LMFDB - Embedded newform 1800.4.a.br.1.1

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.121909.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 87x + 270 $ x^3 - x^2 - 87*x + 270
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.43305$ 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)

$f(q)$	$=$	$q-23.6993 q^{7} +O(q^{10})$	$q-23.6993 q^{7} +9.73221 q^{11} -71.8629 q^{13} +135.994 q^{17} -49.1637 q^{19} +191.326 q^{23} +45.6004 q^{29} +32.1689 q^{31} +218.523 q^{37} -394.123 q^{41} +396.484 q^{43} -33.3221 q^{47} +218.655 q^{49} +150.659 q^{53} +396.510 q^{59} -505.990 q^{61} -552.405 q^{67} -756.113 q^{71} -579.853 q^{73} -230.646 q^{77} +18.6652 q^{79} -1310.63 q^{83} -541.330 q^{89} +1703.10 q^{91} -16.7303 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 9 q^{7}+O(q^{10}) $ 3 * q - 9 * q^7	$ 3 q - 9 q^{7} - 8 q^{11} - 17 q^{13} + 48 q^{17} - 11 q^{19} + 40 q^{23} + 59 q^{31} + 10 q^{37} - 400 q^{41} + 159 q^{43} + 272 q^{47} + 110 q^{49} - 256 q^{53} - 176 q^{59} - 375 q^{61} + 143 q^{67} - 1288 q^{71} - 398 q^{73} + 736 q^{77} - 292 q^{79} - 1424 q^{83} - 928 q^{89} + 851 q^{91} - 697 q^{97}+O(q^{100}) $ 3 * q - 9 * q^7 - 8 * q^11 - 17 * q^13 + 48 * q^17 - 11 * q^19 + 40 * q^23 + 59 * q^31 + 10 * q^37 - 400 * q^41 + 159 * q^43 + 272 * q^47 + 110 * q^49 - 256 * q^53 - 176 * q^59 - 375 * q^61 + 143 * q^67 - 1288 * q^71 - 398 * q^73 + 736 * q^77 - 292 * q^79 - 1424 * q^83 - 928 * q^89 + 851 * q^91 - 697 * q^97

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
$7$	−23.6993	−1.27964	−0.639820	−	0.768525i	$-0.720991\pi$
$7$	−23.6993	−1.27964	−0.639820	+	0.768525i	$0.720991\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	9.73221	0.266761	0.133381	−	0.991065i	$-0.457417\pi$
$11$	9.73221	0.266761	0.133381	+	0.991065i	$0.457417\pi$
$12$	0	0
$13$	−71.8629	−1.53317	−0.766584	−	0.642144i	$-0.778045\pi$
$13$	−71.8629	−1.53317	−0.766584	+	0.642144i	$0.778045\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	135.994	1.94019	0.970097	−	0.242717i	$-0.0780385\pi$
$17$	135.994	1.94019	0.970097	+	0.242717i	$0.0780385\pi$
$18$	0	0
$19$	−49.1637	−0.593627	−0.296814	−	0.954935i	$-0.595924\pi$
$19$	−49.1637	−0.593627	−0.296814	+	0.954935i	$0.595924\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	191.326	1.73453	0.867267	−	0.497843i	$-0.165875\pi$
$23$	191.326	1.73453	0.867267	+	0.497843i	$0.165875\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	45.6004	0.291992	0.145996	−	0.989285i	$-0.453361\pi$
$29$	45.6004	0.291992	0.145996	+	0.989285i	$0.453361\pi$
$30$	0	0
$31$	32.1689	0.186378	0.0931888	−	0.995648i	$-0.470294\pi$
$31$	32.1689	0.186378	0.0931888	+	0.995648i	$0.470294\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	218.523	0.970944	0.485472	−	0.874252i	$-0.338648\pi$
$37$	218.523	0.970944	0.485472	+	0.874252i	$0.338648\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−394.123	−1.50126	−0.750631	−	0.660722i	$-0.770250\pi$
$41$	−394.123	−1.50126	−0.750631	+	0.660722i	$0.770250\pi$
$42$	0	0
$43$	396.484	1.40612	0.703060	−	0.711130i	$-0.251816\pi$
$43$	396.484	1.40612	0.703060	+	0.711130i	$0.251816\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−33.3221	−0.103416	−0.0517078	−	0.998662i	$-0.516466\pi$
$47$	−33.3221	−0.103416	−0.0517078	+	0.998662i	$0.516466\pi$
$48$	0	0
$49$	218.655	0.637477
$50$	0	0
$51$	0	0
$52$	0	0
$53$	150.659	0.390464	0.195232	−	0.980757i	$-0.437454\pi$
$53$	150.659	0.390464	0.195232	+	0.980757i	$0.437454\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	396.510	0.874936	0.437468	−	0.899234i	$-0.355875\pi$
$59$	396.510	0.874936	0.437468	+	0.899234i	$0.355875\pi$
$60$	0	0
$61$	−505.990	−1.06206	−0.531028	−	0.847354i	$-0.678194\pi$
$61$	−505.990	−1.06206	−0.531028	+	0.847354i	$0.678194\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−552.405	−1.00727	−0.503634	−	0.863917i	$-0.668004\pi$
$67$	−552.405	−1.00727	−0.503634	+	0.863917i	$0.668004\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−756.113	−1.26386	−0.631930	−	0.775025i	$-0.717737\pi$
$71$	−756.113	−1.26386	−0.631930	+	0.775025i	$0.717737\pi$
$72$	0	0
$73$	−579.853	−0.929681	−0.464840	−	0.885395i	$-0.653888\pi$
$73$	−579.853	−0.929681	−0.464840	+	0.885395i	$0.653888\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−230.646	−0.341358
$78$	0	0
$79$	18.6652	0.0265823	0.0132911	−	0.999912i	$-0.495769\pi$
$79$	18.6652	0.0265823	0.0132911	+	0.999912i	$0.495769\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1310.63	−1.73325	−0.866627	−	0.498957i	$-0.833716\pi$
$83$	−1310.63	−1.73325	−0.866627	+	0.498957i	$0.833716\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−541.330	−0.644729	−0.322365	−	0.946616i	$-0.604478\pi$
$89$	−541.330	−0.644729	−0.322365	+	0.946616i	$0.604478\pi$
$90$	0	0
$91$	1703.10	1.96190
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−16.7303	−0.0175124	−0.00875620	−	0.999962i	$-0.502787\pi$
$97$	−16.7303	−0.0175124	−0.00875620	+	0.999962i	$0.502787\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	705.796	0.695340	0.347670	−	0.937617i	$-0.386973\pi$
$101$	705.796	0.695340	0.347670	+	0.937617i	$0.386973\pi$
$102$	0	0
$103$	−442.096	−0.422922	−0.211461	−	0.977386i	$-0.567822\pi$
$103$	−442.096	−0.422922	−0.211461	+	0.977386i	$0.567822\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1555.28	1.40518	0.702592	−	0.711593i	$-0.252026\pi$
$107$	1555.28	1.40518	0.702592	+	0.711593i	$0.252026\pi$
$108$	0	0
$109$	−1440.59	−1.26591	−0.632954	−	0.774190i	$-0.718158\pi$
$109$	−1440.59	−1.26591	−0.632954	+	0.774190i	$0.718158\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1623.99	−1.35197	−0.675983	−	0.736917i	$-0.736281\pi$
$113$	−1623.99	−1.35197	−0.675983	+	0.736917i	$0.736281\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3222.95	−2.48275
$120$	0	0
$121$	−1236.28	−0.928839
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1925.52	1.34537	0.672687	−	0.739927i	$-0.265140\pi$
$127$	1925.52	1.34537	0.672687	+	0.739927i	$0.265140\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−738.854	−0.492778	−0.246389	−	0.969171i	$-0.579244\pi$
$131$	−738.854	−0.492778	−0.246389	+	0.969171i	$0.579244\pi$
$132$	0	0
$133$	1165.14	0.759629
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1773.32	1.10587	0.552937	−	0.833223i	$-0.313507\pi$
$137$	1773.32	1.10587	0.552937	+	0.833223i	$0.313507\pi$
$138$	0	0
$139$	1553.23	0.947795	0.473898	−	0.880580i	$-0.342847\pi$
$139$	1553.23	0.947795	0.473898	+	0.880580i	$0.342847\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−699.385	−0.408990
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−868.799	−0.477683	−0.238842	−	0.971059i	$-0.576768\pi$
$149$	−868.799	−0.477683	−0.238842	+	0.971059i	$0.576768\pi$
$150$	0	0
$151$	−1827.37	−0.984828	−0.492414	−	0.870361i	$-0.663885\pi$
$151$	−1827.37	−0.984828	−0.492414	+	0.870361i	$0.663885\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2692.49	−1.36869	−0.684345	−	0.729158i	$-0.739912\pi$
$157$	−2692.49	−1.36869	−0.684345	+	0.729158i	$0.739912\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4534.29	−2.21958
$162$	0	0
$163$	1456.38	0.699833	0.349917	−	0.936781i	$-0.386210\pi$
$163$	1456.38	0.699833	0.349917	+	0.936781i	$0.386210\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−665.911	−0.308561	−0.154281	−	0.988027i	$-0.549306\pi$
$167$	−665.911	−0.308561	−0.154281	+	0.988027i	$0.549306\pi$
$168$	0	0
$169$	2967.28	1.35061
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3091.89	−1.35880	−0.679400	−	0.733769i	$-0.737760\pi$
$173$	−3091.89	−1.35880	−0.679400	+	0.733769i	$0.737760\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2436.11	1.01722	0.508612	−	0.860996i	$-0.330159\pi$
$179$	2436.11	1.01722	0.508612	+	0.860996i	$0.330159\pi$
$180$	0	0
$181$	589.940	0.242264	0.121132	−	0.992636i	$-0.461347\pi$
$181$	589.940	0.242264	0.121132	+	0.992636i	$0.461347\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1323.52	0.517568
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3605.75	−1.36599	−0.682993	−	0.730425i	$-0.739322\pi$
$191$	−3605.75	−1.36599	−0.682993	+	0.730425i	$0.739322\pi$
$192$	0	0
$193$	−2468.64	−0.920707	−0.460354	−	0.887736i	$-0.652277\pi$
$193$	−2468.64	−0.920707	−0.460354	+	0.887736i	$0.652277\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2664.03	0.963473	0.481737	−	0.876316i	$-0.340006\pi$
$197$	2664.03	0.963473	0.481737	+	0.876316i	$0.340006\pi$
$198$	0	0
$199$	−5184.67	−1.84689	−0.923445	−	0.383730i	$-0.874639\pi$
$199$	−5184.67	−1.84689	−0.923445	+	0.383730i	$0.874639\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−1080.70	−0.373645
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−478.471	−0.158357
$210$	0	0
$211$	1788.06	0.583390	0.291695	−	0.956511i	$-0.405781\pi$
$211$	1788.06	0.583390	0.291695	+	0.956511i	$0.405781\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−762.379	−0.238496
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−9772.90	−2.97465
$222$	0	0
$223$	−2550.67	−0.765944	−0.382972	−	0.923760i	$-0.625099\pi$
$223$	−2550.67	−0.765944	−0.382972	+	0.923760i	$0.625099\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4764.65	1.39313	0.696566	−	0.717493i	$-0.254710\pi$
$227$	4764.65	1.39313	0.696566	+	0.717493i	$0.254710\pi$
$228$	0	0
$229$	563.956	0.162739	0.0813695	−	0.996684i	$-0.474071\pi$
$229$	563.956	0.162739	0.0813695	+	0.996684i	$0.474071\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2367.76	0.665738	0.332869	−	0.942973i	$-0.391983\pi$
$233$	2367.76	0.665738	0.332869	+	0.942973i	$0.391983\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1730.67	0.468401	0.234200	−	0.972188i	$-0.424753\pi$
$239$	1730.67	0.468401	0.234200	+	0.972188i	$0.424753\pi$
$240$	0	0
$241$	5219.37	1.39506	0.697530	−	0.716556i	$-0.254282\pi$
$241$	5219.37	1.39506	0.697530	+	0.716556i	$0.254282\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3533.05	0.910131
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7162.29	−1.80111	−0.900557	−	0.434737i	$-0.856841\pi$
$251$	−7162.29	−1.80111	−0.900557	+	0.434737i	$0.856841\pi$
$252$	0	0
$253$	1862.03	0.462706
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1957.29	0.475069	0.237534	−	0.971379i	$-0.423661\pi$
$257$	1957.29	0.475069	0.237534	+	0.971379i	$0.423661\pi$
$258$	0	0
$259$	−5178.83	−1.24246
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2126.00	−0.498458	−0.249229	−	0.968445i	$-0.580177\pi$
$263$	−2126.00	−0.498458	−0.249229	+	0.968445i	$0.580177\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4017.69	−0.910642	−0.455321	−	0.890327i	$-0.650475\pi$
$269$	−4017.69	−0.910642	−0.455321	+	0.890327i	$0.650475\pi$
$270$	0	0
$271$	−2917.11	−0.653882	−0.326941	−	0.945045i	$-0.606018\pi$
$271$	−2917.11	−0.653882	−0.326941	+	0.945045i	$0.606018\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−8080.94	−1.75284	−0.876420	−	0.481547i	$-0.840075\pi$
$277$	−8080.94	−1.75284	−0.876420	+	0.481547i	$0.840075\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3770.27	0.800410	0.400205	−	0.916426i	$-0.368939\pi$
$281$	3770.27	0.800410	0.400205	+	0.916426i	$0.368939\pi$
$282$	0	0
$283$	4073.28	0.855587	0.427794	−	0.903876i	$-0.359291\pi$
$283$	4073.28	0.855587	0.427794	+	0.903876i	$0.359291\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9340.43	1.92107
$288$	0	0
$289$	13581.3	2.76435
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3853.36	−0.768314	−0.384157	−	0.923268i	$-0.625508\pi$
$293$	−3853.36	−0.768314	−0.384157	+	0.923268i	$0.625508\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−13749.3	−2.65933
$300$	0	0
$301$	−9396.37	−1.79933
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5171.63	0.961434	0.480717	−	0.876876i	$-0.340376\pi$
$307$	5171.63	0.961434	0.480717	+	0.876876i	$0.340376\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−8102.83	−1.47739	−0.738696	−	0.674038i	$-0.764558\pi$
$311$	−8102.83	−1.47739	−0.738696	+	0.674038i	$0.764558\pi$
$312$	0	0
$313$	−1642.15	−0.296549	−0.148275	−	0.988946i	$-0.547372\pi$
$313$	−1642.15	−0.296549	−0.148275	+	0.988946i	$0.547372\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8487.78	1.50385	0.751926	−	0.659248i	$-0.229125\pi$
$317$	8487.78	1.50385	0.751926	+	0.659248i	$0.229125\pi$
$318$	0	0
$319$	443.793	0.0778922
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6685.95	−1.15175
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	789.709	0.132335
$330$	0	0
$331$	1224.18	0.203283	0.101642	−	0.994821i	$-0.467590\pi$
$331$	1224.18	0.203283	0.101642	+	0.994821i	$0.467590\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	689.609	0.111470	0.0557350	−	0.998446i	$-0.482250\pi$
$337$	689.609	0.111470	0.0557350	+	0.998446i	$0.482250\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	313.075	0.0497183
$342$	0	0
$343$	2946.89	0.463898
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−7806.29	−1.20768	−0.603838	−	0.797107i	$-0.706363\pi$
$347$	−7806.29	−1.20768	−0.603838	+	0.797107i	$0.706363\pi$
$348$	0	0
$349$	−8753.47	−1.34259	−0.671293	−	0.741192i	$-0.734261\pi$
$349$	−8753.47	−1.34259	−0.671293	+	0.741192i	$0.734261\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9063.79	−1.36662	−0.683310	−	0.730129i	$-0.739460\pi$
$353$	−9063.79	−1.36662	−0.683310	+	0.730129i	$0.739460\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2348.65	0.345284	0.172642	−	0.984985i	$-0.444770\pi$
$359$	2348.65	0.345284	0.172642	+	0.984985i	$0.444770\pi$
$360$	0	0
$361$	−4441.93	−0.647606
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7990.09	1.13646	0.568228	−	0.822871i	$-0.307629\pi$
$367$	7990.09	1.13646	0.568228	+	0.822871i	$0.307629\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3570.50	−0.499653
$372$	0	0
$373$	−2576.54	−0.357663	−0.178832	−	0.983880i	$-0.557232\pi$
$373$	−2576.54	−0.357663	−0.178832	+	0.983880i	$0.557232\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−3276.98	−0.447674
$378$	0	0
$379$	−1055.63	−0.143071	−0.0715357	−	0.997438i	$-0.522790\pi$
$379$	−1055.63	−0.143071	−0.0715357	+	0.997438i	$0.522790\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−13734.2	−1.83234	−0.916168	−	0.400795i	$-0.868734\pi$
$383$	−13734.2	−1.83234	−0.916168	+	0.400795i	$0.868734\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2216.92	−0.288952	−0.144476	−	0.989508i	$-0.546150\pi$
$389$	−2216.92	−0.288952	−0.144476	+	0.989508i	$0.546150\pi$
$390$	0	0
$391$	26019.2	3.36533
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−15180.5	−1.91911	−0.959556	−	0.281519i	$-0.909162\pi$
$397$	−15180.5	−1.91911	−0.959556	+	0.281519i	$0.909162\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14493.3	−1.80489	−0.902447	−	0.430802i	$-0.858231\pi$
$401$	−14493.3	−1.80489	−0.902447	+	0.430802i	$0.858231\pi$
$402$	0	0
$403$	−2311.75	−0.285748
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2126.71	0.259010
$408$	0	0
$409$	−7287.05	−0.880982	−0.440491	−	0.897757i	$-0.645196\pi$
$409$	−7287.05	−0.880982	−0.440491	+	0.897757i	$0.645196\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9397.00	−1.11960
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−99.8425	−0.0116411	−0.00582055	−	0.999983i	$-0.501853\pi$
$419$	−99.8425	−0.0116411	−0.00582055	+	0.999983i	$0.501853\pi$
$420$	0	0
$421$	−9407.33	−1.08904	−0.544519	−	0.838748i	$-0.683288\pi$
$421$	−9407.33	−1.08904	−0.544519	+	0.838748i	$0.683288\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	11991.6	1.35905
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12255.3	1.36964	0.684821	−	0.728711i	$-0.259880\pi$
$431$	12255.3	1.36964	0.684821	+	0.728711i	$0.259880\pi$
$432$	0	0
$433$	−15753.9	−1.74846	−0.874229	−	0.485514i	$-0.838633\pi$
$433$	−15753.9	−1.74846	−0.874229	+	0.485514i	$0.838633\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9406.30	−1.02967
$438$	0	0
$439$	1427.10	0.155152	0.0775758	−	0.996986i	$-0.475282\pi$
$439$	1427.10	0.155152	0.0775758	+	0.996986i	$0.475282\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12990.0	1.39317	0.696586	−	0.717473i	$-0.254701\pi$
$443$	12990.0	1.39317	0.696586	+	0.717473i	$0.254701\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10187.7	−1.07080	−0.535398	−	0.844600i	$-0.679838\pi$
$449$	−10187.7	−1.07080	−0.535398	+	0.844600i	$0.679838\pi$
$450$	0	0
$451$	−3835.69	−0.400478
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7083.80	−0.725090	−0.362545	−	0.931966i	$-0.618092\pi$
$457$	−7083.80	−0.725090	−0.362545	+	0.931966i	$0.618092\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−9638.61	−0.973785	−0.486893	−	0.873462i	$-0.661870\pi$
$461$	−9638.61	−0.973785	−0.486893	+	0.873462i	$0.661870\pi$
$462$	0	0
$463$	10092.2	1.01301	0.506505	−	0.862237i	$-0.330937\pi$
$463$	10092.2	1.01301	0.506505	+	0.862237i	$0.330937\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2824.60	0.279886	0.139943	−	0.990160i	$-0.455308\pi$
$467$	2824.60	0.279886	0.139943	+	0.990160i	$0.455308\pi$
$468$	0	0
$469$	13091.6	1.28894
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3858.66	0.375098
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	9224.52	0.879915	0.439957	−	0.898019i	$-0.354994\pi$
$479$	9224.52	0.879915	0.439957	+	0.898019i	$0.354994\pi$
$480$	0	0
$481$	−15703.7	−1.48862
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−10002.4	−0.930704	−0.465352	−	0.885126i	$-0.654072\pi$
$487$	−10002.4	−0.930704	−0.465352	+	0.885126i	$0.654072\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3410.78	−0.313496	−0.156748	−	0.987639i	$-0.550101\pi$
$491$	−3410.78	−0.313496	−0.156748	+	0.987639i	$0.550101\pi$
$492$	0	0
$493$	6201.36	0.566522
$494$	0	0
$495$	0	0
$496$	0	0
$497$	17919.3	1.61729
$498$	0	0
$499$	−2511.74	−0.225333	−0.112666	−	0.993633i	$-0.535939\pi$
$499$	−2511.74	−0.225333	−0.112666	+	0.993633i	$0.535939\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	6820.38	0.604584	0.302292	−	0.953215i	$-0.402248\pi$
$503$	6820.38	0.604584	0.302292	+	0.953215i	$0.402248\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4487.70	0.390793	0.195397	−	0.980724i	$-0.437401\pi$
$509$	4487.70	0.390793	0.195397	+	0.980724i	$0.437401\pi$
$510$	0	0
$511$	13742.1	1.18966
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−324.298	−0.0275872
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5525.03	0.464599	0.232299	−	0.972644i	$-0.425375\pi$
$521$	5525.03	0.464599	0.232299	+	0.972644i	$0.425375\pi$
$522$	0	0
$523$	17951.3	1.50087	0.750436	−	0.660943i	$-0.229843\pi$
$523$	17951.3	1.50087	0.750436	+	0.660943i	$0.229843\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4374.77	0.361609
$528$	0	0
$529$	24438.7	2.00861
$530$	0	0
$531$	0	0
$532$	0	0
$533$	28322.9	2.30169
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2127.99	0.170054
$540$	0	0
$541$	−4736.05	−0.376375	−0.188188	−	0.982133i	$-0.560261\pi$
$541$	−4736.05	−0.376375	−0.188188	+	0.982133i	$0.560261\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	23541.2	1.84013	0.920063	−	0.391771i	$-0.128138\pi$
$547$	23541.2	1.84013	0.920063	+	0.391771i	$0.128138\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2241.88	−0.173335
$552$	0	0
$553$	−442.351	−0.0340157
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5535.67	0.421102	0.210551	−	0.977583i	$-0.432474\pi$
$557$	5535.67	0.421102	0.210551	+	0.977583i	$0.432474\pi$
$558$	0	0
$559$	−28492.5	−2.15582
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7234.21	0.541537	0.270769	−	0.962644i	$-0.412722\pi$
$563$	7234.21	0.541537	0.270769	+	0.962644i	$0.412722\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	12822.1	0.944691	0.472346	−	0.881413i	$-0.343407\pi$
$569$	12822.1	0.944691	0.472346	+	0.881413i	$0.343407\pi$
$570$	0	0
$571$	2642.42	0.193663	0.0968317	−	0.995301i	$-0.469129\pi$
$571$	2642.42	0.193663	0.0968317	+	0.995301i	$0.469129\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−1505.71	−0.108637	−0.0543184	−	0.998524i	$-0.517299\pi$
$577$	−1505.71	−0.108637	−0.0543184	+	0.998524i	$0.517299\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	31060.9	2.21794
$582$	0	0
$583$	1466.24	0.104161
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5272.11	0.370704	0.185352	−	0.982672i	$-0.440657\pi$
$587$	5272.11	0.370704	0.185352	+	0.982672i	$0.440657\pi$
$588$	0	0
$589$	−1581.54	−0.110639
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−6428.60	−0.445179	−0.222589	−	0.974912i	$-0.571451\pi$
$593$	−6428.60	−0.445179	−0.222589	+	0.974912i	$0.571451\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−17105.8	−1.16682	−0.583408	−	0.812179i	$-0.698281\pi$
$599$	−17105.8	−1.16682	−0.583408	+	0.812179i	$0.698281\pi$
$600$	0	0
$601$	−3684.40	−0.250066	−0.125033	−	0.992153i	$-0.539904\pi$
$601$	−3684.40	−0.250066	−0.125033	+	0.992153i	$0.539904\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	5992.91	0.400733	0.200366	−	0.979721i	$-0.435787\pi$
$607$	5992.91	0.400733	0.200366	+	0.979721i	$0.435787\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2394.63	0.158553
$612$	0	0
$613$	−27808.2	−1.83224	−0.916121	−	0.400902i	$-0.868697\pi$
$613$	−27808.2	−1.83224	−0.916121	+	0.400902i	$0.868697\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22570.2	−1.47267	−0.736337	−	0.676615i	$-0.763446\pi$
$617$	−22570.2	−1.47267	−0.736337	+	0.676615i	$0.763446\pi$
$618$	0	0
$619$	−13319.9	−0.864900	−0.432450	−	0.901658i	$-0.642351\pi$
$619$	−13319.9	−0.864900	−0.432450	+	0.901658i	$0.642351\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12829.1	0.825021
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	29717.7	1.88382
$630$	0	0
$631$	12296.8	0.775797	0.387899	−	0.921702i	$-0.373201\pi$
$631$	12296.8	0.775797	0.387899	+	0.921702i	$0.373201\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−15713.2	−0.977360
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2558.68	0.157663	0.0788313	−	0.996888i	$-0.474881\pi$
$641$	2558.68	0.157663	0.0788313	+	0.996888i	$0.474881\pi$
$642$	0	0
$643$	−12248.9	−0.751244	−0.375622	−	0.926773i	$-0.622571\pi$
$643$	−12248.9	−0.751244	−0.375622	+	0.926773i	$0.622571\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−26518.7	−1.61137	−0.805687	−	0.592342i	$-0.798204\pi$
$647$	−26518.7	−1.61137	−0.805687	+	0.592342i	$0.798204\pi$
$648$	0	0
$649$	3858.92	0.233399
$650$	0	0
$651$	0	0
$652$	0	0
$653$	10023.7	0.600702	0.300351	−	0.953829i	$-0.402896\pi$
$653$	10023.7	0.600702	0.300351	+	0.953829i	$0.402896\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	21392.3	1.26453	0.632265	−	0.774752i	$-0.282125\pi$
$659$	21392.3	1.26453	0.632265	+	0.774752i	$0.282125\pi$
$660$	0	0
$661$	8862.91	0.521524	0.260762	−	0.965403i	$-0.416026\pi$
$661$	8862.91	0.521524	0.260762	+	0.965403i	$0.416026\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	8724.55	0.506471
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4924.41	−0.283315
$672$	0	0
$673$	654.413	0.0374826	0.0187413	−	0.999824i	$-0.494034\pi$
$673$	654.413	0.0374826	0.0187413	+	0.999824i	$0.494034\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−32757.9	−1.85966	−0.929828	−	0.367993i	$-0.880045\pi$
$677$	−32757.9	−1.85966	−0.929828	+	0.367993i	$0.880045\pi$
$678$	0	0
$679$	396.495	0.0224095
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−23977.6	−1.34330	−0.671652	−	0.740867i	$-0.734415\pi$
$683$	−23977.6	−1.34330	−0.671652	+	0.740867i	$0.734415\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−10826.8	−0.598647
$690$	0	0
$691$	−21665.9	−1.19278	−0.596390	−	0.802695i	$-0.703399\pi$
$691$	−21665.9	−1.19278	−0.596390	+	0.802695i	$0.703399\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−53598.3	−2.91274
$698$	0	0
$699$	0	0
$700$	0	0
$701$	16412.4	0.884289	0.442144	−	0.896944i	$-0.354218\pi$
$701$	16412.4	0.884289	0.442144	+	0.896944i	$0.354218\pi$
$702$	0	0
$703$	−10743.4	−0.576379
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−16726.8	−0.889785
$708$	0	0
$709$	27003.9	1.43040	0.715198	−	0.698922i	$-0.246336\pi$
$709$	27003.9	1.43040	0.715198	+	0.698922i	$0.246336\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6154.76	0.323278
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−704.799	−0.0365571	−0.0182786	−	0.999833i	$-0.505819\pi$
$719$	−704.799	−0.0365571	−0.0182786	+	0.999833i	$0.505819\pi$
$720$	0	0
$721$	10477.3	0.541188
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	21007.6	1.07170	0.535852	−	0.844312i	$-0.319991\pi$
$727$	21007.6	1.07170	0.535852	+	0.844312i	$0.319991\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	53919.3	2.72815
$732$	0	0
$733$	16455.2	0.829177	0.414589	−	0.910009i	$-0.363926\pi$
$733$	16455.2	0.829177	0.414589	+	0.910009i	$0.363926\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5376.12	−0.268700
$738$	0	0
$739$	−2024.26	−0.100762	−0.0503812	−	0.998730i	$-0.516044\pi$
$739$	−2024.26	−0.100762	−0.0503812	+	0.998730i	$0.516044\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	18314.6	0.904303	0.452151	−	0.891941i	$-0.350657\pi$
$743$	18314.6	0.904303	0.452151	+	0.891941i	$0.350657\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−36859.0	−1.79813
$750$	0	0
$751$	−17128.2	−0.832248	−0.416124	−	0.909308i	$-0.636612\pi$
$751$	−17128.2	−0.832248	−0.416124	+	0.909308i	$0.636612\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	21267.3	1.02110	0.510550	−	0.859848i	$-0.329442\pi$
$757$	21267.3	1.02110	0.510550	+	0.859848i	$0.329442\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	36412.1	1.73448	0.867240	−	0.497891i	$-0.165892\pi$
$761$	36412.1	1.73448	0.867240	+	0.497891i	$0.165892\pi$
$762$	0	0
$763$	34141.0	1.61991
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−28494.4	−1.34142
$768$	0	0
$769$	−14435.1	−0.676909	−0.338454	−	0.940983i	$-0.609904\pi$
$769$	−14435.1	−0.676909	−0.338454	+	0.940983i	$0.609904\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−17408.5	−0.810012	−0.405006	−	0.914314i	$-0.632731\pi$
$773$	−17408.5	−0.810012	−0.405006	+	0.914314i	$0.632731\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	19376.6	0.891190
$780$	0	0
$781$	−7358.65	−0.337149
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	35750.7	1.61928	0.809642	−	0.586924i	$-0.199661\pi$
$787$	35750.7	1.61928	0.809642	+	0.586924i	$0.199661\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	38487.4	1.73003
$792$	0	0
$793$	36362.0	1.62831
$794$	0	0
$795$	0	0
$796$	0	0
$797$	17326.4	0.770053	0.385026	−	0.922906i	$-0.374192\pi$
$797$	17326.4	0.770053	0.385026	+	0.922906i	$0.374192\pi$
$798$	0	0
$799$	−4531.60	−0.200646
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5643.25	−0.248003
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	13201.9	0.573736	0.286868	−	0.957970i	$-0.407386\pi$
$809$	13201.9	0.573736	0.286868	+	0.957970i	$0.407386\pi$
$810$	0	0
$811$	20131.3	0.871649	0.435824	−	0.900032i	$-0.356457\pi$
$811$	20131.3	0.871649	0.435824	+	0.900032i	$0.356457\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−19492.6	−0.834712
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10722.7	−0.455818	−0.227909	−	0.973682i	$-0.573189\pi$
$821$	−10722.7	−0.455818	−0.227909	+	0.973682i	$0.573189\pi$
$822$	0	0
$823$	−34158.8	−1.44678	−0.723390	−	0.690440i	$-0.757417\pi$
$823$	−34158.8	−1.44678	−0.723390	+	0.690440i	$0.757417\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	13887.4	0.583932	0.291966	−	0.956429i	$-0.405691\pi$
$827$	13887.4	0.583932	0.291966	+	0.956429i	$0.405691\pi$
$828$	0	0
$829$	9875.69	0.413748	0.206874	−	0.978368i	$-0.433671\pi$
$829$	9875.69	0.413748	0.206874	+	0.978368i	$0.433671\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	29735.7	1.23683
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	10593.8	0.435923	0.217962	−	0.975957i	$-0.430059\pi$
$839$	10593.8	0.435923	0.217962	+	0.975957i	$0.430059\pi$
$840$	0	0
$841$	−22309.6	−0.914740
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	29299.0	1.18858
$848$	0	0
$849$	0	0
$850$	0	0
$851$	41809.2	1.68414
$852$	0	0
$853$	−16863.7	−0.676906	−0.338453	−	0.940983i	$-0.609904\pi$
$853$	−16863.7	−0.676906	−0.338453	+	0.940983i	$0.609904\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22175.8	−0.883909	−0.441955	−	0.897037i	$-0.645715\pi$
$857$	−22175.8	−0.883909	−0.441955	+	0.897037i	$0.645715\pi$
$858$	0	0
$859$	40622.9	1.61354	0.806772	−	0.590862i	$-0.201212\pi$
$859$	40622.9	1.61354	0.806772	+	0.590862i	$0.201212\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	42495.5	1.67620	0.838101	−	0.545515i	$-0.183666\pi$
$863$	42495.5	1.67620	0.838101	+	0.545515i	$0.183666\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	181.654	0.00709111
$870$	0	0
$871$	39697.4	1.54431
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−28966.0	−1.11529	−0.557647	−	0.830078i	$-0.688296\pi$
$877$	−28966.0	−1.11529	−0.557647	+	0.830078i	$0.688296\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	35838.1	1.37051	0.685254	−	0.728304i	$-0.259691\pi$
$881$	35838.1	1.37051	0.685254	+	0.728304i	$0.259691\pi$
$882$	0	0
$883$	28283.8	1.07795	0.538973	−	0.842323i	$-0.318812\pi$
$883$	28283.8	1.07795	0.538973	+	0.842323i	$0.318812\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2470.07	0.0935028	0.0467514	−	0.998907i	$-0.485113\pi$
$887$	2470.07	0.0935028	0.0467514	+	0.998907i	$0.485113\pi$
$888$	0	0
$889$	−45633.4	−1.72159
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1638.24	0.0613903
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1466.92	0.0544209
$900$	0	0
$901$	20488.6	0.757576
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−49022.2	−1.79466	−0.897330	−	0.441360i	$-0.854496\pi$
$907$	−49022.2	−1.79466	−0.897330	+	0.441360i	$0.854496\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	5097.39	0.185383	0.0926915	−	0.995695i	$-0.470453\pi$
$911$	5097.39	0.185383	0.0926915	+	0.995695i	$0.470453\pi$
$912$	0	0
$913$	−12755.3	−0.462365
$914$	0	0
$915$	0	0
$916$	0	0
$917$	17510.3	0.630579
$918$	0	0
$919$	−39339.4	−1.41206	−0.706032	−	0.708180i	$-0.749516\pi$
$919$	−39339.4	−1.41206	−0.706032	+	0.708180i	$0.749516\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	54336.5	1.93771
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	40982.4	1.44735	0.723675	−	0.690141i	$-0.242452\pi$
$929$	40982.4	1.44735	0.723675	+	0.690141i	$0.242452\pi$
$930$	0	0
$931$	−10749.9	−0.378424
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	10157.4	0.354138	0.177069	−	0.984198i	$-0.443338\pi$
$937$	10157.4	0.354138	0.177069	+	0.984198i	$0.443338\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−8451.19	−0.292775	−0.146387	−	0.989227i	$-0.546765\pi$
$941$	−8451.19	−0.292775	−0.146387	+	0.989227i	$0.546765\pi$
$942$	0	0
$943$	−75406.1	−2.60399
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29635.8	−1.01693	−0.508466	−	0.861082i	$-0.669787\pi$
$947$	−29635.8	−1.01693	−0.508466	+	0.861082i	$0.669787\pi$
$948$	0	0
$949$	41670.0	1.42536
$950$	0	0
$951$	0	0
$952$	0	0
$953$	27971.8	0.950781	0.475390	−	0.879775i	$-0.342307\pi$
$953$	27971.8	0.950781	0.475390	+	0.879775i	$0.342307\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−42026.3	−1.41512
$960$	0	0
$961$	−28756.2	−0.965263
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−28546.1	−0.949308	−0.474654	−	0.880173i	$-0.657427\pi$
$967$	−28546.1	−0.949308	−0.474654	+	0.880173i	$0.657427\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−50082.2	−1.65521	−0.827607	−	0.561307i	$-0.810299\pi$
$971$	−50082.2	−1.65521	−0.827607	+	0.561307i	$0.810299\pi$
$972$	0	0
$973$	−36810.5	−1.21284
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−25897.7	−0.848046	−0.424023	−	0.905651i	$-0.639382\pi$
$977$	−25897.7	−0.848046	−0.424023	+	0.905651i	$0.639382\pi$
$978$	0	0
$979$	−5268.34	−0.171989
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−32099.3	−1.04151	−0.520756	−	0.853705i	$-0.674350\pi$
$983$	−32099.3	−1.04151	−0.520756	+	0.853705i	$0.674350\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	75857.7	2.43896
$990$	0	0
$991$	−41005.3	−1.31441	−0.657204	−	0.753713i	$-0.728261\pi$
$991$	−41005.3	−1.31441	−0.657204	+	0.753713i	$0.728261\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	40222.7	1.27770	0.638849	−	0.769332i	$-0.279411\pi$
$997$	40222.7	1.27770	0.638849	+	0.769332i	$0.279411\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1800.4.a.br.1.1	✓	3
3.2	odd	2		1800.4.a.bs.1.1	yes	3
5.2	odd	4		1800.4.f.ba.649.1		6
5.3	odd	4		1800.4.f.ba.649.6		6
5.4	even	2		1800.4.a.bt.1.3	yes	3
15.2	even	4		1800.4.f.bb.649.1		6
15.8	even	4		1800.4.f.bb.649.6		6
15.14	odd	2		1800.4.a.bu.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1800.4.a.br.1.1	✓	3	1.1	even	1	trivial
1800.4.a.bs.1.1	yes	3	3.2	odd	2
1800.4.a.bt.1.3	yes	3	5.4	even	2
1800.4.a.bu.1.3	yes	3	15.14	odd	2
1800.4.f.ba.649.1		6	5.2	odd	4
1800.4.f.ba.649.6		6	5.3	odd	4
1800.4.f.bb.649.1		6	15.2	even	4
1800.4.f.bb.649.6		6	15.8	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:	\( N \)	\(=\)	\( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1800.a (trivial)

\(f(q)\)	\(=\)	\(q-23.6993 q^{7} +O(q^{10})\)	\(q-23.6993 q^{7} +9.73221 q^{11} -71.8629 q^{13} +135.994 q^{17} -49.1637 q^{19} +191.326 q^{23} +45.6004 q^{29} +32.1689 q^{31} +218.523 q^{37} -394.123 q^{41} +396.484 q^{43} -33.3221 q^{47} +218.655 q^{49} +150.659 q^{53} +396.510 q^{59} -505.990 q^{61} -552.405 q^{67} -756.113 q^{71} -579.853 q^{73} -230.646 q^{77} +18.6652 q^{79} -1310.63 q^{83} -541.330 q^{89} +1703.10 q^{91} -16.7303 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 9 q^{7}+O(q^{10}) \) 3 * q - 9 * q^7	\( 3 q - 9 q^{7} - 8 q^{11} - 17 q^{13} + 48 q^{17} - 11 q^{19} + 40 q^{23} + 59 q^{31} + 10 q^{37} - 400 q^{41} + 159 q^{43} + 272 q^{47} + 110 q^{49} - 256 q^{53} - 176 q^{59} - 375 q^{61} + 143 q^{67} - 1288 q^{71} - 398 q^{73} + 736 q^{77} - 292 q^{79} - 1424 q^{83} - 928 q^{89} + 851 q^{91} - 697 q^{97}+O(q^{100}) \) 3 * q - 9 * q^7 - 8 * q^11 - 17 * q^13 + 48 * q^17 - 11 * q^19 + 40 * q^23 + 59 * q^31 + 10 * q^37 - 400 * q^41 + 159 * q^43 + 272 * q^47 + 110 * q^49 - 256 * q^53 - 176 * q^59 - 375 * q^61 + 143 * q^67 - 1288 * q^71 - 398 * q^73 + 736 * q^77 - 292 * q^79 - 1424 * q^83 - 928 * q^89 + 851 * q^91 - 697 * q^97

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