LMFDB - Embedded newform 1575.4.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,4,Mod(1,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, 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("1575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1575.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:	$92.9280082590$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{65}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 16 $ x^2 - x - 16
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.53113$ of defining polynomial
Character	$\chi$	$=$	1575.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.53113 q^{2} +12.5311 q^{4} +7.00000 q^{7} +20.5311 q^{8} +O(q^{10})$	$q+4.53113 q^{2} +12.5311 q^{4} +7.00000 q^{7} +20.5311 q^{8} +19.0623 q^{11} +2.93774 q^{13} +31.7179 q^{14} -7.21984 q^{16} -6.49806 q^{17} -5.43580 q^{19} +86.3735 q^{22} +49.3774 q^{23} +13.3113 q^{26} +87.7179 q^{28} +291.494 q^{29} +244.307 q^{31} -196.963 q^{32} -29.4436 q^{34} +193.121 q^{37} -24.6303 q^{38} -315.113 q^{41} +300.996 q^{43} +238.872 q^{44} +223.735 q^{46} +86.5058 q^{47} +49.0000 q^{49} +36.8132 q^{52} +509.677 q^{53} +143.718 q^{56} +1320.80 q^{58} +83.3852 q^{59} -5.25291 q^{61} +1106.99 q^{62} -834.706 q^{64} -205.992 q^{67} -81.4281 q^{68} -1004.31 q^{71} +1007.29 q^{73} +875.055 q^{74} -68.1168 q^{76} +133.436 q^{77} -863.237 q^{79} -1427.82 q^{82} +1334.72 q^{83} +1363.85 q^{86} +391.370 q^{88} -326.249 q^{89} +20.5642 q^{91} +618.755 q^{92} +391.969 q^{94} -1526.77 q^{97} +222.025 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 17 q^{4} + 14 q^{7} + 33 q^{8}+O(q^{10}) $ 2 * q + q^2 + 17 * q^4 + 14 * q^7 + 33 * q^8	$ 2 q + q^{2} + 17 q^{4} + 14 q^{7} + 33 q^{8} + 22 q^{11} + 22 q^{13} + 7 q^{14} - 87 q^{16} + 116 q^{17} + 102 q^{19} + 76 q^{22} + 260 q^{23} - 54 q^{26} + 119 q^{28} + 196 q^{29} + 150 q^{31} - 15 q^{32} - 462 q^{34} + 96 q^{37} - 404 q^{38} + 176 q^{41} + 344 q^{43} + 252 q^{44} - 520 q^{46} + 560 q^{47} + 98 q^{49} + 122 q^{52} + 326 q^{53} + 231 q^{56} + 1658 q^{58} + 844 q^{59} - 204 q^{61} + 1440 q^{62} - 839 q^{64} + 104 q^{67} + 466 q^{68} - 1670 q^{71} + 386 q^{73} + 1218 q^{74} + 412 q^{76} + 154 q^{77} - 888 q^{79} - 3162 q^{82} + 928 q^{83} + 1212 q^{86} + 428 q^{88} - 588 q^{89} + 154 q^{91} + 1560 q^{92} - 1280 q^{94} - 522 q^{97} + 49 q^{98}+O(q^{100}) $ 2 * q + q^2 + 17 * q^4 + 14 * q^7 + 33 * q^8 + 22 * q^11 + 22 * q^13 + 7 * q^14 - 87 * q^16 + 116 * q^17 + 102 * q^19 + 76 * q^22 + 260 * q^23 - 54 * q^26 + 119 * q^28 + 196 * q^29 + 150 * q^31 - 15 * q^32 - 462 * q^34 + 96 * q^37 - 404 * q^38 + 176 * q^41 + 344 * q^43 + 252 * q^44 - 520 * q^46 + 560 * q^47 + 98 * q^49 + 122 * q^52 + 326 * q^53 + 231 * q^56 + 1658 * q^58 + 844 * q^59 - 204 * q^61 + 1440 * q^62 - 839 * q^64 + 104 * q^67 + 466 * q^68 - 1670 * q^71 + 386 * q^73 + 1218 * q^74 + 412 * q^76 + 154 * q^77 - 888 * q^79 - 3162 * q^82 + 928 * q^83 + 1212 * q^86 + 428 * q^88 - 588 * q^89 + 154 * q^91 + 1560 * q^92 - 1280 * q^94 - 522 * q^97 + 49 * q^98

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 $

$2$	4.53113	1.60200	0.800998	−	0.598667i	$-0.204303\pi$
$2$	4.53113	1.60200	0.800998	+	0.598667i	$0.204303\pi$
$3$	0	0
$3$	0	0
$4$	12.5311	1.56639
$5$	0	0
$5$	0	0
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	20.5311	0.907356
$9$	0	0
$10$	0	0
$11$	19.0623	0.522499	0.261249	−	0.965271i	$-0.415865\pi$
$11$	19.0623	0.522499	0.261249	+	0.965271i	$0.415865\pi$
$12$	0	0
$13$	2.93774	0.0626756	0.0313378	−	0.999509i	$-0.490023\pi$
$13$	2.93774	0.0626756	0.0313378	+	0.999509i	$0.490023\pi$
$14$	31.7179	0.605498
$15$	0	0
$16$	−7.21984	−0.112810
$17$	−6.49806	−0.0927066	−0.0463533	−	0.998925i	$-0.514760\pi$
$17$	−6.49806	−0.0927066	−0.0463533	+	0.998925i	$0.514760\pi$
$18$	0	0
$19$	−5.43580	−0.0656347	−0.0328173	−	0.999461i	$-0.510448\pi$
$19$	−5.43580	−0.0656347	−0.0328173	+	0.999461i	$0.510448\pi$
$20$	0	0
$21$	0	0
$22$	86.3735	0.837041
$23$	49.3774	0.447648	0.223824	−	0.974630i	$-0.428146\pi$
$23$	49.3774	0.447648	0.223824	+	0.974630i	$0.428146\pi$
$24$	0	0
$25$	0	0
$26$	13.3113	0.100406
$27$	0	0
$28$	87.7179	0.592040
$29$	291.494	1.86652	0.933261	−	0.359200i	$-0.116950\pi$
$29$	291.494	1.86652	0.933261	+	0.359200i	$0.116950\pi$
$30$	0	0
$31$	244.307	1.41545	0.707724	−	0.706489i	$-0.249722\pi$
$31$	244.307	1.41545	0.707724	+	0.706489i	$0.249722\pi$
$32$	−196.963	−1.08808
$33$	0	0
$34$	−29.4436	−0.148516
$35$	0	0
$36$	0	0
$37$	193.121	0.858077	0.429038	−	0.903286i	$-0.358853\pi$
$37$	193.121	0.858077	0.429038	+	0.903286i	$0.358853\pi$
$38$	−24.6303	−0.105147
$39$	0	0
$40$	0	0
$41$	−315.113	−1.20030	−0.600151	−	0.799887i	$-0.704893\pi$
$41$	−315.113	−1.20030	−0.600151	+	0.799887i	$0.704893\pi$
$42$	0	0
$43$	300.996	1.06748	0.533738	−	0.845650i	$-0.320787\pi$
$43$	300.996	1.06748	0.533738	+	0.845650i	$0.320787\pi$
$44$	238.872	0.818437
$45$	0	0
$46$	223.735	0.717130
$47$	86.5058	0.268472	0.134236	−	0.990949i	$-0.457142\pi$
$47$	86.5058	0.268472	0.134236	+	0.990949i	$0.457142\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	36.8132	0.0981745
$53$	509.677	1.32093	0.660467	−	0.750855i	$-0.270358\pi$
$53$	509.677	1.32093	0.660467	+	0.750855i	$0.270358\pi$
$54$	0	0
$55$	0	0
$56$	143.718	0.342948
$57$	0	0
$58$	1320.80	2.99016
$59$	83.3852	0.183997	0.0919985	−	0.995759i	$-0.470674\pi$
$59$	83.3852	0.183997	0.0919985	+	0.995759i	$0.470674\pi$
$60$	0	0
$61$	−5.25291	−0.0110257	−0.00551283	−	0.999985i	$-0.501755\pi$
$61$	−5.25291	−0.0110257	−0.00551283	+	0.999985i	$0.501755\pi$
$62$	1106.99	2.26754
$63$	0	0
$64$	−834.706	−1.63029
$65$	0	0
$66$	0	0
$67$	−205.992	−0.375611	−0.187806	−	0.982206i	$-0.560138\pi$
$67$	−205.992	−0.375611	−0.187806	+	0.982206i	$0.560138\pi$
$68$	−81.4281	−0.145215
$69$	0	0
$70$	0	0
$71$	−1004.31	−1.67872	−0.839362	−	0.543573i	$-0.817071\pi$
$71$	−1004.31	−1.67872	−0.839362	+	0.543573i	$0.817071\pi$
$72$	0	0
$73$	1007.29	1.61499	0.807494	−	0.589876i	$-0.200823\pi$
$73$	1007.29	1.61499	0.807494	+	0.589876i	$0.200823\pi$
$74$	875.055	1.37464
$75$	0	0
$76$	−68.1168	−0.102810
$77$	133.436	0.197486
$78$	0	0
$79$	−863.237	−1.22939	−0.614695	−	0.788765i	$-0.710721\pi$
$79$	−863.237	−1.22939	−0.614695	+	0.788765i	$0.710721\pi$
$80$	0	0
$81$	0	0
$82$	−1427.82	−1.92288
$83$	1334.72	1.76512	0.882560	−	0.470200i	$-0.155818\pi$
$83$	1334.72	1.76512	0.882560	+	0.470200i	$0.155818\pi$
$84$	0	0
$85$	0	0
$86$	1363.85	1.71009
$87$	0	0
$88$	391.370	0.474093
$89$	−326.249	−0.388565	−0.194283	−	0.980946i	$-0.562238\pi$
$89$	−326.249	−0.388565	−0.194283	+	0.980946i	$0.562238\pi$
$90$	0	0
$91$	20.5642	0.0236892
$92$	618.755	0.701192
$93$	0	0
$94$	391.969	0.430091
$95$	0	0
$96$	0	0
$97$	−1526.77	−1.59815	−0.799075	−	0.601232i	$-0.794677\pi$
$97$	−1526.77	−1.59815	−0.799075	+	0.601232i	$0.794677\pi$
$98$	222.025	0.228857
$99$	0	0
$100$	0	0
$101$	−96.8716	−0.0954365	−0.0477182	−	0.998861i	$-0.515195\pi$
$101$	−96.8716	−0.0954365	−0.0477182	+	0.998861i	$0.515195\pi$
$102$	0	0
$103$	−1321.99	−1.26466	−0.632329	−	0.774700i	$-0.717901\pi$
$103$	−1321.99	−1.26466	−0.632329	+	0.774700i	$0.717901\pi$
$104$	60.3152	0.0568691
$105$	0	0
$106$	2309.41	2.11613
$107$	1745.71	1.57724	0.788619	−	0.614883i	$-0.210797\pi$
$107$	1745.71	1.57724	0.788619	+	0.614883i	$0.210797\pi$
$108$	0	0
$109$	476.856	0.419032	0.209516	−	0.977805i	$-0.432811\pi$
$109$	476.856	0.419032	0.209516	+	0.977805i	$0.432811\pi$
$110$	0	0
$111$	0	0
$112$	−50.5389	−0.0426382
$113$	1641.65	1.36666	0.683332	−	0.730108i	$-0.260530\pi$
$113$	1641.65	1.36666	0.683332	+	0.730108i	$0.260530\pi$
$114$	0	0
$115$	0	0
$116$	3652.75	2.92370
$117$	0	0
$118$	377.829	0.294763
$119$	−45.4864	−0.0350398
$120$	0	0
$121$	−967.630	−0.726995
$122$	−23.8016	−0.0176631
$123$	0	0
$124$	3061.45	2.21715
$125$	0	0
$126$	0	0
$127$	844.016	0.589719	0.294859	−	0.955541i	$-0.404727\pi$
$127$	844.016	0.589719	0.294859	+	0.955541i	$0.404727\pi$
$128$	−2206.46	−1.52363
$129$	0	0
$130$	0	0
$131$	2796.20	1.86493	0.932463	−	0.361265i	$-0.117655\pi$
$131$	2796.20	1.86493	0.932463	+	0.361265i	$0.117655\pi$
$132$	0	0
$133$	−38.0506	−0.0248076
$134$	−933.377	−0.601728
$135$	0	0
$136$	−133.413	−0.0841179
$137$	−2057.13	−1.28287	−0.641433	−	0.767179i	$-0.721660\pi$
$137$	−2057.13	−1.28287	−0.641433	+	0.767179i	$0.721660\pi$
$138$	0	0
$139$	−1745.12	−1.06489	−0.532444	−	0.846465i	$-0.678726\pi$
$139$	−1745.12	−1.06489	−0.532444	+	0.846465i	$0.678726\pi$
$140$	0	0
$141$	0	0
$142$	−4550.65	−2.68931
$143$	56.0000	0.0327479
$144$	0	0
$145$	0	0
$146$	4564.15	2.58720
$147$	0	0
$148$	2420.02	1.34408
$149$	1173.57	0.645254	0.322627	−	0.946526i	$-0.395434\pi$
$149$	1173.57	0.645254	0.322627	+	0.946526i	$0.395434\pi$
$150$	0	0
$151$	1540.07	0.829994	0.414997	−	0.909823i	$-0.363783\pi$
$151$	1540.07	0.829994	0.414997	+	0.909823i	$0.363783\pi$
$152$	−111.603	−0.0595540
$153$	0	0
$154$	604.615	0.316372
$155$	0	0
$156$	0	0
$157$	2544.53	1.29348	0.646738	−	0.762712i	$-0.276133\pi$
$157$	2544.53	1.29348	0.646738	+	0.762712i	$0.276133\pi$
$158$	−3911.44	−1.96948
$159$	0	0
$160$	0	0
$161$	345.642	0.169195
$162$	0	0
$163$	594.708	0.285774	0.142887	−	0.989739i	$-0.454361\pi$
$163$	594.708	0.285774	0.142887	+	0.989739i	$0.454361\pi$
$164$	−3948.72	−1.88014
$165$	0	0
$166$	6047.81	2.82772
$167$	928.498	0.430236	0.215118	−	0.976588i	$-0.430986\pi$
$167$	928.498	0.430236	0.215118	+	0.976588i	$0.430986\pi$
$168$	0	0
$169$	−2188.37	−0.996072
$170$	0	0
$171$	0	0
$172$	3771.82	1.67209
$173$	−315.642	−0.138716	−0.0693578	−	0.997592i	$-0.522095\pi$
$173$	−315.642	−0.138716	−0.0693578	+	0.997592i	$0.522095\pi$
$174$	0	0
$175$	0	0
$176$	−137.626	−0.0589431
$177$	0	0
$178$	−1478.28	−0.622480
$179$	1445.49	0.603581	0.301791	−	0.953374i	$-0.402416\pi$
$179$	1445.49	0.603581	0.301791	+	0.953374i	$0.402416\pi$
$180$	0	0
$181$	−1843.81	−0.757180	−0.378590	−	0.925564i	$-0.623591\pi$
$181$	−1843.81	−0.757180	−0.378590	+	0.925564i	$0.623591\pi$
$182$	93.1790	0.0379499
$183$	0	0
$184$	1013.77	0.406176
$185$	0	0
$186$	0	0
$187$	−123.868	−0.0484391
$188$	1084.02	0.420532
$189$	0	0
$190$	0	0
$191$	244.074	0.0924637	0.0462318	−	0.998931i	$-0.485279\pi$
$191$	244.074	0.0924637	0.0462318	+	0.998931i	$0.485279\pi$
$192$	0	0
$193$	1733.03	0.646355	0.323178	−	0.946338i	$-0.395249\pi$
$193$	1733.03	0.646355	0.323178	+	0.946338i	$0.395249\pi$
$194$	−6918.01	−2.56023
$195$	0	0
$196$	614.025	0.223770
$197$	358.230	0.129557	0.0647787	−	0.997900i	$-0.479366\pi$
$197$	358.230	0.129557	0.0647787	+	0.997900i	$0.479366\pi$
$198$	0	0
$199$	−3203.63	−1.14120	−0.570601	−	0.821227i	$-0.693290\pi$
$199$	−3203.63	−1.14120	−0.570601	+	0.821227i	$0.693290\pi$
$200$	0	0
$201$	0	0
$202$	−438.938	−0.152889
$203$	2040.46	0.705479
$204$	0	0
$205$	0	0
$206$	−5990.12	−2.02598
$207$	0	0
$208$	−21.2100	−0.00707044
$209$	−103.619	−0.0342940
$210$	0	0
$211$	4943.16	1.61280	0.806401	−	0.591369i	$-0.201412\pi$
$211$	4943.16	1.61280	0.806401	+	0.591369i	$0.201412\pi$
$212$	6386.83	2.06910
$213$	0	0
$214$	7910.05	2.52673
$215$	0	0
$216$	0	0
$217$	1710.15	0.534989
$218$	2160.70	0.671288
$219$	0	0
$220$	0	0
$221$	−19.0896	−0.00581044
$222$	0	0
$223$	−3160.15	−0.948965	−0.474482	−	0.880265i	$-0.657365\pi$
$223$	−3160.15	−0.948965	−0.474482	+	0.880265i	$0.657365\pi$
$224$	−1378.74	−0.411255
$225$	0	0
$226$	7438.51	2.18939
$227$	−3651.11	−1.06755	−0.533773	−	0.845628i	$-0.679226\pi$
$227$	−3651.11	−1.06755	−0.533773	+	0.845628i	$0.679226\pi$
$228$	0	0
$229$	−4083.70	−1.17842	−0.589210	−	0.807980i	$-0.700561\pi$
$229$	−4083.70	−1.17842	−0.589210	+	0.807980i	$0.700561\pi$
$230$	0	0
$231$	0	0
$232$	5984.70	1.69360
$233$	−3682.51	−1.03540	−0.517702	−	0.855561i	$-0.673212\pi$
$233$	−3682.51	−1.03540	−0.517702	+	0.855561i	$0.673212\pi$
$234$	0	0
$235$	0	0
$236$	1044.91	0.288211
$237$	0	0
$238$	−206.105	−0.0561336
$239$	−2658.78	−0.719591	−0.359796	−	0.933031i	$-0.617154\pi$
$239$	−2658.78	−0.719591	−0.359796	+	0.933031i	$0.617154\pi$
$240$	0	0
$241$	−4820.39	−1.28842	−0.644209	−	0.764850i	$-0.722813\pi$
$241$	−4820.39	−1.28842	−0.644209	+	0.764850i	$0.722813\pi$
$242$	−4384.46	−1.16464
$243$	0	0
$244$	−65.8249	−0.0172705
$245$	0	0
$246$	0	0
$247$	−15.9690	−0.00411369
$248$	5015.91	1.28432
$249$	0	0
$250$	0	0
$251$	−1672.27	−0.420530	−0.210265	−	0.977644i	$-0.567433\pi$
$251$	−1672.27	−0.420530	−0.210265	+	0.977644i	$0.567433\pi$
$252$	0	0
$253$	941.245	0.233896
$254$	3824.34	0.944727
$255$	0	0
$256$	−3320.09	−0.810569
$257$	−3697.74	−0.897506	−0.448753	−	0.893656i	$-0.648132\pi$
$257$	−3697.74	−0.897506	−0.448753	+	0.893656i	$0.648132\pi$
$258$	0	0
$259$	1351.84	0.324323
$260$	0	0
$261$	0	0
$262$	12670.0	2.98760
$263$	7319.00	1.71600	0.858002	−	0.513646i	$-0.171706\pi$
$263$	7319.00	1.71600	0.858002	+	0.513646i	$0.171706\pi$
$264$	0	0
$265$	0	0
$266$	−172.412	−0.0397416
$267$	0	0
$268$	−2581.32	−0.588354
$269$	815.097	0.184749	0.0923743	−	0.995724i	$-0.470554\pi$
$269$	815.097	0.184749	0.0923743	+	0.995724i	$0.470554\pi$
$270$	0	0
$271$	−5106.02	−1.14453	−0.572267	−	0.820068i	$-0.693936\pi$
$271$	−5106.02	−1.14453	−0.572267	+	0.820068i	$0.693936\pi$
$272$	46.9150	0.0104582
$273$	0	0
$274$	−9321.13	−2.05515
$275$	0	0
$276$	0	0
$277$	−1398.72	−0.303398	−0.151699	−	0.988427i	$-0.548474\pi$
$277$	−1398.72	−0.303398	−0.151699	+	0.988427i	$0.548474\pi$
$278$	−7907.39	−1.70595
$279$	0	0
$280$	0	0
$281$	7102.38	1.50780	0.753901	−	0.656988i	$-0.228170\pi$
$281$	7102.38	1.50780	0.753901	+	0.656988i	$0.228170\pi$
$282$	0	0
$283$	−4465.18	−0.937907	−0.468953	−	0.883223i	$-0.655369\pi$
$283$	−4465.18	−0.937907	−0.468953	+	0.883223i	$0.655369\pi$
$284$	−12585.1	−2.62954
$285$	0	0
$286$	253.743	0.0524621
$287$	−2205.79	−0.453671
$288$	0	0
$289$	−4870.78	−0.991405
$290$	0	0
$291$	0	0
$292$	12622.5	2.52970
$293$	7590.61	1.51348	0.756738	−	0.653718i	$-0.226792\pi$
$293$	7590.61	1.51348	0.756738	+	0.653718i	$0.226792\pi$
$294$	0	0
$295$	0	0
$296$	3964.98	0.778581
$297$	0	0
$298$	5317.61	1.03369
$299$	145.058	0.0280566
$300$	0	0
$301$	2106.97	0.403468
$302$	6978.26	1.32965
$303$	0	0
$304$	39.2456	0.00740425
$305$	0	0
$306$	0	0
$307$	−9480.12	−1.76241	−0.881203	−	0.472737i	$-0.843266\pi$
$307$	−9480.12	−1.76241	−0.881203	+	0.472737i	$0.843266\pi$
$308$	1672.10	0.309340
$309$	0	0
$310$	0	0
$311$	−7078.01	−1.29054	−0.645268	−	0.763956i	$-0.723254\pi$
$311$	−7078.01	−1.29054	−0.645268	+	0.763956i	$0.723254\pi$
$312$	0	0
$313$	−5593.84	−1.01017	−0.505084	−	0.863070i	$-0.668539\pi$
$313$	−5593.84	−1.01017	−0.505084	+	0.863070i	$0.668539\pi$
$314$	11529.6	2.07214
$315$	0	0
$316$	−10817.3	−1.92571
$317$	3567.81	0.632139	0.316070	−	0.948736i	$-0.397637\pi$
$317$	3567.81	0.632139	0.316070	+	0.948736i	$0.397637\pi$
$318$	0	0
$319$	5556.54	0.975255
$320$	0	0
$321$	0	0
$322$	1566.15	0.271050
$323$	35.3222	0.00608477
$324$	0	0
$325$	0	0
$326$	2694.70	0.457809
$327$	0	0
$328$	−6469.62	−1.08910
$329$	605.541	0.101473
$330$	0	0
$331$	−4389.67	−0.728936	−0.364468	−	0.931216i	$-0.618749\pi$
$331$	−4389.67	−0.728936	−0.364468	+	0.931216i	$0.618749\pi$
$332$	16725.6	2.76487
$333$	0	0
$334$	4207.14	0.689236
$335$	0	0
$336$	0	0
$337$	−2348.83	−0.379671	−0.189835	−	0.981816i	$-0.560795\pi$
$337$	−2348.83	−0.379671	−0.189835	+	0.981816i	$0.560795\pi$
$338$	−9915.79	−1.59570
$339$	0	0
$340$	0	0
$341$	4657.05	0.739570
$342$	0	0
$343$	343.000	0.0539949
$344$	6179.79	0.968581
$345$	0	0
$346$	−1430.21	−0.222222
$347$	558.436	0.0863931	0.0431965	−	0.999067i	$-0.486246\pi$
$347$	558.436	0.0863931	0.0431965	+	0.999067i	$0.486246\pi$
$348$	0	0
$349$	3233.89	0.496006	0.248003	−	0.968759i	$-0.420226\pi$
$349$	3233.89	0.496006	0.248003	+	0.968759i	$0.420226\pi$
$350$	0	0
$351$	0	0
$352$	−3754.56	−0.568519
$353$	−7516.35	−1.13330	−0.566650	−	0.823959i	$-0.691761\pi$
$353$	−7516.35	−1.13330	−0.566650	+	0.823959i	$0.691761\pi$
$354$	0	0
$355$	0	0
$356$	−4088.27	−0.608646
$357$	0	0
$358$	6549.70	0.966935
$359$	6577.76	0.967021	0.483511	−	0.875338i	$-0.339361\pi$
$359$	6577.76	0.967021	0.483511	+	0.875338i	$0.339361\pi$
$360$	0	0
$361$	−6829.45	−0.995692
$362$	−8354.56	−1.21300
$363$	0	0
$364$	257.693	0.0371065
$365$	0	0
$366$	0	0
$367$	−8307.17	−1.18155	−0.590777	−	0.806835i	$-0.701179\pi$
$367$	−8307.17	−1.18155	−0.590777	+	0.806835i	$0.701179\pi$
$368$	−356.497	−0.0504992
$369$	0	0
$370$	0	0
$371$	3567.74	0.499266
$372$	0	0
$373$	4551.09	0.631760	0.315880	−	0.948799i	$-0.397700\pi$
$373$	4551.09	0.631760	0.315880	+	0.948799i	$0.397700\pi$
$374$	−561.261	−0.0775992
$375$	0	0
$376$	1776.06	0.243599
$377$	856.335	0.116985
$378$	0	0
$379$	−1788.29	−0.242370	−0.121185	−	0.992630i	$-0.538669\pi$
$379$	−1788.29	−0.242370	−0.121185	+	0.992630i	$0.538669\pi$
$380$	0	0
$381$	0	0
$382$	1105.93	0.148126
$383$	−1358.47	−0.181240	−0.0906199	−	0.995886i	$-0.528885\pi$
$383$	−1358.47	−0.181240	−0.0906199	+	0.995886i	$0.528885\pi$
$384$	0	0
$385$	0	0
$386$	7852.60	1.03546
$387$	0	0
$388$	−19132.2	−2.50333
$389$	−9722.54	−1.26723	−0.633615	−	0.773649i	$-0.718430\pi$
$389$	−9722.54	−1.26723	−0.633615	+	0.773649i	$0.718430\pi$
$390$	0	0
$391$	−320.858	−0.0414999
$392$	1006.03	0.129622
$393$	0	0
$394$	1623.19	0.207551
$395$	0	0
$396$	0	0
$397$	4788.04	0.605302	0.302651	−	0.953101i	$-0.402128\pi$
$397$	4788.04	0.605302	0.302651	+	0.953101i	$0.402128\pi$
$398$	−14516.1	−1.82820
$399$	0	0
$400$	0	0
$401$	−9681.41	−1.20565	−0.602826	−	0.797873i	$-0.705959\pi$
$401$	−9681.41	−1.20565	−0.602826	+	0.797873i	$0.705959\pi$
$402$	0	0
$403$	717.712	0.0887141
$404$	−1213.91	−0.149491
$405$	0	0
$406$	9245.58	1.13017
$407$	3681.32	0.448344
$408$	0	0
$409$	11113.1	1.34353	0.671767	−	0.740763i	$-0.265536\pi$
$409$	11113.1	1.34353	0.671767	+	0.740763i	$0.265536\pi$
$410$	0	0
$411$	0	0
$412$	−16566.1	−1.98095
$413$	583.696	0.0695443
$414$	0	0
$415$	0	0
$416$	−578.627	−0.0681959
$417$	0	0
$418$	−469.510	−0.0549389
$419$	1230.09	0.143421	0.0717107	−	0.997425i	$-0.477154\pi$
$419$	1230.09	0.143421	0.0717107	+	0.997425i	$0.477154\pi$
$420$	0	0
$421$	−12356.5	−1.43044	−0.715222	−	0.698897i	$-0.753674\pi$
$421$	−12356.5	−1.43044	−0.715222	+	0.698897i	$0.753674\pi$
$422$	22398.1	2.58370
$423$	0	0
$424$	10464.2	1.19856
$425$	0	0
$426$	0	0
$427$	−36.7703	−0.00416731
$428$	21875.7	2.47057
$429$	0	0
$430$	0	0
$431$	7375.27	0.824256	0.412128	−	0.911126i	$-0.364786\pi$
$431$	7375.27	0.824256	0.412128	+	0.911126i	$0.364786\pi$
$432$	0	0
$433$	690.067	0.0765877	0.0382939	−	0.999267i	$-0.487808\pi$
$433$	690.067	0.0765877	0.0382939	+	0.999267i	$0.487808\pi$
$434$	7748.92	0.857051
$435$	0	0
$436$	5975.55	0.656369
$437$	−268.406	−0.0293812
$438$	0	0
$439$	8408.79	0.914191	0.457095	−	0.889418i	$-0.348890\pi$
$439$	8408.79	0.914191	0.457095	+	0.889418i	$0.348890\pi$
$440$	0	0
$441$	0	0
$442$	−86.4976	−0.00930830
$443$	6568.55	0.704473	0.352236	−	0.935911i	$-0.385421\pi$
$443$	6568.55	0.704473	0.352236	+	0.935911i	$0.385421\pi$
$444$	0	0
$445$	0	0
$446$	−14319.0	−1.52024
$447$	0	0
$448$	−5842.94	−0.616190
$449$	−2954.55	−0.310543	−0.155271	−	0.987872i	$-0.549625\pi$
$449$	−2954.55	−0.310543	−0.155271	+	0.987872i	$0.549625\pi$
$450$	0	0
$451$	−6006.76	−0.627156
$452$	20571.7	2.14073
$453$	0	0
$454$	−16543.7	−1.71020
$455$	0	0
$456$	0	0
$457$	8144.84	0.833697	0.416849	−	0.908976i	$-0.363135\pi$
$457$	8144.84	0.833697	0.416849	+	0.908976i	$0.363135\pi$
$458$	−18503.8	−1.88782
$459$	0	0
$460$	0	0
$461$	−2495.26	−0.252095	−0.126048	−	0.992024i	$-0.540229\pi$
$461$	−2495.26	−0.252095	−0.126048	+	0.992024i	$0.540229\pi$
$462$	0	0
$463$	5755.66	0.577728	0.288864	−	0.957370i	$-0.406722\pi$
$463$	5755.66	0.577728	0.288864	+	0.957370i	$0.406722\pi$
$464$	−2104.54	−0.210562
$465$	0	0
$466$	−16685.9	−1.65871
$467$	−4143.73	−0.410598	−0.205299	−	0.978699i	$-0.565817\pi$
$467$	−4143.73	−0.410598	−0.205299	+	0.978699i	$0.565817\pi$
$468$	0	0
$469$	−1441.95	−0.141968
$470$	0	0
$471$	0	0
$472$	1711.99	0.166951
$473$	5737.67	0.557755
$474$	0	0
$475$	0	0
$476$	−569.996	−0.0548860
$477$	0	0
$478$	−12047.3	−1.15278
$479$	6765.96	0.645396	0.322698	−	0.946502i	$-0.395410\pi$
$479$	6765.96	0.645396	0.322698	+	0.946502i	$0.395410\pi$
$480$	0	0
$481$	567.339	0.0537805
$482$	−21841.8	−2.06404
$483$	0	0
$484$	−12125.5	−1.13876
$485$	0	0
$486$	0	0
$487$	−6360.42	−0.591824	−0.295912	−	0.955215i	$-0.595623\pi$
$487$	−6360.42	−0.591824	−0.295912	+	0.955215i	$0.595623\pi$
$488$	−107.848	−0.0100042
$489$	0	0
$490$	0	0
$491$	7072.54	0.650060	0.325030	−	0.945704i	$-0.394626\pi$
$491$	7072.54	0.650060	0.325030	+	0.945704i	$0.394626\pi$
$492$	0	0
$493$	−1894.15	−0.173039
$494$	−72.3576	−0.00659012
$495$	0	0
$496$	−1763.86	−0.159677
$497$	−7030.15	−0.634498
$498$	0	0
$499$	−18473.9	−1.65732	−0.828661	−	0.559751i	$-0.810897\pi$
$499$	−18473.9	−1.65732	−0.828661	+	0.559751i	$0.810897\pi$
$500$	0	0
$501$	0	0
$502$	−7577.28	−0.673687
$503$	11379.2	1.00869	0.504347	−	0.863501i	$-0.331733\pi$
$503$	11379.2	1.00869	0.504347	+	0.863501i	$0.331733\pi$
$504$	0	0
$505$	0	0
$506$	4264.90	0.374700
$507$	0	0
$508$	10576.5	0.923730
$509$	−6064.48	−0.528101	−0.264051	−	0.964509i	$-0.585059\pi$
$509$	−6064.48	−0.528101	−0.264051	+	0.964509i	$0.585059\pi$
$510$	0	0
$511$	7051.02	0.610408
$512$	2607.89	0.225105
$513$	0	0
$514$	−16755.0	−1.43780
$515$	0	0
$516$	0	0
$517$	1649.00	0.140276
$518$	6125.38	0.519563
$519$	0	0
$520$	0	0
$521$	−2682.88	−0.225603	−0.112801	−	0.993618i	$-0.535982\pi$
$521$	−2682.88	−0.225603	−0.112801	+	0.993618i	$0.535982\pi$
$522$	0	0
$523$	4309.02	0.360268	0.180134	−	0.983642i	$-0.442347\pi$
$523$	4309.02	0.360268	0.180134	+	0.983642i	$0.442347\pi$
$524$	35039.6	2.92120
$525$	0	0
$526$	33163.3	2.74903
$527$	−1587.52	−0.131221
$528$	0	0
$529$	−9728.87	−0.799611
$530$	0	0
$531$	0	0
$532$	−476.817	−0.0388584
$533$	−925.720	−0.0752296
$534$	0	0
$535$	0	0
$536$	−4229.25	−0.340813
$537$	0	0
$538$	3693.31	0.295966
$539$	934.051	0.0746427
$540$	0	0
$541$	4081.47	0.324355	0.162178	−	0.986762i	$-0.448148\pi$
$541$	4081.47	0.324355	0.162178	+	0.986762i	$0.448148\pi$
$542$	−23136.0	−1.83354
$543$	0	0
$544$	1279.88	0.100872
$545$	0	0
$546$	0	0
$547$	−8844.82	−0.691366	−0.345683	−	0.938351i	$-0.612353\pi$
$547$	−8844.82	−0.691366	−0.345683	+	0.938351i	$0.612353\pi$
$548$	−25778.2	−2.00947
$549$	0	0
$550$	0	0
$551$	−1584.51	−0.122509
$552$	0	0
$553$	−6042.66	−0.464666
$554$	−6337.80	−0.486042
$555$	0	0
$556$	−21868.4	−1.66803
$557$	−11144.7	−0.847787	−0.423894	−	0.905712i	$-0.639337\pi$
$557$	−11144.7	−0.847787	−0.423894	+	0.905712i	$0.639337\pi$
$558$	0	0
$559$	884.249	0.0669047
$560$	0	0
$561$	0	0
$562$	32181.8	2.41549
$563$	−21857.5	−1.63621	−0.818104	−	0.575071i	$-0.804975\pi$
$563$	−21857.5	−1.63621	−0.818104	+	0.575071i	$0.804975\pi$
$564$	0	0
$565$	0	0
$566$	−20232.3	−1.50252
$567$	0	0
$568$	−20619.6	−1.52320
$569$	−23496.4	−1.73115	−0.865573	−	0.500783i	$-0.833046\pi$
$569$	−23496.4	−1.73115	−0.865573	+	0.500783i	$0.833046\pi$
$570$	0	0
$571$	11067.8	0.811164	0.405582	−	0.914059i	$-0.367069\pi$
$571$	11067.8	0.811164	0.405582	+	0.914059i	$0.367069\pi$
$572$	701.743	0.0512961
$573$	0	0
$574$	−9994.72	−0.726780
$575$	0	0
$576$	0	0
$577$	−20482.9	−1.47784	−0.738922	−	0.673791i	$-0.764665\pi$
$577$	−20482.9	−1.47784	−0.738922	+	0.673791i	$0.764665\pi$
$578$	−22070.1	−1.58823
$579$	0	0
$580$	0	0
$581$	9343.07	0.667153
$582$	0	0
$583$	9715.60	0.690187
$584$	20680.8	1.46537
$585$	0	0
$586$	34394.1	2.42458
$587$	−23444.3	−1.64847	−0.824235	−	0.566248i	$-0.808394\pi$
$587$	−23444.3	−1.64847	−0.824235	+	0.566248i	$0.808394\pi$
$588$	0	0
$589$	−1328.01	−0.0929025
$590$	0	0
$591$	0	0
$592$	−1394.30	−0.0967996
$593$	4404.69	0.305024	0.152512	−	0.988302i	$-0.451264\pi$
$593$	4404.69	0.305024	0.152512	+	0.988302i	$0.451264\pi$
$594$	0	0
$595$	0	0
$596$	14706.2	1.01072
$597$	0	0
$598$	657.277	0.0449466
$599$	−3327.05	−0.226945	−0.113472	−	0.993541i	$-0.536197\pi$
$599$	−3327.05	−0.226945	−0.113472	+	0.993541i	$0.536197\pi$
$600$	0	0
$601$	−14244.8	−0.966818	−0.483409	−	0.875395i	$-0.660602\pi$
$601$	−14244.8	−0.966818	−0.483409	+	0.875395i	$0.660602\pi$
$602$	9546.97	0.646354
$603$	0	0
$604$	19298.8	1.30010
$605$	0	0
$606$	0	0
$607$	−11446.5	−0.765402	−0.382701	−	0.923872i	$-0.625006\pi$
$607$	−11446.5	−0.765402	−0.382701	+	0.923872i	$0.625006\pi$
$608$	1070.65	0.0714156
$609$	0	0
$610$	0	0
$611$	254.132	0.0168266
$612$	0	0
$613$	19436.4	1.28063	0.640316	−	0.768111i	$-0.278803\pi$
$613$	19436.4	1.28063	0.640316	+	0.768111i	$0.278803\pi$
$614$	−42955.6	−2.82337
$615$	0	0
$616$	2739.59	0.179190
$617$	−20530.1	−1.33956	−0.669781	−	0.742558i	$-0.733612\pi$
$617$	−20530.1	−1.33956	−0.669781	+	0.742558i	$0.733612\pi$
$618$	0	0
$619$	5833.35	0.378776	0.189388	−	0.981902i	$-0.439350\pi$
$619$	5833.35	0.378776	0.189388	+	0.981902i	$0.439350\pi$
$620$	0	0
$621$	0	0
$622$	−32071.4	−2.06744
$623$	−2283.74	−0.146864
$624$	0	0
$625$	0	0
$626$	−25346.4	−1.61829
$627$	0	0
$628$	31885.9	2.02609
$629$	−1254.91	−0.0795493
$630$	0	0
$631$	24776.6	1.56314	0.781568	−	0.623820i	$-0.214420\pi$
$631$	24776.6	1.56314	0.781568	+	0.623820i	$0.214420\pi$
$632$	−17723.2	−1.11549
$633$	0	0
$634$	16166.2	1.01268
$635$	0	0
$636$	0	0
$637$	143.949	0.00895366
$638$	25177.4	1.56235
$639$	0	0
$640$	0	0
$641$	−27219.4	−1.67723	−0.838613	−	0.544728i	$-0.816633\pi$
$641$	−27219.4	−1.67723	−0.838613	+	0.544728i	$0.816633\pi$
$642$	0	0
$643$	−7091.79	−0.434950	−0.217475	−	0.976066i	$-0.569782\pi$
$643$	−7091.79	−0.434950	−0.217475	+	0.976066i	$0.569782\pi$
$644$	4331.28	0.265026
$645$	0	0
$646$	160.049	0.00974777
$647$	27773.0	1.68758	0.843792	−	0.536670i	$-0.180318\pi$
$647$	27773.0	1.68758	0.843792	+	0.536670i	$0.180318\pi$
$648$	0	0
$649$	1589.51	0.0961382
$650$	0	0
$651$	0	0
$652$	7452.37	0.447634
$653$	21380.4	1.28129	0.640643	−	0.767839i	$-0.278668\pi$
$653$	21380.4	1.28129	0.640643	+	0.767839i	$0.278668\pi$
$654$	0	0
$655$	0	0
$656$	2275.06	0.135406
$657$	0	0
$658$	2743.78	0.162559
$659$	17232.3	1.01863	0.509315	−	0.860580i	$-0.329899\pi$
$659$	17232.3	1.01863	0.509315	+	0.860580i	$0.329899\pi$
$660$	0	0
$661$	26577.7	1.56392	0.781962	−	0.623326i	$-0.214219\pi$
$661$	26577.7	1.56392	0.781962	+	0.623326i	$0.214219\pi$
$662$	−19890.1	−1.16775
$663$	0	0
$664$	27403.4	1.60159
$665$	0	0
$666$	0	0
$667$	14393.2	0.835544
$668$	11635.1	0.673917
$669$	0	0
$670$	0	0
$671$	−100.132	−0.00576090
$672$	0	0
$673$	31695.2	1.81540	0.907698	−	0.419624i	$-0.137838\pi$
$673$	31695.2	1.81540	0.907698	+	0.419624i	$0.137838\pi$
$674$	−10642.9	−0.608231
$675$	0	0
$676$	−27422.7	−1.56024
$677$	20440.3	1.16039	0.580195	−	0.814477i	$-0.302976\pi$
$677$	20440.3	1.16039	0.580195	+	0.814477i	$0.302976\pi$
$678$	0	0
$679$	−10687.4	−0.604044
$680$	0	0
$681$	0	0
$682$	21101.7	1.18479
$683$	−22896.9	−1.28276	−0.641381	−	0.767223i	$-0.721638\pi$
$683$	−22896.9	−1.28276	−0.641381	+	0.767223i	$0.721638\pi$
$684$	0	0
$685$	0	0
$686$	1554.18	0.0864997
$687$	0	0
$688$	−2173.14	−0.120422
$689$	1497.30	0.0827904
$690$	0	0
$691$	−23764.0	−1.30829	−0.654143	−	0.756371i	$-0.726971\pi$
$691$	−23764.0	−1.30829	−0.654143	+	0.756371i	$0.726971\pi$
$692$	−3955.35	−0.217283
$693$	0	0
$694$	2530.34	0.138401
$695$	0	0
$696$	0	0
$697$	2047.62	0.111276
$698$	14653.2	0.794600
$699$	0	0
$700$	0	0
$701$	−26259.5	−1.41485	−0.707423	−	0.706791i	$-0.750142\pi$
$701$	−26259.5	−1.41485	−0.707423	+	0.706791i	$0.750142\pi$
$702$	0	0
$703$	−1049.77	−0.0563196
$704$	−15911.4	−0.851822
$705$	0	0
$706$	−34057.6	−1.81554
$707$	−678.101	−0.0360716
$708$	0	0
$709$	12783.0	0.677116	0.338558	−	0.940945i	$-0.390061\pi$
$709$	12783.0	0.677116	0.338558	+	0.940945i	$0.390061\pi$
$710$	0	0
$711$	0	0
$712$	−6698.26	−0.352567
$713$	12063.3	0.633623
$714$	0	0
$715$	0	0
$716$	18113.6	0.945444
$717$	0	0
$718$	29804.7	1.54916
$719$	27609.0	1.43205	0.716025	−	0.698075i	$-0.245960\pi$
$719$	27609.0	1.43205	0.716025	+	0.698075i	$0.245960\pi$
$720$	0	0
$721$	−9253.95	−0.477996
$722$	−30945.1	−1.59509
$723$	0	0
$724$	−23105.1	−1.18604
$725$	0	0
$726$	0	0
$727$	−31306.2	−1.59709	−0.798544	−	0.601937i	$-0.794396\pi$
$727$	−31306.2	−1.59709	−0.798544	+	0.601937i	$0.794396\pi$
$728$	422.206	0.0214945
$729$	0	0
$730$	0	0
$731$	−1955.89	−0.0989621
$732$	0	0
$733$	15765.8	0.794441	0.397220	−	0.917723i	$-0.369975\pi$
$733$	15765.8	0.794441	0.397220	+	0.917723i	$0.369975\pi$
$734$	−37640.8	−1.89285
$735$	0	0
$736$	−9725.53	−0.487076
$737$	−3926.68	−0.196256
$738$	0	0
$739$	3966.51	0.197443	0.0987216	−	0.995115i	$-0.468525\pi$
$739$	3966.51	0.197443	0.0987216	+	0.995115i	$0.468525\pi$
$740$	0	0
$741$	0	0
$742$	16165.9	0.799823
$743$	8224.50	0.406094	0.203047	−	0.979169i	$-0.434916\pi$
$743$	8224.50	0.406094	0.203047	+	0.979169i	$0.434916\pi$
$744$	0	0
$745$	0	0
$746$	20621.6	1.01208
$747$	0	0
$748$	−1552.20	−0.0758745
$749$	12220.0	0.596140
$750$	0	0
$751$	18929.2	0.919754	0.459877	−	0.887983i	$-0.347894\pi$
$751$	18929.2	0.919754	0.459877	+	0.887983i	$0.347894\pi$
$752$	−624.558	−0.0302863
$753$	0	0
$754$	3880.16	0.187410
$755$	0	0
$756$	0	0
$757$	34906.8	1.67597	0.837984	−	0.545695i	$-0.183734\pi$
$757$	34906.8	1.67597	0.837984	+	0.545695i	$0.183734\pi$
$758$	−8102.96	−0.388276
$759$	0	0
$760$	0	0
$761$	13683.4	0.651803	0.325902	−	0.945404i	$-0.394332\pi$
$761$	13683.4	0.651803	0.325902	+	0.945404i	$0.394332\pi$
$762$	0	0
$763$	3337.99	0.158379
$764$	3058.52	0.144834
$765$	0	0
$766$	−6155.42	−0.290345
$767$	244.964	0.0115321
$768$	0	0
$769$	41837.3	1.96189	0.980943	−	0.194294i	$-0.0622416\pi$
$769$	41837.3	1.96189	0.980943	+	0.194294i	$0.0622416\pi$
$770$	0	0
$771$	0	0
$772$	21716.9	1.01245
$773$	19640.0	0.913843	0.456921	−	0.889507i	$-0.348952\pi$
$773$	19640.0	0.913843	0.456921	+	0.889507i	$0.348952\pi$
$774$	0	0
$775$	0	0
$776$	−31346.4	−1.45009
$777$	0	0
$778$	−44054.1	−2.03010
$779$	1712.89	0.0787814
$780$	0	0
$781$	−19144.4	−0.877131
$782$	−1453.85	−0.0664827
$783$	0	0
$784$	−353.772	−0.0161157
$785$	0	0
$786$	0	0
$787$	−24935.3	−1.12941	−0.564705	−	0.825293i	$-0.691010\pi$
$787$	−24935.3	−1.12941	−0.564705	+	0.825293i	$0.691010\pi$
$788$	4489.02	0.202938
$789$	0	0
$790$	0	0
$791$	11491.5	0.516551
$792$	0	0
$793$	−15.4317	−0.000691041 0
$794$	21695.2	0.969692
$795$	0	0
$796$	−40145.1	−1.78757
$797$	−1168.33	−0.0519251	−0.0259625	−	0.999663i	$-0.508265\pi$
$797$	−1168.33	−0.0519251	−0.0259625	+	0.999663i	$0.508265\pi$
$798$	0	0
$799$	−562.120	−0.0248891
$800$	0	0
$801$	0	0
$802$	−43867.7	−1.93145
$803$	19201.2	0.843829
$804$	0	0
$805$	0	0
$806$	3252.05	0.142120
$807$	0	0
$808$	−1988.88	−0.0865949
$809$	35175.7	1.52869	0.764345	−	0.644807i	$-0.223062\pi$
$809$	35175.7	1.52869	0.764345	+	0.644807i	$0.223062\pi$
$810$	0	0
$811$	−15256.5	−0.660577	−0.330288	−	0.943880i	$-0.607146\pi$
$811$	−15256.5	−0.660577	−0.330288	+	0.943880i	$0.607146\pi$
$812$	25569.3	1.10506
$813$	0	0
$814$	16680.5	0.718245
$815$	0	0
$816$	0	0
$817$	−1636.16	−0.0700635
$818$	50354.7	2.15233
$819$	0	0
$820$	0	0
$821$	15971.9	0.678956	0.339478	−	0.940614i	$-0.389750\pi$
$821$	15971.9	0.678956	0.339478	+	0.940614i	$0.389750\pi$
$822$	0	0
$823$	−2312.41	−0.0979409	−0.0489705	−	0.998800i	$-0.515594\pi$
$823$	−2312.41	−0.0979409	−0.0489705	+	0.998800i	$0.515594\pi$
$824$	−27142.0	−1.14750
$825$	0	0
$826$	2644.80	0.111410
$827$	10422.4	0.438238	0.219119	−	0.975698i	$-0.429682\pi$
$827$	10422.4	0.438238	0.219119	+	0.975698i	$0.429682\pi$
$828$	0	0
$829$	−13213.4	−0.553584	−0.276792	−	0.960930i	$-0.589271\pi$
$829$	−13213.4	−0.553584	−0.276792	+	0.960930i	$0.589271\pi$
$830$	0	0
$831$	0	0
$832$	−2452.15	−0.102179
$833$	−318.405	−0.0132438
$834$	0	0
$835$	0	0
$836$	−1298.46	−0.0537179
$837$	0	0
$838$	5573.67	0.229761
$839$	10119.6	0.416409	0.208205	−	0.978085i	$-0.433238\pi$
$839$	10119.6	0.416409	0.208205	+	0.978085i	$0.433238\pi$
$840$	0	0
$841$	60579.9	2.48390
$842$	−55988.7	−2.29157
$843$	0	0
$844$	61943.4	2.52628
$845$	0	0
$846$	0	0
$847$	−6773.41	−0.274778
$848$	−3679.79	−0.149015
$849$	0	0
$850$	0	0
$851$	9535.80	0.384116
$852$	0	0
$853$	−35378.1	−1.42007	−0.710037	−	0.704165i	$-0.751322\pi$
$853$	−35378.1	−1.42007	−0.710037	+	0.704165i	$0.751322\pi$
$854$	−166.611	−0.00667602
$855$	0	0
$856$	35841.4	1.43112
$857$	6697.57	0.266960	0.133480	−	0.991052i	$-0.457385\pi$
$857$	6697.57	0.266960	0.133480	+	0.991052i	$0.457385\pi$
$858$	0	0
$859$	−24298.4	−0.965135	−0.482568	−	0.875859i	$-0.660296\pi$
$859$	−24298.4	−0.965135	−0.482568	+	0.875859i	$0.660296\pi$
$860$	0	0
$861$	0	0
$862$	33418.3	1.32045
$863$	24942.9	0.983853	0.491926	−	0.870637i	$-0.336293\pi$
$863$	24942.9	0.983853	0.491926	+	0.870637i	$0.336293\pi$
$864$	0	0
$865$	0	0
$866$	3126.78	0.122693
$867$	0	0
$868$	21430.1	0.838002
$869$	−16455.3	−0.642355
$870$	0	0
$871$	−605.152	−0.0235417
$872$	9790.39	0.380212
$873$	0	0
$874$	−1216.18	−0.0470686
$875$	0	0
$876$	0	0
$877$	16276.6	0.626705	0.313353	−	0.949637i	$-0.398548\pi$
$877$	16276.6	0.626705	0.313353	+	0.949637i	$0.398548\pi$
$878$	38101.3	1.46453
$879$	0	0
$880$	0	0
$881$	−26636.5	−1.01862	−0.509311	−	0.860582i	$-0.670100\pi$
$881$	−26636.5	−1.01862	−0.509311	+	0.860582i	$0.670100\pi$
$882$	0	0
$883$	21788.3	0.830392	0.415196	−	0.909732i	$-0.363713\pi$
$883$	21788.3	0.830392	0.415196	+	0.909732i	$0.363713\pi$
$884$	−239.215	−0.00910142
$885$	0	0
$886$	29763.0	1.12856
$887$	−26813.2	−1.01499	−0.507496	−	0.861654i	$-0.669429\pi$
$887$	−26813.2	−1.01499	−0.507496	+	0.861654i	$0.669429\pi$
$888$	0	0
$889$	5908.11	0.222893
$890$	0	0
$891$	0	0
$892$	−39600.2	−1.48645
$893$	−470.229	−0.0176211
$894$	0	0
$895$	0	0
$896$	−15445.2	−0.575879
$897$	0	0
$898$	−13387.4	−0.497488
$899$	71214.2	2.64196
$900$	0	0
$901$	−3311.91	−0.122459
$902$	−27217.4	−1.00470
$903$	0	0
$904$	33704.8	1.24005
$905$	0	0
$906$	0	0
$907$	−15543.0	−0.569014	−0.284507	−	0.958674i	$-0.591830\pi$
$907$	−15543.0	−0.569014	−0.284507	+	0.958674i	$0.591830\pi$
$908$	−45752.6	−1.67219
$909$	0	0
$910$	0	0
$911$	−48711.1	−1.77154	−0.885768	−	0.464128i	$-0.846368\pi$
$911$	−48711.1	−1.77154	−0.885768	+	0.464128i	$0.846368\pi$
$912$	0	0
$913$	25442.8	0.922273
$914$	36905.3	1.33558
$915$	0	0
$916$	−51173.3	−1.84587
$917$	19573.4	0.704876
$918$	0	0
$919$	1030.47	0.0369883	0.0184941	−	0.999829i	$-0.494113\pi$
$919$	1030.47	0.0369883	0.0184941	+	0.999829i	$0.494113\pi$
$920$	0	0
$921$	0	0
$922$	−11306.3	−0.403855
$923$	−2950.40	−0.105215
$924$	0	0
$925$	0	0
$926$	26079.6	0.925518
$927$	0	0
$928$	−57413.6	−2.03092
$929$	−879.756	−0.0310698	−0.0155349	−	0.999879i	$-0.504945\pi$
$929$	−879.756	−0.0310698	−0.0155349	+	0.999879i	$0.504945\pi$
$930$	0	0
$931$	−266.354	−0.00937638
$932$	−46146.0	−1.62185
$933$	0	0
$934$	−18775.8	−0.657776
$935$	0	0
$936$	0	0
$937$	18668.1	0.650864	0.325432	−	0.945565i	$-0.394490\pi$
$937$	18668.1	0.650864	0.325432	+	0.945565i	$0.394490\pi$
$938$	−6533.64	−0.227432
$939$	0	0
$940$	0	0
$941$	−29613.4	−1.02590	−0.512948	−	0.858420i	$-0.671447\pi$
$941$	−29613.4	−1.02590	−0.512948	+	0.858420i	$0.671447\pi$
$942$	0	0
$943$	−15559.5	−0.537313
$944$	−602.028	−0.0207567
$945$	0	0
$946$	25998.1	0.893521
$947$	20738.9	0.711640	0.355820	−	0.934554i	$-0.384202\pi$
$947$	20738.9	0.711640	0.355820	+	0.934554i	$0.384202\pi$
$948$	0	0
$949$	2959.15	0.101220
$950$	0	0
$951$	0	0
$952$	−933.888	−0.0317936
$953$	−45776.5	−1.55598	−0.777988	−	0.628279i	$-0.783760\pi$
$953$	−45776.5	−1.55598	−0.777988	+	0.628279i	$0.783760\pi$
$954$	0	0
$955$	0	0
$956$	−33317.5	−1.12716
$957$	0	0
$958$	30657.4	1.03392
$959$	−14399.9	−0.484878
$960$	0	0
$961$	29895.1	1.00349
$962$	2570.68	0.0861561
$963$	0	0
$964$	−60404.9	−2.01817
$965$	0	0
$966$	0	0
$967$	−34461.0	−1.14601	−0.573005	−	0.819552i	$-0.694222\pi$
$967$	−34461.0	−1.14601	−0.573005	+	0.819552i	$0.694222\pi$
$968$	−19866.5	−0.659643
$969$	0	0
$970$	0	0
$971$	22762.8	0.752309	0.376154	−	0.926557i	$-0.377246\pi$
$971$	22762.8	0.752309	0.376154	+	0.926557i	$0.377246\pi$
$972$	0	0
$973$	−12215.9	−0.402490
$974$	−28819.9	−0.948099
$975$	0	0
$976$	37.9251	0.00124381
$977$	4809.57	0.157494	0.0787470	−	0.996895i	$-0.474908\pi$
$977$	4809.57	0.157494	0.0787470	+	0.996895i	$0.474908\pi$
$978$	0	0
$979$	−6219.04	−0.203025
$980$	0	0
$981$	0	0
$982$	32046.6	1.04139
$983$	−27591.6	−0.895256	−0.447628	−	0.894220i	$-0.647731\pi$
$983$	−27591.6	−0.895256	−0.447628	+	0.894220i	$0.647731\pi$
$984$	0	0
$985$	0	0
$986$	−8582.63	−0.277207
$987$	0	0
$988$	−200.109	−0.00644365
$989$	14862.4	0.477854
$990$	0	0
$991$	−22263.4	−0.713643	−0.356822	−	0.934173i	$-0.616140\pi$
$991$	−22263.4	−0.713643	−0.356822	+	0.934173i	$0.616140\pi$
$992$	−48119.5	−1.54012
$993$	0	0
$994$	−31854.5	−1.01646
$995$	0	0
$996$	0	0
$997$	−30378.2	−0.964983	−0.482491	−	0.875901i	$-0.660268\pi$
$997$	−30378.2	−0.964983	−0.482491	+	0.875901i	$0.660268\pi$
$998$	−83707.4	−2.65502
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.4.a.w.1.2		2
3.2	odd	2		525.4.a.k.1.1		2
5.4	even	2		315.4.a.i.1.1		2
15.2	even	4		525.4.d.h.274.1		4
15.8	even	4		525.4.d.h.274.4		4
15.14	odd	2		105.4.a.f.1.2	✓	2
35.34	odd	2		2205.4.a.z.1.1		2
60.59	even	2		1680.4.a.bg.1.2		2
105.104	even	2		735.4.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.4.a.f.1.2	✓	2	15.14	odd	2
315.4.a.i.1.1		2	5.4	even	2
525.4.a.k.1.1		2	3.2	odd	2
525.4.d.h.274.1		4	15.2	even	4
525.4.d.h.274.4		4	15.8	even	4
735.4.a.p.1.2		2	105.104	even	2
1575.4.a.w.1.2		2	1.1	even	1	trivial
1680.4.a.bg.1.2		2	60.59	even	2
2205.4.a.z.1.1		2	35.34	odd	2

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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1575.a (trivial)

\(f(q)\)	\(=\)	\(q+4.53113 q^{2} +12.5311 q^{4} +7.00000 q^{7} +20.5311 q^{8} +O(q^{10})\)	\(q+4.53113 q^{2} +12.5311 q^{4} +7.00000 q^{7} +20.5311 q^{8} +19.0623 q^{11} +2.93774 q^{13} +31.7179 q^{14} -7.21984 q^{16} -6.49806 q^{17} -5.43580 q^{19} +86.3735 q^{22} +49.3774 q^{23} +13.3113 q^{26} +87.7179 q^{28} +291.494 q^{29} +244.307 q^{31} -196.963 q^{32} -29.4436 q^{34} +193.121 q^{37} -24.6303 q^{38} -315.113 q^{41} +300.996 q^{43} +238.872 q^{44} +223.735 q^{46} +86.5058 q^{47} +49.0000 q^{49} +36.8132 q^{52} +509.677 q^{53} +143.718 q^{56} +1320.80 q^{58} +83.3852 q^{59} -5.25291 q^{61} +1106.99 q^{62} -834.706 q^{64} -205.992 q^{67} -81.4281 q^{68} -1004.31 q^{71} +1007.29 q^{73} +875.055 q^{74} -68.1168 q^{76} +133.436 q^{77} -863.237 q^{79} -1427.82 q^{82} +1334.72 q^{83} +1363.85 q^{86} +391.370 q^{88} -326.249 q^{89} +20.5642 q^{91} +618.755 q^{92} +391.969 q^{94} -1526.77 q^{97} +222.025 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 17 q^{4} + 14 q^{7} + 33 q^{8}+O(q^{10}) \) 2 * q + q^2 + 17 * q^4 + 14 * q^7 + 33 * q^8	\( 2 q + q^{2} + 17 q^{4} + 14 q^{7} + 33 q^{8} + 22 q^{11} + 22 q^{13} + 7 q^{14} - 87 q^{16} + 116 q^{17} + 102 q^{19} + 76 q^{22} + 260 q^{23} - 54 q^{26} + 119 q^{28} + 196 q^{29} + 150 q^{31} - 15 q^{32} - 462 q^{34} + 96 q^{37} - 404 q^{38} + 176 q^{41} + 344 q^{43} + 252 q^{44} - 520 q^{46} + 560 q^{47} + 98 q^{49} + 122 q^{52} + 326 q^{53} + 231 q^{56} + 1658 q^{58} + 844 q^{59} - 204 q^{61} + 1440 q^{62} - 839 q^{64} + 104 q^{67} + 466 q^{68} - 1670 q^{71} + 386 q^{73} + 1218 q^{74} + 412 q^{76} + 154 q^{77} - 888 q^{79} - 3162 q^{82} + 928 q^{83} + 1212 q^{86} + 428 q^{88} - 588 q^{89} + 154 q^{91} + 1560 q^{92} - 1280 q^{94} - 522 q^{97} + 49 q^{98}+O(q^{100}) \) 2 * q + q^2 + 17 * q^4 + 14 * q^7 + 33 * q^8 + 22 * q^11 + 22 * q^13 + 7 * q^14 - 87 * q^16 + 116 * q^17 + 102 * q^19 + 76 * q^22 + 260 * q^23 - 54 * q^26 + 119 * q^28 + 196 * q^29 + 150 * q^31 - 15 * q^32 - 462 * q^34 + 96 * q^37 - 404 * q^38 + 176 * q^41 + 344 * q^43 + 252 * q^44 - 520 * q^46 + 560 * q^47 + 98 * q^49 + 122 * q^52 + 326 * q^53 + 231 * q^56 + 1658 * q^58 + 844 * q^59 - 204 * q^61 + 1440 * q^62 - 839 * q^64 + 104 * q^67 + 466 * q^68 - 1670 * q^71 + 386 * q^73 + 1218 * q^74 + 412 * q^76 + 154 * q^77 - 888 * q^79 - 3162 * q^82 + 928 * q^83 + 1212 * q^86 + 428 * q^88 - 588 * q^89 + 154 * q^91 + 1560 * q^92 - 1280 * q^94 - 522 * q^97 + 49 * q^98

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