LMFDB - Embedded newform 405.4.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, 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("405.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	405.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:	$23.8957735523$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.37228 q^{2} -2.37228 q^{4} +5.00000 q^{5} +7.37228 q^{7} -24.6060 q^{8} +O(q^{10})$	\(q+2.37228 q^{2} -2.37228 q^{4} +5.00000 q^{5} +7.37228 q^{7} -24.6060 q^{8} +11.8614 q^{10} -4.13859 q^{11} -78.9783 q^{13} +17.4891 q^{14} -39.3940 q^{16} -33.3070 q^{17} +89.3070 q^{19} -11.8614 q^{20} -9.81791 q^{22} -199.562 q^{23} +25.0000 q^{25} -187.359 q^{26} -17.4891 q^{28} +50.4810 q^{29} -6.00000 q^{31} +103.394 q^{32} -79.0137 q^{34} +36.8614 q^{35} -290.277 q^{37} +211.861 q^{38} -123.030 q^{40} -53.3369 q^{41} -298.388 q^{43} +9.81791 q^{44} -473.418 q^{46} -418.905 q^{47} -288.649 q^{49} +59.3070 q^{50} +187.359 q^{52} +399.228 q^{53} -20.6930 q^{55} -181.402 q^{56} +119.755 q^{58} +98.2418 q^{59} -683.595 q^{61} -14.2337 q^{62} +560.432 q^{64} -394.891 q^{65} +225.500 q^{67} +79.0137 q^{68} +87.4456 q^{70} +512.951 q^{71} -994.318 q^{73} -688.619 q^{74} -211.861 q^{76} -30.5109 q^{77} -201.707 q^{79} -196.970 q^{80} -126.530 q^{82} +1107.64 q^{83} -166.535 q^{85} -707.861 q^{86} +101.834 q^{88} +372.269 q^{89} -582.250 q^{91} +473.418 q^{92} -993.760 q^{94} +446.535 q^{95} -139.231 q^{97} -684.758 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + q^{4} + 10 q^{5} + 9 q^{7} - 9 q^{8}+O(q^{10}) $ 2 * q - q^2 + q^4 + 10 * q^5 + 9 * q^7 - 9 * q^8	$ 2 q - q^{2} + q^{4} + 10 q^{5} + 9 q^{7} - 9 q^{8} - 5 q^{10} - 37 q^{11} - 112 q^{13} + 12 q^{14} - 119 q^{16} + 77 q^{17} + 35 q^{19} + 5 q^{20} + 101 q^{22} - 267 q^{23} + 50 q^{25} - 76 q^{26} - 12 q^{28} + 325 q^{29} - 12 q^{31} + 247 q^{32} - 451 q^{34} + 45 q^{35} - 638 q^{37} + 395 q^{38} - 45 q^{40} + 238 q^{41} - 97 q^{43} - 101 q^{44} - 246 q^{46} - 901 q^{47} - 629 q^{49} - 25 q^{50} + 76 q^{52} + 224 q^{53} - 185 q^{55} - 156 q^{56} - 806 q^{58} - 85 q^{59} - 247 q^{61} + 6 q^{62} + 713 q^{64} - 560 q^{65} - 606 q^{67} + 451 q^{68} + 60 q^{70} + 394 q^{71} - 811 q^{73} + 484 q^{74} - 395 q^{76} - 84 q^{77} - 840 q^{79} - 595 q^{80} - 1109 q^{82} - 387 q^{83} + 385 q^{85} - 1387 q^{86} - 411 q^{88} - 1065 q^{89} - 636 q^{91} + 246 q^{92} + 632 q^{94} + 175 q^{95} - 1031 q^{97} + 463 q^{98}+O(q^{100}) $ 2 * q - q^2 + q^4 + 10 * q^5 + 9 * q^7 - 9 * q^8 - 5 * q^10 - 37 * q^11 - 112 * q^13 + 12 * q^14 - 119 * q^16 + 77 * q^17 + 35 * q^19 + 5 * q^20 + 101 * q^22 - 267 * q^23 + 50 * q^25 - 76 * q^26 - 12 * q^28 + 325 * q^29 - 12 * q^31 + 247 * q^32 - 451 * q^34 + 45 * q^35 - 638 * q^37 + 395 * q^38 - 45 * q^40 + 238 * q^41 - 97 * q^43 - 101 * q^44 - 246 * q^46 - 901 * q^47 - 629 * q^49 - 25 * q^50 + 76 * q^52 + 224 * q^53 - 185 * q^55 - 156 * q^56 - 806 * q^58 - 85 * q^59 - 247 * q^61 + 6 * q^62 + 713 * q^64 - 560 * q^65 - 606 * q^67 + 451 * q^68 + 60 * q^70 + 394 * q^71 - 811 * q^73 + 484 * q^74 - 395 * q^76 - 84 * q^77 - 840 * q^79 - 595 * q^80 - 1109 * q^82 - 387 * q^83 + 385 * q^85 - 1387 * q^86 - 411 * q^88 - 1065 * q^89 - 636 * q^91 + 246 * q^92 + 632 * q^94 + 175 * q^95 - 1031 * q^97 + 463 * 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 $

$2$	2.37228	0.838728	0.419364	−	0.907818i	$-0.362253\pi$
$2$	2.37228	0.838728	0.419364	+	0.907818i	$0.362253\pi$
$3$	0	0
$3$	0	0
$4$	−2.37228	−0.296535
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	7.37228	0.398066	0.199033	−	0.979993i	$-0.436220\pi$
$7$	7.37228	0.398066	0.199033	+	0.979993i	$0.436220\pi$
$8$	−24.6060	−1.08744
$9$	0	0
$10$	11.8614	0.375091
$11$	−4.13859	−0.113439	−0.0567197	−	0.998390i	$-0.518064\pi$
$11$	−4.13859	−0.113439	−0.0567197	+	0.998390i	$0.518064\pi$
$12$	0	0
$13$	−78.9783	−1.68497	−0.842486	−	0.538719i	$-0.818909\pi$
$13$	−78.9783	−1.68497	−0.842486	+	0.538719i	$0.818909\pi$
$14$	17.4891	0.333869
$15$	0	0
$16$	−39.3940	−0.615532
$17$	−33.3070	−0.475185	−0.237592	−	0.971365i	$-0.576358\pi$
$17$	−33.3070	−0.475185	−0.237592	+	0.971365i	$0.576358\pi$
$18$	0	0
$19$	89.3070	1.07834	0.539169	−	0.842197i	$-0.318738\pi$
$19$	89.3070	1.07834	0.539169	+	0.842197i	$0.318738\pi$
$20$	−11.8614	−0.132615
$21$	0	0
$22$	−9.81791	−0.0951448
$23$	−199.562	−1.80920	−0.904601	−	0.426259i	$-0.859831\pi$
$23$	−199.562	−1.80920	−0.904601	+	0.426259i	$0.859831\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−187.359	−1.41323
$27$	0	0
$28$	−17.4891	−0.118041
$29$	50.4810	0.323245	0.161622	−	0.986853i	$-0.448327\pi$
$29$	50.4810	0.323245	0.161622	+	0.986853i	$0.448327\pi$
$30$	0	0
$31$	−6.00000	−0.0347623	−0.0173812	−	0.999849i	$-0.505533\pi$
$31$	−6.00000	−0.0347623	−0.0173812	+	0.999849i	$0.505533\pi$
$32$	103.394	0.571177
$33$	0	0
$34$	−79.0137	−0.398551
$35$	36.8614	0.178020
$36$	0	0
$37$	−290.277	−1.28976	−0.644882	−	0.764282i	$-0.723094\pi$
$37$	−290.277	−1.28976	−0.644882	+	0.764282i	$0.723094\pi$
$38$	211.861	0.904433
$39$	0	0
$40$	−123.030	−0.486318
$41$	−53.3369	−0.203166	−0.101583	−	0.994827i	$-0.532391\pi$
$41$	−53.3369	−0.203166	−0.101583	+	0.994827i	$0.532391\pi$
$42$	0	0
$43$	−298.388	−1.05823	−0.529114	−	0.848551i	$-0.677476\pi$
$43$	−298.388	−1.05823	−0.529114	+	0.848551i	$0.677476\pi$
$44$	9.81791	0.0336388
$45$	0	0
$46$	−473.418	−1.51743
$47$	−418.905	−1.30008	−0.650038	−	0.759902i	$-0.725247\pi$
$47$	−418.905	−1.30008	−0.650038	+	0.759902i	$0.725247\pi$
$48$	0	0
$49$	−288.649	−0.841544
$50$	59.3070	0.167746
$51$	0	0
$52$	187.359	0.499653
$53$	399.228	1.03468	0.517342	−	0.855779i	$-0.326922\pi$
$53$	399.228	1.03468	0.517342	+	0.855779i	$0.326922\pi$
$54$	0	0
$55$	−20.6930	−0.0507316
$56$	−181.402	−0.432873
$57$	0	0
$58$	119.755	0.271114
$59$	98.2418	0.216780	0.108390	−	0.994108i	$-0.465431\pi$
$59$	98.2418	0.216780	0.108390	+	0.994108i	$0.465431\pi$
$60$	0	0
$61$	−683.595	−1.43484	−0.717421	−	0.696640i	$-0.754678\pi$
$61$	−683.595	−1.43484	−0.717421	+	0.696640i	$0.754678\pi$
$62$	−14.2337	−0.0291561
$63$	0	0
$64$	560.432	1.09459
$65$	−394.891	−0.753542
$66$	0	0
$67$	225.500	0.411182	0.205591	−	0.978638i	$-0.434088\pi$
$67$	225.500	0.411182	0.205591	+	0.978638i	$0.434088\pi$
$68$	79.0137	0.140909
$69$	0	0
$70$	87.4456	0.149311
$71$	512.951	0.857410	0.428705	−	0.903445i	$-0.358970\pi$
$71$	512.951	0.857410	0.428705	+	0.903445i	$0.358970\pi$
$72$	0	0
$73$	−994.318	−1.59419	−0.797096	−	0.603852i	$-0.793632\pi$
$73$	−994.318	−1.59419	−0.797096	+	0.603852i	$0.793632\pi$
$74$	−688.619	−1.08176
$75$	0	0
$76$	−211.861	−0.319765
$77$	−30.5109	−0.0451563
$78$	0	0
$79$	−201.707	−0.287263	−0.143631	−	0.989631i	$-0.545878\pi$
$79$	−201.707	−0.287263	−0.143631	+	0.989631i	$0.545878\pi$
$80$	−196.970	−0.275274
$81$	0	0
$82$	−126.530	−0.170401
$83$	1107.64	1.46482	0.732408	−	0.680866i	$-0.238397\pi$
$83$	1107.64	1.46482	0.732408	+	0.680866i	$0.238397\pi$
$84$	0	0
$85$	−166.535	−0.212509
$86$	−707.861	−0.887566
$87$	0	0
$88$	101.834	0.123359
$89$	372.269	0.443375	0.221688	−	0.975118i	$-0.428843\pi$
$89$	372.269	0.443375	0.221688	+	0.975118i	$0.428843\pi$
$90$	0	0
$91$	−582.250	−0.670729
$92$	473.418	0.536492
$93$	0	0
$94$	−993.760	−1.09041
$95$	446.535	0.482248
$96$	0	0
$97$	−139.231	−0.145740	−0.0728700	−	0.997341i	$-0.523216\pi$
$97$	−139.231	−0.145740	−0.0728700	+	0.997341i	$0.523216\pi$
$98$	−684.758	−0.705826
$99$	0	0
$100$	−59.3070	−0.0593070
$101$	1971.74	1.94253	0.971267	−	0.237994i	$-0.0764898\pi$
$101$	1971.74	1.94253	0.971267	+	0.237994i	$0.0764898\pi$
$102$	0	0
$103$	1812.71	1.73409	0.867045	−	0.498230i	$-0.166017\pi$
$103$	1812.71	1.73409	0.867045	+	0.498230i	$0.166017\pi$
$104$	1943.34	1.83231
$105$	0	0
$106$	947.081	0.867818
$107$	−259.217	−0.234201	−0.117100	−	0.993120i	$-0.537360\pi$
$107$	−259.217	−0.234201	−0.117100	+	0.993120i	$0.537360\pi$
$108$	0	0
$109$	775.556	0.681512	0.340756	−	0.940152i	$-0.389317\pi$
$109$	775.556	0.681512	0.340756	+	0.940152i	$0.389317\pi$
$110$	−49.0895	−0.0425500
$111$	0	0
$112$	−290.424	−0.245022
$113$	−215.255	−0.179199	−0.0895997	−	0.995978i	$-0.528559\pi$
$113$	−215.255	−0.179199	−0.0895997	+	0.995978i	$0.528559\pi$
$114$	0	0
$115$	−997.812	−0.809100
$116$	−119.755	−0.0958534
$117$	0	0
$118$	233.057	0.181819
$119$	−245.549	−0.189155
$120$	0	0
$121$	−1313.87	−0.987132
$122$	−1621.68	−1.20344
$123$	0	0
$124$	14.2337	0.0103082
$125$	125.000	0.0894427
$126$	0	0
$127$	2424.76	1.69420	0.847098	−	0.531436i	$-0.178348\pi$
$127$	2424.76	1.69420	0.847098	+	0.531436i	$0.178348\pi$
$128$	502.350	0.346890
$129$	0	0
$130$	−936.793	−0.632017
$131$	260.495	0.173737	0.0868684	−	0.996220i	$-0.472314\pi$
$131$	260.495	0.173737	0.0868684	+	0.996220i	$0.472314\pi$
$132$	0	0
$133$	658.397	0.429250
$134$	534.949	0.344870
$135$	0	0
$136$	819.552	0.516735
$137$	−404.638	−0.252340	−0.126170	−	0.992009i	$-0.540268\pi$
$137$	−404.638	−0.252340	−0.126170	+	0.992009i	$0.540268\pi$
$138$	0	0
$139$	−1767.68	−1.07865	−0.539327	−	0.842097i	$-0.681321\pi$
$139$	−1767.68	−1.07865	−0.539327	+	0.842097i	$0.681321\pi$
$140$	−87.4456	−0.0527893
$141$	0	0
$142$	1216.86	0.719133
$143$	326.859	0.191142
$144$	0	0
$145$	252.405	0.144559
$146$	−2358.80	−1.33709
$147$	0	0
$148$	688.619	0.382460
$149$	1920.69	1.05603	0.528016	−	0.849234i	$-0.322936\pi$
$149$	1920.69	1.05603	0.528016	+	0.849234i	$0.322936\pi$
$150$	0	0
$151$	−2524.56	−1.36057	−0.680284	−	0.732949i	$-0.738143\pi$
$151$	−2524.56	−1.36057	−0.680284	+	0.732949i	$0.738143\pi$
$152$	−2197.49	−1.17263
$153$	0	0
$154$	−72.3804	−0.0378739
$155$	−30.0000	−0.0155462
$156$	0	0
$157$	1942.70	0.987544	0.493772	−	0.869592i	$-0.335618\pi$
$157$	1942.70	0.987544	0.493772	+	0.869592i	$0.335618\pi$
$158$	−478.505	−0.240935
$159$	0	0
$160$	516.970	0.255438
$161$	−1471.23	−0.720181
$162$	0	0
$163$	−1051.21	−0.505134	−0.252567	−	0.967579i	$-0.581275\pi$
$163$	−1051.21	−0.505134	−0.252567	+	0.967579i	$0.581275\pi$
$164$	126.530	0.0602460
$165$	0	0
$166$	2627.64	1.22858
$167$	−2816.44	−1.30505	−0.652524	−	0.757768i	$-0.726290\pi$
$167$	−2816.44	−1.30505	−0.652524	+	0.757768i	$0.726290\pi$
$168$	0	0
$169$	4040.56	1.83913
$170$	−395.068	−0.178237
$171$	0	0
$172$	707.861	0.313802
$173$	−1442.45	−0.633916	−0.316958	−	0.948440i	$-0.602661\pi$
$173$	−1442.45	−0.633916	−0.316958	+	0.948440i	$0.602661\pi$
$174$	0	0
$175$	184.307	0.0796132
$176$	163.036	0.0698255
$177$	0	0
$178$	883.126	0.371871
$179$	1534.89	0.640912	0.320456	−	0.947263i	$-0.396164\pi$
$179$	1534.89	0.640912	0.320456	+	0.947263i	$0.396164\pi$
$180$	0	0
$181$	−3650.43	−1.49908	−0.749542	−	0.661956i	$-0.769726\pi$
$181$	−3650.43	−1.49908	−0.749542	+	0.661956i	$0.769726\pi$
$182$	−1381.26	−0.562560
$183$	0	0
$184$	4910.43	1.96740
$185$	−1451.39	−0.576800
$186$	0	0
$187$	137.844	0.0539047
$188$	993.760	0.385518
$189$	0	0
$190$	1059.31	0.404475
$191$	1357.29	0.514188	0.257094	−	0.966386i	$-0.417235\pi$
$191$	1357.29	0.514188	0.257094	+	0.966386i	$0.417235\pi$
$192$	0	0
$193$	−1803.04	−0.672465	−0.336232	−	0.941779i	$-0.609153\pi$
$193$	−1803.04	−0.672465	−0.336232	+	0.941779i	$0.609153\pi$
$194$	−330.295	−0.122236
$195$	0	0
$196$	684.758	0.249547
$197$	−263.403	−0.0952624	−0.0476312	−	0.998865i	$-0.515167\pi$
$197$	−263.403	−0.0952624	−0.0476312	+	0.998865i	$0.515167\pi$
$198$	0	0
$199$	492.853	0.175565	0.0877824	−	0.996140i	$-0.472022\pi$
$199$	492.853	0.175565	0.0877824	+	0.996140i	$0.472022\pi$
$200$	−615.149	−0.217488
$201$	0	0
$202$	4677.53	1.62926
$203$	372.160	0.128673
$204$	0	0
$205$	−266.684	−0.0908588
$206$	4300.25	1.45443
$207$	0	0
$208$	3111.27	1.03715
$209$	−369.605	−0.122326
$210$	0	0
$211$	−1001.54	−0.326773	−0.163387	−	0.986562i	$-0.552242\pi$
$211$	−1001.54	−0.326773	−0.163387	+	0.986562i	$0.552242\pi$
$212$	−947.081	−0.306820
$213$	0	0
$214$	−614.937	−0.196431
$215$	−1491.94	−0.473254
$216$	0	0
$217$	−44.2337	−0.0138377
$218$	1839.84	0.571603
$219$	0	0
$220$	49.0895	0.0150437
$221$	2630.53	0.800673
$222$	0	0
$223$	561.149	0.168508	0.0842540	−	0.996444i	$-0.473149\pi$
$223$	561.149	0.168508	0.0842540	+	0.996444i	$0.473149\pi$
$224$	762.250	0.227366
$225$	0	0
$226$	−510.646	−0.150300
$227$	−3906.46	−1.14221	−0.571103	−	0.820878i	$-0.693484\pi$
$227$	−3906.46	−1.14221	−0.571103	+	0.820878i	$0.693484\pi$
$228$	0	0
$229$	2996.91	0.864809	0.432405	−	0.901680i	$-0.357665\pi$
$229$	2996.91	0.864809	0.432405	+	0.901680i	$0.357665\pi$
$230$	−2367.09	−0.678615
$231$	0	0
$232$	−1242.13	−0.351509
$233$	4194.30	1.17930	0.589651	−	0.807658i	$-0.299265\pi$
$233$	4194.30	1.17930	0.589651	+	0.807658i	$0.299265\pi$
$234$	0	0
$235$	−2094.52	−0.581412
$236$	−233.057	−0.0642827
$237$	0	0
$238$	−582.511	−0.158649
$239$	−166.472	−0.0450553	−0.0225276	−	0.999746i	$-0.507171\pi$
$239$	−166.472	−0.0450553	−0.0225276	+	0.999746i	$0.507171\pi$
$240$	0	0
$241$	614.544	0.164258	0.0821291	−	0.996622i	$-0.473828\pi$
$241$	614.544	0.164258	0.0821291	+	0.996622i	$0.473828\pi$
$242$	−3116.87	−0.827935
$243$	0	0
$244$	1621.68	0.425481
$245$	−1443.25	−0.376350
$246$	0	0
$247$	−7053.31	−1.81697
$248$	147.636	0.0378020
$249$	0	0
$250$	296.535	0.0750181
$251$	−6136.16	−1.54307	−0.771536	−	0.636185i	$-0.780511\pi$
$251$	−6136.16	−1.54307	−0.771536	+	0.636185i	$0.780511\pi$
$252$	0	0
$253$	825.908	0.205235
$254$	5752.22	1.42097
$255$	0	0
$256$	−3291.74	−0.803647
$257$	5450.40	1.32291	0.661453	−	0.749987i	$-0.269940\pi$
$257$	5450.40	1.32291	0.661453	+	0.749987i	$0.269940\pi$
$258$	0	0
$259$	−2140.01	−0.513411
$260$	936.793	0.223452
$261$	0	0
$262$	617.967	0.145718
$263$	−783.239	−0.183637	−0.0918186	−	0.995776i	$-0.529268\pi$
$263$	−783.239	−0.183637	−0.0918186	+	0.995776i	$0.529268\pi$
$264$	0	0
$265$	1996.14	0.462724
$266$	1561.90	0.360024
$267$	0	0
$268$	−534.949	−0.121930
$269$	141.019	0.0319632	0.0159816	−	0.999872i	$-0.494913\pi$
$269$	141.019	0.0319632	0.0159816	+	0.999872i	$0.494913\pi$
$270$	0	0
$271$	6375.83	1.42917	0.714583	−	0.699551i	$-0.246616\pi$
$271$	6375.83	1.42917	0.714583	+	0.699551i	$0.246616\pi$
$272$	1312.10	0.292491
$273$	0	0
$274$	−959.916	−0.211645
$275$	−103.465	−0.0226879
$276$	0	0
$277$	6823.17	1.48002	0.740008	−	0.672598i	$-0.234821\pi$
$277$	6823.17	1.48002	0.740008	+	0.672598i	$0.234821\pi$
$278$	−4193.44	−0.904697
$279$	0	0
$280$	−907.011	−0.193587
$281$	589.693	0.125189	0.0625945	−	0.998039i	$-0.480063\pi$
$281$	589.693	0.125189	0.0625945	+	0.998039i	$0.480063\pi$
$282$	0	0
$283$	−5976.03	−1.25526	−0.627629	−	0.778513i	$-0.715974\pi$
$283$	−5976.03	−1.25526	−0.627629	+	0.778513i	$0.715974\pi$
$284$	−1216.86	−0.254252
$285$	0	0
$286$	775.401	0.160316
$287$	−393.214	−0.0808736
$288$	0	0
$289$	−3803.64	−0.774199
$290$	598.776	0.121246
$291$	0	0
$292$	2358.80	0.472734
$293$	8057.49	1.60656	0.803282	−	0.595599i	$-0.203085\pi$
$293$	8057.49	1.60656	0.803282	+	0.595599i	$0.203085\pi$
$294$	0	0
$295$	491.209	0.0969467
$296$	7142.55	1.40254
$297$	0	0
$298$	4556.41	0.885724
$299$	15761.1	3.04845
$300$	0	0
$301$	−2199.80	−0.421244
$302$	−5988.96	−1.14115
$303$	0	0
$304$	−3518.16	−0.663752
$305$	−3417.97	−0.641681
$306$	0	0
$307$	−4546.58	−0.845235	−0.422618	−	0.906308i	$-0.638889\pi$
$307$	−4546.58	−0.845235	−0.422618	+	0.906308i	$0.638889\pi$
$308$	72.3804	0.0133904
$309$	0	0
$310$	−71.1684	−0.0130390
$311$	−4202.79	−0.766298	−0.383149	−	0.923687i	$-0.625160\pi$
$311$	−4202.79	−0.766298	−0.383149	+	0.923687i	$0.625160\pi$
$312$	0	0
$313$	−6973.28	−1.25927	−0.629637	−	0.776889i	$-0.716797\pi$
$313$	−6973.28	−1.25927	−0.629637	+	0.776889i	$0.716797\pi$
$314$	4608.63	0.828281
$315$	0	0
$316$	478.505	0.0851835
$317$	−7360.03	−1.30404	−0.652020	−	0.758202i	$-0.726078\pi$
$317$	−7360.03	−1.30404	−0.652020	+	0.758202i	$0.726078\pi$
$318$	0	0
$319$	−208.920	−0.0366687
$320$	2802.16	0.489517
$321$	0	0
$322$	−3490.17	−0.604036
$323$	−2974.55	−0.512410
$324$	0	0
$325$	−1974.46	−0.336994
$326$	−2493.76	−0.423670
$327$	0	0
$328$	1312.41	0.220931
$329$	−3088.28	−0.517516
$330$	0	0
$331$	−6834.12	−1.13486	−0.567428	−	0.823423i	$-0.692062\pi$
$331$	−6834.12	−1.13486	−0.567428	+	0.823423i	$0.692062\pi$
$332$	−2627.64	−0.434369
$333$	0	0
$334$	−6681.39	−1.09458
$335$	1127.50	0.183886
$336$	0	0
$337$	−6928.48	−1.11994	−0.559968	−	0.828514i	$-0.689186\pi$
$337$	−6928.48	−1.11994	−0.559968	+	0.828514i	$0.689186\pi$
$338$	9585.35	1.54253
$339$	0	0
$340$	395.068	0.0630164
$341$	24.8316	0.00394341
$342$	0	0
$343$	−4656.70	−0.733055
$344$	7342.14	1.15076
$345$	0	0
$346$	−3421.90	−0.531683
$347$	−7540.60	−1.16657	−0.583286	−	0.812267i	$-0.698233\pi$
$347$	−7540.60	−1.16657	−0.583286	+	0.812267i	$0.698233\pi$
$348$	0	0
$349$	−1844.17	−0.282854	−0.141427	−	0.989949i	$-0.545169\pi$
$349$	−1844.17	−0.282854	−0.141427	+	0.989949i	$0.545169\pi$
$350$	437.228	0.0667738
$351$	0	0
$352$	−427.906	−0.0647939
$353$	−7097.49	−1.07015	−0.535073	−	0.844806i	$-0.679716\pi$
$353$	−7097.49	−1.07015	−0.535073	+	0.844806i	$0.679716\pi$
$354$	0	0
$355$	2564.75	0.383445
$356$	−883.126	−0.131476
$357$	0	0
$358$	3641.19	0.537550
$359$	7709.65	1.13342	0.566712	−	0.823916i	$-0.308215\pi$
$359$	7709.65	1.13342	0.566712	+	0.823916i	$0.308215\pi$
$360$	0	0
$361$	1116.75	0.162815
$362$	−8659.85	−1.25732
$363$	0	0
$364$	1381.26	0.198895
$365$	−4971.59	−0.712945
$366$	0	0
$367$	−5256.39	−0.747633	−0.373816	−	0.927503i	$-0.621951\pi$
$367$	−5256.39	−0.747633	−0.373816	+	0.927503i	$0.621951\pi$
$368$	7861.57	1.11362
$369$	0	0
$370$	−3443.10	−0.483778
$371$	2943.22	0.411872
$372$	0	0
$373$	996.454	0.138323	0.0691615	−	0.997605i	$-0.477968\pi$
$373$	996.454	0.138323	0.0691615	+	0.997605i	$0.477968\pi$
$374$	327.005	0.0452114
$375$	0	0
$376$	10307.6	1.41376
$377$	−3986.90	−0.544658
$378$	0	0
$379$	−2735.20	−0.370707	−0.185354	−	0.982672i	$-0.559343\pi$
$379$	−2735.20	−0.370707	−0.185354	+	0.982672i	$0.559343\pi$
$380$	−1059.31	−0.143003
$381$	0	0
$382$	3219.87	0.431264
$383$	−8939.46	−1.19265	−0.596325	−	0.802743i	$-0.703373\pi$
$383$	−8939.46	−1.19265	−0.596325	+	0.802743i	$0.703373\pi$
$384$	0	0
$385$	−152.554	−0.0201945
$386$	−4277.32	−0.564015
$387$	0	0
$388$	330.295	0.0432170
$389$	−11133.9	−1.45118	−0.725590	−	0.688127i	$-0.758433\pi$
$389$	−11133.9	−1.45118	−0.725590	+	0.688127i	$0.758433\pi$
$390$	0	0
$391$	6646.83	0.859705
$392$	7102.50	0.915129
$393$	0	0
$394$	−624.866	−0.0798992
$395$	−1008.53	−0.128468
$396$	0	0
$397$	−9479.40	−1.19838	−0.599191	−	0.800606i	$-0.704511\pi$
$397$	−9479.40	−1.19838	−0.599191	+	0.800606i	$0.704511\pi$
$398$	1169.19	0.147251
$399$	0	0
$400$	−984.851	−0.123106
$401$	9427.60	1.17404	0.587022	−	0.809571i	$-0.300300\pi$
$401$	9427.60	1.17404	0.587022	+	0.809571i	$0.300300\pi$
$402$	0	0
$403$	473.870	0.0585735
$404$	−4677.53	−0.576029
$405$	0	0
$406$	882.869	0.107921
$407$	1201.34	0.146310
$408$	0	0
$409$	408.186	0.0493484	0.0246742	−	0.999696i	$-0.492145\pi$
$409$	408.186	0.0493484	0.0246742	+	0.999696i	$0.492145\pi$
$410$	−632.650	−0.0762058
$411$	0	0
$412$	−4300.25	−0.514219
$413$	724.266	0.0862925
$414$	0	0
$415$	5538.22	0.655085
$416$	−8165.88	−0.962416
$417$	0	0
$418$	−876.808	−0.102598
$419$	8813.88	1.02765	0.513826	−	0.857894i	$-0.328228\pi$
$419$	8813.88	1.02765	0.513826	+	0.857894i	$0.328228\pi$
$420$	0	0
$421$	−2349.37	−0.271974	−0.135987	−	0.990711i	$-0.543421\pi$
$421$	−2349.37	−0.271974	−0.135987	+	0.990711i	$0.543421\pi$
$422$	−2375.94	−0.274074
$423$	0	0
$424$	−9823.40	−1.12516
$425$	−832.676	−0.0950370
$426$	0	0
$427$	−5039.65	−0.571161
$428$	614.937	0.0694488
$429$	0	0
$430$	−3539.31	−0.396931
$431$	4481.16	0.500812	0.250406	−	0.968141i	$-0.419436\pi$
$431$	4481.16	0.500812	0.250406	+	0.968141i	$0.419436\pi$
$432$	0	0
$433$	3422.69	0.379871	0.189935	−	0.981797i	$-0.439172\pi$
$433$	3422.69	0.379871	0.189935	+	0.981797i	$0.439172\pi$
$434$	−104.935	−0.0116061
$435$	0	0
$436$	−1839.84	−0.202092
$437$	−17822.3	−1.95093
$438$	0	0
$439$	−8129.18	−0.883792	−0.441896	−	0.897066i	$-0.645694\pi$
$439$	−8129.18	−0.883792	−0.441896	+	0.897066i	$0.645694\pi$
$440$	509.171	0.0551676
$441$	0	0
$442$	6240.36	0.671547
$443$	2350.78	0.252119	0.126060	−	0.992023i	$-0.459767\pi$
$443$	2350.78	0.252119	0.126060	+	0.992023i	$0.459767\pi$
$444$	0	0
$445$	1861.34	0.198283
$446$	1331.20	0.141332
$447$	0	0
$448$	4131.66	0.435720
$449$	4760.99	0.500412	0.250206	−	0.968193i	$-0.419502\pi$
$449$	4760.99	0.500412	0.250206	+	0.968193i	$0.419502\pi$
$450$	0	0
$451$	220.740	0.0230471
$452$	510.646	0.0531389
$453$	0	0
$454$	−9267.22	−0.958000
$455$	−2911.25	−0.299959
$456$	0	0
$457$	2229.08	0.228167	0.114083	−	0.993471i	$-0.463607\pi$
$457$	2229.08	0.228167	0.114083	+	0.993471i	$0.463607\pi$
$458$	7109.51	0.725340
$459$	0	0
$460$	2367.09	0.239927
$461$	5736.62	0.579569	0.289784	−	0.957092i	$-0.406416\pi$
$461$	5736.62	0.579569	0.289784	+	0.957092i	$0.406416\pi$
$462$	0	0
$463$	−3603.98	−0.361752	−0.180876	−	0.983506i	$-0.557893\pi$
$463$	−3603.98	−0.361752	−0.180876	+	0.983506i	$0.557893\pi$
$464$	−1988.65	−0.198967
$465$	0	0
$466$	9950.05	0.989114
$467$	−3780.37	−0.374593	−0.187296	−	0.982303i	$-0.559972\pi$
$467$	−3780.37	−0.374593	−0.187296	+	0.982303i	$0.559972\pi$
$468$	0	0
$469$	1662.45	0.163677
$470$	−4968.80	−0.487646
$471$	0	0
$472$	−2417.33	−0.235735
$473$	1234.91	0.120045
$474$	0	0
$475$	2232.68	0.215668
$476$	582.511	0.0560911
$477$	0	0
$478$	−394.920	−0.0377891
$479$	−14461.2	−1.37943	−0.689715	−	0.724081i	$-0.742264\pi$
$479$	−14461.2	−1.37943	−0.689715	+	0.724081i	$0.742264\pi$
$480$	0	0
$481$	22925.6	2.17322
$482$	1457.87	0.137768
$483$	0	0
$484$	3116.87	0.292719
$485$	−696.156	−0.0651769
$486$	0	0
$487$	3581.74	0.333273	0.166636	−	0.986018i	$-0.446709\pi$
$487$	3581.74	0.333273	0.166636	+	0.986018i	$0.446709\pi$
$488$	16820.5	1.56031
$489$	0	0
$490$	−3423.79	−0.315655
$491$	−7015.39	−0.644807	−0.322403	−	0.946602i	$-0.604491\pi$
$491$	−7015.39	−0.644807	−0.322403	+	0.946602i	$0.604491\pi$
$492$	0	0
$493$	−1681.37	−0.153601
$494$	−16732.4	−1.52394
$495$	0	0
$496$	236.364	0.0213973
$497$	3781.62	0.341305
$498$	0	0
$499$	2519.79	0.226055	0.113028	−	0.993592i	$-0.463945\pi$
$499$	2519.79	0.226055	0.113028	+	0.993592i	$0.463945\pi$
$500$	−296.535	−0.0265229
$501$	0	0
$502$	−14556.7	−1.29422
$503$	−4989.32	−0.442272	−0.221136	−	0.975243i	$-0.570976\pi$
$503$	−4989.32	−0.442272	−0.221136	+	0.975243i	$0.570976\pi$
$504$	0	0
$505$	9858.72	0.868727
$506$	1959.29	0.172136
$507$	0	0
$508$	−5752.22	−0.502389
$509$	4719.52	0.410980	0.205490	−	0.978659i	$-0.434121\pi$
$509$	4719.52	0.410980	0.205490	+	0.978659i	$0.434121\pi$
$510$	0	0
$511$	−7330.39	−0.634594
$512$	−11827.7	−1.02093
$513$	0	0
$514$	12929.9	1.10956
$515$	9063.53	0.775509
$516$	0	0
$517$	1733.68	0.147480
$518$	−5076.69	−0.430612
$519$	0	0
$520$	9716.68	0.819432
$521$	−10711.0	−0.900682	−0.450341	−	0.892857i	$-0.648698\pi$
$521$	−10711.0	−0.900682	−0.450341	+	0.892857i	$0.648698\pi$
$522$	0	0
$523$	−10566.4	−0.883433	−0.441717	−	0.897155i	$-0.645630\pi$
$523$	−10566.4	−0.883433	−0.441717	+	0.897155i	$0.645630\pi$
$524$	−617.967	−0.0515191
$525$	0	0
$526$	−1858.06	−0.154022
$527$	199.842	0.0165185
$528$	0	0
$529$	27658.2	2.27321
$530$	4735.41	0.388100
$531$	0	0
$532$	−1561.90	−0.127288
$533$	4212.45	0.342329
$534$	0	0
$535$	−1296.09	−0.104738
$536$	−5548.64	−0.447136
$537$	0	0
$538$	334.537	0.0268084
$539$	1194.60	0.0954642
$540$	0	0
$541$	−6595.81	−0.524170	−0.262085	−	0.965045i	$-0.584410\pi$
$541$	−6595.81	−0.524170	−0.262085	+	0.965045i	$0.584410\pi$
$542$	15125.3	1.19868
$543$	0	0
$544$	−3443.75	−0.271415
$545$	3877.78	0.304782
$546$	0	0
$547$	−6388.78	−0.499387	−0.249693	−	0.968325i	$-0.580330\pi$
$547$	−6388.78	−0.499387	−0.249693	+	0.968325i	$0.580330\pi$
$548$	959.916	0.0748277
$549$	0	0
$550$	−245.448	−0.0190290
$551$	4508.31	0.348567
$552$	0	0
$553$	−1487.04	−0.114350
$554$	16186.5	1.24133
$555$	0	0
$556$	4193.44	0.319859
$557$	15992.4	1.21655	0.608276	−	0.793726i	$-0.291862\pi$
$557$	15992.4	1.21655	0.608276	+	0.793726i	$0.291862\pi$
$558$	0	0
$559$	23566.2	1.78308
$560$	−1452.12	−0.109577
$561$	0	0
$562$	1398.92	0.105000
$563$	−6351.12	−0.475431	−0.237716	−	0.971335i	$-0.576399\pi$
$563$	−6351.12	−0.475431	−0.237716	+	0.971335i	$0.576399\pi$
$564$	0	0
$565$	−1076.28	−0.0801404
$566$	−14176.8	−1.05282
$567$	0	0
$568$	−12621.7	−0.932382
$569$	9620.42	0.708803	0.354402	−	0.935093i	$-0.384685\pi$
$569$	9620.42	0.708803	0.354402	+	0.935093i	$0.384685\pi$
$570$	0	0
$571$	−5064.60	−0.371186	−0.185593	−	0.982627i	$-0.559421\pi$
$571$	−5064.60	−0.371186	−0.185593	+	0.982627i	$0.559421\pi$
$572$	−775.401	−0.0566803
$573$	0	0
$574$	−932.815	−0.0678309
$575$	−4989.06	−0.361840
$576$	0	0
$577$	11355.1	0.819273	0.409637	−	0.912249i	$-0.365655\pi$
$577$	11355.1	0.819273	0.409637	+	0.912249i	$0.365655\pi$
$578$	−9023.31	−0.649343
$579$	0	0
$580$	−598.776	−0.0428669
$581$	8165.86	0.583093
$582$	0	0
$583$	−1652.24	−0.117374
$584$	24466.2	1.73359
$585$	0	0
$586$	19114.6	1.34747
$587$	−10032.0	−0.705391	−0.352696	−	0.935738i	$-0.614735\pi$
$587$	−10032.0	−0.705391	−0.352696	+	0.935738i	$0.614735\pi$
$588$	0	0
$589$	−535.842	−0.0374856
$590$	1165.29	0.0813120
$591$	0	0
$592$	11435.2	0.793891
$593$	−1325.12	−0.0917643	−0.0458821	−	0.998947i	$-0.514610\pi$
$593$	−1325.12	−0.0917643	−0.0458821	+	0.998947i	$0.514610\pi$
$594$	0	0
$595$	−1227.74	−0.0845926
$596$	−4556.41	−0.313151
$597$	0	0
$598$	37389.8	2.55682
$599$	16571.6	1.13038	0.565188	−	0.824962i	$-0.308804\pi$
$599$	16571.6	1.13038	0.565188	+	0.824962i	$0.308804\pi$
$600$	0	0
$601$	17209.0	1.16800	0.584001	−	0.811753i	$-0.301486\pi$
$601$	17209.0	1.16800	0.584001	+	0.811753i	$0.301486\pi$
$602$	−5218.55	−0.353310
$603$	0	0
$604$	5988.96	0.403456
$605$	−6569.36	−0.441459
$606$	0	0
$607$	4177.98	0.279373	0.139686	−	0.990196i	$-0.455391\pi$
$607$	4177.98	0.279373	0.139686	+	0.990196i	$0.455391\pi$
$608$	9233.81	0.615922
$609$	0	0
$610$	−8108.40	−0.538196
$611$	33084.4	2.19059
$612$	0	0
$613$	14944.2	0.984649	0.492324	−	0.870412i	$-0.336147\pi$
$613$	14944.2	0.984649	0.492324	+	0.870412i	$0.336147\pi$
$614$	−10785.8	−0.708922
$615$	0	0
$616$	750.750	0.0491048
$617$	20502.9	1.33779	0.668893	−	0.743358i	$-0.266768\pi$
$617$	20502.9	1.33779	0.668893	+	0.743358i	$0.266768\pi$
$618$	0	0
$619$	2728.53	0.177171	0.0885854	−	0.996069i	$-0.471765\pi$
$619$	2728.53	0.177171	0.0885854	+	0.996069i	$0.471765\pi$
$620$	71.1684	0.00460999
$621$	0	0
$622$	−9970.21	−0.642715
$623$	2744.47	0.176493
$624$	0	0
$625$	625.000	0.0400000
$626$	−16542.6	−1.05619
$627$	0	0
$628$	−4608.63	−0.292841
$629$	9668.27	0.612876
$630$	0	0
$631$	−3393.08	−0.214067	−0.107034	−	0.994255i	$-0.534135\pi$
$631$	−3393.08	−0.214067	−0.107034	+	0.994255i	$0.534135\pi$
$632$	4963.19	0.312381
$633$	0	0
$634$	−17460.1	−1.09373
$635$	12123.8	0.757668
$636$	0	0
$637$	22797.0	1.41798
$638$	−495.618	−0.0307550
$639$	0	0
$640$	2511.75	0.155134
$641$	13561.8	0.835659	0.417830	−	0.908525i	$-0.362791\pi$
$641$	13561.8	0.835659	0.417830	+	0.908525i	$0.362791\pi$
$642$	0	0
$643$	12966.9	0.795279	0.397639	−	0.917542i	$-0.369830\pi$
$643$	12966.9	0.795279	0.397639	+	0.917542i	$0.369830\pi$
$644$	3490.17	0.213559
$645$	0	0
$646$	−7056.47	−0.429773
$647$	25837.9	1.57000	0.785001	−	0.619495i	$-0.212663\pi$
$647$	25837.9	1.57000	0.785001	+	0.619495i	$0.212663\pi$
$648$	0	0
$649$	−406.583	−0.0245913
$650$	−4683.97	−0.282647
$651$	0	0
$652$	2493.76	0.149790
$653$	−28266.9	−1.69398	−0.846989	−	0.531611i	$-0.821587\pi$
$653$	−28266.9	−1.69398	−0.846989	+	0.531611i	$0.821587\pi$
$654$	0	0
$655$	1302.47	0.0776975
$656$	2101.15	0.125055
$657$	0	0
$658$	−7326.28	−0.434055
$659$	−15957.8	−0.943292	−0.471646	−	0.881788i	$-0.656340\pi$
$659$	−15957.8	−0.943292	−0.471646	+	0.881788i	$0.656340\pi$
$660$	0	0
$661$	24057.8	1.41564	0.707821	−	0.706392i	$-0.249678\pi$
$661$	24057.8	1.41564	0.707821	+	0.706392i	$0.249678\pi$
$662$	−16212.5	−0.951836
$663$	0	0
$664$	−27254.6	−1.59290
$665$	3291.98	0.191966
$666$	0	0
$667$	−10074.1	−0.584815
$668$	6681.39	0.386992
$669$	0	0
$670$	2674.74	0.154230
$671$	2829.12	0.162768
$672$	0	0
$673$	1253.04	0.0717698	0.0358849	−	0.999356i	$-0.488575\pi$
$673$	1253.04	0.0717698	0.0358849	+	0.999356i	$0.488575\pi$
$674$	−16436.3	−0.939321
$675$	0	0
$676$	−9585.35	−0.545366
$677$	880.728	0.0499987	0.0249994	−	0.999687i	$-0.492042\pi$
$677$	880.728	0.0499987	0.0249994	+	0.999687i	$0.492042\pi$
$678$	0	0
$679$	−1026.45	−0.0580141
$680$	4097.76	0.231091
$681$	0	0
$682$	58.9074	0.00330745
$683$	−26686.4	−1.49506	−0.747531	−	0.664227i	$-0.768761\pi$
$683$	−26686.4	−1.49506	−0.747531	+	0.664227i	$0.768761\pi$
$684$	0	0
$685$	−2023.19	−0.112850
$686$	−11047.0	−0.614834
$687$	0	0
$688$	11754.7	0.651373
$689$	−31530.3	−1.74341
$690$	0	0
$691$	−171.412	−0.00943678	−0.00471839	−	0.999989i	$-0.501502\pi$
$691$	−171.412	−0.00943678	−0.00471839	+	0.999989i	$0.501502\pi$
$692$	3421.90	0.187978
$693$	0	0
$694$	−17888.4	−0.978437
$695$	−8838.41	−0.482388
$696$	0	0
$697$	1776.49	0.0965416
$698$	−4374.88	−0.237238
$699$	0	0
$700$	−437.228	−0.0236081
$701$	−14229.6	−0.766684	−0.383342	−	0.923607i	$-0.625227\pi$
$701$	−14229.6	−0.766684	−0.383342	+	0.923607i	$0.625227\pi$
$702$	0	0
$703$	−25923.8	−1.39080
$704$	−2319.40	−0.124170
$705$	0	0
$706$	−16837.2	−0.897561
$707$	14536.3	0.773256
$708$	0	0
$709$	−7774.08	−0.411794	−0.205897	−	0.978574i	$-0.566011\pi$
$709$	−7774.08	−0.411794	−0.205897	+	0.978574i	$0.566011\pi$
$710$	6084.32	0.321606
$711$	0	0
$712$	−9160.03	−0.482144
$713$	1197.37	0.0628921
$714$	0	0
$715$	1634.29	0.0854813
$716$	−3641.19	−0.190053
$717$	0	0
$718$	18289.4	0.950635
$719$	−22091.8	−1.14588	−0.572939	−	0.819598i	$-0.694197\pi$
$719$	−22091.8	−1.14588	−0.572939	+	0.819598i	$0.694197\pi$
$720$	0	0
$721$	13363.8	0.690282
$722$	2649.24	0.136557
$723$	0	0
$724$	8659.85	0.444531
$725$	1262.03	0.0646489
$726$	0	0
$727$	6643.48	0.338918	0.169459	−	0.985537i	$-0.445798\pi$
$727$	6643.48	0.338918	0.169459	+	0.985537i	$0.445798\pi$
$728$	14326.8	0.729378
$729$	0	0
$730$	−11794.0	−0.597967
$731$	9938.43	0.502854
$732$	0	0
$733$	−3122.81	−0.157358	−0.0786791	−	0.996900i	$-0.525070\pi$
$733$	−3122.81	−0.157358	−0.0786791	+	0.996900i	$0.525070\pi$
$734$	−12469.6	−0.627061
$735$	0	0
$736$	−20633.6	−1.03337
$737$	−933.252	−0.0466442
$738$	0	0
$739$	19549.5	0.973127	0.486563	−	0.873645i	$-0.338250\pi$
$739$	19549.5	0.973127	0.486563	+	0.873645i	$0.338250\pi$
$740$	3443.10	0.171042
$741$	0	0
$742$	6982.15	0.345449
$743$	9391.85	0.463733	0.231866	−	0.972748i	$-0.425517\pi$
$743$	9391.85	0.463733	0.231866	+	0.972748i	$0.425517\pi$
$744$	0	0
$745$	9603.44	0.472272
$746$	2363.87	0.116015
$747$	0	0
$748$	−327.005	−0.0159846
$749$	−1911.02	−0.0932274
$750$	0	0
$751$	−18272.0	−0.887824	−0.443912	−	0.896070i	$-0.646410\pi$
$751$	−18272.0	−0.887824	−0.443912	+	0.896070i	$0.646410\pi$
$752$	16502.4	0.800238
$753$	0	0
$754$	−9458.06	−0.456820
$755$	−12622.8	−0.608464
$756$	0	0
$757$	2016.30	0.0968082	0.0484041	−	0.998828i	$-0.484586\pi$
$757$	2016.30	0.0968082	0.0484041	+	0.998828i	$0.484586\pi$
$758$	−6488.68	−0.310923
$759$	0	0
$760$	−10987.4	−0.524416
$761$	−17692.8	−0.842789	−0.421395	−	0.906877i	$-0.638459\pi$
$761$	−17692.8	−0.842789	−0.421395	+	0.906877i	$0.638459\pi$
$762$	0	0
$763$	5717.62	0.271287
$764$	−3219.87	−0.152475
$765$	0	0
$766$	−21206.9	−1.00031
$767$	−7758.96	−0.365267
$768$	0	0
$769$	−9759.72	−0.457665	−0.228833	−	0.973466i	$-0.573491\pi$
$769$	−9759.72	−0.457665	−0.228833	+	0.973466i	$0.573491\pi$
$770$	−361.902	−0.0169377
$771$	0	0
$772$	4277.32	0.199409
$773$	−10338.3	−0.481038	−0.240519	−	0.970644i	$-0.577318\pi$
$773$	−10338.3	−0.481038	−0.240519	+	0.970644i	$0.577318\pi$
$774$	0	0
$775$	−150.000	−0.00695246
$776$	3425.92	0.158484
$777$	0	0
$778$	−26412.7	−1.21715
$779$	−4763.36	−0.219082
$780$	0	0
$781$	−2122.90	−0.0972640
$782$	15768.2	0.721059
$783$	0	0
$784$	11371.1	0.517997
$785$	9713.50	0.441643
$786$	0	0
$787$	4727.29	0.214116	0.107058	−	0.994253i	$-0.465857\pi$
$787$	4727.29	0.214116	0.107058	+	0.994253i	$0.465857\pi$
$788$	624.866	0.0282487
$789$	0	0
$790$	−2392.52	−0.107750
$791$	−1586.92	−0.0713331
$792$	0	0
$793$	53989.1	2.41767
$794$	−22487.8	−1.00512
$795$	0	0
$796$	−1169.19	−0.0520612
$797$	28198.9	1.25327	0.626634	−	0.779313i	$-0.284432\pi$
$797$	28198.9	1.25327	0.626634	+	0.779313i	$0.284432\pi$
$798$	0	0
$799$	13952.5	0.617776
$800$	2584.85	0.114235
$801$	0	0
$802$	22364.9	0.984704
$803$	4115.08	0.180844
$804$	0	0
$805$	−7356.15	−0.322075
$806$	1124.15	0.0491272
$807$	0	0
$808$	−48516.7	−2.11239
$809$	−27310.5	−1.18688	−0.593439	−	0.804879i	$-0.702230\pi$
$809$	−27310.5	−1.18688	−0.593439	+	0.804879i	$0.702230\pi$
$810$	0	0
$811$	−18045.0	−0.781312	−0.390656	−	0.920537i	$-0.627752\pi$
$811$	−18045.0	−0.781312	−0.390656	+	0.920537i	$0.627752\pi$
$812$	−882.869	−0.0381559
$813$	0	0
$814$	2849.91	0.122714
$815$	−5256.03	−0.225903
$816$	0	0
$817$	−26648.2	−1.14113
$818$	968.333	0.0413899
$819$	0	0
$820$	632.650	0.0269428
$821$	−12577.7	−0.534671	−0.267335	−	0.963604i	$-0.586143\pi$
$821$	−12577.7	−0.534671	−0.267335	+	0.963604i	$0.586143\pi$
$822$	0	0
$823$	−34968.5	−1.48107	−0.740537	−	0.672015i	$-0.765429\pi$
$823$	−34968.5	−1.48107	−0.740537	+	0.672015i	$0.765429\pi$
$824$	−44603.4	−1.88572
$825$	0	0
$826$	1718.16	0.0723759
$827$	−6735.01	−0.283191	−0.141596	−	0.989925i	$-0.545223\pi$
$827$	−6735.01	−0.283191	−0.141596	+	0.989925i	$0.545223\pi$
$828$	0	0
$829$	−2867.97	−0.120155	−0.0600777	−	0.998194i	$-0.519135\pi$
$829$	−2867.97	−0.120155	−0.0600777	+	0.998194i	$0.519135\pi$
$830$	13138.2	0.549438
$831$	0	0
$832$	−44261.9	−1.84436
$833$	9614.06	0.399889
$834$	0	0
$835$	−14082.2	−0.583635
$836$	876.808	0.0362740
$837$	0	0
$838$	20909.0	0.861921
$839$	27855.8	1.14623	0.573116	−	0.819474i	$-0.305734\pi$
$839$	27855.8	1.14623	0.573116	+	0.819474i	$0.305734\pi$
$840$	0	0
$841$	−21840.7	−0.895513
$842$	−5573.36	−0.228113
$843$	0	0
$844$	2375.94	0.0968997
$845$	20202.8	0.822483
$846$	0	0
$847$	−9686.23	−0.392943
$848$	−15727.2	−0.636880
$849$	0	0
$850$	−1975.34	−0.0797102
$851$	57928.4	2.33344
$852$	0	0
$853$	10926.1	0.438573	0.219287	−	0.975661i	$-0.429627\pi$
$853$	10926.1	0.438573	0.219287	+	0.975661i	$0.429627\pi$
$854$	−11955.5	−0.479049
$855$	0	0
$856$	6378.30	0.254680
$857$	45169.3	1.80041	0.900206	−	0.435464i	$-0.143416\pi$
$857$	45169.3	1.80041	0.900206	+	0.435464i	$0.143416\pi$
$858$	0	0
$859$	−7051.90	−0.280102	−0.140051	−	0.990144i	$-0.544727\pi$
$859$	−7051.90	−0.280102	−0.140051	+	0.990144i	$0.544727\pi$
$860$	3539.31	0.140336
$861$	0	0
$862$	10630.6	0.420045
$863$	6882.52	0.271476	0.135738	−	0.990745i	$-0.456659\pi$
$863$	6882.52	0.271476	0.135738	+	0.990745i	$0.456659\pi$
$864$	0	0
$865$	−7212.25	−0.283496
$866$	8119.59	0.318608
$867$	0	0
$868$	104.935	0.00410336
$869$	834.782	0.0325869
$870$	0	0
$871$	−17809.6	−0.692829
$872$	−19083.3	−0.741104
$873$	0	0
$874$	−42279.6	−1.63630
$875$	921.535	0.0356041
$876$	0	0
$877$	37368.7	1.43883	0.719413	−	0.694583i	$-0.244411\pi$
$877$	37368.7	1.43883	0.719413	+	0.694583i	$0.244411\pi$
$878$	−19284.7	−0.741261
$879$	0	0
$880$	815.179	0.0312269
$881$	−23880.5	−0.913229	−0.456614	−	0.889665i	$-0.650938\pi$
$881$	−23880.5	−0.913229	−0.456614	+	0.889665i	$0.650938\pi$
$882$	0	0
$883$	−33107.0	−1.26177	−0.630883	−	0.775878i	$-0.717307\pi$
$883$	−33107.0	−1.26177	−0.630883	+	0.775878i	$0.717307\pi$
$884$	−6240.36	−0.237428
$885$	0	0
$886$	5576.71	0.211460
$887$	11302.1	0.427834	0.213917	−	0.976852i	$-0.431378\pi$
$887$	11302.1	0.427834	0.213917	+	0.976852i	$0.431378\pi$
$888$	0	0
$889$	17876.0	0.674402
$890$	4415.63	0.166306
$891$	0	0
$892$	−1331.20	−0.0499686
$893$	−37411.2	−1.40192
$894$	0	0
$895$	7674.46	0.286624
$896$	3703.47	0.138085
$897$	0	0
$898$	11294.4	0.419710
$899$	−302.886	−0.0112367
$900$	0	0
$901$	−13297.1	−0.491666
$902$	523.657	0.0193302
$903$	0	0
$904$	5296.57	0.194869
$905$	−18252.2	−0.670411
$906$	0	0
$907$	3962.24	0.145054	0.0725270	−	0.997366i	$-0.476894\pi$
$907$	3962.24	0.145054	0.0725270	+	0.997366i	$0.476894\pi$
$908$	9267.22	0.338704
$909$	0	0
$910$	−6906.30	−0.251584
$911$	−34046.2	−1.23820	−0.619100	−	0.785313i	$-0.712502\pi$
$911$	−34046.2	−1.23820	−0.619100	+	0.785313i	$0.712502\pi$
$912$	0	0
$913$	−4584.09	−0.166168
$914$	5288.02	0.191370
$915$	0	0
$916$	−7109.51	−0.256446
$917$	1920.44	0.0691587
$918$	0	0
$919$	−35121.5	−1.26066	−0.630332	−	0.776326i	$-0.717081\pi$
$919$	−35121.5	−1.26066	−0.630332	+	0.776326i	$0.717081\pi$
$920$	24552.1	0.879848
$921$	0	0
$922$	13608.9	0.486101
$923$	−40512.0	−1.44471
$924$	0	0
$925$	−7256.93	−0.257953
$926$	−8549.64	−0.303411
$927$	0	0
$928$	5219.44	0.184630
$929$	24457.8	0.863761	0.431881	−	0.901931i	$-0.357850\pi$
$929$	24457.8	0.863761	0.431881	+	0.901931i	$0.357850\pi$
$930$	0	0
$931$	−25778.4	−0.907469
$932$	−9950.05	−0.349705
$933$	0	0
$934$	−8968.11	−0.314181
$935$	689.221	0.0241069
$936$	0	0
$937$	41877.5	1.46006	0.730032	−	0.683413i	$-0.239505\pi$
$937$	41877.5	1.46006	0.730032	+	0.683413i	$0.239505\pi$
$938$	3943.79	0.137281
$939$	0	0
$940$	4968.80	0.172409
$941$	−9198.97	−0.318680	−0.159340	−	0.987224i	$-0.550937\pi$
$941$	−9198.97	−0.318680	−0.159340	+	0.987224i	$0.550937\pi$
$942$	0	0
$943$	10644.0	0.367569
$944$	−3870.14	−0.133435
$945$	0	0
$946$	2929.55	0.100685
$947$	20963.6	0.719351	0.359675	−	0.933078i	$-0.382887\pi$
$947$	20963.6	0.719351	0.359675	+	0.933078i	$0.382887\pi$
$948$	0	0
$949$	78529.5	2.68617
$950$	5296.54	0.180887
$951$	0	0
$952$	6041.97	0.205695
$953$	−27943.7	−0.949828	−0.474914	−	0.880032i	$-0.657521\pi$
$953$	−27943.7	−0.949828	−0.474914	+	0.880032i	$0.657521\pi$
$954$	0	0
$955$	6786.44	0.229952
$956$	394.920	0.0133605
$957$	0	0
$958$	−34305.9	−1.15697
$959$	−2983.11	−0.100448
$960$	0	0
$961$	−29755.0	−0.998792
$962$	54385.9	1.82274
$963$	0	0
$964$	−1457.87	−0.0487084
$965$	−9015.20	−0.300735
$966$	0	0
$967$	2243.52	0.0746089	0.0373045	−	0.999304i	$-0.488123\pi$
$967$	2243.52	0.0746089	0.0373045	+	0.999304i	$0.488123\pi$
$968$	32329.1	1.07345
$969$	0	0
$970$	−1651.48	−0.0546657
$971$	57345.0	1.89525	0.947626	−	0.319381i	$-0.103475\pi$
$971$	57345.0	1.89525	0.947626	+	0.319381i	$0.103475\pi$
$972$	0	0
$973$	−13031.8	−0.429375
$974$	8496.88	0.279525
$975$	0	0
$976$	26929.6	0.883191
$977$	18790.9	0.615326	0.307663	−	0.951495i	$-0.400453\pi$
$977$	18790.9	0.615326	0.307663	+	0.951495i	$0.400453\pi$
$978$	0	0
$979$	−1540.67	−0.0502962
$980$	3423.79	0.111601
$981$	0	0
$982$	−16642.5	−0.540817
$983$	−34597.7	−1.12258	−0.561289	−	0.827620i	$-0.689694\pi$
$983$	−34597.7	−1.12258	−0.561289	+	0.827620i	$0.689694\pi$
$984$	0	0
$985$	−1317.02	−0.0426026
$986$	−3988.69	−0.128829
$987$	0	0
$988$	16732.4	0.538795
$989$	59547.1	1.91455
$990$	0	0
$991$	−45026.3	−1.44330	−0.721649	−	0.692259i	$-0.756616\pi$
$991$	−45026.3	−1.44330	−0.721649	+	0.692259i	$0.756616\pi$
$992$	−620.364	−0.0198554
$993$	0	0
$994$	8971.06	0.286262
$995$	2464.26	0.0785150
$996$	0	0
$997$	−11972.1	−0.380301	−0.190150	−	0.981755i	$-0.560898\pi$
$997$	−11972.1	−0.380301	−0.190150	+	0.981755i	$0.560898\pi$
$998$	5977.66	0.189599
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.d.1.2		2
3.2	odd	2		405.4.a.e.1.1		2
5.4	even	2		2025.4.a.l.1.1		2
9.2	odd	6		135.4.e.a.91.2		4
9.4	even	3		45.4.e.a.16.1	✓	4
9.5	odd	6		135.4.e.a.46.2		4
9.7	even	3		45.4.e.a.31.1	yes	4
15.14	odd	2		2025.4.a.j.1.2		2
45.4	even	6		225.4.e.a.151.2		4
45.7	odd	12		225.4.k.a.49.3		8
45.13	odd	12		225.4.k.a.124.3		8
45.22	odd	12		225.4.k.a.124.2		8
45.34	even	6		225.4.e.a.76.2		4
45.43	odd	12		225.4.k.a.49.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.a.16.1	✓	4	9.4	even	3
45.4.e.a.31.1	yes	4	9.7	even	3
135.4.e.a.46.2		4	9.5	odd	6
135.4.e.a.91.2		4	9.2	odd	6
225.4.e.a.76.2		4	45.34	even	6
225.4.e.a.151.2		4	45.4	even	6
225.4.k.a.49.2		8	45.43	odd	12
225.4.k.a.49.3		8	45.7	odd	12
225.4.k.a.124.2		8	45.22	odd	12
225.4.k.a.124.3		8	45.13	odd	12
405.4.a.d.1.2		2	1.1	even	1	trivial
405.4.a.e.1.1		2	3.2	odd	2
2025.4.a.j.1.2		2	15.14	odd	2
2025.4.a.l.1.1		2	5.4	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q+2.37228 q^{2} -2.37228 q^{4} +5.00000 q^{5} +7.37228 q^{7} -24.6060 q^{8} +O(q^{10})\)	\(q+2.37228 q^{2} -2.37228 q^{4} +5.00000 q^{5} +7.37228 q^{7} -24.6060 q^{8} +11.8614 q^{10} -4.13859 q^{11} -78.9783 q^{13} +17.4891 q^{14} -39.3940 q^{16} -33.3070 q^{17} +89.3070 q^{19} -11.8614 q^{20} -9.81791 q^{22} -199.562 q^{23} +25.0000 q^{25} -187.359 q^{26} -17.4891 q^{28} +50.4810 q^{29} -6.00000 q^{31} +103.394 q^{32} -79.0137 q^{34} +36.8614 q^{35} -290.277 q^{37} +211.861 q^{38} -123.030 q^{40} -53.3369 q^{41} -298.388 q^{43} +9.81791 q^{44} -473.418 q^{46} -418.905 q^{47} -288.649 q^{49} +59.3070 q^{50} +187.359 q^{52} +399.228 q^{53} -20.6930 q^{55} -181.402 q^{56} +119.755 q^{58} +98.2418 q^{59} -683.595 q^{61} -14.2337 q^{62} +560.432 q^{64} -394.891 q^{65} +225.500 q^{67} +79.0137 q^{68} +87.4456 q^{70} +512.951 q^{71} -994.318 q^{73} -688.619 q^{74} -211.861 q^{76} -30.5109 q^{77} -201.707 q^{79} -196.970 q^{80} -126.530 q^{82} +1107.64 q^{83} -166.535 q^{85} -707.861 q^{86} +101.834 q^{88} +372.269 q^{89} -582.250 q^{91} +473.418 q^{92} -993.760 q^{94} +446.535 q^{95} -139.231 q^{97} -684.758 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + q^{4} + 10 q^{5} + 9 q^{7} - 9 q^{8}+O(q^{10}) \) 2 * q - q^2 + q^4 + 10 * q^5 + 9 * q^7 - 9 * q^8	\( 2 q - q^{2} + q^{4} + 10 q^{5} + 9 q^{7} - 9 q^{8} - 5 q^{10} - 37 q^{11} - 112 q^{13} + 12 q^{14} - 119 q^{16} + 77 q^{17} + 35 q^{19} + 5 q^{20} + 101 q^{22} - 267 q^{23} + 50 q^{25} - 76 q^{26} - 12 q^{28} + 325 q^{29} - 12 q^{31} + 247 q^{32} - 451 q^{34} + 45 q^{35} - 638 q^{37} + 395 q^{38} - 45 q^{40} + 238 q^{41} - 97 q^{43} - 101 q^{44} - 246 q^{46} - 901 q^{47} - 629 q^{49} - 25 q^{50} + 76 q^{52} + 224 q^{53} - 185 q^{55} - 156 q^{56} - 806 q^{58} - 85 q^{59} - 247 q^{61} + 6 q^{62} + 713 q^{64} - 560 q^{65} - 606 q^{67} + 451 q^{68} + 60 q^{70} + 394 q^{71} - 811 q^{73} + 484 q^{74} - 395 q^{76} - 84 q^{77} - 840 q^{79} - 595 q^{80} - 1109 q^{82} - 387 q^{83} + 385 q^{85} - 1387 q^{86} - 411 q^{88} - 1065 q^{89} - 636 q^{91} + 246 q^{92} + 632 q^{94} + 175 q^{95} - 1031 q^{97} + 463 q^{98}+O(q^{100}) \) 2 * q - q^2 + q^4 + 10 * q^5 + 9 * q^7 - 9 * q^8 - 5 * q^10 - 37 * q^11 - 112 * q^13 + 12 * q^14 - 119 * q^16 + 77 * q^17 + 35 * q^19 + 5 * q^20 + 101 * q^22 - 267 * q^23 + 50 * q^25 - 76 * q^26 - 12 * q^28 + 325 * q^29 - 12 * q^31 + 247 * q^32 - 451 * q^34 + 45 * q^35 - 638 * q^37 + 395 * q^38 - 45 * q^40 + 238 * q^41 - 97 * q^43 - 101 * q^44 - 246 * q^46 - 901 * q^47 - 629 * q^49 - 25 * q^50 + 76 * q^52 + 224 * q^53 - 185 * q^55 - 156 * q^56 - 806 * q^58 - 85 * q^59 - 247 * q^61 + 6 * q^62 + 713 * q^64 - 560 * q^65 - 606 * q^67 + 451 * q^68 + 60 * q^70 + 394 * q^71 - 811 * q^73 + 484 * q^74 - 395 * q^76 - 84 * q^77 - 840 * q^79 - 595 * q^80 - 1109 * q^82 - 387 * q^83 + 385 * q^85 - 1387 * q^86 - 411 * q^88 - 1065 * q^89 - 636 * q^91 + 246 * q^92 + 632 * q^94 + 175 * q^95 - 1031 * q^97 + 463 * q^98

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