.514744
$716$	12227.7	0.638227
$717$	0	0
$718$	18727.4	0.973398
$719$	−1118.09	−0.0579939	−0.0289970	−	0.999579i	$-0.509231\pi$
$719$	−1118.09	−0.0579939	−0.0289970	+	0.999579i	$0.509231\pi$
$720$	0	0
$721$	−17984.2	−0.928939
$722$	6270.47	0.323217
$723$	0	0
$724$	−13831.9	−0.710025
$725$	20180.5	1.03377
$726$	0	0
$727$	−1741.54	−0.0888449	−0.0444225	−	0.999013i	$-0.514145\pi$
$727$	−1741.54	−0.0888449	−0.0444225	+	0.999013i	$0.514145\pi$
$728$	−54291.3	−2.76397
$729$	0	0
$730$	−3421.95	−0.173496
$731$	−1238.58	−0.0626683
$732$	0	0
$733$	−9461.10	−0.476744	−0.238372	−	0.971174i	$-0.576614\pi$
$733$	−9461.10	−0.476744	−0.238372	+	0.971174i	$0.576614\pi$
$734$	8092.33	0.406939
$735$	0	0
$736$	459.003	0.0229879
$737$	−4344.54	−0.217141
$738$	0	0
$739$	30605.3	1.52346	0.761728	−	0.647897i	$-0.224351\pi$
$739$	30605.3	1.52346	0.761728	+	0.647897i	$0.224351\pi$
$740$	712.240	0.0353817
$741$	0	0
$742$	−11785.9	−0.583121
$743$	−14556.5	−0.718745	−0.359373	−	0.933194i	$-0.617009\pi$
$743$	−14556.5	−0.718745	−0.359373	+	0.933194i	$0.617009\pi$
$744$	0	0
$745$	−6299.03	−0.309770
$746$	−8117.10	−0.398375
$747$	0	0
$748$	−4867.67	−0.237941
$749$	8361.55	0.407910
$750$	0	0
$751$	−17528.3	−0.851685	−0.425843	−	0.904797i	$-0.640022\pi$
$751$	−17528.3	−0.851685	−0.425843	+	0.904797i	$0.640022\pi$
$752$	−671.364	−0.0325560
$753$	0	0
$754$	28757.7	1.38898
$755$	4030.22	0.194271
$756$	0	0
$757$	−37789.1	−1.81436	−0.907179	−	0.420744i	$-0.861769\pi$
$757$	−37789.1	−1.81436	−0.907179	+	0.420744i	$0.861769\pi$
$758$	1623.48	0.0777934
$759$	0	0
$760$	4082.96	0.194874
$761$	−3292.90	−0.156856	−0.0784281	−	0.996920i	$-0.524990\pi$
$761$	−3292.90	−0.156856	−0.0784281	+	0.996920i	$0.524990\pi$
$762$	0	0
$763$	44101.2	2.09249
$764$	6480.52	0.306881
$765$	0	0
$766$	9352.52	0.441149
$767$	11443.5	0.538722
$768$	0	0
$769$	−28023.9	−1.31413	−0.657065	−	0.753834i	$-0.728202\pi$
$769$	−28023.9	−1.31413	−0.657065	+	0.753834i	$0.728202\pi$
$770$	−5294.61	−0.247798
$771$	0	0
$772$	5716.73	0.266515
$773$	−22131.0	−1.02975	−0.514875	−	0.857265i	$-0.672162\pi$
$773$	−22131.0	−1.02975	−0.514875	+	0.857265i	$0.672162\pi$
$774$	0	0
$775$	−6728.01	−0.311841
$776$	−19003.5	−0.879103
$777$	0	0
$778$	−6314.04	−0.290963
$779$	8400.67	0.386374
$780$	0	0
$781$	−30099.6	−1.37906
$782$	140.050	0.00640431
$783$	0	0
$784$	−2115.83	−0.0963845
$785$	4491.63	0.204220
$786$	0	0
$787$	5390.88	0.244173	0.122087	−	0.992519i	$-0.461041\pi$
$787$	5390.88	0.244173	0.122087	+	0.992519i	$0.461041\pi$
$788$	−22613.6	−1.02231
$789$	0	0
$790$	2081.82	0.0937567
$791$	22327.0	1.00361
$792$	0	0
$793$	−39722.4	−1.77880
$794$	−14106.3	−0.630495
$795$	0	0
$796$	−2533.51	−0.112811
$797$	−43815.4	−1.94733	−0.973664	−	0.227986i	$-0.926786\pi$
$797$	−43815.4	−1.94733	−0.973664	+	0.227986i	$0.926786\pi$
$798$	0	0
$799$	3077.80	0.136276
$800$	20252.6	0.895047
$801$	0	0
$802$	−15526.1	−0.683597
$803$	−22882.5	−1.00561
$804$	0	0
$805$	−204.613	−0.00895860
$806$	−9587.56	−0.418992
$807$	0	0
$808$	−11940.2	−0.519870
$809$	19752.4	0.858415	0.429208	−	0.903206i	$-0.358793\pi$
$809$	19752.4	0.858415	0.429208	+	0.903206i	$0.358793\pi$
$810$	0	0
$811$	21280.3	0.921398	0.460699	−	0.887556i	$-0.347599\pi$
$811$	21280.3	0.921398	0.460699	+	0.887556i	$0.347599\pi$
$812$	−20781.5	−0.898135
$813$	0	0
$814$	−3545.83	−0.152680
$815$	3752.81	0.161295
$816$	0	0
$817$	2531.31	0.108396
$818$	21790.0	0.931383
$819$	0	0
$820$	−1951.78	−0.0831208
$821$	19716.5	0.838136	0.419068	−	0.907955i	$-0.362357\pi$
$821$	19716.5	0.838136	0.419068	+	0.907955i	$0.362357\pi$
$822$	0	0
$823$	16269.0	0.689065	0.344532	−	0.938774i	$-0.388037\pi$
$823$	16269.0	0.689065	0.344532	+	0.938774i	$0.388037\pi$
$824$	16041.3	0.678185
$825$	0	0
$826$	6156.61	0.259341
$827$	27410.0	1.15253	0.576263	−	0.817264i	$-0.304510\pi$
$827$	27410.0	1.15253	0.576263	+	0.817264i	$0.304510\pi$
$828$	0	0
$829$	−25392.8	−1.06385	−0.531923	−	0.846792i	$-0.678531\pi$
$829$	−25392.8	−1.06385	−0.531923	+	0.846792i	$0.678531\pi$
$830$	8008.45	0.334913
$831$	0	0
$832$	33361.3	1.39014
$833$	9699.82	0.403456
$834$	0	0
$835$	8244.99	0.341712
$836$	9948.15	0.411560
$837$	0	0
$838$	−19645.0	−0.809814
$839$	20579.5	0.846823	0.423412	−	0.905937i	$-0.360832\pi$
$839$	20579.5	0.846823	0.423412	+	0.905937i	$0.360832\pi$
$840$	0	0
$841$	5821.76	0.238704
$842$	5795.48	0.237203
$843$	0	0
$844$	−10557.7	−0.430580
$845$	−17360.7	−0.706777
$846$	0	0
$847$	−703.094	−0.0285225
$848$	1537.17	0.0622484
$849$	0	0
$850$	6179.42	0.249356
$851$	−137.031	−0.00551980
$852$	0	0
$853$	6873.98	0.275921	0.137961	−	0.990438i	$-0.455945\pi$
$853$	6873.98	0.275921	0.137961	+	0.990438i	$0.455945\pi$
$854$	−21370.8	−0.856314
$855$	0	0
$856$	−7458.23	−0.297800
$857$	20623.8	0.822049	0.411025	−	0.911624i	$-0.365171\pi$
$857$	20623.8	0.822049	0.411025	+	0.911624i	$0.365171\pi$
$858$	0	0
$859$	40959.1	1.62690	0.813450	−	0.581635i	$-0.197587\pi$
$859$	40959.1	1.62690	0.813450	+	0.581635i	$0.197587\pi$
$860$	−588.115	−0.0233193
$861$	0	0
$862$	−314.819	−0.0124394
$863$	−33784.0	−1.33258	−0.666292	−	0.745691i	$-0.732120\pi$
$863$	−33784.0	−1.33258	−0.666292	+	0.745691i	$0.732120\pi$
$864$	0	0
$865$	2995.01	0.117727
$866$	−3868.52	−0.151799
$867$	0	0
$868$	6928.36	0.270926
$869$	13921.1	0.543430
$870$	0	0
$871$	−10556.8	−0.410680
$872$	−39336.8	−1.52765
$873$	0	0
$874$	−286.223	−0.0110774
$875$	−18748.0	−0.724340
$876$	0	0
$877$	−37125.7	−1.42947	−0.714736	−	0.699395i	$-0.753453\pi$
$877$	−37125.7	−1.42947	−0.714736	+	0.699395i	$0.753453\pi$
$878$	20066.7	0.771319
$879$	0	0
$880$	690.545	0.0264526
$881$	108.651	0.00415499	0.00207749	−	0.999998i	$-0.499339\pi$
$881$	108.651	0.00415499	0.00207749	+	0.999998i	$0.499339\pi$
$882$	0	0
$883$	−19622.4	−0.747843	−0.373921	−	0.927460i	$-0.621987\pi$
$883$	−19622.4	−0.747843	−0.373921	+	0.927460i	$0.621987\pi$
$884$	−11827.9	−0.450017
$885$	0	0
$886$	15255.5	0.578462
$887$	26012.1	0.984668	0.492334	−	0.870406i	$-0.336144\pi$
$887$	26012.1	0.984668	0.492334	+	0.870406i	$0.336144\pi$
$888$	0	0
$889$	58237.6	2.19710
$890$	−3460.74	−0.130342
$891$	0	0
$892$	−11732.5	−0.440397
$893$	−6290.17	−0.235714
$894$	0	0
$895$	7952.40	0.297005
$896$	−18434.3	−0.687327
$897$	0	0
$898$	−12549.9	−0.466365
$899$	−10072.0	−0.373660
$900$	0	0
$901$	−7047.01	−0.260566
$902$	9716.78	0.358684
$903$	0	0
$904$	−19915.0	−0.732702
$905$	−8995.70	−0.330417
$906$	0	0
$907$	−42258.2	−1.54703	−0.773517	−	0.633776i	$-0.781504\pi$
$907$	−42258.2	−1.54703	−0.773517	+	0.633776i	$0.781504\pi$
$908$	16613.5	0.607200
$909$	0	0
$910$	−12865.3	−0.468661
$911$	−6717.75	−0.244313	−0.122156	−	0.992511i	$-0.538981\pi$
$911$	−6717.75	−0.244313	−0.122156	+	0.992511i	$0.538981\pi$
$912$	0	0
$913$	53552.4	1.94121
$914$	−7399.49	−0.267783
$915$	0	0
$916$	−5298.78	−0.191132
$917$	−53486.9	−1.92616
$918$	0	0
$919$	46824.2	1.68073	0.840364	−	0.542023i	$-0.182341\pi$
$919$	46824.2	1.68073	0.840364	+	0.542023i	$0.182341\pi$
$920$	182.509	0.00654036
$921$	0	0
$922$	2151.44	0.0768479
$923$	−73138.7	−2.60822
$924$	0	0
$925$	−6046.21	−0.214917
$926$	−4743.14	−0.168325
$927$	0	0
$928$	30318.7	1.07248
$929$	−14300.0	−0.505025	−0.252513	−	0.967594i	$-0.581257\pi$
$929$	−14300.0	−0.505025	−0.252513	+	0.967594i	$0.581257\pi$
$930$	0	0
$931$	−19823.7	−0.697848
$932$	2420.41	0.0850679
$933$	0	0
$934$	−14143.4	−0.495489
$935$	−3165.73	−0.110728
$936$	0	0
$937$	44630.3	1.55604	0.778019	−	0.628241i	$-0.216225\pi$
$937$	44630.3	1.55604	0.778019	+	0.628241i	$0.216225\pi$
$938$	−5679.56	−0.197702
$939$	0	0
$940$	1461.43	0.0507092
$941$	12069.3	0.418116	0.209058	−	0.977903i	$-0.432960\pi$
$941$	12069.3	0.418116	0.209058	+	0.977903i	$0.432960\pi$
$942$	0	0
$943$	375.511	0.0129675
$944$	−802.970	−0.0276848
$945$	0	0
$946$	2927.89	0.100628
$947$	−23634.0	−0.810984	−0.405492	−	0.914099i	$-0.632900\pi$
$947$	−23634.0	−0.810984	−0.405492	+	0.914099i	$0.632900\pi$
$948$	0	0
$949$	−55602.0	−1.90192
$950$	−12629.0	−0.431304
$951$	0	0
$952$	−17464.4	−0.594564
$953$	31376.9	1.06652	0.533262	−	0.845950i	$-0.320966\pi$
$953$	31376.9	1.06652	0.533262	+	0.845950i	$0.320966\pi$
$954$	0	0
$955$	4214.67	0.142810
$956$	−6151.50	−0.208110
$957$	0	0
$958$	−16175.5	−0.545518
$959$	67643.6	2.27771
$960$	0	0
$961$	−26433.1	−0.887284
$962$	−8615.98	−0.288763
$963$	0	0
$964$	17146.2	0.572865
$965$	3717.93	0.124025
$966$	0	0
$967$	32665.7	1.08631	0.543154	−	0.839633i	$-0.317230\pi$
$967$	32665.7	1.08631	0.543154	+	0.839633i	$0.317230\pi$
$968$	627.137	0.0208233
$969$	0	0
$970$	−4503.22	−0.149062
$971$	41567.8	1.37381	0.686907	−	0.726745i	$-0.258968\pi$
$971$	41567.8	1.37381	0.686907	+	0.726745i	$0.258968\pi$
$972$	0	0
$973$	−29733.6	−0.979665
$974$	17380.6	0.571776
$975$	0	0
$976$	2787.26	0.0914119
$977$	25363.2	0.830542	0.415271	−	0.909698i	$-0.363687\pi$
$977$	25363.2	0.830542	0.415271	+	0.909698i	$0.363687\pi$
$978$	0	0
$979$	−23141.9	−0.755484
$980$	4605.77	0.150128
$981$	0	0
$982$	11755.7	0.382014
$983$	3961.23	0.128528	0.0642642	−	0.997933i	$-0.479530\pi$
$983$	3961.23	0.128528	0.0642642	+	0.997933i	$0.479530\pi$
$984$	0	0
$985$	−14707.0	−0.475740
$986$	9250.76	0.298787
$987$	0	0
$988$	24172.9	0.778384
$989$	113.150	0.00363798
$990$	0	0
$991$	27940.2	0.895610	0.447805	−	0.894131i	$-0.352206\pi$
$991$	27940.2	0.895610	0.447805	+	0.894131i	$0.352206\pi$
$992$	−10108.0	−0.323518
$993$	0	0
$994$	−39348.8	−1.25560
$995$	−1647.69	−0.0524978
$996$	0	0
$997$	−15531.3	−0.493363	−0.246681	−	0.969097i	$-0.579340\pi$
$997$	−15531.3	−0.493363	−0.246681	+	0.969097i	$0.579340\pi$
$998$	−21387.2	−0.678357
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.4.a.h.1.3		6
3.2	odd	2		43.4.a.b.1.4	✓	6
12.11	even	2		688.4.a.i.1.1		6
15.14	odd	2		1075.4.a.b.1.3		6
21.20	even	2		2107.4.a.c.1.4		6
129.128	even	2		1849.4.a.c.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.4.a.b.1.4	✓	6	3.2	odd	2
387.4.a.h.1.3		6	1.1	even	1	trivial
688.4.a.i.1.1		6	12.11	even	2
1075.4.a.b.1.3		6	15.14	odd	2
1849.4.a.c.1.3		6	129.128	even	2
2107.4.a.c.1.4		6	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	387.a (trivial)

\(f(q)\)	\(=\)	\(q-1.84774 q^{2} -4.58586 q^{4} -2.98245 q^{5} -26.0720 q^{7} +23.2554 q^{8} +O(q^{10})\)	\(q-1.84774 q^{2} -4.58586 q^{4} -2.98245 q^{5} -26.0720 q^{7} +23.2554 q^{8} +5.51080 q^{10} +36.8506 q^{11} +89.5430 q^{13} +48.1743 q^{14} -6.28309 q^{16} +28.8042 q^{17} -58.8677 q^{19} +13.6771 q^{20} -68.0904 q^{22} -2.63139 q^{23} -116.105 q^{25} -165.452 q^{26} +119.563 q^{28} -173.812 q^{29} +57.9476 q^{31} -174.434 q^{32} -53.2227 q^{34} +77.7586 q^{35} +52.0754 q^{37} +108.772 q^{38} -69.3581 q^{40} -142.704 q^{41} -43.0000 q^{43} -168.992 q^{44} +4.86213 q^{46} +106.853 q^{47} +336.750 q^{49} +214.532 q^{50} -410.631 q^{52} -244.652 q^{53} -109.905 q^{55} -606.315 q^{56} +321.160 q^{58} +127.799 q^{59} -443.613 q^{61} -107.072 q^{62} +372.573 q^{64} -267.058 q^{65} -117.896 q^{67} -132.092 q^{68} -143.678 q^{70} -816.799 q^{71} -620.953 q^{73} -96.2217 q^{74} +269.959 q^{76} -960.770 q^{77} +377.771 q^{79} +18.7390 q^{80} +263.680 q^{82} +1453.23 q^{83} -85.9072 q^{85} +79.4528 q^{86} +856.975 q^{88} -627.993 q^{89} -2334.57 q^{91} +12.0672 q^{92} -197.436 q^{94} +175.570 q^{95} -817.163 q^{97} -622.227 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8}+O(q^{10}) \) 6 * q - 6 * q^2 + 22 * q^4 - 43 * q^5 + 8 * q^7 - 54 * q^8	\( 6 q - 6 q^{2} + 22 q^{4} - 43 q^{5} + 8 q^{7} - 54 q^{8} + 57 q^{10} + 28 q^{11} + 56 q^{13} + 184 q^{14} - 54 q^{16} - 19 q^{17} - 75 q^{19} - 135 q^{20} - 504 q^{22} - 131 q^{23} + 105 q^{25} - 44 q^{26} - 404 q^{28} - 515 q^{29} + 237 q^{31} - 558 q^{32} - 107 q^{34} - 198 q^{35} + 269 q^{37} - 527 q^{38} + 613 q^{40} - 471 q^{41} - 258 q^{43} + 428 q^{44} - 67 q^{46} - 415 q^{47} + 350 q^{49} - 1335 q^{50} - 8 q^{52} - 450 q^{53} - 1732 q^{55} + 780 q^{56} - 1055 q^{58} - 356 q^{59} - 1328 q^{61} - 1603 q^{62} + 466 q^{64} + 62 q^{65} - 632 q^{67} - 571 q^{68} - 1902 q^{70} + 144 q^{71} + 864 q^{73} - 1207 q^{74} + 1005 q^{76} - 2660 q^{77} - 1613 q^{79} - 2399 q^{80} + 1673 q^{82} + 682 q^{83} + 84 q^{85} + 258 q^{86} - 608 q^{88} - 3378 q^{89} - 3900 q^{91} - 3491 q^{92} + 3197 q^{94} + 79 q^{95} - 55 q^{97} - 2398 q^{98}+O(q^{100}) \) 6 * q - 6 * q^2 + 22 * q^4 - 43 * q^5 + 8 * q^7 - 54 * q^8 + 57 * q^10 + 28 * q^11 + 56 * q^13 + 184 * q^14 - 54 * q^16 - 19 * q^17 - 75 * q^19 - 135 * q^20 - 504 * q^22 - 131 * q^23 + 105 * q^25 - 44 * q^26 - 404 * q^28 - 515 * q^29 + 237 * q^31 - 558 * q^32 - 107 * q^34 - 198 * q^35 + 269 * q^37 - 527 * q^38 + 613 * q^40 - 471 * q^41 - 258 * q^43 + 428 * q^44 - 67 * q^46 - 415 * q^47 + 350 * q^49 - 1335 * q^50 - 8 * q^52 - 450 * q^53 - 1732 * q^55 + 780 * q^56 - 1055 * q^58 - 356 * q^59 - 1328 * q^61 - 1603 * q^62 + 466 * q^64 + 62 * q^65 - 632 * q^67 - 571 * q^68 - 1902 * q^70 + 144 * q^71 + 864 * q^73 - 1207 * q^74 + 1005 * q^76 - 2660 * q^77 - 1613 * q^79 - 2399 * q^80 + 1673 * q^82 + 682 * q^83 + 84 * q^85 + 258 * q^86 - 608 * q^88 - 3378 * q^89 - 3900 * q^91 - 3491 * q^92 + 3197 * q^94 + 79 * q^95 - 55 * q^97 - 2398 * q^98

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