LMFDB - Embedded newform 405.4.a.j.1.1

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:	$3$
Coefficient field:	3.3.2292.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 13x + 1 $ x^3 - x^2 - 13*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.18296$ 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)

$f(q)$	$=$	$q-3.74857 q^{2} +6.05174 q^{4} -5.00000 q^{5} -31.3492 q^{7} +7.30318 q^{8} +O(q^{10})$	\(q-3.74857 q^{2} +6.05174 q^{4} -5.00000 q^{5} -31.3492 q^{7} +7.30318 q^{8} +18.7428 q^{10} +20.8333 q^{11} +59.9310 q^{13} +117.514 q^{14} -75.7904 q^{16} +74.0460 q^{17} -63.8390 q^{19} -30.2587 q^{20} -78.0950 q^{22} +32.8494 q^{23} +25.0000 q^{25} -224.655 q^{26} -189.717 q^{28} +160.009 q^{29} -254.373 q^{31} +225.680 q^{32} -277.566 q^{34} +156.746 q^{35} +215.365 q^{37} +239.305 q^{38} -36.5159 q^{40} -141.681 q^{41} +137.906 q^{43} +126.078 q^{44} -123.138 q^{46} -33.5790 q^{47} +639.771 q^{49} -93.7141 q^{50} +362.687 q^{52} +41.9914 q^{53} -104.166 q^{55} -228.949 q^{56} -599.803 q^{58} -615.142 q^{59} -134.307 q^{61} +953.535 q^{62} -239.652 q^{64} -299.655 q^{65} -857.533 q^{67} +448.107 q^{68} -587.572 q^{70} -588.665 q^{71} -618.191 q^{73} -807.311 q^{74} -386.337 q^{76} -653.107 q^{77} -345.288 q^{79} +378.952 q^{80} +531.102 q^{82} +1093.17 q^{83} -370.230 q^{85} -516.949 q^{86} +152.149 q^{88} -414.849 q^{89} -1878.79 q^{91} +198.796 q^{92} +125.873 q^{94} +319.195 q^{95} -201.411 q^{97} -2398.22 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 11 q^{4} - 15 q^{5} - 43 q^{7} + 27 q^{8}+O(q^{10}) $ 3 * q + q^2 + 11 * q^4 - 15 * q^5 - 43 * q^7 + 27 * q^8	$ 3 q + q^{2} + 11 q^{4} - 15 q^{5} - 43 q^{7} + 27 q^{8} - 5 q^{10} - 14 q^{11} + 40 q^{13} + 27 q^{14} - 13 q^{16} + 166 q^{17} - 164 q^{19} - 55 q^{20} - 376 q^{22} - 171 q^{23} + 75 q^{25} - 434 q^{26} - 517 q^{28} + 335 q^{29} - 352 q^{31} + 77 q^{32} - 52 q^{34} + 215 q^{35} + 402 q^{37} + 178 q^{38} - 135 q^{40} - 187 q^{41} - 602 q^{43} - 982 q^{44} - 201 q^{46} - 665 q^{47} + 430 q^{49} + 25 q^{50} - 456 q^{52} + 730 q^{53} + 70 q^{55} - 705 q^{56} + 217 q^{58} + 298 q^{59} - 1439 q^{61} + 1614 q^{62} - 1569 q^{64} - 200 q^{65} - 1849 q^{67} + 710 q^{68} - 135 q^{70} - 70 q^{71} - 368 q^{73} + 320 q^{74} + 204 q^{76} + 948 q^{77} - 382 q^{79} + 65 q^{80} - 575 q^{82} + 831 q^{83} - 830 q^{85} - 1580 q^{86} - 1428 q^{88} - 1719 q^{89} - 710 q^{91} + 1623 q^{92} - 2077 q^{94} + 820 q^{95} - 282 q^{97} - 2164 q^{98}+O(q^{100}) $ 3 * q + q^2 + 11 * q^4 - 15 * q^5 - 43 * q^7 + 27 * q^8 - 5 * q^10 - 14 * q^11 + 40 * q^13 + 27 * q^14 - 13 * q^16 + 166 * q^17 - 164 * q^19 - 55 * q^20 - 376 * q^22 - 171 * q^23 + 75 * q^25 - 434 * q^26 - 517 * q^28 + 335 * q^29 - 352 * q^31 + 77 * q^32 - 52 * q^34 + 215 * q^35 + 402 * q^37 + 178 * q^38 - 135 * q^40 - 187 * q^41 - 602 * q^43 - 982 * q^44 - 201 * q^46 - 665 * q^47 + 430 * q^49 + 25 * q^50 - 456 * q^52 + 730 * q^53 + 70 * q^55 - 705 * q^56 + 217 * q^58 + 298 * q^59 - 1439 * q^61 + 1614 * q^62 - 1569 * q^64 - 200 * q^65 - 1849 * q^67 + 710 * q^68 - 135 * q^70 - 70 * q^71 - 368 * q^73 + 320 * q^74 + 204 * q^76 + 948 * q^77 - 382 * q^79 + 65 * q^80 - 575 * q^82 + 831 * q^83 - 830 * q^85 - 1580 * q^86 - 1428 * q^88 - 1719 * q^89 - 710 * q^91 + 1623 * q^92 - 2077 * q^94 + 820 * q^95 - 282 * q^97 - 2164 * 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 $

$2$	−3.74857	−1.32532	−0.662659	−	0.748921i	$-0.730572\pi$
$2$	−3.74857	−1.32532	−0.662659	+	0.748921i	$0.730572\pi$
$3$	0	0
$3$	0	0
$4$	6.05174	0.756468
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−31.3492	−1.69270	−0.846348	−	0.532630i	$-0.821204\pi$
$7$	−31.3492	−1.69270	−0.846348	+	0.532630i	$0.821204\pi$
$8$	7.30318	0.322758
$9$	0	0
$10$	18.7428	0.592700
$11$	20.8333	0.571043	0.285522	−	0.958372i	$-0.407833\pi$
$11$	20.8333	0.571043	0.285522	+	0.958372i	$0.407833\pi$
$12$	0	0
$13$	59.9310	1.27861	0.639303	−	0.768955i	$-0.279223\pi$
$13$	59.9310	1.27861	0.639303	+	0.768955i	$0.279223\pi$
$14$	117.514	2.24336
$15$	0	0
$16$	−75.7904	−1.18422
$17$	74.0460	1.05640	0.528200	−	0.849120i	$-0.322867\pi$
$17$	74.0460	1.05640	0.528200	+	0.849120i	$0.322867\pi$
$18$	0	0
$19$	−63.8390	−0.770825	−0.385413	−	0.922744i	$-0.625941\pi$
$19$	−63.8390	−0.770825	−0.385413	+	0.922744i	$0.625941\pi$
$20$	−30.2587	−0.338303
$21$	0	0
$22$	−78.0950	−0.756814
$23$	32.8494	0.297807	0.148904	−	0.988852i	$-0.452426\pi$
$23$	32.8494	0.297807	0.148904	+	0.988852i	$0.452426\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−224.655	−1.69456
$27$	0	0
$28$	−189.717	−1.28047
$29$	160.009	1.02458	0.512291	−	0.858812i	$-0.328797\pi$
$29$	160.009	1.02458	0.512291	+	0.858812i	$0.328797\pi$
$30$	0	0
$31$	−254.373	−1.47377	−0.736884	−	0.676019i	$-0.763704\pi$
$31$	−254.373	−1.47377	−0.736884	+	0.676019i	$0.763704\pi$
$32$	225.680	1.24672
$33$	0	0
$34$	−277.566	−1.40007
$35$	156.746	0.756997
$36$	0	0
$37$	215.365	0.956914	0.478457	−	0.878111i	$-0.341196\pi$
$37$	215.365	0.956914	0.478457	+	0.878111i	$0.341196\pi$
$38$	239.305	1.02159
$39$	0	0
$40$	−36.5159	−0.144342
$41$	−141.681	−0.539681	−0.269841	−	0.962905i	$-0.586971\pi$
$41$	−141.681	−0.539681	−0.269841	+	0.962905i	$0.586971\pi$
$42$	0	0
$43$	137.906	0.489080	0.244540	−	0.969639i	$-0.421363\pi$
$43$	137.906	0.489080	0.244540	+	0.969639i	$0.421363\pi$
$44$	126.078	0.431976
$45$	0	0
$46$	−123.138	−0.394689
$47$	−33.5790	−0.104213	−0.0521064	−	0.998642i	$-0.516593\pi$
$47$	−33.5790	−0.104213	−0.0521064	+	0.998642i	$0.516593\pi$
$48$	0	0
$49$	639.771	1.86522
$50$	−93.7141	−0.265064
$51$	0	0
$52$	362.687	0.967224
$53$	41.9914	0.108829	0.0544147	−	0.998518i	$-0.482671\pi$
$53$	41.9914	0.108829	0.0544147	+	0.998518i	$0.482671\pi$
$54$	0	0
$55$	−104.166	−0.255378
$56$	−228.949	−0.546331
$57$	0	0
$58$	−599.803	−1.35790
$59$	−615.142	−1.35737	−0.678683	−	0.734431i	$-0.737449\pi$
$59$	−615.142	−1.35737	−0.678683	+	0.734431i	$0.737449\pi$
$60$	0	0
$61$	−134.307	−0.281906	−0.140953	−	0.990016i	$-0.545017\pi$
$61$	−134.307	−0.281906	−0.140953	+	0.990016i	$0.545017\pi$
$62$	953.535	1.95321
$63$	0	0
$64$	−239.652	−0.468071
$65$	−299.655	−0.571810
$66$	0	0
$67$	−857.533	−1.56365	−0.781824	−	0.623500i	$-0.785710\pi$
$67$	−857.533	−1.56365	−0.781824	+	0.623500i	$0.785710\pi$
$68$	448.107	0.799132
$69$	0	0
$70$	−587.572	−1.00326
$71$	−588.665	−0.983968	−0.491984	−	0.870604i	$-0.663728\pi$
$71$	−588.665	−0.983968	−0.491984	+	0.870604i	$0.663728\pi$
$72$	0	0
$73$	−618.191	−0.991148	−0.495574	−	0.868566i	$-0.665042\pi$
$73$	−618.191	−0.991148	−0.495574	+	0.868566i	$0.665042\pi$
$74$	−807.311	−1.26822
$75$	0	0
$76$	−386.337	−0.583104
$77$	−653.107	−0.966603
$78$	0	0
$79$	−345.288	−0.491747	−0.245873	−	0.969302i	$-0.579075\pi$
$79$	−345.288	−0.491747	−0.245873	+	0.969302i	$0.579075\pi$
$80$	378.952	0.529601
$81$	0	0
$82$	531.102	0.715249
$83$	1093.17	1.44567	0.722836	−	0.691020i	$-0.242838\pi$
$83$	1093.17	1.44567	0.722836	+	0.691020i	$0.242838\pi$
$84$	0	0
$85$	−370.230	−0.472436
$86$	−516.949	−0.648186
$87$	0	0
$88$	152.149	0.184309
$89$	−414.849	−0.494089	−0.247045	−	0.969004i	$-0.579459\pi$
$89$	−414.849	−0.494089	−0.247045	+	0.969004i	$0.579459\pi$
$90$	0	0
$91$	−1878.79	−2.16429
$92$	198.796	0.225282
$93$	0	0
$94$	125.873	0.138115
$95$	319.195	0.344724
$96$	0	0
$97$	−201.411	−0.210826	−0.105413	−	0.994429i	$-0.533617\pi$
$97$	−201.411	−0.210826	−0.105413	+	0.994429i	$0.533617\pi$
$98$	−2398.22	−2.47201
$99$	0	0
$100$	151.294	0.151294
$101$	−265.383	−0.261451	−0.130726	−	0.991419i	$-0.541731\pi$
$101$	−265.383	−0.261451	−0.130726	+	0.991419i	$0.541731\pi$
$102$	0	0
$103$	−527.618	−0.504735	−0.252368	−	0.967631i	$-0.581209\pi$
$103$	−527.618	−0.504735	−0.252368	+	0.967631i	$0.581209\pi$
$104$	437.687	0.412680
$105$	0	0
$106$	−157.408	−0.144234
$107$	−2084.24	−1.88310	−0.941549	−	0.336877i	$-0.890629\pi$
$107$	−2084.24	−1.88310	−0.941549	+	0.336877i	$0.890629\pi$
$108$	0	0
$109$	925.651	0.813406	0.406703	−	0.913560i	$-0.366678\pi$
$109$	925.651	0.813406	0.406703	+	0.913560i	$0.366678\pi$
$110$	390.475	0.338457
$111$	0	0
$112$	2375.97	2.00453
$113$	−546.019	−0.454559	−0.227279	−	0.973830i	$-0.572983\pi$
$113$	−546.019	−0.454559	−0.227279	+	0.973830i	$0.572983\pi$
$114$	0	0
$115$	−164.247	−0.133183
$116$	968.332	0.775063
$117$	0	0
$118$	2305.90	1.79894
$119$	−2321.28	−1.78816
$120$	0	0
$121$	−896.974	−0.673910
$122$	503.458	0.373614
$123$	0	0
$124$	−1539.40	−1.11486
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−975.972	−0.681918	−0.340959	−	0.940078i	$-0.610752\pi$
$127$	−975.972	−0.681918	−0.340959	+	0.940078i	$0.610752\pi$
$128$	−907.086	−0.626374
$129$	0	0
$130$	1123.28	0.757830
$131$	1629.23	1.08661	0.543307	−	0.839534i	$-0.317172\pi$
$131$	1629.23	1.08661	0.543307	+	0.839534i	$0.317172\pi$
$132$	0	0
$133$	2001.30	1.30477
$134$	3214.52	2.07233
$135$	0	0
$136$	540.771	0.340961
$137$	−661.451	−0.412493	−0.206246	−	0.978500i	$-0.566125\pi$
$137$	−661.451	−0.412493	−0.206246	+	0.978500i	$0.566125\pi$
$138$	0	0
$139$	1382.99	0.843911	0.421955	−	0.906617i	$-0.361344\pi$
$139$	1382.99	0.843911	0.421955	+	0.906617i	$0.361344\pi$
$140$	948.586	0.572644
$141$	0	0
$142$	2206.65	1.30407
$143$	1248.56	0.730139
$144$	0	0
$145$	−800.044	−0.458207
$146$	2317.33	1.31359
$147$	0	0
$148$	1303.33	0.723875
$149$	−3163.49	−1.73935	−0.869676	−	0.493624i	$-0.835672\pi$
$149$	−3163.49	−1.73935	−0.869676	+	0.493624i	$0.835672\pi$
$150$	0	0
$151$	−2558.80	−1.37902	−0.689509	−	0.724277i	$-0.742174\pi$
$151$	−2558.80	−1.37902	−0.689509	+	0.724277i	$0.742174\pi$
$152$	−466.228	−0.248790
$153$	0	0
$154$	2448.21	1.28106
$155$	1271.87	0.659089
$156$	0	0
$157$	3500.96	1.77966	0.889832	−	0.456288i	$-0.150821\pi$
$157$	3500.96	1.77966	0.889832	+	0.456288i	$0.150821\pi$
$158$	1294.34	0.651721
$159$	0	0
$160$	−1128.40	−0.557548
$161$	−1029.80	−0.504097
$162$	0	0
$163$	263.950	0.126835	0.0634176	−	0.997987i	$-0.479800\pi$
$163$	263.950	0.126835	0.0634176	+	0.997987i	$0.479800\pi$
$164$	−857.419	−0.408251
$165$	0	0
$166$	−4097.81	−1.91597
$167$	3589.05	1.66305	0.831524	−	0.555489i	$-0.187469\pi$
$167$	3589.05	1.66305	0.831524	+	0.555489i	$0.187469\pi$
$168$	0	0
$169$	1394.73	0.634834
$170$	1387.83	0.626128
$171$	0	0
$172$	834.570	0.369973
$173$	−370.456	−0.162805	−0.0814024	−	0.996681i	$-0.525940\pi$
$173$	−370.456	−0.162805	−0.0814024	+	0.996681i	$0.525940\pi$
$174$	0	0
$175$	−783.729	−0.338539
$176$	−1578.96	−0.676243
$177$	0	0
$178$	1555.09	0.654825
$179$	−446.898	−0.186607	−0.0933036	−	0.995638i	$-0.529743\pi$
$179$	−446.898	−0.186607	−0.0933036	+	0.995638i	$0.529743\pi$
$180$	0	0
$181$	−904.046	−0.371255	−0.185628	−	0.982620i	$-0.559432\pi$
$181$	−904.046	−0.371255	−0.185628	+	0.982620i	$0.559432\pi$
$182$	7042.76	2.86838
$183$	0	0
$184$	239.905	0.0961196
$185$	−1076.83	−0.427945
$186$	0	0
$187$	1542.62	0.603250
$188$	−203.211	−0.0788336
$189$	0	0
$190$	−1196.52	−0.456868
$191$	599.587	0.227144	0.113572	−	0.993530i	$-0.463771\pi$
$191$	599.587	0.227144	0.113572	+	0.993530i	$0.463771\pi$
$192$	0	0
$193$	−4413.73	−1.64615	−0.823076	−	0.567931i	$-0.807744\pi$
$193$	−4413.73	−1.64615	−0.823076	+	0.567931i	$0.807744\pi$
$194$	755.001	0.279412
$195$	0	0
$196$	3871.73	1.41098
$197$	4807.15	1.73855	0.869277	−	0.494325i	$-0.164585\pi$
$197$	4807.15	1.73855	0.869277	+	0.494325i	$0.164585\pi$
$198$	0	0
$199$	313.833	0.111794	0.0558970	−	0.998437i	$-0.482198\pi$
$199$	313.833	0.111794	0.0558970	+	0.998437i	$0.482198\pi$
$200$	182.579	0.0645516
$201$	0	0
$202$	994.805	0.346506
$203$	−5016.14	−1.73431
$204$	0	0
$205$	708.407	0.241353
$206$	1977.81	0.668935
$207$	0	0
$208$	−4542.20	−1.51416
$209$	−1329.98	−0.440175
$210$	0	0
$211$	2438.82	0.795711	0.397856	−	0.917448i	$-0.369754\pi$
$211$	2438.82	0.795711	0.397856	+	0.917448i	$0.369754\pi$
$212$	254.121	0.0823260
$213$	0	0
$214$	7812.92	2.49570
$215$	−689.529	−0.218723
$216$	0	0
$217$	7974.40	2.49464
$218$	−3469.86	−1.07802
$219$	0	0
$220$	−630.389	−0.193185
$221$	4437.65	1.35072
$222$	0	0
$223$	−2333.41	−0.700703	−0.350352	−	0.936618i	$-0.613938\pi$
$223$	−2333.41	−0.700703	−0.350352	+	0.936618i	$0.613938\pi$
$224$	−7074.87	−2.11031
$225$	0	0
$226$	2046.79	0.602435
$227$	2723.69	0.796378	0.398189	−	0.917303i	$-0.369639\pi$
$227$	2723.69	0.796378	0.398189	+	0.917303i	$0.369639\pi$
$228$	0	0
$229$	−3314.16	−0.956358	−0.478179	−	0.878262i	$-0.658703\pi$
$229$	−3314.16	−0.956358	−0.478179	+	0.878262i	$0.658703\pi$
$230$	615.690	0.176510
$231$	0	0
$232$	1168.57	0.330692
$233$	3175.44	0.892833	0.446416	−	0.894825i	$-0.352700\pi$
$233$	3175.44	0.892833	0.446416	+	0.894825i	$0.352700\pi$
$234$	0	0
$235$	167.895	0.0466054
$236$	−3722.68	−1.02680
$237$	0	0
$238$	8701.47	2.36989
$239$	246.125	0.0666129	0.0333064	−	0.999445i	$-0.489396\pi$
$239$	246.125	0.0666129	0.0333064	+	0.999445i	$0.489396\pi$
$240$	0	0
$241$	−5287.73	−1.41333	−0.706666	−	0.707547i	$-0.749802\pi$
$241$	−5287.73	−1.41333	−0.706666	+	0.707547i	$0.749802\pi$
$242$	3362.36	0.893144
$243$	0	0
$244$	−812.791	−0.213252
$245$	−3198.85	−0.834152
$246$	0	0
$247$	−3825.94	−0.985582
$248$	−1857.73	−0.475670
$249$	0	0
$250$	468.571	0.118540
$251$	−2821.23	−0.709459	−0.354730	−	0.934969i	$-0.615427\pi$
$251$	−2821.23	−0.709459	−0.354730	+	0.934969i	$0.615427\pi$
$252$	0	0
$253$	684.361	0.170061
$254$	3658.50	0.903758
$255$	0	0
$256$	5317.49	1.29821
$257$	1884.19	0.457326	0.228663	−	0.973506i	$-0.426565\pi$
$257$	1884.19	0.457326	0.228663	+	0.973506i	$0.426565\pi$
$258$	0	0
$259$	−6751.52	−1.61977
$260$	−1813.44	−0.432556
$261$	0	0
$262$	−6107.27	−1.44011
$263$	−553.011	−0.129658	−0.0648292	−	0.997896i	$-0.520650\pi$
$263$	−553.011	−0.129658	−0.0648292	+	0.997896i	$0.520650\pi$
$264$	0	0
$265$	−209.957	−0.0486700
$266$	−7502.01	−1.72924
$267$	0	0
$268$	−5189.57	−1.18285
$269$	−3363.48	−0.762361	−0.381180	−	0.924501i	$-0.624482\pi$
$269$	−3363.48	−0.762361	−0.381180	+	0.924501i	$0.624482\pi$
$270$	0	0
$271$	−3333.85	−0.747295	−0.373648	−	0.927571i	$-0.621893\pi$
$271$	−3333.85	−0.747295	−0.373648	+	0.927571i	$0.621893\pi$
$272$	−5611.97	−1.25101
$273$	0	0
$274$	2479.49	0.546684
$275$	520.832	0.114209
$276$	0	0
$277$	−5387.25	−1.16855	−0.584275	−	0.811555i	$-0.698621\pi$
$277$	−5387.25	−1.16855	−0.584275	+	0.811555i	$0.698621\pi$
$278$	−5184.23	−1.11845
$279$	0	0
$280$	1144.74	0.244327
$281$	−3716.70	−0.789038	−0.394519	−	0.918888i	$-0.629089\pi$
$281$	−3716.70	−0.789038	−0.394519	+	0.918888i	$0.629089\pi$
$282$	0	0
$283$	2768.76	0.581574	0.290787	−	0.956788i	$-0.406083\pi$
$283$	2768.76	0.581574	0.290787	+	0.956788i	$0.406083\pi$
$284$	−3562.45	−0.744340
$285$	0	0
$286$	−4680.31	−0.967667
$287$	4441.60	0.913516
$288$	0	0
$289$	569.810	0.115980
$290$	2999.02	0.607270
$291$	0	0
$292$	−3741.13	−0.749771
$293$	3479.99	0.693868	0.346934	−	0.937890i	$-0.387223\pi$
$293$	3479.99	0.693868	0.346934	+	0.937890i	$0.387223\pi$
$294$	0	0
$295$	3075.71	0.607033
$296$	1572.85	0.308852
$297$	0	0
$298$	11858.6	2.30519
$299$	1968.70	0.380778
$300$	0	0
$301$	−4323.23	−0.827864
$302$	9591.81	1.82764
$303$	0	0
$304$	4838.38	0.912830
$305$	671.535	0.126072
$306$	0	0
$307$	−1810.36	−0.336555	−0.168278	−	0.985740i	$-0.553821\pi$
$307$	−1810.36	−0.336555	−0.168278	+	0.985740i	$0.553821\pi$
$308$	−3952.43	−0.731204
$309$	0	0
$310$	−4767.68	−0.873503
$311$	887.298	0.161781	0.0808907	−	0.996723i	$-0.474224\pi$
$311$	887.298	0.161781	0.0808907	+	0.996723i	$0.474224\pi$
$312$	0	0
$313$	−2215.13	−0.400022	−0.200011	−	0.979794i	$-0.564098\pi$
$313$	−2215.13	−0.400022	−0.200011	+	0.979794i	$0.564098\pi$
$314$	−13123.6	−2.35862
$315$	0	0
$316$	−2089.60	−0.371990
$317$	658.046	0.116592	0.0582958	−	0.998299i	$-0.481433\pi$
$317$	658.046	0.116592	0.0582958	+	0.998299i	$0.481433\pi$
$318$	0	0
$319$	3333.51	0.585081
$320$	1198.26	0.209328
$321$	0	0
$322$	3860.28	0.668089
$323$	−4727.03	−0.814299
$324$	0	0
$325$	1498.28	0.255721
$326$	−989.433	−0.168097
$327$	0	0
$328$	−1034.72	−0.174186
$329$	1052.67	0.176401
$330$	0	0
$331$	5276.44	0.876192	0.438096	−	0.898928i	$-0.355653\pi$
$331$	5276.44	0.876192	0.438096	+	0.898928i	$0.355653\pi$
$332$	6615.57	1.09360
$333$	0	0
$334$	−13453.8	−2.20407
$335$	4287.67	0.699284
$336$	0	0
$337$	−3361.27	−0.543324	−0.271662	−	0.962393i	$-0.587573\pi$
$337$	−3361.27	−0.543324	−0.271662	+	0.962393i	$0.587573\pi$
$338$	−5228.24	−0.841357
$339$	0	0
$340$	−2240.54	−0.357383
$341$	−5299.44	−0.841586
$342$	0	0
$343$	−9303.52	−1.46456
$344$	1007.15	0.157854
$345$	0	0
$346$	1388.68	0.215768
$347$	−9355.54	−1.44735	−0.723677	−	0.690139i	$-0.757549\pi$
$347$	−9355.54	−1.44735	−0.723677	+	0.690139i	$0.757549\pi$
$348$	0	0
$349$	7039.30	1.07967	0.539836	−	0.841770i	$-0.318486\pi$
$349$	7039.30	1.07967	0.539836	+	0.841770i	$0.318486\pi$
$350$	2937.86	0.448672
$351$	0	0
$352$	4701.65	0.711929
$353$	−4202.18	−0.633596	−0.316798	−	0.948493i	$-0.602608\pi$
$353$	−4202.18	−0.633596	−0.316798	+	0.948493i	$0.602608\pi$
$354$	0	0
$355$	2943.33	0.440044
$356$	−2510.56	−0.373762
$357$	0	0
$358$	1675.22	0.247314
$359$	−588.013	−0.0864461	−0.0432230	−	0.999065i	$-0.513763\pi$
$359$	−588.013	−0.0864461	−0.0432230	+	0.999065i	$0.513763\pi$
$360$	0	0
$361$	−2783.58	−0.405828
$362$	3388.87	0.492031
$363$	0	0
$364$	−11369.9	−1.63722
$365$	3090.95	0.443255
$366$	0	0
$367$	−7785.11	−1.10730	−0.553650	−	0.832749i	$-0.686766\pi$
$367$	−7785.11	−1.10730	−0.553650	+	0.832749i	$0.686766\pi$
$368$	−2489.67	−0.352671
$369$	0	0
$370$	4036.55	0.567163
$371$	−1316.40	−0.184215
$372$	0	0
$373$	−7824.03	−1.08609	−0.543047	−	0.839702i	$-0.682729\pi$
$373$	−7824.03	−1.08609	−0.543047	+	0.839702i	$0.682729\pi$
$374$	−5782.62	−0.799498
$375$	0	0
$376$	−245.233	−0.0336355
$377$	9589.49	1.31004
$378$	0	0
$379$	−4679.90	−0.634275	−0.317138	−	0.948379i	$-0.602722\pi$
$379$	−4679.90	−0.634275	−0.317138	+	0.948379i	$0.602722\pi$
$380$	1931.69	0.260772
$381$	0	0
$382$	−2247.59	−0.301038
$383$	6727.65	0.897563	0.448782	−	0.893641i	$-0.351858\pi$
$383$	6727.65	0.897563	0.448782	+	0.893641i	$0.351858\pi$
$384$	0	0
$385$	3265.53	0.432278
$386$	16545.2	2.18167
$387$	0	0
$388$	−1218.89	−0.159483
$389$	4772.88	0.622094	0.311047	−	0.950395i	$-0.399320\pi$
$389$	4772.88	0.622094	0.311047	+	0.950395i	$0.399320\pi$
$390$	0	0
$391$	2432.36	0.314603
$392$	4672.36	0.602015
$393$	0	0
$394$	−18019.9	−2.30414
$395$	1726.44	0.219916
$396$	0	0
$397$	−4688.95	−0.592775	−0.296388	−	0.955068i	$-0.595782\pi$
$397$	−4688.95	−0.592775	−0.296388	+	0.955068i	$0.595782\pi$
$398$	−1176.42	−0.148163
$399$	0	0
$400$	−1894.76	−0.236845
$401$	−1533.83	−0.191012	−0.0955061	−	0.995429i	$-0.530447\pi$
$401$	−1533.83	−0.191012	−0.0955061	+	0.995429i	$0.530447\pi$
$402$	0	0
$403$	−15244.9	−1.88437
$404$	−1606.03	−0.197780
$405$	0	0
$406$	18803.3	2.29851
$407$	4486.77	0.546439
$408$	0	0
$409$	−8778.82	−1.06133	−0.530666	−	0.847581i	$-0.678058\pi$
$409$	−8778.82	−1.06133	−0.530666	+	0.847581i	$0.678058\pi$
$410$	−2655.51	−0.319869
$411$	0	0
$412$	−3193.01	−0.381816
$413$	19284.2	2.29761
$414$	0	0
$415$	−5465.84	−0.646524
$416$	13525.2	1.59406
$417$	0	0
$418$	4985.51	0.583371
$419$	4276.05	0.498565	0.249282	−	0.968431i	$-0.419805\pi$
$419$	4276.05	0.498565	0.249282	+	0.968431i	$0.419805\pi$
$420$	0	0
$421$	14463.3	1.67434	0.837169	−	0.546944i	$-0.184209\pi$
$421$	14463.3	1.67434	0.837169	+	0.546944i	$0.184209\pi$
$422$	−9142.07	−1.05457
$423$	0	0
$424$	306.671	0.0351256
$425$	1851.15	0.211280
$426$	0	0
$427$	4210.41	0.477181
$428$	−12613.3	−1.42450
$429$	0	0
$430$	2584.74	0.289878
$431$	2208.11	0.246777	0.123389	−	0.992358i	$-0.460624\pi$
$431$	2208.11	0.246777	0.123389	+	0.992358i	$0.460624\pi$
$432$	0	0
$433$	−10062.3	−1.11677	−0.558386	−	0.829581i	$-0.688579\pi$
$433$	−10062.3	−1.11677	−0.558386	+	0.829581i	$0.688579\pi$
$434$	−29892.6	−3.30619
$435$	0	0
$436$	5601.80	0.615315
$437$	−2097.07	−0.229557
$438$	0	0
$439$	13317.2	1.44783	0.723915	−	0.689889i	$-0.242341\pi$
$439$	13317.2	1.44783	0.723915	+	0.689889i	$0.242341\pi$
$440$	−760.746	−0.0824253
$441$	0	0
$442$	−16634.8	−1.79013
$443$	14275.1	1.53099	0.765497	−	0.643439i	$-0.222493\pi$
$443$	14275.1	1.53099	0.765497	+	0.643439i	$0.222493\pi$
$444$	0	0
$445$	2074.25	0.220963
$446$	8746.95	0.928654
$447$	0	0
$448$	7512.90	0.792302
$449$	1690.02	0.177632	0.0888162	−	0.996048i	$-0.471692\pi$
$449$	1690.02	0.177632	0.0888162	+	0.996048i	$0.471692\pi$
$450$	0	0
$451$	−2951.69	−0.308181
$452$	−3304.36	−0.343859
$453$	0	0
$454$	−10209.9	−1.05545
$455$	9393.94	0.967901
$456$	0	0
$457$	−6664.39	−0.682159	−0.341080	−	0.940034i	$-0.610793\pi$
$457$	−6664.39	−0.682159	−0.341080	+	0.940034i	$0.610793\pi$
$458$	12423.3	1.26748
$459$	0	0
$460$	−993.979	−0.100749
$461$	−1667.42	−0.168459	−0.0842295	−	0.996446i	$-0.526843\pi$
$461$	−1667.42	−0.168459	−0.0842295	+	0.996446i	$0.526843\pi$
$462$	0	0
$463$	−5832.02	−0.585393	−0.292697	−	0.956205i	$-0.594553\pi$
$463$	−5832.02	−0.585393	−0.292697	+	0.956205i	$0.594553\pi$
$464$	−12127.1	−1.21334
$465$	0	0
$466$	−11903.3	−1.18329
$467$	−17410.6	−1.72519	−0.862597	−	0.505892i	$-0.831163\pi$
$467$	−17410.6	−1.72519	−0.862597	+	0.505892i	$0.831163\pi$
$468$	0	0
$469$	26883.0	2.64678
$470$	−629.366	−0.0617670
$471$	0	0
$472$	−4492.49	−0.438101
$473$	2873.03	0.279286
$474$	0	0
$475$	−1595.98	−0.154165
$476$	−14047.8	−1.35269
$477$	0	0
$478$	−922.614	−0.0882832
$479$	3279.34	0.312812	0.156406	−	0.987693i	$-0.450009\pi$
$479$	3279.34	0.312812	0.156406	+	0.987693i	$0.450009\pi$
$480$	0	0
$481$	12907.1	1.22352
$482$	19821.4	1.87311
$483$	0	0
$484$	−5428.25	−0.509791
$485$	1007.05	0.0942844
$486$	0	0
$487$	−10506.7	−0.977624	−0.488812	−	0.872389i	$-0.662570\pi$
$487$	−10506.7	−0.977624	−0.488812	+	0.872389i	$0.662570\pi$
$488$	−980.867	−0.0909872
$489$	0	0
$490$	11991.1	1.10552
$491$	14264.5	1.31110	0.655548	−	0.755154i	$-0.272438\pi$
$491$	14264.5	1.31110	0.655548	+	0.755154i	$0.272438\pi$
$492$	0	0
$493$	11848.0	1.08237
$494$	14341.8	1.30621
$495$	0	0
$496$	19279.1	1.74527
$497$	18454.2	1.66556
$498$	0	0
$499$	9447.40	0.847542	0.423771	−	0.905769i	$-0.360706\pi$
$499$	9447.40	0.847542	0.423771	+	0.905769i	$0.360706\pi$
$500$	−756.468	−0.0676605
$501$	0	0
$502$	10575.6	0.940259
$503$	−14579.2	−1.29235	−0.646177	−	0.763188i	$-0.723633\pi$
$503$	−14579.2	−1.29235	−0.646177	+	0.763188i	$0.723633\pi$
$504$	0	0
$505$	1326.91	0.116925
$506$	−2565.37	−0.225385
$507$	0	0
$508$	−5906.33	−0.515849
$509$	−8410.65	−0.732407	−0.366204	−	0.930535i	$-0.619343\pi$
$509$	−8410.65	−0.732407	−0.366204	+	0.930535i	$0.619343\pi$
$510$	0	0
$511$	19379.8	1.67771
$512$	−12676.3	−1.09417
$513$	0	0
$514$	−7063.02	−0.606102
$515$	2638.09	0.225725
$516$	0	0
$517$	−699.562	−0.0595100
$518$	25308.5	2.14670
$519$	0	0
$520$	−2188.43	−0.184556
$521$	−10058.1	−0.845781	−0.422890	−	0.906181i	$-0.638984\pi$
$521$	−10058.1	−0.845781	−0.422890	+	0.906181i	$0.638984\pi$
$522$	0	0
$523$	20006.3	1.67269	0.836344	−	0.548205i	$-0.184688\pi$
$523$	20006.3	1.67269	0.836344	+	0.548205i	$0.184688\pi$
$524$	9859.67	0.821988
$525$	0	0
$526$	2073.00	0.171838
$527$	−18835.3	−1.55689
$528$	0	0
$529$	−11087.9	−0.911311
$530$	787.038	0.0645033
$531$	0	0
$532$	12111.4	0.987019
$533$	−8491.12	−0.690040
$534$	0	0
$535$	10421.2	0.842147
$536$	−6262.72	−0.504679
$537$	0	0
$538$	12608.2	1.01037
$539$	13328.5	1.06512
$540$	0	0
$541$	−1884.15	−0.149734	−0.0748668	−	0.997194i	$-0.523853\pi$
$541$	−1884.15	−0.149734	−0.0748668	+	0.997194i	$0.523853\pi$
$542$	12497.2	0.990404
$543$	0	0
$544$	16710.7	1.31703
$545$	−4628.25	−0.363766
$546$	0	0
$547$	−17996.5	−1.40672	−0.703358	−	0.710836i	$-0.748317\pi$
$547$	−17996.5	−1.40672	−0.703358	+	0.710836i	$0.748317\pi$
$548$	−4002.93	−0.312038
$549$	0	0
$550$	−1952.37	−0.151363
$551$	−10214.8	−0.789774
$552$	0	0
$553$	10824.5	0.832378
$554$	20194.5	1.54870
$555$	0	0
$556$	8369.49	0.638391
$557$	3615.76	0.275054	0.137527	−	0.990498i	$-0.456085\pi$
$557$	3615.76	0.275054	0.137527	+	0.990498i	$0.456085\pi$
$558$	0	0
$559$	8264.84	0.625341
$560$	−11879.8	−0.896454
$561$	0	0
$562$	13932.3	1.04573
$563$	−14821.8	−1.10953	−0.554765	−	0.832007i	$-0.687192\pi$
$563$	−14821.8	−1.10953	−0.554765	+	0.832007i	$0.687192\pi$
$564$	0	0
$565$	2730.09	0.203285
$566$	−10378.9	−0.770771
$567$	0	0
$568$	−4299.13	−0.317583
$569$	−22449.5	−1.65401	−0.827005	−	0.562195i	$-0.809957\pi$
$569$	−22449.5	−1.65401	−0.827005	+	0.562195i	$0.809957\pi$
$570$	0	0
$571$	−16013.5	−1.17363	−0.586817	−	0.809720i	$-0.699619\pi$
$571$	−16013.5	−1.17363	−0.586817	+	0.809720i	$0.699619\pi$
$572$	7555.97	0.552327
$573$	0	0
$574$	−16649.6	−1.21070
$575$	821.234	0.0595615
$576$	0	0
$577$	3096.97	0.223446	0.111723	−	0.993739i	$-0.464363\pi$
$577$	3096.97	0.223446	0.111723	+	0.993739i	$0.464363\pi$
$578$	−2135.97	−0.153711
$579$	0	0
$580$	−4841.66	−0.346619
$581$	−34269.9	−2.44708
$582$	0	0
$583$	874.820	0.0621464
$584$	−4514.76	−0.319901
$585$	0	0
$586$	−13045.0	−0.919595
$587$	−24624.7	−1.73147	−0.865734	−	0.500504i	$-0.833148\pi$
$587$	−24624.7	−1.73147	−0.865734	+	0.500504i	$0.833148\pi$
$588$	0	0
$589$	16239.0	1.13602
$590$	−11529.5	−0.804511
$591$	0	0
$592$	−16322.6	−1.13320
$593$	−27128.8	−1.87866	−0.939330	−	0.343014i	$-0.888552\pi$
$593$	−27128.8	−1.87866	−0.939330	+	0.343014i	$0.888552\pi$
$594$	0	0
$595$	11606.4	0.799691
$596$	−19144.6	−1.31576
$597$	0	0
$598$	−7379.79	−0.504652
$599$	−5631.75	−0.384152	−0.192076	−	0.981380i	$-0.561522\pi$
$599$	−5631.75	−0.384152	−0.192076	+	0.981380i	$0.561522\pi$
$600$	0	0
$601$	−14751.8	−1.00123	−0.500613	−	0.865671i	$-0.666892\pi$
$601$	−14751.8	−1.00123	−0.500613	+	0.865671i	$0.666892\pi$
$602$	16205.9	1.09718
$603$	0	0
$604$	−15485.2	−1.04318
$605$	4484.87	0.301382
$606$	0	0
$607$	−14782.1	−0.988445	−0.494222	−	0.869336i	$-0.664547\pi$
$607$	−14782.1	−0.988445	−0.494222	+	0.869336i	$0.664547\pi$
$608$	−14407.2	−0.961000
$609$	0	0
$610$	−2517.29	−0.167085
$611$	−2012.43	−0.133247
$612$	0	0
$613$	4947.28	0.325969	0.162984	−	0.986629i	$-0.447888\pi$
$613$	4947.28	0.325969	0.162984	+	0.986629i	$0.447888\pi$
$614$	6786.24	0.446043
$615$	0	0
$616$	−4769.75	−0.311979
$617$	3743.65	0.244268	0.122134	−	0.992514i	$-0.461026\pi$
$617$	3743.65	0.244268	0.122134	+	0.992514i	$0.461026\pi$
$618$	0	0
$619$	6138.74	0.398605	0.199303	−	0.979938i	$-0.436132\pi$
$619$	6138.74	0.398605	0.199303	+	0.979938i	$0.436132\pi$
$620$	7697.01	0.498580
$621$	0	0
$622$	−3326.09	−0.214412
$623$	13005.2	0.836343
$624$	0	0
$625$	625.000	0.0400000
$626$	8303.57	0.530156
$627$	0	0
$628$	21186.9	1.34626
$629$	15946.9	1.01088
$630$	0	0
$631$	−5548.00	−0.350019	−0.175010	−	0.984567i	$-0.555996\pi$
$631$	−5548.00	−0.350019	−0.175010	+	0.984567i	$0.555996\pi$
$632$	−2521.70	−0.158715
$633$	0	0
$634$	−2466.73	−0.154521
$635$	4879.86	0.304963
$636$	0	0
$637$	38342.1	2.38488
$638$	−12495.9	−0.775418
$639$	0	0
$640$	4535.43	0.280123
$641$	11608.1	0.715277	0.357639	−	0.933860i	$-0.383582\pi$
$641$	11608.1	0.715277	0.357639	+	0.933860i	$0.383582\pi$
$642$	0	0
$643$	−12721.7	−0.780240	−0.390120	−	0.920764i	$-0.627567\pi$
$643$	−12721.7	−0.780240	−0.390120	+	0.920764i	$0.627567\pi$
$644$	−6232.09	−0.381333
$645$	0	0
$646$	17719.6	1.07921
$647$	28203.7	1.71376	0.856879	−	0.515518i	$-0.172400\pi$
$647$	28203.7	1.71376	0.856879	+	0.515518i	$0.172400\pi$
$648$	0	0
$649$	−12815.4	−0.775115
$650$	−5616.39	−0.338912
$651$	0	0
$652$	1597.36	0.0959468
$653$	22692.8	1.35993	0.679966	−	0.733243i	$-0.261994\pi$
$653$	22692.8	1.35993	0.679966	+	0.733243i	$0.261994\pi$
$654$	0	0
$655$	−8146.14	−0.485948
$656$	10738.1	0.639103
$657$	0	0
$658$	−3946.02	−0.233787
$659$	−6004.00	−0.354905	−0.177453	−	0.984129i	$-0.556786\pi$
$659$	−6004.00	−0.354905	−0.177453	+	0.984129i	$0.556786\pi$
$660$	0	0
$661$	11916.4	0.701199	0.350600	−	0.936525i	$-0.385978\pi$
$661$	11916.4	0.701199	0.350600	+	0.936525i	$0.385978\pi$
$662$	−19779.1	−1.16123
$663$	0	0
$664$	7983.59	0.466602
$665$	−10006.5	−0.583512
$666$	0	0
$667$	5256.19	0.305128
$668$	21720.0	1.25804
$669$	0	0
$670$	−16072.6	−0.926774
$671$	−2798.06	−0.160980
$672$	0	0
$673$	−16033.7	−0.918356	−0.459178	−	0.888344i	$-0.651856\pi$
$673$	−16033.7	−0.918356	−0.459178	+	0.888344i	$0.651856\pi$
$674$	12599.9	0.720077
$675$	0	0
$676$	8440.54	0.480231
$677$	−11311.9	−0.642173	−0.321086	−	0.947050i	$-0.604048\pi$
$677$	−11311.9	−0.642173	−0.321086	+	0.947050i	$0.604048\pi$
$678$	0	0
$679$	6314.06	0.356865
$680$	−2703.85	−0.152483
$681$	0	0
$682$	19865.3	1.11537
$683$	652.395	0.0365493	0.0182747	−	0.999833i	$-0.494183\pi$
$683$	652.395	0.0365493	0.0182747	+	0.999833i	$0.494183\pi$
$684$	0	0
$685$	3307.25	0.184472
$686$	34874.9	1.94100
$687$	0	0
$688$	−10451.9	−0.579180
$689$	2516.59	0.139150
$690$	0	0
$691$	12537.9	0.690250	0.345125	−	0.938557i	$-0.387837\pi$
$691$	12537.9	0.690250	0.345125	+	0.938557i	$0.387837\pi$
$692$	−2241.90	−0.123157
$693$	0	0
$694$	35069.8	1.91820
$695$	−6914.95	−0.377408
$696$	0	0
$697$	−10490.9	−0.570119
$698$	−26387.3	−1.43091
$699$	0	0
$700$	−4742.93	−0.256094
$701$	−5880.60	−0.316844	−0.158422	−	0.987372i	$-0.550641\pi$
$701$	−5880.60	−0.316844	−0.158422	+	0.987372i	$0.550641\pi$
$702$	0	0
$703$	−13748.7	−0.737614
$704$	−4992.75	−0.267289
$705$	0	0
$706$	15752.1	0.839716
$707$	8319.54	0.442558
$708$	0	0
$709$	6406.66	0.339361	0.169681	−	0.985499i	$-0.445726\pi$
$709$	6406.66	0.339361	0.169681	+	0.985499i	$0.445726\pi$
$710$	−11033.3	−0.583198
$711$	0	0
$712$	−3029.72	−0.159471
$713$	−8356.01	−0.438899
$714$	0	0
$715$	−6242.81	−0.326528
$716$	−2704.51	−0.141162
$717$	0	0
$718$	2204.21	0.114569
$719$	−21907.0	−1.13629	−0.568144	−	0.822929i	$-0.692338\pi$
$719$	−21907.0	−1.13629	−0.568144	+	0.822929i	$0.692338\pi$
$720$	0	0
$721$	16540.4	0.854364
$722$	10434.4	0.537852
$723$	0	0
$724$	−5471.05	−0.280843
$725$	4000.22	0.204916
$726$	0	0
$727$	13370.9	0.682118	0.341059	−	0.940042i	$-0.389214\pi$
$727$	13370.9	0.682118	0.341059	+	0.940042i	$0.389214\pi$
$728$	−13721.1	−0.698542
$729$	0	0
$730$	−11586.6	−0.587453
$731$	10211.4	0.516664
$732$	0	0
$733$	30191.1	1.52133	0.760663	−	0.649147i	$-0.224874\pi$
$733$	30191.1	1.52133	0.760663	+	0.649147i	$0.224874\pi$
$734$	29183.0	1.46752
$735$	0	0
$736$	7413.44	0.371281
$737$	−17865.2	−0.892910
$738$	0	0
$739$	−11624.7	−0.578650	−0.289325	−	0.957231i	$-0.593431\pi$
$739$	−11624.7	−0.578650	−0.289325	+	0.957231i	$0.593431\pi$
$740$	−6516.67	−0.323727
$741$	0	0
$742$	4934.60	0.244144
$743$	5254.37	0.259440	0.129720	−	0.991551i	$-0.458592\pi$
$743$	5254.37	0.259440	0.129720	+	0.991551i	$0.458592\pi$
$744$	0	0
$745$	15817.5	0.777862
$746$	29328.9	1.43942
$747$	0	0
$748$	9335.55	0.456339
$749$	65339.3	3.18751
$750$	0	0
$751$	−29228.8	−1.42020	−0.710102	−	0.704099i	$-0.751351\pi$
$751$	−29228.8	−1.42020	−0.710102	+	0.704099i	$0.751351\pi$
$752$	2544.97	0.123411
$753$	0	0
$754$	−35946.8	−1.73622
$755$	12794.0	0.616716
$756$	0	0
$757$	32885.9	1.57894	0.789470	−	0.613789i	$-0.210355\pi$
$757$	32885.9	1.57894	0.789470	+	0.613789i	$0.210355\pi$
$758$	17542.9	0.840617
$759$	0	0
$760$	2331.14	0.111262
$761$	13268.2	0.632027	0.316014	−	0.948755i	$-0.397655\pi$
$761$	13268.2	0.632027	0.316014	+	0.948755i	$0.397655\pi$
$762$	0	0
$763$	−29018.4	−1.37685
$764$	3628.54	0.171827
$765$	0	0
$766$	−25219.0	−1.18956
$767$	−36866.1	−1.73554
$768$	0	0
$769$	34285.8	1.60777	0.803887	−	0.594782i	$-0.202762\pi$
$769$	34285.8	1.60777	0.803887	+	0.594782i	$0.202762\pi$
$770$	−12241.1	−0.572906
$771$	0	0
$772$	−26710.8	−1.24526
$773$	27987.1	1.30223	0.651117	−	0.758978i	$-0.274301\pi$
$773$	27987.1	1.30223	0.651117	+	0.758978i	$0.274301\pi$
$774$	0	0
$775$	−6359.34	−0.294754
$776$	−1470.94	−0.0680458
$777$	0	0
$778$	−17891.4	−0.824472
$779$	9044.81	0.416000
$780$	0	0
$781$	−12263.8	−0.561888
$782$	−9117.88	−0.416950
$783$	0	0
$784$	−48488.5	−2.20884
$785$	−17504.8	−0.795890
$786$	0	0
$787$	354.799	0.0160702	0.00803509	−	0.999968i	$-0.497442\pi$
$787$	354.799	0.0160702	0.00803509	+	0.999968i	$0.497442\pi$
$788$	29091.6	1.31516
$789$	0	0
$790$	−6471.68	−0.291458
$791$	17117.2	0.769430
$792$	0	0
$793$	−8049.15	−0.360446
$794$	17576.8	0.785616
$795$	0	0
$796$	1899.23	0.0845686
$797$	−12801.3	−0.568940	−0.284470	−	0.958685i	$-0.591818\pi$
$797$	−12801.3	−0.568940	−0.284470	+	0.958685i	$0.591818\pi$
$798$	0	0
$799$	−2486.39	−0.110090
$800$	5641.99	0.249343
$801$	0	0
$802$	5749.67	0.253152
$803$	−12879.0	−0.565988
$804$	0	0
$805$	5149.00	0.225439
$806$	57146.4	2.49739
$807$	0	0
$808$	−1938.14	−0.0843855
$809$	16374.9	0.711632	0.355816	−	0.934556i	$-0.384203\pi$
$809$	16374.9	0.711632	0.355816	+	0.934556i	$0.384203\pi$
$810$	0	0
$811$	−35518.2	−1.53787	−0.768936	−	0.639326i	$-0.779214\pi$
$811$	−35518.2	−1.53787	−0.768936	+	0.639326i	$0.779214\pi$
$812$	−30356.4	−1.31195
$813$	0	0
$814$	−16818.9	−0.724206
$815$	−1319.75	−0.0567225
$816$	0	0
$817$	−8803.77	−0.376995
$818$	32908.0	1.40660
$819$	0	0
$820$	4287.10	0.182576
$821$	−42943.9	−1.82552	−0.912760	−	0.408496i	$-0.866053\pi$
$821$	−42943.9	−1.82552	−0.912760	+	0.408496i	$0.866053\pi$
$822$	0	0
$823$	−7916.93	−0.335318	−0.167659	−	0.985845i	$-0.553621\pi$
$823$	−7916.93	−0.335318	−0.167659	+	0.985845i	$0.553621\pi$
$824$	−3853.29	−0.162907
$825$	0	0
$826$	−72288.0	−3.04506
$827$	−18774.4	−0.789421	−0.394710	−	0.918806i	$-0.629155\pi$
$827$	−18774.4	−0.789421	−0.394710	+	0.918806i	$0.629155\pi$
$828$	0	0
$829$	22166.1	0.928661	0.464330	−	0.885662i	$-0.346295\pi$
$829$	22166.1	0.928661	0.464330	+	0.885662i	$0.346295\pi$
$830$	20489.0	0.856850
$831$	0	0
$832$	−14362.6	−0.598478
$833$	47372.5	1.97042
$834$	0	0
$835$	−17945.3	−0.743738
$836$	−8048.68	−0.332978
$837$	0	0
$838$	−16029.1	−0.660757
$839$	−42590.8	−1.75256	−0.876279	−	0.481804i	$-0.839982\pi$
$839$	−42590.8	−1.75256	−0.876279	+	0.481804i	$0.839982\pi$
$840$	0	0
$841$	1213.82	0.0497690
$842$	−54216.5	−2.21903
$843$	0	0
$844$	14759.1	0.601930
$845$	−6973.65	−0.283906
$846$	0	0
$847$	28119.4	1.14072
$848$	−3182.54	−0.128879
$849$	0	0
$850$	−6939.16	−0.280013
$851$	7074.61	0.284976
$852$	0	0
$853$	26461.1	1.06215	0.531074	−	0.847326i	$-0.321789\pi$
$853$	26461.1	1.06215	0.531074	+	0.847326i	$0.321789\pi$
$854$	−15783.0	−0.632416
$855$	0	0
$856$	−15221.6	−0.607784
$857$	20009.0	0.797543	0.398772	−	0.917050i	$-0.369437\pi$
$857$	20009.0	0.797543	0.398772	+	0.917050i	$0.369437\pi$
$858$	0	0
$859$	−17818.1	−0.707735	−0.353868	−	0.935296i	$-0.615134\pi$
$859$	−17818.1	−0.707735	−0.353868	+	0.935296i	$0.615134\pi$
$860$	−4172.85	−0.165457
$861$	0	0
$862$	−8277.26	−0.327059
$863$	−12769.5	−0.503683	−0.251841	−	0.967769i	$-0.581036\pi$
$863$	−12769.5	−0.503683	−0.251841	+	0.967769i	$0.581036\pi$
$864$	0	0
$865$	1852.28	0.0728085
$866$	37719.1	1.48008
$867$	0	0
$868$	48259.0	1.88712
$869$	−7193.50	−0.280809
$870$	0	0
$871$	−51392.9	−1.99929
$872$	6760.19	0.262533
$873$	0	0
$874$	7861.01	0.304236
$875$	3918.65	0.151399
$876$	0	0
$877$	27477.3	1.05797	0.528986	−	0.848630i	$-0.322572\pi$
$877$	27477.3	1.05797	0.528986	+	0.848630i	$0.322572\pi$
$878$	−49920.5	−1.91883
$879$	0	0
$880$	7894.82	0.302425
$881$	−31509.1	−1.20496	−0.602480	−	0.798134i	$-0.705821\pi$
$881$	−31509.1	−1.20496	−0.602480	+	0.798134i	$0.705821\pi$
$882$	0	0
$883$	5271.46	0.200905	0.100452	−	0.994942i	$-0.467971\pi$
$883$	5271.46	0.200905	0.100452	+	0.994942i	$0.467971\pi$
$884$	26855.5	1.02178
$885$	0	0
$886$	−53511.1	−2.02905
$887$	−10433.8	−0.394963	−0.197481	−	0.980307i	$-0.563276\pi$
$887$	−10433.8	−0.394963	−0.197481	+	0.980307i	$0.563276\pi$
$888$	0	0
$889$	30595.9	1.15428
$890$	−7775.45	−0.292847
$891$	0	0
$892$	−14121.2	−0.530059
$893$	2143.65	0.0803299
$894$	0	0
$895$	2234.49	0.0834533
$896$	28436.4	1.06026
$897$	0	0
$898$	−6335.15	−0.235419
$899$	−40702.0	−1.51000
$900$	0	0
$901$	3109.30	0.114967
$902$	11064.6	0.408438
$903$	0	0
$904$	−3987.67	−0.146712
$905$	4520.23	0.166030
$906$	0	0
$907$	20686.6	0.757317	0.378659	−	0.925536i	$-0.376385\pi$
$907$	20686.6	0.757317	0.378659	+	0.925536i	$0.376385\pi$
$908$	16483.1	0.602434
$909$	0	0
$910$	−35213.8	−1.28278
$911$	23640.1	0.859748	0.429874	−	0.902889i	$-0.358558\pi$
$911$	23640.1	0.859748	0.429874	+	0.902889i	$0.358558\pi$
$912$	0	0
$913$	22774.3	0.825541
$914$	24981.9	0.904078
$915$	0	0
$916$	−20056.4	−0.723454
$917$	−51075.0	−1.83931
$918$	0	0
$919$	27335.2	0.981180	0.490590	−	0.871390i	$-0.336781\pi$
$919$	27335.2	0.981180	0.490590	+	0.871390i	$0.336781\pi$
$920$	−1199.52	−0.0429860
$921$	0	0
$922$	6250.44	0.223262
$923$	−35279.3	−1.25811
$924$	0	0
$925$	5384.13	0.191383
$926$	21861.7	0.775832
$927$	0	0
$928$	36110.7	1.27736
$929$	13228.6	0.467185	0.233593	−	0.972335i	$-0.424952\pi$
$929$	13228.6	0.467185	0.233593	+	0.972335i	$0.424952\pi$
$930$	0	0
$931$	−40842.4	−1.43776
$932$	19216.9	0.675399
$933$	0	0
$934$	65264.7	2.28643
$935$	−7713.11	−0.269782
$936$	0	0
$937$	−18740.4	−0.653387	−0.326693	−	0.945130i	$-0.605934\pi$
$937$	−18740.4	−0.653387	−0.326693	+	0.945130i	$0.605934\pi$
$938$	−100773.	−3.50783
$939$	0	0
$940$	1016.06	0.0352555
$941$	−54489.0	−1.88766	−0.943832	−	0.330427i	$-0.892807\pi$
$941$	−54489.0	−1.88766	−0.943832	+	0.330427i	$0.892807\pi$
$942$	0	0
$943$	−4654.15	−0.160721
$944$	46621.8	1.60743
$945$	0	0
$946$	−10769.7	−0.370142
$947$	11282.7	0.387158	0.193579	−	0.981085i	$-0.437990\pi$
$947$	11282.7	0.387158	0.193579	+	0.981085i	$0.437990\pi$
$948$	0	0
$949$	−37048.8	−1.26729
$950$	5982.62	0.204318
$951$	0	0
$952$	−16952.7	−0.577144
$953$	51190.0	1.73999	0.869994	−	0.493063i	$-0.164123\pi$
$953$	51190.0	1.73999	0.869994	+	0.493063i	$0.164123\pi$
$954$	0	0
$955$	−2997.93	−0.101582
$956$	1489.48	0.0503905
$957$	0	0
$958$	−12292.8	−0.414575
$959$	20735.9	0.698225
$960$	0	0
$961$	34914.9	1.17199
$962$	−48383.0	−1.62155
$963$	0	0
$964$	−32000.0	−1.06914
$965$	22068.7	0.736182
$966$	0	0
$967$	23143.5	0.769643	0.384822	−	0.922991i	$-0.374263\pi$
$967$	23143.5	0.769643	0.384822	+	0.922991i	$0.374263\pi$
$968$	−6550.76	−0.217510
$969$	0	0
$970$	−3775.00	−0.124957
$971$	34124.5	1.12782	0.563908	−	0.825838i	$-0.309297\pi$
$971$	34124.5	1.12782	0.563908	+	0.825838i	$0.309297\pi$
$972$	0	0
$973$	−43355.6	−1.42848
$974$	39384.9	1.29566
$975$	0	0
$976$	10179.2	0.333839
$977$	36463.7	1.19404	0.597020	−	0.802226i	$-0.296351\pi$
$977$	36463.7	1.19404	0.597020	+	0.802226i	$0.296351\pi$
$978$	0	0
$979$	−8642.68	−0.282146
$980$	−19358.6	−0.631009
$981$	0	0
$982$	−53471.4	−1.73762
$983$	51541.6	1.67235	0.836176	−	0.548462i	$-0.184786\pi$
$983$	51541.6	1.67235	0.836176	+	0.548462i	$0.184786\pi$
$984$	0	0
$985$	−24035.7	−0.777505
$986$	−44413.0	−1.43448
$987$	0	0
$988$	−23153.6	−0.745561
$989$	4530.12	0.145652
$990$	0	0
$991$	34299.1	1.09944	0.549721	−	0.835348i	$-0.314734\pi$
$991$	34299.1	1.09944	0.549721	+	0.835348i	$0.314734\pi$
$992$	−57406.9	−1.83737
$993$	0	0
$994$	−69176.7	−2.20740
$995$	−1569.16	−0.0499958
$996$	0	0
$997$	21297.0	0.676512	0.338256	−	0.941054i	$-0.390163\pi$
$997$	21297.0	0.676512	0.338256	+	0.941054i	$0.390163\pi$
$998$	−35414.2	−1.12326
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.a.j.1.1		3
3.2	odd	2		405.4.a.h.1.3		3
5.4	even	2		2025.4.a.q.1.3		3
9.2	odd	6		45.4.e.b.31.1	yes	6
9.4	even	3		135.4.e.b.46.3		6
9.5	odd	6		45.4.e.b.16.1	✓	6
9.7	even	3		135.4.e.b.91.3		6
15.14	odd	2		2025.4.a.s.1.1		3
45.2	even	12		225.4.k.c.49.5		12
45.14	odd	6		225.4.e.c.151.3		6
45.23	even	12		225.4.k.c.124.5		12
45.29	odd	6		225.4.e.c.76.3		6
45.32	even	12		225.4.k.c.124.2		12
45.38	even	12		225.4.k.c.49.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.4.e.b.16.1	✓	6	9.5	odd	6
45.4.e.b.31.1	yes	6	9.2	odd	6
135.4.e.b.46.3		6	9.4	even	3
135.4.e.b.91.3		6	9.7	even	3
225.4.e.c.76.3		6	45.29	odd	6
225.4.e.c.151.3		6	45.14	odd	6
225.4.k.c.49.2		12	45.38	even	12
225.4.k.c.49.5		12	45.2	even	12
225.4.k.c.124.2		12	45.32	even	12
225.4.k.c.124.5		12	45.23	even	12
405.4.a.h.1.3		3	3.2	odd	2
405.4.a.j.1.1		3	1.1	even	1	trivial
2025.4.a.q.1.3		3	5.4	even	2
2025.4.a.s.1.1		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q-3.74857 q^{2} +6.05174 q^{4} -5.00000 q^{5} -31.3492 q^{7} +7.30318 q^{8} +O(q^{10})\)	\(q-3.74857 q^{2} +6.05174 q^{4} -5.00000 q^{5} -31.3492 q^{7} +7.30318 q^{8} +18.7428 q^{10} +20.8333 q^{11} +59.9310 q^{13} +117.514 q^{14} -75.7904 q^{16} +74.0460 q^{17} -63.8390 q^{19} -30.2587 q^{20} -78.0950 q^{22} +32.8494 q^{23} +25.0000 q^{25} -224.655 q^{26} -189.717 q^{28} +160.009 q^{29} -254.373 q^{31} +225.680 q^{32} -277.566 q^{34} +156.746 q^{35} +215.365 q^{37} +239.305 q^{38} -36.5159 q^{40} -141.681 q^{41} +137.906 q^{43} +126.078 q^{44} -123.138 q^{46} -33.5790 q^{47} +639.771 q^{49} -93.7141 q^{50} +362.687 q^{52} +41.9914 q^{53} -104.166 q^{55} -228.949 q^{56} -599.803 q^{58} -615.142 q^{59} -134.307 q^{61} +953.535 q^{62} -239.652 q^{64} -299.655 q^{65} -857.533 q^{67} +448.107 q^{68} -587.572 q^{70} -588.665 q^{71} -618.191 q^{73} -807.311 q^{74} -386.337 q^{76} -653.107 q^{77} -345.288 q^{79} +378.952 q^{80} +531.102 q^{82} +1093.17 q^{83} -370.230 q^{85} -516.949 q^{86} +152.149 q^{88} -414.849 q^{89} -1878.79 q^{91} +198.796 q^{92} +125.873 q^{94} +319.195 q^{95} -201.411 q^{97} -2398.22 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 11 q^{4} - 15 q^{5} - 43 q^{7} + 27 q^{8}+O(q^{10}) \) 3 * q + q^2 + 11 * q^4 - 15 * q^5 - 43 * q^7 + 27 * q^8	\( 3 q + q^{2} + 11 q^{4} - 15 q^{5} - 43 q^{7} + 27 q^{8} - 5 q^{10} - 14 q^{11} + 40 q^{13} + 27 q^{14} - 13 q^{16} + 166 q^{17} - 164 q^{19} - 55 q^{20} - 376 q^{22} - 171 q^{23} + 75 q^{25} - 434 q^{26} - 517 q^{28} + 335 q^{29} - 352 q^{31} + 77 q^{32} - 52 q^{34} + 215 q^{35} + 402 q^{37} + 178 q^{38} - 135 q^{40} - 187 q^{41} - 602 q^{43} - 982 q^{44} - 201 q^{46} - 665 q^{47} + 430 q^{49} + 25 q^{50} - 456 q^{52} + 730 q^{53} + 70 q^{55} - 705 q^{56} + 217 q^{58} + 298 q^{59} - 1439 q^{61} + 1614 q^{62} - 1569 q^{64} - 200 q^{65} - 1849 q^{67} + 710 q^{68} - 135 q^{70} - 70 q^{71} - 368 q^{73} + 320 q^{74} + 204 q^{76} + 948 q^{77} - 382 q^{79} + 65 q^{80} - 575 q^{82} + 831 q^{83} - 830 q^{85} - 1580 q^{86} - 1428 q^{88} - 1719 q^{89} - 710 q^{91} + 1623 q^{92} - 2077 q^{94} + 820 q^{95} - 282 q^{97} - 2164 q^{98}+O(q^{100}) \) 3 * q + q^2 + 11 * q^4 - 15 * q^5 - 43 * q^7 + 27 * q^8 - 5 * q^10 - 14 * q^11 + 40 * q^13 + 27 * q^14 - 13 * q^16 + 166 * q^17 - 164 * q^19 - 55 * q^20 - 376 * q^22 - 171 * q^23 + 75 * q^25 - 434 * q^26 - 517 * q^28 + 335 * q^29 - 352 * q^31 + 77 * q^32 - 52 * q^34 + 215 * q^35 + 402 * q^37 + 178 * q^38 - 135 * q^40 - 187 * q^41 - 602 * q^43 - 982 * q^44 - 201 * q^46 - 665 * q^47 + 430 * q^49 + 25 * q^50 - 456 * q^52 + 730 * q^53 + 70 * q^55 - 705 * q^56 + 217 * q^58 + 298 * q^59 - 1439 * q^61 + 1614 * q^62 - 1569 * q^64 - 200 * q^65 - 1849 * q^67 + 710 * q^68 - 135 * q^70 - 70 * q^71 - 368 * q^73 + 320 * q^74 + 204 * q^76 + 948 * q^77 - 382 * q^79 + 65 * q^80 - 575 * q^82 + 831 * q^83 - 830 * q^85 - 1580 * q^86 - 1428 * q^88 - 1719 * q^89 - 710 * q^91 + 1623 * q^92 - 2077 * q^94 + 820 * q^95 - 282 * q^97 - 2164 * q^98

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