LMFDB - Embedded newform 1200.5.e.f.751.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1200,5,Mod(751,1200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("1200.751");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 1200 = 2^{4} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	1200.e (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:	$124.043955701$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-43})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 10x^{2} - 11x + 121 $ x^4 - x^3 - 10x^2 - 11x + 121
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			751.2
Root			$-2.58945 + 2.07237i$ of defining polynomial
Character	$\chi$	$=$	1200.751
Dual form			1200.5.e.f.751.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.19615i q^{3} +61.8613i q^{7} -27.0000 q^{9} +O(q^{10})$	$q-5.19615i q^{3} +61.8613i q^{7} -27.0000 q^{9} -83.3929i q^{11} +37.0000 q^{13} -164.147 q^{17} -596.351i q^{19} +321.441 q^{21} +326.389i q^{23} +140.296i q^{27} +1089.91 q^{29} +1606.41i q^{31} -433.322 q^{33} -389.706 q^{37} -192.258i q^{39} -2642.26 q^{41} +11.9026i q^{43} +1620.79i q^{47} -1425.82 q^{49} +852.932i q^{51} +1647.03 q^{53} -3098.73 q^{57} -2974.65i q^{59} +1237.76 q^{61} -1670.25i q^{63} -492.264i q^{67} +1695.97 q^{69} -2870.16i q^{71} +4209.23 q^{73} +5158.79 q^{77} -746.145i q^{79} +729.000 q^{81} -9512.21i q^{83} -5663.34i q^{87} -11355.8 q^{89} +2288.87i q^{91} +8347.14 q^{93} +2710.04 q^{97} +2251.61i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 108 q^{9}+O(q^{10}) $ 4 * q - 108 * q^9	$ 4 q - 108 q^{9} + 148 q^{13} - 384 q^{17} + 468 q^{21} + 816 q^{29} + 720 q^{33} - 2104 q^{37} - 2664 q^{41} + 1384 q^{49} + 4680 q^{53} - 1764 q^{57} - 7588 q^{61} - 576 q^{69} + 10840 q^{73} + 15456 q^{77} + 2916 q^{81} - 23616 q^{89} + 14580 q^{93} - 20780 q^{97}+O(q^{100}) $ 4 * q - 108 * q^9 + 148 * q^13 - 384 * q^17 + 468 * q^21 + 816 * q^29 + 720 * q^33 - 2104 * q^37 - 2664 * q^41 + 1384 * q^49 + 4680 * q^53 - 1764 * q^57 - 7588 * q^61 - 576 * q^69 + 10840 * q^73 + 15456 * q^77 + 2916 * q^81 - 23616 * q^89 + 14580 * q^93 - 20780 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$.

$n$	$401$	$577$	$751$	$901$
$\chi(n)$	$1$	$1$	$-1$	$1$

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 $

$2$	0	0
$2$	0	0
$3$	− 5.19615i	− 0.577350i
$3$	− 5.19615i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	61.8613i	1.26248i	0.775589	+	0.631238i	$0.217453\pi$
$7$	61.8613i	1.26248i	−0.775589	+	0.631238i	$0.782547\pi$
$8$	0	0
$9$	−27.0000	−0.333333
$10$	0	0
$11$	− 83.3929i	− 0.689197i	−0.938750	−	0.344599i	$-0.888015\pi$
$11$	− 83.3929i	− 0.689197i	0.938750	−	0.344599i	$-0.111985\pi$
$12$	0	0
$13$	37.0000	0.218935	0.109467	−	0.993990i	$-0.465085\pi$
$13$	37.0000	0.218935	0.109467	+	0.993990i	$0.465085\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−164.147	−0.567982	−0.283991	−	0.958827i	$-0.591659\pi$
$17$	−164.147	−0.567982	−0.283991	+	0.958827i	$0.591659\pi$
$18$	0	0
$19$	− 596.351i	− 1.65194i	−0.563713	−	0.825970i	$-0.690628\pi$
$19$	− 596.351i	− 1.65194i	0.563713	−	0.825970i	$-0.309372\pi$
$20$	0	0
$21$	321.441	0.728890
$22$	0	0
$23$	326.389i	0.616992i	0.951226	+	0.308496i	$0.0998257\pi$
$23$	326.389i	0.616992i	−0.951226	+	0.308496i	$0.900174\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	140.296i	0.192450i
$28$	0	0
$29$	1089.91	1.29597	0.647984	−	0.761654i	$-0.275612\pi$
$29$	1089.91	1.29597	0.647984	+	0.761654i	$0.275612\pi$
$30$	0	0
$31$	1606.41i	1.67160i	0.549034	+	0.835800i	$0.314996\pi$
$31$	1606.41i	1.67160i	−0.549034	+	0.835800i	$0.685004\pi$
$32$	0	0
$33$	−433.322	−0.397908
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−389.706	−0.284665	−0.142332	−	0.989819i	$-0.545460\pi$
$37$	−389.706	−0.284665	−0.142332	+	0.989819i	$0.545460\pi$
$38$	0	0
$39$	− 192.258i	− 0.126402i
$40$	0	0
$41$	−2642.26	−1.57184	−0.785919	−	0.618329i	$-0.787810\pi$
$41$	−2642.26	−1.57184	−0.785919	+	0.618329i	$0.787810\pi$
$42$	0	0
$43$	11.9026i	0.00643732i	0.999995	+	0.00321866i	$0.00102453\pi$
$43$	11.9026i	0.00643732i	−0.999995	+	0.00321866i	$0.998975\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1620.79i	0.733721i	0.930276	+	0.366860i	$0.119567\pi$
$47$	1620.79i	0.733721i	−0.930276	+	0.366860i	$0.880433\pi$
$48$	0	0
$49$	−1425.82	−0.593844
$50$	0	0
$51$	852.932i	0.327925i
$52$	0	0
$53$	1647.03	0.586340	0.293170	−	0.956060i	$-0.405290\pi$
$53$	1647.03	0.586340	0.293170	+	0.956060i	$0.405290\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3098.73	−0.953749
$58$	0	0
$59$	− 2974.65i	− 0.854538i	−0.904125	−	0.427269i	$-0.859476\pi$
$59$	− 2974.65i	− 0.854538i	0.904125	−	0.427269i	$-0.140524\pi$
$60$	0	0
$61$	1237.76	0.332641	0.166321	−	0.986072i	$-0.446811\pi$
$61$	1237.76	0.332641	0.166321	+	0.986072i	$0.446811\pi$
$62$	0	0
$63$	− 1670.25i	− 0.420825i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 492.264i	− 0.109660i	−0.998496	−	0.0548301i	$-0.982538\pi$
$67$	− 492.264i	− 0.109660i	0.998496	−	0.0548301i	$-0.0174617\pi$
$68$	0	0
$69$	1695.97	0.356221
$70$	0	0
$71$	− 2870.16i	− 0.569362i	−0.958622	−	0.284681i	$-0.908112\pi$
$71$	− 2870.16i	− 0.569362i	0.958622	−	0.284681i	$-0.0918877\pi$
$72$	0	0
$73$	4209.23	0.789873	0.394936	−	0.918708i	$-0.370767\pi$
$73$	4209.23	0.789873	0.394936	+	0.918708i	$0.370767\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5158.79	0.870095
$78$	0	0
$79$	− 746.145i	− 0.119555i	−0.998212	−	0.0597777i	$-0.980961\pi$
$79$	− 746.145i	− 0.119555i	0.998212	−	0.0597777i	$-0.0190392\pi$
$80$	0	0
$81$	729.000	0.111111
$82$	0	0
$83$	− 9512.21i	− 1.38078i	−0.723436	−	0.690391i	$-0.757438\pi$
$83$	− 9512.21i	− 1.38078i	0.723436	−	0.690391i	$-0.242562\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 5663.34i	− 0.748228i
$88$	0	0
$89$	−11355.8	−1.43363	−0.716813	−	0.697265i	$-0.754400\pi$
$89$	−11355.8	−1.43363	−0.716813	+	0.697265i	$0.754400\pi$
$90$	0	0
$91$	2288.87i	0.276400i
$92$	0	0
$93$	8347.14	0.965098
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2710.04	0.288026	0.144013	−	0.989576i	$-0.453999\pi$
$97$	2710.04	0.288026	0.144013	+	0.989576i	$0.453999\pi$
$98$	0	0
$99$	2251.61i	0.229732i
$100$	0	0
$101$	−13821.3	−1.35490	−0.677449	−	0.735570i	$-0.736914\pi$
$101$	−13821.3	−1.35490	−0.677449	+	0.735570i	$0.736914\pi$
$102$	0	0
$103$	− 1375.02i	− 0.129609i	−0.997898	−	0.0648044i	$-0.979358\pi$
$103$	− 1375.02i	− 0.129609i	0.997898	−	0.0648044i	$-0.0206423\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 19402.8i	− 1.69471i	−0.531025	−	0.847356i	$-0.678193\pi$
$107$	− 19402.8i	− 1.69471i	0.531025	−	0.847356i	$-0.321807\pi$
$108$	0	0
$109$	−959.542	−0.0807627	−0.0403814	−	0.999184i	$-0.512857\pi$
$109$	−959.542	−0.0807627	−0.0403814	+	0.999184i	$0.512857\pi$
$110$	0	0
$111$	2024.97i	0.164351i
$112$	0	0
$113$	−15052.5	−1.17884	−0.589418	−	0.807829i	$-0.700643\pi$
$113$	−15052.5	−1.17884	−0.589418	+	0.807829i	$0.700643\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−999.000	−0.0729783
$118$	0	0
$119$	− 10154.3i	− 0.717064i
$120$	0	0
$121$	7686.63	0.525007
$122$	0	0
$123$	13729.6i	0.907501i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 16107.1i	− 0.998644i	−0.866416	−	0.499322i	$-0.833582\pi$
$127$	− 16107.1i	− 0.998644i	0.866416	−	0.499322i	$-0.166418\pi$
$128$	0	0
$129$	61.8477	0.00371659
$130$	0	0
$131$	− 14129.9i	− 0.823372i	−0.911326	−	0.411686i	$-0.864940\pi$
$131$	− 14129.9i	− 0.823372i	0.911326	−	0.411686i	$-0.135060\pi$
$132$	0	0
$133$	36891.0	2.08553
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−11319.2	−0.603082	−0.301541	−	0.953453i	$-0.597501\pi$
$137$	−11319.2	−0.603082	−0.301541	+	0.953453i	$0.597501\pi$
$138$	0	0
$139$	− 19054.3i	− 0.986199i	−0.869973	−	0.493099i	$-0.835864\pi$
$139$	− 19054.3i	− 0.986199i	0.869973	−	0.493099i	$-0.164136\pi$
$140$	0	0
$141$	8421.87	0.423614
$142$	0	0
$143$	− 3085.54i	− 0.150889i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	7408.77i	0.342856i
$148$	0	0
$149$	−28887.6	−1.30118	−0.650592	−	0.759428i	$-0.725479\pi$
$149$	−28887.6	−1.30118	−0.650592	+	0.759428i	$0.725479\pi$
$150$	0	0
$151$	− 5543.28i	− 0.243116i	−0.992584	−	0.121558i	$-0.961211\pi$
$151$	− 5543.28i	− 0.243116i	0.992584	−	0.121558i	$-0.0387890\pi$
$152$	0	0
$153$	4431.97	0.189327
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−39894.8	−1.61851	−0.809257	−	0.587454i	$-0.800130\pi$
$157$	−39894.8	−1.61851	−0.809257	+	0.587454i	$0.800130\pi$
$158$	0	0
$159$	− 8558.21i	− 0.338523i
$160$	0	0
$161$	−20190.8	−0.778937
$162$	0	0
$163$	12898.6i	0.485474i	0.970092	+	0.242737i	$0.0780452\pi$
$163$	12898.6i	0.485474i	−0.970092	+	0.242737i	$0.921955\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	38130.4i	1.36722i	0.729847	+	0.683611i	$0.239591\pi$
$167$	38130.4i	1.36722i	−0.729847	+	0.683611i	$0.760409\pi$
$168$	0	0
$169$	−27192.0	−0.952068
$170$	0	0
$171$	16101.5i	0.550647i
$172$	0	0
$173$	41518.0	1.38722	0.693609	−	0.720352i	$-0.256020\pi$
$173$	41518.0	1.38722	0.693609	+	0.720352i	$0.256020\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−15456.7	−0.493368
$178$	0	0
$179$	− 11770.7i	− 0.367364i	−0.982986	−	0.183682i	$-0.941198\pi$
$179$	− 11770.7i	− 0.367364i	0.982986	−	0.183682i	$-0.0588017\pi$
$180$	0	0
$181$	−35937.2	−1.09695	−0.548475	−	0.836167i	$-0.684791\pi$
$181$	−35937.2	−1.09695	−0.548475	+	0.836167i	$0.684791\pi$
$182$	0	0
$183$	− 6431.58i	− 0.192050i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	13688.7i	0.391452i
$188$	0	0
$189$	−8678.90	−0.242963
$190$	0	0
$191$	3982.68i	0.109171i	0.998509	+	0.0545857i	$0.0173838\pi$
$191$	3982.68i	0.109171i	−0.998509	+	0.0545857i	$0.982616\pi$
$192$	0	0
$193$	−44119.7	−1.18445	−0.592226	−	0.805772i	$-0.701751\pi$
$193$	−44119.7	−1.18445	−0.592226	+	0.805772i	$0.701751\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16021.6	0.412833	0.206417	−	0.978464i	$-0.433820\pi$
$197$	16021.6	0.412833	0.206417	+	0.978464i	$0.433820\pi$
$198$	0	0
$199$	− 17872.7i	− 0.451320i	−0.974206	−	0.225660i	$-0.927546\pi$
$199$	− 17872.7i	− 0.451320i	0.974206	−	0.225660i	$-0.0724538\pi$
$200$	0	0
$201$	−2557.88	−0.0633123
$202$	0	0
$203$	67423.2i	1.63613i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 8812.50i	− 0.205664i
$208$	0	0
$209$	−49731.4	−1.13851
$210$	0	0
$211$	10915.5i	0.245176i	0.992458	+	0.122588i	$0.0391194\pi$
$211$	10915.5i	0.245176i	−0.992458	+	0.122588i	$0.960881\pi$
$212$	0	0
$213$	−14913.8	−0.328721
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−99374.4	−2.11035
$218$	0	0
$219$	− 21871.8i	− 0.456033i
$220$	0	0
$221$	−6073.44	−0.124351
$222$	0	0
$223$	80006.4i	1.60885i	0.594056	+	0.804424i	$0.297526\pi$
$223$	80006.4i	1.60885i	−0.594056	+	0.804424i	$0.702474\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 62721.8i	− 1.21721i	−0.793472	−	0.608607i	$-0.791729\pi$
$227$	− 62721.8i	− 1.21721i	0.793472	−	0.608607i	$-0.208271\pi$
$228$	0	0
$229$	−83116.5	−1.58495	−0.792476	−	0.609903i	$-0.791208\pi$
$229$	−83116.5	−1.58495	−0.792476	+	0.609903i	$0.791208\pi$
$230$	0	0
$231$	− 26805.9i	− 0.502349i
$232$	0	0
$233$	79756.4	1.46911	0.734554	−	0.678550i	$-0.237391\pi$
$233$	79756.4	1.46911	0.734554	+	0.678550i	$0.237391\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3877.08	−0.0690253
$238$	0	0
$239$	− 9240.62i	− 0.161773i	−0.996723	−	0.0808864i	$-0.974225\pi$
$239$	− 9240.62i	− 0.161773i	0.996723	−	0.0808864i	$-0.0257751\pi$
$240$	0	0
$241$	−40469.4	−0.696775	−0.348387	−	0.937351i	$-0.613271\pi$
$241$	−40469.4	−0.696775	−0.348387	+	0.937351i	$0.613271\pi$
$242$	0	0
$243$	− 3788.00i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 22065.0i	− 0.361668i
$248$	0	0
$249$	−49426.9	−0.797195
$250$	0	0
$251$	− 88508.8i	− 1.40488i	−0.711743	−	0.702440i	$-0.752094\pi$
$251$	− 88508.8i	− 1.40488i	0.711743	−	0.702440i	$-0.247906\pi$
$252$	0	0
$253$	27218.5	0.425229
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−105642.	−1.59945	−0.799725	−	0.600366i	$-0.795022\pi$
$257$	−105642.	−1.59945	−0.799725	+	0.600366i	$0.795022\pi$
$258$	0	0
$259$	− 24107.7i	− 0.359382i
$260$	0	0
$261$	−29427.6	−0.431990
$262$	0	0
$263$	− 3704.44i	− 0.0535563i	−0.999641	−	0.0267782i	$-0.991475\pi$
$263$	− 3704.44i	− 0.0535563i	0.999641	−	0.0267782i	$-0.00852478\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	59006.2i	0.827704i
$268$	0	0
$269$	−83661.4	−1.15617	−0.578084	−	0.815977i	$-0.696199\pi$
$269$	−83661.4	−1.15617	−0.578084	+	0.815977i	$0.696199\pi$
$270$	0	0
$271$	− 114123.i	− 1.55394i	−0.629538	−	0.776970i	$-0.716756\pi$
$271$	− 114123.i	− 1.55394i	0.629538	−	0.776970i	$-0.283244\pi$
$272$	0	0
$273$	11893.3	0.159580
$274$	0	0
$275$	0	0
$276$	0	0
$277$	113995.	1.48568	0.742842	−	0.669467i	$-0.233477\pi$
$277$	113995.	1.48568	0.742842	+	0.669467i	$0.233477\pi$
$278$	0	0
$279$	− 43373.0i	− 0.557200i
$280$	0	0
$281$	94716.2	1.19953	0.599766	−	0.800175i	$-0.295260\pi$
$281$	94716.2	1.19953	0.599766	+	0.800175i	$0.295260\pi$
$282$	0	0
$283$	− 41774.6i	− 0.521602i	−0.965393	−	0.260801i	$-0.916013\pi$
$283$	− 41774.6i	− 0.521602i	0.965393	−	0.260801i	$-0.0839867\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 163454.i	− 1.98441i
$288$	0	0
$289$	−56576.8	−0.677396
$290$	0	0
$291$	− 14081.8i	− 0.166292i
$292$	0	0
$293$	10439.9	0.121608	0.0608041	−	0.998150i	$-0.480634\pi$
$293$	10439.9	0.121608	0.0608041	+	0.998150i	$0.480634\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	11699.7	0.132636
$298$	0	0
$299$	12076.4i	0.135081i
$300$	0	0
$301$	−736.310	−0.00812695
$302$	0	0
$303$	71817.6i	0.782251i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 55507.0i	− 0.588940i	−0.955661	−	0.294470i	$-0.904857\pi$
$307$	− 55507.0i	− 0.588940i	0.955661	−	0.294470i	$-0.0951432\pi$
$308$	0	0
$309$	−7144.81	−0.0748297
$310$	0	0
$311$	611.666i	0.00632402i	0.999995	+	0.00316201i	$0.00100650\pi$
$311$	611.666i	0.00632402i	−0.999995	+	0.00316201i	$0.998993\pi$
$312$	0	0
$313$	147215.	1.50267	0.751333	−	0.659924i	$-0.229411\pi$
$313$	147215.	1.50267	0.751333	+	0.659924i	$0.229411\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−93749.3	−0.932931	−0.466466	−	0.884539i	$-0.654473\pi$
$317$	−93749.3	−0.932931	−0.466466	+	0.884539i	$0.654473\pi$
$318$	0	0
$319$	− 90890.7i	− 0.893178i
$320$	0	0
$321$	−100820.	−0.978443
$322$	0	0
$323$	97889.1i	0.938273i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4985.93i	0.0466284i
$328$	0	0
$329$	−100264.	−0.926304
$330$	0	0
$331$	90113.4i	0.822495i	0.911524	+	0.411248i	$0.134907\pi$
$331$	90113.4i	0.822495i	−0.911524	+	0.411248i	$0.865093\pi$
$332$	0	0
$333$	10522.1	0.0948883
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−67180.6	−0.591540	−0.295770	−	0.955259i	$-0.595576\pi$
$337$	−67180.6	−0.591540	−0.295770	+	0.955259i	$0.595576\pi$
$338$	0	0
$339$	78215.3i	0.680601i
$340$	0	0
$341$	133963.	1.15206
$342$	0	0
$343$	60325.9i	0.512762i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 187996.i	− 1.56131i	−0.624960	−	0.780657i	$-0.714885\pi$
$347$	− 187996.i	− 1.56131i	0.624960	−	0.780657i	$-0.285115\pi$
$348$	0	0
$349$	−180728.	−1.48380	−0.741900	−	0.670510i	$-0.766075\pi$
$349$	−180728.	−1.48380	−0.741900	+	0.670510i	$0.766075\pi$
$350$	0	0
$351$	5190.96i	0.0421340i
$352$	0	0
$353$	95669.5	0.767758	0.383879	−	0.923383i	$-0.374588\pi$
$353$	95669.5	0.767758	0.383879	+	0.923383i	$0.374588\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−52763.5	−0.413997
$358$	0	0
$359$	79558.5i	0.617302i	0.951175	+	0.308651i	$0.0998775\pi$
$359$	79558.5i	0.617302i	−0.951175	+	0.308651i	$0.900123\pi$
$360$	0	0
$361$	−225313.	−1.72891
$362$	0	0
$363$	− 39940.9i	− 0.303113i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	228720.i	1.69814i	0.528283	+	0.849068i	$0.322836\pi$
$367$	228720.i	1.69814i	−0.528283	+	0.849068i	$0.677164\pi$
$368$	0	0
$369$	71341.0	0.523946
$370$	0	0
$371$	101887.i	0.740239i
$372$	0	0
$373$	−19451.7	−0.139811	−0.0699054	−	0.997554i	$-0.522270\pi$
$373$	−19451.7	−0.139811	−0.0699054	+	0.997554i	$0.522270\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	40326.7	0.283733
$378$	0	0
$379$	− 82749.6i	− 0.576086i	−0.957617	−	0.288043i	$-0.906995\pi$
$379$	− 82749.6i	− 0.576086i	0.957617	−	0.288043i	$-0.0930046\pi$
$380$	0	0
$381$	−83695.1	−0.576568
$382$	0	0
$383$	− 240321.i	− 1.63830i	−0.573578	−	0.819151i	$-0.694445\pi$
$383$	− 240321.i	− 1.63830i	0.573578	−	0.819151i	$-0.305555\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 321.370i	− 0.00214577i
$388$	0	0
$389$	−132179.	−0.873499	−0.436749	−	0.899583i	$-0.643870\pi$
$389$	−132179.	−0.873499	−0.436749	+	0.899583i	$0.643870\pi$
$390$	0	0
$391$	− 53575.7i	− 0.350441i
$392$	0	0
$393$	−73421.1	−0.475374
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−268305.	−1.70234	−0.851172	−	0.524887i	$-0.824108\pi$
$397$	−268305.	−1.70234	−0.851172	+	0.524887i	$0.824108\pi$
$398$	0	0
$399$	− 191691.i	− 1.20408i
$400$	0	0
$401$	−141049.	−0.877164	−0.438582	−	0.898691i	$-0.644519\pi$
$401$	−141049.	−0.877164	−0.438582	+	0.898691i	$0.644519\pi$
$402$	0	0
$403$	59437.1i	0.365971i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	32498.7i	0.196190i
$408$	0	0
$409$	136033.	0.813203	0.406602	−	0.913606i	$-0.366714\pi$
$409$	136033.	0.813203	0.406602	+	0.913606i	$0.366714\pi$
$410$	0	0
$411$	58816.5i	0.348189i
$412$	0	0
$413$	184015.	1.07883
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−99009.3	−0.569382
$418$	0	0
$419$	− 236816.i	− 1.34891i	−0.738315	−	0.674456i	$-0.764378\pi$
$419$	− 236816.i	− 1.34891i	0.738315	−	0.674456i	$-0.235622\pi$
$420$	0	0
$421$	150917.	0.851481	0.425740	−	0.904845i	$-0.360014\pi$
$421$	150917.	0.851481	0.425740	+	0.904845i	$0.360014\pi$
$422$	0	0
$423$	− 43761.3i	− 0.244574i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	76569.3i	0.419951i
$428$	0	0
$429$	−16032.9	−0.0871160
$430$	0	0
$431$	− 182164.i	− 0.980638i	−0.871543	−	0.490319i	$-0.836880\pi$
$431$	− 182164.i	− 0.980638i	0.871543	−	0.490319i	$-0.163120\pi$
$432$	0	0
$433$	−6063.92	−0.0323428	−0.0161714	−	0.999869i	$-0.505148\pi$
$433$	−6063.92	−0.0323428	−0.0161714	+	0.999869i	$0.505148\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	194642.	1.01923
$438$	0	0
$439$	− 352332.i	− 1.82820i	−0.405491	−	0.914099i	$-0.632900\pi$
$439$	− 352332.i	− 1.82820i	0.405491	−	0.914099i	$-0.367100\pi$
$440$	0	0
$441$	38497.1	0.197948
$442$	0	0
$443$	85412.4i	0.435225i	0.976035	+	0.217612i	$0.0698268\pi$
$443$	85412.4i	0.435225i	−0.976035	+	0.217612i	$0.930173\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	150104.i	0.751239i
$448$	0	0
$449$	253197.	1.25593	0.627965	−	0.778242i	$-0.283888\pi$
$449$	253197.	1.25593	0.627965	+	0.778242i	$0.283888\pi$
$450$	0	0
$451$	220346.i	1.08331i
$452$	0	0
$453$	−28803.7	−0.140363
$454$	0	0
$455$	0	0
$456$	0	0
$457$	131482.	0.629554	0.314777	−	0.949166i	$-0.398070\pi$
$457$	131482.	0.629554	0.314777	+	0.949166i	$0.398070\pi$
$458$	0	0
$459$	− 23029.2i	− 0.109308i
$460$	0	0
$461$	−118423.	−0.557232	−0.278616	−	0.960403i	$-0.589876\pi$
$461$	−118423.	−0.557232	−0.278616	+	0.960403i	$0.589876\pi$
$462$	0	0
$463$	− 77570.0i	− 0.361853i	−0.983497	−	0.180926i	$-0.942090\pi$
$463$	− 77570.0i	− 0.361853i	0.983497	−	0.180926i	$-0.0579096\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 151100.i	− 0.692838i	−0.938080	−	0.346419i	$-0.887398\pi$
$467$	− 151100.i	− 0.692838i	0.938080	−	0.346419i	$-0.112602\pi$
$468$	0	0
$469$	30452.1	0.138443
$470$	0	0
$471$	207299.i	0.934450i
$472$	0	0
$473$	992.592	0.00443658
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−44469.8	−0.195447
$478$	0	0
$479$	453553.i	1.97677i	0.151961	+	0.988387i	$0.451441\pi$
$479$	453553.i	1.97677i	−0.151961	+	0.988387i	$0.548559\pi$
$480$	0	0
$481$	−14419.1	−0.0623231
$482$	0	0
$483$	104915.i	0.449720i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 143695.i	− 0.605875i	−0.953010	−	0.302938i	$-0.902033\pi$
$487$	− 143695.i	− 0.605875i	0.953010	−	0.302938i	$-0.0979674\pi$
$488$	0	0
$489$	67022.9	0.280289
$490$	0	0
$491$	− 179466.i	− 0.744420i	−0.928149	−	0.372210i	$-0.878600\pi$
$491$	− 179466.i	− 0.744420i	0.928149	−	0.372210i	$-0.121400\pi$
$492$	0	0
$493$	−178905.	−0.736087
$494$	0	0
$495$	0	0
$496$	0	0
$497$	177552.	0.718806
$498$	0	0
$499$	307441.i	1.23470i	0.786690	+	0.617348i	$0.211793\pi$
$499$	307441.i	1.23470i	−0.786690	+	0.617348i	$0.788207\pi$
$500$	0	0
$501$	198132.	0.789366
$502$	0	0
$503$	− 191607.i	− 0.757313i	−0.925537	−	0.378657i	$-0.876386\pi$
$503$	− 191607.i	− 0.757313i	0.925537	−	0.378657i	$-0.123614\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	141294.i	0.549676i
$508$	0	0
$509$	−162446.	−0.627010	−0.313505	−	0.949587i	$-0.601503\pi$
$509$	−162446.	−0.627010	−0.313505	+	0.949587i	$0.601503\pi$
$510$	0	0
$511$	260389.i	0.997195i
$512$	0	0
$513$	83665.7	0.317916
$514$	0	0
$515$	0	0
$516$	0	0
$517$	135162.	0.505678
$518$	0	0
$519$	− 215734.i	− 0.800911i
$520$	0	0
$521$	−170445.	−0.627927	−0.313964	−	0.949435i	$-0.601657\pi$
$521$	−170445.	−0.627927	−0.313964	+	0.949435i	$0.601657\pi$
$522$	0	0
$523$	− 12730.8i	− 0.0465429i	−0.999729	−	0.0232714i	$-0.992592\pi$
$523$	− 12730.8i	− 0.0465429i	0.999729	−	0.0232714i	$-0.00740820\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 263687.i	− 0.949439i
$528$	0	0
$529$	173311.	0.619321
$530$	0	0
$531$	80315.4i	0.284846i
$532$	0	0
$533$	−97763.6	−0.344130
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−61162.5	−0.212098
$538$	0	0
$539$	118903.i	0.409276i
$540$	0	0
$541$	−91472.6	−0.312534	−0.156267	−	0.987715i	$-0.549946\pi$
$541$	−91472.6	−0.312534	−0.156267	+	0.987715i	$0.549946\pi$
$542$	0	0
$543$	186735.i	0.633324i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 40143.9i	− 0.134167i	−0.997747	−	0.0670834i	$-0.978631\pi$
$547$	− 40143.9i	− 0.134167i	0.997747	−	0.0670834i	$-0.0213694\pi$
$548$	0	0
$549$	−33419.4	−0.110880
$550$	0	0
$551$	− 649968.i	− 2.14086i
$552$	0	0
$553$	46157.5	0.150936
$554$	0	0
$555$	0	0
$556$	0	0
$557$	548703.	1.76859	0.884295	−	0.466928i	$-0.154639\pi$
$557$	548703.	1.76859	0.884295	+	0.466928i	$0.154639\pi$
$558$	0	0
$559$	440.396i	0.00140935i
$560$	0	0
$561$	71128.5	0.226005
$562$	0	0
$563$	− 390357.i	− 1.23153i	−0.787930	−	0.615765i	$-0.788847\pi$
$563$	− 390357.i	− 1.23153i	0.787930	−	0.615765i	$-0.211153\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	45096.9i	0.140275i
$568$	0	0
$569$	−345635.	−1.06756	−0.533781	−	0.845623i	$-0.679229\pi$
$569$	−345635.	−1.06756	−0.533781	+	0.845623i	$0.679229\pi$
$570$	0	0
$571$	179039.i	0.549130i	0.961569	+	0.274565i	$0.0885339\pi$
$571$	179039.i	0.549130i	−0.961569	+	0.274565i	$0.911466\pi$
$572$	0	0
$573$	20694.6	0.0630302
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−483243.	−1.45149	−0.725745	−	0.687964i	$-0.758505\pi$
$577$	−483243.	−1.45149	−0.725745	+	0.687964i	$0.758505\pi$
$578$	0	0
$579$	229253.i	0.683844i
$580$	0	0
$581$	588437.	1.74320
$582$	0	0
$583$	− 137350.i	− 0.404104i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 503384.i	− 1.46091i	−0.682961	−	0.730455i	$-0.739308\pi$
$587$	− 503384.i	− 1.46091i	0.682961	−	0.730455i	$-0.260692\pi$
$588$	0	0
$589$	957982.	2.76138
$590$	0	0
$591$	− 83250.9i	− 0.238349i
$592$	0	0
$593$	−364949.	−1.03782	−0.518910	−	0.854829i	$-0.673662\pi$
$593$	−364949.	−1.03782	−0.518910	+	0.854829i	$0.673662\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−92869.4	−0.260570
$598$	0	0
$599$	535191.i	1.49161i	0.666165	+	0.745805i	$0.267935\pi$
$599$	535191.i	1.49161i	−0.666165	+	0.745805i	$0.732065\pi$
$600$	0	0
$601$	350102.	0.969273	0.484637	−	0.874716i	$-0.338952\pi$
$601$	350102.	0.969273	0.484637	+	0.874716i	$0.338952\pi$
$602$	0	0
$603$	13291.1i	0.0365534i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	547130.i	1.48495i	0.669872	+	0.742477i	$0.266349\pi$
$607$	547130.i	1.48495i	−0.669872	+	0.742477i	$0.733651\pi$
$608$	0	0
$609$	350341.	0.944619
$610$	0	0
$611$	59969.2i	0.160637i
$612$	0	0
$613$	56676.1	0.150827	0.0754135	−	0.997152i	$-0.475972\pi$
$613$	56676.1	0.150827	0.0754135	+	0.997152i	$0.475972\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	702781.	1.84608	0.923039	−	0.384707i	$-0.125697\pi$
$617$	702781.	1.84608	0.923039	+	0.384707i	$0.125697\pi$
$618$	0	0
$619$	439023.i	1.14579i	0.819628	+	0.572896i	$0.194180\pi$
$619$	439023.i	1.14579i	−0.819628	+	0.572896i	$0.805820\pi$
$620$	0	0
$621$	−45791.1	−0.118740
$622$	0	0
$623$	− 702481.i	− 1.80992i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	258412.i	0.657321i
$628$	0	0
$629$	63969.1	0.161685
$630$	0	0
$631$	− 362287.i	− 0.909902i	−0.890516	−	0.454951i	$-0.849657\pi$
$631$	− 362287.i	− 0.909902i	0.890516	−	0.454951i	$-0.150343\pi$
$632$	0	0
$633$	56718.6	0.141553
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−52755.3	−0.130013
$638$	0	0
$639$	77494.2i	0.189787i
$640$	0	0
$641$	537150.	1.30731	0.653656	−	0.756791i	$-0.273234\pi$
$641$	537150.	1.30731	0.653656	+	0.756791i	$0.273234\pi$
$642$	0	0
$643$	504622.i	1.22052i	0.792202	+	0.610259i	$0.208935\pi$
$643$	504622.i	1.22052i	−0.792202	+	0.610259i	$0.791065\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	404053.i	0.965227i	0.875834	+	0.482613i	$0.160312\pi$
$647$	404053.i	0.965227i	−0.875834	+	0.482613i	$0.839688\pi$
$648$	0	0
$649$	−248064.	−0.588945
$650$	0	0
$651$	516365.i	1.21841i
$652$	0	0
$653$	−461059.	−1.08126	−0.540630	−	0.841260i	$-0.681814\pi$
$653$	−461059.	−1.08126	−0.540630	+	0.841260i	$0.681814\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−113649.	−0.263291
$658$	0	0
$659$	329198.i	0.758030i	0.925391	+	0.379015i	$0.123737\pi$
$659$	329198.i	0.758030i	−0.925391	+	0.379015i	$0.876263\pi$
$660$	0	0
$661$	−12874.2	−0.0294657	−0.0147329	−	0.999891i	$-0.504690\pi$
$661$	−12874.2	−0.0294657	−0.0147329	+	0.999891i	$0.504690\pi$
$662$	0	0
$663$	31558.5i	0.0717942i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	355734.i	0.799603i
$668$	0	0
$669$	415725.	0.928869
$670$	0	0
$671$	− 103220.i	− 0.229255i
$672$	0	0
$673$	−740449.	−1.63480	−0.817400	−	0.576070i	$-0.804586\pi$
$673$	−740449.	−1.63480	−0.817400	+	0.576070i	$0.804586\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	516307.	1.12650	0.563250	−	0.826287i	$-0.309551\pi$
$677$	516307.	1.12650	0.563250	+	0.826287i	$0.309551\pi$
$678$	0	0
$679$	167647.i	0.363626i
$680$	0	0
$681$	−325912.	−0.702759
$682$	0	0
$683$	569453.i	1.22072i	0.792124	+	0.610361i	$0.208975\pi$
$683$	569453.i	1.22072i	−0.792124	+	0.610361i	$0.791025\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	431886.i	0.915073i
$688$	0	0
$689$	60940.0	0.128370
$690$	0	0
$691$	383630.i	0.803446i	0.915761	+	0.401723i	$0.131588\pi$
$691$	383630.i	0.803446i	−0.915761	+	0.401723i	$0.868412\pi$
$692$	0	0
$693$	−139287.	−0.290032
$694$	0	0
$695$	0	0
$696$	0	0
$697$	433719.	0.892776
$698$	0	0
$699$	− 414427.i	− 0.848190i
$700$	0	0
$701$	−452921.	−0.921693	−0.460847	−	0.887480i	$-0.652454\pi$
$701$	−452921.	−0.921693	−0.460847	+	0.887480i	$0.652454\pi$
$702$	0	0
$703$	232402.i	0.470250i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 855004.i	− 1.71052i
$708$	0	0
$709$	302501.	0.601774	0.300887	−	0.953660i	$-0.402717\pi$
$709$	302501.	0.601774	0.300887	+	0.953660i	$0.402717\pi$
$710$	0	0
$711$	20145.9i	0.0398518i
$712$	0	0
$713$	−524313.	−1.03136
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−48015.7	−0.0933995
$718$	0	0
$719$	108427.i	0.209739i	0.994486	+	0.104870i	$0.0334425\pi$
$719$	108427.i	0.209739i	−0.994486	+	0.104870i	$0.966557\pi$
$720$	0	0
$721$	85060.5	0.163628
$722$	0	0
$723$	210285.i	0.402283i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	659371.i	1.24756i	0.781600	+	0.623780i	$0.214404\pi$
$727$	659371.i	1.24756i	−0.781600	+	0.623780i	$0.785596\pi$
$728$	0	0
$729$	−19683.0	−0.0370370
$730$	0	0
$731$	− 1953.77i	− 0.00365628i
$732$	0	0
$733$	−448465.	−0.834682	−0.417341	−	0.908750i	$-0.637038\pi$
$733$	−448465.	−0.834682	−0.417341	+	0.908750i	$0.637038\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−41051.3	−0.0755775
$738$	0	0
$739$	− 658790.i	− 1.20631i	−0.797625	−	0.603154i	$-0.793911\pi$
$739$	− 658790.i	− 1.20631i	0.797625	−	0.603154i	$-0.206089\pi$
$740$	0	0
$741$	−114653.	−0.208809
$742$	0	0
$743$	− 117235.i	− 0.212364i	−0.994347	−	0.106182i	$-0.966137\pi$
$743$	− 117235.i	− 0.212364i	0.994347	−	0.106182i	$-0.0338626\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	256830.i	0.460261i
$748$	0	0
$749$	1.20028e6	2.13953
$750$	0	0
$751$	19810.5i	0.0351249i	0.999846	+	0.0175624i	$0.00559058\pi$
$751$	19810.5i	0.0351249i	−0.999846	+	0.0175624i	$0.994409\pi$
$752$	0	0
$753$	−459905.	−0.811108
$754$	0	0
$755$	0	0
$756$	0	0
$757$	156311.	0.272770	0.136385	−	0.990656i	$-0.456452\pi$
$757$	156311.	0.272770	0.136385	+	0.990656i	$0.456452\pi$
$758$	0	0
$759$	− 141432.i	− 0.245506i
$760$	0	0
$761$	890693.	1.53801	0.769004	−	0.639244i	$-0.220753\pi$
$761$	890693.	1.53801	0.769004	+	0.639244i	$0.220753\pi$
$762$	0	0
$763$	− 59358.5i	− 0.101961i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 110062.i	− 0.187088i
$768$	0	0
$769$	1.10673e6	1.87150	0.935750	−	0.352664i	$-0.114724\pi$
$769$	1.10673e6	1.87150	0.935750	+	0.352664i	$0.114724\pi$
$770$	0	0
$771$	548932.i	0.923443i
$772$	0	0
$773$	866276.	1.44976	0.724882	−	0.688873i	$-0.241894\pi$
$773$	866276.	1.44976	0.724882	+	0.688873i	$0.241894\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−125267.	−0.207490
$778$	0	0
$779$	1.57571e6i	2.59658i
$780$	0	0
$781$	−239351.	−0.392403
$782$	0	0
$783$	152910.i	0.249409i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	1.22977e6i	1.98552i	0.120095	+	0.992762i	$0.461680\pi$
$787$	1.22977e6i	1.98552i	−0.120095	+	0.992762i	$0.538320\pi$
$788$	0	0
$789$	−19248.8	−0.0309208
$790$	0	0
$791$	− 931170.i	− 1.48825i
$792$	0	0
$793$	45797.0	0.0728267
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−272215.	−0.428543	−0.214272	−	0.976774i	$-0.568738\pi$
$797$	−272215.	−0.428543	−0.214272	+	0.976774i	$0.568738\pi$
$798$	0	0
$799$	− 266047.i	− 0.416740i
$800$	0	0
$801$	306605.	0.477875
$802$	0	0
$803$	− 351020.i	− 0.544378i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	434718.i	0.667514i
$808$	0	0
$809$	388984.	0.594340	0.297170	−	0.954825i	$-0.403957\pi$
$809$	388984.	0.594340	0.297170	+	0.954825i	$0.403957\pi$
$810$	0	0
$811$	1.06377e6i	1.61736i	0.588248	+	0.808681i	$0.299818\pi$
$811$	1.06377e6i	1.61736i	−0.588248	+	0.808681i	$0.700182\pi$
$812$	0	0
$813$	−593000.	−0.897167
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7098.12	0.0106341
$818$	0	0
$819$	− 61799.4i	− 0.0921333i
$820$	0	0
$821$	73807.8	0.109500	0.0547502	−	0.998500i	$-0.482564\pi$
$821$	73807.8	0.109500	0.0547502	+	0.998500i	$0.482564\pi$
$822$	0	0
$823$	− 1.15603e6i	− 1.70675i	−0.521293	−	0.853377i	$-0.674550\pi$
$823$	− 1.15603e6i	− 1.70675i	0.521293	−	0.853377i	$-0.325450\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1.06525e6i	− 1.55755i	−0.627304	−	0.778775i	$-0.715842\pi$
$827$	− 1.06525e6i	− 1.55755i	0.627304	−	0.778775i	$-0.284158\pi$
$828$	0	0
$829$	−742002.	−1.07968	−0.539841	−	0.841767i	$-0.681516\pi$
$829$	−742002.	−1.07968	−0.539841	+	0.841767i	$0.681516\pi$
$830$	0	0
$831$	− 592336.i	− 0.857760i
$832$	0	0
$833$	234044.	0.337293
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−225373.	−0.321699
$838$	0	0
$839$	167829.i	0.238420i	0.992869	+	0.119210i	$0.0380361\pi$
$839$	167829.i	0.238420i	−0.992869	+	0.119210i	$0.961964\pi$
$840$	0	0
$841$	480622.	0.679535
$842$	0	0
$843$	− 492160.i	− 0.692550i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	475505.i	0.662808i
$848$	0	0
$849$	−217067.	−0.301147
$850$	0	0
$851$	− 127196.i	− 0.175636i
$852$	0	0
$853$	−629066.	−0.864566	−0.432283	−	0.901738i	$-0.642292\pi$
$853$	−629066.	−0.864566	−0.432283	+	0.901738i	$0.642292\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	941559.	1.28199	0.640997	−	0.767544i	$-0.278521\pi$
$857$	941559.	1.28199	0.640997	+	0.767544i	$0.278521\pi$
$858$	0	0
$859$	452274.i	0.612936i	0.951881	+	0.306468i	$0.0991473\pi$
$859$	452274.i	0.612936i	−0.951881	+	0.306468i	$0.900853\pi$
$860$	0	0
$861$	−849330.	−1.14570
$862$	0	0
$863$	− 484361.i	− 0.650350i	−0.945654	−	0.325175i	$-0.894577\pi$
$863$	− 484361.i	− 0.650350i	0.945654	−	0.325175i	$-0.105423\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	293982.i	0.391095i
$868$	0	0
$869$	−62223.2	−0.0823972
$870$	0	0
$871$	− 18213.8i	− 0.0240084i
$872$	0	0
$873$	−73171.1	−0.0960088
$874$	0	0
$875$	0	0
$876$	0	0
$877$	588731.	0.765451	0.382726	−	0.923862i	$-0.374985\pi$
$877$	588731.	0.765451	0.382726	+	0.923862i	$0.374985\pi$
$878$	0	0
$879$	− 54247.5i	− 0.0702105i
$880$	0	0
$881$	1.25076e6	1.61147	0.805733	−	0.592279i	$-0.201772\pi$
$881$	1.25076e6	1.61147	0.805733	+	0.592279i	$0.201772\pi$
$882$	0	0
$883$	837604.i	1.07428i	0.843493	+	0.537140i	$0.180495\pi$
$883$	837604.i	1.07428i	−0.843493	+	0.537140i	$0.819505\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 506950.i	− 0.644344i	−0.946681	−	0.322172i	$-0.895587\pi$
$887$	− 506950.i	− 0.644344i	0.946681	−	0.322172i	$-0.104413\pi$
$888$	0	0
$889$	996408.	1.26076
$890$	0	0
$891$	− 60793.4i	− 0.0765775i
$892$	0	0
$893$	966558.	1.21206
$894$	0	0
$895$	0	0
$896$	0	0
$897$	62750.8	0.0779891
$898$	0	0
$899$	1.75084e6i	2.16634i
$900$	0	0
$901$	−270355.	−0.333031
$902$	0	0
$903$	3825.98i	0.00469210i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.18958e6i	− 1.44604i	−0.690829	−	0.723018i	$-0.742754\pi$
$907$	− 1.18958e6i	− 1.44604i	0.690829	−	0.723018i	$-0.257246\pi$
$908$	0	0
$909$	373175.	0.451633
$910$	0	0
$911$	175647.i	0.211644i	0.994385	+	0.105822i	$0.0337473\pi$
$911$	175647.i	0.211644i	−0.994385	+	0.105822i	$0.966253\pi$
$912$	0	0
$913$	−793250.	−0.951631
$914$	0	0
$915$	0	0
$916$	0	0
$917$	874094.	1.03949
$918$	0	0
$919$	− 684071.i	− 0.809973i	−0.914323	−	0.404986i	$-0.867276\pi$
$919$	− 684071.i	− 0.809973i	0.914323	−	0.404986i	$-0.132724\pi$
$920$	0	0
$921$	−288423.	−0.340025
$922$	0	0
$923$	− 106196.i	− 0.124653i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	37125.5i	0.0432029i
$928$	0	0
$929$	−1.28454e6	−1.48839	−0.744195	−	0.667963i	$-0.767167\pi$
$929$	−1.28454e6	−1.48839	−0.744195	+	0.667963i	$0.767167\pi$
$930$	0	0
$931$	850288.i	0.980995i
$932$	0	0
$933$	3178.31	0.00365118
$934$	0	0
$935$	0	0
$936$	0	0
$937$	542198.	0.617559	0.308780	−	0.951134i	$-0.400079\pi$
$937$	542198.	0.617559	0.308780	+	0.951134i	$0.400079\pi$
$938$	0	0
$939$	− 764950.i	− 0.867564i
$940$	0	0
$941$	1.14252e6	1.29029	0.645144	−	0.764061i	$-0.276798\pi$
$941$	1.14252e6	1.29029	0.645144	+	0.764061i	$0.276798\pi$
$942$	0	0
$943$	− 862404.i	− 0.969812i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.54481e6i	− 1.72256i	−0.508129	−	0.861281i	$-0.669663\pi$
$947$	− 1.54481e6i	− 1.72256i	0.508129	−	0.861281i	$-0.330337\pi$
$948$	0	0
$949$	155742.	0.172931
$950$	0	0
$951$	487136.i	0.538628i
$952$	0	0
$953$	772945.	0.851065	0.425532	−	0.904943i	$-0.360087\pi$
$953$	772945.	0.851065	0.425532	+	0.904943i	$0.360087\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−472282.	−0.515677
$958$	0	0
$959$	− 700223.i	− 0.761376i
$960$	0	0
$961$	−1.65702e6	−1.79424
$962$	0	0
$963$	523875.i	0.564904i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 44995.2i	− 0.0481186i	−0.999711	−	0.0240593i	$-0.992341\pi$
$967$	− 44995.2i	− 0.0481186i	0.999711	−	0.0240593i	$-0.00765906\pi$
$968$	0	0
$969$	508647.	0.541712
$970$	0	0
$971$	1.02778e6i	1.09008i	0.838408	+	0.545042i	$0.183486\pi$
$971$	1.02778e6i	1.09008i	−0.838408	+	0.545042i	$0.816514\pi$
$972$	0	0
$973$	1.17873e6	1.24505
$974$	0	0
$975$	0	0
$976$	0	0
$977$	48136.8	0.0504299	0.0252149	−	0.999682i	$-0.491973\pi$
$977$	48136.8	0.0504299	0.0252149	+	0.999682i	$0.491973\pi$
$978$	0	0
$979$	946989.i	0.988051i
$980$	0	0
$981$	25907.6	0.0269209
$982$	0	0
$983$	− 707744.i	− 0.732435i	−0.930529	−	0.366218i	$-0.880653\pi$
$983$	− 707744.i	− 0.732435i	0.930529	−	0.366218i	$-0.119347\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	520987.i	0.534802i
$988$	0	0
$989$	−3884.87	−0.00397177
$990$	0	0
$991$	− 1.67192e6i	− 1.70242i	−0.524822	−	0.851212i	$-0.675868\pi$
$991$	− 1.67192e6i	− 1.70242i	0.524822	−	0.851212i	$-0.324132\pi$
$992$	0	0
$993$	468243.	0.474868
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.62054e6	−1.63031	−0.815153	−	0.579246i	$-0.803347\pi$
$997$	−1.62054e6	−1.63031	−0.815153	+	0.579246i	$0.803347\pi$
$998$	0	0
$999$	− 54674.3i	− 0.0547838i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1200.5.e.f.751.2	yes	4
4.3	odd	2	inner	1200.5.e.f.751.3	yes	4
5.2	odd	4		1200.5.j.d.799.1		8
5.3	odd	4		1200.5.j.d.799.7		8
5.4	even	2		1200.5.e.e.751.3	yes	4
20.3	even	4		1200.5.j.d.799.2		8
20.7	even	4		1200.5.j.d.799.8		8
20.19	odd	2		1200.5.e.e.751.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1200.5.e.e.751.2	✓	4	20.19	odd	2
1200.5.e.e.751.3	yes	4	5.4	even	2
1200.5.e.f.751.2	yes	4	1.1	even	1	trivial
1200.5.e.f.751.3	yes	4	4.3	odd	2	inner
1200.5.j.d.799.1		8	5.2	odd	4
1200.5.j.d.799.2		8	20.3	even	4
1200.5.j.d.799.7		8	5.3	odd	4
1200.5.j.d.799.8		8	20.7	even	4

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:	\( N \)	\(=\)	\( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	1200.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.19615i q^{3} +61.8613i q^{7} -27.0000 q^{9} +O(q^{10})\)	\(q-5.19615i q^{3} +61.8613i q^{7} -27.0000 q^{9} -83.3929i q^{11} +37.0000 q^{13} -164.147 q^{17} -596.351i q^{19} +321.441 q^{21} +326.389i q^{23} +140.296i q^{27} +1089.91 q^{29} +1606.41i q^{31} -433.322 q^{33} -389.706 q^{37} -192.258i q^{39} -2642.26 q^{41} +11.9026i q^{43} +1620.79i q^{47} -1425.82 q^{49} +852.932i q^{51} +1647.03 q^{53} -3098.73 q^{57} -2974.65i q^{59} +1237.76 q^{61} -1670.25i q^{63} -492.264i q^{67} +1695.97 q^{69} -2870.16i q^{71} +4209.23 q^{73} +5158.79 q^{77} -746.145i q^{79} +729.000 q^{81} -9512.21i q^{83} -5663.34i q^{87} -11355.8 q^{89} +2288.87i q^{91} +8347.14 q^{93} +2710.04 q^{97} +2251.61i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 108 q^{9}+O(q^{10}) \) 4 * q - 108 * q^9	\( 4 q - 108 q^{9} + 148 q^{13} - 384 q^{17} + 468 q^{21} + 816 q^{29} + 720 q^{33} - 2104 q^{37} - 2664 q^{41} + 1384 q^{49} + 4680 q^{53} - 1764 q^{57} - 7588 q^{61} - 576 q^{69} + 10840 q^{73} + 15456 q^{77} + 2916 q^{81} - 23616 q^{89} + 14580 q^{93} - 20780 q^{97}+O(q^{100}) \) 4 * q - 108 * q^9 + 148 * q^13 - 384 * q^17 + 468 * q^21 + 816 * q^29 + 720 * q^33 - 2104 * q^37 - 2664 * q^41 + 1384 * q^49 + 4680 * q^53 - 1764 * q^57 - 7588 * q^61 - 576 * q^69 + 10840 * q^73 + 15456 * q^77 + 2916 * q^81 - 23616 * q^89 + 14580 * q^93 - 20780 * q^97

Introduction

L-functions

Modular forms

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