LMFDB - Embedded newform 405.3.c.a.161.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,3,Mod(161,405)]

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([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("405.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	405.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$11.0354507066$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 48x^{14} + 912x^{12} + 8704x^{10} + 43602x^{8} + 109032x^{6} + 117844x^{4} + 36000x^{2} + 81 $ x^16 + 48x^14 + 912x^12 + 8704x^10 + 43602x^8 + 109032x^6 + 117844x^4 + 36000*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{14} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.3
Root			$-3.09125i$ of defining polynomial
Character	$\chi$	$=$	405.161
Dual form			405.3.c.a.161.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.09125i q^{2} -5.55585 q^{4} +2.23607i q^{5} -2.20591 q^{7} +4.80953i q^{8} +O(q^{10})$	\(q-3.09125i q^{2} -5.55585 q^{4} +2.23607i q^{5} -2.20591 q^{7} +4.80953i q^{8} +6.91225 q^{10} +17.8348i q^{11} -2.51375 q^{13} +6.81903i q^{14} -7.35593 q^{16} +32.6026i q^{17} +7.93398 q^{19} -12.4233i q^{20} +55.1319 q^{22} -21.2944i q^{23} -5.00000 q^{25} +7.77065i q^{26} +12.2557 q^{28} +35.5335i q^{29} +2.02443 q^{31} +41.9772i q^{32} +100.783 q^{34} -4.93257i q^{35} +50.6833 q^{37} -24.5260i q^{38} -10.7544 q^{40} +5.37001i q^{41} +15.5360 q^{43} -99.0875i q^{44} -65.8263 q^{46} +1.88724i q^{47} -44.1340 q^{49} +15.4563i q^{50} +13.9660 q^{52} -62.0293i q^{53} -39.8798 q^{55} -10.6094i q^{56} +109.843 q^{58} +36.1654i q^{59} -42.2485 q^{61} -6.25802i q^{62} +100.338 q^{64} -5.62092i q^{65} -14.7748 q^{67} -181.135i q^{68} -15.2478 q^{70} +105.070i q^{71} -66.9435 q^{73} -156.675i q^{74} -44.0800 q^{76} -39.3420i q^{77} -69.1506 q^{79} -16.4484i q^{80} +16.6001 q^{82} +21.2285i q^{83} -72.9016 q^{85} -48.0256i q^{86} -85.7770 q^{88} +7.16304i q^{89} +5.54511 q^{91} +118.308i q^{92} +5.83394 q^{94} +17.7409i q^{95} +111.341 q^{97} +136.429i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} - 4 q^{7}+O(q^{10}) $ 16 * q - 32 * q^4 - 4 * q^7	$ 16 q - 32 q^{4} - 4 q^{7} + 20 q^{13} + 64 q^{16} - 52 q^{19} + 48 q^{22} - 80 q^{25} + 32 q^{28} - 64 q^{31} - 108 q^{34} + 44 q^{37} + 60 q^{40} + 248 q^{43} - 108 q^{46} + 108 q^{49} - 124 q^{52} - 180 q^{58} - 124 q^{61} + 256 q^{64} - 28 q^{67} + 120 q^{70} - 268 q^{73} + 212 q^{76} + 80 q^{79} - 204 q^{82} - 60 q^{85} - 288 q^{88} + 136 q^{91} + 300 q^{94} + 284 q^{97}+O(q^{100}) $ 16 * q - 32 * q^4 - 4 * q^7 + 20 * q^13 + 64 * q^16 - 52 * q^19 + 48 * q^22 - 80 * q^25 + 32 * q^28 - 64 * q^31 - 108 * q^34 + 44 * q^37 + 60 * q^40 + 248 * q^43 - 108 * q^46 + 108 * q^49 - 124 * q^52 - 180 * q^58 - 124 * q^61 + 256 * q^64 - 28 * q^67 + 120 * q^70 - 268 * q^73 + 212 * q^76 + 80 * q^79 - 204 * q^82 - 60 * q^85 - 288 * q^88 + 136 * q^91 + 300 * q^94 + 284 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 3.09125i	− 1.54563i	−0.634633	−	0.772813i	$-0.718849\pi$
$2$	− 3.09125i	− 1.54563i	0.634633	−	0.772813i	$-0.281151\pi$
$3$	0	0
$3$	0	0
$4$	−5.55585	−1.38896
$5$	2.23607i	0.447214i
$5$	2.23607i	0.447214i
$6$	0	0
$7$	−2.20591	−0.315130	−0.157565	−	0.987509i	$-0.550364\pi$
$7$	−2.20591	−0.315130	−0.157565	+	0.987509i	$0.550364\pi$
$8$	4.80953i	0.601191i
$9$	0	0
$10$	6.91225	0.691225
$11$	17.8348i	1.62135i	0.585500	+	0.810673i	$0.300898\pi$
$11$	17.8348i	1.62135i	−0.585500	+	0.810673i	$0.699102\pi$
$12$	0	0
$13$	−2.51375	−0.193366	−0.0966828	−	0.995315i	$-0.530823\pi$
$13$	−2.51375	−0.193366	−0.0966828	+	0.995315i	$0.530823\pi$
$14$	6.81903i	0.487074i
$15$	0	0
$16$	−7.35593	−0.459746
$17$	32.6026i	1.91780i	0.283746	+	0.958899i	$0.408423\pi$
$17$	32.6026i	1.91780i	−0.283746	+	0.958899i	$0.591577\pi$
$18$	0	0
$19$	7.93398	0.417578	0.208789	−	0.977961i	$-0.433048\pi$
$19$	7.93398	0.417578	0.208789	+	0.977961i	$0.433048\pi$
$20$	− 12.4233i	− 0.621163i
$21$	0	0
$22$	55.1319	2.50599
$23$	− 21.2944i	− 0.925842i	−0.886400	−	0.462921i	$-0.846801\pi$
$23$	− 21.2944i	− 0.925842i	0.886400	−	0.462921i	$-0.153199\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	7.77065i	0.298871i
$27$	0	0
$28$	12.2557	0.437704
$29$	35.5335i	1.22529i	0.790356	+	0.612647i	$0.209895\pi$
$29$	35.5335i	1.22529i	−0.790356	+	0.612647i	$0.790105\pi$
$30$	0	0
$31$	2.02443	0.0653041	0.0326521	−	0.999467i	$-0.489605\pi$
$31$	2.02443	0.0653041	0.0326521	+	0.999467i	$0.489605\pi$
$32$	41.9772i	1.31179i
$33$	0	0
$34$	100.783	2.96420
$35$	− 4.93257i	− 0.140930i
$36$	0	0
$37$	50.6833	1.36982	0.684909	−	0.728628i	$-0.259842\pi$
$37$	50.6833	1.36982	0.684909	+	0.728628i	$0.259842\pi$
$38$	− 24.5260i	− 0.645420i
$39$	0	0
$40$	−10.7544	−0.268861
$41$	5.37001i	0.130976i	0.997853	+	0.0654879i	$0.0208604\pi$
$41$	5.37001i	0.130976i	−0.997853	+	0.0654879i	$0.979140\pi$
$42$	0	0
$43$	15.5360	0.361301	0.180651	−	0.983547i	$-0.442180\pi$
$43$	15.5360	0.361301	0.180651	+	0.983547i	$0.442180\pi$
$44$	− 99.0875i	− 2.25199i
$45$	0	0
$46$	−65.8263	−1.43101
$47$	1.88724i	0.0401540i	0.999798	+	0.0200770i	$0.00639114\pi$
$47$	1.88724i	0.0401540i	−0.999798	+	0.0200770i	$0.993609\pi$
$48$	0	0
$49$	−44.1340	−0.900693
$50$	15.4563i	0.309125i
$51$	0	0
$52$	13.9660	0.268578
$53$	− 62.0293i	− 1.17036i	−0.810902	−	0.585182i	$-0.801023\pi$
$53$	− 62.0293i	− 1.17036i	0.810902	−	0.585182i	$-0.198977\pi$
$54$	0	0
$55$	−39.8798	−0.725088
$56$	− 10.6094i	− 0.189453i
$57$	0	0
$58$	109.843	1.89385
$59$	36.1654i	0.612973i	0.951875	+	0.306487i	$0.0991535\pi$
$59$	36.1654i	0.612973i	−0.951875	+	0.306487i	$0.900847\pi$
$60$	0	0
$61$	−42.2485	−0.692598	−0.346299	−	0.938124i	$-0.612562\pi$
$61$	−42.2485	−0.692598	−0.346299	+	0.938124i	$0.612562\pi$
$62$	− 6.25802i	− 0.100936i
$63$	0	0
$64$	100.338	1.56779
$65$	− 5.62092i	− 0.0864757i
$66$	0	0
$67$	−14.7748	−0.220519	−0.110259	−	0.993903i	$-0.535168\pi$
$67$	−14.7748	−0.220519	−0.110259	+	0.993903i	$0.535168\pi$
$68$	− 181.135i	− 2.66375i
$69$	0	0
$70$	−15.2478	−0.217826
$71$	105.070i	1.47985i	0.672687	+	0.739927i	$0.265140\pi$
$71$	105.070i	1.47985i	−0.672687	+	0.739927i	$0.734860\pi$
$72$	0	0
$73$	−66.9435	−0.917034	−0.458517	−	0.888686i	$-0.651619\pi$
$73$	−66.9435	−0.917034	−0.458517	+	0.888686i	$0.651619\pi$
$74$	− 156.675i	− 2.11723i
$75$	0	0
$76$	−44.0800	−0.580000
$77$	− 39.3420i	− 0.510935i
$78$	0	0
$79$	−69.1506	−0.875324	−0.437662	−	0.899140i	$-0.644193\pi$
$79$	−69.1506	−0.875324	−0.437662	+	0.899140i	$0.644193\pi$
$80$	− 16.4484i	− 0.205604i
$81$	0	0
$82$	16.6001	0.202440
$83$	21.2285i	0.255765i	0.991789	+	0.127883i	$0.0408181\pi$
$83$	21.2285i	0.255765i	−0.991789	+	0.127883i	$0.959182\pi$
$84$	0	0
$85$	−72.9016	−0.857666
$86$	− 48.0256i	− 0.558437i
$87$	0	0
$88$	−85.7770	−0.974738
$89$	7.16304i	0.0804836i	0.999190	+	0.0402418i	$0.0128128\pi$
$89$	7.16304i	0.0804836i	−0.999190	+	0.0402418i	$0.987187\pi$
$90$	0	0
$91$	5.54511	0.0609353
$92$	118.308i	1.28596i
$93$	0	0
$94$	5.83394	0.0620632
$95$	17.7409i	0.186747i
$96$	0	0
$97$	111.341	1.14784	0.573921	−	0.818911i	$-0.305422\pi$
$97$	111.341	1.14784	0.573921	+	0.818911i	$0.305422\pi$
$98$	136.429i	1.39214i
$99$	0	0
$100$	27.7793	0.277793
$101$	− 62.1946i	− 0.615788i	−0.951421	−	0.307894i	$-0.900376\pi$
$101$	− 62.1946i	− 0.615788i	0.951421	−	0.307894i	$-0.0996242\pi$
$102$	0	0
$103$	55.0946	0.534899	0.267449	−	0.963572i	$-0.413819\pi$
$103$	55.0946	0.534899	0.267449	+	0.963572i	$0.413819\pi$
$104$	− 12.0900i	− 0.116250i
$105$	0	0
$106$	−191.748	−1.80895
$107$	85.2584i	0.796807i	0.917210	+	0.398404i	$0.130436\pi$
$107$	85.2584i	0.796807i	−0.917210	+	0.398404i	$0.869564\pi$
$108$	0	0
$109$	−64.0279	−0.587412	−0.293706	−	0.955896i	$-0.594889\pi$
$109$	−64.0279	−0.587412	−0.293706	+	0.955896i	$0.594889\pi$
$110$	123.279i	1.12072i
$111$	0	0
$112$	16.2265	0.144880
$113$	151.492i	1.34064i	0.742074	+	0.670319i	$0.233843\pi$
$113$	151.492i	1.34064i	−0.742074	+	0.670319i	$0.766157\pi$
$114$	0	0
$115$	47.6156	0.414049
$116$	− 197.419i	− 1.70189i
$117$	0	0
$118$	111.796	0.947428
$119$	− 71.9184i	− 0.604356i
$120$	0	0
$121$	−197.080	−1.62876
$122$	130.601i	1.07050i
$123$	0	0
$124$	−11.2474	−0.0907050
$125$	− 11.1803i	− 0.0894427i
$126$	0	0
$127$	42.9132	0.337899	0.168950	−	0.985625i	$-0.445963\pi$
$127$	42.9132	0.337899	0.168950	+	0.985625i	$0.445963\pi$
$128$	− 142.263i	− 1.11143i
$129$	0	0
$130$	−17.3757	−0.133659
$131$	60.8453i	0.464468i	0.972660	+	0.232234i	$0.0746036\pi$
$131$	60.8453i	0.464468i	−0.972660	+	0.232234i	$0.925396\pi$
$132$	0	0
$133$	−17.5017	−0.131591
$134$	45.6726i	0.340840i
$135$	0	0
$136$	−156.803	−1.15296
$137$	208.424i	1.52134i	0.649138	+	0.760670i	$0.275130\pi$
$137$	208.424i	1.52134i	−0.649138	+	0.760670i	$0.724870\pi$
$138$	0	0
$139$	134.489	0.967546	0.483773	−	0.875193i	$-0.339266\pi$
$139$	134.489	0.967546	0.483773	+	0.875193i	$0.339266\pi$
$140$	27.4046i	0.195747i
$141$	0	0
$142$	324.797	2.28730
$143$	− 44.8323i	− 0.313512i
$144$	0	0
$145$	−79.4554	−0.547968
$146$	206.939i	1.41739i
$147$	0	0
$148$	−281.589	−1.90263
$149$	− 121.527i	− 0.815615i	−0.913068	−	0.407807i	$-0.866294\pi$
$149$	− 121.527i	− 0.815615i	0.913068	−	0.407807i	$-0.133706\pi$
$150$	0	0
$151$	−42.0492	−0.278471	−0.139236	−	0.990259i	$-0.544465\pi$
$151$	−42.0492	−0.278471	−0.139236	+	0.990259i	$0.544465\pi$
$152$	38.1587i	0.251044i
$153$	0	0
$154$	−121.616	−0.789714
$155$	4.52676i	0.0292049i
$156$	0	0
$157$	93.5881	0.596103	0.298051	−	0.954550i	$-0.403663\pi$
$157$	93.5881	0.596103	0.298051	+	0.954550i	$0.403663\pi$
$158$	213.762i	1.35292i
$159$	0	0
$160$	−93.8638	−0.586649
$161$	46.9735i	0.291761i
$162$	0	0
$163$	−115.792	−0.710377	−0.355189	−	0.934795i	$-0.615583\pi$
$163$	−115.792	−0.710377	−0.355189	+	0.934795i	$0.615583\pi$
$164$	− 29.8350i	− 0.181920i
$165$	0	0
$166$	65.6228	0.395318
$167$	− 1.62681i	− 0.00974136i	−0.999988	−	0.00487068i	$-0.998450\pi$
$167$	− 1.62681i	− 0.00974136i	0.999988	−	0.00487068i	$-0.00155039\pi$
$168$	0	0
$169$	−162.681	−0.962610
$170$	225.357i	1.32563i
$171$	0	0
$172$	−86.3154	−0.501834
$173$	120.203i	0.694812i	0.937715	+	0.347406i	$0.112937\pi$
$173$	120.203i	0.694812i	−0.937715	+	0.347406i	$0.887063\pi$
$174$	0	0
$175$	11.0296	0.0630260
$176$	− 131.192i	− 0.745406i
$177$	0	0
$178$	22.1428	0.124398
$179$	− 157.742i	− 0.881238i	−0.897694	−	0.440619i	$-0.854759\pi$
$179$	− 157.742i	− 0.881238i	0.897694	−	0.440619i	$-0.145241\pi$
$180$	0	0
$181$	−219.069	−1.21033	−0.605164	−	0.796101i	$-0.706892\pi$
$181$	−219.069	−1.21033	−0.605164	+	0.796101i	$0.706892\pi$
$182$	− 17.1414i	− 0.0941833i
$183$	0	0
$184$	102.416	0.556608
$185$	113.331i	0.612602i
$186$	0	0
$187$	−581.460	−3.10941
$188$	− 10.4852i	− 0.0557725i
$189$	0	0
$190$	54.8417	0.288641
$191$	− 371.552i	− 1.94530i	−0.232284	−	0.972648i	$-0.574620\pi$
$191$	− 371.552i	− 1.94530i	0.232284	−	0.972648i	$-0.425380\pi$
$192$	0	0
$193$	−39.5592	−0.204970	−0.102485	−	0.994735i	$-0.532679\pi$
$193$	−39.5592	−0.204970	−0.102485	+	0.994735i	$0.532679\pi$
$194$	− 344.182i	− 1.77413i
$195$	0	0
$196$	245.202	1.25103
$197$	− 104.454i	− 0.530224i	−0.964218	−	0.265112i	$-0.914591\pi$
$197$	− 104.454i	− 0.530224i	0.964218	−	0.265112i	$-0.0854089\pi$
$198$	0	0
$199$	118.036	0.593147	0.296573	−	0.955010i	$-0.404156\pi$
$199$	118.036	0.593147	0.296573	+	0.955010i	$0.404156\pi$
$200$	− 24.0476i	− 0.120238i
$201$	0	0
$202$	−192.259	−0.951778
$203$	− 78.3838i	− 0.386127i
$204$	0	0
$205$	−12.0077	−0.0585742
$206$	− 170.311i	− 0.826754i
$207$	0	0
$208$	18.4910	0.0888990
$209$	141.501i	0.677038i
$210$	0	0
$211$	136.399	0.646441	0.323220	−	0.946324i	$-0.395234\pi$
$211$	136.399	0.646441	0.323220	+	0.946324i	$0.395234\pi$
$212$	344.625i	1.62559i
$213$	0	0
$214$	263.555	1.23157
$215$	34.7395i	0.161579i
$216$	0	0
$217$	−4.46571	−0.0205793
$218$	197.926i	0.907920i
$219$	0	0
$220$	221.566	1.00712
$221$	− 81.9548i	− 0.370836i
$222$	0	0
$223$	338.327	1.51716	0.758581	−	0.651578i	$-0.225893\pi$
$223$	338.327	1.51716	0.758581	+	0.651578i	$0.225893\pi$
$224$	− 92.5979i	− 0.413383i
$225$	0	0
$226$	468.300	2.07212
$227$	− 153.404i	− 0.675786i	−0.941184	−	0.337893i	$-0.890286\pi$
$227$	− 153.404i	− 0.675786i	0.941184	−	0.337893i	$-0.109714\pi$
$228$	0	0
$229$	−253.376	−1.10645	−0.553223	−	0.833033i	$-0.686602\pi$
$229$	−253.376	−1.10645	−0.553223	+	0.833033i	$0.686602\pi$
$230$	− 147.192i	− 0.639965i
$231$	0	0
$232$	−170.900	−0.736636
$233$	− 309.268i	− 1.32733i	−0.748030	−	0.663664i	$-0.769000\pi$
$233$	− 309.268i	− 1.32733i	0.748030	−	0.663664i	$-0.231000\pi$
$234$	0	0
$235$	−4.22000	−0.0179574
$236$	− 200.930i	− 0.851397i
$237$	0	0
$238$	−222.318	−0.934109
$239$	− 71.8263i	− 0.300528i	−0.988646	−	0.150264i	$-0.951988\pi$
$239$	− 71.8263i	− 0.300528i	0.988646	−	0.150264i	$-0.0480124\pi$
$240$	0	0
$241$	302.959	1.25709	0.628546	−	0.777772i	$-0.283650\pi$
$241$	302.959	1.25709	0.628546	+	0.777772i	$0.283650\pi$
$242$	609.224i	2.51746i
$243$	0	0
$244$	234.726	0.961993
$245$	− 98.6865i	− 0.402802i
$246$	0	0
$247$	−19.9441	−0.0807452
$248$	9.73654i	0.0392602i
$249$	0	0
$250$	−34.5613	−0.138245
$251$	− 108.551i	− 0.432476i	−0.976341	−	0.216238i	$-0.930621\pi$
$251$	− 108.551i	− 0.432476i	0.976341	−	0.216238i	$-0.0693787\pi$
$252$	0	0
$253$	379.781	1.50111
$254$	− 132.656i	− 0.522266i
$255$	0	0
$256$	−38.4165	−0.150065
$257$	− 100.326i	− 0.390373i	−0.980766	−	0.195186i	$-0.937469\pi$
$257$	− 100.326i	− 0.390373i	0.980766	−	0.195186i	$-0.0625312\pi$
$258$	0	0
$259$	−111.803	−0.431671
$260$	31.2290i	0.120112i
$261$	0	0
$262$	188.088	0.717895
$263$	− 431.459i	− 1.64053i	−0.571985	−	0.820264i	$-0.693826\pi$
$263$	− 431.459i	− 1.64053i	0.571985	−	0.820264i	$-0.306174\pi$
$264$	0	0
$265$	138.702	0.523403
$266$	54.1021i	0.203391i
$267$	0	0
$268$	82.0864	0.306293
$269$	233.075i	0.866448i	0.901286	+	0.433224i	$0.142624\pi$
$269$	233.075i	0.866448i	−0.901286	+	0.433224i	$0.857376\pi$
$270$	0	0
$271$	388.155	1.43230	0.716152	−	0.697944i	$-0.245902\pi$
$271$	388.155	1.43230	0.716152	+	0.697944i	$0.245902\pi$
$272$	− 239.822i	− 0.881700i
$273$	0	0
$274$	644.291	2.35143
$275$	− 89.1740i	− 0.324269i
$276$	0	0
$277$	35.9093	0.129636	0.0648182	−	0.997897i	$-0.479353\pi$
$277$	35.9093	0.129636	0.0648182	+	0.997897i	$0.479353\pi$
$278$	− 415.739i	− 1.49547i
$279$	0	0
$280$	23.7233	0.0847261
$281$	301.394i	1.07258i	0.844035	+	0.536289i	$0.180174\pi$
$281$	301.394i	1.07258i	−0.844035	+	0.536289i	$0.819826\pi$
$282$	0	0
$283$	425.469	1.50342	0.751712	−	0.659492i	$-0.229229\pi$
$283$	425.469	1.50342	0.751712	+	0.659492i	$0.229229\pi$
$284$	− 583.751i	− 2.05546i
$285$	0	0
$286$	−138.588	−0.484573
$287$	− 11.8458i	− 0.0412744i
$288$	0	0
$289$	−773.928	−2.67795
$290$	245.617i	0.846955i
$291$	0	0
$292$	371.928	1.27373
$293$	− 384.419i	− 1.31201i	−0.754756	−	0.656006i	$-0.772245\pi$
$293$	− 384.419i	− 1.31201i	0.754756	−	0.656006i	$-0.227755\pi$
$294$	0	0
$295$	−80.8683	−0.274130
$296$	243.763i	0.823523i
$297$	0	0
$298$	−375.669	−1.26064
$299$	53.5288i	0.179026i
$300$	0	0
$301$	−34.2709	−0.113857
$302$	129.985i	0.430413i
$303$	0	0
$304$	−58.3618	−0.191980
$305$	− 94.4705i	− 0.309739i
$306$	0	0
$307$	−130.907	−0.426407	−0.213204	−	0.977008i	$-0.568390\pi$
$307$	−130.907	−0.426407	−0.213204	+	0.977008i	$0.568390\pi$
$308$	218.578i	0.709669i
$309$	0	0
$310$	13.9934	0.0451399
$311$	561.707i	1.80613i	0.429502	+	0.903066i	$0.358689\pi$
$311$	561.707i	1.80613i	−0.429502	+	0.903066i	$0.641311\pi$
$312$	0	0
$313$	40.4794	0.129327	0.0646636	−	0.997907i	$-0.479403\pi$
$313$	40.4794	0.129327	0.0646636	+	0.997907i	$0.479403\pi$
$314$	− 289.305i	− 0.921352i
$315$	0	0
$316$	384.190	1.21579
$317$	130.605i	0.412003i	0.978552	+	0.206001i	$0.0660452\pi$
$317$	130.605i	0.412003i	−0.978552	+	0.206001i	$0.933955\pi$
$318$	0	0
$319$	−633.734	−1.98663
$320$	224.363i	0.701135i
$321$	0	0
$322$	145.207	0.450953
$323$	258.668i	0.800831i
$324$	0	0
$325$	12.5688	0.0386731
$326$	357.941i	1.09798i
$327$	0	0
$328$	−25.8272	−0.0787415
$329$	− 4.16308i	− 0.0126537i
$330$	0	0
$331$	490.683	1.48243	0.741213	−	0.671270i	$-0.234251\pi$
$331$	490.683	1.48243	0.741213	+	0.671270i	$0.234251\pi$
$332$	− 117.943i	− 0.355249i
$333$	0	0
$334$	−5.02887	−0.0150565
$335$	− 33.0374i	− 0.0986191i
$336$	0	0
$337$	−111.987	−0.332305	−0.166152	−	0.986100i	$-0.553134\pi$
$337$	−111.987	−0.332305	−0.166152	+	0.986100i	$0.553134\pi$
$338$	502.888i	1.48784i
$339$	0	0
$340$	405.030	1.19127
$341$	36.1053i	0.105881i
$342$	0	0
$343$	205.445	0.598966
$344$	74.7206i	0.217211i
$345$	0	0
$346$	371.576	1.07392
$347$	196.518i	0.566336i	0.959070	+	0.283168i	$0.0913853\pi$
$347$	196.518i	0.566336i	−0.959070	+	0.283168i	$0.908615\pi$
$348$	0	0
$349$	−219.159	−0.627963	−0.313981	−	0.949429i	$-0.601663\pi$
$349$	−219.159	−0.627963	−0.313981	+	0.949429i	$0.601663\pi$
$350$	− 34.0951i	− 0.0974147i
$351$	0	0
$352$	−748.654	−2.12686
$353$	− 253.433i	− 0.717941i	−0.933349	−	0.358970i	$-0.883128\pi$
$353$	− 253.433i	− 0.717941i	0.933349	−	0.358970i	$-0.116872\pi$
$354$	0	0
$355$	−234.943	−0.661811
$356$	− 39.7968i	− 0.111789i
$357$	0	0
$358$	−487.619	−1.36206
$359$	152.604i	0.425081i	0.977152	+	0.212540i	$0.0681737\pi$
$359$	152.604i	0.425081i	−0.977152	+	0.212540i	$0.931826\pi$
$360$	0	0
$361$	−298.052	−0.825629
$362$	677.199i	1.87071i
$363$	0	0
$364$	−30.8078	−0.0846369
$365$	− 149.690i	− 0.410110i
$366$	0	0
$367$	349.994	0.953663	0.476832	−	0.878995i	$-0.341785\pi$
$367$	349.994	0.953663	0.476832	+	0.878995i	$0.341785\pi$
$368$	156.640i	0.425652i
$369$	0	0
$370$	350.336	0.946853
$371$	136.831i	0.368817i
$372$	0	0
$373$	−405.997	−1.08846	−0.544231	−	0.838935i	$-0.683179\pi$
$373$	−405.997	−1.08846	−0.544231	+	0.838935i	$0.683179\pi$
$374$	1797.44i	4.80599i
$375$	0	0
$376$	−9.07673	−0.0241402
$377$	− 89.3225i	− 0.236930i
$378$	0	0
$379$	410.698	1.08364	0.541818	−	0.840496i	$-0.317736\pi$
$379$	410.698	1.08364	0.541818	+	0.840496i	$0.317736\pi$
$380$	− 98.5659i	− 0.259384i
$381$	0	0
$382$	−1148.56	−3.00670
$383$	− 486.656i	− 1.27064i	−0.772248	−	0.635322i	$-0.780868\pi$
$383$	− 486.656i	− 1.27064i	0.772248	−	0.635322i	$-0.219132\pi$
$384$	0	0
$385$	87.9713	0.228497
$386$	122.288i	0.316807i
$387$	0	0
$388$	−618.592	−1.59431
$389$	− 639.526i	− 1.64403i	−0.569468	−	0.822013i	$-0.692851\pi$
$389$	− 639.526i	− 1.64403i	0.569468	−	0.822013i	$-0.307149\pi$
$390$	0	0
$391$	694.251	1.77558
$392$	− 212.263i	− 0.541489i
$393$	0	0
$394$	−322.894	−0.819528
$395$	− 154.625i	− 0.391457i
$396$	0	0
$397$	295.328	0.743899	0.371950	−	0.928253i	$-0.378689\pi$
$397$	295.328	0.743899	0.371950	+	0.928253i	$0.378689\pi$
$398$	− 364.880i	− 0.916784i
$399$	0	0
$400$	36.7796	0.0919491
$401$	41.4767i	0.103433i	0.998662	+	0.0517166i	$0.0164693\pi$
$401$	41.4767i	0.103433i	−0.998662	+	0.0517166i	$0.983531\pi$
$402$	0	0
$403$	−5.08891	−0.0126276
$404$	345.544i	0.855306i
$405$	0	0
$406$	−242.304	−0.596809
$407$	903.926i	2.22095i
$408$	0	0
$409$	20.9197	0.0511485	0.0255742	−	0.999673i	$-0.491859\pi$
$409$	20.9197	0.0511485	0.0255742	+	0.999673i	$0.491859\pi$
$410$	37.1189i	0.0905338i
$411$	0	0
$412$	−306.097	−0.742954
$413$	− 79.7777i	− 0.193166i
$414$	0	0
$415$	−47.4684	−0.114382
$416$	− 105.520i	− 0.253654i
$417$	0	0
$418$	437.415	1.04645
$419$	− 416.639i	− 0.994364i	−0.867646	−	0.497182i	$-0.834368\pi$
$419$	− 416.639i	− 0.994364i	0.867646	−	0.497182i	$-0.165632\pi$
$420$	0	0
$421$	334.193	0.793807	0.396904	−	0.917860i	$-0.370085\pi$
$421$	334.193	0.793807	0.396904	+	0.917860i	$0.370085\pi$
$422$	− 421.644i	− 0.999156i
$423$	0	0
$424$	298.332	0.703612
$425$	− 163.013i	− 0.383560i
$426$	0	0
$427$	93.1964	0.218259
$428$	− 473.683i	− 1.10674i
$429$	0	0
$430$	107.388	0.249741
$431$	389.685i	0.904142i	0.891982	+	0.452071i	$0.149315\pi$
$431$	389.685i	0.904142i	−0.891982	+	0.452071i	$0.850685\pi$
$432$	0	0
$433$	742.678	1.71519	0.857596	−	0.514324i	$-0.171957\pi$
$433$	742.678	1.71519	0.857596	+	0.514324i	$0.171957\pi$
$434$	13.8046i	0.0318079i
$435$	0	0
$436$	355.729	0.815893
$437$	− 168.949i	− 0.386611i
$438$	0	0
$439$	−221.662	−0.504924	−0.252462	−	0.967607i	$-0.581240\pi$
$439$	−221.662	−0.504924	−0.252462	+	0.967607i	$0.581240\pi$
$440$	− 191.803i	− 0.435916i
$441$	0	0
$442$	−253.343	−0.573174
$443$	174.434i	0.393757i	0.980428	+	0.196878i	$0.0630804\pi$
$443$	174.434i	0.393757i	−0.980428	+	0.196878i	$0.936920\pi$
$444$	0	0
$445$	−16.0171	−0.0359934
$446$	− 1045.86i	− 2.34497i
$447$	0	0
$448$	−221.337	−0.494057
$449$	− 317.066i	− 0.706161i	−0.935593	−	0.353080i	$-0.885134\pi$
$449$	− 317.066i	− 0.706161i	0.935593	−	0.353080i	$-0.114866\pi$
$450$	0	0
$451$	−95.7730	−0.212357
$452$	− 841.667i	− 1.86209i
$453$	0	0
$454$	−474.209	−1.04451
$455$	12.3992i	0.0272511i
$456$	0	0
$457$	−495.243	−1.08368	−0.541842	−	0.840481i	$-0.682273\pi$
$457$	−495.243	−1.08368	−0.541842	+	0.840481i	$0.682273\pi$
$458$	783.250i	1.71015i
$459$	0	0
$460$	−264.545	−0.575099
$461$	214.609i	0.465529i	0.972533	+	0.232764i	$0.0747771\pi$
$461$	214.609i	0.465529i	−0.972533	+	0.232764i	$0.925223\pi$
$462$	0	0
$463$	421.785	0.910983	0.455492	−	0.890240i	$-0.349464\pi$
$463$	421.785	0.910983	0.455492	+	0.890240i	$0.349464\pi$
$464$	− 261.382i	− 0.563324i
$465$	0	0
$466$	−956.025	−2.05156
$467$	− 123.800i	− 0.265097i	−0.991177	−	0.132549i	$-0.957684\pi$
$467$	− 123.800i	− 0.265097i	0.991177	−	0.132549i	$-0.0423160\pi$
$468$	0	0
$469$	32.5918	0.0694922
$470$	13.0451i	0.0277555i
$471$	0	0
$472$	−173.939	−0.368514
$473$	277.081i	0.585794i
$474$	0	0
$475$	−39.6699	−0.0835156
$476$	399.568i	0.839428i
$477$	0	0
$478$	−222.033	−0.464505
$479$	671.584i	1.40205i	0.713135	+	0.701027i	$0.247275\pi$
$479$	671.584i	1.40205i	−0.713135	+	0.701027i	$0.752725\pi$
$480$	0	0
$481$	−127.405	−0.264876
$482$	− 936.524i	− 1.94300i
$483$	0	0
$484$	1094.95	2.26229
$485$	248.965i	0.513330i
$486$	0	0
$487$	134.773	0.276742	0.138371	−	0.990380i	$-0.455813\pi$
$487$	134.773	0.276742	0.138371	+	0.990380i	$0.455813\pi$
$488$	− 203.195i	− 0.416384i
$489$	0	0
$490$	−305.065	−0.622582
$491$	474.707i	0.966817i	0.875395	+	0.483408i	$0.160601\pi$
$491$	474.707i	0.966817i	−0.875395	+	0.483408i	$0.839399\pi$
$492$	0	0
$493$	−1158.49	−2.34987
$494$	61.6522i	0.124802i
$495$	0	0
$496$	−14.8915	−0.0300233
$497$	− 231.774i	− 0.466346i
$498$	0	0
$499$	558.392	1.11902	0.559511	−	0.828823i	$-0.310989\pi$
$499$	558.392	1.11902	0.559511	+	0.828823i	$0.310989\pi$
$500$	62.1163i	0.124233i
$501$	0	0
$502$	−335.560	−0.668446
$503$	681.680i	1.35523i	0.735417	+	0.677614i	$0.236986\pi$
$503$	681.680i	1.35523i	−0.735417	+	0.677614i	$0.763014\pi$
$504$	0	0
$505$	139.071	0.275389
$506$	− 1174.00i	− 2.32015i
$507$	0	0
$508$	−238.419	−0.469329
$509$	534.500i	1.05010i	0.851072	+	0.525049i	$0.175953\pi$
$509$	534.500i	1.05010i	−0.851072	+	0.525049i	$0.824047\pi$
$510$	0	0
$511$	147.671	0.288985
$512$	− 450.295i	− 0.879483i
$513$	0	0
$514$	−310.133	−0.603371
$515$	123.195i	0.239214i
$516$	0	0
$517$	−33.6585	−0.0651036
$518$	345.611i	0.667202i
$519$	0	0
$520$	27.0340	0.0519884
$521$	− 352.483i	− 0.676552i	−0.941047	−	0.338276i	$-0.890156\pi$
$521$	− 352.483i	− 0.676552i	0.941047	−	0.338276i	$-0.109844\pi$
$522$	0	0
$523$	−260.994	−0.499033	−0.249516	−	0.968371i	$-0.580272\pi$
$523$	−260.994	−0.499033	−0.249516	+	0.968371i	$0.580272\pi$
$524$	− 338.048i	− 0.645129i
$525$	0	0
$526$	−1333.75	−2.53564
$527$	66.0016i	0.125240i
$528$	0	0
$529$	75.5501	0.142817
$530$	− 428.762i	− 0.808985i
$531$	0	0
$532$	97.2366	0.182776
$533$	− 13.4989i	− 0.0253262i
$534$	0	0
$535$	−190.643	−0.356343
$536$	− 71.0597i	− 0.132574i
$537$	0	0
$538$	720.493	1.33921
$539$	− 787.120i	− 1.46033i
$540$	0	0
$541$	492.273	0.909932	0.454966	−	0.890509i	$-0.349651\pi$
$541$	492.273	0.909932	0.454966	+	0.890509i	$0.349651\pi$
$542$	− 1199.88i	− 2.21381i
$543$	0	0
$544$	−1368.56	−2.51574
$545$	− 143.171i	− 0.262699i
$546$	0	0
$547$	858.132	1.56880	0.784399	−	0.620257i	$-0.212972\pi$
$547$	858.132	1.56880	0.784399	+	0.620257i	$0.212972\pi$
$548$	− 1157.97i	− 2.11309i
$549$	0	0
$550$	−275.659	−0.501199
$551$	281.923i	0.511656i
$552$	0	0
$553$	152.540	0.275841
$554$	− 111.005i	− 0.200369i
$555$	0	0
$556$	−747.200	−1.34389
$557$	474.887i	0.852581i	0.904586	+	0.426290i	$0.140180\pi$
$557$	474.887i	0.852581i	−0.904586	+	0.426290i	$0.859820\pi$
$558$	0	0
$559$	−39.0535	−0.0698632
$560$	36.2836i	0.0647922i
$561$	0	0
$562$	931.686	1.65780
$563$	− 749.574i	− 1.33139i	−0.746223	−	0.665696i	$-0.768135\pi$
$563$	− 749.574i	− 1.33139i	0.746223	−	0.665696i	$-0.231865\pi$
$564$	0	0
$565$	−338.746	−0.599551
$566$	− 1315.23i	− 2.32373i
$567$	0	0
$568$	−505.335	−0.889675
$569$	498.498i	0.876094i	0.898952	+	0.438047i	$0.144330\pi$
$569$	498.498i	0.876094i	−0.898952	+	0.438047i	$0.855670\pi$
$570$	0	0
$571$	−605.485	−1.06039	−0.530197	−	0.847874i	$-0.677882\pi$
$571$	−605.485	−1.06039	−0.530197	+	0.847874i	$0.677882\pi$
$572$	249.081i	0.435457i
$573$	0	0
$574$	−36.6182	−0.0637948
$575$	106.472i	0.185168i
$576$	0	0
$577$	−898.993	−1.55805	−0.779023	−	0.626995i	$-0.784285\pi$
$577$	−898.993	−1.55805	−0.779023	+	0.626995i	$0.784285\pi$
$578$	2392.41i	4.13912i
$579$	0	0
$580$	441.442	0.761108
$581$	− 46.8282i	− 0.0805994i
$582$	0	0
$583$	1106.28	1.89756
$584$	− 321.966i	− 0.551312i
$585$	0	0
$586$	−1188.34	−2.02788
$587$	− 842.960i	− 1.43605i	−0.696019	−	0.718024i	$-0.745047\pi$
$587$	− 842.960i	− 1.43605i	0.696019	−	0.718024i	$-0.254953\pi$
$588$	0	0
$589$	16.0618	0.0272696
$590$	249.985i	0.423703i
$591$	0	0
$592$	−372.823	−0.629768
$593$	− 238.158i	− 0.401615i	−0.979631	−	0.200808i	$-0.935643\pi$
$593$	− 238.158i	− 0.401615i	0.979631	−	0.200808i	$-0.0643566\pi$
$594$	0	0
$595$	160.814	0.270276
$596$	675.183i	1.13286i
$597$	0	0
$598$	165.471	0.276707
$599$	898.187i	1.49948i	0.661734	+	0.749739i	$0.269821\pi$
$599$	898.187i	1.49948i	−0.661734	+	0.749739i	$0.730179\pi$
$600$	0	0
$601$	−103.884	−0.172852	−0.0864260	−	0.996258i	$-0.527545\pi$
$601$	−103.884	−0.172852	−0.0864260	+	0.996258i	$0.527545\pi$
$602$	105.940i	0.175980i
$603$	0	0
$604$	233.619	0.386786
$605$	− 440.684i	− 0.728404i
$606$	0	0
$607$	1078.63	1.77698	0.888490	−	0.458896i	$-0.151755\pi$
$607$	1078.63	1.77698	0.888490	+	0.458896i	$0.151755\pi$
$608$	333.046i	0.547773i
$609$	0	0
$610$	−292.032	−0.478741
$611$	− 4.74405i	− 0.00776441i
$612$	0	0
$613$	−387.302	−0.631814	−0.315907	−	0.948790i	$-0.602309\pi$
$613$	−387.302	−0.631814	−0.315907	+	0.948790i	$0.602309\pi$
$614$	404.667i	0.659066i
$615$	0	0
$616$	189.216	0.307169
$617$	530.487i	0.859785i	0.902880	+	0.429892i	$0.141449\pi$
$617$	530.487i	0.859785i	−0.902880	+	0.429892i	$0.858551\pi$
$618$	0	0
$619$	642.292	1.03763	0.518814	−	0.854887i	$-0.326374\pi$
$619$	642.292	1.03763	0.518814	+	0.854887i	$0.326374\pi$
$620$	− 25.1500i	− 0.0405645i
$621$	0	0
$622$	1736.38	2.79161
$623$	− 15.8010i	− 0.0253628i
$624$	0	0
$625$	25.0000	0.0400000
$626$	− 125.132i	− 0.199892i
$627$	0	0
$628$	−519.961	−0.827964
$629$	1652.41i	2.62704i
$630$	0	0
$631$	−88.5117	−0.140272	−0.0701361	−	0.997537i	$-0.522343\pi$
$631$	−88.5117	−0.140272	−0.0701361	+	0.997537i	$0.522343\pi$
$632$	− 332.582i	− 0.526237i
$633$	0	0
$634$	403.733	0.636803
$635$	95.9568i	0.151113i
$636$	0	0
$637$	110.942	0.174163
$638$	1959.03i	3.07058i
$639$	0	0
$640$	318.109	0.497045
$641$	− 329.830i	− 0.514555i	−0.966338	−	0.257278i	$-0.917175\pi$
$641$	− 329.830i	− 0.514555i	0.966338	−	0.257278i	$-0.0828255\pi$
$642$	0	0
$643$	−78.4678	−0.122034	−0.0610170	−	0.998137i	$-0.519434\pi$
$643$	−78.4678	−0.122034	−0.0610170	+	0.998137i	$0.519434\pi$
$644$	− 260.978i	− 0.405245i
$645$	0	0
$646$	799.609	1.23779
$647$	68.7558i	0.106269i	0.998587	+	0.0531343i	$0.0169212\pi$
$647$	68.7558i	0.106269i	−0.998587	+	0.0531343i	$0.983079\pi$
$648$	0	0
$649$	−645.003	−0.993841
$650$	− 38.8532i	− 0.0597742i
$651$	0	0
$652$	643.320	0.986688
$653$	960.937i	1.47157i	0.677214	+	0.735786i	$0.263187\pi$
$653$	960.937i	1.47157i	−0.677214	+	0.735786i	$0.736813\pi$
$654$	0	0
$655$	−136.054	−0.207717
$656$	− 39.5014i	− 0.0602155i
$657$	0	0
$658$	−12.8691	−0.0195580
$659$	121.834i	0.184878i	0.995718	+	0.0924389i	$0.0294663\pi$
$659$	121.834i	0.184878i	−0.995718	+	0.0924389i	$0.970534\pi$
$660$	0	0
$661$	836.002	1.26475	0.632377	−	0.774661i	$-0.282080\pi$
$661$	836.002	1.26475	0.632377	+	0.774661i	$0.282080\pi$
$662$	− 1516.83i	− 2.29128i
$663$	0	0
$664$	−102.099	−0.153764
$665$	− 39.1349i	− 0.0588495i
$666$	0	0
$667$	756.664	1.13443
$668$	9.03829i	0.0135304i
$669$	0	0
$670$	−102.127	−0.152428
$671$	− 753.493i	− 1.12294i
$672$	0	0
$673$	343.616	0.510573	0.255287	−	0.966865i	$-0.417830\pi$
$673$	343.616	0.510573	0.255287	+	0.966865i	$0.417830\pi$
$674$	346.179i	0.513619i
$675$	0	0
$676$	903.832	1.33703
$677$	− 385.921i	− 0.570045i	−0.958521	−	0.285023i	$-0.907999\pi$
$677$	− 385.921i	− 0.570045i	0.958521	−	0.285023i	$-0.0920011\pi$
$678$	0	0
$679$	−245.607	−0.361719
$680$	− 350.622i	− 0.515621i
$681$	0	0
$682$	111.611	0.163652
$683$	− 699.925i	− 1.02478i	−0.858753	−	0.512390i	$-0.828760\pi$
$683$	− 699.925i	− 1.02478i	0.858753	−	0.512390i	$-0.171240\pi$
$684$	0	0
$685$	−466.050	−0.680364
$686$	− 635.083i	− 0.925777i
$687$	0	0
$688$	−114.281	−0.166107
$689$	155.926i	0.226308i
$690$	0	0
$691$	−514.130	−0.744037	−0.372019	−	0.928225i	$-0.621334\pi$
$691$	−514.130	−0.744037	−0.372019	+	0.928225i	$0.621334\pi$
$692$	− 667.827i	− 0.965068i
$693$	0	0
$694$	607.488	0.875344
$695$	300.726i	0.432700i
$696$	0	0
$697$	−175.076	−0.251185
$698$	677.476i	0.970596i
$699$	0	0
$700$	−61.2785	−0.0875408
$701$	842.009i	1.20115i	0.799567	+	0.600577i	$0.205062\pi$
$701$	842.009i	1.20115i	−0.799567	+	0.600577i	$0.794938\pi$
$702$	0	0
$703$	402.120	0.572006
$704$	1789.51i	2.54192i
$705$	0	0
$706$	−783.426	−1.10967
$707$	137.196i	0.194053i
$708$	0	0
$709$	−168.580	−0.237771	−0.118885	−	0.992908i	$-0.537932\pi$
$709$	−168.580	−0.237771	−0.118885	+	0.992908i	$0.537932\pi$
$710$	726.268i	1.02291i
$711$	0	0
$712$	−34.4509	−0.0483860
$713$	− 43.1089i	− 0.0604613i
$714$	0	0
$715$	100.248	0.140207
$716$	876.388i	1.22401i
$717$	0	0
$718$	471.738	0.657017
$719$	− 1132.23i	− 1.57472i	−0.616490	−	0.787362i	$-0.711446\pi$
$719$	− 1132.23i	− 1.57472i	0.616490	−	0.787362i	$-0.288554\pi$
$720$	0	0
$721$	−121.534	−0.168563
$722$	921.354i	1.27611i
$723$	0	0
$724$	1217.12	1.68110
$725$	− 177.668i	− 0.245059i
$726$	0	0
$727$	952.356	1.30998	0.654990	−	0.755637i	$-0.272673\pi$
$727$	952.356	1.30998	0.654990	+	0.755637i	$0.272673\pi$
$728$	26.6694i	0.0366338i
$729$	0	0
$730$	−462.730	−0.633877
$731$	506.512i	0.692903i
$732$	0	0
$733$	138.183	0.188518	0.0942588	−	0.995548i	$-0.469952\pi$
$733$	138.183	0.188518	0.0942588	+	0.995548i	$0.469952\pi$
$734$	− 1081.92i	− 1.47401i
$735$	0	0
$736$	893.877	1.21451
$737$	− 263.505i	− 0.357537i
$738$	0	0
$739$	529.373	0.716337	0.358168	−	0.933657i	$-0.383401\pi$
$739$	529.373	0.716337	0.358168	+	0.933657i	$0.383401\pi$
$740$	− 629.652i	− 0.850881i
$741$	0	0
$742$	422.980	0.570053
$743$	620.665i	0.835350i	0.908597	+	0.417675i	$0.137155\pi$
$743$	620.665i	0.835350i	−0.908597	+	0.417675i	$0.862845\pi$
$744$	0	0
$745$	271.742	0.364754
$746$	1255.04i	1.68236i
$747$	0	0
$748$	3230.51	4.31886
$749$	− 188.072i	− 0.251098i
$750$	0	0
$751$	−99.2928	−0.132214	−0.0661070	−	0.997813i	$-0.521058\pi$
$751$	−99.2928	−0.132214	−0.0661070	+	0.997813i	$0.521058\pi$
$752$	− 13.8824i	− 0.0184606i
$753$	0	0
$754$	−276.119	−0.366205
$755$	− 94.0248i	− 0.124536i
$756$	0	0
$757$	395.640	0.522642	0.261321	−	0.965252i	$-0.415842\pi$
$757$	395.640	0.522642	0.261321	+	0.965252i	$0.415842\pi$
$758$	− 1269.57i	− 1.67490i
$759$	0	0
$760$	−85.3255	−0.112270
$761$	1408.29i	1.85058i	0.379258	+	0.925291i	$0.376179\pi$
$761$	1408.29i	1.85058i	−0.379258	+	0.925291i	$0.623821\pi$
$762$	0	0
$763$	141.240	0.185111
$764$	2064.28i	2.70194i
$765$	0	0
$766$	−1504.38	−1.96394
$767$	− 90.9109i	− 0.118528i
$768$	0	0
$769$	764.118	0.993652	0.496826	−	0.867850i	$-0.334499\pi$
$769$	764.118	0.993652	0.496826	+	0.867850i	$0.334499\pi$
$770$	− 271.942i	− 0.353171i
$771$	0	0
$772$	219.785	0.284696
$773$	− 477.095i	− 0.617199i	−0.951192	−	0.308600i	$-0.900140\pi$
$773$	− 477.095i	− 0.617199i	0.951192	−	0.308600i	$-0.0998603\pi$
$774$	0	0
$775$	−10.1221	−0.0130608
$776$	535.496i	0.690072i
$777$	0	0
$778$	−1976.94	−2.54105
$779$	42.6055i	0.0546926i
$780$	0	0
$781$	−1873.90	−2.39935
$782$	− 2146.11i	− 2.74438i
$783$	0	0
$784$	324.646	0.414090
$785$	209.269i	0.266585i
$786$	0	0
$787$	−1230.24	−1.56320	−0.781600	−	0.623779i	$-0.785596\pi$
$787$	−1230.24	−1.56320	−0.781600	+	0.623779i	$0.785596\pi$
$788$	580.331i	0.736461i
$789$	0	0
$790$	−477.987	−0.605046
$791$	− 334.178i	− 0.422475i
$792$	0	0
$793$	106.202	0.133925
$794$	− 912.934i	− 1.14979i
$795$	0	0
$796$	−655.792	−0.823859
$797$	95.3115i	0.119588i	0.998211	+	0.0597939i	$0.0190443\pi$
$797$	95.3115i	0.119588i	−0.998211	+	0.0597939i	$0.980956\pi$
$798$	0	0
$799$	−61.5289	−0.0770074
$800$	− 209.886i	− 0.262357i
$801$	0	0
$802$	128.215	0.159869
$803$	− 1193.92i	− 1.48683i
$804$	0	0
$805$	−105.036	−0.130479
$806$	15.7311i	0.0195175i
$807$	0	0
$808$	299.127	0.370206
$809$	1184.24i	1.46383i	0.681395	+	0.731916i	$0.261374\pi$
$809$	1184.24i	1.46383i	−0.681395	+	0.731916i	$0.738626\pi$
$810$	0	0
$811$	−991.880	−1.22303	−0.611517	−	0.791232i	$-0.709440\pi$
$811$	−991.880	−1.22303	−0.611517	+	0.791232i	$0.709440\pi$
$812$	435.489i	0.536316i
$813$	0	0
$814$	2794.27	3.43276
$815$	− 258.918i	− 0.317690i
$816$	0	0
$817$	123.262	0.150871
$818$	− 64.6682i	− 0.0790565i
$819$	0	0
$820$	66.7130	0.0813573
$821$	1138.81i	1.38711i	0.720406	+	0.693553i	$0.243956\pi$
$821$	1138.81i	1.38711i	−0.720406	+	0.693553i	$0.756044\pi$
$822$	0	0
$823$	762.472	0.926454	0.463227	−	0.886240i	$-0.346691\pi$
$823$	762.472	0.926454	0.463227	+	0.886240i	$0.346691\pi$
$824$	264.979i	0.321576i
$825$	0	0
$826$	−246.613	−0.298563
$827$	1111.94i	1.34455i	0.740301	+	0.672275i	$0.234683\pi$
$827$	1111.94i	1.34455i	−0.740301	+	0.672275i	$0.765317\pi$
$828$	0	0
$829$	−128.075	−0.154493	−0.0772465	−	0.997012i	$-0.524613\pi$
$829$	−128.075	−0.154493	−0.0772465	+	0.997012i	$0.524613\pi$
$830$	146.737i	0.176792i
$831$	0	0
$832$	−252.226	−0.303156
$833$	− 1438.88i	− 1.72735i
$834$	0	0
$835$	3.63765	0.00435647
$836$	− 786.158i	− 0.940381i
$837$	0	0
$838$	−1287.94	−1.53692
$839$	− 690.172i	− 0.822613i	−0.911497	−	0.411307i	$-0.865073\pi$
$839$	− 690.172i	− 0.822613i	0.911497	−	0.411307i	$-0.134927\pi$
$840$	0	0
$841$	−421.633	−0.501347
$842$	− 1033.08i	− 1.22693i
$843$	0	0
$844$	−757.813	−0.897882
$845$	− 363.766i	− 0.430492i
$846$	0	0
$847$	434.741	0.513272
$848$	456.283i	0.538070i
$849$	0	0
$850$	−503.914	−0.592840
$851$	− 1079.27i	− 1.26824i
$852$	0	0
$853$	−42.4195	−0.0497298	−0.0248649	−	0.999691i	$-0.507916\pi$
$853$	−42.4195	−0.0497298	−0.0248649	+	0.999691i	$0.507916\pi$
$854$	− 288.094i	− 0.337346i
$855$	0	0
$856$	−410.052	−0.479033
$857$	149.020i	0.173885i	0.996213	+	0.0869427i	$0.0277097\pi$
$857$	149.020i	0.173885i	−0.996213	+	0.0869427i	$0.972290\pi$
$858$	0	0
$859$	−203.471	−0.236869	−0.118435	−	0.992962i	$-0.537788\pi$
$859$	−203.471	−0.236869	−0.118435	+	0.992962i	$0.537788\pi$
$860$	− 193.007i	− 0.224427i
$861$	0	0
$862$	1204.62	1.39747
$863$	− 923.552i	− 1.07016i	−0.844800	−	0.535082i	$-0.820281\pi$
$863$	− 923.552i	− 1.07016i	0.844800	−	0.535082i	$-0.179719\pi$
$864$	0	0
$865$	−268.781	−0.310729
$866$	− 2295.81i	− 2.65105i
$867$	0	0
$868$	24.8108	0.0285839
$869$	− 1233.29i	− 1.41920i
$870$	0	0
$871$	37.1401	0.0426408
$872$	− 307.944i	− 0.353147i
$873$	0	0
$874$	−522.265	−0.597557
$875$	24.6628i	0.0281861i
$876$	0	0
$877$	−667.142	−0.760710	−0.380355	−	0.924841i	$-0.624198\pi$
$877$	−667.142	−0.760710	−0.380355	+	0.924841i	$0.624198\pi$
$878$	685.213i	0.780425i
$879$	0	0
$880$	293.353	0.333356
$881$	− 837.220i	− 0.950307i	−0.879903	−	0.475153i	$-0.842393\pi$
$881$	− 837.220i	− 0.950307i	0.879903	−	0.475153i	$-0.157607\pi$
$882$	0	0
$883$	−790.539	−0.895287	−0.447644	−	0.894212i	$-0.647737\pi$
$883$	−790.539	−0.895287	−0.447644	+	0.894212i	$0.647737\pi$
$884$	455.329i	0.515078i
$885$	0	0
$886$	539.221	0.608601
$887$	− 388.932i	− 0.438481i	−0.975671	−	0.219240i	$-0.929642\pi$
$887$	− 388.932i	− 0.438481i	0.975671	−	0.219240i	$-0.0703579\pi$
$888$	0	0
$889$	−94.6626	−0.106482
$890$	49.5128i	0.0556323i
$891$	0	0
$892$	−1879.70	−2.10728
$893$	14.9733i	0.0167674i
$894$	0	0
$895$	352.721	0.394101
$896$	313.819i	0.350244i
$897$	0	0
$898$	−980.132	−1.09146
$899$	71.9351i	0.0800168i
$900$	0	0
$901$	2022.31	2.24452
$902$	296.059i	0.328225i
$903$	0	0
$904$	−728.605	−0.805979
$905$	− 489.854i	− 0.541275i
$906$	0	0
$907$	73.2467	0.0807571	0.0403785	−	0.999184i	$-0.487144\pi$
$907$	73.2467	0.0807571	0.0403785	+	0.999184i	$0.487144\pi$
$908$	852.287i	0.938642i
$909$	0	0
$910$	38.3292	0.0421200
$911$	959.012i	1.05270i	0.850267	+	0.526351i	$0.176440\pi$
$911$	959.012i	1.05270i	−0.850267	+	0.526351i	$0.823560\pi$
$912$	0	0
$913$	−378.607	−0.414684
$914$	1530.92i	1.67497i
$915$	0	0
$916$	1407.72	1.53681
$917$	− 134.219i	− 0.146368i
$918$	0	0
$919$	1280.94	1.39384	0.696919	−	0.717149i	$-0.254554\pi$
$919$	1280.94	1.39384	0.696919	+	0.717149i	$0.254554\pi$
$920$	229.009i	0.248923i
$921$	0	0
$922$	663.410	0.719534
$923$	− 264.119i	− 0.286153i
$924$	0	0
$925$	−253.416	−0.273964
$926$	− 1303.85i	− 1.40804i
$927$	0	0
$928$	−1491.60	−1.60732
$929$	− 1695.39i	− 1.82497i	−0.409115	−	0.912483i	$-0.634162\pi$
$929$	− 1695.39i	− 1.82497i	0.409115	−	0.912483i	$-0.365838\pi$
$930$	0	0
$931$	−350.158	−0.376110
$932$	1718.24i	1.84361i
$933$	0	0
$934$	−382.698	−0.409741
$935$	− 1300.19i	− 1.39057i
$936$	0	0
$937$	−1690.96	−1.80465	−0.902326	−	0.431054i	$-0.858142\pi$
$937$	−1690.96	−1.80465	−0.902326	+	0.431054i	$0.858142\pi$
$938$	− 100.750i	− 0.107409i
$939$	0	0
$940$	23.4457	0.0249422
$941$	− 1211.85i	− 1.28783i	−0.765096	−	0.643917i	$-0.777308\pi$
$941$	− 1211.85i	− 1.28783i	0.765096	−	0.643917i	$-0.222692\pi$
$942$	0	0
$943$	114.351	0.121263
$944$	− 266.030i	− 0.281812i
$945$	0	0
$946$	856.527	0.905419
$947$	589.846i	0.622858i	0.950269	+	0.311429i	$0.100808\pi$
$947$	589.846i	0.622858i	−0.950269	+	0.311429i	$0.899192\pi$
$948$	0	0
$949$	168.279	0.177323
$950$	122.630i	0.129084i
$951$	0	0
$952$	345.893	0.363333
$953$	− 1597.56i	− 1.67635i	−0.545402	−	0.838174i	$-0.683623\pi$
$953$	− 1597.56i	− 1.67635i	0.545402	−	0.838174i	$-0.316377\pi$
$954$	0	0
$955$	830.815	0.869963
$956$	399.056i	0.417423i
$957$	0	0
$958$	2076.04	2.16705
$959$	− 459.764i	− 0.479420i
$960$	0	0
$961$	−956.902	−0.995735
$962$	393.842i	0.409399i
$963$	0	0
$964$	−1683.20	−1.74605
$965$	− 88.4571i	− 0.0916654i
$966$	0	0
$967$	626.952	0.648348	0.324174	−	0.945998i	$-0.394914\pi$
$967$	626.952	0.648348	0.324174	+	0.945998i	$0.394914\pi$
$968$	− 947.862i	− 0.979196i
$969$	0	0
$970$	769.614	0.793417
$971$	1907.25i	1.96421i	0.188328	+	0.982106i	$0.439693\pi$
$971$	1907.25i	1.96421i	−0.188328	+	0.982106i	$0.560307\pi$
$972$	0	0
$973$	−296.671	−0.304903
$974$	− 416.619i	− 0.427740i
$975$	0	0
$976$	310.777	0.318419
$977$	− 882.363i	− 0.903135i	−0.892237	−	0.451568i	$-0.850865\pi$
$977$	− 882.363i	− 0.903135i	0.892237	−	0.451568i	$-0.149135\pi$
$978$	0	0
$979$	−127.751	−0.130492
$980$	548.288i	0.559477i
$981$	0	0
$982$	1467.44	1.49434
$983$	− 967.696i	− 0.984432i	−0.870473	−	0.492216i	$-0.836187\pi$
$983$	− 967.696i	− 0.984432i	0.870473	−	0.492216i	$-0.163813\pi$
$984$	0	0
$985$	233.566	0.237123
$986$	3581.17i	3.63202i
$987$	0	0
$988$	110.806	0.112152
$989$	− 330.828i	− 0.334508i
$990$	0	0
$991$	416.462	0.420245	0.210122	−	0.977675i	$-0.432614\pi$
$991$	416.462	0.420245	0.210122	+	0.977675i	$0.432614\pi$
$992$	84.9797i	0.0856650i
$993$	0	0
$994$	−716.473	−0.720798
$995$	263.937i	0.265263i
$996$	0	0
$997$	−877.183	−0.879822	−0.439911	−	0.898041i	$-0.644990\pi$
$997$	−877.183	−0.879822	−0.439911	+	0.898041i	$0.644990\pi$
$998$	− 1726.13i	− 1.72959i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.3.c.a.161.3		16
3.2	odd	2	inner	405.3.c.a.161.14		16
9.2	odd	6		45.3.i.a.41.8	yes	16
9.4	even	3		45.3.i.a.11.8	✓	16
9.5	odd	6		135.3.i.a.116.1		16
9.7	even	3		135.3.i.a.71.1		16
36.7	odd	6		2160.3.bs.c.881.3		16
36.11	even	6		720.3.bs.c.401.8		16
36.23	even	6		2160.3.bs.c.1601.3		16
36.31	odd	6		720.3.bs.c.641.8		16
45.2	even	12		225.3.i.b.149.14		32
45.4	even	6		225.3.j.b.101.1		16
45.7	odd	12		675.3.i.c.449.3		32
45.13	odd	12		225.3.i.b.74.14		32
45.14	odd	6		675.3.j.b.251.8		16
45.22	odd	12		225.3.i.b.74.3		32
45.23	even	12		675.3.i.c.224.3		32
45.29	odd	6		225.3.j.b.176.1		16
45.32	even	12		675.3.i.c.224.14		32
45.34	even	6		675.3.j.b.476.8		16
45.38	even	12		225.3.i.b.149.3		32
45.43	odd	12		675.3.i.c.449.14		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.i.a.11.8	✓	16	9.4	even	3
45.3.i.a.41.8	yes	16	9.2	odd	6
135.3.i.a.71.1		16	9.7	even	3
135.3.i.a.116.1		16	9.5	odd	6
225.3.i.b.74.3		32	45.22	odd	12
225.3.i.b.74.14		32	45.13	odd	12
225.3.i.b.149.3		32	45.38	even	12
225.3.i.b.149.14		32	45.2	even	12
225.3.j.b.101.1		16	45.4	even	6
225.3.j.b.176.1		16	45.29	odd	6
405.3.c.a.161.3		16	1.1	even	1	trivial
405.3.c.a.161.14		16	3.2	odd	2	inner
675.3.i.c.224.3		32	45.23	even	12
675.3.i.c.224.14		32	45.32	even	12
675.3.i.c.449.3		32	45.7	odd	12
675.3.i.c.449.14		32	45.43	odd	12
675.3.j.b.251.8		16	45.14	odd	6
675.3.j.b.476.8		16	45.34	even	6
720.3.bs.c.401.8		16	36.11	even	6
720.3.bs.c.641.8		16	36.31	odd	6
2160.3.bs.c.881.3		16	36.7	odd	6
2160.3.bs.c.1601.3		16	36.23	even	6

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:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	405.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.09125i q^{2} -5.55585 q^{4} +2.23607i q^{5} -2.20591 q^{7} +4.80953i q^{8} +O(q^{10})\)	\(q-3.09125i q^{2} -5.55585 q^{4} +2.23607i q^{5} -2.20591 q^{7} +4.80953i q^{8} +6.91225 q^{10} +17.8348i q^{11} -2.51375 q^{13} +6.81903i q^{14} -7.35593 q^{16} +32.6026i q^{17} +7.93398 q^{19} -12.4233i q^{20} +55.1319 q^{22} -21.2944i q^{23} -5.00000 q^{25} +7.77065i q^{26} +12.2557 q^{28} +35.5335i q^{29} +2.02443 q^{31} +41.9772i q^{32} +100.783 q^{34} -4.93257i q^{35} +50.6833 q^{37} -24.5260i q^{38} -10.7544 q^{40} +5.37001i q^{41} +15.5360 q^{43} -99.0875i q^{44} -65.8263 q^{46} +1.88724i q^{47} -44.1340 q^{49} +15.4563i q^{50} +13.9660 q^{52} -62.0293i q^{53} -39.8798 q^{55} -10.6094i q^{56} +109.843 q^{58} +36.1654i q^{59} -42.2485 q^{61} -6.25802i q^{62} +100.338 q^{64} -5.62092i q^{65} -14.7748 q^{67} -181.135i q^{68} -15.2478 q^{70} +105.070i q^{71} -66.9435 q^{73} -156.675i q^{74} -44.0800 q^{76} -39.3420i q^{77} -69.1506 q^{79} -16.4484i q^{80} +16.6001 q^{82} +21.2285i q^{83} -72.9016 q^{85} -48.0256i q^{86} -85.7770 q^{88} +7.16304i q^{89} +5.54511 q^{91} +118.308i q^{92} +5.83394 q^{94} +17.7409i q^{95} +111.341 q^{97} +136.429i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} - 4 q^{7}+O(q^{10}) \) 16 * q - 32 * q^4 - 4 * q^7	\( 16 q - 32 q^{4} - 4 q^{7} + 20 q^{13} + 64 q^{16} - 52 q^{19} + 48 q^{22} - 80 q^{25} + 32 q^{28} - 64 q^{31} - 108 q^{34} + 44 q^{37} + 60 q^{40} + 248 q^{43} - 108 q^{46} + 108 q^{49} - 124 q^{52} - 180 q^{58} - 124 q^{61} + 256 q^{64} - 28 q^{67} + 120 q^{70} - 268 q^{73} + 212 q^{76} + 80 q^{79} - 204 q^{82} - 60 q^{85} - 288 q^{88} + 136 q^{91} + 300 q^{94} + 284 q^{97}+O(q^{100}) \) 16 * q - 32 * q^4 - 4 * q^7 + 20 * q^13 + 64 * q^16 - 52 * q^19 + 48 * q^22 - 80 * q^25 + 32 * q^28 - 64 * q^31 - 108 * q^34 + 44 * q^37 + 60 * q^40 + 248 * q^43 - 108 * q^46 + 108 * q^49 - 124 * q^52 - 180 * q^58 - 124 * q^61 + 256 * q^64 - 28 * q^67 + 120 * q^70 - 268 * q^73 + 212 * q^76 + 80 * q^79 - 204 * q^82 - 60 * q^85 - 288 * q^88 + 136 * q^91 + 300 * q^94 + 284 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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