LMFDB - Embedded newform 1800.4.a.bw.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1800,4,Mod(1,1800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1800.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1800.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:	$106.203438010$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{6}, \sqrt{46})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 26x^{2} + 100 $ x^4 - 26*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 360)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.16642$ of defining polynomial
Character	$\chi$	$=$	1800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.13228 q^{7} +O(q^{10})$	$q+1.13228 q^{7} +20.7282 q^{11} +38.0596 q^{13} -4.61325 q^{17} -46.4530 q^{19} -17.2265 q^{23} +138.304 q^{29} +151.359 q^{31} -248.300 q^{37} -92.6654 q^{41} +443.127 q^{43} +262.132 q^{47} -341.718 q^{49} +278.425 q^{53} -662.079 q^{59} +365.624 q^{61} -786.101 q^{67} +401.671 q^{71} +290.889 q^{73} +23.4701 q^{77} -185.171 q^{79} -399.889 q^{83} +886.255 q^{89} +43.0940 q^{91} -48.9444 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q + 48 q^{17} + 80 q^{19} + 64 q^{23} - 192 q^{31} + 384 q^{47} + 228 q^{49} - 16 q^{53} - 664 q^{61} + 2752 q^{77} + 1120 q^{79} + 1856 q^{83} + 704 q^{91}+O(q^{100}) $ 4 * q + 48 * q^17 + 80 * q^19 + 64 * q^23 - 192 * q^31 + 384 * q^47 + 228 * q^49 - 16 * q^53 - 664 * q^61 + 2752 * q^77 + 1120 * q^79 + 1856 * q^83 + 704 * q^91

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.13228	0.0611373	0.0305686	−	0.999533i	$-0.490268\pi$
$7$	1.13228	0.0611373	0.0305686	+	0.999533i	$0.490268\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	20.7282	0.568162	0.284081	−	0.958800i	$-0.408311\pi$
$11$	20.7282	0.568162	0.284081	+	0.958800i	$0.408311\pi$
$12$	0	0
$13$	38.0596	0.811986	0.405993	−	0.913876i	$-0.366926\pi$
$13$	38.0596	0.811986	0.405993	+	0.913876i	$0.366926\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.61325	−0.0658163	−0.0329081	−	0.999458i	$-0.510477\pi$
$17$	−4.61325	−0.0658163	−0.0329081	+	0.999458i	$0.510477\pi$
$18$	0	0
$19$	−46.4530	−0.560897	−0.280449	−	0.959869i	$-0.590483\pi$
$19$	−46.4530	−0.560897	−0.280449	+	0.959869i	$0.590483\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−17.2265	−0.156173	−0.0780864	−	0.996947i	$-0.524881\pi$
$23$	−17.2265	−0.156173	−0.0780864	+	0.996947i	$0.524881\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	138.304	0.885598	0.442799	−	0.896621i	$-0.353985\pi$
$29$	138.304	0.885598	0.442799	+	0.896621i	$0.353985\pi$
$30$	0	0
$31$	151.359	0.876931	0.438466	−	0.898748i	$-0.355522\pi$
$31$	151.359	0.876931	0.438466	+	0.898748i	$0.355522\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−248.300	−1.10325	−0.551626	−	0.834091i	$-0.685993\pi$
$37$	−248.300	−1.10325	−0.551626	+	0.834091i	$0.685993\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−92.6654	−0.352973	−0.176487	−	0.984303i	$-0.556473\pi$
$41$	−92.6654	−0.352973	−0.176487	+	0.984303i	$0.556473\pi$
$42$	0	0
$43$	443.127	1.57154	0.785771	−	0.618518i	$-0.212267\pi$
$43$	443.127	1.57154	0.785771	+	0.618518i	$0.212267\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	262.132	0.813531	0.406765	−	0.913533i	$-0.366657\pi$
$47$	262.132	0.813531	0.406765	+	0.913533i	$0.366657\pi$
$48$	0	0
$49$	−341.718	−0.996262
$50$	0	0
$51$	0	0
$52$	0	0
$53$	278.425	0.721597	0.360799	−	0.932644i	$-0.382504\pi$
$53$	278.425	0.721597	0.360799	+	0.932644i	$0.382504\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−662.079	−1.46094	−0.730469	−	0.682946i	$-0.760699\pi$
$59$	−662.079	−1.46094	−0.730469	+	0.682946i	$0.760699\pi$
$60$	0	0
$61$	365.624	0.767432	0.383716	−	0.923451i	$-0.374644\pi$
$61$	365.624	0.767432	0.383716	+	0.923451i	$0.374644\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−786.101	−1.43340	−0.716698	−	0.697383i	$-0.754347\pi$
$67$	−786.101	−1.43340	−0.716698	+	0.697383i	$0.754347\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	401.671	0.671402	0.335701	−	0.941969i	$-0.391027\pi$
$71$	401.671	0.671402	0.335701	+	0.941969i	$0.391027\pi$
$72$	0	0
$73$	290.889	0.466384	0.233192	−	0.972431i	$-0.425083\pi$
$73$	290.889	0.466384	0.233192	+	0.972431i	$0.425083\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	23.4701	0.0347359
$78$	0	0
$79$	−185.171	−0.263713	−0.131857	−	0.991269i	$-0.542094\pi$
$79$	−185.171	−0.263713	−0.131857	+	0.991269i	$0.542094\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−399.889	−0.528837	−0.264419	−	0.964408i	$-0.585180\pi$
$83$	−399.889	−0.528837	−0.264419	+	0.964408i	$0.585180\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	886.255	1.05554	0.527769	−	0.849388i	$-0.323029\pi$
$89$	886.255	1.05554	0.527769	+	0.849388i	$0.323029\pi$
$90$	0	0
$91$	43.0940	0.0496426
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−48.9444	−0.0512325	−0.0256163	−	0.999672i	$-0.508155\pi$
$97$	−48.9444	−0.0512325	−0.0256163	+	0.999672i	$0.508155\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1309.53	−1.29013	−0.645065	−	0.764128i	$-0.723170\pi$
$101$	−1309.53	−1.29013	−0.645065	+	0.764128i	$0.723170\pi$
$102$	0	0
$103$	−625.937	−0.598790	−0.299395	−	0.954129i	$-0.596785\pi$
$103$	−625.937	−0.598790	−0.299395	+	0.954129i	$0.596785\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2062.61	1.86355	0.931774	−	0.363038i	$-0.118260\pi$
$107$	2062.61	1.86355	0.931774	+	0.363038i	$0.118260\pi$
$108$	0	0
$109$	167.436	0.147133	0.0735663	−	0.997290i	$-0.476562\pi$
$109$	167.436	0.147133	0.0735663	+	0.997290i	$0.476562\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1238.96	1.03143	0.515713	−	0.856762i	$-0.327527\pi$
$113$	1238.96	1.03143	0.515713	+	0.856762i	$0.327527\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.22348	−0.00402383
$120$	0	0
$121$	−901.342	−0.677191
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2109.82	1.47415	0.737073	−	0.675814i	$-0.236208\pi$
$127$	2109.82	1.47415	0.737073	+	0.675814i	$0.236208\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1937.11	1.29196	0.645978	−	0.763356i	$-0.276450\pi$
$131$	1937.11	1.29196	0.645978	+	0.763356i	$0.276450\pi$
$132$	0	0
$133$	−52.5977	−0.0342917
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1332.72	0.831112	0.415556	−	0.909568i	$-0.363587\pi$
$137$	1332.72	0.831112	0.415556	+	0.909568i	$0.363587\pi$
$138$	0	0
$139$	1879.51	1.14689	0.573447	−	0.819243i	$-0.305606\pi$
$139$	1879.51	1.14689	0.573447	+	0.819243i	$0.305606\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	788.906	0.461340
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	233.052	0.128137	0.0640684	−	0.997946i	$-0.479592\pi$
$149$	233.052	0.128137	0.0640684	+	0.997946i	$0.479592\pi$
$150$	0	0
$151$	−378.983	−0.204246	−0.102123	−	0.994772i	$-0.532564\pi$
$151$	−378.983	−0.204246	−0.102123	+	0.994772i	$0.532564\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2361.52	1.20044	0.600222	−	0.799833i	$-0.295079\pi$
$157$	2361.52	1.20044	0.600222	+	0.799833i	$0.295079\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−19.5052	−0.00954798
$162$	0	0
$163$	−2633.85	−1.26564	−0.632820	−	0.774299i	$-0.718103\pi$
$163$	−2633.85	−1.26564	−0.632820	+	0.774299i	$0.718103\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2536.70	1.17542	0.587711	−	0.809071i	$-0.300029\pi$
$167$	2536.70	1.17542	0.587711	+	0.809071i	$0.300029\pi$
$168$	0	0
$169$	−748.470	−0.340678
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1370.75	0.602404	0.301202	−	0.953560i	$-0.402612\pi$
$173$	1370.75	0.602404	0.301202	+	0.953560i	$0.402612\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−1540.66	−0.643322	−0.321661	−	0.946855i	$-0.604241\pi$
$179$	−1540.66	−0.643322	−0.321661	+	0.946855i	$0.604241\pi$
$180$	0	0
$181$	−2406.31	−0.988174	−0.494087	−	0.869413i	$-0.664498\pi$
$181$	−2406.31	−0.988174	−0.494087	+	0.869413i	$0.664498\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−95.6243	−0.0373943
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2544.84	−0.964075	−0.482037	−	0.876151i	$-0.660103\pi$
$191$	−2544.84	−0.964075	−0.482037	+	0.876151i	$0.660103\pi$
$192$	0	0
$193$	−2956.81	−1.10278	−0.551388	−	0.834249i	$-0.685902\pi$
$193$	−2956.81	−1.10278	−0.551388	+	0.834249i	$0.685902\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4187.92	1.51460	0.757301	−	0.653066i	$-0.226517\pi$
$197$	4187.92	1.51460	0.757301	+	0.653066i	$0.226517\pi$
$198$	0	0
$199$	3432.95	1.22289	0.611445	−	0.791287i	$-0.290588\pi$
$199$	3432.95	1.22289	0.611445	+	0.791287i	$0.290588\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	156.598	0.0541431
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−962.887	−0.318681
$210$	0	0
$211$	543.205	0.177231	0.0886156	−	0.996066i	$-0.471756\pi$
$211$	543.205	0.177231	0.0886156	+	0.996066i	$0.471756\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	171.381	0.0536132
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−175.578	−0.0534419
$222$	0	0
$223$	3991.03	1.19847	0.599236	−	0.800573i	$-0.295471\pi$
$223$	3991.03	1.19847	0.599236	+	0.800573i	$0.295471\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5391.89	1.57653	0.788265	−	0.615336i	$-0.210980\pi$
$227$	5391.89	1.57653	0.788265	+	0.615336i	$0.210980\pi$
$228$	0	0
$229$	−2827.81	−0.816013	−0.408007	−	0.912979i	$-0.633776\pi$
$229$	−2827.81	−0.816013	−0.408007	+	0.912979i	$0.633776\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	593.686	0.166926	0.0834628	−	0.996511i	$-0.473402\pi$
$233$	593.686	0.166926	0.0834628	+	0.996511i	$0.473402\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3524.46	0.953883	0.476942	−	0.878935i	$-0.341745\pi$
$239$	3524.46	0.953883	0.476942	+	0.878935i	$0.341745\pi$
$240$	0	0
$241$	1707.37	0.456354	0.228177	−	0.973620i	$-0.426724\pi$
$241$	1707.37	0.456354	0.228177	+	0.973620i	$0.426724\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1767.98	−0.455441
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2228.36	0.560371	0.280185	−	0.959946i	$-0.409604\pi$
$251$	2228.36	0.560371	0.280185	+	0.959946i	$0.409604\pi$
$252$	0	0
$253$	−357.074	−0.0887315
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6735.49	1.63482	0.817409	−	0.576058i	$-0.195410\pi$
$257$	6735.49	1.63482	0.817409	+	0.576058i	$0.195410\pi$
$258$	0	0
$259$	−281.145	−0.0674499
$260$	0	0
$261$	0	0
$262$	0	0
$263$	7298.79	1.71127	0.855633	−	0.517584i	$-0.173168\pi$
$263$	7298.79	1.71127	0.855633	+	0.517584i	$0.173168\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1562.43	−0.354139	−0.177069	−	0.984198i	$-0.556662\pi$
$269$	−1562.43	−0.354139	−0.177069	+	0.984198i	$0.556662\pi$
$270$	0	0
$271$	1182.08	0.264967	0.132484	−	0.991185i	$-0.457705\pi$
$271$	1182.08	0.264967	0.132484	+	0.991185i	$0.457705\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7630.10	−1.65505	−0.827524	−	0.561430i	$-0.810251\pi$
$277$	−7630.10	−1.65505	−0.827524	+	0.561430i	$0.810251\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5589.27	1.18658	0.593289	−	0.804990i	$-0.297829\pi$
$281$	5589.27	1.18658	0.593289	+	0.804990i	$0.297829\pi$
$282$	0	0
$283$	1459.49	0.306564	0.153282	−	0.988183i	$-0.451016\pi$
$283$	1459.49	0.306564	0.153282	+	0.988183i	$0.451016\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−104.923	−0.0215798
$288$	0	0
$289$	−4891.72	−0.995668
$290$	0	0
$291$	0	0
$292$	0	0
$293$	907.079	0.180860	0.0904302	−	0.995903i	$-0.471176\pi$
$293$	907.079	0.180860	0.0904302	+	0.995903i	$0.471176\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−655.633	−0.126810
$300$	0	0
$301$	501.744	0.0960798
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1153.47	0.214437	0.107218	−	0.994235i	$-0.465806\pi$
$307$	1153.47	0.214437	0.107218	+	0.994235i	$0.465806\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9248.22	−1.68623	−0.843116	−	0.537732i	$-0.819281\pi$
$311$	−9248.22	−1.68623	−0.843116	+	0.537732i	$0.819281\pi$
$312$	0	0
$313$	9245.92	1.66968	0.834841	−	0.550492i	$-0.185560\pi$
$313$	9245.92	1.66968	0.834841	+	0.550492i	$0.185560\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8760.43	1.55216	0.776080	−	0.630634i	$-0.217205\pi$
$317$	8760.43	1.55216	0.776080	+	0.630634i	$0.217205\pi$
$318$	0	0
$319$	2866.79	0.503164
$320$	0	0
$321$	0	0
$322$	0	0
$323$	214.299	0.0369162
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	296.807	0.0497371
$330$	0	0
$331$	9484.81	1.57502	0.787511	−	0.616300i	$-0.211369\pi$
$331$	9484.81	1.57502	0.787511	+	0.616300i	$0.211369\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	24.3972	0.00394362	0.00197181	−	0.999998i	$-0.499372\pi$
$337$	24.3972	0.00394362	0.00197181	+	0.999998i	$0.499372\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3137.40	0.498239
$342$	0	0
$343$	−775.291	−0.122046
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9098.43	1.40758	0.703788	−	0.710410i	$-0.251490\pi$
$347$	9098.43	1.40758	0.703788	+	0.710410i	$0.251490\pi$
$348$	0	0
$349$	−12438.2	−1.90774	−0.953868	−	0.300226i	$-0.902938\pi$
$349$	−12438.2	−1.90774	−0.953868	+	0.300226i	$0.902938\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2304.10	−0.347408	−0.173704	−	0.984798i	$-0.555574\pi$
$353$	−2304.10	−0.347408	−0.173704	+	0.984798i	$0.555574\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10184.3	1.49723	0.748614	−	0.663006i	$-0.230720\pi$
$359$	10184.3	1.49723	0.748614	+	0.663006i	$0.230720\pi$
$360$	0	0
$361$	−4701.12	−0.685394
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−7438.67	−1.05803	−0.529013	−	0.848614i	$-0.677438\pi$
$367$	−7438.67	−1.05803	−0.529013	+	0.848614i	$0.677438\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	315.255	0.0441165
$372$	0	0
$373$	3352.67	0.465401	0.232701	−	0.972548i	$-0.425244\pi$
$373$	3352.67	0.465401	0.232701	+	0.972548i	$0.425244\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	5263.78	0.719094
$378$	0	0
$379$	11746.2	1.59198	0.795989	−	0.605311i	$-0.206951\pi$
$379$	11746.2	1.59198	0.795989	+	0.605311i	$0.206951\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7443.01	0.993003	0.496502	−	0.868036i	$-0.334618\pi$
$383$	7443.01	0.993003	0.496502	+	0.868036i	$0.334618\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−7607.37	−0.991539	−0.495770	−	0.868454i	$-0.665114\pi$
$389$	−7607.37	−0.991539	−0.495770	+	0.868454i	$0.665114\pi$
$390$	0	0
$391$	79.4701	0.0102787
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8923.98	1.12817	0.564083	−	0.825718i	$-0.309230\pi$
$397$	8923.98	1.12817	0.564083	+	0.825718i	$0.309230\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1065.34	−0.132669	−0.0663346	−	0.997797i	$-0.521130\pi$
$401$	−1065.34	−0.132669	−0.0663346	+	0.997797i	$0.521130\pi$
$402$	0	0
$403$	5760.66	0.712056
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5146.82	−0.626827
$408$	0	0
$409$	−5567.35	−0.673075	−0.336538	−	0.941670i	$-0.609256\pi$
$409$	−5567.35	−0.673075	−0.336538	+	0.941670i	$0.609256\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−749.658	−0.0893178
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−13482.5	−1.57199	−0.785994	−	0.618235i	$-0.787848\pi$
$419$	−13482.5	−1.57199	−0.785994	+	0.618235i	$0.787848\pi$
$420$	0	0
$421$	−851.504	−0.0985743	−0.0492871	−	0.998785i	$-0.515695\pi$
$421$	−851.504	−0.0985743	−0.0492871	+	0.998785i	$0.515695\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	413.988	0.0469187
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5289.63	−0.591166	−0.295583	−	0.955317i	$-0.595514\pi$
$431$	−5289.63	−0.591166	−0.295583	+	0.955317i	$0.595514\pi$
$432$	0	0
$433$	−13063.4	−1.44986	−0.724928	−	0.688825i	$-0.758127\pi$
$433$	−13063.4	−1.44986	−0.724928	+	0.688825i	$0.758127\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	800.222	0.0875968
$438$	0	0
$439$	5326.52	0.579091	0.289546	−	0.957164i	$-0.406496\pi$
$439$	5326.52	0.579091	0.289546	+	0.957164i	$0.406496\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2927.44	−0.313966	−0.156983	−	0.987601i	$-0.550177\pi$
$443$	−2927.44	−0.313966	−0.156983	+	0.987601i	$0.550177\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−7338.81	−0.771358	−0.385679	−	0.922633i	$-0.626033\pi$
$449$	−7338.81	−0.771358	−0.385679	+	0.922633i	$0.626033\pi$
$450$	0	0
$451$	−1920.79	−0.200546
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4373.27	−0.447643	−0.223822	−	0.974630i	$-0.571853\pi$
$457$	−4373.27	−0.447643	−0.223822	+	0.974630i	$0.571853\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2160.28	0.218252	0.109126	−	0.994028i	$-0.465195\pi$
$461$	2160.28	0.218252	0.109126	+	0.994028i	$0.465195\pi$
$462$	0	0
$463$	−14233.5	−1.42870	−0.714349	−	0.699790i	$-0.753277\pi$
$463$	−14233.5	−1.42870	−0.714349	+	0.699790i	$0.753277\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5334.36	0.528575	0.264288	−	0.964444i	$-0.414863\pi$
$467$	5334.36	0.528575	0.264288	+	0.964444i	$0.414863\pi$
$468$	0	0
$469$	−890.086	−0.0876340
$470$	0	0
$471$	0	0
$472$	0	0
$473$	9185.23	0.892891
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	17700.4	1.68842	0.844210	−	0.536013i	$-0.180070\pi$
$479$	17700.4	1.68842	0.844210	+	0.536013i	$0.180070\pi$
$480$	0	0
$481$	−9450.21	−0.895826
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	10985.7	1.02220	0.511100	−	0.859521i	$-0.329238\pi$
$487$	10985.7	1.02220	0.511100	+	0.859521i	$0.329238\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18643.6	1.71359	0.856795	−	0.515657i	$-0.172452\pi$
$491$	18643.6	1.71359	0.856795	+	0.515657i	$0.172452\pi$
$492$	0	0
$493$	−638.029	−0.0582868
$494$	0	0
$495$	0	0
$496$	0	0
$497$	454.803	0.0410477
$498$	0	0
$499$	18067.0	1.62082	0.810411	−	0.585862i	$-0.199244\pi$
$499$	18067.0	1.62082	0.810411	+	0.585862i	$0.199244\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−673.782	−0.0597266	−0.0298633	−	0.999554i	$-0.509507\pi$
$503$	−673.782	−0.0597266	−0.0298633	+	0.999554i	$0.509507\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16508.2	1.43755	0.718776	−	0.695241i	$-0.244702\pi$
$509$	16508.2	1.43755	0.718776	+	0.695241i	$0.244702\pi$
$510$	0	0
$511$	329.367	0.0285134
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	5433.53	0.462218
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14300.6	−1.20254	−0.601269	−	0.799047i	$-0.705338\pi$
$521$	−14300.6	−1.20254	−0.601269	+	0.799047i	$0.705338\pi$
$522$	0	0
$523$	−7989.67	−0.667999	−0.334000	−	0.942573i	$-0.608398\pi$
$523$	−7989.67	−0.667999	−0.334000	+	0.942573i	$0.608398\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−698.256	−0.0577164
$528$	0	0
$529$	−11870.2	−0.975610
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3526.80	−0.286609
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−7083.20	−0.566039
$540$	0	0
$541$	−13233.0	−1.05163	−0.525816	−	0.850598i	$-0.676240\pi$
$541$	−13233.0	−1.05163	−0.525816	+	0.850598i	$0.676240\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	6295.09	0.492063	0.246032	−	0.969262i	$-0.420873\pi$
$547$	6295.09	0.492063	0.246032	+	0.969262i	$0.420873\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6424.62	−0.496730
$552$	0	0
$553$	−209.665	−0.0161227
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−15932.0	−1.21196	−0.605978	−	0.795481i	$-0.707218\pi$
$557$	−15932.0	−1.21196	−0.605978	+	0.795481i	$0.707218\pi$
$558$	0	0
$559$	16865.2	1.27607
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−15021.3	−1.12446	−0.562232	−	0.826979i	$-0.690057\pi$
$563$	−15021.3	−1.12446	−0.562232	+	0.826979i	$0.690057\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7928.26	0.584130	0.292065	−	0.956399i	$-0.405658\pi$
$569$	7928.26	0.584130	0.292065	+	0.956399i	$0.405658\pi$
$570$	0	0
$571$	6990.59	0.512342	0.256171	−	0.966632i	$-0.417539\pi$
$571$	6990.59	0.512342	0.256171	+	0.966632i	$0.417539\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	9822.29	0.708678	0.354339	−	0.935117i	$-0.384706\pi$
$577$	9822.29	0.708678	0.354339	+	0.935117i	$0.384706\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−452.786	−0.0323317
$582$	0	0
$583$	5771.25	0.409984
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3166.66	−0.222661	−0.111330	−	0.993783i	$-0.535511\pi$
$587$	−3166.66	−0.222661	−0.111330	+	0.993783i	$0.535511\pi$
$588$	0	0
$589$	−7031.08	−0.491868
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−8939.61	−0.619065	−0.309533	−	0.950889i	$-0.600173\pi$
$593$	−8939.61	−0.619065	−0.309533	+	0.950889i	$0.600173\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−19944.3	−1.36044	−0.680219	−	0.733009i	$-0.738115\pi$
$599$	−19944.3	−1.36044	−0.680219	+	0.733009i	$0.738115\pi$
$600$	0	0
$601$	3911.45	0.265477	0.132738	−	0.991151i	$-0.457623\pi$
$601$	3911.45	0.265477	0.132738	+	0.991151i	$0.457623\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5176.42	0.346136	0.173068	−	0.984910i	$-0.444632\pi$
$607$	5176.42	0.346136	0.173068	+	0.984910i	$0.444632\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	9976.65	0.660576
$612$	0	0
$613$	−15569.7	−1.02586	−0.512932	−	0.858429i	$-0.671441\pi$
$613$	−15569.7	−1.02586	−0.512932	+	0.858429i	$0.671441\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−25414.5	−1.65827	−0.829133	−	0.559052i	$-0.811165\pi$
$617$	−25414.5	−1.65827	−0.829133	+	0.559052i	$0.811165\pi$
$618$	0	0
$619$	7164.47	0.465209	0.232604	−	0.972571i	$-0.425275\pi$
$619$	7164.47	0.465209	0.232604	+	0.972571i	$0.425275\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1003.49	0.0645327
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1145.47	0.0726120
$630$	0	0
$631$	−7752.28	−0.489086	−0.244543	−	0.969638i	$-0.578638\pi$
$631$	−7752.28	−0.489086	−0.244543	+	0.969638i	$0.578638\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−13005.6	−0.808951
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4370.61	0.269312	0.134656	−	0.990892i	$-0.457007\pi$
$641$	4370.61	0.269312	0.134656	+	0.990892i	$0.457007\pi$
$642$	0	0
$643$	11663.0	0.715306	0.357653	−	0.933855i	$-0.383577\pi$
$643$	11663.0	0.715306	0.357653	+	0.933855i	$0.383577\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15502.0	−0.941960	−0.470980	−	0.882144i	$-0.656100\pi$
$647$	−15502.0	−0.941960	−0.470980	+	0.882144i	$0.656100\pi$
$648$	0	0
$649$	−13723.7	−0.830050
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17771.1	1.06499	0.532494	−	0.846434i	$-0.321255\pi$
$653$	17771.1	1.06499	0.532494	+	0.846434i	$0.321255\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−12389.3	−0.732353	−0.366176	−	0.930545i	$-0.619333\pi$
$659$	−12389.3	−0.732353	−0.366176	+	0.930545i	$0.619333\pi$
$660$	0	0
$661$	−1803.88	−0.106146	−0.0530732	−	0.998591i	$-0.516902\pi$
$661$	−1803.88	−0.106146	−0.0530732	+	0.998591i	$0.516902\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−2382.49	−0.138306
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7578.72	0.436026
$672$	0	0
$673$	9913.75	0.567826	0.283913	−	0.958850i	$-0.408367\pi$
$673$	9913.75	0.567826	0.283913	+	0.958850i	$0.408367\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8328.12	−0.472785	−0.236393	−	0.971658i	$-0.575965\pi$
$677$	−8328.12	−0.472785	−0.236393	+	0.971658i	$0.575965\pi$
$678$	0	0
$679$	−55.4187	−0.00313222
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−18142.1	−1.01638	−0.508189	−	0.861246i	$-0.669685\pi$
$683$	−18142.1	−1.01638	−0.508189	+	0.861246i	$0.669685\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	10596.7	0.585927
$690$	0	0
$691$	27210.7	1.49804	0.749019	−	0.662548i	$-0.230525\pi$
$691$	27210.7	1.49804	0.749019	+	0.662548i	$0.230525\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	427.488	0.0232314
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−15259.0	−0.822147	−0.411073	−	0.911602i	$-0.634846\pi$
$701$	−15259.0	−0.822147	−0.411073	+	0.911602i	$0.634846\pi$
$702$	0	0
$703$	11534.3	0.618811
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1482.75	−0.0788750
$708$	0	0
$709$	−22160.8	−1.17386	−0.586931	−	0.809637i	$-0.699664\pi$
$709$	−22160.8	−1.17386	−0.586931	+	0.809637i	$0.699664\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2607.38	−0.136953
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−5768.33	−0.299196	−0.149598	−	0.988747i	$-0.547798\pi$
$719$	−5768.33	−0.299196	−0.149598	+	0.988747i	$0.547798\pi$
$720$	0	0
$721$	−708.735	−0.0366084
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−33193.6	−1.69337	−0.846687	−	0.532092i	$-0.821406\pi$
$727$	−33193.6	−1.69337	−0.846687	+	0.532092i	$0.821406\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−2044.26	−0.103433
$732$	0	0
$733$	21860.9	1.10157	0.550784	−	0.834648i	$-0.314329\pi$
$733$	21860.9	1.10157	0.550784	+	0.834648i	$0.314329\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16294.5	−0.814402
$738$	0	0
$739$	10235.0	0.509471	0.254736	−	0.967011i	$-0.418012\pi$
$739$	10235.0	0.509471	0.254736	+	0.967011i	$0.418012\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10942.3	−0.540287	−0.270143	−	0.962820i	$-0.587071\pi$
$743$	−10942.3	−0.540287	−0.270143	+	0.962820i	$0.587071\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2335.45	0.113932
$750$	0	0
$751$	152.145	0.00739262	0.00369631	−	0.999993i	$-0.498823\pi$
$751$	152.145	0.00739262	0.00369631	+	0.999993i	$0.498823\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−8890.95	−0.426878	−0.213439	−	0.976956i	$-0.568467\pi$
$757$	−8890.95	−0.426878	−0.213439	+	0.976956i	$0.568467\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−29697.2	−1.41462	−0.707309	−	0.706905i	$-0.750091\pi$
$761$	−29697.2	−1.41462	−0.707309	+	0.706905i	$0.750091\pi$
$762$	0	0
$763$	189.584	0.00899529
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−25198.4	−1.18626
$768$	0	0
$769$	−18476.8	−0.866438	−0.433219	−	0.901289i	$-0.642622\pi$
$769$	−18476.8	−0.866438	−0.433219	+	0.901289i	$0.642622\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−27450.6	−1.27727	−0.638634	−	0.769510i	$-0.720500\pi$
$773$	−27450.6	−1.27727	−0.638634	+	0.769510i	$0.720500\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4304.58	0.197982
$780$	0	0
$781$	8325.91	0.381466
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−7089.82	−0.321124	−0.160562	−	0.987026i	$-0.551331\pi$
$787$	−7089.82	−0.321124	−0.160562	+	0.987026i	$0.551331\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1402.84	0.0630586
$792$	0	0
$793$	13915.5	0.623144
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−33829.2	−1.50350	−0.751750	−	0.659448i	$-0.770790\pi$
$797$	−33829.2	−1.50350	−0.751750	+	0.659448i	$0.770790\pi$
$798$	0	0
$799$	−1209.28	−0.0535436
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6029.61	0.264982
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6866.33	0.298402	0.149201	−	0.988807i	$-0.452330\pi$
$809$	6866.33	0.298402	0.149201	+	0.988807i	$0.452330\pi$
$810$	0	0
$811$	29316.4	1.26934	0.634671	−	0.772782i	$-0.281135\pi$
$811$	29316.4	1.26934	0.634671	+	0.772782i	$0.281135\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−20584.6	−0.881473
$818$	0	0
$819$	0	0
$820$	0	0
$821$	33772.7	1.43566	0.717830	−	0.696218i	$-0.245136\pi$
$821$	33772.7	1.43566	0.717830	+	0.696218i	$0.245136\pi$
$822$	0	0
$823$	1240.44	0.0525385	0.0262693	−	0.999655i	$-0.491637\pi$
$823$	1240.44	0.0525385	0.0262693	+	0.999655i	$0.491637\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−35311.1	−1.48475	−0.742374	−	0.669986i	$-0.766300\pi$
$827$	−35311.1	−1.48475	−0.742374	+	0.669986i	$0.766300\pi$
$828$	0	0
$829$	−5126.89	−0.214794	−0.107397	−	0.994216i	$-0.534252\pi$
$829$	−5126.89	−0.214794	−0.107397	+	0.994216i	$0.534252\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1576.43	0.0655703
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	13093.1	0.538765	0.269382	−	0.963033i	$-0.413180\pi$
$839$	13093.1	0.538765	0.269382	+	0.963033i	$0.413180\pi$
$840$	0	0
$841$	−5261.09	−0.215716
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1020.57	−0.0414017
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4277.35	0.172298
$852$	0	0
$853$	1950.55	0.0782947	0.0391474	−	0.999233i	$-0.487536\pi$
$853$	1950.55	0.0782947	0.0391474	+	0.999233i	$0.487536\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−13702.5	−0.546169	−0.273084	−	0.961990i	$-0.588044\pi$
$857$	−13702.5	−0.546169	−0.273084	+	0.961990i	$0.588044\pi$
$858$	0	0
$859$	7397.00	0.293810	0.146905	−	0.989151i	$-0.453069\pi$
$859$	7397.00	0.293810	0.146905	+	0.989151i	$0.453069\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−22017.8	−0.868475	−0.434238	−	0.900798i	$-0.642982\pi$
$863$	−22017.8	−0.868475	−0.434238	+	0.900798i	$0.642982\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3838.26	−0.149832
$870$	0	0
$871$	−29918.7	−1.16390
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−3294.67	−0.126856	−0.0634282	−	0.997986i	$-0.520203\pi$
$877$	−3294.67	−0.126856	−0.0634282	+	0.997986i	$0.520203\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	42961.2	1.64291	0.821453	−	0.570277i	$-0.193164\pi$
$881$	42961.2	1.64291	0.821453	+	0.570277i	$0.193164\pi$
$882$	0	0
$883$	−24530.0	−0.934882	−0.467441	−	0.884024i	$-0.654824\pi$
$883$	−24530.0	−0.934882	−0.467441	+	0.884024i	$0.654824\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3343.71	−0.126574	−0.0632869	−	0.997995i	$-0.520158\pi$
$887$	−3343.71	−0.126574	−0.0632869	+	0.997995i	$0.520158\pi$
$888$	0	0
$889$	2388.91	0.0901252
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−12176.8	−0.456307
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	20933.5	0.776609
$900$	0	0
$901$	−1284.44	−0.0474928
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	10735.0	0.392999	0.196499	−	0.980504i	$-0.437043\pi$
$907$	10735.0	0.392999	0.196499	+	0.980504i	$0.437043\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−14593.9	−0.530753	−0.265376	−	0.964145i	$-0.585496\pi$
$911$	−14593.9	−0.530753	−0.265376	+	0.964145i	$0.585496\pi$
$912$	0	0
$913$	−8288.98	−0.300466
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2193.35	0.0789867
$918$	0	0
$919$	−8030.38	−0.288246	−0.144123	−	0.989560i	$-0.546036\pi$
$919$	−8030.38	−0.288246	−0.144123	+	0.989560i	$0.546036\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	15287.4	0.545170
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−51303.2	−1.81184	−0.905922	−	0.423446i	$-0.860820\pi$
$929$	−51303.2	−1.81184	−0.905922	+	0.423446i	$0.860820\pi$
$930$	0	0
$931$	15873.8	0.558801
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	13053.9	0.455125	0.227563	−	0.973763i	$-0.426924\pi$
$937$	13053.9	0.455125	0.227563	+	0.973763i	$0.426924\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−34473.8	−1.19427	−0.597137	−	0.802139i	$-0.703695\pi$
$941$	−34473.8	−1.19427	−0.597137	+	0.802139i	$0.703695\pi$
$942$	0	0
$943$	1596.30	0.0551248
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−43587.1	−1.49566	−0.747831	−	0.663889i	$-0.768905\pi$
$947$	−43587.1	−1.49566	−0.747831	+	0.663889i	$0.768905\pi$
$948$	0	0
$949$	11071.1	0.378697
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−21266.3	−0.722857	−0.361429	−	0.932400i	$-0.617711\pi$
$953$	−21266.3	−0.722857	−0.361429	+	0.932400i	$0.617711\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1509.02	0.0508119
$960$	0	0
$961$	−6881.46	−0.230991
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	40621.4	1.35088	0.675438	−	0.737417i	$-0.263955\pi$
$967$	40621.4	1.35088	0.675438	+	0.737417i	$0.263955\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−42340.1	−1.39934	−0.699670	−	0.714466i	$-0.746670\pi$
$971$	−42340.1	−1.39934	−0.699670	+	0.714466i	$0.746670\pi$
$972$	0	0
$973$	2128.13	0.0701180
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5416.55	0.177370	0.0886851	−	0.996060i	$-0.471734\pi$
$977$	5416.55	0.177370	0.0886851	+	0.996060i	$0.471734\pi$
$978$	0	0
$979$	18370.5	0.599717
$980$	0	0
$981$	0	0
$982$	0	0
$983$	13908.1	0.451272	0.225636	−	0.974212i	$-0.427554\pi$
$983$	13908.1	0.451272	0.225636	+	0.974212i	$0.427554\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−7633.53	−0.245432
$990$	0	0
$991$	−33701.7	−1.08029	−0.540146	−	0.841572i	$-0.681631\pi$
$991$	−33701.7	−1.08029	−0.540146	+	0.841572i	$0.681631\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	54466.4	1.73016	0.865080	−	0.501634i	$-0.167268\pi$
$997$	54466.4	1.73016	0.865080	+	0.501634i	$0.167268\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.4.a.bw.1.3		4
3.2	odd	2		1800.4.a.bv.1.3		4
5.2	odd	4		360.4.f.f.289.7	yes	8
5.3	odd	4		360.4.f.f.289.8	yes	8
5.4	even	2		1800.4.a.bv.1.2		4
15.2	even	4		360.4.f.f.289.2	yes	8
15.8	even	4		360.4.f.f.289.1	✓	8
15.14	odd	2	inner	1800.4.a.bw.1.2		4
20.3	even	4		720.4.f.n.289.8		8
20.7	even	4		720.4.f.n.289.7		8
60.23	odd	4		720.4.f.n.289.1		8
60.47	odd	4		720.4.f.n.289.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.4.f.f.289.1	✓	8	15.8	even	4
360.4.f.f.289.2	yes	8	15.2	even	4
360.4.f.f.289.7	yes	8	5.2	odd	4
360.4.f.f.289.8	yes	8	5.3	odd	4
720.4.f.n.289.1		8	60.23	odd	4
720.4.f.n.289.2		8	60.47	odd	4
720.4.f.n.289.7		8	20.7	even	4
720.4.f.n.289.8		8	20.3	even	4
1800.4.a.bv.1.2		4	5.4	even	2
1800.4.a.bv.1.3		4	3.2	odd	2
1800.4.a.bw.1.2		4	15.14	odd	2	inner
1800.4.a.bw.1.3		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1800.a (trivial)

\(f(q)\)	\(=\)	\(q+1.13228 q^{7} +O(q^{10})\)	\(q+1.13228 q^{7} +20.7282 q^{11} +38.0596 q^{13} -4.61325 q^{17} -46.4530 q^{19} -17.2265 q^{23} +138.304 q^{29} +151.359 q^{31} -248.300 q^{37} -92.6654 q^{41} +443.127 q^{43} +262.132 q^{47} -341.718 q^{49} +278.425 q^{53} -662.079 q^{59} +365.624 q^{61} -786.101 q^{67} +401.671 q^{71} +290.889 q^{73} +23.4701 q^{77} -185.171 q^{79} -399.889 q^{83} +886.255 q^{89} +43.0940 q^{91} -48.9444 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q + 48 q^{17} + 80 q^{19} + 64 q^{23} - 192 q^{31} + 384 q^{47} + 228 q^{49} - 16 q^{53} - 664 q^{61} + 2752 q^{77} + 1120 q^{79} + 1856 q^{83} + 704 q^{91}+O(q^{100}) \) 4 * q + 48 * q^17 + 80 * q^19 + 64 * q^23 - 192 * q^31 + 384 * q^47 + 228 * q^49 - 16 * q^53 - 664 * q^61 + 2752 * q^77 + 1120 * q^79 + 1856 * q^83 + 704 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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