LMFDB - Embedded newform 1728.4.d.e.865.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,Mod(865,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1728.865");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1728.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$101.955300490$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2261390379264.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 8x^{6} + 38x^{5} - 38x^{4} + 8x^{3} + 325x^{2} - 322x + 2122 $ x^8 - 4x^7 - 8x^6 + 38x^5 - 38x^4 + 8x^3 + 325x^2 - 322*x + 2122
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			865.6
Root			$1.29746 - 1.88096i$ of defining polynomial
Character	$\chi$	$=$	1728.865
Dual form			1728.4.d.e.865.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.50098i q^{5} +16.0706 q^{7} +O(q^{10})$	$q+6.50098i q^{5} +16.0706 q^{7} -27.9124i q^{11} +35.5735i q^{13} +18.9124 q^{17} -76.7372i q^{19} -190.142 q^{23} +82.7372 q^{25} -138.134i q^{29} -265.085 q^{31} +104.474i q^{35} -14.9791i q^{37} -202.562 q^{41} +133.474i q^{43} +130.020 q^{47} -84.7372 q^{49} -131.633i q^{53} +181.458 q^{55} -477.124i q^{59} +582.906i q^{61} -231.263 q^{65} +20.0000i q^{67} -60.1222 q^{71} -321.212 q^{73} -448.568i q^{77} -210.009 q^{79} -587.737i q^{83} +122.949i q^{85} -55.5767 q^{89} +571.686i q^{91} +498.867 q^{95} -75.9488 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 168 q^{17} - 296 q^{25} - 24 q^{41} + 280 q^{49} - 2808 q^{65} + 304 q^{73} - 6192 q^{89} + 3224 q^{97}+O(q^{100}) $ 8 * q - 168 * q^17 - 296 * q^25 - 24 * q^41 + 280 * q^49 - 2808 * q^65 + 304 * q^73 - 6192 * q^89 + 3224 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	6.50098i	0.581466i	0.956804	+	0.290733i	$0.0938991\pi$
$5$	6.50098i	0.581466i	−0.956804	+	0.290733i	$0.906101\pi$
$6$	0	0
$7$	16.0706	0.867728	0.433864	−	0.900978i	$-0.357150\pi$
$7$	16.0706	0.867728	0.433864	+	0.900978i	$0.357150\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 27.9124i	− 0.765082i	−0.923938	−	0.382541i	$-0.875049\pi$
$11$	− 27.9124i	− 0.765082i	0.923938	−	0.382541i	$-0.124951\pi$
$12$	0	0
$13$	35.5735i	0.758947i	0.925203	+	0.379474i	$0.123895\pi$
$13$	35.5735i	0.758947i	−0.925203	+	0.379474i	$0.876105\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	18.9124	0.269820	0.134910	−	0.990858i	$-0.456926\pi$
$17$	18.9124	0.269820	0.134910	+	0.990858i	$0.456926\pi$
$18$	0	0
$19$	− 76.7372i	− 0.926564i	−0.886211	−	0.463282i	$-0.846672\pi$
$19$	− 76.7372i	− 0.926564i	0.886211	−	0.463282i	$-0.153328\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−190.142	−1.72380	−0.861898	−	0.507081i	$-0.830724\pi$
$23$	−190.142	−1.72380	−0.861898	+	0.507081i	$0.830724\pi$
$24$	0	0
$25$	82.7372	0.661898
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 138.134i	− 0.884512i	−0.896889	−	0.442256i	$-0.854178\pi$
$29$	− 138.134i	− 0.884512i	0.896889	−	0.442256i	$-0.145822\pi$
$30$	0	0
$31$	−265.085	−1.53583	−0.767914	−	0.640553i	$-0.778705\pi$
$31$	−265.085	−1.53583	−0.767914	+	0.640553i	$0.778705\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	104.474i	0.504554i
$36$	0	0
$37$	− 14.9791i	− 0.0665556i	−0.999446	−	0.0332778i	$-0.989405\pi$
$37$	− 14.9791i	− 0.0665556i	0.999446	−	0.0332778i	$-0.0105946\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−202.562	−0.771582	−0.385791	−	0.922586i	$-0.626071\pi$
$41$	−202.562	−0.771582	−0.385791	+	0.922586i	$0.626071\pi$
$42$	0	0
$43$	133.474i	0.473364i	0.971587	+	0.236682i	$0.0760600\pi$
$43$	133.474i	0.473364i	−0.971587	+	0.236682i	$0.923940\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	130.020	0.403517	0.201759	−	0.979435i	$-0.435334\pi$
$47$	130.020	0.403517	0.201759	+	0.979435i	$0.435334\pi$
$48$	0	0
$49$	−84.7372	−0.247047
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 131.633i	− 0.341155i	−0.985344	−	0.170577i	$-0.945437\pi$
$53$	− 131.633i	− 0.341155i	0.985344	−	0.170577i	$-0.0545632\pi$
$54$	0	0
$55$	181.458	0.444869
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 477.124i	− 1.05282i	−0.850231	−	0.526409i	$-0.823538\pi$
$59$	− 477.124i	− 1.05282i	0.850231	−	0.526409i	$-0.176462\pi$
$60$	0	0
$61$	582.906i	1.22350i	0.791052	+	0.611749i	$0.209534\pi$
$61$	582.906i	1.22350i	−0.791052	+	0.611749i	$0.790466\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−231.263	−0.441302
$66$	0	0
$67$	20.0000i	0.0364685i	0.999834	+	0.0182342i	$0.00580446\pi$
$67$	20.0000i	0.0364685i	−0.999834	+	0.0182342i	$0.994196\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−60.1222	−0.100496	−0.0502479	−	0.998737i	$-0.516001\pi$
$71$	−60.1222	−0.100496	−0.0502479	+	0.998737i	$0.516001\pi$
$72$	0	0
$73$	−321.212	−0.515000	−0.257500	−	0.966278i	$-0.582899\pi$
$73$	−321.212	−0.515000	−0.257500	+	0.966278i	$0.582899\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 448.568i	− 0.663884i
$78$	0	0
$79$	−210.009	−0.299086	−0.149543	−	0.988755i	$-0.547780\pi$
$79$	−210.009	−0.299086	−0.149543	+	0.988755i	$0.547780\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 587.737i	− 0.777260i	−0.921394	−	0.388630i	$-0.872949\pi$
$83$	− 587.737i	− 0.777260i	0.921394	−	0.388630i	$-0.127051\pi$
$84$	0	0
$85$	122.949i	0.156891i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−55.5767	−0.0661924	−0.0330962	−	0.999452i	$-0.510537\pi$
$89$	−55.5767	−0.0661924	−0.0330962	+	0.999452i	$0.510537\pi$
$90$	0	0
$91$	571.686i	0.658560i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	498.867	0.538765
$96$	0	0
$97$	−75.9488	−0.0794994	−0.0397497	−	0.999210i	$-0.512656\pi$
$97$	−75.9488	−0.0794994	−0.0397497	+	0.999210i	$0.512656\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1805.57i	1.77882i	0.457114	+	0.889408i	$0.348883\pi$
$101$	1805.57i	1.77882i	−0.457114	+	0.889408i	$0.651117\pi$
$102$	0	0
$103$	−896.519	−0.857637	−0.428819	−	0.903391i	$-0.641070\pi$
$103$	−896.519	−0.857637	−0.428819	+	0.903391i	$0.641070\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 554.226i	− 0.500739i	−0.968150	−	0.250370i	$-0.919448\pi$
$107$	− 554.226i	− 0.500739i	0.968150	−	0.250370i	$-0.0805521\pi$
$108$	0	0
$109$	− 1256.94i	− 1.10452i	−0.833672	−	0.552260i	$-0.813766\pi$
$109$	− 1256.94i	− 1.10452i	0.833672	−	0.552260i	$-0.186234\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1615.69	−1.34505	−0.672526	−	0.740073i	$-0.734791\pi$
$113$	−1615.69	−1.34505	−0.672526	+	0.740073i	$0.734791\pi$
$114$	0	0
$115$	− 1236.11i	− 1.00233i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	303.933	0.234130
$120$	0	0
$121$	551.898	0.414649
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1350.50i	0.966336i
$126$	0	0
$127$	1861.53	1.30066	0.650330	−	0.759652i	$-0.274631\pi$
$127$	1861.53	1.30066	0.650330	+	0.759652i	$0.274631\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1335.53i	− 0.890728i	−0.895350	−	0.445364i	$-0.853074\pi$
$131$	− 1335.53i	− 0.890728i	0.895350	−	0.445364i	$-0.146926\pi$
$132$	0	0
$133$	− 1233.21i	− 0.804006i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2153.20	−1.34277	−0.671387	−	0.741107i	$-0.734301\pi$
$137$	−2153.20	−1.34277	−0.671387	+	0.741107i	$0.734301\pi$
$138$	0	0
$139$	− 97.4744i	− 0.0594797i	−0.999558	−	0.0297398i	$-0.990532\pi$
$139$	− 97.4744i	− 0.0594797i	0.999558	−	0.0297398i	$-0.00946788\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	992.942	0.580657
$144$	0	0
$145$	898.007	0.514313
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 1392.82i	− 0.765803i	−0.923789	−	0.382901i	$-0.874925\pi$
$149$	− 1392.82i	− 0.765803i	0.923789	−	0.382901i	$-0.125075\pi$
$150$	0	0
$151$	838.627	0.451963	0.225982	−	0.974132i	$-0.427441\pi$
$151$	838.627	0.451963	0.225982	+	0.974132i	$0.427441\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1723.31i	− 0.893032i
$156$	0	0
$157$	2738.54i	1.39210i	0.717994	+	0.696049i	$0.245061\pi$
$157$	2738.54i	1.39210i	−0.717994	+	0.696049i	$0.754939\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3055.69	−1.49579
$162$	0	0
$163$	421.788i	0.202681i	0.994852	+	0.101341i	$0.0323132\pi$
$163$	421.788i	0.202681i	−0.994852	+	0.101341i	$0.967687\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3364.09	−1.55881	−0.779405	−	0.626520i	$-0.784479\pi$
$167$	−3364.09	−1.55881	−0.779405	+	0.626520i	$0.784479\pi$
$168$	0	0
$169$	931.526	0.423999
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1550.32i	− 0.681321i	−0.940186	−	0.340660i	$-0.889349\pi$
$173$	− 1550.32i	− 0.681321i	0.940186	−	0.340660i	$-0.110651\pi$
$174$	0	0
$175$	1329.63	0.574347
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 4634.17i	− 1.93505i	−0.252775	−	0.967525i	$-0.581343\pi$
$179$	− 4634.17i	− 1.93505i	0.252775	−	0.967525i	$-0.418657\pi$
$180$	0	0
$181$	− 1688.80i	− 0.693522i	−0.937953	−	0.346761i	$-0.887281\pi$
$181$	− 1688.80i	− 0.693522i	0.937953	−	0.346761i	$-0.112719\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	97.3792	0.0386998
$186$	0	0
$187$	− 527.891i	− 0.206434i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3125.26	1.18396	0.591980	−	0.805953i	$-0.298347\pi$
$191$	3125.26	1.18396	0.591980	+	0.805953i	$0.298347\pi$
$192$	0	0
$193$	−2085.96	−0.777981	−0.388991	−	0.921242i	$-0.627176\pi$
$193$	−2085.96	−0.777981	−0.388991	+	0.921242i	$0.627176\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2356.44i	− 0.852231i	−0.904669	−	0.426115i	$-0.859882\pi$
$197$	− 2356.44i	− 0.852231i	0.904669	−	0.426115i	$-0.140118\pi$
$198$	0	0
$199$	−970.940	−0.345870	−0.172935	−	0.984933i	$-0.555325\pi$
$199$	−970.940	−0.345870	−0.172935	+	0.984933i	$0.555325\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 2219.89i	− 0.767516i
$204$	0	0
$205$	− 1316.85i	− 0.448649i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2141.92	−0.708898
$210$	0	0
$211$	1621.58i	0.529073i	0.964376	+	0.264537i	$0.0852190\pi$
$211$	1621.58i	0.529073i	−0.964376	+	0.264537i	$0.914781\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−867.715	−0.275245
$216$	0	0
$217$	−4260.07	−1.33268
$218$	0	0
$219$	0	0
$220$	0	0
$221$	672.781i	0.204779i
$222$	0	0
$223$	−3391.03	−1.01830	−0.509148	−	0.860679i	$-0.670040\pi$
$223$	−3391.03	−1.01830	−0.509148	+	0.860679i	$0.670040\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3111.93i	− 0.909894i	−0.890519	−	0.454947i	$-0.849658\pi$
$227$	− 3111.93i	− 0.909894i	0.890519	−	0.454947i	$-0.150342\pi$
$228$	0	0
$229$	− 4512.85i	− 1.30226i	−0.758966	−	0.651130i	$-0.774295\pi$
$229$	− 4512.85i	− 1.30226i	0.758966	−	0.651130i	$-0.225705\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6022.61	−1.69337	−0.846683	−	0.532097i	$-0.821404\pi$
$233$	−6022.61	−1.69337	−0.846683	+	0.532097i	$0.821404\pi$
$234$	0	0
$235$	845.256i	0.234632i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2371.15	−0.641745	−0.320872	−	0.947122i	$-0.603976\pi$
$239$	−2371.15	−0.641745	−0.320872	+	0.947122i	$0.603976\pi$
$240$	0	0
$241$	4124.74	1.10248	0.551241	−	0.834346i	$-0.314155\pi$
$241$	4124.74	1.10248	0.551241	+	0.834346i	$0.314155\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 550.875i	− 0.143649i
$246$	0	0
$247$	2729.81	0.703214
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 4354.18i	− 1.09495i	−0.836821	−	0.547476i	$-0.815589\pi$
$251$	− 4354.18i	− 1.09495i	0.836821	−	0.547476i	$-0.184411\pi$
$252$	0	0
$253$	5307.32i	1.31885i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7727.55	−1.87561	−0.937805	−	0.347163i	$-0.887145\pi$
$257$	−7727.55	−1.87561	−0.937805	+	0.347163i	$0.887145\pi$
$258$	0	0
$259$	− 240.723i	− 0.0577522i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3944.29	−0.924774	−0.462387	−	0.886678i	$-0.653007\pi$
$263$	−3944.29	−0.924774	−0.462387	+	0.886678i	$0.653007\pi$
$264$	0	0
$265$	855.744	0.198370
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 290.836i	− 0.0659204i	−0.999457	−	0.0329602i	$-0.989507\pi$
$269$	− 290.836i	− 0.0659204i	0.999457	−	0.0329602i	$-0.0104935\pi$
$270$	0	0
$271$	−2842.92	−0.637252	−0.318626	−	0.947881i	$-0.603221\pi$
$271$	−2842.92	−0.637252	−0.318626	+	0.947881i	$0.603221\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 2309.39i	− 0.506406i
$276$	0	0
$277$	6030.03i	1.30798i	0.756505	+	0.653988i	$0.226906\pi$
$277$	6030.03i	1.30798i	−0.756505	+	0.653988i	$0.773094\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3545.58	−0.752711	−0.376356	−	0.926475i	$-0.622823\pi$
$281$	−3545.58	−0.752711	−0.376356	+	0.926475i	$0.622823\pi$
$282$	0	0
$283$	− 1499.48i	− 0.314964i	−0.987522	−	0.157482i	$-0.949662\pi$
$283$	− 1499.48i	− 0.314964i	0.987522	−	0.157482i	$-0.0503377\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3255.28	−0.669524
$288$	0	0
$289$	−4555.32	−0.927197
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1174.97i	− 0.234275i	−0.993116	−	0.117137i	$-0.962628\pi$
$293$	− 1174.97i	− 0.234275i	0.993116	−	0.117137i	$-0.0373718\pi$
$294$	0	0
$295$	3101.78	0.612177
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 6764.01i	− 1.30827i
$300$	0	0
$301$	2145.01i	0.410752i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−3789.46	−0.711423
$306$	0	0
$307$	5403.90i	1.00462i	0.864689	+	0.502308i	$0.167516\pi$
$307$	5403.90i	1.00462i	−0.864689	+	0.502308i	$0.832484\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9941.05	1.81256	0.906278	−	0.422682i	$-0.138911\pi$
$311$	9941.05	1.81256	0.906278	+	0.422682i	$0.138911\pi$
$312$	0	0
$313$	5905.11	1.06638	0.533189	−	0.845996i	$-0.320993\pi$
$313$	5905.11	1.06638	0.533189	+	0.845996i	$0.320993\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7828.23i	1.38699i	0.720460	+	0.693497i	$0.243931\pi$
$317$	7828.23i	1.38699i	−0.720460	+	0.693497i	$0.756069\pi$
$318$	0	0
$319$	−3855.65	−0.676724
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 1451.29i	− 0.250005i
$324$	0	0
$325$	2943.25i	0.502346i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2089.49	0.350144
$330$	0	0
$331$	− 11665.1i	− 1.93707i	−0.248874	−	0.968536i	$-0.580061\pi$
$331$	− 11665.1i	− 1.93707i	0.248874	−	0.968536i	$-0.419939\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−130.020	−0.0212052
$336$	0	0
$337$	1705.59	0.275696	0.137848	−	0.990453i	$-0.455982\pi$
$337$	1705.59	0.275696	0.137848	+	0.990453i	$0.455982\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7399.16i	1.17504i
$342$	0	0
$343$	−6873.98	−1.08210
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 7701.30i	− 1.19143i	−0.803195	−	0.595717i	$-0.796868\pi$
$347$	− 7701.30i	− 1.19143i	0.803195	−	0.595717i	$-0.203132\pi$
$348$	0	0
$349$	− 6875.08i	− 1.05448i	−0.849715	−	0.527242i	$-0.823226\pi$
$349$	− 6875.08i	− 1.05448i	0.849715	−	0.527242i	$-0.176774\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7008.39	−1.05671	−0.528355	−	0.849023i	$-0.677191\pi$
$353$	−7008.39	−1.05671	−0.528355	+	0.849023i	$0.677191\pi$
$354$	0	0
$355$	− 390.854i	− 0.0584348i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8996.79	1.32265	0.661327	−	0.750098i	$-0.269994\pi$
$359$	8996.79	1.32265	0.661327	+	0.750098i	$0.269994\pi$
$360$	0	0
$361$	970.400	0.141478
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 2088.19i	− 0.299455i
$366$	0	0
$367$	4690.25	0.667109	0.333554	−	0.942731i	$-0.391752\pi$
$367$	4690.25	0.667109	0.333554	+	0.942731i	$0.391752\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 2115.42i	− 0.296029i
$372$	0	0
$373$	− 8072.72i	− 1.12062i	−0.828285	−	0.560308i	$-0.810683\pi$
$373$	− 8072.72i	− 1.12062i	0.828285	−	0.560308i	$-0.189317\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4913.91	0.671298
$378$	0	0
$379$	8379.80i	1.13573i	0.823122	+	0.567864i	$0.192230\pi$
$379$	8379.80i	1.13573i	−0.823122	+	0.567864i	$0.807770\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	869.756	0.116038	0.0580189	−	0.998315i	$-0.481522\pi$
$383$	869.756	0.116038	0.0580189	+	0.998315i	$0.481522\pi$
$384$	0	0
$385$	2916.13	0.386026
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 8524.17i	− 1.11103i	−0.831505	−	0.555517i	$-0.812520\pi$
$389$	− 8524.17i	− 1.11103i	0.831505	−	0.555517i	$-0.187480\pi$
$390$	0	0
$391$	−3596.04	−0.465114
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1365.26i	− 0.173908i
$396$	0	0
$397$	3385.75i	0.428025i	0.976831	+	0.214012i	$0.0686533\pi$
$397$	3385.75i	0.428025i	−0.976831	+	0.214012i	$0.931347\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−13391.3	−1.66766	−0.833829	−	0.552023i	$-0.813856\pi$
$401$	−13391.3	−1.66766	−0.833829	+	0.552023i	$0.813856\pi$
$402$	0	0
$403$	− 9430.01i	− 1.16561i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−418.104	−0.0509205
$408$	0	0
$409$	−7965.98	−0.963062	−0.481531	−	0.876429i	$-0.659919\pi$
$409$	−7965.98	−0.963062	−0.481531	+	0.876429i	$0.659919\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 7667.65i	− 0.913560i
$414$	0	0
$415$	3820.87	0.451950
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 6356.23i	− 0.741103i	−0.928812	−	0.370552i	$-0.879169\pi$
$419$	− 6356.23i	− 0.741103i	0.928812	−	0.370552i	$-0.120831\pi$
$420$	0	0
$421$	6365.81i	0.736937i	0.929640	+	0.368468i	$0.120118\pi$
$421$	6365.81i	0.736937i	−0.929640	+	0.368468i	$0.879882\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1564.76	0.178593
$426$	0	0
$427$	9367.62i	1.06166i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−4612.28	−0.515466	−0.257733	−	0.966216i	$-0.582975\pi$
$431$	−4612.28	−0.515466	−0.257733	+	0.966216i	$0.582975\pi$
$432$	0	0
$433$	−10400.1	−1.15426	−0.577131	−	0.816652i	$-0.695828\pi$
$433$	−10400.1	−1.15426	−0.577131	+	0.816652i	$0.695828\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	14591.0i	1.59721i
$438$	0	0
$439$	−1480.23	−0.160928	−0.0804642	−	0.996757i	$-0.525640\pi$
$439$	−1480.23	−0.160928	−0.0804642	+	0.996757i	$0.525640\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5799.47i	0.621989i	0.950412	+	0.310995i	$0.100662\pi$
$443$	5799.47i	0.621989i	−0.950412	+	0.310995i	$0.899338\pi$
$444$	0	0
$445$	− 361.303i	− 0.0384886i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−6011.34	−0.631832	−0.315916	−	0.948787i	$-0.602312\pi$
$449$	−6011.34	−0.631832	−0.315916	+	0.948787i	$0.602312\pi$
$450$	0	0
$451$	5653.99i	0.590324i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3716.52	−0.382930
$456$	0	0
$457$	7891.93	0.807810	0.403905	−	0.914801i	$-0.367653\pi$
$457$	7891.93	0.807810	0.403905	+	0.914801i	$0.367653\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 8338.72i	− 0.842458i	−0.906954	−	0.421229i	$-0.861599\pi$
$461$	− 8338.72i	− 0.842458i	0.906954	−	0.421229i	$-0.138401\pi$
$462$	0	0
$463$	16445.4	1.65072	0.825360	−	0.564607i	$-0.190972\pi$
$463$	16445.4	1.65072	0.825360	+	0.564607i	$0.190972\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4975.50i	0.493017i	0.969141	+	0.246508i	$0.0792833\pi$
$467$	4975.50i	0.493017i	−0.969141	+	0.246508i	$0.920717\pi$
$468$	0	0
$469$	321.411i	0.0316447i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3725.59	0.362163
$474$	0	0
$475$	− 6349.02i	− 0.613291i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−617.261	−0.0588797	−0.0294398	−	0.999567i	$-0.509372\pi$
$479$	−617.261	−0.0588797	−0.0294398	+	0.999567i	$0.509372\pi$
$480$	0	0
$481$	532.861	0.0505122
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 493.742i	− 0.0462261i
$486$	0	0
$487$	4775.91	0.444389	0.222194	−	0.975002i	$-0.428678\pi$
$487$	4775.91	0.444389	0.222194	+	0.975002i	$0.428678\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8682.25i	0.798014i	0.916948	+	0.399007i	$0.130645\pi$
$491$	8682.25i	0.798014i	−0.916948	+	0.399007i	$0.869355\pi$
$492$	0	0
$493$	− 2612.45i	− 0.238659i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−966.198	−0.0872030
$498$	0	0
$499$	3011.15i	0.270136i	0.990836	+	0.135068i	$0.0431252\pi$
$499$	3011.15i	0.270136i	−0.990836	+	0.135068i	$0.956875\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4948.29	−0.438635	−0.219318	−	0.975654i	$-0.570383\pi$
$503$	−4948.29	−0.438635	−0.219318	+	0.975654i	$0.570383\pi$
$504$	0	0
$505$	−11737.9	−1.03432
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8060.89i	0.701950i	0.936385	+	0.350975i	$0.114150\pi$
$509$	8060.89i	0.701950i	−0.936385	+	0.350975i	$0.885850\pi$
$510$	0	0
$511$	−5162.05	−0.446880
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 5828.25i	− 0.498687i
$516$	0	0
$517$	− 3629.16i	− 0.308724i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18392.8	1.54665	0.773325	−	0.634010i	$-0.218592\pi$
$521$	18392.8	1.54665	0.773325	+	0.634010i	$0.218592\pi$
$522$	0	0
$523$	15531.7i	1.29857i	0.760544	+	0.649286i	$0.224932\pi$
$523$	15531.7i	1.29857i	−0.760544	+	0.649286i	$0.775068\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5013.40	−0.414397
$528$	0	0
$529$	23986.9	1.97148
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 7205.84i	− 0.585590i
$534$	0	0
$535$	3603.02	0.291163
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2365.22i	0.189012i
$540$	0	0
$541$	− 24594.4i	− 1.95452i	−0.212035	−	0.977262i	$-0.568009\pi$
$541$	− 24594.4i	− 1.95452i	0.212035	−	0.977262i	$-0.431991\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8171.32	0.642240
$546$	0	0
$547$	− 15437.8i	− 1.20671i	−0.797471	−	0.603357i	$-0.793830\pi$
$547$	− 15437.8i	− 1.20671i	0.797471	−	0.603357i	$-0.206170\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−10600.0	−0.819557
$552$	0	0
$553$	−3374.96	−0.259526
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7720.89i	− 0.587333i	−0.955908	−	0.293667i	$-0.905124\pi$
$557$	− 7720.89i	− 0.587333i	0.955908	−	0.293667i	$-0.0948756\pi$
$558$	0	0
$559$	−4748.15	−0.359258
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4965.48i	0.371705i	0.982578	+	0.185852i	$0.0595046\pi$
$563$	4965.48i	0.371705i	−0.982578	+	0.185852i	$0.940495\pi$
$564$	0	0
$565$	− 10503.5i	− 0.782102i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14638.2	1.07850	0.539250	−	0.842146i	$-0.318708\pi$
$569$	14638.2	1.07850	0.539250	+	0.842146i	$0.318708\pi$
$570$	0	0
$571$	− 10966.5i	− 0.803735i	−0.915698	−	0.401867i	$-0.868361\pi$
$571$	− 10966.5i	− 0.803735i	0.915698	−	0.401867i	$-0.131639\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−15731.8	−1.14098
$576$	0	0
$577$	−21576.8	−1.55676	−0.778382	−	0.627791i	$-0.783959\pi$
$577$	−21576.8	−1.55676	−0.778382	+	0.627791i	$0.783959\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 9445.26i	− 0.674450i
$582$	0	0
$583$	−3674.19	−0.261011
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7747.02i	0.544725i	0.962195	+	0.272362i	$0.0878050\pi$
$587$	7747.02i	0.544725i	−0.962195	+	0.272362i	$0.912195\pi$
$588$	0	0
$589$	20341.9i	1.42304i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9423.20	−0.652554	−0.326277	−	0.945274i	$-0.605794\pi$
$593$	−9423.20	−0.652554	−0.326277	+	0.945274i	$0.605794\pi$
$594$	0	0
$595$	1975.86i	0.136139i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	16746.0	1.14227	0.571137	−	0.820855i	$-0.306503\pi$
$599$	16746.0	1.14227	0.571137	+	0.820855i	$0.306503\pi$
$600$	0	0
$601$	24440.4	1.65881	0.829405	−	0.558648i	$-0.188680\pi$
$601$	24440.4	1.65881	0.829405	+	0.558648i	$0.188680\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	3587.88i	0.241104i
$606$	0	0
$607$	−24645.3	−1.64798	−0.823988	−	0.566607i	$-0.808256\pi$
$607$	−24645.3	−1.64798	−0.823988	+	0.566607i	$0.808256\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4625.26i	0.306248i
$612$	0	0
$613$	13945.4i	0.918842i	0.888219	+	0.459421i	$0.151943\pi$
$613$	13945.4i	0.918842i	−0.888219	+	0.459421i	$0.848057\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20120.0	1.31281	0.656403	−	0.754410i	$-0.272077\pi$
$617$	20120.0	1.31281	0.656403	+	0.754410i	$0.272077\pi$
$618$	0	0
$619$	− 15834.6i	− 1.02819i	−0.857734	−	0.514093i	$-0.828129\pi$
$619$	− 15834.6i	− 1.02819i	0.857734	−	0.514093i	$-0.171871\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−893.149	−0.0574370
$624$	0	0
$625$	1562.60	0.100006
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 283.292i	− 0.0179580i
$630$	0	0
$631$	25542.9	1.61149	0.805744	−	0.592264i	$-0.201766\pi$
$631$	25542.9	1.61149	0.805744	+	0.592264i	$0.201766\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	12101.8i	0.756289i
$636$	0	0
$637$	− 3014.40i	− 0.187496i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−4368.61	−0.269188	−0.134594	−	0.990901i	$-0.542973\pi$
$641$	−4368.61	−0.269188	−0.134594	+	0.990901i	$0.542973\pi$
$642$	0	0
$643$	6253.46i	0.383534i	0.981440	+	0.191767i	$0.0614218\pi$
$643$	6253.46i	0.383534i	−0.981440	+	0.191767i	$0.938578\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	27111.2	1.64737	0.823687	−	0.567045i	$-0.191913\pi$
$647$	27111.2	1.64737	0.823687	+	0.567045i	$0.191913\pi$
$648$	0	0
$649$	−13317.7	−0.805492
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 11206.8i	− 0.671601i	−0.941933	−	0.335801i	$-0.890993\pi$
$653$	− 11206.8i	− 0.671601i	0.941933	−	0.335801i	$-0.109007\pi$
$654$	0	0
$655$	8682.23	0.517928
$656$	0	0
$657$	0	0
$658$	0	0
$659$	28324.2i	1.67429i	0.546983	+	0.837143i	$0.315776\pi$
$659$	28324.2i	1.67429i	−0.546983	+	0.837143i	$0.684224\pi$
$660$	0	0
$661$	27069.6i	1.59287i	0.604727	+	0.796433i	$0.293282\pi$
$661$	27069.6i	1.59287i	−0.604727	+	0.796433i	$0.706718\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8017.08	0.467502
$666$	0	0
$667$	26265.1i	1.52472i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	16270.3	0.936077
$672$	0	0
$673$	18620.6	1.06652	0.533262	−	0.845950i	$-0.320966\pi$
$673$	18620.6	1.06652	0.533262	+	0.845950i	$0.320966\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	25175.3i	1.42919i	0.699536	+	0.714597i	$0.253390\pi$
$677$	25175.3i	1.42919i	−0.699536	+	0.714597i	$0.746610\pi$
$678$	0	0
$679$	−1220.54	−0.0689839
$680$	0	0
$681$	0	0
$682$	0	0
$683$	13438.8i	0.752886i	0.926440	+	0.376443i	$0.122853\pi$
$683$	13438.8i	0.752886i	−0.926440	+	0.376443i	$0.877147\pi$
$684$	0	0
$685$	− 13997.9i	− 0.780777i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4682.65	0.258918
$690$	0	0
$691$	− 28254.8i	− 1.55552i	−0.628563	−	0.777758i	$-0.716357\pi$
$691$	− 28254.8i	− 1.55552i	0.628563	−	0.777758i	$-0.283643\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	633.680	0.0345854
$696$	0	0
$697$	−3830.93	−0.208188
$698$	0	0
$699$	0	0
$700$	0	0
$701$	32350.0i	1.74300i	0.490397	+	0.871499i	$0.336852\pi$
$701$	32350.0i	1.74300i	−0.490397	+	0.871499i	$0.663148\pi$
$702$	0	0
$703$	−1149.46	−0.0616680
$704$	0	0
$705$	0	0
$706$	0	0
$707$	29016.4i	1.54353i
$708$	0	0
$709$	20041.0i	1.06157i	0.847505	+	0.530787i	$0.178103\pi$
$709$	20041.0i	1.06157i	−0.847505	+	0.530787i	$0.821897\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	50403.8	2.64746
$714$	0	0
$715$	6455.10i	0.337632i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	11518.4	0.597447	0.298723	−	0.954340i	$-0.403439\pi$
$719$	11518.4	0.597447	0.298723	+	0.954340i	$0.403439\pi$
$720$	0	0
$721$	−14407.6	−0.744196
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 11428.8i	− 0.585456i
$726$	0	0
$727$	−16989.3	−0.866710	−0.433355	−	0.901223i	$-0.642670\pi$
$727$	−16989.3	−0.866710	−0.433355	+	0.901223i	$0.642670\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2524.32i	0.127723i
$732$	0	0
$733$	− 11934.0i	− 0.601355i	−0.953726	−	0.300677i	$-0.902787\pi$
$733$	− 11934.0i	− 0.601355i	0.953726	−	0.300677i	$-0.0972127\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	558.248	0.0279014
$738$	0	0
$739$	19610.5i	0.976160i	0.872799	+	0.488080i	$0.162303\pi$
$739$	19610.5i	0.976160i	−0.872799	+	0.488080i	$0.837697\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−25308.6	−1.24964	−0.624820	−	0.780769i	$-0.714828\pi$
$743$	−25308.6	−1.24964	−0.624820	+	0.780769i	$0.714828\pi$
$744$	0	0
$745$	9054.73	0.445288
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 8906.73i	− 0.434506i
$750$	0	0
$751$	−11835.9	−0.575097	−0.287548	−	0.957766i	$-0.592840\pi$
$751$	−11835.9	−0.575097	−0.287548	+	0.957766i	$0.592840\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5451.90i	0.262801i
$756$	0	0
$757$	− 13648.5i	− 0.655299i	−0.944799	−	0.327650i	$-0.893743\pi$
$757$	− 13648.5i	− 0.655299i	0.944799	−	0.327650i	$-0.106257\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	33364.9	1.58932	0.794662	−	0.607052i	$-0.207648\pi$
$761$	33364.9	1.58932	0.794662	+	0.607052i	$0.207648\pi$
$762$	0	0
$763$	− 20199.7i	− 0.958423i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	16973.0	0.799033
$768$	0	0
$769$	−17035.1	−0.798830	−0.399415	−	0.916770i	$-0.630787\pi$
$769$	−17035.1	−0.798830	−0.399415	+	0.916770i	$0.630787\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 18934.3i	− 0.881010i	−0.897750	−	0.440505i	$-0.854799\pi$
$773$	− 18934.3i	− 0.881010i	0.897750	−	0.440505i	$-0.145201\pi$
$774$	0	0
$775$	−21932.4	−1.01656
$776$	0	0
$777$	0	0
$778$	0	0
$779$	15544.0i	0.714921i
$780$	0	0
$781$	1678.16i	0.0768875i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−17803.2	−0.809458
$786$	0	0
$787$	31964.7i	1.44780i	0.689904	+	0.723901i	$0.257653\pi$
$787$	31964.7i	1.44780i	−0.689904	+	0.723901i	$0.742347\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−25965.0	−1.16714
$792$	0	0
$793$	−20736.0	−0.928571
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27500.2i	1.22222i	0.791546	+	0.611110i	$0.209277\pi$
$797$	27500.2i	1.22222i	−0.791546	+	0.611110i	$0.790723\pi$
$798$	0	0
$799$	2458.98	0.108877
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8965.79i	0.394017i
$804$	0	0
$805$	− 19865.0i	− 0.869749i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	41290.0	1.79441	0.897206	−	0.441613i	$-0.145593\pi$
$809$	41290.0	1.79441	0.897206	+	0.441613i	$0.145593\pi$
$810$	0	0
$811$	25905.2i	1.12164i	0.827936	+	0.560822i	$0.189515\pi$
$811$	25905.2i	1.12164i	−0.827936	+	0.560822i	$0.810485\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2742.04	−0.117852
$816$	0	0
$817$	10242.5	0.438602
$818$	0	0
$819$	0	0
$820$	0	0
$821$	9011.17i	0.383060i	0.981487	+	0.191530i	$0.0613449\pi$
$821$	9011.17i	0.383060i	−0.981487	+	0.191530i	$0.938655\pi$
$822$	0	0
$823$	−7843.81	−0.332221	−0.166111	−	0.986107i	$-0.553121\pi$
$823$	−7843.81	−0.332221	−0.166111	+	0.986107i	$0.553121\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3843.55i	0.161612i	0.996730	+	0.0808061i	$0.0257495\pi$
$827$	3843.55i	0.161612i	−0.996730	+	0.0808061i	$0.974251\pi$
$828$	0	0
$829$	36286.9i	1.52026i	0.649770	+	0.760131i	$0.274865\pi$
$829$	36286.9i	1.52026i	−0.649770	+	0.760131i	$0.725135\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1602.58	−0.0666582
$834$	0	0
$835$	− 21869.9i	− 0.906395i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	434.523	0.0178801	0.00894004	−	0.999960i	$-0.497154\pi$
$839$	434.523	0.0178801	0.00894004	+	0.999960i	$0.497154\pi$
$840$	0	0
$841$	5307.99	0.217639
$842$	0	0
$843$	0	0
$844$	0	0
$845$	6055.83i	0.246541i
$846$	0	0
$847$	8869.30	0.359803
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2848.16i	0.114728i
$852$	0	0
$853$	20871.7i	0.837787i	0.908035	+	0.418893i	$0.137582\pi$
$853$	20871.7i	0.837787i	−0.908035	+	0.418893i	$0.862418\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34692.7	1.38282	0.691412	−	0.722460i	$-0.256989\pi$
$857$	34692.7	1.38282	0.691412	+	0.722460i	$0.256989\pi$
$858$	0	0
$859$	− 22844.9i	− 0.907403i	−0.891154	−	0.453702i	$-0.850103\pi$
$859$	− 22844.9i	− 0.907403i	0.891154	−	0.453702i	$-0.149897\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3998.34	0.157712	0.0788558	−	0.996886i	$-0.474873\pi$
$863$	3998.34	0.157712	0.0788558	+	0.996886i	$0.474873\pi$
$864$	0	0
$865$	10078.6	0.396165
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5861.85i	0.228826i
$870$	0	0
$871$	−711.470	−0.0276777
$872$	0	0
$873$	0	0
$874$	0	0
$875$	21703.2i	0.838518i
$876$	0	0
$877$	− 16596.6i	− 0.639028i	−0.947581	−	0.319514i	$-0.896480\pi$
$877$	− 16596.6i	− 0.639028i	0.947581	−	0.319514i	$-0.103520\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	342.063	0.0130811	0.00654053	−	0.999979i	$-0.497918\pi$
$881$	342.063	0.0130811	0.00654053	+	0.999979i	$0.497918\pi$
$882$	0	0
$883$	14529.0i	0.553725i	0.960910	+	0.276862i	$0.0892946\pi$
$883$	14529.0i	0.553725i	−0.960910	+	0.276862i	$0.910705\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−33035.3	−1.25053	−0.625263	−	0.780414i	$-0.715008\pi$
$887$	−33035.3	−1.25053	−0.625263	+	0.780414i	$0.715008\pi$
$888$	0	0
$889$	29915.8	1.12862
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 9977.35i	− 0.373885i
$894$	0	0
$895$	30126.6	1.12517
$896$	0	0
$897$	0	0
$898$	0	0
$899$	36617.3i	1.35846i
$900$	0	0
$901$	− 2489.50i	− 0.0920501i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	10978.9	0.403259
$906$	0	0
$907$	− 52453.7i	− 1.92028i	−0.279515	−	0.960141i	$-0.590174\pi$
$907$	− 52453.7i	− 1.92028i	0.279515	−	0.960141i	$-0.409826\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	48342.7	1.75814	0.879070	−	0.476692i	$-0.158164\pi$
$911$	48342.7	1.75814	0.879070	+	0.476692i	$0.158164\pi$
$912$	0	0
$913$	−16405.2	−0.594668
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 21462.6i	− 0.772910i
$918$	0	0
$919$	21911.9	0.786514	0.393257	−	0.919429i	$-0.371348\pi$
$919$	21911.9	0.786514	0.393257	+	0.919429i	$0.371348\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 2138.76i	− 0.0762710i
$924$	0	0
$925$	− 1239.33i	− 0.0440530i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−25184.7	−0.889433	−0.444717	−	0.895671i	$-0.646696\pi$
$929$	−25184.7	−0.889433	−0.444717	+	0.895671i	$0.646696\pi$
$930$	0	0
$931$	6502.50i	0.228905i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3431.81	0.120034
$936$	0	0
$937$	−37948.3	−1.32307	−0.661536	−	0.749914i	$-0.730095\pi$
$937$	−37948.3	−1.32307	−0.661536	+	0.749914i	$0.730095\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 37907.9i	− 1.31325i	−0.754219	−	0.656623i	$-0.771984\pi$
$941$	− 37907.9i	− 1.31325i	0.754219	−	0.656623i	$-0.228016\pi$
$942$	0	0
$943$	38515.5	1.33005
$944$	0	0
$945$	0	0
$946$	0	0
$947$	20062.2i	0.688419i	0.938893	+	0.344209i	$0.111853\pi$
$947$	20062.2i	0.688419i	−0.938893	+	0.344209i	$0.888147\pi$
$948$	0	0
$949$	− 11426.6i	− 0.390858i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	43232.8	1.46952	0.734758	−	0.678329i	$-0.237296\pi$
$953$	43232.8	1.46952	0.734758	+	0.678329i	$0.237296\pi$
$954$	0	0
$955$	20317.3i	0.688432i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−34603.1	−1.16516
$960$	0	0
$961$	40479.1	1.35877
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 13560.8i	− 0.452369i
$966$	0	0
$967$	−9292.66	−0.309030	−0.154515	−	0.987990i	$-0.549381\pi$
$967$	−9292.66	−0.309030	−0.154515	+	0.987990i	$0.549381\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	38499.0i	1.27239i	0.771528	+	0.636196i	$0.219493\pi$
$971$	38499.0i	1.27239i	−0.771528	+	0.636196i	$0.780507\pi$
$972$	0	0
$973$	− 1566.47i	− 0.0516122i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−9843.93	−0.322349	−0.161175	−	0.986926i	$-0.551528\pi$
$977$	−9843.93	−0.322349	−0.161175	+	0.986926i	$0.551528\pi$
$978$	0	0
$979$	1551.28i	0.0506426i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−43241.9	−1.40305	−0.701527	−	0.712643i	$-0.747498\pi$
$983$	−43241.9	−1.40305	−0.701527	+	0.712643i	$0.747498\pi$
$984$	0	0
$985$	15319.2	0.495543
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 25379.1i	− 0.815984i
$990$	0	0
$991$	−34891.3	−1.11842	−0.559212	−	0.829025i	$-0.688896\pi$
$991$	−34891.3	−1.11842	−0.559212	+	0.829025i	$0.688896\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 6312.07i	− 0.201111i
$996$	0	0
$997$	− 50166.9i	− 1.59358i	−0.604256	−	0.796790i	$-0.706529\pi$
$997$	− 50166.9i	− 1.59358i	0.604256	−	0.796790i	$-0.293471\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.d.e.865.6	yes	8
3.2	odd	2		1728.4.d.h.865.4	yes	8
4.3	odd	2	inner	1728.4.d.e.865.5	yes	8
8.3	odd	2	inner	1728.4.d.e.865.3	✓	8
8.5	even	2	inner	1728.4.d.e.865.4	yes	8
12.11	even	2		1728.4.d.h.865.3	yes	8
24.5	odd	2		1728.4.d.h.865.6	yes	8
24.11	even	2		1728.4.d.h.865.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1728.4.d.e.865.3	✓	8	8.3	odd	2	inner
1728.4.d.e.865.4	yes	8	8.5	even	2	inner
1728.4.d.e.865.5	yes	8	4.3	odd	2	inner
1728.4.d.e.865.6	yes	8	1.1	even	1	trivial
1728.4.d.h.865.3	yes	8	12.11	even	2
1728.4.d.h.865.4	yes	8	3.2	odd	2
1728.4.d.h.865.5	yes	8	24.11	even	2
1728.4.d.h.865.6	yes	8	24.5	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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1728.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+6.50098i q^{5} +16.0706 q^{7} +O(q^{10})\)	\(q+6.50098i q^{5} +16.0706 q^{7} -27.9124i q^{11} +35.5735i q^{13} +18.9124 q^{17} -76.7372i q^{19} -190.142 q^{23} +82.7372 q^{25} -138.134i q^{29} -265.085 q^{31} +104.474i q^{35} -14.9791i q^{37} -202.562 q^{41} +133.474i q^{43} +130.020 q^{47} -84.7372 q^{49} -131.633i q^{53} +181.458 q^{55} -477.124i q^{59} +582.906i q^{61} -231.263 q^{65} +20.0000i q^{67} -60.1222 q^{71} -321.212 q^{73} -448.568i q^{77} -210.009 q^{79} -587.737i q^{83} +122.949i q^{85} -55.5767 q^{89} +571.686i q^{91} +498.867 q^{95} -75.9488 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 168 q^{17} - 296 q^{25} - 24 q^{41} + 280 q^{49} - 2808 q^{65} + 304 q^{73} - 6192 q^{89} + 3224 q^{97}+O(q^{100}) \) 8 * q - 168 * q^17 - 296 * q^25 - 24 * q^41 + 280 * q^49 - 2808 * q^65 + 304 * q^73 - 6192 * q^89 + 3224 * q^97

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