LMFDB - Embedded newform 1620.4.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,4,Mod(1,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1620.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1620.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:	$95.5830942093$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 151x^{4} - 212x^{3} + 4412x^{2} + 8656x - 19148 $ x^6 - 2x^5 - 151x^4 - 212x^3 + 4412x^2 + 8656*x - 19148
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-7.39017$ of defining polynomial
Character	$\chi$	$=$	1620.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{5} -23.3157 q^{7} +O(q^{10})$	$q+5.00000 q^{5} -23.3157 q^{7} +33.7490 q^{11} -88.8938 q^{13} +108.661 q^{17} +21.4799 q^{19} +85.9225 q^{23} +25.0000 q^{25} -18.9781 q^{29} -166.345 q^{31} -116.578 q^{35} -167.664 q^{37} +378.285 q^{41} +380.444 q^{43} +246.829 q^{47} +200.622 q^{49} -284.344 q^{53} +168.745 q^{55} -727.397 q^{59} +470.868 q^{61} -444.469 q^{65} -751.956 q^{67} -864.963 q^{71} -1003.53 q^{73} -786.881 q^{77} -838.501 q^{79} +476.114 q^{83} +543.306 q^{85} +554.055 q^{89} +2072.62 q^{91} +107.399 q^{95} -1047.99 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 30 q^{5} + 12 q^{7}+O(q^{10}) $ 6 * q + 30 * q^5 + 12 * q^7	$ 6 q + 30 q^{5} + 12 q^{7} - 84 q^{13} - 12 q^{17} - 114 q^{19} - 30 q^{23} + 150 q^{25} - 168 q^{29} - 324 q^{31} + 60 q^{35} - 492 q^{37} - 312 q^{41} - 156 q^{43} - 462 q^{47} - 588 q^{49} - 1014 q^{53} - 1008 q^{59} + 36 q^{61} - 420 q^{65} + 144 q^{67} - 1212 q^{71} - 900 q^{73} - 672 q^{77} - 936 q^{79} - 288 q^{83} - 60 q^{85} + 120 q^{89} + 2286 q^{91} - 570 q^{95} - 1188 q^{97}+O(q^{100}) $ 6 * q + 30 * q^5 + 12 * q^7 - 84 * q^13 - 12 * q^17 - 114 * q^19 - 30 * q^23 + 150 * q^25 - 168 * q^29 - 324 * q^31 + 60 * q^35 - 492 * q^37 - 312 * q^41 - 156 * q^43 - 462 * q^47 - 588 * q^49 - 1014 * q^53 - 1008 * q^59 + 36 * q^61 - 420 * q^65 + 144 * q^67 - 1212 * q^71 - 900 * q^73 - 672 * q^77 - 936 * q^79 - 288 * q^83 - 60 * q^85 + 120 * q^89 + 2286 * q^91 - 570 * q^95 - 1188 * 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
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−23.3157	−1.25893	−0.629465	−	0.777029i	$-0.716726\pi$
$7$	−23.3157	−1.25893	−0.629465	+	0.777029i	$0.716726\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	33.7490	0.925063	0.462532	−	0.886603i	$-0.346941\pi$
$11$	33.7490	0.925063	0.462532	+	0.886603i	$0.346941\pi$
$12$	0	0
$13$	−88.8938	−1.89652	−0.948258	−	0.317500i	$-0.897157\pi$
$13$	−88.8938	−1.89652	−0.948258	+	0.317500i	$0.897157\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	108.661	1.55025	0.775124	−	0.631809i	$-0.217687\pi$
$17$	108.661	1.55025	0.775124	+	0.631809i	$0.217687\pi$
$18$	0	0
$19$	21.4799	0.259359	0.129679	−	0.991556i	$-0.458605\pi$
$19$	21.4799	0.259359	0.129679	+	0.991556i	$0.458605\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	85.9225	0.778960	0.389480	−	0.921035i	$-0.372655\pi$
$23$	85.9225	0.778960	0.389480	+	0.921035i	$0.372655\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−18.9781	−0.121522	−0.0607612	−	0.998152i	$-0.519353\pi$
$29$	−18.9781	−0.121522	−0.0607612	+	0.998152i	$0.519353\pi$
$30$	0	0
$31$	−166.345	−0.963755	−0.481877	−	0.876239i	$-0.660045\pi$
$31$	−166.345	−0.963755	−0.481877	+	0.876239i	$0.660045\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−116.578	−0.563010
$36$	0	0
$37$	−167.664	−0.744966	−0.372483	−	0.928039i	$-0.621493\pi$
$37$	−167.664	−0.744966	−0.372483	+	0.928039i	$0.621493\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	378.285	1.44093	0.720466	−	0.693490i	$-0.243928\pi$
$41$	378.285	1.44093	0.720466	+	0.693490i	$0.243928\pi$
$42$	0	0
$43$	380.444	1.34924	0.674618	−	0.738167i	$-0.264308\pi$
$43$	380.444	1.34924	0.674618	+	0.738167i	$0.264308\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	246.829	0.766036	0.383018	−	0.923741i	$-0.374885\pi$
$47$	246.829	0.766036	0.383018	+	0.923741i	$0.374885\pi$
$48$	0	0
$49$	200.622	0.584903
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−284.344	−0.736937	−0.368469	−	0.929640i	$-0.620118\pi$
$53$	−284.344	−0.736937	−0.368469	+	0.929640i	$0.620118\pi$
$54$	0	0
$55$	168.745	0.413701
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−727.397	−1.60507	−0.802534	−	0.596606i	$-0.796516\pi$
$59$	−727.397	−1.60507	−0.802534	+	0.596606i	$0.796516\pi$
$60$	0	0
$61$	470.868	0.988335	0.494168	−	0.869367i	$-0.335473\pi$
$61$	470.868	0.988335	0.494168	+	0.869367i	$0.335473\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−444.469	−0.848148
$66$	0	0
$67$	−751.956	−1.37113	−0.685567	−	0.728009i	$-0.740446\pi$
$67$	−751.956	−1.37113	−0.685567	+	0.728009i	$0.740446\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−864.963	−1.44581	−0.722903	−	0.690949i	$-0.757193\pi$
$71$	−864.963	−1.44581	−0.722903	+	0.690949i	$0.757193\pi$
$72$	0	0
$73$	−1003.53	−1.60896	−0.804479	−	0.593982i	$-0.797555\pi$
$73$	−1003.53	−1.60896	−0.804479	+	0.593982i	$0.797555\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−786.881	−1.16459
$78$	0	0
$79$	−838.501	−1.19416	−0.597081	−	0.802181i	$-0.703673\pi$
$79$	−838.501	−1.19416	−0.597081	+	0.802181i	$0.703673\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	476.114	0.629643	0.314821	−	0.949151i	$-0.398055\pi$
$83$	476.114	0.629643	0.314821	+	0.949151i	$0.398055\pi$
$84$	0	0
$85$	543.306	0.693292
$86$	0	0
$87$	0	0
$88$	0	0
$89$	554.055	0.659884	0.329942	−	0.944001i	$-0.392971\pi$
$89$	554.055	0.659884	0.329942	+	0.944001i	$0.392971\pi$
$90$	0	0
$91$	2072.62	2.38758
$92$	0	0
$93$	0	0
$94$	0	0
$95$	107.399	0.115989
$96$	0	0
$97$	−1047.99	−1.09698	−0.548490	−	0.836157i	$-0.684797\pi$
$97$	−1047.99	−1.09698	−0.548490	+	0.836157i	$0.684797\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−418.398	−0.412199	−0.206100	−	0.978531i	$-0.566077\pi$
$101$	−418.398	−0.412199	−0.206100	+	0.978531i	$0.566077\pi$
$102$	0	0
$103$	−1280.79	−1.22524	−0.612620	−	0.790377i	$-0.709884\pi$
$103$	−1280.79	−1.22524	−0.612620	+	0.790377i	$0.709884\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	351.116	0.317231	0.158615	−	0.987340i	$-0.449297\pi$
$107$	351.116	0.317231	0.158615	+	0.987340i	$0.449297\pi$
$108$	0	0
$109$	1762.43	1.54872	0.774360	−	0.632746i	$-0.218072\pi$
$109$	1762.43	1.54872	0.774360	+	0.632746i	$0.218072\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1001.86	−0.834047	−0.417023	−	0.908896i	$-0.636927\pi$
$113$	−1001.86	−0.834047	−0.417023	+	0.908896i	$0.636927\pi$
$114$	0	0
$115$	429.612	0.348361
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2533.51	−1.95165
$120$	0	0
$121$	−192.007	−0.144258
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	852.753	0.595824	0.297912	−	0.954593i	$-0.403710\pi$
$127$	852.753	0.595824	0.297912	+	0.954593i	$0.403710\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2678.23	1.78625	0.893123	−	0.449812i	$-0.148509\pi$
$131$	2678.23	1.78625	0.893123	+	0.449812i	$0.148509\pi$
$132$	0	0
$133$	−500.818	−0.326514
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−423.993	−0.264410	−0.132205	−	0.991222i	$-0.542206\pi$
$137$	−423.993	−0.264410	−0.132205	+	0.991222i	$0.542206\pi$
$138$	0	0
$139$	−671.515	−0.409764	−0.204882	−	0.978787i	$-0.565681\pi$
$139$	−671.515	−0.409764	−0.204882	+	0.978787i	$0.565681\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3000.08	−1.75440
$144$	0	0
$145$	−94.8907	−0.0543465
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2397.44	−1.31816	−0.659080	−	0.752073i	$-0.729054\pi$
$149$	−2397.44	−1.31816	−0.659080	+	0.752073i	$0.729054\pi$
$150$	0	0
$151$	−302.009	−0.162763	−0.0813813	−	0.996683i	$-0.525933\pi$
$151$	−302.009	−0.162763	−0.0813813	+	0.996683i	$0.525933\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−831.724	−0.431004
$156$	0	0
$157$	−2234.99	−1.13613	−0.568063	−	0.822985i	$-0.692307\pi$
$157$	−2234.99	−1.13613	−0.568063	+	0.822985i	$0.692307\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−2003.34	−0.980655
$162$	0	0
$163$	−45.7276	−0.0219734	−0.0109867	−	0.999940i	$-0.503497\pi$
$163$	−45.7276	−0.0219734	−0.0109867	+	0.999940i	$0.503497\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2476.14	−1.14736	−0.573681	−	0.819079i	$-0.694485\pi$
$167$	−2476.14	−1.14736	−0.573681	+	0.819079i	$0.694485\pi$
$168$	0	0
$169$	5705.11	2.59677
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3149.44	−1.38409	−0.692044	−	0.721856i	$-0.743290\pi$
$173$	−3149.44	−1.38409	−0.692044	+	0.721856i	$0.743290\pi$
$174$	0	0
$175$	−582.892	−0.251786
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4111.57	1.71683	0.858417	−	0.512953i	$-0.171448\pi$
$179$	4111.57	1.71683	0.858417	+	0.512953i	$0.171448\pi$
$180$	0	0
$181$	−1344.52	−0.552138	−0.276069	−	0.961138i	$-0.589032\pi$
$181$	−1344.52	−0.552138	−0.276069	+	0.961138i	$0.589032\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−838.318	−0.333159
$186$	0	0
$187$	3667.20	1.43408
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2529.57	0.958288	0.479144	−	0.877736i	$-0.340947\pi$
$191$	2529.57	0.958288	0.479144	+	0.877736i	$0.340947\pi$
$192$	0	0
$193$	−4009.64	−1.49544	−0.747722	−	0.664012i	$-0.768852\pi$
$193$	−4009.64	−1.49544	−0.747722	+	0.664012i	$0.768852\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2489.65	−0.900408	−0.450204	−	0.892926i	$-0.648649\pi$
$197$	−2489.65	−0.900408	−0.450204	+	0.892926i	$0.648649\pi$
$198$	0	0
$199$	1358.91	0.484073	0.242037	−	0.970267i	$-0.422185\pi$
$199$	1358.91	0.484073	0.242037	+	0.970267i	$0.422185\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	442.488	0.152988
$204$	0	0
$205$	1891.43	0.644405
$206$	0	0
$207$	0	0
$208$	0	0
$209$	724.923	0.239923
$210$	0	0
$211$	−3794.62	−1.23807	−0.619034	−	0.785364i	$-0.712476\pi$
$211$	−3794.62	−1.23807	−0.619034	+	0.785364i	$0.712476\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1902.22	0.603397
$216$	0	0
$217$	3878.44	1.21330
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−9659.31	−2.94007
$222$	0	0
$223$	−1328.72	−0.399002	−0.199501	−	0.979898i	$-0.563932\pi$
$223$	−1328.72	−0.399002	−0.199501	+	0.979898i	$0.563932\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−991.577	−0.289926	−0.144963	−	0.989437i	$-0.546306\pi$
$227$	−991.577	−0.289926	−0.144963	+	0.989437i	$0.546306\pi$
$228$	0	0
$229$	−3975.79	−1.14728	−0.573640	−	0.819107i	$-0.694469\pi$
$229$	−3975.79	−1.14728	−0.573640	+	0.819107i	$0.694469\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4850.18	−1.36372	−0.681858	−	0.731484i	$-0.738828\pi$
$233$	−4850.18	−1.36372	−0.681858	+	0.731484i	$0.738828\pi$
$234$	0	0
$235$	1234.14	0.342582
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3671.01	0.993547	0.496773	−	0.867880i	$-0.334518\pi$
$239$	3671.01	0.993547	0.496773	+	0.867880i	$0.334518\pi$
$240$	0	0
$241$	−2275.77	−0.608280	−0.304140	−	0.952627i	$-0.598369\pi$
$241$	−2275.77	−0.608280	−0.304140	+	0.952627i	$0.598369\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1003.11	0.261577
$246$	0	0
$247$	−1909.43	−0.491878
$248$	0	0
$249$	0	0
$250$	0	0
$251$	816.764	0.205393	0.102697	−	0.994713i	$-0.467253\pi$
$251$	816.764	0.205393	0.102697	+	0.994713i	$0.467253\pi$
$252$	0	0
$253$	2899.80	0.720587
$254$	0	0
$255$	0	0
$256$	0	0
$257$	6294.13	1.52769	0.763847	−	0.645398i	$-0.223308\pi$
$257$	6294.13	1.52769	0.763847	+	0.645398i	$0.223308\pi$
$258$	0	0
$259$	3909.20	0.937859
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1854.99	0.434918	0.217459	−	0.976069i	$-0.430223\pi$
$263$	1854.99	0.434918	0.217459	+	0.976069i	$0.430223\pi$
$264$	0	0
$265$	−1421.72	−0.329568
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3602.16	−0.816460	−0.408230	−	0.912879i	$-0.633854\pi$
$269$	−3602.16	−0.816460	−0.408230	+	0.912879i	$0.633854\pi$
$270$	0	0
$271$	2920.88	0.654727	0.327363	−	0.944898i	$-0.393840\pi$
$271$	2920.88	0.654727	0.327363	+	0.944898i	$0.393840\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	843.724	0.185013
$276$	0	0
$277$	−537.879	−0.116672	−0.0583358	−	0.998297i	$-0.518579\pi$
$277$	−537.879	−0.116672	−0.0583358	+	0.998297i	$0.518579\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4070.53	−0.864154	−0.432077	−	0.901837i	$-0.642219\pi$
$281$	−4070.53	−0.864154	−0.432077	+	0.901837i	$0.642219\pi$
$282$	0	0
$283$	−1611.67	−0.338529	−0.169265	−	0.985571i	$-0.554139\pi$
$283$	−1611.67	−0.338529	−0.169265	+	0.985571i	$0.554139\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8819.98	−1.81403
$288$	0	0
$289$	6894.26	1.40327
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4456.58	0.888587	0.444293	−	0.895881i	$-0.353455\pi$
$293$	4456.58	0.888587	0.444293	+	0.895881i	$0.353455\pi$
$294$	0	0
$295$	−3636.99	−0.717809
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−7637.98	−1.47731
$300$	0	0
$301$	−8870.32	−1.69859
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2354.34	0.441997
$306$	0	0
$307$	−11.9533	−0.00222219	−0.00111109	−	0.999999i	$-0.500354\pi$
$307$	−11.9533	−0.00222219	−0.00111109	+	0.999999i	$0.500354\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9345.13	−1.70390	−0.851951	−	0.523622i	$-0.824581\pi$
$311$	−9345.13	−1.70390	−0.851951	+	0.523622i	$0.824581\pi$
$312$	0	0
$313$	−8768.23	−1.58342	−0.791709	−	0.610899i	$-0.790808\pi$
$313$	−8768.23	−1.58342	−0.791709	+	0.610899i	$0.790808\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1976.17	−0.350136	−0.175068	−	0.984556i	$-0.556014\pi$
$317$	−1976.17	−0.350136	−0.175068	+	0.984556i	$0.556014\pi$
$318$	0	0
$319$	−640.493	−0.112416
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2334.03	0.402070
$324$	0	0
$325$	−2222.35	−0.379303
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5754.98	−0.964385
$330$	0	0
$331$	7288.56	1.21032	0.605159	−	0.796104i	$-0.293109\pi$
$331$	7288.56	1.21032	0.605159	+	0.796104i	$0.293109\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3759.78	−0.613190
$336$	0	0
$337$	7299.78	1.17995	0.589977	−	0.807420i	$-0.299137\pi$
$337$	7299.78	1.17995	0.589977	+	0.807420i	$0.299137\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5613.96	−0.891534
$342$	0	0
$343$	3319.65	0.522577
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6762.36	−1.04617	−0.523087	−	0.852279i	$-0.675220\pi$
$347$	−6762.36	−1.04617	−0.523087	+	0.852279i	$0.675220\pi$
$348$	0	0
$349$	−6872.96	−1.05416	−0.527079	−	0.849816i	$-0.676713\pi$
$349$	−6872.96	−1.05416	−0.527079	+	0.849816i	$0.676713\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−9611.18	−1.44916	−0.724578	−	0.689193i	$-0.757965\pi$
$353$	−9611.18	−1.44916	−0.724578	+	0.689193i	$0.757965\pi$
$354$	0	0
$355$	−4324.82	−0.646584
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−7951.69	−1.16901	−0.584504	−	0.811391i	$-0.698711\pi$
$359$	−7951.69	−1.16901	−0.584504	+	0.811391i	$0.698711\pi$
$360$	0	0
$361$	−6397.62	−0.932733
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5017.63	−0.719548
$366$	0	0
$367$	6388.98	0.908724	0.454362	−	0.890817i	$-0.349867\pi$
$367$	6388.98	0.908724	0.454362	+	0.890817i	$0.349867\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6629.69	0.927752
$372$	0	0
$373$	−9163.15	−1.27198	−0.635992	−	0.771696i	$-0.719409\pi$
$373$	−9163.15	−1.27198	−0.635992	+	0.771696i	$0.719409\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1687.04	0.230469
$378$	0	0
$379$	−10303.5	−1.39646	−0.698228	−	0.715875i	$-0.746028\pi$
$379$	−10303.5	−1.39646	−0.698228	+	0.715875i	$0.746028\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3719.90	−0.496287	−0.248144	−	0.968723i	$-0.579821\pi$
$383$	−3719.90	−0.496287	−0.248144	+	0.968723i	$0.579821\pi$
$384$	0	0
$385$	−3934.40	−0.520820
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4470.47	0.582678	0.291339	−	0.956620i	$-0.405899\pi$
$389$	4470.47	0.582678	0.291339	+	0.956620i	$0.405899\pi$
$390$	0	0
$391$	9336.44	1.20758
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4192.51	−0.534045
$396$	0	0
$397$	4848.35	0.612927	0.306463	−	0.951882i	$-0.400854\pi$
$397$	4848.35	0.612927	0.306463	+	0.951882i	$0.400854\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	149.533	0.0186218	0.00931090	−	0.999957i	$-0.497036\pi$
$401$	149.533	0.0186218	0.00931090	+	0.999957i	$0.497036\pi$
$402$	0	0
$403$	14787.0	1.82778
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5658.48	−0.689141
$408$	0	0
$409$	10649.7	1.28751	0.643756	−	0.765231i	$-0.277375\pi$
$409$	10649.7	1.28751	0.643756	+	0.765231i	$0.277375\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	16959.8	2.02067
$414$	0	0
$415$	2380.57	0.281585
$416$	0	0
$417$	0	0
$418$	0	0
$419$	339.793	0.0396181	0.0198090	−	0.999804i	$-0.493694\pi$
$419$	339.793	0.0396181	0.0198090	+	0.999804i	$0.493694\pi$
$420$	0	0
$421$	2132.04	0.246815	0.123408	−	0.992356i	$-0.460618\pi$
$421$	2132.04	0.246815	0.123408	+	0.992356i	$0.460618\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	2716.53	0.310050
$426$	0	0
$427$	−10978.6	−1.24424
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2509.67	−0.280480	−0.140240	−	0.990118i	$-0.544787\pi$
$431$	−2509.67	−0.280480	−0.140240	+	0.990118i	$0.544787\pi$
$432$	0	0
$433$	2792.17	0.309891	0.154946	−	0.987923i	$-0.450480\pi$
$433$	2792.17	0.309891	0.154946	+	0.987923i	$0.450480\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1845.60	0.202030
$438$	0	0
$439$	−6557.00	−0.712867	−0.356433	−	0.934321i	$-0.616007\pi$
$439$	−6557.00	−0.712867	−0.356433	+	0.934321i	$0.616007\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1899.78	0.203750	0.101875	−	0.994797i	$-0.467516\pi$
$443$	1899.78	0.203750	0.101875	+	0.994797i	$0.467516\pi$
$444$	0	0
$445$	2770.27	0.295109
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−9468.83	−0.995238	−0.497619	−	0.867396i	$-0.665792\pi$
$449$	−9468.83	−0.995238	−0.497619	+	0.867396i	$0.665792\pi$
$450$	0	0
$451$	12766.7	1.33295
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10363.1	1.06776
$456$	0	0
$457$	2899.70	0.296810	0.148405	−	0.988927i	$-0.452586\pi$
$457$	2899.70	0.296810	0.148405	+	0.988927i	$0.452586\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5517.61	0.557442	0.278721	−	0.960372i	$-0.410089\pi$
$461$	5517.61	0.557442	0.278721	+	0.960372i	$0.410089\pi$
$462$	0	0
$463$	−8970.30	−0.900400	−0.450200	−	0.892928i	$-0.648647\pi$
$463$	−8970.30	−0.900400	−0.450200	+	0.892928i	$0.648647\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12314.8	1.22026	0.610130	−	0.792302i	$-0.291117\pi$
$467$	12314.8	1.22026	0.610130	+	0.792302i	$0.291117\pi$
$468$	0	0
$469$	17532.4	1.72616
$470$	0	0
$471$	0	0
$472$	0	0
$473$	12839.6	1.24813
$474$	0	0
$475$	536.996	0.0518718
$476$	0	0
$477$	0	0
$478$	0	0
$479$	12728.9	1.21420	0.607098	−	0.794627i	$-0.292333\pi$
$479$	12728.9	1.21420	0.607098	+	0.794627i	$0.292333\pi$
$480$	0	0
$481$	14904.3	1.41284
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5239.94	−0.490584
$486$	0	0
$487$	−7230.67	−0.672798	−0.336399	−	0.941719i	$-0.609209\pi$
$487$	−7230.67	−0.672798	−0.336399	+	0.941719i	$0.609209\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17606.9	1.61831	0.809153	−	0.587598i	$-0.199926\pi$
$491$	17606.9	1.61831	0.809153	+	0.587598i	$0.199926\pi$
$492$	0	0
$493$	−2062.19	−0.188390
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20167.2	1.82017
$498$	0	0
$499$	−20351.5	−1.82577	−0.912886	−	0.408215i	$-0.866151\pi$
$499$	−20351.5	−1.82577	−0.912886	+	0.408215i	$0.866151\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	14477.1	1.28331	0.641654	−	0.766995i	$-0.278249\pi$
$503$	14477.1	1.28331	0.641654	+	0.766995i	$0.278249\pi$
$504$	0	0
$505$	−2091.99	−0.184341
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3683.62	−0.320773	−0.160387	−	0.987054i	$-0.551274\pi$
$509$	−3683.62	−0.320773	−0.160387	+	0.987054i	$0.551274\pi$
$510$	0	0
$511$	23397.9	2.02556
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6403.94	−0.547944
$516$	0	0
$517$	8330.22	0.708632
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20684.0	1.73931	0.869655	−	0.493660i	$-0.164341\pi$
$521$	20684.0	1.73931	0.869655	+	0.493660i	$0.164341\pi$
$522$	0	0
$523$	−18781.8	−1.57031	−0.785153	−	0.619302i	$-0.787416\pi$
$523$	−18781.8	−1.57031	−0.785153	+	0.619302i	$0.787416\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−18075.2	−1.49406
$528$	0	0
$529$	−4784.33	−0.393222
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−33627.2	−2.73275
$534$	0	0
$535$	1755.58	0.141870
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6770.78	0.541073
$540$	0	0
$541$	9322.54	0.740864	0.370432	−	0.928860i	$-0.379210\pi$
$541$	9322.54	0.740864	0.370432	+	0.928860i	$0.379210\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8812.16	0.692608
$546$	0	0
$547$	−8144.96	−0.636661	−0.318330	−	0.947980i	$-0.603122\pi$
$547$	−8144.96	−0.636661	−0.318330	+	0.947980i	$0.603122\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−407.648	−0.0315179
$552$	0	0
$553$	19550.2	1.50336
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1025.83	−0.0780355	−0.0390177	−	0.999239i	$-0.512423\pi$
$557$	−1025.83	−0.0780355	−0.0390177	+	0.999239i	$0.512423\pi$
$558$	0	0
$559$	−33819.1	−2.55885
$560$	0	0
$561$	0	0
$562$	0	0
$563$	8812.62	0.659694	0.329847	−	0.944034i	$-0.393003\pi$
$563$	8812.62	0.659694	0.329847	+	0.944034i	$0.393003\pi$
$564$	0	0
$565$	−5009.31	−0.372997
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6372.87	0.469534	0.234767	−	0.972052i	$-0.424567\pi$
$569$	6372.87	0.469534	0.234767	+	0.972052i	$0.424567\pi$
$570$	0	0
$571$	10842.0	0.794611	0.397306	−	0.917686i	$-0.369945\pi$
$571$	10842.0	0.794611	0.397306	+	0.917686i	$0.369945\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2148.06	0.155792
$576$	0	0
$577$	−1494.00	−0.107792	−0.0538959	−	0.998547i	$-0.517164\pi$
$577$	−1494.00	−0.107792	−0.0538959	+	0.998547i	$0.517164\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−11100.9	−0.792676
$582$	0	0
$583$	−9596.33	−0.681714
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15659.1	1.10105	0.550527	−	0.834817i	$-0.314427\pi$
$587$	15659.1	1.10105	0.550527	+	0.834817i	$0.314427\pi$
$588$	0	0
$589$	−3573.06	−0.249958
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1722.59	−0.119289	−0.0596444	−	0.998220i	$-0.518997\pi$
$593$	−1722.59	−0.119289	−0.0596444	+	0.998220i	$0.518997\pi$
$594$	0	0
$595$	−12667.6	−0.872806
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6603.71	0.450451	0.225226	−	0.974307i	$-0.427688\pi$
$599$	6603.71	0.450451	0.225226	+	0.974307i	$0.427688\pi$
$600$	0	0
$601$	−6475.96	−0.439534	−0.219767	−	0.975552i	$-0.570530\pi$
$601$	−6475.96	−0.439534	−0.219767	+	0.975552i	$0.570530\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−960.034	−0.0645139
$606$	0	0
$607$	−7931.87	−0.530387	−0.265193	−	0.964195i	$-0.585436\pi$
$607$	−7931.87	−0.530387	−0.265193	+	0.964195i	$0.585436\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−21941.6	−1.45280
$612$	0	0
$613$	−16332.6	−1.07613	−0.538065	−	0.842903i	$-0.680845\pi$
$613$	−16332.6	−1.07613	−0.538065	+	0.842903i	$0.680845\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11671.5	−0.761552	−0.380776	−	0.924667i	$-0.624343\pi$
$617$	−11671.5	−0.761552	−0.380776	+	0.924667i	$0.624343\pi$
$618$	0	0
$619$	−10722.5	−0.696240	−0.348120	−	0.937450i	$-0.613180\pi$
$619$	−10722.5	−0.696240	−0.348120	+	0.937450i	$0.613180\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−12918.2	−0.830748
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−18218.5	−1.15488
$630$	0	0
$631$	9754.15	0.615383	0.307691	−	0.951486i	$-0.400444\pi$
$631$	9754.15	0.615383	0.307691	+	0.951486i	$0.400444\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4263.77	0.266460
$636$	0	0
$637$	−17834.0	−1.10928
$638$	0	0
$639$	0	0
$640$	0	0
$641$	19365.5	1.19328	0.596638	−	0.802511i	$-0.296503\pi$
$641$	19365.5	1.19328	0.596638	+	0.802511i	$0.296503\pi$
$642$	0	0
$643$	29075.3	1.78323	0.891616	−	0.452792i	$-0.149572\pi$
$643$	29075.3	1.78323	0.891616	+	0.452792i	$0.149572\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17400.3	1.05730	0.528651	−	0.848839i	$-0.322698\pi$
$647$	17400.3	1.05730	0.528651	+	0.848839i	$0.322698\pi$
$648$	0	0
$649$	−24548.9	−1.48479
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8467.68	−0.507452	−0.253726	−	0.967276i	$-0.581656\pi$
$653$	−8467.68	−0.507452	−0.253726	+	0.967276i	$0.581656\pi$
$654$	0	0
$655$	13391.2	0.798834
$656$	0	0
$657$	0	0
$658$	0	0
$659$	23982.6	1.41764	0.708822	−	0.705387i	$-0.249227\pi$
$659$	23982.6	1.41764	0.708822	+	0.705387i	$0.249227\pi$
$660$	0	0
$661$	26723.1	1.57248	0.786238	−	0.617923i	$-0.212026\pi$
$661$	26723.1	1.57248	0.786238	+	0.617923i	$0.212026\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2504.09	−0.146022
$666$	0	0
$667$	−1630.65	−0.0946611
$668$	0	0
$669$	0	0
$670$	0	0
$671$	15891.3	0.914273
$672$	0	0
$673$	−15121.4	−0.866104	−0.433052	−	0.901369i	$-0.642563\pi$
$673$	−15121.4	−0.866104	−0.433052	+	0.901369i	$0.642563\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14937.6	−0.848006	−0.424003	−	0.905661i	$-0.639375\pi$
$677$	−14937.6	−0.848006	−0.424003	+	0.905661i	$0.639375\pi$
$678$	0	0
$679$	24434.6	1.38102
$680$	0	0
$681$	0	0
$682$	0	0
$683$	11141.0	0.624158	0.312079	−	0.950056i	$-0.398975\pi$
$683$	11141.0	0.624158	0.312079	+	0.950056i	$0.398975\pi$
$684$	0	0
$685$	−2119.97	−0.118248
$686$	0	0
$687$	0	0
$688$	0	0
$689$	25276.5	1.39761
$690$	0	0
$691$	−11962.8	−0.658594	−0.329297	−	0.944226i	$-0.606812\pi$
$691$	−11962.8	−0.658594	−0.329297	+	0.944226i	$0.606812\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3357.57	−0.183252
$696$	0	0
$697$	41104.9	2.23380
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−17436.0	−0.939442	−0.469721	−	0.882815i	$-0.655646\pi$
$701$	−17436.0	−0.939442	−0.469721	+	0.882815i	$0.655646\pi$
$702$	0	0
$703$	−3601.39	−0.193213
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9755.23	0.518930
$708$	0	0
$709$	11328.4	0.600067	0.300034	−	0.953929i	$-0.403002\pi$
$709$	11328.4	0.600067	0.300034	+	0.953929i	$0.403002\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−14292.7	−0.750726
$714$	0	0
$715$	−15000.4	−0.784591
$716$	0	0
$717$	0	0
$718$	0	0
$719$	36889.9	1.91344	0.956718	−	0.291015i	$-0.0939931\pi$
$719$	36889.9	1.91344	0.956718	+	0.291015i	$0.0939931\pi$
$720$	0	0
$721$	29862.5	1.54249
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−474.453	−0.0243045
$726$	0	0
$727$	10955.8	0.558910	0.279455	−	0.960159i	$-0.409846\pi$
$727$	10955.8	0.558910	0.279455	+	0.960159i	$0.409846\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	41339.5	2.09165
$732$	0	0
$733$	−22382.1	−1.12783	−0.563915	−	0.825833i	$-0.690706\pi$
$733$	−22382.1	−1.12783	−0.563915	+	0.825833i	$0.690706\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−25377.7	−1.26839
$738$	0	0
$739$	27800.0	1.38382	0.691908	−	0.721985i	$-0.256770\pi$
$739$	27800.0	1.38382	0.691908	+	0.721985i	$0.256770\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	17926.3	0.885129	0.442564	−	0.896737i	$-0.354069\pi$
$743$	17926.3	0.885129	0.442564	+	0.896737i	$0.354069\pi$
$744$	0	0
$745$	−11987.2	−0.589499
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−8186.52	−0.399371
$750$	0	0
$751$	24119.8	1.17196	0.585982	−	0.810324i	$-0.300709\pi$
$751$	24119.8	1.17196	0.585982	+	0.810324i	$0.300709\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1510.05	−0.0727896
$756$	0	0
$757$	10582.4	0.508088	0.254044	−	0.967193i	$-0.418239\pi$
$757$	10582.4	0.508088	0.254044	+	0.967193i	$0.418239\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−36583.1	−1.74262	−0.871312	−	0.490730i	$-0.836730\pi$
$761$	−36583.1	−1.74262	−0.871312	+	0.490730i	$0.836730\pi$
$762$	0	0
$763$	−41092.3	−1.94973
$764$	0	0
$765$	0	0
$766$	0	0
$767$	64661.1	3.04404
$768$	0	0
$769$	14776.5	0.692917	0.346459	−	0.938065i	$-0.387384\pi$
$769$	14776.5	0.692917	0.346459	+	0.938065i	$0.387384\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3195.12	0.148668	0.0743340	−	0.997233i	$-0.476317\pi$
$773$	3195.12	0.148668	0.0743340	+	0.997233i	$0.476317\pi$
$774$	0	0
$775$	−4158.62	−0.192751
$776$	0	0
$777$	0	0
$778$	0	0
$779$	8125.51	0.373718
$780$	0	0
$781$	−29191.6	−1.33746
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−11175.0	−0.508091
$786$	0	0
$787$	30987.1	1.40352	0.701761	−	0.712412i	$-0.252397\pi$
$787$	30987.1	1.40352	0.701761	+	0.712412i	$0.252397\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	23359.1	1.05001
$792$	0	0
$793$	−41857.2	−1.87439
$794$	0	0
$795$	0	0
$796$	0	0
$797$	37083.1	1.64812	0.824059	−	0.566503i	$-0.191704\pi$
$797$	37083.1	1.64812	0.824059	+	0.566503i	$0.191704\pi$
$798$	0	0
$799$	26820.7	1.18754
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−33868.0	−1.48839
$804$	0	0
$805$	−10016.7	−0.438562
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−648.034	−0.0281627	−0.0140814	−	0.999901i	$-0.504482\pi$
$809$	−648.034	−0.0281627	−0.0140814	+	0.999901i	$0.504482\pi$
$810$	0	0
$811$	−36017.4	−1.55948	−0.779742	−	0.626101i	$-0.784650\pi$
$811$	−36017.4	−1.55948	−0.779742	+	0.626101i	$0.784650\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−228.638	−0.00982679
$816$	0	0
$817$	8171.88	0.349936
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28808.9	1.22465	0.612326	−	0.790605i	$-0.290234\pi$
$821$	28808.9	1.22465	0.612326	+	0.790605i	$0.290234\pi$
$822$	0	0
$823$	−27554.7	−1.16707	−0.583533	−	0.812090i	$-0.698330\pi$
$823$	−27554.7	−1.16707	−0.583533	+	0.812090i	$0.698330\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	34914.4	1.46807	0.734035	−	0.679111i	$-0.237635\pi$
$827$	34914.4	1.46807	0.734035	+	0.679111i	$0.237635\pi$
$828$	0	0
$829$	749.124	0.0313850	0.0156925	−	0.999877i	$-0.495005\pi$
$829$	749.124	0.0313850	0.0156925	+	0.999877i	$0.495005\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	21799.8	0.906745
$834$	0	0
$835$	−12380.7	−0.513116
$836$	0	0
$837$	0	0
$838$	0	0
$839$	37455.7	1.54126	0.770629	−	0.637284i	$-0.219942\pi$
$839$	37455.7	1.54126	0.770629	+	0.637284i	$0.219942\pi$
$840$	0	0
$841$	−24028.8	−0.985232
$842$	0	0
$843$	0	0
$844$	0	0
$845$	28525.6	1.16131
$846$	0	0
$847$	4476.77	0.181610
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−14406.1	−0.580298
$852$	0	0
$853$	−11806.0	−0.473894	−0.236947	−	0.971523i	$-0.576147\pi$
$853$	−11806.0	−0.473894	−0.236947	+	0.971523i	$0.576147\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−24069.1	−0.959376	−0.479688	−	0.877439i	$-0.659250\pi$
$857$	−24069.1	−0.959376	−0.479688	+	0.877439i	$0.659250\pi$
$858$	0	0
$859$	−11368.0	−0.451536	−0.225768	−	0.974181i	$-0.572489\pi$
$859$	−11368.0	−0.451536	−0.225768	+	0.974181i	$0.572489\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10733.7	0.423382	0.211691	−	0.977337i	$-0.432103\pi$
$863$	10733.7	0.423382	0.211691	+	0.977337i	$0.432103\pi$
$864$	0	0
$865$	−15747.2	−0.618983
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−28298.6	−1.10468
$870$	0	0
$871$	66844.2	2.60038
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2914.46	−0.112602
$876$	0	0
$877$	19551.0	0.752781	0.376391	−	0.926461i	$-0.377165\pi$
$877$	19551.0	0.752781	0.376391	+	0.926461i	$0.377165\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	35047.7	1.34028	0.670140	−	0.742235i	$-0.266234\pi$
$881$	35047.7	1.34028	0.670140	+	0.742235i	$0.266234\pi$
$882$	0	0
$883$	−35769.3	−1.36323	−0.681615	−	0.731711i	$-0.738722\pi$
$883$	−35769.3	−1.36323	−0.681615	+	0.731711i	$0.738722\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	18039.4	0.682867	0.341434	−	0.939906i	$-0.389088\pi$
$887$	18039.4	0.682867	0.341434	+	0.939906i	$0.389088\pi$
$888$	0	0
$889$	−19882.5	−0.750100
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5301.85	0.198678
$894$	0	0
$895$	20557.9	0.767791
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3156.91	0.117118
$900$	0	0
$901$	−30897.2	−1.14244
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6722.58	−0.246924
$906$	0	0
$907$	29050.2	1.06350	0.531750	−	0.846901i	$-0.321534\pi$
$907$	29050.2	1.06350	0.531750	+	0.846901i	$0.321534\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	10546.6	0.383562	0.191781	−	0.981438i	$-0.438574\pi$
$911$	10546.6	0.383562	0.191781	+	0.981438i	$0.438574\pi$
$912$	0	0
$913$	16068.4	0.582459
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−62444.9	−2.24876
$918$	0	0
$919$	2975.12	0.106790	0.0533951	−	0.998573i	$-0.482996\pi$
$919$	2975.12	0.106790	0.0533951	+	0.998573i	$0.482996\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	76889.9	2.74200
$924$	0	0
$925$	−4191.59	−0.148993
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−50304.7	−1.77658	−0.888290	−	0.459283i	$-0.848106\pi$
$929$	−50304.7	−1.77658	−0.888290	+	0.459283i	$0.848106\pi$
$930$	0	0
$931$	4309.33	0.151700
$932$	0	0
$933$	0	0
$934$	0	0
$935$	18336.0	0.641339
$936$	0	0
$937$	−44253.7	−1.54291	−0.771454	−	0.636285i	$-0.780470\pi$
$937$	−44253.7	−1.54291	−0.771454	+	0.636285i	$0.780470\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−47352.1	−1.64042	−0.820210	−	0.572063i	$-0.806143\pi$
$941$	−47352.1	−1.64042	−0.820210	+	0.572063i	$0.806143\pi$
$942$	0	0
$943$	32503.2	1.12243
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−23964.1	−0.822311	−0.411156	−	0.911565i	$-0.634875\pi$
$947$	−23964.1	−0.822311	−0.411156	+	0.911565i	$0.634875\pi$
$948$	0	0
$949$	89207.3	3.05141
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−6697.41	−0.227650	−0.113825	−	0.993501i	$-0.536310\pi$
$953$	−6697.41	−0.227650	−0.113825	+	0.993501i	$0.536310\pi$
$954$	0	0
$955$	12647.8	0.428559
$956$	0	0
$957$	0	0
$958$	0	0
$959$	9885.70	0.332874
$960$	0	0
$961$	−2120.44	−0.0711770
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−20048.2	−0.668782
$966$	0	0
$967$	−36995.7	−1.23030	−0.615150	−	0.788410i	$-0.710905\pi$
$967$	−36995.7	−1.23030	−0.615150	+	0.788410i	$0.710905\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−32987.1	−1.09022	−0.545111	−	0.838364i	$-0.683513\pi$
$971$	−32987.1	−1.09022	−0.545111	+	0.838364i	$0.683513\pi$
$972$	0	0
$973$	15656.8	0.515864
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−19849.6	−0.649994	−0.324997	−	0.945715i	$-0.605363\pi$
$977$	−19849.6	−0.649994	−0.324997	+	0.945715i	$0.605363\pi$
$978$	0	0
$979$	18698.8	0.610435
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−13006.7	−0.422024	−0.211012	−	0.977484i	$-0.567676\pi$
$983$	−13006.7	−0.422024	−0.211012	+	0.977484i	$0.567676\pi$
$984$	0	0
$985$	−12448.3	−0.402675
$986$	0	0
$987$	0	0
$988$	0	0
$989$	32688.7	1.05100
$990$	0	0
$991$	14119.3	0.452588	0.226294	−	0.974059i	$-0.427339\pi$
$991$	14119.3	0.452588	0.226294	+	0.974059i	$0.427339\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	6794.55	0.216484
$996$	0	0
$997$	19916.9	0.632671	0.316336	−	0.948647i	$-0.397547\pi$
$997$	19916.9	0.632671	0.316336	+	0.948647i	$0.397547\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.a.j.1.1	yes	6
3.2	odd	2		1620.4.a.i.1.1	✓	6
9.2	odd	6		1620.4.i.x.1081.6		12
9.4	even	3		1620.4.i.w.541.6		12
9.5	odd	6		1620.4.i.x.541.6		12
9.7	even	3		1620.4.i.w.1081.6		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.a.i.1.1	✓	6	3.2	odd	2
1620.4.a.j.1.1	yes	6	1.1	even	1	trivial
1620.4.i.w.541.6		12	9.4	even	3
1620.4.i.w.1081.6		12	9.7	even	3
1620.4.i.x.541.6		12	9.5	odd	6
1620.4.i.x.1081.6		12	9.2	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1620.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} -23.3157 q^{7} +O(q^{10})\)	\(q+5.00000 q^{5} -23.3157 q^{7} +33.7490 q^{11} -88.8938 q^{13} +108.661 q^{17} +21.4799 q^{19} +85.9225 q^{23} +25.0000 q^{25} -18.9781 q^{29} -166.345 q^{31} -116.578 q^{35} -167.664 q^{37} +378.285 q^{41} +380.444 q^{43} +246.829 q^{47} +200.622 q^{49} -284.344 q^{53} +168.745 q^{55} -727.397 q^{59} +470.868 q^{61} -444.469 q^{65} -751.956 q^{67} -864.963 q^{71} -1003.53 q^{73} -786.881 q^{77} -838.501 q^{79} +476.114 q^{83} +543.306 q^{85} +554.055 q^{89} +2072.62 q^{91} +107.399 q^{95} -1047.99 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 30 q^{5} + 12 q^{7}+O(q^{10}) \) 6 * q + 30 * q^5 + 12 * q^7	\( 6 q + 30 q^{5} + 12 q^{7} - 84 q^{13} - 12 q^{17} - 114 q^{19} - 30 q^{23} + 150 q^{25} - 168 q^{29} - 324 q^{31} + 60 q^{35} - 492 q^{37} - 312 q^{41} - 156 q^{43} - 462 q^{47} - 588 q^{49} - 1014 q^{53} - 1008 q^{59} + 36 q^{61} - 420 q^{65} + 144 q^{67} - 1212 q^{71} - 900 q^{73} - 672 q^{77} - 936 q^{79} - 288 q^{83} - 60 q^{85} + 120 q^{89} + 2286 q^{91} - 570 q^{95} - 1188 q^{97}+O(q^{100}) \) 6 * q + 30 * q^5 + 12 * q^7 - 84 * q^13 - 12 * q^17 - 114 * q^19 - 30 * q^23 + 150 * q^25 - 168 * q^29 - 324 * q^31 + 60 * q^35 - 492 * q^37 - 312 * q^41 - 156 * q^43 - 462 * q^47 - 588 * q^49 - 1014 * q^53 - 1008 * q^59 + 36 * q^61 - 420 * q^65 + 144 * q^67 - 1212 * q^71 - 900 * q^73 - 672 * q^77 - 936 * q^79 - 288 * q^83 - 60 * q^85 + 120 * q^89 + 2286 * q^91 - 570 * q^95 - 1188 * q^97

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