LMFDB - Embedded newform 1620.4.a.d.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:	$3$
Coefficient field:	3.3.560145.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 90x + 192 $ x^3 - x^2 - 90*x + 192
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-9.96505$ 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)

$f(q)$	$=$	$q-5.00000 q^{5} -19.6337 q^{7} +O(q^{10})$	$q-5.00000 q^{5} -19.6337 q^{7} +29.1567 q^{11} -13.6337 q^{13} -39.0460 q^{17} +43.7443 q^{19} +159.214 q^{23} +25.0000 q^{25} -25.1450 q^{29} -116.737 q^{31} +98.1683 q^{35} +329.383 q^{37} +180.802 q^{41} -213.673 q^{43} -179.412 q^{47} +42.4809 q^{49} +60.8635 q^{53} -145.783 q^{55} -382.578 q^{59} +433.682 q^{61} +68.1683 q^{65} +125.107 q^{67} -30.0873 q^{71} +676.299 q^{73} -572.452 q^{77} -833.849 q^{79} -353.458 q^{83} +195.230 q^{85} -948.224 q^{89} +267.679 q^{91} -218.722 q^{95} -285.838 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 15 q^{5} + 15 q^{7}+O(q^{10}) $ 3 * q - 15 * q^5 + 15 * q^7	$ 3 q - 15 q^{5} + 15 q^{7} - 24 q^{11} + 33 q^{13} - 42 q^{17} + 21 q^{19} + 33 q^{23} + 75 q^{25} - 222 q^{29} + 132 q^{31} - 75 q^{35} + 174 q^{37} + 99 q^{41} - 120 q^{43} - 537 q^{47} + 492 q^{49} - 267 q^{53} + 120 q^{55} + 225 q^{59} - 480 q^{61} - 165 q^{65} + 12 q^{67} - 570 q^{71} + 1062 q^{73} - 312 q^{77} - 1026 q^{79} - 702 q^{83} + 210 q^{85} - 1140 q^{89} + 1611 q^{91} - 105 q^{95} - 2556 q^{97}+O(q^{100}) $ 3 * q - 15 * q^5 + 15 * q^7 - 24 * q^11 + 33 * q^13 - 42 * q^17 + 21 * q^19 + 33 * q^23 + 75 * q^25 - 222 * q^29 + 132 * q^31 - 75 * q^35 + 174 * q^37 + 99 * q^41 - 120 * q^43 - 537 * q^47 + 492 * q^49 - 267 * q^53 + 120 * q^55 + 225 * q^59 - 480 * q^61 - 165 * q^65 + 12 * q^67 - 570 * q^71 + 1062 * q^73 - 312 * q^77 - 1026 * q^79 - 702 * q^83 + 210 * q^85 - 1140 * q^89 + 1611 * q^91 - 105 * q^95 - 2556 * 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
$7$	−19.6337	−1.06012	−0.530059	−	0.847961i	$-0.677830\pi$
$7$	−19.6337	−1.06012	−0.530059	+	0.847961i	$0.677830\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	29.1567	0.799187	0.399594	−	0.916692i	$-0.369151\pi$
$11$	29.1567	0.799187	0.399594	+	0.916692i	$0.369151\pi$
$12$	0	0
$13$	−13.6337	−0.290869	−0.145435	−	0.989368i	$-0.546458\pi$
$13$	−13.6337	−0.290869	−0.145435	+	0.989368i	$0.546458\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−39.0460	−0.557061	−0.278531	−	0.960427i	$-0.589847\pi$
$17$	−39.0460	−0.557061	−0.278531	+	0.960427i	$0.589847\pi$
$18$	0	0
$19$	43.7443	0.528192	0.264096	−	0.964496i	$-0.414926\pi$
$19$	43.7443	0.528192	0.264096	+	0.964496i	$0.414926\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	159.214	1.44341	0.721706	−	0.692200i	$-0.243358\pi$
$23$	159.214	1.44341	0.721706	+	0.692200i	$0.243358\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−25.1450	−0.161010	−0.0805052	−	0.996754i	$-0.525653\pi$
$29$	−25.1450	−0.161010	−0.0805052	+	0.996754i	$0.525653\pi$
$30$	0	0
$31$	−116.737	−0.676343	−0.338172	−	0.941084i	$-0.609808\pi$
$31$	−116.737	−0.676343	−0.338172	+	0.941084i	$0.609808\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	98.1683	0.474099
$36$	0	0
$37$	329.383	1.46352	0.731759	−	0.681563i	$-0.238700\pi$
$37$	329.383	1.46352	0.731759	+	0.681563i	$0.238700\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	180.802	0.688696	0.344348	−	0.938842i	$-0.388100\pi$
$41$	180.802	0.688696	0.344348	+	0.938842i	$0.388100\pi$
$42$	0	0
$43$	−213.673	−0.757785	−0.378893	−	0.925441i	$-0.623695\pi$
$43$	−213.673	−0.757785	−0.378893	+	0.925441i	$0.623695\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−179.412	−0.556808	−0.278404	−	0.960464i	$-0.589805\pi$
$47$	−179.412	−0.556808	−0.278404	+	0.960464i	$0.589805\pi$
$48$	0	0
$49$	42.4809	0.123851
$50$	0	0
$51$	0	0
$52$	0	0
$53$	60.8635	0.157740	0.0788702	−	0.996885i	$-0.474869\pi$
$53$	60.8635	0.157740	0.0788702	+	0.996885i	$0.474869\pi$
$54$	0	0
$55$	−145.783	−0.357407
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−382.578	−0.844194	−0.422097	−	0.906551i	$-0.638706\pi$
$59$	−382.578	−0.844194	−0.422097	+	0.906551i	$0.638706\pi$
$60$	0	0
$61$	433.682	0.910283	0.455142	−	0.890419i	$-0.349589\pi$
$61$	433.682	0.910283	0.455142	+	0.890419i	$0.349589\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	68.1683	0.130081
$66$	0	0
$67$	125.107	0.228123	0.114061	−	0.993474i	$-0.463614\pi$
$67$	125.107	0.228123	0.114061	+	0.993474i	$0.463614\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−30.0873	−0.0502917	−0.0251458	−	0.999684i	$-0.508005\pi$
$71$	−30.0873	−0.0502917	−0.0251458	+	0.999684i	$0.508005\pi$
$72$	0	0
$73$	676.299	1.08431	0.542156	−	0.840278i	$-0.317608\pi$
$73$	676.299	1.08431	0.542156	+	0.840278i	$0.317608\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−572.452	−0.847233
$78$	0	0
$79$	−833.849	−1.18754	−0.593768	−	0.804636i	$-0.702360\pi$
$79$	−833.849	−1.18754	−0.593768	+	0.804636i	$0.702360\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−353.458	−0.467434	−0.233717	−	0.972305i	$-0.575089\pi$
$83$	−353.458	−0.467434	−0.233717	+	0.972305i	$0.575089\pi$
$84$	0	0
$85$	195.230	0.249125
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−948.224	−1.12934	−0.564671	−	0.825316i	$-0.690997\pi$
$89$	−948.224	−1.12934	−0.564671	+	0.825316i	$0.690997\pi$
$90$	0	0
$91$	267.679	0.308356
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−218.722	−0.236215
$96$	0	0
$97$	−285.838	−0.299200	−0.149600	−	0.988747i	$-0.547799\pi$
$97$	−285.838	−0.299200	−0.149600	+	0.988747i	$0.547799\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1474.71	−1.45286	−0.726430	−	0.687240i	$-0.758822\pi$
$101$	−1474.71	−1.45286	−0.726430	+	0.687240i	$0.758822\pi$
$102$	0	0
$103$	1755.30	1.67917	0.839586	−	0.543227i	$-0.182798\pi$
$103$	1755.30	1.67917	0.839586	+	0.543227i	$0.182798\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−657.811	−0.594327	−0.297164	−	0.954827i	$-0.596041\pi$
$107$	−657.811	−0.594327	−0.297164	+	0.954827i	$0.596041\pi$
$108$	0	0
$109$	−1042.07	−0.915706	−0.457853	−	0.889028i	$-0.651381\pi$
$109$	−1042.07	−0.915706	−0.457853	+	0.889028i	$0.651381\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	563.526	0.469133	0.234567	−	0.972100i	$-0.424633\pi$
$113$	563.526	0.469133	0.234567	+	0.972100i	$0.424633\pi$
$114$	0	0
$115$	−796.072	−0.645513
$116$	0	0
$117$	0	0
$118$	0	0
$119$	766.616	0.590551
$120$	0	0
$121$	−480.890	−0.361300
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	202.776	0.141681	0.0708406	−	0.997488i	$-0.477432\pi$
$127$	202.776	0.141681	0.0708406	+	0.997488i	$0.477432\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1111.89	−0.741575	−0.370788	−	0.928718i	$-0.620912\pi$
$131$	−1111.89	−0.741575	−0.370788	+	0.928718i	$0.620912\pi$
$132$	0	0
$133$	−858.862	−0.559946
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−545.406	−0.340125	−0.170063	−	0.985433i	$-0.554397\pi$
$137$	−545.406	−0.340125	−0.170063	+	0.985433i	$0.554397\pi$
$138$	0	0
$139$	−1356.57	−0.827792	−0.413896	−	0.910324i	$-0.635832\pi$
$139$	−1356.57	−0.827792	−0.413896	+	0.910324i	$0.635832\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−397.512	−0.232459
$144$	0	0
$145$	125.725	0.0720061
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3472.60	−1.90931	−0.954653	−	0.297719i	$-0.903774\pi$
$149$	−3472.60	−1.90931	−0.954653	+	0.297719i	$0.903774\pi$
$150$	0	0
$151$	−3403.40	−1.83420	−0.917101	−	0.398655i	$-0.869477\pi$
$151$	−3403.40	−1.83420	−0.917101	+	0.398655i	$0.869477\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	583.686	0.302470
$156$	0	0
$157$	−362.193	−0.184116	−0.0920579	−	0.995754i	$-0.529344\pi$
$157$	−362.193	−0.184116	−0.0920579	+	0.995754i	$0.529344\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3125.96	−1.53019
$162$	0	0
$163$	1830.32	0.879520	0.439760	−	0.898115i	$-0.355063\pi$
$163$	1830.32	0.879520	0.439760	+	0.898115i	$0.355063\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	835.518	0.387152	0.193576	−	0.981085i	$-0.437991\pi$
$167$	835.518	0.387152	0.193576	+	0.981085i	$0.437991\pi$
$168$	0	0
$169$	−2011.12	−0.915395
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1719.80	−0.755805	−0.377903	−	0.925845i	$-0.623355\pi$
$173$	−1719.80	−0.755805	−0.377903	+	0.925845i	$0.623355\pi$
$174$	0	0
$175$	−490.842	−0.212024
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2291.42	0.956809	0.478404	−	0.878140i	$-0.341215\pi$
$179$	2291.42	0.956809	0.478404	+	0.878140i	$0.341215\pi$
$180$	0	0
$181$	4228.30	1.73639	0.868197	−	0.496221i	$-0.165279\pi$
$181$	4228.30	1.73639	0.868197	+	0.496221i	$0.165279\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1646.91	−0.654505
$186$	0	0
$187$	−1138.45	−0.445196
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2046.44	−0.775264	−0.387632	−	0.921814i	$-0.626707\pi$
$191$	−2046.44	−0.775264	−0.387632	+	0.921814i	$0.626707\pi$
$192$	0	0
$193$	1096.92	0.409111	0.204555	−	0.978855i	$-0.434425\pi$
$193$	1096.92	0.409111	0.204555	+	0.978855i	$0.434425\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1593.40	−0.576269	−0.288135	−	0.957590i	$-0.593035\pi$
$197$	−1593.40	−0.576269	−0.288135	+	0.957590i	$0.593035\pi$
$198$	0	0
$199$	4936.25	1.75840	0.879201	−	0.476452i	$-0.158077\pi$
$199$	4936.25	1.75840	0.879201	+	0.476452i	$0.158077\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	493.688	0.170690
$204$	0	0
$205$	−904.010	−0.307994
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1275.44	0.422124
$210$	0	0
$211$	1329.42	0.433748	0.216874	−	0.976200i	$-0.430414\pi$
$211$	1329.42	0.433748	0.216874	+	0.976200i	$0.430414\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1068.36	0.338892
$216$	0	0
$217$	2291.98	0.717004
$218$	0	0
$219$	0	0
$220$	0	0
$221$	532.340	0.162032
$222$	0	0
$223$	−1024.82	−0.307745	−0.153872	−	0.988091i	$-0.549174\pi$
$223$	−1024.82	−0.307745	−0.153872	+	0.988091i	$0.549174\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−533.689	−0.156045	−0.0780224	−	0.996952i	$-0.524861\pi$
$227$	−533.689	−0.156045	−0.0780224	+	0.996952i	$0.524861\pi$
$228$	0	0
$229$	−51.0937	−0.0147440	−0.00737198	−	0.999973i	$-0.502347\pi$
$229$	−51.0937	−0.0147440	−0.00737198	+	0.999973i	$0.502347\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−509.227	−0.143178	−0.0715892	−	0.997434i	$-0.522807\pi$
$233$	−509.227	−0.143178	−0.0715892	+	0.997434i	$0.522807\pi$
$234$	0	0
$235$	897.062	0.249012
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2784.76	−0.753686	−0.376843	−	0.926277i	$-0.622990\pi$
$239$	−2784.76	−0.753686	−0.376843	+	0.926277i	$0.622990\pi$
$240$	0	0
$241$	−83.0194	−0.0221898	−0.0110949	−	0.999938i	$-0.503532\pi$
$241$	−83.0194	−0.0221898	−0.0110949	+	0.999938i	$0.503532\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−212.404	−0.0553878
$246$	0	0
$247$	−596.396	−0.153635
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1286.35	−0.323480	−0.161740	−	0.986833i	$-0.551711\pi$
$251$	−1286.35	−0.323480	−0.161740	+	0.986833i	$0.551711\pi$
$252$	0	0
$253$	4642.16	1.15356
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1544.13	−0.374786	−0.187393	−	0.982285i	$-0.560004\pi$
$257$	−1544.13	−0.374786	−0.187393	+	0.982285i	$0.560004\pi$
$258$	0	0
$259$	−6466.99	−1.55150
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8323.57	−1.95153	−0.975767	−	0.218811i	$-0.929782\pi$
$263$	−8323.57	−1.95153	−0.975767	+	0.218811i	$0.929782\pi$
$264$	0	0
$265$	−304.318	−0.0705437
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3203.80	−0.726168	−0.363084	−	0.931756i	$-0.618276\pi$
$269$	−3203.80	−0.726168	−0.363084	+	0.931756i	$0.618276\pi$
$270$	0	0
$271$	−1419.39	−0.318161	−0.159081	−	0.987266i	$-0.550853\pi$
$271$	−1419.39	−0.318161	−0.159081	+	0.987266i	$0.550853\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	728.916	0.159837
$276$	0	0
$277$	−8876.89	−1.92549	−0.962745	−	0.270410i	$-0.912841\pi$
$277$	−8876.89	−1.92549	−0.962745	+	0.270410i	$0.912841\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2606.54	0.553357	0.276679	−	0.960963i	$-0.410766\pi$
$281$	2606.54	0.553357	0.276679	+	0.960963i	$0.410766\pi$
$282$	0	0
$283$	1406.78	0.295492	0.147746	−	0.989025i	$-0.452798\pi$
$283$	1406.78	0.295492	0.147746	+	0.989025i	$0.452798\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3549.81	−0.730099
$288$	0	0
$289$	−3388.41	−0.689683
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−7170.01	−1.42961	−0.714806	−	0.699323i	$-0.753485\pi$
$293$	−7170.01	−1.42961	−0.714806	+	0.699323i	$0.753485\pi$
$294$	0	0
$295$	1912.89	0.377535
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2170.67	−0.419844
$300$	0	0
$301$	4195.18	0.803342
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2168.41	−0.407091
$306$	0	0
$307$	3695.22	0.686963	0.343481	−	0.939160i	$-0.388394\pi$
$307$	3695.22	0.686963	0.343481	+	0.939160i	$0.388394\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6085.66	1.10960	0.554801	−	0.831983i	$-0.312794\pi$
$311$	6085.66	1.10960	0.554801	+	0.831983i	$0.312794\pi$
$312$	0	0
$313$	1774.57	0.320462	0.160231	−	0.987080i	$-0.448776\pi$
$313$	1774.57	0.320462	0.160231	+	0.987080i	$0.448776\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8353.53	−1.48007	−0.740033	−	0.672571i	$-0.765190\pi$
$317$	−8353.53	−1.48007	−0.740033	+	0.672571i	$0.765190\pi$
$318$	0	0
$319$	−733.143	−0.128678
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1708.04	−0.294235
$324$	0	0
$325$	−340.842	−0.0581738
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3522.52	0.590282
$330$	0	0
$331$	−3785.80	−0.628661	−0.314330	−	0.949314i	$-0.601780\pi$
$331$	−3785.80	−0.628661	−0.314330	+	0.949314i	$0.601780\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−625.534	−0.102020
$336$	0	0
$337$	7463.09	1.20635	0.603176	−	0.797608i	$-0.293902\pi$
$337$	7463.09	1.20635	0.603176	+	0.797608i	$0.293902\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3403.67	−0.540525
$342$	0	0
$343$	5900.29	0.928822
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3421.06	−0.529257	−0.264629	−	0.964350i	$-0.585249\pi$
$347$	−3421.06	−0.529257	−0.264629	+	0.964350i	$0.585249\pi$
$348$	0	0
$349$	−2600.81	−0.398906	−0.199453	−	0.979907i	$-0.563917\pi$
$349$	−2600.81	−0.398906	−0.199453	+	0.979907i	$0.563917\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2217.78	0.334392	0.167196	−	0.985924i	$-0.446529\pi$
$353$	2217.78	0.334392	0.167196	+	0.985924i	$0.446529\pi$
$354$	0	0
$355$	150.437	0.0224911
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10423.6	−1.53241	−0.766207	−	0.642594i	$-0.777858\pi$
$359$	−10423.6	−1.53241	−0.766207	+	0.642594i	$0.777858\pi$
$360$	0	0
$361$	−4945.43	−0.721014
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3381.50	−0.484919
$366$	0	0
$367$	31.9363	0.00454239	0.00227120	−	0.999997i	$-0.499277\pi$
$367$	31.9363	0.00454239	0.00227120	+	0.999997i	$0.499277\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1194.97	−0.167224
$372$	0	0
$373$	233.348	0.0323923	0.0161961	−	0.999869i	$-0.494844\pi$
$373$	233.348	0.0323923	0.0161961	+	0.999869i	$0.494844\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	342.818	0.0468330
$378$	0	0
$379$	5558.90	0.753408	0.376704	−	0.926334i	$-0.377057\pi$
$379$	5558.90	0.753408	0.376704	+	0.926334i	$0.377057\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−8164.24	−1.08923	−0.544613	−	0.838688i	$-0.683323\pi$
$383$	−8164.24	−1.08923	−0.544613	+	0.838688i	$0.683323\pi$
$384$	0	0
$385$	2862.26	0.378894
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−10197.1	−1.32908	−0.664542	−	0.747251i	$-0.731373\pi$
$389$	−10197.1	−1.32908	−0.664542	+	0.747251i	$0.731373\pi$
$390$	0	0
$391$	−6216.68	−0.804069
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4169.24	0.531082
$396$	0	0
$397$	−2601.86	−0.328926	−0.164463	−	0.986383i	$-0.552589\pi$
$397$	−2601.86	−0.328926	−0.164463	+	0.986383i	$0.552589\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5185.27	−0.645736	−0.322868	−	0.946444i	$-0.604647\pi$
$401$	−5185.27	−0.645736	−0.322868	+	0.946444i	$0.604647\pi$
$402$	0	0
$403$	1591.56	0.196727
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9603.69	1.16963
$408$	0	0
$409$	7183.47	0.868459	0.434229	−	0.900802i	$-0.357021\pi$
$409$	7183.47	0.868459	0.434229	+	0.900802i	$0.357021\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7511.41	0.894945
$414$	0	0
$415$	1767.29	0.209043
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−683.229	−0.0796609	−0.0398304	−	0.999206i	$-0.512682\pi$
$419$	−683.229	−0.0796609	−0.0398304	+	0.999206i	$0.512682\pi$
$420$	0	0
$421$	−10745.1	−1.24390	−0.621951	−	0.783056i	$-0.713660\pi$
$421$	−10745.1	−1.24390	−0.621951	+	0.783056i	$0.713660\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−976.149	−0.111412
$426$	0	0
$427$	−8514.76	−0.965008
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1055.60	0.117974	0.0589869	−	0.998259i	$-0.481213\pi$
$431$	1055.60	0.117974	0.0589869	+	0.998259i	$0.481213\pi$
$432$	0	0
$433$	−4536.95	−0.503538	−0.251769	−	0.967787i	$-0.581012\pi$
$433$	−4536.95	−0.503538	−0.251769	+	0.967787i	$0.581012\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6964.73	0.762398
$438$	0	0
$439$	7567.85	0.822765	0.411383	−	0.911463i	$-0.365046\pi$
$439$	7567.85	0.822765	0.411383	+	0.911463i	$0.365046\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	10345.1	1.10951	0.554754	−	0.832014i	$-0.312812\pi$
$443$	10345.1	1.10951	0.554754	+	0.832014i	$0.312812\pi$
$444$	0	0
$445$	4741.12	0.505057
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11122.4	1.16904	0.584520	−	0.811379i	$-0.301283\pi$
$449$	11122.4	1.16904	0.584520	+	0.811379i	$0.301283\pi$
$450$	0	0
$451$	5271.58	0.550397
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1338.39	−0.137901
$456$	0	0
$457$	−1738.45	−0.177946	−0.0889728	−	0.996034i	$-0.528358\pi$
$457$	−1738.45	−0.177946	−0.0889728	+	0.996034i	$0.528358\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7572.75	0.765072	0.382536	−	0.923941i	$-0.375051\pi$
$461$	7572.75	0.765072	0.382536	+	0.923941i	$0.375051\pi$
$462$	0	0
$463$	−12717.9	−1.27657	−0.638285	−	0.769801i	$-0.720356\pi$
$463$	−12717.9	−1.27657	−0.638285	+	0.769801i	$0.720356\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4641.96	0.459967	0.229983	−	0.973195i	$-0.426133\pi$
$467$	4641.96	0.459967	0.229983	+	0.973195i	$0.426133\pi$
$468$	0	0
$469$	−2456.31	−0.241837
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6229.98	−0.605612
$474$	0	0
$475$	1093.61	0.105638
$476$	0	0
$477$	0	0
$478$	0	0
$479$	5635.96	0.537607	0.268804	−	0.963195i	$-0.413372\pi$
$479$	5635.96	0.537607	0.268804	+	0.963195i	$0.413372\pi$
$480$	0	0
$481$	−4490.69	−0.425692
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1429.19	0.133807
$486$	0	0
$487$	−1557.90	−0.144960	−0.0724799	−	0.997370i	$-0.523091\pi$
$487$	−1557.90	−0.144960	−0.0724799	+	0.997370i	$0.523091\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6749.03	0.620324	0.310162	−	0.950684i	$-0.399617\pi$
$491$	6749.03	0.620324	0.310162	+	0.950684i	$0.399617\pi$
$492$	0	0
$493$	981.810	0.0896927
$494$	0	0
$495$	0	0
$496$	0	0
$497$	590.724	0.0533151
$498$	0	0
$499$	2229.48	0.200011	0.100005	−	0.994987i	$-0.468114\pi$
$499$	2229.48	0.200011	0.100005	+	0.994987i	$0.468114\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	13685.7	1.21315	0.606577	−	0.795024i	$-0.292542\pi$
$503$	13685.7	1.21315	0.606577	+	0.795024i	$0.292542\pi$
$504$	0	0
$505$	7373.54	0.649739
$506$	0	0
$507$	0	0
$508$	0	0
$509$	13627.4	1.18669	0.593346	−	0.804948i	$-0.297807\pi$
$509$	13627.4	1.18669	0.593346	+	0.804948i	$0.297807\pi$
$510$	0	0
$511$	−13278.2	−1.14950
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8776.49	−0.750948
$516$	0	0
$517$	−5231.06	−0.444994
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−21425.1	−1.80163	−0.900817	−	0.434200i	$-0.857031\pi$
$521$	−21425.1	−1.80163	−0.900817	+	0.434200i	$0.857031\pi$
$522$	0	0
$523$	8954.62	0.748677	0.374338	−	0.927292i	$-0.377870\pi$
$523$	8954.62	0.748677	0.374338	+	0.927292i	$0.377870\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4558.12	0.376764
$528$	0	0
$529$	13182.2	1.08344
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2464.99	−0.200320
$534$	0	0
$535$	3289.06	0.265791
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1238.60	0.0989801
$540$	0	0
$541$	−10982.6	−0.872788	−0.436394	−	0.899756i	$-0.643745\pi$
$541$	−10982.6	−0.872788	−0.436394	+	0.899756i	$0.643745\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5210.34	0.409516
$546$	0	0
$547$	15213.2	1.18916	0.594580	−	0.804036i	$-0.297318\pi$
$547$	15213.2	1.18916	0.594580	+	0.804036i	$0.297318\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1099.95	−0.0850444
$552$	0	0
$553$	16371.5	1.25893
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−12732.7	−0.968588	−0.484294	−	0.874905i	$-0.660923\pi$
$557$	−12732.7	−0.968588	−0.484294	+	0.874905i	$0.660923\pi$
$558$	0	0
$559$	2913.14	0.220416
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10889.2	−0.815144	−0.407572	−	0.913173i	$-0.633624\pi$
$563$	−10889.2	−0.815144	−0.407572	+	0.913173i	$0.633624\pi$
$564$	0	0
$565$	−2817.63	−0.209803
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1568.02	0.115527	0.0577634	−	0.998330i	$-0.481603\pi$
$569$	1568.02	0.115527	0.0577634	+	0.998330i	$0.481603\pi$
$570$	0	0
$571$	−24418.3	−1.78962	−0.894809	−	0.446448i	$-0.852689\pi$
$571$	−24418.3	−1.78962	−0.894809	+	0.446448i	$0.852689\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3980.36	0.288682
$576$	0	0
$577$	−21305.8	−1.53721	−0.768606	−	0.639722i	$-0.779049\pi$
$577$	−21305.8	−1.53721	−0.768606	+	0.639722i	$0.779049\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6939.67	0.495535
$582$	0	0
$583$	1774.58	0.126064
$584$	0	0
$585$	0	0
$586$	0	0
$587$	25462.9	1.79040	0.895202	−	0.445661i	$-0.147031\pi$
$587$	25462.9	1.79040	0.895202	+	0.445661i	$0.147031\pi$
$588$	0	0
$589$	−5106.60	−0.357239
$590$	0	0
$591$	0	0
$592$	0	0
$593$	26914.5	1.86382	0.931911	−	0.362686i	$-0.118140\pi$
$593$	26914.5	1.86382	0.931911	+	0.362686i	$0.118140\pi$
$594$	0	0
$595$	−3833.08	−0.264102
$596$	0	0
$597$	0	0
$598$	0	0
$599$	20952.5	1.42921	0.714604	−	0.699530i	$-0.246607\pi$
$599$	20952.5	1.42921	0.714604	+	0.699530i	$0.246607\pi$
$600$	0	0
$601$	25791.8	1.75053	0.875265	−	0.483644i	$-0.160687\pi$
$601$	25791.8	1.75053	0.875265	+	0.483644i	$0.160687\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2404.45	0.161578
$606$	0	0
$607$	−6536.97	−0.437113	−0.218556	−	0.975824i	$-0.570135\pi$
$607$	−6536.97	−0.437113	−0.218556	+	0.975824i	$0.570135\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2446.05	0.161958
$612$	0	0
$613$	−22102.9	−1.45633	−0.728163	−	0.685404i	$-0.759626\pi$
$613$	−22102.9	−1.45633	−0.728163	+	0.685404i	$0.759626\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2787.94	0.181910	0.0909549	−	0.995855i	$-0.471008\pi$
$617$	2787.94	0.181910	0.0909549	+	0.995855i	$0.471008\pi$
$618$	0	0
$619$	7802.59	0.506644	0.253322	−	0.967382i	$-0.418477\pi$
$619$	7802.59	0.506644	0.253322	+	0.967382i	$0.418477\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	18617.1	1.19724
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−12861.1	−0.815269
$630$	0	0
$631$	−18267.8	−1.15251	−0.576253	−	0.817272i	$-0.695486\pi$
$631$	−18267.8	−1.15251	−0.576253	+	0.817272i	$0.695486\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−1013.88	−0.0633617
$636$	0	0
$637$	−579.170	−0.0360244
$638$	0	0
$639$	0	0
$640$	0	0
$641$	3608.97	0.222380	0.111190	−	0.993799i	$-0.464534\pi$
$641$	3608.97	0.222380	0.111190	+	0.993799i	$0.464534\pi$
$642$	0	0
$643$	19851.3	1.21751	0.608756	−	0.793357i	$-0.291669\pi$
$643$	19851.3	1.21751	0.608756	+	0.793357i	$0.291669\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−9352.11	−0.568268	−0.284134	−	0.958785i	$-0.591706\pi$
$647$	−9352.11	−0.568268	−0.284134	+	0.958785i	$0.591706\pi$
$648$	0	0
$649$	−11154.7	−0.674669
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−26216.5	−1.57111	−0.785553	−	0.618794i	$-0.787622\pi$
$653$	−26216.5	−1.57111	−0.785553	+	0.618794i	$0.787622\pi$
$654$	0	0
$655$	5559.45	0.331642
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−15383.8	−0.909359	−0.454679	−	0.890655i	$-0.650246\pi$
$659$	−15383.8	−0.909359	−0.454679	+	0.890655i	$0.650246\pi$
$660$	0	0
$661$	13711.0	0.806800	0.403400	−	0.915024i	$-0.367828\pi$
$661$	13711.0	0.806800	0.403400	+	0.915024i	$0.367828\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	4294.31	0.250415
$666$	0	0
$667$	−4003.44	−0.232404
$668$	0	0
$669$	0	0
$670$	0	0
$671$	12644.7	0.727487
$672$	0	0
$673$	27796.8	1.59211	0.796054	−	0.605226i	$-0.206917\pi$
$673$	27796.8	1.59211	0.796054	+	0.605226i	$0.206917\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28593.4	−1.62324	−0.811621	−	0.584185i	$-0.801414\pi$
$677$	−28593.4	−1.62324	−0.811621	+	0.584185i	$0.801414\pi$
$678$	0	0
$679$	5612.05	0.317188
$680$	0	0
$681$	0	0
$682$	0	0
$683$	33510.5	1.87737	0.938686	−	0.344774i	$-0.112044\pi$
$683$	33510.5	1.87737	0.938686	+	0.344774i	$0.112044\pi$
$684$	0	0
$685$	2727.03	0.152109
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−829.793	−0.0458818
$690$	0	0
$691$	9569.98	0.526858	0.263429	−	0.964679i	$-0.415146\pi$
$691$	9569.98	0.526858	0.263429	+	0.964679i	$0.415146\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6782.87	0.370200
$696$	0	0
$697$	−7059.59	−0.383646
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−6945.59	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$701$	−6945.59	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$702$	0	0
$703$	14408.6	0.773018
$704$	0	0
$705$	0	0
$706$	0	0
$707$	28953.9	1.54020
$708$	0	0
$709$	5938.26	0.314550	0.157275	−	0.987555i	$-0.449729\pi$
$709$	5938.26	0.314550	0.157275	+	0.987555i	$0.449729\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−18586.2	−0.976242
$714$	0	0
$715$	1987.56	0.103959
$716$	0	0
$717$	0	0
$718$	0	0
$719$	7669.30	0.397798	0.198899	−	0.980020i	$-0.436263\pi$
$719$	7669.30	0.397798	0.198899	+	0.980020i	$0.436263\pi$
$720$	0	0
$721$	−34462.9	−1.78012
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−628.624	−0.0322021
$726$	0	0
$727$	7859.47	0.400951	0.200476	−	0.979699i	$-0.435751\pi$
$727$	7859.47	0.400951	0.200476	+	0.979699i	$0.435751\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	8343.05	0.422133
$732$	0	0
$733$	30208.8	1.52222	0.761111	−	0.648621i	$-0.224654\pi$
$733$	30208.8	1.52222	0.761111	+	0.648621i	$0.224654\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3647.70	0.182313
$738$	0	0
$739$	6648.76	0.330959	0.165480	−	0.986213i	$-0.447083\pi$
$739$	6648.76	0.330959	0.165480	+	0.986213i	$0.447083\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7646.68	0.377563	0.188782	−	0.982019i	$-0.439546\pi$
$743$	7646.68	0.377563	0.188782	+	0.982019i	$0.439546\pi$
$744$	0	0
$745$	17363.0	0.853868
$746$	0	0
$747$	0	0
$748$	0	0
$749$	12915.2	0.630057
$750$	0	0
$751$	−31626.9	−1.53673	−0.768363	−	0.640015i	$-0.778928\pi$
$751$	−31626.9	−1.53673	−0.768363	+	0.640015i	$0.778928\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	17017.0	0.820280
$756$	0	0
$757$	−6859.10	−0.329324	−0.164662	−	0.986350i	$-0.552653\pi$
$757$	−6859.10	−0.329324	−0.164662	+	0.986350i	$0.552653\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17978.6	0.856405	0.428202	−	0.903683i	$-0.359147\pi$
$761$	17978.6	0.856405	0.428202	+	0.903683i	$0.359147\pi$
$762$	0	0
$763$	20459.6	0.970757
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5215.94	0.245550
$768$	0	0
$769$	−15169.0	−0.711326	−0.355663	−	0.934614i	$-0.615745\pi$
$769$	−15169.0	−0.711326	−0.355663	+	0.934614i	$0.615745\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2921.73	0.135947	0.0679736	−	0.997687i	$-0.478347\pi$
$773$	2921.73	0.135947	0.0679736	+	0.997687i	$0.478347\pi$
$774$	0	0
$775$	−2918.43	−0.135269
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7909.07	0.363763
$780$	0	0
$781$	−877.245	−0.0401925
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1810.97	0.0823390
$786$	0	0
$787$	16054.8	0.727180	0.363590	−	0.931559i	$-0.381551\pi$
$787$	16054.8	0.727180	0.363590	+	0.931559i	$0.381551\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−11064.1	−0.497337
$792$	0	0
$793$	−5912.67	−0.264773
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−17435.0	−0.774880	−0.387440	−	0.921895i	$-0.626640\pi$
$797$	−17435.0	−0.774880	−0.387440	+	0.921895i	$0.626640\pi$
$798$	0	0
$799$	7005.33	0.310176
$800$	0	0
$801$	0	0
$802$	0	0
$803$	19718.6	0.866569
$804$	0	0
$805$	15629.8	0.684321
$806$	0	0
$807$	0	0
$808$	0	0
$809$	22185.0	0.964132	0.482066	−	0.876135i	$-0.339887\pi$
$809$	22185.0	0.964132	0.482066	+	0.876135i	$0.339887\pi$
$810$	0	0
$811$	−18738.5	−0.811341	−0.405671	−	0.914019i	$-0.632962\pi$
$811$	−18738.5	−0.811341	−0.405671	+	0.914019i	$0.632962\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9151.61	−0.393333
$816$	0	0
$817$	−9346.97	−0.400256
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16408.7	0.697524	0.348762	−	0.937211i	$-0.386602\pi$
$821$	16408.7	0.697524	0.348762	+	0.937211i	$0.386602\pi$
$822$	0	0
$823$	−7816.24	−0.331053	−0.165527	−	0.986205i	$-0.552932\pi$
$823$	−7816.24	−0.331053	−0.165527	+	0.986205i	$0.552932\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−12444.2	−0.523248	−0.261624	−	0.965170i	$-0.584258\pi$
$827$	−12444.2	−0.523248	−0.261624	+	0.965170i	$0.584258\pi$
$828$	0	0
$829$	−21850.1	−0.915422	−0.457711	−	0.889101i	$-0.651331\pi$
$829$	−21850.1	−0.915422	−0.457711	+	0.889101i	$0.651331\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1658.71	−0.0689925
$834$	0	0
$835$	−4177.59	−0.173139
$836$	0	0
$837$	0	0
$838$	0	0
$839$	38891.4	1.60033	0.800166	−	0.599779i	$-0.204745\pi$
$839$	38891.4	1.60033	0.800166	+	0.599779i	$0.204745\pi$
$840$	0	0
$841$	−23756.7	−0.974076
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10055.6	0.409377
$846$	0	0
$847$	9441.63	0.383020
$848$	0	0
$849$	0	0
$850$	0	0
$851$	52442.4	2.11246
$852$	0	0
$853$	29331.4	1.17736	0.588680	−	0.808366i	$-0.299648\pi$
$853$	29331.4	1.17736	0.588680	+	0.808366i	$0.299648\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	14018.2	0.558755	0.279377	−	0.960181i	$-0.409872\pi$
$857$	14018.2	0.558755	0.279377	+	0.960181i	$0.409872\pi$
$858$	0	0
$859$	−7816.63	−0.310477	−0.155239	−	0.987877i	$-0.549615\pi$
$859$	−7816.63	−0.310477	−0.155239	+	0.987877i	$0.549615\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	39764.9	1.56850	0.784248	−	0.620448i	$-0.213049\pi$
$863$	39764.9	1.56850	0.784248	+	0.620448i	$0.213049\pi$
$864$	0	0
$865$	8599.02	0.338006
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−24312.2	−0.949063
$870$	0	0
$871$	−1705.66	−0.0663539
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2454.21	0.0948199
$876$	0	0
$877$	−36718.0	−1.41377	−0.706887	−	0.707327i	$-0.749901\pi$
$877$	−36718.0	−1.41377	−0.706887	+	0.707327i	$0.749901\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−49326.5	−1.88633	−0.943163	−	0.332331i	$-0.892165\pi$
$881$	−49326.5	−1.88633	−0.943163	+	0.332331i	$0.892165\pi$
$882$	0	0
$883$	32093.1	1.22312	0.611562	−	0.791197i	$-0.290542\pi$
$883$	32093.1	1.22312	0.611562	+	0.791197i	$0.290542\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	21150.8	0.800648	0.400324	−	0.916374i	$-0.368898\pi$
$887$	21150.8	0.800648	0.400324	+	0.916374i	$0.368898\pi$
$888$	0	0
$889$	−3981.25	−0.150199
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−7848.27	−0.294101
$894$	0	0
$895$	−11457.1	−0.427898
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2935.36	0.108898
$900$	0	0
$901$	−2376.47	−0.0878711
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−21141.5	−0.776539
$906$	0	0
$907$	−17582.6	−0.643682	−0.321841	−	0.946794i	$-0.604302\pi$
$907$	−17582.6	−0.643682	−0.321841	+	0.946794i	$0.604302\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	13698.9	0.498205	0.249102	−	0.968477i	$-0.419864\pi$
$911$	13698.9	0.498205	0.249102	+	0.968477i	$0.419864\pi$
$912$	0	0
$913$	−10305.6	−0.373567
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21830.5	0.786157
$918$	0	0
$919$	−14632.7	−0.525231	−0.262616	−	0.964901i	$-0.584585\pi$
$919$	−14632.7	−0.525231	−0.262616	+	0.964901i	$0.584585\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	410.200	0.0146283
$924$	0	0
$925$	8234.57	0.292704
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−15062.5	−0.531953	−0.265976	−	0.963980i	$-0.585694\pi$
$929$	−15062.5	−0.531953	−0.265976	+	0.963980i	$0.585694\pi$
$930$	0	0
$931$	1858.30	0.0654170
$932$	0	0
$933$	0	0
$934$	0	0
$935$	5692.25	0.199098
$936$	0	0
$937$	−52963.9	−1.84659	−0.923296	−	0.384090i	$-0.874515\pi$
$937$	−52963.9	−1.84659	−0.923296	+	0.384090i	$0.874515\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	24862.0	0.861295	0.430647	−	0.902520i	$-0.358285\pi$
$941$	24862.0	0.861295	0.430647	+	0.902520i	$0.358285\pi$
$942$	0	0
$943$	28786.3	0.994072
$944$	0	0
$945$	0	0
$946$	0	0
$947$	29796.4	1.02244	0.511221	−	0.859449i	$-0.329193\pi$
$947$	29796.4	1.02244	0.511221	+	0.859449i	$0.329193\pi$
$948$	0	0
$949$	−9220.44	−0.315393
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10042.2	0.341341	0.170670	−	0.985328i	$-0.445407\pi$
$953$	10042.2	0.341341	0.170670	+	0.985328i	$0.445407\pi$
$954$	0	0
$955$	10232.2	0.346708
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10708.3	0.360573
$960$	0	0
$961$	−16163.4	−0.542560
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5484.62	−0.182960
$966$	0	0
$967$	−25075.4	−0.833890	−0.416945	−	0.908932i	$-0.636899\pi$
$967$	−25075.4	−0.833890	−0.416945	+	0.908932i	$0.636899\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	6632.06	0.219189	0.109595	−	0.993976i	$-0.465045\pi$
$971$	6632.06	0.219189	0.109595	+	0.993976i	$0.465045\pi$
$972$	0	0
$973$	26634.5	0.877557
$974$	0	0
$975$	0	0
$976$	0	0
$977$	40968.0	1.34154	0.670768	−	0.741667i	$-0.265965\pi$
$977$	40968.0	1.34154	0.670768	+	0.741667i	$0.265965\pi$
$978$	0	0
$979$	−27647.0	−0.902556
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−57442.6	−1.86382	−0.931910	−	0.362690i	$-0.881858\pi$
$983$	−57442.6	−1.86382	−0.931910	+	0.362690i	$0.881858\pi$
$984$	0	0
$985$	7967.00	0.257715
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−34019.7	−1.09380
$990$	0	0
$991$	34805.7	1.11568	0.557840	−	0.829948i	$-0.311630\pi$
$991$	34805.7	1.11568	0.557840	+	0.829948i	$0.311630\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−24681.3	−0.786381
$996$	0	0
$997$	56989.6	1.81031	0.905155	−	0.425083i	$-0.139755\pi$
$997$	56989.6	1.81031	0.905155	+	0.425083i	$0.139755\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.4.a.d.1.1	✓	3
3.2	odd	2		1620.4.a.f.1.1	yes	3
9.2	odd	6		1620.4.i.s.1081.3		6
9.4	even	3		1620.4.i.u.541.3		6
9.5	odd	6		1620.4.i.s.541.3		6
9.7	even	3		1620.4.i.u.1081.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.4.a.d.1.1	✓	3	1.1	even	1	trivial
1620.4.a.f.1.1	yes	3	3.2	odd	2
1620.4.i.s.541.3		6	9.5	odd	6
1620.4.i.s.1081.3		6	9.2	odd	6
1620.4.i.u.541.3		6	9.4	even	3
1620.4.i.u.1081.3		6	9.7	even	3

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)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} -19.6337 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} -19.6337 q^{7} +29.1567 q^{11} -13.6337 q^{13} -39.0460 q^{17} +43.7443 q^{19} +159.214 q^{23} +25.0000 q^{25} -25.1450 q^{29} -116.737 q^{31} +98.1683 q^{35} +329.383 q^{37} +180.802 q^{41} -213.673 q^{43} -179.412 q^{47} +42.4809 q^{49} +60.8635 q^{53} -145.783 q^{55} -382.578 q^{59} +433.682 q^{61} +68.1683 q^{65} +125.107 q^{67} -30.0873 q^{71} +676.299 q^{73} -572.452 q^{77} -833.849 q^{79} -353.458 q^{83} +195.230 q^{85} -948.224 q^{89} +267.679 q^{91} -218.722 q^{95} -285.838 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 15 q^{5} + 15 q^{7}+O(q^{10}) \) 3 * q - 15 * q^5 + 15 * q^7	\( 3 q - 15 q^{5} + 15 q^{7} - 24 q^{11} + 33 q^{13} - 42 q^{17} + 21 q^{19} + 33 q^{23} + 75 q^{25} - 222 q^{29} + 132 q^{31} - 75 q^{35} + 174 q^{37} + 99 q^{41} - 120 q^{43} - 537 q^{47} + 492 q^{49} - 267 q^{53} + 120 q^{55} + 225 q^{59} - 480 q^{61} - 165 q^{65} + 12 q^{67} - 570 q^{71} + 1062 q^{73} - 312 q^{77} - 1026 q^{79} - 702 q^{83} + 210 q^{85} - 1140 q^{89} + 1611 q^{91} - 105 q^{95} - 2556 q^{97}+O(q^{100}) \) 3 * q - 15 * q^5 + 15 * q^7 - 24 * q^11 + 33 * q^13 - 42 * q^17 + 21 * q^19 + 33 * q^23 + 75 * q^25 - 222 * q^29 + 132 * q^31 - 75 * q^35 + 174 * q^37 + 99 * q^41 - 120 * q^43 - 537 * q^47 + 492 * q^49 - 267 * q^53 + 120 * q^55 + 225 * q^59 - 480 * q^61 - 165 * q^65 + 12 * q^67 - 570 * q^71 + 1062 * q^73 - 312 * q^77 - 1026 * q^79 - 702 * q^83 + 210 * q^85 - 1140 * q^89 + 1611 * q^91 - 105 * q^95 - 2556 * q^97

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